LMFDB - Embedded newform 864.2.bh.a.815.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,2,Mod(47,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 9, 7]))

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

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

chi := DirichletCharacter("864.47");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	864.bh (of order $18$, degree $6$, not 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:	$6.89907473464$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 216)
Sato-Tate group:	$\mathrm{U}(1)[D_{18}]$

Embedding invariants

Embedding label			815.2
Root			$-1.39273 + 0.245576i$ of defining polynomial
Character	$\chi$	$=$	864.815
Dual form			864.2.bh.a.335.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.456003 - 1.67095i) q^{3} +(-2.58412 + 1.52391i) q^{9} +O(q^{10})$	$q+(-0.456003 - 1.67095i) q^{3} +(-2.58412 + 1.52391i) q^{9} +(-2.18893 + 6.01403i) q^{11} +(7.09286 + 4.09506i) q^{17} +(0.511376 + 0.885729i) q^{19} +(-3.83022 + 3.21394i) q^{25} +(3.72474 + 3.62302i) q^{27} +(11.0473 + 0.915165i) q^{33} +(8.15282 - 9.71615i) q^{41} +(-2.07316 - 0.754568i) q^{43} +(-6.57785 + 2.39414i) q^{49} +(3.60827 - 13.7191i) q^{51} +(1.24682 - 1.25838i) q^{57} +(3.84333 + 10.5595i) q^{59} +(4.53904 + 3.80871i) q^{67} +(4.68819 + 8.12018i) q^{73} +(7.11691 + 4.93453i) q^{75} +(4.35538 - 7.87596i) q^{81} +(10.9291 + 13.0248i) q^{83} +(-11.0505 + 6.38002i) q^{89} +(-14.0731 - 5.12218i) q^{97} +(-3.50840 - 18.8767i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q - 18 q^{11} + 30 q^{27} + 6 q^{33} - 18 q^{41} - 30 q^{43} + 12 q^{51} + 42 q^{57} - 36 q^{59} + 42 q^{67} - 162 q^{89} + 30 q^{97}+O(q^{100}) $ 12 * q - 18 * q^11 + 30 * q^27 + 6 * q^33 - 18 * q^41 - 30 * q^43 + 12 * q^51 + 42 * q^57 - 36 * q^59 + 42 * q^67 - 162 * q^89 + 30 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$.

$n$	$325$	$353$	$703$
$\chi(n)$	$-1$	$e\left(\frac{5}{18}\right)$	$-1$

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
$3$	−0.456003	−	1.67095i	−0.263274	−	0.964721i
$3$	−0.456003	−	1.67095i	−0.263274	−	0.964721i
$4$		0			0
$5$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$5$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$6$		0			0
$7$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$7$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$8$		0			0
$9$	−2.58412	+	1.52391i	−0.861374	+	0.507971i
$10$		0			0
$11$	−2.18893	+	6.01403i	−0.659987	+	1.81330i	−0.0829925	+	0.996550i	$0.526448\pi$
$11$	−2.18893	+	6.01403i	−0.659987	+	1.81330i	−0.576994	+	0.816748i	$0.695774\pi$
$12$		0			0
$13$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$13$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	7.09286	+	4.09506i	1.72027	+	0.993199i	0.918351	+	0.395768i	$0.129521\pi$
$17$	7.09286	+	4.09506i	1.72027	+	0.993199i	0.801920	+	0.597431i	$0.203812\pi$
$18$		0			0
$19$	0.511376	+	0.885729i	0.117318	+	0.203200i	0.918704	−	0.394947i	$-0.129237\pi$
$19$	0.511376	+	0.885729i	0.117318	+	0.203200i	−0.801386	+	0.598147i	$0.795904\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$23$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$24$		0			0
$25$	−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	$-0.722222\pi$
$29$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$30$		0			0
$31$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$31$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$32$		0			0
$33$	11.0473	+	0.915165i	1.92308	+	0.159310i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$37$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	8.15282	−	9.71615i	1.27326	−	1.51741i	0.530831	−	0.847477i	$-0.321880\pi$
$41$	8.15282	−	9.71615i	1.27326	−	1.51741i	0.742424	−	0.669930i	$-0.233676\pi$
$42$		0			0
$43$	−2.07316	−	0.754568i	−0.316154	−	0.115071i	0.179069	−	0.983836i	$-0.442691\pi$
$43$	−2.07316	−	0.754568i	−0.316154	−	0.115071i	−0.495223	+	0.868766i	$0.664914\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$47$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$48$		0			0
$49$	−6.57785	+	2.39414i	−0.939693	+	0.342020i
$50$		0			0
$51$	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			0
$55$		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	0.890925	+	0.454150i	$0.150057\pi$
$59$	3.84333	+	10.5595i	0.500359	+	1.37472i	−0.390567	+	0.920575i	$0.627721\pi$
$60$		0			0
$61$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$61$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	4.53904	+	3.80871i	0.554532	+	0.465308i	0.876472	−	0.481452i	$-0.159891\pi$
$67$	4.53904	+	3.80871i	0.554532	+	0.465308i	−0.321940	+	0.946760i	$0.604335\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$71$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$72$		0			0
$73$	4.68819	+	8.12018i	0.548711	+	0.950395i	0.998363	+	0.0571917i	$0.0182146\pi$
$73$	4.68819	+	8.12018i	0.548711	+	0.950395i	−0.449652	+	0.893204i	$0.648452\pi$
$74$		0			0
$75$	7.11691	+	4.93453i	0.821790	+	0.569790i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$79$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$80$		0			0
$81$	4.35538	−	7.87596i	0.483931	−	0.875106i
$82$		0			0
$83$	10.9291	+	13.0248i	1.19963	+	1.42966i	0.875210	+	0.483743i	$0.160723\pi$
$83$	10.9291	+	13.0248i	1.19963	+	1.42966i	0.324415	+	0.945915i	$0.394833\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−11.0505	+	6.38002i	−1.17135	+	0.676280i	−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−11.0505	+	6.38002i	−1.17135	+	0.676280i	−0.217354	+	0.976093i	$0.569742\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−14.0731	−	5.12218i	−1.42890	−	0.520079i	−0.492287	−	0.870433i	$-0.663839\pi$
$97$	−14.0731	−	5.12218i	−1.42890	−	0.520079i	−0.936617	+	0.350354i	$0.886061\pi$
$98$		0			0
$99$	−3.50840	−	18.8767i	−0.352608	−	1.89718i
$100$		0			0
$101$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$101$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$102$		0			0
$103$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$103$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$104$		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	$-0.184115\pi$
$108$		0			0
$109$		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	0.377048	+	0.926194i	$0.376939\pi$
$113$	7.26632	+	19.9641i	0.683558	+	1.87806i	0.306510	+	0.951867i	$0.400839\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		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	$-0.666667\pi$
$127$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$128$		0			0
$129$	−0.315476	+	3.80822i	−0.0277761	+	0.335295i
$130$		0			0
$131$	8.38799	+	1.47903i	0.732862	+	0.129223i	0.527611	−	0.849486i	$-0.323088\pi$
$131$	8.38799	+	1.47903i	0.732862	+	0.129223i	0.205251	+	0.978709i	$0.434199\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.27249	+	1.51649i	0.108716	+	0.129562i	0.817658	−	0.575704i	$-0.195272\pi$
$137$	1.27249	+	1.51649i	0.108716	+	0.129562i	−0.708942	+	0.705266i	$0.750827\pi$
$138$		0			0
$139$	2.11445	−	11.9916i	0.179345	−	1.01712i	−0.753663	−	0.657262i	$-0.771715\pi$
$139$	2.11445	−	11.9916i	0.179345	−	1.01712i	0.933008	−	0.359856i	$-0.117174\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	7.00000	+	9.89949i	0.577350	+	0.816497i
$148$		0			0
$149$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$149$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$150$		0			0
$151$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$151$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$152$		0			0
$153$	−24.5693	+	0.226755i	−1.98631	+	0.0183321i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$157$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−21.0454			−1.64840			−0.824202	−	0.566296i	$-0.808376\pi$
$163$	−21.0454			−1.64840			−0.824202	+	0.566296i	$0.808376\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$167$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$168$		0			0
$169$	2.25743	+	12.8025i	0.173648	+	0.984808i
$170$		0			0
$171$	−2.67123	−	1.50954i	−0.204274	−	0.115437i
$172$		0			0
$173$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$173$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	15.8917	−	11.2371i	1.19449	−	0.844635i
$178$		0			0
$179$	−22.0732	−	12.7440i	−1.64983	−	0.952529i	−0.977138	−	0.212607i	$-0.931805\pi$
$179$	−22.0732	−	12.7440i	−1.64983	−	0.952529i	−0.672692	−	0.739923i	$-0.734862\pi$
$180$		0			0
$181$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$181$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$182$		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	$-0.722222\pi$
$191$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$192$		0			0
$193$	2.58197	−	14.6431i	0.185854	−	1.05403i	−0.738999	−	0.673707i	$-0.764701\pi$
$193$	2.58197	−	14.6431i	0.185854	−	1.05403i	0.924853	−	0.380325i	$-0.124188\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$197$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$198$		0			0
$199$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$199$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$200$		0			0
$201$	4.29433	−	9.32127i	0.302899	−	0.657472i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		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	$-0.437831\pi$
$211$	14.1381	−	5.14583i	0.973304	−	0.354254i	0.779233	+	0.626734i	$0.215609\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		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	$-0.555556\pi$
$223$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$224$		0			0
$225$	5.00000	−	14.1421i	0.333333	−	0.942809i
$226$		0			0
$227$	−7.33644	+	20.1567i	−0.486937	+	1.33785i	0.416503	+	0.909134i	$0.363255\pi$
$227$	−7.33644	+	20.1567i	−0.486937	+	1.33785i	−0.903440	+	0.428714i	$0.858967\pi$
$228$		0			0
$229$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$229$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−13.4895	−	7.78815i	−0.883725	−	0.510219i	−0.0118403	−	0.999930i	$-0.503769\pi$
$233$	−13.4895	−	7.78815i	−0.883725	−	0.510219i	−0.871885	+	0.489711i	$0.837102\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$239$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$240$		0			0
$241$	16.2613	−	13.6448i	1.04748	−	0.878941i	0.0546547	−	0.998505i	$-0.482594\pi$
$241$	16.2613	−	13.6448i	1.04748	−	0.878941i	0.992826	+	0.119564i	$0.0381497\pi$
$242$		0			0
$243$	−15.1464	−	3.68614i	−0.971640	−	0.236466i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		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	0.513771	−	0.857927i	$-0.328248\pi$
$251$	23.9806	−	13.8452i	1.51364	−	0.873902i	0.999872	−	0.0159750i	$-0.00508522\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	17.7278	−	21.1272i	1.10583	−	1.31788i	0.162247	−	0.986750i	$-0.448126\pi$
$257$	17.7278	−	21.1272i	1.10583	−	1.31788i	0.943585	−	0.331130i	$-0.107430\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$263$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	15.6997	+	15.5555i	0.960808	+	0.951981i
$268$		0			0
$269$		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			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−10.9446	−	30.0702i	−0.659987	−	1.81330i
$276$		0			0
$277$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$277$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−10.1685	+	27.9376i	−0.606600	+	1.66662i	0.131002	+	0.991382i	$0.458181\pi$
$281$	−10.1685	+	27.9376i	−0.606600	+	1.66662i	−0.737601	+	0.675236i	$0.764042\pi$
$282$		0			0
$283$	−25.3143	−	21.2412i	−1.50478	−	1.26266i	−0.873219	−	0.487327i	$-0.837972\pi$
$283$	−25.3143	−	21.2412i	−1.50478	−	1.26266i	−0.631557	−	0.775330i	$-0.717584\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	25.0391	+	43.3690i	1.47289	+	2.55112i
$290$		0			0
$291$	−2.14152	+	25.8511i	−0.125538	+	1.51542i
$292$		0			0
$293$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$293$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−29.9422	+	14.4702i	−1.73742	+	0.839646i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	17.4258	−	30.1824i	0.994545	−	1.72260i	0.406942	−	0.913454i	$-0.366595\pi$
$307$	17.4258	−	30.1824i	0.994545	−	1.72260i	0.587603	−	0.809149i	$-0.300072\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$311$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$312$		0			0
$313$	−19.7387	−	7.18430i	−1.11570	−	0.406080i	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−19.7387	−	7.18430i	−1.11570	−	0.406080i	−0.833080	+	0.553153i	$0.813425\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$317$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$318$		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
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.57072	−	8.90799i	−0.0863345	−	0.489627i	−0.997061	−	0.0766165i	$-0.975588\pi$
$331$	−1.57072	−	8.90799i	−0.0863345	−	0.489627i	0.910726	−	0.413011i	$-0.135523\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	24.9283	+	20.9173i	1.35793	+	1.13944i	0.976616	+	0.214993i	$0.0689729\pi$
$337$	24.9283	+	20.9173i	1.35793	+	1.13944i	0.381314	+	0.924445i	$0.375472\pi$
$338$		0			0
$339$	30.0454	−	21.2453i	1.63184	−	1.15389i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	36.0481	+	6.35625i	1.93516	+	0.341222i	0.999918	−	0.0127797i	$-0.00406802\pi$
$347$	36.0481	+	6.35625i	1.93516	+	0.341222i	0.935245	+	0.354001i	$0.115179\pi$
$348$		0			0
$349$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$349$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.25345	+	1.49380i	0.0667144	+	0.0795071i	0.798369	−	0.602168i	$-0.205696\pi$
$353$	1.25345	+	1.49380i	0.0667144	+	0.0795071i	−0.731655	+	0.681675i	$0.761252\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$359$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$360$		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	$-0.888889\pi$
$367$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$368$		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	$-0.111111\pi$
$373$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	32.8062			1.68514			0.842570	−	0.538587i	$-0.181042\pi$
$379$	32.8062			1.68514			0.842570	+	0.538587i	$0.181042\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$383$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	6.50720	−	1.20942i	0.330779	−	0.0614782i
$388$		0			0
$389$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$389$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−1.35357	−	14.6903i	−0.0682787	−	0.741029i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$397$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	34.6237	+	6.10509i	1.72902	+	0.304874i	0.947679	−	0.319225i	$-0.103423\pi$
$401$	34.6237	+	6.10509i	1.72902	+	0.304874i	0.781345	+	0.624099i	$0.214534\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.861982	−	4.88854i	0.0426223	−	0.241723i	−0.956052	−	0.293197i	$-0.905281\pi$
$409$	0.861982	−	4.88854i	0.0426223	−	0.241723i	0.998674	+	0.0514740i	$0.0163919\pi$
$410$		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
$418$		0			0
$419$	−21.8376	+	26.0250i	−1.06684	+	1.27140i	−0.105976	+	0.994369i	$0.533797\pi$
$419$	−21.8376	+	26.0250i	−1.06684	+	1.27140i	−0.960860	+	0.277036i	$0.910648\pi$
$420$		0			0
$421$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$421$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−40.3285	+	7.11100i	−1.95622	+	0.344934i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	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
$438$		0			0
$439$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$439$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$440$		0			0
$441$	13.3495	−	16.2108i	0.635691	−	0.771944i
$442$		0			0
$443$	5.82211	−	15.9961i	0.276617	−	0.759998i	−0.721124	−	0.692807i	$-0.756374\pi$
$443$	5.82211	−	15.9961i	0.276617	−	0.759998i	0.997740	−	0.0671913i	$-0.0214038\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−34.9698	−	20.1898i	−1.65033	−	0.952816i	−0.976937	−	0.213528i	$-0.931505\pi$
$449$	−34.9698	−	20.1898i	−1.65033	−	0.952816i	−0.673389	−	0.739288i	$-0.735162\pi$
$450$		0			0
$451$	40.5873	+	70.2992i	1.91118	+	3.31026i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−9.82334	+	8.24276i	−0.459516	+	0.385580i	−0.842953	−	0.537987i	$-0.819185\pi$
$457$	−9.82334	+	8.24276i	−0.459516	+	0.385580i	0.383437	+	0.923567i	$0.374740\pi$
$458$		0			0
$459$	11.5826	+	40.9506i	0.540629	+	1.91141i
$460$		0			0
$461$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$461$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$462$		0			0
$463$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$463$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−34.2054	+	19.7485i	−1.58284	+	0.913851i	−0.588393	+	0.808575i	$0.700239\pi$
$467$	−34.2054	+	19.7485i	−1.58284	+	0.913851i	−0.994443	+	0.105276i	$0.966427\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	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	$-0.0555556\pi$
$479$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		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	−0.781419	−	0.624006i	$-0.785504\pi$
$491$	−12.9388	−	35.5491i	−0.583921	−	1.60431i	0.197499	−	0.980303i	$-0.436718\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	12.6754	+	10.6359i	0.567428	+	0.476129i	0.880791	−	0.473504i	$-0.157011\pi$
$499$	12.6754	+	10.6359i	0.567428	+	0.476129i	−0.313363	+	0.949633i	$0.601456\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$503$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$504$		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	$-0.944444\pi$
$509$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−1.30427	+	5.15184i	−0.0575849	+	0.227459i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	26.6827	−	15.4052i	1.16899	−	0.674916i	0.215547	−	0.976493i	$-0.430847\pi$
$521$	26.6827	−	15.4052i	1.16899	−	0.674916i	0.953442	+	0.301577i	$0.0975132\pi$
$522$		0			0
$523$	1.52270	−	2.63740i	0.0665832	−	0.115325i	−0.830812	−	0.556553i	$-0.812124\pi$
$523$	1.52270	−	2.63740i	0.0665832	−	0.115325i	0.897395	+	0.441228i	$0.145457\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	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
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−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
$544$		0			0
$545$		0			0
$546$		0			0
$547$	5.17190	+	29.3313i	0.221135	+	1.25412i	0.869938	+	0.493161i	$0.164159\pi$
$547$	5.17190	+	29.3313i	0.221135	+	1.25412i	−0.648803	+	0.760956i	$0.724730\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$557$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	74.6091	+	51.7304i	3.15000	+	2.18406i
$562$		0			0
$563$	22.8074	+	4.02156i	0.961218	+	0.169489i	0.632175	−	0.774826i	$-0.282163\pi$
$563$	22.8074	+	4.02156i	0.961218	+	0.169489i	0.329044	+	0.944315i	$0.393274\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	4.43393	+	5.28415i	0.185880	+	0.221523i	0.850935	−	0.525271i	$-0.176036\pi$
$569$	4.43393	+	5.28415i	0.185880	+	0.221523i	−0.665055	+	0.746795i	$0.731592\pi$
$570$		0			0
$571$	7.66208	−	43.4538i	0.320648	−	1.81848i	−0.217994	−	0.975950i	$-0.569951\pi$
$571$	7.66208	−	43.4538i	0.320648	−	1.81848i	0.538642	−	0.842535i	$-0.318938\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	19.6648	−	34.0604i	0.818655	−	1.41795i	−0.0880190	−	0.996119i	$-0.528054\pi$
$577$	19.6648	−	34.0604i	0.818655	−	1.41795i	0.906674	−	0.421833i	$-0.138613\pi$
$578$		0			0
$579$	−25.6452	+	2.36296i	−1.06578	+	0.0982011i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−2.40364	+	0.423827i	−0.0992090	+	0.0174932i	−0.223032	−	0.974811i	$-0.571596\pi$
$587$	−2.40364	+	0.423827i	−0.0992090	+	0.0174932i	0.123823	+	0.992304i	$0.460484\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	38.2159i		−	1.56934i	−0.619915	−	0.784669i	$-0.712833\pi$
$593$		−	38.2159i		−	1.56934i	0.619915	−	0.784669i	$-0.287167\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$599$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$600$		0			0
$601$	1.51507	+	8.59237i	0.0618009	+	0.350490i	0.999990	+	0.00436841i	$0.00139051\pi$
$601$	1.51507	+	8.59237i	0.0618009	+	0.350490i	−0.938190	+	0.346122i	$0.887498\pi$
$602$		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	$-0.777778\pi$
$607$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$613$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−35.8670	−	6.32433i	−1.44395	−	0.254608i	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−35.8670	−	6.32433i	−1.44395	−	0.254608i	−0.840076	+	0.542469i	$0.817489\pi$
$618$		0			0
$619$	29.8318	−	25.0319i	1.19904	−	1.00612i	0.199386	−	0.979921i	$-0.436105\pi$
$619$	29.8318	−	25.0319i	1.19904	−	1.00612i	0.999657	−	0.0261952i	$-0.00833914\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	4.34120	−	24.6202i	0.173648	−	0.984808i
$626$		0			0
$627$	4.83873	+	10.2529i	0.193240	+	0.409461i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$631$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$632$		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	0.694127	−	0.719852i	$-0.255791\pi$
$641$	40.3213	−	7.10974i	1.59260	−	0.280818i	0.898470	+	0.439034i	$0.144679\pi$
$642$		0			0
$643$	0.306376	−	0.111512i	0.0120823	−	0.00439759i	−0.335972	−	0.941872i	$-0.609065\pi$
$643$	0.306376	−	0.111512i	0.0120823	−	0.00439759i	0.348054	+	0.937474i	$0.386843\pi$
$644$		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
$649$	−71.9177			−2.82302
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$653$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−24.4893	−	13.8392i	−0.955419	−	0.539917i
$658$		0			0
$659$	−2.89116	+	7.94338i	−0.112623	+	0.309430i	−0.983180	−	0.182637i	$-0.941537\pi$
$659$	−2.89116	+	7.94338i	−0.112623	+	0.309430i	0.870557	+	0.492068i	$0.163759\pi$
$660$		0			0
$661$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$661$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		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	−0.999812	−	0.0194154i	$-0.993820\pi$
$673$	−29.9453	+	25.1271i	−1.15431	+	0.968578i	−0.154495	+	0.987994i	$0.549375\pi$
$674$		0			0
$675$	−25.9108	−	1.90587i	−0.997306	−	0.0733571i
$676$		0			0
$677$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$677$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$678$		0			0
$679$		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	$-0.908165\pi$
$683$	−31.1416	+	17.9796i	−1.19160	+	0.687971i	−0.232932	+	0.972493i	$0.574832\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	42.3288	+	15.4064i	1.61026	+	0.586088i	0.981492	−	0.191501i	$-0.0613355\pi$
$691$	42.3288	+	15.4064i	1.61026	+	0.586088i	0.628772	+	0.777589i	$0.283558\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		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
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$709$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−30.2150	−	20.9496i	−1.12371	−	0.779125i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$727$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$728$		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	$-0.444444\pi$
$733$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−32.8413	+	18.9609i	−1.20973	+	0.698435i
$738$		0			0
$739$	17.9389	−	31.0711i	0.659893	−	1.14297i	−0.320749	−	0.947164i	$-0.603935\pi$
$739$	17.9389	−	31.0711i	0.659893	−	1.14297i	0.980643	−	0.195805i	$-0.0627319\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$743$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−48.0908	−	17.0027i	−1.75955	−	0.622095i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$751$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$752$		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			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−3.86952	−	10.6314i	−0.140270	−	0.385388i	0.849589	−	0.527446i	$-0.176850\pi$
$761$	−3.86952	−	10.6314i	−0.140270	−	0.385388i	−0.989858	+	0.142058i	$0.954628\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−25.3490	−	21.2704i	−0.914110	−	0.767029i	0.0587868	−	0.998271i	$-0.481277\pi$
$769$	−25.3490	−	21.2704i	−0.914110	−	0.767029i	−0.972896	+	0.231242i	$0.925721\pi$
$770$		0			0
$771$	−43.3864	−	19.9882i	−1.56252	−	0.719857i
$772$		0			0
$773$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$773$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		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
$784$		0			0
$785$		0			0
$786$		0			0
$787$	0.513060	−	2.90971i	0.0182886	−	0.103720i	−0.974297	−	0.225267i	$-0.927675\pi$
$787$	0.513060	−	2.90971i	0.0182886	−	0.103720i	0.992586	+	0.121547i	$0.0387856\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$797$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	18.8333	−	33.3268i	0.665441	−	1.17754i
$802$		0			0
$803$	−59.0971	+	10.4204i	−2.08549	+	0.367729i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	47.1907i		−	1.65914i	−0.558404	−	0.829569i	$-0.688586\pi$
$809$		−	47.1907i		−	1.65914i	0.558404	−	0.829569i	$-0.311414\pi$
$810$		0			0
$811$	−48.3805			−1.69887			−0.849434	−	0.527694i	$-0.823057\pi$
$811$	−48.3805			−1.69887			−0.849434	+	0.527694i	$0.823057\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−0.391821	−	2.22213i	−0.0137081	−	0.0777424i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$821$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$822$		0			0
$823$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$823$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$824$		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	$-0.221470\pi$
$827$	17.1464	+	9.89949i	0.596240	+	0.344239i	−0.171321	+	0.985215i	$0.554804\pi$
$828$		0			0
$829$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$829$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−56.4599	−	9.95540i	−1.95622	−	0.344934i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$839$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$840$		0			0
$841$	−5.03580	+	28.5594i	−0.173648	+	0.984808i
$842$		0			0
$843$	51.3191	+	4.25131i	1.76752	+	0.146423i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−23.9495	+	51.9848i	−0.821944	+	1.78411i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$853$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	22.2837	−	3.92921i	0.761195	−	0.134219i	0.220441	−	0.975400i	$-0.429250\pi$
$857$	22.2837	−	3.92921i	0.761195	−	0.134219i	0.540754	+	0.841181i	$0.318139\pi$
$858$		0			0
$859$	−17.3166	+	6.30274i	−0.590836	+	0.215047i	−0.620097	−	0.784525i	$-0.712907\pi$
$859$	−17.3166	+	6.30274i	−0.590836	+	0.215047i	0.0292613	+	0.999572i	$0.490684\pi$
$860$		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
$865$		0			0
$866$		0			0
$867$	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
$876$		0			0
$877$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$877$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−10.9949	−	6.34791i	−0.370428	−	0.213866i	0.303218	−	0.952921i	$-0.401939\pi$
$881$	−10.9949	−	6.34791i	−0.370428	−	0.213866i	−0.673645	+	0.739055i	$0.735272\pi$
$882$		0			0
$883$	19.8559	+	34.3914i	0.668203	+	1.15736i	0.978406	+	0.206691i	$0.0662696\pi$
$883$	19.8559	+	34.3914i	0.668203	+	1.15736i	−0.310203	+	0.950670i	$0.600397\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$887$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$888$		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
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−56.5987	−	20.6002i	−1.87933	−	0.684020i	−0.942311	−	0.334738i	$-0.891352\pi$
$907$	−56.5987	−	20.6002i	−1.87933	−	0.684020i	−0.937018	−	0.349281i	$-0.886426\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$911$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$912$		0			0
$913$	−102.255	+	37.2176i	−3.38413	+	1.23172i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		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	−0.673725	−	0.738982i	$-0.735307\pi$
$929$	9.67379	−	26.5785i	0.317387	−	0.872013i	0.991112	−	0.133031i	$-0.0424710\pi$
$930$		0			0
$931$	−5.48431	−	4.60189i	−0.179741	−	0.150821i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	13.5454	+	23.4613i	0.442509	+	0.766448i	0.997875	−	0.0651578i	$-0.0207551\pi$
$937$	13.5454	+	23.4613i	0.442509	+	0.766448i	−0.555366	+	0.831606i	$0.687422\pi$
$938$		0			0
$939$	−3.00367	+	36.2583i	−0.0980210	+	1.18325i
$940$		0			0
$941$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$941$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−38.8571	−	46.3081i	−1.26269	−	1.50481i	−0.775252	−	0.631652i	$-0.782377\pi$
$947$	−38.8571	−	46.3081i	−1.26269	−	1.50481i	−0.487435	−	0.873160i	$-0.662067\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	42.6261	−	24.6102i	1.38079	−	0.797202i	0.388540	−	0.921432i	$-0.372979\pi$
$953$	42.6261	−	24.6102i	1.38079	−	0.797202i	0.992253	+	0.124230i	$0.0396461\pi$
$954$		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
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$967$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$968$		0			0
$969$	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	$0.833617\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	20.7007	+	56.8748i	0.662275	+	1.81959i	0.566296	+	0.824202i	$0.308376\pi$
$977$	20.7007	+	56.8748i	0.662275	+	1.81959i	0.0959785	+	0.995383i	$0.469402\pi$
$978$		0			0
$979$	−14.1808	−	80.4235i	−0.453221	−	2.57035i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$983$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$991$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$992$		0			0
$993$	−14.1685	+	6.68666i	−0.449624	+	0.212195i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$997$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.2.bh.a.815.2		12
4.3	odd	2		216.2.v.a.59.2	yes	12
8.3	odd	2	CM	864.2.bh.a.815.2		12
8.5	even	2		216.2.v.a.59.2	yes	12
12.11	even	2		648.2.v.a.611.1		12
24.5	odd	2		648.2.v.a.611.1		12
27.11	odd	18	inner	864.2.bh.a.335.2		12
108.11	even	18		216.2.v.a.11.2	✓	12
108.43	odd	18		648.2.v.a.35.1		12
216.11	even	18	inner	864.2.bh.a.335.2		12
216.173	odd	18		216.2.v.a.11.2	✓	12
216.205	even	18		648.2.v.a.35.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.2.v.a.11.2	✓	12	108.11	even	18
216.2.v.a.11.2	✓	12	216.173	odd	18
216.2.v.a.59.2	yes	12	4.3	odd	2
216.2.v.a.59.2	yes	12	8.5	even	2
648.2.v.a.35.1		12	108.43	odd	18
648.2.v.a.35.1		12	216.205	even	18
648.2.v.a.611.1		12	12.11	even	2
648.2.v.a.611.1		12	24.5	odd	2
864.2.bh.a.335.2		12	27.11	odd	18	inner
864.2.bh.a.335.2		12	216.11	even	18	inner
864.2.bh.a.815.2		12	1.1	even	1	trivial
864.2.bh.a.815.2		12	8.3	odd	2	CM

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:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	864.bh (of order \(18\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.456003 - 1.67095i) q^{3} +(-2.58412 + 1.52391i) q^{9} +O(q^{10})\)	\(q+(-0.456003 - 1.67095i) q^{3} +(-2.58412 + 1.52391i) q^{9} +(-2.18893 + 6.01403i) q^{11} +(7.09286 + 4.09506i) q^{17} +(0.511376 + 0.885729i) q^{19} +(-3.83022 + 3.21394i) q^{25} +(3.72474 + 3.62302i) q^{27} +(11.0473 + 0.915165i) q^{33} +(8.15282 - 9.71615i) q^{41} +(-2.07316 - 0.754568i) q^{43} +(-6.57785 + 2.39414i) q^{49} +(3.60827 - 13.7191i) q^{51} +(1.24682 - 1.25838i) q^{57} +(3.84333 + 10.5595i) q^{59} +(4.53904 + 3.80871i) q^{67} +(4.68819 + 8.12018i) q^{73} +(7.11691 + 4.93453i) q^{75} +(4.35538 - 7.87596i) q^{81} +(10.9291 + 13.0248i) q^{83} +(-11.0505 + 6.38002i) q^{89} +(-14.0731 - 5.12218i) q^{97} +(-3.50840 - 18.8767i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q - 18 q^{11} + 30 q^{27} + 6 q^{33} - 18 q^{41} - 30 q^{43} + 12 q^{51} + 42 q^{57} - 36 q^{59} + 42 q^{67} - 162 q^{89} + 30 q^{97}+O(q^{100}) \) 12 * q - 18 * q^11 + 30 * q^27 + 6 * q^33 - 18 * q^41 - 30 * q^43 + 12 * q^51 + 42 * q^57 - 36 * q^59 + 42 * q^67 - 162 * q^89 + 30 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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