LMFDB - Embedded newform 2898.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2898,2,Mod(1,2898)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2898, 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("2898.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2898.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:	$23.1406465058$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 966)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2898.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q+1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -3.00000 q^{10} -4.00000 q^{11} +3.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -3.00000 q^{20} -4.00000 q^{22} +1.00000 q^{23} +4.00000 q^{25} +3.00000 q^{26} +1.00000 q^{28} -1.00000 q^{29} -2.00000 q^{31} +1.00000 q^{32} -3.00000 q^{35} -5.00000 q^{37} -3.00000 q^{40} -5.00000 q^{41} -7.00000 q^{43} -4.00000 q^{44} +1.00000 q^{46} +3.00000 q^{47} +1.00000 q^{49} +4.00000 q^{50} +3.00000 q^{52} -12.0000 q^{53} +12.0000 q^{55} +1.00000 q^{56} -1.00000 q^{58} +2.00000 q^{59} -6.00000 q^{61} -2.00000 q^{62} +1.00000 q^{64} -9.00000 q^{65} -12.0000 q^{67} -3.00000 q^{70} -10.0000 q^{71} -5.00000 q^{74} -4.00000 q^{77} +4.00000 q^{79} -3.00000 q^{80} -5.00000 q^{82} -4.00000 q^{83} -7.00000 q^{86} -4.00000 q^{88} -10.0000 q^{89} +3.00000 q^{91} +1.00000 q^{92} +3.00000 q^{94} +19.0000 q^{97} +1.00000 q^{98} +O(q^{100})$

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
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	−3.00000	−0.948683
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−3.00000	−0.670820
$21$	0	0
$22$	−4.00000	−0.852803
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	4.00000	0.800000
$26$	3.00000	0.588348
$27$	0	0
$28$	1.00000	0.188982
$29$	−1.00000	−0.185695	−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000	−0.185695	−0.0928477	+	0.995680i	$0.529597\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0	0
$35$	−3.00000	−0.507093
$36$	0	0
$37$	−5.00000	−0.821995	−0.410997	−	0.911636i	$-0.634819\pi$
$37$	−5.00000	−0.821995	−0.410997	+	0.911636i	$0.634819\pi$
$38$	0	0
$39$	0	0
$40$	−3.00000	−0.474342
$41$	−5.00000	−0.780869	−0.390434	−	0.920631i	$-0.627675\pi$
$41$	−5.00000	−0.780869	−0.390434	+	0.920631i	$0.627675\pi$
$42$	0	0
$43$	−7.00000	−1.06749	−0.533745	−	0.845645i	$-0.679216\pi$
$43$	−7.00000	−1.06749	−0.533745	+	0.845645i	$0.679216\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	1.00000	0.147442
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	4.00000	0.565685
$51$	0	0
$52$	3.00000	0.416025
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	0	0
$55$	12.0000	1.61808
$56$	1.00000	0.133631
$57$	0	0
$58$	−1.00000	−0.131306
$59$	2.00000	0.260378	0.130189	−	0.991489i	$-0.458442\pi$
$59$	2.00000	0.260378	0.130189	+	0.991489i	$0.458442\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	−2.00000	−0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	−9.00000	−1.11631
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	0	0
$69$	0	0
$70$	−3.00000	−0.358569
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	−5.00000	−0.581238
$75$	0	0
$76$	0	0
$77$	−4.00000	−0.455842
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	−3.00000	−0.335410
$81$	0	0
$82$	−5.00000	−0.552158
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	0	0
$86$	−7.00000	−0.754829
$87$	0	0
$88$	−4.00000	−0.426401
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	3.00000	0.314485
$92$	1.00000	0.104257
$93$	0	0
$94$	3.00000	0.309426
$95$	0	0
$96$	0	0
$97$	19.0000	1.92916	0.964579	−	0.263795i	$-0.0849741\pi$
$97$	19.0000	1.92916	0.964579	+	0.263795i	$0.0849741\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	4.00000	0.400000
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	0	0
$103$	−1.00000	−0.0985329	−0.0492665	−	0.998786i	$-0.515688\pi$
$103$	−1.00000	−0.0985329	−0.0492665	+	0.998786i	$0.515688\pi$
$104$	3.00000	0.294174
$105$	0	0
$106$	−12.0000	−1.16554
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	−1.00000	−0.0957826	−0.0478913	−	0.998853i	$-0.515250\pi$
$109$	−1.00000	−0.0957826	−0.0478913	+	0.998853i	$0.515250\pi$
$110$	12.0000	1.14416
$111$	0	0
$112$	1.00000	0.0944911
$113$	−7.00000	−0.658505	−0.329252	−	0.944242i	$-0.606797\pi$
$113$	−7.00000	−0.658505	−0.329252	+	0.944242i	$0.606797\pi$
$114$	0	0
$115$	−3.00000	−0.279751
$116$	−1.00000	−0.0928477
$117$	0	0
$118$	2.00000	0.184115
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	−6.00000	−0.543214
$123$	0	0
$124$	−2.00000	−0.179605
$125$	3.00000	0.268328
$126$	0	0
$127$	7.00000	0.621150	0.310575	−	0.950549i	$-0.399478\pi$
$127$	7.00000	0.621150	0.310575	+	0.950549i	$0.399478\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−9.00000	−0.789352
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	0	0
$133$	0	0
$134$	−12.0000	−1.03664
$135$	0	0
$136$	0	0
$137$	17.0000	1.45241	0.726204	−	0.687479i	$-0.241283\pi$
$137$	17.0000	1.45241	0.726204	+	0.687479i	$0.241283\pi$
$138$	0	0
$139$	7.00000	0.593732	0.296866	−	0.954919i	$-0.404058\pi$
$139$	7.00000	0.593732	0.296866	+	0.954919i	$0.404058\pi$
$140$	−3.00000	−0.253546
$141$	0	0
$142$	−10.0000	−0.839181
$143$	−12.0000	−1.00349
$144$	0	0
$145$	3.00000	0.249136
$146$	0	0
$147$	0	0
$148$	−5.00000	−0.410997
$149$	−12.0000	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$150$	0	0
$151$	−17.0000	−1.38344	−0.691720	−	0.722166i	$-0.743147\pi$
$151$	−17.0000	−1.38344	−0.691720	+	0.722166i	$0.743147\pi$
$152$	0	0
$153$	0	0
$154$	−4.00000	−0.322329
$155$	6.00000	0.481932
$156$	0	0
$157$	−24.0000	−1.91541	−0.957704	−	0.287754i	$-0.907091\pi$
$157$	−24.0000	−1.91541	−0.957704	+	0.287754i	$0.907091\pi$
$158$	4.00000	0.318223
$159$	0	0
$160$	−3.00000	−0.237171
$161$	1.00000	0.0788110
$162$	0	0
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	−5.00000	−0.390434
$165$	0	0
$166$	−4.00000	−0.310460
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	0	0
$172$	−7.00000	−0.533745
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	−4.00000	−0.301511
$177$	0	0
$178$	−10.0000	−0.749532
$179$	−9.00000	−0.672692	−0.336346	−	0.941739i	$-0.609191\pi$
$179$	−9.00000	−0.672692	−0.336346	+	0.941739i	$0.609191\pi$
$180$	0	0
$181$	18.0000	1.33793	0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000	1.33793	0.668965	+	0.743294i	$0.266738\pi$
$182$	3.00000	0.222375
$183$	0	0
$184$	1.00000	0.0737210
$185$	15.0000	1.10282
$186$	0	0
$187$	0	0
$188$	3.00000	0.218797
$189$	0	0
$190$	0	0
$191$	20.0000	1.44715	0.723575	−	0.690246i	$-0.242498\pi$
$191$	20.0000	1.44715	0.723575	+	0.690246i	$0.242498\pi$
$192$	0	0
$193$	−1.00000	−0.0719816	−0.0359908	−	0.999352i	$-0.511459\pi$
$193$	−1.00000	−0.0719816	−0.0359908	+	0.999352i	$0.511459\pi$
$194$	19.0000	1.36412
$195$	0	0
$196$	1.00000	0.0714286
$197$	21.0000	1.49619	0.748094	−	0.663593i	$-0.230969\pi$
$197$	21.0000	1.49619	0.748094	+	0.663593i	$0.230969\pi$
$198$	0	0
$199$	−23.0000	−1.63043	−0.815213	−	0.579161i	$-0.803380\pi$
$199$	−23.0000	−1.63043	−0.815213	+	0.579161i	$0.803380\pi$
$200$	4.00000	0.282843
$201$	0	0
$202$	−14.0000	−0.985037
$203$	−1.00000	−0.0701862
$204$	0	0
$205$	15.0000	1.04765
$206$	−1.00000	−0.0696733
$207$	0	0
$208$	3.00000	0.208013
$209$	0	0
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	−12.0000	−0.824163
$213$	0	0
$214$	−4.00000	−0.273434
$215$	21.0000	1.43219
$216$	0	0
$217$	−2.00000	−0.135769
$218$	−1.00000	−0.0677285
$219$	0	0
$220$	12.0000	0.809040
$221$	0	0
$222$	0	0
$223$	2.00000	0.133930	0.0669650	−	0.997755i	$-0.478668\pi$
$223$	2.00000	0.133930	0.0669650	+	0.997755i	$0.478668\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−7.00000	−0.465633
$227$	5.00000	0.331862	0.165931	−	0.986137i	$-0.446937\pi$
$227$	5.00000	0.331862	0.165931	+	0.986137i	$0.446937\pi$
$228$	0	0
$229$	16.0000	1.05731	0.528655	−	0.848837i	$-0.322697\pi$
$229$	16.0000	1.05731	0.528655	+	0.848837i	$0.322697\pi$
$230$	−3.00000	−0.197814
$231$	0	0
$232$	−1.00000	−0.0656532
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	−9.00000	−0.587095
$236$	2.00000	0.130189
$237$	0	0
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	9.00000	0.579741	0.289870	−	0.957066i	$-0.406388\pi$
$241$	9.00000	0.579741	0.289870	+	0.957066i	$0.406388\pi$
$242$	5.00000	0.321412
$243$	0	0
$244$	−6.00000	−0.384111
$245$	−3.00000	−0.191663
$246$	0	0
$247$	0	0
$248$	−2.00000	−0.127000
$249$	0	0
$250$	3.00000	0.189737
$251$	−19.0000	−1.19927	−0.599635	−	0.800274i	$-0.704687\pi$
$251$	−19.0000	−1.19927	−0.599635	+	0.800274i	$0.704687\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	7.00000	0.439219
$255$	0	0
$256$	1.00000	0.0625000
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	0	0
$259$	−5.00000	−0.310685
$260$	−9.00000	−0.558156
$261$	0	0
$262$	18.0000	1.11204
$263$	21.0000	1.29492	0.647458	−	0.762101i	$-0.275832\pi$
$263$	21.0000	1.29492	0.647458	+	0.762101i	$0.275832\pi$
$264$	0	0
$265$	36.0000	2.21146
$266$	0	0
$267$	0	0
$268$	−12.0000	−0.733017
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	0	0
$271$	−12.0000	−0.728948	−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000	−0.728948	−0.364474	+	0.931214i	$0.618751\pi$
$272$	0	0
$273$	0	0
$274$	17.0000	1.02701
$275$	−16.0000	−0.964836
$276$	0	0
$277$	−24.0000	−1.44202	−0.721010	−	0.692925i	$-0.756322\pi$
$277$	−24.0000	−1.44202	−0.721010	+	0.692925i	$0.756322\pi$
$278$	7.00000	0.419832
$279$	0	0
$280$	−3.00000	−0.179284
$281$	−1.00000	−0.0596550	−0.0298275	−	0.999555i	$-0.509496\pi$
$281$	−1.00000	−0.0596550	−0.0298275	+	0.999555i	$0.509496\pi$
$282$	0	0
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	−10.0000	−0.593391
$285$	0	0
$286$	−12.0000	−0.709575
$287$	−5.00000	−0.295141
$288$	0	0
$289$	−17.0000	−1.00000
$290$	3.00000	0.176166
$291$	0	0
$292$	0	0
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	0	0
$295$	−6.00000	−0.349334
$296$	−5.00000	−0.290619
$297$	0	0
$298$	−12.0000	−0.695141
$299$	3.00000	0.173494
$300$	0	0
$301$	−7.00000	−0.403473
$302$	−17.0000	−0.978240
$303$	0	0
$304$	0	0
$305$	18.0000	1.03068
$306$	0	0
$307$	25.0000	1.42683	0.713413	−	0.700744i	$-0.247149\pi$
$307$	25.0000	1.42683	0.713413	+	0.700744i	$0.247149\pi$
$308$	−4.00000	−0.227921
$309$	0	0
$310$	6.00000	0.340777
$311$	4.00000	0.226819	0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000	0.226819	0.113410	+	0.993548i	$0.463823\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	−24.0000	−1.35440
$315$	0	0
$316$	4.00000	0.225018
$317$	33.0000	1.85346	0.926732	−	0.375722i	$-0.122605\pi$
$317$	33.0000	1.85346	0.926732	+	0.375722i	$0.122605\pi$
$318$	0	0
$319$	4.00000	0.223957
$320$	−3.00000	−0.167705
$321$	0	0
$322$	1.00000	0.0557278
$323$	0	0
$324$	0	0
$325$	12.0000	0.665640
$326$	−12.0000	−0.664619
$327$	0	0
$328$	−5.00000	−0.276079
$329$	3.00000	0.165395
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	−4.00000	−0.219529
$333$	0	0
$334$	12.0000	0.656611
$335$	36.0000	1.96689
$336$	0	0
$337$	−20.0000	−1.08947	−0.544735	−	0.838608i	$-0.683370\pi$
$337$	−20.0000	−1.08947	−0.544735	+	0.838608i	$0.683370\pi$
$338$	−4.00000	−0.217571
$339$	0	0
$340$	0	0
$341$	8.00000	0.433224
$342$	0	0
$343$	1.00000	0.0539949
$344$	−7.00000	−0.377415
$345$	0	0
$346$	−2.00000	−0.107521
$347$	3.00000	0.161048	0.0805242	−	0.996753i	$-0.474341\pi$
$347$	3.00000	0.161048	0.0805242	+	0.996753i	$0.474341\pi$
$348$	0	0
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	4.00000	0.213809
$351$	0	0
$352$	−4.00000	−0.213201
$353$	17.0000	0.904819	0.452409	−	0.891810i	$-0.350565\pi$
$353$	17.0000	0.904819	0.452409	+	0.891810i	$0.350565\pi$
$354$	0	0
$355$	30.0000	1.59223
$356$	−10.0000	−0.529999
$357$	0	0
$358$	−9.00000	−0.475665
$359$	−9.00000	−0.475002	−0.237501	−	0.971387i	$-0.576328\pi$
$359$	−9.00000	−0.475002	−0.237501	+	0.971387i	$0.576328\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	18.0000	0.946059
$363$	0	0
$364$	3.00000	0.157243
$365$	0	0
$366$	0	0
$367$	21.0000	1.09619	0.548096	−	0.836416i	$-0.315353\pi$
$367$	21.0000	1.09619	0.548096	+	0.836416i	$0.315353\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	15.0000	0.779813
$371$	−12.0000	−0.623009
$372$	0	0
$373$	−2.00000	−0.103556	−0.0517780	−	0.998659i	$-0.516489\pi$
$373$	−2.00000	−0.103556	−0.0517780	+	0.998659i	$0.516489\pi$
$374$	0	0
$375$	0	0
$376$	3.00000	0.154713
$377$	−3.00000	−0.154508
$378$	0	0
$379$	−5.00000	−0.256833	−0.128416	−	0.991720i	$-0.540989\pi$
$379$	−5.00000	−0.256833	−0.128416	+	0.991720i	$0.540989\pi$
$380$	0	0
$381$	0	0
$382$	20.0000	1.02329
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	0	0
$385$	12.0000	0.611577
$386$	−1.00000	−0.0508987
$387$	0	0
$388$	19.0000	0.964579
$389$	14.0000	0.709828	0.354914	−	0.934899i	$-0.384510\pi$
$389$	14.0000	0.709828	0.354914	+	0.934899i	$0.384510\pi$
$390$	0	0
$391$	0	0
$392$	1.00000	0.0505076
$393$	0	0
$394$	21.0000	1.05796
$395$	−12.0000	−0.603786
$396$	0	0
$397$	6.00000	0.301131	0.150566	−	0.988600i	$-0.451890\pi$
$397$	6.00000	0.301131	0.150566	+	0.988600i	$0.451890\pi$
$398$	−23.0000	−1.15289
$399$	0	0
$400$	4.00000	0.200000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	−6.00000	−0.298881
$404$	−14.0000	−0.696526
$405$	0	0
$406$	−1.00000	−0.0496292
$407$	20.0000	0.991363
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	15.0000	0.740797
$411$	0	0
$412$	−1.00000	−0.0492665
$413$	2.00000	0.0984136
$414$	0	0
$415$	12.0000	0.589057
$416$	3.00000	0.147087
$417$	0	0
$418$	0	0
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	0	0
$421$	−1.00000	−0.0487370	−0.0243685	−	0.999703i	$-0.507758\pi$
$421$	−1.00000	−0.0487370	−0.0243685	+	0.999703i	$0.507758\pi$
$422$	0	0
$423$	0	0
$424$	−12.0000	−0.582772
$425$	0	0
$426$	0	0
$427$	−6.00000	−0.290360
$428$	−4.00000	−0.193347
$429$	0	0
$430$	21.0000	1.01271
$431$	−3.00000	−0.144505	−0.0722525	−	0.997386i	$-0.523019\pi$
$431$	−3.00000	−0.144505	−0.0722525	+	0.997386i	$0.523019\pi$
$432$	0	0
$433$	−27.0000	−1.29754	−0.648769	−	0.760986i	$-0.724716\pi$
$433$	−27.0000	−1.29754	−0.648769	+	0.760986i	$0.724716\pi$
$434$	−2.00000	−0.0960031
$435$	0	0
$436$	−1.00000	−0.0478913
$437$	0	0
$438$	0	0
$439$	14.0000	0.668184	0.334092	−	0.942541i	$-0.391570\pi$
$439$	14.0000	0.668184	0.334092	+	0.942541i	$0.391570\pi$
$440$	12.0000	0.572078
$441$	0	0
$442$	0	0
$443$	29.0000	1.37783	0.688916	−	0.724841i	$-0.258087\pi$
$443$	29.0000	1.37783	0.688916	+	0.724841i	$0.258087\pi$
$444$	0	0
$445$	30.0000	1.42214
$446$	2.00000	0.0947027
$447$	0	0
$448$	1.00000	0.0472456
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	20.0000	0.941763
$452$	−7.00000	−0.329252
$453$	0	0
$454$	5.00000	0.234662
$455$	−9.00000	−0.421927
$456$	0	0
$457$	−4.00000	−0.187112	−0.0935561	−	0.995614i	$-0.529823\pi$
$457$	−4.00000	−0.187112	−0.0935561	+	0.995614i	$0.529823\pi$
$458$	16.0000	0.747631
$459$	0	0
$460$	−3.00000	−0.139876
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	5.00000	0.232370	0.116185	−	0.993228i	$-0.462933\pi$
$463$	5.00000	0.232370	0.116185	+	0.993228i	$0.462933\pi$
$464$	−1.00000	−0.0464238
$465$	0	0
$466$	10.0000	0.463241
$467$	27.0000	1.24941	0.624705	−	0.780860i	$-0.285219\pi$
$467$	27.0000	1.24941	0.624705	+	0.780860i	$0.285219\pi$
$468$	0	0
$469$	−12.0000	−0.554109
$470$	−9.00000	−0.415139
$471$	0	0
$472$	2.00000	0.0920575
$473$	28.0000	1.28744
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	−24.0000	−1.09773
$479$	6.00000	0.274147	0.137073	−	0.990561i	$-0.456230\pi$
$479$	6.00000	0.274147	0.137073	+	0.990561i	$0.456230\pi$
$480$	0	0
$481$	−15.0000	−0.683941
$482$	9.00000	0.409939
$483$	0	0
$484$	5.00000	0.227273
$485$	−57.0000	−2.58824
$486$	0	0
$487$	13.0000	0.589086	0.294543	−	0.955638i	$-0.404833\pi$
$487$	13.0000	0.589086	0.294543	+	0.955638i	$0.404833\pi$
$488$	−6.00000	−0.271607
$489$	0	0
$490$	−3.00000	−0.135526
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	−10.0000	−0.448561
$498$	0	0
$499$	−10.0000	−0.447661	−0.223831	−	0.974628i	$-0.571856\pi$
$499$	−10.0000	−0.447661	−0.223831	+	0.974628i	$0.571856\pi$
$500$	3.00000	0.134164
$501$	0	0
$502$	−19.0000	−0.848012
$503$	2.00000	0.0891756	0.0445878	−	0.999005i	$-0.485803\pi$
$503$	2.00000	0.0891756	0.0445878	+	0.999005i	$0.485803\pi$
$504$	0	0
$505$	42.0000	1.86898
$506$	−4.00000	−0.177822
$507$	0	0
$508$	7.00000	0.310575
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	−14.0000	−0.617514
$515$	3.00000	0.132196
$516$	0	0
$517$	−12.0000	−0.527759
$518$	−5.00000	−0.219687
$519$	0	0
$520$	−9.00000	−0.394676
$521$	−34.0000	−1.48957	−0.744784	−	0.667306i	$-0.767447\pi$
$521$	−34.0000	−1.48957	−0.744784	+	0.667306i	$0.767447\pi$
$522$	0	0
$523$	−34.0000	−1.48672	−0.743358	−	0.668894i	$-0.766768\pi$
$523$	−34.0000	−1.48672	−0.743358	+	0.668894i	$0.766768\pi$
$524$	18.0000	0.786334
$525$	0	0
$526$	21.0000	0.915644
$527$	0	0
$528$	0	0
$529$	1.00000	0.0434783
$530$	36.0000	1.56374
$531$	0	0
$532$	0	0
$533$	−15.0000	−0.649722
$534$	0	0
$535$	12.0000	0.518805
$536$	−12.0000	−0.518321
$537$	0	0
$538$	2.00000	0.0862261
$539$	−4.00000	−0.172292
$540$	0	0
$541$	−24.0000	−1.03184	−0.515920	−	0.856637i	$-0.672550\pi$
$541$	−24.0000	−1.03184	−0.515920	+	0.856637i	$0.672550\pi$
$542$	−12.0000	−0.515444
$543$	0	0
$544$	0	0
$545$	3.00000	0.128506
$546$	0	0
$547$	−2.00000	−0.0855138	−0.0427569	−	0.999086i	$-0.513614\pi$
$547$	−2.00000	−0.0855138	−0.0427569	+	0.999086i	$0.513614\pi$
$548$	17.0000	0.726204
$549$	0	0
$550$	−16.0000	−0.682242
$551$	0	0
$552$	0	0
$553$	4.00000	0.170097
$554$	−24.0000	−1.01966
$555$	0	0
$556$	7.00000	0.296866
$557$	−36.0000	−1.52537	−0.762684	−	0.646771i	$-0.776119\pi$
$557$	−36.0000	−1.52537	−0.762684	+	0.646771i	$0.776119\pi$
$558$	0	0
$559$	−21.0000	−0.888205
$560$	−3.00000	−0.126773
$561$	0	0
$562$	−1.00000	−0.0421825
$563$	−39.0000	−1.64365	−0.821827	−	0.569737i	$-0.807045\pi$
$563$	−39.0000	−1.64365	−0.821827	+	0.569737i	$0.807045\pi$
$564$	0	0
$565$	21.0000	0.883477
$566$	14.0000	0.588464
$567$	0	0
$568$	−10.0000	−0.419591
$569$	19.0000	0.796521	0.398261	−	0.917272i	$-0.369614\pi$
$569$	19.0000	0.796521	0.398261	+	0.917272i	$0.369614\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	−12.0000	−0.501745
$573$	0	0
$574$	−5.00000	−0.208696
$575$	4.00000	0.166812
$576$	0	0
$577$	−38.0000	−1.58196	−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000	−1.58196	−0.790980	+	0.611842i	$0.790429\pi$
$578$	−17.0000	−0.707107
$579$	0	0
$580$	3.00000	0.124568
$581$	−4.00000	−0.165948
$582$	0	0
$583$	48.0000	1.98796
$584$	0	0
$585$	0	0
$586$	−14.0000	−0.578335
$587$	−16.0000	−0.660391	−0.330195	−	0.943913i	$-0.607115\pi$
$587$	−16.0000	−0.660391	−0.330195	+	0.943913i	$0.607115\pi$
$588$	0	0
$589$	0	0
$590$	−6.00000	−0.247016
$591$	0	0
$592$	−5.00000	−0.205499
$593$	−9.00000	−0.369586	−0.184793	−	0.982777i	$-0.559161\pi$
$593$	−9.00000	−0.369586	−0.184793	+	0.982777i	$0.559161\pi$
$594$	0	0
$595$	0	0
$596$	−12.0000	−0.491539
$597$	0	0
$598$	3.00000	0.122679
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	−7.00000	−0.285299
$603$	0	0
$604$	−17.0000	−0.691720
$605$	−15.0000	−0.609837
$606$	0	0
$607$	−14.0000	−0.568242	−0.284121	−	0.958788i	$-0.591702\pi$
$607$	−14.0000	−0.568242	−0.284121	+	0.958788i	$0.591702\pi$
$608$	0	0
$609$	0	0
$610$	18.0000	0.728799
$611$	9.00000	0.364101
$612$	0	0
$613$	21.0000	0.848182	0.424091	−	0.905620i	$-0.360594\pi$
$613$	21.0000	0.848182	0.424091	+	0.905620i	$0.360594\pi$
$614$	25.0000	1.00892
$615$	0	0
$616$	−4.00000	−0.161165
$617$	42.0000	1.69086	0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000	1.69086	0.845428	+	0.534089i	$0.179345\pi$
$618$	0	0
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	6.00000	0.240966
$621$	0	0
$622$	4.00000	0.160385
$623$	−10.0000	−0.400642
$624$	0	0
$625$	−29.0000	−1.16000
$626$	14.0000	0.559553
$627$	0	0
$628$	−24.0000	−0.957704
$629$	0	0
$630$	0	0
$631$	−32.0000	−1.27390	−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000	−1.27390	−0.636950	+	0.770905i	$0.719804\pi$
$632$	4.00000	0.159111
$633$	0	0
$634$	33.0000	1.31060
$635$	−21.0000	−0.833360
$636$	0	0
$637$	3.00000	0.118864
$638$	4.00000	0.158362
$639$	0	0
$640$	−3.00000	−0.118585
$641$	−1.00000	−0.0394976	−0.0197488	−	0.999805i	$-0.506287\pi$
$641$	−1.00000	−0.0394976	−0.0197488	+	0.999805i	$0.506287\pi$
$642$	0	0
$643$	34.0000	1.34083	0.670415	−	0.741987i	$-0.266116\pi$
$643$	34.0000	1.34083	0.670415	+	0.741987i	$0.266116\pi$
$644$	1.00000	0.0394055
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	−8.00000	−0.314027
$650$	12.0000	0.470679
$651$	0	0
$652$	−12.0000	−0.469956
$653$	−13.0000	−0.508729	−0.254365	−	0.967108i	$-0.581866\pi$
$653$	−13.0000	−0.508729	−0.254365	+	0.967108i	$0.581866\pi$
$654$	0	0
$655$	−54.0000	−2.10995
$656$	−5.00000	−0.195217
$657$	0	0
$658$	3.00000	0.116952
$659$	8.00000	0.311636	0.155818	−	0.987786i	$-0.450199\pi$
$659$	8.00000	0.311636	0.155818	+	0.987786i	$0.450199\pi$
$660$	0	0
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	4.00000	0.155464
$663$	0	0
$664$	−4.00000	−0.155230
$665$	0	0
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	12.0000	0.464294
$669$	0	0
$670$	36.0000	1.39080
$671$	24.0000	0.926510
$672$	0	0
$673$	19.0000	0.732396	0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000	0.732396	0.366198	+	0.930537i	$0.380659\pi$
$674$	−20.0000	−0.770371
$675$	0	0
$676$	−4.00000	−0.153846
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	0	0
$679$	19.0000	0.729153
$680$	0	0
$681$	0	0
$682$	8.00000	0.306336
$683$	4.00000	0.153056	0.0765279	−	0.997067i	$-0.475617\pi$
$683$	4.00000	0.153056	0.0765279	+	0.997067i	$0.475617\pi$
$684$	0	0
$685$	−51.0000	−1.94861
$686$	1.00000	0.0381802
$687$	0	0
$688$	−7.00000	−0.266872
$689$	−36.0000	−1.37149
$690$	0	0
$691$	21.0000	0.798878	0.399439	−	0.916760i	$-0.369205\pi$
$691$	21.0000	0.798878	0.399439	+	0.916760i	$0.369205\pi$
$692$	−2.00000	−0.0760286
$693$	0	0
$694$	3.00000	0.113878
$695$	−21.0000	−0.796575
$696$	0	0
$697$	0	0
$698$	26.0000	0.984115
$699$	0	0
$700$	4.00000	0.151186
$701$	48.0000	1.81293	0.906467	−	0.422276i	$-0.138769\pi$
$701$	48.0000	1.81293	0.906467	+	0.422276i	$0.138769\pi$
$702$	0	0
$703$	0	0
$704$	−4.00000	−0.150756
$705$	0	0
$706$	17.0000	0.639803
$707$	−14.0000	−0.526524
$708$	0	0
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	30.0000	1.12588
$711$	0	0
$712$	−10.0000	−0.374766
$713$	−2.00000	−0.0749006
$714$	0	0
$715$	36.0000	1.34632
$716$	−9.00000	−0.336346
$717$	0	0
$718$	−9.00000	−0.335877
$719$	43.0000	1.60363	0.801815	−	0.597573i	$-0.203868\pi$
$719$	43.0000	1.60363	0.801815	+	0.597573i	$0.203868\pi$
$720$	0	0
$721$	−1.00000	−0.0372419
$722$	−19.0000	−0.707107
$723$	0	0
$724$	18.0000	0.668965
$725$	−4.00000	−0.148556
$726$	0	0
$727$	−28.0000	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$728$	3.00000	0.111187
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	36.0000	1.32969	0.664845	−	0.746981i	$-0.268498\pi$
$733$	36.0000	1.32969	0.664845	+	0.746981i	$0.268498\pi$
$734$	21.0000	0.775124
$735$	0	0
$736$	1.00000	0.0368605
$737$	48.0000	1.76810
$738$	0	0
$739$	30.0000	1.10357	0.551784	−	0.833987i	$-0.313947\pi$
$739$	30.0000	1.10357	0.551784	+	0.833987i	$0.313947\pi$
$740$	15.0000	0.551411
$741$	0	0
$742$	−12.0000	−0.440534
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	0	0
$745$	36.0000	1.31894
$746$	−2.00000	−0.0732252
$747$	0	0
$748$	0	0
$749$	−4.00000	−0.146157
$750$	0	0
$751$	12.0000	0.437886	0.218943	−	0.975738i	$-0.429739\pi$
$751$	12.0000	0.437886	0.218943	+	0.975738i	$0.429739\pi$
$752$	3.00000	0.109399
$753$	0	0
$754$	−3.00000	−0.109254
$755$	51.0000	1.85608
$756$	0	0
$757$	−22.0000	−0.799604	−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000	−0.799604	−0.399802	+	0.916602i	$0.630921\pi$
$758$	−5.00000	−0.181608
$759$	0	0
$760$	0	0
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	0	0
$763$	−1.00000	−0.0362024
$764$	20.0000	0.723575
$765$	0	0
$766$	16.0000	0.578103
$767$	6.00000	0.216647
$768$	0	0
$769$	−1.00000	−0.0360609	−0.0180305	−	0.999837i	$-0.505740\pi$
$769$	−1.00000	−0.0360609	−0.0180305	+	0.999837i	$0.505740\pi$
$770$	12.0000	0.432450
$771$	0	0
$772$	−1.00000	−0.0359908
$773$	−35.0000	−1.25886	−0.629431	−	0.777056i	$-0.716712\pi$
$773$	−35.0000	−1.25886	−0.629431	+	0.777056i	$0.716712\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	19.0000	0.682060
$777$	0	0
$778$	14.0000	0.501924
$779$	0	0
$780$	0	0
$781$	40.0000	1.43131
$782$	0	0
$783$	0	0
$784$	1.00000	0.0357143
$785$	72.0000	2.56979
$786$	0	0
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	21.0000	0.748094
$789$	0	0
$790$	−12.0000	−0.426941
$791$	−7.00000	−0.248891
$792$	0	0
$793$	−18.0000	−0.639199
$794$	6.00000	0.212932
$795$	0	0
$796$	−23.0000	−0.815213
$797$	−47.0000	−1.66483	−0.832413	−	0.554156i	$-0.813041\pi$
$797$	−47.0000	−1.66483	−0.832413	+	0.554156i	$0.813041\pi$
$798$	0	0
$799$	0	0
$800$	4.00000	0.141421
$801$	0	0
$802$	−18.0000	−0.635602
$803$	0	0
$804$	0	0
$805$	−3.00000	−0.105736
$806$	−6.00000	−0.211341
$807$	0	0
$808$	−14.0000	−0.492518
$809$	−48.0000	−1.68759	−0.843795	−	0.536666i	$-0.819684\pi$
$809$	−48.0000	−1.68759	−0.843795	+	0.536666i	$0.819684\pi$
$810$	0	0
$811$	23.0000	0.807639	0.403820	−	0.914839i	$-0.367682\pi$
$811$	23.0000	0.807639	0.403820	+	0.914839i	$0.367682\pi$
$812$	−1.00000	−0.0350931
$813$	0	0
$814$	20.0000	0.701000
$815$	36.0000	1.26102
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	15.0000	0.523823
$821$	−54.0000	−1.88461	−0.942306	−	0.334751i	$-0.891348\pi$
$821$	−54.0000	−1.88461	−0.942306	+	0.334751i	$0.891348\pi$
$822$	0	0
$823$	31.0000	1.08059	0.540296	−	0.841475i	$-0.318312\pi$
$823$	31.0000	1.08059	0.540296	+	0.841475i	$0.318312\pi$
$824$	−1.00000	−0.0348367
$825$	0	0
$826$	2.00000	0.0695889
$827$	42.0000	1.46048	0.730242	−	0.683189i	$-0.239408\pi$
$827$	42.0000	1.46048	0.730242	+	0.683189i	$0.239408\pi$
$828$	0	0
$829$	−38.0000	−1.31979	−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000	−1.31979	−0.659897	+	0.751356i	$0.729400\pi$
$830$	12.0000	0.416526
$831$	0	0
$832$	3.00000	0.104006
$833$	0	0
$834$	0	0
$835$	−36.0000	−1.24583
$836$	0	0
$837$	0	0
$838$	−4.00000	−0.138178
$839$	42.0000	1.45000	0.725001	−	0.688748i	$-0.241839\pi$
$839$	42.0000	1.45000	0.725001	+	0.688748i	$0.241839\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	−1.00000	−0.0344623
$843$	0	0
$844$	0	0
$845$	12.0000	0.412813
$846$	0	0
$847$	5.00000	0.171802
$848$	−12.0000	−0.412082
$849$	0	0
$850$	0	0
$851$	−5.00000	−0.171398
$852$	0	0
$853$	41.0000	1.40381	0.701907	−	0.712269i	$-0.252332\pi$
$853$	41.0000	1.40381	0.701907	+	0.712269i	$0.252332\pi$
$854$	−6.00000	−0.205316
$855$	0	0
$856$	−4.00000	−0.136717
$857$	3.00000	0.102478	0.0512390	−	0.998686i	$-0.483683\pi$
$857$	3.00000	0.102478	0.0512390	+	0.998686i	$0.483683\pi$
$858$	0	0
$859$	−29.0000	−0.989467	−0.494734	−	0.869045i	$-0.664734\pi$
$859$	−29.0000	−0.989467	−0.494734	+	0.869045i	$0.664734\pi$
$860$	21.0000	0.716094
$861$	0	0
$862$	−3.00000	−0.102180
$863$	48.0000	1.63394	0.816970	−	0.576681i	$-0.195652\pi$
$863$	48.0000	1.63394	0.816970	+	0.576681i	$0.195652\pi$
$864$	0	0
$865$	6.00000	0.204006
$866$	−27.0000	−0.917497
$867$	0	0
$868$	−2.00000	−0.0678844
$869$	−16.0000	−0.542763
$870$	0	0
$871$	−36.0000	−1.21981
$872$	−1.00000	−0.0338643
$873$	0	0
$874$	0	0
$875$	3.00000	0.101419
$876$	0	0
$877$	−20.0000	−0.675352	−0.337676	−	0.941262i	$-0.609641\pi$
$877$	−20.0000	−0.675352	−0.337676	+	0.941262i	$0.609641\pi$
$878$	14.0000	0.472477
$879$	0	0
$880$	12.0000	0.404520
$881$	−38.0000	−1.28025	−0.640126	−	0.768270i	$-0.721118\pi$
$881$	−38.0000	−1.28025	−0.640126	+	0.768270i	$0.721118\pi$
$882$	0	0
$883$	6.00000	0.201916	0.100958	−	0.994891i	$-0.467809\pi$
$883$	6.00000	0.201916	0.100958	+	0.994891i	$0.467809\pi$
$884$	0	0
$885$	0	0
$886$	29.0000	0.974274
$887$	16.0000	0.537227	0.268614	−	0.963248i	$-0.413434\pi$
$887$	16.0000	0.537227	0.268614	+	0.963248i	$0.413434\pi$
$888$	0	0
$889$	7.00000	0.234772
$890$	30.0000	1.00560
$891$	0	0
$892$	2.00000	0.0669650
$893$	0	0
$894$	0	0
$895$	27.0000	0.902510
$896$	1.00000	0.0334077
$897$	0	0
$898$	18.0000	0.600668
$899$	2.00000	0.0667037
$900$	0	0
$901$	0	0
$902$	20.0000	0.665927
$903$	0	0
$904$	−7.00000	−0.232817
$905$	−54.0000	−1.79502
$906$	0	0
$907$	−49.0000	−1.62702	−0.813509	−	0.581552i	$-0.802446\pi$
$907$	−49.0000	−1.62702	−0.813509	+	0.581552i	$0.802446\pi$
$908$	5.00000	0.165931
$909$	0	0
$910$	−9.00000	−0.298347
$911$	55.0000	1.82223	0.911116	−	0.412151i	$-0.135222\pi$
$911$	55.0000	1.82223	0.911116	+	0.412151i	$0.135222\pi$
$912$	0	0
$913$	16.0000	0.529523
$914$	−4.00000	−0.132308
$915$	0	0
$916$	16.0000	0.528655
$917$	18.0000	0.594412
$918$	0	0
$919$	60.0000	1.97922	0.989609	−	0.143787i	$-0.0459280\pi$
$919$	60.0000	1.97922	0.989609	+	0.143787i	$0.0459280\pi$
$920$	−3.00000	−0.0989071
$921$	0	0
$922$	30.0000	0.987997
$923$	−30.0000	−0.987462
$924$	0	0
$925$	−20.0000	−0.657596
$926$	5.00000	0.164310
$927$	0	0
$928$	−1.00000	−0.0328266
$929$	−21.0000	−0.688988	−0.344494	−	0.938789i	$-0.611949\pi$
$929$	−21.0000	−0.688988	−0.344494	+	0.938789i	$0.611949\pi$
$930$	0	0
$931$	0	0
$932$	10.0000	0.327561
$933$	0	0
$934$	27.0000	0.883467
$935$	0	0
$936$	0	0
$937$	−53.0000	−1.73143	−0.865717	−	0.500533i	$-0.833137\pi$
$937$	−53.0000	−1.73143	−0.865717	+	0.500533i	$0.833137\pi$
$938$	−12.0000	−0.391814
$939$	0	0
$940$	−9.00000	−0.293548
$941$	21.0000	0.684580	0.342290	−	0.939594i	$-0.388797\pi$
$941$	21.0000	0.684580	0.342290	+	0.939594i	$0.388797\pi$
$942$	0	0
$943$	−5.00000	−0.162822
$944$	2.00000	0.0650945
$945$	0	0
$946$	28.0000	0.910359
$947$	−3.00000	−0.0974869	−0.0487435	−	0.998811i	$-0.515522\pi$
$947$	−3.00000	−0.0974869	−0.0487435	+	0.998811i	$0.515522\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	34.0000	1.10137	0.550684	−	0.834714i	$-0.314367\pi$
$953$	34.0000	1.10137	0.550684	+	0.834714i	$0.314367\pi$
$954$	0	0
$955$	−60.0000	−1.94155
$956$	−24.0000	−0.776215
$957$	0	0
$958$	6.00000	0.193851
$959$	17.0000	0.548959
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−15.0000	−0.483619
$963$	0	0
$964$	9.00000	0.289870
$965$	3.00000	0.0965734
$966$	0	0
$967$	52.0000	1.67221	0.836104	−	0.548572i	$-0.184828\pi$
$967$	52.0000	1.67221	0.836104	+	0.548572i	$0.184828\pi$
$968$	5.00000	0.160706
$969$	0	0
$970$	−57.0000	−1.83016
$971$	40.0000	1.28366	0.641831	−	0.766846i	$-0.278175\pi$
$971$	40.0000	1.28366	0.641831	+	0.766846i	$0.278175\pi$
$972$	0	0
$973$	7.00000	0.224410
$974$	13.0000	0.416547
$975$	0	0
$976$	−6.00000	−0.192055
$977$	13.0000	0.415907	0.207953	−	0.978139i	$-0.433320\pi$
$977$	13.0000	0.415907	0.207953	+	0.978139i	$0.433320\pi$
$978$	0	0
$979$	40.0000	1.27841
$980$	−3.00000	−0.0958315
$981$	0	0
$982$	−28.0000	−0.893516
$983$	−2.00000	−0.0637901	−0.0318950	−	0.999491i	$-0.510154\pi$
$983$	−2.00000	−0.0637901	−0.0318950	+	0.999491i	$0.510154\pi$
$984$	0	0
$985$	−63.0000	−2.00735
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−7.00000	−0.222587
$990$	0	0
$991$	44.0000	1.39771	0.698853	−	0.715265i	$-0.253694\pi$
$991$	44.0000	1.39771	0.698853	+	0.715265i	$0.253694\pi$
$992$	−2.00000	−0.0635001
$993$	0	0
$994$	−10.0000	−0.317181
$995$	69.0000	2.18745
$996$	0	0
$997$	−50.0000	−1.58352	−0.791758	−	0.610835i	$-0.790834\pi$
$997$	−50.0000	−1.58352	−0.791758	+	0.610835i	$0.790834\pi$
$998$	−10.0000	−0.316544
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2898.2.a.k.1.1		1
3.2	odd	2		966.2.a.d.1.1	✓	1
12.11	even	2		7728.2.a.u.1.1		1
21.20	even	2		6762.2.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.2.a.d.1.1	✓	1	3.2	odd	2
2898.2.a.k.1.1		1	1.1	even	1	trivial
6762.2.a.l.1.1		1	21.20	even	2
7728.2.a.u.1.1		1	12.11	even	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:	\( N \)	\(=\)	\( 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2898.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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