LMFDB - Embedded newform 2520.2.bi.a.361.1

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:	$1$
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 280)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2520.361
Dual form			2520.2.bi.a.1801.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{5} +(-2.50000 + 0.866025i) q^{7} +O(q^{10})$	$q+(-0.500000 - 0.866025i) q^{5} +(-2.50000 + 0.866025i) q^{7} +(1.00000 - 1.73205i) q^{11} +(2.00000 - 3.46410i) q^{17} +(1.00000 + 1.73205i) q^{19} +(0.500000 + 0.866025i) q^{23} +(-0.500000 + 0.866025i) q^{25} -9.00000 q^{29} +(-2.00000 + 3.46410i) q^{31} +(2.00000 + 1.73205i) q^{35} +(-2.00000 - 3.46410i) q^{37} -1.00000 q^{41} +9.00000 q^{43} +(5.50000 - 4.33013i) q^{49} +(-5.00000 + 8.66025i) q^{53} -2.00000 q^{55} +(-5.00000 + 8.66025i) q^{59} +(-4.50000 - 7.79423i) q^{61} +(-2.50000 + 4.33013i) q^{67} -14.0000 q^{71} +(-6.00000 + 10.3923i) q^{73} +(-1.00000 + 5.19615i) q^{77} +(-7.00000 - 12.1244i) q^{79} -11.0000 q^{83} -4.00000 q^{85} +(-7.50000 - 12.9904i) q^{89} +(1.00000 - 1.73205i) q^{95} -18.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{5} - 5 q^{7}+O(q^{10}) $ 2 * q - q^5 - 5 * q^7	$ 2 q - q^{5} - 5 q^{7} + 2 q^{11} + 4 q^{17} + 2 q^{19} + q^{23} - q^{25} - 18 q^{29} - 4 q^{31} + 4 q^{35} - 4 q^{37} - 2 q^{41} + 18 q^{43} + 11 q^{49} - 10 q^{53} - 4 q^{55} - 10 q^{59} - 9 q^{61} - 5 q^{67} - 28 q^{71} - 12 q^{73} - 2 q^{77} - 14 q^{79} - 22 q^{83} - 8 q^{85} - 15 q^{89} + 2 q^{95} - 36 q^{97}+O(q^{100}) $ 2 * q - q^5 - 5 * q^7 + 2 * q^11 + 4 * q^17 + 2 * q^19 + q^23 - q^25 - 18 * q^29 - 4 * q^31 + 4 * q^35 - 4 * q^37 - 2 * q^41 + 18 * q^43 + 11 * q^49 - 10 * q^53 - 4 * q^55 - 10 * q^59 - 9 * q^61 - 5 * q^67 - 28 * q^71 - 12 * q^73 - 2 * q^77 - 14 * q^79 - 22 * q^83 - 8 * q^85 - 15 * q^89 + 2 * q^95 - 36 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$281$	$631$	$1081$	$1261$	$2017$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$1$

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
$7$	−2.50000	+	0.866025i	−0.944911	+	0.327327i
$7$	−2.50000	+	0.866025i	−0.944911	+	0.327327i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.00000	−	1.73205i	0.301511	−	0.522233i	−0.674967	−	0.737848i	$-0.735842\pi$
$11$	1.00000	−	1.73205i	0.301511	−	0.522233i	0.976478	+	0.215615i	$0.0691756\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.00000	−	3.46410i	0.485071	−	0.840168i	−0.514782	−	0.857321i	$-0.672127\pi$
$17$	2.00000	−	3.46410i	0.485071	−	0.840168i	0.999853	+	0.0171533i	$0.00546033\pi$
$18$		0			0
$19$	1.00000	+	1.73205i	0.229416	+	0.397360i	0.957635	−	0.287984i	$-0.0929851\pi$
$19$	1.00000	+	1.73205i	0.229416	+	0.397360i	−0.728219	+	0.685344i	$0.759652\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.500000	+	0.866025i	0.104257	+	0.180579i	0.913434	−	0.406986i	$-0.133420\pi$
$23$	0.500000	+	0.866025i	0.104257	+	0.180579i	−0.809177	+	0.587565i	$0.800087\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−9.00000			−1.67126			−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000			−1.67126			−0.835629	+	0.549294i	$0.814897\pi$
$30$		0			0
$31$	−2.00000	+	3.46410i	−0.359211	+	0.622171i	−0.987829	−	0.155543i	$-0.950287\pi$
$31$	−2.00000	+	3.46410i	−0.359211	+	0.622171i	0.628619	+	0.777714i	$0.283621\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.00000	+	1.73205i	0.338062	+	0.292770i
$36$		0			0
$37$	−2.00000	−	3.46410i	−0.328798	−	0.569495i	0.653476	−	0.756948i	$-0.273310\pi$
$37$	−2.00000	−	3.46410i	−0.328798	−	0.569495i	−0.982274	+	0.187453i	$0.939977\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.00000			−0.156174			−0.0780869	−	0.996947i	$-0.524881\pi$
$41$	−1.00000			−0.156174			−0.0780869	+	0.996947i	$0.524881\pi$
$42$		0			0
$43$	9.00000			1.37249			0.686244	−	0.727372i	$-0.259258\pi$
$43$	9.00000			1.37249			0.686244	+	0.727372i	$0.259258\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		0			0
$49$	5.50000	−	4.33013i	0.785714	−	0.618590i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−5.00000	+	8.66025i	−0.686803	+	1.18958i	0.286064	+	0.958211i	$0.407653\pi$
$53$	−5.00000	+	8.66025i	−0.686803	+	1.18958i	−0.972867	+	0.231367i	$0.925680\pi$
$54$		0			0
$55$	−2.00000			−0.269680
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5.00000	+	8.66025i	−0.650945	+	1.12747i	0.331949	+	0.943297i	$0.392294\pi$
$59$	−5.00000	+	8.66025i	−0.650945	+	1.12747i	−0.982894	+	0.184172i	$0.941040\pi$
$60$		0			0
$61$	−4.50000	−	7.79423i	−0.576166	−	0.997949i	−0.995914	−	0.0903080i	$-0.971215\pi$
$61$	−4.50000	−	7.79423i	−0.576166	−	0.997949i	0.419748	−	0.907641i	$-0.362118\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−2.50000	+	4.33013i	−0.305424	+	0.529009i	−0.977356	−	0.211604i	$-0.932131\pi$
$67$	−2.50000	+	4.33013i	−0.305424	+	0.529009i	0.671932	+	0.740613i	$0.265465\pi$
$68$		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
$73$	−6.00000	+	10.3923i	−0.702247	+	1.21633i	0.265429	+	0.964130i	$0.414486\pi$
$73$	−6.00000	+	10.3923i	−0.702247	+	1.21633i	−0.967676	+	0.252197i	$0.918847\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.00000	+	5.19615i	−0.113961	+	0.592157i
$78$		0			0
$79$	−7.00000	−	12.1244i	−0.787562	−	1.36410i	−0.927457	−	0.373930i	$-0.878010\pi$
$79$	−7.00000	−	12.1244i	−0.787562	−	1.36410i	0.139895	−	0.990166i	$-0.455323\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−11.0000			−1.20741			−0.603703	−	0.797209i	$-0.706309\pi$
$83$	−11.0000			−1.20741			−0.603703	+	0.797209i	$0.706309\pi$
$84$		0			0
$85$	−4.00000			−0.433861
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−7.50000	−	12.9904i	−0.794998	−	1.37698i	−0.922840	−	0.385183i	$-0.874138\pi$
$89$	−7.50000	−	12.9904i	−0.794998	−	1.37698i	0.127842	−	0.991795i	$-0.459195\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.00000	−	1.73205i	0.102598	−	0.177705i
$96$		0			0
$97$	−18.0000			−1.82762			−0.913812	−	0.406138i	$-0.866875\pi$
$97$	−18.0000			−1.82762			−0.913812	+	0.406138i	$0.866875\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.50000	−	2.59808i	0.149256	−	0.258518i	−0.781697	−	0.623658i	$-0.785646\pi$
$101$	1.50000	−	2.59808i	0.149256	−	0.258518i	0.930953	+	0.365140i	$0.118979\pi$
$102$		0			0
$103$	6.50000	+	11.2583i	0.640464	+	1.10932i	0.985329	+	0.170664i	$0.0545913\pi$
$103$	6.50000	+	11.2583i	0.640464	+	1.10932i	−0.344865	+	0.938652i	$0.612075\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	4.50000	+	7.79423i	0.435031	+	0.753497i	0.997298	−	0.0734594i	$-0.0234039\pi$
$107$	4.50000	+	7.79423i	0.435031	+	0.753497i	−0.562267	+	0.826956i	$0.690071\pi$
$108$		0			0
$109$	0.500000	−	0.866025i	0.0478913	−	0.0829502i	−0.841086	−	0.540901i	$-0.818083\pi$
$109$	0.500000	−	0.866025i	0.0478913	−	0.0829502i	0.888977	+	0.457951i	$0.151417\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−2.00000			−0.188144			−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000			−0.188144			−0.0940721	+	0.995565i	$0.529988\pi$
$114$		0			0
$115$	0.500000	−	0.866025i	0.0466252	−	0.0807573i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2.00000	+	10.3923i	−0.183340	+	0.952661i
$120$		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
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−4.00000	−	6.92820i	−0.349482	−	0.605320i	0.636676	−	0.771132i	$-0.280309\pi$
$131$	−4.00000	−	6.92820i	−0.349482	−	0.605320i	−0.986157	+	0.165812i	$0.946976\pi$
$132$		0			0
$133$	−4.00000	−	3.46410i	−0.346844	−	0.300376i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	6.00000	−	10.3923i	0.512615	−	0.887875i	−0.487278	−	0.873247i	$-0.662010\pi$
$137$	6.00000	−	10.3923i	0.512615	−	0.887875i	0.999893	−	0.0146279i	$-0.00465636\pi$
$138$		0			0
$139$	−2.00000			−0.169638			−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000			−0.169638			−0.0848189	+	0.996396i	$0.527031\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	4.50000	+	7.79423i	0.373705	+	0.647275i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−2.50000	−	4.33013i	−0.204808	−	0.354738i	0.745264	−	0.666770i	$-0.232324\pi$
$149$	−2.50000	−	4.33013i	−0.204808	−	0.354738i	−0.950072	+	0.312032i	$0.898990\pi$
$150$		0			0
$151$	−4.00000	+	6.92820i	−0.325515	+	0.563809i	−0.981617	−	0.190864i	$-0.938871\pi$
$151$	−4.00000	+	6.92820i	−0.325515	+	0.563809i	0.656101	+	0.754673i	$0.272204\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	4.00000			0.321288
$156$		0			0
$157$	−1.00000	+	1.73205i	−0.0798087	+	0.138233i	−0.903167	−	0.429289i	$-0.858764\pi$
$157$	−1.00000	+	1.73205i	−0.0798087	+	0.138233i	0.823359	+	0.567521i	$0.192098\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2.00000	−	1.73205i	−0.157622	−	0.136505i
$162$		0			0
$163$	10.0000	+	17.3205i	0.783260	+	1.35665i	0.930033	+	0.367477i	$0.119778\pi$
$163$	10.0000	+	17.3205i	0.783260	+	1.35665i	−0.146772	+	0.989170i	$0.546888\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−17.0000			−1.31550			−0.657750	−	0.753237i	$-0.728492\pi$
$167$	−17.0000			−1.31550			−0.657750	+	0.753237i	$0.728492\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$	8.00000	+	13.8564i	0.608229	+	1.05348i	0.991532	+	0.129861i	$0.0414530\pi$
$173$	8.00000	+	13.8564i	0.608229	+	1.05348i	−0.383304	+	0.923622i	$0.625214\pi$
$174$		0			0
$175$	0.500000	−	2.59808i	0.0377964	−	0.196396i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	6.00000	−	10.3923i	0.448461	−	0.776757i	−0.549825	−	0.835280i	$-0.685306\pi$
$179$	6.00000	−	10.3923i	0.448461	−	0.776757i	0.998286	+	0.0585225i	$0.0186389\pi$
$180$		0			0
$181$	−25.0000			−1.85824			−0.929118	−	0.369784i	$-0.879432\pi$
$181$	−25.0000			−1.85824			−0.929118	+	0.369784i	$0.879432\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2.00000	+	3.46410i	−0.147043	+	0.254686i
$186$		0			0
$187$	−4.00000	−	6.92820i	−0.292509	−	0.506640i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	9.00000	+	15.5885i	0.651217	+	1.12794i	0.982828	+	0.184525i	$0.0590746\pi$
$191$	9.00000	+	15.5885i	0.651217	+	1.12794i	−0.331611	+	0.943416i	$0.607592\pi$
$192$		0			0
$193$	7.00000	−	12.1244i	0.503871	−	0.872730i	−0.496119	−	0.868255i	$-0.665242\pi$
$193$	7.00000	−	12.1244i	0.503871	−	0.872730i	0.999990	−	0.00447566i	$-0.00142465\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−18.0000			−1.28245			−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000			−1.28245			−0.641223	+	0.767354i	$0.721573\pi$
$198$		0			0
$199$	2.00000	−	3.46410i	0.141776	−	0.245564i	−0.786389	−	0.617731i	$-0.788052\pi$
$199$	2.00000	−	3.46410i	0.141776	−	0.245564i	0.928166	+	0.372168i	$0.121385\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	22.5000	−	7.79423i	1.57919	−	0.547048i
$204$		0			0
$205$	0.500000	+	0.866025i	0.0349215	+	0.0604858i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.00000			0.276686
$210$		0			0
$211$	2.00000			0.137686			0.0688428	−	0.997628i	$-0.478069\pi$
$211$	2.00000			0.137686			0.0688428	+	0.997628i	$0.478069\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−4.50000	−	7.79423i	−0.306897	−	0.531562i
$216$		0			0
$217$	2.00000	−	10.3923i	0.135769	−	0.705476i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−4.00000			−0.267860			−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000			−0.267860			−0.133930	+	0.990991i	$0.542760\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−10.0000	+	17.3205i	−0.663723	+	1.14960i	0.315906	+	0.948790i	$0.397691\pi$
$227$	−10.0000	+	17.3205i	−0.663723	+	1.14960i	−0.979630	+	0.200812i	$0.935642\pi$
$228$		0			0
$229$	5.00000	+	8.66025i	0.330409	+	0.572286i	0.982592	−	0.185776i	$-0.0594799\pi$
$229$	5.00000	+	8.66025i	0.330409	+	0.572286i	−0.652183	+	0.758062i	$0.726147\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	4.00000	+	6.92820i	0.262049	+	0.453882i	0.966786	−	0.255586i	$-0.0822686\pi$
$233$	4.00000	+	6.92820i	0.262049	+	0.453882i	−0.704737	+	0.709468i	$0.748935\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	24.0000			1.55243			0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000			1.55243			0.776215	+	0.630468i	$0.217137\pi$
$240$		0			0
$241$	11.0000	−	19.0526i	0.708572	−	1.22728i	−0.256814	−	0.966461i	$-0.582673\pi$
$241$	11.0000	−	19.0526i	0.708572	−	1.22728i	0.965387	−	0.260822i	$-0.0839937\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−6.50000	−	2.59808i	−0.415270	−	0.165985i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−8.00000			−0.504956			−0.252478	−	0.967603i	$-0.581245\pi$
$251$	−8.00000			−0.504956			−0.252478	+	0.967603i	$0.581245\pi$
$252$		0			0
$253$	2.00000			0.125739
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$257$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$258$		0			0
$259$	8.00000	+	6.92820i	0.497096	+	0.430498i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−0.500000	+	0.866025i	−0.0308313	+	0.0534014i	−0.881029	−	0.473062i	$-0.843149\pi$
$263$	−0.500000	+	0.866025i	−0.0308313	+	0.0534014i	0.850198	+	0.526463i	$0.176482\pi$
$264$		0			0
$265$	10.0000			0.614295
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−10.5000	+	18.1865i	−0.640196	+	1.10885i	0.345192	+	0.938532i	$0.387814\pi$
$269$	−10.5000	+	18.1865i	−0.640196	+	1.10885i	−0.985389	+	0.170321i	$0.945520\pi$
$270$		0			0
$271$	−11.0000	−	19.0526i	−0.668202	−	1.15736i	−0.978406	−	0.206691i	$-0.933731\pi$
$271$	−11.0000	−	19.0526i	−0.668202	−	1.15736i	0.310204	−	0.950670i	$-0.399603\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.00000	+	1.73205i	0.0603023	+	0.104447i
$276$		0			0
$277$	14.0000	−	24.2487i	0.841178	−	1.45696i	−0.0477206	−	0.998861i	$-0.515196\pi$
$277$	14.0000	−	24.2487i	0.841178	−	1.45696i	0.888899	−	0.458103i	$-0.151471\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−18.0000			−1.07379			−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000			−1.07379			−0.536895	+	0.843649i	$0.680403\pi$
$282$		0			0
$283$	2.00000	−	3.46410i	0.118888	−	0.205919i	−0.800439	−	0.599414i	$-0.795400\pi$
$283$	2.00000	−	3.46410i	0.118888	−	0.205919i	0.919327	+	0.393494i	$0.128734\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2.50000	−	0.866025i	0.147570	−	0.0511199i
$288$		0			0
$289$	0.500000	+	0.866025i	0.0294118	+	0.0509427i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−12.0000			−0.701047			−0.350524	−	0.936554i	$-0.613996\pi$
$293$	−12.0000			−0.701047			−0.350524	+	0.936554i	$0.613996\pi$
$294$		0			0
$295$	10.0000			0.582223
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−22.5000	+	7.79423i	−1.29688	+	0.449252i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−4.50000	+	7.79423i	−0.257669	+	0.446296i
$306$		0			0
$307$	21.0000			1.19853			0.599267	−	0.800549i	$-0.295459\pi$
$307$	21.0000			1.19853			0.599267	+	0.800549i	$0.295459\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	13.0000	−	22.5167i	0.737162	−	1.27680i	−0.216606	−	0.976259i	$-0.569499\pi$
$311$	13.0000	−	22.5167i	0.737162	−	1.27680i	0.953768	−	0.300544i	$-0.0971681\pi$
$312$		0			0
$313$	−8.00000	−	13.8564i	−0.452187	−	0.783210i	0.546335	−	0.837567i	$-0.316023\pi$
$313$	−8.00000	−	13.8564i	−0.452187	−	0.783210i	−0.998522	+	0.0543564i	$0.982689\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−8.00000	−	13.8564i	−0.449325	−	0.778253i	0.549017	−	0.835811i	$-0.315002\pi$
$317$	−8.00000	−	13.8564i	−0.449325	−	0.778253i	−0.998342	+	0.0575576i	$0.981669\pi$
$318$		0			0
$319$	−9.00000	+	15.5885i	−0.503903	+	0.872786i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	8.00000			0.445132
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$331$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	5.00000			0.273179
$336$		0			0
$337$	−2.00000			−0.108947			−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000			−0.108947			−0.0544735	+	0.998515i	$0.517348\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	4.00000	+	6.92820i	0.216612	+	0.375183i
$342$		0			0
$343$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3.50000	+	6.06218i	−0.187890	+	0.325435i	−0.944547	−	0.328378i	$-0.893498\pi$
$347$	−3.50000	+	6.06218i	−0.187890	+	0.325435i	0.756657	+	0.653812i	$0.226831\pi$
$348$		0			0
$349$	−19.0000			−1.01705			−0.508523	−	0.861048i	$-0.669808\pi$
$349$	−19.0000			−1.01705			−0.508523	+	0.861048i	$0.669808\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	7.00000	−	12.1244i	0.372572	−	0.645314i	−0.617388	−	0.786659i	$-0.711809\pi$
$353$	7.00000	−	12.1244i	0.372572	−	0.645314i	0.989960	+	0.141344i	$0.0451425\pi$
$354$		0			0
$355$	7.00000	+	12.1244i	0.371521	+	0.643494i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	6.00000	+	10.3923i	0.316668	+	0.548485i	0.979791	−	0.200026i	$-0.0641026\pi$
$359$	6.00000	+	10.3923i	0.316668	+	0.548485i	−0.663123	+	0.748511i	$0.730769\pi$
$360$		0			0
$361$	7.50000	−	12.9904i	0.394737	−	0.683704i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	12.0000			0.628109
$366$		0			0
$367$	−3.50000	+	6.06218i	−0.182699	+	0.316443i	−0.942799	−	0.333363i	$-0.891817\pi$
$367$	−3.50000	+	6.06218i	−0.182699	+	0.316443i	0.760100	+	0.649806i	$0.225150\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	5.00000	−	25.9808i	0.259587	−	1.34885i
$372$		0			0
$373$	14.0000	+	24.2487i	0.724893	+	1.25555i	0.959018	+	0.283344i	$0.0914439\pi$
$373$	14.0000	+	24.2487i	0.724893	+	1.25555i	−0.234126	+	0.972206i	$0.575223\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−30.0000			−1.54100			−0.770498	−	0.637442i	$-0.779993\pi$
$379$	−30.0000			−1.54100			−0.770498	+	0.637442i	$0.779993\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	10.5000	+	18.1865i	0.536525	+	0.929288i	0.999088	+	0.0427020i	$0.0135966\pi$
$383$	10.5000	+	18.1865i	0.536525	+	0.929288i	−0.462563	+	0.886586i	$0.653070\pi$
$384$		0			0
$385$	5.00000	−	1.73205i	0.254824	−	0.0882735i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	13.0000	−	22.5167i	0.659126	−	1.14164i	−0.321716	−	0.946836i	$-0.604260\pi$
$389$	13.0000	−	22.5167i	0.659126	−	1.14164i	0.980842	−	0.194804i	$-0.0624070\pi$
$390$		0			0
$391$	4.00000			0.202289
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−7.00000	+	12.1244i	−0.352208	+	0.610043i
$396$		0			0
$397$	−3.00000	−	5.19615i	−0.150566	−	0.260787i	0.780870	−	0.624694i	$-0.214776\pi$
$397$	−3.00000	−	5.19615i	−0.150566	−	0.260787i	−0.931436	+	0.363906i	$0.881443\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−8.50000	−	14.7224i	−0.424470	−	0.735203i	0.571901	−	0.820323i	$-0.306206\pi$
$401$	−8.50000	−	14.7224i	−0.424470	−	0.735203i	−0.996371	+	0.0851195i	$0.972873\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−8.00000			−0.396545
$408$		0			0
$409$	10.5000	−	18.1865i	0.519192	−	0.899266i	−0.480560	−	0.876962i	$-0.659566\pi$
$409$	10.5000	−	18.1865i	0.519192	−	0.899266i	0.999751	−	0.0223042i	$-0.00710022\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	5.00000	−	25.9808i	0.246034	−	1.27843i
$414$		0			0
$415$	5.50000	+	9.52628i	0.269984	+	0.467627i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−16.0000			−0.781651			−0.390826	−	0.920465i	$-0.627810\pi$
$419$	−16.0000			−0.781651			−0.390826	+	0.920465i	$0.627810\pi$
$420$		0			0
$421$	−27.0000			−1.31590			−0.657950	−	0.753062i	$-0.728576\pi$
$421$	−27.0000			−1.31590			−0.657950	+	0.753062i	$0.728576\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2.00000	+	3.46410i	0.0970143	+	0.168034i
$426$		0			0
$427$	18.0000	+	15.5885i	0.871081	+	0.754378i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	7.00000	−	12.1244i	0.337178	−	0.584010i	−0.646723	−	0.762725i	$-0.723861\pi$
$431$	7.00000	−	12.1244i	0.337178	−	0.584010i	0.983901	+	0.178716i	$0.0571942\pi$
$432$		0			0
$433$	30.0000			1.44171			0.720854	−	0.693087i	$-0.243750\pi$
$433$	30.0000			1.44171			0.720854	+	0.693087i	$0.243750\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.00000	+	1.73205i	−0.0478365	+	0.0828552i
$438$		0			0
$439$	14.0000	+	24.2487i	0.668184	+	1.15733i	0.978412	+	0.206666i	$0.0662612\pi$
$439$	14.0000	+	24.2487i	0.668184	+	1.15733i	−0.310228	+	0.950662i	$0.600405\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−5.50000	−	9.52628i	−0.261313	−	0.452607i	0.705278	−	0.708931i	$-0.250822\pi$
$443$	−5.50000	−	9.52628i	−0.261313	−	0.452607i	−0.966591	+	0.256323i	$0.917489\pi$
$444$		0			0
$445$	−7.50000	+	12.9904i	−0.355534	+	0.615803i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	41.0000			1.93491			0.967455	−	0.253044i	$-0.0814317\pi$
$449$	41.0000			1.93491			0.967455	+	0.253044i	$0.0814317\pi$
$450$		0			0
$451$	−1.00000	+	1.73205i	−0.0470882	+	0.0815591i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−4.00000	−	6.92820i	−0.187112	−	0.324088i	0.757174	−	0.653213i	$-0.226579\pi$
$457$	−4.00000	−	6.92820i	−0.187112	−	0.324088i	−0.944286	+	0.329125i	$0.893246\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−2.00000			−0.0931493			−0.0465746	−	0.998915i	$-0.514831\pi$
$461$	−2.00000			−0.0931493			−0.0465746	+	0.998915i	$0.514831\pi$
$462$		0			0
$463$	39.0000			1.81248			0.906242	−	0.422760i	$-0.138939\pi$
$463$	39.0000			1.81248			0.906242	+	0.422760i	$0.138939\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−3.50000	−	6.06218i	−0.161961	−	0.280524i	0.773611	−	0.633661i	$-0.218448\pi$
$467$	−3.50000	−	6.06218i	−0.161961	−	0.280524i	−0.935572	+	0.353137i	$0.885115\pi$
$468$		0			0
$469$	2.50000	−	12.9904i	0.115439	−	0.599840i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	9.00000	−	15.5885i	0.413820	−	0.716758i
$474$		0			0
$475$	−2.00000			−0.0917663
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−8.00000	+	13.8564i	−0.365529	+	0.633115i	−0.988861	−	0.148842i	$-0.952445\pi$
$479$	−8.00000	+	13.8564i	−0.365529	+	0.633115i	0.623332	+	0.781958i	$0.285779\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	9.00000	+	15.5885i	0.408669	+	0.707835i
$486$		0			0
$487$	4.00000	−	6.92820i	0.181257	−	0.313947i	−0.761052	−	0.648691i	$-0.775317\pi$
$487$	4.00000	−	6.92820i	0.181257	−	0.313947i	0.942309	+	0.334744i	$0.108650\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	20.0000			0.902587			0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000			0.902587			0.451294	+	0.892375i	$0.350963\pi$
$492$		0			0
$493$	−18.0000	+	31.1769i	−0.810679	+	1.40414i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	35.0000	−	12.1244i	1.56996	−	0.543852i
$498$		0			0
$499$	19.0000	+	32.9090i	0.850557	+	1.47321i	0.880707	+	0.473662i	$0.157068\pi$
$499$	19.0000	+	32.9090i	0.850557	+	1.47321i	−0.0301498	+	0.999545i	$0.509598\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	23.0000			1.02552			0.512760	−	0.858532i	$-0.328623\pi$
$503$	23.0000			1.02552			0.512760	+	0.858532i	$0.328623\pi$
$504$		0			0
$505$	−3.00000			−0.133498
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−7.50000	−	12.9904i	−0.332432	−	0.575789i	0.650556	−	0.759458i	$-0.274536\pi$
$509$	−7.50000	−	12.9904i	−0.332432	−	0.575789i	−0.982988	+	0.183669i	$0.941202\pi$
$510$		0			0
$511$	6.00000	−	31.1769i	0.265424	−	1.37919i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	6.50000	−	11.2583i	0.286424	−	0.496101i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−21.0000	+	36.3731i	−0.920027	+	1.59353i	−0.120656	+	0.992694i	$0.538500\pi$
$521$	−21.0000	+	36.3731i	−0.920027	+	1.59353i	−0.799370	+	0.600839i	$0.794833\pi$
$522$		0			0
$523$	−14.0000	−	24.2487i	−0.612177	−	1.06032i	−0.990873	−	0.134801i	$-0.956961\pi$
$523$	−14.0000	−	24.2487i	−0.612177	−	1.06032i	0.378695	−	0.925521i	$-0.376373\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	8.00000	+	13.8564i	0.348485	+	0.603595i
$528$		0			0
$529$	11.0000	−	19.0526i	0.478261	−	0.828372i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	4.50000	−	7.79423i	0.194552	−	0.336974i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2.00000	−	13.8564i	−0.0861461	−	0.596838i
$540$		0			0
$541$	6.50000	+	11.2583i	0.279457	+	0.484033i	0.971250	−	0.238062i	$-0.0765123\pi$
$541$	6.50000	+	11.2583i	0.279457	+	0.484033i	−0.691793	+	0.722096i	$0.743179\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1.00000			−0.0428353
$546$		0			0
$547$	−35.0000			−1.49649			−0.748246	−	0.663421i	$-0.769104\pi$
$547$	−35.0000			−1.49649			−0.748246	+	0.663421i	$0.769104\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−9.00000	−	15.5885i	−0.383413	−	0.664091i
$552$		0			0
$553$	28.0000	+	24.2487i	1.19068	+	1.03116i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	15.0000	−	25.9808i	0.635570	−	1.10084i	−0.350824	−	0.936442i	$-0.614098\pi$
$557$	15.0000	−	25.9808i	0.635570	−	1.10084i	0.986394	−	0.164399i	$-0.0525683\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−22.5000	+	38.9711i	−0.948262	+	1.64244i	−0.199177	+	0.979963i	$0.563827\pi$
$563$	−22.5000	+	38.9711i	−0.948262	+	1.64244i	−0.749085	+	0.662474i	$0.769506\pi$
$564$		0			0
$565$	1.00000	+	1.73205i	0.0420703	+	0.0728679i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	23.0000	+	39.8372i	0.964210	+	1.67006i	0.711722	+	0.702461i	$0.247915\pi$
$569$	23.0000	+	39.8372i	0.964210	+	1.67006i	0.252488	+	0.967600i	$0.418751\pi$
$570$		0			0
$571$	13.0000	−	22.5167i	0.544033	−	0.942293i	−0.454634	−	0.890678i	$-0.650230\pi$
$571$	13.0000	−	22.5167i	0.544033	−	0.942293i	0.998667	−	0.0516146i	$-0.0164367\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.00000			−0.0417029
$576$		0			0
$577$	−1.00000	+	1.73205i	−0.0416305	+	0.0721062i	−0.886090	−	0.463513i	$-0.846589\pi$
$577$	−1.00000	+	1.73205i	−0.0416305	+	0.0721062i	0.844459	+	0.535620i	$0.179922\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	27.5000	−	9.52628i	1.14089	−	0.395217i
$582$		0			0
$583$	10.0000	+	17.3205i	0.414158	+	0.717342i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	12.0000			0.495293			0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000			0.495293			0.247647	+	0.968850i	$0.420343\pi$
$588$		0			0
$589$	−8.00000			−0.329634
$590$		0			0
$591$		0			0
$592$		0			0
$593$	9.00000	+	15.5885i	0.369586	+	0.640141i	0.989501	−	0.144528i	$-0.0461663\pi$
$593$	9.00000	+	15.5885i	0.369586	+	0.640141i	−0.619915	+	0.784669i	$0.712833\pi$
$594$		0			0
$595$	10.0000	−	3.46410i	0.409960	−	0.142014i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	2.00000	−	3.46410i	0.0817178	−	0.141539i	−0.822270	−	0.569097i	$-0.807293\pi$
$599$	2.00000	−	3.46410i	0.0817178	−	0.141539i	0.903988	+	0.427558i	$0.140626\pi$
$600$		0			0
$601$	10.0000			0.407909			0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000			0.407909			0.203954	+	0.978980i	$0.434621\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3.50000	−	6.06218i	0.142295	−	0.246463i
$606$		0			0
$607$	−13.5000	−	23.3827i	−0.547948	−	0.949074i	−0.998415	−	0.0562808i	$-0.982076\pi$
$607$	−13.5000	−	23.3827i	−0.547948	−	0.949074i	0.450467	−	0.892793i	$-0.351258\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	10.0000	−	17.3205i	0.403896	−	0.699569i	−0.590296	−	0.807187i	$-0.700989\pi$
$613$	10.0000	−	17.3205i	0.403896	−	0.699569i	0.994192	+	0.107618i	$0.0343224\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	12.0000			0.483102			0.241551	−	0.970388i	$-0.422344\pi$
$617$	12.0000			0.483102			0.241551	+	0.970388i	$0.422344\pi$
$618$		0			0
$619$	17.0000	−	29.4449i	0.683288	−	1.18349i	−0.290684	−	0.956819i	$-0.593883\pi$
$619$	17.0000	−	29.4449i	0.683288	−	1.18349i	0.973972	−	0.226670i	$-0.0727838\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	30.0000	+	25.9808i	1.20192	+	1.04090i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−16.0000			−0.637962
$630$		0			0
$631$	−2.00000			−0.0796187			−0.0398094	−	0.999207i	$-0.512675\pi$
$631$	−2.00000			−0.0796187			−0.0398094	+	0.999207i	$0.512675\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−4.00000	−	6.92820i	−0.158735	−	0.274937i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−9.50000	+	16.4545i	−0.375227	+	0.649913i	−0.990361	−	0.138510i	$-0.955769\pi$
$641$	−9.50000	+	16.4545i	−0.375227	+	0.649913i	0.615134	+	0.788423i	$0.289102\pi$
$642$		0			0
$643$	−4.00000			−0.157745			−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000			−0.157745			−0.0788723	+	0.996885i	$0.525132\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	11.5000	−	19.9186i	0.452112	−	0.783080i	−0.546405	−	0.837521i	$-0.684004\pi$
$647$	11.5000	−	19.9186i	0.452112	−	0.783080i	0.998517	+	0.0544405i	$0.0173375\pi$
$648$		0			0
$649$	10.0000	+	17.3205i	0.392534	+	0.679889i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	18.0000	+	31.1769i	0.704394	+	1.22005i	0.966910	+	0.255119i	$0.0821147\pi$
$653$	18.0000	+	31.1769i	0.704394	+	1.22005i	−0.262515	+	0.964928i	$0.584552\pi$
$654$		0			0
$655$	−4.00000	+	6.92820i	−0.156293	+	0.270707i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−2.00000			−0.0779089			−0.0389545	−	0.999241i	$-0.512403\pi$
$659$	−2.00000			−0.0779089			−0.0389545	+	0.999241i	$0.512403\pi$
$660$		0			0
$661$	1.50000	−	2.59808i	0.0583432	−	0.101053i	−0.835379	−	0.549675i	$-0.814752\pi$
$661$	1.50000	−	2.59808i	0.0583432	−	0.101053i	0.893722	+	0.448622i	$0.148085\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.00000	+	5.19615i	−0.0387783	+	0.201498i
$666$		0			0
$667$	−4.50000	−	7.79423i	−0.174241	−	0.301794i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−18.0000			−0.694882
$672$		0			0
$673$	24.0000			0.925132			0.462566	−	0.886585i	$-0.346929\pi$
$673$	24.0000			0.925132			0.462566	+	0.886585i	$0.346929\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−12.0000	−	20.7846i	−0.461197	−	0.798817i	0.537823	−	0.843057i	$-0.319247\pi$
$677$	−12.0000	−	20.7846i	−0.461197	−	0.798817i	−0.999021	+	0.0442400i	$0.985913\pi$
$678$		0			0
$679$	45.0000	−	15.5885i	1.72694	−	0.598230i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	4.50000	−	7.79423i	0.172188	−	0.298238i	−0.766997	−	0.641651i	$-0.778250\pi$
$683$	4.50000	−	7.79423i	0.172188	−	0.298238i	0.939184	+	0.343413i	$0.111583\pi$
$684$		0			0
$685$	−12.0000			−0.458496
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−25.0000	−	43.3013i	−0.951045	−	1.64726i	−0.743170	−	0.669102i	$-0.766679\pi$
$691$	−25.0000	−	43.3013i	−0.951045	−	1.64726i	−0.207875	−	0.978155i	$-0.566655\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1.00000	+	1.73205i	0.0379322	+	0.0657004i
$696$		0			0
$697$	−2.00000	+	3.46410i	−0.0757554	+	0.131212i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−9.00000			−0.339925			−0.169963	−	0.985451i	$-0.554365\pi$
$701$	−9.00000			−0.339925			−0.169963	+	0.985451i	$0.554365\pi$
$702$		0			0
$703$	4.00000	−	6.92820i	0.150863	−	0.261302i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.50000	+	7.79423i	−0.0564133	+	0.293132i
$708$		0			0
$709$	−10.5000	−	18.1865i	−0.394336	−	0.683010i	0.598680	−	0.800988i	$-0.295692\pi$
$709$	−10.5000	−	18.1865i	−0.394336	−	0.683010i	−0.993016	+	0.117978i	$0.962359\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−4.00000			−0.149801
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−21.0000	−	36.3731i	−0.783168	−	1.35649i	−0.930087	−	0.367338i	$-0.880269\pi$
$719$	−21.0000	−	36.3731i	−0.783168	−	1.35649i	0.146920	−	0.989148i	$-0.453064\pi$
$720$		0			0
$721$	−26.0000	−	22.5167i	−0.968291	−	0.838564i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	4.50000	−	7.79423i	0.167126	−	0.289470i
$726$		0			0
$727$	−31.0000			−1.14973			−0.574863	−	0.818250i	$-0.694945\pi$
$727$	−31.0000			−1.14973			−0.574863	+	0.818250i	$0.694945\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	18.0000	−	31.1769i	0.665754	−	1.15312i
$732$		0			0
$733$	13.0000	+	22.5167i	0.480166	+	0.831672i	0.999741	−	0.0227529i	$-0.00724310\pi$
$733$	13.0000	+	22.5167i	0.480166	+	0.831672i	−0.519575	+	0.854425i	$0.673910\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	5.00000	+	8.66025i	0.184177	+	0.319005i
$738$		0			0
$739$	−5.00000	+	8.66025i	−0.183928	+	0.318573i	−0.943215	−	0.332184i	$-0.892215\pi$
$739$	−5.00000	+	8.66025i	−0.183928	+	0.318573i	0.759287	+	0.650756i	$0.225548\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−27.0000			−0.990534			−0.495267	−	0.868741i	$-0.664930\pi$
$743$	−27.0000			−0.990534			−0.495267	+	0.868741i	$0.664930\pi$
$744$		0			0
$745$	−2.50000	+	4.33013i	−0.0915929	+	0.158644i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−18.0000	−	15.5885i	−0.657706	−	0.569590i
$750$		0			0
$751$	−6.00000	−	10.3923i	−0.218943	−	0.379221i	0.735542	−	0.677479i	$-0.236928\pi$
$751$	−6.00000	−	10.3923i	−0.218943	−	0.379221i	−0.954485	+	0.298259i	$0.903594\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	8.00000			0.291150
$756$		0			0
$757$	−24.0000			−0.872295			−0.436147	−	0.899875i	$-0.643657\pi$
$757$	−24.0000			−0.872295			−0.436147	+	0.899875i	$0.643657\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−3.00000	−	5.19615i	−0.108750	−	0.188360i	0.806514	−	0.591215i	$-0.201351\pi$
$761$	−3.00000	−	5.19615i	−0.108750	−	0.188360i	−0.915264	+	0.402854i	$0.868018\pi$
$762$		0			0
$763$	−0.500000	+	2.59808i	−0.0181012	+	0.0940567i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	34.0000			1.22607			0.613036	−	0.790055i	$-0.289948\pi$
$769$	34.0000			1.22607			0.613036	+	0.790055i	$0.289948\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$773$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$774$		0			0
$775$	−2.00000	−	3.46410i	−0.0718421	−	0.124434i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1.00000	−	1.73205i	−0.0358287	−	0.0620572i
$780$		0			0
$781$	−14.0000	+	24.2487i	−0.500959	+	0.867687i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	2.00000			0.0713831
$786$		0			0
$787$	−22.5000	+	38.9711i	−0.802038	+	1.38917i	0.116234	+	0.993222i	$0.462918\pi$
$787$	−22.5000	+	38.9711i	−0.802038	+	1.38917i	−0.918272	+	0.395949i	$0.870416\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	5.00000	−	1.73205i	0.177780	−	0.0615846i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−8.00000			−0.283375			−0.141687	−	0.989911i	$-0.545253\pi$
$797$	−8.00000			−0.283375			−0.141687	+	0.989911i	$0.545253\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	12.0000	+	20.7846i	0.423471	+	0.733473i
$804$		0			0
$805$	−0.500000	+	2.59808i	−0.0176227	+	0.0915702i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	6.50000	−	11.2583i	0.228528	−	0.395822i	−0.728844	−	0.684680i	$-0.759942\pi$
$809$	6.50000	−	11.2583i	0.228528	−	0.395822i	0.957372	+	0.288858i	$0.0932755\pi$
$810$		0			0
$811$	−44.0000			−1.54505			−0.772524	−	0.634985i	$-0.781006\pi$
$811$	−44.0000			−1.54505			−0.772524	+	0.634985i	$0.781006\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	10.0000	−	17.3205i	0.350285	−	0.606711i
$816$		0			0
$817$	9.00000	+	15.5885i	0.314870	+	0.545371i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	15.0000	+	25.9808i	0.523504	+	0.906735i	0.999626	+	0.0273557i	$0.00870868\pi$
$821$	15.0000	+	25.9808i	0.523504	+	0.906735i	−0.476122	+	0.879379i	$0.657958\pi$
$822$		0			0
$823$	−24.5000	+	42.4352i	−0.854016	+	1.47920i	0.0235383	+	0.999723i	$0.492507\pi$
$823$	−24.5000	+	42.4352i	−0.854016	+	1.47920i	−0.877555	+	0.479477i	$0.840826\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	17.0000			0.591148			0.295574	−	0.955320i	$-0.404489\pi$
$827$	17.0000			0.591148			0.295574	+	0.955320i	$0.404489\pi$
$828$		0			0
$829$	−1.00000	+	1.73205i	−0.0347314	+	0.0601566i	−0.882869	−	0.469620i	$-0.844391\pi$
$829$	−1.00000	+	1.73205i	−0.0347314	+	0.0601566i	0.848137	+	0.529777i	$0.177724\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−4.00000	−	27.7128i	−0.138592	−	0.960192i
$834$		0			0
$835$	8.50000	+	14.7224i	0.294155	+	0.509491i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−10.0000			−0.345238			−0.172619	−	0.984989i	$-0.555223\pi$
$839$	−10.0000			−0.345238			−0.172619	+	0.984989i	$0.555223\pi$
$840$		0			0
$841$	52.0000			1.79310
$842$		0			0
$843$		0			0
$844$		0			0
$845$	6.50000	+	11.2583i	0.223607	+	0.387298i
$846$		0			0
$847$	−14.0000	−	12.1244i	−0.481046	−	0.416598i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2.00000	−	3.46410i	0.0685591	−	0.118748i
$852$		0			0
$853$	38.0000			1.30110			0.650548	−	0.759465i	$-0.274539\pi$
$853$	38.0000			1.30110			0.650548	+	0.759465i	$0.274539\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	15.0000	−	25.9808i	0.512390	−	0.887486i	−0.487507	−	0.873119i	$-0.662093\pi$
$857$	15.0000	−	25.9808i	0.512390	−	0.887486i	0.999897	−	0.0143666i	$-0.00457319\pi$
$858$		0			0
$859$	−20.0000	−	34.6410i	−0.682391	−	1.18194i	−0.974249	−	0.225475i	$-0.927607\pi$
$859$	−20.0000	−	34.6410i	−0.682391	−	1.18194i	0.291858	−	0.956462i	$-0.405727\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−0.500000	−	0.866025i	−0.0170202	−	0.0294798i	0.857390	−	0.514667i	$-0.172085\pi$
$863$	−0.500000	−	0.866025i	−0.0170202	−	0.0294798i	−0.874410	+	0.485188i	$0.838751\pi$
$864$		0			0
$865$	8.00000	−	13.8564i	0.272008	−	0.471132i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−28.0000			−0.949835
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.50000	+	0.866025i	−0.0845154	+	0.0292770i
$876$		0			0
$877$	−3.00000	−	5.19615i	−0.101303	−	0.175462i	0.810919	−	0.585159i	$-0.198968\pi$
$877$	−3.00000	−	5.19615i	−0.101303	−	0.175462i	−0.912222	+	0.409697i	$0.865634\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−17.0000			−0.572745			−0.286372	−	0.958118i	$-0.592449\pi$
$881$	−17.0000			−0.572745			−0.286372	+	0.958118i	$0.592449\pi$
$882$		0			0
$883$	20.0000			0.673054			0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000			0.673054			0.336527	+	0.941674i	$0.390748\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	7.50000	+	12.9904i	0.251825	+	0.436174i	0.964028	−	0.265799i	$-0.0856358\pi$
$887$	7.50000	+	12.9904i	0.251825	+	0.436174i	−0.712203	+	0.701974i	$0.752302\pi$
$888$		0			0
$889$	−20.0000	+	6.92820i	−0.670778	+	0.232364i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−12.0000			−0.401116
$896$		0			0
$897$		0			0
$898$		0			0
$899$	18.0000	−	31.1769i	0.600334	−	1.03981i
$900$		0			0
$901$	20.0000	+	34.6410i	0.666297	+	1.15406i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	12.5000	+	21.6506i	0.415514	+	0.719691i
$906$		0			0
$907$	−3.50000	+	6.06218i	−0.116216	+	0.201291i	−0.918265	−	0.395966i	$-0.870410\pi$
$907$	−3.50000	+	6.06218i	−0.116216	+	0.201291i	0.802049	+	0.597258i	$0.203743\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	6.00000			0.198789			0.0993944	−	0.995048i	$-0.468309\pi$
$911$	6.00000			0.198789			0.0993944	+	0.995048i	$0.468309\pi$
$912$		0			0
$913$	−11.0000	+	19.0526i	−0.364047	+	0.630548i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	16.0000	+	13.8564i	0.528367	+	0.457579i
$918$		0			0
$919$	−21.0000	−	36.3731i	−0.692726	−	1.19984i	−0.970941	−	0.239318i	$-0.923076\pi$
$919$	−21.0000	−	36.3731i	−0.692726	−	1.19984i	0.278215	−	0.960519i	$-0.410257\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	4.00000			0.131519
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−6.50000	−	11.2583i	−0.213258	−	0.369374i	0.739474	−	0.673185i	$-0.235074\pi$
$929$	−6.50000	−	11.2583i	−0.213258	−	0.369374i	−0.952732	+	0.303811i	$0.901741\pi$
$930$		0			0
$931$	13.0000	+	5.19615i	0.426058	+	0.170297i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−4.00000	+	6.92820i	−0.130814	+	0.226576i
$936$		0			0
$937$	4.00000			0.130674			0.0653372	−	0.997863i	$-0.479188\pi$
$937$	4.00000			0.130674			0.0653372	+	0.997863i	$0.479188\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−15.0000	+	25.9808i	−0.488986	+	0.846949i	−0.999920	−	0.0126715i	$-0.995966\pi$
$941$	−15.0000	+	25.9808i	−0.488986	+	0.846949i	0.510934	+	0.859620i	$0.329300\pi$
$942$		0			0
$943$	−0.500000	−	0.866025i	−0.0162822	−	0.0282017i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	2.50000	+	4.33013i	0.0812391	+	0.140710i	0.903782	−	0.427992i	$-0.140779\pi$
$947$	2.50000	+	4.33013i	0.0812391	+	0.140710i	−0.822543	+	0.568702i	$0.807446\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−28.0000			−0.907009			−0.453504	−	0.891254i	$-0.649826\pi$
$953$	−28.0000			−0.907009			−0.453504	+	0.891254i	$0.649826\pi$
$954$		0			0
$955$	9.00000	−	15.5885i	0.291233	−	0.504431i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−6.00000	+	31.1769i	−0.193750	+	1.00676i
$960$		0			0
$961$	7.50000	+	12.9904i	0.241935	+	0.419045i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−14.0000			−0.450676
$966$		0			0
$967$	55.0000			1.76868			0.884340	−	0.466843i	$-0.154609\pi$
$967$	55.0000			1.76868			0.884340	+	0.466843i	$0.154609\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−4.00000	−	6.92820i	−0.128366	−	0.222337i	0.794678	−	0.607032i	$-0.207640\pi$
$971$	−4.00000	−	6.92820i	−0.128366	−	0.222337i	−0.923044	+	0.384695i	$0.874307\pi$
$972$		0			0
$973$	5.00000	−	1.73205i	0.160293	−	0.0555270i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	21.0000	−	36.3731i	0.671850	−	1.16368i	−0.305530	−	0.952183i	$-0.598833\pi$
$977$	21.0000	−	36.3731i	0.671850	−	1.16368i	0.977379	−	0.211495i	$-0.0678332\pi$
$978$		0			0
$979$	−30.0000			−0.958804
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1.50000	−	2.59808i	0.0478426	−	0.0828658i	−0.841112	−	0.540860i	$-0.818099\pi$
$983$	1.50000	−	2.59808i	0.0478426	−	0.0828658i	0.888955	+	0.457995i	$0.151432\pi$
$984$		0			0
$985$	9.00000	+	15.5885i	0.286764	+	0.496690i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	4.50000	+	7.79423i	0.143092	+	0.247842i
$990$		0			0
$991$	20.0000	−	34.6410i	0.635321	−	1.10041i	−0.351126	−	0.936328i	$-0.614201\pi$
$991$	20.0000	−	34.6410i	0.635321	−	1.10041i	0.986447	−	0.164080i	$-0.0524655\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−4.00000			−0.126809
$996$		0			0
$997$	−5.00000	+	8.66025i	−0.158352	+	0.274273i	−0.934274	−	0.356555i	$-0.883951\pi$
$997$	−5.00000	+	8.66025i	−0.158352	+	0.274273i	0.775923	+	0.630828i	$0.217285\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2520.2.bi.a.361.1		2
3.2	odd	2		280.2.q.a.81.1	✓	2
7.2	even	3	inner	2520.2.bi.a.1801.1		2
12.11	even	2		560.2.q.h.81.1		2
15.2	even	4		1400.2.bh.b.249.2		4
15.8	even	4		1400.2.bh.b.249.1		4
15.14	odd	2		1400.2.q.e.1201.1		2
21.2	odd	6		280.2.q.a.121.1	yes	2
21.5	even	6		1960.2.q.k.961.1		2
21.11	odd	6		1960.2.a.i.1.1		1
21.17	even	6		1960.2.a.e.1.1		1
21.20	even	2		1960.2.q.k.361.1		2
84.11	even	6		3920.2.a.m.1.1		1
84.23	even	6		560.2.q.h.401.1		2
84.59	odd	6		3920.2.a.y.1.1		1
105.2	even	12		1400.2.bh.b.849.1		4
105.23	even	12		1400.2.bh.b.849.2		4
105.44	odd	6		1400.2.q.e.401.1		2
105.59	even	6		9800.2.a.bc.1.1		1
105.74	odd	6		9800.2.a.r.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.q.a.81.1	✓	2	3.2	odd	2
280.2.q.a.121.1	yes	2	21.2	odd	6
560.2.q.h.81.1		2	12.11	even	2
560.2.q.h.401.1		2	84.23	even	6
1400.2.q.e.401.1		2	105.44	odd	6
1400.2.q.e.1201.1		2	15.14	odd	2
1400.2.bh.b.249.1		4	15.8	even	4
1400.2.bh.b.249.2		4	15.2	even	4
1400.2.bh.b.849.1		4	105.2	even	12
1400.2.bh.b.849.2		4	105.23	even	12
1960.2.a.e.1.1		1	21.17	even	6
1960.2.a.i.1.1		1	21.11	odd	6
1960.2.q.k.361.1		2	21.20	even	2
1960.2.q.k.961.1		2	21.5	even	6
2520.2.bi.a.361.1		2	1.1	even	1	trivial
2520.2.bi.a.1801.1		2	7.2	even	3	inner
3920.2.a.m.1.1		1	84.11	even	6
3920.2.a.y.1.1		1	84.59	odd	6
9800.2.a.r.1.1		1	105.74	odd	6
9800.2.a.bc.1.1		1	105.59	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)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{5} +(-2.50000 + 0.866025i) q^{7} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{5} +(-2.50000 + 0.866025i) q^{7} +(1.00000 - 1.73205i) q^{11} +(2.00000 - 3.46410i) q^{17} +(1.00000 + 1.73205i) q^{19} +(0.500000 + 0.866025i) q^{23} +(-0.500000 + 0.866025i) q^{25} -9.00000 q^{29} +(-2.00000 + 3.46410i) q^{31} +(2.00000 + 1.73205i) q^{35} +(-2.00000 - 3.46410i) q^{37} -1.00000 q^{41} +9.00000 q^{43} +(5.50000 - 4.33013i) q^{49} +(-5.00000 + 8.66025i) q^{53} -2.00000 q^{55} +(-5.00000 + 8.66025i) q^{59} +(-4.50000 - 7.79423i) q^{61} +(-2.50000 + 4.33013i) q^{67} -14.0000 q^{71} +(-6.00000 + 10.3923i) q^{73} +(-1.00000 + 5.19615i) q^{77} +(-7.00000 - 12.1244i) q^{79} -11.0000 q^{83} -4.00000 q^{85} +(-7.50000 - 12.9904i) q^{89} +(1.00000 - 1.73205i) q^{95} -18.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{5} - 5 q^{7}+O(q^{10}) \) 2 * q - q^5 - 5 * q^7	\( 2 q - q^{5} - 5 q^{7} + 2 q^{11} + 4 q^{17} + 2 q^{19} + q^{23} - q^{25} - 18 q^{29} - 4 q^{31} + 4 q^{35} - 4 q^{37} - 2 q^{41} + 18 q^{43} + 11 q^{49} - 10 q^{53} - 4 q^{55} - 10 q^{59} - 9 q^{61} - 5 q^{67} - 28 q^{71} - 12 q^{73} - 2 q^{77} - 14 q^{79} - 22 q^{83} - 8 q^{85} - 15 q^{89} + 2 q^{95} - 36 q^{97}+O(q^{100}) \) 2 * q - q^5 - 5 * q^7 + 2 * q^11 + 4 * q^17 + 2 * q^19 + q^23 - q^25 - 18 * q^29 - 4 * q^31 + 4 * q^35 - 4 * q^37 - 2 * q^41 + 18 * q^43 + 11 * q^49 - 10 * q^53 - 4 * q^55 - 10 * q^59 - 9 * q^61 - 5 * q^67 - 28 * q^71 - 12 * q^73 - 2 * q^77 - 14 * q^79 - 22 * q^83 - 8 * q^85 - 15 * q^89 + 2 * q^95 - 36 * q^97

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