LMFDB - Embedded newform 3240.2.q.w.2161.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3240,2,Mod(1081,3240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3240, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))

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

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

chi := DirichletCharacter("3240.1081");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3240 = 2^{3} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3240.q (of order $3$, 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:	$25.8715302549$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1080)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			2161.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	3240.2161
Dual form			3240.2.q.w.1081.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{5} +(2.00000 - 3.46410i) q^{7} +O(q^{10})$	$q+(0.500000 + 0.866025i) q^{5} +(2.00000 - 3.46410i) q^{7} +(-1.00000 + 1.73205i) q^{11} +(-2.00000 - 3.46410i) q^{13} +1.00000 q^{17} -5.00000 q^{19} +(-2.50000 - 4.33013i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(-4.00000 + 6.92820i) q^{29} +(-3.50000 - 6.06218i) q^{31} +4.00000 q^{35} -6.00000 q^{37} +(-3.00000 - 5.19615i) q^{41} +(1.00000 - 1.73205i) q^{43} +(-4.00000 + 6.92820i) q^{47} +(-4.50000 - 7.79423i) q^{49} +9.00000 q^{53} -2.00000 q^{55} +(-2.00000 - 3.46410i) q^{59} +(-6.50000 + 11.2583i) q^{61} +(2.00000 - 3.46410i) q^{65} +(5.00000 + 8.66025i) q^{67} -6.00000 q^{71} -6.00000 q^{73} +(4.00000 + 6.92820i) q^{77} +(-4.50000 + 7.79423i) q^{79} +(8.50000 - 14.7224i) q^{83} +(0.500000 + 0.866025i) q^{85} -6.00000 q^{89} -16.0000 q^{91} +(-2.50000 - 4.33013i) q^{95} +(4.00000 - 6.92820i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{5} + 4 q^{7}+O(q^{10}) $ 2 * q + q^5 + 4 * q^7	$ 2 q + q^{5} + 4 q^{7} - 2 q^{11} - 4 q^{13} + 2 q^{17} - 10 q^{19} - 5 q^{23} - q^{25} - 8 q^{29} - 7 q^{31} + 8 q^{35} - 12 q^{37} - 6 q^{41} + 2 q^{43} - 8 q^{47} - 9 q^{49} + 18 q^{53} - 4 q^{55} - 4 q^{59} - 13 q^{61} + 4 q^{65} + 10 q^{67} - 12 q^{71} - 12 q^{73} + 8 q^{77} - 9 q^{79} + 17 q^{83} + q^{85} - 12 q^{89} - 32 q^{91} - 5 q^{95} + 8 q^{97}+O(q^{100}) $ 2 * q + q^5 + 4 * q^7 - 2 * q^11 - 4 * q^13 + 2 * q^17 - 10 * q^19 - 5 * q^23 - q^25 - 8 * q^29 - 7 * q^31 + 8 * q^35 - 12 * q^37 - 6 * q^41 + 2 * q^43 - 8 * q^47 - 9 * q^49 + 18 * q^53 - 4 * q^55 - 4 * q^59 - 13 * q^61 + 4 * q^65 + 10 * q^67 - 12 * q^71 - 12 * q^73 + 8 * q^77 - 9 * q^79 + 17 * q^83 + q^85 - 12 * q^89 - 32 * q^91 - 5 * q^95 + 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times$.

$n$	$1297$	$1621$	$2431$	$3161$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i
$6$		0			0
$7$	2.00000	−	3.46410i	0.755929	−	1.30931i	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	2.00000	−	3.46410i	0.755929	−	1.30931i	0.944911	−	0.327327i	$-0.106148\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	−0.976478	−	0.215615i	$-0.930824\pi$
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	0.674967	+	0.737848i	$0.264158\pi$
$12$		0			0
$13$	−2.00000	−	3.46410i	−0.554700	−	0.960769i	−0.997927	−	0.0643593i	$-0.979500\pi$
$13$	−2.00000	−	3.46410i	−0.554700	−	0.960769i	0.443227	−	0.896410i	$-0.353834\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.00000			0.242536			0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000			0.242536			0.121268	+	0.992620i	$0.461304\pi$
$18$		0			0
$19$	−5.00000			−1.14708			−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000			−1.14708			−0.573539	+	0.819178i	$0.694430\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.50000	−	4.33013i	−0.521286	−	0.902894i	−0.999694	−	0.0247559i	$-0.992119\pi$
$23$	−2.50000	−	4.33013i	−0.521286	−	0.902894i	0.478407	−	0.878138i	$-0.341214\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.00000	+	6.92820i	−0.742781	+	1.28654i	0.208443	+	0.978035i	$0.433160\pi$
$29$	−4.00000	+	6.92820i	−0.742781	+	1.28654i	−0.951224	+	0.308500i	$0.900173\pi$
$30$		0			0
$31$	−3.50000	−	6.06218i	−0.628619	−	1.08880i	−0.987829	−	0.155543i	$-0.950287\pi$
$31$	−3.50000	−	6.06218i	−0.628619	−	1.08880i	0.359211	−	0.933257i	$-0.383046\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.00000			0.676123
$36$		0			0
$37$	−6.00000			−0.986394			−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000			−0.986394			−0.493197	+	0.869918i	$0.664172\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.00000	−	5.19615i	−0.468521	−	0.811503i	0.530831	−	0.847477i	$-0.321880\pi$
$41$	−3.00000	−	5.19615i	−0.468521	−	0.811503i	−0.999353	+	0.0359748i	$0.988546\pi$
$42$		0			0
$43$	1.00000	−	1.73205i	0.152499	−	0.264135i	−0.779647	−	0.626219i	$-0.784601\pi$
$43$	1.00000	−	1.73205i	0.152499	−	0.264135i	0.932145	+	0.362084i	$0.117935\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.00000	+	6.92820i	−0.583460	+	1.01058i	0.411606	+	0.911362i	$0.364968\pi$
$47$	−4.00000	+	6.92820i	−0.583460	+	1.01058i	−0.995066	+	0.0992202i	$0.968365\pi$
$48$		0			0
$49$	−4.50000	−	7.79423i	−0.642857	−	1.11346i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	9.00000			1.23625			0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000			1.23625			0.618123	+	0.786082i	$0.287894\pi$
$54$		0			0
$55$	−2.00000			−0.269680
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2.00000	−	3.46410i	−0.260378	−	0.450988i	0.705965	−	0.708247i	$-0.250514\pi$
$59$	−2.00000	−	3.46410i	−0.260378	−	0.450988i	−0.966342	+	0.257260i	$0.917180\pi$
$60$		0			0
$61$	−6.50000	+	11.2583i	−0.832240	+	1.44148i	0.0640184	+	0.997949i	$0.479608\pi$
$61$	−6.50000	+	11.2583i	−0.832240	+	1.44148i	−0.896258	+	0.443533i	$0.853725\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.00000	−	3.46410i	0.248069	−	0.429669i
$66$		0			0
$67$	5.00000	+	8.66025i	0.610847	+	1.05802i	0.991098	+	0.133135i	$0.0425044\pi$
$67$	5.00000	+	8.66025i	0.610847	+	1.05802i	−0.380251	+	0.924883i	$0.624162\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−6.00000			−0.712069			−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000			−0.712069			−0.356034	+	0.934473i	$0.615871\pi$
$72$		0			0
$73$	−6.00000			−0.702247			−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000			−0.702247			−0.351123	+	0.936329i	$0.614200\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	4.00000	+	6.92820i	0.455842	+	0.789542i
$78$		0			0
$79$	−4.50000	+	7.79423i	−0.506290	+	0.876919i	0.493684	+	0.869641i	$0.335650\pi$
$79$	−4.50000	+	7.79423i	−0.506290	+	0.876919i	−0.999974	+	0.00727784i	$0.997683\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	8.50000	−	14.7224i	0.932996	−	1.61600i	0.154828	−	0.987942i	$-0.450518\pi$
$83$	8.50000	−	14.7224i	0.932996	−	1.61600i	0.778169	−	0.628055i	$-0.216149\pi$
$84$		0			0
$85$	0.500000	+	0.866025i	0.0542326	+	0.0939336i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.00000			−0.635999			−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000			−0.635999			−0.317999	+	0.948091i	$0.603011\pi$
$90$		0			0
$91$	−16.0000			−1.67726
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.50000	−	4.33013i	−0.256495	−	0.444262i
$96$		0			0
$97$	4.00000	−	6.92820i	0.406138	−	0.703452i	−0.588315	−	0.808632i	$-0.700208\pi$
$97$	4.00000	−	6.92820i	0.406138	−	0.703452i	0.994453	+	0.105180i	$0.0335417\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6.00000	+	10.3923i	−0.597022	+	1.03407i	0.396236	+	0.918149i	$0.370316\pi$
$101$	−6.00000	+	10.3923i	−0.597022	+	1.03407i	−0.993258	+	0.115924i	$0.963017\pi$
$102$		0			0
$103$	−2.00000	−	3.46410i	−0.197066	−	0.341328i	0.750510	−	0.660859i	$-0.229808\pi$
$103$	−2.00000	−	3.46410i	−0.197066	−	0.341328i	−0.947576	+	0.319531i	$0.896475\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	12.0000			1.16008			0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000			1.16008			0.580042	+	0.814587i	$0.303036\pi$
$108$		0			0
$109$	17.0000			1.62830			0.814152	−	0.580651i	$-0.197202\pi$
$109$	17.0000			1.62830			0.814152	+	0.580651i	$0.197202\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−3.00000	−	5.19615i	−0.282216	−	0.488813i	0.689714	−	0.724082i	$-0.257736\pi$
$113$	−3.00000	−	5.19615i	−0.282216	−	0.488813i	−0.971930	+	0.235269i	$0.924403\pi$
$114$		0			0
$115$	2.50000	−	4.33013i	0.233126	−	0.403786i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.00000	−	3.46410i	0.183340	−	0.317554i
$120$		0			0
$121$	3.50000	+	6.06218i	0.318182	+	0.551107i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−18.0000			−1.59724			−0.798621	−	0.601834i	$-0.794437\pi$
$127$	−18.0000			−1.59724			−0.798621	+	0.601834i	$0.794437\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.00000	−	5.19615i	−0.262111	−	0.453990i	0.704692	−	0.709514i	$-0.251085\pi$
$131$	−3.00000	−	5.19615i	−0.262111	−	0.453990i	−0.966803	+	0.255524i	$0.917752\pi$
$132$		0			0
$133$	−10.0000	+	17.3205i	−0.867110	+	1.50188i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−5.50000	+	9.52628i	−0.469897	+	0.813885i	−0.999408	−	0.0344182i	$-0.989042\pi$
$137$	−5.50000	+	9.52628i	−0.469897	+	0.813885i	0.529511	+	0.848303i	$0.322376\pi$
$138$		0			0
$139$	−6.00000	−	10.3923i	−0.508913	−	0.881464i	−0.999947	−	0.0103230i	$-0.996714\pi$
$139$	−6.00000	−	10.3923i	−0.508913	−	0.881464i	0.491033	−	0.871141i	$-0.336619\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	8.00000			0.668994
$144$		0			0
$145$	−8.00000			−0.664364
$146$		0			0
$147$		0			0
$148$		0			0
$149$	7.00000	+	12.1244i	0.573462	+	0.993266i	0.996207	+	0.0870170i	$0.0277334\pi$
$149$	7.00000	+	12.1244i	0.573462	+	0.993266i	−0.422744	+	0.906249i	$0.638933\pi$
$150$		0			0
$151$	8.00000	−	13.8564i	0.651031	−	1.12762i	−0.331842	−	0.943335i	$-0.607670\pi$
$151$	8.00000	−	13.8564i	0.651031	−	1.12762i	0.982873	−	0.184284i	$-0.0589965\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	3.50000	−	6.06218i	0.281127	−	0.486926i
$156$		0			0
$157$	−11.0000	−	19.0526i	−0.877896	−	1.52056i	−0.853646	−	0.520854i	$-0.825614\pi$
$157$	−11.0000	−	19.0526i	−0.877896	−	1.52056i	−0.0242497	−	0.999706i	$-0.507720\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−20.0000			−1.57622
$162$		0			0
$163$	−2.00000			−0.156652			−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000			−0.156652			−0.0783260	+	0.996928i	$0.524958\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	7.50000	+	12.9904i	0.580367	+	1.00523i	0.995436	+	0.0954356i	$0.0304244\pi$
$167$	7.50000	+	12.9904i	0.580367	+	1.00523i	−0.415068	+	0.909790i	$0.636242\pi$
$168$		0			0
$169$	−1.50000	+	2.59808i	−0.115385	+	0.199852i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−3.50000	+	6.06218i	−0.266100	+	0.460899i	−0.967851	−	0.251523i	$-0.919068\pi$
$173$	−3.50000	+	6.06218i	−0.266100	+	0.460899i	0.701751	+	0.712422i	$0.252402\pi$
$174$		0			0
$175$	2.00000	+	3.46410i	0.151186	+	0.261861i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	16.0000			1.19590			0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000			1.19590			0.597948	+	0.801535i	$0.295983\pi$
$180$		0			0
$181$	−7.00000			−0.520306			−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000			−0.520306			−0.260153	+	0.965567i	$0.583773\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.00000	−	5.19615i	−0.220564	−	0.382029i
$186$		0			0
$187$	−1.00000	+	1.73205i	−0.0731272	+	0.126660i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	13.0000	−	22.5167i	0.940647	−	1.62925i	0.176406	−	0.984317i	$-0.443553\pi$
$191$	13.0000	−	22.5167i	0.940647	−	1.62925i	0.764241	−	0.644931i	$-0.223114\pi$
$192$		0			0
$193$	−8.00000	−	13.8564i	−0.575853	−	0.997406i	−0.995948	−	0.0899262i	$-0.971337\pi$
$193$	−8.00000	−	13.8564i	−0.575853	−	0.997406i	0.420096	−	0.907480i	$-0.361996\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−25.0000			−1.78118			−0.890588	−	0.454811i	$-0.849707\pi$
$197$	−25.0000			−1.78118			−0.890588	+	0.454811i	$0.849707\pi$
$198$		0			0
$199$	−4.00000			−0.283552			−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000			−0.283552			−0.141776	+	0.989899i	$0.545281\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	16.0000	+	27.7128i	1.12298	+	1.94506i
$204$		0			0
$205$	3.00000	−	5.19615i	0.209529	−	0.362915i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	5.00000	−	8.66025i	0.345857	−	0.599042i
$210$		0			0
$211$	−8.50000	−	14.7224i	−0.585164	−	1.01353i	−0.994855	−	0.101310i	$-0.967697\pi$
$211$	−8.50000	−	14.7224i	−0.585164	−	1.01353i	0.409691	−	0.912224i	$-0.365637\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2.00000			0.136399
$216$		0			0
$217$	−28.0000			−1.90076
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2.00000	−	3.46410i	−0.134535	−	0.233021i
$222$		0			0
$223$	−1.00000	+	1.73205i	−0.0669650	+	0.115987i	−0.897564	−	0.440884i	$-0.854665\pi$
$223$	−1.00000	+	1.73205i	−0.0669650	+	0.115987i	0.830599	+	0.556871i	$0.187998\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	4.50000	−	7.79423i	0.298675	−	0.517321i	−0.677158	−	0.735838i	$-0.736789\pi$
$227$	4.50000	−	7.79423i	0.298675	−	0.517321i	0.975833	+	0.218517i	$0.0701218\pi$
$228$		0			0
$229$	−9.50000	−	16.4545i	−0.627778	−	1.08734i	−0.987997	−	0.154475i	$-0.950631\pi$
$229$	−9.50000	−	16.4545i	−0.627778	−	1.08734i	0.360219	−	0.932868i	$-0.382702\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−6.00000			−0.393073			−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000			−0.393073			−0.196537	+	0.980497i	$0.562969\pi$
$234$		0			0
$235$	−8.00000			−0.521862
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−3.00000	−	5.19615i	−0.194054	−	0.336111i	0.752536	−	0.658551i	$-0.228830\pi$
$239$	−3.00000	−	5.19615i	−0.194054	−	0.336111i	−0.946590	+	0.322440i	$0.895497\pi$
$240$		0			0
$241$	0.500000	−	0.866025i	0.0322078	−	0.0557856i	−0.849472	−	0.527633i	$-0.823079\pi$
$241$	0.500000	−	0.866025i	0.0322078	−	0.0557856i	0.881680	+	0.471848i	$0.156413\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	4.50000	−	7.79423i	0.287494	−	0.497955i
$246$		0			0
$247$	10.0000	+	17.3205i	0.636285	+	1.10208i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$		0			0
$253$	10.0000			0.628695
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.50000	−	2.59808i	−0.0935674	−	0.162064i	0.815442	−	0.578838i	$-0.196494\pi$
$257$	−1.50000	−	2.59808i	−0.0935674	−	0.162064i	−0.909010	+	0.416775i	$0.863160\pi$
$258$		0			0
$259$	−12.0000	+	20.7846i	−0.745644	+	1.29149i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	6.00000	−	10.3923i	0.369976	−	0.640817i	−0.619586	−	0.784929i	$-0.712699\pi$
$263$	6.00000	−	10.3923i	0.369976	−	0.640817i	0.989561	+	0.144112i	$0.0460326\pi$
$264$		0			0
$265$	4.50000	+	7.79423i	0.276433	+	0.478796i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−26.0000			−1.58525			−0.792624	−	0.609711i	$-0.791286\pi$
$269$	−26.0000			−1.58525			−0.792624	+	0.609711i	$0.791286\pi$
$270$		0			0
$271$	15.0000			0.911185			0.455593	−	0.890188i	$-0.349427\pi$
$271$	15.0000			0.911185			0.455593	+	0.890188i	$0.349427\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.00000	−	1.73205i	−0.0603023	−	0.104447i
$276$		0			0
$277$	11.0000	−	19.0526i	0.660926	−	1.14476i	−0.319447	−	0.947604i	$-0.603497\pi$
$277$	11.0000	−	19.0526i	0.660926	−	1.14476i	0.980373	−	0.197153i	$-0.0631696\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	8.00000	−	13.8564i	0.477240	−	0.826604i	−0.522420	−	0.852688i	$-0.674971\pi$
$281$	8.00000	−	13.8564i	0.477240	−	0.826604i	0.999660	+	0.0260845i	$0.00830391\pi$
$282$		0			0
$283$	−7.00000	−	12.1244i	−0.416107	−	0.720718i	0.579437	−	0.815017i	$-0.303272\pi$
$283$	−7.00000	−	12.1244i	−0.416107	−	0.720718i	−0.995544	+	0.0942988i	$0.969939\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−24.0000			−1.41668
$288$		0			0
$289$	−16.0000			−0.941176
$290$		0			0
$291$		0			0
$292$		0			0
$293$	0.500000	+	0.866025i	0.0292103	+	0.0505937i	0.880261	−	0.474490i	$-0.157367\pi$
$293$	0.500000	+	0.866025i	0.0292103	+	0.0505937i	−0.851051	+	0.525084i	$0.824034\pi$
$294$		0			0
$295$	2.00000	−	3.46410i	0.116445	−	0.201688i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−10.0000	+	17.3205i	−0.578315	+	1.00167i
$300$		0			0
$301$	−4.00000	−	6.92820i	−0.230556	−	0.399335i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−13.0000			−0.744378
$306$		0			0
$307$	10.0000			0.570730			0.285365	−	0.958419i	$-0.407885\pi$
$307$	10.0000			0.570730			0.285365	+	0.958419i	$0.407885\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−8.00000	−	13.8564i	−0.453638	−	0.785725i	0.544970	−	0.838455i	$-0.316541\pi$
$311$	−8.00000	−	13.8564i	−0.453638	−	0.785725i	−0.998609	+	0.0527306i	$0.983208\pi$
$312$		0			0
$313$	14.0000	−	24.2487i	0.791327	−	1.37062i	−0.133819	−	0.991006i	$-0.542724\pi$
$313$	14.0000	−	24.2487i	0.791327	−	1.37062i	0.925146	−	0.379612i	$-0.123943\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−16.5000	+	28.5788i	−0.926732	+	1.60515i	−0.137981	+	0.990435i	$0.544061\pi$
$317$	−16.5000	+	28.5788i	−0.926732	+	1.60515i	−0.788751	+	0.614713i	$0.789272\pi$
$318$		0			0
$319$	−8.00000	−	13.8564i	−0.447914	−	0.775810i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−5.00000			−0.278207
$324$		0			0
$325$	4.00000			0.221880
$326$		0			0
$327$		0			0
$328$		0			0
$329$	16.0000	+	27.7128i	0.882109	+	1.52786i
$330$		0			0
$331$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$331$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−5.00000	+	8.66025i	−0.273179	+	0.473160i
$336$		0			0
$337$	10.0000	+	17.3205i	0.544735	+	0.943508i	0.998624	+	0.0524499i	$0.0167030\pi$
$337$	10.0000	+	17.3205i	0.544735	+	0.943508i	−0.453889	+	0.891058i	$0.649964\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	14.0000			0.758143
$342$		0			0
$343$	−8.00000			−0.431959
$344$		0			0
$345$		0			0
$346$		0			0
$347$	10.0000	+	17.3205i	0.536828	+	0.929814i	0.999072	+	0.0430610i	$0.0137110\pi$
$347$	10.0000	+	17.3205i	0.536828	+	0.929814i	−0.462244	+	0.886753i	$0.652956\pi$
$348$		0			0
$349$	−4.50000	+	7.79423i	−0.240879	+	0.417215i	−0.960965	−	0.276670i	$-0.910769\pi$
$349$	−4.50000	+	7.79423i	−0.240879	+	0.417215i	0.720086	+	0.693885i	$0.244103\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−17.0000	+	29.4449i	−0.904819	+	1.56719i	−0.0836583	+	0.996495i	$0.526660\pi$
$353$	−17.0000	+	29.4449i	−0.904819	+	1.56719i	−0.821160	+	0.570697i	$0.806673\pi$
$354$		0			0
$355$	−3.00000	−	5.19615i	−0.159223	−	0.275783i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	30.0000			1.58334			0.791670	−	0.610949i	$-0.209212\pi$
$359$	30.0000			1.58334			0.791670	+	0.610949i	$0.209212\pi$
$360$		0			0
$361$	6.00000			0.315789
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−3.00000	−	5.19615i	−0.157027	−	0.271979i
$366$		0			0
$367$	2.00000	−	3.46410i	0.104399	−	0.180825i	−0.809093	−	0.587680i	$-0.800041\pi$
$367$	2.00000	−	3.46410i	0.104399	−	0.180825i	0.913493	+	0.406855i	$0.133375\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	18.0000	−	31.1769i	0.934513	−	1.61862i
$372$		0			0
$373$	10.0000	+	17.3205i	0.517780	+	0.896822i	0.999787	+	0.0206542i	$0.00657489\pi$
$373$	10.0000	+	17.3205i	0.517780	+	0.896822i	−0.482006	+	0.876168i	$0.660092\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	32.0000			1.64808
$378$		0			0
$379$	13.0000			0.667765			0.333883	−	0.942615i	$-0.391641\pi$
$379$	13.0000			0.667765			0.333883	+	0.942615i	$0.391641\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−10.5000	−	18.1865i	−0.536525	−	0.929288i	−0.999088	−	0.0427020i	$-0.986403\pi$
$383$	−10.5000	−	18.1865i	−0.536525	−	0.929288i	0.462563	−	0.886586i	$-0.346930\pi$
$384$		0			0
$385$	−4.00000	+	6.92820i	−0.203859	+	0.353094i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3.00000	−	5.19615i	0.152106	−	0.263455i	−0.779895	−	0.625910i	$-0.784728\pi$
$389$	3.00000	−	5.19615i	0.152106	−	0.263455i	0.932002	+	0.362454i	$0.118061\pi$
$390$		0			0
$391$	−2.50000	−	4.33013i	−0.126430	−	0.218984i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−9.00000			−0.452839
$396$		0			0
$397$	−32.0000			−1.60603			−0.803017	−	0.595956i	$-0.796773\pi$
$397$	−32.0000			−1.60603			−0.803017	+	0.595956i	$0.796773\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−15.0000	−	25.9808i	−0.749064	−	1.29742i	−0.948272	−	0.317460i	$-0.897170\pi$
$401$	−15.0000	−	25.9808i	−0.749064	−	1.29742i	0.199207	−	0.979957i	$-0.436163\pi$
$402$		0			0
$403$	−14.0000	+	24.2487i	−0.697390	+	1.20791i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	6.00000	−	10.3923i	0.297409	−	0.515127i
$408$		0			0
$409$	11.5000	+	19.9186i	0.568638	+	0.984911i	0.996701	+	0.0811615i	$0.0258630\pi$
$409$	11.5000	+	19.9186i	0.568638	+	0.984911i	−0.428063	+	0.903749i	$0.640804\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−16.0000			−0.787309
$414$		0			0
$415$	17.0000			0.834497
$416$		0			0
$417$		0			0
$418$		0			0
$419$	3.00000	+	5.19615i	0.146560	+	0.253849i	0.929954	−	0.367677i	$-0.119847\pi$
$419$	3.00000	+	5.19615i	0.146560	+	0.253849i	−0.783394	+	0.621525i	$0.786513\pi$
$420$		0			0
$421$	−7.50000	+	12.9904i	−0.365528	+	0.633112i	−0.988861	−	0.148844i	$-0.952445\pi$
$421$	−7.50000	+	12.9904i	−0.365528	+	0.633112i	0.623333	+	0.781956i	$0.285778\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−0.500000	+	0.866025i	−0.0242536	+	0.0420084i
$426$		0			0
$427$	26.0000	+	45.0333i	1.25823	+	2.17932i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−20.0000			−0.963366			−0.481683	−	0.876346i	$-0.659974\pi$
$431$	−20.0000			−0.963366			−0.481683	+	0.876346i	$0.659974\pi$
$432$		0			0
$433$	20.0000			0.961139			0.480569	−	0.876957i	$-0.340430\pi$
$433$	20.0000			0.961139			0.480569	+	0.876957i	$0.340430\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	12.5000	+	21.6506i	0.597956	+	1.03569i
$438$		0			0
$439$	2.50000	−	4.33013i	0.119318	−	0.206666i	−0.800179	−	0.599761i	$-0.795262\pi$
$439$	2.50000	−	4.33013i	0.119318	−	0.206666i	0.919498	+	0.393095i	$0.128596\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1.50000	+	2.59808i	−0.0712672	+	0.123438i	−0.899457	−	0.437009i	$-0.856038\pi$
$443$	−1.50000	+	2.59808i	−0.0712672	+	0.123438i	0.828190	+	0.560448i	$0.189371\pi$
$444$		0			0
$445$	−3.00000	−	5.19615i	−0.142214	−	0.246321i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$	12.0000			0.565058
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−8.00000	−	13.8564i	−0.375046	−	0.649598i
$456$		0			0
$457$	4.00000	−	6.92820i	0.187112	−	0.324088i	−0.757174	−	0.653213i	$-0.773421\pi$
$457$	4.00000	−	6.92820i	0.187112	−	0.324088i	0.944286	+	0.329125i	$0.106754\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	3.00000	−	5.19615i	0.139724	−	0.242009i	−0.787668	−	0.616100i	$-0.788712\pi$
$461$	3.00000	−	5.19615i	0.139724	−	0.242009i	0.927392	+	0.374091i	$0.122045\pi$
$462$		0			0
$463$	−3.00000	−	5.19615i	−0.139422	−	0.241486i	0.787856	−	0.615859i	$-0.211191\pi$
$463$	−3.00000	−	5.19615i	−0.139422	−	0.241486i	−0.927278	+	0.374374i	$0.877858\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	31.0000			1.43451			0.717254	−	0.696811i	$-0.245399\pi$
$467$	31.0000			1.43451			0.717254	+	0.696811i	$0.245399\pi$
$468$		0			0
$469$	40.0000			1.84703
$470$		0			0
$471$		0			0
$472$		0			0
$473$	2.00000	+	3.46410i	0.0919601	+	0.159280i
$474$		0			0
$475$	2.50000	−	4.33013i	0.114708	−	0.198680i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−13.0000	+	22.5167i	−0.593985	+	1.02881i	0.399704	+	0.916644i	$0.369113\pi$
$479$	−13.0000	+	22.5167i	−0.593985	+	1.02881i	−0.993689	+	0.112168i	$0.964220\pi$
$480$		0			0
$481$	12.0000	+	20.7846i	0.547153	+	0.947697i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	8.00000			0.363261
$486$		0			0
$487$	34.0000			1.54069			0.770344	−	0.637629i	$-0.220085\pi$
$487$	34.0000			1.54069			0.770344	+	0.637629i	$0.220085\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−15.0000	−	25.9808i	−0.676941	−	1.17250i	−0.975898	−	0.218229i	$-0.929972\pi$
$491$	−15.0000	−	25.9808i	−0.676941	−	1.17250i	0.298957	−	0.954267i	$-0.403361\pi$
$492$		0			0
$493$	−4.00000	+	6.92820i	−0.180151	+	0.312031i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−12.0000	+	20.7846i	−0.538274	+	0.932317i
$498$		0			0
$499$	0.500000	+	0.866025i	0.0223831	+	0.0387686i	0.877000	−	0.480490i	$-0.159541\pi$
$499$	0.500000	+	0.866025i	0.0223831	+	0.0387686i	−0.854617	+	0.519259i	$0.826208\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−27.0000			−1.20387			−0.601935	−	0.798545i	$-0.705603\pi$
$503$	−27.0000			−1.20387			−0.601935	+	0.798545i	$0.705603\pi$
$504$		0			0
$505$	−12.0000			−0.533993
$506$		0			0
$507$		0			0
$508$		0			0
$509$	14.0000	+	24.2487i	0.620539	+	1.07481i	0.989385	+	0.145315i	$0.0464195\pi$
$509$	14.0000	+	24.2487i	0.620539	+	1.07481i	−0.368846	+	0.929490i	$0.620247\pi$
$510$		0			0
$511$	−12.0000	+	20.7846i	−0.530849	+	0.919457i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2.00000	−	3.46410i	0.0881305	−	0.152647i
$516$		0			0
$517$	−8.00000	−	13.8564i	−0.351840	−	0.609404i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	14.0000			0.613351			0.306676	−	0.951814i	$-0.400783\pi$
$521$	14.0000			0.613351			0.306676	+	0.951814i	$0.400783\pi$
$522$		0			0
$523$	−8.00000			−0.349816			−0.174908	−	0.984585i	$-0.555963\pi$
$523$	−8.00000			−0.349816			−0.174908	+	0.984585i	$0.555963\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3.50000	−	6.06218i	−0.152462	−	0.264073i
$528$		0			0
$529$	−1.00000	+	1.73205i	−0.0434783	+	0.0753066i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−12.0000	+	20.7846i	−0.519778	+	0.900281i
$534$		0			0
$535$	6.00000	+	10.3923i	0.259403	+	0.449299i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	18.0000			0.775315
$540$		0			0
$541$	22.0000			0.945854			0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000			0.945854			0.472927	+	0.881102i	$0.343197\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	8.50000	+	14.7224i	0.364100	+	0.630640i
$546$		0			0
$547$	5.00000	−	8.66025i	0.213785	−	0.370286i	−0.739111	−	0.673583i	$-0.764754\pi$
$547$	5.00000	−	8.66025i	0.213785	−	0.370286i	0.952896	+	0.303298i	$0.0980876\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	20.0000	−	34.6410i	0.852029	−	1.47576i
$552$		0			0
$553$	18.0000	+	31.1769i	0.765438	+	1.32578i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	22.0000			0.932170			0.466085	−	0.884740i	$-0.345664\pi$
$557$	22.0000			0.932170			0.466085	+	0.884740i	$0.345664\pi$
$558$		0			0
$559$	−8.00000			−0.338364
$560$		0			0
$561$		0			0
$562$		0			0
$563$	2.00000	+	3.46410i	0.0842900	+	0.145994i	0.905088	−	0.425223i	$-0.139804\pi$
$563$	2.00000	+	3.46410i	0.0842900	+	0.145994i	−0.820798	+	0.571218i	$0.806471\pi$
$564$		0			0
$565$	3.00000	−	5.19615i	0.126211	−	0.218604i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−14.0000	+	24.2487i	−0.586911	+	1.01656i	0.407724	+	0.913105i	$0.366323\pi$
$569$	−14.0000	+	24.2487i	−0.586911	+	1.01656i	−0.994634	+	0.103454i	$0.967011\pi$
$570$		0			0
$571$	2.50000	+	4.33013i	0.104622	+	0.181210i	0.913584	−	0.406651i	$-0.133303\pi$
$571$	2.50000	+	4.33013i	0.104622	+	0.181210i	−0.808962	+	0.587861i	$0.799970\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	5.00000			0.208514
$576$		0			0
$577$	−10.0000			−0.416305			−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000			−0.416305			−0.208153	+	0.978096i	$0.566745\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−34.0000	−	58.8897i	−1.41056	−	2.44316i
$582$		0			0
$583$	−9.00000	+	15.5885i	−0.372742	+	0.645608i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−16.5000	+	28.5788i	−0.681028	+	1.17957i	0.293640	+	0.955916i	$0.405133\pi$
$587$	−16.5000	+	28.5788i	−0.681028	+	1.17957i	−0.974668	+	0.223659i	$0.928200\pi$
$588$		0			0
$589$	17.5000	+	30.3109i	0.721075	+	1.24894i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	15.0000			0.615976			0.307988	−	0.951390i	$-0.400344\pi$
$593$	15.0000			0.615976			0.307988	+	0.951390i	$0.400344\pi$
$594$		0			0
$595$	4.00000			0.163984
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−18.0000	−	31.1769i	−0.735460	−	1.27385i	−0.954521	−	0.298143i	$-0.903633\pi$
$599$	−18.0000	−	31.1769i	−0.735460	−	1.27385i	0.219061	−	0.975711i	$-0.429701\pi$
$600$		0			0
$601$	23.5000	−	40.7032i	0.958585	−	1.66032i	0.232643	−	0.972562i	$-0.425263\pi$
$601$	23.5000	−	40.7032i	0.958585	−	1.66032i	0.725942	−	0.687756i	$-0.241404\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−3.50000	+	6.06218i	−0.142295	+	0.246463i
$606$		0			0
$607$	−15.0000	−	25.9808i	−0.608831	−	1.05453i	−0.991433	−	0.130613i	$-0.958305\pi$
$607$	−15.0000	−	25.9808i	−0.608831	−	1.05453i	0.382602	−	0.923913i	$-0.375028\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	32.0000			1.29458
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	13.5000	+	23.3827i	0.543490	+	0.941351i	0.998700	+	0.0509678i	$0.0162306\pi$
$617$	13.5000	+	23.3827i	0.543490	+	0.941351i	−0.455211	+	0.890384i	$0.650436\pi$
$618$		0			0
$619$	14.0000	−	24.2487i	0.562708	−	0.974638i	−0.434551	−	0.900647i	$-0.643093\pi$
$619$	14.0000	−	24.2487i	0.562708	−	0.974638i	0.997259	−	0.0739910i	$-0.0235736\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−12.0000	+	20.7846i	−0.480770	+	0.832718i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−6.00000			−0.239236
$630$		0			0
$631$	1.00000			0.0398094			0.0199047	−	0.999802i	$-0.493664\pi$
$631$	1.00000			0.0398094			0.0199047	+	0.999802i	$0.493664\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−9.00000	−	15.5885i	−0.357154	−	0.618609i
$636$		0			0
$637$	−18.0000	+	31.1769i	−0.713186	+	1.23527i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	24.0000	−	41.5692i	0.947943	−	1.64189i	0.198194	−	0.980163i	$-0.436492\pi$
$641$	24.0000	−	41.5692i	0.947943	−	1.64189i	0.749749	−	0.661723i	$-0.230174\pi$
$642$		0			0
$643$	−6.00000	−	10.3923i	−0.236617	−	0.409832i	0.723124	−	0.690718i	$-0.242705\pi$
$643$	−6.00000	−	10.3923i	−0.236617	−	0.409832i	−0.959741	+	0.280885i	$0.909372\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	33.0000			1.29736			0.648682	−	0.761060i	$-0.275321\pi$
$647$	33.0000			1.29736			0.648682	+	0.761060i	$0.275321\pi$
$648$		0			0
$649$	8.00000			0.314027
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−12.5000	−	21.6506i	−0.489163	−	0.847255i	0.510759	−	0.859724i	$-0.329364\pi$
$653$	−12.5000	−	21.6506i	−0.489163	−	0.847255i	−0.999922	+	0.0124688i	$0.996031\pi$
$654$		0			0
$655$	3.00000	−	5.19615i	0.117220	−	0.203030i
$656$		0			0
$657$		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
$661$	9.00000	+	15.5885i	0.350059	+	0.606321i	0.986260	−	0.165203i	$-0.0528281\pi$
$661$	9.00000	+	15.5885i	0.350059	+	0.606321i	−0.636200	+	0.771524i	$0.719495\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−20.0000			−0.775567
$666$		0			0
$667$	40.0000			1.54881
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−13.0000	−	22.5167i	−0.501859	−	0.869246i
$672$		0			0
$673$	18.0000	−	31.1769i	0.693849	−	1.20178i	−0.276718	−	0.960951i	$-0.589247\pi$
$673$	18.0000	−	31.1769i	0.693849	−	1.20178i	0.970567	−	0.240831i	$-0.0774198\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	17.0000	−	29.4449i	0.653363	−	1.13166i	−0.328938	−	0.944351i	$-0.606691\pi$
$677$	17.0000	−	29.4449i	0.653363	−	1.13166i	0.982301	−	0.187307i	$-0.0599758\pi$
$678$		0			0
$679$	−16.0000	−	27.7128i	−0.614024	−	1.06352i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−9.00000			−0.344375			−0.172188	−	0.985064i	$-0.555084\pi$
$683$	−9.00000			−0.344375			−0.172188	+	0.985064i	$0.555084\pi$
$684$		0			0
$685$	−11.0000			−0.420288
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−18.0000	−	31.1769i	−0.685745	−	1.18775i
$690$		0			0
$691$	−8.50000	+	14.7224i	−0.323355	+	0.560068i	−0.981178	−	0.193105i	$-0.938144\pi$
$691$	−8.50000	+	14.7224i	−0.323355	+	0.560068i	0.657823	+	0.753173i	$0.271478\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	6.00000	−	10.3923i	0.227593	−	0.394203i
$696$		0			0
$697$	−3.00000	−	5.19615i	−0.113633	−	0.196818i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	10.0000			0.377695			0.188847	−	0.982006i	$-0.439525\pi$
$701$	10.0000			0.377695			0.188847	+	0.982006i	$0.439525\pi$
$702$		0			0
$703$	30.0000			1.13147
$704$		0			0
$705$		0			0
$706$		0			0
$707$	24.0000	+	41.5692i	0.902613	+	1.56337i
$708$		0			0
$709$	23.0000	−	39.8372i	0.863783	−	1.49612i	−0.00446726	−	0.999990i	$-0.501422\pi$
$709$	23.0000	−	39.8372i	0.863783	−	1.49612i	0.868250	−	0.496126i	$-0.165245\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−17.5000	+	30.3109i	−0.655380	+	1.13515i
$714$		0			0
$715$	4.00000	+	6.92820i	0.149592	+	0.259100i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−34.0000			−1.26799			−0.633993	−	0.773339i	$-0.718585\pi$
$719$	−34.0000			−1.26799			−0.633993	+	0.773339i	$0.718585\pi$
$720$		0			0
$721$	−16.0000			−0.595871
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−4.00000	−	6.92820i	−0.148556	−	0.257307i
$726$		0			0
$727$	2.00000	−	3.46410i	0.0741759	−	0.128476i	−0.826552	−	0.562861i	$-0.809701\pi$
$727$	2.00000	−	3.46410i	0.0741759	−	0.128476i	0.900728	+	0.434384i	$0.143034\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1.00000	−	1.73205i	0.0369863	−	0.0640622i
$732$		0			0
$733$	−12.0000	−	20.7846i	−0.443230	−	0.767697i	0.554697	−	0.832052i	$-0.312834\pi$
$733$	−12.0000	−	20.7846i	−0.443230	−	0.767697i	−0.997927	+	0.0643554i	$0.979501\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−20.0000			−0.736709
$738$		0			0
$739$	9.00000			0.331070			0.165535	−	0.986204i	$-0.447065\pi$
$739$	9.00000			0.331070			0.165535	+	0.986204i	$0.447065\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	24.0000	+	41.5692i	0.880475	+	1.52503i	0.850814	+	0.525467i	$0.176109\pi$
$743$	24.0000	+	41.5692i	0.880475	+	1.52503i	0.0296605	+	0.999560i	$0.490557\pi$
$744$		0			0
$745$	−7.00000	+	12.1244i	−0.256460	+	0.444202i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	24.0000	−	41.5692i	0.876941	−	1.51891i
$750$		0			0
$751$	19.5000	+	33.7750i	0.711565	+	1.23247i	0.964269	+	0.264923i	$0.0853467\pi$
$751$	19.5000	+	33.7750i	0.711565	+	1.23247i	−0.252704	+	0.967544i	$0.581320\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	16.0000			0.582300
$756$		0			0
$757$	38.0000			1.38113			0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000			1.38113			0.690567	+	0.723269i	$0.257361\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	12.0000	+	20.7846i	0.435000	+	0.753442i	0.997296	−	0.0734946i	$-0.0234152\pi$
$761$	12.0000	+	20.7846i	0.435000	+	0.753442i	−0.562296	+	0.826936i	$0.690082\pi$
$762$		0			0
$763$	34.0000	−	58.8897i	1.23088	−	2.13195i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−8.00000	+	13.8564i	−0.288863	+	0.500326i
$768$		0			0
$769$	17.5000	+	30.3109i	0.631066	+	1.09304i	0.987334	+	0.158655i	$0.0507157\pi$
$769$	17.5000	+	30.3109i	0.631066	+	1.09304i	−0.356268	+	0.934384i	$0.615951\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	33.0000			1.18693			0.593464	−	0.804861i	$-0.297760\pi$
$773$	33.0000			1.18693			0.593464	+	0.804861i	$0.297760\pi$
$774$		0			0
$775$	7.00000			0.251447
$776$		0			0
$777$		0			0
$778$		0			0
$779$	15.0000	+	25.9808i	0.537431	+	0.930857i
$780$		0			0
$781$	6.00000	−	10.3923i	0.214697	−	0.371866i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	11.0000	−	19.0526i	0.392607	−	0.680015i
$786$		0			0
$787$	−19.0000	−	32.9090i	−0.677277	−	1.17308i	−0.975798	−	0.218675i	$-0.929827\pi$
$787$	−19.0000	−	32.9090i	−0.677277	−	1.17308i	0.298521	−	0.954403i	$-0.403507\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−24.0000			−0.853342
$792$		0			0
$793$	52.0000			1.84657
$794$		0			0
$795$		0			0
$796$		0			0
$797$	10.5000	+	18.1865i	0.371929

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:	\( N \)	\(=\)	\( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3240.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{5} +(2.00000 - 3.46410i) q^{7} +O(q^{10})\)	\(q+(0.500000 + 0.866025i) q^{5} +(2.00000 - 3.46410i) q^{7} +(-1.00000 + 1.73205i) q^{11} +(-2.00000 - 3.46410i) q^{13} +1.00000 q^{17} -5.00000 q^{19} +(-2.50000 - 4.33013i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(-4.00000 + 6.92820i) q^{29} +(-3.50000 - 6.06218i) q^{31} +4.00000 q^{35} -6.00000 q^{37} +(-3.00000 - 5.19615i) q^{41} +(1.00000 - 1.73205i) q^{43} +(-4.00000 + 6.92820i) q^{47} +(-4.50000 - 7.79423i) q^{49} +9.00000 q^{53} -2.00000 q^{55} +(-2.00000 - 3.46410i) q^{59} +(-6.50000 + 11.2583i) q^{61} +(2.00000 - 3.46410i) q^{65} +(5.00000 + 8.66025i) q^{67} -6.00000 q^{71} -6.00000 q^{73} +(4.00000 + 6.92820i) q^{77} +(-4.50000 + 7.79423i) q^{79} +(8.50000 - 14.7224i) q^{83} +(0.500000 + 0.866025i) q^{85} -6.00000 q^{89} -16.0000 q^{91} +(-2.50000 - 4.33013i) q^{95} +(4.00000 - 6.92820i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{5} + 4 q^{7}+O(q^{10}) \) 2 * q + q^5 + 4 * q^7	\( 2 q + q^{5} + 4 q^{7} - 2 q^{11} - 4 q^{13} + 2 q^{17} - 10 q^{19} - 5 q^{23} - q^{25} - 8 q^{29} - 7 q^{31} + 8 q^{35} - 12 q^{37} - 6 q^{41} + 2 q^{43} - 8 q^{47} - 9 q^{49} + 18 q^{53} - 4 q^{55} - 4 q^{59} - 13 q^{61} + 4 q^{65} + 10 q^{67} - 12 q^{71} - 12 q^{73} + 8 q^{77} - 9 q^{79} + 17 q^{83} + q^{85} - 12 q^{89} - 32 q^{91} - 5 q^{95} + 8 q^{97}+O(q^{100}) \) 2 * q + q^5 + 4 * q^7 - 2 * q^11 - 4 * q^13 + 2 * q^17 - 10 * q^19 - 5 * q^23 - q^25 - 8 * q^29 - 7 * q^31 + 8 * q^35 - 12 * q^37 - 6 * q^41 + 2 * q^43 - 8 * q^47 - 9 * q^49 + 18 * q^53 - 4 * q^55 - 4 * q^59 - 13 * q^61 + 4 * q^65 + 10 * q^67 - 12 * q^71 - 12 * q^73 + 8 * q^77 - 9 * q^79 + 17 * q^83 + q^85 - 12 * q^89 - 32 * q^91 - 5 * q^95 + 8 * q^97

Introduction

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