LMFDB - Embedded newform 8450.2.a.cx.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8450,2,Mod(1,8450)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8450.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8450 = 2 \cdot 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8450.a (trivial)

Newform invariants

comment:select newform

sage:traces = [9,9,-7,9,0,-7,1,9,8,0,-4,-7,0,1,0,9,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$67.4735897080$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 14x^{7} + 17x^{6} + 53x^{5} - 69x^{4} - 33x^{3} + 26x^{2} + 8x - 1 $ x^9 - x^8 - 14x^7 + 17x^6 + 53x^5 - 69x^4 - 33x^3 + 26x^2 + 8*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1690)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.506152$ of defining polynomial
Character	$\chi$	$=$	8450.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} -0.0670024 q^{3} +1.00000 q^{4} -0.0670024 q^{6} -5.03568 q^{7} +1.00000 q^{8} -2.99551 q^{9} +3.28967 q^{11} -0.0670024 q^{12} -5.03568 q^{14} +1.00000 q^{16} -0.516695 q^{17} -2.99551 q^{18} +5.26981 q^{19} +0.337403 q^{21} +3.28967 q^{22} -0.524116 q^{23} -0.0670024 q^{24} +0.401714 q^{27} -5.03568 q^{28} +6.36392 q^{29} +6.84648 q^{31} +1.00000 q^{32} -0.220416 q^{33} -0.516695 q^{34} -2.99551 q^{36} -5.49639 q^{37} +5.26981 q^{38} -5.54869 q^{41} +0.337403 q^{42} -9.35921 q^{43} +3.28967 q^{44} -0.524116 q^{46} -1.65182 q^{47} -0.0670024 q^{48} +18.3580 q^{49} +0.0346198 q^{51} +9.30982 q^{53} +0.401714 q^{54} -5.03568 q^{56} -0.353090 q^{57} +6.36392 q^{58} -12.8188 q^{59} -11.5310 q^{61} +6.84648 q^{62} +15.0844 q^{63} +1.00000 q^{64} -0.220416 q^{66} +3.38856 q^{67} -0.516695 q^{68} +0.0351171 q^{69} +3.86643 q^{71} -2.99551 q^{72} -6.43004 q^{73} -5.49639 q^{74} +5.26981 q^{76} -16.5657 q^{77} -7.28413 q^{79} +8.95962 q^{81} -5.54869 q^{82} +2.57870 q^{83} +0.337403 q^{84} -9.35921 q^{86} -0.426398 q^{87} +3.28967 q^{88} -4.82497 q^{89} -0.524116 q^{92} -0.458731 q^{93} -1.65182 q^{94} -0.0670024 q^{96} -1.88098 q^{97} +18.3580 q^{98} -9.85425 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{2} - 7 q^{3} + 9 q^{4} - 7 q^{6} + q^{7} + 9 q^{8} + 8 q^{9} - 4 q^{11} - 7 q^{12} + q^{14} + 9 q^{16} - 12 q^{17} + 8 q^{18} - 6 q^{19} - 8 q^{21} - 4 q^{22} - 11 q^{23} - 7 q^{24} - 34 q^{27}+ \cdots - 16 q^{99}+O(q^{100}) $ 9 * q + 9 * q^2 - 7 * q^3 + 9 * q^4 - 7 * q^6 + q^7 + 9 * q^8 + 8 * q^9 - 4 * q^11 - 7 * q^12 + q^14 + 9 * q^16 - 12 * q^17 + 8 * q^18 - 6 * q^19 - 8 * q^21 - 4 * q^22 - 11 * q^23 - 7 * q^24 - 34 * q^27 + q^28 + 15 * q^29 + 6 * q^31 + 9 * q^32 - 14 * q^33 - 12 * q^34 + 8 * q^36 - 10 * q^37 - 6 * q^38 - 27 * q^41 - 8 * q^42 - 41 * q^43 - 4 * q^44 - 11 * q^46 + 5 * q^47 - 7 * q^48 - 10 * q^49 + 12 * q^51 - 18 * q^53 - 34 * q^54 + q^56 - 12 * q^57 + 15 * q^58 - 20 * q^59 - 19 * q^61 + 6 * q^62 + 39 * q^63 + 9 * q^64 - 14 * q^66 - q^67 - 12 * q^68 - 14 * q^69 + 10 * q^71 + 8 * q^72 - 12 * q^73 - 10 * q^74 - 6 * q^76 - 30 * q^77 - 14 * q^79 + 53 * q^81 - 27 * q^82 + q^83 - 8 * q^84 - 41 * q^86 - 33 * q^87 - 4 * q^88 - 43 * q^89 - 11 * q^92 - 28 * q^93 + 5 * q^94 - 7 * q^96 + 6 * q^97 - 10 * q^98 - 16 * q^99

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−0.0670024	−0.0386839	−0.0193419	−	0.999813i	$-0.506157\pi$
$3$	−0.0670024	−0.0386839	−0.0193419	+	0.999813i	$0.506157\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−0.0670024	−0.0273536
$7$	−5.03568	−1.90331	−0.951653	−	0.307174i	$-0.900617\pi$
$7$	−5.03568	−1.90331	−0.951653	+	0.307174i	$0.900617\pi$
$8$	1.00000	0.353553
$9$	−2.99551	−0.998504
$10$	0	0
$11$	3.28967	0.991874	0.495937	−	0.868358i	$-0.334825\pi$
$11$	3.28967	0.991874	0.495937	+	0.868358i	$0.334825\pi$
$12$	−0.0670024	−0.0193419
$13$	0	0
$13$	0	0
$14$	−5.03568	−1.34584
$15$	0	0
$16$	1.00000	0.250000
$17$	−0.516695	−0.125317	−0.0626584	−	0.998035i	$-0.519958\pi$
$17$	−0.516695	−0.125317	−0.0626584	+	0.998035i	$0.519958\pi$
$18$	−2.99551	−0.706049
$19$	5.26981	1.20898	0.604489	−	0.796614i	$-0.293377\pi$
$19$	5.26981	1.20898	0.604489	+	0.796614i	$0.293377\pi$
$20$	0	0
$21$	0.337403	0.0736273
$22$	3.28967	0.701361
$23$	−0.524116	−0.109286	−0.0546429	−	0.998506i	$-0.517402\pi$
$23$	−0.524116	−0.109286	−0.0546429	+	0.998506i	$0.517402\pi$
$24$	−0.0670024	−0.0136768
$25$	0	0
$26$	0	0
$27$	0.401714	0.0773099
$28$	−5.03568	−0.951653
$29$	6.36392	1.18175	0.590875	−	0.806763i	$-0.298783\pi$
$29$	6.36392	1.18175	0.590875	+	0.806763i	$0.298783\pi$
$30$	0	0
$31$	6.84648	1.22966	0.614832	−	0.788658i	$-0.289224\pi$
$31$	6.84648	1.22966	0.614832	+	0.788658i	$0.289224\pi$
$32$	1.00000	0.176777
$33$	−0.220416	−0.0383695
$34$	−0.516695	−0.0886124
$35$	0	0
$36$	−2.99551	−0.499252
$37$	−5.49639	−0.903600	−0.451800	−	0.892119i	$-0.649218\pi$
$37$	−5.49639	−0.903600	−0.451800	+	0.892119i	$0.649218\pi$
$38$	5.26981	0.854876
$39$	0	0
$40$	0	0
$41$	−5.54869	−0.866560	−0.433280	−	0.901259i	$-0.642644\pi$
$41$	−5.54869	−0.866560	−0.433280	+	0.901259i	$0.642644\pi$
$42$	0.337403	0.0520624
$43$	−9.35921	−1.42727	−0.713633	−	0.700520i	$-0.752952\pi$
$43$	−9.35921	−1.42727	−0.713633	+	0.700520i	$0.752952\pi$
$44$	3.28967	0.495937
$45$	0	0
$46$	−0.524116	−0.0772767
$47$	−1.65182	−0.240942	−0.120471	−	0.992717i	$-0.538440\pi$
$47$	−1.65182	−0.240942	−0.120471	+	0.992717i	$0.538440\pi$
$48$	−0.0670024	−0.00967097
$49$	18.3580	2.62258
$50$	0	0
$51$	0.0346198	0.00484774
$52$	0	0
$53$	9.30982	1.27880	0.639401	−	0.768873i	$-0.279182\pi$
$53$	9.30982	1.27880	0.639401	+	0.768873i	$0.279182\pi$
$54$	0.401714	0.0546663
$55$	0	0
$56$	−5.03568	−0.672921
$57$	−0.353090	−0.0467679
$58$	6.36392	0.835624
$59$	−12.8188	−1.66887	−0.834434	−	0.551108i	$-0.814205\pi$
$59$	−12.8188	−1.66887	−0.834434	+	0.551108i	$0.814205\pi$
$60$	0	0
$61$	−11.5310	−1.47639	−0.738196	−	0.674586i	$-0.764322\pi$
$61$	−11.5310	−1.47639	−0.738196	+	0.674586i	$0.764322\pi$
$62$	6.84648	0.869503
$63$	15.0844	1.90046
$64$	1.00000	0.125000
$65$	0	0
$66$	−0.220416	−0.0271314
$67$	3.38856	0.413978	0.206989	−	0.978343i	$-0.433634\pi$
$67$	3.38856	0.413978	0.206989	+	0.978343i	$0.433634\pi$
$68$	−0.516695	−0.0626584
$69$	0.0351171	0.00422760
$70$	0	0
$71$	3.86643	0.458861	0.229430	−	0.973325i	$-0.426314\pi$
$71$	3.86643	0.458861	0.229430	+	0.973325i	$0.426314\pi$
$72$	−2.99551	−0.353024
$73$	−6.43004	−0.752579	−0.376290	−	0.926502i	$-0.622800\pi$
$73$	−6.43004	−0.752579	−0.376290	+	0.926502i	$0.622800\pi$
$74$	−5.49639	−0.638942
$75$	0	0
$76$	5.26981	0.604489
$77$	−16.5657	−1.88784
$78$	0	0
$79$	−7.28413	−0.819528	−0.409764	−	0.912191i	$-0.634389\pi$
$79$	−7.28413	−0.819528	−0.409764	+	0.912191i	$0.634389\pi$
$80$	0	0
$81$	8.95962	0.995513
$82$	−5.54869	−0.612750
$83$	2.57870	0.283050	0.141525	−	0.989935i	$-0.454800\pi$
$83$	2.57870	0.283050	0.141525	+	0.989935i	$0.454800\pi$
$84$	0.337403	0.0368136
$85$	0	0
$86$	−9.35921	−1.00923
$87$	−0.426398	−0.0457147
$88$	3.28967	0.350680
$89$	−4.82497	−0.511446	−0.255723	−	0.966750i	$-0.582313\pi$
$89$	−4.82497	−0.511446	−0.255723	+	0.966750i	$0.582313\pi$
$90$	0	0
$91$	0	0
$92$	−0.524116	−0.0546429
$93$	−0.458731	−0.0475681
$94$	−1.65182	−0.170372
$95$	0	0
$96$	−0.0670024	−0.00683841
$97$	−1.88098	−0.190985	−0.0954925	−	0.995430i	$-0.530443\pi$
$97$	−1.88098	−0.190985	−0.0954925	+	0.995430i	$0.530443\pi$
$98$	18.3580	1.85444
$99$	−9.85425	−0.990390
$100$	0	0
$101$	3.10344	0.308804	0.154402	−	0.988008i	$-0.450655\pi$
$101$	3.10344	0.308804	0.154402	+	0.988008i	$0.450655\pi$
$102$	0.0346198	0.00342787
$103$	−14.9509	−1.47316	−0.736580	−	0.676351i	$-0.763560\pi$
$103$	−14.9509	−1.47316	−0.736580	+	0.676351i	$0.763560\pi$
$104$	0	0
$105$	0	0
$106$	9.30982	0.904250
$107$	−15.3210	−1.48113	−0.740567	−	0.671983i	$-0.765443\pi$
$107$	−15.3210	−1.48113	−0.740567	+	0.671983i	$0.765443\pi$
$108$	0.401714	0.0386549
$109$	4.19883	0.402175	0.201087	−	0.979573i	$-0.435553\pi$
$109$	4.19883	0.402175	0.201087	+	0.979573i	$0.435553\pi$
$110$	0	0
$111$	0.368271	0.0349548
$112$	−5.03568	−0.475827
$113$	−11.7363	−1.10406	−0.552029	−	0.833825i	$-0.686146\pi$
$113$	−11.7363	−1.10406	−0.552029	+	0.833825i	$0.686146\pi$
$114$	−0.353090	−0.0330699
$115$	0	0
$116$	6.36392	0.590875
$117$	0	0
$118$	−12.8188	−1.18007
$119$	2.60191	0.238516
$120$	0	0
$121$	−0.178043	−0.0161857
$122$	−11.5310	−1.04397
$123$	0.371776	0.0335219
$124$	6.84648	0.614832
$125$	0	0
$126$	15.0844	1.34383
$127$	9.96441	0.884198	0.442099	−	0.896966i	$-0.354234\pi$
$127$	9.96441	0.884198	0.442099	+	0.896966i	$0.354234\pi$
$128$	1.00000	0.0883883
$129$	0.627090	0.0552122
$130$	0	0
$131$	−3.43248	−0.299897	−0.149948	−	0.988694i	$-0.547911\pi$
$131$	−3.43248	−0.299897	−0.149948	+	0.988694i	$0.547911\pi$
$132$	−0.220416	−0.0191848
$133$	−26.5371	−2.30106
$134$	3.38856	0.292727
$135$	0	0
$136$	−0.516695	−0.0443062
$137$	17.1983	1.46935	0.734677	−	0.678417i	$-0.237334\pi$
$137$	17.1983	1.46935	0.734677	+	0.678417i	$0.237334\pi$
$138$	0.0351171	0.00298936
$139$	−6.56328	−0.556690	−0.278345	−	0.960481i	$-0.589786\pi$
$139$	−6.56328	−0.556690	−0.278345	+	0.960481i	$0.589786\pi$
$140$	0	0
$141$	0.110676	0.00932057
$142$	3.86643	0.324464
$143$	0	0
$144$	−2.99551	−0.249626
$145$	0	0
$146$	−6.43004	−0.532154
$147$	−1.23003	−0.101451
$148$	−5.49639	−0.451800
$149$	−12.7625	−1.04554	−0.522772	−	0.852473i	$-0.675102\pi$
$149$	−12.7625	−1.04554	−0.522772	+	0.852473i	$0.675102\pi$
$150$	0	0
$151$	9.73737	0.792416	0.396208	−	0.918161i	$-0.370326\pi$
$151$	9.73737	0.792416	0.396208	+	0.918161i	$0.370326\pi$
$152$	5.26981	0.427438
$153$	1.54776	0.125129
$154$	−16.5657	−1.33491
$155$	0	0
$156$	0	0
$157$	−21.3156	−1.70117	−0.850587	−	0.525835i	$-0.823753\pi$
$157$	−21.3156	−1.70117	−0.850587	+	0.525835i	$0.823753\pi$
$158$	−7.28413	−0.579494
$159$	−0.623781	−0.0494690
$160$	0	0
$161$	2.63928	0.208004
$162$	8.95962	0.703934
$163$	1.27897	0.100177	0.0500883	−	0.998745i	$-0.484050\pi$
$163$	1.27897	0.100177	0.0500883	+	0.998745i	$0.484050\pi$
$164$	−5.54869	−0.433280
$165$	0	0
$166$	2.57870	0.200146
$167$	1.90682	0.147554	0.0737770	−	0.997275i	$-0.476495\pi$
$167$	1.90682	0.147554	0.0737770	+	0.997275i	$0.476495\pi$
$168$	0.337403	0.0260312
$169$	0	0
$170$	0	0
$171$	−15.7858	−1.20717
$172$	−9.35921	−0.713633
$173$	−0.112802	−0.00857615	−0.00428807	−	0.999991i	$-0.501365\pi$
$173$	−0.112802	−0.00857615	−0.00428807	+	0.999991i	$0.501365\pi$
$174$	−0.426398	−0.0323252
$175$	0	0
$176$	3.28967	0.247969
$177$	0.858892	0.0645583
$178$	−4.82497	−0.361647
$179$	−9.78320	−0.731231	−0.365615	−	0.930766i	$-0.619141\pi$
$179$	−9.78320	−0.731231	−0.365615	+	0.930766i	$0.619141\pi$
$180$	0	0
$181$	8.36184	0.621531	0.310765	−	0.950487i	$-0.399415\pi$
$181$	8.36184	0.621531	0.310765	+	0.950487i	$0.399415\pi$
$182$	0	0
$183$	0.772604	0.0571126
$184$	−0.524116	−0.0386384
$185$	0	0
$186$	−0.458731	−0.0336358
$187$	−1.69976	−0.124299
$188$	−1.65182	−0.120471
$189$	−2.02290	−0.147144
$190$	0	0
$191$	−9.25588	−0.669732	−0.334866	−	0.942266i	$-0.608691\pi$
$191$	−9.25588	−0.669732	−0.334866	+	0.942266i	$0.608691\pi$
$192$	−0.0670024	−0.00483548
$193$	17.5149	1.26075	0.630374	−	0.776292i	$-0.282902\pi$
$193$	17.5149	1.26075	0.630374	+	0.776292i	$0.282902\pi$
$194$	−1.88098	−0.135047
$195$	0	0
$196$	18.3580	1.31129
$197$	−20.2809	−1.44496	−0.722479	−	0.691393i	$-0.756997\pi$
$197$	−20.2809	−1.44496	−0.722479	+	0.691393i	$0.756997\pi$
$198$	−9.85425	−0.700311
$199$	−27.0802	−1.91966	−0.959832	−	0.280577i	$-0.909474\pi$
$199$	−27.0802	−1.91966	−0.959832	+	0.280577i	$0.909474\pi$
$200$	0	0
$201$	−0.227042	−0.0160143
$202$	3.10344	0.218357
$203$	−32.0466	−2.24923
$204$	0.0346198	0.00242387
$205$	0	0
$206$	−14.9509	−1.04168
$207$	1.57000	0.109122
$208$	0	0
$209$	17.3360	1.19915
$210$	0	0
$211$	−11.9828	−0.824928	−0.412464	−	0.910974i	$-0.635332\pi$
$211$	−11.9828	−0.824928	−0.412464	+	0.910974i	$0.635332\pi$
$212$	9.30982	0.639401
$213$	−0.259060	−0.0177505
$214$	−15.3210	−1.04732
$215$	0	0
$216$	0.401714	0.0273332
$217$	−34.4766	−2.34043
$218$	4.19883	0.284380
$219$	0.430828	0.0291127
$220$	0	0
$221$	0	0
$222$	0.368271	0.0247167
$223$	12.7910	0.856546	0.428273	−	0.903649i	$-0.359122\pi$
$223$	12.7910	0.856546	0.428273	+	0.903649i	$0.359122\pi$
$224$	−5.03568	−0.336460
$225$	0	0
$226$	−11.7363	−0.780687
$227$	5.83533	0.387305	0.193652	−	0.981070i	$-0.437967\pi$
$227$	5.83533	0.387305	0.193652	+	0.981070i	$0.437967\pi$
$228$	−0.353090	−0.0233840
$229$	−15.2208	−1.00582	−0.502909	−	0.864340i	$-0.667737\pi$
$229$	−15.2208	−1.00582	−0.502909	+	0.864340i	$0.667737\pi$
$230$	0	0
$231$	1.10994	0.0730290
$232$	6.36392	0.417812
$233$	−18.4629	−1.20955	−0.604773	−	0.796398i	$-0.706736\pi$
$233$	−18.4629	−1.20955	−0.604773	+	0.796398i	$0.706736\pi$
$234$	0	0
$235$	0	0
$236$	−12.8188	−0.834434
$237$	0.488054	0.0317025
$238$	2.60191	0.168657
$239$	−3.76483	−0.243526	−0.121763	−	0.992559i	$-0.538855\pi$
$239$	−3.76483	−0.243526	−0.121763	+	0.992559i	$0.538855\pi$
$240$	0	0
$241$	15.4771	0.996965	0.498482	−	0.866900i	$-0.333891\pi$
$241$	15.4771	0.996965	0.498482	+	0.866900i	$0.333891\pi$
$242$	−0.178043	−0.0114450
$243$	−1.80546	−0.115820
$244$	−11.5310	−0.738196
$245$	0	0
$246$	0.371776	0.0237035
$247$	0	0
$248$	6.84648	0.434752
$249$	−0.172779	−0.0109495
$250$	0	0
$251$	1.26747	0.0800019	0.0400010	−	0.999200i	$-0.487264\pi$
$251$	1.26747	0.0800019	0.0400010	+	0.999200i	$0.487264\pi$
$252$	15.0844	0.950229
$253$	−1.72417	−0.108398
$254$	9.96441	0.625223
$255$	0	0
$256$	1.00000	0.0625000
$257$	−29.7338	−1.85474	−0.927371	−	0.374143i	$-0.877937\pi$
$257$	−29.7338	−1.85474	−0.927371	+	0.374143i	$0.877937\pi$
$258$	0.627090	0.0390409
$259$	27.6780	1.71983
$260$	0	0
$261$	−19.0632	−1.17998
$262$	−3.43248	−0.212059
$263$	−20.2454	−1.24839	−0.624193	−	0.781270i	$-0.714572\pi$
$263$	−20.2454	−1.24839	−0.624193	+	0.781270i	$0.714572\pi$
$264$	−0.220416	−0.0135657
$265$	0	0
$266$	−26.5371	−1.62709
$267$	0.323285	0.0197847
$268$	3.38856	0.206989
$269$	12.2284	0.745576	0.372788	−	0.927917i	$-0.378402\pi$
$269$	12.2284	0.745576	0.372788	+	0.927917i	$0.378402\pi$
$270$	0	0
$271$	−13.3526	−0.811114	−0.405557	−	0.914070i	$-0.632922\pi$
$271$	−13.3526	−0.811114	−0.405557	+	0.914070i	$0.632922\pi$
$272$	−0.516695	−0.0313292
$273$	0	0
$274$	17.1983	1.03899
$275$	0	0
$276$	0.0351171	0.00211380
$277$	9.36534	0.562709	0.281354	−	0.959604i	$-0.409216\pi$
$277$	9.36534	0.562709	0.281354	+	0.959604i	$0.409216\pi$
$278$	−6.56328	−0.393640
$279$	−20.5087	−1.22782
$280$	0	0
$281$	−20.5996	−1.22887	−0.614434	−	0.788968i	$-0.710616\pi$
$281$	−20.5996	−1.22887	−0.614434	+	0.788968i	$0.710616\pi$
$282$	0.110676	0.00659064
$283$	14.4039	0.856225	0.428112	−	0.903726i	$-0.359179\pi$
$283$	14.4039	0.856225	0.428112	+	0.903726i	$0.359179\pi$
$284$	3.86643	0.229430
$285$	0	0
$286$	0	0
$287$	27.9414	1.64933
$288$	−2.99551	−0.176512
$289$	−16.7330	−0.984296
$290$	0	0
$291$	0.126031	0.00738804
$292$	−6.43004	−0.376290
$293$	15.0823	0.881117	0.440559	−	0.897724i	$-0.354780\pi$
$293$	15.0823	0.881117	0.440559	+	0.897724i	$0.354780\pi$
$294$	−1.23003	−0.0717370
$295$	0	0
$296$	−5.49639	−0.319471
$297$	1.32151	0.0766816
$298$	−12.7625	−0.739311
$299$	0	0
$300$	0	0
$301$	47.1300	2.71653
$302$	9.73737	0.560323
$303$	−0.207938	−0.0119457
$304$	5.26981	0.302244
$305$	0	0
$306$	1.54776	0.0884798
$307$	−1.59681	−0.0911346	−0.0455673	−	0.998961i	$-0.514510\pi$
$307$	−1.59681	−0.0911346	−0.0455673	+	0.998961i	$0.514510\pi$
$308$	−16.5657	−0.943920
$309$	1.00175	0.0569875
$310$	0	0
$311$	3.52287	0.199764	0.0998819	−	0.994999i	$-0.468154\pi$
$311$	3.52287	0.199764	0.0998819	+	0.994999i	$0.468154\pi$
$312$	0	0
$313$	−20.7603	−1.17344	−0.586722	−	0.809788i	$-0.699582\pi$
$313$	−20.7603	−1.17344	−0.586722	+	0.809788i	$0.699582\pi$
$314$	−21.3156	−1.20291
$315$	0	0
$316$	−7.28413	−0.409764
$317$	−4.92085	−0.276382	−0.138191	−	0.990406i	$-0.544129\pi$
$317$	−4.92085	−0.276382	−0.138191	+	0.990406i	$0.544129\pi$
$318$	−0.623781	−0.0349799
$319$	20.9352	1.17215
$320$	0	0
$321$	1.02654	0.0572960
$322$	2.63928	0.147081
$323$	−2.72288	−0.151505
$324$	8.95962	0.497756
$325$	0	0
$326$	1.27897	0.0708355
$327$	−0.281332	−0.0155577
$328$	−5.54869	−0.306375
$329$	8.31801	0.458587
$330$	0	0
$331$	14.1568	0.778126	0.389063	−	0.921211i	$-0.372799\pi$
$331$	14.1568	0.778126	0.389063	+	0.921211i	$0.372799\pi$
$332$	2.57870	0.141525
$333$	16.4645	0.902248
$334$	1.90682	0.104336
$335$	0	0
$336$	0.337403	0.0184068
$337$	18.7663	1.02227	0.511133	−	0.859502i	$-0.329226\pi$
$337$	18.7663	1.02227	0.511133	+	0.859502i	$0.329226\pi$
$338$	0	0
$339$	0.786361	0.0427093
$340$	0	0
$341$	22.5227	1.21967
$342$	−15.7858	−0.853597
$343$	−57.1954	−3.08826
$344$	−9.35921	−0.504615
$345$	0	0
$346$	−0.112802	−0.00606425
$347$	32.2646	1.73205	0.866027	−	0.499997i	$-0.166665\pi$
$347$	32.2646	1.73205	0.866027	+	0.499997i	$0.166665\pi$
$348$	−0.426398	−0.0228573
$349$	−8.62949	−0.461926	−0.230963	−	0.972963i	$-0.574188\pi$
$349$	−8.62949	−0.461926	−0.230963	+	0.972963i	$0.574188\pi$
$350$	0	0
$351$	0	0
$352$	3.28967	0.175340
$353$	−19.0692	−1.01495	−0.507474	−	0.861667i	$-0.669421\pi$
$353$	−19.0692	−1.01495	−0.507474	+	0.861667i	$0.669421\pi$
$354$	0.858892	0.0456496
$355$	0	0
$356$	−4.82497	−0.255723
$357$	−0.174334	−0.00922674
$358$	−9.78320	−0.517058
$359$	−2.70589	−0.142811	−0.0714056	−	0.997447i	$-0.522748\pi$
$359$	−2.70589	−0.142811	−0.0714056	+	0.997447i	$0.522748\pi$
$360$	0	0
$361$	8.77090	0.461627
$362$	8.36184	0.439489
$363$	0.0119293	0.000626126 0
$364$	0	0
$365$	0	0
$366$	0.772604	0.0403847
$367$	7.80278	0.407302	0.203651	−	0.979044i	$-0.434719\pi$
$367$	7.80278	0.407302	0.203651	+	0.979044i	$0.434719\pi$
$368$	−0.524116	−0.0273214
$369$	16.6212	0.865263
$370$	0	0
$371$	−46.8812	−2.43395
$372$	−0.458731	−0.0237841
$373$	−14.0835	−0.729218	−0.364609	−	0.931161i	$-0.618797\pi$
$373$	−14.0835	−0.729218	−0.364609	+	0.931161i	$0.618797\pi$
$374$	−1.69976	−0.0878923
$375$	0	0
$376$	−1.65182	−0.0851859
$377$	0	0
$378$	−2.02290	−0.104047
$379$	−0.236252	−0.0121354	−0.00606772	−	0.999982i	$-0.501931\pi$
$379$	−0.236252	−0.0121354	−0.00606772	+	0.999982i	$0.501931\pi$
$380$	0	0
$381$	−0.667640	−0.0342042
$382$	−9.25588	−0.473572
$383$	11.7086	0.598281	0.299140	−	0.954209i	$-0.403300\pi$
$383$	11.7086	0.598281	0.299140	+	0.954209i	$0.403300\pi$
$384$	−0.0670024	−0.00341920
$385$	0	0
$386$	17.5149	0.891483
$387$	28.0356	1.42513
$388$	−1.88098	−0.0954925
$389$	−2.05842	−0.104366	−0.0521830	−	0.998638i	$-0.516618\pi$
$389$	−2.05842	−0.104366	−0.0521830	+	0.998638i	$0.516618\pi$
$390$	0	0
$391$	0.270808	0.0136953
$392$	18.3580	0.927221
$393$	0.229984	0.0116012
$394$	−20.2809	−1.02174
$395$	0	0
$396$	−9.85425	−0.495195
$397$	10.6417	0.534091	0.267046	−	0.963684i	$-0.413953\pi$
$397$	10.6417	0.534091	0.267046	+	0.963684i	$0.413953\pi$
$398$	−27.0802	−1.35741
$399$	1.77805	0.0890137
$400$	0	0
$401$	−7.49704	−0.374384	−0.187192	−	0.982323i	$-0.559939\pi$
$401$	−7.49704	−0.374384	−0.187192	+	0.982323i	$0.559939\pi$
$402$	−0.227042	−0.0113238
$403$	0	0
$404$	3.10344	0.154402
$405$	0	0
$406$	−32.0466	−1.59045
$407$	−18.0813	−0.896258
$408$	0.0346198	0.00171394
$409$	21.5011	1.06316	0.531582	−	0.847007i	$-0.321598\pi$
$409$	21.5011	1.06316	0.531582	+	0.847007i	$0.321598\pi$
$410$	0	0
$411$	−1.15233	−0.0568403
$412$	−14.9509	−0.736580
$413$	64.5514	3.17637
$414$	1.57000	0.0771611
$415$	0	0
$416$	0	0
$417$	0.439756	0.0215349
$418$	17.3360	0.847930
$419$	12.8640	0.628449	0.314224	−	0.949349i	$-0.398256\pi$
$419$	12.8640	0.628449	0.314224	+	0.949349i	$0.398256\pi$
$420$	0	0
$421$	−4.00323	−0.195105	−0.0975527	−	0.995230i	$-0.531101\pi$
$421$	−4.00323	−0.195105	−0.0975527	+	0.995230i	$0.531101\pi$
$422$	−11.9828	−0.583312
$423$	4.94803	0.240581
$424$	9.30982	0.452125
$425$	0	0
$426$	−0.259060	−0.0125515
$427$	58.0663	2.81003
$428$	−15.3210	−0.740567
$429$	0	0
$430$	0	0
$431$	−21.6439	−1.04255	−0.521274	−	0.853389i	$-0.674543\pi$
$431$	−21.6439	−1.04255	−0.521274	+	0.853389i	$0.674543\pi$
$432$	0.401714	0.0193275
$433$	−1.51015	−0.0725733	−0.0362866	−	0.999341i	$-0.511553\pi$
$433$	−1.51015	−0.0725733	−0.0362866	+	0.999341i	$0.511553\pi$
$434$	−34.4766	−1.65493
$435$	0	0
$436$	4.19883	0.201087
$437$	−2.76199	−0.132124
$438$	0.430828	0.0205858
$439$	−16.1800	−0.772229	−0.386114	−	0.922451i	$-0.626183\pi$
$439$	−16.1800	−0.772229	−0.386114	+	0.922451i	$0.626183\pi$
$440$	0	0
$441$	−54.9917	−2.61865
$442$	0	0
$443$	−3.49878	−0.166232	−0.0831161	−	0.996540i	$-0.526487\pi$
$443$	−3.49878	−0.166232	−0.0831161	+	0.996540i	$0.526487\pi$
$444$	0.368271	0.0174774
$445$	0	0
$446$	12.7910	0.605669
$447$	0.855118	0.0404457
$448$	−5.03568	−0.237913
$449$	7.98505	0.376838	0.188419	−	0.982089i	$-0.439664\pi$
$449$	7.98505	0.376838	0.188419	+	0.982089i	$0.439664\pi$
$450$	0	0
$451$	−18.2534	−0.859518
$452$	−11.7363	−0.552029
$453$	−0.652428	−0.0306537
$454$	5.83533	0.273866
$455$	0	0
$456$	−0.353090	−0.0165350
$457$	17.4867	0.817992	0.408996	−	0.912536i	$-0.365879\pi$
$457$	17.4867	0.817992	0.408996	+	0.912536i	$0.365879\pi$
$458$	−15.2208	−0.711220
$459$	−0.207563	−0.00968823
$460$	0	0
$461$	15.0988	0.703222	0.351611	−	0.936146i	$-0.385634\pi$
$461$	15.0988	0.703222	0.351611	+	0.936146i	$0.385634\pi$
$462$	1.10994	0.0516393
$463$	−12.3107	−0.572127	−0.286064	−	0.958211i	$-0.592347\pi$
$463$	−12.3107	−0.572127	−0.286064	+	0.958211i	$0.592347\pi$
$464$	6.36392	0.295438
$465$	0	0
$466$	−18.4629	−0.855278
$467$	−9.27853	−0.429359	−0.214680	−	0.976685i	$-0.568871\pi$
$467$	−9.27853	−0.429359	−0.214680	+	0.976685i	$0.568871\pi$
$468$	0	0
$469$	−17.0637	−0.787928
$470$	0	0
$471$	1.42820	0.0658080
$472$	−12.8188	−0.590034
$473$	−30.7888	−1.41567
$474$	0.488054	0.0224171
$475$	0	0
$476$	2.60191	0.119258
$477$	−27.8877	−1.27689
$478$	−3.76483	−0.172199
$479$	38.6780	1.76724	0.883622	−	0.468201i	$-0.155098\pi$
$479$	38.6780	1.76724	0.883622	+	0.468201i	$0.155098\pi$
$480$	0	0
$481$	0	0
$482$	15.4771	0.704961
$483$	−0.176838	−0.00804642
$484$	−0.178043	−0.00809286
$485$	0	0
$486$	−1.80546	−0.0818972
$487$	−19.3496	−0.876812	−0.438406	−	0.898777i	$-0.644457\pi$
$487$	−19.3496	−0.876812	−0.438406	+	0.898777i	$0.644457\pi$
$488$	−11.5310	−0.521983
$489$	−0.0856940	−0.00387522
$490$	0	0
$491$	27.7335	1.25160	0.625798	−	0.779986i	$-0.284774\pi$
$491$	27.7335	1.25160	0.625798	+	0.779986i	$0.284774\pi$
$492$	0.371776	0.0167609
$493$	−3.28820	−0.148093
$494$	0	0
$495$	0	0
$496$	6.84648	0.307416
$497$	−19.4701	−0.873353
$498$	−0.172779	−0.00774243
$499$	22.2962	0.998116	0.499058	−	0.866569i	$-0.333679\pi$
$499$	22.2962	0.998116	0.499058	+	0.866569i	$0.333679\pi$
$500$	0	0
$501$	−0.127761	−0.00570796
$502$	1.26747	0.0565699
$503$	−5.15651	−0.229918	−0.114959	−	0.993370i	$-0.536674\pi$
$503$	−5.15651	−0.229918	−0.114959	+	0.993370i	$0.536674\pi$
$504$	15.0844	0.671914
$505$	0	0
$506$	−1.72417	−0.0766488
$507$	0	0
$508$	9.96441	0.442099
$509$	−9.11596	−0.404058	−0.202029	−	0.979380i	$-0.564754\pi$
$509$	−9.11596	−0.404058	−0.202029	+	0.979380i	$0.564754\pi$
$510$	0	0
$511$	32.3796	1.43239
$512$	1.00000	0.0441942
$513$	2.11696	0.0934659
$514$	−29.7338	−1.31150
$515$	0	0
$516$	0.627090	0.0276061
$517$	−5.43393	−0.238984
$518$	27.6780	1.21610
$519$	0.00755798	0.000331759 0
$520$	0	0
$521$	−28.0573	−1.22921	−0.614606	−	0.788834i	$-0.710685\pi$
$521$	−28.0573	−1.22921	−0.614606	+	0.788834i	$0.710685\pi$
$522$	−19.0632	−0.834373
$523$	−20.2889	−0.887170	−0.443585	−	0.896232i	$-0.646294\pi$
$523$	−20.2889	−0.887170	−0.443585	+	0.896232i	$0.646294\pi$
$524$	−3.43248	−0.149948
$525$	0	0
$526$	−20.2454	−0.882743
$527$	−3.53754	−0.154098
$528$	−0.220416	−0.00959238
$529$	−22.7253	−0.988057
$530$	0	0
$531$	38.3989	1.66637
$532$	−26.5371	−1.15053
$533$	0	0
$534$	0.323285	0.0139899
$535$	0	0
$536$	3.38856	0.146363
$537$	0.655498	0.0282868
$538$	12.2284	0.527202
$539$	60.3920	2.60127
$540$	0	0
$541$	34.8443	1.49807	0.749036	−	0.662529i	$-0.230517\pi$
$541$	34.8443	1.49807	0.749036	+	0.662529i	$0.230517\pi$
$542$	−13.3526	−0.573544
$543$	−0.560264	−0.0240432
$544$	−0.516695	−0.0221531
$545$	0	0
$546$	0	0
$547$	6.66151	0.284825	0.142413	−	0.989807i	$-0.454514\pi$
$547$	6.66151	0.284825	0.142413	+	0.989807i	$0.454514\pi$
$548$	17.1983	0.734677
$549$	34.5412	1.47418
$550$	0	0
$551$	33.5367	1.42871
$552$	0.0351171	0.00149468
$553$	36.6805	1.55981
$554$	9.36534	0.397895
$555$	0	0
$556$	−6.56328	−0.278345
$557$	1.12641	0.0477277	0.0238638	−	0.999715i	$-0.492403\pi$
$557$	1.12641	0.0477277	0.0238638	+	0.999715i	$0.492403\pi$
$558$	−20.5087	−0.868202
$559$	0	0
$560$	0	0
$561$	0.113888	0.00480835
$562$	−20.5996	−0.868941
$563$	−12.0651	−0.508484	−0.254242	−	0.967141i	$-0.581826\pi$
$563$	−12.0651	−0.508484	−0.254242	+	0.967141i	$0.581826\pi$
$564$	0.110676	0.00466028
$565$	0	0
$566$	14.4039	0.605442
$567$	−45.1177	−1.89477
$568$	3.86643	0.162232
$569$	26.4132	1.10730	0.553650	−	0.832750i	$-0.313235\pi$
$569$	26.4132	1.10730	0.553650	+	0.832750i	$0.313235\pi$
$570$	0	0
$571$	29.7751	1.24605	0.623025	−	0.782202i	$-0.285904\pi$
$571$	29.7751	1.24605	0.623025	+	0.782202i	$0.285904\pi$
$572$	0	0
$573$	0.620167	0.0259078
$574$	27.9414	1.16625
$575$	0	0
$576$	−2.99551	−0.124813
$577$	−28.5908	−1.19025	−0.595126	−	0.803633i	$-0.702898\pi$
$577$	−28.5908	−1.19025	−0.595126	+	0.803633i	$0.702898\pi$
$578$	−16.7330	−0.696002
$579$	−1.17354	−0.0487706
$580$	0	0
$581$	−12.9855	−0.538730
$582$	0.126031	0.00522413
$583$	30.6263	1.26841
$584$	−6.43004	−0.266077
$585$	0	0
$586$	15.0823	0.623044
$587$	1.57376	0.0649562	0.0324781	−	0.999472i	$-0.489660\pi$
$587$	1.57376	0.0649562	0.0324781	+	0.999472i	$0.489660\pi$
$588$	−1.23003	−0.0507257
$589$	36.0796	1.48664
$590$	0	0
$591$	1.35887	0.0558965
$592$	−5.49639	−0.225900
$593$	−31.9506	−1.31205	−0.656027	−	0.754737i	$-0.727764\pi$
$593$	−31.9506	−1.31205	−0.656027	+	0.754737i	$0.727764\pi$
$594$	1.32151	0.0542221
$595$	0	0
$596$	−12.7625	−0.522772
$597$	1.81444	0.0742600
$598$	0	0
$599$	−29.2165	−1.19375	−0.596877	−	0.802332i	$-0.703592\pi$
$599$	−29.2165	−1.19375	−0.596877	+	0.802332i	$0.703592\pi$
$600$	0	0
$601$	24.1056	0.983287	0.491643	−	0.870797i	$-0.336396\pi$
$601$	24.1056	0.983287	0.491643	+	0.870797i	$0.336396\pi$
$602$	47.1300	1.92087
$603$	−10.1505	−0.413359
$604$	9.73737	0.396208
$605$	0	0
$606$	−0.207938	−0.00844690
$607$	22.1386	0.898579	0.449290	−	0.893386i	$-0.351677\pi$
$607$	22.1386	0.898579	0.449290	+	0.893386i	$0.351677\pi$
$608$	5.26981	0.213719
$609$	2.14720	0.0870091
$610$	0	0
$611$	0	0
$612$	1.54776	0.0625647
$613$	−37.1880	−1.50201	−0.751004	−	0.660297i	$-0.770430\pi$
$613$	−37.1880	−1.50201	−0.751004	+	0.660297i	$0.770430\pi$
$614$	−1.59681	−0.0644419
$615$	0	0
$616$	−16.5657	−0.667453
$617$	40.8840	1.64593	0.822963	−	0.568094i	$-0.192319\pi$
$617$	40.8840	1.64593	0.822963	+	0.568094i	$0.192319\pi$
$618$	1.00175	0.0402963
$619$	17.3785	0.698501	0.349250	−	0.937029i	$-0.386436\pi$
$619$	17.3785	0.698501	0.349250	+	0.937029i	$0.386436\pi$
$620$	0	0
$621$	−0.210545	−0.00844887
$622$	3.52287	0.141254
$623$	24.2970	0.973438
$624$	0	0
$625$	0	0
$626$	−20.7603	−0.829750
$627$	−1.16155	−0.0463879
$628$	−21.3156	−0.850587
$629$	2.83995	0.113236
$630$	0	0
$631$	−1.17310	−0.0467005	−0.0233503	−	0.999727i	$-0.507433\pi$
$631$	−1.17310	−0.0467005	−0.0233503	+	0.999727i	$0.507433\pi$
$632$	−7.28413	−0.289747
$633$	0.802875	0.0319114
$634$	−4.92085	−0.195432
$635$	0	0
$636$	−0.623781	−0.0247345
$637$	0	0
$638$	20.9352	0.828834
$639$	−11.5819	−0.458174
$640$	0	0
$641$	−30.0814	−1.18815	−0.594073	−	0.804411i	$-0.702481\pi$
$641$	−30.0814	−1.18815	−0.594073	+	0.804411i	$0.702481\pi$
$642$	1.02654	0.0405144
$643$	39.2988	1.54979	0.774896	−	0.632089i	$-0.217802\pi$
$643$	39.2988	1.54979	0.774896	+	0.632089i	$0.217802\pi$
$644$	2.63928	0.104002
$645$	0	0
$646$	−2.72288	−0.107130
$647$	47.8194	1.87997	0.939987	−	0.341211i	$-0.110837\pi$
$647$	47.8194	1.87997	0.939987	+	0.341211i	$0.110837\pi$
$648$	8.95962	0.351967
$649$	−42.1697	−1.65531
$650$	0	0
$651$	2.31002	0.0905368
$652$	1.27897	0.0500883
$653$	−32.7143	−1.28021	−0.640104	−	0.768288i	$-0.721109\pi$
$653$	−32.7143	−1.28021	−0.640104	+	0.768288i	$0.721109\pi$
$654$	−0.281332	−0.0110009
$655$	0	0
$656$	−5.54869	−0.216640
$657$	19.2613	0.751453
$658$	8.31801	0.324270
$659$	−41.9069	−1.63246	−0.816231	−	0.577725i	$-0.803940\pi$
$659$	−41.9069	−1.63246	−0.816231	+	0.577725i	$0.803940\pi$
$660$	0	0
$661$	21.6745	0.843039	0.421520	−	0.906819i	$-0.361497\pi$
$661$	21.6745	0.843039	0.421520	+	0.906819i	$0.361497\pi$
$662$	14.1568	0.550218
$663$	0	0
$664$	2.57870	0.100073
$665$	0	0
$666$	16.4645	0.637986
$667$	−3.33543	−0.129149
$668$	1.90682	0.0737770
$669$	−0.857025	−0.0331345
$670$	0	0
$671$	−37.9332	−1.46440
$672$	0.337403	0.0130156
$673$	−37.9352	−1.46230	−0.731148	−	0.682219i	$-0.761015\pi$
$673$	−37.9352	−1.46230	−0.731148	+	0.682219i	$0.761015\pi$
$674$	18.7663	0.722851
$675$	0	0
$676$	0	0
$677$	−10.4768	−0.402654	−0.201327	−	0.979524i	$-0.564525\pi$
$677$	−10.4768	−0.402654	−0.201327	+	0.979524i	$0.564525\pi$
$678$	0.786361	0.0302000
$679$	9.47203	0.363503
$680$	0	0
$681$	−0.390981	−0.0149824
$682$	22.5227	0.862438
$683$	24.4591	0.935903	0.467952	−	0.883754i	$-0.344992\pi$
$683$	24.4591	0.935903	0.467952	+	0.883754i	$0.344992\pi$
$684$	−15.7858	−0.603584
$685$	0	0
$686$	−57.1954	−2.18373
$687$	1.01983	0.0389089
$688$	−9.35921	−0.356817
$689$	0	0
$690$	0	0
$691$	20.4414	0.777626	0.388813	−	0.921317i	$-0.372885\pi$
$691$	20.4414	0.777626	0.388813	+	0.921317i	$0.372885\pi$
$692$	−0.112802	−0.00428807
$693$	49.6228	1.88502
$694$	32.2646	1.22475
$695$	0	0
$696$	−0.426398	−0.0161626
$697$	2.86698	0.108594
$698$	−8.62949	−0.326631
$699$	1.23706	0.0467899
$700$	0	0
$701$	11.4168	0.431208	0.215604	−	0.976481i	$-0.430828\pi$
$701$	11.4168	0.431208	0.215604	+	0.976481i	$0.430828\pi$
$702$	0	0
$703$	−28.9649	−1.09243
$704$	3.28967	0.123984
$705$	0	0
$706$	−19.0692	−0.717677
$707$	−15.6279	−0.587748
$708$	0.858892	0.0322791
$709$	−39.7802	−1.49398	−0.746989	−	0.664836i	$-0.768501\pi$
$709$	−39.7802	−1.49398	−0.746989	+	0.664836i	$0.768501\pi$
$710$	0	0
$711$	21.8197	0.818302
$712$	−4.82497	−0.180823
$713$	−3.58835	−0.134385
$714$	−0.174334	−0.00652429
$715$	0	0
$716$	−9.78320	−0.365615
$717$	0.252253	0.00942054
$718$	−2.70589	−0.100983
$719$	21.4776	0.800980	0.400490	−	0.916301i	$-0.368840\pi$
$719$	21.4776	0.800980	0.400490	+	0.916301i	$0.368840\pi$
$720$	0	0
$721$	75.2881	2.80387
$722$	8.77090	0.326419
$723$	−1.03700	−0.0385665
$724$	8.36184	0.310765
$725$	0	0
$726$	0.0119293	0.000442738 0
$727$	26.5625	0.985148	0.492574	−	0.870271i	$-0.336056\pi$
$727$	26.5625	0.985148	0.492574	+	0.870271i	$0.336056\pi$
$728$	0	0
$729$	−26.7579	−0.991033
$730$	0	0
$731$	4.83585	0.178861
$732$	0.772604	0.0285563
$733$	28.8341	1.06501	0.532505	−	0.846427i	$-0.321251\pi$
$733$	28.8341	1.06501	0.532505	+	0.846427i	$0.321251\pi$
$734$	7.80278	0.288006
$735$	0	0
$736$	−0.524116	−0.0193192
$737$	11.1473	0.410614
$738$	16.6212	0.611833
$739$	9.48392	0.348872	0.174436	−	0.984669i	$-0.444190\pi$
$739$	9.48392	0.348872	0.174436	+	0.984669i	$0.444190\pi$
$740$	0	0
$741$	0	0
$742$	−46.8812	−1.72106
$743$	10.2767	0.377017	0.188508	−	0.982072i	$-0.439635\pi$
$743$	10.2767	0.377017	0.188508	+	0.982072i	$0.439635\pi$
$744$	−0.458731	−0.0168179
$745$	0	0
$746$	−14.0835	−0.515635
$747$	−7.72454	−0.282626
$748$	−1.69976	−0.0621493
$749$	77.1514	2.81905
$750$	0	0
$751$	29.7407	1.08525	0.542627	−	0.839974i	$-0.317430\pi$
$751$	29.7407	1.08525	0.542627	+	0.839974i	$0.317430\pi$
$752$	−1.65182	−0.0602355
$753$	−0.0849235	−0.00309478
$754$	0	0
$755$	0	0
$756$	−2.02290	−0.0735722
$757$	2.50703	0.0911197	0.0455599	−	0.998962i	$-0.485493\pi$
$757$	2.50703	0.0911197	0.0455599	+	0.998962i	$0.485493\pi$
$758$	−0.236252	−0.00858106
$759$	0.115524	0.00419324
$760$	0	0
$761$	−36.3569	−1.31794	−0.658969	−	0.752170i	$-0.729007\pi$
$761$	−36.3569	−1.31794	−0.658969	+	0.752170i	$0.729007\pi$
$762$	−0.667640	−0.0241860
$763$	−21.1439	−0.765462
$764$	−9.25588	−0.334866
$765$	0	0
$766$	11.7086	0.423048
$767$	0	0
$768$	−0.0670024	−0.00241774
$769$	9.73400	0.351017	0.175509	−	0.984478i	$-0.443843\pi$
$769$	9.73400	0.351017	0.175509	+	0.984478i	$0.443843\pi$
$770$	0	0
$771$	1.99224	0.0717486
$772$	17.5149	0.630374
$773$	−40.0014	−1.43875	−0.719375	−	0.694621i	$-0.755572\pi$
$773$	−40.0014	−1.43875	−0.719375	+	0.694621i	$0.755572\pi$
$774$	28.0356	1.00772
$775$	0	0
$776$	−1.88098	−0.0675234
$777$	−1.85449	−0.0665296
$778$	−2.05842	−0.0737979
$779$	−29.2405	−1.04765
$780$	0	0
$781$	12.7193	0.455132
$782$	0.270808	0.00968407
$783$	2.55647	0.0913610
$784$	18.3580	0.655644
$785$	0	0
$786$	0.229984	0.00820327
$787$	−46.0334	−1.64091	−0.820456	−	0.571710i	$-0.806280\pi$
$787$	−46.0334	−1.64091	−0.820456	+	0.571710i	$0.806280\pi$
$788$	−20.2809	−0.722479
$789$	1.35649	0.0482924
$790$	0	0
$791$	59.1002	2.10136
$792$	−9.85425	−0.350156
$793$	0	0
$794$	10.6417	0.377660
$795$	0	0
$796$	−27.0802	−0.959832
$797$	5.63521	0.199610	0.0998048	−	0.995007i	$-0.468178\pi$
$797$	5.63521	0.199610	0.0998048	+	0.995007i	$0.468178\pi$
$798$	1.77805	0.0629422
$799$	0.853484	0.0301941
$800$	0	0
$801$	14.4533	0.510680
$802$	−7.49704	−0.264730
$803$	−21.1527	−0.746464
$804$	−0.227042	−0.00800714
$805$	0	0
$806$	0	0
$807$	−0.819330	−0.0288418
$808$	3.10344	0.109179
$809$	−35.5797	−1.25092	−0.625459	−	0.780257i	$-0.715088\pi$
$809$	−35.5797	−1.25092	−0.625459	+	0.780257i	$0.715088\pi$
$810$	0	0
$811$	23.1724	0.813694	0.406847	−	0.913496i	$-0.366628\pi$
$811$	23.1724	0.813694	0.406847	+	0.913496i	$0.366628\pi$
$812$	−32.0466	−1.12462
$813$	0.894658	0.0313770
$814$	−18.0813	−0.633750
$815$	0	0
$816$	0.0346198	0.00121194
$817$	−49.3213	−1.72553
$818$	21.5011	0.751770
$819$	0	0
$820$	0	0
$821$	−22.8480	−0.797401	−0.398701	−	0.917081i	$-0.630539\pi$
$821$	−22.8480	−0.797401	−0.398701	+	0.917081i	$0.630539\pi$
$822$	−1.15233	−0.0401922
$823$	−23.7086	−0.826431	−0.413216	−	0.910633i	$-0.635594\pi$
$823$	−23.7086	−0.826431	−0.413216	+	0.910633i	$0.635594\pi$
$824$	−14.9509	−0.520840
$825$	0	0
$826$	64.5514	2.24603
$827$	47.3108	1.64516	0.822579	−	0.568650i	$-0.192534\pi$
$827$	47.3108	1.64516	0.822579	+	0.568650i	$0.192534\pi$
$828$	1.57000	0.0545611
$829$	43.6231	1.51509	0.757547	−	0.652781i	$-0.226398\pi$
$829$	43.6231	1.51509	0.757547	+	0.652781i	$0.226398\pi$
$830$	0	0
$831$	−0.627501	−0.0217678
$832$	0	0
$833$	−9.48550	−0.328653
$834$	0.439756	0.0152275
$835$	0	0
$836$	17.3360	0.599577
$837$	2.75032	0.0950651
$838$	12.8640	0.444380
$839$	−33.9488	−1.17204	−0.586021	−	0.810296i	$-0.699306\pi$
$839$	−33.9488	−1.17204	−0.586021	+	0.810296i	$0.699306\pi$
$840$	0	0
$841$	11.4995	0.396534
$842$	−4.00323	−0.137960
$843$	1.38022	0.0475374
$844$	−11.9828	−0.412464
$845$	0	0
$846$	4.94803	0.170117
$847$	0.896566	0.0308064
$848$	9.30982	0.319701
$849$	−0.965098	−0.0331221
$850$	0	0
$851$	2.88075	0.0987507
$852$	−0.259060	−0.00887526
$853$	30.4292	1.04187	0.520937	−	0.853595i	$-0.325583\pi$
$853$	30.4292	1.04187	0.520937	+	0.853595i	$0.325583\pi$
$854$	58.0663	1.98699
$855$	0	0
$856$	−15.3210	−0.523660
$857$	−50.9646	−1.74092	−0.870459	−	0.492240i	$-0.836178\pi$
$857$	−50.9646	−1.74092	−0.870459	+	0.492240i	$0.836178\pi$
$858$	0	0
$859$	38.5183	1.31423	0.657114	−	0.753791i	$-0.271777\pi$
$859$	38.5183	1.31423	0.657114	+	0.753791i	$0.271777\pi$
$860$	0	0
$861$	−1.87214	−0.0638024
$862$	−21.6439	−0.737193
$863$	56.6567	1.92862	0.964308	−	0.264784i	$-0.0853008\pi$
$863$	56.6567	1.92862	0.964308	+	0.264784i	$0.0853008\pi$
$864$	0.401714	0.0136666
$865$	0	0
$866$	−1.51015	−0.0513170
$867$	1.12115	0.0380764
$868$	−34.4766	−1.17021
$869$	−23.9624	−0.812869
$870$	0	0
$871$	0	0
$872$	4.19883	0.142190
$873$	5.63451	0.190699
$874$	−2.76199	−0.0934258
$875$	0	0
$876$	0.430828	0.0145563
$877$	−41.9193	−1.41551	−0.707757	−	0.706456i	$-0.750293\pi$
$877$	−41.9193	−1.41551	−0.707757	+	0.706456i	$0.750293\pi$
$878$	−16.1800	−0.546048
$879$	−1.01055	−0.0340850
$880$	0	0
$881$	1.17194	0.0394836	0.0197418	−	0.999805i	$-0.493716\pi$
$881$	1.17194	0.0394836	0.0197418	+	0.999805i	$0.493716\pi$
$882$	−54.9917	−1.85167
$883$	−24.3462	−0.819314	−0.409657	−	0.912240i	$-0.634352\pi$
$883$	−24.3462	−0.819314	−0.409657	+	0.912240i	$0.634352\pi$
$884$	0	0
$885$	0	0
$886$	−3.49878	−0.117544
$887$	−8.57827	−0.288030	−0.144015	−	0.989575i	$-0.546001\pi$
$887$	−8.57827	−0.288030	−0.144015	+	0.989575i	$0.546001\pi$
$888$	0.368271	0.0123584
$889$	−50.1775	−1.68290
$890$	0	0
$891$	29.4742	0.987424
$892$	12.7910	0.428273
$893$	−8.70475	−0.291293
$894$	0.855118	0.0285994
$895$	0	0
$896$	−5.03568	−0.168230
$897$	0	0
$898$	7.98505	0.266465
$899$	43.5704	1.45316
$900$	0	0
$901$	−4.81033	−0.160255
$902$	−18.2534	−0.607771
$903$	−3.15782	−0.105086
$904$	−11.7363	−0.390344
$905$	0	0
$906$	−0.652428	−0.0216755
$907$	−25.2134	−0.837197	−0.418598	−	0.908172i	$-0.637478\pi$
$907$	−25.2134	−0.837197	−0.418598	+	0.908172i	$0.637478\pi$
$908$	5.83533	0.193652
$909$	−9.29638	−0.308341
$910$	0	0
$911$	27.1861	0.900715	0.450358	−	0.892848i	$-0.351296\pi$
$911$	27.1861	0.900715	0.450358	+	0.892848i	$0.351296\pi$
$912$	−0.353090	−0.0116920
$913$	8.48310	0.280750
$914$	17.4867	0.578408
$915$	0	0
$916$	−15.2208	−0.502909
$917$	17.2848	0.570796
$918$	−0.207563	−0.00685061
$919$	13.6082	0.448893	0.224446	−	0.974486i	$-0.427943\pi$
$919$	13.6082	0.448893	0.224446	+	0.974486i	$0.427943\pi$
$920$	0	0
$921$	0.106990	0.00352544
$922$	15.0988	0.497253
$923$	0	0
$924$	1.10994	0.0365145
$925$	0	0
$926$	−12.3107	−0.404555
$927$	44.7857	1.47095
$928$	6.36392	0.208906
$929$	−14.8556	−0.487396	−0.243698	−	0.969851i	$-0.578361\pi$
$929$	−14.8556	−0.487396	−0.243698	+	0.969851i	$0.578361\pi$
$930$	0	0
$931$	96.7434	3.17064
$932$	−18.4629	−0.604773
$933$	−0.236041	−0.00772764
$934$	−9.27853	−0.303603
$935$	0	0
$936$	0	0
$937$	−47.7682	−1.56052	−0.780259	−	0.625457i	$-0.784913\pi$
$937$	−47.7682	−1.56052	−0.780259	+	0.625457i	$0.784913\pi$
$938$	−17.0637	−0.557149
$939$	1.39099	0.0453934
$940$	0	0
$941$	−11.2464	−0.366623	−0.183311	−	0.983055i	$-0.558682\pi$
$941$	−11.2464	−0.366623	−0.183311	+	0.983055i	$0.558682\pi$
$942$	1.42820	0.0465333
$943$	2.90816	0.0947026
$944$	−12.8188	−0.417217
$945$	0	0
$946$	−30.7888	−1.00103
$947$	−35.7394	−1.16137	−0.580687	−	0.814127i	$-0.697216\pi$
$947$	−35.7394	−1.16137	−0.580687	+	0.814127i	$0.697216\pi$
$948$	0.488054	0.0158513
$949$	0	0
$950$	0	0
$951$	0.329709	0.0106915
$952$	2.60191	0.0843283
$953$	−40.0847	−1.29847	−0.649236	−	0.760587i	$-0.724911\pi$
$953$	−40.0847	−1.29847	−0.649236	+	0.760587i	$0.724911\pi$
$954$	−27.8877	−0.902896
$955$	0	0
$956$	−3.76483	−0.121763
$957$	−1.40271	−0.0453432
$958$	38.6780	1.24963
$959$	−86.6053	−2.79663
$960$	0	0
$961$	15.8742	0.512072
$962$	0	0
$963$	45.8941	1.47892
$964$	15.4771	0.498482
$965$	0	0
$966$	−0.176838	−0.00568968
$967$	24.2441	0.779639	0.389820	−	0.920891i	$-0.372537\pi$
$967$	24.2441	0.779639	0.389820	+	0.920891i	$0.372537\pi$
$968$	−0.178043	−0.00572251
$969$	0.182440	0.00586081
$970$	0	0
$971$	−33.3744	−1.07104	−0.535519	−	0.844523i	$-0.679884\pi$
$971$	−33.3744	−1.07104	−0.535519	+	0.844523i	$0.679884\pi$
$972$	−1.80546	−0.0579101
$973$	33.0506	1.05955
$974$	−19.3496	−0.620000
$975$	0	0
$976$	−11.5310	−0.369098
$977$	6.21560	0.198855	0.0994273	−	0.995045i	$-0.468299\pi$
$977$	6.21560	0.198855	0.0994273	+	0.995045i	$0.468299\pi$
$978$	−0.0856940	−0.00274019
$979$	−15.8726	−0.507290
$980$	0	0
$981$	−12.5776	−0.401573
$982$	27.7335	0.885011
$983$	42.7293	1.36285	0.681427	−	0.731886i	$-0.261360\pi$
$983$	42.7293	1.36285	0.681427	+	0.731886i	$0.261360\pi$
$984$	0.371776	0.0118518
$985$	0	0
$986$	−3.28820	−0.104718
$987$	−0.557327	−0.0177399
$988$	0	0
$989$	4.90531	0.155980
$990$	0	0
$991$	40.0026	1.27072	0.635362	−	0.772214i	$-0.280851\pi$
$991$	40.0026	1.27072	0.635362	+	0.772214i	$0.280851\pi$
$992$	6.84648	0.217376
$993$	−0.948538	−0.0301009
$994$	−19.4701	−0.617554
$995$	0	0
$996$	−0.172779	−0.00547473
$997$	15.9741	0.505904	0.252952	−	0.967479i	$-0.418599\pi$
$997$	15.9741	0.505904	0.252952	+	0.967479i	$0.418599\pi$
$998$	22.2962	0.705775
$999$	−2.20797	−0.0698572

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8450.2.a.cx.1.6		9
5.2	odd	4		1690.2.b.f.339.13	yes	18
5.3	odd	4		1690.2.b.f.339.6	✓	18
5.4	even	2		8450.2.a.cw.1.4		9
13.12	even	2		8450.2.a.ct.1.6		9
65.8	even	4		1690.2.c.h.1689.9		18
65.12	odd	4		1690.2.b.g.339.4	yes	18
65.18	even	4		1690.2.c.g.1689.9		18
65.38	odd	4		1690.2.b.g.339.15	yes	18
65.47	even	4		1690.2.c.g.1689.10		18
65.57	even	4		1690.2.c.h.1689.10		18
65.64	even	2		8450.2.a.da.1.4		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1690.2.b.f.339.6	✓	18	5.3	odd	4
1690.2.b.f.339.13	yes	18	5.2	odd	4
1690.2.b.g.339.4	yes	18	65.12	odd	4
1690.2.b.g.339.15	yes	18	65.38	odd	4
1690.2.c.g.1689.9		18	65.18	even	4
1690.2.c.g.1689.10		18	65.47	even	4
1690.2.c.h.1689.9		18	65.8	even	4
1690.2.c.h.1689.10		18	65.57	even	4
8450.2.a.ct.1.6		9	13.12	even	2
8450.2.a.cw.1.4		9	5.4	even	2
8450.2.a.cx.1.6		9	1.1	even	1	trivial
8450.2.a.da.1.4		9	65.64	even	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:	\( N \)	\(=\)	\( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8450.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.0670024 q^{3} +1.00000 q^{4} -0.0670024 q^{6} -5.03568 q^{7} +1.00000 q^{8} -2.99551 q^{9} +3.28967 q^{11} -0.0670024 q^{12} -5.03568 q^{14} +1.00000 q^{16} -0.516695 q^{17} -2.99551 q^{18} +5.26981 q^{19} +0.337403 q^{21} +3.28967 q^{22} -0.524116 q^{23} -0.0670024 q^{24} +0.401714 q^{27} -5.03568 q^{28} +6.36392 q^{29} +6.84648 q^{31} +1.00000 q^{32} -0.220416 q^{33} -0.516695 q^{34} -2.99551 q^{36} -5.49639 q^{37} +5.26981 q^{38} -5.54869 q^{41} +0.337403 q^{42} -9.35921 q^{43} +3.28967 q^{44} -0.524116 q^{46} -1.65182 q^{47} -0.0670024 q^{48} +18.3580 q^{49} +0.0346198 q^{51} +9.30982 q^{53} +0.401714 q^{54} -5.03568 q^{56} -0.353090 q^{57} +6.36392 q^{58} -12.8188 q^{59} -11.5310 q^{61} +6.84648 q^{62} +15.0844 q^{63} +1.00000 q^{64} -0.220416 q^{66} +3.38856 q^{67} -0.516695 q^{68} +0.0351171 q^{69} +3.86643 q^{71} -2.99551 q^{72} -6.43004 q^{73} -5.49639 q^{74} +5.26981 q^{76} -16.5657 q^{77} -7.28413 q^{79} +8.95962 q^{81} -5.54869 q^{82} +2.57870 q^{83} +0.337403 q^{84} -9.35921 q^{86} -0.426398 q^{87} +3.28967 q^{88} -4.82497 q^{89} -0.524116 q^{92} -0.458731 q^{93} -1.65182 q^{94} -0.0670024 q^{96} -1.88098 q^{97} +18.3580 q^{98} -9.85425 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{2} - 7 q^{3} + 9 q^{4} - 7 q^{6} + q^{7} + 9 q^{8} + 8 q^{9} - 4 q^{11} - 7 q^{12} + q^{14} + 9 q^{16} - 12 q^{17} + 8 q^{18} - 6 q^{19} - 8 q^{21} - 4 q^{22} - 11 q^{23} - 7 q^{24} - 34 q^{27}+ \cdots - 16 q^{99}+O(q^{100}) \) 9 * q + 9 * q^2 - 7 * q^3 + 9 * q^4 - 7 * q^6 + q^7 + 9 * q^8 + 8 * q^9 - 4 * q^11 - 7 * q^12 + q^14 + 9 * q^16 - 12 * q^17 + 8 * q^18 - 6 * q^19 - 8 * q^21 - 4 * q^22 - 11 * q^23 - 7 * q^24 - 34 * q^27 + q^28 + 15 * q^29 + 6 * q^31 + 9 * q^32 - 14 * q^33 - 12 * q^34 + 8 * q^36 - 10 * q^37 - 6 * q^38 - 27 * q^41 - 8 * q^42 - 41 * q^43 - 4 * q^44 - 11 * q^46 + 5 * q^47 - 7 * q^48 - 10 * q^49 + 12 * q^51 - 18 * q^53 - 34 * q^54 + q^56 - 12 * q^57 + 15 * q^58 - 20 * q^59 - 19 * q^61 + 6 * q^62 + 39 * q^63 + 9 * q^64 - 14 * q^66 - q^67 - 12 * q^68 - 14 * q^69 + 10 * q^71 + 8 * q^72 - 12 * q^73 - 10 * q^74 - 6 * q^76 - 30 * q^77 - 14 * q^79 + 53 * q^81 - 27 * q^82 + q^83 - 8 * q^84 - 41 * q^86 - 33 * q^87 - 4 * q^88 - 43 * q^89 - 11 * q^92 - 28 * q^93 + 5 * q^94 - 7 * q^96 + 6 * q^97 - 10 * q^98 - 16 * q^99

Introduction

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