LMFDB - Embedded newform 1638.2.a.j.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:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 182)
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} +4.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} -4.00000 q^{10} -1.00000 q^{11} -1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -6.00000 q^{19} +4.00000 q^{20} +1.00000 q^{22} +7.00000 q^{23} +11.0000 q^{25} +1.00000 q^{26} -1.00000 q^{28} +4.00000 q^{29} +7.00000 q^{31} -1.00000 q^{32} -4.00000 q^{35} +9.00000 q^{37} +6.00000 q^{38} -4.00000 q^{40} +3.00000 q^{41} +4.00000 q^{43} -1.00000 q^{44} -7.00000 q^{46} -7.00000 q^{47} +1.00000 q^{49} -11.0000 q^{50} -1.00000 q^{52} -4.00000 q^{55} +1.00000 q^{56} -4.00000 q^{58} +10.0000 q^{59} +1.00000 q^{61} -7.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} +1.00000 q^{67} +4.00000 q^{70} -16.0000 q^{71} +5.00000 q^{73} -9.00000 q^{74} -6.00000 q^{76} +1.00000 q^{77} +11.0000 q^{79} +4.00000 q^{80} -3.00000 q^{82} -4.00000 q^{86} +1.00000 q^{88} +6.00000 q^{89} +1.00000 q^{91} +7.00000 q^{92} +7.00000 q^{94} -24.0000 q^{95} -1.00000 q^{97} -1.00000 q^{98} +O(q^{100})

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
$5$	4.00000	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$5$	4.00000	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−4.00000	−1.26491
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	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
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	4.00000	0.894427
$21$	0	0
$22$	1.00000	0.213201
$23$	7.00000	1.45960	0.729800	−	0.683660i	$-0.239613\pi$
$23$	7.00000	1.45960	0.729800	+	0.683660i	$0.239613\pi$
$24$	0	0
$25$	11.0000	2.20000
$26$	1.00000	0.196116
$27$	0	0
$28$	−1.00000	−0.188982
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0	0
$35$	−4.00000	−0.676123
$36$	0	0
$37$	9.00000	1.47959	0.739795	−	0.672832i	$-0.234922\pi$
$37$	9.00000	1.47959	0.739795	+	0.672832i	$0.234922\pi$
$38$	6.00000	0.973329
$39$	0	0
$40$	−4.00000	−0.632456
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−7.00000	−1.03209
$47$	−7.00000	−1.02105	−0.510527	−	0.859861i	$-0.670550\pi$
$47$	−7.00000	−1.02105	−0.510527	+	0.859861i	$0.670550\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−11.0000	−1.55563
$51$	0	0
$52$	−1.00000	−0.138675
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	1.00000	0.133631
$57$	0	0
$58$	−4.00000	−0.525226
$59$	10.0000	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$60$	0	0
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	−7.00000	−0.889001
$63$	0	0
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	0	0
$67$	1.00000	0.122169	0.0610847	−	0.998133i	$-0.480544\pi$
$67$	1.00000	0.122169	0.0610847	+	0.998133i	$0.480544\pi$
$68$	0	0
$69$	0	0
$70$	4.00000	0.478091
$71$	−16.0000	−1.89885	−0.949425	−	0.313993i	$-0.898333\pi$
$71$	−16.0000	−1.89885	−0.949425	+	0.313993i	$0.898333\pi$
$72$	0	0
$73$	5.00000	0.585206	0.292603	−	0.956234i	$-0.405479\pi$
$73$	5.00000	0.585206	0.292603	+	0.956234i	$0.405479\pi$
$74$	−9.00000	−1.04623
$75$	0	0
$76$	−6.00000	−0.688247
$77$	1.00000	0.113961
$78$	0	0
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	4.00000	0.447214
$81$	0	0
$82$	−3.00000	−0.331295
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	−4.00000	−0.431331
$87$	0	0
$88$	1.00000	0.106600
$89$	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
$92$	7.00000	0.729800
$93$	0	0
$94$	7.00000	0.721995
$95$	−24.0000	−2.46235
$96$	0	0
$97$	−1.00000	−0.101535	−0.0507673	−	0.998711i	$-0.516167\pi$
$97$	−1.00000	−0.101535	−0.0507673	+	0.998711i	$0.516167\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	11.0000	1.10000
$101$	5.00000	0.497519	0.248759	−	0.968565i	$-0.419977\pi$
$101$	5.00000	0.497519	0.248759	+	0.968565i	$0.419977\pi$
$102$	0	0
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	0	0
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	4.00000	0.381385
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	7.00000	0.658505	0.329252	−	0.944242i	$-0.393203\pi$
$113$	7.00000	0.658505	0.329252	+	0.944242i	$0.393203\pi$
$114$	0	0
$115$	28.0000	2.61101
$116$	4.00000	0.371391
$117$	0	0
$118$	−10.0000	−0.920575
$119$	0	0
$120$	0	0
$121$	−10.0000	−0.909091
$122$	−1.00000	−0.0905357
$123$	0	0
$124$	7.00000	0.628619
$125$	24.0000	2.14663
$126$	0	0
$127$	−11.0000	−0.976092	−0.488046	−	0.872818i	$-0.662290\pi$
$127$	−11.0000	−0.976092	−0.488046	+	0.872818i	$0.662290\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	4.00000	0.350823
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	−1.00000	−0.0863868
$135$	0	0
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	−4.00000	−0.338062
$141$	0	0
$142$	16.0000	1.34269
$143$	1.00000	0.0836242
$144$	0	0
$145$	16.0000	1.32873
$146$	−5.00000	−0.413803
$147$	0	0
$148$	9.00000	0.739795
$149$	−9.00000	−0.737309	−0.368654	−	0.929567i	$-0.620181\pi$
$149$	−9.00000	−0.737309	−0.368654	+	0.929567i	$0.620181\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	6.00000	0.486664
$153$	0	0
$154$	−1.00000	−0.0805823
$155$	28.0000	2.24901
$156$	0	0
$157$	5.00000	0.399043	0.199522	−	0.979893i	$-0.436061\pi$
$157$	5.00000	0.399043	0.199522	+	0.979893i	$0.436061\pi$
$158$	−11.0000	−0.875113
$159$	0	0
$160$	−4.00000	−0.316228
$161$	−7.00000	−0.551677
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	3.00000	0.234261
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	4.00000	0.304997
$173$	−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
$175$	−11.0000	−0.831522
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	−6.00000	−0.449719
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	15.0000	1.11494	0.557471	−	0.830197i	$-0.311772\pi$
$181$	15.0000	1.11494	0.557471	+	0.830197i	$0.311772\pi$
$182$	−1.00000	−0.0741249
$183$	0	0
$184$	−7.00000	−0.516047
$185$	36.0000	2.64677
$186$	0	0
$187$	0	0
$188$	−7.00000	−0.510527
$189$	0	0
$190$	24.0000	1.74114
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	20.0000	1.43963	0.719816	−	0.694165i	$-0.244226\pi$
$193$	20.0000	1.43963	0.719816	+	0.694165i	$0.244226\pi$
$194$	1.00000	0.0717958
$195$	0	0
$196$	1.00000	0.0714286
$197$	27.0000	1.92367	0.961835	−	0.273629i	$-0.0882242\pi$
$197$	27.0000	1.92367	0.961835	+	0.273629i	$0.0882242\pi$
$198$	0	0
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	−11.0000	−0.777817
$201$	0	0
$202$	−5.00000	−0.351799
$203$	−4.00000	−0.280745
$204$	0	0
$205$	12.0000	0.838116
$206$	14.0000	0.975426
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	6.00000	0.415029
$210$	0	0
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	0	0
$213$	0	0
$214$	−4.00000	−0.273434
$215$	16.0000	1.09119
$216$	0	0
$217$	−7.00000	−0.475191
$218$	14.0000	0.948200
$219$	0	0
$220$	−4.00000	−0.269680
$221$	0	0
$222$	0	0
$223$	21.0000	1.40626	0.703132	−	0.711059i	$-0.251784\pi$
$223$	21.0000	1.40626	0.703132	+	0.711059i	$0.251784\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−7.00000	−0.465633
$227$	−24.0000	−1.59294	−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000	−1.59294	−0.796468	+	0.604681i	$0.793301\pi$
$228$	0	0
$229$	−24.0000	−1.58596	−0.792982	−	0.609245i	$-0.791473\pi$
$229$	−24.0000	−1.58596	−0.792982	+	0.609245i	$0.791473\pi$
$230$	−28.0000	−1.84627
$231$	0	0
$232$	−4.00000	−0.262613
$233$	5.00000	0.327561	0.163780	−	0.986497i	$-0.447631\pi$
$233$	5.00000	0.327561	0.163780	+	0.986497i	$0.447631\pi$
$234$	0	0
$235$	−28.0000	−1.82652
$236$	10.0000	0.650945
$237$	0	0
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	10.0000	0.642824
$243$	0	0
$244$	1.00000	0.0640184
$245$	4.00000	0.255551
$246$	0	0
$247$	6.00000	0.381771
$248$	−7.00000	−0.444500
$249$	0	0
$250$	−24.0000	−1.51789
$251$	−3.00000	−0.189358	−0.0946792	−	0.995508i	$-0.530183\pi$
$251$	−3.00000	−0.189358	−0.0946792	+	0.995508i	$0.530183\pi$
$252$	0	0
$253$	−7.00000	−0.440086
$254$	11.0000	0.690201
$255$	0	0
$256$	1.00000	0.0625000
$257$	−24.0000	−1.49708	−0.748539	−	0.663090i	$-0.769245\pi$
$257$	−24.0000	−1.49708	−0.748539	+	0.663090i	$0.769245\pi$
$258$	0	0
$259$	−9.00000	−0.559233
$260$	−4.00000	−0.248069
$261$	0	0
$262$	8.00000	0.494242
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	0	0
$266$	−6.00000	−0.367884
$267$	0	0
$268$	1.00000	0.0610847
$269$	−9.00000	−0.548740	−0.274370	−	0.961624i	$-0.588469\pi$
$269$	−9.00000	−0.548740	−0.274370	+	0.961624i	$0.588469\pi$
$270$	0	0
$271$	−23.0000	−1.39715	−0.698575	−	0.715537i	$-0.746182\pi$
$271$	−23.0000	−1.39715	−0.698575	+	0.715537i	$0.746182\pi$
$272$	0	0
$273$	0	0
$274$	−6.00000	−0.362473
$275$	−11.0000	−0.663325
$276$	0	0
$277$	−18.0000	−1.08152	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	−18.0000	−1.08152	−0.540758	+	0.841178i	$0.681862\pi$
$278$	−4.00000	−0.239904
$279$	0	0
$280$	4.00000	0.239046
$281$	8.00000	0.477240	0.238620	−	0.971113i	$-0.423305\pi$
$281$	8.00000	0.477240	0.238620	+	0.971113i	$0.423305\pi$
$282$	0	0
$283$	19.0000	1.12943	0.564716	−	0.825285i	$-0.308986\pi$
$283$	19.0000	1.12943	0.564716	+	0.825285i	$0.308986\pi$
$284$	−16.0000	−0.949425
$285$	0	0
$286$	−1.00000	−0.0591312
$287$	−3.00000	−0.177084
$288$	0	0
$289$	−17.0000	−1.00000
$290$	−16.0000	−0.939552
$291$	0	0
$292$	5.00000	0.292603
$293$	26.0000	1.51894	0.759468	−	0.650545i	$-0.225459\pi$
$293$	26.0000	1.51894	0.759468	+	0.650545i	$0.225459\pi$
$294$	0	0
$295$	40.0000	2.32889
$296$	−9.00000	−0.523114
$297$	0	0
$298$	9.00000	0.521356
$299$	−7.00000	−0.404820
$300$	0	0
$301$	−4.00000	−0.230556
$302$	0	0
$303$	0	0
$304$	−6.00000	−0.344124
$305$	4.00000	0.229039
$306$	0	0
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	1.00000	0.0569803
$309$	0	0
$310$	−28.0000	−1.59029
$311$	−30.0000	−1.70114	−0.850572	−	0.525859i	$-0.823744\pi$
$311$	−30.0000	−1.70114	−0.850572	+	0.525859i	$0.823744\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	−5.00000	−0.282166
$315$	0	0
$316$	11.0000	0.618798
$317$	−21.0000	−1.17948	−0.589739	−	0.807594i	$-0.700769\pi$
$317$	−21.0000	−1.17948	−0.589739	+	0.807594i	$0.700769\pi$
$318$	0	0
$319$	−4.00000	−0.223957
$320$	4.00000	0.223607
$321$	0	0
$322$	7.00000	0.390095
$323$	0	0
$324$	0	0
$325$	−11.0000	−0.610170
$326$	4.00000	0.221540
$327$	0	0
$328$	−3.00000	−0.165647
$329$	7.00000	0.385922
$330$	0	0
$331$	−7.00000	−0.384755	−0.192377	−	0.981321i	$-0.561620\pi$
$331$	−7.00000	−0.384755	−0.192377	+	0.981321i	$0.561620\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	4.00000	0.218543
$336$	0	0
$337$	17.0000	0.926049	0.463025	−	0.886345i	$-0.346764\pi$
$337$	17.0000	0.926049	0.463025	+	0.886345i	$0.346764\pi$
$338$	−1.00000	−0.0543928
$339$	0	0
$340$	0	0
$341$	−7.00000	−0.379071
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−4.00000	−0.215666
$345$	0	0
$346$	2.00000	0.107521
$347$	24.0000	1.28839	0.644194	−	0.764862i	$-0.277193\pi$
$347$	24.0000	1.28839	0.644194	+	0.764862i	$0.277193\pi$
$348$	0	0
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	11.0000	0.587975
$351$	0	0
$352$	1.00000	0.0533002
$353$	−25.0000	−1.33062	−0.665308	−	0.746569i	$-0.731700\pi$
$353$	−25.0000	−1.33062	−0.665308	+	0.746569i	$0.731700\pi$
$354$	0	0
$355$	−64.0000	−3.39677
$356$	6.00000	0.317999
$357$	0	0
$358$	−6.00000	−0.317110
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	−15.0000	−0.788382
$363$	0	0
$364$	1.00000	0.0524142
$365$	20.0000	1.04685
$366$	0	0
$367$	32.0000	1.67039	0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000	1.67039	0.835193	+	0.549957i	$0.185356\pi$
$368$	7.00000	0.364900
$369$	0	0
$370$	−36.0000	−1.87155
$371$	0	0
$372$	0	0
$373$	−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
$375$	0	0
$376$	7.00000	0.360997
$377$	−4.00000	−0.206010
$378$	0	0
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	−24.0000	−1.23117
$381$	0	0
$382$	−8.00000	−0.409316
$383$	15.0000	0.766464	0.383232	−	0.923652i	$-0.374811\pi$
$383$	15.0000	0.766464	0.383232	+	0.923652i	$0.374811\pi$
$384$	0	0
$385$	4.00000	0.203859
$386$	−20.0000	−1.01797
$387$	0	0
$388$	−1.00000	−0.0507673
$389$	36.0000	1.82527	0.912636	−	0.408773i	$-0.134043\pi$
$389$	36.0000	1.82527	0.912636	+	0.408773i	$0.134043\pi$
$390$	0	0
$391$	0	0
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−27.0000	−1.36024
$395$	44.0000	2.21388
$396$	0	0
$397$	−28.0000	−1.40528	−0.702640	−	0.711546i	$-0.747995\pi$
$397$	−28.0000	−1.40528	−0.702640	+	0.711546i	$0.747995\pi$
$398$	4.00000	0.200502
$399$	0	0
$400$	11.0000	0.550000
$401$	−12.0000	−0.599251	−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000	−0.599251	−0.299626	+	0.954057i	$0.596862\pi$
$402$	0	0
$403$	−7.00000	−0.348695
$404$	5.00000	0.248759
$405$	0	0
$406$	4.00000	0.198517
$407$	−9.00000	−0.446113
$408$	0	0
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	−12.0000	−0.592638
$411$	0	0
$412$	−14.0000	−0.689730
$413$	−10.0000	−0.492068
$414$	0	0
$415$	0	0
$416$	1.00000	0.0490290
$417$	0	0
$418$	−6.00000	−0.293470
$419$	−15.0000	−0.732798	−0.366399	−	0.930458i	$-0.619409\pi$
$419$	−15.0000	−0.732798	−0.366399	+	0.930458i	$0.619409\pi$
$420$	0	0
$421$	19.0000	0.926003	0.463002	−	0.886357i	$-0.346772\pi$
$421$	19.0000	0.926003	0.463002	+	0.886357i	$0.346772\pi$
$422$	10.0000	0.486792
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1.00000	−0.0483934
$428$	4.00000	0.193347
$429$	0	0
$430$	−16.0000	−0.771589
$431$	−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
$433$	34.0000	1.63394	0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000	1.63394	0.816968	+	0.576683i	$0.195653\pi$
$434$	7.00000	0.336011
$435$	0	0
$436$	−14.0000	−0.670478
$437$	−42.0000	−2.00913
$438$	0	0
$439$	2.00000	0.0954548	0.0477274	−	0.998860i	$-0.484802\pi$
$439$	2.00000	0.0954548	0.0477274	+	0.998860i	$0.484802\pi$
$440$	4.00000	0.190693
$441$	0	0
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	0	0
$445$	24.0000	1.13771
$446$	−21.0000	−0.994379
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	2.00000	0.0943858	0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000	0.0943858	0.0471929	+	0.998886i	$0.484972\pi$
$450$	0	0
$451$	−3.00000	−0.141264
$452$	7.00000	0.329252
$453$	0	0
$454$	24.0000	1.12638
$455$	4.00000	0.187523
$456$	0	0
$457$	−26.0000	−1.21623	−0.608114	−	0.793849i	$-0.708074\pi$
$457$	−26.0000	−1.21623	−0.608114	+	0.793849i	$0.708074\pi$
$458$	24.0000	1.12145
$459$	0	0
$460$	28.0000	1.30551
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	4.00000	0.185695
$465$	0	0
$466$	−5.00000	−0.231621
$467$	4.00000	0.185098	0.0925490	−	0.995708i	$-0.470499\pi$
$467$	4.00000	0.185098	0.0925490	+	0.995708i	$0.470499\pi$
$468$	0	0
$469$	−1.00000	−0.0461757
$470$	28.0000	1.29154
$471$	0	0
$472$	−10.0000	−0.460287
$473$	−4.00000	−0.183920
$474$	0	0
$475$	−66.0000	−3.02829
$476$	0	0
$477$	0	0
$478$	6.00000	0.274434
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	−9.00000	−0.410365
$482$	18.0000	0.819878
$483$	0	0
$484$	−10.0000	−0.454545
$485$	−4.00000	−0.181631
$486$	0	0
$487$	38.0000	1.72194	0.860972	−	0.508652i	$-0.169856\pi$
$487$	38.0000	1.72194	0.860972	+	0.508652i	$0.169856\pi$
$488$	−1.00000	−0.0452679
$489$	0	0
$490$	−4.00000	−0.180702
$491$	−16.0000	−0.722070	−0.361035	−	0.932552i	$-0.617576\pi$
$491$	−16.0000	−0.722070	−0.361035	+	0.932552i	$0.617576\pi$
$492$	0	0
$493$	0	0
$494$	−6.00000	−0.269953
$495$	0	0
$496$	7.00000	0.314309
$497$	16.0000	0.717698
$498$	0	0
$499$	−5.00000	−0.223831	−0.111915	−	0.993718i	$-0.535699\pi$
$499$	−5.00000	−0.223831	−0.111915	+	0.993718i	$0.535699\pi$
$500$	24.0000	1.07331
$501$	0	0
$502$	3.00000	0.133897
$503$	20.0000	0.891756	0.445878	−	0.895094i	$-0.352892\pi$
$503$	20.0000	0.891756	0.445878	+	0.895094i	$0.352892\pi$
$504$	0	0
$505$	20.0000	0.889988
$506$	7.00000	0.311188
$507$	0	0
$508$	−11.0000	−0.488046
$509$	−10.0000	−0.443242	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000	−0.443242	−0.221621	+	0.975133i	$0.571135\pi$
$510$	0	0
$511$	−5.00000	−0.221187
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	24.0000	1.05859
$515$	−56.0000	−2.46765
$516$	0	0
$517$	7.00000	0.307860
$518$	9.00000	0.395437
$519$	0	0
$520$	4.00000	0.175412
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	27.0000	1.18063	0.590314	−	0.807174i	$-0.299004\pi$
$523$	27.0000	1.18063	0.590314	+	0.807174i	$0.299004\pi$
$524$	−8.00000	−0.349482
$525$	0	0
$526$	24.0000	1.04645
$527$	0	0
$528$	0	0
$529$	26.0000	1.13043
$530$	0	0
$531$	0	0
$532$	6.00000	0.260133
$533$	−3.00000	−0.129944
$534$	0	0
$535$	16.0000	0.691740
$536$	−1.00000	−0.0431934
$537$	0	0
$538$	9.00000	0.388018
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	23.0000	0.987935
$543$	0	0
$544$	0	0
$545$	−56.0000	−2.39878
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	6.00000	0.256307
$549$	0	0
$550$	11.0000	0.469042
$551$	−24.0000	−1.02243
$552$	0	0
$553$	−11.0000	−0.467768
$554$	18.0000	0.764747
$555$	0	0
$556$	4.00000	0.169638
$557$	−9.00000	−0.381342	−0.190671	−	0.981654i	$-0.561066\pi$
$557$	−9.00000	−0.381342	−0.190671	+	0.981654i	$0.561066\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	−4.00000	−0.169031
$561$	0	0
$562$	−8.00000	−0.337460
$563$	−11.0000	−0.463595	−0.231797	−	0.972764i	$-0.574461\pi$
$563$	−11.0000	−0.463595	−0.231797	+	0.972764i	$0.574461\pi$
$564$	0	0
$565$	28.0000	1.17797
$566$	−19.0000	−0.798630
$567$	0	0
$568$	16.0000	0.671345
$569$	35.0000	1.46728	0.733638	−	0.679540i	$-0.237821\pi$
$569$	35.0000	1.46728	0.733638	+	0.679540i	$0.237821\pi$
$570$	0	0
$571$	6.00000	0.251092	0.125546	−	0.992088i	$-0.459932\pi$
$571$	6.00000	0.251092	0.125546	+	0.992088i	$0.459932\pi$
$572$	1.00000	0.0418121
$573$	0	0
$574$	3.00000	0.125218
$575$	77.0000	3.21112
$576$	0	0
$577$	30.0000	1.24892	0.624458	−	0.781058i	$-0.285320\pi$
$577$	30.0000	1.24892	0.624458	+	0.781058i	$0.285320\pi$
$578$	17.0000	0.707107
$579$	0	0
$580$	16.0000	0.664364
$581$	0	0
$582$	0	0
$583$	0	0
$584$	−5.00000	−0.206901
$585$	0	0
$586$	−26.0000	−1.07405
$587$	−14.0000	−0.577842	−0.288921	−	0.957353i	$-0.593296\pi$
$587$	−14.0000	−0.577842	−0.288921	+	0.957353i	$0.593296\pi$
$588$	0	0
$589$	−42.0000	−1.73058
$590$	−40.0000	−1.64677
$591$	0	0
$592$	9.00000	0.369898
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	0	0
$595$	0	0
$596$	−9.00000	−0.368654
$597$	0	0
$598$	7.00000	0.286251
$599$	29.0000	1.18491	0.592454	−	0.805604i	$-0.298159\pi$
$599$	29.0000	1.18491	0.592454	+	0.805604i	$0.298159\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	4.00000	0.163028
$603$	0	0
$604$	0	0
$605$	−40.0000	−1.62623
$606$	0	0
$607$	6.00000	0.243532	0.121766	−	0.992559i	$-0.461144\pi$
$607$	6.00000	0.243532	0.121766	+	0.992559i	$0.461144\pi$
$608$	6.00000	0.243332
$609$	0	0
$610$	−4.00000	−0.161955
$611$	7.00000	0.283190
$612$	0	0
$613$	−49.0000	−1.97909	−0.989546	−	0.144220i	$-0.953933\pi$
$613$	−49.0000	−1.97909	−0.989546	+	0.144220i	$0.953933\pi$
$614$	4.00000	0.161427
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	−22.0000	−0.885687	−0.442843	−	0.896599i	$-0.646030\pi$
$617$	−22.0000	−0.885687	−0.442843	+	0.896599i	$0.646030\pi$
$618$	0	0
$619$	22.0000	0.884255	0.442127	−	0.896952i	$-0.354224\pi$
$619$	22.0000	0.884255	0.442127	+	0.896952i	$0.354224\pi$
$620$	28.0000	1.12451
$621$	0	0
$622$	30.0000	1.20289
$623$	−6.00000	−0.240385
$624$	0	0
$625$	41.0000	1.64000
$626$	−14.0000	−0.559553
$627$	0	0
$628$	5.00000	0.199522
$629$	0	0
$630$	0	0
$631$	−2.00000	−0.0796187	−0.0398094	−	0.999207i	$-0.512675\pi$
$631$	−2.00000	−0.0796187	−0.0398094	+	0.999207i	$0.512675\pi$
$632$	−11.0000	−0.437557
$633$	0	0
$634$	21.0000	0.834017
$635$	−44.0000	−1.74609
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	4.00000	0.158362
$639$	0	0
$640$	−4.00000	−0.158114
$641$	−7.00000	−0.276483	−0.138242	−	0.990399i	$-0.544145\pi$
$641$	−7.00000	−0.276483	−0.138242	+	0.990399i	$0.544145\pi$
$642$	0	0
$643$	−16.0000	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$644$	−7.00000	−0.275839
$645$	0	0
$646$	0	0
$647$	8.00000	0.314512	0.157256	−	0.987558i	$-0.449735\pi$
$647$	8.00000	0.314512	0.157256	+	0.987558i	$0.449735\pi$
$648$	0	0
$649$	−10.0000	−0.392534
$650$	11.0000	0.431455
$651$	0	0
$652$	−4.00000	−0.156652
$653$	36.0000	1.40879	0.704394	−	0.709809i	$-0.251219\pi$
$653$	36.0000	1.40879	0.704394	+	0.709809i	$0.251219\pi$
$654$	0	0
$655$	−32.0000	−1.25034
$656$	3.00000	0.117130
$657$	0	0
$658$	−7.00000	−0.272888
$659$	18.0000	0.701180	0.350590	−	0.936529i	$-0.385981\pi$
$659$	18.0000	0.701180	0.350590	+	0.936529i	$0.385981\pi$
$660$	0	0
$661$	−4.00000	−0.155582	−0.0777910	−	0.996970i	$-0.524787\pi$
$661$	−4.00000	−0.155582	−0.0777910	+	0.996970i	$0.524787\pi$
$662$	7.00000	0.272063
$663$	0	0
$664$	0	0
$665$	24.0000	0.930680
$666$	0	0
$667$	28.0000	1.08416
$668$	0	0
$669$	0	0
$670$	−4.00000	−0.154533
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	7.00000	0.269830	0.134915	−	0.990857i	$-0.456924\pi$
$673$	7.00000	0.269830	0.134915	+	0.990857i	$0.456924\pi$
$674$	−17.0000	−0.654816
$675$	0	0
$676$	1.00000	0.0384615
$677$	−39.0000	−1.49889	−0.749446	−	0.662066i	$-0.769680\pi$
$677$	−39.0000	−1.49889	−0.749446	+	0.662066i	$0.769680\pi$
$678$	0	0
$679$	1.00000	0.0383765
$680$	0	0
$681$	0	0
$682$	7.00000	0.268044
$683$	−1.00000	−0.0382639	−0.0191320	−	0.999817i	$-0.506090\pi$
$683$	−1.00000	−0.0382639	−0.0191320	+	0.999817i	$0.506090\pi$
$684$	0	0
$685$	24.0000	0.916993
$686$	1.00000	0.0381802
$687$	0	0
$688$	4.00000	0.152499
$689$	0	0
$690$	0	0
$691$	−32.0000	−1.21734	−0.608669	−	0.793424i	$-0.708296\pi$
$691$	−32.0000	−1.21734	−0.608669	+	0.793424i	$0.708296\pi$
$692$	−2.00000	−0.0760286
$693$	0	0
$694$	−24.0000	−0.911028
$695$	16.0000	0.606915
$696$	0	0
$697$	0	0
$698$	26.0000	0.984115
$699$	0	0
$700$	−11.0000	−0.415761
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	−54.0000	−2.03665
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	25.0000	0.940887
$707$	−5.00000	−0.188044
$708$	0	0
$709$	−13.0000	−0.488225	−0.244113	−	0.969747i	$-0.578497\pi$
$709$	−13.0000	−0.488225	−0.244113	+	0.969747i	$0.578497\pi$
$710$	64.0000	2.40188
$711$	0	0
$712$	−6.00000	−0.224860
$713$	49.0000	1.83506
$714$	0	0
$715$	4.00000	0.149592
$716$	6.00000	0.224231
$717$	0	0
$718$	12.0000	0.447836
$719$	−2.00000	−0.0745874	−0.0372937	−	0.999304i	$-0.511874\pi$
$719$	−2.00000	−0.0745874	−0.0372937	+	0.999304i	$0.511874\pi$
$720$	0	0
$721$	14.0000	0.521387
$722$	−17.0000	−0.632674
$723$	0	0
$724$	15.0000	0.557471
$725$	44.0000	1.63412
$726$	0	0
$727$	−26.0000	−0.964287	−0.482143	−	0.876092i	$-0.660142\pi$
$727$	−26.0000	−0.964287	−0.482143	+	0.876092i	$0.660142\pi$
$728$	−1.00000	−0.0370625
$729$	0	0
$730$	−20.0000	−0.740233
$731$	0	0
$732$	0	0
$733$	−4.00000	−0.147743	−0.0738717	−	0.997268i	$-0.523536\pi$
$733$	−4.00000	−0.147743	−0.0738717	+	0.997268i	$0.523536\pi$
$734$	−32.0000	−1.18114
$735$	0	0
$736$	−7.00000	−0.258023
$737$	−1.00000	−0.0368355
$738$	0	0
$739$	24.0000	0.882854	0.441427	−	0.897297i	$-0.354472\pi$
$739$	24.0000	0.882854	0.441427	+	0.897297i	$0.354472\pi$
$740$	36.0000	1.32339
$741$	0	0
$742$	0	0
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	0	0
$745$	−36.0000	−1.31894
$746$	16.0000	0.585802
$747$	0	0
$748$	0	0
$749$	−4.00000	−0.146157
$750$	0	0
$751$	−27.0000	−0.985244	−0.492622	−	0.870243i	$-0.663961\pi$
$751$	−27.0000	−0.985244	−0.492622	+	0.870243i	$0.663961\pi$
$752$	−7.00000	−0.255264
$753$	0	0
$754$	4.00000	0.145671
$755$	0	0
$756$	0	0
$757$	54.0000	1.96266	0.981332	−	0.192323i	$-0.0616021\pi$
$757$	54.0000	1.96266	0.981332	+	0.192323i	$0.0616021\pi$
$758$	8.00000	0.290573
$759$	0	0
$760$	24.0000	0.870572
$761$	−45.0000	−1.63125	−0.815624	−	0.578582i	$-0.803606\pi$
$761$	−45.0000	−1.63125	−0.815624	+	0.578582i	$0.803606\pi$
$762$	0	0
$763$	14.0000	0.506834
$764$	8.00000	0.289430
$765$	0	0
$766$	−15.0000	−0.541972
$767$	−10.0000	−0.361079
$768$	0	0
$769$	21.0000	0.757279	0.378640	−	0.925544i	$-0.376392\pi$
$769$	21.0000	0.757279	0.378640	+	0.925544i	$0.376392\pi$
$770$	−4.00000	−0.144150
$771$	0	0
$772$	20.0000	0.719816
$773$	−14.0000	−0.503545	−0.251773	−	0.967786i	$-0.581013\pi$
$773$	−14.0000	−0.503545	−0.251773	+	0.967786i	$0.581013\pi$
$774$	0	0
$775$	77.0000	2.76592
$776$	1.00000	0.0358979
$777$	0	0
$778$	−36.0000	−1.29066
$779$	−18.0000	−0.644917
$780$	0	0
$781$	16.0000	0.572525
$782$	0	0
$783$	0	0
$784$	1.00000	0.0357143
$785$	20.0000	0.713831
$786$	0	0
$787$	32.0000	1.14068	0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000	1.14068	0.570338	+	0.821410i	$0.306812\pi$
$788$	27.0000	0.961835
$789$	0	0
$790$	−44.0000	−1.56545
$791$	−7.00000	−0.248891
$792$	0	0
$793$	−1.00000	−0.0355110
$794$	28.0000	0.993683
$795$	0	0
$796$	−4.00000	−0.141776
$797$	−15.0000	−0.531327	−0.265664	−	0.964066i	$-0.585591\pi$
$797$	−15.0000	−0.531327	−0.265664	+	0.964066i	$0.585591\pi$
$798$	0	0
$799$	0	0
$800$	−11.0000	−0.388909
$801$	0	0
$802$	12.0000	0.423735
$803$	−5.00000	−0.176446
$804$	0	0
$805$	−28.0000	−0.986870
$806$	7.00000	0.246564
$807$	0	0
$808$	−5.00000	−0.175899
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	−4.00000	−0.140372
$813$	0	0
$814$	9.00000	0.315450
$815$	−16.0000	−0.560456
$816$	0	0
$817$	−24.0000	−0.839654
$818$	26.0000	0.909069
$819$	0	0
$820$	12.0000	0.419058
$821$	46.0000	1.60541	0.802706	−	0.596376i	$-0.203393\pi$
$821$	46.0000	1.60541	0.802706	+	0.596376i	$0.203393\pi$
$822$	0	0
$823$	−3.00000	−0.104573	−0.0522867	−	0.998632i	$-0.516651\pi$
$823$	−3.00000	−0.104573	−0.0522867	+	0.998632i	$0.516651\pi$
$824$	14.0000	0.487713
$825$	0	0
$826$	10.0000	0.347945
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	0	0
$831$	0	0
$832$	−1.00000	−0.0346688
$833$	0	0
$834$	0	0
$835$	0	0
$836$	6.00000	0.207514
$837$	0	0
$838$	15.0000	0.518166
$839$	−17.0000	−0.586905	−0.293453	−	0.955974i	$-0.594804\pi$
$839$	−17.0000	−0.586905	−0.293453	+	0.955974i	$0.594804\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	−19.0000	−0.654783
$843$	0	0
$844$	−10.0000	−0.344214
$845$	4.00000	0.137604
$846$	0	0
$847$	10.0000	0.343604
$848$	0	0
$849$	0	0
$850$	0	0
$851$	63.0000	2.15961
$852$	0	0
$853$	16.0000	0.547830	0.273915	−	0.961754i	$-0.411681\pi$
$853$	16.0000	0.547830	0.273915	+	0.961754i	$0.411681\pi$
$854$	1.00000	0.0342193
$855$	0	0
$856$	−4.00000	−0.136717
$857$	−22.0000	−0.751506	−0.375753	−	0.926720i	$-0.622616\pi$
$857$	−22.0000	−0.751506	−0.375753	+	0.926720i	$0.622616\pi$
$858$	0	0
$859$	5.00000	0.170598	0.0852989	−	0.996355i	$-0.472815\pi$
$859$	5.00000	0.170598	0.0852989	+	0.996355i	$0.472815\pi$
$860$	16.0000	0.545595
$861$	0	0
$862$	18.0000	0.613082
$863$	−28.0000	−0.953131	−0.476566	−	0.879139i	$-0.658119\pi$
$863$	−28.0000	−0.953131	−0.476566	+	0.879139i	$0.658119\pi$
$864$	0	0
$865$	−8.00000	−0.272008
$866$	−34.0000	−1.15537
$867$	0	0
$868$	−7.00000	−0.237595
$869$	−11.0000	−0.373149
$870$	0	0
$871$	−1.00000	−0.0338837
$872$	14.0000	0.474100
$873$	0	0
$874$	42.0000	1.42067
$875$	−24.0000	−0.811348
$876$	0	0
$877$	−45.0000	−1.51954	−0.759771	−	0.650191i	$-0.774689\pi$
$877$	−45.0000	−1.51954	−0.759771	+	0.650191i	$0.774689\pi$
$878$	−2.00000	−0.0674967
$879$	0	0
$880$	−4.00000	−0.134840
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	−14.0000	−0.471138	−0.235569	−	0.971858i	$-0.575695\pi$
$883$	−14.0000	−0.471138	−0.235569	+	0.971858i	$0.575695\pi$
$884$	0	0
$885$	0	0
$886$	12.0000	0.403148
$887$	38.0000	1.27592	0.637958	−	0.770072i	$-0.279780\pi$
$887$	38.0000	1.27592	0.637958	+	0.770072i	$0.279780\pi$
$888$	0	0
$889$	11.0000	0.368928
$890$	−24.0000	−0.804482
$891$	0	0
$892$	21.0000	0.703132
$893$	42.0000	1.40548
$894$	0	0
$895$	24.0000	0.802232
$896$	1.00000	0.0334077
$897$	0	0
$898$	−2.00000	−0.0667409
$899$	28.0000	0.933852
$900$	0	0
$901$	0	0
$902$	3.00000	0.0998891
$903$	0	0
$904$	−7.00000	−0.232817
$905$	60.0000	1.99447
$906$	0	0
$907$	−30.0000	−0.996134	−0.498067	−	0.867139i	$-0.665957\pi$
$907$	−30.0000	−0.996134	−0.498067	+	0.867139i	$0.665957\pi$
$908$	−24.0000	−0.796468
$909$	0	0
$910$	−4.00000	−0.132599
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	0	0
$913$	0	0
$914$	26.0000	0.860004
$915$	0	0
$916$	−24.0000	−0.792982
$917$	8.00000	0.264183
$918$	0	0
$919$	27.0000	0.890648	0.445324	−	0.895370i	$-0.353089\pi$
$919$	27.0000	0.890648	0.445324	+	0.895370i	$0.353089\pi$
$920$	−28.0000	−0.923133
$921$	0	0
$922$	12.0000	0.395199
$923$	16.0000	0.526646
$924$	0	0
$925$	99.0000	3.25510
$926$	−16.0000	−0.525793
$927$	0	0
$928$	−4.00000	−0.131306
$929$	15.0000	0.492134	0.246067	−	0.969253i	$-0.420862\pi$
$929$	15.0000	0.492134	0.246067	+	0.969253i	$0.420862\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	5.00000	0.163780
$933$	0	0
$934$	−4.00000	−0.130884
$935$	0	0
$936$	0	0
$937$	−36.0000	−1.17607	−0.588034	−	0.808836i	$-0.700098\pi$
$937$	−36.0000	−1.17607	−0.588034	+	0.808836i	$0.700098\pi$
$938$	1.00000	0.0326512
$939$	0	0
$940$	−28.0000	−0.913259
$941$	50.0000	1.62995	0.814977	−	0.579494i	$-0.196750\pi$
$941$	50.0000	1.62995	0.814977	+	0.579494i	$0.196750\pi$
$942$	0	0
$943$	21.0000	0.683854
$944$	10.0000	0.325472
$945$	0	0
$946$	4.00000	0.130051
$947$	−24.0000	−0.779895	−0.389948	−	0.920837i	$-0.627507\pi$
$947$	−24.0000	−0.779895	−0.389948	+	0.920837i	$0.627507\pi$
$948$	0	0
$949$	−5.00000	−0.162307
$950$	66.0000	2.14132
$951$	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
$955$	32.0000	1.03550
$956$	−6.00000	−0.194054
$957$	0	0
$958$	24.0000	0.775405
$959$	−6.00000	−0.193750
$960$	0	0
$961$	18.0000	0.580645
$962$	9.00000	0.290172
$963$	0	0
$964$	−18.0000	−0.579741
$965$	80.0000	2.57529
$966$	0	0
$967$	−22.0000	−0.707472	−0.353736	−	0.935345i	$-0.615089\pi$
$967$	−22.0000	−0.707472	−0.353736	+	0.935345i	$0.615089\pi$
$968$	10.0000	0.321412
$969$	0	0
$970$	4.00000	0.128432
$971$	15.0000	0.481373	0.240686	−	0.970603i	$-0.422627\pi$
$971$	15.0000	0.481373	0.240686	+	0.970603i	$0.422627\pi$
$972$	0	0
$973$	−4.00000	−0.128234
$974$	−38.0000	−1.21760
$975$	0	0
$976$	1.00000	0.0320092
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	0	0
$979$	−6.00000	−0.191761
$980$	4.00000	0.127775
$981$	0	0
$982$	16.0000	0.510581
$983$	40.0000	1.27580	0.637901	−	0.770118i	$-0.279803\pi$
$983$	40.0000	1.27580	0.637901	+	0.770118i	$0.279803\pi$
$984$	0	0
$985$	108.000	3.44117
$986$	0	0
$987$	0	0
$988$	6.00000	0.190885
$989$	28.0000	0.890348
$990$	0	0
$991$	−57.0000	−1.81066	−0.905332	−	0.424704i	$-0.860378\pi$
$991$	−57.0000	−1.81066	−0.905332	+	0.424704i	$0.860378\pi$
$992$	−7.00000	−0.222250
$993$	0	0
$994$	−16.0000	−0.507489
$995$	−16.0000	−0.507234
$996$	0	0
$997$	5.00000	0.158352	0.0791758	−	0.996861i	$-0.474771\pi$
$997$	5.00000	0.158352	0.0791758	+	0.996861i	$0.474771\pi$
$998$	5.00000	0.158272
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1638.2.a.j.1.1		1
3.2	odd	2		182.2.a.e.1.1	✓	1
12.11	even	2		1456.2.a.a.1.1		1
15.14	odd	2		4550.2.a.a.1.1		1
21.2	odd	6		1274.2.f.b.1145.1		2
21.5	even	6		1274.2.f.k.1145.1		2
21.11	odd	6		1274.2.f.b.79.1		2
21.17	even	6		1274.2.f.k.79.1		2
21.20	even	2		1274.2.a.h.1.1		1
24.5	odd	2		5824.2.a.b.1.1		1
24.11	even	2		5824.2.a.bf.1.1		1
39.5	even	4		2366.2.d.j.337.1		2
39.8	even	4		2366.2.d.j.337.2		2
39.38	odd	2		2366.2.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
182.2.a.e.1.1	✓	1	3.2	odd	2
1274.2.a.h.1.1		1	21.20	even	2
1274.2.f.b.79.1		2	21.11	odd	6
1274.2.f.b.1145.1		2	21.2	odd	6
1274.2.f.k.79.1		2	21.17	even	6
1274.2.f.k.1145.1		2	21.5	even	6
1456.2.a.a.1.1		1	12.11	even	2
1638.2.a.j.1.1		1	1.1	even	1	trivial
2366.2.a.h.1.1		1	39.38	odd	2
2366.2.d.j.337.1		2	39.5	even	4
2366.2.d.j.337.2		2	39.8	even	4
4550.2.a.a.1.1		1	15.14	odd	2
5824.2.a.b.1.1		1	24.5	odd	2
5824.2.a.bf.1.1		1	24.11	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 $\F_{q}$

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1638.2.a.j.1.1

Level	$1638$
Weight	$2$
Character	1638.1
Self dual	yes
Analytic conductor	$13.079$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$