LMFDB - Embedded newform 672.2.p.a.559.5

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			559.5
Root			$-1.20933 - 0.733159i$ of defining polynomial
Character	$\chi$	$=$	672.559
Dual form			672.2.p.a.559.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +1.12786 q^{5} +(-2.11337 + 1.59175i) q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +1.12786 q^{5} +(-2.11337 + 1.59175i) q^{7} -1.00000 q^{9} -5.11151 q^{11} -5.88054 q^{13} -1.12786i q^{15} -3.31623i q^{17} -7.49510i q^{19} +(1.59175 + 2.11337i) q^{21} -1.73845i q^{23} -3.72792 q^{25} +1.00000i q^{27} +5.88054i q^{29} +6.04982 q^{31} +5.11151i q^{33} +(-2.38359 + 1.79528i) q^{35} -1.65381i q^{37} +5.88054i q^{39} -1.45096i q^{41} -1.79528 q^{43} -1.12786 q^{45} -5.56335 q^{47} +(1.93264 - 6.72792i) q^{49} -3.31623 q^{51} +3.62481i q^{53} -5.76510 q^{55} -7.49510 q^{57} +0.767184i q^{59} -0.317194 q^{61} +(2.11337 - 1.59175i) q^{63} -6.63246 q^{65} +6.56247 q^{67} -1.73845 q^{69} +10.1919i q^{71} -6.63246i q^{73} +3.72792i q^{75} +(10.8025 - 8.13627i) q^{77} -3.01423i q^{79} +1.00000 q^{81} +16.0883i q^{83} -3.74026i q^{85} +5.88054 q^{87} +8.08341i q^{89} +(12.4277 - 9.36038i) q^{91} -6.04982i q^{93} -8.45347i q^{95} -0.357752i q^{97} +5.11151 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	1.12786			0.504397			0.252198	−	0.967676i	$-0.418846\pi$
$5$	1.12786			0.504397			0.252198	+	0.967676i	$0.418846\pi$
$6$		0			0
$7$	−2.11337	+	1.59175i	−0.798777	+	0.601627i
$7$	−2.11337	+	1.59175i	−0.798777	+	0.601627i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−5.11151			−1.54118			−0.770590	−	0.637332i	$-0.780038\pi$
$11$	−5.11151			−1.54118			−0.770590	+	0.637332i	$0.780038\pi$
$12$		0			0
$13$	−5.88054			−1.63097			−0.815484	−	0.578779i	$-0.803529\pi$
$13$	−5.88054			−1.63097			−0.815484	+	0.578779i	$0.803529\pi$
$14$		0			0
$15$		−	1.12786i		−	0.291213i
$16$		0			0
$17$		−	3.31623i		−	0.804304i	−0.915573	−	0.402152i	$-0.868262\pi$
$17$		−	3.31623i		−	0.804304i	0.915573	−	0.402152i	$-0.131738\pi$
$18$		0			0
$19$		−	7.49510i		−	1.71949i	−0.510719	−	0.859747i	$-0.670621\pi$
$19$		−	7.49510i		−	1.71949i	0.510719	−	0.859747i	$-0.329379\pi$
$20$		0			0
$21$	1.59175	+	2.11337i	0.347349	+	0.461174i
$22$		0			0
$23$		−	1.73845i		−	0.362492i	−0.983438	−	0.181246i	$-0.941987\pi$
$23$		−	1.73845i		−	0.362492i	0.983438	−	0.181246i	$-0.0580130\pi$
$24$		0			0
$25$	−3.72792			−0.745584
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$			5.88054i			1.09199i	0.837789	+	0.545995i	$0.183848\pi$
$29$			5.88054i			1.09199i	−0.837789	+	0.545995i	$0.816152\pi$
$30$		0			0
$31$	6.04982			1.08658			0.543290	−	0.839545i	$-0.317178\pi$
$31$	6.04982			1.08658			0.543290	+	0.839545i	$0.317178\pi$
$32$		0			0
$33$			5.11151i			0.889800i
$34$		0			0
$35$	−2.38359	+	1.79528i	−0.402901	+	0.303458i
$36$		0			0
$37$		−	1.65381i		−	0.271885i	−0.990717	−	0.135942i	$-0.956594\pi$
$37$		−	1.65381i		−	0.271885i	0.990717	−	0.135942i	$-0.0434062\pi$
$38$		0			0
$39$			5.88054i			0.941640i
$40$		0			0
$41$		−	1.45096i		−	0.226601i	−0.993561	−	0.113301i	$-0.963858\pi$
$41$		−	1.45096i		−	0.226601i	0.993561	−	0.113301i	$-0.0361423\pi$
$42$		0			0
$43$	−1.79528			−0.273778			−0.136889	−	0.990586i	$-0.543710\pi$
$43$	−1.79528			−0.273778			−0.136889	+	0.990586i	$0.543710\pi$
$44$		0			0
$45$	−1.12786			−0.168132
$46$		0			0
$47$	−5.56335			−0.811498			−0.405749	−	0.913985i	$-0.632989\pi$
$47$	−5.56335			−0.811498			−0.405749	+	0.913985i	$0.632989\pi$
$48$		0			0
$49$	1.93264	−	6.72792i	0.276091	−	0.961132i
$50$		0			0
$51$	−3.31623			−0.464365
$52$		0			0
$53$			3.62481i			0.497906i	0.968516	+	0.248953i	$0.0800865\pi$
$53$			3.62481i			0.497906i	−0.968516	+	0.248953i	$0.919913\pi$
$54$		0			0
$55$	−5.76510			−0.777365
$56$		0			0
$57$	−7.49510			−0.992751
$58$		0			0
$59$			0.767184i			0.0998789i	0.998752	+	0.0499394i	$0.0159028\pi$
$59$			0.767184i			0.0998789i	−0.998752	+	0.0499394i	$0.984097\pi$
$60$		0			0
$61$	−0.317194			−0.0406125			−0.0203063	−	0.999794i	$-0.506464\pi$
$61$	−0.317194			−0.0406125			−0.0203063	+	0.999794i	$0.506464\pi$
$62$		0			0
$63$	2.11337	−	1.59175i	0.266259	−	0.200542i
$64$		0			0
$65$	−6.63246			−0.822655
$66$		0			0
$67$	6.56247			0.801733			0.400867	−	0.916136i	$-0.368709\pi$
$67$	6.56247			0.801733			0.400867	+	0.916136i	$0.368709\pi$
$68$		0			0
$69$	−1.73845			−0.209285
$70$		0			0
$71$			10.1919i			1.20956i	0.796393	+	0.604779i	$0.206739\pi$
$71$			10.1919i			1.20956i	−0.796393	+	0.604779i	$0.793261\pi$
$72$		0			0
$73$		−	6.63246i		−	0.776270i	−0.921602	−	0.388135i	$-0.873119\pi$
$73$		−	6.63246i		−	0.776270i	0.921602	−	0.388135i	$-0.126881\pi$
$74$		0			0
$75$			3.72792i			0.430463i
$76$		0			0
$77$	10.8025	−	8.13627i	1.23106	−	0.927214i
$78$		0			0
$79$		−	3.01423i		−	0.339127i	−0.985519	−	0.169564i	$-0.945764\pi$
$79$		−	3.01423i		−	0.339127i	0.985519	−	0.169564i	$-0.0542358\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			16.0883i			1.76592i	0.469448	+	0.882960i	$0.344453\pi$
$83$			16.0883i			1.76592i	−0.469448	+	0.882960i	$0.655547\pi$
$84$		0			0
$85$		−	3.74026i		−	0.405688i
$86$		0			0
$87$	5.88054			0.630460
$88$		0			0
$89$			8.08341i			0.856840i	0.903580	+	0.428420i	$0.140930\pi$
$89$			8.08341i			0.856840i	−0.903580	+	0.428420i	$0.859070\pi$
$90$		0			0
$91$	12.4277	−	9.36038i	1.30278	−	0.981234i
$92$		0			0
$93$		−	6.04982i		−	0.627337i
$94$		0			0
$95$		−	8.45347i		−	0.867307i
$96$		0			0
$97$		−	0.357752i		−	0.0363242i	−0.999835	−	0.0181621i	$-0.994219\pi$
$97$		−	0.357752i		−	0.0363242i	0.999835	−	0.0181621i	$-0.00578150\pi$
$98$		0			0
$99$	5.11151			0.513726
$100$		0			0
$101$	−3.38359			−0.336680			−0.168340	−	0.985729i	$-0.553841\pi$
$101$	−3.38359			−0.336680			−0.168340	+	0.985729i	$0.553841\pi$
$102$		0			0
$103$	−14.5033			−1.42905			−0.714526	−	0.699609i	$-0.753357\pi$
$103$	−14.5033			−1.42905			−0.714526	+	0.699609i	$0.753357\pi$
$104$		0			0
$105$	1.79528	+	2.38359i	0.175202	+	0.232615i
$106$		0			0
$107$	1.87870			0.181621			0.0908103	−	0.995868i	$-0.471054\pi$
$107$	1.87870			0.181621			0.0908103	+	0.995868i	$0.471054\pi$
$108$		0			0
$109$		−	13.6166i		−	1.30424i	−0.758117	−	0.652119i	$-0.773880\pi$
$109$		−	13.6166i		−	1.30424i	0.758117	−	0.652119i	$-0.226120\pi$
$110$		0			0
$111$	−1.65381			−0.156973
$112$		0			0
$113$	4.63246			0.435785			0.217892	−	0.975973i	$-0.430082\pi$
$113$	4.63246			0.435785			0.217892	+	0.975973i	$0.430082\pi$
$114$		0			0
$115$		−	1.96074i		−	0.182840i
$116$		0			0
$117$	5.88054			0.543656
$118$		0			0
$119$	5.27862	+	7.00841i	0.483890	+	0.642460i
$120$		0			0
$121$	15.1276			1.37523
$122$		0			0
$123$	−1.45096			−0.130828
$124$		0			0
$125$	−9.84392			−0.880467
$126$		0			0
$127$		−	9.38124i		−	0.832451i	−0.909261	−	0.416225i	$-0.863353\pi$
$127$		−	9.38124i		−	0.832451i	0.909261	−	0.416225i	$-0.136647\pi$
$128$		0			0
$129$			1.79528i			0.158066i
$130$		0			0
$131$		−	6.22303i		−	0.543708i	−0.962338	−	0.271854i	$-0.912363\pi$
$131$		−	6.22303i		−	0.543708i	0.962338	−	0.271854i	$-0.0876368\pi$
$132$		0			0
$133$	11.9304	+	15.8399i	1.03449	+	1.37349i
$134$		0			0
$135$			1.12786i			0.0970712i
$136$		0			0
$137$	7.45584			0.636996			0.318498	−	0.947924i	$-0.396822\pi$
$137$	7.45584			0.636996			0.318498	+	0.947924i	$0.396822\pi$
$138$		0			0
$139$			2.27471i			0.192938i	0.995336	+	0.0964690i	$0.0307549\pi$
$139$			2.27471i			0.192938i	−0.995336	+	0.0964690i	$0.969245\pi$
$140$		0			0
$141$			5.56335i			0.468518i
$142$		0			0
$143$	30.0585			2.51362
$144$		0			0
$145$			6.63246i			0.550796i
$146$		0			0
$147$	−6.72792	−	1.93264i	−0.554910	−	0.159401i
$148$		0			0
$149$			4.19427i			0.343608i	0.985131	+	0.171804i	$0.0549595\pi$
$149$			4.19427i			0.343608i	−0.985131	+	0.171804i	$0.945040\pi$
$150$		0			0
$151$		−	9.79874i		−	0.797411i	−0.917079	−	0.398705i	$-0.869460\pi$
$151$		−	9.79874i		−	0.797411i	0.917079	−	0.398705i	$-0.130540\pi$
$152$		0			0
$153$			3.31623i			0.268101i
$154$		0			0
$155$	6.82338			0.548067
$156$		0			0
$157$	−4.82865			−0.385369			−0.192684	−	0.981261i	$-0.561719\pi$
$157$	−4.82865			−0.385369			−0.192684	+	0.981261i	$0.561719\pi$
$158$		0			0
$159$	3.62481			0.287466
$160$		0			0
$161$	2.76718	+	3.67398i	0.218085	+	0.289550i
$162$		0			0
$163$	1.43753			0.112596			0.0562981	−	0.998414i	$-0.482070\pi$
$163$	1.43753			0.112596			0.0562981	+	0.998414i	$0.482070\pi$
$164$		0			0
$165$			5.76510i			0.448812i
$166$		0			0
$167$	−16.0417			−1.24134			−0.620670	−	0.784072i	$-0.713139\pi$
$167$	−16.0417			−1.24134			−0.620670	+	0.784072i	$0.713139\pi$
$168$		0			0
$169$	21.5808			1.66006
$170$		0			0
$171$			7.49510i			0.573165i
$172$		0			0
$173$	4.01798			0.305482			0.152741	−	0.988266i	$-0.451190\pi$
$173$	4.01798			0.305482			0.152741	+	0.988266i	$0.451190\pi$
$174$		0			0
$175$	7.87846	−	5.93393i	0.595556	−	0.448563i
$176$		0			0
$177$	0.767184			0.0576651
$178$		0			0
$179$	6.97679			0.521469			0.260735	−	0.965410i	$-0.416035\pi$
$179$	6.97679			0.521469			0.260735	+	0.965410i	$0.416035\pi$
$180$		0			0
$181$	−13.4687			−1.00112			−0.500561	−	0.865701i	$-0.666873\pi$
$181$	−13.4687			−1.00112			−0.500561	+	0.865701i	$0.666873\pi$
$182$		0			0
$183$			0.317194i			0.0234477i
$184$		0			0
$185$		−	1.86527i		−	0.137138i
$186$		0			0
$187$			16.9509i			1.23958i
$188$		0			0
$189$	−1.59175	−	2.11337i	−0.115783	−	0.153725i
$190$		0			0
$191$		−	21.0704i		−	1.52460i	−0.647224	−	0.762300i	$-0.724070\pi$
$191$		−	21.0704i		−	1.52460i	0.647224	−	0.762300i	$-0.275930\pi$
$192$		0			0
$193$	−5.13735			−0.369795			−0.184897	−	0.982758i	$-0.559195\pi$
$193$	−5.13735			−0.369795			−0.184897	+	0.982758i	$0.559195\pi$
$194$		0			0
$195$			6.63246i			0.474960i
$196$		0			0
$197$		−	26.4337i		−	1.88332i	−0.336566	−	0.941660i	$-0.609265\pi$
$197$		−	26.4337i		−	1.88332i	0.336566	−	0.941660i	$-0.390735\pi$
$198$		0			0
$199$	10.8420			0.768567			0.384283	−	0.923215i	$-0.374449\pi$
$199$	10.8420			0.768567			0.384283	+	0.923215i	$0.374449\pi$
$200$		0			0
$201$		−	6.56247i		−	0.462881i
$202$		0			0
$203$	−9.36038	−	12.4277i	−0.656970	−	0.872256i
$204$		0			0
$205$		−	1.63648i		−	0.114297i
$206$		0			0
$207$			1.73845i			0.120831i
$208$		0			0
$209$			38.3113i			2.65005i
$210$		0			0
$211$	−9.79528			−0.674335			−0.337168	−	0.941445i	$-0.609469\pi$
$211$	−9.79528			−0.674335			−0.337168	+	0.941445i	$0.609469\pi$
$212$		0			0
$213$	10.1919			0.698339
$214$		0			0
$215$	−2.02484			−0.138093
$216$		0			0
$217$	−12.7855	+	9.62983i	−0.867936	+	0.653716i
$218$		0			0
$219$	−6.63246			−0.448180
$220$		0			0
$221$			19.5012i			1.31179i
$222$		0			0
$223$	−15.3534			−1.02814			−0.514071	−	0.857748i	$-0.671863\pi$
$223$	−15.3534			−1.02814			−0.514071	+	0.857748i	$0.671863\pi$
$224$		0			0
$225$	3.72792			0.248528
$226$		0			0
$227$		−	5.86527i		−	0.389292i	−0.980874	−	0.194646i	$-0.937644\pi$
$227$		−	5.86527i		−	0.389292i	0.980874	−	0.194646i	$-0.0623558\pi$
$228$		0			0
$229$	2.57292			0.170024			0.0850118	−	0.996380i	$-0.472907\pi$
$229$	2.57292			0.170024			0.0850118	+	0.996380i	$0.472907\pi$
$230$		0			0
$231$	−8.13627	−	10.8025i	−0.535327	−	0.710752i
$232$		0			0
$233$	−19.6227			−1.28552			−0.642762	−	0.766066i	$-0.722212\pi$
$233$	−19.6227			−1.28552			−0.642762	+	0.766066i	$0.722212\pi$
$234$		0			0
$235$	−6.27471			−0.409317
$236$		0			0
$237$	−3.01423			−0.195795
$238$		0			0
$239$		−	0.348000i		−	0.0225102i	−0.999937	−	0.0112551i	$-0.996417\pi$
$239$		−	0.348000i		−	0.0225102i	0.999937	−	0.0112551i	$-0.00358269\pi$
$240$		0			0
$241$		−	20.0883i		−	1.29400i	−0.762490	−	0.647001i	$-0.776023\pi$
$241$		−	20.0883i		−	1.29400i	0.762490	−	0.647001i	$-0.223977\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	2.17975	−	7.58819i	0.139259	−	0.484791i
$246$		0			0
$247$			44.0753i			2.80444i
$248$		0			0
$249$	16.0883			1.01955
$250$		0			0
$251$		−	9.12494i		−	0.575961i	−0.957636	−	0.287980i	$-0.907016\pi$
$251$		−	9.12494i		−	0.575961i	0.957636	−	0.287980i	$-0.0929838\pi$
$252$		0			0
$253$			8.88611i			0.558665i
$254$		0			0
$255$	−3.74026			−0.234224
$256$		0			0
$257$		−	21.8970i		−	1.36590i	−0.730466	−	0.682949i	$-0.760697\pi$
$257$		−	21.8970i		−	1.36590i	0.730466	−	0.682949i	$-0.239303\pi$
$258$		0			0
$259$	2.63246	+	3.49510i	0.163573	+	0.217175i
$260$		0			0
$261$		−	5.88054i		−	0.363996i
$262$		0			0
$263$			28.1507i			1.73585i	0.496696	+	0.867924i	$0.334546\pi$
$263$			28.1507i			1.73585i	−0.496696	+	0.867924i	$0.665454\pi$
$264$		0			0
$265$			4.08830i			0.251142i
$266$		0			0
$267$	8.08341			0.494697
$268$		0			0
$269$	25.6230			1.56226			0.781130	−	0.624368i	$-0.214643\pi$
$269$	25.6230			1.56226			0.781130	+	0.624368i	$0.214643\pi$
$270$		0			0
$271$	19.0147			1.15506			0.577532	−	0.816368i	$-0.304016\pi$
$271$	19.0147			1.15506			0.577532	+	0.816368i	$0.304016\pi$
$272$		0			0
$273$	−9.36038	−	12.4277i	−0.566516	−	0.752161i
$274$		0			0
$275$	19.0553			1.14908
$276$		0			0
$277$		−	18.2649i		−	1.09743i	−0.836009	−	0.548716i	$-0.815117\pi$
$277$		−	18.2649i		−	1.09743i	0.836009	−	0.548716i	$-0.184883\pi$
$278$		0			0
$279$	−6.04982			−0.362193
$280$		0			0
$281$	−19.8136			−1.18198			−0.590990	−	0.806679i	$-0.701263\pi$
$281$	−19.8136			−1.18198			−0.590990	+	0.806679i	$0.701263\pi$
$282$		0			0
$283$		−	13.7698i		−	0.818530i	−0.912416	−	0.409265i	$-0.865785\pi$
$283$		−	13.7698i		−	0.818530i	0.912416	−	0.409265i	$-0.134215\pi$
$284$		0			0
$285$	−8.45347			−0.500740
$286$		0			0
$287$	2.30956	+	3.06640i	0.136329	+	0.181004i
$288$		0			0
$289$	6.00263			0.353096
$290$		0			0
$291$	−0.357752			−0.0209718
$292$		0			0
$293$	−25.2844			−1.47713			−0.738566	−	0.674181i	$-0.764497\pi$
$293$	−25.2844			−1.47713			−0.738566	+	0.674181i	$0.764497\pi$
$294$		0			0
$295$			0.865280i			0.0503786i
$296$		0			0
$297$		−	5.11151i		−	0.296600i
$298$		0			0
$299$			10.2230i			0.591213i
$300$		0			0
$301$	3.79409	−	2.85765i	0.218688	−	0.164712i
$302$		0			0
$303$			3.38359i			0.194382i
$304$		0			0
$305$	−0.357752			−0.0204848
$306$		0			0
$307$			5.96074i			0.340197i	0.985427	+	0.170099i	$0.0544086\pi$
$307$			5.96074i			0.340197i	−0.985427	+	0.170099i	$0.945591\pi$
$308$		0			0
$309$			14.5033i			0.825063i
$310$		0			0
$311$	−26.4123			−1.49770			−0.748852	−	0.662738i	$-0.769394\pi$
$311$	−26.4123			−1.49770			−0.748852	+	0.662738i	$0.769394\pi$
$312$		0			0
$313$		−	9.53437i		−	0.538914i	−0.963012	−	0.269457i	$-0.913156\pi$
$313$		−	9.53437i		−	0.538914i	0.963012	−	0.269457i	$-0.0868443\pi$
$314$		0			0
$315$	2.38359	−	1.79528i	0.134300	−	0.101153i
$316$		0			0
$317$			24.8123i			1.39360i	0.717266	+	0.696799i	$0.245393\pi$
$317$			24.8123i			1.39360i	−0.717266	+	0.696799i	$0.754607\pi$
$318$		0			0
$319$		−	30.0585i		−	1.68295i
$320$		0			0
$321$		−	1.87870i		−	0.104859i
$322$		0			0
$323$	−24.8555			−1.38300
$324$		0			0
$325$	21.9222			1.21602
$326$		0			0
$327$	−13.6166			−0.753002
$328$		0			0
$329$	11.7574	−	8.85548i	0.648206	−	0.488219i
$330$		0			0
$331$	−9.46438			−0.520209			−0.260105	−	0.965580i	$-0.583757\pi$
$331$	−9.46438			−0.520209			−0.260105	+	0.965580i	$0.583757\pi$
$332$		0			0
$333$			1.65381i			0.0906282i
$334$		0			0
$335$	7.40158			0.404391
$336$		0			0
$337$	−35.7110			−1.94530			−0.972650	−	0.232275i	$-0.925383\pi$
$337$	−35.7110			−1.94530			−0.972650	+	0.232275i	$0.925383\pi$
$338$		0			0
$339$		−	4.63246i		−	0.251601i
$340$		0			0
$341$	−30.9237			−1.67461
$342$		0			0
$343$	6.62483	+	17.2948i	0.357707	+	0.933834i
$344$		0			0
$345$	−1.96074			−0.105562
$346$		0			0
$347$	26.3764			1.41596			0.707980	−	0.706232i	$-0.249606\pi$
$347$	26.3764			1.41596			0.707980	+	0.706232i	$0.249606\pi$
$348$		0			0
$349$	−7.90538			−0.423165			−0.211583	−	0.977360i	$-0.567862\pi$
$349$	−7.90538			−0.423165			−0.211583	+	0.977360i	$0.567862\pi$
$350$		0			0
$351$		−	5.88054i		−	0.313880i
$352$		0			0
$353$		−	14.5490i		−	0.774368i	−0.922003	−	0.387184i	$-0.873448\pi$
$353$		−	14.5490i		−	0.774368i	0.922003	−	0.387184i	$-0.126552\pi$
$354$		0			0
$355$			11.4951i			0.610097i
$356$		0			0
$357$	7.00841	−	5.27862i	0.370924	−	0.279374i
$358$		0			0
$359$		−	21.3186i		−	1.12515i	−0.826745	−	0.562577i	$-0.809810\pi$
$359$		−	21.3186i		−	1.12515i	0.826745	−	0.562577i	$-0.190190\pi$
$360$		0			0
$361$	−37.1766			−1.95666
$362$		0			0
$363$		−	15.1276i		−	0.793991i
$364$		0			0
$365$		−	7.48052i		−	0.391548i
$366$		0			0
$367$	−6.89996			−0.360175			−0.180088	−	0.983651i	$-0.557638\pi$
$367$	−6.89996			−0.360175			−0.180088	+	0.983651i	$0.557638\pi$
$368$		0			0
$369$			1.45096i			0.0755337i
$370$		0			0
$371$	−5.76981	−	7.66056i	−0.299554	−	0.397716i
$372$		0			0
$373$			26.5816i			1.37634i	0.725549	+	0.688171i	$0.241586\pi$
$373$			26.5816i			1.37634i	−0.725549	+	0.688171i	$0.758414\pi$
$374$		0			0
$375$			9.84392i			0.508338i
$376$		0			0
$377$		−	34.5808i		−	1.78100i
$378$		0			0
$379$	−5.52583			−0.283843			−0.141921	−	0.989878i	$-0.545328\pi$
$379$	−5.52583			−0.283843			−0.141921	+	0.989878i	$0.545328\pi$
$380$		0			0
$381$	−9.38124			−0.480616
$382$		0			0
$383$	−10.3706			−0.529915			−0.264957	−	0.964260i	$-0.585358\pi$
$383$	−10.3706			−0.529915			−0.264957	+	0.964260i	$0.585358\pi$
$384$		0			0
$385$	12.1838	−	9.17662i	0.620942	−	0.467684i
$386$		0			0
$387$	1.79528			0.0912594
$388$		0			0
$389$		−	12.2302i		−	0.620097i	−0.950721	−	0.310049i	$-0.899655\pi$
$389$		−	12.2302i		−	0.620097i	0.950721	−	0.310049i	$-0.100345\pi$
$390$		0			0
$391$	−5.76510			−0.291553
$392$		0			0
$393$	−6.22303			−0.313910
$394$		0			0
$395$		−	3.39964i		−	0.171055i
$396$		0			0
$397$	−7.50188			−0.376509			−0.188254	−	0.982120i	$-0.560283\pi$
$397$	−7.50188			−0.376509			−0.188254	+	0.982120i	$0.560283\pi$
$398$		0			0
$399$	15.8399	−	11.9304i	0.792987	−	0.597265i
$400$		0			0
$401$	11.4558			0.572077			0.286039	−	0.958218i	$-0.407661\pi$
$401$	11.4558			0.572077			0.286039	+	0.958218i	$0.407661\pi$
$402$		0			0
$403$	−35.5762			−1.77218
$404$		0			0
$405$	1.12786			0.0560441
$406$		0			0
$407$			8.45347i			0.419023i
$408$		0			0
$409$			29.6227i			1.46475i	0.680903	+	0.732373i	$0.261587\pi$
$409$			29.6227i			1.46475i	−0.680903	+	0.732373i	$0.738413\pi$
$410$		0			0
$411$		−	7.45584i		−	0.367770i
$412$		0			0
$413$	−1.22117	−	1.62134i	−0.0600898	−	0.0797810i
$414$		0			0
$415$			18.1454i			0.890724i
$416$		0			0
$417$	2.27471			0.111393
$418$		0			0
$419$		−	17.8136i		−	0.870251i	−0.900370	−	0.435125i	$-0.856704\pi$
$419$		−	17.8136i		−	0.870251i	0.900370	−	0.435125i	$-0.143296\pi$
$420$		0			0
$421$			13.4149i			0.653802i	0.945059	+	0.326901i	$0.106004\pi$
$421$			13.4149i			0.653802i	−0.945059	+	0.326901i	$0.893996\pi$
$422$		0			0
$423$	5.56335			0.270499
$424$		0			0
$425$			12.3626i			0.599676i
$426$		0			0
$427$	0.670347	−	0.504895i	0.0324404	−	0.0244336i
$428$		0			0
$429$		−	30.0585i		−	1.45124i
$430$		0			0
$431$			17.6108i			0.848283i	0.905596	+	0.424142i	$0.139424\pi$
$431$			17.6108i			0.848283i	−0.905596	+	0.424142i	$0.860576\pi$
$432$		0			0
$433$			36.4461i			1.75149i	0.482778	+	0.875743i	$0.339628\pi$
$433$			36.4461i			1.75149i	−0.482778	+	0.875743i	$0.660372\pi$
$434$		0			0
$435$	6.63246			0.318002
$436$		0			0
$437$	−13.0299			−0.623303
$438$		0			0
$439$	−4.79619			−0.228909			−0.114455	−	0.993428i	$-0.536512\pi$
$439$	−4.79619			−0.228909			−0.114455	+	0.993428i	$0.536512\pi$
$440$		0			0
$441$	−1.93264	+	6.72792i	−0.0920303	+	0.320377i
$442$		0			0
$443$	−32.7021			−1.55372			−0.776861	−	0.629672i	$-0.783189\pi$
$443$	−32.7021			−1.55372			−0.776861	+	0.629672i	$0.783189\pi$
$444$		0			0
$445$			9.11700i			0.432187i
$446$		0			0
$447$	4.19427			0.198382
$448$		0			0
$449$	25.3480			1.19624			0.598122	−	0.801405i	$-0.295914\pi$
$449$	25.3480			1.19624			0.598122	+	0.801405i	$0.295914\pi$
$450$		0			0
$451$			7.41658i			0.349233i
$452$		0			0
$453$	−9.79874			−0.460385
$454$		0			0
$455$	14.0168	−	10.5572i	0.657118	−	0.494931i
$456$		0			0
$457$	−1.66910			−0.0780770			−0.0390385	−	0.999238i	$-0.512429\pi$
$457$	−1.66910			−0.0780770			−0.0390385	+	0.999238i	$0.512429\pi$
$458$		0			0
$459$	3.31623			0.154788
$460$		0			0
$461$	33.0246			1.53811			0.769054	−	0.639184i	$-0.220728\pi$
$461$	33.0246			1.53811			0.769054	+	0.639184i	$0.220728\pi$
$462$		0			0
$463$		−	15.6833i		−	0.728866i	−0.931230	−	0.364433i	$-0.881263\pi$
$463$		−	15.6833i		−	0.728866i	0.931230	−	0.364433i	$-0.118737\pi$
$464$		0			0
$465$		−	6.82338i		−	0.316427i
$466$		0			0
$467$			4.85548i			0.224685i	0.993670	+	0.112342i	$0.0358354\pi$
$467$			4.85548i			0.224685i	−0.993670	+	0.112342i	$0.964165\pi$
$468$		0			0
$469$	−13.8689	+	10.4458i	−0.640406	+	0.482344i
$470$		0			0
$471$			4.82865i			0.222493i
$472$		0			0
$473$	9.17662			0.421941
$474$		0			0
$475$			27.9412i			1.28203i
$476$		0			0
$477$		−	3.62481i		−	0.165969i
$478$		0			0
$479$	−7.10575			−0.324670			−0.162335	−	0.986736i	$-0.551903\pi$
$479$	−7.10575			−0.324670			−0.162335	+	0.986736i	$0.551903\pi$
$480$		0			0
$481$			9.72529i			0.443435i
$482$		0			0
$483$	3.67398	−	2.76718i	0.167172	−	0.125911i
$484$		0			0
$485$		−	0.403496i		−	0.0183218i
$486$		0			0
$487$			4.29701i			0.194716i	0.995249	+	0.0973580i	$0.0310392\pi$
$487$			4.29701i			0.194716i	−0.995249	+	0.0973580i	$0.968961\pi$
$488$		0			0
$489$		−	1.43753i		−	0.0650074i
$490$		0			0
$491$	19.3862			0.874888			0.437444	−	0.899246i	$-0.355884\pi$
$491$	19.3862			0.874888			0.437444	+	0.899246i	$0.355884\pi$
$492$		0			0
$493$	19.5012			0.878291
$494$		0			0
$495$	5.76510			0.259122
$496$		0			0
$497$	−16.2230	−	21.5393i	−0.727702	−	0.966168i
$498$		0			0
$499$	30.5722			1.36860			0.684301	−	0.729200i	$-0.260108\pi$
$499$	30.5722			1.36860			0.684301	+	0.729200i	$0.260108\pi$
$500$		0			0
$501$			16.0417i			0.716688i
$502$		0			0
$503$	−26.9817			−1.20306			−0.601528	−	0.798852i	$-0.705441\pi$
$503$	−26.9817			−1.20306			−0.601528	+	0.798852i	$0.705441\pi$
$504$		0			0
$505$	−3.81624			−0.169820
$506$		0			0
$507$		−	21.5808i		−	0.958436i
$508$		0			0
$509$	14.1717			0.628151			0.314075	−	0.949398i	$-0.398306\pi$
$509$	14.1717			0.628151			0.314075	+	0.949398i	$0.398306\pi$
$510$		0			0
$511$	10.5572	+	14.0168i	0.467025	+	0.620067i
$512$		0			0
$513$	7.49510			0.330917
$514$		0			0
$515$	−16.3578			−0.720809
$516$		0			0
$517$	28.4371			1.25066
$518$		0			0
$519$		−	4.01798i		−	0.176370i
$520$		0			0
$521$			25.9487i			1.13683i	0.822741	+	0.568416i	$0.192444\pi$
$521$			25.9487i			1.13683i	−0.822741	+	0.568416i	$0.807556\pi$
$522$		0			0
$523$		−	24.6370i		−	1.07730i	−0.842530	−	0.538650i	$-0.818935\pi$
$523$		−	24.6370i		−	1.07730i	0.842530	−	0.538650i	$-0.181065\pi$
$524$		0			0
$525$	−5.93393	−	7.87846i	−0.258978	−	0.343844i
$526$		0			0
$527$		−	20.0626i		−	0.873940i
$528$		0			0
$529$	19.9778			0.868600
$530$		0			0
$531$		−	0.767184i		−	0.0332930i
$532$		0			0
$533$			8.53241i			0.369579i
$534$		0			0
$535$	2.11892			0.0916088
$536$		0			0
$537$		−	6.97679i		−	0.301071i
$538$		0			0
$539$	−9.87870	+	34.3899i	−0.425506	+	1.48128i
$540$		0			0
$541$			5.59582i			0.240583i	0.992739	+	0.120291i	$0.0383829\pi$
$541$			5.59582i			0.240583i	−0.992739	+	0.120291i	$0.961617\pi$
$542$		0			0
$543$			13.4687i			0.577998i
$544$		0			0
$545$		−	15.3577i		−	0.657853i
$546$		0			0
$547$	2.15304			0.0920572			0.0460286	−	0.998940i	$-0.485343\pi$
$547$	2.15304			0.0920572			0.0460286	+	0.998940i	$0.485343\pi$
$548$		0			0
$549$	0.317194			0.0135375
$550$		0			0
$551$	44.0753			1.87767
$552$		0			0
$553$	4.79791	+	6.37017i	0.204028	+	0.270887i
$554$		0			0
$555$	−1.86527			−0.0791764
$556$		0			0
$557$			31.1317i			1.31909i	0.751664	+	0.659547i	$0.229252\pi$
$557$			31.1317i			1.31909i	−0.751664	+	0.659547i	$0.770748\pi$
$558$		0			0
$559$	10.5572			0.446524
$560$		0			0
$561$	16.9509			0.715670
$562$		0			0
$563$		−	17.6227i		−	0.742707i	−0.928492	−	0.371353i	$-0.878894\pi$
$563$		−	17.6227i		−	0.742707i	0.928492	−	0.371353i	$-0.121106\pi$
$564$		0			0
$565$	5.22479			0.219808
$566$		0			0
$567$	−2.11337	+	1.59175i	−0.0887531	+	0.0668474i
$568$		0			0
$569$	13.5389			0.567580			0.283790	−	0.958887i	$-0.408408\pi$
$569$	13.5389			0.567580			0.283790	+	0.958887i	$0.408408\pi$
$570$		0			0
$571$	−30.2145			−1.26444			−0.632218	−	0.774790i	$-0.717855\pi$
$571$	−30.2145			−1.26444			−0.632218	+	0.774790i	$0.717855\pi$
$572$		0			0
$573$	−21.0704			−0.880228
$574$		0			0
$575$			6.48080i			0.270268i
$576$		0			0
$577$			38.1570i			1.58850i	0.607593	+	0.794249i	$0.292135\pi$
$577$			38.1570i			1.58850i	−0.607593	+	0.794249i	$0.707865\pi$
$578$		0			0
$579$			5.13735i			0.213501i
$580$		0			0
$581$	−25.6086	−	34.0005i	−1.06242	−	1.41058i
$582$		0			0
$583$		−	18.5283i		−	0.767363i
$584$		0			0
$585$	6.63246			0.274218
$586$		0			0
$587$			35.7057i			1.47373i	0.676039	+	0.736866i	$0.263695\pi$
$587$			35.7057i			1.47373i	−0.676039	+	0.736866i	$0.736305\pi$
$588$		0			0
$589$		−	45.3441i		−	1.86837i
$590$		0			0
$591$	−26.4337			−1.08734
$592$		0			0
$593$			16.9630i			0.696587i	0.937386	+	0.348293i	$0.113239\pi$
$593$			16.9630i			0.696587i	−0.937386	+	0.348293i	$0.886761\pi$
$594$		0			0
$595$	5.95357	+	7.90454i	0.244073	+	0.324054i
$596$		0			0
$597$		−	10.8420i		−	0.443732i
$598$		0			0
$599$		−	30.2372i		−	1.23546i	−0.786391	−	0.617729i	$-0.788053\pi$
$599$		−	30.2372i		−	1.23546i	0.786391	−	0.617729i	$-0.211947\pi$
$600$		0			0
$601$		−	29.4558i		−	1.20153i	−0.799426	−	0.600764i	$-0.794863\pi$
$601$		−	29.4558i		−	1.20153i	0.799426	−	0.600764i	$-0.205137\pi$
$602$		0			0
$603$	−6.56247			−0.267244
$604$		0			0
$605$	17.0618			0.693663
$606$		0			0
$607$	21.7031			0.880902			0.440451	−	0.897777i	$-0.354818\pi$
$607$	21.7031			0.880902			0.440451	+	0.897777i	$0.354818\pi$
$608$		0			0
$609$	−12.4277	+	9.36038i	−0.503598	+	0.379302i
$610$		0			0
$611$	32.7155			1.32353
$612$		0			0
$613$		−	41.0159i		−	1.65662i	−0.560273	−	0.828308i	$-0.689304\pi$
$613$		−	41.0159i		−	1.65662i	0.560273	−	0.828308i	$-0.310696\pi$
$614$		0			0
$615$	−1.63648			−0.0659893
$616$		0			0
$617$	42.2551			1.70113			0.850564	−	0.525872i	$-0.176261\pi$
$617$	42.2551			1.70113			0.850564	+	0.525872i	$0.176261\pi$
$618$		0			0
$619$			14.5442i			0.584579i	0.956330	+	0.292290i	$0.0944171\pi$
$619$			14.5442i			0.584579i	−0.956330	+	0.292290i	$0.905583\pi$
$620$		0			0
$621$	1.73845			0.0697616
$622$		0			0
$623$	−12.8668	−	17.0832i	−0.515498	−	0.684425i
$624$		0			0
$625$	7.53699			0.301480
$626$		0			0
$627$	38.3113			1.53001
$628$		0			0
$629$	−5.48441			−0.218678
$630$		0			0
$631$		−	11.6986i		−	0.465713i	−0.972511	−	0.232857i	$-0.925193\pi$
$631$		−	11.6986i		−	0.465713i	0.972511	−	0.232857i	$-0.0748073\pi$
$632$		0			0
$633$			9.79528i			0.389328i
$634$		0			0
$635$		−	10.5808i		−	0.419885i
$636$		0			0
$637$	−11.3650	+	39.5638i	−0.450296	+	1.56758i
$638$		0			0
$639$		−	10.1919i		−	0.403186i
$640$		0			0
$641$	−11.5344			−0.455580			−0.227790	−	0.973710i	$-0.573150\pi$
$641$	−11.5344			−0.455580			−0.227790	+	0.973710i	$0.573150\pi$
$642$		0			0
$643$		−	13.7698i		−	0.543028i	−0.962434	−	0.271514i	$-0.912476\pi$
$643$		−	13.7698i		−	0.543028i	0.962434	−	0.271514i	$-0.0875244\pi$
$644$		0			0
$645$			2.02484i			0.0797279i
$646$		0			0
$647$	−18.7149			−0.735758			−0.367879	−	0.929874i	$-0.619916\pi$
$647$	−18.7149			−0.735758			−0.367879	+	0.929874i	$0.619916\pi$
$648$		0			0
$649$		−	3.92147i		−	0.153931i
$650$		0			0
$651$	9.62983	+	12.7855i	0.377423	+	0.501103i
$652$		0			0
$653$		−	19.5588i		−	0.765395i	−0.923874	−	0.382697i	$-0.874995\pi$
$653$		−	19.5588i		−	0.765395i	0.923874	−	0.382697i	$-0.125005\pi$
$654$		0			0
$655$		−	7.01873i		−	0.274245i
$656$		0			0
$657$			6.63246i			0.258757i
$658$		0			0
$659$	2.56735			0.100010			0.0500050	−	0.998749i	$-0.484076\pi$
$659$	2.56735			0.100010			0.0500050	+	0.998749i	$0.484076\pi$
$660$		0			0
$661$	9.00155			0.350120			0.175060	−	0.984558i	$-0.443988\pi$
$661$	9.00155			0.350120			0.175060	+	0.984558i	$0.443988\pi$
$662$		0			0
$663$	19.5012			0.757365
$664$		0			0
$665$	13.4558	+	17.8653i	0.521795	+	0.692786i
$666$		0			0
$667$	10.2230			0.395837
$668$		0			0
$669$			15.3534i			0.593598i
$670$		0			0
$671$	1.62134			0.0625912
$672$		0			0
$673$	−28.7645			−1.10879			−0.554396	−	0.832253i	$-0.687051\pi$
$673$	−28.7645			−1.10879			−0.554396	+	0.832253i	$0.687051\pi$
$674$		0			0
$675$		−	3.72792i		−	0.143488i
$676$		0			0
$677$	17.2485			0.662912			0.331456	−	0.943471i	$-0.392460\pi$
$677$	17.2485			0.662912			0.331456	+	0.943471i	$0.392460\pi$
$678$		0			0
$679$	0.569453	+	0.756061i	0.0218536	+	0.0290150i
$680$		0			0
$681$	−5.86527			−0.224758
$682$		0			0
$683$	26.2096			1.00288			0.501441	−	0.865192i	$-0.332803\pi$
$683$	26.2096			1.00288			0.501441	+	0.865192i	$0.332803\pi$
$684$		0			0
$685$	8.40918			0.321298
$686$		0			0
$687$		−	2.57292i		−	0.0981632i
$688$		0			0
$689$		−	21.3159i		−	0.812070i
$690$		0			0
$691$			2.99021i			0.113753i	0.998381	+	0.0568765i	$0.0181141\pi$
$691$			2.99021i			0.113753i	−0.998381	+	0.0568765i	$0.981886\pi$
$692$		0			0
$693$	−10.8025	+	8.13627i	−0.410353	+	0.309071i
$694$		0			0
$695$			2.56556i			0.0973173i
$696$		0			0
$697$	−4.81170			−0.182256
$698$		0			0
$699$			19.6227i			0.742197i
$700$		0			0
$701$			18.1971i			0.687294i	0.939099	+	0.343647i	$0.111662\pi$
$701$			18.1971i			0.687294i	−0.939099	+	0.343647i	$0.888338\pi$
$702$		0			0
$703$	−12.3955			−0.467504
$704$		0			0
$705$			6.27471i			0.236319i
$706$		0			0
$707$	7.15078	−	5.38585i	0.268933	−	0.202556i
$708$		0			0
$709$			4.86735i			0.182797i	0.995814	+	0.0913986i	$0.0291337\pi$
$709$			4.86735i			0.182797i	−0.995814	+	0.0913986i	$0.970866\pi$
$710$		0			0
$711$			3.01423i			0.113042i
$712$		0			0
$713$		−	10.5173i		−	0.393876i
$714$		0			0
$715$	33.9019			1.26786
$716$		0			0
$717$	−0.348000			−0.0129963
$718$		0			0
$719$	−7.10575			−0.265000			−0.132500	−	0.991183i	$-0.542300\pi$
$719$	−7.10575			−0.265000			−0.132500	+	0.991183i	$0.542300\pi$
$720$		0			0
$721$	30.6508	−	23.0857i	1.14149	−	0.859755i
$722$		0			0
$723$	−20.0883			−0.747092
$724$		0			0
$725$		−	21.9222i		−	0.814170i
$726$		0			0
$727$	42.5369			1.57761			0.788803	−	0.614645i	$-0.210701\pi$
$727$	42.5369			1.57761			0.788803	+	0.614645i	$0.210701\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$			5.95357i			0.220201i
$732$		0			0
$733$	−38.8291			−1.43419			−0.717093	−	0.696977i	$-0.754528\pi$
$733$	−38.8291			−1.43419			−0.717093	+	0.696977i	$0.754528\pi$
$734$		0			0
$735$	−7.58819	−	2.17975i	−0.279894	−	0.0804014i
$736$		0			0
$737$	−33.5441			−1.23561
$738$		0			0
$739$	27.0085			0.993524			0.496762	−	0.867887i	$-0.334522\pi$
$739$	27.0085			0.993524			0.496762	+	0.867887i	$0.334522\pi$
$740$		0			0
$741$	44.0753			1.61915
$742$		0			0
$743$			28.0242i			1.02811i	0.857758	+	0.514054i	$0.171857\pi$
$743$			28.0242i			1.02811i	−0.857758	+	0.514054i	$0.828143\pi$
$744$		0			0
$745$			4.73057i			0.173315i
$746$		0			0
$747$		−	16.0883i		−	0.588640i
$748$		0			0
$749$	−3.97038	+	2.99042i	−0.145074	+	0.109268i
$750$		0			0
$751$			46.6071i			1.70072i	0.526204	+	0.850358i	$0.323615\pi$
$751$			46.6071i			1.70072i	−0.526204	+	0.850358i	$0.676385\pi$
$752$		0			0
$753$	−9.12494			−0.332531
$754$		0			0
$755$		−	11.0517i		−	0.402211i
$756$		0			0
$757$			18.9934i			0.690326i	0.938543	+	0.345163i	$0.112176\pi$
$757$			18.9934i			0.690326i	−0.938543	+	0.345163i	$0.887824\pi$
$758$		0			0
$759$	8.88611			0.322545
$760$		0			0
$761$		−	25.9487i		−	0.940639i	−0.882496	−	0.470320i	$-0.844139\pi$
$761$		−	25.9487i		−	0.940639i	0.882496	−	0.470320i	$-0.155861\pi$
$762$		0			0
$763$	21.6743	+	28.7770i	0.784664	+	1.04180i
$764$		0			0
$765$			3.74026i			0.135229i
$766$		0			0
$767$		−	4.51146i		−	0.162899i
$768$		0			0
$769$			5.27924i			0.190374i	0.995459	+	0.0951872i	$0.0303450\pi$
$769$			5.27924i			0.190374i	−0.995459	+	0.0951872i	$0.969655\pi$
$770$		0			0
$771$	−21.8970			−0.788601
$772$		0			0
$773$	−5.30076			−0.190655			−0.0953276	−	0.995446i	$-0.530390\pi$
$773$	−5.30076			−0.190655			−0.0953276	+	0.995446i	$0.530390\pi$
$774$		0			0
$775$	−22.5533			−0.810137
$776$		0			0
$777$	3.49510	−	2.63246i	0.125386	−	0.0944389i
$778$		0			0
$779$	−10.8751			−0.389640
$780$		0			0
$781$		−	52.0961i		−	1.86415i
$782$		0			0
$783$	−5.88054			−0.210153
$784$		0			0
$785$	−5.44607			−0.194379
$786$		0			0
$787$			15.7305i			0.560733i	0.959893	+	0.280367i	$0.0904561\pi$
$787$			15.7305i			0.560733i	−0.959893	+	0.280367i	$0.909544\pi$
$788$		0			0
$789$	28.1507			1.00219
$790$		0			0
$791$	−9.79008	+	7.37373i	−0.348095	+	0.262180i
$792$		0			0
$793$	1.86527			0.0662378
$794$		0			0
$795$	4.08830			0.144997
$796$		0			0
$797$	8.71605			0.308738			0.154369	−	0.988013i	$-0.450665\pi$
$797$	8.71605			0.308738			0.154369	+	0.988013i	$0.450665\pi$
$798$		0			0
$799$			18.4493i			0.652691i
$800$		0			0
$801$		−	8.08341i		−	0.285613i
$802$		0			0
$803$			33.9019i			1.19637i
$804$		0			0
$805$	3.12101	+	4.14375i	0.110001	+	0.146048i
$806$		0			0
$807$		−	25.6230i		−	0.901972i
$808$		0			0
$809$	4.28448			0.150634			0.0753171	−	0.997160i	$-0.476003\pi$
$809$	4.28448			0.150634			0.0753171	+	0.997160i	$0.476003\pi$
$810$		0			0
$811$		−	30.7208i		−	1.07875i	−0.842065	−	0.539376i	$-0.818660\pi$
$811$		−	30.7208i		−	1.07875i	0.842065	−	0.539376i	$-0.181340\pi$
$812$		0			0
$813$		−	19.0147i		−	0.666876i
$814$		0			0
$815$	1.62134			0.0567931
$816$		0			0
$817$			13.4558i			0.470760i
$818$		0			0
$819$	−12.4277	+	9.36038i	−0.434260	+	0.327078i
$820$		0			0
$821$		−	27.1897i		−	0.948928i	−0.880275	−	0.474464i	$-0.842642\pi$
$821$		−	27.1897i		−	0.948928i	0.880275	−	0.474464i	$-0.157358\pi$
$822$		0			0
$823$		−	49.5318i		−	1.72657i	−0.504715	−	0.863286i	$-0.668403\pi$
$823$		−	49.5318i		−	1.72657i	0.504715	−	0.863286i	$-0.331597\pi$
$824$		0			0
$825$		−	19.0553i		−	0.663421i
$826$		0			0
$827$	−27.4228			−0.953585			−0.476793	−	0.879016i	$-0.658201\pi$
$827$	−27.4228			−0.953585			−0.476793	+	0.879016i	$0.658201\pi$
$828$		0			0
$829$	−44.3135			−1.53907			−0.769536	−	0.638603i	$-0.779513\pi$
$829$	−44.3135			−1.53907			−0.769536	+	0.638603i	$0.779513\pi$
$830$		0			0
$831$	−18.2649			−0.633603
$832$		0			0
$833$	−22.3113	−	6.40907i	−0.773042	−	0.222061i
$834$		0			0
$835$	−18.0928			−0.626128
$836$		0			0
$837$			6.04982i			0.209112i
$838$		0			0
$839$	52.9322			1.82742			0.913712	−	0.406362i	$-0.133203\pi$
$839$	52.9322			1.82742			0.913712	+	0.406362i	$0.133203\pi$
$840$		0			0
$841$	−5.58078			−0.192441
$842$		0			0
$843$			19.8136i			0.682416i
$844$		0			0
$845$	24.3402			0.837328
$846$		0			0
$847$	−31.9701	+	24.0794i	−1.09851	+	0.827377i
$848$		0			0
$849$	−13.7698			−0.472579
$850$		0			0
$851$	−2.87506			−0.0985559
$852$		0			0
$853$	23.8821			0.817707			0.408854	−	0.912600i	$-0.365929\pi$
$853$	23.8821			0.817707			0.408854	+	0.912600i	$0.365929\pi$
$854$		0			0
$855$			8.45347i			0.289102i
$856$		0			0
$857$			46.2279i			1.57912i	0.613676	+	0.789558i	$0.289690\pi$
$857$			46.2279i			1.57912i	−0.613676	+	0.789558i	$0.710310\pi$
$858$		0			0
$859$		−	18.2159i		−	0.621517i	−0.950489	−	0.310759i	$-0.899417\pi$
$859$		−	18.2159i		−	0.621517i	0.950489	−	0.310759i	$-0.100583\pi$
$860$		0			0
$861$	3.06640	−	2.30956i	0.104503	−	0.0787098i
$862$		0			0
$863$			3.72055i			0.126649i	0.997993	+	0.0633245i	$0.0201703\pi$
$863$			3.72055i			0.126649i	−0.997993	+	0.0633245i	$0.979830\pi$
$864$		0			0
$865$	4.53174			0.154084
$866$		0			0
$867$		−	6.00263i		−	0.203860i
$868$		0			0
$869$			15.4073i			0.522656i
$870$		0			0
$871$	−38.5909			−1.30760
$872$		0			0
$873$			0.357752i			0.0121081i
$874$		0			0
$875$	20.8038	−	15.6691i	0.703297	−	0.529712i
$876$		0			0
$877$		−	19.1011i		−	0.644997i	−0.946570	−	0.322498i	$-0.895477\pi$
$877$		−	19.1011i		−	0.644997i	0.946570	−	0.322498i	$-0.104523\pi$
$878$		0			0
$879$			25.2844i			0.852822i
$880$		0			0
$881$			29.3993i			0.990487i	0.868754	+	0.495243i	$0.164921\pi$
$881$			29.3993i			0.990487i	−0.868754	+	0.495243i	$0.835079\pi$
$882$		0			0
$883$	4.33944			0.146034			0.0730169	−	0.997331i	$-0.476737\pi$
$883$	4.33944			0.146034			0.0730169	+	0.997331i	$0.476737\pi$
$884$		0			0
$885$	0.865280			0.0290861
$886$		0			0
$887$	−8.74929			−0.293773			−0.146886	−	0.989153i	$-0.546925\pi$
$887$	−8.74929			−0.293773			−0.146886	+	0.989153i	$0.546925\pi$
$888$		0			0
$889$	14.9326	+	19.8260i	0.500825	+	0.664943i
$890$		0			0
$891$	−5.11151			−0.171242
$892$		0			0
$893$			41.6979i			1.39537i
$894$		0			0
$895$	7.86887			0.263027
$896$		0			0
$897$	10.2230			0.341337
$898$		0			0
$899$			35.5762i			1.18653i
$900$		0			0
$901$	12.0207			0.400468
$902$		0			0
$903$	−2.85765	−	3.79409i	−0.0950967	−	0.126260i
$904$		0			0
$905$	−15.1909			−0.504963
$906$		0			0
$907$	−40.5946			−1.34792			−0.673960	−	0.738768i	$-0.735408\pi$
$907$	−40.5946			−1.34792			−0.673960	+	0.738768i	$0.735408\pi$
$908$		0			0
$909$	3.38359			0.112227
$910$		0			0
$911$			30.4887i			1.01014i	0.863079	+	0.505068i	$0.168533\pi$
$911$			30.4887i			1.01014i	−0.863079	+	0.505068i	$0.831467\pi$
$912$		0			0
$913$		−	82.2355i		−	2.72160i
$914$		0			0
$915$			0.357752i			0.0118269i
$916$		0			0
$917$	9.90553	+	13.1515i	0.327109	+	0.434302i
$918$		0			0
$919$		−	16.6925i		−	0.550634i	−0.961353	−	0.275317i	$-0.911217\pi$
$919$		−	16.6925i		−	0.550634i	0.961353	−	0.275317i	$-0.0887828\pi$
$920$		0			0
$921$	5.96074			0.196413
$922$		0			0
$923$		−	59.9340i		−	1.97275i
$924$		0			0
$925$			6.16527i			0.202713i
$926$		0			0
$927$	14.5033			0.476351
$928$		0			0
$929$			44.6963i			1.46644i	0.679993	+	0.733219i	$0.261983\pi$
$929$			44.6963i			1.46644i	−0.679993	+	0.733219i	$0.738017\pi$
$930$		0			0
$931$	−50.4265	−	14.4853i	−1.65266	−	0.474737i
$932$		0			0
$933$			26.4123i			0.864699i
$934$		0			0
$935$			19.1184i			0.625238i
$936$		0			0
$937$		−	58.7208i		−	1.91832i	−0.282858	−	0.959162i	$-0.591283\pi$
$937$		−	58.7208i		−	1.91832i	0.282858	−	0.959162i	$-0.408717\pi$
$938$		0			0
$939$	−9.53437			−0.311142
$940$		0			0
$941$	−28.7137			−0.936040			−0.468020	−	0.883718i	$-0.655033\pi$
$941$	−28.7137			−0.936040			−0.468020	+	0.883718i	$0.655033\pi$
$942$		0			0
$943$	−2.52241			−0.0821411
$944$		0			0
$945$	−1.79528	−	2.38359i	−0.0584006	−	0.0775383i
$946$		0			0
$947$	−40.7146			−1.32305			−0.661523	−	0.749925i	$-0.730090\pi$
$947$	−40.7146			−1.32305			−0.661523	+	0.749925i	$0.730090\pi$
$948$		0			0
$949$			39.0024i			1.26607i
$950$		0			0
$951$	24.8123			0.804594
$952$		0			0
$953$	−25.5396			−0.827309			−0.413655	−	0.910434i	$-0.635748\pi$
$953$	−25.5396			−0.827309			−0.413655	+	0.910434i	$0.635748\pi$
$954$		0			0
$955$		−	23.7646i		−	0.769003i
$956$		0			0
$957$	−30.0585			−0.971652
$958$		0			0
$959$	−15.7569	+	11.8679i	−0.508818	+	0.383233i
$960$		0			0
$961$	5.60036			0.180657
$962$		0			0
$963$	−1.87870			−0.0605402
$964$		0			0
$965$	−5.79424			−0.186523
$966$		0			0
$967$		−	20.8465i		−	0.670378i	−0.942151	−	0.335189i	$-0.891200\pi$
$967$		−	20.8465i		−	0.670378i	0.942151	−	0.335189i	$-0.108800\pi$
$968$		0			0
$969$			24.8555i			0.798473i
$970$		0			0
$971$		−	35.1302i		−	1.12738i	−0.825986	−	0.563691i	$-0.809381\pi$
$971$		−	35.1302i		−	1.12738i	0.825986	−	0.563691i	$-0.190619\pi$
$972$		0			0
$973$	−3.62077	−	4.80729i	−0.116077	−	0.154115i
$974$		0			0
$975$		−	21.9222i		−	0.702072i
$976$		0			0
$977$	8.63246			0.276177			0.138088	−	0.990420i	$-0.455904\pi$
$977$	8.63246			0.276177			0.138088	+	0.990420i	$0.455904\pi$
$978$		0			0
$979$		−	41.3185i		−	1.32054i
$980$		0			0
$981$			13.6166i			0.434746i
$982$		0			0
$983$	25.0646			0.799436			0.399718	−	0.916638i	$-0.369108\pi$
$983$	25.0646			0.799436			0.399718	+	0.916638i	$0.369108\pi$
$984$		0			0
$985$		−	29.8136i		−	0.949940i
$986$		0			0
$987$	−8.85548	−	11.7574i	−0.281873	−	0.374242i
$988$		0			0
$989$			3.12101i			0.0992424i
$990$		0			0
$991$		−	10.3855i		−	0.329907i	−0.986301	−	0.164954i	$-0.947253\pi$
$991$		−	10.3855i		−	0.329907i	0.986301	−	0.164954i	$-0.0527474\pi$
$992$		0			0
$993$			9.46438i			0.300343i
$994$		0			0
$995$	12.2283			0.387662
$996$		0			0
$997$	−50.1424			−1.58803			−0.794013	−	0.607900i	$-0.792012\pi$
$997$	−50.1424			−1.58803			−0.794013	+	0.607900i	$0.792012\pi$
$998$		0			0
$999$	1.65381			0.0523242

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.p.a.559.5		16
3.2	odd	2		2016.2.p.g.559.8		16
4.3	odd	2		168.2.p.a.139.10	yes	16
7.6	odd	2	inner	672.2.p.a.559.12		16
8.3	odd	2	inner	672.2.p.a.559.4		16
8.5	even	2		168.2.p.a.139.12	yes	16
12.11	even	2		504.2.p.g.307.7		16
21.20	even	2		2016.2.p.g.559.10		16
24.5	odd	2		504.2.p.g.307.6		16
24.11	even	2		2016.2.p.g.559.9		16
28.27	even	2		168.2.p.a.139.9	✓	16
56.13	odd	2		168.2.p.a.139.11	yes	16
56.27	even	2	inner	672.2.p.a.559.13		16
84.83	odd	2		504.2.p.g.307.8		16
168.83	odd	2		2016.2.p.g.559.7		16
168.125	even	2		504.2.p.g.307.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.p.a.139.9	✓	16	28.27	even	2
168.2.p.a.139.10	yes	16	4.3	odd	2
168.2.p.a.139.11	yes	16	56.13	odd	2
168.2.p.a.139.12	yes	16	8.5	even	2
504.2.p.g.307.5		16	168.125	even	2
504.2.p.g.307.6		16	24.5	odd	2
504.2.p.g.307.7		16	12.11	even	2
504.2.p.g.307.8		16	84.83	odd	2
672.2.p.a.559.4		16	8.3	odd	2	inner
672.2.p.a.559.5		16	1.1	even	1	trivial
672.2.p.a.559.12		16	7.6	odd	2	inner
672.2.p.a.559.13		16	56.27	even	2	inner
2016.2.p.g.559.7		16	168.83	odd	2
2016.2.p.g.559.8		16	3.2	odd	2
2016.2.p.g.559.9		16	24.11	even	2
2016.2.p.g.559.10		16	21.20	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +1.12786 q^{5} +(-2.11337 + 1.59175i) q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +1.12786 q^{5} +(-2.11337 + 1.59175i) q^{7} -1.00000 q^{9} -5.11151 q^{11} -5.88054 q^{13} -1.12786i q^{15} -3.31623i q^{17} -7.49510i q^{19} +(1.59175 + 2.11337i) q^{21} -1.73845i q^{23} -3.72792 q^{25} +1.00000i q^{27} +5.88054i q^{29} +6.04982 q^{31} +5.11151i q^{33} +(-2.38359 + 1.79528i) q^{35} -1.65381i q^{37} +5.88054i q^{39} -1.45096i q^{41} -1.79528 q^{43} -1.12786 q^{45} -5.56335 q^{47} +(1.93264 - 6.72792i) q^{49} -3.31623 q^{51} +3.62481i q^{53} -5.76510 q^{55} -7.49510 q^{57} +0.767184i q^{59} -0.317194 q^{61} +(2.11337 - 1.59175i) q^{63} -6.63246 q^{65} +6.56247 q^{67} -1.73845 q^{69} +10.1919i q^{71} -6.63246i q^{73} +3.72792i q^{75} +(10.8025 - 8.13627i) q^{77} -3.01423i q^{79} +1.00000 q^{81} +16.0883i q^{83} -3.74026i q^{85} +5.88054 q^{87} +8.08341i q^{89} +(12.4277 - 9.36038i) q^{91} -6.04982i q^{93} -8.45347i q^{95} -0.357752i q^{97} +5.11151 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{9}+O(q^{10}) \) 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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