LMFDB - Embedded newform 6930.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6930,2,Mod(1,6930)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6930, 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("6930.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6930.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:	$55.3363286007$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2310)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6930.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} +1.00000 q^{11} -6.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} -4.00000 q^{19} +1.00000 q^{20} -1.00000 q^{22} +4.00000 q^{23} +1.00000 q^{25} +6.00000 q^{26} -1.00000 q^{28} -2.00000 q^{29} -1.00000 q^{32} -6.00000 q^{34} -1.00000 q^{35} +2.00000 q^{37} +4.00000 q^{38} -1.00000 q^{40} +2.00000 q^{41} -4.00000 q^{43} +1.00000 q^{44} -4.00000 q^{46} +8.00000 q^{47} +1.00000 q^{49} -1.00000 q^{50} -6.00000 q^{52} -2.00000 q^{53} +1.00000 q^{55} +1.00000 q^{56} +2.00000 q^{58} -8.00000 q^{59} +2.00000 q^{61} +1.00000 q^{64} -6.00000 q^{65} +4.00000 q^{67} +6.00000 q^{68} +1.00000 q^{70} -8.00000 q^{71} -6.00000 q^{73} -2.00000 q^{74} -4.00000 q^{76} -1.00000 q^{77} +12.0000 q^{79} +1.00000 q^{80} -2.00000 q^{82} +6.00000 q^{85} +4.00000 q^{86} -1.00000 q^{88} -2.00000 q^{89} +6.00000 q^{91} +4.00000 q^{92} -8.00000 q^{94} -4.00000 q^{95} -2.00000 q^{97} -1.00000 q^{98} +O(q^{100})$

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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	1.00000	0.267261
$15$	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	$0.240633\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−1.00000	−0.213201
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	6.00000	1.17670
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−6.00000	−1.02899
$35$	−1.00000	−0.169031
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	4.00000	0.648886
$39$	0	0
$40$	−1.00000	−0.158114
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−4.00000	−0.589768
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−6.00000	−0.832050
$53$	−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
$55$	1.00000	0.134840
$56$	1.00000	0.133631
$57$	0	0
$58$	2.00000	0.262613
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	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
$63$	0	0
$64$	1.00000	0.125000
$65$	−6.00000	−0.744208
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	6.00000	0.727607
$69$	0	0
$70$	1.00000	0.119523
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	−4.00000	−0.458831
$77$	−1.00000	−0.113961
$78$	0	0
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−2.00000	−0.220863
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	6.00000	0.650791
$86$	4.00000	0.431331
$87$	0	0
$88$	−1.00000	−0.106600
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	6.00000	0.628971
$92$	4.00000	0.417029
$93$	0	0
$94$	−8.00000	−0.825137
$95$	−4.00000	−0.410391
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	2.00000	0.194257
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	−1.00000	−0.0953463
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	−2.00000	−0.185695
$117$	0	0
$118$	8.00000	0.736460
$119$	−6.00000	−0.550019
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.00000	−0.181071
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	6.00000	0.526235
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	−4.00000	−0.345547
$135$	0	0
$136$	−6.00000	−0.514496
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	−1.00000	−0.0845154
$141$	0	0
$142$	8.00000	0.671345
$143$	−6.00000	−0.501745
$144$	0	0
$145$	−2.00000	−0.166091
$146$	6.00000	0.496564
$147$	0	0
$148$	2.00000	0.164399
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	4.00000	0.324443
$153$	0	0
$154$	1.00000	0.0805823
$155$	0	0
$156$	0	0
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	−12.0000	−0.954669
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	−4.00000	−0.315244
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	−6.00000	−0.460179
$171$	0	0
$172$	−4.00000	−0.304997
$173$	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
$175$	−1.00000	−0.0755929
$176$	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	$0.231285\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	−6.00000	−0.444750
$183$	0	0
$184$	−4.00000	−0.294884
$185$	2.00000	0.147043
$186$	0	0
$187$	6.00000	0.438763
$188$	8.00000	0.583460
$189$	0	0
$190$	4.00000	0.290191
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	1.00000	0.0714286
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−6.00000	−0.422159
$203$	2.00000	0.140372
$204$	0	0
$205$	2.00000	0.139686
$206$	−8.00000	−0.557386
$207$	0	0
$208$	−6.00000	−0.416025
$209$	−4.00000	−0.276686
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	−2.00000	−0.137361
$213$	0	0
$214$	−12.0000	−0.820303
$215$	−4.00000	−0.272798
$216$	0	0
$217$	0	0
$218$	14.0000	0.948200
$219$	0	0
$220$	1.00000	0.0674200
$221$	−36.0000	−2.42162
$222$	0	0
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−14.0000	−0.931266
$227$	−24.0000	−1.59294	−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000	−1.59294	−0.796468	+	0.604681i	$0.793301\pi$
$228$	0	0
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	−4.00000	−0.263752
$231$	0	0
$232$	2.00000	0.131306
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	−8.00000	−0.520756
$237$	0	0
$238$	6.00000	0.388922
$239$	4.00000	0.258738	0.129369	−	0.991596i	$-0.458705\pi$
$239$	4.00000	0.258738	0.129369	+	0.991596i	$0.458705\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	2.00000	0.128037
$245$	1.00000	0.0638877
$246$	0	0
$247$	24.0000	1.52708
$248$	0	0
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	10.0000	0.623783	0.311891	−	0.950118i	$-0.399037\pi$
$257$	10.0000	0.623783	0.311891	+	0.950118i	$0.399037\pi$
$258$	0	0
$259$	−2.00000	−0.124274
$260$	−6.00000	−0.372104
$261$	0	0
$262$	−12.0000	−0.741362
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	−2.00000	−0.122859
$266$	−4.00000	−0.245256
$267$	0	0
$268$	4.00000	0.244339
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	−24.0000	−1.45790	−0.728948	−	0.684569i	$-0.759990\pi$
$271$	−24.0000	−1.45790	−0.728948	+	0.684569i	$0.759990\pi$
$272$	6.00000	0.363803
$273$	0	0
$274$	−6.00000	−0.362473
$275$	1.00000	0.0603023
$276$	0	0
$277$	14.0000	0.841178	0.420589	−	0.907251i	$-0.361823\pi$
$277$	14.0000	0.841178	0.420589	+	0.907251i	$0.361823\pi$
$278$	12.0000	0.719712
$279$	0	0
$280$	1.00000	0.0597614
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	32.0000	1.90220	0.951101	−	0.308879i	$-0.0999539\pi$
$283$	32.0000	1.90220	0.951101	+	0.308879i	$0.0999539\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	6.00000	0.354787
$287$	−2.00000	−0.118056
$288$	0	0
$289$	19.0000	1.11765
$290$	2.00000	0.117444
$291$	0	0
$292$	−6.00000	−0.351123
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	−2.00000	−0.116248
$297$	0	0
$298$	18.0000	1.04271
$299$	−24.0000	−1.38796
$300$	0	0
$301$	4.00000	0.230556
$302$	4.00000	0.230174
$303$	0	0
$304$	−4.00000	−0.229416
$305$	2.00000	0.114520
$306$	0	0
$307$	−32.0000	−1.82634	−0.913168	−	0.407583i	$-0.866372\pi$
$307$	−32.0000	−1.82634	−0.913168	+	0.407583i	$0.866372\pi$
$308$	−1.00000	−0.0569803
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−26.0000	−1.46961	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000	−1.46961	−0.734803	+	0.678280i	$0.762726\pi$
$314$	−18.0000	−1.01580
$315$	0	0
$316$	12.0000	0.675053
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	0	0
$319$	−2.00000	−0.111979
$320$	1.00000	0.0559017
$321$	0	0
$322$	4.00000	0.222911
$323$	−24.0000	−1.33540
$324$	0	0
$325$	−6.00000	−0.332820
$326$	4.00000	0.221540
$327$	0	0
$328$	−2.00000	−0.110432
$329$	−8.00000	−0.441054
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	4.00000	0.218543
$336$	0	0
$337$	34.0000	1.85210	0.926049	−	0.377403i	$-0.123183\pi$
$337$	34.0000	1.85210	0.926049	+	0.377403i	$0.123183\pi$
$338$	−23.0000	−1.25104
$339$	0	0
$340$	6.00000	0.325396
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	4.00000	0.215666
$345$	0	0
$346$	−6.00000	−0.322562
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	1.00000	0.0534522
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	26.0000	1.38384	0.691920	−	0.721974i	$-0.256765\pi$
$353$	26.0000	1.38384	0.691920	+	0.721974i	$0.256765\pi$
$354$	0	0
$355$	−8.00000	−0.424596
$356$	−2.00000	−0.106000
$357$	0	0
$358$	−20.0000	−1.05703
$359$	20.0000	1.05556	0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000	1.05556	0.527780	+	0.849381i	$0.323025\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	−2.00000	−0.105118
$363$	0	0
$364$	6.00000	0.314485
$365$	−6.00000	−0.314054
$366$	0	0
$367$	24.0000	1.25279	0.626395	−	0.779506i	$-0.284530\pi$
$367$	24.0000	1.25279	0.626395	+	0.779506i	$0.284530\pi$
$368$	4.00000	0.208514
$369$	0	0
$370$	−2.00000	−0.103975
$371$	2.00000	0.103835
$372$	0	0
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	−6.00000	−0.310253
$375$	0	0
$376$	−8.00000	−0.412568
$377$	12.0000	0.618031
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	−4.00000	−0.205196
$381$	0	0
$382$	−24.0000	−1.22795
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	−2.00000	−0.101797
$387$	0	0
$388$	−2.00000	−0.101535
$389$	−22.0000	−1.11544	−0.557722	−	0.830028i	$-0.688325\pi$
$389$	−22.0000	−1.11544	−0.557722	+	0.830028i	$0.688325\pi$
$390$	0	0
$391$	24.0000	1.21373
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−10.0000	−0.503793
$395$	12.0000	0.603786
$396$	0	0
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	−16.0000	−0.802008
$399$	0	0
$400$	1.00000	0.0500000
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	0	0
$404$	6.00000	0.298511
$405$	0	0
$406$	−2.00000	−0.0992583
$407$	2.00000	0.0991363
$408$	0	0
$409$	30.0000	1.48340	0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000	1.48340	0.741702	+	0.670729i	$0.234019\pi$
$410$	−2.00000	−0.0987730
$411$	0	0
$412$	8.00000	0.394132
$413$	8.00000	0.393654
$414$	0	0
$415$	0	0
$416$	6.00000	0.294174
$417$	0	0
$418$	4.00000	0.195646
$419$	24.0000	1.17248	0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000	1.17248	0.586238	+	0.810139i	$0.300608\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	−4.00000	−0.194717
$423$	0	0
$424$	2.00000	0.0971286
$425$	6.00000	0.291043
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	12.0000	0.580042
$429$	0	0
$430$	4.00000	0.192897
$431$	−36.0000	−1.73406	−0.867029	−	0.498257i	$-0.833974\pi$
$431$	−36.0000	−1.73406	−0.867029	+	0.498257i	$0.833974\pi$
$432$	0	0
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	0	0
$435$	0	0
$436$	−14.0000	−0.670478
$437$	−16.0000	−0.765384
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	−1.00000	−0.0476731
$441$	0	0
$442$	36.0000	1.71235
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	0	0
$445$	−2.00000	−0.0948091
$446$	−8.00000	−0.378811
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	0	0
$451$	2.00000	0.0941763
$452$	14.0000	0.658505
$453$	0	0
$454$	24.0000	1.12638
$455$	6.00000	0.281284
$456$	0	0
$457$	42.0000	1.96468	0.982339	−	0.187112i	$-0.0599128\pi$
$457$	42.0000	1.96468	0.982339	+	0.187112i	$0.0599128\pi$
$458$	−2.00000	−0.0934539
$459$	0	0
$460$	4.00000	0.186501
$461$	−2.00000	−0.0931493	−0.0465746	−	0.998915i	$-0.514831\pi$
$461$	−2.00000	−0.0931493	−0.0465746	+	0.998915i	$0.514831\pi$
$462$	0	0
$463$	−36.0000	−1.67306	−0.836531	−	0.547920i	$-0.815420\pi$
$463$	−36.0000	−1.67306	−0.836531	+	0.547920i	$0.815420\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	−6.00000	−0.277945
$467$	−20.0000	−0.925490	−0.462745	−	0.886492i	$-0.653135\pi$
$467$	−20.0000	−0.925490	−0.462745	+	0.886492i	$0.653135\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	−8.00000	−0.369012
$471$	0	0
$472$	8.00000	0.368230
$473$	−4.00000	−0.183920
$474$	0	0
$475$	−4.00000	−0.183533
$476$	−6.00000	−0.275010
$477$	0	0
$478$	−4.00000	−0.182956
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	10.0000	0.455488
$483$	0	0
$484$	1.00000	0.0454545
$485$	−2.00000	−0.0908153
$486$	0	0
$487$	28.0000	1.26880	0.634401	−	0.773004i	$-0.281247\pi$
$487$	28.0000	1.26880	0.634401	+	0.773004i	$0.281247\pi$
$488$	−2.00000	−0.0905357
$489$	0	0
$490$	−1.00000	−0.0451754
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	0	0
$493$	−12.0000	−0.540453
$494$	−24.0000	−1.07981
$495$	0	0
$496$	0	0
$497$	8.00000	0.358849
$498$	0	0
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	0	0
$503$	40.0000	1.78351	0.891756	−	0.452517i	$-0.149474\pi$
$503$	40.0000	1.78351	0.891756	+	0.452517i	$0.149474\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	−4.00000	−0.177822
$507$	0	0
$508$	8.00000	0.354943
$509$	−26.0000	−1.15243	−0.576215	−	0.817298i	$-0.695471\pi$
$509$	−26.0000	−1.15243	−0.576215	+	0.817298i	$0.695471\pi$
$510$	0	0
$511$	6.00000	0.265424
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−10.0000	−0.441081
$515$	8.00000	0.352522
$516$	0	0
$517$	8.00000	0.351840
$518$	2.00000	0.0878750
$519$	0	0
$520$	6.00000	0.263117
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	−8.00000	−0.349816	−0.174908	−	0.984585i	$-0.555963\pi$
$523$	−8.00000	−0.349816	−0.174908	+	0.984585i	$0.555963\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	−24.0000	−1.04645
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	2.00000	0.0868744
$531$	0	0
$532$	4.00000	0.173422
$533$	−12.0000	−0.519778
$534$	0	0
$535$	12.0000	0.518805
$536$	−4.00000	−0.172774
$537$	0	0
$538$	10.0000	0.431131
$539$	1.00000	0.0430730
$540$	0	0
$541$	34.0000	1.46177	0.730887	−	0.682498i	$-0.239107\pi$
$541$	34.0000	1.46177	0.730887	+	0.682498i	$0.239107\pi$
$542$	24.0000	1.03089
$543$	0	0
$544$	−6.00000	−0.257248
$545$	−14.0000	−0.599694
$546$	0	0
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	6.00000	0.256307
$549$	0	0
$550$	−1.00000	−0.0426401
$551$	8.00000	0.340811
$552$	0	0
$553$	−12.0000	−0.510292
$554$	−14.0000	−0.594803
$555$	0	0
$556$	−12.0000	−0.508913
$557$	2.00000	0.0847427	0.0423714	−	0.999102i	$-0.486509\pi$
$557$	2.00000	0.0847427	0.0423714	+	0.999102i	$0.486509\pi$
$558$	0	0
$559$	24.0000	1.01509
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	18.0000	0.759284
$563$	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
$565$	14.0000	0.588984
$566$	−32.0000	−1.34506
$567$	0	0
$568$	8.00000	0.335673
$569$	−42.0000	−1.76073	−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−42.0000	−1.76073	−0.880366	+	0.474295i	$0.842703\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	−6.00000	−0.250873
$573$	0	0
$574$	2.00000	0.0834784
$575$	4.00000	0.166812
$576$	0	0
$577$	22.0000	0.915872	0.457936	−	0.888985i	$-0.348589\pi$
$577$	22.0000	0.915872	0.457936	+	0.888985i	$0.348589\pi$
$578$	−19.0000	−0.790296
$579$	0	0
$580$	−2.00000	−0.0830455
$581$	0	0
$582$	0	0
$583$	−2.00000	−0.0828315
$584$	6.00000	0.248282
$585$	0	0
$586$	−6.00000	−0.247858
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	0	0
$589$	0	0
$590$	8.00000	0.329355
$591$	0	0
$592$	2.00000	0.0821995
$593$	−34.0000	−1.39621	−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000	−1.39621	−0.698106	+	0.715994i	$0.745974\pi$
$594$	0	0
$595$	−6.00000	−0.245976
$596$	−18.0000	−0.737309
$597$	0	0
$598$	24.0000	0.981433
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	−34.0000	−1.38689	−0.693444	−	0.720510i	$-0.743908\pi$
$601$	−34.0000	−1.38689	−0.693444	+	0.720510i	$0.743908\pi$
$602$	−4.00000	−0.163028
$603$	0	0
$604$	−4.00000	−0.162758
$605$	1.00000	0.0406558
$606$	0	0
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	4.00000	0.162221
$609$	0	0
$610$	−2.00000	−0.0809776
$611$	−48.0000	−1.94187
$612$	0	0
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	32.0000	1.29141
$615$	0	0
$616$	1.00000	0.0402911
$617$	30.0000	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$618$	0	0
$619$	8.00000	0.321547	0.160774	−	0.986991i	$-0.448601\pi$
$619$	8.00000	0.321547	0.160774	+	0.986991i	$0.448601\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.00000	0.0801283
$624$	0	0
$625$	1.00000	0.0400000
$626$	26.0000	1.03917
$627$	0	0
$628$	18.0000	0.718278
$629$	12.0000	0.478471
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	−12.0000	−0.477334
$633$	0	0
$634$	2.00000	0.0794301
$635$	8.00000	0.317470
$636$	0	0
$637$	−6.00000	−0.237729
$638$	2.00000	0.0791808
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	−12.0000	−0.473234	−0.236617	−	0.971603i	$-0.576039\pi$
$643$	−12.0000	−0.473234	−0.236617	+	0.971603i	$0.576039\pi$
$644$	−4.00000	−0.157622
$645$	0	0
$646$	24.0000	0.944267
$647$	−8.00000	−0.314512	−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000	−0.314512	−0.157256	+	0.987558i	$0.550265\pi$
$648$	0	0
$649$	−8.00000	−0.314027
$650$	6.00000	0.235339
$651$	0	0
$652$	−4.00000	−0.156652
$653$	−34.0000	−1.33052	−0.665261	−	0.746611i	$-0.731680\pi$
$653$	−34.0000	−1.33052	−0.665261	+	0.746611i	$0.731680\pi$
$654$	0	0
$655$	12.0000	0.468879
$656$	2.00000	0.0780869
$657$	0	0
$658$	8.00000	0.311872
$659$	28.0000	1.09073	0.545363	−	0.838200i	$-0.316392\pi$
$659$	28.0000	1.09073	0.545363	+	0.838200i	$0.316392\pi$
$660$	0	0
$661$	42.0000	1.63361	0.816805	−	0.576913i	$-0.195743\pi$
$661$	42.0000	1.63361	0.816805	+	0.576913i	$0.195743\pi$
$662$	20.0000	0.777322
$663$	0	0
$664$	0	0
$665$	4.00000	0.155113
$666$	0	0
$667$	−8.00000	−0.309761
$668$	0	0
$669$	0	0
$670$	−4.00000	−0.154533
$671$	2.00000	0.0772091
$672$	0	0
$673$	−46.0000	−1.77317	−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000	−1.77317	−0.886585	+	0.462566i	$0.846929\pi$
$674$	−34.0000	−1.30963
$675$	0	0
$676$	23.0000	0.884615
$677$	14.0000	0.538064	0.269032	−	0.963131i	$-0.413296\pi$
$677$	14.0000	0.538064	0.269032	+	0.963131i	$0.413296\pi$
$678$	0	0
$679$	2.00000	0.0767530
$680$	−6.00000	−0.230089
$681$	0	0
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	6.00000	0.229248
$686$	1.00000	0.0381802
$687$	0	0
$688$	−4.00000	−0.152499
$689$	12.0000	0.457164
$690$	0	0
$691$	−40.0000	−1.52167	−0.760836	−	0.648944i	$-0.775211\pi$
$691$	−40.0000	−1.52167	−0.760836	+	0.648944i	$0.775211\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	12.0000	0.455514
$695$	−12.0000	−0.455186
$696$	0	0
$697$	12.0000	0.454532
$698$	−2.00000	−0.0757011
$699$	0	0
$700$	−1.00000	−0.0377964
$701$	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
$703$	−8.00000	−0.301726
$704$	1.00000	0.0376889
$705$	0	0
$706$	−26.0000	−0.978523
$707$	−6.00000	−0.225653
$708$	0	0
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	8.00000	0.300235
$711$	0	0
$712$	2.00000	0.0749532
$713$	0	0
$714$	0	0
$715$	−6.00000	−0.224387
$716$	20.0000	0.747435
$717$	0	0
$718$	−20.0000	−0.746393
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	3.00000	0.111648
$723$	0	0
$724$	2.00000	0.0743294
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	−6.00000	−0.222375
$729$	0	0
$730$	6.00000	0.222070
$731$	−24.0000	−0.887672
$732$	0	0
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	−24.0000	−0.885856
$735$	0	0
$736$	−4.00000	−0.147442
$737$	4.00000	0.147342
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	2.00000	0.0735215
$741$	0	0
$742$	−2.00000	−0.0734223
$743$	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
$745$	−18.0000	−0.659469
$746$	−6.00000	−0.219676
$747$	0	0
$748$	6.00000	0.219382
$749$	−12.0000	−0.438470
$750$	0	0
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	8.00000	0.291730
$753$	0	0
$754$	−12.0000	−0.437014
$755$	−4.00000	−0.145575
$756$	0	0
$757$	−6.00000	−0.218074	−0.109037	−	0.994038i	$-0.534777\pi$
$757$	−6.00000	−0.218074	−0.109037	+	0.994038i	$0.534777\pi$
$758$	−20.0000	−0.726433
$759$	0	0
$760$	4.00000	0.145095
$761$	50.0000	1.81250	0.906249	−	0.422744i	$-0.138933\pi$
$761$	50.0000	1.81250	0.906249	+	0.422744i	$0.138933\pi$
$762$	0	0
$763$	14.0000	0.506834
$764$	24.0000	0.868290
$765$	0	0
$766$	0	0
$767$	48.0000	1.73318
$768$	0	0
$769$	46.0000	1.65880	0.829401	−	0.558653i	$-0.188682\pi$
$769$	46.0000	1.65880	0.829401	+	0.558653i	$0.188682\pi$
$770$	1.00000	0.0360375
$771$	0	0
$772$	2.00000	0.0719816
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	0	0
$775$	0	0
$776$	2.00000	0.0717958
$777$	0	0
$778$	22.0000	0.788738
$779$	−8.00000	−0.286630
$780$	0	0
$781$	−8.00000	−0.286263
$782$	−24.0000	−0.858238
$783$	0	0
$784$	1.00000	0.0357143
$785$	18.0000	0.642448
$786$	0	0
$787$	−8.00000	−0.285169	−0.142585	−	0.989783i	$-0.545541\pi$
$787$	−8.00000	−0.285169	−0.142585	+	0.989783i	$0.545541\pi$
$788$	10.0000	0.356235
$789$	0	0
$790$	−12.0000	−0.426941
$791$	−14.0000	−0.497783
$792$	0	0
$793$	−12.0000	−0.426132
$794$	14.0000	0.496841
$795$	0	0
$796$	16.0000	0.567105
$797$	30.0000	1.06265	0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000	1.06265	0.531327	+	0.847167i	$0.321693\pi$
$798$	0	0
$799$	48.0000	1.69812
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	−30.0000	−1.05934
$803$	−6.00000	−0.211735
$804$	0	0
$805$	−4.00000	−0.140981
$806$	0	0
$807$	0	0
$808$	−6.00000	−0.211079
$809$	−50.0000	−1.75791	−0.878953	−	0.476908i	$-0.841757\pi$
$809$	−50.0000	−1.75791	−0.878953	+	0.476908i	$0.841757\pi$
$810$	0	0
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	2.00000	0.0701862
$813$	0	0
$814$	−2.00000	−0.0701000
$815$	−4.00000	−0.140114
$816$	0	0
$817$	16.0000	0.559769
$818$	−30.0000	−1.04893
$819$	0	0
$820$	2.00000	0.0698430
$821$	22.0000	0.767805	0.383903	−	0.923374i	$-0.374580\pi$
$821$	22.0000	0.767805	0.383903	+	0.923374i	$0.374580\pi$
$822$	0	0
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	−8.00000	−0.278693
$825$	0	0
$826$	−8.00000	−0.278356
$827$	−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
$829$	50.0000	1.73657	0.868286	−	0.496064i	$-0.165222\pi$
$829$	50.0000	1.73657	0.868286	+	0.496064i	$0.165222\pi$
$830$	0	0
$831$	0	0
$832$	−6.00000	−0.208013
$833$	6.00000	0.207888
$834$	0	0
$835$	0	0
$836$	−4.00000	−0.138343
$837$	0	0
$838$	−24.0000	−0.829066
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	10.0000	0.344623
$843$	0	0
$844$	4.00000	0.137686
$845$	23.0000	0.791224
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	−2.00000	−0.0686803
$849$	0	0
$850$	−6.00000	−0.205798
$851$	8.00000	0.274236
$852$	0	0
$853$	−14.0000	−0.479351	−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000	−0.479351	−0.239675	+	0.970853i	$0.577041\pi$
$854$	2.00000	0.0684386
$855$	0	0
$856$	−12.0000	−0.410152
$857$	−2.00000	−0.0683187	−0.0341593	−	0.999416i	$-0.510875\pi$
$857$	−2.00000	−0.0683187	−0.0341593	+	0.999416i	$0.510875\pi$
$858$	0	0
$859$	−32.0000	−1.09183	−0.545913	−	0.837842i	$-0.683817\pi$
$859$	−32.0000	−1.09183	−0.545913	+	0.837842i	$0.683817\pi$
$860$	−4.00000	−0.136399
$861$	0	0
$862$	36.0000	1.22616
$863$	−12.0000	−0.408485	−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000	−0.408485	−0.204242	+	0.978920i	$0.565473\pi$
$864$	0	0
$865$	6.00000	0.204006
$866$	26.0000	0.883516
$867$	0	0
$868$	0	0
$869$	12.0000	0.407072
$870$	0	0
$871$	−24.0000	−0.813209
$872$	14.0000	0.474100
$873$	0	0
$874$	16.0000	0.541208
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	6.00000	0.202606	0.101303	−	0.994856i	$-0.467699\pi$
$877$	6.00000	0.202606	0.101303	+	0.994856i	$0.467699\pi$
$878$	0	0
$879$	0	0
$880$	1.00000	0.0337100
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	−36.0000	−1.21150	−0.605748	−	0.795656i	$-0.707126\pi$
$883$	−36.0000	−1.21150	−0.605748	+	0.795656i	$0.707126\pi$
$884$	−36.0000	−1.21081
$885$	0	0
$886$	−4.00000	−0.134383
$887$	48.0000	1.61168	0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000	1.61168	0.805841	+	0.592132i	$0.201714\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	2.00000	0.0670402
$891$	0	0
$892$	8.00000	0.267860
$893$	−32.0000	−1.07084
$894$	0	0
$895$	20.0000	0.668526
$896$	1.00000	0.0334077
$897$	0	0
$898$	−14.0000	−0.467186
$899$	0	0
$900$	0	0
$901$	−12.0000	−0.399778
$902$	−2.00000	−0.0665927
$903$	0	0
$904$	−14.0000	−0.465633
$905$	2.00000	0.0664822
$906$	0	0
$907$	52.0000	1.72663	0.863316	−	0.504664i	$-0.168384\pi$
$907$	52.0000	1.72663	0.863316	+	0.504664i	$0.168384\pi$
$908$	−24.0000	−0.796468
$909$	0	0
$910$	−6.00000	−0.198898
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	0	0
$913$	0	0
$914$	−42.0000	−1.38924
$915$	0	0
$916$	2.00000	0.0660819
$917$	−12.0000	−0.396275
$918$	0	0
$919$	20.0000	0.659739	0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659739	0.329870	+	0.944027i	$0.392995\pi$
$920$	−4.00000	−0.131876
$921$	0	0
$922$	2.00000	0.0658665
$923$	48.0000	1.57994
$924$	0	0
$925$	2.00000	0.0657596
$926$	36.0000	1.18303
$927$	0	0
$928$	2.00000	0.0656532
$929$	−2.00000	−0.0656179	−0.0328089	−	0.999462i	$-0.510445\pi$
$929$	−2.00000	−0.0656179	−0.0328089	+	0.999462i	$0.510445\pi$
$930$	0	0
$931$	−4.00000	−0.131095
$932$	6.00000	0.196537
$933$	0	0
$934$	20.0000	0.654420
$935$	6.00000	0.196221
$936$	0	0
$937$	−30.0000	−0.980057	−0.490029	−	0.871706i	$-0.663014\pi$
$937$	−30.0000	−0.980057	−0.490029	+	0.871706i	$0.663014\pi$
$938$	4.00000	0.130605
$939$	0	0
$940$	8.00000	0.260931
$941$	−50.0000	−1.62995	−0.814977	−	0.579494i	$-0.803250\pi$
$941$	−50.0000	−1.62995	−0.814977	+	0.579494i	$0.803250\pi$
$942$	0	0
$943$	8.00000	0.260516
$944$	−8.00000	−0.260378
$945$	0	0
$946$	4.00000	0.130051
$947$	60.0000	1.94974	0.974869	−	0.222779i	$-0.0715128\pi$
$947$	60.0000	1.94974	0.974869	+	0.222779i	$0.0715128\pi$
$948$	0	0
$949$	36.0000	1.16861
$950$	4.00000	0.129777
$951$	0	0
$952$	6.00000	0.194461
$953$	−26.0000	−0.842223	−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000	−0.842223	−0.421111	+	0.907009i	$0.638360\pi$
$954$	0	0
$955$	24.0000	0.776622
$956$	4.00000	0.129369
$957$	0	0
$958$	−24.0000	−0.775405
$959$	−6.00000	−0.193750
$960$	0	0
$961$	−31.0000	−1.00000
$962$	12.0000	0.386896
$963$	0	0
$964$	−10.0000	−0.322078
$965$	2.00000	0.0643823
$966$	0	0
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	2.00000	0.0642161
$971$	−40.0000	−1.28366	−0.641831	−	0.766846i	$-0.721825\pi$
$971$	−40.0000	−1.28366	−0.641831	+	0.766846i	$0.721825\pi$
$972$	0	0
$973$	12.0000	0.384702
$974$	−28.0000	−0.897178
$975$	0	0
$976$	2.00000	0.0640184
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	−2.00000	−0.0639203
$980$	1.00000	0.0319438
$981$	0	0
$982$	−20.0000	−0.638226
$983$	8.00000	0.255160	0.127580	−	0.991828i	$-0.459279\pi$
$983$	8.00000	0.255160	0.127580	+	0.991828i	$0.459279\pi$
$984$	0	0
$985$	10.0000	0.318626
$986$	12.0000	0.382158
$987$	0	0
$988$	24.0000	0.763542
$989$	−16.0000	−0.508770
$990$	0	0
$991$	−24.0000	−0.762385	−0.381193	−	0.924496i	$-0.624487\pi$
$991$	−24.0000	−0.762385	−0.381193	+	0.924496i	$0.624487\pi$
$992$	0	0
$993$	0	0
$994$	−8.00000	−0.253745
$995$	16.0000	0.507234
$996$	0	0
$997$	−38.0000	−1.20347	−0.601736	−	0.798695i	$-0.705524\pi$
$997$	−38.0000	−1.20347	−0.601736	+	0.798695i	$0.705524\pi$
$998$	−28.0000	−0.886325
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6930.2.a.l.1.1		1
3.2	odd	2		2310.2.a.q.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2310.2.a.q.1.1	✓	1	3.2	odd	2
6930.2.a.l.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:	\( N \)	\(=\)	\( 6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6930.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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