LMFDB - Embedded newform 648.2.n.h.541.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [648,2,Mod(109,648)]

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([0, 3, 2]))

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

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

chi := DirichletCharacter("648.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	648.n (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 72)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			541.1
Root			$-1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	648.541
Dual form			648.2.n.h.109.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.22474 - 0.707107i) q^{2} +(1.00000 + 1.73205i) q^{4} +(2.44949 - 1.41421i) q^{5} +(-1.00000 + 1.73205i) q^{7} -2.82843i q^{8} +O(q^{10})$	$q+(-1.22474 - 0.707107i) q^{2} +(1.00000 + 1.73205i) q^{4} +(2.44949 - 1.41421i) q^{5} +(-1.00000 + 1.73205i) q^{7} -2.82843i q^{8} -4.00000 q^{10} +(4.89898 + 2.82843i) q^{11} +(2.44949 - 1.41421i) q^{14} +(-2.00000 + 3.46410i) q^{16} +(4.89898 + 2.82843i) q^{20} +(-4.00000 - 6.92820i) q^{22} +(1.50000 - 2.59808i) q^{25} -4.00000 q^{28} +(-2.44949 - 1.41421i) q^{29} +(5.00000 + 8.66025i) q^{31} +(4.89898 - 2.82843i) q^{32} +5.65685i q^{35} +(-4.00000 - 6.92820i) q^{40} +11.3137i q^{44} +(1.50000 + 2.59808i) q^{49} +(-3.67423 + 2.12132i) q^{50} -14.1421i q^{53} +16.0000 q^{55} +(4.89898 + 2.82843i) q^{56} +(2.00000 + 3.46410i) q^{58} +(9.79796 - 5.65685i) q^{59} -14.1421i q^{62} -8.00000 q^{64} +(4.00000 - 6.92820i) q^{70} +14.0000 q^{73} +(-9.79796 + 5.65685i) q^{77} +(5.00000 - 8.66025i) q^{79} +11.3137i q^{80} +(4.89898 + 2.82843i) q^{83} +(8.00000 - 13.8564i) q^{88} +(-1.00000 + 1.73205i) q^{97} -4.24264i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} - 4 q^{7}+O(q^{10}) $ 4 * q + 4 * q^4 - 4 * q^7	$ 4 q + 4 q^{4} - 4 q^{7} - 16 q^{10} - 8 q^{16} - 16 q^{22} + 6 q^{25} - 16 q^{28} + 20 q^{31} - 16 q^{40} + 6 q^{49} + 64 q^{55} + 8 q^{58} - 32 q^{64} + 16 q^{70} + 56 q^{73} + 20 q^{79} + 32 q^{88} - 4 q^{97}+O(q^{100}) $ 4 * q + 4 * q^4 - 4 * q^7 - 16 * q^10 - 8 * q^16 - 16 * q^22 + 6 * q^25 - 16 * q^28 + 20 * q^31 - 16 * q^40 + 6 * q^49 + 64 * q^55 + 8 * q^58 - 32 * q^64 + 16 * q^70 + 56 * q^73 + 20 * q^79 + 32 * q^88 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$325$	$487$	$569$
$\chi(n)$	$-1$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	2.44949	−	1.41421i	1.09545	−	0.632456i	0.160424	−	0.987048i	$-0.448714\pi$
$5$	2.44949	−	1.41421i	1.09545	−	0.632456i	0.935021	+	0.354593i	$0.115380\pi$
$6$		0			0
$7$	−1.00000	+	1.73205i	−0.377964	+	0.654654i	−0.990766	−	0.135583i	$-0.956709\pi$
$7$	−1.00000	+	1.73205i	−0.377964	+	0.654654i	0.612801	+	0.790237i	$0.290043\pi$
$8$		−	2.82843i		−	1.00000i
$9$		0			0
$10$	−4.00000			−1.26491
$11$	4.89898	+	2.82843i	1.47710	+	0.852803i	0.999665	−	0.0258656i	$-0.00823419\pi$
$11$	4.89898	+	2.82843i	1.47710	+	0.852803i	0.477432	+	0.878668i	$0.341568\pi$
$12$		0			0
$13$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$13$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$14$	2.44949	−	1.41421i	0.654654	−	0.377964i
$15$		0			0
$16$	−2.00000	+	3.46410i	−0.500000	+	0.866025i
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$	4.89898	+	2.82843i	1.09545	+	0.632456i
$21$		0			0
$22$	−4.00000	−	6.92820i	−0.852803	−	1.47710i
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$	1.50000	−	2.59808i	0.300000	−	0.519615i
$26$		0			0
$27$		0			0
$28$	−4.00000			−0.755929
$29$	−2.44949	−	1.41421i	−0.454859	−	0.262613i	0.255021	−	0.966935i	$-0.417918\pi$
$29$	−2.44949	−	1.41421i	−0.454859	−	0.262613i	−0.709880	+	0.704323i	$0.751251\pi$
$30$		0			0
$31$	5.00000	+	8.66025i	0.898027	+	1.55543i	0.830014	+	0.557743i	$0.188333\pi$
$31$	5.00000	+	8.66025i	0.898027	+	1.55543i	0.0680129	+	0.997684i	$0.478334\pi$
$32$	4.89898	−	2.82843i	0.866025	−	0.500000i
$33$		0			0
$34$		0			0
$35$			5.65685i			0.956183i
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$	−4.00000	−	6.92820i	−0.632456	−	1.09545i
$41$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$41$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$42$		0			0
$43$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$43$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$44$			11.3137i			1.70561i
$45$		0			0
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		0			0
$49$	1.50000	+	2.59808i	0.214286	+	0.371154i
$50$	−3.67423	+	2.12132i	−0.519615	+	0.300000i
$51$		0			0
$52$		0			0
$53$		−	14.1421i		−	1.94257i	−0.237915	−	0.971286i	$-0.576464\pi$
$53$		−	14.1421i		−	1.94257i	0.237915	−	0.971286i	$-0.423536\pi$
$54$		0			0
$55$	16.0000			2.15744
$56$	4.89898	+	2.82843i	0.654654	+	0.377964i
$57$		0			0
$58$	2.00000	+	3.46410i	0.262613	+	0.454859i
$59$	9.79796	−	5.65685i	1.27559	−	0.736460i	0.299552	−	0.954080i	$-0.403163\pi$
$59$	9.79796	−	5.65685i	1.27559	−	0.736460i	0.976034	+	0.217620i	$0.0698294\pi$
$60$		0			0
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$		−	14.1421i		−	1.79605i
$63$		0			0
$64$	−8.00000			−1.00000
$65$		0			0
$66$		0			0
$67$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$		0			0
$69$		0			0
$70$	4.00000	−	6.92820i	0.478091	−	0.828079i
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	14.0000			1.63858			0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000			1.63858			0.819288	+	0.573382i	$0.194369\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−9.79796	+	5.65685i	−1.11658	+	0.644658i
$78$		0			0
$79$	5.00000	−	8.66025i	0.562544	−	0.974355i	−0.434730	−	0.900561i	$-0.643156\pi$
$79$	5.00000	−	8.66025i	0.562544	−	0.974355i	0.997274	−	0.0737937i	$-0.0235106\pi$
$80$			11.3137i			1.26491i
$81$		0			0
$82$		0			0
$83$	4.89898	+	2.82843i	0.537733	+	0.310460i	0.744160	−	0.668002i	$-0.232850\pi$
$83$	4.89898	+	2.82843i	0.537733	+	0.310460i	−0.206427	+	0.978462i	$0.566184\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	8.00000	−	13.8564i	0.852803	−	1.47710i
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.00000	+	1.73205i	−0.101535	+	0.175863i	−0.912317	−	0.409484i	$-0.865709\pi$
$97$	−1.00000	+	1.73205i	−0.101535	+	0.175863i	0.810782	+	0.585348i	$0.199042\pi$
$98$		−	4.24264i		−	0.428571i
$99$		0			0
$100$	6.00000			0.600000
$101$	−17.1464	−	9.89949i	−1.70613	−	0.985037i	−0.939239	−	0.343263i	$-0.888468\pi$
$101$	−17.1464	−	9.89949i	−1.70613	−	0.985037i	−0.766894	−	0.641774i	$-0.778199\pi$
$102$		0			0
$103$	−7.00000	−	12.1244i	−0.689730	−	1.19465i	−0.971925	−	0.235291i	$-0.924396\pi$
$103$	−7.00000	−	12.1244i	−0.689730	−	1.19465i	0.282194	−	0.959357i	$-0.408938\pi$
$104$		0			0
$105$		0			0
$106$	−10.0000	+	17.3205i	−0.971286	+	1.68232i
$107$			11.3137i			1.09374i	0.837218	+	0.546869i	$0.184180\pi$
$107$			11.3137i			1.09374i	−0.837218	+	0.546869i	$0.815820\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$	−19.5959	−	11.3137i	−1.86840	−	1.07872i
$111$		0			0
$112$	−4.00000	−	6.92820i	−0.377964	−	0.654654i
$113$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$113$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$114$		0			0
$115$		0			0
$116$		−	5.65685i		−	0.525226i
$117$		0			0
$118$	−16.0000			−1.47292
$119$		0			0
$120$		0			0
$121$	10.5000	+	18.1865i	0.954545	+	1.65332i
$122$		0			0
$123$		0			0
$124$	−10.0000	+	17.3205i	−0.898027	+	1.55543i
$125$			5.65685i			0.505964i
$126$		0			0
$127$	−22.0000			−1.95218			−0.976092	−	0.217357i	$-0.930256\pi$
$127$	−22.0000			−1.95218			−0.976092	+	0.217357i	$0.930256\pi$
$128$	9.79796	+	5.65685i	0.866025	+	0.500000i
$129$		0			0
$130$		0			0
$131$	−19.5959	+	11.3137i	−1.71210	+	0.988483i	−0.780387	+	0.625297i	$0.784978\pi$
$131$	−19.5959	+	11.3137i	−1.71210	+	0.988483i	−0.931717	+	0.363186i	$0.881689\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$137$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$138$		0			0
$139$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$139$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$140$	−9.79796	+	5.65685i	−0.828079	+	0.478091i
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−8.00000			−0.664364
$146$	−17.1464	−	9.89949i	−1.41905	−	0.819288i
$147$		0			0
$148$		0			0
$149$	2.44949	−	1.41421i	0.200670	−	0.115857i	−0.396298	−	0.918122i	$-0.629705\pi$
$149$	2.44949	−	1.41421i	0.200670	−	0.115857i	0.596968	+	0.802265i	$0.296372\pi$
$150$		0			0
$151$	−1.00000	+	1.73205i	−0.0813788	+	0.140952i	−0.903842	−	0.427865i	$-0.859266\pi$
$151$	−1.00000	+	1.73205i	−0.0813788	+	0.140952i	0.822464	+	0.568818i	$0.192599\pi$
$152$		0			0
$153$		0			0
$154$	16.0000			1.28932
$155$	24.4949	+	14.1421i	1.96748	+	1.13592i
$156$		0			0
$157$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$157$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$158$	−12.2474	+	7.07107i	−0.974355	+	0.562544i
$159$		0			0
$160$	8.00000	−	13.8564i	0.632456	−	1.09545i
$161$		0			0
$162$		0			0
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$	−4.00000	−	6.92820i	−0.310460	−	0.537733i
$167$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$167$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$168$		0			0
$169$	−6.50000	+	11.2583i	−0.500000	+	0.866025i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−17.1464	−	9.89949i	−1.30362	−	0.752645i	−0.322596	−	0.946537i	$-0.604555\pi$
$173$	−17.1464	−	9.89949i	−1.30362	−	0.752645i	−0.981023	+	0.193892i	$0.937889\pi$
$174$		0			0
$175$	3.00000	+	5.19615i	0.226779	+	0.392792i
$176$	−19.5959	+	11.3137i	−1.47710	+	0.852803i
$177$		0			0
$178$		0			0
$179$			11.3137i			0.845626i	0.906217	+	0.422813i	$0.138957\pi$
$179$			11.3137i			0.845626i	−0.906217	+	0.422813i	$0.861043\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$191$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$192$		0			0
$193$	−13.0000	−	22.5167i	−0.935760	−	1.62078i	−0.773272	−	0.634074i	$-0.781381\pi$
$193$	−13.0000	−	22.5167i	−0.935760	−	1.62078i	−0.162488	−	0.986710i	$-0.551952\pi$
$194$	2.44949	−	1.41421i	0.175863	−	0.101535i
$195$		0			0
$196$	−3.00000	+	5.19615i	−0.214286	+	0.371154i
$197$		−	14.1421i		−	1.00759i	−0.863825	−	0.503793i	$-0.831938\pi$
$197$		−	14.1421i		−	1.00759i	0.863825	−	0.503793i	$-0.168062\pi$
$198$		0			0
$199$	14.0000			0.992434			0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000			0.992434			0.496217	+	0.868199i	$0.334722\pi$
$200$	−7.34847	−	4.24264i	−0.519615	−	0.300000i
$201$		0			0
$202$	14.0000	+	24.2487i	0.985037	+	1.70613i
$203$	4.89898	−	2.82843i	0.343841	−	0.198517i
$204$		0			0
$205$		0			0
$206$			19.7990i			1.37946i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$211$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$212$	24.4949	−	14.1421i	1.68232	−	0.971286i
$213$		0			0
$214$	8.00000	−	13.8564i	0.546869	−	0.947204i
$215$		0			0
$216$		0			0
$217$	−20.0000			−1.35769
$218$		0			0
$219$		0			0
$220$	16.0000	+	27.7128i	1.07872	+	1.86840i
$221$		0			0
$222$		0			0
$223$	−13.0000	+	22.5167i	−0.870544	+	1.50783i	−0.00910984	+	0.999959i	$0.502900\pi$
$223$	−13.0000	+	22.5167i	−0.870544	+	1.50783i	−0.861435	+	0.507869i	$0.830434\pi$
$224$			11.3137i			0.755929i
$225$		0			0
$226$		0			0
$227$	−24.4949	−	14.1421i	−1.62578	−	0.938647i	−0.985332	−	0.170648i	$-0.945414\pi$
$227$	−24.4949	−	14.1421i	−1.62578	−	0.938647i	−0.640451	−	0.767999i	$-0.721253\pi$
$228$		0			0
$229$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$229$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$230$		0			0
$231$		0			0
$232$	−4.00000	+	6.92820i	−0.262613	+	0.454859i
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$		0			0
$236$	19.5959	+	11.3137i	1.27559	+	0.736460i
$237$		0			0
$238$		0			0
$239$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$239$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$240$		0			0
$241$	5.00000	−	8.66025i	0.322078	−	0.557856i	−0.658838	−	0.752285i	$-0.728952\pi$
$241$	5.00000	−	8.66025i	0.322078	−	0.557856i	0.980917	+	0.194429i	$0.0622852\pi$
$242$		−	29.6985i		−	1.90909i
$243$		0			0
$244$		0			0
$245$	7.34847	+	4.24264i	0.469476	+	0.271052i
$246$		0			0
$247$		0			0
$248$	24.4949	−	14.1421i	1.55543	−	0.898027i
$249$		0			0
$250$	4.00000	−	6.92820i	0.252982	−	0.438178i
$251$		−	5.65685i		−	0.357057i	−0.983935	−	0.178529i	$-0.942866\pi$
$251$		−	5.65685i		−	0.357057i	0.983935	−	0.178529i	$-0.0571337\pi$
$252$		0			0
$253$		0			0
$254$	26.9444	+	15.5563i	1.69064	+	0.976092i
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$257$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	32.0000			1.97697
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$	−20.0000	−	34.6410i	−1.22859	−	2.12798i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	31.1127i		−	1.89697i	−0.316815	−	0.948487i	$-0.602613\pi$
$269$		−	31.1127i		−	1.89697i	0.316815	−	0.948487i	$-0.397387\pi$
$270$		0			0
$271$	−22.0000			−1.33640			−0.668202	−	0.743980i	$-0.732936\pi$
$271$	−22.0000			−1.33640			−0.668202	+	0.743980i	$0.732936\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	14.6969	−	8.48528i	0.886259	−	0.511682i
$276$		0			0
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$		0			0
$279$		0			0
$280$	16.0000			0.956183
$281$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$281$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$282$		0			0
$283$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$283$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−17.0000			−1.00000
$290$	9.79796	+	5.65685i	0.575356	+	0.332182i
$291$		0			0
$292$	14.0000	+	24.2487i	0.819288	+	1.41905i
$293$	−12.2474	+	7.07107i	−0.715504	+	0.413096i	−0.813095	−	0.582130i	$-0.802219\pi$
$293$	−12.2474	+	7.07107i	−0.715504	+	0.413096i	0.0975919	+	0.995227i	$0.468886\pi$
$294$		0			0
$295$	16.0000	−	27.7128i	0.931556	−	1.61350i
$296$		0			0
$297$		0			0
$298$	−4.00000			−0.231714
$299$		0			0
$300$		0			0
$301$		0			0
$302$	2.44949	−	1.41421i	0.140952	−	0.0813788i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$	−19.5959	−	11.3137i	−1.11658	−	0.644658i
$309$		0			0
$310$	−20.0000	−	34.6410i	−1.13592	−	1.96748i
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	17.0000	−	29.4449i	0.960897	−	1.66432i	0.240640	−	0.970614i	$-0.422643\pi$
$313$	17.0000	−	29.4449i	0.960897	−	1.66432i	0.720257	−	0.693708i	$-0.244024\pi$
$314$		0			0
$315$		0			0
$316$	20.0000			1.12509
$317$	26.9444	+	15.5563i	1.51335	+	0.873732i	0.999878	+	0.0156238i	$0.00497343\pi$
$317$	26.9444	+	15.5563i	1.51335	+	0.873732i	0.513470	+	0.858108i	$0.328360\pi$
$318$		0			0
$319$	−8.00000	−	13.8564i	−0.447914	−	0.775810i
$320$	−19.5959	+	11.3137i	−1.09545	+	0.632456i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$			11.3137i			0.620920i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	11.0000	+	19.0526i	0.599208	+	1.03786i	0.992938	+	0.118633i	$0.0378512\pi$
$337$	11.0000	+	19.0526i	0.599208	+	1.03786i	−0.393730	+	0.919226i	$0.628816\pi$
$338$	15.9217	−	9.19239i	0.866025	−	0.500000i
$339$		0			0
$340$		0			0
$341$			56.5685i			3.06336i
$342$		0			0
$343$	−20.0000			−1.07990
$344$		0			0
$345$		0			0
$346$	14.0000	+	24.2487i	0.752645	+	1.30362i
$347$	24.4949	−	14.1421i	1.31495	−	0.759190i	0.332043	−	0.943264i	$-0.392262\pi$
$347$	24.4949	−	14.1421i	1.31495	−	0.759190i	0.982912	+	0.184075i	$0.0589288\pi$
$348$		0			0
$349$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$349$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$350$		−	8.48528i		−	0.453557i
$351$		0			0
$352$	32.0000			1.70561
$353$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$353$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	8.00000	−	13.8564i	0.422813	−	0.732334i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	19.0000			1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$	34.2929	−	19.7990i	1.79497	−	1.03633i
$366$		0			0
$367$	−19.0000	+	32.9090i	−0.991792	+	1.71783i	−0.385164	+	0.922848i	$0.625855\pi$
$367$	−19.0000	+	32.9090i	−0.991792	+	1.71783i	−0.606628	+	0.794986i	$0.707478\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	24.4949	+	14.1421i	1.27171	+	0.734223i
$372$		0			0
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$	−16.0000	+	27.7128i	−0.815436	+	1.41238i
$386$			36.7696i			1.87152i
$387$		0			0
$388$	−4.00000			−0.203069
$389$	26.9444	+	15.5563i	1.36613	+	0.788738i	0.990432	−	0.138002i	$-0.0440680\pi$
$389$	26.9444	+	15.5563i	1.36613	+	0.788738i	0.375703	+	0.926740i	$0.377401\pi$
$390$		0			0
$391$		0			0
$392$	7.34847	−	4.24264i	0.371154	−	0.214286i
$393$		0			0
$394$	−10.0000	+	17.3205i	−0.503793	+	0.872595i
$395$		−	28.2843i		−	1.42314i
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$	−17.1464	−	9.89949i	−0.859473	−	0.496217i
$399$		0			0
$400$	6.00000	+	10.3923i	0.300000	+	0.519615i
$401$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$401$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$402$		0			0
$403$		0			0
$404$		−	39.5980i		−	1.97007i
$405$		0			0
$406$	−8.00000			−0.397033
$407$		0			0
$408$		0			0
$409$	5.00000	+	8.66025i	0.247234	+	0.428222i	0.962757	−	0.270367i	$-0.0871450\pi$
$409$	5.00000	+	8.66025i	0.247234	+	0.428222i	−0.715523	+	0.698589i	$0.753812\pi$
$410$		0			0
$411$		0			0
$412$	14.0000	−	24.2487i	0.689730	−	1.19465i
$413$			22.6274i			1.11342i
$414$		0			0
$415$	16.0000			0.785409
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−34.2929	+	19.7990i	−1.67532	+	0.967244i	−0.710734	+	0.703461i	$0.751637\pi$
$419$	−34.2929	+	19.7990i	−1.67532	+	0.967244i	−0.964582	+	0.263783i	$0.915030\pi$
$420$		0			0
$421$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$421$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$422$		0			0
$423$		0			0
$424$	−40.0000			−1.94257
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−19.5959	+	11.3137i	−0.947204	+	0.546869i
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	14.0000			0.672797			0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000			0.672797			0.336399	+	0.941720i	$0.390791\pi$
$434$	24.4949	+	14.1421i	1.17579	+	0.678844i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	17.0000	−	29.4449i	0.811366	−	1.40533i	−0.100543	−	0.994933i	$-0.532058\pi$
$439$	17.0000	−	29.4449i	0.811366	−	1.40533i	0.911908	−	0.410394i	$-0.134609\pi$
$440$		−	45.2548i		−	2.15744i
$441$		0			0
$442$		0			0
$443$	−24.4949	−	14.1421i	−1.16379	−	0.671913i	−0.211579	−	0.977361i	$-0.567861\pi$
$443$	−24.4949	−	14.1421i	−1.16379	−	0.671913i	−0.952209	+	0.305448i	$0.901194\pi$
$444$		0			0
$445$		0			0
$446$	31.8434	−	18.3848i	1.50783	−	0.870544i
$447$		0			0
$448$	8.00000	−	13.8564i	0.377964	−	0.654654i
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$	20.0000	+	34.6410i	0.938647	+	1.62578i
$455$		0			0
$456$		0			0
$457$	−19.0000	+	32.9090i	−0.888783	+	1.53942i	−0.0474665	+	0.998873i	$0.515115\pi$
$457$	−19.0000	+	32.9090i	−0.888783	+	1.53942i	−0.841316	+	0.540544i	$0.818219\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−17.1464	−	9.89949i	−0.798589	−	0.461065i	0.0443887	−	0.999014i	$-0.485866\pi$
$461$	−17.1464	−	9.89949i	−0.798589	−	0.461065i	−0.842977	+	0.537949i	$0.819199\pi$
$462$		0			0
$463$	−13.0000	−	22.5167i	−0.604161	−	1.04644i	−0.992183	−	0.124788i	$-0.960175\pi$
$463$	−13.0000	−	22.5167i	−0.604161	−	1.04644i	0.388022	−	0.921650i	$-0.373158\pi$
$464$	9.79796	−	5.65685i	0.454859	−	0.262613i
$465$		0			0
$466$		0			0
$467$		−	39.5980i		−	1.83238i	−0.400749	−	0.916188i	$-0.631250\pi$
$467$		−	39.5980i		−	1.83238i	0.400749	−	0.916188i	$-0.368750\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−16.0000	−	27.7128i	−0.736460	−	1.27559i
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$		0			0
$482$	−12.2474	+	7.07107i	−0.557856	+	0.322078i
$483$		0			0
$484$	−21.0000	+	36.3731i	−0.954545	+	1.65332i
$485$			5.65685i			0.256865i
$486$		0			0
$487$	2.00000			0.0906287			0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000			0.0906287			0.0453143	+	0.998973i	$0.485571\pi$
$488$		0			0
$489$		0			0
$490$	−6.00000	−	10.3923i	−0.271052	−	0.469476i
$491$	−19.5959	+	11.3137i	−0.884351	+	0.510581i	−0.872091	−	0.489344i	$-0.837236\pi$
$491$	−19.5959	+	11.3137i	−0.884351	+	0.510581i	−0.0122607	+	0.999925i	$0.503903\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−40.0000			−1.79605
$497$		0			0
$498$		0			0
$499$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$499$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$500$	−9.79796	+	5.65685i	−0.438178	+	0.252982i
$501$		0			0
$502$	−4.00000	+	6.92820i	−0.178529	+	0.309221i
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$	−56.0000			−2.49197
$506$		0			0
$507$		0			0
$508$	−22.0000	−	38.1051i	−0.976092	−	1.69064i
$509$	2.44949	−	1.41421i	0.108572	−	0.0626839i	−0.444731	−	0.895664i	$-0.646701\pi$
$509$	2.44949	−	1.41421i	0.108572	−	0.0626839i	0.553303	+	0.832980i	$0.313367\pi$
$510$		0			0
$511$	−14.0000	+	24.2487i	−0.619324	+	1.07270i
$512$			22.6274i			1.00000i
$513$		0			0
$514$		0			0
$515$	−34.2929	−	19.7990i	−1.51112	−	0.872448i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$	−39.1918	−	22.6274i	−1.71210	−	0.988483i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	−	19.9186i	0.500000	−	0.866025i
$530$			56.5685i			2.45718i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	16.0000	+	27.7128i	0.691740	+	1.19813i
$536$		0			0
$537$		0			0
$538$	−22.0000	+	38.1051i	−0.948487	+	1.64283i
$539$			16.9706i			0.730974i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$	26.9444	+	15.5563i	1.15736	+	0.668202i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$547$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$548$		0			0
$549$		0			0
$550$	−24.0000			−1.02336
$551$		0			0
$552$		0			0
$553$	10.0000	+	17.3205i	0.425243	+	0.736543i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	14.1421i		−	0.599222i	−0.954062	−	0.299611i	$-0.903143\pi$
$557$		−	14.1421i		−	0.599222i	0.954062	−	0.299611i	$-0.0968568\pi$
$558$		0			0
$559$		0			0
$560$	−19.5959	−	11.3137i	−0.828079	−	0.478091i
$561$		0			0
$562$		0			0
$563$	24.4949	−	14.1421i	1.03234	−	0.596020i	0.114684	−	0.993402i	$-0.463415\pi$
$563$	24.4949	−	14.1421i	1.03234	−	0.596020i	0.917653	+	0.397382i	$0.130081\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$569$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$570$		0			0
$571$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$571$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	38.0000			1.58196			0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000			1.58196			0.790980	+	0.611842i	$0.209571\pi$
$578$	20.8207	+	12.0208i	0.866025	+	0.500000i
$579$		0			0
$580$	−8.00000	−	13.8564i	−0.332182	−	0.575356i
$581$	−9.79796	+	5.65685i	−0.406488	+	0.234686i
$582$		0			0
$583$	40.0000	−	69.2820i	1.65663	−	2.86937i
$584$		−	39.5980i		−	1.63858i
$585$		0			0
$586$	20.0000			0.826192
$587$	−39.1918	−	22.6274i	−1.61762	−	0.933933i	−0.987534	−	0.157409i	$-0.949686\pi$
$587$	−39.1918	−	22.6274i	−1.61762	−	0.933933i	−0.630087	−	0.776525i	$-0.716981\pi$
$588$		0			0
$589$		0			0
$590$	−39.1918	+	22.6274i	−1.61350	+	0.931556i
$591$		0			0
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$	4.89898	+	2.82843i	0.200670	+	0.115857i
$597$		0			0
$598$		0			0
$599$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$599$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$600$		0			0
$601$	−1.00000	+	1.73205i	−0.0407909	+	0.0706518i	−0.885700	−	0.464258i	$-0.846321\pi$
$601$	−1.00000	+	1.73205i	−0.0407909	+	0.0706518i	0.844909	+	0.534910i	$0.179654\pi$
$602$		0			0
$603$		0			0
$604$	−4.00000			−0.162758
$605$	51.4393	+	29.6985i	2.09130	+	1.20742i
$606$		0			0
$607$	11.0000	+	19.0526i	0.446476	+	0.773320i	0.998154	−	0.0607380i	$-0.0193454\pi$
$607$	11.0000	+	19.0526i	0.446476	+	0.773320i	−0.551678	+	0.834058i	$0.686012\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$	16.0000	+	27.7128i	0.644658	+	1.11658i
$617$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$617$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$618$		0			0
$619$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$619$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$620$			56.5685i			2.27185i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	15.5000	+	26.8468i	0.620000	+	1.07387i
$626$	−41.6413	+	24.0416i	−1.66432	+	0.960897i
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	50.0000			1.99047			0.995234	−	0.0975126i	$-0.0310886\pi$
$631$	50.0000			1.99047			0.995234	+	0.0975126i	$0.0310886\pi$
$632$	−24.4949	−	14.1421i	−0.974355	−	0.562544i
$633$		0			0
$634$	−22.0000	−	38.1051i	−0.873732	−	1.51335i
$635$	−53.8888	+	31.1127i	−2.13851	+	1.23467i
$636$		0			0
$637$		0			0
$638$			22.6274i			0.895828i
$639$		0			0
$640$	32.0000			1.26491
$641$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$641$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$642$		0			0
$643$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$643$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$644$		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
$649$	64.0000			2.51222
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−41.6413	+	24.0416i	−1.62955	+	0.940822i	−0.645325	+	0.763909i	$0.723278\pi$
$653$	−41.6413	+	24.0416i	−1.62955	+	0.940822i	−0.984226	+	0.176913i	$0.943389\pi$
$654$		0			0
$655$	−32.0000	+	55.4256i	−1.25034	+	2.16566i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−39.1918	−	22.6274i	−1.52670	−	0.881439i	−0.999498	−	0.0316976i	$-0.989909\pi$
$659$	−39.1918	−	22.6274i	−1.52670	−	0.881439i	−0.527200	−	0.849741i	$-0.676758\pi$
$660$		0			0
$661$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$661$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$662$		0			0
$663$		0			0
$664$	8.00000	−	13.8564i	0.310460	−	0.537733i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	17.0000	−	29.4449i	0.655302	−	1.13502i	−0.326516	−	0.945192i	$-0.605875\pi$
$673$	17.0000	−	29.4449i	0.655302	−	1.13502i	0.981818	−	0.189824i	$-0.0607919\pi$
$674$		−	31.1127i		−	1.19842i
$675$		0			0
$676$	−26.0000			−1.00000
$677$	−2.44949	−	1.41421i	−0.0941415	−	0.0543526i	0.452190	−	0.891922i	$-0.350643\pi$
$677$	−2.44949	−	1.41421i	−0.0941415	−	0.0543526i	−0.546332	+	0.837569i	$0.683976\pi$
$678$		0			0
$679$	−2.00000	−	3.46410i	−0.0767530	−	0.132940i
$680$		0			0
$681$		0			0
$682$	40.0000	−	69.2820i	1.53168	−	2.65295i
$683$		−	5.65685i		−	0.216454i	−0.994126	−	0.108227i	$-0.965483\pi$
$683$		−	5.65685i		−	0.216454i	0.994126	−	0.108227i	$-0.0345173\pi$
$684$		0			0
$685$		0			0
$686$	24.4949	+	14.1421i	0.935220	+	0.539949i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$691$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$692$		−	39.5980i		−	1.50529i
$693$		0			0
$694$	−40.0000			−1.51838
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−6.00000	+	10.3923i	−0.226779	+	0.392792i
$701$			36.7696i			1.38877i	0.719605	+	0.694383i	$0.244323\pi$
$701$			36.7696i			1.38877i	−0.719605	+	0.694383i	$0.755677\pi$
$702$		0			0
$703$		0			0
$704$	−39.1918	−	22.6274i	−1.47710	−	0.852803i
$705$		0			0
$706$		0			0
$707$	34.2929	−	19.7990i	1.28972	−	0.744618i
$708$		0			0
$709$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−19.5959	+	11.3137i	−0.732334	+	0.422813i
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	28.0000			1.04277
$722$	−23.2702	−	13.4350i	−0.866025	−	0.500000i
$723$		0			0
$724$		0			0
$725$	−7.34847	+	4.24264i	−0.272915	+	0.157568i
$726$		0			0
$727$	−1.00000	+	1.73205i	−0.0370879	+	0.0642382i	−0.883974	−	0.467537i	$-0.845142\pi$
$727$	−1.00000	+	1.73205i	−0.0370879	+	0.0642382i	0.846886	+	0.531775i	$0.178475\pi$
$728$		0			0
$729$		0			0
$730$	−56.0000			−2.07265
$731$		0			0
$732$		0			0
$733$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$733$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$734$	46.5403	−	26.8701i	1.71783	−	0.991792i
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$	−20.0000	−	34.6410i	−0.734223	−	1.27171i
$743$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$743$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$744$		0			0
$745$	4.00000	−	6.92820i	0.146549	−	0.253830i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−19.5959	−	11.3137i	−0.716019	−	0.413394i
$750$		0			0
$751$	5.00000	+	8.66025i	0.182453	+	0.316017i	0.942715	−	0.333599i	$-0.108263\pi$
$751$	5.00000	+	8.66025i	0.182453	+	0.316017i	−0.760263	+	0.649616i	$0.774930\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			5.65685i			0.205874i
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$761$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−13.0000	−	22.5167i	−0.468792	−	0.811972i	0.530572	−	0.847640i	$-0.321977\pi$
$769$	−13.0000	−	22.5167i	−0.468792	−	0.811972i	−0.999364	+	0.0356685i	$0.988644\pi$
$770$	39.1918	−	22.6274i	1.41238	−	0.815436i
$771$		0			0
$772$	26.0000	−	45.0333i	0.935760	−	1.62078i
$773$			19.7990i			0.712120i	0.934463	+	0.356060i	$0.115880\pi$
$773$			19.7990i			0.712120i	−0.934463	+	0.356060i	$0.884120\pi$
$774$		0			0
$775$	30.0000			1.07763
$776$	4.89898	+	2.82843i	0.175863	+	0.101535i
$777$		0			0
$778$	−22.0000	−	38.1051i	−0.788738	−	1.36613i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−12.0000			−0.428571
$785$		0			0
$786$		0			0
$787$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$787$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$788$	24.4949	−	14.1421i	0.872595	−	0.503793i
$789$		0			0
$790$	−20.0000	+	34.6410i	−0.711568	+	1.23247i
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	14.0000	+	24.2487i	0.496217	+	0.859473i
$797$	46.5403	−	26.8701i	1.64854	−	0.951786i	0.670890	−	0.741557i	$-0.265912\pi$
$797$	46.5403	−	26.8701i	1.64854	−	0.951786i	0.977652	−	0.210230i	$-0.0674211\pi$
$798$		0			0
$799$		0			0
$800$		−	16.9706i		−	0.600000i
$801$		0			0
$802$		0			0
$803$	68.5857	+	39.5980i	2.42034	+	1.39738i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−28.0000	+	48.4974i	−0.985037	+	1.70613i
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	9.79796	+	5.65685i	0.343841	+	0.198517i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		−	14.1421i		−	0.494468i
$819$		0			0
$820$		0			0
$821$	41.6413	+	24.0416i	1.45329	+	0.839059i	0.998667	−	0.0516239i	$-0.0164397\pi$
$821$	41.6413	+	24.0416i	1.45329	+	0.839059i	0.454626	+	0.890683i	$0.349773\pi$
$822$		0			0
$823$	23.0000	+	39.8372i	0.801730	+	1.38864i	0.918477	+	0.395475i	$0.129420\pi$
$823$	23.0000	+	39.8372i	0.801730	+	1.38864i	−0.116747	+	0.993162i	$0.537247\pi$
$824$	−34.2929	+	19.7990i	−1.19465	+	0.689730i
$825$		0			0
$826$	16.0000	−	27.7128i	0.556711	−	0.964252i
$827$		−	56.5685i		−	1.96708i	−0.180688	−	0.983540i	$-0.557832\pi$
$827$		−	56.5685i		−	1.96708i	0.180688	−	0.983540i	$-0.442168\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$	−19.5959	−	11.3137i	−0.680184	−	0.392705i
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$	56.0000			1.93449
$839$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$839$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$840$		0			0
$841$	−10.5000	−	18.1865i	−0.362069	−	0.627122i
$842$		0			0
$843$		0			0
$844$		0			0
$845$			36.7696i			1.26491i
$846$		0			0
$847$	−42.0000			−1.44314
$848$	48.9898	+	28.2843i	1.68232	+	0.971286i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$853$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$854$		0			0
$855$		0			0
$856$	32.0000			1.09374
$857$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$857$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$858$		0			0
$859$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$859$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$860$		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
$865$	−56.0000			−1.90406
$866$	−17.1464	−	9.89949i	−0.582659	−	0.336399i
$867$		0			0
$868$	−20.0000	−	34.6410i	−0.678844	−	1.17579i
$869$	48.9898	−	28.2843i	1.66186	−	0.959478i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−9.79796	−	5.65685i	−0.331231	−	0.191237i
$876$		0			0
$877$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$877$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$878$	−41.6413	+	24.0416i	−1.40533	+	0.811366i
$879$		0			0
$880$	−32.0000	+	55.4256i	−1.07872	+	1.86840i
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$	20.0000	+	34.6410i	0.671913	+	1.16379i
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$	22.0000	−	38.1051i	0.737856	−	1.27800i
$890$		0			0
$891$		0			0
$892$	−52.0000			−1.74109
$893$		0			0
$894$		0			0
$895$	16.0000	+	27.7128i	0.534821	+	0.926337i
$896$	−19.5959	+	11.3137i	−0.654654	+	0.377964i
$897$		0			0
$898$		0			0
$899$		−	28.2843i		−	0.943333i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$907$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$908$		−	56.5685i		−	1.87729i
$909$		0			0
$910$		0			0
$911$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$911$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$912$		0			0
$913$	16.0000	+	27.7128i	0.529523	+	0.917160i
$914$	46.5403	−	26.8701i	1.53942	−	0.888783i
$915$		0			0
$916$		0			0
$917$		−	45.2548i		−	1.49445i
$918$		0			0
$919$	50.0000			1.64935			0.824674	−	0.565608i	$-0.191359\pi$
$919$	50.0000			1.64935			0.824674	+	0.565608i	$0.191359\pi$
$920$		0			0
$921$		0			0
$922$	14.0000	+	24.2487i	0.461065	+	0.798589i
$923$		0			0
$924$		0			0
$925$		0			0
$926$			36.7696i			1.20832i
$927$		0			0
$928$	−16.0000			−0.525226
$929$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$929$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$	−28.0000	+	48.4974i	−0.916188	+	1.58688i
$935$		0			0
$936$		0			0
$937$	−58.0000			−1.89478			−0.947389	−	0.320085i	$-0.896288\pi$
$937$	−58.0000			−1.89478			−0.947389	+	0.320085i	$0.896288\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−41.6413	+	24.0416i	−1.35747	+	0.783735i	−0.989282	−	0.146017i	$-0.953354\pi$
$941$	−41.6413	+	24.0416i	−1.35747	+	0.783735i	−0.368186	+	0.929752i	$0.620021\pi$
$942$		0			0
$943$		0			0
$944$			45.2548i			1.47292i
$945$		0			0
$946$		0			0
$947$	48.9898	+	28.2843i	1.59195	+	0.919115i	0.992972	+	0.118354i	$0.0377616\pi$
$947$	48.9898	+	28.2843i	1.59195	+	0.919115i	0.598983	+	0.800762i	$0.295572\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−34.5000	+	59.7558i	−1.11290	+	1.92760i
$962$		0			0
$963$		0			0
$964$	20.0000			0.644157
$965$	−63.6867	−	36.7696i	−2.05015	−	1.18365i
$966$		0			0
$967$	−31.0000	−	53.6936i	−0.996893	−	1.72667i	−0.566663	−	0.823949i	$-0.691766\pi$
$967$	−31.0000	−	53.6936i	−0.996893	−	1.72667i	−0.430229	−	0.902720i	$-0.641567\pi$
$968$	51.4393	−	29.6985i	1.65332	−	0.954545i
$969$		0			0
$970$	4.00000	−	6.92820i	0.128432	−	0.222451i
$971$			62.2254i			1.99691i	0.0555842	+	0.998454i	$0.482298\pi$
$971$			62.2254i			1.99691i	−0.0555842	+	0.998454i	$0.517702\pi$
$972$		0			0
$973$		0			0
$974$	−2.44949	−	1.41421i	−0.0784867	−	0.0453143i
$975$		0			0
$976$		0			0
$977$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$977$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$978$		0			0
$979$		0			0
$980$			16.9706i			0.542105i
$981$		0			0
$982$	32.0000			1.02116
$983$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$983$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$984$		0			0
$985$	−20.0000	−	34.6410i	−0.637253	−	1.10375i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−58.0000			−1.84243			−0.921215	−	0.389053i	$-0.872802\pi$
$991$	−58.0000			−1.84243			−0.921215	+	0.389053i	$0.872802\pi$
$992$	48.9898	+	28.2843i	1.55543	+	0.898027i
$993$		0			0
$994$		0			0
$995$	34.2929	−	19.7990i	1.08716	−	0.627670i
$996$		0			0
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.n.h.541.1		4
3.2	odd	2	inner	648.2.n.h.541.2		4
4.3	odd	2		2592.2.r.i.2161.2		4
8.3	odd	2		2592.2.r.i.2161.1		4
8.5	even	2	inner	648.2.n.h.541.2		4
9.2	odd	6		72.2.d.a.37.1	✓	2
9.4	even	3	inner	648.2.n.h.109.2		4
9.5	odd	6	inner	648.2.n.h.109.1		4
9.7	even	3		72.2.d.a.37.2	yes	2
12.11	even	2		2592.2.r.i.2161.1		4
24.5	odd	2	CM	648.2.n.h.541.1		4
24.11	even	2		2592.2.r.i.2161.2		4
36.7	odd	6		288.2.d.a.145.2		2
36.11	even	6		288.2.d.a.145.1		2
36.23	even	6		2592.2.r.i.433.2		4
36.31	odd	6		2592.2.r.i.433.1		4
45.2	even	12		1800.2.d.n.1549.4		4
45.7	odd	12		1800.2.d.n.1549.2		4
45.29	odd	6		1800.2.k.e.901.2		2
45.34	even	6		1800.2.k.e.901.1		2
45.38	even	12		1800.2.d.n.1549.1		4
45.43	odd	12		1800.2.d.n.1549.3		4
72.5	odd	6	inner	648.2.n.h.109.2		4
72.11	even	6		288.2.d.a.145.2		2
72.13	even	6	inner	648.2.n.h.109.1		4
72.29	odd	6		72.2.d.a.37.2	yes	2
72.43	odd	6		288.2.d.a.145.1		2
72.59	even	6		2592.2.r.i.433.1		4
72.61	even	6		72.2.d.a.37.1	✓	2
72.67	odd	6		2592.2.r.i.433.2		4
144.11	even	12		2304.2.a.y.1.2		2
144.29	odd	12		2304.2.a.q.1.1		2
144.43	odd	12		2304.2.a.y.1.1		2
144.61	even	12		2304.2.a.q.1.2		2
144.83	even	12		2304.2.a.y.1.1		2
144.101	odd	12		2304.2.a.q.1.2		2
144.115	odd	12		2304.2.a.y.1.2		2
144.133	even	12		2304.2.a.q.1.1		2
180.7	even	12		7200.2.d.p.2449.2		4
180.43	even	12		7200.2.d.p.2449.4		4
180.47	odd	12		7200.2.d.p.2449.1		4
180.79	odd	6		7200.2.k.h.3601.2		2
180.83	odd	12		7200.2.d.p.2449.3		4
180.119	even	6		7200.2.k.h.3601.1		2
360.29	odd	6		1800.2.k.e.901.1		2
360.43	even	12		7200.2.d.p.2449.3		4
360.83	odd	12		7200.2.d.p.2449.4		4
360.133	odd	12		1800.2.d.n.1549.1		4
360.173	even	12		1800.2.d.n.1549.3		4
360.187	even	12		7200.2.d.p.2449.1		4
360.227	odd	12		7200.2.d.p.2449.2		4
360.259	odd	6		7200.2.k.h.3601.1		2
360.277	odd	12		1800.2.d.n.1549.4		4
360.299	even	6		7200.2.k.h.3601.2		2
360.317	even	12		1800.2.d.n.1549.2		4
360.349	even	6		1800.2.k.e.901.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.2.d.a.37.1	✓	2	9.2	odd	6
72.2.d.a.37.1	✓	2	72.61	even	6
72.2.d.a.37.2	yes	2	9.7	even	3
72.2.d.a.37.2	yes	2	72.29	odd	6
288.2.d.a.145.1		2	36.11	even	6
288.2.d.a.145.1		2	72.43	odd	6
288.2.d.a.145.2		2	36.7	odd	6
288.2.d.a.145.2		2	72.11	even	6
648.2.n.h.109.1		4	9.5	odd	6	inner
648.2.n.h.109.1		4	72.13	even	6	inner
648.2.n.h.109.2		4	9.4	even	3	inner
648.2.n.h.109.2		4	72.5	odd	6	inner
648.2.n.h.541.1		4	1.1	even	1	trivial
648.2.n.h.541.1		4	24.5	odd	2	CM
648.2.n.h.541.2		4	3.2	odd	2	inner
648.2.n.h.541.2		4	8.5	even	2	inner
1800.2.d.n.1549.1		4	45.38	even	12
1800.2.d.n.1549.1		4	360.133	odd	12
1800.2.d.n.1549.2		4	45.7	odd	12
1800.2.d.n.1549.2		4	360.317	even	12
1800.2.d.n.1549.3		4	45.43	odd	12
1800.2.d.n.1549.3		4	360.173	even	12
1800.2.d.n.1549.4		4	45.2	even	12
1800.2.d.n.1549.4		4	360.277	odd	12
1800.2.k.e.901.1		2	45.34	even	6
1800.2.k.e.901.1		2	360.29	odd	6
1800.2.k.e.901.2		2	45.29	odd	6
1800.2.k.e.901.2		2	360.349	even	6
2304.2.a.q.1.1		2	144.29	odd	12
2304.2.a.q.1.1		2	144.133	even	12
2304.2.a.q.1.2		2	144.61	even	12
2304.2.a.q.1.2		2	144.101	odd	12
2304.2.a.y.1.1		2	144.43	odd	12
2304.2.a.y.1.1		2	144.83	even	12
2304.2.a.y.1.2		2	144.11	even	12
2304.2.a.y.1.2		2	144.115	odd	12
2592.2.r.i.433.1		4	36.31	odd	6
2592.2.r.i.433.1		4	72.59	even	6
2592.2.r.i.433.2		4	36.23	even	6
2592.2.r.i.433.2		4	72.67	odd	6
2592.2.r.i.2161.1		4	8.3	odd	2
2592.2.r.i.2161.1		4	12.11	even	2
2592.2.r.i.2161.2		4	4.3	odd	2
2592.2.r.i.2161.2		4	24.11	even	2
7200.2.d.p.2449.1		4	180.47	odd	12
7200.2.d.p.2449.1		4	360.187	even	12
7200.2.d.p.2449.2		4	180.7	even	12
7200.2.d.p.2449.2		4	360.227	odd	12
7200.2.d.p.2449.3		4	180.83	odd	12
7200.2.d.p.2449.3		4	360.43	even	12
7200.2.d.p.2449.4		4	180.43	even	12
7200.2.d.p.2449.4		4	360.83	odd	12
7200.2.k.h.3601.1		2	180.119	even	6
7200.2.k.h.3601.1		2	360.259	odd	6
7200.2.k.h.3601.2		2	180.79	odd	6
7200.2.k.h.3601.2		2	360.299	even	6

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.n (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.22474 - 0.707107i) q^{2} +(1.00000 + 1.73205i) q^{4} +(2.44949 - 1.41421i) q^{5} +(-1.00000 + 1.73205i) q^{7} -2.82843i q^{8} +O(q^{10})\)	\(q+(-1.22474 - 0.707107i) q^{2} +(1.00000 + 1.73205i) q^{4} +(2.44949 - 1.41421i) q^{5} +(-1.00000 + 1.73205i) q^{7} -2.82843i q^{8} -4.00000 q^{10} +(4.89898 + 2.82843i) q^{11} +(2.44949 - 1.41421i) q^{14} +(-2.00000 + 3.46410i) q^{16} +(4.89898 + 2.82843i) q^{20} +(-4.00000 - 6.92820i) q^{22} +(1.50000 - 2.59808i) q^{25} -4.00000 q^{28} +(-2.44949 - 1.41421i) q^{29} +(5.00000 + 8.66025i) q^{31} +(4.89898 - 2.82843i) q^{32} +5.65685i q^{35} +(-4.00000 - 6.92820i) q^{40} +11.3137i q^{44} +(1.50000 + 2.59808i) q^{49} +(-3.67423 + 2.12132i) q^{50} -14.1421i q^{53} +16.0000 q^{55} +(4.89898 + 2.82843i) q^{56} +(2.00000 + 3.46410i) q^{58} +(9.79796 - 5.65685i) q^{59} -14.1421i q^{62} -8.00000 q^{64} +(4.00000 - 6.92820i) q^{70} +14.0000 q^{73} +(-9.79796 + 5.65685i) q^{77} +(5.00000 - 8.66025i) q^{79} +11.3137i q^{80} +(4.89898 + 2.82843i) q^{83} +(8.00000 - 13.8564i) q^{88} +(-1.00000 + 1.73205i) q^{97} -4.24264i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} - 4 q^{7}+O(q^{10}) \) 4 * q + 4 * q^4 - 4 * q^7	\( 4 q + 4 q^{4} - 4 q^{7} - 16 q^{10} - 8 q^{16} - 16 q^{22} + 6 q^{25} - 16 q^{28} + 20 q^{31} - 16 q^{40} + 6 q^{49} + 64 q^{55} + 8 q^{58} - 32 q^{64} + 16 q^{70} + 56 q^{73} + 20 q^{79} + 32 q^{88} - 4 q^{97}+O(q^{100}) \) 4 * q + 4 * q^4 - 4 * q^7 - 16 * q^10 - 8 * q^16 - 16 * q^22 + 6 * q^25 - 16 * q^28 + 20 * q^31 - 16 * q^40 + 6 * q^49 + 64 * q^55 + 8 * q^58 - 32 * q^64 + 16 * q^70 + 56 * q^73 + 20 * q^79 + 32 * q^88 - 4 * q^97

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