LMFDB - Embedded newform 3240.2.q.e.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.e.1081.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{5} +O(q^{10})$	$q+(-0.500000 - 0.866025i) q^{5} +(-1.00000 + 1.73205i) q^{11} +3.00000 q^{17} -1.00000 q^{19} +(-1.50000 - 2.59808i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(-2.00000 + 3.46410i) q^{29} +(2.50000 + 4.33013i) q^{31} +10.0000 q^{37} +(-3.00000 - 5.19615i) q^{41} +(3.00000 - 5.19615i) q^{43} +(-4.00000 + 6.92820i) q^{47} +(3.50000 + 6.06218i) q^{49} +3.00000 q^{53} +2.00000 q^{55} +(-2.50000 + 4.33013i) q^{61} +(1.00000 + 1.73205i) q^{67} +2.00000 q^{71} +6.00000 q^{73} +(5.50000 - 9.52628i) q^{79} +(-4.50000 + 7.79423i) q^{83} +(-1.50000 - 2.59808i) q^{85} +10.0000 q^{89} +(0.500000 + 0.866025i) q^{95} +(-4.00000 + 6.92820i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{5}+O(q^{10}) $ 2 * q - q^5	$ 2 q - q^{5} - 2 q^{11} + 6 q^{17} - 2 q^{19} - 3 q^{23} - q^{25} - 4 q^{29} + 5 q^{31} + 20 q^{37} - 6 q^{41} + 6 q^{43} - 8 q^{47} + 7 q^{49} + 6 q^{53} + 4 q^{55} - 5 q^{61} + 2 q^{67} + 4 q^{71} + 12 q^{73} + 11 q^{79} - 9 q^{83} - 3 q^{85} + 20 q^{89} + q^{95} - 8 q^{97}+O(q^{100}) $ 2 * q - q^5 - 2 * q^11 + 6 * q^17 - 2 * q^19 - 3 * q^23 - q^25 - 4 * q^29 + 5 * q^31 + 20 * q^37 - 6 * q^41 + 6 * q^43 - 8 * q^47 + 7 * q^49 + 6 * q^53 + 4 * q^55 - 5 * q^61 + 2 * q^67 + 4 * q^71 + 12 * q^73 + 11 * q^79 - 9 * q^83 - 3 * q^85 + 20 * q^89 + 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
$7$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$7$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$8$		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
$13$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$13$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.00000			0.727607			0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000			0.727607			0.363803	+	0.931476i	$0.381478\pi$
$18$		0			0
$19$	−1.00000			−0.229416			−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000			−0.229416			−0.114708	+	0.993399i	$0.536593\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.50000	−	2.59808i	−0.312772	−	0.541736i	0.666190	−	0.745782i	$-0.267924\pi$
$23$	−1.50000	−	2.59808i	−0.312772	−	0.541736i	−0.978961	+	0.204046i	$0.934591\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.00000	+	3.46410i	−0.371391	+	0.643268i	−0.989780	−	0.142605i	$-0.954452\pi$
$29$	−2.00000	+	3.46410i	−0.371391	+	0.643268i	0.618389	+	0.785872i	$0.287786\pi$
$30$		0			0
$31$	2.50000	+	4.33013i	0.449013	+	0.777714i	0.998322	−	0.0579057i	$-0.0184423\pi$
$31$	2.50000	+	4.33013i	0.449013	+	0.777714i	−0.549309	+	0.835619i	$0.685109\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	10.0000			1.64399			0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000			1.64399			0.821995	+	0.569495i	$0.192861\pi$
$38$		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
$43$	3.00000	−	5.19615i	0.457496	−	0.792406i	−0.541332	−	0.840809i	$-0.682080\pi$
$43$	3.00000	−	5.19615i	0.457496	−	0.792406i	0.998828	+	0.0484030i	$0.0154132\pi$
$44$		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
$49$	3.50000	+	6.06218i	0.500000	+	0.866025i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3.00000			0.412082			0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000			0.412082			0.206041	+	0.978543i	$0.433942\pi$
$54$		0			0
$55$	2.00000			0.269680
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$	−2.50000	+	4.33013i	−0.320092	+	0.554416i	−0.980507	−	0.196485i	$-0.937047\pi$
$61$	−2.50000	+	4.33013i	−0.320092	+	0.554416i	0.660415	+	0.750901i	$0.270381\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	1.00000	+	1.73205i	0.122169	+	0.211604i	0.920623	−	0.390453i	$-0.127682\pi$
$67$	1.00000	+	1.73205i	0.122169	+	0.211604i	−0.798454	+	0.602056i	$0.794348\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	2.00000			0.237356			0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000			0.237356			0.118678	+	0.992933i	$0.462134\pi$
$72$		0			0
$73$	6.00000			0.702247			0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000			0.702247			0.351123	+	0.936329i	$0.385800\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	5.50000	−	9.52628i	0.618798	−	1.07179i	−0.370907	−	0.928670i	$-0.620953\pi$
$79$	5.50000	−	9.52628i	0.618798	−	1.07179i	0.989705	−	0.143120i	$-0.0457135\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.50000	+	7.79423i	−0.493939	+	0.855528i	−0.999976	−	0.00698436i	$-0.997777\pi$
$83$	−4.50000	+	7.79423i	−0.493939	+	0.855528i	0.506036	+	0.862512i	$0.331110\pi$
$84$		0			0
$85$	−1.50000	−	2.59808i	−0.162698	−	0.281801i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	10.0000			1.06000			0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000			1.06000			0.529999	+	0.847998i	$0.322192\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.500000	+	0.866025i	0.0512989	+	0.0888523i
$96$		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	$0.299792\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.00000	−	10.3923i	0.597022	−	1.03407i	−0.396236	−	0.918149i	$-0.629684\pi$
$101$	6.00000	−	10.3923i	0.597022	−	1.03407i	0.993258	−	0.115924i	$-0.0369830\pi$
$102$		0			0
$103$	6.00000	+	10.3923i	0.591198	+	1.02398i	0.994071	+	0.108729i	$0.0346780\pi$
$103$	6.00000	+	10.3923i	0.591198	+	1.02398i	−0.402874	+	0.915255i	$0.631989\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−4.00000			−0.386695			−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000			−0.386695			−0.193347	+	0.981130i	$0.561934\pi$
$108$		0			0
$109$	9.00000			0.862044			0.431022	−	0.902342i	$-0.358153\pi$
$109$	9.00000			0.862044			0.431022	+	0.902342i	$0.358153\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.00000	+	5.19615i	0.282216	+	0.488813i	0.971930	−	0.235269i	$-0.0755971\pi$
$113$	3.00000	+	5.19615i	0.282216	+	0.488813i	−0.689714	+	0.724082i	$0.742264\pi$
$114$		0			0
$115$	−1.50000	+	2.59808i	−0.139876	+	0.242272i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		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
$127$	22.0000			1.95218			0.976092	−	0.217357i	$-0.0697436\pi$
$127$	22.0000			1.95218			0.976092	+	0.217357i	$0.0697436\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−5.00000	−	8.66025i	−0.436852	−	0.756650i	0.560593	−	0.828092i	$-0.310573\pi$
$131$	−5.00000	−	8.66025i	−0.436852	−	0.756650i	−0.997445	+	0.0714417i	$0.977240\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	3.50000	−	6.06218i	0.299025	−	0.517927i	−0.676888	−	0.736086i	$-0.736672\pi$
$137$	3.50000	−	6.06218i	0.299025	−	0.517927i	0.975913	+	0.218159i	$0.0700052\pi$
$138$		0			0
$139$	10.0000	+	17.3205i	0.848189	+	1.46911i	0.882823	+	0.469706i	$0.155640\pi$
$139$	10.0000	+	17.3205i	0.848189	+	1.46911i	−0.0346338	+	0.999400i	$0.511026\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	4.00000			0.332182
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.00000	−	1.73205i	−0.0819232	−	0.141895i	0.822153	−	0.569267i	$-0.192773\pi$
$149$	−1.00000	−	1.73205i	−0.0819232	−	0.141895i	−0.904076	+	0.427372i	$0.859440\pi$
$150$		0			0
$151$	4.00000	−	6.92820i	0.325515	−	0.563809i	−0.656101	−	0.754673i	$-0.727796\pi$
$151$	4.00000	−	6.92820i	0.325515	−	0.563809i	0.981617	+	0.190864i	$0.0611289\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.50000	−	4.33013i	0.200805	−	0.347804i
$156$		0			0
$157$	−9.00000	−	15.5885i	−0.718278	−	1.24409i	−0.961681	−	0.274169i	$-0.911597\pi$
$157$	−9.00000	−	15.5885i	−0.718278	−	1.24409i	0.243403	−	0.969925i	$-0.421736\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		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
$167$	8.50000	+	14.7224i	0.657750	+	1.13926i	0.981197	+	0.193010i	$0.0618249\pi$
$167$	8.50000	+	14.7224i	0.657750	+	1.13926i	−0.323447	+	0.946246i	$0.604842\pi$
$168$		0			0
$169$	6.50000	−	11.2583i	0.500000	−	0.866025i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	9.50000	−	16.4545i	0.722272	−	1.25101i	−0.237816	−	0.971310i	$-0.576431\pi$
$173$	9.50000	−	16.4545i	0.722272	−	1.25101i	0.960087	−	0.279701i	$-0.0902353\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−24.0000			−1.79384			−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000			−1.79384			−0.896922	+	0.442189i	$0.854202\pi$
$180$		0			0
$181$	17.0000			1.26360			0.631800	−	0.775131i	$-0.282316\pi$
$181$	17.0000			1.26360			0.631800	+	0.775131i	$0.282316\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−5.00000	−	8.66025i	−0.367607	−	0.636715i
$186$		0			0
$187$	−3.00000	+	5.19615i	−0.219382	+	0.379980i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	3.00000	−	5.19615i	0.217072	−	0.375980i	−0.736839	−	0.676068i	$-0.763683\pi$
$191$	3.00000	−	5.19615i	0.217072	−	0.375980i	0.953912	+	0.300088i	$0.0970159\pi$
$192$		0			0
$193$	10.0000	+	17.3205i	0.719816	+	1.24676i	0.961073	+	0.276296i	$0.0891071\pi$
$193$	10.0000	+	17.3205i	0.719816	+	1.24676i	−0.241257	+	0.970461i	$0.577560\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	5.00000			0.356235			0.178118	−	0.984009i	$-0.442999\pi$
$197$	5.00000			0.356235			0.178118	+	0.984009i	$0.442999\pi$
$198$		0			0
$199$	−12.0000			−0.850657			−0.425329	−	0.905039i	$-0.639842\pi$
$199$	−12.0000			−0.850657			−0.425329	+	0.905039i	$0.639842\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−3.00000	+	5.19615i	−0.209529	+	0.362915i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.00000	−	1.73205i	0.0691714	−	0.119808i
$210$		0			0
$211$	1.50000	+	2.59808i	0.103264	+	0.178859i	0.913028	−	0.407898i	$-0.133738\pi$
$211$	1.50000	+	2.59808i	0.103264	+	0.178859i	−0.809763	+	0.586756i	$0.800405\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−6.00000			−0.409197
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		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
$227$	−8.50000	+	14.7224i	−0.564165	+	0.977162i	0.432962	+	0.901412i	$0.357468\pi$
$227$	−8.50000	+	14.7224i	−0.564165	+	0.977162i	−0.997127	+	0.0757500i	$0.975865\pi$
$228$		0			0
$229$	6.50000	+	11.2583i	0.429532	+	0.743971i	0.996832	−	0.0795401i	$-0.0253452\pi$
$229$	6.50000	+	11.2583i	0.429532	+	0.743971i	−0.567300	+	0.823511i	$0.692012\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−10.0000			−0.655122			−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000			−0.655122			−0.327561	+	0.944830i	$0.606227\pi$
$234$		0			0
$235$	8.00000			0.521862
$236$		0			0
$237$		0			0
$238$		0			0
$239$	15.0000	+	25.9808i	0.970269	+	1.68056i	0.694737	+	0.719264i	$0.255521\pi$
$239$	15.0000	+	25.9808i	0.970269	+	1.68056i	0.275533	+	0.961292i	$0.411146\pi$
$240$		0			0
$241$	8.50000	−	14.7224i	0.547533	−	0.948355i	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	8.50000	−	14.7224i	0.547533	−	0.948355i	0.998443	−	0.0557856i	$-0.0177663\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	3.50000	−	6.06218i	0.223607	−	0.387298i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−20.0000			−1.26239			−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000			−1.26239			−0.631194	+	0.775625i	$0.717435\pi$
$252$		0			0
$253$	6.00000			0.377217
$254$		0			0
$255$		0			0
$256$		0			0
$257$	11.5000	+	19.9186i	0.717350	+	1.24249i	0.962046	+	0.272887i	$0.0879786\pi$
$257$	11.5000	+	19.9186i	0.717350	+	1.24249i	−0.244696	+	0.969600i	$0.578688\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6.00000	+	10.3923i	−0.369976	+	0.640817i	−0.989561	−	0.144112i	$-0.953967\pi$
$263$	−6.00000	+	10.3923i	−0.369976	+	0.640817i	0.619586	+	0.784929i	$0.287301\pi$
$264$		0			0
$265$	−1.50000	−	2.59808i	−0.0921443	−	0.159599i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	18.0000			1.09748			0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000			1.09748			0.548740	+	0.835993i	$0.315108\pi$
$270$		0			0
$271$	3.00000			0.182237			0.0911185	−	0.995840i	$-0.470956\pi$
$271$	3.00000			0.182237			0.0911185	+	0.995840i	$0.470956\pi$
$272$		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	$0.396503\pi$
$277$	−11.0000	+	19.0526i	−0.660926	+	1.14476i	−0.980373	+	0.197153i	$0.936830\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	14.0000	−	24.2487i	0.835170	−	1.44656i	−0.0587220	−	0.998274i	$-0.518703\pi$
$281$	14.0000	−	24.2487i	0.835170	−	1.44656i	0.893892	−	0.448282i	$-0.147964\pi$
$282$		0			0
$283$	−1.00000	−	1.73205i	−0.0594438	−	0.102960i	0.834772	−	0.550596i	$-0.185599\pi$
$283$	−1.00000	−	1.73205i	−0.0594438	−	0.102960i	−0.894216	+	0.447636i	$0.852266\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−8.00000			−0.470588
$290$		0			0
$291$		0			0
$292$		0			0
$293$	5.50000	+	9.52628i	0.321313	+	0.556531i	0.980759	−	0.195221i	$-0.0625424\pi$
$293$	5.50000	+	9.52628i	0.321313	+	0.556531i	−0.659446	+	0.751752i	$0.729209\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	5.00000			0.286299
$306$		0			0
$307$	26.0000			1.48390			0.741949	−	0.670456i	$-0.233902\pi$
$307$	26.0000			1.48390			0.741949	+	0.670456i	$0.233902\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−10.0000	−	17.3205i	−0.567048	−	0.982156i	−0.996856	−	0.0792356i	$-0.974752\pi$
$311$	−10.0000	−	17.3205i	−0.567048	−	0.982156i	0.429808	−	0.902920i	$-0.358581\pi$
$312$		0			0
$313$	10.0000	−	17.3205i	0.565233	−	0.979013i	−0.431795	−	0.901972i	$-0.642119\pi$
$313$	10.0000	−	17.3205i	0.565233	−	0.979013i	0.997028	−	0.0770410i	$-0.0245472\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−5.50000	+	9.52628i	−0.308911	+	0.535049i	−0.978124	−	0.208021i	$-0.933298\pi$
$317$	−5.50000	+	9.52628i	−0.308911	+	0.535049i	0.669214	+	0.743070i	$0.266631\pi$
$318$		0			0
$319$	−4.00000	−	6.92820i	−0.223957	−	0.387905i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−3.00000			−0.166924
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−4.00000	+	6.92820i	−0.219860	+	0.380808i	−0.954765	−	0.297361i	$-0.903893\pi$
$331$	−4.00000	+	6.92820i	−0.219860	+	0.380808i	0.734905	+	0.678170i	$0.237227\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1.00000	−	1.73205i	0.0546358	−	0.0946320i
$336$		0			0
$337$	−4.00000	−	6.92820i	−0.217894	−	0.377403i	0.736270	−	0.676688i	$-0.236585\pi$
$337$	−4.00000	−	6.92820i	−0.217894	−	0.377403i	−0.954164	+	0.299285i	$0.903252\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−10.0000			−0.541530
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6.00000	+	10.3923i	0.322097	+	0.557888i	0.980921	−	0.194409i	$-0.0622790\pi$
$347$	6.00000	+	10.3923i	0.322097	+	0.557888i	−0.658824	+	0.752297i	$0.728946\pi$
$348$		0			0
$349$	3.50000	−	6.06218i	0.187351	−	0.324501i	−0.757015	−	0.653397i	$-0.773343\pi$
$349$	3.50000	−	6.06218i	0.187351	−	0.324501i	0.944366	+	0.328896i	$0.106677\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	5.00000	−	8.66025i	0.266123	−	0.460939i	−0.701734	−	0.712439i	$-0.747591\pi$
$353$	5.00000	−	8.66025i	0.266123	−	0.460939i	0.967857	+	0.251500i	$0.0809239\pi$
$354$		0			0
$355$	−1.00000	−	1.73205i	−0.0530745	−	0.0919277i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−6.00000			−0.316668			−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−6.00000			−0.316668			−0.158334	+	0.987386i	$0.550612\pi$
$360$		0			0
$361$	−18.0000			−0.947368
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−3.00000	−	5.19615i	−0.157027	−	0.271979i
$366$		0			0
$367$	−8.00000	+	13.8564i	−0.417597	+	0.723299i	−0.995697	−	0.0926670i	$-0.970461\pi$
$367$	−8.00000	+	13.8564i	−0.417597	+	0.723299i	0.578101	+	0.815966i	$0.303794\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−2.00000	−	3.46410i	−0.103556	−	0.179364i	0.809591	−	0.586994i	$-0.199689\pi$
$373$	−2.00000	−	3.46410i	−0.103556	−	0.179364i	−0.913147	+	0.407630i	$0.866355\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−23.0000			−1.18143			−0.590715	−	0.806880i	$-0.701154\pi$
$379$	−23.0000			−1.18143			−0.590715	+	0.806880i	$0.701154\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1.50000	−	2.59808i	−0.0766464	−	0.132755i	0.825155	−	0.564907i	$-0.191088\pi$
$383$	−1.50000	−	2.59808i	−0.0766464	−	0.132755i	−0.901801	+	0.432151i	$0.857755\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−9.00000	+	15.5885i	−0.456318	+	0.790366i	−0.998763	−	0.0497253i	$-0.984165\pi$
$389$	−9.00000	+	15.5885i	−0.456318	+	0.790366i	0.542445	+	0.840091i	$0.317499\pi$
$390$		0			0
$391$	−4.50000	−	7.79423i	−0.227575	−	0.394171i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−11.0000			−0.553470
$396$		0			0
$397$	−4.00000			−0.200754			−0.100377	−	0.994949i	$-0.532005\pi$
$397$	−4.00000			−0.200754			−0.100377	+	0.994949i	$0.532005\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−17.0000	−	29.4449i	−0.848939	−	1.47041i	−0.882156	−	0.470958i	$-0.843908\pi$
$401$	−17.0000	−	29.4449i	−0.848939	−	1.47041i	0.0332161	−	0.999448i	$-0.489425\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−10.0000	+	17.3205i	−0.495682	+	0.858546i
$408$		0			0
$409$	15.5000	+	26.8468i	0.766426	+	1.32749i	0.939490	+	0.342578i	$0.111300\pi$
$409$	15.5000	+	26.8468i	0.766426	+	1.32749i	−0.173064	+	0.984911i	$0.555367\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	9.00000			0.441793
$416$		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
$421$	0.500000	−	0.866025i	0.0243685	−	0.0422075i	−0.853584	−	0.520955i	$-0.825576\pi$
$421$	0.500000	−	0.866025i	0.0243685	−	0.0422075i	0.877952	+	0.478748i	$0.158909\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.50000	+	2.59808i	−0.0727607	+	0.126025i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	8.00000			0.385346			0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000			0.385346			0.192673	+	0.981263i	$0.438284\pi$
$432$		0			0
$433$	−24.0000			−1.15337			−0.576683	−	0.816968i	$-0.695653\pi$
$433$	−24.0000			−1.15337			−0.576683	+	0.816968i	$0.695653\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.50000	+	2.59808i	0.0717547	+	0.124283i
$438$		0			0
$439$	−3.50000	+	6.06218i	−0.167046	+	0.289332i	−0.937380	−	0.348309i	$-0.886756\pi$
$439$	−3.50000	+	6.06218i	−0.167046	+	0.289332i	0.770334	+	0.637641i	$0.220089\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−14.5000	+	25.1147i	−0.688916	+	1.19324i	0.283273	+	0.959039i	$0.408580\pi$
$443$	−14.5000	+	25.1147i	−0.688916	+	1.19324i	−0.972189	+	0.234198i	$0.924754\pi$
$444$		0			0
$445$	−5.00000	−	8.66025i	−0.237023	−	0.410535i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	12.0000			0.566315			0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000			0.566315			0.283158	+	0.959073i	$0.408618\pi$
$450$		0			0
$451$	12.0000			0.565058
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	8.00000	−	13.8564i	0.374224	−	0.648175i	−0.615986	−	0.787757i	$-0.711242\pi$
$457$	8.00000	−	13.8564i	0.374224	−	0.648175i	0.990211	+	0.139581i	$0.0445757\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	17.0000	−	29.4449i	0.791769	−	1.37138i	−0.133102	−	0.991102i	$-0.542494\pi$
$461$	17.0000	−	29.4449i	0.791769	−	1.37138i	0.924871	−	0.380282i	$-0.124173\pi$
$462$		0			0
$463$	−7.00000	−	12.1244i	−0.325318	−	0.563467i	0.656259	−	0.754536i	$-0.272138\pi$
$463$	−7.00000	−	12.1244i	−0.325318	−	0.563467i	−0.981577	+	0.191069i	$0.938805\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−7.00000			−0.323921			−0.161961	−	0.986797i	$-0.551782\pi$
$467$	−7.00000			−0.323921			−0.161961	+	0.986797i	$0.551782\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	6.00000	+	10.3923i	0.275880	+	0.477839i
$474$		0			0
$475$	0.500000	−	0.866025i	0.0229416	−	0.0397360i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−5.00000	+	8.66025i	−0.228456	+	0.395697i	−0.957351	−	0.288929i	$-0.906701\pi$
$479$	−5.00000	+	8.66025i	−0.228456	+	0.395697i	0.728895	+	0.684626i	$0.240034\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	8.00000			0.363261
$486$		0			0
$487$	−2.00000			−0.0906287			−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000			−0.0906287			−0.0453143	+	0.998973i	$0.514429\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−21.0000	−	36.3731i	−0.947717	−	1.64149i	−0.750218	−	0.661190i	$-0.770052\pi$
$491$	−21.0000	−	36.3731i	−0.947717	−	1.64149i	−0.197499	−	0.980303i	$-0.563282\pi$
$492$		0			0
$493$	−6.00000	+	10.3923i	−0.270226	+	0.468046i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−21.5000	−	37.2391i	−0.962472	−	1.66705i	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−21.5000	−	37.2391i	−0.962472	−	1.66705i	−0.246214	−	0.969216i	$-0.579187\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	35.0000			1.56057			0.780286	−	0.625422i	$-0.215073\pi$
$503$	35.0000			1.56057			0.780286	+	0.625422i	$0.215073\pi$
$504$		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	$-0.953580\pi$
$509$	−14.0000	−	24.2487i	−0.620539	−	1.07481i	0.368846	−	0.929490i	$-0.379753\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	6.00000	−	10.3923i	0.264392	−	0.457940i
$516$		0			0
$517$	−8.00000	−	13.8564i	−0.351840	−	0.609404i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	30.0000			1.31432			0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000			1.31432			0.657162	+	0.753749i	$0.271757\pi$
$522$		0			0
$523$	−28.0000			−1.22435			−0.612177	−	0.790721i	$-0.709706\pi$
$523$	−28.0000			−1.22435			−0.612177	+	0.790721i	$0.709706\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	7.50000	+	12.9904i	0.326705	+	0.565870i
$528$		0			0
$529$	7.00000	−	12.1244i	0.304348	−	0.527146i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	2.00000	+	3.46410i	0.0864675	+	0.149766i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−14.0000			−0.603023
$540$		0			0
$541$	−18.0000			−0.773880			−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000			−0.773880			−0.386940	+	0.922105i	$0.626468\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−4.50000	−	7.79423i	−0.192759	−	0.333868i
$546$		0			0
$547$	−9.00000	+	15.5885i	−0.384812	+	0.666514i	−0.991743	−	0.128240i	$-0.959067\pi$
$547$	−9.00000	+	15.5885i	−0.384812	+	0.666514i	0.606931	+	0.794755i	$0.292400\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	2.00000	−	3.46410i	0.0852029	−	0.147576i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	18.0000			0.762684			0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000			0.762684			0.381342	+	0.924434i	$0.375462\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−14.0000	−	24.2487i	−0.590030	−	1.02196i	−0.994228	−	0.107290i	$-0.965783\pi$
$563$	−14.0000	−	24.2487i	−0.590030	−	1.02196i	0.404198	−	0.914671i	$-0.367551\pi$
$564$		0			0
$565$	3.00000	−	5.19615i	0.126211	−	0.218604i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−12.0000	+	20.7846i	−0.503066	+	0.871336i	0.496928	+	0.867792i	$0.334461\pi$
$569$	−12.0000	+	20.7846i	−0.503066	+	0.871336i	−0.999994	+	0.00354413i	$0.998872\pi$
$570$		0			0
$571$	4.50000	+	7.79423i	0.188319	+	0.326178i	0.944690	−	0.327965i	$-0.106363\pi$
$571$	4.50000	+	7.79423i	0.188319	+	0.326178i	−0.756371	+	0.654143i	$0.773029\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3.00000			0.125109
$576$		0			0
$577$	−46.0000			−1.91501			−0.957503	−	0.288425i	$-0.906868\pi$
$577$	−46.0000			−1.91501			−0.957503	+	0.288425i	$0.906868\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−3.00000	+	5.19615i	−0.124247	+	0.215203i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−11.5000	+	19.9186i	−0.474656	+	0.822128i	−0.999579	−	0.0290218i	$-0.990761\pi$
$587$	−11.5000	+	19.9186i	−0.474656	+	0.822128i	0.524923	+	0.851150i	$0.324094\pi$
$588$		0			0
$589$	−2.50000	−	4.33013i	−0.103011	−	0.178420i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−3.00000			−0.123195			−0.0615976	−	0.998101i	$-0.519620\pi$
$593$	−3.00000			−0.123195			−0.0615976	+	0.998101i	$0.519620\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	20.0000	+	34.6410i	0.817178	+	1.41539i	0.907754	+	0.419504i	$0.137796\pi$
$599$	20.0000	+	34.6410i	0.817178	+	1.41539i	−0.0905757	+	0.995890i	$0.528871\pi$
$600$		0			0
$601$	−8.50000	+	14.7224i	−0.346722	+	0.600541i	−0.985665	−	0.168714i	$-0.946039\pi$
$601$	−8.50000	+	14.7224i	−0.346722	+	0.600541i	0.638943	+	0.769254i	$0.279372\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3.50000	−	6.06218i	0.142295	−	0.246463i
$606$		0			0
$607$	−5.00000	−	8.66025i	−0.202944	−	0.351509i	0.746532	−	0.665350i	$-0.231718\pi$
$607$	−5.00000	−	8.66025i	−0.202944	−	0.351509i	−0.949476	+	0.313841i	$0.898384\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	16.0000			0.646234			0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000			0.646234			0.323117	+	0.946359i	$0.395269\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	12.5000	+	21.6506i	0.503231	+	0.871622i	0.999993	+	0.00373492i	$0.00118887\pi$
$617$	12.5000	+	21.6506i	0.503231	+	0.871622i	−0.496762	+	0.867887i	$0.665478\pi$
$618$		0			0
$619$	2.00000	−	3.46410i	0.0803868	−	0.139234i	−0.823029	−	0.567999i	$-0.807718\pi$
$619$	2.00000	−	3.46410i	0.0803868	−	0.139234i	0.903416	+	0.428765i	$0.141051\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	30.0000			1.19618
$630$		0			0
$631$	−19.0000			−0.756378			−0.378189	−	0.925728i	$-0.623453\pi$
$631$	−19.0000			−0.756378			−0.378189	+	0.925728i	$0.623453\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−11.0000	−	19.0526i	−0.436522	−	0.756078i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$641$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$642$		0			0
$643$	−20.0000	−	34.6410i	−0.788723	−	1.36611i	−0.926750	−	0.375680i	$-0.877409\pi$
$643$	−20.0000	−	34.6410i	−0.788723	−	1.36611i	0.138027	−	0.990429i	$-0.455924\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	39.0000			1.53325			0.766624	−	0.642096i	$-0.221935\pi$
$647$	39.0000			1.53325			0.766624	+	0.642096i	$0.221935\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−5.50000	−	9.52628i	−0.215232	−	0.372792i	0.738113	−	0.674678i	$-0.235717\pi$
$653$	−5.50000	−	9.52628i	−0.215232	−	0.372792i	−0.953344	+	0.301885i	$0.902384\pi$
$654$		0			0
$655$	−5.00000	+	8.66025i	−0.195366	+	0.338384i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−2.00000	+	3.46410i	−0.0779089	+	0.134942i	−0.902348	−	0.431009i	$-0.858158\pi$
$659$	−2.00000	+	3.46410i	−0.0779089	+	0.134942i	0.824439	+	0.565951i	$0.191491\pi$
$660$		0			0
$661$	−3.00000	−	5.19615i	−0.116686	−	0.202107i	0.801766	−	0.597638i	$-0.203894\pi$
$661$	−3.00000	−	5.19615i	−0.116686	−	0.202107i	−0.918453	+	0.395531i	$0.870561\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	12.0000			0.464642
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−5.00000	−	8.66025i	−0.193023	−	0.334325i
$672$		0			0
$673$	4.00000	−	6.92820i	0.154189	−	0.267063i	−0.778575	−	0.627552i	$-0.784057\pi$
$673$	4.00000	−	6.92820i	0.154189	−	0.267063i	0.932763	+	0.360489i	$0.117390\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−25.0000	+	43.3013i	−0.960828	+	1.66420i	−0.240399	+	0.970674i	$0.577278\pi$
$677$	−25.0000	+	43.3013i	−0.960828	+	1.66420i	−0.720429	+	0.693529i	$0.756055\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−23.0000			−0.880071			−0.440035	−	0.897980i	$-0.645034\pi$
$683$	−23.0000			−0.880071			−0.440035	+	0.897980i	$0.645034\pi$
$684$		0			0
$685$	−7.00000			−0.267456
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−6.50000	+	11.2583i	−0.247272	+	0.428287i	−0.962768	−	0.270330i	$-0.912867\pi$
$691$	−6.50000	+	11.2583i	−0.247272	+	0.428287i	0.715496	+	0.698617i	$0.246201\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	10.0000	−	17.3205i	0.379322	−	0.657004i
$696$		0			0
$697$	−9.00000	−	15.5885i	−0.340899	−	0.590455i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−42.0000			−1.58632			−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000			−1.58632			−0.793159	+	0.609015i	$0.791565\pi$
$702$		0			0
$703$	−10.0000			−0.377157
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−13.0000	+	22.5167i	−0.488225	+	0.845631i	−0.999908	−	0.0135434i	$-0.995689\pi$
$709$	−13.0000	+	22.5167i	−0.488225	+	0.845631i	0.511683	+	0.859174i	$0.329022\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	7.50000	−	12.9904i	0.280877	−	0.486494i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	18.0000			0.671287			0.335643	−	0.941989i	$-0.391046\pi$
$719$	18.0000			0.671287			0.335643	+	0.941989i	$0.391046\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.00000	−	3.46410i	−0.0742781	−	0.128654i
$726$		0			0
$727$	22.0000	−	38.1051i	0.815935	−	1.41324i	−0.0927199	−	0.995692i	$-0.529556\pi$
$727$	22.0000	−	38.1051i	0.815935	−	1.41324i	0.908655	−	0.417548i	$-0.137111\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	9.00000	−	15.5885i	0.332877	−	0.576560i
$732$		0			0
$733$	4.00000	+	6.92820i	0.147743	+	0.255899i	0.930393	−	0.366563i	$-0.119466\pi$
$733$	4.00000	+	6.92820i	0.147743	+	0.255899i	−0.782650	+	0.622462i	$0.786132\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−4.00000			−0.147342
$738$		0			0
$739$	−19.0000			−0.698926			−0.349463	−	0.936950i	$-0.613636\pi$
$739$	−19.0000			−0.698926			−0.349463	+	0.936950i	$0.613636\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	8.00000	+	13.8564i	0.293492	+	0.508342i	0.974633	−	0.223810i	$-0.0718494\pi$
$743$	8.00000	+	13.8564i	0.293492	+	0.508342i	−0.681141	+	0.732152i	$0.738516\pi$
$744$		0			0
$745$	−1.00000	+	1.73205i	−0.0366372	+	0.0634574i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−26.5000	−	45.8993i	−0.966999	−	1.67489i	−0.704146	−	0.710055i	$-0.748670\pi$
$751$	−26.5000	−	45.8993i	−0.966999	−	1.67489i	−0.262852	−	0.964836i	$-0.584663\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−8.00000			−0.291150
$756$		0			0
$757$	−50.0000			−1.81728			−0.908640	−	0.417579i	$-0.862879\pi$
$757$	−50.0000			−1.81728			−0.908640	+	0.417579i	$0.862879\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−22.0000	−	38.1051i	−0.797499	−	1.38131i	−0.921240	−	0.388995i	$-0.872822\pi$
$761$	−22.0000	−	38.1051i	−0.797499	−	1.38131i	0.123741	−	0.992315i	$-0.460511\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−10.5000	−	18.1865i	−0.378640	−	0.655823i	0.612225	−	0.790684i	$-0.290275\pi$
$769$	−10.5000	−	18.1865i	−0.378640	−	0.655823i	−0.990865	+	0.134860i	$0.956941\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	3.00000			0.107903			0.0539513	−	0.998544i	$-0.482818\pi$
$773$	3.00000			0.107903			0.0539513	+	0.998544i	$0.482818\pi$
$774$		0			0
$775$	−5.00000			−0.179605
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.00000	+	5.19615i	0.107486	+	0.186171i
$780$		0			0
$781$	−2.00000	+	3.46410i	−0.0715656	+	0.123955i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−9.00000	+	15.5885i	−0.321224	+	0.556376i
$786$		0			0
$787$	19.0000	+	32.9090i	0.677277	+	1.17308i	0.975798	+	0.218675i	$0.0701734\pi$
$787$	19.0000	+	32.9090i	0.677277	+	1.17308i	−0.298521	+	0.954403i	$0.596493\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	11.5000	+	19.9186i	0.407351	+	0.705552i	0.994592	−	0.103860i	$-0.0331193\pi$
$797$	11.5000	+	19.9186i	0.407351	+	0.705552i	−0.587241	+	0.809412i	$0.699786\pi$
$798$		0			0
$799$	−12.0000	+	20.7846i	−0.424529	+	0.735307i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−6.00000	+	10.3923i	−0.211735	+	0.366736i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−18.0000			−0.632846			−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000			−0.632846			−0.316423	+	0.948618i	$0.602482\pi$
$810$		0			0
$811$	−52.0000			−1.82597			−0.912983	−	0.407997i	$-0.866228\pi$
$811$	−52.0000			−1.82597			−0.912983	+	0.407997i	$0.866228\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.00000	+	1.73205i	0.0350285	+	0.0606711i
$816$		0			0
$817$	−3.00000	+	5.19615i	−0.104957	+	0.181790i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	17.0000	−	29.4449i	0.593304	−	1.02763i	−0.400480	−	0.916306i	$-0.631157\pi$
$821$	17.0000	−	29.4449i	0.593304	−	1.02763i	0.993784	−	0.111327i	$-0.0355102\pi$
$822$		0			0
$823$	3.00000	+	5.19615i	0.104573	+	0.181126i	0.913564	−	0.406695i	$-0.133319\pi$
$823$	3.00000	+	5.19615i	0.104573	+	0.181126i	−0.808990	+	0.587822i	$0.799986\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−21.0000			−0.730242			−0.365121	−	0.930960i	$-0.618972\pi$
$827$	−21.0000			−0.730242			−0.365121	+	0.930960i	$0.618972\pi$
$828$		0			0
$829$	34.0000			1.18087			0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000			1.18087			0.590434	+	0.807086i	$0.298956\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	10.5000	+	18.1865i	0.363803	+	0.630126i
$834$		0			0
$835$	8.50000	−	14.7224i	0.294155	−	0.509491i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	10.0000	−	17.3205i	0.345238	−	0.597970i	−0.640159	−	0.768243i	$-0.721131\pi$
$839$	10.0000	−	17.3205i	0.345238	−	0.597970i	0.985397	+	0.170272i	$0.0544647\pi$
$840$		0			0
$841$	6.50000	+	11.2583i	0.224138	+	0.388218i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−13.0000			−0.447214
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−15.0000	−	25.9808i	−0.514193	−	0.890609i
$852$		0			0
$853$	−4.00000	+	6.92820i	−0.136957	+	0.237217i	−0.926343	−	0.376680i	$-0.877066\pi$
$853$	−4.00000	+	6.92820i	−0.136957	+	0.237217i	0.789386	+	0.613897i	$0.210399\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	0.500000	−	0.866025i	0.0170797	−	0.0295829i	−0.857359	−	0.514718i	$-0.827896\pi$
$857$	0.500000	−	0.866025i	0.0170797	−	0.0295829i	0.874439	+	0.485136i	$0.161230\pi$
$858$		0			0
$859$	−18.5000	−	32.0429i	−0.631212	−	1.09329i	−0.987304	−	0.158840i	$-0.949225\pi$
$859$	−18.5000	−	32.0429i	−0.631212	−	1.09329i	0.356092	−	0.934451i	$-0.384109\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	23.0000			0.782929			0.391465	−	0.920193i	$-0.371969\pi$
$863$	23.0000			0.782929			0.391465	+	0.920193i	$0.371969\pi$
$864$		0			0
$865$	−19.0000			−0.646019
$866$		0			0
$867$		0			0
$868$		0			0
$869$	11.0000	+	19.0526i	0.373149	+	0.646314i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−11.0000	−	19.0526i	−0.371444	−	0.643359i	0.618344	−	0.785907i	$-0.287804\pi$
$877$	−11.0000	−	19.0526i	−0.371444	−	0.643359i	−0.989788	+	0.142548i	$0.954470\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−4.00000			−0.134763			−0.0673817	−	0.997727i	$-0.521465\pi$
$881$	−4.00000			−0.134763			−0.0673817	+	0.997727i	$0.521465\pi$
$882$		0			0
$883$	16.0000			0.538443			0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000			0.538443			0.269221	+	0.963078i	$0.413234\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	16.5000	+	28.5788i	0.554016	+	0.959583i	0.997979	+	0.0635387i	$0.0202386\pi$
$887$	16.5000	+	28.5788i	0.554016	+	0.959583i	−0.443964	+	0.896045i	$0.646428\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	4.00000	−	6.92820i	0.133855	−	0.231843i
$894$		0			0
$895$	12.0000	+	20.7846i	0.401116	+	0.694753i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−20.0000			−0.667037
$900$		0			0
$901$	9.00000			0.299833
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−8.50000	−	14.7224i	−0.282550	−	0.489390i
$906$		0			0
$907$	22.0000	−	38.1051i	0.730498	−	1.26526i	−0.226173	−	0.974087i	$-0.572621\pi$
$907$	22.0000	−	38.1051i	0.730498	−	1.26526i	0.956671	−	0.291172i	$-0.0940453\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	18.0000	−	31.1769i	0.596367	−	1.03294i	−0.396986	−	0.917825i	$-0.629944\pi$
$911$	18.0000	−	31.1769i	0.596367	−	1.03294i	0.993352	−	0.115113i	$-0.0367229\pi$
$912$		0			0
$913$	−9.00000	−	15.5885i	−0.297857	−	0.515903i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−40.0000			−1.31948			−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000			−1.31948			−0.659739	+	0.751495i	$0.729333\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−5.00000	+	8.66025i	−0.164399	+	0.284747i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−3.00000	+	5.19615i	−0.0984268	+	0.170480i	−0.911034	−	0.412332i	$-0.864714\pi$
$929$	−3.00000	+	5.19615i	−0.0984268	+	0.170480i	0.812607	+	0.582812i	$0.198048\pi$
$930$		0			0
$931$	−3.50000	−	6.06218i	−0.114708	−	0.198680i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	6.00000			0.196221
$936$		0			0
$937$	−12.0000			−0.392023			−0.196011	−	0.980602i	$-0.562799\pi$
$937$	−12.0000			−0.392023			−0.196011	+	0.980602i	$0.562799\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−5.00000	−	8.66025i	−0.162995	−	0.282316i	0.772946	−	0.634472i	$-0.218782\pi$
$941$	−5.00000	−	8.66025i	−0.162995	−	0.282316i	−0.935942	+	0.352155i	$0.885449\pi$
$942$		0			0
$943$	−9.00000	+	15.5885i	−0.293080	+	0.507630i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1.50000	−	2.59808i	0.0487435	−	0.0844261i	−0.840624	−	0.541619i	$-0.817812\pi$
$947$	1.50000	−	2.59808i	0.0487435	−	0.0844261i	0.889368	+	0.457193i	$0.151145\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−26.0000			−0.842223			−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000			−0.842223			−0.421111	+	0.907009i	$0.638360\pi$
$954$		0			0
$955$	−6.00000			−0.194155
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	3.00000	−	5.19615i	0.0967742	−	0.167618i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	10.0000	−	17.3205i	0.321911	−	0.557567i
$966$		0			0
$967$	21.0000	+	36.3731i	0.675314	+	1.16968i	0.976377	+	0.216075i	$0.0693254\pi$
$967$	21.0000	+	36.3731i	0.675314	+	1.16968i	−0.301062	+	0.953604i	$0.597341\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−6.00000			−0.192549			−0.0962746	−	0.995355i	$-0.530693\pi$
$971$	−6.00000			−0.192549			−0.0962746	+	0.995355i	$0.530693\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	25.0000	+	43.3013i	0.799821	+	1.38533i	0.919732	+	0.392546i	$0.128406\pi$
$977$	25.0000	+	43.3013i	0.799821	+	1.38533i	−0.119912	+	0.992785i	$0.538261\pi$
$978$		0			0
$979$	−10.0000	+	17.3205i	−0.319601	+	0.553566i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−5.50000	+	9.52628i	−0.175423	+	0.303841i	−0.940307	−	0.340326i	$-0.889463\pi$
$983$	−5.50000	+	9.52628i	−0.175423	+	0.303841i	0.764885	+	0.644167i	$0.222796\pi$
$984$		0			0
$985$	−2.50000	−	4.33013i	−0.0796566	−	0.137969i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−18.0000			−0.572367
$990$		0			0
$991$	1.00000			0.0317660			0.0158830	−	0.999874i	$-0.494944\pi$
$991$	1.00000			0.0317660			0.0158830	+	0.999874i	$0.494944\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	6.00000	+	10.3923i	0.190213	+	0.329458i
$996$		0			0
$997$	9.00000	−	15.5885i	0.285033	−	0.493691i	−0.687584	−	0.726105i	$-0.741329\pi$
$997$	9.00000	−	15.5885i	0.285033	−	0.493691i	0.972617	+	0.232413i	$0.0746622\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3240.2.q.e.2161.1		2
3.2	odd	2		3240.2.q.s.2161.1		2
9.2	odd	6		1080.2.a.d.1.1	✓	1
9.4	even	3	inner	3240.2.q.e.1081.1		2
9.5	odd	6		3240.2.q.s.1081.1		2
9.7	even	3		1080.2.a.j.1.1	yes	1
36.7	odd	6		2160.2.a.r.1.1		1
36.11	even	6		2160.2.a.f.1.1		1
45.2	even	12		5400.2.f.j.649.1		2
45.7	odd	12		5400.2.f.r.649.2		2
45.29	odd	6		5400.2.a.w.1.1		1
45.34	even	6		5400.2.a.x.1.1		1
45.38	even	12		5400.2.f.j.649.2		2
45.43	odd	12		5400.2.f.r.649.1		2
72.11	even	6		8640.2.a.bs.1.1		1
72.29	odd	6		8640.2.a.bt.1.1		1
72.43	odd	6		8640.2.a.p.1.1		1
72.61	even	6		8640.2.a.o.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.2.a.d.1.1	✓	1	9.2	odd	6
1080.2.a.j.1.1	yes	1	9.7	even	3
2160.2.a.f.1.1		1	36.11	even	6
2160.2.a.r.1.1		1	36.7	odd	6
3240.2.q.e.1081.1		2	9.4	even	3	inner
3240.2.q.e.2161.1		2	1.1	even	1	trivial
3240.2.q.s.1081.1		2	9.5	odd	6
3240.2.q.s.2161.1		2	3.2	odd	2
5400.2.a.w.1.1		1	45.29	odd	6
5400.2.a.x.1.1		1	45.34	even	6
5400.2.f.j.649.1		2	45.2	even	12
5400.2.f.j.649.2		2	45.38	even	12
5400.2.f.r.649.1		2	45.43	odd	12
5400.2.f.r.649.2		2	45.7	odd	12
8640.2.a.o.1.1		1	72.61	even	6
8640.2.a.p.1.1		1	72.43	odd	6
8640.2.a.bs.1.1		1	72.11	even	6
8640.2.a.bt.1.1		1	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:	\( N \)	\(=\)	\( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3240.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{5} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{5} +(-1.00000 + 1.73205i) q^{11} +3.00000 q^{17} -1.00000 q^{19} +(-1.50000 - 2.59808i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(-2.00000 + 3.46410i) q^{29} +(2.50000 + 4.33013i) q^{31} +10.0000 q^{37} +(-3.00000 - 5.19615i) q^{41} +(3.00000 - 5.19615i) q^{43} +(-4.00000 + 6.92820i) q^{47} +(3.50000 + 6.06218i) q^{49} +3.00000 q^{53} +2.00000 q^{55} +(-2.50000 + 4.33013i) q^{61} +(1.00000 + 1.73205i) q^{67} +2.00000 q^{71} +6.00000 q^{73} +(5.50000 - 9.52628i) q^{79} +(-4.50000 + 7.79423i) q^{83} +(-1.50000 - 2.59808i) q^{85} +10.0000 q^{89} +(0.500000 + 0.866025i) q^{95} +(-4.00000 + 6.92820i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{5}+O(q^{10}) \) 2 * q - q^5	\( 2 q - q^{5} - 2 q^{11} + 6 q^{17} - 2 q^{19} - 3 q^{23} - q^{25} - 4 q^{29} + 5 q^{31} + 20 q^{37} - 6 q^{41} + 6 q^{43} - 8 q^{47} + 7 q^{49} + 6 q^{53} + 4 q^{55} - 5 q^{61} + 2 q^{67} + 4 q^{71} + 12 q^{73} + 11 q^{79} - 9 q^{83} - 3 q^{85} + 20 q^{89} + q^{95} - 8 q^{97}+O(q^{100}) \) 2 * q - q^5 - 2 * q^11 + 6 * q^17 - 2 * q^19 - 3 * q^23 - q^25 - 4 * q^29 + 5 * q^31 + 20 * q^37 - 6 * q^41 + 6 * q^43 - 8 * q^47 + 7 * q^49 + 6 * q^53 + 4 * q^55 - 5 * q^61 + 2 * q^67 + 4 * q^71 + 12 * q^73 + 11 * q^79 - 9 * q^83 - 3 * q^85 + 20 * q^89 + q^95 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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