LMFDB - Embedded newform 672.2.p.a.559.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [672,2,Mod(559,672)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(672, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0, 1]))

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

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

chi := DirichletCharacter("672.559");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	672.p (of order $2$, degree $1$, not 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:	$5.36594701583$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + x^{14} - 4x^{12} - 4x^{10} + 16x^{8} - 16x^{6} - 64x^{4} + 64x^{2} + 256 $ x^16 + x^14 - 4x^12 - 4x^10 + 16x^8 - 16x^6 - 64x^4 + 64x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{17} $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			559.15
Root			$0.310478 - 1.37971i$ of defining polynomial
Character	$\chi$	$=$	672.559
Dual form			672.2.p.a.559.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +2.33443 q^{5} +(-0.490487 - 2.59989i) q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +2.33443 q^{5} +(-0.490487 - 2.59989i) q^{7} -1.00000 q^{9} +0.304570 q^{11} +5.46072 q^{13} +2.33443i q^{15} -6.37384i q^{17} +0.840438i q^{19} +(2.59989 - 0.490487i) q^{21} +0.111550i q^{23} +0.449578 q^{25} -1.00000i q^{27} +5.46072i q^{29} +7.64576 q^{31} +0.304570i q^{33} +(-1.14501 - 6.06927i) q^{35} -6.44169i q^{37} +5.46072i q^{39} +8.66385i q^{41} -6.06927 q^{43} -2.33443 q^{45} +8.21451 q^{47} +(-6.51885 + 2.55042i) q^{49} +6.37384 q^{51} +10.1296i q^{53} +0.710999 q^{55} -0.840438 q^{57} +1.70998i q^{59} -2.75380 q^{61} +(0.490487 + 2.59989i) q^{63} +12.7477 q^{65} +8.35928 q^{67} -0.111550 q^{69} -2.07350i q^{71} -12.7477i q^{73} +0.449578i q^{75} +(-0.149388 - 0.791849i) q^{77} -7.90670i q^{79} +1.00000 q^{81} +11.6468i q^{83} -14.8793i q^{85} -5.46072 q^{87} +4.08382i q^{89} +(-2.67841 - 14.1973i) q^{91} +7.64576i q^{93} +1.96195i q^{95} +6.42855i q^{97} -0.304570 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{9}+O(q^{10}) $ 16 * q - 16 * q^9	$ 16 q - 16 q^{9} + 8 q^{11} + 16 q^{25} - 24 q^{35} + 8 q^{43} - 8 q^{49} - 16 q^{57} + 40 q^{67} + 16 q^{81} + 56 q^{91} - 8 q^{99}+O(q^{100}) $ 16 * q - 16 * q^9 + 8 * q^11 + 16 * q^25 - 24 * q^35 + 8 * q^43 - 8 * q^49 - 16 * q^57 + 40 * q^67 + 16 * q^81 + 56 * q^91 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$.

$n$	$127$	$421$	$449$	$577$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	2.33443			1.04399			0.521995	−	0.852948i	$-0.325188\pi$
$5$	2.33443			1.04399			0.521995	+	0.852948i	$0.325188\pi$
$6$		0			0
$7$	−0.490487	−	2.59989i	−0.185387	−	0.982666i
$7$	−0.490487	−	2.59989i	−0.185387	−	0.982666i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	0.304570			0.0918314			0.0459157	−	0.998945i	$-0.485379\pi$
$11$	0.304570			0.0918314			0.0459157	+	0.998945i	$0.485379\pi$
$12$		0			0
$13$	5.46072			1.51453			0.757265	−	0.653108i	$-0.226535\pi$
$13$	5.46072			1.51453			0.757265	+	0.653108i	$0.226535\pi$
$14$		0			0
$15$			2.33443i			0.602748i
$16$		0			0
$17$		−	6.37384i		−	1.54588i	−0.634477	−	0.772941i	$-0.718785\pi$
$17$		−	6.37384i		−	1.54588i	0.634477	−	0.772941i	$-0.281215\pi$
$18$		0			0
$19$			0.840438i			0.192810i	0.995342	+	0.0964049i	$0.0307344\pi$
$19$			0.840438i			0.192810i	−0.995342	+	0.0964049i	$0.969266\pi$
$20$		0			0
$21$	2.59989	−	0.490487i	0.567342	−	0.107033i
$22$		0			0
$23$			0.111550i			0.0232597i	0.999932	+	0.0116298i	$0.00370198\pi$
$23$			0.111550i			0.0232597i	−0.999932	+	0.0116298i	$0.996298\pi$
$24$		0			0
$25$	0.449578			0.0899157
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$			5.46072i			1.01403i	0.861937	+	0.507015i	$0.169251\pi$
$29$			5.46072i			1.01403i	−0.861937	+	0.507015i	$0.830749\pi$
$30$		0			0
$31$	7.64576			1.37322			0.686610	−	0.727026i	$-0.259098\pi$
$31$	7.64576			1.37322			0.686610	+	0.727026i	$0.259098\pi$
$32$		0			0
$33$			0.304570i			0.0530189i
$34$		0			0
$35$	−1.14501	−	6.06927i	−0.193542	−	1.02589i
$36$		0			0
$37$		−	6.44169i		−	1.05901i	−0.848308	−	0.529504i	$-0.822378\pi$
$37$		−	6.44169i		−	1.05901i	0.848308	−	0.529504i	$-0.177622\pi$
$38$		0			0
$39$			5.46072i			0.874414i
$40$		0			0
$41$			8.66385i			1.35307i	0.736412	+	0.676533i	$0.236519\pi$
$41$			8.66385i			1.35307i	−0.736412	+	0.676533i	$0.763481\pi$
$42$		0			0
$43$	−6.06927			−0.925555			−0.462777	−	0.886475i	$-0.653147\pi$
$43$	−6.06927			−0.925555			−0.462777	+	0.886475i	$0.653147\pi$
$44$		0			0
$45$	−2.33443			−0.347997
$46$		0			0
$47$	8.21451			1.19821			0.599105	−	0.800671i	$-0.295523\pi$
$47$	8.21451			1.19821			0.599105	+	0.800671i	$0.295523\pi$
$48$		0			0
$49$	−6.51885	+	2.55042i	−0.931264	+	0.364346i
$50$		0			0
$51$	6.37384			0.892516
$52$		0			0
$53$			10.1296i			1.39141i	0.718330	+	0.695703i	$0.244907\pi$
$53$			10.1296i			1.39141i	−0.718330	+	0.695703i	$0.755093\pi$
$54$		0			0
$55$	0.710999			0.0958711
$56$		0			0
$57$	−0.840438			−0.111319
$58$		0			0
$59$			1.70998i			0.222621i	0.993786	+	0.111310i	$0.0355048\pi$
$59$			1.70998i			0.222621i	−0.993786	+	0.111310i	$0.964495\pi$
$60$		0			0
$61$	−2.75380			−0.352587			−0.176294	−	0.984338i	$-0.556411\pi$
$61$	−2.75380			−0.352587			−0.176294	+	0.984338i	$0.556411\pi$
$62$		0			0
$63$	0.490487	+	2.59989i	0.0617955	+	0.327555i
$64$		0			0
$65$	12.7477			1.58115
$66$		0			0
$67$	8.35928			1.02125			0.510625	−	0.859804i	$-0.329414\pi$
$67$	8.35928			1.02125			0.510625	+	0.859804i	$0.329414\pi$
$68$		0			0
$69$	−0.111550			−0.0134290
$70$		0			0
$71$		−	2.07350i		−	0.246079i	−0.992402	−	0.123039i	$-0.960736\pi$
$71$		−	2.07350i		−	0.246079i	0.992402	−	0.123039i	$-0.0392641\pi$
$72$		0			0
$73$		−	12.7477i		−	1.49200i	−0.665945	−	0.746001i	$-0.731971\pi$
$73$		−	12.7477i		−	1.49200i	0.665945	−	0.746001i	$-0.268029\pi$
$74$		0			0
$75$			0.449578i			0.0519129i
$76$		0			0
$77$	−0.149388	−	0.791849i	−0.0170243	−	0.0902395i
$78$		0			0
$79$		−	7.90670i		−	0.889573i	−0.895637	−	0.444787i	$-0.853280\pi$
$79$		−	7.90670i		−	0.889573i	0.895637	−	0.444787i	$-0.146720\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			11.6468i			1.27841i	0.769038	+	0.639203i	$0.220736\pi$
$83$			11.6468i			1.27841i	−0.769038	+	0.639203i	$0.779264\pi$
$84$		0			0
$85$		−	14.8793i		−	1.61389i
$86$		0			0
$87$	−5.46072			−0.585450
$88$		0			0
$89$			4.08382i			0.432884i	0.976295	+	0.216442i	$0.0694452\pi$
$89$			4.08382i			0.432884i	−0.976295	+	0.216442i	$0.930555\pi$
$90$		0			0
$91$	−2.67841	−	14.1973i	−0.280773	−	1.48828i
$92$		0			0
$93$			7.64576i			0.792828i
$94$		0			0
$95$			1.96195i			0.201291i
$96$		0			0
$97$			6.42855i			0.652720i	0.945245	+	0.326360i	$0.105822\pi$
$97$			6.42855i			0.652720i	−0.945245	+	0.326360i	$0.894178\pi$
$98$		0			0
$99$	−0.304570			−0.0306105
$100$		0			0
$101$	−7.00330			−0.696854			−0.348427	−	0.937336i	$-0.613284\pi$
$101$	−7.00330			−0.696854			−0.348427	+	0.937336i	$0.613284\pi$
$102$		0			0
$103$	−9.60771			−0.946676			−0.473338	−	0.880881i	$-0.656951\pi$
$103$	−9.60771			−0.946676			−0.473338	+	0.880881i	$0.656951\pi$
$104$		0			0
$105$	6.06927	−	1.14501i	0.592300	−	0.111741i
$106$		0			0
$107$	−6.01455			−0.581449			−0.290724	−	0.956807i	$-0.593896\pi$
$107$	−6.01455			−0.581449			−0.290724	+	0.956807i	$0.593896\pi$
$108$		0			0
$109$		−	9.85961i		−	0.944379i	−0.881497	−	0.472190i	$-0.843464\pi$
$109$		−	9.85961i		−	0.944379i	0.881497	−	0.472190i	$-0.156536\pi$
$110$		0			0
$111$	6.44169			0.611418
$112$		0			0
$113$	−14.7477			−1.38734			−0.693672	−	0.720291i	$-0.744008\pi$
$113$	−14.7477			−1.38734			−0.693672	+	0.720291i	$0.744008\pi$
$114$		0			0
$115$			0.260405i			0.0242829i
$116$		0			0
$117$	−5.46072			−0.504843
$118$		0			0
$119$	−16.5713	+	3.12628i	−1.51909	+	0.286586i
$120$		0			0
$121$	−10.9072			−0.991567
$122$		0			0
$123$	−8.66385			−0.781193
$124$		0			0
$125$	−10.6227			−0.950119
$126$		0			0
$127$			2.49286i			0.221205i	0.993865	+	0.110603i	$0.0352781\pi$
$127$			2.49286i			0.221205i	−0.993865	+	0.110603i	$0.964722\pi$
$128$		0			0
$129$		−	6.06927i		−	0.534369i
$130$		0			0
$131$		−	4.60914i		−	0.402702i	−0.979519	−	0.201351i	$-0.935467\pi$
$131$		−	4.60914i		−	0.402702i	0.979519	−	0.201351i	$-0.0645333\pi$
$132$		0			0
$133$	2.18505	−	0.412224i	0.189467	−	0.0357443i
$134$		0			0
$135$		−	2.33443i		−	0.200916i
$136$		0			0
$137$	−0.899157			−0.0768202			−0.0384101	−	0.999262i	$-0.512229\pi$
$137$	−0.899157			−0.0768202			−0.0384101	+	0.999262i	$0.512229\pi$
$138$		0			0
$139$			23.1762i			1.96578i	0.184189	+	0.982891i	$0.441034\pi$
$139$			23.1762i			1.96578i	−0.184189	+	0.982891i	$0.558966\pi$
$140$		0			0
$141$			8.21451i			0.691787i
$142$		0			0
$143$	1.66317			0.139081
$144$		0			0
$145$			12.7477i			1.05864i
$146$		0			0
$147$	−2.55042	−	6.51885i	−0.210355	−	0.537665i
$148$		0			0
$149$		−	6.58394i		−	0.539377i	−0.962948	−	0.269689i	$-0.913079\pi$
$149$		−	6.58394i		−	0.539377i	0.962948	−	0.269689i	$-0.0869208\pi$
$150$		0			0
$151$		−	20.5670i		−	1.67372i	−0.547419	−	0.836858i	$-0.684390\pi$
$151$		−	20.5670i		−	1.67372i	0.547419	−	0.836858i	$-0.315610\pi$
$152$		0			0
$153$			6.37384i			0.515294i
$154$		0			0
$155$	17.8485			1.43363
$156$		0			0
$157$	−12.0915			−0.965009			−0.482505	−	0.875893i	$-0.660273\pi$
$157$	−12.0915			−0.965009			−0.482505	+	0.875893i	$0.660273\pi$
$158$		0			0
$159$	−10.1296			−0.803328
$160$		0			0
$161$	0.290017	−	0.0547136i	0.0228565	−	0.00431203i
$162$		0			0
$163$	−0.359284			−0.0281413			−0.0140706	−	0.999901i	$-0.504479\pi$
$163$	−0.359284			−0.0281413			−0.0140706	+	0.999901i	$0.504479\pi$
$164$		0			0
$165$			0.710999i			0.0553512i
$166$		0			0
$167$	−7.91574			−0.612538			−0.306269	−	0.951945i	$-0.599081\pi$
$167$	−7.91574			−0.612538			−0.306269	+	0.951945i	$0.599081\pi$
$168$		0			0
$169$	16.8194			1.29380
$170$		0			0
$171$		−	0.840438i		−	0.0642699i
$172$		0			0
$173$	12.5109			0.951185			0.475593	−	0.879666i	$-0.342234\pi$
$173$	12.5109			0.951185			0.475593	+	0.879666i	$0.342234\pi$
$174$		0			0
$175$	−0.220512	−	1.16885i	−0.0166692	−	0.0883571i
$176$		0			0
$177$	−1.70998			−0.128530
$178$		0			0
$179$	−15.3423			−1.14673			−0.573367	−	0.819298i	$-0.694363\pi$
$179$	−15.3423			−1.14673			−0.573367	+	0.819298i	$0.694363\pi$
$180$		0			0
$181$	−0.493074			−0.0366499			−0.0183249	−	0.999832i	$-0.505833\pi$
$181$	−0.493074			−0.0366499			−0.0183249	+	0.999832i	$0.505833\pi$
$182$		0			0
$183$		−	2.75380i		−	0.203566i
$184$		0			0
$185$		−	15.0377i		−	1.10559i
$186$		0			0
$187$		−	1.94128i		−	0.141961i
$188$		0			0
$189$	−2.59989	+	0.490487i	−0.189114	+	0.0356777i
$190$		0			0
$191$			21.8108i			1.57817i	0.614282	+	0.789087i	$0.289446\pi$
$191$			21.8108i			1.57817i	−0.614282	+	0.789087i	$0.710554\pi$
$192$		0			0
$193$	7.58811			0.546204			0.273102	−	0.961985i	$-0.411950\pi$
$193$	7.58811			0.546204			0.273102	+	0.961985i	$0.411950\pi$
$194$		0			0
$195$			12.7477i			0.912880i
$196$		0			0
$197$			11.7928i			0.840199i	0.907478	+	0.420099i	$0.138005\pi$
$197$			11.7928i			0.840199i	−0.907478	+	0.420099i	$0.861995\pi$
$198$		0			0
$199$	−24.7858			−1.75702			−0.878509	−	0.477726i	$-0.841461\pi$
$199$	−24.7858			−1.75702			−0.878509	+	0.477726i	$0.841461\pi$
$200$		0			0
$201$			8.35928i			0.589618i
$202$		0			0
$203$	14.1973	−	2.67841i	0.996452	−	0.187987i
$204$		0			0
$205$			20.2252i			1.41259i
$206$		0			0
$207$		−	0.111550i		−	0.00775323i
$208$		0			0
$209$			0.255972i			0.0177060i
$210$		0			0
$211$	−14.0693			−0.968568			−0.484284	−	0.874911i	$-0.660920\pi$
$211$	−14.0693			−0.968568			−0.484284	+	0.874911i	$0.660920\pi$
$212$		0			0
$213$	2.07350			0.142074
$214$		0			0
$215$	−14.1683			−0.966270
$216$		0			0
$217$	−3.75014	−	19.8781i	−0.254576	−	1.34942i
$218$		0			0
$219$	12.7477			0.861408
$220$		0			0
$221$		−	34.8057i		−	2.34129i
$222$		0			0
$223$	15.4480			1.03448			0.517239	−	0.855841i	$-0.326960\pi$
$223$	15.4480			1.03448			0.517239	+	0.855841i	$0.326960\pi$
$224$		0			0
$225$	−0.449578			−0.0299719
$226$		0			0
$227$		−	11.0377i		−	0.732597i	−0.930497	−	0.366299i	$-0.880625\pi$
$227$		−	11.0377i		−	0.732597i	0.930497	−	0.366299i	$-0.119375\pi$
$228$		0			0
$229$	7.42266			0.490503			0.245252	−	0.969459i	$-0.421129\pi$
$229$	7.42266			0.490503			0.245252	+	0.969459i	$0.421129\pi$
$230$		0			0
$231$	0.791849	−	0.149388i	0.0520998	−	0.00982898i
$232$		0			0
$233$	13.0668			0.856034			0.428017	−	0.903771i	$-0.359212\pi$
$233$	13.0668			0.856034			0.428017	+	0.903771i	$0.359212\pi$
$234$		0			0
$235$	19.1762			1.25092
$236$		0			0
$237$	7.90670			0.513595
$238$		0			0
$239$		−	8.54916i		−	0.552999i	−0.961014	−	0.276500i	$-0.910826\pi$
$239$		−	8.54916i		−	0.552999i	0.961014	−	0.276500i	$-0.0891744\pi$
$240$		0			0
$241$		−	7.64683i		−	0.492576i	−0.969197	−	0.246288i	$-0.920789\pi$
$241$		−	7.64683i		−	0.492576i	0.969197	−	0.246288i	$-0.0792109\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	−15.2178	+	5.95379i	−0.972230	+	0.380374i
$246$		0			0
$247$			4.58939i			0.292016i
$248$		0			0
$249$	−11.6468			−0.738088
$250$		0			0
$251$			12.7186i			0.802789i	0.915905	+	0.401394i	$0.131474\pi$
$251$			12.7186i			0.802789i	−0.915905	+	0.401394i	$0.868526\pi$
$252$		0			0
$253$			0.0339747i			0.00213597i
$254$		0			0
$255$	14.8793			0.931778
$256$		0			0
$257$			7.44557i			0.464442i	0.972663	+	0.232221i	$0.0745993\pi$
$257$			7.44557i			0.464442i	−0.972663	+	0.232221i	$0.925401\pi$
$258$		0			0
$259$	−16.7477	+	3.15956i	−1.04065	+	0.196326i
$260$		0			0
$261$		−	5.46072i		−	0.338010i
$262$		0			0
$263$			11.5549i			0.712503i	0.934390	+	0.356251i	$0.115945\pi$
$263$			11.5549i			0.712503i	−0.934390	+	0.356251i	$0.884055\pi$
$264$		0			0
$265$			23.6468i			1.45261i
$266$		0			0
$267$	−4.08382			−0.249926
$268$		0			0
$269$	12.2121			0.744586			0.372293	−	0.928115i	$-0.378572\pi$
$269$	12.2121			0.744586			0.372293	+	0.928115i	$0.378572\pi$
$270$		0			0
$271$	18.9454			1.15085			0.575427	−	0.817853i	$-0.304836\pi$
$271$	18.9454			1.15085			0.575427	+	0.817853i	$0.304836\pi$
$272$		0			0
$273$	14.1973	−	2.67841i	0.859257	−	0.162105i
$274$		0			0
$275$	0.136928			0.00825708
$276$		0			0
$277$			18.1876i			1.09278i	0.837529	+	0.546392i	$0.183999\pi$
$277$			18.1876i			1.09278i	−0.837529	+	0.546392i	$0.816001\pi$
$278$		0			0
$279$	−7.64576			−0.457740
$280$		0			0
$281$	−17.5294			−1.04572			−0.522858	−	0.852420i	$-0.675134\pi$
$281$	−17.5294			−1.04572			−0.522858	+	0.852420i	$0.675134\pi$
$282$		0			0
$283$		−	18.3358i		−	1.08995i	−0.838452	−	0.544975i	$-0.816539\pi$
$283$		−	18.3358i		−	1.08995i	0.838452	−	0.544975i	$-0.183461\pi$
$284$		0			0
$285$	−1.96195			−0.116216
$286$		0			0
$287$	22.5251	−	4.24950i	1.32961	−	0.250840i
$288$		0			0
$289$	−23.6258			−1.38975
$290$		0			0
$291$	−6.42855			−0.376848
$292$		0			0
$293$	14.0008			0.817938			0.408969	−	0.912548i	$-0.365888\pi$
$293$	14.0008			0.817938			0.408969	+	0.912548i	$0.365888\pi$
$294$		0			0
$295$			3.99184i			0.232414i
$296$		0			0
$297$		−	0.304570i		−	0.0176730i
$298$		0			0
$299$			0.609140i			0.0352275i
$300$		0			0
$301$	2.97689	+	15.7794i	0.171585	+	0.909511i
$302$		0			0
$303$		−	7.00330i		−	0.402329i
$304$		0			0
$305$	−6.42855			−0.368098
$306$		0			0
$307$		−	4.26041i		−	0.243154i	−0.992582	−	0.121577i	$-0.961205\pi$
$307$		−	4.26041i		−	0.243154i	0.992582	−	0.121577i	$-0.0387952\pi$
$308$		0			0
$309$		−	9.60771i		−	0.546563i
$310$		0			0
$311$	11.6664			0.661541			0.330771	−	0.943711i	$-0.392691\pi$
$311$	11.6664			0.661541			0.330771	+	0.943711i	$0.392691\pi$
$312$		0			0
$313$			4.58003i			0.258879i	0.991587	+	0.129439i	$0.0413178\pi$
$313$			4.58003i			0.258879i	−0.991587	+	0.129439i	$0.958682\pi$
$314$		0			0
$315$	1.14501	+	6.06927i	0.0645139	+	0.341964i
$316$		0			0
$317$		−	12.6315i		−	0.709454i	−0.934970	−	0.354727i	$-0.884574\pi$
$317$		−	12.6315i		−	0.709454i	0.934970	−	0.354727i	$-0.115426\pi$
$318$		0			0
$319$			1.66317i			0.0931197i
$320$		0			0
$321$		−	6.01455i		−	0.335700i
$322$		0			0
$323$	5.35682			0.298061
$324$		0			0
$325$	2.45502			0.136180
$326$		0			0
$327$	9.85961			0.545238
$328$		0			0
$329$	−4.02911	−	21.3568i	−0.222132	−	1.17744i
$330$		0			0
$331$	−25.6870			−1.41188			−0.705942	−	0.708269i	$-0.749476\pi$
$331$	−25.6870			−1.41188			−0.705942	+	0.708269i	$0.749476\pi$
$332$		0			0
$333$			6.44169i			0.353002i
$334$		0			0
$335$	19.5142			1.06617
$336$		0			0
$337$	24.7136			1.34624			0.673119	−	0.739535i	$-0.264954\pi$
$337$	24.7136			1.34624			0.673119	+	0.739535i	$0.264954\pi$
$338$		0			0
$339$		−	14.7477i		−	0.800984i
$340$		0			0
$341$	2.32867			0.126105
$342$		0			0
$343$	9.82822	+	15.6973i	0.530674	+	0.847576i
$344$		0			0
$345$	−0.260405			−0.0140197
$346$		0			0
$347$	−17.7999			−0.955550			−0.477775	−	0.878482i	$-0.658557\pi$
$347$	−17.7999			−0.955550			−0.477775	+	0.878482i	$0.658557\pi$
$348$		0			0
$349$	−8.70758			−0.466106			−0.233053	−	0.972464i	$-0.574872\pi$
$349$	−8.70758			−0.466106			−0.233053	+	0.972464i	$0.574872\pi$
$350$		0			0
$351$		−	5.46072i		−	0.291471i
$352$		0			0
$353$			7.33615i			0.390464i	0.980757	+	0.195232i	$0.0625459\pi$
$353$			7.33615i			0.390464i	−0.980757	+	0.195232i	$0.937454\pi$
$354$		0			0
$355$		−	4.84044i		−	0.256904i
$356$		0			0
$357$	−3.12628	−	16.5713i	−0.165460	−	0.877045i
$358$		0			0
$359$		−	14.3555i		−	0.757656i	−0.925467	−	0.378828i	$-0.876327\pi$
$359$		−	14.3555i		−	0.757656i	0.925467	−	0.378828i	$-0.123673\pi$
$360$		0			0
$361$	18.2937			0.962824
$362$		0			0
$363$		−	10.9072i		−	0.572481i
$364$		0			0
$365$		−	29.7586i		−	1.55764i
$366$		0			0
$367$	17.4100			0.908794			0.454397	−	0.890799i	$-0.349855\pi$
$367$	17.4100			0.908794			0.454397	+	0.890799i	$0.349855\pi$
$368$		0			0
$369$		−	8.66385i		−	0.451022i
$370$		0			0
$371$	26.3358	−	4.96842i	1.36729	−	0.257948i
$372$		0			0
$373$		−	1.44007i		−	0.0745641i	−0.999305	−	0.0372821i	$-0.988130\pi$
$373$		−	1.44007i		−	0.0745641i	0.999305	−	0.0372821i	$-0.0118700\pi$
$374$		0			0
$375$		−	10.6227i		−	0.548552i
$376$		0			0
$377$			29.8194i			1.53578i
$378$		0			0
$379$	24.0061			1.23311			0.616556	−	0.787311i	$-0.288528\pi$
$379$	24.0061			1.23311			0.616556	+	0.787311i	$0.288528\pi$
$380$		0			0
$381$	−2.49286			−0.127713
$382$		0			0
$383$	19.5821			1.00060			0.500300	−	0.865852i	$-0.333223\pi$
$383$	19.5821			1.00060			0.500300	+	0.865852i	$0.333223\pi$
$384$		0			0
$385$	−0.348735	−	1.84852i	−0.0177732	−	0.0942092i
$386$		0			0
$387$	6.06927			0.308518
$388$		0			0
$389$			31.6057i			1.60247i	0.598347	+	0.801237i	$0.295824\pi$
$389$			31.6057i			1.60247i	−0.598347	+	0.801237i	$0.704176\pi$
$390$		0			0
$391$	0.710999			0.0359568
$392$		0			0
$393$	4.60914			0.232500
$394$		0			0
$395$		−	18.4577i		−	0.928706i
$396$		0			0
$397$	6.29944			0.316160			0.158080	−	0.987426i	$-0.449470\pi$
$397$	6.29944			0.316160			0.158080	+	0.987426i	$0.449470\pi$
$398$		0			0
$399$	0.412224	+	2.18505i	0.0206370	+	0.109389i
$400$		0			0
$401$	3.10084			0.154849			0.0774244	−	0.996998i	$-0.475330\pi$
$401$	3.10084			0.154849			0.0774244	+	0.996998i	$0.475330\pi$
$402$		0			0
$403$	41.7513			2.07978
$404$		0			0
$405$	2.33443			0.115999
$406$		0			0
$407$		−	1.96195i		−	0.0972501i
$408$		0			0
$409$			3.06680i			0.151643i	0.997121	+	0.0758217i	$0.0241580\pi$
$409$			3.06680i			0.151643i	−0.997121	+	0.0758217i	$0.975842\pi$
$410$		0			0
$411$		−	0.899157i		−	0.0443521i
$412$		0			0
$413$	4.44577	−	0.838724i	0.218762	−	0.0412709i
$414$		0			0
$415$			27.1888i			1.33464i
$416$		0			0
$417$	−23.1762			−1.13494
$418$		0			0
$419$			15.5294i			0.758661i	0.925261	+	0.379330i	$0.123846\pi$
$419$			15.5294i			0.758661i	−0.925261	+	0.379330i	$0.876154\pi$
$420$		0			0
$421$			17.3631i			0.846227i	0.906077	+	0.423113i	$0.139063\pi$
$421$			17.3631i			0.846227i	−0.906077	+	0.423113i	$0.860937\pi$
$422$		0			0
$423$	−8.21451			−0.399403
$424$		0			0
$425$		−	2.86554i		−	0.138999i
$426$		0			0
$427$	1.35070	+	7.15956i	0.0653649	+	0.346475i
$428$		0			0
$429$			1.66317i			0.0802987i
$430$		0			0
$431$			5.07919i			0.244656i	0.992490	+	0.122328i	$0.0390360\pi$
$431$			5.07919i			0.244656i	−0.992490	+	0.122328i	$0.960964\pi$
$432$		0			0
$433$		−	14.7817i		−	0.710364i	−0.934797	−	0.355182i	$-0.884419\pi$
$433$		−	14.7817i		−	0.710364i	0.934797	−	0.355182i	$-0.115581\pi$
$434$		0			0
$435$	−12.7477			−0.611204
$436$		0			0
$437$	−0.0937505			−0.00448469
$438$		0			0
$439$	−17.6945			−0.844512			−0.422256	−	0.906477i	$-0.638762\pi$
$439$	−17.6945			−0.844512			−0.422256	+	0.906477i	$0.638762\pi$
$440$		0			0
$441$	6.51885	−	2.55042i	0.310421	−	0.121449i
$442$		0			0
$443$	−35.8340			−1.70252			−0.851262	−	0.524742i	$-0.824162\pi$
$443$	−35.8340			−1.70252			−0.851262	+	0.524742i	$0.824162\pi$
$444$		0			0
$445$			9.53341i			0.451927i
$446$		0			0
$447$	6.58394			0.311410
$448$		0			0
$449$	18.1094			0.854637			0.427318	−	0.904101i	$-0.359458\pi$
$449$	18.1094			0.854637			0.427318	+	0.904101i	$0.359458\pi$
$450$		0			0
$451$			2.63875i			0.124254i
$452$		0			0
$453$	20.5670			0.966321
$454$		0			0
$455$	−6.25256	−	33.1425i	−0.293125	−	1.55375i
$456$		0			0
$457$	−13.6177			−0.637010			−0.318505	−	0.947921i	$-0.603181\pi$
$457$	−13.6177			−0.637010			−0.318505	+	0.947921i	$0.603181\pi$
$458$		0			0
$459$	−6.37384			−0.297505
$460$		0			0
$461$	31.7263			1.47764			0.738821	−	0.673902i	$-0.235383\pi$
$461$	31.7263			1.47764			0.738821	+	0.673902i	$0.235383\pi$
$462$		0			0
$463$			24.0983i			1.11994i	0.828511	+	0.559972i	$0.189188\pi$
$463$			24.0983i			1.11994i	−0.828511	+	0.559972i	$0.810812\pi$
$464$		0			0
$465$			17.8485i			0.827705i
$466$		0			0
$467$			25.3568i			1.17337i	0.809814	+	0.586687i	$0.199568\pi$
$467$			25.3568i			1.17337i	−0.809814	+	0.586687i	$0.800432\pi$
$468$		0			0
$469$	−4.10012	−	21.7332i	−0.189326	−	1.00355i
$470$		0			0
$471$		−	12.0915i		−	0.557148i
$472$		0			0
$473$	−1.84852			−0.0849950
$474$		0			0
$475$			0.377843i			0.0173366i
$476$		0			0
$477$		−	10.1296i		−	0.463802i
$478$		0			0
$479$	−40.2196			−1.83768			−0.918839	−	0.394632i	$-0.870872\pi$
$479$	−40.2196			−1.83768			−0.918839	+	0.394632i	$0.870872\pi$
$480$		0			0
$481$		−	35.1762i		−	1.60390i
$482$		0			0
$483$	0.0547136	+	0.290017i	0.00248955	+	0.0131962i
$484$		0			0
$485$			15.0070i			0.681434i
$486$		0			0
$487$			34.9584i			1.58412i	0.610446	+	0.792058i	$0.290990\pi$
$487$			34.9584i			1.58412i	−0.610446	+	0.792058i	$0.709010\pi$
$488$		0			0
$489$		−	0.359284i		−	0.0162474i
$490$		0			0
$491$	−11.4808			−0.518121			−0.259061	−	0.965861i	$-0.583413\pi$
$491$	−11.4808			−0.518121			−0.259061	+	0.965861i	$0.583413\pi$
$492$		0			0
$493$	34.8057			1.56757
$494$		0			0
$495$	−0.710999			−0.0319570
$496$		0			0
$497$	−5.39086	+	1.01702i	−0.241813	+	0.0456197i
$498$		0			0
$499$	1.23326			0.0552084			0.0276042	−	0.999619i	$-0.491212\pi$
$499$	1.23326			0.0552084			0.0276042	+	0.999619i	$0.491212\pi$
$500$		0			0
$501$		−	7.91574i		−	0.353649i
$502$		0			0
$503$	−5.04712			−0.225040			−0.112520	−	0.993649i	$-0.535892\pi$
$503$	−5.04712			−0.225040			−0.112520	+	0.993649i	$0.535892\pi$
$504$		0			0
$505$	−16.3487			−0.727509
$506$		0			0
$507$			16.8194i			0.746976i
$508$		0			0
$509$	−35.6387			−1.57966			−0.789828	−	0.613328i	$-0.789830\pi$
$509$	−35.6387			−1.57966			−0.789828	+	0.613328i	$0.789830\pi$
$510$		0			0
$511$	−33.1425	+	6.25256i	−1.46614	+	0.276597i
$512$		0			0
$513$	0.840438			0.0371062
$514$		0			0
$515$	−22.4286			−0.988320
$516$		0			0
$517$	2.50190			0.110033
$518$		0			0
$519$			12.5109i			0.549167i
$520$		0			0
$521$			3.12151i			0.136756i	0.997659	+	0.0683780i	$0.0217824\pi$
$521$			3.12151i			0.136756i	−0.997659	+	0.0683780i	$0.978218\pi$
$522$		0			0
$523$			33.3779i			1.45951i	0.683706	+	0.729757i	$0.260367\pi$
$523$			33.3779i			1.45951i	−0.683706	+	0.729757i	$0.739633\pi$
$524$		0			0
$525$	1.16885	−	0.220512i	0.0510130	−	0.00962394i
$526$		0			0
$527$		−	48.7328i		−	2.12284i
$528$		0			0
$529$	22.9876			0.999459
$530$		0			0
$531$		−	1.70998i		−	0.0742070i
$532$		0			0
$533$			47.3108i			2.04926i
$534$		0			0
$535$	−14.0406			−0.607027
$536$		0			0
$537$		−	15.3423i		−	0.662067i
$538$		0			0
$539$	−1.98545	+	0.776782i	−0.0855192	+	0.0334584i
$540$		0			0
$541$			13.8175i			0.594060i	0.954868	+	0.297030i	$0.0959961\pi$
$541$			13.8175i			0.594060i	−0.954868	+	0.297030i	$0.904004\pi$
$542$		0			0
$543$		−	0.493074i		−	0.0211598i
$544$		0			0
$545$		−	23.0166i		−	0.985923i
$546$		0			0
$547$	12.4978			0.534368			0.267184	−	0.963646i	$-0.413907\pi$
$547$	12.4978			0.534368			0.267184	+	0.963646i	$0.413907\pi$
$548$		0			0
$549$	2.75380			0.117529
$550$		0			0
$551$	−4.58939			−0.195515
$552$		0			0
$553$	−20.5565	+	3.87813i	−0.874153	+	0.164915i
$554$		0			0
$555$	15.0377			0.638314
$556$		0			0
$557$		−	7.57008i		−	0.320755i	−0.987056	−	0.160377i	$-0.948729\pi$
$557$		−	7.57008i		−	0.320755i	0.987056	−	0.160377i	$-0.0512711\pi$
$558$		0			0
$559$	−33.1425			−1.40178
$560$		0			0
$561$	1.94128			0.0819610
$562$		0			0
$563$		−	15.0668i		−	0.634990i	−0.948260	−	0.317495i	$-0.897158\pi$
$563$		−	15.0668i		−	0.634990i	0.948260	−	0.317495i	$-0.102842\pi$
$564$		0			0
$565$	−34.4275			−1.44837
$566$		0			0
$567$	−0.490487	−	2.59989i	−0.0205985	−	0.109185i
$568$		0			0
$569$	36.7056			1.53878			0.769390	−	0.638780i	$-0.220560\pi$
$569$	36.7056			1.53878			0.769390	+	0.638780i	$0.220560\pi$
$570$		0			0
$571$	5.19529			0.217416			0.108708	−	0.994074i	$-0.465329\pi$
$571$	5.19529			0.217416			0.108708	+	0.994074i	$0.465329\pi$
$572$		0			0
$573$	−21.8108			−0.911159
$574$		0			0
$575$			0.0501503i			0.00209141i
$576$		0			0
$577$			43.9319i			1.82891i	0.404688	+	0.914455i	$0.367380\pi$
$577$			43.9319i			1.82891i	−0.404688	+	0.914455i	$0.632620\pi$
$578$		0			0
$579$			7.58811i			0.315351i
$580$		0			0
$581$	30.2805	−	5.71262i	1.25625	−	0.236999i
$582$		0			0
$583$			3.08517i			0.127775i
$584$		0			0
$585$	−12.7477			−0.527051
$586$		0			0
$587$		−	34.5380i		−	1.42553i	−0.701400	−	0.712767i	$-0.747441\pi$
$587$		−	34.5380i		−	1.42553i	0.701400	−	0.712767i	$-0.252559\pi$
$588$		0			0
$589$			6.42579i			0.264770i
$590$		0			0
$591$	−11.7928			−0.485089
$592$		0			0
$593$		−	29.3232i		−	1.20416i	−0.798436	−	0.602080i	$-0.794339\pi$
$593$		−	29.3232i		−	1.20416i	0.798436	−	0.602080i	$-0.205661\pi$
$594$		0			0
$595$	−38.6845	+	7.29810i	−1.58591	+	0.299193i
$596$		0			0
$597$		−	24.7858i		−	1.01441i
$598$		0			0
$599$		−	19.9925i		−	0.816870i	−0.912787	−	0.408435i	$-0.866075\pi$
$599$		−	19.9925i		−	0.816870i	0.912787	−	0.408435i	$-0.133925\pi$
$600$		0			0
$601$			21.1008i			0.860721i	0.902657	+	0.430361i	$0.141614\pi$
$601$			21.1008i			0.860721i	−0.902657	+	0.430361i	$0.858386\pi$
$602$		0			0
$603$	−8.35928			−0.340416
$604$		0			0
$605$	−25.4622			−1.03519
$606$		0			0
$607$	21.6184			0.877463			0.438732	−	0.898618i	$-0.355428\pi$
$607$	21.6184			0.877463			0.438732	+	0.898618i	$0.355428\pi$
$608$		0			0
$609$	2.67841	+	14.1973i	0.108535	+	0.575302i
$610$		0			0
$611$	44.8571			1.81472
$612$		0			0
$613$		−	27.8723i		−	1.12575i	−0.826541	−	0.562876i	$-0.809695\pi$
$613$		−	27.8723i		−	1.12575i	0.826541	−	0.562876i	$-0.190305\pi$
$614$		0			0
$615$	−20.2252			−0.815558
$616$		0			0
$617$	−9.81447			−0.395116			−0.197558	−	0.980291i	$-0.563301\pi$
$617$	−9.81447			−0.395116			−0.197558	+	0.980291i	$0.563301\pi$
$618$		0			0
$619$		−	22.8992i		−	0.920395i	−0.887817	−	0.460197i	$-0.847779\pi$
$619$		−	22.8992i		−	0.920395i	0.887817	−	0.460197i	$-0.152221\pi$
$620$		0			0
$621$	0.111550			0.00447633
$622$		0			0
$623$	10.6175	−	2.00306i	0.425380	−	0.0802509i
$624$		0			0
$625$	−27.0458			−1.08183
$626$		0			0
$627$	−0.255972			−0.0102226
$628$		0			0
$629$	−41.0583			−1.63710
$630$		0			0
$631$		−	15.4442i		−	0.614823i	−0.951577	−	0.307412i	$-0.900537\pi$
$631$		−	15.4442i		−	0.614823i	0.951577	−	0.307412i	$-0.0994628\pi$
$632$		0			0
$633$		−	14.0693i		−	0.559203i
$634$		0			0
$635$			5.81941i			0.230936i
$636$		0			0
$637$	−35.5976	+	13.9271i	−1.41043	+	0.551813i
$638$		0			0
$639$			2.07350i			0.0820262i
$640$		0			0
$641$	−6.58003			−0.259896			−0.129948	−	0.991521i	$-0.541481\pi$
$641$	−6.58003			−0.259896			−0.129948	+	0.991521i	$0.541481\pi$
$642$		0			0
$643$		−	18.3358i		−	0.723093i	−0.932354	−	0.361546i	$-0.882249\pi$
$643$		−	18.3358i		−	0.723093i	0.932354	−	0.361546i	$-0.117751\pi$
$644$		0			0
$645$		−	14.1683i		−	0.557876i
$646$		0			0
$647$	10.4752			0.411824			0.205912	−	0.978571i	$-0.433984\pi$
$647$	10.4752			0.411824			0.205912	+	0.978571i	$0.433984\pi$
$648$		0			0
$649$			0.520810i			0.0204436i
$650$		0			0
$651$	19.8781	−	3.75014i	0.779085	−	0.146980i
$652$		0			0
$653$		−	37.9262i		−	1.48417i	−0.670307	−	0.742084i	$-0.733838\pi$
$653$		−	37.9262i		−	1.48417i	0.670307	−	0.742084i	$-0.266162\pi$
$654$		0			0
$655$		−	10.7597i		−	0.420417i
$656$		0			0
$657$			12.7477i			0.497334i
$658$		0			0
$659$	−11.2037			−0.436435			−0.218218	−	0.975900i	$-0.570024\pi$
$659$	−11.2037			−0.436435			−0.218218	+	0.975900i	$0.570024\pi$
$660$		0			0
$661$	−4.78369			−0.186064			−0.0930320	−	0.995663i	$-0.529656\pi$
$661$	−4.78369			−0.186064			−0.0930320	+	0.995663i	$0.529656\pi$
$662$		0			0
$663$	34.8057			1.35174
$664$		0			0
$665$	5.10084	−	0.962308i	0.197802	−	0.0373167i
$666$		0			0
$667$	−0.609140			−0.0235860
$668$		0			0
$669$			15.4480i			0.597256i
$670$		0			0
$671$	−0.838724			−0.0323786
$672$		0			0
$673$	−11.4707			−0.442162			−0.221081	−	0.975255i	$-0.570959\pi$
$673$	−11.4707			−0.442162			−0.221081	+	0.975255i	$0.570959\pi$
$674$		0			0
$675$		−	0.449578i		−	0.0173043i
$676$		0			0
$677$	−39.0226			−1.49976			−0.749881	−	0.661573i	$-0.769889\pi$
$677$	−39.0226			−1.49976			−0.749881	+	0.661573i	$0.769889\pi$
$678$		0			0
$679$	16.7135	−	3.15312i	0.641406	−	0.121006i
$680$		0			0
$681$	11.0377			0.422965
$682$		0			0
$683$	6.36772			0.243654			0.121827	−	0.992551i	$-0.461125\pi$
$683$	6.36772			0.243654			0.121827	+	0.992551i	$0.461125\pi$
$684$		0			0
$685$	−2.09902			−0.0801995
$686$		0			0
$687$			7.42266i			0.283192i
$688$		0			0
$689$			55.3148i			2.10732i
$690$		0			0
$691$			10.3191i			0.392558i	0.980548	+	0.196279i	$0.0628858\pi$
$691$			10.3191i			0.392558i	−0.980548	+	0.196279i	$0.937114\pi$
$692$		0			0
$693$	0.149388	+	0.791849i	0.00567477	+	0.0300798i
$694$		0			0
$695$			54.1034i			2.05226i
$696$		0			0
$697$	55.2220			2.09168
$698$		0			0
$699$			13.0668i			0.494232i
$700$		0			0
$701$		−	38.3982i		−	1.45028i	−0.688601	−	0.725141i	$-0.741775\pi$
$701$		−	38.3982i		−	1.45028i	0.688601	−	0.725141i	$-0.258225\pi$
$702$		0			0
$703$	5.41384			0.204187
$704$		0			0
$705$			19.1762i			0.722218i
$706$		0			0
$707$	3.43502	+	18.2078i	0.129187	+	0.684775i
$708$		0			0
$709$		−	8.88381i		−	0.333638i	−0.985988	−	0.166819i	$-0.946650\pi$
$709$		−	8.88381i		−	0.333638i	0.985988	−	0.166819i	$-0.0533496\pi$
$710$		0			0
$711$			7.90670i			0.296524i
$712$		0			0
$713$			0.852881i			0.0319407i
$714$		0			0
$715$	3.88256			0.145200
$716$		0			0
$717$	8.54916			0.319274
$718$		0			0
$719$	−40.2196			−1.49994			−0.749968	−	0.661474i	$-0.769931\pi$
$719$	−40.2196			−1.49994			−0.749968	+	0.661474i	$0.769931\pi$
$720$		0			0
$721$	4.71245	+	24.9790i	0.175501	+	0.930266i
$722$		0			0
$723$	7.64683			0.284389
$724$		0			0
$725$			2.45502i			0.0911772i
$726$		0			0
$727$	−2.89742			−0.107459			−0.0537297	−	0.998556i	$-0.517111\pi$
$727$	−2.89742			−0.107459			−0.0537297	+	0.998556i	$0.517111\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$			38.6845i			1.43080i
$732$		0			0
$733$	−6.37891			−0.235611			−0.117805	−	0.993037i	$-0.537586\pi$
$733$	−6.37891			−0.235611			−0.117805	+	0.993037i	$0.537586\pi$
$734$		0			0
$735$	−5.95379	−	15.2178i	−0.219609	−	0.561317i
$736$		0			0
$737$	2.54599			0.0937827
$738$		0			0
$739$	7.14100			0.262686			0.131343	−	0.991337i	$-0.458071\pi$
$739$	7.14100			0.262686			0.131343	+	0.991337i	$0.458071\pi$
$740$		0			0
$741$	−4.58939			−0.168596
$742$		0			0
$743$		−	22.2570i		−	0.816530i	−0.912864	−	0.408265i	$-0.866134\pi$
$743$		−	22.2570i		−	0.816530i	0.912864	−	0.408265i	$-0.133866\pi$
$744$		0			0
$745$		−	15.3698i		−	0.563105i
$746$		0			0
$747$		−	11.6468i		−	0.426135i
$748$		0			0
$749$	2.95006	+	15.6372i	0.107793	+	0.571370i
$750$		0			0
$751$		−	21.7697i		−	0.794387i	−0.917735	−	0.397193i	$-0.869984\pi$
$751$		−	21.7697i		−	0.794387i	0.917735	−	0.397193i	$-0.130016\pi$
$752$		0			0
$753$	−12.7186			−0.463490
$754$		0			0
$755$		−	48.0122i		−	1.74734i
$756$		0			0
$757$			4.51372i			0.164054i	0.996630	+	0.0820269i	$0.0261393\pi$
$757$			4.51372i			0.164054i	−0.996630	+	0.0820269i	$0.973861\pi$
$758$		0			0
$759$	−0.0339747			−0.00123320
$760$		0			0
$761$		−	3.12151i		−	0.113155i	−0.998398	−	0.0565774i	$-0.981981\pi$
$761$		−	3.12151i		−	0.113155i	0.998398	−	0.0565774i	$-0.0180188\pi$
$762$		0			0
$763$	−25.6339	+	4.83601i	−0.928009	+	0.175075i
$764$		0			0
$765$			14.8793i			0.537962i
$766$		0			0
$767$			9.33773i			0.337166i
$768$		0			0
$769$		−	52.3945i		−	1.88939i	−0.327944	−	0.944697i	$-0.606356\pi$
$769$		−	52.3945i		−	1.88939i	0.327944	−	0.944697i	$-0.393644\pi$
$770$		0			0
$771$	−7.44557			−0.268146
$772$		0			0
$773$	14.5408			0.522996			0.261498	−	0.965204i	$-0.415784\pi$
$773$	14.5408			0.522996			0.261498	+	0.965204i	$0.415784\pi$
$774$		0			0
$775$	3.43737			0.123474
$776$		0			0
$777$	−3.15956	−	16.7477i	−0.113349	−	0.600819i
$778$		0			0
$779$	−7.28143			−0.260884
$780$		0			0
$781$		−	0.631525i		−	0.0225977i
$782$		0			0
$783$	5.46072			0.195150
$784$		0			0
$785$	−28.2269			−1.00746
$786$		0			0
$787$			18.0754i			0.644318i	0.946686	+	0.322159i	$0.104409\pi$
$787$			18.0754i			0.644318i	−0.946686	+	0.322159i	$0.895591\pi$
$788$		0			0
$789$	−11.5549			−0.411364
$790$		0			0
$791$	7.23354	+	38.3423i	0.257195	+	1.36330i
$792$		0			0
$793$	−15.0377			−0.534004
$794$		0			0
$795$	−23.6468			−0.838667
$796$		0			0
$797$	8.28822			0.293584			0.146792	−	0.989167i	$-0.453105\pi$
$797$	8.28822			0.293584			0.146792	+	0.989167i	$0.453105\pi$
$798$		0			0
$799$		−	52.3580i		−	1.85229i
$800$		0			0
$801$		−	4.08382i		−	0.144295i
$802$		0			0
$803$		−	3.88256i		−	0.137013i
$804$		0			0
$805$	0.677024	−	0.127725i	0.0238620	−	0.00450172i
$806$		0			0
$807$			12.2121i			0.429887i
$808$		0			0
$809$	−52.3022			−1.83885			−0.919425	−	0.393266i	$-0.871345\pi$
$809$	−52.3022			−1.83885			−0.919425	+	0.393266i	$0.871345\pi$
$810$		0			0
$811$		−	16.3945i		−	0.575689i	−0.957677	−	0.287845i	$-0.907061\pi$
$811$		−	16.3945i		−	0.575689i	0.957677	−	0.287845i	$-0.0929386\pi$
$812$		0			0
$813$			18.9454i			0.664446i
$814$		0			0
$815$	−0.838724			−0.0293792
$816$		0			0
$817$		−	5.10084i		−	0.178456i
$818$		0			0
$819$	2.67841	+	14.1973i	0.0935911	+	0.496092i
$820$		0			0
$821$			14.9459i			0.521614i	0.965391	+	0.260807i	$0.0839887\pi$
$821$			14.9459i			0.521614i	−0.965391	+	0.260807i	$0.916011\pi$
$822$		0			0
$823$		−	21.3876i		−	0.745526i	−0.927927	−	0.372763i	$-0.878410\pi$
$823$		−	21.3876i		−	0.745526i	0.927927	−	0.372763i	$-0.121590\pi$
$824$		0			0
$825$			0.136928i			0.00476723i
$826$		0			0
$827$	16.5605			0.575866			0.287933	−	0.957650i	$-0.407032\pi$
$827$	16.5605			0.575866			0.287933	+	0.957650i	$0.407032\pi$
$828$		0			0
$829$	−47.4372			−1.64756			−0.823781	−	0.566908i	$-0.808139\pi$
$829$	−47.4372			−1.64756			−0.823781	+	0.566908i	$0.808139\pi$
$830$		0			0
$831$	−18.1876			−0.630920
$832$		0			0
$833$	16.2560	+	41.5501i	0.563236	+	1.43962i
$834$		0			0
$835$	−18.4788			−0.639484
$836$		0			0
$837$		−	7.64576i		−	0.264276i
$838$		0			0
$839$	12.3796			0.427390			0.213695	−	0.976900i	$-0.431450\pi$
$839$	12.3796			0.427390			0.213695	+	0.976900i	$0.431450\pi$
$840$		0			0
$841$	−0.819411			−0.0282555
$842$		0			0
$843$		−	17.5294i		−	0.603744i
$844$		0			0
$845$	39.2638			1.35072
$846$		0			0
$847$	5.34985	+	28.3576i	0.183823	+	0.974379i
$848$		0			0
$849$	18.3358			0.629283
$850$		0			0
$851$	0.718568			0.0246322
$852$		0			0
$853$	27.8292			0.952855			0.476428	−	0.879214i	$-0.341931\pi$
$853$	27.8292			0.952855			0.476428	+	0.879214i	$0.341931\pi$
$854$		0			0
$855$		−	1.96195i		−	0.0670972i
$856$		0			0
$857$		−	19.8278i		−	0.677306i	−0.940911	−	0.338653i	$-0.890029\pi$
$857$		−	19.8278i		−	0.677306i	0.940911	−	0.338653i	$-0.109971\pi$
$858$		0			0
$859$		−	35.5541i		−	1.21309i	−0.795049	−	0.606545i	$-0.792555\pi$
$859$		−	35.5541i		−	1.21309i	0.795049	−	0.606545i	$-0.207445\pi$
$860$		0			0
$861$	4.24950	+	22.5251i	0.144823	+	0.767652i
$862$		0			0
$863$			32.6385i			1.11103i	0.831508	+	0.555513i	$0.187478\pi$
$863$			32.6385i			1.11103i	−0.831508	+	0.555513i	$0.812522\pi$
$864$		0			0
$865$	29.2058			0.993028
$866$		0			0
$867$		−	23.6258i		−	0.802374i
$868$		0			0
$869$		−	2.40814i		−	0.0816907i
$870$		0			0
$871$	45.6477			1.54671
$872$		0			0
$873$		−	6.42855i		−	0.217573i
$874$		0			0
$875$	5.21027	+	27.6177i	0.176139	+	0.933649i
$876$		0			0
$877$			31.1987i			1.05350i	0.850019	+	0.526752i	$0.176590\pi$
$877$			31.1987i			1.05350i	−0.850019	+	0.526752i	$0.823410\pi$
$878$		0			0
$879$			14.0008i			0.472237i
$880$		0			0
$881$		−	51.2309i		−	1.72601i	−0.505192	−	0.863007i	$-0.668578\pi$
$881$		−	51.2309i		−	1.72601i	0.505192	−	0.863007i	$-0.331422\pi$
$882$		0			0
$883$	16.9684			0.571033			0.285516	−	0.958374i	$-0.407835\pi$
$883$	16.9684			0.571033			0.285516	+	0.958374i	$0.407835\pi$
$884$		0			0
$885$	−3.99184			−0.134184
$886$		0			0
$887$	18.7434			0.629342			0.314671	−	0.949201i	$-0.398106\pi$
$887$	18.7434			0.629342			0.314671	+	0.949201i	$0.398106\pi$
$888$		0			0
$889$	6.48115	−	1.22271i	0.217371	−	0.0410085i
$890$		0			0
$891$	0.304570			0.0102035
$892$		0			0
$893$			6.90379i			0.231026i
$894$		0			0
$895$	−35.8155			−1.19718
$896$		0			0
$897$	−0.609140			−0.0203386
$898$		0			0
$899$			41.7513i			1.39248i
$900$		0			0
$901$	64.5643			2.15095
$902$		0			0
$903$	−15.7794	+	2.97689i	−0.525106	+	0.0990648i
$904$		0			0
$905$	−1.15105			−0.0382621
$906$		0			0
$907$	−1.15395			−0.0383163			−0.0191581	−	0.999816i	$-0.506099\pi$
$907$	−1.15395			−0.0383163			−0.0191581	+	0.999816i	$0.506099\pi$
$908$		0			0
$909$	7.00330			0.232285
$910$		0			0
$911$			44.7588i			1.48292i	0.670994	+	0.741462i	$0.265867\pi$
$911$			44.7588i			1.48292i	−0.670994	+	0.741462i	$0.734133\pi$
$912$		0			0
$913$			3.54728i			0.117398i
$914$		0			0
$915$		−	6.42855i		−	0.212521i
$916$		0			0
$917$	−11.9833	+	2.26072i	−0.395722	+	0.0746556i
$918$		0			0
$919$		−	40.3722i		−	1.33176i	−0.746060	−	0.665878i	$-0.768057\pi$
$919$		−	40.3722i		−	1.33176i	0.746060	−	0.665878i	$-0.231943\pi$
$920$		0			0
$921$	4.26041			0.140385
$922$		0			0
$923$		−	11.3228i		−	0.372694i
$924$		0			0
$925$		−	2.89604i		−	0.0952214i
$926$		0			0
$927$	9.60771			0.315559
$928$		0			0
$929$			13.4697i			0.441928i	0.975282	+	0.220964i	$0.0709203\pi$
$929$			13.4697i			0.441928i	−0.975282	+	0.220964i	$0.929080\pi$
$930$		0			0
$931$	−2.14347	−	5.47869i	−0.0702494	−	0.179557i
$932$		0			0
$933$			11.6664i			0.381941i
$934$		0			0
$935$		−	4.53179i		−	0.148205i
$936$		0			0
$937$			11.6055i			0.379135i	0.981868	+	0.189567i	$0.0607086\pi$
$937$			11.6055i			0.379135i	−0.981868	+	0.189567i	$0.939291\pi$
$938$		0			0
$939$	−4.58003			−0.149464
$940$		0			0
$941$	29.2387			0.953154			0.476577	−	0.879133i	$-0.341877\pi$
$941$	29.2387			0.953154			0.476577	+	0.879133i	$0.341877\pi$
$942$		0			0
$943$	−0.966449			−0.0314719
$944$		0			0
$945$	−6.06927	+	1.14501i	−0.197433	+	0.0372471i
$946$		0			0
$947$	24.0096			0.780208			0.390104	−	0.920771i	$-0.372439\pi$
$947$	24.0096			0.780208			0.390104	+	0.920771i	$0.372439\pi$
$948$		0			0
$949$		−	69.6114i		−	2.25968i
$950$		0			0
$951$	12.6315			0.409604
$952$		0			0
$953$	38.6716			1.25270			0.626348	−	0.779544i	$-0.284549\pi$
$953$	38.6716			1.25270			0.626348	+	0.779544i	$0.284549\pi$
$954$		0			0
$955$			50.9158i			1.64760i
$956$		0			0
$957$	−1.66317			−0.0537627
$958$		0			0
$959$	0.441024	+	2.33771i	0.0142414	+	0.0754885i
$960$		0			0
$961$	27.4577			0.885731
$962$		0			0
$963$	6.01455			0.193816
$964$		0			0
$965$	17.7139			0.570232
$966$		0			0
$967$			12.2768i			0.394795i	0.980324	+	0.197397i	$0.0632489\pi$
$967$			12.2768i			0.394795i	−0.980324	+	0.197397i	$0.936751\pi$
$968$		0			0
$969$			5.35682i			0.172086i
$970$		0			0
$971$		−	20.5330i		−	0.658937i	−0.944167	−	0.329468i	$-0.893131\pi$
$971$		−	20.5330i		−	0.658937i	0.944167	−	0.329468i	$-0.106869\pi$
$972$		0			0
$973$	60.2556	−	11.3676i	1.93171	−	0.364429i
$974$		0			0
$975$			2.45502i			0.0786236i
$976$		0			0
$977$	−10.7477			−0.343849			−0.171924	−	0.985110i	$-0.554998\pi$
$977$	−10.7477			−0.343849			−0.171924	+	0.985110i	$0.554998\pi$
$978$		0			0
$979$			1.24381i			0.0397523i
$980$		0			0
$981$			9.85961i			0.314793i
$982$		0			0
$983$	26.5912			0.848128			0.424064	−	0.905632i	$-0.360603\pi$
$983$	26.5912			0.848128			0.424064	+	0.905632i	$0.360603\pi$
$984$		0			0
$985$			27.5294i			0.877159i
$986$		0			0
$987$	21.3568	−	4.02911i	0.679795	−	0.128248i
$988$		0			0
$989$		−	0.677024i		−	0.0215281i
$990$		0			0
$991$		−	30.5203i		−	0.969510i	−0.874650	−	0.484755i	$-0.838909\pi$
$991$		−	30.5203i		−	0.969510i	0.874650	−	0.484755i	$-0.161091\pi$
$992$		0			0
$993$		−	25.6870i		−	0.815152i
$994$		0			0
$995$	−57.8607			−1.83431
$996$		0			0
$997$	23.6105			0.747752			0.373876	−	0.927479i	$-0.378029\pi$
$997$	23.6105			0.747752			0.373876	+	0.927479i	$0.378029\pi$
$998$		0			0
$999$	−6.44169			−0.203806

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.p.a.559.15		16
3.2	odd	2		2016.2.p.g.559.3		16
4.3	odd	2		168.2.p.a.139.1	✓	16
7.6	odd	2	inner	672.2.p.a.559.2		16
8.3	odd	2	inner	672.2.p.a.559.10		16
8.5	even	2		168.2.p.a.139.3	yes	16
12.11	even	2		504.2.p.g.307.15		16
21.20	even	2		2016.2.p.g.559.13		16
24.5	odd	2		504.2.p.g.307.14		16
24.11	even	2		2016.2.p.g.559.14		16
28.27	even	2		168.2.p.a.139.2	yes	16
56.13	odd	2		168.2.p.a.139.4	yes	16
56.27	even	2	inner	672.2.p.a.559.7		16
84.83	odd	2		504.2.p.g.307.16		16
168.83	odd	2		2016.2.p.g.559.4		16
168.125	even	2		504.2.p.g.307.13		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.p.a.139.1	✓	16	4.3	odd	2
168.2.p.a.139.2	yes	16	28.27	even	2
168.2.p.a.139.3	yes	16	8.5	even	2
168.2.p.a.139.4	yes	16	56.13	odd	2
504.2.p.g.307.13		16	168.125	even	2
504.2.p.g.307.14		16	24.5	odd	2
504.2.p.g.307.15		16	12.11	even	2
504.2.p.g.307.16		16	84.83	odd	2
672.2.p.a.559.2		16	7.6	odd	2	inner
672.2.p.a.559.7		16	56.27	even	2	inner
672.2.p.a.559.10		16	8.3	odd	2	inner
672.2.p.a.559.15		16	1.1	even	1	trivial
2016.2.p.g.559.3		16	3.2	odd	2
2016.2.p.g.559.4		16	168.83	odd	2
2016.2.p.g.559.13		16	21.20	even	2
2016.2.p.g.559.14		16	24.11	even	2

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:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	672.p (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +2.33443 q^{5} +(-0.490487 - 2.59989i) q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +2.33443 q^{5} +(-0.490487 - 2.59989i) q^{7} -1.00000 q^{9} +0.304570 q^{11} +5.46072 q^{13} +2.33443i q^{15} -6.37384i q^{17} +0.840438i q^{19} +(2.59989 - 0.490487i) q^{21} +0.111550i q^{23} +0.449578 q^{25} -1.00000i q^{27} +5.46072i q^{29} +7.64576 q^{31} +0.304570i q^{33} +(-1.14501 - 6.06927i) q^{35} -6.44169i q^{37} +5.46072i q^{39} +8.66385i q^{41} -6.06927 q^{43} -2.33443 q^{45} +8.21451 q^{47} +(-6.51885 + 2.55042i) q^{49} +6.37384 q^{51} +10.1296i q^{53} +0.710999 q^{55} -0.840438 q^{57} +1.70998i q^{59} -2.75380 q^{61} +(0.490487 + 2.59989i) q^{63} +12.7477 q^{65} +8.35928 q^{67} -0.111550 q^{69} -2.07350i q^{71} -12.7477i q^{73} +0.449578i q^{75} +(-0.149388 - 0.791849i) q^{77} -7.90670i q^{79} +1.00000 q^{81} +11.6468i q^{83} -14.8793i q^{85} -5.46072 q^{87} +4.08382i q^{89} +(-2.67841 - 14.1973i) q^{91} +7.64576i q^{93} +1.96195i q^{95} +6.42855i q^{97} -0.304570 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{9}+O(q^{10}) \) 16 * q - 16 * q^9	\( 16 q - 16 q^{9} + 8 q^{11} + 16 q^{25} - 24 q^{35} + 8 q^{43} - 8 q^{49} - 16 q^{57} + 40 q^{67} + 16 q^{81} + 56 q^{91} - 8 q^{99}+O(q^{100}) \) 16 * q - 16 * q^9 + 8 * q^11 + 16 * q^25 - 24 * q^35 + 8 * q^43 - 8 * q^49 - 16 * q^57 + 40 * q^67 + 16 * q^81 + 56 * q^91 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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