LMFDB - Embedded newform 7938.2.a.bb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7938,2,Mod(1,7938)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7938, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7938.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7938 = 2 \cdot 3^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7938.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:	$63.3852491245$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 126)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	7938.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{8} +O(q^{10})$

$q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{8} +2.00000 q^{10} -1.00000 q^{11} +6.00000 q^{13} +1.00000 q^{16} -5.00000 q^{17} +7.00000 q^{19} +2.00000 q^{20} -1.00000 q^{22} -4.00000 q^{23} -1.00000 q^{25} +6.00000 q^{26} +4.00000 q^{29} +6.00000 q^{31} +1.00000 q^{32} -5.00000 q^{34} +2.00000 q^{37} +7.00000 q^{38} +2.00000 q^{40} +3.00000 q^{41} -1.00000 q^{43} -1.00000 q^{44} -4.00000 q^{46} -1.00000 q^{50} +6.00000 q^{52} -12.0000 q^{53} -2.00000 q^{55} +4.00000 q^{58} -7.00000 q^{59} +12.0000 q^{61} +6.00000 q^{62} +1.00000 q^{64} +12.0000 q^{65} +13.0000 q^{67} -5.00000 q^{68} +8.00000 q^{71} -1.00000 q^{73} +2.00000 q^{74} +7.00000 q^{76} -6.00000 q^{79} +2.00000 q^{80} +3.00000 q^{82} +16.0000 q^{83} -10.0000 q^{85} -1.00000 q^{86} -1.00000 q^{88} -6.00000 q^{89} -4.00000 q^{92} +14.0000 q^{95} +5.00000 q^{97} +O(q^{100})$

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
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	0	0
$10$	2.00000	0.632456
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0	0
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	0	0
$19$	7.00000	1.60591	0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000	1.60591	0.802955	+	0.596040i	$0.203260\pi$
$20$	2.00000	0.447214
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	6.00000	1.17670
$27$	0	0
$28$	0	0
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−5.00000	−0.857493
$35$	0	0
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	7.00000	1.13555
$39$	0	0
$40$	2.00000	0.316228
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	0	0
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−4.00000	−0.589768
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	0	0
$50$	−1.00000	−0.141421
$51$	0	0
$52$	6.00000	0.832050
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	0	0
$57$	0	0
$58$	4.00000	0.525226
$59$	−7.00000	−0.911322	−0.455661	−	0.890153i	$-0.650597\pi$
$59$	−7.00000	−0.911322	−0.455661	+	0.890153i	$0.650597\pi$
$60$	0	0
$61$	12.0000	1.53644	0.768221	−	0.640184i	$-0.221142\pi$
$61$	12.0000	1.53644	0.768221	+	0.640184i	$0.221142\pi$
$62$	6.00000	0.762001
$63$	0	0
$64$	1.00000	0.125000
$65$	12.0000	1.48842
$66$	0	0
$67$	13.0000	1.58820	0.794101	−	0.607785i	$-0.207942\pi$
$67$	13.0000	1.58820	0.794101	+	0.607785i	$0.207942\pi$
$68$	−5.00000	−0.606339
$69$	0	0
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	−1.00000	−0.117041	−0.0585206	−	0.998286i	$-0.518638\pi$
$73$	−1.00000	−0.117041	−0.0585206	+	0.998286i	$0.518638\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	7.00000	0.802955
$77$	0	0
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	2.00000	0.223607
$81$	0	0
$82$	3.00000	0.331295
$83$	16.0000	1.75623	0.878114	−	0.478451i	$-0.158802\pi$
$83$	16.0000	1.75623	0.878114	+	0.478451i	$0.158802\pi$
$84$	0	0
$85$	−10.0000	−1.08465
$86$	−1.00000	−0.107833
$87$	0	0
$88$	−1.00000	−0.106600
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	0	0
$92$	−4.00000	−0.417029
$93$	0	0
$94$	0	0
$95$	14.0000	1.43637
$96$	0	0
$97$	5.00000	0.507673	0.253837	−	0.967247i	$-0.418307\pi$
$97$	5.00000	0.507673	0.253837	+	0.967247i	$0.418307\pi$
$98$	0	0
$99$	0	0
$100$	−1.00000	−0.100000
$101$	−4.00000	−0.398015	−0.199007	−	0.979998i	$-0.563772\pi$
$101$	−4.00000	−0.398015	−0.199007	+	0.979998i	$0.563772\pi$
$102$	0	0
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	−12.0000	−1.16554
$107$	−3.00000	−0.290021	−0.145010	−	0.989430i	$-0.546322\pi$
$107$	−3.00000	−0.290021	−0.145010	+	0.989430i	$0.546322\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	−2.00000	−0.190693
$111$	0	0
$112$	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
$115$	−8.00000	−0.746004
$116$	4.00000	0.371391
$117$	0	0
$118$	−7.00000	−0.644402
$119$	0	0
$120$	0	0
$121$	−10.0000	−0.909091
$122$	12.0000	1.08643
$123$	0	0
$124$	6.00000	0.538816
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−12.0000	−1.06483	−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000	−1.06483	−0.532414	+	0.846484i	$0.678715\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	12.0000	1.05247
$131$	−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
$133$	0	0
$134$	13.0000	1.12303
$135$	0	0
$136$	−5.00000	−0.428746
$137$	19.0000	1.62328	0.811640	−	0.584158i	$-0.198575\pi$
$137$	19.0000	1.62328	0.811640	+	0.584158i	$0.198575\pi$
$138$	0	0
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	0	0
$141$	0	0
$142$	8.00000	0.671345
$143$	−6.00000	−0.501745
$144$	0	0
$145$	8.00000	0.664364
$146$	−1.00000	−0.0827606
$147$	0	0
$148$	2.00000	0.164399
$149$	24.0000	1.96616	0.983078	−	0.183186i	$-0.0586410\pi$
$149$	24.0000	1.96616	0.983078	+	0.183186i	$0.0586410\pi$
$150$	0	0
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	7.00000	0.567775
$153$	0	0
$154$	0	0
$155$	12.0000	0.963863
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	−6.00000	−0.477334
$159$	0	0
$160$	2.00000	0.158114
$161$	0	0
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	3.00000	0.234261
$165$	0	0
$166$	16.0000	1.24184
$167$	20.0000	1.54765	0.773823	−	0.633402i	$-0.218342\pi$
$167$	20.0000	1.54765	0.773823	+	0.633402i	$0.218342\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	−10.0000	−0.766965
$171$	0	0
$172$	−1.00000	−0.0762493
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	0	0
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	−6.00000	−0.449719
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	0	0
$184$	−4.00000	−0.294884
$185$	4.00000	0.294086
$186$	0	0
$187$	5.00000	0.365636
$188$	0	0
$189$	0	0
$190$	14.0000	1.01567
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	17.0000	1.22369	0.611843	−	0.790979i	$-0.290428\pi$
$193$	17.0000	1.22369	0.611843	+	0.790979i	$0.290428\pi$
$194$	5.00000	0.358979
$195$	0	0
$196$	0	0
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	0	0
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−4.00000	−0.281439
$203$	0	0
$204$	0	0
$205$	6.00000	0.419058
$206$	−14.0000	−0.975426
$207$	0	0
$208$	6.00000	0.416025
$209$	−7.00000	−0.484200
$210$	0	0
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	−12.0000	−0.824163
$213$	0	0
$214$	−3.00000	−0.205076
$215$	−2.00000	−0.136399
$216$	0	0
$217$	0	0
$218$	−2.00000	−0.135457
$219$	0	0
$220$	−2.00000	−0.134840
$221$	−30.0000	−2.01802
$222$	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
$225$	0	0
$226$	10.0000	0.665190
$227$	3.00000	0.199117	0.0995585	−	0.995032i	$-0.468257\pi$
$227$	3.00000	0.199117	0.0995585	+	0.995032i	$0.468257\pi$
$228$	0	0
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	−8.00000	−0.527504
$231$	0	0
$232$	4.00000	0.262613
$233$	29.0000	1.89985	0.949927	−	0.312473i	$-0.101157\pi$
$233$	29.0000	1.89985	0.949927	+	0.312473i	$0.101157\pi$
$234$	0	0
$235$	0	0
$236$	−7.00000	−0.455661
$237$	0	0
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	−23.0000	−1.48156	−0.740780	−	0.671748i	$-0.765544\pi$
$241$	−23.0000	−1.48156	−0.740780	+	0.671748i	$0.765544\pi$
$242$	−10.0000	−0.642824
$243$	0	0
$244$	12.0000	0.768221
$245$	0	0
$246$	0	0
$247$	42.0000	2.67240
$248$	6.00000	0.381000
$249$	0	0
$250$	−12.0000	−0.758947
$251$	3.00000	0.189358	0.0946792	−	0.995508i	$-0.469817\pi$
$251$	3.00000	0.189358	0.0946792	+	0.995508i	$0.469817\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	−12.0000	−0.752947
$255$	0	0
$256$	1.00000	0.0625000
$257$	−15.0000	−0.935674	−0.467837	−	0.883815i	$-0.654967\pi$
$257$	−15.0000	−0.935674	−0.467837	+	0.883815i	$0.654967\pi$
$258$	0	0
$259$	0	0
$260$	12.0000	0.744208
$261$	0	0
$262$	−4.00000	−0.247121
$263$	−18.0000	−1.10993	−0.554964	−	0.831875i	$-0.687268\pi$
$263$	−18.0000	−1.10993	−0.554964	+	0.831875i	$0.687268\pi$
$264$	0	0
$265$	−24.0000	−1.47431
$266$	0	0
$267$	0	0
$268$	13.0000	0.794101
$269$	20.0000	1.21942	0.609711	−	0.792624i	$-0.291286\pi$
$269$	20.0000	1.21942	0.609711	+	0.792624i	$0.291286\pi$
$270$	0	0
$271$	6.00000	0.364474	0.182237	−	0.983255i	$-0.441666\pi$
$271$	6.00000	0.364474	0.182237	+	0.983255i	$0.441666\pi$
$272$	−5.00000	−0.303170
$273$	0	0
$274$	19.0000	1.14783
$275$	1.00000	0.0603023
$276$	0	0
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	5.00000	0.299880
$279$	0	0
$280$	0	0
$281$	−22.0000	−1.31241	−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000	−1.31241	−0.656205	+	0.754583i	$0.727839\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	−6.00000	−0.354787
$287$	0	0
$288$	0	0
$289$	8.00000	0.470588
$290$	8.00000	0.469776
$291$	0	0
$292$	−1.00000	−0.0585206
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	−14.0000	−0.815112
$296$	2.00000	0.116248
$297$	0	0
$298$	24.0000	1.39028
$299$	−24.0000	−1.38796
$300$	0	0
$301$	0	0
$302$	10.0000	0.575435
$303$	0	0
$304$	7.00000	0.401478
$305$	24.0000	1.37424
$306$	0	0
$307$	−7.00000	−0.399511	−0.199756	−	0.979846i	$-0.564015\pi$
$307$	−7.00000	−0.399511	−0.199756	+	0.979846i	$0.564015\pi$
$308$	0	0
$309$	0	0
$310$	12.0000	0.681554
$311$	2.00000	0.113410	0.0567048	−	0.998391i	$-0.481941\pi$
$311$	2.00000	0.113410	0.0567048	+	0.998391i	$0.481941\pi$
$312$	0	0
$313$	17.0000	0.960897	0.480448	−	0.877023i	$-0.340474\pi$
$313$	17.0000	0.960897	0.480448	+	0.877023i	$0.340474\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	−6.00000	−0.337526
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	−4.00000	−0.223957
$320$	2.00000	0.111803
$321$	0	0
$322$	0	0
$323$	−35.0000	−1.94745
$324$	0	0
$325$	−6.00000	−0.332820
$326$	−4.00000	−0.221540
$327$	0	0
$328$	3.00000	0.165647
$329$	0	0
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	16.0000	0.878114
$333$	0	0
$334$	20.0000	1.09435
$335$	26.0000	1.42053
$336$	0	0
$337$	−9.00000	−0.490261	−0.245131	−	0.969490i	$-0.578831\pi$
$337$	−9.00000	−0.490261	−0.245131	+	0.969490i	$0.578831\pi$
$338$	23.0000	1.25104
$339$	0	0
$340$	−10.0000	−0.542326
$341$	−6.00000	−0.324918
$342$	0	0
$343$	0	0
$344$	−1.00000	−0.0539164
$345$	0	0
$346$	2.00000	0.107521
$347$	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
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	0	0
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	−15.0000	−0.798369	−0.399185	−	0.916871i	$-0.630707\pi$
$353$	−15.0000	−0.798369	−0.399185	+	0.916871i	$0.630707\pi$
$354$	0	0
$355$	16.0000	0.849192
$356$	−6.00000	−0.317999
$357$	0	0
$358$	−24.0000	−1.26844
$359$	−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
$361$	30.0000	1.57895
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2.00000	−0.104685
$366$	0	0
$367$	22.0000	1.14839	0.574195	−	0.818718i	$-0.305315\pi$
$367$	22.0000	1.14839	0.574195	+	0.818718i	$0.305315\pi$
$368$	−4.00000	−0.208514
$369$	0	0
$370$	4.00000	0.207950
$371$	0	0
$372$	0	0
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	5.00000	0.258544
$375$	0	0
$376$	0	0
$377$	24.0000	1.23606
$378$	0	0
$379$	−17.0000	−0.873231	−0.436616	−	0.899648i	$-0.643823\pi$
$379$	−17.0000	−0.873231	−0.436616	+	0.899648i	$0.643823\pi$
$380$	14.0000	0.718185
$381$	0	0
$382$	−12.0000	−0.613973
$383$	4.00000	0.204390	0.102195	−	0.994764i	$-0.467413\pi$
$383$	4.00000	0.204390	0.102195	+	0.994764i	$0.467413\pi$
$384$	0	0
$385$	0	0
$386$	17.0000	0.865277
$387$	0	0
$388$	5.00000	0.253837
$389$	−8.00000	−0.405616	−0.202808	−	0.979219i	$-0.565007\pi$
$389$	−8.00000	−0.405616	−0.202808	+	0.979219i	$0.565007\pi$
$390$	0	0
$391$	20.0000	1.01144
$392$	0	0
$393$	0	0
$394$	−10.0000	−0.503793
$395$	−12.0000	−0.603786
$396$	0	0
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	−14.0000	−0.701757
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−9.00000	−0.449439	−0.224719	−	0.974424i	$-0.572147\pi$
$401$	−9.00000	−0.449439	−0.224719	+	0.974424i	$0.572147\pi$
$402$	0	0
$403$	36.0000	1.79329
$404$	−4.00000	−0.199007
$405$	0	0
$406$	0	0
$407$	−2.00000	−0.0991363
$408$	0	0
$409$	−11.0000	−0.543915	−0.271957	−	0.962309i	$-0.587671\pi$
$409$	−11.0000	−0.543915	−0.271957	+	0.962309i	$0.587671\pi$
$410$	6.00000	0.296319
$411$	0	0
$412$	−14.0000	−0.689730
$413$	0	0
$414$	0	0
$415$	32.0000	1.57082
$416$	6.00000	0.294174
$417$	0	0
$418$	−7.00000	−0.342381
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	−12.0000	−0.584844	−0.292422	−	0.956289i	$-0.594461\pi$
$421$	−12.0000	−0.584844	−0.292422	+	0.956289i	$0.594461\pi$
$422$	−16.0000	−0.778868
$423$	0	0
$424$	−12.0000	−0.582772
$425$	5.00000	0.242536
$426$	0	0
$427$	0	0
$428$	−3.00000	−0.145010
$429$	0	0
$430$	−2.00000	−0.0964486
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	25.0000	1.20142	0.600712	−	0.799466i	$-0.294884\pi$
$433$	25.0000	1.20142	0.600712	+	0.799466i	$0.294884\pi$
$434$	0	0
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	−28.0000	−1.33942
$438$	0	0
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	−2.00000	−0.0953463
$441$	0	0
$442$	−30.0000	−1.42695
$443$	−7.00000	−0.332580	−0.166290	−	0.986077i	$-0.553179\pi$
$443$	−7.00000	−0.332580	−0.166290	+	0.986077i	$0.553179\pi$
$444$	0	0
$445$	−12.0000	−0.568855
$446$	4.00000	0.189405
$447$	0	0
$448$	0	0
$449$	−17.0000	−0.802280	−0.401140	−	0.916017i	$-0.631386\pi$
$449$	−17.0000	−0.802280	−0.401140	+	0.916017i	$0.631386\pi$
$450$	0	0
$451$	−3.00000	−0.141264
$452$	10.0000	0.470360
$453$	0	0
$454$	3.00000	0.140797
$455$	0	0
$456$	0	0
$457$	1.00000	0.0467780	0.0233890	−	0.999726i	$-0.492554\pi$
$457$	1.00000	0.0467780	0.0233890	+	0.999726i	$0.492554\pi$
$458$	26.0000	1.21490
$459$	0	0
$460$	−8.00000	−0.373002
$461$	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
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	4.00000	0.185695
$465$	0	0
$466$	29.0000	1.34340
$467$	−13.0000	−0.601568	−0.300784	−	0.953692i	$-0.597248\pi$
$467$	−13.0000	−0.601568	−0.300784	+	0.953692i	$0.597248\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−7.00000	−0.322201
$473$	1.00000	0.0459800
$474$	0	0
$475$	−7.00000	−0.321182
$476$	0	0
$477$	0	0
$478$	−6.00000	−0.274434
$479$	−20.0000	−0.913823	−0.456912	−	0.889512i	$-0.651044\pi$
$479$	−20.0000	−0.913823	−0.456912	+	0.889512i	$0.651044\pi$
$480$	0	0
$481$	12.0000	0.547153
$482$	−23.0000	−1.04762
$483$	0	0
$484$	−10.0000	−0.454545
$485$	10.0000	0.454077
$486$	0	0
$487$	−10.0000	−0.453143	−0.226572	−	0.973995i	$-0.572752\pi$
$487$	−10.0000	−0.453143	−0.226572	+	0.973995i	$0.572752\pi$
$488$	12.0000	0.543214
$489$	0	0
$490$	0	0
$491$	−33.0000	−1.48927	−0.744635	−	0.667472i	$-0.767376\pi$
$491$	−33.0000	−1.48927	−0.744635	+	0.667472i	$0.767376\pi$
$492$	0	0
$493$	−20.0000	−0.900755
$494$	42.0000	1.88967
$495$	0	0
$496$	6.00000	0.269408
$497$	0	0
$498$	0	0
$499$	−29.0000	−1.29822	−0.649109	−	0.760695i	$-0.724858\pi$
$499$	−29.0000	−1.29822	−0.649109	+	0.760695i	$0.724858\pi$
$500$	−12.0000	−0.536656
$501$	0	0
$502$	3.00000	0.133897
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	0	0
$505$	−8.00000	−0.355995
$506$	4.00000	0.177822
$507$	0	0
$508$	−12.0000	−0.532414
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	−15.0000	−0.661622
$515$	−28.0000	−1.23383
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	12.0000	0.526235
$521$	−9.00000	−0.394297	−0.197149	−	0.980374i	$-0.563168\pi$
$521$	−9.00000	−0.394297	−0.197149	+	0.980374i	$0.563168\pi$
$522$	0	0
$523$	28.0000	1.22435	0.612177	−	0.790721i	$-0.290294\pi$
$523$	28.0000	1.22435	0.612177	+	0.790721i	$0.290294\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	−18.0000	−0.784837
$527$	−30.0000	−1.30682
$528$	0	0
$529$	−7.00000	−0.304348
$530$	−24.0000	−1.04249
$531$	0	0
$532$	0	0
$533$	18.0000	0.779667
$534$	0	0
$535$	−6.00000	−0.259403
$536$	13.0000	0.561514
$537$	0	0
$538$	20.0000	0.862261
$539$	0	0
$540$	0	0
$541$	−24.0000	−1.03184	−0.515920	−	0.856637i	$-0.672550\pi$
$541$	−24.0000	−1.03184	−0.515920	+	0.856637i	$0.672550\pi$
$542$	6.00000	0.257722
$543$	0	0
$544$	−5.00000	−0.214373
$545$	−4.00000	−0.171341
$546$	0	0
$547$	−21.0000	−0.897895	−0.448948	−	0.893558i	$-0.648201\pi$
$547$	−21.0000	−0.897895	−0.448948	+	0.893558i	$0.648201\pi$
$548$	19.0000	0.811640
$549$	0	0
$550$	1.00000	0.0426401
$551$	28.0000	1.19284
$552$	0	0
$553$	0	0
$554$	2.00000	0.0849719
$555$	0	0
$556$	5.00000	0.212047
$557$	28.0000	1.18640	0.593199	−	0.805056i	$-0.297865\pi$
$557$	28.0000	1.18640	0.593199	+	0.805056i	$0.297865\pi$
$558$	0	0
$559$	−6.00000	−0.253773
$560$	0	0
$561$	0	0
$562$	−22.0000	−0.928014
$563$	31.0000	1.30649	0.653247	−	0.757145i	$-0.273406\pi$
$563$	31.0000	1.30649	0.653247	+	0.757145i	$0.273406\pi$
$564$	0	0
$565$	20.0000	0.841406
$566$	−4.00000	−0.168133
$567$	0	0
$568$	8.00000	0.335673
$569$	15.0000	0.628833	0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000	0.628833	0.314416	+	0.949285i	$0.398191\pi$
$570$	0	0
$571$	−33.0000	−1.38101	−0.690504	−	0.723329i	$-0.742611\pi$
$571$	−33.0000	−1.38101	−0.690504	+	0.723329i	$0.742611\pi$
$572$	−6.00000	−0.250873
$573$	0	0
$574$	0	0
$575$	4.00000	0.166812
$576$	0	0
$577$	35.0000	1.45707	0.728535	−	0.685009i	$-0.240202\pi$
$577$	35.0000	1.45707	0.728535	+	0.685009i	$0.240202\pi$
$578$	8.00000	0.332756
$579$	0	0
$580$	8.00000	0.332182
$581$	0	0
$582$	0	0
$583$	12.0000	0.496989
$584$	−1.00000	−0.0413803
$585$	0	0
$586$	0	0
$587$	−47.0000	−1.93990	−0.969949	−	0.243309i	$-0.921767\pi$
$587$	−47.0000	−1.93990	−0.969949	+	0.243309i	$0.921767\pi$
$588$	0	0
$589$	42.0000	1.73058
$590$	−14.0000	−0.576371
$591$	0	0
$592$	2.00000	0.0821995
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	0	0
$595$	0	0
$596$	24.0000	0.983078
$597$	0	0
$598$	−24.0000	−0.981433
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	19.0000	0.775026	0.387513	−	0.921864i	$-0.373334\pi$
$601$	19.0000	0.775026	0.387513	+	0.921864i	$0.373334\pi$
$602$	0	0
$603$	0	0
$604$	10.0000	0.406894
$605$	−20.0000	−0.813116
$606$	0	0
$607$	24.0000	0.974130	0.487065	−	0.873366i	$-0.338067\pi$
$607$	24.0000	0.974130	0.487065	+	0.873366i	$0.338067\pi$
$608$	7.00000	0.283887
$609$	0	0
$610$	24.0000	0.971732
$611$	0	0
$612$	0	0
$613$	42.0000	1.69636	0.848182	−	0.529705i	$-0.177697\pi$
$613$	42.0000	1.69636	0.848182	+	0.529705i	$0.177697\pi$
$614$	−7.00000	−0.282497
$615$	0	0
$616$	0	0
$617$	17.0000	0.684394	0.342197	−	0.939628i	$-0.388829\pi$
$617$	17.0000	0.684394	0.342197	+	0.939628i	$0.388829\pi$
$618$	0	0
$619$	−37.0000	−1.48716	−0.743578	−	0.668649i	$-0.766873\pi$
$619$	−37.0000	−1.48716	−0.743578	+	0.668649i	$0.766873\pi$
$620$	12.0000	0.481932
$621$	0	0
$622$	2.00000	0.0801927
$623$	0	0
$624$	0	0
$625$	−19.0000	−0.760000
$626$	17.0000	0.679457
$627$	0	0
$628$	−2.00000	−0.0798087
$629$	−10.0000	−0.398726
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	−6.00000	−0.238667
$633$	0	0
$634$	6.00000	0.238290
$635$	−24.0000	−0.952411
$636$	0	0
$637$	0	0
$638$	−4.00000	−0.158362
$639$	0	0
$640$	2.00000	0.0790569
$641$	−1.00000	−0.0394976	−0.0197488	−	0.999805i	$-0.506287\pi$
$641$	−1.00000	−0.0394976	−0.0197488	+	0.999805i	$0.506287\pi$
$642$	0	0
$643$	7.00000	0.276053	0.138027	−	0.990429i	$-0.455924\pi$
$643$	7.00000	0.276053	0.138027	+	0.990429i	$0.455924\pi$
$644$	0	0
$645$	0	0
$646$	−35.0000	−1.37706
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	0	0
$649$	7.00000	0.274774
$650$	−6.00000	−0.235339
$651$	0	0
$652$	−4.00000	−0.156652
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	0	0
$655$	−8.00000	−0.312586
$656$	3.00000	0.117130
$657$	0	0
$658$	0	0
$659$	−16.0000	−0.623272	−0.311636	−	0.950202i	$-0.600877\pi$
$659$	−16.0000	−0.623272	−0.311636	+	0.950202i	$0.600877\pi$
$660$	0	0
$661$	−28.0000	−1.08907	−0.544537	−	0.838737i	$-0.683295\pi$
$661$	−28.0000	−1.08907	−0.544537	+	0.838737i	$0.683295\pi$
$662$	8.00000	0.310929
$663$	0	0
$664$	16.0000	0.620920
$665$	0	0
$666$	0	0
$667$	−16.0000	−0.619522
$668$	20.0000	0.773823
$669$	0	0
$670$	26.0000	1.00447
$671$	−12.0000	−0.463255
$672$	0	0
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	−9.00000	−0.346667
$675$	0	0
$676$	23.0000	0.884615
$677$	30.0000	1.15299	0.576497	−	0.817099i	$-0.304419\pi$
$677$	30.0000	1.15299	0.576497	+	0.817099i	$0.304419\pi$
$678$	0	0
$679$	0	0
$680$	−10.0000	−0.383482
$681$	0	0
$682$	−6.00000	−0.229752
$683$	−39.0000	−1.49229	−0.746147	−	0.665782i	$-0.768098\pi$
$683$	−39.0000	−1.49229	−0.746147	+	0.665782i	$0.768098\pi$
$684$	0	0
$685$	38.0000	1.45191
$686$	0	0
$687$	0	0
$688$	−1.00000	−0.0381246
$689$	−72.0000	−2.74298
$690$	0	0
$691$	−32.0000	−1.21734	−0.608669	−	0.793424i	$-0.708296\pi$
$691$	−32.0000	−1.21734	−0.608669	+	0.793424i	$0.708296\pi$
$692$	2.00000	0.0760286
$693$	0	0
$694$	3.00000	0.113878
$695$	10.0000	0.379322
$696$	0	0
$697$	−15.0000	−0.568166
$698$	14.0000	0.529908
$699$	0	0
$700$	0	0
$701$	−8.00000	−0.302156	−0.151078	−	0.988522i	$-0.548274\pi$
$701$	−8.00000	−0.302156	−0.151078	+	0.988522i	$0.548274\pi$
$702$	0	0
$703$	14.0000	0.528020
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−15.0000	−0.564532
$707$	0	0
$708$	0	0
$709$	−4.00000	−0.150223	−0.0751116	−	0.997175i	$-0.523931\pi$
$709$	−4.00000	−0.150223	−0.0751116	+	0.997175i	$0.523931\pi$
$710$	16.0000	0.600469
$711$	0	0
$712$	−6.00000	−0.224860
$713$	−24.0000	−0.898807
$714$	0	0
$715$	−12.0000	−0.448775
$716$	−24.0000	−0.896922
$717$	0	0
$718$	−2.00000	−0.0746393
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	0	0
$721$	0	0
$722$	30.0000	1.11648
$723$	0	0
$724$	0	0
$725$	−4.00000	−0.148556
$726$	0	0
$727$	14.0000	0.519231	0.259616	−	0.965712i	$-0.416404\pi$
$727$	14.0000	0.519231	0.259616	+	0.965712i	$0.416404\pi$
$728$	0	0
$729$	0	0
$730$	−2.00000	−0.0740233
$731$	5.00000	0.184932
$732$	0	0
$733$	18.0000	0.664845	0.332423	−	0.943131i	$-0.392134\pi$
$733$	18.0000	0.664845	0.332423	+	0.943131i	$0.392134\pi$
$734$	22.0000	0.812035
$735$	0	0
$736$	−4.00000	−0.147442
$737$	−13.0000	−0.478861
$738$	0	0
$739$	−33.0000	−1.21392	−0.606962	−	0.794731i	$-0.707612\pi$
$739$	−33.0000	−1.21392	−0.606962	+	0.794731i	$0.707612\pi$
$740$	4.00000	0.147043
$741$	0	0
$742$	0	0
$743$	6.00000	0.220119	0.110059	−	0.993925i	$-0.464896\pi$
$743$	6.00000	0.220119	0.110059	+	0.993925i	$0.464896\pi$
$744$	0	0
$745$	48.0000	1.75858
$746$	22.0000	0.805477
$747$	0	0
$748$	5.00000	0.182818
$749$	0	0
$750$	0	0
$751$	−18.0000	−0.656829	−0.328415	−	0.944534i	$-0.606514\pi$
$751$	−18.0000	−0.656829	−0.328415	+	0.944534i	$0.606514\pi$
$752$	0	0
$753$	0	0
$754$	24.0000	0.874028
$755$	20.0000	0.727875
$756$	0	0
$757$	−48.0000	−1.74459	−0.872295	−	0.488980i	$-0.837369\pi$
$757$	−48.0000	−1.74459	−0.872295	+	0.488980i	$0.837369\pi$
$758$	−17.0000	−0.617468
$759$	0	0
$760$	14.0000	0.507833
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	0	0
$763$	0	0
$764$	−12.0000	−0.434145
$765$	0	0
$766$	4.00000	0.144526
$767$	−42.0000	−1.51653
$768$	0	0
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	0	0
$771$	0	0
$772$	17.0000	0.611843
$773$	−52.0000	−1.87031	−0.935155	−	0.354239i	$-0.884740\pi$
$773$	−52.0000	−1.87031	−0.935155	+	0.354239i	$0.884740\pi$
$774$	0	0
$775$	−6.00000	−0.215526
$776$	5.00000	0.179490
$777$	0	0
$778$	−8.00000	−0.286814
$779$	21.0000	0.752403
$780$	0	0
$781$	−8.00000	−0.286263
$782$	20.0000	0.715199
$783$	0	0
$784$	0	0
$785$	−4.00000	−0.142766
$786$	0	0
$787$	12.0000	0.427754	0.213877	−	0.976861i	$-0.431391\pi$
$787$	12.0000	0.427754	0.213877	+	0.976861i	$0.431391\pi$
$788$	−10.0000	−0.356235
$789$	0	0
$790$	−12.0000	−0.426941
$791$	0	0
$792$	0	0
$793$	72.0000	2.55679
$794$	−18.0000	−0.638796
$795$	0	0
$796$	−14.0000	−0.496217
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	0	0
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	−9.00000	−0.317801
$803$	1.00000	0.0352892
$804$	0	0
$805$	0	0
$806$	36.0000	1.26805
$807$	0	0
$808$	−4.00000	−0.140720
$809$	43.0000	1.51180	0.755900	−	0.654687i	$-0.227200\pi$
$809$	43.0000	1.51180	0.755900	+	0.654687i	$0.227200\pi$
$810$	0	0
$811$	−31.0000	−1.08856	−0.544279	−	0.838905i	$-0.683197\pi$
$811$	−31.0000	−1.08856	−0.544279	+	0.838905i	$0.683197\pi$
$812$	0	0
$813$	0	0
$814$	−2.00000	−0.0701000
$815$	−8.00000	−0.280228
$816$	0	0
$817$	−7.00000	−0.244899
$818$	−11.0000	−0.384606
$819$	0	0
$820$	6.00000	0.209529
$821$	−46.0000	−1.60541	−0.802706	−	0.596376i	$-0.796607\pi$
$821$	−46.0000	−1.60541	−0.802706	+	0.596376i	$0.796607\pi$
$822$	0	0
$823$	34.0000	1.18517	0.592583	−	0.805510i	$-0.298108\pi$
$823$	34.0000	1.18517	0.592583	+	0.805510i	$0.298108\pi$
$824$	−14.0000	−0.487713
$825$	0	0
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	−4.00000	−0.138926	−0.0694629	−	0.997585i	$-0.522129\pi$
$829$	−4.00000	−0.138926	−0.0694629	+	0.997585i	$0.522129\pi$
$830$	32.0000	1.11074
$831$	0	0
$832$	6.00000	0.208013
$833$	0	0
$834$	0	0
$835$	40.0000	1.38426
$836$	−7.00000	−0.242100
$837$	0	0
$838$	−12.0000	−0.414533
$839$	−20.0000	−0.690477	−0.345238	−	0.938515i	$-0.612202\pi$
$839$	−20.0000	−0.690477	−0.345238	+	0.938515i	$0.612202\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	−12.0000	−0.413547
$843$	0	0
$844$	−16.0000	−0.550743
$845$	46.0000	1.58245
$846$	0	0
$847$	0	0
$848$	−12.0000	−0.412082
$849$	0	0
$850$	5.00000	0.171499
$851$	−8.00000	−0.274236
$852$	0	0
$853$	−44.0000	−1.50653	−0.753266	−	0.657716i	$-0.771523\pi$
$853$	−44.0000	−1.50653	−0.753266	+	0.657716i	$0.771523\pi$
$854$	0	0
$855$	0	0
$856$	−3.00000	−0.102538
$857$	30.0000	1.02478	0.512390	−	0.858753i	$-0.328760\pi$
$857$	30.0000	1.02478	0.512390	+	0.858753i	$0.328760\pi$
$858$	0	0
$859$	−29.0000	−0.989467	−0.494734	−	0.869045i	$-0.664734\pi$
$859$	−29.0000	−0.989467	−0.494734	+	0.869045i	$0.664734\pi$
$860$	−2.00000	−0.0681994
$861$	0	0
$862$	0	0
$863$	−38.0000	−1.29354	−0.646768	−	0.762687i	$-0.723880\pi$
$863$	−38.0000	−1.29354	−0.646768	+	0.762687i	$0.723880\pi$
$864$	0	0
$865$	4.00000	0.136004
$866$	25.0000	0.849535
$867$	0	0
$868$	0	0
$869$	6.00000	0.203536
$870$	0	0
$871$	78.0000	2.64293
$872$	−2.00000	−0.0677285
$873$	0	0
$874$	−28.0000	−0.947114
$875$	0	0
$876$	0	0
$877$	16.0000	0.540282	0.270141	−	0.962821i	$-0.412930\pi$
$877$	16.0000	0.540282	0.270141	+	0.962821i	$0.412930\pi$
$878$	24.0000	0.809961
$879$	0	0
$880$	−2.00000	−0.0674200
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	0	0
$883$	53.0000	1.78359	0.891796	−	0.452438i	$-0.149446\pi$
$883$	53.0000	1.78359	0.891796	+	0.452438i	$0.149446\pi$
$884$	−30.0000	−1.00901
$885$	0	0
$886$	−7.00000	−0.235170
$887$	−6.00000	−0.201460	−0.100730	−	0.994914i	$-0.532118\pi$
$887$	−6.00000	−0.201460	−0.100730	+	0.994914i	$0.532118\pi$
$888$	0	0
$889$	0	0
$890$	−12.0000	−0.402241
$891$	0	0
$892$	4.00000	0.133930
$893$	0	0
$894$	0	0
$895$	−48.0000	−1.60446
$896$	0	0
$897$	0	0
$898$	−17.0000	−0.567297
$899$	24.0000	0.800445
$900$	0	0
$901$	60.0000	1.99889
$902$	−3.00000	−0.0998891
$903$	0	0
$904$	10.0000	0.332595
$905$	0	0
$906$	0	0
$907$	−27.0000	−0.896520	−0.448260	−	0.893903i	$-0.647956\pi$
$907$	−27.0000	−0.896520	−0.448260	+	0.893903i	$0.647956\pi$
$908$	3.00000	0.0995585
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	−16.0000	−0.529523
$914$	1.00000	0.0330771
$915$	0	0
$916$	26.0000	0.859064
$917$	0	0
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	−8.00000	−0.263752
$921$	0	0
$922$	14.0000	0.461065
$923$	48.0000	1.57994
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	−8.00000	−0.262896
$927$	0	0
$928$	4.00000	0.131306
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	0	0
$932$	29.0000	0.949927
$933$	0	0
$934$	−13.0000	−0.425373
$935$	10.0000	0.327035
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	20.0000	0.651981	0.325991	−	0.945373i	$-0.394302\pi$
$941$	20.0000	0.651981	0.325991	+	0.945373i	$0.394302\pi$
$942$	0	0
$943$	−12.0000	−0.390774
$944$	−7.00000	−0.227831
$945$	0	0
$946$	1.00000	0.0325128
$947$	37.0000	1.20234	0.601169	−	0.799122i	$-0.294702\pi$
$947$	37.0000	1.20234	0.601169	+	0.799122i	$0.294702\pi$
$948$	0	0
$949$	−6.00000	−0.194768
$950$	−7.00000	−0.227110
$951$	0	0
$952$	0	0
$953$	35.0000	1.13376	0.566881	−	0.823800i	$-0.308150\pi$
$953$	35.0000	1.13376	0.566881	+	0.823800i	$0.308150\pi$
$954$	0	0
$955$	−24.0000	−0.776622
$956$	−6.00000	−0.194054
$957$	0	0
$958$	−20.0000	−0.646171
$959$	0	0
$960$	0	0
$961$	5.00000	0.161290
$962$	12.0000	0.386896
$963$	0	0
$964$	−23.0000	−0.740780
$965$	34.0000	1.09450
$966$	0	0
$967$	14.0000	0.450210	0.225105	−	0.974335i	$-0.427728\pi$
$967$	14.0000	0.450210	0.225105	+	0.974335i	$0.427728\pi$
$968$	−10.0000	−0.321412
$969$	0	0
$970$	10.0000	0.321081
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	0	0
$973$	0	0
$974$	−10.0000	−0.320421
$975$	0	0
$976$	12.0000	0.384111
$977$	15.0000	0.479893	0.239946	−	0.970786i	$-0.422870\pi$
$977$	15.0000	0.479893	0.239946	+	0.970786i	$0.422870\pi$
$978$	0	0
$979$	6.00000	0.191761
$980$	0	0
$981$	0	0
$982$	−33.0000	−1.05307
$983$	−60.0000	−1.91370	−0.956851	−	0.290578i	$-0.906153\pi$
$983$	−60.0000	−1.91370	−0.956851	+	0.290578i	$0.906153\pi$
$984$	0	0
$985$	−20.0000	−0.637253
$986$	−20.0000	−0.636930
$987$	0	0
$988$	42.0000	1.33620
$989$	4.00000	0.127193
$990$	0	0
$991$	−28.0000	−0.889449	−0.444725	−	0.895667i	$-0.646698\pi$
$991$	−28.0000	−0.889449	−0.444725	+	0.895667i	$0.646698\pi$
$992$	6.00000	0.190500
$993$	0	0
$994$	0	0
$995$	−28.0000	−0.887660
$996$	0	0
$997$	−2.00000	−0.0633406	−0.0316703	−	0.999498i	$-0.510083\pi$
$997$	−2.00000	−0.0633406	−0.0316703	+	0.999498i	$0.510083\pi$
$998$	−29.0000	−0.917979
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7938.2.a.bb.1.1		1
3.2	odd	2		7938.2.a.e.1.1		1
7.6	odd	2		1134.2.a.f.1.1		1
9.2	odd	6		882.2.f.f.589.1		2
9.4	even	3		2646.2.f.b.883.1		2
9.5	odd	6		882.2.f.f.295.1		2
9.7	even	3		2646.2.f.b.1765.1		2
21.20	even	2		1134.2.a.c.1.1		1
28.27	even	2		9072.2.a.f.1.1		1
63.2	odd	6		882.2.h.g.67.1		2
63.4	even	3		2646.2.h.c.667.1		2
63.5	even	6		882.2.e.a.655.1		2
63.11	odd	6		882.2.e.e.373.1		2
63.13	odd	6		378.2.f.b.127.1		2
63.16	even	3		2646.2.h.c.361.1		2
63.20	even	6		126.2.f.b.85.1	yes	2
63.23	odd	6		882.2.e.e.655.1		2
63.25	even	3		2646.2.e.h.1549.1		2
63.31	odd	6		2646.2.h.b.667.1		2
63.32	odd	6		882.2.h.g.79.1		2
63.34	odd	6		378.2.f.b.253.1		2
63.38	even	6		882.2.e.a.373.1		2
63.40	odd	6		2646.2.e.i.2125.1		2
63.41	even	6		126.2.f.b.43.1	✓	2
63.47	even	6		882.2.h.h.67.1		2
63.52	odd	6		2646.2.e.i.1549.1		2
63.58	even	3		2646.2.e.h.2125.1		2
63.59	even	6		882.2.h.h.79.1		2
63.61	odd	6		2646.2.h.b.361.1		2
84.83	odd	2		9072.2.a.t.1.1		1
252.83	odd	6		1008.2.r.a.337.1		2
252.139	even	6		3024.2.r.c.2017.1		2
252.167	odd	6		1008.2.r.a.673.1		2
252.223	even	6		3024.2.r.c.1009.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.2.f.b.43.1	✓	2	63.41	even	6
126.2.f.b.85.1	yes	2	63.20	even	6
378.2.f.b.127.1		2	63.13	odd	6
378.2.f.b.253.1		2	63.34	odd	6
882.2.e.a.373.1		2	63.38	even	6
882.2.e.a.655.1		2	63.5	even	6
882.2.e.e.373.1		2	63.11	odd	6
882.2.e.e.655.1		2	63.23	odd	6
882.2.f.f.295.1		2	9.5	odd	6
882.2.f.f.589.1		2	9.2	odd	6
882.2.h.g.67.1		2	63.2	odd	6
882.2.h.g.79.1		2	63.32	odd	6
882.2.h.h.67.1		2	63.47	even	6
882.2.h.h.79.1		2	63.59	even	6
1008.2.r.a.337.1		2	252.83	odd	6
1008.2.r.a.673.1		2	252.167	odd	6
1134.2.a.c.1.1		1	21.20	even	2
1134.2.a.f.1.1		1	7.6	odd	2
2646.2.e.h.1549.1		2	63.25	even	3
2646.2.e.h.2125.1		2	63.58	even	3
2646.2.e.i.1549.1		2	63.52	odd	6
2646.2.e.i.2125.1		2	63.40	odd	6
2646.2.f.b.883.1		2	9.4	even	3
2646.2.f.b.1765.1		2	9.7	even	3
2646.2.h.b.361.1		2	63.61	odd	6
2646.2.h.b.667.1		2	63.31	odd	6
2646.2.h.c.361.1		2	63.16	even	3
2646.2.h.c.667.1		2	63.4	even	3
3024.2.r.c.1009.1		2	252.223	even	6
3024.2.r.c.2017.1		2	252.139	even	6
7938.2.a.e.1.1		1	3.2	odd	2
7938.2.a.bb.1.1		1	1.1	even	1	trivial
9072.2.a.f.1.1		1	28.27	even	2
9072.2.a.t.1.1		1	84.83	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 7938 = 2 \cdot 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7938.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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