LMFDB - Embedded newform 1728.3.h.a.161.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,3,Mod(161,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1728.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1728.h (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$47.0845896815$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 9 $ x^4 - 5*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.1
Root			$-1.65831 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1728.161
Dual form			1728.3.h.a.161.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.00000 q^{5} -9.94987 q^{7} +O(q^{10})$	$q-6.00000 q^{5} -9.94987 q^{7} -19.8997 q^{11} -9.94987i q^{13} -19.8997i q^{17} +7.00000i q^{19} -42.0000i q^{23} +11.0000 q^{25} -24.0000 q^{29} -39.7995 q^{31} +59.6992 q^{35} -49.7494i q^{37} +39.7995i q^{41} +50.0000i q^{43} -6.00000i q^{47} +50.0000 q^{49} -84.0000 q^{53} +119.398 q^{55} -19.8997 q^{59} +89.5489i q^{61} +59.6992i q^{65} +53.0000i q^{67} +60.0000i q^{71} +119.000 q^{73} +198.000 q^{77} -49.7494 q^{79} +119.398i q^{85} -59.6992i q^{89} +99.0000i q^{91} -42.0000i q^{95} +13.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 24 q^{5}+O(q^{10}) $ 4 * q - 24 * q^5	$ 4 q - 24 q^{5} + 44 q^{25} - 96 q^{29} + 200 q^{49} - 336 q^{53} + 476 q^{73} + 792 q^{77} + 52 q^{97}+O(q^{100}) $ 4 * q - 24 * q^5 + 44 * q^25 - 96 * q^29 + 200 * q^49 - 336 * q^53 + 476 * q^73 + 792 * q^77 + 52 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−6.00000	−1.20000	−0.600000	−	0.800000i	$-0.704833\pi$
$5$	−6.00000	−1.20000	−0.600000	+	0.800000i	$0.704833\pi$
$6$	0	0
$7$	−9.94987	−1.42141	−0.710705	−	0.703490i	$-0.751624\pi$
$7$	−9.94987	−1.42141	−0.710705	+	0.703490i	$0.751624\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−19.8997	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−19.8997	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	0	0
$13$	− 9.94987i	− 0.765375i	−0.923878	−	0.382687i	$-0.874999\pi$
$13$	− 9.94987i	− 0.765375i	0.923878	−	0.382687i	$-0.125001\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 19.8997i	− 1.17057i	−0.810826	−	0.585287i	$-0.800982\pi$
$17$	− 19.8997i	− 1.17057i	0.810826	−	0.585287i	$-0.199018\pi$
$18$	0	0
$19$	7.00000i	0.368421i	0.982887	+	0.184211i	$0.0589728\pi$
$19$	7.00000i	0.368421i	−0.982887	+	0.184211i	$0.941027\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 42.0000i	− 1.82609i	−0.407862	−	0.913043i	$-0.633726\pi$
$23$	− 42.0000i	− 1.82609i	0.407862	−	0.913043i	$-0.366274\pi$
$24$	0	0
$25$	11.0000	0.440000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−24.0000	−0.827586	−0.413793	−	0.910371i	$-0.635796\pi$
$29$	−24.0000	−0.827586	−0.413793	+	0.910371i	$0.635796\pi$
$30$	0	0
$31$	−39.7995	−1.28385	−0.641927	−	0.766765i	$-0.721865\pi$
$31$	−39.7995	−1.28385	−0.641927	+	0.766765i	$0.721865\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	59.6992	1.70569
$36$	0	0
$37$	− 49.7494i	− 1.34458i	−0.740289	−	0.672289i	$-0.765311\pi$
$37$	− 49.7494i	− 1.34458i	0.740289	−	0.672289i	$-0.234689\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	39.7995i	0.970719i	0.874315	+	0.485360i	$0.161311\pi$
$41$	39.7995i	0.970719i	−0.874315	+	0.485360i	$0.838689\pi$
$42$	0	0
$43$	50.0000i	1.16279i	0.813621	+	0.581395i	$0.197493\pi$
$43$	50.0000i	1.16279i	−0.813621	+	0.581395i	$0.802507\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 6.00000i	− 0.127660i	−0.997961	−	0.0638298i	$-0.979669\pi$
$47$	− 6.00000i	− 0.127660i	0.997961	−	0.0638298i	$-0.0203315\pi$
$48$	0	0
$49$	50.0000	1.02041
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−84.0000	−1.58491	−0.792453	−	0.609933i	$-0.791196\pi$
$53$	−84.0000	−1.58491	−0.792453	+	0.609933i	$0.791196\pi$
$54$	0	0
$55$	119.398	2.17088
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−19.8997	−0.337284	−0.168642	−	0.985677i	$-0.553938\pi$
$59$	−19.8997	−0.337284	−0.168642	+	0.985677i	$0.553938\pi$
$60$	0	0
$61$	89.5489i	1.46801i	0.679142	+	0.734007i	$0.262352\pi$
$61$	89.5489i	1.46801i	−0.679142	+	0.734007i	$0.737648\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	59.6992i	0.918450i
$66$	0	0
$67$	53.0000i	0.791045i	0.918456	+	0.395522i	$0.129436\pi$
$67$	53.0000i	0.791045i	−0.918456	+	0.395522i	$0.870564\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	60.0000i	0.845070i	0.906347	+	0.422535i	$0.138860\pi$
$71$	60.0000i	0.845070i	−0.906347	+	0.422535i	$0.861140\pi$
$72$	0	0
$73$	119.000	1.63014	0.815068	−	0.579365i	$-0.196699\pi$
$73$	119.000	1.63014	0.815068	+	0.579365i	$0.196699\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	198.000	2.57143
$78$	0	0
$79$	−49.7494	−0.629739	−0.314869	−	0.949135i	$-0.601961\pi$
$79$	−49.7494	−0.629739	−0.314869	+	0.949135i	$0.601961\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	119.398i	1.40469i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 59.6992i	− 0.670778i	−0.942080	−	0.335389i	$-0.891132\pi$
$89$	− 59.6992i	− 0.670778i	0.942080	−	0.335389i	$-0.108868\pi$
$90$	0	0
$91$	99.0000i	1.08791i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 42.0000i	− 0.442105i
$96$	0	0
$97$	13.0000	0.134021	0.0670103	−	0.997752i	$-0.478654\pi$
$97$	13.0000	0.134021	0.0670103	+	0.997752i	$0.478654\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−12.0000	−0.118812	−0.0594059	−	0.998234i	$-0.518921\pi$
$101$	−12.0000	−0.118812	−0.0594059	+	0.998234i	$0.518921\pi$
$102$	0	0
$103$	89.5489	0.869406	0.434703	−	0.900574i	$-0.356853\pi$
$103$	89.5489	0.869406	0.434703	+	0.900574i	$0.356853\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−139.298	−1.30185	−0.650926	−	0.759141i	$-0.725619\pi$
$107$	−139.298	−1.30185	−0.650926	+	0.759141i	$0.725619\pi$
$108$	0	0
$109$	− 39.7995i	− 0.365133i	−0.983194	−	0.182567i	$-0.941560\pi$
$109$	− 39.7995i	− 0.365133i	0.983194	−	0.182567i	$-0.0584405\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 179.098i	− 1.58494i	−0.609914	−	0.792468i	$-0.708796\pi$
$113$	− 179.098i	− 1.58494i	0.609914	−	0.792468i	$-0.291204\pi$
$114$	0	0
$115$	252.000i	2.19130i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	198.000i	1.66387i
$120$	0	0
$121$	275.000	2.27273
$122$	0	0
$123$	0	0
$124$	0	0
$125$	84.0000	0.672000
$126$	0	0
$127$	−79.5990	−0.626764	−0.313382	−	0.949627i	$-0.601462\pi$
$127$	−79.5990	−0.626764	−0.313382	+	0.949627i	$0.601462\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	− 69.6491i	− 0.523678i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 139.298i	− 1.01678i	−0.861128	−	0.508388i	$-0.830242\pi$
$137$	− 139.298i	− 1.01678i	0.861128	−	0.508388i	$-0.169758\pi$
$138$	0	0
$139$	55.0000i	0.395683i	0.980234	+	0.197842i	$0.0633932\pi$
$139$	55.0000i	0.395683i	−0.980234	+	0.197842i	$0.936607\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	198.000i	1.38462i
$144$	0	0
$145$	144.000	0.993103
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−36.0000	−0.241611	−0.120805	−	0.992676i	$-0.538548\pi$
$149$	−36.0000	−0.241611	−0.120805	+	0.992676i	$0.538548\pi$
$150$	0	0
$151$	−29.8496	−0.197680	−0.0988398	−	0.995103i	$-0.531513\pi$
$151$	−29.8496	−0.197680	−0.0988398	+	0.995103i	$0.531513\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	238.797	1.54063
$156$	0	0
$157$	− 198.997i	− 1.26750i	−0.773538	−	0.633750i	$-0.781515\pi$
$157$	− 198.997i	− 1.26750i	0.773538	−	0.633750i	$-0.218485\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	417.895i	2.59562i
$162$	0	0
$163$	79.0000i	0.484663i	0.970194	+	0.242331i	$0.0779121\pi$
$163$	79.0000i	0.484663i	−0.970194	+	0.242331i	$0.922088\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 210.000i	− 1.25749i	−0.777614	−	0.628743i	$-0.783570\pi$
$167$	− 210.000i	− 1.25749i	0.777614	−	0.628743i	$-0.216430\pi$
$168$	0	0
$169$	70.0000	0.414201
$170$	0	0
$171$	0	0
$172$	0	0
$173$	84.0000	0.485549	0.242775	−	0.970083i	$-0.421942\pi$
$173$	84.0000	0.485549	0.242775	+	0.970083i	$0.421942\pi$
$174$	0	0
$175$	−109.449	−0.625421
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−119.398	−0.667031	−0.333515	−	0.942745i	$-0.608235\pi$
$179$	−119.398	−0.667031	−0.333515	+	0.942745i	$0.608235\pi$
$180$	0	0
$181$	− 69.6491i	− 0.384802i	−0.981316	−	0.192401i	$-0.938373\pi$
$181$	− 69.6491i	− 0.384802i	0.981316	−	0.192401i	$-0.0616274\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	298.496i	1.61349i
$186$	0	0
$187$	396.000i	2.11765i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 246.000i	− 1.28796i	−0.765043	−	0.643979i	$-0.777282\pi$
$191$	− 246.000i	− 1.28796i	0.765043	−	0.643979i	$-0.222718\pi$
$192$	0	0
$193$	179.000	0.927461	0.463731	−	0.885976i	$-0.346511\pi$
$193$	179.000	0.927461	0.463731	+	0.885976i	$0.346511\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−126.000	−0.639594	−0.319797	−	0.947486i	$-0.603615\pi$
$197$	−126.000	−0.639594	−0.319797	+	0.947486i	$0.603615\pi$
$198$	0	0
$199$	−308.446	−1.54998	−0.774990	−	0.631973i	$-0.782245\pi$
$199$	−308.446	−1.54998	−0.774990	+	0.631973i	$0.782245\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	238.797	1.17634
$204$	0	0
$205$	− 238.797i	− 1.16486i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 139.298i	− 0.666499i
$210$	0	0
$211$	− 55.0000i	− 0.260664i	−0.991470	−	0.130332i	$-0.958396\pi$
$211$	− 55.0000i	− 0.260664i	0.991470	−	0.130332i	$-0.0416042\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 300.000i	− 1.39535i
$216$	0	0
$217$	396.000	1.82488
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−198.000	−0.895928
$222$	0	0
$223$	278.596	1.24931	0.624656	−	0.780900i	$-0.285239\pi$
$223$	278.596	1.24931	0.624656	+	0.780900i	$0.285239\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−278.596	−1.22730	−0.613649	−	0.789579i	$-0.710299\pi$
$227$	−278.596	−1.22730	−0.613649	+	0.789579i	$0.710299\pi$
$228$	0	0
$229$	159.198i	0.695188i	0.937645	+	0.347594i	$0.113001\pi$
$229$	159.198i	0.695188i	−0.937645	+	0.347594i	$0.886999\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 278.596i	− 1.19569i	−0.801611	−	0.597847i	$-0.796023\pi$
$233$	− 278.596i	− 1.19569i	0.801611	−	0.597847i	$-0.203977\pi$
$234$	0	0
$235$	36.0000i	0.153191i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	312.000i	1.30544i	0.757600	+	0.652720i	$0.226372\pi$
$239$	312.000i	1.30544i	−0.757600	+	0.652720i	$0.773628\pi$
$240$	0	0
$241$	−445.000	−1.84647	−0.923237	−	0.384232i	$-0.874466\pi$
$241$	−445.000	−1.84647	−0.923237	+	0.384232i	$0.874466\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−300.000	−1.22449
$246$	0	0
$247$	69.6491	0.281980
$248$	0	0
$249$	0	0
$250$	0	0
$251$	278.596	1.10995	0.554973	−	0.831868i	$-0.312729\pi$
$251$	278.596	1.10995	0.554973	+	0.831868i	$0.312729\pi$
$252$	0	0
$253$	835.789i	3.30352i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 278.596i	− 1.08403i	−0.840368	−	0.542017i	$-0.817661\pi$
$257$	− 278.596i	− 1.08403i	0.840368	−	0.542017i	$-0.182339\pi$
$258$	0	0
$259$	495.000i	1.91120i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 72.0000i	− 0.273764i	−0.990587	−	0.136882i	$-0.956292\pi$
$263$	− 72.0000i	− 0.273764i	0.990587	−	0.136882i	$-0.0437082\pi$
$264$	0	0
$265$	504.000	1.90189
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−126.000	−0.468401	−0.234201	−	0.972188i	$-0.575247\pi$
$269$	−126.000	−0.468401	−0.234201	+	0.972188i	$0.575247\pi$
$270$	0	0
$271$	89.5489	0.330439	0.165219	−	0.986257i	$-0.447167\pi$
$271$	89.5489	0.330439	0.165219	+	0.986257i	$0.447167\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−218.897	−0.795990
$276$	0	0
$277$	39.7995i	0.143680i	0.997416	+	0.0718402i	$0.0228872\pi$
$277$	39.7995i	0.143680i	−0.997416	+	0.0718402i	$0.977113\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	238.797i	0.849811i	0.905238	+	0.424906i	$0.139693\pi$
$281$	238.797i	0.849811i	−0.905238	+	0.424906i	$0.860307\pi$
$282$	0	0
$283$	− 242.000i	− 0.855124i	−0.903986	−	0.427562i	$-0.859373\pi$
$283$	− 242.000i	− 0.855124i	0.903986	−	0.427562i	$-0.140627\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 396.000i	− 1.37979i
$288$	0	0
$289$	−107.000	−0.370242
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−66.0000	−0.225256	−0.112628	−	0.993637i	$-0.535927\pi$
$293$	−66.0000	−0.225256	−0.112628	+	0.993637i	$0.535927\pi$
$294$	0	0
$295$	119.398	0.404741
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−417.895	−1.39764
$300$	0	0
$301$	− 497.494i	− 1.65280i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 537.293i	− 1.76162i
$306$	0	0
$307$	− 358.000i	− 1.16612i	−0.812428	−	0.583062i	$-0.801855\pi$
$307$	− 358.000i	− 1.16612i	0.812428	−	0.583062i	$-0.198145\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	366.000i	1.17685i	0.808552	+	0.588424i	$0.200252\pi$
$311$	366.000i	1.17685i	−0.808552	+	0.588424i	$0.799748\pi$
$312$	0	0
$313$	−203.000	−0.648562	−0.324281	−	0.945961i	$-0.605122\pi$
$313$	−203.000	−0.648562	−0.324281	+	0.945961i	$0.605122\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	288.000	0.908517	0.454259	−	0.890870i	$-0.349904\pi$
$317$	288.000	0.908517	0.454259	+	0.890870i	$0.349904\pi$
$318$	0	0
$319$	477.594	1.49716
$320$	0	0
$321$	0	0
$322$	0	0
$323$	139.298	0.431264
$324$	0	0
$325$	− 109.449i	− 0.336765i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	59.6992i	0.181457i
$330$	0	0
$331$	175.000i	0.528701i	0.964427	+	0.264350i	$0.0851576\pi$
$331$	175.000i	0.528701i	−0.964427	+	0.264350i	$0.914842\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 318.000i	− 0.949254i
$336$	0	0
$337$	−35.0000	−0.103858	−0.0519288	−	0.998651i	$-0.516537\pi$
$337$	−35.0000	−0.103858	−0.0519288	+	0.998651i	$0.516537\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	792.000	2.32258
$342$	0	0
$343$	−9.94987	−0.0290084
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−557.193	−1.60574	−0.802872	−	0.596152i	$-0.796696\pi$
$347$	−557.193	−1.60574	−0.802872	+	0.596152i	$0.796696\pi$
$348$	0	0
$349$	427.845i	1.22592i	0.790116	+	0.612958i	$0.210020\pi$
$349$	427.845i	1.22592i	−0.790116	+	0.612958i	$0.789980\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	557.193i	1.57845i	0.614104	+	0.789225i	$0.289518\pi$
$353$	557.193i	1.57845i	−0.614104	+	0.789225i	$0.710482\pi$
$354$	0	0
$355$	− 360.000i	− 1.01408i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	42.0000i	0.116992i	0.998288	+	0.0584958i	$0.0186304\pi$
$359$	42.0000i	0.116992i	−0.998288	+	0.0584958i	$0.981370\pi$
$360$	0	0
$361$	312.000	0.864266
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−714.000	−1.95616
$366$	0	0
$367$	208.947	0.569339	0.284669	−	0.958626i	$-0.408116\pi$
$367$	208.947	0.569339	0.284669	+	0.958626i	$0.408116\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	835.789	2.25280
$372$	0	0
$373$	348.246i	0.933634i	0.884354	+	0.466817i	$0.154599\pi$
$373$	348.246i	0.933634i	−0.884354	+	0.466817i	$0.845401\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	238.797i	0.633414i
$378$	0	0
$379$	− 43.0000i	− 0.113456i	−0.998390	−	0.0567282i	$-0.981933\pi$
$379$	− 43.0000i	− 0.113456i	0.998390	−	0.0567282i	$-0.0180669\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	420.000i	1.09661i	0.836280	+	0.548303i	$0.184726\pi$
$383$	420.000i	1.09661i	−0.836280	+	0.548303i	$0.815274\pi$
$384$	0	0
$385$	−1188.00	−3.08571
$386$	0	0
$387$	0	0
$388$	0	0
$389$	546.000	1.40360	0.701799	−	0.712375i	$-0.252380\pi$
$389$	546.000	1.40360	0.701799	+	0.712375i	$0.252380\pi$
$390$	0	0
$391$	−835.789	−2.13757
$392$	0	0
$393$	0	0
$394$	0	0
$395$	298.496	0.755687
$396$	0	0
$397$	− 676.591i	− 1.70426i	−0.523330	−	0.852130i	$-0.675310\pi$
$397$	− 676.591i	− 1.70426i	0.523330	−	0.852130i	$-0.324690\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	636.792i	1.58801i	0.607911	+	0.794005i	$0.292008\pi$
$401$	636.792i	1.58801i	−0.607911	+	0.794005i	$0.707992\pi$
$402$	0	0
$403$	396.000i	0.982630i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	990.000i	2.43243i
$408$	0	0
$409$	395.000	0.965770	0.482885	−	0.875684i	$-0.339589\pi$
$409$	395.000	0.965770	0.482885	+	0.875684i	$0.339589\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	198.000	0.479419
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	656.692	1.56728	0.783642	−	0.621213i	$-0.213360\pi$
$419$	656.692	1.56728	0.783642	+	0.621213i	$0.213360\pi$
$420$	0	0
$421$	− 766.140i	− 1.81981i	−0.414816	−	0.909905i	$-0.636154\pi$
$421$	− 766.140i	− 1.81981i	0.414816	−	0.909905i	$-0.363846\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 218.897i	− 0.515052i
$426$	0	0
$427$	− 891.000i	− 2.08665i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	726.000i	1.68445i	0.539122	+	0.842227i	$0.318756\pi$
$431$	726.000i	1.68445i	−0.539122	+	0.842227i	$0.681244\pi$
$432$	0	0
$433$	−470.000	−1.08545	−0.542725	−	0.839910i	$-0.682607\pi$
$433$	−470.000	−1.08545	−0.542725	+	0.839910i	$0.682607\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	294.000	0.672769
$438$	0	0
$439$	−835.789	−1.90385	−0.951924	−	0.306334i	$-0.900898\pi$
$439$	−835.789	−1.90385	−0.951924	+	0.306334i	$0.900898\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−278.596	−0.628886	−0.314443	−	0.949276i	$-0.601818\pi$
$443$	−278.596	−0.628886	−0.314443	+	0.949276i	$0.601818\pi$
$444$	0	0
$445$	358.195i	0.804934i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 99.4987i	− 0.221601i	−0.993843	−	0.110800i	$-0.964659\pi$
$449$	− 99.4987i	− 0.221601i	0.993843	−	0.110800i	$-0.0353414\pi$
$450$	0	0
$451$	− 792.000i	− 1.75610i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 594.000i	− 1.30549i
$456$	0	0
$457$	−418.000	−0.914661	−0.457330	−	0.889297i	$-0.651194\pi$
$457$	−418.000	−0.914661	−0.457330	+	0.889297i	$0.651194\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−270.000	−0.585683	−0.292842	−	0.956161i	$-0.594601\pi$
$461$	−270.000	−0.585683	−0.292842	+	0.956161i	$0.594601\pi$
$462$	0	0
$463$	−49.7494	−0.107450	−0.0537250	−	0.998556i	$-0.517109\pi$
$463$	−49.7494	−0.107450	−0.0537250	+	0.998556i	$0.517109\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−338.296	−0.724402	−0.362201	−	0.932100i	$-0.617975\pi$
$467$	−338.296	−0.724402	−0.362201	+	0.932100i	$0.617975\pi$
$468$	0	0
$469$	− 527.343i	− 1.12440i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 994.987i	− 2.10357i
$474$	0	0
$475$	77.0000i	0.162105i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 828.000i	− 1.72860i	−0.502976	−	0.864301i	$-0.667761\pi$
$479$	− 828.000i	− 1.72860i	0.502976	−	0.864301i	$-0.332239\pi$
$480$	0	0
$481$	−495.000	−1.02911
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−78.0000	−0.160825
$486$	0	0
$487$	427.845	0.878531	0.439266	−	0.898357i	$-0.355239\pi$
$487$	427.845	0.878531	0.439266	+	0.898357i	$0.355239\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	139.298	0.283703	0.141852	−	0.989888i	$-0.454694\pi$
$491$	139.298	0.283703	0.141852	+	0.989888i	$0.454694\pi$
$492$	0	0
$493$	477.594i	0.968750i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 596.992i	− 1.20119i
$498$	0	0
$499$	− 730.000i	− 1.46293i	−0.681881	−	0.731463i	$-0.738838\pi$
$499$	− 730.000i	− 1.46293i	0.681881	−	0.731463i	$-0.261162\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 546.000i	− 1.08549i	−0.839898	−	0.542744i	$-0.817386\pi$
$503$	− 546.000i	− 1.08549i	0.839898	−	0.542744i	$-0.182614\pi$
$504$	0	0
$505$	72.0000	0.142574
$506$	0	0
$507$	0	0
$508$	0	0
$509$	294.000	0.577603	0.288802	−	0.957389i	$-0.406743\pi$
$509$	294.000	0.577603	0.288802	+	0.957389i	$0.406743\pi$
$510$	0	0
$511$	−1184.04	−2.31709
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−537.293	−1.04329
$516$	0	0
$517$	119.398i	0.230945i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	696.491i	1.33684i	0.743786	+	0.668418i	$0.233028\pi$
$521$	696.491i	1.33684i	−0.743786	+	0.668418i	$0.766972\pi$
$522$	0	0
$523$	− 583.000i	− 1.11472i	−0.830270	−	0.557361i	$-0.811814\pi$
$523$	− 583.000i	− 1.11472i	0.830270	−	0.557361i	$-0.188186\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	792.000i	1.50285i
$528$	0	0
$529$	−1235.00	−2.33459
$530$	0	0
$531$	0	0
$532$	0	0
$533$	396.000	0.742964
$534$	0	0
$535$	835.789	1.56222
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−994.987	−1.84599
$540$	0	0
$541$	− 527.343i	− 0.974757i	−0.873191	−	0.487378i	$-0.837953\pi$
$541$	− 527.343i	− 0.974757i	0.873191	−	0.487378i	$-0.162047\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	238.797i	0.438160i
$546$	0	0
$547$	715.000i	1.30713i	0.756870	+	0.653565i	$0.226727\pi$
$547$	715.000i	1.30713i	−0.756870	+	0.653565i	$0.773273\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 168.000i	− 0.304900i
$552$	0	0
$553$	495.000	0.895118
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−702.000	−1.26032	−0.630162	−	0.776464i	$-0.717011\pi$
$557$	−702.000	−1.26032	−0.630162	+	0.776464i	$0.717011\pi$
$558$	0	0
$559$	497.494	0.889971
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−278.596	−0.494843	−0.247421	−	0.968908i	$-0.579583\pi$
$563$	−278.596	−0.494843	−0.247421	+	0.968908i	$0.579583\pi$
$564$	0	0
$565$	1074.59i	1.90192i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 338.296i	− 0.594544i	−0.954793	−	0.297272i	$-0.903923\pi$
$569$	− 338.296i	− 0.594544i	0.954793	−	0.297272i	$-0.0960769\pi$
$570$	0	0
$571$	− 665.000i	− 1.16462i	−0.812966	−	0.582312i	$-0.802148\pi$
$571$	− 665.000i	− 1.16462i	0.812966	−	0.582312i	$-0.197852\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 462.000i	− 0.803478i
$576$	0	0
$577$	−479.000	−0.830156	−0.415078	−	0.909786i	$-0.636246\pi$
$577$	−479.000	−0.830156	−0.415078	+	0.909786i	$0.636246\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1671.58	2.86720
$584$	0	0
$585$	0	0
$586$	0	0
$587$	457.694	0.779718	0.389859	−	0.920875i	$-0.372524\pi$
$587$	457.694	0.779718	0.389859	+	0.920875i	$0.372524\pi$
$588$	0	0
$589$	− 278.596i	− 0.472999i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	477.594i	0.805386i	0.915335	+	0.402693i	$0.131926\pi$
$593$	477.594i	0.805386i	−0.915335	+	0.402693i	$0.868074\pi$
$594$	0	0
$595$	− 1188.00i	− 1.99664i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 672.000i	− 1.12187i	−0.827860	−	0.560935i	$-0.810442\pi$
$599$	− 672.000i	− 1.12187i	0.827860	−	0.560935i	$-0.189558\pi$
$600$	0	0
$601$	−706.000	−1.17471	−0.587354	−	0.809330i	$-0.699831\pi$
$601$	−706.000	−1.17471	−0.587354	+	0.809330i	$0.699831\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1650.00	−2.72727
$606$	0	0
$607$	−487.544	−0.803202	−0.401601	−	0.915815i	$-0.631546\pi$
$607$	−487.544	−0.803202	−0.401601	+	0.915815i	$0.631546\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−59.6992	−0.0977074
$612$	0	0
$613$	447.744i	0.730415i	0.930926	+	0.365207i	$0.119002\pi$
$613$	447.744i	0.730415i	−0.930926	+	0.365207i	$0.880998\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 99.4987i	− 0.161262i	−0.996744	−	0.0806311i	$-0.974306\pi$
$617$	− 99.4987i	− 0.161262i	0.996744	−	0.0806311i	$-0.0256936\pi$
$618$	0	0
$619$	− 437.000i	− 0.705977i	−0.935628	−	0.352989i	$-0.885165\pi$
$619$	− 437.000i	− 0.705977i	0.935628	−	0.352989i	$-0.114835\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	594.000i	0.953451i
$624$	0	0
$625$	−779.000	−1.24640
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−990.000	−1.57393
$630$	0	0
$631$	129.348	0.204989	0.102495	−	0.994734i	$-0.467318\pi$
$631$	129.348	0.204989	0.102495	+	0.994734i	$0.467318\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	477.594	0.752116
$636$	0	0
$637$	− 497.494i	− 0.780995i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 994.987i	− 1.55224i	−0.630584	−	0.776121i	$-0.717184\pi$
$641$	− 994.987i	− 1.55224i	0.630584	−	0.776121i	$-0.282816\pi$
$642$	0	0
$643$	− 314.000i	− 0.488336i	−0.969733	−	0.244168i	$-0.921485\pi$
$643$	− 314.000i	− 0.488336i	0.969733	−	0.244168i	$-0.0785148\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	420.000i	0.649150i	0.945860	+	0.324575i	$0.105221\pi$
$647$	420.000i	0.649150i	−0.945860	+	0.324575i	$0.894779\pi$
$648$	0	0
$649$	396.000	0.610169
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1008.00	1.54364	0.771822	−	0.635838i	$-0.219345\pi$
$653$	1008.00	1.54364	0.771822	+	0.635838i	$0.219345\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	596.992	0.905907	0.452953	−	0.891534i	$-0.350370\pi$
$659$	596.992	0.905907	0.452953	+	0.891534i	$0.350370\pi$
$660$	0	0
$661$	208.947i	0.316108i	0.987430	+	0.158054i	$0.0505220\pi$
$661$	208.947i	0.316108i	−0.987430	+	0.158054i	$0.949478\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	417.895i	0.628413i
$666$	0	0
$667$	1008.00i	1.51124i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 1782.00i	− 2.65574i
$672$	0	0
$673$	−1211.00	−1.79941	−0.899703	−	0.436503i	$-0.856217\pi$
$673$	−1211.00	−1.79941	−0.899703	+	0.436503i	$0.856217\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−294.000	−0.434269	−0.217134	−	0.976142i	$-0.569671\pi$
$677$	−294.000	−0.434269	−0.217134	+	0.976142i	$0.569671\pi$
$678$	0	0
$679$	−129.348	−0.190498
$680$	0	0
$681$	0	0
$682$	0	0
$683$	676.591	0.990617	0.495309	−	0.868717i	$-0.335055\pi$
$683$	676.591	0.990617	0.495309	+	0.868717i	$0.335055\pi$
$684$	0	0
$685$	835.789i	1.22013i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	835.789i	1.21305i
$690$	0	0
$691$	− 194.000i	− 0.280753i	−0.990098	−	0.140376i	$-0.955169\pi$
$691$	− 194.000i	− 0.280753i	0.990098	−	0.140376i	$-0.0448312\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 330.000i	− 0.474820i
$696$	0	0
$697$	792.000	1.13630
$698$	0	0
$699$	0	0
$700$	0	0
$701$	378.000	0.539230	0.269615	−	0.962968i	$-0.413104\pi$
$701$	378.000	0.539230	0.269615	+	0.962968i	$0.413104\pi$
$702$	0	0
$703$	348.246	0.495371
$704$	0	0
$705$	0	0
$706$	0	0
$707$	119.398	0.168880
$708$	0	0
$709$	467.644i	0.659583i	0.944054	+	0.329791i	$0.106978\pi$
$709$	467.644i	0.659583i	−0.944054	+	0.329791i	$0.893022\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1671.58i	2.34443i
$714$	0	0
$715$	− 1188.00i	− 1.66154i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	216.000i	0.300417i	0.988654	+	0.150209i	$0.0479945\pi$
$719$	216.000i	0.300417i	−0.988654	+	0.150209i	$0.952005\pi$
$720$	0	0
$721$	−891.000	−1.23578
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−264.000	−0.364138
$726$	0	0
$727$	−358.195	−0.492704	−0.246352	−	0.969180i	$-0.579232\pi$
$727$	−358.195	−0.492704	−0.246352	+	0.969180i	$0.579232\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	994.987	1.36113
$732$	0	0
$733$	676.591i	0.923044i	0.887129	+	0.461522i	$0.152697\pi$
$733$	676.591i	0.923044i	−0.887129	+	0.461522i	$0.847303\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1054.69i	− 1.43105i
$738$	0	0
$739$	− 934.000i	− 1.26387i	−0.775021	−	0.631935i	$-0.782261\pi$
$739$	− 934.000i	− 1.26387i	0.775021	−	0.631935i	$-0.217739\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 66.0000i	− 0.0888291i	−0.999013	−	0.0444145i	$-0.985858\pi$
$743$	− 66.0000i	− 0.0888291i	0.999013	−	0.0444145i	$-0.0141422\pi$
$744$	0	0
$745$	216.000	0.289933
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1386.00	1.85047
$750$	0	0
$751$	−487.544	−0.649193	−0.324596	−	0.945853i	$-0.605228\pi$
$751$	−487.544	−0.649193	−0.324596	+	0.945853i	$0.605228\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	179.098	0.237216
$756$	0	0
$757$	149.248i	0.197157i	0.995129	+	0.0985787i	$0.0314296\pi$
$757$	149.248i	0.197157i	−0.995129	+	0.0985787i	$0.968570\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 417.895i	− 0.549139i	−0.961567	−	0.274569i	$-0.911465\pi$
$761$	− 417.895i	− 0.549139i	0.961567	−	0.274569i	$-0.0885353\pi$
$762$	0	0
$763$	396.000i	0.519004i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	198.000i	0.258149i
$768$	0	0
$769$	215.000	0.279584	0.139792	−	0.990181i	$-0.455357\pi$
$769$	215.000	0.279584	0.139792	+	0.990181i	$0.455357\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	168.000	0.217335	0.108668	−	0.994078i	$-0.465342\pi$
$773$	168.000	0.217335	0.108668	+	0.994078i	$0.465342\pi$
$774$	0	0
$775$	−437.794	−0.564896
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−278.596	−0.357633
$780$	0	0
$781$	− 1193.98i	− 1.52879i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1193.98i	1.52100i
$786$	0	0
$787$	487.000i	0.618806i	0.950931	+	0.309403i	$0.100129\pi$
$787$	487.000i	0.618806i	−0.950931	+	0.309403i	$0.899871\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1782.00i	2.25284i
$792$	0	0
$793$	891.000	1.12358
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−60.0000	−0.0752823	−0.0376412	−	0.999291i	$-0.511984\pi$
$797$	−60.0000	−0.0752823	−0.0376412	+	0.999291i	$0.511984\pi$
$798$	0	0
$799$	−119.398	−0.149435
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−2368.07	−2.94903
$804$	0	0
$805$	− 2507.37i	− 3.11474i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 1392.98i	− 1.72186i	−0.508726	−	0.860929i	$-0.669883\pi$
$809$	− 1392.98i	− 1.72186i	0.508726	−	0.860929i	$-0.330117\pi$
$810$	0	0
$811$	− 70.0000i	− 0.0863132i	−0.999068	−	0.0431566i	$-0.986259\pi$
$811$	− 70.0000i	− 0.0863132i	0.999068	−	0.0431566i	$-0.0137414\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 474.000i	− 0.581595i
$816$	0	0
$817$	−350.000	−0.428397
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−42.0000	−0.0511571	−0.0255786	−	0.999673i	$-0.508143\pi$
$821$	−42.0000	−0.0511571	−0.0255786	+	0.999673i	$0.508143\pi$
$822$	0	0
$823$	169.148	0.205526	0.102763	−	0.994706i	$-0.467232\pi$
$823$	169.148	0.205526	0.102763	+	0.994706i	$0.467232\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	417.895	0.505314	0.252657	−	0.967556i	$-0.418696\pi$
$827$	417.895	0.505314	0.252657	+	0.967556i	$0.418696\pi$
$828$	0	0
$829$	− 348.246i	− 0.420079i	−0.977693	−	0.210040i	$-0.932641\pi$
$829$	− 348.246i	− 0.420079i	0.977693	−	0.210040i	$-0.0673593\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 994.987i	− 1.19446i
$834$	0	0
$835$	1260.00i	1.50898i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1092.00i	1.30155i	0.759271	+	0.650775i	$0.225556\pi$
$839$	1092.00i	1.30155i	−0.759271	+	0.650775i	$0.774444\pi$
$840$	0	0
$841$	−265.000	−0.315101
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−420.000	−0.497041
$846$	0	0
$847$	−2736.22	−3.23048
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−2089.47	−2.45532
$852$	0	0
$853$	− 1044.74i	− 1.22478i	−0.790556	−	0.612390i	$-0.790208\pi$
$853$	− 1044.74i	− 1.22478i	0.790556	−	0.612390i	$-0.209792\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	119.398i	0.139321i	0.997571	+	0.0696607i	$0.0221917\pi$
$857$	119.398i	0.139321i	−0.997571	+	0.0696607i	$0.977808\pi$
$858$	0	0
$859$	485.000i	0.564610i	0.959325	+	0.282305i	$0.0910990\pi$
$859$	485.000i	0.564610i	−0.959325	+	0.282305i	$0.908901\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1050.00i	1.21669i	0.793674	+	0.608343i	$0.208165\pi$
$863$	1050.00i	1.21669i	−0.793674	+	0.608343i	$0.791835\pi$
$864$	0	0
$865$	−504.000	−0.582659
$866$	0	0
$867$	0	0
$868$	0	0
$869$	990.000	1.13924
$870$	0	0
$871$	527.343	0.605446
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−835.789	−0.955188
$876$	0	0
$877$	1024.84i	1.16857i	0.811548	+	0.584286i	$0.198625\pi$
$877$	1024.84i	1.16857i	−0.811548	+	0.584286i	$0.801375\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 417.895i	− 0.474341i	−0.971468	−	0.237171i	$-0.923780\pi$
$881$	− 417.895i	− 0.474341i	0.971468	−	0.237171i	$-0.0762201\pi$
$882$	0	0
$883$	− 245.000i	− 0.277463i	−0.990330	−	0.138732i	$-0.955697\pi$
$883$	− 245.000i	− 0.277463i	0.990330	−	0.138732i	$-0.0443025\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	504.000i	0.568207i	0.958794	+	0.284104i	$0.0916960\pi$
$887$	504.000i	0.568207i	−0.958794	+	0.284104i	$0.908304\pi$
$888$	0	0
$889$	792.000	0.890889
$890$	0	0
$891$	0	0
$892$	0	0
$893$	42.0000	0.0470325
$894$	0	0
$895$	716.391	0.800437
$896$	0	0
$897$	0	0
$898$	0	0
$899$	955.188	1.06250
$900$	0	0
$901$	1671.58i	1.85525i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	417.895i	0.461762i
$906$	0	0
$907$	− 1039.00i	− 1.14553i	−0.819718	−	0.572767i	$-0.805870\pi$
$907$	− 1039.00i	− 1.14553i	0.819718	−	0.572767i	$-0.194130\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	444.000i	0.487377i	0.969854	+	0.243688i	$0.0783574\pi$
$911$	444.000i	0.487377i	−0.969854	+	0.243688i	$0.921643\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−119.398	−0.129922	−0.0649611	−	0.997888i	$-0.520692\pi$
$919$	−119.398	−0.129922	−0.0649611	+	0.997888i	$0.520692\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	596.992	0.646796
$924$	0	0
$925$	− 547.243i	− 0.591614i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 477.594i	− 0.514095i	−0.966399	−	0.257047i	$-0.917250\pi$
$929$	− 477.594i	− 0.514095i	0.966399	−	0.257047i	$-0.0827496\pi$
$930$	0	0
$931$	350.000i	0.375940i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 2376.00i	− 2.54118i
$936$	0	0
$937$	925.000	0.987193	0.493597	−	0.869691i	$-0.335682\pi$
$937$	925.000	0.987193	0.493597	+	0.869691i	$0.335682\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1650.00	1.75345	0.876727	−	0.480989i	$-0.159722\pi$
$941$	1650.00	1.75345	0.876727	+	0.480989i	$0.159722\pi$
$942$	0	0
$943$	1671.58	1.77262
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1333.28	1.40790	0.703951	−	0.710249i	$-0.251417\pi$
$947$	1333.28	1.40790	0.703951	+	0.710249i	$0.251417\pi$
$948$	0	0
$949$	− 1184.04i	− 1.24767i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	139.298i	0.146168i	0.997326	+	0.0730841i	$0.0232841\pi$
$953$	139.298i	0.146168i	−0.997326	+	0.0730841i	$0.976716\pi$
$954$	0	0
$955$	1476.00i	1.54555i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1386.00i	1.44526i
$960$	0	0
$961$	623.000	0.648283
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1074.00	−1.11295
$966$	0	0
$967$	268.647	0.277814	0.138907	−	0.990305i	$-0.455641\pi$
$967$	268.647	0.277814	0.138907	+	0.990305i	$0.455641\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−417.895	−0.430376	−0.215188	−	0.976573i	$-0.569036\pi$
$971$	−417.895	−0.430376	−0.215188	+	0.976573i	$0.569036\pi$
$972$	0	0
$973$	− 547.243i	− 0.562429i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1114.39i	1.14062i	0.821429	+	0.570310i	$0.193177\pi$
$977$	1114.39i	1.14062i	−0.821429	+	0.570310i	$0.806823\pi$
$978$	0	0
$979$	1188.00i	1.21348i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1206.00i	− 1.22686i	−0.789750	−	0.613428i	$-0.789790\pi$
$983$	− 1206.00i	− 1.22686i	0.789750	−	0.613428i	$-0.210210\pi$
$984$	0	0
$985$	756.000	0.767513
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2100.00	2.12336
$990$	0	0
$991$	447.744	0.451811	0.225905	−	0.974149i	$-0.427466\pi$
$991$	447.744	0.451811	0.225905	+	0.974149i	$0.427466\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1850.68	1.85998
$996$	0	0
$997$	596.992i	0.598789i	0.954129	+	0.299394i	$0.0967846\pi$
$997$	596.992i	0.598789i	−0.954129	+	0.299394i	$0.903215\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.h.a.161.1	✓	4
3.2	odd	2		1728.3.h.f.161.1	yes	4
4.3	odd	2	inner	1728.3.h.a.161.3	yes	4
8.3	odd	2		1728.3.h.f.161.4	yes	4
8.5	even	2		1728.3.h.f.161.2	yes	4
12.11	even	2		1728.3.h.f.161.3	yes	4
24.5	odd	2	inner	1728.3.h.a.161.2	yes	4
24.11	even	2	inner	1728.3.h.a.161.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1728.3.h.a.161.1	✓	4	1.1	even	1	trivial
1728.3.h.a.161.2	yes	4	24.5	odd	2	inner
1728.3.h.a.161.3	yes	4	4.3	odd	2	inner
1728.3.h.a.161.4	yes	4	24.11	even	2	inner
1728.3.h.f.161.1	yes	4	3.2	odd	2
1728.3.h.f.161.2	yes	4	8.5	even	2
1728.3.h.f.161.3	yes	4	12.11	even	2
1728.3.h.f.161.4	yes	4	8.3	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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-6.00000 q^{5} -9.94987 q^{7} +O(q^{10})\)	\(q-6.00000 q^{5} -9.94987 q^{7} -19.8997 q^{11} -9.94987i q^{13} -19.8997i q^{17} +7.00000i q^{19} -42.0000i q^{23} +11.0000 q^{25} -24.0000 q^{29} -39.7995 q^{31} +59.6992 q^{35} -49.7494i q^{37} +39.7995i q^{41} +50.0000i q^{43} -6.00000i q^{47} +50.0000 q^{49} -84.0000 q^{53} +119.398 q^{55} -19.8997 q^{59} +89.5489i q^{61} +59.6992i q^{65} +53.0000i q^{67} +60.0000i q^{71} +119.000 q^{73} +198.000 q^{77} -49.7494 q^{79} +119.398i q^{85} -59.6992i q^{89} +99.0000i q^{91} -42.0000i q^{95} +13.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 24 q^{5}+O(q^{10}) \) 4 * q - 24 * q^5	\( 4 q - 24 q^{5} + 44 q^{25} - 96 q^{29} + 200 q^{49} - 336 q^{53} + 476 q^{73} + 792 q^{77} + 52 q^{97}+O(q^{100}) \) 4 * q - 24 * q^5 + 44 * q^25 - 96 * q^29 + 200 * q^49 - 336 * q^53 + 476 * q^73 + 792 * q^77 + 52 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more