LMFDB - Embedded newform 83.13.b.c.82.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [83,13,Mod(82,83)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(83, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 13, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("83.82");

S:= CuspForms(chi, 13);

N := Newforms(S);

Level:	$ N $	$=$	$ 83 $
Weight:	$ k $	$=$	$ 13 $
Character orbit:	$[\chi]$	$=$	83.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$75.8614868339$
Analytic rank:	$0$
Dimension:	$80$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			82.2
Character	$\chi$	$=$	83.82
Dual form			83.13.b.c.82.79

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-124.655i q^{2} -510.562 q^{3} -11442.8 q^{4} +12605.4i q^{5} +63644.0i q^{6} +14598.4 q^{7} +915813. i q^{8} -270767. q^{9} +O(q^{10})$	\(q-124.655i q^{2} -510.562 q^{3} -11442.8 q^{4} +12605.4i q^{5} +63644.0i q^{6} +14598.4 q^{7} +915813. i q^{8} -270767. q^{9} +1.57132e6 q^{10} +979745. q^{11} +5.84226e6 q^{12} +103633. i q^{13} -1.81976e6i q^{14} -6.43585e6i q^{15} +6.72907e7 q^{16} +1.33368e7 q^{17} +3.37524e7i q^{18} -4.09900e6i q^{19} -1.44241e8i q^{20} -7.45338e6 q^{21} -1.22130e8i q^{22} -1.59085e8 q^{23} -4.67580e8i q^{24} +8.52442e7 q^{25} +1.29183e7 q^{26} +4.09577e8 q^{27} -1.67046e8 q^{28} -4.65977e8 q^{29} -8.02259e8 q^{30} +6.14661e8 q^{31} -4.63693e9i q^{32} -5.00221e8 q^{33} -1.66250e9i q^{34} +1.84019e8i q^{35} +3.09833e9 q^{36} +1.36542e9 q^{37} -5.10960e8 q^{38} -5.29109e7i q^{39} -1.15442e10 q^{40} -1.90883e9 q^{41} +9.29099e8i q^{42} +9.43397e9i q^{43} -1.12110e10 q^{44} -3.41313e9i q^{45} +1.98307e10i q^{46} +6.02806e9i q^{47} -3.43561e10 q^{48} -1.36282e10 q^{49} -1.06261e10i q^{50} -6.80927e9 q^{51} -1.18585e9i q^{52} +7.68473e8i q^{53} -5.10557e10i q^{54} +1.23501e10i q^{55} +1.33694e10i q^{56} +2.09280e9i q^{57} +5.80862e10i q^{58} +6.42192e9 q^{59} +7.36441e10i q^{60} +6.61068e10 q^{61} -7.66204e10i q^{62} -3.95276e9 q^{63} -3.02393e11 q^{64} -1.30633e9 q^{65} +6.23549e10i q^{66} -9.18774e10i q^{67} -1.52610e11 q^{68} +8.12230e10 q^{69} +2.29388e10 q^{70} -1.29863e11i q^{71} -2.47972e11i q^{72} -6.98789e10i q^{73} -1.70206e11i q^{74} -4.35225e10 q^{75} +4.69041e10i q^{76} +1.43027e10 q^{77} -6.59559e9 q^{78} +2.84341e11i q^{79} +8.48227e11i q^{80} -6.52180e10 q^{81} +2.37945e11i q^{82} +(-8.94013e10 - 3.14480e11i) q^{83} +8.52875e10 q^{84} +1.68116e11i q^{85} +1.17599e12 q^{86} +2.37910e11 q^{87} +8.97263e11i q^{88} +5.52990e10i q^{89} -4.25463e11 q^{90} +1.51287e9i q^{91} +1.82038e12 q^{92} -3.13823e11 q^{93} +7.51426e11 q^{94} +5.16696e10 q^{95} +2.36744e12i q^{96} -7.09395e11i q^{97} +1.69882e12i q^{98} -2.65283e11 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q + 918 q^{3} - 181150 q^{4} + 103918 q^{7} + 14902142 q^{9}+O(q^{10}) $ 80 * q + 918 * q^3 - 181150 * q^4 + 103918 * q^7 + 14902142 * q^9	\( 80 q + 918 q^{3} - 181150 q^{4} + 103918 q^{7} + 14902142 q^{9} + 1177482 q^{10} + 510406 q^{11} - 9260402 q^{12} + 203791850 q^{16} + 5571718 q^{17} + 46565862 q^{21} + 804389464 q^{23} - 4263713272 q^{25} + 18061794 q^{26} - 2326165338 q^{27} - 204811652 q^{28} + 1486960270 q^{29} - 2621683648 q^{30} - 665743010 q^{31} + 135224502 q^{33} - 54936709824 q^{36} - 280627202 q^{37} + 6646145310 q^{38} - 7119722058 q^{40} + 51318109072 q^{41} - 11512674650 q^{44} + 85368259738 q^{48} + 148785395094 q^{49} - 100143389562 q^{51} - 63584241050 q^{59} - 29216180978 q^{61} - 332932206620 q^{63} - 323596534090 q^{64} + 362112989184 q^{65} - 86115426752 q^{68} + 272383417100 q^{69} + 105630718656 q^{70} + 785418808326 q^{75} - 663355117738 q^{77} + 1483841884620 q^{78} + 2430778545148 q^{81} + 837315119192 q^{83} - 3013574788354 q^{84} + 452180651958 q^{86} - 682263689498 q^{87} - 499714512022 q^{90} + 997428187414 q^{92} - 1487992716298 q^{93} + 3169817690580 q^{94} + 1542762610848 q^{95} + 2021347267420 q^{99}+O(q^{100}) \) 80 * q + 918 * q^3 - 181150 * q^4 + 103918 * q^7 + 14902142 * q^9 + 1177482 * q^10 + 510406 * q^11 - 9260402 * q^12 + 203791850 * q^16 + 5571718 * q^17 + 46565862 * q^21 + 804389464 * q^23 - 4263713272 * q^25 + 18061794 * q^26 - 2326165338 * q^27 - 204811652 * q^28 + 1486960270 * q^29 - 2621683648 * q^30 - 665743010 * q^31 + 135224502 * q^33 - 54936709824 * q^36 - 280627202 * q^37 + 6646145310 * q^38 - 7119722058 * q^40 + 51318109072 * q^41 - 11512674650 * q^44 + 85368259738 * q^48 + 148785395094 * q^49 - 100143389562 * q^51 - 63584241050 * q^59 - 29216180978 * q^61 - 332932206620 * q^63 - 323596534090 * q^64 + 362112989184 * q^65 - 86115426752 * q^68 + 272383417100 * q^69 + 105630718656 * q^70 + 785418808326 * q^75 - 663355117738 * q^77 + 1483841884620 * q^78 + 2430778545148 * q^81 + 837315119192 * q^83 - 3013574788354 * q^84 + 452180651958 * q^86 - 682263689498 * q^87 - 499714512022 * q^90 + 997428187414 * q^92 - 1487992716298 * q^93 + 3169817690580 * q^94 + 1542762610848 * q^95 + 2021347267420 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/83\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	124.655i		−	1.94773i	−0.227128	−	0.973865i	$-0.572934\pi$
$2$		−	124.655i		−	1.94773i	0.227128	−	0.973865i	$-0.427066\pi$
$3$	−510.562			−0.700360			−0.350180	−	0.936682i	$-0.613880\pi$
$3$	−510.562			−0.700360			−0.350180	+	0.936682i	$0.613880\pi$
$4$	−11442.8			−2.79365
$5$			12605.4i			0.806746i	0.915036	+	0.403373i	$0.132162\pi$
$5$			12605.4i			0.806746i	−0.915036	+	0.403373i	$0.867838\pi$
$6$			63644.0i			1.36411i
$7$	14598.4			0.124084			0.0620421	−	0.998074i	$-0.480239\pi$
$7$	14598.4			0.124084			0.0620421	+	0.998074i	$0.480239\pi$
$8$			915813.i			3.49355i
$9$	−270767.			−0.509496
$10$	1.57132e6			1.57132
$11$	979745.			0.553040			0.276520	−	0.961008i	$-0.410819\pi$
$11$	979745.			0.553040			0.276520	+	0.961008i	$0.410819\pi$
$12$	5.84226e6			1.95656
$13$			103633.i			0.0214702i	0.999942	+	0.0107351i	$0.00341716\pi$
$13$			103633.i			0.0214702i	−0.999942	+	0.0107351i	$0.996583\pi$
$14$		−	1.81976e6i		−	0.241682i
$15$		−	6.43585e6i		−	0.565013i
$16$	6.72907e7			4.01084
$17$	1.33368e7			0.552533			0.276267	−	0.961081i	$-0.410903\pi$
$17$	1.33368e7			0.552533			0.276267	+	0.961081i	$0.410903\pi$
$18$			3.37524e7i			0.992361i
$19$		−	4.09900e6i		−	0.0871278i	−0.999051	−	0.0435639i	$-0.986129\pi$
$19$		−	4.09900e6i		−	0.0871278i	0.999051	−	0.0435639i	$-0.0138712\pi$
$20$		−	1.44241e8i		−	2.25377i
$21$	−7.45338e6			−0.0869036
$22$		−	1.22130e8i		−	1.07717i
$23$	−1.59085e8			−1.07464			−0.537320	−	0.843378i	$-0.680563\pi$
$23$	−1.59085e8			−1.07464			−0.537320	+	0.843378i	$0.680563\pi$
$24$		−	4.67580e8i		−	2.44674i
$25$	8.52442e7			0.349160
$26$	1.29183e7			0.0418182
$27$	4.09577e8			1.05719
$28$	−1.67046e8			−0.346648
$29$	−4.65977e8			−0.783387			−0.391694	−	0.920096i	$-0.628111\pi$
$29$	−4.65977e8			−0.783387			−0.391694	+	0.920096i	$0.628111\pi$
$30$	−8.02259e8			−1.10049
$31$	6.14661e8			0.692573			0.346286	−	0.938129i	$-0.387443\pi$
$31$	6.14661e8			0.692573			0.346286	+	0.938129i	$0.387443\pi$
$32$		−	4.63693e9i		−	4.31848i
$33$	−5.00221e8			−0.387327
$34$		−	1.66250e9i		−	1.07619i
$35$			1.84019e8i			0.100104i
$36$	3.09833e9			1.42335
$37$	1.36542e9			0.532178			0.266089	−	0.963948i	$-0.414269\pi$
$37$	1.36542e9			0.532178			0.266089	+	0.963948i	$0.414269\pi$
$38$	−5.10960e8			−0.169701
$39$		−	5.29109e7i		−	0.0150369i
$40$	−1.15442e10			−2.81841
$41$	−1.90883e9			−0.401850			−0.200925	−	0.979607i	$-0.564395\pi$
$41$	−1.90883e9			−0.401850			−0.200925	+	0.979607i	$0.564395\pi$
$42$			9.29099e8i			0.169265i
$43$			9.43397e9i			1.49239i	0.665725	+	0.746197i	$0.268122\pi$
$43$			9.43397e9i			1.49239i	−0.665725	+	0.746197i	$0.731878\pi$
$44$	−1.12110e10			−1.54500
$45$		−	3.41313e9i		−	0.411034i
$46$			1.98307e10i			2.09311i
$47$			6.02806e9i			0.559230i	0.960112	+	0.279615i	$0.0902068\pi$
$47$			6.02806e9i			0.559230i	−0.960112	+	0.279615i	$0.909793\pi$
$48$	−3.43561e10			−2.80903
$49$	−1.36282e10			−0.984603
$50$		−	1.06261e10i		−	0.680070i
$51$	−6.80927e9			−0.386972
$52$		−	1.18585e9i		−	0.0599803i
$53$			7.68473e8i			0.0346716i	0.999850	+	0.0173358i	$0.00551843\pi$
$53$			7.68473e8i			0.0346716i	−0.999850	+	0.0173358i	$0.994482\pi$
$54$		−	5.10557e10i		−	2.05912i
$55$			1.23501e10i			0.446163i
$56$			1.33694e10i			0.433494i
$57$			2.09280e9i			0.0610208i
$58$			5.80862e10i			1.52583i
$59$	6.42192e9			0.152248			0.0761242	−	0.997098i	$-0.475745\pi$
$59$	6.42192e9			0.152248			0.0761242	+	0.997098i	$0.475745\pi$
$60$			7.36441e10i			1.57845i
$61$	6.61068e10			1.28312			0.641560	−	0.767073i	$-0.278288\pi$
$61$	6.61068e10			1.28312			0.641560	+	0.767073i	$0.278288\pi$
$62$		−	7.66204e10i		−	1.34894i
$63$	−3.95276e9			−0.0632204
$64$	−3.02393e11			−4.40039
$65$	−1.30633e9			−0.0173210
$66$			6.23549e10i			0.754409i
$67$		−	9.18774e10i		−	1.01569i	−0.861449	−	0.507844i	$-0.830443\pi$
$67$		−	9.18774e10i		−	1.01569i	0.861449	−	0.507844i	$-0.169557\pi$
$68$	−1.52610e11			−1.54359
$69$	8.12230e10			0.752635
$70$	2.29388e10			0.194976
$71$		−	1.29863e11i		−	1.01376i	−0.862017	−	0.506879i	$-0.830799\pi$
$71$		−	1.29863e11i		−	1.01376i	0.862017	−	0.506879i	$-0.169201\pi$
$72$		−	2.47972e11i		−	1.77995i
$73$		−	6.98789e10i		−	0.461752i	−0.972983	−	0.230876i	$-0.925841\pi$
$73$		−	6.98789e10i		−	0.461752i	0.972983	−	0.230876i	$-0.0741592\pi$
$74$		−	1.70206e11i		−	1.03654i
$75$	−4.35225e10			−0.244538
$76$			4.69041e10i			0.243405i
$77$	1.43027e10			0.0686236
$78$	−6.59559e9			−0.0292878
$79$			2.84341e11i			1.16971i	0.811139	+	0.584854i	$0.198848\pi$
$79$			2.84341e11i			1.16971i	−0.811139	+	0.584854i	$0.801152\pi$
$80$			8.48227e11i			3.23573i
$81$	−6.52180e10			−0.230918
$82$			2.37945e11i			0.782695i
$83$	−8.94013e10	−	3.14480e11i	−0.273448	−	0.961887i
$83$	−8.94013e10	−	3.14480e11i	−0.273448	−	0.961887i
$84$	8.52875e10			0.242778
$85$			1.68116e11i			0.445754i
$86$	1.17599e12			2.90678
$87$	2.37910e11			0.548653
$88$			8.97263e11i			1.93207i
$89$			5.52990e10i			0.111270i	0.998451	+	0.0556349i	$0.0177183\pi$
$89$			5.52990e10i			0.111270i	−0.998451	+	0.0556349i	$0.982282\pi$
$90$	−4.25463e11			−0.800583
$91$			1.51287e9i			0.00266411i
$92$	1.82038e12			3.00217
$93$	−3.13823e11			−0.485050
$94$	7.51426e11			1.08923
$95$	5.16696e10			0.0702900
$96$			2.36744e12i			3.02449i
$97$		−	7.09395e11i		−	0.851643i	−0.904807	−	0.425821i	$-0.859985\pi$
$97$		−	7.09395e11i		−	0.851643i	0.904807	−	0.425821i	$-0.140015\pi$
$98$			1.69882e12i			1.91774i
$99$	−2.65283e11			−0.281772
$100$	−9.75432e11			−0.975432
$101$			1.29110e12i			1.21628i	0.793832	+	0.608138i	$0.208083\pi$
$101$			1.29110e12i			1.21628i	−0.793832	+	0.608138i	$0.791917\pi$
$102$			8.48808e11i			0.753717i
$103$		−	1.08521e12i		−	0.908844i	−0.890786	−	0.454422i	$-0.849846\pi$
$103$		−	1.08521e12i		−	0.908844i	0.890786	−	0.454422i	$-0.150154\pi$
$104$	−9.49081e10			−0.0750072
$105$		−	9.39530e10i		−	0.0701091i
$106$	9.57938e10			0.0675309
$107$			1.00607e12i			0.670388i	0.942149	+	0.335194i	$0.108802\pi$
$107$			1.00607e12i			0.670388i	−0.942149	+	0.335194i	$0.891198\pi$
$108$	−4.68671e12			−2.95342
$109$	−2.17977e12			−1.29972			−0.649862	−	0.760052i	$-0.725173\pi$
$109$	−2.17977e12			−1.29972			−0.649862	+	0.760052i	$0.725173\pi$
$110$	1.53950e12			0.869006
$111$	−6.97133e11			−0.372716
$112$	9.82335e11			0.497681
$113$	3.16572e12			1.52055			0.760276	−	0.649600i	$-0.225064\pi$
$113$	3.16572e12			1.52055			0.760276	+	0.649600i	$0.225064\pi$
$114$	2.60877e11			0.118852
$115$		−	2.00534e12i		−	0.866962i
$116$	5.33208e12			2.18851
$117$		−	2.80603e10i		−	0.0109390i
$118$		−	8.00523e11i		−	0.296539i
$119$	1.94696e11			0.0685606
$120$	5.89403e12			1.97390
$121$	−2.17853e12			−0.694146
$122$		−	8.24052e12i		−	2.49917i
$123$	9.74576e11			0.281440
$124$	−7.03344e12			−1.93481
$125$			4.15203e12i			1.08843i
$126$			4.92730e11i			0.123136i
$127$	−2.90895e12			−0.693289			−0.346645	−	0.937997i	$-0.612679\pi$
$127$	−2.90895e12			−0.693289			−0.346645	+	0.937997i	$0.612679\pi$
$128$			1.87018e13i			4.25230i
$129$		−	4.81663e12i		−	1.04521i
$130$			1.62840e11i			0.0337367i
$131$	−9.30051e12			−1.84026			−0.920130	−	0.391614i	$-0.871917\pi$
$131$	−9.30051e12			−1.84026			−0.920130	+	0.391614i	$0.871917\pi$
$132$	5.72393e12			1.08206
$133$		−	5.98388e10i		−	0.0108112i
$134$	−1.14530e13			−1.97828
$135$			5.16289e12i			0.852885i
$136$			1.22140e13i			1.93030i
$137$		−	2.82659e12i		−	0.427503i	−0.976888	−	0.213751i	$-0.931432\pi$
$137$		−	2.82659e12i		−	0.427503i	0.976888	−	0.213751i	$-0.0685683\pi$
$138$		−	1.01248e13i		−	1.46593i
$139$			3.85766e12i			0.534854i	0.963578	+	0.267427i	$0.0861734\pi$
$139$			3.85766e12i			0.534854i	−0.963578	+	0.267427i	$0.913827\pi$
$140$		−	2.10569e12i		−	0.279657i
$141$		−	3.07770e12i		−	0.391662i
$142$	−1.61880e13			−1.97453
$143$			1.01534e11i			0.0118739i
$144$	−1.82201e13			−2.04351
$145$		−	5.87383e12i		−	0.631995i
$146$	−8.71073e12			−0.899368
$147$	6.95803e12			0.689577
$148$	−1.56243e13			−1.48672
$149$			2.47085e12i			0.225803i	0.993606	+	0.112901i	$0.0360144\pi$
$149$			2.47085e12i			0.225803i	−0.993606	+	0.112901i	$0.963986\pi$
$150$			5.42528e12i			0.476294i
$151$	−1.77164e13			−1.49456			−0.747280	−	0.664509i	$-0.768641\pi$
$151$	−1.77164e13			−1.49456			−0.747280	+	0.664509i	$0.768641\pi$
$152$	3.75392e12			0.304385
$153$	−3.61117e12			−0.281513
$154$		−	1.78290e12i		−	0.133660i
$155$			7.74805e12i			0.558731i
$156$			6.05449e11i			0.0420078i
$157$		−	1.47129e13i		−	0.982428i	−0.871039	−	0.491214i	$-0.836553\pi$
$157$		−	1.47129e13i		−	0.982428i	0.871039	−	0.491214i	$-0.163447\pi$
$158$	3.54445e13			2.27827
$159$		−	3.92354e11i		−	0.0242826i
$160$	5.84505e13			3.48392
$161$	−2.32239e12			−0.133346
$162$			8.12973e12i			0.449766i
$163$			2.14021e13i			1.14112i	0.821257	+	0.570559i	$0.193273\pi$
$163$			2.14021e13i			1.14112i	−0.821257	+	0.570559i	$0.806727\pi$
$164$	2.18423e13			1.12263
$165$		−	6.30549e12i		−	0.312475i
$166$	−3.92014e13	+	1.11443e13i	−1.87350	+	0.532603i
$167$	1.14040e13			0.525725			0.262862	−	0.964833i	$-0.415333\pi$
$167$	1.14040e13			0.525725			0.262862	+	0.964833i	$0.415333\pi$
$168$		−	6.82590e12i		−	0.303602i
$169$	2.32873e13			0.999539
$170$	2.09565e13			0.868209
$171$			1.10988e12i			0.0443913i
$172$		−	1.07951e14i		−	4.16923i
$173$	−1.59252e13			−0.594030			−0.297015	−	0.954873i	$-0.595991\pi$
$173$	−1.59252e13			−0.594030			−0.297015	+	0.954873i	$0.595991\pi$
$174$		−	2.96566e13i		−	1.06863i
$175$	1.24443e12			0.0433252
$176$	6.59277e13			2.21816
$177$	−3.27879e12			−0.106629
$178$	6.89328e12			0.216723
$179$		−	6.49794e12i		−	0.197541i	−0.995110	−	0.0987706i	$-0.968509\pi$
$179$		−	6.49794e12i		−	0.197541i	0.995110	−	0.0987706i	$-0.0314910\pi$
$180$			3.90558e13i			1.14829i
$181$		−	6.60480e12i		−	0.187840i	−0.995580	−	0.0939201i	$-0.970060\pi$
$181$		−	6.60480e12i		−	0.187840i	0.995580	−	0.0939201i	$-0.0299398\pi$
$182$	1.88586e11			0.00518897
$183$	−3.37516e13			−0.898645
$184$		−	1.45692e14i		−	3.75431i
$185$			1.72117e13i			0.429333i
$186$			3.91195e13i			0.944747i
$187$	1.30667e13			0.305573
$188$		−	6.89779e13i		−	1.56229i
$189$	5.97916e12			0.131181
$190$		−	6.44086e12i		−	0.136906i
$191$	−5.56902e13			−1.14704			−0.573520	−	0.819191i	$-0.694423\pi$
$191$	−5.56902e13			−1.14704			−0.573520	+	0.819191i	$0.694423\pi$
$192$	1.54390e14			3.08186
$193$	−8.46667e13			−1.63821			−0.819104	−	0.573645i	$-0.805529\pi$
$193$	−8.46667e13			−1.63821			−0.819104	+	0.573645i	$0.805529\pi$
$194$	−8.84294e13			−1.65877
$195$	6.66964e11			0.0121309
$196$	1.55944e14			2.75064
$197$	−3.02651e13			−0.517779			−0.258890	−	0.965907i	$-0.583357\pi$
$197$	−3.02651e13			−0.517779			−0.258890	+	0.965907i	$0.583357\pi$
$198$			3.30687e13i			0.548816i
$199$	2.34778e13			0.378040			0.189020	−	0.981973i	$-0.439469\pi$
$199$	2.34778e13			0.378040			0.189020	+	0.981973i	$0.439469\pi$
$200$			7.80677e13i			1.21981i
$201$			4.69092e13i			0.711347i
$202$	1.60942e14			2.36898
$203$	−6.80251e12			−0.0972059
$204$	7.79171e13			1.08107
$205$		−	2.40616e13i		−	0.324191i
$206$	−1.35276e14			−1.77018
$207$	4.30751e13			0.547525
$208$			6.97351e12i			0.0861135i
$209$		−	4.01598e12i		−	0.0481852i
$210$	−1.17117e13			−0.136554
$211$		−	1.53508e13i		−	0.173955i	−0.996210	−	0.0869776i	$-0.972279\pi$
$211$		−	1.53508e13i		−	0.173955i	0.996210	−	0.0869776i	$-0.0277209\pi$
$212$		−	8.79348e12i		−	0.0968603i
$213$			6.63030e13i			0.709995i
$214$	1.25412e14			1.30573
$215$	−1.18919e14			−1.20398
$216$			3.75096e14i			3.69335i
$217$	8.97305e12			0.0859373
$218$			2.71718e14i			2.53151i
$219$			3.56775e13i			0.323393i
$220$		−	1.41320e14i		−	1.24643i
$221$			1.38213e12i			0.0118630i
$222$			8.69010e13i			0.725950i
$223$		−	1.00942e14i		−	0.820811i	−0.911903	−	0.410405i	$-0.865387\pi$
$223$		−	1.00942e14i		−	0.820811i	0.911903	−	0.410405i	$-0.134613\pi$
$224$		−	6.76917e13i		−	0.535855i
$225$	−2.30813e13			−0.177896
$226$		−	3.94621e14i		−	2.96163i
$227$	−2.18418e14			−1.59637			−0.798185	−	0.602413i	$-0.794206\pi$
$227$	−2.18418e14			−1.59637			−0.798185	+	0.602413i	$0.794206\pi$
$228$		−	2.39475e13i		−	0.170471i
$229$	−7.04090e13			−0.488219			−0.244110	−	0.969748i	$-0.578496\pi$
$229$	−7.04090e13			−0.488219			−0.244110	+	0.969748i	$0.578496\pi$
$230$	−2.49975e14			−1.68861
$231$	−7.30241e12			−0.0480612
$232$		−	4.26748e14i		−	2.73680i
$233$		−	2.44142e14i		−	1.52583i	−0.646496	−	0.762917i	$-0.723766\pi$
$233$		−	2.44142e14i		−	1.52583i	0.646496	−	0.762917i	$-0.276234\pi$
$234$	−3.49785e12			−0.0213062
$235$	−7.59862e13			−0.451157
$236$	−7.34848e13			−0.425329
$237$		−	1.45174e14i		−	0.819216i
$238$		−	2.42697e13i		−	0.133538i
$239$		−	9.99201e13i		−	0.536124i	−0.963402	−	0.268062i	$-0.913617\pi$
$239$		−	9.99201e13i		−	0.536124i	0.963402	−	0.268062i	$-0.0863832\pi$
$240$		−	4.33073e14i		−	2.26618i
$241$	−1.69184e14			−0.863490			−0.431745	−	0.901996i	$-0.642102\pi$
$241$	−1.69184e14			−0.863490			−0.431745	+	0.901996i	$0.642102\pi$
$242$			2.71564e14i			1.35201i
$243$	−1.84368e14			−0.895465
$244$	−7.56446e14			−3.58459
$245$		−	1.71789e14i		−	0.794325i
$246$		−	1.21486e14i		−	0.548168i
$247$	4.24790e11			0.00187065
$248$			5.62914e14i			2.41954i
$249$	4.56449e13	+	1.60561e14i	0.191512	+	0.673667i
$250$	5.17570e14			2.11997
$251$		−	1.36110e14i		−	0.544312i	−0.962253	−	0.272156i	$-0.912263\pi$
$251$		−	1.36110e14i		−	0.544312i	0.962253	−	0.272156i	$-0.0877367\pi$
$252$	4.52306e13			0.176616
$253$	−1.55863e14			−0.594319
$254$			3.62615e14i			1.35034i
$255$		−	8.58337e13i		−	0.312188i
$256$	1.09267e15			3.88194
$257$		−	2.22793e14i		−	0.773218i	−0.922244	−	0.386609i	$-0.873646\pi$
$257$		−	2.22793e14i		−	0.773218i	0.922244	−	0.386609i	$-0.126354\pi$
$258$	−6.00415e14			−2.03579
$259$	1.99330e13			0.0660348
$260$	1.49481e13			0.0483889
$261$	1.26171e14			0.399133
$262$			1.15935e15i			3.58433i
$263$			1.85861e14i			0.561634i	0.959761	+	0.280817i	$0.0906053\pi$
$263$			1.85861e14i			0.561634i	−0.959761	+	0.280817i	$0.909395\pi$
$264$		−	4.58109e14i		−	1.35315i
$265$	−9.68692e12			−0.0279712
$266$	−7.45919e12			−0.0210573
$267$		−	2.82336e13i		−	0.0779289i
$268$			1.05133e15i			2.83748i
$269$		−	6.36423e14i		−	1.67970i	−0.542818	−	0.839850i	$-0.682643\pi$
$269$		−	6.36423e14i		−	1.67970i	0.542818	−	0.839850i	$-0.317357\pi$
$270$	6.43579e14			1.66119
$271$		−	7.57330e14i		−	1.91192i	−0.293498	−	0.955960i	$-0.594819\pi$
$271$		−	7.57330e14i		−	1.91192i	0.293498	−	0.955960i	$-0.405181\pi$
$272$	8.97443e14			2.21612
$273$		−	7.72413e11i		−	0.00186584i
$274$	−3.52347e14			−0.832660
$275$	8.35176e13			0.193100
$276$	−9.29418e14			−2.10260
$277$	3.57138e14			0.790601			0.395301	−	0.918552i	$-0.370640\pi$
$277$	3.57138e14			0.790601			0.395301	+	0.918552i	$0.370640\pi$
$278$	4.80876e14			1.04175
$279$	−1.66430e14			−0.352863
$280$	−1.68527e14			−0.349720
$281$			6.30135e14i			1.27996i	0.768392	+	0.639979i	$0.221057\pi$
$281$			6.30135e14i			1.27996i	−0.768392	+	0.639979i	$0.778943\pi$
$282$	−3.83650e14			−0.762852
$283$			4.42149e14i			0.860696i	0.902663	+	0.430348i	$0.141609\pi$
$283$			4.42149e14i			0.860696i	−0.902663	+	0.430348i	$0.858391\pi$
$284$			1.48599e15i			2.83209i
$285$	−2.63806e13			−0.0492283
$286$	1.26566e13			0.0231271
$287$	−2.78658e13			−0.0498632
$288$			1.25553e15i			2.20025i
$289$	−4.04752e14			−0.694707
$290$	−7.32201e14			−1.23096
$291$			3.62190e14i			0.596456i
$292$			7.99610e14i			1.28997i
$293$	3.68169e13			0.0581891			0.0290946	−	0.999577i	$-0.490738\pi$
$293$	3.68169e13			0.0581891			0.0290946	+	0.999577i	$0.490738\pi$
$294$		−	8.67352e14i		−	1.34311i
$295$			8.09510e13i			0.122826i
$296$			1.25047e15i			1.85919i
$297$	4.01281e14			0.584669
$298$	3.08003e14			0.439803
$299$		−	1.64864e13i		−	0.0230727i
$300$	4.98019e14			0.683153
$301$			1.37721e14i			0.185183i
$302$			2.20843e15i			2.91100i
$303$		−	6.59187e14i		−	0.851830i
$304$		−	2.75825e14i		−	0.349456i
$305$			8.33303e14i			1.03515i
$306$			4.50149e14i			0.548312i
$307$		−	5.01771e14i		−	0.599342i	−0.954043	−	0.299671i	$-0.903123\pi$
$307$		−	5.01771e14i		−	0.599342i	0.954043	−	0.299671i	$-0.0968770\pi$
$308$	−1.63663e14			−0.191710
$309$			5.54066e14i			0.636518i
$310$	9.65831e14			1.08826
$311$		−	1.56927e15i		−	1.73435i	−0.498008	−	0.867173i	$-0.665935\pi$
$311$		−	1.56927e15i		−	1.73435i	0.498008	−	0.867173i	$-0.334065\pi$
$312$	4.84565e13			0.0525321
$313$	1.47749e15			1.57130			0.785648	−	0.618673i	$-0.212330\pi$
$313$	1.47749e15			1.57130			0.785648	+	0.618673i	$0.212330\pi$
$314$	−1.83403e15			−1.91350
$315$		−	4.98262e13i		−	0.0510028i
$316$		−	3.25366e15i		−	3.26775i
$317$	1.60842e15			1.58505			0.792527	−	0.609837i	$-0.208765\pi$
$317$	1.60842e15			1.58505			0.792527	+	0.609837i	$0.208765\pi$
$318$	−4.89087e13			−0.0472959
$319$	−4.56539e14			−0.433245
$320$		−	3.81179e15i		−	3.55000i
$321$		−	5.13662e14i		−	0.469513i
$322$			2.89497e14i			0.259722i
$323$		−	5.46676e13i		−	0.0481410i
$324$	7.46277e14			0.645104
$325$			8.83408e12i			0.00749654i
$326$	2.66787e15			2.22259
$327$	1.11291e15			0.910275
$328$		−	1.74813e15i		−	1.40388i
$329$			8.79999e13i			0.0693916i
$330$	−7.86009e14			−0.608617
$331$		−	1.72139e14i		−	0.130892i	−0.997856	−	0.0654459i	$-0.979153\pi$
$331$		−	1.72139e14i		−	0.130892i	0.997856	−	0.0654459i	$-0.0208470\pi$
$332$	1.02300e15	+	3.59853e15i	0.763919	+	2.68718i
$333$	−3.69711e14			−0.271142
$334$		−	1.42156e15i		−	1.02397i
$335$	1.15815e15			0.819402
$336$	−5.01543e14			−0.348556
$337$			1.33764e14i			0.0913184i	0.998957	+	0.0456592i	$0.0145388\pi$
$337$			1.33764e14i			0.0913184i	−0.998957	+	0.0456592i	$0.985461\pi$
$338$		−	2.90288e15i		−	1.94683i
$339$	−1.61630e15			−1.06493
$340$		−	1.92372e15i		−	1.24528i
$341$	6.02211e14			0.383021
$342$	1.38351e14			0.0864622
$343$	−4.01010e14			−0.246258
$344$	−8.63975e15			−5.21375
$345$			1.02385e15i			0.607185i
$346$			1.98515e15i			1.15701i
$347$		−	1.11446e15i		−	0.638390i	−0.947689	−	0.319195i	$-0.896587\pi$
$347$		−	1.11446e15i		−	0.638390i	0.947689	−	0.319195i	$-0.103413\pi$
$348$	−2.72236e15			−1.53275
$349$	−9.30485e13			−0.0514940			−0.0257470	−	0.999668i	$-0.508196\pi$
$349$	−9.30485e13			−0.0514940			−0.0257470	+	0.999668i	$0.508196\pi$
$350$		−	1.55124e14i		−	0.0843859i
$351$			4.24456e13i			0.0226981i
$352$		−	4.54301e15i		−	2.38829i
$353$	−2.79157e15			−1.44278			−0.721389	−	0.692530i	$-0.756496\pi$
$353$	−2.79157e15			−1.44278			−0.721389	+	0.692530i	$0.756496\pi$
$354$			4.08717e14i			0.207684i
$355$	1.63697e15			0.817846
$356$		−	6.32775e14i		−	0.310849i
$357$	−9.94043e13			−0.0480171
$358$	−8.09999e14			−0.384757
$359$	2.62947e15			1.22829			0.614147	−	0.789192i	$-0.289500\pi$
$359$	2.62947e15			1.22829			0.614147	+	0.789192i	$0.289500\pi$
$360$	3.12579e15			1.43597
$361$	2.19651e15			0.992409
$362$	−8.23320e14			−0.365862
$363$	1.11227e15			0.486152
$364$		−	1.73114e13i		−	0.00744260i
$365$	8.80852e14			0.372517
$366$			4.20730e15i			1.75032i
$367$			2.12359e15i			0.869107i	0.900646	+	0.434554i	$0.143094\pi$
$367$			2.12359e15i			0.869107i	−0.900646	+	0.434554i	$0.856906\pi$
$368$	−1.07050e16			−4.31021
$369$	5.16848e14			0.204741
$370$	2.14552e15			0.836224
$371$			1.12185e13i			0.00430219i
$372$	3.59101e15			1.35506
$373$	−7.70165e14			−0.285977			−0.142988	−	0.989724i	$-0.545671\pi$
$373$	−7.70165e14			−0.285977			−0.142988	+	0.989724i	$0.545671\pi$
$374$		−	1.62882e15i		−	0.595174i
$375$		−	2.11987e15i		−	0.762293i
$376$	−5.52058e15			−1.95370
$377$		−	4.82904e13i		−	0.0168195i
$378$		−	7.45331e14i		−	0.255504i
$379$			3.72341e15i			1.25633i	0.778078	+	0.628167i	$0.216195\pi$
$379$			3.72341e15i			1.25633i	−0.778078	+	0.628167i	$0.783805\pi$
$380$	−5.91245e14			−0.196366
$381$	1.48520e15			0.485552
$382$			6.94205e15i			2.23412i
$383$	−4.00619e15			−1.26923			−0.634613	−	0.772830i	$-0.718840\pi$
$383$	−4.00619e15			−1.26923			−0.634613	+	0.772830i	$0.718840\pi$
$384$		−	9.54844e15i		−	2.97814i
$385$			1.80291e14i			0.0553618i
$386$			1.05541e16i			3.19079i
$387$		−	2.55441e15i		−	0.760369i
$388$			8.11746e15i			2.37919i
$389$		−	3.51997e15i		−	1.01588i	−0.861393	−	0.507939i	$-0.830407\pi$
$389$		−	3.51997e15i		−	1.01588i	0.861393	−	0.507939i	$-0.169593\pi$
$390$		−	8.31402e13i		−	0.0236278i
$391$	−2.12169e15			−0.593774
$392$		−	1.24809e16i		−	3.43976i
$393$	4.74849e15			1.28884
$394$			3.77269e15i			1.00849i
$395$	−3.58424e15			−0.943657
$396$	3.03558e15			0.787173
$397$	4.04496e14			0.103317			0.0516584	−	0.998665i	$-0.483549\pi$
$397$	4.04496e14			0.103317			0.0516584	+	0.998665i	$0.483549\pi$
$398$		−	2.92661e15i		−	0.736321i
$399$			3.05514e13i			0.00757172i
$400$	5.73614e15			1.40042
$401$	1.01301e15			0.243640			0.121820	−	0.992552i	$-0.461127\pi$
$401$	1.01301e15			0.243640			0.121820	+	0.992552i	$0.461127\pi$
$402$	5.84745e15			1.38551
$403$			6.36989e13i			0.0148697i
$404$		−	1.47738e16i		−	3.39785i
$405$		−	8.22100e14i		−	0.186292i
$406$			8.47965e14i			0.189331i
$407$	1.33777e15			0.294316
$408$		−	6.23602e15i		−	1.35191i
$409$	−1.59192e15			−0.340079			−0.170040	−	0.985437i	$-0.554390\pi$
$409$	−1.59192e15			−0.340079			−0.170040	+	0.985437i	$0.554390\pi$
$410$	−2.99939e15			−0.631436
$411$			1.44315e15i			0.299406i
$412$			1.24178e16i			2.53899i
$413$	9.37496e13			0.0188916
$414$		−	5.36951e15i		−	1.06643i
$415$	3.96415e15	−	1.12694e15i	0.775999	−	0.220603i
$416$	4.80537e14			0.0927187
$417$		−	1.96958e15i		−	0.374590i
$418$	−5.00611e14			−0.0938518
$419$	6.16159e15			1.13870			0.569349	−	0.822096i	$-0.307195\pi$
$419$	6.16159e15			1.13870			0.569349	+	0.822096i	$0.307195\pi$
$420$			1.07508e15i			0.195861i
$421$			1.97775e15i			0.355206i	0.984102	+	0.177603i	$0.0568342\pi$
$421$			1.97775e15i			0.355206i	−0.984102	+	0.177603i	$0.943166\pi$
$422$	−1.91356e15			−0.338818
$423$		−	1.63220e15i		−	0.284925i
$424$	−7.03778e14			−0.121127
$425$	1.13689e15			0.192923
$426$	8.26498e15			1.38288
$427$	9.65052e14			0.159215
$428$		−	1.15123e16i		−	1.87283i
$429$		−	5.18392e13i		−	0.00831600i
$430$			1.48238e16i			2.34504i
$431$	−8.14465e15			−1.27060			−0.635300	−	0.772266i	$-0.719124\pi$
$431$	−8.14465e15			−1.27060			−0.635300	+	0.772266i	$0.719124\pi$
$432$	2.75607e16			4.24022
$433$			7.80835e15i			1.18476i	0.805657	+	0.592382i	$0.201812\pi$
$433$			7.80835e15i			1.18476i	−0.805657	+	0.592382i	$0.798188\pi$
$434$		−	1.11853e15i		−	0.167383i
$435$			2.99896e15i			0.442624i
$436$	2.49427e16			3.63098
$437$			6.52091e14i			0.0936310i
$438$	4.44737e15			0.629881
$439$		−	1.23112e16i		−	1.71994i	−0.510345	−	0.859970i	$-0.670482\pi$
$439$		−	1.23112e16i		−	1.71994i	0.510345	−	0.859970i	$-0.329518\pi$
$440$	−1.13104e16			−1.55869
$441$	3.69006e15			0.501651
$442$	1.72289e14			0.0231059
$443$	1.21804e16			1.61154			0.805769	−	0.592230i	$-0.201752\pi$
$443$	1.21804e16			1.61154			0.805769	+	0.592230i	$0.201752\pi$
$444$	7.97716e15			1.04124
$445$	−6.97067e14			−0.0897665
$446$	−1.25829e16			−1.59872
$447$		−	1.26152e15i		−	0.158143i
$448$	−4.41444e15			−0.546019
$449$		−	1.19649e16i		−	1.46026i	−0.683307	−	0.730131i	$-0.739459\pi$
$449$		−	1.19649e16i		−	1.46026i	0.683307	−	0.730131i	$-0.260541\pi$
$450$			2.87719e15i			0.346493i
$451$	−1.87017e15			−0.222239
$452$	−3.62247e16			−4.24789
$453$	9.04532e15			1.04673
$454$			2.72268e16i			3.10930i
$455$	−1.90703e13			−0.00214926
$456$	−1.91661e15			−0.213179
$457$		−	5.20157e15i		−	0.571001i	−0.958379	−	0.285501i	$-0.907840\pi$
$457$		−	5.20157e15i		−	0.571001i	0.958379	−	0.285501i	$-0.0921599\pi$
$458$			8.77681e15i			0.950920i
$459$	5.46245e15			0.584133
$460$			2.29467e16i			2.42199i
$461$			8.20988e15i			0.855325i	0.903938	+	0.427663i	$0.140663\pi$
$461$			8.20988e15i			0.855325i	−0.903938	+	0.427663i	$0.859337\pi$
$462$			9.10280e14i			0.0936102i
$463$	1.14783e16			1.16518			0.582590	−	0.812766i	$-0.302039\pi$
$463$	1.14783e16			1.16518			0.582590	+	0.812766i	$0.302039\pi$
$464$	−3.13559e16			−3.14204
$465$		−	3.95587e15i		−	0.391313i
$466$	−3.04335e16			−2.97191
$467$		−	1.93815e16i		−	1.86847i	−0.356652	−	0.934237i	$-0.616082\pi$
$467$		−	1.93815e16i		−	1.86847i	0.356652	−	0.934237i	$-0.383918\pi$
$468$			3.21088e14i			0.0305597i
$469$		−	1.34126e15i		−	0.126031i
$470$			9.47204e15i			0.878732i
$471$			7.51186e15i			0.688053i
$472$			5.88128e15i			0.531888i
$473$			9.24288e15i			0.825355i
$474$	−1.80966e16			−1.59561
$475$		−	3.49416e14i		−	0.0304216i
$476$	−2.22786e15			−0.191534
$477$		−	2.08077e14i		−	0.0176650i
$478$	−1.24555e16			−1.04423
$479$	−1.67218e16			−1.38443			−0.692214	−	0.721692i	$-0.743365\pi$
$479$	−1.67218e16			−1.38443			−0.692214	+	0.721692i	$0.743365\pi$
$480$	−2.98426e16			−2.44000
$481$			1.41502e14i			0.0114260i
$482$			2.10896e16i			1.68184i
$483$	1.18572e15			0.0933901
$484$	2.49285e16			1.93920
$485$	8.94221e15			0.687060
$486$			2.29824e16i			1.74412i
$487$		−	1.25596e16i		−	0.941457i	−0.882278	−	0.470729i	$-0.843991\pi$
$487$		−	1.25596e16i		−	0.941457i	0.882278	−	0.470729i	$-0.156009\pi$
$488$			6.05414e16i			4.48264i
$489$		−	1.09271e16i		−	0.799193i
$490$	−2.14143e16			−1.54713
$491$		−	2.72294e16i		−	1.94334i	−0.236333	−	0.971672i	$-0.575946\pi$
$491$		−	2.72294e16i		−	1.94334i	0.236333	−	0.971672i	$-0.424054\pi$
$492$	−1.11519e16			−0.786244
$493$	−6.21465e15			−0.432847
$494$		−	5.29521e13i		−	0.00364353i
$495$		−	3.34400e15i		−	0.227318i
$496$	4.13610e16			2.77780
$497$		−	1.89578e15i		−	0.125791i
$498$	2.00147e16	−	5.68985e15i	1.31212	−	0.373014i
$499$	1.41235e16			0.914824			0.457412	−	0.889255i	$-0.348776\pi$
$499$	1.41235e16			0.914824			0.457412	+	0.889255i	$0.348776\pi$
$500$		−	4.75109e16i		−	3.04069i
$501$	−5.82245e15			−0.368196
$502$	−1.69668e16			−1.06017
$503$			5.38944e14i			0.0332763i	0.999862	+	0.0166382i	$0.00529634\pi$
$503$			5.38944e14i			0.0332763i	−0.999862	+	0.0166382i	$0.994704\pi$
$504$		−	3.61999e15i		−	0.220863i
$505$	−1.62749e16			−0.981226
$506$			1.94291e16i			1.15757i
$507$	−1.18896e16			−0.700037
$508$	3.32866e16			1.93681
$509$	−4.43625e15			−0.255099			−0.127550	−	0.991832i	$-0.540711\pi$
$509$	−4.43625e15			−0.255099			−0.127550	+	0.991832i	$0.540711\pi$
$510$	−1.06996e16			−0.608059
$511$		−	1.02012e15i		−	0.0572961i
$512$		−	5.96036e16i		−	3.30867i
$513$		−	1.67886e15i		−	0.0921107i
$514$	−2.77722e16			−1.50602
$515$	1.36795e16			0.733207
$516$			5.51157e16i			2.91996i
$517$			5.90596e15i			0.309277i
$518$		−	2.48474e15i		−	0.128618i
$519$	8.13080e15			0.416035
$520$		−	1.19636e15i		−	0.0605118i
$521$	1.30131e15			0.0650661			0.0325330	−	0.999471i	$-0.489643\pi$
$521$	1.30131e15			0.0650661			0.0325330	+	0.999471i	$0.489643\pi$
$522$		−	1.57278e16i		−	0.777403i
$523$	−1.05660e16			−0.516297			−0.258149	−	0.966105i	$-0.583112\pi$
$523$	−1.05660e16			−0.516297			−0.258149	+	0.966105i	$0.583112\pi$
$524$	1.06424e17			5.14104
$525$	−6.35357e14			−0.0303433
$526$	2.31684e16			1.09391
$527$	8.19762e15			0.382669
$528$	−3.36602e16			−1.55351
$529$	3.39351e15			0.154851
$530$			1.20752e15i			0.0544803i
$531$	−1.73884e15			−0.0775700
$532$			6.84723e14i			0.0302027i
$533$		−	1.97817e14i		−	0.00862780i
$534$	−3.51945e15			−0.151784
$535$	−1.26819e16			−0.540833
$536$	8.41425e16			3.54835
$537$			3.31760e15i			0.138350i
$538$	−7.93331e16			−3.27160
$539$	−1.33521e16			−0.544525
$540$		−	5.90779e16i		−	2.38266i
$541$		−	2.67192e16i		−	1.06571i	−0.846205	−	0.532857i	$-0.821118\pi$
$541$		−	2.67192e16i		−	1.06571i	0.846205	−	0.532857i	$-0.178882\pi$
$542$	−9.44048e16			−3.72390
$543$			3.37216e15i			0.131556i
$544$		−	6.18419e16i		−	2.38610i
$545$		−	2.74769e16i		−	1.04855i
$546$	−9.62850e13			−0.00363415
$547$	−3.54608e16			−1.32381			−0.661904	−	0.749589i	$-0.730251\pi$
$547$	−3.54608e16			−1.32381			−0.661904	+	0.749589i	$0.730251\pi$
$548$			3.23441e16i			1.19429i
$549$	−1.78995e16			−0.653744
$550$		−	1.04109e16i		−	0.376106i
$551$			1.91004e15i			0.0682548i
$552$			7.43850e16i			2.62937i
$553$			4.15092e15i			0.145142i
$554$		−	4.45189e16i		−	1.53988i
$555$		−	8.78765e15i		−	0.300687i
$556$		−	4.41424e16i		−	1.49420i
$557$	−5.52205e16			−1.84914			−0.924569	−	0.381013i	$-0.875575\pi$
$557$	−5.52205e16			−1.84914			−0.924569	+	0.381013i	$0.875575\pi$
$558$			2.07463e16i			0.687282i
$559$	−9.77667e14			−0.0320420
$560$			1.23827e16i			0.401503i
$561$	−6.67135e15			−0.214011
$562$	7.85493e16			2.49301
$563$	−3.88023e16			−1.21845			−0.609225	−	0.792998i	$-0.708519\pi$
$563$	−3.88023e16			−1.21845			−0.609225	+	0.792998i	$0.708519\pi$
$564$			3.52175e16i			1.09417i
$565$			3.99052e16i			1.22670i
$566$	5.51159e16			1.67640
$567$	−9.52077e14			−0.0286532
$568$	1.18930e17			3.54161
$569$			5.71371e16i			1.68362i	0.539773	+	0.841811i	$0.318510\pi$
$569$			5.71371e16i			1.68362i	−0.539773	+	0.841811i	$0.681490\pi$
$570$			3.28846e15i			0.0958835i
$571$		−	2.81864e16i		−	0.813248i	−0.913596	−	0.406624i	$-0.866706\pi$
$571$		−	2.81864e16i		−	0.813248i	0.913596	−	0.406624i	$-0.133294\pi$
$572$		−	1.16183e15i		−	0.0331715i
$573$	2.84333e16			0.803341
$574$			3.47360e15i			0.0971201i
$575$	−1.35611e16			−0.375222
$576$	8.18780e16			2.24198
$577$		−	2.11746e16i		−	0.573800i	−0.957961	−	0.286900i	$-0.907375\pi$
$577$		−	2.11746e16i		−	0.573800i	0.957961	−	0.286900i	$-0.0926247\pi$
$578$			5.04542e16i			1.35310i
$579$	4.32277e16			1.14734
$580$			6.72131e16i			1.76557i
$581$	−1.30511e15	−	4.59089e15i	−0.0339306	−	0.119355i
$582$	4.51487e16			1.16174
$583$			7.52908e14i			0.0191748i
$584$	6.39960e16			1.61315
$585$	3.53712e14			0.00882499
$586$		−	4.58940e15i		−	0.113337i
$587$			2.53641e16i			0.619999i	0.950737	+	0.309999i	$0.100329\pi$
$587$			2.53641e16i			0.619999i	−0.950737	+	0.309999i	$0.899671\pi$
$588$	−7.96194e16			−1.92644
$589$		−	2.51950e15i		−	0.0603423i
$590$	1.00909e16			0.239232
$591$	1.54522e16			0.362632
$592$	9.18802e16			2.13448
$593$	−6.33209e16			−1.45619			−0.728096	−	0.685475i	$-0.759594\pi$
$593$	−6.33209e16			−1.45619			−0.728096	+	0.685475i	$0.759594\pi$
$594$		−	5.00216e16i		−	1.13878i
$595$			2.45422e15i			0.0553110i
$596$		−	2.82735e16i		−	0.630814i
$597$	−1.19869e16			−0.264764
$598$	−2.05511e15			−0.0449395
$599$		−	4.96510e16i		−	1.07490i	−0.843297	−	0.537448i	$-0.819388\pi$
$599$		−	4.96510e16i		−	1.07490i	0.843297	−	0.537448i	$-0.180612\pi$
$600$		−	3.98584e16i		−	0.854305i
$601$		−	1.00830e16i		−	0.213964i	−0.994261	−	0.106982i	$-0.965881\pi$
$601$		−	1.00830e16i		−	0.213964i	0.994261	−	0.106982i	$-0.0341187\pi$
$602$	1.71675e16			0.360685
$603$			2.48774e16i			0.517489i
$604$	2.02725e17			4.17528
$605$		−	2.74612e16i		−	0.560000i
$606$	−8.21708e16			−1.65914
$607$	2.84690e15			0.0569167			0.0284583	−	0.999595i	$-0.490940\pi$
$607$	2.84690e15			0.0569167			0.0284583	+	0.999595i	$0.490940\pi$
$608$	−1.90068e16			−0.376260
$609$	3.47310e15			0.0680791
$610$	1.03875e17			2.01620
$611$	−6.24704e14			−0.0120068
$612$	4.13219e16			0.786451
$613$		−	2.99189e16i		−	0.563874i	−0.959433	−	0.281937i	$-0.909023\pi$
$613$		−	2.99189e16i		−	0.563874i	0.959433	−	0.281937i	$-0.0909769\pi$
$614$	−6.25481e16			−1.16736
$615$			1.22849e16i			0.227050i
$616$			1.30986e16i			0.239740i
$617$	−6.38333e16			−1.15701			−0.578504	−	0.815680i	$-0.696363\pi$
$617$	−6.38333e16			−1.15701			−0.578504	+	0.815680i	$0.696363\pi$
$618$	6.90670e16			1.23977
$619$	3.57647e16			0.635785			0.317892	−	0.948127i	$-0.397025\pi$
$619$	3.57647e16			0.635785			0.317892	+	0.948127i	$0.397025\pi$
$620$		−	8.86594e16i		−	1.56090i
$621$	−6.51577e16			−1.13610
$622$	−1.95617e17			−3.37804
$623$			8.07275e14i			0.0138068i
$624$		−	3.56041e15i		−	0.0603105i
$625$	−3.15265e16			−0.528927
$626$		−	1.84176e17i		−	3.06046i
$627$			2.05041e15i			0.0337470i
$628$			1.68357e17i			2.74456i
$629$	1.82104e16			0.294046
$630$	−6.21107e15			−0.0993397
$631$			5.66894e15i			0.0898102i	0.998991	+	0.0449051i	$0.0142985\pi$
$631$			5.66894e15i			0.0898102i	−0.998991	+	0.0449051i	$0.985701\pi$
$632$	−2.60403e17			−4.08643
$633$			7.83756e15i			0.121831i
$634$		−	2.00497e17i		−	3.08726i
$635$		−	3.66685e16i		−	0.559308i
$636$			4.48962e15i			0.0678371i
$637$		−	1.41232e15i		−	0.0211396i
$638$			5.69097e16i			0.843844i
$639$			3.51625e16i			0.516506i
$640$	−2.35744e17			−3.43053
$641$			6.51126e16i			0.938678i	0.883018	+	0.469339i	$0.155508\pi$
$641$			6.51126e16i			0.938678i	−0.883018	+	0.469339i	$0.844492\pi$
$642$	−6.40304e16			−0.914484
$643$		−	3.85714e16i		−	0.545757i	−0.962048	−	0.272879i	$-0.912024\pi$
$643$		−	3.85714e16i		−	0.545757i	0.962048	−	0.272879i	$-0.0879758\pi$
$644$	2.65746e16			0.372522
$645$	6.07156e16			0.843222
$646$	−6.81458e15			−0.0937657
$647$			7.08265e16i			0.965540i	0.875747	+	0.482770i	$0.160369\pi$
$647$			7.08265e16i			0.965540i	−0.875747	+	0.482770i	$0.839631\pi$
$648$		−	5.97275e16i		−	0.806723i
$649$	6.29185e15			0.0841996
$650$	1.10121e15			0.0146012
$651$	−4.58130e15			−0.0601870
$652$		−	2.44900e17i		−	3.18789i
$653$			9.88149e16i			1.27451i	0.770653	+	0.637255i	$0.219930\pi$
$653$			9.88149e16i			1.27451i	−0.770653	+	0.637255i	$0.780070\pi$
$654$		−	1.38729e17i		−	1.77297i
$655$		−	1.17237e17i		−	1.48462i
$656$	−1.28446e17			−1.61175
$657$			1.89209e16i			0.235261i
$658$	1.09696e16			0.135156
$659$	6.09770e16			0.744481			0.372241	−	0.928136i	$-0.378590\pi$
$659$	6.09770e16			0.744481			0.372241	+	0.928136i	$0.378590\pi$
$660$			7.21525e16i			0.872946i
$661$			1.26819e17i			1.52046i	0.649655	+	0.760229i	$0.274913\pi$
$661$			1.26819e17i			1.52046i	−0.649655	+	0.760229i	$0.725087\pi$
$662$	−2.14580e16			−0.254942
$663$		−	7.05663e14i		−	0.00830837i
$664$	2.88004e17	−	8.18748e16i	3.36040	−	0.955305i
$665$	7.54293e14			0.00872188
$666$			4.60863e16i			0.528112i
$667$	7.41301e16			0.841859
$668$	−1.30494e17			−1.46869
$669$			5.15372e16i			0.574863i
$670$		−	1.44369e17i		−	1.59597i
$671$	6.47678e16			0.709617
$672$			3.45608e16i			0.375291i
$673$	5.68003e16			0.611308			0.305654	−	0.952143i	$-0.401125\pi$
$673$	5.68003e16			0.611308			0.305654	+	0.952143i	$0.401125\pi$
$674$	1.66743e16			0.177864
$675$	3.49141e16			0.369129
$676$	−2.66472e17			−2.79236
$677$			3.31428e16i			0.344237i	0.985076	+	0.172118i	$0.0550611\pi$
$677$			3.31428e16i			0.344237i	−0.985076	+	0.172118i	$0.944939\pi$
$678$			2.01479e17i			2.07420i
$679$		−	1.03560e16i		−	0.105675i
$680$	−1.53963e17			−1.55726
$681$	1.11516e17			1.11803
$682$		−	7.50684e16i		−	0.746021i
$683$			1.52740e17i			1.50463i	0.658805	+	0.752314i	$0.271062\pi$
$683$			1.52740e17i			1.50463i	−0.658805	+	0.752314i	$0.728938\pi$
$684$		−	1.27001e16i		−	0.124014i
$685$	3.56303e16			0.344886
$686$			4.99877e16i			0.479644i
$687$	3.59482e16			0.341929
$688$			6.34818e17i			5.98575i
$689$	−7.96389e13			−0.000744406
$690$	1.27628e17			1.18263
$691$	−3.98692e16			−0.366243			−0.183121	−	0.983090i	$-0.558620\pi$
$691$	−3.98692e16			−0.366243			−0.183121	+	0.983090i	$0.558620\pi$
$692$	1.82229e17			1.65951
$693$	−3.87270e15			−0.0349634
$694$	−1.38922e17			−1.24341
$695$	−4.86274e16			−0.431492
$696$			2.17881e17i			1.91675i
$697$	−2.54577e16			−0.222035
$698$			1.15989e16i			0.100296i
$699$			1.24650e17i			1.06863i
$700$	−1.42397e16			−0.121036
$701$	−1.08300e17			−0.912685			−0.456342	−	0.889804i	$-0.650841\pi$
$701$	−1.08300e17			−0.912685			−0.456342	+	0.889804i	$0.650841\pi$
$702$	5.29104e15			0.0442098
$703$		−	5.59687e15i		−	0.0463675i
$704$	−2.96268e17			−2.43360
$705$	3.87957e16			0.315972
$706$			3.47982e17i			2.81014i
$707$			1.88480e16i			0.150920i
$708$	3.75185e16			0.297884
$709$			4.52249e15i			0.0356041i	0.999842	+	0.0178021i	$0.00566687\pi$
$709$			4.52249e15i			0.0356041i	−0.999842	+	0.0178021i	$0.994333\pi$
$710$		−	2.04056e17i		−	1.59294i
$711$		−	7.69902e16i		−	0.595961i
$712$	−5.06435e16			−0.388726
$713$	−9.77835e16			−0.744266
$714$			1.23912e16i			0.0935244i
$715$	−1.27987e15			−0.00957922
$716$			7.43546e16i			0.551861i
$717$			5.10154e16i			0.375480i
$718$		−	3.27776e17i		−	2.39238i
$719$		−	2.71080e17i		−	1.96211i	−0.193720	−	0.981057i	$-0.562055\pi$
$719$		−	2.71080e17i		−	1.96211i	0.193720	−	0.981057i	$-0.437945\pi$
$720$		−	2.29672e17i		−	1.64859i
$721$		−	1.58423e16i		−	0.112773i
$722$		−	2.73806e17i		−	1.93294i
$723$	8.63790e16			0.604754
$724$			7.55774e16i			0.524760i
$725$	−3.97218e16			−0.273528
$726$		−	1.38650e17i		−	0.946893i
$727$	1.46819e17			0.994431			0.497216	−	0.867627i	$-0.334356\pi$
$727$	1.46819e17			0.994431			0.497216	+	0.867627i	$0.334356\pi$
$728$	−1.38550e15			−0.00930721
$729$	1.28791e17			0.858066
$730$		−	1.09802e17i		−	0.725562i
$731$			1.25819e17i			0.824598i
$732$	3.86213e17			2.51050
$733$	1.83452e17			1.18277			0.591383	−	0.806391i	$-0.298582\pi$
$733$	1.83452e17			1.18277			0.591383	+	0.806391i	$0.298582\pi$
$734$	2.64715e17			1.69279
$735$			8.77089e16i			0.556313i
$736$			7.37668e17i			4.64081i
$737$		−	9.00164e16i		−	0.561716i
$738$		−	6.44275e16i		−	0.398780i
$739$	−2.37473e17			−1.45797			−0.728983	−	0.684531i	$-0.760007\pi$
$739$	−2.37473e17			−1.45797			−0.728983	+	0.684531i	$0.760007\pi$
$740$		−	1.96950e17i		−	1.19941i
$741$	−2.16882e14			−0.00131013
$742$	1.39843e15			0.00837951
$743$		−	3.78539e16i		−	0.224998i	−0.993652	−	0.112499i	$-0.964115\pi$
$743$		−	3.78539e16i		−	0.224998i	0.993652	−	0.112499i	$-0.0358854\pi$
$744$		−	2.87403e17i		−	1.69455i
$745$	−3.11461e16			−0.182166
$746$			9.60047e16i			0.557006i
$747$	2.42069e16	+	8.51507e16i	0.139321	+	0.490077i
$748$	−1.49519e17			−0.853665
$749$			1.46870e16i			0.0831845i
$750$	−2.64252e17			−1.48474
$751$	1.69956e17			0.947323			0.473661	−	0.880707i	$-0.342932\pi$
$751$	1.69956e17			0.947323			0.473661	+	0.880707i	$0.342932\pi$
$752$			4.05632e17i			2.24298i
$753$			6.94927e16i			0.381214i
$754$	−6.01963e15			−0.0327598
$755$		−	2.23322e17i		−	1.20573i
$756$	−6.84184e16			−0.366473
$757$	−1.21611e17			−0.646248			−0.323124	−	0.946357i	$-0.604733\pi$
$757$	−1.21611e17			−0.646248			−0.323124	+	0.946357i	$0.604733\pi$
$758$	4.64140e17			2.44700
$759$	7.95778e16			0.416238
$760$			4.73197e16i			0.245562i
$761$			3.49102e17i			1.79740i	0.438566	+	0.898699i	$0.355486\pi$
$761$			3.49102e17i			1.79740i	−0.438566	+	0.898699i	$0.644514\pi$
$762$		−	1.85137e17i		−	0.945724i
$763$	−3.18211e16			−0.161275
$764$	6.37252e17			3.20443
$765$		−	4.55203e16i		−	0.227110i
$766$			4.99390e17i			2.47211i
$767$			6.65520e14i			0.00326881i
$768$	−5.57875e17			−2.71875
$769$		−	2.99651e17i		−	1.44897i	−0.689293	−	0.724483i	$-0.742079\pi$
$769$		−	2.99651e17i		−	1.44897i	0.689293	−	0.724483i	$-0.257921\pi$
$770$	2.24742e16			0.107830
$771$			1.13750e17i			0.541531i
$772$	9.68824e17			4.57658
$773$	1.45471e17			0.681869			0.340934	−	0.940087i	$-0.389257\pi$
$773$	1.45471e17			0.681869			0.340934	+	0.940087i	$0.389257\pi$
$774$	−3.18419e17			−1.48099
$775$	5.23963e16			0.241819
$776$	6.49673e17			2.97526
$777$	−1.01770e16			−0.0462481
$778$	−4.38781e17			−1.97866
$779$			7.82430e15i			0.0350123i
$780$	−7.63193e15			−0.0338896
$781$		−	1.27232e17i		−	0.560649i
$782$			2.64479e17i			1.15651i
$783$	−1.90854e17			−0.828189
$784$	−9.17049e17			−3.94908
$785$	1.85462e17			0.792570
$786$		−	5.91922e17i		−	2.51032i
$787$	5.03958e16			0.212103			0.106051	−	0.994361i	$-0.466179\pi$
$787$	5.03958e16			0.212103			0.106051	+	0.994361i	$0.466179\pi$
$788$	3.46317e17			1.44650
$789$		−	9.48935e16i		−	0.393346i
$790$			4.46792e17i			1.83799i
$791$	4.62143e16			0.188676
$792$		−	2.42949e17i		−	0.984384i
$793$			6.85082e15i			0.0275488i
$794$		−	5.04223e16i		−	0.201233i
$795$	4.94578e15			0.0195899
$796$	−2.68651e17			−1.05611
$797$			3.43948e16i			0.134197i	0.997746	+	0.0670985i	$0.0213742\pi$
$797$			3.43948e16i			0.134197i	−0.997746	+	0.0670985i	$0.978626\pi$
$798$	3.80838e15			0.0147477
$799$			8.03951e16i			0.308993i
$800$		−	3.95272e17i		−	1.50784i
$801$		−	1.49731e16i		−	0.0566915i
$802$		−	1.26277e17i		−	0.474544i
$803$		−	6.84635e16i		−	0.255368i
$804$		−	5.36772e17i		−	1.98725i
$805$		−	2.92746e16i		−	0.107576i
$806$	7.94037e15			0.0289621
$807$			3.24933e17i			1.17640i
$808$	−1.18241e18			−4.24912
$809$		−	1.37089e17i		−	0.489005i	−0.969649	−	0.244502i	$-0.921375\pi$
$809$		−	1.37089e17i		−	0.489005i	0.969649	−	0.244502i	$-0.0786246\pi$
$810$	−1.02479e17			−0.362847
$811$	−1.23531e17			−0.434162			−0.217081	−	0.976154i	$-0.569654\pi$
$811$	−1.23531e17			−0.434162			−0.217081	+	0.976154i	$0.569654\pi$
$812$	7.78397e16			0.271560
$813$			3.86664e17i			1.33903i
$814$		−	1.66759e17i		−	0.573248i
$815$	−2.69782e17			−0.920593
$816$	−4.58201e17			−1.55208
$817$	3.86699e16			0.130029
$818$			1.98440e17i			0.662383i
$819$		−	4.09635e14i		−	0.00135735i
$820$			2.75332e17i			0.905677i
$821$			1.08316e17i			0.353699i	0.984238	+	0.176849i	$0.0565906\pi$
$821$			1.08316e17i			0.353699i	−0.984238	+	0.176849i	$0.943409\pi$
$822$	1.79895e17			0.583162
$823$			1.81057e17i			0.582661i	0.956622	+	0.291331i	$0.0940980\pi$
$823$			1.81057e17i			0.582661i	−0.956622	+	0.291331i	$0.905902\pi$
$824$	9.93847e17			3.17509
$825$	−4.26409e16			−0.135239
$826$		−	1.16863e16i		−	0.0367958i
$827$			3.64781e17i			1.14025i	0.821559	+	0.570124i	$0.193105\pi$
$827$			3.64781e17i			1.14025i	−0.821559	+	0.570124i	$0.806895\pi$
$828$	−4.92899e17			−1.52959
$829$			3.71436e17i			1.14434i	0.820134	+	0.572172i	$0.193899\pi$
$829$			3.71436e17i			1.14434i	−0.820134	+	0.572172i	$0.806101\pi$
$830$	−1.40478e17	−	4.94149e17i	−0.429676	−	1.51144i
$831$	−1.82341e17			−0.553705
$832$		−	3.13378e16i		−	0.0944774i
$833$	−1.81756e17			−0.544026
$834$	−2.45517e17			−0.729601
$835$			1.43752e17i			0.424126i
$836$			4.59540e16i			0.134613i
$837$	2.51751e17			0.732181
$838$		−	7.68071e17i		−	2.21788i
$839$	−7.55106e15			−0.0216489			−0.0108245	−	0.999941i	$-0.503446\pi$
$839$	−7.55106e15			−0.0216489			−0.0108245	+	0.999941i	$0.503446\pi$
$840$	8.60433e16			0.244930
$841$	−1.36680e17			−0.386305
$842$	2.46536e17			0.691845
$843$		−	3.21723e17i		−	0.896432i
$844$			1.75657e17i			0.485970i
$845$			2.93547e17i			0.806375i
$846$	−2.03461e17			−0.554958
$847$	−3.18030e16			−0.0861325
$848$			5.17111e16i			0.139062i
$849$		−	2.25745e17i		−	0.602797i
$850$		−	1.41718e17i		−	0.375761i
$851$	−2.17219e17			−0.571900
$852$		−	7.58692e17i		−	1.98348i
$853$	1.08352e17			0.281283			0.140641	−	0.990061i	$-0.455084\pi$
$853$	1.08352e17			0.281283			0.140641	+	0.990061i	$0.455084\pi$
$854$		−	1.20298e17i		−	0.310107i
$855$	−1.39904e16			−0.0358125
$856$	−9.21373e17			−2.34203
$857$	2.70741e17			0.683392			0.341696	−	0.939811i	$-0.388999\pi$
$857$	2.70741e17			0.683392			0.341696	+	0.939811i	$0.388999\pi$
$858$	−6.46200e15			−0.0161973
$859$	−2.60948e17			−0.649523			−0.324761	−	0.945796i	$-0.605284\pi$
$859$	−2.60948e17			−0.649523			−0.324761	+	0.945796i	$0.605284\pi$
$860$	1.36077e18			3.36351
$861$	1.42272e16			0.0349222
$862$			1.01527e18i			2.47479i
$863$	2.21781e17			0.536859			0.268430	−	0.963299i	$-0.413495\pi$
$863$	2.21781e17			0.536859			0.268430	+	0.963299i	$0.413495\pi$
$864$		−	1.89918e18i		−	4.56546i
$865$		−	2.00744e17i		−	0.479231i
$866$	9.73348e17			2.30760
$867$	2.06651e17			0.486545
$868$	−1.02677e17			−0.240079
$869$			2.78582e17i			0.646895i
$870$	3.73834e17			0.862112
$871$	9.52150e15			0.0218070
$872$		−	1.99626e18i		−	4.54065i
$873$			1.92081e17i			0.433909i
$874$	8.12863e16			0.182368
$875$			6.06129e16i			0.135057i
$876$		−	4.08251e17i		−	0.903446i
$877$			1.04686e17i			0.230085i	0.993361	+	0.115043i	$0.0367005\pi$
$877$			1.04686e17i			0.230085i	−0.993361	+	0.115043i	$0.963300\pi$
$878$	−1.53465e18			−3.34998
$879$	−1.87973e16			−0.0407533
$880$			8.31046e17i			1.78949i
$881$	−5.24397e17			−1.12151			−0.560757	−	0.827980i	$-0.689490\pi$
$881$	−5.24397e17			−1.12151			−0.560757	+	0.827980i	$0.689490\pi$
$882$		−	4.59983e17i		−	0.977081i
$883$		−	1.20210e17i		−	0.253617i	−0.991927	−	0.126808i	$-0.959527\pi$
$883$		−	1.20210e17i		−	0.253617i	0.991927	−	0.126808i	$-0.0404733\pi$
$884$		−	1.58154e16i		−	0.0331411i
$885$		−	4.13305e16i		−	0.0860223i
$886$		−	1.51835e18i		−	3.13884i
$887$		−	6.70569e17i		−	1.37690i	−0.725285	−	0.688449i	$-0.758292\pi$
$887$		−	6.70569e17i		−	1.37690i	0.725285	−	0.688449i	$-0.241708\pi$
$888$		−	6.38444e17i		−	1.30210i
$889$	−4.24660e16			−0.0860262
$890$			8.68926e16i			0.174841i
$891$	−6.38970e16			−0.127707
$892$			1.15506e18i			2.29306i
$893$	2.47090e16			0.0487245
$894$	−1.57255e17			−0.308020
$895$	8.19092e16			0.159366
$896$			2.73016e17i			0.527643i
$897$			8.41735e15i			0.0161592i
$898$	−1.49148e18			−2.84420
$899$	−2.86418e17			−0.542553
$900$	2.64115e17			0.496979
$901$			1.02490e16i			0.0191572i
$902$			2.33125e17i			0.432862i
$903$		−	7.03150e16i		−	0.129694i
$904$			2.89920e18i			5.31212i
$905$	8.32563e16			0.151539
$906$		−	1.12754e18i		−	2.03875i
$907$	3.47921e17			0.624938			0.312469	−	0.949928i	$-0.398844\pi$
$907$	3.47921e17			0.624938			0.312469	+	0.949928i	$0.398844\pi$
$908$	2.49931e18			4.45970
$909$		−	3.49587e17i		−	0.619687i
$910$			2.37721e15i			0.00418618i
$911$	4.37556e17			0.765461			0.382731	−	0.923860i	$-0.374984\pi$
$911$	4.37556e17			0.765461			0.382731	+	0.923860i	$0.374984\pi$
$912$			1.40826e17i			0.244745i
$913$	−8.75904e16	−	3.08110e17i	−0.151228	−	0.531962i
$914$	−6.48400e17			−1.11216
$915$		−	4.25453e17i		−	0.724979i
$916$	8.05675e17			1.36391
$917$	−1.35772e17			−0.228347
$918$		−	6.80921e17i		−	1.13773i
$919$			4.46413e17i			0.741043i	0.928824	+	0.370522i	$0.120821\pi$
$919$			4.46413e17i			0.741043i	−0.928824	+	0.370522i	$0.879179\pi$
$920$	1.83651e18			3.02877
$921$			2.56185e17i			0.419755i
$922$	1.02340e18			1.66594
$923$	1.34580e16			0.0217656
$924$	8.35600e16			0.134266
$925$	1.16394e17			0.185815
$926$		−	1.43083e18i		−	2.26946i
$927$			2.93838e17i			0.463052i
$928$			2.16070e18i			3.38304i
$929$	6.69789e17			1.04194			0.520972	−	0.853574i	$-0.325570\pi$
$929$	6.69789e17			1.04194			0.520972	+	0.853574i	$0.325570\pi$
$930$	−4.93117e17			−0.762171
$931$			5.58619e16i			0.0857863i
$932$			2.79367e18i			4.26265i
$933$			8.01211e17i			1.21467i
$934$	−2.41600e18			−3.63928
$935$			1.64711e17i			0.246520i
$936$	2.56980e16			0.0382159
$937$			6.18214e17i			0.913485i	0.889599	+	0.456743i	$0.150984\pi$
$937$			6.18214e17i			0.913485i	−0.889599	+	0.456743i	$0.849016\pi$
$938$	−1.67195e17			−0.245474
$939$	−7.54350e17			−1.10047
$940$	8.69495e17			1.26037
$941$	−8.34104e17			−1.20139			−0.600693	−	0.799480i	$-0.705109\pi$
$941$	−8.34104e17			−1.20139			−0.600693	+	0.799480i	$0.705109\pi$
$942$	9.36389e17			1.34014
$943$	3.03667e17			0.431844
$944$	4.32136e17			0.610644
$945$			7.53698e16i			0.105829i
$946$	1.15217e18			1.60757
$947$		−	1.13491e18i		−	1.57348i	−0.617287	−	0.786738i	$-0.711768\pi$
$947$		−	1.13491e18i		−	1.57348i	0.617287	−	0.786738i	$-0.288232\pi$
$948$			1.66120e18i			2.28860i
$949$	7.24173e15			0.00991391
$950$	−4.35564e16			−0.0592530
$951$	−8.21199e17			−1.11011
$952$			1.78305e17i			0.239520i
$953$	−7.63159e17			−1.01873			−0.509363	−	0.860552i	$-0.670119\pi$
$953$	−7.63159e17			−1.01873			−0.509363	+	0.860552i	$0.670119\pi$
$954$	−2.59378e16			−0.0344067
$955$		−	7.01998e17i		−	0.925371i
$956$			1.14337e18i			1.49774i
$957$	2.33091e17			0.303427
$958$			2.08445e18i			2.69649i
$959$		−	4.12636e16i		−	0.0530463i
$960$			1.94615e18i			2.48628i
$961$	−4.09855e17			−0.520343
$962$	1.76389e16			0.0222547
$963$		−	2.72411e17i		−	0.341560i
$964$	1.93594e18			2.41229
$965$		−	1.06726e18i		−	1.32162i
$966$		−	1.47806e17i		−	0.181899i
$967$			1.51169e18i			1.84886i	0.381347	+	0.924432i	$0.375460\pi$
$967$			1.51169e18i			1.84886i	−0.381347	+	0.924432i	$0.624540\pi$
$968$		−	1.99512e18i		−	2.42503i
$969$			2.79112e16i			0.0337160i
$970$		−	1.11469e18i		−	1.33821i
$971$			6.98614e16i			0.0833531i	0.999131	+	0.0416765i	$0.0132699\pi$
$971$			6.98614e16i			0.0833531i	−0.999131	+	0.0416765i	$0.986730\pi$
$972$	2.10969e18			2.50162
$973$			5.63156e16i			0.0663669i
$974$	−1.56561e18			−1.83370
$975$		−	4.51035e15i		−	0.00525028i
$976$	4.44837e18			5.14638
$977$	4.18055e17			0.480691			0.240346	−	0.970687i	$-0.422739\pi$
$977$	4.18055e17			0.480691			0.240346	+	0.970687i	$0.422739\pi$
$978$	−1.36211e18			−1.55661
$979$			5.41789e16i			0.0615367i
$980$			1.96574e18i			2.21907i
$981$	5.90210e17			0.662205
$982$	−3.39428e18			−3.78511
$983$	−1.76368e18			−1.95479			−0.977394	−	0.211424i	$-0.932190\pi$
$983$	−1.76368e18			−1.95479			−0.977394	+	0.211424i	$0.932190\pi$
$984$			8.92530e17i			0.983223i
$985$		−	3.81504e17i		−	0.417717i
$986$			7.74685e17i			0.843070i
$987$		−	4.49294e16i		−	0.0485991i
$988$	−4.86079e15			−0.00522595
$989$		−	1.50081e18i		−	1.60379i
$990$	−4.16845e17			−0.442755
$991$	−5.72015e17			−0.603900			−0.301950	−	0.953324i	$-0.597638\pi$
$991$	−5.72015e17			−0.603900			−0.301950	+	0.953324i	$0.597638\pi$
$992$		−	2.85014e18i		−	2.99086i
$993$			8.78879e16i			0.0916714i
$994$	−2.36318e17			−0.245007
$995$			2.95947e17i			0.304983i
$996$	−5.22306e17	−	1.83727e18i	−0.535018	−	1.88199i
$997$	−5.68640e17			−0.578984			−0.289492	−	0.957180i	$-0.593486\pi$
$997$	−5.68640e17			−0.578984			−0.289492	+	0.957180i	$0.593486\pi$
$998$		−	1.76056e18i		−	1.78183i
$999$	5.59246e17			0.562613

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	83.13.b.c.82.2	✓	80
83.82	odd	2	inner	83.13.b.c.82.79	yes	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
83.13.b.c.82.2	✓	80	1.1	even	1	trivial
83.13.b.c.82.79	yes	80	83.82	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 83 \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	83.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-124.655i q^{2} -510.562 q^{3} -11442.8 q^{4} +12605.4i q^{5} +63644.0i q^{6} +14598.4 q^{7} +915813. i q^{8} -270767. q^{9} +O(q^{10})\)	\(q-124.655i q^{2} -510.562 q^{3} -11442.8 q^{4} +12605.4i q^{5} +63644.0i q^{6} +14598.4 q^{7} +915813. i q^{8} -270767. q^{9} +1.57132e6 q^{10} +979745. q^{11} +5.84226e6 q^{12} +103633. i q^{13} -1.81976e6i q^{14} -6.43585e6i q^{15} +6.72907e7 q^{16} +1.33368e7 q^{17} +3.37524e7i q^{18} -4.09900e6i q^{19} -1.44241e8i q^{20} -7.45338e6 q^{21} -1.22130e8i q^{22} -1.59085e8 q^{23} -4.67580e8i q^{24} +8.52442e7 q^{25} +1.29183e7 q^{26} +4.09577e8 q^{27} -1.67046e8 q^{28} -4.65977e8 q^{29} -8.02259e8 q^{30} +6.14661e8 q^{31} -4.63693e9i q^{32} -5.00221e8 q^{33} -1.66250e9i q^{34} +1.84019e8i q^{35} +3.09833e9 q^{36} +1.36542e9 q^{37} -5.10960e8 q^{38} -5.29109e7i q^{39} -1.15442e10 q^{40} -1.90883e9 q^{41} +9.29099e8i q^{42} +9.43397e9i q^{43} -1.12110e10 q^{44} -3.41313e9i q^{45} +1.98307e10i q^{46} +6.02806e9i q^{47} -3.43561e10 q^{48} -1.36282e10 q^{49} -1.06261e10i q^{50} -6.80927e9 q^{51} -1.18585e9i q^{52} +7.68473e8i q^{53} -5.10557e10i q^{54} +1.23501e10i q^{55} +1.33694e10i q^{56} +2.09280e9i q^{57} +5.80862e10i q^{58} +6.42192e9 q^{59} +7.36441e10i q^{60} +6.61068e10 q^{61} -7.66204e10i q^{62} -3.95276e9 q^{63} -3.02393e11 q^{64} -1.30633e9 q^{65} +6.23549e10i q^{66} -9.18774e10i q^{67} -1.52610e11 q^{68} +8.12230e10 q^{69} +2.29388e10 q^{70} -1.29863e11i q^{71} -2.47972e11i q^{72} -6.98789e10i q^{73} -1.70206e11i q^{74} -4.35225e10 q^{75} +4.69041e10i q^{76} +1.43027e10 q^{77} -6.59559e9 q^{78} +2.84341e11i q^{79} +8.48227e11i q^{80} -6.52180e10 q^{81} +2.37945e11i q^{82} +(-8.94013e10 - 3.14480e11i) q^{83} +8.52875e10 q^{84} +1.68116e11i q^{85} +1.17599e12 q^{86} +2.37910e11 q^{87} +8.97263e11i q^{88} +5.52990e10i q^{89} -4.25463e11 q^{90} +1.51287e9i q^{91} +1.82038e12 q^{92} -3.13823e11 q^{93} +7.51426e11 q^{94} +5.16696e10 q^{95} +2.36744e12i q^{96} -7.09395e11i q^{97} +1.69882e12i q^{98} -2.65283e11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 918 q^{3} - 181150 q^{4} + 103918 q^{7} + 14902142 q^{9}+O(q^{10}) \) 80 * q + 918 * q^3 - 181150 * q^4 + 103918 * q^7 + 14902142 * q^9	\( 80 q + 918 q^{3} - 181150 q^{4} + 103918 q^{7} + 14902142 q^{9} + 1177482 q^{10} + 510406 q^{11} - 9260402 q^{12} + 203791850 q^{16} + 5571718 q^{17} + 46565862 q^{21} + 804389464 q^{23} - 4263713272 q^{25} + 18061794 q^{26} - 2326165338 q^{27} - 204811652 q^{28} + 1486960270 q^{29} - 2621683648 q^{30} - 665743010 q^{31} + 135224502 q^{33} - 54936709824 q^{36} - 280627202 q^{37} + 6646145310 q^{38} - 7119722058 q^{40} + 51318109072 q^{41} - 11512674650 q^{44} + 85368259738 q^{48} + 148785395094 q^{49} - 100143389562 q^{51} - 63584241050 q^{59} - 29216180978 q^{61} - 332932206620 q^{63} - 323596534090 q^{64} + 362112989184 q^{65} - 86115426752 q^{68} + 272383417100 q^{69} + 105630718656 q^{70} + 785418808326 q^{75} - 663355117738 q^{77} + 1483841884620 q^{78} + 2430778545148 q^{81} + 837315119192 q^{83} - 3013574788354 q^{84} + 452180651958 q^{86} - 682263689498 q^{87} - 499714512022 q^{90} + 997428187414 q^{92} - 1487992716298 q^{93} + 3169817690580 q^{94} + 1542762610848 q^{95} + 2021347267420 q^{99}+O(q^{100}) \) 80 * q + 918 * q^3 - 181150 * q^4 + 103918 * q^7 + 14902142 * q^9 + 1177482 * q^10 + 510406 * q^11 - 9260402 * q^12 + 203791850 * q^16 + 5571718 * q^17 + 46565862 * q^21 + 804389464 * q^23 - 4263713272 * q^25 + 18061794 * q^26 - 2326165338 * q^27 - 204811652 * q^28 + 1486960270 * q^29 - 2621683648 * q^30 - 665743010 * q^31 + 135224502 * q^33 - 54936709824 * q^36 - 280627202 * q^37 + 6646145310 * q^38 - 7119722058 * q^40 + 51318109072 * q^41 - 11512674650 * q^44 + 85368259738 * q^48 + 148785395094 * q^49 - 100143389562 * q^51 - 63584241050 * q^59 - 29216180978 * q^61 - 332932206620 * q^63 - 323596534090 * q^64 + 362112989184 * q^65 - 86115426752 * q^68 + 272383417100 * q^69 + 105630718656 * q^70 + 785418808326 * q^75 - 663355117738 * q^77 + 1483841884620 * q^78 + 2430778545148 * q^81 + 837315119192 * q^83 - 3013574788354 * q^84 + 452180651958 * q^86 - 682263689498 * q^87 - 499714512022 * q^90 + 997428187414 * q^92 - 1487992716298 * q^93 + 3169817690580 * q^94 + 1542762610848 * q^95 + 2021347267420 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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