LMFDB - Embedded newform 704.2.w.a.289.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [704,2,Mod(97,704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(704, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 5, 6]))

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

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

chi := DirichletCharacter("704.97");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 704 = 2^{6} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	704.w (of order $10$, degree $4$, 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:	$5.62146830230$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{10})$
Coefficient field:	$\Q(\zeta_{40})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{12} + x^{8} - x^{4} + 1 $ x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{10}]$

Embedding invariants

Embedding label			289.1
Root			$0.156434 + 0.987688i$ of defining polynomial
Character	$\chi$	$=$	704.289
Dual form			704.2.w.a.609.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.11998 + 1.01374i) q^{3} +(6.27955 - 4.56236i) q^{9} +O(q^{10})$	$q+(-3.11998 + 1.01374i) q^{3} +(6.27955 - 4.56236i) q^{9} +(-3.29019 - 0.417946i) q^{11} +(-2.85250 - 2.07246i) q^{17} +(8.26279 - 2.68474i) q^{19} +(1.54508 + 4.75528i) q^{25} +(-9.18226 + 12.6383i) q^{27} +(10.6890 - 2.03142i) q^{33} +(3.55497 + 10.9411i) q^{41} +13.0777i q^{43} +(5.66312 + 4.11450i) q^{49} +(11.0007 + 3.57434i) q^{51} +(-23.0581 + 16.7527i) q^{57} +(14.5550 + 4.72922i) q^{59} -3.74374i q^{67} +(2.58246 - 7.94799i) q^{73} +(-9.64127 - 13.2701i) q^{75} +(8.64075 - 26.5935i) q^{81} +(9.53671 - 13.1261i) q^{83} -11.2373 q^{89} +(-1.52490 + 1.10790i) q^{97} +(-22.5677 + 12.3865i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 4 q^{9}+O(q^{10}) $ 16 * q + 4 * q^9	$ 16 q + 4 q^{9} + 24 q^{17} - 20 q^{25} + 76 q^{33} + 24 q^{41} + 28 q^{49} - 124 q^{57} - 8 q^{73} - 16 q^{81} - 72 q^{89} + 60 q^{97}+O(q^{100}) $ 16 * q + 4 * q^9 + 24 * q^17 - 20 * q^25 + 76 * q^33 + 24 * q^41 + 28 * q^49 - 124 * q^57 - 8 * q^73 - 16 * q^81 - 72 * q^89 + 60 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/704\mathbb{Z}\right)^\times$.

$n$	$133$	$321$	$639$
$\chi(n)$	$-1$	$e\left(\frac{4}{5}\right)$	$1$

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
$3$	−3.11998	+	1.01374i	−1.80132	+	0.585285i	−0.999918	−	0.0128385i	$-0.995913\pi$
$3$	−3.11998	+	1.01374i	−1.80132	+	0.585285i	−0.801404	+	0.598123i	$0.795913\pi$
$4$		0			0
$5$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$5$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$6$		0			0
$7$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$7$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$8$		0			0
$9$	6.27955	−	4.56236i	2.09318	−	1.52079i
$10$		0			0
$11$	−3.29019	−	0.417946i	−0.992028	−	0.126015i
$11$	−3.29019	−	0.417946i	−0.992028	−	0.126015i
$12$		0			0
$13$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$13$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.85250	−	2.07246i	−0.691833	−	0.502646i	0.185429	−	0.982658i	$-0.440633\pi$
$17$	−2.85250	−	2.07246i	−0.691833	−	0.502646i	−0.877262	+	0.480011i	$0.840633\pi$
$18$		0			0
$19$	8.26279	−	2.68474i	1.89562	−	0.615923i	0.922287	−	0.386507i	$-0.126318\pi$
$19$	8.26279	−	2.68474i	1.89562	−	0.615923i	0.973329	−	0.229416i	$-0.0736815\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	1.54508	+	4.75528i	0.309017	+	0.951057i
$26$		0			0
$27$	−9.18226	+	12.6383i	−1.76713	+	2.43224i
$28$		0			0
$29$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$29$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$30$		0			0
$31$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$31$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$32$		0			0
$33$	10.6890	−	2.03142i	1.86072	−	0.353625i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$37$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.55497	+	10.9411i	0.555193	+	1.70871i	0.695432	+	0.718592i	$0.255213\pi$
$41$	3.55497	+	10.9411i	0.555193	+	1.70871i	−0.140238	+	0.990118i	$0.544787\pi$
$42$		0			0
$43$			13.0777i			1.99433i	0.0752492	+	0.997165i	$0.476025\pi$
$43$			13.0777i			1.99433i	−0.0752492	+	0.997165i	$0.523975\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$47$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$48$		0			0
$49$	5.66312	+	4.11450i	0.809017	+	0.587785i
$50$		0			0
$51$	11.0007	+	3.57434i	1.54041	+	0.500508i
$52$		0			0
$53$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$53$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−23.0581	+	16.7527i	−3.05412	+	2.21895i
$58$		0			0
$59$	14.5550	+	4.72922i	1.89490	+	0.615692i	0.974330	+	0.225125i	$0.0722791\pi$
$59$	14.5550	+	4.72922i	1.89490	+	0.615692i	0.920575	+	0.390567i	$0.127721\pi$
$60$		0			0
$61$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$61$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	3.74374i		−	0.457371i	−0.973500	−	0.228686i	$-0.926557\pi$
$67$		−	3.74374i		−	0.457371i	0.973500	−	0.228686i	$-0.0734428\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$71$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$72$		0			0
$73$	2.58246	−	7.94799i	0.302254	−	0.930242i	−0.678434	−	0.734662i	$-0.737341\pi$
$73$	2.58246	−	7.94799i	0.302254	−	0.930242i	0.980688	−	0.195580i	$-0.0626591\pi$
$74$		0			0
$75$	−9.64127	−	13.2701i	−1.11328	−	1.53230i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$79$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$80$		0			0
$81$	8.64075	−	26.5935i	0.960084	−	2.95483i
$82$		0			0
$83$	9.53671	−	13.1261i	1.04679	−	1.44078i	0.155230	−	0.987878i	$-0.450388\pi$
$83$	9.53671	−	13.1261i	1.04679	−	1.44078i	0.891559	−	0.452904i	$-0.149612\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−11.2373			−1.19115			−0.595575	−	0.803300i	$-0.703076\pi$
$89$	−11.2373			−1.19115			−0.595575	+	0.803300i	$0.703076\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.52490	+	1.10790i	−0.154830	+	0.112490i	−0.662503	−	0.749059i	$-0.730506\pi$
$97$	−1.52490	+	1.10790i	−0.154830	+	0.112490i	0.507673	+	0.861550i	$0.330506\pi$
$98$		0			0
$99$	−22.5677	+	12.3865i	−2.26814	+	1.24489i
$100$		0			0
$101$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$101$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$102$		0			0
$103$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$103$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.45145	+	2.09620i	−0.623685	+	0.202647i	−0.603776	−	0.797154i	$-0.706338\pi$
$107$	−6.45145	+	2.09620i	−0.623685	+	0.202647i	−0.0199092	+	0.999802i	$0.506338\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	5.04386	+	15.5234i	0.474487	+	1.46032i	0.846649	+	0.532152i	$0.178617\pi$
$113$	5.04386	+	15.5234i	0.474487	+	1.46032i	−0.372162	+	0.928168i	$0.621383\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	10.6506	+	2.75024i	0.968240	+	0.250022i
$122$		0			0
$123$	−22.1829	−	30.5321i	−2.00016	−	2.75299i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$127$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$128$		0			0
$129$	−13.2574	−	40.8021i	−1.16725	−	3.59243i
$130$		0			0
$131$		−	19.0123i		−	1.66111i	−0.556936	−	0.830555i	$-0.688023\pi$
$131$		−	19.0123i		−	1.66111i	0.556936	−	0.830555i	$-0.311977\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	6.83293	+	4.96441i	0.583776	+	0.424138i	0.840083	−	0.542457i	$-0.182506\pi$
$137$	6.83293	+	4.96441i	0.583776	+	0.424138i	−0.256307	+	0.966595i	$0.582506\pi$
$138$		0			0
$139$	20.9232	+	6.79837i	1.77469	+	0.576631i	0.998545	−	0.0539282i	$-0.0171742\pi$
$139$	20.9232	+	6.79837i	1.77469	+	0.576631i	0.776142	+	0.630559i	$0.217174\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−21.8399	−	7.09620i	−1.80132	−	0.585285i
$148$		0			0
$149$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$149$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$150$		0			0
$151$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$151$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$152$		0			0
$153$	−27.3678			−2.21255
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$157$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	14.5935	+	20.0863i	1.14305	+	1.57328i	0.760493	+	0.649347i	$0.224958\pi$
$163$	14.5935	+	20.0863i	1.14305	+	1.57328i	0.382560	+	0.923931i	$0.375042\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$167$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$168$		0			0
$169$	4.01722	−	12.3637i	0.309017	−	0.951057i
$170$		0			0
$171$	39.6379	−	54.5569i	3.03118	−	4.17207i
$172$		0			0
$173$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$173$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−50.2056			−3.77369
$178$		0			0
$179$	−12.6183	+	4.09993i	−0.943135	+	0.306443i	−0.739923	−	0.672692i	$-0.765138\pi$
$179$	−12.6183	+	4.09993i	−0.943135	+	0.306443i	−0.203212	+	0.979135i	$0.565138\pi$
$180$		0			0
$181$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$181$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	8.51908	+	8.01098i	0.622977	+	0.585821i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$191$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$192$		0			0
$193$	17.7984	+	12.9313i	1.28115	+	0.930814i	0.999587	−	0.0287278i	$-0.00914559\pi$
$193$	17.7984	+	12.9313i	1.28115	+	0.930814i	0.281568	+	0.959541i	$0.409146\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$	3.79520	+	11.6804i	0.267692	+	0.823873i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−28.3082	+	5.37991i	−1.95812	+	0.372136i
$210$		0			0
$211$	−2.13738	−	2.94185i	−0.147143	−	0.202526i	0.729083	−	0.684425i	$-0.239947\pi$
$211$	−2.13738	−	2.94185i	−0.147143	−	0.202526i	−0.876226	+	0.481900i	$0.839947\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$			27.4155i			1.85257i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$223$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$224$		0			0
$225$	31.3978	+	22.8118i	2.09318	+	1.52079i
$226$		0			0
$227$	11.3751	+	3.69600i	0.754993	+	0.245312i	0.661128	−	0.750273i	$-0.270078\pi$
$227$	11.3751	+	3.69600i	0.754993	+	0.245312i	0.0938647	+	0.995585i	$0.470078\pi$
$228$		0			0
$229$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$229$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−16.9453	+	12.3115i	−1.11012	+	0.806550i	−0.982683	−	0.185296i	$-0.940675\pi$
$233$	−16.9453	+	12.3115i	−1.11012	+	0.806550i	−0.127438	+	0.991847i	$0.540675\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$239$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$240$		0			0
$241$	24.1744			1.55721			0.778605	−	0.627514i	$-0.215928\pi$
$241$	24.1744			1.55721			0.778605	+	0.627514i	$0.215928\pi$
$242$		0			0
$243$			44.8654i			2.87811i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−16.4478	+	50.6211i	−1.04234	+	3.20798i
$250$		0			0
$251$	−3.52671	−	4.85410i	−0.222604	−	0.306388i	0.683078	−	0.730345i	$-0.260641\pi$
$251$	−3.52671	−	4.85410i	−0.222604	−	0.306388i	−0.905682	+	0.423957i	$0.860641\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	9.55497	−	29.4072i	0.596023	−	1.83437i	0.0464552	−	0.998920i	$-0.485208\pi$
$257$	9.55497	−	29.4072i	0.596023	−	1.83437i	0.549568	−	0.835449i	$-0.314792\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	35.0601	−	11.3917i	2.14564	−	0.697162i
$268$		0			0
$269$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$269$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$270$		0			0
$271$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$271$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3.09616	−	16.2915i	−0.186706	−	0.982416i
$276$		0			0
$277$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$277$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.66882	−	1.21247i	−0.0995533	−	0.0723297i	0.536895	−	0.843649i	$-0.319597\pi$
$281$	−1.66882	−	1.21247i	−0.0995533	−	0.0723297i	−0.636448	+	0.771319i	$0.719597\pi$
$282$		0			0
$283$	−20.9232	+	6.79837i	−1.24376	+	0.404121i	−0.855680	−	0.517505i	$-0.826861\pi$
$283$	−20.9232	+	6.79837i	−1.24376	+	0.404121i	−0.388078	+	0.921627i	$0.626861\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1.41163	−	4.34455i	−0.0830370	−	0.255562i
$290$		0			0
$291$	3.63452	−	5.00249i	0.213059	−	0.293251i
$292$		0			0
$293$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$293$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	35.4935	−	37.7446i	2.05954	−	2.19017i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$			22.5191i			1.28523i	0.766189	+	0.642615i	$0.222151\pi$
$307$			22.5191i			1.28523i	−0.766189	+	0.642615i	$0.777849\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$311$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$312$		0			0
$313$	−28.6150	−	20.7900i	−1.61742	−	1.17512i	−0.824951	−	0.565204i	$-0.808798\pi$
$313$	−28.6150	−	20.7900i	−1.61742	−	1.17512i	−0.792465	−	0.609918i	$-0.791202\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$317$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	18.0034	−	13.0802i	1.00485	−	0.730067i
$322$		0			0
$323$	−29.1337	−	9.46610i	−1.62104	−	0.526708i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$			6.07187i			0.333740i	0.985979	+	0.166870i	$0.0533661\pi$
$331$			6.07187i			0.333740i	−0.985979	+	0.166870i	$0.946634\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	8.63817	−	26.5855i	0.470551	−	1.44821i	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	8.63817	−	26.5855i	0.470551	−	1.44821i	0.851865	−	0.523761i	$-0.175472\pi$
$338$		0			0
$339$	−31.4735	−	43.3196i	−1.70941	−	2.35280i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−19.4650	+	26.7913i	−1.04494	+	1.43823i	−0.151817	+	0.988409i	$0.548512\pi$
$347$	−19.4650	+	26.7913i	−1.04494	+	1.43823i	−0.893118	+	0.449822i	$0.851488\pi$
$348$		0			0
$349$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$349$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	12.2494			0.651972			0.325986	−	0.945375i	$-0.394304\pi$
$353$	12.2494			0.651972			0.325986	+	0.945375i	$0.394304\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$359$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$360$		0			0
$361$	45.6946	−	33.1991i	2.40498	−	1.74732i
$362$		0			0
$363$	−36.0178	+	2.21632i	−1.89045	+	0.116327i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$367$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$368$		0			0
$369$	72.2408	+	52.4860i	3.76071	+	2.73231i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−15.1385	+	20.8363i	−0.777612	+	1.07029i	0.217930	+	0.975964i	$0.430070\pi$
$379$	−15.1385	+	20.8363i	−0.777612	+	1.07029i	−0.995541	+	0.0943260i	$0.969930\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$383$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	59.6652	+	82.1221i	3.03295	+	4.17450i
$388$		0			0
$389$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$389$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	19.2736	+	59.3179i	0.972223	+	2.99219i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−22.7570	−	16.5339i	−1.13643	−	0.825665i	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−22.7570	−	16.5339i	−1.13643	−	0.825665i	−0.986618	+	0.163049i	$0.947867\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	17.7984	−	12.9313i	0.880073	−	0.639410i	−0.0531978	−	0.998584i	$-0.516941\pi$
$409$	17.7984	−	12.9313i	0.880073	−	0.639410i	0.933271	+	0.359174i	$0.116941\pi$
$410$		0			0
$411$	−26.3512	−	8.56203i	−1.29981	−	0.422334i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−72.1719			−3.53427
$418$		0			0
$419$		−	7.05029i		−	0.344429i	−0.985059	−	0.172215i	$-0.944908\pi$
$419$		−	7.05029i		−	0.344429i	0.985059	−	0.172215i	$-0.0550923\pi$
$420$		0			0
$421$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$421$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	5.44779	−	16.7666i	0.264257	−	0.813299i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$431$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$432$		0			0
$433$	−8.61620	+	26.5179i	−0.414068	+	1.27437i	0.499014	+	0.866594i	$0.333696\pi$
$433$	−8.61620	+	26.5179i	−0.414068	+	1.27437i	−0.913082	+	0.407777i	$0.866304\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	54.3337			2.58732
$442$		0			0
$443$	−30.7345	+	9.98626i	−1.46024	+	0.474461i	−0.928144	−	0.372221i	$-0.878596\pi$
$443$	−30.7345	+	9.98626i	−1.46024	+	0.474461i	−0.532098	+	0.846683i	$0.678596\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−14.8525	+	10.7910i	−0.700933	+	0.509258i	−0.880236	−	0.474536i	$-0.842616\pi$
$449$	−14.8525	+	10.7910i	−0.700933	+	0.509258i	0.179303	+	0.983794i	$0.442616\pi$
$450$		0			0
$451$	−7.12374	−	37.4840i	−0.335444	−	1.76505i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.877257	+	0.637364i	0.0410363	+	0.0298146i	0.608114	−	0.793849i	$-0.291926\pi$
$457$	0.877257	+	0.637364i	0.0410363	+	0.0298146i	−0.567078	+	0.823664i	$0.691926\pi$
$458$		0			0
$459$	52.3848	−	17.0209i	2.44511	−	0.794466i
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	17.6336	−	24.2705i	0.815984	−	1.12311i	−0.174389	−	0.984677i	$-0.555795\pi$
$467$	17.6336	−	24.2705i	0.815984	−	1.12311i	0.990372	−	0.138428i	$-0.0442051\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	5.46577	−	43.0280i	0.251316	−	1.97843i
$474$		0			0
$475$	25.5334	+	35.1438i	1.17155	+	1.61251i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$479$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$487$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$488$		0			0
$489$	−65.8938	−	47.8747i	−2.97982	−	2.16497i
$490$		0			0
$491$	24.4100	+	7.93128i	1.10161	+	0.357934i	0.802721	−	0.596355i	$-0.203385\pi$
$491$	24.4100	+	7.93128i	1.10161	+	0.357934i	0.298886	+	0.954289i	$0.403385\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	12.9452	+	4.20615i	0.579506	+	0.188293i	0.584079	−	0.811697i	$-0.301456\pi$
$499$	12.9452	+	4.20615i	0.579506	+	0.188293i	−0.00457310	+	0.999990i	$0.501456\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$503$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			42.6470i			1.89402i
$508$		0			0
$509$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$509$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−41.9405	+	129.080i	−1.85172	+	5.69900i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−13.8730	+	42.6967i	−0.607788	+	1.87058i	−0.131432	+	0.991325i	$0.541958\pi$
$521$	−13.8730	+	42.6967i	−0.607788	+	1.87058i	−0.476355	+	0.879253i	$0.658042\pi$
$522$		0			0
$523$	−7.32810	+	10.0863i	−0.320435	+	0.441041i	−0.938600	−	0.345007i	$-0.887876\pi$
$523$	−7.32810	+	10.0863i	−0.320435	+	0.441041i	0.618165	+	0.786049i	$0.287876\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$	112.976	−	36.7080i	4.90272	−	1.59299i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	35.2125	−	25.5834i	1.51953	−	1.10401i
$538$		0			0
$539$	−16.9131	−	15.9043i	−0.728498	−	0.685048i
$540$		0			0
$541$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$541$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−21.1941	+	6.88637i	−0.906193	+	0.294440i	−0.724791	−	0.688969i	$-0.758064\pi$
$547$	−21.1941	+	6.88637i	−0.906193	+	0.294440i	−0.181402	+	0.983409i	$0.558064\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$557$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−34.7004	−	16.3579i	−1.46505	−	0.690633i
$562$		0			0
$563$	1.56228	+	2.15029i	0.0658422	+	0.0906241i	0.840668	−	0.541551i	$-0.182163\pi$
$563$	1.56228	+	2.15029i	0.0658422	+	0.0906241i	−0.774826	+	0.632175i	$0.782163\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	10.6607	+	32.8102i	0.446919	+	1.37547i	0.880366	+	0.474295i	$0.157297\pi$
$569$	10.6607	+	32.8102i	0.446919	+	1.37547i	−0.433447	+	0.901179i	$0.642703\pi$
$570$		0			0
$571$			22.0000i			0.920671i	0.887745	+	0.460336i	$0.152271\pi$
$571$			22.0000i			0.920671i	−0.887745	+	0.460336i	$0.847729\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	17.6150	+	12.7981i	0.733322	+	0.532790i	0.890613	−	0.454762i	$-0.150276\pi$
$577$	17.6150	+	12.7981i	0.733322	+	0.532790i	−0.157290	+	0.987552i	$0.550276\pi$
$578$		0			0
$579$	−68.6396	−	22.3023i	−2.85256	−	0.926854i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−41.7284	−	13.5584i	−1.72231	−	0.559614i	−0.730010	−	0.683437i	$-0.760484\pi$
$587$	−41.7284	−	13.5584i	−1.72231	−	0.559614i	−0.992304	+	0.123823i	$0.960484\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	48.6022			1.99585			0.997927	−	0.0643593i	$-0.0205004\pi$
$593$	48.6022			1.99585			0.997927	+	0.0643593i	$0.0205004\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$599$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$600$		0			0
$601$	8.41754	−	25.9065i	0.343359	−	1.05675i	−0.619098	−	0.785314i	$-0.712502\pi$
$601$	8.41754	−	25.9065i	0.343359	−	1.05675i	0.962457	−	0.271436i	$-0.0874984\pi$
$602$		0			0
$603$	−17.0803	−	23.5090i	−0.695564	−	0.957362i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$607$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$613$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−46.9304			−1.88935			−0.944674	−	0.328011i	$-0.893622\pi$
$617$	−46.9304			−1.88935			−0.944674	+	0.328011i	$0.893622\pi$
$618$		0			0
$619$	−30.7338	+	9.98603i	−1.23530	+	0.401372i	−0.852631	−	0.522514i	$-0.824994\pi$
$619$	−30.7338	+	9.98603i	−1.23530	+	0.401372i	−0.382667	+	0.923887i	$0.624994\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−20.2254	+	14.6946i	−0.809017	+	0.587785i
$626$		0			0
$627$	82.8672	−	45.4825i	3.30940	−	1.81639i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$631$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$632$		0			0
$633$	9.65088	+	7.01177i	0.383588	+	0.278693i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−15.6374	−	48.1271i	−0.617641	−	1.90090i	−0.342714	−	0.939440i	$-0.611346\pi$
$641$	−15.6374	−	48.1271i	−0.617641	−	1.90090i	−0.274928	−	0.961465i	$-0.588654\pi$
$642$		0			0
$643$	26.7080	−	36.7604i	1.05326	−	1.44969i	0.167313	−	0.985904i	$-0.446491\pi$
$643$	26.7080	−	36.7604i	1.05326	−	1.44969i	0.885948	−	0.463786i	$-0.153509\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$647$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$648$		0			0
$649$	−45.9122	−	21.6432i	−1.80221	−	0.849571i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$653$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−20.0449	−	61.6919i	−0.782027	−	2.40683i
$658$		0			0
$659$			40.1676i			1.56471i	0.622834	+	0.782354i	$0.285981\pi$
$659$			40.1676i			1.56471i	−0.622834	+	0.782354i	$0.714019\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	36.6725	−	26.6441i	1.41362	−	1.02706i	0.420838	−	0.907136i	$-0.361736\pi$
$673$	36.6725	−	26.6441i	1.41362	−	1.02706i	0.992784	−	0.119920i	$-0.0382640\pi$
$674$		0			0
$675$	−74.2860	−	24.1370i	−2.85927	−	0.929033i
$676$		0			0
$677$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$677$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−39.2369			−1.50356
$682$		0			0
$683$		−	42.0000i		−	1.60709i	−0.595247	−	0.803543i	$-0.702946\pi$
$683$		−	42.0000i		−	1.60709i	0.595247	−	0.803543i	$-0.297054\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−22.5855	−	31.0863i	−0.859192	−	1.18258i	−0.981761	−	0.190117i	$-0.939113\pi$
$691$	−22.5855	−	31.0863i	−0.859192	−	1.18258i	0.122569	−	0.992460i	$-0.460887\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	12.5344	−	38.5770i	0.474775	−	1.46121i
$698$		0			0
$699$	40.3882	−	55.5896i	1.52762	−	2.10259i
$700$		0			0
$701$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$701$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$709$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$719$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−75.4237	+	24.5066i	−2.80504	+	0.911412i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	−19.5597	−	60.1986i	−0.724433	−	2.22958i
$730$		0			0
$731$	27.1030	−	37.3041i	1.00244	−	1.37974i
$732$		0			0
$733$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$733$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.56468	+	12.3176i	−0.0576358	+	0.453725i
$738$		0			0
$739$	−17.5415	−	24.1438i	−0.645273	−	0.888143i	0.353610	−	0.935393i	$-0.384954\pi$
$739$	−17.5415	−	24.1438i	−0.645273	−	0.888143i	−0.998883	+	0.0472504i	$0.984954\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$743$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		−	125.936i		−	4.60777i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$751$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$752$		0			0
$753$	15.9241	+	11.5695i	0.580306	+	0.421617i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$757$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	22.2050	−	16.1329i	0.804931	−	0.584816i	−0.107426	−	0.994213i	$-0.534261\pi$
$761$	22.2050	−	16.1329i	0.804931	−	0.584816i	0.912356	+	0.409397i	$0.134261\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−22.0000			−0.793340			−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000			−0.793340			−0.396670	+	0.917961i	$0.629834\pi$
$770$		0			0
$771$			101.436i			3.65313i
$772$		0			0
$773$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$773$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	58.7480	+	80.8597i	2.10487	+	2.89710i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−23.3120	+	32.0863i	−0.830984	+	1.14375i	0.156755	+	0.987638i	$0.449897\pi$
$787$	−23.3120	+	32.0863i	−0.830984	+	1.14375i	−0.987739	+	0.156114i	$0.950103\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$797$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−70.5652	+	51.2686i	−2.49330	+	1.81149i
$802$		0			0
$803$	−11.8186	+	25.0710i	−0.417069	+	0.884738i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	45.0250	+	32.7126i	1.58300	+	1.15011i	0.913158	+	0.407605i	$0.133636\pi$
$809$	45.0250	+	32.7126i	1.58300	+	1.15011i	0.669837	+	0.742508i	$0.266364\pi$
$810$		0			0
$811$	5.52091	−	1.79385i	0.193865	−	0.0629907i	−0.210475	−	0.977599i	$-0.567501\pi$
$811$	5.52091	−	1.79385i	0.193865	−	0.0629907i	0.404340	+	0.914609i	$0.367501\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	35.1103	+	108.058i	1.22835	+	3.78048i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$821$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$822$		0			0
$823$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$823$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$824$		0			0
$825$	26.1754	+	47.6905i	0.911310	+	1.66037i
$826$		0			0
$827$	1.25965	+	1.73376i	0.0438022	+	0.0602886i	0.830356	−	0.557233i	$-0.188137\pi$
$827$	1.25965	+	1.73376i	0.0438022	+	0.0602886i	−0.786554	+	0.617521i	$0.788137\pi$
$828$		0			0
$829$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$829$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−7.62691	−	23.4732i	−0.264257	−	0.813299i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$839$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$840$		0			0
$841$	−23.4615	−	17.0458i	−0.809017	−	0.587785i
$842$		0			0
$843$	6.43581	+	2.09112i	0.221661	+	0.0720220i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	58.3883	−	42.4216i	2.00388	−	1.45591i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$853$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	30.3869			1.03800			0.518998	−	0.854776i	$-0.326305\pi$
$857$	30.3869			1.03800			0.518998	+	0.854776i	$0.326305\pi$
$858$		0			0
$859$		−	9.85300i		−	0.336180i	−0.985772	−	0.168090i	$-0.946240\pi$
$859$		−	9.85300i		−	0.336180i	0.985772	−	0.168090i	$-0.0537599\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$863$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	8.80851	+	12.1239i	0.299153	+	0.411748i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−4.52102	+	13.9143i	−0.153013	+	0.470927i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$877$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−47.8125			−1.61084			−0.805421	−	0.592703i	$-0.798061\pi$
$881$	−47.8125			−1.61084			−0.805421	+	0.592703i	$0.798061\pi$
$882$		0			0
$883$	31.6651	−	10.2886i	1.06561	−	0.346239i	0.276836	−	0.960917i	$-0.410714\pi$
$883$	31.6651	−	10.2886i	1.06561	−	0.346239i	0.788778	+	0.614678i	$0.210714\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$887$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−39.5443	+	83.8862i	−1.32479	+	2.81029i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	31.3876	−	43.2013i	1.04221	−	1.43447i	0.146832	−	0.989161i	$-0.453092\pi$
$907$	31.3876	−	43.2013i	1.04221	−	1.43447i	0.895375	−	0.445313i	$-0.146908\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$911$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$912$		0			0
$913$	−36.8636	+	39.2016i	−1.22001	+	1.29738i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$919$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$920$		0			0
$921$	−22.8285	−	70.2590i	−0.752226	−	2.31511i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−8.26251	−	6.00306i	−0.271084	−	0.196954i	0.443935	−	0.896059i	$-0.353582\pi$
$929$	−8.26251	−	6.00306i	−0.271084	−	0.196954i	−0.715019	+	0.699105i	$0.753582\pi$
$930$		0			0
$931$	57.8396	+	18.7932i	1.89562	+	0.615923i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−47.6725	+	34.6361i	−1.55739	+	1.13151i	−0.619287	+	0.785165i	$0.712578\pi$
$937$	−47.6725	+	34.6361i	−1.55739	+	1.13151i	−0.938106	+	0.346348i	$0.887422\pi$
$938$		0			0
$939$	110.354	+	35.8562i	3.60127	+	1.17012i
$940$		0			0
$941$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$941$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	41.8394i		−	1.35960i	−0.733399	−	0.679799i	$-0.762067\pi$
$947$		−	41.8394i		−	1.35960i	0.733399	−	0.679799i	$-0.237933\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−2.28011	+	7.01746i	−0.0738600	+	0.227318i	−0.981171	−	0.193143i	$-0.938132\pi$
$953$	−2.28011	+	7.01746i	−0.0738600	+	0.227318i	0.907311	+	0.420461i	$0.138132\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−9.57953	+	29.4828i	−0.309017	+	0.951057i
$962$		0			0
$963$	−30.9486	+	42.5970i	−0.997304	+	1.37267i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$		0			0
$969$	100.493			3.22829
$970$		0			0
$971$	−51.3571	+	16.6869i	−1.64813	+	0.535509i	−0.978333	−	0.207039i	$-0.933617\pi$
$971$	−51.3571	+	16.6869i	−1.64813	+	0.535509i	−0.669793	+	0.742547i	$0.733617\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	4.85410	−	3.52671i	0.155296	−	0.112829i	−0.507423	−	0.861697i	$-0.669402\pi$
$977$	4.85410	−	3.52671i	0.155296	−	0.112829i	0.662720	+	0.748867i	$0.269402\pi$
$978$		0			0
$979$	36.9728	+	4.69658i	1.18165	+	0.150103i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$983$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$	−6.15532	−	18.9441i	−0.195333	−	0.601174i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$997$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	704.2.w.a.289.1	✓	16
4.3	odd	2	inner	704.2.w.a.289.4	yes	16
8.3	odd	2	CM	704.2.w.a.289.1	✓	16
8.5	even	2	inner	704.2.w.a.289.4	yes	16
11.4	even	5	inner	704.2.w.a.609.4	yes	16
44.15	odd	10	inner	704.2.w.a.609.1	yes	16
88.37	even	10	inner	704.2.w.a.609.1	yes	16
88.59	odd	10	inner	704.2.w.a.609.4	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
704.2.w.a.289.1	✓	16	1.1	even	1	trivial
704.2.w.a.289.1	✓	16	8.3	odd	2	CM
704.2.w.a.289.4	yes	16	4.3	odd	2	inner
704.2.w.a.289.4	yes	16	8.5	even	2	inner
704.2.w.a.609.1	yes	16	44.15	odd	10	inner
704.2.w.a.609.1	yes	16	88.37	even	10	inner
704.2.w.a.609.4	yes	16	11.4	even	5	inner
704.2.w.a.609.4	yes	16	88.59	odd	10	inner

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:	\( N \)	\(=\)	\( 704 = 2^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	704.w (of order \(10\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(-3.11998 + 1.01374i) q^{3} +(6.27955 - 4.56236i) q^{9} +O(q^{10})\)	\(q+(-3.11998 + 1.01374i) q^{3} +(6.27955 - 4.56236i) q^{9} +(-3.29019 - 0.417946i) q^{11} +(-2.85250 - 2.07246i) q^{17} +(8.26279 - 2.68474i) q^{19} +(1.54508 + 4.75528i) q^{25} +(-9.18226 + 12.6383i) q^{27} +(10.6890 - 2.03142i) q^{33} +(3.55497 + 10.9411i) q^{41} +13.0777i q^{43} +(5.66312 + 4.11450i) q^{49} +(11.0007 + 3.57434i) q^{51} +(-23.0581 + 16.7527i) q^{57} +(14.5550 + 4.72922i) q^{59} -3.74374i q^{67} +(2.58246 - 7.94799i) q^{73} +(-9.64127 - 13.2701i) q^{75} +(8.64075 - 26.5935i) q^{81} +(9.53671 - 13.1261i) q^{83} -11.2373 q^{89} +(-1.52490 + 1.10790i) q^{97} +(-22.5677 + 12.3865i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{9}+O(q^{10}) \) 16 * q + 4 * q^9	\( 16 q + 4 q^{9} + 24 q^{17} - 20 q^{25} + 76 q^{33} + 24 q^{41} + 28 q^{49} - 124 q^{57} - 8 q^{73} - 16 q^{81} - 72 q^{89} + 60 q^{97}+O(q^{100}) \) 16 * q + 4 * q^9 + 24 * q^17 - 20 * q^25 + 76 * q^33 + 24 * q^41 + 28 * q^49 - 124 * q^57 - 8 * q^73 - 16 * q^81 - 72 * q^89 + 60 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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