LMFDB - Embedded newform 418.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [418,2,Mod(1,418)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(418, 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("418.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 418 = 2 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	418.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:	$3.33774680449$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	418.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} +2.00000 q^{7} +1.00000 q^{8} -3.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} +2.00000 q^{7} +1.00000 q^{8} -3.00000 q^{9} +2.00000 q^{10} +1.00000 q^{11} -2.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} -3.00000 q^{18} +1.00000 q^{19} +2.00000 q^{20} +1.00000 q^{22} -8.00000 q^{23} -1.00000 q^{25} -2.00000 q^{26} +2.00000 q^{28} -6.00000 q^{29} +6.00000 q^{31} +1.00000 q^{32} +6.00000 q^{34} +4.00000 q^{35} -3.00000 q^{36} +8.00000 q^{37} +1.00000 q^{38} +2.00000 q^{40} +6.00000 q^{41} -8.00000 q^{43} +1.00000 q^{44} -6.00000 q^{45} -8.00000 q^{46} -8.00000 q^{47} -3.00000 q^{49} -1.00000 q^{50} -2.00000 q^{52} +12.0000 q^{53} +2.00000 q^{55} +2.00000 q^{56} -6.00000 q^{58} -8.00000 q^{61} +6.00000 q^{62} -6.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} -8.00000 q^{67} +6.00000 q^{68} +4.00000 q^{70} -6.00000 q^{71} -3.00000 q^{72} -14.0000 q^{73} +8.00000 q^{74} +1.00000 q^{76} +2.00000 q^{77} -12.0000 q^{79} +2.00000 q^{80} +9.00000 q^{81} +6.00000 q^{82} -12.0000 q^{83} +12.0000 q^{85} -8.00000 q^{86} +1.00000 q^{88} +2.00000 q^{89} -6.00000 q^{90} -4.00000 q^{91} -8.00000 q^{92} -8.00000 q^{94} +2.00000 q^{95} -2.00000 q^{97} -3.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	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
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	1.00000	0.353553
$9$	−3.00000	−1.00000
$10$	2.00000	0.632456
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	−3.00000	−0.707107
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	2.00000	0.447214
$21$	0	0
$22$	1.00000	0.213201
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	−2.00000	−0.392232
$27$	0	0
$28$	2.00000	0.377964
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	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
$34$	6.00000	1.02899
$35$	4.00000	0.676123
$36$	−3.00000	−0.500000
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	2.00000	0.316228
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	1.00000	0.150756
$45$	−6.00000	−0.894427
$46$	−8.00000	−1.17954
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−2.00000	−0.277350
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	2.00000	0.267261
$57$	0	0
$58$	−6.00000	−0.787839
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	6.00000	0.762001
$63$	−6.00000	−0.755929
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	6.00000	0.727607
$69$	0	0
$70$	4.00000	0.478091
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	−3.00000	−0.353553
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	8.00000	0.929981
$75$	0	0
$76$	1.00000	0.114708
$77$	2.00000	0.227921
$78$	0	0
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	2.00000	0.223607
$81$	9.00000	1.00000
$82$	6.00000	0.662589
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	12.0000	1.30158
$86$	−8.00000	−0.862662
$87$	0	0
$88$	1.00000	0.106600
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	−6.00000	−0.632456
$91$	−4.00000	−0.419314
$92$	−8.00000	−0.834058
$93$	0	0
$94$	−8.00000	−0.825137
$95$	2.00000	0.205196
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	−3.00000	−0.303046
$99$	−3.00000	−0.301511
$100$	−1.00000	−0.100000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	12.0000	1.16554
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	2.00000	0.190693
$111$	0	0
$112$	2.00000	0.188982
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	0	0
$115$	−16.0000	−1.49201
$116$	−6.00000	−0.557086
$117$	6.00000	0.554700
$118$	0	0
$119$	12.0000	1.10004
$120$	0	0
$121$	1.00000	0.0909091
$122$	−8.00000	−0.724286
$123$	0	0
$124$	6.00000	0.538816
$125$	−12.0000	−1.07331
$126$	−6.00000	−0.534522
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−4.00000	−0.350823
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	2.00000	0.173422
$134$	−8.00000	−0.691095
$135$	0	0
$136$	6.00000	0.514496
$137$	22.0000	1.87959	0.939793	−	0.341743i	$-0.111017\pi$
$137$	22.0000	1.87959	0.939793	+	0.341743i	$0.111017\pi$
$138$	0	0
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	4.00000	0.338062
$141$	0	0
$142$	−6.00000	−0.503509
$143$	−2.00000	−0.167248
$144$	−3.00000	−0.250000
$145$	−12.0000	−0.996546
$146$	−14.0000	−1.15865
$147$	0	0
$148$	8.00000	0.657596
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	1.00000	0.0811107
$153$	−18.0000	−1.45521
$154$	2.00000	0.161165
$155$	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	$0.525431\pi$
$158$	−12.0000	−0.954669
$159$	0	0
$160$	2.00000	0.158114
$161$	−16.0000	−1.26098
$162$	9.00000	0.707107
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	−12.0000	−0.931381
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	12.0000	0.920358
$171$	−3.00000	−0.229416
$172$	−8.00000	−0.609994
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	1.00000	0.0753778
$177$	0	0
$178$	2.00000	0.149906
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	−6.00000	−0.447214
$181$	8.00000	0.594635	0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000	0.594635	0.297318	+	0.954779i	$0.403908\pi$
$182$	−4.00000	−0.296500
$183$	0	0
$184$	−8.00000	−0.589768
$185$	16.0000	1.17634
$186$	0	0
$187$	6.00000	0.438763
$188$	−8.00000	−0.583460
$189$	0	0
$190$	2.00000	0.145095
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	−18.0000	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	−3.00000	−0.214286
$197$	12.0000	0.854965	0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000	0.854965	0.427482	+	0.904024i	$0.359401\pi$
$198$	−3.00000	−0.213201
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	0	0
$203$	−12.0000	−0.842235
$204$	0	0
$205$	12.0000	0.838116
$206$	14.0000	0.975426
$207$	24.0000	1.66812
$208$	−2.00000	−0.138675
$209$	1.00000	0.0691714
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	12.0000	0.824163
$213$	0	0
$214$	−12.0000	−0.820303
$215$	−16.0000	−1.09119
$216$	0	0
$217$	12.0000	0.814613
$218$	10.0000	0.677285
$219$	0	0
$220$	2.00000	0.134840
$221$	−12.0000	−0.807207
$222$	0	0
$223$	−10.0000	−0.669650	−0.334825	−	0.942280i	$-0.608677\pi$
$223$	−10.0000	−0.669650	−0.334825	+	0.942280i	$0.608677\pi$
$224$	2.00000	0.133631
$225$	3.00000	0.200000
$226$	18.0000	1.19734
$227$	−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
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	−16.0000	−1.05501
$231$	0	0
$232$	−6.00000	−0.393919
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	6.00000	0.392232
$235$	−16.0000	−1.04372
$236$	0	0
$237$	0	0
$238$	12.0000	0.777844
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	−8.00000	−0.512148
$245$	−6.00000	−0.383326
$246$	0	0
$247$	−2.00000	−0.127257
$248$	6.00000	0.381000
$249$	0	0
$250$	−12.0000	−0.758947
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	−6.00000	−0.377964
$253$	−8.00000	−0.502956
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	26.0000	1.62184	0.810918	−	0.585160i	$-0.198968\pi$
$257$	26.0000	1.62184	0.810918	+	0.585160i	$0.198968\pi$
$258$	0	0
$259$	16.0000	0.994192
$260$	−4.00000	−0.248069
$261$	18.0000	1.11417
$262$	4.00000	0.247121
$263$	−6.00000	−0.369976	−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000	−0.369976	−0.184988	+	0.982741i	$0.559225\pi$
$264$	0	0
$265$	24.0000	1.47431
$266$	2.00000	0.122628
$267$	0	0
$268$	−8.00000	−0.488678
$269$	−16.0000	−0.975537	−0.487769	−	0.872973i	$-0.662189\pi$
$269$	−16.0000	−0.975537	−0.487769	+	0.872973i	$0.662189\pi$
$270$	0	0
$271$	−6.00000	−0.364474	−0.182237	−	0.983255i	$-0.558334\pi$
$271$	−6.00000	−0.364474	−0.182237	+	0.983255i	$0.558334\pi$
$272$	6.00000	0.363803
$273$	0	0
$274$	22.0000	1.32907
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	16.0000	0.959616
$279$	−18.0000	−1.07763
$280$	4.00000	0.239046
$281$	26.0000	1.55103	0.775515	−	0.631329i	$-0.217490\pi$
$281$	26.0000	1.55103	0.775515	+	0.631329i	$0.217490\pi$
$282$	0	0
$283$	28.0000	1.66443	0.832214	−	0.554455i	$-0.187073\pi$
$283$	28.0000	1.66443	0.832214	+	0.554455i	$0.187073\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	−2.00000	−0.118262
$287$	12.0000	0.708338
$288$	−3.00000	−0.176777
$289$	19.0000	1.11765
$290$	−12.0000	−0.704664
$291$	0	0
$292$	−14.0000	−0.819288
$293$	10.0000	0.584206	0.292103	−	0.956387i	$-0.405645\pi$
$293$	10.0000	0.584206	0.292103	+	0.956387i	$0.405645\pi$
$294$	0	0
$295$	0	0
$296$	8.00000	0.464991
$297$	0	0
$298$	0	0
$299$	16.0000	0.925304
$300$	0	0
$301$	−16.0000	−0.922225
$302$	12.0000	0.690522
$303$	0	0
$304$	1.00000	0.0573539
$305$	−16.0000	−0.916157
$306$	−18.0000	−1.02899
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	2.00000	0.113961
$309$	0	0
$310$	12.0000	0.681554
$311$	−28.0000	−1.58773	−0.793867	−	0.608091i	$-0.791935\pi$
$311$	−28.0000	−1.58773	−0.793867	+	0.608091i	$0.791935\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	−2.00000	−0.112867
$315$	−12.0000	−0.676123
$316$	−12.0000	−0.675053
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	0	0
$319$	−6.00000	−0.335936
$320$	2.00000	0.111803
$321$	0	0
$322$	−16.0000	−0.891645
$323$	6.00000	0.333849
$324$	9.00000	0.500000
$325$	2.00000	0.110940
$326$	12.0000	0.664619
$327$	0	0
$328$	6.00000	0.331295
$329$	−16.0000	−0.882109
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	−12.0000	−0.658586
$333$	−24.0000	−1.31519
$334$	8.00000	0.437741
$335$	−16.0000	−0.874173
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	−9.00000	−0.489535
$339$	0	0
$340$	12.0000	0.650791
$341$	6.00000	0.324918
$342$	−3.00000	−0.162221
$343$	−20.0000	−1.07990
$344$	−8.00000	−0.431331
$345$	0	0
$346$	14.0000	0.752645
$347$	24.0000	1.28839	0.644194	−	0.764862i	$-0.277193\pi$
$347$	24.0000	1.28839	0.644194	+	0.764862i	$0.277193\pi$
$348$	0	0
$349$	−8.00000	−0.428230	−0.214115	−	0.976808i	$-0.568687\pi$
$349$	−8.00000	−0.428230	−0.214115	+	0.976808i	$0.568687\pi$
$350$	−2.00000	−0.106904
$351$	0	0
$352$	1.00000	0.0533002
$353$	−2.00000	−0.106449	−0.0532246	−	0.998583i	$-0.516950\pi$
$353$	−2.00000	−0.106449	−0.0532246	+	0.998583i	$0.516950\pi$
$354$	0	0
$355$	−12.0000	−0.636894
$356$	2.00000	0.106000
$357$	0	0
$358$	20.0000	1.05703
$359$	−14.0000	−0.738892	−0.369446	−	0.929252i	$-0.620452\pi$
$359$	−14.0000	−0.738892	−0.369446	+	0.929252i	$0.620452\pi$
$360$	−6.00000	−0.316228
$361$	1.00000	0.0526316
$362$	8.00000	0.420471
$363$	0	0
$364$	−4.00000	−0.209657
$365$	−28.0000	−1.46559
$366$	0	0
$367$	−4.00000	−0.208798	−0.104399	−	0.994535i	$-0.533292\pi$
$367$	−4.00000	−0.208798	−0.104399	+	0.994535i	$0.533292\pi$
$368$	−8.00000	−0.417029
$369$	−18.0000	−0.937043
$370$	16.0000	0.831800
$371$	24.0000	1.24602
$372$	0	0
$373$	−34.0000	−1.76045	−0.880227	−	0.474554i	$-0.842610\pi$
$373$	−34.0000	−1.76045	−0.880227	+	0.474554i	$0.842610\pi$
$374$	6.00000	0.310253
$375$	0	0
$376$	−8.00000	−0.412568
$377$	12.0000	0.618031
$378$	0	0
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	2.00000	0.102598
$381$	0	0
$382$	12.0000	0.613973
$383$	−6.00000	−0.306586	−0.153293	−	0.988181i	$-0.548988\pi$
$383$	−6.00000	−0.306586	−0.153293	+	0.988181i	$0.548988\pi$
$384$	0	0
$385$	4.00000	0.203859
$386$	−18.0000	−0.916176
$387$	24.0000	1.21999
$388$	−2.00000	−0.101535
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	−48.0000	−2.42746
$392$	−3.00000	−0.151523
$393$	0	0
$394$	12.0000	0.604551
$395$	−24.0000	−1.20757
$396$	−3.00000	−0.150756
$397$	38.0000	1.90717	0.953583	−	0.301131i	$-0.0973643\pi$
$397$	38.0000	1.90717	0.953583	+	0.301131i	$0.0973643\pi$
$398$	−24.0000	−1.20301
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−10.0000	−0.499376	−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000	−0.499376	−0.249688	+	0.968326i	$0.580328\pi$
$402$	0	0
$403$	−12.0000	−0.597763
$404$	0	0
$405$	18.0000	0.894427
$406$	−12.0000	−0.595550
$407$	8.00000	0.396545
$408$	0	0
$409$	2.00000	0.0988936	0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000	0.0988936	0.0494468	+	0.998777i	$0.484254\pi$
$410$	12.0000	0.592638
$411$	0	0
$412$	14.0000	0.689730
$413$	0	0
$414$	24.0000	1.17954
$415$	−24.0000	−1.17811
$416$	−2.00000	−0.0980581
$417$	0	0
$418$	1.00000	0.0489116
$419$	−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
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	−12.0000	−0.584151
$423$	24.0000	1.16692
$424$	12.0000	0.582772
$425$	−6.00000	−0.291043
$426$	0	0
$427$	−16.0000	−0.774294
$428$	−12.0000	−0.580042
$429$	0	0
$430$	−16.0000	−0.771589
$431$	−20.0000	−0.963366	−0.481683	−	0.876346i	$-0.659974\pi$
$431$	−20.0000	−0.963366	−0.481683	+	0.876346i	$0.659974\pi$
$432$	0	0
$433$	22.0000	1.05725	0.528626	−	0.848855i	$-0.322707\pi$
$433$	22.0000	1.05725	0.528626	+	0.848855i	$0.322707\pi$
$434$	12.0000	0.576018
$435$	0	0
$436$	10.0000	0.478913
$437$	−8.00000	−0.382692
$438$	0	0
$439$	−24.0000	−1.14546	−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000	−1.14546	−0.572729	+	0.819745i	$0.694115\pi$
$440$	2.00000	0.0953463
$441$	9.00000	0.428571
$442$	−12.0000	−0.570782
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	0	0
$445$	4.00000	0.189618
$446$	−10.0000	−0.473514
$447$	0	0
$448$	2.00000	0.0944911
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	3.00000	0.141421
$451$	6.00000	0.282529
$452$	18.0000	0.846649
$453$	0	0
$454$	−4.00000	−0.187729
$455$	−8.00000	−0.375046
$456$	0	0
$457$	−26.0000	−1.21623	−0.608114	−	0.793849i	$-0.708074\pi$
$457$	−26.0000	−1.21623	−0.608114	+	0.793849i	$0.708074\pi$
$458$	2.00000	0.0934539
$459$	0	0
$460$	−16.0000	−0.746004
$461$	−16.0000	−0.745194	−0.372597	−	0.927993i	$-0.621533\pi$
$461$	−16.0000	−0.745194	−0.372597	+	0.927993i	$0.621533\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	18.0000	0.833834
$467$	20.0000	0.925490	0.462745	−	0.886492i	$-0.346865\pi$
$467$	20.0000	0.925490	0.462745	+	0.886492i	$0.346865\pi$
$468$	6.00000	0.277350
$469$	−16.0000	−0.738811
$470$	−16.0000	−0.738025
$471$	0	0
$472$	0	0
$473$	−8.00000	−0.367840
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	12.0000	0.550019
$477$	−36.0000	−1.64833
$478$	6.00000	0.274434
$479$	14.0000	0.639676	0.319838	−	0.947472i	$-0.396371\pi$
$479$	14.0000	0.639676	0.319838	+	0.947472i	$0.396371\pi$
$480$	0	0
$481$	−16.0000	−0.729537
$482$	−22.0000	−1.00207
$483$	0	0
$484$	1.00000	0.0454545
$485$	−4.00000	−0.181631
$486$	0	0
$487$	−18.0000	−0.815658	−0.407829	−	0.913058i	$-0.633714\pi$
$487$	−18.0000	−0.815658	−0.407829	+	0.913058i	$0.633714\pi$
$488$	−8.00000	−0.362143
$489$	0	0
$490$	−6.00000	−0.271052
$491$	−8.00000	−0.361035	−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000	−0.361035	−0.180517	+	0.983572i	$0.557777\pi$
$492$	0	0
$493$	−36.0000	−1.62136
$494$	−2.00000	−0.0899843
$495$	−6.00000	−0.269680
$496$	6.00000	0.269408
$497$	−12.0000	−0.538274
$498$	0	0
$499$	36.0000	1.61158	0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000	1.61158	0.805791	+	0.592200i	$0.201741\pi$
$500$	−12.0000	−0.536656
$501$	0	0
$502$	12.0000	0.535586
$503$	−26.0000	−1.15928	−0.579641	−	0.814872i	$-0.696807\pi$
$503$	−26.0000	−1.15928	−0.579641	+	0.814872i	$0.696807\pi$
$504$	−6.00000	−0.267261
$505$	0	0
$506$	−8.00000	−0.355643
$507$	0	0
$508$	−8.00000	−0.354943
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	0	0
$511$	−28.0000	−1.23865
$512$	1.00000	0.0441942
$513$	0	0
$514$	26.0000	1.14681
$515$	28.0000	1.23383
$516$	0	0
$517$	−8.00000	−0.351840
$518$	16.0000	0.703000
$519$	0	0
$520$	−4.00000	−0.175412
$521$	−34.0000	−1.48957	−0.744784	−	0.667306i	$-0.767447\pi$
$521$	−34.0000	−1.48957	−0.744784	+	0.667306i	$0.767447\pi$
$522$	18.0000	0.787839
$523$	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
$526$	−6.00000	−0.261612
$527$	36.0000	1.56818
$528$	0	0
$529$	41.0000	1.78261
$530$	24.0000	1.04249
$531$	0	0
$532$	2.00000	0.0867110
$533$	−12.0000	−0.519778
$534$	0	0
$535$	−24.0000	−1.03761
$536$	−8.00000	−0.345547
$537$	0	0
$538$	−16.0000	−0.689809
$539$	−3.00000	−0.129219
$540$	0	0
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	−6.00000	−0.257722
$543$	0	0
$544$	6.00000	0.257248
$545$	20.0000	0.856706
$546$	0	0
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	22.0000	0.939793
$549$	24.0000	1.02430
$550$	−1.00000	−0.0426401
$551$	−6.00000	−0.255609
$552$	0	0
$553$	−24.0000	−1.02058
$554$	8.00000	0.339887
$555$	0	0
$556$	16.0000	0.678551
$557$	−8.00000	−0.338971	−0.169485	−	0.985533i	$-0.554211\pi$
$557$	−8.00000	−0.338971	−0.169485	+	0.985533i	$0.554211\pi$
$558$	−18.0000	−0.762001
$559$	16.0000	0.676728
$560$	4.00000	0.169031
$561$	0	0
$562$	26.0000	1.09674
$563$	28.0000	1.18006	0.590030	−	0.807382i	$-0.299116\pi$
$563$	28.0000	1.18006	0.590030	+	0.807382i	$0.299116\pi$
$564$	0	0
$565$	36.0000	1.51453
$566$	28.0000	1.17693
$567$	18.0000	0.755929
$568$	−6.00000	−0.251754
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	−2.00000	−0.0836242
$573$	0	0
$574$	12.0000	0.500870
$575$	8.00000	0.333623
$576$	−3.00000	−0.125000
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	19.0000	0.790296
$579$	0	0
$580$	−12.0000	−0.498273
$581$	−24.0000	−0.995688
$582$	0	0
$583$	12.0000	0.496989
$584$	−14.0000	−0.579324
$585$	12.0000	0.496139
$586$	10.0000	0.413096
$587$	4.00000	0.165098	0.0825488	−	0.996587i	$-0.473694\pi$
$587$	4.00000	0.165098	0.0825488	+	0.996587i	$0.473694\pi$
$588$	0	0
$589$	6.00000	0.247226
$590$	0	0
$591$	0	0
$592$	8.00000	0.328798
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	0	0
$595$	24.0000	0.983904
$596$	0	0
$597$	0	0
$598$	16.0000	0.654289
$599$	30.0000	1.22577	0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000	1.22577	0.612883	+	0.790173i	$0.290010\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	−16.0000	−0.652111
$603$	24.0000	0.977356
$604$	12.0000	0.488273
$605$	2.00000	0.0813116
$606$	0	0
$607$	28.0000	1.13648	0.568242	−	0.822861i	$-0.307624\pi$
$607$	28.0000	1.13648	0.568242	+	0.822861i	$0.307624\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	−16.0000	−0.647821
$611$	16.0000	0.647291
$612$	−18.0000	−0.727607
$613$	4.00000	0.161558	0.0807792	−	0.996732i	$-0.474259\pi$
$613$	4.00000	0.161558	0.0807792	+	0.996732i	$0.474259\pi$
$614$	4.00000	0.161427
$615$	0	0
$616$	2.00000	0.0805823
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	12.0000	0.481932
$621$	0	0
$622$	−28.0000	−1.12270
$623$	4.00000	0.160257
$624$	0	0
$625$	−19.0000	−0.760000
$626$	6.00000	0.239808
$627$	0	0
$628$	−2.00000	−0.0798087
$629$	48.0000	1.91389
$630$	−12.0000	−0.478091
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	−12.0000	−0.477334
$633$	0	0
$634$	12.0000	0.476581
$635$	−16.0000	−0.634941
$636$	0	0
$637$	6.00000	0.237729
$638$	−6.00000	−0.237542
$639$	18.0000	0.712069
$640$	2.00000	0.0790569
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	12.0000	0.473234	0.236617	−	0.971603i	$-0.423961\pi$
$643$	12.0000	0.473234	0.236617	+	0.971603i	$0.423961\pi$
$644$	−16.0000	−0.630488
$645$	0	0
$646$	6.00000	0.236067
$647$	−32.0000	−1.25805	−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000	−1.25805	−0.629025	+	0.777385i	$0.716546\pi$
$648$	9.00000	0.353553
$649$	0	0
$650$	2.00000	0.0784465
$651$	0	0
$652$	12.0000	0.469956
$653$	42.0000	1.64359	0.821794	−	0.569785i	$-0.192974\pi$
$653$	42.0000	1.64359	0.821794	+	0.569785i	$0.192974\pi$
$654$	0	0
$655$	8.00000	0.312586
$656$	6.00000	0.234261
$657$	42.0000	1.63858
$658$	−16.0000	−0.623745
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	0	0
$661$	−28.0000	−1.08907	−0.544537	−	0.838737i	$-0.683295\pi$
$661$	−28.0000	−1.08907	−0.544537	+	0.838737i	$0.683295\pi$
$662$	−12.0000	−0.466393
$663$	0	0
$664$	−12.0000	−0.465690
$665$	4.00000	0.155113
$666$	−24.0000	−0.929981
$667$	48.0000	1.85857
$668$	8.00000	0.309529
$669$	0	0
$670$	−16.0000	−0.618134
$671$	−8.00000	−0.308837
$672$	0	0
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	−14.0000	−0.539260
$675$	0	0
$676$	−9.00000	−0.346154
$677$	−30.0000	−1.15299	−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000	−1.15299	−0.576497	+	0.817099i	$0.695581\pi$
$678$	0	0
$679$	−4.00000	−0.153506
$680$	12.0000	0.460179
$681$	0	0
$682$	6.00000	0.229752
$683$	−20.0000	−0.765279	−0.382639	−	0.923898i	$-0.624985\pi$
$683$	−20.0000	−0.765279	−0.382639	+	0.923898i	$0.624985\pi$
$684$	−3.00000	−0.114708
$685$	44.0000	1.68115
$686$	−20.0000	−0.763604
$687$	0	0
$688$	−8.00000	−0.304997
$689$	−24.0000	−0.914327
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	14.0000	0.532200
$693$	−6.00000	−0.227921
$694$	24.0000	0.911028
$695$	32.0000	1.21383
$696$	0	0
$697$	36.0000	1.36360
$698$	−8.00000	−0.302804
$699$	0	0
$700$	−2.00000	−0.0755929
$701$	−20.0000	−0.755390	−0.377695	−	0.925930i	$-0.623283\pi$
$701$	−20.0000	−0.755390	−0.377695	+	0.925930i	$0.623283\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	1.00000	0.0376889
$705$	0	0
$706$	−2.00000	−0.0752710
$707$	0	0
$708$	0	0
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	−12.0000	−0.450352
$711$	36.0000	1.35011
$712$	2.00000	0.0749532
$713$	−48.0000	−1.79761
$714$	0	0
$715$	−4.00000	−0.149592
$716$	20.0000	0.747435
$717$	0	0
$718$	−14.0000	−0.522475
$719$	−44.0000	−1.64092	−0.820462	−	0.571702i	$-0.806283\pi$
$719$	−44.0000	−1.64092	−0.820462	+	0.571702i	$0.806283\pi$
$720$	−6.00000	−0.223607
$721$	28.0000	1.04277
$722$	1.00000	0.0372161
$723$	0	0
$724$	8.00000	0.297318
$725$	6.00000	0.222834
$726$	0	0
$727$	−4.00000	−0.148352	−0.0741759	−	0.997245i	$-0.523633\pi$
$727$	−4.00000	−0.148352	−0.0741759	+	0.997245i	$0.523633\pi$
$728$	−4.00000	−0.148250
$729$	−27.0000	−1.00000
$730$	−28.0000	−1.03633
$731$	−48.0000	−1.77534
$732$	0	0
$733$	−48.0000	−1.77292	−0.886460	−	0.462805i	$-0.846843\pi$
$733$	−48.0000	−1.77292	−0.886460	+	0.462805i	$0.846843\pi$
$734$	−4.00000	−0.147643
$735$	0	0
$736$	−8.00000	−0.294884
$737$	−8.00000	−0.294684
$738$	−18.0000	−0.662589
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	16.0000	0.588172
$741$	0	0
$742$	24.0000	0.881068
$743$	48.0000	1.76095	0.880475	−	0.474093i	$-0.157224\pi$
$743$	48.0000	1.76095	0.880475	+	0.474093i	$0.157224\pi$
$744$	0	0
$745$	0	0
$746$	−34.0000	−1.24483
$747$	36.0000	1.31717
$748$	6.00000	0.219382
$749$	−24.0000	−0.876941
$750$	0	0
$751$	−6.00000	−0.218943	−0.109472	−	0.993990i	$-0.534916\pi$
$751$	−6.00000	−0.218943	−0.109472	+	0.993990i	$0.534916\pi$
$752$	−8.00000	−0.291730
$753$	0	0
$754$	12.0000	0.437014
$755$	24.0000	0.873449
$756$	0	0
$757$	−6.00000	−0.218074	−0.109037	−	0.994038i	$-0.534777\pi$
$757$	−6.00000	−0.218074	−0.109037	+	0.994038i	$0.534777\pi$
$758$	−4.00000	−0.145287
$759$	0	0
$760$	2.00000	0.0725476
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	0	0
$763$	20.0000	0.724049
$764$	12.0000	0.434145
$765$	−36.0000	−1.30158
$766$	−6.00000	−0.216789
$767$	0	0
$768$	0	0
$769$	−18.0000	−0.649097	−0.324548	−	0.945869i	$-0.605212\pi$
$769$	−18.0000	−0.649097	−0.324548	+	0.945869i	$0.605212\pi$
$770$	4.00000	0.144150
$771$	0	0
$772$	−18.0000	−0.647834
$773$	−8.00000	−0.287740	−0.143870	−	0.989597i	$-0.545955\pi$
$773$	−8.00000	−0.287740	−0.143870	+	0.989597i	$0.545955\pi$
$774$	24.0000	0.862662
$775$	−6.00000	−0.215526
$776$	−2.00000	−0.0717958
$777$	0	0
$778$	−6.00000	−0.215110
$779$	6.00000	0.214972
$780$	0	0
$781$	−6.00000	−0.214697
$782$	−48.0000	−1.71648
$783$	0	0
$784$	−3.00000	−0.107143
$785$	−4.00000	−0.142766
$786$	0	0
$787$	20.0000	0.712923	0.356462	−	0.934310i	$-0.383983\pi$
$787$	20.0000	0.712923	0.356462	+	0.934310i	$0.383983\pi$
$788$	12.0000	0.427482
$789$	0	0
$790$	−24.0000	−0.853882
$791$	36.0000	1.28001
$792$	−3.00000	−0.106600
$793$	16.0000	0.568177
$794$	38.0000	1.34857
$795$	0	0
$796$	−24.0000	−0.850657
$797$	40.0000	1.41687	0.708436	−	0.705775i	$-0.249401\pi$
$797$	40.0000	1.41687	0.708436	+	0.705775i	$0.249401\pi$
$798$	0	0
$799$	−48.0000	−1.69812
$800$	−1.00000	−0.0353553
$801$	−6.00000	−0.212000
$802$	−10.0000	−0.353112
$803$	−14.0000	−0.494049
$804$	0	0
$805$	−32.0000	−1.12785
$806$	−12.0000	−0.422682
$807$	0	0
$808$	0	0
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	18.0000	0.632456
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	−12.0000	−0.421117
$813$	0	0
$814$	8.00000	0.280400
$815$	24.0000	0.840683
$816$	0	0
$817$	−8.00000	−0.279885
$818$	2.00000	0.0699284
$819$	12.0000	0.419314
$820$	12.0000	0.419058
$821$	8.00000	0.279202	0.139601	−	0.990208i	$-0.455418\pi$
$821$	8.00000	0.279202	0.139601	+	0.990208i	$0.455418\pi$
$822$	0	0
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	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	$0.566904\pi$
$828$	24.0000	0.834058
$829$	−20.0000	−0.694629	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000	−0.694629	−0.347314	+	0.937749i	$0.612906\pi$
$830$	−24.0000	−0.833052
$831$	0	0
$832$	−2.00000	−0.0693375
$833$	−18.0000	−0.623663
$834$	0	0
$835$	16.0000	0.553703
$836$	1.00000	0.0345857
$837$	0	0
$838$	−12.0000	−0.414533
$839$	−30.0000	−1.03572	−0.517858	−	0.855467i	$-0.673270\pi$
$839$	−30.0000	−1.03572	−0.517858	+	0.855467i	$0.673270\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	8.00000	0.275698
$843$	0	0
$844$	−12.0000	−0.413057
$845$	−18.0000	−0.619219
$846$	24.0000	0.825137
$847$	2.00000	0.0687208
$848$	12.0000	0.412082
$849$	0	0
$850$	−6.00000	−0.205798
$851$	−64.0000	−2.19389
$852$	0	0
$853$	−40.0000	−1.36957	−0.684787	−	0.728743i	$-0.740105\pi$
$853$	−40.0000	−1.36957	−0.684787	+	0.728743i	$0.740105\pi$
$854$	−16.0000	−0.547509
$855$	−6.00000	−0.205196
$856$	−12.0000	−0.410152
$857$	−14.0000	−0.478231	−0.239115	−	0.970991i	$-0.576857\pi$
$857$	−14.0000	−0.478231	−0.239115	+	0.970991i	$0.576857\pi$
$858$	0	0
$859$	28.0000	0.955348	0.477674	−	0.878537i	$-0.341480\pi$
$859$	28.0000	0.955348	0.477674	+	0.878537i	$0.341480\pi$
$860$	−16.0000	−0.545595
$861$	0	0
$862$	−20.0000	−0.681203
$863$	−54.0000	−1.83818	−0.919091	−	0.394046i	$-0.871075\pi$
$863$	−54.0000	−1.83818	−0.919091	+	0.394046i	$0.871075\pi$
$864$	0	0
$865$	28.0000	0.952029
$866$	22.0000	0.747590
$867$	0	0
$868$	12.0000	0.407307
$869$	−12.0000	−0.407072
$870$	0	0
$871$	16.0000	0.542139
$872$	10.0000	0.338643
$873$	6.00000	0.203069
$874$	−8.00000	−0.270604
$875$	−24.0000	−0.811348
$876$	0	0
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	−24.0000	−0.809961
$879$	0	0
$880$	2.00000	0.0674200
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	9.00000	0.303046
$883$	−36.0000	−1.21150	−0.605748	−	0.795656i	$-0.707126\pi$
$883$	−36.0000	−1.21150	−0.605748	+	0.795656i	$0.707126\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	36.0000	1.20944
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	4.00000	0.134080
$891$	9.00000	0.301511
$892$	−10.0000	−0.334825
$893$	−8.00000	−0.267710
$894$	0	0
$895$	40.0000	1.33705
$896$	2.00000	0.0668153
$897$	0	0
$898$	6.00000	0.200223
$899$	−36.0000	−1.20067
$900$	3.00000	0.100000
$901$	72.0000	2.39867
$902$	6.00000	0.199778
$903$	0	0
$904$	18.0000	0.598671
$905$	16.0000	0.531858
$906$	0	0
$907$	−32.0000	−1.06254	−0.531271	−	0.847202i	$-0.678286\pi$
$907$	−32.0000	−1.06254	−0.531271	+	0.847202i	$0.678286\pi$
$908$	−4.00000	−0.132745
$909$	0	0
$910$	−8.00000	−0.265197
$911$	−2.00000	−0.0662630	−0.0331315	−	0.999451i	$-0.510548\pi$
$911$	−2.00000	−0.0662630	−0.0331315	+	0.999451i	$0.510548\pi$
$912$	0	0
$913$	−12.0000	−0.397142
$914$	−26.0000	−0.860004
$915$	0	0
$916$	2.00000	0.0660819
$917$	8.00000	0.264183
$918$	0	0
$919$	38.0000	1.25350	0.626752	−	0.779219i	$-0.284384\pi$
$919$	38.0000	1.25350	0.626752	+	0.779219i	$0.284384\pi$
$920$	−16.0000	−0.527504
$921$	0	0
$922$	−16.0000	−0.526932
$923$	12.0000	0.394985
$924$	0	0
$925$	−8.00000	−0.263038
$926$	8.00000	0.262896
$927$	−42.0000	−1.37946
$928$	−6.00000	−0.196960
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	0	0
$931$	−3.00000	−0.0983210
$932$	18.0000	0.589610
$933$	0	0
$934$	20.0000	0.654420
$935$	12.0000	0.392442
$936$	6.00000	0.196116
$937$	18.0000	0.588034	0.294017	−	0.955800i	$-0.405008\pi$
$937$	18.0000	0.588034	0.294017	+	0.955800i	$0.405008\pi$
$938$	−16.0000	−0.522419
$939$	0	0
$940$	−16.0000	−0.521862
$941$	−34.0000	−1.10837	−0.554184	−	0.832394i	$-0.686970\pi$
$941$	−34.0000	−1.10837	−0.554184	+	0.832394i	$0.686970\pi$
$942$	0	0
$943$	−48.0000	−1.56310
$944$	0	0
$945$	0	0
$946$	−8.00000	−0.260102
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	0	0
$949$	28.0000	0.908918
$950$	−1.00000	−0.0324443
$951$	0	0
$952$	12.0000	0.388922
$953$	38.0000	1.23094	0.615470	−	0.788160i	$-0.288966\pi$
$953$	38.0000	1.23094	0.615470	+	0.788160i	$0.288966\pi$
$954$	−36.0000	−1.16554
$955$	24.0000	0.776622
$956$	6.00000	0.194054
$957$	0	0
$958$	14.0000	0.452319
$959$	44.0000	1.42083
$960$	0	0
$961$	5.00000	0.161290
$962$	−16.0000	−0.515861
$963$	36.0000	1.16008
$964$	−22.0000	−0.708572
$965$	−36.0000	−1.15888
$966$	0	0
$967$	−50.0000	−1.60789	−0.803946	−	0.594703i	$-0.797270\pi$
$967$	−50.0000	−1.60789	−0.803946	+	0.594703i	$0.797270\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	−4.00000	−0.128432
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	0	0
$973$	32.0000	1.02587
$974$	−18.0000	−0.576757
$975$	0	0
$976$	−8.00000	−0.256074
$977$	62.0000	1.98356	0.991778	−	0.127971i	$-0.0408466\pi$
$977$	62.0000	1.98356	0.991778	+	0.127971i	$0.0408466\pi$
$978$	0	0
$979$	2.00000	0.0639203
$980$	−6.00000	−0.191663
$981$	−30.0000	−0.957826
$982$	−8.00000	−0.255290
$983$	10.0000	0.318950	0.159475	−	0.987202i	$-0.449020\pi$
$983$	10.0000	0.318950	0.159475	+	0.987202i	$0.449020\pi$
$984$	0	0
$985$	24.0000	0.764704
$986$	−36.0000	−1.14647
$987$	0	0
$988$	−2.00000	−0.0636285
$989$	64.0000	2.03508
$990$	−6.00000	−0.190693
$991$	22.0000	0.698853	0.349427	−	0.936964i	$-0.386376\pi$
$991$	22.0000	0.698853	0.349427	+	0.936964i	$0.386376\pi$
$992$	6.00000	0.190500
$993$	0	0
$994$	−12.0000	−0.380617
$995$	−48.0000	−1.52170
$996$	0	0
$997$	48.0000	1.52018	0.760088	−	0.649821i	$-0.225156\pi$
$997$	48.0000	1.52018	0.760088	+	0.649821i	$0.225156\pi$
$998$	36.0000	1.13956
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	418.2.a.b.1.1	✓	1
3.2	odd	2		3762.2.a.c.1.1		1
4.3	odd	2		3344.2.a.g.1.1		1
11.10	odd	2		4598.2.a.f.1.1		1
19.18	odd	2		7942.2.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.a.b.1.1	✓	1	1.1	even	1	trivial
3344.2.a.g.1.1		1	4.3	odd	2
3762.2.a.c.1.1		1	3.2	odd	2
4598.2.a.f.1.1		1	11.10	odd	2
7942.2.a.g.1.1		1	19.18	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:	\( N \)	\(=\)	\( 418 = 2 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	418.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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