LMFDB - Embedded newform 2340.2.bp.c.1513.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2340,2,Mod(1477,2340)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2340, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2340.1477"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2340.bp (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,4,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$18.6849940730$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1513.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2340.1513
Dual form			2340.2.bp.c.1477.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.00000 - 1.00000i) q^{5} -4.00000 q^{7} +(-2.00000 - 2.00000i) q^{11} +(-3.00000 - 2.00000i) q^{13} +(3.00000 + 3.00000i) q^{17} +(2.00000 + 2.00000i) q^{19} +(2.00000 - 2.00000i) q^{23} +(3.00000 - 4.00000i) q^{25} +(-8.00000 + 4.00000i) q^{35} -4.00000 q^{37} +(-7.00000 + 7.00000i) q^{41} +(-8.00000 + 8.00000i) q^{43} -4.00000 q^{47} +9.00000 q^{49} +(-7.00000 - 7.00000i) q^{53} +(-6.00000 - 2.00000i) q^{55} +(-10.0000 + 10.0000i) q^{59} +(-8.00000 - 1.00000i) q^{65} -4.00000i q^{67} +(-8.00000 + 8.00000i) q^{71} +10.0000i q^{73} +(8.00000 + 8.00000i) q^{77} -4.00000i q^{79} -16.0000 q^{83} +(9.00000 + 3.00000i) q^{85} +(7.00000 - 7.00000i) q^{89} +(12.0000 + 8.00000i) q^{91} +(6.00000 + 2.00000i) q^{95} -12.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{5} - 8 q^{7} - 4 q^{11} - 6 q^{13} + 6 q^{17} + 4 q^{19} + 4 q^{23} + 6 q^{25} - 16 q^{35} - 8 q^{37} - 14 q^{41} - 16 q^{43} - 8 q^{47} + 18 q^{49} - 14 q^{53} - 12 q^{55} - 20 q^{59} - 16 q^{65}+ \cdots + 12 q^{95}+O(q^{100}) $ 2 * q + 4 * q^5 - 8 * q^7 - 4 * q^11 - 6 * q^13 + 6 * q^17 + 4 * q^19 + 4 * q^23 + 6 * q^25 - 16 * q^35 - 8 * q^37 - 14 * q^41 - 16 * q^43 - 8 * q^47 + 18 * q^49 - 14 * q^53 - 12 * q^55 - 20 * q^59 - 16 * q^65 - 16 * q^71 + 16 * q^77 - 32 * q^83 + 18 * q^85 + 14 * q^89 + 24 * q^91 + 12 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2340\mathbb{Z}\right)^\times$.

$n$	$937$	$1081$	$1171$	$2081$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$e\left(\frac{3}{4}\right)$	$1$	$1$

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
$5$	2.00000	−	1.00000i	0.894427	−	0.447214i
$5$	2.00000	−	1.00000i	0.894427	−	0.447214i
$6$		0			0
$7$	−4.00000			−1.51186			−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000			−1.51186			−0.755929	+	0.654654i	$0.772814\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.00000	−	2.00000i	−0.603023	−	0.603023i	0.338091	−	0.941113i	$-0.390219\pi$
$11$	−2.00000	−	2.00000i	−0.603023	−	0.603023i	−0.941113	+	0.338091i	$0.890219\pi$
$12$		0			0
$13$	−3.00000	−	2.00000i	−0.832050	−	0.554700i
$13$	−3.00000	−	2.00000i	−0.832050	−	0.554700i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.00000	+	3.00000i	0.727607	+	0.727607i	0.970143	−	0.242536i	$-0.0779791\pi$
$17$	3.00000	+	3.00000i	0.727607	+	0.727607i	−0.242536	+	0.970143i	$0.577979\pi$
$18$		0			0
$19$	2.00000	+	2.00000i	0.458831	+	0.458831i	0.898272	−	0.439440i	$-0.144823\pi$
$19$	2.00000	+	2.00000i	0.458831	+	0.458831i	−0.439440	+	0.898272i	$0.644823\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.00000	−	2.00000i	0.417029	−	0.417029i	−0.467150	−	0.884178i	$-0.654719\pi$
$23$	2.00000	−	2.00000i	0.417029	−	0.417029i	0.884178	+	0.467150i	$0.154719\pi$
$24$		0			0
$25$	3.00000	−	4.00000i	0.600000	−	0.800000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$31$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−8.00000	+	4.00000i	−1.35225	+	0.676123i
$36$		0			0
$37$	−4.00000			−0.657596			−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000			−0.657596			−0.328798	+	0.944400i	$0.606644\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−7.00000	+	7.00000i	−1.09322	+	1.09322i	−0.0980332	+	0.995183i	$0.531255\pi$
$41$	−7.00000	+	7.00000i	−1.09322	+	1.09322i	−0.995183	+	0.0980332i	$0.968745\pi$
$42$		0			0
$43$	−8.00000	+	8.00000i	−1.21999	+	1.21999i	−0.252353	+	0.967635i	$0.581205\pi$
$43$	−8.00000	+	8.00000i	−1.21999	+	1.21999i	−0.967635	+	0.252353i	$0.918795\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.00000			−0.583460			−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000			−0.583460			−0.291730	+	0.956501i	$0.594231\pi$
$48$		0			0
$49$	9.00000			1.28571
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−7.00000	−	7.00000i	−0.961524	−	0.961524i	0.0377628	−	0.999287i	$-0.487977\pi$
$53$	−7.00000	−	7.00000i	−0.961524	−	0.961524i	−0.999287	+	0.0377628i	$0.987977\pi$
$54$		0			0
$55$	−6.00000	−	2.00000i	−0.809040	−	0.269680i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−10.0000	+	10.0000i	−1.30189	+	1.30189i	−0.374772	+	0.927117i	$0.622279\pi$
$59$	−10.0000	+	10.0000i	−1.30189	+	1.30189i	−0.927117	+	0.374772i	$0.877721\pi$
$60$		0			0
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−8.00000	−	1.00000i	−0.992278	−	0.124035i
$66$		0			0
$67$		−	4.00000i		−	0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$		−	4.00000i		−	0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−8.00000	+	8.00000i	−0.949425	+	0.949425i	−0.998781	−	0.0493559i	$-0.984283\pi$
$71$	−8.00000	+	8.00000i	−0.949425	+	0.949425i	0.0493559	+	0.998781i	$0.484283\pi$
$72$		0			0
$73$			10.0000i			1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$			10.0000i			1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	8.00000	+	8.00000i	0.911685	+	0.911685i
$78$		0			0
$79$		−	4.00000i		−	0.450035i	−0.974355	−	0.225018i	$-0.927756\pi$
$79$		−	4.00000i		−	0.450035i	0.974355	−	0.225018i	$-0.0722440\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−16.0000			−1.75623			−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000			−1.75623			−0.878114	+	0.478451i	$0.841198\pi$
$84$		0			0
$85$	9.00000	+	3.00000i	0.976187	+	0.325396i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	7.00000	−	7.00000i	0.741999	−	0.741999i	−0.230964	−	0.972962i	$-0.574188\pi$
$89$	7.00000	−	7.00000i	0.741999	−	0.741999i	0.972962	+	0.230964i	$0.0741879\pi$
$90$		0			0
$91$	12.0000	+	8.00000i	1.25794	+	0.838628i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	6.00000	+	2.00000i	0.615587	+	0.205196i
$96$		0			0
$97$		−	12.0000i		−	1.21842i	−0.793011	−	0.609208i	$-0.791488\pi$
$97$		−	12.0000i		−	1.21842i	0.793011	−	0.609208i	$-0.208512\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	6.00000i		−	0.597022i	−0.954406	−	0.298511i	$-0.903510\pi$
$101$		−	6.00000i		−	0.597022i	0.954406	−	0.298511i	$-0.0964900\pi$
$102$		0			0
$103$	6.00000	−	6.00000i	0.591198	−	0.591198i	−0.346757	−	0.937955i	$-0.612717\pi$
$103$	6.00000	−	6.00000i	0.591198	−	0.591198i	0.937955	+	0.346757i	$0.112717\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−12.0000	+	12.0000i	−1.16008	+	1.16008i	−0.175627	+	0.984457i	$0.556195\pi$
$107$	−12.0000	+	12.0000i	−1.16008	+	1.16008i	−0.984457	+	0.175627i	$0.943805\pi$
$108$		0			0
$109$	13.0000	+	13.0000i	1.24517	+	1.24517i	0.957826	+	0.287348i	$0.0927736\pi$
$109$	13.0000	+	13.0000i	1.24517	+	1.24517i	0.287348	+	0.957826i	$0.407226\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.00000	+	1.00000i	0.0940721	+	0.0940721i	0.752577	−	0.658505i	$-0.228811\pi$
$113$	1.00000	+	1.00000i	0.0940721	+	0.0940721i	−0.658505	+	0.752577i	$0.728811\pi$
$114$		0			0
$115$	2.00000	−	6.00000i	0.186501	−	0.559503i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−12.0000	−	12.0000i	−1.10004	−	1.10004i
$120$		0			0
$121$		−	3.00000i		−	0.272727i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	2.00000	−	11.0000i	0.178885	−	0.983870i
$126$		0			0
$127$	−10.0000	−	10.0000i	−0.887357	−	0.887357i	0.106912	−	0.994268i	$-0.465904\pi$
$127$	−10.0000	−	10.0000i	−0.887357	−	0.887357i	−0.994268	+	0.106912i	$0.965904\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−8.00000			−0.698963			−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000			−0.698963			−0.349482	+	0.936943i	$0.613642\pi$
$132$		0			0
$133$	−8.00000	−	8.00000i	−0.693688	−	0.693688i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−12.0000			−1.02523			−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000			−1.02523			−0.512615	+	0.858619i	$0.671323\pi$
$138$		0			0
$139$		−	12.0000i		−	1.01783i	−0.860818	−	0.508913i	$-0.830047\pi$
$139$		−	12.0000i		−	1.01783i	0.860818	−	0.508913i	$-0.169953\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2.00000	+	10.0000i	0.167248	+	0.836242i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	17.0000	+	17.0000i	1.39269	+	1.39269i	0.819232	+	0.573462i	$0.194400\pi$
$149$	17.0000	+	17.0000i	1.39269	+	1.39269i	0.573462	+	0.819232i	$0.305600\pi$
$150$		0			0
$151$	4.00000	+	4.00000i	0.325515	+	0.325515i	0.850878	−	0.525363i	$-0.176070\pi$
$151$	4.00000	+	4.00000i	0.325515	+	0.325515i	−0.525363	+	0.850878i	$0.676070\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−3.00000	+	3.00000i	−0.239426	+	0.239426i	−0.816612	−	0.577186i	$-0.804151\pi$
$157$	−3.00000	+	3.00000i	−0.239426	+	0.239426i	0.577186	+	0.816612i	$0.304151\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−8.00000	+	8.00000i	−0.630488	+	0.630488i
$162$		0			0
$163$			16.0000i			1.25322i	0.779334	+	0.626608i	$0.215557\pi$
$163$			16.0000i			1.25322i	−0.779334	+	0.626608i	$0.784443\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−12.0000			−0.928588			−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000			−0.928588			−0.464294	+	0.885681i	$0.653692\pi$
$168$		0			0
$169$	5.00000	+	12.0000i	0.384615	+	0.923077i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	15.0000	−	15.0000i	1.14043	−	1.14043i	0.152057	−	0.988372i	$-0.451410\pi$
$173$	15.0000	−	15.0000i	1.14043	−	1.14043i	0.988372	−	0.152057i	$-0.0485898\pi$
$174$		0			0
$175$	−12.0000	+	16.0000i	−0.907115	+	1.20949i
$176$		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
$181$		−	10.0000i		−	0.743294i	−0.928374	−	0.371647i	$-0.878793\pi$
$181$		−	10.0000i		−	0.743294i	0.928374	−	0.371647i	$-0.121207\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−8.00000	+	4.00000i	−0.588172	+	0.294086i
$186$		0			0
$187$		−	12.0000i		−	0.877527i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	12.0000			0.868290			0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000			0.868290			0.434145	+	0.900843i	$0.357051\pi$
$192$		0			0
$193$		−	4.00000i		−	0.287926i	−0.989583	−	0.143963i	$-0.954015\pi$
$193$		−	4.00000i		−	0.287926i	0.989583	−	0.143963i	$-0.0459847\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			6.00000i			0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$			6.00000i			0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$		0			0
$199$	8.00000			0.567105			0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000			0.567105			0.283552	+	0.958957i	$0.408487\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−7.00000	+	21.0000i	−0.488901	+	1.46670i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	8.00000i		−	0.553372i
$210$		0			0
$211$	−16.0000			−1.10149			−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000			−1.10149			−0.550743	+	0.834675i	$0.685655\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−8.00000	+	24.0000i	−0.545595	+	1.63679i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−3.00000	−	15.0000i	−0.201802	−	1.00901i
$222$		0			0
$223$	8.00000			0.535720			0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000			0.535720			0.267860	+	0.963458i	$0.413684\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	8.00000i		−	0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$		−	8.00000i		−	0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$		0			0
$229$	−13.0000	+	13.0000i	−0.859064	+	0.859064i	−0.991228	−	0.132164i	$-0.957808\pi$
$229$	−13.0000	+	13.0000i	−0.859064	+	0.859064i	0.132164	+	0.991228i	$0.457808\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5.00000	+	5.00000i	−0.327561	+	0.327561i	−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−5.00000	+	5.00000i	−0.327561	+	0.327561i	0.524097	+	0.851658i	$0.324403\pi$
$234$		0			0
$235$	−8.00000	+	4.00000i	−0.521862	+	0.260931i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−8.00000	−	8.00000i	−0.517477	−	0.517477i	0.399330	−	0.916807i	$-0.369243\pi$
$239$	−8.00000	−	8.00000i	−0.517477	−	0.517477i	−0.916807	+	0.399330i	$0.869243\pi$
$240$		0			0
$241$	−11.0000	−	11.0000i	−0.708572	−	0.708572i	0.257663	−	0.966235i	$-0.417048\pi$
$241$	−11.0000	−	11.0000i	−0.708572	−	0.708572i	−0.966235	+	0.257663i	$0.917048\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	18.0000	−	9.00000i	1.14998	−	0.574989i
$246$		0			0
$247$	−2.00000	−	10.0000i	−0.127257	−	0.636285i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			4.00000i			0.252478i	0.992000	+	0.126239i	$0.0402906\pi$
$251$			4.00000i			0.252478i	−0.992000	+	0.126239i	$0.959709\pi$
$252$		0			0
$253$	−8.00000			−0.502956
$254$		0			0
$255$		0			0
$256$		0			0
$257$	7.00000	+	7.00000i	0.436648	+	0.436648i	0.890882	−	0.454234i	$-0.150087\pi$
$257$	7.00000	+	7.00000i	0.436648	+	0.436648i	−0.454234	+	0.890882i	$0.650087\pi$
$258$		0			0
$259$	16.0000			0.994192
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−10.0000	−	10.0000i	−0.616626	−	0.616626i	0.328038	−	0.944664i	$-0.393613\pi$
$263$	−10.0000	−	10.0000i	−0.616626	−	0.616626i	−0.944664	+	0.328038i	$0.893613\pi$
$264$		0			0
$265$	−21.0000	−	7.00000i	−1.29002	−	0.430007i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			2.00000i			0.121942i	0.998140	+	0.0609711i	$0.0194197\pi$
$269$			2.00000i			0.121942i	−0.998140	+	0.0609711i	$0.980580\pi$
$270$		0			0
$271$	−12.0000	−	12.0000i	−0.728948	−	0.728948i	0.241462	−	0.970410i	$-0.422373\pi$
$271$	−12.0000	−	12.0000i	−0.728948	−	0.728948i	−0.970410	+	0.241462i	$0.922373\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−14.0000	+	2.00000i	−0.844232	+	0.120605i
$276$		0			0
$277$	9.00000	+	9.00000i	0.540758	+	0.540758i	0.923751	−	0.382993i	$-0.125107\pi$
$277$	9.00000	+	9.00000i	0.540758	+	0.540758i	−0.382993	+	0.923751i	$0.625107\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−7.00000	−	7.00000i	−0.417585	−	0.417585i	0.466786	−	0.884371i	$-0.345412\pi$
$281$	−7.00000	−	7.00000i	−0.417585	−	0.417585i	−0.884371	+	0.466786i	$0.845412\pi$
$282$		0			0
$283$	12.0000	−	12.0000i	0.713326	−	0.713326i	−0.253904	−	0.967230i	$-0.581715\pi$
$283$	12.0000	−	12.0000i	0.713326	−	0.713326i	0.967230	+	0.253904i	$0.0817146\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	28.0000	−	28.0000i	1.65279	−	1.65279i
$288$		0			0
$289$			1.00000i			0.0588235i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			28.0000i			1.63578i	0.575376	+	0.817889i	$0.304856\pi$
$293$			28.0000i			1.63578i	−0.575376	+	0.817889i	$0.695144\pi$
$294$		0			0
$295$	−10.0000	+	30.0000i	−0.582223	+	1.74667i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−10.0000	+	2.00000i	−0.578315	+	0.115663i
$300$		0			0
$301$	32.0000	−	32.0000i	1.84445	−	1.84445i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	16.0000			0.913168			0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000			0.913168			0.456584	+	0.889680i	$0.349073\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	12.0000i		−	0.680458i	−0.940343	−	0.340229i	$-0.889495\pi$
$311$		−	12.0000i		−	0.680458i	0.940343	−	0.340229i	$-0.110505\pi$
$312$		0			0
$313$	−9.00000	−	9.00000i	−0.508710	−	0.508710i	0.405420	−	0.914130i	$-0.367125\pi$
$313$	−9.00000	−	9.00000i	−0.508710	−	0.508710i	−0.914130	+	0.405420i	$0.867125\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	12.0000i		−	0.673987i	−0.941507	−	0.336994i	$-0.890590\pi$
$317$		−	12.0000i		−	0.673987i	0.941507	−	0.336994i	$-0.109410\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$			12.0000i			0.667698i
$324$		0			0
$325$	−17.0000	+	6.00000i	−0.942990	+	0.332820i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	16.0000			0.882109
$330$		0			0
$331$	18.0000	−	18.0000i	0.989369	−	0.989369i	−0.0105746	−	0.999944i	$-0.503366\pi$
$331$	18.0000	−	18.0000i	0.989369	−	0.989369i	0.999944	+	0.0105746i	$0.00336607\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−4.00000	−	8.00000i	−0.218543	−	0.437087i
$336$		0			0
$337$	−9.00000	−	9.00000i	−0.490261	−	0.490261i	0.418127	−	0.908388i	$-0.362687\pi$
$337$	−9.00000	−	9.00000i	−0.490261	−	0.490261i	−0.908388	+	0.418127i	$0.862687\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−8.00000			−0.431959
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−8.00000	+	8.00000i	−0.429463	+	0.429463i	−0.888445	−	0.458983i	$-0.848214\pi$
$347$	−8.00000	+	8.00000i	−0.429463	+	0.429463i	0.458983	+	0.888445i	$0.348214\pi$
$348$		0			0
$349$	7.00000	−	7.00000i	0.374701	−	0.374701i	−0.494485	−	0.869186i	$-0.664643\pi$
$349$	7.00000	−	7.00000i	0.374701	−	0.374701i	0.869186	+	0.494485i	$0.164643\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−34.0000			−1.80964			−0.904819	−	0.425797i	$-0.859994\pi$
$353$	−34.0000			−1.80964			−0.904819	+	0.425797i	$0.859994\pi$
$354$		0			0
$355$	−8.00000	+	24.0000i	−0.424596	+	1.27379i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−8.00000	+	8.00000i	−0.422224	+	0.422224i	−0.885969	−	0.463745i	$-0.846505\pi$
$359$	−8.00000	+	8.00000i	−0.422224	+	0.422224i	0.463745	+	0.885969i	$0.346505\pi$
$360$		0			0
$361$		−	11.0000i		−	0.578947i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	10.0000	+	20.0000i	0.523424	+	1.04685i
$366$		0			0
$367$	−2.00000	+	2.00000i	−0.104399	+	0.104399i	−0.757377	−	0.652978i	$-0.773519\pi$
$367$	−2.00000	+	2.00000i	−0.104399	+	0.104399i	0.652978	+	0.757377i	$0.273519\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	28.0000	+	28.0000i	1.45369	+	1.45369i
$372$		0			0
$373$	−15.0000	−	15.0000i	−0.776671	−	0.776671i	0.202593	−	0.979263i	$-0.435063\pi$
$373$	−15.0000	−	15.0000i	−0.776671	−	0.776671i	−0.979263	+	0.202593i	$0.935063\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	18.0000	+	18.0000i	0.924598	+	0.924598i	0.997350	−	0.0727522i	$-0.0231782\pi$
$379$	18.0000	+	18.0000i	0.924598	+	0.924598i	−0.0727522	+	0.997350i	$0.523178\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	16.0000			0.817562			0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000			0.817562			0.408781	+	0.912633i	$0.365954\pi$
$384$		0			0
$385$	24.0000	+	8.00000i	1.22315	+	0.407718i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−8.00000			−0.405616			−0.202808	−	0.979219i	$-0.565007\pi$
$389$	−8.00000			−0.405616			−0.202808	+	0.979219i	$0.565007\pi$
$390$		0			0
$391$	12.0000			0.606866
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−4.00000	−	8.00000i	−0.201262	−	0.402524i
$396$		0			0
$397$	28.0000			1.40528			0.702640	−	0.711546i	$-0.252005\pi$
$397$	28.0000			1.40528			0.702640	+	0.711546i	$0.252005\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.00000	+	1.00000i	0.0499376	+	0.0499376i	0.731635	−	0.681697i	$-0.238758\pi$
$401$	1.00000	+	1.00000i	0.0499376	+	0.0499376i	−0.681697	+	0.731635i	$0.738758\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	8.00000	+	8.00000i	0.396545	+	0.396545i
$408$		0			0
$409$	21.0000	+	21.0000i	1.03838	+	1.03838i	0.999233	+	0.0391498i	$0.0124650\pi$
$409$	21.0000	+	21.0000i	1.03838	+	1.03838i	0.0391498	+	0.999233i	$0.487535\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	40.0000	−	40.0000i	1.96827	−	1.96827i
$414$		0			0
$415$	−32.0000	+	16.0000i	−1.57082	+	0.785409i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	−19.0000	+	19.0000i	−0.926003	+	0.926003i	−0.997445	−	0.0714415i	$-0.977240\pi$
$421$	−19.0000	+	19.0000i	−0.926003	+	0.926003i	0.0714415	+	0.997445i	$0.477240\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	21.0000	−	3.00000i	1.01865	−	0.145521i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	24.0000	−	24.0000i	1.15604	−	1.15604i	0.170720	−	0.985320i	$-0.445391\pi$
$431$	24.0000	−	24.0000i	1.15604	−	1.15604i	0.985320	−	0.170720i	$-0.0546093\pi$
$432$		0			0
$433$	−27.0000	+	27.0000i	−1.29754	+	1.29754i	−0.367523	+	0.930015i	$0.619794\pi$
$433$	−27.0000	+	27.0000i	−1.29754	+	1.29754i	−0.930015	+	0.367523i	$0.880206\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	8.00000			0.382692
$438$		0			0
$439$	4.00000			0.190910			0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000			0.190910			0.0954548	+	0.995434i	$0.469569\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$443$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$444$		0			0
$445$	7.00000	−	21.0000i	0.331832	−	0.995495i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	7.00000	−	7.00000i	0.330350	−	0.330350i	−0.522369	−	0.852720i	$-0.674952\pi$
$449$	7.00000	−	7.00000i	0.330350	−	0.330350i	0.852720	+	0.522369i	$0.174952\pi$
$450$		0			0
$451$	28.0000			1.31847
$452$		0			0
$453$		0			0
$454$		0			0
$455$	32.0000	+	4.00000i	1.50018	+	0.187523i
$456$		0			0
$457$		−	4.00000i		−	0.187112i	−0.995614	−	0.0935561i	$-0.970177\pi$
$457$		−	4.00000i		−	0.187112i	0.995614	−	0.0935561i	$-0.0298234\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−11.0000	+	11.0000i	−0.512321	+	0.512321i	−0.915237	−	0.402916i	$-0.867997\pi$
$461$	−11.0000	+	11.0000i	−0.512321	+	0.512321i	0.402916	+	0.915237i	$0.367997\pi$
$462$		0			0
$463$		−	20.0000i		−	0.929479i	−0.885448	−	0.464739i	$-0.846148\pi$
$463$		−	20.0000i		−	0.929479i	0.885448	−	0.464739i	$-0.153852\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−20.0000	−	20.0000i	−0.925490	−	0.925490i	0.0719207	−	0.997410i	$-0.477087\pi$
$467$	−20.0000	−	20.0000i	−0.925490	−	0.925490i	−0.997410	+	0.0719207i	$0.977087\pi$
$468$		0			0
$469$			16.0000i			0.738811i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	32.0000			1.47136
$474$		0			0
$475$	14.0000	−	2.00000i	0.642364	−	0.0917663i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−8.00000	+	8.00000i	−0.365529	+	0.365529i	−0.865844	−	0.500314i	$-0.833218\pi$
$479$	−8.00000	+	8.00000i	−0.365529	+	0.365529i	0.500314	+	0.865844i	$0.333218\pi$
$480$		0			0
$481$	12.0000	+	8.00000i	0.547153	+	0.364769i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−12.0000	−	24.0000i	−0.544892	−	1.08978i
$486$		0			0
$487$		−	16.0000i		−	0.725029i	−0.931978	−	0.362515i	$-0.881918\pi$
$487$		−	16.0000i		−	0.725029i	0.931978	−	0.362515i	$-0.118082\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	24.0000i		−	1.08310i	−0.840667	−	0.541552i	$-0.817837\pi$
$491$		−	24.0000i		−	1.08310i	0.840667	−	0.541552i	$-0.182163\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	32.0000	−	32.0000i	1.43540	−	1.43540i
$498$		0			0
$499$	2.00000	+	2.00000i	0.0895323	+	0.0895323i	0.750454	−	0.660922i	$-0.229835\pi$
$499$	2.00000	+	2.00000i	0.0895323	+	0.0895323i	−0.660922	+	0.750454i	$0.729835\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	18.0000	+	18.0000i	0.802580	+	0.802580i	0.983498	−	0.180918i	$-0.0579068\pi$
$503$	18.0000	+	18.0000i	0.802580	+	0.802580i	−0.180918	+	0.983498i	$0.557907\pi$
$504$		0			0
$505$	−6.00000	−	12.0000i	−0.266996	−	0.533993i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−9.00000	−	9.00000i	−0.398918	−	0.398918i	0.478933	−	0.877851i	$-0.341024\pi$
$509$	−9.00000	−	9.00000i	−0.398918	−	0.398918i	−0.877851	+	0.478933i	$0.841024\pi$
$510$		0			0
$511$		−	40.0000i		−	1.76950i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	6.00000	−	18.0000i	0.264392	−	0.793175i
$516$		0			0
$517$	8.00000	+	8.00000i	0.351840	+	0.351840i
$518$		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
$523$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$523$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$			15.0000i			0.652174i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	35.0000	−	7.00000i	1.51602	−	0.303204i
$534$		0			0
$535$	−12.0000	+	36.0000i	−0.518805	+	1.55642i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−18.0000	−	18.0000i	−0.775315	−	0.775315i
$540$		0			0
$541$	−29.0000	−	29.0000i	−1.24681	−	1.24681i	−0.957122	−	0.289685i	$-0.906449\pi$
$541$	−29.0000	−	29.0000i	−1.24681	−	1.24681i	−0.289685	−	0.957122i	$-0.593551\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	39.0000	+	13.0000i	1.67058	+	0.556859i
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$			16.0000i			0.680389i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	26.0000			1.10166			0.550828	−	0.834619i	$-0.314312\pi$
$557$	26.0000			1.10166			0.550828	+	0.834619i	$0.314312\pi$
$558$		0			0
$559$	40.0000	−	8.00000i	1.69182	−	0.338364i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−20.0000	+	20.0000i	−0.842900	+	0.842900i	−0.989235	−	0.146336i	$-0.953252\pi$
$563$	−20.0000	+	20.0000i	−0.842900	+	0.842900i	0.146336	+	0.989235i	$0.453252\pi$
$564$		0			0
$565$	3.00000	+	1.00000i	0.126211	+	0.0420703i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	18.0000			0.754599			0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000			0.754599			0.377300	+	0.926091i	$0.376853\pi$
$570$		0			0
$571$			8.00000i			0.334790i	0.985890	+	0.167395i	$0.0535355\pi$
$571$			8.00000i			0.334790i	−0.985890	+	0.167395i	$0.946465\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−2.00000	−	14.0000i	−0.0834058	−	0.583840i
$576$		0			0
$577$		−	30.0000i		−	1.24892i	−0.781058	−	0.624458i	$-0.785320\pi$
$577$		−	30.0000i		−	1.24892i	0.781058	−	0.624458i	$-0.214680\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	64.0000			2.65517
$582$		0			0
$583$			28.0000i			1.15964i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			12.0000i			0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$			12.0000i			0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	18.0000i		−	0.739171i	−0.929197	−	0.369586i	$-0.879500\pi$
$593$		−	18.0000i		−	0.739171i	0.929197	−	0.369586i	$-0.120500\pi$
$594$		0			0
$595$	−36.0000	−	12.0000i	−1.47586	−	0.491952i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			48.0000i			1.96123i	0.195952	+	0.980613i	$0.437220\pi$
$599$			48.0000i			1.96123i	−0.195952	+	0.980613i	$0.562780\pi$
$600$		0			0
$601$	2.00000			0.0815817			0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000			0.0815817			0.0407909	+	0.999168i	$0.487012\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−3.00000	−	6.00000i	−0.121967	−	0.243935i
$606$		0			0
$607$	2.00000	−	2.00000i	0.0811775	−	0.0811775i	−0.665352	−	0.746530i	$-0.731719\pi$
$607$	2.00000	−	2.00000i	0.0811775	−	0.0811775i	0.746530	+	0.665352i	$0.231719\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	12.0000	+	8.00000i	0.485468	+	0.323645i
$612$		0			0
$613$	−6.00000			−0.242338			−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000			−0.242338			−0.121169	+	0.992632i	$0.538664\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	10.0000i		−	0.402585i	−0.979531	−	0.201292i	$-0.935486\pi$
$617$		−	10.0000i		−	0.402585i	0.979531	−	0.201292i	$-0.0645141\pi$
$618$		0			0
$619$	18.0000	−	18.0000i	0.723481	−	0.723481i	−0.245831	−	0.969313i	$-0.579061\pi$
$619$	18.0000	−	18.0000i	0.723481	−	0.723481i	0.969313	+	0.245831i	$0.0790610\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−28.0000	+	28.0000i	−1.12180	+	1.12180i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−12.0000	−	12.0000i	−0.478471	−	0.478471i
$630$		0			0
$631$	−4.00000	−	4.00000i	−0.159237	−	0.159237i	0.622991	−	0.782229i	$-0.285917\pi$
$631$	−4.00000	−	4.00000i	−0.159237	−	0.159237i	−0.782229	+	0.622991i	$0.785917\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−30.0000	−	10.0000i	−1.19051	−	0.396838i
$636$		0			0
$637$	−27.0000	−	18.0000i	−1.06978	−	0.713186i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	26.0000i		−	1.02694i	−0.858108	−	0.513469i	$-0.828360\pi$
$641$		−	26.0000i		−	1.02694i	0.858108	−	0.513469i	$-0.171640\pi$
$642$		0			0
$643$	4.00000			0.157745			0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000			0.157745			0.0788723	+	0.996885i	$0.474868\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−22.0000	−	22.0000i	−0.864909	−	0.864909i	0.126994	−	0.991903i	$-0.459467\pi$
$647$	−22.0000	−	22.0000i	−0.864909	−	0.864909i	−0.991903	+	0.126994i	$0.959467\pi$
$648$		0			0
$649$	40.0000			1.57014
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−7.00000	−	7.00000i	−0.273931	−	0.273931i	0.556749	−	0.830681i	$-0.312048\pi$
$653$	−7.00000	−	7.00000i	−0.273931	−	0.273931i	−0.830681	+	0.556749i	$0.812048\pi$
$654$		0			0
$655$	−16.0000	+	8.00000i	−0.625172	+	0.312586i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	8.00000i		−	0.311636i	−0.987786	−	0.155818i	$-0.950199\pi$
$659$		−	8.00000i		−	0.311636i	0.987786	−	0.155818i	$-0.0498013\pi$
$660$		0			0
$661$	−5.00000	−	5.00000i	−0.194477	−	0.194477i	0.603150	−	0.797628i	$-0.293912\pi$
$661$	−5.00000	−	5.00000i	−0.194477	−	0.194477i	−0.797628	+	0.603150i	$0.793912\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−24.0000	−	8.00000i	−0.930680	−	0.310227i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−17.0000	+	17.0000i	−0.655302	+	0.655302i	−0.954265	−	0.298963i	$-0.903359\pi$
$673$	−17.0000	+	17.0000i	−0.655302	+	0.655302i	0.298963	+	0.954265i	$0.403359\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−15.0000	+	15.0000i	−0.576497	+	0.576497i	−0.933936	−	0.357439i	$-0.883650\pi$
$677$	−15.0000	+	15.0000i	−0.576497	+	0.576497i	0.357439	+	0.933936i	$0.383650\pi$
$678$		0			0
$679$			48.0000i			1.84207i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			24.0000i			0.918334i	0.888350	+	0.459167i	$0.151852\pi$
$683$			24.0000i			0.918334i	−0.888350	+	0.459167i	$0.848148\pi$
$684$		0			0
$685$	−24.0000	+	12.0000i	−0.916993	+	0.458496i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	7.00000	+	35.0000i	0.266679	+	1.33339i
$690$		0			0
$691$	26.0000	−	26.0000i	0.989087	−	0.989087i	−0.0108545	−	0.999941i	$-0.503455\pi$
$691$	26.0000	−	26.0000i	0.989087	−	0.989087i	0.999941	+	0.0108545i	$0.00345515\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−12.0000	−	24.0000i	−0.455186	−	0.910372i
$696$		0			0
$697$	−42.0000			−1.59086
$698$		0			0
$699$		0			0
$700$		0			0
$701$			46.0000i			1.73740i	0.495342	+	0.868698i	$0.335043\pi$
$701$			46.0000i			1.73740i	−0.495342	+	0.868698i	$0.664957\pi$
$702$		0			0
$703$	−8.00000	−	8.00000i	−0.301726	−	0.301726i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			24.0000i			0.902613i
$708$		0			0
$709$	−29.0000	+	29.0000i	−1.08912	+	1.08912i	−0.0934984	+	0.995619i	$0.529805\pi$
$709$	−29.0000	+	29.0000i	−1.08912	+	1.08912i	−0.995619	+	0.0934984i	$0.970195\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	14.0000	+	18.0000i	0.523570	+	0.673162i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−20.0000			−0.745874			−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000			−0.745874			−0.372937	+	0.927857i	$0.621649\pi$
$720$		0			0
$721$	−24.0000	+	24.0000i	−0.893807	+	0.893807i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−30.0000	−	30.0000i	−1.11264	−	1.11264i	−0.992793	−	0.119846i	$-0.961760\pi$
$727$	−30.0000	−	30.0000i	−1.11264	−	1.11264i	−0.119846	−	0.992793i	$-0.538240\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−48.0000			−1.77534
$732$		0			0
$733$	42.0000			1.55131			0.775653	−	0.631160i	$-0.217421\pi$
$733$	42.0000			1.55131			0.775653	+	0.631160i	$0.217421\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−8.00000	+	8.00000i	−0.294684	+	0.294684i
$738$		0			0
$739$	18.0000	−	18.0000i	0.662141	−	0.662141i	−0.293744	−	0.955884i	$-0.594901\pi$
$739$	18.0000	−	18.0000i	0.662141	−	0.662141i	0.955884	+	0.293744i	$0.0949012\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	8.00000			0.293492			0.146746	−	0.989174i	$-0.453120\pi$
$743$	8.00000			0.293492			0.146746	+	0.989174i	$0.453120\pi$
$744$		0			0
$745$	51.0000	+	17.0000i	1.86850	+	0.622832i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	48.0000	−	48.0000i	1.75388	−	1.75388i
$750$		0			0
$751$			32.0000i			1.16770i	0.811863	+	0.583848i	$0.198454\pi$
$751$			32.0000i			1.16770i	−0.811863	+	0.583848i	$0.801546\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	12.0000	+	4.00000i	0.436725	+	0.145575i
$756$		0			0
$757$	21.0000	−	21.0000i	0.763258	−	0.763258i	−0.213652	−	0.976910i	$-0.568536\pi$
$757$	21.0000	−	21.0000i	0.763258	−	0.763258i	0.976910	+	0.213652i	$0.0685358\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	13.0000	+	13.0000i	0.471250	+	0.471250i	0.902319	−	0.431069i	$-0.141864\pi$
$761$	13.0000	+	13.0000i	0.471250	+	0.471250i	−0.431069	+	0.902319i	$0.641864\pi$
$762$		0			0
$763$	−52.0000	−	52.0000i	−1.88253	−	1.88253i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	50.0000	−	10.0000i	1.80540	−	0.361079i
$768$		0			0
$769$	−27.0000	−	27.0000i	−0.973645	−	0.973645i	0.0260166	−	0.999662i	$-0.491718\pi$
$769$	−27.0000	−	27.0000i	−0.973645	−	0.973645i	−0.999662	+	0.0260166i	$0.991718\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−50.0000			−1.79838			−0.899188	−	0.437564i	$-0.855842\pi$
$773$	−50.0000			−1.79838			−0.899188	+	0.437564i	$0.855842\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−28.0000			−1.00320
$780$		0			0
$781$	32.0000			1.14505
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−3.00000	+	9.00000i	−0.107075	+	0.321224i
$786$		0			0
$787$	−32.0000			−1.14068			−0.570338	−	0.821410i	$-0.693188\pi$
$787$	−32.0000			−1.14068			−0.570338	+	0.821410i	$0.693188\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−4.00000	−	4.00000i	−0.142224	−	0.142224i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−13.0000	−	13.0000i	−0.460484	−	0.460484i	0.438330	−	0.898814i	$-0.355570\pi$
$797$	−13.0000	−	13.0000i	−0.460484	−	0.460484i	−0.898814	+	0.438330i	$0.855570\pi$
$798$		0			0
$799$	−12.0000	−	12.0000i	−0.424529	−	0.424529i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	20.0000	−	20.0000i	0.705785	−	0.705785i
$804$		0			0
$805$	−8.00000	+	24.0000i	−0.281963	+	0.845889i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	48.0000i		−	1.68759i	−0.536666	−	0.843795i	$-0.680316\pi$
$809$		−	48.0000i		−	1.68759i	0.536666	−	0.843795i	$-0.319684\pi$
$810$		0			0
$811$	−22.0000	+	22.0000i	−0.772524	+	0.772524i	−0.978547	−	0.206023i	$-0.933948\pi$
$811$	−22.0000	+	22.0000i	−0.772524	+	0.772524i	0.206023	+	0.978547i	$0.433948\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	16.0000	+	32.0000i	0.560456	+	1.12091i
$816$		0			0
$817$	−32.0000			−1.11954
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−29.0000	+	29.0000i	−1.01211	+	1.01211i	−0.0121812	+	0.999926i	$0.503877\pi$
$821$	−29.0000	+	29.0000i	−1.01211	+	1.01211i	−0.999926	+	0.0121812i	$0.996123\pi$
$822$		0			0
$823$	−6.00000	+	6.00000i	−0.209147	+	0.209147i	−0.803905	−	0.594758i	$-0.797248\pi$
$823$	−6.00000	+	6.00000i	−0.209147	+	0.209147i	0.594758	+	0.803905i	$0.297248\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−36.0000			−1.25184			−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000			−1.25184			−0.625921	+	0.779886i	$0.715277\pi$
$828$		0			0
$829$	40.0000			1.38926			0.694629	−	0.719368i	$-0.255569\pi$
$829$	40.0000			1.38926			0.694629	+	0.719368i	$0.255569\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	27.0000	+	27.0000i	0.935495	+	0.935495i
$834$		0			0
$835$	−24.0000	+	12.0000i	−0.830554	+	0.415277i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$839$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$	22.0000	+	19.0000i	0.756823	+	0.653620i
$846$		0			0
$847$			12.0000i			0.412325i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−8.00000	+	8.00000i	−0.274236	+	0.274236i
$852$		0			0
$853$			14.0000i			0.479351i	0.970853	+	0.239675i	$0.0770410\pi$
$853$			14.0000i			0.479351i	−0.970853	+	0.239675i	$0.922959\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−11.0000	−	11.0000i	−0.375753	−	0.375753i	0.493814	−	0.869567i	$-0.335602\pi$
$857$	−11.0000	−	11.0000i	−0.375753	−	0.375753i	−0.869567	+	0.493814i	$0.835602\pi$
$858$		0			0
$859$			16.0000i			0.545913i	0.962026	+	0.272956i	$0.0880015\pi$
$859$			16.0000i			0.545913i	−0.962026	+	0.272956i	$0.911998\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−4.00000			−0.136162			−0.0680808	−	0.997680i	$-0.521688\pi$
$863$	−4.00000			−0.136162			−0.0680808	+	0.997680i	$0.521688\pi$
$864$		0			0
$865$	15.0000	−	45.0000i	0.510015	−	1.53005i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−8.00000	+	8.00000i	−0.271381	+	0.271381i
$870$		0			0
$871$	−8.00000	+	12.0000i	−0.271070	+	0.406604i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−8.00000	+	44.0000i	−0.270449	+	1.48747i
$876$		0			0
$877$		−	20.0000i		−	0.675352i	−0.941262	−	0.337676i	$-0.890359\pi$
$877$		−	20.0000i		−	0.675352i	0.941262	−	0.337676i	$-0.109641\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	56.0000i		−	1.88669i	−0.331816	−	0.943344i	$-0.607661\pi$
$881$		−	56.0000i		−	1.88669i	0.331816	−	0.943344i	$-0.392339\pi$
$882$		0			0
$883$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$883$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−10.0000	+	10.0000i	−0.335767	+	0.335767i	−0.854772	−	0.519004i	$-0.826303\pi$
$887$	−10.0000	+	10.0000i	−0.335767	+	0.335767i	0.519004	+	0.854772i	$0.326303\pi$
$888$		0			0
$889$	40.0000	+	40.0000i	1.34156	+	1.34156i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−8.00000	−	8.00000i	−0.267710	−	0.267710i
$894$		0			0
$895$	32.0000	−	16.0000i	1.06964	−	0.534821i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		−	42.0000i		−	1.39922i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−10.0000	−	20.0000i	−0.332411	−	0.664822i
$906$		0			0
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	20.0000			0.662630			0.331315	−	0.943520i	$-0.392508\pi$
$911$	20.0000			0.662630			0.331315	+	0.943520i	$0.392508\pi$
$912$		0			0
$913$	32.0000	+	32.0000i	1.05905	+	1.05905i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	32.0000			1.05673
$918$		0			0
$919$		−	40.0000i		−	1.31948i	−0.751495	−	0.659739i	$-0.770667\pi$
$919$		−	40.0000i		−	1.31948i	0.751495	−	0.659739i	$-0.229333\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	40.0000	−	8.00000i	1.31662	−	0.263323i
$924$		0			0
$925$	−12.0000	+	16.0000i	−0.394558	+	0.526077i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−15.0000	−	15.0000i	−0.492134	−	0.492134i	0.416844	−	0.908978i	$-0.363136\pi$
$929$	−15.0000	−	15.0000i	−0.492134	−	0.492134i	−0.908978	+	0.416844i	$0.863136\pi$
$930$		0			0
$931$	18.0000	+	18.0000i	0.589926	+	0.589926i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−12.0000	−	24.0000i	−0.392442	−	0.784884i
$936$		0			0
$937$	−11.0000	+	11.0000i	−0.359354	+	0.359354i	−0.863575	−	0.504221i	$-0.831780\pi$
$937$	−11.0000	+	11.0000i	−0.359354	+	0.359354i	0.504221	+	0.863575i	$0.331780\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	5.00000	−	5.00000i	0.162995	−	0.162995i	−0.620897	−	0.783892i	$-0.713231\pi$
$941$	5.00000	−	5.00000i	0.162995	−	0.162995i	0.783892	+	0.620897i	$0.213231\pi$
$942$		0			0
$943$			28.0000i			0.911805i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−4.00000			−0.129983			−0.0649913	−	0.997886i	$-0.520702\pi$
$947$	−4.00000			−0.129983			−0.0649913	+	0.997886i	$0.520702\pi$
$948$		0			0
$949$	20.0000	−	30.0000i	0.649227	−	0.973841i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	11.0000	−	11.0000i	0.356325	−	0.356325i	−0.506131	−	0.862456i	$-0.668925\pi$
$953$	11.0000	−	11.0000i	0.356325	−	0.356325i	0.862456	+	0.506131i	$0.168925\pi$
$954$		0			0
$955$	24.0000	−	12.0000i	0.776622	−	0.388311i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	48.0000			1.55000
$960$		0			0
$961$			31.0000i			1.00000i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−4.00000	−	8.00000i	−0.128765	−	0.257529i
$966$		0			0
$967$		−	60.0000i		−	1.92947i	−0.263223	−	0.964735i	$-0.584786\pi$
$967$		−	60.0000i		−	1.92947i	0.263223	−	0.964735i	$-0.415214\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−60.0000			−1.92549			−0.962746	−	0.270408i	$-0.912841\pi$
$971$	−60.0000			−1.92549			−0.962746	+	0.270408i	$0.912841\pi$
$972$		0			0
$973$			48.0000i			1.53881i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			52.0000i			1.66363i	0.555055	+	0.831814i	$0.312697\pi$
$977$			52.0000i			1.66363i	−0.555055	+	0.831814i	$0.687303\pi$
$978$		0			0
$979$	−28.0000			−0.894884
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$	6.00000	+	12.0000i	0.191176	+	0.382352i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			32.0000i			1.01754i
$990$		0			0
$991$	40.0000			1.27064			0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000			1.27064			0.635321	+	0.772248i	$0.280868\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	16.0000	−	8.00000i	0.507234	−	0.253617i
$996$		0			0
$997$	29.0000	−	29.0000i	0.918439	−	0.918439i	−0.0784767	−	0.996916i	$-0.525006\pi$
$997$	29.0000	−	29.0000i	0.918439	−	0.918439i	0.996916	+	0.0784767i	$0.0250056\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2340.2.bp.c.1513.1	yes	2
3.2	odd	2		2340.2.bp.a.1513.1	yes	2
5.2	odd	4		2340.2.u.b.577.1	yes	2
13.8	odd	4		2340.2.u.b.73.1	yes	2
15.2	even	4		2340.2.u.a.577.1	yes	2
39.8	even	4		2340.2.u.a.73.1	✓	2
65.47	even	4	inner	2340.2.bp.c.1477.1	yes	2
195.47	odd	4		2340.2.bp.a.1477.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2340.2.u.a.73.1	✓	2	39.8	even	4
2340.2.u.a.577.1	yes	2	15.2	even	4
2340.2.u.b.73.1	yes	2	13.8	odd	4
2340.2.u.b.577.1	yes	2	5.2	odd	4
2340.2.bp.a.1477.1	yes	2	195.47	odd	4
2340.2.bp.a.1513.1	yes	2	3.2	odd	2
2340.2.bp.c.1477.1	yes	2	65.47	even	4	inner
2340.2.bp.c.1513.1	yes	2	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2340.bp (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.00000 - 1.00000i) q^{5} -4.00000 q^{7} +(-2.00000 - 2.00000i) q^{11} +(-3.00000 - 2.00000i) q^{13} +(3.00000 + 3.00000i) q^{17} +(2.00000 + 2.00000i) q^{19} +(2.00000 - 2.00000i) q^{23} +(3.00000 - 4.00000i) q^{25} +(-8.00000 + 4.00000i) q^{35} -4.00000 q^{37} +(-7.00000 + 7.00000i) q^{41} +(-8.00000 + 8.00000i) q^{43} -4.00000 q^{47} +9.00000 q^{49} +(-7.00000 - 7.00000i) q^{53} +(-6.00000 - 2.00000i) q^{55} +(-10.0000 + 10.0000i) q^{59} +(-8.00000 - 1.00000i) q^{65} -4.00000i q^{67} +(-8.00000 + 8.00000i) q^{71} +10.0000i q^{73} +(8.00000 + 8.00000i) q^{77} -4.00000i q^{79} -16.0000 q^{83} +(9.00000 + 3.00000i) q^{85} +(7.00000 - 7.00000i) q^{89} +(12.0000 + 8.00000i) q^{91} +(6.00000 + 2.00000i) q^{95} -12.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{5} - 8 q^{7} - 4 q^{11} - 6 q^{13} + 6 q^{17} + 4 q^{19} + 4 q^{23} + 6 q^{25} - 16 q^{35} - 8 q^{37} - 14 q^{41} - 16 q^{43} - 8 q^{47} + 18 q^{49} - 14 q^{53} - 12 q^{55} - 20 q^{59} - 16 q^{65}+ \cdots + 12 q^{95}+O(q^{100}) \) 2 * q + 4 * q^5 - 8 * q^7 - 4 * q^11 - 6 * q^13 + 6 * q^17 + 4 * q^19 + 4 * q^23 + 6 * q^25 - 16 * q^35 - 8 * q^37 - 14 * q^41 - 16 * q^43 - 8 * q^47 + 18 * q^49 - 14 * q^53 - 12 * q^55 - 20 * q^59 - 16 * q^65 - 16 * q^71 + 16 * q^77 - 32 * q^83 + 18 * q^85 + 14 * q^89 + 24 * q^91 + 12 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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