LMFDB - Embedded newform 3234.2.a.bj.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3234,2,Mod(1,3234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3234, 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("3234.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3234.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:	$25.8236200137$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.14336.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 8x^{2} + 14 $ x^4 - 8*x^2 + 14
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.60804$ of defining polynomial
Character	$\chi$	$=$	3234.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.27411 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.27411 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +2.27411 q^{10} -1.00000 q^{11} -1.00000 q^{12} -1.80186 q^{13} +2.27411 q^{15} +1.00000 q^{16} -5.10254 q^{17} -1.00000 q^{18} +6.90440 q^{19} -2.27411 q^{20} +1.00000 q^{22} +8.59272 q^{23} +1.00000 q^{24} +0.171573 q^{25} +1.80186 q^{26} -1.00000 q^{27} -2.82843 q^{29} -2.27411 q^{30} -3.14010 q^{31} -1.00000 q^{32} +1.00000 q^{33} +5.10254 q^{34} +1.00000 q^{36} +4.44078 q^{37} -6.90440 q^{38} +1.80186 q^{39} +2.27411 q^{40} +5.10254 q^{41} +6.93587 q^{43} -1.00000 q^{44} -2.27411 q^{45} -8.59272 q^{46} +0.0759718 q^{47} -1.00000 q^{48} -0.171573 q^{50} +5.10254 q^{51} -1.80186 q^{52} -6.87293 q^{53} +1.00000 q^{54} +2.27411 q^{55} -6.90440 q^{57} +2.82843 q^{58} -5.17157 q^{59} +2.27411 q^{60} -8.63029 q^{61} +3.14010 q^{62} +1.00000 q^{64} +4.09763 q^{65} -1.00000 q^{66} +13.4841 q^{67} -5.10254 q^{68} -8.59272 q^{69} -9.26058 q^{71} -1.00000 q^{72} +3.04941 q^{73} -4.44078 q^{74} -0.171573 q^{75} +6.90440 q^{76} -1.80186 q^{78} +9.26921 q^{79} -2.27411 q^{80} +1.00000 q^{81} -5.10254 q^{82} +1.96853 q^{83} +11.6037 q^{85} -6.93587 q^{86} +2.82843 q^{87} +1.00000 q^{88} -15.0624 q^{89} +2.27411 q^{90} +8.59272 q^{92} +3.14010 q^{93} -0.0759718 q^{94} -15.7014 q^{95} +1.00000 q^{96} +9.01794 q^{97} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{6} - 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 4 * q^6 - 4 * q^8 + 4 * q^9	$ 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{6} - 4 q^{8} + 4 q^{9} - 4 q^{11} - 4 q^{12} + 4 q^{16} - 4 q^{18} + 4 q^{22} - 8 q^{23} + 4 q^{24} + 12 q^{25} - 4 q^{27} - 16 q^{31} - 4 q^{32} + 4 q^{33} + 4 q^{36} + 8 q^{37} + 8 q^{43} - 4 q^{44} + 8 q^{46} - 16 q^{47} - 4 q^{48} - 12 q^{50} + 8 q^{53} + 4 q^{54} - 32 q^{59} - 16 q^{61} + 16 q^{62} + 4 q^{64} - 16 q^{65} - 4 q^{66} + 16 q^{67} + 8 q^{69} - 4 q^{72} - 8 q^{74} - 12 q^{75} + 16 q^{79} + 4 q^{81} + 32 q^{85} - 8 q^{86} + 4 q^{88} - 16 q^{89} - 8 q^{92} + 16 q^{93} + 16 q^{94} - 16 q^{95} + 4 q^{96} + 16 q^{97} - 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 4 * q^6 - 4 * q^8 + 4 * q^9 - 4 * q^11 - 4 * q^12 + 4 * q^16 - 4 * q^18 + 4 * q^22 - 8 * q^23 + 4 * q^24 + 12 * q^25 - 4 * q^27 - 16 * q^31 - 4 * q^32 + 4 * q^33 + 4 * q^36 + 8 * q^37 + 8 * q^43 - 4 * q^44 + 8 * q^46 - 16 * q^47 - 4 * q^48 - 12 * q^50 + 8 * q^53 + 4 * q^54 - 32 * q^59 - 16 * q^61 + 16 * q^62 + 4 * q^64 - 16 * q^65 - 4 * q^66 + 16 * q^67 + 8 * q^69 - 4 * q^72 - 8 * q^74 - 12 * q^75 + 16 * q^79 + 4 * q^81 + 32 * q^85 - 8 * q^86 + 4 * q^88 - 16 * q^89 - 8 * q^92 + 16 * q^93 + 16 * q^94 - 16 * q^95 + 4 * q^96 + 16 * q^97 - 4 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−2.27411	−1.01701	−0.508506	−	0.861058i	$-0.669802\pi$
$5$	−2.27411	−1.01701	−0.508506	+	0.861058i	$0.669802\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	2.27411	0.719136
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−1.00000	−0.288675
$13$	−1.80186	−0.499747	−0.249873	−	0.968279i	$-0.580389\pi$
$13$	−1.80186	−0.499747	−0.249873	+	0.968279i	$0.580389\pi$
$14$	0	0
$15$	2.27411	0.587172
$16$	1.00000	0.250000
$17$	−5.10254	−1.23755	−0.618773	−	0.785570i	$-0.712370\pi$
$17$	−5.10254	−1.23755	−0.618773	+	0.785570i	$0.712370\pi$
$18$	−1.00000	−0.235702
$19$	6.90440	1.58398	0.791989	−	0.610536i	$-0.209046\pi$
$19$	6.90440	1.58398	0.791989	+	0.610536i	$0.209046\pi$
$20$	−2.27411	−0.508506
$21$	0	0
$22$	1.00000	0.213201
$23$	8.59272	1.79171	0.895853	−	0.444350i	$-0.146565\pi$
$23$	8.59272	1.79171	0.895853	+	0.444350i	$0.146565\pi$
$24$	1.00000	0.204124
$25$	0.171573	0.0343146
$26$	1.80186	0.353374
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.82843	−0.525226	−0.262613	−	0.964901i	$-0.584584\pi$
$29$	−2.82843	−0.525226	−0.262613	+	0.964901i	$0.584584\pi$
$30$	−2.27411	−0.415194
$31$	−3.14010	−0.563979	−0.281990	−	0.959417i	$-0.590994\pi$
$31$	−3.14010	−0.563979	−0.281990	+	0.959417i	$0.590994\pi$
$32$	−1.00000	−0.176777
$33$	1.00000	0.174078
$34$	5.10254	0.875078
$35$	0	0
$36$	1.00000	0.166667
$37$	4.44078	0.730059	0.365030	−	0.930996i	$-0.381059\pi$
$37$	4.44078	0.730059	0.365030	+	0.930996i	$0.381059\pi$
$38$	−6.90440	−1.12004
$39$	1.80186	0.288529
$40$	2.27411	0.359568
$41$	5.10254	0.796882	0.398441	−	0.917194i	$-0.369551\pi$
$41$	5.10254	0.796882	0.398441	+	0.917194i	$0.369551\pi$
$42$	0	0
$43$	6.93587	1.05771	0.528855	−	0.848712i	$-0.322622\pi$
$43$	6.93587	1.05771	0.528855	+	0.848712i	$0.322622\pi$
$44$	−1.00000	−0.150756
$45$	−2.27411	−0.339004
$46$	−8.59272	−1.26693
$47$	0.0759718	0.0110816	0.00554082	−	0.999985i	$-0.498236\pi$
$47$	0.0759718	0.0110816	0.00554082	+	0.999985i	$0.498236\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	−0.171573	−0.0242641
$51$	5.10254	0.714498
$52$	−1.80186	−0.249873
$53$	−6.87293	−0.944070	−0.472035	−	0.881580i	$-0.656480\pi$
$53$	−6.87293	−0.944070	−0.472035	+	0.881580i	$0.656480\pi$
$54$	1.00000	0.136083
$55$	2.27411	0.306641
$56$	0	0
$57$	−6.90440	−0.914510
$58$	2.82843	0.371391
$59$	−5.17157	−0.673281	−0.336641	−	0.941633i	$-0.609291\pi$
$59$	−5.17157	−0.673281	−0.336641	+	0.941633i	$0.609291\pi$
$60$	2.27411	0.293586
$61$	−8.63029	−1.10500	−0.552498	−	0.833514i	$-0.686325\pi$
$61$	−8.63029	−1.10500	−0.552498	+	0.833514i	$0.686325\pi$
$62$	3.14010	0.398794
$63$	0	0
$64$	1.00000	0.125000
$65$	4.09763	0.508249
$66$	−1.00000	−0.123091
$67$	13.4841	1.64734	0.823672	−	0.567067i	$-0.191922\pi$
$67$	13.4841	1.64734	0.823672	+	0.567067i	$0.191922\pi$
$68$	−5.10254	−0.618773
$69$	−8.59272	−1.03444
$70$	0	0
$71$	−9.26058	−1.09903	−0.549514	−	0.835484i	$-0.685187\pi$
$71$	−9.26058	−1.09903	−0.549514	+	0.835484i	$0.685187\pi$
$72$	−1.00000	−0.117851
$73$	3.04941	0.356906	0.178453	−	0.983948i	$-0.442891\pi$
$73$	3.04941	0.356906	0.178453	+	0.983948i	$0.442891\pi$
$74$	−4.44078	−0.516230
$75$	−0.171573	−0.0198115
$76$	6.90440	0.791989
$77$	0	0
$78$	−1.80186	−0.204021
$79$	9.26921	1.04287	0.521434	−	0.853292i	$-0.325397\pi$
$79$	9.26921	1.04287	0.521434	+	0.853292i	$0.325397\pi$
$80$	−2.27411	−0.254253
$81$	1.00000	0.111111
$82$	−5.10254	−0.563481
$83$	1.96853	0.216074	0.108037	−	0.994147i	$-0.465543\pi$
$83$	1.96853	0.216074	0.108037	+	0.994147i	$0.465543\pi$
$84$	0	0
$85$	11.6037	1.25860
$86$	−6.93587	−0.747914
$87$	2.82843	0.303239
$88$	1.00000	0.106600
$89$	−15.0624	−1.59662	−0.798308	−	0.602250i	$-0.794271\pi$
$89$	−15.0624	−1.59662	−0.798308	+	0.602250i	$0.794271\pi$
$90$	2.27411	0.239712
$91$	0	0
$92$	8.59272	0.895853
$93$	3.14010	0.325614
$94$	−0.0759718	−0.00783590
$95$	−15.7014	−1.61093
$96$	1.00000	0.102062
$97$	9.01794	0.915633	0.457816	−	0.889047i	$-0.348632\pi$
$97$	9.01794	0.915633	0.457816	+	0.889047i	$0.348632\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0.171573	0.0171573
$101$	11.2230	1.11673	0.558366	−	0.829595i	$-0.311429\pi$
$101$	11.2230	1.11673	0.558366	+	0.829595i	$0.311429\pi$
$102$	−5.10254	−0.505226
$103$	−17.0118	−1.67623	−0.838113	−	0.545496i	$-0.816341\pi$
$103$	−17.0118	−1.67623	−0.838113	+	0.545496i	$0.816341\pi$
$104$	1.80186	0.176687
$105$	0	0
$106$	6.87293	0.667558
$107$	11.6482	1.12608	0.563038	−	0.826431i	$-0.309632\pi$
$107$	11.6482	1.12608	0.563038	+	0.826431i	$0.309632\pi$
$108$	−1.00000	−0.0962250
$109$	−14.0890	−1.34948	−0.674741	−	0.738055i	$-0.735745\pi$
$109$	−14.0890	−1.34948	−0.674741	+	0.738055i	$0.735745\pi$
$110$	−2.27411	−0.216828
$111$	−4.44078	−0.421500
$112$	0	0
$113$	−5.71979	−0.538073	−0.269036	−	0.963130i	$-0.586705\pi$
$113$	−5.71979	−0.538073	−0.269036	+	0.963130i	$0.586705\pi$
$114$	6.90440	0.646656
$115$	−19.5408	−1.82219
$116$	−2.82843	−0.262613
$117$	−1.80186	−0.166582
$118$	5.17157	0.476082
$119$	0	0
$120$	−2.27411	−0.207597
$121$	1.00000	0.0909091
$122$	8.63029	0.781350
$123$	−5.10254	−0.460080
$124$	−3.14010	−0.281990
$125$	10.9804	0.982114
$126$	0	0
$127$	6.60492	0.586092	0.293046	−	0.956098i	$-0.405331\pi$
$127$	6.60492	0.586092	0.293046	+	0.956098i	$0.405331\pi$
$128$	−1.00000	−0.0883883
$129$	−6.93587	−0.610669
$130$	−4.09763	−0.359386
$131$	−5.96853	−0.521473	−0.260737	−	0.965410i	$-0.583965\pi$
$131$	−5.96853	−0.521473	−0.260737	+	0.965410i	$0.583965\pi$
$132$	1.00000	0.0870388
$133$	0	0
$134$	−13.4841	−1.16485
$135$	2.27411	0.195724
$136$	5.10254	0.437539
$137$	11.2606	0.962056	0.481028	−	0.876705i	$-0.340263\pi$
$137$	11.2606	0.962056	0.481028	+	0.876705i	$0.340263\pi$
$138$	8.59272	0.731461
$139$	−8.62419	−0.731494	−0.365747	−	0.930714i	$-0.619187\pi$
$139$	−8.62419	−0.731494	−0.365747	+	0.930714i	$0.619187\pi$
$140$	0	0
$141$	−0.0759718	−0.00639798
$142$	9.26058	0.777131
$143$	1.80186	0.150679
$144$	1.00000	0.0833333
$145$	6.43215	0.534161
$146$	−3.04941	−0.252371
$147$	0	0
$148$	4.44078	0.365030
$149$	−12.3211	−1.00939	−0.504694	−	0.863299i	$-0.668394\pi$
$149$	−12.3211	−1.00939	−0.504694	+	0.863299i	$0.668394\pi$
$150$	0.171573	0.0140089
$151$	−11.5286	−0.938183	−0.469092	−	0.883149i	$-0.655419\pi$
$151$	−11.5286	−0.938183	−0.469092	+	0.883149i	$0.655419\pi$
$152$	−6.90440	−0.560021
$153$	−5.10254	−0.412516
$154$	0	0
$155$	7.14094	0.573574
$156$	1.80186	0.144264
$157$	−1.49881	−0.119618	−0.0598091	−	0.998210i	$-0.519049\pi$
$157$	−1.49881	−0.119618	−0.0598091	+	0.998210i	$0.519049\pi$
$158$	−9.26921	−0.737418
$159$	6.87293	0.545059
$160$	2.27411	0.179784
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	13.6569	1.06969	0.534844	−	0.844951i	$-0.320370\pi$
$163$	13.6569	1.06969	0.534844	+	0.844951i	$0.320370\pi$
$164$	5.10254	0.398441
$165$	−2.27411	−0.177039
$166$	−1.96853	−0.152788
$167$	−19.6927	−1.52387	−0.761935	−	0.647654i	$-0.775750\pi$
$167$	−19.6927	−1.52387	−0.761935	+	0.647654i	$0.775750\pi$
$168$	0	0
$169$	−9.75329	−0.750253
$170$	−11.6037	−0.889965
$171$	6.90440	0.527993
$172$	6.93587	0.528855
$173$	14.5107	1.10322	0.551612	−	0.834101i	$-0.314013\pi$
$173$	14.5107	1.10322	0.551612	+	0.834101i	$0.314013\pi$
$174$	−2.82843	−0.214423
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	5.17157	0.388719
$178$	15.0624	1.12898
$179$	−22.3656	−1.67169	−0.835843	−	0.548968i	$-0.815021\pi$
$179$	−22.3656	−1.67169	−0.835843	+	0.548968i	$0.815021\pi$
$180$	−2.27411	−0.169502
$181$	5.73976	0.426633	0.213317	−	0.976983i	$-0.431573\pi$
$181$	5.73976	0.426633	0.213317	+	0.976983i	$0.431573\pi$
$182$	0	0
$183$	8.63029	0.637969
$184$	−8.59272	−0.633464
$185$	−10.0988	−0.742480
$186$	−3.14010	−0.230244
$187$	5.10254	0.373134
$188$	0.0759718	0.00554082
$189$	0	0
$190$	15.7014	1.13910
$191$	−11.7112	−0.847390	−0.423695	−	0.905805i	$-0.639267\pi$
$191$	−11.7112	−0.847390	−0.423695	+	0.905805i	$0.639267\pi$
$192$	−1.00000	−0.0721688
$193$	−12.7533	−0.918002	−0.459001	−	0.888436i	$-0.651793\pi$
$193$	−12.7533	−0.918002	−0.459001	+	0.888436i	$0.651793\pi$
$194$	−9.01794	−0.647450
$195$	−4.09763	−0.293438
$196$	0	0
$197$	−23.2606	−1.65725	−0.828624	−	0.559806i	$-0.810876\pi$
$197$	−23.2606	−1.65725	−0.828624	+	0.559806i	$0.810876\pi$
$198$	1.00000	0.0710669
$199$	−1.48325	−0.105145	−0.0525724	−	0.998617i	$-0.516742\pi$
$199$	−1.48325	−0.105145	−0.0525724	+	0.998617i	$0.516742\pi$
$200$	−0.171573	−0.0121320
$201$	−13.4841	−0.951094
$202$	−11.2230	−0.789648
$203$	0	0
$204$	5.10254	0.357249
$205$	−11.6037	−0.810439
$206$	17.0118	1.18527
$207$	8.59272	0.597235
$208$	−1.80186	−0.124937
$209$	−6.90440	−0.477587
$210$	0	0
$211$	0.503715	0.0346772	0.0173386	−	0.999850i	$-0.494481\pi$
$211$	0.503715	0.0346772	0.0173386	+	0.999850i	$0.494481\pi$
$212$	−6.87293	−0.472035
$213$	9.26058	0.634524
$214$	−11.6482	−0.796257
$215$	−15.7729	−1.07570
$216$	1.00000	0.0680414
$217$	0	0
$218$	14.0890	0.954228
$219$	−3.04941	−0.206060
$220$	2.27411	0.153320
$221$	9.19407	0.618460
$222$	4.44078	0.298046
$223$	−20.0044	−1.33959	−0.669797	−	0.742544i	$-0.733619\pi$
$223$	−20.0044	−1.33959	−0.669797	+	0.742544i	$0.733619\pi$
$224$	0	0
$225$	0.171573	0.0114382
$226$	5.71979	0.380475
$227$	−13.5601	−0.900013	−0.450006	−	0.893025i	$-0.648578\pi$
$227$	−13.5601	−0.900013	−0.450006	+	0.893025i	$0.648578\pi$
$228$	−6.90440	−0.457255
$229$	5.10254	0.337185	0.168593	−	0.985686i	$-0.446078\pi$
$229$	5.10254	0.337185	0.168593	+	0.985686i	$0.446078\pi$
$230$	19.5408	1.28848
$231$	0	0
$232$	2.82843	0.185695
$233$	−0.343146	−0.0224802	−0.0112401	−	0.999937i	$-0.503578\pi$
$233$	−0.343146	−0.0224802	−0.0112401	+	0.999937i	$0.503578\pi$
$234$	1.80186	0.117791
$235$	−0.172768	−0.0112702
$236$	−5.17157	−0.336641
$237$	−9.26921	−0.602100
$238$	0	0
$239$	−18.4322	−1.19228	−0.596138	−	0.802882i	$-0.703299\pi$
$239$	−18.4322	−1.19228	−0.596138	+	0.802882i	$0.703299\pi$
$240$	2.27411	0.146793
$241$	−24.5682	−1.58258	−0.791288	−	0.611444i	$-0.790589\pi$
$241$	−24.5682	−1.58258	−0.791288	+	0.611444i	$0.790589\pi$
$242$	−1.00000	−0.0642824
$243$	−1.00000	−0.0641500
$244$	−8.63029	−0.552498
$245$	0	0
$246$	5.10254	0.325326
$247$	−12.4408	−0.791588
$248$	3.14010	0.199397
$249$	−1.96853	−0.124751
$250$	−10.9804	−0.694460
$251$	−1.50491	−0.0949891	−0.0474946	−	0.998871i	$-0.515124\pi$
$251$	−1.50491	−0.0949891	−0.0474946	+	0.998871i	$0.515124\pi$
$252$	0	0
$253$	−8.59272	−0.540220
$254$	−6.60492	−0.414430
$255$	−11.6037	−0.726653
$256$	1.00000	0.0625000
$257$	−10.8476	−0.676652	−0.338326	−	0.941029i	$-0.609861\pi$
$257$	−10.8476	−0.676652	−0.338326	+	0.941029i	$0.609861\pi$
$258$	6.93587	0.431808
$259$	0	0
$260$	4.09763	0.254124
$261$	−2.82843	−0.175075
$262$	5.96853	0.368737
$263$	6.25938	0.385970	0.192985	−	0.981202i	$-0.438183\pi$
$263$	6.25938	0.385970	0.192985	+	0.981202i	$0.438183\pi$
$264$	−1.00000	−0.0615457
$265$	15.6298	0.960131
$266$	0	0
$267$	15.0624	0.921806
$268$	13.4841	0.823672
$269$	−14.8345	−0.904477	−0.452239	−	0.891897i	$-0.649374\pi$
$269$	−14.8345	−0.904477	−0.452239	+	0.891897i	$0.649374\pi$
$270$	−2.27411	−0.138398
$271$	2.91743	0.177221	0.0886107	−	0.996066i	$-0.471757\pi$
$271$	2.91743	0.177221	0.0886107	+	0.996066i	$0.471757\pi$
$272$	−5.10254	−0.309387
$273$	0	0
$274$	−11.2606	−0.680276
$275$	−0.171573	−0.0103462
$276$	−8.59272	−0.517221
$277$	12.1012	0.727091	0.363545	−	0.931576i	$-0.381566\pi$
$277$	12.1012	0.727091	0.363545	+	0.931576i	$0.381566\pi$
$278$	8.62419	0.517245
$279$	−3.14010	−0.187993
$280$	0	0
$281$	7.74586	0.462079	0.231040	−	0.972944i	$-0.425787\pi$
$281$	7.74586	0.462079	0.231040	+	0.972944i	$0.425787\pi$
$282$	0.0759718	0.00452406
$283$	−17.5948	−1.04590	−0.522950	−	0.852364i	$-0.675168\pi$
$283$	−17.5948	−1.04590	−0.522950	+	0.852364i	$0.675168\pi$
$284$	−9.26058	−0.549514
$285$	15.7014	0.930068
$286$	−1.80186	−0.106546
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	9.03588	0.531522
$290$	−6.43215	−0.377709
$291$	−9.01794	−0.528641
$292$	3.04941	0.178453
$293$	16.1675	0.944516	0.472258	−	0.881460i	$-0.343439\pi$
$293$	16.1675	0.944516	0.472258	+	0.881460i	$0.343439\pi$
$294$	0	0
$295$	11.7607	0.684736
$296$	−4.44078	−0.258115
$297$	1.00000	0.0580259
$298$	12.3211	0.713744
$299$	−15.4829	−0.895399
$300$	−0.171573	−0.00990576
$301$	0	0
$302$	11.5286	0.663396
$303$	−11.2230	−0.644745
$304$	6.90440	0.395994
$305$	19.6262	1.12379
$306$	5.10254	0.291693
$307$	27.0563	1.54419	0.772094	−	0.635509i	$-0.219210\pi$
$307$	27.0563	1.54419	0.772094	+	0.635509i	$0.219210\pi$
$308$	0	0
$309$	17.0118	0.967770
$310$	−7.14094	−0.405578
$311$	24.5751	1.39353	0.696764	−	0.717301i	$-0.254623\pi$
$311$	24.5751	1.39353	0.696764	+	0.717301i	$0.254623\pi$
$312$	−1.80186	−0.102010
$313$	−13.0809	−0.739375	−0.369687	−	0.929156i	$-0.620535\pi$
$313$	−13.0809	−0.739375	−0.369687	+	0.929156i	$0.620535\pi$
$314$	1.49881	0.0845828
$315$	0	0
$316$	9.26921	0.521434
$317$	20.6557	1.16014	0.580069	−	0.814568i	$-0.303026\pi$
$317$	20.6557	1.16014	0.580069	+	0.814568i	$0.303026\pi$
$318$	−6.87293	−0.385415
$319$	2.82843	0.158362
$320$	−2.27411	−0.127127
$321$	−11.6482	−0.650141
$322$	0	0
$323$	−35.2299	−1.96025
$324$	1.00000	0.0555556
$325$	−0.309151	−0.0171486
$326$	−13.6569	−0.756383
$327$	14.0890	0.779124
$328$	−5.10254	−0.281740
$329$	0	0
$330$	2.27411	0.125186
$331$	−19.7014	−1.08288	−0.541442	−	0.840738i	$-0.682122\pi$
$331$	−19.7014	−1.08288	−0.541442	+	0.840738i	$0.682122\pi$
$332$	1.96853	0.108037
$333$	4.44078	0.243353
$334$	19.6927	1.07754
$335$	−30.6643	−1.67537
$336$	0	0
$337$	7.88036	0.429271	0.214635	−	0.976694i	$-0.431144\pi$
$337$	7.88036	0.429271	0.214635	+	0.976694i	$0.431144\pi$
$338$	9.75329	0.530509
$339$	5.71979	0.310656
$340$	11.6037	0.629300
$341$	3.14010	0.170046
$342$	−6.90440	−0.373347
$343$	0	0
$344$	−6.93587	−0.373957
$345$	19.5408	1.05204
$346$	−14.5107	−0.780097
$347$	−31.1768	−1.67366	−0.836830	−	0.547463i	$-0.815594\pi$
$347$	−31.1768	−1.67366	−0.836830	+	0.547463i	$0.815594\pi$
$348$	2.82843	0.151620
$349$	−2.96124	−0.158511	−0.0792557	−	0.996854i	$-0.525254\pi$
$349$	−2.96124	−0.158511	−0.0792557	+	0.996854i	$0.525254\pi$
$350$	0	0
$351$	1.80186	0.0961763
$352$	1.00000	0.0533002
$353$	12.9514	0.689335	0.344667	−	0.938725i	$-0.387992\pi$
$353$	12.9514	0.689335	0.344667	+	0.938725i	$0.387992\pi$
$354$	−5.17157	−0.274866
$355$	21.0596	1.11773
$356$	−15.0624	−0.798308
$357$	0	0
$358$	22.3656	1.18206
$359$	25.6621	1.35439	0.677197	−	0.735802i	$-0.263195\pi$
$359$	25.6621	1.35439	0.677197	+	0.735802i	$0.263195\pi$
$360$	2.27411	0.119856
$361$	28.6707	1.50899
$362$	−5.73976	−0.301675
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−6.93469	−0.362978
$366$	−8.63029	−0.451112
$367$	−28.3885	−1.48187	−0.740933	−	0.671578i	$-0.765617\pi$
$367$	−28.3885	−1.48187	−0.740933	+	0.671578i	$0.765617\pi$
$368$	8.59272	0.447927
$369$	5.10254	0.265627
$370$	10.0988	0.525012
$371$	0	0
$372$	3.14010	0.162807
$373$	14.5743	0.754628	0.377314	−	0.926085i	$-0.376848\pi$
$373$	14.5743	0.754628	0.377314	+	0.926085i	$0.376848\pi$
$374$	−5.10254	−0.263846
$375$	−10.9804	−0.567024
$376$	−0.0759718	−0.00391795
$377$	5.09644	0.262480
$378$	0	0
$379$	6.64703	0.341435	0.170718	−	0.985320i	$-0.445391\pi$
$379$	6.64703	0.341435	0.170718	+	0.985320i	$0.445391\pi$
$380$	−15.7014	−0.805463
$381$	−6.60492	−0.338380
$382$	11.7112	0.599195
$383$	−27.7108	−1.41596	−0.707978	−	0.706234i	$-0.750393\pi$
$383$	−27.7108	−1.41596	−0.707978	+	0.706234i	$0.750393\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	12.7533	0.649125
$387$	6.93587	0.352570
$388$	9.01794	0.457816
$389$	−10.6250	−0.538710	−0.269355	−	0.963041i	$-0.586811\pi$
$389$	−10.6250	−0.538710	−0.269355	+	0.963041i	$0.586811\pi$
$390$	4.09763	0.207492
$391$	−43.8447	−2.21732
$392$	0	0
$393$	5.96853	0.301073
$394$	23.2606	1.17185
$395$	−21.0792	−1.06061
$396$	−1.00000	−0.0502519
$397$	−2.14824	−0.107817	−0.0539084	−	0.998546i	$-0.517168\pi$
$397$	−2.14824	−0.107817	−0.0539084	+	0.998546i	$0.517168\pi$
$398$	1.48325	0.0743486
$399$	0	0
$400$	0.171573	0.00857864
$401$	−34.5743	−1.72656	−0.863279	−	0.504727i	$-0.831593\pi$
$401$	−34.5743	−1.72656	−0.863279	+	0.504727i	$0.831593\pi$
$402$	13.4841	0.672525
$403$	5.65804	0.281847
$404$	11.2230	0.558366
$405$	−2.27411	−0.113001
$406$	0	0
$407$	−4.44078	−0.220121
$408$	−5.10254	−0.252613
$409$	−38.5323	−1.90530	−0.952650	−	0.304069i	$-0.901655\pi$
$409$	−38.5323	−1.90530	−0.952650	+	0.304069i	$0.901655\pi$
$410$	11.6037	0.573067
$411$	−11.2606	−0.555443
$412$	−17.0118	−0.838113
$413$	0	0
$414$	−8.59272	−0.422309
$415$	−4.47666	−0.219750
$416$	1.80186	0.0883436
$417$	8.62419	0.422328
$418$	6.90440	0.337705
$419$	−11.7729	−0.575145	−0.287572	−	0.957759i	$-0.592848\pi$
$419$	−11.7729	−0.575145	−0.287572	+	0.957759i	$0.592848\pi$
$420$	0	0
$421$	−14.3039	−0.697129	−0.348564	−	0.937285i	$-0.613331\pi$
$421$	−14.3039	−0.697129	−0.348564	+	0.937285i	$0.613331\pi$
$422$	−0.503715	−0.0245205
$423$	0.0759718	0.00369388
$424$	6.87293	0.333779
$425$	−0.875457	−0.0424659
$426$	−9.26058	−0.448677
$427$	0	0
$428$	11.6482	0.563038
$429$	−1.80186	−0.0869947
$430$	15.7729	0.760638
$431$	0.172768	0.00832195	0.00416098	−	0.999991i	$-0.498676\pi$
$431$	0.172768	0.00832195	0.00416098	+	0.999991i	$0.498676\pi$
$432$	−1.00000	−0.0481125
$433$	−19.7712	−0.950145	−0.475072	−	0.879947i	$-0.657578\pi$
$433$	−19.7712	−0.950145	−0.475072	+	0.879947i	$0.657578\pi$
$434$	0	0
$435$	−6.43215	−0.308398
$436$	−14.0890	−0.674741
$437$	59.3276	2.83802
$438$	3.04941	0.145706
$439$	1.17157	0.0559161	0.0279581	−	0.999609i	$-0.491100\pi$
$439$	1.17157	0.0559161	0.0279581	+	0.999609i	$0.491100\pi$
$440$	−2.27411	−0.108414
$441$	0	0
$442$	−9.19407	−0.437317
$443$	−33.6621	−1.59933	−0.799667	−	0.600443i	$-0.794991\pi$
$443$	−33.6621	−1.59933	−0.799667	+	0.600443i	$0.794991\pi$
$444$	−4.44078	−0.210750
$445$	34.2536	1.62378
$446$	20.0044	0.947236
$447$	12.3211	0.582770
$448$	0	0
$449$	7.27039	0.343111	0.171555	−	0.985174i	$-0.445121\pi$
$449$	7.27039	0.343111	0.171555	+	0.985174i	$0.445121\pi$
$450$	−0.171573	−0.00808802
$451$	−5.10254	−0.240269
$452$	−5.71979	−0.269036
$453$	11.5286	0.541660
$454$	13.5601	0.636405
$455$	0	0
$456$	6.90440	0.323328
$457$	−31.0596	−1.45291	−0.726453	−	0.687216i	$-0.758832\pi$
$457$	−31.0596	−1.45291	−0.726453	+	0.687216i	$0.758832\pi$
$458$	−5.10254	−0.238426
$459$	5.10254	0.238166
$460$	−19.5408	−0.911094
$461$	10.5228	0.490098	0.245049	−	0.969511i	$-0.421196\pi$
$461$	10.5228	0.490098	0.245049	+	0.969511i	$0.421196\pi$
$462$	0	0
$463$	−8.30389	−0.385914	−0.192957	−	0.981207i	$-0.561808\pi$
$463$	−8.30389	−0.385914	−0.192957	+	0.981207i	$0.561808\pi$
$464$	−2.82843	−0.131306
$465$	−7.14094	−0.331153
$466$	0.343146	0.0158959
$467$	−11.5530	−0.534608	−0.267304	−	0.963612i	$-0.586133\pi$
$467$	−11.5530	−0.534608	−0.267304	+	0.963612i	$0.586133\pi$
$468$	−1.80186	−0.0832911
$469$	0	0
$470$	0.172768	0.00796920
$471$	1.49881	0.0690616
$472$	5.17157	0.238041
$473$	−6.93587	−0.318912
$474$	9.26921	0.425749
$475$	1.18461	0.0543535
$476$	0	0
$477$	−6.87293	−0.314690
$478$	18.4322	0.843067
$479$	10.7482	0.491100	0.245550	−	0.969384i	$-0.421032\pi$
$479$	10.7482	0.491100	0.245550	+	0.969384i	$0.421032\pi$
$480$	−2.27411	−0.103798
$481$	−8.00167	−0.364845
$482$	24.5682	1.11905
$483$	0	0
$484$	1.00000	0.0454545
$485$	−20.5078	−0.931210
$486$	1.00000	0.0453609
$487$	−38.6164	−1.74988	−0.874938	−	0.484235i	$-0.839098\pi$
$487$	−38.6164	−1.74988	−0.874938	+	0.484235i	$0.839098\pi$
$488$	8.63029	0.390675
$489$	−13.6569	−0.617584
$490$	0	0
$491$	−8.68629	−0.392007	−0.196003	−	0.980603i	$-0.562796\pi$
$491$	−8.68629	−0.392007	−0.196003	+	0.980603i	$0.562796\pi$
$492$	−5.10254	−0.230040
$493$	14.4322	0.649991
$494$	12.4408	0.559737
$495$	2.27411	0.102214
$496$	−3.14010	−0.140995
$497$	0	0
$498$	1.96853	0.0882120
$499$	43.1041	1.92960	0.964802	−	0.262979i	$-0.0847049\pi$
$499$	43.1041	1.92960	0.964802	+	0.262979i	$0.0847049\pi$
$500$	10.9804	0.491057
$501$	19.6927	0.879806
$502$	1.50491	0.0671674
$503$	24.7174	1.10210	0.551048	−	0.834474i	$-0.314228\pi$
$503$	24.7174	1.10210	0.551048	+	0.834474i	$0.314228\pi$
$504$	0	0
$505$	−25.5224	−1.13573
$506$	8.59272	0.381993
$507$	9.75329	0.433159
$508$	6.60492	0.293046
$509$	−21.4619	−0.951284	−0.475642	−	0.879639i	$-0.657784\pi$
$509$	−21.4619	−0.951284	−0.475642	+	0.879639i	$0.657784\pi$
$510$	11.6037	0.513822
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−6.90440	−0.304837
$514$	10.8476	0.478465
$515$	38.6868	1.70474
$516$	−6.93587	−0.305335
$517$	−0.0759718	−0.00334124
$518$	0	0
$519$	−14.5107	−0.636947
$520$	−4.09763	−0.179693
$521$	19.2773	0.844555	0.422277	−	0.906467i	$-0.361231\pi$
$521$	19.2773	0.844555	0.422277	+	0.906467i	$0.361231\pi$
$522$	2.82843	0.123797
$523$	12.7304	0.556664	0.278332	−	0.960485i	$-0.410219\pi$
$523$	12.7304	0.556664	0.278332	+	0.960485i	$0.410219\pi$
$524$	−5.96853	−0.260737
$525$	0	0
$526$	−6.25938	−0.272922
$527$	16.0225	0.697951
$528$	1.00000	0.0435194
$529$	50.8349	2.21021
$530$	−15.6298	−0.678915
$531$	−5.17157	−0.224427
$532$	0	0
$533$	−9.19407	−0.398239
$534$	−15.0624	−0.651816
$535$	−26.4893	−1.14523
$536$	−13.4841	−0.582424
$537$	22.3656	0.965149
$538$	14.8345	0.639562
$539$	0	0
$540$	2.27411	0.0978621
$541$	−7.36684	−0.316725	−0.158363	−	0.987381i	$-0.550621\pi$
$541$	−7.36684	−0.316725	−0.158363	+	0.987381i	$0.550621\pi$
$542$	−2.91743	−0.125315
$543$	−5.73976	−0.246317
$544$	5.10254	0.218769
$545$	32.0399	1.37244
$546$	0	0
$547$	13.7320	0.587138	0.293569	−	0.955938i	$-0.405157\pi$
$547$	13.7320	0.587138	0.293569	+	0.955938i	$0.405157\pi$
$548$	11.2606	0.481028
$549$	−8.63029	−0.368332
$550$	0.171573	0.00731589
$551$	−19.5286	−0.831946
$552$	8.59272	0.365731
$553$	0	0
$554$	−12.1012	−0.514131
$555$	10.0988	0.428671
$556$	−8.62419	−0.365747
$557$	0.845679	0.0358326	0.0179163	−	0.999839i	$-0.494297\pi$
$557$	0.845679	0.0358326	0.0179163	+	0.999839i	$0.494297\pi$
$558$	3.14010	0.132931
$559$	−12.4975	−0.528587
$560$	0	0
$561$	−5.10254	−0.215429
$562$	−7.74586	−0.326739
$563$	18.1009	0.762860	0.381430	−	0.924398i	$-0.375432\pi$
$563$	18.1009	0.762860	0.381430	+	0.924398i	$0.375432\pi$
$564$	−0.0759718	−0.00319899
$565$	13.0074	0.547227
$566$	17.5948	0.739563
$567$	0	0
$568$	9.26058	0.388565
$569$	−34.4819	−1.44556	−0.722778	−	0.691080i	$-0.757135\pi$
$569$	−34.4819	−1.44556	−0.722778	+	0.691080i	$0.757135\pi$
$570$	−15.7014	−0.657657
$571$	−35.6927	−1.49369	−0.746847	−	0.664996i	$-0.768433\pi$
$571$	−35.6927	−1.49369	−0.746847	+	0.664996i	$0.768433\pi$
$572$	1.80186	0.0753397
$573$	11.7112	0.489241
$574$	0	0
$575$	1.47428	0.0614816
$576$	1.00000	0.0416667
$577$	1.64129	0.0683279	0.0341640	−	0.999416i	$-0.489123\pi$
$577$	1.64129	0.0683279	0.0341640	+	0.999416i	$0.489123\pi$
$578$	−9.03588	−0.375843
$579$	12.7533	0.530009
$580$	6.43215	0.267081
$581$	0	0
$582$	9.01794	0.373806
$583$	6.87293	0.284648
$584$	−3.04941	−0.126185
$585$	4.09763	0.169416
$586$	−16.1675	−0.667873
$587$	42.4582	1.75244	0.876219	−	0.481913i	$-0.160058\pi$
$587$	42.4582	1.75244	0.876219	+	0.481913i	$0.160058\pi$
$588$	0	0
$589$	−21.6805	−0.893331
$590$	−11.7607	−0.484181
$591$	23.2606	0.956812
$592$	4.44078	0.182515
$593$	23.4457	0.962799	0.481399	−	0.876501i	$-0.340129\pi$
$593$	23.4457	0.962799	0.481399	+	0.876501i	$0.340129\pi$
$594$	−1.00000	−0.0410305
$595$	0	0
$596$	−12.3211	−0.504694
$597$	1.48325	0.0607054
$598$	15.4829	0.633143
$599$	47.4166	1.93739	0.968695	−	0.248256i	$-0.0798573\pi$
$599$	47.4166	1.93739	0.968695	+	0.248256i	$0.0798573\pi$
$600$	0.171573	0.00700443
$601$	31.9570	1.30355	0.651777	−	0.758410i	$-0.274024\pi$
$601$	31.9570	1.30355	0.651777	+	0.758410i	$0.274024\pi$
$602$	0	0
$603$	13.4841	0.549114
$604$	−11.5286	−0.469092
$605$	−2.27411	−0.0924557
$606$	11.2230	0.455904
$607$	21.0596	0.854782	0.427391	−	0.904067i	$-0.359433\pi$
$607$	21.0596	0.854782	0.427391	+	0.904067i	$0.359433\pi$
$608$	−6.90440	−0.280010
$609$	0	0
$610$	−19.6262	−0.794642
$611$	−0.136891	−0.00553801
$612$	−5.10254	−0.206258
$613$	22.6013	0.912860	0.456430	−	0.889759i	$-0.349128\pi$
$613$	22.6013	0.912860	0.456430	+	0.889759i	$0.349128\pi$
$614$	−27.0563	−1.09191
$615$	11.6037	0.467907
$616$	0	0
$617$	−29.6447	−1.19345	−0.596724	−	0.802446i	$-0.703531\pi$
$617$	−29.6447	−1.19345	−0.596724	+	0.802446i	$0.703531\pi$
$618$	−17.0118	−0.684317
$619$	−1.10864	−0.0445598	−0.0222799	−	0.999752i	$-0.507093\pi$
$619$	−1.10864	−0.0445598	−0.0222799	+	0.999752i	$0.507093\pi$
$620$	7.14094	0.286787
$621$	−8.59272	−0.344814
$622$	−24.5751	−0.985373
$623$	0	0
$624$	1.80186	0.0721322
$625$	−25.8284	−1.03314
$626$	13.0809	0.522817
$627$	6.90440	0.275735
$628$	−1.49881	−0.0598091
$629$	−22.6592	−0.903483
$630$	0	0
$631$	−8.76192	−0.348806	−0.174403	−	0.984674i	$-0.555800\pi$
$631$	−8.76192	−0.348806	−0.174403	+	0.984674i	$0.555800\pi$
$632$	−9.26921	−0.368709
$633$	−0.503715	−0.0200209
$634$	−20.6557	−0.820341
$635$	−15.0203	−0.596063
$636$	6.87293	0.272529
$637$	0	0
$638$	−2.82843	−0.111979
$639$	−9.26058	−0.366343
$640$	2.27411	0.0898921
$641$	47.2474	1.86616	0.933080	−	0.359669i	$-0.117110\pi$
$641$	47.2474	1.86616	0.933080	+	0.359669i	$0.117110\pi$
$642$	11.6482	0.459719
$643$	−4.71741	−0.186037	−0.0930183	−	0.995664i	$-0.529652\pi$
$643$	−4.71741	−0.186037	−0.0930183	+	0.995664i	$0.529652\pi$
$644$	0	0
$645$	15.7729	0.621058
$646$	35.2299	1.38610
$647$	28.0391	1.10233	0.551165	−	0.834396i	$-0.314183\pi$
$647$	28.0391	1.10233	0.551165	+	0.834396i	$0.314183\pi$
$648$	−1.00000	−0.0392837
$649$	5.17157	0.203002
$650$	0.309151	0.0121259
$651$	0	0
$652$	13.6569	0.534844
$653$	27.0596	1.05892	0.529461	−	0.848334i	$-0.322394\pi$
$653$	27.0596	1.05892	0.529461	+	0.848334i	$0.322394\pi$
$654$	−14.0890	−0.550924
$655$	13.5731	0.530345
$656$	5.10254	0.199221
$657$	3.04941	0.118969
$658$	0	0
$659$	15.7741	0.614472	0.307236	−	0.951633i	$-0.400596\pi$
$659$	15.7741	0.614472	0.307236	+	0.951633i	$0.400596\pi$
$660$	−2.27411	−0.0885196
$661$	−20.1597	−0.784122	−0.392061	−	0.919939i	$-0.628238\pi$
$661$	−20.1597	−0.784122	−0.392061	+	0.919939i	$0.628238\pi$
$662$	19.7014	0.765715
$663$	−9.19407	−0.357068
$664$	−1.96853	−0.0763938
$665$	0	0
$666$	−4.44078	−0.172077
$667$	−24.3039	−0.941050
$668$	−19.6927	−0.761935
$669$	20.0044	0.773415
$670$	30.6643	1.18466
$671$	8.63029	0.333169
$672$	0	0
$673$	7.77410	0.299670	0.149835	−	0.988711i	$-0.452126\pi$
$673$	7.77410	0.299670	0.149835	+	0.988711i	$0.452126\pi$
$674$	−7.88036	−0.303540
$675$	−0.171573	−0.00660384
$676$	−9.75329	−0.375127
$677$	−49.6568	−1.90847	−0.954234	−	0.299062i	$-0.903326\pi$
$677$	−49.6568	−1.90847	−0.954234	+	0.299062i	$0.903326\pi$
$678$	−5.71979	−0.219667
$679$	0	0
$680$	−11.6037	−0.444983
$681$	13.5601	0.519623
$682$	−3.14010	−0.120241
$683$	−22.9261	−0.877241	−0.438621	−	0.898672i	$-0.644533\pi$
$683$	−22.9261	−0.877241	−0.438621	+	0.898672i	$0.644533\pi$
$684$	6.90440	0.263996
$685$	−25.6078	−0.978423
$686$	0	0
$687$	−5.10254	−0.194674
$688$	6.93587	0.264427
$689$	12.3841	0.471796
$690$	−19.5408	−0.743905
$691$	34.1299	1.29836	0.649182	−	0.760633i	$-0.275111\pi$
$691$	34.1299	1.29836	0.649182	+	0.760633i	$0.275111\pi$
$692$	14.5107	0.551612
$693$	0	0
$694$	31.1768	1.18346
$695$	19.6124	0.743939
$696$	−2.82843	−0.107211
$697$	−26.0359	−0.986179
$698$	2.96124	0.112084
$699$	0.343146	0.0129790
$700$	0	0
$701$	12.1929	0.460518	0.230259	−	0.973129i	$-0.426043\pi$
$701$	12.1929	0.460518	0.230259	+	0.973129i	$0.426043\pi$
$702$	−1.80186	−0.0680069
$703$	30.6609	1.15640
$704$	−1.00000	−0.0376889
$705$	0.172768	0.00650683
$706$	−12.9514	−0.487433
$707$	0	0
$708$	5.17157	0.194360
$709$	−21.0902	−0.792059	−0.396030	−	0.918238i	$-0.629612\pi$
$709$	−21.0902	−0.792059	−0.396030	+	0.918238i	$0.629612\pi$
$710$	−21.0596	−0.790352
$711$	9.26921	0.347622
$712$	15.0624	0.564489
$713$	−26.9820	−1.01049
$714$	0	0
$715$	−4.09763	−0.153243
$716$	−22.3656	−0.835843
$717$	18.4322	0.688361
$718$	−25.6621	−0.957701
$719$	22.7403	0.848068	0.424034	−	0.905646i	$-0.360614\pi$
$719$	22.7403	0.848068	0.424034	+	0.905646i	$0.360614\pi$
$720$	−2.27411	−0.0847510
$721$	0	0
$722$	−28.6707	−1.06701
$723$	24.5682	0.913701
$724$	5.73976	0.213317
$725$	−0.485281	−0.0180229
$726$	1.00000	0.0371135
$727$	1.83046	0.0678879	0.0339440	−	0.999424i	$-0.489193\pi$
$727$	1.83046	0.0678879	0.0339440	+	0.999424i	$0.489193\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	6.93469	0.256664
$731$	−35.3905	−1.30897
$732$	8.63029	0.318985
$733$	36.0191	1.33040	0.665199	−	0.746667i	$-0.268347\pi$
$733$	36.0191	1.33040	0.665199	+	0.746667i	$0.268347\pi$
$734$	28.3885	1.04784
$735$	0	0
$736$	−8.59272	−0.316732
$737$	−13.4841	−0.496693
$738$	−5.10254	−0.187827
$739$	26.5719	0.977463	0.488732	−	0.872434i	$-0.337460\pi$
$739$	26.5719	0.977463	0.488732	+	0.872434i	$0.337460\pi$
$740$	−10.0988	−0.371240
$741$	12.4408	0.457023
$742$	0	0
$743$	−10.5580	−0.387336	−0.193668	−	0.981067i	$-0.562039\pi$
$743$	−10.5580	−0.387336	−0.193668	+	0.981067i	$0.562039\pi$
$744$	−3.14010	−0.115122
$745$	28.0196	1.02656
$746$	−14.5743	−0.533603
$747$	1.96853	0.0720248
$748$	5.10254	0.186567
$749$	0	0
$750$	10.9804	0.400946
$751$	−31.3003	−1.14217	−0.571083	−	0.820893i	$-0.693476\pi$
$751$	−31.3003	−1.14217	−0.571083	+	0.820893i	$0.693476\pi$
$752$	0.0759718	0.00277041
$753$	1.50491	0.0548420
$754$	−5.09644	−0.185601
$755$	26.2173	0.954144
$756$	0	0
$757$	−21.3223	−0.774973	−0.387487	−	0.921875i	$-0.626657\pi$
$757$	−21.3223	−0.774973	−0.387487	+	0.921875i	$0.626657\pi$
$758$	−6.64703	−0.241431
$759$	8.59272	0.311896
$760$	15.7014	0.569548
$761$	6.48899	0.235226	0.117613	−	0.993060i	$-0.462476\pi$
$761$	6.48899	0.235226	0.117613	+	0.993060i	$0.462476\pi$
$762$	6.60492	0.239271
$763$	0	0
$764$	−11.7112	−0.423695
$765$	11.6037	0.419534
$766$	27.7108	1.00123
$767$	9.31846	0.336470
$768$	−1.00000	−0.0360844
$769$	41.9719	1.51355	0.756773	−	0.653678i	$-0.226775\pi$
$769$	41.9719	1.51355	0.756773	+	0.653678i	$0.226775\pi$
$770$	0	0
$771$	10.8476	0.390665
$772$	−12.7533	−0.459001
$773$	29.8951	1.07525	0.537626	−	0.843184i	$-0.319321\pi$
$773$	29.8951	1.07525	0.537626	+	0.843184i	$0.319321\pi$
$774$	−6.93587	−0.249305
$775$	−0.538757	−0.0193527
$776$	−9.01794	−0.323725
$777$	0	0
$778$	10.6250	0.380926
$779$	35.2299	1.26224
$780$	−4.09763	−0.146719
$781$	9.26058	0.331370
$782$	43.8447	1.56788
$783$	2.82843	0.101080
$784$	0	0
$785$	3.40846	0.121653
$786$	−5.96853	−0.212890
$787$	47.3602	1.68821	0.844105	−	0.536178i	$-0.180132\pi$
$787$	47.3602	1.68821	0.844105	+	0.536178i	$0.180132\pi$
$788$	−23.2606	−0.828624
$789$	−6.25938	−0.222840
$790$	21.0792	0.749964
$791$	0	0
$792$	1.00000	0.0355335
$793$	15.5506	0.552218
$794$	2.14824	0.0762380
$795$	−15.6298	−0.554332
$796$	−1.48325	−0.0525724
$797$	−26.7421	−0.947255	−0.473628	−	0.880725i	$-0.657056\pi$
$797$	−26.7421	−0.947255	−0.473628	+	0.880725i	$0.657056\pi$
$798$	0	0
$799$	−0.387649	−0.0137140
$800$	−0.171573	−0.00606602
$801$	−15.0624	−0.532205
$802$	34.5743	1.22086
$803$	−3.04941	−0.107611
$804$	−13.4841	−0.475547
$805$	0	0
$806$	−5.65804	−0.199296
$807$	14.8345	0.522200
$808$	−11.2230	−0.394824
$809$	−9.75924	−0.343117	−0.171558	−	0.985174i	$-0.554880\pi$
$809$	−9.75924	−0.343117	−0.171558	+	0.985174i	$0.554880\pi$
$810$	2.27411	0.0799041
$811$	43.2255	1.51785	0.758927	−	0.651176i	$-0.225724\pi$
$811$	43.2255	1.51785	0.758927	+	0.651176i	$0.225724\pi$
$812$	0	0
$813$	−2.91743	−0.102319
$814$	4.44078	0.155649
$815$	−31.0572	−1.08789
$816$	5.10254	0.178625
$817$	47.8880	1.67539
$818$	38.5323	1.34725
$819$	0	0
$820$	−11.6037	−0.405220
$821$	−15.9631	−0.557117	−0.278559	−	0.960419i	$-0.589857\pi$
$821$	−15.9631	−0.557117	−0.278559	+	0.960419i	$0.589857\pi$
$822$	11.2606	0.392758
$823$	−51.4980	−1.79511	−0.897553	−	0.440907i	$-0.854657\pi$
$823$	−51.4980	−1.79511	−0.897553	+	0.440907i	$0.854657\pi$
$824$	17.0118	0.592636
$825$	0.171573	0.00597340
$826$	0	0
$827$	−15.7679	−0.548302	−0.274151	−	0.961687i	$-0.588397\pi$
$827$	−15.7679	−0.548302	−0.274151	+	0.961687i	$0.588397\pi$
$828$	8.59272	0.298618
$829$	47.0437	1.63389	0.816947	−	0.576713i	$-0.195665\pi$
$829$	47.0437	1.63389	0.816947	+	0.576713i	$0.195665\pi$
$830$	4.47666	0.155387
$831$	−12.1012	−0.419786
$832$	−1.80186	−0.0624683
$833$	0	0
$834$	−8.62419	−0.298631
$835$	44.7834	1.54979
$836$	−6.90440	−0.238794
$837$	3.14010	0.108538
$838$	11.7729	0.406689
$839$	−27.6046	−0.953015	−0.476508	−	0.879170i	$-0.658098\pi$
$839$	−27.6046	−0.953015	−0.476508	+	0.879170i	$0.658098\pi$
$840$	0	0
$841$	−21.0000	−0.724138
$842$	14.3039	0.492945
$843$	−7.74586	−0.266782
$844$	0.503715	0.0173386
$845$	22.1800	0.763017
$846$	−0.0759718	−0.00261197
$847$	0	0
$848$	−6.87293	−0.236017
$849$	17.5948	0.603850
$850$	0.875457	0.0300279
$851$	38.1584	1.30805
$852$	9.26058	0.317262
$853$	47.4171	1.62353	0.811765	−	0.583985i	$-0.198507\pi$
$853$	47.4171	1.62353	0.811765	+	0.583985i	$0.198507\pi$
$854$	0	0
$855$	−15.7014	−0.536975
$856$	−11.6482	−0.398128
$857$	57.6474	1.96920	0.984598	−	0.174831i	$-0.0559380\pi$
$857$	57.6474	1.96920	0.984598	+	0.174831i	$0.0559380\pi$
$858$	1.80186	0.0615146
$859$	−33.3865	−1.13913	−0.569566	−	0.821946i	$-0.692889\pi$
$859$	−33.3865	−1.13913	−0.569566	+	0.821946i	$0.692889\pi$
$860$	−15.7729	−0.537852
$861$	0	0
$862$	−0.172768	−0.00588451
$863$	−7.13113	−0.242747	−0.121373	−	0.992607i	$-0.538730\pi$
$863$	−7.13113	−0.242747	−0.121373	+	0.992607i	$0.538730\pi$
$864$	1.00000	0.0340207
$865$	−32.9988	−1.12199
$866$	19.7712	0.671854
$867$	−9.03588	−0.306874
$868$	0	0
$869$	−9.26921	−0.314436
$870$	6.43215	0.218070
$871$	−24.2965	−0.823254
$872$	14.0890	0.477114
$873$	9.01794	0.305211
$874$	−59.3276	−2.00679
$875$	0	0
$876$	−3.04941	−0.103030
$877$	−9.46398	−0.319576	−0.159788	−	0.987151i	$-0.551081\pi$
$877$	−9.46398	−0.319576	−0.159788	+	0.987151i	$0.551081\pi$
$878$	−1.17157	−0.0395387
$879$	−16.1675	−0.545316
$880$	2.27411	0.0766602
$881$	−18.7846	−0.632870	−0.316435	−	0.948614i	$-0.602486\pi$
$881$	−18.7846	−0.632870	−0.316435	+	0.948614i	$0.602486\pi$
$882$	0	0
$883$	32.2843	1.08645	0.543226	−	0.839586i	$-0.317203\pi$
$883$	32.2843	1.08645	0.543226	+	0.839586i	$0.317203\pi$
$884$	9.19407	0.309230
$885$	−11.7607	−0.395332
$886$	33.6621	1.13090
$887$	−10.5384	−0.353845	−0.176923	−	0.984225i	$-0.556614\pi$
$887$	−10.5384	−0.353845	−0.176923	+	0.984225i	$0.556614\pi$
$888$	4.44078	0.149023
$889$	0	0
$890$	−34.2536	−1.14818
$891$	−1.00000	−0.0335013
$892$	−20.0044	−0.669797
$893$	0.524540	0.0175531
$894$	−12.3211	−0.412081
$895$	50.8619	1.70013
$896$	0	0
$897$	15.4829	0.516959
$898$	−7.27039	−0.242616
$899$	8.88156	0.296216
$900$	0.171573	0.00571910
$901$	35.0694	1.16833
$902$	5.10254	0.169896
$903$	0	0
$904$	5.71979	0.190237
$905$	−13.0528	−0.433891
$906$	−11.5286	−0.383012
$907$	−2.49866	−0.0829667	−0.0414834	−	0.999139i	$-0.513208\pi$
$907$	−2.49866	−0.0829667	−0.0414834	+	0.999139i	$0.513208\pi$
$908$	−13.5601	−0.450006
$909$	11.2230	0.372244
$910$	0	0
$911$	−14.4645	−0.479229	−0.239614	−	0.970868i	$-0.577021\pi$
$911$	−14.4645	−0.479229	−0.239614	+	0.970868i	$0.577021\pi$
$912$	−6.90440	−0.228627
$913$	−1.96853	−0.0651489
$914$	31.0596	1.02736
$915$	−19.6262	−0.648823
$916$	5.10254	0.168593
$917$	0	0
$918$	−5.10254	−0.168409
$919$	−3.29646	−0.108740	−0.0543700	−	0.998521i	$-0.517315\pi$
$919$	−3.29646	−0.108740	−0.0543700	+	0.998521i	$0.517315\pi$
$920$	19.5408	0.644241
$921$	−27.0563	−0.891537
$922$	−10.5228	−0.346552
$923$	16.6863	0.549236
$924$	0	0
$925$	0.761917	0.0250517
$926$	8.30389	0.272883
$927$	−17.0118	−0.558742
$928$	2.82843	0.0928477
$929$	29.7650	0.976558	0.488279	−	0.872688i	$-0.337625\pi$
$929$	29.7650	0.976558	0.488279	+	0.872688i	$0.337625\pi$
$930$	7.14094	0.234161
$931$	0	0
$932$	−0.343146	−0.0112401
$933$	−24.5751	−0.804553
$934$	11.5530	0.378025
$935$	−11.6037	−0.379482
$936$	1.80186	0.0588957
$937$	−42.4521	−1.38685	−0.693425	−	0.720529i	$-0.743899\pi$
$937$	−42.4521	−1.38685	−0.693425	+	0.720529i	$0.743899\pi$
$938$	0	0
$939$	13.0809	0.426878
$940$	−0.172768	−0.00563508
$941$	−34.4494	−1.12302	−0.561509	−	0.827471i	$-0.689779\pi$
$941$	−34.4494	−1.12302	−0.561509	+	0.827471i	$0.689779\pi$
$942$	−1.49881	−0.0488339
$943$	43.8447	1.42778
$944$	−5.17157	−0.168320
$945$	0	0
$946$	6.93587	0.225505
$947$	−51.3611	−1.66901	−0.834505	−	0.551000i	$-0.814246\pi$
$947$	−51.3611	−1.66901	−0.834505	+	0.551000i	$0.814246\pi$
$948$	−9.26921	−0.301050
$949$	−5.49461	−0.178363
$950$	−1.18461	−0.0384337
$951$	−20.6557	−0.669806
$952$	0	0
$953$	50.4015	1.63267	0.816333	−	0.577582i	$-0.196004\pi$
$953$	50.4015	1.63267	0.816333	+	0.577582i	$0.196004\pi$
$954$	6.87293	0.222519
$955$	26.6325	0.861806
$956$	−18.4322	−0.596138
$957$	−2.82843	−0.0914301
$958$	−10.7482	−0.347260
$959$	0	0
$960$	2.27411	0.0733966
$961$	−21.1397	−0.681927
$962$	8.00167	0.257984
$963$	11.6482	0.375359
$964$	−24.5682	−0.791288
$965$	29.0024	0.933620
$966$	0	0
$967$	30.9141	0.994129	0.497064	−	0.867714i	$-0.334411\pi$
$967$	30.9141	0.994129	0.497064	+	0.867714i	$0.334411\pi$
$968$	−1.00000	−0.0321412
$969$	35.2299	1.13175
$970$	20.5078	0.658465
$971$	25.2623	0.810704	0.405352	−	0.914161i	$-0.367149\pi$
$971$	25.2623	0.810704	0.405352	+	0.914161i	$0.367149\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	38.6164	1.23735
$975$	0.309151	0.00990075
$976$	−8.63029	−0.276249
$977$	−8.97798	−0.287231	−0.143616	−	0.989634i	$-0.545873\pi$
$977$	−8.97798	−0.287231	−0.143616	+	0.989634i	$0.545873\pi$
$978$	13.6569	0.436698
$979$	15.0624	0.481398
$980$	0	0
$981$	−14.0890	−0.449827
$982$	8.68629	0.277191
$983$	−19.2638	−0.614420	−0.307210	−	0.951642i	$-0.599395\pi$
$983$	−19.2638	−0.614420	−0.307210	+	0.951642i	$0.599395\pi$
$984$	5.10254	0.162663
$985$	52.8971	1.68544
$986$	−14.4322	−0.459613
$987$	0	0
$988$	−12.4408	−0.395794
$989$	59.5980	1.89511
$990$	−2.27411	−0.0722759
$991$	−46.5858	−1.47985	−0.739923	−	0.672692i	$-0.765138\pi$
$991$	−46.5858	−1.47985	−0.739923	+	0.672692i	$0.765138\pi$
$992$	3.14010	0.0996984
$993$	19.7014	0.625204
$994$	0	0
$995$	3.37307	0.106934
$996$	−1.96853	−0.0623753
$997$	−31.0330	−0.982825	−0.491412	−	0.870927i	$-0.663519\pi$
$997$	−31.0330	−0.982825	−0.491412	+	0.870927i	$0.663519\pi$
$998$	−43.1041	−1.36444
$999$	−4.44078	−0.140500

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3234.2.a.bj.1.2	✓	4
3.2	odd	2		9702.2.a.eb.1.3		4
7.6	odd	2		3234.2.a.bk.1.3	yes	4
21.20	even	2		9702.2.a.ec.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3234.2.a.bj.1.2	✓	4	1.1	even	1	trivial
3234.2.a.bk.1.3	yes	4	7.6	odd	2
9702.2.a.eb.1.3		4	3.2	odd	2
9702.2.a.ec.1.2		4	21.20	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3234.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.27411 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.27411 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +2.27411 q^{10} -1.00000 q^{11} -1.00000 q^{12} -1.80186 q^{13} +2.27411 q^{15} +1.00000 q^{16} -5.10254 q^{17} -1.00000 q^{18} +6.90440 q^{19} -2.27411 q^{20} +1.00000 q^{22} +8.59272 q^{23} +1.00000 q^{24} +0.171573 q^{25} +1.80186 q^{26} -1.00000 q^{27} -2.82843 q^{29} -2.27411 q^{30} -3.14010 q^{31} -1.00000 q^{32} +1.00000 q^{33} +5.10254 q^{34} +1.00000 q^{36} +4.44078 q^{37} -6.90440 q^{38} +1.80186 q^{39} +2.27411 q^{40} +5.10254 q^{41} +6.93587 q^{43} -1.00000 q^{44} -2.27411 q^{45} -8.59272 q^{46} +0.0759718 q^{47} -1.00000 q^{48} -0.171573 q^{50} +5.10254 q^{51} -1.80186 q^{52} -6.87293 q^{53} +1.00000 q^{54} +2.27411 q^{55} -6.90440 q^{57} +2.82843 q^{58} -5.17157 q^{59} +2.27411 q^{60} -8.63029 q^{61} +3.14010 q^{62} +1.00000 q^{64} +4.09763 q^{65} -1.00000 q^{66} +13.4841 q^{67} -5.10254 q^{68} -8.59272 q^{69} -9.26058 q^{71} -1.00000 q^{72} +3.04941 q^{73} -4.44078 q^{74} -0.171573 q^{75} +6.90440 q^{76} -1.80186 q^{78} +9.26921 q^{79} -2.27411 q^{80} +1.00000 q^{81} -5.10254 q^{82} +1.96853 q^{83} +11.6037 q^{85} -6.93587 q^{86} +2.82843 q^{87} +1.00000 q^{88} -15.0624 q^{89} +2.27411 q^{90} +8.59272 q^{92} +3.14010 q^{93} -0.0759718 q^{94} -15.7014 q^{95} +1.00000 q^{96} +9.01794 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{6} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 4 * q^6 - 4 * q^8 + 4 * q^9	\( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{6} - 4 q^{8} + 4 q^{9} - 4 q^{11} - 4 q^{12} + 4 q^{16} - 4 q^{18} + 4 q^{22} - 8 q^{23} + 4 q^{24} + 12 q^{25} - 4 q^{27} - 16 q^{31} - 4 q^{32} + 4 q^{33} + 4 q^{36} + 8 q^{37} + 8 q^{43} - 4 q^{44} + 8 q^{46} - 16 q^{47} - 4 q^{48} - 12 q^{50} + 8 q^{53} + 4 q^{54} - 32 q^{59} - 16 q^{61} + 16 q^{62} + 4 q^{64} - 16 q^{65} - 4 q^{66} + 16 q^{67} + 8 q^{69} - 4 q^{72} - 8 q^{74} - 12 q^{75} + 16 q^{79} + 4 q^{81} + 32 q^{85} - 8 q^{86} + 4 q^{88} - 16 q^{89} - 8 q^{92} + 16 q^{93} + 16 q^{94} - 16 q^{95} + 4 q^{96} + 16 q^{97} - 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 4 * q^6 - 4 * q^8 + 4 * q^9 - 4 * q^11 - 4 * q^12 + 4 * q^16 - 4 * q^18 + 4 * q^22 - 8 * q^23 + 4 * q^24 + 12 * q^25 - 4 * q^27 - 16 * q^31 - 4 * q^32 + 4 * q^33 + 4 * q^36 + 8 * q^37 + 8 * q^43 - 4 * q^44 + 8 * q^46 - 16 * q^47 - 4 * q^48 - 12 * q^50 + 8 * q^53 + 4 * q^54 - 32 * q^59 - 16 * q^61 + 16 * q^62 + 4 * q^64 - 16 * q^65 - 4 * q^66 + 16 * q^67 + 8 * q^69 - 4 * q^72 - 8 * q^74 - 12 * q^75 + 16 * q^79 + 4 * q^81 + 32 * q^85 - 8 * q^86 + 4 * q^88 - 16 * q^89 - 8 * q^92 + 16 * q^93 + 16 * q^94 - 16 * q^95 + 4 * q^96 + 16 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

Properties

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