LMFDB - Embedded newform 8012.2.a.b.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8012,2,Mod(1,8012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8012, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8012 = 2^{2} \cdot 2003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8012.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:	$63.9761420994$
Analytic rank:	$0$
Dimension:	$88$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	8012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.71734 q^{3} -0.520933 q^{5} -1.18624 q^{7} +4.38393 q^{9} +O(q^{10})$	$q-2.71734 q^{3} -0.520933 q^{5} -1.18624 q^{7} +4.38393 q^{9} -1.20530 q^{11} -5.96065 q^{13} +1.41555 q^{15} +2.66061 q^{17} -5.13476 q^{19} +3.22341 q^{21} -5.66417 q^{23} -4.72863 q^{25} -3.76061 q^{27} -1.33306 q^{29} +5.32086 q^{31} +3.27521 q^{33} +0.617951 q^{35} -5.43080 q^{37} +16.1971 q^{39} -5.85556 q^{41} +1.00695 q^{43} -2.28374 q^{45} -6.90110 q^{47} -5.59284 q^{49} -7.22978 q^{51} -4.91014 q^{53} +0.627881 q^{55} +13.9529 q^{57} +4.97599 q^{59} -14.8040 q^{61} -5.20038 q^{63} +3.10510 q^{65} +3.75799 q^{67} +15.3915 q^{69} +6.55504 q^{71} +3.10618 q^{73} +12.8493 q^{75} +1.42977 q^{77} -8.64639 q^{79} -2.93295 q^{81} +15.0476 q^{83} -1.38600 q^{85} +3.62239 q^{87} -9.88486 q^{89} +7.07075 q^{91} -14.4586 q^{93} +2.67487 q^{95} +8.01128 q^{97} -5.28395 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 88 q + 19 q^{3} + 44 q^{7} + 99 q^{9}+O(q^{10}) $ 88 * q + 19 * q^3 + 44 * q^7 + 99 * q^9	$ 88 q + 19 q^{3} + 44 q^{7} + 99 q^{9} + 16 q^{11} - 3 q^{13} + 21 q^{15} + 32 q^{17} + 49 q^{19} + 7 q^{21} + 24 q^{23} + 114 q^{25} + 82 q^{27} + 2 q^{29} + 40 q^{31} + 31 q^{33} + 26 q^{35} + 37 q^{39} + 22 q^{41} + 98 q^{43} + 12 q^{45} + 48 q^{47} + 132 q^{49} + 55 q^{51} - q^{53} + 116 q^{55} + 62 q^{57} + 34 q^{59} + 132 q^{63} + 39 q^{65} + 75 q^{67} + 15 q^{69} + 24 q^{71} + 104 q^{73} + 87 q^{75} + 4 q^{77} + 111 q^{79} + 128 q^{81} + 64 q^{83} + 7 q^{85} + 115 q^{87} + 14 q^{89} + 73 q^{91} - 7 q^{93} + 51 q^{95} + 117 q^{97} + 72 q^{99}+O(q^{100}) $ 88 * q + 19 * q^3 + 44 * q^7 + 99 * q^9 + 16 * q^11 - 3 * q^13 + 21 * q^15 + 32 * q^17 + 49 * q^19 + 7 * q^21 + 24 * q^23 + 114 * q^25 + 82 * q^27 + 2 * q^29 + 40 * q^31 + 31 * q^33 + 26 * q^35 + 37 * q^39 + 22 * q^41 + 98 * q^43 + 12 * q^45 + 48 * q^47 + 132 * q^49 + 55 * q^51 - q^53 + 116 * q^55 + 62 * q^57 + 34 * q^59 + 132 * q^63 + 39 * q^65 + 75 * q^67 + 15 * q^69 + 24 * q^71 + 104 * q^73 + 87 * q^75 + 4 * q^77 + 111 * q^79 + 128 * q^81 + 64 * q^83 + 7 * q^85 + 115 * q^87 + 14 * q^89 + 73 * q^91 - 7 * q^93 + 51 * q^95 + 117 * q^97 + 72 * 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
$3$	−2.71734	−1.56886	−0.784428	−	0.620220i	$-0.787043\pi$
$3$	−2.71734	−1.56886	−0.784428	+	0.620220i	$0.787043\pi$
$4$	0	0
$5$	−0.520933	−0.232969	−0.116484	−	0.993193i	$-0.537162\pi$
$5$	−0.520933	−0.232969	−0.116484	+	0.993193i	$0.537162\pi$
$6$	0	0
$7$	−1.18624	−0.448356	−0.224178	−	0.974548i	$-0.571970\pi$
$7$	−1.18624	−0.448356	−0.224178	+	0.974548i	$0.571970\pi$
$8$	0	0
$9$	4.38393	1.46131
$10$	0	0
$11$	−1.20530	−0.363411	−0.181706	−	0.983353i	$-0.558162\pi$
$11$	−1.20530	−0.363411	−0.181706	+	0.983353i	$0.558162\pi$
$12$	0	0
$13$	−5.96065	−1.65319	−0.826594	−	0.562799i	$-0.809725\pi$
$13$	−5.96065	−1.65319	−0.826594	+	0.562799i	$0.809725\pi$
$14$	0	0
$15$	1.41555	0.365494
$16$	0	0
$17$	2.66061	0.645292	0.322646	−	0.946520i	$-0.395428\pi$
$17$	2.66061	0.645292	0.322646	+	0.946520i	$0.395428\pi$
$18$	0	0
$19$	−5.13476	−1.17800	−0.588998	−	0.808135i	$-0.700477\pi$
$19$	−5.13476	−1.17800	−0.588998	+	0.808135i	$0.700477\pi$
$20$	0	0
$21$	3.22341	0.703406
$22$	0	0
$23$	−5.66417	−1.18106	−0.590530	−	0.807015i	$-0.701081\pi$
$23$	−5.66417	−1.18106	−0.590530	+	0.807015i	$0.701081\pi$
$24$	0	0
$25$	−4.72863	−0.945726
$26$	0	0
$27$	−3.76061	−0.723729
$28$	0	0
$29$	−1.33306	−0.247544	−0.123772	−	0.992311i	$-0.539499\pi$
$29$	−1.33306	−0.247544	−0.123772	+	0.992311i	$0.539499\pi$
$30$	0	0
$31$	5.32086	0.955654	0.477827	−	0.878454i	$-0.341425\pi$
$31$	5.32086	0.955654	0.477827	+	0.878454i	$0.341425\pi$
$32$	0	0
$33$	3.27521	0.570140
$34$	0	0
$35$	0.617951	0.104453
$36$	0	0
$37$	−5.43080	−0.892817	−0.446409	−	0.894829i	$-0.647297\pi$
$37$	−5.43080	−0.892817	−0.446409	+	0.894829i	$0.647297\pi$
$38$	0	0
$39$	16.1971	2.59361
$40$	0	0
$41$	−5.85556	−0.914484	−0.457242	−	0.889342i	$-0.651163\pi$
$41$	−5.85556	−0.914484	−0.457242	+	0.889342i	$0.651163\pi$
$42$	0	0
$43$	1.00695	0.153558	0.0767790	−	0.997048i	$-0.475536\pi$
$43$	1.00695	0.153558	0.0767790	+	0.997048i	$0.475536\pi$
$44$	0	0
$45$	−2.28374	−0.340439
$46$	0	0
$47$	−6.90110	−1.00663	−0.503315	−	0.864103i	$-0.667886\pi$
$47$	−6.90110	−1.00663	−0.503315	+	0.864103i	$0.667886\pi$
$48$	0	0
$49$	−5.59284	−0.798977
$50$	0	0
$51$	−7.22978	−1.01237
$52$	0	0
$53$	−4.91014	−0.674459	−0.337230	−	0.941422i	$-0.609490\pi$
$53$	−4.91014	−0.674459	−0.337230	+	0.941422i	$0.609490\pi$
$54$	0	0
$55$	0.627881	0.0846634
$56$	0	0
$57$	13.9529	1.84811
$58$	0	0
$59$	4.97599	0.647818	0.323909	−	0.946088i	$-0.395003\pi$
$59$	4.97599	0.647818	0.323909	+	0.946088i	$0.395003\pi$
$60$	0	0
$61$	−14.8040	−1.89546	−0.947728	−	0.319080i	$-0.896626\pi$
$61$	−14.8040	−1.89546	−0.947728	+	0.319080i	$0.896626\pi$
$62$	0	0
$63$	−5.20038	−0.655187
$64$	0	0
$65$	3.10510	0.385141
$66$	0	0
$67$	3.75799	0.459111	0.229556	−	0.973296i	$-0.426273\pi$
$67$	3.75799	0.459111	0.229556	+	0.973296i	$0.426273\pi$
$68$	0	0
$69$	15.3915	1.85291
$70$	0	0
$71$	6.55504	0.777940	0.388970	−	0.921250i	$-0.372831\pi$
$71$	6.55504	0.777940	0.388970	+	0.921250i	$0.372831\pi$
$72$	0	0
$73$	3.10618	0.363550	0.181775	−	0.983340i	$-0.441816\pi$
$73$	3.10618	0.363550	0.181775	+	0.983340i	$0.441816\pi$
$74$	0	0
$75$	12.8493	1.48371
$76$	0	0
$77$	1.42977	0.162938
$78$	0	0
$79$	−8.64639	−0.972795	−0.486397	−	0.873738i	$-0.661689\pi$
$79$	−8.64639	−0.972795	−0.486397	+	0.873738i	$0.661689\pi$
$80$	0	0
$81$	−2.93295	−0.325883
$82$	0	0
$83$	15.0476	1.65169	0.825843	−	0.563900i	$-0.190700\pi$
$83$	15.0476	1.65169	0.825843	+	0.563900i	$0.190700\pi$
$84$	0	0
$85$	−1.38600	−0.150333
$86$	0	0
$87$	3.62239	0.388361
$88$	0	0
$89$	−9.88486	−1.04779	−0.523897	−	0.851782i	$-0.675522\pi$
$89$	−9.88486	−1.04779	−0.523897	+	0.851782i	$0.675522\pi$
$90$	0	0
$91$	7.07075	0.741216
$92$	0	0
$93$	−14.4586	−1.49928
$94$	0	0
$95$	2.67487	0.274436
$96$	0	0
$97$	8.01128	0.813422	0.406711	−	0.913557i	$-0.366676\pi$
$97$	8.01128	0.813422	0.406711	+	0.913557i	$0.366676\pi$
$98$	0	0
$99$	−5.28395	−0.531057
$100$	0	0
$101$	−19.5347	−1.94377	−0.971887	−	0.235448i	$-0.924344\pi$
$101$	−19.5347	−1.94377	−0.971887	+	0.235448i	$0.924344\pi$
$102$	0	0
$103$	−14.9388	−1.47196	−0.735980	−	0.677003i	$-0.763278\pi$
$103$	−14.9388	−1.47196	−0.735980	+	0.677003i	$0.763278\pi$
$104$	0	0
$105$	−1.67918	−0.163871
$106$	0	0
$107$	−7.65560	−0.740095	−0.370047	−	0.929013i	$-0.620659\pi$
$107$	−7.65560	−0.740095	−0.370047	+	0.929013i	$0.620659\pi$
$108$	0	0
$109$	−3.95312	−0.378641	−0.189320	−	0.981915i	$-0.560628\pi$
$109$	−3.95312	−0.378641	−0.189320	+	0.981915i	$0.560628\pi$
$110$	0	0
$111$	14.7573	1.40070
$112$	0	0
$113$	−6.76745	−0.636628	−0.318314	−	0.947985i	$-0.603117\pi$
$113$	−6.76745	−0.636628	−0.318314	+	0.947985i	$0.603117\pi$
$114$	0	0
$115$	2.95065	0.275150
$116$	0	0
$117$	−26.1311	−2.41582
$118$	0	0
$119$	−3.15611	−0.289321
$120$	0	0
$121$	−9.54725	−0.867932
$122$	0	0
$123$	15.9115	1.43469
$124$	0	0
$125$	5.06797	0.453293
$126$	0	0
$127$	5.65867	0.502126	0.251063	−	0.967971i	$-0.419220\pi$
$127$	5.65867	0.502126	0.251063	+	0.967971i	$0.419220\pi$
$128$	0	0
$129$	−2.73622	−0.240910
$130$	0	0
$131$	−0.506494	−0.0442526	−0.0221263	−	0.999755i	$-0.507044\pi$
$131$	−0.506494	−0.0442526	−0.0221263	+	0.999755i	$0.507044\pi$
$132$	0	0
$133$	6.09105	0.528161
$134$	0	0
$135$	1.95903	0.168606
$136$	0	0
$137$	10.2716	0.877559	0.438779	−	0.898595i	$-0.355411\pi$
$137$	10.2716	0.877559	0.438779	+	0.898595i	$0.355411\pi$
$138$	0	0
$139$	−10.4579	−0.887026	−0.443513	−	0.896268i	$-0.646268\pi$
$139$	−10.4579	−0.887026	−0.443513	+	0.896268i	$0.646268\pi$
$140$	0	0
$141$	18.7526	1.57926
$142$	0	0
$143$	7.18437	0.600787
$144$	0	0
$145$	0.694438	0.0576699
$146$	0	0
$147$	15.1976	1.25348
$148$	0	0
$149$	−15.3624	−1.25854	−0.629270	−	0.777187i	$-0.716646\pi$
$149$	−15.3624	−1.25854	−0.629270	+	0.777187i	$0.716646\pi$
$150$	0	0
$151$	7.17063	0.583537	0.291769	−	0.956489i	$-0.405756\pi$
$151$	7.17063	0.583537	0.291769	+	0.956489i	$0.405756\pi$
$152$	0	0
$153$	11.6639	0.942972
$154$	0	0
$155$	−2.77181	−0.222637
$156$	0	0
$157$	−7.88096	−0.628969	−0.314484	−	0.949263i	$-0.601832\pi$
$157$	−7.88096	−0.628969	−0.314484	+	0.949263i	$0.601832\pi$
$158$	0	0
$159$	13.3425	1.05813
$160$	0	0
$161$	6.71905	0.529535
$162$	0	0
$163$	−21.1321	−1.65519	−0.827595	−	0.561325i	$-0.810292\pi$
$163$	−21.1321	−1.65519	−0.827595	+	0.561325i	$0.810292\pi$
$164$	0	0
$165$	−1.70616	−0.132825
$166$	0	0
$167$	4.42470	0.342394	0.171197	−	0.985237i	$-0.445237\pi$
$167$	4.42470	0.342394	0.171197	+	0.985237i	$0.445237\pi$
$168$	0	0
$169$	22.5294	1.73303
$170$	0	0
$171$	−22.5104	−1.72142
$172$	0	0
$173$	21.8847	1.66387	0.831933	−	0.554876i	$-0.187234\pi$
$173$	21.8847	1.66387	0.831933	+	0.554876i	$0.187234\pi$
$174$	0	0
$175$	5.60928	0.424021
$176$	0	0
$177$	−13.5214	−1.01633
$178$	0	0
$179$	12.7491	0.952916	0.476458	−	0.879197i	$-0.341920\pi$
$179$	12.7491	0.952916	0.476458	+	0.879197i	$0.341920\pi$
$180$	0	0
$181$	−17.2363	−1.28117	−0.640584	−	0.767888i	$-0.721308\pi$
$181$	−17.2363	−1.28117	−0.640584	+	0.767888i	$0.721308\pi$
$182$	0	0
$183$	40.2274	2.97370
$184$	0	0
$185$	2.82908	0.207998
$186$	0	0
$187$	−3.20683	−0.234507
$188$	0	0
$189$	4.46097	0.324488
$190$	0	0
$191$	−12.5227	−0.906113	−0.453057	−	0.891482i	$-0.649666\pi$
$191$	−12.5227	−0.906113	−0.453057	+	0.891482i	$0.649666\pi$
$192$	0	0
$193$	5.33697	0.384164	0.192082	−	0.981379i	$-0.438476\pi$
$193$	5.33697	0.384164	0.192082	+	0.981379i	$0.438476\pi$
$194$	0	0
$195$	−8.43762	−0.604230
$196$	0	0
$197$	−24.4599	−1.74270	−0.871349	−	0.490664i	$-0.836754\pi$
$197$	−24.4599	−1.74270	−0.871349	+	0.490664i	$0.836754\pi$
$198$	0	0
$199$	−20.2771	−1.43741	−0.718703	−	0.695317i	$-0.755264\pi$
$199$	−20.2771	−1.43741	−0.718703	+	0.695317i	$0.755264\pi$
$200$	0	0
$201$	−10.2117	−0.720279
$202$	0	0
$203$	1.58133	0.110988
$204$	0	0
$205$	3.05036	0.213046
$206$	0	0
$207$	−24.8313	−1.72590
$208$	0	0
$209$	6.18893	0.428097
$210$	0	0
$211$	−6.92429	−0.476688	−0.238344	−	0.971181i	$-0.576605\pi$
$211$	−6.92429	−0.476688	−0.238344	+	0.971181i	$0.576605\pi$
$212$	0	0
$213$	−17.8123	−1.22048
$214$	0	0
$215$	−0.524552	−0.0357742
$216$	0	0
$217$	−6.31180	−0.428473
$218$	0	0
$219$	−8.44053	−0.570358
$220$	0	0
$221$	−15.8590	−1.06679
$222$	0	0
$223$	18.4258	1.23389	0.616943	−	0.787008i	$-0.288371\pi$
$223$	18.4258	1.23389	0.616943	+	0.787008i	$0.288371\pi$
$224$	0	0
$225$	−20.7300	−1.38200
$226$	0	0
$227$	13.0828	0.868336	0.434168	−	0.900832i	$-0.357042\pi$
$227$	13.0828	0.868336	0.434168	+	0.900832i	$0.357042\pi$
$228$	0	0
$229$	−1.47392	−0.0973992	−0.0486996	−	0.998813i	$-0.515508\pi$
$229$	−1.47392	−0.0973992	−0.0486996	+	0.998813i	$0.515508\pi$
$230$	0	0
$231$	−3.88517	−0.255626
$232$	0	0
$233$	13.8499	0.907337	0.453668	−	0.891171i	$-0.350115\pi$
$233$	13.8499	0.907337	0.453668	+	0.891171i	$0.350115\pi$
$234$	0	0
$235$	3.59501	0.234513
$236$	0	0
$237$	23.4952	1.52618
$238$	0	0
$239$	−27.4390	−1.77488	−0.887441	−	0.460922i	$-0.847519\pi$
$239$	−27.4390	−1.77488	−0.887441	+	0.460922i	$0.847519\pi$
$240$	0	0
$241$	−23.2700	−1.49896	−0.749478	−	0.662030i	$-0.769695\pi$
$241$	−23.2700	−1.49896	−0.749478	+	0.662030i	$0.769695\pi$
$242$	0	0
$243$	19.2516	1.23499
$244$	0	0
$245$	2.91350	0.186137
$246$	0	0
$247$	30.6065	1.94745
$248$	0	0
$249$	−40.8894	−2.59126
$250$	0	0
$251$	−16.7075	−1.05457	−0.527285	−	0.849688i	$-0.676790\pi$
$251$	−16.7075	−1.05457	−0.527285	+	0.849688i	$0.676790\pi$
$252$	0	0
$253$	6.82702	0.429211
$254$	0	0
$255$	3.76623	0.235851
$256$	0	0
$257$	11.7993	0.736020	0.368010	−	0.929822i	$-0.380039\pi$
$257$	11.7993	0.736020	0.368010	+	0.929822i	$0.380039\pi$
$258$	0	0
$259$	6.44221	0.400300
$260$	0	0
$261$	−5.84406	−0.361738
$262$	0	0
$263$	16.2364	1.00118	0.500590	−	0.865684i	$-0.333116\pi$
$263$	16.2364	1.00118	0.500590	+	0.865684i	$0.333116\pi$
$264$	0	0
$265$	2.55785	0.157128
$266$	0	0
$267$	26.8605	1.64384
$268$	0	0
$269$	−6.44090	−0.392709	−0.196354	−	0.980533i	$-0.562910\pi$
$269$	−6.44090	−0.392709	−0.196354	+	0.980533i	$0.562910\pi$
$270$	0	0
$271$	15.7898	0.959160	0.479580	−	0.877498i	$-0.340789\pi$
$271$	15.7898	0.959160	0.479580	+	0.877498i	$0.340789\pi$
$272$	0	0
$273$	−19.2136	−1.16286
$274$	0	0
$275$	5.69941	0.343688
$276$	0	0
$277$	−15.3712	−0.923569	−0.461784	−	0.886992i	$-0.652791\pi$
$277$	−15.3712	−0.923569	−0.461784	+	0.886992i	$0.652791\pi$
$278$	0	0
$279$	23.3263	1.39651
$280$	0	0
$281$	−17.9682	−1.07189	−0.535947	−	0.844252i	$-0.680045\pi$
$281$	−17.9682	−1.07189	−0.535947	+	0.844252i	$0.680045\pi$
$282$	0	0
$283$	−11.4916	−0.683102	−0.341551	−	0.939863i	$-0.610952\pi$
$283$	−11.4916	−0.683102	−0.341551	+	0.939863i	$0.610952\pi$
$284$	0	0
$285$	−7.26853	−0.430551
$286$	0	0
$287$	6.94608	0.410014
$288$	0	0
$289$	−9.92116	−0.583598
$290$	0	0
$291$	−21.7694	−1.27614
$292$	0	0
$293$	29.3012	1.71179	0.855897	−	0.517146i	$-0.173006\pi$
$293$	29.3012	1.71179	0.855897	+	0.517146i	$0.173006\pi$
$294$	0	0
$295$	−2.59216	−0.150921
$296$	0	0
$297$	4.53266	0.263011
$298$	0	0
$299$	33.7621	1.95252
$300$	0	0
$301$	−1.19448	−0.0688486
$302$	0	0
$303$	53.0824	3.04950
$304$	0	0
$305$	7.71189	0.441581
$306$	0	0
$307$	−27.5418	−1.57189	−0.785946	−	0.618295i	$-0.787824\pi$
$307$	−27.5418	−1.57189	−0.785946	+	0.618295i	$0.787824\pi$
$308$	0	0
$309$	40.5937	2.30929
$310$	0	0
$311$	21.5072	1.21956	0.609782	−	0.792570i	$-0.291257\pi$
$311$	21.5072	1.21956	0.609782	+	0.792570i	$0.291257\pi$
$312$	0	0
$313$	24.6873	1.39541	0.697704	−	0.716386i	$-0.254205\pi$
$313$	24.6873	1.39541	0.697704	+	0.716386i	$0.254205\pi$
$314$	0	0
$315$	2.70905	0.152638
$316$	0	0
$317$	−22.1204	−1.24241	−0.621203	−	0.783650i	$-0.713356\pi$
$317$	−22.1204	−1.24241	−0.621203	+	0.783650i	$0.713356\pi$
$318$	0	0
$319$	1.60674	0.0899602
$320$	0	0
$321$	20.8029	1.16110
$322$	0	0
$323$	−13.6616	−0.760152
$324$	0	0
$325$	28.1857	1.56346
$326$	0	0
$327$	10.7420	0.594033
$328$	0	0
$329$	8.18635	0.451328
$330$	0	0
$331$	−11.8753	−0.652724	−0.326362	−	0.945245i	$-0.605823\pi$
$331$	−11.8753	−0.652724	−0.326362	+	0.945245i	$0.605823\pi$
$332$	0	0
$333$	−23.8082	−1.30468
$334$	0	0
$335$	−1.95766	−0.106958
$336$	0	0
$337$	22.1848	1.20848	0.604240	−	0.796802i	$-0.293477\pi$
$337$	22.1848	1.20848	0.604240	+	0.796802i	$0.293477\pi$
$338$	0	0
$339$	18.3895	0.998779
$340$	0	0
$341$	−6.41323	−0.347296
$342$	0	0
$343$	14.9381	0.806582
$344$	0	0
$345$	−8.01793	−0.431671
$346$	0	0
$347$	−24.6816	−1.32498	−0.662488	−	0.749072i	$-0.730500\pi$
$347$	−24.6816	−1.32498	−0.662488	+	0.749072i	$0.730500\pi$
$348$	0	0
$349$	32.3371	1.73096	0.865482	−	0.500941i	$-0.167013\pi$
$349$	32.3371	1.73096	0.865482	+	0.500941i	$0.167013\pi$
$350$	0	0
$351$	22.4157	1.19646
$352$	0	0
$353$	32.0690	1.70686	0.853429	−	0.521209i	$-0.174519\pi$
$353$	32.0690	1.70686	0.853429	+	0.521209i	$0.174519\pi$
$354$	0	0
$355$	−3.41474	−0.181235
$356$	0	0
$357$	8.57623	0.453902
$358$	0	0
$359$	19.5366	1.03110	0.515551	−	0.856859i	$-0.327587\pi$
$359$	19.5366	1.03110	0.515551	+	0.856859i	$0.327587\pi$
$360$	0	0
$361$	7.36581	0.387674
$362$	0	0
$363$	25.9431	1.36166
$364$	0	0
$365$	−1.61811	−0.0846958
$366$	0	0
$367$	13.9347	0.727387	0.363693	−	0.931519i	$-0.381516\pi$
$367$	13.9347	0.727387	0.363693	+	0.931519i	$0.381516\pi$
$368$	0	0
$369$	−25.6704	−1.33635
$370$	0	0
$371$	5.82459	0.302398
$372$	0	0
$373$	−26.2518	−1.35927	−0.679633	−	0.733552i	$-0.737861\pi$
$373$	−26.2518	−1.35927	−0.679633	+	0.733552i	$0.737861\pi$
$374$	0	0
$375$	−13.7714	−0.711151
$376$	0	0
$377$	7.94593	0.409236
$378$	0	0
$379$	1.93769	0.0995326	0.0497663	−	0.998761i	$-0.484152\pi$
$379$	1.93769	0.0995326	0.0497663	+	0.998761i	$0.484152\pi$
$380$	0	0
$381$	−15.3765	−0.787764
$382$	0	0
$383$	−14.7271	−0.752521	−0.376260	−	0.926514i	$-0.622790\pi$
$383$	−14.7271	−0.752521	−0.376260	+	0.926514i	$0.622790\pi$
$384$	0	0
$385$	−0.744816	−0.0379593
$386$	0	0
$387$	4.41438	0.224396
$388$	0	0
$389$	−7.33509	−0.371904	−0.185952	−	0.982559i	$-0.559537\pi$
$389$	−7.33509	−0.371904	−0.185952	+	0.982559i	$0.559537\pi$
$390$	0	0
$391$	−15.0701	−0.762130
$392$	0	0
$393$	1.37632	0.0694259
$394$	0	0
$395$	4.50419	0.226631
$396$	0	0
$397$	14.1708	0.711212	0.355606	−	0.934636i	$-0.384275\pi$
$397$	14.1708	0.711212	0.355606	+	0.934636i	$0.384275\pi$
$398$	0	0
$399$	−16.5514	−0.828609
$400$	0	0
$401$	26.2129	1.30901	0.654505	−	0.756058i	$-0.272877\pi$
$401$	26.2129	1.30901	0.654505	+	0.756058i	$0.272877\pi$
$402$	0	0
$403$	−31.7158	−1.57988
$404$	0	0
$405$	1.52787	0.0759205
$406$	0	0
$407$	6.54573	0.324460
$408$	0	0
$409$	37.8361	1.87087	0.935436	−	0.353496i	$-0.115007\pi$
$409$	37.8361	1.87087	0.935436	+	0.353496i	$0.115007\pi$
$410$	0	0
$411$	−27.9113	−1.37676
$412$	0	0
$413$	−5.90270	−0.290453
$414$	0	0
$415$	−7.83879	−0.384791
$416$	0	0
$417$	28.4176	1.39162
$418$	0	0
$419$	15.5825	0.761255	0.380628	−	0.924728i	$-0.375708\pi$
$419$	15.5825	0.761255	0.380628	+	0.924728i	$0.375708\pi$
$420$	0	0
$421$	−2.19838	−0.107143	−0.0535713	−	0.998564i	$-0.517060\pi$
$421$	−2.19838	−0.107143	−0.0535713	+	0.998564i	$0.517060\pi$
$422$	0	0
$423$	−30.2540	−1.47100
$424$	0	0
$425$	−12.5810	−0.610270
$426$	0	0
$427$	17.5610	0.849838
$428$	0	0
$429$	−19.5224	−0.942549
$430$	0	0
$431$	21.8900	1.05440	0.527202	−	0.849740i	$-0.323241\pi$
$431$	21.8900	1.05440	0.527202	+	0.849740i	$0.323241\pi$
$432$	0	0
$433$	13.2625	0.637354	0.318677	−	0.947863i	$-0.396761\pi$
$433$	13.2625	0.637354	0.318677	+	0.947863i	$0.396761\pi$
$434$	0	0
$435$	−1.88702	−0.0904758
$436$	0	0
$437$	29.0842	1.39128
$438$	0	0
$439$	−1.95699	−0.0934021	−0.0467010	−	0.998909i	$-0.514871\pi$
$439$	−1.95699	−0.0934021	−0.0467010	+	0.998909i	$0.514871\pi$
$440$	0	0
$441$	−24.5186	−1.16755
$442$	0	0
$443$	−1.50201	−0.0713628	−0.0356814	−	0.999363i	$-0.511360\pi$
$443$	−1.50201	−0.0713628	−0.0356814	+	0.999363i	$0.511360\pi$
$444$	0	0
$445$	5.14935	0.244103
$446$	0	0
$447$	41.7449	1.97447
$448$	0	0
$449$	11.2610	0.531439	0.265719	−	0.964050i	$-0.414391\pi$
$449$	11.2610	0.531439	0.265719	+	0.964050i	$0.414391\pi$
$450$	0	0
$451$	7.05770	0.332334
$452$	0	0
$453$	−19.4850	−0.915486
$454$	0	0
$455$	−3.68339	−0.172680
$456$	0	0
$457$	−1.35116	−0.0632047	−0.0316024	−	0.999501i	$-0.510061\pi$
$457$	−1.35116	−0.0632047	−0.0316024	+	0.999501i	$0.510061\pi$
$458$	0	0
$459$	−10.0055	−0.467017
$460$	0	0
$461$	−24.7553	−1.15297	−0.576484	−	0.817109i	$-0.695576\pi$
$461$	−24.7553	−1.15297	−0.576484	+	0.817109i	$0.695576\pi$
$462$	0	0
$463$	0.749799	0.0348461	0.0174231	−	0.999848i	$-0.494454\pi$
$463$	0.749799	0.0348461	0.0174231	+	0.999848i	$0.494454\pi$
$464$	0	0
$465$	7.53195	0.349286
$466$	0	0
$467$	5.29981	0.245246	0.122623	−	0.992453i	$-0.460869\pi$
$467$	5.29981	0.245246	0.122623	+	0.992453i	$0.460869\pi$
$468$	0	0
$469$	−4.45786	−0.205845
$470$	0	0
$471$	21.4152	0.986762
$472$	0	0
$473$	−1.21367	−0.0558047
$474$	0	0
$475$	24.2804	1.11406
$476$	0	0
$477$	−21.5257	−0.985594
$478$	0	0
$479$	1.36716	0.0624672	0.0312336	−	0.999512i	$-0.490056\pi$
$479$	1.36716	0.0624672	0.0312336	+	0.999512i	$0.490056\pi$
$480$	0	0
$481$	32.3711	1.47599
$482$	0	0
$483$	−18.2579	−0.830765
$484$	0	0
$485$	−4.17334	−0.189502
$486$	0	0
$487$	6.21827	0.281777	0.140888	−	0.990025i	$-0.455004\pi$
$487$	6.21827	0.281777	0.140888	+	0.990025i	$0.455004\pi$
$488$	0	0
$489$	57.4230	2.59676
$490$	0	0
$491$	38.3658	1.73142	0.865712	−	0.500543i	$-0.166866\pi$
$491$	38.3658	1.73142	0.865712	+	0.500543i	$0.166866\pi$
$492$	0	0
$493$	−3.54676	−0.159738
$494$	0	0
$495$	2.75259	0.123720
$496$	0	0
$497$	−7.77583	−0.348794
$498$	0	0
$499$	12.0874	0.541108	0.270554	−	0.962705i	$-0.412793\pi$
$499$	12.0874	0.541108	0.270554	+	0.962705i	$0.412793\pi$
$500$	0	0
$501$	−12.0234	−0.537167
$502$	0	0
$503$	−13.2379	−0.590250	−0.295125	−	0.955459i	$-0.595361\pi$
$503$	−13.2379	−0.590250	−0.295125	+	0.955459i	$0.595361\pi$
$504$	0	0
$505$	10.1763	0.452838
$506$	0	0
$507$	−61.2200	−2.71887
$508$	0	0
$509$	−15.8225	−0.701319	−0.350659	−	0.936503i	$-0.614042\pi$
$509$	−15.8225	−0.701319	−0.350659	+	0.936503i	$0.614042\pi$
$510$	0	0
$511$	−3.68466	−0.163000
$512$	0	0
$513$	19.3098	0.852550
$514$	0	0
$515$	7.78210	0.342921
$516$	0	0
$517$	8.31790	0.365821
$518$	0	0
$519$	−59.4683	−2.61037
$520$	0	0
$521$	−9.09248	−0.398349	−0.199174	−	0.979964i	$-0.563826\pi$
$521$	−9.09248	−0.398349	−0.199174	+	0.979964i	$0.563826\pi$
$522$	0	0
$523$	−5.01602	−0.219335	−0.109668	−	0.993968i	$-0.534979\pi$
$523$	−5.01602	−0.219335	−0.109668	+	0.993968i	$0.534979\pi$
$524$	0	0
$525$	−15.2423	−0.665229
$526$	0	0
$527$	14.1567	0.616676
$528$	0	0
$529$	9.08281	0.394905
$530$	0	0
$531$	21.8144	0.946664
$532$	0	0
$533$	34.9029	1.51181
$534$	0	0
$535$	3.98806	0.172419
$536$	0	0
$537$	−34.6437	−1.49499
$538$	0	0
$539$	6.74105	0.290357
$540$	0	0
$541$	−10.8297	−0.465604	−0.232802	−	0.972524i	$-0.574789\pi$
$541$	−10.8297	−0.465604	−0.232802	+	0.972524i	$0.574789\pi$
$542$	0	0
$543$	46.8370	2.00997
$544$	0	0
$545$	2.05931	0.0882113
$546$	0	0
$547$	17.0443	0.728761	0.364381	−	0.931250i	$-0.381281\pi$
$547$	17.0443	0.728761	0.364381	+	0.931250i	$0.381281\pi$
$548$	0	0
$549$	−64.8996	−2.76985
$550$	0	0
$551$	6.84497	0.291606
$552$	0	0
$553$	10.2567	0.436158
$554$	0	0
$555$	−7.68758	−0.326319
$556$	0	0
$557$	42.0447	1.78149	0.890746	−	0.454502i	$-0.150183\pi$
$557$	42.0447	1.78149	0.890746	+	0.454502i	$0.150183\pi$
$558$	0	0
$559$	−6.00206	−0.253860
$560$	0	0
$561$	8.71405	0.367907
$562$	0	0
$563$	16.6580	0.702052	0.351026	−	0.936366i	$-0.385833\pi$
$563$	16.6580	0.702052	0.351026	+	0.936366i	$0.385833\pi$
$564$	0	0
$565$	3.52539	0.148314
$566$	0	0
$567$	3.47917	0.146112
$568$	0	0
$569$	5.87687	0.246371	0.123186	−	0.992384i	$-0.460689\pi$
$569$	5.87687	0.246371	0.123186	+	0.992384i	$0.460689\pi$
$570$	0	0
$571$	−19.6723	−0.823260	−0.411630	−	0.911351i	$-0.635040\pi$
$571$	−19.6723	−0.823260	−0.411630	+	0.911351i	$0.635040\pi$
$572$	0	0
$573$	34.0285	1.42156
$574$	0	0
$575$	26.7838	1.11696
$576$	0	0
$577$	−10.2886	−0.428319	−0.214160	−	0.976799i	$-0.568701\pi$
$577$	−10.2886	−0.428319	−0.214160	+	0.976799i	$0.568701\pi$
$578$	0	0
$579$	−14.5024	−0.602697
$580$	0	0
$581$	−17.8500	−0.740543
$582$	0	0
$583$	5.91818	0.245106
$584$	0	0
$585$	13.6126	0.562810
$586$	0	0
$587$	−9.38465	−0.387346	−0.193673	−	0.981066i	$-0.562040\pi$
$587$	−9.38465	−0.387346	−0.193673	+	0.981066i	$0.562040\pi$
$588$	0	0
$589$	−27.3213	−1.12576
$590$	0	0
$591$	66.4659	2.73404
$592$	0	0
$593$	−34.6714	−1.42378	−0.711891	−	0.702290i	$-0.752161\pi$
$593$	−34.6714	−1.42378	−0.711891	+	0.702290i	$0.752161\pi$
$594$	0	0
$595$	1.64413	0.0674026
$596$	0	0
$597$	55.0998	2.25508
$598$	0	0
$599$	29.9382	1.22324	0.611621	−	0.791151i	$-0.290518\pi$
$599$	29.9382	1.22324	0.611621	+	0.791151i	$0.290518\pi$
$600$	0	0
$601$	11.3561	0.463227	0.231613	−	0.972808i	$-0.425600\pi$
$601$	11.3561	0.463227	0.231613	+	0.972808i	$0.425600\pi$
$602$	0	0
$603$	16.4747	0.670904
$604$	0	0
$605$	4.97348	0.202201
$606$	0	0
$607$	23.4174	0.950483	0.475242	−	0.879855i	$-0.342361\pi$
$607$	23.4174	0.950483	0.475242	+	0.879855i	$0.342361\pi$
$608$	0	0
$609$	−4.29701	−0.174124
$610$	0	0
$611$	41.1351	1.66415
$612$	0	0
$613$	20.3716	0.822800	0.411400	−	0.911455i	$-0.365040\pi$
$613$	20.3716	0.822800	0.411400	+	0.911455i	$0.365040\pi$
$614$	0	0
$615$	−8.28885	−0.334239
$616$	0	0
$617$	−35.8831	−1.44460	−0.722299	−	0.691581i	$-0.756914\pi$
$617$	−35.8831	−1.44460	−0.722299	+	0.691581i	$0.756914\pi$
$618$	0	0
$619$	−15.8012	−0.635103	−0.317552	−	0.948241i	$-0.602861\pi$
$619$	−15.8012	−0.635103	−0.317552	+	0.948241i	$0.602861\pi$
$620$	0	0
$621$	21.3007	0.854768
$622$	0	0
$623$	11.7258	0.469784
$624$	0	0
$625$	21.0031	0.840123
$626$	0	0
$627$	−16.8174	−0.671623
$628$	0	0
$629$	−14.4492	−0.576128
$630$	0	0
$631$	−29.3464	−1.16826	−0.584131	−	0.811660i	$-0.698564\pi$
$631$	−29.3464	−1.16826	−0.584131	+	0.811660i	$0.698564\pi$
$632$	0	0
$633$	18.8156	0.747855
$634$	0	0
$635$	−2.94779	−0.116980
$636$	0	0
$637$	33.3370	1.32086
$638$	0	0
$639$	28.7368	1.13681
$640$	0	0
$641$	20.6129	0.814162	0.407081	−	0.913392i	$-0.366547\pi$
$641$	20.6129	0.814162	0.407081	+	0.913392i	$0.366547\pi$
$642$	0	0
$643$	28.4297	1.12116	0.560579	−	0.828101i	$-0.310579\pi$
$643$	28.4297	1.12116	0.560579	+	0.828101i	$0.310579\pi$
$644$	0	0
$645$	1.42539	0.0561245
$646$	0	0
$647$	−36.9860	−1.45407	−0.727036	−	0.686600i	$-0.759102\pi$
$647$	−36.9860	−1.45407	−0.727036	+	0.686600i	$0.759102\pi$
$648$	0	0
$649$	−5.99756	−0.235425
$650$	0	0
$651$	17.1513	0.672212
$652$	0	0
$653$	39.6840	1.55296	0.776478	−	0.630145i	$-0.217005\pi$
$653$	39.6840	1.55296	0.776478	+	0.630145i	$0.217005\pi$
$654$	0	0
$655$	0.263850	0.0103095
$656$	0	0
$657$	13.6173	0.531260
$658$	0	0
$659$	23.8385	0.928614	0.464307	−	0.885674i	$-0.346303\pi$
$659$	23.8385	0.928614	0.464307	+	0.885674i	$0.346303\pi$
$660$	0	0
$661$	−18.9449	−0.736869	−0.368435	−	0.929654i	$-0.620106\pi$
$661$	−18.9449	−0.736869	−0.368435	+	0.929654i	$0.620106\pi$
$662$	0	0
$663$	43.0942	1.67364
$664$	0	0
$665$	−3.17303	−0.123045
$666$	0	0
$667$	7.55070	0.292364
$668$	0	0
$669$	−50.0692	−1.93579
$670$	0	0
$671$	17.8432	0.688830
$672$	0	0
$673$	−34.6728	−1.33654	−0.668269	−	0.743920i	$-0.732964\pi$
$673$	−34.6728	−1.33654	−0.668269	+	0.743920i	$0.732964\pi$
$674$	0	0
$675$	17.7825	0.684449
$676$	0	0
$677$	−35.3683	−1.35931	−0.679657	−	0.733530i	$-0.737872\pi$
$677$	−35.3683	−1.35931	−0.679657	+	0.733530i	$0.737872\pi$
$678$	0	0
$679$	−9.50328	−0.364702
$680$	0	0
$681$	−35.5504	−1.36230
$682$	0	0
$683$	−16.6095	−0.635545	−0.317773	−	0.948167i	$-0.602935\pi$
$683$	−16.6095	−0.635545	−0.317773	+	0.948167i	$0.602935\pi$
$684$	0	0
$685$	−5.35080	−0.204444
$686$	0	0
$687$	4.00513	0.152805
$688$	0	0
$689$	29.2676	1.11501
$690$	0	0
$691$	27.7841	1.05696	0.528479	−	0.848947i	$-0.322763\pi$
$691$	27.7841	1.05696	0.528479	+	0.848947i	$0.322763\pi$
$692$	0	0
$693$	6.26802	0.238102
$694$	0	0
$695$	5.44786	0.206649
$696$	0	0
$697$	−15.5793	−0.590110
$698$	0	0
$699$	−37.6349	−1.42348
$700$	0	0
$701$	−35.4076	−1.33733	−0.668663	−	0.743566i	$-0.733133\pi$
$701$	−35.4076	−1.33733	−0.668663	+	0.743566i	$0.733133\pi$
$702$	0	0
$703$	27.8859	1.05174
$704$	0	0
$705$	−9.76887	−0.367917
$706$	0	0
$707$	23.1728	0.871502
$708$	0	0
$709$	−9.58839	−0.360099	−0.180050	−	0.983658i	$-0.557626\pi$
$709$	−9.58839	−0.360099	−0.180050	+	0.983658i	$0.557626\pi$
$710$	0	0
$711$	−37.9052	−1.42155
$712$	0	0
$713$	−30.1382	−1.12869
$714$	0	0
$715$	−3.74258	−0.139965
$716$	0	0
$717$	74.5611	2.78453
$718$	0	0
$719$	44.8107	1.67116	0.835578	−	0.549372i	$-0.185133\pi$
$719$	44.8107	1.67116	0.835578	+	0.549372i	$0.185133\pi$
$720$	0	0
$721$	17.7209	0.659962
$722$	0	0
$723$	63.2326	2.35164
$724$	0	0
$725$	6.30356	0.234108
$726$	0	0
$727$	−30.9752	−1.14881	−0.574404	−	0.818572i	$-0.694766\pi$
$727$	−30.9752	−1.14881	−0.574404	+	0.818572i	$0.694766\pi$
$728$	0	0
$729$	−43.5144	−1.61164
$730$	0	0
$731$	2.67909	0.0990898
$732$	0	0
$733$	37.1306	1.37145	0.685724	−	0.727861i	$-0.259486\pi$
$733$	37.1306	1.37145	0.685724	+	0.727861i	$0.259486\pi$
$734$	0	0
$735$	−7.91696	−0.292021
$736$	0	0
$737$	−4.52950	−0.166846
$738$	0	0
$739$	33.0311	1.21507	0.607534	−	0.794294i	$-0.292159\pi$
$739$	33.0311	1.21507	0.607534	+	0.794294i	$0.292159\pi$
$740$	0	0
$741$	−83.1684	−3.05527
$742$	0	0
$743$	−9.11379	−0.334353	−0.167176	−	0.985927i	$-0.553465\pi$
$743$	−9.11379	−0.334353	−0.167176	+	0.985927i	$0.553465\pi$
$744$	0	0
$745$	8.00280	0.293200
$746$	0	0
$747$	65.9675	2.41363
$748$	0	0
$749$	9.08136	0.331826
$750$	0	0
$751$	15.9971	0.583742	0.291871	−	0.956458i	$-0.405722\pi$
$751$	15.9971	0.583742	0.291871	+	0.956458i	$0.405722\pi$
$752$	0	0
$753$	45.4000	1.65447
$754$	0	0
$755$	−3.73542	−0.135946
$756$	0	0
$757$	−41.4562	−1.50675	−0.753375	−	0.657591i	$-0.771576\pi$
$757$	−41.4562	−1.50675	−0.753375	+	0.657591i	$0.771576\pi$
$758$	0	0
$759$	−18.5513	−0.673370
$760$	0	0
$761$	−17.9481	−0.650618	−0.325309	−	0.945608i	$-0.605468\pi$
$761$	−17.9481	−0.650618	−0.325309	+	0.945608i	$0.605468\pi$
$762$	0	0
$763$	4.68934	0.169766
$764$	0	0
$765$	−6.07613	−0.219683
$766$	0	0
$767$	−29.6601	−1.07097
$768$	0	0
$769$	39.9631	1.44111	0.720554	−	0.693399i	$-0.243888\pi$
$769$	39.9631	1.44111	0.720554	+	0.693399i	$0.243888\pi$
$770$	0	0
$771$	−32.0627	−1.15471
$772$	0	0
$773$	−31.0898	−1.11822	−0.559111	−	0.829093i	$-0.688857\pi$
$773$	−31.0898	−1.11822	−0.559111	+	0.829093i	$0.688857\pi$
$774$	0	0
$775$	−25.1604	−0.903787
$776$	0	0
$777$	−17.5057	−0.628013
$778$	0	0
$779$	30.0669	1.07726
$780$	0	0
$781$	−7.90078	−0.282712
$782$	0	0
$783$	5.01313	0.179155
$784$	0	0
$785$	4.10545	0.146530
$786$	0	0
$787$	43.6435	1.55572	0.777862	−	0.628436i	$-0.216304\pi$
$787$	43.6435	1.55572	0.777862	+	0.628436i	$0.216304\pi$
$788$	0	0
$789$	−44.1199	−1.57071
$790$	0	0
$791$	8.02781	0.285436
$792$	0	0
$793$	88.2414	3.13354
$794$	0	0
$795$	−6.95056	−0.246511
$796$	0	0
$797$	−7.67501	−0.271863	−0.135931	−	0.990718i	$-0.543403\pi$
$797$	−7.67501	−0.271863	−0.135931	+	0.990718i	$0.543403\pi$
$798$	0	0
$799$	−18.3611	−0.649570
$800$	0	0
$801$	−43.3345	−1.53115
$802$	0	0
$803$	−3.74387	−0.132118
$804$	0	0
$805$	−3.50018	−0.123365
$806$	0	0
$807$	17.5021	0.616103
$808$	0	0
$809$	13.2836	0.467028	0.233514	−	0.972353i	$-0.424978\pi$
$809$	13.2836	0.467028	0.233514	+	0.972353i	$0.424978\pi$
$810$	0	0
$811$	−36.4400	−1.27958	−0.639791	−	0.768549i	$-0.720979\pi$
$811$	−36.4400	−1.27958	−0.639791	+	0.768549i	$0.720979\pi$
$812$	0	0
$813$	−42.9061	−1.50478
$814$	0	0
$815$	11.0084	0.385607
$816$	0	0
$817$	−5.17043	−0.180891
$818$	0	0
$819$	30.9977	1.08315
$820$	0	0
$821$	−24.1977	−0.844507	−0.422253	−	0.906478i	$-0.638761\pi$
$821$	−24.1977	−0.844507	−0.422253	+	0.906478i	$0.638761\pi$
$822$	0	0
$823$	−38.5625	−1.34420	−0.672101	−	0.740459i	$-0.734608\pi$
$823$	−38.5625	−1.34420	−0.672101	+	0.740459i	$0.734608\pi$
$824$	0	0
$825$	−15.4872	−0.539196
$826$	0	0
$827$	−32.9458	−1.14564	−0.572819	−	0.819682i	$-0.694150\pi$
$827$	−32.9458	−1.14564	−0.572819	+	0.819682i	$0.694150\pi$
$828$	0	0
$829$	−13.9315	−0.483860	−0.241930	−	0.970294i	$-0.577780\pi$
$829$	−13.9315	−0.483860	−0.241930	+	0.970294i	$0.577780\pi$
$830$	0	0
$831$	41.7689	1.44895
$832$	0	0
$833$	−14.8804	−0.515574
$834$	0	0
$835$	−2.30498	−0.0797670
$836$	0	0
$837$	−20.0097	−0.691635
$838$	0	0
$839$	−32.8187	−1.13303	−0.566514	−	0.824052i	$-0.691708\pi$
$839$	−32.8187	−1.13303	−0.566514	+	0.824052i	$0.691708\pi$
$840$	0	0
$841$	−27.2229	−0.938722
$842$	0	0
$843$	48.8257	1.68165
$844$	0	0
$845$	−11.7363	−0.403741
$846$	0	0
$847$	11.3253	0.389142
$848$	0	0
$849$	31.2264	1.07169
$850$	0	0
$851$	30.7609	1.05447
$852$	0	0
$853$	38.0226	1.30187	0.650934	−	0.759135i	$-0.274378\pi$
$853$	38.0226	1.30187	0.650934	+	0.759135i	$0.274378\pi$
$854$	0	0
$855$	11.7264	0.401036
$856$	0	0
$857$	−43.5232	−1.48672	−0.743362	−	0.668890i	$-0.766770\pi$
$857$	−43.5232	−1.48672	−0.743362	+	0.668890i	$0.766770\pi$
$858$	0	0
$859$	28.8837	0.985498	0.492749	−	0.870171i	$-0.335992\pi$
$859$	28.8837	0.985498	0.492749	+	0.870171i	$0.335992\pi$
$860$	0	0
$861$	−18.8749	−0.643253
$862$	0	0
$863$	−17.1354	−0.583296	−0.291648	−	0.956526i	$-0.594204\pi$
$863$	−17.1354	−0.583296	−0.291648	+	0.956526i	$0.594204\pi$
$864$	0	0
$865$	−11.4005	−0.387628
$866$	0	0
$867$	26.9592	0.915581
$868$	0	0
$869$	10.4215	0.353525
$870$	0	0
$871$	−22.4000	−0.758997
$872$	0	0
$873$	35.1209	1.18866
$874$	0	0
$875$	−6.01181	−0.203236
$876$	0	0
$877$	52.3846	1.76890	0.884450	−	0.466634i	$-0.154534\pi$
$877$	52.3846	1.76890	0.884450	+	0.466634i	$0.154534\pi$
$878$	0	0
$879$	−79.6213	−2.68556
$880$	0	0
$881$	−18.2737	−0.615658	−0.307829	−	0.951442i	$-0.599602\pi$
$881$	−18.2737	−0.615658	−0.307829	+	0.951442i	$0.599602\pi$
$882$	0	0
$883$	−47.5701	−1.60086	−0.800430	−	0.599426i	$-0.795396\pi$
$883$	−47.5701	−1.60086	−0.800430	+	0.599426i	$0.795396\pi$
$884$	0	0
$885$	7.04377	0.236774
$886$	0	0
$887$	−33.7056	−1.13172	−0.565861	−	0.824501i	$-0.691456\pi$
$887$	−33.7056	−1.13172	−0.565861	+	0.824501i	$0.691456\pi$
$888$	0	0
$889$	−6.71253	−0.225131
$890$	0	0
$891$	3.53508	0.118430
$892$	0	0
$893$	35.4355	1.18581
$894$	0	0
$895$	−6.64146	−0.221999
$896$	0	0
$897$	−91.7432	−3.06322
$898$	0	0
$899$	−7.09304	−0.236566
$900$	0	0
$901$	−13.0640	−0.435223
$902$	0	0
$903$	3.24580	0.108014
$904$	0	0
$905$	8.97899	0.298472
$906$	0	0
$907$	49.9229	1.65766	0.828831	−	0.559499i	$-0.189006\pi$
$907$	49.9229	1.65766	0.828831	+	0.559499i	$0.189006\pi$
$908$	0	0
$909$	−85.6387	−2.84046
$910$	0	0
$911$	−45.6377	−1.51204	−0.756022	−	0.654546i	$-0.772860\pi$
$911$	−45.6377	−1.51204	−0.756022	+	0.654546i	$0.772860\pi$
$912$	0	0
$913$	−18.1368	−0.600242
$914$	0	0
$915$	−20.9558	−0.692778
$916$	0	0
$917$	0.600822	0.0198409
$918$	0	0
$919$	7.60383	0.250827	0.125414	−	0.992105i	$-0.459974\pi$
$919$	7.60383	0.250827	0.125414	+	0.992105i	$0.459974\pi$
$920$	0	0
$921$	74.8404	2.46607
$922$	0	0
$923$	−39.0723	−1.28608
$924$	0	0
$925$	25.6802	0.844360
$926$	0	0
$927$	−65.4905	−2.15099
$928$	0	0
$929$	−46.1491	−1.51410	−0.757052	−	0.653355i	$-0.773361\pi$
$929$	−46.1491	−1.51410	−0.757052	+	0.653355i	$0.773361\pi$
$930$	0	0
$931$	28.7179	0.941192
$932$	0	0
$933$	−58.4424	−1.91332
$934$	0	0
$935$	1.67055	0.0546327
$936$	0	0
$937$	16.2631	0.531292	0.265646	−	0.964071i	$-0.414415\pi$
$937$	16.2631	0.531292	0.265646	+	0.964071i	$0.414415\pi$
$938$	0	0
$939$	−67.0837	−2.18919
$940$	0	0
$941$	−36.6074	−1.19337	−0.596683	−	0.802477i	$-0.703515\pi$
$941$	−36.6074	−1.19337	−0.596683	+	0.802477i	$0.703515\pi$
$942$	0	0
$943$	33.1669	1.08006
$944$	0	0
$945$	−2.32387	−0.0755955
$946$	0	0
$947$	2.26356	0.0735560	0.0367780	−	0.999323i	$-0.488291\pi$
$947$	2.26356	0.0735560	0.0367780	+	0.999323i	$0.488291\pi$
$948$	0	0
$949$	−18.5148	−0.601017
$950$	0	0
$951$	60.1086	1.94916
$952$	0	0
$953$	−11.5415	−0.373867	−0.186933	−	0.982373i	$-0.559855\pi$
$953$	−11.5415	−0.373867	−0.186933	+	0.982373i	$0.559855\pi$
$954$	0	0
$955$	6.52351	0.211096
$956$	0	0
$957$	−4.36606	−0.141135
$958$	0	0
$959$	−12.1845	−0.393459
$960$	0	0
$961$	−2.68848	−0.0867253
$962$	0	0
$963$	−33.5616	−1.08151
$964$	0	0
$965$	−2.78021	−0.0894980
$966$	0	0
$967$	17.1288	0.550824	0.275412	−	0.961326i	$-0.411186\pi$
$967$	17.1288	0.550824	0.275412	+	0.961326i	$0.411186\pi$
$968$	0	0
$969$	37.1232	1.19257
$970$	0	0
$971$	26.7415	0.858177	0.429089	−	0.903262i	$-0.358835\pi$
$971$	26.7415	0.858177	0.429089	+	0.903262i	$0.358835\pi$
$972$	0	0
$973$	12.4055	0.397703
$974$	0	0
$975$	−76.5901	−2.45285
$976$	0	0
$977$	49.3487	1.57880	0.789402	−	0.613877i	$-0.210391\pi$
$977$	49.3487	1.57880	0.789402	+	0.613877i	$0.210391\pi$
$978$	0	0
$979$	11.9142	0.380780
$980$	0	0
$981$	−17.3302	−0.553311
$982$	0	0
$983$	6.62409	0.211276	0.105638	−	0.994405i	$-0.466312\pi$
$983$	6.62409	0.211276	0.105638	+	0.994405i	$0.466312\pi$
$984$	0	0
$985$	12.7420	0.405994
$986$	0	0
$987$	−22.2451	−0.708069
$988$	0	0
$989$	−5.70352	−0.181361
$990$	0	0
$991$	40.4721	1.28564	0.642820	−	0.766018i	$-0.277764\pi$
$991$	40.4721	1.28564	0.642820	+	0.766018i	$0.277764\pi$
$992$	0	0
$993$	32.2691	1.02403
$994$	0	0
$995$	10.5630	0.334870
$996$	0	0
$997$	−6.02480	−0.190807	−0.0954036	−	0.995439i	$-0.530414\pi$
$997$	−6.02480	−0.190807	−0.0954036	+	0.995439i	$0.530414\pi$
$998$	0	0
$999$	20.4231	0.646158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8012.2.a.b.1.8	✓	88

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8012.2.a.b.1.8	✓	88	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8012 = 2^{2} \cdot 2003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8012.a (trivial)

\(f(q)\)	\(=\)	\(q-2.71734 q^{3} -0.520933 q^{5} -1.18624 q^{7} +4.38393 q^{9} +O(q^{10})\)	\(q-2.71734 q^{3} -0.520933 q^{5} -1.18624 q^{7} +4.38393 q^{9} -1.20530 q^{11} -5.96065 q^{13} +1.41555 q^{15} +2.66061 q^{17} -5.13476 q^{19} +3.22341 q^{21} -5.66417 q^{23} -4.72863 q^{25} -3.76061 q^{27} -1.33306 q^{29} +5.32086 q^{31} +3.27521 q^{33} +0.617951 q^{35} -5.43080 q^{37} +16.1971 q^{39} -5.85556 q^{41} +1.00695 q^{43} -2.28374 q^{45} -6.90110 q^{47} -5.59284 q^{49} -7.22978 q^{51} -4.91014 q^{53} +0.627881 q^{55} +13.9529 q^{57} +4.97599 q^{59} -14.8040 q^{61} -5.20038 q^{63} +3.10510 q^{65} +3.75799 q^{67} +15.3915 q^{69} +6.55504 q^{71} +3.10618 q^{73} +12.8493 q^{75} +1.42977 q^{77} -8.64639 q^{79} -2.93295 q^{81} +15.0476 q^{83} -1.38600 q^{85} +3.62239 q^{87} -9.88486 q^{89} +7.07075 q^{91} -14.4586 q^{93} +2.67487 q^{95} +8.01128 q^{97} -5.28395 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 19 q^{3} + 44 q^{7} + 99 q^{9}+O(q^{10}) \) 88 * q + 19 * q^3 + 44 * q^7 + 99 * q^9	\( 88 q + 19 q^{3} + 44 q^{7} + 99 q^{9} + 16 q^{11} - 3 q^{13} + 21 q^{15} + 32 q^{17} + 49 q^{19} + 7 q^{21} + 24 q^{23} + 114 q^{25} + 82 q^{27} + 2 q^{29} + 40 q^{31} + 31 q^{33} + 26 q^{35} + 37 q^{39} + 22 q^{41} + 98 q^{43} + 12 q^{45} + 48 q^{47} + 132 q^{49} + 55 q^{51} - q^{53} + 116 q^{55} + 62 q^{57} + 34 q^{59} + 132 q^{63} + 39 q^{65} + 75 q^{67} + 15 q^{69} + 24 q^{71} + 104 q^{73} + 87 q^{75} + 4 q^{77} + 111 q^{79} + 128 q^{81} + 64 q^{83} + 7 q^{85} + 115 q^{87} + 14 q^{89} + 73 q^{91} - 7 q^{93} + 51 q^{95} + 117 q^{97} + 72 q^{99}+O(q^{100}) \) 88 * q + 19 * q^3 + 44 * q^7 + 99 * q^9 + 16 * q^11 - 3 * q^13 + 21 * q^15 + 32 * q^17 + 49 * q^19 + 7 * q^21 + 24 * q^23 + 114 * q^25 + 82 * q^27 + 2 * q^29 + 40 * q^31 + 31 * q^33 + 26 * q^35 + 37 * q^39 + 22 * q^41 + 98 * q^43 + 12 * q^45 + 48 * q^47 + 132 * q^49 + 55 * q^51 - q^53 + 116 * q^55 + 62 * q^57 + 34 * q^59 + 132 * q^63 + 39 * q^65 + 75 * q^67 + 15 * q^69 + 24 * q^71 + 104 * q^73 + 87 * q^75 + 4 * q^77 + 111 * q^79 + 128 * q^81 + 64 * q^83 + 7 * q^85 + 115 * q^87 + 14 * q^89 + 73 * q^91 - 7 * q^93 + 51 * q^95 + 117 * q^97 + 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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