LMFDB - Embedded newform 8036.2.a.m.1.4

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			1.4
Root			$-0.308117$ 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)

$f(q)$	$=$	$q+0.308117 q^{3} -3.30124 q^{5} -2.90506 q^{9} +O(q^{10})$	$q+0.308117 q^{3} -3.30124 q^{5} -2.90506 q^{9} +1.51731 q^{11} +1.10922 q^{13} -1.01717 q^{15} +1.60677 q^{17} +0.880518 q^{19} +0.682060 q^{23} +5.89819 q^{25} -1.81945 q^{27} -1.88250 q^{29} +4.71409 q^{31} +0.467508 q^{33} -3.22136 q^{37} +0.341769 q^{39} +1.00000 q^{41} -9.57315 q^{43} +9.59031 q^{45} +4.78592 q^{47} +0.495073 q^{51} -1.63217 q^{53} -5.00899 q^{55} +0.271303 q^{57} +1.01005 q^{59} -0.380672 q^{61} -3.66180 q^{65} +8.48699 q^{67} +0.210154 q^{69} +7.90937 q^{71} +1.56461 q^{73} +1.81733 q^{75} -8.41469 q^{79} +8.15459 q^{81} +1.55396 q^{83} -5.30433 q^{85} -0.580030 q^{87} +13.9216 q^{89} +1.45249 q^{93} -2.90680 q^{95} +0.601041 q^{97} -4.40787 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.308117	0.177891	0.0889457	−	0.996036i	$-0.471650\pi$
$3$	0.308117	0.177891	0.0889457	+	0.996036i	$0.471650\pi$
$4$	0	0
$5$	−3.30124	−1.47636	−0.738180	−	0.674604i	$-0.764314\pi$
$5$	−3.30124	−1.47636	−0.738180	+	0.674604i	$0.764314\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.90506	−0.968355
$10$	0	0
$11$	1.51731	0.457485	0.228743	−	0.973487i	$-0.426539\pi$
$11$	1.51731	0.457485	0.228743	+	0.973487i	$0.426539\pi$
$12$	0	0
$13$	1.10922	0.307642	0.153821	−	0.988099i	$-0.450842\pi$
$13$	1.10922	0.307642	0.153821	+	0.988099i	$0.450842\pi$
$14$	0	0
$15$	−1.01717	−0.262632
$16$	0	0
$17$	1.60677	0.389699	0.194850	−	0.980833i	$-0.437578\pi$
$17$	1.60677	0.389699	0.194850	+	0.980833i	$0.437578\pi$
$18$	0	0
$19$	0.880518	0.202005	0.101002	−	0.994886i	$-0.467795\pi$
$19$	0.880518	0.202005	0.101002	+	0.994886i	$0.467795\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.682060	0.142219	0.0711097	−	0.997469i	$-0.477346\pi$
$23$	0.682060	0.142219	0.0711097	+	0.997469i	$0.477346\pi$
$24$	0	0
$25$	5.89819	1.17964
$26$	0	0
$27$	−1.81945	−0.350153
$28$	0	0
$29$	−1.88250	−0.349572	−0.174786	−	0.984606i	$-0.555923\pi$
$29$	−1.88250	−0.349572	−0.174786	+	0.984606i	$0.555923\pi$
$30$	0	0
$31$	4.71409	0.846675	0.423337	−	0.905972i	$-0.360858\pi$
$31$	4.71409	0.846675	0.423337	+	0.905972i	$0.360858\pi$
$32$	0	0
$33$	0.467508	0.0813826
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.22136	−0.529588	−0.264794	−	0.964305i	$-0.585304\pi$
$37$	−3.22136	−0.529588	−0.264794	+	0.964305i	$0.585304\pi$
$38$	0	0
$39$	0.341769	0.0547268
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−9.57315	−1.45989	−0.729946	−	0.683505i	$-0.760455\pi$
$43$	−9.57315	−1.45989	−0.729946	+	0.683505i	$0.760455\pi$
$44$	0	0
$45$	9.59031	1.42964
$46$	0	0
$47$	4.78592	0.698098	0.349049	−	0.937105i	$-0.386505\pi$
$47$	4.78592	0.698098	0.349049	+	0.937105i	$0.386505\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0.495073	0.0693241
$52$	0	0
$53$	−1.63217	−0.224196	−0.112098	−	0.993697i	$-0.535757\pi$
$53$	−1.63217	−0.224196	−0.112098	+	0.993697i	$0.535757\pi$
$54$	0	0
$55$	−5.00899	−0.675412
$56$	0	0
$57$	0.271303	0.0359349
$58$	0	0
$59$	1.01005	0.131497	0.0657485	−	0.997836i	$-0.479056\pi$
$59$	1.01005	0.131497	0.0657485	+	0.997836i	$0.479056\pi$
$60$	0	0
$61$	−0.380672	−0.0487400	−0.0243700	−	0.999703i	$-0.507758\pi$
$61$	−0.380672	−0.0487400	−0.0243700	+	0.999703i	$0.507758\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.66180	−0.454190
$66$	0	0
$67$	8.48699	1.03685	0.518425	−	0.855123i	$-0.326518\pi$
$67$	8.48699	1.03685	0.518425	+	0.855123i	$0.326518\pi$
$68$	0	0
$69$	0.210154	0.0252996
$70$	0	0
$71$	7.90937	0.938669	0.469335	−	0.883020i	$-0.344494\pi$
$71$	7.90937	0.938669	0.469335	+	0.883020i	$0.344494\pi$
$72$	0	0
$73$	1.56461	0.183124	0.0915618	−	0.995799i	$-0.470814\pi$
$73$	1.56461	0.183124	0.0915618	+	0.995799i	$0.470814\pi$
$74$	0	0
$75$	1.81733	0.209847
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−8.41469	−0.946727	−0.473364	−	0.880867i	$-0.656960\pi$
$79$	−8.41469	−0.946727	−0.473364	+	0.880867i	$0.656960\pi$
$80$	0	0
$81$	8.15459	0.906065
$82$	0	0
$83$	1.55396	0.170570	0.0852848	−	0.996357i	$-0.472820\pi$
$83$	1.55396	0.170570	0.0852848	+	0.996357i	$0.472820\pi$
$84$	0	0
$85$	−5.30433	−0.575336
$86$	0	0
$87$	−0.580030	−0.0621858
$88$	0	0
$89$	13.9216	1.47568	0.737841	−	0.674974i	$-0.235845\pi$
$89$	13.9216	1.47568	0.737841	+	0.674974i	$0.235845\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.45249	0.150616
$94$	0	0
$95$	−2.90680	−0.298232
$96$	0	0
$97$	0.601041	0.0610264	0.0305132	−	0.999534i	$-0.490286\pi$
$97$	0.601041	0.0610264	0.0305132	+	0.999534i	$0.490286\pi$
$98$	0	0
$99$	−4.40787	−0.443008
$100$	0	0
$101$	3.79423	0.377540	0.188770	−	0.982021i	$-0.439550\pi$
$101$	3.79423	0.377540	0.188770	+	0.982021i	$0.439550\pi$
$102$	0	0
$103$	0.520009	0.0512380	0.0256190	−	0.999672i	$-0.491844\pi$
$103$	0.520009	0.0512380	0.0256190	+	0.999672i	$0.491844\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.35377	−0.904263	−0.452132	−	0.891951i	$-0.649336\pi$
$107$	−9.35377	−0.904263	−0.452132	+	0.891951i	$0.649336\pi$
$108$	0	0
$109$	5.25348	0.503192	0.251596	−	0.967832i	$-0.419045\pi$
$109$	5.25348	0.503192	0.251596	+	0.967832i	$0.419045\pi$
$110$	0	0
$111$	−0.992554	−0.0942091
$112$	0	0
$113$	−0.945705	−0.0889644	−0.0444822	−	0.999010i	$-0.514164\pi$
$113$	−0.945705	−0.0889644	−0.0444822	+	0.999010i	$0.514164\pi$
$114$	0	0
$115$	−2.25164	−0.209967
$116$	0	0
$117$	−3.22235	−0.297906
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−8.69778	−0.790707
$122$	0	0
$123$	0.308117	0.0277820
$124$	0	0
$125$	−2.96513	−0.265210
$126$	0	0
$127$	11.5966	1.02903	0.514517	−	0.857480i	$-0.327971\pi$
$127$	11.5966	1.02903	0.514517	+	0.857480i	$0.327971\pi$
$128$	0	0
$129$	−2.94965	−0.259702
$130$	0	0
$131$	−8.01947	−0.700664	−0.350332	−	0.936626i	$-0.613931\pi$
$131$	−8.01947	−0.700664	−0.350332	+	0.936626i	$0.613931\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	6.00644	0.516952
$136$	0	0
$137$	−0.817226	−0.0698203	−0.0349102	−	0.999390i	$-0.511115\pi$
$137$	−0.817226	−0.0698203	−0.0349102	+	0.999390i	$0.511115\pi$
$138$	0	0
$139$	2.61528	0.221825	0.110912	−	0.993830i	$-0.464623\pi$
$139$	2.61528	0.221825	0.110912	+	0.993830i	$0.464623\pi$
$140$	0	0
$141$	1.47462	0.124186
$142$	0	0
$143$	1.68302	0.140742
$144$	0	0
$145$	6.21459	0.516093
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−16.0183	−1.31227	−0.656134	−	0.754644i	$-0.727809\pi$
$149$	−16.0183	−1.31227	−0.656134	+	0.754644i	$0.727809\pi$
$150$	0	0
$151$	−9.67825	−0.787605	−0.393802	−	0.919195i	$-0.628841\pi$
$151$	−9.67825	−0.787605	−0.393802	+	0.919195i	$0.628841\pi$
$152$	0	0
$153$	−4.66777	−0.377367
$154$	0	0
$155$	−15.5623	−1.25000
$156$	0	0
$157$	8.78060	0.700768	0.350384	−	0.936606i	$-0.386051\pi$
$157$	8.78060	0.700768	0.350384	+	0.936606i	$0.386051\pi$
$158$	0	0
$159$	−0.502900	−0.0398825
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−11.5969	−0.908335	−0.454168	−	0.890916i	$-0.650063\pi$
$163$	−11.5969	−0.908335	−0.454168	+	0.890916i	$0.650063\pi$
$164$	0	0
$165$	−1.54336	−0.120150
$166$	0	0
$167$	−3.23013	−0.249955	−0.124977	−	0.992160i	$-0.539886\pi$
$167$	−3.23013	−0.249955	−0.124977	+	0.992160i	$0.539886\pi$
$168$	0	0
$169$	−11.7696	−0.905356
$170$	0	0
$171$	−2.55796	−0.195612
$172$	0	0
$173$	9.85729	0.749436	0.374718	−	0.927139i	$-0.377740\pi$
$173$	9.85729	0.749436	0.374718	+	0.927139i	$0.377740\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0.311213	0.0233922
$178$	0	0
$179$	−20.0945	−1.50193	−0.750967	−	0.660339i	$-0.770412\pi$
$179$	−20.0945	−1.50193	−0.750967	+	0.660339i	$0.770412\pi$
$180$	0	0
$181$	3.60880	0.268240	0.134120	−	0.990965i	$-0.457179\pi$
$181$	3.60880	0.268240	0.134120	+	0.990965i	$0.457179\pi$
$182$	0	0
$183$	−0.117291	−0.00867042
$184$	0	0
$185$	10.6345	0.781862
$186$	0	0
$187$	2.43796	0.178282
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−19.5419	−1.41400	−0.707000	−	0.707214i	$-0.749952\pi$
$191$	−19.5419	−1.41400	−0.707000	+	0.707214i	$0.749952\pi$
$192$	0	0
$193$	−26.9641	−1.94092	−0.970459	−	0.241266i	$-0.922437\pi$
$193$	−26.9641	−1.94092	−0.970459	+	0.241266i	$0.922437\pi$
$194$	0	0
$195$	−1.12826	−0.0807965
$196$	0	0
$197$	10.5882	0.754381	0.377191	−	0.926136i	$-0.376890\pi$
$197$	10.5882	0.754381	0.377191	+	0.926136i	$0.376890\pi$
$198$	0	0
$199$	−19.8558	−1.40754	−0.703770	−	0.710428i	$-0.748501\pi$
$199$	−19.8558	−1.40754	−0.703770	+	0.710428i	$0.748501\pi$
$200$	0	0
$201$	2.61498	0.184447
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−3.30124	−0.230569
$206$	0	0
$207$	−1.98143	−0.137719
$208$	0	0
$209$	1.33602	0.0924142
$210$	0	0
$211$	9.05365	0.623279	0.311639	−	0.950200i	$-0.399122\pi$
$211$	9.05365	0.623279	0.311639	+	0.950200i	$0.399122\pi$
$212$	0	0
$213$	2.43701	0.166981
$214$	0	0
$215$	31.6033	2.15532
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0.482082	0.0325761
$220$	0	0
$221$	1.78226	0.119888
$222$	0	0
$223$	−11.4325	−0.765580	−0.382790	−	0.923835i	$-0.625037\pi$
$223$	−11.4325	−0.765580	−0.382790	+	0.923835i	$0.625037\pi$
$224$	0	0
$225$	−17.1346	−1.14231
$226$	0	0
$227$	−21.2978	−1.41358	−0.706792	−	0.707421i	$-0.749858\pi$
$227$	−21.2978	−1.41358	−0.706792	+	0.707421i	$0.749858\pi$
$228$	0	0
$229$	−21.1127	−1.39517	−0.697583	−	0.716504i	$-0.745741\pi$
$229$	−21.1127	−1.39517	−0.697583	+	0.716504i	$0.745741\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	22.6782	1.48570	0.742849	−	0.669459i	$-0.233474\pi$
$233$	22.6782	1.48570	0.742849	+	0.669459i	$0.233474\pi$
$234$	0	0
$235$	−15.7995	−1.03064
$236$	0	0
$237$	−2.59271	−0.168415
$238$	0	0
$239$	17.7226	1.14638	0.573191	−	0.819422i	$-0.305705\pi$
$239$	17.7226	1.14638	0.573191	+	0.819422i	$0.305705\pi$
$240$	0	0
$241$	−20.1713	−1.29935	−0.649675	−	0.760212i	$-0.725095\pi$
$241$	−20.1713	−1.29935	−0.649675	+	0.760212i	$0.725095\pi$
$242$	0	0
$243$	7.97092	0.511335
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.976687	0.0621451
$248$	0	0
$249$	0.478802	0.0303429
$250$	0	0
$251$	−16.5113	−1.04219	−0.521093	−	0.853500i	$-0.674476\pi$
$251$	−16.5113	−1.04219	−0.521093	+	0.853500i	$0.674476\pi$
$252$	0	0
$253$	1.03489	0.0650632
$254$	0	0
$255$	−1.63436	−0.102347
$256$	0	0
$257$	17.1369	1.06897	0.534485	−	0.845178i	$-0.320506\pi$
$257$	17.1369	1.06897	0.534485	+	0.845178i	$0.320506\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	5.46878	0.338509
$262$	0	0
$263$	−17.2443	−1.06333	−0.531665	−	0.846955i	$-0.678433\pi$
$263$	−17.2443	−1.06333	−0.531665	+	0.846955i	$0.678433\pi$
$264$	0	0
$265$	5.38819	0.330994
$266$	0	0
$267$	4.28947	0.262511
$268$	0	0
$269$	−14.5073	−0.884524	−0.442262	−	0.896886i	$-0.645824\pi$
$269$	−14.5073	−0.884524	−0.442262	+	0.896886i	$0.645824\pi$
$270$	0	0
$271$	−23.8779	−1.45048	−0.725240	−	0.688496i	$-0.758271\pi$
$271$	−23.8779	−1.45048	−0.725240	+	0.688496i	$0.758271\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	8.94936	0.539667
$276$	0	0
$277$	−15.6351	−0.939422	−0.469711	−	0.882820i	$-0.655642\pi$
$277$	−15.6351	−0.939422	−0.469711	+	0.882820i	$0.655642\pi$
$278$	0	0
$279$	−13.6947	−0.819881
$280$	0	0
$281$	5.17005	0.308419	0.154210	−	0.988038i	$-0.450717\pi$
$281$	5.17005	0.308419	0.154210	+	0.988038i	$0.450717\pi$
$282$	0	0
$283$	−10.0611	−0.598071	−0.299035	−	0.954242i	$-0.596665\pi$
$283$	−10.0611	−0.598071	−0.299035	+	0.954242i	$0.596665\pi$
$284$	0	0
$285$	−0.895635	−0.0530528
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−14.4183	−0.848135
$290$	0	0
$291$	0.185191	0.0108561
$292$	0	0
$293$	−9.32341	−0.544680	−0.272340	−	0.962201i	$-0.587797\pi$
$293$	−9.32341	−0.544680	−0.272340	+	0.962201i	$0.587797\pi$
$294$	0	0
$295$	−3.33441	−0.194137
$296$	0	0
$297$	−2.76066	−0.160190
$298$	0	0
$299$	0.756554	0.0437526
$300$	0	0
$301$	0	0
$302$	0	0
$303$	1.16906	0.0671610
$304$	0	0
$305$	1.25669	0.0719578
$306$	0	0
$307$	−10.3822	−0.592544	−0.296272	−	0.955104i	$-0.595743\pi$
$307$	−10.3822	−0.592544	−0.296272	+	0.955104i	$0.595743\pi$
$308$	0	0
$309$	0.160224	0.00911480
$310$	0	0
$311$	31.7534	1.80057	0.900284	−	0.435302i	$-0.143358\pi$
$311$	31.7534	1.80057	0.900284	+	0.435302i	$0.143358\pi$
$312$	0	0
$313$	−24.5198	−1.38594	−0.692971	−	0.720966i	$-0.743699\pi$
$313$	−24.5198	−1.38594	−0.692971	+	0.720966i	$0.743699\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−7.89454	−0.443402	−0.221701	−	0.975115i	$-0.571161\pi$
$317$	−7.89454	−0.443402	−0.221701	+	0.975115i	$0.571161\pi$
$318$	0	0
$319$	−2.85633	−0.159924
$320$	0	0
$321$	−2.88206	−0.160861
$322$	0	0
$323$	1.41479	0.0787211
$324$	0	0
$325$	6.54238	0.362906
$326$	0	0
$327$	1.61868	0.0895135
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−14.1270	−0.776489	−0.388244	−	0.921557i	$-0.626918\pi$
$331$	−14.1270	−0.776489	−0.388244	+	0.921557i	$0.626918\pi$
$332$	0	0
$333$	9.35824	0.512829
$334$	0	0
$335$	−28.0176	−1.53076
$336$	0	0
$337$	16.5644	0.902319	0.451160	−	0.892443i	$-0.351011\pi$
$337$	16.5644	0.902319	0.451160	+	0.892443i	$0.351011\pi$
$338$	0	0
$339$	−0.291388	−0.0158260
$340$	0	0
$341$	7.15271	0.387341
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−0.693769	−0.0373513
$346$	0	0
$347$	−13.1015	−0.703328	−0.351664	−	0.936126i	$-0.614384\pi$
$347$	−13.1015	−0.703328	−0.351664	+	0.936126i	$0.614384\pi$
$348$	0	0
$349$	22.0563	1.18065	0.590324	−	0.807167i	$-0.299000\pi$
$349$	22.0563	1.18065	0.590324	+	0.807167i	$0.299000\pi$
$350$	0	0
$351$	−2.01817	−0.107722
$352$	0	0
$353$	20.6032	1.09660	0.548298	−	0.836283i	$-0.315276\pi$
$353$	20.6032	1.09660	0.548298	+	0.836283i	$0.315276\pi$
$354$	0	0
$355$	−26.1107	−1.38581
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−15.1830	−0.801328	−0.400664	−	0.916225i	$-0.631220\pi$
$359$	−15.1830	−0.801328	−0.400664	+	0.916225i	$0.631220\pi$
$360$	0	0
$361$	−18.2247	−0.959194
$362$	0	0
$363$	−2.67993	−0.140660
$364$	0	0
$365$	−5.16515	−0.270356
$366$	0	0
$367$	1.02831	0.0536773	0.0268386	−	0.999640i	$-0.491456\pi$
$367$	1.02831	0.0536773	0.0268386	+	0.999640i	$0.491456\pi$
$368$	0	0
$369$	−2.90506	−0.151232
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−25.0723	−1.29820	−0.649098	−	0.760704i	$-0.724854\pi$
$373$	−25.0723	−1.29820	−0.649098	+	0.760704i	$0.724854\pi$
$374$	0	0
$375$	−0.913607	−0.0471785
$376$	0	0
$377$	−2.08810	−0.107543
$378$	0	0
$379$	7.20297	0.369992	0.184996	−	0.982739i	$-0.440773\pi$
$379$	7.20297	0.369992	0.184996	+	0.982739i	$0.440773\pi$
$380$	0	0
$381$	3.57311	0.183056
$382$	0	0
$383$	−10.9882	−0.561469	−0.280734	−	0.959786i	$-0.590578\pi$
$383$	−10.9882	−0.561469	−0.280734	+	0.959786i	$0.590578\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	27.8106	1.41369
$388$	0	0
$389$	6.82819	0.346203	0.173102	−	0.984904i	$-0.444621\pi$
$389$	6.82819	0.346203	0.173102	+	0.984904i	$0.444621\pi$
$390$	0	0
$391$	1.09591	0.0554227
$392$	0	0
$393$	−2.47093	−0.124642
$394$	0	0
$395$	27.7789	1.39771
$396$	0	0
$397$	1.82458	0.0915733	0.0457866	−	0.998951i	$-0.485421\pi$
$397$	1.82458	0.0915733	0.0457866	+	0.998951i	$0.485421\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0.510011	0.0254687	0.0127344	−	0.999919i	$-0.495946\pi$
$401$	0.510011	0.0254687	0.0127344	+	0.999919i	$0.495946\pi$
$402$	0	0
$403$	5.22895	0.260473
$404$	0	0
$405$	−26.9203	−1.33768
$406$	0	0
$407$	−4.88778	−0.242278
$408$	0	0
$409$	23.4467	1.15936	0.579681	−	0.814843i	$-0.303177\pi$
$409$	23.4467	1.15936	0.579681	+	0.814843i	$0.303177\pi$
$410$	0	0
$411$	−0.251801	−0.0124204
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−5.13001	−0.251822
$416$	0	0
$417$	0.805811	0.0394607
$418$	0	0
$419$	−38.8743	−1.89913	−0.949566	−	0.313567i	$-0.898476\pi$
$419$	−38.8743	−1.89913	−0.949566	+	0.313567i	$0.898476\pi$
$420$	0	0
$421$	3.62394	0.176620	0.0883101	−	0.996093i	$-0.471853\pi$
$421$	3.62394	0.176620	0.0883101	+	0.996093i	$0.471853\pi$
$422$	0	0
$423$	−13.9034	−0.676006
$424$	0	0
$425$	9.47703	0.459704
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0.518568	0.0250367
$430$	0	0
$431$	−8.37794	−0.403551	−0.201776	−	0.979432i	$-0.564671\pi$
$431$	−8.37794	−0.403551	−0.201776	+	0.979432i	$0.564671\pi$
$432$	0	0
$433$	−5.27478	−0.253490	−0.126745	−	0.991935i	$-0.540453\pi$
$433$	−5.27478	−0.253490	−0.126745	+	0.991935i	$0.540453\pi$
$434$	0	0
$435$	1.91482	0.0918085
$436$	0	0
$437$	0.600566	0.0287290
$438$	0	0
$439$	28.5702	1.36358	0.681790	−	0.731548i	$-0.261202\pi$
$439$	28.5702	1.36358	0.681790	+	0.731548i	$0.261202\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−12.3310	−0.585863	−0.292932	−	0.956133i	$-0.594631\pi$
$443$	−12.3310	−0.585863	−0.292932	+	0.956133i	$0.594631\pi$
$444$	0	0
$445$	−45.9584	−2.17864
$446$	0	0
$447$	−4.93550	−0.233441
$448$	0	0
$449$	−36.7971	−1.73656	−0.868280	−	0.496074i	$-0.834775\pi$
$449$	−36.7971	−1.73656	−0.868280	+	0.496074i	$0.834775\pi$
$450$	0	0
$451$	1.51731	0.0714472
$452$	0	0
$453$	−2.98203	−0.140108
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4.46379	0.208807	0.104404	−	0.994535i	$-0.466707\pi$
$457$	4.46379	0.208807	0.104404	+	0.994535i	$0.466707\pi$
$458$	0	0
$459$	−2.92344	−0.136454
$460$	0	0
$461$	10.8591	0.505757	0.252878	−	0.967498i	$-0.418623\pi$
$461$	10.8591	0.505757	0.252878	+	0.967498i	$0.418623\pi$
$462$	0	0
$463$	17.7680	0.825747	0.412874	−	0.910788i	$-0.364525\pi$
$463$	17.7680	0.825747	0.412874	+	0.910788i	$0.364525\pi$
$464$	0	0
$465$	−4.79502	−0.222364
$466$	0	0
$467$	−3.65317	−0.169048	−0.0845242	−	0.996421i	$-0.526937\pi$
$467$	−3.65317	−0.169048	−0.0845242	+	0.996421i	$0.526937\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	2.70545	0.124661
$472$	0	0
$473$	−14.5254	−0.667878
$474$	0	0
$475$	5.19346	0.238292
$476$	0	0
$477$	4.74156	0.217101
$478$	0	0
$479$	1.84559	0.0843270	0.0421635	−	0.999111i	$-0.486575\pi$
$479$	1.84559	0.0843270	0.0421635	+	0.999111i	$0.486575\pi$
$480$	0	0
$481$	−3.57319	−0.162923
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.98418	−0.0900970
$486$	0	0
$487$	6.98021	0.316304	0.158152	−	0.987415i	$-0.449446\pi$
$487$	6.98021	0.316304	0.158152	+	0.987415i	$0.449446\pi$
$488$	0	0
$489$	−3.57319	−0.161585
$490$	0	0
$491$	4.53099	0.204481	0.102240	−	0.994760i	$-0.467399\pi$
$491$	4.53099	0.204481	0.102240	+	0.994760i	$0.467399\pi$
$492$	0	0
$493$	−3.02475	−0.136228
$494$	0	0
$495$	14.5514	0.654039
$496$	0	0
$497$	0	0
$498$	0	0
$499$	28.8012	1.28932	0.644659	−	0.764471i	$-0.277000\pi$
$499$	28.8012	1.28932	0.644659	+	0.764471i	$0.277000\pi$
$500$	0	0
$501$	−0.995257	−0.0444648
$502$	0	0
$503$	−12.0511	−0.537330	−0.268665	−	0.963234i	$-0.586583\pi$
$503$	−12.0511	−0.537330	−0.268665	+	0.963234i	$0.586583\pi$
$504$	0	0
$505$	−12.5256	−0.557384
$506$	0	0
$507$	−3.62642	−0.161055
$508$	0	0
$509$	15.2708	0.676868	0.338434	−	0.940990i	$-0.390103\pi$
$509$	15.2708	0.676868	0.338434	+	0.940990i	$0.390103\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−1.60206	−0.0707326
$514$	0	0
$515$	−1.71668	−0.0756458
$516$	0	0
$517$	7.26170	0.319369
$518$	0	0
$519$	3.03720	0.133318
$520$	0	0
$521$	22.1754	0.971522	0.485761	−	0.874092i	$-0.338542\pi$
$521$	22.1754	0.971522	0.485761	+	0.874092i	$0.338542\pi$
$522$	0	0
$523$	−34.0779	−1.49012	−0.745062	−	0.666995i	$-0.767580\pi$
$523$	−34.0779	−1.49012	−0.745062	+	0.666995i	$0.767580\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.57445	0.329948
$528$	0	0
$529$	−22.5348	−0.979774
$530$	0	0
$531$	−2.93425	−0.127336
$532$	0	0
$533$	1.10922	0.0480456
$534$	0	0
$535$	30.8791	1.33502
$536$	0	0
$537$	−6.19146	−0.267181
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−29.3752	−1.26294	−0.631470	−	0.775400i	$-0.717548\pi$
$541$	−29.3752	−1.26294	−0.631470	+	0.775400i	$0.717548\pi$
$542$	0	0
$543$	1.11193	0.0477176
$544$	0	0
$545$	−17.3430	−0.742892
$546$	0	0
$547$	−4.60982	−0.197102	−0.0985509	−	0.995132i	$-0.531421\pi$
$547$	−4.60982	−0.197102	−0.0985509	+	0.995132i	$0.531421\pi$
$548$	0	0
$549$	1.10588	0.0471976
$550$	0	0
$551$	−1.65758	−0.0706151
$552$	0	0
$553$	0	0
$554$	0	0
$555$	3.27666	0.139086
$556$	0	0
$557$	−1.15369	−0.0488836	−0.0244418	−	0.999701i	$-0.507781\pi$
$557$	−1.15369	−0.0488836	−0.0244418	+	0.999701i	$0.507781\pi$
$558$	0	0
$559$	−10.6187	−0.449124
$560$	0	0
$561$	0.751178	0.0317147
$562$	0	0
$563$	−0.652565	−0.0275024	−0.0137512	−	0.999905i	$-0.504377\pi$
$563$	−0.652565	−0.0275024	−0.0137512	+	0.999905i	$0.504377\pi$
$564$	0	0
$565$	3.12200	0.131343
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−4.56467	−0.191361	−0.0956804	−	0.995412i	$-0.530503\pi$
$569$	−4.56467	−0.191361	−0.0956804	+	0.995412i	$0.530503\pi$
$570$	0	0
$571$	32.6522	1.36645	0.683226	−	0.730207i	$-0.260577\pi$
$571$	32.6522	1.36645	0.683226	+	0.730207i	$0.260577\pi$
$572$	0	0
$573$	−6.02118	−0.251538
$574$	0	0
$575$	4.02292	0.167767
$576$	0	0
$577$	−15.9542	−0.664183	−0.332091	−	0.943247i	$-0.607754\pi$
$577$	−15.9542	−0.664183	−0.332091	+	0.943247i	$0.607754\pi$
$578$	0	0
$579$	−8.30809	−0.345273
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−2.47650	−0.102566
$584$	0	0
$585$	10.6378	0.439817
$586$	0	0
$587$	7.90734	0.326371	0.163185	−	0.986595i	$-0.447823\pi$
$587$	7.90734	0.326371	0.163185	+	0.986595i	$0.447823\pi$
$588$	0	0
$589$	4.15084	0.171032
$590$	0	0
$591$	3.26242	0.134198
$592$	0	0
$593$	27.1920	1.11664	0.558320	−	0.829625i	$-0.311446\pi$
$593$	27.1920	1.11664	0.558320	+	0.829625i	$0.311446\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−6.11790	−0.250389
$598$	0	0
$599$	24.0880	0.984211	0.492105	−	0.870536i	$-0.336228\pi$
$599$	24.0880	0.984211	0.492105	+	0.870536i	$0.336228\pi$
$600$	0	0
$601$	−13.6546	−0.556984	−0.278492	−	0.960439i	$-0.589835\pi$
$601$	−13.6546	−0.556984	−0.278492	+	0.960439i	$0.589835\pi$
$602$	0	0
$603$	−24.6552	−1.00404
$604$	0	0
$605$	28.7135	1.16737
$606$	0	0
$607$	15.3900	0.624662	0.312331	−	0.949973i	$-0.398890\pi$
$607$	15.3900	0.624662	0.312331	+	0.949973i	$0.398890\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.30863	0.214764
$612$	0	0
$613$	10.7887	0.435753	0.217876	−	0.975976i	$-0.430087\pi$
$613$	10.7887	0.435753	0.217876	+	0.975976i	$0.430087\pi$
$614$	0	0
$615$	−1.01717	−0.0410162
$616$	0	0
$617$	−26.2621	−1.05727	−0.528637	−	0.848848i	$-0.677297\pi$
$617$	−26.2621	−1.05727	−0.528637	+	0.848848i	$0.677297\pi$
$618$	0	0
$619$	16.0257	0.644127	0.322063	−	0.946718i	$-0.395624\pi$
$619$	16.0257	0.644127	0.322063	+	0.946718i	$0.395624\pi$
$620$	0	0
$621$	−1.24097	−0.0497986
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−19.7023	−0.788093
$626$	0	0
$627$	0.411649	0.0164397
$628$	0	0
$629$	−5.17598	−0.206380
$630$	0	0
$631$	−3.37257	−0.134260	−0.0671299	−	0.997744i	$-0.521384\pi$
$631$	−3.37257	−0.134260	−0.0671299	+	0.997744i	$0.521384\pi$
$632$	0	0
$633$	2.78958	0.110876
$634$	0	0
$635$	−38.2832	−1.51922
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−22.9772	−0.908965
$640$	0	0
$641$	35.4637	1.40073	0.700365	−	0.713785i	$-0.253021\pi$
$641$	35.4637	1.40073	0.700365	+	0.713785i	$0.253021\pi$
$642$	0	0
$643$	30.4094	1.19923	0.599615	−	0.800288i	$-0.295320\pi$
$643$	30.4094	1.19923	0.599615	+	0.800288i	$0.295320\pi$
$644$	0	0
$645$	9.73750	0.383414
$646$	0	0
$647$	30.5807	1.20225	0.601126	−	0.799154i	$-0.294719\pi$
$647$	30.5807	1.20225	0.601126	+	0.799154i	$0.294719\pi$
$648$	0	0
$649$	1.53255	0.0601579
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−47.1664	−1.84576	−0.922882	−	0.385083i	$-0.874173\pi$
$653$	−47.1664	−1.84576	−0.922882	+	0.385083i	$0.874173\pi$
$654$	0	0
$655$	26.4742	1.03443
$656$	0	0
$657$	−4.54529	−0.177329
$658$	0	0
$659$	2.62840	0.102388	0.0511940	−	0.998689i	$-0.483697\pi$
$659$	2.62840	0.102388	0.0511940	+	0.998689i	$0.483697\pi$
$660$	0	0
$661$	23.7133	0.922341	0.461171	−	0.887311i	$-0.347430\pi$
$661$	23.7133	0.922341	0.461171	+	0.887311i	$0.347430\pi$
$662$	0	0
$663$	0.549144	0.0213270
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.28398	−0.0497158
$668$	0	0
$669$	−3.52256	−0.136190
$670$	0	0
$671$	−0.577595	−0.0222978
$672$	0	0
$673$	−22.4850	−0.866734	−0.433367	−	0.901218i	$-0.642675\pi$
$673$	−22.4850	−0.866734	−0.433367	+	0.901218i	$0.642675\pi$
$674$	0	0
$675$	−10.7315	−0.413054
$676$	0	0
$677$	25.4832	0.979397	0.489699	−	0.871892i	$-0.337107\pi$
$677$	25.4832	0.979397	0.489699	+	0.871892i	$0.337107\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−6.56221	−0.251464
$682$	0	0
$683$	10.0754	0.385525	0.192762	−	0.981245i	$-0.438255\pi$
$683$	10.0754	0.385525	0.192762	+	0.981245i	$0.438255\pi$
$684$	0	0
$685$	2.69786	0.103080
$686$	0	0
$687$	−6.50518	−0.248188
$688$	0	0
$689$	−1.81043	−0.0689721
$690$	0	0
$691$	−21.0068	−0.799138	−0.399569	−	0.916703i	$-0.630840\pi$
$691$	−21.0068	−0.799138	−0.399569	+	0.916703i	$0.630840\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−8.63366	−0.327493
$696$	0	0
$697$	1.60677	0.0608608
$698$	0	0
$699$	6.98754	0.264293
$700$	0	0
$701$	48.8524	1.84513	0.922564	−	0.385844i	$-0.126090\pi$
$701$	48.8524	1.84513	0.922564	+	0.385844i	$0.126090\pi$
$702$	0	0
$703$	−2.83646	−0.106979
$704$	0	0
$705$	−4.86808	−0.183342
$706$	0	0
$707$	0	0
$708$	0	0
$709$	41.0714	1.54247	0.771234	−	0.636551i	$-0.219640\pi$
$709$	41.0714	1.54247	0.771234	+	0.636551i	$0.219640\pi$
$710$	0	0
$711$	24.4452	0.916768
$712$	0	0
$713$	3.21529	0.120414
$714$	0	0
$715$	−5.55607	−0.207785
$716$	0	0
$717$	5.46065	0.203932
$718$	0	0
$719$	35.1717	1.31168	0.655841	−	0.754899i	$-0.272314\pi$
$719$	35.1717	1.31168	0.655841	+	0.754899i	$0.272314\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−6.21513	−0.231143
$724$	0	0
$725$	−11.1033	−0.412368
$726$	0	0
$727$	−31.0287	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$727$	−31.0287	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$728$	0	0
$729$	−22.0078	−0.815103
$730$	0	0
$731$	−15.3818	−0.568918
$732$	0	0
$733$	2.72105	0.100504	0.0502522	−	0.998737i	$-0.483997\pi$
$733$	2.72105	0.100504	0.0502522	+	0.998737i	$0.483997\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	12.8774	0.474344
$738$	0	0
$739$	21.0421	0.774045	0.387022	−	0.922070i	$-0.373504\pi$
$739$	21.0421	0.774045	0.387022	+	0.922070i	$0.373504\pi$
$740$	0	0
$741$	0.300934	0.0110551
$742$	0	0
$743$	30.2470	1.10965	0.554827	−	0.831966i	$-0.312785\pi$
$743$	30.2470	1.10965	0.554827	+	0.831966i	$0.312785\pi$
$744$	0	0
$745$	52.8802	1.93738
$746$	0	0
$747$	−4.51436	−0.165172
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−47.0853	−1.71817	−0.859083	−	0.511836i	$-0.828965\pi$
$751$	−47.0853	−1.71817	−0.859083	+	0.511836i	$0.828965\pi$
$752$	0	0
$753$	−5.08742	−0.185396
$754$	0	0
$755$	31.9502	1.16279
$756$	0	0
$757$	−20.1878	−0.733738	−0.366869	−	0.930273i	$-0.619570\pi$
$757$	−20.1878	−0.733738	−0.366869	+	0.930273i	$0.619570\pi$
$758$	0	0
$759$	0.318868	0.0115742
$760$	0	0
$761$	51.2177	1.85664	0.928320	−	0.371782i	$-0.121253\pi$
$761$	51.2177	1.85664	0.928320	+	0.371782i	$0.121253\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	15.4094	0.557129
$766$	0	0
$767$	1.12036	0.0404540
$768$	0	0
$769$	0.804472	0.0290100	0.0145050	−	0.999895i	$-0.495383\pi$
$769$	0.804472	0.0290100	0.0145050	+	0.999895i	$0.495383\pi$
$770$	0	0
$771$	5.28017	0.190160
$772$	0	0
$773$	−40.0107	−1.43909	−0.719543	−	0.694448i	$-0.755649\pi$
$773$	−40.0107	−1.43909	−0.719543	+	0.694448i	$0.755649\pi$
$774$	0	0
$775$	27.8046	0.998769
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0.880518	0.0315478
$780$	0	0
$781$	12.0009	0.429427
$782$	0	0
$783$	3.42511	0.122404
$784$	0	0
$785$	−28.9869	−1.03459
$786$	0	0
$787$	−45.1844	−1.61065	−0.805326	−	0.592833i	$-0.798010\pi$
$787$	−45.1844	−1.61065	−0.805326	+	0.592833i	$0.798010\pi$
$788$	0	0
$789$	−5.31326	−0.189157
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−0.422248	−0.0149945
$794$	0	0
$795$	1.66019	0.0588809
$796$	0	0
$797$	42.7382	1.51386	0.756932	−	0.653494i	$-0.226697\pi$
$797$	42.7382	1.51386	0.756932	+	0.653494i	$0.226697\pi$
$798$	0	0
$799$	7.68987	0.272048
$800$	0	0
$801$	−40.4430	−1.42898
$802$	0	0
$803$	2.37399	0.0837763
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−4.46993	−0.157349
$808$	0	0
$809$	13.0164	0.457632	0.228816	−	0.973470i	$-0.426515\pi$
$809$	13.0164	0.457632	0.228816	+	0.973470i	$0.426515\pi$
$810$	0	0
$811$	−21.9400	−0.770418	−0.385209	−	0.922829i	$-0.625871\pi$
$811$	−21.9400	−0.770418	−0.385209	+	0.922829i	$0.625871\pi$
$812$	0	0
$813$	−7.35719	−0.258028
$814$	0	0
$815$	38.2840	1.34103
$816$	0	0
$817$	−8.42933	−0.294905
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−27.5081	−0.960040	−0.480020	−	0.877257i	$-0.659371\pi$
$821$	−27.5081	−0.960040	−0.480020	+	0.877257i	$0.659371\pi$
$822$	0	0
$823$	−29.8968	−1.04214	−0.521068	−	0.853515i	$-0.674466\pi$
$823$	−29.8968	−1.04214	−0.521068	+	0.853515i	$0.674466\pi$
$824$	0	0
$825$	2.75745	0.0960020
$826$	0	0
$827$	8.18193	0.284513	0.142257	−	0.989830i	$-0.454564\pi$
$827$	8.18193	0.284513	0.142257	+	0.989830i	$0.454564\pi$
$828$	0	0
$829$	−56.3549	−1.95729	−0.978643	−	0.205565i	$-0.934097\pi$
$829$	−56.3549	−1.95729	−0.978643	+	0.205565i	$0.934097\pi$
$830$	0	0
$831$	−4.81744	−0.167115
$832$	0	0
$833$	0	0
$834$	0	0
$835$	10.6634	0.369023
$836$	0	0
$837$	−8.57704	−0.296466
$838$	0	0
$839$	26.0609	0.899724	0.449862	−	0.893098i	$-0.351473\pi$
$839$	26.0609	0.899724	0.449862	+	0.893098i	$0.351473\pi$
$840$	0	0
$841$	−25.4562	−0.877800
$842$	0	0
$843$	1.59298	0.0548651
$844$	0	0
$845$	38.8544	1.33663
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−3.10000	−0.106392
$850$	0	0
$851$	−2.19716	−0.0753176
$852$	0	0
$853$	34.0375	1.16542	0.582712	−	0.812679i	$-0.301992\pi$
$853$	34.0375	1.16542	0.582712	+	0.812679i	$0.301992\pi$
$854$	0	0
$855$	8.44445	0.288794
$856$	0	0
$857$	−40.0483	−1.36802	−0.684011	−	0.729471i	$-0.739766\pi$
$857$	−40.0483	−1.36802	−0.684011	+	0.729471i	$0.739766\pi$
$858$	0	0
$859$	−25.4840	−0.869504	−0.434752	−	0.900550i	$-0.643164\pi$
$859$	−25.4840	−0.869504	−0.434752	+	0.900550i	$0.643164\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−0.747306	−0.0254386	−0.0127193	−	0.999919i	$-0.504049\pi$
$863$	−0.747306	−0.0254386	−0.0127193	+	0.999919i	$0.504049\pi$
$864$	0	0
$865$	−32.5413	−1.10644
$866$	0	0
$867$	−4.44252	−0.150876
$868$	0	0
$869$	−12.7677	−0.433114
$870$	0	0
$871$	9.41393	0.318979
$872$	0	0
$873$	−1.74606	−0.0590952
$874$	0	0
$875$	0	0
$876$	0	0
$877$	16.5402	0.558523	0.279262	−	0.960215i	$-0.409910\pi$
$877$	16.5402	0.558523	0.279262	+	0.960215i	$0.409910\pi$
$878$	0	0
$879$	−2.87270	−0.0968938
$880$	0	0
$881$	−50.7950	−1.71133	−0.855663	−	0.517533i	$-0.826850\pi$
$881$	−50.7950	−1.71133	−0.855663	+	0.517533i	$0.826850\pi$
$882$	0	0
$883$	−26.1612	−0.880394	−0.440197	−	0.897901i	$-0.645091\pi$
$883$	−26.1612	−0.880394	−0.440197	+	0.897901i	$0.645091\pi$
$884$	0	0
$885$	−1.02739	−0.0345353
$886$	0	0
$887$	35.5282	1.19292	0.596461	−	0.802642i	$-0.296573\pi$
$887$	35.5282	1.19292	0.596461	+	0.802642i	$0.296573\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	12.3730	0.414511
$892$	0	0
$893$	4.21409	0.141019
$894$	0	0
$895$	66.3368	2.21740
$896$	0	0
$897$	0.233107	0.00778321
$898$	0	0
$899$	−8.87427	−0.295973
$900$	0	0
$901$	−2.62252	−0.0873690
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−11.9135	−0.396019
$906$	0	0
$907$	29.0487	0.964545	0.482273	−	0.876021i	$-0.339811\pi$
$907$	29.0487	0.964545	0.482273	+	0.876021i	$0.339811\pi$
$908$	0	0
$909$	−11.0225	−0.365592
$910$	0	0
$911$	10.5596	0.349855	0.174927	−	0.984581i	$-0.444031\pi$
$911$	10.5596	0.349855	0.174927	+	0.984581i	$0.444031\pi$
$912$	0	0
$913$	2.35784	0.0780331
$914$	0	0
$915$	0.387207	0.0128007
$916$	0	0
$917$	0	0
$918$	0	0
$919$	15.5083	0.511572	0.255786	−	0.966733i	$-0.417666\pi$
$919$	15.5083	0.511572	0.255786	+	0.966733i	$0.417666\pi$
$920$	0	0
$921$	−3.19893	−0.105408
$922$	0	0
$923$	8.77322	0.288774
$924$	0	0
$925$	−19.0002	−0.624721
$926$	0	0
$927$	−1.51066	−0.0496166
$928$	0	0
$929$	5.10650	0.167539	0.0837694	−	0.996485i	$-0.473304\pi$
$929$	5.10650	0.167539	0.0837694	+	0.996485i	$0.473304\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	9.78375	0.320306
$934$	0	0
$935$	−8.04830	−0.263208
$936$	0	0
$937$	−6.24701	−0.204081	−0.102040	−	0.994780i	$-0.532537\pi$
$937$	−6.24701	−0.204081	−0.102040	+	0.994780i	$0.532537\pi$
$938$	0	0
$939$	−7.55497	−0.246547
$940$	0	0
$941$	−30.3734	−0.990145	−0.495073	−	0.868852i	$-0.664859\pi$
$941$	−30.3734	−0.990145	−0.495073	+	0.868852i	$0.664859\pi$
$942$	0	0
$943$	0.682060	0.0222109
$944$	0	0
$945$	0	0
$946$	0	0
$947$	42.1003	1.36808	0.684038	−	0.729446i	$-0.260222\pi$
$947$	42.1003	1.36808	0.684038	+	0.729446i	$0.260222\pi$
$948$	0	0
$949$	1.73549	0.0563365
$950$	0	0
$951$	−2.43244	−0.0788773
$952$	0	0
$953$	−48.9720	−1.58636	−0.793180	−	0.608988i	$-0.791576\pi$
$953$	−48.9720	−1.58636	−0.793180	+	0.608988i	$0.791576\pi$
$954$	0	0
$955$	64.5124	2.08757
$956$	0	0
$957$	−0.880083	−0.0284491
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−8.77739	−0.283142
$962$	0	0
$963$	27.1733	0.875648
$964$	0	0
$965$	89.0150	2.86549
$966$	0	0
$967$	33.8769	1.08941	0.544704	−	0.838628i	$-0.316642\pi$
$967$	33.8769	1.08941	0.544704	+	0.838628i	$0.316642\pi$
$968$	0	0
$969$	0.435921	0.0140038
$970$	0	0
$971$	−0.943295	−0.0302718	−0.0151359	−	0.999885i	$-0.504818\pi$
$971$	−0.943295	−0.0302718	−0.0151359	+	0.999885i	$0.504818\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	2.01582	0.0645578
$976$	0	0
$977$	−17.1014	−0.547124	−0.273562	−	0.961854i	$-0.588202\pi$
$977$	−17.1014	−0.547124	−0.273562	+	0.961854i	$0.588202\pi$
$978$	0	0
$979$	21.1233	0.675103
$980$	0	0
$981$	−15.2617	−0.487268
$982$	0	0
$983$	1.88552	0.0601388	0.0300694	−	0.999548i	$-0.490427\pi$
$983$	1.88552	0.0601388	0.0300694	+	0.999548i	$0.490427\pi$
$984$	0	0
$985$	−34.9544	−1.11374
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6.52946	−0.207625
$990$	0	0
$991$	22.9252	0.728244	0.364122	−	0.931351i	$-0.381369\pi$
$991$	22.9252	0.728244	0.364122	+	0.931351i	$0.381369\pi$
$992$	0	0
$993$	−4.35276	−0.138131
$994$	0	0
$995$	65.5487	2.07803
$996$	0	0
$997$	−48.2258	−1.52733	−0.763663	−	0.645615i	$-0.776601\pi$
$997$	−48.2258	−1.52733	−0.763663	+	0.645615i	$0.776601\pi$
$998$	0	0
$999$	5.86110	0.185437

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.m.1.4		8
7.2	even	3		1148.2.i.d.165.5	✓	16
7.4	even	3		1148.2.i.d.821.5	yes	16
7.6	odd	2		8036.2.a.n.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.i.d.165.5	✓	16	7.2	even	3
1148.2.i.d.821.5	yes	16	7.4	even	3
8036.2.a.m.1.4		8	1.1	even	1	trivial
8036.2.a.n.1.5		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)

\(f(q)\)	\(=\)	\(q+0.308117 q^{3} -3.30124 q^{5} -2.90506 q^{9} +O(q^{10})\)	\(q+0.308117 q^{3} -3.30124 q^{5} -2.90506 q^{9} +1.51731 q^{11} +1.10922 q^{13} -1.01717 q^{15} +1.60677 q^{17} +0.880518 q^{19} +0.682060 q^{23} +5.89819 q^{25} -1.81945 q^{27} -1.88250 q^{29} +4.71409 q^{31} +0.467508 q^{33} -3.22136 q^{37} +0.341769 q^{39} +1.00000 q^{41} -9.57315 q^{43} +9.59031 q^{45} +4.78592 q^{47} +0.495073 q^{51} -1.63217 q^{53} -5.00899 q^{55} +0.271303 q^{57} +1.01005 q^{59} -0.380672 q^{61} -3.66180 q^{65} +8.48699 q^{67} +0.210154 q^{69} +7.90937 q^{71} +1.56461 q^{73} +1.81733 q^{75} -8.41469 q^{79} +8.15459 q^{81} +1.55396 q^{83} -5.30433 q^{85} -0.580030 q^{87} +13.9216 q^{89} +1.45249 q^{93} -2.90680 q^{95} +0.601041 q^{97} -4.40787 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{9}+O(q^{10}) \) 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

L-functions

Modular forms

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Database

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