LMFDB - Embedded newform 5265.2.a.ba.1.2

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			1.2
Root			$2.25662$ 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)

$f(q)$	$=$	$q-2.25662 q^{2} +3.09235 q^{4} +1.00000 q^{5} +0.706571 q^{7} -2.46502 q^{8} +O(q^{10})$	\(q-2.25662 q^{2} +3.09235 q^{4} +1.00000 q^{5} +0.706571 q^{7} -2.46502 q^{8} -2.25662 q^{10} +2.33021 q^{11} -1.00000 q^{13} -1.59446 q^{14} -0.622073 q^{16} -6.04736 q^{17} -7.91884 q^{19} +3.09235 q^{20} -5.25841 q^{22} +8.95599 q^{23} +1.00000 q^{25} +2.25662 q^{26} +2.18496 q^{28} +2.40680 q^{29} -3.06432 q^{31} +6.33383 q^{32} +13.6466 q^{34} +0.706571 q^{35} -2.38079 q^{37} +17.8698 q^{38} -2.46502 q^{40} +9.45889 q^{41} +11.9359 q^{43} +7.20582 q^{44} -20.2103 q^{46} -9.72867 q^{47} -6.50076 q^{49} -2.25662 q^{50} -3.09235 q^{52} -5.90968 q^{53} +2.33021 q^{55} -1.74171 q^{56} -5.43123 q^{58} -2.97437 q^{59} -9.24156 q^{61} +6.91501 q^{62} -13.0489 q^{64} -1.00000 q^{65} -1.69448 q^{67} -18.7006 q^{68} -1.59446 q^{70} +12.4439 q^{71} +4.88899 q^{73} +5.37255 q^{74} -24.4878 q^{76} +1.64646 q^{77} -11.0254 q^{79} -0.622073 q^{80} -21.3452 q^{82} +0.521691 q^{83} -6.04736 q^{85} -26.9348 q^{86} -5.74402 q^{88} -10.7608 q^{89} -0.706571 q^{91} +27.6950 q^{92} +21.9539 q^{94} -7.91884 q^{95} -2.96306 q^{97} +14.6698 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.25662	−1.59567	−0.797837	−	0.602873i	$-0.794022\pi$
$2$	−2.25662	−1.59567	−0.797837	+	0.602873i	$0.794022\pi$
$3$	0	0
$3$	0	0
$4$	3.09235	1.54617
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.706571	0.267059	0.133529	−	0.991045i	$-0.457369\pi$
$7$	0.706571	0.267059	0.133529	+	0.991045i	$0.457369\pi$
$8$	−2.46502	−0.871517
$9$	0	0
$10$	−2.25662	−0.713607
$11$	2.33021	0.702585	0.351292	−	0.936266i	$-0.385742\pi$
$11$	2.33021	0.702585	0.351292	+	0.936266i	$0.385742\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−1.59446	−0.426138
$15$	0	0
$16$	−0.622073	−0.155518
$17$	−6.04736	−1.46670	−0.733351	−	0.679851i	$-0.762045\pi$
$17$	−6.04736	−1.46670	−0.733351	+	0.679851i	$0.762045\pi$
$18$	0	0
$19$	−7.91884	−1.81671	−0.908353	−	0.418204i	$-0.862660\pi$
$19$	−7.91884	−1.81671	−0.908353	+	0.418204i	$0.862660\pi$
$20$	3.09235	0.691470
$21$	0	0
$22$	−5.25841	−1.12110
$23$	8.95599	1.86745	0.933726	−	0.357988i	$-0.116537\pi$
$23$	8.95599	1.86745	0.933726	+	0.357988i	$0.116537\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.25662	0.442560
$27$	0	0
$28$	2.18496	0.412919
$29$	2.40680	0.446931	0.223465	−	0.974712i	$-0.428263\pi$
$29$	2.40680	0.446931	0.223465	+	0.974712i	$0.428263\pi$
$30$	0	0
$31$	−3.06432	−0.550367	−0.275184	−	0.961392i	$-0.588739\pi$
$31$	−3.06432	−0.550367	−0.275184	+	0.961392i	$0.588739\pi$
$32$	6.33383	1.11967
$33$	0	0
$34$	13.6466	2.34038
$35$	0.706571	0.119432
$36$	0	0
$37$	−2.38079	−0.391400	−0.195700	−	0.980664i	$-0.562698\pi$
$37$	−2.38079	−0.391400	−0.195700	+	0.980664i	$0.562698\pi$
$38$	17.8698	2.89887
$39$	0	0
$40$	−2.46502	−0.389754
$41$	9.45889	1.47723	0.738616	−	0.674127i	$-0.235480\pi$
$41$	9.45889	1.47723	0.738616	+	0.674127i	$0.235480\pi$
$42$	0	0
$43$	11.9359	1.82021	0.910103	−	0.414381i	$-0.136002\pi$
$43$	11.9359	1.82021	0.910103	+	0.414381i	$0.136002\pi$
$44$	7.20582	1.08632
$45$	0	0
$46$	−20.2103	−2.97985
$47$	−9.72867	−1.41907	−0.709536	−	0.704669i	$-0.751095\pi$
$47$	−9.72867	−1.41907	−0.709536	+	0.704669i	$0.751095\pi$
$48$	0	0
$49$	−6.50076	−0.928680
$50$	−2.25662	−0.319135
$51$	0	0
$52$	−3.09235	−0.428832
$53$	−5.90968	−0.811756	−0.405878	−	0.913927i	$-0.633034\pi$
$53$	−5.90968	−0.811756	−0.405878	+	0.913927i	$0.633034\pi$
$54$	0	0
$55$	2.33021	0.314205
$56$	−1.74171	−0.232746
$57$	0	0
$58$	−5.43123	−0.713156
$59$	−2.97437	−0.387230	−0.193615	−	0.981078i	$-0.562021\pi$
$59$	−2.97437	−0.387230	−0.193615	+	0.981078i	$0.562021\pi$
$60$	0	0
$61$	−9.24156	−1.18326	−0.591630	−	0.806209i	$-0.701516\pi$
$61$	−9.24156	−1.18326	−0.591630	+	0.806209i	$0.701516\pi$
$62$	6.91501	0.878207
$63$	0	0
$64$	−13.0489	−1.63111
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−1.69448	−0.207014	−0.103507	−	0.994629i	$-0.533006\pi$
$67$	−1.69448	−0.207014	−0.103507	+	0.994629i	$0.533006\pi$
$68$	−18.7006	−2.26778
$69$	0	0
$70$	−1.59446	−0.190575
$71$	12.4439	1.47682	0.738410	−	0.674352i	$-0.235577\pi$
$71$	12.4439	1.47682	0.738410	+	0.674352i	$0.235577\pi$
$72$	0	0
$73$	4.88899	0.572213	0.286106	−	0.958198i	$-0.407639\pi$
$73$	4.88899	0.572213	0.286106	+	0.958198i	$0.407639\pi$
$74$	5.37255	0.624546
$75$	0	0
$76$	−24.4878	−2.80895
$77$	1.64646	0.187631
$78$	0	0
$79$	−11.0254	−1.24045	−0.620227	−	0.784423i	$-0.712959\pi$
$79$	−11.0254	−1.24045	−0.620227	+	0.784423i	$0.712959\pi$
$80$	−0.622073	−0.0695499
$81$	0	0
$82$	−21.3452	−2.35718
$83$	0.521691	0.0572630	0.0286315	−	0.999590i	$-0.490885\pi$
$83$	0.521691	0.0572630	0.0286315	+	0.999590i	$0.490885\pi$
$84$	0	0
$85$	−6.04736	−0.655929
$86$	−26.9348	−2.90446
$87$	0	0
$88$	−5.74402	−0.612314
$89$	−10.7608	−1.14065	−0.570323	−	0.821420i	$-0.693182\pi$
$89$	−10.7608	−1.14065	−0.570323	+	0.821420i	$0.693182\pi$
$90$	0	0
$91$	−0.706571	−0.0740687
$92$	27.6950	2.88741
$93$	0	0
$94$	21.9539	2.26438
$95$	−7.91884	−0.812456
$96$	0	0
$97$	−2.96306	−0.300853	−0.150427	−	0.988621i	$-0.548065\pi$
$97$	−2.96306	−0.300853	−0.150427	+	0.988621i	$0.548065\pi$
$98$	14.6698	1.48187
$99$	0	0
$100$	3.09235	0.309235
$101$	−13.6715	−1.36036	−0.680181	−	0.733044i	$-0.738099\pi$
$101$	−13.6715	−1.36036	−0.680181	+	0.733044i	$0.738099\pi$
$102$	0	0
$103$	−7.13071	−0.702610	−0.351305	−	0.936261i	$-0.614262\pi$
$103$	−7.13071	−0.702610	−0.351305	+	0.936261i	$0.614262\pi$
$104$	2.46502	0.241715
$105$	0	0
$106$	13.3359	1.29530
$107$	1.70776	0.165095	0.0825477	−	0.996587i	$-0.473694\pi$
$107$	1.70776	0.165095	0.0825477	+	0.996587i	$0.473694\pi$
$108$	0	0
$109$	−3.23628	−0.309979	−0.154989	−	0.987916i	$-0.549534\pi$
$109$	−3.23628	−0.309979	−0.154989	+	0.987916i	$0.549534\pi$
$110$	−5.25841	−0.501369
$111$	0	0
$112$	−0.439539	−0.0415325
$113$	0.678440	0.0638223	0.0319112	−	0.999491i	$-0.489841\pi$
$113$	0.678440	0.0638223	0.0319112	+	0.999491i	$0.489841\pi$
$114$	0	0
$115$	8.95599	0.835150
$116$	7.44266	0.691033
$117$	0	0
$118$	6.71204	0.617893
$119$	−4.27289	−0.391695
$120$	0	0
$121$	−5.57012	−0.506375
$122$	20.8547	1.88810
$123$	0	0
$124$	−9.47594	−0.850964
$125$	1.00000	0.0894427
$126$	0	0
$127$	14.6889	1.30343	0.651714	−	0.758465i	$-0.274050\pi$
$127$	14.6889	1.30343	0.651714	+	0.758465i	$0.274050\pi$
$128$	16.7788	1.48305
$129$	0	0
$130$	2.25662	0.197919
$131$	9.06388	0.791915	0.395957	−	0.918269i	$-0.370413\pi$
$131$	9.06388	0.791915	0.395957	+	0.918269i	$0.370413\pi$
$132$	0	0
$133$	−5.59522	−0.485167
$134$	3.82381	0.330327
$135$	0	0
$136$	14.9069	1.27825
$137$	2.02984	0.173421	0.0867104	−	0.996234i	$-0.472365\pi$
$137$	2.02984	0.173421	0.0867104	+	0.996234i	$0.472365\pi$
$138$	0	0
$139$	−2.85318	−0.242003	−0.121002	−	0.992652i	$-0.538611\pi$
$139$	−2.85318	−0.242003	−0.121002	+	0.992652i	$0.538611\pi$
$140$	2.18496	0.184663
$141$	0	0
$142$	−28.0812	−2.35652
$143$	−2.33021	−0.194862
$144$	0	0
$145$	2.40680	0.199874
$146$	−11.0326	−0.913065
$147$	0	0
$148$	−7.36224	−0.605173
$149$	−8.98225	−0.735854	−0.367927	−	0.929855i	$-0.619932\pi$
$149$	−8.98225	−0.735854	−0.367927	+	0.929855i	$0.619932\pi$
$150$	0	0
$151$	−2.75483	−0.224185	−0.112092	−	0.993698i	$-0.535755\pi$
$151$	−2.75483	−0.224185	−0.112092	+	0.993698i	$0.535755\pi$
$152$	19.5201	1.58329
$153$	0	0
$154$	−3.71544	−0.299398
$155$	−3.06432	−0.246132
$156$	0	0
$157$	24.4986	1.95520	0.977601	−	0.210468i	$-0.0674989\pi$
$157$	24.4986	1.95520	0.977601	+	0.210468i	$0.0674989\pi$
$158$	24.8802	1.97936
$159$	0	0
$160$	6.33383	0.500733
$161$	6.32804	0.498719
$162$	0	0
$163$	−18.2489	−1.42937	−0.714684	−	0.699448i	$-0.753429\pi$
$163$	−18.2489	−1.42937	−0.714684	+	0.699448i	$0.753429\pi$
$164$	29.2502	2.28406
$165$	0	0
$166$	−1.17726	−0.0913730
$167$	−5.60131	−0.433442	−0.216721	−	0.976234i	$-0.569536\pi$
$167$	−5.60131	−0.433442	−0.216721	+	0.976234i	$0.569536\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	13.6466	1.04665
$171$	0	0
$172$	36.9100	2.81436
$173$	3.60260	0.273900	0.136950	−	0.990578i	$-0.456270\pi$
$173$	3.60260	0.273900	0.136950	+	0.990578i	$0.456270\pi$
$174$	0	0
$175$	0.706571	0.0534117
$176$	−1.44956	−0.109265
$177$	0	0
$178$	24.2832	1.82010
$179$	−15.3123	−1.14449	−0.572246	−	0.820082i	$-0.693928\pi$
$179$	−15.3123	−1.14449	−0.572246	+	0.820082i	$0.693928\pi$
$180$	0	0
$181$	−7.23734	−0.537947	−0.268974	−	0.963148i	$-0.586684\pi$
$181$	−7.23734	−0.537947	−0.268974	+	0.963148i	$0.586684\pi$
$182$	1.59446	0.118190
$183$	0	0
$184$	−22.0767	−1.62752
$185$	−2.38079	−0.175039
$186$	0	0
$187$	−14.0916	−1.03048
$188$	−30.0844	−2.19413
$189$	0	0
$190$	17.8698	1.29641
$191$	19.3679	1.40141	0.700705	−	0.713452i	$-0.252869\pi$
$191$	19.3679	1.40141	0.700705	+	0.713452i	$0.252869\pi$
$192$	0	0
$193$	5.05546	0.363900	0.181950	−	0.983308i	$-0.441759\pi$
$193$	5.05546	0.363900	0.181950	+	0.983308i	$0.441759\pi$
$194$	6.68651	0.480064
$195$	0	0
$196$	−20.1026	−1.43590
$197$	−6.35430	−0.452725	−0.226363	−	0.974043i	$-0.572683\pi$
$197$	−6.35430	−0.452725	−0.226363	+	0.974043i	$0.572683\pi$
$198$	0	0
$199$	−16.7816	−1.18961	−0.594807	−	0.803868i	$-0.702772\pi$
$199$	−16.7816	−1.18961	−0.594807	+	0.803868i	$0.702772\pi$
$200$	−2.46502	−0.174303
$201$	0	0
$202$	30.8514	2.17069
$203$	1.70057	0.119357
$204$	0	0
$205$	9.45889	0.660638
$206$	16.0913	1.12114
$207$	0	0
$208$	0.622073	0.0431330
$209$	−18.4526	−1.27639
$210$	0	0
$211$	−11.3631	−0.782268	−0.391134	−	0.920334i	$-0.627917\pi$
$211$	−11.3631	−0.782268	−0.391134	+	0.920334i	$0.627917\pi$
$212$	−18.2748	−1.25512
$213$	0	0
$214$	−3.85377	−0.263439
$215$	11.9359	0.814021
$216$	0	0
$217$	−2.16516	−0.146980
$218$	7.30305	0.494625
$219$	0	0
$220$	7.20582	0.485817
$221$	6.04736	0.406790
$222$	0	0
$223$	−20.0857	−1.34504	−0.672518	−	0.740080i	$-0.734787\pi$
$223$	−20.0857	−1.34504	−0.672518	+	0.740080i	$0.734787\pi$
$224$	4.47530	0.299018
$225$	0	0
$226$	−1.53098	−0.101840
$227$	11.3482	0.753210	0.376605	−	0.926374i	$-0.377091\pi$
$227$	11.3482	0.753210	0.376605	+	0.926374i	$0.377091\pi$
$228$	0	0
$229$	−3.68135	−0.243271	−0.121635	−	0.992575i	$-0.538814\pi$
$229$	−3.68135	−0.243271	−0.121635	+	0.992575i	$0.538814\pi$
$230$	−20.2103	−1.33263
$231$	0	0
$232$	−5.93281	−0.389508
$233$	−12.5286	−0.820778	−0.410389	−	0.911911i	$-0.634607\pi$
$233$	−12.5286	−0.820778	−0.410389	+	0.911911i	$0.634607\pi$
$234$	0	0
$235$	−9.72867	−0.634628
$236$	−9.19779	−0.598725
$237$	0	0
$238$	9.64230	0.625018
$239$	−3.32887	−0.215327	−0.107663	−	0.994187i	$-0.534337\pi$
$239$	−3.32887	−0.215327	−0.107663	+	0.994187i	$0.534337\pi$
$240$	0	0
$241$	−26.7838	−1.72530	−0.862649	−	0.505804i	$-0.831196\pi$
$241$	−26.7838	−1.72530	−0.862649	+	0.505804i	$0.831196\pi$
$242$	12.5697	0.808009
$243$	0	0
$244$	−28.5781	−1.82953
$245$	−6.50076	−0.415318
$246$	0	0
$247$	7.91884	0.503864
$248$	7.55361	0.479654
$249$	0	0
$250$	−2.25662	−0.142721
$251$	6.05718	0.382326	0.191163	−	0.981558i	$-0.438774\pi$
$251$	6.05718	0.382326	0.191163	+	0.981558i	$0.438774\pi$
$252$	0	0
$253$	20.8693	1.31204
$254$	−33.1473	−2.07985
$255$	0	0
$256$	−11.7657	−0.735356
$257$	3.95270	0.246563	0.123281	−	0.992372i	$-0.460658\pi$
$257$	3.95270	0.246563	0.123281	+	0.992372i	$0.460658\pi$
$258$	0	0
$259$	−1.68220	−0.104527
$260$	−3.09235	−0.191779
$261$	0	0
$262$	−20.4538	−1.26364
$263$	−11.3011	−0.696859	−0.348429	−	0.937335i	$-0.613285\pi$
$263$	−11.3011	−0.696859	−0.348429	+	0.937335i	$0.613285\pi$
$264$	0	0
$265$	−5.90968	−0.363029
$266$	12.6263	0.774169
$267$	0	0
$268$	−5.23993	−0.320080
$269$	20.7773	1.26681	0.633407	−	0.773819i	$-0.281656\pi$
$269$	20.7773	1.26681	0.633407	+	0.773819i	$0.281656\pi$
$270$	0	0
$271$	−1.72567	−0.104827	−0.0524136	−	0.998625i	$-0.516691\pi$
$271$	−1.72567	−0.104827	−0.0524136	+	0.998625i	$0.516691\pi$
$272$	3.76190	0.228099
$273$	0	0
$274$	−4.58058	−0.276723
$275$	2.33021	0.140517
$276$	0	0
$277$	−3.46722	−0.208325	−0.104163	−	0.994560i	$-0.533216\pi$
$277$	−3.46722	−0.208325	−0.104163	+	0.994560i	$0.533216\pi$
$278$	6.43855	0.386158
$279$	0	0
$280$	−1.74171	−0.104087
$281$	−1.17263	−0.0699534	−0.0349767	−	0.999388i	$-0.511136\pi$
$281$	−1.17263	−0.0699534	−0.0349767	+	0.999388i	$0.511136\pi$
$282$	0	0
$283$	21.8122	1.29660	0.648301	−	0.761384i	$-0.275480\pi$
$283$	21.8122	1.29660	0.648301	+	0.761384i	$0.275480\pi$
$284$	38.4809	2.28342
$285$	0	0
$286$	5.25841	0.310936
$287$	6.68338	0.394507
$288$	0	0
$289$	19.5706	1.15121
$290$	−5.43123	−0.318933
$291$	0	0
$292$	15.1185	0.884741
$293$	27.8919	1.62946	0.814732	−	0.579838i	$-0.196884\pi$
$293$	27.8919	1.62946	0.814732	+	0.579838i	$0.196884\pi$
$294$	0	0
$295$	−2.97437	−0.173175
$296$	5.86870	0.341112
$297$	0	0
$298$	20.2695	1.17418
$299$	−8.95599	−0.517938
$300$	0	0
$301$	8.43355	0.486102
$302$	6.21661	0.357726
$303$	0	0
$304$	4.92610	0.282531
$305$	−9.24156	−0.529170
$306$	0	0
$307$	−0.574330	−0.0327787	−0.0163894	−	0.999866i	$-0.505217\pi$
$307$	−0.574330	−0.0327787	−0.0163894	+	0.999866i	$0.505217\pi$
$308$	5.09142	0.290111
$309$	0	0
$310$	6.91501	0.392746
$311$	−23.1071	−1.31029	−0.655143	−	0.755505i	$-0.727392\pi$
$311$	−23.1071	−1.31029	−0.655143	+	0.755505i	$0.727392\pi$
$312$	0	0
$313$	25.4854	1.44052	0.720260	−	0.693704i	$-0.244022\pi$
$313$	25.4854	1.44052	0.720260	+	0.693704i	$0.244022\pi$
$314$	−55.2841	−3.11986
$315$	0	0
$316$	−34.0944	−1.91796
$317$	−32.3557	−1.81728	−0.908638	−	0.417585i	$-0.862877\pi$
$317$	−32.3557	−1.81728	−0.908638	+	0.417585i	$0.862877\pi$
$318$	0	0
$319$	5.60834	0.314007
$320$	−13.0489	−0.729457
$321$	0	0
$322$	−14.2800	−0.795793
$323$	47.8881	2.66457
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	41.1810	2.28080
$327$	0	0
$328$	−23.3164	−1.28743
$329$	−6.87399	−0.378976
$330$	0	0
$331$	30.1696	1.65827	0.829135	−	0.559048i	$-0.188833\pi$
$331$	30.1696	1.65827	0.829135	+	0.559048i	$0.188833\pi$
$332$	1.61325	0.0885386
$333$	0	0
$334$	12.6400	0.691633
$335$	−1.69448	−0.0925795
$336$	0	0
$337$	6.57838	0.358347	0.179174	−	0.983817i	$-0.442658\pi$
$337$	6.57838	0.358347	0.179174	+	0.983817i	$0.442658\pi$
$338$	−2.25662	−0.122744
$339$	0	0
$340$	−18.7006	−1.01418
$341$	−7.14050	−0.386680
$342$	0	0
$343$	−9.53924	−0.515071
$344$	−29.4222	−1.58634
$345$	0	0
$346$	−8.12970	−0.437055
$347$	2.01063	0.107936	0.0539682	−	0.998543i	$-0.482813\pi$
$347$	2.01063	0.107936	0.0539682	+	0.998543i	$0.482813\pi$
$348$	0	0
$349$	14.0039	0.749610	0.374805	−	0.927104i	$-0.377710\pi$
$349$	14.0039	0.749610	0.374805	+	0.927104i	$0.377710\pi$
$350$	−1.59446	−0.0852277
$351$	0	0
$352$	14.7591	0.786665
$353$	−34.5998	−1.84156	−0.920780	−	0.390081i	$-0.872447\pi$
$353$	−34.5998	−1.84156	−0.920780	+	0.390081i	$0.872447\pi$
$354$	0	0
$355$	12.4439	0.660454
$356$	−33.2763	−1.76364
$357$	0	0
$358$	34.5540	1.82624
$359$	−22.7679	−1.20164	−0.600822	−	0.799382i	$-0.705160\pi$
$359$	−22.7679	−1.20164	−0.600822	+	0.799382i	$0.705160\pi$
$360$	0	0
$361$	43.7080	2.30042
$362$	16.3319	0.858388
$363$	0	0
$364$	−2.18496	−0.114523
$365$	4.88899	0.255901
$366$	0	0
$367$	−11.3939	−0.594758	−0.297379	−	0.954760i	$-0.596112\pi$
$367$	−11.3939	−0.594758	−0.297379	+	0.954760i	$0.596112\pi$
$368$	−5.57128	−0.290423
$369$	0	0
$370$	5.37255	0.279306
$371$	−4.17560	−0.216787
$372$	0	0
$373$	2.59519	0.134374	0.0671870	−	0.997740i	$-0.478598\pi$
$373$	2.59519	0.134374	0.0671870	+	0.997740i	$0.478598\pi$
$374$	31.7995	1.64431
$375$	0	0
$376$	23.9814	1.23675
$377$	−2.40680	−0.123956
$378$	0	0
$379$	2.38710	0.122617	0.0613084	−	0.998119i	$-0.480473\pi$
$379$	2.38710	0.122617	0.0613084	+	0.998119i	$0.480473\pi$
$380$	−24.4878	−1.25620
$381$	0	0
$382$	−43.7060	−2.23619
$383$	32.2726	1.64905	0.824525	−	0.565825i	$-0.191442\pi$
$383$	32.2726	1.64905	0.824525	+	0.565825i	$0.191442\pi$
$384$	0	0
$385$	1.64646	0.0839113
$386$	−11.4083	−0.580666
$387$	0	0
$388$	−9.16282	−0.465172
$389$	−7.50105	−0.380318	−0.190159	−	0.981753i	$-0.560900\pi$
$389$	−7.50105	−0.380318	−0.190159	+	0.981753i	$0.560900\pi$
$390$	0	0
$391$	−54.1601	−2.73899
$392$	16.0245	0.809360
$393$	0	0
$394$	14.3393	0.722402
$395$	−11.0254	−0.554748
$396$	0	0
$397$	−22.5214	−1.13032	−0.565158	−	0.824983i	$-0.691185\pi$
$397$	−22.5214	−1.13032	−0.565158	+	0.824983i	$0.691185\pi$
$398$	37.8697	1.89824
$399$	0	0
$400$	−0.622073	−0.0311036
$401$	−2.53261	−0.126472	−0.0632362	−	0.997999i	$-0.520142\pi$
$401$	−2.53261	−0.126472	−0.0632362	+	0.997999i	$0.520142\pi$
$402$	0	0
$403$	3.06432	0.152644
$404$	−42.2770	−2.10336
$405$	0	0
$406$	−3.83755	−0.190454
$407$	−5.54775	−0.274992
$408$	0	0
$409$	−6.37084	−0.315018	−0.157509	−	0.987518i	$-0.550346\pi$
$409$	−6.37084	−0.315018	−0.157509	+	0.987518i	$0.550346\pi$
$410$	−21.3452	−1.05416
$411$	0	0
$412$	−22.0506	−1.08636
$413$	−2.10160	−0.103413
$414$	0	0
$415$	0.521691	0.0256088
$416$	−6.33383	−0.310541
$417$	0	0
$418$	41.6405	2.03670
$419$	−30.9358	−1.51131	−0.755657	−	0.654968i	$-0.772682\pi$
$419$	−30.9358	−1.51131	−0.755657	+	0.654968i	$0.772682\pi$
$420$	0	0
$421$	−13.9644	−0.680584	−0.340292	−	0.940320i	$-0.610526\pi$
$421$	−13.9644	−0.680584	−0.340292	+	0.940320i	$0.610526\pi$
$422$	25.6422	1.24824
$423$	0	0
$424$	14.5675	0.707459
$425$	−6.04736	−0.293340
$426$	0	0
$427$	−6.52982	−0.316000
$428$	5.28099	0.255266
$429$	0	0
$430$	−26.9348	−1.29891
$431$	−20.2027	−0.973131	−0.486566	−	0.873644i	$-0.661751\pi$
$431$	−20.2027	−0.973131	−0.486566	+	0.873644i	$0.661751\pi$
$432$	0	0
$433$	−13.7657	−0.661537	−0.330768	−	0.943712i	$-0.607308\pi$
$433$	−13.7657	−0.661537	−0.330768	+	0.943712i	$0.607308\pi$
$434$	4.88594	0.234533
$435$	0	0
$436$	−10.0077	−0.479282
$437$	−70.9210	−3.39261
$438$	0	0
$439$	7.76756	0.370725	0.185363	−	0.982670i	$-0.440654\pi$
$439$	7.76756	0.370725	0.185363	+	0.982670i	$0.440654\pi$
$440$	−5.74402	−0.273835
$441$	0	0
$442$	−13.6466	−0.649104
$443$	−16.4315	−0.780682	−0.390341	−	0.920670i	$-0.627643\pi$
$443$	−16.4315	−0.780682	−0.390341	+	0.920670i	$0.627643\pi$
$444$	0	0
$445$	−10.7608	−0.510113
$446$	45.3258	2.14624
$447$	0	0
$448$	−9.21998	−0.435603
$449$	−9.97715	−0.470851	−0.235425	−	0.971892i	$-0.575648\pi$
$449$	−9.97715	−0.470851	−0.235425	+	0.971892i	$0.575648\pi$
$450$	0	0
$451$	22.0412	1.03788
$452$	2.09797	0.0986804
$453$	0	0
$454$	−25.6087	−1.20188
$455$	−0.706571	−0.0331245
$456$	0	0
$457$	−17.3560	−0.811879	−0.405939	−	0.913900i	$-0.633056\pi$
$457$	−17.3560	−0.811879	−0.405939	+	0.913900i	$0.633056\pi$
$458$	8.30743	0.388181
$459$	0	0
$460$	27.6950	1.29129
$461$	10.5929	0.493361	0.246681	−	0.969097i	$-0.420660\pi$
$461$	10.5929	0.493361	0.246681	+	0.969097i	$0.420660\pi$
$462$	0	0
$463$	−3.60744	−0.167652	−0.0838260	−	0.996480i	$-0.526714\pi$
$463$	−3.60744	−0.167652	−0.0838260	+	0.996480i	$0.526714\pi$
$464$	−1.49720	−0.0695059
$465$	0	0
$466$	28.2724	1.30969
$467$	−6.96611	−0.322353	−0.161177	−	0.986926i	$-0.551529\pi$
$467$	−6.96611	−0.322353	−0.161177	+	0.986926i	$0.551529\pi$
$468$	0	0
$469$	−1.19727	−0.0552849
$470$	21.9539	1.01266
$471$	0	0
$472$	7.33189	0.337478
$473$	27.8131	1.27885
$474$	0	0
$475$	−7.91884	−0.363341
$476$	−13.2133	−0.605629
$477$	0	0
$478$	7.51200	0.343591
$479$	22.0606	1.00797	0.503987	−	0.863711i	$-0.331866\pi$
$479$	22.0606	1.00797	0.503987	+	0.863711i	$0.331866\pi$
$480$	0	0
$481$	2.38079	0.108555
$482$	60.4410	2.75301
$483$	0	0
$484$	−17.2248	−0.782944
$485$	−2.96306	−0.134546
$486$	0	0
$487$	−5.19759	−0.235525	−0.117763	−	0.993042i	$-0.537572\pi$
$487$	−5.19759	−0.235525	−0.117763	+	0.993042i	$0.537572\pi$
$488$	22.7807	1.03123
$489$	0	0
$490$	14.6698	0.662712
$491$	2.23657	0.100935	0.0504675	−	0.998726i	$-0.483929\pi$
$491$	2.23657	0.100935	0.0504675	+	0.998726i	$0.483929\pi$
$492$	0	0
$493$	−14.5548	−0.655514
$494$	−17.8698	−0.804002
$495$	0	0
$496$	1.90623	0.0855922
$497$	8.79250	0.394398
$498$	0	0
$499$	−21.2893	−0.953038	−0.476519	−	0.879164i	$-0.658102\pi$
$499$	−21.2893	−0.953038	−0.476519	+	0.879164i	$0.658102\pi$
$500$	3.09235	0.138294
$501$	0	0
$502$	−13.6688	−0.610068
$503$	33.5371	1.49534	0.747672	−	0.664068i	$-0.231171\pi$
$503$	33.5371	1.49534	0.747672	+	0.664068i	$0.231171\pi$
$504$	0	0
$505$	−13.6715	−0.608372
$506$	−47.0942	−2.09359
$507$	0	0
$508$	45.4232	2.01533
$509$	3.07482	0.136289	0.0681444	−	0.997675i	$-0.478292\pi$
$509$	3.07482	0.136289	0.0681444	+	0.997675i	$0.478292\pi$
$510$	0	0
$511$	3.45442	0.152814
$512$	−7.00695	−0.309666
$513$	0	0
$514$	−8.91975	−0.393433
$515$	−7.13071	−0.314217
$516$	0	0
$517$	−22.6698	−0.997018
$518$	3.79609	0.166791
$519$	0	0
$520$	2.46502	0.108098
$521$	−32.8855	−1.44074	−0.720371	−	0.693589i	$-0.756028\pi$
$521$	−32.8855	−1.44074	−0.720371	+	0.693589i	$0.756028\pi$
$522$	0	0
$523$	19.0090	0.831206	0.415603	−	0.909546i	$-0.363571\pi$
$523$	19.0090	0.831206	0.415603	+	0.909546i	$0.363571\pi$
$524$	28.0287	1.22444
$525$	0	0
$526$	25.5024	1.11196
$527$	18.5310	0.807224
$528$	0	0
$529$	57.2097	2.48738
$530$	13.3359	0.579275
$531$	0	0
$532$	−17.3024	−0.750153
$533$	−9.45889	−0.409710
$534$	0	0
$535$	1.70776	0.0738330
$536$	4.17694	0.180416
$537$	0	0
$538$	−46.8865	−2.02142
$539$	−15.1481	−0.652476
$540$	0	0
$541$	−13.4399	−0.577825	−0.288912	−	0.957356i	$-0.593294\pi$
$541$	−13.4399	−0.577825	−0.288912	+	0.957356i	$0.593294\pi$
$542$	3.89420	0.167270
$543$	0	0
$544$	−38.3030	−1.64223
$545$	−3.23628	−0.138627
$546$	0	0
$547$	11.3254	0.484239	0.242119	−	0.970246i	$-0.422157\pi$
$547$	11.3254	0.484239	0.242119	+	0.970246i	$0.422157\pi$
$548$	6.27697	0.268139
$549$	0	0
$550$	−5.25841	−0.224219
$551$	−19.0590	−0.811943
$552$	0	0
$553$	−7.79022	−0.331274
$554$	7.82421	0.332419
$555$	0	0
$556$	−8.82302	−0.374179
$557$	−4.71355	−0.199720	−0.0998598	−	0.995002i	$-0.531839\pi$
$557$	−4.71355	−0.199720	−0.0998598	+	0.995002i	$0.531839\pi$
$558$	0	0
$559$	−11.9359	−0.504835
$560$	−0.439539	−0.0185739
$561$	0	0
$562$	2.64619	0.111623
$563$	−35.3159	−1.48839	−0.744193	−	0.667964i	$-0.767166\pi$
$563$	−35.3159	−1.48839	−0.744193	+	0.667964i	$0.767166\pi$
$564$	0	0
$565$	0.678440	0.0285422
$566$	−49.2220	−2.06895
$567$	0	0
$568$	−30.6745	−1.28707
$569$	26.5359	1.11244	0.556222	−	0.831034i	$-0.312251\pi$
$569$	26.5359	1.11244	0.556222	+	0.831034i	$0.312251\pi$
$570$	0	0
$571$	−33.7542	−1.41257	−0.706284	−	0.707928i	$-0.749630\pi$
$571$	−33.7542	−1.41257	−0.706284	+	0.707928i	$0.749630\pi$
$572$	−7.20582	−0.301291
$573$	0	0
$574$	−15.0819	−0.629505
$575$	8.95599	0.373491
$576$	0	0
$577$	−30.7531	−1.28027	−0.640133	−	0.768264i	$-0.721121\pi$
$577$	−30.7531	−1.28027	−0.640133	+	0.768264i	$0.721121\pi$
$578$	−44.1635	−1.83696
$579$	0	0
$580$	7.44266	0.309040
$581$	0.368611	0.0152926
$582$	0	0
$583$	−13.7708	−0.570328
$584$	−12.0515	−0.498693
$585$	0	0
$586$	−62.9416	−2.60009
$587$	−5.23724	−0.216164	−0.108082	−	0.994142i	$-0.534471\pi$
$587$	−5.23724	−0.216164	−0.108082	+	0.994142i	$0.534471\pi$
$588$	0	0
$589$	24.2658	0.999856
$590$	6.71204	0.276330
$591$	0	0
$592$	1.48103	0.0608698
$593$	−15.8490	−0.650839	−0.325420	−	0.945570i	$-0.605506\pi$
$593$	−15.8490	−0.650839	−0.325420	+	0.945570i	$0.605506\pi$
$594$	0	0
$595$	−4.27289	−0.175171
$596$	−27.7762	−1.13776
$597$	0	0
$598$	20.2103	0.826460
$599$	−26.3235	−1.07555	−0.537775	−	0.843088i	$-0.680735\pi$
$599$	−26.3235	−1.07555	−0.537775	+	0.843088i	$0.680735\pi$
$600$	0	0
$601$	1.80372	0.0735754	0.0367877	−	0.999323i	$-0.488287\pi$
$601$	1.80372	0.0735754	0.0367877	+	0.999323i	$0.488287\pi$
$602$	−19.0314	−0.775660
$603$	0	0
$604$	−8.51889	−0.346629
$605$	−5.57012	−0.226458
$606$	0	0
$607$	12.5387	0.508929	0.254464	−	0.967082i	$-0.418101\pi$
$607$	12.5387	0.508929	0.254464	+	0.967082i	$0.418101\pi$
$608$	−50.1566	−2.03412
$609$	0	0
$610$	20.8547	0.844383
$611$	9.72867	0.393580
$612$	0	0
$613$	9.01785	0.364228	0.182114	−	0.983277i	$-0.441706\pi$
$613$	9.01785	0.364228	0.182114	+	0.983277i	$0.441706\pi$
$614$	1.29605	0.0523042
$615$	0	0
$616$	−4.05856	−0.163524
$617$	−14.6819	−0.591071	−0.295536	−	0.955332i	$-0.595498\pi$
$617$	−14.6819	−0.591071	−0.295536	+	0.955332i	$0.595498\pi$
$618$	0	0
$619$	−41.6709	−1.67490	−0.837448	−	0.546517i	$-0.815953\pi$
$619$	−41.6709	−1.67490	−0.837448	+	0.546517i	$0.815953\pi$
$620$	−9.47594	−0.380563
$621$	0	0
$622$	52.1441	2.09079
$623$	−7.60330	−0.304620
$624$	0	0
$625$	1.00000	0.0400000
$626$	−57.5110	−2.29860
$627$	0	0
$628$	75.7582	3.02308
$629$	14.3975	0.574067
$630$	0	0
$631$	−19.8681	−0.790936	−0.395468	−	0.918480i	$-0.629418\pi$
$631$	−19.8681	−0.790936	−0.395468	+	0.918480i	$0.629418\pi$
$632$	27.1778	1.08108
$633$	0	0
$634$	73.0146	2.89978
$635$	14.6889	0.582911
$636$	0	0
$637$	6.50076	0.257569
$638$	−12.6559	−0.501053
$639$	0	0
$640$	16.7788	0.663242
$641$	−22.3153	−0.881401	−0.440700	−	0.897654i	$-0.645270\pi$
$641$	−22.3153	−0.881401	−0.440700	+	0.897654i	$0.645270\pi$
$642$	0	0
$643$	15.4444	0.609070	0.304535	−	0.952501i	$-0.401499\pi$
$643$	15.4444	0.609070	0.304535	+	0.952501i	$0.401499\pi$
$644$	19.5685	0.771107
$645$	0	0
$646$	−108.065	−4.25178
$647$	25.0635	0.985349	0.492675	−	0.870214i	$-0.336019\pi$
$647$	25.0635	0.985349	0.492675	+	0.870214i	$0.336019\pi$
$648$	0	0
$649$	−6.93091	−0.272062
$650$	2.25662	0.0885121
$651$	0	0
$652$	−56.4321	−2.21005
$653$	2.98488	0.116807	0.0584036	−	0.998293i	$-0.481399\pi$
$653$	2.98488	0.116807	0.0584036	+	0.998293i	$0.481399\pi$
$654$	0	0
$655$	9.06388	0.354155
$656$	−5.88412	−0.229736
$657$	0	0
$658$	15.5120	0.604721
$659$	37.4330	1.45818	0.729092	−	0.684416i	$-0.239943\pi$
$659$	37.4330	1.45818	0.729092	+	0.684416i	$0.239943\pi$
$660$	0	0
$661$	15.3726	0.597925	0.298963	−	0.954265i	$-0.403359\pi$
$661$	15.3726	0.597925	0.298963	+	0.954265i	$0.403359\pi$
$662$	−68.0814	−2.64606
$663$	0	0
$664$	−1.28598	−0.0499057
$665$	−5.59522	−0.216973
$666$	0	0
$667$	21.5552	0.834622
$668$	−17.3212	−0.670178
$669$	0	0
$670$	3.82381	0.147727
$671$	−21.5348	−0.831341
$672$	0	0
$673$	−0.543557	−0.0209526	−0.0104763	−	0.999945i	$-0.503335\pi$
$673$	−0.543557	−0.0209526	−0.0104763	+	0.999945i	$0.503335\pi$
$674$	−14.8449	−0.571805
$675$	0	0
$676$	3.09235	0.118937
$677$	20.0587	0.770917	0.385458	−	0.922725i	$-0.374043\pi$
$677$	20.0587	0.770917	0.385458	+	0.922725i	$0.374043\pi$
$678$	0	0
$679$	−2.09361	−0.0803454
$680$	14.9069	0.571653
$681$	0	0
$682$	16.1134	0.617015
$683$	−35.5213	−1.35919	−0.679593	−	0.733589i	$-0.737844\pi$
$683$	−35.5213	−1.35919	−0.679593	+	0.733589i	$0.737844\pi$
$684$	0	0
$685$	2.02984	0.0775561
$686$	21.5265	0.821885
$687$	0	0
$688$	−7.42500	−0.283075
$689$	5.90968	0.225141
$690$	0	0
$691$	−51.3512	−1.95349	−0.976746	−	0.214401i	$-0.931220\pi$
$691$	−51.3512	−1.95349	−0.976746	+	0.214401i	$0.931220\pi$
$692$	11.1405	0.423498
$693$	0	0
$694$	−4.53724	−0.172231
$695$	−2.85318	−0.108227
$696$	0	0
$697$	−57.2014	−2.16666
$698$	−31.6015	−1.19613
$699$	0	0
$700$	2.18496	0.0825839
$701$	−26.2661	−0.992057	−0.496029	−	0.868306i	$-0.665209\pi$
$701$	−26.2661	−0.992057	−0.496029	+	0.868306i	$0.665209\pi$
$702$	0	0
$703$	18.8531	0.711059
$704$	−30.4067	−1.14600
$705$	0	0
$706$	78.0787	2.93853
$707$	−9.65986	−0.363296
$708$	0	0
$709$	−25.7362	−0.966545	−0.483273	−	0.875470i	$-0.660552\pi$
$709$	−25.7362	−0.966545	−0.483273	+	0.875470i	$0.660552\pi$
$710$	−28.0812	−1.05387
$711$	0	0
$712$	26.5257	0.994093
$713$	−27.4440	−1.02778
$714$	0	0
$715$	−2.33021	−0.0871449
$716$	−47.3509	−1.76959
$717$	0	0
$718$	51.3786	1.91743
$719$	21.6829	0.808634	0.404317	−	0.914619i	$-0.367509\pi$
$719$	21.6829	0.808634	0.404317	+	0.914619i	$0.367509\pi$
$720$	0	0
$721$	−5.03835	−0.187638
$722$	−98.6326	−3.67073
$723$	0	0
$724$	−22.3804	−0.831760
$725$	2.40680	0.0893862
$726$	0	0
$727$	−49.3288	−1.82950	−0.914752	−	0.404016i	$-0.867614\pi$
$727$	−49.3288	−1.82950	−0.914752	+	0.404016i	$0.867614\pi$
$728$	1.74171	0.0645522
$729$	0	0
$730$	−11.0326	−0.408335
$731$	−72.1807	−2.66970
$732$	0	0
$733$	26.1165	0.964636	0.482318	−	0.875996i	$-0.339795\pi$
$733$	26.1165	0.964636	0.482318	+	0.875996i	$0.339795\pi$
$734$	25.7118	0.949039
$735$	0	0
$736$	56.7257	2.09094
$737$	−3.94850	−0.145445
$738$	0	0
$739$	−20.3143	−0.747275	−0.373637	−	0.927575i	$-0.621890\pi$
$739$	−20.3143	−0.747275	−0.373637	+	0.927575i	$0.621890\pi$
$740$	−7.36224	−0.270641
$741$	0	0
$742$	9.42277	0.345921
$743$	−9.47288	−0.347526	−0.173763	−	0.984787i	$-0.555593\pi$
$743$	−9.47288	−0.347526	−0.173763	+	0.984787i	$0.555593\pi$
$744$	0	0
$745$	−8.98225	−0.329084
$746$	−5.85637	−0.214417
$747$	0	0
$748$	−43.5762	−1.59330
$749$	1.20665	0.0440902
$750$	0	0
$751$	−6.05844	−0.221076	−0.110538	−	0.993872i	$-0.535257\pi$
$751$	−6.05844	−0.221076	−0.110538	+	0.993872i	$0.535257\pi$
$752$	6.05194	0.220692
$753$	0	0
$754$	5.43123	0.197794
$755$	−2.75483	−0.100258
$756$	0	0
$757$	19.3009	0.701503	0.350752	−	0.936469i	$-0.385926\pi$
$757$	19.3009	0.701503	0.350752	+	0.936469i	$0.385926\pi$
$758$	−5.38678	−0.195657
$759$	0	0
$760$	19.5201	0.708069
$761$	31.5087	1.14219	0.571095	−	0.820884i	$-0.306519\pi$
$761$	31.5087	1.14219	0.571095	+	0.820884i	$0.306519\pi$
$762$	0	0
$763$	−2.28666	−0.0827826
$764$	59.8922	2.16682
$765$	0	0
$766$	−72.8270	−2.63135
$767$	2.97437	0.107398
$768$	0	0
$769$	9.31924	0.336060	0.168030	−	0.985782i	$-0.446259\pi$
$769$	9.31924	0.336060	0.168030	+	0.985782i	$0.446259\pi$
$770$	−3.71544	−0.133895
$771$	0	0
$772$	15.6332	0.562653
$773$	−24.3144	−0.874529	−0.437265	−	0.899333i	$-0.644053\pi$
$773$	−24.3144	−0.874529	−0.437265	+	0.899333i	$0.644053\pi$
$774$	0	0
$775$	−3.06432	−0.110073
$776$	7.30401	0.262199
$777$	0	0
$778$	16.9270	0.606864
$779$	−74.9035	−2.68370
$780$	0	0
$781$	28.9969	1.03759
$782$	122.219	4.37054
$783$	0	0
$784$	4.04395	0.144427
$785$	24.4986	0.874393
$786$	0	0
$787$	−42.8300	−1.52672	−0.763362	−	0.645970i	$-0.776453\pi$
$787$	−42.8300	−1.52672	−0.763362	+	0.645970i	$0.776453\pi$
$788$	−19.6497	−0.699992
$789$	0	0
$790$	24.8802	0.885196
$791$	0.479366	0.0170443
$792$	0	0
$793$	9.24156	0.328177
$794$	50.8223	1.80362
$795$	0	0
$796$	−51.8945	−1.83935
$797$	30.8397	1.09240	0.546199	−	0.837656i	$-0.316074\pi$
$797$	30.8397	1.09240	0.546199	+	0.837656i	$0.316074\pi$
$798$	0	0
$799$	58.8328	2.08135
$800$	6.33383	0.223935
$801$	0	0
$802$	5.71514	0.201809
$803$	11.3924	0.402028
$804$	0	0
$805$	6.32804	0.223034
$806$	−6.91501	−0.243571
$807$	0	0
$808$	33.7005	1.18558
$809$	14.4671	0.508636	0.254318	−	0.967121i	$-0.418149\pi$
$809$	14.4671	0.508636	0.254318	+	0.967121i	$0.418149\pi$
$810$	0	0
$811$	40.9902	1.43936	0.719681	−	0.694305i	$-0.244288\pi$
$811$	40.9902	1.43936	0.719681	+	0.694305i	$0.244288\pi$
$812$	5.25876	0.184546
$813$	0	0
$814$	12.5192	0.438797
$815$	−18.2489	−0.639233
$816$	0	0
$817$	−94.5184	−3.30678
$818$	14.3766	0.502666
$819$	0	0
$820$	29.2502	1.02146
$821$	47.2815	1.65014	0.825068	−	0.565033i	$-0.191137\pi$
$821$	47.2815	1.65014	0.825068	+	0.565033i	$0.191137\pi$
$822$	0	0
$823$	29.8208	1.03949	0.519744	−	0.854322i	$-0.326027\pi$
$823$	29.8208	1.03949	0.519744	+	0.854322i	$0.326027\pi$
$824$	17.5774	0.612336
$825$	0	0
$826$	4.74253	0.165014
$827$	−3.90807	−0.135897	−0.0679484	−	0.997689i	$-0.521645\pi$
$827$	−3.90807	−0.135897	−0.0679484	+	0.997689i	$0.521645\pi$
$828$	0	0
$829$	−7.76600	−0.269724	−0.134862	−	0.990864i	$-0.543059\pi$
$829$	−7.76600	−0.269724	−0.134862	+	0.990864i	$0.543059\pi$
$830$	−1.17726	−0.0408633
$831$	0	0
$832$	13.0489	0.452390
$833$	39.3124	1.36210
$834$	0	0
$835$	−5.60131	−0.193841
$836$	−57.0618	−1.97352
$837$	0	0
$838$	69.8105	2.41156
$839$	−15.6099	−0.538914	−0.269457	−	0.963012i	$-0.586844\pi$
$839$	−15.6099	−0.538914	−0.269457	+	0.963012i	$0.586844\pi$
$840$	0	0
$841$	−23.2073	−0.800253
$842$	31.5124	1.08599
$843$	0	0
$844$	−35.1387	−1.20952
$845$	1.00000	0.0344010
$846$	0	0
$847$	−3.93569	−0.135232
$848$	3.67625	0.126243
$849$	0	0
$850$	13.6466	0.468075
$851$	−21.3223	−0.730921
$852$	0	0
$853$	−12.3460	−0.422719	−0.211360	−	0.977408i	$-0.567789\pi$
$853$	−12.3460	−0.422719	−0.211360	+	0.977408i	$0.567789\pi$
$854$	14.7353	0.504233
$855$	0	0
$856$	−4.20967	−0.143884
$857$	36.9006	1.26050	0.630251	−	0.776392i	$-0.282952\pi$
$857$	36.9006	1.26050	0.630251	+	0.776392i	$0.282952\pi$
$858$	0	0
$859$	−28.3978	−0.968920	−0.484460	−	0.874814i	$-0.660984\pi$
$859$	−28.3978	−0.968920	−0.484460	+	0.874814i	$0.660984\pi$
$860$	36.9100	1.25862
$861$	0	0
$862$	45.5900	1.55280
$863$	26.6238	0.906283	0.453142	−	0.891438i	$-0.350303\pi$
$863$	26.6238	0.906283	0.453142	+	0.891438i	$0.350303\pi$
$864$	0	0
$865$	3.60260	0.122492
$866$	31.0640	1.05560
$867$	0	0
$868$	−6.69542	−0.227257
$869$	−25.6915	−0.871524
$870$	0	0
$871$	1.69448	0.0574153
$872$	7.97749	0.270152
$873$	0	0
$874$	160.042	5.41350
$875$	0.706571	0.0238865
$876$	0	0
$877$	−51.2039	−1.72903	−0.864516	−	0.502605i	$-0.832375\pi$
$877$	−51.2039	−1.72903	−0.864516	+	0.502605i	$0.832375\pi$
$878$	−17.5285	−0.591557
$879$	0	0
$880$	−1.44956	−0.0488647
$881$	29.0213	0.977751	0.488876	−	0.872354i	$-0.337407\pi$
$881$	29.0213	0.977751	0.488876	+	0.872354i	$0.337407\pi$
$882$	0	0
$883$	21.5185	0.724154	0.362077	−	0.932148i	$-0.382068\pi$
$883$	21.5185	0.724154	0.362077	+	0.932148i	$0.382068\pi$
$884$	18.7006	0.628968
$885$	0	0
$886$	37.0796	1.24571
$887$	−19.2022	−0.644745	−0.322373	−	0.946613i	$-0.604480\pi$
$887$	−19.2022	−0.644745	−0.322373	+	0.946613i	$0.604480\pi$
$888$	0	0
$889$	10.3787	0.348092
$890$	24.2832	0.813974
$891$	0	0
$892$	−62.1120	−2.07966
$893$	77.0398	2.57804
$894$	0	0
$895$	−15.3123	−0.511833
$896$	11.8554	0.396062
$897$	0	0
$898$	22.5147	0.751324
$899$	−7.37519	−0.245976
$900$	0	0
$901$	35.7380	1.19060
$902$	−49.7387	−1.65612
$903$	0	0
$904$	−1.67237	−0.0556222
$905$	−7.23734	−0.240577
$906$	0	0
$907$	47.4831	1.57665	0.788326	−	0.615258i	$-0.210948\pi$
$907$	47.4831	1.57665	0.788326	+	0.615258i	$0.210948\pi$
$908$	35.0927	1.16459
$909$	0	0
$910$	1.59446	0.0528560
$911$	14.0104	0.464186	0.232093	−	0.972694i	$-0.425443\pi$
$911$	14.0104	0.464186	0.232093	+	0.972694i	$0.425443\pi$
$912$	0	0
$913$	1.21565	0.0402321
$914$	39.1659	1.29549
$915$	0	0
$916$	−11.3840	−0.376139
$917$	6.40427	0.211488
$918$	0	0
$919$	47.7699	1.57579	0.787893	−	0.615813i	$-0.211172\pi$
$919$	47.7699	1.57579	0.787893	+	0.615813i	$0.211172\pi$
$920$	−22.0767	−0.727847
$921$	0	0
$922$	−23.9042	−0.787243
$923$	−12.4439	−0.409596
$924$	0	0
$925$	−2.38079	−0.0782800
$926$	8.14064	0.267518
$927$	0	0
$928$	15.2442	0.500417
$929$	−36.2956	−1.19082	−0.595409	−	0.803423i	$-0.703010\pi$
$929$	−36.2956	−1.19082	−0.595409	+	0.803423i	$0.703010\pi$
$930$	0	0
$931$	51.4785	1.68714
$932$	−38.7429	−1.26907
$933$	0	0
$934$	15.7199	0.514371
$935$	−14.0916	−0.460845
$936$	0	0
$937$	−16.6768	−0.544808	−0.272404	−	0.962183i	$-0.587819\pi$
$937$	−16.6768	−0.544808	−0.272404	+	0.962183i	$0.587819\pi$
$938$	2.70179	0.0882166
$939$	0	0
$940$	−30.0844	−0.981246
$941$	−48.8290	−1.59178	−0.795890	−	0.605441i	$-0.792997\pi$
$941$	−48.8290	−1.59178	−0.795890	+	0.605441i	$0.792997\pi$
$942$	0	0
$943$	84.7137	2.75866
$944$	1.85028	0.0602213
$945$	0	0
$946$	−62.7638	−2.04063
$947$	10.0714	0.327277	0.163639	−	0.986520i	$-0.447677\pi$
$947$	10.0714	0.327277	0.163639	+	0.986520i	$0.447677\pi$
$948$	0	0
$949$	−4.88899	−0.158703
$950$	17.8698	0.579774
$951$	0	0
$952$	10.5328	0.341369
$953$	−19.2233	−0.622703	−0.311351	−	0.950295i	$-0.600782\pi$
$953$	−19.2233	−0.622703	−0.311351	+	0.950295i	$0.600782\pi$
$954$	0	0
$955$	19.3679	0.626729
$956$	−10.2940	−0.332933
$957$	0	0
$958$	−49.7825	−1.60840
$959$	1.43422	0.0463135
$960$	0	0
$961$	−21.6100	−0.697096
$962$	−5.37255	−0.173218
$963$	0	0
$964$	−82.8249	−2.66761
$965$	5.05546	0.162741
$966$	0	0
$967$	−23.9667	−0.770717	−0.385358	−	0.922767i	$-0.625922\pi$
$967$	−23.9667	−0.770717	−0.385358	+	0.922767i	$0.625922\pi$
$968$	13.7305	0.441314
$969$	0	0
$970$	6.68651	0.214691
$971$	−22.5951	−0.725111	−0.362556	−	0.931962i	$-0.618096\pi$
$971$	−22.5951	−0.725111	−0.362556	+	0.931962i	$0.618096\pi$
$972$	0	0
$973$	−2.01597	−0.0646291
$974$	11.7290	0.375821
$975$	0	0
$976$	5.74893	0.184019
$977$	9.54650	0.305420	0.152710	−	0.988271i	$-0.451200\pi$
$977$	9.54650	0.305420	0.152710	+	0.988271i	$0.451200\pi$
$978$	0	0
$979$	−25.0750	−0.801401
$980$	−20.1026	−0.642155
$981$	0	0
$982$	−5.04709	−0.161059
$983$	−29.8467	−0.951963	−0.475981	−	0.879455i	$-0.657907\pi$
$983$	−29.8467	−0.951963	−0.475981	+	0.879455i	$0.657907\pi$
$984$	0	0
$985$	−6.35430	−0.202465
$986$	32.8447	1.04599
$987$	0	0
$988$	24.4878	0.779061
$989$	106.898	3.39915
$990$	0	0
$991$	−24.1499	−0.767146	−0.383573	−	0.923511i	$-0.625307\pi$
$991$	−24.1499	−0.767146	−0.383573	+	0.923511i	$0.625307\pi$
$992$	−19.4088	−0.616232
$993$	0	0
$994$	−19.8414	−0.629330
$995$	−16.7816	−0.532012
$996$	0	0
$997$	31.9786	1.01277	0.506387	−	0.862306i	$-0.330981\pi$
$997$	31.9786	1.01277	0.506387	+	0.862306i	$0.330981\pi$
$998$	48.0419	1.52074
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5265.2.a.ba.1.2		8
3.2	odd	2		5265.2.a.bf.1.7		8
9.2	odd	6		585.2.i.e.391.2	yes	16
9.4	even	3		1755.2.i.f.586.7		16
9.5	odd	6		585.2.i.e.196.2	✓	16
9.7	even	3		1755.2.i.f.1171.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.i.e.196.2	✓	16	9.5	odd	6
585.2.i.e.391.2	yes	16	9.2	odd	6
1755.2.i.f.586.7		16	9.4	even	3
1755.2.i.f.1171.7		16	9.7	even	3
5265.2.a.ba.1.2		8	1.1	even	1	trivial
5265.2.a.bf.1.7		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)

\(f(q)\)	\(=\)	\(q-2.25662 q^{2} +3.09235 q^{4} +1.00000 q^{5} +0.706571 q^{7} -2.46502 q^{8} +O(q^{10})\)	\(q-2.25662 q^{2} +3.09235 q^{4} +1.00000 q^{5} +0.706571 q^{7} -2.46502 q^{8} -2.25662 q^{10} +2.33021 q^{11} -1.00000 q^{13} -1.59446 q^{14} -0.622073 q^{16} -6.04736 q^{17} -7.91884 q^{19} +3.09235 q^{20} -5.25841 q^{22} +8.95599 q^{23} +1.00000 q^{25} +2.25662 q^{26} +2.18496 q^{28} +2.40680 q^{29} -3.06432 q^{31} +6.33383 q^{32} +13.6466 q^{34} +0.706571 q^{35} -2.38079 q^{37} +17.8698 q^{38} -2.46502 q^{40} +9.45889 q^{41} +11.9359 q^{43} +7.20582 q^{44} -20.2103 q^{46} -9.72867 q^{47} -6.50076 q^{49} -2.25662 q^{50} -3.09235 q^{52} -5.90968 q^{53} +2.33021 q^{55} -1.74171 q^{56} -5.43123 q^{58} -2.97437 q^{59} -9.24156 q^{61} +6.91501 q^{62} -13.0489 q^{64} -1.00000 q^{65} -1.69448 q^{67} -18.7006 q^{68} -1.59446 q^{70} +12.4439 q^{71} +4.88899 q^{73} +5.37255 q^{74} -24.4878 q^{76} +1.64646 q^{77} -11.0254 q^{79} -0.622073 q^{80} -21.3452 q^{82} +0.521691 q^{83} -6.04736 q^{85} -26.9348 q^{86} -5.74402 q^{88} -10.7608 q^{89} -0.706571 q^{91} +27.6950 q^{92} +21.9539 q^{94} -7.91884 q^{95} -2.96306 q^{97} +14.6698 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 3 q^{2} + 9 q^{4} + 8 q^{5} - 11 q^{7} + 6 q^{8}+O(q^{10}) \) 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

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