LMFDB - Embedded newform 5265.2.a.bc.1.6

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:	$0$
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} - 11x^{6} - 2x^{5} + 38x^{4} + 14x^{3} - 39x^{2} - 22x - 2 $ x^8 - 11x^6 - 2x^5 + 38x^4 + 14x^3 - 39x^2 - 22x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.46324$ 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)

$f(q)$	$=$	$q+1.46324 q^{2} +0.141066 q^{4} -1.00000 q^{5} -4.95241 q^{7} -2.72006 q^{8} +O(q^{10})$	\(q+1.46324 q^{2} +0.141066 q^{4} -1.00000 q^{5} -4.95241 q^{7} -2.72006 q^{8} -1.46324 q^{10} -1.59098 q^{11} -1.00000 q^{13} -7.24655 q^{14} -4.26223 q^{16} -5.55131 q^{17} -4.74454 q^{19} -0.141066 q^{20} -2.32799 q^{22} +0.0764816 q^{23} +1.00000 q^{25} -1.46324 q^{26} -0.698616 q^{28} +3.53848 q^{29} +1.59217 q^{31} -0.796535 q^{32} -8.12288 q^{34} +4.95241 q^{35} +4.13079 q^{37} -6.94240 q^{38} +2.72006 q^{40} -4.90631 q^{41} +4.71684 q^{43} -0.224434 q^{44} +0.111911 q^{46} +11.5366 q^{47} +17.5264 q^{49} +1.46324 q^{50} -0.141066 q^{52} -1.29315 q^{53} +1.59098 q^{55} +13.4709 q^{56} +5.17764 q^{58} +4.66770 q^{59} -10.6244 q^{61} +2.32972 q^{62} +7.35894 q^{64} +1.00000 q^{65} -1.55688 q^{67} -0.783100 q^{68} +7.24655 q^{70} -2.51438 q^{71} +2.90174 q^{73} +6.04434 q^{74} -0.669293 q^{76} +7.87921 q^{77} +3.32589 q^{79} +4.26223 q^{80} -7.17911 q^{82} -5.35683 q^{83} +5.55131 q^{85} +6.90186 q^{86} +4.32758 q^{88} -0.225989 q^{89} +4.95241 q^{91} +0.0107889 q^{92} +16.8808 q^{94} +4.74454 q^{95} +18.8550 q^{97} +25.6452 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 6 q^{4} - 8 q^{5} + 4 q^{7} + 6 q^{8}+O(q^{10}) $ 8 * q + 6 * q^4 - 8 * q^5 + 4 * q^7 + 6 * q^8	$ 8 q + 6 q^{4} - 8 q^{5} + 4 q^{7} + 6 q^{8} + 6 q^{11} - 8 q^{13} + 4 q^{14} - 10 q^{16} - 2 q^{17} - 10 q^{19} - 6 q^{20} + 30 q^{23} + 8 q^{25} + 2 q^{28} + 8 q^{29} - 10 q^{31} - 8 q^{32} - 4 q^{34} - 4 q^{35} - 8 q^{37} + 24 q^{38} - 6 q^{40} + 6 q^{41} + 34 q^{44} + 16 q^{46} + 18 q^{47} + 26 q^{49} - 6 q^{52} + 14 q^{53} - 6 q^{55} + 28 q^{56} + 30 q^{59} - 18 q^{61} + 10 q^{62} - 36 q^{64} + 8 q^{65} - 6 q^{67} - 8 q^{68} - 4 q^{70} + 16 q^{71} + 12 q^{73} + 20 q^{74} - 2 q^{76} + 8 q^{77} - 30 q^{79} + 10 q^{80} + 20 q^{82} + 26 q^{83} + 2 q^{85} + 30 q^{86} + 10 q^{88} + 14 q^{89} - 4 q^{91} + 26 q^{92} - 16 q^{94} + 10 q^{95} + 44 q^{97} + 60 q^{98}+O(q^{100}) $ 8 * q + 6 * q^4 - 8 * q^5 + 4 * q^7 + 6 * q^8 + 6 * q^11 - 8 * q^13 + 4 * q^14 - 10 * q^16 - 2 * q^17 - 10 * q^19 - 6 * q^20 + 30 * q^23 + 8 * q^25 + 2 * q^28 + 8 * q^29 - 10 * q^31 - 8 * q^32 - 4 * q^34 - 4 * q^35 - 8 * q^37 + 24 * q^38 - 6 * q^40 + 6 * q^41 + 34 * q^44 + 16 * q^46 + 18 * q^47 + 26 * q^49 - 6 * q^52 + 14 * q^53 - 6 * q^55 + 28 * q^56 + 30 * q^59 - 18 * q^61 + 10 * q^62 - 36 * q^64 + 8 * q^65 - 6 * q^67 - 8 * q^68 - 4 * q^70 + 16 * q^71 + 12 * q^73 + 20 * q^74 - 2 * q^76 + 8 * q^77 - 30 * q^79 + 10 * q^80 + 20 * q^82 + 26 * q^83 + 2 * q^85 + 30 * q^86 + 10 * q^88 + 14 * q^89 - 4 * q^91 + 26 * q^92 - 16 * q^94 + 10 * q^95 + 44 * q^97 + 60 * 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 $

$2$	1.46324	1.03467	0.517333	−	0.855784i	$-0.326925\pi$
$2$	1.46324	1.03467	0.517333	+	0.855784i	$0.326925\pi$
$3$	0	0
$3$	0	0
$4$	0.141066	0.0705330
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.95241	−1.87183	−0.935917	−	0.352220i	$-0.885427\pi$
$7$	−4.95241	−1.87183	−0.935917	+	0.352220i	$0.885427\pi$
$8$	−2.72006	−0.961688
$9$	0	0
$10$	−1.46324	−0.462717
$11$	−1.59098	−0.479700	−0.239850	−	0.970810i	$-0.577098\pi$
$11$	−1.59098	−0.479700	−0.239850	+	0.970810i	$0.577098\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−7.24655	−1.93672
$15$	0	0
$16$	−4.26223	−1.06556
$17$	−5.55131	−1.34639	−0.673195	−	0.739465i	$-0.735078\pi$
$17$	−5.55131	−1.34639	−0.673195	+	0.739465i	$0.735078\pi$
$18$	0	0
$19$	−4.74454	−1.08847	−0.544236	−	0.838932i	$-0.683181\pi$
$19$	−4.74454	−1.08847	−0.544236	+	0.838932i	$0.683181\pi$
$20$	−0.141066	−0.0315433
$21$	0	0
$22$	−2.32799	−0.496329
$23$	0.0764816	0.0159475	0.00797376	−	0.999968i	$-0.497462\pi$
$23$	0.0764816	0.0159475	0.00797376	+	0.999968i	$0.497462\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.46324	−0.286965
$27$	0	0
$28$	−0.698616	−0.132026
$29$	3.53848	0.657079	0.328539	−	0.944490i	$-0.393444\pi$
$29$	3.53848	0.657079	0.328539	+	0.944490i	$0.393444\pi$
$30$	0	0
$31$	1.59217	0.285962	0.142981	−	0.989725i	$-0.454331\pi$
$31$	1.59217	0.285962	0.142981	+	0.989725i	$0.454331\pi$
$32$	−0.796535	−0.140809
$33$	0	0
$34$	−8.12288	−1.39306
$35$	4.95241	0.837110
$36$	0	0
$37$	4.13079	0.679099	0.339549	−	0.940588i	$-0.389725\pi$
$37$	4.13079	0.679099	0.339549	+	0.940588i	$0.389725\pi$
$38$	−6.94240	−1.12621
$39$	0	0
$40$	2.72006	0.430080
$41$	−4.90631	−0.766237	−0.383119	−	0.923699i	$-0.625150\pi$
$41$	−4.90631	−0.766237	−0.383119	+	0.923699i	$0.625150\pi$
$42$	0	0
$43$	4.71684	0.719311	0.359656	−	0.933085i	$-0.382894\pi$
$43$	4.71684	0.719311	0.359656	+	0.933085i	$0.382894\pi$
$44$	−0.224434	−0.0338347
$45$	0	0
$46$	0.111911	0.0165003
$47$	11.5366	1.68279	0.841395	−	0.540421i	$-0.181735\pi$
$47$	11.5366	1.68279	0.841395	+	0.540421i	$0.181735\pi$
$48$	0	0
$49$	17.5264	2.50377
$50$	1.46324	0.206933
$51$	0	0
$52$	−0.141066	−0.0195623
$53$	−1.29315	−0.177628	−0.0888139	−	0.996048i	$-0.528308\pi$
$53$	−1.29315	−0.177628	−0.0888139	+	0.996048i	$0.528308\pi$
$54$	0	0
$55$	1.59098	0.214528
$56$	13.4709	1.80012
$57$	0	0
$58$	5.17764	0.679857
$59$	4.66770	0.607683	0.303841	−	0.952723i	$-0.401731\pi$
$59$	4.66770	0.607683	0.303841	+	0.952723i	$0.401731\pi$
$60$	0	0
$61$	−10.6244	−1.36032	−0.680160	−	0.733063i	$-0.738090\pi$
$61$	−10.6244	−1.36032	−0.680160	+	0.733063i	$0.738090\pi$
$62$	2.32972	0.295875
$63$	0	0
$64$	7.35894	0.919868
$65$	1.00000	0.124035
$66$	0	0
$67$	−1.55688	−0.190203	−0.0951014	−	0.995468i	$-0.530318\pi$
$67$	−1.55688	−0.190203	−0.0951014	+	0.995468i	$0.530318\pi$
$68$	−0.783100	−0.0949649
$69$	0	0
$70$	7.24655	0.866129
$71$	−2.51438	−0.298402	−0.149201	−	0.988807i	$-0.547670\pi$
$71$	−2.51438	−0.298402	−0.149201	+	0.988807i	$0.547670\pi$
$72$	0	0
$73$	2.90174	0.339624	0.169812	−	0.985477i	$-0.445684\pi$
$73$	2.90174	0.339624	0.169812	+	0.985477i	$0.445684\pi$
$74$	6.04434	0.702640
$75$	0	0
$76$	−0.669293	−0.0767732
$77$	7.87921	0.897919
$78$	0	0
$79$	3.32589	0.374192	0.187096	−	0.982342i	$-0.440092\pi$
$79$	3.32589	0.374192	0.187096	+	0.982342i	$0.440092\pi$
$80$	4.26223	0.476532
$81$	0	0
$82$	−7.17911	−0.792800
$83$	−5.35683	−0.587988	−0.293994	−	0.955807i	$-0.594985\pi$
$83$	−5.35683	−0.587988	−0.293994	+	0.955807i	$0.594985\pi$
$84$	0	0
$85$	5.55131	0.602124
$86$	6.90186	0.744247
$87$	0	0
$88$	4.32758	0.461322
$89$	−0.225989	−0.0239548	−0.0119774	−	0.999928i	$-0.503813\pi$
$89$	−0.225989	−0.0239548	−0.0119774	+	0.999928i	$0.503813\pi$
$90$	0	0
$91$	4.95241	0.519154
$92$	0.0107889	0.00112483
$93$	0	0
$94$	16.8808	1.74112
$95$	4.74454	0.486780
$96$	0	0
$97$	18.8550	1.91444	0.957219	−	0.289363i	$-0.0934435\pi$
$97$	18.8550	1.91444	0.957219	+	0.289363i	$0.0934435\pi$
$98$	25.6452	2.59056
$99$	0	0
$100$	0.141066	0.0141066
$101$	−10.3079	−1.02567	−0.512837	−	0.858486i	$-0.671406\pi$
$101$	−10.3079	−1.02567	−0.512837	+	0.858486i	$0.671406\pi$
$102$	0	0
$103$	−2.65848	−0.261948	−0.130974	−	0.991386i	$-0.541810\pi$
$103$	−2.65848	−0.261948	−0.130974	+	0.991386i	$0.541810\pi$
$104$	2.72006	0.266724
$105$	0	0
$106$	−1.89219	−0.183785
$107$	−17.5939	−1.70087	−0.850433	−	0.526084i	$-0.823660\pi$
$107$	−17.5939	−1.70087	−0.850433	+	0.526084i	$0.823660\pi$
$108$	0	0
$109$	−17.2053	−1.64797	−0.823985	−	0.566612i	$-0.808254\pi$
$109$	−17.2053	−1.64797	−0.823985	+	0.566612i	$0.808254\pi$
$110$	2.32799	0.221965
$111$	0	0
$112$	21.1083	1.99455
$113$	−16.3236	−1.53560	−0.767799	−	0.640690i	$-0.778648\pi$
$113$	−16.3236	−1.53560	−0.767799	+	0.640690i	$0.778648\pi$
$114$	0	0
$115$	−0.0764816	−0.00713195
$116$	0.499159	0.0463457
$117$	0	0
$118$	6.82996	0.628748
$119$	27.4923	2.52022
$120$	0	0
$121$	−8.46877	−0.769888
$122$	−15.5461	−1.40748
$123$	0	0
$124$	0.224601	0.0201698
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−11.1380	−0.988342	−0.494171	−	0.869365i	$-0.664528\pi$
$127$	−11.1380	−0.988342	−0.494171	+	0.869365i	$0.664528\pi$
$128$	12.3610	1.09256
$129$	0	0
$130$	1.46324	0.128334
$131$	9.15691	0.800043	0.400022	−	0.916506i	$-0.369003\pi$
$131$	9.15691	0.800043	0.400022	+	0.916506i	$0.369003\pi$
$132$	0	0
$133$	23.4969	2.03744
$134$	−2.27808	−0.196796
$135$	0	0
$136$	15.0999	1.29481
$137$	14.3061	1.22225	0.611125	−	0.791534i	$-0.290717\pi$
$137$	14.3061	1.22225	0.611125	+	0.791534i	$0.290717\pi$
$138$	0	0
$139$	2.68341	0.227604	0.113802	−	0.993503i	$-0.463697\pi$
$139$	2.68341	0.227604	0.113802	+	0.993503i	$0.463697\pi$
$140$	0.698616	0.0590439
$141$	0	0
$142$	−3.67913	−0.308746
$143$	1.59098	0.133045
$144$	0	0
$145$	−3.53848	−0.293855
$146$	4.24594	0.351397
$147$	0	0
$148$	0.582715	0.0478988
$149$	−8.10911	−0.664324	−0.332162	−	0.943222i	$-0.607778\pi$
$149$	−8.10911	−0.664324	−0.332162	+	0.943222i	$0.607778\pi$
$150$	0	0
$151$	−9.49289	−0.772520	−0.386260	−	0.922390i	$-0.626233\pi$
$151$	−9.49289	−0.772520	−0.386260	+	0.922390i	$0.626233\pi$
$152$	12.9055	1.04677
$153$	0	0
$154$	11.5292	0.929046
$155$	−1.59217	−0.127886
$156$	0	0
$157$	−1.37156	−0.109463	−0.0547314	−	0.998501i	$-0.517430\pi$
$157$	−1.37156	−0.109463	−0.0547314	+	0.998501i	$0.517430\pi$
$158$	4.86657	0.387164
$159$	0	0
$160$	0.796535	0.0629716
$161$	−0.378768	−0.0298511
$162$	0	0
$163$	−8.03430	−0.629295	−0.314648	−	0.949209i	$-0.601886\pi$
$163$	−8.03430	−0.629295	−0.314648	+	0.949209i	$0.601886\pi$
$164$	−0.692114	−0.0540450
$165$	0	0
$166$	−7.83832	−0.608371
$167$	23.4981	1.81834	0.909171	−	0.416424i	$-0.136717\pi$
$167$	23.4981	1.81834	0.909171	+	0.416424i	$0.136717\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	8.12288	0.622997
$171$	0	0
$172$	0.665386	0.0507352
$173$	1.84099	0.139968	0.0699839	−	0.997548i	$-0.477705\pi$
$173$	1.84099	0.139968	0.0699839	+	0.997548i	$0.477705\pi$
$174$	0	0
$175$	−4.95241	−0.374367
$176$	6.78115	0.511148
$177$	0	0
$178$	−0.330676	−0.0247852
$179$	−12.4946	−0.933888	−0.466944	−	0.884287i	$-0.654645\pi$
$179$	−12.4946	−0.933888	−0.466944	+	0.884287i	$0.654645\pi$
$180$	0	0
$181$	−23.3250	−1.73374	−0.866868	−	0.498538i	$-0.833870\pi$
$181$	−23.3250	−1.73374	−0.866868	+	0.498538i	$0.833870\pi$
$182$	7.24655	0.537150
$183$	0	0
$184$	−0.208035	−0.0153365
$185$	−4.13079	−0.303702
$186$	0	0
$187$	8.83204	0.645863
$188$	1.62742	0.118692
$189$	0	0
$190$	6.94240	0.503654
$191$	19.5264	1.41288	0.706439	−	0.707774i	$-0.250300\pi$
$191$	19.5264	1.41288	0.706439	+	0.707774i	$0.250300\pi$
$192$	0	0
$193$	23.3207	1.67866	0.839332	−	0.543620i	$-0.182947\pi$
$193$	23.3207	1.67866	0.839332	+	0.543620i	$0.182947\pi$
$194$	27.5894	1.98080
$195$	0	0
$196$	2.47237	0.176598
$197$	9.68456	0.689996	0.344998	−	0.938603i	$-0.387880\pi$
$197$	9.68456	0.689996	0.344998	+	0.938603i	$0.387880\pi$
$198$	0	0
$199$	−12.9358	−0.916993	−0.458497	−	0.888696i	$-0.651612\pi$
$199$	−12.9358	−0.916993	−0.458497	+	0.888696i	$0.651612\pi$
$200$	−2.72006	−0.192338
$201$	0	0
$202$	−15.0829	−1.06123
$203$	−17.5240	−1.22994
$204$	0	0
$205$	4.90631	0.342672
$206$	−3.88999	−0.271028
$207$	0	0
$208$	4.26223	0.295533
$209$	7.54850	0.522140
$210$	0	0
$211$	17.0435	1.17332	0.586661	−	0.809833i	$-0.300442\pi$
$211$	17.0435	1.17332	0.586661	+	0.809833i	$0.300442\pi$
$212$	−0.182419	−0.0125286
$213$	0	0
$214$	−25.7441	−1.75983
$215$	−4.71684	−0.321686
$216$	0	0
$217$	−7.88508	−0.535274
$218$	−25.1755	−1.70510
$219$	0	0
$220$	0.224434	0.0151313
$221$	5.55131	0.373421
$222$	0	0
$223$	10.2204	0.684407	0.342203	−	0.939626i	$-0.388827\pi$
$223$	10.2204	0.684407	0.342203	+	0.939626i	$0.388827\pi$
$224$	3.94476	0.263571
$225$	0	0
$226$	−23.8854	−1.58883
$227$	21.9465	1.45664	0.728319	−	0.685238i	$-0.240302\pi$
$227$	21.9465	1.45664	0.728319	+	0.685238i	$0.240302\pi$
$228$	0	0
$229$	−23.2117	−1.53387	−0.766937	−	0.641722i	$-0.778220\pi$
$229$	−23.2117	−1.53387	−0.766937	+	0.641722i	$0.778220\pi$
$230$	−0.111911	−0.00737918
$231$	0	0
$232$	−9.62488	−0.631905
$233$	−14.7926	−0.969096	−0.484548	−	0.874765i	$-0.661016\pi$
$233$	−14.7926	−0.969096	−0.484548	+	0.874765i	$0.661016\pi$
$234$	0	0
$235$	−11.5366	−0.752566
$236$	0.658453	0.0428617
$237$	0	0
$238$	40.2278	2.60758
$239$	−2.62092	−0.169533	−0.0847666	−	0.996401i	$-0.527014\pi$
$239$	−2.62092	−0.169533	−0.0847666	+	0.996401i	$0.527014\pi$
$240$	0	0
$241$	−4.32645	−0.278691	−0.139346	−	0.990244i	$-0.544500\pi$
$241$	−4.32645	−0.278691	−0.139346	+	0.990244i	$0.544500\pi$
$242$	−12.3918	−0.796577
$243$	0	0
$244$	−1.49875	−0.0959475
$245$	−17.5264	−1.11972
$246$	0	0
$247$	4.74454	0.301888
$248$	−4.33081	−0.275006
$249$	0	0
$250$	−1.46324	−0.0925433
$251$	10.8607	0.685520	0.342760	−	0.939423i	$-0.388638\pi$
$251$	10.8607	0.685520	0.342760	+	0.939423i	$0.388638\pi$
$252$	0	0
$253$	−0.121681	−0.00765002
$254$	−16.2976	−1.02260
$255$	0	0
$256$	3.36914	0.210571
$257$	7.01308	0.437464	0.218732	−	0.975785i	$-0.429808\pi$
$257$	7.01308	0.437464	0.218732	+	0.975785i	$0.429808\pi$
$258$	0	0
$259$	−20.4574	−1.27116
$260$	0.141066	0.00874854
$261$	0	0
$262$	13.3987	0.827777
$263$	−8.84407	−0.545349	−0.272674	−	0.962106i	$-0.587908\pi$
$263$	−8.84407	−0.545349	−0.272674	+	0.962106i	$0.587908\pi$
$264$	0	0
$265$	1.29315	0.0794375
$266$	34.3816	2.10807
$267$	0	0
$268$	−0.219622	−0.0134156
$269$	25.3530	1.54580	0.772899	−	0.634529i	$-0.218806\pi$
$269$	25.3530	1.54580	0.772899	+	0.634529i	$0.218806\pi$
$270$	0	0
$271$	20.7057	1.25778	0.628892	−	0.777493i	$-0.283509\pi$
$271$	20.7057	1.25778	0.628892	+	0.777493i	$0.283509\pi$
$272$	23.6610	1.43466
$273$	0	0
$274$	20.9332	1.26462
$275$	−1.59098	−0.0959400
$276$	0	0
$277$	20.1257	1.20924	0.604620	−	0.796514i	$-0.293325\pi$
$277$	20.1257	1.20924	0.604620	+	0.796514i	$0.293325\pi$
$278$	3.92647	0.235494
$279$	0	0
$280$	−13.4709	−0.805038
$281$	14.2406	0.849523	0.424762	−	0.905305i	$-0.360358\pi$
$281$	14.2406	0.849523	0.424762	+	0.905305i	$0.360358\pi$
$282$	0	0
$283$	26.4221	1.57063	0.785315	−	0.619096i	$-0.212501\pi$
$283$	26.4221	1.57063	0.785315	+	0.619096i	$0.212501\pi$
$284$	−0.354693	−0.0210472
$285$	0	0
$286$	2.32799	0.137657
$287$	24.2981	1.43427
$288$	0	0
$289$	13.8170	0.812765
$290$	−5.17764	−0.304041
$291$	0	0
$292$	0.409337	0.0239547
$293$	−17.0177	−0.994183	−0.497091	−	0.867698i	$-0.665599\pi$
$293$	−17.0177	−0.994183	−0.497091	+	0.867698i	$0.665599\pi$
$294$	0	0
$295$	−4.66770	−0.271764
$296$	−11.2360	−0.653081
$297$	0	0
$298$	−11.8656	−0.687354
$299$	−0.0764816	−0.00442304
$300$	0	0
$301$	−23.3597	−1.34643
$302$	−13.8904	−0.799300
$303$	0	0
$304$	20.2223	1.15983
$305$	10.6244	0.608354
$306$	0	0
$307$	17.3829	0.992095	0.496048	−	0.868295i	$-0.334784\pi$
$307$	17.3829	0.992095	0.496048	+	0.868295i	$0.334784\pi$
$308$	1.11149	0.0633329
$309$	0	0
$310$	−2.32972	−0.132319
$311$	13.6360	0.773225	0.386612	−	0.922242i	$-0.373645\pi$
$311$	13.6360	0.773225	0.386612	+	0.922242i	$0.373645\pi$
$312$	0	0
$313$	8.12157	0.459058	0.229529	−	0.973302i	$-0.426281\pi$
$313$	8.12157	0.459058	0.229529	+	0.973302i	$0.426281\pi$
$314$	−2.00693	−0.113257
$315$	0	0
$316$	0.469170	0.0263929
$317$	18.6219	1.04591	0.522955	−	0.852361i	$-0.324830\pi$
$317$	18.6219	1.04591	0.522955	+	0.852361i	$0.324830\pi$
$318$	0	0
$319$	−5.62967	−0.315201
$320$	−7.35894	−0.411378
$321$	0	0
$322$	−0.554228	−0.0308859
$323$	26.3384	1.46551
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	−11.7561	−0.651110
$327$	0	0
$328$	13.3455	0.736881
$329$	−57.1341	−3.14990
$330$	0	0
$331$	−28.7903	−1.58246	−0.791229	−	0.611519i	$-0.790559\pi$
$331$	−28.7903	−1.58246	−0.791229	+	0.611519i	$0.790559\pi$
$332$	−0.755666	−0.0414726
$333$	0	0
$334$	34.3834	1.88138
$335$	1.55688	0.0850613
$336$	0	0
$337$	−11.8343	−0.644658	−0.322329	−	0.946628i	$-0.604466\pi$
$337$	−11.8343	−0.644658	−0.322329	+	0.946628i	$0.604466\pi$
$338$	1.46324	0.0795897
$339$	0	0
$340$	0.783100	0.0424696
$341$	−2.53312	−0.137176
$342$	0	0
$343$	−52.1308	−2.81480
$344$	−12.8301	−0.691753
$345$	0	0
$346$	2.69380	0.144820
$347$	31.9068	1.71285	0.856423	−	0.516275i	$-0.172682\pi$
$347$	31.9068	1.71285	0.856423	+	0.516275i	$0.172682\pi$
$348$	0	0
$349$	31.2577	1.67319	0.836594	−	0.547824i	$-0.184544\pi$
$349$	31.2577	1.67319	0.836594	+	0.547824i	$0.184544\pi$
$350$	−7.24655	−0.387345
$351$	0	0
$352$	1.26727	0.0675459
$353$	10.2084	0.543336	0.271668	−	0.962391i	$-0.412425\pi$
$353$	10.2084	0.543336	0.271668	+	0.962391i	$0.412425\pi$
$354$	0	0
$355$	2.51438	0.133449
$356$	−0.0318794	−0.00168960
$357$	0	0
$358$	−18.2825	−0.966261
$359$	8.03454	0.424047	0.212023	−	0.977265i	$-0.431995\pi$
$359$	8.03454	0.424047	0.212023	+	0.977265i	$0.431995\pi$
$360$	0	0
$361$	3.51068	0.184773
$362$	−34.1301	−1.79384
$363$	0	0
$364$	0.698616	0.0366174
$365$	−2.90174	−0.151884
$366$	0	0
$367$	−23.5639	−1.23002	−0.615012	−	0.788518i	$-0.710849\pi$
$367$	−23.5639	−1.23002	−0.615012	+	0.788518i	$0.710849\pi$
$368$	−0.325982	−0.0169930
$369$	0	0
$370$	−6.04434	−0.314230
$371$	6.40420	0.332490
$372$	0	0
$373$	−12.3593	−0.639942	−0.319971	−	0.947427i	$-0.603673\pi$
$373$	−12.3593	−0.639942	−0.319971	+	0.947427i	$0.603673\pi$
$374$	12.9234	0.668252
$375$	0	0
$376$	−31.3803	−1.61832
$377$	−3.53848	−0.182241
$378$	0	0
$379$	−30.6886	−1.57637	−0.788183	−	0.615442i	$-0.788978\pi$
$379$	−30.6886	−1.57637	−0.788183	+	0.615442i	$0.788978\pi$
$380$	0.669293	0.0343340
$381$	0	0
$382$	28.5717	1.46186
$383$	23.2200	1.18649	0.593244	−	0.805023i	$-0.297847\pi$
$383$	23.2200	1.18649	0.593244	+	0.805023i	$0.297847\pi$
$384$	0	0
$385$	−7.87921	−0.401562
$386$	34.1238	1.73686
$387$	0	0
$388$	2.65980	0.135031
$389$	−23.0785	−1.17013	−0.585064	−	0.810987i	$-0.698931\pi$
$389$	−23.0785	−1.17013	−0.585064	+	0.810987i	$0.698931\pi$
$390$	0	0
$391$	−0.424573	−0.0214716
$392$	−47.6728	−2.40784
$393$	0	0
$394$	14.1708	0.713916
$395$	−3.32589	−0.167344
$396$	0	0
$397$	16.3847	0.822325	0.411163	−	0.911562i	$-0.365123\pi$
$397$	16.3847	0.822325	0.411163	+	0.911562i	$0.365123\pi$
$398$	−18.9281	−0.948781
$399$	0	0
$400$	−4.26223	−0.213112
$401$	19.8915	0.993335	0.496668	−	0.867941i	$-0.334557\pi$
$401$	19.8915	0.993335	0.496668	+	0.867941i	$0.334557\pi$
$402$	0	0
$403$	−1.59217	−0.0793117
$404$	−1.45409	−0.0723439
$405$	0	0
$406$	−25.6418	−1.27258
$407$	−6.57203	−0.325764
$408$	0	0
$409$	−33.2385	−1.64354	−0.821769	−	0.569820i	$-0.807013\pi$
$409$	−33.2385	−1.64354	−0.821769	+	0.569820i	$0.807013\pi$
$410$	7.17911	0.354551
$411$	0	0
$412$	−0.375021	−0.0184759
$413$	−23.1164	−1.13748
$414$	0	0
$415$	5.35683	0.262956
$416$	0.796535	0.0390533
$417$	0	0
$418$	11.0452	0.540241
$419$	−7.92162	−0.386996	−0.193498	−	0.981101i	$-0.561983\pi$
$419$	−7.92162	−0.386996	−0.193498	+	0.981101i	$0.561983\pi$
$420$	0	0
$421$	1.23413	0.0601477	0.0300738	−	0.999548i	$-0.490426\pi$
$421$	1.23413	0.0601477	0.0300738	+	0.999548i	$0.490426\pi$
$422$	24.9387	1.21400
$423$	0	0
$424$	3.51745	0.170822
$425$	−5.55131	−0.269278
$426$	0	0
$427$	52.6166	2.54630
$428$	−2.48190	−0.119967
$429$	0	0
$430$	−6.90186	−0.332837
$431$	−5.34263	−0.257345	−0.128673	−	0.991687i	$-0.541072\pi$
$431$	−5.34263	−0.257345	−0.128673	+	0.991687i	$0.541072\pi$
$432$	0	0
$433$	−21.4233	−1.02954	−0.514770	−	0.857328i	$-0.672123\pi$
$433$	−21.4233	−1.02954	−0.514770	+	0.857328i	$0.672123\pi$
$434$	−11.5378	−0.553830
$435$	0	0
$436$	−2.42708	−0.116236
$437$	−0.362870	−0.0173584
$438$	0	0
$439$	−18.8049	−0.897510	−0.448755	−	0.893655i	$-0.648132\pi$
$439$	−18.8049	−0.897510	−0.448755	+	0.893655i	$0.648132\pi$
$440$	−4.32758	−0.206309
$441$	0	0
$442$	8.12288	0.386366
$443$	−15.0986	−0.717355	−0.358678	−	0.933461i	$-0.616772\pi$
$443$	−15.0986	−0.717355	−0.358678	+	0.933461i	$0.616772\pi$
$444$	0	0
$445$	0.225989	0.0107129
$446$	14.9548	0.708132
$447$	0	0
$448$	−36.4445	−1.72184
$449$	−6.11019	−0.288358	−0.144179	−	0.989552i	$-0.546054\pi$
$449$	−6.11019	−0.288358	−0.144179	+	0.989552i	$0.546054\pi$
$450$	0	0
$451$	7.80587	0.367564
$452$	−2.30271	−0.108310
$453$	0	0
$454$	32.1129	1.50713
$455$	−4.95241	−0.232173
$456$	0	0
$457$	−36.2218	−1.69438	−0.847192	−	0.531287i	$-0.821709\pi$
$457$	−36.2218	−1.69438	−0.847192	+	0.531287i	$0.821709\pi$
$458$	−33.9643	−1.58705
$459$	0	0
$460$	−0.0107889	−0.000503037 0
$461$	−15.2538	−0.710441	−0.355220	−	0.934783i	$-0.615594\pi$
$461$	−15.2538	−0.710441	−0.355220	+	0.934783i	$0.615594\pi$
$462$	0	0
$463$	7.34655	0.341423	0.170712	−	0.985321i	$-0.445393\pi$
$463$	7.34655	0.341423	0.170712	+	0.985321i	$0.445393\pi$
$464$	−15.0818	−0.700156
$465$	0	0
$466$	−21.6451	−1.00269
$467$	−2.11495	−0.0978682	−0.0489341	−	0.998802i	$-0.515582\pi$
$467$	−2.11495	−0.0978682	−0.0489341	+	0.998802i	$0.515582\pi$
$468$	0	0
$469$	7.71029	0.356028
$470$	−16.8808	−0.778655
$471$	0	0
$472$	−12.6964	−0.584401
$473$	−7.50442	−0.345054
$474$	0	0
$475$	−4.74454	−0.217695
$476$	3.87823	0.177759
$477$	0	0
$478$	−3.83503	−0.175410
$479$	19.6744	0.898944	0.449472	−	0.893294i	$-0.351612\pi$
$479$	19.6744	0.898944	0.449472	+	0.893294i	$0.351612\pi$
$480$	0	0
$481$	−4.13079	−0.188348
$482$	−6.33063	−0.288352
$483$	0	0
$484$	−1.19465	−0.0543025
$485$	−18.8550	−0.856163
$486$	0	0
$487$	19.4574	0.881698	0.440849	−	0.897581i	$-0.354677\pi$
$487$	19.4574	0.881698	0.440849	+	0.897581i	$0.354677\pi$
$488$	28.8992	1.30820
$489$	0	0
$490$	−25.6452	−1.15853
$491$	−14.0281	−0.633077	−0.316539	−	0.948580i	$-0.602521\pi$
$491$	−14.0281	−0.633077	−0.316539	+	0.948580i	$0.602521\pi$
$492$	0	0
$493$	−19.6432	−0.884684
$494$	6.94240	0.312353
$495$	0	0
$496$	−6.78620	−0.304709
$497$	12.4522	0.558559
$498$	0	0
$499$	−6.41203	−0.287042	−0.143521	−	0.989647i	$-0.545842\pi$
$499$	−6.41203	−0.287042	−0.143521	+	0.989647i	$0.545842\pi$
$500$	−0.141066	−0.00630866
$501$	0	0
$502$	15.8918	0.709284
$503$	1.58905	0.0708521	0.0354260	−	0.999372i	$-0.488721\pi$
$503$	1.58905	0.0708521	0.0354260	+	0.999372i	$0.488721\pi$
$504$	0	0
$505$	10.3079	0.458696
$506$	−0.178048	−0.00791522
$507$	0	0
$508$	−1.57120	−0.0697107
$509$	40.4369	1.79233	0.896167	−	0.443717i	$-0.146341\pi$
$509$	40.4369	1.79233	0.896167	+	0.443717i	$0.146341\pi$
$510$	0	0
$511$	−14.3706	−0.635719
$512$	−19.7921	−0.874694
$513$	0	0
$514$	10.2618	0.452629
$515$	2.65848	0.117147
$516$	0	0
$517$	−18.3546	−0.807234
$518$	−29.9340	−1.31523
$519$	0	0
$520$	−2.72006	−0.119283
$521$	20.3148	0.890007	0.445003	−	0.895529i	$-0.353203\pi$
$521$	20.3148	0.890007	0.445003	+	0.895529i	$0.353203\pi$
$522$	0	0
$523$	−9.84446	−0.430468	−0.215234	−	0.976563i	$-0.569051\pi$
$523$	−9.84446	−0.430468	−0.215234	+	0.976563i	$0.569051\pi$
$524$	1.29173	0.0564294
$525$	0	0
$526$	−12.9410	−0.564254
$527$	−8.83863	−0.385017
$528$	0	0
$529$	−22.9942	−0.999746
$530$	1.89219	0.0821913
$531$	0	0
$532$	3.31461	0.143707
$533$	4.90631	0.212516
$534$	0	0
$535$	17.5939	0.760650
$536$	4.23480	0.182916
$537$	0	0
$538$	37.0974	1.59938
$539$	−27.8842	−1.20106
$540$	0	0
$541$	−19.7146	−0.847596	−0.423798	−	0.905757i	$-0.639303\pi$
$541$	−19.7146	−0.847596	−0.423798	+	0.905757i	$0.639303\pi$
$542$	30.2974	1.30139
$543$	0	0
$544$	4.42181	0.189583
$545$	17.2053	0.736994
$546$	0	0
$547$	15.7344	0.672756	0.336378	−	0.941727i	$-0.390798\pi$
$547$	15.7344	0.672756	0.336378	+	0.941727i	$0.390798\pi$
$548$	2.01810	0.0862090
$549$	0	0
$550$	−2.32799	−0.0992658
$551$	−16.7885	−0.715212
$552$	0	0
$553$	−16.4712	−0.700426
$554$	29.4488	1.25116
$555$	0	0
$556$	0.378538	0.0160536
$557$	8.29587	0.351507	0.175754	−	0.984434i	$-0.443764\pi$
$557$	8.29587	0.351507	0.175754	+	0.984434i	$0.443764\pi$
$558$	0	0
$559$	−4.71684	−0.199501
$560$	−21.1083	−0.891989
$561$	0	0
$562$	20.8374	0.878973
$563$	42.1827	1.77779	0.888893	−	0.458114i	$-0.151475\pi$
$563$	42.1827	1.77779	0.888893	+	0.458114i	$0.151475\pi$
$564$	0	0
$565$	16.3236	0.686741
$566$	38.6618	1.62508
$567$	0	0
$568$	6.83927	0.286969
$569$	−7.79477	−0.326774	−0.163387	−	0.986562i	$-0.552242\pi$
$569$	−7.79477	−0.326774	−0.163387	+	0.986562i	$0.552242\pi$
$570$	0	0
$571$	−29.3684	−1.22903	−0.614516	−	0.788905i	$-0.710649\pi$
$571$	−29.3684	−1.22903	−0.614516	+	0.788905i	$0.710649\pi$
$572$	0.224434	0.00938405
$573$	0	0
$574$	35.5539	1.48399
$575$	0.0764816	0.00318950
$576$	0	0
$577$	−40.7191	−1.69516	−0.847580	−	0.530668i	$-0.821941\pi$
$577$	−40.7191	−1.69516	−0.847580	+	0.530668i	$0.821941\pi$
$578$	20.2176	0.840940
$579$	0	0
$580$	−0.499159	−0.0207264
$581$	26.5292	1.10062
$582$	0	0
$583$	2.05738	0.0852080
$584$	−7.89293	−0.326612
$585$	0	0
$586$	−24.9009	−1.02865
$587$	25.7170	1.06145	0.530726	−	0.847543i	$-0.321919\pi$
$587$	25.7170	1.06145	0.530726	+	0.847543i	$0.321919\pi$
$588$	0	0
$589$	−7.55412	−0.311262
$590$	−6.82996	−0.281185
$591$	0	0
$592$	−17.6064	−0.723619
$593$	−10.7998	−0.443497	−0.221748	−	0.975104i	$-0.571176\pi$
$593$	−10.7998	−0.443497	−0.221748	+	0.975104i	$0.571176\pi$
$594$	0	0
$595$	−27.4923	−1.12708
$596$	−1.14392	−0.0468568
$597$	0	0
$598$	−0.111911	−0.00457637
$599$	14.7325	0.601954	0.300977	−	0.953631i	$-0.402687\pi$
$599$	14.7325	0.601954	0.300977	+	0.953631i	$0.402687\pi$
$600$	0	0
$601$	−17.1542	−0.699734	−0.349867	−	0.936799i	$-0.613773\pi$
$601$	−17.1542	−0.699734	−0.349867	+	0.936799i	$0.613773\pi$
$602$	−34.1808	−1.39311
$603$	0	0
$604$	−1.33912	−0.0544882
$605$	8.46877	0.344304
$606$	0	0
$607$	−32.2003	−1.30697	−0.653486	−	0.756939i	$-0.726694\pi$
$607$	−32.2003	−1.30697	−0.653486	+	0.756939i	$0.726694\pi$
$608$	3.77919	0.153266
$609$	0	0
$610$	15.5461	0.629443
$611$	−11.5366	−0.466722
$612$	0	0
$613$	14.3120	0.578058	0.289029	−	0.957320i	$-0.406668\pi$
$613$	14.3120	0.578058	0.289029	+	0.957320i	$0.406668\pi$
$614$	25.4353	1.02649
$615$	0	0
$616$	−21.4319	−0.863518
$617$	44.8862	1.80705	0.903525	−	0.428536i	$-0.140970\pi$
$617$	44.8862	1.80705	0.903525	+	0.428536i	$0.140970\pi$
$618$	0	0
$619$	1.99365	0.0801317	0.0400659	−	0.999197i	$-0.487243\pi$
$619$	1.99365	0.0801317	0.0400659	+	0.999197i	$0.487243\pi$
$620$	−0.224601	−0.00902020
$621$	0	0
$622$	19.9527	0.800029
$623$	1.11919	0.0448394
$624$	0	0
$625$	1.00000	0.0400000
$626$	11.8838	0.474972
$627$	0	0
$628$	−0.193481	−0.00772073
$629$	−22.9313	−0.914331
$630$	0	0
$631$	33.9679	1.35224	0.676121	−	0.736791i	$-0.263660\pi$
$631$	33.9679	1.35224	0.676121	+	0.736791i	$0.263660\pi$
$632$	−9.04664	−0.359856
$633$	0	0
$634$	27.2482	1.08217
$635$	11.1380	0.442000
$636$	0	0
$637$	−17.5264	−0.694420
$638$	−8.23754	−0.326127
$639$	0	0
$640$	−12.3610	−0.488610
$641$	−46.5675	−1.83931	−0.919653	−	0.392732i	$-0.871530\pi$
$641$	−46.5675	−1.83931	−0.919653	+	0.392732i	$0.871530\pi$
$642$	0	0
$643$	1.04675	0.0412799	0.0206400	−	0.999787i	$-0.493430\pi$
$643$	1.04675	0.0412799	0.0206400	+	0.999787i	$0.493430\pi$
$644$	−0.0534313	−0.00210549
$645$	0	0
$646$	38.5394	1.51631
$647$	38.3293	1.50688	0.753440	−	0.657517i	$-0.228393\pi$
$647$	38.3293	1.50688	0.753440	+	0.657517i	$0.228393\pi$
$648$	0	0
$649$	−7.42624	−0.291505
$650$	−1.46324	−0.0573929
$651$	0	0
$652$	−1.13337	−0.0443861
$653$	13.8726	0.542878	0.271439	−	0.962456i	$-0.412500\pi$
$653$	13.8726	0.542878	0.271439	+	0.962456i	$0.412500\pi$
$654$	0	0
$655$	−9.15691	−0.357790
$656$	20.9118	0.816471
$657$	0	0
$658$	−83.6008	−3.25910
$659$	4.74266	0.184748	0.0923740	−	0.995724i	$-0.470554\pi$
$659$	4.74266	0.184748	0.0923740	+	0.995724i	$0.470554\pi$
$660$	0	0
$661$	−6.74129	−0.262206	−0.131103	−	0.991369i	$-0.541852\pi$
$661$	−6.74129	−0.262206	−0.131103	+	0.991369i	$0.541852\pi$
$662$	−42.1271	−1.63732
$663$	0	0
$664$	14.5709	0.565461
$665$	−23.4969	−0.911171
$666$	0	0
$667$	0.270628	0.0104788
$668$	3.31479	0.128253
$669$	0	0
$670$	2.27808	0.0880100
$671$	16.9033	0.652546
$672$	0	0
$673$	−2.58340	−0.0995828	−0.0497914	−	0.998760i	$-0.515856\pi$
$673$	−2.58340	−0.0995828	−0.0497914	+	0.998760i	$0.515856\pi$
$674$	−17.3165	−0.667005
$675$	0	0
$676$	0.141066	0.00542561
$677$	−42.7696	−1.64377	−0.821884	−	0.569654i	$-0.807077\pi$
$677$	−42.7696	−1.64377	−0.821884	+	0.569654i	$0.807077\pi$
$678$	0	0
$679$	−93.3779	−3.58351
$680$	−15.0999	−0.579055
$681$	0	0
$682$	−3.70656	−0.141931
$683$	14.8065	0.566553	0.283277	−	0.959038i	$-0.408579\pi$
$683$	14.8065	0.566553	0.283277	+	0.959038i	$0.408579\pi$
$684$	0	0
$685$	−14.3061	−0.546607
$686$	−76.2798	−2.91238
$687$	0	0
$688$	−20.1043	−0.766468
$689$	1.29315	0.0492651
$690$	0	0
$691$	15.7360	0.598624	0.299312	−	0.954155i	$-0.403243\pi$
$691$	15.7360	0.598624	0.299312	+	0.954155i	$0.403243\pi$
$692$	0.259701	0.00987234
$693$	0	0
$694$	46.6872	1.77222
$695$	−2.68341	−0.101788
$696$	0	0
$697$	27.2364	1.03165
$698$	45.7375	1.73119
$699$	0	0
$700$	−0.698616	−0.0264052
$701$	31.5607	1.19203	0.596015	−	0.802973i	$-0.296750\pi$
$701$	31.5607	1.19203	0.596015	+	0.802973i	$0.296750\pi$
$702$	0	0
$703$	−19.5987	−0.739180
$704$	−11.7080	−0.441261
$705$	0	0
$706$	14.9373	0.562172
$707$	51.0490	1.91989
$708$	0	0
$709$	40.4938	1.52078	0.760388	−	0.649469i	$-0.225009\pi$
$709$	40.4938	1.52078	0.760388	+	0.649469i	$0.225009\pi$
$710$	3.67913	0.138075
$711$	0	0
$712$	0.614705	0.0230370
$713$	0.121772	0.00456039
$714$	0	0
$715$	−1.59098	−0.0594995
$716$	−1.76256	−0.0658699
$717$	0	0
$718$	11.7564	0.438747
$719$	34.6714	1.29302	0.646512	−	0.762904i	$-0.276227\pi$
$719$	34.6714	1.29302	0.646512	+	0.762904i	$0.276227\pi$
$720$	0	0
$721$	13.1659	0.490323
$722$	5.13697	0.191178
$723$	0	0
$724$	−3.29037	−0.122286
$725$	3.53848	0.131416
$726$	0	0
$727$	36.1748	1.34165	0.670824	−	0.741616i	$-0.265940\pi$
$727$	36.1748	1.34165	0.670824	+	0.741616i	$0.265940\pi$
$728$	−13.4709	−0.499264
$729$	0	0
$730$	−4.24594	−0.157149
$731$	−26.1846	−0.968473
$732$	0	0
$733$	−47.2468	−1.74510	−0.872549	−	0.488526i	$-0.837535\pi$
$733$	−47.2468	−1.74510	−0.872549	+	0.488526i	$0.837535\pi$
$734$	−34.4796	−1.27266
$735$	0	0
$736$	−0.0609202	−0.00224555
$737$	2.47697	0.0912403
$738$	0	0
$739$	23.4117	0.861215	0.430607	−	0.902539i	$-0.358299\pi$
$739$	23.4117	0.861215	0.430607	+	0.902539i	$0.358299\pi$
$740$	−0.582715	−0.0214210
$741$	0	0
$742$	9.37088	0.344016
$743$	53.8155	1.97430	0.987150	−	0.159797i	$-0.0510839\pi$
$743$	53.8155	1.97430	0.987150	+	0.159797i	$0.0510839\pi$
$744$	0	0
$745$	8.10911	0.297095
$746$	−18.0847	−0.662126
$747$	0	0
$748$	1.24590	0.0455546
$749$	87.1321	3.18374
$750$	0	0
$751$	35.6661	1.30148	0.650738	−	0.759302i	$-0.274460\pi$
$751$	35.6661	1.30148	0.650738	+	0.759302i	$0.274460\pi$
$752$	−49.1718	−1.79311
$753$	0	0
$754$	−5.17764	−0.188558
$755$	9.49289	0.345482
$756$	0	0
$757$	46.6061	1.69393	0.846963	−	0.531651i	$-0.178428\pi$
$757$	46.6061	1.69393	0.846963	+	0.531651i	$0.178428\pi$
$758$	−44.9047	−1.63101
$759$	0	0
$760$	−12.9055	−0.468130
$761$	18.2267	0.660716	0.330358	−	0.943856i	$-0.392831\pi$
$761$	18.2267	0.660716	0.330358	+	0.943856i	$0.392831\pi$
$762$	0	0
$763$	85.2077	3.08473
$764$	2.75451	0.0996545
$765$	0	0
$766$	33.9764	1.22762
$767$	−4.66770	−0.168541
$768$	0	0
$769$	10.2917	0.371128	0.185564	−	0.982632i	$-0.440589\pi$
$769$	10.2917	0.371128	0.185564	+	0.982632i	$0.440589\pi$
$770$	−11.5292	−0.415482
$771$	0	0
$772$	3.28976	0.118401
$773$	−33.1268	−1.19149	−0.595745	−	0.803174i	$-0.703143\pi$
$773$	−33.1268	−1.19149	−0.595745	+	0.803174i	$0.703143\pi$
$774$	0	0
$775$	1.59217	0.0571925
$776$	−51.2869	−1.84109
$777$	0	0
$778$	−33.7694	−1.21069
$779$	23.2782	0.834029
$780$	0	0
$781$	4.00034	0.143143
$782$	−0.621251	−0.0222159
$783$	0	0
$784$	−74.7014	−2.66791
$785$	1.37156	0.0489532
$786$	0	0
$787$	4.11339	0.146626	0.0733132	−	0.997309i	$-0.476643\pi$
$787$	4.11339	0.146626	0.0733132	+	0.997309i	$0.476643\pi$
$788$	1.36616	0.0486675
$789$	0	0
$790$	−4.86657	−0.173145
$791$	80.8413	2.87439
$792$	0	0
$793$	10.6244	0.377285
$794$	23.9747	0.850832
$795$	0	0
$796$	−1.82480	−0.0646783
$797$	−31.2433	−1.10670	−0.553348	−	0.832950i	$-0.686650\pi$
$797$	−31.2433	−1.10670	−0.553348	+	0.832950i	$0.686650\pi$
$798$	0	0
$799$	−64.0433	−2.26569
$800$	−0.796535	−0.0281617
$801$	0	0
$802$	29.1060	1.02777
$803$	−4.61663	−0.162917
$804$	0	0
$805$	0.378768	0.0133498
$806$	−2.32972	−0.0820611
$807$	0	0
$808$	28.0382	0.986379
$809$	24.4812	0.860712	0.430356	−	0.902659i	$-0.358388\pi$
$809$	24.4812	0.860712	0.430356	+	0.902659i	$0.358388\pi$
$810$	0	0
$811$	27.5036	0.965782	0.482891	−	0.875681i	$-0.339587\pi$
$811$	27.5036	0.965782	0.482891	+	0.875681i	$0.339587\pi$
$812$	−2.47204	−0.0867516
$813$	0	0
$814$	−9.61645	−0.337056
$815$	8.03430	0.281429
$816$	0	0
$817$	−22.3793	−0.782951
$818$	−48.6359	−1.70051
$819$	0	0
$820$	0.692114	0.0241697
$821$	19.2162	0.670649	0.335325	−	0.942103i	$-0.391154\pi$
$821$	19.2162	0.670649	0.335325	+	0.942103i	$0.391154\pi$
$822$	0	0
$823$	−50.8253	−1.77166	−0.885829	−	0.464012i	$-0.846409\pi$
$823$	−50.8253	−1.77166	−0.885829	+	0.464012i	$0.846409\pi$
$824$	7.23123	0.251912
$825$	0	0
$826$	−33.8247	−1.17691
$827$	−56.5580	−1.96671	−0.983357	−	0.181684i	$-0.941845\pi$
$827$	−56.5580	−1.96671	−0.983357	+	0.181684i	$0.941845\pi$
$828$	0	0
$829$	25.9253	0.900424	0.450212	−	0.892922i	$-0.351348\pi$
$829$	25.9253	0.900424	0.450212	+	0.892922i	$0.351348\pi$
$830$	7.83832	0.272072
$831$	0	0
$832$	−7.35894	−0.255126
$833$	−97.2942	−3.37104
$834$	0	0
$835$	−23.4981	−0.813187
$836$	1.06484	0.0368281
$837$	0	0
$838$	−11.5912	−0.400412
$839$	−2.90284	−0.100217	−0.0501086	−	0.998744i	$-0.515957\pi$
$839$	−2.90284	−0.100217	−0.0501086	+	0.998744i	$0.515957\pi$
$840$	0	0
$841$	−16.4792	−0.568247
$842$	1.80582	0.0622327
$843$	0	0
$844$	2.40425	0.0827578
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	41.9408	1.44110
$848$	5.51170	0.189273
$849$	0	0
$850$	−8.12288	−0.278613
$851$	0.315930	0.0108299
$852$	0	0
$853$	33.4101	1.14394	0.571971	−	0.820274i	$-0.306179\pi$
$853$	33.4101	1.14394	0.571971	+	0.820274i	$0.306179\pi$
$854$	76.9906	2.63456
$855$	0	0
$856$	47.8565	1.63570
$857$	17.5740	0.600316	0.300158	−	0.953890i	$-0.402961\pi$
$857$	17.5740	0.600316	0.300158	+	0.953890i	$0.402961\pi$
$858$	0	0
$859$	−1.25578	−0.0428466	−0.0214233	−	0.999770i	$-0.506820\pi$
$859$	−1.25578	−0.0428466	−0.0214233	+	0.999770i	$0.506820\pi$
$860$	−0.665386	−0.0226895
$861$	0	0
$862$	−7.81754	−0.266266
$863$	−16.8692	−0.574235	−0.287118	−	0.957895i	$-0.592697\pi$
$863$	−16.8692	−0.574235	−0.287118	+	0.957895i	$0.592697\pi$
$864$	0	0
$865$	−1.84099	−0.0625955
$866$	−31.3474	−1.06523
$867$	0	0
$868$	−1.11232	−0.0377545
$869$	−5.29145	−0.179500
$870$	0	0
$871$	1.55688	0.0527528
$872$	46.7995	1.58483
$873$	0	0
$874$	−0.530965	−0.0179602
$875$	4.95241	0.167422
$876$	0	0
$877$	−17.9517	−0.606186	−0.303093	−	0.952961i	$-0.598019\pi$
$877$	−17.9517	−0.606186	−0.303093	+	0.952961i	$0.598019\pi$
$878$	−27.5161	−0.928623
$879$	0	0
$880$	−6.78115	−0.228592
$881$	33.1498	1.11685	0.558423	−	0.829556i	$-0.311407\pi$
$881$	33.1498	1.11685	0.558423	+	0.829556i	$0.311407\pi$
$882$	0	0
$883$	−24.9328	−0.839055	−0.419528	−	0.907743i	$-0.637804\pi$
$883$	−24.9328	−0.839055	−0.419528	+	0.907743i	$0.637804\pi$
$884$	0.783100	0.0263385
$885$	0	0
$886$	−22.0928	−0.742223
$887$	−54.6710	−1.83567	−0.917835	−	0.396961i	$-0.870065\pi$
$887$	−54.6710	−1.83567	−0.917835	+	0.396961i	$0.870065\pi$
$888$	0	0
$889$	55.1602	1.85001
$890$	0.330676	0.0110843
$891$	0	0
$892$	1.44175	0.0482732
$893$	−54.7360	−1.83167
$894$	0	0
$895$	12.4946	0.417647
$896$	−61.2165	−2.04510
$897$	0	0
$898$	−8.94067	−0.298354
$899$	5.63386	0.187900
$900$	0	0
$901$	7.17867	0.239156
$902$	11.4218	0.380306
$903$	0	0
$904$	44.4013	1.47677
$905$	23.3250	0.775350
$906$	0	0
$907$	50.8770	1.68934	0.844671	−	0.535286i	$-0.179796\pi$
$907$	50.8770	1.68934	0.844671	+	0.535286i	$0.179796\pi$
$908$	3.09590	0.102741
$909$	0	0
$910$	−7.24655	−0.240221
$911$	−37.0508	−1.22755	−0.613773	−	0.789482i	$-0.710349\pi$
$911$	−37.0508	−1.22755	−0.613773	+	0.789482i	$0.710349\pi$
$912$	0	0
$913$	8.52263	0.282058
$914$	−53.0011	−1.75312
$915$	0	0
$916$	−3.27439	−0.108189
$917$	−45.3488	−1.49755
$918$	0	0
$919$	3.88820	0.128260	0.0641300	−	0.997942i	$-0.479573\pi$
$919$	3.88820	0.128260	0.0641300	+	0.997942i	$0.479573\pi$
$920$	0.208035	0.00685870
$921$	0	0
$922$	−22.3200	−0.735069
$923$	2.51438	0.0827618
$924$	0	0
$925$	4.13079	0.135820
$926$	10.7498	0.353259
$927$	0	0
$928$	−2.81852	−0.0925225
$929$	−10.2003	−0.334661	−0.167330	−	0.985901i	$-0.553515\pi$
$929$	−10.2003	−0.334661	−0.167330	+	0.985901i	$0.553515\pi$
$930$	0	0
$931$	−83.1545	−2.72528
$932$	−2.08673	−0.0683532
$933$	0	0
$934$	−3.09467	−0.101261
$935$	−8.83204	−0.288839
$936$	0	0
$937$	48.7823	1.59365	0.796825	−	0.604211i	$-0.206511\pi$
$937$	48.7823	1.59365	0.796825	+	0.604211i	$0.206511\pi$
$938$	11.2820	0.368370
$939$	0	0
$940$	−1.62742	−0.0530807
$941$	45.7324	1.49083	0.745416	−	0.666599i	$-0.232251\pi$
$941$	45.7324	1.49083	0.745416	+	0.666599i	$0.232251\pi$
$942$	0	0
$943$	−0.375243	−0.0122196
$944$	−19.8948	−0.647521
$945$	0	0
$946$	−10.9808	−0.357015
$947$	14.7312	0.478699	0.239350	−	0.970933i	$-0.423066\pi$
$947$	14.7312	0.478699	0.239350	+	0.970933i	$0.423066\pi$
$948$	0	0
$949$	−2.90174	−0.0941946
$950$	−6.94240	−0.225241
$951$	0	0
$952$	−74.7809	−2.42366
$953$	−57.2024	−1.85297	−0.926484	−	0.376334i	$-0.877184\pi$
$953$	−57.2024	−1.85297	−0.926484	+	0.376334i	$0.877184\pi$
$954$	0	0
$955$	−19.5264	−0.631858
$956$	−0.369722	−0.0119577
$957$	0	0
$958$	28.7883	0.930107
$959$	−70.8495	−2.28785
$960$	0	0
$961$	−28.4650	−0.918226
$962$	−6.04434	−0.194877
$963$	0	0
$964$	−0.610315	−0.0196569
$965$	−23.3207	−0.750721
$966$	0	0
$967$	10.2741	0.330391	0.165196	−	0.986261i	$-0.447174\pi$
$967$	10.2741	0.330391	0.165196	+	0.986261i	$0.447174\pi$
$968$	23.0356	0.740392
$969$	0	0
$970$	−27.5894	−0.885843
$971$	1.15531	0.0370757	0.0185378	−	0.999828i	$-0.494099\pi$
$971$	1.15531	0.0370757	0.0185378	+	0.999828i	$0.494099\pi$
$972$	0	0
$973$	−13.2893	−0.426037
$974$	28.4708	0.912263
$975$	0	0
$976$	45.2839	1.44950
$977$	20.7840	0.664940	0.332470	−	0.943114i	$-0.392118\pi$
$977$	20.7840	0.664940	0.332470	+	0.943114i	$0.392118\pi$
$978$	0	0
$979$	0.359545	0.0114911
$980$	−2.47237	−0.0789770
$981$	0	0
$982$	−20.5264	−0.655023
$983$	44.1555	1.40834	0.704171	−	0.710030i	$-0.251319\pi$
$983$	44.1555	1.40834	0.704171	+	0.710030i	$0.251319\pi$
$984$	0	0
$985$	−9.68456	−0.308576
$986$	−28.7426	−0.915352
$987$	0	0
$988$	0.669293	0.0212931
$989$	0.360752	0.0114712
$990$	0	0
$991$	−3.86612	−0.122811	−0.0614056	−	0.998113i	$-0.519558\pi$
$991$	−3.86612	−0.122811	−0.0614056	+	0.998113i	$0.519558\pi$
$992$	−1.26822	−0.0402660
$993$	0	0
$994$	18.2206	0.577922
$995$	12.9358	0.410092
$996$	0	0
$997$	50.3098	1.59333	0.796663	−	0.604424i	$-0.206597\pi$
$997$	50.3098	1.59333	0.796663	+	0.604424i	$0.206597\pi$
$998$	−9.38233	−0.296993
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5265.2.a.bc.1.6	✓	8
3.2	odd	2		5265.2.a.bd.1.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5265.2.a.bc.1.6	✓	8	1.1	even	1	trivial
5265.2.a.bd.1.3	yes	8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.46324 q^{2} +0.141066 q^{4} -1.00000 q^{5} -4.95241 q^{7} -2.72006 q^{8} +O(q^{10})\)	\(q+1.46324 q^{2} +0.141066 q^{4} -1.00000 q^{5} -4.95241 q^{7} -2.72006 q^{8} -1.46324 q^{10} -1.59098 q^{11} -1.00000 q^{13} -7.24655 q^{14} -4.26223 q^{16} -5.55131 q^{17} -4.74454 q^{19} -0.141066 q^{20} -2.32799 q^{22} +0.0764816 q^{23} +1.00000 q^{25} -1.46324 q^{26} -0.698616 q^{28} +3.53848 q^{29} +1.59217 q^{31} -0.796535 q^{32} -8.12288 q^{34} +4.95241 q^{35} +4.13079 q^{37} -6.94240 q^{38} +2.72006 q^{40} -4.90631 q^{41} +4.71684 q^{43} -0.224434 q^{44} +0.111911 q^{46} +11.5366 q^{47} +17.5264 q^{49} +1.46324 q^{50} -0.141066 q^{52} -1.29315 q^{53} +1.59098 q^{55} +13.4709 q^{56} +5.17764 q^{58} +4.66770 q^{59} -10.6244 q^{61} +2.32972 q^{62} +7.35894 q^{64} +1.00000 q^{65} -1.55688 q^{67} -0.783100 q^{68} +7.24655 q^{70} -2.51438 q^{71} +2.90174 q^{73} +6.04434 q^{74} -0.669293 q^{76} +7.87921 q^{77} +3.32589 q^{79} +4.26223 q^{80} -7.17911 q^{82} -5.35683 q^{83} +5.55131 q^{85} +6.90186 q^{86} +4.32758 q^{88} -0.225989 q^{89} +4.95241 q^{91} +0.0107889 q^{92} +16.8808 q^{94} +4.74454 q^{95} +18.8550 q^{97} +25.6452 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{4} - 8 q^{5} + 4 q^{7} + 6 q^{8}+O(q^{10}) \) 8 * q + 6 * q^4 - 8 * q^5 + 4 * q^7 + 6 * q^8	\( 8 q + 6 q^{4} - 8 q^{5} + 4 q^{7} + 6 q^{8} + 6 q^{11} - 8 q^{13} + 4 q^{14} - 10 q^{16} - 2 q^{17} - 10 q^{19} - 6 q^{20} + 30 q^{23} + 8 q^{25} + 2 q^{28} + 8 q^{29} - 10 q^{31} - 8 q^{32} - 4 q^{34} - 4 q^{35} - 8 q^{37} + 24 q^{38} - 6 q^{40} + 6 q^{41} + 34 q^{44} + 16 q^{46} + 18 q^{47} + 26 q^{49} - 6 q^{52} + 14 q^{53} - 6 q^{55} + 28 q^{56} + 30 q^{59} - 18 q^{61} + 10 q^{62} - 36 q^{64} + 8 q^{65} - 6 q^{67} - 8 q^{68} - 4 q^{70} + 16 q^{71} + 12 q^{73} + 20 q^{74} - 2 q^{76} + 8 q^{77} - 30 q^{79} + 10 q^{80} + 20 q^{82} + 26 q^{83} + 2 q^{85} + 30 q^{86} + 10 q^{88} + 14 q^{89} - 4 q^{91} + 26 q^{92} - 16 q^{94} + 10 q^{95} + 44 q^{97} + 60 q^{98}+O(q^{100}) \) 8 * q + 6 * q^4 - 8 * q^5 + 4 * q^7 + 6 * q^8 + 6 * q^11 - 8 * q^13 + 4 * q^14 - 10 * q^16 - 2 * q^17 - 10 * q^19 - 6 * q^20 + 30 * q^23 + 8 * q^25 + 2 * q^28 + 8 * q^29 - 10 * q^31 - 8 * q^32 - 4 * q^34 - 4 * q^35 - 8 * q^37 + 24 * q^38 - 6 * q^40 + 6 * q^41 + 34 * q^44 + 16 * q^46 + 18 * q^47 + 26 * q^49 - 6 * q^52 + 14 * q^53 - 6 * q^55 + 28 * q^56 + 30 * q^59 - 18 * q^61 + 10 * q^62 - 36 * q^64 + 8 * q^65 - 6 * q^67 - 8 * q^68 - 4 * q^70 + 16 * q^71 + 12 * q^73 + 20 * q^74 - 2 * q^76 + 8 * q^77 - 30 * q^79 + 10 * q^80 + 20 * q^82 + 26 * q^83 + 2 * q^85 + 30 * q^86 + 10 * q^88 + 14 * q^89 - 4 * q^91 + 26 * q^92 - 16 * q^94 + 10 * q^95 + 44 * q^97 + 60 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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