LMFDB - Embedded newform 2475.4.a.a.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 825)
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)

$f(q)$

$=$

$q-5.00000 q^{2} +17.0000 q^{4} +3.00000 q^{7} -45.0000 q^{8} +O(q^{10})$

$q-5.00000 q^{2} +17.0000 q^{4} +3.00000 q^{7} -45.0000 q^{8} +11.0000 q^{11} +32.0000 q^{13} -15.0000 q^{14} +89.0000 q^{16} -33.0000 q^{17} +47.0000 q^{19} -55.0000 q^{22} -113.000 q^{23} -160.000 q^{26} +51.0000 q^{28} +54.0000 q^{29} +178.000 q^{31} -85.0000 q^{32} +165.000 q^{34} +19.0000 q^{37} -235.000 q^{38} -139.000 q^{41} -308.000 q^{43} +187.000 q^{44} +565.000 q^{46} -195.000 q^{47} -334.000 q^{49} +544.000 q^{52} -152.000 q^{53} -135.000 q^{56} -270.000 q^{58} +625.000 q^{59} +320.000 q^{61} -890.000 q^{62} -287.000 q^{64} +200.000 q^{67} -561.000 q^{68} +947.000 q^{71} -448.000 q^{73} -95.0000 q^{74} +799.000 q^{76} +33.0000 q^{77} -721.000 q^{79} +695.000 q^{82} -142.000 q^{83} +1540.00 q^{86} -495.000 q^{88} -404.000 q^{89} +96.0000 q^{91} -1921.00 q^{92} +975.000 q^{94} +79.0000 q^{97} +1670.00 q^{98} +O(q^{100})$

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 $

$2$	−5.00000	−1.76777	−0.883883	−	0.467707i	$-0.845080\pi$
$2$	−5.00000	−1.76777	−0.883883	+	0.467707i	$0.845080\pi$
$3$	0	0
$3$	0	0
$4$	17.0000	2.12500
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.00000	0.161985	0.0809924	−	0.996715i	$-0.474191\pi$
$7$	3.00000	0.161985	0.0809924	+	0.996715i	$0.474191\pi$
$8$	−45.0000	−1.98874
$9$	0	0
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	32.0000	0.682708	0.341354	−	0.939935i	$-0.389115\pi$
$13$	32.0000	0.682708	0.341354	+	0.939935i	$0.389115\pi$
$14$	−15.0000	−0.286351
$15$	0	0
$16$	89.0000	1.39062
$17$	−33.0000	−0.470804	−0.235402	−	0.971898i	$-0.575641\pi$
$17$	−33.0000	−0.470804	−0.235402	+	0.971898i	$0.575641\pi$
$18$	0	0
$19$	47.0000	0.567502	0.283751	−	0.958898i	$-0.408421\pi$
$19$	47.0000	0.567502	0.283751	+	0.958898i	$0.408421\pi$
$20$	0	0
$21$	0	0
$22$	−55.0000	−0.533002
$23$	−113.000	−1.02444	−0.512220	−	0.858854i	$-0.671177\pi$
$23$	−113.000	−1.02444	−0.512220	+	0.858854i	$0.671177\pi$
$24$	0	0
$25$	0	0
$26$	−160.000	−1.20687
$27$	0	0
$28$	51.0000	0.344218
$29$	54.0000	0.345778	0.172889	−	0.984941i	$-0.444690\pi$
$29$	54.0000	0.345778	0.172889	+	0.984941i	$0.444690\pi$
$30$	0	0
$31$	178.000	1.03128	0.515641	−	0.856805i	$-0.327554\pi$
$31$	178.000	1.03128	0.515641	+	0.856805i	$0.327554\pi$
$32$	−85.0000	−0.469563
$33$	0	0
$34$	165.000	0.832273
$35$	0	0
$36$	0	0
$37$	19.0000	0.0844211	0.0422106	−	0.999109i	$-0.486560\pi$
$37$	19.0000	0.0844211	0.0422106	+	0.999109i	$0.486560\pi$
$38$	−235.000	−1.00321
$39$	0	0
$40$	0	0
$41$	−139.000	−0.529467	−0.264734	−	0.964322i	$-0.585284\pi$
$41$	−139.000	−0.529467	−0.264734	+	0.964322i	$0.585284\pi$
$42$	0	0
$43$	−308.000	−1.09232	−0.546158	−	0.837682i	$-0.683910\pi$
$43$	−308.000	−1.09232	−0.546158	+	0.837682i	$0.683910\pi$
$44$	187.000	0.640712
$45$	0	0
$46$	565.000	1.81097
$47$	−195.000	−0.605185	−0.302592	−	0.953120i	$-0.597852\pi$
$47$	−195.000	−0.605185	−0.302592	+	0.953120i	$0.597852\pi$
$48$	0	0
$49$	−334.000	−0.973761
$50$	0	0
$51$	0	0
$52$	544.000	1.45075
$53$	−152.000	−0.393940	−0.196970	−	0.980410i	$-0.563110\pi$
$53$	−152.000	−0.393940	−0.196970	+	0.980410i	$0.563110\pi$
$54$	0	0
$55$	0	0
$56$	−135.000	−0.322145
$57$	0	0
$58$	−270.000	−0.611254
$59$	625.000	1.37912	0.689560	−	0.724229i	$-0.257804\pi$
$59$	625.000	1.37912	0.689560	+	0.724229i	$0.257804\pi$
$60$	0	0
$61$	320.000	0.671669	0.335834	−	0.941921i	$-0.390982\pi$
$61$	320.000	0.671669	0.335834	+	0.941921i	$0.390982\pi$
$62$	−890.000	−1.82307
$63$	0	0
$64$	−287.000	−0.560547
$65$	0	0
$66$	0	0
$67$	200.000	0.364685	0.182342	−	0.983235i	$-0.441632\pi$
$67$	200.000	0.364685	0.182342	+	0.983235i	$0.441632\pi$
$68$	−561.000	−1.00046
$69$	0	0
$70$	0	0
$71$	947.000	1.58293	0.791466	−	0.611213i	$-0.209318\pi$
$71$	947.000	1.58293	0.791466	+	0.611213i	$0.209318\pi$
$72$	0	0
$73$	−448.000	−0.718280	−0.359140	−	0.933284i	$-0.616930\pi$
$73$	−448.000	−0.718280	−0.359140	+	0.933284i	$0.616930\pi$
$74$	−95.0000	−0.149237
$75$	0	0
$76$	799.000	1.20594
$77$	33.0000	0.0488402
$78$	0	0
$79$	−721.000	−1.02682	−0.513410	−	0.858143i	$-0.671618\pi$
$79$	−721.000	−1.02682	−0.513410	+	0.858143i	$0.671618\pi$
$80$	0	0
$81$	0	0
$82$	695.000	0.935975
$83$	−142.000	−0.187789	−0.0938947	−	0.995582i	$-0.529932\pi$
$83$	−142.000	−0.187789	−0.0938947	+	0.995582i	$0.529932\pi$
$84$	0	0
$85$	0	0
$86$	1540.00	1.93096
$87$	0	0
$88$	−495.000	−0.599627
$89$	−404.000	−0.481168	−0.240584	−	0.970628i	$-0.577339\pi$
$89$	−404.000	−0.481168	−0.240584	+	0.970628i	$0.577339\pi$
$90$	0	0
$91$	96.0000	0.110588
$92$	−1921.00	−2.17694
$93$	0	0
$94$	975.000	1.06983
$95$	0	0
$96$	0	0
$97$	79.0000	0.0826931	0.0413466	−	0.999145i	$-0.486835\pi$
$97$	79.0000	0.0826931	0.0413466	+	0.999145i	$0.486835\pi$
$98$	1670.00	1.72138
$99$	0	0
$100$	0	0
$101$	545.000	0.536926	0.268463	−	0.963290i	$-0.413484\pi$
$101$	545.000	0.536926	0.268463	+	0.963290i	$0.413484\pi$
$102$	0	0
$103$	−1306.00	−1.24936	−0.624680	−	0.780881i	$-0.714770\pi$
$103$	−1306.00	−1.24936	−0.624680	+	0.780881i	$0.714770\pi$
$104$	−1440.00	−1.35773
$105$	0	0
$106$	760.000	0.696394
$107$	−1938.00	−1.75097	−0.875484	−	0.483247i	$-0.839457\pi$
$107$	−1938.00	−1.75097	−0.875484	+	0.483247i	$0.839457\pi$
$108$	0	0
$109$	−576.000	−0.506154	−0.253077	−	0.967446i	$-0.581443\pi$
$109$	−576.000	−0.506154	−0.253077	+	0.967446i	$0.581443\pi$
$110$	0	0
$111$	0	0
$112$	267.000	0.225260
$113$	1104.00	0.919076	0.459538	−	0.888158i	$-0.348015\pi$
$113$	1104.00	0.919076	0.459538	+	0.888158i	$0.348015\pi$
$114$	0	0
$115$	0	0
$116$	918.000	0.734777
$117$	0	0
$118$	−3125.00	−2.43796
$119$	−99.0000	−0.0762632
$120$	0	0
$121$	121.000	0.0909091
$122$	−1600.00	−1.18735
$123$	0	0
$124$	3026.00	2.19147
$125$	0	0
$126$	0	0
$127$	−1739.00	−1.21505	−0.607525	−	0.794301i	$-0.707837\pi$
$127$	−1739.00	−1.21505	−0.607525	+	0.794301i	$0.707837\pi$
$128$	2115.00	1.46048
$129$	0	0
$130$	0	0
$131$	−1818.00	−1.21251	−0.606257	−	0.795269i	$-0.707330\pi$
$131$	−1818.00	−1.21251	−0.606257	+	0.795269i	$0.707330\pi$
$132$	0	0
$133$	141.000	0.0919267
$134$	−1000.00	−0.644678
$135$	0	0
$136$	1485.00	0.936307
$137$	−870.000	−0.542548	−0.271274	−	0.962502i	$-0.587445\pi$
$137$	−870.000	−0.542548	−0.271274	+	0.962502i	$0.587445\pi$
$138$	0	0
$139$	−636.000	−0.388092	−0.194046	−	0.980992i	$-0.562161\pi$
$139$	−636.000	−0.388092	−0.194046	+	0.980992i	$0.562161\pi$
$140$	0	0
$141$	0	0
$142$	−4735.00	−2.79826
$143$	352.000	0.205844
$144$	0	0
$145$	0	0
$146$	2240.00	1.26975
$147$	0	0
$148$	323.000	0.179395
$149$	239.000	0.131407	0.0657035	−	0.997839i	$-0.479071\pi$
$149$	239.000	0.131407	0.0657035	+	0.997839i	$0.479071\pi$
$150$	0	0
$151$	1208.00	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	1208.00	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	−2115.00	−1.12861
$153$	0	0
$154$	−165.000	−0.0863382
$155$	0	0
$156$	0	0
$157$	−1874.00	−0.952621	−0.476310	−	0.879277i	$-0.658026\pi$
$157$	−1874.00	−0.952621	−0.476310	+	0.879277i	$0.658026\pi$
$158$	3605.00	1.81518
$159$	0	0
$160$	0	0
$161$	−339.000	−0.165944
$162$	0	0
$163$	−1904.00	−0.914925	−0.457463	−	0.889229i	$-0.651242\pi$
$163$	−1904.00	−0.914925	−0.457463	+	0.889229i	$0.651242\pi$
$164$	−2363.00	−1.12512
$165$	0	0
$166$	710.000	0.331968
$167$	1180.00	0.546773	0.273387	−	0.961904i	$-0.411856\pi$
$167$	1180.00	0.546773	0.273387	+	0.961904i	$0.411856\pi$
$168$	0	0
$169$	−1173.00	−0.533910
$170$	0	0
$171$	0	0
$172$	−5236.00	−2.32117
$173$	3177.00	1.39620	0.698101	−	0.716000i	$-0.254029\pi$
$173$	3177.00	1.39620	0.698101	+	0.716000i	$0.254029\pi$
$174$	0	0
$175$	0	0
$176$	979.000	0.419289
$177$	0	0
$178$	2020.00	0.850592
$179$	−1787.00	−0.746182	−0.373091	−	0.927795i	$-0.621702\pi$
$179$	−1787.00	−0.746182	−0.373091	+	0.927795i	$0.621702\pi$
$180$	0	0
$181$	−835.000	−0.342901	−0.171450	−	0.985193i	$-0.554845\pi$
$181$	−835.000	−0.342901	−0.171450	+	0.985193i	$0.554845\pi$
$182$	−480.000	−0.195494
$183$	0	0
$184$	5085.00	2.03734
$185$	0	0
$186$	0	0
$187$	−363.000	−0.141953
$188$	−3315.00	−1.28602
$189$	0	0
$190$	0	0
$191$	−3613.00	−1.36873	−0.684365	−	0.729139i	$-0.739921\pi$
$191$	−3613.00	−1.36873	−0.684365	+	0.729139i	$0.739921\pi$
$192$	0	0
$193$	4204.00	1.56793	0.783965	−	0.620805i	$-0.213194\pi$
$193$	4204.00	1.56793	0.783965	+	0.620805i	$0.213194\pi$
$194$	−395.000	−0.146182
$195$	0	0
$196$	−5678.00	−2.06924
$197$	4517.00	1.63362	0.816809	−	0.576908i	$-0.195741\pi$
$197$	4517.00	1.63362	0.816809	+	0.576908i	$0.195741\pi$
$198$	0	0
$199$	4164.00	1.48331	0.741654	−	0.670783i	$-0.234042\pi$
$199$	4164.00	1.48331	0.741654	+	0.670783i	$0.234042\pi$
$200$	0	0
$201$	0	0
$202$	−2725.00	−0.949160
$203$	162.000	0.0560107
$204$	0	0
$205$	0	0
$206$	6530.00	2.20858
$207$	0	0
$208$	2848.00	0.949391
$209$	517.000	0.171108
$210$	0	0
$211$	4660.00	1.52042	0.760208	−	0.649680i	$-0.225097\pi$
$211$	4660.00	1.52042	0.760208	+	0.649680i	$0.225097\pi$
$212$	−2584.00	−0.837122
$213$	0	0
$214$	9690.00	3.09530
$215$	0	0
$216$	0	0
$217$	534.000	0.167052
$218$	2880.00	0.894762
$219$	0	0
$220$	0	0
$221$	−1056.00	−0.321422
$222$	0	0
$223$	3560.00	1.06904	0.534518	−	0.845157i	$-0.320493\pi$
$223$	3560.00	1.06904	0.534518	+	0.845157i	$0.320493\pi$
$224$	−255.000	−0.0760621
$225$	0	0
$226$	−5520.00	−1.62471
$227$	4678.00	1.36780	0.683898	−	0.729577i	$-0.260283\pi$
$227$	4678.00	1.36780	0.683898	+	0.729577i	$0.260283\pi$
$228$	0	0
$229$	−4447.00	−1.28326	−0.641629	−	0.767015i	$-0.721741\pi$
$229$	−4447.00	−1.28326	−0.641629	+	0.767015i	$0.721741\pi$
$230$	0	0
$231$	0	0
$232$	−2430.00	−0.687661
$233$	−411.000	−0.115560	−0.0577801	−	0.998329i	$-0.518402\pi$
$233$	−411.000	−0.115560	−0.0577801	+	0.998329i	$0.518402\pi$
$234$	0	0
$235$	0	0
$236$	10625.0	2.93063
$237$	0	0
$238$	495.000	0.134815
$239$	6380.00	1.72673	0.863364	−	0.504582i	$-0.168353\pi$
$239$	6380.00	1.72673	0.863364	+	0.504582i	$0.168353\pi$
$240$	0	0
$241$	7282.00	1.94637	0.973184	−	0.230027i	$-0.0738813\pi$
$241$	7282.00	1.94637	0.973184	+	0.230027i	$0.0738813\pi$
$242$	−605.000	−0.160706
$243$	0	0
$244$	5440.00	1.42730
$245$	0	0
$246$	0	0
$247$	1504.00	0.387438
$248$	−8010.00	−2.05095
$249$	0	0
$250$	0	0
$251$	4728.00	1.18896	0.594480	−	0.804111i	$-0.297358\pi$
$251$	4728.00	1.18896	0.594480	+	0.804111i	$0.297358\pi$
$252$	0	0
$253$	−1243.00	−0.308880
$254$	8695.00	2.14792
$255$	0	0
$256$	−8279.00	−2.02124
$257$	−5418.00	−1.31504	−0.657521	−	0.753437i	$-0.728395\pi$
$257$	−5418.00	−1.31504	−0.657521	+	0.753437i	$0.728395\pi$
$258$	0	0
$259$	57.0000	0.0136749
$260$	0	0
$261$	0	0
$262$	9090.00	2.14344
$263$	3354.00	0.786375	0.393187	−	0.919458i	$-0.371372\pi$
$263$	3354.00	0.786375	0.393187	+	0.919458i	$0.371372\pi$
$264$	0	0
$265$	0	0
$266$	−705.000	−0.162505
$267$	0	0
$268$	3400.00	0.774955
$269$	−1062.00	−0.240711	−0.120356	−	0.992731i	$-0.538403\pi$
$269$	−1062.00	−0.240711	−0.120356	+	0.992731i	$0.538403\pi$
$270$	0	0
$271$	−4821.00	−1.08065	−0.540323	−	0.841458i	$-0.681698\pi$
$271$	−4821.00	−1.08065	−0.540323	+	0.841458i	$0.681698\pi$
$272$	−2937.00	−0.654712
$273$	0	0
$274$	4350.00	0.959099
$275$	0	0
$276$	0	0
$277$	4.00000	0.000867642 0	0.000433821	−	1.00000i	$-0.499862\pi$
$277$	4.00000	0.000867642 0	0.000433821		1.00000i	$0.499862\pi$
$278$	3180.00	0.686057
$279$	0	0
$280$	0	0
$281$	−4647.00	−0.986537	−0.493268	−	0.869877i	$-0.664198\pi$
$281$	−4647.00	−0.986537	−0.493268	+	0.869877i	$0.664198\pi$
$282$	0	0
$283$	−4283.00	−0.899639	−0.449820	−	0.893119i	$-0.648512\pi$
$283$	−4283.00	−0.899639	−0.449820	+	0.893119i	$0.648512\pi$
$284$	16099.0	3.36373
$285$	0	0
$286$	−1760.00	−0.363885
$287$	−417.000	−0.0857656
$288$	0	0
$289$	−3824.00	−0.778343
$290$	0	0
$291$	0	0
$292$	−7616.00	−1.52634
$293$	6811.00	1.35803	0.679015	−	0.734124i	$-0.262407\pi$
$293$	6811.00	1.35803	0.679015	+	0.734124i	$0.262407\pi$
$294$	0	0
$295$	0	0
$296$	−855.000	−0.167891
$297$	0	0
$298$	−1195.00	−0.232297
$299$	−3616.00	−0.699394
$300$	0	0
$301$	−924.000	−0.176938
$302$	−6040.00	−1.15087
$303$	0	0
$304$	4183.00	0.789183
$305$	0	0
$306$	0	0
$307$	−460.000	−0.0855166	−0.0427583	−	0.999085i	$-0.513615\pi$
$307$	−460.000	−0.0855166	−0.0427583	+	0.999085i	$0.513615\pi$
$308$	561.000	0.103786
$309$	0	0
$310$	0	0
$311$	−8328.00	−1.51845	−0.759224	−	0.650829i	$-0.774421\pi$
$311$	−8328.00	−1.51845	−0.759224	+	0.650829i	$0.774421\pi$
$312$	0	0
$313$	−5929.00	−1.07069	−0.535346	−	0.844633i	$-0.679819\pi$
$313$	−5929.00	−1.07069	−0.535346	+	0.844633i	$0.679819\pi$
$314$	9370.00	1.68401
$315$	0	0
$316$	−12257.0	−2.18199
$317$	−5040.00	−0.892980	−0.446490	−	0.894789i	$-0.647326\pi$
$317$	−5040.00	−0.892980	−0.446490	+	0.894789i	$0.647326\pi$
$318$	0	0
$319$	594.000	0.104256
$320$	0	0
$321$	0	0
$322$	1695.00	0.293350
$323$	−1551.00	−0.267183
$324$	0	0
$325$	0	0
$326$	9520.00	1.61737
$327$	0	0
$328$	6255.00	1.05297
$329$	−585.000	−0.0980307
$330$	0	0
$331$	10396.0	1.72633	0.863166	−	0.504920i	$-0.168478\pi$
$331$	10396.0	1.72633	0.863166	+	0.504920i	$0.168478\pi$
$332$	−2414.00	−0.399053
$333$	0	0
$334$	−5900.00	−0.966568
$335$	0	0
$336$	0	0
$337$	−7236.00	−1.16964	−0.584822	−	0.811162i	$-0.698836\pi$
$337$	−7236.00	−1.16964	−0.584822	+	0.811162i	$0.698836\pi$
$338$	5865.00	0.943828
$339$	0	0
$340$	0	0
$341$	1958.00	0.310943
$342$	0	0
$343$	−2031.00	−0.319719
$344$	13860.0	2.17233
$345$	0	0
$346$	−15885.0	−2.46816
$347$	−1468.00	−0.227108	−0.113554	−	0.993532i	$-0.536223\pi$
$347$	−1468.00	−0.227108	−0.113554	+	0.993532i	$0.536223\pi$
$348$	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
$352$	−935.000	−0.141579
$353$	−5376.00	−0.810582	−0.405291	−	0.914188i	$-0.632830\pi$
$353$	−5376.00	−0.810582	−0.405291	+	0.914188i	$0.632830\pi$
$354$	0	0
$355$	0	0
$356$	−6868.00	−1.02248
$357$	0	0
$358$	8935.00	1.31908
$359$	−3734.00	−0.548950	−0.274475	−	0.961594i	$-0.588504\pi$
$359$	−3734.00	−0.548950	−0.274475	+	0.961594i	$0.588504\pi$
$360$	0	0
$361$	−4650.00	−0.677941
$362$	4175.00	0.606169
$363$	0	0
$364$	1632.00	0.235000
$365$	0	0
$366$	0	0
$367$	−10274.0	−1.46130	−0.730652	−	0.682750i	$-0.760784\pi$
$367$	−10274.0	−1.46130	−0.730652	+	0.682750i	$0.760784\pi$
$368$	−10057.0	−1.42461
$369$	0	0
$370$	0	0
$371$	−456.000	−0.0638122
$372$	0	0
$373$	13662.0	1.89649	0.948246	−	0.317537i	$-0.102856\pi$
$373$	13662.0	1.89649	0.948246	+	0.317537i	$0.102856\pi$
$374$	1815.00	0.250940
$375$	0	0
$376$	8775.00	1.20355
$377$	1728.00	0.236065
$378$	0	0
$379$	−7906.00	−1.07151	−0.535757	−	0.844372i	$-0.679974\pi$
$379$	−7906.00	−1.07151	−0.535757	+	0.844372i	$0.679974\pi$
$380$	0	0
$381$	0	0
$382$	18065.0	2.41960
$383$	3168.00	0.422656	0.211328	−	0.977415i	$-0.432221\pi$
$383$	3168.00	0.422656	0.211328	+	0.977415i	$0.432221\pi$
$384$	0	0
$385$	0	0
$386$	−21020.0	−2.77174
$387$	0	0
$388$	1343.00	0.175723
$389$	−10770.0	−1.40375	−0.701877	−	0.712298i	$-0.747655\pi$
$389$	−10770.0	−1.40375	−0.701877	+	0.712298i	$0.747655\pi$
$390$	0	0
$391$	3729.00	0.482311
$392$	15030.0	1.93656
$393$	0	0
$394$	−22585.0	−2.88786
$395$	0	0
$396$	0	0
$397$	5670.00	0.716799	0.358399	−	0.933568i	$-0.383323\pi$
$397$	5670.00	0.716799	0.358399	+	0.933568i	$0.383323\pi$
$398$	−20820.0	−2.62214
$399$	0	0
$400$	0	0
$401$	−832.000	−0.103611	−0.0518056	−	0.998657i	$-0.516498\pi$
$401$	−832.000	−0.103611	−0.0518056	+	0.998657i	$0.516498\pi$
$402$	0	0
$403$	5696.00	0.704064
$404$	9265.00	1.14097
$405$	0	0
$406$	−810.000	−0.0990139
$407$	209.000	0.0254539
$408$	0	0
$409$	−5712.00	−0.690563	−0.345281	−	0.938499i	$-0.612217\pi$
$409$	−5712.00	−0.690563	−0.345281	+	0.938499i	$0.612217\pi$
$410$	0	0
$411$	0	0
$412$	−22202.0	−2.65489
$413$	1875.00	0.223396
$414$	0	0
$415$	0	0
$416$	−2720.00	−0.320574
$417$	0	0
$418$	−2585.00	−0.302480
$419$	4559.00	0.531555	0.265778	−	0.964034i	$-0.414371\pi$
$419$	4559.00	0.531555	0.265778	+	0.964034i	$0.414371\pi$
$420$	0	0
$421$	6855.00	0.793568	0.396784	−	0.917912i	$-0.370126\pi$
$421$	6855.00	0.793568	0.396784	+	0.917912i	$0.370126\pi$
$422$	−23300.0	−2.68774
$423$	0	0
$424$	6840.00	0.783443
$425$	0	0
$426$	0	0
$427$	960.000	0.108800
$428$	−32946.0	−3.72081
$429$	0	0
$430$	0	0
$431$	−10770.0	−1.20365	−0.601824	−	0.798628i	$-0.705559\pi$
$431$	−10770.0	−1.20365	−0.601824	+	0.798628i	$0.705559\pi$
$432$	0	0
$433$	8498.00	0.943159	0.471579	−	0.881824i	$-0.343684\pi$
$433$	8498.00	0.943159	0.471579	+	0.881824i	$0.343684\pi$
$434$	−2670.00	−0.295309
$435$	0	0
$436$	−9792.00	−1.07558
$437$	−5311.00	−0.581372
$438$	0	0
$439$	9835.00	1.06925	0.534623	−	0.845091i	$-0.320454\pi$
$439$	9835.00	1.06925	0.534623	+	0.845091i	$0.320454\pi$
$440$	0	0
$441$	0	0
$442$	5280.00	0.568199
$443$	10745.0	1.15239	0.576197	−	0.817311i	$-0.304536\pi$
$443$	10745.0	1.15239	0.576197	+	0.817311i	$0.304536\pi$
$444$	0	0
$445$	0	0
$446$	−17800.0	−1.88981
$447$	0	0
$448$	−861.000	−0.0908001
$449$	−8356.00	−0.878272	−0.439136	−	0.898421i	$-0.644715\pi$
$449$	−8356.00	−0.878272	−0.439136	+	0.898421i	$0.644715\pi$
$450$	0	0
$451$	−1529.00	−0.159640
$452$	18768.0	1.95304
$453$	0	0
$454$	−23390.0	−2.41795
$455$	0	0
$456$	0	0
$457$	−7058.00	−0.722449	−0.361225	−	0.932479i	$-0.617641\pi$
$457$	−7058.00	−0.722449	−0.361225	+	0.932479i	$0.617641\pi$
$458$	22235.0	2.26850
$459$	0	0
$460$	0	0
$461$	−646.000	−0.0652651	−0.0326326	−	0.999467i	$-0.510389\pi$
$461$	−646.000	−0.0652651	−0.0326326	+	0.999467i	$0.510389\pi$
$462$	0	0
$463$	−8982.00	−0.901574	−0.450787	−	0.892631i	$-0.648857\pi$
$463$	−8982.00	−0.901574	−0.450787	+	0.892631i	$0.648857\pi$
$464$	4806.00	0.480847
$465$	0	0
$466$	2055.00	0.204283
$467$	13476.0	1.33532	0.667661	−	0.744466i	$-0.267296\pi$
$467$	13476.0	1.33532	0.667661	+	0.744466i	$0.267296\pi$
$468$	0	0
$469$	600.000	0.0590734
$470$	0	0
$471$	0	0
$472$	−28125.0	−2.74271
$473$	−3388.00	−0.329345
$474$	0	0
$475$	0	0
$476$	−1683.00	−0.162059
$477$	0	0
$478$	−31900.0	−3.05245
$479$	−12996.0	−1.23967	−0.619835	−	0.784732i	$-0.712801\pi$
$479$	−12996.0	−1.23967	−0.619835	+	0.784732i	$0.712801\pi$
$480$	0	0
$481$	608.000	0.0576350
$482$	−36410.0	−3.44073
$483$	0	0
$484$	2057.00	0.193182
$485$	0	0
$486$	0	0
$487$	−6026.00	−0.560707	−0.280353	−	0.959897i	$-0.590452\pi$
$487$	−6026.00	−0.560707	−0.280353	+	0.959897i	$0.590452\pi$
$488$	−14400.0	−1.33577
$489$	0	0
$490$	0	0
$491$	−11698.0	−1.07520	−0.537600	−	0.843200i	$-0.680669\pi$
$491$	−11698.0	−1.07520	−0.537600	+	0.843200i	$0.680669\pi$
$492$	0	0
$493$	−1782.00	−0.162794
$494$	−7520.00	−0.684900
$495$	0	0
$496$	15842.0	1.43413
$497$	2841.00	0.256411
$498$	0	0
$499$	−17052.0	−1.52976	−0.764882	−	0.644170i	$-0.777203\pi$
$499$	−17052.0	−1.52976	−0.764882	+	0.644170i	$0.777203\pi$
$500$	0	0
$501$	0	0
$502$	−23640.0	−2.10180
$503$	932.000	0.0826160	0.0413080	−	0.999146i	$-0.486848\pi$
$503$	932.000	0.0826160	0.0413080	+	0.999146i	$0.486848\pi$
$504$	0	0
$505$	0	0
$506$	6215.00	0.546029
$507$	0	0
$508$	−29563.0	−2.58198
$509$	−4384.00	−0.381763	−0.190882	−	0.981613i	$-0.561135\pi$
$509$	−4384.00	−0.381763	−0.190882	+	0.981613i	$0.561135\pi$
$510$	0	0
$511$	−1344.00	−0.116350
$512$	24475.0	2.11260
$513$	0	0
$514$	27090.0	2.32469
$515$	0	0
$516$	0	0
$517$	−2145.00	−0.182470
$518$	−285.000	−0.0241741
$519$	0	0
$520$	0	0
$521$	2322.00	0.195257	0.0976283	−	0.995223i	$-0.468874\pi$
$521$	2322.00	0.195257	0.0976283	+	0.995223i	$0.468874\pi$
$522$	0	0
$523$	−9749.00	−0.815094	−0.407547	−	0.913184i	$-0.633616\pi$
$523$	−9749.00	−0.815094	−0.407547	+	0.913184i	$0.633616\pi$
$524$	−30906.0	−2.57659
$525$	0	0
$526$	−16770.0	−1.39013
$527$	−5874.00	−0.485532
$528$	0	0
$529$	602.000	0.0494781
$530$	0	0
$531$	0	0
$532$	2397.00	0.195344
$533$	−4448.00	−0.361471
$534$	0	0
$535$	0	0
$536$	−9000.00	−0.725263
$537$	0	0
$538$	5310.00	0.425521
$539$	−3674.00	−0.293600
$540$	0	0
$541$	4208.00	0.334410	0.167205	−	0.985922i	$-0.446526\pi$
$541$	4208.00	0.334410	0.167205	+	0.985922i	$0.446526\pi$
$542$	24105.0	1.91033
$543$	0	0
$544$	2805.00	0.221072
$545$	0	0
$546$	0	0
$547$	10179.0	0.795654	0.397827	−	0.917461i	$-0.369764\pi$
$547$	10179.0	0.795654	0.397827	+	0.917461i	$0.369764\pi$
$548$	−14790.0	−1.15292
$549$	0	0
$550$	0	0
$551$	2538.00	0.196229
$552$	0	0
$553$	−2163.00	−0.166329
$554$	−20.0000	−0.00153379
$555$	0	0
$556$	−10812.0	−0.824696
$557$	2314.00	0.176028	0.0880138	−	0.996119i	$-0.471948\pi$
$557$	2314.00	0.176028	0.0880138	+	0.996119i	$0.471948\pi$
$558$	0	0
$559$	−9856.00	−0.745732
$560$	0	0
$561$	0	0
$562$	23235.0	1.74397
$563$	−24330.0	−1.82129	−0.910646	−	0.413188i	$-0.864415\pi$
$563$	−24330.0	−1.82129	−0.910646	+	0.413188i	$0.864415\pi$
$564$	0	0
$565$	0	0
$566$	21415.0	1.59035
$567$	0	0
$568$	−42615.0	−3.14804
$569$	−3445.00	−0.253817	−0.126909	−	0.991914i	$-0.540505\pi$
$569$	−3445.00	−0.253817	−0.126909	+	0.991914i	$0.540505\pi$
$570$	0	0
$571$	−13056.0	−0.956877	−0.478438	−	0.878121i	$-0.658797\pi$
$571$	−13056.0	−0.956877	−0.478438	+	0.878121i	$0.658797\pi$
$572$	5984.00	0.437419
$573$	0	0
$574$	2085.00	0.151614
$575$	0	0
$576$	0	0
$577$	−17347.0	−1.25159	−0.625793	−	0.779989i	$-0.715225\pi$
$577$	−17347.0	−1.25159	−0.625793	+	0.779989i	$0.715225\pi$
$578$	19120.0	1.37593
$579$	0	0
$580$	0	0
$581$	−426.000	−0.0304190
$582$	0	0
$583$	−1672.00	−0.118777
$584$	20160.0	1.42847
$585$	0	0
$586$	−34055.0	−2.40068
$587$	−8379.00	−0.589162	−0.294581	−	0.955626i	$-0.595180\pi$
$587$	−8379.00	−0.589162	−0.294581	+	0.955626i	$0.595180\pi$
$588$	0	0
$589$	8366.00	0.585255
$590$	0	0
$591$	0	0
$592$	1691.00	0.117398
$593$	−1958.00	−0.135591	−0.0677955	−	0.997699i	$-0.521597\pi$
$593$	−1958.00	−0.135591	−0.0677955	+	0.997699i	$0.521597\pi$
$594$	0	0
$595$	0	0
$596$	4063.00	0.279240
$597$	0	0
$598$	18080.0	1.23636
$599$	−23583.0	−1.60864	−0.804320	−	0.594196i	$-0.797470\pi$
$599$	−23583.0	−1.60864	−0.804320	+	0.594196i	$0.797470\pi$
$600$	0	0
$601$	−15328.0	−1.04034	−0.520168	−	0.854064i	$-0.674131\pi$
$601$	−15328.0	−1.04034	−0.520168	+	0.854064i	$0.674131\pi$
$602$	4620.00	0.312786
$603$	0	0
$604$	20536.0	1.38344
$605$	0	0
$606$	0	0
$607$	160.000	0.0106988	0.00534942	−	0.999986i	$-0.498297\pi$
$607$	160.000	0.0106988	0.00534942	+	0.999986i	$0.498297\pi$
$608$	−3995.00	−0.266478
$609$	0	0
$610$	0	0
$611$	−6240.00	−0.413164
$612$	0	0
$613$	−5948.00	−0.391904	−0.195952	−	0.980613i	$-0.562780\pi$
$613$	−5948.00	−0.391904	−0.195952	+	0.980613i	$0.562780\pi$
$614$	2300.00	0.151173
$615$	0	0
$616$	−1485.00	−0.0971304
$617$	−334.000	−0.0217931	−0.0108965	−	0.999941i	$-0.503469\pi$
$617$	−334.000	−0.0217931	−0.0108965	+	0.999941i	$0.503469\pi$
$618$	0	0
$619$	−7202.00	−0.467646	−0.233823	−	0.972279i	$-0.575124\pi$
$619$	−7202.00	−0.467646	−0.233823	+	0.972279i	$0.575124\pi$
$620$	0	0
$621$	0	0
$622$	41640.0	2.68426
$623$	−1212.00	−0.0779418
$624$	0	0
$625$	0	0
$626$	29645.0	1.89274
$627$	0	0
$628$	−31858.0	−2.02432
$629$	−627.000	−0.0397458
$630$	0	0
$631$	10306.0	0.650199	0.325099	−	0.945680i	$-0.394602\pi$
$631$	10306.0	0.650199	0.325099	+	0.945680i	$0.394602\pi$
$632$	32445.0	2.04208
$633$	0	0
$634$	25200.0	1.57858
$635$	0	0
$636$	0	0
$637$	−10688.0	−0.664794
$638$	−2970.00	−0.184300
$639$	0	0
$640$	0	0
$641$	1228.00	0.0756678	0.0378339	−	0.999284i	$-0.487954\pi$
$641$	1228.00	0.0756678	0.0378339	+	0.999284i	$0.487954\pi$
$642$	0	0
$643$	−18454.0	−1.13181	−0.565906	−	0.824470i	$-0.691473\pi$
$643$	−18454.0	−1.13181	−0.565906	+	0.824470i	$0.691473\pi$
$644$	−5763.00	−0.352630
$645$	0	0
$646$	7755.00	0.472316
$647$	−17647.0	−1.07230	−0.536148	−	0.844124i	$-0.680121\pi$
$647$	−17647.0	−1.07230	−0.536148	+	0.844124i	$0.680121\pi$
$648$	0	0
$649$	6875.00	0.415820
$650$	0	0
$651$	0	0
$652$	−32368.0	−1.94422
$653$	−25918.0	−1.55322	−0.776608	−	0.629984i	$-0.783061\pi$
$653$	−25918.0	−1.55322	−0.776608	+	0.629984i	$0.783061\pi$
$654$	0	0
$655$	0	0
$656$	−12371.0	−0.736290
$657$	0	0
$658$	2925.00	0.173295
$659$	−12864.0	−0.760410	−0.380205	−	0.924902i	$-0.624147\pi$
$659$	−12864.0	−0.760410	−0.380205	+	0.924902i	$0.624147\pi$
$660$	0	0
$661$	−11419.0	−0.671933	−0.335966	−	0.941874i	$-0.609063\pi$
$661$	−11419.0	−0.671933	−0.335966	+	0.941874i	$0.609063\pi$
$662$	−51980.0	−3.05175
$663$	0	0
$664$	6390.00	0.373464
$665$	0	0
$666$	0	0
$667$	−6102.00	−0.354228
$668$	20060.0	1.16189
$669$	0	0
$670$	0	0
$671$	3520.00	0.202516
$672$	0	0
$673$	15784.0	0.904054	0.452027	−	0.892004i	$-0.350701\pi$
$673$	15784.0	0.904054	0.452027	+	0.892004i	$0.350701\pi$
$674$	36180.0	2.06766
$675$	0	0
$676$	−19941.0	−1.13456
$677$	26050.0	1.47885	0.739426	−	0.673238i	$-0.235097\pi$
$677$	26050.0	1.47885	0.739426	+	0.673238i	$0.235097\pi$
$678$	0	0
$679$	237.000	0.0133950
$680$	0	0
$681$	0	0
$682$	−9790.00	−0.549675
$683$	15095.0	0.845672	0.422836	−	0.906206i	$-0.361035\pi$
$683$	15095.0	0.845672	0.422836	+	0.906206i	$0.361035\pi$
$684$	0	0
$685$	0	0
$686$	10155.0	0.565189
$687$	0	0
$688$	−27412.0	−1.51900
$689$	−4864.00	−0.268946
$690$	0	0
$691$	15896.0	0.875126	0.437563	−	0.899188i	$-0.355842\pi$
$691$	15896.0	0.875126	0.437563	+	0.899188i	$0.355842\pi$
$692$	54009.0	2.96693
$693$	0	0
$694$	7340.00	0.401473
$695$	0	0
$696$	0	0
$697$	4587.00	0.249275
$698$	−28450.0	−1.54276
$699$	0	0
$700$	0	0
$701$	−10529.0	−0.567296	−0.283648	−	0.958928i	$-0.591545\pi$
$701$	−10529.0	−0.567296	−0.283648	+	0.958928i	$0.591545\pi$
$702$	0	0
$703$	893.000	0.0479092
$704$	−3157.00	−0.169011
$705$	0	0
$706$	26880.0	1.43292
$707$	1635.00	0.0869738
$708$	0	0
$709$	−16087.0	−0.852130	−0.426065	−	0.904693i	$-0.640100\pi$
$709$	−16087.0	−0.852130	−0.426065	+	0.904693i	$0.640100\pi$
$710$	0	0
$711$	0	0
$712$	18180.0	0.956916
$713$	−20114.0	−1.05649
$714$	0	0
$715$	0	0
$716$	−30379.0	−1.58564
$717$	0	0
$718$	18670.0	0.970415
$719$	−24336.0	−1.26228	−0.631140	−	0.775669i	$-0.717413\pi$
$719$	−24336.0	−1.26228	−0.631140	+	0.775669i	$0.717413\pi$
$720$	0	0
$721$	−3918.00	−0.202377
$722$	23250.0	1.19844
$723$	0	0
$724$	−14195.0	−0.728664
$725$	0	0
$726$	0	0
$727$	13960.0	0.712170	0.356085	−	0.934454i	$-0.384111\pi$
$727$	13960.0	0.712170	0.356085	+	0.934454i	$0.384111\pi$
$728$	−4320.00	−0.219931
$729$	0	0
$730$	0	0
$731$	10164.0	0.514267
$732$	0	0
$733$	9252.00	0.466208	0.233104	−	0.972452i	$-0.425112\pi$
$733$	9252.00	0.466208	0.233104	+	0.972452i	$0.425112\pi$
$734$	51370.0	2.58324
$735$	0	0
$736$	9605.00	0.481039
$737$	2200.00	0.109957
$738$	0	0
$739$	28453.0	1.41632	0.708160	−	0.706052i	$-0.249526\pi$
$739$	28453.0	1.41632	0.708160	+	0.706052i	$0.249526\pi$
$740$	0	0
$741$	0	0
$742$	2280.00	0.112805
$743$	−512.000	−0.0252806	−0.0126403	−	0.999920i	$-0.504024\pi$
$743$	−512.000	−0.0252806	−0.0126403	+	0.999920i	$0.504024\pi$
$744$	0	0
$745$	0	0
$746$	−68310.0	−3.35256
$747$	0	0
$748$	−6171.00	−0.301650
$749$	−5814.00	−0.283630
$750$	0	0
$751$	772.000	0.0375109	0.0187554	−	0.999824i	$-0.494030\pi$
$751$	772.000	0.0375109	0.0187554	+	0.999824i	$0.494030\pi$
$752$	−17355.0	−0.841585
$753$	0	0
$754$	−8640.00	−0.417308
$755$	0	0
$756$	0	0
$757$	8058.00	0.386886	0.193443	−	0.981111i	$-0.438034\pi$
$757$	8058.00	0.386886	0.193443	+	0.981111i	$0.438034\pi$
$758$	39530.0	1.89419
$759$	0	0
$760$	0	0
$761$	−18650.0	−0.888386	−0.444193	−	0.895931i	$-0.646510\pi$
$761$	−18650.0	−0.888386	−0.444193	+	0.895931i	$0.646510\pi$
$762$	0	0
$763$	−1728.00	−0.0819893
$764$	−61421.0	−2.90855
$765$	0	0
$766$	−15840.0	−0.747157
$767$	20000.0	0.941536
$768$	0	0
$769$	−7144.00	−0.335005	−0.167503	−	0.985872i	$-0.553570\pi$
$769$	−7144.00	−0.335005	−0.167503	+	0.985872i	$0.553570\pi$
$770$	0	0
$771$	0	0
$772$	71468.0	3.33185
$773$	1904.00	0.0885927	0.0442963	−	0.999018i	$-0.485895\pi$
$773$	1904.00	0.0885927	0.0442963	+	0.999018i	$0.485895\pi$
$774$	0	0
$775$	0	0
$776$	−3555.00	−0.164455
$777$	0	0
$778$	53850.0	2.48151
$779$	−6533.00	−0.300474
$780$	0	0
$781$	10417.0	0.477272
$782$	−18645.0	−0.852614
$783$	0	0
$784$	−29726.0	−1.35414
$785$	0	0
$786$	0	0
$787$	7555.00	0.342194	0.171097	−	0.985254i	$-0.445269\pi$
$787$	7555.00	0.342194	0.171097	+	0.985254i	$0.445269\pi$
$788$	76789.0	3.47144
$789$	0	0
$790$	0	0
$791$	3312.00	0.148876
$792$	0	0
$793$	10240.0	0.458554
$794$	−28350.0	−1.26713
$795$	0	0
$796$	70788.0	3.15203
$797$	−24950.0	−1.10888	−0.554438	−	0.832225i	$-0.687067\pi$
$797$	−24950.0	−1.10888	−0.554438	+	0.832225i	$0.687067\pi$
$798$	0	0
$799$	6435.00	0.284924
$800$	0	0
$801$	0	0
$802$	4160.00	0.183160
$803$	−4928.00	−0.216570
$804$	0	0
$805$	0	0
$806$	−28480.0	−1.24462
$807$	0	0
$808$	−24525.0	−1.06781
$809$	−19893.0	−0.864525	−0.432262	−	0.901748i	$-0.642285\pi$
$809$	−19893.0	−0.864525	−0.432262	+	0.901748i	$0.642285\pi$
$810$	0	0
$811$	34503.0	1.49391	0.746957	−	0.664872i	$-0.231514\pi$
$811$	34503.0	1.49391	0.746957	+	0.664872i	$0.231514\pi$
$812$	2754.00	0.119023
$813$	0	0
$814$	−1045.00	−0.0449966
$815$	0	0
$816$	0	0
$817$	−14476.0	−0.619891
$818$	28560.0	1.22075
$819$	0	0
$820$	0	0
$821$	−16890.0	−0.717984	−0.358992	−	0.933341i	$-0.616880\pi$
$821$	−16890.0	−0.717984	−0.358992	+	0.933341i	$0.616880\pi$
$822$	0	0
$823$	34692.0	1.46936	0.734682	−	0.678411i	$-0.237331\pi$
$823$	34692.0	1.46936	0.734682	+	0.678411i	$0.237331\pi$
$824$	58770.0	2.48465
$825$	0	0
$826$	−9375.00	−0.394913
$827$	−41424.0	−1.74178	−0.870891	−	0.491476i	$-0.836457\pi$
$827$	−41424.0	−1.74178	−0.870891	+	0.491476i	$0.836457\pi$
$828$	0	0
$829$	−18494.0	−0.774817	−0.387408	−	0.921908i	$-0.626630\pi$
$829$	−18494.0	−0.774817	−0.387408	+	0.921908i	$0.626630\pi$
$830$	0	0
$831$	0	0
$832$	−9184.00	−0.382690
$833$	11022.0	0.458451
$834$	0	0
$835$	0	0
$836$	8789.00	0.363605
$837$	0	0
$838$	−22795.0	−0.939666
$839$	−6680.00	−0.274874	−0.137437	−	0.990511i	$-0.543886\pi$
$839$	−6680.00	−0.274874	−0.137437	+	0.990511i	$0.543886\pi$
$840$	0	0
$841$	−21473.0	−0.880438
$842$	−34275.0	−1.40284
$843$	0	0
$844$	79220.0	3.23088
$845$	0	0
$846$	0	0
$847$	363.000	0.0147259
$848$	−13528.0	−0.547822
$849$	0	0
$850$	0	0
$851$	−2147.00	−0.0864844
$852$	0	0
$853$	43358.0	1.74039	0.870193	−	0.492711i	$-0.163994\pi$
$853$	43358.0	1.74039	0.870193	+	0.492711i	$0.163994\pi$
$854$	−4800.00	−0.192333
$855$	0	0
$856$	87210.0	3.48222
$857$	−15585.0	−0.621206	−0.310603	−	0.950540i	$-0.600531\pi$
$857$	−15585.0	−0.621206	−0.310603	+	0.950540i	$0.600531\pi$
$858$	0	0
$859$	−17036.0	−0.676672	−0.338336	−	0.941025i	$-0.609864\pi$
$859$	−17036.0	−0.676672	−0.338336	+	0.941025i	$0.609864\pi$
$860$	0	0
$861$	0	0
$862$	53850.0	2.12777
$863$	−28064.0	−1.10696	−0.553482	−	0.832861i	$-0.686701\pi$
$863$	−28064.0	−1.10696	−0.553482	+	0.832861i	$0.686701\pi$
$864$	0	0
$865$	0	0
$866$	−42490.0	−1.66729
$867$	0	0
$868$	9078.00	0.354985
$869$	−7931.00	−0.309598
$870$	0	0
$871$	6400.00	0.248973
$872$	25920.0	1.00661
$873$	0	0
$874$	26555.0	1.02773
$875$	0	0
$876$	0	0
$877$	−22654.0	−0.872259	−0.436130	−	0.899884i	$-0.643651\pi$
$877$	−22654.0	−0.872259	−0.436130	+	0.899884i	$0.643651\pi$
$878$	−49175.0	−1.89018
$879$	0	0
$880$	0	0
$881$	22380.0	0.855847	0.427924	−	0.903815i	$-0.359245\pi$
$881$	22380.0	0.855847	0.427924	+	0.903815i	$0.359245\pi$
$882$	0	0
$883$	35174.0	1.34054	0.670271	−	0.742116i	$-0.266178\pi$
$883$	35174.0	1.34054	0.670271	+	0.742116i	$0.266178\pi$
$884$	−17952.0	−0.683022
$885$	0	0
$886$	−53725.0	−2.03716
$887$	−30868.0	−1.16848	−0.584242	−	0.811579i	$-0.698608\pi$
$887$	−30868.0	−1.16848	−0.584242	+	0.811579i	$0.698608\pi$
$888$	0	0
$889$	−5217.00	−0.196820
$890$	0	0
$891$	0	0
$892$	60520.0	2.27170
$893$	−9165.00	−0.343443
$894$	0	0
$895$	0	0
$896$	6345.00	0.236575
$897$	0	0
$898$	41780.0	1.55258
$899$	9612.00	0.356594
$900$	0	0
$901$	5016.00	0.185469
$902$	7645.00	0.282207
$903$	0	0
$904$	−49680.0	−1.82780
$905$	0	0
$906$	0	0
$907$	10070.0	0.368654	0.184327	−	0.982865i	$-0.440990\pi$
$907$	10070.0	0.368654	0.184327	+	0.982865i	$0.440990\pi$
$908$	79526.0	2.90657
$909$	0	0
$910$	0	0
$911$	−1885.00	−0.0685542	−0.0342771	−	0.999412i	$-0.510913\pi$
$911$	−1885.00	−0.0685542	−0.0342771	+	0.999412i	$0.510913\pi$
$912$	0	0
$913$	−1562.00	−0.0566207
$914$	35290.0	1.27712
$915$	0	0
$916$	−75599.0	−2.72692
$917$	−5454.00	−0.196409
$918$	0	0
$919$	23703.0	0.850805	0.425403	−	0.905004i	$-0.360133\pi$
$919$	23703.0	0.850805	0.425403	+	0.905004i	$0.360133\pi$
$920$	0	0
$921$	0	0
$922$	3230.00	0.115374
$923$	30304.0	1.08068
$924$	0	0
$925$	0	0
$926$	44910.0	1.59377
$927$	0	0
$928$	−4590.00	−0.162364
$929$	−53804.0	−1.90016	−0.950082	−	0.312001i	$-0.899001\pi$
$929$	−53804.0	−1.90016	−0.950082	+	0.312001i	$0.899001\pi$
$930$	0	0
$931$	−15698.0	−0.552611
$932$	−6987.00	−0.245565
$933$	0	0
$934$	−67380.0	−2.36054
$935$	0	0
$936$	0	0
$937$	1326.00	0.0462311	0.0231155	−	0.999733i	$-0.492641\pi$
$937$	1326.00	0.0462311	0.0231155	+	0.999733i	$0.492641\pi$
$938$	−3000.00	−0.104428
$939$	0	0
$940$	0	0
$941$	27109.0	0.939137	0.469569	−	0.882896i	$-0.344409\pi$
$941$	27109.0	0.939137	0.469569	+	0.882896i	$0.344409\pi$
$942$	0	0
$943$	15707.0	0.542408
$944$	55625.0	1.91784
$945$	0	0
$946$	16940.0	0.582206
$947$	−31143.0	−1.06865	−0.534325	−	0.845279i	$-0.679434\pi$
$947$	−31143.0	−1.06865	−0.534325	+	0.845279i	$0.679434\pi$
$948$	0	0
$949$	−14336.0	−0.490375
$950$	0	0
$951$	0	0
$952$	4455.00	0.151667
$953$	879.000	0.0298779	0.0149389	−	0.999888i	$-0.495245\pi$
$953$	879.000	0.0298779	0.0149389	+	0.999888i	$0.495245\pi$
$954$	0	0
$955$	0	0
$956$	108460.	3.66930
$957$	0	0
$958$	64980.0	2.19145
$959$	−2610.00	−0.0878846
$960$	0	0
$961$	1893.00	0.0635427
$962$	−3040.00	−0.101885
$963$	0	0
$964$	123794.	4.13603
$965$	0	0
$966$	0	0
$967$	−14824.0	−0.492976	−0.246488	−	0.969146i	$-0.579277\pi$
$967$	−14824.0	−0.492976	−0.246488	+	0.969146i	$0.579277\pi$
$968$	−5445.00	−0.180794
$969$	0	0
$970$	0	0
$971$	−34089.0	−1.12664	−0.563320	−	0.826239i	$-0.690476\pi$
$971$	−34089.0	−1.12664	−0.563320	+	0.826239i	$0.690476\pi$
$972$	0	0
$973$	−1908.00	−0.0628650
$974$	30130.0	0.991199
$975$	0	0
$976$	28480.0	0.934040
$977$	−33446.0	−1.09522	−0.547611	−	0.836733i	$-0.684463\pi$
$977$	−33446.0	−1.09522	−0.547611	+	0.836733i	$0.684463\pi$
$978$	0	0
$979$	−4444.00	−0.145077
$980$	0	0
$981$	0	0
$982$	58490.0	1.90070
$983$	52025.0	1.68804	0.844018	−	0.536315i	$-0.180184\pi$
$983$	52025.0	1.68804	0.844018	+	0.536315i	$0.180184\pi$
$984$	0	0
$985$	0	0
$986$	8910.00	0.287781
$987$	0	0
$988$	25568.0	0.823306
$989$	34804.0	1.11901
$990$	0	0
$991$	−41260.0	−1.32257	−0.661285	−	0.750135i	$-0.729989\pi$
$991$	−41260.0	−1.32257	−0.661285	+	0.750135i	$0.729989\pi$
$992$	−15130.0	−0.484252
$993$	0	0
$994$	−14205.0	−0.453275
$995$	0	0
$996$	0	0
$997$	−190.000	−0.00603547	−0.00301773	−	0.999995i	$-0.500961\pi$
$997$	−190.000	−0.00603547	−0.00301773	+	0.999995i	$0.500961\pi$
$998$	85260.0	2.70427
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.a.1.1		1
3.2	odd	2		825.4.a.j.1.1	yes	1
5.4	even	2		2475.4.a.k.1.1		1
15.2	even	4		825.4.c.b.199.2		2
15.8	even	4		825.4.c.b.199.1		2
15.14	odd	2		825.4.a.a.1.1	✓	1

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

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L-functions

Modular forms

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Representations

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Database

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