LMFDB - Embedded newform 6897.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6897,2,Mod(1,6897)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6897, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6897.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6897 = 3 \cdot 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6897.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:	$55.0728222741$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 57)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6897.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -1.00000 q^{12} -6.00000 q^{13} -2.00000 q^{15} -1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} +1.00000 q^{19} +2.00000 q^{20} +4.00000 q^{23} +3.00000 q^{24} -1.00000 q^{25} +6.00000 q^{26} +1.00000 q^{27} -2.00000 q^{29} +2.00000 q^{30} +8.00000 q^{31} -5.00000 q^{32} -6.00000 q^{34} -1.00000 q^{36} -10.0000 q^{37} -1.00000 q^{38} -6.00000 q^{39} -6.00000 q^{40} +2.00000 q^{41} +4.00000 q^{43} -2.00000 q^{45} -4.00000 q^{46} +12.0000 q^{47} -1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} +6.00000 q^{51} +6.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} +1.00000 q^{57} +2.00000 q^{58} -12.0000 q^{59} +2.00000 q^{60} +2.00000 q^{61} -8.00000 q^{62} +7.00000 q^{64} +12.0000 q^{65} -4.00000 q^{67} -6.00000 q^{68} +4.00000 q^{69} +3.00000 q^{72} -10.0000 q^{73} +10.0000 q^{74} -1.00000 q^{75} -1.00000 q^{76} +6.00000 q^{78} +2.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -16.0000 q^{83} -12.0000 q^{85} -4.00000 q^{86} -2.00000 q^{87} -2.00000 q^{89} +2.00000 q^{90} -4.00000 q^{92} +8.00000 q^{93} -12.0000 q^{94} -2.00000 q^{95} -5.00000 q^{96} +10.0000 q^{97} +7.00000 q^{98} +O(q^{100})\)

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	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.00000	−0.500000
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	−1.00000	−0.408248
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	3.00000	1.06066
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	0	0
$11$	0	0
$12$	−1.00000	−0.288675
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	−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	$0.240633\pi$
$18$	−1.00000	−0.235702
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	2.00000	0.447214
$21$	0	0
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	3.00000	0.612372
$25$	−1.00000	−0.200000
$26$	6.00000	1.17670
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	2.00000	0.365148
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	−5.00000	−0.883883
$33$	0	0
$34$	−6.00000	−1.02899
$35$	0	0
$36$	−1.00000	−0.166667
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	−1.00000	−0.162221
$39$	−6.00000	−0.960769
$40$	−6.00000	−0.948683
$41$	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	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	−4.00000	−0.589768
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	−1.00000	−0.144338
$49$	−7.00000	−1.00000
$50$	1.00000	0.141421
$51$	6.00000	0.840168
$52$	6.00000	0.832050
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	1.00000	0.132453
$58$	2.00000	0.262613
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	2.00000	0.258199
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−8.00000	−1.01600
$63$	0	0
$64$	7.00000	0.875000
$65$	12.0000	1.48842
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−6.00000	−0.727607
$69$	4.00000	0.481543
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	3.00000	0.353553
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	10.0000	1.16248
$75$	−1.00000	−0.115470
$76$	−1.00000	−0.114708
$77$	0	0
$78$	6.00000	0.679366
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	2.00000	0.223607
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	−16.0000	−1.75623	−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000	−1.75623	−0.878114	+	0.478451i	$0.841198\pi$
$84$	0	0
$85$	−12.0000	−1.30158
$86$	−4.00000	−0.431331
$87$	−2.00000	−0.214423
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	2.00000	0.210819
$91$	0	0
$92$	−4.00000	−0.417029
$93$	8.00000	0.829561
$94$	−12.0000	−1.23771
$95$	−2.00000	−0.205196
$96$	−5.00000	−0.510310
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	7.00000	0.707107
$99$	0	0
$100$	1.00000	0.100000
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	−6.00000	−0.594089
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	−18.0000	−1.76505
$105$	0	0
$106$	6.00000	0.582772
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	−1.00000	−0.0962250
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	−10.0000	−0.949158
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	−1.00000	−0.0936586
$115$	−8.00000	−0.746004
$116$	2.00000	0.185695
$117$	−6.00000	−0.554700
$118$	12.0000	1.10469
$119$	0	0
$120$	−6.00000	−0.547723
$121$	0	0
$122$	−2.00000	−0.181071
$123$	2.00000	0.180334
$124$	−8.00000	−0.718421
$125$	12.0000	1.07331
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	3.00000	0.265165
$129$	4.00000	0.352180
$130$	−12.0000	−1.05247
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	0	0
$133$	0	0
$134$	4.00000	0.345547
$135$	−2.00000	−0.172133
$136$	18.0000	1.54349
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	−4.00000	−0.340503
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	0	0
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	4.00000	0.332182
$146$	10.0000	0.827606
$147$	−7.00000	−0.577350
$148$	10.0000	0.821995
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	1.00000	0.0816497
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	3.00000	0.243332
$153$	6.00000	0.485071
$154$	0	0
$155$	−16.0000	−1.28515
$156$	6.00000	0.480384
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	10.0000	0.790569
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−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
$166$	16.0000	1.24184
$167$	−24.0000	−1.85718	−0.928588	−	0.371113i	$-0.878976\pi$
$167$	−24.0000	−1.85718	−0.928588	+	0.371113i	$0.878976\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	12.0000	0.920358
$171$	1.00000	0.0764719
$172$	−4.00000	−0.304997
$173$	22.0000	1.67263	0.836315	−	0.548250i	$-0.184706\pi$
$173$	22.0000	1.67263	0.836315	+	0.548250i	$0.184706\pi$
$174$	2.00000	0.151620
$175$	0	0
$176$	0	0
$177$	−12.0000	−0.901975
$178$	2.00000	0.149906
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	2.00000	0.149071
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	12.0000	0.884652
$185$	20.0000	1.47043
$186$	−8.00000	−0.586588
$187$	0	0
$188$	−12.0000	−0.875190
$189$	0	0
$190$	2.00000	0.145095
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	7.00000	0.505181
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	−10.0000	−0.717958
$195$	12.0000	0.859338
$196$	7.00000	0.500000
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	−3.00000	−0.212132
$201$	−4.00000	−0.282138
$202$	−10.0000	−0.703598
$203$	0	0
$204$	−6.00000	−0.420084
$205$	−4.00000	−0.279372
$206$	−8.00000	−0.557386
$207$	4.00000	0.278019
$208$	6.00000	0.416025
$209$	0	0
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	4.00000	0.273434
$215$	−8.00000	−0.545595
$216$	3.00000	0.204124
$217$	0	0
$218$	−10.0000	−0.677285
$219$	−10.0000	−0.675737
$220$	0	0
$221$	−36.0000	−2.42162
$222$	10.0000	0.671156
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	−6.00000	−0.399114
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	−1.00000	−0.0662266
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	8.00000	0.527504
$231$	0	0
$232$	−6.00000	−0.393919
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	6.00000	0.392232
$235$	−24.0000	−1.56559
$236$	12.0000	0.781133
$237$	0	0
$238$	0	0
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	2.00000	0.129099
$241$	6.00000	0.386494	0.193247	−	0.981150i	$-0.438098\pi$
$241$	6.00000	0.386494	0.193247	+	0.981150i	$0.438098\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	−2.00000	−0.128037
$245$	14.0000	0.894427
$246$	−2.00000	−0.127515
$247$	−6.00000	−0.381771
$248$	24.0000	1.52400
$249$	−16.0000	−1.01396
$250$	−12.0000	−0.758947
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	0	0
$253$	0	0
$254$	−8.00000	−0.501965
$255$	−12.0000	−0.751469
$256$	−17.0000	−1.06250
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	−4.00000	−0.249029
$259$	0	0
$260$	−12.0000	−0.744208
$261$	−2.00000	−0.123797
$262$	8.00000	0.494242
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	12.0000	0.737154
$266$	0	0
$267$	−2.00000	−0.122398
$268$	4.00000	0.244339
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	2.00000	0.121716
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	−6.00000	−0.363803
$273$	0	0
$274$	−18.0000	−1.08742
$275$	0	0
$276$	−4.00000	−0.240772
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	4.00000	0.239904
$279$	8.00000	0.478947
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	−12.0000	−0.714590
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	0	0
$285$	−2.00000	−0.118470
$286$	0	0
$287$	0	0
$288$	−5.00000	−0.294628
$289$	19.0000	1.11765
$290$	−4.00000	−0.234888
$291$	10.0000	0.586210
$292$	10.0000	0.585206
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	7.00000	0.408248
$295$	24.0000	1.39733
$296$	−30.0000	−1.74371
$297$	0	0
$298$	6.00000	0.347571
$299$	−24.0000	−1.38796
$300$	1.00000	0.0577350
$301$	0	0
$302$	−8.00000	−0.460348
$303$	10.0000	0.574485
$304$	−1.00000	−0.0573539
$305$	−4.00000	−0.229039
$306$	−6.00000	−0.342997
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	16.0000	0.908739
$311$	4.00000	0.226819	0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000	0.226819	0.113410	+	0.993548i	$0.463823\pi$
$312$	−18.0000	−1.01905
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	2.00000	0.112867
$315$	0	0
$316$	0	0
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	6.00000	0.336463
$319$	0	0
$320$	−14.0000	−0.782624
$321$	−4.00000	−0.223258
$322$	0	0
$323$	6.00000	0.333849
$324$	−1.00000	−0.0555556
$325$	6.00000	0.332820
$326$	4.00000	0.221540
$327$	10.0000	0.553001
$328$	6.00000	0.331295
$329$	0	0
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	16.0000	0.878114
$333$	−10.0000	−0.547997
$334$	24.0000	1.31322
$335$	8.00000	0.437087
$336$	0	0
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	−23.0000	−1.25104
$339$	6.00000	0.325875
$340$	12.0000	0.650791
$341$	0	0
$342$	−1.00000	−0.0540738
$343$	0	0
$344$	12.0000	0.646997
$345$	−8.00000	−0.430706
$346$	−22.0000	−1.18273
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	2.00000	0.107211
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	−6.00000	−0.320256
$352$	0	0
$353$	−22.0000	−1.17094	−0.585471	−	0.810693i	$-0.699090\pi$
$353$	−22.0000	−1.17094	−0.585471	+	0.810693i	$0.699090\pi$
$354$	12.0000	0.637793
$355$	0	0
$356$	2.00000	0.106000
$357$	0	0
$358$	4.00000	0.211407
$359$	20.0000	1.05556	0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000	1.05556	0.527780	+	0.849381i	$0.323025\pi$
$360$	−6.00000	−0.316228
$361$	1.00000	0.0526316
$362$	−14.0000	−0.735824
$363$	0	0
$364$	0	0
$365$	20.0000	1.04685
$366$	−2.00000	−0.104542
$367$	32.0000	1.67039	0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000	1.67039	0.835193	+	0.549957i	$0.185356\pi$
$368$	−4.00000	−0.208514
$369$	2.00000	0.104116
$370$	−20.0000	−1.03975
$371$	0	0
$372$	−8.00000	−0.414781
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	0	0
$375$	12.0000	0.619677
$376$	36.0000	1.85656
$377$	12.0000	0.618031
$378$	0	0
$379$	12.0000	0.616399	0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000	0.616399	0.308199	+	0.951322i	$0.400274\pi$
$380$	2.00000	0.102598
$381$	8.00000	0.409852
$382$	12.0000	0.613973
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	3.00000	0.153093
$385$	0	0
$386$	−14.0000	−0.712581
$387$	4.00000	0.203331
$388$	−10.0000	−0.507673
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	−12.0000	−0.607644
$391$	24.0000	1.21373
$392$	−21.0000	−1.06066
$393$	−8.00000	−0.403547
$394$	−2.00000	−0.100759
$395$	0	0
$396$	0	0
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	8.00000	0.401004
$399$	0	0
$400$	1.00000	0.0500000
$401$	38.0000	1.89763	0.948815	−	0.315833i	$-0.102284\pi$
$401$	38.0000	1.89763	0.948815	+	0.315833i	$0.102284\pi$
$402$	4.00000	0.199502
$403$	−48.0000	−2.39105
$404$	−10.0000	−0.497519
$405$	−2.00000	−0.0993808
$406$	0	0
$407$	0	0
$408$	18.0000	0.891133
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	4.00000	0.197546
$411$	18.0000	0.887875
$412$	−8.00000	−0.394132
$413$	0	0
$414$	−4.00000	−0.196589
$415$	32.0000	1.57082
$416$	30.0000	1.47087
$417$	−4.00000	−0.195881
$418$	0	0
$419$	8.00000	0.390826	0.195413	−	0.980721i	$-0.437395\pi$
$419$	8.00000	0.390826	0.195413	+	0.980721i	$0.437395\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	−4.00000	−0.194717
$423$	12.0000	0.583460
$424$	−18.0000	−0.874157
$425$	−6.00000	−0.291043
$426$	0	0
$427$	0	0
$428$	4.00000	0.193347
$429$	0	0
$430$	8.00000	0.385794
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	−1.00000	−0.0481125
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	4.00000	0.191785
$436$	−10.0000	−0.478913
$437$	4.00000	0.191346
$438$	10.0000	0.477818
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	36.0000	1.71235
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	10.0000	0.474579
$445$	4.00000	0.189618
$446$	−16.0000	−0.757622
$447$	−6.00000	−0.283790
$448$	0	0
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	1.00000	0.0471405
$451$	0	0
$452$	−6.00000	−0.282216
$453$	8.00000	0.375873
$454$	−12.0000	−0.563188
$455$	0	0
$456$	3.00000	0.140488
$457$	6.00000	0.280668	0.140334	−	0.990104i	$-0.455182\pi$
$457$	6.00000	0.280668	0.140334	+	0.990104i	$0.455182\pi$
$458$	−6.00000	−0.280362
$459$	6.00000	0.280056
$460$	8.00000	0.373002
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	32.0000	1.48717	0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000	1.48717	0.743583	+	0.668644i	$0.233125\pi$
$464$	2.00000	0.0928477
$465$	−16.0000	−0.741982
$466$	10.0000	0.463241
$467$	−32.0000	−1.48078	−0.740392	−	0.672176i	$-0.765360\pi$
$467$	−32.0000	−1.48078	−0.740392	+	0.672176i	$0.765360\pi$
$468$	6.00000	0.277350
$469$	0	0
$470$	24.0000	1.10704
$471$	−2.00000	−0.0921551
$472$	−36.0000	−1.65703
$473$	0	0
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	−6.00000	−0.274721
$478$	−12.0000	−0.548867
$479$	−20.0000	−0.913823	−0.456912	−	0.889512i	$-0.651044\pi$
$479$	−20.0000	−0.913823	−0.456912	+	0.889512i	$0.651044\pi$
$480$	10.0000	0.456435
$481$	60.0000	2.73576
$482$	−6.00000	−0.273293
$483$	0	0
$484$	0	0
$485$	−20.0000	−0.908153
$486$	−1.00000	−0.0453609
$487$	32.0000	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$488$	6.00000	0.271607
$489$	−4.00000	−0.180886
$490$	−14.0000	−0.632456
$491$	32.0000	1.44414	0.722070	−	0.691820i	$-0.243191\pi$
$491$	32.0000	1.44414	0.722070	+	0.691820i	$0.243191\pi$
$492$	−2.00000	−0.0901670
$493$	−12.0000	−0.540453
$494$	6.00000	0.269953
$495$	0	0
$496$	−8.00000	−0.359211
$497$	0	0
$498$	16.0000	0.716977
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	−12.0000	−0.536656
$501$	−24.0000	−1.07224
$502$	24.0000	1.07117
$503$	−12.0000	−0.535054	−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000	−0.535054	−0.267527	+	0.963550i	$0.586206\pi$
$504$	0	0
$505$	−20.0000	−0.889988
$506$	0	0
$507$	23.0000	1.02147
$508$	−8.00000	−0.354943
$509$	−22.0000	−0.975133	−0.487566	−	0.873086i	$-0.662115\pi$
$509$	−22.0000	−0.975133	−0.487566	+	0.873086i	$0.662115\pi$
$510$	12.0000	0.531369
$511$	0	0
$512$	11.0000	0.486136
$513$	1.00000	0.0441511
$514$	−14.0000	−0.617514
$515$	−16.0000	−0.705044
$516$	−4.00000	−0.176090
$517$	0	0
$518$	0	0
$519$	22.0000	0.965693
$520$	36.0000	1.57870
$521$	14.0000	0.613351	0.306676	−	0.951814i	$-0.400783\pi$
$521$	14.0000	0.613351	0.306676	+	0.951814i	$0.400783\pi$
$522$	2.00000	0.0875376
$523$	28.0000	1.22435	0.612177	−	0.790721i	$-0.290294\pi$
$523$	28.0000	1.22435	0.612177	+	0.790721i	$0.290294\pi$
$524$	8.00000	0.349482
$525$	0	0
$526$	12.0000	0.523225
$527$	48.0000	2.09091
$528$	0	0
$529$	−7.00000	−0.304348
$530$	−12.0000	−0.521247
$531$	−12.0000	−0.520756
$532$	0	0
$533$	−12.0000	−0.519778
$534$	2.00000	0.0865485
$535$	8.00000	0.345870
$536$	−12.0000	−0.518321
$537$	−4.00000	−0.172613
$538$	6.00000	0.258678
$539$	0	0
$540$	2.00000	0.0860663
$541$	−14.0000	−0.601907	−0.300954	−	0.953639i	$-0.597305\pi$
$541$	−14.0000	−0.601907	−0.300954	+	0.953639i	$0.597305\pi$
$542$	0	0
$543$	14.0000	0.600798
$544$	−30.0000	−1.28624
$545$	−20.0000	−0.856706
$546$	0	0
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	−18.0000	−0.768922
$549$	2.00000	0.0853579
$550$	0	0
$551$	−2.00000	−0.0852029
$552$	12.0000	0.510754
$553$	0	0
$554$	22.0000	0.934690
$555$	20.0000	0.848953
$556$	4.00000	0.169638
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	−8.00000	−0.338667
$559$	−24.0000	−1.01509
$560$	0	0
$561$	0	0
$562$	−10.0000	−0.421825
$563$	−20.0000	−0.842900	−0.421450	−	0.906852i	$-0.638479\pi$
$563$	−20.0000	−0.842900	−0.421450	+	0.906852i	$0.638479\pi$
$564$	−12.0000	−0.505291
$565$	−12.0000	−0.504844
$566$	−20.0000	−0.840663
$567$	0	0
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	2.00000	0.0837708
$571$	−44.0000	−1.84134	−0.920671	−	0.390339i	$-0.872358\pi$
$571$	−44.0000	−1.84134	−0.920671	+	0.390339i	$0.872358\pi$
$572$	0	0
$573$	−12.0000	−0.501307
$574$	0	0
$575$	−4.00000	−0.166812
$576$	7.00000	0.291667
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	−19.0000	−0.790296
$579$	14.0000	0.581820
$580$	−4.00000	−0.166091
$581$	0	0
$582$	−10.0000	−0.414513
$583$	0	0
$584$	−30.0000	−1.24141
$585$	12.0000	0.496139
$586$	−14.0000	−0.578335
$587$	8.00000	0.330195	0.165098	−	0.986277i	$-0.447206\pi$
$587$	8.00000	0.330195	0.165098	+	0.986277i	$0.447206\pi$
$588$	7.00000	0.288675
$589$	8.00000	0.329634
$590$	−24.0000	−0.988064
$591$	2.00000	0.0822690
$592$	10.0000	0.410997
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	0	0
$595$	0	0
$596$	6.00000	0.245770
$597$	−8.00000	−0.327418
$598$	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	$0.663115\pi$
$600$	−3.00000	−0.122474
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	−8.00000	−0.325515
$605$	0	0
$606$	−10.0000	−0.406222
$607$	−24.0000	−0.974130	−0.487065	−	0.873366i	$-0.661933\pi$
$607$	−24.0000	−0.974130	−0.487065	+	0.873366i	$0.661933\pi$
$608$	−5.00000	−0.202777
$609$	0	0
$610$	4.00000	0.161955
$611$	−72.0000	−2.91281
$612$	−6.00000	−0.242536
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	−12.0000	−0.484281
$615$	−4.00000	−0.161296
$616$	0	0
$617$	2.00000	0.0805170	0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000	0.0805170	0.0402585	+	0.999189i	$0.487182\pi$
$618$	−8.00000	−0.321807
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	16.0000	0.642575
$621$	4.00000	0.160514
$622$	−4.00000	−0.160385
$623$	0	0
$624$	6.00000	0.240192
$625$	−19.0000	−0.760000
$626$	22.0000	0.879297
$627$	0	0
$628$	2.00000	0.0798087
$629$	−60.0000	−2.39236
$630$	0	0
$631$	−32.0000	−1.27390	−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000	−1.27390	−0.636950	+	0.770905i	$0.719804\pi$
$632$	0	0
$633$	4.00000	0.158986
$634$	6.00000	0.238290
$635$	−16.0000	−0.634941
$636$	6.00000	0.237915
$637$	42.0000	1.66410
$638$	0	0
$639$	0	0
$640$	−6.00000	−0.237171
$641$	38.0000	1.50091	0.750455	−	0.660922i	$-0.229834\pi$
$641$	38.0000	1.50091	0.750455	+	0.660922i	$0.229834\pi$
$642$	4.00000	0.157867
$643$	20.0000	0.788723	0.394362	−	0.918955i	$-0.370966\pi$
$643$	20.0000	0.788723	0.394362	+	0.918955i	$0.370966\pi$
$644$	0	0
$645$	−8.00000	−0.315000
$646$	−6.00000	−0.236067
$647$	36.0000	1.41531	0.707653	−	0.706560i	$-0.249754\pi$
$647$	36.0000	1.41531	0.707653	+	0.706560i	$0.249754\pi$
$648$	3.00000	0.117851
$649$	0	0
$650$	−6.00000	−0.235339
$651$	0	0
$652$	4.00000	0.156652
$653$	14.0000	0.547862	0.273931	−	0.961749i	$-0.411676\pi$
$653$	14.0000	0.547862	0.273931	+	0.961749i	$0.411676\pi$
$654$	−10.0000	−0.391031
$655$	16.0000	0.625172
$656$	−2.00000	−0.0780869
$657$	−10.0000	−0.390137
$658$	0	0
$659$	44.0000	1.71400	0.856998	−	0.515319i	$-0.172327\pi$
$659$	44.0000	1.71400	0.856998	+	0.515319i	$0.172327\pi$
$660$	0	0
$661$	6.00000	0.233373	0.116686	−	0.993169i	$-0.462773\pi$
$661$	6.00000	0.233373	0.116686	+	0.993169i	$0.462773\pi$
$662$	−12.0000	−0.466393
$663$	−36.0000	−1.39812
$664$	−48.0000	−1.86276
$665$	0	0
$666$	10.0000	0.387492
$667$	−8.00000	−0.309761
$668$	24.0000	0.928588
$669$	16.0000	0.618596
$670$	−8.00000	−0.309067
$671$	0	0
$672$	0	0
$673$	46.0000	1.77317	0.886585	−	0.462566i	$-0.153071\pi$
$673$	46.0000	1.77317	0.886585	+	0.462566i	$0.153071\pi$
$674$	−22.0000	−0.847408
$675$	−1.00000	−0.0384900
$676$	−23.0000	−0.884615
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	−6.00000	−0.230429
$679$	0	0
$680$	−36.0000	−1.38054
$681$	12.0000	0.459841
$682$	0	0
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	−1.00000	−0.0382360
$685$	−36.0000	−1.37549
$686$	0	0
$687$	6.00000	0.228914
$688$	−4.00000	−0.152499
$689$	36.0000	1.37149
$690$	8.00000	0.304555
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	−22.0000	−0.836315
$693$	0	0
$694$	0	0
$695$	8.00000	0.303457
$696$	−6.00000	−0.227429
$697$	12.0000	0.454532
$698$	−2.00000	−0.0757011
$699$	−10.0000	−0.378235
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	6.00000	0.226455
$703$	−10.0000	−0.377157
$704$	0	0
$705$	−24.0000	−0.903892
$706$	22.0000	0.827981
$707$	0	0
$708$	12.0000	0.450988
$709$	38.0000	1.42712	0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000	1.42712	0.713560	+	0.700594i	$0.247082\pi$
$710$	0	0
$711$	0	0
$712$	−6.00000	−0.224860
$713$	32.0000	1.19841
$714$	0	0
$715$	0	0
$716$	4.00000	0.149487
$717$	12.0000	0.448148
$718$	−20.0000	−0.746393
$719$	−20.0000	−0.745874	−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000	−0.745874	−0.372937	+	0.927857i	$0.621649\pi$
$720$	2.00000	0.0745356
$721$	0	0
$722$	−1.00000	−0.0372161
$723$	6.00000	0.223142
$724$	−14.0000	−0.520306
$725$	2.00000	0.0742781
$726$	0	0
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−20.0000	−0.740233
$731$	24.0000	0.887672
$732$	−2.00000	−0.0739221
$733$	−46.0000	−1.69905	−0.849524	−	0.527549i	$-0.823111\pi$
$733$	−46.0000	−1.69905	−0.849524	+	0.527549i	$0.823111\pi$
$734$	−32.0000	−1.18114
$735$	14.0000	0.516398
$736$	−20.0000	−0.737210
$737$	0	0
$738$	−2.00000	−0.0736210
$739$	36.0000	1.32428	0.662141	−	0.749380i	$-0.269648\pi$
$739$	36.0000	1.32428	0.662141	+	0.749380i	$0.269648\pi$
$740$	−20.0000	−0.735215
$741$	−6.00000	−0.220416
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	24.0000	0.879883
$745$	12.0000	0.439646
$746$	−10.0000	−0.366126
$747$	−16.0000	−0.585409
$748$	0	0
$749$	0	0
$750$	−12.0000	−0.438178
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	−12.0000	−0.437595
$753$	−24.0000	−0.874609
$754$	−12.0000	−0.437014
$755$	−16.0000	−0.582300
$756$	0	0
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	−12.0000	−0.435860
$759$	0	0
$760$	−6.00000	−0.217643
$761$	−50.0000	−1.81250	−0.906249	−	0.422744i	$-0.861067\pi$
$761$	−50.0000	−1.81250	−0.906249	+	0.422744i	$0.861067\pi$
$762$	−8.00000	−0.289809
$763$	0	0
$764$	12.0000	0.434145
$765$	−12.0000	−0.433861
$766$	−8.00000	−0.289052
$767$	72.0000	2.59977
$768$	−17.0000	−0.613435
$769$	−18.0000	−0.649097	−0.324548	−	0.945869i	$-0.605212\pi$
$769$	−18.0000	−0.649097	−0.324548	+	0.945869i	$0.605212\pi$
$770$	0	0
$771$	14.0000	0.504198
$772$	−14.0000	−0.503871
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	−4.00000	−0.143777
$775$	−8.00000	−0.287368
$776$	30.0000	1.07694
$777$	0	0
$778$	−30.0000	−1.07555
$779$	2.00000	0.0716574
$780$	−12.0000	−0.429669
$781$	0	0
$782$	−24.0000	−0.858238
$783$	−2.00000	−0.0714742
$784$	7.00000	0.250000
$785$	4.00000	0.142766
$786$	8.00000	0.285351
$787$	−44.0000	−1.56843	−0.784215	−	0.620489i	$-0.786934\pi$
$787$	−44.0000	−1.56843	−0.784215	+	0.620489i	$0.786934\pi$
$788$	−2.00000	−0.0712470
$789$	−12.0000	−0.427211
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−12.0000	−0.426132
$794$	−14.0000	−0.496841
$795$	12.0000	0.425596
$796$	8.00000	0.283552
$797$	−6.00000	−0.212531	−0.106265	−	0.994338i	$-0.533889\pi$
$797$	−6.00000	−0.212531	−0.106265	+	0.994338i	$0.533889\pi$
$798$	0	0
$799$	72.0000	2.54718
$800$	5.00000	0.176777
$801$	−2.00000	−0.0706665
$802$	−38.0000	−1.34183
$803$	0	0
$804$	4.00000	0.141069
$805$	0	0
$806$	48.0000	1.69073
$807$	−6.00000	−0.211210
$808$	30.0000	1.05540
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	2.00000	0.0702728
$811$	−44.0000	−1.54505	−0.772524	−	0.634985i	$-0.781006\pi$
$811$	−44.0000	−1.54505	−0.772524	+	0.634985i	$0.781006\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	8.00000	0.280228
$816$	−6.00000	−0.210042
$817$	4.00000	0.139942
$818$	−14.0000	−0.489499
$819$	0	0
$820$	4.00000	0.139686
$821$	42.0000	1.46581	0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000	1.46581	0.732905	+	0.680331i	$0.238164\pi$
$822$	−18.0000	−0.627822
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	24.0000	0.836080
$825$	0	0
$826$	0	0
$827$	28.0000	0.973655	0.486828	−	0.873498i	$-0.338154\pi$
$827$	28.0000	0.973655	0.486828	+	0.873498i	$0.338154\pi$
$828$	−4.00000	−0.139010
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	−32.0000	−1.11074
$831$	−22.0000	−0.763172
$832$	−42.0000	−1.45609
$833$	−42.0000	−1.45521
$834$	4.00000	0.138509
$835$	48.0000	1.66111
$836$	0	0
$837$	8.00000	0.276520
$838$	−8.00000	−0.276355
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−14.0000	−0.482472
$843$	10.0000	0.344418
$844$	−4.00000	−0.137686
$845$	−46.0000	−1.58245
$846$	−12.0000	−0.412568
$847$	0	0
$848$	6.00000	0.206041
$849$	20.0000	0.686398
$850$	6.00000	0.205798
$851$	−40.0000	−1.37118
$852$	0	0
$853$	−22.0000	−0.753266	−0.376633	−	0.926363i	$-0.622918\pi$
$853$	−22.0000	−0.753266	−0.376633	+	0.926363i	$0.622918\pi$
$854$	0	0
$855$	−2.00000	−0.0683986
$856$	−12.0000	−0.410152
$857$	−30.0000	−1.02478	−0.512390	−	0.858753i	$-0.671240\pi$
$857$	−30.0000	−1.02478	−0.512390	+	0.858753i	$0.671240\pi$
$858$	0	0
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	8.00000	0.272798
$861$	0	0
$862$	−24.0000	−0.817443
$863$	−40.0000	−1.36162	−0.680808	−	0.732462i	$-0.738371\pi$
$863$	−40.0000	−1.36162	−0.680808	+	0.732462i	$0.738371\pi$
$864$	−5.00000	−0.170103
$865$	−44.0000	−1.49604
$866$	14.0000	0.475739
$867$	19.0000	0.645274
$868$	0	0
$869$	0	0
$870$	−4.00000	−0.135613
$871$	24.0000	0.813209
$872$	30.0000	1.01593
$873$	10.0000	0.338449
$874$	−4.00000	−0.135302
$875$	0	0
$876$	10.0000	0.337869
$877$	34.0000	1.14810	0.574049	−	0.818821i	$-0.305372\pi$
$877$	34.0000	1.14810	0.574049	+	0.818821i	$0.305372\pi$
$878$	−8.00000	−0.269987
$879$	14.0000	0.472208
$880$	0	0
$881$	−38.0000	−1.28025	−0.640126	−	0.768270i	$-0.721118\pi$
$881$	−38.0000	−1.28025	−0.640126	+	0.768270i	$0.721118\pi$
$882$	7.00000	0.235702
$883$	−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
$885$	24.0000	0.806751
$886$	0	0
$887$	40.0000	1.34307	0.671534	−	0.740973i	$-0.265636\pi$
$887$	40.0000	1.34307	0.671534	+	0.740973i	$0.265636\pi$
$888$	−30.0000	−1.00673
$889$	0	0
$890$	−4.00000	−0.134080
$891$	0	0
$892$	−16.0000	−0.535720
$893$	12.0000	0.401565
$894$	6.00000	0.200670
$895$	8.00000	0.267411
$896$	0	0
$897$	−24.0000	−0.801337
$898$	2.00000	0.0667409
$899$	−16.0000	−0.533630
$900$	1.00000	0.0333333
$901$	−36.0000	−1.19933
$902$	0	0
$903$	0	0
$904$	18.0000	0.598671
$905$	−28.0000	−0.930751
$906$	−8.00000	−0.265782
$907$	4.00000	0.132818	0.0664089	−	0.997792i	$-0.478846\pi$
$907$	4.00000	0.132818	0.0664089	+	0.997792i	$0.478846\pi$
$908$	−12.0000	−0.398234
$909$	10.0000	0.331679
$910$	0	0
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	−1.00000	−0.0331133
$913$	0	0
$914$	−6.00000	−0.198462
$915$	−4.00000	−0.132236
$916$	−6.00000	−0.198246
$917$	0	0
$918$	−6.00000	−0.198030
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	−24.0000	−0.791257
$921$	12.0000	0.395413
$922$	−18.0000	−0.592798
$923$	0	0
$924$	0	0
$925$	10.0000	0.328798
$926$	−32.0000	−1.05159
$927$	8.00000	0.262754
$928$	10.0000	0.328266
$929$	34.0000	1.11550	0.557752	−	0.830008i	$-0.311664\pi$
$929$	34.0000	1.11550	0.557752	+	0.830008i	$0.311664\pi$
$930$	16.0000	0.524661
$931$	−7.00000	−0.229416
$932$	10.0000	0.327561
$933$	4.00000	0.130954
$934$	32.0000	1.04707
$935$	0	0
$936$	−18.0000	−0.588348
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	0	0
$939$	−22.0000	−0.717943
$940$	24.0000	0.782794
$941$	22.0000	0.717180	0.358590	−	0.933495i	$-0.383258\pi$
$941$	22.0000	0.717180	0.358590	+	0.933495i	$0.383258\pi$
$942$	2.00000	0.0651635
$943$	8.00000	0.260516
$944$	12.0000	0.390567
$945$	0	0
$946$	0	0
$947$	−8.00000	−0.259965	−0.129983	−	0.991516i	$-0.541492\pi$
$947$	−8.00000	−0.259965	−0.129983	+	0.991516i	$0.541492\pi$
$948$	0	0
$949$	60.0000	1.94768
$950$	1.00000	0.0324443
$951$	−6.00000	−0.194563
$952$	0	0
$953$	−38.0000	−1.23094	−0.615470	−	0.788160i	$-0.711034\pi$
$953$	−38.0000	−1.23094	−0.615470	+	0.788160i	$0.711034\pi$
$954$	6.00000	0.194257
$955$	24.0000	0.776622
$956$	−12.0000	−0.388108
$957$	0	0
$958$	20.0000	0.646171
$959$	0	0
$960$	−14.0000	−0.451848
$961$	33.0000	1.06452
$962$	−60.0000	−1.93448
$963$	−4.00000	−0.128898
$964$	−6.00000	−0.193247
$965$	−28.0000	−0.901352
$966$	0	0
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	0	0
$969$	6.00000	0.192748
$970$	20.0000	0.642161
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−32.0000	−1.02535
$975$	6.00000	0.192154
$976$	−2.00000	−0.0640184
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	4.00000	0.127906
$979$	0	0
$980$	−14.0000	−0.447214
$981$	10.0000	0.319275
$982$	−32.0000	−1.02116
$983$	−8.00000	−0.255160	−0.127580	−	0.991828i	$-0.540721\pi$
$983$	−8.00000	−0.255160	−0.127580	+	0.991828i	$0.540721\pi$
$984$	6.00000	0.191273
$985$	−4.00000	−0.127451
$986$	12.0000	0.382158
$987$	0	0
$988$	6.00000	0.190885
$989$	16.0000	0.508770
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	−40.0000	−1.27000
$993$	12.0000	0.380808
$994$	0	0
$995$	16.0000	0.507234
$996$	16.0000	0.506979
$997$	58.0000	1.83688	0.918439	−	0.395562i	$-0.129450\pi$
$997$	58.0000	1.83688	0.918439	+	0.395562i	$0.129450\pi$
$998$	−28.0000	−0.886325
$999$	−10.0000	−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6897.2.a.a.1.1		1
11.10	odd	2		57.2.a.c.1.1	✓	1
33.32	even	2		171.2.a.a.1.1		1
44.43	even	2		912.2.a.b.1.1		1
55.32	even	4		1425.2.c.g.799.2		2
55.43	even	4		1425.2.c.g.799.1		2
55.54	odd	2		1425.2.a.a.1.1		1
77.76	even	2		2793.2.a.i.1.1		1
88.21	odd	2		3648.2.a.o.1.1		1
88.43	even	2		3648.2.a.bf.1.1		1
132.131	odd	2		2736.2.a.s.1.1		1
143.142	odd	2		9633.2.a.h.1.1		1
165.164	even	2		4275.2.a.m.1.1		1
209.208	even	2		1083.2.a.a.1.1		1
231.230	odd	2		8379.2.a.e.1.1		1
627.626	odd	2		3249.2.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.2.a.c.1.1	✓	1	11.10	odd	2
171.2.a.a.1.1		1	33.32	even	2
912.2.a.b.1.1		1	44.43	even	2
1083.2.a.a.1.1		1	209.208	even	2
1425.2.a.a.1.1		1	55.54	odd	2
1425.2.c.g.799.1		2	55.43	even	4
1425.2.c.g.799.2		2	55.32	even	4
2736.2.a.s.1.1		1	132.131	odd	2
2793.2.a.i.1.1		1	77.76	even	2
3249.2.a.g.1.1		1	627.626	odd	2
3648.2.a.o.1.1		1	88.21	odd	2
3648.2.a.bf.1.1		1	88.43	even	2
4275.2.a.m.1.1		1	165.164	even	2
6897.2.a.a.1.1		1	1.1	even	1	trivial
8379.2.a.e.1.1		1	231.230	odd	2
9633.2.a.h.1.1		1	143.142	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:	\( N \)	\(=\)	\( 6897 = 3 \cdot 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6897.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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