LMFDB - Embedded newform 9610.2.a.bz.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9610,2,Mod(1,9610)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9610, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("9610.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9610 = 2 \cdot 5 \cdot 31^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9610.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:	$76.7362363425$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.14623232.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 13x^{4} + 14x^{3} + 26x^{2} - 28x + 1 $ x^6 - 2x^5 - 13x^4 + 14x^3 + 26x^2 - 28*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$4.06531$ of defining polynomial
Character	$\chi$	$=$	9610.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +2.33500 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.33500 q^{6} +2.54778 q^{7} +1.00000 q^{8} +2.45222 q^{9} +1.00000 q^{10} +3.74921 q^{11} +2.33500 q^{12} -2.97453 q^{13} +2.54778 q^{14} +2.33500 q^{15} +1.00000 q^{16} +2.45222 q^{18} -4.20662 q^{19} +1.00000 q^{20} +5.94907 q^{21} +3.74921 q^{22} +6.08421 q^{23} +2.33500 q^{24} +1.00000 q^{25} -2.97453 q^{26} -1.27907 q^{27} +2.54778 q^{28} +0.493428 q^{29} +2.33500 q^{30} +1.00000 q^{32} +8.75441 q^{33} +2.54778 q^{35} +2.45222 q^{36} +5.59078 q^{37} -4.20662 q^{38} -6.94553 q^{39} +1.00000 q^{40} +6.54778 q^{41} +5.94907 q^{42} +5.16343 q^{43} +3.74921 q^{44} +2.45222 q^{45} +6.08421 q^{46} +6.05659 q^{47} +2.33500 q^{48} -0.508811 q^{49} +1.00000 q^{50} -2.97453 q^{52} -7.64453 q^{53} -1.27907 q^{54} +3.74921 q^{55} +2.54778 q^{56} -9.82246 q^{57} +0.493428 q^{58} +5.30219 q^{59} +2.33500 q^{60} -13.6487 q^{61} +6.24772 q^{63} +1.00000 q^{64} -2.97453 q^{65} +8.75441 q^{66} +3.30219 q^{67} +14.2066 q^{69} +2.54778 q^{70} -5.50881 q^{71} +2.45222 q^{72} -11.3937 q^{73} +5.59078 q^{74} +2.33500 q^{75} -4.20662 q^{76} +9.55217 q^{77} -6.94553 q^{78} +12.6728 q^{79} +1.00000 q^{80} -10.3433 q^{81} +6.54778 q^{82} -13.9409 q^{83} +5.94907 q^{84} +5.16343 q^{86} +1.15215 q^{87} +3.74921 q^{88} -11.4489 q^{89} +2.45222 q^{90} -7.57846 q^{91} +6.08421 q^{92} +6.05659 q^{94} -4.20662 q^{95} +2.33500 q^{96} +1.24559 q^{97} -0.508811 q^{98} +9.19389 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} + 6 q^{4} + 6 q^{5} + 8 q^{7} + 6 q^{8} + 22 q^{9} + 6 q^{10} + 8 q^{14} + 6 q^{16} + 22 q^{18} - 12 q^{19} + 6 q^{20} + 6 q^{25} + 8 q^{28} + 6 q^{32} + 32 q^{33} + 8 q^{35} + 22 q^{36} - 12 q^{38}+ \cdots + 38 q^{98}+O(q^{100}) $ 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^5 + 8 * q^7 + 6 * q^8 + 22 * q^9 + 6 * q^10 + 8 * q^14 + 6 * q^16 + 22 * q^18 - 12 * q^19 + 6 * q^20 + 6 * q^25 + 8 * q^28 + 6 * q^32 + 32 * q^33 + 8 * q^35 + 22 * q^36 - 12 * q^38 + 8 * q^39 + 6 * q^40 + 32 * q^41 + 22 * q^45 - 12 * q^47 + 38 * q^49 + 6 * q^50 + 8 * q^56 + 4 * q^59 - 40 * q^63 + 6 * q^64 + 32 * q^66 - 8 * q^67 + 72 * q^69 + 8 * q^70 + 8 * q^71 + 22 * q^72 - 12 * q^76 + 8 * q^78 + 6 * q^80 + 30 * q^81 + 32 * q^82 - 56 * q^87 + 22 * q^90 - 12 * q^94 - 12 * q^95 + 28 * q^97 + 38 * q^98

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
$3$	2.33500	1.34811	0.674056	−	0.738680i	$-0.264551\pi$
$3$	2.33500	1.34811	0.674056	+	0.738680i	$0.264551\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	2.33500	0.953259
$7$	2.54778	0.962971	0.481485	−	0.876454i	$-0.340097\pi$
$7$	2.54778	0.962971	0.481485	+	0.876454i	$0.340097\pi$
$8$	1.00000	0.353553
$9$	2.45222	0.817406
$10$	1.00000	0.316228
$11$	3.74921	1.13043	0.565215	−	0.824944i	$-0.308793\pi$
$11$	3.74921	1.13043	0.565215	+	0.824944i	$0.308793\pi$
$12$	2.33500	0.674056
$13$	−2.97453	−0.824987	−0.412493	−	0.910961i	$-0.635342\pi$
$13$	−2.97453	−0.824987	−0.412493	+	0.910961i	$0.635342\pi$
$14$	2.54778	0.680923
$15$	2.33500	0.602894
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	2.45222	0.577994
$19$	−4.20662	−0.965066	−0.482533	−	0.875878i	$-0.660283\pi$
$19$	−4.20662	−0.965066	−0.482533	+	0.875878i	$0.660283\pi$
$20$	1.00000	0.223607
$21$	5.94907	1.29819
$22$	3.74921	0.799335
$23$	6.08421	1.26865	0.634323	−	0.773068i	$-0.281279\pi$
$23$	6.08421	1.26865	0.634323	+	0.773068i	$0.281279\pi$
$24$	2.33500	0.476630
$25$	1.00000	0.200000
$26$	−2.97453	−0.583354
$27$	−1.27907	−0.246157
$28$	2.54778	0.481485
$29$	0.493428	0.0916274	0.0458137	−	0.998950i	$-0.485412\pi$
$29$	0.493428	0.0916274	0.0458137	+	0.998950i	$0.485412\pi$
$30$	2.33500	0.426310
$31$	0	0
$31$	0	0
$32$	1.00000	0.176777
$33$	8.75441	1.52395
$34$	0	0
$35$	2.54778	0.430654
$36$	2.45222	0.408703
$37$	5.59078	0.919119	0.459559	−	0.888147i	$-0.348007\pi$
$37$	5.59078	0.919119	0.459559	+	0.888147i	$0.348007\pi$
$38$	−4.20662	−0.682405
$39$	−6.94553	−1.11217
$40$	1.00000	0.158114
$41$	6.54778	1.02259	0.511296	−	0.859405i	$-0.329166\pi$
$41$	6.54778	1.02259	0.511296	+	0.859405i	$0.329166\pi$
$42$	5.94907	0.917961
$43$	5.16343	0.787415	0.393708	−	0.919236i	$-0.371192\pi$
$43$	5.16343	0.787415	0.393708	+	0.919236i	$0.371192\pi$
$44$	3.74921	0.565215
$45$	2.45222	0.365555
$46$	6.08421	0.897068
$47$	6.05659	0.883445	0.441722	−	0.897152i	$-0.354368\pi$
$47$	6.05659	0.883445	0.441722	+	0.897152i	$0.354368\pi$
$48$	2.33500	0.337028
$49$	−0.508811	−0.0726873
$50$	1.00000	0.141421
$51$	0	0
$52$	−2.97453	−0.412493
$53$	−7.64453	−1.05006	−0.525028	−	0.851085i	$-0.675945\pi$
$53$	−7.64453	−1.05006	−0.525028	+	0.851085i	$0.675945\pi$
$54$	−1.27907	−0.174059
$55$	3.74921	0.505544
$56$	2.54778	0.340462
$57$	−9.82246	−1.30102
$58$	0.493428	0.0647903
$59$	5.30219	0.690286	0.345143	−	0.938550i	$-0.387830\pi$
$59$	5.30219	0.690286	0.345143	+	0.938550i	$0.387830\pi$
$60$	2.33500	0.301447
$61$	−13.6487	−1.74754	−0.873769	−	0.486341i	$-0.838331\pi$
$61$	−13.6487	−1.74754	−0.873769	+	0.486341i	$0.838331\pi$
$62$	0	0
$63$	6.24772	0.787138
$64$	1.00000	0.125000
$65$	−2.97453	−0.368945
$66$	8.75441	1.07759
$67$	3.30219	0.403426	0.201713	−	0.979445i	$-0.435349\pi$
$67$	3.30219	0.403426	0.201713	+	0.979445i	$0.435349\pi$
$68$	0	0
$69$	14.2066	1.71028
$70$	2.54778	0.304518
$71$	−5.50881	−0.653776	−0.326888	−	0.945063i	$-0.606000\pi$
$71$	−5.50881	−0.653776	−0.326888	+	0.945063i	$0.606000\pi$
$72$	2.45222	0.288997
$73$	−11.3937	−1.33354	−0.666768	−	0.745265i	$-0.732323\pi$
$73$	−11.3937	−1.33354	−0.666768	+	0.745265i	$0.732323\pi$
$74$	5.59078	0.649915
$75$	2.33500	0.269622
$76$	−4.20662	−0.482533
$77$	9.55217	1.08857
$78$	−6.94553	−0.786426
$79$	12.6728	1.42580	0.712901	−	0.701264i	$-0.247381\pi$
$79$	12.6728	1.42580	0.712901	+	0.701264i	$0.247381\pi$
$80$	1.00000	0.111803
$81$	−10.3433	−1.14925
$82$	6.54778	0.723081
$83$	−13.9409	−1.53021	−0.765107	−	0.643903i	$-0.777314\pi$
$83$	−13.9409	−1.53021	−0.765107	+	0.643903i	$0.777314\pi$
$84$	5.94907	0.649096
$85$	0	0
$86$	5.16343	0.556787
$87$	1.15215	0.123524
$88$	3.74921	0.399667
$89$	−11.4489	−1.21358	−0.606788	−	0.794864i	$-0.707542\pi$
$89$	−11.4489	−1.21358	−0.606788	+	0.794864i	$0.707542\pi$
$90$	2.45222	0.258487
$91$	−7.57846	−0.794438
$92$	6.08421	0.634323
$93$	0	0
$94$	6.05659	0.624690
$95$	−4.20662	−0.431591
$96$	2.33500	0.238315
$97$	1.24559	0.126471	0.0632355	−	0.997999i	$-0.479858\pi$
$97$	1.24559	0.126471	0.0632355	+	0.997999i	$0.479858\pi$
$98$	−0.508811	−0.0513977
$99$	9.19389	0.924021
$100$	1.00000	0.100000
$101$	−17.0176	−1.69332	−0.846658	−	0.532137i	$-0.821389\pi$
$101$	−17.0176	−1.69332	−0.846658	+	0.532137i	$0.821389\pi$
$102$	0	0
$103$	9.50881	0.936931	0.468466	−	0.883482i	$-0.344807\pi$
$103$	9.50881	0.936931	0.468466	+	0.883482i	$0.344807\pi$
$104$	−2.97453	−0.291677
$105$	5.94907	0.580569
$106$	−7.64453	−0.742502
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	−1.27907	−0.123078
$109$	17.4154	1.66809	0.834045	−	0.551696i	$-0.186019\pi$
$109$	17.4154	1.66809	0.834045	+	0.551696i	$0.186019\pi$
$110$	3.74921	0.357473
$111$	13.0545	1.23908
$112$	2.54778	0.240743
$113$	12.2632	1.15363	0.576813	−	0.816876i	$-0.304296\pi$
$113$	12.2632	1.15363	0.576813	+	0.816876i	$0.304296\pi$
$114$	−9.82246	−0.919958
$115$	6.08421	0.567356
$116$	0.493428	0.0458137
$117$	−7.29421	−0.674350
$118$	5.30219	0.488106
$119$	0	0
$120$	2.33500	0.213155
$121$	3.05659	0.277872
$122$	−13.6487	−1.23570
$123$	15.2891	1.37857
$124$	0	0
$125$	1.00000	0.0894427
$126$	6.24772	0.556591
$127$	13.5026	1.19816	0.599081	−	0.800688i	$-0.295533\pi$
$127$	13.5026	1.19816	0.599081	+	0.800688i	$0.295533\pi$
$128$	1.00000	0.0883883
$129$	12.0566	1.06152
$130$	−2.97453	−0.260884
$131$	3.79338	0.331429	0.165714	−	0.986174i	$-0.447007\pi$
$131$	3.79338	0.331429	0.165714	+	0.986174i	$0.447007\pi$
$132$	8.75441	0.761973
$133$	−10.7176	−0.929330
$134$	3.30219	0.285265
$135$	−1.27907	−0.110085
$136$	0	0
$137$	16.7584	1.43177	0.715883	−	0.698221i	$-0.246025\pi$
$137$	16.7584	1.43177	0.715883	+	0.698221i	$0.246025\pi$
$138$	14.2066	1.20935
$139$	17.6792	1.49953	0.749764	−	0.661706i	$-0.230167\pi$
$139$	17.6792	1.49953	0.749764	+	0.661706i	$0.230167\pi$
$140$	2.54778	0.215327
$141$	14.1421	1.19098
$142$	−5.50881	−0.462289
$143$	−11.1522	−0.932590
$144$	2.45222	0.204352
$145$	0.493428	0.0409770
$146$	−11.3937	−0.942953
$147$	−1.18807	−0.0979907
$148$	5.59078	0.459559
$149$	−13.3022	−1.08976	−0.544879	−	0.838515i	$-0.683424\pi$
$149$	−13.3022	−1.08976	−0.544879	+	0.838515i	$0.683424\pi$
$150$	2.33500	0.190652
$151$	−21.4284	−1.74382	−0.871909	−	0.489669i	$-0.837118\pi$
$151$	−21.4284	−1.74382	−0.871909	+	0.489669i	$0.837118\pi$
$152$	−4.20662	−0.341202
$153$	0	0
$154$	9.55217	0.769736
$155$	0	0
$156$	−6.94553	−0.556087
$157$	9.01762	0.719685	0.359842	−	0.933013i	$-0.382830\pi$
$157$	9.01762	0.719685	0.359842	+	0.933013i	$0.382830\pi$
$158$	12.6728	1.00819
$159$	−17.8500	−1.41559
$160$	1.00000	0.0790569
$161$	15.5012	1.22167
$162$	−10.3433	−0.812645
$163$	−12.3977	−0.971067	−0.485533	−	0.874218i	$-0.661375\pi$
$163$	−12.3977	−0.971067	−0.485533	+	0.874218i	$0.661375\pi$
$164$	6.54778	0.511296
$165$	8.75441	0.681530
$166$	−13.9409	−1.08203
$167$	−23.0548	−1.78403	−0.892016	−	0.452004i	$-0.850709\pi$
$167$	−23.0548	−1.78403	−0.892016	+	0.452004i	$0.850709\pi$
$168$	5.94907	0.458980
$169$	−4.15215	−0.319397
$170$	0	0
$171$	−10.3156	−0.788851
$172$	5.16343	0.393708
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	1.15215	0.0873446
$175$	2.54778	0.192594
$176$	3.74921	0.282608
$177$	12.3806	0.930583
$178$	−11.4489	−0.858128
$179$	7.64453	0.571379	0.285690	−	0.958322i	$-0.407777\pi$
$179$	7.64453	0.571379	0.285690	+	0.958322i	$0.407777\pi$
$180$	2.45222	0.182778
$181$	−22.9887	−1.70874	−0.854368	−	0.519668i	$-0.826056\pi$
$181$	−22.9887	−1.70874	−0.854368	+	0.519668i	$0.826056\pi$
$182$	−7.57846	−0.561753
$183$	−31.8697	−2.35588
$184$	6.08421	0.448534
$185$	5.59078	0.411043
$186$	0	0
$187$	0	0
$188$	6.05659	0.441722
$189$	−3.25879	−0.237042
$190$	−4.20662	−0.305181
$191$	5.84997	0.423289	0.211644	−	0.977347i	$-0.432118\pi$
$191$	5.84997	0.423289	0.211644	+	0.977347i	$0.432118\pi$
$192$	2.33500	0.168514
$193$	19.2087	1.38268	0.691338	−	0.722532i	$-0.257022\pi$
$193$	19.2087	1.38268	0.691338	+	0.722532i	$0.257022\pi$
$194$	1.24559	0.0894285
$195$	−6.94553	−0.497380
$196$	−0.508811	−0.0363437
$197$	−5.02828	−0.358250	−0.179125	−	0.983826i	$-0.557327\pi$
$197$	−5.02828	−0.358250	−0.179125	+	0.983826i	$0.557327\pi$
$198$	9.19389	0.653381
$199$	19.9591	1.41486	0.707430	−	0.706783i	$-0.249854\pi$
$199$	19.9591	1.41486	0.707430	+	0.706783i	$0.249854\pi$
$200$	1.00000	0.0707107
$201$	7.71060	0.543864
$202$	−17.0176	−1.19736
$203$	1.25715	0.0882345
$204$	0	0
$205$	6.54778	0.457317
$206$	9.50881	0.662510
$207$	14.9198	1.03700
$208$	−2.97453	−0.206247
$209$	−15.7715	−1.09094
$210$	5.94907	0.410525
$211$	17.6220	1.21315	0.606574	−	0.795027i	$-0.292543\pi$
$211$	17.6220	1.21315	0.606574	+	0.795027i	$0.292543\pi$
$212$	−7.64453	−0.525028
$213$	−12.8631	−0.881363
$214$	8.00000	0.546869
$215$	5.16343	0.352143
$216$	−1.27907	−0.0870296
$217$	0	0
$218$	17.4154	1.17952
$219$	−26.6044	−1.79776
$220$	3.74921	0.252772
$221$	0	0
$222$	13.0545	0.876159
$223$	9.39511	0.629143	0.314571	−	0.949234i	$-0.398139\pi$
$223$	9.39511	0.629143	0.314571	+	0.949234i	$0.398139\pi$
$224$	2.54778	0.170231
$225$	2.45222	0.163481
$226$	12.2632	0.815737
$227$	−20.1287	−1.33599	−0.667994	−	0.744167i	$-0.732847\pi$
$227$	−20.1287	−1.33599	−0.667994	+	0.744167i	$0.732847\pi$
$228$	−9.82246	−0.650508
$229$	4.57900	0.302589	0.151295	−	0.988489i	$-0.451656\pi$
$229$	4.57900	0.302589	0.151295	+	0.988489i	$0.451656\pi$
$230$	6.08421	0.401181
$231$	22.3043	1.46752
$232$	0.493428	0.0323952
$233$	−15.8500	−1.03837	−0.519183	−	0.854663i	$-0.673764\pi$
$233$	−15.8500	−1.03837	−0.519183	+	0.854663i	$0.673764\pi$
$234$	−7.29421	−0.476837
$235$	6.05659	0.395089
$236$	5.30219	0.345143
$237$	29.5910	1.92214
$238$	0	0
$239$	−13.8499	−0.895877	−0.447939	−	0.894064i	$-0.647842\pi$
$239$	−13.8499	−0.895877	−0.447939	+	0.894064i	$0.647842\pi$
$240$	2.33500	0.150724
$241$	−8.40825	−0.541623	−0.270811	−	0.962632i	$-0.587292\pi$
$241$	−8.40825	−0.541623	−0.270811	+	0.962632i	$0.587292\pi$
$242$	3.05659	0.196485
$243$	−20.3143	−1.30317
$244$	−13.6487	−0.873769
$245$	−0.508811	−0.0325068
$246$	15.2891	0.974795
$247$	12.5127	0.796167
$248$	0	0
$249$	−32.5520	−2.06290
$250$	1.00000	0.0632456
$251$	−19.7329	−1.24553	−0.622765	−	0.782409i	$-0.713991\pi$
$251$	−19.7329	−1.24553	−0.622765	+	0.782409i	$0.713991\pi$
$252$	6.24772	0.393569
$253$	22.8110	1.43412
$254$	13.5026	0.847229
$255$	0	0
$256$	1.00000	0.0625000
$257$	0.604374	0.0376998	0.0188499	−	0.999822i	$-0.494000\pi$
$257$	0.604374	0.0376998	0.0188499	+	0.999822i	$0.494000\pi$
$258$	12.0566	0.750611
$259$	14.2441	0.885085
$260$	−2.97453	−0.184473
$261$	1.20999	0.0748968
$262$	3.79338	0.234356
$263$	−29.3542	−1.81006	−0.905028	−	0.425353i	$-0.860150\pi$
$263$	−29.3542	−1.81006	−0.905028	+	0.425353i	$0.860150\pi$
$264$	8.75441	0.538796
$265$	−7.64453	−0.469600
$266$	−10.7176	−0.657136
$267$	−26.7331	−1.63604
$268$	3.30219	0.201713
$269$	1.75058	0.106735	0.0533673	−	0.998575i	$-0.483005\pi$
$269$	1.75058	0.106735	0.0533673	+	0.998575i	$0.483005\pi$
$270$	−1.27907	−0.0778416
$271$	−12.9431	−0.786237	−0.393119	−	0.919488i	$-0.628604\pi$
$271$	−12.9431	−0.786237	−0.393119	+	0.919488i	$0.628604\pi$
$272$	0	0
$273$	−17.6957	−1.07099
$274$	16.7584	1.01241
$275$	3.74921	0.226086
$276$	14.2066	0.855138
$277$	−22.0570	−1.32527	−0.662637	−	0.748941i	$-0.730563\pi$
$277$	−22.0570	−1.32527	−0.662637	+	0.748941i	$0.730563\pi$
$278$	17.6792	1.06033
$279$	0	0
$280$	2.54778	0.152259
$281$	10.7955	0.644005	0.322003	−	0.946739i	$-0.395644\pi$
$281$	10.7955	0.644005	0.322003	+	0.946739i	$0.395644\pi$
$282$	14.1421	0.842152
$283$	−13.7934	−0.819931	−0.409966	−	0.912101i	$-0.634459\pi$
$283$	−13.7934	−0.819931	−0.409966	+	0.912101i	$0.634459\pi$
$284$	−5.50881	−0.326888
$285$	−9.82246	−0.581832
$286$	−11.1522	−0.659441
$287$	16.6823	0.984726
$288$	2.45222	0.144498
$289$	−17.0000	−1.00000
$290$	0.493428	0.0289751
$291$	2.90846	0.170497
$292$	−11.3937	−0.666768
$293$	−11.0021	−0.642751	−0.321375	−	0.946952i	$-0.604145\pi$
$293$	−11.0021	−0.642751	−0.321375	+	0.946952i	$0.604145\pi$
$294$	−1.18807	−0.0692899
$295$	5.30219	0.308705
$296$	5.59078	0.324958
$297$	−4.79550	−0.278263
$298$	−13.3022	−0.770575
$299$	−18.0977	−1.04662
$300$	2.33500	0.134811
$301$	13.1553	0.758258
$302$	−21.4284	−1.23306
$303$	−39.7361	−2.28278
$304$	−4.20662	−0.241266
$305$	−13.6487	−0.781523
$306$	0	0
$307$	15.3177	0.874227	0.437113	−	0.899406i	$-0.356001\pi$
$307$	15.3177	0.874227	0.437113	+	0.899406i	$0.356001\pi$
$308$	9.55217	0.544286
$309$	22.2031	1.26309
$310$	0	0
$311$	7.58675	0.430205	0.215103	−	0.976591i	$-0.430991\pi$
$311$	7.58675	0.430205	0.215103	+	0.976591i	$0.430991\pi$
$312$	−6.94553	−0.393213
$313$	14.4124	0.814639	0.407319	−	0.913286i	$-0.366464\pi$
$313$	14.4124	0.814639	0.407319	+	0.913286i	$0.366464\pi$
$314$	9.01762	0.508894
$315$	6.24772	0.352019
$316$	12.6728	0.712901
$317$	−17.4933	−0.982522	−0.491261	−	0.871012i	$-0.663464\pi$
$317$	−17.4933	−0.982522	−0.491261	+	0.871012i	$0.663464\pi$
$318$	−17.8500	−1.00098
$319$	1.84997	0.103578
$320$	1.00000	0.0559017
$321$	18.6800	1.04262
$322$	15.5012	0.863850
$323$	0	0
$324$	−10.3433	−0.574627
$325$	−2.97453	−0.164997
$326$	−12.3977	−0.686648
$327$	40.6649	2.24877
$328$	6.54778	0.361541
$329$	15.4309	0.850732
$330$	8.75441	0.481914
$331$	8.92360	0.490485	0.245243	−	0.969462i	$-0.421132\pi$
$331$	8.92360	0.490485	0.245243	+	0.969462i	$0.421132\pi$
$332$	−13.9409	−0.765107
$333$	13.7098	0.751294
$334$	−23.0548	−1.26150
$335$	3.30219	0.180418
$336$	5.94907	0.324548
$337$	−1.84157	−0.100317	−0.0501584	−	0.998741i	$-0.515973\pi$
$337$	−1.84157	−0.100317	−0.0501584	+	0.998741i	$0.515973\pi$
$338$	−4.15215	−0.225847
$339$	28.6346	1.55522
$340$	0	0
$341$	0	0
$342$	−10.3156	−0.557802
$343$	−19.1308	−1.03297
$344$	5.16343	0.278393
$345$	14.2066	0.764859
$346$	14.0000	0.752645
$347$	14.9278	0.801365	0.400682	−	0.916217i	$-0.368773\pi$
$347$	14.9278	0.801365	0.400682	+	0.916217i	$0.368773\pi$
$348$	1.15215	0.0617620
$349$	−33.1153	−1.77262	−0.886311	−	0.463091i	$-0.846740\pi$
$349$	−33.1153	−1.77262	−0.886311	+	0.463091i	$0.846740\pi$
$350$	2.54778	0.136185
$351$	3.80463	0.203076
$352$	3.74921	0.199834
$353$	−19.4547	−1.03547	−0.517734	−	0.855542i	$-0.673224\pi$
$353$	−19.4547	−1.03547	−0.517734	+	0.855542i	$0.673224\pi$
$354$	12.3806	0.658021
$355$	−5.50881	−0.292377
$356$	−11.4489	−0.606788
$357$	0	0
$358$	7.64453	0.404026
$359$	22.6455	1.19518	0.597591	−	0.801801i	$-0.296125\pi$
$359$	22.6455	1.19518	0.597591	+	0.801801i	$0.296125\pi$
$360$	2.45222	0.129243
$361$	−1.30431	−0.0686479
$362$	−22.9887	−1.20826
$363$	7.13714	0.374603
$364$	−7.57846	−0.397219
$365$	−11.3937	−0.596376
$366$	−31.8697	−1.66586
$367$	−2.67136	−0.139444	−0.0697220	−	0.997566i	$-0.522211\pi$
$367$	−2.67136	−0.139444	−0.0697220	+	0.997566i	$0.522211\pi$
$368$	6.08421	0.317161
$369$	16.0566	0.835873
$370$	5.59078	0.290651
$371$	−19.4766	−1.01117
$372$	0	0
$373$	16.1287	0.835112	0.417556	−	0.908651i	$-0.362887\pi$
$373$	16.1287	0.835112	0.417556	+	0.908651i	$0.362887\pi$
$374$	0	0
$375$	2.33500	0.120579
$376$	6.05659	0.312345
$377$	−1.46772	−0.0755914
$378$	−3.25879	−0.167614
$379$	6.28456	0.322816	0.161408	−	0.986888i	$-0.448396\pi$
$379$	6.28456	0.322816	0.161408	+	0.986888i	$0.448396\pi$
$380$	−4.20662	−0.215795
$381$	31.5286	1.61526
$382$	5.84997	0.299310
$383$	−5.17739	−0.264552	−0.132276	−	0.991213i	$-0.542229\pi$
$383$	−5.17739	−0.264552	−0.132276	+	0.991213i	$0.542229\pi$
$384$	2.33500	0.119157
$385$	9.55217	0.486824
$386$	19.2087	0.977699
$387$	12.6619	0.643638
$388$	1.24559	0.0632355
$389$	2.04279	0.103573	0.0517867	−	0.998658i	$-0.483508\pi$
$389$	2.04279	0.103573	0.0517867	+	0.998658i	$0.483508\pi$
$390$	−6.94553	−0.351701
$391$	0	0
$392$	−0.508811	−0.0256989
$393$	8.85753	0.446803
$394$	−5.02828	−0.253321
$395$	12.6728	0.637638
$396$	9.19389	0.462010
$397$	−31.0021	−1.55595	−0.777976	−	0.628294i	$-0.783753\pi$
$397$	−31.0021	−1.55595	−0.777976	+	0.628294i	$0.783753\pi$
$398$	19.9591	1.00046
$399$	−25.0255	−1.25284
$400$	1.00000	0.0500000
$401$	−31.7802	−1.58703	−0.793513	−	0.608554i	$-0.791750\pi$
$401$	−31.7802	−1.58703	−0.793513	+	0.608554i	$0.791750\pi$
$402$	7.71060	0.384570
$403$	0	0
$404$	−17.0176	−0.846658
$405$	−10.3433	−0.513962
$406$	1.25715	0.0623912
$407$	20.9610	1.03900
$408$	0	0
$409$	−24.3338	−1.20323	−0.601616	−	0.798786i	$-0.705476\pi$
$409$	−24.3338	−1.20323	−0.601616	+	0.798786i	$0.705476\pi$
$410$	6.54778	0.323372
$411$	39.1308	1.93018
$412$	9.50881	0.468466
$413$	13.5088	0.664725
$414$	14.9198	0.733269
$415$	−13.9409	−0.684333
$416$	−2.97453	−0.145838
$417$	41.2808	2.02153
$418$	−15.7715	−0.771411
$419$	3.90656	0.190848	0.0954240	−	0.995437i	$-0.469579\pi$
$419$	3.90656	0.190848	0.0954240	+	0.995437i	$0.469579\pi$
$420$	5.94907	0.290285
$421$	3.79338	0.184878	0.0924389	−	0.995718i	$-0.470534\pi$
$421$	3.79338	0.184878	0.0924389	+	0.995718i	$0.470534\pi$
$422$	17.6220	0.857825
$423$	14.8521	0.722133
$424$	−7.64453	−0.371251
$425$	0	0
$426$	−12.8631	−0.623218
$427$	−34.7739	−1.68283
$428$	8.00000	0.386695
$429$	−26.0403	−1.25724
$430$	5.16343	0.249003
$431$	32.4543	1.56327	0.781635	−	0.623736i	$-0.214386\pi$
$431$	32.4543	1.56327	0.781635	+	0.623736i	$0.214386\pi$
$432$	−1.27907	−0.0615392
$433$	−16.4881	−0.792367	−0.396184	−	0.918171i	$-0.629666\pi$
$433$	−16.4881	−0.792367	−0.396184	+	0.918171i	$0.629666\pi$
$434$	0	0
$435$	1.15215	0.0552416
$436$	17.4154	0.834045
$437$	−25.5940	−1.22433
$438$	−26.6044	−1.27121
$439$	13.1677	0.628458	0.314229	−	0.949347i	$-0.398254\pi$
$439$	13.1677	0.628458	0.314229	+	0.949347i	$0.398254\pi$
$440$	3.74921	0.178737
$441$	−1.24772	−0.0594151
$442$	0	0
$443$	−36.9087	−1.75358	−0.876792	−	0.480869i	$-0.840321\pi$
$443$	−36.9087	−1.75358	−0.876792	+	0.480869i	$0.840321\pi$
$444$	13.0545	0.619538
$445$	−11.4489	−0.542728
$446$	9.39511	0.444871
$447$	−31.0606	−1.46912
$448$	2.54778	0.120371
$449$	8.05792	0.380277	0.190138	−	0.981757i	$-0.439106\pi$
$449$	8.05792	0.380277	0.190138	+	0.981757i	$0.439106\pi$
$450$	2.45222	0.115599
$451$	24.5490	1.15597
$452$	12.2632	0.576813
$453$	−50.0352	−2.35086
$454$	−20.1287	−0.944686
$455$	−7.57846	−0.355284
$456$	−9.82246	−0.459979
$457$	−30.7903	−1.44031	−0.720155	−	0.693814i	$-0.755929\pi$
$457$	−30.7903	−1.44031	−0.720155	+	0.693814i	$0.755929\pi$
$458$	4.57900	0.213963
$459$	0	0
$460$	6.08421	0.283678
$461$	−13.8088	−0.643139	−0.321569	−	0.946886i	$-0.604210\pi$
$461$	−13.8088	−0.643139	−0.321569	+	0.946886i	$0.604210\pi$
$462$	22.3043	1.03769
$463$	19.7220	0.916557	0.458279	−	0.888809i	$-0.348466\pi$
$463$	19.7220	0.916557	0.458279	+	0.888809i	$0.348466\pi$
$464$	0.493428	0.0229068
$465$	0	0
$466$	−15.8500	−0.734236
$467$	−14.6044	−0.675810	−0.337905	−	0.941180i	$-0.609718\pi$
$467$	−14.6044	−0.675810	−0.337905	+	0.941180i	$0.609718\pi$
$468$	−7.29421	−0.337175
$469$	8.41325	0.388488
$470$	6.05659	0.279370
$471$	21.0561	0.970216
$472$	5.30219	0.244053
$473$	19.3588	0.890118
$474$	29.5910	1.35916
$475$	−4.20662	−0.193013
$476$	0	0
$477$	−18.7461	−0.858323
$478$	−13.8499	−0.633481
$479$	13.8911	0.634699	0.317349	−	0.948309i	$-0.397207\pi$
$479$	13.8911	0.634699	0.317349	+	0.948309i	$0.397207\pi$
$480$	2.33500	0.106578
$481$	−16.6300	−0.758261
$482$	−8.40825	−0.382985
$483$	36.1954	1.64695
$484$	3.05659	0.138936
$485$	1.24559	0.0565595
$486$	−20.3143	−0.921477
$487$	−3.62803	−0.164402	−0.0822008	−	0.996616i	$-0.526195\pi$
$487$	−3.62803	−0.164402	−0.0822008	+	0.996616i	$0.526195\pi$
$488$	−13.6487	−0.617848
$489$	−28.9487	−1.30911
$490$	−0.508811	−0.0229858
$491$	−25.7620	−1.16262	−0.581312	−	0.813681i	$-0.697460\pi$
$491$	−25.7620	−1.16262	−0.581312	+	0.813681i	$0.697460\pi$
$492$	15.2891	0.689284
$493$	0	0
$494$	12.5127	0.562975
$495$	9.19389	0.413235
$496$	0	0
$497$	−14.0352	−0.629567
$498$	−32.5520	−1.45869
$499$	19.4626	0.871267	0.435633	−	0.900124i	$-0.356524\pi$
$499$	19.4626	0.871267	0.435633	+	0.900124i	$0.356524\pi$
$500$	1.00000	0.0447214
$501$	−53.8329	−2.40508
$502$	−19.7329	−0.880723
$503$	38.8307	1.73138	0.865689	−	0.500583i	$-0.166881\pi$
$503$	38.8307	1.73138	0.865689	+	0.500583i	$0.166881\pi$
$504$	6.24772	0.278295
$505$	−17.0176	−0.757274
$506$	22.8110	1.01407
$507$	−9.69528	−0.430582
$508$	13.5026	0.599081
$509$	5.14151	0.227893	0.113947	−	0.993487i	$-0.463651\pi$
$509$	5.14151	0.227893	0.113947	+	0.993487i	$0.463651\pi$
$510$	0	0
$511$	−29.0288	−1.28416
$512$	1.00000	0.0441942
$513$	5.38056	0.237558
$514$	0.604374	0.0266578
$515$	9.50881	0.419008
$516$	12.0566	0.530762
$517$	22.7075	0.998673
$518$	14.2441	0.625849
$519$	32.6900	1.43493
$520$	−2.97453	−0.130442
$521$	13.8089	0.604978	0.302489	−	0.953153i	$-0.402182\pi$
$521$	13.8089	0.604978	0.302489	+	0.953153i	$0.402182\pi$
$522$	1.20999	0.0529600
$523$	23.8434	1.04260	0.521300	−	0.853374i	$-0.325447\pi$
$523$	23.8434	1.04260	0.521300	+	0.853374i	$0.325447\pi$
$524$	3.79338	0.165714
$525$	5.94907	0.259639
$526$	−29.3542	−1.27990
$527$	0	0
$528$	8.75441	0.380987
$529$	14.0176	0.609462
$530$	−7.64453	−0.332057
$531$	13.0021	0.564244
$532$	−10.7176	−0.464665
$533$	−19.4766	−0.843625
$534$	−26.7331	−1.15685
$535$	8.00000	0.345870
$536$	3.30219	0.142633
$537$	17.8500	0.770283
$538$	1.75058	0.0754727
$539$	−1.90764	−0.0821680
$540$	−1.27907	−0.0550423
$541$	−2.39775	−0.103087	−0.0515436	−	0.998671i	$-0.516414\pi$
$541$	−2.39775	−0.103087	−0.0515436	+	0.998671i	$0.516414\pi$
$542$	−12.9431	−0.555954
$543$	−53.6786	−2.30357
$544$	0	0
$545$	17.4154	0.745993
$546$	−17.6957	−0.757306
$547$	7.60225	0.325049	0.162524	−	0.986705i	$-0.448036\pi$
$547$	7.60225	0.325049	0.162524	+	0.986705i	$0.448036\pi$
$548$	16.7584	0.715883
$549$	−33.4696	−1.42845
$550$	3.74921	0.159867
$551$	−2.07567	−0.0884264
$552$	14.2066	0.604674
$553$	32.2875	1.37301
$554$	−22.0570	−0.937110
$555$	13.0545	0.554131
$556$	17.6792	0.749764
$557$	−15.9176	−0.674452	−0.337226	−	0.941424i	$-0.609489\pi$
$557$	−15.9176	−0.674452	−0.337226	+	0.941424i	$0.609489\pi$
$558$	0	0
$559$	−15.3588	−0.649607
$560$	2.54778	0.107663
$561$	0	0
$562$	10.7955	0.455381
$563$	2.31981	0.0977683	0.0488842	−	0.998804i	$-0.484433\pi$
$563$	2.31981	0.0977683	0.0488842	+	0.998804i	$0.484433\pi$
$564$	14.1421	0.595491
$565$	12.2632	0.515917
$566$	−13.7934	−0.579779
$567$	−26.3524	−1.10670
$568$	−5.50881	−0.231145
$569$	−12.5958	−0.528043	−0.264021	−	0.964517i	$-0.585049\pi$
$569$	−12.5958	−0.528043	−0.264021	+	0.964517i	$0.585049\pi$
$570$	−9.82246	−0.411418
$571$	17.3069	0.724272	0.362136	−	0.932125i	$-0.382048\pi$
$571$	17.3069	0.724272	0.362136	+	0.932125i	$0.382048\pi$
$572$	−11.1522	−0.466295
$573$	13.6597	0.570641
$574$	16.6823	0.696306
$575$	6.08421	0.253729
$576$	2.45222	0.102176
$577$	34.5675	1.43906	0.719532	−	0.694459i	$-0.244356\pi$
$577$	34.5675	1.43906	0.719532	+	0.694459i	$0.244356\pi$
$578$	−17.0000	−0.707107
$579$	44.8524	1.86400
$580$	0.493428	0.0204885
$581$	−35.5184	−1.47355
$582$	2.90846	0.120560
$583$	−28.6610	−1.18702
$584$	−11.3937	−0.471476
$585$	−7.29421	−0.301578
$586$	−11.0021	−0.454493
$587$	33.3375	1.37598	0.687992	−	0.725718i	$-0.258492\pi$
$587$	33.3375	1.37598	0.687992	+	0.725718i	$0.258492\pi$
$588$	−1.18807	−0.0489953
$589$	0	0
$590$	5.30219	0.218288
$591$	−11.7410	−0.482961
$592$	5.59078	0.229780
$593$	−9.47196	−0.388967	−0.194483	−	0.980906i	$-0.562303\pi$
$593$	−9.47196	−0.388967	−0.194483	+	0.980906i	$0.562303\pi$
$594$	−4.79550	−0.196762
$595$	0	0
$596$	−13.3022	−0.544879
$597$	46.6044	1.90739
$598$	−18.0977	−0.740069
$599$	42.6765	1.74371	0.871857	−	0.489761i	$-0.162916\pi$
$599$	42.6765	1.74371	0.871857	+	0.489761i	$0.162916\pi$
$600$	2.33500	0.0953259
$601$	−18.7351	−0.764221	−0.382110	−	0.924117i	$-0.624803\pi$
$601$	−18.7351	−0.764221	−0.382110	+	0.924117i	$0.624803\pi$
$602$	13.1553	0.536169
$603$	8.09768	0.329763
$604$	−21.4284	−0.871909
$605$	3.05659	0.124268
$606$	−39.7361	−1.61417
$607$	−14.1132	−0.572837	−0.286418	−	0.958105i	$-0.592465\pi$
$607$	−14.1132	−0.572837	−0.286418	+	0.958105i	$0.592465\pi$
$608$	−4.20662	−0.170601
$609$	2.93544	0.118950
$610$	−13.6487	−0.552620
$611$	−18.0155	−0.728830
$612$	0	0
$613$	−14.2082	−0.573864	−0.286932	−	0.957951i	$-0.592635\pi$
$613$	−14.2082	−0.573864	−0.286932	+	0.957951i	$0.592635\pi$
$614$	15.3177	0.618172
$615$	15.2891	0.616514
$616$	9.55217	0.384868
$617$	23.8911	0.961818	0.480909	−	0.876771i	$-0.340307\pi$
$617$	23.8911	0.961818	0.480909	+	0.876771i	$0.340307\pi$
$618$	22.2031	0.893138
$619$	−3.03265	−0.121892	−0.0609462	−	0.998141i	$-0.519412\pi$
$619$	−3.03265	−0.121892	−0.0609462	+	0.998141i	$0.519412\pi$
$620$	0	0
$621$	−7.78212	−0.312286
$622$	7.58675	0.304201
$623$	−29.1692	−1.16864
$624$	−6.94553	−0.278044
$625$	1.00000	0.0400000
$626$	14.4124	0.576036
$627$	−36.8265	−1.47071
$628$	9.01762	0.359842
$629$	0	0
$630$	6.24772	0.248915
$631$	26.8149	1.06749	0.533743	−	0.845647i	$-0.320785\pi$
$631$	26.8149	1.06749	0.533743	+	0.845647i	$0.320785\pi$
$632$	12.6728	0.504097
$633$	41.1473	1.63546
$634$	−17.4933	−0.694748
$635$	13.5026	0.535834
$636$	−17.8500	−0.707797
$637$	1.51348	0.0599661
$638$	1.84997	0.0732409
$639$	−13.5088	−0.534400
$640$	1.00000	0.0395285
$641$	−32.3147	−1.27636	−0.638178	−	0.769889i	$-0.720311\pi$
$641$	−32.3147	−1.27636	−0.638178	+	0.769889i	$0.720311\pi$
$642$	18.6800	0.737240
$643$	36.4581	1.43777	0.718884	−	0.695130i	$-0.244653\pi$
$643$	36.4581	1.43777	0.718884	+	0.695130i	$0.244653\pi$
$644$	15.5012	0.610834
$645$	12.0566	0.474728
$646$	0	0
$647$	35.0911	1.37957	0.689786	−	0.724013i	$-0.257705\pi$
$647$	35.0911	1.37957	0.689786	+	0.724013i	$0.257705\pi$
$648$	−10.3433	−0.406322
$649$	19.8790	0.780320
$650$	−2.97453	−0.116671
$651$	0	0
$652$	−12.3977	−0.485533
$653$	17.5868	0.688223	0.344111	−	0.938929i	$-0.388180\pi$
$653$	17.5868	0.688223	0.344111	+	0.938929i	$0.388180\pi$
$654$	40.6649	1.59012
$655$	3.79338	0.148219
$656$	6.54778	0.255648
$657$	−27.9400	−1.09004
$658$	15.4309	0.601558
$659$	45.2285	1.76185	0.880926	−	0.473254i	$-0.156921\pi$
$659$	45.2285	1.76185	0.880926	+	0.473254i	$0.156921\pi$
$660$	8.75441	0.340765
$661$	−17.3022	−0.672977	−0.336489	−	0.941688i	$-0.609239\pi$
$661$	−17.3022	−0.672977	−0.336489	+	0.941688i	$0.609239\pi$
$662$	8.92360	0.346825
$663$	0	0
$664$	−13.9409	−0.541013
$665$	−10.7176	−0.415609
$666$	13.7098	0.531245
$667$	3.00212	0.116243
$668$	−23.0548	−0.892016
$669$	21.9376	0.848155
$670$	3.30219	0.127575
$671$	−51.1719	−1.97547
$672$	5.94907	0.229490
$673$	41.6797	1.60663	0.803316	−	0.595553i	$-0.203067\pi$
$673$	41.6797	1.60663	0.803316	+	0.595553i	$0.203067\pi$
$674$	−1.84157	−0.0709346
$675$	−1.27907	−0.0492314
$676$	−4.15215	−0.159698
$677$	0.124185	0.00477282	0.00238641	−	0.999997i	$-0.499240\pi$
$677$	0.124185	0.00477282	0.00238641	+	0.999997i	$0.499240\pi$
$678$	28.6346	1.09971
$679$	3.17350	0.121788
$680$	0	0
$681$	−47.0005	−1.80106
$682$	0	0
$683$	31.3999	1.20148	0.600741	−	0.799443i	$-0.294872\pi$
$683$	31.3999	1.20148	0.600741	+	0.799443i	$0.294872\pi$
$684$	−10.3156	−0.394425
$685$	16.7584	0.640305
$686$	−19.1308	−0.730418
$687$	10.6920	0.407924
$688$	5.16343	0.196854
$689$	22.7389	0.866283
$690$	14.2066	0.540837
$691$	21.3219	0.811125	0.405562	−	0.914067i	$-0.367076\pi$
$691$	21.3219	0.811125	0.405562	+	0.914067i	$0.367076\pi$
$692$	14.0000	0.532200
$693$	23.4240	0.889805
$694$	14.9278	0.566651
$695$	17.6792	0.670609
$696$	1.15215	0.0436723
$697$	0	0
$698$	−33.1153	−1.25343
$699$	−37.0097	−1.39983
$700$	2.54778	0.0962971
$701$	−22.8265	−0.862145	−0.431072	−	0.902317i	$-0.641865\pi$
$701$	−22.8265	−0.862145	−0.431072	+	0.902317i	$0.641865\pi$
$702$	3.80463	0.143597
$703$	−23.5183	−0.887010
$704$	3.74921	0.141304
$705$	14.1421	0.532624
$706$	−19.4547	−0.732186
$707$	−43.3572	−1.63061
$708$	12.3806	0.465291
$709$	−12.7940	−0.480489	−0.240244	−	0.970712i	$-0.577228\pi$
$709$	−12.7940	−0.480489	−0.240244	+	0.970712i	$0.577228\pi$
$710$	−5.50881	−0.206742
$711$	31.0765	1.16546
$712$	−11.4489	−0.429064
$713$	0	0
$714$	0	0
$715$	−11.1522	−0.417067
$716$	7.64453	0.285690
$717$	−32.3396	−1.20774
$718$	22.6455	0.845121
$719$	−13.7178	−0.511587	−0.255793	−	0.966731i	$-0.582337\pi$
$719$	−13.7178	−0.511587	−0.255793	+	0.966731i	$0.582337\pi$
$720$	2.45222	0.0913888
$721$	24.2264	0.902237
$722$	−1.30431	−0.0485414
$723$	−19.6333	−0.730168
$724$	−22.9887	−0.854368
$725$	0.493428	0.0183255
$726$	7.13714	0.264884
$727$	19.8911	0.737719	0.368859	−	0.929485i	$-0.379748\pi$
$727$	19.8911	0.737719	0.368859	+	0.929485i	$0.379748\pi$
$728$	−7.57846	−0.280876
$729$	−16.4041	−0.607560
$730$	−11.3937	−0.421701
$731$	0	0
$732$	−31.8697	−1.17794
$733$	37.1932	1.37376	0.686882	−	0.726769i	$-0.258979\pi$
$733$	37.1932	1.37376	0.686882	+	0.726769i	$0.258979\pi$
$734$	−2.67136	−0.0986017
$735$	−1.18807	−0.0438228
$736$	6.08421	0.224267
$737$	12.3806	0.456045
$738$	16.0566	0.591051
$739$	−34.6497	−1.27461	−0.637306	−	0.770611i	$-0.719951\pi$
$739$	−34.6497	−1.27461	−0.637306	+	0.770611i	$0.719951\pi$
$740$	5.59078	0.205521
$741$	29.2172	1.07332
$742$	−19.4766	−0.715008
$743$	−22.7626	−0.835077	−0.417539	−	0.908659i	$-0.637107\pi$
$743$	−22.7626	−0.835077	−0.417539	+	0.908659i	$0.637107\pi$
$744$	0	0
$745$	−13.3022	−0.487354
$746$	16.1287	0.590513
$747$	−34.1862	−1.25081
$748$	0	0
$749$	20.3822	0.744751
$750$	2.33500	0.0852621
$751$	25.8500	0.943279	0.471639	−	0.881792i	$-0.343662\pi$
$751$	25.8500	0.943279	0.471639	+	0.881792i	$0.343662\pi$
$752$	6.05659	0.220861
$753$	−46.0763	−1.67912
$754$	−1.46772	−0.0534512
$755$	−21.4284	−0.779859
$756$	−3.25879	−0.118521
$757$	14.3683	0.522224	0.261112	−	0.965309i	$-0.415911\pi$
$757$	14.3683	0.522224	0.261112	+	0.965309i	$0.415911\pi$
$758$	6.28456	0.228266
$759$	53.2637	1.93335
$760$	−4.20662	−0.152590
$761$	−39.4107	−1.42864	−0.714319	−	0.699820i	$-0.753263\pi$
$761$	−39.4107	−1.42864	−0.714319	+	0.699820i	$0.753263\pi$
$762$	31.5286	1.14216
$763$	44.3706	1.60632
$764$	5.84997	0.211644
$765$	0	0
$766$	−5.17739	−0.187067
$767$	−15.7715	−0.569477
$768$	2.33500	0.0842570
$769$	−18.8831	−0.680942	−0.340471	−	0.940255i	$-0.610586\pi$
$769$	−18.8831	−0.680942	−0.340471	+	0.940255i	$0.610586\pi$
$770$	9.55217	0.344236
$771$	1.41121	0.0508235
$772$	19.2087	0.691338
$773$	20.3974	0.733643	0.366821	−	0.930291i	$-0.380446\pi$
$773$	20.3974	0.733643	0.366821	+	0.930291i	$0.380446\pi$
$774$	12.6619	0.455121
$775$	0	0
$776$	1.24559	0.0447142
$777$	33.2599	1.19319
$778$	2.04279	0.0732375
$779$	−27.5441	−0.986868
$780$	−6.94553	−0.248690
$781$	−20.6537	−0.739048
$782$	0	0
$783$	−0.631129	−0.0225547
$784$	−0.508811	−0.0181718
$785$	9.01762	0.321853
$786$	8.85753	0.315938
$787$	7.13714	0.254412	0.127206	−	0.991876i	$-0.459399\pi$
$787$	7.13714	0.254412	0.127206	+	0.991876i	$0.459399\pi$
$788$	−5.02828	−0.179125
$789$	−68.5419	−2.44016
$790$	12.6728	0.450878
$791$	31.2440	1.11091
$792$	9.19389	0.326691
$793$	40.5985	1.44170
$794$	−31.0021	−1.10022
$795$	−17.8500	−0.633073
$796$	19.9591	0.707430
$797$	−25.9523	−0.919277	−0.459638	−	0.888106i	$-0.652021\pi$
$797$	−25.9523	−0.919277	−0.459638	+	0.888106i	$0.652021\pi$
$798$	−25.0255	−0.885893
$799$	0	0
$800$	1.00000	0.0353553
$801$	−28.0751	−0.991985
$802$	−31.7802	−1.12220
$803$	−42.7176	−1.50747
$804$	7.71060	0.271932
$805$	15.5012	0.546347
$806$	0	0
$807$	4.08759	0.143890
$808$	−17.0176	−0.598678
$809$	20.4966	0.720623	0.360312	−	0.932832i	$-0.382670\pi$
$809$	20.4966	0.720623	0.360312	+	0.932832i	$0.382670\pi$
$810$	−10.3433	−0.363426
$811$	−28.2888	−0.993354	−0.496677	−	0.867935i	$-0.665447\pi$
$811$	−28.2888	−0.993354	−0.496677	+	0.867935i	$0.665447\pi$
$812$	1.25715	0.0441172
$813$	−30.2221	−1.05994
$814$	20.9610	0.734684
$815$	−12.3977	−0.434274
$816$	0	0
$817$	−21.7206	−0.759907
$818$	−24.3338	−0.850813
$819$	−18.5840	−0.649379
$820$	6.54778	0.228658
$821$	20.5846	0.718409	0.359204	−	0.933259i	$-0.383048\pi$
$821$	20.5846	0.718409	0.359204	+	0.933259i	$0.383048\pi$
$822$	39.1308	1.36484
$823$	6.37642	0.222268	0.111134	−	0.993805i	$-0.464552\pi$
$823$	6.37642	0.222268	0.111134	+	0.993805i	$0.464552\pi$
$824$	9.50881	0.331255
$825$	8.75441	0.304789
$826$	13.5088	0.470032
$827$	−7.99185	−0.277904	−0.138952	−	0.990299i	$-0.544373\pi$
$827$	−7.99185	−0.277904	−0.138952	+	0.990299i	$0.544373\pi$
$828$	14.9198	0.518499
$829$	−34.4625	−1.19693	−0.598466	−	0.801148i	$-0.704223\pi$
$829$	−34.4625	−1.19693	−0.598466	+	0.801148i	$0.704223\pi$
$830$	−13.9409	−0.483896
$831$	−51.5030	−1.78662
$832$	−2.97453	−0.103123
$833$	0	0
$834$	41.2808	1.42944
$835$	−23.0548	−0.797843
$836$	−15.7715	−0.545470
$837$	0	0
$838$	3.90656	0.134950
$839$	−17.1367	−0.591623	−0.295812	−	0.955246i	$-0.595590\pi$
$839$	−17.1367	−0.591623	−0.295812	+	0.955246i	$0.595590\pi$
$840$	5.94907	0.205262
$841$	−28.7565	−0.991604
$842$	3.79338	0.130728
$843$	25.2075	0.868192
$844$	17.6220	0.606574
$845$	−4.15215	−0.142838
$846$	14.8521	0.510625
$847$	7.78753	0.267583
$848$	−7.64453	−0.262514
$849$	−32.2075	−1.10536
$850$	0	0
$851$	34.0155	1.16604
$852$	−12.8631	−0.440681
$853$	−30.1132	−1.03106	−0.515528	−	0.856873i	$-0.672404\pi$
$853$	−30.1132	−1.03106	−0.515528	+	0.856873i	$0.672404\pi$
$854$	−34.7739	−1.18994
$855$	−10.3156	−0.352785
$856$	8.00000	0.273434
$857$	−40.8307	−1.39475	−0.697376	−	0.716706i	$-0.745649\pi$
$857$	−40.8307	−1.39475	−0.697376	+	0.716706i	$0.745649\pi$
$858$	−26.0403	−0.889000
$859$	−20.1354	−0.687009	−0.343505	−	0.939151i	$-0.611614\pi$
$859$	−20.1354	−0.687009	−0.343505	+	0.939151i	$0.611614\pi$
$860$	5.16343	0.176071
$861$	38.9532	1.32752
$862$	32.4543	1.10540
$863$	56.8336	1.93464	0.967319	−	0.253564i	$-0.0816029\pi$
$863$	56.8336	1.93464	0.967319	+	0.253564i	$0.0816029\pi$
$864$	−1.27907	−0.0435148
$865$	14.0000	0.476014
$866$	−16.4881	−0.560288
$867$	−39.6950	−1.34811
$868$	0	0
$869$	47.5131	1.61177
$870$	1.15215	0.0390617
$871$	−9.82246	−0.332821
$872$	17.4154	0.589759
$873$	3.05447	0.103378
$874$	−25.5940	−0.865730
$875$	2.54778	0.0861307
$876$	−26.6044	−0.898878
$877$	−11.6844	−0.394555	−0.197278	−	0.980348i	$-0.563210\pi$
$877$	−11.6844	−0.394555	−0.197278	+	0.980348i	$0.563210\pi$
$878$	13.1677	0.444387
$879$	−25.6899	−0.866500
$880$	3.74921	0.126386
$881$	−5.17739	−0.174431	−0.0872153	−	0.996189i	$-0.527797\pi$
$881$	−5.17739	−0.174431	−0.0872153	+	0.996189i	$0.527797\pi$
$882$	−1.24772	−0.0420128
$883$	−44.6293	−1.50189	−0.750947	−	0.660362i	$-0.770403\pi$
$883$	−44.6293	−1.50189	−0.750947	+	0.660362i	$0.770403\pi$
$884$	0	0
$885$	12.3806	0.416169
$886$	−36.9087	−1.23997
$887$	52.1740	1.75183	0.875916	−	0.482463i	$-0.160258\pi$
$887$	52.1740	1.75183	0.875916	+	0.482463i	$0.160258\pi$
$888$	13.0545	0.438079
$889$	34.4017	1.15380
$890$	−11.4489	−0.383766
$891$	−38.7791	−1.29915
$892$	9.39511	0.314571
$893$	−25.4778	−0.852582
$894$	−31.0606	−1.03882
$895$	7.64453	0.255528
$896$	2.54778	0.0851154
$897$	−42.2581	−1.41096
$898$	8.05792	0.268896
$899$	0	0
$900$	2.45222	0.0817406
$901$	0	0
$902$	24.5490	0.817393
$903$	30.7176	1.02222
$904$	12.2632	0.407869
$905$	−22.9887	−0.764170
$906$	−50.0352	−1.66231
$907$	−31.0176	−1.02992	−0.514962	−	0.857213i	$-0.672194\pi$
$907$	−31.0176	−1.02992	−0.514962	+	0.857213i	$0.672194\pi$
$908$	−20.1287	−0.667994
$909$	−41.7309	−1.38413
$910$	−7.57846	−0.251223
$911$	19.7469	0.654243	0.327122	−	0.944982i	$-0.393921\pi$
$911$	19.7469	0.654243	0.327122	+	0.944982i	$0.393921\pi$
$912$	−9.82246	−0.325254
$913$	−52.2675	−1.72980
$914$	−30.7903	−1.01845
$915$	−31.8697	−1.05358
$916$	4.57900	0.151295
$917$	9.66469	0.319156
$918$	0	0
$919$	−56.3396	−1.85847	−0.929235	−	0.369489i	$-0.879533\pi$
$919$	−56.3396	−1.85847	−0.929235	+	0.369489i	$0.879533\pi$
$920$	6.08421	0.200590
$921$	35.7668	1.17856
$922$	−13.8088	−0.454768
$923$	16.3861	0.539356
$924$	22.3043	0.733758
$925$	5.59078	0.183824
$926$	19.7220	0.648104
$927$	23.3177	0.765853
$928$	0.493428	0.0161976
$929$	−54.4076	−1.78505	−0.892527	−	0.450993i	$-0.851070\pi$
$929$	−54.4076	−1.78505	−0.892527	+	0.450993i	$0.851070\pi$
$930$	0	0
$931$	2.14038	0.0701481
$932$	−15.8500	−0.519183
$933$	17.7151	0.579965
$934$	−14.6044	−0.477870
$935$	0	0
$936$	−7.29421	−0.238419
$937$	26.7955	0.875371	0.437685	−	0.899128i	$-0.355798\pi$
$937$	26.7955	0.875371	0.437685	+	0.899128i	$0.355798\pi$
$938$	8.41325	0.274702
$939$	33.6530	1.09822
$940$	6.05659	0.197544
$941$	23.6833	0.772055	0.386028	−	0.922487i	$-0.373847\pi$
$941$	23.6833	0.772055	0.386028	+	0.922487i	$0.373847\pi$
$942$	21.0561	0.686046
$943$	39.8381	1.29731
$944$	5.30219	0.172571
$945$	−3.25879	−0.106008
$946$	19.3588	0.629408
$947$	29.8144	0.968838	0.484419	−	0.874836i	$-0.339031\pi$
$947$	29.8144	0.968838	0.484419	+	0.874836i	$0.339031\pi$
$948$	29.5910	0.961071
$949$	33.8911	1.10015
$950$	−4.20662	−0.136481
$951$	−40.8469	−1.32455
$952$	0	0
$953$	−12.1103	−0.392291	−0.196146	−	0.980575i	$-0.562843\pi$
$953$	−12.1103	−0.392291	−0.196146	+	0.980575i	$0.562843\pi$
$954$	−18.7461	−0.606926
$955$	5.84997	0.189301
$956$	−13.8499	−0.447939
$957$	4.31967	0.139635
$958$	13.8911	0.448800
$959$	42.6967	1.37875
$960$	2.33500	0.0753618
$961$	0	0
$962$	−16.6300	−0.536172
$963$	19.6178	0.632173
$964$	−8.40825	−0.270811
$965$	19.2087	0.618351
$966$	36.1954	1.16457
$967$	44.3510	1.42623	0.713116	−	0.701046i	$-0.247283\pi$
$967$	44.3510	1.42623	0.713116	+	0.701046i	$0.247283\pi$
$968$	3.05659	0.0982426
$969$	0	0
$970$	1.24559	0.0399936
$971$	−54.6044	−1.75234	−0.876169	−	0.482004i	$-0.839909\pi$
$971$	−54.6044	−1.75234	−0.876169	+	0.482004i	$0.839909\pi$
$972$	−20.3143	−0.651583
$973$	45.0427	1.44400
$974$	−3.62803	−0.116250
$975$	−6.94553	−0.222435
$976$	−13.6487	−0.436884
$977$	−33.7309	−1.07915	−0.539574	−	0.841938i	$-0.681415\pi$
$977$	−33.7309	−1.07915	−0.539574	+	0.841938i	$0.681415\pi$
$978$	−28.9487	−0.925678
$979$	−42.9242	−1.37186
$980$	−0.508811	−0.0162534
$981$	42.7063	1.36351
$982$	−25.7620	−0.822099
$983$	−9.78927	−0.312229	−0.156115	−	0.987739i	$-0.549897\pi$
$983$	−9.78927	−0.312229	−0.156115	+	0.987739i	$0.549897\pi$
$984$	15.2891	0.487397
$985$	−5.02828	−0.160214
$986$	0	0
$987$	36.0311	1.14688
$988$	12.5127	0.398083
$989$	31.4154	0.998951
$990$	9.19389	0.292201
$991$	−8.16288	−0.259302	−0.129651	−	0.991560i	$-0.541386\pi$
$991$	−8.16288	−0.259302	−0.129651	+	0.991560i	$0.541386\pi$
$992$	0	0
$993$	20.8366	0.661229
$994$	−14.0352	−0.445171
$995$	19.9591	0.632745
$996$	−32.5520	−1.03145
$997$	−32.5419	−1.03061	−0.515307	−	0.857006i	$-0.672322\pi$
$997$	−32.5419	−1.03061	−0.515307	+	0.857006i	$0.672322\pi$
$998$	19.4626	0.616079
$999$	−7.15099	−0.226247

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9610.2.a.bz.1.5	yes	6
31.30	odd	2	inner	9610.2.a.bz.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9610.2.a.bz.1.2	✓	6	31.30	odd	2	inner
9610.2.a.bz.1.5	yes	6	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:	\( N \)	\(=\)	\( 9610 = 2 \cdot 5 \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9610.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.33500 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.33500 q^{6} +2.54778 q^{7} +1.00000 q^{8} +2.45222 q^{9} +1.00000 q^{10} +3.74921 q^{11} +2.33500 q^{12} -2.97453 q^{13} +2.54778 q^{14} +2.33500 q^{15} +1.00000 q^{16} +2.45222 q^{18} -4.20662 q^{19} +1.00000 q^{20} +5.94907 q^{21} +3.74921 q^{22} +6.08421 q^{23} +2.33500 q^{24} +1.00000 q^{25} -2.97453 q^{26} -1.27907 q^{27} +2.54778 q^{28} +0.493428 q^{29} +2.33500 q^{30} +1.00000 q^{32} +8.75441 q^{33} +2.54778 q^{35} +2.45222 q^{36} +5.59078 q^{37} -4.20662 q^{38} -6.94553 q^{39} +1.00000 q^{40} +6.54778 q^{41} +5.94907 q^{42} +5.16343 q^{43} +3.74921 q^{44} +2.45222 q^{45} +6.08421 q^{46} +6.05659 q^{47} +2.33500 q^{48} -0.508811 q^{49} +1.00000 q^{50} -2.97453 q^{52} -7.64453 q^{53} -1.27907 q^{54} +3.74921 q^{55} +2.54778 q^{56} -9.82246 q^{57} +0.493428 q^{58} +5.30219 q^{59} +2.33500 q^{60} -13.6487 q^{61} +6.24772 q^{63} +1.00000 q^{64} -2.97453 q^{65} +8.75441 q^{66} +3.30219 q^{67} +14.2066 q^{69} +2.54778 q^{70} -5.50881 q^{71} +2.45222 q^{72} -11.3937 q^{73} +5.59078 q^{74} +2.33500 q^{75} -4.20662 q^{76} +9.55217 q^{77} -6.94553 q^{78} +12.6728 q^{79} +1.00000 q^{80} -10.3433 q^{81} +6.54778 q^{82} -13.9409 q^{83} +5.94907 q^{84} +5.16343 q^{86} +1.15215 q^{87} +3.74921 q^{88} -11.4489 q^{89} +2.45222 q^{90} -7.57846 q^{91} +6.08421 q^{92} +6.05659 q^{94} -4.20662 q^{95} +2.33500 q^{96} +1.24559 q^{97} -0.508811 q^{98} +9.19389 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + 6 q^{4} + 6 q^{5} + 8 q^{7} + 6 q^{8} + 22 q^{9} + 6 q^{10} + 8 q^{14} + 6 q^{16} + 22 q^{18} - 12 q^{19} + 6 q^{20} + 6 q^{25} + 8 q^{28} + 6 q^{32} + 32 q^{33} + 8 q^{35} + 22 q^{36} - 12 q^{38}+ \cdots + 38 q^{98}+O(q^{100}) \) 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^5 + 8 * q^7 + 6 * q^8 + 22 * q^9 + 6 * q^10 + 8 * q^14 + 6 * q^16 + 22 * q^18 - 12 * q^19 + 6 * q^20 + 6 * q^25 + 8 * q^28 + 6 * q^32 + 32 * q^33 + 8 * q^35 + 22 * q^36 - 12 * q^38 + 8 * q^39 + 6 * q^40 + 32 * q^41 + 22 * q^45 - 12 * q^47 + 38 * q^49 + 6 * q^50 + 8 * q^56 + 4 * q^59 - 40 * q^63 + 6 * q^64 + 32 * q^66 - 8 * q^67 + 72 * q^69 + 8 * q^70 + 8 * q^71 + 22 * q^72 - 12 * q^76 + 8 * q^78 + 6 * q^80 + 30 * q^81 + 32 * q^82 - 56 * q^87 + 22 * q^90 - 12 * q^94 - 12 * q^95 + 28 * q^97 + 38 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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