LMFDB - Embedded newform 8036.2.a.m.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8036,2,Mod(1,8036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8036, 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("8036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8036 = 2^{2} \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8036.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.1677830643$
Analytic rank:	$1$
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} - 14x^{6} - 4x^{5} + 60x^{4} + 31x^{3} - 75x^{2} - 60x - 11 $ x^8 - 14x^6 - 4x^5 + 60x^4 + 31x^3 - 75x^2 - 60x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1148)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-2.28881$ of defining polynomial
Character	$\chi$	$=$	8036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.28881 q^{3} +0.350035 q^{5} +2.23863 q^{9} +O(q^{10})$	$q+2.28881 q^{3} +0.350035 q^{5} +2.23863 q^{9} +3.11014 q^{11} -3.97253 q^{13} +0.801161 q^{15} -2.70114 q^{17} -0.682733 q^{19} -4.11511 q^{23} -4.87748 q^{25} -1.74262 q^{27} -6.74476 q^{29} +0.465267 q^{31} +7.11852 q^{33} -4.33767 q^{37} -9.09236 q^{39} +1.00000 q^{41} -2.58476 q^{43} +0.783600 q^{45} +5.96732 q^{47} -6.18238 q^{51} +2.04555 q^{53} +1.08866 q^{55} -1.56264 q^{57} -10.3899 q^{59} +2.87314 q^{61} -1.39052 q^{65} -3.15397 q^{67} -9.41870 q^{69} -3.23326 q^{71} -0.599331 q^{73} -11.1636 q^{75} -0.856150 q^{79} -10.7044 q^{81} -12.9143 q^{83} -0.945491 q^{85} -15.4375 q^{87} +14.0293 q^{89} +1.06490 q^{93} -0.238980 q^{95} -3.40759 q^{97} +6.96247 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{9}+O(q^{10}) $ 8 * q + 4 * q^9	$ 8 q + 4 q^{9} - 8 q^{11} - 7 q^{13} - q^{15} - q^{17} + 4 q^{19} - 3 q^{23} - 4 q^{25} - 12 q^{27} - 4 q^{29} + 4 q^{31} + 23 q^{33} - 31 q^{37} + 5 q^{39} + 8 q^{41} - 8 q^{43} + q^{45} + 24 q^{47} - 23 q^{51} - q^{53} + 2 q^{55} - 15 q^{57} + 4 q^{59} - 4 q^{61} - 24 q^{65} - 21 q^{69} + 8 q^{71} + 11 q^{73} - 15 q^{75} + 14 q^{79} - 28 q^{81} - 42 q^{83} - 20 q^{85} + 25 q^{87} - 11 q^{89} - 27 q^{93} - 15 q^{95} - 16 q^{97} - 8 q^{99}+O(q^{100}) $ 8 * q + 4 * q^9 - 8 * q^11 - 7 * q^13 - q^15 - q^17 + 4 * q^19 - 3 * q^23 - 4 * q^25 - 12 * q^27 - 4 * q^29 + 4 * q^31 + 23 * q^33 - 31 * q^37 + 5 * q^39 + 8 * q^41 - 8 * q^43 + q^45 + 24 * q^47 - 23 * q^51 - q^53 + 2 * q^55 - 15 * q^57 + 4 * q^59 - 4 * q^61 - 24 * q^65 - 21 * q^69 + 8 * q^71 + 11 * q^73 - 15 * q^75 + 14 * q^79 - 28 * q^81 - 42 * q^83 - 20 * q^85 + 25 * q^87 - 11 * q^89 - 27 * q^93 - 15 * q^95 - 16 * q^97 - 8 * 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 $

$2$	0	0
$2$	0	0
$3$	2.28881	1.32144	0.660721	−	0.750631i	$-0.270250\pi$
$3$	2.28881	1.32144	0.660721	+	0.750631i	$0.270250\pi$
$4$	0	0
$5$	0.350035	0.156540	0.0782701	−	0.996932i	$-0.475060\pi$
$5$	0.350035	0.156540	0.0782701	+	0.996932i	$0.475060\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.23863	0.746212
$10$	0	0
$11$	3.11014	0.937743	0.468872	−	0.883266i	$-0.344661\pi$
$11$	3.11014	0.937743	0.468872	+	0.883266i	$0.344661\pi$
$12$	0	0
$13$	−3.97253	−1.10178	−0.550891	−	0.834577i	$-0.685712\pi$
$13$	−3.97253	−1.10178	−0.550891	+	0.834577i	$0.685712\pi$
$14$	0	0
$15$	0.801161	0.206859
$16$	0	0
$17$	−2.70114	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$17$	−2.70114	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$18$	0	0
$19$	−0.682733	−0.156630	−0.0783149	−	0.996929i	$-0.524954\pi$
$19$	−0.682733	−0.156630	−0.0783149	+	0.996929i	$0.524954\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.11511	−0.858060	−0.429030	−	0.903290i	$-0.641145\pi$
$23$	−4.11511	−0.858060	−0.429030	+	0.903290i	$0.641145\pi$
$24$	0	0
$25$	−4.87748	−0.975495
$26$	0	0
$27$	−1.74262	−0.335367
$28$	0	0
$29$	−6.74476	−1.25247	−0.626236	−	0.779634i	$-0.715405\pi$
$29$	−6.74476	−1.25247	−0.626236	+	0.779634i	$0.715405\pi$
$30$	0	0
$31$	0.465267	0.0835643	0.0417822	−	0.999127i	$-0.486696\pi$
$31$	0.465267	0.0835643	0.0417822	+	0.999127i	$0.486696\pi$
$32$	0	0
$33$	7.11852	1.23917
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.33767	−0.713108	−0.356554	−	0.934275i	$-0.616048\pi$
$37$	−4.33767	−0.713108	−0.356554	+	0.934275i	$0.616048\pi$
$38$	0	0
$39$	−9.09236	−1.45594
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−2.58476	−0.394172	−0.197086	−	0.980386i	$-0.563148\pi$
$43$	−2.58476	−0.394172	−0.197086	+	0.980386i	$0.563148\pi$
$44$	0	0
$45$	0.783600	0.116812
$46$	0	0
$47$	5.96732	0.870423	0.435211	−	0.900328i	$-0.356674\pi$
$47$	5.96732	0.870423	0.435211	+	0.900328i	$0.356674\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−6.18238	−0.865706
$52$	0	0
$53$	2.04555	0.280977	0.140489	−	0.990082i	$-0.455133\pi$
$53$	2.04555	0.280977	0.140489	+	0.990082i	$0.455133\pi$
$54$	0	0
$55$	1.08866	0.146795
$56$	0	0
$57$	−1.56264	−0.206977
$58$	0	0
$59$	−10.3899	−1.35265	−0.676324	−	0.736604i	$-0.736428\pi$
$59$	−10.3899	−1.35265	−0.676324	+	0.736604i	$0.736428\pi$
$60$	0	0
$61$	2.87314	0.367868	0.183934	−	0.982939i	$-0.441117\pi$
$61$	2.87314	0.367868	0.183934	+	0.982939i	$0.441117\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.39052	−0.172473
$66$	0	0
$67$	−3.15397	−0.385318	−0.192659	−	0.981266i	$-0.561711\pi$
$67$	−3.15397	−0.385318	−0.192659	+	0.981266i	$0.561711\pi$
$68$	0	0
$69$	−9.41870	−1.13388
$70$	0	0
$71$	−3.23326	−0.383717	−0.191859	−	0.981423i	$-0.561452\pi$
$71$	−3.23326	−0.383717	−0.191859	+	0.981423i	$0.561452\pi$
$72$	0	0
$73$	−0.599331	−0.0701464	−0.0350732	−	0.999385i	$-0.511166\pi$
$73$	−0.599331	−0.0701464	−0.0350732	+	0.999385i	$0.511166\pi$
$74$	0	0
$75$	−11.1636	−1.28906
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−0.856150	−0.0963244	−0.0481622	−	0.998840i	$-0.515336\pi$
$79$	−0.856150	−0.0963244	−0.0481622	+	0.998840i	$0.515336\pi$
$80$	0	0
$81$	−10.7044	−1.18938
$82$	0	0
$83$	−12.9143	−1.41753	−0.708764	−	0.705445i	$-0.750747\pi$
$83$	−12.9143	−1.41753	−0.708764	+	0.705445i	$0.750747\pi$
$84$	0	0
$85$	−0.945491	−0.102553
$86$	0	0
$87$	−15.4375	−1.65507
$88$	0	0
$89$	14.0293	1.48710	0.743551	−	0.668680i	$-0.233140\pi$
$89$	14.0293	1.48710	0.743551	+	0.668680i	$0.233140\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.06490	0.110426
$94$	0	0
$95$	−0.238980	−0.0245189
$96$	0	0
$97$	−3.40759	−0.345988	−0.172994	−	0.984923i	$-0.555344\pi$
$97$	−3.40759	−0.345988	−0.172994	+	0.984923i	$0.555344\pi$
$98$	0	0
$99$	6.96247	0.699755
$100$	0	0
$101$	−8.90018	−0.885601	−0.442801	−	0.896620i	$-0.646015\pi$
$101$	−8.90018	−0.885601	−0.442801	+	0.896620i	$0.646015\pi$
$102$	0	0
$103$	−4.65227	−0.458402	−0.229201	−	0.973379i	$-0.573611\pi$
$103$	−4.65227	−0.458402	−0.229201	+	0.973379i	$0.573611\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.38956	−0.521029	−0.260514	−	0.965470i	$-0.583892\pi$
$107$	−5.38956	−0.521029	−0.260514	+	0.965470i	$0.583892\pi$
$108$	0	0
$109$	7.20589	0.690199	0.345100	−	0.938566i	$-0.387845\pi$
$109$	7.20589	0.690199	0.345100	+	0.938566i	$0.387845\pi$
$110$	0	0
$111$	−9.92808	−0.942332
$112$	0	0
$113$	−4.45820	−0.419392	−0.209696	−	0.977767i	$-0.567247\pi$
$113$	−4.45820	−0.419392	−0.209696	+	0.977767i	$0.567247\pi$
$114$	0	0
$115$	−1.44043	−0.134321
$116$	0	0
$117$	−8.89305	−0.822163
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1.32701	−0.120637
$122$	0	0
$123$	2.28881	0.206375
$124$	0	0
$125$	−3.45746	−0.309244
$126$	0	0
$127$	14.3286	1.27146	0.635730	−	0.771912i	$-0.280699\pi$
$127$	14.3286	1.27146	0.635730	+	0.771912i	$0.280699\pi$
$128$	0	0
$129$	−5.91602	−0.520876
$130$	0	0
$131$	0.547873	0.0478678	0.0239339	−	0.999714i	$-0.492381\pi$
$131$	0.547873	0.0478678	0.0239339	+	0.999714i	$0.492381\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−0.609976	−0.0524984
$136$	0	0
$137$	10.3889	0.887584	0.443792	−	0.896130i	$-0.353633\pi$
$137$	10.3889	0.887584	0.443792	+	0.896130i	$0.353633\pi$
$138$	0	0
$139$	−16.4437	−1.39474	−0.697368	−	0.716713i	$-0.745646\pi$
$139$	−16.4437	−1.39474	−0.697368	+	0.716713i	$0.745646\pi$
$140$	0	0
$141$	13.6580	1.15021
$142$	0	0
$143$	−12.3551	−1.03319
$144$	0	0
$145$	−2.36090	−0.196062
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3.01767	−0.247217	−0.123608	−	0.992331i	$-0.539447\pi$
$149$	−3.01767	−0.247217	−0.123608	+	0.992331i	$0.539447\pi$
$150$	0	0
$151$	12.7279	1.03578	0.517892	−	0.855446i	$-0.326717\pi$
$151$	12.7279	1.03578	0.517892	+	0.855446i	$0.326717\pi$
$152$	0	0
$153$	−6.04686	−0.488859
$154$	0	0
$155$	0.162859	0.0130812
$156$	0	0
$157$	24.2932	1.93881	0.969406	−	0.245462i	$-0.0789398\pi$
$157$	24.2932	1.93881	0.969406	+	0.245462i	$0.0789398\pi$
$158$	0	0
$159$	4.68186	0.371295
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−17.0804	−1.33784	−0.668921	−	0.743333i	$-0.733244\pi$
$163$	−17.0804	−1.33784	−0.668921	+	0.743333i	$0.733244\pi$
$164$	0	0
$165$	2.49173	0.193981
$166$	0	0
$167$	11.3763	0.880323	0.440162	−	0.897919i	$-0.354921\pi$
$167$	11.3763	0.880323	0.440162	+	0.897919i	$0.354921\pi$
$168$	0	0
$169$	2.78101	0.213924
$170$	0	0
$171$	−1.52839	−0.116879
$172$	0	0
$173$	1.76915	0.134506	0.0672531	−	0.997736i	$-0.478577\pi$
$173$	1.76915	0.134506	0.0672531	+	0.997736i	$0.478577\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−23.7804	−1.78745
$178$	0	0
$179$	−7.27631	−0.543857	−0.271929	−	0.962317i	$-0.587661\pi$
$179$	−7.27631	−0.543857	−0.271929	+	0.962317i	$0.587661\pi$
$180$	0	0
$181$	3.49738	0.259958	0.129979	−	0.991517i	$-0.458509\pi$
$181$	3.49738	0.259958	0.129979	+	0.991517i	$0.458509\pi$
$182$	0	0
$183$	6.57606	0.486116
$184$	0	0
$185$	−1.51833	−0.111630
$186$	0	0
$187$	−8.40092	−0.614336
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−0.894964	−0.0647573	−0.0323786	−	0.999476i	$-0.510308\pi$
$191$	−0.894964	−0.0647573	−0.0323786	+	0.999476i	$0.510308\pi$
$192$	0	0
$193$	12.5454	0.903039	0.451519	−	0.892261i	$-0.350882\pi$
$193$	12.5454	0.903039	0.451519	+	0.892261i	$0.350882\pi$
$194$	0	0
$195$	−3.18264	−0.227914
$196$	0	0
$197$	9.91618	0.706499	0.353249	−	0.935529i	$-0.385077\pi$
$197$	9.91618	0.706499	0.353249	+	0.935529i	$0.385077\pi$
$198$	0	0
$199$	21.9822	1.55827	0.779137	−	0.626853i	$-0.215657\pi$
$199$	21.9822	1.55827	0.779137	+	0.626853i	$0.215657\pi$
$200$	0	0
$201$	−7.21882	−0.509176
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0.350035	0.0244475
$206$	0	0
$207$	−9.21223	−0.640294
$208$	0	0
$209$	−2.12340	−0.146878
$210$	0	0
$211$	−21.4721	−1.47820	−0.739101	−	0.673595i	$-0.764749\pi$
$211$	−21.4721	−1.47820	−0.739101	+	0.673595i	$0.764749\pi$
$212$	0	0
$213$	−7.40031	−0.507061
$214$	0	0
$215$	−0.904756	−0.0617038
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−1.37175	−0.0926945
$220$	0	0
$221$	10.7303	0.721801
$222$	0	0
$223$	−7.30791	−0.489374	−0.244687	−	0.969602i	$-0.578685\pi$
$223$	−7.30791	−0.489374	−0.244687	+	0.969602i	$0.578685\pi$
$224$	0	0
$225$	−10.9189	−0.727926
$226$	0	0
$227$	18.4580	1.22510	0.612552	−	0.790430i	$-0.290143\pi$
$227$	18.4580	1.22510	0.612552	+	0.790430i	$0.290143\pi$
$228$	0	0
$229$	5.01695	0.331529	0.165765	−	0.986165i	$-0.446991\pi$
$229$	5.01695	0.331529	0.165765	+	0.986165i	$0.446991\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−26.7556	−1.75281	−0.876407	−	0.481570i	$-0.840067\pi$
$233$	−26.7556	−1.75281	−0.876407	+	0.481570i	$0.840067\pi$
$234$	0	0
$235$	2.08877	0.136256
$236$	0	0
$237$	−1.95956	−0.127287
$238$	0	0
$239$	18.7665	1.21390	0.606951	−	0.794739i	$-0.292392\pi$
$239$	18.7665	1.21390	0.606951	+	0.794739i	$0.292392\pi$
$240$	0	0
$241$	23.2788	1.49952	0.749761	−	0.661709i	$-0.230168\pi$
$241$	23.2788	1.49952	0.749761	+	0.661709i	$0.230168\pi$
$242$	0	0
$243$	−19.2725	−1.23633
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.71218	0.172572
$248$	0	0
$249$	−29.5583	−1.87318
$250$	0	0
$251$	−20.6641	−1.30430	−0.652152	−	0.758088i	$-0.726134\pi$
$251$	−20.6641	−1.30430	−0.652152	+	0.758088i	$0.726134\pi$
$252$	0	0
$253$	−12.7986	−0.804640
$254$	0	0
$255$	−2.16405	−0.135518
$256$	0	0
$257$	9.11654	0.568674	0.284337	−	0.958724i	$-0.408227\pi$
$257$	9.11654	0.568674	0.284337	+	0.958724i	$0.408227\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−15.0991	−0.934609
$262$	0	0
$263$	11.9857	0.739068	0.369534	−	0.929217i	$-0.379517\pi$
$263$	11.9857	0.739068	0.369534	+	0.929217i	$0.379517\pi$
$264$	0	0
$265$	0.716012	0.0439842
$266$	0	0
$267$	32.1103	1.96512
$268$	0	0
$269$	5.37311	0.327604	0.163802	−	0.986493i	$-0.447624\pi$
$269$	5.37311	0.327604	0.163802	+	0.986493i	$0.447624\pi$
$270$	0	0
$271$	−21.9279	−1.33203	−0.666014	−	0.745939i	$-0.732001\pi$
$271$	−21.9279	−1.33203	−0.666014	+	0.745939i	$0.732001\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−15.1696	−0.914764
$276$	0	0
$277$	−5.42553	−0.325989	−0.162994	−	0.986627i	$-0.552115\pi$
$277$	−5.42553	−0.325989	−0.162994	+	0.986627i	$0.552115\pi$
$278$	0	0
$279$	1.04156	0.0623567
$280$	0	0
$281$	−17.6364	−1.05210	−0.526049	−	0.850454i	$-0.676327\pi$
$281$	−17.6364	−1.05210	−0.526049	+	0.850454i	$0.676327\pi$
$282$	0	0
$283$	14.6715	0.872132	0.436066	−	0.899915i	$-0.356371\pi$
$283$	14.6715	0.872132	0.436066	+	0.899915i	$0.356371\pi$
$284$	0	0
$285$	−0.546979	−0.0324003
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−9.70387	−0.570816
$290$	0	0
$291$	−7.79930	−0.457203
$292$	0	0
$293$	7.13441	0.416797	0.208398	−	0.978044i	$-0.433175\pi$
$293$	7.13441	0.416797	0.208398	+	0.978044i	$0.433175\pi$
$294$	0	0
$295$	−3.63682	−0.211744
$296$	0	0
$297$	−5.41979	−0.314488
$298$	0	0
$299$	16.3474	0.945396
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−20.3708	−1.17027
$304$	0	0
$305$	1.00570	0.0575861
$306$	0	0
$307$	17.4342	0.995024	0.497512	−	0.867457i	$-0.334247\pi$
$307$	17.4342	0.995024	0.497512	+	0.867457i	$0.334247\pi$
$308$	0	0
$309$	−10.6482	−0.605752
$310$	0	0
$311$	0.294179	0.0166813	0.00834067	−	0.999965i	$-0.497345\pi$
$311$	0.294179	0.0166813	0.00834067	+	0.999965i	$0.497345\pi$
$312$	0	0
$313$	32.0838	1.81348	0.906742	−	0.421686i	$-0.138561\pi$
$313$	32.0838	1.81348	0.906742	+	0.421686i	$0.138561\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8.20370	−0.460766	−0.230383	−	0.973100i	$-0.573998\pi$
$317$	−8.20370	−0.460766	−0.230383	+	0.973100i	$0.573998\pi$
$318$	0	0
$319$	−20.9772	−1.17450
$320$	0	0
$321$	−12.3357	−0.688509
$322$	0	0
$323$	1.84415	0.102612
$324$	0	0
$325$	19.3759	1.07478
$326$	0	0
$327$	16.4929	0.912059
$328$	0	0
$329$	0	0
$330$	0	0
$331$	12.4423	0.683892	0.341946	−	0.939720i	$-0.388914\pi$
$331$	12.4423	0.683892	0.341946	+	0.939720i	$0.388914\pi$
$332$	0	0
$333$	−9.71045	−0.532130
$334$	0	0
$335$	−1.10400	−0.0603178
$336$	0	0
$337$	17.6330	0.960530	0.480265	−	0.877123i	$-0.340540\pi$
$337$	17.6330	0.960530	0.480265	+	0.877123i	$0.340540\pi$
$338$	0	0
$339$	−10.2040	−0.554203
$340$	0	0
$341$	1.44705	0.0783619
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−3.29687	−0.177497
$346$	0	0
$347$	−6.29714	−0.338048	−0.169024	−	0.985612i	$-0.554062\pi$
$347$	−6.29714	−0.338048	−0.169024	+	0.985612i	$0.554062\pi$
$348$	0	0
$349$	1.11133	0.0594879	0.0297440	−	0.999558i	$-0.490531\pi$
$349$	1.11133	0.0594879	0.0297440	+	0.999558i	$0.490531\pi$
$350$	0	0
$351$	6.92260	0.369501
$352$	0	0
$353$	−33.7307	−1.79530	−0.897651	−	0.440708i	$-0.854728\pi$
$353$	−33.7307	−1.79530	−0.897651	+	0.440708i	$0.854728\pi$
$354$	0	0
$355$	−1.13175	−0.0600672
$356$	0	0
$357$	0	0
$358$	0	0
$359$	24.0483	1.26922	0.634611	−	0.772832i	$-0.281160\pi$
$359$	24.0483	1.26922	0.634611	+	0.772832i	$0.281160\pi$
$360$	0	0
$361$	−18.5339	−0.975467
$362$	0	0
$363$	−3.03727	−0.159415
$364$	0	0
$365$	−0.209787	−0.0109807
$366$	0	0
$367$	−28.8283	−1.50483	−0.752414	−	0.658691i	$-0.771111\pi$
$367$	−28.8283	−1.50483	−0.752414	+	0.658691i	$0.771111\pi$
$368$	0	0
$369$	2.23863	0.116539
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−18.3669	−0.951002	−0.475501	−	0.879715i	$-0.657733\pi$
$373$	−18.3669	−0.951002	−0.475501	+	0.879715i	$0.657733\pi$
$374$	0	0
$375$	−7.91345	−0.408649
$376$	0	0
$377$	26.7938	1.37995
$378$	0	0
$379$	−20.6497	−1.06070	−0.530352	−	0.847778i	$-0.677940\pi$
$379$	−20.6497	−1.06070	−0.530352	+	0.847778i	$0.677940\pi$
$380$	0	0
$381$	32.7954	1.68016
$382$	0	0
$383$	−11.7692	−0.601377	−0.300688	−	0.953722i	$-0.597216\pi$
$383$	−11.7692	−0.601377	−0.300688	+	0.953722i	$0.597216\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−5.78634	−0.294136
$388$	0	0
$389$	−24.2787	−1.23098	−0.615489	−	0.788146i	$-0.711041\pi$
$389$	−24.2787	−1.23098	−0.615489	+	0.788146i	$0.711041\pi$
$390$	0	0
$391$	11.1155	0.562134
$392$	0	0
$393$	1.25397	0.0632546
$394$	0	0
$395$	−0.299682	−0.0150786
$396$	0	0
$397$	−20.6490	−1.03634	−0.518171	−	0.855277i	$-0.673387\pi$
$397$	−20.6490	−1.03634	−0.518171	+	0.855277i	$0.673387\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	32.3762	1.61679	0.808396	−	0.588639i	$-0.200336\pi$
$401$	32.3762	1.61679	0.808396	+	0.588639i	$0.200336\pi$
$402$	0	0
$403$	−1.84829	−0.0920697
$404$	0	0
$405$	−3.74692	−0.186186
$406$	0	0
$407$	−13.4908	−0.668712
$408$	0	0
$409$	12.8064	0.633238	0.316619	−	0.948553i	$-0.397452\pi$
$409$	12.8064	0.633238	0.316619	+	0.948553i	$0.397452\pi$
$410$	0	0
$411$	23.7782	1.17289
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−4.52045	−0.221900
$416$	0	0
$417$	−37.6364	−1.84306
$418$	0	0
$419$	21.8220	1.06608	0.533038	−	0.846092i	$-0.321050\pi$
$419$	21.8220	1.06608	0.533038	+	0.846092i	$0.321050\pi$
$420$	0	0
$421$	−4.59928	−0.224155	−0.112078	−	0.993699i	$-0.535751\pi$
$421$	−4.59928	−0.224155	−0.112078	+	0.993699i	$0.535751\pi$
$422$	0	0
$423$	13.3586	0.649520
$424$	0	0
$425$	13.1747	0.639068
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−28.2785	−1.36530
$430$	0	0
$431$	−14.9859	−0.721847	−0.360923	−	0.932595i	$-0.617538\pi$
$431$	−14.9859	−0.721847	−0.360923	+	0.932595i	$0.617538\pi$
$432$	0	0
$433$	−30.5080	−1.46612	−0.733060	−	0.680164i	$-0.761909\pi$
$433$	−30.5080	−1.46612	−0.733060	+	0.680164i	$0.761909\pi$
$434$	0	0
$435$	−5.40364	−0.259085
$436$	0	0
$437$	2.80952	0.134398
$438$	0	0
$439$	−14.3877	−0.686689	−0.343345	−	0.939210i	$-0.611560\pi$
$439$	−14.3877	−0.686689	−0.343345	+	0.939210i	$0.611560\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−10.5194	−0.499794	−0.249897	−	0.968272i	$-0.580397\pi$
$443$	−10.5194	−0.499794	−0.249897	+	0.968272i	$0.580397\pi$
$444$	0	0
$445$	4.91074	0.232791
$446$	0	0
$447$	−6.90686	−0.326683
$448$	0	0
$449$	−5.21224	−0.245981	−0.122990	−	0.992408i	$-0.539248\pi$
$449$	−5.21224	−0.245981	−0.122990	+	0.992408i	$0.539248\pi$
$450$	0	0
$451$	3.11014	0.146451
$452$	0	0
$453$	29.1318	1.36873
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−10.2557	−0.479739	−0.239870	−	0.970805i	$-0.577105\pi$
$457$	−10.2557	−0.479739	−0.239870	+	0.970805i	$0.577105\pi$
$458$	0	0
$459$	4.70705	0.219706
$460$	0	0
$461$	−3.59542	−0.167456	−0.0837278	−	0.996489i	$-0.526683\pi$
$461$	−3.59542	−0.167456	−0.0837278	+	0.996489i	$0.526683\pi$
$462$	0	0
$463$	−19.8730	−0.923575	−0.461788	−	0.886991i	$-0.652792\pi$
$463$	−19.8730	−0.923575	−0.461788	+	0.886991i	$0.652792\pi$
$464$	0	0
$465$	0.372754	0.0172860
$466$	0	0
$467$	−38.1902	−1.76723	−0.883617	−	0.468211i	$-0.844899\pi$
$467$	−38.1902	−1.76723	−0.883617	+	0.468211i	$0.844899\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	55.6025	2.56203
$472$	0	0
$473$	−8.03898	−0.369633
$474$	0	0
$475$	3.33001	0.152792
$476$	0	0
$477$	4.57923	0.209668
$478$	0	0
$479$	19.2981	0.881754	0.440877	−	0.897567i	$-0.354667\pi$
$479$	19.2981	0.881754	0.440877	+	0.897567i	$0.354667\pi$
$480$	0	0
$481$	17.2315	0.785690
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.19277	−0.0541610
$486$	0	0
$487$	41.7601	1.89233	0.946166	−	0.323681i	$-0.104921\pi$
$487$	41.7601	1.89233	0.946166	+	0.323681i	$0.104921\pi$
$488$	0	0
$489$	−39.0938	−1.76788
$490$	0	0
$491$	−34.1057	−1.53917	−0.769584	−	0.638546i	$-0.779536\pi$
$491$	−34.1057	−1.53917	−0.769584	+	0.638546i	$0.779536\pi$
$492$	0	0
$493$	18.2185	0.820521
$494$	0	0
$495$	2.43711	0.109540
$496$	0	0
$497$	0	0
$498$	0	0
$499$	22.0459	0.986910	0.493455	−	0.869771i	$-0.335734\pi$
$499$	22.0459	0.986910	0.493455	+	0.869771i	$0.335734\pi$
$500$	0	0
$501$	26.0381	1.16330
$502$	0	0
$503$	−2.01342	−0.0897738	−0.0448869	−	0.998992i	$-0.514293\pi$
$503$	−2.01342	−0.0897738	−0.0448869	+	0.998992i	$0.514293\pi$
$504$	0	0
$505$	−3.11537	−0.138632
$506$	0	0
$507$	6.36520	0.282689
$508$	0	0
$509$	14.2774	0.632836	0.316418	−	0.948620i	$-0.397520\pi$
$509$	14.2774	0.632836	0.316418	+	0.948620i	$0.397520\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	1.18974	0.0525284
$514$	0	0
$515$	−1.62846	−0.0717584
$516$	0	0
$517$	18.5592	0.816233
$518$	0	0
$519$	4.04925	0.177742
$520$	0	0
$521$	−11.6837	−0.511874	−0.255937	−	0.966693i	$-0.582384\pi$
$521$	−11.6837	−0.511874	−0.255937	+	0.966693i	$0.582384\pi$
$522$	0	0
$523$	31.4279	1.37424	0.687122	−	0.726542i	$-0.258874\pi$
$523$	31.4279	1.37424	0.687122	+	0.726542i	$0.258874\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.25675	−0.0547448
$528$	0	0
$529$	−6.06585	−0.263733
$530$	0	0
$531$	−23.2592	−1.00936
$532$	0	0
$533$	−3.97253	−0.172069
$534$	0	0
$535$	−1.88653	−0.0815619
$536$	0	0
$537$	−16.6541	−0.718676
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−41.9942	−1.80547	−0.902736	−	0.430195i	$-0.858445\pi$
$541$	−41.9942	−1.80547	−0.902736	+	0.430195i	$0.858445\pi$
$542$	0	0
$543$	8.00482	0.343520
$544$	0	0
$545$	2.52231	0.108044
$546$	0	0
$547$	−32.0258	−1.36933	−0.684663	−	0.728860i	$-0.740050\pi$
$547$	−32.0258	−1.36933	−0.684663	+	0.728860i	$0.740050\pi$
$548$	0	0
$549$	6.43191	0.274507
$550$	0	0
$551$	4.60487	0.196174
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−3.47517	−0.147513
$556$	0	0
$557$	34.8563	1.47691	0.738455	−	0.674303i	$-0.235556\pi$
$557$	34.8563	1.47691	0.738455	+	0.674303i	$0.235556\pi$
$558$	0	0
$559$	10.2680	0.434292
$560$	0	0
$561$	−19.2281	−0.811810
$562$	0	0
$563$	4.63359	0.195283	0.0976413	−	0.995222i	$-0.468870\pi$
$563$	4.63359	0.195283	0.0976413	+	0.995222i	$0.468870\pi$
$564$	0	0
$565$	−1.56052	−0.0656517
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−11.0085	−0.461500	−0.230750	−	0.973013i	$-0.574118\pi$
$569$	−11.0085	−0.461500	−0.230750	+	0.973013i	$0.574118\pi$
$570$	0	0
$571$	13.7068	0.573613	0.286807	−	0.957989i	$-0.407406\pi$
$571$	13.7068	0.573613	0.286807	+	0.957989i	$0.407406\pi$
$572$	0	0
$573$	−2.04840	−0.0855731
$574$	0	0
$575$	20.0714	0.837034
$576$	0	0
$577$	8.56801	0.356691	0.178345	−	0.983968i	$-0.442926\pi$
$577$	8.56801	0.356691	0.178345	+	0.983968i	$0.442926\pi$
$578$	0	0
$579$	28.7140	1.19331
$580$	0	0
$581$	0	0
$582$	0	0
$583$	6.36194	0.263485
$584$	0	0
$585$	−3.11287	−0.128702
$586$	0	0
$587$	0.718126	0.0296402	0.0148201	−	0.999890i	$-0.495282\pi$
$587$	0.718126	0.0296402	0.0148201	+	0.999890i	$0.495282\pi$
$588$	0	0
$589$	−0.317653	−0.0130887
$590$	0	0
$591$	22.6962	0.933598
$592$	0	0
$593$	11.5165	0.472928	0.236464	−	0.971640i	$-0.424012\pi$
$593$	11.5165	0.472928	0.236464	+	0.971640i	$0.424012\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	50.3129	2.05917
$598$	0	0
$599$	−44.4224	−1.81505	−0.907525	−	0.419997i	$-0.862031\pi$
$599$	−44.4224	−1.81505	−0.907525	+	0.419997i	$0.862031\pi$
$600$	0	0
$601$	−27.2986	−1.11353	−0.556766	−	0.830669i	$-0.687958\pi$
$601$	−27.2986	−1.11353	−0.556766	+	0.830669i	$0.687958\pi$
$602$	0	0
$603$	−7.06058	−0.287529
$604$	0	0
$605$	−0.464500	−0.0188846
$606$	0	0
$607$	−12.6979	−0.515391	−0.257696	−	0.966226i	$-0.582963\pi$
$607$	−12.6979	−0.515391	−0.257696	+	0.966226i	$0.582963\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−23.7054	−0.959017
$612$	0	0
$613$	−0.363102	−0.0146656	−0.00733278	−	0.999973i	$-0.502334\pi$
$613$	−0.363102	−0.0146656	−0.00733278	+	0.999973i	$0.502334\pi$
$614$	0	0
$615$	0.801161	0.0323059
$616$	0	0
$617$	9.82313	0.395464	0.197732	−	0.980256i	$-0.436642\pi$
$617$	9.82313	0.395464	0.197732	+	0.980256i	$0.436642\pi$
$618$	0	0
$619$	38.9379	1.56505	0.782524	−	0.622621i	$-0.213932\pi$
$619$	38.9379	1.56505	0.782524	+	0.622621i	$0.213932\pi$
$620$	0	0
$621$	7.17107	0.287765
$622$	0	0
$623$	0	0
$624$	0	0
$625$	23.1771	0.927086
$626$	0	0
$627$	−4.86005	−0.194092
$628$	0	0
$629$	11.7166	0.467173
$630$	0	0
$631$	−21.3256	−0.848959	−0.424479	−	0.905438i	$-0.639543\pi$
$631$	−21.3256	−0.848959	−0.424479	+	0.905438i	$0.639543\pi$
$632$	0	0
$633$	−49.1455	−1.95336
$634$	0	0
$635$	5.01551	0.199035
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−7.23809	−0.286334
$640$	0	0
$641$	−43.3605	−1.71264	−0.856318	−	0.516449i	$-0.827254\pi$
$641$	−43.3605	−1.71264	−0.856318	+	0.516449i	$0.827254\pi$
$642$	0	0
$643$	−0.783742	−0.0309078	−0.0154539	−	0.999881i	$-0.504919\pi$
$643$	−0.783742	−0.0309078	−0.0154539	+	0.999881i	$0.504919\pi$
$644$	0	0
$645$	−2.07081	−0.0815381
$646$	0	0
$647$	28.9601	1.13854	0.569269	−	0.822151i	$-0.307226\pi$
$647$	28.9601	1.13854	0.569269	+	0.822151i	$0.307226\pi$
$648$	0	0
$649$	−32.3140	−1.26844
$650$	0	0
$651$	0	0
$652$	0	0
$653$	22.0336	0.862243	0.431122	−	0.902294i	$-0.358118\pi$
$653$	22.0336	0.862243	0.431122	+	0.902294i	$0.358118\pi$
$654$	0	0
$655$	0.191774	0.00749324
$656$	0	0
$657$	−1.34168	−0.0523441
$658$	0	0
$659$	−11.7364	−0.457186	−0.228593	−	0.973522i	$-0.573413\pi$
$659$	−11.7364	−0.457186	−0.228593	+	0.973522i	$0.573413\pi$
$660$	0	0
$661$	−25.2679	−0.982806	−0.491403	−	0.870932i	$-0.663516\pi$
$661$	−25.2679	−0.982806	−0.491403	+	0.870932i	$0.663516\pi$
$662$	0	0
$663$	24.5597	0.953819
$664$	0	0
$665$	0	0
$666$	0	0
$667$	27.7555	1.07470
$668$	0	0
$669$	−16.7264	−0.646680
$670$	0	0
$671$	8.93587	0.344965
$672$	0	0
$673$	9.10303	0.350896	0.175448	−	0.984489i	$-0.443863\pi$
$673$	9.10303	0.350896	0.175448	+	0.984489i	$0.443863\pi$
$674$	0	0
$675$	8.49957	0.327149
$676$	0	0
$677$	37.1002	1.42588	0.712938	−	0.701227i	$-0.247364\pi$
$677$	37.1002	1.42588	0.712938	+	0.701227i	$0.247364\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	42.2469	1.61890
$682$	0	0
$683$	15.0203	0.574735	0.287367	−	0.957820i	$-0.407220\pi$
$683$	15.0203	0.574735	0.287367	+	0.957820i	$0.407220\pi$
$684$	0	0
$685$	3.63647	0.138943
$686$	0	0
$687$	11.4828	0.438097
$688$	0	0
$689$	−8.12600	−0.309576
$690$	0	0
$691$	−41.4728	−1.57770	−0.788849	−	0.614587i	$-0.789323\pi$
$691$	−41.4728	−1.57770	−0.788849	+	0.614587i	$0.789323\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−5.75586	−0.218332
$696$	0	0
$697$	−2.70114	−0.102313
$698$	0	0
$699$	−61.2383	−2.31624
$700$	0	0
$701$	−15.7161	−0.593589	−0.296794	−	0.954941i	$-0.595918\pi$
$701$	−15.7161	−0.593589	−0.296794	+	0.954941i	$0.595918\pi$
$702$	0	0
$703$	2.96147	0.111694
$704$	0	0
$705$	4.78079	0.180055
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−39.5914	−1.48689	−0.743444	−	0.668799i	$-0.766809\pi$
$709$	−39.5914	−1.48689	−0.743444	+	0.668799i	$0.766809\pi$
$710$	0	0
$711$	−1.91661	−0.0718784
$712$	0	0
$713$	−1.91462	−0.0717032
$714$	0	0
$715$	−4.32473	−0.161736
$716$	0	0
$717$	42.9528	1.60410
$718$	0	0
$719$	−11.3052	−0.421614	−0.210807	−	0.977528i	$-0.567609\pi$
$719$	−11.3052	−0.421614	−0.210807	+	0.977528i	$0.567609\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	53.2808	1.98153
$724$	0	0
$725$	32.8974	1.22178
$726$	0	0
$727$	28.9056	1.07205	0.536025	−	0.844202i	$-0.319925\pi$
$727$	28.9056	1.07205	0.536025	+	0.844202i	$0.319925\pi$
$728$	0	0
$729$	−11.9977	−0.444361
$730$	0	0
$731$	6.98179	0.258231
$732$	0	0
$733$	23.0034	0.849652	0.424826	−	0.905275i	$-0.360335\pi$
$733$	23.0034	0.849652	0.424826	+	0.905275i	$0.360335\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−9.80929	−0.361330
$738$	0	0
$739$	−30.0797	−1.10650	−0.553250	−	0.833015i	$-0.686613\pi$
$739$	−30.0797	−1.10650	−0.553250	+	0.833015i	$0.686613\pi$
$740$	0	0
$741$	6.20765	0.228044
$742$	0	0
$743$	1.24979	0.0458503	0.0229251	−	0.999737i	$-0.492702\pi$
$743$	1.24979	0.0458503	0.0229251	+	0.999737i	$0.492702\pi$
$744$	0	0
$745$	−1.05629	−0.0386994
$746$	0	0
$747$	−28.9104	−1.05778
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−8.52731	−0.311166	−0.155583	−	0.987823i	$-0.549726\pi$
$751$	−8.52731	−0.311166	−0.155583	+	0.987823i	$0.549726\pi$
$752$	0	0
$753$	−47.2961	−1.72356
$754$	0	0
$755$	4.45521	0.162142
$756$	0	0
$757$	11.0979	0.403359	0.201679	−	0.979452i	$-0.435360\pi$
$757$	11.0979	0.403359	0.201679	+	0.979452i	$0.435360\pi$
$758$	0	0
$759$	−29.2935	−1.06329
$760$	0	0
$761$	31.0298	1.12483	0.562416	−	0.826855i	$-0.309872\pi$
$761$	31.0298	1.12483	0.562416	+	0.826855i	$0.309872\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−2.11661	−0.0765261
$766$	0	0
$767$	41.2742	1.49032
$768$	0	0
$769$	46.3590	1.67175	0.835874	−	0.548922i	$-0.184962\pi$
$769$	46.3590	1.67175	0.835874	+	0.548922i	$0.184962\pi$
$770$	0	0
$771$	20.8660	0.751470
$772$	0	0
$773$	−0.840164	−0.0302186	−0.0151093	−	0.999886i	$-0.504810\pi$
$773$	−0.840164	−0.0302186	−0.0151093	+	0.999886i	$0.504810\pi$
$774$	0	0
$775$	−2.26933	−0.0815166
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−0.682733	−0.0244615
$780$	0	0
$781$	−10.0559	−0.359828
$782$	0	0
$783$	11.7535	0.420037
$784$	0	0
$785$	8.50348	0.303502
$786$	0	0
$787$	10.3231	0.367979	0.183990	−	0.982928i	$-0.441099\pi$
$787$	10.3231	0.367979	0.183990	+	0.982928i	$0.441099\pi$
$788$	0	0
$789$	27.4329	0.976637
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−11.4136	−0.405310
$794$	0	0
$795$	1.63881	0.0581227
$796$	0	0
$797$	−14.5828	−0.516550	−0.258275	−	0.966071i	$-0.583154\pi$
$797$	−14.5828	−0.516550	−0.258275	+	0.966071i	$0.583154\pi$
$798$	0	0
$799$	−16.1185	−0.570233
$800$	0	0
$801$	31.4064	1.10969
$802$	0	0
$803$	−1.86401	−0.0657793
$804$	0	0
$805$	0	0
$806$	0	0
$807$	12.2980	0.432910
$808$	0	0
$809$	−41.7496	−1.46784	−0.733919	−	0.679237i	$-0.762311\pi$
$809$	−41.7496	−1.46784	−0.733919	+	0.679237i	$0.762311\pi$
$810$	0	0
$811$	−6.25270	−0.219562	−0.109781	−	0.993956i	$-0.535015\pi$
$811$	−6.25270	−0.219562	−0.109781	+	0.993956i	$0.535015\pi$
$812$	0	0
$813$	−50.1888	−1.76020
$814$	0	0
$815$	−5.97874	−0.209426
$816$	0	0
$817$	1.76470	0.0617391
$818$	0	0
$819$	0	0
$820$	0	0
$821$	22.1640	0.773530	0.386765	−	0.922178i	$-0.373592\pi$
$821$	22.1640	0.773530	0.386765	+	0.922178i	$0.373592\pi$
$822$	0	0
$823$	15.8352	0.551979	0.275990	−	0.961161i	$-0.410994\pi$
$823$	15.8352	0.551979	0.275990	+	0.961161i	$0.410994\pi$
$824$	0	0
$825$	−34.7204	−1.20881
$826$	0	0
$827$	43.9832	1.52945	0.764723	−	0.644359i	$-0.222876\pi$
$827$	43.9832	1.52945	0.764723	+	0.644359i	$0.222876\pi$
$828$	0	0
$829$	24.0080	0.833832	0.416916	−	0.908945i	$-0.363111\pi$
$829$	24.0080	0.833832	0.416916	+	0.908945i	$0.363111\pi$
$830$	0	0
$831$	−12.4180	−0.430776
$832$	0	0
$833$	0	0
$834$	0	0
$835$	3.98209	0.137806
$836$	0	0
$837$	−0.810782	−0.0280247
$838$	0	0
$839$	−22.4393	−0.774690	−0.387345	−	0.921935i	$-0.626608\pi$
$839$	−22.4393	−0.774690	−0.387345	+	0.921935i	$0.626608\pi$
$840$	0	0
$841$	16.4918	0.568684
$842$	0	0
$843$	−40.3663	−1.39029
$844$	0	0
$845$	0.973451	0.0334877
$846$	0	0
$847$	0	0
$848$	0	0
$849$	33.5803	1.15247
$850$	0	0
$851$	17.8500	0.611890
$852$	0	0
$853$	55.6852	1.90663	0.953313	−	0.301984i	$-0.0976491\pi$
$853$	55.6852	1.90663	0.953313	+	0.301984i	$0.0976491\pi$
$854$	0	0
$855$	−0.534989	−0.0182963
$856$	0	0
$857$	5.09534	0.174053	0.0870267	−	0.996206i	$-0.472263\pi$
$857$	5.09534	0.174053	0.0870267	+	0.996206i	$0.472263\pi$
$858$	0	0
$859$	12.2332	0.417391	0.208696	−	0.977981i	$-0.433078\pi$
$859$	12.2332	0.417391	0.208696	+	0.977981i	$0.433078\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	12.6995	0.432298	0.216149	−	0.976360i	$-0.430650\pi$
$863$	12.6995	0.432298	0.216149	+	0.976360i	$0.430650\pi$
$864$	0	0
$865$	0.619265	0.0210556
$866$	0	0
$867$	−22.2103	−0.754300
$868$	0	0
$869$	−2.66275	−0.0903275
$870$	0	0
$871$	12.5292	0.424537
$872$	0	0
$873$	−7.62834	−0.258180
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−35.5287	−1.19972	−0.599859	−	0.800106i	$-0.704777\pi$
$877$	−35.5287	−1.19972	−0.599859	+	0.800106i	$0.704777\pi$
$878$	0	0
$879$	16.3293	0.550773
$880$	0	0
$881$	10.8665	0.366102	0.183051	−	0.983103i	$-0.441403\pi$
$881$	10.8665	0.366102	0.183051	+	0.983103i	$0.441403\pi$
$882$	0	0
$883$	17.7307	0.596686	0.298343	−	0.954459i	$-0.403566\pi$
$883$	17.7307	0.596686	0.298343	+	0.954459i	$0.403566\pi$
$884$	0	0
$885$	−8.32397	−0.279807
$886$	0	0
$887$	−15.0636	−0.505787	−0.252894	−	0.967494i	$-0.581382\pi$
$887$	−15.0636	−0.505787	−0.252894	+	0.967494i	$0.581382\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−33.2923	−1.11533
$892$	0	0
$893$	−4.07409	−0.136334
$894$	0	0
$895$	−2.54696	−0.0851355
$896$	0	0
$897$	37.4161	1.24929
$898$	0	0
$899$	−3.13811	−0.104662
$900$	0	0
$901$	−5.52529	−0.184074
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.22420	0.0406939
$906$	0	0
$907$	−33.3910	−1.10873	−0.554366	−	0.832273i	$-0.687039\pi$
$907$	−33.3910	−1.10873	−0.554366	+	0.832273i	$0.687039\pi$
$908$	0	0
$909$	−19.9243	−0.660846
$910$	0	0
$911$	−46.2548	−1.53249	−0.766244	−	0.642549i	$-0.777877\pi$
$911$	−46.2548	−1.53249	−0.766244	+	0.642549i	$0.777877\pi$
$912$	0	0
$913$	−40.1653	−1.32928
$914$	0	0
$915$	2.30185	0.0760967
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−22.2188	−0.732932	−0.366466	−	0.930432i	$-0.619432\pi$
$919$	−22.2188	−0.732932	−0.366466	+	0.930432i	$0.619432\pi$
$920$	0	0
$921$	39.9036	1.31487
$922$	0	0
$923$	12.8442	0.422773
$924$	0	0
$925$	21.1569	0.695634
$926$	0	0
$927$	−10.4147	−0.342065
$928$	0	0
$929$	36.3680	1.19319	0.596597	−	0.802541i	$-0.296519\pi$
$929$	36.3680	1.19319	0.596597	+	0.802541i	$0.296519\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0.673318	0.0220434
$934$	0	0
$935$	−2.94061	−0.0961683
$936$	0	0
$937$	50.1107	1.63704	0.818522	−	0.574475i	$-0.194794\pi$
$937$	50.1107	1.63704	0.818522	+	0.574475i	$0.194794\pi$
$938$	0	0
$939$	73.4336	2.39642
$940$	0	0
$941$	−31.8951	−1.03975	−0.519876	−	0.854242i	$-0.674022\pi$
$941$	−31.8951	−1.03975	−0.519876	+	0.854242i	$0.674022\pi$
$942$	0	0
$943$	−4.11511	−0.134006
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−21.6045	−0.702052	−0.351026	−	0.936366i	$-0.614167\pi$
$947$	−21.6045	−0.702052	−0.351026	+	0.936366i	$0.614167\pi$
$948$	0	0
$949$	2.38086	0.0772861
$950$	0	0
$951$	−18.7767	−0.608876
$952$	0	0
$953$	24.8795	0.805928	0.402964	−	0.915216i	$-0.367980\pi$
$953$	24.8795	0.805928	0.402964	+	0.915216i	$0.367980\pi$
$954$	0	0
$955$	−0.313268	−0.0101371
$956$	0	0
$957$	−48.0127	−1.55203
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−30.7835	−0.993017
$962$	0	0
$963$	−12.0653	−0.388798
$964$	0	0
$965$	4.39133	0.141362
$966$	0	0
$967$	40.4487	1.30074	0.650371	−	0.759616i	$-0.274613\pi$
$967$	40.4487	1.30074	0.650371	+	0.759616i	$0.274613\pi$
$968$	0	0
$969$	4.22091	0.135595
$970$	0	0
$971$	−23.3494	−0.749318	−0.374659	−	0.927163i	$-0.622240\pi$
$971$	−23.3494	−0.749318	−0.374659	+	0.927163i	$0.622240\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	44.3478	1.42026
$976$	0	0
$977$	−60.7351	−1.94309	−0.971544	−	0.236861i	$-0.923882\pi$
$977$	−60.7351	−1.94309	−0.971544	+	0.236861i	$0.923882\pi$
$978$	0	0
$979$	43.6331	1.39452
$980$	0	0
$981$	16.1314	0.515035
$982$	0	0
$983$	48.5598	1.54882	0.774409	−	0.632685i	$-0.218047\pi$
$983$	48.5598	1.54882	0.774409	+	0.632685i	$0.218047\pi$
$984$	0	0
$985$	3.47101	0.110595
$986$	0	0
$987$	0	0
$988$	0	0
$989$	10.6366	0.338224
$990$	0	0
$991$	17.9816	0.571205	0.285603	−	0.958348i	$-0.407806\pi$
$991$	17.9816	0.571205	0.285603	+	0.958348i	$0.407806\pi$
$992$	0	0
$993$	28.4781	0.903724
$994$	0	0
$995$	7.69452	0.243933
$996$	0	0
$997$	−32.2058	−1.01997	−0.509983	−	0.860184i	$-0.670348\pi$
$997$	−32.2058	−1.01997	−0.509983	+	0.860184i	$0.670348\pi$
$998$	0	0
$999$	7.55890	0.239153

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.m.1.8		8
7.2	even	3		1148.2.i.d.165.1	✓	16
7.4	even	3		1148.2.i.d.821.1	yes	16
7.6	odd	2		8036.2.a.n.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.i.d.165.1	✓	16	7.2	even	3
1148.2.i.d.821.1	yes	16	7.4	even	3
8036.2.a.m.1.8		8	1.1	even	1	trivial
8036.2.a.n.1.1		8	7.6	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:	\( N \)	\(=\)	\( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8036.a (trivial)

\(f(q)\)	\(=\)	\(q+2.28881 q^{3} +0.350035 q^{5} +2.23863 q^{9} +O(q^{10})\)	\(q+2.28881 q^{3} +0.350035 q^{5} +2.23863 q^{9} +3.11014 q^{11} -3.97253 q^{13} +0.801161 q^{15} -2.70114 q^{17} -0.682733 q^{19} -4.11511 q^{23} -4.87748 q^{25} -1.74262 q^{27} -6.74476 q^{29} +0.465267 q^{31} +7.11852 q^{33} -4.33767 q^{37} -9.09236 q^{39} +1.00000 q^{41} -2.58476 q^{43} +0.783600 q^{45} +5.96732 q^{47} -6.18238 q^{51} +2.04555 q^{53} +1.08866 q^{55} -1.56264 q^{57} -10.3899 q^{59} +2.87314 q^{61} -1.39052 q^{65} -3.15397 q^{67} -9.41870 q^{69} -3.23326 q^{71} -0.599331 q^{73} -11.1636 q^{75} -0.856150 q^{79} -10.7044 q^{81} -12.9143 q^{83} -0.945491 q^{85} -15.4375 q^{87} +14.0293 q^{89} +1.06490 q^{93} -0.238980 q^{95} -3.40759 q^{97} +6.96247 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{9}+O(q^{10}) \) 8 * q + 4 * q^9	\( 8 q + 4 q^{9} - 8 q^{11} - 7 q^{13} - q^{15} - q^{17} + 4 q^{19} - 3 q^{23} - 4 q^{25} - 12 q^{27} - 4 q^{29} + 4 q^{31} + 23 q^{33} - 31 q^{37} + 5 q^{39} + 8 q^{41} - 8 q^{43} + q^{45} + 24 q^{47} - 23 q^{51} - q^{53} + 2 q^{55} - 15 q^{57} + 4 q^{59} - 4 q^{61} - 24 q^{65} - 21 q^{69} + 8 q^{71} + 11 q^{73} - 15 q^{75} + 14 q^{79} - 28 q^{81} - 42 q^{83} - 20 q^{85} + 25 q^{87} - 11 q^{89} - 27 q^{93} - 15 q^{95} - 16 q^{97} - 8 q^{99}+O(q^{100}) \) 8 * q + 4 * q^9 - 8 * q^11 - 7 * q^13 - q^15 - q^17 + 4 * q^19 - 3 * q^23 - 4 * q^25 - 12 * q^27 - 4 * q^29 + 4 * q^31 + 23 * q^33 - 31 * q^37 + 5 * q^39 + 8 * q^41 - 8 * q^43 + q^45 + 24 * q^47 - 23 * q^51 - q^53 + 2 * q^55 - 15 * q^57 + 4 * q^59 - 4 * q^61 - 24 * q^65 - 21 * q^69 + 8 * q^71 + 11 * q^73 - 15 * q^75 + 14 * q^79 - 28 * q^81 - 42 * q^83 - 20 * q^85 + 25 * q^87 - 11 * q^89 - 27 * q^93 - 15 * q^95 - 16 * q^97 - 8 * q^99

Introduction

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