LMFDB - Embedded newform 4000.2.a.p.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4000,2,Mod(1,4000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4000, 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("4000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$4000 = 2^{5} \cdot 5^{3}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4000.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:	$31.9401608085$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.26208800000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - 11x^{6} + 34x^{4} - 30x^{2} + 5$ x^8 - 11x^6 + 34x^4 - 30*x^2 + 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$2^{4}$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.53025$ of defining polynomial
Character	$\chi$	$=$	4000.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.420293 q^{3} -2.86781 q^{7} -2.82335 q^{9} -3.96815 q^{11} +3.36296 q^{13} -4.80435 q^{17} -7.66856 q^{19} +1.20532 q^{21} -8.44782 q^{23} +2.44752 q^{27} +4.18632 q^{29} +6.15286 q^{31} +1.66779 q^{33} -1.44139 q^{37} -1.41343 q^{39} +1.77568 q^{41} +7.76777 q^{43} +11.0608 q^{47} +1.22432 q^{49} +2.01923 q^{51} +10.4721 q^{53} +3.22304 q^{57} +7.19188 q^{59} -14.7546 q^{61} +8.09684 q^{63} -9.39256 q^{67} +3.55056 q^{69} -12.5735 q^{71} +16.7179 q^{73} +11.3799 q^{77} -6.42060 q^{79} +7.44139 q^{81} +4.07230 q^{83} -1.75948 q^{87} -0.854102 q^{89} -9.64433 q^{91} -2.58600 q^{93} +5.04768 q^{97} +11.2035 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$8 q + 8 q^{9} + 8 q^{13} + 20 q^{17} - 12 q^{21} - 16 q^{29} + 36 q^{33} + 28 q^{37} + 8 q^{41} + 16 q^{49} + 48 q^{53} + 20 q^{57} - 28 q^{61} - 28 q^{69} + 44 q^{77} + 20 q^{81} + 20 q^{89} + 4 q^{93}+ \cdots + 16 q^{97}+O(q^{100})$ 8 * q + 8 * q^9 + 8 * q^13 + 20 * q^17 - 12 * q^21 - 16 * q^29 + 36 * q^33 + 28 * q^37 + 8 * q^41 + 16 * q^49 + 48 * q^53 + 20 * q^57 - 28 * q^61 - 28 * q^69 + 44 * q^77 + 20 * q^81 + 20 * q^89 + 4 * q^93 + 16 * q^97

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
$3$	−0.420293	−0.242656	−0.121328	−	0.992612i	$-0.538715\pi$
$3$	−0.420293	−0.242656	−0.121328	+	0.992612i	$0.538715\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.86781	−1.08393	−0.541965	−	0.840401i	$-0.682319\pi$
$7$	−2.86781	−1.08393	−0.541965	+	0.840401i	$0.682319\pi$
$8$	0	0
$9$	−2.82335	−0.941118
$10$	0	0
$11$	−3.96815	−1.19644	−0.598221	−	0.801331i	$-0.704126\pi$
$11$	−3.96815	−1.19644	−0.598221	+	0.801331i	$0.704126\pi$
$12$	0	0
$13$	3.36296	0.932718	0.466359	−	0.884596i	$-0.345566\pi$
$13$	3.36296	0.932718	0.466359	+	0.884596i	$0.345566\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.80435	−1.16523	−0.582613	−	0.812750i	$-0.697970\pi$
$17$	−4.80435	−1.16523	−0.582613	+	0.812750i	$0.697970\pi$
$18$	0	0
$19$	−7.66856	−1.75929	−0.879644	−	0.475633i	$-0.842219\pi$
$19$	−7.66856	−1.75929	−0.879644	+	0.475633i	$0.842219\pi$
$20$	0	0
$21$	1.20532	0.263022
$22$	0	0
$23$	−8.44782	−1.76149	−0.880746	−	0.473588i	$-0.842959\pi$
$23$	−8.44782	−1.76149	−0.880746	+	0.473588i	$0.842959\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2.44752	0.471024
$28$	0	0
$29$	4.18632	0.777379	0.388690	−	0.921369i	$-0.372928\pi$
$29$	4.18632	0.777379	0.388690	+	0.921369i	$0.372928\pi$
$30$	0	0
$31$	6.15286	1.10509	0.552543	−	0.833484i	$-0.313658\pi$
$31$	6.15286	1.10509	0.552543	+	0.833484i	$0.313658\pi$
$32$	0	0
$33$	1.66779	0.290324
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.44139	−0.236963	−0.118481	−	0.992956i	$-0.537803\pi$
$37$	−1.44139	−0.236963	−0.118481	+	0.992956i	$0.537803\pi$
$38$	0	0
$39$	−1.41343	−0.226330
$40$	0	0
$41$	1.77568	0.277314	0.138657	−	0.990340i	$-0.455721\pi$
$41$	1.77568	0.277314	0.138657	+	0.990340i	$0.455721\pi$
$42$	0	0
$43$	7.76777	1.18457	0.592287	−	0.805727i	$-0.298225\pi$
$43$	7.76777	1.18457	0.592287	+	0.805727i	$0.298225\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.0608	1.61339	0.806693	−	0.590971i	$-0.201255\pi$
$47$	11.0608	1.61339	0.806693	+	0.590971i	$0.201255\pi$
$48$	0	0
$49$	1.22432	0.174903
$50$	0	0
$51$	2.01923	0.282749
$52$	0	0
$53$	10.4721	1.43846	0.719229	−	0.694773i	$-0.244495\pi$
$53$	10.4721	1.43846	0.719229	+	0.694773i	$0.244495\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.22304	0.426902
$58$	0	0
$59$	7.19188	0.936303	0.468152	−	0.883648i	$-0.344920\pi$
$59$	7.19188	0.936303	0.468152	+	0.883648i	$0.344920\pi$
$60$	0	0
$61$	−14.7546	−1.88913	−0.944566	−	0.328320i	$-0.893517\pi$
$61$	−14.7546	−1.88913	−0.944566	+	0.328320i	$0.893517\pi$
$62$	0	0
$63$	8.09684	1.02011
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−9.39256	−1.14748	−0.573742	−	0.819036i	$-0.694509\pi$
$67$	−9.39256	−1.14748	−0.573742	+	0.819036i	$0.694509\pi$
$68$	0	0
$69$	3.55056	0.427437
$70$	0	0
$71$	−12.5735	−1.49220	−0.746098	−	0.665837i	$-0.768075\pi$
$71$	−12.5735	−1.49220	−0.746098	+	0.665837i	$0.768075\pi$
$72$	0	0
$73$	16.7179	1.95668	0.978340	−	0.207006i	$-0.0663721\pi$
$73$	16.7179	1.95668	0.978340	+	0.207006i	$0.0663721\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	11.3799	1.29686
$78$	0	0
$79$	−6.42060	−0.722374	−0.361187	−	0.932493i	$-0.617628\pi$
$79$	−6.42060	−0.722374	−0.361187	+	0.932493i	$0.617628\pi$
$80$	0	0
$81$	7.44139	0.826821
$82$	0	0
$83$	4.07230	0.446993	0.223497	−	0.974705i	$-0.428253\pi$
$83$	4.07230	0.446993	0.223497	+	0.974705i	$0.428253\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−1.75948	−0.188636
$88$	0	0
$89$	−0.854102	−0.0905346	−0.0452673	−	0.998975i	$-0.514414\pi$
$89$	−0.854102	−0.0905346	−0.0452673	+	0.998975i	$0.514414\pi$
$90$	0	0
$91$	−9.64433	−1.01100
$92$	0	0
$93$	−2.58600	−0.268156
$94$	0	0
$95$	0	0
$96$	0	0
$97$	5.04768	0.512514	0.256257	−	0.966609i	$-0.417511\pi$
$97$	5.04768	0.512514	0.256257	+	0.966609i	$0.417511\pi$
$98$	0	0
$99$	11.2035	1.12599
$100$	0	0
$101$	−2.39499	−0.238311	−0.119155	−	0.992876i	$-0.538019\pi$
$101$	−2.39499	−0.238311	−0.119155	+	0.992876i	$0.538019\pi$
$102$	0	0
$103$	0.417242	0.0411121	0.0205561	−	0.999789i	$-0.493456\pi$
$103$	0.417242	0.0411121	0.0205561	+	0.999789i	$0.493456\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.19382	−0.308758	−0.154379	−	0.988012i	$-0.549338\pi$
$107$	−3.19382	−0.308758	−0.154379	+	0.988012i	$0.549338\pi$
$108$	0	0
$109$	−12.2647	−1.17475	−0.587375	−	0.809315i	$-0.699838\pi$
$109$	−12.2647	−1.17475	−0.587375	+	0.809315i	$0.699838\pi$
$110$	0	0
$111$	0.605805	0.0575005
$112$	0	0
$113$	15.7736	1.48386	0.741928	−	0.670480i	$-0.233912\pi$
$113$	15.7736	1.48386	0.741928	+	0.670480i	$0.233912\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−9.49483	−0.877798
$118$	0	0
$119$	13.7780	1.26302
$120$	0	0
$121$	4.74621	0.431474
$122$	0	0
$123$	−0.746305	−0.0672920
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−10.2202	−0.906898	−0.453449	−	0.891282i	$-0.649807\pi$
$127$	−10.2202	−0.906898	−0.453449	+	0.891282i	$0.649807\pi$
$128$	0	0
$129$	−3.26474	−0.287444
$130$	0	0
$131$	8.10177	0.707855	0.353928	−	0.935273i	$-0.384846\pi$
$131$	8.10177	0.707855	0.353928	+	0.935273i	$0.384846\pi$
$132$	0	0
$133$	21.9919	1.90694
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.3937	1.05887	0.529433	−	0.848352i	$-0.322405\pi$
$137$	12.3937	1.05887	0.529433	+	0.848352i	$0.322405\pi$
$138$	0	0
$139$	10.4910	0.889837	0.444918	−	0.895571i	$-0.353233\pi$
$139$	10.4910	0.889837	0.444918	+	0.895571i	$0.353233\pi$
$140$	0	0
$141$	−4.64878	−0.391498
$142$	0	0
$143$	−13.3447	−1.11594
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−0.514575	−0.0424414
$148$	0	0
$149$	4.15167	0.340118	0.170059	−	0.985434i	$-0.445604\pi$
$149$	4.15167	0.340118	0.170059	+	0.985434i	$0.445604\pi$
$150$	0	0
$151$	8.91651	0.725616	0.362808	−	0.931864i	$-0.381818\pi$
$151$	8.91651	0.725616	0.362808	+	0.931864i	$0.381818\pi$
$152$	0	0
$153$	13.5644	1.09662
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−13.7486	−1.09726	−0.548630	−	0.836065i	$-0.684850\pi$
$157$	−13.7486	−1.09726	−0.548630	+	0.836065i	$0.684850\pi$
$158$	0	0
$159$	−4.40137	−0.349051
$160$	0	0
$161$	24.2267	1.90933
$162$	0	0
$163$	−8.92450	−0.699021	−0.349510	−	0.936932i	$-0.613652\pi$
$163$	−8.92450	−0.699021	−0.349510	+	0.936932i	$0.613652\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.91177	0.302702	0.151351	−	0.988480i	$-0.451638\pi$
$167$	3.91177	0.302702	0.151351	+	0.988480i	$0.451638\pi$
$168$	0	0
$169$	−1.69048	−0.130037
$170$	0	0
$171$	21.6510	1.65570
$172$	0	0
$173$	10.8658	0.826115	0.413058	−	0.910705i	$-0.364461\pi$
$173$	10.8658	0.826115	0.413058	+	0.910705i	$0.364461\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−3.02270	−0.227200
$178$	0	0
$179$	−21.3833	−1.59826	−0.799132	−	0.601156i	$-0.794707\pi$
$179$	−21.3833	−1.59826	−0.799132	+	0.601156i	$0.794707\pi$
$180$	0	0
$181$	−7.93253	−0.589620	−0.294810	−	0.955556i	$-0.595256\pi$
$181$	−7.93253	−0.589620	−0.294810	+	0.955556i	$0.595256\pi$
$182$	0	0
$183$	6.20126	0.458410
$184$	0	0
$185$	0	0
$186$	0	0
$187$	19.0644	1.39413
$188$	0	0
$189$	−7.01900	−0.510557
$190$	0	0
$191$	−15.4662	−1.11910	−0.559549	−	0.828797i	$-0.689026\pi$
$191$	−15.4662	−1.11910	−0.559549	+	0.828797i	$0.689026\pi$
$192$	0	0
$193$	−7.37989	−0.531216	−0.265608	−	0.964081i	$-0.585573\pi$
$193$	−7.37989	−0.531216	−0.265608	+	0.964081i	$0.585573\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.13992	0.579945	0.289973	−	0.957035i	$-0.406354\pi$
$197$	8.13992	0.579945	0.289973	+	0.957035i	$0.406354\pi$
$198$	0	0
$199$	14.5658	1.03254	0.516272	−	0.856424i	$-0.327319\pi$
$199$	14.5658	1.03254	0.516272	+	0.856424i	$0.327319\pi$
$200$	0	0
$201$	3.94763	0.278444
$202$	0	0
$203$	−12.0056	−0.842625
$204$	0	0
$205$	0	0
$206$	0	0
$207$	23.8512	1.65777
$208$	0	0
$209$	30.4300	2.10489
$210$	0	0
$211$	−0.476677	−0.0328158	−0.0164079	−	0.999865i	$-0.505223\pi$
$211$	−0.476677	−0.0328158	−0.0164079	+	0.999865i	$0.505223\pi$
$212$	0	0
$213$	5.28454	0.362091
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−17.6452	−1.19784
$218$	0	0
$219$	−7.02641	−0.474801
$220$	0	0
$221$	−16.1569	−1.08683
$222$	0	0
$223$	11.1141	0.744258	0.372129	−	0.928181i	$-0.378628\pi$
$223$	11.1141	0.744258	0.372129	+	0.928181i	$0.378628\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2.70233	−0.179360	−0.0896801	−	0.995971i	$-0.528584\pi$
$227$	−2.70233	−0.179360	−0.0896801	+	0.995971i	$0.528584\pi$
$228$	0	0
$229$	2.80563	0.185401	0.0927007	−	0.995694i	$-0.470450\pi$
$229$	2.80563	0.185401	0.0927007	+	0.995694i	$0.470450\pi$
$230$	0	0
$231$	−4.78289	−0.314691
$232$	0	0
$233$	22.3322	1.46303	0.731516	−	0.681824i	$-0.238813\pi$
$233$	22.3322	1.46303	0.731516	+	0.681824i	$0.238813\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2.69853	0.175289
$238$	0	0
$239$	2.88567	0.186658	0.0933292	−	0.995635i	$-0.470249\pi$
$239$	2.88567	0.186658	0.0933292	+	0.995635i	$0.470249\pi$
$240$	0	0
$241$	16.7061	1.07614	0.538068	−	0.842901i	$-0.319154\pi$
$241$	16.7061	1.07614	0.538068	+	0.842901i	$0.319154\pi$
$242$	0	0
$243$	−10.4701	−0.671658
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−25.7891	−1.64092
$248$	0	0
$249$	−1.71156	−0.108466
$250$	0	0
$251$	−14.1914	−0.895755	−0.447877	−	0.894095i	$-0.647820\pi$
$251$	−14.1914	−0.895755	−0.447877	+	0.894095i	$0.647820\pi$
$252$	0	0
$253$	33.5222	2.10752
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−12.0808	−0.753582	−0.376791	−	0.926298i	$-0.622972\pi$
$257$	−12.0808	−0.753582	−0.376791	+	0.926298i	$0.622972\pi$
$258$	0	0
$259$	4.13362	0.256851
$260$	0	0
$261$	−11.8195	−0.731606
$262$	0	0
$263$	11.8222	0.728989	0.364495	−	0.931205i	$-0.381242\pi$
$263$	11.8222	0.728989	0.364495	+	0.931205i	$0.381242\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0.358973	0.0219688
$268$	0	0
$269$	−8.19229	−0.499493	−0.249746	−	0.968311i	$-0.580347\pi$
$269$	−8.19229	−0.499493	−0.249746	+	0.968311i	$0.580347\pi$
$270$	0	0
$271$	7.62510	0.463192	0.231596	−	0.972812i	$-0.425605\pi$
$271$	7.62510	0.463192	0.231596	+	0.972812i	$0.425605\pi$
$272$	0	0
$273$	4.05345	0.245326
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−0.550561	−0.0330800	−0.0165400	−	0.999863i	$-0.505265\pi$
$277$	−0.550561	−0.0330800	−0.0165400	+	0.999863i	$0.505265\pi$
$278$	0	0
$279$	−17.3717	−1.04002
$280$	0	0
$281$	−6.84107	−0.408104	−0.204052	−	0.978960i	$-0.565411\pi$
$281$	−6.84107	−0.408104	−0.204052	+	0.978960i	$0.565411\pi$
$282$	0	0
$283$	−24.1942	−1.43820	−0.719098	−	0.694909i	$-0.755445\pi$
$283$	−24.1942	−1.43820	−0.719098	+	0.694909i	$0.755445\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.09230	−0.300589
$288$	0	0
$289$	6.08178	0.357752
$290$	0	0
$291$	−2.12150	−0.124365
$292$	0	0
$293$	0.0273913	0.00160022	0.000800109	−	1.00000i	$-0.499745\pi$
$293$	0.0273913	0.00160022	0.000800109		1.00000i	$0.499745\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−9.71211	−0.563554
$298$	0	0
$299$	−28.4097	−1.64298
$300$	0	0
$301$	−22.2765	−1.28400
$302$	0	0
$303$	1.00660	0.0578276
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−21.1076	−1.20467	−0.602337	−	0.798242i	$-0.705763\pi$
$307$	−21.1076	−1.20467	−0.602337	+	0.798242i	$0.705763\pi$
$308$	0	0
$309$	−0.175364	−0.00997611
$310$	0	0
$311$	8.87305	0.503145	0.251572	−	0.967839i	$-0.419052\pi$
$311$	8.87305	0.503145	0.251572	+	0.967839i	$0.419052\pi$
$312$	0	0
$313$	−20.3346	−1.14938	−0.574691	−	0.818371i	$-0.694878\pi$
$313$	−20.3346	−1.14938	−0.574691	+	0.818371i	$0.694878\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.97502	−0.110928	−0.0554641	−	0.998461i	$-0.517664\pi$
$317$	−1.97502	−0.110928	−0.0554641	+	0.998461i	$0.517664\pi$
$318$	0	0
$319$	−16.6119	−0.930089
$320$	0	0
$321$	1.34234	0.0749221
$322$	0	0
$323$	36.8424	2.04997
$324$	0	0
$325$	0	0
$326$	0	0
$327$	5.15479	0.285060
$328$	0	0
$329$	−31.7203	−1.74880
$330$	0	0
$331$	19.9040	1.09402	0.547010	−	0.837126i	$-0.315766\pi$
$331$	19.9040	1.09402	0.547010	+	0.837126i	$0.315766\pi$
$332$	0	0
$333$	4.06955	0.223010
$334$	0	0
$335$	0	0
$336$	0	0
$337$	21.2684	1.15857	0.579283	−	0.815127i	$-0.303333\pi$
$337$	21.2684	1.15857	0.579283	+	0.815127i	$0.303333\pi$
$338$	0	0
$339$	−6.62954	−0.360067
$340$	0	0
$341$	−24.4155	−1.32217
$342$	0	0
$343$	16.5635	0.894347
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−10.9866	−0.589790	−0.294895	−	0.955530i	$-0.595285\pi$
$347$	−10.9866	−0.589790	−0.294895	+	0.955530i	$0.595285\pi$
$348$	0	0
$349$	−6.62529	−0.354644	−0.177322	−	0.984153i	$-0.556743\pi$
$349$	−6.62529	−0.354644	−0.177322	+	0.984153i	$0.556743\pi$
$350$	0	0
$351$	8.23090	0.439333
$352$	0	0
$353$	30.4026	1.61817	0.809083	−	0.587694i	$-0.199964\pi$
$353$	30.4026	1.61817	0.809083	+	0.587694i	$0.199964\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−5.79078	−0.306480
$358$	0	0
$359$	−9.68779	−0.511302	−0.255651	−	0.966769i	$-0.582290\pi$
$359$	−9.68779	−0.511302	−0.255651	+	0.966769i	$0.582290\pi$
$360$	0	0
$361$	39.8068	2.09509
$362$	0	0
$363$	−1.99480	−0.104700
$364$	0	0
$365$	0	0
$366$	0	0
$367$	17.2186	0.898804	0.449402	−	0.893330i	$-0.351637\pi$
$367$	17.2186	0.898804	0.449402	+	0.893330i	$0.351637\pi$
$368$	0	0
$369$	−5.01336	−0.260985
$370$	0	0
$371$	−30.0321	−1.55919
$372$	0	0
$373$	−10.6894	−0.553477	−0.276738	−	0.960945i	$-0.589254\pi$
$373$	−10.6894	−0.553477	−0.276738	+	0.960945i	$0.589254\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	14.0784	0.725076
$378$	0	0
$379$	33.8980	1.74122	0.870611	−	0.491973i	$-0.163724\pi$
$379$	33.8980	1.74122	0.870611	+	0.491973i	$0.163724\pi$
$380$	0	0
$381$	4.29549	0.220065
$382$	0	0
$383$	20.6524	1.05529	0.527645	−	0.849465i	$-0.323075\pi$
$383$	20.6524	1.05529	0.527645	+	0.849465i	$0.323075\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−21.9312	−1.11482
$388$	0	0
$389$	18.9527	0.960938	0.480469	−	0.877012i	$-0.340466\pi$
$389$	18.9527	0.960938	0.480469	+	0.877012i	$0.340466\pi$
$390$	0	0
$391$	40.5863	2.05254
$392$	0	0
$393$	−3.40512	−0.171766
$394$	0	0
$395$	0	0
$396$	0	0
$397$	15.6976	0.787839	0.393920	−	0.919145i	$-0.371119\pi$
$397$	15.6976	0.787839	0.393920	+	0.919145i	$0.371119\pi$
$398$	0	0
$399$	−9.24306	−0.462732
$400$	0	0
$401$	27.8937	1.39295	0.696473	−	0.717583i	$-0.254752\pi$
$401$	27.8937	1.39295	0.696473	+	0.717583i	$0.254752\pi$
$402$	0	0
$403$	20.6918	1.03073
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5.71964	0.283512
$408$	0	0
$409$	3.12092	0.154319	0.0771597	−	0.997019i	$-0.475415\pi$
$409$	3.12092	0.154319	0.0771597	+	0.997019i	$0.475415\pi$
$410$	0	0
$411$	−5.20899	−0.256941
$412$	0	0
$413$	−20.6249	−1.01489
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−4.40930	−0.215924
$418$	0	0
$419$	33.7913	1.65081	0.825406	−	0.564539i	$-0.190946\pi$
$419$	33.7913	1.65081	0.825406	+	0.564539i	$0.190946\pi$
$420$	0	0
$421$	0.675381	0.0329161	0.0164580	−	0.999865i	$-0.494761\pi$
$421$	0.675381	0.0329161	0.0164580	+	0.999865i	$0.494761\pi$
$422$	0	0
$423$	−31.2286	−1.51839
$424$	0	0
$425$	0	0
$426$	0	0
$427$	42.3134	2.04769
$428$	0	0
$429$	5.60870	0.270791
$430$	0	0
$431$	1.65431	0.0796854	0.0398427	−	0.999206i	$-0.487314\pi$
$431$	1.65431	0.0796854	0.0398427	+	0.999206i	$0.487314\pi$
$432$	0	0
$433$	22.6394	1.08798	0.543991	−	0.839091i	$-0.316912\pi$
$433$	22.6394	1.08798	0.543991	+	0.839091i	$0.316912\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	64.7826	3.09897
$438$	0	0
$439$	−1.41343	−0.0674593	−0.0337297	−	0.999431i	$-0.510739\pi$
$439$	−1.41343	−0.0674593	−0.0337297	+	0.999431i	$0.510739\pi$
$440$	0	0
$441$	−3.45670	−0.164605
$442$	0	0
$443$	2.49780	0.118674	0.0593370	−	0.998238i	$-0.481101\pi$
$443$	2.49780	0.118674	0.0593370	+	0.998238i	$0.481101\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−1.74492	−0.0825317
$448$	0	0
$449$	−27.8464	−1.31415	−0.657076	−	0.753824i	$-0.728207\pi$
$449$	−27.8464	−1.31415	−0.657076	+	0.753824i	$0.728207\pi$
$450$	0	0
$451$	−7.04615	−0.331790
$452$	0	0
$453$	−3.74755	−0.176075
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−26.4721	−1.23831	−0.619157	−	0.785267i	$-0.712526\pi$
$457$	−26.4721	−1.23831	−0.619157	+	0.785267i	$0.712526\pi$
$458$	0	0
$459$	−11.7587	−0.548850
$460$	0	0
$461$	−24.5873	−1.14514	−0.572572	−	0.819854i	$-0.694054\pi$
$461$	−24.5873	−1.14514	−0.572572	+	0.819854i	$0.694054\pi$
$462$	0	0
$463$	−37.4409	−1.74003	−0.870013	−	0.493030i	$-0.835889\pi$
$463$	−37.4409	−1.74003	−0.870013	+	0.493030i	$0.835889\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	30.3442	1.40416	0.702080	−	0.712098i	$-0.252255\pi$
$467$	30.3442	1.40416	0.702080	+	0.712098i	$0.252255\pi$
$468$	0	0
$469$	26.9361	1.24379
$470$	0	0
$471$	5.77845	0.266257
$472$	0	0
$473$	−30.8237	−1.41727
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−29.5665	−1.35376
$478$	0	0
$479$	38.1338	1.74238	0.871190	−	0.490946i	$-0.163349\pi$
$479$	38.1338	1.74238	0.871190	+	0.490946i	$0.163349\pi$
$480$	0	0
$481$	−4.84733	−0.221019
$482$	0	0
$483$	−10.1823	−0.463312
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−7.55947	−0.342552	−0.171276	−	0.985223i	$-0.554789\pi$
$487$	−7.55947	−0.342552	−0.171276	+	0.985223i	$0.554789\pi$
$488$	0	0
$489$	3.75091	0.169622
$490$	0	0
$491$	17.7896	0.802832	0.401416	−	0.915896i	$-0.368518\pi$
$491$	17.7896	0.802832	0.401416	+	0.915896i	$0.368518\pi$
$492$	0	0
$493$	−20.1125	−0.905823
$494$	0	0
$495$	0	0
$496$	0	0
$497$	36.0583	1.61743
$498$	0	0
$499$	0.598690	0.0268011	0.0134005	−	0.999910i	$-0.495734\pi$
$499$	0.598690	0.0268011	0.0134005	+	0.999910i	$0.495734\pi$
$500$	0	0
$501$	−1.64409	−0.0734524
$502$	0	0
$503$	4.41037	0.196649	0.0983243	−	0.995154i	$-0.468652\pi$
$503$	4.41037	0.196649	0.0983243	+	0.995154i	$0.468652\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0.710498	0.0315543
$508$	0	0
$509$	−15.4779	−0.686046	−0.343023	−	0.939327i	$-0.611451\pi$
$509$	−15.4779	−0.686046	−0.343023	+	0.939327i	$0.611451\pi$
$510$	0	0
$511$	−47.9437	−2.12090
$512$	0	0
$513$	−18.7689	−0.828668
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−43.8910	−1.93032
$518$	0	0
$519$	−4.56684	−0.200462
$520$	0	0
$521$	31.1710	1.36563	0.682813	−	0.730593i	$-0.260756\pi$
$521$	31.1710	1.36563	0.682813	+	0.730593i	$0.260756\pi$
$522$	0	0
$523$	−12.1630	−0.531853	−0.265926	−	0.963993i	$-0.585678\pi$
$523$	−12.1630	−0.531853	−0.265926	+	0.963993i	$0.585678\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−29.5605	−1.28768
$528$	0	0
$529$	48.3657	2.10286
$530$	0	0
$531$	−20.3052	−0.881172
$532$	0	0
$533$	5.97153	0.258656
$534$	0	0
$535$	0	0
$536$	0	0
$537$	8.98725	0.387829
$538$	0	0
$539$	−4.85830	−0.209262
$540$	0	0
$541$	21.9139	0.942150	0.471075	−	0.882093i	$-0.343866\pi$
$541$	21.9139	0.942150	0.471075	+	0.882093i	$0.343866\pi$
$542$	0	0
$543$	3.33399	0.143075
$544$	0	0
$545$	0	0
$546$	0	0
$547$	36.0638	1.54198	0.770988	−	0.636849i	$-0.219763\pi$
$547$	36.0638	1.54198	0.770988	+	0.636849i	$0.219763\pi$
$548$	0	0
$549$	41.6575	1.77790
$550$	0	0
$551$	−32.1030	−1.36763
$552$	0	0
$553$	18.4131	0.783003
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−18.0363	−0.764221	−0.382111	−	0.924117i	$-0.624803\pi$
$557$	−18.0363	−0.764221	−0.382111	+	0.924117i	$0.624803\pi$
$558$	0	0
$559$	26.1227	1.10487
$560$	0	0
$561$	−8.01263	−0.338293
$562$	0	0
$563$	−21.9028	−0.923094	−0.461547	−	0.887116i	$-0.652705\pi$
$563$	−21.9028	−0.923094	−0.461547	+	0.887116i	$0.652705\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−21.3405	−0.896216
$568$	0	0
$569$	−29.1921	−1.22380	−0.611898	−	0.790937i	$-0.709594\pi$
$569$	−29.1921	−1.22380	−0.611898	+	0.790937i	$0.709594\pi$
$570$	0	0
$571$	−2.92473	−0.122396	−0.0611981	−	0.998126i	$-0.519492\pi$
$571$	−2.92473	−0.122396	−0.0611981	+	0.998126i	$0.519492\pi$
$572$	0	0
$573$	6.50035	0.271556
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−3.13656	−0.130577	−0.0652884	−	0.997866i	$-0.520797\pi$
$577$	−3.13656	−0.130577	−0.0652884	+	0.997866i	$0.520797\pi$
$578$	0	0
$579$	3.10172	0.128903
$580$	0	0
$581$	−11.6786	−0.484509
$582$	0	0
$583$	−41.5550	−1.72103
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−30.9848	−1.27888	−0.639440	−	0.768841i	$-0.720834\pi$
$587$	−30.9848	−1.27888	−0.639440	+	0.768841i	$0.720834\pi$
$588$	0	0
$589$	−47.1835	−1.94416
$590$	0	0
$591$	−3.42115	−0.140727
$592$	0	0
$593$	−43.4868	−1.78579	−0.892894	−	0.450267i	$-0.851329\pi$
$593$	−43.4868	−1.78579	−0.892894	+	0.450267i	$0.851329\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−6.12192	−0.250553
$598$	0	0
$599$	−2.66139	−0.108741	−0.0543706	−	0.998521i	$-0.517315\pi$
$599$	−2.66139	−0.108741	−0.0543706	+	0.998521i	$0.517315\pi$
$600$	0	0
$601$	20.5824	0.839575	0.419788	−	0.907622i	$-0.362105\pi$
$601$	20.5824	0.839575	0.419788	+	0.907622i	$0.362105\pi$
$602$	0	0
$603$	26.5185	1.07992
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−18.6471	−0.756864	−0.378432	−	0.925629i	$-0.623537\pi$
$607$	−18.6471	−0.756864	−0.378432	+	0.925629i	$0.623537\pi$
$608$	0	0
$609$	5.04585	0.204468
$610$	0	0
$611$	37.1971	1.50483
$612$	0	0
$613$	−43.7510	−1.76709	−0.883544	−	0.468349i	$-0.844849\pi$
$613$	−43.7510	−1.76709	−0.883544	+	0.468349i	$0.844849\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	14.4891	0.583308	0.291654	−	0.956524i	$-0.405794\pi$
$617$	14.4891	0.583308	0.291654	+	0.956524i	$0.405794\pi$
$618$	0	0
$619$	−0.566742	−0.0227793	−0.0113896	−	0.999935i	$-0.503626\pi$
$619$	−0.566742	−0.0227793	−0.0113896	+	0.999935i	$0.503626\pi$
$620$	0	0
$621$	−20.6762	−0.829706
$622$	0	0
$623$	2.44940	0.0981332
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−12.7895	−0.510764
$628$	0	0
$629$	6.92493	0.276115
$630$	0	0
$631$	−22.2178	−0.884476	−0.442238	−	0.896898i	$-0.645815\pi$
$631$	−22.2178	−0.884476	−0.442238	+	0.896898i	$0.645815\pi$
$632$	0	0
$633$	0.200344	0.00796296
$634$	0	0
$635$	0	0
$636$	0	0
$637$	4.11735	0.163135
$638$	0	0
$639$	35.4993	1.40433
$640$	0	0
$641$	−23.8330	−0.941348	−0.470674	−	0.882307i	$-0.655989\pi$
$641$	−23.8330	−0.941348	−0.470674	+	0.882307i	$0.655989\pi$
$642$	0	0
$643$	21.2951	0.839798	0.419899	−	0.907571i	$-0.362066\pi$
$643$	21.2951	0.839798	0.419899	+	0.907571i	$0.362066\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	22.9099	0.900681	0.450340	−	0.892857i	$-0.351303\pi$
$647$	22.9099	0.900681	0.450340	+	0.892857i	$0.351303\pi$
$648$	0	0
$649$	−28.5385	−1.12023
$650$	0	0
$651$	7.41616	0.290662
$652$	0	0
$653$	1.99112	0.0779186	0.0389593	−	0.999241i	$-0.487596\pi$
$653$	1.99112	0.0779186	0.0389593	+	0.999241i	$0.487596\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−47.2005	−1.84147
$658$	0	0
$659$	−5.78285	−0.225268	−0.112634	−	0.993637i	$-0.535929\pi$
$659$	−5.78285	−0.225268	−0.112634	+	0.993637i	$0.535929\pi$
$660$	0	0
$661$	−41.3279	−1.60747	−0.803734	−	0.594989i	$-0.797156\pi$
$661$	−41.3279	−1.60747	−0.803734	+	0.594989i	$0.797156\pi$
$662$	0	0
$663$	6.79061	0.263726
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−35.3653	−1.36935
$668$	0	0
$669$	−4.67120	−0.180599
$670$	0	0
$671$	58.5485	2.26024
$672$	0	0
$673$	−29.4583	−1.13553	−0.567767	−	0.823189i	$-0.692193\pi$
$673$	−29.4583	−1.13553	−0.567767	+	0.823189i	$0.692193\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2.43828	0.0937107	0.0468553	−	0.998902i	$-0.485080\pi$
$677$	2.43828	0.0937107	0.0468553	+	0.998902i	$0.485080\pi$
$678$	0	0
$679$	−14.4758	−0.555529
$680$	0	0
$681$	1.13577	0.0435229
$682$	0	0
$683$	35.7141	1.36656	0.683280	−	0.730156i	$-0.260553\pi$
$683$	35.7141	1.36656	0.683280	+	0.730156i	$0.260553\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−1.17919	−0.0449888
$688$	0	0
$689$	35.2174	1.34168
$690$	0	0
$691$	31.7401	1.20745	0.603725	−	0.797192i	$-0.293682\pi$
$691$	31.7401	1.20745	0.603725	+	0.797192i	$0.293682\pi$
$692$	0	0
$693$	−32.1295	−1.22050
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−8.53097	−0.323134
$698$	0	0
$699$	−9.38607	−0.355014
$700$	0	0
$701$	−18.4576	−0.697135	−0.348567	−	0.937284i	$-0.613332\pi$
$701$	−18.4576	−0.697135	−0.348567	+	0.937284i	$0.613332\pi$
$702$	0	0
$703$	11.0534	0.416886
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.86838	0.258312
$708$	0	0
$709$	−36.2179	−1.36019	−0.680096	−	0.733123i	$-0.738062\pi$
$709$	−36.2179	−1.36019	−0.680096	+	0.733123i	$0.738062\pi$
$710$	0	0
$711$	18.1276	0.679839
$712$	0	0
$713$	−51.9783	−1.94660
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−1.21283	−0.0452938
$718$	0	0
$719$	11.3058	0.421634	0.210817	−	0.977526i	$-0.432388\pi$
$719$	11.3058	0.421634	0.210817	+	0.977526i	$0.432388\pi$
$720$	0	0
$721$	−1.19657	−0.0445626
$722$	0	0
$723$	−7.02147	−0.261131
$724$	0	0
$725$	0	0
$726$	0	0
$727$	29.3501	1.08854	0.544268	−	0.838911i	$-0.316807\pi$
$727$	29.3501	1.08854	0.544268	+	0.838911i	$0.316807\pi$
$728$	0	0
$729$	−17.9236	−0.663839
$730$	0	0
$731$	−37.3191	−1.38030
$732$	0	0
$733$	2.95232	0.109047	0.0545233	−	0.998513i	$-0.482636\pi$
$733$	2.95232	0.109047	0.0545233	+	0.998513i	$0.482636\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	37.2711	1.37290
$738$	0	0
$739$	−18.3954	−0.676685	−0.338342	−	0.941023i	$-0.609866\pi$
$739$	−18.3954	−0.676685	−0.338342	+	0.941023i	$0.609866\pi$
$740$	0	0
$741$	10.8390	0.398179
$742$	0	0
$743$	−25.3718	−0.930802	−0.465401	−	0.885100i	$-0.654090\pi$
$743$	−25.3718	−0.930802	−0.465401	+	0.885100i	$0.654090\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−11.4976	−0.420673
$748$	0	0
$749$	9.15926	0.334672
$750$	0	0
$751$	−36.3460	−1.32628	−0.663142	−	0.748493i	$-0.730778\pi$
$751$	−36.3460	−1.32628	−0.663142	+	0.748493i	$0.730778\pi$
$752$	0	0
$753$	5.96456	0.217361
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−23.7736	−0.864066	−0.432033	−	0.901858i	$-0.642204\pi$
$757$	−23.7736	−0.864066	−0.432033	+	0.901858i	$0.642204\pi$
$758$	0	0
$759$	−14.0892	−0.511404
$760$	0	0
$761$	5.83223	0.211418	0.105709	−	0.994397i	$-0.466289\pi$
$761$	5.83223	0.211418	0.105709	+	0.994397i	$0.466289\pi$
$762$	0	0
$763$	35.1729	1.27335
$764$	0	0
$765$	0	0
$766$	0	0
$767$	24.1860	0.873307
$768$	0	0
$769$	29.7572	1.07307	0.536535	−	0.843878i	$-0.319733\pi$
$769$	29.7572	1.07307	0.536535	+	0.843878i	$0.319733\pi$
$770$	0	0
$771$	5.07749	0.182861
$772$	0	0
$773$	−8.27984	−0.297805	−0.148903	−	0.988852i	$-0.547574\pi$
$773$	−8.27984	−0.297805	−0.148903	+	0.988852i	$0.547574\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−1.73733	−0.0623265
$778$	0	0
$779$	−13.6169	−0.487875
$780$	0	0
$781$	49.8934	1.78533
$782$	0	0
$783$	10.2461	0.366165
$784$	0	0
$785$	0	0
$786$	0	0
$787$	47.8995	1.70743	0.853716	−	0.520739i	$-0.174344\pi$
$787$	47.8995	1.70743	0.853716	+	0.520739i	$0.174344\pi$
$788$	0	0
$789$	−4.96880	−0.176894
$790$	0	0
$791$	−45.2357	−1.60840
$792$	0	0
$793$	−49.6192	−1.76203
$794$	0	0
$795$	0	0
$796$	0	0
$797$	17.5313	0.620992	0.310496	−	0.950575i	$-0.399505\pi$
$797$	17.5313	0.620992	0.310496	+	0.950575i	$0.399505\pi$
$798$	0	0
$799$	−53.1400	−1.87996
$800$	0	0
$801$	2.41143	0.0852038
$802$	0	0
$803$	−66.3390	−2.34105
$804$	0	0
$805$	0	0
$806$	0	0
$807$	3.44316	0.121205
$808$	0	0
$809$	−26.5936	−0.934981	−0.467491	−	0.883998i	$-0.654842\pi$
$809$	−26.5936	−0.934981	−0.467491	+	0.883998i	$0.654842\pi$
$810$	0	0
$811$	−19.1666	−0.673032	−0.336516	−	0.941678i	$-0.609249\pi$
$811$	−19.1666	−0.673032	−0.336516	+	0.941678i	$0.609249\pi$
$812$	0	0
$813$	−3.20477	−0.112396
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−59.5676	−2.08401
$818$	0	0
$819$	27.2294	0.951471
$820$	0	0
$821$	14.2705	0.498044	0.249022	−	0.968498i	$-0.419891\pi$
$821$	14.2705	0.498044	0.249022	+	0.968498i	$0.419891\pi$
$822$	0	0
$823$	22.6319	0.788898	0.394449	−	0.918918i	$-0.370935\pi$
$823$	22.6319	0.788898	0.394449	+	0.918918i	$0.370935\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−28.2844	−0.983543	−0.491772	−	0.870724i	$-0.663651\pi$
$827$	−28.2844	−0.983543	−0.491772	+	0.870724i	$0.663651\pi$
$828$	0	0
$829$	−23.4274	−0.813666	−0.406833	−	0.913503i	$-0.633367\pi$
$829$	−23.4274	−0.813666	−0.406833	+	0.913503i	$0.633367\pi$
$830$	0	0
$831$	0.231397	0.00802707
$832$	0	0
$833$	−5.88208	−0.203802
$834$	0	0
$835$	0	0
$836$	0	0
$837$	15.0592	0.520523
$838$	0	0
$839$	−23.1936	−0.800732	−0.400366	−	0.916355i	$-0.631117\pi$
$839$	−23.1936	−0.800732	−0.400366	+	0.916355i	$0.631117\pi$
$840$	0	0
$841$	−11.4748	−0.395681
$842$	0	0
$843$	2.87526	0.0990291
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−13.6112	−0.467687
$848$	0	0
$849$	10.1687	0.348987
$850$	0	0
$851$	12.1766	0.417408
$852$	0	0
$853$	−5.65086	−0.193482	−0.0967408	−	0.995310i	$-0.530842\pi$
$853$	−5.65086	−0.193482	−0.0967408	+	0.995310i	$0.530842\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	20.8747	0.713067	0.356534	−	0.934283i	$-0.383959\pi$
$857$	20.8747	0.713067	0.356534	+	0.934283i	$0.383959\pi$
$858$	0	0
$859$	−37.1536	−1.26767	−0.633833	−	0.773470i	$-0.718519\pi$
$859$	−37.1536	−1.26767	−0.633833	+	0.773470i	$0.718519\pi$
$860$	0	0
$861$	2.14026	0.0729398
$862$	0	0
$863$	52.5025	1.78721	0.893603	−	0.448858i	$-0.148169\pi$
$863$	52.5025	1.78721	0.893603	+	0.448858i	$0.148169\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−2.55613	−0.0868107
$868$	0	0
$869$	25.4779	0.864279
$870$	0	0
$871$	−31.5868	−1.07028
$872$	0	0
$873$	−14.2514	−0.482336
$874$	0	0
$875$	0	0
$876$	0	0
$877$	36.6926	1.23902	0.619511	−	0.784988i	$-0.287331\pi$
$877$	36.6926	1.23902	0.619511	+	0.784988i	$0.287331\pi$
$878$	0	0
$879$	−0.0115124	−0.000388303 0
$880$	0	0
$881$	−4.64878	−0.156621	−0.0783107	−	0.996929i	$-0.524953\pi$
$881$	−4.64878	−0.156621	−0.0783107	+	0.996929i	$0.524953\pi$
$882$	0	0
$883$	44.3640	1.49297	0.746484	−	0.665404i	$-0.231741\pi$
$883$	44.3640	1.49297	0.746484	+	0.665404i	$0.231741\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	33.4881	1.12442	0.562210	−	0.826995i	$-0.309951\pi$
$887$	33.4881	1.12442	0.562210	+	0.826995i	$0.309951\pi$
$888$	0	0
$889$	29.3096	0.983014
$890$	0	0
$891$	−29.5285	−0.989243
$892$	0	0
$893$	−84.8205	−2.83841
$894$	0	0
$895$	0	0
$896$	0	0
$897$	11.9404	0.398678
$898$	0	0
$899$	25.7578	0.859071
$900$	0	0
$901$	−50.3118	−1.67613
$902$	0	0
$903$	9.36265	0.311570
$904$	0	0
$905$	0	0
$906$	0	0
$907$	49.9807	1.65958	0.829792	−	0.558073i	$-0.188459\pi$
$907$	49.9807	1.65958	0.829792	+	0.558073i	$0.188459\pi$
$908$	0	0
$909$	6.76191	0.224278
$910$	0	0
$911$	48.7149	1.61400	0.806999	−	0.590553i	$-0.201090\pi$
$911$	48.7149	1.61400	0.806999	+	0.590553i	$0.201090\pi$
$912$	0	0
$913$	−16.1595	−0.534802
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−23.2343	−0.767265
$918$	0	0
$919$	10.0578	0.331776	0.165888	−	0.986145i	$-0.446951\pi$
$919$	10.0578	0.331776	0.165888	+	0.986145i	$0.446951\pi$
$920$	0	0
$921$	8.87137	0.292321
$922$	0	0
$923$	−42.2841	−1.39180
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1.17802	−0.0386913
$928$	0	0
$929$	19.8662	0.651788	0.325894	−	0.945406i	$-0.394335\pi$
$929$	19.8662	0.651788	0.325894	+	0.945406i	$0.394335\pi$
$930$	0	0
$931$	−9.38879	−0.307705
$932$	0	0
$933$	−3.72928	−0.122091
$934$	0	0
$935$	0	0
$936$	0	0
$937$	12.4155	0.405595	0.202798	−	0.979221i	$-0.434997\pi$
$937$	12.4155	0.405595	0.202798	+	0.979221i	$0.434997\pi$
$938$	0	0
$939$	8.54650	0.278905
$940$	0	0
$941$	49.8818	1.62610	0.813051	−	0.582193i	$-0.197805\pi$
$941$	49.8818	1.62610	0.813051	+	0.582193i	$0.197805\pi$
$942$	0	0
$943$	−15.0006	−0.488487
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−15.9361	−0.517854	−0.258927	−	0.965897i	$-0.583369\pi$
$947$	−15.9361	−0.517854	−0.258927	+	0.965897i	$0.583369\pi$
$948$	0	0
$949$	56.2216	1.82503
$950$	0	0
$951$	0.830087	0.0269174
$952$	0	0
$953$	19.3904	0.628115	0.314058	−	0.949404i	$-0.398311\pi$
$953$	19.3904	0.628115	0.314058	+	0.949404i	$0.398311\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	6.98188	0.225692
$958$	0	0
$959$	−35.5428	−1.14774
$960$	0	0
$961$	6.85767	0.221215
$962$	0	0
$963$	9.01728	0.290578
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−38.6503	−1.24291	−0.621455	−	0.783450i	$-0.713458\pi$
$967$	−38.6503	−1.24291	−0.621455	+	0.783450i	$0.713458\pi$
$968$	0	0
$969$	−15.4846	−0.497438
$970$	0	0
$971$	8.27436	0.265537	0.132768	−	0.991147i	$-0.457613\pi$
$971$	8.27436	0.265537	0.132768	+	0.991147i	$0.457613\pi$
$972$	0	0
$973$	−30.0862	−0.964520
$974$	0	0
$975$	0	0
$976$	0	0
$977$	38.0832	1.21839	0.609196	−	0.793020i	$-0.291492\pi$
$977$	38.0832	1.21839	0.609196	+	0.793020i	$0.291492\pi$
$978$	0	0
$979$	3.38920	0.108319
$980$	0	0
$981$	34.6277	1.10558
$982$	0	0
$983$	−28.2844	−0.902131	−0.451065	−	0.892491i	$-0.648956\pi$
$983$	−28.2844	−0.902131	−0.451065	+	0.892491i	$0.648956\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	13.3318	0.424356
$988$	0	0
$989$	−65.6208	−2.08662
$990$	0	0
$991$	29.7331	0.944502	0.472251	−	0.881464i	$-0.343442\pi$
$991$	29.7331	0.944502	0.472251	+	0.881464i	$0.343442\pi$
$992$	0	0
$993$	−8.36549	−0.265471
$994$	0	0
$995$	0	0
$996$	0	0
$997$	21.3219	0.675271	0.337635	−	0.941277i	$-0.390373\pi$
$997$	21.3219	0.675271	0.337635	+	0.941277i	$0.390373\pi$
$998$	0	0
$999$	−3.52782	−0.111615

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4000.2.a.p.1.4	yes	8
4.3	odd	2	inner	4000.2.a.p.1.5	yes	8
5.2	odd	4		4000.2.c.i.1249.9		16
5.3	odd	4		4000.2.c.i.1249.7		16
5.4	even	2		4000.2.a.o.1.5	yes	8
8.3	odd	2		8000.2.a.by.1.4		8
8.5	even	2		8000.2.a.by.1.5		8
20.3	even	4		4000.2.c.i.1249.10		16
20.7	even	4		4000.2.c.i.1249.8		16
20.19	odd	2		4000.2.a.o.1.4	✓	8
40.19	odd	2		8000.2.a.bz.1.5		8
40.29	even	2		8000.2.a.bz.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4000.2.a.o.1.4	✓	8	20.19	odd	2
4000.2.a.o.1.5	yes	8	5.4	even	2
4000.2.a.p.1.4	yes	8	1.1	even	1	trivial
4000.2.a.p.1.5	yes	8	4.3	odd	2	inner
4000.2.c.i.1249.7		16	5.3	odd	4
4000.2.c.i.1249.8		16	20.7	even	4
4000.2.c.i.1249.9		16	5.2	odd	4
4000.2.c.i.1249.10		16	20.3	even	4
8000.2.a.by.1.4		8	8.3	odd	2
8000.2.a.by.1.5		8	8.5	even	2
8000.2.a.bz.1.4		8	40.29	even	2
8000.2.a.bz.1.5		8	40.19	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 $\F_{q}$

Introduction

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Modular forms

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Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	4000.2.a.p.1.4

Level	$4000$
Weight	$2$
Character	4000.1
Self dual	yes
Analytic conductor	$31.940$
Analytic rank	$0$
Dimension	$8$
Inner twists	$2$