LMFDB - Embedded newform 8025.2.a.s.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8025,2,Mod(1,8025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8025 = 3 \cdot 5^{2} \cdot 107 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8025.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.0799476221$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 321)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	8025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} +1.61803 q^{6} +2.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} +1.61803 q^{6} +2.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} -0.763932 q^{11} +0.618034 q^{12} +1.00000 q^{13} +3.23607 q^{14} -4.85410 q^{16} -5.47214 q^{17} +1.61803 q^{18} -6.23607 q^{19} +2.00000 q^{21} -1.23607 q^{22} +6.47214 q^{23} -2.23607 q^{24} +1.61803 q^{26} +1.00000 q^{27} +1.23607 q^{28} -3.23607 q^{29} -6.00000 q^{31} -3.38197 q^{32} -0.763932 q^{33} -8.85410 q^{34} +0.618034 q^{36} -1.00000 q^{37} -10.0902 q^{38} +1.00000 q^{39} -8.94427 q^{41} +3.23607 q^{42} +0.763932 q^{43} -0.472136 q^{44} +10.4721 q^{46} -0.763932 q^{47} -4.85410 q^{48} -3.00000 q^{49} -5.47214 q^{51} +0.618034 q^{52} -4.76393 q^{53} +1.61803 q^{54} -4.47214 q^{56} -6.23607 q^{57} -5.23607 q^{58} +12.9443 q^{59} -12.4164 q^{61} -9.70820 q^{62} +2.00000 q^{63} +4.23607 q^{64} -1.23607 q^{66} -2.76393 q^{67} -3.38197 q^{68} +6.47214 q^{69} -6.23607 q^{71} -2.23607 q^{72} -12.1803 q^{73} -1.61803 q^{74} -3.85410 q^{76} -1.52786 q^{77} +1.61803 q^{78} +11.4164 q^{79} +1.00000 q^{81} -14.4721 q^{82} +10.4721 q^{83} +1.23607 q^{84} +1.23607 q^{86} -3.23607 q^{87} +1.70820 q^{88} -2.76393 q^{89} +2.00000 q^{91} +4.00000 q^{92} -6.00000 q^{93} -1.23607 q^{94} -3.38197 q^{96} -5.70820 q^{97} -4.85410 q^{98} -0.763932 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 2 q^{3} - q^{4} + q^{6} + 4 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + q^2 + 2 * q^3 - q^4 + q^6 + 4 * q^7 + 2 * q^9	$ 2 q + q^{2} + 2 q^{3} - q^{4} + q^{6} + 4 q^{7} + 2 q^{9} - 6 q^{11} - q^{12} + 2 q^{13} + 2 q^{14} - 3 q^{16} - 2 q^{17} + q^{18} - 8 q^{19} + 4 q^{21} + 2 q^{22} + 4 q^{23} + q^{26} + 2 q^{27} - 2 q^{28} - 2 q^{29} - 12 q^{31} - 9 q^{32} - 6 q^{33} - 11 q^{34} - q^{36} - 2 q^{37} - 9 q^{38} + 2 q^{39} + 2 q^{42} + 6 q^{43} + 8 q^{44} + 12 q^{46} - 6 q^{47} - 3 q^{48} - 6 q^{49} - 2 q^{51} - q^{52} - 14 q^{53} + q^{54} - 8 q^{57} - 6 q^{58} + 8 q^{59} + 2 q^{61} - 6 q^{62} + 4 q^{63} + 4 q^{64} + 2 q^{66} - 10 q^{67} - 9 q^{68} + 4 q^{69} - 8 q^{71} - 2 q^{73} - q^{74} - q^{76} - 12 q^{77} + q^{78} - 4 q^{79} + 2 q^{81} - 20 q^{82} + 12 q^{83} - 2 q^{84} - 2 q^{86} - 2 q^{87} - 10 q^{88} - 10 q^{89} + 4 q^{91} + 8 q^{92} - 12 q^{93} + 2 q^{94} - 9 q^{96} + 2 q^{97} - 3 q^{98} - 6 q^{99}+O(q^{100}) $ 2 * q + q^2 + 2 * q^3 - q^4 + q^6 + 4 * q^7 + 2 * q^9 - 6 * q^11 - q^12 + 2 * q^13 + 2 * q^14 - 3 * q^16 - 2 * q^17 + q^18 - 8 * q^19 + 4 * q^21 + 2 * q^22 + 4 * q^23 + q^26 + 2 * q^27 - 2 * q^28 - 2 * q^29 - 12 * q^31 - 9 * q^32 - 6 * q^33 - 11 * q^34 - q^36 - 2 * q^37 - 9 * q^38 + 2 * q^39 + 2 * q^42 + 6 * q^43 + 8 * q^44 + 12 * q^46 - 6 * q^47 - 3 * q^48 - 6 * q^49 - 2 * q^51 - q^52 - 14 * q^53 + q^54 - 8 * q^57 - 6 * q^58 + 8 * q^59 + 2 * q^61 - 6 * q^62 + 4 * q^63 + 4 * q^64 + 2 * q^66 - 10 * q^67 - 9 * q^68 + 4 * q^69 - 8 * q^71 - 2 * q^73 - q^74 - q^76 - 12 * q^77 + q^78 - 4 * q^79 + 2 * q^81 - 20 * q^82 + 12 * q^83 - 2 * q^84 - 2 * q^86 - 2 * q^87 - 10 * q^88 - 10 * q^89 + 4 * q^91 + 8 * q^92 - 12 * q^93 + 2 * q^94 - 9 * q^96 + 2 * q^97 - 3 * q^98 - 6 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	1.61803	1.14412	0.572061	−	0.820211i	$-0.306144\pi$
$2$	1.61803	1.14412	0.572061	+	0.820211i	$0.306144\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0.618034	0.309017
$5$	0	0
$5$	0	0
$6$	1.61803	0.660560
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	−2.23607	−0.790569
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.763932	−0.230334	−0.115167	−	0.993346i	$-0.536740\pi$
$11$	−0.763932	−0.230334	−0.115167	+	0.993346i	$0.536740\pi$
$12$	0.618034	0.178411
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	3.23607	0.864876
$15$	0	0
$16$	−4.85410	−1.21353
$17$	−5.47214	−1.32719	−0.663594	−	0.748093i	$-0.730970\pi$
$17$	−5.47214	−1.32719	−0.663594	+	0.748093i	$0.730970\pi$
$18$	1.61803	0.381374
$19$	−6.23607	−1.43065	−0.715326	−	0.698791i	$-0.753722\pi$
$19$	−6.23607	−1.43065	−0.715326	+	0.698791i	$0.753722\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	−1.23607	−0.263531
$23$	6.47214	1.34953	0.674767	−	0.738031i	$-0.264244\pi$
$23$	6.47214	1.34953	0.674767	+	0.738031i	$0.264244\pi$
$24$	−2.23607	−0.456435
$25$	0	0
$26$	1.61803	0.317323
$27$	1.00000	0.192450
$28$	1.23607	0.233595
$29$	−3.23607	−0.600923	−0.300461	−	0.953794i	$-0.597141\pi$
$29$	−3.23607	−0.600923	−0.300461	+	0.953794i	$0.597141\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	−3.38197	−0.597853
$33$	−0.763932	−0.132983
$34$	−8.85410	−1.51847
$35$	0	0
$36$	0.618034	0.103006
$37$	−1.00000	−0.164399	−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000	−0.164399	−0.0821995	+	0.996616i	$0.526194\pi$
$38$	−10.0902	−1.63684
$39$	1.00000	0.160128
$40$	0	0
$41$	−8.94427	−1.39686	−0.698430	−	0.715678i	$-0.746118\pi$
$41$	−8.94427	−1.39686	−0.698430	+	0.715678i	$0.746118\pi$
$42$	3.23607	0.499336
$43$	0.763932	0.116499	0.0582493	−	0.998302i	$-0.481448\pi$
$43$	0.763932	0.116499	0.0582493	+	0.998302i	$0.481448\pi$
$44$	−0.472136	−0.0711772
$45$	0	0
$46$	10.4721	1.54403
$47$	−0.763932	−0.111431	−0.0557155	−	0.998447i	$-0.517744\pi$
$47$	−0.763932	−0.111431	−0.0557155	+	0.998447i	$0.517744\pi$
$48$	−4.85410	−0.700629
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−5.47214	−0.766252
$52$	0.618034	0.0857059
$53$	−4.76393	−0.654376	−0.327188	−	0.944959i	$-0.606101\pi$
$53$	−4.76393	−0.654376	−0.327188	+	0.944959i	$0.606101\pi$
$54$	1.61803	0.220187
$55$	0	0
$56$	−4.47214	−0.597614
$57$	−6.23607	−0.825987
$58$	−5.23607	−0.687529
$59$	12.9443	1.68520	0.842600	−	0.538539i	$-0.181024\pi$
$59$	12.9443	1.68520	0.842600	+	0.538539i	$0.181024\pi$
$60$	0	0
$61$	−12.4164	−1.58976	−0.794879	−	0.606768i	$-0.792466\pi$
$61$	−12.4164	−1.58976	−0.794879	+	0.606768i	$0.792466\pi$
$62$	−9.70820	−1.23294
$63$	2.00000	0.251976
$64$	4.23607	0.529508
$65$	0	0
$66$	−1.23607	−0.152149
$67$	−2.76393	−0.337668	−0.168834	−	0.985644i	$-0.554000\pi$
$67$	−2.76393	−0.337668	−0.168834	+	0.985644i	$0.554000\pi$
$68$	−3.38197	−0.410124
$69$	6.47214	0.779154
$70$	0	0
$71$	−6.23607	−0.740085	−0.370043	−	0.929015i	$-0.620657\pi$
$71$	−6.23607	−0.740085	−0.370043	+	0.929015i	$0.620657\pi$
$72$	−2.23607	−0.263523
$73$	−12.1803	−1.42560	−0.712800	−	0.701367i	$-0.752573\pi$
$73$	−12.1803	−1.42560	−0.712800	+	0.701367i	$0.752573\pi$
$74$	−1.61803	−0.188093
$75$	0	0
$76$	−3.85410	−0.442096
$77$	−1.52786	−0.174116
$78$	1.61803	0.183206
$79$	11.4164	1.28445	0.642223	−	0.766518i	$-0.278012\pi$
$79$	11.4164	1.28445	0.642223	+	0.766518i	$0.278012\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−14.4721	−1.59818
$83$	10.4721	1.14947	0.574733	−	0.818341i	$-0.305106\pi$
$83$	10.4721	1.14947	0.574733	+	0.818341i	$0.305106\pi$
$84$	1.23607	0.134866
$85$	0	0
$86$	1.23607	0.133289
$87$	−3.23607	−0.346943
$88$	1.70820	0.182095
$89$	−2.76393	−0.292976	−0.146488	−	0.989212i	$-0.546797\pi$
$89$	−2.76393	−0.292976	−0.146488	+	0.989212i	$0.546797\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	4.00000	0.417029
$93$	−6.00000	−0.622171
$94$	−1.23607	−0.127491
$95$	0	0
$96$	−3.38197	−0.345170
$97$	−5.70820	−0.579580	−0.289790	−	0.957090i	$-0.593586\pi$
$97$	−5.70820	−0.579580	−0.289790	+	0.957090i	$0.593586\pi$
$98$	−4.85410	−0.490338
$99$	−0.763932	−0.0767781
$100$	0	0
$101$	1.70820	0.169973	0.0849863	−	0.996382i	$-0.472915\pi$
$101$	1.70820	0.169973	0.0849863	+	0.996382i	$0.472915\pi$
$102$	−8.85410	−0.876687
$103$	11.2361	1.10712	0.553561	−	0.832808i	$-0.313268\pi$
$103$	11.2361	1.10712	0.553561	+	0.832808i	$0.313268\pi$
$104$	−2.23607	−0.219265
$105$	0	0
$106$	−7.70820	−0.748687
$107$	1.00000	0.0966736
$107$	1.00000	0.0966736
$108$	0.618034	0.0594703
$109$	2.94427	0.282010	0.141005	−	0.990009i	$-0.454967\pi$
$109$	2.94427	0.282010	0.141005	+	0.990009i	$0.454967\pi$
$110$	0	0
$111$	−1.00000	−0.0949158
$112$	−9.70820	−0.917339
$113$	3.47214	0.326631	0.163316	−	0.986574i	$-0.447781\pi$
$113$	3.47214	0.326631	0.163316	+	0.986574i	$0.447781\pi$
$114$	−10.0902	−0.945031
$115$	0	0
$116$	−2.00000	−0.185695
$117$	1.00000	0.0924500
$118$	20.9443	1.92808
$119$	−10.9443	−1.00326
$120$	0	0
$121$	−10.4164	−0.946946
$122$	−20.0902	−1.81888
$123$	−8.94427	−0.806478
$124$	−3.70820	−0.333007
$125$	0	0
$126$	3.23607	0.288292
$127$	5.70820	0.506521	0.253261	−	0.967398i	$-0.418497\pi$
$127$	5.70820	0.506521	0.253261	+	0.967398i	$0.418497\pi$
$128$	13.6180	1.20368
$129$	0.763932	0.0672605
$130$	0	0
$131$	7.76393	0.678338	0.339169	−	0.940725i	$-0.389854\pi$
$131$	7.76393	0.678338	0.339169	+	0.940725i	$0.389854\pi$
$132$	−0.472136	−0.0410942
$133$	−12.4721	−1.08147
$134$	−4.47214	−0.386334
$135$	0	0
$136$	12.2361	1.04923
$137$	−18.1803	−1.55325	−0.776626	−	0.629962i	$-0.783070\pi$
$137$	−18.1803	−1.55325	−0.776626	+	0.629962i	$0.783070\pi$
$138$	10.4721	0.891447
$139$	0.472136	0.0400460	0.0200230	−	0.999800i	$-0.493626\pi$
$139$	0.472136	0.0400460	0.0200230	+	0.999800i	$0.493626\pi$
$140$	0	0
$141$	−0.763932	−0.0643347
$142$	−10.0902	−0.846748
$143$	−0.763932	−0.0638832
$144$	−4.85410	−0.404508
$145$	0	0
$146$	−19.7082	−1.63106
$147$	−3.00000	−0.247436
$148$	−0.618034	−0.0508021
$149$	3.52786	0.289014	0.144507	−	0.989504i	$-0.453840\pi$
$149$	3.52786	0.289014	0.144507	+	0.989504i	$0.453840\pi$
$150$	0	0
$151$	−15.1803	−1.23536	−0.617679	−	0.786430i	$-0.711927\pi$
$151$	−15.1803	−1.23536	−0.617679	+	0.786430i	$0.711927\pi$
$152$	13.9443	1.13103
$153$	−5.47214	−0.442396
$154$	−2.47214	−0.199210
$155$	0	0
$156$	0.618034	0.0494823
$157$	2.29180	0.182905	0.0914526	−	0.995809i	$-0.470849\pi$
$157$	2.29180	0.182905	0.0914526	+	0.995809i	$0.470849\pi$
$158$	18.4721	1.46956
$159$	−4.76393	−0.377804
$160$	0	0
$161$	12.9443	1.02015
$162$	1.61803	0.127125
$163$	−4.94427	−0.387265	−0.193633	−	0.981074i	$-0.562027\pi$
$163$	−4.94427	−0.387265	−0.193633	+	0.981074i	$0.562027\pi$
$164$	−5.52786	−0.431654
$165$	0	0
$166$	16.9443	1.31513
$167$	−1.29180	−0.0999622	−0.0499811	−	0.998750i	$-0.515916\pi$
$167$	−1.29180	−0.0999622	−0.0499811	+	0.998750i	$0.515916\pi$
$168$	−4.47214	−0.345033
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−6.23607	−0.476884
$172$	0.472136	0.0360000
$173$	−5.00000	−0.380143	−0.190071	−	0.981770i	$-0.560872\pi$
$173$	−5.00000	−0.380143	−0.190071	+	0.981770i	$0.560872\pi$
$174$	−5.23607	−0.396945
$175$	0	0
$176$	3.70820	0.279516
$177$	12.9443	0.972951
$178$	−4.47214	−0.335201
$179$	−26.2361	−1.96098	−0.980488	−	0.196579i	$-0.937017\pi$
$179$	−26.2361	−1.96098	−0.980488	+	0.196579i	$0.937017\pi$
$180$	0	0
$181$	1.70820	0.126970	0.0634849	−	0.997983i	$-0.479779\pi$
$181$	1.70820	0.126970	0.0634849	+	0.997983i	$0.479779\pi$
$182$	3.23607	0.239873
$183$	−12.4164	−0.917847
$184$	−14.4721	−1.06690
$185$	0	0
$186$	−9.70820	−0.711840
$187$	4.18034	0.305697
$188$	−0.472136	−0.0344341
$189$	2.00000	0.145479
$190$	0	0
$191$	−9.76393	−0.706493	−0.353247	−	0.935530i	$-0.614922\pi$
$191$	−9.76393	−0.706493	−0.353247	+	0.935530i	$0.614922\pi$
$192$	4.23607	0.305712
$193$	14.9443	1.07571	0.537856	−	0.843037i	$-0.319234\pi$
$193$	14.9443	1.07571	0.537856	+	0.843037i	$0.319234\pi$
$194$	−9.23607	−0.663111
$195$	0	0
$196$	−1.85410	−0.132436
$197$	−14.0000	−0.997459	−0.498729	−	0.866758i	$-0.666200\pi$
$197$	−14.0000	−0.997459	−0.498729	+	0.866758i	$0.666200\pi$
$198$	−1.23607	−0.0878435
$199$	17.6525	1.25135	0.625675	−	0.780084i	$-0.284823\pi$
$199$	17.6525	1.25135	0.625675	+	0.780084i	$0.284823\pi$
$200$	0	0
$201$	−2.76393	−0.194953
$202$	2.76393	0.194470
$203$	−6.47214	−0.454255
$204$	−3.38197	−0.236785
$205$	0	0
$206$	18.1803	1.26668
$207$	6.47214	0.449845
$208$	−4.85410	−0.336571
$209$	4.76393	0.329528
$210$	0	0
$211$	26.9443	1.85492	0.927460	−	0.373922i	$-0.121987\pi$
$211$	26.9443	1.85492	0.927460	+	0.373922i	$0.121987\pi$
$212$	−2.94427	−0.202213
$213$	−6.23607	−0.427288
$214$	1.61803	0.110607
$215$	0	0
$216$	−2.23607	−0.152145
$217$	−12.0000	−0.814613
$218$	4.76393	0.322654
$219$	−12.1803	−0.823071
$220$	0	0
$221$	−5.47214	−0.368096
$222$	−1.61803	−0.108595
$223$	15.1803	1.01655	0.508275	−	0.861195i	$-0.330283\pi$
$223$	15.1803	1.01655	0.508275	+	0.861195i	$0.330283\pi$
$224$	−6.76393	−0.451934
$225$	0	0
$226$	5.61803	0.373706
$227$	−25.7082	−1.70631	−0.853157	−	0.521655i	$-0.825315\pi$
$227$	−25.7082	−1.70631	−0.853157	+	0.521655i	$0.825315\pi$
$228$	−3.85410	−0.255244
$229$	−0.180340	−0.0119172	−0.00595860	−	0.999982i	$-0.501897\pi$
$229$	−0.180340	−0.0119172	−0.00595860	+	0.999982i	$0.501897\pi$
$230$	0	0
$231$	−1.52786	−0.100526
$232$	7.23607	0.475071
$233$	−11.4164	−0.747914	−0.373957	−	0.927446i	$-0.621999\pi$
$233$	−11.4164	−0.747914	−0.373957	+	0.927446i	$0.621999\pi$
$234$	1.61803	0.105774
$235$	0	0
$236$	8.00000	0.520756
$237$	11.4164	0.741575
$238$	−17.7082	−1.14785
$239$	21.1246	1.36644	0.683219	−	0.730214i	$-0.260580\pi$
$239$	21.1246	1.36644	0.683219	+	0.730214i	$0.260580\pi$
$240$	0	0
$241$	−2.41641	−0.155655	−0.0778273	−	0.996967i	$-0.524798\pi$
$241$	−2.41641	−0.155655	−0.0778273	+	0.996967i	$0.524798\pi$
$242$	−16.8541	−1.08342
$243$	1.00000	0.0641500
$244$	−7.67376	−0.491262
$245$	0	0
$246$	−14.4721	−0.922710
$247$	−6.23607	−0.396792
$248$	13.4164	0.851943
$249$	10.4721	0.663645
$250$	0	0
$251$	−4.76393	−0.300697	−0.150348	−	0.988633i	$-0.548040\pi$
$251$	−4.76393	−0.300697	−0.150348	+	0.988633i	$0.548040\pi$
$252$	1.23607	0.0778650
$253$	−4.94427	−0.310844
$254$	9.23607	0.579522
$255$	0	0
$256$	13.5623	0.847644
$257$	−25.4164	−1.58543	−0.792716	−	0.609591i	$-0.791334\pi$
$257$	−25.4164	−1.58543	−0.792716	+	0.609591i	$0.791334\pi$
$258$	1.23607	0.0769542
$259$	−2.00000	−0.124274
$260$	0	0
$261$	−3.23607	−0.200308
$262$	12.5623	0.776102
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	1.70820	0.105133
$265$	0	0
$266$	−20.1803	−1.23734
$267$	−2.76393	−0.169150
$268$	−1.70820	−0.104345
$269$	−10.4164	−0.635100	−0.317550	−	0.948242i	$-0.602860\pi$
$269$	−10.4164	−0.635100	−0.317550	+	0.948242i	$0.602860\pi$
$270$	0	0
$271$	11.6525	0.707837	0.353919	−	0.935276i	$-0.384849\pi$
$271$	11.6525	0.707837	0.353919	+	0.935276i	$0.384849\pi$
$272$	26.5623	1.61058
$273$	2.00000	0.121046
$274$	−29.4164	−1.77711
$275$	0	0
$276$	4.00000	0.240772
$277$	−1.70820	−0.102636	−0.0513180	−	0.998682i	$-0.516342\pi$
$277$	−1.70820	−0.102636	−0.0513180	+	0.998682i	$0.516342\pi$
$278$	0.763932	0.0458176
$279$	−6.00000	−0.359211
$280$	0	0
$281$	−0.472136	−0.0281653	−0.0140826	−	0.999901i	$-0.504483\pi$
$281$	−0.472136	−0.0281653	−0.0140826	+	0.999901i	$0.504483\pi$
$282$	−1.23607	−0.0736068
$283$	−15.7639	−0.937068	−0.468534	−	0.883445i	$-0.655218\pi$
$283$	−15.7639	−0.937068	−0.468534	+	0.883445i	$0.655218\pi$
$284$	−3.85410	−0.228699
$285$	0	0
$286$	−1.23607	−0.0730902
$287$	−17.8885	−1.05593
$288$	−3.38197	−0.199284
$289$	12.9443	0.761428
$290$	0	0
$291$	−5.70820	−0.334621
$292$	−7.52786	−0.440535
$293$	−16.2918	−0.951777	−0.475888	−	0.879506i	$-0.657873\pi$
$293$	−16.2918	−0.951777	−0.475888	+	0.879506i	$0.657873\pi$
$294$	−4.85410	−0.283097
$295$	0	0
$296$	2.23607	0.129969
$297$	−0.763932	−0.0443278
$298$	5.70820	0.330667
$299$	6.47214	0.374293
$300$	0	0
$301$	1.52786	0.0880646
$302$	−24.5623	−1.41340
$303$	1.70820	0.0981338
$304$	30.2705	1.73613
$305$	0	0
$306$	−8.85410	−0.506155
$307$	19.8885	1.13510	0.567550	−	0.823339i	$-0.307892\pi$
$307$	19.8885	1.13510	0.567550	+	0.823339i	$0.307892\pi$
$308$	−0.944272	−0.0538049
$309$	11.2361	0.639198
$310$	0	0
$311$	21.1803	1.20103	0.600513	−	0.799615i	$-0.294963\pi$
$311$	21.1803	1.20103	0.600513	+	0.799615i	$0.294963\pi$
$312$	−2.23607	−0.126592
$313$	24.8885	1.40678	0.703392	−	0.710802i	$-0.251668\pi$
$313$	24.8885	1.40678	0.703392	+	0.710802i	$0.251668\pi$
$314$	3.70820	0.209266
$315$	0	0
$316$	7.05573	0.396916
$317$	−14.3607	−0.806576	−0.403288	−	0.915073i	$-0.632133\pi$
$317$	−14.3607	−0.806576	−0.403288	+	0.915073i	$0.632133\pi$
$318$	−7.70820	−0.432255
$319$	2.47214	0.138413
$320$	0	0
$321$	1.00000	0.0558146
$322$	20.9443	1.16718
$323$	34.1246	1.89874
$324$	0.618034	0.0343352
$325$	0	0
$326$	−8.00000	−0.443079
$327$	2.94427	0.162819
$328$	20.0000	1.10432
$329$	−1.52786	−0.0842339
$330$	0	0
$331$	−16.5967	−0.912240	−0.456120	−	0.889918i	$-0.650761\pi$
$331$	−16.5967	−0.912240	−0.456120	+	0.889918i	$0.650761\pi$
$332$	6.47214	0.355205
$333$	−1.00000	−0.0547997
$334$	−2.09017	−0.114369
$335$	0	0
$336$	−9.70820	−0.529626
$337$	16.4164	0.894259	0.447129	−	0.894469i	$-0.352446\pi$
$337$	16.4164	0.894259	0.447129	+	0.894469i	$0.352446\pi$
$338$	−19.4164	−1.05611
$339$	3.47214	0.188581
$340$	0	0
$341$	4.58359	0.248215
$342$	−10.0902	−0.545614
$343$	−20.0000	−1.07990
$344$	−1.70820	−0.0921002
$345$	0	0
$346$	−8.09017	−0.434930
$347$	−6.70820	−0.360115	−0.180058	−	0.983656i	$-0.557628\pi$
$347$	−6.70820	−0.360115	−0.180058	+	0.983656i	$0.557628\pi$
$348$	−2.00000	−0.107211
$349$	26.0689	1.39544	0.697718	−	0.716373i	$-0.254199\pi$
$349$	26.0689	1.39544	0.697718	+	0.716373i	$0.254199\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	2.58359	0.137706
$353$	18.3607	0.977240	0.488620	−	0.872497i	$-0.337500\pi$
$353$	18.3607	0.977240	0.488620	+	0.872497i	$0.337500\pi$
$354$	20.9443	1.11318
$355$	0	0
$356$	−1.70820	−0.0905346
$357$	−10.9443	−0.579232
$358$	−42.4508	−2.24360
$359$	−33.1803	−1.75119	−0.875596	−	0.483045i	$-0.839531\pi$
$359$	−33.1803	−1.75119	−0.875596	+	0.483045i	$0.839531\pi$
$360$	0	0
$361$	19.8885	1.04677
$362$	2.76393	0.145269
$363$	−10.4164	−0.546720
$364$	1.23607	0.0647876
$365$	0	0
$366$	−20.0902	−1.05013
$367$	29.1246	1.52029	0.760146	−	0.649752i	$-0.225127\pi$
$367$	29.1246	1.52029	0.760146	+	0.649752i	$0.225127\pi$
$368$	−31.4164	−1.63769
$369$	−8.94427	−0.465620
$370$	0	0
$371$	−9.52786	−0.494662
$372$	−3.70820	−0.192261
$373$	34.4164	1.78201	0.891007	−	0.453990i	$-0.150000\pi$
$373$	34.4164	1.78201	0.891007	+	0.453990i	$0.150000\pi$
$374$	6.76393	0.349755
$375$	0	0
$376$	1.70820	0.0880939
$377$	−3.23607	−0.166666
$378$	3.23607	0.166445
$379$	−6.18034	−0.317463	−0.158731	−	0.987322i	$-0.550740\pi$
$379$	−6.18034	−0.317463	−0.158731	+	0.987322i	$0.550740\pi$
$380$	0	0
$381$	5.70820	0.292440
$382$	−15.7984	−0.808315
$383$	34.8328	1.77987	0.889937	−	0.456084i	$-0.150748\pi$
$383$	34.8328	1.77987	0.889937	+	0.456084i	$0.150748\pi$
$384$	13.6180	0.694942
$385$	0	0
$386$	24.1803	1.23075
$387$	0.763932	0.0388328
$388$	−3.52786	−0.179100
$389$	−6.94427	−0.352089	−0.176044	−	0.984382i	$-0.556330\pi$
$389$	−6.94427	−0.352089	−0.176044	+	0.984382i	$0.556330\pi$
$390$	0	0
$391$	−35.4164	−1.79108
$392$	6.70820	0.338815
$393$	7.76393	0.391639
$394$	−22.6525	−1.14122
$395$	0	0
$396$	−0.472136	−0.0237257
$397$	28.8328	1.44708	0.723539	−	0.690284i	$-0.242514\pi$
$397$	28.8328	1.44708	0.723539	+	0.690284i	$0.242514\pi$
$398$	28.5623	1.43170
$399$	−12.4721	−0.624388
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	−4.47214	−0.223050
$403$	−6.00000	−0.298881
$404$	1.05573	0.0525244
$405$	0	0
$406$	−10.4721	−0.519723
$407$	0.763932	0.0378667
$408$	12.2361	0.605776
$409$	25.2361	1.24784	0.623922	−	0.781487i	$-0.285538\pi$
$409$	25.2361	1.24784	0.623922	+	0.781487i	$0.285538\pi$
$410$	0	0
$411$	−18.1803	−0.896770
$412$	6.94427	0.342120
$413$	25.8885	1.27389
$414$	10.4721	0.514677
$415$	0	0
$416$	−3.38197	−0.165815
$417$	0.472136	0.0231206
$418$	7.70820	0.377021
$419$	−5.65248	−0.276142	−0.138071	−	0.990422i	$-0.544090\pi$
$419$	−5.65248	−0.276142	−0.138071	+	0.990422i	$0.544090\pi$
$420$	0	0
$421$	−29.0000	−1.41337	−0.706687	−	0.707527i	$-0.749811\pi$
$421$	−29.0000	−1.41337	−0.706687	+	0.707527i	$0.749811\pi$
$422$	43.5967	2.12226
$423$	−0.763932	−0.0371436
$424$	10.6525	0.517330
$425$	0	0
$426$	−10.0902	−0.488870
$427$	−24.8328	−1.20174
$428$	0.618034	0.0298738
$429$	−0.763932	−0.0368830
$430$	0	0
$431$	3.70820	0.178618	0.0893089	−	0.996004i	$-0.471534\pi$
$431$	3.70820	0.178618	0.0893089	+	0.996004i	$0.471534\pi$
$432$	−4.85410	−0.233543
$433$	−19.1246	−0.919070	−0.459535	−	0.888160i	$-0.651984\pi$
$433$	−19.1246	−0.919070	−0.459535	+	0.888160i	$0.651984\pi$
$434$	−19.4164	−0.932017
$435$	0	0
$436$	1.81966	0.0871459
$437$	−40.3607	−1.93071
$438$	−19.7082	−0.941694
$439$	−34.4721	−1.64527	−0.822633	−	0.568573i	$-0.807496\pi$
$439$	−34.4721	−1.64527	−0.822633	+	0.568573i	$0.807496\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	−8.85410	−0.421147
$443$	−26.8328	−1.27487	−0.637433	−	0.770506i	$-0.720004\pi$
$443$	−26.8328	−1.27487	−0.637433	+	0.770506i	$0.720004\pi$
$444$	−0.618034	−0.0293306
$445$	0	0
$446$	24.5623	1.16306
$447$	3.52786	0.166862
$448$	8.47214	0.400271
$449$	17.8328	0.841583	0.420791	−	0.907157i	$-0.361752\pi$
$449$	17.8328	0.841583	0.420791	+	0.907157i	$0.361752\pi$
$450$	0	0
$451$	6.83282	0.321745
$452$	2.14590	0.100935
$453$	−15.1803	−0.713235
$454$	−41.5967	−1.95223
$455$	0	0
$456$	13.9443	0.653000
$457$	−22.3050	−1.04338	−0.521691	−	0.853135i	$-0.674699\pi$
$457$	−22.3050	−1.04338	−0.521691	+	0.853135i	$0.674699\pi$
$458$	−0.291796	−0.0136347
$459$	−5.47214	−0.255417
$460$	0	0
$461$	23.7082	1.10420	0.552101	−	0.833778i	$-0.313826\pi$
$461$	23.7082	1.10420	0.552101	+	0.833778i	$0.313826\pi$
$462$	−2.47214	−0.115014
$463$	−1.88854	−0.0877681	−0.0438840	−	0.999037i	$-0.513973\pi$
$463$	−1.88854	−0.0877681	−0.0438840	+	0.999037i	$0.513973\pi$
$464$	15.7082	0.729235
$465$	0	0
$466$	−18.4721	−0.855705
$467$	−8.65248	−0.400389	−0.200194	−	0.979756i	$-0.564157\pi$
$467$	−8.65248	−0.400389	−0.200194	+	0.979756i	$0.564157\pi$
$468$	0.618034	0.0285686
$469$	−5.52786	−0.255253
$470$	0	0
$471$	2.29180	0.105600
$472$	−28.9443	−1.33227
$473$	−0.583592	−0.0268336
$474$	18.4721	0.848453
$475$	0	0
$476$	−6.76393	−0.310024
$477$	−4.76393	−0.218125
$478$	34.1803	1.56337
$479$	20.1246	0.919517	0.459758	−	0.888044i	$-0.347936\pi$
$479$	20.1246	0.919517	0.459758	+	0.888044i	$0.347936\pi$
$480$	0	0
$481$	−1.00000	−0.0455961
$482$	−3.90983	−0.178088
$483$	12.9443	0.588985
$484$	−6.43769	−0.292622
$485$	0	0
$486$	1.61803	0.0733955
$487$	−35.8885	−1.62627	−0.813133	−	0.582079i	$-0.802240\pi$
$487$	−35.8885	−1.62627	−0.813133	+	0.582079i	$0.802240\pi$
$488$	27.7639	1.25681
$489$	−4.94427	−0.223588
$490$	0	0
$491$	−13.7639	−0.621158	−0.310579	−	0.950548i	$-0.600523\pi$
$491$	−13.7639	−0.621158	−0.310579	+	0.950548i	$0.600523\pi$
$492$	−5.52786	−0.249215
$493$	17.7082	0.797537
$494$	−10.0902	−0.453978
$495$	0	0
$496$	29.1246	1.30773
$497$	−12.4721	−0.559452
$498$	16.9443	0.759291
$499$	−24.1803	−1.08246	−0.541230	−	0.840874i	$-0.682041\pi$
$499$	−24.1803	−1.08246	−0.541230	+	0.840874i	$0.682041\pi$
$500$	0	0
$501$	−1.29180	−0.0577132
$502$	−7.70820	−0.344034
$503$	28.0689	1.25153	0.625765	−	0.780012i	$-0.284787\pi$
$503$	28.0689	1.25153	0.625765	+	0.780012i	$0.284787\pi$
$504$	−4.47214	−0.199205
$505$	0	0
$506$	−8.00000	−0.355643
$507$	−12.0000	−0.532939
$508$	3.52786	0.156524
$509$	6.94427	0.307799	0.153900	−	0.988086i	$-0.450817\pi$
$509$	6.94427	0.307799	0.153900	+	0.988086i	$0.450817\pi$
$510$	0	0
$511$	−24.3607	−1.07765
$512$	−5.29180	−0.233867
$513$	−6.23607	−0.275329
$514$	−41.1246	−1.81393
$515$	0	0
$516$	0.472136	0.0207846
$517$	0.583592	0.0256664
$518$	−3.23607	−0.142185
$519$	−5.00000	−0.219476
$520$	0	0
$521$	−20.4164	−0.894459	−0.447230	−	0.894419i	$-0.647589\pi$
$521$	−20.4164	−0.894459	−0.447230	+	0.894419i	$0.647589\pi$
$522$	−5.23607	−0.229176
$523$	−3.34752	−0.146377	−0.0731885	−	0.997318i	$-0.523317\pi$
$523$	−3.34752	−0.146377	−0.0731885	+	0.997318i	$0.523317\pi$
$524$	4.79837	0.209618
$525$	0	0
$526$	6.47214	0.282199
$527$	32.8328	1.43022
$528$	3.70820	0.161379
$529$	18.8885	0.821241
$530$	0	0
$531$	12.9443	0.561734
$532$	−7.70820	−0.334193
$533$	−8.94427	−0.387419
$534$	−4.47214	−0.193528
$535$	0	0
$536$	6.18034	0.266950
$537$	−26.2361	−1.13217
$538$	−16.8541	−0.726632
$539$	2.29180	0.0987146
$540$	0	0
$541$	−22.4721	−0.966153	−0.483076	−	0.875578i	$-0.660481\pi$
$541$	−22.4721	−0.966153	−0.483076	+	0.875578i	$0.660481\pi$
$542$	18.8541	0.809853
$543$	1.70820	0.0733060
$544$	18.5066	0.793463
$545$	0	0
$546$	3.23607	0.138491
$547$	−25.7639	−1.10159	−0.550793	−	0.834642i	$-0.685675\pi$
$547$	−25.7639	−1.10159	−0.550793	+	0.834642i	$0.685675\pi$
$548$	−11.2361	−0.479981
$549$	−12.4164	−0.529919
$550$	0	0
$551$	20.1803	0.859711
$552$	−14.4721	−0.615975
$553$	22.8328	0.970950
$554$	−2.76393	−0.117428
$555$	0	0
$556$	0.291796	0.0123749
$557$	−9.11146	−0.386065	−0.193032	−	0.981192i	$-0.561832\pi$
$557$	−9.11146	−0.386065	−0.193032	+	0.981192i	$0.561832\pi$
$558$	−9.70820	−0.410981
$559$	0.763932	0.0323109
$560$	0	0
$561$	4.18034	0.176494
$562$	−0.763932	−0.0322245
$563$	−32.2361	−1.35859	−0.679294	−	0.733866i	$-0.737714\pi$
$563$	−32.2361	−1.35859	−0.679294	+	0.733866i	$0.737714\pi$
$564$	−0.472136	−0.0198805
$565$	0	0
$566$	−25.5066	−1.07212
$567$	2.00000	0.0839921
$568$	13.9443	0.585089
$569$	−41.7082	−1.74850	−0.874249	−	0.485477i	$-0.838646\pi$
$569$	−41.7082	−1.74850	−0.874249	+	0.485477i	$0.838646\pi$
$570$	0	0
$571$	15.1803	0.635277	0.317639	−	0.948212i	$-0.397110\pi$
$571$	15.1803	0.635277	0.317639	+	0.948212i	$0.397110\pi$
$572$	−0.472136	−0.0197410
$573$	−9.76393	−0.407894
$574$	−28.9443	−1.20811
$575$	0	0
$576$	4.23607	0.176503
$577$	45.9443	1.91269	0.956343	−	0.292248i	$-0.0944033\pi$
$577$	45.9443	1.91269	0.956343	+	0.292248i	$0.0944033\pi$
$578$	20.9443	0.871167
$579$	14.9443	0.621063
$580$	0	0
$581$	20.9443	0.868915
$582$	−9.23607	−0.382847
$583$	3.63932	0.150725
$584$	27.2361	1.12704
$585$	0	0
$586$	−26.3607	−1.08895
$587$	2.65248	0.109479	0.0547397	−	0.998501i	$-0.482567\pi$
$587$	2.65248	0.109479	0.0547397	+	0.998501i	$0.482567\pi$
$588$	−1.85410	−0.0764619
$589$	37.4164	1.54172
$590$	0	0
$591$	−14.0000	−0.575883
$592$	4.85410	0.199502
$593$	16.0557	0.659330	0.329665	−	0.944098i	$-0.393064\pi$
$593$	16.0557	0.659330	0.329665	+	0.944098i	$0.393064\pi$
$594$	−1.23607	−0.0507165
$595$	0	0
$596$	2.18034	0.0893102
$597$	17.6525	0.722468
$598$	10.4721	0.428237
$599$	10.0000	0.408589	0.204294	−	0.978909i	$-0.434510\pi$
$599$	10.0000	0.408589	0.204294	+	0.978909i	$0.434510\pi$
$600$	0	0
$601$	−15.8197	−0.645297	−0.322649	−	0.946519i	$-0.604573\pi$
$601$	−15.8197	−0.645297	−0.322649	+	0.946519i	$0.604573\pi$
$602$	2.47214	0.100757
$603$	−2.76393	−0.112556
$604$	−9.38197	−0.381747
$605$	0	0
$606$	2.76393	0.112277
$607$	−15.2361	−0.618413	−0.309206	−	0.950995i	$-0.600063\pi$
$607$	−15.2361	−0.618413	−0.309206	+	0.950995i	$0.600063\pi$
$608$	21.0902	0.855319
$609$	−6.47214	−0.262264
$610$	0	0
$611$	−0.763932	−0.0309054
$612$	−3.38197	−0.136708
$613$	46.2492	1.86799	0.933994	−	0.357288i	$-0.116299\pi$
$613$	46.2492	1.86799	0.933994	+	0.357288i	$0.116299\pi$
$614$	32.1803	1.29869
$615$	0	0
$616$	3.41641	0.137651
$617$	15.8328	0.637405	0.318703	−	0.947855i	$-0.396753\pi$
$617$	15.8328	0.637405	0.318703	+	0.947855i	$0.396753\pi$
$618$	18.1803	0.731321
$619$	32.6525	1.31241	0.656207	−	0.754581i	$-0.272160\pi$
$619$	32.6525	1.31241	0.656207	+	0.754581i	$0.272160\pi$
$620$	0	0
$621$	6.47214	0.259718
$622$	34.2705	1.37412
$623$	−5.52786	−0.221469
$624$	−4.85410	−0.194320
$625$	0	0
$626$	40.2705	1.60953
$627$	4.76393	0.190253
$628$	1.41641	0.0565208
$629$	5.47214	0.218188
$630$	0	0
$631$	5.81966	0.231677	0.115839	−	0.993268i	$-0.463044\pi$
$631$	5.81966	0.231677	0.115839	+	0.993268i	$0.463044\pi$
$632$	−25.5279	−1.01544
$633$	26.9443	1.07094
$634$	−23.2361	−0.922822
$635$	0	0
$636$	−2.94427	−0.116748
$637$	−3.00000	−0.118864
$638$	4.00000	0.158362
$639$	−6.23607	−0.246695
$640$	0	0
$641$	−25.9443	−1.02474	−0.512369	−	0.858766i	$-0.671232\pi$
$641$	−25.9443	−1.02474	−0.512369	+	0.858766i	$0.671232\pi$
$642$	1.61803	0.0638587
$643$	−23.0689	−0.909748	−0.454874	−	0.890556i	$-0.650316\pi$
$643$	−23.0689	−0.909748	−0.454874	+	0.890556i	$0.650316\pi$
$644$	8.00000	0.315244
$645$	0	0
$646$	55.2148	2.17240
$647$	30.8328	1.21216	0.606082	−	0.795403i	$-0.292741\pi$
$647$	30.8328	1.21216	0.606082	+	0.795403i	$0.292741\pi$
$648$	−2.23607	−0.0878410
$649$	−9.88854	−0.388159
$650$	0	0
$651$	−12.0000	−0.470317
$652$	−3.05573	−0.119672
$653$	21.2361	0.831032	0.415516	−	0.909586i	$-0.363601\pi$
$653$	21.2361	0.831032	0.415516	+	0.909586i	$0.363601\pi$
$654$	4.76393	0.186284
$655$	0	0
$656$	43.4164	1.69513
$657$	−12.1803	−0.475200
$658$	−2.47214	−0.0963739
$659$	−10.1115	−0.393886	−0.196943	−	0.980415i	$-0.563101\pi$
$659$	−10.1115	−0.393886	−0.196943	+	0.980415i	$0.563101\pi$
$660$	0	0
$661$	10.9443	0.425683	0.212841	−	0.977087i	$-0.431728\pi$
$661$	10.9443	0.425683	0.212841	+	0.977087i	$0.431728\pi$
$662$	−26.8541	−1.04371
$663$	−5.47214	−0.212520
$664$	−23.4164	−0.908733
$665$	0	0
$666$	−1.61803	−0.0626975
$667$	−20.9443	−0.810965
$668$	−0.798374	−0.0308900
$669$	15.1803	0.586906
$670$	0	0
$671$	9.48529	0.366176
$672$	−6.76393	−0.260924
$673$	−22.7639	−0.877485	−0.438743	−	0.898613i	$-0.644576\pi$
$673$	−22.7639	−0.877485	−0.438743	+	0.898613i	$0.644576\pi$
$674$	26.5623	1.02314
$675$	0	0
$676$	−7.41641	−0.285246
$677$	−28.4721	−1.09427	−0.547137	−	0.837043i	$-0.684282\pi$
$677$	−28.4721	−1.09427	−0.547137	+	0.837043i	$0.684282\pi$
$678$	5.61803	0.215759
$679$	−11.4164	−0.438122
$680$	0	0
$681$	−25.7082	−0.985141
$682$	7.41641	0.283989
$683$	2.94427	0.112659	0.0563297	−	0.998412i	$-0.482060\pi$
$683$	2.94427	0.112659	0.0563297	+	0.998412i	$0.482060\pi$
$684$	−3.85410	−0.147365
$685$	0	0
$686$	−32.3607	−1.23554
$687$	−0.180340	−0.00688040
$688$	−3.70820	−0.141374
$689$	−4.76393	−0.181491
$690$	0	0
$691$	37.1803	1.41441	0.707203	−	0.707010i	$-0.249957\pi$
$691$	37.1803	1.41441	0.707203	+	0.707010i	$0.249957\pi$
$692$	−3.09017	−0.117471
$693$	−1.52786	−0.0580388
$694$	−10.8541	−0.412016
$695$	0	0
$696$	7.23607	0.274282
$697$	48.9443	1.85390
$698$	42.1803	1.59655
$699$	−11.4164	−0.431808
$700$	0	0
$701$	36.3050	1.37122	0.685610	−	0.727969i	$-0.259536\pi$
$701$	36.3050	1.37122	0.685610	+	0.727969i	$0.259536\pi$
$702$	1.61803	0.0610688
$703$	6.23607	0.235198
$704$	−3.23607	−0.121964
$705$	0	0
$706$	29.7082	1.11808
$707$	3.41641	0.128487
$708$	8.00000	0.300658
$709$	−21.3050	−0.800124	−0.400062	−	0.916488i	$-0.631011\pi$
$709$	−21.3050	−0.800124	−0.400062	+	0.916488i	$0.631011\pi$
$710$	0	0
$711$	11.4164	0.428149
$712$	6.18034	0.231618
$713$	−38.8328	−1.45430
$714$	−17.7082	−0.662713
$715$	0	0
$716$	−16.2148	−0.605975
$717$	21.1246	0.788913
$718$	−53.6869	−2.00358
$719$	14.1246	0.526759	0.263380	−	0.964692i	$-0.415163\pi$
$719$	14.1246	0.526759	0.263380	+	0.964692i	$0.415163\pi$
$720$	0	0
$721$	22.4721	0.836906
$722$	32.1803	1.19763
$723$	−2.41641	−0.0898672
$724$	1.05573	0.0392358
$725$	0	0
$726$	−16.8541	−0.625514
$727$	42.1246	1.56232	0.781158	−	0.624334i	$-0.214629\pi$
$727$	42.1246	1.56232	0.781158	+	0.624334i	$0.214629\pi$
$728$	−4.47214	−0.165748
$729$	1.00000	0.0370370
$730$	0	0
$731$	−4.18034	−0.154615
$732$	−7.67376	−0.283630
$733$	−24.0689	−0.889005	−0.444502	−	0.895778i	$-0.646619\pi$
$733$	−24.0689	−0.889005	−0.444502	+	0.895778i	$0.646619\pi$
$734$	47.1246	1.73940
$735$	0	0
$736$	−21.8885	−0.806822
$737$	2.11146	0.0777765
$738$	−14.4721	−0.532727
$739$	25.7082	0.945692	0.472846	−	0.881145i	$-0.343227\pi$
$739$	25.7082	0.945692	0.472846	+	0.881145i	$0.343227\pi$
$740$	0	0
$741$	−6.23607	−0.229088
$742$	−15.4164	−0.565954
$743$	46.9443	1.72222	0.861109	−	0.508420i	$-0.169770\pi$
$743$	46.9443	1.72222	0.861109	+	0.508420i	$0.169770\pi$
$744$	13.4164	0.491869
$745$	0	0
$746$	55.6869	2.03884
$747$	10.4721	0.383155
$748$	2.58359	0.0944655
$749$	2.00000	0.0730784
$750$	0	0
$751$	−23.5967	−0.861058	−0.430529	−	0.902577i	$-0.641673\pi$
$751$	−23.5967	−0.861058	−0.430529	+	0.902577i	$0.641673\pi$
$752$	3.70820	0.135224
$753$	−4.76393	−0.173607
$754$	−5.23607	−0.190686
$755$	0	0
$756$	1.23607	0.0449554
$757$	13.7082	0.498233	0.249117	−	0.968474i	$-0.419860\pi$
$757$	13.7082	0.498233	0.249117	+	0.968474i	$0.419860\pi$
$758$	−10.0000	−0.363216
$759$	−4.94427	−0.179466
$760$	0	0
$761$	23.4164	0.848844	0.424422	−	0.905464i	$-0.360477\pi$
$761$	23.4164	0.848844	0.424422	+	0.905464i	$0.360477\pi$
$762$	9.23607	0.334587
$763$	5.88854	0.213180
$764$	−6.03444	−0.218318
$765$	0	0
$766$	56.3607	2.03639
$767$	12.9443	0.467391
$768$	13.5623	0.489388
$769$	−4.83282	−0.174276	−0.0871379	−	0.996196i	$-0.527772\pi$
$769$	−4.83282	−0.174276	−0.0871379	+	0.996196i	$0.527772\pi$
$770$	0	0
$771$	−25.4164	−0.915350
$772$	9.23607	0.332413
$773$	−50.7214	−1.82432	−0.912160	−	0.409834i	$-0.865587\pi$
$773$	−50.7214	−1.82432	−0.912160	+	0.409834i	$0.865587\pi$
$774$	1.23607	0.0444295
$775$	0	0
$776$	12.7639	0.458198
$777$	−2.00000	−0.0717496
$778$	−11.2361	−0.402833
$779$	55.7771	1.99842
$780$	0	0
$781$	4.76393	0.170467
$782$	−57.3050	−2.04922
$783$	−3.23607	−0.115648
$784$	14.5623	0.520082
$785$	0	0
$786$	12.5623	0.448083
$787$	31.4164	1.11987	0.559937	−	0.828535i	$-0.310825\pi$
$787$	31.4164	1.11987	0.559937	+	0.828535i	$0.310825\pi$
$788$	−8.65248	−0.308232
$789$	4.00000	0.142404
$790$	0	0
$791$	6.94427	0.246910
$792$	1.70820	0.0606984
$793$	−12.4164	−0.440920
$794$	46.6525	1.65563
$795$	0	0
$796$	10.9098	0.386689
$797$	−26.6525	−0.944079	−0.472040	−	0.881577i	$-0.656482\pi$
$797$	−26.6525	−0.944079	−0.472040	+	0.881577i	$0.656482\pi$
$798$	−20.1803	−0.714376
$799$	4.18034	0.147890
$800$	0	0
$801$	−2.76393	−0.0976587
$802$	−29.1246	−1.02843
$803$	9.30495	0.328365
$804$	−1.70820	−0.0602437
$805$	0	0
$806$	−9.70820	−0.341957
$807$	−10.4164	−0.366675
$808$	−3.81966	−0.134375
$809$	−41.2492	−1.45025	−0.725123	−	0.688620i	$-0.758217\pi$
$809$	−41.2492	−1.45025	−0.725123	+	0.688620i	$0.758217\pi$
$810$	0	0
$811$	36.0000	1.26413	0.632065	−	0.774915i	$-0.282207\pi$
$811$	36.0000	1.26413	0.632065	+	0.774915i	$0.282207\pi$
$812$	−4.00000	−0.140372
$813$	11.6525	0.408670
$814$	1.23607	0.0433242
$815$	0	0
$816$	26.5623	0.929867
$817$	−4.76393	−0.166669
$818$	40.8328	1.42769
$819$	2.00000	0.0698857
$820$	0	0
$821$	−37.9443	−1.32426	−0.662132	−	0.749387i	$-0.730348\pi$
$821$	−37.9443	−1.32426	−0.662132	+	0.749387i	$0.730348\pi$
$822$	−29.4164	−1.02602
$823$	32.9443	1.14837	0.574183	−	0.818727i	$-0.305320\pi$
$823$	32.9443	1.14837	0.574183	+	0.818727i	$0.305320\pi$
$824$	−25.1246	−0.875257
$825$	0	0
$826$	41.8885	1.45749
$827$	34.1246	1.18663	0.593315	−	0.804971i	$-0.297819\pi$
$827$	34.1246	1.18663	0.593315	+	0.804971i	$0.297819\pi$
$828$	4.00000	0.139010
$829$	−28.7639	−0.999013	−0.499506	−	0.866310i	$-0.666485\pi$
$829$	−28.7639	−0.999013	−0.499506	+	0.866310i	$0.666485\pi$
$830$	0	0
$831$	−1.70820	−0.0592569
$832$	4.23607	0.146859
$833$	16.4164	0.568795
$834$	0.763932	0.0264528
$835$	0	0
$836$	2.94427	0.101830
$837$	−6.00000	−0.207390
$838$	−9.14590	−0.315940
$839$	0.652476	0.0225260	0.0112630	−	0.999937i	$-0.496415\pi$
$839$	0.652476	0.0225260	0.0112630	+	0.999937i	$0.496415\pi$
$840$	0	0
$841$	−18.5279	−0.638892
$842$	−46.9230	−1.61707
$843$	−0.472136	−0.0162612
$844$	16.6525	0.573202
$845$	0	0
$846$	−1.23607	−0.0424969
$847$	−20.8328	−0.715824
$848$	23.1246	0.794102
$849$	−15.7639	−0.541017
$850$	0	0
$851$	−6.47214	−0.221862
$852$	−3.85410	−0.132039
$853$	−49.1246	−1.68199	−0.840997	−	0.541039i	$-0.818031\pi$
$853$	−49.1246	−1.68199	−0.840997	+	0.541039i	$0.818031\pi$
$854$	−40.1803	−1.37494
$855$	0	0
$856$	−2.23607	−0.0764272
$857$	16.2918	0.556517	0.278259	−	0.960506i	$-0.410243\pi$
$857$	16.2918	0.556517	0.278259	+	0.960506i	$0.410243\pi$
$858$	−1.23607	−0.0421987
$859$	29.2918	0.999423	0.499712	−	0.866192i	$-0.333439\pi$
$859$	29.2918	0.999423	0.499712	+	0.866192i	$0.333439\pi$
$860$	0	0
$861$	−17.8885	−0.609640
$862$	6.00000	0.204361
$863$	21.7639	0.740853	0.370426	−	0.928862i	$-0.379212\pi$
$863$	21.7639	0.740853	0.370426	+	0.928862i	$0.379212\pi$
$864$	−3.38197	−0.115057
$865$	0	0
$866$	−30.9443	−1.05153
$867$	12.9443	0.439611
$868$	−7.41641	−0.251729
$869$	−8.72136	−0.295852
$870$	0	0
$871$	−2.76393	−0.0936523
$872$	−6.58359	−0.222949
$873$	−5.70820	−0.193193
$874$	−65.3050	−2.20897
$875$	0	0
$876$	−7.52786	−0.254343
$877$	49.0132	1.65506	0.827528	−	0.561424i	$-0.189746\pi$
$877$	49.0132	1.65506	0.827528	+	0.561424i	$0.189746\pi$
$878$	−55.7771	−1.88239
$879$	−16.2918	−0.549509
$880$	0	0
$881$	−38.2492	−1.28865	−0.644325	−	0.764752i	$-0.722861\pi$
$881$	−38.2492	−1.28865	−0.644325	+	0.764752i	$0.722861\pi$
$882$	−4.85410	−0.163446
$883$	−37.2918	−1.25497	−0.627484	−	0.778629i	$-0.715915\pi$
$883$	−37.2918	−1.25497	−0.627484	+	0.778629i	$0.715915\pi$
$884$	−3.38197	−0.113748
$885$	0	0
$886$	−43.4164	−1.45860
$887$	9.76393	0.327841	0.163920	−	0.986474i	$-0.447586\pi$
$887$	9.76393	0.327841	0.163920	+	0.986474i	$0.447586\pi$
$888$	2.23607	0.0750375
$889$	11.4164	0.382894
$890$	0	0
$891$	−0.763932	−0.0255927
$892$	9.38197	0.314131
$893$	4.76393	0.159419
$894$	5.70820	0.190911
$895$	0	0
$896$	27.2361	0.909893
$897$	6.47214	0.216098
$898$	28.8541	0.962874
$899$	19.4164	0.647573
$900$	0	0
$901$	26.0689	0.868480
$902$	11.0557	0.368115
$903$	1.52786	0.0508441
$904$	−7.76393	−0.258225
$905$	0	0
$906$	−24.5623	−0.816028
$907$	−20.3607	−0.676065	−0.338033	−	0.941134i	$-0.609761\pi$
$907$	−20.3607	−0.676065	−0.338033	+	0.941134i	$0.609761\pi$
$908$	−15.8885	−0.527280
$909$	1.70820	0.0566575
$910$	0	0
$911$	−44.9443	−1.48907	−0.744535	−	0.667583i	$-0.767329\pi$
$911$	−44.9443	−1.48907	−0.744535	+	0.667583i	$0.767329\pi$
$912$	30.2705	1.00236
$913$	−8.00000	−0.264761
$914$	−36.0902	−1.19376
$915$	0	0
$916$	−0.111456	−0.00368262
$917$	15.5279	0.512775
$918$	−8.85410	−0.292229
$919$	−0.472136	−0.0155743	−0.00778716	−	0.999970i	$-0.502479\pi$
$919$	−0.472136	−0.0155743	−0.00778716	+	0.999970i	$0.502479\pi$
$920$	0	0
$921$	19.8885	0.655350
$922$	38.3607	1.26334
$923$	−6.23607	−0.205263
$924$	−0.944272	−0.0310643
$925$	0	0
$926$	−3.05573	−0.100417
$927$	11.2361	0.369041
$928$	10.9443	0.359263
$929$	25.8328	0.847547	0.423774	−	0.905768i	$-0.360705\pi$
$929$	25.8328	0.847547	0.423774	+	0.905768i	$0.360705\pi$
$930$	0	0
$931$	18.7082	0.613137
$932$	−7.05573	−0.231118
$933$	21.1803	0.693413
$934$	−14.0000	−0.458094
$935$	0	0
$936$	−2.23607	−0.0730882
$937$	7.47214	0.244104	0.122052	−	0.992524i	$-0.461053\pi$
$937$	7.47214	0.244104	0.122052	+	0.992524i	$0.461053\pi$
$938$	−8.94427	−0.292041
$939$	24.8885	0.812207
$940$	0	0
$941$	−43.7082	−1.42485	−0.712423	−	0.701750i	$-0.752402\pi$
$941$	−43.7082	−1.42485	−0.712423	+	0.701750i	$0.752402\pi$
$942$	3.70820	0.120820
$943$	−57.8885	−1.88511
$944$	−62.8328	−2.04503
$945$	0	0
$946$	−0.944272	−0.0307009
$947$	−2.11146	−0.0686131	−0.0343066	−	0.999411i	$-0.510922\pi$
$947$	−2.11146	−0.0686131	−0.0343066	+	0.999411i	$0.510922\pi$
$948$	7.05573	0.229159
$949$	−12.1803	−0.395391
$950$	0	0
$951$	−14.3607	−0.465677
$952$	24.4721	0.793146
$953$	29.7771	0.964574	0.482287	−	0.876013i	$-0.339806\pi$
$953$	29.7771	0.964574	0.482287	+	0.876013i	$0.339806\pi$
$954$	−7.70820	−0.249562
$955$	0	0
$956$	13.0557	0.422252
$957$	2.47214	0.0799128
$958$	32.5623	1.05204
$959$	−36.3607	−1.17415
$960$	0	0
$961$	5.00000	0.161290
$962$	−1.61803	−0.0521675
$963$	1.00000	0.0322245
$964$	−1.49342	−0.0480999
$965$	0	0
$966$	20.9443	0.673871
$967$	−18.1246	−0.582848	−0.291424	−	0.956594i	$-0.594129\pi$
$967$	−18.1246	−0.582848	−0.291424	+	0.956594i	$0.594129\pi$
$968$	23.2918	0.748627
$969$	34.1246	1.09624
$970$	0	0
$971$	−46.7082	−1.49894	−0.749469	−	0.662040i	$-0.769691\pi$
$971$	−46.7082	−1.49894	−0.749469	+	0.662040i	$0.769691\pi$
$972$	0.618034	0.0198234
$973$	0.944272	0.0302720
$974$	−58.0689	−1.86065
$975$	0	0
$976$	60.2705	1.92921
$977$	53.7771	1.72048	0.860241	−	0.509888i	$-0.170313\pi$
$977$	53.7771	1.72048	0.860241	+	0.509888i	$0.170313\pi$
$978$	−8.00000	−0.255812
$979$	2.11146	0.0674824
$980$	0	0
$981$	2.94427	0.0940034
$982$	−22.2705	−0.710681
$983$	−27.2918	−0.870473	−0.435237	−	0.900316i	$-0.643335\pi$
$983$	−27.2918	−0.870473	−0.435237	+	0.900316i	$0.643335\pi$
$984$	20.0000	0.637577
$985$	0	0
$986$	28.6525	0.912481
$987$	−1.52786	−0.0486324
$988$	−3.85410	−0.122615
$989$	4.94427	0.157219
$990$	0	0
$991$	−34.0689	−1.08223	−0.541117	−	0.840947i	$-0.681998\pi$
$991$	−34.0689	−1.08223	−0.541117	+	0.840947i	$0.681998\pi$
$992$	20.2918	0.644265
$993$	−16.5967	−0.526682
$994$	−20.1803	−0.640082
$995$	0	0
$996$	6.47214	0.205077
$997$	59.4721	1.88350	0.941751	−	0.336312i	$-0.109180\pi$
$997$	59.4721	1.88350	0.941751	+	0.336312i	$0.109180\pi$
$998$	−39.1246	−1.23847
$999$	−1.00000	−0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8025.2.a.s.1.2		2
5.4	even	2		321.2.a.a.1.1	✓	2
15.14	odd	2		963.2.a.a.1.2		2
20.19	odd	2		5136.2.a.y.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
321.2.a.a.1.1	✓	2	5.4	even	2
963.2.a.a.1.2		2	15.14	odd	2
5136.2.a.y.1.1		2	20.19	odd	2
8025.2.a.s.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8025 = 3 \cdot 5^{2} \cdot 107 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8025.a (trivial)

\(f(q)\)	\(=\)	\(q+1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} +1.61803 q^{6} +2.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} +1.61803 q^{6} +2.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} -0.763932 q^{11} +0.618034 q^{12} +1.00000 q^{13} +3.23607 q^{14} -4.85410 q^{16} -5.47214 q^{17} +1.61803 q^{18} -6.23607 q^{19} +2.00000 q^{21} -1.23607 q^{22} +6.47214 q^{23} -2.23607 q^{24} +1.61803 q^{26} +1.00000 q^{27} +1.23607 q^{28} -3.23607 q^{29} -6.00000 q^{31} -3.38197 q^{32} -0.763932 q^{33} -8.85410 q^{34} +0.618034 q^{36} -1.00000 q^{37} -10.0902 q^{38} +1.00000 q^{39} -8.94427 q^{41} +3.23607 q^{42} +0.763932 q^{43} -0.472136 q^{44} +10.4721 q^{46} -0.763932 q^{47} -4.85410 q^{48} -3.00000 q^{49} -5.47214 q^{51} +0.618034 q^{52} -4.76393 q^{53} +1.61803 q^{54} -4.47214 q^{56} -6.23607 q^{57} -5.23607 q^{58} +12.9443 q^{59} -12.4164 q^{61} -9.70820 q^{62} +2.00000 q^{63} +4.23607 q^{64} -1.23607 q^{66} -2.76393 q^{67} -3.38197 q^{68} +6.47214 q^{69} -6.23607 q^{71} -2.23607 q^{72} -12.1803 q^{73} -1.61803 q^{74} -3.85410 q^{76} -1.52786 q^{77} +1.61803 q^{78} +11.4164 q^{79} +1.00000 q^{81} -14.4721 q^{82} +10.4721 q^{83} +1.23607 q^{84} +1.23607 q^{86} -3.23607 q^{87} +1.70820 q^{88} -2.76393 q^{89} +2.00000 q^{91} +4.00000 q^{92} -6.00000 q^{93} -1.23607 q^{94} -3.38197 q^{96} -5.70820 q^{97} -4.85410 q^{98} -0.763932 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 2 q^{3} - q^{4} + q^{6} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + q^2 + 2 * q^3 - q^4 + q^6 + 4 * q^7 + 2 * q^9	\( 2 q + q^{2} + 2 q^{3} - q^{4} + q^{6} + 4 q^{7} + 2 q^{9} - 6 q^{11} - q^{12} + 2 q^{13} + 2 q^{14} - 3 q^{16} - 2 q^{17} + q^{18} - 8 q^{19} + 4 q^{21} + 2 q^{22} + 4 q^{23} + q^{26} + 2 q^{27} - 2 q^{28} - 2 q^{29} - 12 q^{31} - 9 q^{32} - 6 q^{33} - 11 q^{34} - q^{36} - 2 q^{37} - 9 q^{38} + 2 q^{39} + 2 q^{42} + 6 q^{43} + 8 q^{44} + 12 q^{46} - 6 q^{47} - 3 q^{48} - 6 q^{49} - 2 q^{51} - q^{52} - 14 q^{53} + q^{54} - 8 q^{57} - 6 q^{58} + 8 q^{59} + 2 q^{61} - 6 q^{62} + 4 q^{63} + 4 q^{64} + 2 q^{66} - 10 q^{67} - 9 q^{68} + 4 q^{69} - 8 q^{71} - 2 q^{73} - q^{74} - q^{76} - 12 q^{77} + q^{78} - 4 q^{79} + 2 q^{81} - 20 q^{82} + 12 q^{83} - 2 q^{84} - 2 q^{86} - 2 q^{87} - 10 q^{88} - 10 q^{89} + 4 q^{91} + 8 q^{92} - 12 q^{93} + 2 q^{94} - 9 q^{96} + 2 q^{97} - 3 q^{98} - 6 q^{99}+O(q^{100}) \) 2 * q + q^2 + 2 * q^3 - q^4 + q^6 + 4 * q^7 + 2 * q^9 - 6 * q^11 - q^12 + 2 * q^13 + 2 * q^14 - 3 * q^16 - 2 * q^17 + q^18 - 8 * q^19 + 4 * q^21 + 2 * q^22 + 4 * q^23 + q^26 + 2 * q^27 - 2 * q^28 - 2 * q^29 - 12 * q^31 - 9 * q^32 - 6 * q^33 - 11 * q^34 - q^36 - 2 * q^37 - 9 * q^38 + 2 * q^39 + 2 * q^42 + 6 * q^43 + 8 * q^44 + 12 * q^46 - 6 * q^47 - 3 * q^48 - 6 * q^49 - 2 * q^51 - q^52 - 14 * q^53 + q^54 - 8 * q^57 - 6 * q^58 + 8 * q^59 + 2 * q^61 - 6 * q^62 + 4 * q^63 + 4 * q^64 + 2 * q^66 - 10 * q^67 - 9 * q^68 + 4 * q^69 - 8 * q^71 - 2 * q^73 - q^74 - q^76 - 12 * q^77 + q^78 - 4 * q^79 + 2 * q^81 - 20 * q^82 + 12 * q^83 - 2 * q^84 - 2 * q^86 - 2 * q^87 - 10 * q^88 - 10 * q^89 + 4 * q^91 + 8 * q^92 - 12 * q^93 + 2 * q^94 - 9 * q^96 + 2 * q^97 - 3 * q^98 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more