LMFDB - Embedded newform 3726.2.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3726,2,Mod(1,3726)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3726, 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("3726.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3726 = 2 \cdot 3^{4} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3726.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:	$29.7522597931$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 34x^{6} - 4x^{5} + 370x^{4} + 104x^{3} - 1466x^{2} - 564x + 1401 $ x^8 - 34x^6 - 4x^5 + 370x^4 + 104x^3 - 1466x^2 - 564x + 1401
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.14814$ of defining polynomial
Character	$\chi$	$=$	3726.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -4.14814 q^{5} +4.86591 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -4.14814 q^{5} +4.86591 q^{7} -1.00000 q^{8} +4.14814 q^{10} -1.41795 q^{11} -3.73684 q^{13} -4.86591 q^{14} +1.00000 q^{16} +3.73820 q^{17} -2.34040 q^{19} -4.14814 q^{20} +1.41795 q^{22} -1.00000 q^{23} +12.2071 q^{25} +3.73684 q^{26} +4.86591 q^{28} +3.63885 q^{29} -4.87070 q^{31} -1.00000 q^{32} -3.73820 q^{34} -20.1845 q^{35} +4.49900 q^{37} +2.34040 q^{38} +4.14814 q^{40} +0.129068 q^{41} -1.00665 q^{43} -1.41795 q^{44} +1.00000 q^{46} -9.34333 q^{47} +16.6771 q^{49} -12.2071 q^{50} -3.73684 q^{52} -10.4832 q^{53} +5.88187 q^{55} -4.86591 q^{56} -3.63885 q^{58} +8.77750 q^{59} -3.02444 q^{61} +4.87070 q^{62} +1.00000 q^{64} +15.5010 q^{65} +0.452693 q^{67} +3.73820 q^{68} +20.1845 q^{70} +14.6842 q^{71} -6.63132 q^{73} -4.49900 q^{74} -2.34040 q^{76} -6.89963 q^{77} -11.0765 q^{79} -4.14814 q^{80} -0.129068 q^{82} +13.3401 q^{83} -15.5066 q^{85} +1.00665 q^{86} +1.41795 q^{88} +12.9522 q^{89} -18.1831 q^{91} -1.00000 q^{92} +9.34333 q^{94} +9.70832 q^{95} +17.7751 q^{97} -16.6771 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 8 q^{4} + 6 q^{7} - 8 q^{8}+O(q^{10}) $ 8 * q - 8 * q^2 + 8 * q^4 + 6 * q^7 - 8 * q^8	$ 8 q - 8 q^{2} + 8 q^{4} + 6 q^{7} - 8 q^{8} + 6 q^{13} - 6 q^{14} + 8 q^{16} + 10 q^{17} + 12 q^{19} - 8 q^{23} + 28 q^{25} - 6 q^{26} + 6 q^{28} - 6 q^{29} + 16 q^{31} - 8 q^{32} - 10 q^{34} - 18 q^{35} + 4 q^{37} - 12 q^{38} + 4 q^{41} + 6 q^{43} + 8 q^{46} - 14 q^{47} + 28 q^{49} - 28 q^{50} + 6 q^{52} - 20 q^{53} + 16 q^{55} - 6 q^{56} + 6 q^{58} - 6 q^{59} + 12 q^{61} - 16 q^{62} + 8 q^{64} + 10 q^{65} + 10 q^{67} + 10 q^{68} + 18 q^{70} + 18 q^{71} + 44 q^{73} - 4 q^{74} + 12 q^{76} - 28 q^{77} + 22 q^{79} - 4 q^{82} - 4 q^{83} + 22 q^{85} - 6 q^{86} + 18 q^{89} + 14 q^{91} - 8 q^{92} + 14 q^{94} + 20 q^{95} + 36 q^{97} - 28 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 + 8 * q^4 + 6 * q^7 - 8 * q^8 + 6 * q^13 - 6 * q^14 + 8 * q^16 + 10 * q^17 + 12 * q^19 - 8 * q^23 + 28 * q^25 - 6 * q^26 + 6 * q^28 - 6 * q^29 + 16 * q^31 - 8 * q^32 - 10 * q^34 - 18 * q^35 + 4 * q^37 - 12 * q^38 + 4 * q^41 + 6 * q^43 + 8 * q^46 - 14 * q^47 + 28 * q^49 - 28 * q^50 + 6 * q^52 - 20 * q^53 + 16 * q^55 - 6 * q^56 + 6 * q^58 - 6 * q^59 + 12 * q^61 - 16 * q^62 + 8 * q^64 + 10 * q^65 + 10 * q^67 + 10 * q^68 + 18 * q^70 + 18 * q^71 + 44 * q^73 - 4 * q^74 + 12 * q^76 - 28 * q^77 + 22 * q^79 - 4 * q^82 - 4 * q^83 + 22 * q^85 - 6 * q^86 + 18 * q^89 + 14 * q^91 - 8 * q^92 + 14 * q^94 + 20 * q^95 + 36 * q^97 - 28 * q^98

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
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−4.14814	−1.85511	−0.927553	−	0.373691i	$-0.878092\pi$
$5$	−4.14814	−1.85511	−0.927553	+	0.373691i	$0.878092\pi$
$6$	0	0
$7$	4.86591	1.83914	0.919570	−	0.392926i	$-0.128537\pi$
$7$	4.86591	1.83914	0.919570	+	0.392926i	$0.128537\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	4.14814	1.31176
$11$	−1.41795	−0.427529	−0.213765	−	0.976885i	$-0.568573\pi$
$11$	−1.41795	−0.427529	−0.213765	+	0.976885i	$0.568573\pi$
$12$	0	0
$13$	−3.73684	−1.03641	−0.518207	−	0.855256i	$-0.673400\pi$
$13$	−3.73684	−1.03641	−0.518207	+	0.855256i	$0.673400\pi$
$14$	−4.86591	−1.30047
$15$	0	0
$16$	1.00000	0.250000
$17$	3.73820	0.906647	0.453324	−	0.891346i	$-0.350238\pi$
$17$	3.73820	0.906647	0.453324	+	0.891346i	$0.350238\pi$
$18$	0	0
$19$	−2.34040	−0.536925	−0.268462	−	0.963290i	$-0.586516\pi$
$19$	−2.34040	−0.536925	−0.268462	+	0.963290i	$0.586516\pi$
$20$	−4.14814	−0.927553
$21$	0	0
$22$	1.41795	0.302309
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	12.2071	2.44142
$26$	3.73684	0.732855
$27$	0	0
$28$	4.86591	0.919570
$29$	3.63885	0.675718	0.337859	−	0.941197i	$-0.390297\pi$
$29$	3.63885	0.675718	0.337859	+	0.941197i	$0.390297\pi$
$30$	0	0
$31$	−4.87070	−0.874803	−0.437402	−	0.899266i	$-0.644101\pi$
$31$	−4.87070	−0.874803	−0.437402	+	0.899266i	$0.644101\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−3.73820	−0.641096
$35$	−20.1845	−3.41180
$36$	0	0
$37$	4.49900	0.739632	0.369816	−	0.929105i	$-0.379421\pi$
$37$	4.49900	0.739632	0.369816	+	0.929105i	$0.379421\pi$
$38$	2.34040	0.379663
$39$	0	0
$40$	4.14814	0.655879
$41$	0.129068	0.0201570	0.0100785	−	0.999949i	$-0.496792\pi$
$41$	0.129068	0.0201570	0.0100785	+	0.999949i	$0.496792\pi$
$42$	0	0
$43$	−1.00665	−0.153513	−0.0767564	−	0.997050i	$-0.524456\pi$
$43$	−1.00665	−0.153513	−0.0767564	+	0.997050i	$0.524456\pi$
$44$	−1.41795	−0.213765
$45$	0	0
$46$	1.00000	0.147442
$47$	−9.34333	−1.36286	−0.681432	−	0.731881i	$-0.738643\pi$
$47$	−9.34333	−1.36286	−0.681432	+	0.731881i	$0.738643\pi$
$48$	0	0
$49$	16.6771	2.38244
$50$	−12.2071	−1.72634
$51$	0	0
$52$	−3.73684	−0.518207
$53$	−10.4832	−1.43998	−0.719988	−	0.693987i	$-0.755853\pi$
$53$	−10.4832	−1.43998	−0.719988	+	0.693987i	$0.755853\pi$
$54$	0	0
$55$	5.88187	0.793112
$56$	−4.86591	−0.650234
$57$	0	0
$58$	−3.63885	−0.477804
$59$	8.77750	1.14273	0.571366	−	0.820695i	$-0.306414\pi$
$59$	8.77750	1.14273	0.571366	+	0.820695i	$0.306414\pi$
$60$	0	0
$61$	−3.02444	−0.387240	−0.193620	−	0.981077i	$-0.562023\pi$
$61$	−3.02444	−0.387240	−0.193620	+	0.981077i	$0.562023\pi$
$62$	4.87070	0.618579
$63$	0	0
$64$	1.00000	0.125000
$65$	15.5010	1.92266
$66$	0	0
$67$	0.452693	0.0553052	0.0276526	−	0.999618i	$-0.491197\pi$
$67$	0.452693	0.0553052	0.0276526	+	0.999618i	$0.491197\pi$
$68$	3.73820	0.453324
$69$	0	0
$70$	20.1845	2.41251
$71$	14.6842	1.74270	0.871349	−	0.490664i	$-0.163246\pi$
$71$	14.6842	1.74270	0.871349	+	0.490664i	$0.163246\pi$
$72$	0	0
$73$	−6.63132	−0.776137	−0.388069	−	0.921630i	$-0.626858\pi$
$73$	−6.63132	−0.776137	−0.388069	+	0.921630i	$0.626858\pi$
$74$	−4.49900	−0.522998
$75$	0	0
$76$	−2.34040	−0.268462
$77$	−6.89963	−0.786286
$78$	0	0
$79$	−11.0765	−1.24620	−0.623102	−	0.782140i	$-0.714128\pi$
$79$	−11.0765	−1.24620	−0.623102	+	0.782140i	$0.714128\pi$
$80$	−4.14814	−0.463777
$81$	0	0
$82$	−0.129068	−0.0142531
$83$	13.3401	1.46426	0.732131	−	0.681164i	$-0.238526\pi$
$83$	13.3401	1.46426	0.732131	+	0.681164i	$0.238526\pi$
$84$	0	0
$85$	−15.5066	−1.68193
$86$	1.00665	0.108550
$87$	0	0
$88$	1.41795	0.151154
$89$	12.9522	1.37293	0.686464	−	0.727164i	$-0.259162\pi$
$89$	12.9522	1.37293	0.686464	+	0.727164i	$0.259162\pi$
$90$	0	0
$91$	−18.1831	−1.90611
$92$	−1.00000	−0.104257
$93$	0	0
$94$	9.34333	0.963691
$95$	9.70832	0.996053
$96$	0	0
$97$	17.7751	1.80478	0.902392	−	0.430917i	$-0.141810\pi$
$97$	17.7751	1.80478	0.902392	+	0.430917i	$0.141810\pi$
$98$	−16.6771	−1.68464
$99$	0	0
$100$	12.2071	1.22071
$101$	−3.63015	−0.361213	−0.180607	−	0.983555i	$-0.557806\pi$
$101$	−3.63015	−0.361213	−0.180607	+	0.983555i	$0.557806\pi$
$102$	0	0
$103$	15.6747	1.54447	0.772237	−	0.635334i	$-0.219138\pi$
$103$	15.6747	1.54447	0.772237	+	0.635334i	$0.219138\pi$
$104$	3.73684	0.366427
$105$	0	0
$106$	10.4832	1.01822
$107$	−14.0291	−1.35625	−0.678123	−	0.734948i	$-0.737206\pi$
$107$	−14.0291	−1.35625	−0.678123	+	0.734948i	$0.737206\pi$
$108$	0	0
$109$	−19.3902	−1.85725	−0.928623	−	0.371026i	$-0.879006\pi$
$109$	−19.3902	−1.85725	−0.928623	+	0.371026i	$0.879006\pi$
$110$	−5.88187	−0.560815
$111$	0	0
$112$	4.86591	0.459785
$113$	−4.62045	−0.434655	−0.217328	−	0.976099i	$-0.569734\pi$
$113$	−4.62045	−0.434655	−0.217328	+	0.976099i	$0.569734\pi$
$114$	0	0
$115$	4.14814	0.386816
$116$	3.63885	0.337859
$117$	0	0
$118$	−8.77750	−0.808034
$119$	18.1898	1.66745
$120$	0	0
$121$	−8.98941	−0.817219
$122$	3.02444	0.273820
$123$	0	0
$124$	−4.87070	−0.437402
$125$	−29.8961	−2.67398
$126$	0	0
$127$	0.660958	0.0586506	0.0293253	−	0.999570i	$-0.490664\pi$
$127$	0.660958	0.0586506	0.0293253	+	0.999570i	$0.490664\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−15.5010	−1.35952
$131$	2.94787	0.257557	0.128778	−	0.991673i	$-0.458894\pi$
$131$	2.94787	0.257557	0.128778	+	0.991673i	$0.458894\pi$
$132$	0	0
$133$	−11.3882	−0.987480
$134$	−0.452693	−0.0391067
$135$	0	0
$136$	−3.73820	−0.320548
$137$	20.9533	1.79016	0.895082	−	0.445903i	$-0.147117\pi$
$137$	20.9533	1.79016	0.895082	+	0.445903i	$0.147117\pi$
$138$	0	0
$139$	3.57589	0.303303	0.151651	−	0.988434i	$-0.451541\pi$
$139$	3.57589	0.303303	0.151651	+	0.988434i	$0.451541\pi$
$140$	−20.1845	−1.70590
$141$	0	0
$142$	−14.6842	−1.23227
$143$	5.29867	0.443097
$144$	0	0
$145$	−15.0945	−1.25353
$146$	6.63132	0.548812
$147$	0	0
$148$	4.49900	0.369816
$149$	−17.3178	−1.41873	−0.709366	−	0.704841i	$-0.751019\pi$
$149$	−17.3178	−1.41873	−0.709366	+	0.704841i	$0.751019\pi$
$150$	0	0
$151$	−8.91536	−0.725522	−0.362761	−	0.931882i	$-0.618166\pi$
$151$	−8.91536	−0.725522	−0.362761	+	0.931882i	$0.618166\pi$
$152$	2.34040	0.189832
$153$	0	0
$154$	6.89963	0.555988
$155$	20.2044	1.62285
$156$	0	0
$157$	10.9308	0.872376	0.436188	−	0.899855i	$-0.356328\pi$
$157$	10.9308	0.872376	0.436188	+	0.899855i	$0.356328\pi$
$158$	11.0765	0.881200
$159$	0	0
$160$	4.14814	0.327940
$161$	−4.86591	−0.383487
$162$	0	0
$163$	12.4529	0.975383	0.487691	−	0.873016i	$-0.337839\pi$
$163$	12.4529	0.975383	0.487691	+	0.873016i	$0.337839\pi$
$164$	0.129068	0.0100785
$165$	0	0
$166$	−13.3401	−1.03539
$167$	−10.9306	−0.845838	−0.422919	−	0.906168i	$-0.638994\pi$
$167$	−10.9306	−0.845838	−0.422919	+	0.906168i	$0.638994\pi$
$168$	0	0
$169$	0.963977	0.0741521
$170$	15.5066	1.18930
$171$	0	0
$172$	−1.00665	−0.0767564
$173$	19.3161	1.46858	0.734288	−	0.678838i	$-0.237516\pi$
$173$	19.3161	1.46858	0.734288	+	0.678838i	$0.237516\pi$
$174$	0	0
$175$	59.3986	4.49011
$176$	−1.41795	−0.106882
$177$	0	0
$178$	−12.9522	−0.970807
$179$	10.2640	0.767167	0.383583	−	0.923506i	$-0.374690\pi$
$179$	10.2640	0.767167	0.383583	+	0.923506i	$0.374690\pi$
$180$	0	0
$181$	7.51068	0.558264	0.279132	−	0.960253i	$-0.409953\pi$
$181$	7.51068	0.558264	0.279132	+	0.960253i	$0.409953\pi$
$182$	18.1831	1.34782
$183$	0	0
$184$	1.00000	0.0737210
$185$	−18.6625	−1.37209
$186$	0	0
$187$	−5.30060	−0.387618
$188$	−9.34333	−0.681432
$189$	0	0
$190$	−9.70832	−0.704316
$191$	7.02099	0.508021	0.254011	−	0.967201i	$-0.418250\pi$
$191$	7.02099	0.508021	0.254011	+	0.967201i	$0.418250\pi$
$192$	0	0
$193$	−4.52844	−0.325964	−0.162982	−	0.986629i	$-0.552111\pi$
$193$	−4.52844	−0.325964	−0.162982	+	0.986629i	$0.552111\pi$
$194$	−17.7751	−1.27617
$195$	0	0
$196$	16.6771	1.19122
$197$	20.4925	1.46003	0.730015	−	0.683431i	$-0.239513\pi$
$197$	20.4925	1.46003	0.730015	+	0.683431i	$0.239513\pi$
$198$	0	0
$199$	25.7203	1.82326	0.911632	−	0.411008i	$-0.134823\pi$
$199$	25.7203	1.82326	0.911632	+	0.411008i	$0.134823\pi$
$200$	−12.2071	−0.863172
$201$	0	0
$202$	3.63015	0.255416
$203$	17.7063	1.24274
$204$	0	0
$205$	−0.535391	−0.0373933
$206$	−15.6747	−1.09211
$207$	0	0
$208$	−3.73684	−0.259103
$209$	3.31858	0.229551
$210$	0	0
$211$	7.11231	0.489632	0.244816	−	0.969570i	$-0.421272\pi$
$211$	7.11231	0.489632	0.244816	+	0.969570i	$0.421272\pi$
$212$	−10.4832	−0.719988
$213$	0	0
$214$	14.0291	0.959011
$215$	4.17573	0.284782
$216$	0	0
$217$	−23.7004	−1.60889
$218$	19.3902	1.31327
$219$	0	0
$220$	5.88187	0.396556
$221$	−13.9691	−0.939661
$222$	0	0
$223$	6.73644	0.451106	0.225553	−	0.974231i	$-0.427581\pi$
$223$	6.73644	0.451106	0.225553	+	0.974231i	$0.427581\pi$
$224$	−4.86591	−0.325117
$225$	0	0
$226$	4.62045	0.307348
$227$	11.5896	0.769232	0.384616	−	0.923077i	$-0.374334\pi$
$227$	11.5896	0.769232	0.384616	+	0.923077i	$0.374334\pi$
$228$	0	0
$229$	17.4743	1.15473	0.577367	−	0.816485i	$-0.304080\pi$
$229$	17.4743	1.15473	0.577367	+	0.816485i	$0.304080\pi$
$230$	−4.14814	−0.273520
$231$	0	0
$232$	−3.63885	−0.238902
$233$	23.0522	1.51020	0.755099	−	0.655611i	$-0.227589\pi$
$233$	23.0522	1.51020	0.755099	+	0.655611i	$0.227589\pi$
$234$	0	0
$235$	38.7575	2.52826
$236$	8.77750	0.571366
$237$	0	0
$238$	−18.1898	−1.17907
$239$	21.2911	1.37721	0.688604	−	0.725138i	$-0.258224\pi$
$239$	21.2911	1.37721	0.688604	+	0.725138i	$0.258224\pi$
$240$	0	0
$241$	11.0289	0.710431	0.355216	−	0.934784i	$-0.384407\pi$
$241$	11.0289	0.710431	0.355216	+	0.934784i	$0.384407\pi$
$242$	8.98941	0.577861
$243$	0	0
$244$	−3.02444	−0.193620
$245$	−69.1788	−4.41967
$246$	0	0
$247$	8.74571	0.556476
$248$	4.87070	0.309290
$249$	0	0
$250$	29.8961	1.89079
$251$	−12.8777	−0.812832	−0.406416	−	0.913688i	$-0.633222\pi$
$251$	−12.8777	−0.812832	−0.406416	+	0.913688i	$0.633222\pi$
$252$	0	0
$253$	1.41795	0.0891460
$254$	−0.660958	−0.0414722
$255$	0	0
$256$	1.00000	0.0625000
$257$	1.27756	0.0796917	0.0398459	−	0.999206i	$-0.487313\pi$
$257$	1.27756	0.0796917	0.0398459	+	0.999206i	$0.487313\pi$
$258$	0	0
$259$	21.8917	1.36029
$260$	15.5010	0.961328
$261$	0	0
$262$	−2.94787	−0.182120
$263$	2.79063	0.172078	0.0860389	−	0.996292i	$-0.472579\pi$
$263$	2.79063	0.172078	0.0860389	+	0.996292i	$0.472579\pi$
$264$	0	0
$265$	43.4857	2.67131
$266$	11.3882	0.698254
$267$	0	0
$268$	0.452693	0.0276526
$269$	17.9799	1.09625	0.548126	−	0.836396i	$-0.315341\pi$
$269$	17.9799	1.09625	0.548126	+	0.836396i	$0.315341\pi$
$270$	0	0
$271$	−3.97419	−0.241415	−0.120707	−	0.992688i	$-0.538516\pi$
$271$	−3.97419	−0.241415	−0.120707	+	0.992688i	$0.538516\pi$
$272$	3.73820	0.226662
$273$	0	0
$274$	−20.9533	−1.26584
$275$	−17.3091	−1.04378
$276$	0	0
$277$	−2.88526	−0.173359	−0.0866793	−	0.996236i	$-0.527626\pi$
$277$	−2.88526	−0.173359	−0.0866793	+	0.996236i	$0.527626\pi$
$278$	−3.57589	−0.214467
$279$	0	0
$280$	20.1845	1.20625
$281$	−10.1115	−0.603202	−0.301601	−	0.953434i	$-0.597521\pi$
$281$	−10.1115	−0.603202	−0.301601	+	0.953434i	$0.597521\pi$
$282$	0	0
$283$	16.3424	0.971453	0.485726	−	0.874111i	$-0.338555\pi$
$283$	16.3424	0.971453	0.485726	+	0.874111i	$0.338555\pi$
$284$	14.6842	0.871349
$285$	0	0
$286$	−5.29867	−0.313317
$287$	0.628031	0.0370715
$288$	0	0
$289$	−3.02584	−0.177991
$290$	15.0945	0.886378
$291$	0	0
$292$	−6.63132	−0.388069
$293$	5.77282	0.337252	0.168626	−	0.985680i	$-0.446067\pi$
$293$	5.77282	0.337252	0.168626	+	0.985680i	$0.446067\pi$
$294$	0	0
$295$	−36.4103	−2.11989
$296$	−4.49900	−0.261499
$297$	0	0
$298$	17.3178	1.00319
$299$	3.73684	0.216107
$300$	0	0
$301$	−4.89827	−0.282332
$302$	8.91536	0.513021
$303$	0	0
$304$	−2.34040	−0.134231
$305$	12.5458	0.718372
$306$	0	0
$307$	23.0450	1.31525	0.657624	−	0.753347i	$-0.271562\pi$
$307$	23.0450	1.31525	0.657624	+	0.753347i	$0.271562\pi$
$308$	−6.89963	−0.393143
$309$	0	0
$310$	−20.2044	−1.14753
$311$	5.12004	0.290331	0.145165	−	0.989407i	$-0.453629\pi$
$311$	5.12004	0.290331	0.145165	+	0.989407i	$0.453629\pi$
$312$	0	0
$313$	−29.2617	−1.65397	−0.826985	−	0.562223i	$-0.809946\pi$
$313$	−29.2617	−1.65397	−0.826985	+	0.562223i	$0.809946\pi$
$314$	−10.9308	−0.616863
$315$	0	0
$316$	−11.0765	−0.623102
$317$	10.4161	0.585027	0.292514	−	0.956261i	$-0.405508\pi$
$317$	10.4161	0.585027	0.292514	+	0.956261i	$0.405508\pi$
$318$	0	0
$319$	−5.15972	−0.288889
$320$	−4.14814	−0.231888
$321$	0	0
$322$	4.86591	0.271166
$323$	−8.74889	−0.486801
$324$	0	0
$325$	−45.6160	−2.53032
$326$	−12.4529	−0.689700
$327$	0	0
$328$	−0.129068	−0.00712656
$329$	−45.4638	−2.50650
$330$	0	0
$331$	−26.9487	−1.48123	−0.740617	−	0.671928i	$-0.765467\pi$
$331$	−26.9487	−1.48123	−0.740617	+	0.671928i	$0.765467\pi$
$332$	13.3401	0.732131
$333$	0	0
$334$	10.9306	0.598098
$335$	−1.87783	−0.102597
$336$	0	0
$337$	−3.56600	−0.194252	−0.0971261	−	0.995272i	$-0.530965\pi$
$337$	−3.56600	−0.194252	−0.0971261	+	0.995272i	$0.530965\pi$
$338$	−0.963977	−0.0524334
$339$	0	0
$340$	−15.5066	−0.840964
$341$	6.90642	0.374004
$342$	0	0
$343$	47.0877	2.54250
$344$	1.00665	0.0542750
$345$	0	0
$346$	−19.3161	−1.03844
$347$	−20.9710	−1.12578	−0.562892	−	0.826530i	$-0.690311\pi$
$347$	−20.9710	−1.12578	−0.562892	+	0.826530i	$0.690311\pi$
$348$	0	0
$349$	18.6452	0.998056	0.499028	−	0.866586i	$-0.333691\pi$
$349$	18.6452	0.998056	0.499028	+	0.866586i	$0.333691\pi$
$350$	−59.3986	−3.17499
$351$	0	0
$352$	1.41795	0.0755772
$353$	4.15881	0.221351	0.110676	−	0.993857i	$-0.464699\pi$
$353$	4.15881	0.221351	0.110676	+	0.993857i	$0.464699\pi$
$354$	0	0
$355$	−60.9123	−3.23289
$356$	12.9522	0.686464
$357$	0	0
$358$	−10.2640	−0.542469
$359$	−8.48135	−0.447629	−0.223814	−	0.974632i	$-0.571851\pi$
$359$	−8.48135	−0.447629	−0.223814	+	0.974632i	$0.571851\pi$
$360$	0	0
$361$	−13.5225	−0.711712
$362$	−7.51068	−0.394752
$363$	0	0
$364$	−18.1831	−0.953055
$365$	27.5077	1.43982
$366$	0	0
$367$	36.9005	1.92619	0.963096	−	0.269160i	$-0.0867460\pi$
$367$	36.9005	1.92619	0.963096	+	0.269160i	$0.0867460\pi$
$368$	−1.00000	−0.0521286
$369$	0	0
$370$	18.6625	0.970218
$371$	−51.0102	−2.64832
$372$	0	0
$373$	−8.52660	−0.441490	−0.220745	−	0.975332i	$-0.570849\pi$
$373$	−8.52660	−0.441490	−0.220745	+	0.975332i	$0.570849\pi$
$374$	5.30060	0.274087
$375$	0	0
$376$	9.34333	0.481845
$377$	−13.5978	−0.700322
$378$	0	0
$379$	38.1466	1.95946	0.979729	−	0.200329i	$-0.0642010\pi$
$379$	38.1466	1.95946	0.979729	+	0.200329i	$0.0642010\pi$
$380$	9.70832	0.498026
$381$	0	0
$382$	−7.02099	−0.359225
$383$	10.6321	0.543274	0.271637	−	0.962400i	$-0.412435\pi$
$383$	10.6321	0.543274	0.271637	+	0.962400i	$0.412435\pi$
$384$	0	0
$385$	28.6207	1.45864
$386$	4.52844	0.230491
$387$	0	0
$388$	17.7751	0.902392
$389$	13.6640	0.692792	0.346396	−	0.938088i	$-0.387405\pi$
$389$	13.6640	0.692792	0.346396	+	0.938088i	$0.387405\pi$
$390$	0	0
$391$	−3.73820	−0.189049
$392$	−16.6771	−0.842319
$393$	0	0
$394$	−20.4925	−1.03240
$395$	45.9469	2.31184
$396$	0	0
$397$	23.5500	1.18194	0.590971	−	0.806693i	$-0.298745\pi$
$397$	23.5500	1.18194	0.590971	+	0.806693i	$0.298745\pi$
$398$	−25.7203	−1.28924
$399$	0	0
$400$	12.2071	0.610355
$401$	−0.331172	−0.0165379	−0.00826897	−	0.999966i	$-0.502632\pi$
$401$	−0.331172	−0.0165379	−0.00826897	+	0.999966i	$0.502632\pi$
$402$	0	0
$403$	18.2010	0.906657
$404$	−3.63015	−0.180607
$405$	0	0
$406$	−17.7063	−0.878749
$407$	−6.37938	−0.316214
$408$	0	0
$409$	5.04986	0.249700	0.124850	−	0.992176i	$-0.460155\pi$
$409$	5.04986	0.249700	0.124850	+	0.992176i	$0.460155\pi$
$410$	0.535391	0.0264411
$411$	0	0
$412$	15.6747	0.772237
$413$	42.7105	2.10165
$414$	0	0
$415$	−55.3365	−2.71636
$416$	3.73684	0.183214
$417$	0	0
$418$	−3.31858	−0.162317
$419$	3.32908	0.162636	0.0813180	−	0.996688i	$-0.474087\pi$
$419$	3.32908	0.162636	0.0813180	+	0.996688i	$0.474087\pi$
$420$	0	0
$421$	13.9129	0.678073	0.339036	−	0.940773i	$-0.389899\pi$
$421$	13.9129	0.678073	0.339036	+	0.940773i	$0.389899\pi$
$422$	−7.11231	−0.346222
$423$	0	0
$424$	10.4832	0.509108
$425$	45.6326	2.21351
$426$	0	0
$427$	−14.7167	−0.712189
$428$	−14.0291	−0.678123
$429$	0	0
$430$	−4.17573	−0.201372
$431$	10.3637	0.499203	0.249602	−	0.968349i	$-0.419700\pi$
$431$	10.3637	0.499203	0.249602	+	0.968349i	$0.419700\pi$
$432$	0	0
$433$	40.5689	1.94962	0.974808	−	0.223044i	$-0.0715994\pi$
$433$	40.5689	1.94962	0.974808	+	0.223044i	$0.0715994\pi$
$434$	23.7004	1.13765
$435$	0	0
$436$	−19.3902	−0.928623
$437$	2.34040	0.111957
$438$	0	0
$439$	−3.79709	−0.181225	−0.0906125	−	0.995886i	$-0.528882\pi$
$439$	−3.79709	−0.181225	−0.0906125	+	0.995886i	$0.528882\pi$
$440$	−5.88187	−0.280407
$441$	0	0
$442$	13.9691	0.664441
$443$	−3.20569	−0.152307	−0.0761535	−	0.997096i	$-0.524264\pi$
$443$	−3.20569	−0.152307	−0.0761535	+	0.997096i	$0.524264\pi$
$444$	0	0
$445$	−53.7275	−2.54693
$446$	−6.73644	−0.318980
$447$	0	0
$448$	4.86591	0.229893
$449$	27.6166	1.30331	0.651654	−	0.758516i	$-0.274075\pi$
$449$	27.6166	1.30331	0.651654	+	0.758516i	$0.274075\pi$
$450$	0	0
$451$	−0.183012	−0.00861769
$452$	−4.62045	−0.217328
$453$	0	0
$454$	−11.5896	−0.543929
$455$	75.4262	3.53603
$456$	0	0
$457$	−18.5467	−0.867578	−0.433789	−	0.901014i	$-0.642824\pi$
$457$	−18.5467	−0.867578	−0.433789	+	0.901014i	$0.642824\pi$
$458$	−17.4743	−0.816520
$459$	0	0
$460$	4.14814	0.193408
$461$	6.71116	0.312570	0.156285	−	0.987712i	$-0.450048\pi$
$461$	6.71116	0.312570	0.156285	+	0.987712i	$0.450048\pi$
$462$	0	0
$463$	−0.386694	−0.0179712	−0.00898560	−	0.999960i	$-0.502860\pi$
$463$	−0.386694	−0.0179712	−0.00898560	+	0.999960i	$0.502860\pi$
$464$	3.63885	0.168929
$465$	0	0
$466$	−23.0522	−1.06787
$467$	−19.9825	−0.924680	−0.462340	−	0.886703i	$-0.652990\pi$
$467$	−19.9825	−0.924680	−0.462340	+	0.886703i	$0.652990\pi$
$468$	0	0
$469$	2.20276	0.101714
$470$	−38.7575	−1.78775
$471$	0	0
$472$	−8.77750	−0.404017
$473$	1.42738	0.0656312
$474$	0	0
$475$	−28.5695	−1.31086
$476$	18.1898	0.833726
$477$	0	0
$478$	−21.2911	−0.973833
$479$	−4.67131	−0.213438	−0.106719	−	0.994289i	$-0.534034\pi$
$479$	−4.67131	−0.213438	−0.106719	+	0.994289i	$0.534034\pi$
$480$	0	0
$481$	−16.8121	−0.766564
$482$	−11.0289	−0.502351
$483$	0	0
$484$	−8.98941	−0.408609
$485$	−73.7335	−3.34806
$486$	0	0
$487$	−20.3583	−0.922523	−0.461261	−	0.887264i	$-0.652603\pi$
$487$	−20.3583	−0.922523	−0.461261	+	0.887264i	$0.652603\pi$
$488$	3.02444	0.136910
$489$	0	0
$490$	69.1788	3.12518
$491$	−24.6441	−1.11217	−0.556087	−	0.831124i	$-0.687698\pi$
$491$	−24.6441	−1.11217	−0.556087	+	0.831124i	$0.687698\pi$
$492$	0	0
$493$	13.6028	0.612637
$494$	−8.74571	−0.393488
$495$	0	0
$496$	−4.87070	−0.218701
$497$	71.4521	3.20507
$498$	0	0
$499$	29.1624	1.30549	0.652745	−	0.757578i	$-0.273617\pi$
$499$	29.1624	1.30549	0.652745	+	0.757578i	$0.273617\pi$
$500$	−29.8961	−1.33699
$501$	0	0
$502$	12.8777	0.574759
$503$	−40.0746	−1.78684	−0.893420	−	0.449222i	$-0.851701\pi$
$503$	−40.0746	−1.78684	−0.893420	+	0.449222i	$0.851701\pi$
$504$	0	0
$505$	15.0584	0.670089
$506$	−1.41795	−0.0630357
$507$	0	0
$508$	0.660958	0.0293253
$509$	5.13505	0.227607	0.113804	−	0.993503i	$-0.463697\pi$
$509$	5.13505	0.227607	0.113804	+	0.993503i	$0.463697\pi$
$510$	0	0
$511$	−32.2674	−1.42743
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−1.27756	−0.0563506
$515$	−65.0209	−2.86516
$516$	0	0
$517$	13.2484	0.582664
$518$	−21.8917	−0.961868
$519$	0	0
$520$	−15.5010	−0.679762
$521$	8.27776	0.362655	0.181328	−	0.983423i	$-0.441961\pi$
$521$	8.27776	0.362655	0.181328	+	0.983423i	$0.441961\pi$
$522$	0	0
$523$	−14.7312	−0.644150	−0.322075	−	0.946714i	$-0.604380\pi$
$523$	−14.7312	−0.644150	−0.322075	+	0.946714i	$0.604380\pi$
$524$	2.94787	0.128778
$525$	0	0
$526$	−2.79063	−0.121677
$527$	−18.2077	−0.793138
$528$	0	0
$529$	1.00000	0.0434783
$530$	−43.4857	−1.88890
$531$	0	0
$532$	−11.3882	−0.493740
$533$	−0.482305	−0.0208909
$534$	0	0
$535$	58.1948	2.51598
$536$	−0.452693	−0.0195534
$537$	0	0
$538$	−17.9799	−0.775167
$539$	−23.6473	−1.01856
$540$	0	0
$541$	−18.1301	−0.779473	−0.389737	−	0.920926i	$-0.627434\pi$
$541$	−18.1301	−0.779473	−0.389737	+	0.920926i	$0.627434\pi$
$542$	3.97419	0.170706
$543$	0	0
$544$	−3.73820	−0.160274
$545$	80.4334	3.44539
$546$	0	0
$547$	−28.3366	−1.21159	−0.605793	−	0.795622i	$-0.707144\pi$
$547$	−28.3366	−1.21159	−0.605793	+	0.795622i	$0.707144\pi$
$548$	20.9533	0.895082
$549$	0	0
$550$	17.3091	0.738062
$551$	−8.51637	−0.362810
$552$	0	0
$553$	−53.8973	−2.29195
$554$	2.88526	0.122583
$555$	0	0
$556$	3.57589	0.151651
$557$	12.0024	0.508556	0.254278	−	0.967131i	$-0.418162\pi$
$557$	12.0024	0.508556	0.254278	+	0.967131i	$0.418162\pi$
$558$	0	0
$559$	3.76169	0.159103
$560$	−20.1845	−0.852950
$561$	0	0
$562$	10.1115	0.426529
$563$	3.80731	0.160459	0.0802294	−	0.996776i	$-0.474435\pi$
$563$	3.80731	0.160459	0.0802294	+	0.996776i	$0.474435\pi$
$564$	0	0
$565$	19.1663	0.806331
$566$	−16.3424	−0.686921
$567$	0	0
$568$	−14.6842	−0.616137
$569$	−6.61795	−0.277439	−0.138720	−	0.990332i	$-0.544299\pi$
$569$	−6.61795	−0.277439	−0.138720	+	0.990332i	$0.544299\pi$
$570$	0	0
$571$	−1.66576	−0.0697098	−0.0348549	−	0.999392i	$-0.511097\pi$
$571$	−1.66576	−0.0697098	−0.0348549	+	0.999392i	$0.511097\pi$
$572$	5.29867	0.221548
$573$	0	0
$574$	−0.628031	−0.0262135
$575$	−12.2071	−0.509071
$576$	0	0
$577$	25.7029	1.07002	0.535012	−	0.844844i	$-0.320307\pi$
$577$	25.7029	1.07002	0.535012	+	0.844844i	$0.320307\pi$
$578$	3.02584	0.125858
$579$	0	0
$580$	−15.0945	−0.626764
$581$	64.9115	2.69298
$582$	0	0
$583$	14.8647	0.615631
$584$	6.63132	0.274406
$585$	0	0
$586$	−5.77282	−0.238473
$587$	18.1885	0.750721	0.375360	−	0.926879i	$-0.377519\pi$
$587$	18.1885	0.750721	0.375360	+	0.926879i	$0.377519\pi$
$588$	0	0
$589$	11.3994	0.469704
$590$	36.4103	1.49899
$591$	0	0
$592$	4.49900	0.184908
$593$	1.77894	0.0730525	0.0365262	−	0.999333i	$-0.488371\pi$
$593$	1.77894	0.0730525	0.0365262	+	0.999333i	$0.488371\pi$
$594$	0	0
$595$	−75.4537	−3.09330
$596$	−17.3178	−0.709366
$597$	0	0
$598$	−3.73684	−0.152811
$599$	−0.0930288	−0.00380105	−0.00190053	−	0.999998i	$-0.500605\pi$
$599$	−0.0930288	−0.00380105	−0.00190053	+	0.999998i	$0.500605\pi$
$600$	0	0
$601$	−27.4030	−1.11779	−0.558896	−	0.829237i	$-0.688775\pi$
$601$	−27.4030	−1.11779	−0.558896	+	0.829237i	$0.688775\pi$
$602$	4.89827	0.199639
$603$	0	0
$604$	−8.91536	−0.362761
$605$	37.2894	1.51603
$606$	0	0
$607$	−47.2008	−1.91582	−0.957910	−	0.287069i	$-0.907319\pi$
$607$	−47.2008	−1.91582	−0.957910	+	0.287069i	$0.907319\pi$
$608$	2.34040	0.0949158
$609$	0	0
$610$	−12.5458	−0.507966
$611$	34.9145	1.41249
$612$	0	0
$613$	−47.9126	−1.93517	−0.967586	−	0.252543i	$-0.918733\pi$
$613$	−47.9126	−1.93517	−0.967586	+	0.252543i	$0.918733\pi$
$614$	−23.0450	−0.930020
$615$	0	0
$616$	6.89963	0.277994
$617$	−30.2470	−1.21770	−0.608850	−	0.793285i	$-0.708369\pi$
$617$	−30.2470	−1.21770	−0.608850	+	0.793285i	$0.708369\pi$
$618$	0	0
$619$	42.8285	1.72142	0.860712	−	0.509092i	$-0.170019\pi$
$619$	42.8285	1.72142	0.860712	+	0.509092i	$0.170019\pi$
$620$	20.2044	0.811426
$621$	0	0
$622$	−5.12004	−0.205295
$623$	63.0241	2.52501
$624$	0	0
$625$	62.9777	2.51911
$626$	29.2617	1.16953
$627$	0	0
$628$	10.9308	0.436188
$629$	16.8182	0.670585
$630$	0	0
$631$	16.9338	0.674123	0.337061	−	0.941483i	$-0.390567\pi$
$631$	16.9338	0.674123	0.337061	+	0.941483i	$0.390567\pi$
$632$	11.0765	0.440600
$633$	0	0
$634$	−10.4161	−0.413677
$635$	−2.74175	−0.108803
$636$	0	0
$637$	−62.3195	−2.46919
$638$	5.15972	0.204275
$639$	0	0
$640$	4.14814	0.163970
$641$	12.3809	0.489016	0.244508	−	0.969647i	$-0.421374\pi$
$641$	12.3809	0.489016	0.244508	+	0.969647i	$0.421374\pi$
$642$	0	0
$643$	3.47253	0.136943	0.0684716	−	0.997653i	$-0.478188\pi$
$643$	3.47253	0.136943	0.0684716	+	0.997653i	$0.478188\pi$
$644$	−4.86591	−0.191744
$645$	0	0
$646$	8.74889	0.344221
$647$	−7.65913	−0.301111	−0.150556	−	0.988602i	$-0.548106\pi$
$647$	−7.65913	−0.301111	−0.150556	+	0.988602i	$0.548106\pi$
$648$	0	0
$649$	−12.4461	−0.488551
$650$	45.6160	1.78921
$651$	0	0
$652$	12.4529	0.487691
$653$	−20.2513	−0.792495	−0.396248	−	0.918144i	$-0.629688\pi$
$653$	−20.2513	−0.792495	−0.396248	+	0.918144i	$0.629688\pi$
$654$	0	0
$655$	−12.2282	−0.477795
$656$	0.129068	0.00503924
$657$	0	0
$658$	45.4638	1.77236
$659$	−7.82860	−0.304959	−0.152479	−	0.988307i	$-0.548726\pi$
$659$	−7.82860	−0.304959	−0.152479	+	0.988307i	$0.548726\pi$
$660$	0	0
$661$	−16.0939	−0.625979	−0.312990	−	0.949757i	$-0.601331\pi$
$661$	−16.0939	−0.625979	−0.312990	+	0.949757i	$0.601331\pi$
$662$	26.9487	1.04739
$663$	0	0
$664$	−13.3401	−0.517695
$665$	47.2398	1.83188
$666$	0	0
$667$	−3.63885	−0.140897
$668$	−10.9306	−0.422919
$669$	0	0
$670$	1.87783	0.0725471
$671$	4.28852	0.165556
$672$	0	0
$673$	5.34379	0.205988	0.102994	−	0.994682i	$-0.467158\pi$
$673$	5.34379	0.205988	0.102994	+	0.994682i	$0.467158\pi$
$674$	3.56600	0.137357
$675$	0	0
$676$	0.963977	0.0370760
$677$	−47.4702	−1.82443	−0.912215	−	0.409712i	$-0.865629\pi$
$677$	−47.4702	−1.82443	−0.912215	+	0.409712i	$0.865629\pi$
$678$	0	0
$679$	86.4918	3.31925
$680$	15.5066	0.594651
$681$	0	0
$682$	−6.90642	−0.264461
$683$	−23.3339	−0.892845	−0.446422	−	0.894822i	$-0.647302\pi$
$683$	−23.3339	−0.892845	−0.446422	+	0.894822i	$0.647302\pi$
$684$	0	0
$685$	−86.9174	−3.32094
$686$	−47.0877	−1.79782
$687$	0	0
$688$	−1.00665	−0.0383782
$689$	39.1740	1.49241
$690$	0	0
$691$	43.2344	1.64471	0.822357	−	0.568972i	$-0.192659\pi$
$691$	43.2344	1.64471	0.822357	+	0.568972i	$0.192659\pi$
$692$	19.3161	0.734288
$693$	0	0
$694$	20.9710	0.796050
$695$	−14.8333	−0.562659
$696$	0	0
$697$	0.482481	0.0182753
$698$	−18.6452	−0.705732
$699$	0	0
$700$	59.3986	2.24506
$701$	45.4362	1.71610	0.858050	−	0.513566i	$-0.171676\pi$
$701$	45.4362	1.71610	0.858050	+	0.513566i	$0.171676\pi$
$702$	0	0
$703$	−10.5295	−0.397127
$704$	−1.41795	−0.0534411
$705$	0	0
$706$	−4.15881	−0.156519
$707$	−17.6640	−0.664322
$708$	0	0
$709$	−50.6178	−1.90099	−0.950495	−	0.310739i	$-0.899423\pi$
$709$	−50.6178	−1.90099	−0.950495	+	0.310739i	$0.899423\pi$
$710$	60.9123	2.28600
$711$	0	0
$712$	−12.9522	−0.485403
$713$	4.87070	0.182409
$714$	0	0
$715$	−21.9796	−0.821991
$716$	10.2640	0.383583
$717$	0	0
$718$	8.48135	0.316521
$719$	7.18463	0.267941	0.133971	−	0.990985i	$-0.457227\pi$
$719$	7.18463	0.267941	0.133971	+	0.990985i	$0.457227\pi$
$720$	0	0
$721$	76.2717	2.84051
$722$	13.5225	0.503256
$723$	0	0
$724$	7.51068	0.279132
$725$	44.4198	1.64971
$726$	0	0
$727$	3.70118	0.137269	0.0686346	−	0.997642i	$-0.478136\pi$
$727$	3.70118	0.137269	0.0686346	+	0.997642i	$0.478136\pi$
$728$	18.1831	0.673911
$729$	0	0
$730$	−27.5077	−1.01810
$731$	−3.76306	−0.139182
$732$	0	0
$733$	−36.7030	−1.35566	−0.677828	−	0.735221i	$-0.737079\pi$
$733$	−36.7030	−1.35566	−0.677828	+	0.735221i	$0.737079\pi$
$734$	−36.9005	−1.36202
$735$	0	0
$736$	1.00000	0.0368605
$737$	−0.641897	−0.0236446
$738$	0	0
$739$	30.6126	1.12610	0.563052	−	0.826421i	$-0.309627\pi$
$739$	30.6126	1.12610	0.563052	+	0.826421i	$0.309627\pi$
$740$	−18.6625	−0.686047
$741$	0	0
$742$	51.0102	1.87264
$743$	4.56705	0.167549	0.0837743	−	0.996485i	$-0.473303\pi$
$743$	4.56705	0.167549	0.0837743	+	0.996485i	$0.473303\pi$
$744$	0	0
$745$	71.8368	2.63190
$746$	8.52660	0.312181
$747$	0	0
$748$	−5.30060	−0.193809
$749$	−68.2644	−2.49433
$750$	0	0
$751$	−12.5071	−0.456389	−0.228195	−	0.973616i	$-0.573282\pi$
$751$	−12.5071	−0.456389	−0.228195	+	0.973616i	$0.573282\pi$
$752$	−9.34333	−0.340716
$753$	0	0
$754$	13.5978	0.495203
$755$	36.9822	1.34592
$756$	0	0
$757$	41.2277	1.49845	0.749224	−	0.662317i	$-0.230427\pi$
$757$	41.2277	1.49845	0.749224	+	0.662317i	$0.230427\pi$
$758$	−38.1466	−1.38555
$759$	0	0
$760$	−9.70832	−0.352158
$761$	16.6794	0.604629	0.302314	−	0.953208i	$-0.402241\pi$
$761$	16.6794	0.604629	0.302314	+	0.953208i	$0.402241\pi$
$762$	0	0
$763$	−94.3510	−3.41573
$764$	7.02099	0.254011
$765$	0	0
$766$	−10.6321	−0.384153
$767$	−32.8001	−1.18434
$768$	0	0
$769$	−11.3844	−0.410530	−0.205265	−	0.978706i	$-0.565806\pi$
$769$	−11.3844	−0.410530	−0.205265	+	0.978706i	$0.565806\pi$
$770$	−28.6207	−1.03142
$771$	0	0
$772$	−4.52844	−0.162982
$773$	29.4934	1.06080	0.530402	−	0.847746i	$-0.322041\pi$
$773$	29.4934	1.06080	0.530402	+	0.847746i	$0.322041\pi$
$774$	0	0
$775$	−59.4571	−2.13576
$776$	−17.7751	−0.638087
$777$	0	0
$778$	−13.6640	−0.489878
$779$	−0.302070	−0.0108228
$780$	0	0
$781$	−20.8216	−0.745054
$782$	3.73820	0.133678
$783$	0	0
$784$	16.6771	0.595609
$785$	−45.3427	−1.61835
$786$	0	0
$787$	20.3757	0.726316	0.363158	−	0.931728i	$-0.381699\pi$
$787$	20.3757	0.726316	0.363158	+	0.931728i	$0.381699\pi$
$788$	20.4925	0.730015
$789$	0	0
$790$	−45.9469	−1.63472
$791$	−22.4827	−0.799392
$792$	0	0
$793$	11.3019	0.401341
$794$	−23.5500	−0.835759
$795$	0	0
$796$	25.7203	0.911632
$797$	−19.2196	−0.680793	−0.340396	−	0.940282i	$-0.610561\pi$
$797$	−19.2196	−0.680793	−0.340396	+	0.940282i	$0.610561\pi$
$798$	0	0
$799$	−34.9273	−1.23564
$800$	−12.2071	−0.431586
$801$	0	0
$802$	0.331172	0.0116941
$803$	9.40290	0.331821
$804$	0	0
$805$	20.1845	0.711410
$806$	−18.2010	−0.641104
$807$	0	0
$808$	3.63015	0.127708
$809$	−36.0656	−1.26800	−0.634000	−	0.773333i	$-0.718588\pi$
$809$	−36.0656	−1.26800	−0.634000	+	0.773333i	$0.718588\pi$
$810$	0	0
$811$	22.6201	0.794301	0.397150	−	0.917754i	$-0.369999\pi$
$811$	22.6201	0.794301	0.397150	+	0.917754i	$0.369999\pi$
$812$	17.7063	0.621370
$813$	0	0
$814$	6.37938	0.223597
$815$	−51.6562	−1.80944
$816$	0	0
$817$	2.35597	0.0824248
$818$	−5.04986	−0.176564
$819$	0	0
$820$	−0.535391	−0.0186967
$821$	−45.7695	−1.59737	−0.798683	−	0.601752i	$-0.794470\pi$
$821$	−45.7695	−1.59737	−0.798683	+	0.601752i	$0.794470\pi$
$822$	0	0
$823$	2.37908	0.0829295	0.0414647	−	0.999140i	$-0.486798\pi$
$823$	2.37908	0.0829295	0.0414647	+	0.999140i	$0.486798\pi$
$824$	−15.6747	−0.546054
$825$	0	0
$826$	−42.7105	−1.48609
$827$	−39.8058	−1.38418	−0.692092	−	0.721810i	$-0.743311\pi$
$827$	−39.8058	−1.38418	−0.692092	+	0.721810i	$0.743311\pi$
$828$	0	0
$829$	−13.1328	−0.456121	−0.228060	−	0.973647i	$-0.573238\pi$
$829$	−13.1328	−0.456121	−0.228060	+	0.973647i	$0.573238\pi$
$830$	55.3365	1.92076
$831$	0	0
$832$	−3.73684	−0.129552
$833$	62.3422	2.16003
$834$	0	0
$835$	45.3418	1.56912
$836$	3.31858	0.114775
$837$	0	0
$838$	−3.32908	−0.115001
$839$	10.8958	0.376165	0.188082	−	0.982153i	$-0.439773\pi$
$839$	10.8958	0.376165	0.188082	+	0.982153i	$0.439773\pi$
$840$	0	0
$841$	−15.7588	−0.543406
$842$	−13.9129	−0.479470
$843$	0	0
$844$	7.11231	0.244816
$845$	−3.99872	−0.137560
$846$	0	0
$847$	−43.7416	−1.50298
$848$	−10.4832	−0.359994
$849$	0	0
$850$	−45.6326	−1.56518
$851$	−4.49900	−0.154224
$852$	0	0
$853$	−19.8093	−0.678256	−0.339128	−	0.940740i	$-0.610132\pi$
$853$	−19.8093	−0.678256	−0.339128	+	0.940740i	$0.610132\pi$
$854$	14.7167	0.503594
$855$	0	0
$856$	14.0291	0.479506
$857$	36.6345	1.25141	0.625706	−	0.780059i	$-0.284811\pi$
$857$	36.6345	1.25141	0.625706	+	0.780059i	$0.284811\pi$
$858$	0	0
$859$	28.1523	0.960543	0.480271	−	0.877120i	$-0.340538\pi$
$859$	28.1523	0.960543	0.480271	+	0.877120i	$0.340538\pi$
$860$	4.17573	0.142391
$861$	0	0
$862$	−10.3637	−0.352990
$863$	11.0141	0.374924	0.187462	−	0.982272i	$-0.439974\pi$
$863$	11.0141	0.374924	0.187462	+	0.982272i	$0.439974\pi$
$864$	0	0
$865$	−80.1259	−2.72436
$866$	−40.5689	−1.37859
$867$	0	0
$868$	−23.7004	−0.804443
$869$	15.7060	0.532789
$870$	0	0
$871$	−1.69164	−0.0573191
$872$	19.3902	0.656635
$873$	0	0
$874$	−2.34040	−0.0791653
$875$	−145.471	−4.91783
$876$	0	0
$877$	−33.4803	−1.13055	−0.565274	−	0.824903i	$-0.691230\pi$
$877$	−33.4803	−1.13055	−0.565274	+	0.824903i	$0.691230\pi$
$878$	3.79709	0.128145
$879$	0	0
$880$	5.88187	0.198278
$881$	−18.6391	−0.627968	−0.313984	−	0.949428i	$-0.601664\pi$
$881$	−18.6391	−0.627968	−0.313984	+	0.949428i	$0.601664\pi$
$882$	0	0
$883$	−36.0524	−1.21326	−0.606630	−	0.794985i	$-0.707479\pi$
$883$	−36.0524	−1.21326	−0.606630	+	0.794985i	$0.707479\pi$
$884$	−13.9691	−0.469831
$885$	0	0
$886$	3.20569	0.107697
$887$	−42.9554	−1.44230	−0.721150	−	0.692779i	$-0.756386\pi$
$887$	−42.9554	−1.44230	−0.721150	+	0.692779i	$0.756386\pi$
$888$	0	0
$889$	3.21616	0.107867
$890$	53.7275	1.80095
$891$	0	0
$892$	6.73644	0.225553
$893$	21.8671	0.731756
$894$	0	0
$895$	−42.5765	−1.42318
$896$	−4.86591	−0.162559
$897$	0	0
$898$	−27.6166	−0.921578
$899$	−17.7237	−0.591120
$900$	0	0
$901$	−39.1882	−1.30555
$902$	0.183012	0.00609362
$903$	0	0
$904$	4.62045	0.153674
$905$	−31.1554	−1.03564
$906$	0	0
$907$	−16.2562	−0.539778	−0.269889	−	0.962891i	$-0.586987\pi$
$907$	−16.2562	−0.539778	−0.269889	+	0.962891i	$0.586987\pi$
$908$	11.5896	0.384616
$909$	0	0
$910$	−75.4262	−2.50035
$911$	41.5670	1.37718	0.688588	−	0.725153i	$-0.258231\pi$
$911$	41.5670	1.37718	0.688588	+	0.725153i	$0.258231\pi$
$912$	0	0
$913$	−18.9156	−0.626014
$914$	18.5467	0.613470
$915$	0	0
$916$	17.4743	0.577367
$917$	14.3441	0.473683
$918$	0	0
$919$	−50.7647	−1.67457	−0.837286	−	0.546766i	$-0.815859\pi$
$919$	−50.7647	−1.67457	−0.837286	+	0.546766i	$0.815859\pi$
$920$	−4.14814	−0.136760
$921$	0	0
$922$	−6.71116	−0.221020
$923$	−54.8726	−1.80615
$924$	0	0
$925$	54.9198	1.80575
$926$	0.386694	0.0127076
$927$	0	0
$928$	−3.63885	−0.119451
$929$	17.4778	0.573430	0.286715	−	0.958016i	$-0.407437\pi$
$929$	17.4778	0.573430	0.286715	+	0.958016i	$0.407437\pi$
$930$	0	0
$931$	−39.0310	−1.27919
$932$	23.0522	0.755099
$933$	0	0
$934$	19.9825	0.653848
$935$	21.9876	0.719073
$936$	0	0
$937$	23.7046	0.774397	0.387198	−	0.921996i	$-0.373443\pi$
$937$	23.7046	0.774397	0.387198	+	0.921996i	$0.373443\pi$
$938$	−2.20276	−0.0719227
$939$	0	0
$940$	38.7575	1.26413
$941$	11.5307	0.375889	0.187945	−	0.982180i	$-0.439817\pi$
$941$	11.5307	0.375889	0.187945	+	0.982180i	$0.439817\pi$
$942$	0	0
$943$	−0.129068	−0.00420302
$944$	8.77750	0.285683
$945$	0	0
$946$	−1.42738	−0.0464082
$947$	−17.3404	−0.563487	−0.281744	−	0.959490i	$-0.590913\pi$
$947$	−17.3404	−0.563487	−0.281744	+	0.959490i	$0.590913\pi$
$948$	0	0
$949$	24.7802	0.804399
$950$	28.5695	0.926917
$951$	0	0
$952$	−18.1898	−0.589533
$953$	−38.7092	−1.25391	−0.626957	−	0.779054i	$-0.715700\pi$
$953$	−38.7092	−1.25391	−0.626957	+	0.779054i	$0.715700\pi$
$954$	0	0
$955$	−29.1241	−0.942433
$956$	21.2911	0.688604
$957$	0	0
$958$	4.67131	0.150923
$959$	101.957	3.29236
$960$	0	0
$961$	−7.27630	−0.234719
$962$	16.8121	0.542042
$963$	0	0
$964$	11.0289	0.355216
$965$	18.7846	0.604698
$966$	0	0
$967$	24.7223	0.795016	0.397508	−	0.917599i	$-0.369875\pi$
$967$	24.7223	0.795016	0.397508	+	0.917599i	$0.369875\pi$
$968$	8.98941	0.288931
$969$	0	0
$970$	73.7335	2.36744
$971$	36.3761	1.16736	0.583682	−	0.811982i	$-0.301611\pi$
$971$	36.3761	1.16736	0.583682	+	0.811982i	$0.301611\pi$
$972$	0	0
$973$	17.3999	0.557816
$974$	20.3583	0.652322
$975$	0	0
$976$	−3.02444	−0.0968101
$977$	−39.6334	−1.26798	−0.633992	−	0.773340i	$-0.718585\pi$
$977$	−39.6334	−1.26798	−0.633992	+	0.773340i	$0.718585\pi$
$978$	0	0
$979$	−18.3656	−0.586967
$980$	−69.1788	−2.20984
$981$	0	0
$982$	24.6441	0.786426
$983$	−54.3232	−1.73264	−0.866321	−	0.499488i	$-0.833522\pi$
$983$	−54.3232	−1.73264	−0.866321	+	0.499488i	$0.833522\pi$
$984$	0	0
$985$	−85.0058	−2.70851
$986$	−13.6028	−0.433200
$987$	0	0
$988$	8.74571	0.278238
$989$	1.00665	0.0320096
$990$	0	0
$991$	−4.74734	−0.150804	−0.0754020	−	0.997153i	$-0.524024\pi$
$991$	−4.74734	−0.150804	−0.0754020	+	0.997153i	$0.524024\pi$
$992$	4.87070	0.154645
$993$	0	0
$994$	−71.4521	−2.26632
$995$	−106.691	−3.38235
$996$	0	0
$997$	−17.8726	−0.566030	−0.283015	−	0.959116i	$-0.591335\pi$
$997$	−17.8726	−0.566030	−0.283015	+	0.959116i	$0.591335\pi$
$998$	−29.1624	−0.923121
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3726.2.a.y.1.1	✓	8
3.2	odd	2		3726.2.a.z.1.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3726.2.a.y.1.1	✓	8	1.1	even	1	trivial
3726.2.a.z.1.8	yes	8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3726 = 2 \cdot 3^{4} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3726.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -4.14814 q^{5} +4.86591 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -4.14814 q^{5} +4.86591 q^{7} -1.00000 q^{8} +4.14814 q^{10} -1.41795 q^{11} -3.73684 q^{13} -4.86591 q^{14} +1.00000 q^{16} +3.73820 q^{17} -2.34040 q^{19} -4.14814 q^{20} +1.41795 q^{22} -1.00000 q^{23} +12.2071 q^{25} +3.73684 q^{26} +4.86591 q^{28} +3.63885 q^{29} -4.87070 q^{31} -1.00000 q^{32} -3.73820 q^{34} -20.1845 q^{35} +4.49900 q^{37} +2.34040 q^{38} +4.14814 q^{40} +0.129068 q^{41} -1.00665 q^{43} -1.41795 q^{44} +1.00000 q^{46} -9.34333 q^{47} +16.6771 q^{49} -12.2071 q^{50} -3.73684 q^{52} -10.4832 q^{53} +5.88187 q^{55} -4.86591 q^{56} -3.63885 q^{58} +8.77750 q^{59} -3.02444 q^{61} +4.87070 q^{62} +1.00000 q^{64} +15.5010 q^{65} +0.452693 q^{67} +3.73820 q^{68} +20.1845 q^{70} +14.6842 q^{71} -6.63132 q^{73} -4.49900 q^{74} -2.34040 q^{76} -6.89963 q^{77} -11.0765 q^{79} -4.14814 q^{80} -0.129068 q^{82} +13.3401 q^{83} -15.5066 q^{85} +1.00665 q^{86} +1.41795 q^{88} +12.9522 q^{89} -18.1831 q^{91} -1.00000 q^{92} +9.34333 q^{94} +9.70832 q^{95} +17.7751 q^{97} -16.6771 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 8 q^{4} + 6 q^{7} - 8 q^{8}+O(q^{10}) \) 8 * q - 8 * q^2 + 8 * q^4 + 6 * q^7 - 8 * q^8	\( 8 q - 8 q^{2} + 8 q^{4} + 6 q^{7} - 8 q^{8} + 6 q^{13} - 6 q^{14} + 8 q^{16} + 10 q^{17} + 12 q^{19} - 8 q^{23} + 28 q^{25} - 6 q^{26} + 6 q^{28} - 6 q^{29} + 16 q^{31} - 8 q^{32} - 10 q^{34} - 18 q^{35} + 4 q^{37} - 12 q^{38} + 4 q^{41} + 6 q^{43} + 8 q^{46} - 14 q^{47} + 28 q^{49} - 28 q^{50} + 6 q^{52} - 20 q^{53} + 16 q^{55} - 6 q^{56} + 6 q^{58} - 6 q^{59} + 12 q^{61} - 16 q^{62} + 8 q^{64} + 10 q^{65} + 10 q^{67} + 10 q^{68} + 18 q^{70} + 18 q^{71} + 44 q^{73} - 4 q^{74} + 12 q^{76} - 28 q^{77} + 22 q^{79} - 4 q^{82} - 4 q^{83} + 22 q^{85} - 6 q^{86} + 18 q^{89} + 14 q^{91} - 8 q^{92} + 14 q^{94} + 20 q^{95} + 36 q^{97} - 28 q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^4 + 6 * q^7 - 8 * q^8 + 6 * q^13 - 6 * q^14 + 8 * q^16 + 10 * q^17 + 12 * q^19 - 8 * q^23 + 28 * q^25 - 6 * q^26 + 6 * q^28 - 6 * q^29 + 16 * q^31 - 8 * q^32 - 10 * q^34 - 18 * q^35 + 4 * q^37 - 12 * q^38 + 4 * q^41 + 6 * q^43 + 8 * q^46 - 14 * q^47 + 28 * q^49 - 28 * q^50 + 6 * q^52 - 20 * q^53 + 16 * q^55 - 6 * q^56 + 6 * q^58 - 6 * q^59 + 12 * q^61 - 16 * q^62 + 8 * q^64 + 10 * q^65 + 10 * q^67 + 10 * q^68 + 18 * q^70 + 18 * q^71 + 44 * q^73 - 4 * q^74 + 12 * q^76 - 28 * q^77 + 22 * q^79 - 4 * q^82 - 4 * q^83 + 22 * q^85 - 6 * q^86 + 18 * q^89 + 14 * q^91 - 8 * q^92 + 14 * q^94 + 20 * q^95 + 36 * q^97 - 28 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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