LMFDB - Embedded newform 6042.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6042,2,Mod(1,6042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6042, 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("6042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6042 = 2 \cdot 3 \cdot 19 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6042.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:	$48.2456129013$
Analytic rank:	$2$
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	$\chi$	$=$	6042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{10} -4.00000 q^{11} +1.00000 q^{12} -6.00000 q^{13} +3.00000 q^{14} -3.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} -1.00000 q^{19} -3.00000 q^{20} -3.00000 q^{21} +4.00000 q^{22} -5.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} +6.00000 q^{26} +1.00000 q^{27} -3.00000 q^{28} -5.00000 q^{29} +3.00000 q^{30} -1.00000 q^{31} -1.00000 q^{32} -4.00000 q^{33} +9.00000 q^{35} +1.00000 q^{36} +1.00000 q^{38} -6.00000 q^{39} +3.00000 q^{40} -10.0000 q^{41} +3.00000 q^{42} -1.00000 q^{43} -4.00000 q^{44} -3.00000 q^{45} +5.00000 q^{46} +1.00000 q^{48} +2.00000 q^{49} -4.00000 q^{50} -6.00000 q^{52} -1.00000 q^{53} -1.00000 q^{54} +12.0000 q^{55} +3.00000 q^{56} -1.00000 q^{57} +5.00000 q^{58} -13.0000 q^{59} -3.00000 q^{60} +2.00000 q^{61} +1.00000 q^{62} -3.00000 q^{63} +1.00000 q^{64} +18.0000 q^{65} +4.00000 q^{66} -3.00000 q^{67} -5.00000 q^{69} -9.00000 q^{70} -6.00000 q^{71} -1.00000 q^{72} -4.00000 q^{73} +4.00000 q^{75} -1.00000 q^{76} +12.0000 q^{77} +6.00000 q^{78} -8.00000 q^{79} -3.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} +12.0000 q^{83} -3.00000 q^{84} +1.00000 q^{86} -5.00000 q^{87} +4.00000 q^{88} -7.00000 q^{89} +3.00000 q^{90} +18.0000 q^{91} -5.00000 q^{92} -1.00000 q^{93} +3.00000 q^{95} -1.00000 q^{96} +2.00000 q^{97} -2.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	−1.00000	−0.408248
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	3.00000	0.948683
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	1.00000	0.288675
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	3.00000	0.801784
$15$	−3.00000	−0.774597
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−1.00000	−0.235702
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−3.00000	−0.670820
$21$	−3.00000	−0.654654
$22$	4.00000	0.852803
$23$	−5.00000	−1.04257	−0.521286	−	0.853382i	$-0.674548\pi$
$23$	−5.00000	−1.04257	−0.521286	+	0.853382i	$0.674548\pi$
$24$	−1.00000	−0.204124
$25$	4.00000	0.800000
$26$	6.00000	1.17670
$27$	1.00000	0.192450
$28$	−3.00000	−0.566947
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	3.00000	0.547723
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	−1.00000	−0.176777
$33$	−4.00000	−0.696311
$34$	0	0
$35$	9.00000	1.52128
$36$	1.00000	0.166667
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	1.00000	0.162221
$39$	−6.00000	−0.960769
$40$	3.00000	0.474342
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	3.00000	0.462910
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	−4.00000	−0.603023
$45$	−3.00000	−0.447214
$46$	5.00000	0.737210
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	1.00000	0.144338
$49$	2.00000	0.285714
$50$	−4.00000	−0.565685
$51$	0	0
$52$	−6.00000	−0.832050
$53$	−1.00000	−0.137361
$53$	−1.00000	−0.137361
$54$	−1.00000	−0.136083
$55$	12.0000	1.61808
$56$	3.00000	0.400892
$57$	−1.00000	−0.132453
$58$	5.00000	0.656532
$59$	−13.0000	−1.69246	−0.846228	−	0.532821i	$-0.821132\pi$
$59$	−13.0000	−1.69246	−0.846228	+	0.532821i	$0.821132\pi$
$60$	−3.00000	−0.387298
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	1.00000	0.127000
$63$	−3.00000	−0.377964
$64$	1.00000	0.125000
$65$	18.0000	2.23263
$66$	4.00000	0.492366
$67$	−3.00000	−0.366508	−0.183254	−	0.983066i	$-0.558663\pi$
$67$	−3.00000	−0.366508	−0.183254	+	0.983066i	$0.558663\pi$
$68$	0	0
$69$	−5.00000	−0.601929
$70$	−9.00000	−1.07571
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	−1.00000	−0.117851
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	0	0
$75$	4.00000	0.461880
$76$	−1.00000	−0.114708
$77$	12.0000	1.36753
$78$	6.00000	0.679366
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	−3.00000	−0.335410
$81$	1.00000	0.111111
$82$	10.0000	1.10432
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	−3.00000	−0.327327
$85$	0	0
$86$	1.00000	0.107833
$87$	−5.00000	−0.536056
$88$	4.00000	0.426401
$89$	−7.00000	−0.741999	−0.370999	−	0.928633i	$-0.620985\pi$
$89$	−7.00000	−0.741999	−0.370999	+	0.928633i	$0.620985\pi$
$90$	3.00000	0.316228
$91$	18.0000	1.88691
$92$	−5.00000	−0.521286
$93$	−1.00000	−0.103695
$94$	0	0
$95$	3.00000	0.307794
$96$	−1.00000	−0.102062
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	−2.00000	−0.202031
$99$	−4.00000	−0.402015
$100$	4.00000	0.400000
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	0	0
$103$	−13.0000	−1.28093	−0.640464	−	0.767988i	$-0.721258\pi$
$103$	−13.0000	−1.28093	−0.640464	+	0.767988i	$0.721258\pi$
$104$	6.00000	0.588348
$105$	9.00000	0.878310
$106$	1.00000	0.0971286
$107$	−11.0000	−1.06341	−0.531705	−	0.846930i	$-0.678449\pi$
$107$	−11.0000	−1.06341	−0.531705	+	0.846930i	$0.678449\pi$
$108$	1.00000	0.0962250
$109$	9.00000	0.862044	0.431022	−	0.902342i	$-0.358153\pi$
$109$	9.00000	0.862044	0.431022	+	0.902342i	$0.358153\pi$
$110$	−12.0000	−1.14416
$111$	0	0
$112$	−3.00000	−0.283473
$113$	11.0000	1.03479	0.517396	−	0.855746i	$-0.326901\pi$
$113$	11.0000	1.03479	0.517396	+	0.855746i	$0.326901\pi$
$114$	1.00000	0.0936586
$115$	15.0000	1.39876
$116$	−5.00000	−0.464238
$117$	−6.00000	−0.554700
$118$	13.0000	1.19675
$119$	0	0
$120$	3.00000	0.273861
$121$	5.00000	0.454545
$122$	−2.00000	−0.181071
$123$	−10.0000	−0.901670
$124$	−1.00000	−0.0898027
$125$	3.00000	0.268328
$126$	3.00000	0.267261
$127$	17.0000	1.50851	0.754253	−	0.656584i	$-0.227999\pi$
$127$	17.0000	1.50851	0.754253	+	0.656584i	$0.227999\pi$
$128$	−1.00000	−0.0883883
$129$	−1.00000	−0.0880451
$130$	−18.0000	−1.57870
$131$	14.0000	1.22319	0.611593	−	0.791173i	$-0.290529\pi$
$131$	14.0000	1.22319	0.611593	+	0.791173i	$0.290529\pi$
$132$	−4.00000	−0.348155
$133$	3.00000	0.260133
$134$	3.00000	0.259161
$135$	−3.00000	−0.258199
$136$	0	0
$137$	−15.0000	−1.28154	−0.640768	−	0.767734i	$-0.721384\pi$
$137$	−15.0000	−1.28154	−0.640768	+	0.767734i	$0.721384\pi$
$138$	5.00000	0.425628
$139$	18.0000	1.52674	0.763370	−	0.645961i	$-0.223543\pi$
$139$	18.0000	1.52674	0.763370	+	0.645961i	$0.223543\pi$
$140$	9.00000	0.760639
$141$	0	0
$142$	6.00000	0.503509
$143$	24.0000	2.00698
$144$	1.00000	0.0833333
$145$	15.0000	1.24568
$146$	4.00000	0.331042
$147$	2.00000	0.164957
$148$	0	0
$149$	−4.00000	−0.327693	−0.163846	−	0.986486i	$-0.552390\pi$
$149$	−4.00000	−0.327693	−0.163846	+	0.986486i	$0.552390\pi$
$150$	−4.00000	−0.326599
$151$	−7.00000	−0.569652	−0.284826	−	0.958579i	$-0.591936\pi$
$151$	−7.00000	−0.569652	−0.284826	+	0.958579i	$0.591936\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	−12.0000	−0.966988
$155$	3.00000	0.240966
$156$	−6.00000	−0.480384
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	8.00000	0.636446
$159$	−1.00000	−0.0793052
$160$	3.00000	0.237171
$161$	15.0000	1.18217
$162$	−1.00000	−0.0785674
$163$	−7.00000	−0.548282	−0.274141	−	0.961689i	$-0.588394\pi$
$163$	−7.00000	−0.548282	−0.274141	+	0.961689i	$0.588394\pi$
$164$	−10.0000	−0.780869
$165$	12.0000	0.934199
$166$	−12.0000	−0.931381
$167$	−22.0000	−1.70241	−0.851206	−	0.524832i	$-0.824128\pi$
$167$	−22.0000	−1.70241	−0.851206	+	0.524832i	$0.824128\pi$
$168$	3.00000	0.231455
$169$	23.0000	1.76923
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	−1.00000	−0.0762493
$173$	−4.00000	−0.304114	−0.152057	−	0.988372i	$-0.548590\pi$
$173$	−4.00000	−0.304114	−0.152057	+	0.988372i	$0.548590\pi$
$174$	5.00000	0.379049
$175$	−12.0000	−0.907115
$176$	−4.00000	−0.301511
$177$	−13.0000	−0.977140
$178$	7.00000	0.524672
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	−3.00000	−0.223607
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	−18.0000	−1.33425
$183$	2.00000	0.147844
$184$	5.00000	0.368605
$185$	0	0
$186$	1.00000	0.0733236
$187$	0	0
$188$	0	0
$189$	−3.00000	−0.218218
$190$	−3.00000	−0.217643
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	1.00000	0.0721688
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	−2.00000	−0.143592
$195$	18.0000	1.28901
$196$	2.00000	0.142857
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	4.00000	0.284268
$199$	23.0000	1.63043	0.815213	−	0.579161i	$-0.196620\pi$
$199$	23.0000	1.63043	0.815213	+	0.579161i	$0.196620\pi$
$200$	−4.00000	−0.282843
$201$	−3.00000	−0.211604
$202$	3.00000	0.211079
$203$	15.0000	1.05279
$204$	0	0
$205$	30.0000	2.09529
$206$	13.0000	0.905753
$207$	−5.00000	−0.347524
$208$	−6.00000	−0.416025
$209$	4.00000	0.276686
$210$	−9.00000	−0.621059
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	−1.00000	−0.0686803
$213$	−6.00000	−0.411113
$214$	11.0000	0.751945
$215$	3.00000	0.204598
$216$	−1.00000	−0.0680414
$217$	3.00000	0.203653
$218$	−9.00000	−0.609557
$219$	−4.00000	−0.270295
$220$	12.0000	0.809040
$221$	0	0
$222$	0	0
$223$	14.0000	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$224$	3.00000	0.200446
$225$	4.00000	0.266667
$226$	−11.0000	−0.731709
$227$	−7.00000	−0.464606	−0.232303	−	0.972643i	$-0.574626\pi$
$227$	−7.00000	−0.464606	−0.232303	+	0.972643i	$0.574626\pi$
$228$	−1.00000	−0.0662266
$229$	1.00000	0.0660819	0.0330409	−	0.999454i	$-0.489481\pi$
$229$	1.00000	0.0660819	0.0330409	+	0.999454i	$0.489481\pi$
$230$	−15.0000	−0.989071
$231$	12.0000	0.789542
$232$	5.00000	0.328266
$233$	22.0000	1.44127	0.720634	−	0.693316i	$-0.243851\pi$
$233$	22.0000	1.44127	0.720634	+	0.693316i	$0.243851\pi$
$234$	6.00000	0.392232
$235$	0	0
$236$	−13.0000	−0.846228
$237$	−8.00000	−0.519656
$238$	0	0
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	−3.00000	−0.193649
$241$	12.0000	0.772988	0.386494	−	0.922292i	$-0.373686\pi$
$241$	12.0000	0.772988	0.386494	+	0.922292i	$0.373686\pi$
$242$	−5.00000	−0.321412
$243$	1.00000	0.0641500
$244$	2.00000	0.128037
$245$	−6.00000	−0.383326
$246$	10.0000	0.637577
$247$	6.00000	0.381771
$248$	1.00000	0.0635001
$249$	12.0000	0.760469
$250$	−3.00000	−0.189737
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	−3.00000	−0.188982
$253$	20.0000	1.25739
$254$	−17.0000	−1.06667
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	1.00000	0.0622573
$259$	0	0
$260$	18.0000	1.11631
$261$	−5.00000	−0.309492
$262$	−14.0000	−0.864923
$263$	7.00000	0.431638	0.215819	−	0.976433i	$-0.430758\pi$
$263$	7.00000	0.431638	0.215819	+	0.976433i	$0.430758\pi$
$264$	4.00000	0.246183
$265$	3.00000	0.184289
$266$	−3.00000	−0.183942
$267$	−7.00000	−0.428393
$268$	−3.00000	−0.183254
$269$	−15.0000	−0.914566	−0.457283	−	0.889321i	$-0.651177\pi$
$269$	−15.0000	−0.914566	−0.457283	+	0.889321i	$0.651177\pi$
$270$	3.00000	0.182574
$271$	−7.00000	−0.425220	−0.212610	−	0.977137i	$-0.568196\pi$
$271$	−7.00000	−0.425220	−0.212610	+	0.977137i	$0.568196\pi$
$272$	0	0
$273$	18.0000	1.08941
$274$	15.0000	0.906183
$275$	−16.0000	−0.964836
$276$	−5.00000	−0.300965
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	−18.0000	−1.07957
$279$	−1.00000	−0.0598684
$280$	−9.00000	−0.537853
$281$	−17.0000	−1.01413	−0.507067	−	0.861906i	$-0.669271\pi$
$281$	−17.0000	−1.01413	−0.507067	+	0.861906i	$0.669271\pi$
$282$	0	0
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	−6.00000	−0.356034
$285$	3.00000	0.177705
$286$	−24.0000	−1.41915
$287$	30.0000	1.77084
$288$	−1.00000	−0.0589256
$289$	−17.0000	−1.00000
$290$	−15.0000	−0.880830
$291$	2.00000	0.117242
$292$	−4.00000	−0.234082
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	−2.00000	−0.116642
$295$	39.0000	2.27067
$296$	0	0
$297$	−4.00000	−0.232104
$298$	4.00000	0.231714
$299$	30.0000	1.73494
$300$	4.00000	0.230940
$301$	3.00000	0.172917
$302$	7.00000	0.402805
$303$	−3.00000	−0.172345
$304$	−1.00000	−0.0573539
$305$	−6.00000	−0.343559
$306$	0	0
$307$	26.0000	1.48390	0.741949	−	0.670456i	$-0.233902\pi$
$307$	26.0000	1.48390	0.741949	+	0.670456i	$0.233902\pi$
$308$	12.0000	0.683763
$309$	−13.0000	−0.739544
$310$	−3.00000	−0.170389
$311$	−6.00000	−0.340229	−0.170114	−	0.985424i	$-0.554414\pi$
$311$	−6.00000	−0.340229	−0.170114	+	0.985424i	$0.554414\pi$
$312$	6.00000	0.339683
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	14.0000	0.790066
$315$	9.00000	0.507093
$316$	−8.00000	−0.450035
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	1.00000	0.0560772
$319$	20.0000	1.11979
$320$	−3.00000	−0.167705
$321$	−11.0000	−0.613960
$322$	−15.0000	−0.835917
$323$	0	0
$324$	1.00000	0.0555556
$325$	−24.0000	−1.33128
$326$	7.00000	0.387694
$327$	9.00000	0.497701
$328$	10.0000	0.552158
$329$	0	0
$330$	−12.0000	−0.660578
$331$	2.00000	0.109930	0.0549650	−	0.998488i	$-0.482495\pi$
$331$	2.00000	0.109930	0.0549650	+	0.998488i	$0.482495\pi$
$332$	12.0000	0.658586
$333$	0	0
$334$	22.0000	1.20379
$335$	9.00000	0.491723
$336$	−3.00000	−0.163663
$337$	−35.0000	−1.90657	−0.953286	−	0.302070i	$-0.902322\pi$
$337$	−35.0000	−1.90657	−0.953286	+	0.302070i	$0.902322\pi$
$338$	−23.0000	−1.25104
$339$	11.0000	0.597438
$340$	0	0
$341$	4.00000	0.216612
$342$	1.00000	0.0540738
$343$	15.0000	0.809924
$344$	1.00000	0.0539164
$345$	15.0000	0.807573
$346$	4.00000	0.215041
$347$	−2.00000	−0.107366	−0.0536828	−	0.998558i	$-0.517096\pi$
$347$	−2.00000	−0.107366	−0.0536828	+	0.998558i	$0.517096\pi$
$348$	−5.00000	−0.268028
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	12.0000	0.641427
$351$	−6.00000	−0.320256
$352$	4.00000	0.213201
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	13.0000	0.690942
$355$	18.0000	0.955341
$356$	−7.00000	−0.370999
$357$	0	0
$358$	−12.0000	−0.634220
$359$	−3.00000	−0.158334	−0.0791670	−	0.996861i	$-0.525226\pi$
$359$	−3.00000	−0.158334	−0.0791670	+	0.996861i	$0.525226\pi$
$360$	3.00000	0.158114
$361$	1.00000	0.0526316
$362$	7.00000	0.367912
$363$	5.00000	0.262432
$364$	18.0000	0.943456
$365$	12.0000	0.628109
$366$	−2.00000	−0.104542
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	−5.00000	−0.260643
$369$	−10.0000	−0.520579
$370$	0	0
$371$	3.00000	0.155752
$372$	−1.00000	−0.0518476
$373$	−29.0000	−1.50156	−0.750782	−	0.660551i	$-0.770323\pi$
$373$	−29.0000	−1.50156	−0.750782	+	0.660551i	$0.770323\pi$
$374$	0	0
$375$	3.00000	0.154919
$376$	0	0
$377$	30.0000	1.54508
$378$	3.00000	0.154303
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	3.00000	0.153897
$381$	17.0000	0.870936
$382$	4.00000	0.204658
$383$	−30.0000	−1.53293	−0.766464	−	0.642287i	$-0.777986\pi$
$383$	−30.0000	−1.53293	−0.766464	+	0.642287i	$0.777986\pi$
$384$	−1.00000	−0.0510310
$385$	−36.0000	−1.83473
$386$	−10.0000	−0.508987
$387$	−1.00000	−0.0508329
$388$	2.00000	0.101535
$389$	−25.0000	−1.26755	−0.633775	−	0.773517i	$-0.718496\pi$
$389$	−25.0000	−1.26755	−0.633775	+	0.773517i	$0.718496\pi$
$390$	−18.0000	−0.911465
$391$	0	0
$392$	−2.00000	−0.101015
$393$	14.0000	0.706207
$394$	18.0000	0.906827
$395$	24.0000	1.20757
$396$	−4.00000	−0.201008
$397$	12.0000	0.602263	0.301131	−	0.953583i	$-0.402636\pi$
$397$	12.0000	0.602263	0.301131	+	0.953583i	$0.402636\pi$
$398$	−23.0000	−1.15289
$399$	3.00000	0.150188
$400$	4.00000	0.200000
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	3.00000	0.149626
$403$	6.00000	0.298881
$404$	−3.00000	−0.149256
$405$	−3.00000	−0.149071
$406$	−15.0000	−0.744438
$407$	0	0
$408$	0	0
$409$	28.0000	1.38451	0.692255	−	0.721653i	$-0.256617\pi$
$409$	28.0000	1.38451	0.692255	+	0.721653i	$0.256617\pi$
$410$	−30.0000	−1.48159
$411$	−15.0000	−0.739895
$412$	−13.0000	−0.640464
$413$	39.0000	1.91906
$414$	5.00000	0.245737
$415$	−36.0000	−1.76717
$416$	6.00000	0.294174
$417$	18.0000	0.881464
$418$	−4.00000	−0.195646
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	9.00000	0.439155
$421$	23.0000	1.12095	0.560476	−	0.828171i	$-0.310618\pi$
$421$	23.0000	1.12095	0.560476	+	0.828171i	$0.310618\pi$
$422$	0	0
$423$	0	0
$424$	1.00000	0.0485643
$425$	0	0
$426$	6.00000	0.290701
$427$	−6.00000	−0.290360
$428$	−11.0000	−0.531705
$429$	24.0000	1.15873
$430$	−3.00000	−0.144673
$431$	−17.0000	−0.818861	−0.409431	−	0.912341i	$-0.634273\pi$
$431$	−17.0000	−0.818861	−0.409431	+	0.912341i	$0.634273\pi$
$432$	1.00000	0.0481125
$433$	−38.0000	−1.82616	−0.913082	−	0.407777i	$-0.866304\pi$
$433$	−38.0000	−1.82616	−0.913082	+	0.407777i	$0.866304\pi$
$434$	−3.00000	−0.144005
$435$	15.0000	0.719195
$436$	9.00000	0.431022
$437$	5.00000	0.239182
$438$	4.00000	0.191127
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	−12.0000	−0.572078
$441$	2.00000	0.0952381
$442$	0	0
$443$	−15.0000	−0.712672	−0.356336	−	0.934358i	$-0.615974\pi$
$443$	−15.0000	−0.712672	−0.356336	+	0.934358i	$0.615974\pi$
$444$	0	0
$445$	21.0000	0.995495
$446$	−14.0000	−0.662919
$447$	−4.00000	−0.189194
$448$	−3.00000	−0.141737
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	−4.00000	−0.188562
$451$	40.0000	1.88353
$452$	11.0000	0.517396
$453$	−7.00000	−0.328889
$454$	7.00000	0.328526
$455$	−54.0000	−2.53156
$456$	1.00000	0.0468293
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	−1.00000	−0.0467269
$459$	0	0
$460$	15.0000	0.699379
$461$	−16.0000	−0.745194	−0.372597	−	0.927993i	$-0.621533\pi$
$461$	−16.0000	−0.745194	−0.372597	+	0.927993i	$0.621533\pi$
$462$	−12.0000	−0.558291
$463$	−2.00000	−0.0929479	−0.0464739	−	0.998920i	$-0.514798\pi$
$463$	−2.00000	−0.0929479	−0.0464739	+	0.998920i	$0.514798\pi$
$464$	−5.00000	−0.232119
$465$	3.00000	0.139122
$466$	−22.0000	−1.01913
$467$	32.0000	1.48078	0.740392	−	0.672176i	$-0.234640\pi$
$467$	32.0000	1.48078	0.740392	+	0.672176i	$0.234640\pi$
$468$	−6.00000	−0.277350
$469$	9.00000	0.415581
$470$	0	0
$471$	−14.0000	−0.645086
$472$	13.0000	0.598374
$473$	4.00000	0.183920
$474$	8.00000	0.367452
$475$	−4.00000	−0.183533
$476$	0	0
$477$	−1.00000	−0.0457869
$478$	20.0000	0.914779
$479$	−27.0000	−1.23366	−0.616831	−	0.787096i	$-0.711584\pi$
$479$	−27.0000	−1.23366	−0.616831	+	0.787096i	$0.711584\pi$
$480$	3.00000	0.136931
$481$	0	0
$482$	−12.0000	−0.546585
$483$	15.0000	0.682524
$484$	5.00000	0.227273
$485$	−6.00000	−0.272446
$486$	−1.00000	−0.0453609
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	−2.00000	−0.0905357
$489$	−7.00000	−0.316551
$490$	6.00000	0.271052
$491$	−37.0000	−1.66979	−0.834893	−	0.550412i	$-0.814471\pi$
$491$	−37.0000	−1.66979	−0.834893	+	0.550412i	$0.814471\pi$
$492$	−10.0000	−0.450835
$493$	0	0
$494$	−6.00000	−0.269953
$495$	12.0000	0.539360
$496$	−1.00000	−0.0449013
$497$	18.0000	0.807410
$498$	−12.0000	−0.537733
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	3.00000	0.134164
$501$	−22.0000	−0.982888
$502$	12.0000	0.535586
$503$	−21.0000	−0.936344	−0.468172	−	0.883637i	$-0.655087\pi$
$503$	−21.0000	−0.936344	−0.468172	+	0.883637i	$0.655087\pi$
$504$	3.00000	0.133631
$505$	9.00000	0.400495
$506$	−20.0000	−0.889108
$507$	23.0000	1.02147
$508$	17.0000	0.754253
$509$	−20.0000	−0.886484	−0.443242	−	0.896402i	$-0.646172\pi$
$509$	−20.0000	−0.886484	−0.443242	+	0.896402i	$0.646172\pi$
$510$	0	0
$511$	12.0000	0.530849
$512$	−1.00000	−0.0441942
$513$	−1.00000	−0.0441511
$514$	−6.00000	−0.264649
$515$	39.0000	1.71855
$516$	−1.00000	−0.0440225
$517$	0	0
$518$	0	0
$519$	−4.00000	−0.175581
$520$	−18.0000	−0.789352
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	5.00000	0.218844
$523$	28.0000	1.22435	0.612177	−	0.790721i	$-0.290294\pi$
$523$	28.0000	1.22435	0.612177	+	0.790721i	$0.290294\pi$
$524$	14.0000	0.611593
$525$	−12.0000	−0.523723
$526$	−7.00000	−0.305215
$527$	0	0
$528$	−4.00000	−0.174078
$529$	2.00000	0.0869565
$530$	−3.00000	−0.130312
$531$	−13.0000	−0.564152
$532$	3.00000	0.130066
$533$	60.0000	2.59889
$534$	7.00000	0.302920
$535$	33.0000	1.42671
$536$	3.00000	0.129580
$537$	12.0000	0.517838
$538$	15.0000	0.646696
$539$	−8.00000	−0.344584
$540$	−3.00000	−0.129099
$541$	−29.0000	−1.24681	−0.623404	−	0.781900i	$-0.714251\pi$
$541$	−29.0000	−1.24681	−0.623404	+	0.781900i	$0.714251\pi$
$542$	7.00000	0.300676
$543$	−7.00000	−0.300399
$544$	0	0
$545$	−27.0000	−1.15655
$546$	−18.0000	−0.770329
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	−15.0000	−0.640768
$549$	2.00000	0.0853579
$550$	16.0000	0.682242
$551$	5.00000	0.213007
$552$	5.00000	0.212814
$553$	24.0000	1.02058
$554$	−22.0000	−0.934690
$555$	0	0
$556$	18.0000	0.763370
$557$	−23.0000	−0.974541	−0.487271	−	0.873251i	$-0.662007\pi$
$557$	−23.0000	−0.974541	−0.487271	+	0.873251i	$0.662007\pi$
$558$	1.00000	0.0423334
$559$	6.00000	0.253773
$560$	9.00000	0.380319
$561$	0	0
$562$	17.0000	0.717102
$563$	18.0000	0.758610	0.379305	−	0.925272i	$-0.376163\pi$
$563$	18.0000	0.758610	0.379305	+	0.925272i	$0.376163\pi$
$564$	0	0
$565$	−33.0000	−1.38832
$566$	−20.0000	−0.840663
$567$	−3.00000	−0.125988
$568$	6.00000	0.251754
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	−3.00000	−0.125656
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	24.0000	1.00349
$573$	−4.00000	−0.167102
$574$	−30.0000	−1.25218
$575$	−20.0000	−0.834058
$576$	1.00000	0.0416667
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	17.0000	0.707107
$579$	10.0000	0.415586
$580$	15.0000	0.622841
$581$	−36.0000	−1.49353
$582$	−2.00000	−0.0829027
$583$	4.00000	0.165663
$584$	4.00000	0.165521
$585$	18.0000	0.744208
$586$	−6.00000	−0.247858
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	2.00000	0.0824786
$589$	1.00000	0.0412043
$590$	−39.0000	−1.60560
$591$	−18.0000	−0.740421
$592$	0	0
$593$	44.0000	1.80686	0.903432	−	0.428732i	$-0.141040\pi$
$593$	44.0000	1.80686	0.903432	+	0.428732i	$0.141040\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	−4.00000	−0.163846
$597$	23.0000	0.941327
$598$	−30.0000	−1.22679
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	−4.00000	−0.163299
$601$	−13.0000	−0.530281	−0.265141	−	0.964210i	$-0.585418\pi$
$601$	−13.0000	−0.530281	−0.265141	+	0.964210i	$0.585418\pi$
$602$	−3.00000	−0.122271
$603$	−3.00000	−0.122169
$604$	−7.00000	−0.284826
$605$	−15.0000	−0.609837
$606$	3.00000	0.121867
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	1.00000	0.0405554
$609$	15.0000	0.607831
$610$	6.00000	0.242933
$611$	0	0
$612$	0	0
$613$	16.0000	0.646234	0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000	0.646234	0.323117	+	0.946359i	$0.395269\pi$
$614$	−26.0000	−1.04927
$615$	30.0000	1.20972
$616$	−12.0000	−0.483494
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	13.0000	0.522937
$619$	13.0000	0.522514	0.261257	−	0.965269i	$-0.415863\pi$
$619$	13.0000	0.522514	0.261257	+	0.965269i	$0.415863\pi$
$620$	3.00000	0.120483
$621$	−5.00000	−0.200643
$622$	6.00000	0.240578
$623$	21.0000	0.841347
$624$	−6.00000	−0.240192
$625$	−29.0000	−1.16000
$626$	6.00000	0.239808
$627$	4.00000	0.159745
$628$	−14.0000	−0.558661
$629$	0	0
$630$	−9.00000	−0.358569
$631$	−46.0000	−1.83123	−0.915616	−	0.402055i	$-0.868296\pi$
$631$	−46.0000	−1.83123	−0.915616	+	0.402055i	$0.868296\pi$
$632$	8.00000	0.318223
$633$	0	0
$634$	−30.0000	−1.19145
$635$	−51.0000	−2.02387
$636$	−1.00000	−0.0396526
$637$	−12.0000	−0.475457
$638$	−20.0000	−0.791808
$639$	−6.00000	−0.237356
$640$	3.00000	0.118585
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	11.0000	0.434135
$643$	−19.0000	−0.749287	−0.374643	−	0.927169i	$-0.622235\pi$
$643$	−19.0000	−0.749287	−0.374643	+	0.927169i	$0.622235\pi$
$644$	15.0000	0.591083
$645$	3.00000	0.118125
$646$	0	0
$647$	48.0000	1.88707	0.943537	−	0.331266i	$-0.107476\pi$
$647$	48.0000	1.88707	0.943537	+	0.331266i	$0.107476\pi$
$648$	−1.00000	−0.0392837
$649$	52.0000	2.04118
$650$	24.0000	0.941357
$651$	3.00000	0.117579
$652$	−7.00000	−0.274141
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	−9.00000	−0.351928
$655$	−42.0000	−1.64108
$656$	−10.0000	−0.390434
$657$	−4.00000	−0.156055
$658$	0	0
$659$	16.0000	0.623272	0.311636	−	0.950202i	$-0.399123\pi$
$659$	16.0000	0.623272	0.311636	+	0.950202i	$0.399123\pi$
$660$	12.0000	0.467099
$661$	−20.0000	−0.777910	−0.388955	−	0.921257i	$-0.627164\pi$
$661$	−20.0000	−0.777910	−0.388955	+	0.921257i	$0.627164\pi$
$662$	−2.00000	−0.0777322
$663$	0	0
$664$	−12.0000	−0.465690
$665$	−9.00000	−0.349005
$666$	0	0
$667$	25.0000	0.968004
$668$	−22.0000	−0.851206
$669$	14.0000	0.541271
$670$	−9.00000	−0.347700
$671$	−8.00000	−0.308837
$672$	3.00000	0.115728
$673$	34.0000	1.31060	0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000	1.31060	0.655302	+	0.755367i	$0.272541\pi$
$674$	35.0000	1.34815
$675$	4.00000	0.153960
$676$	23.0000	0.884615
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	−11.0000	−0.422452
$679$	−6.00000	−0.230259
$680$	0	0
$681$	−7.00000	−0.268241
$682$	−4.00000	−0.153168
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	−1.00000	−0.0382360
$685$	45.0000	1.71936
$686$	−15.0000	−0.572703
$687$	1.00000	0.0381524
$688$	−1.00000	−0.0381246
$689$	6.00000	0.228582
$690$	−15.0000	−0.571040
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	−4.00000	−0.152057
$693$	12.0000	0.455842
$694$	2.00000	0.0759190
$695$	−54.0000	−2.04834
$696$	5.00000	0.189525
$697$	0	0
$698$	14.0000	0.529908
$699$	22.0000	0.832116
$700$	−12.0000	−0.453557
$701$	−25.0000	−0.944237	−0.472118	−	0.881535i	$-0.656511\pi$
$701$	−25.0000	−0.944237	−0.472118	+	0.881535i	$0.656511\pi$
$702$	6.00000	0.226455
$703$	0	0
$704$	−4.00000	−0.150756
$705$	0	0
$706$	18.0000	0.677439
$707$	9.00000	0.338480
$708$	−13.0000	−0.488570
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	−18.0000	−0.675528
$711$	−8.00000	−0.300023
$712$	7.00000	0.262336
$713$	5.00000	0.187251
$714$	0	0
$715$	−72.0000	−2.69265
$716$	12.0000	0.448461
$717$	−20.0000	−0.746914
$718$	3.00000	0.111959
$719$	27.0000	1.00693	0.503465	−	0.864016i	$-0.332058\pi$
$719$	27.0000	1.00693	0.503465	+	0.864016i	$0.332058\pi$
$720$	−3.00000	−0.111803
$721$	39.0000	1.45244
$722$	−1.00000	−0.0372161
$723$	12.0000	0.446285
$724$	−7.00000	−0.260153
$725$	−20.0000	−0.742781
$726$	−5.00000	−0.185567
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	−18.0000	−0.667124
$729$	1.00000	0.0370370
$730$	−12.0000	−0.444140
$731$	0	0
$732$	2.00000	0.0739221
$733$	33.0000	1.21888	0.609441	−	0.792831i	$-0.291394\pi$
$733$	33.0000	1.21888	0.609441	+	0.792831i	$0.291394\pi$
$734$	0	0
$735$	−6.00000	−0.221313
$736$	5.00000	0.184302
$737$	12.0000	0.442026
$738$	10.0000	0.368105
$739$	44.0000	1.61857	0.809283	−	0.587419i	$-0.199856\pi$
$739$	44.0000	1.61857	0.809283	+	0.587419i	$0.199856\pi$
$740$	0	0
$741$	6.00000	0.220416
$742$	−3.00000	−0.110133
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	1.00000	0.0366618
$745$	12.0000	0.439646
$746$	29.0000	1.06177
$747$	12.0000	0.439057
$748$	0	0
$749$	33.0000	1.20579
$750$	−3.00000	−0.109545
$751$	−2.00000	−0.0729810	−0.0364905	−	0.999334i	$-0.511618\pi$
$751$	−2.00000	−0.0729810	−0.0364905	+	0.999334i	$0.511618\pi$
$752$	0	0
$753$	−12.0000	−0.437304
$754$	−30.0000	−1.09254
$755$	21.0000	0.764268
$756$	−3.00000	−0.109109
$757$	−33.0000	−1.19941	−0.599703	−	0.800223i	$-0.704714\pi$
$757$	−33.0000	−1.19941	−0.599703	+	0.800223i	$0.704714\pi$
$758$	20.0000	0.726433
$759$	20.0000	0.725954
$760$	−3.00000	−0.108821
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	−17.0000	−0.615845
$763$	−27.0000	−0.977466
$764$	−4.00000	−0.144715
$765$	0	0
$766$	30.0000	1.08394
$767$	78.0000	2.81642
$768$	1.00000	0.0360844
$769$	−8.00000	−0.288487	−0.144244	−	0.989542i	$-0.546075\pi$
$769$	−8.00000	−0.288487	−0.144244	+	0.989542i	$0.546075\pi$
$770$	36.0000	1.29735
$771$	6.00000	0.216085
$772$	10.0000	0.359908
$773$	4.00000	0.143870	0.0719350	−	0.997409i	$-0.477083\pi$
$773$	4.00000	0.143870	0.0719350	+	0.997409i	$0.477083\pi$
$774$	1.00000	0.0359443
$775$	−4.00000	−0.143684
$776$	−2.00000	−0.0717958
$777$	0	0
$778$	25.0000	0.896293
$779$	10.0000	0.358287
$780$	18.0000	0.644503
$781$	24.0000	0.858788
$782$	0	0
$783$	−5.00000	−0.178685
$784$	2.00000	0.0714286
$785$	42.0000	1.49904
$786$	−14.0000	−0.499363
$787$	−33.0000	−1.17632	−0.588161	−	0.808744i	$-0.700148\pi$
$787$	−33.0000	−1.17632	−0.588161	+	0.808744i	$0.700148\pi$
$788$	−18.0000	−0.641223
$789$	7.00000	0.249207
$790$	−24.0000	−0.853882
$791$	−33.0000	−1.17334
$792$	4.00000	0.142134
$793$	−12.0000	−0.426132
$794$	−12.0000	−0.425864
$795$	3.00000	0.106399
$796$	23.0000	0.815213
$797$	12.0000	0.425062	0.212531	−	0.977154i	$-0.431829\pi$
$797$	12.0000	0.425062	0.212531	+	0.977154i	$0.431829\pi$
$798$	−3.00000	−0.106199
$799$	0	0
$800$	−4.00000	−0.141421
$801$	−7.00000	−0.247333
$802$	6.00000	0.211867
$803$	16.0000	0.564628
$804$	−3.00000	−0.105802
$805$	−45.0000	−1.58604
$806$	−6.00000	−0.211341
$807$	−15.0000	−0.528025
$808$	3.00000	0.105540
$809$	−21.0000	−0.738321	−0.369160	−	0.929366i	$-0.620355\pi$
$809$	−21.0000	−0.738321	−0.369160	+	0.929366i	$0.620355\pi$
$810$	3.00000	0.105409
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	15.0000	0.526397
$813$	−7.00000	−0.245501
$814$	0	0
$815$	21.0000	0.735598
$816$	0	0
$817$	1.00000	0.0349856
$818$	−28.0000	−0.978997
$819$	18.0000	0.628971
$820$	30.0000	1.04765
$821$	−3.00000	−0.104701	−0.0523504	−	0.998629i	$-0.516671\pi$
$821$	−3.00000	−0.104701	−0.0523504	+	0.998629i	$0.516671\pi$
$822$	15.0000	0.523185
$823$	−24.0000	−0.836587	−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000	−0.836587	−0.418294	+	0.908312i	$0.637372\pi$
$824$	13.0000	0.452876
$825$	−16.0000	−0.557048
$826$	−39.0000	−1.35698
$827$	22.0000	0.765015	0.382507	−	0.923952i	$-0.375061\pi$
$827$	22.0000	0.765015	0.382507	+	0.923952i	$0.375061\pi$
$828$	−5.00000	−0.173762
$829$	−17.0000	−0.590434	−0.295217	−	0.955430i	$-0.595392\pi$
$829$	−17.0000	−0.590434	−0.295217	+	0.955430i	$0.595392\pi$
$830$	36.0000	1.24958
$831$	22.0000	0.763172
$832$	−6.00000	−0.208013
$833$	0	0
$834$	−18.0000	−0.623289
$835$	66.0000	2.28402
$836$	4.00000	0.138343
$837$	−1.00000	−0.0345651
$838$	−4.00000	−0.138178
$839$	−17.0000	−0.586905	−0.293453	−	0.955974i	$-0.594804\pi$
$839$	−17.0000	−0.586905	−0.293453	+	0.955974i	$0.594804\pi$
$840$	−9.00000	−0.310530
$841$	−4.00000	−0.137931
$842$	−23.0000	−0.792632
$843$	−17.0000	−0.585511
$844$	0	0
$845$	−69.0000	−2.37367
$846$	0	0
$847$	−15.0000	−0.515406
$848$	−1.00000	−0.0343401
$849$	20.0000	0.686398
$850$	0	0
$851$	0	0
$852$	−6.00000	−0.205557
$853$	20.0000	0.684787	0.342393	−	0.939557i	$-0.388762\pi$
$853$	20.0000	0.684787	0.342393	+	0.939557i	$0.388762\pi$
$854$	6.00000	0.205316
$855$	3.00000	0.102598
$856$	11.0000	0.375972
$857$	−3.00000	−0.102478	−0.0512390	−	0.998686i	$-0.516317\pi$
$857$	−3.00000	−0.102478	−0.0512390	+	0.998686i	$0.516317\pi$
$858$	−24.0000	−0.819346
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	3.00000	0.102299
$861$	30.0000	1.02240
$862$	17.0000	0.579022
$863$	−45.0000	−1.53182	−0.765909	−	0.642949i	$-0.777711\pi$
$863$	−45.0000	−1.53182	−0.765909	+	0.642949i	$0.777711\pi$
$864$	−1.00000	−0.0340207
$865$	12.0000	0.408012
$866$	38.0000	1.29129
$867$	−17.0000	−0.577350
$868$	3.00000	0.101827
$869$	32.0000	1.08553
$870$	−15.0000	−0.508548
$871$	18.0000	0.609907
$872$	−9.00000	−0.304778
$873$	2.00000	0.0676897
$874$	−5.00000	−0.169128
$875$	−9.00000	−0.304256
$876$	−4.00000	−0.135147
$877$	−14.0000	−0.472746	−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000	−0.472746	−0.236373	+	0.971662i	$0.575959\pi$
$878$	32.0000	1.07995
$879$	6.00000	0.202375
$880$	12.0000	0.404520
$881$	−35.0000	−1.17918	−0.589590	−	0.807703i	$-0.700711\pi$
$881$	−35.0000	−1.17918	−0.589590	+	0.807703i	$0.700711\pi$
$882$	−2.00000	−0.0673435
$883$	6.00000	0.201916	0.100958	−	0.994891i	$-0.467809\pi$
$883$	6.00000	0.201916	0.100958	+	0.994891i	$0.467809\pi$
$884$	0	0
$885$	39.0000	1.31097
$886$	15.0000	0.503935
$887$	32.0000	1.07445	0.537227	−	0.843437i	$-0.319472\pi$
$887$	32.0000	1.07445	0.537227	+	0.843437i	$0.319472\pi$
$888$	0	0
$889$	−51.0000	−1.71049
$890$	−21.0000	−0.703922
$891$	−4.00000	−0.134005
$892$	14.0000	0.468755
$893$	0	0
$894$	4.00000	0.133780
$895$	−36.0000	−1.20335
$896$	3.00000	0.100223
$897$	30.0000	1.00167
$898$	10.0000	0.333704
$899$	5.00000	0.166759
$900$	4.00000	0.133333
$901$	0	0
$902$	−40.0000	−1.33185
$903$	3.00000	0.0998337
$904$	−11.0000	−0.365855
$905$	21.0000	0.698064
$906$	7.00000	0.232559
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	−7.00000	−0.232303
$909$	−3.00000	−0.0995037
$910$	54.0000	1.79008
$911$	39.0000	1.29213	0.646064	−	0.763283i	$-0.276414\pi$
$911$	39.0000	1.29213	0.646064	+	0.763283i	$0.276414\pi$
$912$	−1.00000	−0.0331133
$913$	−48.0000	−1.58857
$914$	−22.0000	−0.727695
$915$	−6.00000	−0.198354
$916$	1.00000	0.0330409
$917$	−42.0000	−1.38696
$918$	0	0
$919$	−26.0000	−0.857661	−0.428830	−	0.903385i	$-0.641074\pi$
$919$	−26.0000	−0.857661	−0.428830	+	0.903385i	$0.641074\pi$
$920$	−15.0000	−0.494535
$921$	26.0000	0.856729
$922$	16.0000	0.526932
$923$	36.0000	1.18495
$924$	12.0000	0.394771
$925$	0	0
$926$	2.00000	0.0657241
$927$	−13.0000	−0.426976
$928$	5.00000	0.164133
$929$	36.0000	1.18112	0.590561	−	0.806993i	$-0.298907\pi$
$929$	36.0000	1.18112	0.590561	+	0.806993i	$0.298907\pi$
$930$	−3.00000	−0.0983739
$931$	−2.00000	−0.0655474
$932$	22.0000	0.720634
$933$	−6.00000	−0.196431
$934$	−32.0000	−1.04707
$935$	0	0
$936$	6.00000	0.196116
$937$	−37.0000	−1.20874	−0.604369	−	0.796705i	$-0.706575\pi$
$937$	−37.0000	−1.20874	−0.604369	+	0.796705i	$0.706575\pi$
$938$	−9.00000	−0.293860
$939$	−6.00000	−0.195803
$940$	0	0
$941$	7.00000	0.228193	0.114097	−	0.993470i	$-0.463603\pi$
$941$	7.00000	0.228193	0.114097	+	0.993470i	$0.463603\pi$
$942$	14.0000	0.456145
$943$	50.0000	1.62822
$944$	−13.0000	−0.423114
$945$	9.00000	0.292770
$946$	−4.00000	−0.130051
$947$	−28.0000	−0.909878	−0.454939	−	0.890523i	$-0.650339\pi$
$947$	−28.0000	−0.909878	−0.454939	+	0.890523i	$0.650339\pi$
$948$	−8.00000	−0.259828
$949$	24.0000	0.779073
$950$	4.00000	0.129777
$951$	30.0000	0.972817
$952$	0	0
$953$	−15.0000	−0.485898	−0.242949	−	0.970039i	$-0.578115\pi$
$953$	−15.0000	−0.485898	−0.242949	+	0.970039i	$0.578115\pi$
$954$	1.00000	0.0323762
$955$	12.0000	0.388311
$956$	−20.0000	−0.646846
$957$	20.0000	0.646508
$958$	27.0000	0.872330
$959$	45.0000	1.45313
$960$	−3.00000	−0.0968246
$961$	−30.0000	−0.967742
$962$	0	0
$963$	−11.0000	−0.354470
$964$	12.0000	0.386494
$965$	−30.0000	−0.965734
$966$	−15.0000	−0.482617
$967$	16.0000	0.514525	0.257263	−	0.966342i	$-0.417179\pi$
$967$	16.0000	0.514525	0.257263	+	0.966342i	$0.417179\pi$
$968$	−5.00000	−0.160706
$969$	0	0
$970$	6.00000	0.192648
$971$	47.0000	1.50830	0.754151	−	0.656701i	$-0.228049\pi$
$971$	47.0000	1.50830	0.754151	+	0.656701i	$0.228049\pi$
$972$	1.00000	0.0320750
$973$	−54.0000	−1.73116
$974$	16.0000	0.512673
$975$	−24.0000	−0.768615
$976$	2.00000	0.0640184
$977$	20.0000	0.639857	0.319928	−	0.947442i	$-0.396341\pi$
$977$	20.0000	0.639857	0.319928	+	0.947442i	$0.396341\pi$
$978$	7.00000	0.223835
$979$	28.0000	0.894884
$980$	−6.00000	−0.191663
$981$	9.00000	0.287348
$982$	37.0000	1.18072
$983$	−31.0000	−0.988746	−0.494373	−	0.869250i	$-0.664602\pi$
$983$	−31.0000	−0.988746	−0.494373	+	0.869250i	$0.664602\pi$
$984$	10.0000	0.318788
$985$	54.0000	1.72058
$986$	0	0
$987$	0	0
$988$	6.00000	0.190885
$989$	5.00000	0.158991
$990$	−12.0000	−0.381385
$991$	38.0000	1.20711	0.603555	−	0.797321i	$-0.293750\pi$
$991$	38.0000	1.20711	0.603555	+	0.797321i	$0.293750\pi$
$992$	1.00000	0.0317500
$993$	2.00000	0.0634681
$994$	−18.0000	−0.570925
$995$	−69.0000	−2.18745
$996$	12.0000	0.380235
$997$	21.0000	0.665077	0.332538	−	0.943090i	$-0.392095\pi$
$997$	21.0000	0.665077	0.332538	+	0.943090i	$0.392095\pi$
$998$	24.0000	0.759707
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6042.2.a.e.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6042.2.a.e.1.1	✓	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6042.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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