LMFDB - Embedded newform 324.2.h.e.107.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,2,Mod(107,324)]

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, 1]))

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

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

chi := DirichletCharacter("324.107");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	324.h (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:	$2.58715302549$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			107.2
Root			$0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	324.107
Dual form			324.2.h.e.215.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.22474 - 0.707107i) q^{2} +(1.00000 + 1.73205i) q^{4} +(3.79435 - 2.19067i) q^{5} -2.82843i q^{8} +O(q^{10})$	$q+(-1.22474 - 0.707107i) q^{2} +(1.00000 + 1.73205i) q^{4} +(3.79435 - 2.19067i) q^{5} -2.82843i q^{8} -6.19615 q^{10} +(1.59808 + 2.76795i) q^{13} +(-2.00000 + 3.46410i) q^{16} +0.138701i q^{17} +(7.58871 + 4.38134i) q^{20} +(7.09808 - 12.2942i) q^{25} -4.52004i q^{26} +(-7.46859 - 4.31199i) q^{29} +(4.89898 - 2.82843i) q^{32} +(0.0980762 - 0.169873i) q^{34} +9.39230 q^{37} +(-6.19615 - 10.7321i) q^{40} +(-1.22474 + 0.707107i) q^{41} +(-3.50000 - 6.06218i) q^{49} +(-17.3867 + 10.0382i) q^{50} +(-3.19615 + 5.53590i) q^{52} +7.07107i q^{53} +(6.09808 + 10.5622i) q^{58} +(-7.69615 + 13.3301i) q^{61} -8.00000 q^{64} +(12.1273 + 7.00172i) q^{65} +(-0.240237 + 0.138701i) q^{68} +2.80385 q^{73} +(-11.5032 - 6.64136i) q^{74} +17.5254i q^{80} +2.00000 q^{82} +(0.303848 + 0.526279i) q^{85} +12.8666i q^{89} +(-4.00000 + 6.92820i) q^{97} +9.89949i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{4}+O(q^{10}) $ 8 * q + 8 * q^4	$ 8 q + 8 q^{4} - 8 q^{10} - 8 q^{13} - 16 q^{16} + 36 q^{25} - 20 q^{34} - 8 q^{37} - 8 q^{40} - 28 q^{49} + 16 q^{52} + 28 q^{58} - 20 q^{61} - 64 q^{64} + 64 q^{73} + 16 q^{82} + 44 q^{85} - 32 q^{97}+O(q^{100}) $ 8 * q + 8 * q^4 - 8 * q^10 - 8 * q^13 - 16 * q^16 + 36 * q^25 - 20 * q^34 - 8 * q^37 - 8 * q^40 - 28 * q^49 + 16 * q^52 + 28 * q^58 - 20 * q^61 - 64 * q^64 + 64 * q^73 + 16 * q^82 + 44 * q^85 - 32 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$163$	$245$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$

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
$5$	3.79435	−	2.19067i	1.69689	−	0.979698i	0.748203	−	0.663470i	$-0.230917\pi$
$5$	3.79435	−	2.19067i	1.69689	−	0.979698i	0.948683	−	0.316228i	$-0.102416\pi$
$6$		0			0
$7$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$7$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$8$		−	2.82843i		−	1.00000i
$9$		0			0
$10$	−6.19615			−1.95940
$11$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$11$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$12$		0			0
$13$	1.59808	+	2.76795i	0.443227	+	0.767691i	0.997927	−	0.0643593i	$-0.0205004\pi$
$13$	1.59808	+	2.76795i	0.443227	+	0.767691i	−0.554700	+	0.832050i	$0.687167\pi$
$14$		0			0
$15$		0			0
$16$	−2.00000	+	3.46410i	−0.500000	+	0.866025i
$17$			0.138701i			0.0336399i	0.999859	+	0.0168199i	$0.00535420\pi$
$17$			0.138701i			0.0336399i	−0.999859	+	0.0168199i	$0.994646\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$	7.58871	+	4.38134i	1.69689	+	0.979698i
$21$		0			0
$22$		0			0
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$	7.09808	−	12.2942i	1.41962	−	2.45885i
$26$		−	4.52004i		−	0.886453i
$27$		0			0
$28$		0			0
$29$	−7.46859	−	4.31199i	−1.38688	−	0.800717i	−0.393919	−	0.919145i	$-0.628881\pi$
$29$	−7.46859	−	4.31199i	−1.38688	−	0.800717i	−0.992963	+	0.118428i	$0.962214\pi$
$30$		0			0
$31$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$31$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$32$	4.89898	−	2.82843i	0.866025	−	0.500000i
$33$		0			0
$34$	0.0980762	−	0.169873i	0.0168199	−	0.0291330i
$35$		0			0
$36$		0			0
$37$	9.39230			1.54409			0.772043	−	0.635571i	$-0.219235\pi$
$37$	9.39230			1.54409			0.772043	+	0.635571i	$0.219235\pi$
$38$		0			0
$39$		0			0
$40$	−6.19615	−	10.7321i	−0.979698	−	1.69689i
$41$	−1.22474	+	0.707107i	−0.191273	+	0.110432i	−0.592578	−	0.805513i	$-0.701890\pi$
$41$	−1.22474	+	0.707107i	−0.191273	+	0.110432i	0.401305	+	0.915944i	$0.368557\pi$
$42$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		0			0
$49$	−3.50000	−	6.06218i	−0.500000	−	0.866025i
$50$	−17.3867	+	10.0382i	−2.45885	+	1.41962i
$51$		0			0
$52$	−3.19615	+	5.53590i	−0.443227	+	0.767691i
$53$			7.07107i			0.971286i	0.874157	+	0.485643i	$0.161414\pi$
$53$			7.07107i			0.971286i	−0.874157	+	0.485643i	$0.838586\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	6.09808	+	10.5622i	0.800717	+	1.38688i
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$	−7.69615	+	13.3301i	−0.985391	+	1.70675i	−0.345207	+	0.938527i	$0.612191\pi$
$61$	−7.69615	+	13.3301i	−0.985391	+	1.70675i	−0.640184	+	0.768221i	$0.721142\pi$
$62$		0			0
$63$		0			0
$64$	−8.00000			−1.00000
$65$	12.1273	+	7.00172i	1.50421	+	0.868456i
$66$		0			0
$67$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$	−0.240237	+	0.138701i	−0.0291330	+	0.0168199i
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	2.80385			0.328166			0.164083	−	0.986447i	$-0.447534\pi$
$73$	2.80385			0.328166			0.164083	+	0.986447i	$0.447534\pi$
$74$	−11.5032	−	6.64136i	−1.33722	−	0.772043i
$75$		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	$0.666667\pi$
$80$			17.5254i			1.95940i
$81$		0			0
$82$	2.00000			0.220863
$83$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$83$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$84$		0			0
$85$	0.303848	+	0.526279i	0.0329569	+	0.0570830i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			12.8666i			1.36386i	0.731418	+	0.681930i	$0.238859\pi$
$89$			12.8666i			1.36386i	−0.731418	+	0.681930i	$0.761141\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−4.00000	+	6.92820i	−0.406138	+	0.703452i	−0.994453	−	0.105180i	$-0.966458\pi$
$97$	−4.00000	+	6.92820i	−0.406138	+	0.703452i	0.588315	+	0.808632i	$0.299792\pi$
$98$			9.89949i			1.00000i
$99$		0			0
$100$	28.3923			2.83923
$101$	−13.4722	−	7.77817i	−1.34053	−	0.773957i	−0.353648	−	0.935379i	$-0.615059\pi$
$101$	−13.4722	−	7.77817i	−1.34053	−	0.773957i	−0.986886	+	0.161421i	$0.948392\pi$
$102$		0			0
$103$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$103$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$104$	7.82894	−	4.52004i	0.767691	−	0.443227i
$105$		0			0
$106$	5.00000	−	8.66025i	0.485643	−	0.841158i
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$	−15.1962			−1.45553			−0.727764	−	0.685828i	$-0.759440\pi$
$109$	−15.1962			−1.45553			−0.727764	+	0.685828i	$0.759440\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−15.2975	+	8.83203i	−1.43907	+	0.830848i	−0.997785	−	0.0665190i	$-0.978811\pi$
$113$	−15.2975	+	8.83203i	−1.43907	+	0.830848i	−0.441285	+	0.897367i	$0.645477\pi$
$114$		0			0
$115$		0			0
$116$		−	17.2480i		−	1.60143i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.50000	+	9.52628i	0.500000	+	0.866025i
$122$	18.8516	−	10.8840i	1.70675	−	0.985391i
$123$		0			0
$124$		0			0
$125$		−	40.2915i		−	3.60378i
$126$		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
$130$	−9.90192	−	17.1506i	−0.868456	−	1.50421i
$131$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$131$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	0.392305			0.0336399
$137$	11.6233	+	6.71071i	0.993045	+	0.573335i	0.906183	−	0.422885i	$-0.138983\pi$
$137$	11.6233	+	6.71071i	0.993045	+	0.573335i	0.0868620	+	0.996220i	$0.472316\pi$
$138$		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
$145$	−37.7846			−3.13784
$146$	−3.43400	−	1.98262i	−0.284200	−	0.164083i
$147$		0			0
$148$	9.39230	+	16.2679i	0.772043	+	1.33722i
$149$	−7.22835	+	4.17329i	−0.592170	+	0.341889i	−0.765955	−	0.642894i	$-0.777733\pi$
$149$	−7.22835	+	4.17329i	−0.592170	+	0.341889i	0.173785	+	0.984784i	$0.444400\pi$
$150$		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
$157$	−10.6962	−	18.5263i	−0.853646	−	1.47856i	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−10.6962	−	18.5263i	−0.853646	−	1.47856i	0.0242497	−	0.999706i	$-0.492280\pi$
$158$		0			0
$159$		0			0
$160$	12.3923	−	21.4641i	0.979698	−	1.69689i
$161$		0			0
$162$		0			0
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$	−2.44949	−	1.41421i	−0.191273	−	0.110432i
$165$		0			0
$166$		0			0
$167$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$167$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$168$		0			0
$169$	1.39230	−	2.41154i	0.107100	−	0.185503i
$170$		−	0.859411i		−	0.0659138i
$171$		0			0
$172$		0			0
$173$	22.6460	+	13.0747i	1.72174	+	0.994049i	0.915338	+	0.402685i	$0.131923\pi$
$173$	22.6460	+	13.0747i	1.72174	+	0.994049i	0.806405	+	0.591364i	$0.201410\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	9.09808	−	15.7583i	0.681930	−	1.18114i
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	20.0000			1.48659			0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000			1.48659			0.743294	+	0.668965i	$0.233262\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	35.6377	−	20.5754i	2.62014	−	1.51274i
$186$		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
$193$	13.8923	+	24.0622i	0.999990	+	1.73203i	0.503871	+	0.863779i	$0.331909\pi$
$193$	13.8923	+	24.0622i	0.999990	+	1.73203i	0.496119	+	0.868255i	$0.334758\pi$
$194$	9.79796	−	5.65685i	0.703452	−	0.406138i
$195$		0			0
$196$	7.00000	−	12.1244i	0.500000	−	0.866025i
$197$		−	9.17878i		−	0.653961i	−0.945031	−	0.326981i	$-0.893969\pi$
$197$		−	9.17878i		−	0.653961i	0.945031	−	0.326981i	$-0.106031\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−34.7733	−	20.0764i	−2.45885	−	1.41962i
$201$		0			0
$202$	11.0000	+	19.0526i	0.773957	+	1.34053i
$203$		0			0
$204$		0			0
$205$	−3.09808	+	5.36603i	−0.216379	+	0.374779i
$206$		0			0
$207$		0			0
$208$	−12.7846			−0.886453
$209$		0			0
$210$		0			0
$211$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$211$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$212$	−12.2474	+	7.07107i	−0.841158	+	0.485643i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	18.6114	+	10.7453i	1.26052	+	0.727764i
$219$		0			0
$220$		0			0
$221$	−0.383917	+	0.221654i	−0.0258250	+	0.0149101i
$222$		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
$226$	24.9808			1.66170
$227$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$227$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$228$		0			0
$229$	−13.9904	−	24.2321i	−0.924510	−	1.60130i	−0.792347	−	0.610071i	$-0.791141\pi$
$229$	−13.9904	−	24.2321i	−0.924510	−	1.60130i	−0.132164	−	0.991228i	$-0.542192\pi$
$230$		0			0
$231$		0			0
$232$	−12.1962	+	21.1244i	−0.800717	+	1.38688i
$233$			22.1841i			1.45333i	0.686992	+	0.726665i	$0.258931\pi$
$233$			22.1841i			1.45333i	−0.686992	+	0.726665i	$0.741069\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		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
$241$	−10.9904	+	19.0359i	−0.707953	+	1.22621i	0.257663	+	0.966235i	$0.417048\pi$
$241$	−10.9904	+	19.0359i	−0.707953	+	1.22621i	−0.965615	+	0.259975i	$0.916286\pi$
$242$		−	15.5563i		−	1.00000i
$243$		0			0
$244$	−30.7846			−1.97078
$245$	−26.5605	−	15.3347i	−1.69689	−	0.979698i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−28.4904	+	49.3468i	−1.80189	+	3.12096i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$	−26.3202	+	15.1960i	−1.64181	+	0.947900i	−0.661622	+	0.749838i	$0.730131\pi$
$257$	−26.3202	+	15.1960i	−1.64181	+	0.947900i	−0.980189	+	0.198062i	$0.936535\pi$
$258$		0			0
$259$		0			0
$260$			28.0069i			1.73691i
$261$		0			0
$262$		0			0
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$	15.4904	+	26.8301i	0.951567	+	1.64816i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	12.5892i		−	0.767578i	−0.923421	−	0.383789i	$-0.874619\pi$
$269$		−	12.5892i		−	0.767578i	0.923421	−	0.383789i	$-0.125381\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−0.480473	−	0.277401i	−0.0291330	−	0.0168199i
$273$		0			0
$274$	−9.49038	−	16.4378i	−0.573335	−	0.993045i
$275$		0			0
$276$		0			0
$277$	14.0000	−	24.2487i	0.841178	−	1.45696i	−0.0477206	−	0.998861i	$-0.515196\pi$
$277$	14.0000	−	24.2487i	0.841178	−	1.45696i	0.888899	−	0.458103i	$-0.151471\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−15.5378	−	8.97073i	−0.926905	−	0.535149i	−0.0410739	−	0.999156i	$-0.513078\pi$
$281$	−15.5378	−	8.97073i	−0.926905	−	0.535149i	−0.885832	+	0.464007i	$0.846411\pi$
$282$		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
$289$	16.9808			0.998868
$290$	46.2765	+	26.7178i	2.71745	+	1.56892i
$291$		0			0
$292$	2.80385	+	4.85641i	0.164083	+	0.284200i
$293$	−4.27483	+	2.46807i	−0.249738	+	0.144186i	−0.619644	−	0.784883i	$-0.712723\pi$
$293$	−4.27483	+	2.46807i	−0.249738	+	0.144186i	0.369906	+	0.929069i	$0.379390\pi$
$294$		0			0
$295$		0			0
$296$		−	26.5654i		−	1.54409i
$297$		0			0
$298$	11.8038			0.683779
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$			67.4389i			3.86154i
$306$		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		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	16.8923	−	29.2583i	0.954810	−	1.65378i	0.220006	−	0.975499i	$-0.429392\pi$
$313$	16.8923	−	29.2583i	0.954810	−	1.65378i	0.734803	−	0.678280i	$-0.237274\pi$
$314$			30.2533i			1.70729i
$315$		0			0
$316$		0			0
$317$	−18.4913	−	10.6760i	−1.03857	−	0.599621i	−0.119145	−	0.992877i	$-0.538015\pi$
$317$	−18.4913	−	10.6760i	−1.03857	−	0.599621i	−0.919429	+	0.393256i	$0.871349\pi$
$318$		0			0
$319$		0			0
$320$	−30.3548	+	17.5254i	−1.69689	+	0.979698i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	45.3731			2.51684
$326$		0			0
$327$		0			0
$328$	2.00000	+	3.46410i	0.110432	+	0.191273i
$329$		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
$337$	−16.0000	−	27.7128i	−0.871576	−	1.50961i	−0.860366	−	0.509676i	$-0.829765\pi$
$337$	−16.0000	−	27.7128i	−0.871576	−	1.50961i	−0.0112091	−	0.999937i	$-0.503568\pi$
$338$	−3.41044	+	1.96902i	−0.185503	+	0.107100i
$339$		0			0
$340$	−0.607695	+	1.05256i	−0.0329569	+	0.0570830i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−18.4904	−	32.0263i	−0.994049	−	1.72174i
$347$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$347$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$348$		0			0
$349$	5.00000	−	8.66025i	0.267644	−	0.463573i	−0.700609	−	0.713545i	$-0.747088\pi$
$349$	5.00000	−	8.66025i	0.267644	−	0.463573i	0.968253	+	0.249973i	$0.0804216\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	30.6186	+	17.6777i	1.62966	+	0.940887i	0.984192	+	0.177104i	$0.0566729\pi$
$353$	30.6186	+	17.6777i	1.62966	+	0.940887i	0.645473	+	0.763783i	$0.276660\pi$
$354$		0			0
$355$		0			0
$356$	−22.2856	+	12.8666i	−1.18114	+	0.681930i
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	19.0000			1.00000
$362$	−24.4949	−	14.1421i	−1.28742	−	0.743294i
$363$		0			0
$364$		0			0
$365$	10.6388	−	6.14231i	0.556860	−	0.321503i
$366$		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
$370$	−58.1962			−3.02547
$371$		0			0
$372$		0			0
$373$	−7.00000	−	12.1244i	−0.362446	−	0.627775i	0.625917	−	0.779890i	$-0.284725\pi$
$373$	−7.00000	−	12.1244i	−0.362446	−	0.627775i	−0.988363	+	0.152115i	$0.951392\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	27.5636i		−	1.41960i
$378$		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
$385$		0			0
$386$		−	39.2934i		−	1.99998i
$387$		0			0
$388$	−16.0000			−0.812277
$389$	8.57321	+	4.94975i	0.434679	+	0.250962i	0.701338	−	0.712829i	$-0.252586\pi$
$389$	8.57321	+	4.94975i	0.434679	+	0.250962i	−0.266659	+	0.963791i	$0.585920\pi$
$390$		0			0
$391$		0			0
$392$	−17.1464	+	9.89949i	−0.866025	+	0.500000i
$393$		0			0
$394$	−6.49038	+	11.2417i	−0.326981	+	0.566347i
$395$		0			0
$396$		0			0
$397$	−8.60770			−0.432008			−0.216004	−	0.976392i	$-0.569302\pi$
$397$	−8.60770			−0.432008			−0.216004	+	0.976392i	$0.569302\pi$
$398$		0			0
$399$		0			0
$400$	28.3923	+	49.1769i	1.41962	+	2.45885i
$401$	33.9089	−	19.5773i	1.69333	−	0.977645i	0.741536	−	0.670913i	$-0.234098\pi$
$401$	33.9089	−	19.5773i	1.69333	−	0.977645i	0.951796	−	0.306732i	$-0.0992356\pi$
$402$		0			0
$403$		0			0
$404$		−	31.1127i		−	1.54791i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−7.40192	−	12.8205i	−0.366002	−	0.633933i	0.622935	−	0.782274i	$-0.285940\pi$
$409$	−7.40192	−	12.8205i	−0.366002	−	0.633933i	−0.988936	+	0.148340i	$0.952607\pi$
$410$	7.58871	−	4.38134i	0.374779	−	0.216379i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	15.6579	+	9.04008i	0.767691	+	0.443227i
$417$		0			0
$418$		0			0
$419$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$419$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$420$		0			0
$421$	−19.9904	+	34.6244i	−0.974272	+	1.68749i	−0.291953	+	0.956433i	$0.594305\pi$
$421$	−19.9904	+	34.6244i	−0.974272	+	1.68749i	−0.682318	+	0.731055i	$0.739028\pi$
$422$		0			0
$423$		0			0
$424$	20.0000			0.971286
$425$	1.70522	+	0.984508i	0.0827152	+	0.0477557i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	−3.78461			−0.181877			−0.0909384	−	0.995857i	$-0.528987\pi$
$433$	−3.78461			−0.181877			−0.0909384	+	0.995857i	$0.528987\pi$
$434$		0			0
$435$		0			0
$436$	−15.1962	−	26.3205i	−0.727764	−	1.26052i
$437$		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
$442$	0.626933			0.0298202
$443$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$443$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$444$		0			0
$445$	28.1865	+	48.8205i	1.33617	+	2.31431i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	18.3848i		−	0.867631i	−0.901002	−	0.433816i	$-0.857167\pi$
$449$		−	18.3848i		−	0.867631i	0.901002	−	0.433816i	$-0.142833\pi$
$450$		0			0
$451$		0			0
$452$	−30.5951	−	17.6641i	−1.43907	−	0.830848i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	20.1865	−	34.9641i	0.944286	−	1.63555i	0.187112	−	0.982339i	$-0.440087\pi$
$457$	20.1865	−	34.9641i	0.944286	−	1.63555i	0.757174	−	0.653213i	$-0.226579\pi$
$458$			39.5708i			1.84902i
$459$		0			0
$460$		0			0
$461$	−35.5176	−	20.5061i	−1.65422	−	0.955064i	−0.975309	−	0.220843i	$-0.929119\pi$
$461$	−35.5176	−	20.5061i	−1.65422	−	0.955064i	−0.678910	−	0.734221i	$-0.737547\pi$
$462$		0			0
$463$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$463$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$464$	29.8744	−	17.2480i	1.38688	−	0.800717i
$465$		0			0
$466$	15.6865	−	27.1699i	0.726665	−	1.25862i
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$	15.0096	+	25.9974i	0.684380	+	1.18538i
$482$	26.9208	−	15.5427i	1.22621	−	0.707953i
$483$		0			0
$484$	−11.0000	+	19.0526i	−0.500000	+	0.866025i
$485$			35.0507i			1.59157i
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$	37.7033	+	21.7680i	1.70675	+	0.985391i
$489$		0			0
$490$	21.6865	+	37.5622i	0.979698	+	1.69689i
$491$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$491$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$492$		0			0
$493$	0.598076	−	1.03590i	0.0269360	−	0.0466545i
$494$		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	$0.333333\pi$
$500$	69.7869	−	40.2915i	3.12096	−	1.80189i
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$	−68.1577			−3.03298
$506$		0			0
$507$		0			0
$508$		0			0
$509$	20.8207	−	12.0208i	0.922860	−	0.532813i	0.0383134	−	0.999266i	$-0.487801\pi$
$509$	20.8207	−	12.0208i	0.922860	−	0.532813i	0.884546	+	0.466453i	$0.154468\pi$
$510$		0			0
$511$		0			0
$512$			22.6274i			1.00000i
$513$		0			0
$514$	42.9808			1.89580
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	19.8038	−	34.3013i	0.868456	−	1.50421i
$521$		−	43.8406i		−	1.92069i	−0.278810	−	0.960346i	$-0.589940\pi$
$521$		−	43.8406i		−	1.92069i	0.278810	−	0.960346i	$-0.410060\pi$
$522$		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
$529$	11.5000	−	19.9186i	0.500000	−	0.866025i
$530$		−	43.8134i		−	1.90313i
$531$		0			0
$532$		0			0
$533$	−3.91447	−	2.26002i	−0.169555	−	0.0978924i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−8.90192	+	15.4186i	−0.383789	+	0.664742i
$539$		0			0
$540$		0			0
$541$	−46.3731			−1.99373			−0.996867	−	0.0790969i	$-0.974796\pi$
$541$	−46.3731			−1.99373			−0.996867	+	0.0790969i	$0.974796\pi$
$542$		0			0
$543$		0			0
$544$	0.392305	+	0.679492i	0.0168199	+	0.0291330i
$545$	−57.6596	+	33.2898i	−2.46986	+	1.42598i
$546$		0			0
$547$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$547$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$548$			26.8429i			1.14667i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−34.2929	+	19.7990i	−1.45696	+	0.841178i
$555$		0			0
$556$		0			0
$557$		−	43.9521i		−	1.86231i	−0.364622	−	0.931155i	$-0.618802\pi$
$557$		−	43.9521i		−	1.86231i	0.364622	−	0.931155i	$-0.381198\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	12.6865	+	21.9737i	0.535149	+	0.926905i
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$	−38.6962	+	67.0237i	−1.62796	+	2.81971i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	30.7152	+	17.7334i	1.28765	+	0.743423i	0.978234	−	0.207504i	$-0.0665341\pi$
$569$	30.7152	+	17.7334i	1.28765	+	0.743423i	0.309413	+	0.950928i	$0.399867\pi$
$570$		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
$577$	40.5692			1.68892			0.844459	−	0.535620i	$-0.179922\pi$
$577$	40.5692			1.68892			0.844459	+	0.535620i	$0.179922\pi$
$578$	−20.7971	−	12.0072i	−0.865045	−	0.499434i
$579$		0			0
$580$	−37.7846	−	65.4449i	−1.56892	−	2.71745i
$581$		0			0
$582$		0			0
$583$		0			0
$584$		−	7.93048i		−	0.328166i
$585$		0			0
$586$	6.98076			0.288373
$587$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$587$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−18.7846	+	32.5359i	−0.772043	+	1.33722i
$593$			3.54914i			0.145746i	0.997341	+	0.0728728i	$0.0232167\pi$
$593$			3.54914i			0.145746i	−0.997341	+	0.0728728i	$0.976783\pi$
$594$		0			0
$595$		0			0
$596$	−14.4567	−	8.34658i	−0.592170	−	0.341889i
$597$		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
$601$	−23.2846	+	40.3301i	−0.949799	+	1.64510i	−0.203954	+	0.978980i	$0.565379\pi$
$601$	−23.2846	+	40.3301i	−0.949799	+	1.64510i	−0.745845	+	0.666120i	$0.767954\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	41.7379	+	24.0974i	1.69689	+	0.979698i
$606$		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
$610$	47.6865	−	82.5955i	1.93077	−	3.34419i
$611$		0			0
$612$		0			0
$613$	−34.0000			−1.37325			−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000			−1.37325			−0.686624	+	0.727013i	$0.740908\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−18.2511	+	10.5373i	−0.734760	+	0.424214i	−0.820161	−	0.572133i	$-0.806116\pi$
$617$	−18.2511	+	10.5373i	−0.734760	+	0.424214i	0.0854011	+	0.996347i	$0.472783\pi$
$618$		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
$625$	−52.7750	−	91.4090i	−2.11100	−	3.65636i
$626$	−41.3775	+	23.8893i	−1.65378	+	0.954810i
$627$		0			0
$628$	21.3923	−	37.0526i	0.853646	−	1.47856i
$629$			1.30272i			0.0519428i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$	15.0981	+	26.1506i	0.599621	+	1.03857i
$635$		0			0
$636$		0			0
$637$	11.1865	−	19.3756i	0.443227	−	0.767691i
$638$		0			0
$639$		0			0
$640$	49.5692			1.95940
$641$	−4.51506	−	2.60677i	−0.178334	−	0.102961i	0.408176	−	0.912903i	$-0.366165\pi$
$641$	−4.51506	−	2.60677i	−0.178334	−	0.102961i	−0.586510	+	0.809942i	$0.699498\pi$
$642$		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
$649$		0			0
$650$	−55.5704	−	32.0836i	−2.17965	−	1.25842i
$651$		0			0
$652$		0			0
$653$	42.8661	−	24.7487i	1.67748	−	0.968493i	0.714219	−	0.699922i	$-0.246782\pi$
$653$	42.8661	−	24.7487i	1.67748	−	0.968493i	0.963260	−	0.268571i	$-0.0865513\pi$
$654$		0			0
$655$		0			0
$656$		−	5.65685i		−	0.220863i
$657$		0			0
$658$		0			0
$659$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$659$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$660$		0			0
$661$	7.30385	+	12.6506i	0.284087	+	0.492053i	0.972387	−	0.233373i	$-0.0749763\pi$
$661$	7.30385	+	12.6506i	0.284087	+	0.492053i	−0.688301	+	0.725426i	$0.741643\pi$
$662$		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
$673$	−1.10770	+	1.91858i	−0.0426985	+	0.0739560i	−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−1.10770	+	1.91858i	−0.0426985	+	0.0739560i	0.843886	+	0.536522i	$0.180262\pi$
$674$			45.2548i			1.74315i
$675$		0			0
$676$	5.56922			0.214201
$677$	30.6186	+	17.6777i	1.17677	+	0.679408i	0.955265	−	0.295751i	$-0.0955700\pi$
$677$	30.6186	+	17.6777i	1.17677	+	0.679408i	0.221504	+	0.975159i	$0.428903\pi$
$678$		0			0
$679$		0			0
$680$	1.48854	−	0.859411i	0.0570830	−	0.0329569i
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$	58.8038			2.24678
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−19.5724	+	11.3001i	−0.745647	+	0.430500i
$690$		0			0
$691$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$691$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$692$			52.2987i			1.98810i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−0.0980762	−	0.169873i	−0.00371490	−	0.00643440i
$698$	−12.2474	+	7.07107i	−0.463573	+	0.267644i
$699$		0			0
$700$		0			0
$701$			47.6400i			1.79934i	0.436575	+	0.899668i	$0.356192\pi$
$701$			47.6400i			1.79934i	−0.436575	+	0.899668i	$0.643808\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−25.0000	−	43.3013i	−0.940887	−	1.62966i
$707$		0			0
$708$		0			0
$709$	−1.99038	+	3.44744i	−0.0747503	+	0.129471i	−0.900978	−	0.433865i	$-0.857149\pi$
$709$	−1.99038	+	3.44744i	−0.0747503	+	0.129471i	0.826227	+	0.563337i	$0.190483\pi$
$710$		0			0
$711$		0			0
$712$	36.3923			1.36386
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		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
$722$	−23.2702	−	13.4350i	−0.866025	−	0.500000i
$723$		0			0
$724$	20.0000	+	34.6410i	0.743294	+	1.28742i
$725$	−106.025	+	61.2137i	−3.93768	+	2.27342i
$726$		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
$730$	−17.3731			−0.643006
$731$		0			0
$732$		0			0
$733$	2.00000	+	3.46410i	0.0738717	+	0.127950i	0.900595	−	0.434659i	$-0.143131\pi$
$733$	2.00000	+	3.46410i	0.0738717	+	0.127950i	−0.826723	+	0.562609i	$0.809798\pi$
$734$		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			$\pi$
$740$	71.2754	+	41.1509i	2.62014	+	1.51274i
$741$		0			0
$742$		0			0
$743$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$743$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$744$		0			0
$745$	−18.2846	+	31.6699i	−0.669896	+	1.16029i
$746$			19.7990i			0.724893i
$747$		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
$754$	−19.4904	+	33.7583i	−0.709798	+	1.22941i
$755$		0			0
$756$		0			0
$757$	−52.0000			−1.88997			−0.944986	−	0.327111i	$-0.893925\pi$
$757$	−52.0000			−1.88997			−0.944986	+	0.327111i	$0.893925\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	41.9781	−	24.2361i	1.52171	−	0.878557i	0.522034	−	0.852925i	$-0.325173\pi$
$761$	41.9781	−	24.2361i	1.52171	−	0.878557i	0.999671	−	0.0256326i	$-0.00816000\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	22.8923	+	39.6506i	0.825518	+	1.42984i	0.901523	+	0.432731i	$0.142450\pi$
$769$	22.8923	+	39.6506i	0.825518	+	1.42984i	−0.0760054	+	0.997107i	$0.524217\pi$
$770$		0			0
$771$		0			0
$772$	−27.7846	+	48.1244i	−0.999990	+	1.73203i
$773$			44.2295i			1.59083i	0.606068	+	0.795413i	$0.292746\pi$
$773$			44.2295i			1.59083i	−0.606068	+	0.795413i	$0.707254\pi$
$774$		0			0
$775$		0			0
$776$	19.5959	+	11.3137i	0.703452	+	0.406138i
$777$		0			0
$778$	−7.00000	−	12.1244i	−0.250962	−	0.434679i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	28.0000			1.00000
$785$	−81.1700	−	46.8635i	−2.89708	−	1.67263i
$786$		0			0
$787$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$787$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$788$	15.8981	−	9.17878i	0.566347	−	0.326981i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−49.1962			−1.74701
$794$	10.5422	+	6.08656i	0.374130	+	0.216004i
$795$		0			0
$796$		0			0
$797$	6.74788	−	3.89589i	0.239022	−	0.137999i	−0.375705	−	0.926739i	$-0.622599\pi$
$797$	6.74788	−	3.89589i	0.239022	−	0.137999i	0.614727	+	0.788740i	$0.289266\pi$
$798$		0			0
$799$		0			0
$800$		−	80.3056i		−	2.83923i
$801$		0			0
$802$	−55.3731			−1.95529
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−22.0000	+	38.1051i	−0.773957	+	1.34053i
$809$		−	56.6800i		−	1.99276i	−0.0849879	−	0.996382i	$-0.527085\pi$
$809$		−	56.6800i		−	1.99276i	0.0849879	−	0.996382i	$-0.472915\pi$
$810$		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
$818$			20.9358i			0.732003i
$819$		0			0
$820$	−12.3923			−0.432758
$821$	−34.6296	−	19.9934i	−1.20858	−	0.697776i	−0.246133	−	0.969236i	$-0.579160\pi$
$821$	−34.6296	−	19.9934i	−1.20858	−	0.697776i	−0.962450	+	0.271460i	$0.912493\pi$
$822$		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			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$	20.0000			0.694629			0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000			0.694629			0.347314	+	0.937749i	$0.387094\pi$
$830$		0			0
$831$		0			0
$832$	−12.7846	−	22.1436i	−0.443227	−	0.767691i
$833$	0.840828	−	0.485452i	0.0291330	−	0.0168199i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$839$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$840$		0			0
$841$	22.6865	+	39.2942i	0.782294	+	1.35497i
$842$	48.9662	−	28.2707i	1.68749	−	0.974272i
$843$		0			0
$844$		0			0
$845$		−	12.2003i		−	0.419704i
$846$		0			0
$847$		0			0
$848$	−24.4949	−	14.1421i	−0.841158	−	0.485643i
$849$		0			0
$850$	−1.39230	−	2.41154i	−0.0477557	−	0.0827152i
$851$		0			0
$852$		0			0
$853$	23.0000	−	39.8372i	0.787505	−	1.36400i	−0.139986	−	0.990153i	$-0.544706\pi$
$853$	23.0000	−	39.8372i	0.787505	−	1.36400i	0.927491	−	0.373845i	$-0.121961\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	19.6925	+	11.3695i	0.672682	+	0.388373i	0.797092	−	0.603858i	$-0.206370\pi$
$857$	19.6925	+	11.3695i	0.672682	+	0.388373i	−0.124410	+	0.992231i	$0.539704\pi$
$858$		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
$865$	114.569			3.89547
$866$	4.63518	+	2.67612i	0.157510	+	0.0909384i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$			42.9812i			1.45553i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−19.6962	−	34.1147i	−0.665092	−	1.15197i	−0.979260	−	0.202606i	$-0.935059\pi$
$877$	−19.6962	−	34.1147i	−0.665092	−	1.15197i	0.314169	−	0.949367i	$-0.398274\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			57.9828i			1.95349i	0.214407	+	0.976744i	$0.431218\pi$
$881$			57.9828i			1.95349i	−0.214407	+	0.976744i	$0.568782\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$	−0.767833	−	0.443309i	−0.0258250	−	0.0149101i
$885$		0			0
$886$		0			0
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$		0			0
$890$		−	79.7236i		−	2.67234i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−13.0000	+	22.5167i	−0.433816	+	0.751391i
$899$		0			0
$900$		0			0
$901$	−0.980762			−0.0326739
$902$		0			0
$903$		0			0
$904$	24.9808	+	43.2679i	0.830848	+	1.43907i
$905$	75.8871	−	43.8134i	2.52257	−	1.45641i
$906$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$911$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$912$		0			0
$913$		0			0
$914$	−49.4467	+	28.5481i	−1.63555	+	0.944286i
$915$		0			0
$916$	27.9808	−	48.4641i	0.924510	−	1.60130i
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$	29.0000	+	50.2295i	0.955064	+	1.65422i
$923$		0			0
$924$		0			0
$925$	66.6673	−	115.471i	2.19201	−	3.79667i
$926$		0			0
$927$		0			0
$928$	−48.7846			−1.60143
$929$	−29.5140	−	17.0399i	−0.968323	−	0.559061i	−0.0695983	−	0.997575i	$-0.522172\pi$
$929$	−29.5140	−	17.0399i	−0.968323	−	0.559061i	−0.898725	+	0.438514i	$0.855505\pi$
$930$		0			0
$931$		0			0
$932$	−38.4240	+	22.1841i	−1.25862	+	0.726665i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	22.5692			0.737304			0.368652	−	0.929567i	$-0.379819\pi$
$937$	22.5692			0.737304			0.368652	+	0.929567i	$0.379819\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	53.0008	−	30.6000i	1.72778	−	0.997533i	0.828788	−	0.559563i	$-0.189031\pi$
$941$	53.0008	−	30.6000i	1.72778	−	0.997533i	0.898990	−	0.437969i	$-0.144302\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$947$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$948$		0			0
$949$	4.48076	+	7.76091i	0.145452	+	0.251930i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	47.3626i		−	1.53422i	−0.641513	−	0.767112i	$-0.721693\pi$
$953$		−	47.3626i		−	1.53422i	0.641513	−	0.767112i	$-0.278307\pi$
$954$		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
$962$		−	42.4536i		−	1.36876i
$963$		0			0
$964$	−43.9615			−1.41591
$965$	105.425	+	60.8669i	3.39374	+	1.95938i
$966$		0			0
$967$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$967$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$968$	26.9444	−	15.5563i	0.866025	−	0.500000i
$969$		0			0
$970$	24.7846	−	42.9282i	0.795786	−	1.37834i
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−30.7846	−	53.3205i	−0.985391	−	1.70675i
$977$	42.8661	−	24.7487i	1.37141	−	0.791782i	0.380302	−	0.924862i	$-0.375820\pi$
$977$	42.8661	−	24.7487i	1.37141	−	0.791782i	0.991105	+	0.133080i	$0.0424868\pi$
$978$		0			0
$979$		0			0
$980$		−	61.3388i		−	1.95940i
$981$		0			0
$982$		0			0
$983$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$983$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$984$		0			0
$985$	−20.1077	−	34.8275i	−0.640684	−	1.10970i
$986$	−1.46498	+	0.845807i	−0.0466545	+	0.0269360i
$987$		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
$997$	10.3038	−	17.8468i	0.326326	−	0.565213i	−0.655454	−	0.755235i	$-0.727523\pi$
$997$	10.3038	−	17.8468i	0.326326	−	0.565213i	0.981780	+	0.190022i	$0.0608559\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.2.h.e.107.2		8
3.2	odd	2	inner	324.2.h.e.107.3		8
4.3	odd	2	CM	324.2.h.e.107.2		8
9.2	odd	6		324.2.b.a.323.1	✓	4
9.4	even	3	inner	324.2.h.e.215.3		8
9.5	odd	6	inner	324.2.h.e.215.2		8
9.7	even	3		324.2.b.a.323.4	yes	4
12.11	even	2	inner	324.2.h.e.107.3		8
36.7	odd	6		324.2.b.a.323.4	yes	4
36.11	even	6		324.2.b.a.323.1	✓	4
36.23	even	6	inner	324.2.h.e.215.2		8
36.31	odd	6	inner	324.2.h.e.215.3		8
72.11	even	6		5184.2.c.e.5183.4		4
72.29	odd	6		5184.2.c.e.5183.4		4
72.43	odd	6		5184.2.c.e.5183.1		4
72.61	even	6		5184.2.c.e.5183.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.2.b.a.323.1	✓	4	9.2	odd	6
324.2.b.a.323.1	✓	4	36.11	even	6
324.2.b.a.323.4	yes	4	9.7	even	3
324.2.b.a.323.4	yes	4	36.7	odd	6
324.2.h.e.107.2		8	1.1	even	1	trivial
324.2.h.e.107.2		8	4.3	odd	2	CM
324.2.h.e.107.3		8	3.2	odd	2	inner
324.2.h.e.107.3		8	12.11	even	2	inner
324.2.h.e.215.2		8	9.5	odd	6	inner
324.2.h.e.215.2		8	36.23	even	6	inner
324.2.h.e.215.3		8	9.4	even	3	inner
324.2.h.e.215.3		8	36.31	odd	6	inner
5184.2.c.e.5183.1		4	72.43	odd	6
5184.2.c.e.5183.1		4	72.61	even	6
5184.2.c.e.5183.4		4	72.11	even	6
5184.2.c.e.5183.4		4	72.29	odd	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:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	324.h (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.22474 - 0.707107i) q^{2} +(1.00000 + 1.73205i) q^{4} +(3.79435 - 2.19067i) q^{5} -2.82843i q^{8} +O(q^{10})\)	\(q+(-1.22474 - 0.707107i) q^{2} +(1.00000 + 1.73205i) q^{4} +(3.79435 - 2.19067i) q^{5} -2.82843i q^{8} -6.19615 q^{10} +(1.59808 + 2.76795i) q^{13} +(-2.00000 + 3.46410i) q^{16} +0.138701i q^{17} +(7.58871 + 4.38134i) q^{20} +(7.09808 - 12.2942i) q^{25} -4.52004i q^{26} +(-7.46859 - 4.31199i) q^{29} +(4.89898 - 2.82843i) q^{32} +(0.0980762 - 0.169873i) q^{34} +9.39230 q^{37} +(-6.19615 - 10.7321i) q^{40} +(-1.22474 + 0.707107i) q^{41} +(-3.50000 - 6.06218i) q^{49} +(-17.3867 + 10.0382i) q^{50} +(-3.19615 + 5.53590i) q^{52} +7.07107i q^{53} +(6.09808 + 10.5622i) q^{58} +(-7.69615 + 13.3301i) q^{61} -8.00000 q^{64} +(12.1273 + 7.00172i) q^{65} +(-0.240237 + 0.138701i) q^{68} +2.80385 q^{73} +(-11.5032 - 6.64136i) q^{74} +17.5254i q^{80} +2.00000 q^{82} +(0.303848 + 0.526279i) q^{85} +12.8666i q^{89} +(-4.00000 + 6.92820i) q^{97} +9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{4}+O(q^{10}) \) 8 * q + 8 * q^4	\( 8 q + 8 q^{4} - 8 q^{10} - 8 q^{13} - 16 q^{16} + 36 q^{25} - 20 q^{34} - 8 q^{37} - 8 q^{40} - 28 q^{49} + 16 q^{52} + 28 q^{58} - 20 q^{61} - 64 q^{64} + 64 q^{73} + 16 q^{82} + 44 q^{85} - 32 q^{97}+O(q^{100}) \) 8 * q + 8 * q^4 - 8 * q^10 - 8 * q^13 - 16 * q^16 + 36 * q^25 - 20 * q^34 - 8 * q^37 - 8 * q^40 - 28 * q^49 + 16 * q^52 + 28 * q^58 - 20 * q^61 - 64 * q^64 + 64 * q^73 + 16 * q^82 + 44 * q^85 - 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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