LMFDB - Embedded newform 324.2.h.e.107.4

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			107.4
Root			$-0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	324.107
Dual form			324.2.h.e.215.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 + 0.707107i) q^{2} +(1.00000 + 1.73205i) q^{4} +(2.56961 - 1.48356i) q^{5} +2.82843i q^{8} +O(q^{10})$	$q+(1.22474 + 0.707107i) q^{2} +(1.00000 + 1.73205i) q^{4} +(2.56961 - 1.48356i) q^{5} +2.82843i q^{8} +4.19615 q^{10} +(-3.59808 - 6.23205i) q^{13} +(-2.00000 + 3.46410i) q^{16} +7.20977i q^{17} +(5.13922 + 2.96713i) q^{20} +(1.90192 - 3.29423i) q^{25} -10.1769i q^{26} +(1.10463 + 0.637756i) q^{29} +(-4.89898 + 2.82843i) q^{32} +(-5.09808 + 8.83013i) q^{34} -11.3923 q^{37} +(4.19615 + 7.26795i) q^{40} +(1.22474 - 0.707107i) q^{41} +(-3.50000 - 6.06218i) q^{49} +(4.65874 - 2.68973i) q^{50} +(7.19615 - 12.4641i) q^{52} -7.07107i q^{53} +(0.901924 + 1.56218i) q^{58} +(2.69615 - 4.66987i) q^{61} -8.00000 q^{64} +(-18.4913 - 10.6760i) q^{65} +(-12.4877 + 7.20977i) q^{68} +13.1962 q^{73} +(-13.9527 - 8.05558i) q^{74} +11.8685i q^{80} +2.00000 q^{82} +(10.6962 + 18.5263i) q^{85} -5.51815i q^{89} +(-4.00000 + 6.92820i) q^{97} -9.89949i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.56961	−	1.48356i	1.14916	−	0.663470i	0.200480	−	0.979698i	$-0.435750\pi$
$5$	2.56961	−	1.48356i	1.14916	−	0.663470i	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
$10$	4.19615			1.32694
$11$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$11$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$12$		0			0
$13$	−3.59808	−	6.23205i	−0.997927	−	1.72846i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−3.59808	−	6.23205i	−0.997927	−	1.72846i	−0.443227	−	0.896410i	$-0.646166\pi$
$14$		0			0
$15$		0			0
$16$	−2.00000	+	3.46410i	−0.500000	+	0.866025i
$17$			7.20977i			1.74863i	0.485363	+	0.874313i	$0.338688\pi$
$17$			7.20977i			1.74863i	−0.485363	+	0.874313i	$0.661312\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$	5.13922	+	2.96713i	1.14916	+	0.663470i
$21$		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
$25$	1.90192	−	3.29423i	0.380385	−	0.658846i
$26$		−	10.1769i		−	1.99585i
$27$		0			0
$28$		0			0
$29$	1.10463	+	0.637756i	0.205124	+	0.118428i	0.599043	−	0.800717i	$-0.295548\pi$
$29$	1.10463	+	0.637756i	0.205124	+	0.118428i	−0.393919	+	0.919145i	$0.628881\pi$
$30$		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
$34$	−5.09808	+	8.83013i	−0.874313	+	1.51435i
$35$		0			0
$36$		0			0
$37$	−11.3923			−1.87288			−0.936442	−	0.350823i	$-0.885902\pi$
$37$	−11.3923			−1.87288			−0.936442	+	0.350823i	$0.885902\pi$
$38$		0			0
$39$		0			0
$40$	4.19615	+	7.26795i	0.663470	+	1.14916i
$41$	1.22474	−	0.707107i	0.191273	−	0.110432i	−0.401305	−	0.915944i	$-0.631443\pi$
$41$	1.22474	−	0.707107i	0.191273	−	0.110432i	0.592578	+	0.805513i	$0.298110\pi$
$42$		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
$50$	4.65874	−	2.68973i	0.658846	−	0.380385i
$51$		0			0
$52$	7.19615	−	12.4641i	0.997927	−	1.72846i
$53$		−	7.07107i		−	0.971286i	−0.874157	−	0.485643i	$-0.838586\pi$
$53$		−	7.07107i		−	0.971286i	0.874157	−	0.485643i	$-0.161414\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	0.901924	+	1.56218i	0.118428	+	0.205124i
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$	2.69615	−	4.66987i	0.345207	−	0.597916i	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	2.69615	−	4.66987i	0.345207	−	0.597916i	0.985391	+	0.170305i	$0.0544754\pi$
$62$		0			0
$63$		0			0
$64$	−8.00000			−1.00000
$65$	−18.4913	−	10.6760i	−2.29356	−	1.32419i
$66$		0			0
$67$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$	−12.4877	+	7.20977i	−1.51435	+	0.874313i
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	13.1962			1.54449			0.772246	−	0.635323i	$-0.219133\pi$
$73$	13.1962			1.54449			0.772246	+	0.635323i	$0.219133\pi$
$74$	−13.9527	−	8.05558i	−1.62196	−	0.936442i
$75$		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	$0.666667\pi$
$80$			11.8685i			1.32694i
$81$		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
$85$	10.6962	+	18.5263i	1.16016	+	2.00946i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	5.51815i		−	0.584923i	−0.956277	−	0.292462i	$-0.905526\pi$
$89$		−	5.51815i		−	0.584923i	0.956277	−	0.292462i	$-0.0944744\pi$
$90$		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
$100$	7.60770			0.760770
$101$	13.4722	+	7.77817i	1.34053	+	0.773957i	0.986886	−	0.161421i	$-0.0516078\pi$
$101$	13.4722	+	7.77817i	1.34053	+	0.773957i	0.353648	+	0.935379i	$0.384941\pi$
$102$		0			0
$103$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$103$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$104$	17.6269	−	10.1769i	1.72846	−	0.997927i
$105$		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
$109$	−4.80385			−0.460125			−0.230063	−	0.973176i	$-0.573893\pi$
$109$	−4.80385			−0.460125			−0.230063	+	0.973176i	$0.573893\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−16.5223	+	9.53914i	−1.55428	+	0.897367i	−0.556500	+	0.830848i	$0.687856\pi$
$113$	−16.5223	+	9.53914i	−1.55428	+	0.897367i	−0.997785	+	0.0665190i	$0.978811\pi$
$114$		0			0
$115$		0			0
$116$			2.55103i			0.236857i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.50000	+	9.52628i	0.500000	+	0.866025i
$122$	6.60420	−	3.81294i	0.597916	−	0.345207i
$123$		0			0
$124$		0			0
$125$			3.54914i			0.317444i
$126$		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
$130$	−15.0981	−	26.1506i	−1.32419	−	2.29356i
$131$		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
$136$	−20.3923			−1.74863
$137$	20.1965	+	11.6605i	1.72550	+	0.996220i	0.906183	+	0.422885i	$0.138983\pi$
$137$	20.1965	+	11.6605i	1.72550	+	0.996220i	0.819321	+	0.573335i	$0.194351\pi$
$138$		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
$145$	3.78461			0.314295
$146$	16.1619	+	9.33109i	1.33757	+	0.772246i
$147$		0			0
$148$	−11.3923	−	19.7321i	−0.936442	−	1.62196i
$149$	13.5923	−	7.84752i	1.11353	−	0.642894i	0.173785	−	0.984784i	$-0.444400\pi$
$149$	13.5923	−	7.84752i	1.11353	−	0.642894i	0.939740	+	0.341889i	$0.111067\pi$
$150$		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
$157$	−0.303848	−	0.526279i	−0.0242497	−	0.0420017i	0.853646	−	0.520854i	$-0.174386\pi$
$157$	−0.303848	−	0.526279i	−0.0242497	−	0.0420017i	−0.877896	+	0.478852i	$0.841053\pi$
$158$		0			0
$159$		0			0
$160$	−8.39230	+	14.5359i	−0.663470	+	1.14916i
$161$		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
$169$	−19.3923	+	33.5885i	−1.49172	+	2.58373i
$170$			30.2533i			2.32032i
$171$		0			0
$172$		0			0
$173$	9.17381	+	5.29650i	0.697472	+	0.402685i	0.806405	−	0.591364i	$-0.201410\pi$
$173$	9.17381	+	5.29650i	0.697472	+	0.402685i	−0.108933	+	0.994049i	$0.534744\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	3.90192	−	6.75833i	0.292462	−	0.506558i
$179$		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
$185$	−29.2738	+	16.9012i	−2.15225	+	1.24260i
$186$		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
$193$	−6.89230	−	11.9378i	−0.496119	−	0.859303i	0.503871	−	0.863779i	$-0.331909\pi$
$193$	−6.89230	−	11.9378i	−0.496119	−	0.859303i	−0.999990	+	0.00447566i	$0.998575\pi$
$194$	−9.79796	+	5.65685i	−0.703452	+	0.406138i
$195$		0			0
$196$	7.00000	−	12.1244i	0.500000	−	0.866025i
$197$		−	27.5636i		−	1.96382i	−0.189342	−	0.981911i	$-0.560635\pi$
$197$		−	27.5636i		−	1.96382i	0.189342	−	0.981911i	$-0.439365\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	9.31749	+	5.37945i	0.658846	+	0.380385i
$201$		0			0
$202$	11.0000	+	19.0526i	0.773957	+	1.34053i
$203$		0			0
$204$		0			0
$205$	2.09808	−	3.63397i	0.146536	−	0.253808i
$206$		0			0
$207$		0			0
$208$	28.7846			1.99585
$209$		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
$218$	−5.88349	−	3.39683i	−0.398480	−	0.230063i
$219$		0			0
$220$		0			0
$221$	44.9316	−	25.9413i	3.02243	−	1.74500i
$222$		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
$226$	−26.9808			−1.79473
$227$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$227$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$228$		0			0
$229$	11.9904	+	20.7679i	0.792347	+	1.37238i	0.924510	+	0.381157i	$0.124474\pi$
$229$	11.9904	+	20.7679i	0.792347	+	1.37238i	−0.132164	+	0.991228i	$0.542192\pi$
$230$		0			0
$231$		0			0
$232$	−1.80385	+	3.12436i	−0.118428	+	0.205124i
$233$			29.2552i			1.91657i	0.285814	+	0.958285i	$0.407736\pi$
$233$			29.2552i			1.91657i	−0.285814	+	0.958285i	$0.592264\pi$
$234$		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
$241$	14.9904	−	25.9641i	0.965615	−	1.67249i	0.257663	−	0.966235i	$-0.417048\pi$
$241$	14.9904	−	25.9641i	0.965615	−	1.67249i	0.707953	−	0.706260i	$-0.249619\pi$
$242$			15.5563i			1.00000i
$243$		0			0
$244$	10.7846			0.690414
$245$	−17.9873	−	10.3849i	−1.14916	−	0.663470i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−2.50962	+	4.34679i	−0.158722	+	0.274915i
$251$		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
$257$	−5.49957	+	3.17518i	−0.343054	+	0.198062i	−0.661622	−	0.749838i	$-0.730131\pi$
$257$	−5.49957	+	3.17518i	−0.343054	+	0.198062i	0.318568	+	0.947900i	$0.396798\pi$
$258$		0			0
$259$		0			0
$260$		−	42.7038i		−	2.64838i
$261$		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
$265$	−10.4904	−	18.1699i	−0.644419	−	1.11617i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			19.9377i			1.21562i	0.794082	+	0.607811i	$0.207952\pi$
$269$			19.9377i			1.21562i	−0.794082	+	0.607811i	$0.792048\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−24.9754	−	14.4195i	−1.51435	−	0.874313i
$273$		0			0
$274$	16.4904	+	28.5622i	0.996220	+	1.72550i
$275$		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
$281$	−29.0100	−	16.7489i	−1.73059	−	0.999156i	−0.885832	−	0.464007i	$-0.846411\pi$
$281$	−29.0100	−	16.7489i	−1.73059	−	0.999156i	−0.844758	−	0.535149i	$-0.820255\pi$
$282$		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
$289$	−34.9808			−2.05769
$290$	4.63518	+	2.67612i	0.272187	+	0.157147i
$291$		0			0
$292$	13.1962	+	22.8564i	0.772246	+	1.33757i
$293$	−27.5450	+	15.9031i	−1.60919	+	0.929069i	−0.619644	+	0.784883i	$0.712723\pi$
$293$	−27.5450	+	15.9031i	−1.60919	+	0.929069i	−0.989551	+	0.144186i	$0.953944\pi$
$294$		0			0
$295$		0			0
$296$		−	32.2223i		−	1.87288i
$297$		0			0
$298$	22.1962			1.28579
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	15.9997i		−	0.916138i
$306$		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
$313$	−3.89230	+	6.74167i	−0.220006	+	0.381062i	−0.954810	−	0.297218i	$-0.903941\pi$
$313$	−3.89230	+	6.74167i	−0.220006	+	0.381062i	0.734803	+	0.678280i	$0.237274\pi$
$314$		−	0.859411i		−	0.0484994i
$315$		0			0
$316$		0			0
$317$	12.1273	+	7.00172i	0.681139	+	0.393256i	0.800284	−	0.599621i	$-0.204682\pi$
$317$	12.1273	+	7.00172i	0.681139	+	0.393256i	−0.119145	+	0.992877i	$0.538015\pi$
$318$		0			0
$319$		0			0
$320$	−20.5569	+	11.8685i	−1.14916	+	0.663470i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−27.3731			−1.51838
$326$		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	$-0.503568\pi$
$338$	−47.5013	+	27.4249i	−2.58373	+	1.49172i
$339$		0			0
$340$	−21.3923	+	37.0526i	−1.16016	+	2.00946i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	7.49038	+	12.9737i	0.402685	+	0.697472i
$347$		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	$-0.943327\pi$
$353$	−30.6186	−	17.6777i	−1.62966	−	0.940887i	−0.645473	−	0.763783i	$-0.723340\pi$
$354$		0			0
$355$		0			0
$356$	9.55772	−	5.51815i	0.506558	−	0.292462i
$357$		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
$365$	33.9089	−	19.5773i	1.77488	−	1.02472i
$366$		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
$370$	−47.8038			−2.48520
$371$		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
$377$		−	9.17878i		−	0.472731i
$378$		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
$386$		−	19.4944i		−	0.992238i
$387$		0			0
$388$	−16.0000			−0.812277
$389$	−8.57321	−	4.94975i	−0.434679	−	0.250962i	0.266659	−	0.963791i	$-0.414080\pi$
$389$	−8.57321	−	4.94975i	−0.434679	−	0.250962i	−0.701338	+	0.712829i	$0.747414\pi$
$390$		0			0
$391$		0			0
$392$	17.1464	−	9.89949i	0.866025	−	0.500000i
$393$		0			0
$394$	19.4904	−	33.7583i	0.981911	−	1.70072i
$395$		0			0
$396$		0			0
$397$	−29.3923			−1.47516			−0.737579	−	0.675261i	$-0.764031\pi$
$397$	−29.3923			−1.47516			−0.737579	+	0.675261i	$0.764031\pi$
$398$		0			0
$399$		0			0
$400$	7.60770	+	13.1769i	0.380385	+	0.658846i
$401$	10.6388	−	6.14231i	0.531276	−	0.306732i	−0.210260	−	0.977645i	$-0.567431\pi$
$401$	10.6388	−	6.14231i	0.531276	−	0.306732i	0.741536	+	0.670913i	$0.234098\pi$
$402$		0			0
$403$		0			0
$404$			31.1127i			1.54791i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−12.5981	−	21.8205i	−0.622935	−	1.07895i	−0.988936	−	0.148340i	$-0.952607\pi$
$409$	−12.5981	−	21.8205i	−0.622935	−	1.07895i	0.366002	−	0.930614i	$-0.380726\pi$
$410$	5.13922	−	2.96713i	0.253808	−	0.146536i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	35.2538	+	20.3538i	1.72846	+	0.997927i
$417$		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
$421$	5.99038	−	10.3756i	0.291953	−	0.505678i	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	5.99038	−	10.3756i	0.291953	−	0.505678i	0.974272	+	0.225377i	$0.0723615\pi$
$422$		0			0
$423$		0			0
$424$	20.0000			0.971286
$425$	23.7506	+	13.7124i	1.15207	+	0.665151i
$426$		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
$433$	37.7846			1.81581			0.907906	−	0.419173i	$-0.137680\pi$
$433$	37.7846			1.81581			0.907906	+	0.419173i	$0.137680\pi$
$434$		0			0
$435$		0			0
$436$	−4.80385	−	8.32051i	−0.230063	−	0.398480i
$437$		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
$442$	73.3731			3.49000
$443$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$443$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$444$		0			0
$445$	−8.18653	−	14.1795i	−0.388079	−	0.672172i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			18.3848i			0.867631i	0.901002	+	0.433816i	$0.142833\pi$
$449$			18.3848i			0.867631i	−0.901002	+	0.433816i	$0.857167\pi$
$450$		0			0
$451$		0			0
$452$	−33.0446	−	19.0783i	−1.55428	−	0.897367i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−16.1865	+	28.0359i	−0.757174	+	1.31146i	0.187112	+	0.982339i	$0.440087\pi$
$457$	−16.1865	+	28.0359i	−0.757174	+	1.31146i	−0.944286	+	0.329125i	$0.893246\pi$
$458$			33.9139i			1.58469i
$459$		0			0
$460$		0			0
$461$	35.5176	+	20.5061i	1.65422	+	0.955064i	0.975309	+	0.220843i	$0.0708808\pi$
$461$	35.5176	+	20.5061i	1.65422	+	0.955064i	0.678910	+	0.734221i	$0.262453\pi$
$462$		0			0
$463$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$463$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$464$	−4.41851	+	2.55103i	−0.205124	+	0.118428i
$465$		0			0
$466$	−20.6865	+	35.8301i	−0.958285	+	1.65980i
$467$		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
$481$	40.9904	+	70.9974i	1.86900	+	3.23720i
$482$	36.7188	−	21.1996i	1.67249	−	0.965615i
$483$		0			0
$484$	−11.0000	+	19.0526i	−0.500000	+	0.866025i
$485$			23.7370i			1.07784i
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$	13.2084	+	7.62587i	0.597916	+	0.345207i
$489$		0			0
$490$	−14.6865	−	25.4378i	−0.663470	−	1.14916i
$491$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$491$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$492$		0			0
$493$	−4.59808	+	7.96410i	−0.207087	+	0.358685i
$494$		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	$0.333333\pi$
$500$	−6.14729	+	3.54914i	−0.274915	+	0.158722i
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$	46.1577			2.05399
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−20.8207	+	12.0208i	−0.922860	+	0.532813i	−0.884546	−	0.466453i	$-0.845532\pi$
$509$	−20.8207	+	12.0208i	−0.922860	+	0.532813i	−0.0383134	+	0.999266i	$0.512199\pi$
$510$		0			0
$511$		0			0
$512$		−	22.6274i		−	1.00000i
$513$		0			0
$514$	−8.98076			−0.396124
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	30.1962	−	52.3013i	1.32419	−	2.29356i
$521$			43.8406i			1.92069i	0.278810	+	0.960346i	$0.410060\pi$
$521$			43.8406i			1.92069i	−0.278810	+	0.960346i	$0.589940\pi$
$522$		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
$530$		−	29.6713i		−	1.28884i
$531$		0			0
$532$		0			0
$533$	−8.81345	−	5.08845i	−0.381753	−	0.220405i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−14.0981	+	24.4186i	−0.607811	+	1.05276i
$539$		0			0
$540$		0			0
$541$	26.3731			1.13387			0.566933	−	0.823764i	$-0.308130\pi$
$541$	26.3731			1.13387			0.566933	+	0.823764i	$0.308130\pi$
$542$		0			0
$543$		0			0
$544$	−20.3923	−	35.3205i	−0.874313	−	1.51435i
$545$	−12.3440	+	7.12681i	−0.528759	+	0.305279i
$546$		0			0
$547$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$547$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$548$			46.6418i			1.99244i
$549$		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
$557$		−	36.8810i		−	1.56270i	−0.624093	−	0.781350i	$-0.714531\pi$
$557$		−	36.8810i		−	1.56270i	0.624093	−	0.781350i	$-0.285469\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−23.6865	−	41.0263i	−0.999156	−	1.73059i
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$	−28.3038	+	49.0237i	−1.19075	+	2.06244i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	39.2884	+	22.6832i	1.64706	+	0.950928i	0.978234	+	0.207504i	$0.0665341\pi$
$569$	39.2884	+	22.6832i	1.64706	+	0.950928i	0.668821	+	0.743423i	$0.266799\pi$
$570$		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
$577$	−42.5692			−1.77218			−0.886090	−	0.463513i	$-0.846589\pi$
$577$	−42.5692			−1.77218			−0.886090	+	0.463513i	$0.846589\pi$
$578$	−42.8425	−	24.7351i	−1.78201	−	1.02885i
$579$		0			0
$580$	3.78461	+	6.55514i	0.157147	+	0.272187i
$581$		0			0
$582$		0			0
$583$		0			0
$584$			37.3244i			1.54449i
$585$		0			0
$586$	−44.9808			−1.85814
$587$		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
$592$	22.7846	−	39.4641i	0.936442	−	1.62196i
$593$		−	40.2915i		−	1.65457i	−0.561780	−	0.827286i	$-0.689883\pi$
$593$		−	40.2915i		−	1.65457i	0.561780	−	0.827286i	$-0.310117\pi$
$594$		0			0
$595$		0			0
$596$	27.1846	+	15.6950i	1.11353	+	0.642894i
$597$		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
$601$	18.2846	−	31.6699i	0.745845	−	1.29184i	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	18.2846	−	31.6699i	0.745845	−	1.29184i	0.949799	−	0.312861i	$-0.101287\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	28.2657	+	16.3192i	1.14916	+	0.663470i
$606$		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
$610$	11.3135	−	19.5955i	0.458069	−	0.793399i
$611$		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
$617$	24.6150	−	14.2115i	0.990963	−	0.572133i	0.0854011	−	0.996347i	$-0.472783\pi$
$617$	24.6150	−	14.2115i	0.990963	−	0.572133i	0.905562	+	0.424214i	$0.139449\pi$
$618$		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
$625$	14.7750	+	25.5910i	0.591000	+	1.02364i
$626$	−9.53416	+	5.50455i	−0.381062	+	0.220006i
$627$		0			0
$628$	0.607695	−	1.05256i	0.0242497	−	0.0420017i
$629$		−	82.1359i		−	3.27497i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$	9.90192	+	17.1506i	0.393256	+	0.681139i
$635$		0			0
$636$		0			0
$637$	−25.1865	+	43.6244i	−0.997927	+	1.72846i
$638$		0			0
$639$		0			0
$640$	−33.5692			−1.32694
$641$	−40.0327	−	23.1129i	−1.58120	−	0.912903i	−0.994685	−	0.102961i	$-0.967168\pi$
$641$	−40.0327	−	23.1129i	−1.58120	−	0.912903i	−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
$650$	−33.5250	−	19.3557i	−1.31496	−	0.759192i
$651$		0			0
$652$		0			0
$653$	−42.8661	+	24.7487i	−1.67748	+	0.968493i	−0.714219	+	0.699922i	$0.753218\pi$
$653$	−42.8661	+	24.7487i	−1.67748	+	0.968493i	−0.963260	+	0.268571i	$0.913449\pi$
$654$		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
$661$	17.6962	+	30.6506i	0.688301	+	1.19217i	0.972387	+	0.233373i	$0.0749763\pi$
$661$	17.6962	+	30.6506i	0.688301	+	1.19217i	−0.284087	+	0.958799i	$0.591690\pi$
$662$		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
$673$	−21.8923	+	37.9186i	−0.843886	+	1.46165i	0.0426985	+	0.999088i	$0.486405\pi$
$673$	−21.8923	+	37.9186i	−0.843886	+	1.46165i	−0.886585	+	0.462566i	$0.846929\pi$
$674$		−	45.2548i		−	1.74315i
$675$		0			0
$676$	−77.5692			−2.98343
$677$	−30.6186	−	17.6777i	−1.17677	−	0.679408i	−0.221504	−	0.975159i	$-0.571097\pi$
$677$	−30.6186	−	17.6777i	−1.17677	−	0.679408i	−0.955265	+	0.295751i	$0.904430\pi$
$678$		0			0
$679$		0			0
$680$	−52.4002	+	30.2533i	−2.00946	+	1.16016i
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$	69.1962			2.64385
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−44.0673	+	25.4422i	−1.67883	+	0.969272i
$690$		0			0
$691$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$691$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$692$			21.1860i			0.805371i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	5.09808	+	8.83013i	0.193103	+	0.334465i
$698$	12.2474	−	7.07107i	0.463573	−	0.267644i
$699$		0			0
$700$		0			0
$701$			3.79933i			0.143499i	0.997423	+	0.0717494i	$0.0228582\pi$
$701$			3.79933i			0.143499i	−0.997423	+	0.0717494i	$0.977142\pi$
$702$		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
$709$	23.9904	−	41.5526i	0.900978	−	1.56054i	0.0747503	−	0.997202i	$-0.476184\pi$
$709$	23.9904	−	41.5526i	0.900978	−	1.56054i	0.826227	−	0.563337i	$-0.190483\pi$
$710$		0			0
$711$		0			0
$712$	15.6077			0.584923
$713$		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
$725$	4.20183	−	2.42593i	0.156052	−	0.0900967i
$726$		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
$730$	55.3731			2.04945
$731$		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			$\pi$
$740$	−58.5475	−	33.8024i	−2.15225	−	1.24260i
$741$		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
$745$	23.2846	−	40.3301i	0.853082	−	1.47758i
$746$		−	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
$754$	6.49038	−	11.2417i	0.236366	−	0.409397i
$755$		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
$761$	40.7534	−	23.5290i	1.47731	−	0.852925i	0.477637	−	0.878557i	$-0.341493\pi$
$761$	40.7534	−	23.5290i	1.47731	−	0.852925i	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
$769$	2.10770	+	3.65064i	0.0760054	+	0.131645i	0.901523	−	0.432731i	$-0.142450\pi$
$769$	2.10770	+	3.65064i	0.0760054	+	0.131645i	−0.825518	+	0.564376i	$0.809117\pi$
$770$		0			0
$771$		0			0
$772$	13.7846	−	23.8756i	0.496119	−	0.859303i
$773$			51.3006i			1.84515i	0.385813	+	0.922577i	$0.373921\pi$
$773$			51.3006i			1.84515i	−0.385813	+	0.922577i	$0.626079\pi$
$774$		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
$785$	−1.56154	−	0.901555i	−0.0557337	−	0.0321779i
$786$		0			0
$787$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$787$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$788$	47.7415	−	27.5636i	1.70072	−	0.981911i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−38.8038			−1.37797
$794$	−35.9981	−	20.7835i	−1.27752	−	0.737579i
$795$		0			0
$796$		0			0
$797$	−38.5677	+	22.2671i	−1.36614	+	0.788740i	−0.990432	−	0.137999i	$-0.955933\pi$
$797$	−38.5677	+	22.2671i	−1.36614	+	0.788740i	−0.375705	+	0.926739i	$0.622599\pi$
$798$		0			0
$799$		0			0
$800$			21.5178i			0.760770i
$801$		0			0
$802$	17.3731			0.613464
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−22.0000	+	38.1051i	−0.773957	+	1.34053i
$809$		−	24.1531i		−	0.849179i	−0.905386	−	0.424589i	$-0.860418\pi$
$809$		−	24.1531i		−	0.849179i	0.905386	−	0.424589i	$-0.139582\pi$
$810$		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
$818$		−	35.6327i		−	1.24587i
$819$		0			0
$820$	8.39230			0.293072
$821$	−48.1018	−	27.7716i	−1.67877	−	0.969236i	−0.962450	−	0.271460i	$-0.912493\pi$
$821$	−48.1018	−	27.7716i	−1.67877	−	0.969236i	−0.716316	−	0.697776i	$-0.754173\pi$
$822$		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
$832$	28.7846	+	49.8564i	0.997927	+	1.72846i
$833$	43.7069	−	25.2342i	1.51435	−	0.874313i
$834$		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
$841$	−13.6865	−	23.7058i	−0.471949	−	0.817440i
$842$	14.6734	−	8.47168i	0.505678	−	0.291953i
$843$		0			0
$844$		0			0
$845$			115.079i			3.95883i
$846$		0			0
$847$		0			0
$848$	24.4949	+	14.1421i	0.841158	+	0.485643i
$849$		0			0
$850$	19.3923	+	33.5885i	0.665151	+	1.15207i
$851$		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
$857$	50.3111	+	29.0471i	1.71859	+	0.992231i	0.921502	+	0.388373i	$0.126963\pi$
$857$	50.3111	+	29.0471i	1.71859	+	0.992231i	0.797092	+	0.603858i	$0.206370\pi$
$858$		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
$865$	31.4308			1.06868
$866$	46.2765	+	26.7178i	1.57254	+	0.907906i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		−	13.5873i		−	0.460125i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−9.30385	−	16.1147i	−0.314169	−	0.544156i	0.665092	−	0.746762i	$-0.268392\pi$
$877$	−9.30385	−	16.1147i	−0.314169	−	0.544156i	−0.979260	+	0.202606i	$0.935059\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	57.9828i		−	1.95349i	−0.214407	−	0.976744i	$-0.568782\pi$
$881$		−	57.9828i		−	1.95349i	0.214407	−	0.976744i	$-0.431218\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$	89.8633	+	51.8826i	3.02243	+	1.74500i
$885$		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
$890$		−	23.1550i		−	0.776158i
$891$		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
$901$	50.9808			1.69842
$902$		0			0
$903$		0			0
$904$	−26.9808	−	46.7321i	−0.897367	−	1.55428i
$905$	51.3922	−	29.6713i	1.70833	−	0.986307i
$906$		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
$914$	−39.6487	+	22.8912i	−1.31146	+	0.757174i
$915$		0			0
$916$	−23.9808	+	41.5359i	−0.792347	+	1.37238i
$917$		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
$925$	−21.6673	+	37.5289i	−0.712416	+	1.23394i
$926$		0			0
$927$		0			0
$928$	−7.21539			−0.236857
$929$	23.1500	+	13.3657i	0.759528	+	0.438514i	0.829126	−	0.559061i	$-0.188838\pi$
$929$	23.1500	+	13.3657i	0.759528	+	0.438514i	−0.0695983	+	0.997575i	$0.522172\pi$
$930$		0			0
$931$		0			0
$932$	−50.6715	+	29.2552i	−1.65980	+	0.958285i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−60.5692			−1.97871			−0.989355	−	0.145522i	$-0.953514\pi$
$937$	−60.5692			−1.97871			−0.989355	+	0.145522i	$0.953514\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	29.7307	−	17.1650i	0.969192	−	0.559563i	0.0702023	−	0.997533i	$-0.477636\pi$
$941$	29.7307	−	17.1650i	0.969192	−	0.559563i	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
$949$	−47.4808	−	82.2391i	−1.54129	−	2.66959i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			10.6202i			0.344022i	0.985095	+	0.172011i	$0.0550265\pi$
$953$			10.6202i			0.344022i	−0.985095	+	0.172011i	$0.944974\pi$
$954$		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
$962$			115.938i			3.73800i
$963$		0			0
$964$	59.9615			1.93123
$965$	−35.4210	−	20.4503i	−1.14024	−	0.658320i
$966$		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
$970$	−16.7846	+	29.0718i	−0.538921	+	0.933439i
$971$		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
$976$	10.7846	+	18.6795i	0.345207	+	0.597916i
$977$	−42.8661	+	24.7487i	−1.37141	+	0.791782i	−0.991105	−	0.133080i	$-0.957513\pi$
$977$	−42.8661	+	24.7487i	−1.37141	+	0.791782i	−0.380302	+	0.924862i	$0.624180\pi$
$978$		0			0
$979$		0			0
$980$		−	41.5398i		−	1.32694i
$981$		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
$985$	−40.8923	−	70.8275i	−1.30294	−	2.25675i
$986$	−11.2629	+	6.50266i	−0.358685	+	0.207087i
$987$		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
$997$	20.6962	−	35.8468i	0.655454	−	1.13528i	−0.326326	−	0.945257i	$-0.605811\pi$
$997$	20.6962	−	35.8468i	0.655454	−	1.13528i	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.107.4		8
3.2	odd	2	inner	324.2.h.e.107.1		8
4.3	odd	2	CM	324.2.h.e.107.4		8
9.2	odd	6		324.2.b.a.323.3	yes	4
9.4	even	3	inner	324.2.h.e.215.1		8
9.5	odd	6	inner	324.2.h.e.215.4		8
9.7	even	3		324.2.b.a.323.2	✓	4
12.11	even	2	inner	324.2.h.e.107.1		8
36.7	odd	6		324.2.b.a.323.2	✓	4
36.11	even	6		324.2.b.a.323.3	yes	4
36.23	even	6	inner	324.2.h.e.215.4		8
36.31	odd	6	inner	324.2.h.e.215.1		8
72.11	even	6		5184.2.c.e.5183.3		4
72.29	odd	6		5184.2.c.e.5183.3		4
72.43	odd	6		5184.2.c.e.5183.2		4
72.61	even	6		5184.2.c.e.5183.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.2.b.a.323.2	✓	4	9.7	even	3
324.2.b.a.323.2	✓	4	36.7	odd	6
324.2.b.a.323.3	yes	4	9.2	odd	6
324.2.b.a.323.3	yes	4	36.11	even	6
324.2.h.e.107.1		8	3.2	odd	2	inner
324.2.h.e.107.1		8	12.11	even	2	inner
324.2.h.e.107.4		8	1.1	even	1	trivial
324.2.h.e.107.4		8	4.3	odd	2	CM
324.2.h.e.215.1		8	9.4	even	3	inner
324.2.h.e.215.1		8	36.31	odd	6	inner
324.2.h.e.215.4		8	9.5	odd	6	inner
324.2.h.e.215.4		8	36.23	even	6	inner
5184.2.c.e.5183.2		4	72.43	odd	6
5184.2.c.e.5183.2		4	72.61	even	6
5184.2.c.e.5183.3		4	72.11	even	6
5184.2.c.e.5183.3		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)

\(f(q)\)	\(=\)	\(q+(1.22474 + 0.707107i) q^{2} +(1.00000 + 1.73205i) q^{4} +(2.56961 - 1.48356i) q^{5} +2.82843i q^{8} +O(q^{10})\)	\(q+(1.22474 + 0.707107i) q^{2} +(1.00000 + 1.73205i) q^{4} +(2.56961 - 1.48356i) q^{5} +2.82843i q^{8} +4.19615 q^{10} +(-3.59808 - 6.23205i) q^{13} +(-2.00000 + 3.46410i) q^{16} +7.20977i q^{17} +(5.13922 + 2.96713i) q^{20} +(1.90192 - 3.29423i) q^{25} -10.1769i q^{26} +(1.10463 + 0.637756i) q^{29} +(-4.89898 + 2.82843i) q^{32} +(-5.09808 + 8.83013i) q^{34} -11.3923 q^{37} +(4.19615 + 7.26795i) q^{40} +(1.22474 - 0.707107i) q^{41} +(-3.50000 - 6.06218i) q^{49} +(4.65874 - 2.68973i) q^{50} +(7.19615 - 12.4641i) q^{52} -7.07107i q^{53} +(0.901924 + 1.56218i) q^{58} +(2.69615 - 4.66987i) q^{61} -8.00000 q^{64} +(-18.4913 - 10.6760i) q^{65} +(-12.4877 + 7.20977i) q^{68} +13.1962 q^{73} +(-13.9527 - 8.05558i) q^{74} +11.8685i q^{80} +2.00000 q^{82} +(10.6962 + 18.5263i) q^{85} -5.51815i q^{89} +(-4.00000 + 6.92820i) q^{97} -9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{4}+O(q^{10}) \) 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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