LMFDB - Embedded newform 6897.2.a.f.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:	$1$
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)

$f(q)$

$=$

$q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} -3.00000 q^{5} -2.00000 q^{6} +5.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} -3.00000 q^{5} -2.00000 q^{6} +5.00000 q^{7} +1.00000 q^{9} -6.00000 q^{10} -2.00000 q^{12} -2.00000 q^{13} +10.0000 q^{14} +3.00000 q^{15} -4.00000 q^{16} +1.00000 q^{17} +2.00000 q^{18} +1.00000 q^{19} -6.00000 q^{20} -5.00000 q^{21} -4.00000 q^{23} +4.00000 q^{25} -4.00000 q^{26} -1.00000 q^{27} +10.0000 q^{28} +2.00000 q^{29} +6.00000 q^{30} -6.00000 q^{31} -8.00000 q^{32} +2.00000 q^{34} -15.0000 q^{35} +2.00000 q^{36} +2.00000 q^{38} +2.00000 q^{39} -10.0000 q^{42} +1.00000 q^{43} -3.00000 q^{45} -8.00000 q^{46} -9.00000 q^{47} +4.00000 q^{48} +18.0000 q^{49} +8.00000 q^{50} -1.00000 q^{51} -4.00000 q^{52} +10.0000 q^{53} -2.00000 q^{54} -1.00000 q^{57} +4.00000 q^{58} -8.00000 q^{59} +6.00000 q^{60} +1.00000 q^{61} -12.0000 q^{62} +5.00000 q^{63} -8.00000 q^{64} +6.00000 q^{65} +8.00000 q^{67} +2.00000 q^{68} +4.00000 q^{69} -30.0000 q^{70} -12.0000 q^{71} +11.0000 q^{73} -4.00000 q^{75} +2.00000 q^{76} +4.00000 q^{78} -16.0000 q^{79} +12.0000 q^{80} +1.00000 q^{81} -12.0000 q^{83} -10.0000 q^{84} -3.00000 q^{85} +2.00000 q^{86} -2.00000 q^{87} -6.00000 q^{89} -6.00000 q^{90} -10.0000 q^{91} -8.00000 q^{92} +6.00000 q^{93} -18.0000 q^{94} -3.00000 q^{95} +8.00000 q^{96} -10.0000 q^{97} +36.0000 q^{98} +O(q^{100})\)

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 $

$2$	2.00000	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$2$	2.00000	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.00000	1.00000
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	−2.00000	−0.816497
$7$	5.00000	1.88982	0.944911	−	0.327327i	$-0.106148\pi$
$7$	5.00000	1.88982	0.944911	+	0.327327i	$0.106148\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	−6.00000	−1.89737
$11$	0	0
$11$	0	0
$12$	−2.00000	−0.577350
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	10.0000	2.67261
$15$	3.00000	0.774597
$16$	−4.00000	−1.00000
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	2.00000	0.471405
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−6.00000	−1.34164
$21$	−5.00000	−1.09109
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	−4.00000	−0.784465
$27$	−1.00000	−0.192450
$28$	10.0000	1.88982
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	6.00000	1.09545
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	−8.00000	−1.41421
$33$	0	0
$34$	2.00000	0.342997
$35$	−15.0000	−2.53546
$36$	2.00000	0.333333
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	2.00000	0.324443
$39$	2.00000	0.320256
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	−10.0000	−1.54303
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	0	0
$45$	−3.00000	−0.447214
$46$	−8.00000	−1.17954
$47$	−9.00000	−1.31278	−0.656392	−	0.754420i	$-0.727918\pi$
$47$	−9.00000	−1.31278	−0.656392	+	0.754420i	$0.727918\pi$
$48$	4.00000	0.577350
$49$	18.0000	2.57143
$50$	8.00000	1.13137
$51$	−1.00000	−0.140028
$52$	−4.00000	−0.554700
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	−2.00000	−0.272166
$55$	0	0
$56$	0	0
$57$	−1.00000	−0.132453
$58$	4.00000	0.525226
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	6.00000	0.774597
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	−12.0000	−1.52400
$63$	5.00000	0.629941
$64$	−8.00000	−1.00000
$65$	6.00000	0.744208
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	2.00000	0.242536
$69$	4.00000	0.481543
$70$	−30.0000	−3.58569
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	11.0000	1.28745	0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000	1.28745	0.643726	+	0.765256i	$0.277388\pi$
$74$	0	0
$75$	−4.00000	−0.461880
$76$	2.00000	0.229416
$77$	0	0
$78$	4.00000	0.452911
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	12.0000	1.34164
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	−10.0000	−1.09109
$85$	−3.00000	−0.325396
$86$	2.00000	0.215666
$87$	−2.00000	−0.214423
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	−6.00000	−0.632456
$91$	−10.0000	−1.04828
$92$	−8.00000	−0.834058
$93$	6.00000	0.622171
$94$	−18.0000	−1.85656
$95$	−3.00000	−0.307794
$96$	8.00000	0.816497
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	36.0000	3.63655
$99$	0	0
$100$	8.00000	0.800000
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	−2.00000	−0.198030
$103$	−2.00000	−0.197066	−0.0985329	−	0.995134i	$-0.531415\pi$
$103$	−2.00000	−0.197066	−0.0985329	+	0.995134i	$0.531415\pi$
$104$	0	0
$105$	15.0000	1.46385
$106$	20.0000	1.94257
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	−2.00000	−0.192450
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	0	0
$111$	0	0
$112$	−20.0000	−1.88982
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	−2.00000	−0.187317
$115$	12.0000	1.11901
$116$	4.00000	0.371391
$117$	−2.00000	−0.184900
$118$	−16.0000	−1.47292
$119$	5.00000	0.458349
$120$	0	0
$121$	0	0
$122$	2.00000	0.181071
$123$	0	0
$124$	−12.0000	−1.07763
$125$	3.00000	0.268328
$126$	10.0000	0.890871
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	0	0
$129$	−1.00000	−0.0880451
$130$	12.0000	1.05247
$131$	−7.00000	−0.611593	−0.305796	−	0.952097i	$-0.598923\pi$
$131$	−7.00000	−0.611593	−0.305796	+	0.952097i	$0.598923\pi$
$132$	0	0
$133$	5.00000	0.433555
$134$	16.0000	1.38219
$135$	3.00000	0.258199
$136$	0	0
$137$	−9.00000	−0.768922	−0.384461	−	0.923141i	$-0.625613\pi$
$137$	−9.00000	−0.768922	−0.384461	+	0.923141i	$0.625613\pi$
$138$	8.00000	0.681005
$139$	13.0000	1.10265	0.551323	−	0.834292i	$-0.314123\pi$
$139$	13.0000	1.10265	0.551323	+	0.834292i	$0.314123\pi$
$140$	−30.0000	−2.53546
$141$	9.00000	0.757937
$142$	−24.0000	−2.01404
$143$	0	0
$144$	−4.00000	−0.333333
$145$	−6.00000	−0.498273
$146$	22.0000	1.82073
$147$	−18.0000	−1.48461
$148$	0	0
$149$	21.0000	1.72039	0.860194	−	0.509968i	$-0.170343\pi$
$149$	21.0000	1.72039	0.860194	+	0.509968i	$0.170343\pi$
$150$	−8.00000	−0.653197
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	1.00000	0.0808452
$154$	0	0
$155$	18.0000	1.44579
$156$	4.00000	0.320256
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	−32.0000	−2.54578
$159$	−10.0000	−0.793052
$160$	24.0000	1.89737
$161$	−20.0000	−1.57622
$162$	2.00000	0.157135
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	−24.0000	−1.86276
$167$	−10.0000	−0.773823	−0.386912	−	0.922117i	$-0.626458\pi$
$167$	−10.0000	−0.773823	−0.386912	+	0.922117i	$0.626458\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−6.00000	−0.460179
$171$	1.00000	0.0764719
$172$	2.00000	0.152499
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	−4.00000	−0.303239
$175$	20.0000	1.51186
$176$	0	0
$177$	8.00000	0.601317
$178$	−12.0000	−0.899438
$179$	−18.0000	−1.34538	−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−18.0000	−1.34538	−0.672692	+	0.739923i	$0.734862\pi$
$180$	−6.00000	−0.447214
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	−20.0000	−1.48250
$183$	−1.00000	−0.0739221
$184$	0	0
$185$	0	0
$186$	12.0000	0.879883
$187$	0	0
$188$	−18.0000	−1.31278
$189$	−5.00000	−0.363696
$190$	−6.00000	−0.435286
$191$	9.00000	0.651217	0.325609	−	0.945505i	$-0.394431\pi$
$191$	9.00000	0.651217	0.325609	+	0.945505i	$0.394431\pi$
$192$	8.00000	0.577350
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	−20.0000	−1.43592
$195$	−6.00000	−0.429669
$196$	36.0000	2.57143
$197$	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
$199$	−21.0000	−1.48865	−0.744325	−	0.667817i	$-0.767229\pi$
$199$	−21.0000	−1.48865	−0.744325	+	0.667817i	$0.767229\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	−4.00000	−0.281439
$203$	10.0000	0.701862
$204$	−2.00000	−0.140028
$205$	0	0
$206$	−4.00000	−0.278693
$207$	−4.00000	−0.278019
$208$	8.00000	0.554700
$209$	0	0
$210$	30.0000	2.07020
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	20.0000	1.37361
$213$	12.0000	0.822226
$214$	−12.0000	−0.820303
$215$	−3.00000	−0.204598
$216$	0	0
$217$	−30.0000	−2.03653
$218$	−8.00000	−0.541828
$219$	−11.0000	−0.743311
$220$	0	0
$221$	−2.00000	−0.134535
$222$	0	0
$223$	12.0000	0.803579	0.401790	−	0.915732i	$-0.368388\pi$
$223$	12.0000	0.803579	0.401790	+	0.915732i	$0.368388\pi$
$224$	−40.0000	−2.67261
$225$	4.00000	0.266667
$226$	4.00000	0.266076
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	−2.00000	−0.132453
$229$	25.0000	1.65205	0.826023	−	0.563636i	$-0.190598\pi$
$229$	25.0000	1.65205	0.826023	+	0.563636i	$0.190598\pi$
$230$	24.0000	1.58251
$231$	0	0
$232$	0	0
$233$	−9.00000	−0.589610	−0.294805	−	0.955557i	$-0.595255\pi$
$233$	−9.00000	−0.589610	−0.294805	+	0.955557i	$0.595255\pi$
$234$	−4.00000	−0.261488
$235$	27.0000	1.76129
$236$	−16.0000	−1.04151
$237$	16.0000	1.03931
$238$	10.0000	0.648204
$239$	3.00000	0.194054	0.0970269	−	0.995282i	$-0.469067\pi$
$239$	3.00000	0.194054	0.0970269	+	0.995282i	$0.469067\pi$
$240$	−12.0000	−0.774597
$241$	−20.0000	−1.28831	−0.644157	−	0.764894i	$-0.722792\pi$
$241$	−20.0000	−1.28831	−0.644157	+	0.764894i	$0.722792\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	2.00000	0.128037
$245$	−54.0000	−3.44993
$246$	0	0
$247$	−2.00000	−0.127257
$248$	0	0
$249$	12.0000	0.760469
$250$	6.00000	0.379473
$251$	7.00000	0.441836	0.220918	−	0.975292i	$-0.429095\pi$
$251$	7.00000	0.441836	0.220918	+	0.975292i	$0.429095\pi$
$252$	10.0000	0.629941
$253$	0	0
$254$	4.00000	0.250982
$255$	3.00000	0.187867
$256$	16.0000	1.00000
$257$	−8.00000	−0.499026	−0.249513	−	0.968371i	$-0.580271\pi$
$257$	−8.00000	−0.499026	−0.249513	+	0.968371i	$0.580271\pi$
$258$	−2.00000	−0.124515
$259$	0	0
$260$	12.0000	0.744208
$261$	2.00000	0.123797
$262$	−14.0000	−0.864923
$263$	−23.0000	−1.41824	−0.709120	−	0.705087i	$-0.750908\pi$
$263$	−23.0000	−1.41824	−0.709120	+	0.705087i	$0.750908\pi$
$264$	0	0
$265$	−30.0000	−1.84289
$266$	10.0000	0.613139
$267$	6.00000	0.367194
$268$	16.0000	0.977356
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	6.00000	0.365148
$271$	−12.0000	−0.728948	−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000	−0.728948	−0.364474	+	0.931214i	$0.618751\pi$
$272$	−4.00000	−0.242536
$273$	10.0000	0.605228
$274$	−18.0000	−1.08742
$275$	0	0
$276$	8.00000	0.481543
$277$	11.0000	0.660926	0.330463	−	0.943819i	$-0.392795\pi$
$277$	11.0000	0.660926	0.330463	+	0.943819i	$0.392795\pi$
$278$	26.0000	1.55938
$279$	−6.00000	−0.359211
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	18.0000	1.07188
$283$	13.0000	0.772770	0.386385	−	0.922338i	$-0.373724\pi$
$283$	13.0000	0.772770	0.386385	+	0.922338i	$0.373724\pi$
$284$	−24.0000	−1.42414
$285$	3.00000	0.177705
$286$	0	0
$287$	0	0
$288$	−8.00000	−0.471405
$289$	−16.0000	−0.941176
$290$	−12.0000	−0.704664
$291$	10.0000	0.586210
$292$	22.0000	1.28745
$293$	28.0000	1.63578	0.817889	−	0.575376i	$-0.195144\pi$
$293$	28.0000	1.63578	0.817889	+	0.575376i	$0.195144\pi$
$294$	−36.0000	−2.09956
$295$	24.0000	1.39733
$296$	0	0
$297$	0	0
$298$	42.0000	2.43299
$299$	8.00000	0.462652
$300$	−8.00000	−0.461880
$301$	5.00000	0.288195
$302$	0	0
$303$	2.00000	0.114897
$304$	−4.00000	−0.229416
$305$	−3.00000	−0.171780
$306$	2.00000	0.114332
$307$	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
$309$	2.00000	0.113776
$310$	36.0000	2.04466
$311$	−21.0000	−1.19080	−0.595400	−	0.803429i	$-0.703007\pi$
$311$	−21.0000	−1.19080	−0.595400	+	0.803429i	$0.703007\pi$
$312$	0	0
$313$	−2.00000	−0.113047	−0.0565233	−	0.998401i	$-0.518002\pi$
$313$	−2.00000	−0.113047	−0.0565233	+	0.998401i	$0.518002\pi$
$314$	−36.0000	−2.03160
$315$	−15.0000	−0.845154
$316$	−32.0000	−1.80014
$317$	−4.00000	−0.224662	−0.112331	−	0.993671i	$-0.535832\pi$
$317$	−4.00000	−0.224662	−0.112331	+	0.993671i	$0.535832\pi$
$318$	−20.0000	−1.12154
$319$	0	0
$320$	24.0000	1.34164
$321$	6.00000	0.334887
$322$	−40.0000	−2.22911
$323$	1.00000	0.0556415
$324$	2.00000	0.111111
$325$	−8.00000	−0.443760
$326$	0	0
$327$	4.00000	0.221201
$328$	0	0
$329$	−45.0000	−2.48093
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	−24.0000	−1.31717
$333$	0	0
$334$	−20.0000	−1.09435
$335$	−24.0000	−1.31126
$336$	20.0000	1.09109
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	−18.0000	−0.979071
$339$	−2.00000	−0.108625
$340$	−6.00000	−0.325396
$341$	0	0
$342$	2.00000	0.108148
$343$	55.0000	2.96972
$344$	0	0
$345$	−12.0000	−0.646058
$346$	−12.0000	−0.645124
$347$	25.0000	1.34207	0.671035	−	0.741426i	$-0.265850\pi$
$347$	25.0000	1.34207	0.671035	+	0.741426i	$0.265850\pi$
$348$	−4.00000	−0.214423
$349$	−9.00000	−0.481759	−0.240879	−	0.970555i	$-0.577436\pi$
$349$	−9.00000	−0.481759	−0.240879	+	0.970555i	$0.577436\pi$
$350$	40.0000	2.13809
$351$	2.00000	0.106752
$352$	0	0
$353$	−2.00000	−0.106449	−0.0532246	−	0.998583i	$-0.516950\pi$
$353$	−2.00000	−0.106449	−0.0532246	+	0.998583i	$0.516950\pi$
$354$	16.0000	0.850390
$355$	36.0000	1.91068
$356$	−12.0000	−0.635999
$357$	−5.00000	−0.264628
$358$	−36.0000	−1.90266
$359$	−37.0000	−1.95279	−0.976393	−	0.216003i	$-0.930698\pi$
$359$	−37.0000	−1.95279	−0.976393	+	0.216003i	$0.930698\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−28.0000	−1.47165
$363$	0	0
$364$	−20.0000	−1.04828
$365$	−33.0000	−1.72730
$366$	−2.00000	−0.104542
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	16.0000	0.834058
$369$	0	0
$370$	0	0
$371$	50.0000	2.59587
$372$	12.0000	0.622171
$373$	−16.0000	−0.828449	−0.414224	−	0.910175i	$-0.635947\pi$
$373$	−16.0000	−0.828449	−0.414224	+	0.910175i	$0.635947\pi$
$374$	0	0
$375$	−3.00000	−0.154919
$376$	0	0
$377$	−4.00000	−0.206010
$378$	−10.0000	−0.514344
$379$	34.0000	1.74646	0.873231	−	0.487306i	$-0.162020\pi$
$379$	34.0000	1.74646	0.873231	+	0.487306i	$0.162020\pi$
$380$	−6.00000	−0.307794
$381$	−2.00000	−0.102463
$382$	18.0000	0.920960
$383$	−34.0000	−1.73732	−0.868659	−	0.495410i	$-0.835018\pi$
$383$	−34.0000	−1.73732	−0.868659	+	0.495410i	$0.835018\pi$
$384$	0	0
$385$	0	0
$386$	−8.00000	−0.407189
$387$	1.00000	0.0508329
$388$	−20.0000	−1.01535
$389$	−27.0000	−1.36895	−0.684477	−	0.729034i	$-0.739969\pi$
$389$	−27.0000	−1.36895	−0.684477	+	0.729034i	$0.739969\pi$
$390$	−12.0000	−0.607644
$391$	−4.00000	−0.202289
$392$	0	0
$393$	7.00000	0.353103
$394$	4.00000	0.201517
$395$	48.0000	2.41514
$396$	0	0
$397$	25.0000	1.25471	0.627357	−	0.778732i	$-0.284137\pi$
$397$	25.0000	1.25471	0.627357	+	0.778732i	$0.284137\pi$
$398$	−42.0000	−2.10527
$399$	−5.00000	−0.250313
$400$	−16.0000	−0.800000
$401$	36.0000	1.79775	0.898877	−	0.438201i	$-0.144384\pi$
$401$	36.0000	1.79775	0.898877	+	0.438201i	$0.144384\pi$
$402$	−16.0000	−0.798007
$403$	12.0000	0.597763
$404$	−4.00000	−0.199007
$405$	−3.00000	−0.149071
$406$	20.0000	0.992583
$407$	0	0
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	9.00000	0.443937
$412$	−4.00000	−0.197066
$413$	−40.0000	−1.96827
$414$	−8.00000	−0.393179
$415$	36.0000	1.76717
$416$	16.0000	0.784465
$417$	−13.0000	−0.636613
$418$	0	0
$419$	28.0000	1.36789	0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000	1.36789	0.683945	+	0.729534i	$0.260263\pi$
$420$	30.0000	1.46385
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	−24.0000	−1.16830
$423$	−9.00000	−0.437595
$424$	0	0
$425$	4.00000	0.194029
$426$	24.0000	1.16280
$427$	5.00000	0.241967
$428$	−12.0000	−0.580042
$429$	0	0
$430$	−6.00000	−0.289346
$431$	34.0000	1.63772	0.818861	−	0.573992i	$-0.194606\pi$
$431$	34.0000	1.63772	0.818861	+	0.573992i	$0.194606\pi$
$432$	4.00000	0.192450
$433$	6.00000	0.288342	0.144171	−	0.989553i	$-0.453949\pi$
$433$	6.00000	0.288342	0.144171	+	0.989553i	$0.453949\pi$
$434$	−60.0000	−2.88009
$435$	6.00000	0.287678
$436$	−8.00000	−0.383131
$437$	−4.00000	−0.191346
$438$	−22.0000	−1.05120
$439$	−26.0000	−1.24091	−0.620456	−	0.784241i	$-0.713053\pi$
$439$	−26.0000	−1.24091	−0.620456	+	0.784241i	$0.713053\pi$
$440$	0	0
$441$	18.0000	0.857143
$442$	−4.00000	−0.190261
$443$	−5.00000	−0.237557	−0.118779	−	0.992921i	$-0.537898\pi$
$443$	−5.00000	−0.237557	−0.118779	+	0.992921i	$0.537898\pi$
$444$	0	0
$445$	18.0000	0.853282
$446$	24.0000	1.13643
$447$	−21.0000	−0.993266
$448$	−40.0000	−1.88982
$449$	−36.0000	−1.69895	−0.849473	−	0.527633i	$-0.823080\pi$
$449$	−36.0000	−1.69895	−0.849473	+	0.527633i	$0.823080\pi$
$450$	8.00000	0.377124
$451$	0	0
$452$	4.00000	0.188144
$453$	0	0
$454$	−36.0000	−1.68956
$455$	30.0000	1.40642
$456$	0	0
$457$	29.0000	1.35656	0.678281	−	0.734802i	$-0.262725\pi$
$457$	29.0000	1.35656	0.678281	+	0.734802i	$0.262725\pi$
$458$	50.0000	2.33635
$459$	−1.00000	−0.0466760
$460$	24.0000	1.11901
$461$	−27.0000	−1.25752	−0.628758	−	0.777601i	$-0.716436\pi$
$461$	−27.0000	−1.25752	−0.628758	+	0.777601i	$0.716436\pi$
$462$	0	0
$463$	17.0000	0.790057	0.395029	−	0.918669i	$-0.370735\pi$
$463$	17.0000	0.790057	0.395029	+	0.918669i	$0.370735\pi$
$464$	−8.00000	−0.371391
$465$	−18.0000	−0.834730
$466$	−18.0000	−0.833834
$467$	−5.00000	−0.231372	−0.115686	−	0.993286i	$-0.536907\pi$
$467$	−5.00000	−0.231372	−0.115686	+	0.993286i	$0.536907\pi$
$468$	−4.00000	−0.184900
$469$	40.0000	1.84703
$470$	54.0000	2.49083
$471$	18.0000	0.829396
$472$	0	0
$473$	0	0
$474$	32.0000	1.46981
$475$	4.00000	0.183533
$476$	10.0000	0.458349
$477$	10.0000	0.457869
$478$	6.00000	0.274434
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	−24.0000	−1.09545
$481$	0	0
$482$	−40.0000	−1.82195
$483$	20.0000	0.910032
$484$	0	0
$485$	30.0000	1.36223
$486$	−2.00000	−0.0907218
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	0	0
$489$	0	0
$490$	−108.000	−4.87894
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	2.00000	0.0900755
$494$	−4.00000	−0.179969
$495$	0	0
$496$	24.0000	1.07763
$497$	−60.0000	−2.69137
$498$	24.0000	1.07547
$499$	5.00000	0.223831	0.111915	−	0.993718i	$-0.464301\pi$
$499$	5.00000	0.223831	0.111915	+	0.993718i	$0.464301\pi$
$500$	6.00000	0.268328
$501$	10.0000	0.446767
$502$	14.0000	0.624851
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	0	0
$507$	9.00000	0.399704
$508$	4.00000	0.177471
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	6.00000	0.265684
$511$	55.0000	2.43306
$512$	32.0000	1.41421
$513$	−1.00000	−0.0441511
$514$	−16.0000	−0.705730
$515$	6.00000	0.264392
$516$	−2.00000	−0.0880451
$517$	0	0
$518$	0	0
$519$	6.00000	0.263371
$520$	0	0
$521$	36.0000	1.57719	0.788594	−	0.614914i	$-0.210809\pi$
$521$	36.0000	1.57719	0.788594	+	0.614914i	$0.210809\pi$
$522$	4.00000	0.175075
$523$	34.0000	1.48672	0.743358	−	0.668894i	$-0.233232\pi$
$523$	34.0000	1.48672	0.743358	+	0.668894i	$0.233232\pi$
$524$	−14.0000	−0.611593
$525$	−20.0000	−0.872872
$526$	−46.0000	−2.00570
$527$	−6.00000	−0.261364
$528$	0	0
$529$	−7.00000	−0.304348
$530$	−60.0000	−2.60623
$531$	−8.00000	−0.347170
$532$	10.0000	0.433555
$533$	0	0
$534$	12.0000	0.519291
$535$	18.0000	0.778208
$536$	0	0
$537$	18.0000	0.776757
$538$	−28.0000	−1.20717
$539$	0	0
$540$	6.00000	0.258199
$541$	−3.00000	−0.128980	−0.0644900	−	0.997918i	$-0.520542\pi$
$541$	−3.00000	−0.128980	−0.0644900	+	0.997918i	$0.520542\pi$
$542$	−24.0000	−1.03089
$543$	14.0000	0.600798
$544$	−8.00000	−0.342997
$545$	12.0000	0.514024
$546$	20.0000	0.855921
$547$	26.0000	1.11168	0.555840	−	0.831289i	$-0.312397\pi$
$547$	26.0000	1.11168	0.555840	+	0.831289i	$0.312397\pi$
$548$	−18.0000	−0.768922
$549$	1.00000	0.0426790
$550$	0	0
$551$	2.00000	0.0852029
$552$	0	0
$553$	−80.0000	−3.40195
$554$	22.0000	0.934690
$555$	0	0
$556$	26.0000	1.10265
$557$	41.0000	1.73723	0.868613	−	0.495491i	$-0.165012\pi$
$557$	41.0000	1.73723	0.868613	+	0.495491i	$0.165012\pi$
$558$	−12.0000	−0.508001
$559$	−2.00000	−0.0845910
$560$	60.0000	2.53546
$561$	0	0
$562$	−20.0000	−0.843649
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	18.0000	0.757937
$565$	−6.00000	−0.252422
$566$	26.0000	1.09286
$567$	5.00000	0.209980
$568$	0	0
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	6.00000	0.251312
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	0	0
$573$	−9.00000	−0.375980
$574$	0	0
$575$	−16.0000	−0.667246
$576$	−8.00000	−0.333333
$577$	27.0000	1.12402	0.562012	−	0.827129i	$-0.310027\pi$
$577$	27.0000	1.12402	0.562012	+	0.827129i	$0.310027\pi$
$578$	−32.0000	−1.33102
$579$	4.00000	0.166234
$580$	−12.0000	−0.498273
$581$	−60.0000	−2.48922
$582$	20.0000	0.829027
$583$	0	0
$584$	0	0
$585$	6.00000	0.248069
$586$	56.0000	2.31334
$587$	7.00000	0.288921	0.144460	−	0.989511i	$-0.453855\pi$
$587$	7.00000	0.288921	0.144460	+	0.989511i	$0.453855\pi$
$588$	−36.0000	−1.48461
$589$	−6.00000	−0.247226
$590$	48.0000	1.97613
$591$	−2.00000	−0.0822690
$592$	0	0
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	−15.0000	−0.614940
$596$	42.0000	1.72039
$597$	21.0000	0.859473
$598$	16.0000	0.654289
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	10.0000	0.407570
$603$	8.00000	0.325785
$604$	0	0
$605$	0	0
$606$	4.00000	0.162489
$607$	−26.0000	−1.05531	−0.527654	−	0.849460i	$-0.676928\pi$
$607$	−26.0000	−1.05531	−0.527654	+	0.849460i	$0.676928\pi$
$608$	−8.00000	−0.324443
$609$	−10.0000	−0.405220
$610$	−6.00000	−0.242933
$611$	18.0000	0.728202
$612$	2.00000	0.0808452
$613$	−33.0000	−1.33286	−0.666429	−	0.745569i	$-0.732178\pi$
$613$	−33.0000	−1.33286	−0.666429	+	0.745569i	$0.732178\pi$
$614$	24.0000	0.968561
$615$	0	0
$616$	0	0
$617$	27.0000	1.08698	0.543490	−	0.839416i	$-0.317103\pi$
$617$	27.0000	1.08698	0.543490	+	0.839416i	$0.317103\pi$
$618$	4.00000	0.160904
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	36.0000	1.44579
$621$	4.00000	0.160514
$622$	−42.0000	−1.68405
$623$	−30.0000	−1.20192
$624$	−8.00000	−0.320256
$625$	−29.0000	−1.16000
$626$	−4.00000	−0.159872
$627$	0	0
$628$	−36.0000	−1.43656
$629$	0	0
$630$	−30.0000	−1.19523
$631$	15.0000	0.597141	0.298570	−	0.954388i	$-0.403490\pi$
$631$	15.0000	0.597141	0.298570	+	0.954388i	$0.403490\pi$
$632$	0	0
$633$	12.0000	0.476957
$634$	−8.00000	−0.317721
$635$	−6.00000	−0.238103
$636$	−20.0000	−0.793052
$637$	−36.0000	−1.42637
$638$	0	0
$639$	−12.0000	−0.474713
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	12.0000	0.473602
$643$	−1.00000	−0.0394362	−0.0197181	−	0.999806i	$-0.506277\pi$
$643$	−1.00000	−0.0394362	−0.0197181	+	0.999806i	$0.506277\pi$
$644$	−40.0000	−1.57622
$645$	3.00000	0.118125
$646$	2.00000	0.0786889
$647$	−39.0000	−1.53325	−0.766624	−	0.642096i	$-0.778065\pi$
$647$	−39.0000	−1.53325	−0.766624	+	0.642096i	$0.778065\pi$
$648$	0	0
$649$	0	0
$650$	−16.0000	−0.627572
$651$	30.0000	1.17579
$652$	0	0
$653$	3.00000	0.117399	0.0586995	−	0.998276i	$-0.481305\pi$
$653$	3.00000	0.117399	0.0586995	+	0.998276i	$0.481305\pi$
$654$	8.00000	0.312825
$655$	21.0000	0.820538
$656$	0	0
$657$	11.0000	0.429151
$658$	−90.0000	−3.50857
$659$	14.0000	0.545363	0.272681	−	0.962104i	$-0.412090\pi$
$659$	14.0000	0.545363	0.272681	+	0.962104i	$0.412090\pi$
$660$	0	0
$661$	12.0000	0.466746	0.233373	−	0.972387i	$-0.425024\pi$
$661$	12.0000	0.466746	0.233373	+	0.972387i	$0.425024\pi$
$662$	−8.00000	−0.310929
$663$	2.00000	0.0776736
$664$	0	0
$665$	−15.0000	−0.581675
$666$	0	0
$667$	−8.00000	−0.309761
$668$	−20.0000	−0.773823
$669$	−12.0000	−0.463947
$670$	−48.0000	−1.85440
$671$	0	0
$672$	40.0000	1.54303
$673$	24.0000	0.925132	0.462566	−	0.886585i	$-0.346929\pi$
$673$	24.0000	0.925132	0.462566	+	0.886585i	$0.346929\pi$
$674$	28.0000	1.07852
$675$	−4.00000	−0.153960
$676$	−18.0000	−0.692308
$677$	−34.0000	−1.30673	−0.653363	−	0.757045i	$-0.726642\pi$
$677$	−34.0000	−1.30673	−0.653363	+	0.757045i	$0.726642\pi$
$678$	−4.00000	−0.153619
$679$	−50.0000	−1.91882
$680$	0	0
$681$	18.0000	0.689761
$682$	0	0
$683$	−6.00000	−0.229584	−0.114792	−	0.993390i	$-0.536620\pi$
$683$	−6.00000	−0.229584	−0.114792	+	0.993390i	$0.536620\pi$
$684$	2.00000	0.0764719
$685$	27.0000	1.03162
$686$	110.000	4.19982
$687$	−25.0000	−0.953809
$688$	−4.00000	−0.152499
$689$	−20.0000	−0.761939
$690$	−24.0000	−0.913664
$691$	−31.0000	−1.17930	−0.589648	−	0.807661i	$-0.700733\pi$
$691$	−31.0000	−1.17930	−0.589648	+	0.807661i	$0.700733\pi$
$692$	−12.0000	−0.456172
$693$	0	0
$694$	50.0000	1.89797
$695$	−39.0000	−1.47935
$696$	0	0
$697$	0	0
$698$	−18.0000	−0.681310
$699$	9.00000	0.340411
$700$	40.0000	1.51186
$701$	22.0000	0.830929	0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000	0.830929	0.415464	+	0.909610i	$0.363619\pi$
$702$	4.00000	0.150970
$703$	0	0
$704$	0	0
$705$	−27.0000	−1.01688
$706$	−4.00000	−0.150542
$707$	−10.0000	−0.376089
$708$	16.0000	0.601317
$709$	−42.0000	−1.57734	−0.788672	−	0.614815i	$-0.789231\pi$
$709$	−42.0000	−1.57734	−0.788672	+	0.614815i	$0.789231\pi$
$710$	72.0000	2.70211
$711$	−16.0000	−0.600047
$712$	0	0
$713$	24.0000	0.898807
$714$	−10.0000	−0.374241
$715$	0	0
$716$	−36.0000	−1.34538
$717$	−3.00000	−0.112037
$718$	−74.0000	−2.76166
$719$	33.0000	1.23069	0.615346	−	0.788257i	$-0.289016\pi$
$719$	33.0000	1.23069	0.615346	+	0.788257i	$0.289016\pi$
$720$	12.0000	0.447214
$721$	−10.0000	−0.372419
$722$	2.00000	0.0744323
$723$	20.0000	0.743808
$724$	−28.0000	−1.04061
$725$	8.00000	0.297113
$726$	0	0
$727$	−23.0000	−0.853023	−0.426511	−	0.904482i	$-0.640258\pi$
$727$	−23.0000	−0.853023	−0.426511	+	0.904482i	$0.640258\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−66.0000	−2.44277
$731$	1.00000	0.0369863
$732$	−2.00000	−0.0739221
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	−16.0000	−0.590571
$735$	54.0000	1.99182
$736$	32.0000	1.17954
$737$	0	0
$738$	0	0
$739$	5.00000	0.183928	0.0919640	−	0.995762i	$-0.470686\pi$
$739$	5.00000	0.183928	0.0919640	+	0.995762i	$0.470686\pi$
$740$	0	0
$741$	2.00000	0.0734718
$742$	100.000	3.67112
$743$	8.00000	0.293492	0.146746	−	0.989174i	$-0.453120\pi$
$743$	8.00000	0.293492	0.146746	+	0.989174i	$0.453120\pi$
$744$	0	0
$745$	−63.0000	−2.30814
$746$	−32.0000	−1.17160
$747$	−12.0000	−0.439057
$748$	0	0
$749$	−30.0000	−1.09618
$750$	−6.00000	−0.219089
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	36.0000	1.31278
$753$	−7.00000	−0.255094
$754$	−8.00000	−0.291343
$755$	0	0
$756$	−10.0000	−0.363696
$757$	−17.0000	−0.617876	−0.308938	−	0.951082i	$-0.599973\pi$
$757$	−17.0000	−0.617876	−0.308938	+	0.951082i	$0.599973\pi$
$758$	68.0000	2.46987
$759$	0	0
$760$	0	0
$761$	−15.0000	−0.543750	−0.271875	−	0.962333i	$-0.587644\pi$
$761$	−15.0000	−0.543750	−0.271875	+	0.962333i	$0.587644\pi$
$762$	−4.00000	−0.144905
$763$	−20.0000	−0.724049
$764$	18.0000	0.651217
$765$	−3.00000	−0.108465
$766$	−68.0000	−2.45694
$767$	16.0000	0.577727
$768$	−16.0000	−0.577350
$769$	−11.0000	−0.396670	−0.198335	−	0.980134i	$-0.563553\pi$
$769$	−11.0000	−0.396670	−0.198335	+	0.980134i	$0.563553\pi$
$770$	0	0
$771$	8.00000	0.288113
$772$	−8.00000	−0.287926
$773$	20.0000	0.719350	0.359675	−	0.933078i	$-0.382888\pi$
$773$	20.0000	0.719350	0.359675	+	0.933078i	$0.382888\pi$
$774$	2.00000	0.0718885
$775$	−24.0000	−0.862105
$776$	0	0
$777$	0	0
$778$	−54.0000	−1.93599
$779$	0	0
$780$	−12.0000	−0.429669
$781$	0	0
$782$	−8.00000	−0.286079
$783$	−2.00000	−0.0714742
$784$	−72.0000	−2.57143
$785$	54.0000	1.92734
$786$	14.0000	0.499363
$787$	−40.0000	−1.42585	−0.712923	−	0.701242i	$-0.752629\pi$
$787$	−40.0000	−1.42585	−0.712923	+	0.701242i	$0.752629\pi$
$788$	4.00000	0.142494
$789$	23.0000	0.818822
$790$	96.0000	3.41553
$791$	10.0000	0.355559
$792$	0	0
$793$	−2.00000	−0.0710221
$794$	50.0000	1.77443
$795$	30.0000	1.06399
$796$	−42.0000	−1.48865
$797$	−44.0000	−1.55856	−0.779280	−	0.626676i	$-0.784415\pi$
$797$	−44.0000	−1.55856	−0.779280	+	0.626676i	$0.784415\pi$
$798$	−10.0000	−0.353996
$799$	−9.00000	−0.318397
$800$	−32.0000	−1.13137
$801$	−6.00000	−0.212000
$802$	72.0000	2.54241
$803$	0	0
$804$	−16.0000	−0.564276
$805$	60.0000	2.11472
$806$	24.0000	0.845364
$807$	14.0000	0.492823
$808$	0	0
$809$	55.0000	1.93370	0.966849	−	0.255351i	$-0.0821909\pi$
$809$	55.0000	1.93370	0.966849	+	0.255351i	$0.0821909\pi$
$810$	−6.00000	−0.210819
$811$	38.0000	1.33436	0.667180	−	0.744896i	$-0.267501\pi$
$811$	38.0000	1.33436	0.667180	+	0.744896i	$0.267501\pi$
$812$	20.0000	0.701862
$813$	12.0000	0.420858
$814$	0	0
$815$	0	0
$816$	4.00000	0.140028
$817$	1.00000	0.0349856
$818$	28.0000	0.978997
$819$	−10.0000	−0.349428
$820$	0	0
$821$	45.0000	1.57051	0.785255	−	0.619172i	$-0.212532\pi$
$821$	45.0000	1.57051	0.785255	+	0.619172i	$0.212532\pi$
$822$	18.0000	0.627822
$823$	43.0000	1.49889	0.749443	−	0.662069i	$-0.230321\pi$
$823$	43.0000	1.49889	0.749443	+	0.662069i	$0.230321\pi$
$824$	0	0
$825$	0	0
$826$	−80.0000	−2.78356
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	−8.00000	−0.278019
$829$	52.0000	1.80603	0.903017	−	0.429604i	$-0.141347\pi$
$829$	52.0000	1.80603	0.903017	+	0.429604i	$0.141347\pi$
$830$	72.0000	2.49916
$831$	−11.0000	−0.381586
$832$	16.0000	0.554700
$833$	18.0000	0.623663
$834$	−26.0000	−0.900306
$835$	30.0000	1.03819
$836$	0	0
$837$	6.00000	0.207390
$838$	56.0000	1.93449
$839$	54.0000	1.86429	0.932144	−	0.362089i	$-0.117936\pi$
$839$	54.0000	1.86429	0.932144	+	0.362089i	$0.117936\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	52.0000	1.79204
$843$	10.0000	0.344418
$844$	−24.0000	−0.826114
$845$	27.0000	0.928828
$846$	−18.0000	−0.618853
$847$	0	0
$848$	−40.0000	−1.37361
$849$	−13.0000	−0.446159
$850$	8.00000	0.274398
$851$	0	0
$852$	24.0000	0.822226
$853$	14.0000	0.479351	0.239675	−	0.970853i	$-0.422959\pi$
$853$	14.0000	0.479351	0.239675	+	0.970853i	$0.422959\pi$
$854$	10.0000	0.342193
$855$	−3.00000	−0.102598
$856$	0	0
$857$	8.00000	0.273275	0.136637	−	0.990621i	$-0.456370\pi$
$857$	8.00000	0.273275	0.136637	+	0.990621i	$0.456370\pi$
$858$	0	0
$859$	27.0000	0.921228	0.460614	−	0.887601i	$-0.347629\pi$
$859$	27.0000	0.921228	0.460614	+	0.887601i	$0.347629\pi$
$860$	−6.00000	−0.204598
$861$	0	0
$862$	68.0000	2.31609
$863$	−44.0000	−1.49778	−0.748889	−	0.662696i	$-0.769412\pi$
$863$	−44.0000	−1.49778	−0.748889	+	0.662696i	$0.769412\pi$
$864$	8.00000	0.272166
$865$	18.0000	0.612018
$866$	12.0000	0.407777
$867$	16.0000	0.543388
$868$	−60.0000	−2.03653
$869$	0	0
$870$	12.0000	0.406838
$871$	−16.0000	−0.542139
$872$	0	0
$873$	−10.0000	−0.338449
$874$	−8.00000	−0.270604
$875$	15.0000	0.507093
$876$	−22.0000	−0.743311
$877$	6.00000	0.202606	0.101303	−	0.994856i	$-0.467699\pi$
$877$	6.00000	0.202606	0.101303	+	0.994856i	$0.467699\pi$
$878$	−52.0000	−1.75491
$879$	−28.0000	−0.944417
$880$	0	0
$881$	−37.0000	−1.24656	−0.623281	−	0.781998i	$-0.714201\pi$
$881$	−37.0000	−1.24656	−0.623281	+	0.781998i	$0.714201\pi$
$882$	36.0000	1.21218
$883$	35.0000	1.17784	0.588922	−	0.808190i	$-0.299553\pi$
$883$	35.0000	1.17784	0.588922	+	0.808190i	$0.299553\pi$
$884$	−4.00000	−0.134535
$885$	−24.0000	−0.806751
$886$	−10.0000	−0.335957
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	0	0
$889$	10.0000	0.335389
$890$	36.0000	1.20672
$891$	0	0
$892$	24.0000	0.803579
$893$	−9.00000	−0.301174
$894$	−42.0000	−1.40469
$895$	54.0000	1.80502
$896$	0	0
$897$	−8.00000	−0.267112
$898$	−72.0000	−2.40267
$899$	−12.0000	−0.400222
$900$	8.00000	0.266667
$901$	10.0000	0.333148
$902$	0	0
$903$	−5.00000	−0.166390
$904$	0	0
$905$	42.0000	1.39613
$906$	0	0
$907$	−26.0000	−0.863316	−0.431658	−	0.902037i	$-0.642071\pi$
$907$	−26.0000	−0.863316	−0.431658	+	0.902037i	$0.642071\pi$
$908$	−36.0000	−1.19470
$909$	−2.00000	−0.0663358
$910$	60.0000	1.98898
$911$	−6.00000	−0.198789	−0.0993944	−	0.995048i	$-0.531691\pi$
$911$	−6.00000	−0.198789	−0.0993944	+	0.995048i	$0.531691\pi$
$912$	4.00000	0.132453
$913$	0	0
$914$	58.0000	1.91847
$915$	3.00000	0.0991769
$916$	50.0000	1.65205
$917$	−35.0000	−1.15580
$918$	−2.00000	−0.0660098
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	−12.0000	−0.395413
$922$	−54.0000	−1.77840
$923$	24.0000	0.789970
$924$	0	0
$925$	0	0
$926$	34.0000	1.11731
$927$	−2.00000	−0.0656886
$928$	−16.0000	−0.525226
$929$	−2.00000	−0.0656179	−0.0328089	−	0.999462i	$-0.510445\pi$
$929$	−2.00000	−0.0656179	−0.0328089	+	0.999462i	$0.510445\pi$
$930$	−36.0000	−1.18049
$931$	18.0000	0.589926
$932$	−18.0000	−0.589610
$933$	21.0000	0.687509
$934$	−10.0000	−0.327210
$935$	0	0
$936$	0	0
$937$	−21.0000	−0.686040	−0.343020	−	0.939328i	$-0.611450\pi$
$937$	−21.0000	−0.686040	−0.343020	+	0.939328i	$0.611450\pi$
$938$	80.0000	2.61209
$939$	2.00000	0.0652675
$940$	54.0000	1.76129
$941$	−42.0000	−1.36916	−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000	−1.36916	−0.684580	+	0.728937i	$0.740015\pi$
$942$	36.0000	1.17294
$943$	0	0
$944$	32.0000	1.04151
$945$	15.0000	0.487950
$946$	0	0
$947$	−28.0000	−0.909878	−0.454939	−	0.890523i	$-0.650339\pi$
$947$	−28.0000	−0.909878	−0.454939	+	0.890523i	$0.650339\pi$
$948$	32.0000	1.03931
$949$	−22.0000	−0.714150
$950$	8.00000	0.259554
$951$	4.00000	0.129709
$952$	0	0
$953$	32.0000	1.03658	0.518291	−	0.855204i	$-0.326568\pi$
$953$	32.0000	1.03658	0.518291	+	0.855204i	$0.326568\pi$
$954$	20.0000	0.647524
$955$	−27.0000	−0.873699
$956$	6.00000	0.194054
$957$	0	0
$958$	32.0000	1.03387
$959$	−45.0000	−1.45313
$960$	−24.0000	−0.774597
$961$	5.00000	0.161290
$962$	0	0
$963$	−6.00000	−0.193347
$964$	−40.0000	−1.28831
$965$	12.0000	0.386294
$966$	40.0000	1.28698
$967$	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
$969$	−1.00000	−0.0321246
$970$	60.0000	1.92648
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	−2.00000	−0.0641500
$973$	65.0000	2.08380
$974$	−32.0000	−1.02535
$975$	8.00000	0.256205
$976$	−4.00000	−0.128037
$977$	54.0000	1.72761	0.863807	−	0.503824i	$-0.168074\pi$
$977$	54.0000	1.72761	0.863807	+	0.503824i	$0.168074\pi$
$978$	0	0
$979$	0	0
$980$	−108.000	−3.44993
$981$	−4.00000	−0.127710
$982$	0	0
$983$	−44.0000	−1.40338	−0.701691	−	0.712481i	$-0.747571\pi$
$983$	−44.0000	−1.40338	−0.701691	+	0.712481i	$0.747571\pi$
$984$	0	0
$985$	−6.00000	−0.191176
$986$	4.00000	0.127386
$987$	45.0000	1.43237
$988$	−4.00000	−0.127257
$989$	−4.00000	−0.127193
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	48.0000	1.52400
$993$	4.00000	0.126936
$994$	−120.000	−3.80617
$995$	63.0000	1.99723
$996$	24.0000	0.760469
$997$	47.0000	1.48850	0.744252	−	0.667898i	$-0.232806\pi$
$997$	47.0000	1.48850	0.744252	+	0.667898i	$0.232806\pi$
$998$	10.0000	0.316544
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.2.a.a.1.1	✓	1	11.10	odd	2
171.2.a.d.1.1		1	33.32	even	2
912.2.a.g.1.1		1	44.43	even	2
1083.2.a.e.1.1		1	209.208	even	2
1425.2.a.j.1.1		1	55.54	odd	2
1425.2.c.b.799.1		2	55.32	even	4
1425.2.c.b.799.2		2	55.43	even	4
2736.2.a.v.1.1		1	132.131	odd	2
2793.2.a.b.1.1		1	77.76	even	2
3249.2.a.b.1.1		1	627.626	odd	2
3648.2.a.r.1.1		1	88.43	even	2
3648.2.a.bh.1.1		1	88.21	odd	2
4275.2.a.b.1.1		1	165.164	even	2
6897.2.a.f.1.1		1	1.1	even	1	trivial
8379.2.a.p.1.1		1	231.230	odd	2
9633.2.a.o.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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