LMFDB - Embedded newform 2475.4.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,4,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2475.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2475.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:	$146.029727264$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -7.00000 q^{4} +9.00000 q^{7} -15.0000 q^{8} +O(q^{10})$

$q+1.00000 q^{2} -7.00000 q^{4} +9.00000 q^{7} -15.0000 q^{8} -11.0000 q^{11} -2.00000 q^{13} +9.00000 q^{14} +41.0000 q^{16} +21.0000 q^{17} -85.0000 q^{19} -11.0000 q^{22} +22.0000 q^{23} -2.00000 q^{26} -63.0000 q^{28} +165.000 q^{29} -83.0000 q^{31} +161.000 q^{32} +21.0000 q^{34} -1.00000 q^{37} -85.0000 q^{38} +478.000 q^{41} +8.00000 q^{43} +77.0000 q^{44} +22.0000 q^{46} +126.000 q^{47} -262.000 q^{49} +14.0000 q^{52} -683.000 q^{53} -135.000 q^{56} +165.000 q^{58} +290.000 q^{59} +257.000 q^{61} -83.0000 q^{62} -167.000 q^{64} -776.000 q^{67} -147.000 q^{68} +313.000 q^{71} -902.000 q^{73} -1.00000 q^{74} +595.000 q^{76} -99.0000 q^{77} +830.000 q^{79} +478.000 q^{82} +842.000 q^{83} +8.00000 q^{86} +165.000 q^{88} -25.0000 q^{89} -18.0000 q^{91} -154.000 q^{92} +126.000 q^{94} +1784.00 q^{97} -262.000 q^{98} +O(q^{100})$

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 $

$2$	1.00000	0.353553	0.176777	−	0.984251i	$-0.443433\pi$
$2$	1.00000	0.353553	0.176777	+	0.984251i	$0.443433\pi$
$3$	0	0
$3$	0	0
$4$	−7.00000	−0.875000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	9.00000	0.485954	0.242977	−	0.970032i	$-0.421876\pi$
$7$	9.00000	0.485954	0.242977	+	0.970032i	$0.421876\pi$
$8$	−15.0000	−0.662913
$9$	0	0
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−2.00000	−0.0426692	−0.0213346	−	0.999772i	$-0.506792\pi$
$13$	−2.00000	−0.0426692	−0.0213346	+	0.999772i	$0.506792\pi$
$14$	9.00000	0.171811
$15$	0	0
$16$	41.0000	0.640625
$17$	21.0000	0.299603	0.149801	−	0.988716i	$-0.452137\pi$
$17$	21.0000	0.299603	0.149801	+	0.988716i	$0.452137\pi$
$18$	0	0
$19$	−85.0000	−1.02633	−0.513167	−	0.858289i	$-0.671528\pi$
$19$	−85.0000	−1.02633	−0.513167	+	0.858289i	$0.671528\pi$
$20$	0	0
$21$	0	0
$22$	−11.0000	−0.106600
$23$	22.0000	0.199449	0.0997243	−	0.995015i	$-0.468204\pi$
$23$	22.0000	0.199449	0.0997243	+	0.995015i	$0.468204\pi$
$24$	0	0
$25$	0	0
$26$	−2.00000	−0.0150859
$27$	0	0
$28$	−63.0000	−0.425210
$29$	165.000	1.05654	0.528271	−	0.849076i	$-0.322840\pi$
$29$	165.000	1.05654	0.528271	+	0.849076i	$0.322840\pi$
$30$	0	0
$31$	−83.0000	−0.480879	−0.240439	−	0.970664i	$-0.577292\pi$
$31$	−83.0000	−0.480879	−0.240439	+	0.970664i	$0.577292\pi$
$32$	161.000	0.889408
$33$	0	0
$34$	21.0000	0.105926
$35$	0	0
$36$	0	0
$37$	−1.00000	−0.00444322	−0.00222161	−	0.999998i	$-0.500707\pi$
$37$	−1.00000	−0.00444322	−0.00222161	+	0.999998i	$0.500707\pi$
$38$	−85.0000	−0.362864
$39$	0	0
$40$	0	0
$41$	478.000	1.82076	0.910379	−	0.413776i	$-0.135790\pi$
$41$	478.000	1.82076	0.910379	+	0.413776i	$0.135790\pi$
$42$	0	0
$43$	8.00000	0.0283718	0.0141859	−	0.999899i	$-0.495484\pi$
$43$	8.00000	0.0283718	0.0141859	+	0.999899i	$0.495484\pi$
$44$	77.0000	0.263822
$45$	0	0
$46$	22.0000	0.0705157
$47$	126.000	0.391042	0.195521	−	0.980699i	$-0.437360\pi$
$47$	126.000	0.391042	0.195521	+	0.980699i	$0.437360\pi$
$48$	0	0
$49$	−262.000	−0.763848
$50$	0	0
$51$	0	0
$52$	14.0000	0.0373356
$53$	−683.000	−1.77014	−0.885069	−	0.465461i	$-0.845889\pi$
$53$	−683.000	−1.77014	−0.885069	+	0.465461i	$0.845889\pi$
$54$	0	0
$55$	0	0
$56$	−135.000	−0.322145
$57$	0	0
$58$	165.000	0.373544
$59$	290.000	0.639912	0.319956	−	0.947432i	$-0.396332\pi$
$59$	290.000	0.639912	0.319956	+	0.947432i	$0.396332\pi$
$60$	0	0
$61$	257.000	0.539434	0.269717	−	0.962940i	$-0.413070\pi$
$61$	257.000	0.539434	0.269717	+	0.962940i	$0.413070\pi$
$62$	−83.0000	−0.170016
$63$	0	0
$64$	−167.000	−0.326172
$65$	0	0
$66$	0	0
$67$	−776.000	−1.41498	−0.707489	−	0.706725i	$-0.750172\pi$
$67$	−776.000	−1.41498	−0.707489	+	0.706725i	$0.750172\pi$
$68$	−147.000	−0.262152
$69$	0	0
$70$	0	0
$71$	313.000	0.523187	0.261593	−	0.965178i	$-0.415752\pi$
$71$	313.000	0.523187	0.261593	+	0.965178i	$0.415752\pi$
$72$	0	0
$73$	−902.000	−1.44618	−0.723090	−	0.690754i	$-0.757279\pi$
$73$	−902.000	−1.44618	−0.723090	+	0.690754i	$0.757279\pi$
$74$	−1.00000	−0.00157091
$75$	0	0
$76$	595.000	0.898042
$77$	−99.0000	−0.146521
$78$	0	0
$79$	830.000	1.18205	0.591027	−	0.806652i	$-0.298723\pi$
$79$	830.000	1.18205	0.591027	+	0.806652i	$0.298723\pi$
$80$	0	0
$81$	0	0
$82$	478.000	0.643735
$83$	842.000	1.11351	0.556756	−	0.830676i	$-0.312046\pi$
$83$	842.000	1.11351	0.556756	+	0.830676i	$0.312046\pi$
$84$	0	0
$85$	0	0
$86$	8.00000	0.0100310
$87$	0	0
$88$	165.000	0.199876
$89$	−25.0000	−0.0297752	−0.0148876	−	0.999889i	$-0.504739\pi$
$89$	−25.0000	−0.0297752	−0.0148876	+	0.999889i	$0.504739\pi$
$90$	0	0
$91$	−18.0000	−0.0207353
$92$	−154.000	−0.174517
$93$	0	0
$94$	126.000	0.138254
$95$	0	0
$96$	0	0
$97$	1784.00	1.86740	0.933700	−	0.358057i	$-0.116561\pi$
$97$	1784.00	1.86740	0.933700	+	0.358057i	$0.116561\pi$
$98$	−262.000	−0.270061
$99$	0	0
$100$	0	0
$101$	298.000	0.293585	0.146793	−	0.989167i	$-0.453105\pi$
$101$	298.000	0.293585	0.146793	+	0.989167i	$0.453105\pi$
$102$	0	0
$103$	−1832.00	−1.75255	−0.876273	−	0.481814i	$-0.839978\pi$
$103$	−1832.00	−1.75255	−0.876273	+	0.481814i	$0.839978\pi$
$104$	30.0000	0.0282860
$105$	0	0
$106$	−683.000	−0.625838
$107$	−1404.00	−1.26850	−0.634251	−	0.773127i	$-0.718692\pi$
$107$	−1404.00	−1.26850	−0.634251	+	0.773127i	$0.718692\pi$
$108$	0	0
$109$	1050.00	0.922677	0.461338	−	0.887224i	$-0.347369\pi$
$109$	1050.00	0.922677	0.461338	+	0.887224i	$0.347369\pi$
$110$	0	0
$111$	0	0
$112$	369.000	0.311314
$113$	−1668.00	−1.38860	−0.694302	−	0.719684i	$-0.744287\pi$
$113$	−1668.00	−1.38860	−0.694302	+	0.719684i	$0.744287\pi$
$114$	0	0
$115$	0	0
$116$	−1155.00	−0.924475
$117$	0	0
$118$	290.000	0.226243
$119$	189.000	0.145593
$120$	0	0
$121$	121.000	0.0909091
$122$	257.000	0.190719
$123$	0	0
$124$	581.000	0.420769
$125$	0	0
$126$	0	0
$127$	2384.00	1.66571	0.832857	−	0.553488i	$-0.186703\pi$
$127$	2384.00	1.66571	0.832857	+	0.553488i	$0.186703\pi$
$128$	−1455.00	−1.00473
$129$	0	0
$130$	0	0
$131$	−1167.00	−0.778330	−0.389165	−	0.921168i	$-0.627236\pi$
$131$	−1167.00	−0.778330	−0.389165	+	0.921168i	$0.627236\pi$
$132$	0	0
$133$	−765.000	−0.498751
$134$	−776.000	−0.500270
$135$	0	0
$136$	−315.000	−0.198610
$137$	−1164.00	−0.725892	−0.362946	−	0.931810i	$-0.618229\pi$
$137$	−1164.00	−0.725892	−0.362946	+	0.931810i	$0.618229\pi$
$138$	0	0
$139$	1260.00	0.768862	0.384431	−	0.923154i	$-0.374398\pi$
$139$	1260.00	0.768862	0.384431	+	0.923154i	$0.374398\pi$
$140$	0	0
$141$	0	0
$142$	313.000	0.184974
$143$	22.0000	0.0128653
$144$	0	0
$145$	0	0
$146$	−902.000	−0.511302
$147$	0	0
$148$	7.00000	0.00388781
$149$	55.0000	0.0302401	0.0151201	−	0.999886i	$-0.495187\pi$
$149$	55.0000	0.0302401	0.0151201	+	0.999886i	$0.495187\pi$
$150$	0	0
$151$	−598.000	−0.322282	−0.161141	−	0.986931i	$-0.551517\pi$
$151$	−598.000	−0.322282	−0.161141	+	0.986931i	$0.551517\pi$
$152$	1275.00	0.680369
$153$	0	0
$154$	−99.0000	−0.0518029
$155$	0	0
$156$	0	0
$157$	−1321.00	−0.671511	−0.335756	−	0.941949i	$-0.608992\pi$
$157$	−1321.00	−0.671511	−0.335756	+	0.941949i	$0.608992\pi$
$158$	830.000	0.417919
$159$	0	0
$160$	0	0
$161$	198.000	0.0969229
$162$	0	0
$163$	−577.000	−0.277265	−0.138632	−	0.990344i	$-0.544271\pi$
$163$	−577.000	−0.277265	−0.138632	+	0.990344i	$0.544271\pi$
$164$	−3346.00	−1.59316
$165$	0	0
$166$	842.000	0.393686
$167$	−1169.00	−0.541676	−0.270838	−	0.962625i	$-0.587301\pi$
$167$	−1169.00	−0.541676	−0.270838	+	0.962625i	$0.587301\pi$
$168$	0	0
$169$	−2193.00	−0.998179
$170$	0	0
$171$	0	0
$172$	−56.0000	−0.0248253
$173$	1542.00	0.677665	0.338833	−	0.940847i	$-0.389968\pi$
$173$	1542.00	0.677665	0.338833	+	0.940847i	$0.389968\pi$
$174$	0	0
$175$	0	0
$176$	−451.000	−0.193156
$177$	0	0
$178$	−25.0000	−0.0105271
$179$	560.000	0.233834	0.116917	−	0.993142i	$-0.462699\pi$
$179$	560.000	0.233834	0.116917	+	0.993142i	$0.462699\pi$
$180$	0	0
$181$	−3058.00	−1.25580	−0.627899	−	0.778295i	$-0.716085\pi$
$181$	−3058.00	−1.25580	−0.627899	+	0.778295i	$0.716085\pi$
$182$	−18.0000	−0.00733104
$183$	0	0
$184$	−330.000	−0.132217
$185$	0	0
$186$	0	0
$187$	−231.000	−0.0903337
$188$	−882.000	−0.342162
$189$	0	0
$190$	0	0
$191$	1828.00	0.692510	0.346255	−	0.938140i	$-0.387453\pi$
$191$	1828.00	0.692510	0.346255	+	0.938140i	$0.387453\pi$
$192$	0	0
$193$	−577.000	−0.215199	−0.107599	−	0.994194i	$-0.534316\pi$
$193$	−577.000	−0.215199	−0.107599	+	0.994194i	$0.534316\pi$
$194$	1784.00	0.660225
$195$	0	0
$196$	1834.00	0.668367
$197$	−2164.00	−0.782633	−0.391316	−	0.920256i	$-0.627980\pi$
$197$	−2164.00	−0.782633	−0.391316	+	0.920256i	$0.627980\pi$
$198$	0	0
$199$	−4425.00	−1.57628	−0.788141	−	0.615495i	$-0.788956\pi$
$199$	−4425.00	−1.57628	−0.788141	+	0.615495i	$0.788956\pi$
$200$	0	0
$201$	0	0
$202$	298.000	0.103798
$203$	1485.00	0.513431
$204$	0	0
$205$	0	0
$206$	−1832.00	−0.619619
$207$	0	0
$208$	−82.0000	−0.0273350
$209$	935.000	0.309451
$210$	0	0
$211$	−1793.00	−0.585001	−0.292500	−	0.956265i	$-0.594487\pi$
$211$	−1793.00	−0.585001	−0.292500	+	0.956265i	$0.594487\pi$
$212$	4781.00	1.54887
$213$	0	0
$214$	−1404.00	−0.448483
$215$	0	0
$216$	0	0
$217$	−747.000	−0.233685
$218$	1050.00	0.326215
$219$	0	0
$220$	0	0
$221$	−42.0000	−0.0127838
$222$	0	0
$223$	−1142.00	−0.342933	−0.171466	−	0.985190i	$-0.554850\pi$
$223$	−1142.00	−0.342933	−0.171466	+	0.985190i	$0.554850\pi$
$224$	1449.00	0.432212
$225$	0	0
$226$	−1668.00	−0.490946
$227$	4906.00	1.43446	0.717231	−	0.696836i	$-0.245409\pi$
$227$	4906.00	1.43446	0.717231	+	0.696836i	$0.245409\pi$
$228$	0	0
$229$	1130.00	0.326081	0.163040	−	0.986619i	$-0.447870\pi$
$229$	1130.00	0.326081	0.163040	+	0.986619i	$0.447870\pi$
$230$	0	0
$231$	0	0
$232$	−2475.00	−0.700395
$233$	−5403.00	−1.51915	−0.759576	−	0.650419i	$-0.774593\pi$
$233$	−5403.00	−1.51915	−0.759576	+	0.650419i	$0.774593\pi$
$234$	0	0
$235$	0	0
$236$	−2030.00	−0.559923
$237$	0	0
$238$	189.000	0.0514750
$239$	−5090.00	−1.37759	−0.688797	−	0.724955i	$-0.741861\pi$
$239$	−5090.00	−1.37759	−0.688797	+	0.724955i	$0.741861\pi$
$240$	0	0
$241$	−6578.00	−1.75820	−0.879100	−	0.476637i	$-0.841856\pi$
$241$	−6578.00	−1.75820	−0.879100	+	0.476637i	$0.841856\pi$
$242$	121.000	0.0321412
$243$	0	0
$244$	−1799.00	−0.472005
$245$	0	0
$246$	0	0
$247$	170.000	0.0437929
$248$	1245.00	0.318781
$249$	0	0
$250$	0	0
$251$	−3102.00	−0.780066	−0.390033	−	0.920801i	$-0.627536\pi$
$251$	−3102.00	−0.780066	−0.390033	+	0.920801i	$0.627536\pi$
$252$	0	0
$253$	−242.000	−0.0601360
$254$	2384.00	0.588919
$255$	0	0
$256$	−119.000	−0.0290527
$257$	4866.00	1.18106	0.590531	−	0.807015i	$-0.298918\pi$
$257$	4866.00	1.18106	0.590531	+	0.807015i	$0.298918\pi$
$258$	0	0
$259$	−9.00000	−0.00215920
$260$	0	0
$261$	0	0
$262$	−1167.00	−0.275181
$263$	−2163.00	−0.507134	−0.253567	−	0.967318i	$-0.581604\pi$
$263$	−2163.00	−0.507134	−0.253567	+	0.967318i	$0.581604\pi$
$264$	0	0
$265$	0	0
$266$	−765.000	−0.176335
$267$	0	0
$268$	5432.00	1.23811
$269$	−7020.00	−1.59114	−0.795571	−	0.605861i	$-0.792829\pi$
$269$	−7020.00	−1.59114	−0.795571	+	0.605861i	$0.792829\pi$
$270$	0	0
$271$	4812.00	1.07863	0.539314	−	0.842105i	$-0.318684\pi$
$271$	4812.00	1.07863	0.539314	+	0.842105i	$0.318684\pi$
$272$	861.000	0.191933
$273$	0	0
$274$	−1164.00	−0.256642
$275$	0	0
$276$	0	0
$277$	−5176.00	−1.12273	−0.561364	−	0.827569i	$-0.689723\pi$
$277$	−5176.00	−1.12273	−0.561364	+	0.827569i	$0.689723\pi$
$278$	1260.00	0.271834
$279$	0	0
$280$	0	0
$281$	−1242.00	−0.263671	−0.131835	−	0.991272i	$-0.542087\pi$
$281$	−1242.00	−0.263671	−0.131835	+	0.991272i	$0.542087\pi$
$282$	0	0
$283$	−7402.00	−1.55478	−0.777391	−	0.629018i	$-0.783457\pi$
$283$	−7402.00	−1.55478	−0.777391	+	0.629018i	$0.783457\pi$
$284$	−2191.00	−0.457788
$285$	0	0
$286$	22.0000	0.00454856
$287$	4302.00	0.884805
$288$	0	0
$289$	−4472.00	−0.910238
$290$	0	0
$291$	0	0
$292$	6314.00	1.26541
$293$	−3578.00	−0.713410	−0.356705	−	0.934217i	$-0.616100\pi$
$293$	−3578.00	−0.713410	−0.356705	+	0.934217i	$0.616100\pi$
$294$	0	0
$295$	0	0
$296$	15.0000	0.00294546
$297$	0	0
$298$	55.0000	0.0106915
$299$	−44.0000	−0.00851032
$300$	0	0
$301$	72.0000	0.0137874
$302$	−598.000	−0.113944
$303$	0	0
$304$	−3485.00	−0.657495
$305$	0	0
$306$	0	0
$307$	9094.00	1.69063	0.845313	−	0.534272i	$-0.179414\pi$
$307$	9094.00	1.69063	0.845313	+	0.534272i	$0.179414\pi$
$308$	693.000	0.128206
$309$	0	0
$310$	0	0
$311$	7443.00	1.35709	0.678543	−	0.734561i	$-0.262612\pi$
$311$	7443.00	1.35709	0.678543	+	0.734561i	$0.262612\pi$
$312$	0	0
$313$	−2822.00	−0.509613	−0.254807	−	0.966992i	$-0.582012\pi$
$313$	−2822.00	−0.509613	−0.254807	+	0.966992i	$0.582012\pi$
$314$	−1321.00	−0.237415
$315$	0	0
$316$	−5810.00	−1.03430
$317$	−1869.00	−0.331147	−0.165573	−	0.986197i	$-0.552947\pi$
$317$	−1869.00	−0.331147	−0.165573	+	0.986197i	$0.552947\pi$
$318$	0	0
$319$	−1815.00	−0.318560
$320$	0	0
$321$	0	0
$322$	198.000	0.0342674
$323$	−1785.00	−0.307492
$324$	0	0
$325$	0	0
$326$	−577.000	−0.0980278
$327$	0	0
$328$	−7170.00	−1.20700
$329$	1134.00	0.190029
$330$	0	0
$331$	−11408.0	−1.89438	−0.947191	−	0.320670i	$-0.896092\pi$
$331$	−11408.0	−1.89438	−0.947191	+	0.320670i	$0.896092\pi$
$332$	−5894.00	−0.974323
$333$	0	0
$334$	−1169.00	−0.191511
$335$	0	0
$336$	0	0
$337$	−10251.0	−1.65700	−0.828498	−	0.559992i	$-0.810804\pi$
$337$	−10251.0	−1.65700	−0.828498	+	0.559992i	$0.810804\pi$
$338$	−2193.00	−0.352910
$339$	0	0
$340$	0	0
$341$	913.000	0.144990
$342$	0	0
$343$	−5445.00	−0.857150
$344$	−120.000	−0.0188080
$345$	0	0
$346$	1542.00	0.239591
$347$	−11494.0	−1.77819	−0.889093	−	0.457727i	$-0.848664\pi$
$347$	−11494.0	−1.77819	−0.889093	+	0.457727i	$0.848664\pi$
$348$	0	0
$349$	5690.00	0.872718	0.436359	−	0.899773i	$-0.356268\pi$
$349$	5690.00	0.872718	0.436359	+	0.899773i	$0.356268\pi$
$350$	0	0
$351$	0	0
$352$	−1771.00	−0.268167
$353$	−4398.00	−0.663122	−0.331561	−	0.943434i	$-0.607575\pi$
$353$	−4398.00	−0.663122	−0.331561	+	0.943434i	$0.607575\pi$
$354$	0	0
$355$	0	0
$356$	175.000	0.0260533
$357$	0	0
$358$	560.000	0.0826730
$359$	8840.00	1.29960	0.649801	−	0.760104i	$-0.274852\pi$
$359$	8840.00	1.29960	0.649801	+	0.760104i	$0.274852\pi$
$360$	0	0
$361$	366.000	0.0533605
$362$	−3058.00	−0.443991
$363$	0	0
$364$	126.000	0.0181434
$365$	0	0
$366$	0	0
$367$	5564.00	0.791385	0.395693	−	0.918383i	$-0.370505\pi$
$367$	5564.00	0.791385	0.395693	+	0.918383i	$0.370505\pi$
$368$	902.000	0.127772
$369$	0	0
$370$	0	0
$371$	−6147.00	−0.860206
$372$	0	0
$373$	−9222.00	−1.28015	−0.640076	−	0.768311i	$-0.721097\pi$
$373$	−9222.00	−1.28015	−0.640076	+	0.768311i	$0.721097\pi$
$374$	−231.000	−0.0319378
$375$	0	0
$376$	−1890.00	−0.259227
$377$	−330.000	−0.0450819
$378$	0	0
$379$	4670.00	0.632933	0.316467	−	0.948604i	$-0.397503\pi$
$379$	4670.00	0.632933	0.316467	+	0.948604i	$0.397503\pi$
$380$	0	0
$381$	0	0
$382$	1828.00	0.244839
$383$	−378.000	−0.0504305	−0.0252153	−	0.999682i	$-0.508027\pi$
$383$	−378.000	−0.0504305	−0.0252153	+	0.999682i	$0.508027\pi$
$384$	0	0
$385$	0	0
$386$	−577.000	−0.0760843
$387$	0	0
$388$	−12488.0	−1.63397
$389$	−10110.0	−1.31773	−0.658865	−	0.752261i	$-0.728963\pi$
$389$	−10110.0	−1.31773	−0.658865	+	0.752261i	$0.728963\pi$
$390$	0	0
$391$	462.000	0.0597554
$392$	3930.00	0.506365
$393$	0	0
$394$	−2164.00	−0.276702
$395$	0	0
$396$	0	0
$397$	−12186.0	−1.54055	−0.770274	−	0.637713i	$-0.779881\pi$
$397$	−12186.0	−1.54055	−0.770274	+	0.637713i	$0.779881\pi$
$398$	−4425.00	−0.557300
$399$	0	0
$400$	0	0
$401$	4573.00	0.569488	0.284744	−	0.958604i	$-0.408091\pi$
$401$	4573.00	0.569488	0.284744	+	0.958604i	$0.408091\pi$
$402$	0	0
$403$	166.000	0.0205187
$404$	−2086.00	−0.256887
$405$	0	0
$406$	1485.00	0.181525
$407$	11.0000	0.00133968
$408$	0	0
$409$	2280.00	0.275645	0.137822	−	0.990457i	$-0.455990\pi$
$409$	2280.00	0.275645	0.137822	+	0.990457i	$0.455990\pi$
$410$	0	0
$411$	0	0
$412$	12824.0	1.53348
$413$	2610.00	0.310968
$414$	0	0
$415$	0	0
$416$	−322.000	−0.0379504
$417$	0	0
$418$	935.000	0.109408
$419$	3700.00	0.431401	0.215700	−	0.976460i	$-0.430797\pi$
$419$	3700.00	0.431401	0.215700	+	0.976460i	$0.430797\pi$
$420$	0	0
$421$	3612.00	0.418143	0.209071	−	0.977900i	$-0.432956\pi$
$421$	3612.00	0.418143	0.209071	+	0.977900i	$0.432956\pi$
$422$	−1793.00	−0.206829
$423$	0	0
$424$	10245.0	1.17345
$425$	0	0
$426$	0	0
$427$	2313.00	0.262140
$428$	9828.00	1.10994
$429$	0	0
$430$	0	0
$431$	−792.000	−0.0885135	−0.0442567	−	0.999020i	$-0.514092\pi$
$431$	−792.000	−0.0885135	−0.0442567	+	0.999020i	$0.514092\pi$
$432$	0	0
$433$	4888.00	0.542500	0.271250	−	0.962509i	$-0.412563\pi$
$433$	4888.00	0.542500	0.271250	+	0.962509i	$0.412563\pi$
$434$	−747.000	−0.0826202
$435$	0	0
$436$	−7350.00	−0.807342
$437$	−1870.00	−0.204701
$438$	0	0
$439$	15100.0	1.64165	0.820824	−	0.571181i	$-0.193515\pi$
$439$	15100.0	1.64165	0.820824	+	0.571181i	$0.193515\pi$
$440$	0	0
$441$	0	0
$442$	−42.0000	−0.00451977
$443$	−11188.0	−1.19991	−0.599953	−	0.800036i	$-0.704814\pi$
$443$	−11188.0	−1.19991	−0.599953	+	0.800036i	$0.704814\pi$
$444$	0	0
$445$	0	0
$446$	−1142.00	−0.121245
$447$	0	0
$448$	−1503.00	−0.158505
$449$	12070.0	1.26864	0.634319	−	0.773071i	$-0.281281\pi$
$449$	12070.0	1.26864	0.634319	+	0.773071i	$0.281281\pi$
$450$	0	0
$451$	−5258.00	−0.548979
$452$	11676.0	1.21503
$453$	0	0
$454$	4906.00	0.507159
$455$	0	0
$456$	0	0
$457$	12449.0	1.27427	0.637133	−	0.770754i	$-0.280120\pi$
$457$	12449.0	1.27427	0.637133	+	0.770754i	$0.280120\pi$
$458$	1130.00	0.115287
$459$	0	0
$460$	0	0
$461$	−2957.00	−0.298745	−0.149372	−	0.988781i	$-0.547725\pi$
$461$	−2957.00	−0.298745	−0.149372	+	0.988781i	$0.547725\pi$
$462$	0	0
$463$	9738.00	0.977458	0.488729	−	0.872436i	$-0.337461\pi$
$463$	9738.00	0.977458	0.488729	+	0.872436i	$0.337461\pi$
$464$	6765.00	0.676848
$465$	0	0
$466$	−5403.00	−0.537101
$467$	−13779.0	−1.36534	−0.682672	−	0.730725i	$-0.739182\pi$
$467$	−13779.0	−1.36534	−0.682672	+	0.730725i	$0.739182\pi$
$468$	0	0
$469$	−6984.00	−0.687614
$470$	0	0
$471$	0	0
$472$	−4350.00	−0.424205
$473$	−88.0000	−0.00855443
$474$	0	0
$475$	0	0
$476$	−1323.00	−0.127394
$477$	0	0
$478$	−5090.00	−0.487053
$479$	−16320.0	−1.55674	−0.778371	−	0.627804i	$-0.783954\pi$
$479$	−16320.0	−1.55674	−0.778371	+	0.627804i	$0.783954\pi$
$480$	0	0
$481$	2.00000	0.000189589 0
$482$	−6578.00	−0.621618
$483$	0	0
$484$	−847.000	−0.0795455
$485$	0	0
$486$	0	0
$487$	2744.00	0.255323	0.127662	−	0.991818i	$-0.459253\pi$
$487$	2744.00	0.255323	0.127662	+	0.991818i	$0.459253\pi$
$488$	−3855.00	−0.357598
$489$	0	0
$490$	0	0
$491$	853.000	0.0784019	0.0392010	−	0.999231i	$-0.487519\pi$
$491$	853.000	0.0784019	0.0392010	+	0.999231i	$0.487519\pi$
$492$	0	0
$493$	3465.00	0.316543
$494$	170.000	0.0154831
$495$	0	0
$496$	−3403.00	−0.308063
$497$	2817.00	0.254245
$498$	0	0
$499$	−6840.00	−0.613628	−0.306814	−	0.951769i	$-0.599263\pi$
$499$	−6840.00	−0.613628	−0.306814	+	0.951769i	$0.599263\pi$
$500$	0	0
$501$	0	0
$502$	−3102.00	−0.275795
$503$	−5128.00	−0.454565	−0.227283	−	0.973829i	$-0.572984\pi$
$503$	−5128.00	−0.454565	−0.227283	+	0.973829i	$0.572984\pi$
$504$	0	0
$505$	0	0
$506$	−242.000	−0.0212613
$507$	0	0
$508$	−16688.0	−1.45750
$509$	18160.0	1.58139	0.790695	−	0.612210i	$-0.209719\pi$
$509$	18160.0	1.58139	0.790695	+	0.612210i	$0.209719\pi$
$510$	0	0
$511$	−8118.00	−0.702777
$512$	11521.0	0.994455
$513$	0	0
$514$	4866.00	0.417568
$515$	0	0
$516$	0	0
$517$	−1386.00	−0.117904
$518$	−9.00000	−0.000763392 0
$519$	0	0
$520$	0	0
$521$	−6462.00	−0.543388	−0.271694	−	0.962384i	$-0.587584\pi$
$521$	−6462.00	−0.543388	−0.271694	+	0.962384i	$0.587584\pi$
$522$	0	0
$523$	−5932.00	−0.495962	−0.247981	−	0.968765i	$-0.579767\pi$
$523$	−5932.00	−0.495962	−0.247981	+	0.968765i	$0.579767\pi$
$524$	8169.00	0.681039
$525$	0	0
$526$	−2163.00	−0.179299
$527$	−1743.00	−0.144073
$528$	0	0
$529$	−11683.0	−0.960220
$530$	0	0
$531$	0	0
$532$	5355.00	0.436407
$533$	−956.000	−0.0776904
$534$	0	0
$535$	0	0
$536$	11640.0	0.938006
$537$	0	0
$538$	−7020.00	−0.562553
$539$	2882.00	0.230309
$540$	0	0
$541$	12647.0	1.00506	0.502530	−	0.864560i	$-0.332403\pi$
$541$	12647.0	1.00506	0.502530	+	0.864560i	$0.332403\pi$
$542$	4812.00	0.381353
$543$	0	0
$544$	3381.00	0.266469
$545$	0	0
$546$	0	0
$547$	13524.0	1.05712	0.528560	−	0.848896i	$-0.322732\pi$
$547$	13524.0	1.05712	0.528560	+	0.848896i	$0.322732\pi$
$548$	8148.00	0.635156
$549$	0	0
$550$	0	0
$551$	−14025.0	−1.08436
$552$	0	0
$553$	7470.00	0.574424
$554$	−5176.00	−0.396944
$555$	0	0
$556$	−8820.00	−0.672754
$557$	25066.0	1.90679	0.953394	−	0.301729i	$-0.0975639\pi$
$557$	25066.0	1.90679	0.953394	+	0.301729i	$0.0975639\pi$
$558$	0	0
$559$	−16.0000	−0.00121060
$560$	0	0
$561$	0	0
$562$	−1242.00	−0.0932217
$563$	12102.0	0.905930	0.452965	−	0.891528i	$-0.350366\pi$
$563$	12102.0	0.905930	0.452965	+	0.891528i	$0.350366\pi$
$564$	0	0
$565$	0	0
$566$	−7402.00	−0.549698
$567$	0	0
$568$	−4695.00	−0.346827
$569$	20860.0	1.53690	0.768451	−	0.639909i	$-0.221028\pi$
$569$	20860.0	1.53690	0.768451	+	0.639909i	$0.221028\pi$
$570$	0	0
$571$	20637.0	1.51249	0.756245	−	0.654289i	$-0.227032\pi$
$571$	20637.0	1.51249	0.756245	+	0.654289i	$0.227032\pi$
$572$	−154.000	−0.0112571
$573$	0	0
$574$	4302.00	0.312826
$575$	0	0
$576$	0	0
$577$	−3266.00	−0.235642	−0.117821	−	0.993035i	$-0.537591\pi$
$577$	−3266.00	−0.235642	−0.117821	+	0.993035i	$0.537591\pi$
$578$	−4472.00	−0.321818
$579$	0	0
$580$	0	0
$581$	7578.00	0.541116
$582$	0	0
$583$	7513.00	0.533716
$584$	13530.0	0.958691
$585$	0	0
$586$	−3578.00	−0.252228
$587$	3351.00	0.235623	0.117811	−	0.993036i	$-0.462412\pi$
$587$	3351.00	0.235623	0.117811	+	0.993036i	$0.462412\pi$
$588$	0	0
$589$	7055.00	0.493542
$590$	0	0
$591$	0	0
$592$	−41.0000	−0.00284644
$593$	−20258.0	−1.40286	−0.701430	−	0.712738i	$-0.747455\pi$
$593$	−20258.0	−1.40286	−0.701430	+	0.712738i	$0.747455\pi$
$594$	0	0
$595$	0	0
$596$	−385.000	−0.0264601
$597$	0	0
$598$	−44.0000	−0.00300885
$599$	−20445.0	−1.39459	−0.697296	−	0.716784i	$-0.745613\pi$
$599$	−20445.0	−1.39459	−0.697296	+	0.716784i	$0.745613\pi$
$600$	0	0
$601$	−1498.00	−0.101672	−0.0508359	−	0.998707i	$-0.516189\pi$
$601$	−1498.00	−0.101672	−0.0508359	+	0.998707i	$0.516189\pi$
$602$	72.0000	0.00487459
$603$	0	0
$604$	4186.00	0.281997
$605$	0	0
$606$	0	0
$607$	−23461.0	−1.56879	−0.784393	−	0.620265i	$-0.787025\pi$
$607$	−23461.0	−1.56879	−0.784393	+	0.620265i	$0.787025\pi$
$608$	−13685.0	−0.912829
$609$	0	0
$610$	0	0
$611$	−252.000	−0.0166855
$612$	0	0
$613$	1718.00	0.113196	0.0565982	−	0.998397i	$-0.481975\pi$
$613$	1718.00	0.113196	0.0565982	+	0.998397i	$0.481975\pi$
$614$	9094.00	0.597726
$615$	0	0
$616$	1485.00	0.0971304
$617$	26276.0	1.71448	0.857238	−	0.514920i	$-0.172178\pi$
$617$	26276.0	1.71448	0.857238	+	0.514920i	$0.172178\pi$
$618$	0	0
$619$	−15560.0	−1.01035	−0.505177	−	0.863016i	$-0.668573\pi$
$619$	−15560.0	−1.01035	−0.505177	+	0.863016i	$0.668573\pi$
$620$	0	0
$621$	0	0
$622$	7443.00	0.479802
$623$	−225.000	−0.0144694
$624$	0	0
$625$	0	0
$626$	−2822.00	−0.180175
$627$	0	0
$628$	9247.00	0.587572
$629$	−21.0000	−0.00133120
$630$	0	0
$631$	−28433.0	−1.79382	−0.896910	−	0.442214i	$-0.854193\pi$
$631$	−28433.0	−1.79382	−0.896910	+	0.442214i	$0.854193\pi$
$632$	−12450.0	−0.783599
$633$	0	0
$634$	−1869.00	−0.117078
$635$	0	0
$636$	0	0
$637$	524.000	0.0325928
$638$	−1815.00	−0.112628
$639$	0	0
$640$	0	0
$641$	−9847.00	−0.606760	−0.303380	−	0.952870i	$-0.598115\pi$
$641$	−9847.00	−0.606760	−0.303380	+	0.952870i	$0.598115\pi$
$642$	0	0
$643$	−2177.00	−0.133519	−0.0667593	−	0.997769i	$-0.521266\pi$
$643$	−2177.00	−0.133519	−0.0667593	+	0.997769i	$0.521266\pi$
$644$	−1386.00	−0.0848075
$645$	0	0
$646$	−1785.00	−0.108715
$647$	4466.00	0.271370	0.135685	−	0.990752i	$-0.456676\pi$
$647$	4466.00	0.271370	0.135685	+	0.990752i	$0.456676\pi$
$648$	0	0
$649$	−3190.00	−0.192941
$650$	0	0
$651$	0	0
$652$	4039.00	0.242607
$653$	11507.0	0.689592	0.344796	−	0.938678i	$-0.387948\pi$
$653$	11507.0	0.689592	0.344796	+	0.938678i	$0.387948\pi$
$654$	0	0
$655$	0	0
$656$	19598.0	1.16642
$657$	0	0
$658$	1134.00	0.0671853
$659$	−12825.0	−0.758105	−0.379052	−	0.925375i	$-0.623750\pi$
$659$	−12825.0	−0.758105	−0.379052	+	0.925375i	$0.623750\pi$
$660$	0	0
$661$	−8818.00	−0.518881	−0.259441	−	0.965759i	$-0.583538\pi$
$661$	−8818.00	−0.518881	−0.259441	+	0.965759i	$0.583538\pi$
$662$	−11408.0	−0.669765
$663$	0	0
$664$	−12630.0	−0.738161
$665$	0	0
$666$	0	0
$667$	3630.00	0.210726
$668$	8183.00	0.473967
$669$	0	0
$670$	0	0
$671$	−2827.00	−0.162645
$672$	0	0
$673$	9263.00	0.530553	0.265277	−	0.964172i	$-0.414537\pi$
$673$	9263.00	0.530553	0.265277	+	0.964172i	$0.414537\pi$
$674$	−10251.0	−0.585836
$675$	0	0
$676$	15351.0	0.873407
$677$	−1184.00	−0.0672154	−0.0336077	−	0.999435i	$-0.510700\pi$
$677$	−1184.00	−0.0672154	−0.0336077	+	0.999435i	$0.510700\pi$
$678$	0	0
$679$	16056.0	0.907471
$680$	0	0
$681$	0	0
$682$	913.000	0.0512618
$683$	−1693.00	−0.0948475	−0.0474238	−	0.998875i	$-0.515101\pi$
$683$	−1693.00	−0.0948475	−0.0474238	+	0.998875i	$0.515101\pi$
$684$	0	0
$685$	0	0
$686$	−5445.00	−0.303048
$687$	0	0
$688$	328.000	0.0181757
$689$	1366.00	0.0755304
$690$	0	0
$691$	13022.0	0.716903	0.358452	−	0.933548i	$-0.383305\pi$
$691$	13022.0	0.716903	0.358452	+	0.933548i	$0.383305\pi$
$692$	−10794.0	−0.592957
$693$	0	0
$694$	−11494.0	−0.628683
$695$	0	0
$696$	0	0
$697$	10038.0	0.545504
$698$	5690.00	0.308553
$699$	0	0
$700$	0	0
$701$	−1177.00	−0.0634161	−0.0317080	−	0.999497i	$-0.510095\pi$
$701$	−1177.00	−0.0634161	−0.0317080	+	0.999497i	$0.510095\pi$
$702$	0	0
$703$	85.0000	0.00456022
$704$	1837.00	0.0983445
$705$	0	0
$706$	−4398.00	−0.234449
$707$	2682.00	0.142669
$708$	0	0
$709$	−24130.0	−1.27817	−0.639084	−	0.769137i	$-0.720686\pi$
$709$	−24130.0	−1.27817	−0.639084	+	0.769137i	$0.720686\pi$
$710$	0	0
$711$	0	0
$712$	375.000	0.0197384
$713$	−1826.00	−0.0959106
$714$	0	0
$715$	0	0
$716$	−3920.00	−0.204605
$717$	0	0
$718$	8840.00	0.459479
$719$	−13785.0	−0.715012	−0.357506	−	0.933911i	$-0.616373\pi$
$719$	−13785.0	−0.715012	−0.357506	+	0.933911i	$0.616373\pi$
$720$	0	0
$721$	−16488.0	−0.851658
$722$	366.000	0.0188658
$723$	0	0
$724$	21406.0	1.09882
$725$	0	0
$726$	0	0
$727$	17654.0	0.900620	0.450310	−	0.892872i	$-0.351314\pi$
$727$	17654.0	0.900620	0.450310	+	0.892872i	$0.351314\pi$
$728$	270.000	0.0137457
$729$	0	0
$730$	0	0
$731$	168.000	0.00850028
$732$	0	0
$733$	−14412.0	−0.726220	−0.363110	−	0.931746i	$-0.618285\pi$
$733$	−14412.0	−0.726220	−0.363110	+	0.931746i	$0.618285\pi$
$734$	5564.00	0.279797
$735$	0	0
$736$	3542.00	0.177391
$737$	8536.00	0.426632
$738$	0	0
$739$	16480.0	0.820334	0.410167	−	0.912011i	$-0.365470\pi$
$739$	16480.0	0.820334	0.410167	+	0.912011i	$0.365470\pi$
$740$	0	0
$741$	0	0
$742$	−6147.00	−0.304129
$743$	30127.0	1.48755	0.743777	−	0.668428i	$-0.233033\pi$
$743$	30127.0	1.48755	0.743777	+	0.668428i	$0.233033\pi$
$744$	0	0
$745$	0	0
$746$	−9222.00	−0.452602
$747$	0	0
$748$	1617.00	0.0790419
$749$	−12636.0	−0.616434
$750$	0	0
$751$	36577.0	1.77725	0.888624	−	0.458636i	$-0.151662\pi$
$751$	36577.0	1.77725	0.888624	+	0.458636i	$0.151662\pi$
$752$	5166.00	0.250511
$753$	0	0
$754$	−330.000	−0.0159388
$755$	0	0
$756$	0	0
$757$	−19386.0	−0.930774	−0.465387	−	0.885107i	$-0.654085\pi$
$757$	−19386.0	−0.930774	−0.465387	+	0.885107i	$0.654085\pi$
$758$	4670.00	0.223776
$759$	0	0
$760$	0	0
$761$	18218.0	0.867808	0.433904	−	0.900959i	$-0.357136\pi$
$761$	18218.0	0.867808	0.433904	+	0.900959i	$0.357136\pi$
$762$	0	0
$763$	9450.00	0.448379
$764$	−12796.0	−0.605946
$765$	0	0
$766$	−378.000	−0.0178299
$767$	−580.000	−0.0273045
$768$	0	0
$769$	9650.00	0.452520	0.226260	−	0.974067i	$-0.427350\pi$
$769$	9650.00	0.452520	0.226260	+	0.974067i	$0.427350\pi$
$770$	0	0
$771$	0	0
$772$	4039.00	0.188299
$773$	12617.0	0.587066	0.293533	−	0.955949i	$-0.405169\pi$
$773$	12617.0	0.587066	0.293533	+	0.955949i	$0.405169\pi$
$774$	0	0
$775$	0	0
$776$	−26760.0	−1.23792
$777$	0	0
$778$	−10110.0	−0.465888
$779$	−40630.0	−1.86870
$780$	0	0
$781$	−3443.00	−0.157747
$782$	462.000	0.0211267
$783$	0	0
$784$	−10742.0	−0.489340
$785$	0	0
$786$	0	0
$787$	22364.0	1.01295	0.506474	−	0.862255i	$-0.330949\pi$
$787$	22364.0	1.01295	0.506474	+	0.862255i	$0.330949\pi$
$788$	15148.0	0.684803
$789$	0	0
$790$	0	0
$791$	−15012.0	−0.674798
$792$	0	0
$793$	−514.000	−0.0230172
$794$	−12186.0	−0.544666
$795$	0	0
$796$	30975.0	1.37925
$797$	−17594.0	−0.781947	−0.390973	−	0.920402i	$-0.627862\pi$
$797$	−17594.0	−0.781947	−0.390973	+	0.920402i	$0.627862\pi$
$798$	0	0
$799$	2646.00	0.117157
$800$	0	0
$801$	0	0
$802$	4573.00	0.201344
$803$	9922.00	0.436040
$804$	0	0
$805$	0	0
$806$	166.000	0.00725447
$807$	0	0
$808$	−4470.00	−0.194621
$809$	45030.0	1.95695	0.978474	−	0.206371i	$-0.0661655\pi$
$809$	45030.0	1.95695	0.978474	+	0.206371i	$0.0661655\pi$
$810$	0	0
$811$	−2943.00	−0.127426	−0.0637131	−	0.997968i	$-0.520294\pi$
$811$	−2943.00	−0.127426	−0.0637131	+	0.997968i	$0.520294\pi$
$812$	−10395.0	−0.449252
$813$	0	0
$814$	11.0000	0.000473648 0
$815$	0	0
$816$	0	0
$817$	−680.000	−0.0291190
$818$	2280.00	0.0974552
$819$	0	0
$820$	0	0
$821$	4038.00	0.171653	0.0858265	−	0.996310i	$-0.472647\pi$
$821$	4038.00	0.171653	0.0858265	+	0.996310i	$0.472647\pi$
$822$	0	0
$823$	5688.00	0.240913	0.120456	−	0.992719i	$-0.461564\pi$
$823$	5688.00	0.240913	0.120456	+	0.992719i	$0.461564\pi$
$824$	27480.0	1.16179
$825$	0	0
$826$	2610.00	0.109944
$827$	−16824.0	−0.707410	−0.353705	−	0.935357i	$-0.615078\pi$
$827$	−16824.0	−0.707410	−0.353705	+	0.935357i	$0.615078\pi$
$828$	0	0
$829$	−22940.0	−0.961085	−0.480542	−	0.876972i	$-0.659560\pi$
$829$	−22940.0	−0.961085	−0.480542	+	0.876972i	$0.659560\pi$
$830$	0	0
$831$	0	0
$832$	334.000	0.0139175
$833$	−5502.00	−0.228851
$834$	0	0
$835$	0	0
$836$	−6545.00	−0.270770
$837$	0	0
$838$	3700.00	0.152523
$839$	−12040.0	−0.495431	−0.247716	−	0.968833i	$-0.579680\pi$
$839$	−12040.0	−0.495431	−0.247716	+	0.968833i	$0.579680\pi$
$840$	0	0
$841$	2836.00	0.116282
$842$	3612.00	0.147836
$843$	0	0
$844$	12551.0	0.511876
$845$	0	0
$846$	0	0
$847$	1089.00	0.0441777
$848$	−28003.0	−1.13399
$849$	0	0
$850$	0	0
$851$	−22.0000	−0.000886193 0
$852$	0	0
$853$	14338.0	0.575526	0.287763	−	0.957702i	$-0.407088\pi$
$853$	14338.0	0.575526	0.287763	+	0.957702i	$0.407088\pi$
$854$	2313.00	0.0926806
$855$	0	0
$856$	21060.0	0.840907
$857$	−17619.0	−0.702280	−0.351140	−	0.936323i	$-0.614206\pi$
$857$	−17619.0	−0.702280	−0.351140	+	0.936323i	$0.614206\pi$
$858$	0	0
$859$	−25550.0	−1.01485	−0.507424	−	0.861696i	$-0.669402\pi$
$859$	−25550.0	−1.01485	−0.507424	+	0.861696i	$0.669402\pi$
$860$	0	0
$861$	0	0
$862$	−792.000	−0.0312942
$863$	36922.0	1.45636	0.728180	−	0.685385i	$-0.240366\pi$
$863$	36922.0	1.45636	0.728180	+	0.685385i	$0.240366\pi$
$864$	0	0
$865$	0	0
$866$	4888.00	0.191803
$867$	0	0
$868$	5229.00	0.204474
$869$	−9130.00	−0.356403
$870$	0	0
$871$	1552.00	0.0603760
$872$	−15750.0	−0.611654
$873$	0	0
$874$	−1870.00	−0.0723726
$875$	0	0
$876$	0	0
$877$	−11486.0	−0.442252	−0.221126	−	0.975245i	$-0.570973\pi$
$877$	−11486.0	−0.442252	−0.221126	+	0.975245i	$0.570973\pi$
$878$	15100.0	0.580410
$879$	0	0
$880$	0	0
$881$	2958.00	0.113119	0.0565593	−	0.998399i	$-0.481987\pi$
$881$	2958.00	0.113119	0.0565593	+	0.998399i	$0.481987\pi$
$882$	0	0
$883$	5173.00	0.197152	0.0985761	−	0.995130i	$-0.468571\pi$
$883$	5173.00	0.197152	0.0985761	+	0.995130i	$0.468571\pi$
$884$	294.000	0.0111858
$885$	0	0
$886$	−11188.0	−0.424230
$887$	−33424.0	−1.26524	−0.632620	−	0.774462i	$-0.718021\pi$
$887$	−33424.0	−1.26524	−0.632620	+	0.774462i	$0.718021\pi$
$888$	0	0
$889$	21456.0	0.809461
$890$	0	0
$891$	0	0
$892$	7994.00	0.300066
$893$	−10710.0	−0.401340
$894$	0	0
$895$	0	0
$896$	−13095.0	−0.488251
$897$	0	0
$898$	12070.0	0.448531
$899$	−13695.0	−0.508069
$900$	0	0
$901$	−14343.0	−0.530338
$902$	−5258.00	−0.194093
$903$	0	0
$904$	25020.0	0.920523
$905$	0	0
$906$	0	0
$907$	−27101.0	−0.992143	−0.496072	−	0.868282i	$-0.665225\pi$
$907$	−27101.0	−0.992143	−0.496072	+	0.868282i	$0.665225\pi$
$908$	−34342.0	−1.25515
$909$	0	0
$910$	0	0
$911$	23893.0	0.868947	0.434473	−	0.900685i	$-0.356935\pi$
$911$	23893.0	0.868947	0.434473	+	0.900685i	$0.356935\pi$
$912$	0	0
$913$	−9262.00	−0.335737
$914$	12449.0	0.450521
$915$	0	0
$916$	−7910.00	−0.285321
$917$	−10503.0	−0.378233
$918$	0	0
$919$	−34980.0	−1.25559	−0.627793	−	0.778380i	$-0.716042\pi$
$919$	−34980.0	−1.25559	−0.627793	+	0.778380i	$0.716042\pi$
$920$	0	0
$921$	0	0
$922$	−2957.00	−0.105622
$923$	−626.000	−0.0223240
$924$	0	0
$925$	0	0
$926$	9738.00	0.345584
$927$	0	0
$928$	26565.0	0.939697
$929$	−21595.0	−0.762658	−0.381329	−	0.924439i	$-0.624533\pi$
$929$	−21595.0	−0.762658	−0.381329	+	0.924439i	$0.624533\pi$
$930$	0	0
$931$	22270.0	0.783963
$932$	37821.0	1.32926
$933$	0	0
$934$	−13779.0	−0.482722
$935$	0	0
$936$	0	0
$937$	−31446.0	−1.09637	−0.548184	−	0.836358i	$-0.684680\pi$
$937$	−31446.0	−1.09637	−0.548184	+	0.836358i	$0.684680\pi$
$938$	−6984.00	−0.243108
$939$	0	0
$940$	0	0
$941$	24353.0	0.843661	0.421831	−	0.906675i	$-0.361388\pi$
$941$	24353.0	0.843661	0.421831	+	0.906675i	$0.361388\pi$
$942$	0	0
$943$	10516.0	0.363147
$944$	11890.0	0.409943
$945$	0	0
$946$	−88.0000	−0.00302445
$947$	−22089.0	−0.757968	−0.378984	−	0.925403i	$-0.623727\pi$
$947$	−22089.0	−0.757968	−0.378984	+	0.925403i	$0.623727\pi$
$948$	0	0
$949$	1804.00	0.0617074
$950$	0	0
$951$	0	0
$952$	−2835.00	−0.0965156
$953$	−37893.0	−1.28801	−0.644006	−	0.765021i	$-0.722729\pi$
$953$	−37893.0	−1.28801	−0.644006	+	0.765021i	$0.722729\pi$
$954$	0	0
$955$	0	0
$956$	35630.0	1.20539
$957$	0	0
$958$	−16320.0	−0.550392
$959$	−10476.0	−0.352750
$960$	0	0
$961$	−22902.0	−0.768756
$962$	2.00000	6.70297e−5 0
$963$	0	0
$964$	46046.0	1.53843
$965$	0	0
$966$	0	0
$967$	−40601.0	−1.35020	−0.675098	−	0.737728i	$-0.735899\pi$
$967$	−40601.0	−1.35020	−0.675098	+	0.737728i	$0.735899\pi$
$968$	−1815.00	−0.0602648
$969$	0	0
$970$	0	0
$971$	1188.00	0.0392634	0.0196317	−	0.999807i	$-0.493751\pi$
$971$	1188.00	0.0392634	0.0196317	+	0.999807i	$0.493751\pi$
$972$	0	0
$973$	11340.0	0.373632
$974$	2744.00	0.0902705
$975$	0	0
$976$	10537.0	0.345575
$977$	−17024.0	−0.557468	−0.278734	−	0.960368i	$-0.589915\pi$
$977$	−17024.0	−0.557468	−0.278734	+	0.960368i	$0.589915\pi$
$978$	0	0
$979$	275.000	0.00897757
$980$	0	0
$981$	0	0
$982$	853.000	0.0277193
$983$	43262.0	1.40371	0.701853	−	0.712322i	$-0.252356\pi$
$983$	43262.0	1.40371	0.701853	+	0.712322i	$0.252356\pi$
$984$	0	0
$985$	0	0
$986$	3465.00	0.111915
$987$	0	0
$988$	−1190.00	−0.0383188
$989$	176.000	0.00565872
$990$	0	0
$991$	−18328.0	−0.587496	−0.293748	−	0.955883i	$-0.594903\pi$
$991$	−18328.0	−0.587496	−0.293748	+	0.955883i	$0.594903\pi$
$992$	−13363.0	−0.427697
$993$	0	0
$994$	2817.00	0.0898891
$995$	0	0
$996$	0	0
$997$	−34196.0	−1.08626	−0.543128	−	0.839650i	$-0.682760\pi$
$997$	−34196.0	−1.08626	−0.543128	+	0.839650i	$0.682760\pi$
$998$	−6840.00	−0.216950
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.h.1.1		1
3.2	odd	2		275.4.a.a.1.1		1
5.4	even	2		495.4.a.a.1.1		1
15.2	even	4		275.4.b.a.199.1		2
15.8	even	4		275.4.b.a.199.2		2
15.14	odd	2		55.4.a.a.1.1	✓	1
60.59	even	2		880.4.a.j.1.1		1
165.164	even	2		605.4.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.4.a.a.1.1	✓	1	15.14	odd	2
275.4.a.a.1.1		1	3.2	odd	2
275.4.b.a.199.1		2	15.2	even	4
275.4.b.a.199.2		2	15.8	even	4
495.4.a.a.1.1		1	5.4	even	2
605.4.a.b.1.1		1	165.164	even	2
880.4.a.j.1.1		1	60.59	even	2
2475.4.a.h.1.1		1	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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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