LMFDB - Embedded newform 216.2.d.a.109.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [216,2,Mod(109,216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(216, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("216.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 216 = 2^{3} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	216.d (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:	$1.72476868366$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			109.3
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	216.109
Dual form			216.2.d.a.109.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -2.00000 q^{4} -4.41421i q^{5} +3.24264 q^{7} -2.82843i q^{8} +O(q^{10})$	$q+1.41421i q^{2} -2.00000 q^{4} -4.41421i q^{5} +3.24264 q^{7} -2.82843i q^{8} +6.24264 q^{10} -0.171573i q^{11} +4.58579i q^{14} +4.00000 q^{16} +8.82843i q^{20} +0.242641 q^{22} -14.4853 q^{25} -6.48528 q^{28} +2.82843i q^{29} +9.24264 q^{31} +5.65685i q^{32} -14.3137i q^{35} -12.4853 q^{40} +0.343146i q^{44} +3.51472 q^{49} -20.4853i q^{50} +4.07107i q^{53} -0.757359 q^{55} -9.17157i q^{56} -4.00000 q^{58} +11.3137i q^{59} +13.0711i q^{62} -8.00000 q^{64} +20.2426 q^{70} -15.4853 q^{73} -0.556349i q^{77} -10.0000 q^{79} -17.6569i q^{80} +17.8284i q^{83} -0.485281 q^{88} +15.9706 q^{97} +4.97056i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4} - 4 q^{7}+O(q^{10}) $ 4 * q - 8 * q^4 - 4 * q^7	$ 4 q - 8 q^{4} - 4 q^{7} + 8 q^{10} + 16 q^{16} - 16 q^{22} - 24 q^{25} + 8 q^{28} + 20 q^{31} - 16 q^{40} + 48 q^{49} - 20 q^{55} - 16 q^{58} - 32 q^{64} + 64 q^{70} - 28 q^{73} - 40 q^{79} + 32 q^{88} - 4 q^{97}+O(q^{100}) $ 4 * q - 8 * q^4 - 4 * q^7 + 8 * q^10 + 16 * q^16 - 16 * q^22 - 24 * q^25 + 8 * q^28 + 20 * q^31 - 16 * q^40 + 48 * q^49 - 20 * q^55 - 16 * q^58 - 32 * q^64 + 64 * q^70 - 28 * q^73 - 40 * q^79 + 32 * q^88 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/216\mathbb{Z}\right)^\times$.

$n$	$55$	$109$	$137$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	1.41421i	1.00000i
$2$	1.41421i	1.00000i
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	− 4.41421i	− 1.97410i	−0.160424	−	0.987048i	$-0.551286\pi$
$5$	− 4.41421i	− 1.97410i	0.160424	−	0.987048i	$-0.448714\pi$
$6$	0	0
$7$	3.24264	1.22560	0.612801	−	0.790237i	$-0.290043\pi$
$7$	3.24264	1.22560	0.612801	+	0.790237i	$0.290043\pi$
$8$	− 2.82843i	− 1.00000i
$9$	0	0
$10$	6.24264	1.97410
$11$	− 0.171573i	− 0.0517312i	−0.999665	−	0.0258656i	$-0.991766\pi$
$11$	− 0.171573i	− 0.0517312i	0.999665	−	0.0258656i	$-0.00823419\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	4.58579i	1.22560i
$15$	0	0
$16$	4.00000	1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	8.82843i	1.97410i
$21$	0	0
$22$	0.242641	0.0517312
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−14.4853	−2.89706
$26$	0	0
$27$	0	0
$28$	−6.48528	−1.22560
$29$	2.82843i	0.525226i	0.964901	+	0.262613i	$0.0845842\pi$
$29$	2.82843i	0.525226i	−0.964901	+	0.262613i	$0.915416\pi$
$30$	0	0
$31$	9.24264	1.66003	0.830014	−	0.557743i	$-0.188333\pi$
$31$	9.24264	1.66003	0.830014	+	0.557743i	$0.188333\pi$
$32$	5.65685i	1.00000i
$33$	0	0
$34$	0	0
$35$	− 14.3137i	− 2.41946i
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	−12.4853	−1.97410
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0.343146i	0.0517312i
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	3.51472	0.502103
$50$	− 20.4853i	− 2.89706i
$51$	0	0
$52$	0	0
$53$	4.07107i	0.559204i	0.960116	+	0.279602i	$0.0902025\pi$
$53$	4.07107i	0.559204i	−0.960116	+	0.279602i	$0.909797\pi$
$54$	0	0
$55$	−0.757359	−0.102122
$56$	− 9.17157i	− 1.22560i
$57$	0	0
$58$	−4.00000	−0.525226
$59$	11.3137i	1.47292i	0.676481	+	0.736460i	$0.263504\pi$
$59$	11.3137i	1.47292i	−0.676481	+	0.736460i	$0.736496\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	13.0711i	1.66003i
$63$	0	0
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	20.2426	2.41946
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−15.4853	−1.81242	−0.906208	−	0.422833i	$-0.861036\pi$
$73$	−15.4853	−1.81242	−0.906208	+	0.422833i	$0.861036\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 0.556349i	− 0.0634019i
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	− 17.6569i	− 1.97410i
$81$	0	0
$82$	0	0
$83$	17.8284i	1.95692i	0.206427	+	0.978462i	$0.433816\pi$
$83$	17.8284i	1.95692i	−0.206427	+	0.978462i	$0.566184\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	−0.485281	−0.0517312
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	15.9706	1.62156	0.810782	−	0.585348i	$-0.199042\pi$
$97$	15.9706	1.62156	0.810782	+	0.585348i	$0.199042\pi$
$98$	4.97056i	0.502103i
$99$	0	0
$100$	28.9706	2.89706
$101$	− 12.8995i	− 1.28355i	−0.766894	−	0.641774i	$-0.778199\pi$
$101$	− 12.8995i	− 1.28355i	0.766894	−	0.641774i	$-0.221801\pi$
$102$	0	0
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	0	0
$105$	0	0
$106$	−5.75736	−0.559204
$107$	9.34315i	0.903236i	0.892211	+	0.451618i	$0.149153\pi$
$107$	9.34315i	0.903236i	−0.892211	+	0.451618i	$0.850847\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	− 1.07107i	− 0.102122i
$111$	0	0
$112$	12.9706	1.22560
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	− 5.65685i	− 0.525226i
$117$	0	0
$118$	−16.0000	−1.47292
$119$	0	0
$120$	0	0
$121$	10.9706	0.997324
$122$	0	0
$123$	0	0
$124$	−18.4853	−1.66003
$125$	41.8701i	3.74497i
$126$	0	0
$127$	15.2426	1.35257	0.676283	−	0.736642i	$-0.263590\pi$
$127$	15.2426	1.35257	0.676283	+	0.736642i	$0.263590\pi$
$128$	− 11.3137i	− 1.00000i
$129$	0	0
$130$	0	0
$131$	8.31371i	0.726372i	0.931717	+	0.363186i	$0.118311\pi$
$131$	8.31371i	0.726372i	−0.931717	+	0.363186i	$0.881689\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	28.6274i	2.41946i
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	12.4853	1.03685
$146$	− 21.8995i	− 1.81242i
$147$	0	0
$148$	0	0
$149$	− 22.4142i	− 1.83624i	−0.396298	−	0.918122i	$-0.629705\pi$
$149$	− 22.4142i	− 1.83624i	0.396298	−	0.918122i	$-0.370295\pi$
$150$	0	0
$151$	−22.2132	−1.80768	−0.903842	−	0.427865i	$-0.859266\pi$
$151$	−22.2132	−1.80768	−0.903842	+	0.427865i	$0.859266\pi$
$152$	0	0
$153$	0	0
$154$	0.786797	0.0634019
$155$	− 40.7990i	− 3.27705i
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	− 14.1421i	− 1.12509i
$159$	0	0
$160$	24.9706	1.97410
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	−25.2132	−1.95692
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	5.10051i	0.387784i	0.981023	+	0.193892i	$0.0621112\pi$
$173$	5.10051i	0.387784i	−0.981023	+	0.193892i	$0.937889\pi$
$174$	0	0
$175$	−46.9706	−3.55064
$176$	− 0.686292i	− 0.0517312i
$177$	0	0
$178$	0	0
$179$	− 26.6569i	− 1.99243i	−0.0869415	−	0.996213i	$-0.527709\pi$
$179$	− 26.6569i	− 1.99243i	0.0869415	−	0.996213i	$-0.472291\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−21.4853	−1.54654	−0.773272	−	0.634074i	$-0.781381\pi$
$193$	−21.4853	−1.54654	−0.773272	+	0.634074i	$0.781381\pi$
$194$	22.5858i	1.62156i
$195$	0	0
$196$	−7.02944	−0.502103
$197$	− 13.9289i	− 0.992395i	−0.868210	−	0.496198i	$-0.834729\pi$
$197$	− 13.9289i	− 0.992395i	0.868210	−	0.496198i	$-0.165271\pi$
$198$	0	0
$199$	−28.2132	−1.99998	−0.999990	−	0.00436292i	$-0.998611\pi$
$199$	−28.2132	−1.99998	−0.999990	+	0.00436292i	$0.998611\pi$
$200$	40.9706i	2.89706i
$201$	0	0
$202$	18.2426	1.28355
$203$	9.17157i	0.643718i
$204$	0	0
$205$	0	0
$206$	19.7990i	1.37946i
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	− 8.14214i	− 0.559204i
$213$	0	0
$214$	−13.2132	−0.903236
$215$	0	0
$216$	0	0
$217$	29.9706	2.03453
$218$	0	0
$219$	0	0
$220$	1.51472	0.102122
$221$	0	0
$222$	0	0
$223$	26.0000	1.74109	0.870544	−	0.492090i	$-0.163767\pi$
$223$	26.0000	1.74109	0.870544	+	0.492090i	$0.163767\pi$
$224$	18.3431i	1.22560i
$225$	0	0
$226$	0	0
$227$	28.2843i	1.87729i	0.344881	+	0.938647i	$0.387919\pi$
$227$	28.2843i	1.87729i	−0.344881	+	0.938647i	$0.612081\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	8.00000	0.525226
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	0	0
$236$	− 22.6274i	− 1.47292i
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	15.5147i	0.997324i
$243$	0	0
$244$	0	0
$245$	− 15.5147i	− 0.991199i
$246$	0	0
$247$	0	0
$248$	− 26.1421i	− 1.66003i
$249$	0	0
$250$	−59.2132	−3.74497
$251$	− 5.65685i	− 0.357057i	−0.983935	−	0.178529i	$-0.942866\pi$
$251$	− 5.65685i	− 0.357057i	0.983935	−	0.178529i	$-0.0571337\pi$
$252$	0	0
$253$	0	0
$254$	21.5563i	1.35257i
$255$	0	0
$256$	16.0000	1.00000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	−11.7574	−0.726372
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	17.9706	1.10392
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 31.1127i	− 1.89697i	−0.316815	−	0.948487i	$-0.602613\pi$
$269$	− 31.1127i	− 1.89697i	0.316815	−	0.948487i	$-0.397387\pi$
$270$	0	0
$271$	−10.2132	−0.620408	−0.310204	−	0.950670i	$-0.600397\pi$
$271$	−10.2132	−0.620408	−0.310204	+	0.950670i	$0.600397\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.48528i	0.149868i
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	−40.4853	−2.41946
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	17.6569i	1.03685i
$291$	0	0
$292$	30.9706	1.81242
$293$	− 14.1421i	− 0.826192i	−0.910687	−	0.413096i	$-0.864447\pi$
$293$	− 14.1421i	− 0.826192i	0.910687	−	0.413096i	$-0.135553\pi$
$294$	0	0
$295$	49.9411	2.90768
$296$	0	0
$297$	0	0
$298$	31.6985	1.83624
$299$	0	0
$300$	0	0
$301$	0	0
$302$	− 31.4142i	− 1.80768i
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	1.11270i	0.0634019i
$309$	0	0
$310$	57.6985	3.27705
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	8.51472	0.481280	0.240640	−	0.970614i	$-0.422643\pi$
$313$	8.51472	0.481280	0.240640	+	0.970614i	$0.422643\pi$
$314$	0	0
$315$	0	0
$316$	20.0000	1.12509
$317$	30.5563i	1.71622i	0.513470	+	0.858108i	$0.328360\pi$
$317$	30.5563i	1.71622i	−0.513470	+	0.858108i	$0.671640\pi$
$318$	0	0
$319$	0.485281	0.0271705
$320$	35.3137i	1.97410i
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	− 35.6569i	− 1.95692i
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	18.3848i	1.00000i
$339$	0	0
$340$	0	0
$341$	− 1.58579i	− 0.0858752i
$342$	0	0
$343$	−11.3015	−0.610224
$344$	0	0
$345$	0	0
$346$	−7.21320	−0.387784
$347$	− 35.1421i	− 1.88653i	−0.332043	−	0.943264i	$-0.607738\pi$
$347$	− 35.1421i	− 1.88653i	0.332043	−	0.943264i	$-0.392262\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	− 66.4264i	− 3.55064i
$351$	0	0
$352$	0.970563	0.0517312
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	37.6985	1.99243
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	68.3553i	3.57788i
$366$	0	0
$367$	−14.7574	−0.770328	−0.385164	−	0.922848i	$-0.625855\pi$
$367$	−14.7574	−0.770328	−0.385164	+	0.922848i	$0.625855\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	13.2010i	0.685362i
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	−2.45584	−0.125161
$386$	− 30.3848i	− 1.54654i
$387$	0	0
$388$	−31.9411	−1.62156
$389$	− 5.44365i	− 0.276004i	−0.990432	−	0.138002i	$-0.955932\pi$
$389$	− 5.44365i	− 0.276004i	0.990432	−	0.138002i	$-0.0440680\pi$
$390$	0	0
$391$	0	0
$392$	− 9.94113i	− 0.502103i
$393$	0	0
$394$	19.6985	0.992395
$395$	44.1421i	2.22103i
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	− 39.8995i	− 1.99998i
$399$	0	0
$400$	−57.9411	−2.89706
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	0	0
$404$	25.7990i	1.28355i
$405$	0	0
$406$	−12.9706	−0.643718
$407$	0	0
$408$	0	0
$409$	−28.9411	−1.43105	−0.715523	−	0.698589i	$-0.753812\pi$
$409$	−28.9411	−1.43105	−0.715523	+	0.698589i	$0.753812\pi$
$410$	0	0
$411$	0	0
$412$	−28.0000	−1.37946
$413$	36.6863i	1.80521i
$414$	0	0
$415$	78.6985	3.86316
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 39.5980i	− 1.93449i	−0.253849	−	0.967244i	$-0.581697\pi$
$419$	− 39.5980i	− 1.93449i	0.253849	−	0.967244i	$-0.418303\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	11.5147	0.559204
$425$	0	0
$426$	0	0
$427$	0	0
$428$	− 18.6863i	− 0.903236i
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	−40.9411	−1.96750	−0.983752	−	0.179530i	$-0.942542\pi$
$433$	−40.9411	−1.96750	−0.983752	+	0.179530i	$0.942542\pi$
$434$	42.3848i	2.03453i
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−4.21320	−0.201085	−0.100543	−	0.994933i	$-0.532058\pi$
$439$	−4.21320	−0.201085	−0.100543	+	0.994933i	$0.532058\pi$
$440$	2.14214i	0.102122i
$441$	0	0
$442$	0	0
$443$	28.2843i	1.34383i	0.740630	+	0.671913i	$0.234527\pi$
$443$	28.2843i	1.34383i	−0.740630	+	0.671913i	$0.765473\pi$
$444$	0	0
$445$	0	0
$446$	36.7696i	1.74109i
$447$	0	0
$448$	−25.9411	−1.22560
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	−40.0000	−1.87729
$455$	0	0
$456$	0	0
$457$	−2.02944	−0.0949331	−0.0474665	−	0.998873i	$-0.515115\pi$
$457$	−2.02944	−0.0949331	−0.0474665	+	0.998873i	$0.515115\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	23.1005i	1.07590i	0.842977	+	0.537949i	$0.180801\pi$
$461$	23.1005i	1.07590i	−0.842977	+	0.537949i	$0.819199\pi$
$462$	0	0
$463$	16.6985	0.776044	0.388022	−	0.921650i	$-0.373158\pi$
$463$	16.6985	0.776044	0.388022	+	0.921650i	$0.373158\pi$
$464$	11.3137i	0.525226i
$465$	0	0
$466$	0	0
$467$	34.7990i	1.61031i	0.593068	+	0.805153i	$0.297917\pi$
$467$	34.7990i	1.61031i	−0.593068	+	0.805153i	$0.702083\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	32.0000	1.47292
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	− 14.1421i	− 0.644157i
$483$	0	0
$484$	−21.9411	−0.997324
$485$	− 70.4975i	− 3.20113i
$486$	0	0
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	0	0
$489$	0	0
$490$	21.9411	0.991199
$491$	44.3137i	1.99985i	0.0122607	+	0.999925i	$0.496097\pi$
$491$	44.3137i	1.99985i	−0.0122607	+	0.999925i	$0.503903\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	36.9706	1.66003
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	− 83.7401i	− 3.74497i
$501$	0	0
$502$	8.00000	0.357057
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	−56.9411	−2.53385
$506$	0	0
$507$	0	0
$508$	−30.4853	−1.35257
$509$	− 40.4142i	− 1.79133i	−0.444731	−	0.895664i	$-0.646701\pi$
$509$	− 40.4142i	− 1.79133i	0.444731	−	0.895664i	$-0.353299\pi$
$510$	0	0
$511$	−50.2132	−2.22130
$512$	22.6274i	1.00000i
$513$	0	0
$514$	0	0
$515$	− 61.7990i	− 2.72319i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	− 16.6274i	− 0.726372i
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−23.0000	−1.00000
$530$	25.4142i	1.10392i
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	41.2426	1.78307
$536$	0	0
$537$	0	0
$538$	44.0000	1.89697
$539$	− 0.603030i	− 0.0259744i
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	− 14.4437i	− 0.620408i
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	−3.51472	−0.149868
$551$	0	0
$552$	0	0
$553$	−32.4264	−1.37891
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 31.9289i	− 1.35287i	−0.736501	−	0.676436i	$-0.763523\pi$
$557$	− 31.9289i	− 1.35287i	0.736501	−	0.676436i	$-0.236477\pi$
$558$	0	0
$559$	0	0
$560$	− 57.2548i	− 2.41946i
$561$	0	0
$562$	0	0
$563$	18.8579i	0.794764i	0.917653	+	0.397382i	$0.130081\pi$
$563$	18.8579i	0.794764i	−0.917653	+	0.397382i	$0.869919\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	− 24.0416i	− 1.00000i
$579$	0	0
$580$	−24.9706	−1.03685
$581$	57.8112i	2.39841i
$582$	0	0
$583$	0.698485	0.0289283
$584$	43.7990i	1.81242i
$585$	0	0
$586$	20.0000	0.826192
$587$	− 7.62742i	− 0.314817i	−0.987534	−	0.157409i	$-0.949686\pi$
$587$	− 7.62742i	− 0.314817i	0.987534	−	0.157409i	$-0.0503140\pi$
$588$	0	0
$589$	0	0
$590$	70.6274i	2.90768i
$591$	0	0
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	44.8284i	1.83624i
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	41.4264	1.68982	0.844909	−	0.534910i	$-0.179654\pi$
$601$	41.4264	1.68982	0.844909	+	0.534910i	$0.179654\pi$
$602$	0	0
$603$	0	0
$604$	44.4264	1.80768
$605$	− 48.4264i	− 1.96881i
$606$	0	0
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	−1.57359	−0.0634019
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	81.5980i	3.27705i
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	112.397	4.49588
$626$	12.0416i	0.481280i
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−20.7574	−0.826337	−0.413169	−	0.910654i	$-0.635578\pi$
$631$	−20.7574	−0.826337	−0.413169	+	0.910654i	$0.635578\pi$
$632$	28.2843i	1.12509i
$633$	0	0
$634$	−43.2132	−1.71622
$635$	− 67.2843i	− 2.67009i
$636$	0	0
$637$	0	0
$638$	0.686292i	0.0271705i
$639$	0	0
$640$	−49.9411	−1.97410
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	1.94113	0.0761958
$650$	0	0
$651$	0	0
$652$	0	0
$653$	39.0416i	1.52782i	0.645325	+	0.763909i	$0.276722\pi$
$653$	39.0416i	1.52782i	−0.645325	+	0.763909i	$0.723278\pi$
$654$	0	0
$655$	36.6985	1.43393
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 43.6274i	− 1.69948i	−0.527200	−	0.849741i	$-0.676758\pi$
$659$	− 43.6274i	− 1.69948i	0.527200	−	0.849741i	$-0.323242\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	50.4264	1.95692
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−16.9411	−0.653032	−0.326516	−	0.945192i	$-0.605875\pi$
$673$	−16.9411	−0.653032	−0.326516	+	0.945192i	$0.605875\pi$
$674$	− 31.1127i	− 1.19842i
$675$	0	0
$676$	−26.0000	−1.00000
$677$	2.82843i	0.108705i	0.998522	+	0.0543526i	$0.0173095\pi$
$677$	2.82843i	0.108705i	−0.998522	+	0.0543526i	$0.982690\pi$
$678$	0	0
$679$	51.7868	1.98739
$680$	0	0
$681$	0	0
$682$	2.24264	0.0858752
$683$	− 5.65685i	− 0.216454i	−0.994126	−	0.108227i	$-0.965483\pi$
$683$	− 5.65685i	− 0.216454i	0.994126	−	0.108227i	$-0.0345173\pi$
$684$	0	0
$685$	0	0
$686$	− 15.9828i	− 0.610224i
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	− 10.2010i	− 0.387784i
$693$	0	0
$694$	49.6985	1.88653
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	93.9411	3.55064
$701$	14.6152i	0.552009i	0.961156	+	0.276005i	$0.0890105\pi$
$701$	14.6152i	0.552009i	−0.961156	+	0.276005i	$0.910989\pi$
$702$	0	0
$703$	0	0
$704$	1.37258i	0.0517312i
$705$	0	0
$706$	0	0
$707$	− 41.8284i	− 1.57312i
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	53.3137i	1.99243i
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	45.3970	1.69067
$722$	26.8701i	1.00000i
$723$	0	0
$724$	0	0
$725$	− 40.9706i	− 1.52161i
$726$	0	0
$727$	−47.6690	−1.76795	−0.883974	−	0.467537i	$-0.845142\pi$
$727$	−47.6690	−1.76795	−0.883974	+	0.467537i	$0.845142\pi$
$728$	0	0
$729$	0	0
$730$	−96.6690	−3.57788
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	− 20.8701i	− 0.770328i
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	−18.6690	−0.685362
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	−98.9411	−3.62492
$746$	0	0
$747$	0	0
$748$	0	0
$749$	30.2965i	1.10701i
$750$	0	0
$751$	−41.6690	−1.52053	−0.760263	−	0.649616i	$-0.774930\pi$
$751$	−41.6690	−1.52053	−0.760263	+	0.649616i	$0.774930\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	98.0538i	3.56854i
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	29.4264	1.06114	0.530572	−	0.847640i	$-0.321977\pi$
$769$	29.4264	1.06114	0.530572	+	0.847640i	$0.321977\pi$
$770$	− 3.47309i	− 0.125161i
$771$	0	0
$772$	42.9706	1.54654
$773$	19.7990i	0.712120i	0.934463	+	0.356060i	$0.115880\pi$
$773$	19.7990i	0.712120i	−0.934463	+	0.356060i	$0.884120\pi$
$774$	0	0
$775$	−133.882	−4.80919
$776$	− 45.1716i	− 1.62156i
$777$	0	0
$778$	7.69848	0.276004
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	14.0589	0.502103
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	27.8579i	0.992395i
$789$	0	0
$790$	−62.4264	−2.22103
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	56.4264	1.99998
$797$	− 11.8701i	− 0.420459i	−0.977652	−	0.210230i	$-0.932579\pi$
$797$	− 11.8701i	− 0.420459i	0.977652	−	0.210230i	$-0.0674211\pi$
$798$	0	0
$799$	0	0
$800$	− 81.9411i	− 2.89706i
$801$	0	0
$802$	0	0
$803$	2.65685i	0.0937584i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−36.4853	−1.28355
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	− 18.3431i	− 0.643718i
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	− 40.9289i	− 1.43105i
$819$	0	0
$820$	0	0
$821$	− 48.0833i	− 1.67812i	−0.544041	−	0.839059i	$-0.683106\pi$
$821$	− 48.0833i	− 1.67812i	0.544041	−	0.839059i	$-0.316894\pi$
$822$	0	0
$823$	52.6985	1.83695	0.918477	−	0.395475i	$-0.129420\pi$
$823$	52.6985	1.83695	0.918477	+	0.395475i	$0.129420\pi$
$824$	− 39.5980i	− 1.37946i
$825$	0	0
$826$	−51.8823	−1.80521
$827$	− 56.5685i	− 1.96708i	−0.180688	−	0.983540i	$-0.557832\pi$
$827$	− 56.5685i	− 1.96708i	0.180688	−	0.983540i	$-0.442168\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	111.296i	3.86316i
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	56.0000	1.93449
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	21.0000	0.724138
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 57.3848i	− 1.97410i
$846$	0	0
$847$	35.5736	1.22232
$848$	16.2843i	0.559204i
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	26.4264	0.903236
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	22.5147	0.765523
$866$	− 57.8995i	− 1.96750i
$867$	0	0
$868$	−59.9411	−2.03453
$869$	1.71573i	0.0582021i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	135.770i	4.58985i
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	− 5.95837i	− 0.201085i
$879$	0	0
$880$	−3.02944	−0.102122
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	−40.0000	−1.34383
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	49.4264	1.65771
$890$	0	0
$891$	0	0
$892$	−52.0000	−1.74109
$893$	0	0
$894$	0	0
$895$	−117.669	−3.93324
$896$	− 36.6863i	− 1.22560i
$897$	0	0
$898$	0	0
$899$	26.1421i	0.871889i
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	− 56.5685i	− 1.87729i
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	3.05887	0.101234
$914$	− 2.87006i	− 0.0949331i
$915$	0	0
$916$	0	0
$917$	26.9584i	0.890244i
$918$	0	0
$919$	4.69848	0.154989	0.0774944	−	0.996993i	$-0.475308\pi$
$919$	4.69848	0.154989	0.0774944	+	0.996993i	$0.475308\pi$
$920$	0	0
$921$	0	0
$922$	−32.6690	−1.07590
$923$	0	0
$924$	0	0
$925$	0	0
$926$	23.6152i	0.776044i
$927$	0	0
$928$	−16.0000	−0.525226
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	−49.2132	−1.61031
$935$	0	0
$936$	0	0
$937$	45.9706	1.50179	0.750896	−	0.660420i	$-0.229622\pi$
$937$	45.9706	1.50179	0.750896	+	0.660420i	$0.229622\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	57.0416i	1.85950i	0.368186	+	0.929752i	$0.379979\pi$
$941$	57.0416i	1.85950i	−0.368186	+	0.929752i	$0.620021\pi$
$942$	0	0
$943$	0	0
$944$	45.2548i	1.47292i
$945$	0	0
$946$	0	0
$947$	7.28427i	0.236707i	0.992972	+	0.118354i	$0.0377616\pi$
$947$	7.28427i	0.236707i	−0.992972	+	0.118354i	$0.962238\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	54.4264	1.75569
$962$	0	0
$963$	0	0
$964$	20.0000	0.644157
$965$	94.8406i	3.05303i
$966$	0	0
$967$	−26.7574	−0.860459	−0.430229	−	0.902720i	$-0.641567\pi$
$967$	−26.7574	−0.860459	−0.430229	+	0.902720i	$0.641567\pi$
$968$	− 31.0294i	− 0.997324i
$969$	0	0
$970$	99.6985	3.20113
$971$	− 34.1127i	− 1.09473i	−0.836894	−	0.547364i	$-0.815631\pi$
$971$	− 34.1127i	− 1.09473i	0.836894	−	0.547364i	$-0.184369\pi$
$972$	0	0
$973$	0	0
$974$	2.82843i	0.0906287i
$975$	0	0
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	0	0
$980$	31.0294i	0.991199i
$981$	0	0
$982$	−62.6690	−1.99985
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	−61.4853	−1.95908
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	7.78680	0.247356	0.123678	−	0.992322i	$-0.460531\pi$
$991$	7.78680	0.247356	0.123678	+	0.992322i	$0.460531\pi$
$992$	52.2843i	1.66003i
$993$	0	0
$994$	0	0
$995$	124.539i	3.94816i
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.2.d.a.109.3	yes	4
3.2	odd	2	inner	216.2.d.a.109.2	✓	4
4.3	odd	2		864.2.d.b.433.1		4
8.3	odd	2		864.2.d.b.433.4		4
8.5	even	2	inner	216.2.d.a.109.2	✓	4
9.2	odd	6		648.2.n.p.109.1		8
9.4	even	3		648.2.n.p.541.1		8
9.5	odd	6		648.2.n.p.541.4		8
9.7	even	3		648.2.n.p.109.4		8
12.11	even	2		864.2.d.b.433.4		4
16.3	odd	4		6912.2.a.y.1.1		2
16.5	even	4		6912.2.a.bz.1.2		2
16.11	odd	4		6912.2.a.by.1.2		2
16.13	even	4		6912.2.a.z.1.1		2
24.5	odd	2	CM	216.2.d.a.109.3	yes	4
24.11	even	2		864.2.d.b.433.1		4
36.7	odd	6		2592.2.r.o.433.4		8
36.11	even	6		2592.2.r.o.433.1		8
36.23	even	6		2592.2.r.o.2161.4		8
36.31	odd	6		2592.2.r.o.2161.1		8
48.5	odd	4		6912.2.a.z.1.1		2
48.11	even	4		6912.2.a.y.1.1		2
48.29	odd	4		6912.2.a.bz.1.2		2
48.35	even	4		6912.2.a.by.1.2		2
72.5	odd	6		648.2.n.p.541.1		8
72.11	even	6		2592.2.r.o.433.4		8
72.13	even	6		648.2.n.p.541.4		8
72.29	odd	6		648.2.n.p.109.4		8
72.43	odd	6		2592.2.r.o.433.1		8
72.59	even	6		2592.2.r.o.2161.1		8
72.61	even	6		648.2.n.p.109.1		8
72.67	odd	6		2592.2.r.o.2161.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.2.d.a.109.2	✓	4	3.2	odd	2	inner
216.2.d.a.109.2	✓	4	8.5	even	2	inner
216.2.d.a.109.3	yes	4	1.1	even	1	trivial
216.2.d.a.109.3	yes	4	24.5	odd	2	CM
648.2.n.p.109.1		8	9.2	odd	6
648.2.n.p.109.1		8	72.61	even	6
648.2.n.p.109.4		8	9.7	even	3
648.2.n.p.109.4		8	72.29	odd	6
648.2.n.p.541.1		8	9.4	even	3
648.2.n.p.541.1		8	72.5	odd	6
648.2.n.p.541.4		8	9.5	odd	6
648.2.n.p.541.4		8	72.13	even	6
864.2.d.b.433.1		4	4.3	odd	2
864.2.d.b.433.1		4	24.11	even	2
864.2.d.b.433.4		4	8.3	odd	2
864.2.d.b.433.4		4	12.11	even	2
2592.2.r.o.433.1		8	36.11	even	6
2592.2.r.o.433.1		8	72.43	odd	6
2592.2.r.o.433.4		8	36.7	odd	6
2592.2.r.o.433.4		8	72.11	even	6
2592.2.r.o.2161.1		8	36.31	odd	6
2592.2.r.o.2161.1		8	72.59	even	6
2592.2.r.o.2161.4		8	36.23	even	6
2592.2.r.o.2161.4		8	72.67	odd	6
6912.2.a.y.1.1		2	16.3	odd	4
6912.2.a.y.1.1		2	48.11	even	4
6912.2.a.z.1.1		2	16.13	even	4
6912.2.a.z.1.1		2	48.5	odd	4
6912.2.a.by.1.2		2	16.11	odd	4
6912.2.a.by.1.2		2	48.35	even	4
6912.2.a.bz.1.2		2	16.5	even	4
6912.2.a.bz.1.2		2	48.29	odd	4

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:	\( N \)	\(=\)	\( 216 = 2^{3} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	216.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} -4.41421i q^{5} +3.24264 q^{7} -2.82843i q^{8} +O(q^{10})\)	\(q+1.41421i q^{2} -2.00000 q^{4} -4.41421i q^{5} +3.24264 q^{7} -2.82843i q^{8} +6.24264 q^{10} -0.171573i q^{11} +4.58579i q^{14} +4.00000 q^{16} +8.82843i q^{20} +0.242641 q^{22} -14.4853 q^{25} -6.48528 q^{28} +2.82843i q^{29} +9.24264 q^{31} +5.65685i q^{32} -14.3137i q^{35} -12.4853 q^{40} +0.343146i q^{44} +3.51472 q^{49} -20.4853i q^{50} +4.07107i q^{53} -0.757359 q^{55} -9.17157i q^{56} -4.00000 q^{58} +11.3137i q^{59} +13.0711i q^{62} -8.00000 q^{64} +20.2426 q^{70} -15.4853 q^{73} -0.556349i q^{77} -10.0000 q^{79} -17.6569i q^{80} +17.8284i q^{83} -0.485281 q^{88} +15.9706 q^{97} +4.97056i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} - 4 q^{7}+O(q^{10}) \) 4 * q - 8 * q^4 - 4 * q^7	\( 4 q - 8 q^{4} - 4 q^{7} + 8 q^{10} + 16 q^{16} - 16 q^{22} - 24 q^{25} + 8 q^{28} + 20 q^{31} - 16 q^{40} + 48 q^{49} - 20 q^{55} - 16 q^{58} - 32 q^{64} + 64 q^{70} - 28 q^{73} - 40 q^{79} + 32 q^{88} - 4 q^{97}+O(q^{100}) \) 4 * q - 8 * q^4 - 4 * q^7 + 8 * q^10 + 16 * q^16 - 16 * q^22 - 24 * q^25 + 8 * q^28 + 20 * q^31 - 16 * q^40 + 48 * q^49 - 20 * q^55 - 16 * q^58 - 32 * q^64 + 64 * q^70 - 28 * q^73 - 40 * q^79 + 32 * q^88 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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