LMFDB - Embedded newform 2520.2.bi.c.1801.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2520,2,Mod(361,2520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2520, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 4]))

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

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

chi := DirichletCharacter("2520.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2520.bi (of order $3$, degree $2$, 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:	$20.1223013094$
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 840)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1801.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2520.1801
Dual form			2520.2.bi.c.361.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{5} +(0.500000 + 2.59808i) q^{7} +(-1.00000 - 1.73205i) q^{11} +1.00000 q^{13} +(-1.50000 + 2.59808i) q^{19} +(-0.500000 - 0.866025i) q^{25} +(4.50000 + 7.79423i) q^{31} +(-2.50000 - 0.866025i) q^{35} +(-1.50000 + 2.59808i) q^{37} -2.00000 q^{41} +3.00000 q^{43} +(-3.00000 + 5.19615i) q^{47} +(-6.50000 + 2.59808i) q^{49} +2.00000 q^{55} +(-2.00000 - 3.46410i) q^{59} +(-1.00000 + 1.73205i) q^{61} +(-0.500000 + 0.866025i) q^{65} +(-2.50000 - 4.33013i) q^{67} -14.0000 q^{71} +(-0.500000 - 0.866025i) q^{73} +(4.00000 - 3.46410i) q^{77} +(-4.50000 + 7.79423i) q^{79} +6.00000 q^{83} +(-2.00000 + 3.46410i) q^{89} +(0.500000 + 2.59808i) q^{91} +(-1.50000 - 2.59808i) q^{95} -14.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{5} + q^{7} - 2 q^{11} + 2 q^{13} - 3 q^{19} - q^{25} + 9 q^{31} - 5 q^{35} - 3 q^{37} - 4 q^{41} + 6 q^{43} - 6 q^{47} - 13 q^{49} + 4 q^{55} - 4 q^{59} - 2 q^{61} - q^{65} - 5 q^{67} - 28 q^{71}+ \cdots - 28 q^{97}+O(q^{100}) $ 2 * q - q^5 + q^7 - 2 * q^11 + 2 * q^13 - 3 * q^19 - q^25 + 9 * q^31 - 5 * q^35 - 3 * q^37 - 4 * q^41 + 6 * q^43 - 6 * q^47 - 13 * q^49 + 4 * q^55 - 4 * q^59 - 2 * q^61 - q^65 - 5 * q^67 - 28 * q^71 - q^73 + 8 * q^77 - 9 * q^79 + 12 * q^83 - 4 * q^89 + q^91 - 3 * q^95 - 28 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2520\mathbb{Z}\right)^\times$.

$n$	$281$	$631$	$1081$	$1261$	$2017$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$	$1$	$1$

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
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$6$		0			0
$7$	0.500000	+	2.59808i	0.188982	+	0.981981i
$7$	0.500000	+	2.59808i	0.188982	+	0.981981i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.00000	−	1.73205i	−0.301511	−	0.522233i	0.674967	−	0.737848i	$-0.264158\pi$
$11$	−1.00000	−	1.73205i	−0.301511	−	0.522233i	−0.976478	+	0.215615i	$0.930824\pi$
$12$		0			0
$13$	1.00000			0.277350			0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000			0.277350			0.138675	+	0.990338i	$0.455716\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$		0			0
$19$	−1.50000	+	2.59808i	−0.344124	+	0.596040i	−0.985194	−	0.171442i	$-0.945157\pi$
$19$	−1.50000	+	2.59808i	−0.344124	+	0.596040i	0.641071	+	0.767482i	$0.278491\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	4.50000	+	7.79423i	0.808224	+	1.39988i	0.914093	+	0.405505i	$0.132904\pi$
$31$	4.50000	+	7.79423i	0.808224	+	1.39988i	−0.105869	+	0.994380i	$0.533762\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.50000	−	0.866025i	−0.422577	−	0.146385i
$36$		0			0
$37$	−1.50000	+	2.59808i	−0.246598	+	0.427121i	−0.962580	−	0.270998i	$-0.912646\pi$
$37$	−1.50000	+	2.59808i	−0.246598	+	0.427121i	0.715981	+	0.698119i	$0.245980\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.00000			−0.312348			−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000			−0.312348			−0.156174	+	0.987730i	$0.549916\pi$
$42$		0			0
$43$	3.00000			0.457496			0.228748	−	0.973486i	$-0.426537\pi$
$43$	3.00000			0.457496			0.228748	+	0.973486i	$0.426537\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.00000	+	5.19615i	−0.437595	+	0.757937i	−0.997503	−	0.0706177i	$-0.977503\pi$
$47$	−3.00000	+	5.19615i	−0.437595	+	0.757937i	0.559908	+	0.828554i	$0.310836\pi$
$48$		0			0
$49$	−6.50000	+	2.59808i	−0.928571	+	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$53$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$54$		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
$61$	−1.00000	+	1.73205i	−0.128037	+	0.221766i	−0.922916	−	0.385002i	$-0.874201\pi$
$61$	−1.00000	+	1.73205i	−0.128037	+	0.221766i	0.794879	+	0.606768i	$0.207534\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.500000	+	0.866025i	−0.0620174	+	0.107417i
$66$		0			0
$67$	−2.50000	−	4.33013i	−0.305424	−	0.529009i	0.671932	−	0.740613i	$-0.265465\pi$
$67$	−2.50000	−	4.33013i	−0.305424	−	0.529009i	−0.977356	+	0.211604i	$0.932131\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−14.0000			−1.66149			−0.830747	−	0.556650i	$-0.812086\pi$
$71$	−14.0000			−1.66149			−0.830747	+	0.556650i	$0.812086\pi$
$72$		0			0
$73$	−0.500000	−	0.866025i	−0.0585206	−	0.101361i	0.835281	−	0.549823i	$-0.185305\pi$
$73$	−0.500000	−	0.866025i	−0.0585206	−	0.101361i	−0.893801	+	0.448463i	$0.851972\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	4.00000	−	3.46410i	0.455842	−	0.394771i
$78$		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
$83$	6.00000			0.658586			0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000			0.658586			0.329293	+	0.944228i	$0.393190\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−2.00000	+	3.46410i	−0.212000	+	0.367194i	−0.952340	−	0.305038i	$-0.901331\pi$
$89$	−2.00000	+	3.46410i	−0.212000	+	0.367194i	0.740341	+	0.672232i	$0.234664\pi$
$90$		0			0
$91$	0.500000	+	2.59808i	0.0524142	+	0.272352i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.50000	−	2.59808i	−0.153897	−	0.266557i
$96$		0			0
$97$	−14.0000			−1.42148			−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000			−1.42148			−0.710742	+	0.703452i	$0.751641\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.00000	−	1.73205i	−0.0995037	−	0.172345i	0.811976	−	0.583691i	$-0.198392\pi$
$101$	−1.00000	−	1.73205i	−0.0995037	−	0.172345i	−0.911479	+	0.411346i	$0.865059\pi$
$102$		0			0
$103$	−1.50000	+	2.59808i	−0.147799	+	0.255996i	−0.930414	−	0.366511i	$-0.880552\pi$
$103$	−1.50000	+	2.59808i	−0.147799	+	0.255996i	0.782614	+	0.622507i	$0.213886\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−8.00000	+	13.8564i	−0.773389	+	1.33955i	0.162306	+	0.986740i	$0.448107\pi$
$107$	−8.00000	+	13.8564i	−0.773389	+	1.33955i	−0.935695	+	0.352809i	$0.885227\pi$
$108$		0			0
$109$	−6.50000	−	11.2583i	−0.622587	−	1.07835i	−0.989002	−	0.147901i	$-0.952748\pi$
$109$	−6.50000	−	11.2583i	−0.622587	−	1.07835i	0.366415	−	0.930451i	$-0.380585\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$		0			0
$115$		0			0
$116$		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
$127$	11.0000			0.976092			0.488046	−	0.872818i	$-0.337710\pi$
$127$	11.0000			0.976092			0.488046	+	0.872818i	$0.337710\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−9.00000	+	15.5885i	−0.786334	+	1.36197i	0.141865	+	0.989886i	$0.454690\pi$
$131$	−9.00000	+	15.5885i	−0.786334	+	1.36197i	−0.928199	+	0.372084i	$0.878643\pi$
$132$		0			0
$133$	−7.50000	−	2.59808i	−0.650332	−	0.225282i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	4.00000	+	6.92820i	0.341743	+	0.591916i	0.984757	−	0.173939i	$-0.0556494\pi$
$137$	4.00000	+	6.92820i	0.341743	+	0.591916i	−0.643013	+	0.765855i	$0.722316\pi$
$138$		0			0
$139$	−9.00000			−0.763370			−0.381685	−	0.924292i	$-0.624656\pi$
$139$	−9.00000			−0.763370			−0.381685	+	0.924292i	$0.624656\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.00000	−	1.73205i	−0.0836242	−	0.144841i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−10.0000	+	17.3205i	−0.819232	+	1.41895i	0.0870170	+	0.996207i	$0.472267\pi$
$149$	−10.0000	+	17.3205i	−0.819232	+	1.41895i	−0.906249	+	0.422744i	$0.861067\pi$
$150$		0			0
$151$	8.00000	+	13.8564i	0.651031	+	1.12762i	0.982873	+	0.184284i	$0.0589965\pi$
$151$	8.00000	+	13.8564i	0.651031	+	1.12762i	−0.331842	+	0.943335i	$0.607670\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−9.00000			−0.722897
$156$		0			0
$157$	−5.00000	−	8.66025i	−0.399043	−	0.691164i	0.594565	−	0.804048i	$-0.297324\pi$
$157$	−5.00000	−	8.66025i	−0.399043	−	0.691164i	−0.993608	+	0.112884i	$0.963991\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	6.00000	−	10.3923i	0.469956	−	0.813988i	−0.529454	−	0.848339i	$-0.677603\pi$
$163$	6.00000	−	10.3923i	0.469956	−	0.813988i	0.999410	+	0.0343508i	$0.0109363\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	18.0000			1.39288			0.696441	−	0.717614i	$-0.254766\pi$
$167$	18.0000			1.39288			0.696441	+	0.717614i	$0.254766\pi$
$168$		0			0
$169$	−12.0000			−0.923077
$170$		0			0
$171$		0			0
$172$		0			0
$173$	2.00000	−	3.46410i	0.152057	−	0.263371i	−0.779926	−	0.625871i	$-0.784744\pi$
$173$	2.00000	−	3.46410i	0.152057	−	0.263371i	0.931984	+	0.362500i	$0.118077\pi$
$174$		0			0
$175$	2.00000	−	1.73205i	0.151186	−	0.130931i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	5.00000	+	8.66025i	0.373718	+	0.647298i	0.990134	−	0.140122i	$-0.0447496\pi$
$179$	5.00000	+	8.66025i	0.373718	+	0.647298i	−0.616417	+	0.787420i	$0.711416\pi$
$180$		0			0
$181$	25.0000			1.85824			0.929118	−	0.369784i	$-0.120568\pi$
$181$	25.0000			1.85824			0.929118	+	0.369784i	$0.120568\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.50000	−	2.59808i	−0.110282	−	0.191014i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−11.0000	+	19.0526i	−0.795932	+	1.37859i	0.126314	+	0.991990i	$0.459685\pi$
$191$	−11.0000	+	19.0526i	−0.795932	+	1.37859i	−0.922246	+	0.386604i	$0.873648\pi$
$192$		0			0
$193$	−11.5000	−	19.9186i	−0.827788	−	1.43377i	−0.899770	−	0.436365i	$-0.856266\pi$
$193$	−11.5000	−	19.9186i	−0.827788	−	1.43377i	0.0719816	−	0.997406i	$-0.477068\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	12.0000	+	20.7846i	0.850657	+	1.47338i	0.880616	+	0.473831i	$0.157129\pi$
$199$	12.0000	+	20.7846i	0.850657	+	1.47338i	−0.0299585	+	0.999551i	$0.509538\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	1.00000	−	1.73205i	0.0698430	−	0.120972i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	6.00000			0.415029
$210$		0			0
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1.50000	+	2.59808i	−0.102299	+	0.177187i
$216$		0			0
$217$	−18.0000	+	15.5885i	−1.22192	+	1.05821i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−16.0000			−1.07144			−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000			−1.07144			−0.535720	+	0.844396i	$0.679960\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	13.0000	+	22.5167i	0.862840	+	1.49448i	0.869176	+	0.494503i	$0.164650\pi$
$227$	13.0000	+	22.5167i	0.862840	+	1.49448i	−0.00633544	+	0.999980i	$0.502017\pi$
$228$		0			0
$229$	11.5000	−	19.9186i	0.759941	−	1.31626i	−0.182939	−	0.983124i	$-0.558561\pi$
$229$	11.5000	−	19.9186i	0.759941	−	1.31626i	0.942880	−	0.333133i	$-0.108106\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−11.0000	+	19.0526i	−0.720634	+	1.24817i	0.240112	+	0.970745i	$0.422816\pi$
$233$	−11.0000	+	19.0526i	−0.720634	+	1.24817i	−0.960746	+	0.277429i	$0.910518\pi$
$234$		0			0
$235$	−3.00000	−	5.19615i	−0.195698	−	0.338960i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	22.0000			1.42306			0.711531	−	0.702655i	$-0.248002\pi$
$239$	22.0000			1.42306			0.711531	+	0.702655i	$0.248002\pi$
$240$		0			0
$241$	9.00000	+	15.5885i	0.579741	+	1.00414i	0.995509	+	0.0946700i	$0.0301796\pi$
$241$	9.00000	+	15.5885i	0.579741	+	1.00414i	−0.415768	+	0.909471i	$0.636487\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.00000	−	6.92820i	0.0638877	−	0.442627i
$246$		0			0
$247$	−1.50000	+	2.59808i	−0.0954427	+	0.165312i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	16.0000			1.00991			0.504956	−	0.863145i	$-0.331509\pi$
$251$	16.0000			1.00991			0.504956	+	0.863145i	$0.331509\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	9.00000	−	15.5885i	0.561405	−	0.972381i	−0.435970	−	0.899961i	$-0.643595\pi$
$257$	9.00000	−	15.5885i	0.561405	−	0.972381i	0.997374	−	0.0724199i	$-0.0230722\pi$
$258$		0			0
$259$	−7.50000	−	2.59808i	−0.466027	−	0.161437i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6.00000	−	10.3923i	−0.369976	−	0.640817i	0.619586	−	0.784929i	$-0.287301\pi$
$263$	−6.00000	−	10.3923i	−0.369976	−	0.640817i	−0.989561	+	0.144112i	$0.953967\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.00000	−	1.73205i	−0.0609711	−	0.105605i	0.833929	−	0.551872i	$-0.186086\pi$
$269$	−1.00000	−	1.73205i	−0.0609711	−	0.105605i	−0.894900	+	0.446267i	$0.852753\pi$
$270$		0			0
$271$	−8.00000	+	13.8564i	−0.485965	+	0.841717i	−0.999870	−	0.0161307i	$-0.994865\pi$
$271$	−8.00000	+	13.8564i	−0.485965	+	0.841717i	0.513905	+	0.857847i	$0.328199\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.00000	+	1.73205i	−0.0603023	+	0.104447i
$276$		0			0
$277$	−2.50000	−	4.33013i	−0.150210	−	0.260172i	0.781094	−	0.624413i	$-0.214662\pi$
$277$	−2.50000	−	4.33013i	−0.150210	−	0.260172i	−0.931305	+	0.364241i	$0.881328\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−8.00000			−0.477240			−0.238620	−	0.971113i	$-0.576695\pi$
$281$	−8.00000			−0.477240			−0.238620	+	0.971113i	$0.576695\pi$
$282$		0			0
$283$	−5.50000	−	9.52628i	−0.326941	−	0.566279i	0.654962	−	0.755662i	$-0.272685\pi$
$283$	−5.50000	−	9.52628i	−0.326941	−	0.566279i	−0.981903	+	0.189383i	$0.939351\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1.00000	−	5.19615i	−0.0590281	−	0.306719i
$288$		0			0
$289$	8.50000	−	14.7224i	0.500000	−	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−8.00000			−0.467365			−0.233682	−	0.972313i	$-0.575078\pi$
$293$	−8.00000			−0.467365			−0.233682	+	0.972313i	$0.575078\pi$
$294$		0			0
$295$	4.00000			0.232889
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	1.50000	+	7.79423i	0.0864586	+	0.449252i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.00000	−	1.73205i	−0.0572598	−	0.0991769i
$306$		0			0
$307$	−15.0000			−0.856095			−0.428048	−	0.903756i	$-0.640798\pi$
$307$	−15.0000			−0.856095			−0.428048	+	0.903756i	$0.640798\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−7.00000	−	12.1244i	−0.396934	−	0.687509i	0.596412	−	0.802678i	$-0.296592\pi$
$311$	−7.00000	−	12.1244i	−0.396934	−	0.687509i	−0.993346	+	0.115169i	$0.963259\pi$
$312$		0			0
$313$	6.50000	−	11.2583i	0.367402	−	0.636358i	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	6.50000	−	11.2583i	0.367402	−	0.636358i	0.989158	+	0.146852i	$0.0469141\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	6.00000	−	10.3923i	0.336994	−	0.583690i	−0.646872	−	0.762598i	$-0.723923\pi$
$317$	6.00000	−	10.3923i	0.336994	−	0.583690i	0.983866	+	0.178908i	$0.0572566\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−0.500000	−	0.866025i	−0.0277350	−	0.0480384i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−15.0000	−	5.19615i	−0.826977	−	0.286473i
$330$		0			0
$331$	−6.50000	+	11.2583i	−0.357272	+	0.618814i	−0.987504	−	0.157593i	$-0.949627\pi$
$331$	−6.50000	+	11.2583i	−0.357272	+	0.618814i	0.630232	+	0.776407i	$0.282960\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	5.00000			0.273179
$336$		0			0
$337$	1.00000			0.0544735			0.0272367	−	0.999629i	$-0.491329\pi$
$337$	1.00000			0.0544735			0.0272367	+	0.999629i	$0.491329\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	9.00000	−	15.5885i	0.487377	−	0.844162i
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−2.00000	−	3.46410i	−0.107366	−	0.185963i	0.807337	−	0.590091i	$-0.200908\pi$
$347$	−2.00000	−	3.46410i	−0.107366	−	0.185963i	−0.914702	+	0.404128i	$0.867575\pi$
$348$		0			0
$349$	26.0000			1.39175			0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000			1.39175			0.695874	+	0.718164i	$0.255017\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−15.0000	−	25.9808i	−0.798369	−	1.38282i	−0.920677	−	0.390324i	$-0.872363\pi$
$353$	−15.0000	−	25.9808i	−0.798369	−	1.38282i	0.122308	−	0.992492i	$-0.460970\pi$
$354$		0			0
$355$	7.00000	−	12.1244i	0.371521	−	0.643494i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$359$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$360$		0			0
$361$	5.00000	+	8.66025i	0.263158	+	0.455803i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.00000			0.0523424
$366$		0			0
$367$	17.5000	+	30.3109i	0.913493	+	1.58222i	0.809093	+	0.587680i	$0.199959\pi$
$367$	17.5000	+	30.3109i	0.913493	+	1.58222i	0.104399	+	0.994535i	$0.466708\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	16.5000	−	28.5788i	0.854338	−	1.47976i	−0.0229205	−	0.999737i	$-0.507296\pi$
$373$	16.5000	−	28.5788i	0.854338	−	1.47976i	0.877258	−	0.480019i	$-0.159370\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.00000			0.0513665			0.0256833	−	0.999670i	$-0.491824\pi$
$379$	1.00000			0.0513665			0.0256833	+	0.999670i	$0.491824\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	12.0000	−	20.7846i	0.613171	−	1.06204i	−0.377531	−	0.925997i	$-0.623227\pi$
$383$	12.0000	−	20.7846i	0.613171	−	1.06204i	0.990702	−	0.136047i	$-0.0434398\pi$
$384$		0			0
$385$	1.00000	+	5.19615i	0.0509647	+	0.264820i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−7.00000	−	12.1244i	−0.354914	−	0.614729i	0.632189	−	0.774814i	$-0.282157\pi$
$389$	−7.00000	−	12.1244i	−0.354914	−	0.614729i	−0.987103	+	0.160085i	$0.948823\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−4.50000	−	7.79423i	−0.226420	−	0.392170i
$396$		0			0
$397$	−3.50000	+	6.06218i	−0.175660	+	0.304252i	−0.940389	−	0.340099i	$-0.889539\pi$
$397$	−3.50000	+	6.06218i	−0.175660	+	0.304252i	0.764730	+	0.644351i	$0.222873\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−12.0000	+	20.7846i	−0.599251	+	1.03793i	0.393680	+	0.919247i	$0.371202\pi$
$401$	−12.0000	+	20.7846i	−0.599251	+	1.03793i	−0.992932	+	0.118686i	$0.962132\pi$
$402$		0			0
$403$	4.50000	+	7.79423i	0.224161	+	0.388258i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	6.00000			0.297409
$408$		0			0
$409$	9.50000	+	16.4545i	0.469745	+	0.813622i	0.999402	−	0.0345902i	$-0.0110126\pi$
$409$	9.50000	+	16.4545i	0.469745	+	0.813622i	−0.529657	+	0.848212i	$0.677679\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	8.00000	−	6.92820i	0.393654	−	0.340915i
$414$		0			0
$415$	−3.00000	+	5.19615i	−0.147264	+	0.255069i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	30.0000			1.46560			0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000			1.46560			0.732798	+	0.680446i	$0.238214\pi$
$420$		0			0
$421$	5.00000			0.243685			0.121843	−	0.992549i	$-0.461120\pi$
$421$	5.00000			0.243685			0.121843	+	0.992549i	$0.461120\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−5.00000	−	1.73205i	−0.241967	−	0.0838198i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−17.0000	−	29.4449i	−0.818861	−	1.41831i	−0.906522	−	0.422159i	$-0.861272\pi$
$431$	−17.0000	−	29.4449i	−0.818861	−	1.41831i	0.0876607	−	0.996150i	$-0.472061\pi$
$432$		0			0
$433$	3.00000			0.144171			0.0720854	−	0.997398i	$-0.477035\pi$
$433$	3.00000			0.144171			0.0720854	+	0.997398i	$0.477035\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−4.00000	+	6.92820i	−0.190910	+	0.330665i	−0.945552	−	0.325471i	$-0.894477\pi$
$439$	−4.00000	+	6.92820i	−0.190910	+	0.330665i	0.754642	+	0.656136i	$0.227810\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$443$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$444$		0			0
$445$	−2.00000	−	3.46410i	−0.0948091	−	0.164214i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	6.00000			0.283158			0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000			0.283158			0.141579	+	0.989927i	$0.454782\pi$
$450$		0			0
$451$	2.00000	+	3.46410i	0.0941763	+	0.163118i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2.50000	−	0.866025i	−0.117202	−	0.0405999i
$456$		0			0
$457$	−8.50000	+	14.7224i	−0.397613	+	0.688686i	−0.993431	−	0.114433i	$-0.963495\pi$
$457$	−8.50000	+	14.7224i	−0.397613	+	0.688686i	0.595818	+	0.803120i	$0.296828\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	24.0000			1.11779			0.558896	−	0.829238i	$-0.311225\pi$
$461$	24.0000			1.11779			0.558896	+	0.829238i	$0.311225\pi$
$462$		0			0
$463$	−19.0000			−0.883005			−0.441502	−	0.897260i	$-0.645554\pi$
$463$	−19.0000			−0.883005			−0.441502	+	0.897260i	$0.645554\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−9.00000	+	15.5885i	−0.416470	+	0.721348i	−0.995582	−	0.0939008i	$-0.970066\pi$
$467$	−9.00000	+	15.5885i	−0.416470	+	0.721348i	0.579111	+	0.815249i	$0.303400\pi$
$468$		0			0
$469$	10.0000	−	8.66025i	0.461757	−	0.399893i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−3.00000	−	5.19615i	−0.137940	−	0.238919i
$474$		0			0
$475$	3.00000			0.137649
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−12.0000	−	20.7846i	−0.548294	−	0.949673i	−0.998392	−	0.0566937i	$-0.981944\pi$
$479$	−12.0000	−	20.7846i	−0.548294	−	0.949673i	0.450098	−	0.892979i	$-0.351389\pi$
$480$		0			0
$481$	−1.50000	+	2.59808i	−0.0683941	+	0.118462i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	7.00000	−	12.1244i	0.317854	−	0.550539i
$486$		0			0
$487$	−2.50000	−	4.33013i	−0.113286	−	0.196217i	0.803807	−	0.594890i	$-0.202804\pi$
$487$	−2.50000	−	4.33013i	−0.113286	−	0.196217i	−0.917093	+	0.398673i	$0.869471\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	12.0000			0.541552			0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000			0.541552			0.270776	+	0.962642i	$0.412720\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−7.00000	−	36.3731i	−0.313993	−	1.63156i
$498$		0			0
$499$	0.500000	−	0.866025i	0.0223831	−	0.0387686i	−0.854617	−	0.519259i	$-0.826208\pi$
$499$	0.500000	−	0.866025i	0.0223831	−	0.0387686i	0.877000	+	0.480490i	$0.159541\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	22.0000			0.980932			0.490466	−	0.871460i	$-0.336827\pi$
$503$	22.0000			0.980932			0.490466	+	0.871460i	$0.336827\pi$
$504$		0			0
$505$	2.00000			0.0889988
$506$		0			0
$507$		0			0
$508$		0			0
$509$	15.0000	−	25.9808i	0.664863	−	1.15158i	−0.314459	−	0.949271i	$-0.601823\pi$
$509$	15.0000	−	25.9808i	0.664863	−	1.15158i	0.979322	−	0.202306i	$-0.0648436\pi$
$510$		0			0
$511$	2.00000	−	1.73205i	0.0884748	−	0.0766214i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.50000	−	2.59808i	−0.0660979	−	0.114485i
$516$		0			0
$517$	12.0000			0.527759
$518$		0			0
$519$		0			0
$520$		0			0
$521$	6.00000	+	10.3923i	0.262865	+	0.455295i	0.967002	−	0.254769i	$-0.0819994\pi$
$521$	6.00000	+	10.3923i	0.262865	+	0.455295i	−0.704137	+	0.710064i	$0.748666\pi$
$522$		0			0
$523$	−0.500000	+	0.866025i	−0.0218635	+	0.0378686i	−0.876750	−	0.480946i	$-0.840293\pi$
$523$	−0.500000	+	0.866025i	−0.0218635	+	0.0378686i	0.854887	+	0.518815i	$0.173627\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	+	19.9186i	0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−2.00000			−0.0866296
$534$		0			0
$535$	−8.00000	−	13.8564i	−0.345870	−	0.599065i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	11.0000	+	8.66025i	0.473804	+	0.373024i
$540$		0			0
$541$	−16.5000	+	28.5788i	−0.709390	+	1.22870i	0.255693	+	0.966758i	$0.417696\pi$
$541$	−16.5000	+	28.5788i	−0.709390	+	1.22870i	−0.965084	+	0.261942i	$0.915637\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	13.0000			0.556859
$546$		0			0
$547$	44.0000			1.88130			0.940652	−	0.339372i	$-0.110215\pi$
$547$	44.0000			1.88130			0.940652	+	0.339372i	$0.110215\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−22.5000	−	7.79423i	−0.956797	−	0.331444i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	3.00000	+	5.19615i	0.127114	+	0.220168i	0.922557	−	0.385860i	$-0.126095\pi$
$557$	3.00000	+	5.19615i	0.127114	+	0.220168i	−0.795443	+	0.606028i	$0.792762\pi$
$558$		0			0
$559$	3.00000			0.126886
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.00000	−	1.73205i	−0.0421450	−	0.0729972i	0.844183	−	0.536054i	$-0.180086\pi$
$563$	−1.00000	−	1.73205i	−0.0421450	−	0.0729972i	−0.886328	+	0.463057i	$0.846752\pi$
$564$		0			0
$565$	3.00000	−	5.19615i	0.126211	−	0.218604i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−19.0000	+	32.9090i	−0.796521	+	1.37962i	0.125347	+	0.992113i	$0.459996\pi$
$569$	−19.0000	+	32.9090i	−0.796521	+	1.37962i	−0.921869	+	0.387503i	$0.873338\pi$
$570$		0			0
$571$	−7.50000	−	12.9904i	−0.313865	−	0.543631i	0.665330	−	0.746549i	$-0.268291\pi$
$571$	−7.50000	−	12.9904i	−0.313865	−	0.543631i	−0.979196	+	0.202919i	$0.934957\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	6.50000	+	11.2583i	0.270599	+	0.468690i	0.969015	−	0.247001i	$-0.0794451\pi$
$577$	6.50000	+	11.2583i	0.270599	+	0.468690i	−0.698417	+	0.715691i	$0.746112\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	3.00000	+	15.5885i	0.124461	+	0.646718i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	28.0000			1.15568			0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000			1.15568			0.577842	+	0.816149i	$0.303895\pi$
$588$		0			0
$589$	−27.0000			−1.11252
$590$		0			0
$591$		0			0
$592$		0			0
$593$	19.0000	−	32.9090i	0.780236	−	1.35141i	−0.151567	−	0.988447i	$-0.548432\pi$
$593$	19.0000	−	32.9090i	0.780236	−	1.35141i	0.931804	−	0.362962i	$-0.118235\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−6.00000	−	10.3923i	−0.245153	−	0.424618i	0.717021	−	0.697051i	$-0.245505\pi$
$599$	−6.00000	−	10.3923i	−0.245153	−	0.424618i	−0.962175	+	0.272433i	$0.912172\pi$
$600$		0			0
$601$	23.0000			0.938190			0.469095	−	0.883148i	$-0.344580\pi$
$601$	23.0000			0.938190			0.469095	+	0.883148i	$0.344580\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3.50000	+	6.06218i	0.142295	+	0.246463i
$606$		0			0
$607$	21.5000	−	37.2391i	0.872658	−	1.51149i	0.0134214	−	0.999910i	$-0.495728\pi$
$607$	21.5000	−	37.2391i	0.872658	−	1.51149i	0.859237	−	0.511578i	$-0.170939\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−3.00000	+	5.19615i	−0.121367	+	0.210214i
$612$		0			0
$613$	−13.0000	−	22.5167i	−0.525065	−	0.909439i	−0.999574	−	0.0291886i	$-0.990708\pi$
$613$	−13.0000	−	22.5167i	−0.525065	−	0.909439i	0.474509	−	0.880251i	$-0.342626\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−34.0000			−1.36879			−0.684394	−	0.729112i	$-0.739933\pi$
$617$	−34.0000			−1.36879			−0.684394	+	0.729112i	$0.739933\pi$
$618$		0			0
$619$	15.5000	+	26.8468i	0.622998	+	1.07906i	0.988924	+	0.148420i	$0.0474187\pi$
$619$	15.5000	+	26.8468i	0.622998	+	1.07906i	−0.365927	+	0.930644i	$0.619248\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−10.0000	−	3.46410i	−0.400642	−	0.138786i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	24.0000			0.955425			0.477712	−	0.878516i	$-0.341466\pi$
$631$	24.0000			0.955425			0.477712	+	0.878516i	$0.341466\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−5.50000	+	9.52628i	−0.218261	+	0.378039i
$636$		0			0
$637$	−6.50000	+	2.59808i	−0.257539	+	0.102940i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	20.0000	+	34.6410i	0.789953	+	1.36824i	0.925995	+	0.377535i	$0.123228\pi$
$641$	20.0000	+	34.6410i	0.789953	+	1.36824i	−0.136043	+	0.990703i	$0.543438\pi$
$642$		0			0
$643$	11.0000			0.433798			0.216899	−	0.976194i	$-0.430406\pi$
$643$	11.0000			0.433798			0.216899	+	0.976194i	$0.430406\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	9.00000	+	15.5885i	0.353827	+	0.612845i	0.986916	−	0.161233i	$-0.0515470\pi$
$647$	9.00000	+	15.5885i	0.353827	+	0.612845i	−0.633090	+	0.774078i	$0.718214\pi$
$648$		0			0
$649$	−4.00000	+	6.92820i	−0.157014	+	0.271956i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	3.00000	−	5.19615i	0.117399	−	0.203341i	−0.801337	−	0.598213i	$-0.795878\pi$
$653$	3.00000	−	5.19615i	0.117399	−	0.203341i	0.918736	+	0.394872i	$0.129211\pi$
$654$		0			0
$655$	−9.00000	−	15.5885i	−0.351659	−	0.609091i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	40.0000			1.55818			0.779089	−	0.626913i	$-0.215682\pi$
$659$	40.0000			1.55818			0.779089	+	0.626913i	$0.215682\pi$
$660$		0			0
$661$	−17.5000	−	30.3109i	−0.680671	−	1.17896i	−0.974776	−	0.223184i	$-0.928355\pi$
$661$	−17.5000	−	30.3109i	−0.680671	−	1.17896i	0.294105	−	0.955773i	$-0.404978\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	6.00000	−	5.19615i	0.232670	−	0.201498i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	4.00000			0.154418
$672$		0			0
$673$	−13.0000			−0.501113			−0.250557	−	0.968102i	$-0.580614\pi$
$673$	−13.0000			−0.501113			−0.250557	+	0.968102i	$0.580614\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$677$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$678$		0			0
$679$	−7.00000	−	36.3731i	−0.268635	−	1.39587i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−22.0000	−	38.1051i	−0.841807	−	1.45805i	−0.888366	−	0.459136i	$-0.848159\pi$
$683$	−22.0000	−	38.1051i	−0.841807	−	1.45805i	0.0465592	−	0.998916i	$-0.485174\pi$
$684$		0			0
$685$	−8.00000			−0.305664
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		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
$695$	4.50000	−	7.79423i	0.170695	−	0.295652i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	48.0000			1.81293			0.906467	−	0.422276i	$-0.138769\pi$
$701$	48.0000			1.81293			0.906467	+	0.422276i	$0.138769\pi$
$702$		0			0
$703$	−4.50000	−	7.79423i	−0.169721	−	0.293965i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	4.00000	−	3.46410i	0.150435	−	0.130281i
$708$		0			0
$709$	13.0000	−	22.5167i	0.488225	−	0.845631i	−0.511683	−	0.859174i	$-0.670978\pi$
$709$	13.0000	−	22.5167i	0.488225	−	0.845631i	0.999908	+	0.0135434i	$0.00431112\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	2.00000			0.0747958
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1.00000	−	1.73205i	0.0372937	−	0.0645946i	−0.846776	−	0.531949i	$-0.821460\pi$
$719$	1.00000	−	1.73205i	0.0372937	−	0.0645946i	0.884070	+	0.467355i	$0.154793\pi$
$720$		0			0
$721$	−7.50000	−	2.59808i	−0.279315	−	0.0967574i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	17.0000			0.630495			0.315248	−	0.949009i	$-0.397912\pi$
$727$	17.0000			0.630495			0.315248	+	0.949009i	$0.397912\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	15.5000	−	26.8468i	0.572506	−	0.991609i	−0.423802	−	0.905755i	$-0.639305\pi$
$733$	15.5000	−	26.8468i	0.572506	−	0.991609i	0.996308	−	0.0858539i	$-0.0273618\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−5.00000	+	8.66025i	−0.184177	+	0.319005i
$738$		0			0
$739$	−5.50000	−	9.52628i	−0.202321	−	0.350430i	0.746955	−	0.664875i	$-0.231515\pi$
$739$	−5.50000	−	9.52628i	−0.202321	−	0.350430i	−0.949276	+	0.314445i	$0.898182\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	30.0000			1.10059			0.550297	−	0.834969i	$-0.314515\pi$
$743$	30.0000			1.10059			0.550297	+	0.834969i	$0.314515\pi$
$744$		0			0
$745$	−10.0000	−	17.3205i	−0.366372	−	0.634574i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−40.0000	−	13.8564i	−1.46157	−	0.506302i
$750$		0			0
$751$	−9.50000	+	16.4545i	−0.346660	+	0.600433i	−0.985654	−	0.168779i	$-0.946018\pi$
$751$	−9.50000	+	16.4545i	−0.346660	+	0.600433i	0.638994	+	0.769212i	$0.279351\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−16.0000			−0.582300
$756$		0			0
$757$	26.0000			0.944986			0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000			0.944986			0.472493	+	0.881334i	$0.343354\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	4.00000	−	6.92820i	0.145000	−	0.251147i	−0.784373	−	0.620289i	$-0.787015\pi$
$761$	4.00000	−	6.92820i	0.145000	−	0.251147i	0.929373	+	0.369142i	$0.120348\pi$
$762$		0			0
$763$	26.0000	−	22.5167i	0.941263	−	0.815158i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.00000	−	3.46410i	−0.0722158	−	0.125081i
$768$		0			0
$769$	−25.0000			−0.901523			−0.450762	−	0.892644i	$-0.648848\pi$
$769$	−25.0000			−0.901523			−0.450762	+	0.892644i	$0.648848\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	17.0000	+	29.4449i	0.611448	+	1.05906i	0.990997	+	0.133887i	$0.0427458\pi$
$773$	17.0000	+	29.4449i	0.611448	+	1.05906i	−0.379549	+	0.925172i	$0.623921\pi$
$774$		0			0
$775$	4.50000	−	7.79423i	0.161645	−	0.279977i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.00000	−	5.19615i	0.107486	−	0.186171i
$780$		0			0
$781$	14.0000	+	24.2487i	0.500959	+	0.867687i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	10.0000			0.356915
$786$		0			0
$787$	−4.00000	−	6.92820i	−0.142585	−	0.246964i	0.785885	−	0.618373i	$-0.212208\pi$
$787$	−4.00000	−	6.92820i	−0.142585	−	0.246964i	−0.928469	+	0.371409i	$0.878875\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−3.00000	−	15.5885i	−0.106668	−	0.554262i
$792$		0			0
$793$	−1.00000	+	1.73205i	−0.0355110	+	0.0615069i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−32.0000			−1.13350			−0.566749	−	0.823890i	$-0.691799\pi$
$797$	−32.0000			−1.13350			−0.566749	+	0.823890i	$0.691799\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1.00000	+	1.73205i	−0.0352892	+	0.0611227i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	5.00000	+	8.66025i	0.175791	+	0.304478i	0.940435	−	0.339975i	$-0.110418\pi$
$809$	5.00000	+	8.66025i	0.175791	+	0.304478i	−0.764644	+	0.644453i	$0.777085\pi$
$810$		0			0
$811$	−40.0000			−1.40459			−0.702295	−	0.711886i	$-0.747841\pi$
$811$	−40.0000			−1.40459			−0.702295	+	0.711886i	$0.747841\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	6.00000	+	10.3923i	0.210171	+	0.364027i
$816$		0			0
$817$	−4.50000	+	7.79423i	−0.157435	+	0.272686i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.00000	−	1.73205i	0.0349002	−	0.0604490i	−0.848048	−	0.529920i	$-0.822222\pi$
$821$	1.00000	−	1.73205i	0.0349002	−	0.0604490i	0.882948	+	0.469471i	$0.155555\pi$
$822$		0			0
$823$	−16.0000	−	27.7128i	−0.557725	−	0.966008i	−0.997686	−	0.0679910i	$-0.978341\pi$
$823$	−16.0000	−	27.7128i	−0.557725	−	0.966008i	0.439961	−	0.898017i	$-0.354992\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−50.0000			−1.73867			−0.869335	−	0.494223i	$-0.835453\pi$
$827$	−50.0000			−1.73867			−0.869335	+	0.494223i	$0.835453\pi$
$828$		0			0
$829$	5.50000	+	9.52628i	0.191023	+	0.330861i	0.945589	−	0.325362i	$-0.105486\pi$
$829$	5.50000	+	9.52628i	0.191023	+	0.330861i	−0.754567	+	0.656223i	$0.772153\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−9.00000	+	15.5885i	−0.311458	+	0.539461i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−40.0000			−1.38095			−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000			−1.38095			−0.690477	+	0.723355i	$0.742599\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$	6.00000	−	10.3923i	0.206406	−	0.357506i
$846$		0			0
$847$	17.5000	+	6.06218i	0.601307	+	0.208299i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	43.0000			1.47229			0.736146	−	0.676823i	$-0.236644\pi$
$853$	43.0000			1.47229			0.736146	+	0.676823i	$0.236644\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	26.0000	+	45.0333i	0.888143	+	1.53831i	0.842068	+	0.539371i	$0.181338\pi$
$857$	26.0000	+	45.0333i	0.888143	+	1.53831i	0.0460748	+	0.998938i	$0.485329\pi$
$858$		0			0
$859$	8.00000	−	13.8564i	0.272956	−	0.472774i	−0.696661	−	0.717400i	$-0.745332\pi$
$859$	8.00000	−	13.8564i	0.272956	−	0.472774i	0.969618	+	0.244626i	$0.0786652\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−17.0000	+	29.4449i	−0.578687	+	1.00231i	0.416944	+	0.908932i	$0.363101\pi$
$863$	−17.0000	+	29.4449i	−0.578687	+	1.00231i	−0.995630	+	0.0933825i	$0.970232\pi$
$864$		0			0
$865$	2.00000	+	3.46410i	0.0680020	+	0.117783i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	18.0000			0.610608
$870$		0			0
$871$	−2.50000	−	4.33013i	−0.0847093	−	0.146721i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	0.500000	+	2.59808i	0.0169031	+	0.0878310i
$876$		0			0
$877$	−1.00000	+	1.73205i	−0.0337676	+	0.0584872i	−0.882415	−	0.470471i	$-0.844084\pi$
$877$	−1.00000	+	1.73205i	−0.0337676	+	0.0584872i	0.848648	+	0.528958i	$0.177417\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	52.0000			1.75192			0.875962	−	0.482380i	$-0.160227\pi$
$881$	52.0000			1.75192			0.875962	+	0.482380i	$0.160227\pi$
$882$		0			0
$883$	29.0000			0.975928			0.487964	−	0.872864i	$-0.337740\pi$
$883$	29.0000			0.975928			0.487964	+	0.872864i	$0.337740\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1.00000	+	1.73205i	−0.0335767	+	0.0581566i	−0.882325	−	0.470640i	$-0.844023\pi$
$887$	−1.00000	+	1.73205i	−0.0335767	+	0.0581566i	0.848749	+	0.528796i	$0.177356\pi$
$888$		0			0
$889$	5.50000	+	28.5788i	0.184464	+	0.958503i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−9.00000	−	15.5885i	−0.301174	−	0.521648i
$894$		0			0
$895$	−10.0000			−0.334263
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−12.5000	+	21.6506i	−0.415514	+	0.719691i
$906$		0			0
$907$	1.50000	+	2.59808i	0.0498067	+	0.0862677i	0.889854	−	0.456246i	$-0.150806\pi$
$907$	1.50000	+	2.59808i	0.0498067	+	0.0862677i	−0.840047	+	0.542513i	$0.817473\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−8.00000			−0.265052			−0.132526	−	0.991180i	$-0.542309\pi$
$911$	−8.00000			−0.265052			−0.132526	+	0.991180i	$0.542309\pi$
$912$		0			0
$913$	−6.00000	−	10.3923i	−0.198571	−	0.343935i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−45.0000	−	15.5885i	−1.48603	−	0.514776i
$918$		0			0
$919$	−12.5000	+	21.6506i	−0.412337	+	0.714189i	−0.995145	−	0.0984214i	$-0.968621\pi$
$919$	−12.5000	+	21.6506i	−0.412337	+	0.714189i	0.582808	+	0.812610i	$0.301954\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−14.0000			−0.460816
$924$		0			0
$925$	3.00000			0.0986394
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−9.00000	+	15.5885i	−0.295280	+	0.511441i	−0.975050	−	0.221985i	$-0.928746\pi$
$929$	−9.00000	+	15.5885i	−0.295280	+	0.511441i	0.679770	+	0.733426i	$0.262080\pi$
$930$		0			0
$931$	3.00000	−	20.7846i	0.0983210	−	0.681188i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−5.00000			−0.163343			−0.0816714	−	0.996659i	$-0.526026\pi$
$937$	−5.00000			−0.163343			−0.0816714	+	0.996659i	$0.526026\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	26.0000	+	45.0333i	0.847576	+	1.46804i	0.883365	+	0.468685i	$0.155272\pi$
$941$	26.0000	+	45.0333i	0.847576	+	1.46804i	−0.0357896	+	0.999359i	$0.511395\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	19.0000	−	32.9090i	0.617417	−	1.06940i	−0.372538	−	0.928017i	$-0.621512\pi$
$947$	19.0000	−	32.9090i	0.617417	−	1.06940i	0.989955	−	0.141381i	$-0.0451542\pi$
$948$		0			0
$949$	−0.500000	−	0.866025i	−0.0162307	−	0.0281124i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	16.0000			0.518291			0.259145	−	0.965838i	$-0.416559\pi$
$953$	16.0000			0.518291			0.259145	+	0.965838i	$0.416559\pi$
$954$		0			0
$955$	−11.0000	−	19.0526i	−0.355952	−	0.616526i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−16.0000	+	13.8564i	−0.516667	+	0.447447i
$960$		0			0
$961$	−25.0000	+	43.3013i	−0.806452	+	1.39682i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	23.0000			0.740396
$966$		0			0
$967$	49.0000			1.57573			0.787867	−	0.615846i	$-0.211185\pi$
$967$	49.0000			1.57573			0.787867	+	0.615846i	$0.211185\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	8.00000	−	13.8564i	0.256732	−	0.444673i	−0.708632	−	0.705578i	$-0.750687\pi$
$971$	8.00000	−	13.8564i	0.256732	−	0.444673i	0.965365	+	0.260905i	$0.0840208\pi$
$972$		0			0
$973$	−4.50000	−	23.3827i	−0.144263	−	0.749614i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−3.00000	−	5.19615i	−0.0959785	−	0.166240i	0.814038	−	0.580812i	$-0.197265\pi$
$977$	−3.00000	−	5.19615i	−0.0959785	−	0.166240i	−0.910017	+	0.414572i	$0.863931\pi$
$978$		0			0
$979$	8.00000			0.255681
$980$		0			0
$981$		0			0
$982$		0			0
$983$	18.0000	+	31.1769i	0.574111	+	0.994389i	0.996138	+	0.0878058i	$0.0279855\pi$
$983$	18.0000	+	31.1769i	0.574111	+	0.994389i	−0.422027	+	0.906583i	$0.638681\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−11.5000	−	19.9186i	−0.365310	−	0.632735i	0.623516	−	0.781810i	$-0.285704\pi$
$991$	−11.5000	−	19.9186i	−0.365310	−	0.632735i	−0.988826	+	0.149076i	$0.952370\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−24.0000			−0.760851
$996$		0			0
$997$	−5.50000	−	9.52628i	−0.174187	−	0.301700i	0.765693	−	0.643206i	$-0.222396\pi$
$997$	−5.50000	−	9.52628i	−0.174187	−	0.301700i	−0.939880	+	0.341506i	$0.889063\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2520.2.bi.c.1801.1		2
3.2	odd	2		840.2.bg.e.121.1	✓	2
7.4	even	3	inner	2520.2.bi.c.361.1		2
12.11	even	2		1680.2.bg.h.961.1		2
21.2	odd	6		5880.2.a.c.1.1		1
21.5	even	6		5880.2.a.bc.1.1		1
21.11	odd	6		840.2.bg.e.361.1	yes	2
84.11	even	6		1680.2.bg.h.1201.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.bg.e.121.1	✓	2	3.2	odd	2
840.2.bg.e.361.1	yes	2	21.11	odd	6
1680.2.bg.h.961.1		2	12.11	even	2
1680.2.bg.h.1201.1		2	84.11	even	6
2520.2.bi.c.361.1		2	7.4	even	3	inner
2520.2.bi.c.1801.1		2	1.1	even	1	trivial
5880.2.a.c.1.1		1	21.2	odd	6
5880.2.a.bc.1.1		1	21.5	even	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:	\( N \)	\(=\)	\( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2520.bi (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{5} +(0.500000 + 2.59808i) q^{7} +(-1.00000 - 1.73205i) q^{11} +1.00000 q^{13} +(-1.50000 + 2.59808i) q^{19} +(-0.500000 - 0.866025i) q^{25} +(4.50000 + 7.79423i) q^{31} +(-2.50000 - 0.866025i) q^{35} +(-1.50000 + 2.59808i) q^{37} -2.00000 q^{41} +3.00000 q^{43} +(-3.00000 + 5.19615i) q^{47} +(-6.50000 + 2.59808i) q^{49} +2.00000 q^{55} +(-2.00000 - 3.46410i) q^{59} +(-1.00000 + 1.73205i) q^{61} +(-0.500000 + 0.866025i) q^{65} +(-2.50000 - 4.33013i) q^{67} -14.0000 q^{71} +(-0.500000 - 0.866025i) q^{73} +(4.00000 - 3.46410i) q^{77} +(-4.50000 + 7.79423i) q^{79} +6.00000 q^{83} +(-2.00000 + 3.46410i) q^{89} +(0.500000 + 2.59808i) q^{91} +(-1.50000 - 2.59808i) q^{95} -14.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{5} + q^{7} - 2 q^{11} + 2 q^{13} - 3 q^{19} - q^{25} + 9 q^{31} - 5 q^{35} - 3 q^{37} - 4 q^{41} + 6 q^{43} - 6 q^{47} - 13 q^{49} + 4 q^{55} - 4 q^{59} - 2 q^{61} - q^{65} - 5 q^{67} - 28 q^{71}+ \cdots - 28 q^{97}+O(q^{100}) \) 2 * q - q^5 + q^7 - 2 * q^11 + 2 * q^13 - 3 * q^19 - q^25 + 9 * q^31 - 5 * q^35 - 3 * q^37 - 4 * q^41 + 6 * q^43 - 6 * q^47 - 13 * q^49 + 4 * q^55 - 4 * q^59 - 2 * q^61 - q^65 - 5 * q^67 - 28 * q^71 - q^73 + 8 * q^77 - 9 * q^79 + 12 * q^83 - 4 * q^89 + q^91 - 3 * q^95 - 28 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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