LMFDB - Embedded newform 5265.2.a.bf.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5265,2,Mod(1,5265)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5265, 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("5265.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5265 = 3^{4} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5265.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:	$42.0412366642$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 8x^{6} + 31x^{5} - x^{4} - 70x^{3} + 66x^{2} - 19x + 1 $ x^8 - 3x^7 - 8x^6 + 31x^5 - x^4 - 70x^3 + 66x^2 - 19x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 585)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.58610$ of defining polynomial
Character	$\chi$	$=$	5265.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.58610 q^{2} +4.68793 q^{4} -1.00000 q^{5} -4.27489 q^{7} -6.95128 q^{8} +O(q^{10})$	\(q-2.58610 q^{2} +4.68793 q^{4} -1.00000 q^{5} -4.27489 q^{7} -6.95128 q^{8} +2.58610 q^{10} +4.38446 q^{11} -1.00000 q^{13} +11.0553 q^{14} +8.60086 q^{16} +0.619151 q^{17} -2.61288 q^{19} -4.68793 q^{20} -11.3387 q^{22} +5.42887 q^{23} +1.00000 q^{25} +2.58610 q^{26} -20.0404 q^{28} +6.95774 q^{29} -6.88609 q^{31} -8.34015 q^{32} -1.60119 q^{34} +4.27489 q^{35} -4.02164 q^{37} +6.75717 q^{38} +6.95128 q^{40} -4.07427 q^{41} -10.0369 q^{43} +20.5541 q^{44} -14.0396 q^{46} -0.405987 q^{47} +11.2747 q^{49} -2.58610 q^{50} -4.68793 q^{52} -8.71205 q^{53} -4.38446 q^{55} +29.7160 q^{56} -17.9934 q^{58} -10.7215 q^{59} +5.07217 q^{61} +17.8081 q^{62} +4.36679 q^{64} +1.00000 q^{65} +14.5579 q^{67} +2.90254 q^{68} -11.0553 q^{70} +6.57523 q^{71} +16.7313 q^{73} +10.4004 q^{74} -12.2490 q^{76} -18.7431 q^{77} +14.5458 q^{79} -8.60086 q^{80} +10.5365 q^{82} -3.04113 q^{83} -0.619151 q^{85} +25.9565 q^{86} -30.4776 q^{88} +7.27070 q^{89} +4.27489 q^{91} +25.4502 q^{92} +1.04992 q^{94} +2.61288 q^{95} -9.63475 q^{97} -29.1576 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 3 q^{2} + 9 q^{4} - 8 q^{5} - 11 q^{7} - 6 q^{8}+O(q^{10}) $ 8 * q + 3 * q^2 + 9 * q^4 - 8 * q^5 - 11 * q^7 - 6 * q^8	$ 8 q + 3 q^{2} + 9 q^{4} - 8 q^{5} - 11 q^{7} - 6 q^{8} - 3 q^{10} + 6 q^{11} - 8 q^{13} + 10 q^{14} + 11 q^{16} - 2 q^{17} - 10 q^{19} - 9 q^{20} + 3 q^{22} + 6 q^{23} + 8 q^{25} - 3 q^{26} - 34 q^{28} + 14 q^{29} - 31 q^{31} + q^{32} - 7 q^{34} + 11 q^{35} + q^{37} + 9 q^{38} + 6 q^{40} - 12 q^{41} + 15 q^{43} + 16 q^{44} - 32 q^{46} - 18 q^{47} + 17 q^{49} + 3 q^{50} - 9 q^{52} + 2 q^{53} - 6 q^{55} + 16 q^{56} - 42 q^{58} + 24 q^{59} - 9 q^{61} - 20 q^{62} - 30 q^{64} + 8 q^{65} - 18 q^{67} - 14 q^{68} - 10 q^{70} + 10 q^{71} + 6 q^{73} - 37 q^{74} - 53 q^{76} - 34 q^{77} - 3 q^{79} - 11 q^{80} - 34 q^{82} - 10 q^{83} + 2 q^{85} + 60 q^{86} - 14 q^{88} - 13 q^{89} + 11 q^{91} + 5 q^{92} + 17 q^{94} + 10 q^{95} - 34 q^{97} - 30 q^{98}+O(q^{100}) $ 8 * q + 3 * q^2 + 9 * q^4 - 8 * q^5 - 11 * q^7 - 6 * q^8 - 3 * q^10 + 6 * q^11 - 8 * q^13 + 10 * q^14 + 11 * q^16 - 2 * q^17 - 10 * q^19 - 9 * q^20 + 3 * q^22 + 6 * q^23 + 8 * q^25 - 3 * q^26 - 34 * q^28 + 14 * q^29 - 31 * q^31 + q^32 - 7 * q^34 + 11 * q^35 + q^37 + 9 * q^38 + 6 * q^40 - 12 * q^41 + 15 * q^43 + 16 * q^44 - 32 * q^46 - 18 * q^47 + 17 * q^49 + 3 * q^50 - 9 * q^52 + 2 * q^53 - 6 * q^55 + 16 * q^56 - 42 * q^58 + 24 * q^59 - 9 * q^61 - 20 * q^62 - 30 * q^64 + 8 * q^65 - 18 * q^67 - 14 * q^68 - 10 * q^70 + 10 * q^71 + 6 * q^73 - 37 * q^74 - 53 * q^76 - 34 * q^77 - 3 * q^79 - 11 * q^80 - 34 * q^82 - 10 * q^83 + 2 * q^85 + 60 * q^86 - 14 * q^88 - 13 * q^89 + 11 * q^91 + 5 * q^92 + 17 * q^94 + 10 * q^95 - 34 * q^97 - 30 * q^98

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 $

$2$	−2.58610	−1.82865	−0.914326	−	0.404979i	$-0.867279\pi$
$2$	−2.58610	−1.82865	−0.914326	+	0.404979i	$0.867279\pi$
$3$	0	0
$3$	0	0
$4$	4.68793	2.34397
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.27489	−1.61576	−0.807879	−	0.589349i	$-0.799384\pi$
$7$	−4.27489	−1.61576	−0.807879	+	0.589349i	$0.799384\pi$
$8$	−6.95128	−2.45765
$9$	0	0
$10$	2.58610	0.817798
$11$	4.38446	1.32197	0.660983	−	0.750401i	$-0.270140\pi$
$11$	4.38446	1.32197	0.660983	+	0.750401i	$0.270140\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	11.0553	2.95466
$15$	0	0
$16$	8.60086	2.15021
$17$	0.619151	0.150166	0.0750831	−	0.997177i	$-0.476078\pi$
$17$	0.619151	0.150166	0.0750831	+	0.997177i	$0.476078\pi$
$18$	0	0
$19$	−2.61288	−0.599435	−0.299718	−	0.954028i	$-0.596892\pi$
$19$	−2.61288	−0.599435	−0.299718	+	0.954028i	$0.596892\pi$
$20$	−4.68793	−1.04825
$21$	0	0
$22$	−11.3387	−2.41741
$23$	5.42887	1.13200	0.565998	−	0.824406i	$-0.308491\pi$
$23$	5.42887	1.13200	0.565998	+	0.824406i	$0.308491\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.58610	0.507177
$27$	0	0
$28$	−20.0404	−3.78728
$29$	6.95774	1.29202	0.646010	−	0.763329i	$-0.276436\pi$
$29$	6.95774	1.29202	0.646010	+	0.763329i	$0.276436\pi$
$30$	0	0
$31$	−6.88609	−1.23678	−0.618389	−	0.785872i	$-0.712214\pi$
$31$	−6.88609	−1.23678	−0.618389	+	0.785872i	$0.712214\pi$
$32$	−8.34015	−1.47434
$33$	0	0
$34$	−1.60119	−0.274602
$35$	4.27489	0.722589
$36$	0	0
$37$	−4.02164	−0.661154	−0.330577	−	0.943779i	$-0.607243\pi$
$37$	−4.02164	−0.661154	−0.330577	+	0.943779i	$0.607243\pi$
$38$	6.75717	1.09616
$39$	0	0
$40$	6.95128	1.09909
$41$	−4.07427	−0.636295	−0.318147	−	0.948041i	$-0.603061\pi$
$41$	−4.07427	−0.636295	−0.318147	+	0.948041i	$0.603061\pi$
$42$	0	0
$43$	−10.0369	−1.53061	−0.765307	−	0.643666i	$-0.777413\pi$
$43$	−10.0369	−1.53061	−0.765307	+	0.643666i	$0.777413\pi$
$44$	20.5541	3.09864
$45$	0	0
$46$	−14.0396	−2.07003
$47$	−0.405987	−0.0592192	−0.0296096	−	0.999562i	$-0.509426\pi$
$47$	−0.405987	−0.0592192	−0.0296096	+	0.999562i	$0.509426\pi$
$48$	0	0
$49$	11.2747	1.61067
$50$	−2.58610	−0.365730
$51$	0	0
$52$	−4.68793	−0.650099
$53$	−8.71205	−1.19669	−0.598346	−	0.801238i	$-0.704175\pi$
$53$	−8.71205	−1.19669	−0.598346	+	0.801238i	$0.704175\pi$
$54$	0	0
$55$	−4.38446	−0.591201
$56$	29.7160	3.97096
$57$	0	0
$58$	−17.9934	−2.36265
$59$	−10.7215	−1.39581	−0.697907	−	0.716188i	$-0.745885\pi$
$59$	−10.7215	−1.39581	−0.697907	+	0.716188i	$0.745885\pi$
$60$	0	0
$61$	5.07217	0.649425	0.324713	−	0.945813i	$-0.394732\pi$
$61$	5.07217	0.649425	0.324713	+	0.945813i	$0.394732\pi$
$62$	17.8081	2.26164
$63$	0	0
$64$	4.36679	0.545849
$65$	1.00000	0.124035
$66$	0	0
$67$	14.5579	1.77853	0.889264	−	0.457395i	$-0.151217\pi$
$67$	14.5579	1.77853	0.889264	+	0.457395i	$0.151217\pi$
$68$	2.90254	0.351985
$69$	0	0
$70$	−11.0553	−1.32136
$71$	6.57523	0.780336	0.390168	−	0.920744i	$-0.372417\pi$
$71$	6.57523	0.780336	0.390168	+	0.920744i	$0.372417\pi$
$72$	0	0
$73$	16.7313	1.95825	0.979125	−	0.203257i	$-0.0651527\pi$
$73$	16.7313	1.95825	0.979125	+	0.203257i	$0.0651527\pi$
$74$	10.4004	1.20902
$75$	0	0
$76$	−12.2490	−1.40506
$77$	−18.7431	−2.13598
$78$	0	0
$79$	14.5458	1.63653	0.818263	−	0.574844i	$-0.194937\pi$
$79$	14.5458	1.63653	0.818263	+	0.574844i	$0.194937\pi$
$80$	−8.60086	−0.961605
$81$	0	0
$82$	10.5365	1.16356
$83$	−3.04113	−0.333808	−0.166904	−	0.985973i	$-0.553377\pi$
$83$	−3.04113	−0.333808	−0.166904	+	0.985973i	$0.553377\pi$
$84$	0	0
$85$	−0.619151	−0.0671564
$86$	25.9565	2.79896
$87$	0	0
$88$	−30.4776	−3.24893
$89$	7.27070	0.770693	0.385347	−	0.922772i	$-0.374082\pi$
$89$	7.27070	0.770693	0.385347	+	0.922772i	$0.374082\pi$
$90$	0	0
$91$	4.27489	0.448131
$92$	25.4502	2.65336
$93$	0	0
$94$	1.04992	0.108291
$95$	2.61288	0.268076
$96$	0	0
$97$	−9.63475	−0.978261	−0.489130	−	0.872211i	$-0.662686\pi$
$97$	−9.63475	−0.978261	−0.489130	+	0.872211i	$0.662686\pi$
$98$	−29.1576	−2.94536
$99$	0	0
$100$	4.68793	0.468793
$101$	7.79952	0.776081	0.388041	−	0.921642i	$-0.373152\pi$
$101$	7.79952	0.776081	0.388041	+	0.921642i	$0.373152\pi$
$102$	0	0
$103$	0.993241	0.0978670	0.0489335	−	0.998802i	$-0.484418\pi$
$103$	0.993241	0.0978670	0.0489335	+	0.998802i	$0.484418\pi$
$104$	6.95128	0.681629
$105$	0	0
$106$	22.5303	2.18833
$107$	4.24171	0.410062	0.205031	−	0.978755i	$-0.434270\pi$
$107$	4.24171	0.410062	0.205031	+	0.978755i	$0.434270\pi$
$108$	0	0
$109$	4.72247	0.452330	0.226165	−	0.974089i	$-0.427381\pi$
$109$	4.72247	0.452330	0.226165	+	0.974089i	$0.427381\pi$
$110$	11.3387	1.08110
$111$	0	0
$112$	−36.7677	−3.47422
$113$	−3.55408	−0.334339	−0.167170	−	0.985928i	$-0.553463\pi$
$113$	−3.55408	−0.334339	−0.167170	+	0.985928i	$0.553463\pi$
$114$	0	0
$115$	−5.42887	−0.506244
$116$	32.6174	3.02845
$117$	0	0
$118$	27.7268	2.55246
$119$	−2.64681	−0.242632
$120$	0	0
$121$	8.22353	0.747594
$122$	−13.1172	−1.18757
$123$	0	0
$124$	−32.2815	−2.89897
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−13.6413	−1.21047	−0.605234	−	0.796048i	$-0.706920\pi$
$127$	−13.6413	−1.21047	−0.605234	+	0.796048i	$0.706920\pi$
$128$	5.38732	0.476177
$129$	0	0
$130$	−2.58610	−0.226816
$131$	8.36237	0.730623	0.365312	−	0.930885i	$-0.380962\pi$
$131$	8.36237	0.730623	0.365312	+	0.930885i	$0.380962\pi$
$132$	0	0
$133$	11.1698	0.968542
$134$	−37.6482	−3.25231
$135$	0	0
$136$	−4.30389	−0.369056
$137$	−6.75088	−0.576767	−0.288383	−	0.957515i	$-0.593118\pi$
$137$	−6.75088	−0.576767	−0.288383	+	0.957515i	$0.593118\pi$
$138$	0	0
$139$	3.50577	0.297355	0.148678	−	0.988886i	$-0.452498\pi$
$139$	3.50577	0.297355	0.148678	+	0.988886i	$0.452498\pi$
$140$	20.0404	1.69372
$141$	0	0
$142$	−17.0042	−1.42696
$143$	−4.38446	−0.366647
$144$	0	0
$145$	−6.95774	−0.577809
$146$	−43.2689	−3.58096
$147$	0	0
$148$	−18.8532	−1.54972
$149$	12.2790	1.00594	0.502968	−	0.864305i	$-0.332241\pi$
$149$	12.2790	1.00594	0.502968	+	0.864305i	$0.332241\pi$
$150$	0	0
$151$	−16.6964	−1.35873	−0.679367	−	0.733798i	$-0.737746\pi$
$151$	−16.6964	−1.35873	−0.679367	+	0.733798i	$0.737746\pi$
$152$	18.1628	1.47320
$153$	0	0
$154$	48.4717	3.90596
$155$	6.88609	0.553104
$156$	0	0
$157$	7.99688	0.638220	0.319110	−	0.947718i	$-0.396616\pi$
$157$	7.99688	0.638220	0.319110	+	0.947718i	$0.396616\pi$
$158$	−37.6168	−2.99264
$159$	0	0
$160$	8.34015	0.659347
$161$	−23.2078	−1.82903
$162$	0	0
$163$	−17.3723	−1.36071	−0.680353	−	0.732885i	$-0.738173\pi$
$163$	−17.3723	−1.36071	−0.680353	+	0.732885i	$0.738173\pi$
$164$	−19.0999	−1.49145
$165$	0	0
$166$	7.86468	0.610418
$167$	4.27589	0.330879	0.165439	−	0.986220i	$-0.447096\pi$
$167$	4.27589	0.330879	0.165439	+	0.986220i	$0.447096\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	1.60119	0.122806
$171$	0	0
$172$	−47.0523	−3.58771
$173$	12.9559	0.985016	0.492508	−	0.870308i	$-0.336080\pi$
$173$	12.9559	0.985016	0.492508	+	0.870308i	$0.336080\pi$
$174$	0	0
$175$	−4.27489	−0.323152
$176$	37.7101	2.84251
$177$	0	0
$178$	−18.8028	−1.40933
$179$	18.8530	1.40914	0.704571	−	0.709633i	$-0.251139\pi$
$179$	18.8530	1.40914	0.704571	+	0.709633i	$0.251139\pi$
$180$	0	0
$181$	−21.2593	−1.58019	−0.790097	−	0.612982i	$-0.789970\pi$
$181$	−21.2593	−1.58019	−0.790097	+	0.612982i	$0.789970\pi$
$182$	−11.0553	−0.819475
$183$	0	0
$184$	−37.7375	−2.78205
$185$	4.02164	0.295677
$186$	0	0
$187$	2.71465	0.198515
$188$	−1.90324	−0.138808
$189$	0	0
$190$	−6.75717	−0.490217
$191$	−15.4170	−1.11553	−0.557767	−	0.829998i	$-0.688342\pi$
$191$	−15.4170	−1.11553	−0.557767	+	0.829998i	$0.688342\pi$
$192$	0	0
$193$	1.88802	0.135902	0.0679512	−	0.997689i	$-0.478354\pi$
$193$	1.88802	0.135902	0.0679512	+	0.997689i	$0.478354\pi$
$194$	24.9165	1.78890
$195$	0	0
$196$	52.8551	3.77536
$197$	−10.8597	−0.773723	−0.386862	−	0.922138i	$-0.626441\pi$
$197$	−10.8597	−0.773723	−0.386862	+	0.922138i	$0.626441\pi$
$198$	0	0
$199$	−1.20110	−0.0851436	−0.0425718	−	0.999093i	$-0.513555\pi$
$199$	−1.20110	−0.0851436	−0.0425718	+	0.999093i	$0.513555\pi$
$200$	−6.95128	−0.491529
$201$	0	0
$202$	−20.1704	−1.41918
$203$	−29.7436	−2.08759
$204$	0	0
$205$	4.07427	0.284560
$206$	−2.56863	−0.178965
$207$	0	0
$208$	−8.60086	−0.596362
$209$	−11.4561	−0.792433
$210$	0	0
$211$	−0.235212	−0.0161926	−0.00809632	−	0.999967i	$-0.502577\pi$
$211$	−0.235212	−0.0161926	−0.00809632	+	0.999967i	$0.502577\pi$
$212$	−40.8415	−2.80501
$213$	0	0
$214$	−10.9695	−0.749861
$215$	10.0369	0.684511
$216$	0	0
$217$	29.4373	1.99833
$218$	−12.2128	−0.827154
$219$	0	0
$220$	−20.5541	−1.38576
$221$	−0.619151	−0.0416486
$222$	0	0
$223$	−16.7549	−1.12199	−0.560997	−	0.827818i	$-0.689582\pi$
$223$	−16.7549	−1.12199	−0.560997	+	0.827818i	$0.689582\pi$
$224$	35.6533	2.38218
$225$	0	0
$226$	9.19121	0.611390
$227$	22.8003	1.51331	0.756656	−	0.653813i	$-0.226832\pi$
$227$	22.8003	1.51331	0.756656	+	0.653813i	$0.226832\pi$
$228$	0	0
$229$	−6.97478	−0.460906	−0.230453	−	0.973083i	$-0.574021\pi$
$229$	−6.97478	−0.460906	−0.230453	+	0.973083i	$0.574021\pi$
$230$	14.0396	0.925745
$231$	0	0
$232$	−48.3652	−3.17533
$233$	−8.63201	−0.565502	−0.282751	−	0.959193i	$-0.591247\pi$
$233$	−8.63201	−0.565502	−0.282751	+	0.959193i	$0.591247\pi$
$234$	0	0
$235$	0.405987	0.0264836
$236$	−50.2615	−3.27174
$237$	0	0
$238$	6.84491	0.443690
$239$	−3.07101	−0.198647	−0.0993235	−	0.995055i	$-0.531668\pi$
$239$	−3.07101	−0.198647	−0.0993235	+	0.995055i	$0.531668\pi$
$240$	0	0
$241$	5.83268	0.375716	0.187858	−	0.982196i	$-0.439846\pi$
$241$	5.83268	0.375716	0.187858	+	0.982196i	$0.439846\pi$
$242$	−21.2669	−1.36709
$243$	0	0
$244$	23.7780	1.52223
$245$	−11.2747	−0.720315
$246$	0	0
$247$	2.61288	0.166253
$248$	47.8671	3.03956
$249$	0	0
$250$	2.58610	0.163560
$251$	−23.7101	−1.49657	−0.748285	−	0.663378i	$-0.769122\pi$
$251$	−23.7101	−1.49657	−0.748285	+	0.663378i	$0.769122\pi$
$252$	0	0
$253$	23.8027	1.49646
$254$	35.2777	2.21352
$255$	0	0
$256$	−22.6658	−1.41661
$257$	19.3077	1.20438	0.602190	−	0.798353i	$-0.294295\pi$
$257$	19.3077	1.20438	0.602190	+	0.798353i	$0.294295\pi$
$258$	0	0
$259$	17.1921	1.06826
$260$	4.68793	0.290733
$261$	0	0
$262$	−21.6260	−1.33606
$263$	−11.7197	−0.722667	−0.361334	−	0.932437i	$-0.617678\pi$
$263$	−11.7197	−0.722667	−0.361334	+	0.932437i	$0.617678\pi$
$264$	0	0
$265$	8.71205	0.535177
$266$	−28.8862	−1.77113
$267$	0	0
$268$	68.2464	4.16881
$269$	−17.0088	−1.03705	−0.518524	−	0.855063i	$-0.673518\pi$
$269$	−17.0088	−1.03705	−0.518524	+	0.855063i	$0.673518\pi$
$270$	0	0
$271$	−12.4840	−0.758348	−0.379174	−	0.925325i	$-0.623792\pi$
$271$	−12.4840	−0.758348	−0.379174	+	0.925325i	$0.623792\pi$
$272$	5.32523	0.322890
$273$	0	0
$274$	17.4585	1.05471
$275$	4.38446	0.264393
$276$	0	0
$277$	20.2887	1.21903	0.609516	−	0.792774i	$-0.291364\pi$
$277$	20.2887	1.21903	0.609516	+	0.792774i	$0.291364\pi$
$278$	−9.06628	−0.543759
$279$	0	0
$280$	−29.7160	−1.77587
$281$	−8.37449	−0.499580	−0.249790	−	0.968300i	$-0.580362\pi$
$281$	−8.37449	−0.499580	−0.249790	+	0.968300i	$0.580362\pi$
$282$	0	0
$283$	−2.22562	−0.132300	−0.0661498	−	0.997810i	$-0.521072\pi$
$283$	−2.22562	−0.132300	−0.0661498	+	0.997810i	$0.521072\pi$
$284$	30.8242	1.82908
$285$	0	0
$286$	11.3387	0.670470
$287$	17.4171	1.02810
$288$	0	0
$289$	−16.6167	−0.977450
$290$	17.9934	1.05661
$291$	0	0
$292$	78.4352	4.59008
$293$	−2.53887	−0.148323	−0.0741613	−	0.997246i	$-0.523628\pi$
$293$	−2.53887	−0.148323	−0.0741613	+	0.997246i	$0.523628\pi$
$294$	0	0
$295$	10.7215	0.624227
$296$	27.9555	1.62488
$297$	0	0
$298$	−31.7548	−1.83951
$299$	−5.42887	−0.313959
$300$	0	0
$301$	42.9067	2.47310
$302$	43.1787	2.48465
$303$	0	0
$304$	−22.4730	−1.28891
$305$	−5.07217	−0.290432
$306$	0	0
$307$	19.2343	1.09776	0.548881	−	0.835901i	$-0.315054\pi$
$307$	19.2343	1.09776	0.548881	+	0.835901i	$0.315054\pi$
$308$	−87.8665	−5.00666
$309$	0	0
$310$	−17.8081	−1.01143
$311$	8.76042	0.496758	0.248379	−	0.968663i	$-0.420102\pi$
$311$	8.76042	0.496758	0.248379	+	0.968663i	$0.420102\pi$
$312$	0	0
$313$	−12.1782	−0.688353	−0.344176	−	0.938905i	$-0.611842\pi$
$313$	−12.1782	−0.688353	−0.344176	+	0.938905i	$0.611842\pi$
$314$	−20.6808	−1.16708
$315$	0	0
$316$	68.1896	3.83596
$317$	17.3245	0.973043	0.486521	−	0.873669i	$-0.338266\pi$
$317$	17.3245	0.973043	0.486521	+	0.873669i	$0.338266\pi$
$318$	0	0
$319$	30.5060	1.70801
$320$	−4.36679	−0.244111
$321$	0	0
$322$	60.0178	3.34466
$323$	−1.61777	−0.0900149
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	44.9266	2.48826
$327$	0	0
$328$	28.3214	1.56379
$329$	1.73555	0.0956839
$330$	0	0
$331$	2.46074	0.135254	0.0676272	−	0.997711i	$-0.478457\pi$
$331$	2.46074	0.135254	0.0676272	+	0.997711i	$0.478457\pi$
$332$	−14.2566	−0.782434
$333$	0	0
$334$	−11.0579	−0.605062
$335$	−14.5579	−0.795382
$336$	0	0
$337$	20.7486	1.13025	0.565123	−	0.825007i	$-0.308829\pi$
$337$	20.7486	1.13025	0.565123	+	0.825007i	$0.308829\pi$
$338$	−2.58610	−0.140666
$339$	0	0
$340$	−2.90254	−0.157412
$341$	−30.1918	−1.63498
$342$	0	0
$343$	−18.2739	−0.986700
$344$	69.7693	3.76171
$345$	0	0
$346$	−33.5052	−1.80125
$347$	−12.5877	−0.675741	−0.337871	−	0.941193i	$-0.609707\pi$
$347$	−12.5877	−0.675741	−0.337871	+	0.941193i	$0.609707\pi$
$348$	0	0
$349$	−19.9728	−1.06912	−0.534561	−	0.845130i	$-0.679523\pi$
$349$	−19.9728	−1.06912	−0.534561	+	0.845130i	$0.679523\pi$
$350$	11.0553	0.590932
$351$	0	0
$352$	−36.5671	−1.94903
$353$	5.88934	0.313458	0.156729	−	0.987642i	$-0.449905\pi$
$353$	5.88934	0.313458	0.156729	+	0.987642i	$0.449905\pi$
$354$	0	0
$355$	−6.57523	−0.348977
$356$	34.0846	1.80648
$357$	0	0
$358$	−48.7559	−2.57683
$359$	17.8334	0.941209	0.470605	−	0.882344i	$-0.344036\pi$
$359$	17.8334	0.941209	0.470605	+	0.882344i	$0.344036\pi$
$360$	0	0
$361$	−12.1729	−0.640677
$362$	54.9788	2.88962
$363$	0	0
$364$	20.0404	1.05040
$365$	−16.7313	−0.875756
$366$	0	0
$367$	−29.8329	−1.55727	−0.778633	−	0.627480i	$-0.784086\pi$
$367$	−29.8329	−1.55727	−0.778633	+	0.627480i	$0.784086\pi$
$368$	46.6929	2.43404
$369$	0	0
$370$	−10.4004	−0.540690
$371$	37.2431	1.93357
$372$	0	0
$373$	−14.5794	−0.754892	−0.377446	−	0.926032i	$-0.623198\pi$
$373$	−14.5794	−0.754892	−0.377446	+	0.926032i	$0.623198\pi$
$374$	−7.02036	−0.363014
$375$	0	0
$376$	2.82212	0.145540
$377$	−6.95774	−0.358342
$378$	0	0
$379$	20.7048	1.06353	0.531766	−	0.846891i	$-0.321529\pi$
$379$	20.7048	1.06353	0.531766	+	0.846891i	$0.321529\pi$
$380$	12.2490	0.628360
$381$	0	0
$382$	39.8699	2.03992
$383$	−25.5665	−1.30639	−0.653193	−	0.757191i	$-0.726571\pi$
$383$	−25.5665	−1.30639	−0.653193	+	0.757191i	$0.726571\pi$
$384$	0	0
$385$	18.7431	0.955238
$386$	−4.88260	−0.248518
$387$	0	0
$388$	−45.1671	−2.29301
$389$	−17.1304	−0.868546	−0.434273	−	0.900781i	$-0.642995\pi$
$389$	−17.1304	−0.868546	−0.434273	+	0.900781i	$0.642995\pi$
$390$	0	0
$391$	3.36129	0.169988
$392$	−78.3736	−3.95847
$393$	0	0
$394$	28.0844	1.41487
$395$	−14.5458	−0.731877
$396$	0	0
$397$	−15.8337	−0.794669	−0.397334	−	0.917674i	$-0.630065\pi$
$397$	−15.8337	−0.794669	−0.397334	+	0.917674i	$0.630065\pi$
$398$	3.10617	0.155698
$399$	0	0
$400$	8.60086	0.430043
$401$	−34.1108	−1.70341	−0.851706	−	0.524021i	$-0.824431\pi$
$401$	−34.1108	−1.70341	−0.851706	+	0.524021i	$0.824431\pi$
$402$	0	0
$403$	6.88609	0.343021
$404$	36.5636	1.81911
$405$	0	0
$406$	76.9200	3.81748
$407$	−17.6327	−0.874023
$408$	0	0
$409$	−19.1453	−0.946673	−0.473336	−	0.880882i	$-0.656950\pi$
$409$	−19.1453	−0.946673	−0.473336	+	0.880882i	$0.656950\pi$
$410$	−10.5365	−0.520360
$411$	0	0
$412$	4.65625	0.229397
$413$	45.8331	2.25530
$414$	0	0
$415$	3.04113	0.149283
$416$	8.34015	0.408910
$417$	0	0
$418$	29.6266	1.44908
$419$	7.13499	0.348567	0.174284	−	0.984696i	$-0.444239\pi$
$419$	7.13499	0.348567	0.174284	+	0.984696i	$0.444239\pi$
$420$	0	0
$421$	−26.1618	−1.27505	−0.637524	−	0.770431i	$-0.720041\pi$
$421$	−26.1618	−1.27505	−0.637524	+	0.770431i	$0.720041\pi$
$422$	0.608282	0.0296107
$423$	0	0
$424$	60.5599	2.94105
$425$	0.619151	0.0300332
$426$	0	0
$427$	−21.6830	−1.04931
$428$	19.8849	0.961172
$429$	0	0
$430$	−25.9565	−1.25173
$431$	−12.4211	−0.598302	−0.299151	−	0.954206i	$-0.596703\pi$
$431$	−12.4211	−0.598302	−0.299151	+	0.954206i	$0.596703\pi$
$432$	0	0
$433$	−6.04541	−0.290524	−0.145262	−	0.989393i	$-0.546402\pi$
$433$	−6.04541	−0.290524	−0.145262	+	0.989393i	$0.546402\pi$
$434$	−76.1279	−3.65426
$435$	0	0
$436$	22.1386	1.06025
$437$	−14.1850	−0.678559
$438$	0	0
$439$	−28.6764	−1.36865	−0.684326	−	0.729177i	$-0.739903\pi$
$439$	−28.6764	−1.36865	−0.684326	+	0.729177i	$0.739903\pi$
$440$	30.4776	1.45296
$441$	0	0
$442$	1.60119	0.0761608
$443$	−9.14996	−0.434728	−0.217364	−	0.976091i	$-0.569746\pi$
$443$	−9.14996	−0.434728	−0.217364	+	0.976091i	$0.569746\pi$
$444$	0	0
$445$	−7.27070	−0.344664
$446$	43.3300	2.05174
$447$	0	0
$448$	−18.6676	−0.881960
$449$	35.4645	1.67367	0.836837	−	0.547452i	$-0.184402\pi$
$449$	35.4645	1.67367	0.836837	+	0.547452i	$0.184402\pi$
$450$	0	0
$451$	−17.8635	−0.841160
$452$	−16.6613	−0.783680
$453$	0	0
$454$	−58.9641	−2.76732
$455$	−4.27489	−0.200410
$456$	0	0
$457$	9.41261	0.440303	0.220152	−	0.975466i	$-0.429345\pi$
$457$	9.41261	0.440303	0.220152	+	0.975466i	$0.429345\pi$
$458$	18.0375	0.842837
$459$	0	0
$460$	−25.4502	−1.18662
$461$	−20.5026	−0.954899	−0.477450	−	0.878659i	$-0.658439\pi$
$461$	−20.5026	−0.954899	−0.477450	+	0.878659i	$0.658439\pi$
$462$	0	0
$463$	−37.7831	−1.75593	−0.877965	−	0.478724i	$-0.841099\pi$
$463$	−37.7831	−1.75593	−0.877965	+	0.478724i	$0.841099\pi$
$464$	59.8425	2.77812
$465$	0	0
$466$	22.3233	1.03411
$467$	−12.8882	−0.596394	−0.298197	−	0.954504i	$-0.596385\pi$
$467$	−12.8882	−0.596394	−0.298197	+	0.954504i	$0.596385\pi$
$468$	0	0
$469$	−62.2334	−2.87367
$470$	−1.04992	−0.0484294
$471$	0	0
$472$	74.5278	3.43042
$473$	−44.0064	−2.02342
$474$	0	0
$475$	−2.61288	−0.119887
$476$	−12.4080	−0.568722
$477$	0	0
$478$	7.94194	0.363256
$479$	20.1880	0.922415	0.461207	−	0.887292i	$-0.347416\pi$
$479$	20.1880	0.922415	0.461207	+	0.887292i	$0.347416\pi$
$480$	0	0
$481$	4.02164	0.183371
$482$	−15.0839	−0.687053
$483$	0	0
$484$	38.5514	1.75233
$485$	9.63475	0.437492
$486$	0	0
$487$	15.1480	0.686423	0.343212	−	0.939258i	$-0.388485\pi$
$487$	15.1480	0.686423	0.343212	+	0.939258i	$0.388485\pi$
$488$	−35.2581	−1.59606
$489$	0	0
$490$	29.1576	1.31721
$491$	−11.4875	−0.518423	−0.259211	−	0.965821i	$-0.583463\pi$
$491$	−11.4875	−0.518423	−0.259211	+	0.965821i	$0.583463\pi$
$492$	0	0
$493$	4.30789	0.194018
$494$	−6.75717	−0.304020
$495$	0	0
$496$	−59.2263	−2.65934
$497$	−28.1084	−1.26083
$498$	0	0
$499$	−13.1201	−0.587336	−0.293668	−	0.955907i	$-0.594876\pi$
$499$	−13.1201	−0.587336	−0.293668	+	0.955907i	$0.594876\pi$
$500$	−4.68793	−0.209651
$501$	0	0
$502$	61.3168	2.73670
$503$	7.80396	0.347961	0.173981	−	0.984749i	$-0.444337\pi$
$503$	7.80396	0.347961	0.173981	+	0.984749i	$0.444337\pi$
$504$	0	0
$505$	−7.79952	−0.347074
$506$	−61.5562	−2.73651
$507$	0	0
$508$	−63.9494	−2.83729
$509$	21.9837	0.974411	0.487205	−	0.873287i	$-0.338016\pi$
$509$	21.9837	0.974411	0.487205	+	0.873287i	$0.338016\pi$
$510$	0	0
$511$	−71.5245	−3.16406
$512$	47.8414	2.11431
$513$	0	0
$514$	−49.9316	−2.20239
$515$	−0.993241	−0.0437674
$516$	0	0
$517$	−1.78003	−0.0782858
$518$	−44.4605	−1.95348
$519$	0	0
$520$	−6.95128	−0.304834
$521$	12.1706	0.533205	0.266603	−	0.963807i	$-0.414099\pi$
$521$	12.1706	0.533205	0.266603	+	0.963807i	$0.414099\pi$
$522$	0	0
$523$	32.5468	1.42317	0.711587	−	0.702598i	$-0.247977\pi$
$523$	32.5468	1.42317	0.711587	+	0.702598i	$0.247977\pi$
$524$	39.2022	1.71256
$525$	0	0
$526$	30.3083	1.32151
$527$	−4.26353	−0.185722
$528$	0	0
$529$	6.47259	0.281417
$530$	−22.5303	−0.978653
$531$	0	0
$532$	52.3632	2.27023
$533$	4.07427	0.176476
$534$	0	0
$535$	−4.24171	−0.183385
$536$	−101.196	−4.37099
$537$	0	0
$538$	43.9866	1.89640
$539$	49.4336	2.12925
$540$	0	0
$541$	−16.3865	−0.704511	−0.352256	−	0.935904i	$-0.614585\pi$
$541$	−16.3865	−0.704511	−0.352256	+	0.935904i	$0.614585\pi$
$542$	32.2849	1.38676
$543$	0	0
$544$	−5.16382	−0.221397
$545$	−4.72247	−0.202288
$546$	0	0
$547$	19.0667	0.815233	0.407616	−	0.913153i	$-0.366360\pi$
$547$	19.0667	0.815233	0.407616	+	0.913153i	$0.366360\pi$
$548$	−31.6477	−1.35192
$549$	0	0
$550$	−11.3387	−0.483483
$551$	−18.1797	−0.774482
$552$	0	0
$553$	−62.1816	−2.64423
$554$	−52.4687	−2.22918
$555$	0	0
$556$	16.4348	0.696991
$557$	−38.2913	−1.62245	−0.811227	−	0.584732i	$-0.801200\pi$
$557$	−38.2913	−1.62245	−0.811227	+	0.584732i	$0.801200\pi$
$558$	0	0
$559$	10.0369	0.424516
$560$	36.7677	1.55372
$561$	0	0
$562$	21.6573	0.913559
$563$	10.3408	0.435813	0.217906	−	0.975970i	$-0.430077\pi$
$563$	10.3408	0.435813	0.217906	+	0.975970i	$0.430077\pi$
$564$	0	0
$565$	3.55408	0.149521
$566$	5.75569	0.241930
$567$	0	0
$568$	−45.7062	−1.91779
$569$	−26.0267	−1.09110	−0.545548	−	0.838080i	$-0.683678\pi$
$569$	−26.0267	−1.09110	−0.545548	+	0.838080i	$0.683678\pi$
$570$	0	0
$571$	−32.3239	−1.35271	−0.676357	−	0.736574i	$-0.736442\pi$
$571$	−32.3239	−1.35271	−0.676357	+	0.736574i	$0.736442\pi$
$572$	−20.5541	−0.859409
$573$	0	0
$574$	−45.0424	−1.88003
$575$	5.42887	0.226399
$576$	0	0
$577$	12.4808	0.519581	0.259790	−	0.965665i	$-0.416347\pi$
$577$	12.4808	0.519581	0.259790	+	0.965665i	$0.416347\pi$
$578$	42.9724	1.78742
$579$	0	0
$580$	−32.6174	−1.35436
$581$	13.0005	0.539352
$582$	0	0
$583$	−38.1977	−1.58199
$584$	−116.304	−4.81269
$585$	0	0
$586$	6.56579	0.271230
$587$	−25.1950	−1.03991	−0.519955	−	0.854194i	$-0.674051\pi$
$587$	−25.1950	−1.03991	−0.519955	+	0.854194i	$0.674051\pi$
$588$	0	0
$589$	17.9925	0.741368
$590$	−27.7268	−1.14149
$591$	0	0
$592$	−34.5896	−1.42162
$593$	−38.4456	−1.57877	−0.789385	−	0.613898i	$-0.789601\pi$
$593$	−38.4456	−1.57877	−0.789385	+	0.613898i	$0.789601\pi$
$594$	0	0
$595$	2.64681	0.108508
$596$	57.5632	2.35788
$597$	0	0
$598$	14.0396	0.574122
$599$	−47.3236	−1.93359	−0.966795	−	0.255553i	$-0.917743\pi$
$599$	−47.3236	−1.93359	−0.966795	+	0.255553i	$0.917743\pi$
$600$	0	0
$601$	40.0656	1.63431	0.817156	−	0.576417i	$-0.195550\pi$
$601$	40.0656	1.63431	0.817156	+	0.576417i	$0.195550\pi$
$602$	−110.961	−4.52244
$603$	0	0
$604$	−78.2717	−3.18483
$605$	−8.22353	−0.334334
$606$	0	0
$607$	20.6119	0.836612	0.418306	−	0.908306i	$-0.362624\pi$
$607$	20.6119	0.836612	0.418306	+	0.908306i	$0.362624\pi$
$608$	21.7918	0.883774
$609$	0	0
$610$	13.1172	0.531098
$611$	0.405987	0.0164245
$612$	0	0
$613$	−4.02465	−0.162554	−0.0812770	−	0.996692i	$-0.525900\pi$
$613$	−4.02465	−0.162554	−0.0812770	+	0.996692i	$0.525900\pi$
$614$	−49.7420	−2.00742
$615$	0	0
$616$	130.289	5.24948
$617$	29.8240	1.20067	0.600334	−	0.799749i	$-0.295034\pi$
$617$	29.8240	1.20067	0.600334	+	0.799749i	$0.295034\pi$
$618$	0	0
$619$	−47.5386	−1.91074	−0.955368	−	0.295419i	$-0.904541\pi$
$619$	−47.5386	−1.91074	−0.955368	+	0.295419i	$0.904541\pi$
$620$	32.2815	1.29646
$621$	0	0
$622$	−22.6554	−0.908397
$623$	−31.0815	−1.24525
$624$	0	0
$625$	1.00000	0.0400000
$626$	31.4941	1.25876
$627$	0	0
$628$	37.4888	1.49597
$629$	−2.49000	−0.0992830
$630$	0	0
$631$	32.2213	1.28271	0.641354	−	0.767245i	$-0.278373\pi$
$631$	32.2213	1.28271	0.641354	+	0.767245i	$0.278373\pi$
$632$	−101.112	−4.02200
$633$	0	0
$634$	−44.8030	−1.77936
$635$	13.6413	0.541337
$636$	0	0
$637$	−11.2747	−0.446720
$638$	−78.8916	−3.12335
$639$	0	0
$640$	−5.38732	−0.212953
$641$	3.50215	0.138326	0.0691632	−	0.997605i	$-0.477967\pi$
$641$	3.50215	0.138326	0.0691632	+	0.997605i	$0.477967\pi$
$642$	0	0
$643$	0.0231395	0.000912533 0	0.000456266	−	1.00000i	$-0.499855\pi$
$643$	0.0231395	0.000912533 0	0.000456266		1.00000i	$0.499855\pi$
$644$	−108.797	−4.28719
$645$	0	0
$646$	4.18371	0.164606
$647$	−20.8611	−0.820136	−0.410068	−	0.912055i	$-0.634495\pi$
$647$	−20.8611	−0.820136	−0.410068	+	0.912055i	$0.634495\pi$
$648$	0	0
$649$	−47.0078	−1.84522
$650$	2.58610	0.101435
$651$	0	0
$652$	−81.4403	−3.18945
$653$	10.9565	0.428760	0.214380	−	0.976750i	$-0.431227\pi$
$653$	10.9565	0.428760	0.214380	+	0.976750i	$0.431227\pi$
$654$	0	0
$655$	−8.36237	−0.326745
$656$	−35.0422	−1.36817
$657$	0	0
$658$	−4.48831	−0.174973
$659$	−34.1093	−1.32871	−0.664354	−	0.747418i	$-0.731293\pi$
$659$	−34.1093	−1.32871	−0.664354	+	0.747418i	$0.731293\pi$
$660$	0	0
$661$	−24.1801	−0.940498	−0.470249	−	0.882534i	$-0.655836\pi$
$661$	−24.1801	−0.940498	−0.470249	+	0.882534i	$0.655836\pi$
$662$	−6.36373	−0.247333
$663$	0	0
$664$	21.1397	0.820381
$665$	−11.1698	−0.433145
$666$	0	0
$667$	37.7726	1.46256
$668$	20.0451	0.775569
$669$	0	0
$670$	37.6482	1.45448
$671$	22.2388	0.858518
$672$	0	0
$673$	19.3215	0.744788	0.372394	−	0.928075i	$-0.378537\pi$
$673$	19.3215	0.744788	0.372394	+	0.928075i	$0.378537\pi$
$674$	−53.6579	−2.06683
$675$	0	0
$676$	4.68793	0.180305
$677$	−38.2734	−1.47097	−0.735483	−	0.677543i	$-0.763045\pi$
$677$	−38.2734	−1.47097	−0.735483	+	0.677543i	$0.763045\pi$
$678$	0	0
$679$	41.1875	1.58063
$680$	4.30389	0.165047
$681$	0	0
$682$	78.0792	2.98981
$683$	0.692198	0.0264862	0.0132431	−	0.999912i	$-0.495784\pi$
$683$	0.692198	0.0264862	0.0132431	+	0.999912i	$0.495784\pi$
$684$	0	0
$685$	6.75088	0.257938
$686$	47.2583	1.80433
$687$	0	0
$688$	−86.3260	−3.29115
$689$	8.71205	0.331903
$690$	0	0
$691$	−32.9911	−1.25504	−0.627520	−	0.778600i	$-0.715930\pi$
$691$	−32.9911	−1.25504	−0.627520	+	0.778600i	$0.715930\pi$
$692$	60.7363	2.30885
$693$	0	0
$694$	32.5530	1.23569
$695$	−3.50577	−0.132981
$696$	0	0
$697$	−2.52259	−0.0955500
$698$	51.6518	1.95505
$699$	0	0
$700$	−20.0404	−0.757457
$701$	−18.7776	−0.709219	−0.354610	−	0.935014i	$-0.615386\pi$
$701$	−18.7776	−0.709219	−0.354610	+	0.935014i	$0.615386\pi$
$702$	0	0
$703$	10.5081	0.396319
$704$	19.1461	0.721594
$705$	0	0
$706$	−15.2304	−0.573205
$707$	−33.3421	−1.25396
$708$	0	0
$709$	8.09442	0.303992	0.151996	−	0.988381i	$-0.451430\pi$
$709$	8.09442	0.303992	0.151996	+	0.988381i	$0.451430\pi$
$710$	17.0042	0.638157
$711$	0	0
$712$	−50.5407	−1.89409
$713$	−37.3837	−1.40003
$714$	0	0
$715$	4.38446	0.163970
$716$	88.3818	3.30298
$717$	0	0
$718$	−46.1190	−1.72114
$719$	31.8467	1.18768	0.593840	−	0.804583i	$-0.297611\pi$
$719$	31.8467	1.18768	0.593840	+	0.804583i	$0.297611\pi$
$720$	0	0
$721$	−4.24600	−0.158129
$722$	31.4803	1.17158
$723$	0	0
$724$	−99.6623	−3.70392
$725$	6.95774	0.258404
$726$	0	0
$727$	−20.3092	−0.753227	−0.376613	−	0.926371i	$-0.622911\pi$
$727$	−20.3092	−0.753227	−0.376613	+	0.926371i	$0.622911\pi$
$728$	−29.7160	−1.10135
$729$	0	0
$730$	43.2689	1.60145
$731$	−6.21436	−0.229846
$732$	0	0
$733$	−14.3193	−0.528896	−0.264448	−	0.964400i	$-0.585190\pi$
$733$	−14.3193	−0.528896	−0.264448	+	0.964400i	$0.585190\pi$
$734$	77.1510	2.84770
$735$	0	0
$736$	−45.2776	−1.66895
$737$	63.8285	2.35115
$738$	0	0
$739$	−32.7316	−1.20405	−0.602025	−	0.798477i	$-0.705639\pi$
$739$	−32.7316	−1.20405	−0.602025	+	0.798477i	$0.705639\pi$
$740$	18.8532	0.693057
$741$	0	0
$742$	−96.3145	−3.53582
$743$	38.9507	1.42896	0.714481	−	0.699655i	$-0.246663\pi$
$743$	38.9507	1.42896	0.714481	+	0.699655i	$0.246663\pi$
$744$	0	0
$745$	−12.2790	−0.449868
$746$	37.7038	1.38044
$747$	0	0
$748$	12.7261	0.465312
$749$	−18.1329	−0.662561
$750$	0	0
$751$	−23.0151	−0.839832	−0.419916	−	0.907563i	$-0.637940\pi$
$751$	−23.0151	−0.839832	−0.419916	+	0.907563i	$0.637940\pi$
$752$	−3.49183	−0.127334
$753$	0	0
$754$	17.9934	0.655282
$755$	16.6964	0.607645
$756$	0	0
$757$	41.6474	1.51370	0.756850	−	0.653588i	$-0.226737\pi$
$757$	41.6474	1.51370	0.756850	+	0.653588i	$0.226737\pi$
$758$	−53.5447	−1.94483
$759$	0	0
$760$	−18.1628	−0.658835
$761$	6.88861	0.249712	0.124856	−	0.992175i	$-0.460153\pi$
$761$	6.88861	0.249712	0.124856	+	0.992175i	$0.460153\pi$
$762$	0	0
$763$	−20.1880	−0.730856
$764$	−72.2738	−2.61477
$765$	0	0
$766$	66.1176	2.38893
$767$	10.7215	0.387129
$768$	0	0
$769$	−9.56774	−0.345021	−0.172511	−	0.985008i	$-0.555188\pi$
$769$	−9.56774	−0.345021	−0.172511	+	0.985008i	$0.555188\pi$
$770$	−48.4717	−1.74680
$771$	0	0
$772$	8.85089	0.318551
$773$	−10.5131	−0.378131	−0.189066	−	0.981964i	$-0.560546\pi$
$773$	−10.5131	−0.378131	−0.189066	+	0.981964i	$0.560546\pi$
$774$	0	0
$775$	−6.88609	−0.247356
$776$	66.9738	2.40422
$777$	0	0
$778$	44.3010	1.58827
$779$	10.6456	0.381417
$780$	0	0
$781$	28.8289	1.03158
$782$	−8.69264	−0.310848
$783$	0	0
$784$	96.9722	3.46329
$785$	−7.99688	−0.285421
$786$	0	0
$787$	−36.1200	−1.28754	−0.643769	−	0.765220i	$-0.722630\pi$
$787$	−36.1200	−1.28754	−0.643769	+	0.765220i	$0.722630\pi$
$788$	−50.9097	−1.81358
$789$	0	0
$790$	37.6168	1.33835
$791$	15.1933	0.540211
$792$	0	0
$793$	−5.07217	−0.180118
$794$	40.9475	1.45317
$795$	0	0
$796$	−5.63067	−0.199574
$797$	−14.7492	−0.522445	−0.261223	−	0.965279i	$-0.584126\pi$
$797$	−14.7492	−0.522445	−0.261223	+	0.965279i	$0.584126\pi$
$798$	0	0
$799$	−0.251367	−0.00889273
$800$	−8.34015	−0.294869
$801$	0	0
$802$	88.2140	3.11495
$803$	73.3578	2.58874
$804$	0	0
$805$	23.2078	0.817968
$806$	−17.8081	−0.627265
$807$	0	0
$808$	−54.2166	−1.90733
$809$	1.81657	0.0638672	0.0319336	−	0.999490i	$-0.489833\pi$
$809$	1.81657	0.0638672	0.0319336	+	0.999490i	$0.489833\pi$
$810$	0	0
$811$	−33.3279	−1.17030	−0.585151	−	0.810924i	$-0.698965\pi$
$811$	−33.3279	−1.17030	−0.585151	+	0.810924i	$0.698965\pi$
$812$	−139.436	−4.89324
$813$	0	0
$814$	45.6001	1.59828
$815$	17.3723	0.608526
$816$	0	0
$817$	26.2252	0.917504
$818$	49.5117	1.73113
$819$	0	0
$820$	19.0999	0.666998
$821$	35.4878	1.23853	0.619266	−	0.785181i	$-0.287430\pi$
$821$	35.4878	1.23853	0.619266	+	0.785181i	$0.287430\pi$
$822$	0	0
$823$	−16.5800	−0.577941	−0.288971	−	0.957338i	$-0.593313\pi$
$823$	−16.5800	−0.577941	−0.288971	+	0.957338i	$0.593313\pi$
$824$	−6.90429	−0.240523
$825$	0	0
$826$	−118.529	−4.12415
$827$	8.93071	0.310551	0.155276	−	0.987871i	$-0.450373\pi$
$827$	8.93071	0.310551	0.155276	+	0.987871i	$0.450373\pi$
$828$	0	0
$829$	12.9059	0.448240	0.224120	−	0.974562i	$-0.428049\pi$
$829$	12.9059	0.448240	0.224120	+	0.974562i	$0.428049\pi$
$830$	−7.86468	−0.272987
$831$	0	0
$832$	−4.36679	−0.151391
$833$	6.98075	0.241869
$834$	0	0
$835$	−4.27589	−0.147973
$836$	−53.7053	−1.85744
$837$	0	0
$838$	−18.4518	−0.637408
$839$	34.9919	1.20805	0.604027	−	0.796964i	$-0.293562\pi$
$839$	34.9919	1.20805	0.604027	+	0.796964i	$0.293562\pi$
$840$	0	0
$841$	19.4101	0.669315
$842$	67.6571	2.33162
$843$	0	0
$844$	−1.10266	−0.0379550
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−35.1547	−1.20793
$848$	−74.9311	−2.57314
$849$	0	0
$850$	−1.60119	−0.0549204
$851$	−21.8330	−0.748424
$852$	0	0
$853$	−21.1568	−0.724395	−0.362198	−	0.932101i	$-0.617973\pi$
$853$	−21.1568	−0.724395	−0.362198	+	0.932101i	$0.617973\pi$
$854$	56.0745	1.91883
$855$	0	0
$856$	−29.4853	−1.00779
$857$	−11.6305	−0.397290	−0.198645	−	0.980072i	$-0.563654\pi$
$857$	−11.6305	−0.397290	−0.198645	+	0.980072i	$0.563654\pi$
$858$	0	0
$859$	1.78067	0.0607557	0.0303778	−	0.999538i	$-0.490329\pi$
$859$	1.78067	0.0607557	0.0303778	+	0.999538i	$0.490329\pi$
$860$	47.0523	1.60447
$861$	0	0
$862$	32.1222	1.09409
$863$	−7.65256	−0.260496	−0.130248	−	0.991481i	$-0.541577\pi$
$863$	−7.65256	−0.260496	−0.130248	+	0.991481i	$0.541577\pi$
$864$	0	0
$865$	−12.9559	−0.440513
$866$	15.6341	0.531267
$867$	0	0
$868$	138.000	4.68403
$869$	63.7754	2.16343
$870$	0	0
$871$	−14.5579	−0.493275
$872$	−32.8272	−1.11167
$873$	0	0
$874$	36.6838	1.24085
$875$	4.27489	0.144518
$876$	0	0
$877$	6.79223	0.229357	0.114679	−	0.993403i	$-0.463416\pi$
$877$	6.79223	0.229357	0.114679	+	0.993403i	$0.463416\pi$
$878$	74.1602	2.50279
$879$	0	0
$880$	−37.7101	−1.27121
$881$	8.06104	0.271583	0.135792	−	0.990737i	$-0.456642\pi$
$881$	8.06104	0.271583	0.135792	+	0.990737i	$0.456642\pi$
$882$	0	0
$883$	15.7753	0.530882	0.265441	−	0.964127i	$-0.414482\pi$
$883$	15.7753	0.530882	0.265441	+	0.964127i	$0.414482\pi$
$884$	−2.90254	−0.0976230
$885$	0	0
$886$	23.6627	0.794965
$887$	−17.7631	−0.596428	−0.298214	−	0.954499i	$-0.596391\pi$
$887$	−17.7631	−0.596428	−0.298214	+	0.954499i	$0.596391\pi$
$888$	0	0
$889$	58.3150	1.95582
$890$	18.8028	0.630271
$891$	0	0
$892$	−78.5461	−2.62992
$893$	1.06079	0.0354981
$894$	0	0
$895$	−18.8530	−0.630188
$896$	−23.0302	−0.769386
$897$	0	0
$898$	−91.7149	−3.06057
$899$	−47.9116	−1.59794
$900$	0	0
$901$	−5.39408	−0.179703
$902$	46.1969	1.53819
$903$	0	0
$904$	24.7054	0.821688
$905$	21.2593	0.706684
$906$	0	0
$907$	48.6359	1.61493	0.807464	−	0.589916i	$-0.200839\pi$
$907$	48.6359	1.61493	0.807464	+	0.589916i	$0.200839\pi$
$908$	106.887	3.54715
$909$	0	0
$910$	11.0553	0.366480
$911$	18.6599	0.618229	0.309115	−	0.951025i	$-0.399967\pi$
$911$	18.6599	0.618229	0.309115	+	0.951025i	$0.399967\pi$
$912$	0	0
$913$	−13.3337	−0.441282
$914$	−24.3420	−0.805162
$915$	0	0
$916$	−32.6973	−1.08035
$917$	−35.7482	−1.18051
$918$	0	0
$919$	−13.8812	−0.457898	−0.228949	−	0.973438i	$-0.573529\pi$
$919$	−13.8812	−0.457898	−0.228949	+	0.973438i	$0.573529\pi$
$920$	37.7375	1.24417
$921$	0	0
$922$	53.0217	1.74618
$923$	−6.57523	−0.216426
$924$	0	0
$925$	−4.02164	−0.132231
$926$	97.7111	3.21099
$927$	0	0
$928$	−58.0286	−1.90488
$929$	17.6717	0.579791	0.289895	−	0.957058i	$-0.406379\pi$
$929$	17.6717	0.579791	0.289895	+	0.957058i	$0.406379\pi$
$930$	0	0
$931$	−29.4594	−0.965494
$932$	−40.4663	−1.32552
$933$	0	0
$934$	33.3302	1.09060
$935$	−2.71465	−0.0887784
$936$	0	0
$937$	33.4041	1.09126	0.545632	−	0.838025i	$-0.316290\pi$
$937$	33.4041	1.09126	0.545632	+	0.838025i	$0.316290\pi$
$938$	160.942	5.25494
$939$	0	0
$940$	1.90324	0.0620768
$941$	−57.8947	−1.88731	−0.943656	−	0.330928i	$-0.892638\pi$
$941$	−57.8947	−1.88731	−0.943656	+	0.330928i	$0.892638\pi$
$942$	0	0
$943$	−22.1187	−0.720283
$944$	−92.2136	−3.00130
$945$	0	0
$946$	113.805	3.70013
$947$	−54.6285	−1.77519	−0.887593	−	0.460628i	$-0.847624\pi$
$947$	−54.6285	−1.77519	−0.887593	+	0.460628i	$0.847624\pi$
$948$	0	0
$949$	−16.7313	−0.543121
$950$	6.75717	0.219232
$951$	0	0
$952$	18.3987	0.596305
$953$	19.7207	0.638815	0.319407	−	0.947618i	$-0.396516\pi$
$953$	19.7207	0.638815	0.319407	+	0.947618i	$0.396516\pi$
$954$	0	0
$955$	15.4170	0.498882
$956$	−14.3967	−0.465622
$957$	0	0
$958$	−52.2083	−1.68678
$959$	28.8593	0.931915
$960$	0	0
$961$	16.4182	0.529620
$962$	−10.4004	−0.335322
$963$	0	0
$964$	27.3432	0.880665
$965$	−1.88802	−0.0607774
$966$	0	0
$967$	−35.2175	−1.13252	−0.566259	−	0.824227i	$-0.691610\pi$
$967$	−35.2175	−1.13252	−0.566259	+	0.824227i	$0.691610\pi$
$968$	−57.1640	−1.83732
$969$	0	0
$970$	−24.9165	−0.800020
$971$	−13.9711	−0.448354	−0.224177	−	0.974548i	$-0.571969\pi$
$971$	−13.9711	−0.448354	−0.224177	+	0.974548i	$0.571969\pi$
$972$	0	0
$973$	−14.9868	−0.480454
$974$	−39.1744	−1.25523
$975$	0	0
$976$	43.6250	1.39640
$977$	36.4905	1.16744	0.583718	−	0.811957i	$-0.301597\pi$
$977$	36.4905	1.16744	0.583718	+	0.811957i	$0.301597\pi$
$978$	0	0
$979$	31.8781	1.01883
$980$	−52.8551	−1.68839
$981$	0	0
$982$	29.7078	0.948015
$983$	−17.5005	−0.558180	−0.279090	−	0.960265i	$-0.590033\pi$
$983$	−17.5005	−0.558180	−0.279090	+	0.960265i	$0.590033\pi$
$984$	0	0
$985$	10.8597	0.346020
$986$	−11.1407	−0.354791
$987$	0	0
$988$	12.2490	0.389693
$989$	−54.4890	−1.73265
$990$	0	0
$991$	18.8114	0.597565	0.298783	−	0.954321i	$-0.403419\pi$
$991$	18.8114	0.597565	0.298783	+	0.954321i	$0.403419\pi$
$992$	57.4310	1.82344
$993$	0	0
$994$	72.6912	2.30563
$995$	1.20110	0.0380774
$996$	0	0
$997$	3.53423	0.111930	0.0559651	−	0.998433i	$-0.482176\pi$
$997$	3.53423	0.111930	0.0559651	+	0.998433i	$0.482176\pi$
$998$	33.9299	1.07403
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5265.2.a.bf.1.1		8
3.2	odd	2		5265.2.a.ba.1.8		8
9.2	odd	6		1755.2.i.f.1171.1		16
9.4	even	3		585.2.i.e.196.8	✓	16
9.5	odd	6		1755.2.i.f.586.1		16
9.7	even	3		585.2.i.e.391.8	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.i.e.196.8	✓	16	9.4	even	3
585.2.i.e.391.8	yes	16	9.7	even	3
1755.2.i.f.586.1		16	9.5	odd	6
1755.2.i.f.1171.1		16	9.2	odd	6
5265.2.a.ba.1.8		8	3.2	odd	2
5265.2.a.bf.1.1		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5265 = 3^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5265.a (trivial)

\(f(q)\)	\(=\)	\(q-2.58610 q^{2} +4.68793 q^{4} -1.00000 q^{5} -4.27489 q^{7} -6.95128 q^{8} +O(q^{10})\)	\(q-2.58610 q^{2} +4.68793 q^{4} -1.00000 q^{5} -4.27489 q^{7} -6.95128 q^{8} +2.58610 q^{10} +4.38446 q^{11} -1.00000 q^{13} +11.0553 q^{14} +8.60086 q^{16} +0.619151 q^{17} -2.61288 q^{19} -4.68793 q^{20} -11.3387 q^{22} +5.42887 q^{23} +1.00000 q^{25} +2.58610 q^{26} -20.0404 q^{28} +6.95774 q^{29} -6.88609 q^{31} -8.34015 q^{32} -1.60119 q^{34} +4.27489 q^{35} -4.02164 q^{37} +6.75717 q^{38} +6.95128 q^{40} -4.07427 q^{41} -10.0369 q^{43} +20.5541 q^{44} -14.0396 q^{46} -0.405987 q^{47} +11.2747 q^{49} -2.58610 q^{50} -4.68793 q^{52} -8.71205 q^{53} -4.38446 q^{55} +29.7160 q^{56} -17.9934 q^{58} -10.7215 q^{59} +5.07217 q^{61} +17.8081 q^{62} +4.36679 q^{64} +1.00000 q^{65} +14.5579 q^{67} +2.90254 q^{68} -11.0553 q^{70} +6.57523 q^{71} +16.7313 q^{73} +10.4004 q^{74} -12.2490 q^{76} -18.7431 q^{77} +14.5458 q^{79} -8.60086 q^{80} +10.5365 q^{82} -3.04113 q^{83} -0.619151 q^{85} +25.9565 q^{86} -30.4776 q^{88} +7.27070 q^{89} +4.27489 q^{91} +25.4502 q^{92} +1.04992 q^{94} +2.61288 q^{95} -9.63475 q^{97} -29.1576 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 3 q^{2} + 9 q^{4} - 8 q^{5} - 11 q^{7} - 6 q^{8}+O(q^{10}) \) 8 * q + 3 * q^2 + 9 * q^4 - 8 * q^5 - 11 * q^7 - 6 * q^8	\( 8 q + 3 q^{2} + 9 q^{4} - 8 q^{5} - 11 q^{7} - 6 q^{8} - 3 q^{10} + 6 q^{11} - 8 q^{13} + 10 q^{14} + 11 q^{16} - 2 q^{17} - 10 q^{19} - 9 q^{20} + 3 q^{22} + 6 q^{23} + 8 q^{25} - 3 q^{26} - 34 q^{28} + 14 q^{29} - 31 q^{31} + q^{32} - 7 q^{34} + 11 q^{35} + q^{37} + 9 q^{38} + 6 q^{40} - 12 q^{41} + 15 q^{43} + 16 q^{44} - 32 q^{46} - 18 q^{47} + 17 q^{49} + 3 q^{50} - 9 q^{52} + 2 q^{53} - 6 q^{55} + 16 q^{56} - 42 q^{58} + 24 q^{59} - 9 q^{61} - 20 q^{62} - 30 q^{64} + 8 q^{65} - 18 q^{67} - 14 q^{68} - 10 q^{70} + 10 q^{71} + 6 q^{73} - 37 q^{74} - 53 q^{76} - 34 q^{77} - 3 q^{79} - 11 q^{80} - 34 q^{82} - 10 q^{83} + 2 q^{85} + 60 q^{86} - 14 q^{88} - 13 q^{89} + 11 q^{91} + 5 q^{92} + 17 q^{94} + 10 q^{95} - 34 q^{97} - 30 q^{98}+O(q^{100}) \) 8 * q + 3 * q^2 + 9 * q^4 - 8 * q^5 - 11 * q^7 - 6 * q^8 - 3 * q^10 + 6 * q^11 - 8 * q^13 + 10 * q^14 + 11 * q^16 - 2 * q^17 - 10 * q^19 - 9 * q^20 + 3 * q^22 + 6 * q^23 + 8 * q^25 - 3 * q^26 - 34 * q^28 + 14 * q^29 - 31 * q^31 + q^32 - 7 * q^34 + 11 * q^35 + q^37 + 9 * q^38 + 6 * q^40 - 12 * q^41 + 15 * q^43 + 16 * q^44 - 32 * q^46 - 18 * q^47 + 17 * q^49 + 3 * q^50 - 9 * q^52 + 2 * q^53 - 6 * q^55 + 16 * q^56 - 42 * q^58 + 24 * q^59 - 9 * q^61 - 20 * q^62 - 30 * q^64 + 8 * q^65 - 18 * q^67 - 14 * q^68 - 10 * q^70 + 10 * q^71 + 6 * q^73 - 37 * q^74 - 53 * q^76 - 34 * q^77 - 3 * q^79 - 11 * q^80 - 34 * q^82 - 10 * q^83 + 2 * q^85 + 60 * q^86 - 14 * q^88 - 13 * q^89 + 11 * q^91 + 5 * q^92 + 17 * q^94 + 10 * q^95 - 34 * q^97 - 30 * q^98

Introduction

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