LMFDB - Embedded newform 8036.2.a.l.1.3

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:	$5$
Coefficient field:	5.5.470117.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 6x^{3} + 8x^{2} + 7x + 1 $ x^5 - 2x^4 - 6x^3 + 8x^2 + 7x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
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			1.3
Root			$-0.189142$ 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)

$f(q)$	$=$	$q-0.189142 q^{3} +1.51194 q^{5} -2.96423 q^{9} +O(q^{10})$	$q-0.189142 q^{3} +1.51194 q^{5} -2.96423 q^{9} +5.99368 q^{11} +2.57300 q^{13} -0.285972 q^{15} +2.45228 q^{17} -7.57962 q^{19} -4.57857 q^{23} -2.71403 q^{25} +1.12809 q^{27} -7.11620 q^{29} +6.03822 q^{31} -1.13366 q^{33} -4.63823 q^{37} -0.486663 q^{39} -1.00000 q^{41} -12.9042 q^{43} -4.48174 q^{45} +8.18840 q^{47} -0.463830 q^{51} -2.35121 q^{53} +9.06211 q^{55} +1.43363 q^{57} -6.08600 q^{59} -9.42174 q^{61} +3.89023 q^{65} -8.16869 q^{67} +0.866001 q^{69} -16.1895 q^{71} +0.0799738 q^{73} +0.513337 q^{75} +7.86526 q^{79} +8.67931 q^{81} -13.0912 q^{83} +3.70771 q^{85} +1.34597 q^{87} +5.44776 q^{89} -1.14208 q^{93} -11.4600 q^{95} +11.4936 q^{97} -17.7666 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{3} + q^{5} + q^{9}+O(q^{10}) $ 5 * q + 2 * q^3 + q^5 + q^9	$ 5 q + 2 q^{3} + q^{5} + q^{9} - 2 q^{11} + q^{13} - 9 q^{15} + 3 q^{17} + 4 q^{19} - 8 q^{23} - 6 q^{25} + 8 q^{27} - 9 q^{29} + 11 q^{31} - 5 q^{33} - 11 q^{37} - 17 q^{39} - 5 q^{41} - 27 q^{43} + 3 q^{45} + 3 q^{47} - 3 q^{51} - 19 q^{53} + 13 q^{55} - 11 q^{57} + 15 q^{59} + 7 q^{65} - 21 q^{67} - 14 q^{69} - 16 q^{71} + 10 q^{73} - 12 q^{75} - 14 q^{79} - 7 q^{81} + 2 q^{83} - 21 q^{85} + 36 q^{87} - 6 q^{89} + 17 q^{93} + 9 q^{95} - 20 q^{97} - 17 q^{99}+O(q^{100}) $ 5 * q + 2 * q^3 + q^5 + q^9 - 2 * q^11 + q^13 - 9 * q^15 + 3 * q^17 + 4 * q^19 - 8 * q^23 - 6 * q^25 + 8 * q^27 - 9 * q^29 + 11 * q^31 - 5 * q^33 - 11 * q^37 - 17 * q^39 - 5 * q^41 - 27 * q^43 + 3 * q^45 + 3 * q^47 - 3 * q^51 - 19 * q^53 + 13 * q^55 - 11 * q^57 + 15 * q^59 + 7 * q^65 - 21 * q^67 - 14 * q^69 - 16 * q^71 + 10 * q^73 - 12 * q^75 - 14 * q^79 - 7 * q^81 + 2 * q^83 - 21 * q^85 + 36 * q^87 - 6 * q^89 + 17 * q^93 + 9 * q^95 - 20 * q^97 - 17 * 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
$3$	−0.189142	−0.109201	−0.0546007	−	0.998508i	$-0.517389\pi$
$3$	−0.189142	−0.109201	−0.0546007	+	0.998508i	$0.517389\pi$
$4$	0	0
$5$	1.51194	0.676162	0.338081	−	0.941117i	$-0.390222\pi$
$5$	1.51194	0.676162	0.338081	+	0.941117i	$0.390222\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.96423	−0.988075
$10$	0	0
$11$	5.99368	1.80716	0.903582	−	0.428416i	$-0.140928\pi$
$11$	5.99368	1.80716	0.903582	+	0.428416i	$0.140928\pi$
$12$	0	0
$13$	2.57300	0.713621	0.356811	−	0.934177i	$-0.383864\pi$
$13$	2.57300	0.713621	0.356811	+	0.934177i	$0.383864\pi$
$14$	0	0
$15$	−0.285972	−0.0738377
$16$	0	0
$17$	2.45228	0.594766	0.297383	−	0.954758i	$-0.403886\pi$
$17$	2.45228	0.594766	0.297383	+	0.954758i	$0.403886\pi$
$18$	0	0
$19$	−7.57962	−1.73889	−0.869443	−	0.494034i	$-0.835522\pi$
$19$	−7.57962	−1.73889	−0.869443	+	0.494034i	$0.835522\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.57857	−0.954698	−0.477349	−	0.878714i	$-0.658402\pi$
$23$	−4.57857	−0.954698	−0.477349	+	0.878714i	$0.658402\pi$
$24$	0	0
$25$	−2.71403	−0.542806
$26$	0	0
$27$	1.12809	0.217100
$28$	0	0
$29$	−7.11620	−1.32144	−0.660722	−	0.750630i	$-0.729750\pi$
$29$	−7.11620	−1.32144	−0.660722	+	0.750630i	$0.729750\pi$
$30$	0	0
$31$	6.03822	1.08450	0.542248	−	0.840218i	$-0.317573\pi$
$31$	6.03822	1.08450	0.542248	+	0.840218i	$0.317573\pi$
$32$	0	0
$33$	−1.13366	−0.197345
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.63823	−0.762521	−0.381260	−	0.924468i	$-0.624510\pi$
$37$	−4.63823	−0.762521	−0.381260	+	0.924468i	$0.624510\pi$
$38$	0	0
$39$	−0.486663	−0.0779284
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−12.9042	−1.96788	−0.983938	−	0.178511i	$-0.942872\pi$
$43$	−12.9042	−1.96788	−0.983938	+	0.178511i	$0.942872\pi$
$44$	0	0
$45$	−4.48174	−0.668098
$46$	0	0
$47$	8.18840	1.19440	0.597200	−	0.802092i	$-0.296280\pi$
$47$	8.18840	1.19440	0.597200	+	0.802092i	$0.296280\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−0.463830	−0.0649492
$52$	0	0
$53$	−2.35121	−0.322963	−0.161481	−	0.986876i	$-0.551627\pi$
$53$	−2.35121	−0.322963	−0.161481	+	0.986876i	$0.551627\pi$
$54$	0	0
$55$	9.06211	1.22193
$56$	0	0
$57$	1.43363	0.189889
$58$	0	0
$59$	−6.08600	−0.792329	−0.396165	−	0.918180i	$-0.629659\pi$
$59$	−6.08600	−0.792329	−0.396165	+	0.918180i	$0.629659\pi$
$60$	0	0
$61$	−9.42174	−1.20633	−0.603165	−	0.797616i	$-0.706094\pi$
$61$	−9.42174	−1.20633	−0.603165	+	0.797616i	$0.706094\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.89023	0.482523
$66$	0	0
$67$	−8.16869	−0.997964	−0.498982	−	0.866612i	$-0.666293\pi$
$67$	−8.16869	−0.997964	−0.498982	+	0.866612i	$0.666293\pi$
$68$	0	0
$69$	0.866001	0.104254
$70$	0	0
$71$	−16.1895	−1.92133	−0.960667	−	0.277702i	$-0.910427\pi$
$71$	−16.1895	−1.92133	−0.960667	+	0.277702i	$0.910427\pi$
$72$	0	0
$73$	0.0799738	0.00936023	0.00468011	−	0.999989i	$-0.498510\pi$
$73$	0.0799738	0.00936023	0.00468011	+	0.999989i	$0.498510\pi$
$74$	0	0
$75$	0.513337	0.0592751
$76$	0	0
$77$	0	0
$78$	0	0
$79$	7.86526	0.884910	0.442455	−	0.896791i	$-0.354108\pi$
$79$	7.86526	0.884910	0.442455	+	0.896791i	$0.354108\pi$
$80$	0	0
$81$	8.67931	0.964367
$82$	0	0
$83$	−13.0912	−1.43695	−0.718474	−	0.695553i	$-0.755159\pi$
$83$	−13.0912	−1.43695	−0.718474	+	0.695553i	$0.755159\pi$
$84$	0	0
$85$	3.70771	0.402158
$86$	0	0
$87$	1.34597	0.144304
$88$	0	0
$89$	5.44776	0.577462	0.288731	−	0.957410i	$-0.406767\pi$
$89$	5.44776	0.577462	0.288731	+	0.957410i	$0.406767\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−1.14208	−0.118428
$94$	0	0
$95$	−11.4600	−1.17577
$96$	0	0
$97$	11.4936	1.16700	0.583499	−	0.812114i	$-0.301683\pi$
$97$	11.4936	1.16700	0.583499	+	0.812114i	$0.301683\pi$
$98$	0	0
$99$	−17.7666	−1.78561
$100$	0	0
$101$	−11.1713	−1.11158	−0.555792	−	0.831322i	$-0.687585\pi$
$101$	−11.1713	−1.11158	−0.555792	+	0.831322i	$0.687585\pi$
$102$	0	0
$103$	2.61580	0.257743	0.128871	−	0.991661i	$-0.458865\pi$
$103$	2.61580	0.257743	0.128871	+	0.991661i	$0.458865\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.67625	0.838765	0.419383	−	0.907810i	$-0.362247\pi$
$107$	8.67625	0.838765	0.419383	+	0.907810i	$0.362247\pi$
$108$	0	0
$109$	−0.708796	−0.0678903	−0.0339452	−	0.999424i	$-0.510807\pi$
$109$	−0.708796	−0.0678903	−0.0339452	+	0.999424i	$0.510807\pi$
$110$	0	0
$111$	0.877286	0.0832683
$112$	0	0
$113$	12.4140	1.16781	0.583907	−	0.811821i	$-0.301523\pi$
$113$	12.4140	1.16781	0.583907	+	0.811821i	$0.301523\pi$
$114$	0	0
$115$	−6.92254	−0.645530
$116$	0	0
$117$	−7.62695	−0.705112
$118$	0	0
$119$	0	0
$120$	0	0
$121$	24.9242	2.26584
$122$	0	0
$123$	0.189142	0.0170544
$124$	0	0
$125$	−11.6632	−1.04319
$126$	0	0
$127$	0.390482	0.0346497	0.0173248	−	0.999850i	$-0.494485\pi$
$127$	0.390482	0.0346497	0.0173248	+	0.999850i	$0.494485\pi$
$128$	0	0
$129$	2.44073	0.214895
$130$	0	0
$131$	−0.937891	−0.0819439	−0.0409720	−	0.999160i	$-0.513045\pi$
$131$	−0.937891	−0.0819439	−0.0409720	+	0.999160i	$0.513045\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	1.70560	0.146795
$136$	0	0
$137$	−20.0992	−1.71719	−0.858597	−	0.512651i	$-0.828664\pi$
$137$	−20.0992	−1.71719	−0.858597	+	0.512651i	$0.828664\pi$
$138$	0	0
$139$	15.2022	1.28943	0.644717	−	0.764422i	$-0.276975\pi$
$139$	15.2022	1.28943	0.644717	+	0.764422i	$0.276975\pi$
$140$	0	0
$141$	−1.54877	−0.130430
$142$	0	0
$143$	15.4217	1.28963
$144$	0	0
$145$	−10.7593	−0.893510
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−8.58520	−0.703327	−0.351663	−	0.936127i	$-0.614384\pi$
$149$	−8.58520	−0.703327	−0.351663	+	0.936127i	$0.614384\pi$
$150$	0	0
$151$	5.88272	0.478729	0.239364	−	0.970930i	$-0.423061\pi$
$151$	5.88272	0.478729	0.239364	+	0.970930i	$0.423061\pi$
$152$	0	0
$153$	−7.26912	−0.587673
$154$	0	0
$155$	9.12945	0.733295
$156$	0	0
$157$	9.93083	0.792566	0.396283	−	0.918128i	$-0.370300\pi$
$157$	9.93083	0.792566	0.396283	+	0.918128i	$0.370300\pi$
$158$	0	0
$159$	0.444712	0.0352680
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0.479942	0.0375919	0.0187960	−	0.999823i	$-0.494017\pi$
$163$	0.479942	0.0375919	0.0187960	+	0.999823i	$0.494017\pi$
$164$	0	0
$165$	−1.71403	−0.133437
$166$	0	0
$167$	−13.5498	−1.04852	−0.524258	−	0.851559i	$-0.675657\pi$
$167$	−13.5498	−1.04852	−0.524258	+	0.851559i	$0.675657\pi$
$168$	0	0
$169$	−6.37968	−0.490745
$170$	0	0
$171$	22.4677	1.71815
$172$	0	0
$173$	0.684951	0.0520758	0.0260379	−	0.999661i	$-0.491711\pi$
$173$	0.684951	0.0520758	0.0260379	+	0.999661i	$0.491711\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.15112	0.0865234
$178$	0	0
$179$	0.0838879	0.00627008	0.00313504	−	0.999995i	$-0.499002\pi$
$179$	0.0838879	0.00627008	0.00313504	+	0.999995i	$0.499002\pi$
$180$	0	0
$181$	−12.8162	−0.952618	−0.476309	−	0.879278i	$-0.658026\pi$
$181$	−12.8162	−0.952618	−0.476309	+	0.879278i	$0.658026\pi$
$182$	0	0
$183$	1.78205	0.131733
$184$	0	0
$185$	−7.01274	−0.515587
$186$	0	0
$187$	14.6982	1.07484
$188$	0	0
$189$	0	0
$190$	0	0
$191$	23.6007	1.70769	0.853844	−	0.520528i	$-0.174265\pi$
$191$	23.6007	1.70769	0.853844	+	0.520528i	$0.174265\pi$
$192$	0	0
$193$	−19.5769	−1.40917	−0.704587	−	0.709617i	$-0.748868\pi$
$193$	−19.5769	−1.40917	−0.704587	+	0.709617i	$0.748868\pi$
$194$	0	0
$195$	−0.735806	−0.0526922
$196$	0	0
$197$	0.723514	0.0515482	0.0257741	−	0.999668i	$-0.491795\pi$
$197$	0.723514	0.0515482	0.0257741	+	0.999668i	$0.491795\pi$
$198$	0	0
$199$	26.1583	1.85431	0.927157	−	0.374674i	$-0.122245\pi$
$199$	26.1583	1.85431	0.927157	+	0.374674i	$0.122245\pi$
$200$	0	0
$201$	1.54504	0.108979
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1.51194	−0.105599
$206$	0	0
$207$	13.5719	0.943313
$208$	0	0
$209$	−45.4299	−3.14245
$210$	0	0
$211$	−6.14912	−0.423323	−0.211661	−	0.977343i	$-0.567887\pi$
$211$	−6.14912	−0.423323	−0.211661	+	0.977343i	$0.567887\pi$
$212$	0	0
$213$	3.06211	0.209812
$214$	0	0
$215$	−19.5105	−1.33060
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−0.0151264	−0.00102215
$220$	0	0
$221$	6.30972	0.424438
$222$	0	0
$223$	−16.0367	−1.07390	−0.536950	−	0.843614i	$-0.680423\pi$
$223$	−16.0367	−1.07390	−0.536950	+	0.843614i	$0.680423\pi$
$224$	0	0
$225$	8.04499	0.536333
$226$	0	0
$227$	−25.8470	−1.71552	−0.857762	−	0.514046i	$-0.828146\pi$
$227$	−25.8470	−1.71552	−0.857762	+	0.514046i	$0.828146\pi$
$228$	0	0
$229$	8.68209	0.573729	0.286864	−	0.957971i	$-0.407387\pi$
$229$	8.68209	0.573729	0.286864	+	0.957971i	$0.407387\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	16.8859	1.10623	0.553116	−	0.833104i	$-0.313438\pi$
$233$	16.8859	1.10623	0.553116	+	0.833104i	$0.313438\pi$
$234$	0	0
$235$	12.3804	0.807608
$236$	0	0
$237$	−1.48765	−0.0966334
$238$	0	0
$239$	−12.2054	−0.789501	−0.394750	−	0.918788i	$-0.629169\pi$
$239$	−12.2054	−0.789501	−0.394750	+	0.918788i	$0.629169\pi$
$240$	0	0
$241$	1.10393	0.0711103	0.0355552	−	0.999368i	$-0.488680\pi$
$241$	1.10393	0.0711103	0.0355552	+	0.999368i	$0.488680\pi$
$242$	0	0
$243$	−5.02588	−0.322411
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−19.5024	−1.24091
$248$	0	0
$249$	2.47610	0.156917
$250$	0	0
$251$	20.2063	1.27541	0.637705	−	0.770281i	$-0.279884\pi$
$251$	20.2063	1.27541	0.637705	+	0.770281i	$0.279884\pi$
$252$	0	0
$253$	−27.4425	−1.72530
$254$	0	0
$255$	−0.701285	−0.0439162
$256$	0	0
$257$	−3.52848	−0.220101	−0.110050	−	0.993926i	$-0.535101\pi$
$257$	−3.52848	−0.220101	−0.110050	+	0.993926i	$0.535101\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	21.0940	1.30569
$262$	0	0
$263$	6.85883	0.422934	0.211467	−	0.977385i	$-0.432176\pi$
$263$	6.85883	0.422934	0.211467	+	0.977385i	$0.432176\pi$
$264$	0	0
$265$	−3.55489	−0.218375
$266$	0	0
$267$	−1.03040	−0.0630596
$268$	0	0
$269$	−1.40104	−0.0854232	−0.0427116	−	0.999087i	$-0.513600\pi$
$269$	−1.40104	−0.0854232	−0.0427116	+	0.999087i	$0.513600\pi$
$270$	0	0
$271$	−19.5833	−1.18960	−0.594799	−	0.803874i	$-0.702769\pi$
$271$	−19.5833	−1.18960	−0.594799	+	0.803874i	$0.702769\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−16.2670	−0.980938
$276$	0	0
$277$	3.29060	0.197713	0.0988565	−	0.995102i	$-0.468482\pi$
$277$	3.29060	0.197713	0.0988565	+	0.995102i	$0.468482\pi$
$278$	0	0
$279$	−17.8987	−1.07156
$280$	0	0
$281$	19.9937	1.19272	0.596362	−	0.802716i	$-0.296612\pi$
$281$	19.9937	1.19272	0.596362	+	0.802716i	$0.296612\pi$
$282$	0	0
$283$	19.0681	1.13348	0.566739	−	0.823897i	$-0.308205\pi$
$283$	19.0681	1.13348	0.566739	+	0.823897i	$0.308205\pi$
$284$	0	0
$285$	2.16756	0.128395
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−10.9863	−0.646254
$290$	0	0
$291$	−2.17392	−0.127438
$292$	0	0
$293$	−1.20077	−0.0701496	−0.0350748	−	0.999385i	$-0.511167\pi$
$293$	−1.20077	−0.0701496	−0.0350748	+	0.999385i	$0.511167\pi$
$294$	0	0
$295$	−9.20168	−0.535743
$296$	0	0
$297$	6.76140	0.392336
$298$	0	0
$299$	−11.7807	−0.681293
$300$	0	0
$301$	0	0
$302$	0	0
$303$	2.11296	0.121386
$304$	0	0
$305$	−14.2451	−0.815674
$306$	0	0
$307$	32.7439	1.86880	0.934398	−	0.356232i	$-0.115939\pi$
$307$	32.7439	1.86880	0.934398	+	0.356232i	$0.115939\pi$
$308$	0	0
$309$	−0.494759	−0.0281459
$310$	0	0
$311$	−19.2651	−1.09242	−0.546211	−	0.837647i	$-0.683930\pi$
$311$	−19.2651	−1.09242	−0.546211	+	0.837647i	$0.683930\pi$
$312$	0	0
$313$	−29.9633	−1.69362	−0.846812	−	0.531893i	$-0.821481\pi$
$313$	−29.9633	−1.69362	−0.846812	+	0.531893i	$0.821481\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−11.7544	−0.660191	−0.330095	−	0.943948i	$-0.607081\pi$
$317$	−11.7544	−0.660191	−0.330095	+	0.943948i	$0.607081\pi$
$318$	0	0
$319$	−42.6522	−2.38807
$320$	0	0
$321$	−1.64105	−0.0915943
$322$	0	0
$323$	−18.5874	−1.03423
$324$	0	0
$325$	−6.98319	−0.387358
$326$	0	0
$327$	0.134063	0.00741372
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−14.5852	−0.801675	−0.400837	−	0.916149i	$-0.631281\pi$
$331$	−14.5852	−0.801675	−0.400837	+	0.916149i	$0.631281\pi$
$332$	0	0
$333$	13.7488	0.753428
$334$	0	0
$335$	−12.3506	−0.674785
$336$	0	0
$337$	4.20359	0.228984	0.114492	−	0.993424i	$-0.463476\pi$
$337$	4.20359	0.228984	0.114492	+	0.993424i	$0.463476\pi$
$338$	0	0
$339$	−2.34802	−0.127527
$340$	0	0
$341$	36.1912	1.95986
$342$	0	0
$343$	0	0
$344$	0	0
$345$	1.30934	0.0704927
$346$	0	0
$347$	21.6744	1.16354	0.581771	−	0.813353i	$-0.302360\pi$
$347$	21.6744	1.16354	0.581771	+	0.813353i	$0.302360\pi$
$348$	0	0
$349$	−11.3736	−0.608813	−0.304407	−	0.952542i	$-0.598458\pi$
$349$	−11.3736	−0.608813	−0.304407	+	0.952542i	$0.598458\pi$
$350$	0	0
$351$	2.90257	0.154928
$352$	0	0
$353$	−13.5784	−0.722707	−0.361354	−	0.932429i	$-0.617685\pi$
$353$	−13.5784	−0.722707	−0.361354	+	0.932429i	$0.617685\pi$
$354$	0	0
$355$	−24.4775	−1.29913
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5.12110	−0.270281	−0.135141	−	0.990826i	$-0.543149\pi$
$359$	−5.12110	−0.270281	−0.135141	+	0.990826i	$0.543149\pi$
$360$	0	0
$361$	38.4507	2.02372
$362$	0	0
$363$	−4.71423	−0.247433
$364$	0	0
$365$	0.120916	0.00632903
$366$	0	0
$367$	19.1708	1.00071	0.500353	−	0.865822i	$-0.333204\pi$
$367$	19.1708	1.00071	0.500353	+	0.865822i	$0.333204\pi$
$368$	0	0
$369$	2.96423	0.154311
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−11.7642	−0.609130	−0.304565	−	0.952492i	$-0.598511\pi$
$373$	−11.7642	−0.609130	−0.304565	+	0.952492i	$0.598511\pi$
$374$	0	0
$375$	2.20600	0.113917
$376$	0	0
$377$	−18.3100	−0.943011
$378$	0	0
$379$	−1.09400	−0.0561951	−0.0280975	−	0.999605i	$-0.508945\pi$
$379$	−1.09400	−0.0561951	−0.0280975	+	0.999605i	$0.508945\pi$
$380$	0	0
$381$	−0.0738566	−0.00378379
$382$	0	0
$383$	−36.0148	−1.84027	−0.920134	−	0.391604i	$-0.871920\pi$
$383$	−36.0148	−1.84027	−0.920134	+	0.391604i	$0.871920\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	38.2510	1.94441
$388$	0	0
$389$	−7.34101	−0.372204	−0.186102	−	0.982530i	$-0.559585\pi$
$389$	−7.34101	−0.372204	−0.186102	+	0.982530i	$0.559585\pi$
$390$	0	0
$391$	−11.2279	−0.567822
$392$	0	0
$393$	0.177395	0.00894838
$394$	0	0
$395$	11.8918	0.598342
$396$	0	0
$397$	−28.5849	−1.43464	−0.717318	−	0.696746i	$-0.754630\pi$
$397$	−28.5849	−1.43464	−0.717318	+	0.696746i	$0.754630\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−11.1836	−0.558483	−0.279241	−	0.960221i	$-0.590083\pi$
$401$	−11.1836	−0.558483	−0.279241	+	0.960221i	$0.590083\pi$
$402$	0	0
$403$	15.5363	0.773920
$404$	0	0
$405$	13.1226	0.652068
$406$	0	0
$407$	−27.8001	−1.37800
$408$	0	0
$409$	0.810658	0.0400845	0.0200422	−	0.999799i	$-0.493620\pi$
$409$	0.810658	0.0400845	0.0200422	+	0.999799i	$0.493620\pi$
$410$	0	0
$411$	3.80162	0.187520
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−19.7932	−0.971610
$416$	0	0
$417$	−2.87538	−0.140808
$418$	0	0
$419$	−1.81425	−0.0886319	−0.0443159	−	0.999018i	$-0.514111\pi$
$419$	−1.81425	−0.0886319	−0.0443159	+	0.999018i	$0.514111\pi$
$420$	0	0
$421$	−30.3311	−1.47825	−0.739124	−	0.673569i	$-0.764760\pi$
$421$	−30.3311	−1.47825	−0.739124	+	0.673569i	$0.764760\pi$
$422$	0	0
$423$	−24.2723	−1.18016
$424$	0	0
$425$	−6.65556	−0.322842
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−2.91690	−0.140829
$430$	0	0
$431$	12.7769	0.615444	0.307722	−	0.951476i	$-0.400433\pi$
$431$	12.7769	0.615444	0.307722	+	0.951476i	$0.400433\pi$
$432$	0	0
$433$	−13.1695	−0.632885	−0.316442	−	0.948612i	$-0.602488\pi$
$433$	−13.1695	−0.632885	−0.316442	+	0.948612i	$0.602488\pi$
$434$	0	0
$435$	2.03504	0.0975725
$436$	0	0
$437$	34.7038	1.66011
$438$	0	0
$439$	26.8576	1.28184	0.640922	−	0.767606i	$-0.278552\pi$
$439$	26.8576	1.28184	0.640922	+	0.767606i	$0.278552\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−0.791018	−0.0375824	−0.0187912	−	0.999823i	$-0.505982\pi$
$443$	−0.791018	−0.0375824	−0.0187912	+	0.999823i	$0.505982\pi$
$444$	0	0
$445$	8.23671	0.390457
$446$	0	0
$447$	1.62382	0.0768042
$448$	0	0
$449$	−15.0146	−0.708582	−0.354291	−	0.935135i	$-0.615278\pi$
$449$	−15.0146	−0.708582	−0.354291	+	0.935135i	$0.615278\pi$
$450$	0	0
$451$	−5.99368	−0.282232
$452$	0	0
$453$	−1.11267	−0.0522778
$454$	0	0
$455$	0	0
$456$	0	0
$457$	23.9943	1.12241	0.561203	−	0.827678i	$-0.310339\pi$
$457$	23.9943	1.12241	0.561203	+	0.827678i	$0.310339\pi$
$458$	0	0
$459$	2.76639	0.129124
$460$	0	0
$461$	0.689846	0.0321294	0.0160647	−	0.999871i	$-0.494886\pi$
$461$	0.689846	0.0321294	0.0160647	+	0.999871i	$0.494886\pi$
$462$	0	0
$463$	−19.7359	−0.917206	−0.458603	−	0.888641i	$-0.651650\pi$
$463$	−19.7359	−0.917206	−0.458603	+	0.888641i	$0.651650\pi$
$464$	0	0
$465$	−1.72676	−0.0800768
$466$	0	0
$467$	18.0069	0.833262	0.416631	−	0.909076i	$-0.363211\pi$
$467$	18.0069	0.833262	0.416631	+	0.909076i	$0.363211\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−1.87834	−0.0865493
$472$	0	0
$473$	−77.3438	−3.55627
$474$	0	0
$475$	20.5713	0.943876
$476$	0	0
$477$	6.96950	0.319112
$478$	0	0
$479$	21.8707	0.999297	0.499648	−	0.866228i	$-0.333463\pi$
$479$	21.8707	0.999297	0.499648	+	0.866228i	$0.333463\pi$
$480$	0	0
$481$	−11.9342	−0.544151
$482$	0	0
$483$	0	0
$484$	0	0
$485$	17.3777	0.789079
$486$	0	0
$487$	11.9588	0.541906	0.270953	−	0.962593i	$-0.412661\pi$
$487$	11.9588	0.541906	0.270953	+	0.962593i	$0.412661\pi$
$488$	0	0
$489$	−0.0907772	−0.00410509
$490$	0	0
$491$	−39.6236	−1.78819	−0.894094	−	0.447879i	$-0.852179\pi$
$491$	−39.6236	−1.78819	−0.894094	+	0.447879i	$0.852179\pi$
$492$	0	0
$493$	−17.4509	−0.785950
$494$	0	0
$495$	−26.8621	−1.20736
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−36.5734	−1.63725	−0.818624	−	0.574329i	$-0.805263\pi$
$499$	−36.5734	−1.63725	−0.818624	+	0.574329i	$0.805263\pi$
$500$	0	0
$501$	2.56284	0.114499
$502$	0	0
$503$	38.0403	1.69614	0.848068	−	0.529888i	$-0.177766\pi$
$503$	38.0403	1.69614	0.848068	+	0.529888i	$0.177766\pi$
$504$	0	0
$505$	−16.8903	−0.751610
$506$	0	0
$507$	1.20667	0.0535900
$508$	0	0
$509$	−7.25323	−0.321494	−0.160747	−	0.986996i	$-0.551390\pi$
$509$	−7.25323	−0.321494	−0.160747	+	0.986996i	$0.551390\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−8.55048	−0.377513
$514$	0	0
$515$	3.95495	0.174276
$516$	0	0
$517$	49.0787	2.15848
$518$	0	0
$519$	−0.129553	−0.00568675
$520$	0	0
$521$	10.8304	0.474490	0.237245	−	0.971450i	$-0.423756\pi$
$521$	10.8304	0.474490	0.237245	+	0.971450i	$0.423756\pi$
$522$	0	0
$523$	−19.4652	−0.851155	−0.425578	−	0.904922i	$-0.639929\pi$
$523$	−19.4652	−0.851155	−0.425578	+	0.904922i	$0.639929\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	14.8074	0.645022
$528$	0	0
$529$	−2.03669	−0.0885519
$530$	0	0
$531$	18.0403	0.782881
$532$	0	0
$533$	−2.57300	−0.111449
$534$	0	0
$535$	13.1180	0.567141
$536$	0	0
$537$	−0.0158667	−0.000684701 0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	27.8232	1.19621	0.598106	−	0.801417i	$-0.295920\pi$
$541$	27.8232	1.19621	0.598106	+	0.801417i	$0.295920\pi$
$542$	0	0
$543$	2.42408	0.104027
$544$	0	0
$545$	−1.07166	−0.0459048
$546$	0	0
$547$	40.0668	1.71313	0.856565	−	0.516039i	$-0.172594\pi$
$547$	40.0668	1.71313	0.856565	+	0.516039i	$0.172594\pi$
$548$	0	0
$549$	27.9282	1.19194
$550$	0	0
$551$	53.9381	2.29784
$552$	0	0
$553$	0	0
$554$	0	0
$555$	1.32641	0.0563028
$556$	0	0
$557$	9.47042	0.401275	0.200637	−	0.979666i	$-0.435699\pi$
$557$	9.47042	0.401275	0.200637	+	0.979666i	$0.435699\pi$
$558$	0	0
$559$	−33.2025	−1.40432
$560$	0	0
$561$	−2.78005	−0.117374
$562$	0	0
$563$	8.28311	0.349092	0.174546	−	0.984649i	$-0.444154\pi$
$563$	8.28311	0.349092	0.174546	+	0.984649i	$0.444154\pi$
$564$	0	0
$565$	18.7693	0.789631
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7.73238	0.324158	0.162079	−	0.986778i	$-0.448180\pi$
$569$	7.73238	0.324158	0.162079	+	0.986778i	$0.448180\pi$
$570$	0	0
$571$	−13.0410	−0.545750	−0.272875	−	0.962050i	$-0.587974\pi$
$571$	−13.0410	−0.545750	−0.272875	+	0.962050i	$0.587974\pi$
$572$	0	0
$573$	−4.46389	−0.186482
$574$	0	0
$575$	12.4264	0.518215
$576$	0	0
$577$	0.227296	0.00946246	0.00473123	−	0.999989i	$-0.498494\pi$
$577$	0.227296	0.00946246	0.00473123	+	0.999989i	$0.498494\pi$
$578$	0	0
$579$	3.70282	0.153884
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−14.0924	−0.583647
$584$	0	0
$585$	−11.5315	−0.476769
$586$	0	0
$587$	−34.7688	−1.43506	−0.717531	−	0.696527i	$-0.754728\pi$
$587$	−34.7688	−1.43506	−0.717531	+	0.696527i	$0.754728\pi$
$588$	0	0
$589$	−45.7675	−1.88582
$590$	0	0
$591$	−0.136847	−0.00562913
$592$	0	0
$593$	2.70910	0.111249	0.0556247	−	0.998452i	$-0.482285\pi$
$593$	2.70910	0.111249	0.0556247	+	0.998452i	$0.482285\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−4.94764	−0.202493
$598$	0	0
$599$	39.0649	1.59615	0.798074	−	0.602559i	$-0.205852\pi$
$599$	39.0649	1.59615	0.798074	+	0.602559i	$0.205852\pi$
$600$	0	0
$601$	17.0323	0.694761	0.347381	−	0.937724i	$-0.387071\pi$
$601$	17.0323	0.694761	0.347381	+	0.937724i	$0.387071\pi$
$602$	0	0
$603$	24.2138	0.986064
$604$	0	0
$605$	37.6840	1.53207
$606$	0	0
$607$	37.4924	1.52177	0.760885	−	0.648887i	$-0.224765\pi$
$607$	37.4924	1.52177	0.760885	+	0.648887i	$0.224765\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	21.0687	0.852350
$612$	0	0
$613$	1.39616	0.0563905	0.0281953	−	0.999602i	$-0.491024\pi$
$613$	1.39616	0.0563905	0.0281953	+	0.999602i	$0.491024\pi$
$614$	0	0
$615$	0.285972	0.0115315
$616$	0	0
$617$	−8.47566	−0.341217	−0.170609	−	0.985339i	$-0.554573\pi$
$617$	−8.47566	−0.341217	−0.170609	+	0.985339i	$0.554573\pi$
$618$	0	0
$619$	17.0607	0.685727	0.342863	−	0.939385i	$-0.388603\pi$
$619$	17.0607	0.685727	0.342863	+	0.939385i	$0.388603\pi$
$620$	0	0
$621$	−5.16503	−0.207265
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−4.06392	−0.162557
$626$	0	0
$627$	8.59271	0.343160
$628$	0	0
$629$	−11.3743	−0.453521
$630$	0	0
$631$	2.70985	0.107877	0.0539387	−	0.998544i	$-0.482822\pi$
$631$	2.70985	0.107877	0.0539387	+	0.998544i	$0.482822\pi$
$632$	0	0
$633$	1.16306	0.0462274
$634$	0	0
$635$	0.590387	0.0234288
$636$	0	0
$637$	0	0
$638$	0	0
$639$	47.9892	1.89842
$640$	0	0
$641$	−37.9859	−1.50035	−0.750177	−	0.661237i	$-0.770032\pi$
$641$	−37.9859	−1.50035	−0.750177	+	0.661237i	$0.770032\pi$
$642$	0	0
$643$	34.5744	1.36348	0.681741	−	0.731594i	$-0.261223\pi$
$643$	34.5744	1.36348	0.681741	+	0.731594i	$0.261223\pi$
$644$	0	0
$645$	3.69025	0.145303
$646$	0	0
$647$	2.34308	0.0921158	0.0460579	−	0.998939i	$-0.485334\pi$
$647$	2.34308	0.0921158	0.0460579	+	0.998939i	$0.485334\pi$
$648$	0	0
$649$	−36.4775	−1.43187
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−24.2508	−0.949009	−0.474504	−	0.880253i	$-0.657373\pi$
$653$	−24.2508	−0.949009	−0.474504	+	0.880253i	$0.657373\pi$
$654$	0	0
$655$	−1.41804	−0.0554073
$656$	0	0
$657$	−0.237060	−0.00924861
$658$	0	0
$659$	0.292606	0.0113983	0.00569915	−	0.999984i	$-0.498186\pi$
$659$	0.292606	0.0113983	0.00569915	+	0.999984i	$0.498186\pi$
$660$	0	0
$661$	46.1353	1.79446	0.897228	−	0.441568i	$-0.145578\pi$
$661$	46.1353	1.79446	0.897228	+	0.441568i	$0.145578\pi$
$662$	0	0
$663$	−1.19343	−0.0463491
$664$	0	0
$665$	0	0
$666$	0	0
$667$	32.5820	1.26158
$668$	0	0
$669$	3.03322	0.117271
$670$	0	0
$671$	−56.4709	−2.18004
$672$	0	0
$673$	17.4042	0.670883	0.335441	−	0.942061i	$-0.391115\pi$
$673$	17.4042	0.670883	0.335441	+	0.942061i	$0.391115\pi$
$674$	0	0
$675$	−3.06166	−0.117843
$676$	0	0
$677$	−23.8391	−0.916210	−0.458105	−	0.888898i	$-0.651472\pi$
$677$	−23.8391	−0.916210	−0.458105	+	0.888898i	$0.651472\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	4.88876	0.187338
$682$	0	0
$683$	−26.0894	−0.998282	−0.499141	−	0.866521i	$-0.666351\pi$
$683$	−26.0894	−0.998282	−0.499141	+	0.866521i	$0.666351\pi$
$684$	0	0
$685$	−30.3889	−1.16110
$686$	0	0
$687$	−1.64215	−0.0626520
$688$	0	0
$689$	−6.04965	−0.230473
$690$	0	0
$691$	−19.1131	−0.727097	−0.363548	−	0.931575i	$-0.618435\pi$
$691$	−19.1131	−0.727097	−0.363548	+	0.931575i	$0.618435\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	22.9849	0.871865
$696$	0	0
$697$	−2.45228	−0.0928868
$698$	0	0
$699$	−3.19384	−0.120802
$700$	0	0
$701$	14.0707	0.531442	0.265721	−	0.964050i	$-0.414390\pi$
$701$	14.0707	0.531442	0.265721	+	0.964050i	$0.414390\pi$
$702$	0	0
$703$	35.1561	1.32594
$704$	0	0
$705$	−2.34166	−0.0881918
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−49.0189	−1.84094	−0.920472	−	0.390808i	$-0.872196\pi$
$709$	−49.0189	−1.84094	−0.920472	+	0.390808i	$0.872196\pi$
$710$	0	0
$711$	−23.3144	−0.874358
$712$	0	0
$713$	−27.6464	−1.03537
$714$	0	0
$715$	23.3168	0.871999
$716$	0	0
$717$	2.30855	0.0862145
$718$	0	0
$719$	−50.7846	−1.89394	−0.946972	−	0.321315i	$-0.895875\pi$
$719$	−50.7846	−1.89394	−0.946972	+	0.321315i	$0.895875\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−0.208800	−0.00776534
$724$	0	0
$725$	19.3136	0.717288
$726$	0	0
$727$	20.4041	0.756745	0.378373	−	0.925653i	$-0.376484\pi$
$727$	20.4041	0.756745	0.378373	+	0.925653i	$0.376484\pi$
$728$	0	0
$729$	−25.0873	−0.929160
$730$	0	0
$731$	−31.6448	−1.17043
$732$	0	0
$733$	10.0457	0.371048	0.185524	−	0.982640i	$-0.440602\pi$
$733$	10.0457	0.371048	0.185524	+	0.982640i	$0.440602\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−48.9605	−1.80348
$738$	0	0
$739$	−33.7459	−1.24136	−0.620681	−	0.784063i	$-0.713144\pi$
$739$	−33.7459	−1.24136	−0.620681	+	0.784063i	$0.713144\pi$
$740$	0	0
$741$	3.68872	0.135509
$742$	0	0
$743$	−45.0786	−1.65377	−0.826886	−	0.562369i	$-0.809890\pi$
$743$	−45.0786	−1.65377	−0.826886	+	0.562369i	$0.809890\pi$
$744$	0	0
$745$	−12.9803	−0.475562
$746$	0	0
$747$	38.8053	1.41981
$748$	0	0
$749$	0	0
$750$	0	0
$751$	36.3502	1.32644	0.663218	−	0.748426i	$-0.269190\pi$
$751$	36.3502	1.32644	0.663218	+	0.748426i	$0.269190\pi$
$752$	0	0
$753$	−3.82186	−0.139276
$754$	0	0
$755$	8.89433	0.323698
$756$	0	0
$757$	−49.6889	−1.80598	−0.902988	−	0.429667i	$-0.858631\pi$
$757$	−49.6889	−1.80598	−0.902988	+	0.429667i	$0.858631\pi$
$758$	0	0
$759$	5.19054	0.188405
$760$	0	0
$761$	10.0592	0.364646	0.182323	−	0.983239i	$-0.441638\pi$
$761$	10.0592	0.364646	0.182323	+	0.983239i	$0.441638\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−10.9905	−0.397362
$766$	0	0
$767$	−15.6593	−0.565423
$768$	0	0
$769$	−45.9624	−1.65745	−0.828723	−	0.559659i	$-0.810932\pi$
$769$	−45.9624	−1.65745	−0.828723	+	0.559659i	$0.810932\pi$
$770$	0	0
$771$	0.667385	0.0240353
$772$	0	0
$773$	−4.23830	−0.152441	−0.0762206	−	0.997091i	$-0.524285\pi$
$773$	−4.23830	−0.152441	−0.0762206	+	0.997091i	$0.524285\pi$
$774$	0	0
$775$	−16.3879	−0.588671
$776$	0	0
$777$	0	0
$778$	0	0
$779$	7.57962	0.271568
$780$	0	0
$781$	−97.0344	−3.47217
$782$	0	0
$783$	−8.02769	−0.286886
$784$	0	0
$785$	15.0149	0.535903
$786$	0	0
$787$	−32.0054	−1.14087	−0.570434	−	0.821343i	$-0.693225\pi$
$787$	−32.0054	−1.14087	−0.570434	+	0.821343i	$0.693225\pi$
$788$	0	0
$789$	−1.29729	−0.0461849
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−24.2421	−0.860863
$794$	0	0
$795$	0.672380	0.0238469
$796$	0	0
$797$	41.0575	1.45433	0.727165	−	0.686463i	$-0.240837\pi$
$797$	41.0575	1.45433	0.727165	+	0.686463i	$0.240837\pi$
$798$	0	0
$799$	20.0803	0.710389
$800$	0	0
$801$	−16.1484	−0.570576
$802$	0	0
$803$	0.479338	0.0169155
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0.264997	0.00932833
$808$	0	0
$809$	39.2825	1.38110	0.690549	−	0.723285i	$-0.257369\pi$
$809$	39.2825	1.38110	0.690549	+	0.723285i	$0.257369\pi$
$810$	0	0
$811$	−0.408316	−0.0143379	−0.00716895	−	0.999974i	$-0.502282\pi$
$811$	−0.408316	−0.0143379	−0.00716895	+	0.999974i	$0.502282\pi$
$812$	0	0
$813$	3.70402	0.129906
$814$	0	0
$815$	0.725645	0.0254182
$816$	0	0
$817$	97.8092	3.42191
$818$	0	0
$819$	0	0
$820$	0	0
$821$	26.8315	0.936425	0.468212	−	0.883616i	$-0.344898\pi$
$821$	26.8315	0.936425	0.468212	+	0.883616i	$0.344898\pi$
$822$	0	0
$823$	−27.9696	−0.974959	−0.487480	−	0.873134i	$-0.662084\pi$
$823$	−27.9696	−0.974959	−0.487480	+	0.873134i	$0.662084\pi$
$824$	0	0
$825$	3.07678	0.107120
$826$	0	0
$827$	2.18680	0.0760425	0.0380213	−	0.999277i	$-0.487895\pi$
$827$	2.18680	0.0760425	0.0380213	+	0.999277i	$0.487895\pi$
$828$	0	0
$829$	−26.5207	−0.921103	−0.460551	−	0.887633i	$-0.652348\pi$
$829$	−26.5207	−0.921103	−0.460551	+	0.887633i	$0.652348\pi$
$830$	0	0
$831$	−0.622391	−0.0215905
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−20.4866	−0.708967
$836$	0	0
$837$	6.81164	0.235445
$838$	0	0
$839$	41.5129	1.43319	0.716593	−	0.697492i	$-0.245701\pi$
$839$	41.5129	1.43319	0.716593	+	0.697492i	$0.245701\pi$
$840$	0	0
$841$	21.6403	0.746217
$842$	0	0
$843$	−3.78165	−0.130247
$844$	0	0
$845$	−9.64571	−0.331823
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−3.60658	−0.123777
$850$	0	0
$851$	21.2365	0.727977
$852$	0	0
$853$	−32.8885	−1.12608	−0.563040	−	0.826430i	$-0.690368\pi$
$853$	−32.8885	−1.12608	−0.563040	+	0.826430i	$0.690368\pi$
$854$	0	0
$855$	33.9699	1.16175
$856$	0	0
$857$	−48.2059	−1.64668	−0.823341	−	0.567548i	$-0.807892\pi$
$857$	−48.2059	−1.64668	−0.823341	+	0.567548i	$0.807892\pi$
$858$	0	0
$859$	49.7378	1.69703	0.848515	−	0.529171i	$-0.177497\pi$
$859$	49.7378	1.69703	0.848515	+	0.529171i	$0.177497\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−4.93209	−0.167890	−0.0839451	−	0.996470i	$-0.526752\pi$
$863$	−4.93209	−0.167890	−0.0839451	+	0.996470i	$0.526752\pi$
$864$	0	0
$865$	1.03561	0.0352117
$866$	0	0
$867$	2.07798	0.0705718
$868$	0	0
$869$	47.1419	1.59918
$870$	0	0
$871$	−21.0180	−0.712169
$872$	0	0
$873$	−34.0696	−1.15308
$874$	0	0
$875$	0	0
$876$	0	0
$877$	5.09426	0.172021	0.0860104	−	0.996294i	$-0.472588\pi$
$877$	5.09426	0.172021	0.0860104	+	0.996294i	$0.472588\pi$
$878$	0	0
$879$	0.227116	0.00766042
$880$	0	0
$881$	−6.60204	−0.222429	−0.111214	−	0.993796i	$-0.535474\pi$
$881$	−6.60204	−0.222429	−0.111214	+	0.993796i	$0.535474\pi$
$882$	0	0
$883$	−8.16388	−0.274737	−0.137368	−	0.990520i	$-0.543864\pi$
$883$	−8.16388	−0.274737	−0.137368	+	0.990520i	$0.543864\pi$
$884$	0	0
$885$	1.74043	0.0585038
$886$	0	0
$887$	−10.2461	−0.344031	−0.172015	−	0.985094i	$-0.555028\pi$
$887$	−10.2461	−0.344031	−0.172015	+	0.985094i	$0.555028\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	52.0210	1.74277
$892$	0	0
$893$	−62.0650	−2.07693
$894$	0	0
$895$	0.126834	0.00423958
$896$	0	0
$897$	2.22822	0.0743981
$898$	0	0
$899$	−42.9692	−1.43310
$900$	0	0
$901$	−5.76582	−0.192087
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−19.3773	−0.644123
$906$	0	0
$907$	48.5478	1.61200	0.806002	−	0.591913i	$-0.201627\pi$
$907$	48.5478	1.61200	0.806002	+	0.591913i	$0.201627\pi$
$908$	0	0
$909$	33.1142	1.09833
$910$	0	0
$911$	−5.66134	−0.187569	−0.0937843	−	0.995593i	$-0.529896\pi$
$911$	−5.66134	−0.187569	−0.0937843	+	0.995593i	$0.529896\pi$
$912$	0	0
$913$	−78.4647	−2.59680
$914$	0	0
$915$	2.69436	0.0890727
$916$	0	0
$917$	0	0
$918$	0	0
$919$	20.0271	0.660635	0.330317	−	0.943870i	$-0.392844\pi$
$919$	20.0271	0.660635	0.330317	+	0.943870i	$0.392844\pi$
$920$	0	0
$921$	−6.19326	−0.204075
$922$	0	0
$923$	−41.6554	−1.37111
$924$	0	0
$925$	12.5883	0.413900
$926$	0	0
$927$	−7.75383	−0.254669
$928$	0	0
$929$	−15.7004	−0.515115	−0.257557	−	0.966263i	$-0.582918\pi$
$929$	−15.7004	−0.515115	−0.257557	+	0.966263i	$0.582918\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	3.64384	0.119294
$934$	0	0
$935$	22.2228	0.726765
$936$	0	0
$937$	11.6447	0.380414	0.190207	−	0.981744i	$-0.439084\pi$
$937$	11.6447	0.380414	0.190207	+	0.981744i	$0.439084\pi$
$938$	0	0
$939$	5.66732	0.184946
$940$	0	0
$941$	−15.6441	−0.509984	−0.254992	−	0.966943i	$-0.582073\pi$
$941$	−15.6441	−0.509984	−0.254992	+	0.966943i	$0.582073\pi$
$942$	0	0
$943$	4.57857	0.149099
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7.35951	0.239152	0.119576	−	0.992825i	$-0.461846\pi$
$947$	7.35951	0.239152	0.119576	+	0.992825i	$0.461846\pi$
$948$	0	0
$949$	0.205773	0.00667966
$950$	0	0
$951$	2.22325	0.0720937
$952$	0	0
$953$	−35.4898	−1.14963	−0.574814	−	0.818284i	$-0.694926\pi$
$953$	−35.4898	−1.14963	−0.574814	+	0.818284i	$0.694926\pi$
$954$	0	0
$955$	35.6830	1.15467
$956$	0	0
$957$	8.06734	0.260780
$958$	0	0
$959$	0	0
$960$	0	0
$961$	5.46013	0.176133
$962$	0	0
$963$	−25.7184	−0.828763
$964$	0	0
$965$	−29.5991	−0.952830
$966$	0	0
$967$	−47.1692	−1.51686	−0.758429	−	0.651755i	$-0.774033\pi$
$967$	−47.1692	−1.51686	−0.758429	+	0.651755i	$0.774033\pi$
$968$	0	0
$969$	3.51566	0.112939
$970$	0	0
$971$	−30.2619	−0.971150	−0.485575	−	0.874195i	$-0.661390\pi$
$971$	−30.2619	−0.971150	−0.485575	+	0.874195i	$0.661390\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	1.32082	0.0423000
$976$	0	0
$977$	43.8390	1.40253	0.701267	−	0.712899i	$-0.252618\pi$
$977$	43.8390	1.40253	0.701267	+	0.712899i	$0.252618\pi$
$978$	0	0
$979$	32.6522	1.04357
$980$	0	0
$981$	2.10103	0.0670808
$982$	0	0
$983$	23.3235	0.743904	0.371952	−	0.928252i	$-0.378689\pi$
$983$	23.3235	0.743904	0.371952	+	0.928252i	$0.378689\pi$
$984$	0	0
$985$	1.09391	0.0348549
$986$	0	0
$987$	0	0
$988$	0	0
$989$	59.0829	1.87873
$990$	0	0
$991$	35.6864	1.13361	0.566807	−	0.823850i	$-0.308178\pi$
$991$	35.6864	1.13361	0.566807	+	0.823850i	$0.308178\pi$
$992$	0	0
$993$	2.75868	0.0875440
$994$	0	0
$995$	39.5499	1.25382
$996$	0	0
$997$	−46.7142	−1.47945	−0.739727	−	0.672907i	$-0.765045\pi$
$997$	−46.7142	−1.47945	−0.739727	+	0.672907i	$0.765045\pi$
$998$	0	0
$999$	−5.23233	−0.165544

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.l.1.3		5
7.6	odd	2		1148.2.a.c.1.3	✓	5
28.27	even	2		4592.2.a.be.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.a.c.1.3	✓	5	7.6	odd	2
4592.2.a.be.1.3		5	28.27	even	2
8036.2.a.l.1.3		5	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:	\( N \)	\(=\)	\( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8036.a (trivial)

\(f(q)\)	\(=\)	\(q-0.189142 q^{3} +1.51194 q^{5} -2.96423 q^{9} +O(q^{10})\)	\(q-0.189142 q^{3} +1.51194 q^{5} -2.96423 q^{9} +5.99368 q^{11} +2.57300 q^{13} -0.285972 q^{15} +2.45228 q^{17} -7.57962 q^{19} -4.57857 q^{23} -2.71403 q^{25} +1.12809 q^{27} -7.11620 q^{29} +6.03822 q^{31} -1.13366 q^{33} -4.63823 q^{37} -0.486663 q^{39} -1.00000 q^{41} -12.9042 q^{43} -4.48174 q^{45} +8.18840 q^{47} -0.463830 q^{51} -2.35121 q^{53} +9.06211 q^{55} +1.43363 q^{57} -6.08600 q^{59} -9.42174 q^{61} +3.89023 q^{65} -8.16869 q^{67} +0.866001 q^{69} -16.1895 q^{71} +0.0799738 q^{73} +0.513337 q^{75} +7.86526 q^{79} +8.67931 q^{81} -13.0912 q^{83} +3.70771 q^{85} +1.34597 q^{87} +5.44776 q^{89} -1.14208 q^{93} -11.4600 q^{95} +11.4936 q^{97} -17.7666 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{3} + q^{5} + q^{9}+O(q^{10}) \) 5 * q + 2 * q^3 + q^5 + q^9	\( 5 q + 2 q^{3} + q^{5} + q^{9} - 2 q^{11} + q^{13} - 9 q^{15} + 3 q^{17} + 4 q^{19} - 8 q^{23} - 6 q^{25} + 8 q^{27} - 9 q^{29} + 11 q^{31} - 5 q^{33} - 11 q^{37} - 17 q^{39} - 5 q^{41} - 27 q^{43} + 3 q^{45} + 3 q^{47} - 3 q^{51} - 19 q^{53} + 13 q^{55} - 11 q^{57} + 15 q^{59} + 7 q^{65} - 21 q^{67} - 14 q^{69} - 16 q^{71} + 10 q^{73} - 12 q^{75} - 14 q^{79} - 7 q^{81} + 2 q^{83} - 21 q^{85} + 36 q^{87} - 6 q^{89} + 17 q^{93} + 9 q^{95} - 20 q^{97} - 17 q^{99}+O(q^{100}) \) 5 * q + 2 * q^3 + q^5 + q^9 - 2 * q^11 + q^13 - 9 * q^15 + 3 * q^17 + 4 * q^19 - 8 * q^23 - 6 * q^25 + 8 * q^27 - 9 * q^29 + 11 * q^31 - 5 * q^33 - 11 * q^37 - 17 * q^39 - 5 * q^41 - 27 * q^43 + 3 * q^45 + 3 * q^47 - 3 * q^51 - 19 * q^53 + 13 * q^55 - 11 * q^57 + 15 * q^59 + 7 * q^65 - 21 * q^67 - 14 * q^69 - 16 * q^71 + 10 * q^73 - 12 * q^75 - 14 * q^79 - 7 * q^81 + 2 * q^83 - 21 * q^85 + 36 * q^87 - 6 * q^89 + 17 * q^93 + 9 * q^95 - 20 * q^97 - 17 * q^99

Introduction

L-functions

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Representations

Groups

Database

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