LMFDB - Embedded newform 2016.2.s.g.289.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2016,2,Mod(289,2016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2016, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("2016.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2016 = 2^{5} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2016.s (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:	$16.0978410475$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 672)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2016.289
Dual form			2016.2.s.g.865.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 2.59808i) q^{7} +O(q^{10})$	$q+(0.500000 + 2.59808i) q^{7} +(1.00000 + 1.73205i) q^{11} +5.00000 q^{13} +(-1.00000 - 1.73205i) q^{17} +(-1.50000 + 2.59808i) q^{19} +(1.00000 - 1.73205i) q^{23} +(2.50000 + 4.33013i) q^{25} -8.00000 q^{29} +(0.500000 + 0.866025i) q^{31} +(2.50000 - 4.33013i) q^{37} -2.00000 q^{41} +7.00000 q^{43} +(-4.00000 + 6.92820i) q^{47} +(-6.50000 + 2.59808i) q^{49} +(-1.00000 - 1.73205i) q^{53} +(5.00000 + 8.66025i) q^{59} +(1.00000 - 1.73205i) q^{61} +(5.50000 + 9.52628i) q^{67} -12.0000 q^{71} +(1.50000 + 2.59808i) q^{73} +(-4.00000 + 3.46410i) q^{77} +(-8.50000 + 14.7224i) q^{79} +16.0000 q^{83} +(6.00000 - 10.3923i) q^{89} +(2.50000 + 12.9904i) q^{91} -14.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{7}+O(q^{10}) $ 2 * q + q^7	$ 2 q + q^{7} + 2 q^{11} + 10 q^{13} - 2 q^{17} - 3 q^{19} + 2 q^{23} + 5 q^{25} - 16 q^{29} + q^{31} + 5 q^{37} - 4 q^{41} + 14 q^{43} - 8 q^{47} - 13 q^{49} - 2 q^{53} + 10 q^{59} + 2 q^{61} + 11 q^{67} - 24 q^{71} + 3 q^{73} - 8 q^{77} - 17 q^{79} + 32 q^{83} + 12 q^{89} + 5 q^{91} - 28 q^{97}+O(q^{100}) $ 2 * q + q^7 + 2 * q^11 + 10 * q^13 - 2 * q^17 - 3 * q^19 + 2 * q^23 + 5 * q^25 - 16 * q^29 + q^31 + 5 * q^37 - 4 * q^41 + 14 * q^43 - 8 * q^47 - 13 * q^49 - 2 * q^53 + 10 * q^59 + 2 * q^61 + 11 * q^67 - 24 * q^71 + 3 * q^73 - 8 * q^77 - 17 * q^79 + 32 * q^83 + 12 * q^89 + 5 * q^91 - 28 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1765$	$1793$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$	$1$

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			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$		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	0.976478	−	0.215615i	$-0.0691756\pi$
$11$	1.00000	+	1.73205i	0.301511	+	0.522233i	−0.674967	+	0.737848i	$0.735842\pi$
$12$		0			0
$13$	5.00000			1.38675			0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000			1.38675			0.693375	+	0.720577i	$0.256123\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.00000	−	1.73205i	−0.242536	−	0.420084i	0.718900	−	0.695113i	$-0.244646\pi$
$17$	−1.00000	−	1.73205i	−0.242536	−	0.420084i	−0.961436	+	0.275029i	$0.911312\pi$
$18$		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
$23$	1.00000	−	1.73205i	0.208514	−	0.361158i	−0.742732	−	0.669588i	$-0.766471\pi$
$23$	1.00000	−	1.73205i	0.208514	−	0.361158i	0.951247	+	0.308431i	$0.0998038\pi$
$24$		0			0
$25$	2.50000	+	4.33013i	0.500000	+	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−8.00000			−1.48556			−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000			−1.48556			−0.742781	+	0.669534i	$0.766494\pi$
$30$		0			0
$31$	0.500000	+	0.866025i	0.0898027	+	0.155543i	0.907428	−	0.420208i	$-0.138043\pi$
$31$	0.500000	+	0.866025i	0.0898027	+	0.155543i	−0.817625	+	0.575751i	$0.804710\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	2.50000	−	4.33013i	0.410997	−	0.711868i	−0.584002	−	0.811752i	$-0.698514\pi$
$37$	2.50000	−	4.33013i	0.410997	−	0.711868i	0.994999	+	0.0998840i	$0.0318472\pi$
$38$		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
$43$	7.00000			1.06749			0.533745	−	0.845645i	$-0.320784\pi$
$43$	7.00000			1.06749			0.533745	+	0.845645i	$0.320784\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.00000	+	6.92820i	−0.583460	+	1.01058i	0.411606	+	0.911362i	$0.364968\pi$
$47$	−4.00000	+	6.92820i	−0.583460	+	1.01058i	−0.995066	+	0.0992202i	$0.968365\pi$
$48$		0			0
$49$	−6.50000	+	2.59808i	−0.928571	+	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.00000	−	1.73205i	−0.137361	−	0.237915i	0.789136	−	0.614218i	$-0.210529\pi$
$53$	−1.00000	−	1.73205i	−0.137361	−	0.237915i	−0.926497	+	0.376303i	$0.877195\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	5.00000	+	8.66025i	0.650945	+	1.12747i	0.982894	+	0.184172i	$0.0589603\pi$
$59$	5.00000	+	8.66025i	0.650945	+	1.12747i	−0.331949	+	0.943297i	$0.607706\pi$
$60$		0			0
$61$	1.00000	−	1.73205i	0.128037	−	0.221766i	−0.794879	−	0.606768i	$-0.792466\pi$
$61$	1.00000	−	1.73205i	0.128037	−	0.221766i	0.922916	+	0.385002i	$0.125799\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	5.50000	+	9.52628i	0.671932	+	1.16382i	0.977356	+	0.211604i	$0.0678686\pi$
$67$	5.50000	+	9.52628i	0.671932	+	1.16382i	−0.305424	+	0.952217i	$0.598798\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−12.0000			−1.42414			−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000			−1.42414			−0.712069	+	0.702109i	$0.752242\pi$
$72$		0			0
$73$	1.50000	+	2.59808i	0.175562	+	0.304082i	0.940356	−	0.340193i	$-0.110493\pi$
$73$	1.50000	+	2.59808i	0.175562	+	0.304082i	−0.764794	+	0.644275i	$0.777159\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−4.00000	+	3.46410i	−0.455842	+	0.394771i
$78$		0			0
$79$	−8.50000	+	14.7224i	−0.956325	+	1.65640i	−0.225018	+	0.974355i	$0.572244\pi$
$79$	−8.50000	+	14.7224i	−0.956325	+	1.65640i	−0.731307	+	0.682048i	$0.761089\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	16.0000			1.75623			0.878114	−	0.478451i	$-0.158802\pi$
$83$	16.0000			1.75623			0.878114	+	0.478451i	$0.158802\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	6.00000	−	10.3923i	0.635999	−	1.10158i	−0.350304	−	0.936636i	$-0.613922\pi$
$89$	6.00000	−	10.3923i	0.635999	−	1.10158i	0.986303	−	0.164946i	$-0.0527450\pi$
$90$		0			0
$91$	2.50000	+	12.9904i	0.262071	+	1.36176i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		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
$101$	9.00000	+	15.5885i	0.895533	+	1.55111i	0.833143	+	0.553058i	$0.186539\pi$
$101$	9.00000	+	15.5885i	0.895533	+	1.55111i	0.0623905	+	0.998052i	$0.480128\pi$
$102$		0			0
$103$	0.500000	−	0.866025i	0.0492665	−	0.0853320i	−0.840341	−	0.542059i	$-0.817645\pi$
$103$	0.500000	−	0.866025i	0.0492665	−	0.0853320i	0.889607	+	0.456727i	$0.150978\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1.00000	−	1.73205i	0.0966736	−	0.167444i	−0.813632	−	0.581380i	$-0.802513\pi$
$107$	1.00000	−	1.73205i	0.0966736	−	0.167444i	0.910306	+	0.413936i	$0.135846\pi$
$108$		0			0
$109$	−0.500000	−	0.866025i	−0.0478913	−	0.0829502i	0.841086	−	0.540901i	$-0.181917\pi$
$109$	−0.500000	−	0.866025i	−0.0478913	−	0.0829502i	−0.888977	+	0.457951i	$0.848583\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	12.0000			1.12887			0.564433	−	0.825479i	$-0.309095\pi$
$113$	12.0000			1.12887			0.564433	+	0.825479i	$0.309095\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	4.00000	−	3.46410i	0.366679	−	0.317554i
$120$		0			0
$121$	3.50000	−	6.06218i	0.318182	−	0.551107i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		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
$131$	−3.00000	+	5.19615i	−0.262111	+	0.453990i	−0.966803	−	0.255524i	$-0.917752\pi$
$131$	−3.00000	+	5.19615i	−0.262111	+	0.453990i	0.704692	+	0.709514i	$0.251085\pi$
$132$		0			0
$133$	−7.50000	−	2.59808i	−0.650332	−	0.225282i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2.00000	+	3.46410i	0.170872	+	0.295958i	0.938725	−	0.344668i	$-0.112008\pi$
$137$	2.00000	+	3.46410i	0.170872	+	0.295958i	−0.767853	+	0.640626i	$0.778675\pi$
$138$		0			0
$139$	−1.00000			−0.0848189			−0.0424094	−	0.999100i	$-0.513503\pi$
$139$	−1.00000			−0.0848189			−0.0424094	+	0.999100i	$0.513503\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	5.00000	+	8.66025i	0.418121	+	0.724207i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.00000	−	10.3923i	0.491539	−	0.851371i	−0.508413	−	0.861113i	$-0.669768\pi$
$149$	6.00000	−	10.3923i	0.491539	−	0.851371i	0.999953	+	0.00974235i	$0.00310113\pi$
$150$		0			0
$151$	−4.00000	−	6.92820i	−0.325515	−	0.563809i	0.656101	−	0.754673i	$-0.272204\pi$
$151$	−4.00000	−	6.92820i	−0.325515	−	0.563809i	−0.981617	+	0.190864i	$0.938871\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	3.00000	+	5.19615i	0.239426	+	0.414698i	0.960550	−	0.278108i	$-0.0897074\pi$
$157$	3.00000	+	5.19615i	0.239426	+	0.414698i	−0.721124	+	0.692806i	$0.756374\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	5.00000	+	1.73205i	0.394055	+	0.136505i
$162$		0			0
$163$	−2.00000	+	3.46410i	−0.156652	+	0.271329i	−0.933659	−	0.358162i	$-0.883403\pi$
$163$	−2.00000	+	3.46410i	−0.156652	+	0.271329i	0.777007	+	0.629492i	$0.216737\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−20.0000			−1.54765			−0.773823	−	0.633402i	$-0.781658\pi$
$167$	−20.0000			−1.54765			−0.773823	+	0.633402i	$0.781658\pi$
$168$		0			0
$169$	12.0000			0.923077
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−3.00000	+	5.19615i	−0.228086	+	0.395056i	−0.957241	−	0.289292i	$-0.906580\pi$
$173$	−3.00000	+	5.19615i	−0.228086	+	0.395056i	0.729155	+	0.684349i	$0.239913\pi$
$174$		0			0
$175$	−10.0000	+	8.66025i	−0.755929	+	0.654654i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	6.00000	+	10.3923i	0.448461	+	0.776757i	0.998286	−	0.0585225i	$-0.0186389\pi$
$179$	6.00000	+	10.3923i	0.448461	+	0.776757i	−0.549825	+	0.835280i	$0.685306\pi$
$180$		0			0
$181$	−15.0000			−1.11494			−0.557471	−	0.830197i	$-0.688228\pi$
$181$	−15.0000			−1.11494			−0.557471	+	0.830197i	$0.688228\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	2.00000	−	3.46410i	0.146254	−	0.253320i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	6.00000	−	10.3923i	0.434145	−	0.751961i	−0.563081	−	0.826402i	$-0.690384\pi$
$191$	6.00000	−	10.3923i	0.434145	−	0.751961i	0.997225	+	0.0744412i	$0.0237173\pi$
$192$		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
$197$	18.0000			1.28245			0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000			1.28245			0.641223	+	0.767354i	$0.278427\pi$
$198$		0			0
$199$	−2.00000	−	3.46410i	−0.141776	−	0.245564i	0.786389	−	0.617731i	$-0.211948\pi$
$199$	−2.00000	−	3.46410i	−0.141776	−	0.245564i	−0.928166	+	0.372168i	$0.878615\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−4.00000	−	20.7846i	−0.280745	−	1.45879i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−6.00000			−0.415029
$210$		0			0
$211$	20.0000			1.37686			0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000			1.37686			0.688428	+	0.725304i	$0.258301\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−2.00000	+	1.73205i	−0.135769	+	0.117579i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−5.00000	−	8.66025i	−0.336336	−	0.582552i
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		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
$229$	−6.50000	+	11.2583i	−0.429532	+	0.743971i	−0.996832	−	0.0795401i	$-0.974655\pi$
$229$	−6.50000	+	11.2583i	−0.429532	+	0.743971i	0.567300	+	0.823511i	$0.307988\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−6.00000	+	10.3923i	−0.393073	+	0.680823i	−0.992853	−	0.119342i	$-0.961921\pi$
$233$	−6.00000	+	10.3923i	−0.393073	+	0.680823i	0.599780	+	0.800165i	$0.295255\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−30.0000			−1.94054			−0.970269	−	0.242028i	$-0.922188\pi$
$239$	−30.0000			−1.94054			−0.970269	+	0.242028i	$0.922188\pi$
$240$		0			0
$241$	11.0000	+	19.0526i	0.708572	+	1.22728i	0.965387	+	0.260822i	$0.0839937\pi$
$241$	11.0000	+	19.0526i	0.708572	+	1.22728i	−0.256814	+	0.966461i	$0.582673\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−7.50000	+	12.9904i	−0.477214	+	0.826558i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	22.0000			1.38863			0.694314	−	0.719672i	$-0.255708\pi$
$251$	22.0000			1.38863			0.694314	+	0.719672i	$0.255708\pi$
$252$		0			0
$253$	4.00000			0.251478
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−8.00000	+	13.8564i	−0.499026	+	0.864339i	−0.999999	−	0.00112398i	$-0.999642\pi$
$257$	−8.00000	+	13.8564i	−0.499026	+	0.864339i	0.500973	+	0.865463i	$0.332976\pi$
$258$		0			0
$259$	12.5000	+	4.33013i	0.776712	+	0.269061i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	4.00000	+	6.92820i	0.246651	+	0.427211i	0.962594	−	0.270947i	$-0.0873367\pi$
$263$	4.00000	+	6.92820i	0.246651	+	0.427211i	−0.715944	+	0.698158i	$0.754003\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−9.00000	−	15.5885i	−0.548740	−	0.950445i	−0.998361	−	0.0572259i	$-0.981774\pi$
$269$	−9.00000	−	15.5885i	−0.548740	−	0.950445i	0.449622	−	0.893219i	$-0.351559\pi$
$270$		0			0
$271$	8.00000	−	13.8564i	0.485965	−	0.841717i	−0.513905	−	0.857847i	$-0.671801\pi$
$271$	8.00000	−	13.8564i	0.485965	−	0.841717i	0.999870	+	0.0161307i	$0.00513477\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−5.00000	+	8.66025i	−0.301511	+	0.522233i
$276$		0			0
$277$	−8.50000	−	14.7224i	−0.510716	−	0.884585i	−0.999923	−	0.0124177i	$-0.996047\pi$
$277$	−8.50000	−	14.7224i	−0.510716	−	0.884585i	0.489207	−	0.872167i	$-0.337286\pi$
$278$		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
$283$	−15.5000	−	26.8468i	−0.921379	−	1.59588i	−0.797283	−	0.603606i	$-0.793730\pi$
$283$	−15.5000	−	26.8468i	−0.921379	−	1.59588i	−0.124096	−	0.992270i	$-0.539603\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1.00000	−	5.19615i	−0.0590281	−	0.306719i
$288$		0			0
$289$	6.50000	−	11.2583i	0.382353	−	0.662255i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	4.00000			0.233682			0.116841	−	0.993151i	$-0.462723\pi$
$293$	4.00000			0.233682			0.116841	+	0.993151i	$0.462723\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	5.00000	−	8.66025i	0.289157	−	0.500835i
$300$		0			0
$301$	3.50000	+	18.1865i	0.201737	+	1.04825i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−23.0000			−1.31268			−0.656340	−	0.754466i	$-0.727896\pi$
$307$	−23.0000			−1.31268			−0.656340	+	0.754466i	$0.727896\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−15.0000	−	25.9808i	−0.850572	−	1.47323i	−0.880693	−	0.473688i	$-0.842923\pi$
$311$	−15.0000	−	25.9808i	−0.850572	−	1.47323i	0.0301210	−	0.999546i	$-0.490411\pi$
$312$		0			0
$313$	8.50000	−	14.7224i	0.480448	−	0.832161i	−0.519300	−	0.854592i	$-0.673807\pi$
$313$	8.50000	−	14.7224i	0.480448	−	0.832161i	0.999748	+	0.0224310i	$0.00714060\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	10.0000	−	17.3205i	0.561656	−	0.972817i	−0.435696	−	0.900094i	$-0.643498\pi$
$317$	10.0000	−	17.3205i	0.561656	−	0.972817i	0.997352	−	0.0727229i	$-0.0231689\pi$
$318$		0			0
$319$	−8.00000	−	13.8564i	−0.447914	−	0.775810i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	6.00000			0.333849
$324$		0			0
$325$	12.5000	+	21.6506i	0.693375	+	1.20096i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−20.0000	−	6.92820i	−1.10264	−	0.381964i
$330$		0			0
$331$	7.50000	−	12.9904i	0.412237	−	0.714016i	−0.582897	−	0.812546i	$-0.698081\pi$
$331$	7.50000	−	12.9904i	0.412237	−	0.714016i	0.995134	+	0.0985303i	$0.0314141\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	9.00000			0.490261			0.245131	−	0.969490i	$-0.421169\pi$
$337$	9.00000			0.490261			0.245131	+	0.969490i	$0.421169\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.00000	+	1.73205i	−0.0541530	+	0.0937958i
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−15.0000	−	25.9808i	−0.805242	−	1.39472i	−0.916127	−	0.400887i	$-0.868702\pi$
$347$	−15.0000	−	25.9808i	−0.805242	−	1.39472i	0.110885	−	0.993833i	$-0.464631\pi$
$348$		0			0
$349$	2.00000			0.107058			0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000			0.107058			0.0535288	+	0.998566i	$0.482953\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	12.0000	+	20.7846i	0.638696	+	1.10625i	0.985719	+	0.168397i	$0.0538590\pi$
$353$	12.0000	+	20.7846i	0.638696	+	1.10625i	−0.347024	+	0.937856i	$0.612808\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	16.0000	−	27.7128i	0.844448	−	1.46263i	−0.0416523	−	0.999132i	$-0.513262\pi$
$359$	16.0000	−	27.7128i	0.844448	−	1.46263i	0.886100	−	0.463494i	$-0.153404\pi$
$360$		0			0
$361$	5.00000	+	8.66025i	0.263158	+	0.455803i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−8.50000	−	14.7224i	−0.443696	−	0.768505i	0.554264	−	0.832341i	$-0.313000\pi$
$367$	−8.50000	−	14.7224i	−0.443696	−	0.768505i	−0.997960	+	0.0638362i	$0.979666\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	4.00000	−	3.46410i	0.207670	−	0.179847i
$372$		0			0
$373$	10.5000	−	18.1865i	0.543669	−	0.941663i	−0.455020	−	0.890481i	$-0.650368\pi$
$373$	10.5000	−	18.1865i	0.543669	−	0.941663i	0.998689	−	0.0511818i	$-0.0162988\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−40.0000			−2.06010
$378$		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
$383$	−11.0000	+	19.0526i	−0.562074	+	0.973540i	0.435242	+	0.900314i	$0.356663\pi$
$383$	−11.0000	+	19.0526i	−0.562074	+	0.973540i	−0.997315	+	0.0732266i	$0.976670\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−12.0000	−	20.7846i	−0.608424	−	1.05382i	−0.991500	−	0.130105i	$-0.958469\pi$
$389$	−12.0000	−	20.7846i	−0.608424	−	1.05382i	0.383076	−	0.923717i	$-0.374865\pi$
$390$		0			0
$391$	−4.00000			−0.202289
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	6.50000	−	11.2583i	0.326226	−	0.565039i	−0.655534	−	0.755166i	$-0.727556\pi$
$397$	6.50000	−	11.2583i	0.326226	−	0.565039i	0.981760	+	0.190126i	$0.0608897\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	5.00000	−	8.66025i	0.249688	−	0.432472i	−0.713751	−	0.700399i	$-0.753005\pi$
$401$	5.00000	−	8.66025i	0.249688	−	0.432472i	0.963439	+	0.267927i	$0.0863386\pi$
$402$		0			0
$403$	2.50000	+	4.33013i	0.124534	+	0.215699i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	10.0000			0.495682
$408$		0			0
$409$	17.5000	+	30.3109i	0.865319	+	1.49878i	0.866730	+	0.498778i	$0.166218\pi$
$409$	17.5000	+	30.3109i	0.865319	+	1.49878i	−0.00141047	+	0.999999i	$0.500449\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−20.0000	+	17.3205i	−0.984136	+	0.852286i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−18.0000			−0.879358			−0.439679	−	0.898155i	$-0.644908\pi$
$419$	−18.0000			−0.879358			−0.439679	+	0.898155i	$0.644908\pi$
$420$		0			0
$421$	9.00000			0.438633			0.219317	−	0.975654i	$-0.429617\pi$
$421$	9.00000			0.438633			0.219317	+	0.975654i	$0.429617\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	5.00000	−	8.66025i	0.242536	−	0.420084i
$426$		0			0
$427$	5.00000	+	1.73205i	0.241967	+	0.0838198i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	10.0000	+	17.3205i	0.481683	+	0.834300i	0.999779	−	0.0210230i	$-0.00669232\pi$
$431$	10.0000	+	17.3205i	0.481683	+	0.834300i	−0.518096	+	0.855323i	$0.673359\pi$
$432$		0			0
$433$	−17.0000			−0.816968			−0.408484	−	0.912766i	$-0.633942\pi$
$433$	−17.0000			−0.816968			−0.408484	+	0.912766i	$0.633942\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3.00000	+	5.19615i	0.143509	+	0.248566i
$438$		0			0
$439$	12.0000	−	20.7846i	0.572729	−	0.991995i	−0.423556	−	0.905870i	$-0.639218\pi$
$439$	12.0000	−	20.7846i	0.572729	−	0.991995i	0.996284	−	0.0861252i	$-0.0274485\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	3.00000	−	5.19615i	0.142534	−	0.246877i	−0.785916	−	0.618333i	$-0.787808\pi$
$443$	3.00000	−	5.19615i	0.142534	−	0.246877i	0.928450	+	0.371457i	$0.121142\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	24.0000			1.13263			0.566315	−	0.824189i	$-0.308369\pi$
$449$	24.0000			1.13263			0.566315	+	0.824189i	$0.308369\pi$
$450$		0			0
$451$	−2.00000	−	3.46410i	−0.0941763	−	0.163118i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	11.5000	−	19.9186i	0.537947	−	0.931752i	−0.461067	−	0.887365i	$-0.652533\pi$
$457$	11.5000	−	19.9186i	0.537947	−	0.931752i	0.999014	−	0.0443868i	$-0.0141334\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	20.0000			0.931493			0.465746	−	0.884918i	$-0.345786\pi$
$461$	20.0000			0.931493			0.465746	+	0.884918i	$0.345786\pi$
$462$		0			0
$463$	9.00000			0.418265			0.209133	−	0.977887i	$-0.432936\pi$
$463$	9.00000			0.418265			0.209133	+	0.977887i	$0.432936\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	6.00000	−	10.3923i	0.277647	−	0.480899i	−0.693153	−	0.720791i	$-0.743779\pi$
$467$	6.00000	−	10.3923i	0.277647	−	0.480899i	0.970799	+	0.239892i	$0.0771121\pi$
$468$		0			0
$469$	−22.0000	+	19.0526i	−1.01587	+	0.879765i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	7.00000	+	12.1244i	0.321860	+	0.557478i
$474$		0			0
$475$	−15.0000			−0.688247
$476$		0			0
$477$		0			0
$478$		0			0
$479$	9.00000	+	15.5885i	0.411220	+	0.712255i	0.995023	−	0.0996406i	$-0.0317693\pi$
$479$	9.00000	+	15.5885i	0.411220	+	0.712255i	−0.583803	+	0.811895i	$0.698436\pi$
$480$		0			0
$481$	12.5000	−	21.6506i	0.569951	−	0.987184i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−10.5000	−	18.1865i	−0.475800	−	0.824110i	0.523815	−	0.851832i	$-0.324508\pi$
$487$	−10.5000	−	18.1865i	−0.475800	−	0.824110i	−0.999616	+	0.0277214i	$0.991175\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−32.0000			−1.44414			−0.722070	−	0.691820i	$-0.756809\pi$
$491$	−32.0000			−1.44414			−0.722070	+	0.691820i	$0.756809\pi$
$492$		0			0
$493$	8.00000	+	13.8564i	0.360302	+	0.624061i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−6.00000	−	31.1769i	−0.269137	−	1.39848i
$498$		0			0
$499$	20.5000	−	35.5070i	0.917706	−	1.58951i	0.114816	−	0.993387i	$-0.463372\pi$
$499$	20.5000	−	35.5070i	0.917706	−	1.58951i	0.802890	−	0.596127i	$-0.203294\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−14.0000			−0.624229			−0.312115	−	0.950044i	$-0.601037\pi$
$503$	−14.0000			−0.624229			−0.312115	+	0.950044i	$0.601037\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−2.00000	+	3.46410i	−0.0886484	+	0.153544i	−0.906940	−	0.421260i	$-0.861588\pi$
$509$	−2.00000	+	3.46410i	−0.0886484	+	0.153544i	0.818292	+	0.574803i	$0.194921\pi$
$510$		0			0
$511$	−6.00000	+	5.19615i	−0.265424	+	0.229864i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−16.0000			−0.703679
$518$		0			0
$519$		0			0
$520$		0			0
$521$	16.0000	+	27.7128i	0.700973	+	1.21412i	0.968125	+	0.250466i	$0.0805839\pi$
$521$	16.0000	+	27.7128i	0.700973	+	1.21412i	−0.267153	+	0.963654i	$0.586083\pi$
$522$		0			0
$523$	5.50000	−	9.52628i	0.240498	−	0.416555i	−0.720358	−	0.693602i	$-0.756023\pi$
$523$	5.50000	−	9.52628i	0.240498	−	0.416555i	0.960856	+	0.277047i	$0.0893559\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1.00000	−	1.73205i	0.0435607	−	0.0754493i
$528$		0			0
$529$	9.50000	+	16.4545i	0.413043	+	0.715412i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−10.0000			−0.433148
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−11.0000	−	8.66025i	−0.473804	−	0.373024i
$540$		0			0
$541$	13.5000	−	23.3827i	0.580410	−	1.00530i	−0.415020	−	0.909812i	$-0.636226\pi$
$541$	13.5000	−	23.3827i	0.580410	−	1.00530i	0.995431	−	0.0954880i	$-0.0304412\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−44.0000			−1.88130			−0.940652	−	0.339372i	$-0.889785\pi$
$547$	−44.0000			−1.88130			−0.940652	+	0.339372i	$0.889785\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	12.0000	−	20.7846i	0.511217	−	0.885454i
$552$		0			0
$553$	−42.5000	−	14.7224i	−1.80728	−	0.626061i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	9.00000	+	15.5885i	0.381342	+	0.660504i	0.991254	−	0.131965i	$-0.0421286\pi$
$557$	9.00000	+	15.5885i	0.381342	+	0.660504i	−0.609912	+	0.792469i	$0.708795\pi$
$558$		0			0
$559$	35.0000			1.48034
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−10.0000	−	17.3205i	−0.421450	−	0.729972i	0.574632	−	0.818412i	$-0.305145\pi$
$563$	−10.0000	−	17.3205i	−0.421450	−	0.729972i	−0.996082	+	0.0884397i	$0.971812\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	21.0000	−	36.3731i	0.880366	−	1.52484i	0.0294311	−	0.999567i	$-0.490630\pi$
$569$	21.0000	−	36.3731i	0.880366	−	1.52484i	0.850935	−	0.525271i	$-0.176036\pi$
$570$		0			0
$571$	−11.5000	−	19.9186i	−0.481260	−	0.833567i	0.518509	−	0.855072i	$-0.326487\pi$
$571$	−11.5000	−	19.9186i	−0.481260	−	0.833567i	−0.999769	+	0.0215055i	$0.993154\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	10.0000			0.417029
$576$		0			0
$577$	−3.50000	−	6.06218i	−0.145707	−	0.252372i	0.783930	−	0.620850i	$-0.213212\pi$
$577$	−3.50000	−	6.06218i	−0.145707	−	0.252372i	−0.929636	+	0.368478i	$0.879879\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	8.00000	+	41.5692i	0.331896	+	1.72458i
$582$		0			0
$583$	2.00000	−	3.46410i	0.0828315	−	0.143468i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	24.0000			0.990586			0.495293	−	0.868726i	$-0.335061\pi$
$587$	24.0000			0.990586			0.495293	+	0.868726i	$0.335061\pi$
$588$		0			0
$589$	−3.00000			−0.123613
$590$		0			0
$591$		0			0
$592$		0			0
$593$	15.0000	−	25.9808i	0.615976	−	1.06690i	−0.374236	−	0.927333i	$-0.622095\pi$
$593$	15.0000	−	25.9808i	0.615976	−	1.06690i	0.990212	−	0.139569i	$-0.0445716\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−18.0000	−	31.1769i	−0.735460	−	1.27385i	−0.954521	−	0.298143i	$-0.903633\pi$
$599$	−18.0000	−	31.1769i	−0.735460	−	1.27385i	0.219061	−	0.975711i	$-0.429701\pi$
$600$		0			0
$601$	−37.0000			−1.50926			−0.754631	−	0.656150i	$-0.772184\pi$
$601$	−37.0000			−1.50926			−0.754631	+	0.656150i	$0.772184\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−16.5000	+	28.5788i	−0.669714	+	1.15998i	0.308270	+	0.951299i	$0.400250\pi$
$607$	−16.5000	+	28.5788i	−0.669714	+	1.15998i	−0.977984	+	0.208680i	$0.933083\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−20.0000	+	34.6410i	−0.809113	+	1.40143i
$612$		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
$617$	14.0000			0.563619			0.281809	−	0.959470i	$-0.409065\pi$
$617$	14.0000			0.563619			0.281809	+	0.959470i	$0.409065\pi$
$618$		0			0
$619$	−2.50000	−	4.33013i	−0.100483	−	0.174042i	0.811400	−	0.584491i	$-0.198706\pi$
$619$	−2.50000	−	4.33013i	−0.100483	−	0.174042i	−0.911884	+	0.410448i	$0.865372\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	30.0000	+	10.3923i	1.20192	+	0.416359i
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−10.0000			−0.398726
$630$		0			0
$631$	−4.00000			−0.159237			−0.0796187	−	0.996825i	$-0.525370\pi$
$631$	−4.00000			−0.159237			−0.0796187	+	0.996825i	$0.525370\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−32.5000	+	12.9904i	−1.28770	+	0.514698i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	13.0000	+	22.5167i	0.513469	+	0.889355i	0.999878	+	0.0156233i	$0.00497325\pi$
$641$	13.0000	+	22.5167i	0.513469	+	0.889355i	−0.486409	+	0.873731i	$0.661693\pi$
$642$		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	0.633090	−	0.774078i	$-0.281786\pi$
$647$	−9.00000	−	15.5885i	−0.353827	−	0.612845i	−0.986916	+	0.161233i	$0.948453\pi$
$648$		0			0
$649$	−10.0000	+	17.3205i	−0.392534	+	0.679889i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−7.00000	+	12.1244i	−0.273931	+	0.474463i	−0.969865	−	0.243643i	$-0.921657\pi$
$653$	−7.00000	+	12.1244i	−0.273931	+	0.474463i	0.695934	+	0.718106i	$0.254991\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	42.0000			1.63609			0.818044	−	0.575156i	$-0.195059\pi$
$659$	42.0000			1.63609			0.818044	+	0.575156i	$0.195059\pi$
$660$		0			0
$661$	−1.50000	−	2.59808i	−0.0583432	−	0.101053i	0.835379	−	0.549675i	$-0.185248\pi$
$661$	−1.50000	−	2.59808i	−0.0583432	−	0.101053i	−0.893722	+	0.448622i	$0.851915\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−8.00000	+	13.8564i	−0.309761	+	0.536522i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	4.00000			0.154418
$672$		0			0
$673$	−21.0000			−0.809491			−0.404745	−	0.914429i	$-0.632640\pi$
$673$	−21.0000			−0.809491			−0.404745	+	0.914429i	$0.632640\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−6.00000	+	10.3923i	−0.230599	+	0.399409i	−0.957984	−	0.286820i	$-0.907402\pi$
$677$	−6.00000	+	10.3923i	−0.230599	+	0.399409i	0.727386	+	0.686229i	$0.240735\pi$
$678$		0			0
$679$	−7.00000	−	36.3731i	−0.268635	−	1.39587i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−15.0000	−	25.9808i	−0.573959	−	0.994126i	−0.996154	−	0.0876211i	$-0.972074\pi$
$683$	−15.0000	−	25.9808i	−0.573959	−	0.994126i	0.422195	−	0.906505i	$-0.361260\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−5.00000	−	8.66025i	−0.190485	−	0.329929i
$690$		0			0
$691$	−2.50000	+	4.33013i	−0.0951045	+	0.164726i	−0.909652	−	0.415371i	$-0.863652\pi$
$691$	−2.50000	+	4.33013i	−0.0951045	+	0.164726i	0.814548	+	0.580097i	$0.196985\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	2.00000	+	3.46410i	0.0757554	+	0.131212i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	6.00000			0.226617			0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000			0.226617			0.113308	+	0.993560i	$0.463855\pi$
$702$		0			0
$703$	7.50000	+	12.9904i	0.282868	+	0.489942i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−36.0000	+	31.1769i	−1.35392	+	1.17253i
$708$		0			0
$709$	11.0000	−	19.0526i	0.413114	−	0.715534i	−0.582115	−	0.813107i	$-0.697775\pi$
$709$	11.0000	−	19.0526i	0.413114	−	0.715534i	0.995228	+	0.0975728i	$0.0311079\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	2.00000			0.0749006
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−3.00000	+	5.19615i	−0.111881	+	0.193784i	−0.916529	−	0.399969i	$-0.869021\pi$
$719$	−3.00000	+	5.19615i	−0.111881	+	0.193784i	0.804648	+	0.593753i	$0.202354\pi$
$720$		0			0
$721$	2.50000	+	0.866025i	0.0931049	+	0.0322525i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−20.0000	−	34.6410i	−0.742781	−	1.28654i
$726$		0			0
$727$	13.0000			0.482143			0.241072	−	0.970507i	$-0.422501\pi$
$727$	13.0000			0.482143			0.241072	+	0.970507i	$0.422501\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−7.00000	−	12.1244i	−0.258904	−	0.448435i
$732$		0			0
$733$	9.50000	−	16.4545i	0.350891	−	0.607760i	−0.635515	−	0.772088i	$-0.719212\pi$
$733$	9.50000	−	16.4545i	0.350891	−	0.607760i	0.986406	+	0.164328i	$0.0525456\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−11.0000	+	19.0526i	−0.405190	+	0.701810i
$738$		0			0
$739$	8.50000	+	14.7224i	0.312678	+	0.541573i	0.978941	−	0.204143i	$-0.0654407\pi$
$739$	8.50000	+	14.7224i	0.312678	+	0.541573i	−0.666264	+	0.745716i	$0.732107\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	14.0000			0.513610			0.256805	−	0.966463i	$-0.417330\pi$
$743$	14.0000			0.513610			0.256805	+	0.966463i	$0.417330\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	5.00000	+	1.73205i	0.182696	+	0.0632878i
$750$		0			0
$751$	−13.5000	+	23.3827i	−0.492622	+	0.853246i	−0.999964	−	0.00849853i	$-0.997295\pi$
$751$	−13.5000	+	23.3827i	−0.492622	+	0.853246i	0.507342	+	0.861745i	$0.330628\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	22.0000			0.799604			0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000			0.799604			0.399802	+	0.916602i	$0.369079\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	15.0000	−	25.9808i	0.543750	−	0.941802i	−0.454935	−	0.890525i	$-0.650337\pi$
$761$	15.0000	−	25.9808i	0.543750	−	0.941802i	0.998684	−	0.0512772i	$-0.0163292\pi$
$762$		0			0
$763$	2.00000	−	1.73205i	0.0724049	−	0.0627044i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	25.0000	+	43.3013i	0.902698	+	1.56352i
$768$		0			0
$769$	−9.00000			−0.324548			−0.162274	−	0.986746i	$-0.551883\pi$
$769$	−9.00000			−0.324548			−0.162274	+	0.986746i	$0.551883\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	9.00000	+	15.5885i	0.323708	+	0.560678i	0.981250	−	0.192740i	$-0.0617373\pi$
$773$	9.00000	+	15.5885i	0.323708	+	0.560678i	−0.657542	+	0.753418i	$0.728404\pi$
$774$		0			0
$775$	−2.50000	+	4.33013i	−0.0898027	+	0.155543i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.00000	−	5.19615i	0.107486	−	0.186171i
$780$		0			0
$781$	−12.0000	−	20.7846i	−0.429394	−	0.743732i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	20.0000	+	34.6410i	0.712923	+	1.23482i	0.963755	+	0.266788i	$0.0859624\pi$
$787$	20.0000	+	34.6410i	0.712923	+	1.23482i	−0.250832	+	0.968031i	$0.580704\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	6.00000	+	31.1769i	0.213335	+	1.10852i
$792$		0			0
$793$	5.00000	−	8.66025i	0.177555	−	0.307535i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	34.0000			1.20434			0.602171	−	0.798367i	$-0.294303\pi$
$797$	34.0000			1.20434			0.602171	+	0.798367i	$0.294303\pi$
$798$		0			0
$799$	16.0000			0.566039
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−3.00000	+	5.19615i	−0.105868	+	0.183368i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	2.00000	+	3.46410i	0.0703163	+	0.121791i	0.899040	−	0.437867i	$-0.144266\pi$
$809$	2.00000	+	3.46410i	0.0703163	+	0.121791i	−0.828724	+	0.559658i	$0.810932\pi$
$810$		0			0
$811$	−20.0000			−0.702295			−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000			−0.702295			−0.351147	+	0.936320i	$0.614208\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−10.5000	+	18.1865i	−0.367348	+	0.636266i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	8.00000	−	13.8564i	0.279202	−	0.483592i	−0.691985	−	0.721912i	$-0.743263\pi$
$821$	8.00000	−	13.8564i	0.279202	−	0.483592i	0.971187	+	0.238320i	$0.0765968\pi$
$822$		0			0
$823$	20.0000	+	34.6410i	0.697156	+	1.20751i	0.969448	+	0.245295i	$0.0788849\pi$
$823$	20.0000	+	34.6410i	0.697156	+	1.20751i	−0.272292	+	0.962215i	$0.587782\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−24.0000			−0.834562			−0.417281	−	0.908778i	$-0.637017\pi$
$827$	−24.0000			−0.834562			−0.417281	+	0.908778i	$0.637017\pi$
$828$		0			0
$829$	13.5000	+	23.3827i	0.468874	+	0.812114i	0.999367	−	0.0355753i	$-0.0113264\pi$
$829$	13.5000	+	23.3827i	0.468874	+	0.812114i	−0.530493	+	0.847690i	$0.677993\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	11.0000	+	8.66025i	0.381127	+	0.300060i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	28.0000			0.966667			0.483334	−	0.875436i	$-0.339426\pi$
$839$	28.0000			0.966667			0.483334	+	0.875436i	$0.339426\pi$
$840$		0			0
$841$	35.0000			1.20690
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	17.5000	+	6.06218i	0.601307	+	0.208299i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−5.00000	−	8.66025i	−0.171398	−	0.296870i
$852$		0			0
$853$	39.0000			1.33533			0.667667	−	0.744460i	$-0.267293\pi$
$853$	39.0000			1.33533			0.667667	+	0.744460i	$0.267293\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	12.0000	+	20.7846i	0.409912	+	0.709989i	0.994880	−	0.101068i	$-0.0322260\pi$
$857$	12.0000	+	20.7846i	0.409912	+	0.709989i	−0.584967	+	0.811057i	$0.698893\pi$
$858$		0			0
$859$	−2.00000	+	3.46410i	−0.0682391	+	0.118194i	−0.898126	−	0.439738i	$-0.855071\pi$
$859$	−2.00000	+	3.46410i	−0.0682391	+	0.118194i	0.829887	+	0.557931i	$0.188405\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−3.00000	+	5.19615i	−0.102121	+	0.176879i	−0.912558	−	0.408946i	$-0.865896\pi$
$863$	−3.00000	+	5.19615i	−0.102121	+	0.176879i	0.810437	+	0.585826i	$0.199230\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−34.0000			−1.15337
$870$		0			0
$871$	27.5000	+	47.6314i	0.931802	+	1.61393i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	19.0000	−	32.9090i	0.641584	−	1.11126i	−0.343495	−	0.939155i	$-0.611611\pi$
$877$	19.0000	−	32.9090i	0.641584	−	1.11126i	0.985079	−	0.172102i	$-0.0550559\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−58.0000			−1.95407			−0.977035	−	0.213080i	$-0.931651\pi$
$881$	−58.0000			−1.95407			−0.977035	+	0.213080i	$0.931651\pi$
$882$		0			0
$883$	−43.0000			−1.44707			−0.723533	−	0.690290i	$-0.757483\pi$
$883$	−43.0000			−1.44707			−0.723533	+	0.690290i	$0.757483\pi$
$884$		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
$893$	−12.0000	−	20.7846i	−0.401565	−	0.695530i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−4.00000	−	6.92820i	−0.133407	−	0.231069i
$900$		0			0
$901$	−2.00000	+	3.46410i	−0.0666297	+	0.115406i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	3.50000	+	6.06218i	0.116216	+	0.201291i	0.918265	−	0.395966i	$-0.129590\pi$
$907$	3.50000	+	6.06218i	0.116216	+	0.201291i	−0.802049	+	0.597258i	$0.796257\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	50.0000			1.65657			0.828287	−	0.560304i	$-0.189316\pi$
$911$	50.0000			1.65657			0.828287	+	0.560304i	$0.189316\pi$
$912$		0			0
$913$	16.0000	+	27.7128i	0.529523	+	0.917160i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−15.0000	−	5.19615i	−0.495344	−	0.171592i
$918$		0			0
$919$	27.5000	−	47.6314i	0.907141	−	1.57121i	0.0891245	−	0.996020i	$-0.471593\pi$
$919$	27.5000	−	47.6314i	0.907141	−	1.57121i	0.818017	−	0.575194i	$-0.195074\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−60.0000			−1.97492
$924$		0			0
$925$	25.0000			0.821995
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$929$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$930$		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
$937$	−13.0000			−0.424691			−0.212346	−	0.977195i	$-0.568110\pi$
$937$	−13.0000			−0.424691			−0.212346	+	0.977195i	$0.568110\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.00000	+	1.73205i	0.0325991	+	0.0564632i	0.881865	−	0.471503i	$-0.156288\pi$
$941$	1.00000	+	1.73205i	0.0325991	+	0.0564632i	−0.849266	+	0.527966i	$0.822955\pi$
$942$		0			0
$943$	−2.00000	+	3.46410i	−0.0651290	+	0.112807i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−17.0000	+	29.4449i	−0.552426	+	0.956830i	0.445673	+	0.895196i	$0.352964\pi$
$947$	−17.0000	+	29.4449i	−0.552426	+	0.956830i	−0.998099	+	0.0616337i	$0.980369\pi$
$948$		0			0
$949$	7.50000	+	12.9904i	0.243460	+	0.421686i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	34.0000			1.10137			0.550684	−	0.834714i	$-0.314367\pi$
$953$	34.0000			1.10137			0.550684	+	0.834714i	$0.314367\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−8.00000	+	6.92820i	−0.258333	+	0.223723i
$960$		0			0
$961$	15.0000	−	25.9808i	0.483871	−	0.838089i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−47.0000			−1.51142			−0.755709	−	0.654907i	$-0.772708\pi$
$967$	−47.0000			−1.51142			−0.755709	+	0.654907i	$0.772708\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	3.00000	−	5.19615i	0.0962746	−	0.166752i	−0.813865	−	0.581054i	$-0.802641\pi$
$971$	3.00000	−	5.19615i	0.0962746	−	0.166752i	0.910140	+	0.414301i	$0.135974\pi$
$972$		0			0
$973$	−0.500000	−	2.59808i	−0.0160293	−	0.0832905i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−17.0000	−	29.4449i	−0.543878	−	0.942025i	−0.998677	−	0.0514302i	$-0.983622\pi$
$977$	−17.0000	−	29.4449i	−0.543878	−	0.942025i	0.454798	−	0.890594i	$-0.349711\pi$
$978$		0			0
$979$	24.0000			0.767043
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−3.00000	−	5.19615i	−0.0956851	−	0.165732i	0.814209	−	0.580572i	$-0.197171\pi$
$983$	−3.00000	−	5.19615i	−0.0956851	−	0.165732i	−0.909894	+	0.414840i	$0.863838\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	7.00000	−	12.1244i	0.222587	−	0.385532i
$990$		0			0
$991$	6.50000	+	11.2583i	0.206479	+	0.357633i	0.950603	−	0.310409i	$-0.100466\pi$
$991$	6.50000	+	11.2583i	0.206479	+	0.357633i	−0.744124	+	0.668042i	$0.767133\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	18.5000	+	32.0429i	0.585901	+	1.01481i	0.994762	+	0.102214i	$0.0325925\pi$
$997$	18.5000	+	32.0429i	0.585901	+	1.01481i	−0.408862	+	0.912596i	$0.634074\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2016.2.s.g.289.1		2
3.2	odd	2		672.2.q.c.289.1	yes	2
4.3	odd	2		2016.2.s.f.289.1		2
7.4	even	3	inner	2016.2.s.g.865.1		2
12.11	even	2		672.2.q.i.289.1	yes	2
21.2	odd	6		4704.2.a.bb.1.1		1
21.5	even	6		4704.2.a.i.1.1		1
21.11	odd	6		672.2.q.c.193.1	✓	2
24.5	odd	2		1344.2.q.p.961.1		2
24.11	even	2		1344.2.q.f.961.1		2
28.11	odd	6		2016.2.s.f.865.1		2
84.11	even	6		672.2.q.i.193.1	yes	2
84.23	even	6		4704.2.a.h.1.1		1
84.47	odd	6		4704.2.a.x.1.1		1
168.5	even	6		9408.2.a.cj.1.1		1
168.11	even	6		1344.2.q.f.193.1		2
168.53	odd	6		1344.2.q.p.193.1		2
168.107	even	6		9408.2.a.cm.1.1		1
168.131	odd	6		9408.2.a.y.1.1		1
168.149	odd	6		9408.2.a.s.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.q.c.193.1	✓	2	21.11	odd	6
672.2.q.c.289.1	yes	2	3.2	odd	2
672.2.q.i.193.1	yes	2	84.11	even	6
672.2.q.i.289.1	yes	2	12.11	even	2
1344.2.q.f.193.1		2	168.11	even	6
1344.2.q.f.961.1		2	24.11	even	2
1344.2.q.p.193.1		2	168.53	odd	6
1344.2.q.p.961.1		2	24.5	odd	2
2016.2.s.f.289.1		2	4.3	odd	2
2016.2.s.f.865.1		2	28.11	odd	6
2016.2.s.g.289.1		2	1.1	even	1	trivial
2016.2.s.g.865.1		2	7.4	even	3	inner
4704.2.a.h.1.1		1	84.23	even	6
4704.2.a.i.1.1		1	21.5	even	6
4704.2.a.x.1.1		1	84.47	odd	6
4704.2.a.bb.1.1		1	21.2	odd	6
9408.2.a.s.1.1		1	168.149	odd	6
9408.2.a.y.1.1		1	168.131	odd	6
9408.2.a.cj.1.1		1	168.5	even	6
9408.2.a.cm.1.1		1	168.107	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:	\( N \)	\(=\)	\( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2016.s (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 2.59808i) q^{7} +O(q^{10})\)	\(q+(0.500000 + 2.59808i) q^{7} +(1.00000 + 1.73205i) q^{11} +5.00000 q^{13} +(-1.00000 - 1.73205i) q^{17} +(-1.50000 + 2.59808i) q^{19} +(1.00000 - 1.73205i) q^{23} +(2.50000 + 4.33013i) q^{25} -8.00000 q^{29} +(0.500000 + 0.866025i) q^{31} +(2.50000 - 4.33013i) q^{37} -2.00000 q^{41} +7.00000 q^{43} +(-4.00000 + 6.92820i) q^{47} +(-6.50000 + 2.59808i) q^{49} +(-1.00000 - 1.73205i) q^{53} +(5.00000 + 8.66025i) q^{59} +(1.00000 - 1.73205i) q^{61} +(5.50000 + 9.52628i) q^{67} -12.0000 q^{71} +(1.50000 + 2.59808i) q^{73} +(-4.00000 + 3.46410i) q^{77} +(-8.50000 + 14.7224i) q^{79} +16.0000 q^{83} +(6.00000 - 10.3923i) q^{89} +(2.50000 + 12.9904i) q^{91} -14.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{7}+O(q^{10}) \) 2 * q + q^7	\( 2 q + q^{7} + 2 q^{11} + 10 q^{13} - 2 q^{17} - 3 q^{19} + 2 q^{23} + 5 q^{25} - 16 q^{29} + q^{31} + 5 q^{37} - 4 q^{41} + 14 q^{43} - 8 q^{47} - 13 q^{49} - 2 q^{53} + 10 q^{59} + 2 q^{61} + 11 q^{67} - 24 q^{71} + 3 q^{73} - 8 q^{77} - 17 q^{79} + 32 q^{83} + 12 q^{89} + 5 q^{91} - 28 q^{97}+O(q^{100}) \) 2 * q + q^7 + 2 * q^11 + 10 * q^13 - 2 * q^17 - 3 * q^19 + 2 * q^23 + 5 * q^25 - 16 * q^29 + q^31 + 5 * q^37 - 4 * q^41 + 14 * q^43 - 8 * q^47 - 13 * q^49 - 2 * q^53 + 10 * q^59 + 2 * q^61 + 11 * q^67 - 24 * q^71 + 3 * q^73 - 8 * q^77 - 17 * q^79 + 32 * q^83 + 12 * q^89 + 5 * q^91 - 28 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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