LMFDB - Embedded newform 2205.4.a.f.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:	$2$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 315)
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)

$f(q)$

$=$

$q-3.00000 q^{2} +1.00000 q^{4} +5.00000 q^{5} +21.0000 q^{8} +O(q^{10})$

$q-3.00000 q^{2} +1.00000 q^{4} +5.00000 q^{5} +21.0000 q^{8} -15.0000 q^{10} -60.0000 q^{11} -38.0000 q^{13} -71.0000 q^{16} -84.0000 q^{17} -110.000 q^{19} +5.00000 q^{20} +180.000 q^{22} -120.000 q^{23} +25.0000 q^{25} +114.000 q^{26} -162.000 q^{29} -236.000 q^{31} +45.0000 q^{32} +252.000 q^{34} -376.000 q^{37} +330.000 q^{38} +105.000 q^{40} -126.000 q^{41} -34.0000 q^{43} -60.0000 q^{44} +360.000 q^{46} -6.00000 q^{47} -75.0000 q^{50} -38.0000 q^{52} -582.000 q^{53} -300.000 q^{55} +486.000 q^{58} +492.000 q^{59} +880.000 q^{61} +708.000 q^{62} +433.000 q^{64} -190.000 q^{65} -826.000 q^{67} -84.0000 q^{68} +666.000 q^{71} +826.000 q^{73} +1128.00 q^{74} -110.000 q^{76} -592.000 q^{79} -355.000 q^{80} +378.000 q^{82} +792.000 q^{83} -420.000 q^{85} +102.000 q^{86} -1260.00 q^{88} +1002.00 q^{89} -120.000 q^{92} +18.0000 q^{94} -550.000 q^{95} -1442.00 q^{97} +O(q^{100})$

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 $

$2$	−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
$11$	−60.0000	−1.64461	−0.822304	−	0.569049i	$-0.807311\pi$
$11$	−60.0000	−1.64461	−0.822304	+	0.569049i	$0.807311\pi$
$12$	0	0
$13$	−38.0000	−0.810716	−0.405358	−	0.914158i	$-0.632853\pi$
$13$	−38.0000	−0.810716	−0.405358	+	0.914158i	$0.632853\pi$
$14$	0	0
$15$	0	0
$16$	−71.0000	−1.10938
$17$	−84.0000	−1.19841	−0.599206	−	0.800595i	$-0.704517\pi$
$17$	−84.0000	−1.19841	−0.599206	+	0.800595i	$0.704517\pi$
$18$	0	0
$19$	−110.000	−1.32820	−0.664098	−	0.747645i	$-0.731184\pi$
$19$	−110.000	−1.32820	−0.664098	+	0.747645i	$0.731184\pi$
$20$	5.00000	0.0559017
$21$	0	0
$22$	180.000	1.74437
$23$	−120.000	−1.08790	−0.543951	−	0.839117i	$-0.683072\pi$
$23$	−120.000	−1.08790	−0.543951	+	0.839117i	$0.683072\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	114.000	0.859894
$27$	0	0
$28$	0	0
$29$	−162.000	−1.03733	−0.518666	−	0.854977i	$-0.673571\pi$
$29$	−162.000	−1.03733	−0.518666	+	0.854977i	$0.673571\pi$
$30$	0	0
$31$	−236.000	−1.36732	−0.683659	−	0.729802i	$-0.739612\pi$
$31$	−236.000	−1.36732	−0.683659	+	0.729802i	$0.739612\pi$
$32$	45.0000	0.248592
$33$	0	0
$34$	252.000	1.27111
$35$	0	0
$36$	0	0
$37$	−376.000	−1.67065	−0.835325	−	0.549757i	$-0.814720\pi$
$37$	−376.000	−1.67065	−0.835325	+	0.549757i	$0.814720\pi$
$38$	330.000	1.40876
$39$	0	0
$40$	105.000	0.415049
$41$	−126.000	−0.479949	−0.239974	−	0.970779i	$-0.577139\pi$
$41$	−126.000	−0.479949	−0.239974	+	0.970779i	$0.577139\pi$
$42$	0	0
$43$	−34.0000	−0.120580	−0.0602901	−	0.998181i	$-0.519203\pi$
$43$	−34.0000	−0.120580	−0.0602901	+	0.998181i	$0.519203\pi$
$44$	−60.0000	−0.205576
$45$	0	0
$46$	360.000	1.15389
$47$	−6.00000	−0.0186211	−0.00931053	−	0.999957i	$-0.502964\pi$
$47$	−6.00000	−0.0186211	−0.00931053	+	0.999957i	$0.502964\pi$
$48$	0	0
$49$	0	0
$50$	−75.0000	−0.212132
$51$	0	0
$52$	−38.0000	−0.101339
$53$	−582.000	−1.50837	−0.754187	−	0.656659i	$-0.771969\pi$
$53$	−582.000	−1.50837	−0.754187	+	0.656659i	$0.771969\pi$
$54$	0	0
$55$	−300.000	−0.735491
$56$	0	0
$57$	0	0
$58$	486.000	1.10026
$59$	492.000	1.08564	0.542822	−	0.839848i	$-0.317356\pi$
$59$	492.000	1.08564	0.542822	+	0.839848i	$0.317356\pi$
$60$	0	0
$61$	880.000	1.84709	0.923545	−	0.383491i	$-0.125278\pi$
$61$	880.000	1.84709	0.923545	+	0.383491i	$0.125278\pi$
$62$	708.000	1.45026
$63$	0	0
$64$	433.000	0.845703
$65$	−190.000	−0.362563
$66$	0	0
$67$	−826.000	−1.50615	−0.753074	−	0.657935i	$-0.771430\pi$
$67$	−826.000	−1.50615	−0.753074	+	0.657935i	$0.771430\pi$
$68$	−84.0000	−0.149801
$69$	0	0
$70$	0	0
$71$	666.000	1.11323	0.556617	−	0.830769i	$-0.312099\pi$
$71$	666.000	1.11323	0.556617	+	0.830769i	$0.312099\pi$
$72$	0	0
$73$	826.000	1.32433	0.662164	−	0.749359i	$-0.269638\pi$
$73$	826.000	1.32433	0.662164	+	0.749359i	$0.269638\pi$
$74$	1128.00	1.77199
$75$	0	0
$76$	−110.000	−0.166025
$77$	0	0
$78$	0	0
$79$	−592.000	−0.843104	−0.421552	−	0.906804i	$-0.638514\pi$
$79$	−592.000	−0.843104	−0.421552	+	0.906804i	$0.638514\pi$
$80$	−355.000	−0.496128
$81$	0	0
$82$	378.000	0.509062
$83$	792.000	1.04739	0.523695	−	0.851906i	$-0.324553\pi$
$83$	792.000	1.04739	0.523695	+	0.851906i	$0.324553\pi$
$84$	0	0
$85$	−420.000	−0.535946
$86$	102.000	0.127895
$87$	0	0
$88$	−1260.00	−1.52632
$89$	1002.00	1.19339	0.596695	−	0.802468i	$-0.296480\pi$
$89$	1002.00	1.19339	0.596695	+	0.802468i	$0.296480\pi$
$90$	0	0
$91$	0	0
$92$	−120.000	−0.135988
$93$	0	0
$94$	18.0000	0.0197506
$95$	−550.000	−0.593987
$96$	0	0
$97$	−1442.00	−1.50941	−0.754706	−	0.656063i	$-0.772220\pi$
$97$	−1442.00	−1.50941	−0.754706	+	0.656063i	$0.772220\pi$
$98$	0	0
$99$	0	0
$100$	25.0000	0.0250000
$101$	−1182.00	−1.16449	−0.582245	−	0.813014i	$-0.697825\pi$
$101$	−1182.00	−1.16449	−0.582245	+	0.813014i	$0.697825\pi$
$102$	0	0
$103$	−56.0000	−0.0535713	−0.0267857	−	0.999641i	$-0.508527\pi$
$103$	−56.0000	−0.0535713	−0.0267857	+	0.999641i	$0.508527\pi$
$104$	−798.000	−0.752407
$105$	0	0
$106$	1746.00	1.59987
$107$	1572.00	1.42029	0.710145	−	0.704056i	$-0.248629\pi$
$107$	1572.00	1.42029	0.710145	+	0.704056i	$0.248629\pi$
$108$	0	0
$109$	−1402.00	−1.23199	−0.615997	−	0.787749i	$-0.711246\pi$
$109$	−1402.00	−1.23199	−0.615997	+	0.787749i	$0.711246\pi$
$110$	900.000	0.780106
$111$	0	0
$112$	0	0
$113$	846.000	0.704292	0.352146	−	0.935945i	$-0.385452\pi$
$113$	846.000	0.704292	0.352146	+	0.935945i	$0.385452\pi$
$114$	0	0
$115$	−600.000	−0.486524
$116$	−162.000	−0.129667
$117$	0	0
$118$	−1476.00	−1.15150
$119$	0	0
$120$	0	0
$121$	2269.00	1.70473
$122$	−2640.00	−1.95913
$123$	0	0
$124$	−236.000	−0.170915
$125$	125.000	0.0894427
$126$	0	0
$127$	−1096.00	−0.765782	−0.382891	−	0.923794i	$-0.625071\pi$
$127$	−1096.00	−0.765782	−0.382891	+	0.923794i	$0.625071\pi$
$128$	−1659.00	−1.14560
$129$	0	0
$130$	570.000	0.384556
$131$	−492.000	−0.328139	−0.164070	−	0.986449i	$-0.552462\pi$
$131$	−492.000	−0.328139	−0.164070	+	0.986449i	$0.552462\pi$
$132$	0	0
$133$	0	0
$134$	2478.00	1.59751
$135$	0	0
$136$	−1764.00	−1.11222
$137$	282.000	0.175860	0.0879302	−	0.996127i	$-0.471975\pi$
$137$	282.000	0.175860	0.0879302	+	0.996127i	$0.471975\pi$
$138$	0	0
$139$	−2.00000	−0.00122042	−0.000610208	−	1.00000i	$-0.500194\pi$
$139$	−2.00000	−0.00122042	−0.000610208		1.00000i	$0.500194\pi$
$140$	0	0
$141$	0	0
$142$	−1998.00	−1.18076
$143$	2280.00	1.33331
$144$	0	0
$145$	−810.000	−0.463909
$146$	−2478.00	−1.40466
$147$	0	0
$148$	−376.000	−0.208831
$149$	−606.000	−0.333191	−0.166595	−	0.986025i	$-0.553277\pi$
$149$	−606.000	−0.333191	−0.166595	+	0.986025i	$0.553277\pi$
$150$	0	0
$151$	−2176.00	−1.17272	−0.586359	−	0.810051i	$-0.699439\pi$
$151$	−2176.00	−1.17272	−0.586359	+	0.810051i	$0.699439\pi$
$152$	−2310.00	−1.23267
$153$	0	0
$154$	0	0
$155$	−1180.00	−0.611483
$156$	0	0
$157$	−146.000	−0.0742170	−0.0371085	−	0.999311i	$-0.511815\pi$
$157$	−146.000	−0.0742170	−0.0371085	+	0.999311i	$0.511815\pi$
$158$	1776.00	0.894247
$159$	0	0
$160$	225.000	0.111174
$161$	0	0
$162$	0	0
$163$	−826.000	−0.396916	−0.198458	−	0.980109i	$-0.563593\pi$
$163$	−826.000	−0.396916	−0.198458	+	0.980109i	$0.563593\pi$
$164$	−126.000	−0.0599936
$165$	0	0
$166$	−2376.00	−1.11092
$167$	1206.00	0.558821	0.279410	−	0.960172i	$-0.409861\pi$
$167$	1206.00	0.558821	0.279410	+	0.960172i	$0.409861\pi$
$168$	0	0
$169$	−753.000	−0.342740
$170$	1260.00	0.568456
$171$	0	0
$172$	−34.0000	−0.0150725
$173$	1398.00	0.614381	0.307191	−	0.951648i	$-0.400611\pi$
$173$	1398.00	0.614381	0.307191	+	0.951648i	$0.400611\pi$
$174$	0	0
$175$	0	0
$176$	4260.00	1.82449
$177$	0	0
$178$	−3006.00	−1.26578
$179$	2424.00	1.01217	0.506085	−	0.862484i	$-0.331092\pi$
$179$	2424.00	1.01217	0.506085	+	0.862484i	$0.331092\pi$
$180$	0	0
$181$	−3728.00	−1.53094	−0.765470	−	0.643472i	$-0.777493\pi$
$181$	−3728.00	−1.53094	−0.765470	+	0.643472i	$0.777493\pi$
$182$	0	0
$183$	0	0
$184$	−2520.00	−1.00966
$185$	−1880.00	−0.747137
$186$	0	0
$187$	5040.00	1.97092
$188$	−6.00000	−0.00232763
$189$	0	0
$190$	1650.00	0.630019
$191$	−2550.00	−0.966029	−0.483014	−	0.875612i	$-0.660458\pi$
$191$	−2550.00	−0.966029	−0.483014	+	0.875612i	$0.660458\pi$
$192$	0	0
$193$	−1978.00	−0.737718	−0.368859	−	0.929485i	$-0.620251\pi$
$193$	−1978.00	−0.737718	−0.368859	+	0.929485i	$0.620251\pi$
$194$	4326.00	1.60097
$195$	0	0
$196$	0	0
$197$	1170.00	0.423142	0.211571	−	0.977363i	$-0.432142\pi$
$197$	1170.00	0.423142	0.211571	+	0.977363i	$0.432142\pi$
$198$	0	0
$199$	−3584.00	−1.27670	−0.638349	−	0.769747i	$-0.720382\pi$
$199$	−3584.00	−1.27670	−0.638349	+	0.769747i	$0.720382\pi$
$200$	525.000	0.185616
$201$	0	0
$202$	3546.00	1.23513
$203$	0	0
$204$	0	0
$205$	−630.000	−0.214640
$206$	168.000	0.0568209
$207$	0	0
$208$	2698.00	0.899388
$209$	6600.00	2.18436
$210$	0	0
$211$	1640.00	0.535082	0.267541	−	0.963547i	$-0.413789\pi$
$211$	1640.00	0.535082	0.267541	+	0.963547i	$0.413789\pi$
$212$	−582.000	−0.188547
$213$	0	0
$214$	−4716.00	−1.50644
$215$	−170.000	−0.0539251
$216$	0	0
$217$	0	0
$218$	4206.00	1.30673
$219$	0	0
$220$	−300.000	−0.0919363
$221$	3192.00	0.971571
$222$	0	0
$223$	1888.00	0.566950	0.283475	−	0.958980i	$-0.408513\pi$
$223$	1888.00	0.566950	0.283475	+	0.958980i	$0.408513\pi$
$224$	0	0
$225$	0	0
$226$	−2538.00	−0.747014
$227$	−2244.00	−0.656121	−0.328061	−	0.944657i	$-0.606395\pi$
$227$	−2244.00	−0.656121	−0.328061	+	0.944657i	$0.606395\pi$
$228$	0	0
$229$	4084.00	1.17851	0.589254	−	0.807948i	$-0.299422\pi$
$229$	4084.00	1.17851	0.589254	+	0.807948i	$0.299422\pi$
$230$	1800.00	0.516037
$231$	0	0
$232$	−3402.00	−0.962725
$233$	−4026.00	−1.13198	−0.565991	−	0.824411i	$-0.691506\pi$
$233$	−4026.00	−1.13198	−0.565991	+	0.824411i	$0.691506\pi$
$234$	0	0
$235$	−30.0000	−0.00832759
$236$	492.000	0.135705
$237$	0	0
$238$	0	0
$239$	4590.00	1.24227	0.621135	−	0.783704i	$-0.286672\pi$
$239$	4590.00	1.24227	0.621135	+	0.783704i	$0.286672\pi$
$240$	0	0
$241$	−1946.00	−0.520136	−0.260068	−	0.965590i	$-0.583745\pi$
$241$	−1946.00	−0.520136	−0.260068	+	0.965590i	$0.583745\pi$
$242$	−6807.00	−1.80814
$243$	0	0
$244$	880.000	0.230886
$245$	0	0
$246$	0	0
$247$	4180.00	1.07679
$248$	−4956.00	−1.26898
$249$	0	0
$250$	−375.000	−0.0948683
$251$	3636.00	0.914352	0.457176	−	0.889376i	$-0.348861\pi$
$251$	3636.00	0.914352	0.457176	+	0.889376i	$0.348861\pi$
$252$	0	0
$253$	7200.00	1.78917
$254$	3288.00	0.812234
$255$	0	0
$256$	1513.00	0.369385
$257$	4152.00	1.00776	0.503881	−	0.863773i	$-0.331905\pi$
$257$	4152.00	1.00776	0.503881	+	0.863773i	$0.331905\pi$
$258$	0	0
$259$	0	0
$260$	−190.000	−0.0453204
$261$	0	0
$262$	1476.00	0.348044
$263$	−5388.00	−1.26326	−0.631632	−	0.775269i	$-0.717615\pi$
$263$	−5388.00	−1.26326	−0.631632	+	0.775269i	$0.717615\pi$
$264$	0	0
$265$	−2910.00	−0.674566
$266$	0	0
$267$	0	0
$268$	−826.000	−0.188269
$269$	1194.00	0.270630	0.135315	−	0.990803i	$-0.456795\pi$
$269$	1194.00	0.270630	0.135315	+	0.990803i	$0.456795\pi$
$270$	0	0
$271$	−6788.00	−1.52156	−0.760778	−	0.649012i	$-0.775182\pi$
$271$	−6788.00	−1.52156	−0.760778	+	0.649012i	$0.775182\pi$
$272$	5964.00	1.32949
$273$	0	0
$274$	−846.000	−0.186528
$275$	−1500.00	−0.328921
$276$	0	0
$277$	−8908.00	−1.93224	−0.966119	−	0.258098i	$-0.916904\pi$
$277$	−8908.00	−1.93224	−0.966119	+	0.258098i	$0.916904\pi$
$278$	6.00000	0.00129445
$279$	0	0
$280$	0	0
$281$	−6408.00	−1.36039	−0.680194	−	0.733032i	$-0.738105\pi$
$281$	−6408.00	−1.36039	−0.680194	+	0.733032i	$0.738105\pi$
$282$	0	0
$283$	3652.00	0.767098	0.383549	−	0.923520i	$-0.374702\pi$
$283$	3652.00	0.767098	0.383549	+	0.923520i	$0.374702\pi$
$284$	666.000	0.139154
$285$	0	0
$286$	−6840.00	−1.41419
$287$	0	0
$288$	0	0
$289$	2143.00	0.436190
$290$	2430.00	0.492050
$291$	0	0
$292$	826.000	0.165541
$293$	−2214.00	−0.441445	−0.220722	−	0.975337i	$-0.570841\pi$
$293$	−2214.00	−0.441445	−0.220722	+	0.975337i	$0.570841\pi$
$294$	0	0
$295$	2460.00	0.485514
$296$	−7896.00	−1.55049
$297$	0	0
$298$	1818.00	0.353402
$299$	4560.00	0.881979
$300$	0	0
$301$	0	0
$302$	6528.00	1.24385
$303$	0	0
$304$	7810.00	1.47347
$305$	4400.00	0.826043
$306$	0	0
$307$	5452.00	1.01356	0.506779	−	0.862076i	$-0.330836\pi$
$307$	5452.00	1.01356	0.506779	+	0.862076i	$0.330836\pi$
$308$	0	0
$309$	0	0
$310$	3540.00	0.648576
$311$	−300.000	−0.0546992	−0.0273496	−	0.999626i	$-0.508707\pi$
$311$	−300.000	−0.0546992	−0.0273496	+	0.999626i	$0.508707\pi$
$312$	0	0
$313$	3994.00	0.721260	0.360630	−	0.932709i	$-0.382562\pi$
$313$	3994.00	0.721260	0.360630	+	0.932709i	$0.382562\pi$
$314$	438.000	0.0787190
$315$	0	0
$316$	−592.000	−0.105388
$317$	−2586.00	−0.458184	−0.229092	−	0.973405i	$-0.573576\pi$
$317$	−2586.00	−0.458184	−0.229092	+	0.973405i	$0.573576\pi$
$318$	0	0
$319$	9720.00	1.70600
$320$	2165.00	0.378210
$321$	0	0
$322$	0	0
$323$	9240.00	1.59173
$324$	0	0
$325$	−950.000	−0.162143
$326$	2478.00	0.420993
$327$	0	0
$328$	−2646.00	−0.445430
$329$	0	0
$330$	0	0
$331$	6248.00	1.03753	0.518763	−	0.854918i	$-0.326393\pi$
$331$	6248.00	1.03753	0.518763	+	0.854918i	$0.326393\pi$
$332$	792.000	0.130924
$333$	0	0
$334$	−3618.00	−0.592719
$335$	−4130.00	−0.673570
$336$	0	0
$337$	3062.00	0.494949	0.247474	−	0.968894i	$-0.420399\pi$
$337$	3062.00	0.494949	0.247474	+	0.968894i	$0.420399\pi$
$338$	2259.00	0.363531
$339$	0	0
$340$	−420.000	−0.0669932
$341$	14160.0	2.24870
$342$	0	0
$343$	0	0
$344$	−714.000	−0.111908
$345$	0	0
$346$	−4194.00	−0.651650
$347$	5196.00	0.803850	0.401925	−	0.915673i	$-0.368341\pi$
$347$	5196.00	0.803850	0.401925	+	0.915673i	$0.368341\pi$
$348$	0	0
$349$	−8660.00	−1.32825	−0.664125	−	0.747622i	$-0.731196\pi$
$349$	−8660.00	−1.32825	−0.664125	+	0.747622i	$0.731196\pi$
$350$	0	0
$351$	0	0
$352$	−2700.00	−0.408837
$353$	−1128.00	−0.170078	−0.0850388	−	0.996378i	$-0.527101\pi$
$353$	−1128.00	−0.170078	−0.0850388	+	0.996378i	$0.527101\pi$
$354$	0	0
$355$	3330.00	0.497854
$356$	1002.00	0.149174
$357$	0	0
$358$	−7272.00	−1.07357
$359$	−6618.00	−0.972938	−0.486469	−	0.873698i	$-0.661715\pi$
$359$	−6618.00	−0.972938	−0.486469	+	0.873698i	$0.661715\pi$
$360$	0	0
$361$	5241.00	0.764106
$362$	11184.0	1.62381
$363$	0	0
$364$	0	0
$365$	4130.00	0.592258
$366$	0	0
$367$	−1568.00	−0.223022	−0.111511	−	0.993763i	$-0.535569\pi$
$367$	−1568.00	−0.223022	−0.111511	+	0.993763i	$0.535569\pi$
$368$	8520.00	1.20689
$369$	0	0
$370$	5640.00	0.792458
$371$	0	0
$372$	0	0
$373$	−5200.00	−0.721839	−0.360919	−	0.932597i	$-0.617537\pi$
$373$	−5200.00	−0.721839	−0.360919	+	0.932597i	$0.617537\pi$
$374$	−15120.0	−2.09047
$375$	0	0
$376$	−126.000	−0.0172818
$377$	6156.00	0.840982
$378$	0	0
$379$	−1096.00	−0.148543	−0.0742714	−	0.997238i	$-0.523663\pi$
$379$	−1096.00	−0.148543	−0.0742714	+	0.997238i	$0.523663\pi$
$380$	−550.000	−0.0742484
$381$	0	0
$382$	7650.00	1.02463
$383$	−12318.0	−1.64340	−0.821698	−	0.569924i	$-0.806973\pi$
$383$	−12318.0	−1.64340	−0.821698	+	0.569924i	$0.806973\pi$
$384$	0	0
$385$	0	0
$386$	5934.00	0.782468
$387$	0	0
$388$	−1442.00	−0.188676
$389$	−6558.00	−0.854766	−0.427383	−	0.904071i	$-0.640564\pi$
$389$	−6558.00	−0.854766	−0.427383	+	0.904071i	$0.640564\pi$
$390$	0	0
$391$	10080.0	1.30375
$392$	0	0
$393$	0	0
$394$	−3510.00	−0.448810
$395$	−2960.00	−0.377048
$396$	0	0
$397$	1258.00	0.159036	0.0795179	−	0.996833i	$-0.474662\pi$
$397$	1258.00	0.159036	0.0795179	+	0.996833i	$0.474662\pi$
$398$	10752.0	1.35414
$399$	0	0
$400$	−1775.00	−0.221875
$401$	2976.00	0.370609	0.185305	−	0.982681i	$-0.440673\pi$
$401$	2976.00	0.370609	0.185305	+	0.982681i	$0.440673\pi$
$402$	0	0
$403$	8968.00	1.10851
$404$	−1182.00	−0.145561
$405$	0	0
$406$	0	0
$407$	22560.0	2.74756
$408$	0	0
$409$	9070.00	1.09653	0.548267	−	0.836303i	$-0.315288\pi$
$409$	9070.00	1.09653	0.548267	+	0.836303i	$0.315288\pi$
$410$	1890.00	0.227660
$411$	0	0
$412$	−56.0000	−0.00669641
$413$	0	0
$414$	0	0
$415$	3960.00	0.468407
$416$	−1710.00	−0.201538
$417$	0	0
$418$	−19800.0	−2.31687
$419$	6156.00	0.717757	0.358879	−	0.933384i	$-0.383159\pi$
$419$	6156.00	0.717757	0.358879	+	0.933384i	$0.383159\pi$
$420$	0	0
$421$	−6874.00	−0.795768	−0.397884	−	0.917436i	$-0.630255\pi$
$421$	−6874.00	−0.795768	−0.397884	+	0.917436i	$0.630255\pi$
$422$	−4920.00	−0.567540
$423$	0	0
$424$	−12222.0	−1.39989
$425$	−2100.00	−0.239682
$426$	0	0
$427$	0	0
$428$	1572.00	0.177536
$429$	0	0
$430$	510.000	0.0571962
$431$	10218.0	1.14196	0.570979	−	0.820965i	$-0.306564\pi$
$431$	10218.0	1.14196	0.570979	+	0.820965i	$0.306564\pi$
$432$	0	0
$433$	5830.00	0.647048	0.323524	−	0.946220i	$-0.395132\pi$
$433$	5830.00	0.647048	0.323524	+	0.946220i	$0.395132\pi$
$434$	0	0
$435$	0	0
$436$	−1402.00	−0.153999
$437$	13200.0	1.44495
$438$	0	0
$439$	−8588.00	−0.933674	−0.466837	−	0.884343i	$-0.654607\pi$
$439$	−8588.00	−0.933674	−0.466837	+	0.884343i	$0.654607\pi$
$440$	−6300.00	−0.682593
$441$	0	0
$442$	−9576.00	−1.03051
$443$	12660.0	1.35778	0.678888	−	0.734242i	$-0.262462\pi$
$443$	12660.0	1.35778	0.678888	+	0.734242i	$0.262462\pi$
$444$	0	0
$445$	5010.00	0.533701
$446$	−5664.00	−0.601341
$447$	0	0
$448$	0	0
$449$	−3636.00	−0.382168	−0.191084	−	0.981574i	$-0.561200\pi$
$449$	−3636.00	−0.382168	−0.191084	+	0.981574i	$0.561200\pi$
$450$	0	0
$451$	7560.00	0.789327
$452$	846.000	0.0880365
$453$	0	0
$454$	6732.00	0.695922
$455$	0	0
$456$	0	0
$457$	−3022.00	−0.309329	−0.154664	−	0.987967i	$-0.549430\pi$
$457$	−3022.00	−0.309329	−0.154664	+	0.987967i	$0.549430\pi$
$458$	−12252.0	−1.25000
$459$	0	0
$460$	−600.000	−0.0608155
$461$	−14742.0	−1.48938	−0.744689	−	0.667411i	$-0.767402\pi$
$461$	−14742.0	−1.48938	−0.744689	+	0.667411i	$0.767402\pi$
$462$	0	0
$463$	−9268.00	−0.930282	−0.465141	−	0.885237i	$-0.653996\pi$
$463$	−9268.00	−0.930282	−0.465141	+	0.885237i	$0.653996\pi$
$464$	11502.0	1.15079
$465$	0	0
$466$	12078.0	1.20065
$467$	−13920.0	−1.37932	−0.689658	−	0.724135i	$-0.742239\pi$
$467$	−13920.0	−1.37932	−0.689658	+	0.724135i	$0.742239\pi$
$468$	0	0
$469$	0	0
$470$	90.0000	0.00883275
$471$	0	0
$472$	10332.0	1.00756
$473$	2040.00	0.198307
$474$	0	0
$475$	−2750.00	−0.265639
$476$	0	0
$477$	0	0
$478$	−13770.0	−1.31763
$479$	−8220.00	−0.784095	−0.392047	−	0.919945i	$-0.628233\pi$
$479$	−8220.00	−0.784095	−0.392047	+	0.919945i	$0.628233\pi$
$480$	0	0
$481$	14288.0	1.35442
$482$	5838.00	0.551688
$483$	0	0
$484$	2269.00	0.213092
$485$	−7210.00	−0.675029
$486$	0	0
$487$	15752.0	1.46569	0.732845	−	0.680395i	$-0.238192\pi$
$487$	15752.0	1.46569	0.732845	+	0.680395i	$0.238192\pi$
$488$	18480.0	1.71424
$489$	0	0
$490$	0	0
$491$	−6552.00	−0.602215	−0.301108	−	0.953590i	$-0.597356\pi$
$491$	−6552.00	−0.602215	−0.301108	+	0.953590i	$0.597356\pi$
$492$	0	0
$493$	13608.0	1.24315
$494$	−12540.0	−1.14211
$495$	0	0
$496$	16756.0	1.51687
$497$	0	0
$498$	0	0
$499$	4160.00	0.373201	0.186600	−	0.982436i	$-0.440253\pi$
$499$	4160.00	0.373201	0.186600	+	0.982436i	$0.440253\pi$
$500$	125.000	0.0111803
$501$	0	0
$502$	−10908.0	−0.969816
$503$	5166.00	0.457934	0.228967	−	0.973434i	$-0.426465\pi$
$503$	5166.00	0.457934	0.228967	+	0.973434i	$0.426465\pi$
$504$	0	0
$505$	−5910.00	−0.520775
$506$	−21600.0	−1.89770
$507$	0	0
$508$	−1096.00	−0.0957227
$509$	−12066.0	−1.05072	−0.525360	−	0.850880i	$-0.676069\pi$
$509$	−12066.0	−1.05072	−0.525360	+	0.850880i	$0.676069\pi$
$510$	0	0
$511$	0	0
$512$	8733.00	0.753804
$513$	0	0
$514$	−12456.0	−1.06889
$515$	−280.000	−0.0239578
$516$	0	0
$517$	360.000	0.0306243
$518$	0	0
$519$	0	0
$520$	−3990.00	−0.336487
$521$	−17886.0	−1.50403	−0.752015	−	0.659146i	$-0.770918\pi$
$521$	−17886.0	−1.50403	−0.752015	+	0.659146i	$0.770918\pi$
$522$	0	0
$523$	−5168.00	−0.432086	−0.216043	−	0.976384i	$-0.569315\pi$
$523$	−5168.00	−0.432086	−0.216043	+	0.976384i	$0.569315\pi$
$524$	−492.000	−0.0410174
$525$	0	0
$526$	16164.0	1.33989
$527$	19824.0	1.63861
$528$	0	0
$529$	2233.00	0.183529
$530$	8730.00	0.715485
$531$	0	0
$532$	0	0
$533$	4788.00	0.389102
$534$	0	0
$535$	7860.00	0.635173
$536$	−17346.0	−1.39782
$537$	0	0
$538$	−3582.00	−0.287046
$539$	0	0
$540$	0	0
$541$	7850.00	0.623841	0.311920	−	0.950108i	$-0.399028\pi$
$541$	7850.00	0.623841	0.311920	+	0.950108i	$0.399028\pi$
$542$	20364.0	1.61385
$543$	0	0
$544$	−3780.00	−0.297916
$545$	−7010.00	−0.550964
$546$	0	0
$547$	2990.00	0.233717	0.116858	−	0.993149i	$-0.462718\pi$
$547$	2990.00	0.233717	0.116858	+	0.993149i	$0.462718\pi$
$548$	282.000	0.0219826
$549$	0	0
$550$	4500.00	0.348874
$551$	17820.0	1.37778
$552$	0	0
$553$	0	0
$554$	26724.0	2.04945
$555$	0	0
$556$	−2.00000	−0.000152552 0
$557$	12486.0	0.949818	0.474909	−	0.880035i	$-0.342481\pi$
$557$	12486.0	0.949818	0.474909	+	0.880035i	$0.342481\pi$
$558$	0	0
$559$	1292.00	0.0977563
$560$	0	0
$561$	0	0
$562$	19224.0	1.44291
$563$	−732.000	−0.0547960	−0.0273980	−	0.999625i	$-0.508722\pi$
$563$	−732.000	−0.0547960	−0.0273980	+	0.999625i	$0.508722\pi$
$564$	0	0
$565$	4230.00	0.314969
$566$	−10956.0	−0.813631
$567$	0	0
$568$	13986.0	1.03317
$569$	456.000	0.0335967	0.0167983	−	0.999859i	$-0.494653\pi$
$569$	456.000	0.0335967	0.0167983	+	0.999859i	$0.494653\pi$
$570$	0	0
$571$	−14092.0	−1.03281	−0.516403	−	0.856346i	$-0.672729\pi$
$571$	−14092.0	−1.03281	−0.516403	+	0.856346i	$0.672729\pi$
$572$	2280.00	0.166664
$573$	0	0
$574$	0	0
$575$	−3000.00	−0.217580
$576$	0	0
$577$	−10190.0	−0.735208	−0.367604	−	0.929982i	$-0.619822\pi$
$577$	−10190.0	−0.735208	−0.367604	+	0.929982i	$0.619822\pi$
$578$	−6429.00	−0.462649
$579$	0	0
$580$	−810.000	−0.0579887
$581$	0	0
$582$	0	0
$583$	34920.0	2.48068
$584$	17346.0	1.22908
$585$	0	0
$586$	6642.00	0.468223
$587$	23220.0	1.63270	0.816348	−	0.577561i	$-0.195995\pi$
$587$	23220.0	1.63270	0.816348	+	0.577561i	$0.195995\pi$
$588$	0	0
$589$	25960.0	1.81607
$590$	−7380.00	−0.514966
$591$	0	0
$592$	26696.0	1.85338
$593$	14916.0	1.03293	0.516464	−	0.856309i	$-0.327248\pi$
$593$	14916.0	1.03293	0.516464	+	0.856309i	$0.327248\pi$
$594$	0	0
$595$	0	0
$596$	−606.000	−0.0416489
$597$	0	0
$598$	−13680.0	−0.935480
$599$	16914.0	1.15374	0.576868	−	0.816838i	$-0.304275\pi$
$599$	16914.0	1.15374	0.576868	+	0.816838i	$0.304275\pi$
$600$	0	0
$601$	−5762.00	−0.391076	−0.195538	−	0.980696i	$-0.562645\pi$
$601$	−5762.00	−0.391076	−0.195538	+	0.980696i	$0.562645\pi$
$602$	0	0
$603$	0	0
$604$	−2176.00	−0.146590
$605$	11345.0	0.762380
$606$	0	0
$607$	28528.0	1.90760	0.953802	−	0.300435i	$-0.0971320\pi$
$607$	28528.0	1.90760	0.953802	+	0.300435i	$0.0971320\pi$
$608$	−4950.00	−0.330179
$609$	0	0
$610$	−13200.0	−0.876151
$611$	228.000	0.0150964
$612$	0	0
$613$	−12976.0	−0.854969	−0.427484	−	0.904023i	$-0.640600\pi$
$613$	−12976.0	−0.854969	−0.427484	+	0.904023i	$0.640600\pi$
$614$	−16356.0	−1.07504
$615$	0	0
$616$	0	0
$617$	−8874.00	−0.579017	−0.289509	−	0.957175i	$-0.593492\pi$
$617$	−8874.00	−0.579017	−0.289509	+	0.957175i	$0.593492\pi$
$618$	0	0
$619$	17170.0	1.11490	0.557448	−	0.830212i	$-0.311781\pi$
$619$	17170.0	1.11490	0.557448	+	0.830212i	$0.311781\pi$
$620$	−1180.00	−0.0764354
$621$	0	0
$622$	900.000	0.0580172
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	−11982.0	−0.765011
$627$	0	0
$628$	−146.000	−0.00927712
$629$	31584.0	2.00212
$630$	0	0
$631$	−26728.0	−1.68625	−0.843126	−	0.537716i	$-0.819287\pi$
$631$	−26728.0	−1.68625	−0.843126	+	0.537716i	$0.819287\pi$
$632$	−12432.0	−0.782466
$633$	0	0
$634$	7758.00	0.485977
$635$	−5480.00	−0.342468
$636$	0	0
$637$	0	0
$638$	−29160.0	−1.80949
$639$	0	0
$640$	−8295.00	−0.512326
$641$	24492.0	1.50917	0.754583	−	0.656204i	$-0.227839\pi$
$641$	24492.0	1.50917	0.754583	+	0.656204i	$0.227839\pi$
$642$	0	0
$643$	1888.00	0.115794	0.0578969	−	0.998323i	$-0.481561\pi$
$643$	1888.00	0.115794	0.0578969	+	0.998323i	$0.481561\pi$
$644$	0	0
$645$	0	0
$646$	−27720.0	−1.68828
$647$	−13134.0	−0.798069	−0.399035	−	0.916936i	$-0.630655\pi$
$647$	−13134.0	−0.798069	−0.399035	+	0.916936i	$0.630655\pi$
$648$	0	0
$649$	−29520.0	−1.78546
$650$	2850.00	0.171979
$651$	0	0
$652$	−826.000	−0.0496145
$653$	−12882.0	−0.771993	−0.385997	−	0.922500i	$-0.626142\pi$
$653$	−12882.0	−0.771993	−0.385997	+	0.922500i	$0.626142\pi$
$654$	0	0
$655$	−2460.00	−0.146748
$656$	8946.00	0.532443
$657$	0	0
$658$	0	0
$659$	−7380.00	−0.436243	−0.218121	−	0.975922i	$-0.569993\pi$
$659$	−7380.00	−0.436243	−0.218121	+	0.975922i	$0.569993\pi$
$660$	0	0
$661$	−560.000	−0.0329523	−0.0164762	−	0.999864i	$-0.505245\pi$
$661$	−560.000	−0.0329523	−0.0164762	+	0.999864i	$0.505245\pi$
$662$	−18744.0	−1.10046
$663$	0	0
$664$	16632.0	0.972058
$665$	0	0
$666$	0	0
$667$	19440.0	1.12852
$668$	1206.00	0.0698526
$669$	0	0
$670$	12390.0	0.714429
$671$	−52800.0	−3.03774
$672$	0	0
$673$	−5722.00	−0.327737	−0.163868	−	0.986482i	$-0.552397\pi$
$673$	−5722.00	−0.327737	−0.163868	+	0.986482i	$0.552397\pi$
$674$	−9186.00	−0.524973
$675$	0	0
$676$	−753.000	−0.0428425
$677$	−13650.0	−0.774907	−0.387454	−	0.921889i	$-0.626645\pi$
$677$	−13650.0	−0.774907	−0.387454	+	0.921889i	$0.626645\pi$
$678$	0	0
$679$	0	0
$680$	−8820.00	−0.497399
$681$	0	0
$682$	−42480.0	−2.38511
$683$	−1140.00	−0.0638666	−0.0319333	−	0.999490i	$-0.510166\pi$
$683$	−1140.00	−0.0638666	−0.0319333	+	0.999490i	$0.510166\pi$
$684$	0	0
$685$	1410.00	0.0786472
$686$	0	0
$687$	0	0
$688$	2414.00	0.133769
$689$	22116.0	1.22286
$690$	0	0
$691$	−32510.0	−1.78978	−0.894891	−	0.446286i	$-0.852746\pi$
$691$	−32510.0	−1.78978	−0.894891	+	0.446286i	$0.852746\pi$
$692$	1398.00	0.0767977
$693$	0	0
$694$	−15588.0	−0.852612
$695$	−10.0000	−0.000545787 0
$696$	0	0
$697$	10584.0	0.575176
$698$	25980.0	1.40882
$699$	0	0
$700$	0	0
$701$	−21402.0	−1.15313	−0.576564	−	0.817052i	$-0.695607\pi$
$701$	−21402.0	−1.15313	−0.576564	+	0.817052i	$0.695607\pi$
$702$	0	0
$703$	41360.0	2.21895
$704$	−25980.0	−1.39085
$705$	0	0
$706$	3384.00	0.180395
$707$	0	0
$708$	0	0
$709$	12170.0	0.644646	0.322323	−	0.946630i	$-0.395536\pi$
$709$	12170.0	0.644646	0.322323	+	0.946630i	$0.395536\pi$
$710$	−9990.00	−0.528054
$711$	0	0
$712$	21042.0	1.10756
$713$	28320.0	1.48751
$714$	0	0
$715$	11400.0	0.596274
$716$	2424.00	0.126521
$717$	0	0
$718$	19854.0	1.03196
$719$	−12012.0	−0.623049	−0.311524	−	0.950238i	$-0.600840\pi$
$719$	−12012.0	−0.623049	−0.311524	+	0.950238i	$0.600840\pi$
$720$	0	0
$721$	0	0
$722$	−15723.0	−0.810456
$723$	0	0
$724$	−3728.00	−0.191367
$725$	−4050.00	−0.207467
$726$	0	0
$727$	−8480.00	−0.432608	−0.216304	−	0.976326i	$-0.569400\pi$
$727$	−8480.00	−0.432608	−0.216304	+	0.976326i	$0.569400\pi$
$728$	0	0
$729$	0	0
$730$	−12390.0	−0.628184
$731$	2856.00	0.144505
$732$	0	0
$733$	10906.0	0.549553	0.274776	−	0.961508i	$-0.411396\pi$
$733$	10906.0	0.549553	0.274776	+	0.961508i	$0.411396\pi$
$734$	4704.00	0.236550
$735$	0	0
$736$	−5400.00	−0.270444
$737$	49560.0	2.47702
$738$	0	0
$739$	−11104.0	−0.552730	−0.276365	−	0.961053i	$-0.589130\pi$
$739$	−11104.0	−0.552730	−0.276365	+	0.961053i	$0.589130\pi$
$740$	−1880.00	−0.0933921
$741$	0	0
$742$	0	0
$743$	−25416.0	−1.25494	−0.627471	−	0.778640i	$-0.715910\pi$
$743$	−25416.0	−1.25494	−0.627471	+	0.778640i	$0.715910\pi$
$744$	0	0
$745$	−3030.00	−0.149008
$746$	15600.0	0.765625
$747$	0	0
$748$	5040.00	0.246365
$749$	0	0
$750$	0	0
$751$	−30904.0	−1.50160	−0.750801	−	0.660529i	$-0.770332\pi$
$751$	−30904.0	−1.50160	−0.750801	+	0.660529i	$0.770332\pi$
$752$	426.000	0.0206577
$753$	0	0
$754$	−18468.0	−0.891996
$755$	−10880.0	−0.524455
$756$	0	0
$757$	−16216.0	−0.778574	−0.389287	−	0.921117i	$-0.627279\pi$
$757$	−16216.0	−0.778574	−0.389287	+	0.921117i	$0.627279\pi$
$758$	3288.00	0.157553
$759$	0	0
$760$	−11550.0	−0.551266
$761$	7230.00	0.344399	0.172199	−	0.985062i	$-0.444913\pi$
$761$	7230.00	0.344399	0.172199	+	0.985062i	$0.444913\pi$
$762$	0	0
$763$	0	0
$764$	−2550.00	−0.120754
$765$	0	0
$766$	36954.0	1.74308
$767$	−18696.0	−0.880148
$768$	0	0
$769$	−4790.00	−0.224619	−0.112309	−	0.993673i	$-0.535825\pi$
$769$	−4790.00	−0.224619	−0.112309	+	0.993673i	$0.535825\pi$
$770$	0	0
$771$	0	0
$772$	−1978.00	−0.0922147
$773$	36114.0	1.68038	0.840188	−	0.542296i	$-0.182445\pi$
$773$	36114.0	1.68038	0.840188	+	0.542296i	$0.182445\pi$
$774$	0	0
$775$	−5900.00	−0.273464
$776$	−30282.0	−1.40085
$777$	0	0
$778$	19674.0	0.906616
$779$	13860.0	0.637466
$780$	0	0
$781$	−39960.0	−1.83083
$782$	−30240.0	−1.38284
$783$	0	0
$784$	0	0
$785$	−730.000	−0.0331909
$786$	0	0
$787$	−13556.0	−0.614002	−0.307001	−	0.951709i	$-0.599325\pi$
$787$	−13556.0	−0.614002	−0.307001	+	0.951709i	$0.599325\pi$
$788$	1170.00	0.0528928
$789$	0	0
$790$	8880.00	0.399919
$791$	0	0
$792$	0	0
$793$	−33440.0	−1.49746
$794$	−3774.00	−0.168683
$795$	0	0
$796$	−3584.00	−0.159587
$797$	39006.0	1.73358	0.866790	−	0.498673i	$-0.166179\pi$
$797$	39006.0	1.73358	0.866790	+	0.498673i	$0.166179\pi$
$798$	0	0
$799$	504.000	0.0223157
$800$	1125.00	0.0497184
$801$	0	0
$802$	−8928.00	−0.393091
$803$	−49560.0	−2.17800
$804$	0	0
$805$	0	0
$806$	−26904.0	−1.17575
$807$	0	0
$808$	−24822.0	−1.08074
$809$	−21480.0	−0.933494	−0.466747	−	0.884391i	$-0.654574\pi$
$809$	−21480.0	−0.933494	−0.466747	+	0.884391i	$0.654574\pi$
$810$	0	0
$811$	−146.000	−0.00632152	−0.00316076	−	0.999995i	$-0.501006\pi$
$811$	−146.000	−0.00632152	−0.00316076	+	0.999995i	$0.501006\pi$
$812$	0	0
$813$	0	0
$814$	−67680.0	−2.91423
$815$	−4130.00	−0.177506
$816$	0	0
$817$	3740.00	0.160154
$818$	−27210.0	−1.16305
$819$	0	0
$820$	−630.000	−0.0268299
$821$	−35562.0	−1.51172	−0.755860	−	0.654733i	$-0.772781\pi$
$821$	−35562.0	−1.51172	−0.755860	+	0.654733i	$0.772781\pi$
$822$	0	0
$823$	−33964.0	−1.43853	−0.719265	−	0.694736i	$-0.755521\pi$
$823$	−33964.0	−1.43853	−0.719265	+	0.694736i	$0.755521\pi$
$824$	−1176.00	−0.0497183
$825$	0	0
$826$	0	0
$827$	23292.0	0.979374	0.489687	−	0.871898i	$-0.337111\pi$
$827$	23292.0	0.979374	0.489687	+	0.871898i	$0.337111\pi$
$828$	0	0
$829$	−33716.0	−1.41255	−0.706276	−	0.707937i	$-0.749626\pi$
$829$	−33716.0	−1.41255	−0.706276	+	0.707937i	$0.749626\pi$
$830$	−11880.0	−0.496820
$831$	0	0
$832$	−16454.0	−0.685625
$833$	0	0
$834$	0	0
$835$	6030.00	0.249912
$836$	6600.00	0.273045
$837$	0	0
$838$	−18468.0	−0.761297
$839$	−35280.0	−1.45173	−0.725865	−	0.687838i	$-0.758560\pi$
$839$	−35280.0	−1.45173	−0.725865	+	0.687838i	$0.758560\pi$
$840$	0	0
$841$	1855.00	0.0760589
$842$	20622.0	0.844039
$843$	0	0
$844$	1640.00	0.0668852
$845$	−3765.00	−0.153278
$846$	0	0
$847$	0	0
$848$	41322.0	1.67335
$849$	0	0
$850$	6300.00	0.254221
$851$	45120.0	1.81750
$852$	0	0
$853$	−36902.0	−1.48124	−0.740622	−	0.671922i	$-0.765469\pi$
$853$	−36902.0	−1.48124	−0.740622	+	0.671922i	$0.765469\pi$
$854$	0	0
$855$	0	0
$856$	33012.0	1.31814
$857$	−30648.0	−1.22161	−0.610803	−	0.791783i	$-0.709153\pi$
$857$	−30648.0	−1.22161	−0.610803	+	0.791783i	$0.709153\pi$
$858$	0	0
$859$	−20918.0	−0.830865	−0.415432	−	0.909624i	$-0.636370\pi$
$859$	−20918.0	−0.830865	−0.415432	+	0.909624i	$0.636370\pi$
$860$	−170.000	−0.00674064
$861$	0	0
$862$	−30654.0	−1.21123
$863$	−19812.0	−0.781470	−0.390735	−	0.920503i	$-0.627779\pi$
$863$	−19812.0	−0.781470	−0.390735	+	0.920503i	$0.627779\pi$
$864$	0	0
$865$	6990.00	0.274760
$866$	−17490.0	−0.686298
$867$	0	0
$868$	0	0
$869$	35520.0	1.38657
$870$	0	0
$871$	31388.0	1.22106
$872$	−29442.0	−1.14339
$873$	0	0
$874$	−39600.0	−1.53260
$875$	0	0
$876$	0	0
$877$	38900.0	1.49779	0.748894	−	0.662690i	$-0.230585\pi$
$877$	38900.0	1.49779	0.748894	+	0.662690i	$0.230585\pi$
$878$	25764.0	0.990311
$879$	0	0
$880$	21300.0	0.815935
$881$	16398.0	0.627086	0.313543	−	0.949574i	$-0.398484\pi$
$881$	16398.0	0.627086	0.313543	+	0.949574i	$0.398484\pi$
$882$	0	0
$883$	30422.0	1.15944	0.579718	−	0.814817i	$-0.303163\pi$
$883$	30422.0	1.15944	0.579718	+	0.814817i	$0.303163\pi$
$884$	3192.00	0.121446
$885$	0	0
$886$	−37980.0	−1.44014
$887$	6222.00	0.235529	0.117765	−	0.993042i	$-0.462427\pi$
$887$	6222.00	0.235529	0.117765	+	0.993042i	$0.462427\pi$
$888$	0	0
$889$	0	0
$890$	−15030.0	−0.566075
$891$	0	0
$892$	1888.00	0.0708687
$893$	660.000	0.0247324
$894$	0	0
$895$	12120.0	0.452656
$896$	0	0
$897$	0	0
$898$	10908.0	0.405350
$899$	38232.0	1.41836
$900$	0	0
$901$	48888.0	1.80765
$902$	−22680.0	−0.837208
$903$	0	0
$904$	17766.0	0.653638
$905$	−18640.0	−0.684657
$906$	0	0
$907$	21926.0	0.802691	0.401346	−	0.915927i	$-0.368543\pi$
$907$	21926.0	0.802691	0.401346	+	0.915927i	$0.368543\pi$
$908$	−2244.00	−0.0820151
$909$	0	0
$910$	0	0
$911$	−40662.0	−1.47881	−0.739403	−	0.673263i	$-0.764892\pi$
$911$	−40662.0	−1.47881	−0.739403	+	0.673263i	$0.764892\pi$
$912$	0	0
$913$	−47520.0	−1.72254
$914$	9066.00	0.328093
$915$	0	0
$916$	4084.00	0.147313
$917$	0	0
$918$	0	0
$919$	25688.0	0.922055	0.461028	−	0.887386i	$-0.347481\pi$
$919$	25688.0	0.922055	0.461028	+	0.887386i	$0.347481\pi$
$920$	−12600.0	−0.451532
$921$	0	0
$922$	44226.0	1.57972
$923$	−25308.0	−0.902517
$924$	0	0
$925$	−9400.00	−0.334130
$926$	27804.0	0.986713
$927$	0	0
$928$	−7290.00	−0.257873
$929$	−16026.0	−0.565981	−0.282990	−	0.959123i	$-0.591326\pi$
$929$	−16026.0	−0.565981	−0.282990	+	0.959123i	$0.591326\pi$
$930$	0	0
$931$	0	0
$932$	−4026.00	−0.141498
$933$	0	0
$934$	41760.0	1.46299
$935$	25200.0	0.881420
$936$	0	0
$937$	−18362.0	−0.640193	−0.320096	−	0.947385i	$-0.603715\pi$
$937$	−18362.0	−0.640193	−0.320096	+	0.947385i	$0.603715\pi$
$938$	0	0
$939$	0	0
$940$	−30.0000	−0.00104095
$941$	8034.00	0.278322	0.139161	−	0.990270i	$-0.455559\pi$
$941$	8034.00	0.278322	0.139161	+	0.990270i	$0.455559\pi$
$942$	0	0
$943$	15120.0	0.522137
$944$	−34932.0	−1.20439
$945$	0	0
$946$	−6120.00	−0.210337
$947$	30732.0	1.05455	0.527273	−	0.849696i	$-0.323214\pi$
$947$	30732.0	1.05455	0.527273	+	0.849696i	$0.323214\pi$
$948$	0	0
$949$	−31388.0	−1.07365
$950$	8250.00	0.281753
$951$	0	0
$952$	0	0
$953$	−28242.0	−0.959967	−0.479983	−	0.877278i	$-0.659357\pi$
$953$	−28242.0	−0.959967	−0.479983	+	0.877278i	$0.659357\pi$
$954$	0	0
$955$	−12750.0	−0.432021
$956$	4590.00	0.155284
$957$	0	0
$958$	24660.0	0.831658
$959$	0	0
$960$	0	0
$961$	25905.0	0.869558
$962$	−42864.0	−1.43658
$963$	0	0
$964$	−1946.00	−0.0650171
$965$	−9890.00	−0.329917
$966$	0	0
$967$	37496.0	1.24694	0.623470	−	0.781848i	$-0.285723\pi$
$967$	37496.0	1.24694	0.623470	+	0.781848i	$0.285723\pi$
$968$	47649.0	1.58212
$969$	0	0
$970$	21630.0	0.715977
$971$	−27204.0	−0.899092	−0.449546	−	0.893257i	$-0.648414\pi$
$971$	−27204.0	−0.899092	−0.449546	+	0.893257i	$0.648414\pi$
$972$	0	0
$973$	0	0
$974$	−47256.0	−1.55460
$975$	0	0
$976$	−62480.0	−2.04911
$977$	−42954.0	−1.40657	−0.703286	−	0.710907i	$-0.748285\pi$
$977$	−42954.0	−1.40657	−0.703286	+	0.710907i	$0.748285\pi$
$978$	0	0
$979$	−60120.0	−1.96266
$980$	0	0
$981$	0	0
$982$	19656.0	0.638746
$983$	−10182.0	−0.330372	−0.165186	−	0.986262i	$-0.552822\pi$
$983$	−10182.0	−0.330372	−0.165186	+	0.986262i	$0.552822\pi$
$984$	0	0
$985$	5850.00	0.189235
$986$	−40824.0	−1.31856
$987$	0	0
$988$	4180.00	0.134599
$989$	4080.00	0.131179
$990$	0	0
$991$	30008.0	0.961893	0.480946	−	0.876750i	$-0.340293\pi$
$991$	30008.0	0.961893	0.480946	+	0.876750i	$0.340293\pi$
$992$	−10620.0	−0.339905
$993$	0	0
$994$	0	0
$995$	−17920.0	−0.570957
$996$	0	0
$997$	47554.0	1.51058	0.755291	−	0.655390i	$-0.227496\pi$
$997$	47554.0	1.51058	0.755291	+	0.655390i	$0.227496\pi$
$998$	−12480.0	−0.395839
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.f.1.1		1
3.2	odd	2		2205.4.a.p.1.1		1
7.6	odd	2		315.4.a.b.1.1	✓	1
21.20	even	2		315.4.a.e.1.1	yes	1
35.34	odd	2		1575.4.a.i.1.1		1
105.104	even	2		1575.4.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.4.a.b.1.1	✓	1	7.6	odd	2
315.4.a.e.1.1	yes	1	21.20	even	2
1575.4.a.c.1.1		1	105.104	even	2
1575.4.a.i.1.1		1	35.34	odd	2
2205.4.a.f.1.1		1	1.1	even	1	trivial
2205.4.a.p.1.1		1	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2205.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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