LMFDB - Embedded newform 6045.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6045,2,Mod(1,6045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6045 = 3 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6045.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:	$48.2695680219$
Analytic rank:	$1$
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	$\chi$	$=$	6045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} -5.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q+1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} -5.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -2.00000 q^{12} +1.00000 q^{13} -1.00000 q^{15} +4.00000 q^{16} +5.00000 q^{17} -6.00000 q^{19} +2.00000 q^{20} -5.00000 q^{21} +3.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +10.0000 q^{28} -1.00000 q^{31} +1.00000 q^{33} +5.00000 q^{35} -2.00000 q^{36} +1.00000 q^{37} +1.00000 q^{39} +7.00000 q^{41} -2.00000 q^{43} -2.00000 q^{44} -1.00000 q^{45} +4.00000 q^{47} +4.00000 q^{48} +18.0000 q^{49} +5.00000 q^{51} -2.00000 q^{52} +5.00000 q^{53} -1.00000 q^{55} -6.00000 q^{57} -4.00000 q^{59} +2.00000 q^{60} +3.00000 q^{61} -5.00000 q^{63} -8.00000 q^{64} -1.00000 q^{65} -12.0000 q^{67} -10.0000 q^{68} +3.00000 q^{69} -1.00000 q^{71} -14.0000 q^{73} +1.00000 q^{75} +12.0000 q^{76} -5.00000 q^{77} -5.00000 q^{79} -4.00000 q^{80} +1.00000 q^{81} +14.0000 q^{83} +10.0000 q^{84} -5.00000 q^{85} -5.00000 q^{89} -5.00000 q^{91} -6.00000 q^{92} -1.00000 q^{93} +6.00000 q^{95} -19.0000 q^{97} +1.00000 q^{99} +O(q^{100})$

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 $

$2$	0	0		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−2.00000	−1.00000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−5.00000	−1.88982	−0.944911	−	0.327327i	$-0.893852\pi$
$7$	−5.00000	−1.88982	−0.944911	+	0.327327i	$0.893852\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	−2.00000	−0.577350
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−1.00000	−0.258199
$16$	4.00000	1.00000
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	2.00000	0.447214
$21$	−5.00000	−1.09109
$22$	0	0
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	10.0000	1.88982
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	5.00000	0.845154
$36$	−2.00000	−0.333333
$37$	1.00000	0.164399	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	1.00000	0.164399	0.0821995	+	0.996616i	$0.473806\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	7.00000	1.09322	0.546608	−	0.837389i	$-0.315919\pi$
$41$	7.00000	1.09322	0.546608	+	0.837389i	$0.315919\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	−2.00000	−0.301511
$45$	−1.00000	−0.149071
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	4.00000	0.577350
$49$	18.0000	2.57143
$50$	0	0
$51$	5.00000	0.700140
$52$	−2.00000	−0.277350
$53$	5.00000	0.686803	0.343401	−	0.939189i	$-0.388421\pi$
$53$	5.00000	0.686803	0.343401	+	0.939189i	$0.388421\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	−6.00000	−0.794719
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	2.00000	0.258199
$61$	3.00000	0.384111	0.192055	−	0.981384i	$-0.438485\pi$
$61$	3.00000	0.384111	0.192055	+	0.981384i	$0.438485\pi$
$62$	0	0
$63$	−5.00000	−0.629941
$64$	−8.00000	−1.00000
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	−10.0000	−1.21268
$69$	3.00000	0.361158
$70$	0	0
$71$	−1.00000	−0.118678	−0.0593391	−	0.998238i	$-0.518899\pi$
$71$	−1.00000	−0.118678	−0.0593391	+	0.998238i	$0.518899\pi$
$72$	0	0
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	12.0000	1.37649
$77$	−5.00000	−0.569803
$78$	0	0
$79$	−5.00000	−0.562544	−0.281272	−	0.959628i	$-0.590756\pi$
$79$	−5.00000	−0.562544	−0.281272	+	0.959628i	$0.590756\pi$
$80$	−4.00000	−0.447214
$81$	1.00000	0.111111
$82$	0	0
$83$	14.0000	1.53670	0.768350	−	0.640030i	$-0.221078\pi$
$83$	14.0000	1.53670	0.768350	+	0.640030i	$0.221078\pi$
$84$	10.0000	1.09109
$85$	−5.00000	−0.542326
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−5.00000	−0.529999	−0.264999	−	0.964249i	$-0.585372\pi$
$89$	−5.00000	−0.529999	−0.264999	+	0.964249i	$0.585372\pi$
$90$	0	0
$91$	−5.00000	−0.524142
$92$	−6.00000	−0.625543
$93$	−1.00000	−0.103695
$94$	0	0
$95$	6.00000	0.615587
$96$	0	0
$97$	−19.0000	−1.92916	−0.964579	−	0.263795i	$-0.915026\pi$
$97$	−19.0000	−1.92916	−0.964579	+	0.263795i	$0.915026\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	−2.00000	−0.200000
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	0	0
$103$	−12.0000	−1.18240	−0.591198	−	0.806527i	$-0.701345\pi$
$103$	−12.0000	−1.18240	−0.591198	+	0.806527i	$0.701345\pi$
$104$	0	0
$105$	5.00000	0.487950
$106$	0	0
$107$	13.0000	1.25676	0.628379	−	0.777908i	$-0.283719\pi$
$107$	13.0000	1.25676	0.628379	+	0.777908i	$0.283719\pi$
$108$	−2.00000	−0.192450
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	1.00000	0.0949158
$112$	−20.0000	−1.88982
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	−3.00000	−0.279751
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−25.0000	−2.29175
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	7.00000	0.631169
$124$	2.00000	0.179605
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	0	0
$129$	−2.00000	−0.176090
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	−2.00000	−0.174078
$133$	30.0000	2.60133
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	−23.0000	−1.95083	−0.975417	−	0.220366i	$-0.929275\pi$
$139$	−23.0000	−1.95083	−0.975417	+	0.220366i	$0.929275\pi$
$140$	−10.0000	−0.845154
$141$	4.00000	0.336861
$142$	0	0
$143$	1.00000	0.0836242
$144$	4.00000	0.333333
$145$	0	0
$146$	0	0
$147$	18.0000	1.48461
$148$	−2.00000	−0.164399
$149$	21.0000	1.72039	0.860194	−	0.509968i	$-0.170343\pi$
$149$	21.0000	1.72039	0.860194	+	0.509968i	$0.170343\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	5.00000	0.404226
$154$	0	0
$155$	1.00000	0.0803219
$156$	−2.00000	−0.160128
$157$	−20.0000	−1.59617	−0.798087	−	0.602542i	$-0.794154\pi$
$157$	−20.0000	−1.59617	−0.798087	+	0.602542i	$0.794154\pi$
$158$	0	0
$159$	5.00000	0.396526
$160$	0	0
$161$	−15.0000	−1.18217
$162$	0	0
$163$	−1.00000	−0.0783260	−0.0391630	−	0.999233i	$-0.512469\pi$
$163$	−1.00000	−0.0783260	−0.0391630	+	0.999233i	$0.512469\pi$
$164$	−14.0000	−1.09322
$165$	−1.00000	−0.0778499
$166$	0	0
$167$	−20.0000	−1.54765	−0.773823	−	0.633402i	$-0.781658\pi$
$167$	−20.0000	−1.54765	−0.773823	+	0.633402i	$0.781658\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−6.00000	−0.458831
$172$	4.00000	0.304997
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	−5.00000	−0.377964
$176$	4.00000	0.301511
$177$	−4.00000	−0.300658
$178$	0	0
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	2.00000	0.149071
$181$	7.00000	0.520306	0.260153	−	0.965567i	$-0.416227\pi$
$181$	7.00000	0.520306	0.260153	+	0.965567i	$0.416227\pi$
$182$	0	0
$183$	3.00000	0.221766
$184$	0	0
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	5.00000	0.365636
$188$	−8.00000	−0.583460
$189$	−5.00000	−0.363696
$190$	0	0
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	−8.00000	−0.577350
$193$	19.0000	1.36765	0.683825	−	0.729646i	$-0.260315\pi$
$193$	19.0000	1.36765	0.683825	+	0.729646i	$0.260315\pi$
$194$	0	0
$195$	−1.00000	−0.0716115
$196$	−36.0000	−2.57143
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	0	0
$203$	0	0
$204$	−10.0000	−0.700140
$205$	−7.00000	−0.488901
$206$	0	0
$207$	3.00000	0.208514
$208$	4.00000	0.277350
$209$	−6.00000	−0.415029
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	−10.0000	−0.686803
$213$	−1.00000	−0.0685189
$214$	0	0
$215$	2.00000	0.136399
$216$	0	0
$217$	5.00000	0.339422
$218$	0	0
$219$	−14.0000	−0.946032
$220$	2.00000	0.134840
$221$	5.00000	0.336336
$222$	0	0
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	10.0000	0.663723	0.331862	−	0.943328i	$-0.392323\pi$
$227$	10.0000	0.663723	0.331862	+	0.943328i	$0.392323\pi$
$228$	12.0000	0.794719
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	−5.00000	−0.328976
$232$	0	0
$233$	−19.0000	−1.24473	−0.622366	−	0.782727i	$-0.713828\pi$
$233$	−19.0000	−1.24473	−0.622366	+	0.782727i	$0.713828\pi$
$234$	0	0
$235$	−4.00000	−0.260931
$236$	8.00000	0.520756
$237$	−5.00000	−0.324785
$238$	0	0
$239$	−19.0000	−1.22901	−0.614504	−	0.788914i	$-0.710644\pi$
$239$	−19.0000	−1.22901	−0.614504	+	0.788914i	$0.710644\pi$
$240$	−4.00000	−0.258199
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	−6.00000	−0.384111
$245$	−18.0000	−1.14998
$246$	0	0
$247$	−6.00000	−0.381771
$248$	0	0
$249$	14.0000	0.887214
$250$	0	0
$251$	28.0000	1.76734	0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000	1.76734	0.883672	+	0.468106i	$0.155064\pi$
$252$	10.0000	0.629941
$253$	3.00000	0.188608
$254$	0	0
$255$	−5.00000	−0.313112
$256$	16.0000	1.00000
$257$	30.0000	1.87135	0.935674	−	0.352865i	$-0.114792\pi$
$257$	30.0000	1.87135	0.935674	+	0.352865i	$0.114792\pi$
$258$	0	0
$259$	−5.00000	−0.310685
$260$	2.00000	0.124035
$261$	0	0
$262$	0	0
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	−5.00000	−0.307148
$266$	0	0
$267$	−5.00000	−0.305995
$268$	24.0000	1.46603
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	0	0
$271$	−24.0000	−1.45790	−0.728948	−	0.684569i	$-0.759990\pi$
$271$	−24.0000	−1.45790	−0.728948	+	0.684569i	$0.759990\pi$
$272$	20.0000	1.21268
$273$	−5.00000	−0.302614
$274$	0	0
$275$	1.00000	0.0603023
$276$	−6.00000	−0.361158
$277$	−16.0000	−0.961347	−0.480673	−	0.876900i	$-0.659608\pi$
$277$	−16.0000	−0.961347	−0.480673	+	0.876900i	$0.659608\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	2.00000	0.118678
$285$	6.00000	0.355409
$286$	0	0
$287$	−35.0000	−2.06598
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	−19.0000	−1.11380
$292$	28.0000	1.63858
$293$	−4.00000	−0.233682	−0.116841	−	0.993151i	$-0.537277\pi$
$293$	−4.00000	−0.233682	−0.116841	+	0.993151i	$0.537277\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	3.00000	0.173494
$300$	−2.00000	−0.115470
$301$	10.0000	0.576390
$302$	0	0
$303$	18.0000	1.03407
$304$	−24.0000	−1.37649
$305$	−3.00000	−0.171780
$306$	0	0
$307$	−25.0000	−1.42683	−0.713413	−	0.700744i	$-0.752851\pi$
$307$	−25.0000	−1.42683	−0.713413	+	0.700744i	$0.752851\pi$
$308$	10.0000	0.569803
$309$	−12.0000	−0.682656
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	0	0
$315$	5.00000	0.281718
$316$	10.0000	0.562544
$317$	−8.00000	−0.449325	−0.224662	−	0.974437i	$-0.572128\pi$
$317$	−8.00000	−0.449325	−0.224662	+	0.974437i	$0.572128\pi$
$318$	0	0
$319$	0	0
$320$	8.00000	0.447214
$321$	13.0000	0.725589
$322$	0	0
$323$	−30.0000	−1.66924
$324$	−2.00000	−0.111111
$325$	1.00000	0.0554700
$326$	0	0
$327$	10.0000	0.553001
$328$	0	0
$329$	−20.0000	−1.10264
$330$	0	0
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	−28.0000	−1.53670
$333$	1.00000	0.0547997
$334$	0	0
$335$	12.0000	0.655630
$336$	−20.0000	−1.09109
$337$	−4.00000	−0.217894	−0.108947	−	0.994048i	$-0.534748\pi$
$337$	−4.00000	−0.217894	−0.108947	+	0.994048i	$0.534748\pi$
$338$	0	0
$339$	6.00000	0.325875
$340$	10.0000	0.542326
$341$	−1.00000	−0.0541530
$342$	0	0
$343$	−55.0000	−2.96972
$344$	0	0
$345$	−3.00000	−0.161515
$346$	0	0
$347$	3.00000	0.161048	0.0805242	−	0.996753i	$-0.474341\pi$
$347$	3.00000	0.161048	0.0805242	+	0.996753i	$0.474341\pi$
$348$	0	0
$349$	6.00000	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	4.00000	0.212899	0.106449	−	0.994318i	$-0.466052\pi$
$353$	4.00000	0.212899	0.106449	+	0.994318i	$0.466052\pi$
$354$	0	0
$355$	1.00000	0.0530745
$356$	10.0000	0.529999
$357$	−25.0000	−1.32314
$358$	0	0
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	−10.0000	−0.524864
$364$	10.0000	0.524142
$365$	14.0000	0.732793
$366$	0	0
$367$	−10.0000	−0.521996	−0.260998	−	0.965339i	$-0.584052\pi$
$367$	−10.0000	−0.521996	−0.260998	+	0.965339i	$0.584052\pi$
$368$	12.0000	0.625543
$369$	7.00000	0.364405
$370$	0	0
$371$	−25.0000	−1.29794
$372$	2.00000	0.103695
$373$	−26.0000	−1.34623	−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000	−1.34623	−0.673114	+	0.739538i	$0.735044\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	0	0
$378$	0	0
$379$	10.0000	0.513665	0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000	0.513665	0.256833	+	0.966456i	$0.417321\pi$
$380$	−12.0000	−0.615587
$381$	12.0000	0.614779
$382$	0	0
$383$	−4.00000	−0.204390	−0.102195	−	0.994764i	$-0.532587\pi$
$383$	−4.00000	−0.204390	−0.102195	+	0.994764i	$0.532587\pi$
$384$	0	0
$385$	5.00000	0.254824
$386$	0	0
$387$	−2.00000	−0.101666
$388$	38.0000	1.92916
$389$	20.0000	1.01404	0.507020	−	0.861934i	$-0.330747\pi$
$389$	20.0000	1.01404	0.507020	+	0.861934i	$0.330747\pi$
$390$	0	0
$391$	15.0000	0.758583
$392$	0	0
$393$	−6.00000	−0.302660
$394$	0	0
$395$	5.00000	0.251577
$396$	−2.00000	−0.100504
$397$	23.0000	1.15434	0.577168	−	0.816625i	$-0.304158\pi$
$397$	23.0000	1.15434	0.577168	+	0.816625i	$0.304158\pi$
$398$	0	0
$399$	30.0000	1.50188
$400$	4.00000	0.200000
$401$	14.0000	0.699127	0.349563	−	0.936913i	$-0.386330\pi$
$401$	14.0000	0.699127	0.349563	+	0.936913i	$0.386330\pi$
$402$	0	0
$403$	−1.00000	−0.0498135
$404$	−36.0000	−1.79107
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	1.00000	0.0495682
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	24.0000	1.18240
$413$	20.0000	0.984136
$414$	0	0
$415$	−14.0000	−0.687233
$416$	0	0
$417$	−23.0000	−1.12631
$418$	0	0
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	−10.0000	−0.487950
$421$	−16.0000	−0.779792	−0.389896	−	0.920859i	$-0.627489\pi$
$421$	−16.0000	−0.779792	−0.389896	+	0.920859i	$0.627489\pi$
$422$	0	0
$423$	4.00000	0.194487
$424$	0	0
$425$	5.00000	0.242536
$426$	0	0
$427$	−15.0000	−0.725901
$428$	−26.0000	−1.25676
$429$	1.00000	0.0482805
$430$	0	0
$431$	32.0000	1.54139	0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000	1.54139	0.770693	+	0.637207i	$0.219910\pi$
$432$	4.00000	0.192450
$433$	24.0000	1.15337	0.576683	−	0.816968i	$-0.304347\pi$
$433$	24.0000	1.15337	0.576683	+	0.816968i	$0.304347\pi$
$434$	0	0
$435$	0	0
$436$	−20.0000	−0.957826
$437$	−18.0000	−0.861057
$438$	0	0
$439$	15.0000	0.715911	0.357955	−	0.933739i	$-0.383474\pi$
$439$	15.0000	0.715911	0.357955	+	0.933739i	$0.383474\pi$
$440$	0	0
$441$	18.0000	0.857143
$442$	0	0
$443$	21.0000	0.997740	0.498870	−	0.866677i	$-0.333748\pi$
$443$	21.0000	0.997740	0.498870	+	0.866677i	$0.333748\pi$
$444$	−2.00000	−0.0949158
$445$	5.00000	0.237023
$446$	0	0
$447$	21.0000	0.993266
$448$	40.0000	1.88982
$449$	−19.0000	−0.896665	−0.448333	−	0.893867i	$-0.647982\pi$
$449$	−19.0000	−0.896665	−0.448333	+	0.893867i	$0.647982\pi$
$450$	0	0
$451$	7.00000	0.329617
$452$	−12.0000	−0.564433
$453$	0	0
$454$	0	0
$455$	5.00000	0.234404
$456$	0	0
$457$	11.0000	0.514558	0.257279	−	0.966337i	$-0.417174\pi$
$457$	11.0000	0.514558	0.257279	+	0.966337i	$0.417174\pi$
$458$	0	0
$459$	5.00000	0.233380
$460$	6.00000	0.279751
$461$	9.00000	0.419172	0.209586	−	0.977790i	$-0.432788\pi$
$461$	9.00000	0.419172	0.209586	+	0.977790i	$0.432788\pi$
$462$	0	0
$463$	13.0000	0.604161	0.302081	−	0.953282i	$-0.402319\pi$
$463$	13.0000	0.604161	0.302081	+	0.953282i	$0.402319\pi$
$464$	0	0
$465$	1.00000	0.0463739
$466$	0	0
$467$	37.0000	1.71216	0.856078	−	0.516847i	$-0.172894\pi$
$467$	37.0000	1.71216	0.856078	+	0.516847i	$0.172894\pi$
$468$	−2.00000	−0.0924500
$469$	60.0000	2.77054
$470$	0	0
$471$	−20.0000	−0.921551
$472$	0	0
$473$	−2.00000	−0.0919601
$474$	0	0
$475$	−6.00000	−0.275299
$476$	50.0000	2.29175
$477$	5.00000	0.228934
$478$	0	0
$479$	−15.0000	−0.685367	−0.342684	−	0.939451i	$-0.611336\pi$
$479$	−15.0000	−0.685367	−0.342684	+	0.939451i	$0.611336\pi$
$480$	0	0
$481$	1.00000	0.0455961
$482$	0	0
$483$	−15.0000	−0.682524
$484$	20.0000	0.909091
$485$	19.0000	0.862746
$486$	0	0
$487$	11.0000	0.498458	0.249229	−	0.968445i	$-0.419823\pi$
$487$	11.0000	0.498458	0.249229	+	0.968445i	$0.419823\pi$
$488$	0	0
$489$	−1.00000	−0.0452216
$490$	0	0
$491$	−24.0000	−1.08310	−0.541552	−	0.840667i	$-0.682163\pi$
$491$	−24.0000	−1.08310	−0.541552	+	0.840667i	$0.682163\pi$
$492$	−14.0000	−0.631169
$493$	0	0
$494$	0	0
$495$	−1.00000	−0.0449467
$496$	−4.00000	−0.179605
$497$	5.00000	0.224281
$498$	0	0
$499$	24.0000	1.07439	0.537194	−	0.843459i	$-0.319484\pi$
$499$	24.0000	1.07439	0.537194	+	0.843459i	$0.319484\pi$
$500$	2.00000	0.0894427
$501$	−20.0000	−0.893534
$502$	0	0
$503$	28.0000	1.24846	0.624229	−	0.781241i	$-0.285413\pi$
$503$	28.0000	1.24846	0.624229	+	0.781241i	$0.285413\pi$
$504$	0	0
$505$	−18.0000	−0.800989
$506$	0	0
$507$	1.00000	0.0444116
$508$	−24.0000	−1.06483
$509$	−25.0000	−1.10811	−0.554053	−	0.832482i	$-0.686919\pi$
$509$	−25.0000	−1.10811	−0.554053	+	0.832482i	$0.686919\pi$
$510$	0	0
$511$	70.0000	3.09662
$512$	0	0
$513$	−6.00000	−0.264906
$514$	0	0
$515$	12.0000	0.528783
$516$	4.00000	0.176090
$517$	4.00000	0.175920
$518$	0	0
$519$	−6.00000	−0.263371
$520$	0	0
$521$	14.0000	0.613351	0.306676	−	0.951814i	$-0.400783\pi$
$521$	14.0000	0.613351	0.306676	+	0.951814i	$0.400783\pi$
$522$	0	0
$523$	−34.0000	−1.48672	−0.743358	−	0.668894i	$-0.766768\pi$
$523$	−34.0000	−1.48672	−0.743358	+	0.668894i	$0.766768\pi$
$524$	12.0000	0.524222
$525$	−5.00000	−0.218218
$526$	0	0
$527$	−5.00000	−0.217803
$528$	4.00000	0.174078
$529$	−14.0000	−0.608696
$530$	0	0
$531$	−4.00000	−0.173585
$532$	−60.0000	−2.60133
$533$	7.00000	0.303204
$534$	0	0
$535$	−13.0000	−0.562039
$536$	0	0
$537$	−24.0000	−1.03568
$538$	0	0
$539$	18.0000	0.775315
$540$	2.00000	0.0860663
$541$	32.0000	1.37579	0.687894	−	0.725811i	$-0.258536\pi$
$541$	32.0000	1.37579	0.687894	+	0.725811i	$0.258536\pi$
$542$	0	0
$543$	7.00000	0.300399
$544$	0	0
$545$	−10.0000	−0.428353
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	24.0000	1.02523
$549$	3.00000	0.128037
$550$	0	0
$551$	0	0
$552$	0	0
$553$	25.0000	1.06311
$554$	0	0
$555$	−1.00000	−0.0424476
$556$	46.0000	1.95083
$557$	20.0000	0.847427	0.423714	−	0.905796i	$-0.360726\pi$
$557$	20.0000	0.847427	0.423714	+	0.905796i	$0.360726\pi$
$558$	0	0
$559$	−2.00000	−0.0845910
$560$	20.0000	0.845154
$561$	5.00000	0.211100
$562$	0	0
$563$	−31.0000	−1.30649	−0.653247	−	0.757145i	$-0.726594\pi$
$563$	−31.0000	−1.30649	−0.653247	+	0.757145i	$0.726594\pi$
$564$	−8.00000	−0.336861
$565$	−6.00000	−0.252422
$566$	0	0
$567$	−5.00000	−0.209980
$568$	0	0
$569$	−26.0000	−1.08998	−0.544988	−	0.838444i	$-0.683466\pi$
$569$	−26.0000	−1.08998	−0.544988	+	0.838444i	$0.683466\pi$
$570$	0	0
$571$	−41.0000	−1.71580	−0.857898	−	0.513820i	$-0.828230\pi$
$571$	−41.0000	−1.71580	−0.857898	+	0.513820i	$0.828230\pi$
$572$	−2.00000	−0.0836242
$573$	−6.00000	−0.250654
$574$	0	0
$575$	3.00000	0.125109
$576$	−8.00000	−0.333333
$577$	−19.0000	−0.790980	−0.395490	−	0.918470i	$-0.629425\pi$
$577$	−19.0000	−0.790980	−0.395490	+	0.918470i	$0.629425\pi$
$578$	0	0
$579$	19.0000	0.789613
$580$	0	0
$581$	−70.0000	−2.90409
$582$	0	0
$583$	5.00000	0.207079
$584$	0	0
$585$	−1.00000	−0.0413449
$586$	0	0
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	−36.0000	−1.48461
$589$	6.00000	0.247226
$590$	0	0
$591$	−12.0000	−0.493614
$592$	4.00000	0.164399
$593$	−4.00000	−0.164260	−0.0821302	−	0.996622i	$-0.526172\pi$
$593$	−4.00000	−0.164260	−0.0821302	+	0.996622i	$0.526172\pi$
$594$	0	0
$595$	25.0000	1.02490
$596$	−42.0000	−1.72039
$597$	−8.00000	−0.327418
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	−23.0000	−0.938190	−0.469095	−	0.883148i	$-0.655420\pi$
$601$	−23.0000	−0.938190	−0.469095	+	0.883148i	$0.655420\pi$
$602$	0	0
$603$	−12.0000	−0.488678
$604$	0	0
$605$	10.0000	0.406558
$606$	0	0
$607$	−2.00000	−0.0811775	−0.0405887	−	0.999176i	$-0.512923\pi$
$607$	−2.00000	−0.0811775	−0.0405887	+	0.999176i	$0.512923\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.00000	0.161823
$612$	−10.0000	−0.404226
$613$	−5.00000	−0.201948	−0.100974	−	0.994889i	$-0.532196\pi$
$613$	−5.00000	−0.201948	−0.100974	+	0.994889i	$0.532196\pi$
$614$	0	0
$615$	−7.00000	−0.282267
$616$	0	0
$617$	−4.00000	−0.161034	−0.0805170	−	0.996753i	$-0.525657\pi$
$617$	−4.00000	−0.161034	−0.0805170	+	0.996753i	$0.525657\pi$
$618$	0	0
$619$	−30.0000	−1.20580	−0.602901	−	0.797816i	$-0.705989\pi$
$619$	−30.0000	−1.20580	−0.602901	+	0.797816i	$0.705989\pi$
$620$	−2.00000	−0.0803219
$621$	3.00000	0.120386
$622$	0	0
$623$	25.0000	1.00160
$624$	4.00000	0.160128
$625$	1.00000	0.0400000
$626$	0	0
$627$	−6.00000	−0.239617
$628$	40.0000	1.59617
$629$	5.00000	0.199363
$630$	0	0
$631$	4.00000	0.159237	0.0796187	−	0.996825i	$-0.474630\pi$
$631$	4.00000	0.159237	0.0796187	+	0.996825i	$0.474630\pi$
$632$	0	0
$633$	−20.0000	−0.794929
$634$	0	0
$635$	−12.0000	−0.476205
$636$	−10.0000	−0.396526
$637$	18.0000	0.713186
$638$	0	0
$639$	−1.00000	−0.0395594
$640$	0	0
$641$	6.00000	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$642$	0	0
$643$	−3.00000	−0.118308	−0.0591542	−	0.998249i	$-0.518840\pi$
$643$	−3.00000	−0.118308	−0.0591542	+	0.998249i	$0.518840\pi$
$644$	30.0000	1.18217
$645$	2.00000	0.0787499
$646$	0	0
$647$	−31.0000	−1.21874	−0.609368	−	0.792888i	$-0.708577\pi$
$647$	−31.0000	−1.21874	−0.609368	+	0.792888i	$0.708577\pi$
$648$	0	0
$649$	−4.00000	−0.157014
$650$	0	0
$651$	5.00000	0.195965
$652$	2.00000	0.0783260
$653$	−50.0000	−1.95665	−0.978326	−	0.207072i	$-0.933606\pi$
$653$	−50.0000	−1.95665	−0.978326	+	0.207072i	$0.933606\pi$
$654$	0	0
$655$	6.00000	0.234439
$656$	28.0000	1.09322
$657$	−14.0000	−0.546192
$658$	0	0
$659$	14.0000	0.545363	0.272681	−	0.962104i	$-0.412090\pi$
$659$	14.0000	0.545363	0.272681	+	0.962104i	$0.412090\pi$
$660$	2.00000	0.0778499
$661$	24.0000	0.933492	0.466746	−	0.884391i	$-0.345426\pi$
$661$	24.0000	0.933492	0.466746	+	0.884391i	$0.345426\pi$
$662$	0	0
$663$	5.00000	0.194184
$664$	0	0
$665$	−30.0000	−1.16335
$666$	0	0
$667$	0	0
$668$	40.0000	1.54765
$669$	8.00000	0.309298
$670$	0	0
$671$	3.00000	0.115814
$672$	0	0
$673$	44.0000	1.69608	0.848038	−	0.529936i	$-0.177784\pi$
$673$	44.0000	1.69608	0.848038	+	0.529936i	$0.177784\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	−2.00000	−0.0769231
$677$	−23.0000	−0.883962	−0.441981	−	0.897024i	$-0.645724\pi$
$677$	−23.0000	−0.883962	−0.441981	+	0.897024i	$0.645724\pi$
$678$	0	0
$679$	95.0000	3.64577
$680$	0	0
$681$	10.0000	0.383201
$682$	0	0
$683$	−32.0000	−1.22445	−0.612223	−	0.790685i	$-0.709725\pi$
$683$	−32.0000	−1.22445	−0.612223	+	0.790685i	$0.709725\pi$
$684$	12.0000	0.458831
$685$	12.0000	0.458496
$686$	0	0
$687$	6.00000	0.228914
$688$	−8.00000	−0.304997
$689$	5.00000	0.190485
$690$	0	0
$691$	−42.0000	−1.59776	−0.798878	−	0.601494i	$-0.794573\pi$
$691$	−42.0000	−1.59776	−0.798878	+	0.601494i	$0.794573\pi$
$692$	12.0000	0.456172
$693$	−5.00000	−0.189934
$694$	0	0
$695$	23.0000	0.872440
$696$	0	0
$697$	35.0000	1.32572
$698$	0	0
$699$	−19.0000	−0.718646
$700$	10.0000	0.377964
$701$	24.0000	0.906467	0.453234	−	0.891392i	$-0.350270\pi$
$701$	24.0000	0.906467	0.453234	+	0.891392i	$0.350270\pi$
$702$	0	0
$703$	−6.00000	−0.226294
$704$	−8.00000	−0.301511
$705$	−4.00000	−0.150649
$706$	0	0
$707$	−90.0000	−3.38480
$708$	8.00000	0.300658
$709$	34.0000	1.27690	0.638448	−	0.769665i	$-0.279577\pi$
$709$	34.0000	1.27690	0.638448	+	0.769665i	$0.279577\pi$
$710$	0	0
$711$	−5.00000	−0.187515
$712$	0	0
$713$	−3.00000	−0.112351
$714$	0	0
$715$	−1.00000	−0.0373979
$716$	48.0000	1.79384
$717$	−19.0000	−0.709568
$718$	0	0
$719$	46.0000	1.71551	0.857755	−	0.514058i	$-0.171858\pi$
$719$	46.0000	1.71551	0.857755	+	0.514058i	$0.171858\pi$
$720$	−4.00000	−0.149071
$721$	60.0000	2.23452
$722$	0	0
$723$	−26.0000	−0.966950
$724$	−14.0000	−0.520306
$725$	0	0
$726$	0	0
$727$	28.0000	1.03846	0.519231	−	0.854634i	$-0.326218\pi$
$727$	28.0000	1.03846	0.519231	+	0.854634i	$0.326218\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−10.0000	−0.369863
$732$	−6.00000	−0.221766
$733$	−5.00000	−0.184679	−0.0923396	−	0.995728i	$-0.529435\pi$
$733$	−5.00000	−0.184679	−0.0923396	+	0.995728i	$0.529435\pi$
$734$	0	0
$735$	−18.0000	−0.663940
$736$	0	0
$737$	−12.0000	−0.442026
$738$	0	0
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	2.00000	0.0735215
$741$	−6.00000	−0.220416
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	−21.0000	−0.769380
$746$	0	0
$747$	14.0000	0.512233
$748$	−10.0000	−0.365636
$749$	−65.0000	−2.37505
$750$	0	0
$751$	−17.0000	−0.620339	−0.310169	−	0.950681i	$-0.600386\pi$
$751$	−17.0000	−0.620339	−0.310169	+	0.950681i	$0.600386\pi$
$752$	16.0000	0.583460
$753$	28.0000	1.02038
$754$	0	0
$755$	0	0
$756$	10.0000	0.363696
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	0	0
$759$	3.00000	0.108893
$760$	0	0
$761$	−54.0000	−1.95750	−0.978749	−	0.205061i	$-0.934261\pi$
$761$	−54.0000	−1.95750	−0.978749	+	0.205061i	$0.934261\pi$
$762$	0	0
$763$	−50.0000	−1.81012
$764$	12.0000	0.434145
$765$	−5.00000	−0.180775
$766$	0	0
$767$	−4.00000	−0.144432
$768$	16.0000	0.577350
$769$	16.0000	0.576975	0.288487	−	0.957484i	$-0.406848\pi$
$769$	16.0000	0.576975	0.288487	+	0.957484i	$0.406848\pi$
$770$	0	0
$771$	30.0000	1.08042
$772$	−38.0000	−1.36765
$773$	32.0000	1.15096	0.575480	−	0.817816i	$-0.304815\pi$
$773$	32.0000	1.15096	0.575480	+	0.817816i	$0.304815\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	0	0
$777$	−5.00000	−0.179374
$778$	0	0
$779$	−42.0000	−1.50481
$780$	2.00000	0.0716115
$781$	−1.00000	−0.0357828
$782$	0	0
$783$	0	0
$784$	72.0000	2.57143
$785$	20.0000	0.713831
$786$	0	0
$787$	−20.0000	−0.712923	−0.356462	−	0.934310i	$-0.616017\pi$
$787$	−20.0000	−0.712923	−0.356462	+	0.934310i	$0.616017\pi$
$788$	24.0000	0.854965
$789$	16.0000	0.569615
$790$	0	0
$791$	−30.0000	−1.06668
$792$	0	0
$793$	3.00000	0.106533
$794$	0	0
$795$	−5.00000	−0.177332
$796$	16.0000	0.567105
$797$	−13.0000	−0.460484	−0.230242	−	0.973133i	$-0.573952\pi$
$797$	−13.0000	−0.460484	−0.230242	+	0.973133i	$0.573952\pi$
$798$	0	0
$799$	20.0000	0.707549
$800$	0	0
$801$	−5.00000	−0.176666
$802$	0	0
$803$	−14.0000	−0.494049
$804$	24.0000	0.846415
$805$	15.0000	0.528681
$806$	0	0
$807$	−24.0000	−0.844840
$808$	0	0
$809$	−12.0000	−0.421898	−0.210949	−	0.977497i	$-0.567655\pi$
$809$	−12.0000	−0.421898	−0.210949	+	0.977497i	$0.567655\pi$
$810$	0	0
$811$	−18.0000	−0.632065	−0.316033	−	0.948748i	$-0.602351\pi$
$811$	−18.0000	−0.632065	−0.316033	+	0.948748i	$0.602351\pi$
$812$	0	0
$813$	−24.0000	−0.841717
$814$	0	0
$815$	1.00000	0.0350285
$816$	20.0000	0.700140
$817$	12.0000	0.419827
$818$	0	0
$819$	−5.00000	−0.174714
$820$	14.0000	0.488901
$821$	1.00000	0.0349002	0.0174501	−	0.999848i	$-0.494445\pi$
$821$	1.00000	0.0349002	0.0174501	+	0.999848i	$0.494445\pi$
$822$	0	0
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	0	0
$825$	1.00000	0.0348155
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	−6.00000	−0.208514
$829$	−26.0000	−0.903017	−0.451509	−	0.892267i	$-0.649114\pi$
$829$	−26.0000	−0.903017	−0.451509	+	0.892267i	$0.649114\pi$
$830$	0	0
$831$	−16.0000	−0.555034
$832$	−8.00000	−0.277350
$833$	90.0000	3.11832
$834$	0	0
$835$	20.0000	0.692129
$836$	12.0000	0.415029
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	−25.0000	−0.863096	−0.431548	−	0.902090i	$-0.642032\pi$
$839$	−25.0000	−0.863096	−0.431548	+	0.902090i	$0.642032\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	−18.0000	−0.619953
$844$	40.0000	1.37686
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	50.0000	1.71802
$848$	20.0000	0.686803
$849$	4.00000	0.137280
$850$	0	0
$851$	3.00000	0.102839
$852$	2.00000	0.0685189
$853$	−7.00000	−0.239675	−0.119838	−	0.992793i	$-0.538237\pi$
$853$	−7.00000	−0.239675	−0.119838	+	0.992793i	$0.538237\pi$
$854$	0	0
$855$	6.00000	0.205196
$856$	0	0
$857$	21.0000	0.717346	0.358673	−	0.933463i	$-0.383229\pi$
$857$	21.0000	0.717346	0.358673	+	0.933463i	$0.383229\pi$
$858$	0	0
$859$	41.0000	1.39890	0.699451	−	0.714681i	$-0.253428\pi$
$859$	41.0000	1.39890	0.699451	+	0.714681i	$0.253428\pi$
$860$	−4.00000	−0.136399
$861$	−35.0000	−1.19280
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	6.00000	0.204006
$866$	0	0
$867$	8.00000	0.271694
$868$	−10.0000	−0.339422
$869$	−5.00000	−0.169613
$870$	0	0
$871$	−12.0000	−0.406604
$872$	0	0
$873$	−19.0000	−0.643053
$874$	0	0
$875$	5.00000	0.169031
$876$	28.0000	0.946032
$877$	46.0000	1.55331	0.776655	−	0.629926i	$-0.216915\pi$
$877$	46.0000	1.55331	0.776655	+	0.629926i	$0.216915\pi$
$878$	0	0
$879$	−4.00000	−0.134917
$880$	−4.00000	−0.134840
$881$	−48.0000	−1.61716	−0.808581	−	0.588386i	$-0.799764\pi$
$881$	−48.0000	−1.61716	−0.808581	+	0.588386i	$0.799764\pi$
$882$	0	0
$883$	−46.0000	−1.54802	−0.774012	−	0.633171i	$-0.781753\pi$
$883$	−46.0000	−1.54802	−0.774012	+	0.633171i	$0.781753\pi$
$884$	−10.0000	−0.336336
$885$	4.00000	0.134459
$886$	0	0
$887$	3.00000	0.100730	0.0503651	−	0.998731i	$-0.483962\pi$
$887$	3.00000	0.100730	0.0503651	+	0.998731i	$0.483962\pi$
$888$	0	0
$889$	−60.0000	−2.01234
$890$	0	0
$891$	1.00000	0.0335013
$892$	−16.0000	−0.535720
$893$	−24.0000	−0.803129
$894$	0	0
$895$	24.0000	0.802232
$896$	0	0
$897$	3.00000	0.100167
$898$	0	0
$899$	0	0
$900$	−2.00000	−0.0666667
$901$	25.0000	0.832871
$902$	0	0
$903$	10.0000	0.332779
$904$	0	0
$905$	−7.00000	−0.232688
$906$	0	0
$907$	10.0000	0.332045	0.166022	−	0.986122i	$-0.446908\pi$
$907$	10.0000	0.332045	0.166022	+	0.986122i	$0.446908\pi$
$908$	−20.0000	−0.663723
$909$	18.0000	0.597022
$910$	0	0
$911$	−38.0000	−1.25900	−0.629498	−	0.777002i	$-0.716739\pi$
$911$	−38.0000	−1.25900	−0.629498	+	0.777002i	$0.716739\pi$
$912$	−24.0000	−0.794719
$913$	14.0000	0.463332
$914$	0	0
$915$	−3.00000	−0.0991769
$916$	−12.0000	−0.396491
$917$	30.0000	0.990687
$918$	0	0
$919$	−7.00000	−0.230909	−0.115454	−	0.993313i	$-0.536832\pi$
$919$	−7.00000	−0.230909	−0.115454	+	0.993313i	$0.536832\pi$
$920$	0	0
$921$	−25.0000	−0.823778
$922$	0	0
$923$	−1.00000	−0.0329154
$924$	10.0000	0.328976
$925$	1.00000	0.0328798
$926$	0	0
$927$	−12.0000	−0.394132
$928$	0	0
$929$	43.0000	1.41078	0.705392	−	0.708817i	$-0.250771\pi$
$929$	43.0000	1.41078	0.705392	+	0.708817i	$0.250771\pi$
$930$	0	0
$931$	−108.000	−3.53956
$932$	38.0000	1.24473
$933$	8.00000	0.261908
$934$	0	0
$935$	−5.00000	−0.163517
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	14.0000	0.456873
$940$	8.00000	0.260931
$941$	19.0000	0.619382	0.309691	−	0.950837i	$-0.399774\pi$
$941$	19.0000	0.619382	0.309691	+	0.950837i	$0.399774\pi$
$942$	0	0
$943$	21.0000	0.683854
$944$	−16.0000	−0.520756
$945$	5.00000	0.162650
$946$	0	0
$947$	10.0000	0.324956	0.162478	−	0.986712i	$-0.448051\pi$
$947$	10.0000	0.324956	0.162478	+	0.986712i	$0.448051\pi$
$948$	10.0000	0.324785
$949$	−14.0000	−0.454459
$950$	0	0
$951$	−8.00000	−0.259418
$952$	0	0
$953$	21.0000	0.680257	0.340128	−	0.940379i	$-0.389529\pi$
$953$	21.0000	0.680257	0.340128	+	0.940379i	$0.389529\pi$
$954$	0	0
$955$	6.00000	0.194155
$956$	38.0000	1.22901
$957$	0	0
$958$	0	0
$959$	60.0000	1.93750
$960$	8.00000	0.258199
$961$	1.00000	0.0322581
$962$	0	0
$963$	13.0000	0.418919
$964$	52.0000	1.67481
$965$	−19.0000	−0.611632
$966$	0	0
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	0	0
$969$	−30.0000	−0.963739
$970$	0	0
$971$	−52.0000	−1.66876	−0.834380	−	0.551190i	$-0.814174\pi$
$971$	−52.0000	−1.66876	−0.834380	+	0.551190i	$0.814174\pi$
$972$	−2.00000	−0.0641500
$973$	115.000	3.68673
$974$	0	0
$975$	1.00000	0.0320256
$976$	12.0000	0.384111
$977$	−24.0000	−0.767828	−0.383914	−	0.923369i	$-0.625424\pi$
$977$	−24.0000	−0.767828	−0.383914	+	0.923369i	$0.625424\pi$
$978$	0	0
$979$	−5.00000	−0.159801
$980$	36.0000	1.14998
$981$	10.0000	0.319275
$982$	0	0
$983$	6.00000	0.191370	0.0956851	−	0.995412i	$-0.469496\pi$
$983$	6.00000	0.191370	0.0956851	+	0.995412i	$0.469496\pi$
$984$	0	0
$985$	12.0000	0.382352
$986$	0	0
$987$	−20.0000	−0.636607
$988$	12.0000	0.381771
$989$	−6.00000	−0.190789
$990$	0	0
$991$	45.0000	1.42947	0.714736	−	0.699394i	$-0.246547\pi$
$991$	45.0000	1.42947	0.714736	+	0.699394i	$0.246547\pi$
$992$	0	0
$993$	−8.00000	−0.253872
$994$	0	0
$995$	8.00000	0.253617
$996$	−28.0000	−0.887214
$997$	62.0000	1.96356	0.981780	−	0.190022i	$-0.0608559\pi$
$997$	62.0000	1.96356	0.981780	+	0.190022i	$0.0608559\pi$
$998$	0	0
$999$	1.00000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6045.2.a.e.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6045.2.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:	\( N \)	\(=\)	\( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6045.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more