LMFDB - Embedded newform 864.2.bh.a.239.2

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			239.2
Root			$-0.909039 - 1.08335i$ of defining polynomial
Character	$\chi$	$=$	864.239
Dual form			864.2.bh.a.47.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.56638 + 0.739232i) q^{3} +(1.90707 + 2.31583i) q^{9} +O(q^{10})$	$q+(1.56638 + 0.739232i) q^{3} +(1.90707 + 2.31583i) q^{9} +(-5.93192 + 1.04596i) q^{11} +(6.38072 + 3.68391i) q^{17} +(4.35183 + 7.53760i) q^{19} +(4.69846 + 1.71010i) q^{25} +(1.27526 + 5.03723i) q^{27} +(-10.0648 - 2.74670i) q^{33} +(-1.34901 - 3.70637i) q^{41} +(-1.12781 - 6.39612i) q^{43} +(1.21554 - 6.89365i) q^{49} +(7.27135 + 10.4872i) q^{51} +(1.24458 + 15.0237i) q^{57} +(-8.23753 - 1.45250i) q^{59} +(4.95255 - 1.80258i) q^{67} +(-1.96244 - 3.39905i) q^{73} +(6.09540 + 6.15192i) q^{75} +(-1.72616 + 8.83292i) q^{81} +(4.84788 - 13.3194i) q^{83} +(-15.9495 + 9.20844i) q^{89} +(0.564013 + 3.19868i) q^{97} +(-13.7349 - 11.7426i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$.

$n$	$325$	$353$	$703$
$\chi(n)$	$-1$	$e\left(\frac{11}{18}\right)$	$-1$

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
$3$	1.56638	+	0.739232i	0.904348	+	0.426796i
$3$	1.56638	+	0.739232i	0.904348	+	0.426796i
$4$		0			0
$5$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$5$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$6$		0			0
$7$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$7$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$8$		0			0
$9$	1.90707	+	2.31583i	0.635691	+	0.771944i
$10$		0			0
$11$	−5.93192	+	1.04596i	−1.78854	+	0.315368i	−0.966999	−	0.254781i	$-0.917997\pi$
$11$	−5.93192	+	1.04596i	−1.78854	+	0.315368i	−0.821541	+	0.570149i	$0.806886\pi$
$12$		0			0
$13$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$13$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	6.38072	+	3.68391i	1.54755	+	0.893480i	0.998328	+	0.0578055i	$0.0184103\pi$
$17$	6.38072	+	3.68391i	1.54755	+	0.893480i	0.549225	+	0.835675i	$0.314923\pi$
$18$		0			0
$19$	4.35183	+	7.53760i	0.998379	+	1.72924i	0.548478	+	0.836165i	$0.315207\pi$
$19$	4.35183	+	7.53760i	0.998379	+	1.72924i	0.449901	+	0.893079i	$0.351459\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$23$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$24$		0			0
$25$	4.69846	+	1.71010i	0.939693	+	0.342020i
$26$		0			0
$27$	1.27526	+	5.03723i	0.245423	+	0.969416i
$28$		0			0
$29$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$29$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$30$		0			0
$31$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$31$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$32$		0			0
$33$	−10.0648	−	2.74670i	−1.75206	−	0.478139i
$34$		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
$41$	−1.34901	−	3.70637i	−0.210680	−	0.578838i	0.788673	−	0.614813i	$-0.210769\pi$
$41$	−1.34901	−	3.70637i	−0.210680	−	0.578838i	−0.999353	+	0.0359748i	$0.988546\pi$
$42$		0			0
$43$	−1.12781	−	6.39612i	−0.171989	−	0.975399i	−0.941562	−	0.336840i	$-0.890642\pi$
$43$	−1.12781	−	6.39612i	−0.171989	−	0.975399i	0.769573	−	0.638559i	$-0.220469\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$47$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$48$		0			0
$49$	1.21554	−	6.89365i	0.173648	−	0.984808i
$50$		0			0
$51$	7.27135	+	10.4872i	1.01819	+	1.46851i
$52$		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
$57$	1.24458	+	15.0237i	0.164848	+	1.98994i
$58$		0			0
$59$	−8.23753	−	1.45250i	−1.07243	−	0.189099i	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−8.23753	−	1.45250i	−1.07243	−	0.189099i	−0.681868	+	0.731475i	$0.738832\pi$
$60$		0			0
$61$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$61$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	4.95255	−	1.80258i	0.605050	−	0.220220i	−0.0212861	−	0.999773i	$-0.506776\pi$
$67$	4.95255	−	1.80258i	0.605050	−	0.220220i	0.626336	+	0.779553i	$0.284554\pi$
$68$		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
$73$	−1.96244	−	3.39905i	−0.229687	−	0.397829i	0.728028	−	0.685547i	$-0.240437\pi$
$73$	−1.96244	−	3.39905i	−0.229687	−	0.397829i	−0.957715	+	0.287718i	$0.907104\pi$
$74$		0			0
$75$	6.09540	+	6.15192i	0.703836	+	0.710362i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$79$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$80$		0			0
$81$	−1.72616	+	8.83292i	−0.191795	+	0.981435i
$82$		0			0
$83$	4.84788	−	13.3194i	0.532124	−	1.46200i	−0.324415	−	0.945915i	$-0.605167\pi$
$83$	4.84788	−	13.3194i	0.532124	−	1.46200i	0.856539	−	0.516083i	$-0.172610\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−15.9495	+	9.20844i	−1.69064	+	0.976093i	−0.736644	+	0.676280i	$0.763591\pi$
$89$	−15.9495	+	9.20844i	−1.69064	+	0.976093i	−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
$96$		0			0
$97$	0.564013	+	3.19868i	0.0572669	+	0.324777i	0.999961	−	0.00888289i	$-0.00282755\pi$
$97$	0.564013	+	3.19868i	0.0572669	+	0.324777i	−0.942694	+	0.333659i	$0.891716\pi$
$98$		0			0
$99$	−13.7349	−	11.7426i	−1.38040	−	1.18018i
$100$		0			0
$101$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$101$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$102$		0			0
$103$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$103$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	16.5528i		−	1.60022i	−0.599851	−	0.800112i	$-0.704773\pi$
$107$		−	16.5528i		−	1.60022i	0.599851	−	0.800112i	$-0.295227\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−9.78072	−	1.72460i	−0.920093	−	0.162237i	−0.306510	−	0.951867i	$-0.599161\pi$
$113$	−9.78072	−	1.72460i	−0.920093	−	0.162237i	−0.613583	+	0.789630i	$0.710272\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	23.7570	−	8.64684i	2.15973	−	0.786076i
$122$		0			0
$123$	0.626813	−	6.80281i	0.0565179	−	0.613389i
$124$		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
$129$	2.96164	−	10.8524i	0.260758	−	0.955504i
$130$		0			0
$131$	−14.5653	+	17.3582i	−1.27257	+	1.51659i	−0.527611	+	0.849486i	$0.676912\pi$
$131$	−14.5653	+	17.3582i	−1.27257	+	1.51659i	−0.744962	+	0.667107i	$0.767532\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	4.60936	−	12.6641i	0.393805	−	1.08197i	−0.571445	−	0.820640i	$-0.693617\pi$
$137$	4.60936	−	12.6641i	0.393805	−	1.08197i	0.965250	−	0.261329i	$-0.0841608\pi$
$138$		0			0
$139$	17.0883	+	14.3388i	1.44941	+	1.21620i	0.933008	+	0.359856i	$0.117174\pi$
$139$	17.0883	+	14.3388i	1.44941	+	1.21620i	0.516404	+	0.856345i	$0.327270\pi$
$140$		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		−0.342020	−	0.939693i	$-0.611111\pi$
$149$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$150$		0			0
$151$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$151$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$152$		0			0
$153$	3.63718	+	21.8022i	0.294048	+	1.76260i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$157$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	23.0454			1.80506			0.902528	−	0.430632i	$-0.141709\pi$
$163$	23.0454			1.80506			0.902528	+	0.430632i	$0.141709\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$167$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$168$		0			0
$169$	9.95858	−	8.35624i	0.766044	−	0.642788i
$170$		0			0
$171$	−9.15655	+	24.4529i	−0.700219	+	1.86996i
$172$		0			0
$173$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$173$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−11.8293	−	8.36460i	−0.889147	−	0.628722i
$178$		0			0
$179$	−4.92679	−	2.84448i	−0.368245	−	0.212607i	0.304446	−	0.952529i	$-0.401529\pi$
$179$	−4.92679	−	2.84448i	−0.368245	−	0.212607i	−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
$187$	−41.7031	−	15.1787i	−3.04964	−	1.10998i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$191$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$192$		0			0
$193$	−15.7292	−	13.1984i	−1.13221	−	0.950039i	−0.133056	−	0.991109i	$-0.542479\pi$
$193$	−15.7292	−	13.1984i	−1.13221	−	0.950039i	−0.999156	+	0.0410699i	$0.986923\pi$
$194$		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
$201$	9.09008	+	0.837563i	0.641164	+	0.0590771i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−33.6987	−	40.1606i	−2.33099	−	2.77796i
$210$		0			0
$211$	5.04368	−	28.6041i	0.347221	−	1.96919i	0.153151	−	0.988203i	$-0.451058\pi$
$211$	5.04368	−	28.6041i	0.347221	−	1.96919i	0.194071	−	0.980988i	$-0.437831\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−0.561238	−	6.77490i	−0.0379249	−	0.457805i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$223$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$224$		0			0
$225$	5.00000	+	14.1421i	0.333333	+	0.942809i
$226$		0			0
$227$	12.7222	−	2.24327i	0.844402	−	0.148891i	0.265320	−	0.964160i	$-0.414522\pi$
$227$	12.7222	−	2.24327i	0.844402	−	0.148891i	0.579082	+	0.815270i	$0.303411\pi$
$228$		0			0
$229$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$229$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	24.7354	+	14.2810i	1.62047	+	0.935577i	0.986795	+	0.161976i	$0.0517866\pi$
$233$	24.7354	+	14.2810i	1.62047	+	0.935577i	0.633672	+	0.773602i	$0.281547\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$239$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$240$		0			0
$241$	−28.9666	−	10.5430i	−1.86590	−	0.679133i	−0.973845	−	0.227213i	$-0.927039\pi$
$241$	−28.9666	−	10.5430i	−1.86590	−	0.679133i	−0.892058	−	0.451920i	$-0.850739\pi$
$242$		0			0
$243$	−9.23338	+	12.5596i	−0.592322	+	0.805701i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	17.4398	−	17.2795i	1.10520	−	1.09505i
$250$		0			0
$251$	27.0966	−	15.6442i	1.71032	−	0.987456i	0.776215	−	0.630468i	$-0.217137\pi$
$251$	27.0966	−	15.6442i	1.71032	−	0.987456i	0.934109	−	0.356988i	$-0.116196\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3.63117	−	9.97655i	−0.226506	−	0.622320i	0.773427	−	0.633885i	$-0.218541\pi$
$257$	−3.63117	−	9.97655i	−0.226506	−	0.622320i	−0.999933	+	0.0115651i	$0.996319\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$263$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−31.7901	+	2.63351i	−1.94552	+	0.161168i
$268$		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
$275$	−29.6596	−	5.22979i	−1.78854	−	0.315368i
$276$		0			0
$277$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$277$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.42435	−	0.251151i	0.0849695	−	0.0149824i	−0.131002	−	0.991382i	$-0.541819\pi$
$281$	1.42435	−	0.251151i	0.0849695	−	0.0149824i	0.215971	+	0.976400i	$0.430708\pi$
$282$		0			0
$283$	−10.3793	+	3.77775i	−0.616985	+	0.224564i	−0.631557	−	0.775330i	$-0.717584\pi$
$283$	−10.3793	+	3.77775i	−0.616985	+	0.224564i	0.0145720	+	0.999894i	$0.495361\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	18.6424	+	32.2896i	1.09661	+	1.89939i
$290$		0			0
$291$	−1.48111	+	5.42727i	−0.0868241	+	0.318152i
$292$		0			0
$293$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$293$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−12.8334	−	28.5466i	−0.744672	−	1.65644i
$298$		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
$307$	1.22617	−	2.12378i	0.0699810	−	0.121211i	−0.828912	−	0.559379i	$-0.811039\pi$
$307$	1.22617	−	2.12378i	0.0699810	−	0.121211i	0.898893	+	0.438169i	$0.144373\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$311$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$312$		0			0
$313$	5.50274	+	31.2076i	0.311033	+	1.76396i	0.593650	+	0.804723i	$0.297686\pi$
$313$	5.50274	+	31.2076i	0.311033	+	1.76396i	−0.282617	+	0.959233i	$0.591202\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$317$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	12.2364	−	25.9280i	0.682969	−	1.44716i
$322$		0			0
$323$			64.1271i			3.56813i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	26.8463	−	22.5268i	1.47561	−	1.23818i	0.564882	−	0.825172i	$-0.308922\pi$
$331$	26.8463	−	22.5268i	1.47561	−	1.23818i	0.910726	−	0.413011i	$-0.135523\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	23.2708	−	8.46987i	1.26764	−	0.461383i	0.381314	−	0.924445i	$-0.375472\pi$
$337$	23.2708	−	8.46987i	1.26764	−	0.461383i	0.886326	+	0.463062i	$0.153249\pi$
$338$		0			0
$339$	−14.0454	−	9.93160i	−0.762842	−	0.539411i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−0.306044	+	0.364729i	−0.0164293	+	0.0195797i	−0.774197	−	0.632945i	$-0.781846\pi$
$347$	−0.306044	+	0.364729i	−0.0164293	+	0.0195797i	0.757767	+	0.652525i	$0.226290\pi$
$348$		0			0
$349$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$349$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−3.76298	+	10.3387i	−0.200283	+	0.550273i	−0.998653	−	0.0518946i	$-0.983474\pi$
$353$	−3.76298	+	10.3387i	−0.200283	+	0.550273i	0.798369	+	0.602168i	$0.205696\pi$
$354$		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
$361$	−28.3769	+	49.1503i	−1.49352	+	2.58686i
$362$		0			0
$363$	43.6044	+	4.01773i	2.28864	+	0.210876i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$367$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$368$		0			0
$369$	6.01068	−	10.1924i	0.312904	−	0.530595i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$373$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−14.9550			−0.768187			−0.384093	−	0.923294i	$-0.625486\pi$
$379$	−14.9550			−0.768187			−0.384093	+	0.923294i	$0.625486\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$383$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	12.6615	−	14.8097i	0.643621	−	0.752818i
$388$		0			0
$389$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$389$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−35.6464	+	16.4224i	−1.79812	+	0.828399i
$394$		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
$401$	−8.21802	+	9.79385i	−0.410388	+	0.489082i	−0.931158	−	0.364615i	$-0.881200\pi$
$401$	−8.21802	+	9.79385i	−0.410388	+	0.489082i	0.520770	+	0.853697i	$0.325645\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	6.94395	+	5.82667i	0.343356	+	0.288110i	0.798116	−	0.602504i	$-0.205830\pi$
$409$	6.94395	+	5.82667i	0.343356	+	0.288110i	−0.454759	+	0.890614i	$0.650275\pi$
$410$		0			0
$411$	16.5817	−	16.4294i	0.817916	−	0.810403i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	16.1670	+	35.0922i	0.791703	+	1.71847i
$418$		0			0
$419$	0.956397	+	2.62768i	0.0467231	+	0.128371i	0.960860	−	0.277036i	$-0.0893522\pi$
$419$	0.956397	+	2.62768i	0.0467231	+	0.128371i	−0.914136	+	0.405407i	$0.867130\pi$
$420$		0			0
$421$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$421$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	23.6797	+	28.2204i	1.14864	+	1.36889i
$426$		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
$433$	−29.9040			−1.43710			−0.718548	−	0.695477i	$-0.755193\pi$
$433$	−29.9040			−1.43710			−0.718548	+	0.695477i	$0.755193\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$439$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$440$		0			0
$441$	18.2827	−	10.3317i	0.870603	−	0.491986i
$442$		0			0
$443$	40.2499	−	7.09714i	1.91233	−	0.337195i	0.914588	−	0.404386i	$-0.132515\pi$
$443$	40.2499	−	7.09714i	1.91233	−	0.337195i	0.997740	+	0.0671913i	$0.0214038\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	19.6273	+	11.3318i	0.926271	+	0.534783i	0.885630	−	0.464391i	$-0.153727\pi$
$449$	19.6273	+	11.3318i	0.926271	+	0.534783i	0.0406404	+	0.999174i	$0.487060\pi$
$450$		0			0
$451$	11.8789	+	20.5749i	0.559357	+	0.968834i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−35.6523	−	12.9764i	−1.66774	−	0.607008i	−0.676191	−	0.736726i	$-0.736371\pi$
$457$	−35.6523	−	12.9764i	−1.66774	−	0.607008i	−0.991551	+	0.129718i	$0.958593\pi$
$458$		0			0
$459$	−10.4197	+	36.8391i	−0.486349	+	1.71950i
$460$		0			0
$461$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$461$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$462$		0			0
$463$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$463$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	20.9072	−	12.0708i	0.967470	−	0.558569i	0.0690063	−	0.997616i	$-0.478017\pi$
$467$	20.9072	−	12.0708i	0.967470	−	0.558569i	0.898464	+	0.439047i	$0.144684\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	13.3801	+	36.7616i	0.615219	+	1.69030i
$474$		0			0
$475$	7.55688	+	42.8572i	0.346733	+	1.96642i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$479$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$480$		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
$489$	36.0978	+	17.0359i	1.63240	+	0.770390i
$490$		0			0
$491$	27.2340	+	4.80209i	1.22905	+	0.216715i	0.750218	−	0.661190i	$-0.229948\pi$
$491$	27.2340	+	4.80209i	1.22905	+	0.216715i	0.478834	+	0.877905i	$0.341060\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	1.27327	−	0.463434i	0.0569995	−	0.0207461i	−0.313363	−	0.949633i	$-0.601456\pi$
$499$	1.27327	−	0.463434i	0.0569995	−	0.0207461i	0.370363	+	0.928887i	$0.379233\pi$
$500$		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
$507$	21.7761	−	5.72732i	0.967110	−	0.254359i
$508$		0			0
$509$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$509$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−32.4189	+	31.5336i	−1.43133	+	1.39224i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	35.0511	−	20.2368i	1.53562	−	0.886588i	0.536528	−	0.843882i	$-0.319735\pi$
$521$	35.0511	−	20.2368i	1.53562	−	0.886588i	0.999088	−	0.0427062i	$-0.0135979\pi$
$522$		0			0
$523$	−20.5227	+	35.5464i	−0.897395	+	1.55433i	−0.0665832	+	0.997781i	$0.521210\pi$
$523$	−20.5227	+	35.5464i	−0.897395	+	1.55433i	−0.830812	+	0.556553i	$0.812124\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−3.99391	−	22.6506i	−0.173648	−	0.984808i
$530$		0			0
$531$	−12.3458	−	21.8467i	−0.535763	−	0.948068i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−5.61447	−	8.09757i	−0.242282	−	0.349436i
$538$		0			0
$539$			42.1640i			1.81613i
$540$		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
$547$	−30.8898	+	25.9196i	−1.32075	+	1.10824i	−0.334606	+	0.942358i	$0.608603\pi$
$547$	−30.8898	+	25.9196i	−1.32075	+	1.10824i	−0.986145	+	0.165883i	$0.946952\pi$
$548$		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
$561$	−54.1022	−	54.6039i	−2.28420	−	2.30538i
$562$		0			0
$563$	−5.71019	+	6.80514i	−0.240656	+	0.286803i	−0.872831	−	0.488023i	$-0.837718\pi$
$563$	−5.71019	+	6.80514i	−0.240656	+	0.286803i	0.632175	+	0.774826i	$0.282163\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	15.4905	−	42.5597i	0.649395	−	1.78420i	0.0294311	−	0.999567i	$-0.490630\pi$
$569$	15.4905	−	42.5597i	0.649395	−	1.78420i	0.619964	−	0.784631i	$-0.287147\pi$
$570$		0			0
$571$	−26.9524	−	22.6158i	−1.12792	−	0.946441i	−0.128947	−	0.991651i	$-0.541160\pi$
$571$	−26.9524	−	22.6158i	−1.12792	−	0.946441i	−0.998978	+	0.0452101i	$0.985604\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	23.9312	−	41.4501i	0.996270	−	1.72559i	0.423403	−	0.905941i	$-0.360835\pi$
$577$	23.9312	−	41.4501i	0.996270	−	1.72559i	0.572866	−	0.819649i	$-0.305831\pi$
$578$		0			0
$579$	−14.8812	−	32.3011i	−0.618441	−	1.34239i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−21.1973	−	25.2620i	−0.874907	−	1.04267i	−0.998731	−	0.0503697i	$-0.983960\pi$
$587$	−21.1973	−	25.2620i	−0.874907	−	1.04267i	0.123823	−	0.992304i	$-0.460484\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			7.03896i			0.289055i	0.989501	+	0.144528i	$0.0461663\pi$
$593$			7.03896i			0.289055i	−0.989501	+	0.144528i	$0.953834\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$599$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$600$		0			0
$601$	−18.6375	+	15.6388i	−0.760241	+	0.637918i	−0.938190	−	0.346122i	$-0.887498\pi$
$601$	−18.6375	+	15.6388i	−0.760241	+	0.637918i	0.177949	+	0.984040i	$0.443054\pi$
$602$		0			0
$603$	13.6193	+	8.03161i	0.554622	+	0.327073i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$607$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$608$		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
$617$	−31.8935	+	38.0092i	−1.28398	+	1.53019i	−0.603877	+	0.797077i	$0.706378\pi$
$617$	−31.8935	+	38.0092i	−1.28398	+	1.53019i	−0.680106	+	0.733114i	$0.738066\pi$
$618$		0			0
$619$	−35.0195	−	12.7461i	−1.40755	−	0.512308i	−0.477143	−	0.878826i	$-0.658328\pi$
$619$	−35.0195	−	12.7461i	−1.40755	−	0.512308i	−0.930411	+	0.366518i	$0.880550\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	19.1511	+	16.0697i	0.766044	+	0.642788i
$626$		0			0
$627$	−23.0969	−	87.8178i	−0.922401	−	3.50710i
$628$		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
$633$	29.0454	−	41.0764i	1.15445	−	1.63264i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	23.4299	+	27.9226i	0.925424	+	1.10288i	0.994444	+	0.105263i	$0.0335683\pi$
$641$	23.4299	+	27.9226i	0.925424	+	1.10288i	−0.0690201	+	0.997615i	$0.521987\pi$
$642$		0			0
$643$	−5.70399	+	32.3490i	−0.224944	+	1.27572i	0.637850	+	0.770161i	$0.279824\pi$
$643$	−5.70399	+	32.3490i	−0.224944	+	1.27572i	−0.862793	+	0.505557i	$0.831287\pi$
$644$		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
$649$	50.3836			1.97773
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$653$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	4.12911	−	11.0269i	0.161092	−	0.430202i
$658$		0			0
$659$	−39.0280	+	6.88169i	−1.52032	+	0.268073i	−0.870557	−	0.492068i	$-0.836241\pi$
$659$	−39.0280	+	6.88169i	−1.52032	+	0.268073i	−0.649759	+	0.760140i	$0.725130\pi$
$660$		0			0
$661$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$661$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$662$		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
$673$	−46.1303	−	16.7900i	−1.77819	−	0.647209i	−0.999812	−	0.0194154i	$-0.993820\pi$
$673$	−46.1303	−	16.7900i	−1.77819	−	0.647209i	−0.778380	−	0.627793i	$-0.783958\pi$
$674$		0			0
$675$	−2.62244	+	25.8481i	−0.100938	+	0.994893i
$676$		0			0
$677$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$677$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	21.5860	+	5.89086i	0.827179	+	0.225738i
$682$		0			0
$683$	2.73956	−	1.58168i	0.104826	−	0.0605215i	−0.446670	−	0.894699i	$-0.647390\pi$
$683$	2.73956	−	1.58168i	0.104826	−	0.0605215i	0.551497	+	0.834177i	$0.314057\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−0.165763	−	0.940090i	−0.00630593	−	0.0357627i	0.981492	−	0.191501i	$-0.0613355\pi$
$691$	−0.165763	−	0.940090i	−0.00630593	−	0.0357627i	−0.987798	+	0.155738i	$0.950224\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	5.04630	−	28.6190i	0.191142	−	1.08402i
$698$		0			0
$699$	28.1879	+	40.6545i	1.06617	+	1.53770i
$700$		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		0.766044	−	0.642788i	$-0.222222\pi$
$709$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$710$		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
$723$	−37.5789	−	37.9273i	−1.39757	−	1.41053i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$727$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$728$		0			0
$729$	−23.7474	+	12.8475i	−0.879535	+	0.475834i
$730$		0			0
$731$	16.3665	−	44.9666i	0.605337	−	1.66315i
$732$		0			0
$733$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$733$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−27.4927	+	15.8729i	−1.01271	+	0.584686i
$738$		0			0
$739$	0.612829	−	1.06145i	0.0225433	−	0.0390461i	−0.854534	−	0.519396i	$-0.826157\pi$
$739$	0.612829	−	1.06145i	0.0225433	−	0.0390461i	0.877077	+	0.480350i	$0.159490\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$743$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	40.0908	−	14.1742i	1.46685	−	0.518608i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$751$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$752$		0			0
$753$	54.0083	−	4.47408i	1.96817	−	0.163045i
$754$		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
$761$	−11.1418	−	1.96460i	−0.403891	−	0.0712169i	−0.0319875	−	0.999488i	$-0.510184\pi$
$761$	−11.1418	−	1.96460i	−0.403891	−	0.0712169i	−0.371903	+	0.928271i	$0.621295\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−51.7684	+	18.8422i	−1.86682	+	0.679466i	−0.893921	+	0.448224i	$0.852057\pi$
$769$	−51.7684	+	18.8422i	−1.86682	+	0.679466i	−0.972896	+	0.231242i	$0.925721\pi$
$770$		0			0
$771$	1.68721	−	18.3113i	0.0607634	−	0.659466i
$772$		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
$779$	22.0665	−	26.2978i	0.790614	−	0.942217i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	36.0389	+	30.2402i	1.28465	+	1.07795i	0.992586	+	0.121547i	$0.0387856\pi$
$787$	36.0389	+	30.2402i	1.28465	+	1.07795i	0.292061	+	0.956400i	$0.405659\pi$
$788$		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		−0.342020	−	0.939693i	$-0.611111\pi$
$797$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−51.7420	−	19.3752i	−1.82821	−	0.684588i
$802$		0			0
$803$	15.1963	+	18.1103i	0.536267	+	0.639098i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	55.2092i		−	1.94105i	−0.240999	−	0.970525i	$-0.577475\pi$
$809$		−	55.2092i		−	1.94105i	0.240999	−	0.970525i	$-0.422525\pi$
$810$		0			0
$811$	−56.3809			−1.97980			−0.989900	−	0.141768i	$-0.954721\pi$
$811$	−56.3809			−1.97980			−0.989900	+	0.141768i	$0.954721\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	43.3033	−	36.3358i	1.51499	−	1.27123i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$821$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$822$		0			0
$823$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$823$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$824$		0			0
$825$	−42.5921	−	30.1171i	−1.48287	−	1.04854i
$826$		0			0
$827$	−17.1464	−	9.89949i	−0.596240	−	0.344239i	0.171321	−	0.985215i	$-0.445196\pi$
$827$	−17.1464	−	9.89949i	−0.596240	−	0.344239i	−0.767561	+	0.640976i	$0.778530\pi$
$828$		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
$833$	33.1516	−	39.5086i	1.14864	−	1.36889i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$839$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$840$		0			0
$841$	−22.2153	−	18.6408i	−0.766044	−	0.642788i
$842$		0			0
$843$	2.41673	+	0.659527i	0.0832365	+	0.0227153i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−19.0505	−	1.75532i	−0.653812	−	0.0602425i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$853$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	14.5446	+	17.3336i	0.496835	+	0.592105i	0.954942	−	0.296792i	$-0.0959169\pi$
$857$	14.5446	+	17.3336i	0.496835	+	0.592105i	−0.458107	+	0.888897i	$0.651472\pi$
$858$		0			0
$859$	8.66241	−	49.1269i	0.295557	−	1.67619i	−0.369370	−	0.929282i	$-0.620427\pi$
$859$	8.66241	−	49.1269i	0.295557	−	1.67619i	0.664928	−	0.746908i	$-0.268462\pi$
$860$		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
$867$	5.33153	+	64.3588i	0.181068	+	2.18574i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−6.33199	+	7.40627i	−0.214305	+	0.250664i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$877$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	37.9949	+	21.9364i	1.28008	+	0.739055i	0.976863	−	0.213866i	$-0.0686057\pi$
$881$	37.9949	+	21.9364i	1.28008	+	0.739055i	0.303218	+	0.952921i	$0.401939\pi$
$882$		0			0
$883$	−11.0972	−	19.2209i	−0.373450	−	0.646834i	0.616644	−	0.787242i	$-0.288492\pi$
$883$	−11.0972	−	19.2209i	−0.373450	−	0.646834i	−0.990094	+	0.140408i	$0.955158\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$887$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	1.00056	−	54.2016i	0.0335200	−	1.81582i
$892$		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
$907$	7.96005	+	45.1437i	0.264309	+	1.49897i	0.770996	+	0.636841i	$0.219759\pi$
$907$	7.96005	+	45.1437i	0.264309	+	1.49897i	−0.506687	+	0.862130i	$0.669130\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$911$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$912$		0			0
$913$	−14.8257	+	84.0804i	−0.490657	+	2.78266i
$914$		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
$921$	3.49061	−	2.42022i	0.115019	−	0.0797490i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	27.8546	−	4.91151i	0.913879	−	0.161141i	0.303115	−	0.952954i	$-0.401973\pi$
$929$	27.8546	−	4.91151i	0.913879	−	0.161141i	0.610764	+	0.791813i	$0.290862\pi$
$930$		0			0
$931$	57.2514	−	20.8378i	1.87634	−	0.682932i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−30.5454	−	52.9062i	−0.997875	−	1.72837i	−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−30.5454	−	52.9062i	−0.997875	−	1.72837i	−0.442509	−	0.896764i	$-0.645912\pi$
$938$		0			0
$939$	−14.4503	+	52.9506i	−0.471567	+	1.72798i
$940$		0			0
$941$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$941$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−20.7811	+	57.0955i	−0.675294	+	1.85536i	−0.187860	+	0.982196i	$0.560155\pi$
$947$	−20.7811	+	57.0955i	−0.675294	+	1.85536i	−0.487435	+	0.873160i	$0.662067\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−53.4028	+	30.8321i	−1.72989	+	0.998751i	−0.839964	+	0.542643i	$0.817424\pi$
$953$	−53.4028	+	30.8321i	−1.72989	+	0.998751i	−0.889924	+	0.456109i	$0.849243\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	5.38309	+	30.5290i	0.173648	+	0.984808i
$962$		0			0
$963$	38.3336	−	31.5675i	1.23528	−	1.01725i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$967$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$968$		0			0
$969$	−47.4048	+	100.447i	−1.52286	+	3.22683i
$970$		0			0
$971$		−	31.1127i		−	0.998454i	−0.866471	−	0.499227i	$-0.833617\pi$
$971$		−	31.1127i		−	0.998454i	0.866471	−	0.499227i	$-0.166383\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−4.82209	−	0.850265i	−0.154272	−	0.0272024i	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	−4.82209	−	0.850265i	−0.154272	−	0.0272024i	−0.250251	+	0.968181i	$0.580513\pi$
$978$		0			0
$979$	84.9794	−	71.3062i	2.71595	−	2.27896i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$983$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$984$		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
$993$	58.7040	−	15.4397i	1.86291	−	0.489964i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$997$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$998$		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.239.2		12
4.3	odd	2		216.2.v.a.131.1	✓	12
8.3	odd	2	CM	864.2.bh.a.239.2		12
8.5	even	2		216.2.v.a.131.1	✓	12
12.11	even	2		648.2.v.a.179.2		12
24.5	odd	2		648.2.v.a.179.2		12
27.20	odd	18	inner	864.2.bh.a.47.2		12
108.7	odd	18		648.2.v.a.467.2		12
108.47	even	18		216.2.v.a.155.1	yes	12
216.61	even	18		648.2.v.a.467.2		12
216.101	odd	18		216.2.v.a.155.1	yes	12
216.155	even	18	inner	864.2.bh.a.47.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.2.v.a.131.1	✓	12	4.3	odd	2
216.2.v.a.131.1	✓	12	8.5	even	2
216.2.v.a.155.1	yes	12	108.47	even	18
216.2.v.a.155.1	yes	12	216.101	odd	18
648.2.v.a.179.2		12	12.11	even	2
648.2.v.a.179.2		12	24.5	odd	2
648.2.v.a.467.2		12	108.7	odd	18
648.2.v.a.467.2		12	216.61	even	18
864.2.bh.a.47.2		12	27.20	odd	18	inner
864.2.bh.a.47.2		12	216.155	even	18	inner
864.2.bh.a.239.2		12	1.1	even	1	trivial
864.2.bh.a.239.2		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)

\(f(q)\)	\(=\)	\(q+(1.56638 + 0.739232i) q^{3} +(1.90707 + 2.31583i) q^{9} +O(q^{10})\)	\(q+(1.56638 + 0.739232i) q^{3} +(1.90707 + 2.31583i) q^{9} +(-5.93192 + 1.04596i) q^{11} +(6.38072 + 3.68391i) q^{17} +(4.35183 + 7.53760i) q^{19} +(4.69846 + 1.71010i) q^{25} +(1.27526 + 5.03723i) q^{27} +(-10.0648 - 2.74670i) q^{33} +(-1.34901 - 3.70637i) q^{41} +(-1.12781 - 6.39612i) q^{43} +(1.21554 - 6.89365i) q^{49} +(7.27135 + 10.4872i) q^{51} +(1.24458 + 15.0237i) q^{57} +(-8.23753 - 1.45250i) q^{59} +(4.95255 - 1.80258i) q^{67} +(-1.96244 - 3.39905i) q^{73} +(6.09540 + 6.15192i) q^{75} +(-1.72616 + 8.83292i) q^{81} +(4.84788 - 13.3194i) q^{83} +(-15.9495 + 9.20844i) q^{89} +(0.564013 + 3.19868i) q^{97} +(-13.7349 - 11.7426i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 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

Related objects

Downloads

Learn more