LMFDB - Embedded newform 6040.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6040,2,Mod(1,6040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6040, 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("6040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6040 = 2^{3} \cdot 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6040.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:	$48.2296428209$
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	$\chi$	$=$	6040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} +1.00000 q^{5} -4.00000 q^{7} -2.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} +1.00000 q^{5} -4.00000 q^{7} -2.00000 q^{9} -5.00000 q^{11} -7.00000 q^{13} -1.00000 q^{15} -2.00000 q^{17} +5.00000 q^{19} +4.00000 q^{21} -3.00000 q^{23} +1.00000 q^{25} +5.00000 q^{27} -3.00000 q^{29} -9.00000 q^{31} +5.00000 q^{33} -4.00000 q^{35} -2.00000 q^{37} +7.00000 q^{39} -4.00000 q^{41} -6.00000 q^{43} -2.00000 q^{45} -6.00000 q^{47} +9.00000 q^{49} +2.00000 q^{51} -2.00000 q^{53} -5.00000 q^{55} -5.00000 q^{57} -3.00000 q^{59} +2.00000 q^{61} +8.00000 q^{63} -7.00000 q^{65} -9.00000 q^{67} +3.00000 q^{69} -10.0000 q^{71} -7.00000 q^{73} -1.00000 q^{75} +20.0000 q^{77} -4.00000 q^{79} +1.00000 q^{81} +11.0000 q^{83} -2.00000 q^{85} +3.00000 q^{87} +8.00000 q^{89} +28.0000 q^{91} +9.00000 q^{93} +5.00000 q^{95} +12.0000 q^{97} +10.0000 q^{99} +O(q^{100})$

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
$2$	0	0
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	0	0
$13$	−7.00000	−1.94145	−0.970725	−	0.240192i	$-0.922790\pi$
$13$	−7.00000	−1.94145	−0.970725	+	0.240192i	$0.922790\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	0	0
$21$	4.00000	0.872872
$22$	0	0
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.00000	0.962250
$28$	0	0
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	−9.00000	−1.61645	−0.808224	−	0.588875i	$-0.799571\pi$
$31$	−9.00000	−1.61645	−0.808224	+	0.588875i	$0.799571\pi$
$32$	0	0
$33$	5.00000	0.870388
$34$	0	0
$35$	−4.00000	−0.676123
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	7.00000	1.12090
$40$	0	0
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	2.00000	0.280056
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	−5.00000	−0.674200
$56$	0	0
$57$	−5.00000	−0.662266
$58$	0	0
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	8.00000	1.00791
$64$	0	0
$65$	−7.00000	−0.868243
$66$	0	0
$67$	−9.00000	−1.09952	−0.549762	−	0.835321i	$-0.685282\pi$
$67$	−9.00000	−1.09952	−0.549762	+	0.835321i	$0.685282\pi$
$68$	0	0
$69$	3.00000	0.361158
$70$	0	0
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	0	0
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	20.0000	2.27921
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.0000	1.20741	0.603703	−	0.797209i	$-0.293691\pi$
$83$	11.0000	1.20741	0.603703	+	0.797209i	$0.293691\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	0	0
$87$	3.00000	0.321634
$88$	0	0
$89$	8.00000	0.847998	0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000	0.847998	0.423999	+	0.905663i	$0.360626\pi$
$90$	0	0
$91$	28.0000	2.93520
$92$	0	0
$93$	9.00000	0.933257
$94$	0	0
$95$	5.00000	0.512989
$96$	0	0
$97$	12.0000	1.21842	0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000	1.21842	0.609208	+	0.793011i	$0.291488\pi$
$98$	0	0
$99$	10.0000	1.00504
$100$	0	0
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	0	0
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	0	0
$105$	4.00000	0.390360
$106$	0	0
$107$	−17.0000	−1.64345	−0.821726	−	0.569883i	$-0.806989\pi$
$107$	−17.0000	−1.64345	−0.821726	+	0.569883i	$0.806989\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	0	0
$115$	−3.00000	−0.279751
$116$	0	0
$117$	14.0000	1.29430
$118$	0	0
$119$	8.00000	0.733359
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	4.00000	0.360668
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	0	0
$129$	6.00000	0.528271
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	−20.0000	−1.73422
$134$	0	0
$135$	5.00000	0.430331
$136$	0	0
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	0	0
$143$	35.0000	2.92685
$144$	0	0
$145$	−3.00000	−0.249136
$146$	0	0
$147$	−9.00000	−0.742307
$148$	0	0
$149$	4.00000	0.327693	0.163846	−	0.986486i	$-0.447610\pi$
$149$	4.00000	0.327693	0.163846	+	0.986486i	$0.447610\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	0	0
$153$	4.00000	0.323381
$154$	0	0
$155$	−9.00000	−0.722897
$156$	0	0
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	0	0
$159$	2.00000	0.158610
$160$	0	0
$161$	12.0000	0.945732
$162$	0	0
$163$	−9.00000	−0.704934	−0.352467	−	0.935824i	$-0.614657\pi$
$163$	−9.00000	−0.704934	−0.352467	+	0.935824i	$0.614657\pi$
$164$	0	0
$165$	5.00000	0.389249
$166$	0	0
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	36.0000	2.76923
$170$	0	0
$171$	−10.0000	−0.764719
$172$	0	0
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	0	0
$177$	3.00000	0.225494
$178$	0	0
$179$	−18.0000	−1.34538	−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−18.0000	−1.34538	−0.672692	+	0.739923i	$0.734862\pi$
$180$	0	0
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	0	0
$185$	−2.00000	−0.147043
$186$	0	0
$187$	10.0000	0.731272
$188$	0	0
$189$	−20.0000	−1.45479
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	0	0
$195$	7.00000	0.501280
$196$	0	0
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	0	0
$201$	9.00000	0.634811
$202$	0	0
$203$	12.0000	0.842235
$204$	0	0
$205$	−4.00000	−0.279372
$206$	0	0
$207$	6.00000	0.417029
$208$	0	0
$209$	−25.0000	−1.72929
$210$	0	0
$211$	26.0000	1.78991	0.894957	−	0.446153i	$-0.147206\pi$
$211$	26.0000	1.78991	0.894957	+	0.446153i	$0.147206\pi$
$212$	0	0
$213$	10.0000	0.685189
$214$	0	0
$215$	−6.00000	−0.409197
$216$	0	0
$217$	36.0000	2.44384
$218$	0	0
$219$	7.00000	0.473016
$220$	0	0
$221$	14.0000	0.941742
$222$	0	0
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	0	0
$225$	−2.00000	−0.133333
$226$	0	0
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	−17.0000	−1.12339	−0.561696	−	0.827344i	$-0.689851\pi$
$229$	−17.0000	−1.12339	−0.561696	+	0.827344i	$0.689851\pi$
$230$	0	0
$231$	−20.0000	−1.31590
$232$	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
$235$	−6.00000	−0.391397
$236$	0	0
$237$	4.00000	0.259828
$238$	0	0
$239$	−9.00000	−0.582162	−0.291081	−	0.956698i	$-0.594015\pi$
$239$	−9.00000	−0.582162	−0.291081	+	0.956698i	$0.594015\pi$
$240$	0	0
$241$	−11.0000	−0.708572	−0.354286	−	0.935137i	$-0.615276\pi$
$241$	−11.0000	−0.708572	−0.354286	+	0.935137i	$0.615276\pi$
$242$	0	0
$243$	−16.0000	−1.02640
$244$	0	0
$245$	9.00000	0.574989
$246$	0	0
$247$	−35.0000	−2.22700
$248$	0	0
$249$	−11.0000	−0.697097
$250$	0	0
$251$	−17.0000	−1.07303	−0.536515	−	0.843891i	$-0.680260\pi$
$251$	−17.0000	−1.07303	−0.536515	+	0.843891i	$0.680260\pi$
$252$	0	0
$253$	15.0000	0.943042
$254$	0	0
$255$	2.00000	0.125245
$256$	0	0
$257$	−13.0000	−0.810918	−0.405459	−	0.914113i	$-0.632888\pi$
$257$	−13.0000	−0.810918	−0.405459	+	0.914113i	$0.632888\pi$
$258$	0	0
$259$	8.00000	0.497096
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	−27.0000	−1.66489	−0.832446	−	0.554107i	$-0.813060\pi$
$263$	−27.0000	−1.66489	−0.832446	+	0.554107i	$0.813060\pi$
$264$	0	0
$265$	−2.00000	−0.122859
$266$	0	0
$267$	−8.00000	−0.489592
$268$	0	0
$269$	−9.00000	−0.548740	−0.274370	−	0.961624i	$-0.588469\pi$
$269$	−9.00000	−0.548740	−0.274370	+	0.961624i	$0.588469\pi$
$270$	0	0
$271$	10.0000	0.607457	0.303728	−	0.952759i	$-0.401768\pi$
$271$	10.0000	0.607457	0.303728	+	0.952759i	$0.401768\pi$
$272$	0	0
$273$	−28.0000	−1.69464
$274$	0	0
$275$	−5.00000	−0.301511
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	0	0
$279$	18.0000	1.07763
$280$	0	0
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	0	0
$283$	9.00000	0.534994	0.267497	−	0.963559i	$-0.413803\pi$
$283$	9.00000	0.534994	0.267497	+	0.963559i	$0.413803\pi$
$284$	0	0
$285$	−5.00000	−0.296174
$286$	0	0
$287$	16.0000	0.944450
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−12.0000	−0.703452
$292$	0	0
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	0	0
$295$	−3.00000	−0.174667
$296$	0	0
$297$	−25.0000	−1.45065
$298$	0	0
$299$	21.0000	1.21446
$300$	0	0
$301$	24.0000	1.38334
$302$	0	0
$303$	−8.00000	−0.459588
$304$	0	0
$305$	2.00000	0.114520
$306$	0	0
$307$	10.0000	0.570730	0.285365	−	0.958419i	$-0.407885\pi$
$307$	10.0000	0.570730	0.285365	+	0.958419i	$0.407885\pi$
$308$	0	0
$309$	14.0000	0.796432
$310$	0	0
$311$	−1.00000	−0.0567048	−0.0283524	−	0.999598i	$-0.509026\pi$
$311$	−1.00000	−0.0567048	−0.0283524	+	0.999598i	$0.509026\pi$
$312$	0	0
$313$	−16.0000	−0.904373	−0.452187	−	0.891923i	$-0.649356\pi$
$313$	−16.0000	−0.904373	−0.452187	+	0.891923i	$0.649356\pi$
$314$	0	0
$315$	8.00000	0.450749
$316$	0	0
$317$	5.00000	0.280828	0.140414	−	0.990093i	$-0.455157\pi$
$317$	5.00000	0.280828	0.140414	+	0.990093i	$0.455157\pi$
$318$	0	0
$319$	15.0000	0.839839
$320$	0	0
$321$	17.0000	0.948847
$322$	0	0
$323$	−10.0000	−0.556415
$324$	0	0
$325$	−7.00000	−0.388290
$326$	0	0
$327$	2.00000	0.110600
$328$	0	0
$329$	24.0000	1.32316
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	0	0
$333$	4.00000	0.219199
$334$	0	0
$335$	−9.00000	−0.491723
$336$	0	0
$337$	−5.00000	−0.272367	−0.136184	−	0.990684i	$-0.543484\pi$
$337$	−5.00000	−0.272367	−0.136184	+	0.990684i	$0.543484\pi$
$338$	0	0
$339$	−10.0000	−0.543125
$340$	0	0
$341$	45.0000	2.43689
$342$	0	0
$343$	−8.00000	−0.431959
$344$	0	0
$345$	3.00000	0.161515
$346$	0	0
$347$	30.0000	1.61048	0.805242	−	0.592946i	$-0.202035\pi$
$347$	30.0000	1.61048	0.805242	+	0.592946i	$0.202035\pi$
$348$	0	0
$349$	7.00000	0.374701	0.187351	−	0.982293i	$-0.440010\pi$
$349$	7.00000	0.374701	0.187351	+	0.982293i	$0.440010\pi$
$350$	0	0
$351$	−35.0000	−1.86816
$352$	0	0
$353$	5.00000	0.266123	0.133062	−	0.991108i	$-0.457519\pi$
$353$	5.00000	0.266123	0.133062	+	0.991108i	$0.457519\pi$
$354$	0	0
$355$	−10.0000	−0.530745
$356$	0	0
$357$	−8.00000	−0.423405
$358$	0	0
$359$	−2.00000	−0.105556	−0.0527780	−	0.998606i	$-0.516808\pi$
$359$	−2.00000	−0.105556	−0.0527780	+	0.998606i	$0.516808\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	0	0
$363$	−14.0000	−0.734809
$364$	0	0
$365$	−7.00000	−0.366397
$366$	0	0
$367$	28.0000	1.46159	0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000	1.46159	0.730794	+	0.682598i	$0.239150\pi$
$368$	0	0
$369$	8.00000	0.416463
$370$	0	0
$371$	8.00000	0.415339
$372$	0	0
$373$	−27.0000	−1.39801	−0.699004	−	0.715118i	$-0.746373\pi$
$373$	−27.0000	−1.39801	−0.699004	+	0.715118i	$0.746373\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	21.0000	1.08156
$378$	0	0
$379$	−34.0000	−1.74646	−0.873231	−	0.487306i	$-0.837980\pi$
$379$	−34.0000	−1.74646	−0.873231	+	0.487306i	$0.837980\pi$
$380$	0	0
$381$	16.0000	0.819705
$382$	0	0
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	0	0
$385$	20.0000	1.01929
$386$	0	0
$387$	12.0000	0.609994
$388$	0	0
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	6.00000	0.303433
$392$	0	0
$393$	4.00000	0.201773
$394$	0	0
$395$	−4.00000	−0.201262
$396$	0	0
$397$	−36.0000	−1.80679	−0.903394	−	0.428811i	$-0.858933\pi$
$397$	−36.0000	−1.80679	−0.903394	+	0.428811i	$0.858933\pi$
$398$	0	0
$399$	20.0000	1.00125
$400$	0	0
$401$	−14.0000	−0.699127	−0.349563	−	0.936913i	$-0.613670\pi$
$401$	−14.0000	−0.699127	−0.349563	+	0.936913i	$0.613670\pi$
$402$	0	0
$403$	63.0000	3.13825
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	10.0000	0.495682
$408$	0	0
$409$	−20.0000	−0.988936	−0.494468	−	0.869196i	$-0.664637\pi$
$409$	−20.0000	−0.988936	−0.494468	+	0.869196i	$0.664637\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	0	0
$413$	12.0000	0.590481
$414$	0	0
$415$	11.0000	0.539969
$416$	0	0
$417$	−12.0000	−0.587643
$418$	0	0
$419$	6.00000	0.293119	0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000	0.293119	0.146560	+	0.989202i	$0.453180\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	0	0
$423$	12.0000	0.583460
$424$	0	0
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	−8.00000	−0.387147
$428$	0	0
$429$	−35.0000	−1.68982
$430$	0	0
$431$	34.0000	1.63772	0.818861	−	0.573992i	$-0.194606\pi$
$431$	34.0000	1.63772	0.818861	+	0.573992i	$0.194606\pi$
$432$	0	0
$433$	34.0000	1.63394	0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000	1.63394	0.816968	+	0.576683i	$0.195653\pi$
$434$	0	0
$435$	3.00000	0.143839
$436$	0	0
$437$	−15.0000	−0.717547
$438$	0	0
$439$	21.0000	1.00228	0.501138	−	0.865368i	$-0.332915\pi$
$439$	21.0000	1.00228	0.501138	+	0.865368i	$0.332915\pi$
$440$	0	0
$441$	−18.0000	−0.857143
$442$	0	0
$443$	−20.0000	−0.950229	−0.475114	−	0.879924i	$-0.657593\pi$
$443$	−20.0000	−0.950229	−0.475114	+	0.879924i	$0.657593\pi$
$444$	0	0
$445$	8.00000	0.379236
$446$	0	0
$447$	−4.00000	−0.189194
$448$	0	0
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	20.0000	0.941763
$452$	0	0
$453$	−1.00000	−0.0469841
$454$	0	0
$455$	28.0000	1.31266
$456$	0	0
$457$	−32.0000	−1.49690	−0.748448	−	0.663193i	$-0.769201\pi$
$457$	−32.0000	−1.49690	−0.748448	+	0.663193i	$0.769201\pi$
$458$	0	0
$459$	−10.0000	−0.466760
$460$	0	0
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	0	0
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	0	0
$465$	9.00000	0.417365
$466$	0	0
$467$	−36.0000	−1.66588	−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000	−1.66588	−0.832941	+	0.553362i	$0.813345\pi$
$468$	0	0
$469$	36.0000	1.66233
$470$	0	0
$471$	13.0000	0.599008
$472$	0	0
$473$	30.0000	1.37940
$474$	0	0
$475$	5.00000	0.229416
$476$	0	0
$477$	4.00000	0.183147
$478$	0	0
$479$	18.0000	0.822441	0.411220	−	0.911536i	$-0.365103\pi$
$479$	18.0000	0.822441	0.411220	+	0.911536i	$0.365103\pi$
$480$	0	0
$481$	14.0000	0.638345
$482$	0	0
$483$	−12.0000	−0.546019
$484$	0	0
$485$	12.0000	0.544892
$486$	0	0
$487$	26.0000	1.17817	0.589086	−	0.808070i	$-0.299488\pi$
$487$	26.0000	1.17817	0.589086	+	0.808070i	$0.299488\pi$
$488$	0	0
$489$	9.00000	0.406994
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	6.00000	0.270226
$494$	0	0
$495$	10.0000	0.449467
$496$	0	0
$497$	40.0000	1.79425
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	0	0
$503$	−42.0000	−1.87269	−0.936344	−	0.351085i	$-0.885813\pi$
$503$	−42.0000	−1.87269	−0.936344	+	0.351085i	$0.885813\pi$
$504$	0	0
$505$	8.00000	0.355995
$506$	0	0
$507$	−36.0000	−1.59882
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	28.0000	1.23865
$512$	0	0
$513$	25.0000	1.10378
$514$	0	0
$515$	−14.0000	−0.616914
$516$	0	0
$517$	30.0000	1.31940
$518$	0	0
$519$	−6.00000	−0.263371
$520$	0	0
$521$	17.0000	0.744784	0.372392	−	0.928076i	$-0.378538\pi$
$521$	17.0000	0.744784	0.372392	+	0.928076i	$0.378538\pi$
$522$	0	0
$523$	−3.00000	−0.131181	−0.0655904	−	0.997847i	$-0.520893\pi$
$523$	−3.00000	−0.131181	−0.0655904	+	0.997847i	$0.520893\pi$
$524$	0	0
$525$	4.00000	0.174574
$526$	0	0
$527$	18.0000	0.784092
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	6.00000	0.260378
$532$	0	0
$533$	28.0000	1.21281
$534$	0	0
$535$	−17.0000	−0.734974
$536$	0	0
$537$	18.0000	0.776757
$538$	0	0
$539$	−45.0000	−1.93829
$540$	0	0
$541$	−31.0000	−1.33279	−0.666397	−	0.745597i	$-0.732164\pi$
$541$	−31.0000	−1.33279	−0.666397	+	0.745597i	$0.732164\pi$
$542$	0	0
$543$	−22.0000	−0.944110
$544$	0	0
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	2.00000	0.0855138	0.0427569	−	0.999086i	$-0.486386\pi$
$547$	2.00000	0.0855138	0.0427569	+	0.999086i	$0.486386\pi$
$548$	0	0
$549$	−4.00000	−0.170716
$550$	0	0
$551$	−15.0000	−0.639021
$552$	0	0
$553$	16.0000	0.680389
$554$	0	0
$555$	2.00000	0.0848953
$556$	0	0
$557$	1.00000	0.0423714	0.0211857	−	0.999776i	$-0.493256\pi$
$557$	1.00000	0.0423714	0.0211857	+	0.999776i	$0.493256\pi$
$558$	0	0
$559$	42.0000	1.77641
$560$	0	0
$561$	−10.0000	−0.422200
$562$	0	0
$563$	32.0000	1.34864	0.674320	−	0.738440i	$-0.264437\pi$
$563$	32.0000	1.34864	0.674320	+	0.738440i	$0.264437\pi$
$564$	0	0
$565$	10.0000	0.420703
$566$	0	0
$567$	−4.00000	−0.167984
$568$	0	0
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	0	0
$571$	−43.0000	−1.79949	−0.899747	−	0.436412i	$-0.856249\pi$
$571$	−43.0000	−1.79949	−0.899747	+	0.436412i	$0.856249\pi$
$572$	0	0
$573$	−8.00000	−0.334205
$574$	0	0
$575$	−3.00000	−0.125109
$576$	0	0
$577$	−24.0000	−0.999133	−0.499567	−	0.866276i	$-0.666507\pi$
$577$	−24.0000	−0.999133	−0.499567	+	0.866276i	$0.666507\pi$
$578$	0	0
$579$	−14.0000	−0.581820
$580$	0	0
$581$	−44.0000	−1.82543
$582$	0	0
$583$	10.0000	0.414158
$584$	0	0
$585$	14.0000	0.578829
$586$	0	0
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	−45.0000	−1.85419
$590$	0	0
$591$	6.00000	0.246807
$592$	0	0
$593$	−9.00000	−0.369586	−0.184793	−	0.982777i	$-0.559161\pi$
$593$	−9.00000	−0.369586	−0.184793	+	0.982777i	$0.559161\pi$
$594$	0	0
$595$	8.00000	0.327968
$596$	0	0
$597$	−2.00000	−0.0818546
$598$	0	0
$599$	8.00000	0.326871	0.163436	−	0.986554i	$-0.447742\pi$
$599$	8.00000	0.326871	0.163436	+	0.986554i	$0.447742\pi$
$600$	0	0
$601$	25.0000	1.01977	0.509886	−	0.860242i	$-0.329688\pi$
$601$	25.0000	1.01977	0.509886	+	0.860242i	$0.329688\pi$
$602$	0	0
$603$	18.0000	0.733017
$604$	0	0
$605$	14.0000	0.569181
$606$	0	0
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	0	0
$609$	−12.0000	−0.486265
$610$	0	0
$611$	42.0000	1.69914
$612$	0	0
$613$	6.00000	0.242338	0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000	0.242338	0.121169	+	0.992632i	$0.461336\pi$
$614$	0	0
$615$	4.00000	0.161296
$616$	0	0
$617$	−3.00000	−0.120775	−0.0603877	−	0.998175i	$-0.519234\pi$
$617$	−3.00000	−0.120775	−0.0603877	+	0.998175i	$0.519234\pi$
$618$	0	0
$619$	−26.0000	−1.04503	−0.522514	−	0.852631i	$-0.675006\pi$
$619$	−26.0000	−1.04503	−0.522514	+	0.852631i	$0.675006\pi$
$620$	0	0
$621$	−15.0000	−0.601929
$622$	0	0
$623$	−32.0000	−1.28205
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	25.0000	0.998404
$628$	0	0
$629$	4.00000	0.159490
$630$	0	0
$631$	2.00000	0.0796187	0.0398094	−	0.999207i	$-0.487325\pi$
$631$	2.00000	0.0796187	0.0398094	+	0.999207i	$0.487325\pi$
$632$	0	0
$633$	−26.0000	−1.03341
$634$	0	0
$635$	−16.0000	−0.634941
$636$	0	0
$637$	−63.0000	−2.49615
$638$	0	0
$639$	20.0000	0.791188
$640$	0	0
$641$	−3.00000	−0.118493	−0.0592464	−	0.998243i	$-0.518870\pi$
$641$	−3.00000	−0.118493	−0.0592464	+	0.998243i	$0.518870\pi$
$642$	0	0
$643$	44.0000	1.73519	0.867595	−	0.497271i	$-0.165665\pi$
$643$	44.0000	1.73519	0.867595	+	0.497271i	$0.165665\pi$
$644$	0	0
$645$	6.00000	0.236250
$646$	0	0
$647$	−38.0000	−1.49393	−0.746967	−	0.664861i	$-0.768491\pi$
$647$	−38.0000	−1.49393	−0.746967	+	0.664861i	$0.768491\pi$
$648$	0	0
$649$	15.0000	0.588802
$650$	0	0
$651$	−36.0000	−1.41095
$652$	0	0
$653$	24.0000	0.939193	0.469596	−	0.882881i	$-0.344399\pi$
$653$	24.0000	0.939193	0.469596	+	0.882881i	$0.344399\pi$
$654$	0	0
$655$	−4.00000	−0.156293
$656$	0	0
$657$	14.0000	0.546192
$658$	0	0
$659$	19.0000	0.740135	0.370067	−	0.929005i	$-0.379335\pi$
$659$	19.0000	0.740135	0.370067	+	0.929005i	$0.379335\pi$
$660$	0	0
$661$	40.0000	1.55582	0.777910	−	0.628376i	$-0.216280\pi$
$661$	40.0000	1.55582	0.777910	+	0.628376i	$0.216280\pi$
$662$	0	0
$663$	−14.0000	−0.543715
$664$	0	0
$665$	−20.0000	−0.775567
$666$	0	0
$667$	9.00000	0.348481
$668$	0	0
$669$	−4.00000	−0.154649
$670$	0	0
$671$	−10.0000	−0.386046
$672$	0	0
$673$	10.0000	0.385472	0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000	0.385472	0.192736	+	0.981251i	$0.438264\pi$
$674$	0	0
$675$	5.00000	0.192450
$676$	0	0
$677$	−50.0000	−1.92166	−0.960828	−	0.277145i	$-0.910612\pi$
$677$	−50.0000	−1.92166	−0.960828	+	0.277145i	$0.910612\pi$
$678$	0	0
$679$	−48.0000	−1.84207
$680$	0	0
$681$	4.00000	0.153280
$682$	0	0
$683$	9.00000	0.344375	0.172188	−	0.985064i	$-0.444916\pi$
$683$	9.00000	0.344375	0.172188	+	0.985064i	$0.444916\pi$
$684$	0	0
$685$	18.0000	0.687745
$686$	0	0
$687$	17.0000	0.648590
$688$	0	0
$689$	14.0000	0.533358
$690$	0	0
$691$	−30.0000	−1.14125	−0.570627	−	0.821209i	$-0.693300\pi$
$691$	−30.0000	−1.14125	−0.570627	+	0.821209i	$0.693300\pi$
$692$	0	0
$693$	−40.0000	−1.51947
$694$	0	0
$695$	12.0000	0.455186
$696$	0	0
$697$	8.00000	0.303022
$698$	0	0
$699$	19.0000	0.718646
$700$	0	0
$701$	6.00000	0.226617	0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000	0.226617	0.113308	+	0.993560i	$0.463855\pi$
$702$	0	0
$703$	−10.0000	−0.377157
$704$	0	0
$705$	6.00000	0.225973
$706$	0	0
$707$	−32.0000	−1.20348
$708$	0	0
$709$	−39.0000	−1.46468	−0.732338	−	0.680941i	$-0.761571\pi$
$709$	−39.0000	−1.46468	−0.732338	+	0.680941i	$0.761571\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	0	0
$713$	27.0000	1.01116
$714$	0	0
$715$	35.0000	1.30893
$716$	0	0
$717$	9.00000	0.336111
$718$	0	0
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	0	0
$721$	56.0000	2.08555
$722$	0	0
$723$	11.0000	0.409094
$724$	0	0
$725$	−3.00000	−0.111417
$726$	0	0
$727$	−48.0000	−1.78022	−0.890111	−	0.455744i	$-0.849373\pi$
$727$	−48.0000	−1.78022	−0.890111	+	0.455744i	$0.849373\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	12.0000	0.443836
$732$	0	0
$733$	−31.0000	−1.14501	−0.572506	−	0.819901i	$-0.694029\pi$
$733$	−31.0000	−1.14501	−0.572506	+	0.819901i	$0.694029\pi$
$734$	0	0
$735$	−9.00000	−0.331970
$736$	0	0
$737$	45.0000	1.65760
$738$	0	0
$739$	−30.0000	−1.10357	−0.551784	−	0.833987i	$-0.686053\pi$
$739$	−30.0000	−1.10357	−0.551784	+	0.833987i	$0.686053\pi$
$740$	0	0
$741$	35.0000	1.28576
$742$	0	0
$743$	20.0000	0.733729	0.366864	−	0.930274i	$-0.380431\pi$
$743$	20.0000	0.733729	0.366864	+	0.930274i	$0.380431\pi$
$744$	0	0
$745$	4.00000	0.146549
$746$	0	0
$747$	−22.0000	−0.804938
$748$	0	0
$749$	68.0000	2.48467
$750$	0	0
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	0	0
$753$	17.0000	0.619514
$754$	0	0
$755$	1.00000	0.0363937
$756$	0	0
$757$	8.00000	0.290765	0.145382	−	0.989376i	$-0.453559\pi$
$757$	8.00000	0.290765	0.145382	+	0.989376i	$0.453559\pi$
$758$	0	0
$759$	−15.0000	−0.544466
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	8.00000	0.289619
$764$	0	0
$765$	4.00000	0.144620
$766$	0	0
$767$	21.0000	0.758266
$768$	0	0
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	13.0000	0.468184
$772$	0	0
$773$	−24.0000	−0.863220	−0.431610	−	0.902060i	$-0.642054\pi$
$773$	−24.0000	−0.863220	−0.431610	+	0.902060i	$0.642054\pi$
$774$	0	0
$775$	−9.00000	−0.323290
$776$	0	0
$777$	−8.00000	−0.286998
$778$	0	0
$779$	−20.0000	−0.716574
$780$	0	0
$781$	50.0000	1.78914
$782$	0	0
$783$	−15.0000	−0.536056
$784$	0	0
$785$	−13.0000	−0.463990
$786$	0	0
$787$	−24.0000	−0.855508	−0.427754	−	0.903895i	$-0.640695\pi$
$787$	−24.0000	−0.855508	−0.427754	+	0.903895i	$0.640695\pi$
$788$	0	0
$789$	27.0000	0.961225
$790$	0	0
$791$	−40.0000	−1.42224
$792$	0	0
$793$	−14.0000	−0.497155
$794$	0	0
$795$	2.00000	0.0709327
$796$	0	0
$797$	−52.0000	−1.84193	−0.920967	−	0.389640i	$-0.872599\pi$
$797$	−52.0000	−1.84193	−0.920967	+	0.389640i	$0.872599\pi$
$798$	0	0
$799$	12.0000	0.424529
$800$	0	0
$801$	−16.0000	−0.565332
$802$	0	0
$803$	35.0000	1.23512
$804$	0	0
$805$	12.0000	0.422944
$806$	0	0
$807$	9.00000	0.316815
$808$	0	0
$809$	4.00000	0.140633	0.0703163	−	0.997525i	$-0.477599\pi$
$809$	4.00000	0.140633	0.0703163	+	0.997525i	$0.477599\pi$
$810$	0	0
$811$	44.0000	1.54505	0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000	1.54505	0.772524	+	0.634985i	$0.218994\pi$
$812$	0	0
$813$	−10.0000	−0.350715
$814$	0	0
$815$	−9.00000	−0.315256
$816$	0	0
$817$	−30.0000	−1.04957
$818$	0	0
$819$	−56.0000	−1.95680
$820$	0	0
$821$	38.0000	1.32621	0.663105	−	0.748527i	$-0.269238\pi$
$821$	38.0000	1.32621	0.663105	+	0.748527i	$0.269238\pi$
$822$	0	0
$823$	−42.0000	−1.46403	−0.732014	−	0.681290i	$-0.761419\pi$
$823$	−42.0000	−1.46403	−0.732014	+	0.681290i	$0.761419\pi$
$824$	0	0
$825$	5.00000	0.174078
$826$	0	0
$827$	−26.0000	−0.904109	−0.452054	−	0.891990i	$-0.649309\pi$
$827$	−26.0000	−0.904109	−0.452054	+	0.891990i	$0.649309\pi$
$828$	0	0
$829$	33.0000	1.14614	0.573069	−	0.819507i	$-0.305753\pi$
$829$	33.0000	1.14614	0.573069	+	0.819507i	$0.305753\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	0	0
$833$	−18.0000	−0.623663
$834$	0	0
$835$	−12.0000	−0.415277
$836$	0	0
$837$	−45.0000	−1.55543
$838$	0	0
$839$	20.0000	0.690477	0.345238	−	0.938515i	$-0.387798\pi$
$839$	20.0000	0.690477	0.345238	+	0.938515i	$0.387798\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	12.0000	0.413302
$844$	0	0
$845$	36.0000	1.23844
$846$	0	0
$847$	−56.0000	−1.92418
$848$	0	0
$849$	−9.00000	−0.308879
$850$	0	0
$851$	6.00000	0.205677
$852$	0	0
$853$	18.0000	0.616308	0.308154	−	0.951336i	$-0.400289\pi$
$853$	18.0000	0.616308	0.308154	+	0.951336i	$0.400289\pi$
$854$	0	0
$855$	−10.0000	−0.341993
$856$	0	0
$857$	7.00000	0.239115	0.119558	−	0.992827i	$-0.461852\pi$
$857$	7.00000	0.239115	0.119558	+	0.992827i	$0.461852\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	−16.0000	−0.545279
$862$	0	0
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	0	0
$865$	6.00000	0.204006
$866$	0	0
$867$	13.0000	0.441503
$868$	0	0
$869$	20.0000	0.678454
$870$	0	0
$871$	63.0000	2.13467
$872$	0	0
$873$	−24.0000	−0.812277
$874$	0	0
$875$	−4.00000	−0.135225
$876$	0	0
$877$	−47.0000	−1.58708	−0.793539	−	0.608520i	$-0.791764\pi$
$877$	−47.0000	−1.58708	−0.793539	+	0.608520i	$0.791764\pi$
$878$	0	0
$879$	14.0000	0.472208
$880$	0	0
$881$	28.0000	0.943344	0.471672	−	0.881774i	$-0.343651\pi$
$881$	28.0000	0.943344	0.471672	+	0.881774i	$0.343651\pi$
$882$	0	0
$883$	−44.0000	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	0	0
$885$	3.00000	0.100844
$886$	0	0
$887$	13.0000	0.436497	0.218249	−	0.975893i	$-0.429966\pi$
$887$	13.0000	0.436497	0.218249	+	0.975893i	$0.429966\pi$
$888$	0	0
$889$	64.0000	2.14649
$890$	0	0
$891$	−5.00000	−0.167506
$892$	0	0
$893$	−30.0000	−1.00391
$894$	0	0
$895$	−18.0000	−0.601674
$896$	0	0
$897$	−21.0000	−0.701170
$898$	0	0
$899$	27.0000	0.900500
$900$	0	0
$901$	4.00000	0.133259
$902$	0	0
$903$	−24.0000	−0.798670
$904$	0	0
$905$	22.0000	0.731305
$906$	0	0
$907$	−48.0000	−1.59381	−0.796907	−	0.604102i	$-0.793532\pi$
$907$	−48.0000	−1.59381	−0.796907	+	0.604102i	$0.793532\pi$
$908$	0	0
$909$	−16.0000	−0.530687
$910$	0	0
$911$	51.0000	1.68971	0.844853	−	0.534999i	$-0.179688\pi$
$911$	51.0000	1.68971	0.844853	+	0.534999i	$0.179688\pi$
$912$	0	0
$913$	−55.0000	−1.82023
$914$	0	0
$915$	−2.00000	−0.0661180
$916$	0	0
$917$	16.0000	0.528367
$918$	0	0
$919$	6.00000	0.197922	0.0989609	−	0.995091i	$-0.468448\pi$
$919$	6.00000	0.197922	0.0989609	+	0.995091i	$0.468448\pi$
$920$	0	0
$921$	−10.0000	−0.329511
$922$	0	0
$923$	70.0000	2.30408
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	0	0
$927$	28.0000	0.919641
$928$	0	0
$929$	48.0000	1.57483	0.787414	−	0.616424i	$-0.211419\pi$
$929$	48.0000	1.57483	0.787414	+	0.616424i	$0.211419\pi$
$930$	0	0
$931$	45.0000	1.47482
$932$	0	0
$933$	1.00000	0.0327385
$934$	0	0
$935$	10.0000	0.327035
$936$	0	0
$937$	6.00000	0.196011	0.0980057	−	0.995186i	$-0.468754\pi$
$937$	6.00000	0.196011	0.0980057	+	0.995186i	$0.468754\pi$
$938$	0	0
$939$	16.0000	0.522140
$940$	0	0
$941$	−20.0000	−0.651981	−0.325991	−	0.945373i	$-0.605698\pi$
$941$	−20.0000	−0.651981	−0.325991	+	0.945373i	$0.605698\pi$
$942$	0	0
$943$	12.0000	0.390774
$944$	0	0
$945$	−20.0000	−0.650600
$946$	0	0
$947$	−15.0000	−0.487435	−0.243717	−	0.969846i	$-0.578367\pi$
$947$	−15.0000	−0.487435	−0.243717	+	0.969846i	$0.578367\pi$
$948$	0	0
$949$	49.0000	1.59061
$950$	0	0
$951$	−5.00000	−0.162136
$952$	0	0
$953$	−30.0000	−0.971795	−0.485898	−	0.874016i	$-0.661507\pi$
$953$	−30.0000	−0.971795	−0.485898	+	0.874016i	$0.661507\pi$
$954$	0	0
$955$	8.00000	0.258874
$956$	0	0
$957$	−15.0000	−0.484881
$958$	0	0
$959$	−72.0000	−2.32500
$960$	0	0
$961$	50.0000	1.61290
$962$	0	0
$963$	34.0000	1.09563
$964$	0	0
$965$	14.0000	0.450676
$966$	0	0
$967$	−57.0000	−1.83300	−0.916498	−	0.400039i	$-0.868997\pi$
$967$	−57.0000	−1.83300	−0.916498	+	0.400039i	$0.868997\pi$
$968$	0	0
$969$	10.0000	0.321246
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	−48.0000	−1.53881
$974$	0	0
$975$	7.00000	0.224179
$976$	0	0
$977$	38.0000	1.21573	0.607864	−	0.794041i	$-0.292027\pi$
$977$	38.0000	1.21573	0.607864	+	0.794041i	$0.292027\pi$
$978$	0	0
$979$	−40.0000	−1.27841
$980$	0	0
$981$	4.00000	0.127710
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	−6.00000	−0.191176
$986$	0	0
$987$	−24.0000	−0.763928
$988$	0	0
$989$	18.0000	0.572367
$990$	0	0
$991$	−17.0000	−0.540023	−0.270011	−	0.962857i	$-0.587027\pi$
$991$	−17.0000	−0.540023	−0.270011	+	0.962857i	$0.587027\pi$
$992$	0	0
$993$	−4.00000	−0.126936
$994$	0	0
$995$	2.00000	0.0634043
$996$	0	0
$997$	−16.0000	−0.506725	−0.253363	−	0.967371i	$-0.581537\pi$
$997$	−16.0000	−0.506725	−0.253363	+	0.967371i	$0.581537\pi$
$998$	0	0
$999$	−10.0000	−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.c.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.c.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:	\( N \)	\(=\)	\( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6040.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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