LMFDB - Embedded newform 31.7.b.c.30.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [31,7,Mod(30,31)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(31, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 7, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("31.30"); S:= CuspForms(chi, 7); N := Newforms(S);

Level:	$ N $	$=$	$ 31 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	31.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$7.13167659222$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 7208 x^{10} + 19859688 x^{8} + 26566749360 x^{6} + 17884354852944 x^{4} + \cdots + 59\!\cdots\!00 $ x^12 + 7208x^10 + 19859688x^8 + 26566749360x^6 + 17884354852944x^4 + 5570285336959680*x^2 + 590986232936064000
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{2}\cdot 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			30.3
Root			$39.6325i$ of defining polynomial
Character	$\chi$	$=$	31.30
Dual form			31.7.b.c.30.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-5.88527 q^{2} -39.6325i q^{3} -29.3637 q^{4} -192.768 q^{5} +233.248i q^{6} +330.261 q^{7} +549.470 q^{8} -841.735 q^{9} +1134.49 q^{10} -487.165i q^{11} +1163.75i q^{12} +3875.41i q^{13} -1943.67 q^{14} +7639.86i q^{15} -1354.50 q^{16} -5095.71i q^{17} +4953.83 q^{18} -9888.66 q^{19} +5660.36 q^{20} -13089.1i q^{21} +2867.09i q^{22} +18711.6i q^{23} -21776.9i q^{24} +21534.3 q^{25} -22807.8i q^{26} +4467.96i q^{27} -9697.66 q^{28} -9663.04i q^{29} -44962.6i q^{30} +(-1475.62 + 29754.4i) q^{31} -27194.5 q^{32} -19307.6 q^{33} +29989.6i q^{34} -63663.5 q^{35} +24716.4 q^{36} -22210.9i q^{37} +58197.4 q^{38} +153592. q^{39} -105920. q^{40} -32327.0 q^{41} +77032.6i q^{42} +90033.4i q^{43} +14304.9i q^{44} +162259. q^{45} -110123. i q^{46} -2523.71 q^{47} +53682.3i q^{48} -8576.89 q^{49} -126735. q^{50} -201956. q^{51} -113796. i q^{52} +21169.4i q^{53} -26295.1i q^{54} +93909.6i q^{55} +181468. q^{56} +391912. i q^{57} +56869.6i q^{58} -12229.6 q^{59} -224334. i q^{60} +248340. i q^{61} +(8684.43 - 175113. i) q^{62} -277992. q^{63} +246735. q^{64} -747053. i q^{65} +113630. q^{66} -413987. q^{67} +149629. i q^{68} +741586. q^{69} +374677. q^{70} -78313.0 q^{71} -462508. q^{72} +316585. i q^{73} +130717. i q^{74} -853459. i q^{75} +290367. q^{76} -160891. i q^{77} -903931. q^{78} -2574.72i q^{79} +261104. q^{80} -436548. q^{81} +190253. q^{82} -727400. i q^{83} +384342. i q^{84} +982288. i q^{85} -529870. i q^{86} -382970. q^{87} -267682. i q^{88} -279841. i q^{89} -954938. q^{90} +1.27990e6i q^{91} -549440. i q^{92} +(1.17924e6 + 58482.6i) q^{93} +14852.7 q^{94} +1.90621e6 q^{95} +1.07778e6i q^{96} +1.08152e6 q^{97} +50477.3 q^{98} +410064. i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 2 q^{2} + 122 q^{4} - 146 q^{5} + 6 q^{7} + 1142 q^{8} - 5668 q^{9} - 3004 q^{10} - 1312 q^{14} - 1102 q^{16} - 16382 q^{18} + 10430 q^{19} - 8052 q^{20} + 29618 q^{25} + 136504 q^{28} + 90076 q^{31}+ \cdots + 408810 q^{98}+O(q^{100}) $ 12 * q - 2 * q^2 + 122 * q^4 - 146 * q^5 + 6 * q^7 + 1142 * q^8 - 5668 * q^9 - 3004 * q^10 - 1312 * q^14 - 1102 * q^16 - 16382 * q^18 + 10430 * q^19 - 8052 * q^20 + 29618 * q^25 + 136504 * q^28 + 90076 * q^31 + 38406 * q^32 - 166248 * q^33 - 198898 * q^35 + 26982 * q^36 + 448196 * q^38 + 208224 * q^39 - 354932 * q^40 - 116690 * q^41 - 363818 * q^45 - 221120 * q^47 - 580182 * q^49 - 60318 * q^50 + 179784 * q^51 + 750496 * q^56 + 799510 * q^59 - 73070 * q^62 - 469874 * q^63 - 322286 * q^64 - 511884 * q^66 - 13744 * q^67 + 475128 * q^69 - 71312 * q^70 + 640678 * q^71 - 118870 * q^72 + 2044196 * q^76 + 1373868 * q^78 + 1356020 * q^80 - 678724 * q^81 - 488056 * q^82 - 1051200 * q^87 - 2598532 * q^90 - 2260440 * q^93 - 5045848 * q^94 + 3828206 * q^95 + 3708966 * q^97 + 408810 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/31\mathbb{Z}\right)^\times$.

$n$	$3$
$\chi(n)$	$-1$

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 $

$2$	−5.88527			−0.735658			−0.367829	−	0.929893i	$-0.619899\pi$
$2$	−5.88527			−0.735658			−0.367829	+	0.929893i	$0.619899\pi$
$3$		−	39.6325i		−	1.46787i	−0.679220	−	0.733935i	$-0.737682\pi$
$3$		−	39.6325i		−	1.46787i	0.679220	−	0.733935i	$-0.262318\pi$
$4$	−29.3637			−0.458807
$5$	−192.768			−1.54214			−0.771070	−	0.636750i	$-0.780278\pi$
$5$	−192.768			−1.54214			−0.771070	+	0.636750i	$0.780278\pi$
$6$			233.248i			1.07985i
$7$	330.261			0.962859			0.481430	−	0.876485i	$-0.340118\pi$
$7$	330.261			0.962859			0.481430	+	0.876485i	$0.340118\pi$
$8$	549.470			1.07318
$9$	−841.735			−1.15464
$10$	1134.49			1.13449
$11$		−	487.165i		−	0.366014i	−0.983112	−	0.183007i	$-0.941417\pi$
$11$		−	487.165i		−	0.366014i	0.983112	−	0.183007i	$-0.0585831\pi$
$12$			1163.75i			0.673469i
$13$			3875.41i			1.76396i	0.471291	+	0.881978i	$0.343788\pi$
$13$			3875.41i			1.76396i	−0.471291	+	0.881978i	$0.656212\pi$
$14$	−1943.67			−0.708335
$15$			7639.86i			2.26366i
$16$	−1354.50			−0.330689
$17$		−	5095.71i		−	1.03719i	−0.855020	−	0.518595i	$-0.826455\pi$
$17$		−	5095.71i		−	1.03719i	0.855020	−	0.518595i	$-0.173545\pi$
$18$	4953.83			0.849423
$19$	−9888.66			−1.44171			−0.720853	−	0.693088i	$-0.756250\pi$
$19$	−9888.66			−1.44171			−0.720853	+	0.693088i	$0.756250\pi$
$20$	5660.36			0.707545
$21$		−	13089.1i		−	1.41335i
$22$			2867.09i			0.269261i
$23$			18711.6i			1.53789i	0.639312	+	0.768947i	$0.279219\pi$
$23$			18711.6i			1.53789i	−0.639312	+	0.768947i	$0.720781\pi$
$24$		−	21776.9i		−	1.57529i
$25$	21534.3			1.37820
$26$		−	22807.8i		−	1.29767i
$27$			4467.96i			0.226996i
$28$	−9697.66			−0.441767
$29$		−	9663.04i		−	0.396205i	−0.980181	−	0.198102i	$-0.936522\pi$
$29$		−	9663.04i		−	0.396205i	0.980181	−	0.198102i	$-0.0634779\pi$
$30$		−	44962.6i		−	1.66528i
$31$	−1475.62	+	29754.4i	−0.0495325	+	0.998773i
$31$	−1475.62	+	29754.4i	−0.0495325	+	0.998773i
$32$	−27194.5			−0.829909
$33$	−19307.6			−0.537261
$34$			29989.6i			0.763017i
$35$	−63663.5			−1.48486
$36$	24716.4			0.529758
$37$		−	22210.9i		−	0.438491i	−0.975670	−	0.219246i	$-0.929640\pi$
$37$		−	22210.9i		−	0.438491i	0.975670	−	0.219246i	$-0.0703596\pi$
$38$	58197.4			1.06060
$39$	153592.			2.58926
$40$	−105920.			−1.65500
$41$	−32327.0			−0.469044			−0.234522	−	0.972111i	$-0.575353\pi$
$41$	−32327.0			−0.469044			−0.234522	+	0.972111i	$0.575353\pi$
$42$			77032.6i			1.03974i
$43$			90033.4i			1.13240i	0.824269	+	0.566198i	$0.191586\pi$
$43$			90033.4i			1.13240i	−0.824269	+	0.566198i	$0.808414\pi$
$44$			14304.9i			0.167930i
$45$	162259.			1.78062
$46$		−	110123.i		−	1.13136i
$47$	−2523.71			−0.0243079			−0.0121539	−	0.999926i	$-0.503869\pi$
$47$	−2523.71			−0.0243079			−0.0121539	+	0.999926i	$0.503869\pi$
$48$			53682.3i			0.485408i
$49$	−8576.89			−0.0729024
$50$	−126735.			−1.01388
$51$	−201956.			−1.52246
$52$		−	113796.i		−	0.809316i
$53$			21169.4i			0.142194i	0.997469	+	0.0710971i	$0.0226500\pi$
$53$			21169.4i			0.142194i	−0.997469	+	0.0710971i	$0.977350\pi$
$54$		−	26295.1i		−	0.166991i
$55$			93909.6i			0.564445i
$56$	181468.			1.03332
$57$			391912.i			2.11624i
$58$			56869.6i			0.291471i
$59$	−12229.6			−0.0595465			−0.0297732	−	0.999557i	$-0.509479\pi$
$59$	−12229.6			−0.0595465			−0.0297732	+	0.999557i	$0.509479\pi$
$60$		−	224334.i		−	1.03858i
$61$			248340.i			1.09410i	0.837100	+	0.547049i	$0.184249\pi$
$61$			248340.i			1.09410i	−0.837100	+	0.547049i	$0.815751\pi$
$62$	8684.43	−	175113.i	0.0364390	−	0.734755i
$63$	−277992.			−1.11176
$64$	246735.			0.941218
$65$		−	747053.i		−	2.72027i
$66$	113630.			0.395241
$67$	−413987.			−1.37645			−0.688227	−	0.725495i	$-0.741611\pi$
$67$	−413987.			−1.37645			−0.688227	+	0.725495i	$0.741611\pi$
$68$			149629.i			0.475870i
$69$	741586.			2.25743
$70$	374677.			1.09235
$71$	−78313.0			−0.218806			−0.109403	−	0.993997i	$-0.534894\pi$
$71$	−78313.0			−0.218806			−0.109403	+	0.993997i	$0.534894\pi$
$72$	−462508.			−1.23914
$73$			316585.i			0.813807i	0.913471	+	0.406904i	$0.133392\pi$
$73$			316585.i			0.813807i	−0.913471	+	0.406904i	$0.866608\pi$
$74$			130717.i			0.322580i
$75$		−	853459.i		−	2.02301i
$76$	290367.			0.661465
$77$		−	160891.i		−	0.352420i
$78$	−903931.			−1.90481
$79$		−	2574.72i		−	0.00522215i	−0.999997	−	0.00261107i	$-0.999169\pi$
$79$		−	2574.72i		−	0.00522215i	0.999997	−	0.00261107i	$-0.000831131\pi$
$80$	261104.			0.509969
$81$	−436548.			−0.821443
$82$	190253.			0.345056
$83$		−	727400.i		−	1.27215i	−0.771626	−	0.636076i	$-0.780556\pi$
$83$		−	727400.i		−	1.27215i	0.771626	−	0.636076i	$-0.219444\pi$
$84$			384342.i			0.648456i
$85$			982288.i			1.59949i
$86$		−	529870.i		−	0.833056i
$87$	−382970.			−0.581577
$88$		−	267682.i		−	0.392800i
$89$		−	279841.i		−	0.396954i	−0.980105	−	0.198477i	$-0.936400\pi$
$89$		−	279841.i		−	0.396954i	0.980105	−	0.198477i	$-0.0635996\pi$
$90$	−954938.			−1.30993
$91$			1.27990e6i			1.69844i
$92$		−	549440.i		−	0.705597i
$93$	1.17924e6	+	58482.6i	1.46607	+	0.0727073i
$94$	14852.7			0.0178823
$95$	1.90621e6			2.22331
$96$			1.07778e6i			1.21820i
$97$	1.08152e6			1.18500			0.592499	−	0.805571i	$-0.298141\pi$
$97$	1.08152e6			1.18500			0.592499	+	0.805571i	$0.298141\pi$
$98$	50477.3			0.0536312
$99$			410064.i			0.422616i
$100$	−632326.			−0.632326
$101$	−1.77983e6			−1.72749			−0.863745	−	0.503930i	$-0.831887\pi$
$101$	−1.77983e6			−1.72749			−0.863745	+	0.503930i	$0.831887\pi$
$102$	1.18856e6			1.12001
$103$	−1.00154e6			−0.916547			−0.458273	−	0.888811i	$-0.651532\pi$
$103$	−1.00154e6			−0.916547			−0.458273	+	0.888811i	$0.651532\pi$
$104$			2.12942e6i			1.89305i
$105$			2.52314e6i			2.17959i
$106$		−	124588.i		−	0.104606i
$107$	713784.			0.582660			0.291330	−	0.956623i	$-0.405902\pi$
$107$	713784.			0.582660			0.291330	+	0.956623i	$0.405902\pi$
$108$		−	131196.i		−	0.104147i
$109$	30026.3			0.0231858			0.0115929	−	0.999933i	$-0.496310\pi$
$109$	30026.3			0.0231858			0.0115929	+	0.999933i	$0.496310\pi$
$110$		−	552683.i		−	0.415239i
$111$	−880274.			−0.643648
$112$	−447339.			−0.318407
$113$	−2.01059e6			−1.39344			−0.696721	−	0.717342i	$-0.745359\pi$
$113$	−2.01059e6			−1.39344			−0.696721	+	0.717342i	$0.745359\pi$
$114$		−	2.30651e6i		−	1.55683i
$115$		−	3.60698e6i		−	2.37165i
$116$			283742.i			0.181782i
$117$		−	3.26207e6i		−	2.03674i
$118$	71974.4			0.0438059
$119$		−	1.68291e6i		−	0.998668i
$120$			4.19787e6i			2.42932i
$121$	1.53423e6			0.866034
$122$		−	1.46154e6i		−	0.804882i
$123$			1.28120e6i			0.688496i
$124$	43329.7	−	873699.i	0.0227259	−	0.458244i
$125$	−1.13913e6			−0.583232
$126$	1.63606e6			0.817874
$127$		−	1.50624e6i		−	0.735332i	−0.929958	−	0.367666i	$-0.880157\pi$
$127$		−	1.50624e6i		−	0.735332i	0.929958	−	0.367666i	$-0.119843\pi$
$128$	288346.			0.137494
$129$	3.56825e6			1.66221
$130$			4.39661e6i			2.00119i
$131$	−1.43346e6			−0.637636			−0.318818	−	0.947816i	$-0.603286\pi$
$131$	−1.43346e6			−0.637636			−0.318818	+	0.947816i	$0.603286\pi$
$132$	566941.			0.246499
$133$	−3.26584e6			−1.38816
$134$	2.43642e6			1.01260
$135$		−	861278.i		−	0.350060i
$136$		−	2.79994e6i		−	1.11309i
$137$		−	2.16763e6i		−	0.842993i	−0.906830	−	0.421497i	$-0.861505\pi$
$137$		−	2.16763e6i		−	0.842993i	0.906830	−	0.421497i	$-0.138495\pi$
$138$	−4.36443e6			−1.66070
$139$		−	2.40552e6i		−	0.895705i	−0.894107	−	0.447853i	$-0.852189\pi$
$139$		−	2.40552e6i		−	0.895705i	0.894107	−	0.447853i	$-0.147811\pi$
$140$	1.86939e6			0.681266
$141$			100021.i			0.0356808i
$142$	460893.			0.160966
$143$	1.88796e6			0.645633
$144$	1.14013e6			0.381828
$145$			1.86272e6i			0.611004i
$146$		−	1.86319e6i		−	0.598684i
$147$			339924.i			0.107011i
$148$			652193.i			0.201183i
$149$	−1.10206e6			−0.333155			−0.166577	−	0.986028i	$-0.553272\pi$
$149$	−1.10206e6			−0.333155			−0.166577	+	0.986028i	$0.553272\pi$
$150$			5.02283e6i			1.48825i
$151$			4.93438e6i			1.43318i	0.697493	+	0.716591i	$0.254299\pi$
$151$			4.93438e6i			1.43318i	−0.697493	+	0.716591i	$0.745701\pi$
$152$	−5.43352e6			−1.54722
$153$			4.28924e6i			1.19758i
$154$			946889.i			0.259261i
$155$	284452.	−	5.73569e6i	0.0763860	−	1.54025i
$156$	−4.51003e6			−1.18797
$157$	−3.41153e6			−0.881557			−0.440778	−	0.897616i	$-0.645298\pi$
$157$	−3.41153e6			−0.881557			−0.440778	+	0.897616i	$0.645298\pi$
$158$			15152.9i			0.00384171i
$159$	838998.			0.208723
$160$	5.24221e6			1.27984
$161$			6.17969e6i			1.48078i
$162$	2.56920e6			0.604301
$163$	2.46450e6			0.569071			0.284536	−	0.958665i	$-0.408161\pi$
$163$	2.46450e6			0.569071			0.284536	+	0.958665i	$0.408161\pi$
$164$	949239.			0.215201
$165$	3.72187e6			0.828532
$166$			4.28094e6i			0.935869i
$167$			5.34081e6i			1.14672i	0.819303	+	0.573360i	$0.194360\pi$
$167$			5.34081e6i			1.14672i	−0.819303	+	0.573360i	$0.805640\pi$
$168$		−	7.19204e6i		−	1.51679i
$169$	−1.01920e7			−2.11154
$170$		−	5.78103e6i		−	1.17668i
$171$	8.32363e6			1.66466
$172$		−	2.64371e6i		−	0.519551i
$173$	−3.01771e6			−0.582827			−0.291414	−	0.956597i	$-0.594126\pi$
$173$	−3.01771e6			−0.582827			−0.291414	+	0.956597i	$0.594126\pi$
$174$	2.25388e6			0.427842
$175$	7.11194e6			1.32701
$176$			659866.i			0.121037i
$177$			484689.i			0.0874065i
$178$			1.64694e6i			0.292023i
$179$			1.58938e6i			0.277121i	0.990354	+	0.138560i	$0.0442475\pi$
$179$			1.58938e6i			0.277121i	−0.990354	+	0.138560i	$0.955752\pi$
$180$	−4.76452e6			−0.816962
$181$		−	9.23498e6i		−	1.55740i	−0.627397	−	0.778700i	$-0.715879\pi$
$181$		−	9.23498e6i		−	1.55740i	0.627397	−	0.778700i	$-0.284121\pi$
$182$		−	7.53253e6i		−	1.24947i
$183$	9.84232e6			1.60599
$184$			1.02814e7i			1.65044i
$185$			4.28154e6i			0.676215i
$186$	−6.94015e6	−	344186.i	−1.07853	−	0.0534877i
$187$	−2.48245e6			−0.379626
$188$	74105.5			0.0111526
$189$			1.47559e6i			0.218565i
$190$	−1.12186e7			−1.63560
$191$	2.10873e6			0.302636			0.151318	−	0.988485i	$-0.451648\pi$
$191$	2.10873e6			0.302636			0.151318	+	0.988485i	$0.451648\pi$
$192$		−	9.77872e6i		−	1.38159i
$193$	1.10931e7			1.54305			0.771526	−	0.636198i	$-0.219494\pi$
$193$	1.10931e7			1.54305			0.771526	+	0.636198i	$0.219494\pi$
$194$	−6.36501e6			−0.871753
$195$	−2.96076e7			−3.99300
$196$	251849.			0.0334481
$197$			5.00480e6i			0.654619i	0.944917	+	0.327309i	$0.106142\pi$
$197$			5.00480e6i			0.654619i	−0.944917	+	0.327309i	$0.893858\pi$
$198$		−	2.41333e6i		−	0.310901i
$199$		−	1.26621e6i		−	0.160674i	−0.996768	−	0.0803369i	$-0.974400\pi$
$199$		−	1.26621e6i		−	0.160674i	0.996768	−	0.0803369i	$-0.0255996\pi$
$200$	1.18325e7			1.47906
$201$			1.64073e7i			2.02046i
$202$	1.04748e7			1.27084
$203$		−	3.19132e6i		−	0.381489i
$204$	5.93016e6			0.698515
$205$	6.23160e6			0.723332
$206$	5.89430e6			0.674265
$207$		−	1.57502e7i		−	1.77572i
$208$		−	5.24925e6i		−	0.583321i
$209$			4.81741e6i			0.527685i
$210$		−	1.48494e7i		−	1.60343i
$211$	−1.18260e7			−1.25890			−0.629451	−	0.777040i	$-0.716720\pi$
$211$	−1.18260e7			−1.25890			−0.629451	+	0.777040i	$0.716720\pi$
$212$		−	621612.i		−	0.0652397i
$213$			3.10374e6i			0.321178i
$214$	−4.20081e6			−0.428639
$215$		−	1.73555e7i		−	1.74631i
$216$			2.45501e6i			0.243608i
$217$	−487340.	+	9.82672e6i	−0.0476928	+	0.961677i
$218$	−176713.			−0.0170568
$219$	1.25471e7			1.19456
$220$		−	2.75753e6i		−	0.258971i
$221$	1.97480e7			1.82956
$222$	5.18064e6			0.473505
$223$			196617.i			0.0177299i	0.999961	+	0.00886496i	$0.00282184\pi$
$223$			196617.i			0.0177299i	−0.999961	+	0.00886496i	$0.997178\pi$
$224$	−8.98126e6			−0.799086
$225$	−1.81262e7			−1.59133
$226$	1.18329e7			1.02510
$227$	−1.75289e7			−1.49857			−0.749284	−	0.662248i	$-0.769602\pi$
$227$	−1.75289e7			−1.49857			−0.749284	+	0.662248i	$0.769602\pi$
$228$		−	1.15080e7i		−	0.970945i
$229$		−	3.38621e6i		−	0.281973i	−0.990011	−	0.140987i	$-0.954973\pi$
$229$		−	3.38621e6i		−	0.281973i	0.990011	−	0.140987i	$-0.0450275\pi$
$230$			2.12280e7i			1.74472i
$231$	−6.37653e6			−0.517307
$232$		−	5.30955e6i		−	0.425200i
$233$	5.73177e6			0.453128			0.226564	−	0.973996i	$-0.427251\pi$
$233$	5.73177e6			0.453128			0.226564	+	0.973996i	$0.427251\pi$
$234$			1.91981e7i			1.49834i
$235$	486490.			0.0374861
$236$	359106.			0.0273203
$237$	−102043.			−0.00766543
$238$			9.90439e6i			0.734678i
$239$			1.94664e7i			1.42591i	0.701209	+	0.712955i	$0.252644\pi$
$239$			1.94664e7i			1.42591i	−0.701209	+	0.712955i	$0.747356\pi$
$240$		−	1.03482e7i		−	0.748568i
$241$			1.88617e7i			1.34750i	0.738959	+	0.673750i	$0.235318\pi$
$241$			1.88617e7i			1.34750i	−0.738959	+	0.673750i	$0.764682\pi$
$242$	−9.02936e6			−0.637105
$243$			2.05586e7i			1.43277i
$244$		−	7.29216e6i		−	0.501980i
$245$	1.65335e6			0.112426
$246$		−	7.54020e6i		−	0.506498i
$247$		−	3.83226e7i		−	2.54311i
$248$	−810810.	+	1.63492e7i	−0.0531574	+	1.07187i
$249$	−2.88287e7			−1.86735
$250$	6.70406e6			0.429060
$251$		−	506851.i		−	0.0320523i	−0.999872	−	0.0160262i	$-0.994898\pi$
$251$		−	506851.i		−	0.0320523i	0.999872	−	0.0160262i	$-0.00510150\pi$
$252$	8.16286e6			0.510083
$253$	9.11562e6			0.562891
$254$			8.86464e6i			0.540953i
$255$	3.89305e7			2.34785
$256$	−1.74880e7			−1.04237
$257$	3.19186e6			0.188038			0.0940189	−	0.995570i	$-0.470029\pi$
$257$	3.19186e6			0.188038			0.0940189	+	0.995570i	$0.470029\pi$
$258$	−2.10001e7			−1.22282
$259$		−	7.33539e6i		−	0.422205i
$260$			2.19362e7i			1.24808i
$261$			8.13372e6i			0.457475i
$262$	8.43631e6			0.469082
$263$		−	1.40279e7i		−	0.771129i	−0.922681	−	0.385564i	$-0.874007\pi$
$263$		−	1.40279e7i		−	0.771129i	0.922681	−	0.385564i	$-0.125993\pi$
$264$	−1.06089e7			−0.576580
$265$		−	4.08078e6i		−	0.219283i
$266$	1.92203e7			1.02121
$267$	−1.10908e7			−0.582678
$268$	1.21562e7			0.631527
$269$			2.47587e7i			1.27195i	0.771709	+	0.635976i	$0.219402\pi$
$269$			2.47587e7i			1.27195i	−0.771709	+	0.635976i	$0.780598\pi$
$270$			5.06885e6i			0.257524i
$271$		−	649621.i		−	0.0326402i	−0.999867	−	0.0163201i	$-0.994805\pi$
$271$		−	649621.i		−	0.0326402i	0.999867	−	0.0163201i	$-0.00519507\pi$
$272$			6.90215e6i			0.342987i
$273$	5.07255e7			2.49309
$274$			1.27571e7i			0.620155i
$275$		−	1.04908e7i		−	0.504440i
$276$	−2.17757e7			−1.03572
$277$		−	2.68623e7i		−	1.26387i	−0.775020	−	0.631937i	$-0.782260\pi$
$277$		−	2.68623e7i		−	1.26387i	0.775020	−	0.631937i	$-0.217740\pi$
$278$			1.41571e7i			0.658933i
$279$	1.24208e6	−	2.50453e7i	0.0571923	−	1.15323i
$280$	−3.49812e7			−1.59353
$281$	3.68160e7			1.65927			0.829636	−	0.558305i	$-0.188548\pi$
$281$	3.68160e7			1.65927			0.829636	+	0.558305i	$0.188548\pi$
$282$		−	588651.i		−	0.0262488i
$283$	−2.57920e7			−1.13796			−0.568979	−	0.822352i	$-0.692661\pi$
$283$	−2.57920e7			−1.13796			−0.568979	+	0.822352i	$0.692661\pi$
$284$	2.29956e6			0.100390
$285$		−	7.55480e7i		−	3.26354i
$286$	−1.11112e7			−0.474965
$287$	−1.06763e7			−0.451624
$288$	2.28905e7			0.958249
$289$	−1.82872e6			−0.0757624
$290$		−	1.09626e7i		−	0.449490i
$291$		−	4.28632e7i		−	1.73942i
$292$		−	9.29609e6i		−	0.373381i
$293$	−4.26083e7			−1.69392			−0.846958	−	0.531660i	$-0.821568\pi$
$293$	−4.26083e7			−1.69392			−0.846958	+	0.531660i	$0.821568\pi$
$294$		−	2.00054e6i		−	0.0787237i
$295$	2.35747e6			0.0918290
$296$		−	1.22042e7i		−	0.470582i
$297$	2.17663e6			0.0830837
$298$	6.48591e6			0.245088
$299$	−7.25150e7			−2.71278
$300$			2.50607e7i			0.928173i
$301$			2.97345e7i			1.09034i
$302$		−	2.90401e7i		−	1.05433i
$303$			7.05393e7i			2.53573i
$304$	1.33942e7			0.476756
$305$		−	4.78718e7i		−	1.68725i
$306$		−	2.52433e7i		−	0.881012i
$307$	9.10595e6			0.314710			0.157355	−	0.987542i	$-0.449703\pi$
$307$	9.10595e6			0.314710			0.157355	+	0.987542i	$0.449703\pi$
$308$			4.72436e6i			0.161693i
$309$			3.96933e7i			1.34537i
$310$	−1.67408e6	+	3.37560e7i	−0.0561940	+	1.13310i
$311$	−1.34073e7			−0.445717			−0.222858	−	0.974851i	$-0.571539\pi$
$311$	−1.34073e7			−0.445717			−0.222858	+	0.974851i	$0.571539\pi$
$312$	8.43943e7			2.77875
$313$		−	8.81172e6i		−	0.287361i	−0.989624	−	0.143680i	$-0.954106\pi$
$313$		−	8.81172e6i		−	0.287361i	0.989624	−	0.143680i	$-0.0458937\pi$
$314$	2.00778e7			0.648524
$315$	5.35878e7			1.71449
$316$			75603.2i			0.00239596i
$317$	1.23463e7			0.387578			0.193789	−	0.981043i	$-0.437922\pi$
$317$	1.23463e7			0.387578			0.193789	+	0.981043i	$0.437922\pi$
$318$	−4.93773e6			−0.153549
$319$	−4.70749e6			−0.145017
$320$	−4.75625e7			−1.45149
$321$		−	2.82890e7i		−	0.855270i
$322$		−	3.63691e7i		−	1.08934i
$323$			5.03898e7i			1.49532i
$324$	1.28187e7			0.376884
$325$			8.34544e7i			2.43108i
$326$	−1.45043e7			−0.418642
$327$		−	1.19002e6i		−	0.0340338i
$328$	−1.77627e7			−0.503370
$329$	−833483.			−0.0234050
$330$	−2.19042e7			−0.609517
$331$		−	3.35851e7i		−	0.926109i	−0.886330	−	0.463054i	$-0.846753\pi$
$331$		−	3.35851e7i		−	0.926109i	0.886330	−	0.463054i	$-0.153247\pi$
$332$			2.13591e7i			0.583672i
$333$			1.86957e7i			0.506301i
$334$		−	3.14321e7i		−	0.843594i
$335$	7.98032e7			2.12269
$336$			1.77292e7i			0.467380i
$337$		−	3.61455e7i		−	0.944419i	−0.881486	−	0.472209i	$-0.843457\pi$
$337$		−	3.61455e7i		−	0.944419i	0.881486	−	0.472209i	$-0.156543\pi$
$338$	5.99826e7			1.55337
$339$			7.96849e7i			2.04539i
$340$		−	2.88436e7i		−	0.733858i
$341$	1.44953e7	+	718871.i	0.365565	+	0.0181296i
$342$	−4.89868e7			−1.22462
$343$	−4.16874e7			−1.03305
$344$			4.94706e7i			1.21527i
$345$	−1.42954e8			−3.48127
$346$	1.77600e7			0.428762
$347$			5.03735e7i			1.20563i	0.797881	+	0.602814i	$0.205954\pi$
$347$			5.03735e7i			1.20563i	−0.797881	+	0.602814i	$0.794046\pi$
$348$	1.12454e7			0.266832
$349$	1.45096e7			0.341333			0.170667	−	0.985329i	$-0.445408\pi$
$349$	1.45096e7			0.341333			0.170667	+	0.985329i	$0.445408\pi$
$350$	−4.18557e7			−0.976225
$351$	−1.73152e7			−0.400411
$352$			1.32482e7i			0.303759i
$353$		−	1.97053e7i		−	0.447981i	−0.974591	−	0.223990i	$-0.928092\pi$
$353$		−	1.97053e7i		−	0.447981i	0.974591	−	0.223990i	$-0.0719085\pi$
$354$		−	2.85253e6i		−	0.0643013i
$355$	1.50962e7			0.337429
$356$			8.21714e6i			0.182126i
$357$	−6.66981e7			−1.46591
$358$		−	9.35393e6i		−	0.203866i
$359$	6.43138e7			1.39002			0.695009	−	0.719001i	$-0.255400\pi$
$359$	6.43138e7			1.39002			0.695009	+	0.719001i	$0.255400\pi$
$360$	8.91565e7			1.91093
$361$	5.07398e7			1.07852
$362$			5.43503e7i			1.14571i
$363$		−	6.08054e7i		−	1.27122i
$364$		−	3.75824e7i		−	0.779257i
$365$		−	6.10273e7i		−	1.25501i
$366$	−5.79246e7			−1.18146
$367$		−	1.95113e7i		−	0.394718i	−0.980331	−	0.197359i	$-0.936763\pi$
$367$		−	1.95113e7i		−	0.394718i	0.980331	−	0.197359i	$-0.0632365\pi$
$368$		−	2.53449e7i		−	0.508565i
$369$	2.72108e7			0.541579
$370$		−	2.51980e7i		−	0.497463i
$371$			6.99144e6i			0.136913i
$372$	−3.46269e7	−	1.71726e6i	−0.672643	−	0.0333586i
$373$	−1.81517e7			−0.349776			−0.174888	−	0.984588i	$-0.555956\pi$
$373$	−1.81517e7			−0.349776			−0.174888	+	0.984588i	$0.555956\pi$
$374$	1.46099e7			0.279275
$375$			4.51464e7i			0.856109i
$376$	−1.38670e6			−0.0260868
$377$	3.74483e7			0.698888
$378$		−	8.68425e6i		−	0.160789i
$379$	8.86585e7			1.62856			0.814278	−	0.580475i	$-0.197133\pi$
$379$	8.86585e7			1.62856			0.814278	+	0.580475i	$0.197133\pi$
$380$	−5.59734e7			−1.02007
$381$	−5.96961e7			−1.07937
$382$	−1.24104e7			−0.222636
$383$		−	1.11145e8i		−	1.97831i	−0.146892	−	0.989153i	$-0.546927\pi$
$383$		−	1.11145e8i		−	1.97831i	0.146892	−	0.989153i	$-0.453073\pi$
$384$		−	1.14279e7i		−	0.201824i
$385$			3.10146e7i			0.543481i
$386$	−6.52857e7			−1.13516
$387$		−	7.57842e7i		−	1.30751i
$388$	−3.17573e7			−0.543686
$389$			8.03857e7i			1.36562i	0.730596	+	0.682810i	$0.239242\pi$
$389$			8.03857e7i			1.36562i	−0.730596	+	0.682810i	$0.760758\pi$
$390$	1.74249e8			2.93748
$391$	9.53488e7			1.59509
$392$	−4.71274e6			−0.0782376
$393$			5.68118e7i			0.935967i
$394$		−	2.94546e7i		−	0.481575i
$395$			496323.i			0.00805328i
$396$		−	1.20410e7i		−	0.193899i
$397$	−4.77926e6			−0.0763816			−0.0381908	−	0.999270i	$-0.512159\pi$
$397$	−4.77926e6			−0.0763816			−0.0381908	+	0.999270i	$0.512159\pi$
$398$			7.45195e6i			0.118201i
$399$			1.29433e8i			2.03764i
$400$	−2.91683e7			−0.455754
$401$		−	8.62081e7i		−	1.33695i	−0.743735	−	0.668475i	$-0.766947\pi$
$401$		−	8.62081e7i		−	1.33695i	0.743735	−	0.668475i	$-0.233053\pi$
$402$		−	9.65615e7i		−	1.48637i
$403$	−1.15311e8	−	5.71864e6i	−1.76179	−	0.0873731i
$404$	5.22624e7			0.792584
$405$	8.41523e7			1.26678
$406$			1.87818e7i			0.280646i
$407$	−1.08204e7			−0.160494
$408$	−1.10969e8			−1.63388
$409$		−	1.08176e7i		−	0.158111i	−0.996870	−	0.0790557i	$-0.974810\pi$
$409$		−	1.08176e7i		−	0.158111i	0.996870	−	0.0790557i	$-0.0251905\pi$
$410$	−3.66746e7			−0.532125
$411$	−8.59087e7			−1.23740
$412$	2.94087e7			0.420518
$413$	−4.03895e6			−0.0573349
$414$			9.26940e7i			1.30632i
$415$			1.40219e8i			1.96184i
$416$		−	1.05390e8i		−	1.46392i
$417$	−9.53369e7			−1.31478
$418$		−	2.83517e7i		−	0.388196i
$419$	2.34155e7			0.318319			0.159159	−	0.987253i	$-0.449122\pi$
$419$	2.34155e7			0.318319			0.159159	+	0.987253i	$0.449122\pi$
$420$		−	7.40888e7i		−	1.00001i
$421$	2.47698e7			0.331952			0.165976	−	0.986130i	$-0.446922\pi$
$421$	2.47698e7			0.331952			0.165976	+	0.986130i	$0.446922\pi$
$422$	6.95994e7			0.926122
$423$	2.12430e6			0.0280669
$424$			1.16320e7i			0.152600i
$425$		−	1.09733e8i		−	1.42945i
$426$		−	1.82663e7i		−	0.236278i
$427$			8.20168e7i			1.05346i
$428$	−2.09593e7			−0.267329
$429$		−	7.48247e7i		−	0.947705i
$430$			1.02142e8i			1.28469i
$431$	−1.09881e7			−0.137243			−0.0686217	−	0.997643i	$-0.521860\pi$
$431$	−1.09881e7			−0.137243			−0.0686217	+	0.997643i	$0.521860\pi$
$432$		−	6.05186e6i		−	0.0750650i
$433$			1.48292e8i			1.82664i	0.407241	+	0.913321i	$0.366491\pi$
$433$			1.48292e8i			1.82664i	−0.407241	+	0.913321i	$0.633509\pi$
$434$	2.86812e6	−	5.78328e7i	0.0350856	−	0.707466i
$435$	7.38243e7			0.896874
$436$	−881682.			−0.0106378
$437$		−	1.85032e8i		−	2.21719i
$438$	−7.38427e7			−0.878790
$439$	5.66761e7			0.669894			0.334947	−	0.942237i	$-0.391282\pi$
$439$	5.66761e7			0.669894			0.334947	+	0.942237i	$0.391282\pi$
$440$			5.16005e7i			0.605753i
$441$	7.21947e6			0.0841762
$442$	−1.16222e8			−1.34593
$443$	−4.73407e7			−0.544532			−0.272266	−	0.962222i	$-0.587773\pi$
$443$	−4.73407e7			−0.544532			−0.272266	+	0.962222i	$0.587773\pi$
$444$	2.58480e7			0.295310
$445$			5.39442e7i			0.612159i
$446$		−	1.15714e6i		−	0.0130432i
$447$			4.36773e7i			0.489028i
$448$	8.14868e7			0.906261
$449$			6.53871e7i			0.722359i	0.932496	+	0.361179i	$0.117626\pi$
$449$			6.53871e7i			0.722359i	−0.932496	+	0.361179i	$0.882374\pi$
$450$	1.06677e8			1.17067
$451$			1.57486e7i			0.171677i
$452$	5.90384e7			0.639321
$453$	1.95562e8			2.10373
$454$	1.03162e8			1.10243
$455$		−	2.46722e8i		−	2.61923i
$456$			2.15344e8i			2.27111i
$457$			7.47026e7i			0.782686i	0.920245	+	0.391343i	$0.127989\pi$
$457$			7.47026e7i			0.782686i	−0.920245	+	0.391343i	$0.872011\pi$
$458$			1.99288e7i			0.207436i
$459$	2.27674e7			0.235438
$460$			1.05914e8i			1.08813i
$461$		−	2.55947e7i		−	0.261245i	−0.991432	−	0.130622i	$-0.958302\pi$
$461$		−	2.55947e7i		−	0.261245i	0.991432	−	0.130622i	$-0.0416975\pi$
$462$	3.75276e7			0.380561
$463$			1.04209e8i			1.04994i	0.851121	+	0.524969i	$0.175923\pi$
$463$			1.04209e8i			1.04994i	−0.851121	+	0.524969i	$0.824077\pi$
$464$			1.30886e7i			0.131021i
$465$	−2.27320e8	−	1.12735e7i	−2.26088	−	0.112125i
$466$	−3.37330e7			−0.333347
$467$	−9.18593e7			−0.901929			−0.450965	−	0.892542i	$-0.648920\pi$
$467$	−9.18593e7			−0.901929			−0.450965	+	0.892542i	$0.648920\pi$
$468$			9.57862e7i			0.934471i
$469$	−1.36724e8			−1.32533
$470$	−2.86312e6			−0.0275770
$471$			1.35207e8i			1.29401i
$472$	−6.71980e6			−0.0639043
$473$	4.38611e7			0.414473
$474$	600548.			0.00563914
$475$	−2.12946e8			−1.98695
$476$			4.94165e7i			0.458196i
$477$		−	1.78191e7i		−	0.164184i
$478$		−	1.14565e8i		−	1.04898i
$479$	−1.49592e7			−0.136113			−0.0680567	−	0.997681i	$-0.521680\pi$
$479$	−1.49592e7			−0.136113			−0.0680567	+	0.997681i	$0.521680\pi$
$480$		−	2.07762e8i		−	1.87863i
$481$	8.60764e7			0.773479
$482$		−	1.11006e8i		−	0.991299i
$483$	2.44917e8			2.17359
$484$	−4.50506e7			−0.397342
$485$	−2.08481e8			−1.82743
$486$		−	1.20993e8i		−	1.05403i
$487$		−	9.53784e7i		−	0.825777i	−0.910781	−	0.412889i	$-0.864520\pi$
$487$		−	9.53784e7i		−	0.825777i	0.910781	−	0.412889i	$-0.135480\pi$
$488$			1.36455e8i			1.17417i
$489$		−	9.76744e7i		−	0.835323i
$490$	−9.73038e6			−0.0827069
$491$			9.35853e7i			0.790612i	0.918550	+	0.395306i	$0.129361\pi$
$491$			9.35853e7i			0.790612i	−0.918550	+	0.395306i	$0.870639\pi$
$492$		−	3.76207e7i		−	0.315887i
$493$	−4.92401e7			−0.410940
$494$			2.25539e8i			1.87086i
$495$		−	7.90470e7i		−	0.651733i
$496$	1.99873e6	−	4.03024e7i	0.0163798	−	0.330283i
$497$	−2.58637e7			−0.210679
$498$	1.69664e8			1.37373
$499$		−	2.07866e8i		−	1.67294i	−0.548009	−	0.836472i	$-0.684614\pi$
$499$		−	2.07866e8i		−	1.67294i	0.548009	−	0.836472i	$-0.315386\pi$
$500$	3.34489e7			0.267591
$501$	2.11670e8			1.68324
$502$			2.98295e6i			0.0235795i
$503$	−2.37963e8			−1.86984			−0.934922	−	0.354853i	$-0.884531\pi$
$503$	−2.37963e8			−1.86984			−0.934922	+	0.354853i	$0.884531\pi$
$504$	−1.52748e8			−1.19312
$505$	3.43094e8			2.66403
$506$	−5.36478e7			−0.414096
$507$			4.03935e8i			3.09947i
$508$			4.42288e7i			0.337376i
$509$		−	2.02835e8i		−	1.53812i	−0.639177	−	0.769060i	$-0.720725\pi$
$509$		−	2.02835e8i		−	1.53812i	0.639177	−	0.769060i	$-0.279275\pi$
$510$	−2.29116e8			−1.72721
$511$			1.04556e8i			0.783582i
$512$	8.44675e7			0.629332
$513$		−	4.41822e7i		−	0.327261i
$514$	−1.87850e7			−0.138332
$515$	1.93063e8			1.41344
$516$	−1.04777e8			−0.762634
$517$			1.22946e6i			0.00889702i
$518$			4.31707e7i			0.310599i
$519$			1.19600e8i			0.855515i
$520$		−	4.10483e8i		−	2.91935i
$521$	2.21686e8			1.56756			0.783781	−	0.621037i	$-0.213288\pi$
$521$	2.21686e8			1.56756			0.783781	+	0.621037i	$0.213288\pi$
$522$		−	4.78691e7i		−	0.336545i
$523$		−	2.48932e8i		−	1.74011i	−0.492958	−	0.870053i	$-0.664084\pi$
$523$		−	2.48932e8i		−	1.74011i	0.492958	−	0.870053i	$-0.335916\pi$
$524$	4.20917e7			0.292552
$525$		−	2.81864e8i		−	1.94788i
$526$			8.25582e7i			0.567287i
$527$	1.51620e8	+	7.51935e6i	1.03592	+	0.0513746i
$528$	2.61521e7			0.177666
$529$	−2.02087e8			−1.36512
$530$			2.40165e7i			0.161318i
$531$	1.02941e7			0.0687549
$532$	9.58969e7			0.636898
$533$		−	1.25280e8i		−	0.827373i
$534$	6.52722e7			0.428652
$535$	−1.37594e8			−0.898544
$536$	−2.27473e8			−1.47719
$537$	6.29912e7			0.406777
$538$		−	1.45711e8i		−	0.935722i
$539$			4.17836e6i			0.0266833i
$540$			2.52903e7i			0.160610i
$541$	2.63770e7			0.166584			0.0832922	−	0.996525i	$-0.473457\pi$
$541$	2.63770e7			0.166584			0.0832922	+	0.996525i	$0.473457\pi$
$542$			3.82319e6i			0.0240120i
$543$	−3.66005e8			−2.28606
$544$			1.38575e8i			0.860773i
$545$	−5.78810e6			−0.0357558
$546$	−2.98533e8			−1.83406
$547$	1.99610e8			1.21961			0.609805	−	0.792552i	$-0.291248\pi$
$547$	1.99610e8			1.21961			0.609805	+	0.792552i	$0.291248\pi$
$548$			6.36496e7i			0.386771i
$549$		−	2.09036e8i		−	1.26329i
$550$			6.17409e7i			0.371095i
$551$			9.55546e7i			0.571211i
$552$	4.07479e8			2.42264
$553$		−	850329.i		−	0.00502819i
$554$			1.58092e8i			0.929779i
$555$	1.69688e8			0.992596
$556$			7.06349e7i			0.410956i
$557$		−	2.98300e7i		−	0.172618i	−0.996268	−	0.0863092i	$-0.972493\pi$
$557$		−	2.98300e7i		−	0.172618i	0.996268	−	0.0863092i	$-0.0275073\pi$
$558$	−7.30998e6	+	1.47398e8i	−0.0420740	+	0.848380i
$559$	−3.48916e8			−1.99750
$560$	8.62324e7			0.491028
$561$			9.83858e7i			0.557242i
$562$	−2.16672e8			−1.22066
$563$	−1.55878e8			−0.873494			−0.436747	−	0.899584i	$-0.643870\pi$
$563$	−1.55878e8			−0.873494			−0.436747	+	0.899584i	$0.643870\pi$
$564$		−	2.93698e6i		−	0.0163706i
$565$	3.87577e8			2.14888
$566$	1.51793e8			0.837148
$567$	−1.44175e8			−0.790933
$568$	−4.30306e7			−0.234819
$569$			1.72498e8i			0.936370i	0.883630	+	0.468185i	$0.155092\pi$
$569$			1.72498e8i			0.936370i	−0.883630	+	0.468185i	$0.844908\pi$
$570$			4.44620e8i			2.40085i
$571$			8.70403e7i			0.467533i	0.972293	+	0.233766i	$0.0751051\pi$
$571$			8.70403e7i			0.467533i	−0.972293	+	0.233766i	$0.924895\pi$
$572$	−5.54375e7			−0.296221
$573$		−	8.35740e7i		−	0.444230i
$574$	6.28331e7			0.332241
$575$			4.02941e8i			2.11952i
$576$	−2.07685e8			−1.08677
$577$	−1.89831e8			−0.988187			−0.494094	−	0.869409i	$-0.664500\pi$
$577$	−1.89831e8			−0.988187			−0.494094	+	0.869409i	$0.664500\pi$
$578$	1.07625e7			0.0557353
$579$		−	4.39647e8i		−	2.26500i
$580$		−	5.46963e7i		−	0.280333i
$581$		−	2.40232e8i		−	1.22490i
$582$			2.52261e8i			1.27962i
$583$	1.03130e7			0.0520451
$584$			1.73954e8i			0.873364i
$585$			6.28821e8i			3.14094i
$586$	2.50761e8			1.24614
$587$			1.18312e7i			0.0584946i	0.999572	+	0.0292473i	$0.00931103\pi$
$587$			1.18312e7i			0.0584946i	−0.999572	+	0.0292473i	$0.990689\pi$
$588$		−	9.98140e6i		−	0.0490975i
$589$	1.45919e7	−	2.94232e8i	0.0714113	−	1.43994i
$590$	−1.38743e7			−0.0675548
$591$	1.98353e8			0.960895
$592$			3.00847e7i			0.145004i
$593$	1.07406e8			0.515069			0.257534	−	0.966269i	$-0.417090\pi$
$593$	1.07406e8			0.515069			0.257534	+	0.966269i	$0.417090\pi$
$594$	−1.28101e7			−0.0611212
$595$			3.24411e8i			1.54009i
$596$	3.23605e7			0.152854
$597$	−5.01829e7			−0.235848
$598$	4.26770e8			1.99568
$599$	2.92891e7			0.136278			0.0681389	−	0.997676i	$-0.478294\pi$
$599$	2.92891e7			0.136278			0.0681389	+	0.997676i	$0.478294\pi$
$600$		−	4.68950e8i		−	2.17106i
$601$		−	9.95699e7i		−	0.458675i	−0.973347	−	0.229337i	$-0.926344\pi$
$601$		−	9.95699e7i		−	0.458675i	0.973347	−	0.229337i	$-0.0736559\pi$
$602$		−	1.74995e8i		−	0.802115i
$603$	3.48467e8			1.58931
$604$		−	1.44891e8i		−	0.657554i
$605$	−2.95750e8			−1.33555
$606$		−	4.15142e8i		−	1.86543i
$607$	−3.10423e8			−1.38800			−0.693998	−	0.719977i	$-0.744153\pi$
$607$	−3.10423e8			−1.38800			−0.693998	+	0.719977i	$0.744153\pi$
$608$	2.68917e8			1.19649
$609$	−1.26480e8			−0.559977
$610$			2.81738e8i			1.24124i
$611$		−	9.78043e6i		−	0.0428780i
$612$		−	1.25948e8i		−	0.549460i
$613$			1.26594e8i			0.549583i	0.961504	+	0.274791i	$0.0886088\pi$
$613$			1.26594e8i			0.549583i	−0.961504	+	0.274791i	$0.911391\pi$
$614$	−5.35909e7			−0.231519
$615$		−	2.46974e8i		−	1.06176i
$616$		−	8.84050e7i		−	0.378211i
$617$	−1.23498e8			−0.525780			−0.262890	−	0.964826i	$-0.584676\pi$
$617$	−1.23498e8			−0.525780			−0.262890	+	0.964826i	$0.584676\pi$
$618$		−	2.33606e8i		−	0.989734i
$619$			4.88743e7i			0.206067i	0.994678	+	0.103034i	$0.0328549\pi$
$619$			4.88743e7i			0.206067i	−0.994678	+	0.103034i	$0.967145\pi$
$620$	−8.35255e6	+	1.68421e8i	−0.0350465	+	0.706676i
$621$	−8.36025e7			−0.349096
$622$	7.89053e7			0.327895
$623$		−	9.24203e7i		−	0.382211i
$624$	−2.08041e8			−0.856239
$625$	−1.16887e8			−0.478771
$626$			5.18593e7i			0.211399i
$627$	1.90926e8			0.774573
$628$	1.00175e8			0.404464
$629$	−1.13180e8			−0.454799
$630$	−3.15378e8			−1.26128
$631$		−	3.26706e8i		−	1.30038i	−0.759774	−	0.650188i	$-0.774690\pi$
$631$		−	3.26706e8i		−	1.30038i	0.759774	−	0.650188i	$-0.225310\pi$
$632$		−	1.41473e6i		−	0.00560432i
$633$			4.68695e8i			1.84791i
$634$	−7.26613e7			−0.285125
$635$			2.90355e8i			1.13399i
$636$	−2.46361e7			−0.0957634
$637$		−	3.32390e7i		−	0.128597i
$638$	2.77049e7			0.106683
$639$	6.59188e7			0.252643
$640$	−5.55838e7			−0.212035
$641$			4.41982e8i			1.67815i	0.544017	+	0.839074i	$0.316903\pi$
$641$			4.41982e8i			1.67815i	−0.544017	+	0.839074i	$0.683097\pi$
$642$			1.66489e8i			0.629186i
$643$		−	8.65353e7i		−	0.325507i	−0.986667	−	0.162754i	$-0.947962\pi$
$643$		−	8.65353e7i		−	0.325507i	0.986667	−	0.162754i	$-0.0520376\pi$
$644$		−	1.81458e8i		−	0.679390i
$645$	−6.87842e8			−2.56336
$646$		−	2.96557e8i		−	1.10005i
$647$		−	2.49200e8i		−	0.920100i	−0.887893	−	0.460050i	$-0.847831\pi$
$647$		−	2.49200e8i		−	0.920100i	0.887893	−	0.460050i	$-0.152169\pi$
$648$	−2.39870e8			−0.881558
$649$			5.95783e6i			0.0217949i
$650$		−	4.91151e8i		−	1.78844i
$651$	3.89457e8	+	1.93145e7i	1.41162	+	0.0700068i
$652$	−7.23668e7			−0.261094
$653$	4.39958e8			1.58005			0.790026	−	0.613074i	$-0.210067\pi$
$653$	4.39958e8			1.58005			0.790026	+	0.613074i	$0.210067\pi$
$654$			7.00357e6i			0.0250372i
$655$	2.76325e8			0.983324
$656$	4.37870e7			0.155108
$657$		−	2.66481e8i		−	0.939657i
$658$	4.90527e6			0.0172181
$659$	−1.51407e8			−0.529042			−0.264521	−	0.964380i	$-0.585214\pi$
$659$	−1.51407e8			−0.529042			−0.264521	+	0.964380i	$0.585214\pi$
$660$	−1.09288e8			−0.380137
$661$	2.69858e7			0.0934397			0.0467199	−	0.998908i	$-0.485123\pi$
$661$	2.69858e7			0.0934397			0.0467199	+	0.998908i	$0.485123\pi$
$662$			1.97657e8i			0.681300i
$663$		−	7.82662e8i		−	2.68555i
$664$		−	3.99684e8i		−	1.36525i
$665$	6.29547e8			2.14074
$666$		−	1.10029e8i		−	0.372464i
$667$	1.80811e8			0.609321
$668$		−	1.56826e8i		−	0.526124i
$669$	7.79243e6			0.0260252
$670$	−4.69663e8			−1.56157
$671$	1.20982e8			0.400456
$672$			3.55950e8i			1.17295i
$673$			1.80249e8i			0.591328i	0.955292	+	0.295664i	$0.0955409\pi$
$673$			1.80249e8i			0.591328i	−0.955292	+	0.295664i	$0.904459\pi$
$674$			2.12726e8i			0.694769i
$675$			9.62145e7i			0.312845i
$676$	2.99274e8			0.968790
$677$			2.11700e8i			0.682268i	0.940015	+	0.341134i	$0.110811\pi$
$677$			2.11700e8i			0.682268i	−0.940015	+	0.341134i	$0.889189\pi$
$678$		−	4.68967e8i		−	1.50471i
$679$	3.57182e8			1.14099
$680$			5.39738e8i			1.71655i
$681$			6.94713e8i			2.19970i
$682$	−8.53088e7	−	4.23075e6i	−0.268931	−	0.0133372i
$683$	−3.38255e8			−1.06165			−0.530825	−	0.847481i	$-0.678118\pi$
$683$	−3.38255e8			−1.06165			−0.530825	+	0.847481i	$0.678118\pi$
$684$	−2.44412e8			−0.763756
$685$			4.17849e8i			1.30001i
$686$	2.45342e8			0.759974
$687$	−1.34204e8			−0.413900
$688$		−	1.21950e8i		−	0.374471i
$689$	−8.20403e7			−0.250824
$690$	8.41320e8			2.56103
$691$	−2.07869e8			−0.630022			−0.315011	−	0.949088i	$-0.602008\pi$
$691$	−2.07869e8			−0.630022			−0.315011	+	0.949088i	$0.602008\pi$
$692$	8.86111e7			0.267405
$693$			1.35428e8i			0.406919i
$694$		−	2.96461e8i		−	0.886931i
$695$			4.63707e8i			1.38130i
$696$	−2.10431e8			−0.624139
$697$			1.64729e8i			0.486488i
$698$	−8.53927e7			−0.251104
$699$		−	2.27164e8i		−	0.665133i
$700$	−2.08833e8			−0.608841
$701$	−1.65521e8			−0.480506			−0.240253	−	0.970710i	$-0.577230\pi$
$701$	−1.65521e8			−0.480506			−0.240253	+	0.970710i	$0.577230\pi$
$702$	1.01904e8			0.294565
$703$			2.19636e8i			0.632176i
$704$		−	1.20201e8i		−	0.344499i
$705$		−	1.92808e7i		−	0.0550248i
$706$			1.15971e8i			0.329561i
$707$	−5.87809e8			−1.66333
$708$		−	1.42323e7i		−	0.0401027i
$709$		−	3.45695e8i		−	0.969963i	−0.874525	−	0.484981i	$-0.838826\pi$
$709$		−	3.45695e8i		−	0.969963i	0.874525	−	0.484981i	$-0.161174\pi$
$710$	−8.88452e7			−0.248233
$711$			2.16723e6i			0.00602971i
$712$		−	1.53764e8i		−	0.426005i
$713$	−5.56752e8	−	2.76112e7i	−1.53601	−	0.0761757i
$714$	3.92536e8			1.07841
$715$	−3.63938e8			−0.995657
$716$		−	4.66700e7i		−	0.127145i
$717$	7.71503e8			2.09305
$718$	−3.78504e8			−1.02258
$719$		−	3.94733e8i		−	1.06198i	−0.847378	−	0.530991i	$-0.821820\pi$
$719$		−	3.94733e8i		−	1.06198i	0.847378	−	0.530991i	$-0.178180\pi$
$720$	−2.19780e8			−0.588832
$721$	−3.30768e8			−0.882505
$722$	−2.98617e8			−0.793420
$723$	7.47535e8			1.97796
$724$			2.71173e8i			0.714546i
$725$		−	2.08087e8i		−	0.546048i
$726$			3.57856e8i			0.935187i
$727$	1.47049e8			0.382701			0.191351	−	0.981522i	$-0.438713\pi$
$727$	1.47049e8			0.382701			0.191351	+	0.981522i	$0.438713\pi$
$728$			7.03264e8i			1.82274i
$729$	4.96547e8			1.28167
$730$			3.59162e8i			0.923255i
$731$	4.58784e8			1.17451
$732$	−2.89006e8			−0.736842
$733$	2.61208e8			0.663247			0.331624	−	0.943412i	$-0.392404\pi$
$733$	2.61208e8			0.663247			0.331624	+	0.943412i	$0.392404\pi$
$734$			1.14829e8i			0.290378i
$735$		−	6.55262e7i		−	0.165026i
$736$		−	5.08851e8i		−	1.27631i
$737$			2.01680e8i			0.503802i
$738$	−1.60143e8			−0.398417
$739$		−	1.08837e8i		−	0.269677i	−0.990868	−	0.134838i	$-0.956948\pi$
$739$		−	1.08837e8i		−	0.269677i	0.990868	−	0.134838i	$-0.0430515\pi$
$740$		−	1.25722e8i		−	0.310252i
$741$	−1.51882e9			−3.73295
$742$		−	4.11465e7i		−	0.100721i
$743$		−	1.97010e7i		−	0.0480310i	−0.999712	−	0.0240155i	$-0.992355\pi$
$743$		−	1.97010e7i		−	0.0480310i	0.999712	−	0.0240155i	$-0.00764510\pi$
$744$	6.47958e8	+	3.21344e7i	1.57336	+	0.0780282i
$745$	2.12441e8			0.513771
$746$	1.06827e8			0.257316
$747$			6.12278e8i			1.46888i
$748$	7.28939e7			0.174175
$749$	2.35735e8			0.561020
$750$		−	2.65698e8i		−	0.629804i
$751$	−2.87905e8			−0.679719			−0.339859	−	0.940476i	$-0.610380\pi$
$751$	−2.87905e8			−0.679719			−0.339859	+	0.940476i	$0.610380\pi$
$752$	3.41838e6			0.00803834
$753$	−2.00878e7			−0.0470486
$754$	−2.20393e8			−0.514143
$755$		−	9.51188e8i		−	2.21017i
$756$		−	4.33288e7i		−	0.100279i
$757$			5.62246e8i			1.29610i	0.761597	+	0.648051i	$0.224416\pi$
$757$			5.62246e8i			1.29610i	−0.761597	+	0.648051i	$0.775584\pi$
$758$	−5.21779e8			−1.19806
$759$		−	3.61275e8i		−	0.826251i
$760$	1.04741e9			2.38602
$761$			2.09388e8i			0.475114i	0.971374	+	0.237557i	$0.0763467\pi$
$761$			2.09388e8i			0.475114i	−0.971374	+	0.237557i	$0.923653\pi$
$762$	3.51328e8			0.794049
$763$	9.91651e6			0.0223247
$764$	−6.19199e7			−0.138851
$765$		−	8.26826e8i		−	1.84684i
$766$			6.54117e8i			1.45536i
$767$		−	4.73947e7i		−	0.105037i
$768$			6.93094e8i			1.53006i
$769$	3.14781e8			0.692197			0.346098	−	0.938198i	$-0.387506\pi$
$769$	3.14781e8			0.692197			0.346098	+	0.938198i	$0.387506\pi$
$770$		−	1.82529e8i		−	0.399816i
$771$		−	1.26502e8i		−	0.276015i
$772$	−3.25733e8			−0.707963
$773$		−	3.58451e8i		−	0.776052i	−0.921648	−	0.388026i	$-0.873157\pi$
$773$		−	3.58451e8i		−	0.776052i	0.921648	−	0.388026i	$-0.126843\pi$
$774$			4.46010e8i			0.961882i
$775$	−3.17765e7	+	6.40742e8i	−0.0682655	+	1.37650i
$776$	5.94260e8			1.27172
$777$	−2.90720e8			−0.619743
$778$		−	4.73091e8i		−	1.00463i
$779$	3.19671e8			0.676224
$780$	8.69387e8			1.83202
$781$			3.81513e7i			0.0800860i
$782$	−5.61153e8			−1.17344
$783$	4.31741e7			0.0899369
$784$	1.16174e7			0.0241080
$785$	6.57632e8			1.35948
$786$		−	3.34352e8i		−	0.688552i
$787$			4.48722e8i			0.920561i	0.887773	+	0.460281i	$0.152251\pi$
$787$			4.48722e8i			0.920561i	−0.887773	+	0.460281i	$0.847749\pi$
$788$		−	1.46959e8i		−	0.300344i
$789$	−5.55963e8			−1.13192
$790$		−	2.92099e6i		−	0.00592446i
$791$	−6.64020e8			−1.34169
$792$			2.25318e8i			0.453544i
$793$	−9.62418e8			−1.92994
$794$	2.81272e7			0.0561908
$795$	−1.61732e8			−0.321880
$796$			3.71804e7i			0.0737182i
$797$		−	1.71891e8i		−	0.339530i	−0.985485	−	0.169765i	$-0.945699\pi$
$797$		−	1.71891e8i		−	0.339530i	0.985485	−	0.169765i	$-0.0543009\pi$
$798$		−	7.61749e8i		−	1.49901i
$799$			1.28601e7i			0.0252119i
$800$	−5.85614e8			−1.14378
$801$			2.35552e8i			0.458341i
$802$			5.07358e8i			0.983538i
$803$	1.54229e8			0.297865
$804$		−	4.81779e8i		−	0.927000i
$805$		−	1.19124e9i		−	2.28356i
$806$	6.78634e8	+	3.36557e7i	1.29608	+	0.0642767i
$807$	9.81249e8			1.86706
$808$	−9.77965e8			−1.85391
$809$		−	7.94596e8i		−	1.50072i	−0.661027	−	0.750362i	$-0.729879\pi$
$809$		−	7.94596e8i		−	1.50072i	0.661027	−	0.750362i	$-0.270121\pi$
$810$	−4.95259e8			−0.931917
$811$	1.93998e8			0.363693			0.181846	−	0.983327i	$-0.441793\pi$
$811$	1.93998e8			0.363693			0.181846	+	0.983327i	$0.441793\pi$
$812$			9.37089e7i			0.175030i
$813$	−2.57461e7			−0.0479115
$814$	6.36808e7			0.118069
$815$	−4.75076e8			−0.877588
$816$	2.73550e8			0.503461
$817$		−	8.90310e8i		−	1.63258i
$818$			6.36647e7i			0.116316i
$819$		−	1.07733e9i		−	1.96109i
$820$	−1.82982e8			−0.331870
$821$		−	2.18486e8i		−	0.394815i	−0.980321	−	0.197408i	$-0.936748\pi$
$821$		−	2.18486e8i		−	0.394815i	0.980321	−	0.197408i	$-0.0632523\pi$
$822$	5.05596e8			0.910307
$823$			4.25819e8i			0.763882i	0.924187	+	0.381941i	$0.124744\pi$
$823$			4.25819e8i			0.763882i	−0.924187	+	0.381941i	$0.875256\pi$
$824$	−5.50313e8			−0.983623
$825$	−4.15775e8			−0.740452
$826$	2.37703e7			0.0421789
$827$			1.03592e9i			1.83151i	0.401738	+	0.915755i	$0.368406\pi$
$827$			1.03592e9i			1.83151i	−0.401738	+	0.915755i	$0.631594\pi$
$828$			4.62483e8i			0.814713i
$829$			8.23718e8i			1.44582i	0.690941	+	0.722911i	$0.257196\pi$
$829$			8.23718e8i			1.44582i	−0.690941	+	0.722911i	$0.742804\pi$
$830$		−	8.25227e8i		−	1.44324i
$831$	−1.06462e9			−1.85520
$832$			9.56199e8i			1.66027i
$833$			4.37054e7i			0.0756136i
$834$	5.61083e8			0.967228
$835$		−	1.02953e9i		−	1.76840i
$836$		−	1.41457e8i		−	0.242106i
$837$	−1.32942e8	−	6.59302e6i	−0.226717	−	0.0112437i
$838$	−1.37807e8			−0.234174
$839$	−2.05710e8			−0.348314			−0.174157	−	0.984718i	$-0.555720\pi$
$839$	−2.05710e8			−0.348314			−0.174157	+	0.984718i	$0.555720\pi$
$840$			1.38639e9i			2.33910i
$841$	5.01449e8			0.843022
$842$	−1.45777e8			−0.244204
$843$		−	1.45911e9i		−	2.43560i
$844$	3.47256e8			0.577593
$845$	1.96469e9			3.25629
$846$	−1.25021e7			−0.0206476
$847$	5.06696e8			0.833868
$848$		−	2.86741e7i		−	0.0470221i
$849$			1.02220e9i			1.67037i
$850$			6.45806e8i			1.05159i
$851$	4.15601e8			0.674354
$852$		−	9.11371e7i		−	0.147359i
$853$	1.02787e9			1.65611			0.828055	−	0.560646i	$-0.189447\pi$
$853$	1.02787e9			1.65611			0.828055	+	0.560646i	$0.189447\pi$
$854$		−	4.82691e8i		−	0.774988i
$855$	−1.60453e9			−2.56713
$856$	3.92203e8			0.625301
$857$	4.23199e8			0.672360			0.336180	−	0.941798i	$-0.390865\pi$
$857$	4.23199e8			0.672360			0.336180	+	0.941798i	$0.390865\pi$
$858$			4.40363e8i			0.697187i
$859$			2.00776e8i			0.316762i	0.987378	+	0.158381i	$0.0506274\pi$
$859$			2.00776e8i			0.316762i	−0.987378	+	0.158381i	$0.949373\pi$
$860$			5.09621e8i			0.801221i
$861$			4.23130e8i			0.662925i
$862$	6.46680e7			0.100964
$863$		−	9.98067e7i		−	0.155284i	−0.996981	−	0.0776421i	$-0.975261\pi$
$863$		−	9.98067e7i		−	0.155284i	0.996981	−	0.0776421i	$-0.0247391\pi$
$864$		−	1.21504e8i		−	0.188386i
$865$	5.81717e8			0.898801
$866$		−	8.72736e8i		−	1.34378i
$867$			7.24768e7i			0.111209i
$868$	1.43101e7	−	2.88548e8i	0.0218818	−	0.441224i
$869$	−1.25431e6			−0.00191138
$870$	−4.34475e8			−0.659793
$871$		−	1.60437e9i		−	2.42801i
$872$	1.64985e7			0.0248826
$873$	−9.10349e8			−1.36825
$874$			1.08896e9i			1.63110i
$875$	−3.76208e8			−0.561570
$876$	−3.68427e8			−0.548074
$877$	4.25671e8			0.631067			0.315534	−	0.948914i	$-0.397817\pi$
$877$	4.25671e8			0.631067			0.315534	+	0.948914i	$0.397817\pi$
$878$	−3.33554e8			−0.492813
$879$			1.68868e9i			2.48645i
$880$		−	1.27201e8i		−	0.186656i
$881$			6.54693e8i			0.957436i	0.877969	+	0.478718i	$0.158898\pi$
$881$			6.54693e8i			0.957436i	−0.877969	+	0.478718i	$0.841102\pi$
$882$	−4.24885e7			−0.0619249
$883$			5.11690e8i			0.743233i	0.928386	+	0.371616i	$0.121196\pi$
$883$			5.11690e8i			0.743233i	−0.928386	+	0.371616i	$0.878804\pi$
$884$	−5.79873e8			−0.839414
$885$		−	9.34324e7i		−	0.134793i
$886$	2.78612e8			0.400589
$887$	−2.55940e8			−0.366747			−0.183373	−	0.983043i	$-0.558702\pi$
$887$	−2.55940e8			−0.366747			−0.183373	+	0.983043i	$0.558702\pi$
$888$	−4.83684e8			−0.690753
$889$		−	4.97453e8i		−	0.708022i
$890$		−	3.17476e8i		−	0.450340i
$891$			2.12671e8i			0.300660i
$892$		−	5.77340e6i		−	0.00813461i
$893$	2.49562e7			0.0350448
$894$		−	2.57053e8i		−	0.359757i
$895$		−	3.06381e8i		−	0.427359i
$896$	9.52295e7			0.132388
$897$			2.87395e9i			3.98201i
$898$		−	3.84820e8i		−	0.531409i
$899$	2.87518e8	+	1.42590e7i	0.395719	+	0.0196250i
$900$	5.32251e8			0.730111
$901$	1.07873e8			0.147482
$902$		−	9.26846e7i		−	0.126295i
$903$	1.17845e9			1.60047
$904$	−1.10476e9			−1.49542
$905$			1.78020e9i			2.40173i
$906$	−1.15093e9			−1.54762
$907$	−1.12142e9			−1.50296			−0.751478	−	0.659758i	$-0.770659\pi$
$907$	−1.12142e9			−1.50296			−0.751478	+	0.659758i	$0.770659\pi$
$908$	5.14712e8			0.687554
$909$	1.49815e9			1.99463
$910$			1.45203e9i			1.92686i
$911$		−	1.23872e8i		−	0.163839i	−0.996639	−	0.0819196i	$-0.973895\pi$
$911$		−	1.23872e8i		−	0.163839i	0.996639	−	0.0819196i	$-0.0261051\pi$
$912$		−	5.30846e8i		−	0.699817i
$913$	−3.54364e8			−0.465626
$914$		−	4.39645e8i		−	0.575789i
$915$	−1.89728e9			−2.47667
$916$			9.94316e7i			0.129371i
$917$	−4.73417e8			−0.613954
$918$	−1.33992e8			−0.173202
$919$	2.92250e8			0.376538			0.188269	−	0.982118i	$-0.439712\pi$
$919$	2.92250e8			0.376538			0.188269	+	0.982118i	$0.439712\pi$
$920$		−	1.98193e9i		−	2.54521i
$921$		−	3.60892e8i		−	0.461953i
$922$			1.50632e8i			0.192187i
$923$		−	3.03495e8i		−	0.385964i
$924$	1.87238e8			0.237344
$925$		−	4.78297e8i		−	0.604327i
$926$		−	6.13300e8i		−	0.772396i
$927$	8.43027e8			1.05828
$928$			2.62781e8i			0.328814i
$929$			4.46017e8i			0.556294i	0.960538	+	0.278147i	$0.0897203\pi$
$929$			4.46017e8i			0.556294i	−0.960538	+	0.278147i	$0.910280\pi$
$930$	1.33784e9	+	6.63478e7i	1.66324	+	0.0824855i
$931$	8.48140e7			0.105104
$932$	−1.68306e8			−0.207898
$933$			5.31363e8i			0.654254i
$934$	5.40616e8			0.663512
$935$	4.78536e8			0.585437
$936$		−	1.79241e9i		−	2.18579i
$937$	4.58660e8			0.557535			0.278768	−	0.960359i	$-0.410074\pi$
$937$	4.58660e8			0.557535			0.278768	+	0.960359i	$0.410074\pi$
$938$	8.04654e8			0.974991
$939$	−3.49230e8			−0.421808
$940$	−1.42851e7			−0.0171989
$941$			5.57846e8i			0.669492i	0.942308	+	0.334746i	$0.108651\pi$
$941$			5.57846e8i			0.669492i	−0.942308	+	0.334746i	$0.891349\pi$
$942$		−	7.95732e8i		−	0.951950i
$943$		−	6.04889e8i		−	0.721341i
$944$	1.65650e7			0.0196914
$945$		−	2.84446e8i		−	0.337058i
$946$	−2.58134e8			−0.304910
$947$		−	9.67253e8i		−	1.13891i	−0.822022	−	0.569456i	$-0.807154\pi$
$947$		−	9.67253e8i		−	1.13891i	0.822022	−	0.569456i	$-0.192846\pi$
$948$	2.99634e6			0.00351695
$949$	−1.22690e9			−1.43552
$950$	1.25324e9			1.46172
$951$		−	4.89315e8i		−	0.568914i
$952$		−	9.24710e8i		−	1.07175i
$953$		−	9.15856e8i		−	1.05815i	−0.848574	−	0.529077i	$-0.822538\pi$
$953$		−	9.15856e8i		−	1.05815i	0.848574	−	0.529077i	$-0.177462\pi$
$954$			1.04870e8i			0.120783i
$955$	−4.06494e8			−0.466706
$956$		−	5.71605e8i		−	0.654218i
$957$			1.86570e8i			0.212866i
$958$	8.80386e7			0.100133
$959$		−	7.15884e8i		−	0.811684i
$960$			1.88502e9i			2.13060i
$961$	−8.83149e8	−	8.78126e7i	−0.995093	−	0.0989434i
$962$	−5.06582e8			−0.569016
$963$	−6.00817e8			−0.672765
$964$		−	5.53847e8i		−	0.618243i
$965$	−2.13839e9			−2.37960
$966$	−1.44140e9			−1.59902
$967$			3.49409e8i			0.386415i	0.981158	+	0.193208i	$0.0618891\pi$
$967$			3.49409e8i			0.386415i	−0.981158	+	0.193208i	$0.938111\pi$
$968$	8.43014e8			0.929413
$969$	1.99707e9			2.19494
$970$	1.22697e9			1.34437
$971$	3.96326e8			0.432907			0.216454	−	0.976293i	$-0.430551\pi$
$971$	3.96326e8			0.432907			0.216454	+	0.976293i	$0.430551\pi$
$972$		−	6.03677e8i		−	0.657364i
$973$		−	7.94450e8i		−	0.862438i
$974$			5.61327e8i			0.607490i
$975$	3.30750e9			3.56851
$976$		−	3.36376e8i		−	0.361806i
$977$	−1.02422e9			−1.09827			−0.549134	−	0.835734i	$-0.685042\pi$
$977$	−1.02422e9			−1.09827			−0.549134	+	0.835734i	$0.685042\pi$
$978$			5.74840e8i			0.614512i
$979$	−1.36329e8			−0.145291
$980$	−4.85483e7			−0.0515817
$981$	−2.52742e7			−0.0267713
$982$		−	5.50774e8i		−	0.581620i
$983$			1.81002e9i			1.90556i	0.303666	+	0.952779i	$0.401789\pi$
$983$			1.81002e9i			1.90556i	−0.303666	+	0.952779i	$0.598211\pi$
$984$			7.03981e8i			0.738883i
$985$		−	9.64764e8i		−	1.00951i
$986$	2.89791e8			0.302311
$987$			3.30330e7i			0.0343556i
$988$			1.12529e9i			1.16680i
$989$	−1.68467e9			−1.74151
$990$			4.65212e8i			0.479453i
$991$			7.59937e8i			0.780831i	0.920639	+	0.390415i	$0.127669\pi$
$991$			7.59937e8i			0.780831i	−0.920639	+	0.390415i	$0.872331\pi$
$992$	4.01288e7	−	8.09156e8i	0.0411075	−	0.828891i
$993$	−1.33106e9			−1.35941
$994$	1.52215e8			0.154988
$995$			2.44083e8i			0.247781i
$996$	8.46515e8			0.856755
$997$	−7.52427e8			−0.759240			−0.379620	−	0.925143i	$-0.623945\pi$
$997$	−7.52427e8			−0.759240			−0.379620	+	0.925143i	$0.623945\pi$
$998$			1.22335e9i			1.23072i
$999$	9.92374e7			0.0995357

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	31.7.b.c.30.3	✓	12
3.2	odd	2		279.7.d.f.154.10		12
4.3	odd	2		496.7.e.c.433.11		12
31.30	odd	2	inner	31.7.b.c.30.4	yes	12
93.92	even	2		279.7.d.f.154.9		12
124.123	even	2		496.7.e.c.433.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
31.7.b.c.30.3	✓	12	1.1	even	1	trivial
31.7.b.c.30.4	yes	12	31.30	odd	2	inner
279.7.d.f.154.9		12	93.92	even	2
279.7.d.f.154.10		12	3.2	odd	2
496.7.e.c.433.2		12	124.123	even	2
496.7.e.c.433.11		12	4.3	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 31 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	31.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.88527 q^{2} -39.6325i q^{3} -29.3637 q^{4} -192.768 q^{5} +233.248i q^{6} +330.261 q^{7} +549.470 q^{8} -841.735 q^{9} +1134.49 q^{10} -487.165i q^{11} +1163.75i q^{12} +3875.41i q^{13} -1943.67 q^{14} +7639.86i q^{15} -1354.50 q^{16} -5095.71i q^{17} +4953.83 q^{18} -9888.66 q^{19} +5660.36 q^{20} -13089.1i q^{21} +2867.09i q^{22} +18711.6i q^{23} -21776.9i q^{24} +21534.3 q^{25} -22807.8i q^{26} +4467.96i q^{27} -9697.66 q^{28} -9663.04i q^{29} -44962.6i q^{30} +(-1475.62 + 29754.4i) q^{31} -27194.5 q^{32} -19307.6 q^{33} +29989.6i q^{34} -63663.5 q^{35} +24716.4 q^{36} -22210.9i q^{37} +58197.4 q^{38} +153592. q^{39} -105920. q^{40} -32327.0 q^{41} +77032.6i q^{42} +90033.4i q^{43} +14304.9i q^{44} +162259. q^{45} -110123. i q^{46} -2523.71 q^{47} +53682.3i q^{48} -8576.89 q^{49} -126735. q^{50} -201956. q^{51} -113796. i q^{52} +21169.4i q^{53} -26295.1i q^{54} +93909.6i q^{55} +181468. q^{56} +391912. i q^{57} +56869.6i q^{58} -12229.6 q^{59} -224334. i q^{60} +248340. i q^{61} +(8684.43 - 175113. i) q^{62} -277992. q^{63} +246735. q^{64} -747053. i q^{65} +113630. q^{66} -413987. q^{67} +149629. i q^{68} +741586. q^{69} +374677. q^{70} -78313.0 q^{71} -462508. q^{72} +316585. i q^{73} +130717. i q^{74} -853459. i q^{75} +290367. q^{76} -160891. i q^{77} -903931. q^{78} -2574.72i q^{79} +261104. q^{80} -436548. q^{81} +190253. q^{82} -727400. i q^{83} +384342. i q^{84} +982288. i q^{85} -529870. i q^{86} -382970. q^{87} -267682. i q^{88} -279841. i q^{89} -954938. q^{90} +1.27990e6i q^{91} -549440. i q^{92} +(1.17924e6 + 58482.6i) q^{93} +14852.7 q^{94} +1.90621e6 q^{95} +1.07778e6i q^{96} +1.08152e6 q^{97} +50477.3 q^{98} +410064. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 2 q^{2} + 122 q^{4} - 146 q^{5} + 6 q^{7} + 1142 q^{8} - 5668 q^{9} - 3004 q^{10} - 1312 q^{14} - 1102 q^{16} - 16382 q^{18} + 10430 q^{19} - 8052 q^{20} + 29618 q^{25} + 136504 q^{28} + 90076 q^{31}+ \cdots + 408810 q^{98}+O(q^{100}) \) 12 * q - 2 * q^2 + 122 * q^4 - 146 * q^5 + 6 * q^7 + 1142 * q^8 - 5668 * q^9 - 3004 * q^10 - 1312 * q^14 - 1102 * q^16 - 16382 * q^18 + 10430 * q^19 - 8052 * q^20 + 29618 * q^25 + 136504 * q^28 + 90076 * q^31 + 38406 * q^32 - 166248 * q^33 - 198898 * q^35 + 26982 * q^36 + 448196 * q^38 + 208224 * q^39 - 354932 * q^40 - 116690 * q^41 - 363818 * q^45 - 221120 * q^47 - 580182 * q^49 - 60318 * q^50 + 179784 * q^51 + 750496 * q^56 + 799510 * q^59 - 73070 * q^62 - 469874 * q^63 - 322286 * q^64 - 511884 * q^66 - 13744 * q^67 + 475128 * q^69 - 71312 * q^70 + 640678 * q^71 - 118870 * q^72 + 2044196 * q^76 + 1373868 * q^78 + 1356020 * q^80 - 678724 * q^81 - 488056 * q^82 - 1051200 * q^87 - 2598532 * q^90 - 2260440 * q^93 - 5045848 * q^94 + 3828206 * q^95 + 3708966 * q^97 + 408810 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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