LMFDB - Embedded newform 2205.4.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2205,4,Mod(1,2205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, 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("2205.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$2205 = 3^{2} \cdot 5 \cdot 7^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	2205.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:	$130.099211563$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2205.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q-3.00000 q^{2} +1.00000 q^{4} +5.00000 q^{5} +21.0000 q^{8} -15.0000 q^{10} +45.0000 q^{11} -59.0000 q^{13} -71.0000 q^{16} -54.0000 q^{17} +121.000 q^{19} +5.00000 q^{20} -135.000 q^{22} -69.0000 q^{23} +25.0000 q^{25} +177.000 q^{26} +162.000 q^{29} +88.0000 q^{31} +45.0000 q^{32} +162.000 q^{34} -259.000 q^{37} -363.000 q^{38} +105.000 q^{40} +195.000 q^{41} -286.000 q^{43} +45.0000 q^{44} +207.000 q^{46} +45.0000 q^{47} -75.0000 q^{50} -59.0000 q^{52} -597.000 q^{53} +225.000 q^{55} -486.000 q^{58} -360.000 q^{59} -392.000 q^{61} -264.000 q^{62} +433.000 q^{64} -295.000 q^{65} -280.000 q^{67} -54.0000 q^{68} -48.0000 q^{71} -668.000 q^{73} +777.000 q^{74} +121.000 q^{76} +782.000 q^{79} -355.000 q^{80} -585.000 q^{82} +768.000 q^{83} -270.000 q^{85} +858.000 q^{86} +945.000 q^{88} -1194.00 q^{89} -69.0000 q^{92} -135.000 q^{94} +605.000 q^{95} -902.000 q^{97} +O(q^{100})

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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	−3.00000	−1.06066	−0.530330	−	0.847791i	$-0.677932\pi$
$2$	−3.00000	−1.06066	−0.530330	+	0.847791i	$0.677932\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.125000
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	21.0000	0.928078
$9$	0	0
$10$	−15.0000	−0.474342
$11$	45.0000	1.23346	0.616728	−	0.787177i	$-0.288458\pi$
$11$	45.0000	1.23346	0.616728	+	0.787177i	$0.288458\pi$
$12$	0	0
$13$	−59.0000	−1.25874	−0.629371	−	0.777105i	$-0.716688\pi$
$13$	−59.0000	−1.25874	−0.629371	+	0.777105i	$0.716688\pi$
$14$	0	0
$15$	0	0
$16$	−71.0000	−1.10938
$17$	−54.0000	−0.770407	−0.385204	−	0.922832i	$-0.625869\pi$
$17$	−54.0000	−0.770407	−0.385204	+	0.922832i	$0.625869\pi$
$18$	0	0
$19$	121.000	1.46102	0.730508	−	0.682904i	$-0.239283\pi$
$19$	121.000	1.46102	0.730508	+	0.682904i	$0.239283\pi$
$20$	5.00000	0.0559017
$21$	0	0
$22$	−135.000	−1.30828
$23$	−69.0000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−69.0000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	177.000	1.33510
$27$	0	0
$28$	0	0
$29$	162.000	1.03733	0.518666	−	0.854977i	$-0.326429\pi$
$29$	162.000	1.03733	0.518666	+	0.854977i	$0.326429\pi$
$30$	0	0
$31$	88.0000	0.509847	0.254924	−	0.966961i	$-0.417950\pi$
$31$	88.0000	0.509847	0.254924	+	0.966961i	$0.417950\pi$
$32$	45.0000	0.248592
$33$	0	0
$34$	162.000	0.817140
$35$	0	0
$36$	0	0
$37$	−259.000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−259.000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	−363.000	−1.54964
$39$	0	0
$40$	105.000	0.415049
$41$	195.000	0.742778	0.371389	−	0.928477i	$-0.378882\pi$
$41$	195.000	0.742778	0.371389	+	0.928477i	$0.378882\pi$
$42$	0	0
$43$	−286.000	−1.01429	−0.507146	−	0.861860i	$-0.669300\pi$
$43$	−286.000	−1.01429	−0.507146	+	0.861860i	$0.669300\pi$
$44$	45.0000	0.154182
$45$	0	0
$46$	207.000	0.663489
$47$	45.0000	0.139658	0.0698290	−	0.997559i	$-0.477755\pi$
$47$	45.0000	0.139658	0.0698290	+	0.997559i	$0.477755\pi$
$48$	0	0
$49$	0	0
$50$	−75.0000	−0.212132
$51$	0	0
$52$	−59.0000	−0.157343
$53$	−597.000	−1.54725	−0.773625	−	0.633644i	$-0.781559\pi$
$53$	−597.000	−1.54725	−0.773625	+	0.633644i	$0.781559\pi$
$54$	0	0
$55$	225.000	0.551618
$56$	0	0
$57$	0	0
$58$	−486.000	−1.10026
$59$	−360.000	−0.794373	−0.397187	−	0.917738i	$-0.630013\pi$
$59$	−360.000	−0.794373	−0.397187	+	0.917738i	$0.630013\pi$
$60$	0	0
$61$	−392.000	−0.822794	−0.411397	−	0.911456i	$-0.634959\pi$
$61$	−392.000	−0.822794	−0.411397	+	0.911456i	$0.634959\pi$
$62$	−264.000	−0.540775
$63$	0	0
$64$	433.000	0.845703
$65$	−295.000	−0.562927
$66$	0	0
$67$	−280.000	−0.510559	−0.255279	−	0.966867i	$-0.582167\pi$
$67$	−280.000	−0.510559	−0.255279	+	0.966867i	$0.582167\pi$
$68$	−54.0000	−0.0963009
$69$	0	0
$70$	0	0
$71$	−48.0000	−0.0802331	−0.0401166	−	0.999195i	$-0.512773\pi$
$71$	−48.0000	−0.0802331	−0.0401166	+	0.999195i	$0.512773\pi$
$72$	0	0
$73$	−668.000	−1.07101	−0.535503	−	0.844533i	$-0.679878\pi$
$73$	−668.000	−1.07101	−0.535503	+	0.844533i	$0.679878\pi$
$74$	777.000	1.22060
$75$	0	0
$76$	121.000	0.182627
$77$	0	0
$78$	0	0
$79$	782.000	1.11369	0.556847	−	0.830615i	$-0.312011\pi$
$79$	782.000	1.11369	0.556847	+	0.830615i	$0.312011\pi$
$80$	−355.000	−0.496128
$81$	0	0
$82$	−585.000	−0.787835
$83$	768.000	1.01565	0.507825	−	0.861460i	$-0.330450\pi$
$83$	768.000	1.01565	0.507825	+	0.861460i	$0.330450\pi$
$84$	0	0
$85$	−270.000	−0.344537
$86$	858.000	1.07582
$87$	0	0
$88$	945.000	1.14474
$89$	−1194.00	−1.42206	−0.711032	−	0.703159i	$-0.751772\pi$
$89$	−1194.00	−1.42206	−0.711032	+	0.703159i	$0.751772\pi$
$90$	0	0
$91$	0	0
$92$	−69.0000	−0.0781929
$93$	0	0
$94$	−135.000	−0.148130
$95$	605.000	0.653386
$96$	0	0
$97$	−902.000	−0.944167	−0.472084	−	0.881554i	$-0.656498\pi$
$97$	−902.000	−0.944167	−0.472084	+	0.881554i	$0.656498\pi$
$98$	0	0
$99$	0	0
$100$	25.0000	0.0250000
$101$	684.000	0.673867	0.336933	−	0.941528i	$-0.390610\pi$
$101$	684.000	0.673867	0.336933	+	0.941528i	$0.390610\pi$
$102$	0	0
$103$	1516.00	1.45025	0.725126	−	0.688616i	$-0.241782\pi$
$103$	1516.00	1.45025	0.725126	+	0.688616i	$0.241782\pi$
$104$	−1239.00	−1.16821
$105$	0	0
$106$	1791.00	1.64111
$107$	732.000	0.661356	0.330678	−	0.943744i	$-0.392723\pi$
$107$	732.000	0.661356	0.330678	+	0.943744i	$0.392723\pi$
$108$	0	0
$109$	−1600.00	−1.40598	−0.702992	−	0.711198i	$-0.748153\pi$
$109$	−1600.00	−1.40598	−0.702992	+	0.711198i	$0.748153\pi$
$110$	−675.000	−0.585079
$111$	0	0
$112$	0	0
$113$	1392.00	1.15883	0.579417	−	0.815031i	$-0.303280\pi$
$113$	1392.00	1.15883	0.579417	+	0.815031i	$0.303280\pi$
$114$	0	0
$115$	−345.000	−0.279751
$116$	162.000	0.129667
$117$	0	0
$118$	1080.00	0.842560
$119$	0	0
$120$	0	0
$121$	694.000	0.521412
$122$	1176.00	0.872705
$123$	0	0
$124$	88.0000	0.0637309
$125$	125.000	0.0894427
$126$	0	0
$127$	803.000	0.561061	0.280530	−	0.959845i	$-0.409490\pi$
$127$	803.000	0.561061	0.280530	+	0.959845i	$0.409490\pi$
$128$	−1659.00	−1.14560
$129$	0	0
$130$	885.000	0.597074
$131$	2019.00	1.34657	0.673286	−	0.739382i	$-0.264882\pi$
$131$	2019.00	1.34657	0.673286	+	0.739382i	$0.264882\pi$
$132$	0	0
$133$	0	0
$134$	840.000	0.541529
$135$	0	0
$136$	−1134.00	−0.714998
$137$	−60.0000	−0.0374171	−0.0187086	−	0.999825i	$-0.505955\pi$
$137$	−60.0000	−0.0374171	−0.0187086	+	0.999825i	$0.505955\pi$
$138$	0	0
$139$	1708.00	1.04224	0.521118	−	0.853485i	$-0.325515\pi$
$139$	1708.00	1.04224	0.521118	+	0.853485i	$0.325515\pi$
$140$	0	0
$141$	0	0
$142$	144.000	0.0851001
$143$	−2655.00	−1.55260
$144$	0	0
$145$	810.000	0.463909
$146$	2004.00	1.13597
$147$	0	0
$148$	−259.000	−0.143849
$149$	1086.00	0.597105	0.298552	−	0.954393i	$-0.403496\pi$
$149$	1086.00	0.597105	0.298552	+	0.954393i	$0.403496\pi$
$150$	0	0
$151$	−2866.00	−1.54458	−0.772291	−	0.635269i	$-0.780889\pi$
$151$	−2866.00	−1.54458	−0.772291	+	0.635269i	$0.780889\pi$
$152$	2541.00	1.35594
$153$	0	0
$154$	0	0
$155$	440.000	0.228011
$156$	0	0
$157$	229.000	0.116409	0.0582044	−	0.998305i	$-0.481462\pi$
$157$	229.000	0.116409	0.0582044	+	0.998305i	$0.481462\pi$
$158$	−2346.00	−1.18125
$159$	0	0
$160$	225.000	0.111174
$161$	0	0
$162$	0	0
$163$	−1228.00	−0.590088	−0.295044	−	0.955484i	$-0.595334\pi$
$163$	−1228.00	−0.590088	−0.295044	+	0.955484i	$0.595334\pi$
$164$	195.000	0.0928472
$165$	0	0
$166$	−2304.00	−1.07726
$167$	−1929.00	−0.893835	−0.446918	−	0.894575i	$-0.647478\pi$
$167$	−1929.00	−0.893835	−0.446918	+	0.894575i	$0.647478\pi$
$168$	0	0
$169$	1284.00	0.584433
$170$	810.000	0.365436
$171$	0	0
$172$	−286.000	−0.126787
$173$	−699.000	−0.307191	−0.153595	−	0.988134i	$-0.549085\pi$
$173$	−699.000	−0.307191	−0.153595	+	0.988134i	$0.549085\pi$
$174$	0	0
$175$	0	0
$176$	−3195.00	−1.36836
$177$	0	0
$178$	3582.00	1.50833
$179$	−3117.00	−1.30154	−0.650770	−	0.759275i	$-0.725554\pi$
$179$	−3117.00	−1.30154	−0.650770	+	0.759275i	$0.725554\pi$
$180$	0	0
$181$	1798.00	0.738366	0.369183	−	0.929357i	$-0.379638\pi$
$181$	1798.00	0.738366	0.369183	+	0.929357i	$0.379638\pi$
$182$	0	0
$183$	0	0
$184$	−1449.00	−0.580553
$185$	−1295.00	−0.514650
$186$	0	0
$187$	−2430.00	−0.950263
$188$	45.0000	0.0174572
$189$	0	0
$190$	−1815.00	−0.693021
$191$	2388.00	0.904658	0.452329	−	0.891851i	$-0.350593\pi$
$191$	2388.00	0.904658	0.452329	+	0.891851i	$0.350593\pi$
$192$	0	0
$193$	272.000	0.101446	0.0507228	−	0.998713i	$-0.483848\pi$
$193$	272.000	0.101446	0.0507228	+	0.998713i	$0.483848\pi$
$194$	2706.00	1.00144
$195$	0	0
$196$	0	0
$197$	2109.00	0.762741	0.381371	−	0.924422i	$-0.375452\pi$
$197$	2109.00	0.762741	0.381371	+	0.924422i	$0.375452\pi$
$198$	0	0
$199$	−1424.00	−0.507260	−0.253630	−	0.967301i	$-0.581625\pi$
$199$	−1424.00	−0.507260	−0.253630	+	0.967301i	$0.581625\pi$
$200$	525.000	0.185616
$201$	0	0
$202$	−2052.00	−0.714744
$203$	0	0
$204$	0	0
$205$	975.000	0.332180
$206$	−4548.00	−1.53822
$207$	0	0
$208$	4189.00	1.39642
$209$	5445.00	1.80210
$210$	0	0
$211$	−3625.00	−1.18273	−0.591363	−	0.806405i	$-0.701410\pi$
$211$	−3625.00	−1.18273	−0.591363	+	0.806405i	$0.701410\pi$
$212$	−597.000	−0.193406
$213$	0	0
$214$	−2196.00	−0.701474
$215$	−1430.00	−0.453606
$216$	0	0
$217$	0	0
$218$	4800.00	1.49127
$219$	0	0
$220$	225.000	0.0689523
$221$	3186.00	0.969745
$222$	0	0
$223$	4960.00	1.48944	0.744722	−	0.667374i	$-0.232582\pi$
$223$	4960.00	1.48944	0.744722	+	0.667374i	$0.232582\pi$
$224$	0	0
$225$	0	0
$226$	−4176.00	−1.22913
$227$	−1500.00	−0.438584	−0.219292	−	0.975659i	$-0.570375\pi$
$227$	−1500.00	−0.438584	−0.219292	+	0.975659i	$0.570375\pi$
$228$	0	0
$229$	−6092.00	−1.75795	−0.878975	−	0.476867i	$-0.841772\pi$
$229$	−6092.00	−1.75795	−0.878975	+	0.476867i	$0.841772\pi$
$230$	1035.00	0.296721
$231$	0	0
$232$	3402.00	0.962725
$233$	−138.000	−0.0388012	−0.0194006	−	0.999812i	$-0.506176\pi$
$233$	−138.000	−0.0388012	−0.0194006	+	0.999812i	$0.506176\pi$
$234$	0	0
$235$	225.000	0.0624569
$236$	−360.000	−0.0992966
$237$	0	0
$238$	0	0
$239$	5502.00	1.48910	0.744550	−	0.667567i	$-0.232664\pi$
$239$	5502.00	1.48910	0.744550	+	0.667567i	$0.232664\pi$
$240$	0	0
$241$	−3551.00	−0.949129	−0.474564	−	0.880221i	$-0.657394\pi$
$241$	−3551.00	−0.949129	−0.474564	+	0.880221i	$0.657394\pi$
$242$	−2082.00	−0.553041
$243$	0	0
$244$	−392.000	−0.102849
$245$	0	0
$246$	0	0
$247$	−7139.00	−1.83904
$248$	1848.00	0.473178
$249$	0	0
$250$	−375.000	−0.0948683
$251$	7065.00	1.77665	0.888324	−	0.459216i	$-0.151870\pi$
$251$	7065.00	1.77665	0.888324	+	0.459216i	$0.151870\pi$
$252$	0	0
$253$	−3105.00	−0.771580
$254$	−2409.00	−0.595095
$255$	0	0
$256$	1513.00	0.369385
$257$	−4080.00	−0.990286	−0.495143	−	0.868812i	$-0.664884\pi$
$257$	−4080.00	−0.990286	−0.495143	+	0.868812i	$0.664884\pi$
$258$	0	0
$259$	0	0
$260$	−295.000	−0.0703659
$261$	0	0
$262$	−6057.00	−1.42825
$263$	3288.00	0.770900	0.385450	−	0.922729i	$-0.374046\pi$
$263$	3288.00	0.770900	0.385450	+	0.922729i	$0.374046\pi$
$264$	0	0
$265$	−2985.00	−0.691951
$266$	0	0
$267$	0	0
$268$	−280.000	−0.0638199
$269$	−3264.00	−0.739813	−0.369906	−	0.929069i	$-0.620610\pi$
$269$	−3264.00	−0.739813	−0.369906	+	0.929069i	$0.620610\pi$
$270$	0	0
$271$	2752.00	0.616871	0.308436	−	0.951245i	$-0.400195\pi$
$271$	2752.00	0.616871	0.308436	+	0.951245i	$0.400195\pi$
$272$	3834.00	0.854671
$273$	0	0
$274$	180.000	0.0396869
$275$	1125.00	0.246691
$276$	0	0
$277$	−4690.00	−1.01731	−0.508655	−	0.860971i	$-0.669857\pi$
$277$	−4690.00	−1.01731	−0.508655	+	0.860971i	$0.669857\pi$
$278$	−5124.00	−1.10546
$279$	0	0
$280$	0	0
$281$	−7821.00	−1.66036	−0.830181	−	0.557494i	$-0.811763\pi$
$281$	−7821.00	−1.66036	−0.830181	+	0.557494i	$0.811763\pi$
$282$	0	0
$283$	658.000	0.138212	0.0691061	−	0.997609i	$-0.477985\pi$
$283$	658.000	0.138212	0.0691061	+	0.997609i	$0.477985\pi$
$284$	−48.0000	−0.0100291
$285$	0	0
$286$	7965.00	1.64678
$287$	0	0
$288$	0	0
$289$	−1997.00	−0.406473
$290$	−2430.00	−0.492050
$291$	0	0
$292$	−668.000	−0.133876
$293$	−5997.00	−1.19573	−0.597864	−	0.801597i	$-0.703984\pi$
$293$	−5997.00	−1.19573	−0.597864	+	0.801597i	$0.703984\pi$
$294$	0	0
$295$	−1800.00	−0.355254
$296$	−5439.00	−1.06803
$297$	0	0
$298$	−3258.00	−0.633325
$299$	4071.00	0.787398
$300$	0	0
$301$	0	0
$302$	8598.00	1.63828
$303$	0	0
$304$	−8591.00	−1.62081
$305$	−1960.00	−0.367965
$306$	0	0
$307$	6226.00	1.15745	0.578724	−	0.815523i	$-0.303551\pi$
$307$	6226.00	1.15745	0.578724	+	0.815523i	$0.303551\pi$
$308$	0	0
$309$	0	0
$310$	−1320.00	−0.241842
$311$	4680.00	0.853307	0.426653	−	0.904415i	$-0.359692\pi$
$311$	4680.00	0.853307	0.426653	+	0.904415i	$0.359692\pi$
$312$	0	0
$313$	−1028.00	−0.185642	−0.0928211	−	0.995683i	$-0.529588\pi$
$313$	−1028.00	−0.185642	−0.0928211	+	0.995683i	$0.529588\pi$
$314$	−687.000	−0.123470
$315$	0	0
$316$	782.000	0.139212
$317$	−8622.00	−1.52763	−0.763817	−	0.645433i	$-0.776677\pi$
$317$	−8622.00	−1.52763	−0.763817	+	0.645433i	$0.776677\pi$
$318$	0	0
$319$	7290.00	1.27950
$320$	2165.00	0.378210
$321$	0	0
$322$	0	0
$323$	−6534.00	−1.12558
$324$	0	0
$325$	−1475.00	−0.251749
$326$	3684.00	0.625883
$327$	0	0
$328$	4095.00	0.689355
$329$	0	0
$330$	0	0
$331$	−1999.00	−0.331949	−0.165974	−	0.986130i	$-0.553077\pi$
$331$	−1999.00	−0.331949	−0.165974	+	0.986130i	$0.553077\pi$
$332$	768.000	0.126956
$333$	0	0
$334$	5787.00	0.948056
$335$	−1400.00	−0.228329
$336$	0	0
$337$	5114.00	0.826639	0.413319	−	0.910586i	$-0.364369\pi$
$337$	5114.00	0.826639	0.413319	+	0.910586i	$0.364369\pi$
$338$	−3852.00	−0.619885
$339$	0	0
$340$	−270.000	−0.0430671
$341$	3960.00	0.628874
$342$	0	0
$343$	0	0
$344$	−6006.00	−0.941342
$345$	0	0
$346$	2097.00	0.325825
$347$	−4320.00	−0.668328	−0.334164	−	0.942515i	$-0.608454\pi$
$347$	−4320.00	−0.668328	−0.334164	+	0.942515i	$0.608454\pi$
$348$	0	0
$349$	−7922.00	−1.21506	−0.607529	−	0.794298i	$-0.707839\pi$
$349$	−7922.00	−1.21506	−0.607529	+	0.794298i	$0.707839\pi$
$350$	0	0
$351$	0	0
$352$	2025.00	0.306627
$353$	828.000	0.124844	0.0624221	−	0.998050i	$-0.480118\pi$
$353$	828.000	0.124844	0.0624221	+	0.998050i	$0.480118\pi$
$354$	0	0
$355$	−240.000	−0.0358813
$356$	−1194.00	−0.177758
$357$	0	0
$358$	9351.00	1.38049
$359$	1350.00	0.198469	0.0992344	−	0.995064i	$-0.468361\pi$
$359$	1350.00	0.198469	0.0992344	+	0.995064i	$0.468361\pi$
$360$	0	0
$361$	7782.00	1.13457
$362$	−5394.00	−0.783156
$363$	0	0
$364$	0	0
$365$	−3340.00	−0.478969
$366$	0	0
$367$	−2801.00	−0.398395	−0.199198	−	0.979959i	$-0.563834\pi$
$367$	−2801.00	−0.398395	−0.199198	+	0.979959i	$0.563834\pi$
$368$	4899.00	0.693962
$369$	0	0
$370$	3885.00	0.545869
$371$	0	0
$372$	0	0
$373$	6602.00	0.916457	0.458229	−	0.888834i	$-0.348484\pi$
$373$	6602.00	0.916457	0.458229	+	0.888834i	$0.348484\pi$
$374$	7290.00	1.00791
$375$	0	0
$376$	945.000	0.129613
$377$	−9558.00	−1.30573
$378$	0	0
$379$	−8305.00	−1.12559	−0.562796	−	0.826596i	$-0.690274\pi$
$379$	−8305.00	−1.12559	−0.562796	+	0.826596i	$0.690274\pi$
$380$	605.000	0.0816733
$381$	0	0
$382$	−7164.00	−0.959534
$383$	945.000	0.126076	0.0630382	−	0.998011i	$-0.479921\pi$
$383$	945.000	0.126076	0.0630382	+	0.998011i	$0.479921\pi$
$384$	0	0
$385$	0	0
$386$	−816.000	−0.107599
$387$	0	0
$388$	−902.000	−0.118021
$389$	−12036.0	−1.56876	−0.784382	−	0.620278i	$-0.787020\pi$
$389$	−12036.0	−1.56876	−0.784382	+	0.620278i	$0.787020\pi$
$390$	0	0
$391$	3726.00	0.481923
$392$	0	0
$393$	0	0
$394$	−6327.00	−0.809009
$395$	3910.00	0.498059
$396$	0	0
$397$	2698.00	0.341080	0.170540	−	0.985351i	$-0.445449\pi$
$397$	2698.00	0.341080	0.170540	+	0.985351i	$0.445449\pi$
$398$	4272.00	0.538030
$399$	0	0
$400$	−1775.00	−0.221875
$401$	−7053.00	−0.878329	−0.439165	−	0.898407i	$-0.644726\pi$
$401$	−7053.00	−0.878329	−0.439165	+	0.898407i	$0.644726\pi$
$402$	0	0
$403$	−5192.00	−0.641767
$404$	684.000	0.0842333
$405$	0	0
$406$	0	0
$407$	−11655.0	−1.41945
$408$	0	0
$409$	10870.0	1.31415	0.657074	−	0.753826i	$-0.271794\pi$
$409$	10870.0	1.31415	0.657074	+	0.753826i	$0.271794\pi$
$410$	−2925.00	−0.352330
$411$	0	0
$412$	1516.00	0.181281
$413$	0	0
$414$	0	0
$415$	3840.00	0.454212
$416$	−2655.00	−0.312914
$417$	0	0
$418$	−16335.0	−1.91141
$419$	−9729.00	−1.13435	−0.567175	−	0.823597i	$-0.691964\pi$
$419$	−9729.00	−1.13435	−0.567175	+	0.823597i	$0.691964\pi$
$420$	0	0
$421$	−12550.0	−1.45285	−0.726425	−	0.687246i	$-0.758819\pi$
$421$	−12550.0	−1.45285	−0.726425	+	0.687246i	$0.758819\pi$
$422$	10875.0	1.25447
$423$	0	0
$424$	−12537.0	−1.43597
$425$	−1350.00	−0.154081
$426$	0	0
$427$	0	0
$428$	732.000	0.0826695
$429$	0	0
$430$	4290.00	0.481121
$431$	−2988.00	−0.333937	−0.166969	−	0.985962i	$-0.553398\pi$
$431$	−2988.00	−0.333937	−0.166969	+	0.985962i	$0.553398\pi$
$432$	0	0
$433$	−16616.0	−1.84414	−0.922072	−	0.387019i	$-0.873505\pi$
$433$	−16616.0	−1.84414	−0.922072	+	0.387019i	$0.873505\pi$
$434$	0	0
$435$	0	0
$436$	−1600.00	−0.175748
$437$	−8349.00	−0.913929
$438$	0	0
$439$	−7346.00	−0.798646	−0.399323	−	0.916810i	$-0.630755\pi$
$439$	−7346.00	−0.798646	−0.399323	+	0.916810i	$0.630755\pi$
$440$	4725.00	0.511944
$441$	0	0
$442$	−9558.00	−1.02857
$443$	−12.0000	−0.00128699	−0.000643496	−	1.00000i	$-0.500205\pi$
$443$	−12.0000	−0.00128699	−0.000643496		1.00000i	$0.500205\pi$
$444$	0	0
$445$	−5970.00	−0.635967
$446$	−14880.0	−1.57979
$447$	0	0
$448$	0	0
$449$	−9669.00	−1.01628	−0.508138	−	0.861275i	$-0.669666\pi$
$449$	−9669.00	−1.01628	−0.508138	+	0.861275i	$0.669666\pi$
$450$	0	0
$451$	8775.00	0.916183
$452$	1392.00	0.144854
$453$	0	0
$454$	4500.00	0.465188
$455$	0	0
$456$	0	0
$457$	−9634.00	−0.986126	−0.493063	−	0.869994i	$-0.664123\pi$
$457$	−9634.00	−0.986126	−0.493063	+	0.869994i	$0.664123\pi$
$458$	18276.0	1.86459
$459$	0	0
$460$	−345.000	−0.0349689
$461$	−342.000	−0.0345521	−0.0172761	−	0.999851i	$-0.505499\pi$
$461$	−342.000	−0.0345521	−0.0172761	+	0.999851i	$0.505499\pi$
$462$	0	0
$463$	2411.00	0.242006	0.121003	−	0.992652i	$-0.461389\pi$
$463$	2411.00	0.242006	0.121003	+	0.992652i	$0.461389\pi$
$464$	−11502.0	−1.15079
$465$	0	0
$466$	414.000	0.0411549
$467$	−1206.00	−0.119501	−0.0597506	−	0.998213i	$-0.519031\pi$
$467$	−1206.00	−0.119501	−0.0597506	+	0.998213i	$0.519031\pi$
$468$	0	0
$469$	0	0
$470$	−675.000	−0.0662456
$471$	0	0
$472$	−7560.00	−0.737240
$473$	−12870.0	−1.25109
$474$	0	0
$475$	3025.00	0.292203
$476$	0	0
$477$	0	0
$478$	−16506.0	−1.57943
$479$	−432.000	−0.0412079	−0.0206039	−	0.999788i	$-0.506559\pi$
$479$	−432.000	−0.0412079	−0.0206039	+	0.999788i	$0.506559\pi$
$480$	0	0
$481$	15281.0	1.44855
$482$	10653.0	1.00670
$483$	0	0
$484$	694.000	0.0651766
$485$	−4510.00	−0.422244
$486$	0	0
$487$	−11896.0	−1.10690	−0.553449	−	0.832883i	$-0.686689\pi$
$487$	−11896.0	−1.10690	−0.553449	+	0.832883i	$0.686689\pi$
$488$	−8232.00	−0.763617
$489$	0	0
$490$	0	0
$491$	12276.0	1.12833	0.564163	−	0.825663i	$-0.309199\pi$
$491$	12276.0	1.12833	0.564163	+	0.825663i	$0.309199\pi$
$492$	0	0
$493$	−8748.00	−0.799169
$494$	21417.0	1.95060
$495$	0	0
$496$	−6248.00	−0.565612
$497$	0	0
$498$	0	0
$499$	−10876.0	−0.975705	−0.487852	−	0.872926i	$-0.662220\pi$
$499$	−10876.0	−0.975705	−0.487852	+	0.872926i	$0.662220\pi$
$500$	125.000	0.0111803
$501$	0	0
$502$	−21195.0	−1.88442
$503$	12000.0	1.06372	0.531862	−	0.846831i	$-0.321492\pi$
$503$	12000.0	1.06372	0.531862	+	0.846831i	$0.321492\pi$
$504$	0	0
$505$	3420.00	0.301362
$506$	9315.00	0.818384
$507$	0	0
$508$	803.000	0.0701326
$509$	−11682.0	−1.01728	−0.508640	−	0.860979i	$-0.669852\pi$
$509$	−11682.0	−1.01728	−0.508640	+	0.860979i	$0.669852\pi$
$510$	0	0
$511$	0	0
$512$	8733.00	0.753804
$513$	0	0
$514$	12240.0	1.05036
$515$	7580.00	0.648572
$516$	0	0
$517$	2025.00	0.172262
$518$	0	0
$519$	0	0
$520$	−6195.00	−0.522440
$521$	9609.00	0.808019	0.404010	−	0.914755i	$-0.367616\pi$
$521$	9609.00	0.808019	0.404010	+	0.914755i	$0.367616\pi$
$522$	0	0
$523$	−21188.0	−1.77148	−0.885742	−	0.464177i	$-0.846350\pi$
$523$	−21188.0	−1.77148	−0.885742	+	0.464177i	$0.846350\pi$
$524$	2019.00	0.168321
$525$	0	0
$526$	−9864.00	−0.817663
$527$	−4752.00	−0.392790
$528$	0	0
$529$	−7406.00	−0.608696
$530$	8955.00	0.733925
$531$	0	0
$532$	0	0
$533$	−11505.0	−0.934966
$534$	0	0
$535$	3660.00	0.295767
$536$	−5880.00	−0.473838
$537$	0	0
$538$	9792.00	0.784690
$539$	0	0
$540$	0	0
$541$	8072.00	0.641483	0.320742	−	0.947167i	$-0.396068\pi$
$541$	8072.00	0.641483	0.320742	+	0.947167i	$0.396068\pi$
$542$	−8256.00	−0.654291
$543$	0	0
$544$	−2430.00	−0.191517
$545$	−8000.00	−0.628775
$546$	0	0
$547$	344.000	0.0268892	0.0134446	−	0.999910i	$-0.495720\pi$
$547$	344.000	0.0268892	0.0134446	+	0.999910i	$0.495720\pi$
$548$	−60.0000	−0.00467714
$549$	0	0
$550$	−3375.00	−0.261655
$551$	19602.0	1.51556
$552$	0	0
$553$	0	0
$554$	14070.0	1.07902
$555$	0	0
$556$	1708.00	0.130279
$557$	−18363.0	−1.39689	−0.698443	−	0.715666i	$-0.746123\pi$
$557$	−18363.0	−1.39689	−0.698443	+	0.715666i	$0.746123\pi$
$558$	0	0
$559$	16874.0	1.27673
$560$	0	0
$561$	0	0
$562$	23463.0	1.76108
$563$	−6294.00	−0.471155	−0.235578	−	0.971856i	$-0.575698\pi$
$563$	−6294.00	−0.471155	−0.235578	+	0.971856i	$0.575698\pi$
$564$	0	0
$565$	6960.00	0.518247
$566$	−1974.00	−0.146596
$567$	0	0
$568$	−1008.00	−0.0744626
$569$	−11733.0	−0.864452	−0.432226	−	0.901765i	$-0.642272\pi$
$569$	−11733.0	−0.864452	−0.432226	+	0.901765i	$0.642272\pi$
$570$	0	0
$571$	1052.00	0.0771013	0.0385506	−	0.999257i	$-0.487726\pi$
$571$	1052.00	0.0771013	0.0385506	+	0.999257i	$0.487726\pi$
$572$	−2655.00	−0.194075
$573$	0	0
$574$	0	0
$575$	−1725.00	−0.125109
$576$	0	0
$577$	13156.0	0.949205	0.474603	−	0.880200i	$-0.342592\pi$
$577$	13156.0	0.949205	0.474603	+	0.880200i	$0.342592\pi$
$578$	5991.00	0.431129
$579$	0	0
$580$	810.000	0.0579887
$581$	0	0
$582$	0	0
$583$	−26865.0	−1.90846
$584$	−14028.0	−0.993977
$585$	0	0
$586$	17991.0	1.26826
$587$	−13368.0	−0.939960	−0.469980	−	0.882677i	$-0.655739\pi$
$587$	−13368.0	−0.939960	−0.469980	+	0.882677i	$0.655739\pi$
$588$	0	0
$589$	10648.0	0.744895
$590$	5400.00	0.376804
$591$	0	0
$592$	18389.0	1.27666
$593$	26664.0	1.84647	0.923237	−	0.384231i	$-0.125533\pi$
$593$	26664.0	1.84647	0.923237	+	0.384231i	$0.125533\pi$
$594$	0	0
$595$	0	0
$596$	1086.00	0.0746381
$597$	0	0
$598$	−12213.0	−0.835162
$599$	−7614.00	−0.519365	−0.259682	−	0.965694i	$-0.583618\pi$
$599$	−7614.00	−0.519365	−0.259682	+	0.965694i	$0.583618\pi$
$600$	0	0
$601$	−6410.00	−0.435057	−0.217529	−	0.976054i	$-0.569800\pi$
$601$	−6410.00	−0.435057	−0.217529	+	0.976054i	$0.569800\pi$
$602$	0	0
$603$	0	0
$604$	−2866.00	−0.193073
$605$	3470.00	0.233183
$606$	0	0
$607$	21469.0	1.43558	0.717792	−	0.696257i	$-0.245153\pi$
$607$	21469.0	1.43558	0.717792	+	0.696257i	$0.245153\pi$
$608$	5445.00	0.363197
$609$	0	0
$610$	5880.00	0.390286
$611$	−2655.00	−0.175793
$612$	0	0
$613$	3737.00	0.246225	0.123113	−	0.992393i	$-0.460712\pi$
$613$	3737.00	0.246225	0.123113	+	0.992393i	$0.460712\pi$
$614$	−18678.0	−1.22766
$615$	0	0
$616$	0	0
$617$	−18078.0	−1.17957	−0.589784	−	0.807561i	$-0.700787\pi$
$617$	−18078.0	−1.17957	−0.589784	+	0.807561i	$0.700787\pi$
$618$	0	0
$619$	−12287.0	−0.797829	−0.398915	−	0.916988i	$-0.630613\pi$
$619$	−12287.0	−0.797829	−0.398915	+	0.916988i	$0.630613\pi$
$620$	440.000	0.0285013
$621$	0	0
$622$	−14040.0	−0.905069
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	3084.00	0.196903
$627$	0	0
$628$	229.000	0.0145511
$629$	13986.0	0.886579
$630$	0	0
$631$	−9580.00	−0.604396	−0.302198	−	0.953245i	$-0.597720\pi$
$631$	−9580.00	−0.604396	−0.302198	+	0.953245i	$0.597720\pi$
$632$	16422.0	1.03360
$633$	0	0
$634$	25866.0	1.62030
$635$	4015.00	0.250914
$636$	0	0
$637$	0	0
$638$	−21870.0	−1.35712
$639$	0	0
$640$	−8295.00	−0.512326
$641$	−10779.0	−0.664189	−0.332094	−	0.943246i	$-0.607755\pi$
$641$	−10779.0	−0.664189	−0.332094	+	0.943246i	$0.607755\pi$
$642$	0	0
$643$	−8882.00	−0.544746	−0.272373	−	0.962192i	$-0.587809\pi$
$643$	−8882.00	−0.544746	−0.272373	+	0.962192i	$0.587809\pi$
$644$	0	0
$645$	0	0
$646$	19602.0	1.19386
$647$	−11019.0	−0.669554	−0.334777	−	0.942297i	$-0.608661\pi$
$647$	−11019.0	−0.669554	−0.334777	+	0.942297i	$0.608661\pi$
$648$	0	0
$649$	−16200.0	−0.979824
$650$	4425.00	0.267020
$651$	0	0
$652$	−1228.00	−0.0737610
$653$	−22323.0	−1.33777	−0.668887	−	0.743364i	$-0.733229\pi$
$653$	−22323.0	−1.33777	−0.668887	+	0.743364i	$0.733229\pi$
$654$	0	0
$655$	10095.0	0.602205
$656$	−13845.0	−0.824019
$657$	0	0
$658$	0	0
$659$	11856.0	0.700826	0.350413	−	0.936595i	$-0.386041\pi$
$659$	11856.0	0.700826	0.350413	+	0.936595i	$0.386041\pi$
$660$	0	0
$661$	33244.0	1.95619	0.978095	−	0.208158i	$-0.0667470\pi$
$661$	33244.0	1.95619	0.978095	+	0.208158i	$0.0667470\pi$
$662$	5997.00	0.352085
$663$	0	0
$664$	16128.0	0.942602
$665$	0	0
$666$	0	0
$667$	−11178.0	−0.648896
$668$	−1929.00	−0.111729
$669$	0	0
$670$	4200.00	0.242179
$671$	−17640.0	−1.01488
$672$	0	0
$673$	−12322.0	−0.705763	−0.352881	−	0.935668i	$-0.614798\pi$
$673$	−12322.0	−0.705763	−0.352881	+	0.935668i	$0.614798\pi$
$674$	−15342.0	−0.876783
$675$	0	0
$676$	1284.00	0.0730542
$677$	−12597.0	−0.715129	−0.357564	−	0.933889i	$-0.616393\pi$
$677$	−12597.0	−0.715129	−0.357564	+	0.933889i	$0.616393\pi$
$678$	0	0
$679$	0	0
$680$	−5670.00	−0.319757
$681$	0	0
$682$	−11880.0	−0.667022
$683$	8340.00	0.467235	0.233617	−	0.972329i	$-0.424944\pi$
$683$	8340.00	0.467235	0.233617	+	0.972329i	$0.424944\pi$
$684$	0	0
$685$	−300.000	−0.0167334
$686$	0	0
$687$	0	0
$688$	20306.0	1.12523
$689$	35223.0	1.94759
$690$	0	0
$691$	20200.0	1.11208	0.556038	−	0.831157i	$-0.312321\pi$
$691$	20200.0	1.11208	0.556038	+	0.831157i	$0.312321\pi$
$692$	−699.000	−0.0383988
$693$	0	0
$694$	12960.0	0.708869
$695$	8540.00	0.466102
$696$	0	0
$697$	−10530.0	−0.572241
$698$	23766.0	1.28876
$699$	0	0
$700$	0	0
$701$	−474.000	−0.0255388	−0.0127694	−	0.999918i	$-0.504065\pi$
$701$	−474.000	−0.0255388	−0.0127694	+	0.999918i	$0.504065\pi$
$702$	0	0
$703$	−31339.0	−1.68133
$704$	19485.0	1.04314
$705$	0	0
$706$	−2484.00	−0.132417
$707$	0	0
$708$	0	0
$709$	−25126.0	−1.33093	−0.665463	−	0.746431i	$-0.731766\pi$
$709$	−25126.0	−1.33093	−0.665463	+	0.746431i	$0.731766\pi$
$710$	720.000	0.0380579
$711$	0	0
$712$	−25074.0	−1.31979
$713$	−6072.00	−0.318932
$714$	0	0
$715$	−13275.0	−0.694345
$716$	−3117.00	−0.162692
$717$	0	0
$718$	−4050.00	−0.210508
$719$	−7296.00	−0.378435	−0.189218	−	0.981935i	$-0.560595\pi$
$719$	−7296.00	−0.378435	−0.189218	+	0.981935i	$0.560595\pi$
$720$	0	0
$721$	0	0
$722$	−23346.0	−1.20339
$723$	0	0
$724$	1798.00	0.0922958
$725$	4050.00	0.207467
$726$	0	0
$727$	15421.0	0.786703	0.393352	−	0.919388i	$-0.371316\pi$
$727$	15421.0	0.786703	0.393352	+	0.919388i	$0.371316\pi$
$728$	0	0
$729$	0	0
$730$	10020.0	0.508023
$731$	15444.0	0.781419
$732$	0	0
$733$	29167.0	1.46972	0.734862	−	0.678217i	$-0.237247\pi$
$733$	29167.0	1.46972	0.734862	+	0.678217i	$0.237247\pi$
$734$	8403.00	0.422562
$735$	0	0
$736$	−3105.00	−0.155505
$737$	−12600.0	−0.629752
$738$	0	0
$739$	−13381.0	−0.666073	−0.333037	−	0.942914i	$-0.608073\pi$
$739$	−13381.0	−0.666073	−0.333037	+	0.942914i	$0.608073\pi$
$740$	−1295.00	−0.0643313
$741$	0	0
$742$	0	0
$743$	−5487.00	−0.270927	−0.135463	−	0.990782i	$-0.543252\pi$
$743$	−5487.00	−0.270927	−0.135463	+	0.990782i	$0.543252\pi$
$744$	0	0
$745$	5430.00	0.267033
$746$	−19806.0	−0.972050
$747$	0	0
$748$	−2430.00	−0.118783
$749$	0	0
$750$	0	0
$751$	6638.00	0.322535	0.161268	−	0.986911i	$-0.448442\pi$
$751$	6638.00	0.322535	0.161268	+	0.986911i	$0.448442\pi$
$752$	−3195.00	−0.154933
$753$	0	0
$754$	28674.0	1.38494
$755$	−14330.0	−0.690758
$756$	0	0
$757$	14846.0	0.712797	0.356398	−	0.934334i	$-0.384005\pi$
$757$	14846.0	0.712797	0.356398	+	0.934334i	$0.384005\pi$
$758$	24915.0	1.19387
$759$	0	0
$760$	12705.0	0.606393
$761$	−3651.00	−0.173914	−0.0869571	−	0.996212i	$-0.527714\pi$
$761$	−3651.00	−0.173914	−0.0869571	+	0.996212i	$0.527714\pi$
$762$	0	0
$763$	0	0
$764$	2388.00	0.113082
$765$	0	0
$766$	−2835.00	−0.133724
$767$	21240.0	0.999911
$768$	0	0
$769$	−29855.0	−1.40000	−0.699999	−	0.714144i	$-0.746816\pi$
$769$	−29855.0	−1.40000	−0.699999	+	0.714144i	$0.746816\pi$
$770$	0	0
$771$	0	0
$772$	272.000	0.0126807
$773$	−6519.00	−0.303327	−0.151664	−	0.988432i	$-0.548463\pi$
$773$	−6519.00	−0.303327	−0.151664	+	0.988432i	$0.548463\pi$
$774$	0	0
$775$	2200.00	0.101969
$776$	−18942.0	−0.876261
$777$	0	0
$778$	36108.0	1.66393
$779$	23595.0	1.08521
$780$	0	0
$781$	−2160.00	−0.0989640
$782$	−11178.0	−0.511157
$783$	0	0
$784$	0	0
$785$	1145.00	0.0520596
$786$	0	0
$787$	−35114.0	−1.59044	−0.795222	−	0.606319i	$-0.792646\pi$
$787$	−35114.0	−1.59044	−0.795222	+	0.606319i	$0.792646\pi$
$788$	2109.00	0.0953427
$789$	0	0
$790$	−11730.0	−0.528272
$791$	0	0
$792$	0	0
$793$	23128.0	1.03569
$794$	−8094.00	−0.361770
$795$	0	0
$796$	−1424.00	−0.0634075
$797$	20910.0	0.929323	0.464661	−	0.885488i	$-0.346176\pi$
$797$	20910.0	0.929323	0.464661	+	0.885488i	$0.346176\pi$
$798$	0	0
$799$	−2430.00	−0.107594
$800$	1125.00	0.0497184
$801$	0	0
$802$	21159.0	0.931609
$803$	−30060.0	−1.32104
$804$	0	0
$805$	0	0
$806$	15576.0	0.680696
$807$	0	0
$808$	14364.0	0.625401
$809$	−4431.00	−0.192566	−0.0962829	−	0.995354i	$-0.530695\pi$
$809$	−4431.00	−0.192566	−0.0962829	+	0.995354i	$0.530695\pi$
$810$	0	0
$811$	9577.00	0.414666	0.207333	−	0.978270i	$-0.433522\pi$
$811$	9577.00	0.414666	0.207333	+	0.978270i	$0.433522\pi$
$812$	0	0
$813$	0	0
$814$	34965.0	1.50556
$815$	−6140.00	−0.263895
$816$	0	0
$817$	−34606.0	−1.48190
$818$	−32610.0	−1.39387
$819$	0	0
$820$	975.000	0.0415225
$821$	10938.0	0.464968	0.232484	−	0.972600i	$-0.425315\pi$
$821$	10938.0	0.464968	0.232484	+	0.972600i	$0.425315\pi$
$822$	0	0
$823$	11540.0	0.488772	0.244386	−	0.969678i	$-0.421414\pi$
$823$	11540.0	0.488772	0.244386	+	0.969678i	$0.421414\pi$
$824$	31836.0	1.34595
$825$	0	0
$826$	0	0
$827$	18762.0	0.788898	0.394449	−	0.918918i	$-0.370935\pi$
$827$	18762.0	0.788898	0.394449	+	0.918918i	$0.370935\pi$
$828$	0	0
$829$	39610.0	1.65948	0.829742	−	0.558147i	$-0.188488\pi$
$829$	39610.0	1.65948	0.829742	+	0.558147i	$0.188488\pi$
$830$	−11520.0	−0.481765
$831$	0	0
$832$	−25547.0	−1.06452
$833$	0	0
$834$	0	0
$835$	−9645.00	−0.399735
$836$	5445.00	0.225262
$837$	0	0
$838$	29187.0	1.20316
$839$	39162.0	1.61147	0.805734	−	0.592277i	$-0.201771\pi$
$839$	39162.0	1.61147	0.805734	+	0.592277i	$0.201771\pi$
$840$	0	0
$841$	1855.00	0.0760589
$842$	37650.0	1.54098
$843$	0	0
$844$	−3625.00	−0.147841
$845$	6420.00	0.261367
$846$	0	0
$847$	0	0
$848$	42387.0	1.71648
$849$	0	0
$850$	4050.00	0.163428
$851$	17871.0	0.719871
$852$	0	0
$853$	11527.0	0.462693	0.231346	−	0.972871i	$-0.425687\pi$
$853$	11527.0	0.462693	0.231346	+	0.972871i	$0.425687\pi$
$854$	0	0
$855$	0	0
$856$	15372.0	0.613790
$857$	−41826.0	−1.66715	−0.833576	−	0.552405i	$-0.813710\pi$
$857$	−41826.0	−1.66715	−0.833576	+	0.552405i	$0.813710\pi$
$858$	0	0
$859$	−35192.0	−1.39783	−0.698915	−	0.715205i	$-0.746333\pi$
$859$	−35192.0	−1.39783	−0.698915	+	0.715205i	$0.746333\pi$
$860$	−1430.00	−0.0567007
$861$	0	0
$862$	8964.00	0.354194
$863$	9063.00	0.357483	0.178742	−	0.983896i	$-0.442797\pi$
$863$	9063.00	0.357483	0.178742	+	0.983896i	$0.442797\pi$
$864$	0	0
$865$	−3495.00	−0.137380
$866$	49848.0	1.95601
$867$	0	0
$868$	0	0
$869$	35190.0	1.37369
$870$	0	0
$871$	16520.0	0.642662
$872$	−33600.0	−1.30486
$873$	0	0
$874$	25047.0	0.969368
$875$	0	0
$876$	0	0
$877$	28439.0	1.09500	0.547501	−	0.836805i	$-0.315579\pi$
$877$	28439.0	1.09500	0.547501	+	0.836805i	$0.315579\pi$
$878$	22038.0	0.847092
$879$	0	0
$880$	−15975.0	−0.611951
$881$	−9303.00	−0.355762	−0.177881	−	0.984052i	$-0.556924\pi$
$881$	−9303.00	−0.355762	−0.177881	+	0.984052i	$0.556924\pi$
$882$	0	0
$883$	−14728.0	−0.561310	−0.280655	−	0.959809i	$-0.590552\pi$
$883$	−14728.0	−0.561310	−0.280655	+	0.959809i	$0.590552\pi$
$884$	3186.00	0.121218
$885$	0	0
$886$	36.0000	0.00136506
$887$	−17016.0	−0.644128	−0.322064	−	0.946718i	$-0.604377\pi$
$887$	−17016.0	−0.644128	−0.322064	+	0.946718i	$0.604377\pi$
$888$	0	0
$889$	0	0
$890$	17910.0	0.674544
$891$	0	0
$892$	4960.00	0.186181
$893$	5445.00	0.204043
$894$	0	0
$895$	−15585.0	−0.582066
$896$	0	0
$897$	0	0
$898$	29007.0	1.07792
$899$	14256.0	0.528881
$900$	0	0
$901$	32238.0	1.19201
$902$	−26325.0	−0.971759
$903$	0	0
$904$	29232.0	1.07549
$905$	8990.00	0.330207
$906$	0	0
$907$	−24922.0	−0.912372	−0.456186	−	0.889884i	$-0.650785\pi$
$907$	−24922.0	−0.912372	−0.456186	+	0.889884i	$0.650785\pi$
$908$	−1500.00	−0.0548230
$909$	0	0
$910$	0	0
$911$	−30714.0	−1.11701	−0.558507	−	0.829500i	$-0.688626\pi$
$911$	−30714.0	−1.11701	−0.558507	+	0.829500i	$0.688626\pi$
$912$	0	0
$913$	34560.0	1.25276
$914$	28902.0	1.04594
$915$	0	0
$916$	−6092.00	−0.219744
$917$	0	0
$918$	0	0
$919$	17426.0	0.625496	0.312748	−	0.949836i	$-0.398750\pi$
$919$	17426.0	0.625496	0.312748	+	0.949836i	$0.398750\pi$
$920$	−7245.00	−0.259631
$921$	0	0
$922$	1026.00	0.0366481
$923$	2832.00	0.100993
$924$	0	0
$925$	−6475.00	−0.230159
$926$	−7233.00	−0.256686
$927$	0	0
$928$	7290.00	0.257873
$929$	26649.0	0.941147	0.470573	−	0.882361i	$-0.344047\pi$
$929$	26649.0	0.941147	0.470573	+	0.882361i	$0.344047\pi$
$930$	0	0
$931$	0	0
$932$	−138.000	−0.00485015
$933$	0	0
$934$	3618.00	0.126750
$935$	−12150.0	−0.424971
$936$	0	0
$937$	−27686.0	−0.965274	−0.482637	−	0.875820i	$-0.660321\pi$
$937$	−27686.0	−0.965274	−0.482637	+	0.875820i	$0.660321\pi$
$938$	0	0
$939$	0	0
$940$	225.000	0.00780712
$941$	−17808.0	−0.616923	−0.308461	−	0.951237i	$-0.599814\pi$
$941$	−17808.0	−0.616923	−0.308461	+	0.951237i	$0.599814\pi$
$942$	0	0
$943$	−13455.0	−0.464640
$944$	25560.0	0.881258
$945$	0	0
$946$	38610.0	1.32698
$947$	6906.00	0.236974	0.118487	−	0.992956i	$-0.462196\pi$
$947$	6906.00	0.236974	0.118487	+	0.992956i	$0.462196\pi$
$948$	0	0
$949$	39412.0	1.34812
$950$	−9075.00	−0.309928
$951$	0	0
$952$	0	0
$953$	20940.0	0.711766	0.355883	−	0.934530i	$-0.384180\pi$
$953$	20940.0	0.711766	0.355883	+	0.934530i	$0.384180\pi$
$954$	0	0
$955$	11940.0	0.404575
$956$	5502.00	0.186137
$957$	0	0
$958$	1296.00	0.0437076
$959$	0	0
$960$	0	0
$961$	−22047.0	−0.740056
$962$	−45843.0	−1.53642
$963$	0	0
$964$	−3551.00	−0.118641
$965$	1360.00	0.0453678
$966$	0	0
$967$	9176.00	0.305150	0.152575	−	0.988292i	$-0.451243\pi$
$967$	9176.00	0.305150	0.152575	+	0.988292i	$0.451243\pi$
$968$	14574.0	0.483911
$969$	0	0
$970$	13530.0	0.447858
$971$	29763.0	0.983666	0.491833	−	0.870689i	$-0.336327\pi$
$971$	29763.0	0.983666	0.491833	+	0.870689i	$0.336327\pi$
$972$	0	0
$973$	0	0
$974$	35688.0	1.17404
$975$	0	0
$976$	27832.0	0.912788
$977$	38490.0	1.26039	0.630197	−	0.776436i	$-0.282974\pi$
$977$	38490.0	1.26039	0.630197	+	0.776436i	$0.282974\pi$
$978$	0	0
$979$	−53730.0	−1.75405
$980$	0	0
$981$	0	0
$982$	−36828.0	−1.19677
$983$	−12609.0	−0.409120	−0.204560	−	0.978854i	$-0.565576\pi$
$983$	−12609.0	−0.409120	−0.204560	+	0.978854i	$0.565576\pi$
$984$	0	0
$985$	10545.0	0.341108
$986$	26244.0	0.847646
$987$	0	0
$988$	−7139.00	−0.229880
$989$	19734.0	0.634484
$990$	0	0
$991$	19820.0	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	19820.0	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	3960.00	0.126744
$993$	0	0
$994$	0	0
$995$	−7120.00	−0.226853
$996$	0	0
$997$	−46034.0	−1.46230	−0.731149	−	0.682218i	$-0.761016\pi$
$997$	−46034.0	−1.46230	−0.731149	+	0.682218i	$0.761016\pi$
$998$	32628.0	1.03489
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.g.1.1		1
3.2	odd	2		245.4.a.f.1.1		1
7.3	odd	6		315.4.j.b.226.1		2
7.5	odd	6		315.4.j.b.46.1		2
7.6	odd	2		2205.4.a.e.1.1		1
15.14	odd	2		1225.4.a.a.1.1		1
21.2	odd	6		245.4.e.a.116.1		2
21.5	even	6		35.4.e.a.11.1	✓	2
21.11	odd	6		245.4.e.a.226.1		2
21.17	even	6		35.4.e.a.16.1	yes	2
21.20	even	2		245.4.a.e.1.1		1
84.47	odd	6		560.4.q.b.81.1		2
84.59	odd	6		560.4.q.b.401.1		2
105.17	odd	12		175.4.k.b.149.1		4
105.38	odd	12		175.4.k.b.149.2		4
105.47	odd	12		175.4.k.b.74.2		4
105.59	even	6		175.4.e.b.51.1		2
105.68	odd	12		175.4.k.b.74.1		4
105.89	even	6		175.4.e.b.151.1		2
105.104	even	2		1225.4.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.e.a.11.1	✓	2	21.5	even	6
35.4.e.a.16.1	yes	2	21.17	even	6
175.4.e.b.51.1		2	105.59	even	6
175.4.e.b.151.1		2	105.89	even	6
175.4.k.b.74.1		4	105.68	odd	12
175.4.k.b.74.2		4	105.47	odd	12
175.4.k.b.149.1		4	105.17	odd	12
175.4.k.b.149.2		4	105.38	odd	12
245.4.a.e.1.1		1	21.20	even	2
245.4.a.f.1.1		1	3.2	odd	2
245.4.e.a.116.1		2	21.2	odd	6
245.4.e.a.226.1		2	21.11	odd	6
315.4.j.b.46.1		2	7.5	odd	6
315.4.j.b.226.1		2	7.3	odd	6
560.4.q.b.81.1		2	84.47	odd	6
560.4.q.b.401.1		2	84.59	odd	6
1225.4.a.a.1.1		1	15.14	odd	2
1225.4.a.b.1.1		1	105.104	even	2
2205.4.a.e.1.1		1	7.6	odd	2
2205.4.a.g.1.1		1	1.1	even	1	trivial

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Classical	Maass
Hilbert	Bianchi

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

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2205.4.a.g.1.1

Level	$2205$
Weight	$4$
Character	2205.1
Self dual	yes
Analytic conductor	$130.099$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$