LMFDB - Embedded newform 546.2.a.c.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)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +3.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +5.00000 q^{17} -1.00000 q^{18} +1.00000 q^{19} +1.00000 q^{20} -1.00000 q^{21} -3.00000 q^{22} +3.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} +1.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +5.00000 q^{29} -1.00000 q^{30} +4.00000 q^{31} -1.00000 q^{32} +3.00000 q^{33} -5.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} -5.00000 q^{37} -1.00000 q^{38} -1.00000 q^{39} -1.00000 q^{40} -8.00000 q^{41} +1.00000 q^{42} -1.00000 q^{43} +3.00000 q^{44} +1.00000 q^{45} -3.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +4.00000 q^{50} +5.00000 q^{51} -1.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} +3.00000 q^{55} +1.00000 q^{56} +1.00000 q^{57} -5.00000 q^{58} +1.00000 q^{60} +13.0000 q^{61} -4.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -1.00000 q^{65} -3.00000 q^{66} -10.0000 q^{67} +5.00000 q^{68} +3.00000 q^{69} +1.00000 q^{70} +8.00000 q^{71} -1.00000 q^{72} -15.0000 q^{73} +5.00000 q^{74} -4.00000 q^{75} +1.00000 q^{76} -3.00000 q^{77} +1.00000 q^{78} +6.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +8.00000 q^{82} -2.00000 q^{83} -1.00000 q^{84} +5.00000 q^{85} +1.00000 q^{86} +5.00000 q^{87} -3.00000 q^{88} -2.00000 q^{89} -1.00000 q^{90} +1.00000 q^{91} +3.00000 q^{92} +4.00000 q^{93} -8.00000 q^{94} +1.00000 q^{95} -1.00000 q^{96} -2.00000 q^{97} -1.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	−1.00000	−0.408248
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	1.00000	0.288675
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	1.00000	0.267261
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	−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	$0.463407\pi$
$20$	1.00000	0.223607
$21$	−1.00000	−0.218218
$22$	−3.00000	−0.639602
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	−1.00000	−0.204124
$25$	−4.00000	−0.800000
$26$	1.00000	0.196116
$27$	1.00000	0.192450
$28$	−1.00000	−0.188982
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	−1.00000	−0.182574
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−1.00000	−0.176777
$33$	3.00000	0.522233
$34$	−5.00000	−0.857493
$35$	−1.00000	−0.169031
$36$	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
$40$	−1.00000	−0.158114
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	1.00000	0.154303
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	3.00000	0.452267
$45$	1.00000	0.149071
$46$	−3.00000	−0.442326
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	4.00000	0.565685
$51$	5.00000	0.700140
$52$	−1.00000	−0.138675
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	−1.00000	−0.136083
$55$	3.00000	0.404520
$56$	1.00000	0.133631
$57$	1.00000	0.132453
$58$	−5.00000	−0.656532
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	1.00000	0.129099
$61$	13.0000	1.66448	0.832240	−	0.554416i	$-0.187058\pi$
$61$	13.0000	1.66448	0.832240	+	0.554416i	$0.187058\pi$
$62$	−4.00000	−0.508001
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	−3.00000	−0.369274
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	5.00000	0.606339
$69$	3.00000	0.361158
$70$	1.00000	0.119523
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	−1.00000	−0.117851
$73$	−15.0000	−1.75562	−0.877809	−	0.479012i	$-0.840995\pi$
$73$	−15.0000	−1.75562	−0.877809	+	0.479012i	$0.840995\pi$
$74$	5.00000	0.581238
$75$	−4.00000	−0.461880
$76$	1.00000	0.114708
$77$	−3.00000	−0.341882
$78$	1.00000	0.113228
$79$	6.00000	0.675053	0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000	0.675053	0.337526	+	0.941316i	$0.390410\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	8.00000	0.883452
$83$	−2.00000	−0.219529	−0.109764	−	0.993958i	$-0.535010\pi$
$83$	−2.00000	−0.219529	−0.109764	+	0.993958i	$0.535010\pi$
$84$	−1.00000	−0.109109
$85$	5.00000	0.542326
$86$	1.00000	0.107833
$87$	5.00000	0.536056
$88$	−3.00000	−0.319801
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	−1.00000	−0.105409
$91$	1.00000	0.104828
$92$	3.00000	0.312772
$93$	4.00000	0.414781
$94$	−8.00000	−0.825137
$95$	1.00000	0.102598
$96$	−1.00000	−0.102062
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	−1.00000	−0.101015
$99$	3.00000	0.301511
$100$	−4.00000	−0.400000
$101$	−16.0000	−1.59206	−0.796030	−	0.605257i	$-0.793070\pi$
$101$	−16.0000	−1.59206	−0.796030	+	0.605257i	$0.793070\pi$
$102$	−5.00000	−0.495074
$103$	1.00000	0.0985329	0.0492665	−	0.998786i	$-0.484312\pi$
$103$	1.00000	0.0985329	0.0492665	+	0.998786i	$0.484312\pi$
$104$	1.00000	0.0980581
$105$	−1.00000	−0.0975900
$106$	−6.00000	−0.582772
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	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
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	−1.00000	−0.0936586
$115$	3.00000	0.279751
$116$	5.00000	0.464238
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−5.00000	−0.458349
$120$	−1.00000	−0.0912871
$121$	−2.00000	−0.181818
$122$	−13.0000	−1.17696
$123$	−8.00000	−0.721336
$124$	4.00000	0.359211
$125$	−9.00000	−0.804984
$126$	1.00000	0.0890871
$127$	−18.0000	−1.59724	−0.798621	−	0.601834i	$-0.794437\pi$
$127$	−18.0000	−1.59724	−0.798621	+	0.601834i	$0.794437\pi$
$128$	−1.00000	−0.0883883
$129$	−1.00000	−0.0880451
$130$	1.00000	0.0877058
$131$	17.0000	1.48530	0.742648	−	0.669681i	$-0.233569\pi$
$131$	17.0000	1.48530	0.742648	+	0.669681i	$0.233569\pi$
$132$	3.00000	0.261116
$133$	−1.00000	−0.0867110
$134$	10.0000	0.863868
$135$	1.00000	0.0860663
$136$	−5.00000	−0.428746
$137$	−15.0000	−1.28154	−0.640768	−	0.767734i	$-0.721384\pi$
$137$	−15.0000	−1.28154	−0.640768	+	0.767734i	$0.721384\pi$
$138$	−3.00000	−0.255377
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	−1.00000	−0.0845154
$141$	8.00000	0.673722
$142$	−8.00000	−0.671345
$143$	−3.00000	−0.250873
$144$	1.00000	0.0833333
$145$	5.00000	0.415227
$146$	15.0000	1.24141
$147$	1.00000	0.0824786
$148$	−5.00000	−0.410997
$149$	−16.0000	−1.31077	−0.655386	−	0.755295i	$-0.727494\pi$
$149$	−16.0000	−1.31077	−0.655386	+	0.755295i	$0.727494\pi$
$150$	4.00000	0.326599
$151$	−5.00000	−0.406894	−0.203447	−	0.979086i	$-0.565214\pi$
$151$	−5.00000	−0.406894	−0.203447	+	0.979086i	$0.565214\pi$
$152$	−1.00000	−0.0811107
$153$	5.00000	0.404226
$154$	3.00000	0.241747
$155$	4.00000	0.321288
$156$	−1.00000	−0.0800641
$157$	9.00000	0.718278	0.359139	−	0.933284i	$-0.383070\pi$
$157$	9.00000	0.718278	0.359139	+	0.933284i	$0.383070\pi$
$158$	−6.00000	−0.477334
$159$	6.00000	0.475831
$160$	−1.00000	−0.0790569
$161$	−3.00000	−0.236433
$162$	−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	$0.371911\pi$
$164$	−8.00000	−0.624695
$165$	3.00000	0.233550
$166$	2.00000	0.155230
$167$	−15.0000	−1.16073	−0.580367	−	0.814355i	$-0.697091\pi$
$167$	−15.0000	−1.16073	−0.580367	+	0.814355i	$0.697091\pi$
$168$	1.00000	0.0771517
$169$	1.00000	0.0769231
$170$	−5.00000	−0.383482
$171$	1.00000	0.0764719
$172$	−1.00000	−0.0762493
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	−5.00000	−0.379049
$175$	4.00000	0.302372
$176$	3.00000	0.226134
$177$	0	0
$178$	2.00000	0.149906
$179$	−16.0000	−1.19590	−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000	−1.19590	−0.597948	+	0.801535i	$0.704017\pi$
$180$	1.00000	0.0745356
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	−1.00000	−0.0741249
$183$	13.0000	0.960988
$184$	−3.00000	−0.221163
$185$	−5.00000	−0.367607
$186$	−4.00000	−0.293294
$187$	15.0000	1.09691
$188$	8.00000	0.583460
$189$	−1.00000	−0.0727393
$190$	−1.00000	−0.0725476
$191$	11.0000	0.795932	0.397966	−	0.917400i	$-0.369716\pi$
$191$	11.0000	0.795932	0.397966	+	0.917400i	$0.369716\pi$
$192$	1.00000	0.0721688
$193$	−20.0000	−1.43963	−0.719816	−	0.694165i	$-0.755774\pi$
$193$	−20.0000	−1.43963	−0.719816	+	0.694165i	$0.755774\pi$
$194$	2.00000	0.143592
$195$	−1.00000	−0.0716115
$196$	1.00000	0.0714286
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	−3.00000	−0.213201
$199$	−5.00000	−0.354441	−0.177220	−	0.984171i	$-0.556711\pi$
$199$	−5.00000	−0.354441	−0.177220	+	0.984171i	$0.556711\pi$
$200$	4.00000	0.282843
$201$	−10.0000	−0.705346
$202$	16.0000	1.12576
$203$	−5.00000	−0.350931
$204$	5.00000	0.350070
$205$	−8.00000	−0.558744
$206$	−1.00000	−0.0696733
$207$	3.00000	0.208514
$208$	−1.00000	−0.0693375
$209$	3.00000	0.207514
$210$	1.00000	0.0690066
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	6.00000	0.412082
$213$	8.00000	0.548151
$214$	6.00000	0.410152
$215$	−1.00000	−0.0681994
$216$	−1.00000	−0.0680414
$217$	−4.00000	−0.271538
$218$	7.00000	0.474100
$219$	−15.0000	−1.01361
$220$	3.00000	0.202260
$221$	−5.00000	−0.336336
$222$	5.00000	0.335578
$223$	26.0000	1.74109	0.870544	−	0.492090i	$-0.163767\pi$
$223$	26.0000	1.74109	0.870544	+	0.492090i	$0.163767\pi$
$224$	1.00000	0.0668153
$225$	−4.00000	−0.266667
$226$	6.00000	0.399114
$227$	2.00000	0.132745	0.0663723	−	0.997795i	$-0.478857\pi$
$227$	2.00000	0.132745	0.0663723	+	0.997795i	$0.478857\pi$
$228$	1.00000	0.0662266
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	−3.00000	−0.197814
$231$	−3.00000	−0.197386
$232$	−5.00000	−0.328266
$233$	−4.00000	−0.262049	−0.131024	−	0.991379i	$-0.541827\pi$
$233$	−4.00000	−0.262049	−0.131024	+	0.991379i	$0.541827\pi$
$234$	1.00000	0.0653720
$235$	8.00000	0.521862
$236$	0	0
$237$	6.00000	0.389742
$238$	5.00000	0.324102
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	1.00000	0.0645497
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	2.00000	0.128565
$243$	1.00000	0.0641500
$244$	13.0000	0.832240
$245$	1.00000	0.0638877
$246$	8.00000	0.510061
$247$	−1.00000	−0.0636285
$248$	−4.00000	−0.254000
$249$	−2.00000	−0.126745
$250$	9.00000	0.569210
$251$	21.0000	1.32551	0.662754	−	0.748837i	$-0.269387\pi$
$251$	21.0000	1.32551	0.662754	+	0.748837i	$0.269387\pi$
$252$	−1.00000	−0.0629941
$253$	9.00000	0.565825
$254$	18.0000	1.12942
$255$	5.00000	0.313112
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	1.00000	0.0622573
$259$	5.00000	0.310685
$260$	−1.00000	−0.0620174
$261$	5.00000	0.309492
$262$	−17.0000	−1.05026
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	−3.00000	−0.184637
$265$	6.00000	0.368577
$266$	1.00000	0.0613139
$267$	−2.00000	−0.122398
$268$	−10.0000	−0.610847
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	−1.00000	−0.0608581
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	5.00000	0.303170
$273$	1.00000	0.0605228
$274$	15.0000	0.906183
$275$	−12.0000	−0.723627
$276$	3.00000	0.180579
$277$	−18.0000	−1.08152	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	−18.0000	−1.08152	−0.540758	+	0.841178i	$0.681862\pi$
$278$	16.0000	0.959616
$279$	4.00000	0.239474
$280$	1.00000	0.0597614
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	−8.00000	−0.476393
$283$	−8.00000	−0.475551	−0.237775	−	0.971320i	$-0.576418\pi$
$283$	−8.00000	−0.475551	−0.237775	+	0.971320i	$0.576418\pi$
$284$	8.00000	0.474713
$285$	1.00000	0.0592349
$286$	3.00000	0.177394
$287$	8.00000	0.472225
$288$	−1.00000	−0.0589256
$289$	8.00000	0.470588
$290$	−5.00000	−0.293610
$291$	−2.00000	−0.117242
$292$	−15.0000	−0.877809
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	−1.00000	−0.0583212
$295$	0	0
$296$	5.00000	0.290619
$297$	3.00000	0.174078
$298$	16.0000	0.926855
$299$	−3.00000	−0.173494
$300$	−4.00000	−0.230940
$301$	1.00000	0.0576390
$302$	5.00000	0.287718
$303$	−16.0000	−0.919176
$304$	1.00000	0.0573539
$305$	13.0000	0.744378
$306$	−5.00000	−0.285831
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	−3.00000	−0.170941
$309$	1.00000	0.0568880
$310$	−4.00000	−0.227185
$311$	26.0000	1.47432	0.737162	−	0.675716i	$-0.236165\pi$
$311$	26.0000	1.47432	0.737162	+	0.675716i	$0.236165\pi$
$312$	1.00000	0.0566139
$313$	28.0000	1.58265	0.791327	−	0.611393i	$-0.209391\pi$
$313$	28.0000	1.58265	0.791327	+	0.611393i	$0.209391\pi$
$314$	−9.00000	−0.507899
$315$	−1.00000	−0.0563436
$316$	6.00000	0.337526
$317$	−24.0000	−1.34797	−0.673987	−	0.738743i	$-0.735420\pi$
$317$	−24.0000	−1.34797	−0.673987	+	0.738743i	$0.735420\pi$
$318$	−6.00000	−0.336463
$319$	15.0000	0.839839
$320$	1.00000	0.0559017
$321$	−6.00000	−0.334887
$322$	3.00000	0.167183
$323$	5.00000	0.278207
$324$	1.00000	0.0555556
$325$	4.00000	0.221880
$326$	−10.0000	−0.553849
$327$	−7.00000	−0.387101
$328$	8.00000	0.441726
$329$	−8.00000	−0.441054
$330$	−3.00000	−0.165145
$331$	34.0000	1.86881	0.934405	−	0.356214i	$-0.115932\pi$
$331$	34.0000	1.86881	0.934405	+	0.356214i	$0.115932\pi$
$332$	−2.00000	−0.109764
$333$	−5.00000	−0.273998
$334$	15.0000	0.820763
$335$	−10.0000	−0.546358
$336$	−1.00000	−0.0545545
$337$	−29.0000	−1.57973	−0.789865	−	0.613280i	$-0.789850\pi$
$337$	−29.0000	−1.57973	−0.789865	+	0.613280i	$0.789850\pi$
$338$	−1.00000	−0.0543928
$339$	−6.00000	−0.325875
$340$	5.00000	0.271163
$341$	12.0000	0.649836
$342$	−1.00000	−0.0540738
$343$	−1.00000	−0.0539949
$344$	1.00000	0.0539164
$345$	3.00000	0.161515
$346$	−6.00000	−0.322562
$347$	32.0000	1.71785	0.858925	−	0.512101i	$-0.171133\pi$
$347$	32.0000	1.71785	0.858925	+	0.512101i	$0.171133\pi$
$348$	5.00000	0.268028
$349$	6.00000	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$350$	−4.00000	−0.213809
$351$	−1.00000	−0.0533761
$352$	−3.00000	−0.159901
$353$	2.00000	0.106449	0.0532246	−	0.998583i	$-0.483050\pi$
$353$	2.00000	0.106449	0.0532246	+	0.998583i	$0.483050\pi$
$354$	0	0
$355$	8.00000	0.424596
$356$	−2.00000	−0.106000
$357$	−5.00000	−0.264628
$358$	16.0000	0.845626
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	−1.00000	−0.0527046
$361$	−18.0000	−0.947368
$362$	−6.00000	−0.315353
$363$	−2.00000	−0.104973
$364$	1.00000	0.0524142
$365$	−15.0000	−0.785136
$366$	−13.0000	−0.679521
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	3.00000	0.156386
$369$	−8.00000	−0.416463
$370$	5.00000	0.259938
$371$	−6.00000	−0.311504
$372$	4.00000	0.207390
$373$	−32.0000	−1.65690	−0.828449	−	0.560065i	$-0.810776\pi$
$373$	−32.0000	−1.65690	−0.828449	+	0.560065i	$0.810776\pi$
$374$	−15.0000	−0.775632
$375$	−9.00000	−0.464758
$376$	−8.00000	−0.412568
$377$	−5.00000	−0.257513
$378$	1.00000	0.0514344
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	1.00000	0.0512989
$381$	−18.0000	−0.922168
$382$	−11.0000	−0.562809
$383$	1.00000	0.0510976	0.0255488	−	0.999674i	$-0.491867\pi$
$383$	1.00000	0.0510976	0.0255488	+	0.999674i	$0.491867\pi$
$384$	−1.00000	−0.0510310
$385$	−3.00000	−0.152894
$386$	20.0000	1.01797
$387$	−1.00000	−0.0508329
$388$	−2.00000	−0.101535
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	1.00000	0.0506370
$391$	15.0000	0.758583
$392$	−1.00000	−0.0505076
$393$	17.0000	0.857537
$394$	0	0
$395$	6.00000	0.301893
$396$	3.00000	0.150756
$397$	30.0000	1.50566	0.752828	−	0.658217i	$-0.228689\pi$
$397$	30.0000	1.50566	0.752828	+	0.658217i	$0.228689\pi$
$398$	5.00000	0.250627
$399$	−1.00000	−0.0500626
$400$	−4.00000	−0.200000
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	10.0000	0.498755
$403$	−4.00000	−0.199254
$404$	−16.0000	−0.796030
$405$	1.00000	0.0496904
$406$	5.00000	0.248146
$407$	−15.0000	−0.743522
$408$	−5.00000	−0.247537
$409$	3.00000	0.148340	0.0741702	−	0.997246i	$-0.476369\pi$
$409$	3.00000	0.148340	0.0741702	+	0.997246i	$0.476369\pi$
$410$	8.00000	0.395092
$411$	−15.0000	−0.739895
$412$	1.00000	0.0492665
$413$	0	0
$414$	−3.00000	−0.147442
$415$	−2.00000	−0.0981761
$416$	1.00000	0.0490290
$417$	−16.0000	−0.783523
$418$	−3.00000	−0.146735
$419$	−5.00000	−0.244266	−0.122133	−	0.992514i	$-0.538973\pi$
$419$	−5.00000	−0.244266	−0.122133	+	0.992514i	$0.538973\pi$
$420$	−1.00000	−0.0487950
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	−5.00000	−0.243396
$423$	8.00000	0.388973
$424$	−6.00000	−0.291386
$425$	−20.0000	−0.970143
$426$	−8.00000	−0.387601
$427$	−13.0000	−0.629114
$428$	−6.00000	−0.290021
$429$	−3.00000	−0.144841
$430$	1.00000	0.0482243
$431$	10.0000	0.481683	0.240842	−	0.970564i	$-0.422577\pi$
$431$	10.0000	0.481683	0.240842	+	0.970564i	$0.422577\pi$
$432$	1.00000	0.0481125
$433$	20.0000	0.961139	0.480569	−	0.876957i	$-0.340430\pi$
$433$	20.0000	0.961139	0.480569	+	0.876957i	$0.340430\pi$
$434$	4.00000	0.192006
$435$	5.00000	0.239732
$436$	−7.00000	−0.335239
$437$	3.00000	0.143509
$438$	15.0000	0.716728
$439$	−11.0000	−0.525001	−0.262501	−	0.964932i	$-0.584547\pi$
$439$	−11.0000	−0.525001	−0.262501	+	0.964932i	$0.584547\pi$
$440$	−3.00000	−0.143019
$441$	1.00000	0.0476190
$442$	5.00000	0.237826
$443$	−18.0000	−0.855206	−0.427603	−	0.903967i	$-0.640642\pi$
$443$	−18.0000	−0.855206	−0.427603	+	0.903967i	$0.640642\pi$
$444$	−5.00000	−0.237289
$445$	−2.00000	−0.0948091
$446$	−26.0000	−1.23114
$447$	−16.0000	−0.756774
$448$	−1.00000	−0.0472456
$449$	13.0000	0.613508	0.306754	−	0.951789i	$-0.400757\pi$
$449$	13.0000	0.613508	0.306754	+	0.951789i	$0.400757\pi$
$450$	4.00000	0.188562
$451$	−24.0000	−1.13012
$452$	−6.00000	−0.282216
$453$	−5.00000	−0.234920
$454$	−2.00000	−0.0938647
$455$	1.00000	0.0468807
$456$	−1.00000	−0.0468293
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	6.00000	0.280362
$459$	5.00000	0.233380
$460$	3.00000	0.139876
$461$	−13.0000	−0.605470	−0.302735	−	0.953075i	$-0.597900\pi$
$461$	−13.0000	−0.605470	−0.302735	+	0.953075i	$0.597900\pi$
$462$	3.00000	0.139573
$463$	−13.0000	−0.604161	−0.302081	−	0.953282i	$-0.597681\pi$
$463$	−13.0000	−0.604161	−0.302081	+	0.953282i	$0.597681\pi$
$464$	5.00000	0.232119
$465$	4.00000	0.185496
$466$	4.00000	0.185296
$467$	21.0000	0.971764	0.485882	−	0.874024i	$-0.338498\pi$
$467$	21.0000	0.971764	0.485882	+	0.874024i	$0.338498\pi$
$468$	−1.00000	−0.0462250
$469$	10.0000	0.461757
$470$	−8.00000	−0.369012
$471$	9.00000	0.414698
$472$	0	0
$473$	−3.00000	−0.137940
$474$	−6.00000	−0.275589
$475$	−4.00000	−0.183533
$476$	−5.00000	−0.229175
$477$	6.00000	0.274721
$478$	12.0000	0.548867
$479$	27.0000	1.23366	0.616831	−	0.787096i	$-0.288416\pi$
$479$	27.0000	1.23366	0.616831	+	0.787096i	$0.288416\pi$
$480$	−1.00000	−0.0456435
$481$	5.00000	0.227980
$482$	−22.0000	−1.00207
$483$	−3.00000	−0.136505
$484$	−2.00000	−0.0909091
$485$	−2.00000	−0.0908153
$486$	−1.00000	−0.0453609
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	−13.0000	−0.588482
$489$	10.0000	0.452216
$490$	−1.00000	−0.0451754
$491$	−38.0000	−1.71492	−0.857458	−	0.514554i	$-0.827958\pi$
$491$	−38.0000	−1.71492	−0.857458	+	0.514554i	$0.827958\pi$
$492$	−8.00000	−0.360668
$493$	25.0000	1.12594
$494$	1.00000	0.0449921
$495$	3.00000	0.134840
$496$	4.00000	0.179605
$497$	−8.00000	−0.358849
$498$	2.00000	0.0896221
$499$	8.00000	0.358129	0.179065	−	0.983837i	$-0.442693\pi$
$499$	8.00000	0.358129	0.179065	+	0.983837i	$0.442693\pi$
$500$	−9.00000	−0.402492
$501$	−15.0000	−0.670151
$502$	−21.0000	−0.937276
$503$	−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
$505$	−16.0000	−0.711991
$506$	−9.00000	−0.400099
$507$	1.00000	0.0444116
$508$	−18.0000	−0.798621
$509$	11.0000	0.487566	0.243783	−	0.969830i	$-0.421611\pi$
$509$	11.0000	0.487566	0.243783	+	0.969830i	$0.421611\pi$
$510$	−5.00000	−0.221404
$511$	15.0000	0.663561
$512$	−1.00000	−0.0441942
$513$	1.00000	0.0441511
$514$	18.0000	0.793946
$515$	1.00000	0.0440653
$516$	−1.00000	−0.0440225
$517$	24.0000	1.05552
$518$	−5.00000	−0.219687
$519$	6.00000	0.263371
$520$	1.00000	0.0438529
$521$	39.0000	1.70862	0.854311	−	0.519763i	$-0.173980\pi$
$521$	39.0000	1.70862	0.854311	+	0.519763i	$0.173980\pi$
$522$	−5.00000	−0.218844
$523$	−24.0000	−1.04945	−0.524723	−	0.851273i	$-0.675831\pi$
$523$	−24.0000	−1.04945	−0.524723	+	0.851273i	$0.675831\pi$
$524$	17.0000	0.742648
$525$	4.00000	0.174574
$526$	−12.0000	−0.523225
$527$	20.0000	0.871214
$528$	3.00000	0.130558
$529$	−14.0000	−0.608696
$530$	−6.00000	−0.260623
$531$	0	0
$532$	−1.00000	−0.0433555
$533$	8.00000	0.346518
$534$	2.00000	0.0865485
$535$	−6.00000	−0.259403
$536$	10.0000	0.431934
$537$	−16.0000	−0.690451
$538$	−6.00000	−0.258678
$539$	3.00000	0.129219
$540$	1.00000	0.0430331
$541$	41.0000	1.76273	0.881364	−	0.472438i	$-0.156626\pi$
$541$	41.0000	1.76273	0.881364	+	0.472438i	$0.156626\pi$
$542$	−2.00000	−0.0859074
$543$	6.00000	0.257485
$544$	−5.00000	−0.214373
$545$	−7.00000	−0.299847
$546$	−1.00000	−0.0427960
$547$	−44.0000	−1.88130	−0.940652	−	0.339372i	$-0.889785\pi$
$547$	−44.0000	−1.88130	−0.940652	+	0.339372i	$0.889785\pi$
$548$	−15.0000	−0.640768
$549$	13.0000	0.554826
$550$	12.0000	0.511682
$551$	5.00000	0.213007
$552$	−3.00000	−0.127688
$553$	−6.00000	−0.255146
$554$	18.0000	0.764747
$555$	−5.00000	−0.212238
$556$	−16.0000	−0.678551
$557$	−24.0000	−1.01691	−0.508456	−	0.861088i	$-0.669784\pi$
$557$	−24.0000	−1.01691	−0.508456	+	0.861088i	$0.669784\pi$
$558$	−4.00000	−0.169334
$559$	1.00000	0.0422955
$560$	−1.00000	−0.0422577
$561$	15.0000	0.633300
$562$	−10.0000	−0.421825
$563$	9.00000	0.379305	0.189652	−	0.981851i	$-0.439264\pi$
$563$	9.00000	0.379305	0.189652	+	0.981851i	$0.439264\pi$
$564$	8.00000	0.336861
$565$	−6.00000	−0.252422
$566$	8.00000	0.336265
$567$	−1.00000	−0.0419961
$568$	−8.00000	−0.335673
$569$	34.0000	1.42535	0.712677	−	0.701492i	$-0.247483\pi$
$569$	34.0000	1.42535	0.712677	+	0.701492i	$0.247483\pi$
$570$	−1.00000	−0.0418854
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	−3.00000	−0.125436
$573$	11.0000	0.459532
$574$	−8.00000	−0.333914
$575$	−12.0000	−0.500435
$576$	1.00000	0.0416667
$577$	−34.0000	−1.41544	−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000	−1.41544	−0.707719	+	0.706494i	$0.750276\pi$
$578$	−8.00000	−0.332756
$579$	−20.0000	−0.831172
$580$	5.00000	0.207614
$581$	2.00000	0.0829740
$582$	2.00000	0.0829027
$583$	18.0000	0.745484
$584$	15.0000	0.620704
$585$	−1.00000	−0.0413449
$586$	6.00000	0.247858
$587$	−26.0000	−1.07313	−0.536567	−	0.843857i	$-0.680279\pi$
$587$	−26.0000	−1.07313	−0.536567	+	0.843857i	$0.680279\pi$
$588$	1.00000	0.0412393
$589$	4.00000	0.164817
$590$	0	0
$591$	0	0
$592$	−5.00000	−0.205499
$593$	38.0000	1.56047	0.780236	−	0.625485i	$-0.215099\pi$
$593$	38.0000	1.56047	0.780236	+	0.625485i	$0.215099\pi$
$594$	−3.00000	−0.123091
$595$	−5.00000	−0.204980
$596$	−16.0000	−0.655386
$597$	−5.00000	−0.204636
$598$	3.00000	0.122679
$599$	37.0000	1.51178	0.755890	−	0.654699i	$-0.227205\pi$
$599$	37.0000	1.51178	0.755890	+	0.654699i	$0.227205\pi$
$600$	4.00000	0.163299
$601$	20.0000	0.815817	0.407909	−	0.913023i	$-0.366258\pi$
$601$	20.0000	0.815817	0.407909	+	0.913023i	$0.366258\pi$
$602$	−1.00000	−0.0407570
$603$	−10.0000	−0.407231
$604$	−5.00000	−0.203447
$605$	−2.00000	−0.0813116
$606$	16.0000	0.649956
$607$	−27.0000	−1.09590	−0.547948	−	0.836512i	$-0.684591\pi$
$607$	−27.0000	−1.09590	−0.547948	+	0.836512i	$0.684591\pi$
$608$	−1.00000	−0.0405554
$609$	−5.00000	−0.202610
$610$	−13.0000	−0.526355
$611$	−8.00000	−0.323645
$612$	5.00000	0.202113
$613$	41.0000	1.65597	0.827987	−	0.560747i	$-0.189486\pi$
$613$	41.0000	1.65597	0.827987	+	0.560747i	$0.189486\pi$
$614$	0	0
$615$	−8.00000	−0.322591
$616$	3.00000	0.120873
$617$	43.0000	1.73111	0.865557	−	0.500810i	$-0.166964\pi$
$617$	43.0000	1.73111	0.865557	+	0.500810i	$0.166964\pi$
$618$	−1.00000	−0.0402259
$619$	−31.0000	−1.24600	−0.622998	−	0.782224i	$-0.714085\pi$
$619$	−31.0000	−1.24600	−0.622998	+	0.782224i	$0.714085\pi$
$620$	4.00000	0.160644
$621$	3.00000	0.120386
$622$	−26.0000	−1.04251
$623$	2.00000	0.0801283
$624$	−1.00000	−0.0400320
$625$	11.0000	0.440000
$626$	−28.0000	−1.11911
$627$	3.00000	0.119808
$628$	9.00000	0.359139
$629$	−25.0000	−0.996815
$630$	1.00000	0.0398410
$631$	−9.00000	−0.358284	−0.179142	−	0.983823i	$-0.557332\pi$
$631$	−9.00000	−0.358284	−0.179142	+	0.983823i	$0.557332\pi$
$632$	−6.00000	−0.238667
$633$	5.00000	0.198732
$634$	24.0000	0.953162
$635$	−18.0000	−0.714308
$636$	6.00000	0.237915
$637$	−1.00000	−0.0396214
$638$	−15.0000	−0.593856
$639$	8.00000	0.316475
$640$	−1.00000	−0.0395285
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	6.00000	0.236801
$643$	−1.00000	−0.0394362	−0.0197181	−	0.999806i	$-0.506277\pi$
$643$	−1.00000	−0.0394362	−0.0197181	+	0.999806i	$0.506277\pi$
$644$	−3.00000	−0.118217
$645$	−1.00000	−0.0393750
$646$	−5.00000	−0.196722
$647$	28.0000	1.10079	0.550397	−	0.834903i	$-0.314476\pi$
$647$	28.0000	1.10079	0.550397	+	0.834903i	$0.314476\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	−4.00000	−0.156893
$651$	−4.00000	−0.156772
$652$	10.0000	0.391630
$653$	3.00000	0.117399	0.0586995	−	0.998276i	$-0.481305\pi$
$653$	3.00000	0.117399	0.0586995	+	0.998276i	$0.481305\pi$
$654$	7.00000	0.273722
$655$	17.0000	0.664245
$656$	−8.00000	−0.312348
$657$	−15.0000	−0.585206
$658$	8.00000	0.311872
$659$	6.00000	0.233727	0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000	0.233727	0.116863	+	0.993148i	$0.462716\pi$
$660$	3.00000	0.116775
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	−34.0000	−1.32145
$663$	−5.00000	−0.194184
$664$	2.00000	0.0776151
$665$	−1.00000	−0.0387783
$666$	5.00000	0.193746
$667$	15.0000	0.580802
$668$	−15.0000	−0.580367
$669$	26.0000	1.00522
$670$	10.0000	0.386334
$671$	39.0000	1.50558
$672$	1.00000	0.0385758
$673$	−1.00000	−0.0385472	−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000	−0.0385472	−0.0192736	+	0.999814i	$0.506135\pi$
$674$	29.0000	1.11704
$675$	−4.00000	−0.153960
$676$	1.00000	0.0384615
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	6.00000	0.230429
$679$	2.00000	0.0767530
$680$	−5.00000	−0.191741
$681$	2.00000	0.0766402
$682$	−12.0000	−0.459504
$683$	33.0000	1.26271	0.631355	−	0.775494i	$-0.282499\pi$
$683$	33.0000	1.26271	0.631355	+	0.775494i	$0.282499\pi$
$684$	1.00000	0.0382360
$685$	−15.0000	−0.573121
$686$	1.00000	0.0381802
$687$	−6.00000	−0.228914
$688$	−1.00000	−0.0381246
$689$	−6.00000	−0.228582
$690$	−3.00000	−0.114208
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	6.00000	0.228086
$693$	−3.00000	−0.113961
$694$	−32.0000	−1.21470
$695$	−16.0000	−0.606915
$696$	−5.00000	−0.189525
$697$	−40.0000	−1.51511
$698$	−6.00000	−0.227103
$699$	−4.00000	−0.151294
$700$	4.00000	0.151186
$701$	−14.0000	−0.528773	−0.264386	−	0.964417i	$-0.585169\pi$
$701$	−14.0000	−0.528773	−0.264386	+	0.964417i	$0.585169\pi$
$702$	1.00000	0.0377426
$703$	−5.00000	−0.188579
$704$	3.00000	0.113067
$705$	8.00000	0.301297
$706$	−2.00000	−0.0752710
$707$	16.0000	0.601742
$708$	0	0
$709$	50.0000	1.87779	0.938895	−	0.344204i	$-0.111851\pi$
$709$	50.0000	1.87779	0.938895	+	0.344204i	$0.111851\pi$
$710$	−8.00000	−0.300235
$711$	6.00000	0.225018
$712$	2.00000	0.0749532
$713$	12.0000	0.449404
$714$	5.00000	0.187120
$715$	−3.00000	−0.112194
$716$	−16.0000	−0.597948
$717$	−12.0000	−0.448148
$718$	−24.0000	−0.895672
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	1.00000	0.0372678
$721$	−1.00000	−0.0372419
$722$	18.0000	0.669891
$723$	22.0000	0.818189
$724$	6.00000	0.222988
$725$	−20.0000	−0.742781
$726$	2.00000	0.0742270
$727$	−27.0000	−1.00137	−0.500687	−	0.865628i	$-0.666919\pi$
$727$	−27.0000	−1.00137	−0.500687	+	0.865628i	$0.666919\pi$
$728$	−1.00000	−0.0370625
$729$	1.00000	0.0370370
$730$	15.0000	0.555175
$731$	−5.00000	−0.184932
$732$	13.0000	0.480494
$733$	20.0000	0.738717	0.369358	−	0.929287i	$-0.379577\pi$
$733$	20.0000	0.738717	0.369358	+	0.929287i	$0.379577\pi$
$734$	−8.00000	−0.295285
$735$	1.00000	0.0368856
$736$	−3.00000	−0.110581
$737$	−30.0000	−1.10506
$738$	8.00000	0.294484
$739$	−26.0000	−0.956425	−0.478213	−	0.878244i	$-0.658715\pi$
$739$	−26.0000	−0.956425	−0.478213	+	0.878244i	$0.658715\pi$
$740$	−5.00000	−0.183804
$741$	−1.00000	−0.0367359
$742$	6.00000	0.220267
$743$	−42.0000	−1.54083	−0.770415	−	0.637542i	$-0.779951\pi$
$743$	−42.0000	−1.54083	−0.770415	+	0.637542i	$0.779951\pi$
$744$	−4.00000	−0.146647
$745$	−16.0000	−0.586195
$746$	32.0000	1.17160
$747$	−2.00000	−0.0731762
$748$	15.0000	0.548454
$749$	6.00000	0.219235
$750$	9.00000	0.328634
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	8.00000	0.291730
$753$	21.0000	0.765283
$754$	5.00000	0.182089
$755$	−5.00000	−0.181969
$756$	−1.00000	−0.0363696
$757$	−34.0000	−1.23575	−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000	−1.23575	−0.617876	+	0.786276i	$0.712006\pi$
$758$	8.00000	0.290573
$759$	9.00000	0.326679
$760$	−1.00000	−0.0362738
$761$	−24.0000	−0.869999	−0.435000	−	0.900431i	$-0.643252\pi$
$761$	−24.0000	−0.869999	−0.435000	+	0.900431i	$0.643252\pi$
$762$	18.0000	0.652071
$763$	7.00000	0.253417
$764$	11.0000	0.397966
$765$	5.00000	0.180775
$766$	−1.00000	−0.0361315
$767$	0	0
$768$	1.00000	0.0360844
$769$	43.0000	1.55062	0.775310	−	0.631581i	$-0.217594\pi$
$769$	43.0000	1.55062	0.775310	+	0.631581i	$0.217594\pi$
$770$	3.00000	0.108112
$771$	−18.0000	−0.648254
$772$	−20.0000	−0.719816
$773$	11.0000	0.395643	0.197821	−	0.980238i	$-0.436613\pi$
$773$	11.0000	0.395643	0.197821	+	0.980238i	$0.436613\pi$
$774$	1.00000	0.0359443
$775$	−16.0000	−0.574737
$776$	2.00000	0.0717958
$777$	5.00000	0.179374
$778$	18.0000	0.645331
$779$	−8.00000	−0.286630
$780$	−1.00000	−0.0358057
$781$	24.0000	0.858788
$782$	−15.0000	−0.536399
$783$	5.00000	0.178685
$784$	1.00000	0.0357143
$785$	9.00000	0.321224
$786$	−17.0000	−0.606370
$787$	−47.0000	−1.67537	−0.837685	−	0.546154i	$-0.816091\pi$
$787$	−47.0000	−1.67537	−0.837685	+	0.546154i	$0.816091\pi$
$788$	0	0
$789$	12.0000	0.427211
$790$	−6.00000	−0.213470
$791$	6.00000	0.213335
$792$	−3.00000	−0.106600
$793$	−13.0000	−0.461644
$794$	−30.0000	−1.06466
$795$	6.00000	0.212798
$796$	−5.00000	−0.177220
$797$	−28.0000	−0.991811	−0.495905	−	0.868377i	$-0.665164\pi$
$797$	−28.0000	−0.991811	−0.495905	+	0.868377i	$0.665164\pi$
$798$	1.00000	0.0353996
$799$	40.0000	1.41510
$800$	4.00000	0.141421
$801$	−2.00000	−0.0706665
$802$	30.0000	1.05934
$803$	−45.0000	−1.58802
$804$	−10.0000	−0.352673
$805$	−3.00000	−0.105736
$806$	4.00000	0.140894
$807$	6.00000	0.211210
$808$	16.0000	0.562878
$809$	10.0000	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$810$	−1.00000	−0.0351364
$811$	−19.0000	−0.667180	−0.333590	−	0.942718i	$-0.608260\pi$
$811$	−19.0000	−0.667180	−0.333590	+	0.942718i	$0.608260\pi$
$812$	−5.00000	−0.175466
$813$	2.00000	0.0701431
$814$	15.0000	0.525750
$815$	10.0000	0.350285
$816$	5.00000	0.175035
$817$	−1.00000	−0.0349856
$818$	−3.00000	−0.104893
$819$	1.00000	0.0349428
$820$	−8.00000	−0.279372
$821$	24.0000	0.837606	0.418803	−	0.908077i	$-0.362450\pi$
$821$	24.0000	0.837606	0.418803	+	0.908077i	$0.362450\pi$
$822$	15.0000	0.523185
$823$	24.0000	0.836587	0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000	0.836587	0.418294	+	0.908312i	$0.362628\pi$
$824$	−1.00000	−0.0348367
$825$	−12.0000	−0.417786
$826$	0	0
$827$	43.0000	1.49526	0.747628	−	0.664117i	$-0.231193\pi$
$827$	43.0000	1.49526	0.747628	+	0.664117i	$0.231193\pi$
$828$	3.00000	0.104257
$829$	−19.0000	−0.659897	−0.329949	−	0.943999i	$-0.607031\pi$
$829$	−19.0000	−0.659897	−0.329949	+	0.943999i	$0.607031\pi$
$830$	2.00000	0.0694210
$831$	−18.0000	−0.624413
$832$	−1.00000	−0.0346688
$833$	5.00000	0.173240
$834$	16.0000	0.554035
$835$	−15.0000	−0.519096
$836$	3.00000	0.103757
$837$	4.00000	0.138260
$838$	5.00000	0.172722
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	1.00000	0.0345033
$841$	−4.00000	−0.137931
$842$	22.0000	0.758170
$843$	10.0000	0.344418
$844$	5.00000	0.172107
$845$	1.00000	0.0344010
$846$	−8.00000	−0.275046
$847$	2.00000	0.0687208
$848$	6.00000	0.206041
$849$	−8.00000	−0.274559
$850$	20.0000	0.685994
$851$	−15.0000	−0.514193
$852$	8.00000	0.274075
$853$	−28.0000	−0.958702	−0.479351	−	0.877623i	$-0.659128\pi$
$853$	−28.0000	−0.958702	−0.479351	+	0.877623i	$0.659128\pi$
$854$	13.0000	0.444851
$855$	1.00000	0.0341993
$856$	6.00000	0.205076
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	3.00000	0.102418
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	−1.00000	−0.0340997
$861$	8.00000	0.272639
$862$	−10.0000	−0.340601
$863$	−2.00000	−0.0680808	−0.0340404	−	0.999420i	$-0.510837\pi$
$863$	−2.00000	−0.0680808	−0.0340404	+	0.999420i	$0.510837\pi$
$864$	−1.00000	−0.0340207
$865$	6.00000	0.204006
$866$	−20.0000	−0.679628
$867$	8.00000	0.271694
$868$	−4.00000	−0.135769
$869$	18.0000	0.610608
$870$	−5.00000	−0.169516
$871$	10.0000	0.338837
$872$	7.00000	0.237050
$873$	−2.00000	−0.0676897
$874$	−3.00000	−0.101477
$875$	9.00000	0.304256
$876$	−15.0000	−0.506803
$877$	34.0000	1.14810	0.574049	−	0.818821i	$-0.305372\pi$
$877$	34.0000	1.14810	0.574049	+	0.818821i	$0.305372\pi$
$878$	11.0000	0.371232
$879$	−6.00000	−0.202375
$880$	3.00000	0.101130
$881$	37.0000	1.24656	0.623281	−	0.781998i	$-0.285799\pi$
$881$	37.0000	1.24656	0.623281	+	0.781998i	$0.285799\pi$
$882$	−1.00000	−0.0336718
$883$	−41.0000	−1.37976	−0.689880	−	0.723924i	$-0.742337\pi$
$883$	−41.0000	−1.37976	−0.689880	+	0.723924i	$0.742337\pi$
$884$	−5.00000	−0.168168
$885$	0	0
$886$	18.0000	0.604722
$887$	−20.0000	−0.671534	−0.335767	−	0.941945i	$-0.608996\pi$
$887$	−20.0000	−0.671534	−0.335767	+	0.941945i	$0.608996\pi$
$888$	5.00000	0.167789
$889$	18.0000	0.603701
$890$	2.00000	0.0670402
$891$	3.00000	0.100504
$892$	26.0000	0.870544
$893$	8.00000	0.267710
$894$	16.0000	0.535120
$895$	−16.0000	−0.534821
$896$	1.00000	0.0334077
$897$	−3.00000	−0.100167
$898$	−13.0000	−0.433816
$899$	20.0000	0.667037
$900$	−4.00000	−0.133333
$901$	30.0000	0.999445
$902$	24.0000	0.799113
$903$	1.00000	0.0332779
$904$	6.00000	0.199557
$905$	6.00000	0.199447
$906$	5.00000	0.166114
$907$	−20.0000	−0.664089	−0.332045	−	0.943264i	$-0.607738\pi$
$907$	−20.0000	−0.664089	−0.332045	+	0.943264i	$0.607738\pi$
$908$	2.00000	0.0663723
$909$	−16.0000	−0.530687
$910$	−1.00000	−0.0331497
$911$	29.0000	0.960813	0.480406	−	0.877046i	$-0.340489\pi$
$911$	29.0000	0.960813	0.480406	+	0.877046i	$0.340489\pi$
$912$	1.00000	0.0331133
$913$	−6.00000	−0.198571
$914$	−8.00000	−0.264616
$915$	13.0000	0.429767
$916$	−6.00000	−0.198246
$917$	−17.0000	−0.561389
$918$	−5.00000	−0.165025
$919$	−50.0000	−1.64935	−0.824674	−	0.565608i	$-0.808641\pi$
$919$	−50.0000	−1.64935	−0.824674	+	0.565608i	$0.808641\pi$
$920$	−3.00000	−0.0989071
$921$	0	0
$922$	13.0000	0.428132
$923$	−8.00000	−0.263323
$924$	−3.00000	−0.0986928
$925$	20.0000	0.657596
$926$	13.0000	0.427207
$927$	1.00000	0.0328443
$928$	−5.00000	−0.164133
$929$	−4.00000	−0.131236	−0.0656179	−	0.997845i	$-0.520902\pi$
$929$	−4.00000	−0.131236	−0.0656179	+	0.997845i	$0.520902\pi$
$930$	−4.00000	−0.131165
$931$	1.00000	0.0327737
$932$	−4.00000	−0.131024
$933$	26.0000	0.851202
$934$	−21.0000	−0.687141
$935$	15.0000	0.490552
$936$	1.00000	0.0326860
$937$	−14.0000	−0.457360	−0.228680	−	0.973502i	$-0.573441\pi$
$937$	−14.0000	−0.457360	−0.228680	+	0.973502i	$0.573441\pi$
$938$	−10.0000	−0.326512
$939$	28.0000	0.913745
$940$	8.00000	0.260931
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	−9.00000	−0.293236
$943$	−24.0000	−0.781548
$944$	0	0
$945$	−1.00000	−0.0325300
$946$	3.00000	0.0975384
$947$	47.0000	1.52729	0.763647	−	0.645634i	$-0.223407\pi$
$947$	47.0000	1.52729	0.763647	+	0.645634i	$0.223407\pi$
$948$	6.00000	0.194871
$949$	15.0000	0.486921
$950$	4.00000	0.129777
$951$	−24.0000	−0.778253
$952$	5.00000	0.162051
$953$	−44.0000	−1.42530	−0.712650	−	0.701520i	$-0.752505\pi$
$953$	−44.0000	−1.42530	−0.712650	+	0.701520i	$0.752505\pi$
$954$	−6.00000	−0.194257
$955$	11.0000	0.355952
$956$	−12.0000	−0.388108
$957$	15.0000	0.484881
$958$	−27.0000	−0.872330
$959$	15.0000	0.484375
$960$	1.00000	0.0322749
$961$	−15.0000	−0.483871
$962$	−5.00000	−0.161206
$963$	−6.00000	−0.193347
$964$	22.0000	0.708572
$965$	−20.0000	−0.643823
$966$	3.00000	0.0965234
$967$	−31.0000	−0.996893	−0.498446	−	0.866921i	$-0.666096\pi$
$967$	−31.0000	−0.996893	−0.498446	+	0.866921i	$0.666096\pi$
$968$	2.00000	0.0642824
$969$	5.00000	0.160623
$970$	2.00000	0.0642161
$971$	−32.0000	−1.02693	−0.513464	−	0.858111i	$-0.671638\pi$
$971$	−32.0000	−1.02693	−0.513464	+	0.858111i	$0.671638\pi$
$972$	1.00000	0.0320750
$973$	16.0000	0.512936
$974$	−8.00000	−0.256337
$975$	4.00000	0.128103
$976$	13.0000	0.416120
$977$	−25.0000	−0.799821	−0.399910	−	0.916554i	$-0.630959\pi$
$977$	−25.0000	−0.799821	−0.399910	+	0.916554i	$0.630959\pi$
$978$	−10.0000	−0.319765
$979$	−6.00000	−0.191761
$980$	1.00000	0.0319438
$981$	−7.00000	−0.223493
$982$	38.0000	1.21263
$983$	−5.00000	−0.159475	−0.0797376	−	0.996816i	$-0.525408\pi$
$983$	−5.00000	−0.159475	−0.0797376	+	0.996816i	$0.525408\pi$
$984$	8.00000	0.255031
$985$	0	0
$986$	−25.0000	−0.796162
$987$	−8.00000	−0.254643
$988$	−1.00000	−0.0318142
$989$	−3.00000	−0.0953945
$990$	−3.00000	−0.0953463
$991$	−30.0000	−0.952981	−0.476491	−	0.879180i	$-0.658091\pi$
$991$	−30.0000	−0.952981	−0.476491	+	0.879180i	$0.658091\pi$
$992$	−4.00000	−0.127000
$993$	34.0000	1.07896
$994$	8.00000	0.253745
$995$	−5.00000	−0.158511
$996$	−2.00000	−0.0633724
$997$	−50.0000	−1.58352	−0.791758	−	0.610835i	$-0.790834\pi$
$997$	−50.0000	−1.58352	−0.791758	+	0.610835i	$0.790834\pi$
$998$	−8.00000	−0.253236
$999$	−5.00000	−0.158193

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.a.c.1.1	✓	1	1.1	even	1	trivial
1638.2.a.o.1.1		1	3.2	odd	2
3822.2.a.e.1.1		1	7.6	odd	2
4368.2.a.h.1.1		1	4.3	odd	2
7098.2.a.z.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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