LMFDB - Embedded newform 5265.2.a.bd.1.5

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} - 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			1.5
Root			$-0.475999$ 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)

$f(q)$	$=$	$q+0.475999 q^{2} -1.77343 q^{4} +1.00000 q^{5} -2.21142 q^{7} -1.79615 q^{8} +O(q^{10})$	\(q+0.475999 q^{2} -1.77343 q^{4} +1.00000 q^{5} -2.21142 q^{7} -1.79615 q^{8} +0.475999 q^{10} +0.0413745 q^{11} -1.00000 q^{13} -1.05263 q^{14} +2.69189 q^{16} +3.33374 q^{17} +4.12386 q^{19} -1.77343 q^{20} +0.0196942 q^{22} -3.36196 q^{23} +1.00000 q^{25} -0.475999 q^{26} +3.92179 q^{28} -0.807605 q^{29} -6.11466 q^{31} +4.87363 q^{32} +1.58686 q^{34} -2.21142 q^{35} -3.50591 q^{37} +1.96295 q^{38} -1.79615 q^{40} +8.06249 q^{41} +1.54041 q^{43} -0.0733746 q^{44} -1.60029 q^{46} +1.29984 q^{47} -2.10961 q^{49} +0.475999 q^{50} +1.77343 q^{52} +5.00878 q^{53} +0.0413745 q^{55} +3.97204 q^{56} -0.384419 q^{58} +10.6225 q^{59} -7.44982 q^{61} -2.91057 q^{62} -3.06394 q^{64} -1.00000 q^{65} -6.95654 q^{67} -5.91214 q^{68} -1.05263 q^{70} -10.0893 q^{71} +8.22743 q^{73} -1.66881 q^{74} -7.31336 q^{76} -0.0914965 q^{77} -12.8308 q^{79} +2.69189 q^{80} +3.83774 q^{82} -7.75798 q^{83} +3.33374 q^{85} +0.733234 q^{86} -0.0743146 q^{88} -5.56773 q^{89} +2.21142 q^{91} +5.96218 q^{92} +0.618724 q^{94} +4.12386 q^{95} +17.9000 q^{97} -1.00417 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.475999	0.336582	0.168291	−	0.985737i	$-0.446175\pi$
$2$	0.475999	0.336582	0.168291	+	0.985737i	$0.446175\pi$
$3$	0	0
$3$	0	0
$4$	−1.77343	−0.886713
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.21142	−0.835839	−0.417920	−	0.908484i	$-0.637241\pi$
$7$	−2.21142	−0.835839	−0.417920	+	0.908484i	$0.637241\pi$
$8$	−1.79615	−0.635033
$9$	0	0
$10$	0.475999	0.150524
$11$	0.0413745	0.0124749	0.00623744	−	0.999981i	$-0.498015\pi$
$11$	0.0413745	0.0124749	0.00623744	+	0.999981i	$0.498015\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−1.05263	−0.281328
$15$	0	0
$16$	2.69189	0.672972
$17$	3.33374	0.808551	0.404275	−	0.914637i	$-0.367524\pi$
$17$	3.33374	0.808551	0.404275	+	0.914637i	$0.367524\pi$
$18$	0	0
$19$	4.12386	0.946078	0.473039	−	0.881041i	$-0.343157\pi$
$19$	4.12386	0.946078	0.473039	+	0.881041i	$0.343157\pi$
$20$	−1.77343	−0.396550
$21$	0	0
$22$	0.0196942	0.00419882
$23$	−3.36196	−0.701017	−0.350508	−	0.936560i	$-0.613991\pi$
$23$	−3.36196	−0.701017	−0.350508	+	0.936560i	$0.613991\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−0.475999	−0.0933510
$27$	0	0
$28$	3.92179	0.741149
$29$	−0.807605	−0.149969	−0.0749843	−	0.997185i	$-0.523891\pi$
$29$	−0.807605	−0.149969	−0.0749843	+	0.997185i	$0.523891\pi$
$30$	0	0
$31$	−6.11466	−1.09822	−0.549112	−	0.835748i	$-0.685034\pi$
$31$	−6.11466	−1.09822	−0.549112	+	0.835748i	$0.685034\pi$
$32$	4.87363	0.861543
$33$	0	0
$34$	1.58686	0.272144
$35$	−2.21142	−0.373799
$36$	0	0
$37$	−3.50591	−0.576369	−0.288184	−	0.957575i	$-0.593052\pi$
$37$	−3.50591	−0.576369	−0.288184	+	0.957575i	$0.593052\pi$
$38$	1.96295	0.318433
$39$	0	0
$40$	−1.79615	−0.283996
$41$	8.06249	1.25915	0.629575	−	0.776940i	$-0.283229\pi$
$41$	8.06249	1.25915	0.629575	+	0.776940i	$0.283229\pi$
$42$	0	0
$43$	1.54041	0.234911	0.117455	−	0.993078i	$-0.462526\pi$
$43$	1.54041	0.234911	0.117455	+	0.993078i	$0.462526\pi$
$44$	−0.0733746	−0.0110616
$45$	0	0
$46$	−1.60029	−0.235950
$47$	1.29984	0.189602	0.0948008	−	0.995496i	$-0.469779\pi$
$47$	1.29984	0.189602	0.0948008	+	0.995496i	$0.469779\pi$
$48$	0	0
$49$	−2.10961	−0.301373
$50$	0.475999	0.0673164
$51$	0	0
$52$	1.77343	0.245930
$53$	5.00878	0.688009	0.344004	−	0.938968i	$-0.388216\pi$
$53$	5.00878	0.688009	0.344004	+	0.938968i	$0.388216\pi$
$54$	0	0
$55$	0.0413745	0.00557894
$56$	3.97204	0.530786
$57$	0	0
$58$	−0.384419	−0.0504767
$59$	10.6225	1.38293	0.691463	−	0.722412i	$-0.256966\pi$
$59$	10.6225	1.38293	0.691463	+	0.722412i	$0.256966\pi$
$60$	0	0
$61$	−7.44982	−0.953851	−0.476926	−	0.878944i	$-0.658249\pi$
$61$	−7.44982	−0.953851	−0.476926	+	0.878944i	$0.658249\pi$
$62$	−2.91057	−0.369643
$63$	0	0
$64$	−3.06394	−0.382992
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−6.95654	−0.849877	−0.424938	−	0.905222i	$-0.639704\pi$
$67$	−6.95654	−0.849877	−0.424938	+	0.905222i	$0.639704\pi$
$68$	−5.91214	−0.716952
$69$	0	0
$70$	−1.05263	−0.125814
$71$	−10.0893	−1.19738	−0.598692	−	0.800979i	$-0.704313\pi$
$71$	−10.0893	−1.19738	−0.598692	+	0.800979i	$0.704313\pi$
$72$	0	0
$73$	8.22743	0.962948	0.481474	−	0.876460i	$-0.340102\pi$
$73$	8.22743	0.962948	0.481474	+	0.876460i	$0.340102\pi$
$74$	−1.66881	−0.193995
$75$	0	0
$76$	−7.31336	−0.838900
$77$	−0.0914965	−0.0104270
$78$	0	0
$79$	−12.8308	−1.44357	−0.721787	−	0.692115i	$-0.756679\pi$
$79$	−12.8308	−1.44357	−0.721787	+	0.692115i	$0.756679\pi$
$80$	2.69189	0.300962
$81$	0	0
$82$	3.83774	0.423807
$83$	−7.75798	−0.851549	−0.425774	−	0.904829i	$-0.639998\pi$
$83$	−7.75798	−0.851549	−0.425774	+	0.904829i	$0.639998\pi$
$84$	0	0
$85$	3.33374	0.361595
$86$	0.733234	0.0790667
$87$	0	0
$88$	−0.0743146	−0.00792196
$89$	−5.56773	−0.590178	−0.295089	−	0.955470i	$-0.595349\pi$
$89$	−5.56773	−0.590178	−0.295089	+	0.955470i	$0.595349\pi$
$90$	0	0
$91$	2.21142	0.231820
$92$	5.96218	0.621600
$93$	0	0
$94$	0.618724	0.0638165
$95$	4.12386	0.423099
$96$	0	0
$97$	17.9000	1.81747	0.908734	−	0.417375i	$-0.137050\pi$
$97$	17.9000	1.81747	0.908734	+	0.417375i	$0.137050\pi$
$98$	−1.00417	−0.101437
$99$	0	0
$100$	−1.77343	−0.177343
$101$	−2.19166	−0.218078	−0.109039	−	0.994037i	$-0.534777\pi$
$101$	−2.19166	−0.218078	−0.109039	+	0.994037i	$0.534777\pi$
$102$	0	0
$103$	−8.64383	−0.851702	−0.425851	−	0.904793i	$-0.640025\pi$
$103$	−8.64383	−0.851702	−0.425851	+	0.904793i	$0.640025\pi$
$104$	1.79615	0.176127
$105$	0	0
$106$	2.38417	0.231571
$107$	6.88579	0.665675	0.332837	−	0.942984i	$-0.391994\pi$
$107$	6.88579	0.665675	0.332837	+	0.942984i	$0.391994\pi$
$108$	0	0
$109$	8.93028	0.855366	0.427683	−	0.903929i	$-0.359330\pi$
$109$	8.93028	0.855366	0.427683	+	0.903929i	$0.359330\pi$
$110$	0.0196942	0.00187777
$111$	0	0
$112$	−5.95290	−0.562496
$113$	−17.3006	−1.62751	−0.813754	−	0.581209i	$-0.802580\pi$
$113$	−17.3006	−1.62751	−0.813754	+	0.581209i	$0.802580\pi$
$114$	0	0
$115$	−3.36196	−0.313504
$116$	1.43223	0.132979
$117$	0	0
$118$	5.05628	0.465468
$119$	−7.37231	−0.675819
$120$	0	0
$121$	−10.9983	−0.999844
$122$	−3.54610	−0.321049
$123$	0	0
$124$	10.8439	0.973810
$125$	1.00000	0.0894427
$126$	0	0
$127$	15.9963	1.41944	0.709719	−	0.704484i	$-0.248822\pi$
$127$	15.9963	1.41944	0.709719	+	0.704484i	$0.248822\pi$
$128$	−11.2057	−0.990452
$129$	0	0
$130$	−0.475999	−0.0417478
$131$	−10.0048	−0.874128	−0.437064	−	0.899431i	$-0.643982\pi$
$131$	−10.0048	−0.874128	−0.437064	+	0.899431i	$0.643982\pi$
$132$	0	0
$133$	−9.11960	−0.790770
$134$	−3.31130	−0.286053
$135$	0	0
$136$	−5.98788	−0.513457
$137$	−5.79386	−0.495002	−0.247501	−	0.968888i	$-0.579609\pi$
$137$	−5.79386	−0.495002	−0.247501	+	0.968888i	$0.579609\pi$
$138$	0	0
$139$	−2.72661	−0.231268	−0.115634	−	0.993292i	$-0.536890\pi$
$139$	−2.72661	−0.231268	−0.115634	+	0.993292i	$0.536890\pi$
$140$	3.92179	0.331452
$141$	0	0
$142$	−4.80251	−0.403018
$143$	−0.0413745	−0.00345991
$144$	0	0
$145$	−0.807605	−0.0670680
$146$	3.91625	0.324111
$147$	0	0
$148$	6.21747	0.511073
$149$	−24.3870	−1.99786	−0.998930	−	0.0462385i	$-0.985277\pi$
$149$	−24.3870	−1.99786	−0.998930	+	0.0462385i	$0.985277\pi$
$150$	0	0
$151$	−18.5965	−1.51336	−0.756679	−	0.653786i	$-0.773179\pi$
$151$	−18.5965	−1.51336	−0.756679	+	0.653786i	$0.773179\pi$
$152$	−7.40705	−0.600791
$153$	0	0
$154$	−0.0435522	−0.00350954
$155$	−6.11466	−0.491141
$156$	0	0
$157$	−5.62791	−0.449156	−0.224578	−	0.974456i	$-0.572100\pi$
$157$	−5.62791	−0.449156	−0.224578	+	0.974456i	$0.572100\pi$
$158$	−6.10743	−0.485881
$159$	0	0
$160$	4.87363	0.385294
$161$	7.43471	0.585937
$162$	0	0
$163$	−16.5953	−1.29985	−0.649923	−	0.760000i	$-0.725199\pi$
$163$	−16.5953	−1.29985	−0.649923	+	0.760000i	$0.725199\pi$
$164$	−14.2982	−1.11650
$165$	0	0
$166$	−3.69279	−0.286616
$167$	−9.84509	−0.761836	−0.380918	−	0.924609i	$-0.624392\pi$
$167$	−9.84509	−0.761836	−0.380918	+	0.924609i	$0.624392\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	1.58686	0.121706
$171$	0	0
$172$	−2.73181	−0.208298
$173$	−0.827624	−0.0629231	−0.0314615	−	0.999505i	$-0.510016\pi$
$173$	−0.827624	−0.0629231	−0.0314615	+	0.999505i	$0.510016\pi$
$174$	0	0
$175$	−2.21142	−0.167168
$176$	0.111376	0.00839525
$177$	0	0
$178$	−2.65023	−0.198643
$179$	−2.95500	−0.220867	−0.110433	−	0.993884i	$-0.535224\pi$
$179$	−2.95500	−0.220867	−0.110433	+	0.993884i	$0.535224\pi$
$180$	0	0
$181$	2.99042	0.222276	0.111138	−	0.993805i	$-0.464550\pi$
$181$	2.99042	0.222276	0.111138	+	0.993805i	$0.464550\pi$
$182$	1.05263	0.0780264
$183$	0	0
$184$	6.03856	0.445169
$185$	−3.50591	−0.257760
$186$	0	0
$187$	0.137932	0.0100866
$188$	−2.30517	−0.168122
$189$	0	0
$190$	1.96295	0.142407
$191$	−11.0413	−0.798920	−0.399460	−	0.916751i	$-0.630802\pi$
$191$	−11.0413	−0.798920	−0.399460	+	0.916751i	$0.630802\pi$
$192$	0	0
$193$	−2.30907	−0.166211	−0.0831053	−	0.996541i	$-0.526484\pi$
$193$	−2.30907	−0.166211	−0.0831053	+	0.996541i	$0.526484\pi$
$194$	8.52037	0.611727
$195$	0	0
$196$	3.74123	0.267231
$197$	−15.6983	−1.11846	−0.559228	−	0.829014i	$-0.688902\pi$
$197$	−15.6983	−1.11846	−0.559228	+	0.829014i	$0.688902\pi$
$198$	0	0
$199$	15.4210	1.09317	0.546583	−	0.837405i	$-0.315928\pi$
$199$	15.4210	1.09317	0.546583	+	0.837405i	$0.315928\pi$
$200$	−1.79615	−0.127007
$201$	0	0
$202$	−1.04323	−0.0734012
$203$	1.78596	0.125350
$204$	0	0
$205$	8.06249	0.563109
$206$	−4.11445	−0.286667
$207$	0	0
$208$	−2.69189	−0.186649
$209$	0.170623	0.0118022
$210$	0	0
$211$	−19.0264	−1.30983	−0.654915	−	0.755703i	$-0.727296\pi$
$211$	−19.0264	−1.30983	−0.654915	+	0.755703i	$0.727296\pi$
$212$	−8.88270	−0.610066
$213$	0	0
$214$	3.27763	0.224054
$215$	1.54041	0.105055
$216$	0	0
$217$	13.5221	0.917940
$218$	4.25080	0.287901
$219$	0	0
$220$	−0.0733746	−0.00494691
$221$	−3.33374	−0.224252
$222$	0	0
$223$	5.63566	0.377392	0.188696	−	0.982036i	$-0.439574\pi$
$223$	5.63566	0.377392	0.188696	+	0.982036i	$0.439574\pi$
$224$	−10.7776	−0.720112
$225$	0	0
$226$	−8.23509	−0.547790
$227$	6.89084	0.457361	0.228681	−	0.973501i	$-0.426559\pi$
$227$	6.89084	0.457361	0.228681	+	0.973501i	$0.426559\pi$
$228$	0	0
$229$	8.03975	0.531282	0.265641	−	0.964072i	$-0.414416\pi$
$229$	8.03975	0.531282	0.265641	+	0.964072i	$0.414416\pi$
$230$	−1.60029	−0.105520
$231$	0	0
$232$	1.45058	0.0952350
$233$	23.8477	1.56231	0.781157	−	0.624335i	$-0.214630\pi$
$233$	23.8477	1.56231	0.781157	+	0.624335i	$0.214630\pi$
$234$	0	0
$235$	1.29984	0.0847924
$236$	−18.8381	−1.22626
$237$	0	0
$238$	−3.50921	−0.227468
$239$	5.05127	0.326740	0.163370	−	0.986565i	$-0.447764\pi$
$239$	5.05127	0.326740	0.163370	+	0.986565i	$0.447764\pi$
$240$	0	0
$241$	−15.0733	−0.970958	−0.485479	−	0.874248i	$-0.661355\pi$
$241$	−15.0733	−0.970958	−0.485479	+	0.874248i	$0.661355\pi$
$242$	−5.23517	−0.336529
$243$	0	0
$244$	13.2117	0.845792
$245$	−2.10961	−0.134778
$246$	0	0
$247$	−4.12386	−0.262395
$248$	10.9828	0.697409
$249$	0	0
$250$	0.475999	0.0301048
$251$	0.841751	0.0531309	0.0265654	−	0.999647i	$-0.491543\pi$
$251$	0.841751	0.0531309	0.0265654	+	0.999647i	$0.491543\pi$
$252$	0	0
$253$	−0.139099	−0.00874510
$254$	7.61420	0.477757
$255$	0	0
$256$	0.793985	0.0496241
$257$	6.45568	0.402694	0.201347	−	0.979520i	$-0.435468\pi$
$257$	6.45568	0.402694	0.201347	+	0.979520i	$0.435468\pi$
$258$	0	0
$259$	7.75306	0.481751
$260$	1.77343	0.109983
$261$	0	0
$262$	−4.76229	−0.294215
$263$	−23.7942	−1.46721	−0.733607	−	0.679574i	$-0.762165\pi$
$263$	−23.7942	−1.46721	−0.733607	+	0.679574i	$0.762165\pi$
$264$	0	0
$265$	5.00878	0.307687
$266$	−4.34092	−0.266159
$267$	0	0
$268$	12.3369	0.753596
$269$	−25.9881	−1.58452	−0.792260	−	0.610184i	$-0.791096\pi$
$269$	−25.9881	−1.58452	−0.792260	+	0.610184i	$0.791096\pi$
$270$	0	0
$271$	−10.3537	−0.628940	−0.314470	−	0.949267i	$-0.601827\pi$
$271$	−10.3537	−0.628940	−0.314470	+	0.949267i	$0.601827\pi$
$272$	8.97406	0.544132
$273$	0	0
$274$	−2.75787	−0.166609
$275$	0.0413745	0.00249498
$276$	0	0
$277$	4.77594	0.286958	0.143479	−	0.989653i	$-0.454171\pi$
$277$	4.77594	0.286958	0.143479	+	0.989653i	$0.454171\pi$
$278$	−1.29786	−0.0778405
$279$	0	0
$280$	3.97204	0.237375
$281$	−14.1393	−0.843479	−0.421739	−	0.906717i	$-0.638580\pi$
$281$	−14.1393	−0.843479	−0.421739	+	0.906717i	$0.638580\pi$
$282$	0	0
$283$	−3.56819	−0.212107	−0.106053	−	0.994360i	$-0.533821\pi$
$283$	−3.56819	−0.212107	−0.106053	+	0.994360i	$0.533821\pi$
$284$	17.8927	1.06174
$285$	0	0
$286$	−0.0196942	−0.00116454
$287$	−17.8296	−1.05245
$288$	0	0
$289$	−5.88618	−0.346246
$290$	−0.384419	−0.0225739
$291$	0	0
$292$	−14.5907	−0.853858
$293$	−23.0768	−1.34816	−0.674082	−	0.738657i	$-0.735461\pi$
$293$	−23.0768	−1.34816	−0.674082	+	0.738657i	$0.735461\pi$
$294$	0	0
$295$	10.6225	0.618464
$296$	6.29713	0.366013
$297$	0	0
$298$	−11.6082	−0.672444
$299$	3.36196	0.194427
$300$	0	0
$301$	−3.40650	−0.196348
$302$	−8.85189	−0.509369
$303$	0	0
$304$	11.1010	0.636684
$305$	−7.44982	−0.426575
$306$	0	0
$307$	0.659473	0.0376381	0.0188190	−	0.999823i	$-0.494009\pi$
$307$	0.659473	0.0376381	0.0188190	+	0.999823i	$0.494009\pi$
$308$	0.162262	0.00924575
$309$	0	0
$310$	−2.91057	−0.165309
$311$	−6.61711	−0.375222	−0.187611	−	0.982243i	$-0.560074\pi$
$311$	−6.61711	−0.375222	−0.187611	+	0.982243i	$0.560074\pi$
$312$	0	0
$313$	20.0915	1.13564	0.567820	−	0.823152i	$-0.307787\pi$
$313$	20.0915	1.13564	0.567820	+	0.823152i	$0.307787\pi$
$314$	−2.67888	−0.151178
$315$	0	0
$316$	22.7544	1.28004
$317$	−0.00553274	−0.000310750 0	−0.000155375	−	1.00000i	$-0.500049\pi$
$317$	−0.00553274	−0.000310750 0	−0.000155375		1.00000i	$0.500049\pi$
$318$	0	0
$319$	−0.0334143	−0.00187084
$320$	−3.06394	−0.171279
$321$	0	0
$322$	3.53891	0.197216
$323$	13.7479	0.764952
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	−7.89935	−0.437505
$327$	0	0
$328$	−14.4814	−0.799602
$329$	−2.87450	−0.158476
$330$	0	0
$331$	18.9510	1.04164	0.520820	−	0.853667i	$-0.325626\pi$
$331$	18.9510	1.04164	0.520820	+	0.853667i	$0.325626\pi$
$332$	13.7582	0.755079
$333$	0	0
$334$	−4.68625	−0.256420
$335$	−6.95654	−0.380076
$336$	0	0
$337$	19.5409	1.06446	0.532229	−	0.846600i	$-0.321354\pi$
$337$	19.5409	1.06446	0.532229	+	0.846600i	$0.321354\pi$
$338$	0.475999	0.0258909
$339$	0	0
$340$	−5.91214	−0.320631
$341$	−0.252991	−0.0137002
$342$	0	0
$343$	20.1452	1.08774
$344$	−2.76681	−0.149176
$345$	0	0
$346$	−0.393948	−0.0211788
$347$	−10.9729	−0.589057	−0.294529	−	0.955643i	$-0.595163\pi$
$347$	−10.9729	−0.589057	−0.294529	+	0.955643i	$0.595163\pi$
$348$	0	0
$349$	21.6498	1.15889	0.579445	−	0.815011i	$-0.303269\pi$
$349$	21.6498	1.15889	0.579445	+	0.815011i	$0.303269\pi$
$350$	−1.05263	−0.0562657
$351$	0	0
$352$	0.201644	0.0107477
$353$	5.83088	0.310347	0.155173	−	0.987887i	$-0.450406\pi$
$353$	5.83088	0.310347	0.155173	+	0.987887i	$0.450406\pi$
$354$	0	0
$355$	−10.0893	−0.535486
$356$	9.87395	0.523318
$357$	0	0
$358$	−1.40658	−0.0743398
$359$	−6.82799	−0.360367	−0.180184	−	0.983633i	$-0.557669\pi$
$359$	−6.82799	−0.360367	−0.180184	+	0.983633i	$0.557669\pi$
$360$	0	0
$361$	−1.99378	−0.104936
$362$	1.42343	0.0748140
$363$	0	0
$364$	−3.92179	−0.205558
$365$	8.22743	0.430644
$366$	0	0
$367$	21.9739	1.14703	0.573515	−	0.819195i	$-0.305579\pi$
$367$	21.9739	1.14703	0.573515	+	0.819195i	$0.305579\pi$
$368$	−9.05001	−0.471765
$369$	0	0
$370$	−1.66881	−0.0867573
$371$	−11.0765	−0.575065
$372$	0	0
$373$	4.68525	0.242593	0.121297	−	0.992616i	$-0.461295\pi$
$373$	4.68525	0.242593	0.121297	+	0.992616i	$0.461295\pi$
$374$	0.0656554	0.00339496
$375$	0	0
$376$	−2.33471	−0.120403
$377$	0.807605	0.0415938
$378$	0	0
$379$	20.0543	1.03012	0.515060	−	0.857154i	$-0.327770\pi$
$379$	20.0543	1.03012	0.515060	+	0.857154i	$0.327770\pi$
$380$	−7.31336	−0.375167
$381$	0	0
$382$	−5.25564	−0.268902
$383$	−17.9388	−0.916630	−0.458315	−	0.888790i	$-0.651547\pi$
$383$	−17.9388	−0.916630	−0.458315	+	0.888790i	$0.651547\pi$
$384$	0	0
$385$	−0.0914965	−0.00466309
$386$	−1.09911	−0.0559435
$387$	0	0
$388$	−31.7443	−1.61157
$389$	12.4061	0.629014	0.314507	−	0.949255i	$-0.398161\pi$
$389$	12.4061	0.629014	0.314507	+	0.949255i	$0.398161\pi$
$390$	0	0
$391$	−11.2079	−0.566808
$392$	3.78916	0.191382
$393$	0	0
$394$	−7.47235	−0.376452
$395$	−12.8308	−0.645586
$396$	0	0
$397$	−4.38227	−0.219940	−0.109970	−	0.993935i	$-0.535075\pi$
$397$	−4.38227	−0.219940	−0.109970	+	0.993935i	$0.535075\pi$
$398$	7.34038	0.367940
$399$	0	0
$400$	2.69189	0.134594
$401$	32.7025	1.63309	0.816543	−	0.577284i	$-0.195887\pi$
$401$	32.7025	1.63309	0.816543	+	0.577284i	$0.195887\pi$
$402$	0	0
$403$	6.11466	0.304593
$404$	3.88675	0.193373
$405$	0	0
$406$	0.850113	0.0421904
$407$	−0.145055	−0.00719013
$408$	0	0
$409$	8.90328	0.440239	0.220119	−	0.975473i	$-0.429355\pi$
$409$	8.90328	0.440239	0.220119	+	0.975473i	$0.429355\pi$
$410$	3.83774	0.189532
$411$	0	0
$412$	15.3292	0.755215
$413$	−23.4908	−1.15590
$414$	0	0
$415$	−7.75798	−0.380824
$416$	−4.87363	−0.238949
$417$	0	0
$418$	0.0812161	0.00397241
$419$	−10.0451	−0.490737	−0.245368	−	0.969430i	$-0.578909\pi$
$419$	−10.0451	−0.490737	−0.245368	+	0.969430i	$0.578909\pi$
$420$	0	0
$421$	−35.1652	−1.71385	−0.856923	−	0.515445i	$-0.827627\pi$
$421$	−35.1652	−1.71385	−0.856923	+	0.515445i	$0.827627\pi$
$422$	−9.05653	−0.440865
$423$	0	0
$424$	−8.99650	−0.436908
$425$	3.33374	0.161710
$426$	0	0
$427$	16.4747	0.797266
$428$	−12.2114	−0.590262
$429$	0	0
$430$	0.733234	0.0353597
$431$	−3.57604	−0.172252	−0.0861259	−	0.996284i	$-0.527449\pi$
$431$	−3.57604	−0.172252	−0.0861259	+	0.996284i	$0.527449\pi$
$432$	0	0
$433$	0.500930	0.0240732	0.0120366	−	0.999928i	$-0.496169\pi$
$433$	0.500930	0.0240732	0.0120366	+	0.999928i	$0.496169\pi$
$434$	6.43650	0.308962
$435$	0	0
$436$	−15.8372	−0.758463
$437$	−13.8642	−0.663217
$438$	0	0
$439$	3.79826	0.181281	0.0906406	−	0.995884i	$-0.471109\pi$
$439$	3.79826	0.181281	0.0906406	+	0.995884i	$0.471109\pi$
$440$	−0.0743146	−0.00354281
$441$	0	0
$442$	−1.58686	−0.0754790
$443$	10.4213	0.495131	0.247566	−	0.968871i	$-0.420369\pi$
$443$	10.4213	0.495131	0.247566	+	0.968871i	$0.420369\pi$
$444$	0	0
$445$	−5.56773	−0.263936
$446$	2.68257	0.127023
$447$	0	0
$448$	6.77566	0.320120
$449$	−0.431631	−0.0203699	−0.0101850	−	0.999948i	$-0.503242\pi$
$449$	−0.431631	−0.0203699	−0.0101850	+	0.999948i	$0.503242\pi$
$450$	0	0
$451$	0.333582	0.0157077
$452$	30.6814	1.44313
$453$	0	0
$454$	3.28003	0.153940
$455$	2.21142	0.103673
$456$	0	0
$457$	−31.7484	−1.48513	−0.742563	−	0.669776i	$-0.766390\pi$
$457$	−31.7484	−1.48513	−0.742563	+	0.669776i	$0.766390\pi$
$458$	3.82691	0.178820
$459$	0	0
$460$	5.96218	0.277988
$461$	−19.8485	−0.924435	−0.462218	−	0.886767i	$-0.652946\pi$
$461$	−19.8485	−0.924435	−0.462218	+	0.886767i	$0.652946\pi$
$462$	0	0
$463$	−19.9740	−0.928272	−0.464136	−	0.885764i	$-0.653635\pi$
$463$	−19.9740	−0.928272	−0.464136	+	0.885764i	$0.653635\pi$
$464$	−2.17398	−0.100925
$465$	0	0
$466$	11.3515	0.525846
$467$	−2.18775	−0.101237	−0.0506185	−	0.998718i	$-0.516119\pi$
$467$	−2.18775	−0.101237	−0.0506185	+	0.998718i	$0.516119\pi$
$468$	0	0
$469$	15.3839	0.710360
$470$	0.618724	0.0285396
$471$	0	0
$472$	−19.0795	−0.878204
$473$	0.0637338	0.00293048
$474$	0	0
$475$	4.12386	0.189216
$476$	13.0742	0.599257
$477$	0	0
$478$	2.40440	0.109975
$479$	−2.11871	−0.0968063	−0.0484031	−	0.998828i	$-0.515413\pi$
$479$	−2.11871	−0.0968063	−0.0484031	+	0.998828i	$0.515413\pi$
$480$	0	0
$481$	3.50591	0.159856
$482$	−7.17488	−0.326807
$483$	0	0
$484$	19.5046	0.886575
$485$	17.9000	0.812797
$486$	0	0
$487$	1.14567	0.0519151	0.0259575	−	0.999663i	$-0.491737\pi$
$487$	1.14567	0.0519151	0.0259575	+	0.999663i	$0.491737\pi$
$488$	13.3810	0.605727
$489$	0	0
$490$	−1.00417	−0.0453638
$491$	−7.58821	−0.342451	−0.171225	−	0.985232i	$-0.554773\pi$
$491$	−7.58821	−0.342451	−0.171225	+	0.985232i	$0.554773\pi$
$492$	0	0
$493$	−2.69235	−0.121257
$494$	−1.96295	−0.0883174
$495$	0	0
$496$	−16.4600	−0.739075
$497$	22.3118	1.00082
$498$	0	0
$499$	3.78474	0.169428	0.0847141	−	0.996405i	$-0.473002\pi$
$499$	3.78474	0.169428	0.0847141	+	0.996405i	$0.473002\pi$
$500$	−1.77343	−0.0793100
$501$	0	0
$502$	0.400672	0.0178829
$503$	−37.5852	−1.67584	−0.837921	−	0.545792i	$-0.816229\pi$
$503$	−37.5852	−1.67584	−0.837921	+	0.545792i	$0.816229\pi$
$504$	0	0
$505$	−2.19166	−0.0975276
$506$	−0.0662111	−0.00294344
$507$	0	0
$508$	−28.3682	−1.25863
$509$	0.300055	0.0132997	0.00664985	−	0.999978i	$-0.497883\pi$
$509$	0.300055	0.0132997	0.00664985	+	0.999978i	$0.497883\pi$
$510$	0	0
$511$	−18.1943	−0.804870
$512$	22.7893	1.00715
$513$	0	0
$514$	3.07289	0.135540
$515$	−8.64383	−0.380893
$516$	0	0
$517$	0.0537804	0.00236526
$518$	3.69044	0.162149
$519$	0	0
$520$	1.79615	0.0787662
$521$	23.4362	1.02676	0.513379	−	0.858162i	$-0.328394\pi$
$521$	23.4362	1.02676	0.513379	+	0.858162i	$0.328394\pi$
$522$	0	0
$523$	2.95026	0.129006	0.0645030	−	0.997918i	$-0.479454\pi$
$523$	2.95026	0.129006	0.0645030	+	0.997918i	$0.479454\pi$
$524$	17.7429	0.775100
$525$	0	0
$526$	−11.3260	−0.493838
$527$	−20.3847	−0.887971
$528$	0	0
$529$	−11.6972	−0.508576
$530$	2.38417	0.103562
$531$	0	0
$532$	16.1729	0.701185
$533$	−8.06249	−0.349225
$534$	0	0
$535$	6.88579	0.297699
$536$	12.4950	0.539700
$537$	0	0
$538$	−12.3703	−0.533321
$539$	−0.0872840	−0.00375959
$540$	0	0
$541$	24.3262	1.04587	0.522933	−	0.852374i	$-0.324838\pi$
$541$	24.3262	1.04587	0.522933	+	0.852374i	$0.324838\pi$
$542$	−4.92833	−0.211690
$543$	0	0
$544$	16.2474	0.696602
$545$	8.93028	0.382531
$546$	0	0
$547$	−32.7081	−1.39850	−0.699248	−	0.714879i	$-0.746482\pi$
$547$	−32.7081	−1.39850	−0.699248	+	0.714879i	$0.746482\pi$
$548$	10.2750	0.438925
$549$	0	0
$550$	0.0196942	0.000839764 0
$551$	−3.33045	−0.141882
$552$	0	0
$553$	28.3743	1.20660
$554$	2.27334	0.0965850
$555$	0	0
$556$	4.83543	0.205068
$557$	−25.5223	−1.08141	−0.540706	−	0.841211i	$-0.681843\pi$
$557$	−25.5223	−1.08141	−0.540706	+	0.841211i	$0.681843\pi$
$558$	0	0
$559$	−1.54041	−0.0651525
$560$	−5.95290	−0.251556
$561$	0	0
$562$	−6.73028	−0.283900
$563$	−21.3444	−0.899557	−0.449779	−	0.893140i	$-0.648497\pi$
$563$	−21.3444	−0.899557	−0.449779	+	0.893140i	$0.648497\pi$
$564$	0	0
$565$	−17.3006	−0.727844
$566$	−1.69845	−0.0713913
$567$	0	0
$568$	18.1219	0.760378
$569$	20.5672	0.862223	0.431111	−	0.902299i	$-0.358122\pi$
$569$	20.5672	0.862223	0.431111	+	0.902299i	$0.358122\pi$
$570$	0	0
$571$	28.1034	1.17609	0.588046	−	0.808827i	$-0.299897\pi$
$571$	28.1034	1.17609	0.588046	+	0.808827i	$0.299897\pi$
$572$	0.0733746	0.00306795
$573$	0	0
$574$	−8.48686	−0.354235
$575$	−3.36196	−0.140203
$576$	0	0
$577$	20.0791	0.835904	0.417952	−	0.908469i	$-0.362748\pi$
$577$	20.0791	0.835904	0.417952	+	0.908469i	$0.362748\pi$
$578$	−2.80181	−0.116540
$579$	0	0
$580$	1.43223	0.0594700
$581$	17.1562	0.711758
$582$	0	0
$583$	0.207236	0.00858283
$584$	−14.7777	−0.611504
$585$	0	0
$586$	−10.9845	−0.453767
$587$	−20.5222	−0.847042	−0.423521	−	0.905886i	$-0.639206\pi$
$587$	−20.5222	−0.847042	−0.423521	+	0.905886i	$0.639206\pi$
$588$	0	0
$589$	−25.2160	−1.03901
$590$	5.05628	0.208164
$591$	0	0
$592$	−9.43752	−0.387880
$593$	25.5432	1.04893	0.524467	−	0.851431i	$-0.324265\pi$
$593$	25.5432	1.04893	0.524467	+	0.851431i	$0.324265\pi$
$594$	0	0
$595$	−7.37231	−0.302235
$596$	43.2485	1.77153
$597$	0	0
$598$	1.60029	0.0654406
$599$	−5.89407	−0.240825	−0.120413	−	0.992724i	$-0.538422\pi$
$599$	−5.89407	−0.240825	−0.120413	+	0.992724i	$0.538422\pi$
$600$	0	0
$601$	0.758043	0.0309212	0.0154606	−	0.999880i	$-0.495079\pi$
$601$	0.758043	0.0309212	0.0154606	+	0.999880i	$0.495079\pi$
$602$	−1.62149	−0.0660871
$603$	0	0
$604$	32.9794	1.34191
$605$	−10.9983	−0.447144
$606$	0	0
$607$	25.0862	1.01822	0.509109	−	0.860702i	$-0.329975\pi$
$607$	25.0862	1.01822	0.509109	+	0.860702i	$0.329975\pi$
$608$	20.0982	0.815088
$609$	0	0
$610$	−3.54610	−0.143577
$611$	−1.29984	−0.0525860
$612$	0	0
$613$	−18.4934	−0.746943	−0.373471	−	0.927642i	$-0.621833\pi$
$613$	−18.4934	−0.746943	−0.373471	+	0.927642i	$0.621833\pi$
$614$	0.313908	0.0126683
$615$	0	0
$616$	0.164341	0.00662149
$617$	−22.9983	−0.925876	−0.462938	−	0.886391i	$-0.653205\pi$
$617$	−22.9983	−0.925876	−0.462938	+	0.886391i	$0.653205\pi$
$618$	0	0
$619$	18.4707	0.742400	0.371200	−	0.928553i	$-0.378946\pi$
$619$	18.4707	0.742400	0.371200	+	0.928553i	$0.378946\pi$
$620$	10.8439	0.435501
$621$	0	0
$622$	−3.14974	−0.126293
$623$	12.3126	0.493294
$624$	0	0
$625$	1.00000	0.0400000
$626$	9.56354	0.382236
$627$	0	0
$628$	9.98068	0.398272
$629$	−11.6878	−0.466023
$630$	0	0
$631$	−31.8336	−1.26728	−0.633638	−	0.773630i	$-0.718439\pi$
$631$	−31.8336	−1.26728	−0.633638	+	0.773630i	$0.718439\pi$
$632$	23.0459	0.916718
$633$	0	0
$634$	−0.00263358	−0.000104593 0
$635$	15.9963	0.634792
$636$	0	0
$637$	2.10961	0.0835857
$638$	−0.0159051	−0.000629691 0
$639$	0	0
$640$	−11.2057	−0.442943
$641$	−39.8893	−1.57553	−0.787766	−	0.615975i	$-0.788762\pi$
$641$	−39.8893	−1.57553	−0.787766	+	0.615975i	$0.788762\pi$
$642$	0	0
$643$	10.4971	0.413964	0.206982	−	0.978345i	$-0.433636\pi$
$643$	10.4971	0.413964	0.206982	+	0.978345i	$0.433636\pi$
$644$	−13.1849	−0.519558
$645$	0	0
$646$	6.54397	0.257469
$647$	20.7809	0.816981	0.408491	−	0.912763i	$-0.366055\pi$
$647$	20.7809	0.816981	0.408491	+	0.912763i	$0.366055\pi$
$648$	0	0
$649$	0.439499	0.0172518
$650$	−0.475999	−0.0186702
$651$	0	0
$652$	29.4306	1.15259
$653$	22.6355	0.885794	0.442897	−	0.896573i	$-0.353951\pi$
$653$	22.6355	0.885794	0.442897	+	0.896573i	$0.353951\pi$
$654$	0	0
$655$	−10.0048	−0.390922
$656$	21.7033	0.847373
$657$	0	0
$658$	−1.36826	−0.0533403
$659$	−16.5108	−0.643169	−0.321585	−	0.946881i	$-0.604215\pi$
$659$	−16.5108	−0.643169	−0.321585	+	0.946881i	$0.604215\pi$
$660$	0	0
$661$	−12.7470	−0.495802	−0.247901	−	0.968785i	$-0.579741\pi$
$661$	−12.7470	−0.495802	−0.247901	+	0.968785i	$0.579741\pi$
$662$	9.02063	0.350597
$663$	0	0
$664$	13.9345	0.540762
$665$	−9.11960	−0.353643
$666$	0	0
$667$	2.71514	0.105130
$668$	17.4595	0.675529
$669$	0	0
$670$	−3.31130	−0.127927
$671$	−0.308232	−0.0118992
$672$	0	0
$673$	−15.7068	−0.605451	−0.302725	−	0.953078i	$-0.597897\pi$
$673$	−15.7068	−0.605451	−0.302725	+	0.953078i	$0.597897\pi$
$674$	9.30143	0.358278
$675$	0	0
$676$	−1.77343	−0.0682087
$677$	44.0623	1.69345	0.846725	−	0.532031i	$-0.178571\pi$
$677$	44.0623	1.69345	0.846725	+	0.532031i	$0.178571\pi$
$678$	0	0
$679$	−39.5844	−1.51911
$680$	−5.98788	−0.229625
$681$	0	0
$682$	−0.120423	−0.00461125
$683$	−14.4849	−0.554248	−0.277124	−	0.960834i	$-0.589381\pi$
$683$	−14.4849	−0.554248	−0.277124	+	0.960834i	$0.589381\pi$
$684$	0	0
$685$	−5.79386	−0.221372
$686$	9.58909	0.366113
$687$	0	0
$688$	4.14662	0.158088
$689$	−5.00878	−0.190819
$690$	0	0
$691$	−28.0137	−1.06569	−0.532846	−	0.846212i	$-0.678878\pi$
$691$	−28.0137	−1.06569	−0.532846	+	0.846212i	$0.678878\pi$
$692$	1.46773	0.0557947
$693$	0	0
$694$	−5.22310	−0.198266
$695$	−2.72661	−0.103426
$696$	0	0
$697$	26.8783	1.01809
$698$	10.3053	0.390061
$699$	0	0
$700$	3.92179	0.148230
$701$	31.5907	1.19317	0.596583	−	0.802552i	$-0.296525\pi$
$701$	31.5907	1.19317	0.596583	+	0.802552i	$0.296525\pi$
$702$	0	0
$703$	−14.4579	−0.545290
$704$	−0.126769	−0.00477778
$705$	0	0
$706$	2.77549	0.104457
$707$	4.84669	0.182279
$708$	0	0
$709$	−26.5638	−0.997626	−0.498813	−	0.866710i	$-0.666231\pi$
$709$	−26.5638	−0.997626	−0.498813	+	0.866710i	$0.666231\pi$
$710$	−4.80251	−0.180235
$711$	0	0
$712$	10.0005	0.374783
$713$	20.5572	0.769874
$714$	0	0
$715$	−0.0413745	−0.00154732
$716$	5.24047	0.195846
$717$	0	0
$718$	−3.25011	−0.121293
$719$	−8.25557	−0.307881	−0.153940	−	0.988080i	$-0.549196\pi$
$719$	−8.25557	−0.307881	−0.153940	+	0.988080i	$0.549196\pi$
$720$	0	0
$721$	19.1152	0.711886
$722$	−0.949035	−0.0353194
$723$	0	0
$724$	−5.30328	−0.197095
$725$	−0.807605	−0.0299937
$726$	0	0
$727$	40.1060	1.48745	0.743724	−	0.668487i	$-0.233058\pi$
$727$	40.1060	1.48745	0.743724	+	0.668487i	$0.233058\pi$
$728$	−3.97204	−0.147213
$729$	0	0
$730$	3.91625	0.144947
$731$	5.13534	0.189937
$732$	0	0
$733$	45.9542	1.69736	0.848678	−	0.528909i	$-0.177399\pi$
$733$	45.9542	1.69736	0.848678	+	0.528909i	$0.177399\pi$
$734$	10.4596	0.386070
$735$	0	0
$736$	−16.3849	−0.603956
$737$	−0.287823	−0.0106021
$738$	0	0
$739$	5.41873	0.199331	0.0996655	−	0.995021i	$-0.468223\pi$
$739$	5.41873	0.199331	0.0996655	+	0.995021i	$0.468223\pi$
$740$	6.21747	0.228559
$741$	0	0
$742$	−5.27241	−0.193556
$743$	−48.4349	−1.77691	−0.888453	−	0.458968i	$-0.848219\pi$
$743$	−48.4349	−1.77691	−0.888453	+	0.458968i	$0.848219\pi$
$744$	0	0
$745$	−24.3870	−0.893471
$746$	2.23017	0.0816524
$747$	0	0
$748$	−0.244612	−0.00894389
$749$	−15.2274	−0.556397
$750$	0	0
$751$	0.0894558	0.00326429	0.00163214	−	0.999999i	$-0.499480\pi$
$751$	0.0894558	0.00326429	0.00163214	+	0.999999i	$0.499480\pi$
$752$	3.49903	0.127597
$753$	0	0
$754$	0.384419	0.0139997
$755$	−18.5965	−0.676794
$756$	0	0
$757$	48.3945	1.75893	0.879464	−	0.475966i	$-0.157902\pi$
$757$	48.3945	1.75893	0.879464	+	0.475966i	$0.157902\pi$
$758$	9.54581	0.346719
$759$	0	0
$760$	−7.40705	−0.268682
$761$	−14.8551	−0.538499	−0.269249	−	0.963070i	$-0.586776\pi$
$761$	−14.8551	−0.538499	−0.269249	+	0.963070i	$0.586776\pi$
$762$	0	0
$763$	−19.7486	−0.714948
$764$	19.5809	0.708412
$765$	0	0
$766$	−8.53885	−0.308521
$767$	−10.6225	−0.383555
$768$	0	0
$769$	16.4378	0.592763	0.296382	−	0.955070i	$-0.404220\pi$
$769$	16.4378	0.592763	0.296382	+	0.955070i	$0.404220\pi$
$770$	−0.0435522	−0.00156951
$771$	0	0
$772$	4.09496	0.147381
$773$	−35.1175	−1.26309	−0.631544	−	0.775340i	$-0.717578\pi$
$773$	−35.1175	−1.26309	−0.631544	+	0.775340i	$0.717578\pi$
$774$	0	0
$775$	−6.11466	−0.219645
$776$	−32.1510	−1.15415
$777$	0	0
$778$	5.90528	0.211715
$779$	33.2486	1.19125
$780$	0	0
$781$	−0.417441	−0.0149372
$782$	−5.33494	−0.190777
$783$	0	0
$784$	−5.67883	−0.202815
$785$	−5.62791	−0.200869
$786$	0	0
$787$	52.3619	1.86650	0.933250	−	0.359227i	$-0.116960\pi$
$787$	52.3619	1.86650	0.933250	+	0.359227i	$0.116960\pi$
$788$	27.8397	0.991748
$789$	0	0
$790$	−6.10743	−0.217293
$791$	38.2591	1.36034
$792$	0	0
$793$	7.44982	0.264551
$794$	−2.08596	−0.0740278
$795$	0	0
$796$	−27.3480	−0.969324
$797$	4.90564	0.173767	0.0868833	−	0.996218i	$-0.472309\pi$
$797$	4.90564	0.173767	0.0868833	+	0.996218i	$0.472309\pi$
$798$	0	0
$799$	4.33334	0.153303
$800$	4.87363	0.172309
$801$	0	0
$802$	15.5664	0.549667
$803$	0.340406	0.0120127
$804$	0	0
$805$	7.43471	0.262039
$806$	2.91057	0.102520
$807$	0	0
$808$	3.93654	0.138487
$809$	52.9069	1.86011	0.930053	−	0.367424i	$-0.119760\pi$
$809$	52.9069	1.86011	0.930053	+	0.367424i	$0.119760\pi$
$810$	0	0
$811$	15.1381	0.531571	0.265785	−	0.964032i	$-0.414369\pi$
$811$	15.1381	0.531571	0.265785	+	0.964032i	$0.414369\pi$
$812$	−3.16726	−0.111149
$813$	0	0
$814$	−0.0690462	−0.00242007
$815$	−16.5953	−0.581309
$816$	0	0
$817$	6.35245	0.222244
$818$	4.23795	0.148176
$819$	0	0
$820$	−14.2982	−0.499316
$821$	−27.3865	−0.955794	−0.477897	−	0.878416i	$-0.658601\pi$
$821$	−27.3865	−0.955794	−0.477897	+	0.878416i	$0.658601\pi$
$822$	0	0
$823$	−7.99428	−0.278663	−0.139332	−	0.990246i	$-0.544495\pi$
$823$	−7.99428	−0.278663	−0.139332	+	0.990246i	$0.544495\pi$
$824$	15.5256	0.540859
$825$	0	0
$826$	−11.1816	−0.389056
$827$	−28.5421	−0.992505	−0.496252	−	0.868178i	$-0.665291\pi$
$827$	−28.5421	−0.992505	−0.496252	+	0.868178i	$0.665291\pi$
$828$	0	0
$829$	−12.7230	−0.441886	−0.220943	−	0.975287i	$-0.570914\pi$
$829$	−12.7230	−0.441886	−0.220943	+	0.975287i	$0.570914\pi$
$830$	−3.69279	−0.128178
$831$	0	0
$832$	3.06394	0.106223
$833$	−7.03289	−0.243675
$834$	0	0
$835$	−9.84509	−0.340703
$836$	−0.302587	−0.0104652
$837$	0	0
$838$	−4.78147	−0.165173
$839$	52.2764	1.80478	0.902391	−	0.430917i	$-0.141810\pi$
$839$	52.2764	1.80478	0.902391	+	0.430917i	$0.141810\pi$
$840$	0	0
$841$	−28.3478	−0.977509
$842$	−16.7386	−0.576849
$843$	0	0
$844$	33.7419	1.16144
$845$	1.00000	0.0344010
$846$	0	0
$847$	24.3219	0.835709
$848$	13.4831	0.463011
$849$	0	0
$850$	1.58686	0.0544287
$851$	11.7867	0.404044
$852$	0	0
$853$	−7.15457	−0.244968	−0.122484	−	0.992470i	$-0.539086\pi$
$853$	−7.15457	−0.244968	−0.122484	+	0.992470i	$0.539086\pi$
$854$	7.84193	0.268345
$855$	0	0
$856$	−12.3679	−0.422725
$857$	57.1869	1.95347	0.976734	−	0.214455i	$-0.0687977\pi$
$857$	57.1869	1.95347	0.976734	+	0.214455i	$0.0687977\pi$
$858$	0	0
$859$	42.0315	1.43410	0.717048	−	0.697023i	$-0.245493\pi$
$859$	42.0315	1.43410	0.717048	+	0.697023i	$0.245493\pi$
$860$	−2.73181	−0.0931539
$861$	0	0
$862$	−1.70219	−0.0579769
$863$	−4.88307	−0.166222	−0.0831109	−	0.996540i	$-0.526486\pi$
$863$	−4.88307	−0.166222	−0.0831109	+	0.996540i	$0.526486\pi$
$864$	0	0
$865$	−0.827624	−0.0281401
$866$	0.238442	0.00810259
$867$	0	0
$868$	−23.9804	−0.813949
$869$	−0.530867	−0.0180084
$870$	0	0
$871$	6.95654	0.235713
$872$	−16.0401	−0.543186
$873$	0	0
$874$	−6.59936	−0.223227
$875$	−2.21142	−0.0747597
$876$	0	0
$877$	−3.58110	−0.120925	−0.0604625	−	0.998170i	$-0.519258\pi$
$877$	−3.58110	−0.120925	−0.0604625	+	0.998170i	$0.519258\pi$
$878$	1.80797	0.0610160
$879$	0	0
$880$	0.111376	0.00375447
$881$	−13.8568	−0.466848	−0.233424	−	0.972375i	$-0.574993\pi$
$881$	−13.8568	−0.466848	−0.233424	+	0.972375i	$0.574993\pi$
$882$	0	0
$883$	−24.3108	−0.818123	−0.409061	−	0.912507i	$-0.634144\pi$
$883$	−24.3108	−0.818123	−0.409061	+	0.912507i	$0.634144\pi$
$884$	5.91214	0.198847
$885$	0	0
$886$	4.96053	0.166652
$887$	−0.124409	−0.00417725	−0.00208863	−	0.999998i	$-0.500665\pi$
$887$	−0.124409	−0.00417725	−0.00208863	+	0.999998i	$0.500665\pi$
$888$	0	0
$889$	−35.3745	−1.18642
$890$	−2.65023	−0.0888360
$891$	0	0
$892$	−9.99442	−0.334638
$893$	5.36037	0.179378
$894$	0	0
$895$	−2.95500	−0.0987747
$896$	24.7805	0.827858
$897$	0	0
$898$	−0.205456	−0.00685615
$899$	4.93823	0.164699
$900$	0	0
$901$	16.6980	0.556290
$902$	0.158784	0.00528694
$903$	0	0
$904$	31.0745	1.03352
$905$	2.99042	0.0994048
$906$	0	0
$907$	35.5324	1.17983	0.589917	−	0.807464i	$-0.299161\pi$
$907$	35.5324	1.17983	0.589917	+	0.807464i	$0.299161\pi$
$908$	−12.2204	−0.405548
$909$	0	0
$910$	1.05263	0.0348945
$911$	−4.65132	−0.154105	−0.0770525	−	0.997027i	$-0.524551\pi$
$911$	−4.65132	−0.154105	−0.0770525	+	0.997027i	$0.524551\pi$
$912$	0	0
$913$	−0.320982	−0.0106230
$914$	−15.1122	−0.499867
$915$	0	0
$916$	−14.2579	−0.471094
$917$	22.1250	0.730630
$918$	0	0
$919$	−55.6527	−1.83581	−0.917907	−	0.396795i	$-0.870122\pi$
$919$	−55.6527	−1.83581	−0.917907	+	0.396795i	$0.870122\pi$
$920$	6.03856	0.199086
$921$	0	0
$922$	−9.44784	−0.311148
$923$	10.0893	0.332094
$924$	0	0
$925$	−3.50591	−0.115274
$926$	−9.50761	−0.312439
$927$	0	0
$928$	−3.93597	−0.129204
$929$	−12.2114	−0.400644	−0.200322	−	0.979730i	$-0.564199\pi$
$929$	−12.2114	−0.400644	−0.200322	+	0.979730i	$0.564199\pi$
$930$	0	0
$931$	−8.69973	−0.285122
$932$	−42.2921	−1.38532
$933$	0	0
$934$	−1.04137	−0.0340746
$935$	0.137932	0.00451085
$936$	0	0
$937$	3.36999	0.110093	0.0550464	−	0.998484i	$-0.482469\pi$
$937$	3.36999	0.110093	0.0550464	+	0.998484i	$0.482469\pi$
$938$	7.32269	0.239094
$939$	0	0
$940$	−2.30517	−0.0751865
$941$	2.67820	0.0873069	0.0436535	−	0.999047i	$-0.486100\pi$
$941$	2.67820	0.0873069	0.0436535	+	0.999047i	$0.486100\pi$
$942$	0	0
$943$	−27.1058	−0.882685
$944$	28.5945	0.930671
$945$	0	0
$946$	0.0303372	0.000986348 0
$947$	23.9620	0.778660	0.389330	−	0.921098i	$-0.372707\pi$
$947$	23.9620	0.778660	0.389330	+	0.921098i	$0.372707\pi$
$948$	0	0
$949$	−8.22743	−0.267074
$950$	1.96295	0.0636866
$951$	0	0
$952$	13.2417	0.429167
$953$	−26.4066	−0.855394	−0.427697	−	0.903922i	$-0.640675\pi$
$953$	−26.4066	−0.855394	−0.427697	+	0.903922i	$0.640675\pi$
$954$	0	0
$955$	−11.0413	−0.357288
$956$	−8.95805	−0.289724
$957$	0	0
$958$	−1.00850	−0.0325832
$959$	12.8127	0.413743
$960$	0	0
$961$	6.38904	0.206098
$962$	1.66881	0.0538046
$963$	0	0
$964$	26.7314	0.860960
$965$	−2.30907	−0.0743316
$966$	0	0
$967$	30.6510	0.985670	0.492835	−	0.870123i	$-0.335961\pi$
$967$	30.6510	0.985670	0.492835	+	0.870123i	$0.335961\pi$
$968$	19.7545	0.634934
$969$	0	0
$970$	8.52037	0.273573
$971$	2.10146	0.0674392	0.0337196	−	0.999431i	$-0.489265\pi$
$971$	2.10146	0.0674392	0.0337196	+	0.999431i	$0.489265\pi$
$972$	0	0
$973$	6.02968	0.193303
$974$	0.545335	0.0174737
$975$	0	0
$976$	−20.0541	−0.641915
$977$	29.5306	0.944769	0.472384	−	0.881393i	$-0.343393\pi$
$977$	29.5306	0.944769	0.472384	+	0.881393i	$0.343393\pi$
$978$	0	0
$979$	−0.230362	−0.00736240
$980$	3.74123	0.119509
$981$	0	0
$982$	−3.61198	−0.115263
$983$	−34.6201	−1.10421	−0.552105	−	0.833774i	$-0.686175\pi$
$983$	−34.6201	−1.10421	−0.552105	+	0.833774i	$0.686175\pi$
$984$	0	0
$985$	−15.6983	−0.500188
$986$	−1.28155	−0.0408130
$987$	0	0
$988$	7.31336	0.232669
$989$	−5.17880	−0.164676
$990$	0	0
$991$	−47.2716	−1.50163	−0.750815	−	0.660512i	$-0.770339\pi$
$991$	−47.2716	−1.50163	−0.750815	+	0.660512i	$0.770339\pi$
$992$	−29.8006	−0.946168
$993$	0	0
$994$	10.6204	0.336858
$995$	15.4210	0.488879
$996$	0	0
$997$	−47.9841	−1.51967	−0.759836	−	0.650114i	$-0.774721\pi$
$997$	−47.9841	−1.51967	−0.759836	+	0.650114i	$0.774721\pi$
$998$	1.80153	0.0570265
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5265.2.a.bd.1.5	yes	8
3.2	odd	2		5265.2.a.bc.1.4	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5265.2.a.bc.1.4	✓	8	3.2	odd	2
5265.2.a.bd.1.5	yes	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)

\(f(q)\)	\(=\)	\(q+0.475999 q^{2} -1.77343 q^{4} +1.00000 q^{5} -2.21142 q^{7} -1.79615 q^{8} +O(q^{10})\)	\(q+0.475999 q^{2} -1.77343 q^{4} +1.00000 q^{5} -2.21142 q^{7} -1.79615 q^{8} +0.475999 q^{10} +0.0413745 q^{11} -1.00000 q^{13} -1.05263 q^{14} +2.69189 q^{16} +3.33374 q^{17} +4.12386 q^{19} -1.77343 q^{20} +0.0196942 q^{22} -3.36196 q^{23} +1.00000 q^{25} -0.475999 q^{26} +3.92179 q^{28} -0.807605 q^{29} -6.11466 q^{31} +4.87363 q^{32} +1.58686 q^{34} -2.21142 q^{35} -3.50591 q^{37} +1.96295 q^{38} -1.79615 q^{40} +8.06249 q^{41} +1.54041 q^{43} -0.0733746 q^{44} -1.60029 q^{46} +1.29984 q^{47} -2.10961 q^{49} +0.475999 q^{50} +1.77343 q^{52} +5.00878 q^{53} +0.0413745 q^{55} +3.97204 q^{56} -0.384419 q^{58} +10.6225 q^{59} -7.44982 q^{61} -2.91057 q^{62} -3.06394 q^{64} -1.00000 q^{65} -6.95654 q^{67} -5.91214 q^{68} -1.05263 q^{70} -10.0893 q^{71} +8.22743 q^{73} -1.66881 q^{74} -7.31336 q^{76} -0.0914965 q^{77} -12.8308 q^{79} +2.69189 q^{80} +3.83774 q^{82} -7.75798 q^{83} +3.33374 q^{85} +0.733234 q^{86} -0.0743146 q^{88} -5.56773 q^{89} +2.21142 q^{91} +5.96218 q^{92} +0.618724 q^{94} +4.12386 q^{95} +17.9000 q^{97} -1.00417 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{4} + 8 q^{5} + 4 q^{7} - 6 q^{8}+O(q^{10}) \) 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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