LMFDB - Embedded newform 324.3.f.e.55.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,3,Mod(55,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 4]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("324.55");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	324.f (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:	$8.82836056527$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 36)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			55.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	324.55
Dual form			324.3.f.e.271.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(4.00000 + 6.92820i) q^{5} +8.00000 q^{8} -16.0000 q^{10} +(5.00000 + 8.66025i) q^{13} +(-8.00000 + 13.8564i) q^{16} +16.0000 q^{17} +(16.0000 - 27.7128i) q^{20} +(-19.5000 + 33.7750i) q^{25} -20.0000 q^{26} +(-20.0000 + 34.6410i) q^{29} +(-16.0000 - 27.7128i) q^{32} +(-16.0000 + 27.7128i) q^{34} -70.0000 q^{37} +(32.0000 + 55.4256i) q^{40} +(40.0000 + 69.2820i) q^{41} +(-24.5000 - 42.4352i) q^{49} +(-39.0000 - 67.5500i) q^{50} +(20.0000 - 34.6410i) q^{52} -56.0000 q^{53} +(-40.0000 - 69.2820i) q^{58} +(11.0000 - 19.0526i) q^{61} +64.0000 q^{64} +(-40.0000 + 69.2820i) q^{65} +(-32.0000 - 55.4256i) q^{68} +110.000 q^{73} +(70.0000 - 121.244i) q^{74} -128.000 q^{80} -160.000 q^{82} +(64.0000 + 110.851i) q^{85} +160.000 q^{89} +(65.0000 - 112.583i) q^{97} +98.0000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 4 q^{4} + 8 q^{5} + 16 q^{8} - 32 q^{10} + 10 q^{13} - 16 q^{16} + 32 q^{17} + 32 q^{20} - 39 q^{25} - 40 q^{26} - 40 q^{29} - 32 q^{32} - 32 q^{34} - 140 q^{37} + 64 q^{40} + 80 q^{41}+ \cdots + 196 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 - 4 * q^4 + 8 * q^5 + 16 * q^8 - 32 * q^10 + 10 * q^13 - 16 * q^16 + 32 * q^17 + 32 * q^20 - 39 * q^25 - 40 * q^26 - 40 * q^29 - 32 * q^32 - 32 * q^34 - 140 * q^37 + 64 * q^40 + 80 * q^41 - 49 * q^49 - 78 * q^50 + 40 * q^52 - 112 * q^53 - 80 * q^58 + 22 * q^61 + 128 * q^64 - 80 * q^65 - 64 * q^68 + 220 * q^73 + 140 * q^74 - 256 * q^80 - 320 * q^82 + 128 * q^85 + 320 * q^89 + 130 * q^97 + 196 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$

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 $

$2$	−1.00000	+	1.73205i	−0.500000	+	0.866025i
$2$	−1.00000	+	1.73205i	−0.500000	+	0.866025i
$3$		0			0
$3$		0			0
$4$	−2.00000	−	3.46410i	−0.500000	−	0.866025i
$5$	4.00000	+	6.92820i	0.800000	+	1.38564i	0.919615	+	0.392820i	$0.128501\pi$
$5$	4.00000	+	6.92820i	0.800000	+	1.38564i	−0.119615	+	0.992820i	$0.538166\pi$
$6$		0			0
$7$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$7$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$8$	8.00000			1.00000
$9$		0			0
$10$	−16.0000			−1.60000
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$		0			0
$13$	5.00000	+	8.66025i	0.384615	+	0.666173i	0.991716	−	0.128452i	$-0.0410008\pi$
$13$	5.00000	+	8.66025i	0.384615	+	0.666173i	−0.607100	+	0.794625i	$0.707667\pi$
$14$		0			0
$15$		0			0
$16$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$17$	16.0000			0.941176			0.470588	−	0.882353i	$-0.344042\pi$
$17$	16.0000			0.941176			0.470588	+	0.882353i	$0.344042\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$	16.0000	−	27.7128i	0.800000	−	1.38564i
$21$		0			0
$22$		0			0
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$		0			0
$25$	−19.5000	+	33.7750i	−0.780000	+	1.35100i
$26$	−20.0000			−0.769231
$27$		0			0
$28$		0			0
$29$	−20.0000	+	34.6410i	−0.689655	+	1.19452i	0.282294	+	0.959328i	$0.408905\pi$
$29$	−20.0000	+	34.6410i	−0.689655	+	1.19452i	−0.971949	+	0.235190i	$0.924429\pi$
$30$		0			0
$31$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$31$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$32$	−16.0000	−	27.7128i	−0.500000	−	0.866025i
$33$		0			0
$34$	−16.0000	+	27.7128i	−0.470588	+	0.815083i
$35$		0			0
$36$		0			0
$37$	−70.0000			−1.89189			−0.945946	−	0.324324i	$-0.894863\pi$
$37$	−70.0000			−1.89189			−0.945946	+	0.324324i	$0.894863\pi$
$38$		0			0
$39$		0			0
$40$	32.0000	+	55.4256i	0.800000	+	1.38564i
$41$	40.0000	+	69.2820i	0.975610	+	1.68981i	0.677908	+	0.735147i	$0.262887\pi$
$41$	40.0000	+	69.2820i	0.975610	+	1.68981i	0.297702	+	0.954659i	$0.403780\pi$
$42$		0			0
$43$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$43$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$47$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$48$		0			0
$49$	−24.5000	−	42.4352i	−0.500000	−	0.866025i
$50$	−39.0000	−	67.5500i	−0.780000	−	1.35100i
$51$		0			0
$52$	20.0000	−	34.6410i	0.384615	−	0.666173i
$53$	−56.0000			−1.05660			−0.528302	−	0.849057i	$-0.677171\pi$
$53$	−56.0000			−1.05660			−0.528302	+	0.849057i	$0.677171\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−40.0000	−	69.2820i	−0.689655	−	1.19452i
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$		0			0
$61$	11.0000	−	19.0526i	0.180328	−	0.312337i	−0.761664	−	0.647972i	$-0.775617\pi$
$61$	11.0000	−	19.0526i	0.180328	−	0.312337i	0.941992	+	0.335635i	$0.108951\pi$
$62$		0			0
$63$		0			0
$64$	64.0000			1.00000
$65$	−40.0000	+	69.2820i	−0.615385	+	1.06588i
$66$		0			0
$67$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$	−32.0000	−	55.4256i	−0.470588	−	0.815083i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	110.000			1.50685			0.753425	−	0.657534i	$-0.228401\pi$
$73$	110.000			1.50685			0.753425	+	0.657534i	$0.228401\pi$
$74$	70.0000	−	121.244i	0.945946	−	1.63843i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$79$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$80$	−128.000			−1.60000
$81$		0			0
$82$	−160.000			−1.95122
$83$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$83$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$84$		0			0
$85$	64.0000	+	110.851i	0.752941	+	1.30413i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	160.000			1.79775			0.898876	−	0.438202i	$-0.144385\pi$
$89$	160.000			1.79775			0.898876	+	0.438202i	$0.144385\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	65.0000	−	112.583i	0.670103	−	1.16065i	−0.307771	−	0.951460i	$-0.599583\pi$
$97$	65.0000	−	112.583i	0.670103	−	1.16065i	0.977875	−	0.209192i	$-0.0670835\pi$
$98$	98.0000			1.00000
$99$		0			0
$100$	156.000			1.56000
$101$	−20.0000	+	34.6410i	−0.198020	+	0.342980i	−0.947886	−	0.318609i	$-0.896784\pi$
$101$	−20.0000	+	34.6410i	−0.198020	+	0.342980i	0.749866	+	0.661589i	$0.230118\pi$
$102$		0			0
$103$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$103$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$104$	40.0000	+	69.2820i	0.384615	+	0.666173i
$105$		0			0
$106$	56.0000	−	96.9948i	0.528302	−	0.915046i
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$	182.000			1.66972			0.834862	−	0.550459i	$-0.185547\pi$
$109$	182.000			1.66972			0.834862	+	0.550459i	$0.185547\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	112.000	+	193.990i	0.991150	+	1.71672i	0.610534	+	0.791990i	$0.290955\pi$
$113$	112.000	+	193.990i	0.991150	+	1.71672i	0.380616	+	0.924733i	$0.375712\pi$
$114$		0			0
$115$		0			0
$116$	160.000			1.37931
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−60.5000	−	104.789i	−0.500000	−	0.866025i
$122$	22.0000	+	38.1051i	0.180328	+	0.312337i
$123$		0			0
$124$		0			0
$125$	−112.000			−0.896000
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	−64.0000	+	110.851i	−0.500000	+	0.866025i
$129$		0			0
$130$	−80.0000	−	138.564i	−0.615385	−	1.06588i
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	128.000			0.941176
$137$	88.0000	−	152.420i	0.642336	−	1.11256i	−0.342574	−	0.939491i	$-0.611299\pi$
$137$	88.0000	−	152.420i	0.642336	−	1.11256i	0.984910	−	0.173067i	$-0.0553679\pi$
$138$		0			0
$139$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$139$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−320.000			−2.20690
$146$	−110.000	+	190.526i	−0.753425	+	1.30497i
$147$		0			0
$148$	140.000	+	242.487i	0.945946	+	1.63843i
$149$	−140.000	−	242.487i	−0.939597	−	1.62743i	−0.766223	−	0.642574i	$-0.777866\pi$
$149$	−140.000	−	242.487i	−0.939597	−	1.62743i	−0.173374	−	0.984856i	$-0.555467\pi$
$150$		0			0
$151$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$151$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−85.0000	−	147.224i	−0.541401	−	0.937735i	−0.998824	−	0.0484851i	$-0.984561\pi$
$157$	−85.0000	−	147.224i	−0.541401	−	0.937735i	0.457423	−	0.889249i	$-0.348773\pi$
$158$		0			0
$159$		0			0
$160$	128.000	−	221.703i	0.800000	−	1.38564i
$161$		0			0
$162$		0			0
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$	160.000	−	277.128i	0.975610	−	1.68981i
$165$		0			0
$166$		0			0
$167$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$167$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$168$		0			0
$169$	34.5000	−	59.7558i	0.204142	−	0.353584i
$170$	−256.000			−1.50588
$171$		0			0
$172$		0			0
$173$	52.0000	−	90.0666i	0.300578	−	0.520616i	−0.675689	−	0.737187i	$-0.736154\pi$
$173$	52.0000	−	90.0666i	0.300578	−	0.520616i	0.976267	+	0.216570i	$0.0694871\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−160.000	+	277.128i	−0.898876	+	1.55690i
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$	38.0000			0.209945			0.104972	−	0.994475i	$-0.466525\pi$
$181$	38.0000			0.209945			0.104972	+	0.994475i	$0.466525\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−280.000	−	484.974i	−1.51351	−	2.62148i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$		0			0
$193$	95.0000	+	164.545i	0.492228	+	0.852564i	0.999960	−	0.00895123i	$-0.00284930\pi$
$193$	95.0000	+	164.545i	0.492228	+	0.852564i	−0.507732	+	0.861515i	$0.669516\pi$
$194$	130.000	+	225.167i	0.670103	+	1.16065i
$195$		0			0
$196$	−98.0000	+	169.741i	−0.500000	+	0.866025i
$197$	−56.0000			−0.284264			−0.142132	−	0.989848i	$-0.545396\pi$
$197$	−56.0000			−0.284264			−0.142132	+	0.989848i	$0.545396\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−156.000	+	270.200i	−0.780000	+	1.35100i
$201$		0			0
$202$	−40.0000	−	69.2820i	−0.198020	−	0.342980i
$203$		0			0
$204$		0			0
$205$	−320.000	+	554.256i	−1.56098	+	2.70369i
$206$		0			0
$207$		0			0
$208$	−160.000			−0.769231
$209$		0			0
$210$		0			0
$211$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$211$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$212$	112.000	+	193.990i	0.528302	+	0.915046i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−182.000	+	315.233i	−0.834862	+	1.44602i
$219$		0			0
$220$		0			0
$221$	80.0000	+	138.564i	0.361991	+	0.626987i
$222$		0			0
$223$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$223$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$224$		0			0
$225$		0			0
$226$	−448.000			−1.98230
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$		0			0
$229$	221.000	+	382.783i	0.965066	+	1.67154i	0.709439	+	0.704767i	$0.248948\pi$
$229$	221.000	+	382.783i	0.965066	+	1.67154i	0.255627	+	0.966776i	$0.417718\pi$
$230$		0			0
$231$		0			0
$232$	−160.000	+	277.128i	−0.689655	+	1.19452i
$233$	−416.000			−1.78541			−0.892704	−	0.450644i	$-0.851194\pi$
$233$	−416.000			−1.78541			−0.892704	+	0.450644i	$0.851194\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$239$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$240$		0			0
$241$	209.000	−	361.999i	0.867220	−	1.50207i	0.00239399	−	0.999997i	$-0.499238\pi$
$241$	209.000	−	361.999i	0.867220	−	1.50207i	0.864826	−	0.502072i	$-0.167429\pi$
$242$	242.000			1.00000
$243$		0			0
$244$	−88.0000			−0.360656
$245$	196.000	−	339.482i	0.800000	−	1.38564i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	112.000	−	193.990i	0.448000	−	0.775959i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−128.000	−	221.703i	−0.500000	−	0.866025i
$257$	−32.0000	−	55.4256i	−0.124514	−	0.215664i	0.797029	−	0.603941i	$-0.206404\pi$
$257$	−32.0000	−	55.4256i	−0.124514	−	0.215664i	−0.921543	+	0.388277i	$0.873070\pi$
$258$		0			0
$259$		0			0
$260$	320.000			1.23077
$261$		0			0
$262$		0			0
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$	−224.000	−	387.979i	−0.845283	−	1.46407i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	520.000			1.93309			0.966543	−	0.256506i	$-0.0825712\pi$
$269$	520.000			1.93309			0.966543	+	0.256506i	$0.0825712\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−128.000	+	221.703i	−0.470588	+	0.815083i
$273$		0			0
$274$	176.000	+	304.841i	0.642336	+	1.11256i
$275$		0			0
$276$		0			0
$277$	−115.000	+	199.186i	−0.415162	+	0.719082i	−0.995445	−	0.0953324i	$-0.969609\pi$
$277$	−115.000	+	199.186i	−0.415162	+	0.719082i	0.580283	+	0.814415i	$0.302942\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	160.000	−	277.128i	0.569395	−	0.986221i	−0.427231	−	0.904143i	$-0.640511\pi$
$281$	160.000	−	277.128i	0.569395	−	0.986221i	0.996626	−	0.0820785i	$-0.0261558\pi$
$282$		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
$289$	−33.0000			−0.114187
$290$	320.000	−	554.256i	1.10345	−	1.91123i
$291$		0			0
$292$	−220.000	−	381.051i	−0.753425	−	1.30497i
$293$	−68.0000	−	117.779i	−0.232082	−	0.401978i	0.726339	−	0.687337i	$-0.241220\pi$
$293$	−68.0000	−	117.779i	−0.232082	−	0.401978i	−0.958421	+	0.285359i	$0.907887\pi$
$294$		0			0
$295$		0			0
$296$	−560.000			−1.89189
$297$		0			0
$298$	560.000			1.87919
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	176.000			0.577049
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$	−25.0000	+	43.3013i	−0.0798722	+	0.138343i	−0.903195	−	0.429231i	$-0.858785\pi$
$313$	−25.0000	+	43.3013i	−0.0798722	+	0.138343i	0.823322	+	0.567574i	$0.192118\pi$
$314$	340.000			1.08280
$315$		0			0
$316$		0			0
$317$	−308.000	+	533.472i	−0.971609	+	1.68288i	−0.280909	+	0.959734i	$0.590636\pi$
$317$	−308.000	+	533.472i	−0.971609	+	1.68288i	−0.690700	+	0.723141i	$0.742697\pi$
$318$		0			0
$319$		0			0
$320$	256.000	+	443.405i	0.800000	+	1.38564i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−390.000			−1.20000
$326$		0			0
$327$		0			0
$328$	320.000	+	554.256i	0.975610	+	1.68981i
$329$		0			0
$330$		0			0
$331$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−175.000	−	303.109i	−0.519288	−	0.899433i	−0.999749	−	0.0224168i	$-0.992864\pi$
$337$	−175.000	−	303.109i	−0.519288	−	0.899433i	0.480461	−	0.877016i	$-0.340469\pi$
$338$	69.0000	+	119.512i	0.204142	+	0.353584i
$339$		0			0
$340$	256.000	−	443.405i	0.752941	−	1.30413i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	104.000	+	180.133i	0.300578	+	0.520616i
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$	299.000	−	517.883i	0.856734	−	1.48391i	−0.0182939	−	0.999833i	$-0.505823\pi$
$349$	299.000	−	517.883i	0.856734	−	1.48391i	0.875027	−	0.484073i	$-0.160843\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−272.000	+	471.118i	−0.770538	+	1.33461i	0.166730	+	0.986003i	$0.446679\pi$
$353$	−272.000	+	471.118i	−0.770538	+	1.33461i	−0.937268	+	0.348609i	$0.886654\pi$
$354$		0			0
$355$		0			0
$356$	−320.000	−	554.256i	−0.898876	−	1.55690i
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	361.000			1.00000
$362$	−38.0000	+	65.8179i	−0.104972	+	0.181817i
$363$		0			0
$364$		0			0
$365$	440.000	+	762.102i	1.20548	+	2.08795i
$366$		0			0
$367$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$367$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$368$		0			0
$369$		0			0
$370$	1120.00			3.02703
$371$		0			0
$372$		0			0
$373$	275.000	+	476.314i	0.737265	+	1.27698i	0.953722	+	0.300689i	$0.0972166\pi$
$373$	275.000	+	476.314i	0.737265	+	1.27698i	−0.216457	+	0.976292i	$0.569450\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−400.000			−1.06101
$378$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$383$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$384$		0			0
$385$		0			0
$386$	−380.000			−0.984456
$387$		0			0
$388$	−520.000			−1.34021
$389$	340.000	−	588.897i	0.874036	−	1.51387i	0.0162499	−	0.999868i	$-0.494827\pi$
$389$	340.000	−	588.897i	0.874036	−	1.51387i	0.857786	−	0.514007i	$-0.171839\pi$
$390$		0			0
$391$		0			0
$392$	−196.000	−	339.482i	−0.500000	−	0.866025i
$393$		0			0
$394$	56.0000	−	96.9948i	0.142132	−	0.246180i
$395$		0			0
$396$		0			0
$397$	650.000			1.63728			0.818640	−	0.574307i	$-0.194729\pi$
$397$	650.000			1.63728			0.818640	+	0.574307i	$0.194729\pi$
$398$		0			0
$399$		0			0
$400$	−312.000	−	540.400i	−0.780000	−	1.35100i
$401$	40.0000	+	69.2820i	0.0997506	+	0.172773i	0.911581	−	0.411120i	$-0.134862\pi$
$401$	40.0000	+	69.2820i	0.0997506	+	0.172773i	−0.811831	+	0.583893i	$0.801529\pi$
$402$		0			0
$403$		0			0
$404$	160.000			0.396040
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−391.000	−	677.232i	−0.955990	−	1.65582i	−0.732086	−	0.681213i	$-0.761453\pi$
$409$	−391.000	−	677.232i	−0.955990	−	1.65582i	−0.223905	−	0.974611i	$-0.571880\pi$
$410$	−640.000	−	1108.51i	−1.56098	−	2.70369i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	160.000	−	277.128i	0.384615	−	0.666173i
$417$		0			0
$418$		0			0
$419$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$419$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$420$		0			0
$421$	29.0000	−	50.2295i	0.0688836	−	0.119310i	−0.829527	−	0.558467i	$-0.811390\pi$
$421$	29.0000	−	50.2295i	0.0688836	−	0.119310i	0.898410	+	0.439157i	$0.144723\pi$
$422$		0			0
$423$		0			0
$424$	−448.000			−1.05660
$425$	−312.000	+	540.400i	−0.734118	+	1.27153i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	290.000			0.669746			0.334873	−	0.942263i	$-0.391307\pi$
$433$	290.000			0.669746			0.334873	+	0.942263i	$0.391307\pi$
$434$		0			0
$435$		0			0
$436$	−364.000	−	630.466i	−0.834862	−	1.44602i
$437$		0			0
$438$		0			0
$439$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$		0			0
$442$	−320.000			−0.723982
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$		0			0
$445$	640.000	+	1108.51i	1.43820	+	2.49104i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−560.000			−1.24722			−0.623608	−	0.781737i	$-0.714334\pi$
$449$	−560.000			−1.24722			−0.623608	+	0.781737i	$0.714334\pi$
$450$		0			0
$451$		0			0
$452$	448.000	−	775.959i	0.991150	−	1.71672i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	425.000	−	736.122i	0.929978	−	1.61077i	0.146625	−	0.989192i	$-0.453159\pi$
$457$	425.000	−	736.122i	0.929978	−	1.61077i	0.783353	−	0.621577i	$-0.213508\pi$
$458$	−884.000			−1.93013
$459$		0			0
$460$		0			0
$461$	−380.000	+	658.179i	−0.824295	+	1.42772i	0.0781619	+	0.996941i	$0.475095\pi$
$461$	−380.000	+	658.179i	−0.824295	+	1.42772i	−0.902457	+	0.430780i	$0.858238\pi$
$462$		0			0
$463$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$463$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$464$	−320.000	−	554.256i	−0.689655	−	1.19452i
$465$		0			0
$466$	416.000	−	720.533i	0.892704	−	1.54621i
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$	−350.000	−	606.218i	−0.727651	−	1.26033i
$482$	418.000	+	723.997i	0.867220	+	1.50207i
$483$		0			0
$484$	−242.000	+	419.156i	−0.500000	+	0.866025i
$485$	1040.00			2.14433
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$	88.0000	−	152.420i	0.180328	−	0.312337i
$489$		0			0
$490$	392.000	+	678.964i	0.800000	+	1.38564i
$491$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$491$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$492$		0			0
$493$	−320.000	+	554.256i	−0.649087	+	1.12425i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$499$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$500$	224.000	+	387.979i	0.448000	+	0.775959i
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$	−320.000			−0.633663
$506$		0			0
$507$		0			0
$508$		0			0
$509$	220.000	+	381.051i	0.432220	+	0.748627i	0.997064	−	0.0765706i	$-0.0243970\pi$
$509$	220.000	+	381.051i	0.432220	+	0.748627i	−0.564844	+	0.825198i	$0.691064\pi$
$510$		0			0
$511$		0			0
$512$	512.000			1.00000
$513$		0			0
$514$	128.000			0.249027
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−320.000	+	554.256i	−0.615385	+	1.06588i
$521$	880.000			1.68906			0.844530	−	0.535509i	$-0.179880\pi$
$521$	880.000			1.68906			0.844530	+	0.535509i	$0.179880\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−264.500	+	458.127i	−0.500000	+	0.866025i
$530$	896.000			1.69057
$531$		0			0
$532$		0			0
$533$	−400.000	+	692.820i	−0.750469	+	1.29985i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−520.000	+	900.666i	−0.966543	+	1.67410i
$539$		0			0
$540$		0			0
$541$	−682.000			−1.26063			−0.630314	−	0.776340i	$-0.717074\pi$
$541$	−682.000			−1.26063			−0.630314	+	0.776340i	$0.717074\pi$
$542$		0			0
$543$		0			0
$544$	−256.000	−	443.405i	−0.470588	−	0.815083i
$545$	728.000	+	1260.93i	1.33578	+	2.31364i
$546$		0			0
$547$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$547$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$548$	−704.000			−1.28467
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−230.000	−	398.372i	−0.415162	−	0.719082i
$555$		0			0
$556$		0			0
$557$	−1064.00			−1.91023			−0.955117	−	0.296230i	$-0.904271\pi$
$557$	−1064.00			−1.91023			−0.955117	+	0.296230i	$0.904271\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	320.000	+	554.256i	0.569395	+	0.986221i
$563$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$563$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$564$		0			0
$565$	−896.000	+	1551.92i	−1.58584	+	2.74676i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	520.000	−	900.666i	0.913884	−	1.58289i	0.105357	−	0.994434i	$-0.466401\pi$
$569$	520.000	−	900.666i	0.913884	−	1.58289i	0.808527	−	0.588459i	$-0.200265\pi$
$570$		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
$577$	−1150.00			−1.99307			−0.996534	−	0.0831889i	$-0.973490\pi$
$577$	−1150.00			−1.99307			−0.996534	+	0.0831889i	$0.973490\pi$
$578$	33.0000	−	57.1577i	0.0570934	−	0.0988887i
$579$		0			0
$580$	640.000	+	1108.51i	1.10345	+	1.91123i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	880.000			1.50685
$585$		0			0
$586$	272.000			0.464164
$587$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$587$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	560.000	−	969.948i	0.945946	−	1.63843i
$593$	736.000			1.24115			0.620573	−	0.784148i	$-0.286900\pi$
$593$	736.000			1.24115			0.620573	+	0.784148i	$0.286900\pi$
$594$		0			0
$595$		0			0
$596$	−560.000	+	969.948i	−0.939597	+	1.62743i
$597$		0			0
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$		0			0
$601$	551.000	−	954.360i	0.916805	−	1.58795i	0.112569	−	0.993644i	$-0.464092\pi$
$601$	551.000	−	954.360i	0.916805	−	1.58795i	0.804236	−	0.594309i	$-0.202575\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	484.000	−	838.313i	0.800000	−	1.38564i
$606$		0			0
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$		0			0
$609$		0			0
$610$	−176.000	+	304.841i	−0.288525	+	0.499739i
$611$		0			0
$612$		0			0
$613$	−70.0000			−0.114192			−0.0570962	−	0.998369i	$-0.518184\pi$
$613$	−70.0000			−0.114192			−0.0570962	+	0.998369i	$0.518184\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−608.000	−	1053.09i	−0.985413	−	1.70679i	−0.640085	−	0.768304i	$-0.721101\pi$
$617$	−608.000	−	1053.09i	−0.985413	−	1.70679i	−0.345328	−	0.938482i	$-0.612232\pi$
$618$		0			0
$619$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$619$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	39.5000	+	68.4160i	0.0632000	+	0.109466i
$626$	−50.0000	−	86.6025i	−0.0798722	−	0.138343i
$627$		0			0
$628$	−340.000	+	588.897i	−0.541401	+	0.937735i
$629$	−1120.00			−1.78060
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$	−616.000	−	1066.94i	−0.971609	−	1.68288i
$635$		0			0
$636$		0			0
$637$	245.000	−	424.352i	0.384615	−	0.666173i
$638$		0			0
$639$		0			0
$640$	−1024.00			−1.60000
$641$	−200.000	+	346.410i	−0.312012	+	0.540421i	−0.978798	−	0.204828i	$-0.934336\pi$
$641$	−200.000	+	346.410i	−0.312012	+	0.540421i	0.666785	+	0.745250i	$0.267670\pi$
$642$		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		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$		0			0
$650$	390.000	−	675.500i	0.600000	−	1.03923i
$651$		0			0
$652$		0			0
$653$	−572.000	−	990.733i	−0.875957	−	1.51720i	−0.855740	−	0.517407i	$-0.826897\pi$
$653$	−572.000	−	990.733i	−0.875957	−	1.51720i	−0.0202175	−	0.999796i	$-0.506436\pi$
$654$		0			0
$655$		0			0
$656$	−1280.00			−1.95122
$657$		0			0
$658$		0			0
$659$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$659$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$660$		0			0
$661$	−589.000	−	1020.18i	−0.891074	−	1.54339i	−0.838589	−	0.544764i	$-0.816619\pi$
$661$	−589.000	−	1020.18i	−0.891074	−	1.54339i	−0.0524847	−	0.998622i	$-0.516714\pi$
$662$		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
$673$	−385.000	+	666.840i	−0.572065	+	0.990846i	0.424288	+	0.905527i	$0.360524\pi$
$673$	−385.000	+	666.840i	−0.572065	+	0.990846i	−0.996354	+	0.0853191i	$0.972809\pi$
$674$	700.000			1.03858
$675$		0			0
$676$	−276.000			−0.408284
$677$	52.0000	−	90.0666i	0.0768095	−	0.133038i	−0.825062	−	0.565042i	$-0.808860\pi$
$677$	52.0000	−	90.0666i	0.0768095	−	0.133038i	0.901872	+	0.432004i	$0.142193\pi$
$678$		0			0
$679$		0			0
$680$	512.000	+	886.810i	0.752941	+	1.30413i
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$	1408.00			2.05547
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−280.000	−	484.974i	−0.406386	−	0.703881i
$690$		0			0
$691$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$691$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$692$	−416.000			−0.601156
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	640.000	+	1108.51i	0.918221	+	1.59041i
$698$	598.000	+	1035.77i	0.856734	+	1.48391i
$699$		0			0
$700$		0			0
$701$	520.000			0.741797			0.370899	−	0.928673i	$-0.379050\pi$
$701$	520.000			0.741797			0.370899	+	0.928673i	$0.379050\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−544.000	−	942.236i	−0.770538	−	1.33461i
$707$		0			0
$708$		0			0
$709$	−259.000	+	448.601i	−0.365303	+	0.632724i	−0.988825	−	0.149082i	$-0.952368\pi$
$709$	−259.000	+	448.601i	−0.365303	+	0.632724i	0.623522	+	0.781806i	$0.285701\pi$
$710$		0			0
$711$		0			0
$712$	1280.00			1.79775
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$	−361.000	+	625.270i	−0.500000	+	0.866025i
$723$		0			0
$724$	−76.0000	−	131.636i	−0.104972	−	0.181817i
$725$	−780.000	−	1351.00i	−1.07586	−	1.86345i
$726$		0			0
$727$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$727$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$728$		0			0
$729$		0			0
$730$	−1760.00			−2.41096
$731$		0			0
$732$		0			0
$733$	725.000	+	1255.74i	0.989086	+	1.71315i	0.622143	+	0.782904i	$0.286262\pi$
$733$	725.000	+	1255.74i	0.989086	+	1.71315i	0.366943	+	0.930243i	$0.380404\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$	−1120.00	+	1939.90i	−1.51351	+	2.62148i
$741$		0			0
$742$		0			0
$743$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$743$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$744$		0			0
$745$	1120.00	−	1939.90i	1.50336	−	2.60389i
$746$	−1100.00			−1.47453
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$751$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$752$		0			0
$753$		0			0
$754$	400.000	−	692.820i	0.530504	−	0.918860i
$755$		0			0
$756$		0			0
$757$	1190.00			1.57199			0.785997	−	0.618230i	$-0.212150\pi$
$757$	1190.00			1.57199			0.785997	+	0.618230i	$0.212150\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	760.000	+	1316.36i	0.998686	+	1.72977i	0.543725	+	0.839263i	$0.317013\pi$
$761$	760.000	+	1316.36i	0.998686	+	1.72977i	0.454961	+	0.890512i	$0.349653\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−481.000	−	833.116i	−0.625488	−	1.08338i	−0.988446	−	0.151571i	$-0.951567\pi$
$769$	−481.000	−	833.116i	−0.625488	−	1.08338i	0.362959	−	0.931805i	$-0.381767\pi$
$770$		0			0
$771$		0			0
$772$	380.000	−	658.179i	0.492228	−	0.852564i
$773$	−1496.00			−1.93532			−0.967658	−	0.252264i	$-0.918825\pi$
$773$	−1496.00			−1.93532			−0.967658	+	0.252264i	$0.918825\pi$
$774$		0			0
$775$		0			0
$776$	520.000	−	900.666i	0.670103	−	1.16065i
$777$		0			0
$778$	680.000	+	1177.79i	0.874036	+	1.51387i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	784.000			1.00000
$785$	680.000	−	1177.79i	0.866242	−	1.50038i
$786$		0			0
$787$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$787$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$788$	112.000	+	193.990i	0.142132	+	0.246180i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	220.000			0.277427
$794$	−650.000	+	1125.83i	−0.818640	+	1.41793i
$795$		0			0
$796$		0			0
$797$	−572.000	−	990.733i	−0.717691	−	1.24308i	−0.961912	−	0.273358i	$-0.911866\pi$
$797$	−572.000	−	990.733i	−0.717691	−	1.24308i	0.244221	−	0.969720i	$-0.421468\pi$
$798$		0			0
$799$		0			0
$800$	1248.00			1.56000
$801$		0			0
$802$	−160.000			−0.199501
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−160.000	+	277.128i	−0.198020	+	0.342980i
$809$	−560.000			−0.692213			−0.346106	−	0.938195i	$-0.612496\pi$
$809$	−560.000			−0.692213			−0.346106	+	0.938195i	$0.612496\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	1564.00			1.91198
$819$		0			0
$820$	2560.00			3.12195
$821$	700.000	−	1212.44i	0.852619	−	1.47678i	−0.0262179	−	0.999656i	$-0.508346\pi$
$821$	700.000	−	1212.44i	0.852619	−	1.47678i	0.878837	−	0.477123i	$-0.158320\pi$
$822$		0			0
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	−1258.00			−1.51749			−0.758745	−	0.651387i	$-0.774187\pi$
$829$	−1258.00			−1.51749			−0.758745	+	0.651387i	$0.774187\pi$
$830$		0			0
$831$		0			0
$832$	320.000	+	554.256i	0.384615	+	0.666173i
$833$	−392.000	−	678.964i	−0.470588	−	0.815083i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$839$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$840$		0			0
$841$	−379.500	−	657.313i	−0.451249	−	0.781585i
$842$	58.0000	+	100.459i	0.0688836	+	0.119310i
$843$		0			0
$844$		0			0
$845$	552.000			0.653254
$846$		0			0
$847$		0			0
$848$	448.000	−	775.959i	0.528302	−	0.915046i
$849$		0			0
$850$	−624.000	−	1080.80i	−0.734118	−	1.27153i
$851$		0			0
$852$		0			0
$853$	−205.000	+	355.070i	−0.240328	+	0.416261i	−0.960808	−	0.277215i	$-0.910588\pi$
$853$	−205.000	+	355.070i	−0.240328	+	0.416261i	0.720480	+	0.693476i	$0.243922\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	232.000	−	401.836i	0.270712	−	0.468887i	−0.698333	−	0.715774i	$-0.746074\pi$
$857$	232.000	−	401.836i	0.270712	−	0.468887i	0.969044	+	0.246887i	$0.0794076\pi$
$858$		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		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$	832.000			0.961850
$866$	−290.000	+	502.295i	−0.334873	+	0.580017i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	1456.00			1.66972
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−805.000	−	1394.30i	−0.917902	−	1.58985i	−0.802596	−	0.596523i	$-0.796549\pi$
$877$	−805.000	−	1394.30i	−0.917902	−	1.58985i	−0.115306	−	0.993330i	$-0.536785\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1600.00			1.81612			0.908059	−	0.418842i	$-0.137564\pi$
$881$	1600.00			1.81612			0.908059	+	0.418842i	$0.137564\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$	320.000	−	554.256i	0.361991	−	0.626987i
$885$		0			0
$886$		0			0
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$		0			0
$890$	−2560.00			−2.87640
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	560.000	−	969.948i	0.623608	−	1.08012i
$899$		0			0
$900$		0			0
$901$	−896.000			−0.994451
$902$		0			0
$903$		0			0
$904$	896.000	+	1551.92i	0.991150	+	1.71672i
$905$	152.000	+	263.272i	0.167956	+	0.290908i
$906$		0			0
$907$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$907$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$911$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$912$		0			0
$913$		0			0
$914$	850.000	+	1472.24i	0.929978	+	1.61077i
$915$		0			0
$916$	884.000	−	1531.13i	0.965066	−	1.67154i
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$	−760.000	−	1316.36i	−0.824295	−	1.42772i
$923$		0			0
$924$		0			0
$925$	1365.00	−	2364.25i	1.47568	−	2.55595i
$926$		0			0
$927$		0			0
$928$	1280.00			1.37931
$929$	−920.000	+	1593.49i	−0.990312	+	1.71527i	−0.374901	+	0.927065i	$0.622323\pi$
$929$	−920.000	+	1593.49i	−0.990312	+	1.71527i	−0.615411	+	0.788206i	$0.711010\pi$
$930$		0			0
$931$		0			0
$932$	832.000	+	1441.07i	0.892704	+	1.54621i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−430.000			−0.458911			−0.229456	−	0.973319i	$-0.573695\pi$
$937$	−430.000			−0.458911			−0.229456	+	0.973319i	$0.573695\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	580.000	+	1004.59i	0.616366	+	1.06758i	0.990143	+	0.140058i	$0.0447290\pi$
$941$	580.000	+	1004.59i	0.616366	+	1.06758i	−0.373778	+	0.927518i	$0.621938\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$947$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$948$		0			0
$949$	550.000	+	952.628i	0.579557	+	1.00382i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1456.00			1.52781			0.763903	−	0.645331i	$-0.223280\pi$
$953$	1456.00			1.52781			0.763903	+	0.645331i	$0.223280\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−480.500	+	832.250i	−0.500000	+	0.866025i
$962$	1400.00			1.45530
$963$		0			0
$964$	−1672.00			−1.73444
$965$	−760.000	+	1316.36i	−0.787565	+	1.36410i
$966$		0			0
$967$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$967$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$968$	−484.000	−	838.313i	−0.500000	−	0.866025i
$969$		0			0
$970$	−1040.00	+	1801.33i	−1.07216	+	1.85704i
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	176.000	+	304.841i	0.180328	+	0.312337i
$977$	−248.000	−	429.549i	−0.253838	−	0.439661i	0.710741	−	0.703454i	$-0.248360\pi$
$977$	−248.000	−	429.549i	−0.253838	−	0.439661i	−0.964579	+	0.263793i	$0.915026\pi$
$978$		0			0
$979$		0			0
$980$	−1568.00			−1.60000
$981$		0			0
$982$		0			0
$983$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$983$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$984$		0			0
$985$	−224.000	−	387.979i	−0.227411	−	0.393888i
$986$	−640.000	−	1108.51i	−0.649087	−	1.12425i
$987$		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
$997$	−925.000	+	1602.15i	−0.927783	+	1.60697i	−0.140761	+	0.990044i	$0.544955\pi$
$997$	−925.000	+	1602.15i	−0.927783	+	1.60697i	−0.787023	+	0.616924i	$0.788378\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.3.f.e.55.1		2
3.2	odd	2		324.3.f.f.55.1		2
4.3	odd	2	CM	324.3.f.e.55.1		2
9.2	odd	6		36.3.d.a.19.1	✓	1
9.4	even	3	inner	324.3.f.e.271.1		2
9.5	odd	6		324.3.f.f.271.1		2
9.7	even	3		36.3.d.b.19.1	yes	1
12.11	even	2		324.3.f.f.55.1		2
36.7	odd	6		36.3.d.b.19.1	yes	1
36.11	even	6		36.3.d.a.19.1	✓	1
36.23	even	6		324.3.f.f.271.1		2
36.31	odd	6	inner	324.3.f.e.271.1		2
45.2	even	12		900.3.f.b.199.1		2
45.7	odd	12		900.3.f.a.199.2		2
45.29	odd	6		900.3.c.c.451.1		1
45.34	even	6		900.3.c.b.451.1		1
45.38	even	12		900.3.f.b.199.2		2
45.43	odd	12		900.3.f.a.199.1		2
72.11	even	6		576.3.g.a.127.1		1
72.29	odd	6		576.3.g.a.127.1		1
72.43	odd	6		576.3.g.c.127.1		1
72.61	even	6		576.3.g.c.127.1		1
144.11	even	12		2304.3.b.d.127.2		2
144.29	odd	12		2304.3.b.d.127.1		2
144.43	odd	12		2304.3.b.e.127.1		2
144.61	even	12		2304.3.b.e.127.2		2
144.83	even	12		2304.3.b.d.127.1		2
144.101	odd	12		2304.3.b.d.127.2		2
144.115	odd	12		2304.3.b.e.127.2		2
144.133	even	12		2304.3.b.e.127.1		2
180.7	even	12		900.3.f.a.199.2		2
180.43	even	12		900.3.f.a.199.1		2
180.47	odd	12		900.3.f.b.199.1		2
180.79	odd	6		900.3.c.b.451.1		1
180.83	odd	12		900.3.f.b.199.2		2
180.119	even	6		900.3.c.c.451.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.3.d.a.19.1	✓	1	9.2	odd	6
36.3.d.a.19.1	✓	1	36.11	even	6
36.3.d.b.19.1	yes	1	9.7	even	3
36.3.d.b.19.1	yes	1	36.7	odd	6
324.3.f.e.55.1		2	1.1	even	1	trivial
324.3.f.e.55.1		2	4.3	odd	2	CM
324.3.f.e.271.1		2	9.4	even	3	inner
324.3.f.e.271.1		2	36.31	odd	6	inner
324.3.f.f.55.1		2	3.2	odd	2
324.3.f.f.55.1		2	12.11	even	2
324.3.f.f.271.1		2	9.5	odd	6
324.3.f.f.271.1		2	36.23	even	6
576.3.g.a.127.1		1	72.11	even	6
576.3.g.a.127.1		1	72.29	odd	6
576.3.g.c.127.1		1	72.43	odd	6
576.3.g.c.127.1		1	72.61	even	6
900.3.c.b.451.1		1	45.34	even	6
900.3.c.b.451.1		1	180.79	odd	6
900.3.c.c.451.1		1	45.29	odd	6
900.3.c.c.451.1		1	180.119	even	6
900.3.f.a.199.1		2	45.43	odd	12
900.3.f.a.199.1		2	180.43	even	12
900.3.f.a.199.2		2	45.7	odd	12
900.3.f.a.199.2		2	180.7	even	12
900.3.f.b.199.1		2	45.2	even	12
900.3.f.b.199.1		2	180.47	odd	12
900.3.f.b.199.2		2	45.38	even	12
900.3.f.b.199.2		2	180.83	odd	12
2304.3.b.d.127.1		2	144.29	odd	12
2304.3.b.d.127.1		2	144.83	even	12
2304.3.b.d.127.2		2	144.11	even	12
2304.3.b.d.127.2		2	144.101	odd	12
2304.3.b.e.127.1		2	144.43	odd	12
2304.3.b.e.127.1		2	144.133	even	12
2304.3.b.e.127.2		2	144.61	even	12
2304.3.b.e.127.2		2	144.115	odd	12

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:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	324.f (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(4.00000 + 6.92820i) q^{5} +8.00000 q^{8} -16.0000 q^{10} +(5.00000 + 8.66025i) q^{13} +(-8.00000 + 13.8564i) q^{16} +16.0000 q^{17} +(16.0000 - 27.7128i) q^{20} +(-19.5000 + 33.7750i) q^{25} -20.0000 q^{26} +(-20.0000 + 34.6410i) q^{29} +(-16.0000 - 27.7128i) q^{32} +(-16.0000 + 27.7128i) q^{34} -70.0000 q^{37} +(32.0000 + 55.4256i) q^{40} +(40.0000 + 69.2820i) q^{41} +(-24.5000 - 42.4352i) q^{49} +(-39.0000 - 67.5500i) q^{50} +(20.0000 - 34.6410i) q^{52} -56.0000 q^{53} +(-40.0000 - 69.2820i) q^{58} +(11.0000 - 19.0526i) q^{61} +64.0000 q^{64} +(-40.0000 + 69.2820i) q^{65} +(-32.0000 - 55.4256i) q^{68} +110.000 q^{73} +(70.0000 - 121.244i) q^{74} -128.000 q^{80} -160.000 q^{82} +(64.0000 + 110.851i) q^{85} +160.000 q^{89} +(65.0000 - 112.583i) q^{97} +98.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 4 q^{4} + 8 q^{5} + 16 q^{8} - 32 q^{10} + 10 q^{13} - 16 q^{16} + 32 q^{17} + 32 q^{20} - 39 q^{25} - 40 q^{26} - 40 q^{29} - 32 q^{32} - 32 q^{34} - 140 q^{37} + 64 q^{40} + 80 q^{41}+ \cdots + 196 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 - 4 * q^4 + 8 * q^5 + 16 * q^8 - 32 * q^10 + 10 * q^13 - 16 * q^16 + 32 * q^17 + 32 * q^20 - 39 * q^25 - 40 * q^26 - 40 * q^29 - 32 * q^32 - 32 * q^34 - 140 * q^37 + 64 * q^40 + 80 * q^41 - 49 * q^49 - 78 * q^50 + 40 * q^52 - 112 * q^53 - 80 * q^58 + 22 * q^61 + 128 * q^64 - 80 * q^65 - 64 * q^68 + 220 * q^73 + 140 * q^74 - 256 * q^80 - 320 * q^82 + 128 * q^85 + 320 * q^89 + 130 * q^97 + 196 * q^98

Introduction

L-functions

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Database

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