LMFDB - Embedded newform 1638.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1638,2,Mod(1,1638)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1638, 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("1638.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1638.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:	$13.0794958511$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1638.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q+1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -3.00000 q^{10} -3.00000 q^{11} +1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} -7.00000 q^{19} -3.00000 q^{20} -3.00000 q^{22} -9.00000 q^{23} +4.00000 q^{25} +1.00000 q^{26} +1.00000 q^{28} +9.00000 q^{29} -4.00000 q^{31} +1.00000 q^{32} +3.00000 q^{34} -3.00000 q^{35} -7.00000 q^{37} -7.00000 q^{38} -3.00000 q^{40} -12.0000 q^{41} -1.00000 q^{43} -3.00000 q^{44} -9.00000 q^{46} +1.00000 q^{49} +4.00000 q^{50} +1.00000 q^{52} +6.00000 q^{53} +9.00000 q^{55} +1.00000 q^{56} +9.00000 q^{58} -12.0000 q^{59} -1.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} -3.00000 q^{65} +14.0000 q^{67} +3.00000 q^{68} -3.00000 q^{70} -12.0000 q^{71} -7.00000 q^{73} -7.00000 q^{74} -7.00000 q^{76} -3.00000 q^{77} -10.0000 q^{79} -3.00000 q^{80} -12.0000 q^{82} +6.00000 q^{83} -9.00000 q^{85} -1.00000 q^{86} -3.00000 q^{88} +6.00000 q^{89} +1.00000 q^{91} -9.00000 q^{92} +21.0000 q^{95} -10.0000 q^{97} +1.00000 q^{98} +O(q^{100})$

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
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	−7.00000	−1.60591	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.00000	−1.60591	−0.802955	+	0.596040i	$0.796740\pi$
$20$	−3.00000	−0.670820
$21$	0	0
$22$	−3.00000	−0.639602
$23$	−9.00000	−1.87663	−0.938315	−	0.345782i	$-0.887614\pi$
$23$	−9.00000	−1.87663	−0.938315	+	0.345782i	$0.887614\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	1.00000	0.196116
$27$	0	0
$28$	1.00000	0.188982
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	3.00000	0.514496
$35$	−3.00000	−0.507093
$36$	0	0
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	−7.00000	−1.13555
$39$	0	0
$40$	−3.00000	−0.474342
$41$	−12.0000	−1.87409	−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000	−1.87409	−0.937043	+	0.349215i	$0.886448\pi$
$42$	0	0
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	−3.00000	−0.452267
$45$	0	0
$46$	−9.00000	−1.32698
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	4.00000	0.565685
$51$	0	0
$52$	1.00000	0.138675
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	9.00000	1.21356
$56$	1.00000	0.133631
$57$	0	0
$58$	9.00000	1.18176
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	−3.00000	−0.372104
$66$	0	0
$67$	14.0000	1.71037	0.855186	−	0.518321i	$-0.173443\pi$
$67$	14.0000	1.71037	0.855186	+	0.518321i	$0.173443\pi$
$68$	3.00000	0.363803
$69$	0	0
$70$	−3.00000	−0.358569
$71$	−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
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	−7.00000	−0.813733
$75$	0	0
$76$	−7.00000	−0.802955
$77$	−3.00000	−0.341882
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	−3.00000	−0.335410
$81$	0	0
$82$	−12.0000	−1.32518
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	−9.00000	−0.976187
$86$	−1.00000	−0.107833
$87$	0	0
$88$	−3.00000	−0.319801
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	−9.00000	−0.938315
$93$	0	0
$94$	0	0
$95$	21.0000	2.15455
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	4.00000	0.400000
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	−13.0000	−1.28093	−0.640464	−	0.767988i	$-0.721258\pi$
$103$	−13.0000	−1.28093	−0.640464	+	0.767988i	$0.721258\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	6.00000	0.582772
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	0	0
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	9.00000	0.858116
$111$	0	0
$112$	1.00000	0.0944911
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	0	0
$115$	27.0000	2.51776
$116$	9.00000	0.835629
$117$	0	0
$118$	−12.0000	−1.10469
$119$	3.00000	0.275010
$120$	0	0
$121$	−2.00000	−0.181818
$122$	−1.00000	−0.0905357
$123$	0	0
$124$	−4.00000	−0.359211
$125$	3.00000	0.268328
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−3.00000	−0.263117
$131$	15.0000	1.31056	0.655278	−	0.755388i	$-0.272551\pi$
$131$	15.0000	1.31056	0.655278	+	0.755388i	$0.272551\pi$
$132$	0	0
$133$	−7.00000	−0.606977
$134$	14.0000	1.20942
$135$	0	0
$136$	3.00000	0.257248
$137$	−9.00000	−0.768922	−0.384461	−	0.923141i	$-0.625613\pi$
$137$	−9.00000	−0.768922	−0.384461	+	0.923141i	$0.625613\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	−3.00000	−0.253546
$141$	0	0
$142$	−12.0000	−1.00702
$143$	−3.00000	−0.250873
$144$	0	0
$145$	−27.0000	−2.24223
$146$	−7.00000	−0.579324
$147$	0	0
$148$	−7.00000	−0.575396
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	0	0
$151$	17.0000	1.38344	0.691720	−	0.722166i	$-0.256853\pi$
$151$	17.0000	1.38344	0.691720	+	0.722166i	$0.256853\pi$
$152$	−7.00000	−0.567775
$153$	0	0
$154$	−3.00000	−0.241747
$155$	12.0000	0.963863
$156$	0	0
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	−10.0000	−0.795557
$159$	0	0
$160$	−3.00000	−0.237171
$161$	−9.00000	−0.709299
$162$	0	0
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	−12.0000	−0.937043
$165$	0	0
$166$	6.00000	0.465690
$167$	−3.00000	−0.232147	−0.116073	−	0.993241i	$-0.537031\pi$
$167$	−3.00000	−0.232147	−0.116073	+	0.993241i	$0.537031\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−9.00000	−0.690268
$171$	0	0
$172$	−1.00000	−0.0762493
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	−3.00000	−0.226134
$177$	0	0
$178$	6.00000	0.449719
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	1.00000	0.0741249
$183$	0	0
$184$	−9.00000	−0.663489
$185$	21.0000	1.54395
$186$	0	0
$187$	−9.00000	−0.658145
$188$	0	0
$189$	0	0
$190$	21.0000	1.52350
$191$	15.0000	1.08536	0.542681	−	0.839939i	$-0.317409\pi$
$191$	15.0000	1.08536	0.542681	+	0.839939i	$0.317409\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	1.00000	0.0714286
$197$	12.0000	0.854965	0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000	0.854965	0.427482	+	0.904024i	$0.359401\pi$
$198$	0	0
$199$	−7.00000	−0.496217	−0.248108	−	0.968732i	$-0.579809\pi$
$199$	−7.00000	−0.496217	−0.248108	+	0.968732i	$0.579809\pi$
$200$	4.00000	0.282843
$201$	0	0
$202$	−12.0000	−0.844317
$203$	9.00000	0.631676
$204$	0	0
$205$	36.0000	2.51435
$206$	−13.0000	−0.905753
$207$	0	0
$208$	1.00000	0.0693375
$209$	21.0000	1.45260
$210$	0	0
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	6.00000	0.410152
$215$	3.00000	0.204598
$216$	0	0
$217$	−4.00000	−0.271538
$218$	11.0000	0.745014
$219$	0	0
$220$	9.00000	0.606780
$221$	3.00000	0.201802
$222$	0	0
$223$	26.0000	1.74109	0.870544	−	0.492090i	$-0.163767\pi$
$223$	26.0000	1.74109	0.870544	+	0.492090i	$0.163767\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	18.0000	1.19734
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	0	0
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	27.0000	1.78033
$231$	0	0
$232$	9.00000	0.590879
$233$	−24.0000	−1.57229	−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000	−1.57229	−0.786146	+	0.618041i	$0.787927\pi$
$234$	0	0
$235$	0	0
$236$	−12.0000	−0.781133
$237$	0	0
$238$	3.00000	0.194461
$239$	−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
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	−2.00000	−0.128565
$243$	0	0
$244$	−1.00000	−0.0640184
$245$	−3.00000	−0.191663
$246$	0	0
$247$	−7.00000	−0.445399
$248$	−4.00000	−0.254000
$249$	0	0
$250$	3.00000	0.189737
$251$	−21.0000	−1.32551	−0.662754	−	0.748837i	$-0.730613\pi$
$251$	−21.0000	−1.32551	−0.662754	+	0.748837i	$0.730613\pi$
$252$	0	0
$253$	27.0000	1.69748
$254$	2.00000	0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	−7.00000	−0.434959
$260$	−3.00000	−0.186052
$261$	0	0
$262$	15.0000	0.926703
$263$	−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
$265$	−18.0000	−1.10573
$266$	−7.00000	−0.429198
$267$	0	0
$268$	14.0000	0.855186
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	3.00000	0.181902
$273$	0	0
$274$	−9.00000	−0.543710
$275$	−12.0000	−0.723627
$276$	0	0
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	−4.00000	−0.239904
$279$	0	0
$280$	−3.00000	−0.179284
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	−3.00000	−0.177394
$287$	−12.0000	−0.708338
$288$	0	0
$289$	−8.00000	−0.470588
$290$	−27.0000	−1.58549
$291$	0	0
$292$	−7.00000	−0.409644
$293$	18.0000	1.05157	0.525786	−	0.850617i	$-0.323771\pi$
$293$	18.0000	1.05157	0.525786	+	0.850617i	$0.323771\pi$
$294$	0	0
$295$	36.0000	2.09600
$296$	−7.00000	−0.406867
$297$	0	0
$298$	12.0000	0.695141
$299$	−9.00000	−0.520483
$300$	0	0
$301$	−1.00000	−0.0576390
$302$	17.0000	0.978240
$303$	0	0
$304$	−7.00000	−0.401478
$305$	3.00000	0.171780
$306$	0	0
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	−3.00000	−0.170941
$309$	0	0
$310$	12.0000	0.681554
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	0	0
$313$	8.00000	0.452187	0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	−13.0000	−0.733632
$315$	0	0
$316$	−10.0000	−0.562544
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	−27.0000	−1.51171
$320$	−3.00000	−0.167705
$321$	0	0
$322$	−9.00000	−0.501550
$323$	−21.0000	−1.16847
$324$	0	0
$325$	4.00000	0.221880
$326$	2.00000	0.110770
$327$	0	0
$328$	−12.0000	−0.662589
$329$	0	0
$330$	0	0
$331$	−10.0000	−0.549650	−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000	−0.549650	−0.274825	+	0.961494i	$0.588620\pi$
$332$	6.00000	0.329293
$333$	0	0
$334$	−3.00000	−0.164153
$335$	−42.0000	−2.29471
$336$	0	0
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	1.00000	0.0543928
$339$	0	0
$340$	−9.00000	−0.488094
$341$	12.0000	0.649836
$342$	0	0
$343$	1.00000	0.0539949
$344$	−1.00000	−0.0539164
$345$	0	0
$346$	6.00000	0.322562
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	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
$352$	−3.00000	−0.159901
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	36.0000	1.91068
$356$	6.00000	0.317999
$357$	0	0
$358$	0	0
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	2.00000	0.105118
$363$	0	0
$364$	1.00000	0.0524142
$365$	21.0000	1.09919
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	−9.00000	−0.469157
$369$	0	0
$370$	21.0000	1.09174
$371$	6.00000	0.311504
$372$	0	0
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	−9.00000	−0.465379
$375$	0	0
$376$	0	0
$377$	9.00000	0.463524
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	21.0000	1.07728
$381$	0	0
$382$	15.0000	0.767467
$383$	21.0000	1.07305	0.536525	−	0.843884i	$-0.319737\pi$
$383$	21.0000	1.07305	0.536525	+	0.843884i	$0.319737\pi$
$384$	0	0
$385$	9.00000	0.458682
$386$	−4.00000	−0.203595
$387$	0	0
$388$	−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	$0.224939\pi$
$390$	0	0
$391$	−27.0000	−1.36545
$392$	1.00000	0.0505076
$393$	0	0
$394$	12.0000	0.604551
$395$	30.0000	1.50946
$396$	0	0
$397$	−34.0000	−1.70641	−0.853206	−	0.521575i	$-0.825345\pi$
$397$	−34.0000	−1.70641	−0.853206	+	0.521575i	$0.825345\pi$
$398$	−7.00000	−0.350878
$399$	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
$403$	−4.00000	−0.199254
$404$	−12.0000	−0.597022
$405$	0	0
$406$	9.00000	0.446663
$407$	21.0000	1.04093
$408$	0	0
$409$	−13.0000	−0.642809	−0.321404	−	0.946942i	$-0.604155\pi$
$409$	−13.0000	−0.642809	−0.321404	+	0.946942i	$0.604155\pi$
$410$	36.0000	1.77791
$411$	0	0
$412$	−13.0000	−0.640464
$413$	−12.0000	−0.590481
$414$	0	0
$415$	−18.0000	−0.883585
$416$	1.00000	0.0490290
$417$	0	0
$418$	21.0000	1.02714
$419$	−3.00000	−0.146560	−0.0732798	−	0.997311i	$-0.523347\pi$
$419$	−3.00000	−0.146560	−0.0732798	+	0.997311i	$0.523347\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	5.00000	0.243396
$423$	0	0
$424$	6.00000	0.291386
$425$	12.0000	0.582086
$426$	0	0
$427$	−1.00000	−0.0483934
$428$	6.00000	0.290021
$429$	0	0
$430$	3.00000	0.144673
$431$	−18.0000	−0.867029	−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000	−0.867029	−0.433515	+	0.901146i	$0.642727\pi$
$432$	0	0
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	11.0000	0.526804
$437$	63.0000	3.01370
$438$	0	0
$439$	−1.00000	−0.0477274	−0.0238637	−	0.999715i	$-0.507597\pi$
$439$	−1.00000	−0.0477274	−0.0238637	+	0.999715i	$0.507597\pi$
$440$	9.00000	0.429058
$441$	0	0
$442$	3.00000	0.142695
$443$	−18.0000	−0.855206	−0.427603	−	0.903967i	$-0.640642\pi$
$443$	−18.0000	−0.855206	−0.427603	+	0.903967i	$0.640642\pi$
$444$	0	0
$445$	−18.0000	−0.853282
$446$	26.0000	1.23114
$447$	0	0
$448$	1.00000	0.0472456
$449$	−21.0000	−0.991051	−0.495526	−	0.868593i	$-0.665025\pi$
$449$	−21.0000	−0.991051	−0.495526	+	0.868593i	$0.665025\pi$
$450$	0	0
$451$	36.0000	1.69517
$452$	18.0000	0.846649
$453$	0	0
$454$	18.0000	0.844782
$455$	−3.00000	−0.140642
$456$	0	0
$457$	−28.0000	−1.30978	−0.654892	−	0.755722i	$-0.727286\pi$
$457$	−28.0000	−1.30978	−0.654892	+	0.755722i	$0.727286\pi$
$458$	14.0000	0.654177
$459$	0	0
$460$	27.0000	1.25888
$461$	−33.0000	−1.53696	−0.768482	−	0.639872i	$-0.778987\pi$
$461$	−33.0000	−1.53696	−0.768482	+	0.639872i	$0.778987\pi$
$462$	0	0
$463$	41.0000	1.90543	0.952716	−	0.303863i	$-0.0982765\pi$
$463$	41.0000	1.90543	0.952716	+	0.303863i	$0.0982765\pi$
$464$	9.00000	0.417815
$465$	0	0
$466$	−24.0000	−1.11178
$467$	3.00000	0.138823	0.0694117	−	0.997588i	$-0.477888\pi$
$467$	3.00000	0.138823	0.0694117	+	0.997588i	$0.477888\pi$
$468$	0	0
$469$	14.0000	0.646460
$470$	0	0
$471$	0	0
$472$	−12.0000	−0.552345
$473$	3.00000	0.137940
$474$	0	0
$475$	−28.0000	−1.28473
$476$	3.00000	0.137505
$477$	0	0
$478$	−24.0000	−1.09773
$479$	15.0000	0.685367	0.342684	−	0.939451i	$-0.388664\pi$
$479$	15.0000	0.685367	0.342684	+	0.939451i	$0.388664\pi$
$480$	0	0
$481$	−7.00000	−0.319173
$482$	−10.0000	−0.455488
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	30.0000	1.36223
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	−1.00000	−0.0452679
$489$	0	0
$490$	−3.00000	−0.135526
$491$	−30.0000	−1.35388	−0.676941	−	0.736038i	$-0.736695\pi$
$491$	−30.0000	−1.35388	−0.676941	+	0.736038i	$0.736695\pi$
$492$	0	0
$493$	27.0000	1.21602
$494$	−7.00000	−0.314945
$495$	0	0
$496$	−4.00000	−0.179605
$497$	−12.0000	−0.538274
$498$	0	0
$499$	32.0000	1.43252	0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000	1.43252	0.716258	+	0.697835i	$0.245853\pi$
$500$	3.00000	0.134164
$501$	0	0
$502$	−21.0000	−0.937276
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	0	0
$505$	36.0000	1.60198
$506$	27.0000	1.20030
$507$	0	0
$508$	2.00000	0.0887357
$509$	−9.00000	−0.398918	−0.199459	−	0.979906i	$-0.563918\pi$
$509$	−9.00000	−0.398918	−0.199459	+	0.979906i	$0.563918\pi$
$510$	0	0
$511$	−7.00000	−0.309662
$512$	1.00000	0.0441942
$513$	0	0
$514$	18.0000	0.793946
$515$	39.0000	1.71855
$516$	0	0
$517$	0	0
$518$	−7.00000	−0.307562
$519$	0	0
$520$	−3.00000	−0.131559
$521$	33.0000	1.44576	0.722878	−	0.690976i	$-0.242819\pi$
$521$	33.0000	1.44576	0.722878	+	0.690976i	$0.242819\pi$
$522$	0	0
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	15.0000	0.655278
$525$	0	0
$526$	−12.0000	−0.523225
$527$	−12.0000	−0.522728
$528$	0	0
$529$	58.0000	2.52174
$530$	−18.0000	−0.781870
$531$	0	0
$532$	−7.00000	−0.303488
$533$	−12.0000	−0.519778
$534$	0	0
$535$	−18.0000	−0.778208
$536$	14.0000	0.604708
$537$	0	0
$538$	−6.00000	−0.258678
$539$	−3.00000	−0.129219
$540$	0	0
$541$	11.0000	0.472927	0.236463	−	0.971640i	$-0.424012\pi$
$541$	11.0000	0.472927	0.236463	+	0.971640i	$0.424012\pi$
$542$	2.00000	0.0859074
$543$	0	0
$544$	3.00000	0.128624
$545$	−33.0000	−1.41356
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	−9.00000	−0.384461
$549$	0	0
$550$	−12.0000	−0.511682
$551$	−63.0000	−2.68389
$552$	0	0
$553$	−10.0000	−0.425243
$554$	26.0000	1.10463
$555$	0	0
$556$	−4.00000	−0.169638
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	−1.00000	−0.0422955
$560$	−3.00000	−0.126773
$561$	0	0
$562$	6.00000	0.253095
$563$	15.0000	0.632175	0.316087	−	0.948730i	$-0.397631\pi$
$563$	15.0000	0.632175	0.316087	+	0.948730i	$0.397631\pi$
$564$	0	0
$565$	−54.0000	−2.27180
$566$	−4.00000	−0.168133
$567$	0	0
$568$	−12.0000	−0.503509
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	−3.00000	−0.125436
$573$	0	0
$574$	−12.0000	−0.500870
$575$	−36.0000	−1.50130
$576$	0	0
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	−8.00000	−0.332756
$579$	0	0
$580$	−27.0000	−1.12111
$581$	6.00000	0.248922
$582$	0	0
$583$	−18.0000	−0.745484
$584$	−7.00000	−0.289662
$585$	0	0
$586$	18.0000	0.743573
$587$	18.0000	0.742940	0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000	0.742940	0.371470	+	0.928445i	$0.378854\pi$
$588$	0	0
$589$	28.0000	1.15372
$590$	36.0000	1.48210
$591$	0	0
$592$	−7.00000	−0.287698
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	0	0
$595$	−9.00000	−0.368964
$596$	12.0000	0.491539
$597$	0	0
$598$	−9.00000	−0.368037
$599$	9.00000	0.367730	0.183865	−	0.982952i	$-0.441139\pi$
$599$	9.00000	0.367730	0.183865	+	0.982952i	$0.441139\pi$
$600$	0	0
$601$	8.00000	0.326327	0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000	0.326327	0.163163	+	0.986599i	$0.447830\pi$
$602$	−1.00000	−0.0407570
$603$	0	0
$604$	17.0000	0.691720
$605$	6.00000	0.243935
$606$	0	0
$607$	23.0000	0.933541	0.466771	−	0.884378i	$-0.345417\pi$
$607$	23.0000	0.933541	0.466771	+	0.884378i	$0.345417\pi$
$608$	−7.00000	−0.283887
$609$	0	0
$610$	3.00000	0.121466
$611$	0	0
$612$	0	0
$613$	11.0000	0.444286	0.222143	−	0.975014i	$-0.428695\pi$
$613$	11.0000	0.444286	0.222143	+	0.975014i	$0.428695\pi$
$614$	−16.0000	−0.645707
$615$	0	0
$616$	−3.00000	−0.120873
$617$	−27.0000	−1.08698	−0.543490	−	0.839416i	$-0.682897\pi$
$617$	−27.0000	−1.08698	−0.543490	+	0.839416i	$0.682897\pi$
$618$	0	0
$619$	17.0000	0.683288	0.341644	−	0.939829i	$-0.389016\pi$
$619$	17.0000	0.683288	0.341644	+	0.939829i	$0.389016\pi$
$620$	12.0000	0.481932
$621$	0	0
$622$	−18.0000	−0.721734
$623$	6.00000	0.240385
$624$	0	0
$625$	−29.0000	−1.16000
$626$	8.00000	0.319744
$627$	0	0
$628$	−13.0000	−0.518756
$629$	−21.0000	−0.837325
$630$	0	0
$631$	29.0000	1.15447	0.577236	−	0.816577i	$-0.304131\pi$
$631$	29.0000	1.15447	0.577236	+	0.816577i	$0.304131\pi$
$632$	−10.0000	−0.397779
$633$	0	0
$634$	0	0
$635$	−6.00000	−0.238103
$636$	0	0
$637$	1.00000	0.0396214
$638$	−27.0000	−1.06894
$639$	0	0
$640$	−3.00000	−0.118585
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	−49.0000	−1.93237	−0.966186	−	0.257847i	$-0.916987\pi$
$643$	−49.0000	−1.93237	−0.966186	+	0.257847i	$0.916987\pi$
$644$	−9.00000	−0.354650
$645$	0	0
$646$	−21.0000	−0.826234
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	36.0000	1.41312
$650$	4.00000	0.156893
$651$	0	0
$652$	2.00000	0.0783260
$653$	−9.00000	−0.352197	−0.176099	−	0.984373i	$-0.556348\pi$
$653$	−9.00000	−0.352197	−0.176099	+	0.984373i	$0.556348\pi$
$654$	0	0
$655$	−45.0000	−1.75830
$656$	−12.0000	−0.468521
$657$	0	0
$658$	0	0
$659$	−6.00000	−0.233727	−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000	−0.233727	−0.116863	+	0.993148i	$0.537284\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	−10.0000	−0.388661
$663$	0	0
$664$	6.00000	0.232845
$665$	21.0000	0.814345
$666$	0	0
$667$	−81.0000	−3.13633
$668$	−3.00000	−0.116073
$669$	0	0
$670$	−42.0000	−1.62260
$671$	3.00000	0.115814
$672$	0	0
$673$	−1.00000	−0.0385472	−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000	−0.0385472	−0.0192736	+	0.999814i	$0.506135\pi$
$674$	−13.0000	−0.500741
$675$	0	0
$676$	1.00000	0.0384615
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	−10.0000	−0.383765
$680$	−9.00000	−0.345134
$681$	0	0
$682$	12.0000	0.459504
$683$	−33.0000	−1.26271	−0.631355	−	0.775494i	$-0.717501\pi$
$683$	−33.0000	−1.26271	−0.631355	+	0.775494i	$0.717501\pi$
$684$	0	0
$685$	27.0000	1.03162
$686$	1.00000	0.0381802
$687$	0	0
$688$	−1.00000	−0.0381246
$689$	6.00000	0.228582
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	−12.0000	−0.455514
$695$	12.0000	0.455186
$696$	0	0
$697$	−36.0000	−1.36360
$698$	26.0000	0.984115
$699$	0	0
$700$	4.00000	0.151186
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	49.0000	1.84807
$704$	−3.00000	−0.113067
$705$	0	0
$706$	6.00000	0.225813
$707$	−12.0000	−0.451306
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	36.0000	1.35106
$711$	0	0
$712$	6.00000	0.224860
$713$	36.0000	1.34821
$714$	0	0
$715$	9.00000	0.336581
$716$	0	0
$717$	0	0
$718$	−24.0000	−0.895672
$719$	−18.0000	−0.671287	−0.335643	−	0.941989i	$-0.608954\pi$
$719$	−18.0000	−0.671287	−0.335643	+	0.941989i	$0.608954\pi$
$720$	0	0
$721$	−13.0000	−0.484145
$722$	30.0000	1.11648
$723$	0	0
$724$	2.00000	0.0743294
$725$	36.0000	1.33701
$726$	0	0
$727$	−1.00000	−0.0370879	−0.0185440	−	0.999828i	$-0.505903\pi$
$727$	−1.00000	−0.0370879	−0.0185440	+	0.999828i	$0.505903\pi$
$728$	1.00000	0.0370625
$729$	0	0
$730$	21.0000	0.777245
$731$	−3.00000	−0.110959
$732$	0	0
$733$	32.0000	1.18195	0.590973	−	0.806691i	$-0.298744\pi$
$733$	32.0000	1.18195	0.590973	+	0.806691i	$0.298744\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	−9.00000	−0.331744
$737$	−42.0000	−1.54709
$738$	0	0
$739$	−34.0000	−1.25071	−0.625355	−	0.780340i	$-0.715046\pi$
$739$	−34.0000	−1.25071	−0.625355	+	0.780340i	$0.715046\pi$
$740$	21.0000	0.771975
$741$	0	0
$742$	6.00000	0.220267
$743$	−30.0000	−1.10059	−0.550297	−	0.834969i	$-0.685485\pi$
$743$	−30.0000	−1.10059	−0.550297	+	0.834969i	$0.685485\pi$
$744$	0	0
$745$	−36.0000	−1.31894
$746$	−4.00000	−0.146450
$747$	0	0
$748$	−9.00000	−0.329073
$749$	6.00000	0.219235
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	0	0
$753$	0	0
$754$	9.00000	0.327761
$755$	−51.0000	−1.85608
$756$	0	0
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	−16.0000	−0.581146
$759$	0	0
$760$	21.0000	0.761750
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	0	0
$763$	11.0000	0.398227
$764$	15.0000	0.542681
$765$	0	0
$766$	21.0000	0.758761
$767$	−12.0000	−0.433295
$768$	0	0
$769$	−13.0000	−0.468792	−0.234396	−	0.972141i	$-0.575311\pi$
$769$	−13.0000	−0.468792	−0.234396	+	0.972141i	$0.575311\pi$
$770$	9.00000	0.324337
$771$	0	0
$772$	−4.00000	−0.143963
$773$	−33.0000	−1.18693	−0.593464	−	0.804861i	$-0.702240\pi$
$773$	−33.0000	−1.18693	−0.593464	+	0.804861i	$0.702240\pi$
$774$	0	0
$775$	−16.0000	−0.574737
$776$	−10.0000	−0.358979
$777$	0	0
$778$	30.0000	1.07555
$779$	84.0000	3.00961
$780$	0	0
$781$	36.0000	1.28818
$782$	−27.0000	−0.965518
$783$	0	0
$784$	1.00000	0.0357143
$785$	39.0000	1.39197
$786$	0	0
$787$	−31.0000	−1.10503	−0.552515	−	0.833503i	$-0.686332\pi$
$787$	−31.0000	−1.10503	−0.552515	+	0.833503i	$0.686332\pi$
$788$	12.0000	0.427482
$789$	0	0
$790$	30.0000	1.06735
$791$	18.0000	0.640006
$792$	0	0
$793$	−1.00000	−0.0355110
$794$	−34.0000	−1.20661
$795$	0	0
$796$	−7.00000	−0.248108
$797$	−36.0000	−1.27519	−0.637593	−	0.770374i	$-0.720070\pi$
$797$	−36.0000	−1.27519	−0.637593	+	0.770374i	$0.720070\pi$
$798$	0	0
$799$	0	0
$800$	4.00000	0.141421
$801$	0	0
$802$	−18.0000	−0.635602
$803$	21.0000	0.741074
$804$	0	0
$805$	27.0000	0.951625
$806$	−4.00000	−0.140894
$807$	0	0
$808$	−12.0000	−0.422159
$809$	−54.0000	−1.89854	−0.949269	−	0.314464i	$-0.898175\pi$
$809$	−54.0000	−1.89854	−0.949269	+	0.314464i	$0.898175\pi$
$810$	0	0
$811$	−43.0000	−1.50993	−0.754967	−	0.655763i	$-0.772347\pi$
$811$	−43.0000	−1.50993	−0.754967	+	0.655763i	$0.772347\pi$
$812$	9.00000	0.315838
$813$	0	0
$814$	21.0000	0.736050
$815$	−6.00000	−0.210171
$816$	0	0
$817$	7.00000	0.244899
$818$	−13.0000	−0.454534
$819$	0	0
$820$	36.0000	1.25717
$821$	−24.0000	−0.837606	−0.418803	−	0.908077i	$-0.637550\pi$
$821$	−24.0000	−0.837606	−0.418803	+	0.908077i	$0.637550\pi$
$822$	0	0
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	−13.0000	−0.452876
$825$	0	0
$826$	−12.0000	−0.417533
$827$	−27.0000	−0.938882	−0.469441	−	0.882964i	$-0.655545\pi$
$827$	−27.0000	−0.938882	−0.469441	+	0.882964i	$0.655545\pi$
$828$	0	0
$829$	−25.0000	−0.868286	−0.434143	−	0.900844i	$-0.642949\pi$
$829$	−25.0000	−0.868286	−0.434143	+	0.900844i	$0.642949\pi$
$830$	−18.0000	−0.624789
$831$	0	0
$832$	1.00000	0.0346688
$833$	3.00000	0.103944
$834$	0	0
$835$	9.00000	0.311458
$836$	21.0000	0.726300
$837$	0	0
$838$	−3.00000	−0.103633
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	−10.0000	−0.344623
$843$	0	0
$844$	5.00000	0.172107
$845$	−3.00000	−0.103203
$846$	0	0
$847$	−2.00000	−0.0687208
$848$	6.00000	0.206041
$849$	0	0
$850$	12.0000	0.411597
$851$	63.0000	2.15961
$852$	0	0
$853$	44.0000	1.50653	0.753266	−	0.657716i	$-0.228477\pi$
$853$	44.0000	1.50653	0.753266	+	0.657716i	$0.228477\pi$
$854$	−1.00000	−0.0342193
$855$	0	0
$856$	6.00000	0.205076
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	3.00000	0.102299
$861$	0	0
$862$	−18.0000	−0.613082
$863$	−6.00000	−0.204242	−0.102121	−	0.994772i	$-0.532563\pi$
$863$	−6.00000	−0.204242	−0.102121	+	0.994772i	$0.532563\pi$
$864$	0	0
$865$	−18.0000	−0.612018
$866$	−16.0000	−0.543702
$867$	0	0
$868$	−4.00000	−0.135769
$869$	30.0000	1.01768
$870$	0	0
$871$	14.0000	0.474372
$872$	11.0000	0.372507
$873$	0	0
$874$	63.0000	2.13101
$875$	3.00000	0.101419
$876$	0	0
$877$	14.0000	0.472746	0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000	0.472746	0.236373	+	0.971662i	$0.424041\pi$
$878$	−1.00000	−0.0337484
$879$	0	0
$880$	9.00000	0.303390
$881$	3.00000	0.101073	0.0505363	−	0.998722i	$-0.483907\pi$
$881$	3.00000	0.101073	0.0505363	+	0.998722i	$0.483907\pi$
$882$	0	0
$883$	−25.0000	−0.841317	−0.420658	−	0.907219i	$-0.638201\pi$
$883$	−25.0000	−0.841317	−0.420658	+	0.907219i	$0.638201\pi$
$884$	3.00000	0.100901
$885$	0	0
$886$	−18.0000	−0.604722
$887$	−36.0000	−1.20876	−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000	−1.20876	−0.604381	+	0.796696i	$0.706579\pi$
$888$	0	0
$889$	2.00000	0.0670778
$890$	−18.0000	−0.603361
$891$	0	0
$892$	26.0000	0.870544
$893$	0	0
$894$	0	0
$895$	0	0
$896$	1.00000	0.0334077
$897$	0	0
$898$	−21.0000	−0.700779
$899$	−36.0000	−1.20067
$900$	0	0
$901$	18.0000	0.599667
$902$	36.0000	1.19867
$903$	0	0
$904$	18.0000	0.598671
$905$	−6.00000	−0.199447
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	18.0000	0.597351
$909$	0	0
$910$	−3.00000	−0.0994490
$911$	33.0000	1.09334	0.546669	−	0.837349i	$-0.315895\pi$
$911$	33.0000	1.09334	0.546669	+	0.837349i	$0.315895\pi$
$912$	0	0
$913$	−18.0000	−0.595713
$914$	−28.0000	−0.926158
$915$	0	0
$916$	14.0000	0.462573
$917$	15.0000	0.495344
$918$	0	0
$919$	−34.0000	−1.12156	−0.560778	−	0.827966i	$-0.689498\pi$
$919$	−34.0000	−1.12156	−0.560778	+	0.827966i	$0.689498\pi$
$920$	27.0000	0.890164
$921$	0	0
$922$	−33.0000	−1.08680
$923$	−12.0000	−0.394985
$924$	0	0
$925$	−28.0000	−0.920634
$926$	41.0000	1.34734
$927$	0	0
$928$	9.00000	0.295439
$929$	−12.0000	−0.393707	−0.196854	−	0.980433i	$-0.563072\pi$
$929$	−12.0000	−0.393707	−0.196854	+	0.980433i	$0.563072\pi$
$930$	0	0
$931$	−7.00000	−0.229416
$932$	−24.0000	−0.786146
$933$	0	0
$934$	3.00000	0.0981630
$935$	27.0000	0.882994
$936$	0	0
$937$	−34.0000	−1.11073	−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000	−1.11073	−0.555366	+	0.831606i	$0.687422\pi$
$938$	14.0000	0.457116
$939$	0	0
$940$	0	0
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	0	0
$943$	108.000	3.51696
$944$	−12.0000	−0.390567
$945$	0	0
$946$	3.00000	0.0975384
$947$	57.0000	1.85225	0.926126	−	0.377215i	$-0.123118\pi$
$947$	57.0000	1.85225	0.926126	+	0.377215i	$0.123118\pi$
$948$	0	0
$949$	−7.00000	−0.227230
$950$	−28.0000	−0.908440
$951$	0	0
$952$	3.00000	0.0972306
$953$	24.0000	0.777436	0.388718	−	0.921357i	$-0.372918\pi$
$953$	24.0000	0.777436	0.388718	+	0.921357i	$0.372918\pi$
$954$	0	0
$955$	−45.0000	−1.45617
$956$	−24.0000	−0.776215
$957$	0	0
$958$	15.0000	0.484628
$959$	−9.00000	−0.290625
$960$	0	0
$961$	−15.0000	−0.483871
$962$	−7.00000	−0.225689
$963$	0	0
$964$	−10.0000	−0.322078
$965$	12.0000	0.386294
$966$	0	0
$967$	−13.0000	−0.418052	−0.209026	−	0.977910i	$-0.567029\pi$
$967$	−13.0000	−0.418052	−0.209026	+	0.977910i	$0.567029\pi$
$968$	−2.00000	−0.0642824
$969$	0	0
$970$	30.0000	0.963242
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	−4.00000	−0.128234
$974$	−16.0000	−0.512673
$975$	0	0
$976$	−1.00000	−0.0320092
$977$	−15.0000	−0.479893	−0.239946	−	0.970786i	$-0.577130\pi$
$977$	−15.0000	−0.479893	−0.239946	+	0.970786i	$0.577130\pi$
$978$	0	0
$979$	−18.0000	−0.575282
$980$	−3.00000	−0.0958315
$981$	0	0
$982$	−30.0000	−0.957338
$983$	39.0000	1.24391	0.621953	−	0.783054i	$-0.286339\pi$
$983$	39.0000	1.24391	0.621953	+	0.783054i	$0.286339\pi$
$984$	0	0
$985$	−36.0000	−1.14706
$986$	27.0000	0.859855
$987$	0	0
$988$	−7.00000	−0.222700
$989$	9.00000	0.286183
$990$	0	0
$991$	38.0000	1.20711	0.603555	−	0.797321i	$-0.293750\pi$
$991$	38.0000	1.20711	0.603555	+	0.797321i	$0.293750\pi$
$992$	−4.00000	−0.127000
$993$	0	0
$994$	−12.0000	−0.380617
$995$	21.0000	0.665745
$996$	0	0
$997$	−46.0000	−1.45683	−0.728417	−	0.685134i	$-0.759744\pi$
$997$	−46.0000	−1.45683	−0.728417	+	0.685134i	$0.759744\pi$
$998$	32.0000	1.01294
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1638.2.a.l.1.1		1
3.2	odd	2		546.2.a.d.1.1	✓	1
12.11	even	2		4368.2.a.l.1.1		1
21.20	even	2		3822.2.a.a.1.1		1
39.38	odd	2		7098.2.a.w.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.a.d.1.1	✓	1	3.2	odd	2
1638.2.a.l.1.1		1	1.1	even	1	trivial
3822.2.a.a.1.1		1	21.20	even	2
4368.2.a.l.1.1		1	12.11	even	2
7098.2.a.w.1.1		1	39.38	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:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1638.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more