LMFDB - Embedded newform 546.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [546,2,Mod(1,546)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(546, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("546.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 546 = 2 \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	546.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:	$4.35983195036$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	546.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} -1.00000 q^{13} -1.00000 q^{14} -3.00000 q^{15} +1.00000 q^{16} +7.00000 q^{17} +1.00000 q^{18} +1.00000 q^{19} +3.00000 q^{20} +1.00000 q^{21} +1.00000 q^{22} -7.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} -1.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} +3.00000 q^{29} -3.00000 q^{30} +1.00000 q^{32} -1.00000 q^{33} +7.00000 q^{34} -3.00000 q^{35} +1.00000 q^{36} -5.00000 q^{37} +1.00000 q^{38} +1.00000 q^{39} +3.00000 q^{40} +4.00000 q^{41} +1.00000 q^{42} +11.0000 q^{43} +1.00000 q^{44} +3.00000 q^{45} -7.00000 q^{46} -1.00000 q^{48} +1.00000 q^{49} +4.00000 q^{50} -7.00000 q^{51} -1.00000 q^{52} -14.0000 q^{53} -1.00000 q^{54} +3.00000 q^{55} -1.00000 q^{56} -1.00000 q^{57} +3.00000 q^{58} +4.00000 q^{59} -3.00000 q^{60} +1.00000 q^{61} -1.00000 q^{63} +1.00000 q^{64} -3.00000 q^{65} -1.00000 q^{66} -6.00000 q^{67} +7.00000 q^{68} +7.00000 q^{69} -3.00000 q^{70} -12.0000 q^{71} +1.00000 q^{72} +5.00000 q^{73} -5.00000 q^{74} -4.00000 q^{75} +1.00000 q^{76} -1.00000 q^{77} +1.00000 q^{78} -10.0000 q^{79} +3.00000 q^{80} +1.00000 q^{81} +4.00000 q^{82} -14.0000 q^{83} +1.00000 q^{84} +21.0000 q^{85} +11.0000 q^{86} -3.00000 q^{87} +1.00000 q^{88} -6.00000 q^{89} +3.00000 q^{90} +1.00000 q^{91} -7.00000 q^{92} +3.00000 q^{95} -1.00000 q^{96} +6.00000 q^{97} +1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	−1.00000	−0.408248
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	3.00000	0.948683
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	−1.00000	−0.288675
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−1.00000	−0.267261
$15$	−3.00000	−0.774597
$16$	1.00000	0.250000
$17$	7.00000	1.69775	0.848875	−	0.528594i	$-0.177281\pi$
$17$	7.00000	1.69775	0.848875	+	0.528594i	$0.177281\pi$
$18$	1.00000	0.235702
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	3.00000	0.670820
$21$	1.00000	0.218218
$22$	1.00000	0.213201
$23$	−7.00000	−1.45960	−0.729800	−	0.683660i	$-0.760387\pi$
$23$	−7.00000	−1.45960	−0.729800	+	0.683660i	$0.760387\pi$
$24$	−1.00000	−0.204124
$25$	4.00000	0.800000
$26$	−1.00000	−0.196116
$27$	−1.00000	−0.192450
$28$	−1.00000	−0.188982
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	−3.00000	−0.547723
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	7.00000	1.20049
$35$	−3.00000	−0.507093
$36$	1.00000	0.166667
$37$	−5.00000	−0.821995	−0.410997	−	0.911636i	$-0.634819\pi$
$37$	−5.00000	−0.821995	−0.410997	+	0.911636i	$0.634819\pi$
$38$	1.00000	0.162221
$39$	1.00000	0.160128
$40$	3.00000	0.474342
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	1.00000	0.154303
$43$	11.0000	1.67748	0.838742	−	0.544529i	$-0.183292\pi$
$43$	11.0000	1.67748	0.838742	+	0.544529i	$0.183292\pi$
$44$	1.00000	0.150756
$45$	3.00000	0.447214
$46$	−7.00000	−1.03209
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	4.00000	0.565685
$51$	−7.00000	−0.980196
$52$	−1.00000	−0.138675
$53$	−14.0000	−1.92305	−0.961524	−	0.274721i	$-0.911414\pi$
$53$	−14.0000	−1.92305	−0.961524	+	0.274721i	$0.911414\pi$
$54$	−1.00000	−0.136083
$55$	3.00000	0.404520
$56$	−1.00000	−0.133631
$57$	−1.00000	−0.132453
$58$	3.00000	0.393919
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	−3.00000	−0.387298
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	−3.00000	−0.372104
$66$	−1.00000	−0.123091
$67$	−6.00000	−0.733017	−0.366508	−	0.930415i	$-0.619447\pi$
$67$	−6.00000	−0.733017	−0.366508	+	0.930415i	$0.619447\pi$
$68$	7.00000	0.848875
$69$	7.00000	0.842701
$70$	−3.00000	−0.358569
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	1.00000	0.117851
$73$	5.00000	0.585206	0.292603	−	0.956234i	$-0.405479\pi$
$73$	5.00000	0.585206	0.292603	+	0.956234i	$0.405479\pi$
$74$	−5.00000	−0.581238
$75$	−4.00000	−0.461880
$76$	1.00000	0.114708
$77$	−1.00000	−0.113961
$78$	1.00000	0.113228
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	3.00000	0.335410
$81$	1.00000	0.111111
$82$	4.00000	0.441726
$83$	−14.0000	−1.53670	−0.768350	−	0.640030i	$-0.778922\pi$
$83$	−14.0000	−1.53670	−0.768350	+	0.640030i	$0.778922\pi$
$84$	1.00000	0.109109
$85$	21.0000	2.27777
$86$	11.0000	1.18616
$87$	−3.00000	−0.321634
$88$	1.00000	0.106600
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	3.00000	0.316228
$91$	1.00000	0.104828
$92$	−7.00000	−0.729800
$93$	0	0
$94$	0	0
$95$	3.00000	0.307794
$96$	−1.00000	−0.102062
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	1.00000	0.101015
$99$	1.00000	0.100504
$100$	4.00000	0.400000
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	−7.00000	−0.693103
$103$	−7.00000	−0.689730	−0.344865	−	0.938652i	$-0.612075\pi$
$103$	−7.00000	−0.689730	−0.344865	+	0.938652i	$0.612075\pi$
$104$	−1.00000	−0.0980581
$105$	3.00000	0.292770
$106$	−14.0000	−1.35980
$107$	−18.0000	−1.74013	−0.870063	−	0.492941i	$-0.835922\pi$
$107$	−18.0000	−1.74013	−0.870063	+	0.492941i	$0.835922\pi$
$108$	−1.00000	−0.0962250
$109$	−7.00000	−0.670478	−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000	−0.670478	−0.335239	+	0.942133i	$0.608817\pi$
$110$	3.00000	0.286039
$111$	5.00000	0.474579
$112$	−1.00000	−0.0944911
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	−1.00000	−0.0936586
$115$	−21.0000	−1.95826
$116$	3.00000	0.278543
$117$	−1.00000	−0.0924500
$118$	4.00000	0.368230
$119$	−7.00000	−0.641689
$120$	−3.00000	−0.273861
$121$	−10.0000	−0.909091
$122$	1.00000	0.0905357
$123$	−4.00000	−0.360668
$124$	0	0
$125$	−3.00000	−0.268328
$126$	−1.00000	−0.0890871
$127$	10.0000	0.887357	0.443678	−	0.896186i	$-0.353673\pi$
$127$	10.0000	0.887357	0.443678	+	0.896186i	$0.353673\pi$
$128$	1.00000	0.0883883
$129$	−11.0000	−0.968496
$130$	−3.00000	−0.263117
$131$	15.0000	1.31056	0.655278	−	0.755388i	$-0.272551\pi$
$131$	15.0000	1.31056	0.655278	+	0.755388i	$0.272551\pi$
$132$	−1.00000	−0.0870388
$133$	−1.00000	−0.0867110
$134$	−6.00000	−0.518321
$135$	−3.00000	−0.258199
$136$	7.00000	0.600245
$137$	15.0000	1.28154	0.640768	−	0.767734i	$-0.278616\pi$
$137$	15.0000	1.28154	0.640768	+	0.767734i	$0.278616\pi$
$138$	7.00000	0.595880
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	−3.00000	−0.253546
$141$	0	0
$142$	−12.0000	−1.00702
$143$	−1.00000	−0.0836242
$144$	1.00000	0.0833333
$145$	9.00000	0.747409
$146$	5.00000	0.413803
$147$	−1.00000	−0.0824786
$148$	−5.00000	−0.410997
$149$	16.0000	1.31077	0.655386	−	0.755295i	$-0.272506\pi$
$149$	16.0000	1.31077	0.655386	+	0.755295i	$0.272506\pi$
$150$	−4.00000	−0.326599
$151$	7.00000	0.569652	0.284826	−	0.958579i	$-0.408064\pi$
$151$	7.00000	0.569652	0.284826	+	0.958579i	$0.408064\pi$
$152$	1.00000	0.0811107
$153$	7.00000	0.565916
$154$	−1.00000	−0.0805823
$155$	0	0
$156$	1.00000	0.0800641
$157$	5.00000	0.399043	0.199522	−	0.979893i	$-0.436061\pi$
$157$	5.00000	0.399043	0.199522	+	0.979893i	$0.436061\pi$
$158$	−10.0000	−0.795557
$159$	14.0000	1.11027
$160$	3.00000	0.237171
$161$	7.00000	0.551677
$162$	1.00000	0.0785674
$163$	10.0000	0.783260	0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000	0.783260	0.391630	+	0.920123i	$0.371911\pi$
$164$	4.00000	0.312348
$165$	−3.00000	−0.233550
$166$	−14.0000	−1.08661
$167$	7.00000	0.541676	0.270838	−	0.962625i	$-0.412699\pi$
$167$	7.00000	0.541676	0.270838	+	0.962625i	$0.412699\pi$
$168$	1.00000	0.0771517
$169$	1.00000	0.0769231
$170$	21.0000	1.61063
$171$	1.00000	0.0764719
$172$	11.0000	0.838742
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	−3.00000	−0.227429
$175$	−4.00000	−0.302372
$176$	1.00000	0.0753778
$177$	−4.00000	−0.300658
$178$	−6.00000	−0.449719
$179$	8.00000	0.597948	0.298974	−	0.954261i	$-0.403356\pi$
$179$	8.00000	0.597948	0.298974	+	0.954261i	$0.403356\pi$
$180$	3.00000	0.223607
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	1.00000	0.0741249
$183$	−1.00000	−0.0739221
$184$	−7.00000	−0.516047
$185$	−15.0000	−1.10282
$186$	0	0
$187$	7.00000	0.511891
$188$	0	0
$189$	1.00000	0.0727393
$190$	3.00000	0.217643
$191$	−15.0000	−1.08536	−0.542681	−	0.839939i	$-0.682591\pi$
$191$	−15.0000	−1.08536	−0.542681	+	0.839939i	$0.682591\pi$
$192$	−1.00000	−0.0721688
$193$	−8.00000	−0.575853	−0.287926	−	0.957653i	$-0.592966\pi$
$193$	−8.00000	−0.575853	−0.287926	+	0.957653i	$0.592966\pi$
$194$	6.00000	0.430775
$195$	3.00000	0.214834
$196$	1.00000	0.0714286
$197$	−20.0000	−1.42494	−0.712470	−	0.701702i	$-0.752424\pi$
$197$	−20.0000	−1.42494	−0.712470	+	0.701702i	$0.752424\pi$
$198$	1.00000	0.0710669
$199$	3.00000	0.212664	0.106332	−	0.994331i	$-0.466089\pi$
$199$	3.00000	0.212664	0.106332	+	0.994331i	$0.466089\pi$
$200$	4.00000	0.282843
$201$	6.00000	0.423207
$202$	−12.0000	−0.844317
$203$	−3.00000	−0.210559
$204$	−7.00000	−0.490098
$205$	12.0000	0.838116
$206$	−7.00000	−0.487713
$207$	−7.00000	−0.486534
$208$	−1.00000	−0.0693375
$209$	1.00000	0.0691714
$210$	3.00000	0.207020
$211$	25.0000	1.72107	0.860535	−	0.509390i	$-0.170129\pi$
$211$	25.0000	1.72107	0.860535	+	0.509390i	$0.170129\pi$
$212$	−14.0000	−0.961524
$213$	12.0000	0.822226
$214$	−18.0000	−1.23045
$215$	33.0000	2.25058
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−7.00000	−0.474100
$219$	−5.00000	−0.337869
$220$	3.00000	0.202260
$221$	−7.00000	−0.470871
$222$	5.00000	0.335578
$223$	14.0000	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$224$	−1.00000	−0.0668153
$225$	4.00000	0.266667
$226$	−14.0000	−0.931266
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	−1.00000	−0.0662266
$229$	18.0000	1.18947	0.594737	−	0.803921i	$-0.297256\pi$
$229$	18.0000	1.18947	0.594737	+	0.803921i	$0.297256\pi$
$230$	−21.0000	−1.38470
$231$	1.00000	0.0657952
$232$	3.00000	0.196960
$233$	−12.0000	−0.786146	−0.393073	−	0.919507i	$-0.628588\pi$
$233$	−12.0000	−0.786146	−0.393073	+	0.919507i	$0.628588\pi$
$234$	−1.00000	−0.0653720
$235$	0	0
$236$	4.00000	0.260378
$237$	10.0000	0.649570
$238$	−7.00000	−0.453743
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	−3.00000	−0.193649
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	−10.0000	−0.642824
$243$	−1.00000	−0.0641500
$244$	1.00000	0.0640184
$245$	3.00000	0.191663
$246$	−4.00000	−0.255031
$247$	−1.00000	−0.0636285
$248$	0	0
$249$	14.0000	0.887214
$250$	−3.00000	−0.189737
$251$	3.00000	0.189358	0.0946792	−	0.995508i	$-0.469817\pi$
$251$	3.00000	0.189358	0.0946792	+	0.995508i	$0.469817\pi$
$252$	−1.00000	−0.0629941
$253$	−7.00000	−0.440086
$254$	10.0000	0.627456
$255$	−21.0000	−1.31507
$256$	1.00000	0.0625000
$257$	10.0000	0.623783	0.311891	−	0.950118i	$-0.399037\pi$
$257$	10.0000	0.623783	0.311891	+	0.950118i	$0.399037\pi$
$258$	−11.0000	−0.684830
$259$	5.00000	0.310685
$260$	−3.00000	−0.186052
$261$	3.00000	0.185695
$262$	15.0000	0.926703
$263$	−4.00000	−0.246651	−0.123325	−	0.992366i	$-0.539356\pi$
$263$	−4.00000	−0.246651	−0.123325	+	0.992366i	$0.539356\pi$
$264$	−1.00000	−0.0615457
$265$	−42.0000	−2.58004
$266$	−1.00000	−0.0613139
$267$	6.00000	0.367194
$268$	−6.00000	−0.366508
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	−3.00000	−0.182574
$271$	−30.0000	−1.82237	−0.911185	−	0.411997i	$-0.864831\pi$
$271$	−30.0000	−1.82237	−0.911185	+	0.411997i	$0.864831\pi$
$272$	7.00000	0.424437
$273$	−1.00000	−0.0605228
$274$	15.0000	0.906183
$275$	4.00000	0.241209
$276$	7.00000	0.421350
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	4.00000	0.239904
$279$	0	0
$280$	−3.00000	−0.179284
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	−12.0000	−0.712069
$285$	−3.00000	−0.177705
$286$	−1.00000	−0.0591312
$287$	−4.00000	−0.236113
$288$	1.00000	0.0589256
$289$	32.0000	1.88235
$290$	9.00000	0.528498
$291$	−6.00000	−0.351726
$292$	5.00000	0.292603
$293$	30.0000	1.75262	0.876309	−	0.481749i	$-0.159998\pi$
$293$	30.0000	1.75262	0.876309	+	0.481749i	$0.159998\pi$
$294$	−1.00000	−0.0583212
$295$	12.0000	0.698667
$296$	−5.00000	−0.290619
$297$	−1.00000	−0.0580259
$298$	16.0000	0.926855
$299$	7.00000	0.404820
$300$	−4.00000	−0.230940
$301$	−11.0000	−0.634029
$302$	7.00000	0.402805
$303$	12.0000	0.689382
$304$	1.00000	0.0573539
$305$	3.00000	0.171780
$306$	7.00000	0.400163
$307$	−32.0000	−1.82634	−0.913168	−	0.407583i	$-0.866372\pi$
$307$	−32.0000	−1.82634	−0.913168	+	0.407583i	$0.866372\pi$
$308$	−1.00000	−0.0569803
$309$	7.00000	0.398216
$310$	0	0
$311$	2.00000	0.113410	0.0567048	−	0.998391i	$-0.481941\pi$
$311$	2.00000	0.113410	0.0567048	+	0.998391i	$0.481941\pi$
$312$	1.00000	0.0566139
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	5.00000	0.282166
$315$	−3.00000	−0.169031
$316$	−10.0000	−0.562544
$317$	−28.0000	−1.57264	−0.786318	−	0.617822i	$-0.788015\pi$
$317$	−28.0000	−1.57264	−0.786318	+	0.617822i	$0.788015\pi$
$318$	14.0000	0.785081
$319$	3.00000	0.167968
$320$	3.00000	0.167705
$321$	18.0000	1.00466
$322$	7.00000	0.390095
$323$	7.00000	0.389490
$324$	1.00000	0.0555556
$325$	−4.00000	−0.221880
$326$	10.0000	0.553849
$327$	7.00000	0.387101
$328$	4.00000	0.220863
$329$	0	0
$330$	−3.00000	−0.165145
$331$	−14.0000	−0.769510	−0.384755	−	0.923019i	$-0.625714\pi$
$331$	−14.0000	−0.769510	−0.384755	+	0.923019i	$0.625714\pi$
$332$	−14.0000	−0.768350
$333$	−5.00000	−0.273998
$334$	7.00000	0.383023
$335$	−18.0000	−0.983445
$336$	1.00000	0.0545545
$337$	3.00000	0.163420	0.0817102	−	0.996656i	$-0.473962\pi$
$337$	3.00000	0.163420	0.0817102	+	0.996656i	$0.473962\pi$
$338$	1.00000	0.0543928
$339$	14.0000	0.760376
$340$	21.0000	1.13888
$341$	0	0
$342$	1.00000	0.0540738
$343$	−1.00000	−0.0539949
$344$	11.0000	0.593080
$345$	21.0000	1.13060
$346$	2.00000	0.107521
$347$	32.0000	1.71785	0.858925	−	0.512101i	$-0.171133\pi$
$347$	32.0000	1.71785	0.858925	+	0.512101i	$0.171133\pi$
$348$	−3.00000	−0.160817
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	−4.00000	−0.213809
$351$	1.00000	0.0533761
$352$	1.00000	0.0533002
$353$	−10.0000	−0.532246	−0.266123	−	0.963939i	$-0.585743\pi$
$353$	−10.0000	−0.532246	−0.266123	+	0.963939i	$0.585743\pi$
$354$	−4.00000	−0.212598
$355$	−36.0000	−1.91068
$356$	−6.00000	−0.317999
$357$	7.00000	0.370479
$358$	8.00000	0.422813
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	3.00000	0.158114
$361$	−18.0000	−0.947368
$362$	22.0000	1.15629
$363$	10.0000	0.524864
$364$	1.00000	0.0524142
$365$	15.0000	0.785136
$366$	−1.00000	−0.0522708
$367$	32.0000	1.67039	0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000	1.67039	0.835193	+	0.549957i	$0.185356\pi$
$368$	−7.00000	−0.364900
$369$	4.00000	0.208232
$370$	−15.0000	−0.779813
$371$	14.0000	0.726844
$372$	0	0
$373$	12.0000	0.621336	0.310668	−	0.950518i	$-0.399447\pi$
$373$	12.0000	0.621336	0.310668	+	0.950518i	$0.399447\pi$
$374$	7.00000	0.361961
$375$	3.00000	0.154919
$376$	0	0
$377$	−3.00000	−0.154508
$378$	1.00000	0.0514344
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	3.00000	0.153897
$381$	−10.0000	−0.512316
$382$	−15.0000	−0.767467
$383$	−1.00000	−0.0510976	−0.0255488	−	0.999674i	$-0.508133\pi$
$383$	−1.00000	−0.0510976	−0.0255488	+	0.999674i	$0.508133\pi$
$384$	−1.00000	−0.0510310
$385$	−3.00000	−0.152894
$386$	−8.00000	−0.407189
$387$	11.0000	0.559161
$388$	6.00000	0.304604
$389$	34.0000	1.72387	0.861934	−	0.507020i	$-0.169253\pi$
$389$	34.0000	1.72387	0.861934	+	0.507020i	$0.169253\pi$
$390$	3.00000	0.151911
$391$	−49.0000	−2.47804
$392$	1.00000	0.0505076
$393$	−15.0000	−0.756650
$394$	−20.0000	−1.00759
$395$	−30.0000	−1.50946
$396$	1.00000	0.0502519
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	3.00000	0.150376
$399$	1.00000	0.0500626
$400$	4.00000	0.200000
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	6.00000	0.299253
$403$	0	0
$404$	−12.0000	−0.597022
$405$	3.00000	0.149071
$406$	−3.00000	−0.148888
$407$	−5.00000	−0.247841
$408$	−7.00000	−0.346552
$409$	23.0000	1.13728	0.568638	−	0.822588i	$-0.307470\pi$
$409$	23.0000	1.13728	0.568638	+	0.822588i	$0.307470\pi$
$410$	12.0000	0.592638
$411$	−15.0000	−0.739895
$412$	−7.00000	−0.344865
$413$	−4.00000	−0.196827
$414$	−7.00000	−0.344031
$415$	−42.0000	−2.06170
$416$	−1.00000	−0.0490290
$417$	−4.00000	−0.195881
$418$	1.00000	0.0489116
$419$	29.0000	1.41674	0.708371	−	0.705840i	$-0.249430\pi$
$419$	29.0000	1.41674	0.708371	+	0.705840i	$0.249430\pi$
$420$	3.00000	0.146385
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	25.0000	1.21698
$423$	0	0
$424$	−14.0000	−0.679900
$425$	28.0000	1.35820
$426$	12.0000	0.581402
$427$	−1.00000	−0.0483934
$428$	−18.0000	−0.870063
$429$	1.00000	0.0482805
$430$	33.0000	1.59140
$431$	−10.0000	−0.481683	−0.240842	−	0.970564i	$-0.577423\pi$
$431$	−10.0000	−0.481683	−0.240842	+	0.970564i	$0.577423\pi$
$432$	−1.00000	−0.0481125
$433$	−8.00000	−0.384455	−0.192228	−	0.981350i	$-0.561571\pi$
$433$	−8.00000	−0.384455	−0.192228	+	0.981350i	$0.561571\pi$
$434$	0	0
$435$	−9.00000	−0.431517
$436$	−7.00000	−0.335239
$437$	−7.00000	−0.334855
$438$	−5.00000	−0.238909
$439$	−19.0000	−0.906821	−0.453410	−	0.891302i	$-0.649793\pi$
$439$	−19.0000	−0.906821	−0.453410	+	0.891302i	$0.649793\pi$
$440$	3.00000	0.143019
$441$	1.00000	0.0476190
$442$	−7.00000	−0.332956
$443$	26.0000	1.23530	0.617649	−	0.786454i	$-0.288085\pi$
$443$	26.0000	1.23530	0.617649	+	0.786454i	$0.288085\pi$
$444$	5.00000	0.237289
$445$	−18.0000	−0.853282
$446$	14.0000	0.662919
$447$	−16.0000	−0.756774
$448$	−1.00000	−0.0472456
$449$	19.0000	0.896665	0.448333	−	0.893867i	$-0.352018\pi$
$449$	19.0000	0.896665	0.448333	+	0.893867i	$0.352018\pi$
$450$	4.00000	0.188562
$451$	4.00000	0.188353
$452$	−14.0000	−0.658505
$453$	−7.00000	−0.328889
$454$	−18.0000	−0.844782
$455$	3.00000	0.140642
$456$	−1.00000	−0.0468293
$457$	−40.0000	−1.87112	−0.935561	−	0.353166i	$-0.885105\pi$
$457$	−40.0000	−1.87112	−0.935561	+	0.353166i	$0.885105\pi$
$458$	18.0000	0.841085
$459$	−7.00000	−0.326732
$460$	−21.0000	−0.979130
$461$	−23.0000	−1.07122	−0.535608	−	0.844466i	$-0.679918\pi$
$461$	−23.0000	−1.07122	−0.535608	+	0.844466i	$0.679918\pi$
$462$	1.00000	0.0465242
$463$	23.0000	1.06890	0.534450	−	0.845200i	$-0.320519\pi$
$463$	23.0000	1.06890	0.534450	+	0.845200i	$0.320519\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	−12.0000	−0.555889
$467$	3.00000	0.138823	0.0694117	−	0.997588i	$-0.477888\pi$
$467$	3.00000	0.138823	0.0694117	+	0.997588i	$0.477888\pi$
$468$	−1.00000	−0.0462250
$469$	6.00000	0.277054
$470$	0	0
$471$	−5.00000	−0.230388
$472$	4.00000	0.184115
$473$	11.0000	0.505781
$474$	10.0000	0.459315
$475$	4.00000	0.183533
$476$	−7.00000	−0.320844
$477$	−14.0000	−0.641016
$478$	20.0000	0.914779
$479$	−11.0000	−0.502603	−0.251301	−	0.967909i	$-0.580859\pi$
$479$	−11.0000	−0.502603	−0.251301	+	0.967909i	$0.580859\pi$
$480$	−3.00000	−0.136931
$481$	5.00000	0.227980
$482$	−18.0000	−0.819878
$483$	−7.00000	−0.318511
$484$	−10.0000	−0.454545
$485$	18.0000	0.817338
$486$	−1.00000	−0.0453609
$487$	−32.0000	−1.45006	−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000	−1.45006	−0.725029	+	0.688718i	$0.758174\pi$
$488$	1.00000	0.0452679
$489$	−10.0000	−0.452216
$490$	3.00000	0.135526
$491$	2.00000	0.0902587	0.0451294	−	0.998981i	$-0.485630\pi$
$491$	2.00000	0.0902587	0.0451294	+	0.998981i	$0.485630\pi$
$492$	−4.00000	−0.180334
$493$	21.0000	0.945792
$494$	−1.00000	−0.0449921
$495$	3.00000	0.134840
$496$	0	0
$497$	12.0000	0.538274
$498$	14.0000	0.627355
$499$	16.0000	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$500$	−3.00000	−0.134164
$501$	−7.00000	−0.312737
$502$	3.00000	0.133897
$503$	−6.00000	−0.267527	−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000	−0.267527	−0.133763	+	0.991013i	$0.542706\pi$
$504$	−1.00000	−0.0445435
$505$	−36.0000	−1.60198
$506$	−7.00000	−0.311188
$507$	−1.00000	−0.0444116
$508$	10.0000	0.443678
$509$	−39.0000	−1.72864	−0.864322	−	0.502938i	$-0.832252\pi$
$509$	−39.0000	−1.72864	−0.864322	+	0.502938i	$0.832252\pi$
$510$	−21.0000	−0.929896
$511$	−5.00000	−0.221187
$512$	1.00000	0.0441942
$513$	−1.00000	−0.0441511
$514$	10.0000	0.441081
$515$	−21.0000	−0.925371
$516$	−11.0000	−0.484248
$517$	0	0
$518$	5.00000	0.219687
$519$	−2.00000	−0.0877903
$520$	−3.00000	−0.131559
$521$	−11.0000	−0.481919	−0.240959	−	0.970535i	$-0.577462\pi$
$521$	−11.0000	−0.481919	−0.240959	+	0.970535i	$0.577462\pi$
$522$	3.00000	0.131306
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	15.0000	0.655278
$525$	4.00000	0.174574
$526$	−4.00000	−0.174408
$527$	0	0
$528$	−1.00000	−0.0435194
$529$	26.0000	1.13043
$530$	−42.0000	−1.82436
$531$	4.00000	0.173585
$532$	−1.00000	−0.0433555
$533$	−4.00000	−0.173259
$534$	6.00000	0.259645
$535$	−54.0000	−2.33462
$536$	−6.00000	−0.259161
$537$	−8.00000	−0.345225
$538$	2.00000	0.0862261
$539$	1.00000	0.0430730
$540$	−3.00000	−0.129099
$541$	25.0000	1.07483	0.537417	−	0.843317i	$-0.319400\pi$
$541$	25.0000	1.07483	0.537417	+	0.843317i	$0.319400\pi$
$542$	−30.0000	−1.28861
$543$	−22.0000	−0.944110
$544$	7.00000	0.300123
$545$	−21.0000	−0.899541
$546$	−1.00000	−0.0427960
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	15.0000	0.640768
$549$	1.00000	0.0426790
$550$	4.00000	0.170561
$551$	3.00000	0.127804
$552$	7.00000	0.297940
$553$	10.0000	0.425243
$554$	10.0000	0.424859
$555$	15.0000	0.636715
$556$	4.00000	0.169638
$557$	−12.0000	−0.508456	−0.254228	−	0.967144i	$-0.581821\pi$
$557$	−12.0000	−0.508456	−0.254228	+	0.967144i	$0.581821\pi$
$558$	0	0
$559$	−11.0000	−0.465250
$560$	−3.00000	−0.126773
$561$	−7.00000	−0.295540
$562$	6.00000	0.253095
$563$	39.0000	1.64365	0.821827	−	0.569737i	$-0.192955\pi$
$563$	39.0000	1.64365	0.821827	+	0.569737i	$0.192955\pi$
$564$	0	0
$565$	−42.0000	−1.76695
$566$	−16.0000	−0.672530
$567$	−1.00000	−0.0419961
$568$	−12.0000	−0.503509
$569$	14.0000	0.586911	0.293455	−	0.955973i	$-0.405195\pi$
$569$	14.0000	0.586911	0.293455	+	0.955973i	$0.405195\pi$
$570$	−3.00000	−0.125656
$571$	−8.00000	−0.334790	−0.167395	−	0.985890i	$-0.553535\pi$
$571$	−8.00000	−0.334790	−0.167395	+	0.985890i	$0.553535\pi$
$572$	−1.00000	−0.0418121
$573$	15.0000	0.626634
$574$	−4.00000	−0.166957
$575$	−28.0000	−1.16768
$576$	1.00000	0.0416667
$577$	−26.0000	−1.08239	−0.541197	−	0.840896i	$-0.682029\pi$
$577$	−26.0000	−1.08239	−0.541197	+	0.840896i	$0.682029\pi$
$578$	32.0000	1.33102
$579$	8.00000	0.332469
$580$	9.00000	0.373705
$581$	14.0000	0.580818
$582$	−6.00000	−0.248708
$583$	−14.0000	−0.579821
$584$	5.00000	0.206901
$585$	−3.00000	−0.124035
$586$	30.0000	1.23929
$587$	42.0000	1.73353	0.866763	−	0.498721i	$-0.166197\pi$
$587$	42.0000	1.73353	0.866763	+	0.498721i	$0.166197\pi$
$588$	−1.00000	−0.0412393
$589$	0	0
$590$	12.0000	0.494032
$591$	20.0000	0.822690
$592$	−5.00000	−0.205499
$593$	−42.0000	−1.72473	−0.862367	−	0.506284i	$-0.831019\pi$
$593$	−42.0000	−1.72473	−0.862367	+	0.506284i	$0.831019\pi$
$594$	−1.00000	−0.0410305
$595$	−21.0000	−0.860916
$596$	16.0000	0.655386
$597$	−3.00000	−0.122782
$598$	7.00000	0.286251
$599$	−1.00000	−0.0408589	−0.0204294	−	0.999791i	$-0.506503\pi$
$599$	−1.00000	−0.0408589	−0.0204294	+	0.999791i	$0.506503\pi$
$600$	−4.00000	−0.163299
$601$	16.0000	0.652654	0.326327	−	0.945257i	$-0.394189\pi$
$601$	16.0000	0.652654	0.326327	+	0.945257i	$0.394189\pi$
$602$	−11.0000	−0.448327
$603$	−6.00000	−0.244339
$604$	7.00000	0.284826
$605$	−30.0000	−1.21967
$606$	12.0000	0.487467
$607$	13.0000	0.527654	0.263827	−	0.964570i	$-0.415015\pi$
$607$	13.0000	0.527654	0.263827	+	0.964570i	$0.415015\pi$
$608$	1.00000	0.0405554
$609$	3.00000	0.121566
$610$	3.00000	0.121466
$611$	0	0
$612$	7.00000	0.282958
$613$	−7.00000	−0.282727	−0.141364	−	0.989958i	$-0.545149\pi$
$613$	−7.00000	−0.282727	−0.141364	+	0.989958i	$0.545149\pi$
$614$	−32.0000	−1.29141
$615$	−12.0000	−0.483887
$616$	−1.00000	−0.0402911
$617$	−27.0000	−1.08698	−0.543490	−	0.839416i	$-0.682897\pi$
$617$	−27.0000	−1.08698	−0.543490	+	0.839416i	$0.682897\pi$
$618$	7.00000	0.281581
$619$	1.00000	0.0401934	0.0200967	−	0.999798i	$-0.493603\pi$
$619$	1.00000	0.0401934	0.0200967	+	0.999798i	$0.493603\pi$
$620$	0	0
$621$	7.00000	0.280900
$622$	2.00000	0.0801927
$623$	6.00000	0.240385
$624$	1.00000	0.0400320
$625$	−29.0000	−1.16000
$626$	0	0
$627$	−1.00000	−0.0399362
$628$	5.00000	0.199522
$629$	−35.0000	−1.39554
$630$	−3.00000	−0.119523
$631$	−37.0000	−1.47295	−0.736473	−	0.676467i	$-0.763510\pi$
$631$	−37.0000	−1.47295	−0.736473	+	0.676467i	$0.763510\pi$
$632$	−10.0000	−0.397779
$633$	−25.0000	−0.993661
$634$	−28.0000	−1.11202
$635$	30.0000	1.19051
$636$	14.0000	0.555136
$637$	−1.00000	−0.0396214
$638$	3.00000	0.118771
$639$	−12.0000	−0.474713
$640$	3.00000	0.118585
$641$	42.0000	1.65890	0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000	1.65890	0.829450	+	0.558581i	$0.188654\pi$
$642$	18.0000	0.710403
$643$	−9.00000	−0.354925	−0.177463	−	0.984128i	$-0.556789\pi$
$643$	−9.00000	−0.354925	−0.177463	+	0.984128i	$0.556789\pi$
$644$	7.00000	0.275839
$645$	−33.0000	−1.29937
$646$	7.00000	0.275411
$647$	−36.0000	−1.41531	−0.707653	−	0.706560i	$-0.750246\pi$
$647$	−36.0000	−1.41531	−0.707653	+	0.706560i	$0.750246\pi$
$648$	1.00000	0.0392837
$649$	4.00000	0.157014
$650$	−4.00000	−0.156893
$651$	0	0
$652$	10.0000	0.391630
$653$	−43.0000	−1.68272	−0.841360	−	0.540475i	$-0.818245\pi$
$653$	−43.0000	−1.68272	−0.841360	+	0.540475i	$0.818245\pi$
$654$	7.00000	0.273722
$655$	45.0000	1.75830
$656$	4.00000	0.156174
$657$	5.00000	0.195069
$658$	0	0
$659$	−18.0000	−0.701180	−0.350590	−	0.936529i	$-0.614019\pi$
$659$	−18.0000	−0.701180	−0.350590	+	0.936529i	$0.614019\pi$
$660$	−3.00000	−0.116775
$661$	38.0000	1.47803	0.739014	−	0.673690i	$-0.235292\pi$
$661$	38.0000	1.47803	0.739014	+	0.673690i	$0.235292\pi$
$662$	−14.0000	−0.544125
$663$	7.00000	0.271857
$664$	−14.0000	−0.543305
$665$	−3.00000	−0.116335
$666$	−5.00000	−0.193746
$667$	−21.0000	−0.813123
$668$	7.00000	0.270838
$669$	−14.0000	−0.541271
$670$	−18.0000	−0.695401
$671$	1.00000	0.0386046
$672$	1.00000	0.0385758
$673$	7.00000	0.269830	0.134915	−	0.990857i	$-0.456924\pi$
$673$	7.00000	0.269830	0.134915	+	0.990857i	$0.456924\pi$
$674$	3.00000	0.115556
$675$	−4.00000	−0.153960
$676$	1.00000	0.0384615
$677$	−38.0000	−1.46046	−0.730229	−	0.683202i	$-0.760587\pi$
$677$	−38.0000	−1.46046	−0.730229	+	0.683202i	$0.760587\pi$
$678$	14.0000	0.537667
$679$	−6.00000	−0.230259
$680$	21.0000	0.805313
$681$	18.0000	0.689761
$682$	0	0
$683$	−13.0000	−0.497431	−0.248716	−	0.968577i	$-0.580008\pi$
$683$	−13.0000	−0.497431	−0.248716	+	0.968577i	$0.580008\pi$
$684$	1.00000	0.0382360
$685$	45.0000	1.71936
$686$	−1.00000	−0.0381802
$687$	−18.0000	−0.686743
$688$	11.0000	0.419371
$689$	14.0000	0.533358
$690$	21.0000	0.799456
$691$	−4.00000	−0.152167	−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000	−0.152167	−0.0760836	+	0.997101i	$0.524242\pi$
$692$	2.00000	0.0760286
$693$	−1.00000	−0.0379869
$694$	32.0000	1.21470
$695$	12.0000	0.455186
$696$	−3.00000	−0.113715
$697$	28.0000	1.06058
$698$	2.00000	0.0757011
$699$	12.0000	0.453882
$700$	−4.00000	−0.151186
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	1.00000	0.0377426
$703$	−5.00000	−0.188579
$704$	1.00000	0.0376889
$705$	0	0
$706$	−10.0000	−0.376355
$707$	12.0000	0.451306
$708$	−4.00000	−0.150329
$709$	50.0000	1.87779	0.938895	−	0.344204i	$-0.111851\pi$
$709$	50.0000	1.87779	0.938895	+	0.344204i	$0.111851\pi$
$710$	−36.0000	−1.35106
$711$	−10.0000	−0.375029
$712$	−6.00000	−0.224860
$713$	0	0
$714$	7.00000	0.261968
$715$	−3.00000	−0.112194
$716$	8.00000	0.298974
$717$	−20.0000	−0.746914
$718$	12.0000	0.447836
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	3.00000	0.111803
$721$	7.00000	0.260694
$722$	−18.0000	−0.669891
$723$	18.0000	0.669427
$724$	22.0000	0.817624
$725$	12.0000	0.445669
$726$	10.0000	0.371135
$727$	37.0000	1.37225	0.686127	−	0.727482i	$-0.259309\pi$
$727$	37.0000	1.37225	0.686127	+	0.727482i	$0.259309\pi$
$728$	1.00000	0.0370625
$729$	1.00000	0.0370370
$730$	15.0000	0.555175
$731$	77.0000	2.84795
$732$	−1.00000	−0.0369611
$733$	−4.00000	−0.147743	−0.0738717	−	0.997268i	$-0.523536\pi$
$733$	−4.00000	−0.147743	−0.0738717	+	0.997268i	$0.523536\pi$
$734$	32.0000	1.18114
$735$	−3.00000	−0.110657
$736$	−7.00000	−0.258023
$737$	−6.00000	−0.221013
$738$	4.00000	0.147242
$739$	38.0000	1.39785	0.698926	−	0.715194i	$-0.253662\pi$
$739$	38.0000	1.39785	0.698926	+	0.715194i	$0.253662\pi$
$740$	−15.0000	−0.551411
$741$	1.00000	0.0367359
$742$	14.0000	0.513956
$743$	−26.0000	−0.953847	−0.476924	−	0.878945i	$-0.658248\pi$
$743$	−26.0000	−0.953847	−0.476924	+	0.878945i	$0.658248\pi$
$744$	0	0
$745$	48.0000	1.75858
$746$	12.0000	0.439351
$747$	−14.0000	−0.512233
$748$	7.00000	0.255945
$749$	18.0000	0.657706
$750$	3.00000	0.109545
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	0	0
$753$	−3.00000	−0.109326
$754$	−3.00000	−0.109254
$755$	21.0000	0.764268
$756$	1.00000	0.0363696
$757$	26.0000	0.944986	0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000	0.944986	0.472493	+	0.881334i	$0.343354\pi$
$758$	20.0000	0.726433
$759$	7.00000	0.254084
$760$	3.00000	0.108821
$761$	−32.0000	−1.16000	−0.580000	−	0.814617i	$-0.696947\pi$
$761$	−32.0000	−1.16000	−0.580000	+	0.814617i	$0.696947\pi$
$762$	−10.0000	−0.362262
$763$	7.00000	0.253417
$764$	−15.0000	−0.542681
$765$	21.0000	0.759257
$766$	−1.00000	−0.0361315
$767$	−4.00000	−0.144432
$768$	−1.00000	−0.0360844
$769$	−49.0000	−1.76699	−0.883493	−	0.468445i	$-0.844814\pi$
$769$	−49.0000	−1.76699	−0.883493	+	0.468445i	$0.844814\pi$
$770$	−3.00000	−0.108112
$771$	−10.0000	−0.360141
$772$	−8.00000	−0.287926
$773$	−7.00000	−0.251773	−0.125886	−	0.992045i	$-0.540177\pi$
$773$	−7.00000	−0.251773	−0.125886	+	0.992045i	$0.540177\pi$
$774$	11.0000	0.395387
$775$	0	0
$776$	6.00000	0.215387
$777$	−5.00000	−0.179374
$778$	34.0000	1.21896
$779$	4.00000	0.143315
$780$	3.00000	0.107417
$781$	−12.0000	−0.429394
$782$	−49.0000	−1.75224
$783$	−3.00000	−0.107211
$784$	1.00000	0.0357143
$785$	15.0000	0.535373
$786$	−15.0000	−0.535032
$787$	−31.0000	−1.10503	−0.552515	−	0.833503i	$-0.686332\pi$
$787$	−31.0000	−1.10503	−0.552515	+	0.833503i	$0.686332\pi$
$788$	−20.0000	−0.712470
$789$	4.00000	0.142404
$790$	−30.0000	−1.06735
$791$	14.0000	0.497783
$792$	1.00000	0.0355335
$793$	−1.00000	−0.0355110
$794$	−14.0000	−0.496841
$795$	42.0000	1.48959
$796$	3.00000	0.106332
$797$	36.0000	1.27519	0.637593	−	0.770374i	$-0.279930\pi$
$797$	36.0000	1.27519	0.637593	+	0.770374i	$0.279930\pi$
$798$	1.00000	0.0353996
$799$	0	0
$800$	4.00000	0.141421
$801$	−6.00000	−0.212000
$802$	−2.00000	−0.0706225
$803$	5.00000	0.176446
$804$	6.00000	0.211604
$805$	21.0000	0.740153
$806$	0	0
$807$	−2.00000	−0.0704033
$808$	−12.0000	−0.422159
$809$	−50.0000	−1.75791	−0.878953	−	0.476908i	$-0.841757\pi$
$809$	−50.0000	−1.75791	−0.878953	+	0.476908i	$0.841757\pi$
$810$	3.00000	0.105409
$811$	21.0000	0.737410	0.368705	−	0.929547i	$-0.379801\pi$
$811$	21.0000	0.737410	0.368705	+	0.929547i	$0.379801\pi$
$812$	−3.00000	−0.105279
$813$	30.0000	1.05215
$814$	−5.00000	−0.175250
$815$	30.0000	1.05085
$816$	−7.00000	−0.245049
$817$	11.0000	0.384841
$818$	23.0000	0.804176
$819$	1.00000	0.0349428
$820$	12.0000	0.419058
$821$	−32.0000	−1.11681	−0.558404	−	0.829569i	$-0.688586\pi$
$821$	−32.0000	−1.11681	−0.558404	+	0.829569i	$0.688586\pi$
$822$	−15.0000	−0.523185
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	−7.00000	−0.243857
$825$	−4.00000	−0.139262
$826$	−4.00000	−0.139178
$827$	9.00000	0.312961	0.156480	−	0.987681i	$-0.449985\pi$
$827$	9.00000	0.312961	0.156480	+	0.987681i	$0.449985\pi$
$828$	−7.00000	−0.243267
$829$	49.0000	1.70184	0.850920	−	0.525295i	$-0.176045\pi$
$829$	49.0000	1.70184	0.850920	+	0.525295i	$0.176045\pi$
$830$	−42.0000	−1.45784
$831$	−10.0000	−0.346896
$832$	−1.00000	−0.0346688
$833$	7.00000	0.242536
$834$	−4.00000	−0.138509
$835$	21.0000	0.726735
$836$	1.00000	0.0345857
$837$	0	0
$838$	29.0000	1.00179
$839$	−4.00000	−0.138095	−0.0690477	−	0.997613i	$-0.521996\pi$
$839$	−4.00000	−0.138095	−0.0690477	+	0.997613i	$0.521996\pi$
$840$	3.00000	0.103510
$841$	−20.0000	−0.689655
$842$	26.0000	0.896019
$843$	−6.00000	−0.206651
$844$	25.0000	0.860535
$845$	3.00000	0.103203
$846$	0	0
$847$	10.0000	0.343604
$848$	−14.0000	−0.480762
$849$	16.0000	0.549119
$850$	28.0000	0.960392
$851$	35.0000	1.19978
$852$	12.0000	0.411113
$853$	44.0000	1.50653	0.753266	−	0.657716i	$-0.228477\pi$
$853$	44.0000	1.50653	0.753266	+	0.657716i	$0.228477\pi$
$854$	−1.00000	−0.0342193
$855$	3.00000	0.102598
$856$	−18.0000	−0.615227
$857$	22.0000	0.751506	0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000	0.751506	0.375753	+	0.926720i	$0.377384\pi$
$858$	1.00000	0.0341394
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	33.0000	1.12529
$861$	4.00000	0.136320
$862$	−10.0000	−0.340601
$863$	−42.0000	−1.42970	−0.714848	−	0.699280i	$-0.753504\pi$
$863$	−42.0000	−1.42970	−0.714848	+	0.699280i	$0.753504\pi$
$864$	−1.00000	−0.0340207
$865$	6.00000	0.204006
$866$	−8.00000	−0.271851
$867$	−32.0000	−1.08678
$868$	0	0
$869$	−10.0000	−0.339227
$870$	−9.00000	−0.305129
$871$	6.00000	0.203302
$872$	−7.00000	−0.237050
$873$	6.00000	0.203069
$874$	−7.00000	−0.236779
$875$	3.00000	0.101419
$876$	−5.00000	−0.168934
$877$	18.0000	0.607817	0.303908	−	0.952701i	$-0.401708\pi$
$877$	18.0000	0.607817	0.303908	+	0.952701i	$0.401708\pi$
$878$	−19.0000	−0.641219
$879$	−30.0000	−1.01187
$880$	3.00000	0.101130
$881$	−17.0000	−0.572745	−0.286372	−	0.958118i	$-0.592449\pi$
$881$	−17.0000	−0.572745	−0.286372	+	0.958118i	$0.592449\pi$
$882$	1.00000	0.0336718
$883$	35.0000	1.17784	0.588922	−	0.808190i	$-0.299553\pi$
$883$	35.0000	1.17784	0.588922	+	0.808190i	$0.299553\pi$
$884$	−7.00000	−0.235435
$885$	−12.0000	−0.403376
$886$	26.0000	0.873487
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	5.00000	0.167789
$889$	−10.0000	−0.335389
$890$	−18.0000	−0.603361
$891$	1.00000	0.0335013
$892$	14.0000	0.468755
$893$	0	0
$894$	−16.0000	−0.535120
$895$	24.0000	0.802232
$896$	−1.00000	−0.0334077
$897$	−7.00000	−0.233723
$898$	19.0000	0.634038
$899$	0	0
$900$	4.00000	0.133333
$901$	−98.0000	−3.26485
$902$	4.00000	0.133185
$903$	11.0000	0.366057
$904$	−14.0000	−0.465633
$905$	66.0000	2.19391
$906$	−7.00000	−0.232559
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	−18.0000	−0.597351
$909$	−12.0000	−0.398015
$910$	3.00000	0.0994490
$911$	15.0000	0.496972	0.248486	−	0.968635i	$-0.420067\pi$
$911$	15.0000	0.496972	0.248486	+	0.968635i	$0.420067\pi$
$912$	−1.00000	−0.0331133
$913$	−14.0000	−0.463332
$914$	−40.0000	−1.32308
$915$	−3.00000	−0.0991769
$916$	18.0000	0.594737
$917$	−15.0000	−0.495344
$918$	−7.00000	−0.231034
$919$	−50.0000	−1.64935	−0.824674	−	0.565608i	$-0.808641\pi$
$919$	−50.0000	−1.64935	−0.824674	+	0.565608i	$0.808641\pi$
$920$	−21.0000	−0.692349
$921$	32.0000	1.05444
$922$	−23.0000	−0.757465
$923$	12.0000	0.394985
$924$	1.00000	0.0328976
$925$	−20.0000	−0.657596
$926$	23.0000	0.755827
$927$	−7.00000	−0.229910
$928$	3.00000	0.0984798
$929$	48.0000	1.57483	0.787414	−	0.616424i	$-0.211419\pi$
$929$	48.0000	1.57483	0.787414	+	0.616424i	$0.211419\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	−12.0000	−0.393073
$933$	−2.00000	−0.0654771
$934$	3.00000	0.0981630
$935$	21.0000	0.686773
$936$	−1.00000	−0.0326860
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	6.00000	0.195907
$939$	0	0
$940$	0	0
$941$	34.0000	1.10837	0.554184	−	0.832394i	$-0.313030\pi$
$941$	34.0000	1.10837	0.554184	+	0.832394i	$0.313030\pi$
$942$	−5.00000	−0.162909
$943$	−28.0000	−0.911805
$944$	4.00000	0.130189
$945$	3.00000	0.0975900
$946$	11.0000	0.357641
$947$	−11.0000	−0.357452	−0.178726	−	0.983899i	$-0.557198\pi$
$947$	−11.0000	−0.357452	−0.178726	+	0.983899i	$0.557198\pi$
$948$	10.0000	0.324785
$949$	−5.00000	−0.162307
$950$	4.00000	0.129777
$951$	28.0000	0.907962
$952$	−7.00000	−0.226871
$953$	36.0000	1.16615	0.583077	−	0.812417i	$-0.301849\pi$
$953$	36.0000	1.16615	0.583077	+	0.812417i	$0.301849\pi$
$954$	−14.0000	−0.453267
$955$	−45.0000	−1.45617
$956$	20.0000	0.646846
$957$	−3.00000	−0.0969762
$958$	−11.0000	−0.355394
$959$	−15.0000	−0.484375
$960$	−3.00000	−0.0968246
$961$	−31.0000	−1.00000
$962$	5.00000	0.161206
$963$	−18.0000	−0.580042
$964$	−18.0000	−0.579741
$965$	−24.0000	−0.772587
$966$	−7.00000	−0.225221
$967$	13.0000	0.418052	0.209026	−	0.977910i	$-0.432971\pi$
$967$	13.0000	0.418052	0.209026	+	0.977910i	$0.432971\pi$
$968$	−10.0000	−0.321412
$969$	−7.00000	−0.224872
$970$	18.0000	0.577945
$971$	48.0000	1.54039	0.770197	−	0.637806i	$-0.220158\pi$
$971$	48.0000	1.54039	0.770197	+	0.637806i	$0.220158\pi$
$972$	−1.00000	−0.0320750
$973$	−4.00000	−0.128234
$974$	−32.0000	−1.02535
$975$	4.00000	0.128103
$976$	1.00000	0.0320092
$977$	−7.00000	−0.223950	−0.111975	−	0.993711i	$-0.535718\pi$
$977$	−7.00000	−0.223950	−0.111975	+	0.993711i	$0.535718\pi$
$978$	−10.0000	−0.319765
$979$	−6.00000	−0.191761
$980$	3.00000	0.0958315
$981$	−7.00000	−0.223493
$982$	2.00000	0.0638226
$983$	−19.0000	−0.606006	−0.303003	−	0.952990i	$-0.597989\pi$
$983$	−19.0000	−0.606006	−0.303003	+	0.952990i	$0.597989\pi$
$984$	−4.00000	−0.127515
$985$	−60.0000	−1.91176
$986$	21.0000	0.668776
$987$	0	0
$988$	−1.00000	−0.0318142
$989$	−77.0000	−2.44846
$990$	3.00000	0.0953463
$991$	34.0000	1.08005	0.540023	−	0.841650i	$-0.318416\pi$
$991$	34.0000	1.08005	0.540023	+	0.841650i	$0.318416\pi$
$992$	0	0
$993$	14.0000	0.444277
$994$	12.0000	0.380617
$995$	9.00000	0.285319
$996$	14.0000	0.443607
$997$	54.0000	1.71020	0.855099	−	0.518465i	$-0.173497\pi$
$997$	54.0000	1.71020	0.855099	+	0.518465i	$0.173497\pi$
$998$	16.0000	0.506471
$999$	5.00000	0.158193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	546.2.a.e.1.1	✓	1
3.2	odd	2		1638.2.a.a.1.1		1
4.3	odd	2		4368.2.a.z.1.1		1
7.6	odd	2		3822.2.a.bc.1.1		1
13.12	even	2		7098.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.a.e.1.1	✓	1	1.1	even	1	trivial
1638.2.a.a.1.1		1	3.2	odd	2
3822.2.a.bc.1.1		1	7.6	odd	2
4368.2.a.z.1.1		1	4.3	odd	2
7098.2.a.b.1.1		1	13.12	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:	\( N \)	\(=\)	\( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	546.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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