LMFDB - Embedded newform 216.2.v.a.11.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [216,2,Mod(11,216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(216, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 9, 13]))

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

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

chi := DirichletCharacter("216.11");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 216 = 2^{3} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	216.v (of order $18$, degree $6$, 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:	$12$
Relative dimension:	$2$ over $\Q(\zeta_{18})$
Coefficient field:	12.0.101559956668416.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 8x^{6} + 64 $ x^12 - 8*x^6 + 64
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{18}]$

Embedding invariants

Embedding label			11.2
Root			$-0.483690 + 1.32893i$ of defining polynomial
Character	$\chi$	$=$	216.11
Dual form			216.2.v.a.59.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.909039 - 1.08335i) q^{2} +(0.456003 - 1.67095i) q^{3} +(-0.347296 - 1.96962i) q^{4} +(-1.39570 - 2.01297i) q^{6} +(-2.44949 - 1.41421i) q^{8} +(-2.58412 - 1.52391i) q^{9} +O(q^{10})$	\(q+(0.909039 - 1.08335i) q^{2} +(0.456003 - 1.67095i) q^{3} +(-0.347296 - 1.96962i) q^{4} +(-1.39570 - 2.01297i) q^{6} +(-2.44949 - 1.41421i) q^{8} +(-2.58412 - 1.52391i) q^{9} +(2.18893 + 6.01403i) q^{11} +(-3.44949 - 0.317837i) q^{12} +(-3.75877 + 1.36808i) q^{16} +(7.09286 - 4.09506i) q^{17} +(-4.00000 + 1.41421i) q^{18} +(-0.511376 + 0.885729i) q^{19} +(8.50512 + 3.09561i) q^{22} +(-3.48005 + 3.44808i) q^{24} +(-3.83022 - 3.21394i) q^{25} +(-3.72474 + 3.62302i) q^{27} +(-1.93476 + 5.31570i) q^{32} +(11.0473 - 0.915165i) q^{33} +(2.01130 - 11.4066i) q^{34} +(-2.10407 + 5.61898i) q^{36} +(0.494694 + 1.35916i) q^{38} +(8.15282 + 9.71615i) q^{41} +(2.07316 - 0.754568i) q^{43} +(11.0851 - 6.40000i) q^{44} +(0.571978 + 6.90455i) q^{48} +(-6.57785 - 2.39414i) q^{49} +(-6.96364 + 1.22788i) q^{50} +(-3.60827 - 13.7191i) q^{51} +(0.539062 + 7.32867i) q^{54} +(1.24682 + 1.25838i) q^{57} +(-3.84333 + 10.5595i) q^{59} +(4.00000 + 6.92820i) q^{64} +(9.05096 - 12.8000i) q^{66} +(-4.53904 + 3.80871i) q^{67} +(-10.5290 - 12.5480i) q^{68} +(4.17464 + 7.38731i) q^{72} +(4.68819 - 8.12018i) q^{73} +(-7.11691 + 4.93453i) q^{75} +(1.92214 + 0.699604i) q^{76} +(4.35538 + 7.87596i) q^{81} +17.9372 q^{82} +(-10.9291 + 13.0248i) q^{83} +(1.06712 - 2.93189i) q^{86} +(3.14337 - 17.8269i) q^{88} +(-11.0505 - 6.38002i) q^{89} +(8.00000 + 5.65685i) q^{96} +(-14.0731 + 5.12218i) q^{97} +(-8.57321 + 4.94975i) q^{98} +(3.50840 - 18.8767i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q + 18 q^{11} - 12 q^{12} - 48 q^{18} + 12 q^{22} - 30 q^{27} + 6 q^{33} - 24 q^{34} + 72 q^{38} - 18 q^{41} + 30 q^{43} - 12 q^{51} + 42 q^{57} + 36 q^{59} + 48 q^{64} - 42 q^{67} - 72 q^{68} - 24 q^{76} + 36 q^{86} + 48 q^{88} - 162 q^{89} + 96 q^{96} + 30 q^{97}+O(q^{100}) $ 12 * q + 18 * q^11 - 12 * q^12 - 48 * q^18 + 12 * q^22 - 30 * q^27 + 6 * q^33 - 24 * q^34 + 72 * q^38 - 18 * q^41 + 30 * q^43 - 12 * q^51 + 42 * q^57 + 36 * q^59 + 48 * q^64 - 42 * q^67 - 72 * q^68 - 24 * q^76 + 36 * q^86 + 48 * q^88 - 162 * q^89 + 96 * q^96 + 30 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$55$	$109$	$137$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{13}{18}\right)$

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 $

$2$	0.909039	−	1.08335i	0.642788	−	0.766044i
$2$	0.909039	−	1.08335i	0.642788	−	0.766044i
$3$	0.456003	−	1.67095i	0.263274	−	0.964721i
$3$	0.456003	−	1.67095i	0.263274	−	0.964721i
$4$	−0.347296	−	1.96962i	−0.173648	−	0.984808i
$5$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$5$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$6$	−1.39570	−	2.01297i	−0.569790	−	0.821790i
$7$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$7$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$8$	−2.44949	−	1.41421i	−0.866025	−	0.500000i
$9$	−2.58412	−	1.52391i	−0.861374	−	0.507971i
$10$		0			0
$11$	2.18893	+	6.01403i	0.659987	+	1.81330i	0.576994	+	0.816748i	$0.304226\pi$
$11$	2.18893	+	6.01403i	0.659987	+	1.81330i	0.0829925	+	0.996550i	$0.473552\pi$
$12$	−3.44949	−	0.317837i	−0.995782	−	0.0917517i
$13$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$13$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$14$		0			0
$15$		0			0
$16$	−3.75877	+	1.36808i	−0.939693	+	0.342020i
$17$	7.09286	−	4.09506i	1.72027	−	0.993199i	0.801920	−	0.597431i	$-0.203812\pi$
$17$	7.09286	−	4.09506i	1.72027	−	0.993199i	0.918351	−	0.395768i	$-0.129521\pi$
$18$	−4.00000	+	1.41421i	−0.942809	+	0.333333i
$19$	−0.511376	+	0.885729i	−0.117318	+	0.203200i	−0.918704	−	0.394947i	$-0.870763\pi$
$19$	−0.511376	+	0.885729i	−0.117318	+	0.203200i	0.801386	+	0.598147i	$0.204096\pi$
$20$		0			0
$21$		0			0
$22$	8.50512	+	3.09561i	1.81330	+	0.659987i
$23$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$23$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$24$	−3.48005	+	3.44808i	−0.710362	+	0.703836i
$25$	−3.83022	−	3.21394i	−0.766044	−	0.642788i
$26$		0			0
$27$	−3.72474	+	3.62302i	−0.716827	+	0.697251i
$28$		0			0
$29$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$29$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$30$		0			0
$31$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$31$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$32$	−1.93476	+	5.31570i	−0.342020	+	0.939693i
$33$	11.0473	−	0.915165i	1.92308	−	0.159310i
$34$	2.01130	−	11.4066i	0.344934	−	1.95622i
$35$		0			0
$36$	−2.10407	+	5.61898i	−0.350678	+	0.936496i
$37$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$37$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$38$	0.494694	+	1.35916i	0.0802500	+	0.220485i
$39$		0			0
$40$		0			0
$41$	8.15282	+	9.71615i	1.27326	+	1.51741i	0.742424	+	0.669930i	$0.233676\pi$
$41$	8.15282	+	9.71615i	1.27326	+	1.51741i	0.530831	+	0.847477i	$0.321880\pi$
$42$		0			0
$43$	2.07316	−	0.754568i	0.316154	−	0.115071i	−0.179069	−	0.983836i	$-0.557309\pi$
$43$	2.07316	−	0.754568i	0.316154	−	0.115071i	0.495223	+	0.868766i	$0.335086\pi$
$44$	11.0851	−	6.40000i	1.67114	−	0.964836i
$45$		0			0
$46$		0			0
$47$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$47$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$48$	0.571978	+	6.90455i	0.0825579	+	0.996586i
$49$	−6.57785	−	2.39414i	−0.939693	−	0.342020i
$50$	−6.96364	+	1.22788i	−0.984808	+	0.173648i
$51$	−3.60827	−	13.7191i	−0.505258	−	1.92106i
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	0.539062	+	7.32867i	0.0733571	+	0.997306i
$55$		0			0
$56$		0			0
$57$	1.24682	+	1.25838i	0.165145	+	0.166676i
$58$		0			0
$59$	−3.84333	+	10.5595i	−0.500359	+	1.37472i	0.390567	+	0.920575i	$0.372279\pi$
$59$	−3.84333	+	10.5595i	−0.500359	+	1.37472i	−0.890925	+	0.454150i	$0.849943\pi$
$60$		0			0
$61$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$61$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$62$		0			0
$63$		0			0
$64$	4.00000	+	6.92820i	0.500000	+	0.866025i
$65$		0			0
$66$	9.05096	−	12.8000i	1.11410	−	1.57557i
$67$	−4.53904	+	3.80871i	−0.554532	+	0.465308i	−0.876472	−	0.481452i	$-0.840109\pi$
$67$	−4.53904	+	3.80871i	−0.554532	+	0.465308i	0.321940	+	0.946760i	$0.395665\pi$
$68$	−10.5290	−	12.5480i	−1.27683	−	1.52167i
$69$		0			0
$70$		0			0
$71$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$71$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$72$	4.17464	+	7.38731i	0.491986	+	0.870603i
$73$	4.68819	−	8.12018i	0.548711	−	0.950395i	−0.449652	−	0.893204i	$-0.648452\pi$
$73$	4.68819	−	8.12018i	0.548711	−	0.950395i	0.998363	−	0.0571917i	$-0.0182146\pi$
$74$		0			0
$75$	−7.11691	+	4.93453i	−0.821790	+	0.569790i
$76$	1.92214	+	0.699604i	0.220485	+	0.0802500i
$77$		0			0
$78$		0			0
$79$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$79$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$80$		0			0
$81$	4.35538	+	7.87596i	0.483931	+	0.875106i
$82$	17.9372			1.98083
$83$	−10.9291	+	13.0248i	−1.19963	+	1.42966i	−0.324415	+	0.945915i	$0.605167\pi$
$83$	−10.9291	+	13.0248i	−1.19963	+	1.42966i	−0.875210	+	0.483743i	$0.839277\pi$
$84$		0			0
$85$		0			0
$86$	1.06712	−	2.93189i	0.115071	−	0.316154i
$87$		0			0
$88$	3.14337	−	17.8269i	0.335084	−	1.90036i
$89$	−11.0505	−	6.38002i	−1.17135	−	0.676280i	−0.217354	−	0.976093i	$-0.569742\pi$
$89$	−11.0505	−	6.38002i	−1.17135	−	0.676280i	−0.953998	+	0.299813i	$0.903076\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	8.00000	+	5.65685i	0.816497	+	0.577350i
$97$	−14.0731	+	5.12218i	−1.42890	+	0.520079i	−0.936617	−	0.350354i	$-0.886061\pi$
$97$	−14.0731	+	5.12218i	−1.42890	+	0.520079i	−0.492287	+	0.870433i	$0.663839\pi$
$98$	−8.57321	+	4.94975i	−0.866025	+	0.500000i
$99$	3.50840	−	18.8767i	0.352608	−	1.89718i
$100$	−5.00000	+	8.66025i	−0.500000	+	0.866025i
$101$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$101$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$102$	−18.1427	−	8.56222i	−1.79639	−	0.847786i
$103$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$103$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	11.3102i		−	1.09340i	−0.837330	−	0.546698i	$-0.815885\pi$
$107$		−	11.3102i		−	1.09340i	0.837330	−	0.546698i	$-0.184115\pi$
$108$	8.42955	+	6.07805i	0.811134	+	0.584861i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	7.26632	−	19.9641i	0.683558	−	1.87806i	0.306510	−	0.951867i	$-0.400839\pi$
$113$	7.26632	−	19.9641i	0.683558	−	1.87806i	0.377048	−	0.926194i	$-0.376939\pi$
$114$	2.49667	−	0.206826i	0.233834	−	0.0193710i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	7.94586	+	13.7626i	0.731475	+	1.26695i
$119$		0			0
$120$		0			0
$121$	−22.9507	+	19.2579i	−2.08642	+	1.75072i
$122$		0			0
$123$	19.9529	−	9.19232i	1.79909	−	0.828844i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$127$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$128$	11.1418	+	1.96460i	0.984808	+	0.173648i
$129$	−0.315476	−	3.80822i	−0.0277761	−	0.335295i
$130$		0			0
$131$	−8.38799	+	1.47903i	−0.732862	+	0.129223i	−0.527611	−	0.849486i	$-0.676912\pi$
$131$	−8.38799	+	1.47903i	−0.732862	+	0.129223i	−0.205251	+	0.978709i	$0.565801\pi$
$132$	−5.63920	−	21.4411i	−0.490830	−	1.86620i
$133$		0			0
$134$			8.37963i			0.723890i
$135$		0			0
$136$	−23.1652			−1.98640
$137$	1.27249	−	1.51649i	0.108716	−	0.129562i	−0.708942	−	0.705266i	$-0.750827\pi$
$137$	1.27249	−	1.51649i	0.108716	−	0.129562i	0.817658	+	0.575704i	$0.195272\pi$
$138$		0			0
$139$	−2.11445	−	11.9916i	−0.179345	−	1.01712i	−0.933008	−	0.359856i	$-0.882826\pi$
$139$	−2.11445	−	11.9916i	−0.179345	−	1.01712i	0.753663	−	0.657262i	$-0.228285\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	11.7980	+	2.19275i	0.983163	+	0.182729i
$145$		0			0
$146$	−4.53526	−	12.4605i	−0.375340	−	1.03124i
$147$	−7.00000	+	9.89949i	−0.577350	+	0.816497i
$148$		0			0
$149$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$149$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$150$	−1.12372	+	12.1958i	−0.0917517	+	0.995782i
$151$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$151$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$152$	2.50522	−	1.44639i	0.203200	−	0.117318i
$153$	−24.5693	−	0.226755i	−1.98631	−	0.0183321i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$157$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	12.4916	+	2.44115i	0.981435	+	0.191795i
$163$	21.0454			1.64840			0.824202	−	0.566296i	$-0.191624\pi$
$163$	21.0454			1.64840			0.824202	+	0.566296i	$0.191624\pi$
$164$	16.3056	−	19.4323i	1.27326	−	1.51741i
$165$		0			0
$166$	4.17544	+	23.6801i	0.324077	+	1.83793i
$167$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$167$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$168$		0			0
$169$	2.25743	−	12.8025i	0.173648	−	0.984808i
$170$		0			0
$171$	2.67123	−	1.50954i	0.204274	−	0.115437i
$172$	−2.20621	−	3.82127i	−0.168222	−	0.291369i
$173$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$173$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$174$		0			0
$175$		0			0
$176$	−16.4554	−	19.6107i	−1.24037	−	1.47821i
$177$	15.8917	+	11.2371i	1.19449	+	0.844635i
$178$	−16.9571	+	6.17189i	−1.27099	+	0.462603i
$179$	22.0732	−	12.7440i	1.64983	−	0.952529i	0.672692	−	0.739923i	$-0.265138\pi$
$179$	22.0732	−	12.7440i	1.64983	−	0.952529i	0.977138	−	0.212607i	$-0.0681952\pi$
$180$		0			0
$181$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$181$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	40.1536	+	33.6929i	2.93632	+	2.46387i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$191$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$192$	13.4007	−	3.52450i	0.967110	−	0.254359i
$193$	2.58197	+	14.6431i	0.185854	+	1.05403i	0.924853	+	0.380325i	$0.124188\pi$
$193$	2.58197	+	14.6431i	0.185854	+	1.05403i	−0.738999	+	0.673707i	$0.764701\pi$
$194$	−7.24386	+	19.9023i	−0.520079	+	1.42890i
$195$		0			0
$196$	−2.43107	+	13.7873i	−0.173648	+	0.984808i
$197$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$197$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$198$	−17.2608	−	20.9605i	−1.22667	−	1.48960i
$199$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$199$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$200$	4.83690	+	13.2893i	0.342020	+	0.939693i
$201$	4.29433	+	9.32127i	0.302899	+	0.657472i
$202$		0			0
$203$		0			0
$204$	−25.7683	+	11.8715i	−1.80414	+	0.831172i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−6.44617	−	1.13663i	−0.445891	−	0.0786226i
$210$		0			0
$211$	−14.1381	−	5.14583i	−0.973304	−	0.354254i	−0.194071	−	0.980988i	$-0.562169\pi$
$211$	−14.1381	−	5.14583i	−0.973304	−	0.354254i	−0.779233	+	0.626734i	$0.784391\pi$
$212$		0			0
$213$		0			0
$214$	−12.2529	−	10.2814i	−0.837590	−	0.702822i
$215$		0			0
$216$	14.2474	−	3.60697i	0.969416	−	0.245423i
$217$		0			0
$218$		0			0
$219$	−11.4306	−	11.5365i	−0.772406	−	0.779567i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$223$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$224$		0			0
$225$	5.00000	+	14.1421i	0.333333	+	0.942809i
$226$	−15.0227	−	26.0201i	−0.999295	−	1.73083i
$227$	7.33644	+	20.1567i	0.486937	+	1.33785i	0.903440	+	0.428714i	$0.141033\pi$
$227$	7.33644	+	20.1567i	0.486937	+	1.33785i	−0.416503	+	0.909134i	$0.636745\pi$
$228$	2.04550	−	2.89278i	0.135467	−	0.191579i
$229$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$229$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−13.4895	+	7.78815i	−0.883725	+	0.510219i	−0.871885	−	0.489711i	$-0.837102\pi$
$233$	−13.4895	+	7.78815i	−0.883725	+	0.510219i	−0.0118403	+	0.999930i	$0.503769\pi$
$234$		0			0
$235$		0			0
$236$	22.1328	+	3.90262i	1.44073	+	0.254039i
$237$		0			0
$238$		0			0
$239$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$239$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$240$		0			0
$241$	16.2613	+	13.6448i	1.04748	+	0.878941i	0.992826	−	0.119564i	$-0.0381497\pi$
$241$	16.2613	+	13.6448i	1.04748	+	0.878941i	0.0546547	+	0.998505i	$0.482594\pi$
$242$			42.3698i			2.72363i
$243$	15.1464	−	3.68614i	0.971640	−	0.236466i
$244$		0			0
$245$		0			0
$246$	8.17943	−	29.9721i	0.521501	−	1.91095i
$247$		0			0
$248$		0			0
$249$	16.7800	+	24.2013i	1.06339	+	1.53369i
$250$		0			0
$251$	−23.9806	−	13.8452i	−1.51364	−	0.873902i	−0.999872	−	0.0159750i	$-0.994915\pi$
$251$	−23.9806	−	13.8452i	−1.51364	−	0.873902i	−0.513771	−	0.857927i	$-0.671752\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	12.2567	−	10.2846i	0.766044	−	0.642788i
$257$	17.7278	+	21.1272i	1.10583	+	1.31788i	0.943585	+	0.331130i	$0.107430\pi$
$257$	17.7278	+	21.1272i	1.10583	+	1.31788i	0.162247	+	0.986750i	$0.448126\pi$
$258$	−4.41242	−	3.12005i	−0.274705	−	0.194246i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−6.02270	+	10.4316i	−0.372084	+	0.644468i
$263$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$263$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$264$	−28.3544	−	13.3815i	−1.74509	−	0.823576i
$265$		0			0
$266$		0			0
$267$	−15.6997	+	15.5555i	−0.960808	+	0.951981i
$268$	9.07808	+	7.61741i	0.554532	+	0.465308i
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−21.0580	+	25.0960i	−1.27683	+	1.52167i
$273$		0			0
$274$	−0.486151	−	2.75710i	−0.0293694	−	0.166562i
$275$	10.9446	−	30.0702i	0.659987	−	1.81330i
$276$		0			0
$277$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$277$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$278$	−14.9133	−	8.61018i	−0.894438	−	0.516404i
$279$		0			0
$280$		0			0
$281$	−10.1685	−	27.9376i	−0.606600	−	1.66662i	−0.737601	−	0.675236i	$-0.764042\pi$
$281$	−10.1685	−	27.9376i	−0.606600	−	1.66662i	0.131002	−	0.991382i	$-0.458181\pi$
$282$		0			0
$283$	25.3143	−	21.2412i	1.50478	−	1.26266i	0.631557	−	0.775330i	$-0.282416\pi$
$283$	25.3143	−	21.2412i	1.50478	−	1.26266i	0.873219	−	0.487327i	$-0.162028\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	13.1003	−	10.7880i	0.771944	−	0.635691i
$289$	25.0391	−	43.3690i	1.47289	−	2.55112i
$290$		0			0
$291$	2.14152	+	25.8511i	0.125538	+	1.51542i
$292$	−17.6218	−	6.41382i	−1.03124	−	0.375340i
$293$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$293$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$294$	4.36135	+	16.5825i	0.254359	+	0.967110i
$295$		0			0
$296$		0			0
$297$	−29.9422	−	14.4702i	−1.73742	−	0.839646i
$298$		0			0
$299$		0			0
$300$	12.1908	+	12.3038i	0.703836	+	0.710362i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	0.710396	−	4.02886i	0.0407440	−	0.231071i
$305$		0			0
$306$	−22.5801	+	26.4111i	−1.29082	+	1.50982i
$307$	−17.4258	−	30.1824i	−0.994545	−	1.72260i	−0.587603	−	0.809149i	$-0.699928\pi$
$307$	−17.4258	−	30.1824i	−0.994545	−	1.72260i	−0.406942	−	0.913454i	$-0.633405\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$311$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$312$		0			0
$313$	−19.7387	+	7.18430i	−1.11570	+	0.406080i	−0.833080	−	0.553153i	$-0.813425\pi$
$313$	−19.7387	+	7.18430i	−1.11570	+	0.406080i	−0.282617	+	0.959233i	$0.591202\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$317$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−18.8987	−	5.15748i	−1.05482	−	0.287862i
$322$		0			0
$323$			8.37647i			0.466079i
$324$	14.0000	−	11.3137i	0.777778	−	0.628539i
$325$		0			0
$326$	19.1311	−	22.7996i	1.05957	−	1.26275i
$327$		0			0
$328$	−6.22953	−	35.3294i	−0.343968	−	1.95074i
$329$		0			0
$330$		0			0
$331$	1.57072	−	8.90799i	0.0863345	−	0.489627i	−0.910726	−	0.413011i	$-0.864477\pi$
$331$	1.57072	−	8.90799i	0.0863345	−	0.489627i	0.997061	−	0.0766165i	$-0.0244117\pi$
$332$	29.4495	+	17.0027i	1.61625	+	0.933143i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	24.9283	−	20.9173i	1.35793	−	1.13944i	0.381314	−	0.924445i	$-0.375472\pi$
$337$	24.9283	−	20.9173i	1.35793	−	1.13944i	0.976616	−	0.214993i	$-0.0689729\pi$
$338$	−11.8175	−	14.0836i	−0.642788	−	0.766044i
$339$	−30.0454	−	21.2453i	−1.63184	−	1.15389i
$340$		0			0
$341$		0			0
$342$	0.792894	−	4.26611i	0.0428748	−	0.230685i
$343$		0			0
$344$	−6.14530	−	1.08358i	−0.331333	−	0.0584229i
$345$		0			0
$346$		0			0
$347$	−36.0481	+	6.35625i	−1.93516	+	0.341222i	−0.999918	−	0.0127797i	$-0.995932\pi$
$347$	−36.0481	+	6.35625i	−1.93516	+	0.341222i	−0.935245	+	0.354001i	$0.884821\pi$
$348$		0			0
$349$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$349$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$350$		0			0
$351$		0			0
$352$	−36.2039			−1.92967
$353$	1.25345	−	1.49380i	0.0667144	−	0.0795071i	−0.731655	−	0.681675i	$-0.761252\pi$
$353$	1.25345	−	1.49380i	0.0667144	−	0.0795071i	0.798369	+	0.602168i	$0.205696\pi$
$354$	26.6199	−	7.00130i	1.41483	−	0.372115i
$355$		0			0
$356$	−8.72837	+	23.9810i	−0.462603	+	1.27099i
$357$		0			0
$358$	6.25922	−	35.4978i	0.330810	−	1.87612i
$359$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$359$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$360$		0			0
$361$	8.97699	+	15.5486i	0.472473	+	0.818347i
$362$		0			0
$363$	21.7133	+	47.1310i	1.13965	+	2.47374i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$367$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$368$		0			0
$369$	−6.26131	−	37.5319i	−0.325951	−	1.95383i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$373$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$374$	73.0024	−	12.8723i	3.77486	−	0.665610i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−32.8062			−1.68514			−0.842570	−	0.538587i	$-0.818958\pi$
$379$	−32.8062			−1.68514			−0.842570	+	0.538587i	$0.818958\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$383$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$384$	8.36346	−	17.7215i	0.426796	−	0.904348i
$385$		0			0
$386$	18.2107	+	10.5139i	0.926900	+	0.535146i
$387$	−6.50720	−	1.20942i	−0.330779	−	0.0614782i
$388$	14.9763	+	25.9396i	0.760304	+	1.31689i
$389$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$389$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$390$		0			0
$391$		0			0
$392$	12.7265	+	15.1669i	0.642788	+	0.766044i
$393$	−1.35357	+	14.6903i	−0.0682787	+	0.741029i
$394$		0			0
$395$		0			0
$396$	−38.3984	−	0.354386i	−1.92959	−	0.0178086i
$397$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$397$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$398$		0			0
$399$		0			0
$400$	18.7939	+	6.84040i	0.939693	+	0.342020i
$401$	34.6237	−	6.10509i	1.72902	−	0.304874i	0.781345	−	0.624099i	$-0.214534\pi$
$401$	34.6237	−	6.10509i	1.72902	−	0.304874i	0.947679	+	0.319225i	$0.103423\pi$
$402$	14.0019	+	3.82114i	0.698352	+	0.190581i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−10.5634	+	38.7078i	−0.522966	+	1.91632i
$409$	0.861982	+	4.88854i	0.0426223	+	0.241723i	0.998674	−	0.0514740i	$-0.0163919\pi$
$409$	0.861982	+	4.88854i	0.0426223	+	0.241723i	−0.956052	+	0.293197i	$0.905281\pi$
$410$		0			0
$411$	−1.95372	−	2.81778i	−0.0963697	−	0.138991i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−21.0016	−	1.93509i	−1.02845	−	0.0947619i
$418$	−7.09119	+	5.95021i	−0.346841	+	0.291034i
$419$	21.8376	+	26.0250i	1.06684	+	1.27140i	0.960860	+	0.277036i	$0.0893522\pi$
$419$	21.8376	+	26.0250i	1.06684	+	1.27140i	0.105976	+	0.994369i	$0.466203\pi$
$420$		0			0
$421$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$421$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$422$	−18.4268	+	10.6387i	−0.897002	+	0.517884i
$423$		0			0
$424$		0			0
$425$	−40.3285	−	7.11100i	−1.95622	−	0.344934i
$426$		0			0
$427$		0			0
$428$	−22.2767	+	3.92798i	−1.07679	+	0.189866i
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	9.04388	−	18.7139i	0.435124	−	0.900371i
$433$	18.2012			0.874695			0.437347	−	0.899293i	$-0.355918\pi$
$433$	18.2012			0.874695			0.437347	+	0.899293i	$0.355918\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−22.8889	+	1.89614i	−1.09368	+	0.0906009i
$439$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$439$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$440$		0			0
$441$	13.3495	+	16.2108i	0.635691	+	0.771944i
$442$		0			0
$443$	−5.82211	−	15.9961i	−0.276617	−	0.759998i	−0.997740	−	0.0671913i	$-0.978596\pi$
$443$	−5.82211	−	15.9961i	−0.276617	−	0.759998i	0.721124	−	0.692807i	$-0.243626\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−34.9698	+	20.1898i	−1.65033	+	0.952816i	−0.673389	+	0.739288i	$0.735162\pi$
$449$	−34.9698	+	20.1898i	−1.65033	+	0.952816i	−0.976937	+	0.213528i	$0.931505\pi$
$450$	19.8661	+	7.43900i	0.936496	+	0.350678i
$451$	−40.5873	+	70.2992i	−1.91118	+	3.31026i
$452$	−41.8451	−	7.37842i	−1.96823	−	0.347052i
$453$		0			0
$454$	28.5059	+	10.3753i	1.33785	+	0.486937i
$455$		0			0
$456$	−1.27445	−	4.84565i	−0.0596816	−	0.226918i
$457$	−9.82334	−	8.24276i	−0.459516	−	0.385580i	0.383437	−	0.923567i	$-0.374740\pi$
$457$	−9.82334	−	8.24276i	−0.459516	−	0.385580i	−0.842953	+	0.537987i	$0.819185\pi$
$458$		0			0
$459$	−11.5826	+	40.9506i	−0.540629	+	1.91141i
$460$		0			0
$461$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$461$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$462$		0			0
$463$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$463$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$464$		0			0
$465$		0			0
$466$	−3.82516	+	21.6936i	−0.177197	+	1.00494i
$467$	34.2054	+	19.7485i	1.58284	+	0.913851i	0.994443	+	0.105276i	$0.0335727\pi$
$467$	34.2054	+	19.7485i	1.58284	+	0.913851i	0.588393	+	0.808575i	$0.299761\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	24.3475	−	20.4300i	1.12069	−	0.940367i
$473$	9.07600	+	10.8164i	0.417315	+	0.497336i
$474$		0			0
$475$	4.80536	−	1.74901i	0.220485	−	0.0802500i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$479$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$480$		0			0
$481$		0			0
$482$	29.5643	−	5.21298i	1.34662	−	0.237445i
$483$		0			0
$484$	45.9013	+	38.5158i	2.08642	+	1.75072i
$485$		0			0
$486$	9.77526	−	19.7597i	0.443415	−	0.896317i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$	9.59677	−	35.1657i	0.433981	−	1.59025i
$490$		0			0
$491$	12.9388	−	35.5491i	0.583921	−	1.60431i	−0.197499	−	0.980303i	$-0.563282\pi$
$491$	12.9388	−	35.5491i	0.583921	−	1.60431i	0.781419	−	0.624006i	$-0.214496\pi$
$492$	−25.0349	−	36.1070i	−1.12866	−	1.62783i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	41.4722	+	3.82126i	1.85841	+	0.171235i
$499$	−12.6754	+	10.6359i	−0.567428	+	0.476129i	−0.880791	−	0.473504i	$-0.842989\pi$
$499$	−12.6754	+	10.6359i	−0.567428	+	0.476129i	0.313363	+	0.949633i	$0.398544\pi$
$500$		0			0
$501$		0			0
$502$	−36.7985	+	13.3936i	−1.64240	+	0.597784i
$503$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$503$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−20.3629	−	9.61002i	−0.904348	−	0.426796i
$508$		0			0
$509$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$509$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$510$		0			0
$511$		0			0
$512$		−	22.6274i		−	1.00000i
$513$	−1.30427	−	5.15184i	−0.0575849	−	0.227459i
$514$	39.0035			1.72037
$515$		0			0
$516$	−7.39117	+	1.94395i	−0.325378	+	0.0855776i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	26.6827	+	15.4052i	1.16899	+	0.674916i	0.953442	−	0.301577i	$-0.0975132\pi$
$521$	26.6827	+	15.4052i	1.16899	+	0.674916i	0.215547	+	0.976493i	$0.430847\pi$
$522$		0			0
$523$	−1.52270	−	2.63740i	−0.0665832	−	0.115325i	0.830812	−	0.556553i	$-0.187876\pi$
$523$	−1.52270	−	2.63740i	−0.0665832	−	0.115325i	−0.897395	+	0.441228i	$0.854543\pi$
$524$	5.82624	+	16.0075i	0.254520	+	0.699289i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−40.2722	+	18.5535i	−1.75262	+	0.807436i
$529$	21.6129	−	7.86646i	0.939693	−	0.342020i
$530$		0			0
$531$	26.0233	−	21.4300i	1.12932	−	0.929984i
$532$		0			0
$533$		0			0
$534$	2.58039	+	31.1489i	0.111665	+	1.34794i
$535$		0			0
$536$	16.5047	−	2.91022i	0.712893	−	0.125702i
$537$	−11.2290	−	42.6944i	−0.484569	−	1.84240i
$538$		0			0
$539$		−	44.8000i		−	1.92967i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$	8.04518	+	45.6265i	0.344934	+	1.95622i
$545$		0			0
$546$		0			0
$547$	−5.17190	+	29.3313i	−0.221135	+	1.25412i	0.648803	+	0.760956i	$0.275270\pi$
$547$	−5.17190	+	29.3313i	−0.221135	+	1.25412i	−0.869938	+	0.493161i	$0.835841\pi$
$548$	−3.42883	−	1.97964i	−0.146472	−	0.0845659i
$549$		0			0
$550$	−22.6274	−	39.1918i	−0.964836	−	1.67114i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−22.8846	+	8.32931i	−0.970522	+	0.353241i
$557$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$557$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	74.6091	−	51.7304i	3.15000	−	2.18406i
$562$	−39.5098	−	14.3804i	−1.66662	−	0.606600i
$563$	−22.8074	+	4.02156i	−0.961218	+	0.169489i	−0.632175	−	0.774826i	$-0.717837\pi$
$563$	−22.8074	+	4.02156i	−0.961218	+	0.169489i	−0.329044	+	0.944315i	$0.606726\pi$
$564$		0			0
$565$		0			0
$566$		−	46.7333i		−	1.96435i
$567$		0			0
$568$		0			0
$569$	4.43393	−	5.28415i	0.185880	−	0.221523i	−0.665055	−	0.746795i	$-0.731592\pi$
$569$	4.43393	−	5.28415i	0.185880	−	0.221523i	0.850935	+	0.525271i	$0.176036\pi$
$570$		0			0
$571$	−7.66208	−	43.4538i	−0.320648	−	1.81848i	−0.538642	−	0.842535i	$-0.681062\pi$
$571$	−7.66208	−	43.4538i	−0.320648	−	1.81848i	0.217994	−	0.975950i	$-0.430049\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.221491	−	23.9990i	0.00922881	−	0.999957i
$577$	19.6648	+	34.0604i	0.818655	+	1.41795i	0.906674	+	0.421833i	$0.138613\pi$
$577$	19.6648	+	34.0604i	0.818655	+	1.41795i	−0.0880190	+	0.996119i	$0.528054\pi$
$578$	−24.2223	−	66.5502i	−1.00751	−	2.76812i
$579$	25.6452	+	2.36296i	1.06578	+	0.0982011i
$580$		0			0
$581$		0			0
$582$	29.9525	+	21.1796i	1.24157	+	0.877924i
$583$		0			0
$584$	−22.9673	+	13.2602i	−0.950395	+	0.548711i
$585$		0			0
$586$		0			0
$587$	2.40364	+	0.423827i	0.0992090	+	0.0174932i	0.223032	−	0.974811i	$-0.428404\pi$
$587$	2.40364	+	0.423827i	0.0992090	+	0.0174932i	−0.123823	+	0.992304i	$0.539516\pi$
$588$	21.9293	+	10.3493i	0.904348	+	0.426796i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			38.2159i			1.56934i	0.619915	+	0.784669i	$0.287167\pi$
$593$			38.2159i			1.56934i	−0.619915	+	0.784669i	$0.712833\pi$
$594$	−42.8949	+	19.2839i	−1.76000	+	0.791227i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$599$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$600$	24.4113	−	2.02225i	0.996586	−	0.0825579i
$601$	1.51507	−	8.59237i	0.0618009	−	0.350490i	−0.938190	−	0.346122i	$-0.887498\pi$
$601$	1.51507	−	8.59237i	0.0618009	−	0.350490i	0.999990	−	0.00436841i	$-0.00139051\pi$
$602$		0			0
$603$	17.5336	−	2.92506i	0.714022	−	0.119118i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$607$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$608$	−3.71889	−	4.43199i	−0.150821	−	0.179741i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	8.08622	+	48.4709i	0.326866	+	1.95932i
$613$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$613$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$614$	−48.5389	−	8.55872i	−1.95887	−	0.345402i
$615$		0			0
$616$		0			0
$617$	−35.8670	+	6.32433i	−1.44395	+	0.254608i	−0.840076	−	0.542469i	$-0.817489\pi$
$617$	−35.8670	+	6.32433i	−1.44395	+	0.254608i	−0.603877	+	0.797077i	$0.706378\pi$
$618$		0			0
$619$	−29.8318	−	25.0319i	−1.19904	−	1.00612i	−0.999657	−	0.0261952i	$-0.991661\pi$
$619$	−29.8318	−	25.0319i	−1.19904	−	1.00612i	−0.199386	−	0.979921i	$-0.563895\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	4.34120	+	24.6202i	0.173648	+	0.984808i
$626$	−10.1601	+	27.9147i	−0.406080	+	1.11570i
$627$	−4.83873	+	10.2529i	−0.193240	+	0.409461i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$631$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$632$		0			0
$633$	−15.0454	+	21.2774i	−0.598001	+	0.845702i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	40.3213	+	7.10974i	1.59260	+	0.280818i	0.898470	−	0.439034i	$-0.144679\pi$
$641$	40.3213	+	7.10974i	1.59260	+	0.280818i	0.694127	+	0.719852i	$0.255791\pi$
$642$	−22.7670	+	15.7856i	−0.898542	+	0.623007i
$643$	−0.306376	−	0.111512i	−0.0120823	−	0.00439759i	0.335972	−	0.941872i	$-0.390935\pi$
$643$	−0.306376	−	0.111512i	−0.0120823	−	0.00439759i	−0.348054	+	0.937474i	$0.613157\pi$
$644$		0			0
$645$		0			0
$646$	9.07465	+	7.61454i	0.357037	+	0.299590i
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	0.469834	−	25.4515i	0.0184568	−	0.999830i
$649$	−71.9177			−2.82302
$650$		0			0
$651$		0			0
$652$	−7.30899	−	41.4514i	−0.286242	−	1.62336i
$653$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$653$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$654$		0			0
$655$		0			0
$656$	−43.9370	−	25.3671i	−1.71545	−	0.990417i
$657$	−24.4893	+	13.8392i	−0.955419	+	0.539917i
$658$		0			0
$659$	2.89116	+	7.94338i	0.112623	+	0.309430i	0.983180	−	0.182637i	$-0.0584634\pi$
$659$	2.89116	+	7.94338i	0.112623	+	0.309430i	−0.870557	+	0.492068i	$0.836241\pi$
$660$		0			0
$661$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$661$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$662$	−8.22263	−	9.79935i	−0.319582	−	0.380862i
$663$		0			0
$664$	45.1906	−	16.4480i	1.75373	−	0.638307i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−29.9453	−	25.1271i	−1.15431	−	0.968578i	−0.154495	−	0.987994i	$-0.549375\pi$
$673$	−29.9453	−	25.1271i	−1.15431	−	0.968578i	−0.999812	+	0.0194154i	$0.993820\pi$
$674$		−	46.0207i		−	1.77265i
$675$	25.9108	−	1.90587i	0.997306	−	0.0733571i
$676$	−26.0000			−1.00000
$677$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$677$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$678$	−50.3286	+	13.2369i	−1.93286	+	0.508360i
$679$		0			0
$680$		0			0
$681$	37.0262	−	3.06728i	1.41885	−	0.117538i
$682$		0			0
$683$	31.1416	+	17.9796i	1.19160	+	0.687971i	0.958670	−	0.284522i	$-0.0918347\pi$
$683$	31.1416	+	17.9796i	1.19160	+	0.687971i	0.232932	+	0.972493i	$0.425168\pi$
$684$	−3.90092	−	4.73704i	−0.149155	−	0.181125i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−6.76022	+	5.67250i	−0.257731	+	0.216262i
$689$		0			0
$690$		0			0
$691$	−42.3288	+	15.4064i	−1.61026	+	0.586088i	−0.981492	−	0.191501i	$-0.938665\pi$
$691$	−42.3288	+	15.4064i	−1.61026	+	0.586088i	−0.628772	+	0.777589i	$0.716442\pi$
$692$		0			0
$693$		0			0
$694$	−25.8831	+	44.8308i	−0.982508	+	1.70175i
$695$		0			0
$696$		0			0
$697$	97.6150	+	35.5290i	3.69743	+	1.34576i
$698$		0			0
$699$	6.86234	+	26.0916i	0.259558	+	0.986876i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$	−32.9107	+	39.2215i	−1.24037	+	1.47821i
$705$		0			0
$706$	−0.478878	−	2.71585i	−0.0180228	−	0.102212i
$707$		0			0
$708$	16.6137	−	35.2032i	0.624381	−	1.32302i
$709$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$709$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$710$		0			0
$711$		0			0
$712$	18.0454	+	31.2556i	0.676280	+	1.17135i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−32.7667	−	39.0498i	−1.22455	−	1.45936i
$717$		0			0
$718$		0			0
$719$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$		0			0
$722$	25.0050	+	4.40906i	0.930590	+	0.164088i
$723$	30.2150	−	20.9496i	1.12371	−	0.779125i
$724$		0			0
$725$		0			0
$726$	70.7977	+	19.3208i	2.62755	+	0.717061i
$727$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$727$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$728$		0			0
$729$	0.747449	−	26.9897i	0.0276833	−	0.999617i
$730$		0			0
$731$	11.6146	−	13.8418i	0.429582	−	0.511956i
$732$		0			0
$733$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$733$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−32.8413	−	18.9609i	−1.20973	−	0.698435i
$738$	−46.3520	−	27.3348i	−1.70624	−	1.00621i
$739$	−17.9389	−	31.0711i	−0.659893	−	1.14297i	−0.980643	−	0.195805i	$-0.937268\pi$
$739$	−17.9389	−	31.0711i	−0.659893	−	1.14297i	0.320749	−	0.947164i	$-0.396065\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$743$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	48.0908	−	17.0027i	1.75955	−	0.622095i
$748$	52.4168	−	90.7886i	1.91655	−	3.31956i
$749$		0			0
$750$		0			0
$751$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$751$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$752$		0			0
$753$	−34.0699	+	33.7569i	−1.24157	+	1.23017i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	−29.8221	+	35.5406i	−1.08319	+	1.29089i
$759$		0			0
$760$		0			0
$761$	−3.86952	+	10.6314i	−0.140270	+	0.385388i	−0.989858	−	0.142058i	$-0.954628\pi$
$761$	−3.86952	+	10.6314i	−0.140270	+	0.385388i	0.849589	+	0.527446i	$0.176850\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−11.5959	−	25.1701i	−0.418432	−	0.908248i
$769$	−25.3490	+	21.2704i	−0.914110	+	0.767029i	−0.972896	−	0.231242i	$-0.925721\pi$
$769$	−25.3490	+	21.2704i	−0.914110	+	0.767029i	0.0587868	+	0.998271i	$0.481277\pi$
$770$		0			0
$771$	43.3864	−	19.9882i	1.56252	−	0.719857i
$772$	27.9445	−	10.1710i	1.00574	−	0.366061i
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$	−7.22552	+	5.95016i	−0.259716	+	0.213874i
$775$		0			0
$776$	41.7157	+	7.35561i	1.49751	+	0.264051i
$777$		0			0
$778$		0			0
$779$	−12.7750	+	2.25258i	−0.457713	+	0.0807071i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	28.0000			1.00000
$785$		0			0
$786$	14.6843	+	14.8205i	0.523772	+	0.528629i
$787$	−0.513060	−	2.90971i	−0.0182886	−	0.103720i	0.974297	−	0.225267i	$-0.0723255\pi$
$787$	−0.513060	−	2.90971i	−0.0182886	−	0.103720i	−0.992586	+	0.121547i	$0.961214\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−35.2895	+	41.2767i	−1.25396	+	1.46670i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$797$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$798$		0			0
$799$		0			0
$800$	24.4949	−	14.1421i	0.866025	−	0.500000i
$801$	18.8333	+	33.3268i	0.665441	+	1.17754i
$802$	24.8603	−	43.0593i	0.877849	−	1.52048i
$803$	59.0971	+	10.4204i	2.08549	+	0.367729i
$804$	16.8679	−	11.6954i	0.594886	−	0.412466i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			47.1907i			1.65914i	0.558404	+	0.829569i	$0.311414\pi$
$809$			47.1907i			1.65914i	−0.558404	+	0.829569i	$0.688586\pi$
$810$		0			0
$811$	48.3805			1.69887			0.849434	−	0.527694i	$-0.176943\pi$
$811$	48.3805			1.69887			0.849434	+	0.527694i	$0.176943\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	32.3315	+	46.6307i	1.13183	+	1.63240i
$817$	−0.391821	+	2.22213i	−0.0137081	+	0.0777424i
$818$	6.07958	+	3.51005i	0.212568	+	0.122726i
$819$		0			0
$820$		0			0
$821$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$821$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$822$	−4.82865	−	0.444913i	−0.168418	−	0.0155181i
$823$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$823$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$824$		0			0
$825$	−45.2548	−	32.0000i	−1.57557	−	1.11410i
$826$		0			0
$827$	−17.1464	+	9.89949i	−0.596240	+	0.344239i	−0.767561	−	0.640976i	$-0.778530\pi$
$827$	−17.1464	+	9.89949i	−0.596240	+	0.344239i	0.171321	+	0.985215i	$0.445196\pi$
$828$		0			0
$829$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$829$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−56.4599	+	9.95540i	−1.95622	+	0.344934i
$834$	−21.1876	+	20.9930i	−0.733668	+	0.726928i
$835$		0			0
$836$			13.0912i			0.452769i
$837$		0			0
$838$	48.0454			1.65970
$839$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$839$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$840$		0			0
$841$	−5.03580	−	28.5594i	−0.173648	−	0.984808i
$842$		0			0
$843$	−51.3191	+	4.25131i	−1.76752	+	0.146423i
$844$	−5.22522	+	29.6337i	−0.179859	+	1.02003i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−23.9495	−	51.9848i	−0.821944	−	1.78411i
$850$	−44.3639	+	37.2257i	−1.52167	+	1.27683i
$851$		0			0
$852$		0			0
$853$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$853$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$854$		0			0
$855$		0			0
$856$	−15.9950	+	27.7042i	−0.546698	+	0.946909i
$857$	22.2837	+	3.92921i	0.761195	+	0.134219i	0.540754	−	0.841181i	$-0.318139\pi$
$857$	22.2837	+	3.92921i	0.761195	+	0.134219i	0.220441	+	0.975400i	$0.429250\pi$
$858$		0			0
$859$	17.3166	+	6.30274i	0.590836	+	0.215047i	0.620097	−	0.784525i	$-0.287093\pi$
$859$	17.3166	+	6.30274i	0.590836	+	0.215047i	−0.0292613	+	0.999572i	$0.509316\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$	−12.0524	−	26.8093i	−0.410032	−	0.912071i
$865$		0			0
$866$	16.5456	−	19.7183i	0.562243	−	0.670055i
$867$	−61.0493	−	61.6154i	−2.07334	−	2.09257i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	44.1723	+	8.20981i	1.49501	+	0.277860i
$874$		0			0
$875$		0			0
$876$	−18.7528	+	26.5204i	−0.633597	+	0.896041i
$877$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$877$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−10.9949	+	6.34791i	−0.370428	+	0.213866i	−0.673645	−	0.739055i	$-0.735272\pi$
$881$	−10.9949	+	6.34791i	−0.370428	+	0.213866i	0.303218	+	0.952921i	$0.401939\pi$
$882$	29.6972	+	0.274082i	0.999957	+	0.00922881i
$883$	−19.8559	+	34.3914i	−0.668203	+	1.15736i	0.310203	+	0.950670i	$0.399603\pi$
$883$	−19.8559	+	34.3914i	−0.668203	+	1.15736i	−0.978406	+	0.206691i	$0.933730\pi$
$884$		0			0
$885$		0			0
$886$	−22.6219	−	8.23370i	−0.759998	−	0.276617i
$887$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$887$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−37.8326	+	43.4333i	−1.26744	+	1.45507i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−9.91625	+	56.2379i	−0.330910	+	1.87668i
$899$		0			0
$900$	26.1181	−	14.7596i	0.870603	−	0.491986i
$901$		0			0
$902$	39.2633	+	107.875i	1.30732	+	3.59184i
$903$		0			0
$904$	−46.0322	+	38.6256i	−1.53101	+	1.28467i
$905$		0			0
$906$		0			0
$907$	56.5987	−	20.6002i	1.87933	−	0.684020i	0.937018	−	0.349281i	$-0.113574\pi$
$907$	56.5987	−	20.6002i	1.87933	−	0.684020i	0.942311	−	0.334738i	$-0.108648\pi$
$908$	37.1531	−	21.4503i	1.23297	−	0.711854i
$909$		0			0
$910$		0			0
$911$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$911$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$912$	−6.40806	−	3.02420i	−0.212192	−	0.100141i
$913$	−102.255	−	37.2176i	−3.38413	−	1.23172i
$914$	−17.8596	+	3.14913i	−0.590743	+	0.104164i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	33.8349	+	49.7737i	1.11672	+	1.64278i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	−58.3795	+	15.3544i	−1.92367	+	0.505943i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	9.67379	+	26.5785i	0.317387	+	0.872013i	0.991112	+	0.133031i	$0.0424710\pi$
$929$	9.67379	+	26.5785i	0.317387	+	0.872013i	−0.673725	+	0.738982i	$0.735307\pi$
$930$		0			0
$931$	5.48431	−	4.60189i	0.179741	−	0.150821i
$932$	20.0245	+	23.8643i	0.655925	+	0.781701i
$933$		0			0
$934$	52.4886	−	19.1043i	1.71748	−	0.625111i
$935$		0			0
$936$		0			0
$937$	13.5454	−	23.4613i	0.442509	−	0.766448i	−0.555366	−	0.831606i	$-0.687422\pi$
$937$	13.5454	−	23.4613i	0.442509	−	0.766448i	0.997875	+	0.0651578i	$0.0207551\pi$
$938$		0			0
$939$	3.00367	+	36.2583i	0.0980210	+	1.18325i
$940$		0			0
$941$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$941$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$942$		0			0
$943$		0			0
$944$		−	44.9486i		−	1.46295i
$945$		0			0
$946$	19.9683			0.649226
$947$	38.8571	−	46.3081i	1.26269	−	1.50481i	0.487435	−	0.873160i	$-0.337933\pi$
$947$	38.8571	−	46.3081i	1.26269	−	1.50481i	0.775252	−	0.631652i	$-0.217623\pi$
$948$		0			0
$949$		0			0
$950$	2.47347	−	6.79581i	0.0802500	−	0.220485i
$951$		0			0
$952$		0			0
$953$	42.6261	+	24.6102i	1.38079	+	0.797202i	0.992253	−	0.124230i	$-0.0396461\pi$
$953$	42.6261	+	24.6102i	1.38079	+	0.797202i	0.388540	+	0.921432i	$0.372979\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−29.1305	+	10.6026i	−0.939693	+	0.342020i
$962$		0			0
$963$	−17.2357	+	29.2269i	−0.555414	+	0.941823i
$964$	21.2276	−	36.7673i	0.683695	−	1.18419i
$965$		0			0
$966$		0			0
$967$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$967$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$968$	83.4522	−	14.7149i	2.68226	−	0.472954i
$969$	13.9966	+	3.81970i	0.449636	+	0.122706i
$970$		0			0
$971$			31.1127i			0.998454i	0.866471	+	0.499227i	$0.166383\pi$
$971$			31.1127i			0.998454i	−0.866471	+	0.499227i	$0.833617\pi$
$972$	−12.5206	−	28.5523i	−0.401597	−	0.915817i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	20.7007	−	56.8748i	0.662275	−	1.81959i	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	20.7007	−	56.8748i	0.662275	−	1.81959i	0.566296	−	0.824202i	$-0.308376\pi$
$978$	−29.3730	−	42.3637i	−0.939245	−	1.35464i
$979$	14.1808	−	80.4235i	0.453221	−	2.57035i
$980$		0			0
$981$		0			0
$982$	−26.7503	−	46.3328i	−0.853635	−	1.47854i
$983$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$983$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$984$	−61.8743	−	5.70112i	−1.97248	−	0.181745i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$991$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$992$		0			0
$993$	−14.1685	−	6.68666i	−0.449624	−	0.212195i
$994$		0			0
$995$		0			0
$996$	41.8396	−	41.4552i	1.32574	−	1.31356i
$997$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$997$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$998$			23.4004i			0.740725i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.2.v.a.11.2	✓	12
3.2	odd	2		648.2.v.a.35.1		12
4.3	odd	2		864.2.bh.a.335.2		12
8.3	odd	2	CM	216.2.v.a.11.2	✓	12
8.5	even	2		864.2.bh.a.335.2		12
24.11	even	2		648.2.v.a.35.1		12
27.5	odd	18	inner	216.2.v.a.59.2	yes	12
27.22	even	9		648.2.v.a.611.1		12
108.59	even	18		864.2.bh.a.815.2		12
216.5	odd	18		864.2.bh.a.815.2		12
216.59	even	18	inner	216.2.v.a.59.2	yes	12
216.211	odd	18		648.2.v.a.611.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.2.v.a.11.2	✓	12	1.1	even	1	trivial
216.2.v.a.11.2	✓	12	8.3	odd	2	CM
216.2.v.a.59.2	yes	12	27.5	odd	18	inner
216.2.v.a.59.2	yes	12	216.59	even	18	inner
648.2.v.a.35.1		12	3.2	odd	2
648.2.v.a.35.1		12	24.11	even	2
648.2.v.a.611.1		12	27.22	even	9
648.2.v.a.611.1		12	216.211	odd	18
864.2.bh.a.335.2		12	4.3	odd	2
864.2.bh.a.335.2		12	8.5	even	2
864.2.bh.a.815.2		12	108.59	even	18
864.2.bh.a.815.2		12	216.5	odd	18

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.v (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(0.909039 - 1.08335i) q^{2} +(0.456003 - 1.67095i) q^{3} +(-0.347296 - 1.96962i) q^{4} +(-1.39570 - 2.01297i) q^{6} +(-2.44949 - 1.41421i) q^{8} +(-2.58412 - 1.52391i) q^{9} +O(q^{10})\)	\(q+(0.909039 - 1.08335i) q^{2} +(0.456003 - 1.67095i) q^{3} +(-0.347296 - 1.96962i) q^{4} +(-1.39570 - 2.01297i) q^{6} +(-2.44949 - 1.41421i) q^{8} +(-2.58412 - 1.52391i) q^{9} +(2.18893 + 6.01403i) q^{11} +(-3.44949 - 0.317837i) q^{12} +(-3.75877 + 1.36808i) q^{16} +(7.09286 - 4.09506i) q^{17} +(-4.00000 + 1.41421i) q^{18} +(-0.511376 + 0.885729i) q^{19} +(8.50512 + 3.09561i) q^{22} +(-3.48005 + 3.44808i) q^{24} +(-3.83022 - 3.21394i) q^{25} +(-3.72474 + 3.62302i) q^{27} +(-1.93476 + 5.31570i) q^{32} +(11.0473 - 0.915165i) q^{33} +(2.01130 - 11.4066i) q^{34} +(-2.10407 + 5.61898i) q^{36} +(0.494694 + 1.35916i) q^{38} +(8.15282 + 9.71615i) q^{41} +(2.07316 - 0.754568i) q^{43} +(11.0851 - 6.40000i) q^{44} +(0.571978 + 6.90455i) q^{48} +(-6.57785 - 2.39414i) q^{49} +(-6.96364 + 1.22788i) q^{50} +(-3.60827 - 13.7191i) q^{51} +(0.539062 + 7.32867i) q^{54} +(1.24682 + 1.25838i) q^{57} +(-3.84333 + 10.5595i) q^{59} +(4.00000 + 6.92820i) q^{64} +(9.05096 - 12.8000i) q^{66} +(-4.53904 + 3.80871i) q^{67} +(-10.5290 - 12.5480i) q^{68} +(4.17464 + 7.38731i) q^{72} +(4.68819 - 8.12018i) q^{73} +(-7.11691 + 4.93453i) q^{75} +(1.92214 + 0.699604i) q^{76} +(4.35538 + 7.87596i) q^{81} +17.9372 q^{82} +(-10.9291 + 13.0248i) q^{83} +(1.06712 - 2.93189i) q^{86} +(3.14337 - 17.8269i) q^{88} +(-11.0505 - 6.38002i) q^{89} +(8.00000 + 5.65685i) q^{96} +(-14.0731 + 5.12218i) q^{97} +(-8.57321 + 4.94975i) q^{98} +(3.50840 - 18.8767i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q + 18 q^{11} - 12 q^{12} - 48 q^{18} + 12 q^{22} - 30 q^{27} + 6 q^{33} - 24 q^{34} + 72 q^{38} - 18 q^{41} + 30 q^{43} - 12 q^{51} + 42 q^{57} + 36 q^{59} + 48 q^{64} - 42 q^{67} - 72 q^{68} - 24 q^{76} + 36 q^{86} + 48 q^{88} - 162 q^{89} + 96 q^{96} + 30 q^{97}+O(q^{100}) \) 12 * q + 18 * q^11 - 12 * q^12 - 48 * q^18 + 12 * q^22 - 30 * q^27 + 6 * q^33 - 24 * q^34 + 72 * q^38 - 18 * q^41 + 30 * q^43 - 12 * q^51 + 42 * q^57 + 36 * q^59 + 48 * q^64 - 42 * q^67 - 72 * q^68 - 24 * q^76 + 36 * q^86 + 48 * q^88 - 162 * q^89 + 96 * q^96 + 30 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more