LMFDB - Embedded newform 4508.2.a.g.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4508,2,Mod(1,4508)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4508, 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("4508.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4508 = 2^{2} \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4508.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:	$35.9965612312$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.8580816.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 12x^{3} + 10x^{2} + 20x + 6 $ x^5 - x^4 - 12x^3 + 10x^2 + 20*x + 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 644)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-3.18838$ of defining polynomial
Character	$\chi$	$=$	4508.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.18838 q^{3} -3.43522 q^{5} +7.16574 q^{9} +O(q^{10})$	$q+3.18838 q^{3} -3.43522 q^{5} +7.16574 q^{9} -2.57602 q^{11} +1.58972 q^{13} -10.9528 q^{15} +4.77810 q^{17} -8.65835 q^{19} +1.00000 q^{23} +6.80073 q^{25} +13.2819 q^{27} +7.67205 q^{29} +5.02494 q^{31} -8.21332 q^{33} +3.71840 q^{37} +5.06863 q^{39} -11.9665 q^{41} +7.46104 q^{43} -24.6159 q^{45} +4.43610 q^{47} +15.2344 q^{51} +8.06775 q^{53} +8.84919 q^{55} -27.6061 q^{57} +1.88025 q^{59} +5.71682 q^{61} -5.46104 q^{65} +13.2618 q^{67} +3.18838 q^{69} +5.96647 q^{71} +10.8369 q^{73} +21.6833 q^{75} +2.01282 q^{79} +20.8506 q^{81} -3.43364 q^{83} -16.4138 q^{85} +24.4614 q^{87} +6.82337 q^{89} +16.0214 q^{93} +29.7433 q^{95} -10.0258 q^{97} -18.4591 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{3} - 4 q^{5} + 10 q^{9}+O(q^{10}) $ 5 * q - q^3 - 4 * q^5 + 10 * q^9	$ 5 q - q^{3} - 4 q^{5} + 10 q^{9} + 2 q^{11} - 3 q^{13} - 6 q^{15} - 4 q^{17} - 6 q^{19} + 5 q^{23} + 15 q^{25} - q^{27} + 5 q^{29} + q^{31} + 22 q^{37} - q^{39} - 15 q^{41} + 12 q^{43} - 36 q^{45} + 21 q^{47} + 24 q^{51} + 2 q^{53} + 22 q^{55} - 34 q^{57} + 12 q^{61} - 2 q^{65} + 22 q^{67} - q^{69} - 15 q^{71} - 17 q^{73} + 47 q^{75} + 2 q^{79} + 37 q^{81} + 16 q^{83} - 8 q^{85} + 33 q^{87} + 24 q^{89} + 5 q^{93} - 8 q^{95} - 38 q^{97} - 36 q^{99}+O(q^{100}) $ 5 * q - q^3 - 4 * q^5 + 10 * q^9 + 2 * q^11 - 3 * q^13 - 6 * q^15 - 4 * q^17 - 6 * q^19 + 5 * q^23 + 15 * q^25 - q^27 + 5 * q^29 + q^31 + 22 * q^37 - q^39 - 15 * q^41 + 12 * q^43 - 36 * q^45 + 21 * q^47 + 24 * q^51 + 2 * q^53 + 22 * q^55 - 34 * q^57 + 12 * q^61 - 2 * q^65 + 22 * q^67 - q^69 - 15 * q^71 - 17 * q^73 + 47 * q^75 + 2 * q^79 + 37 * q^81 + 16 * q^83 - 8 * q^85 + 33 * q^87 + 24 * q^89 + 5 * q^93 - 8 * q^95 - 38 * q^97 - 36 * 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
$3$	3.18838	1.84081	0.920405	−	0.390967i	$-0.127859\pi$
$3$	3.18838	1.84081	0.920405	+	0.390967i	$0.127859\pi$
$4$	0	0
$5$	−3.43522	−1.53628	−0.768138	−	0.640284i	$-0.778817\pi$
$5$	−3.43522	−1.53628	−0.768138	+	0.640284i	$0.778817\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	7.16574	2.38858
$10$	0	0
$11$	−2.57602	−0.776699	−0.388349	−	0.921512i	$-0.626955\pi$
$11$	−2.57602	−0.776699	−0.388349	+	0.921512i	$0.626955\pi$
$12$	0	0
$13$	1.58972	0.440909	0.220455	−	0.975397i	$-0.429246\pi$
$13$	1.58972	0.440909	0.220455	+	0.975397i	$0.429246\pi$
$14$	0	0
$15$	−10.9528	−2.82799
$16$	0	0
$17$	4.77810	1.15886	0.579429	−	0.815022i	$-0.303275\pi$
$17$	4.77810	1.15886	0.579429	+	0.815022i	$0.303275\pi$
$18$	0	0
$19$	−8.65835	−1.98636	−0.993181	−	0.116583i	$-0.962806\pi$
$19$	−8.65835	−1.98636	−0.993181	+	0.116583i	$0.962806\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	6.80073	1.36015
$26$	0	0
$27$	13.2819	2.55611
$28$	0	0
$29$	7.67205	1.42466	0.712332	−	0.701842i	$-0.247639\pi$
$29$	7.67205	1.42466	0.712332	+	0.701842i	$0.247639\pi$
$30$	0	0
$31$	5.02494	0.902506	0.451253	−	0.892396i	$-0.350977\pi$
$31$	5.02494	0.902506	0.451253	+	0.892396i	$0.350977\pi$
$32$	0	0
$33$	−8.21332	−1.42975
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.71840	0.611301	0.305651	−	0.952144i	$-0.401126\pi$
$37$	3.71840	0.611301	0.305651	+	0.952144i	$0.401126\pi$
$38$	0	0
$39$	5.06863	0.811630
$40$	0	0
$41$	−11.9665	−1.86885	−0.934425	−	0.356161i	$-0.884085\pi$
$41$	−11.9665	−1.86885	−0.934425	+	0.356161i	$0.884085\pi$
$42$	0	0
$43$	7.46104	1.13780	0.568899	−	0.822407i	$-0.307369\pi$
$43$	7.46104	1.13780	0.568899	+	0.822407i	$0.307369\pi$
$44$	0	0
$45$	−24.6159	−3.66952
$46$	0	0
$47$	4.43610	0.647072	0.323536	−	0.946216i	$-0.395128\pi$
$47$	4.43610	0.647072	0.323536	+	0.946216i	$0.395128\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	15.2344	2.13324
$52$	0	0
$53$	8.06775	1.10819	0.554095	−	0.832453i	$-0.313064\pi$
$53$	8.06775	1.10819	0.554095	+	0.832453i	$0.313064\pi$
$54$	0	0
$55$	8.84919	1.19322
$56$	0	0
$57$	−27.6061	−3.65651
$58$	0	0
$59$	1.88025	0.244788	0.122394	−	0.992482i	$-0.460943\pi$
$59$	1.88025	0.244788	0.122394	+	0.992482i	$0.460943\pi$
$60$	0	0
$61$	5.71682	0.731964	0.365982	−	0.930622i	$-0.380733\pi$
$61$	5.71682	0.731964	0.365982	+	0.930622i	$0.380733\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5.46104	−0.677359
$66$	0	0
$67$	13.2618	1.62018	0.810092	−	0.586303i	$-0.199417\pi$
$67$	13.2618	1.62018	0.810092	+	0.586303i	$0.199417\pi$
$68$	0	0
$69$	3.18838	0.383835
$70$	0	0
$71$	5.96647	0.708090	0.354045	−	0.935228i	$-0.384806\pi$
$71$	5.96647	0.708090	0.354045	+	0.935228i	$0.384806\pi$
$72$	0	0
$73$	10.8369	1.26836	0.634182	−	0.773184i	$-0.281337\pi$
$73$	10.8369	1.26836	0.634182	+	0.773184i	$0.281337\pi$
$74$	0	0
$75$	21.6833	2.50377
$76$	0	0
$77$	0	0
$78$	0	0
$79$	2.01282	0.226460	0.113230	−	0.993569i	$-0.463880\pi$
$79$	2.01282	0.226460	0.113230	+	0.993569i	$0.463880\pi$
$80$	0	0
$81$	20.8506	2.31673
$82$	0	0
$83$	−3.43364	−0.376891	−0.188445	−	0.982084i	$-0.560345\pi$
$83$	−3.43364	−0.376891	−0.188445	+	0.982084i	$0.560345\pi$
$84$	0	0
$85$	−16.4138	−1.78033
$86$	0	0
$87$	24.4614	2.62254
$88$	0	0
$89$	6.82337	0.723276	0.361638	−	0.932319i	$-0.382218\pi$
$89$	6.82337	0.723276	0.361638	+	0.932319i	$0.382218\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	16.0214	1.66134
$94$	0	0
$95$	29.7433	3.05160
$96$	0	0
$97$	−10.0258	−1.01797	−0.508984	−	0.860776i	$-0.669979\pi$
$97$	−10.0258	−1.01797	−0.508984	+	0.860776i	$0.669979\pi$
$98$	0	0
$99$	−18.4591	−1.85521
$100$	0	0
$101$	−3.95473	−0.393510	−0.196755	−	0.980453i	$-0.563040\pi$
$101$	−3.95473	−0.393510	−0.196755	+	0.980453i	$0.563040\pi$
$102$	0	0
$103$	1.31424	0.129496	0.0647482	−	0.997902i	$-0.479376\pi$
$103$	1.31424	0.129496	0.0647482	+	0.997902i	$0.479376\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.45928	−0.914463	−0.457231	−	0.889348i	$-0.651159\pi$
$107$	−9.45928	−0.914463	−0.457231	+	0.889348i	$0.651159\pi$
$108$	0	0
$109$	−0.376752	−0.0360863	−0.0180431	−	0.999837i	$-0.505744\pi$
$109$	−0.376752	−0.0360863	−0.0180431	+	0.999837i	$0.505744\pi$
$110$	0	0
$111$	11.8557	1.12529
$112$	0	0
$113$	−15.5741	−1.46508	−0.732542	−	0.680722i	$-0.761666\pi$
$113$	−15.5741	−1.46508	−0.732542	+	0.680722i	$0.761666\pi$
$114$	0	0
$115$	−3.43522	−0.320336
$116$	0	0
$117$	11.3915	1.05315
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−4.36413	−0.396739
$122$	0	0
$123$	−38.1536	−3.44020
$124$	0	0
$125$	−6.18591	−0.553285
$126$	0	0
$127$	18.8095	1.66907	0.834537	−	0.550952i	$-0.185735\pi$
$127$	18.8095	1.66907	0.834537	+	0.550952i	$0.185735\pi$
$128$	0	0
$129$	23.7886	2.09447
$130$	0	0
$131$	1.42348	0.124370	0.0621848	−	0.998065i	$-0.480193\pi$
$131$	1.42348	0.124370	0.0621848	+	0.998065i	$0.480193\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−45.6264	−3.92690
$136$	0	0
$137$	−16.2145	−1.38530	−0.692651	−	0.721273i	$-0.743557\pi$
$137$	−16.2145	−1.38530	−0.692651	+	0.721273i	$0.743557\pi$
$138$	0	0
$139$	−7.59253	−0.643990	−0.321995	−	0.946741i	$-0.604353\pi$
$139$	−7.59253	−0.643990	−0.321995	+	0.946741i	$0.604353\pi$
$140$	0	0
$141$	14.1440	1.19114
$142$	0	0
$143$	−4.09515	−0.342454
$144$	0	0
$145$	−26.3552	−2.18868
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.68576	0.711565	0.355782	−	0.934569i	$-0.384214\pi$
$149$	8.68576	0.711565	0.355782	+	0.934569i	$0.384214\pi$
$150$	0	0
$151$	14.3158	1.16500	0.582502	−	0.812829i	$-0.302074\pi$
$151$	14.3158	1.16500	0.582502	+	0.812829i	$0.302074\pi$
$152$	0	0
$153$	34.2386	2.76803
$154$	0	0
$155$	−17.2618	−1.38650
$156$	0	0
$157$	0.962781	0.0768383	0.0384191	−	0.999262i	$-0.487768\pi$
$157$	0.962781	0.0768383	0.0384191	+	0.999262i	$0.487768\pi$
$158$	0	0
$159$	25.7230	2.03997
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−1.37763	−0.107905	−0.0539523	−	0.998544i	$-0.517182\pi$
$163$	−1.37763	−0.107905	−0.0539523	+	0.998544i	$0.517182\pi$
$164$	0	0
$165$	28.2145	2.19650
$166$	0	0
$167$	6.00158	0.464416	0.232208	−	0.972666i	$-0.425405\pi$
$167$	6.00158	0.464416	0.232208	+	0.972666i	$0.425405\pi$
$168$	0	0
$169$	−10.4728	−0.805599
$170$	0	0
$171$	−62.0435	−4.74458
$172$	0	0
$173$	9.08429	0.690666	0.345333	−	0.938480i	$-0.387766\pi$
$173$	9.08429	0.690666	0.345333	+	0.938480i	$0.387766\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5.99495	0.450608
$178$	0	0
$179$	−1.56232	−0.116773	−0.0583865	−	0.998294i	$-0.518596\pi$
$179$	−1.56232	−0.116773	−0.0583865	+	0.998294i	$0.518596\pi$
$180$	0	0
$181$	−0.446616	−0.0331967	−0.0165984	−	0.999862i	$-0.505284\pi$
$181$	−0.446616	−0.0331967	−0.0165984	+	0.999862i	$0.505284\pi$
$182$	0	0
$183$	18.2274	1.34741
$184$	0	0
$185$	−12.7735	−0.939128
$186$	0	0
$187$	−12.3085	−0.900084
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−18.0225	−1.30406	−0.652030	−	0.758193i	$-0.726083\pi$
$191$	−18.0225	−1.30406	−0.652030	+	0.758193i	$0.726083\pi$
$192$	0	0
$193$	−12.6701	−0.912013	−0.456007	−	0.889976i	$-0.650721\pi$
$193$	−12.6701	−0.912013	−0.456007	+	0.889976i	$0.650721\pi$
$194$	0	0
$195$	−17.4119	−1.24689
$196$	0	0
$197$	4.46016	0.317773	0.158887	−	0.987297i	$-0.449210\pi$
$197$	4.46016	0.317773	0.158887	+	0.987297i	$0.449210\pi$
$198$	0	0
$199$	−13.4835	−0.955821	−0.477911	−	0.878408i	$-0.658606\pi$
$199$	−13.4835	−0.955821	−0.477911	+	0.878408i	$0.658606\pi$
$200$	0	0
$201$	42.2835	2.98245
$202$	0	0
$203$	0	0
$204$	0	0
$205$	41.1075	2.87107
$206$	0	0
$207$	7.16574	0.498053
$208$	0	0
$209$	22.3041	1.54281
$210$	0	0
$211$	20.3493	1.40091	0.700453	−	0.713698i	$-0.252981\pi$
$211$	20.3493	1.40091	0.700453	+	0.713698i	$0.252981\pi$
$212$	0	0
$213$	19.0234	1.30346
$214$	0	0
$215$	−25.6303	−1.74797
$216$	0	0
$217$	0	0
$218$	0	0
$219$	34.5521	2.33482
$220$	0	0
$221$	7.59584	0.510952
$222$	0	0
$223$	−16.0253	−1.07313	−0.536566	−	0.843858i	$-0.680279\pi$
$223$	−16.0253	−1.07313	−0.536566	+	0.843858i	$0.680279\pi$
$224$	0	0
$225$	48.7323	3.24882
$226$	0	0
$227$	16.4266	1.09027	0.545137	−	0.838347i	$-0.316478\pi$
$227$	16.4266	1.09027	0.545137	+	0.838347i	$0.316478\pi$
$228$	0	0
$229$	13.5316	0.894193	0.447097	−	0.894486i	$-0.352458\pi$
$229$	13.5316	0.894193	0.447097	+	0.894486i	$0.352458\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3.50524	0.229636	0.114818	−	0.993387i	$-0.463372\pi$
$233$	3.50524	0.229636	0.114818	+	0.993387i	$0.463372\pi$
$234$	0	0
$235$	−15.2390	−0.994081
$236$	0	0
$237$	6.41763	0.416870
$238$	0	0
$239$	−20.7418	−1.34167	−0.670836	−	0.741605i	$-0.734065\pi$
$239$	−20.7418	−1.34167	−0.670836	+	0.741605i	$0.734065\pi$
$240$	0	0
$241$	3.95754	0.254928	0.127464	−	0.991843i	$-0.459316\pi$
$241$	3.95754	0.254928	0.127464	+	0.991843i	$0.459316\pi$
$242$	0	0
$243$	26.6338	1.70856
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−13.7644	−0.875806
$248$	0	0
$249$	−10.9477	−0.693784
$250$	0	0
$251$	29.8150	1.88191	0.940953	−	0.338537i	$-0.109932\pi$
$251$	29.8150	1.88191	0.940953	+	0.338537i	$0.109932\pi$
$252$	0	0
$253$	−2.57602	−0.161953
$254$	0	0
$255$	−52.3334	−3.27724
$256$	0	0
$257$	−3.44930	−0.215161	−0.107581	−	0.994196i	$-0.534310\pi$
$257$	−3.44930	−0.215161	−0.107581	+	0.994196i	$0.534310\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	54.9759	3.40293
$262$	0	0
$263$	−8.05493	−0.496688	−0.248344	−	0.968672i	$-0.579886\pi$
$263$	−8.05493	−0.496688	−0.248344	+	0.968672i	$0.579886\pi$
$264$	0	0
$265$	−27.7145	−1.70249
$266$	0	0
$267$	21.7555	1.33141
$268$	0	0
$269$	−4.43452	−0.270377	−0.135189	−	0.990820i	$-0.543164\pi$
$269$	−4.43452	−0.270377	−0.135189	+	0.990820i	$0.543164\pi$
$270$	0	0
$271$	24.0253	1.45943	0.729716	−	0.683750i	$-0.239652\pi$
$271$	24.0253	1.45943	0.729716	+	0.683750i	$0.239652\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−17.5188	−1.05642
$276$	0	0
$277$	−0.330404	−0.0198520	−0.00992602	−	0.999951i	$-0.503160\pi$
$277$	−0.330404	−0.0198520	−0.00992602	+	0.999951i	$0.503160\pi$
$278$	0	0
$279$	36.0074	2.15571
$280$	0	0
$281$	25.7483	1.53601	0.768006	−	0.640442i	$-0.221249\pi$
$281$	25.7483	1.53601	0.768006	+	0.640442i	$0.221249\pi$
$282$	0	0
$283$	21.8104	1.29649	0.648247	−	0.761431i	$-0.275503\pi$
$283$	21.8104	1.29649	0.648247	+	0.761431i	$0.275503\pi$
$284$	0	0
$285$	94.8329	5.61742
$286$	0	0
$287$	0	0
$288$	0	0
$289$	5.83021	0.342954
$290$	0	0
$291$	−31.9661	−1.87389
$292$	0	0
$293$	15.1487	0.884996	0.442498	−	0.896769i	$-0.354092\pi$
$293$	15.1487	0.884996	0.442498	+	0.896769i	$0.354092\pi$
$294$	0	0
$295$	−6.45908	−0.376062
$296$	0	0
$297$	−34.2145	−1.98533
$298$	0	0
$299$	1.58972	0.0919360
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−12.6092	−0.724377
$304$	0	0
$305$	−19.6385	−1.12450
$306$	0	0
$307$	11.3183	0.645969	0.322984	−	0.946404i	$-0.395314\pi$
$307$	11.3183	0.645969	0.322984	+	0.946404i	$0.395314\pi$
$308$	0	0
$309$	4.19031	0.238378
$310$	0	0
$311$	3.13357	0.177688	0.0888441	−	0.996046i	$-0.471683\pi$
$311$	3.13357	0.177688	0.0888441	+	0.996046i	$0.471683\pi$
$312$	0	0
$313$	9.12744	0.515914	0.257957	−	0.966156i	$-0.416951\pi$
$313$	9.12744	0.515914	0.257957	+	0.966156i	$0.416951\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−33.8032	−1.89858	−0.949288	−	0.314407i	$-0.898194\pi$
$317$	−33.8032	−1.89858	−0.949288	+	0.314407i	$0.898194\pi$
$318$	0	0
$319$	−19.7634	−1.10654
$320$	0	0
$321$	−30.1597	−1.68335
$322$	0	0
$323$	−41.3704	−2.30191
$324$	0	0
$325$	10.8113	0.599701
$326$	0	0
$327$	−1.20123	−0.0664279
$328$	0	0
$329$	0	0
$330$	0	0
$331$	10.6448	0.585094	0.292547	−	0.956251i	$-0.405497\pi$
$331$	10.6448	0.585094	0.292547	+	0.956251i	$0.405497\pi$
$332$	0	0
$333$	26.6451	1.46014
$334$	0	0
$335$	−45.5571	−2.48905
$336$	0	0
$337$	−33.8838	−1.84577	−0.922883	−	0.385081i	$-0.874174\pi$
$337$	−33.8838	−1.84577	−0.922883	+	0.385081i	$0.874174\pi$
$338$	0	0
$339$	−49.6560	−2.69694
$340$	0	0
$341$	−12.9443	−0.700975
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−10.9528	−0.589677
$346$	0	0
$347$	8.91254	0.478450	0.239225	−	0.970964i	$-0.423107\pi$
$347$	8.91254	0.478450	0.239225	+	0.970964i	$0.423107\pi$
$348$	0	0
$349$	−8.03353	−0.430025	−0.215012	−	0.976611i	$-0.568979\pi$
$349$	−8.03353	−0.430025	−0.215012	+	0.976611i	$0.568979\pi$
$350$	0	0
$351$	21.1146	1.12701
$352$	0	0
$353$	−19.8221	−1.05503	−0.527513	−	0.849547i	$-0.676875\pi$
$353$	−19.8221	−1.05503	−0.527513	+	0.849547i	$0.676875\pi$
$354$	0	0
$355$	−20.4961	−1.08782
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−12.1547	−0.641500	−0.320750	−	0.947164i	$-0.603935\pi$
$359$	−12.1547	−0.641500	−0.320750	+	0.947164i	$0.603935\pi$
$360$	0	0
$361$	55.9670	2.94563
$362$	0	0
$363$	−13.9145	−0.730320
$364$	0	0
$365$	−37.2272	−1.94856
$366$	0	0
$367$	2.57274	0.134296	0.0671479	−	0.997743i	$-0.478610\pi$
$367$	2.57274	0.134296	0.0671479	+	0.997743i	$0.478610\pi$
$368$	0	0
$369$	−85.7486	−4.46390
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−8.59521	−0.445043	−0.222522	−	0.974928i	$-0.571429\pi$
$373$	−8.59521	−0.445043	−0.222522	+	0.974928i	$0.571429\pi$
$374$	0	0
$375$	−19.7230	−1.01849
$376$	0	0
$377$	12.1964	0.628148
$378$	0	0
$379$	23.8780	1.22653	0.613266	−	0.789877i	$-0.289855\pi$
$379$	23.8780	1.22653	0.613266	+	0.789877i	$0.289855\pi$
$380$	0	0
$381$	59.9718	3.07245
$382$	0	0
$383$	−14.0902	−0.719977	−0.359988	−	0.932957i	$-0.617219\pi$
$383$	−14.0902	−0.719977	−0.359988	+	0.932957i	$0.617219\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	53.4639	2.71772
$388$	0	0
$389$	−28.1429	−1.42690	−0.713450	−	0.700706i	$-0.752869\pi$
$389$	−28.1429	−1.42690	−0.713450	+	0.700706i	$0.752869\pi$
$390$	0	0
$391$	4.77810	0.241639
$392$	0	0
$393$	4.53858	0.228941
$394$	0	0
$395$	−6.91448	−0.347905
$396$	0	0
$397$	−25.7590	−1.29281	−0.646403	−	0.762996i	$-0.723728\pi$
$397$	−25.7590	−1.29281	−0.646403	+	0.762996i	$0.723728\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−23.1745	−1.15728	−0.578640	−	0.815583i	$-0.696416\pi$
$401$	−23.1745	−1.15728	−0.578640	+	0.815583i	$0.696416\pi$
$402$	0	0
$403$	7.98826	0.397923
$404$	0	0
$405$	−71.6264	−3.55915
$406$	0	0
$407$	−9.57867	−0.474797
$408$	0	0
$409$	10.9443	0.541161	0.270581	−	0.962697i	$-0.412784\pi$
$409$	10.9443	0.541161	0.270581	+	0.962697i	$0.412784\pi$
$410$	0	0
$411$	−51.6981	−2.55008
$412$	0	0
$413$	0	0
$414$	0	0
$415$	11.7953	0.579008
$416$	0	0
$417$	−24.2078	−1.18546
$418$	0	0
$419$	19.0867	0.932449	0.466224	−	0.884667i	$-0.345614\pi$
$419$	19.0867	0.932449	0.466224	+	0.884667i	$0.345614\pi$
$420$	0	0
$421$	30.7325	1.49781	0.748905	−	0.662678i	$-0.230580\pi$
$421$	30.7325	1.49781	0.748905	+	0.662678i	$0.230580\pi$
$422$	0	0
$423$	31.7879	1.54558
$424$	0	0
$425$	32.4946	1.57622
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−13.0569	−0.630392
$430$	0	0
$431$	−11.5758	−0.557588	−0.278794	−	0.960351i	$-0.589935\pi$
$431$	−11.5758	−0.557588	−0.278794	+	0.960351i	$0.589935\pi$
$432$	0	0
$433$	−36.8000	−1.76850	−0.884249	−	0.467017i	$-0.845329\pi$
$433$	−36.8000	−1.76850	−0.884249	+	0.467017i	$0.845329\pi$
$434$	0	0
$435$	−84.0302	−4.02894
$436$	0	0
$437$	−8.65835	−0.414185
$438$	0	0
$439$	32.6519	1.55839	0.779194	−	0.626782i	$-0.215628\pi$
$439$	32.6519	1.55839	0.779194	+	0.626782i	$0.215628\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−0.881489	−0.0418808	−0.0209404	−	0.999781i	$-0.506666\pi$
$443$	−0.881489	−0.0418808	−0.0209404	+	0.999781i	$0.506666\pi$
$444$	0	0
$445$	−23.4398	−1.11115
$446$	0	0
$447$	27.6935	1.30986
$448$	0	0
$449$	−10.2990	−0.486041	−0.243021	−	0.970021i	$-0.578138\pi$
$449$	−10.2990	−0.486041	−0.243021	+	0.970021i	$0.578138\pi$
$450$	0	0
$451$	30.8259	1.45153
$452$	0	0
$453$	45.6442	2.14455
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−5.57798	−0.260927	−0.130463	−	0.991453i	$-0.541647\pi$
$457$	−5.57798	−0.260927	−0.130463	+	0.991453i	$0.541647\pi$
$458$	0	0
$459$	63.4624	2.96217
$460$	0	0
$461$	−40.9542	−1.90742	−0.953712	−	0.300720i	$-0.902773\pi$
$461$	−40.9542	−1.90742	−0.953712	+	0.300720i	$0.902773\pi$
$462$	0	0
$463$	2.44626	0.113687	0.0568437	−	0.998383i	$-0.481896\pi$
$463$	2.44626	0.113687	0.0568437	+	0.998383i	$0.481896\pi$
$464$	0	0
$465$	−55.0370	−2.55228
$466$	0	0
$467$	−40.1401	−1.85746	−0.928731	−	0.370754i	$-0.879099\pi$
$467$	−40.1401	−1.85746	−0.928731	+	0.370754i	$0.879099\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	3.06971	0.141445
$472$	0	0
$473$	−19.2198	−0.883727
$474$	0	0
$475$	−58.8831	−2.70174
$476$	0	0
$477$	57.8114	2.64700
$478$	0	0
$479$	−28.4010	−1.29767	−0.648837	−	0.760927i	$-0.724744\pi$
$479$	−28.4010	−1.29767	−0.648837	+	0.760927i	$0.724744\pi$
$480$	0	0
$481$	5.91122	0.269529
$482$	0	0
$483$	0	0
$484$	0	0
$485$	34.4409	1.56388
$486$	0	0
$487$	−23.2832	−1.05506	−0.527531	−	0.849536i	$-0.676882\pi$
$487$	−23.2832	−1.05506	−0.527531	+	0.849536i	$0.676882\pi$
$488$	0	0
$489$	−4.39241	−0.198632
$490$	0	0
$491$	−14.8105	−0.668389	−0.334195	−	0.942504i	$-0.608464\pi$
$491$	−14.8105	−0.668389	−0.334195	+	0.942504i	$0.608464\pi$
$492$	0	0
$493$	36.6578	1.65099
$494$	0	0
$495$	63.4110	2.85011
$496$	0	0
$497$	0	0
$498$	0	0
$499$	20.2189	0.905122	0.452561	−	0.891733i	$-0.350510\pi$
$499$	20.2189	0.905122	0.452561	+	0.891733i	$0.350510\pi$
$500$	0	0
$501$	19.1353	0.854902
$502$	0	0
$503$	12.9852	0.578982	0.289491	−	0.957181i	$-0.406514\pi$
$503$	12.9852	0.578982	0.289491	+	0.957181i	$0.406514\pi$
$504$	0	0
$505$	13.5854	0.604541
$506$	0	0
$507$	−33.3912	−1.48295
$508$	0	0
$509$	16.0213	0.710131	0.355065	−	0.934841i	$-0.384459\pi$
$509$	16.0213	0.710131	0.355065	+	0.934841i	$0.384459\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−115.000	−5.07736
$514$	0	0
$515$	−4.51472	−0.198942
$516$	0	0
$517$	−11.4275	−0.502580
$518$	0	0
$519$	28.9641	1.27138
$520$	0	0
$521$	−18.5855	−0.814245	−0.407123	−	0.913374i	$-0.633468\pi$
$521$	−18.5855	−0.814245	−0.407123	+	0.913374i	$0.633468\pi$
$522$	0	0
$523$	−3.32718	−0.145488	−0.0727438	−	0.997351i	$-0.523176\pi$
$523$	−3.32718	−0.145488	−0.0727438	+	0.997351i	$0.523176\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	24.0097	1.04588
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	13.4734	0.584696
$532$	0	0
$533$	−19.0234	−0.823993
$534$	0	0
$535$	32.4947	1.40487
$536$	0	0
$537$	−4.98125	−0.214957
$538$	0	0
$539$	0	0
$540$	0	0
$541$	33.1858	1.42677	0.713385	−	0.700772i	$-0.247161\pi$
$541$	33.1858	1.42677	0.713385	+	0.700772i	$0.247161\pi$
$542$	0	0
$543$	−1.42398	−0.0611089
$544$	0	0
$545$	1.29422	0.0554385
$546$	0	0
$547$	−23.3057	−0.996478	−0.498239	−	0.867040i	$-0.666020\pi$
$547$	−23.3057	−0.996478	−0.498239	+	0.867040i	$0.666020\pi$
$548$	0	0
$549$	40.9652	1.74835
$550$	0	0
$551$	−66.4273	−2.82990
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−40.7268	−1.72876
$556$	0	0
$557$	−5.60608	−0.237537	−0.118769	−	0.992922i	$-0.537895\pi$
$557$	−5.60608	−0.237537	−0.118769	+	0.992922i	$0.537895\pi$
$558$	0	0
$559$	11.8610	0.501666
$560$	0	0
$561$	−39.2440	−1.65688
$562$	0	0
$563$	−10.3041	−0.434265	−0.217133	−	0.976142i	$-0.569670\pi$
$563$	−10.3041	−0.434265	−0.217133	+	0.976142i	$0.569670\pi$
$564$	0	0
$565$	53.5003	2.25078
$566$	0	0
$567$	0	0
$568$	0	0
$569$	6.54318	0.274304	0.137152	−	0.990550i	$-0.456205\pi$
$569$	6.54318	0.274304	0.137152	+	0.990550i	$0.456205\pi$
$570$	0	0
$571$	−2.80730	−0.117482	−0.0587410	−	0.998273i	$-0.518709\pi$
$571$	−2.80730	−0.117482	−0.0587410	+	0.998273i	$0.518709\pi$
$572$	0	0
$573$	−57.4624	−2.40053
$574$	0	0
$575$	6.80073	0.283610
$576$	0	0
$577$	6.48194	0.269847	0.134923	−	0.990856i	$-0.456921\pi$
$577$	6.48194	0.269847	0.134923	+	0.990856i	$0.456921\pi$
$578$	0	0
$579$	−40.3970	−1.67884
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−20.7827	−0.860730
$584$	0	0
$585$	−39.1324	−1.61793
$586$	0	0
$587$	26.0813	1.07649	0.538245	−	0.842788i	$-0.319087\pi$
$587$	26.0813	1.07649	0.538245	+	0.842788i	$0.319087\pi$
$588$	0	0
$589$	−43.5077	−1.79270
$590$	0	0
$591$	14.2207	0.584960
$592$	0	0
$593$	−1.51092	−0.0620462	−0.0310231	−	0.999519i	$-0.509877\pi$
$593$	−1.51092	−0.0620462	−0.0310231	+	0.999519i	$0.509877\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−42.9905	−1.75949
$598$	0	0
$599$	−41.3940	−1.69131	−0.845656	−	0.533728i	$-0.820791\pi$
$599$	−41.3940	−1.69131	−0.845656	+	0.533728i	$0.820791\pi$
$600$	0	0
$601$	−28.4401	−1.16010	−0.580049	−	0.814582i	$-0.696967\pi$
$601$	−28.4401	−1.16010	−0.580049	+	0.814582i	$0.696967\pi$
$602$	0	0
$603$	95.0304	3.86994
$604$	0	0
$605$	14.9917	0.609501
$606$	0	0
$607$	3.95069	0.160354	0.0801768	−	0.996781i	$-0.474452\pi$
$607$	3.95069	0.160354	0.0801768	+	0.996781i	$0.474452\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	7.05216	0.285300
$612$	0	0
$613$	36.9389	1.49195	0.745974	−	0.665975i	$-0.231984\pi$
$613$	36.9389	1.49195	0.745974	+	0.665975i	$0.231984\pi$
$614$	0	0
$615$	131.066	5.28509
$616$	0	0
$617$	5.69561	0.229297	0.114648	−	0.993406i	$-0.463426\pi$
$617$	5.69561	0.229297	0.114648	+	0.993406i	$0.463426\pi$
$618$	0	0
$619$	−19.0162	−0.764327	−0.382163	−	0.924095i	$-0.624821\pi$
$619$	−19.0162	−0.764327	−0.382163	+	0.924095i	$0.624821\pi$
$620$	0	0
$621$	13.2819	0.532986
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−12.7537	−0.510148
$626$	0	0
$627$	71.1138	2.84001
$628$	0	0
$629$	17.7669	0.708412
$630$	0	0
$631$	−29.5641	−1.17693	−0.588464	−	0.808524i	$-0.700267\pi$
$631$	−29.5641	−1.17693	−0.588464	+	0.808524i	$0.700267\pi$
$632$	0	0
$633$	64.8814	2.57880
$634$	0	0
$635$	−64.6148	−2.56416
$636$	0	0
$637$	0	0
$638$	0	0
$639$	42.7542	1.69133
$640$	0	0
$641$	6.70647	0.264890	0.132445	−	0.991190i	$-0.457717\pi$
$641$	6.70647	0.264890	0.132445	+	0.991190i	$0.457717\pi$
$642$	0	0
$643$	22.0912	0.871193	0.435597	−	0.900142i	$-0.356537\pi$
$643$	22.0912	0.871193	0.435597	+	0.900142i	$0.356537\pi$
$644$	0	0
$645$	−81.7191	−3.21769
$646$	0	0
$647$	−14.4343	−0.567472	−0.283736	−	0.958902i	$-0.591574\pi$
$647$	−14.4343	−0.567472	−0.283736	+	0.958902i	$0.591574\pi$
$648$	0	0
$649$	−4.84357	−0.190127
$650$	0	0
$651$	0	0
$652$	0	0
$653$	9.94368	0.389126	0.194563	−	0.980890i	$-0.437671\pi$
$653$	9.94368	0.389126	0.194563	+	0.980890i	$0.437671\pi$
$654$	0	0
$655$	−4.88995	−0.191066
$656$	0	0
$657$	77.6545	3.02959
$658$	0	0
$659$	−29.8406	−1.16243	−0.581213	−	0.813751i	$-0.697422\pi$
$659$	−29.8406	−1.16243	−0.581213	+	0.813751i	$0.697422\pi$
$660$	0	0
$661$	−38.4960	−1.49732	−0.748660	−	0.662954i	$-0.769302\pi$
$661$	−38.4960	−1.49732	−0.748660	+	0.662954i	$0.769302\pi$
$662$	0	0
$663$	24.2184	0.940565
$664$	0	0
$665$	0	0
$666$	0	0
$667$	7.67205	0.297063
$668$	0	0
$669$	−51.0946	−1.97543
$670$	0	0
$671$	−14.7266	−0.568515
$672$	0	0
$673$	−25.9336	−0.999668	−0.499834	−	0.866121i	$-0.666606\pi$
$673$	−25.9336	−0.999668	−0.499834	+	0.866121i	$0.666606\pi$
$674$	0	0
$675$	90.3270	3.47669
$676$	0	0
$677$	−2.75584	−0.105915	−0.0529577	−	0.998597i	$-0.516865\pi$
$677$	−2.75584	−0.105915	−0.0529577	+	0.998597i	$0.516865\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	52.3743	2.00699
$682$	0	0
$683$	36.4205	1.39359	0.696796	−	0.717269i	$-0.254608\pi$
$683$	36.4205	1.39359	0.696796	+	0.717269i	$0.254608\pi$
$684$	0	0
$685$	55.7005	2.12821
$686$	0	0
$687$	43.1438	1.64604
$688$	0	0
$689$	12.8255	0.488612
$690$	0	0
$691$	2.47424	0.0941245	0.0470622	−	0.998892i	$-0.485014\pi$
$691$	2.47424	0.0941245	0.0470622	+	0.998892i	$0.485014\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	26.0820	0.989347
$696$	0	0
$697$	−57.1770	−2.16573
$698$	0	0
$699$	11.1760	0.422715
$700$	0	0
$701$	−35.6538	−1.34663	−0.673313	−	0.739358i	$-0.735129\pi$
$701$	−35.6538	−1.34663	−0.673313	+	0.739358i	$0.735129\pi$
$702$	0	0
$703$	−32.1952	−1.21427
$704$	0	0
$705$	−48.5876	−1.82991
$706$	0	0
$707$	0	0
$708$	0	0
$709$	33.9263	1.27413	0.637064	−	0.770811i	$-0.280149\pi$
$709$	33.9263	1.27413	0.637064	+	0.770811i	$0.280149\pi$
$710$	0	0
$711$	14.4234	0.540918
$712$	0	0
$713$	5.02494	0.188186
$714$	0	0
$715$	14.0677	0.526104
$716$	0	0
$717$	−66.1325	−2.46976
$718$	0	0
$719$	20.1883	0.752898	0.376449	−	0.926437i	$-0.377145\pi$
$719$	20.1883	0.752898	0.376449	+	0.926437i	$0.377145\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	12.6181	0.469273
$724$	0	0
$725$	52.1756	1.93775
$726$	0	0
$727$	−14.7092	−0.545536	−0.272768	−	0.962080i	$-0.587939\pi$
$727$	−14.7092	−0.545536	−0.272768	+	0.962080i	$0.587939\pi$
$728$	0	0
$729$	22.3666	0.828392
$730$	0	0
$731$	35.6496	1.31855
$732$	0	0
$733$	−25.3029	−0.934583	−0.467292	−	0.884103i	$-0.654770\pi$
$733$	−25.3029	−0.934583	−0.467292	+	0.884103i	$0.654770\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−34.1626	−1.25839
$738$	0	0
$739$	3.12066	0.114796	0.0573978	−	0.998351i	$-0.481720\pi$
$739$	3.12066	0.114796	0.0573978	+	0.998351i	$0.481720\pi$
$740$	0	0
$741$	−43.8860	−1.61219
$742$	0	0
$743$	4.89456	0.179564	0.0897820	−	0.995961i	$-0.471383\pi$
$743$	4.89456	0.179564	0.0897820	+	0.995961i	$0.471383\pi$
$744$	0	0
$745$	−29.8375	−1.09316
$746$	0	0
$747$	−24.6046	−0.900233
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−14.9050	−0.543893	−0.271946	−	0.962312i	$-0.587667\pi$
$751$	−14.9050	−0.543893	−0.271946	+	0.962312i	$0.587667\pi$
$752$	0	0
$753$	95.0614	3.46423
$754$	0	0
$755$	−49.1780	−1.78977
$756$	0	0
$757$	3.57059	0.129775	0.0648877	−	0.997893i	$-0.479331\pi$
$757$	3.57059	0.129775	0.0648877	+	0.997893i	$0.479331\pi$
$758$	0	0
$759$	−8.21332	−0.298125
$760$	0	0
$761$	1.13853	0.0412717	0.0206359	−	0.999787i	$-0.493431\pi$
$761$	1.13853	0.0412717	0.0206359	+	0.999787i	$0.493431\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−117.617	−4.25246
$766$	0	0
$767$	2.98908	0.107929
$768$	0	0
$769$	27.3160	0.985040	0.492520	−	0.870301i	$-0.336076\pi$
$769$	27.3160	0.985040	0.492520	+	0.870301i	$0.336076\pi$
$770$	0	0
$771$	−10.9977	−0.396071
$772$	0	0
$773$	−8.23845	−0.296316	−0.148158	−	0.988964i	$-0.547334\pi$
$773$	−8.23845	−0.296316	−0.148158	+	0.988964i	$0.547334\pi$
$774$	0	0
$775$	34.1733	1.22754
$776$	0	0
$777$	0	0
$778$	0	0
$779$	103.610	3.71221
$780$	0	0
$781$	−15.3697	−0.549973
$782$	0	0
$783$	101.900	3.64160
$784$	0	0
$785$	−3.30736	−0.118045
$786$	0	0
$787$	28.3493	1.01055	0.505273	−	0.862960i	$-0.331392\pi$
$787$	28.3493	1.01055	0.505273	+	0.862960i	$0.331392\pi$
$788$	0	0
$789$	−25.6821	−0.914308
$790$	0	0
$791$	0	0
$792$	0	0
$793$	9.08815	0.322730
$794$	0	0
$795$	−88.3642	−3.13395
$796$	0	0
$797$	28.7795	1.01942	0.509712	−	0.860345i	$-0.329752\pi$
$797$	28.7795	1.01942	0.509712	+	0.860345i	$0.329752\pi$
$798$	0	0
$799$	21.1961	0.749865
$800$	0	0
$801$	48.8945	1.72760
$802$	0	0
$803$	−27.9161	−0.985137
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−14.1389	−0.497713
$808$	0	0
$809$	25.5797	0.899334	0.449667	−	0.893196i	$-0.351543\pi$
$809$	25.5797	0.899334	0.449667	+	0.893196i	$0.351543\pi$
$810$	0	0
$811$	34.4158	1.20850	0.604250	−	0.796794i	$-0.293473\pi$
$811$	34.4158	1.20850	0.604250	+	0.796794i	$0.293473\pi$
$812$	0	0
$813$	76.6016	2.68654
$814$	0	0
$815$	4.73247	0.165771
$816$	0	0
$817$	−64.6003	−2.26008
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−36.2951	−1.26671	−0.633354	−	0.773862i	$-0.718322\pi$
$821$	−36.2951	−1.26671	−0.633354	+	0.773862i	$0.718322\pi$
$822$	0	0
$823$	−9.63499	−0.335855	−0.167927	−	0.985799i	$-0.553707\pi$
$823$	−9.63499	−0.335855	−0.167927	+	0.985799i	$0.553707\pi$
$824$	0	0
$825$	−55.8566	−1.94468
$826$	0	0
$827$	−18.0345	−0.627120	−0.313560	−	0.949568i	$-0.601522\pi$
$827$	−18.0345	−0.627120	−0.313560	+	0.949568i	$0.601522\pi$
$828$	0	0
$829$	31.5070	1.09428	0.547142	−	0.837040i	$-0.315716\pi$
$829$	31.5070	1.09428	0.547142	+	0.837040i	$0.315716\pi$
$830$	0	0
$831$	−1.05345	−0.0365438
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−20.6168	−0.713472
$836$	0	0
$837$	66.7410	2.30691
$838$	0	0
$839$	7.88698	0.272289	0.136144	−	0.990689i	$-0.456529\pi$
$839$	7.88698	0.272289	0.136144	+	0.990689i	$0.456529\pi$
$840$	0	0
$841$	29.8604	1.02967
$842$	0	0
$843$	82.0951	2.82751
$844$	0	0
$845$	35.9763	1.23762
$846$	0	0
$847$	0	0
$848$	0	0
$849$	69.5397	2.38660
$850$	0	0
$851$	3.71840	0.127465
$852$	0	0
$853$	27.4196	0.938830	0.469415	−	0.882978i	$-0.344465\pi$
$853$	27.4196	0.938830	0.469415	+	0.882978i	$0.344465\pi$
$854$	0	0
$855$	213.133	7.28900
$856$	0	0
$857$	−50.6512	−1.73021	−0.865105	−	0.501590i	$-0.832749\pi$
$857$	−50.6512	−1.73021	−0.865105	+	0.501590i	$0.832749\pi$
$858$	0	0
$859$	−52.0138	−1.77469	−0.887345	−	0.461107i	$-0.847453\pi$
$859$	−52.0138	−1.77469	−0.887345	+	0.461107i	$0.847453\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−14.8208	−0.504506	−0.252253	−	0.967661i	$-0.581172\pi$
$863$	−14.8208	−0.504506	−0.252253	+	0.967661i	$0.581172\pi$
$864$	0	0
$865$	−31.2065	−1.06105
$866$	0	0
$867$	18.5889	0.631313
$868$	0	0
$869$	−5.18506	−0.175891
$870$	0	0
$871$	21.0825	0.714354
$872$	0	0
$873$	−71.8424	−2.43150
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−26.9784	−0.910996	−0.455498	−	0.890237i	$-0.650539\pi$
$877$	−26.9784	−0.910996	−0.455498	+	0.890237i	$0.650539\pi$
$878$	0	0
$879$	48.2997	1.62911
$880$	0	0
$881$	−6.39919	−0.215594	−0.107797	−	0.994173i	$-0.534380\pi$
$881$	−6.39919	−0.215594	−0.107797	+	0.994173i	$0.534380\pi$
$882$	0	0
$883$	29.5755	0.995295	0.497648	−	0.867379i	$-0.334197\pi$
$883$	29.5755	0.995295	0.497648	+	0.867379i	$0.334197\pi$
$884$	0	0
$885$	−20.5940	−0.692259
$886$	0	0
$887$	−17.8147	−0.598159	−0.299080	−	0.954228i	$-0.596680\pi$
$887$	−17.8147	−0.598159	−0.299080	+	0.954228i	$0.596680\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−53.7116	−1.79941
$892$	0	0
$893$	−38.4093	−1.28532
$894$	0	0
$895$	5.36690	0.179396
$896$	0	0
$897$	5.06863	0.169237
$898$	0	0
$899$	38.5516	1.28577
$900$	0	0
$901$	38.5485	1.28424
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.53423	0.0509994
$906$	0	0
$907$	46.3725	1.53977	0.769887	−	0.638180i	$-0.220312\pi$
$907$	46.3725	1.53977	0.769887	+	0.638180i	$0.220312\pi$
$908$	0	0
$909$	−28.3386	−0.939931
$910$	0	0
$911$	−16.1122	−0.533821	−0.266910	−	0.963721i	$-0.586003\pi$
$911$	−16.1122	−0.533821	−0.266910	+	0.963721i	$0.586003\pi$
$912$	0	0
$913$	8.84511	0.292731
$914$	0	0
$915$	−62.6150	−2.06999
$916$	0	0
$917$	0	0
$918$	0	0
$919$	31.6441	1.04384	0.521922	−	0.852993i	$-0.325215\pi$
$919$	31.6441	1.04384	0.521922	+	0.852993i	$0.325215\pi$
$920$	0	0
$921$	36.0869	1.18911
$922$	0	0
$923$	9.48503	0.312204
$924$	0	0
$925$	25.2879	0.831459
$926$	0	0
$927$	9.41753	0.309312
$928$	0	0
$929$	18.2983	0.600349	0.300175	−	0.953884i	$-0.402955\pi$
$929$	18.2983	0.600349	0.300175	+	0.953884i	$0.402955\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	9.99099	0.327090
$934$	0	0
$935$	42.2823	1.38278
$936$	0	0
$937$	−21.6984	−0.708856	−0.354428	−	0.935083i	$-0.615324\pi$
$937$	−21.6984	−0.708856	−0.354428	+	0.935083i	$0.615324\pi$
$938$	0	0
$939$	29.1017	0.949699
$940$	0	0
$941$	50.9481	1.66086	0.830430	−	0.557123i	$-0.188095\pi$
$941$	50.9481	1.66086	0.830430	+	0.557123i	$0.188095\pi$
$942$	0	0
$943$	−11.9665	−0.389682
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−48.6818	−1.58195	−0.790973	−	0.611852i	$-0.790425\pi$
$947$	−48.6818	−1.58195	−0.790973	+	0.611852i	$0.790425\pi$
$948$	0	0
$949$	17.2277	0.559234
$950$	0	0
$951$	−107.777	−3.49492
$952$	0	0
$953$	18.3462	0.594291	0.297146	−	0.954832i	$-0.403965\pi$
$953$	18.3462	0.594291	0.297146	+	0.954832i	$0.403965\pi$
$954$	0	0
$955$	61.9112	2.00340
$956$	0	0
$957$	−63.0130	−2.03692
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−5.74997	−0.185483
$962$	0	0
$963$	−67.7827	−2.18427
$964$	0	0
$965$	43.5246	1.40111
$966$	0	0
$967$	−25.7034	−0.826567	−0.413283	−	0.910602i	$-0.635618\pi$
$967$	−25.7034	−0.826567	−0.413283	+	0.910602i	$0.635618\pi$
$968$	0	0
$969$	−131.905	−4.23738
$970$	0	0
$971$	−9.27074	−0.297512	−0.148756	−	0.988874i	$-0.547527\pi$
$971$	−9.27074	−0.297512	−0.148756	+	0.988874i	$0.547527\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	34.4704	1.10394
$976$	0	0
$977$	25.7630	0.824230	0.412115	−	0.911132i	$-0.364790\pi$
$977$	25.7630	0.824230	0.412115	+	0.911132i	$0.364790\pi$
$978$	0	0
$979$	−17.5771	−0.561767
$980$	0	0
$981$	−2.69970	−0.0861949
$982$	0	0
$983$	−1.11509	−0.0355660	−0.0177830	−	0.999842i	$-0.505661\pi$
$983$	−1.11509	−0.0355660	−0.0177830	+	0.999842i	$0.505661\pi$
$984$	0	0
$985$	−15.3216	−0.488188
$986$	0	0
$987$	0	0
$988$	0	0
$989$	7.46104	0.237247
$990$	0	0
$991$	56.9530	1.80917	0.904586	−	0.426291i	$-0.140180\pi$
$991$	56.9530	1.80917	0.904586	+	0.426291i	$0.140180\pi$
$992$	0	0
$993$	33.9398	1.07705
$994$	0	0
$995$	46.3188	1.46841
$996$	0	0
$997$	26.6313	0.843423	0.421711	−	0.906730i	$-0.361430\pi$
$997$	26.6313	0.843423	0.421711	+	0.906730i	$0.361430\pi$
$998$	0	0
$999$	49.3876	1.56255

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4508.2.a.g.1.5		5
7.6	odd	2		644.2.a.c.1.1	✓	5
21.20	even	2		5796.2.a.s.1.1		5
28.27	even	2		2576.2.a.bc.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
644.2.a.c.1.1	✓	5	7.6	odd	2
2576.2.a.bc.1.5		5	28.27	even	2
4508.2.a.g.1.5		5	1.1	even	1	trivial
5796.2.a.s.1.1		5	21.20	even	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:	\( N \)	\(=\)	\( 4508 = 2^{2} \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4508.a (trivial)

\(f(q)\)	\(=\)	\(q+3.18838 q^{3} -3.43522 q^{5} +7.16574 q^{9} +O(q^{10})\)	\(q+3.18838 q^{3} -3.43522 q^{5} +7.16574 q^{9} -2.57602 q^{11} +1.58972 q^{13} -10.9528 q^{15} +4.77810 q^{17} -8.65835 q^{19} +1.00000 q^{23} +6.80073 q^{25} +13.2819 q^{27} +7.67205 q^{29} +5.02494 q^{31} -8.21332 q^{33} +3.71840 q^{37} +5.06863 q^{39} -11.9665 q^{41} +7.46104 q^{43} -24.6159 q^{45} +4.43610 q^{47} +15.2344 q^{51} +8.06775 q^{53} +8.84919 q^{55} -27.6061 q^{57} +1.88025 q^{59} +5.71682 q^{61} -5.46104 q^{65} +13.2618 q^{67} +3.18838 q^{69} +5.96647 q^{71} +10.8369 q^{73} +21.6833 q^{75} +2.01282 q^{79} +20.8506 q^{81} -3.43364 q^{83} -16.4138 q^{85} +24.4614 q^{87} +6.82337 q^{89} +16.0214 q^{93} +29.7433 q^{95} -10.0258 q^{97} -18.4591 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{3} - 4 q^{5} + 10 q^{9}+O(q^{10}) \) 5 * q - q^3 - 4 * q^5 + 10 * q^9	\( 5 q - q^{3} - 4 q^{5} + 10 q^{9} + 2 q^{11} - 3 q^{13} - 6 q^{15} - 4 q^{17} - 6 q^{19} + 5 q^{23} + 15 q^{25} - q^{27} + 5 q^{29} + q^{31} + 22 q^{37} - q^{39} - 15 q^{41} + 12 q^{43} - 36 q^{45} + 21 q^{47} + 24 q^{51} + 2 q^{53} + 22 q^{55} - 34 q^{57} + 12 q^{61} - 2 q^{65} + 22 q^{67} - q^{69} - 15 q^{71} - 17 q^{73} + 47 q^{75} + 2 q^{79} + 37 q^{81} + 16 q^{83} - 8 q^{85} + 33 q^{87} + 24 q^{89} + 5 q^{93} - 8 q^{95} - 38 q^{97} - 36 q^{99}+O(q^{100}) \) 5 * q - q^3 - 4 * q^5 + 10 * q^9 + 2 * q^11 - 3 * q^13 - 6 * q^15 - 4 * q^17 - 6 * q^19 + 5 * q^23 + 15 * q^25 - q^27 + 5 * q^29 + q^31 + 22 * q^37 - q^39 - 15 * q^41 + 12 * q^43 - 36 * q^45 + 21 * q^47 + 24 * q^51 + 2 * q^53 + 22 * q^55 - 34 * q^57 + 12 * q^61 - 2 * q^65 + 22 * q^67 - q^69 - 15 * q^71 - 17 * q^73 + 47 * q^75 + 2 * q^79 + 37 * q^81 + 16 * q^83 - 8 * q^85 + 33 * q^87 + 24 * q^89 + 5 * q^93 - 8 * q^95 - 38 * q^97 - 36 * q^99

Introduction

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