LMFDB - Embedded newform 648.2.l.b.539.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [648,2,Mod(107,648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(648, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 1]))

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

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

chi := DirichletCharacter("648.107");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	648.l (of order $6$, degree $2$, 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:	$5.17430605098$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			539.2
Root			$1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	648.539
Dual form			648.2.l.b.107.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 - 0.707107i) q^{2} +(1.00000 - 1.73205i) q^{4} -2.82843i q^{8} +O(q^{10})$	$q+(1.22474 - 0.707107i) q^{2} +(1.00000 - 1.73205i) q^{4} -2.82843i q^{8} +(2.44949 - 1.41421i) q^{11} +(-2.00000 - 3.46410i) q^{16} -5.65685i q^{17} +2.00000 q^{19} +(2.00000 - 3.46410i) q^{22} +(2.50000 + 4.33013i) q^{25} +(-4.89898 - 2.82843i) q^{32} +(-4.00000 - 6.92820i) q^{34} +(2.44949 - 1.41421i) q^{38} +(-9.79796 - 5.65685i) q^{41} +(5.00000 + 8.66025i) q^{43} -5.65685i q^{44} +(-3.50000 + 6.06218i) q^{49} +(6.12372 + 3.53553i) q^{50} +(12.2474 + 7.07107i) q^{59} -8.00000 q^{64} +(-7.00000 + 12.1244i) q^{67} +(-9.79796 - 5.65685i) q^{68} +2.00000 q^{73} +(2.00000 - 3.46410i) q^{76} -16.0000 q^{82} +(2.44949 - 1.41421i) q^{83} +(12.2474 + 7.07107i) q^{86} +(-4.00000 - 6.92820i) q^{88} -5.65685i q^{89} +(5.00000 + 8.66025i) q^{97} +9.89949i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4}+O(q^{10}) $ 4 * q + 4 * q^4	$ 4 q + 4 q^{4} - 8 q^{16} + 8 q^{19} + 8 q^{22} + 10 q^{25} - 16 q^{34} + 20 q^{43} - 14 q^{49} - 32 q^{64} - 28 q^{67} + 8 q^{73} + 8 q^{76} - 64 q^{82} - 16 q^{88} + 20 q^{97}+O(q^{100}) $ 4 * q + 4 * q^4 - 8 * q^16 + 8 * q^19 + 8 * q^22 + 10 * q^25 - 16 * q^34 + 20 * q^43 - 14 * q^49 - 32 * q^64 - 28 * q^67 + 8 * q^73 + 8 * q^76 - 64 * q^82 - 16 * q^88 + 20 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/648\mathbb{Z}\right)^\times$.

$n$	$325$	$487$	$569$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{5}{6}\right)$

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 $

$2$	1.22474	−	0.707107i	0.866025	−	0.500000i
$2$	1.22474	−	0.707107i	0.866025	−	0.500000i
$3$		0			0
$3$		0			0
$4$	1.00000	−	1.73205i	0.500000	−	0.866025i
$5$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$		0			0
$7$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$7$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$8$		−	2.82843i		−	1.00000i
$9$		0			0
$10$		0			0
$11$	2.44949	−	1.41421i	0.738549	−	0.426401i	−0.0829925	−	0.996550i	$-0.526448\pi$
$11$	2.44949	−	1.41421i	0.738549	−	0.426401i	0.821541	+	0.570149i	$0.193114\pi$
$12$		0			0
$13$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$13$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$14$		0			0
$15$		0			0
$16$	−2.00000	−	3.46410i	−0.500000	−	0.866025i
$17$		−	5.65685i		−	1.37199i	−0.727607	−	0.685994i	$-0.759367\pi$
$17$		−	5.65685i		−	1.37199i	0.727607	−	0.685994i	$-0.240633\pi$
$18$		0			0
$19$	2.00000			0.458831			0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000			0.458831			0.229416	+	0.973329i	$0.426318\pi$
$20$		0			0
$21$		0			0
$22$	2.00000	−	3.46410i	0.426401	−	0.738549i
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$		0			0
$25$	2.50000	+	4.33013i	0.500000	+	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$29$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$30$		0			0
$31$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$31$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$32$	−4.89898	−	2.82843i	−0.866025	−	0.500000i
$33$		0			0
$34$	−4.00000	−	6.92820i	−0.685994	−	1.18818i
$35$		0			0
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	2.44949	−	1.41421i	0.397360	−	0.229416i
$39$		0			0
$40$		0			0
$41$	−9.79796	−	5.65685i	−1.53018	−	0.883452i	−0.999353	−	0.0359748i	$-0.988546\pi$
$41$	−9.79796	−	5.65685i	−1.53018	−	0.883452i	−0.530831	−	0.847477i	$-0.678120\pi$
$42$		0			0
$43$	5.00000	+	8.66025i	0.762493	+	1.32068i	0.941562	+	0.336840i	$0.109358\pi$
$43$	5.00000	+	8.66025i	0.762493	+	1.32068i	−0.179069	+	0.983836i	$0.557309\pi$
$44$		−	5.65685i		−	0.852803i
$45$		0			0
$46$		0			0
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		0			0
$49$	−3.50000	+	6.06218i	−0.500000	+	0.866025i
$50$	6.12372	+	3.53553i	0.866025	+	0.500000i
$51$		0			0
$52$		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
$57$		0			0
$58$		0			0
$59$	12.2474	+	7.07107i	1.59448	+	0.920575i	0.992524	+	0.122047i	$0.0389457\pi$
$59$	12.2474	+	7.07107i	1.59448	+	0.920575i	0.601958	+	0.798528i	$0.294388\pi$
$60$		0			0
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$		0			0
$63$		0			0
$64$	−8.00000			−1.00000
$65$		0			0
$66$		0			0
$67$	−7.00000	+	12.1244i	−0.855186	+	1.48123i	0.0212861	+	0.999773i	$0.493224\pi$
$67$	−7.00000	+	12.1244i	−0.855186	+	1.48123i	−0.876472	+	0.481452i	$0.840109\pi$
$68$	−9.79796	−	5.65685i	−1.18818	−	0.685994i
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	2.00000			0.234082			0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000			0.234082			0.117041	+	0.993127i	$0.462659\pi$
$74$		0			0
$75$		0			0
$76$	2.00000	−	3.46410i	0.229416	−	0.397360i
$77$		0			0
$78$		0			0
$79$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$79$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$80$		0			0
$81$		0			0
$82$	−16.0000			−1.76690
$83$	2.44949	−	1.41421i	0.268866	−	0.155230i	−0.359506	−	0.933143i	$-0.617055\pi$
$83$	2.44949	−	1.41421i	0.268866	−	0.155230i	0.628372	+	0.777913i	$0.283721\pi$
$84$		0			0
$85$		0			0
$86$	12.2474	+	7.07107i	1.32068	+	0.762493i
$87$		0			0
$88$	−4.00000	−	6.92820i	−0.426401	−	0.738549i
$89$		−	5.65685i		−	0.599625i	−0.953998	−	0.299813i	$-0.903076\pi$
$89$		−	5.65685i		−	0.599625i	0.953998	−	0.299813i	$-0.0969242\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	5.00000	+	8.66025i	0.507673	+	0.879316i	0.999961	+	0.00888289i	$0.00282755\pi$
$97$	5.00000	+	8.66025i	0.507673	+	0.879316i	−0.492287	+	0.870433i	$0.663839\pi$
$98$			9.89949i			1.00000i
$99$		0			0
$100$	10.0000			1.00000
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$103$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			19.7990i			1.91404i	0.290021	+	0.957020i	$0.406338\pi$
$107$			19.7990i			1.91404i	−0.290021	+	0.957020i	$0.593662\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−9.79796	−	5.65685i	−0.921714	−	0.532152i	−0.0375328	−	0.999295i	$-0.511950\pi$
$113$	−9.79796	−	5.65685i	−0.921714	−	0.532152i	−0.884182	+	0.467143i	$0.845283\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	20.0000			1.84115
$119$		0			0
$120$		0			0
$121$	−1.50000	+	2.59808i	−0.136364	+	0.236189i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	−9.79796	+	5.65685i	−0.866025	+	0.500000i
$129$		0			0
$130$		0			0
$131$	12.2474	+	7.07107i	1.07006	+	0.617802i	0.928199	−	0.372084i	$-0.121357\pi$
$131$	12.2474	+	7.07107i	1.07006	+	0.617802i	0.141865	+	0.989886i	$0.454690\pi$
$132$		0			0
$133$		0			0
$134$			19.7990i			1.71037i
$135$		0			0
$136$	−16.0000			−1.37199
$137$	−19.5959	+	11.3137i	−1.67419	+	0.966595i	−0.708942	+	0.705266i	$0.750827\pi$
$137$	−19.5959	+	11.3137i	−1.67419	+	0.966595i	−0.965250	+	0.261329i	$0.915839\pi$
$138$		0			0
$139$	11.0000	−	19.0526i	0.933008	−	1.61602i	0.154859	−	0.987937i	$-0.450508\pi$
$139$	11.0000	−	19.0526i	0.933008	−	1.61602i	0.778148	−	0.628080i	$-0.216159\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$	2.44949	−	1.41421i	0.202721	−	0.117041i
$147$		0			0
$148$		0			0
$149$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$149$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$150$		0			0
$151$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$151$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$152$		−	5.65685i		−	0.458831i
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$157$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	2.00000			0.156652			0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000			0.156652			0.0783260	+	0.996928i	$0.475042\pi$
$164$	−19.5959	+	11.3137i	−1.53018	+	0.883452i
$165$		0			0
$166$	2.00000	−	3.46410i	0.155230	−	0.268866i
$167$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$167$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$168$		0			0
$169$	−6.50000	−	11.2583i	−0.500000	−	0.866025i
$170$		0			0
$171$		0			0
$172$	20.0000			1.52499
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$		0			0
$176$	−9.79796	−	5.65685i	−0.738549	−	0.426401i
$177$		0			0
$178$	−4.00000	−	6.92820i	−0.299813	−	0.519291i
$179$			19.7990i			1.47985i	0.672692	+	0.739923i	$0.265138\pi$
$179$			19.7990i			1.47985i	−0.672692	+	0.739923i	$0.734862\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−8.00000	−	13.8564i	−0.585018	−	1.01328i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$191$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$192$		0			0
$193$	11.0000	−	19.0526i	0.791797	−	1.37143i	−0.133056	−	0.991109i	$-0.542479\pi$
$193$	11.0000	−	19.0526i	0.791797	−	1.37143i	0.924853	−	0.380325i	$-0.124188\pi$
$194$	12.2474	+	7.07107i	0.879316	+	0.507673i
$195$		0			0
$196$	7.00000	+	12.1244i	0.500000	+	0.866025i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	12.2474	−	7.07107i	0.866025	−	0.500000i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.89898	−	2.82843i	0.338869	−	0.195646i
$210$		0			0
$211$	−7.00000	+	12.1244i	−0.481900	+	0.834675i	−0.999784	−	0.0207756i	$-0.993386\pi$
$211$	−7.00000	+	12.1244i	−0.481900	+	0.834675i	0.517884	+	0.855451i	$0.326720\pi$
$212$		0			0
$213$		0			0
$214$	14.0000	+	24.2487i	0.957020	+	1.65761i
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$223$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$224$		0			0
$225$		0			0
$226$	−16.0000			−1.06430
$227$	2.44949	−	1.41421i	0.162578	−	0.0938647i	−0.416503	−	0.909134i	$-0.636745\pi$
$227$	2.44949	−	1.41421i	0.162578	−	0.0938647i	0.579082	+	0.815270i	$0.303411\pi$
$228$		0			0
$229$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$229$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	5.65685i		−	0.370593i	−0.982683	−	0.185296i	$-0.940675\pi$
$233$		−	5.65685i		−	0.370593i	0.982683	−	0.185296i	$-0.0593245\pi$
$234$		0			0
$235$		0			0
$236$	24.4949	−	14.1421i	1.59448	−	0.920575i
$237$		0			0
$238$		0			0
$239$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$239$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$240$		0			0
$241$	−13.0000	−	22.5167i	−0.837404	−	1.45043i	−0.892058	−	0.451920i	$-0.850739\pi$
$241$	−13.0000	−	22.5167i	−0.837404	−	1.45043i	0.0546547	−	0.998505i	$-0.482594\pi$
$242$			4.24264i			0.272727i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	31.1127i		−	1.96382i	−0.189358	−	0.981908i	$-0.560641\pi$
$251$		−	31.1127i		−	1.96382i	0.189358	−	0.981908i	$-0.439359\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$	−9.79796	−	5.65685i	−0.611180	−	0.352865i	0.162247	−	0.986750i	$-0.448126\pi$
$257$	−9.79796	−	5.65685i	−0.611180	−	0.352865i	−0.773427	+	0.633885i	$0.781459\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	20.0000			1.23560
$263$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$263$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	14.0000	+	24.2487i	0.855186	+	1.48123i
$269$		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			$\pi$
$272$	−19.5959	+	11.3137i	−1.18818	+	0.685994i
$273$		0			0
$274$	−16.0000	+	27.7128i	−0.966595	+	1.67419i
$275$	12.2474	+	7.07107i	0.738549	+	0.426401i
$276$		0			0
$277$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$277$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$278$		−	31.1127i		−	1.86602i
$279$		0			0
$280$		0			0
$281$	24.4949	−	14.1421i	1.46124	−	0.843649i	0.462174	−	0.886789i	$-0.347070\pi$
$281$	24.4949	−	14.1421i	1.46124	−	0.843649i	0.999069	+	0.0431402i	$0.0137362\pi$
$282$		0			0
$283$	11.0000	−	19.0526i	0.653882	−	1.13256i	−0.328291	−	0.944577i	$-0.606473\pi$
$283$	11.0000	−	19.0526i	0.653882	−	1.13256i	0.982173	−	0.187980i	$-0.0601941\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−15.0000			−0.882353
$290$		0			0
$291$		0			0
$292$	2.00000	−	3.46410i	0.117041	−	0.202721i
$293$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$293$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−4.00000	−	6.92820i	−0.229416	−	0.397360i
$305$		0			0
$306$		0			0
$307$	−34.0000			−1.94048			−0.970241	−	0.242140i	$-0.922151\pi$
$307$	−34.0000			−1.94048			−0.970241	+	0.242140i	$0.922151\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$	5.00000	+	8.66025i	0.282617	+	0.489506i	0.972028	−	0.234863i	$-0.0754642\pi$
$313$	5.00000	+	8.66025i	0.282617	+	0.489506i	−0.689412	+	0.724370i	$0.742131\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$317$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	11.3137i		−	0.629512i
$324$		0			0
$325$		0			0
$326$	2.44949	−	1.41421i	0.135665	−	0.0783260i
$327$		0			0
$328$	−16.0000	+	27.7128i	−0.883452	+	1.53018i
$329$		0			0
$330$		0			0
$331$	−13.0000	−	22.5167i	−0.714545	−	1.23763i	−0.963135	−	0.269019i	$-0.913301\pi$
$331$	−13.0000	−	22.5167i	−0.714545	−	1.23763i	0.248590	−	0.968609i	$-0.420033\pi$
$332$		−	5.65685i		−	0.310460i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−7.00000	+	12.1244i	−0.381314	+	0.660456i	−0.991250	−	0.131995i	$-0.957862\pi$
$337$	−7.00000	+	12.1244i	−0.381314	+	0.660456i	0.609936	+	0.792451i	$0.291195\pi$
$338$	−15.9217	−	9.19239i	−0.866025	−	0.500000i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	24.4949	−	14.1421i	1.32068	−	0.762493i
$345$		0			0
$346$		0			0
$347$	−31.8434	−	18.3848i	−1.70944	−	0.986947i	−0.935245	−	0.354001i	$-0.884821\pi$
$347$	−31.8434	−	18.3848i	−1.70944	−	0.986947i	−0.774197	−	0.632945i	$-0.781846\pi$
$348$		0			0
$349$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$349$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$350$		0			0
$351$		0			0
$352$	−16.0000			−0.852803
$353$	−19.5959	+	11.3137i	−1.04299	+	0.602168i	−0.920677	−	0.390324i	$-0.872363\pi$
$353$	−19.5959	+	11.3137i	−1.04299	+	0.602168i	−0.122308	+	0.992492i	$0.539030\pi$
$354$		0			0
$355$		0			0
$356$	−9.79796	−	5.65685i	−0.519291	−	0.299813i
$357$		0			0
$358$	14.0000	+	24.2487i	0.739923	+	1.28158i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$367$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$373$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$374$	−19.5959	−	11.3137i	−1.01328	−	0.585018i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	38.0000			1.95193			0.975964	−	0.217930i	$-0.0699304\pi$
$379$	38.0000			1.95193			0.975964	+	0.217930i	$0.0699304\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$		0			0
$385$		0			0
$386$		−	31.1127i		−	1.58359i
$387$		0			0
$388$	20.0000			1.01535
$389$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$389$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$390$		0			0
$391$		0			0
$392$	17.1464	+	9.89949i	0.866025	+	0.500000i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	10.0000	−	17.3205i	0.500000	−	0.866025i
$401$	34.2929	+	19.7990i	1.71250	+	0.988714i	0.931158	+	0.364615i	$0.118800\pi$
$401$	34.2929	+	19.7990i	1.71250	+	0.988714i	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
$409$	11.0000	−	19.0526i	0.543915	−	0.942088i	−0.454759	−	0.890614i	$-0.650275\pi$
$409$	11.0000	−	19.0526i	0.543915	−	0.942088i	0.998674	−	0.0514740i	$-0.0163919\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	4.00000	−	6.92820i	0.195646	−	0.338869i
$419$	−31.8434	−	18.3848i	−1.55565	−	0.898155i	−0.997665	−	0.0683046i	$-0.978241\pi$
$419$	−31.8434	−	18.3848i	−1.55565	−	0.898155i	−0.557986	−	0.829851i	$-0.688426\pi$
$420$		0			0
$421$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$421$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$422$			19.7990i			0.963800i
$423$		0			0
$424$		0			0
$425$	24.4949	−	14.1421i	1.18818	−	0.685994i
$426$		0			0
$427$		0			0
$428$	34.2929	+	19.7990i	1.65761	+	0.957020i
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	38.0000			1.82616			0.913082	−	0.407777i	$-0.133696\pi$
$433$	38.0000			1.82616			0.913082	+	0.407777i	$0.133696\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2.44949	−	1.41421i	0.116379	−	0.0671913i	−0.440681	−	0.897664i	$-0.645263\pi$
$443$	2.44949	−	1.41421i	0.116379	−	0.0671913i	0.557059	+	0.830473i	$0.311930\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	5.65685i		−	0.266963i	−0.991051	−	0.133482i	$-0.957384\pi$
$449$		−	5.65685i		−	0.266963i	0.991051	−	0.133482i	$-0.0426157\pi$
$450$		0			0
$451$	−32.0000			−1.50682
$452$	−19.5959	+	11.3137i	−0.921714	+	0.532152i
$453$		0			0
$454$	2.00000	−	3.46410i	0.0938647	−	0.162578i
$455$		0			0
$456$		0			0
$457$	−13.0000	−	22.5167i	−0.608114	−	1.05328i	−0.991551	−	0.129718i	$-0.958593\pi$
$457$	−13.0000	−	22.5167i	−0.608114	−	1.05328i	0.383437	−	0.923567i	$-0.374740\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$461$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$462$		0			0
$463$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$463$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$464$		0			0
$465$		0			0
$466$	−4.00000	−	6.92820i	−0.185296	−	0.320943i
$467$		−	31.1127i		−	1.43972i	−0.694117	−	0.719862i	$-0.744205\pi$
$467$		−	31.1127i		−	1.43972i	0.694117	−	0.719862i	$-0.255795\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	20.0000	−	34.6410i	0.920575	−	1.59448i
$473$	24.4949	+	14.1421i	1.12628	+	0.650256i
$474$		0			0
$475$	5.00000	+	8.66025i	0.229416	+	0.397360i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$		0			0
$482$	−31.8434	−	18.3848i	−1.45043	−	0.837404i
$483$		0			0
$484$	3.00000	+	5.19615i	0.136364	+	0.236189i
$485$		0			0
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	12.2474	+	7.07107i	0.552720	+	0.319113i	0.750218	−	0.661190i	$-0.229948\pi$
$491$	12.2474	+	7.07107i	0.552720	+	0.319113i	−0.197499	+	0.980303i	$0.563282\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−7.00000	+	12.1244i	−0.313363	+	0.542761i	−0.979088	−	0.203436i	$-0.934789\pi$
$499$	−7.00000	+	12.1244i	−0.313363	+	0.542761i	0.665725	+	0.746197i	$0.268122\pi$
$500$		0			0
$501$		0			0
$502$	−22.0000	−	38.1051i	−0.981908	−	1.70071i
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		0			0
$511$		0			0
$512$			22.6274i			1.00000i
$513$		0			0
$514$	−16.0000			−0.705730
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$			45.2548i			1.98265i	0.131432	+	0.991325i	$0.458042\pi$
$521$			45.2548i			1.98265i	−0.131432	+	0.991325i	$0.541958\pi$
$522$		0			0
$523$	38.0000			1.66162			0.830812	−	0.556553i	$-0.187876\pi$
$523$	38.0000			1.66162			0.830812	+	0.556553i	$0.187876\pi$
$524$	24.4949	−	14.1421i	1.07006	−	0.617802i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	+	19.9186i	0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	34.2929	+	19.7990i	1.48123	+	0.855186i
$537$		0			0
$538$		0			0
$539$			19.7990i			0.852803i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$	−16.0000	+	27.7128i	−0.685994	+	1.18818i
$545$		0			0
$546$		0			0
$547$	23.0000	+	39.8372i	0.983409	+	1.70331i	0.648803	+	0.760956i	$0.275270\pi$
$547$	23.0000	+	39.8372i	0.983409	+	1.70331i	0.334606	+	0.942358i	$0.391397\pi$
$548$			45.2548i			1.93319i
$549$		0			0
$550$	20.0000			0.852803
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−22.0000	−	38.1051i	−0.933008	−	1.61602i
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	20.0000	−	34.6410i	0.843649	−	1.46124i
$563$	−31.8434	−	18.3848i	−1.34204	−	0.774826i	−0.354932	−	0.934892i	$-0.615496\pi$
$563$	−31.8434	−	18.3848i	−1.34204	−	0.774826i	−0.987106	+	0.160066i	$0.948829\pi$
$564$		0			0
$565$		0			0
$566$		−	31.1127i		−	1.30776i
$567$		0			0
$568$		0			0
$569$	−19.5959	+	11.3137i	−0.821504	+	0.474295i	−0.850935	−	0.525271i	$-0.823964\pi$
$569$	−19.5959	+	11.3137i	−0.821504	+	0.474295i	0.0294311	+	0.999567i	$0.490630\pi$
$570$		0			0
$571$	11.0000	−	19.0526i	0.460336	−	0.797325i	−0.538642	−	0.842535i	$-0.681062\pi$
$571$	11.0000	−	19.0526i	0.460336	−	0.797325i	0.998978	+	0.0452101i	$0.0143957\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−34.0000			−1.41544			−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000			−1.41544			−0.707719	+	0.706494i	$0.750276\pi$
$578$	−18.3712	+	10.6066i	−0.764140	+	0.441176i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		−	5.65685i		−	0.234082i
$585$		0			0
$586$		0			0
$587$	−41.6413	+	24.0416i	−1.71872	+	0.992304i	−0.797449	+	0.603386i	$0.793818\pi$
$587$	−41.6413	+	24.0416i	−1.71872	+	0.992304i	−0.921272	+	0.388918i	$0.872849\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			45.2548i			1.85839i	0.369586	+	0.929197i	$0.379500\pi$
$593$			45.2548i			1.85839i	−0.369586	+	0.929197i	$0.620500\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$599$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$600$		0			0
$601$	23.0000	+	39.8372i	0.938190	+	1.62499i	0.768845	+	0.639435i	$0.220832\pi$
$601$	23.0000	+	39.8372i	0.938190	+	1.62499i	0.169344	+	0.985557i	$0.445835\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$	−9.79796	−	5.65685i	−0.397360	−	0.229416i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$	−41.6413	+	24.0416i	−1.68051	+	0.970241i
$615$		0			0
$616$		0			0
$617$	34.2929	+	19.7990i	1.38058	+	0.797077i	0.992228	−	0.124434i	$-0.0397116\pi$
$617$	34.2929	+	19.7990i	1.38058	+	0.797077i	0.388351	+	0.921512i	$0.373045\pi$
$618$		0			0
$619$	−13.0000	−	22.5167i	−0.522514	−	0.905021i	−0.999657	−	0.0261952i	$-0.991661\pi$
$619$	−13.0000	−	22.5167i	−0.522514	−	0.905021i	0.477143	−	0.878826i	$-0.341672\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$626$	12.2474	+	7.07107i	0.489506	+	0.282617i
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	24.4949	−	14.1421i	0.967490	−	0.558581i	0.0690201	−	0.997615i	$-0.478013\pi$
$641$	24.4949	−	14.1421i	0.967490	−	0.558581i	0.898470	+	0.439034i	$0.144679\pi$
$642$		0			0
$643$	−25.0000	+	43.3013i	−0.985904	+	1.70764i	−0.348054	+	0.937474i	$0.613157\pi$
$643$	−25.0000	+	43.3013i	−0.985904	+	1.70764i	−0.637850	+	0.770161i	$0.720176\pi$
$644$		0			0
$645$		0			0
$646$	−8.00000	−	13.8564i	−0.314756	−	0.545173i
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$	40.0000			1.57014
$650$		0			0
$651$		0			0
$652$	2.00000	−	3.46410i	0.0783260	−	0.135665i
$653$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$653$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$654$		0			0
$655$		0			0
$656$			45.2548i			1.76690i
$657$		0			0
$658$		0			0
$659$	−41.6413	+	24.0416i	−1.62212	+	0.936529i	−0.635763	+	0.771885i	$0.719314\pi$
$659$	−41.6413	+	24.0416i	−1.62212	+	0.936529i	−0.986353	+	0.164644i	$0.947352\pi$
$660$		0			0
$661$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$661$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$662$	−31.8434	−	18.3848i	−1.23763	−	0.714545i
$663$		0			0
$664$	−4.00000	−	6.92820i	−0.155230	−	0.268866i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	5.00000	+	8.66025i	0.192736	+	0.333828i	0.946156	−	0.323711i	$-0.104931\pi$
$673$	5.00000	+	8.66025i	0.192736	+	0.333828i	−0.753420	+	0.657539i	$0.771597\pi$
$674$			19.7990i			0.762629i
$675$		0			0
$676$	−26.0000			−1.00000
$677$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$677$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	31.1127i		−	1.19049i	−0.803543	−	0.595247i	$-0.797054\pi$
$683$		−	31.1127i		−	1.19049i	0.803543	−	0.595247i	$-0.202946\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	20.0000	−	34.6410i	0.762493	−	1.32068i
$689$		0			0
$690$		0			0
$691$	23.0000	+	39.8372i	0.874961	+	1.51548i	0.856804	+	0.515642i	$0.172447\pi$
$691$	23.0000	+	39.8372i	0.874961	+	1.51548i	0.0181572	+	0.999835i	$0.494220\pi$
$692$		0			0
$693$		0			0
$694$	−52.0000			−1.97389
$695$		0			0
$696$		0			0
$697$	−32.0000	+	55.4256i	−1.21209	+	2.09940i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$	−19.5959	+	11.3137i	−0.738549	+	0.426401i
$705$		0			0
$706$	−16.0000	+	27.7128i	−0.602168	+	1.04299i
$707$		0			0
$708$		0			0
$709$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$709$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$710$		0			0
$711$		0			0
$712$	−16.0000			−0.599625
$713$		0			0
$714$		0			0
$715$		0			0
$716$	34.2929	+	19.7990i	1.28158	+	0.739923i
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	−18.3712	+	10.6066i	−0.683704	+	0.394737i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$727$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	48.9898	−	28.2843i	1.81195	−	1.04613i
$732$		0			0
$733$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$733$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			39.5980i			1.45861i
$738$		0			0
$739$	−34.0000			−1.25071			−0.625355	−	0.780340i	$-0.715046\pi$
$739$	−34.0000			−1.25071			−0.625355	+	0.780340i	$0.715046\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$743$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$	−32.0000			−1.17004
$749$		0			0
$750$		0			0
$751$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$751$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	46.5403	−	26.8701i	1.69042	−	0.975964i
$759$		0			0
$760$		0			0
$761$	−9.79796	−	5.65685i	−0.355176	−	0.205061i	0.311787	−	0.950152i	$-0.399073\pi$
$761$	−9.79796	−	5.65685i	−0.355176	−	0.205061i	−0.666962	+	0.745091i	$0.732406\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	11.0000	−	19.0526i	0.396670	−	0.687053i	−0.596643	−	0.802507i	$-0.703499\pi$
$769$	11.0000	−	19.0526i	0.396670	−	0.687053i	0.993313	+	0.115454i	$0.0368323\pi$
$770$		0			0
$771$		0			0
$772$	−22.0000	−	38.1051i	−0.791797	−	1.37143i
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$	24.4949	−	14.1421i	0.879316	−	0.507673i
$777$		0			0
$778$		0			0
$779$	−19.5959	−	11.3137i	−0.702097	−	0.405356i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	28.0000			1.00000
$785$		0			0
$786$		0			0
$787$	−25.0000	+	43.3013i	−0.891154	+	1.54352i	−0.0526599	+	0.998613i	$0.516770\pi$
$787$	−25.0000	+	43.3013i	−0.891154	+	1.54352i	−0.838494	+	0.544911i	$0.816563\pi$
$788$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$797$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$798$		0			0
$799$		0			0
$800$		−	28.2843i		−	1.00000i
$801$		0			0
$802$	56.0000			1.97743
$803$	4.89898	−	2.82843i	0.172881	−	0.0998130i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	56.5685i		−	1.98884i	−0.105474	−	0.994422i	$-0.533636\pi$
$809$		−	56.5685i		−	1.98884i	0.105474	−	0.994422i	$-0.466364\pi$
$810$		0			0
$811$	38.0000			1.33436			0.667180	−	0.744896i	$-0.267501\pi$
$811$	38.0000			1.33436			0.667180	+	0.744896i	$0.267501\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	10.0000	+	17.3205i	0.349856	+	0.605968i
$818$		−	31.1127i		−	1.08783i
$819$		0			0
$820$		0			0
$821$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$821$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$822$		0			0
$823$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$823$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			19.7990i			0.688478i	0.938882	+	0.344239i	$0.111863\pi$
$827$			19.7990i			0.688478i	−0.938882	+	0.344239i	$0.888137\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	34.2929	+	19.7990i	1.18818	+	0.685994i
$834$		0			0
$835$		0			0
$836$		−	11.3137i		−	0.391293i
$837$		0			0
$838$	−52.0000			−1.79631
$839$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$839$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$840$		0			0
$841$	14.5000	−	25.1147i	0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$	14.0000	+	24.2487i	0.481900	+	0.834675i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$	20.0000	−	34.6410i	0.685994	−	1.18818i
$851$		0			0
$852$		0			0
$853$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$853$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$854$		0			0
$855$		0			0
$856$	56.0000			1.91404
$857$	−19.5959	+	11.3137i	−0.669384	+	0.386469i	−0.795843	−	0.605503i	$-0.792972\pi$
$857$	−19.5959	+	11.3137i	−0.669384	+	0.386469i	0.126459	+	0.991972i	$0.459639\pi$
$858$		0			0
$859$	29.0000	−	50.2295i	0.989467	−	1.71381i	0.369370	−	0.929282i	$-0.379573\pi$
$859$	29.0000	−	50.2295i	0.989467	−	1.71381i	0.620097	−	0.784525i	$-0.287093\pi$
$860$		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
$866$	46.5403	−	26.8701i	1.58150	−	0.913082i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$877$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	56.5685i		−	1.90584i	−0.303218	−	0.952921i	$-0.598061\pi$
$881$		−	56.5685i		−	1.90584i	0.303218	−	0.952921i	$-0.401939\pi$
$882$		0			0
$883$	2.00000			0.0673054			0.0336527	−	0.999434i	$-0.489286\pi$
$883$	2.00000			0.0673054			0.0336527	+	0.999434i	$0.489286\pi$
$884$		0			0
$885$		0			0
$886$	2.00000	−	3.46410i	0.0671913	−	0.116379i
$887$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$887$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−4.00000	−	6.92820i	−0.133482	−	0.231197i
$899$		0			0
$900$		0			0
$901$		0			0
$902$	−39.1918	+	22.6274i	−1.30495	+	0.753411i
$903$		0			0
$904$	−16.0000	+	27.7128i	−0.532152	+	0.921714i
$905$		0			0
$906$		0			0
$907$	5.00000	+	8.66025i	0.166022	+	0.287559i	0.937018	−	0.349281i	$-0.113574\pi$
$907$	5.00000	+	8.66025i	0.166022	+	0.287559i	−0.770996	+	0.636841i	$0.780241\pi$
$908$		−	5.65685i		−	0.187729i
$909$		0			0
$910$		0			0
$911$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$911$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$912$		0			0
$913$	4.00000	−	6.92820i	0.132381	−	0.229290i
$914$	−31.8434	−	18.3848i	−1.05328	−	0.608114i
$915$		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
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	24.4949	−	14.1421i	0.803652	−	0.463988i	−0.0410949	−	0.999155i	$-0.513085\pi$
$929$	24.4949	−	14.1421i	0.803652	−	0.463988i	0.844746	+	0.535167i	$0.179751\pi$
$930$		0			0
$931$	−7.00000	+	12.1244i	−0.229416	+	0.397360i
$932$	−9.79796	−	5.65685i	−0.320943	−	0.185296i
$933$		0			0
$934$	−22.0000	−	38.1051i	−0.719862	−	1.24684i
$935$		0			0
$936$		0			0
$937$	−34.0000			−1.11073			−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000			−1.11073			−0.555366	+	0.831606i	$0.687422\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$941$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$942$		0			0
$943$		0			0
$944$		−	56.5685i		−	1.84115i
$945$		0			0
$946$	40.0000			1.30051
$947$	46.5403	−	26.8701i	1.51236	−	0.873160i	0.512461	−	0.858710i	$-0.328734\pi$
$947$	46.5403	−	26.8701i	1.51236	−	0.873160i	0.999896	−	0.0144491i	$-0.00459946\pi$
$948$		0			0
$949$		0			0
$950$	12.2474	+	7.07107i	0.397360	+	0.229416i
$951$		0			0
$952$		0			0
$953$			45.2548i			1.46595i	0.680257	+	0.732974i	$0.261868\pi$
$953$			45.2548i			1.46595i	−0.680257	+	0.732974i	$0.738132\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−15.5000	−	26.8468i	−0.500000	−	0.866025i
$962$		0			0
$963$		0			0
$964$	−52.0000			−1.67481
$965$		0			0
$966$		0			0
$967$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$967$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$968$	7.34847	+	4.24264i	0.236189	+	0.136364i
$969$		0			0
$970$		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
$977$	−53.8888	−	31.1127i	−1.72405	−	0.995383i	−0.910017	−	0.414572i	$-0.863931\pi$
$977$	−53.8888	−	31.1127i	−1.72405	−	0.995383i	−0.814038	−	0.580812i	$-0.802735\pi$
$978$		0			0
$979$	−8.00000	−	13.8564i	−0.255681	−	0.442853i
$980$		0			0
$981$		0			0
$982$	20.0000			0.638226
$983$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$983$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$997$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$998$			19.7990i			0.626726i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.l.b.539.2		4
3.2	odd	2	inner	648.2.l.b.539.1		4
4.3	odd	2		2592.2.p.b.2159.1		4
8.3	odd	2	CM	648.2.l.b.539.2		4
8.5	even	2		2592.2.p.b.2159.1		4
9.2	odd	6	inner	648.2.l.b.107.2		4
9.4	even	3		24.2.f.a.11.2	yes	2
9.5	odd	6		24.2.f.a.11.1	✓	2
9.7	even	3	inner	648.2.l.b.107.1		4
12.11	even	2		2592.2.p.b.2159.2		4
24.5	odd	2		2592.2.p.b.2159.2		4
24.11	even	2	inner	648.2.l.b.539.1		4
36.7	odd	6		2592.2.p.b.431.2		4
36.11	even	6		2592.2.p.b.431.1		4
36.23	even	6		96.2.f.a.47.1		2
36.31	odd	6		96.2.f.a.47.2		2
45.4	even	6		600.2.b.a.251.1		2
45.13	odd	12		600.2.m.a.299.3		4
45.14	odd	6		600.2.b.a.251.2		2
45.22	odd	12		600.2.m.a.299.2		4
45.23	even	12		600.2.m.a.299.1		4
45.32	even	12		600.2.m.a.299.4		4
72.5	odd	6		96.2.f.a.47.1		2
72.11	even	6	inner	648.2.l.b.107.2		4
72.13	even	6		96.2.f.a.47.2		2
72.29	odd	6		2592.2.p.b.431.1		4
72.43	odd	6	inner	648.2.l.b.107.1		4
72.59	even	6		24.2.f.a.11.1	✓	2
72.61	even	6		2592.2.p.b.431.2		4
72.67	odd	6		24.2.f.a.11.2	yes	2
144.5	odd	12		768.2.c.h.767.4		4
144.13	even	12		768.2.c.h.767.3		4
144.59	even	12		768.2.c.h.767.1		4
144.67	odd	12		768.2.c.h.767.2		4
144.77	odd	12		768.2.c.h.767.1		4
144.85	even	12		768.2.c.h.767.2		4
144.131	even	12		768.2.c.h.767.4		4
144.139	odd	12		768.2.c.h.767.3		4
180.23	odd	12		2400.2.m.a.1199.4		4
180.59	even	6		2400.2.b.a.2351.2		2
180.67	even	12		2400.2.m.a.1199.3		4
180.103	even	12		2400.2.m.a.1199.2		4
180.139	odd	6		2400.2.b.a.2351.1		2
180.167	odd	12		2400.2.m.a.1199.1		4
360.13	odd	12		2400.2.m.a.1199.2		4
360.59	even	6		600.2.b.a.251.2		2
360.67	even	12		600.2.m.a.299.2		4
360.77	even	12		2400.2.m.a.1199.1		4
360.139	odd	6		600.2.b.a.251.1		2
360.149	odd	6		2400.2.b.a.2351.2		2
360.157	odd	12		2400.2.m.a.1199.3		4
360.203	odd	12		600.2.m.a.299.1		4
360.229	even	6		2400.2.b.a.2351.1		2
360.283	even	12		600.2.m.a.299.3		4
360.293	even	12		2400.2.m.a.1199.4		4
360.347	odd	12		600.2.m.a.299.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.2.f.a.11.1	✓	2	9.5	odd	6
24.2.f.a.11.1	✓	2	72.59	even	6
24.2.f.a.11.2	yes	2	9.4	even	3
24.2.f.a.11.2	yes	2	72.67	odd	6
96.2.f.a.47.1		2	36.23	even	6
96.2.f.a.47.1		2	72.5	odd	6
96.2.f.a.47.2		2	36.31	odd	6
96.2.f.a.47.2		2	72.13	even	6
600.2.b.a.251.1		2	45.4	even	6
600.2.b.a.251.1		2	360.139	odd	6
600.2.b.a.251.2		2	45.14	odd	6
600.2.b.a.251.2		2	360.59	even	6
600.2.m.a.299.1		4	45.23	even	12
600.2.m.a.299.1		4	360.203	odd	12
600.2.m.a.299.2		4	45.22	odd	12
600.2.m.a.299.2		4	360.67	even	12
600.2.m.a.299.3		4	45.13	odd	12
600.2.m.a.299.3		4	360.283	even	12
600.2.m.a.299.4		4	45.32	even	12
600.2.m.a.299.4		4	360.347	odd	12
648.2.l.b.107.1		4	9.7	even	3	inner
648.2.l.b.107.1		4	72.43	odd	6	inner
648.2.l.b.107.2		4	9.2	odd	6	inner
648.2.l.b.107.2		4	72.11	even	6	inner
648.2.l.b.539.1		4	3.2	odd	2	inner
648.2.l.b.539.1		4	24.11	even	2	inner
648.2.l.b.539.2		4	1.1	even	1	trivial
648.2.l.b.539.2		4	8.3	odd	2	CM
768.2.c.h.767.1		4	144.59	even	12
768.2.c.h.767.1		4	144.77	odd	12
768.2.c.h.767.2		4	144.67	odd	12
768.2.c.h.767.2		4	144.85	even	12
768.2.c.h.767.3		4	144.13	even	12
768.2.c.h.767.3		4	144.139	odd	12
768.2.c.h.767.4		4	144.5	odd	12
768.2.c.h.767.4		4	144.131	even	12
2400.2.b.a.2351.1		2	180.139	odd	6
2400.2.b.a.2351.1		2	360.229	even	6
2400.2.b.a.2351.2		2	180.59	even	6
2400.2.b.a.2351.2		2	360.149	odd	6
2400.2.m.a.1199.1		4	180.167	odd	12
2400.2.m.a.1199.1		4	360.77	even	12
2400.2.m.a.1199.2		4	180.103	even	12
2400.2.m.a.1199.2		4	360.13	odd	12
2400.2.m.a.1199.3		4	180.67	even	12
2400.2.m.a.1199.3		4	360.157	odd	12
2400.2.m.a.1199.4		4	180.23	odd	12
2400.2.m.a.1199.4		4	360.293	even	12
2592.2.p.b.431.1		4	36.11	even	6
2592.2.p.b.431.1		4	72.29	odd	6
2592.2.p.b.431.2		4	36.7	odd	6
2592.2.p.b.431.2		4	72.61	even	6
2592.2.p.b.2159.1		4	4.3	odd	2
2592.2.p.b.2159.1		4	8.5	even	2
2592.2.p.b.2159.2		4	12.11	even	2
2592.2.p.b.2159.2		4	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	648.l (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.22474 - 0.707107i) q^{2} +(1.00000 - 1.73205i) q^{4} -2.82843i q^{8} +O(q^{10})\)	\(q+(1.22474 - 0.707107i) q^{2} +(1.00000 - 1.73205i) q^{4} -2.82843i q^{8} +(2.44949 - 1.41421i) q^{11} +(-2.00000 - 3.46410i) q^{16} -5.65685i q^{17} +2.00000 q^{19} +(2.00000 - 3.46410i) q^{22} +(2.50000 + 4.33013i) q^{25} +(-4.89898 - 2.82843i) q^{32} +(-4.00000 - 6.92820i) q^{34} +(2.44949 - 1.41421i) q^{38} +(-9.79796 - 5.65685i) q^{41} +(5.00000 + 8.66025i) q^{43} -5.65685i q^{44} +(-3.50000 + 6.06218i) q^{49} +(6.12372 + 3.53553i) q^{50} +(12.2474 + 7.07107i) q^{59} -8.00000 q^{64} +(-7.00000 + 12.1244i) q^{67} +(-9.79796 - 5.65685i) q^{68} +2.00000 q^{73} +(2.00000 - 3.46410i) q^{76} -16.0000 q^{82} +(2.44949 - 1.41421i) q^{83} +(12.2474 + 7.07107i) q^{86} +(-4.00000 - 6.92820i) q^{88} -5.65685i q^{89} +(5.00000 + 8.66025i) q^{97} +9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4}+O(q^{10}) \) 4 * q + 4 * q^4	\( 4 q + 4 q^{4} - 8 q^{16} + 8 q^{19} + 8 q^{22} + 10 q^{25} - 16 q^{34} + 20 q^{43} - 14 q^{49} - 32 q^{64} - 28 q^{67} + 8 q^{73} + 8 q^{76} - 64 q^{82} - 16 q^{88} + 20 q^{97}+O(q^{100}) \) 4 * q + 4 * q^4 - 8 * q^16 + 8 * q^19 + 8 * q^22 + 10 * q^25 - 16 * q^34 + 20 * q^43 - 14 * q^49 - 32 * q^64 - 28 * q^67 + 8 * q^73 + 8 * q^76 - 64 * q^82 - 16 * q^88 + 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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