LMFDB - Embedded newform 252.2.t.a.89.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,2,Mod(17,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 1]))

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

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

chi := DirichletCharacter("252.17");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	252.t (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$2.01223013094$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			89.2
Root			$-1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	252.89
Dual form			252.2.t.a.17.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 - 2.12132i) q^{5} +(0.500000 + 2.59808i) q^{7} +O(q^{10})$	$q+(1.22474 - 2.12132i) q^{5} +(0.500000 + 2.59808i) q^{7} +(3.67423 - 2.12132i) q^{11} -1.73205i q^{13} +(-2.44949 - 4.24264i) q^{17} +(4.50000 + 2.59808i) q^{19} +(-0.500000 - 0.866025i) q^{25} +8.48528i q^{29} +(-1.50000 + 0.866025i) q^{31} +(6.12372 + 2.12132i) q^{35} +(2.50000 - 4.33013i) q^{37} -12.2474 q^{41} -11.0000 q^{43} +(1.22474 - 2.12132i) q^{47} +(-6.50000 + 2.59808i) q^{49} +(-7.34847 + 4.24264i) q^{53} -10.3923i q^{55} +(2.44949 + 4.24264i) q^{59} +(-3.00000 - 1.73205i) q^{61} +(-3.67423 - 2.12132i) q^{65} +(3.50000 + 6.06218i) q^{67} +12.7279i q^{71} +(13.5000 - 7.79423i) q^{73} +(7.34847 + 8.48528i) q^{77} +(-5.50000 + 9.52628i) q^{79} -12.2474 q^{83} -12.0000 q^{85} +(7.34847 - 12.7279i) q^{89} +(4.50000 - 0.866025i) q^{91} +(11.0227 - 6.36396i) q^{95} -3.46410i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{7}+O(q^{10}) $ 4 * q + 2 * q^7	$ 4 q + 2 q^{7} + 18 q^{19} - 2 q^{25} - 6 q^{31} + 10 q^{37} - 44 q^{43} - 26 q^{49} - 12 q^{61} + 14 q^{67} + 54 q^{73} - 22 q^{79} - 48 q^{85} + 18 q^{91}+O(q^{100}) $ 4 * q + 2 * q^7 + 18 * q^19 - 2 * q^25 - 6 * q^31 + 10 * q^37 - 44 * q^43 - 26 * q^49 - 12 * q^61 + 14 * q^67 + 54 * q^73 - 22 * q^79 - 48 * q^85 + 18 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.22474	−	2.12132i	0.547723	−	0.948683i	−0.450708	−	0.892672i	$-0.648828\pi$
$5$	1.22474	−	2.12132i	0.547723	−	0.948683i	0.998430	−	0.0560116i	$-0.0178384\pi$
$6$		0			0
$7$	0.500000	+	2.59808i	0.188982	+	0.981981i
$7$	0.500000	+	2.59808i	0.188982	+	0.981981i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.67423	−	2.12132i	1.10782	−	0.639602i	0.169559	−	0.985520i	$-0.445766\pi$
$11$	3.67423	−	2.12132i	1.10782	−	0.639602i	0.938265	+	0.345918i	$0.112432\pi$
$12$		0			0
$13$		−	1.73205i		−	0.480384i	−0.970725	−	0.240192i	$-0.922790\pi$
$13$		−	1.73205i		−	0.480384i	0.970725	−	0.240192i	$-0.0772105\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.44949	−	4.24264i	−0.594089	−	1.02899i	−0.993675	−	0.112296i	$-0.964180\pi$
$17$	−2.44949	−	4.24264i	−0.594089	−	1.02899i	0.399586	−	0.916696i	$-0.369154\pi$
$18$		0			0
$19$	4.50000	+	2.59808i	1.03237	+	0.596040i	0.917663	−	0.397360i	$-0.130073\pi$
$19$	4.50000	+	2.59808i	1.03237	+	0.596040i	0.114708	+	0.993399i	$0.463407\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			8.48528i			1.57568i	0.615882	+	0.787839i	$0.288800\pi$
$29$			8.48528i			1.57568i	−0.615882	+	0.787839i	$0.711200\pi$
$30$		0			0
$31$	−1.50000	+	0.866025i	−0.269408	+	0.155543i	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−1.50000	+	0.866025i	−0.269408	+	0.155543i	0.359211	+	0.933257i	$0.383046\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	6.12372	+	2.12132i	1.03510	+	0.358569i
$36$		0			0
$37$	2.50000	−	4.33013i	0.410997	−	0.711868i	−0.584002	−	0.811752i	$-0.698514\pi$
$37$	2.50000	−	4.33013i	0.410997	−	0.711868i	0.994999	+	0.0998840i	$0.0318472\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−12.2474			−1.91273			−0.956365	−	0.292174i	$-0.905621\pi$
$41$	−12.2474			−1.91273			−0.956365	+	0.292174i	$0.905621\pi$
$42$		0			0
$43$	−11.0000			−1.67748			−0.838742	−	0.544529i	$-0.816708\pi$
$43$	−11.0000			−1.67748			−0.838742	+	0.544529i	$0.816708\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.22474	−	2.12132i	0.178647	−	0.309426i	−0.762770	−	0.646670i	$-0.776161\pi$
$47$	1.22474	−	2.12132i	0.178647	−	0.309426i	0.941417	+	0.337243i	$0.109495\pi$
$48$		0			0
$49$	−6.50000	+	2.59808i	−0.928571	+	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−7.34847	+	4.24264i	−1.00939	+	0.582772i	−0.911013	−	0.412378i	$-0.864698\pi$
$53$	−7.34847	+	4.24264i	−1.00939	+	0.582772i	−0.0983769	+	0.995149i	$0.531365\pi$
$54$		0			0
$55$		−	10.3923i		−	1.40130i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.44949	+	4.24264i	0.318896	+	0.552345i	0.980258	−	0.197722i	$-0.0633545\pi$
$59$	2.44949	+	4.24264i	0.318896	+	0.552345i	−0.661362	+	0.750067i	$0.730021\pi$
$60$		0			0
$61$	−3.00000	−	1.73205i	−0.384111	−	0.221766i	0.295495	−	0.955344i	$-0.404516\pi$
$61$	−3.00000	−	1.73205i	−0.384111	−	0.221766i	−0.679605	+	0.733578i	$0.737849\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.67423	−	2.12132i	−0.455733	−	0.263117i
$66$		0			0
$67$	3.50000	+	6.06218i	0.427593	+	0.740613i	0.996659	−	0.0816792i	$-0.0260283\pi$
$67$	3.50000	+	6.06218i	0.427593	+	0.740613i	−0.569066	+	0.822292i	$0.692695\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			12.7279i			1.51053i	0.655422	+	0.755263i	$0.272491\pi$
$71$			12.7279i			1.51053i	−0.655422	+	0.755263i	$0.727509\pi$
$72$		0			0
$73$	13.5000	−	7.79423i	1.58006	−	0.912245i	0.585206	−	0.810885i	$-0.301014\pi$
$73$	13.5000	−	7.79423i	1.58006	−	0.912245i	0.994850	−	0.101361i	$-0.0323196\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	7.34847	+	8.48528i	0.837436	+	0.966988i
$78$		0			0
$79$	−5.50000	+	9.52628i	−0.618798	+	1.07179i	0.370907	+	0.928670i	$0.379047\pi$
$79$	−5.50000	+	9.52628i	−0.618798	+	1.07179i	−0.989705	+	0.143120i	$0.954286\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−12.2474			−1.34433			−0.672166	−	0.740400i	$-0.734636\pi$
$83$	−12.2474			−1.34433			−0.672166	+	0.740400i	$0.734636\pi$
$84$		0			0
$85$	−12.0000			−1.30158
$86$		0			0
$87$		0			0
$88$		0			0
$89$	7.34847	−	12.7279i	0.778936	−	1.34916i	−0.153619	−	0.988130i	$-0.549093\pi$
$89$	7.34847	−	12.7279i	0.778936	−	1.34916i	0.932555	−	0.361027i	$-0.117574\pi$
$90$		0			0
$91$	4.50000	−	0.866025i	0.471728	−	0.0907841i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	11.0227	−	6.36396i	1.13091	−	0.652929i
$96$		0			0
$97$		−	3.46410i		−	0.351726i	−0.984415	−	0.175863i	$-0.943728\pi$
$97$		−	3.46410i		−	0.351726i	0.984415	−	0.175863i	$-0.0562716\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−3.67423	−	6.36396i	−0.365600	−	0.633238i	0.623272	−	0.782005i	$-0.285803\pi$
$101$	−3.67423	−	6.36396i	−0.365600	−	0.633238i	−0.988872	+	0.148767i	$0.952470\pi$
$102$		0			0
$103$	7.50000	+	4.33013i	0.738997	+	0.426660i	0.821705	−	0.569914i	$-0.193023\pi$
$103$	7.50000	+	4.33013i	0.738997	+	0.426660i	−0.0827075	+	0.996574i	$0.526357\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.34847	−	4.24264i	−0.710403	−	0.410152i	0.100807	−	0.994906i	$-0.467858\pi$
$107$	−7.34847	−	4.24264i	−0.710403	−	0.410152i	−0.811210	+	0.584754i	$0.801191\pi$
$108$		0			0
$109$	2.50000	+	4.33013i	0.239457	+	0.414751i	0.960558	−	0.278078i	$-0.0896974\pi$
$109$	2.50000	+	4.33013i	0.239457	+	0.414751i	−0.721102	+	0.692829i	$0.756364\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	4.24264i		−	0.399114i	−0.979886	−	0.199557i	$-0.936050\pi$
$113$		−	4.24264i		−	0.399114i	0.979886	−	0.199557i	$-0.0639503\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	9.79796	−	8.48528i	0.898177	−	0.777844i
$120$		0			0
$121$	3.50000	−	6.06218i	0.318182	−	0.551107i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	9.79796			0.876356
$126$		0			0
$127$	5.00000			0.443678			0.221839	−	0.975083i	$-0.428794\pi$
$127$	5.00000			0.443678			0.221839	+	0.975083i	$0.428794\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	3.67423	−	6.36396i	0.321019	−	0.556022i	−0.659679	−	0.751547i	$-0.729308\pi$
$131$	3.67423	−	6.36396i	0.321019	−	0.556022i	0.980699	+	0.195525i	$0.0626412\pi$
$132$		0			0
$133$	−4.50000	+	12.9904i	−0.390199	+	1.12641i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$		0			0
$139$			1.73205i			0.146911i	0.997299	+	0.0734553i	$0.0234026\pi$
$139$			1.73205i			0.146911i	−0.997299	+	0.0734553i	$0.976597\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.67423	−	6.36396i	−0.307255	−	0.532181i
$144$		0			0
$145$	18.0000	+	10.3923i	1.49482	+	0.863034i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	14.6969	+	8.48528i	1.20402	+	0.695141i	0.961447	−	0.274992i	$-0.0886751\pi$
$149$	14.6969	+	8.48528i	1.20402	+	0.695141i	0.242574	+	0.970133i	$0.422008\pi$
$150$		0			0
$151$	1.00000	+	1.73205i	0.0813788	+	0.140952i	0.903842	−	0.427865i	$-0.140734\pi$
$151$	1.00000	+	1.73205i	0.0813788	+	0.140952i	−0.822464	+	0.568818i	$0.807401\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			4.24264i			0.340777i
$156$		0			0
$157$	−9.00000	+	5.19615i	−0.718278	+	0.414698i	−0.814119	−	0.580699i	$-0.802779\pi$
$157$	−9.00000	+	5.19615i	−0.718278	+	0.414698i	0.0958404	+	0.995397i	$0.469446\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	11.0000	−	19.0526i	0.861586	−	1.49231i	−0.00881059	−	0.999961i	$-0.502805\pi$
$163$	11.0000	−	19.0526i	0.861586	−	1.49231i	0.870397	−	0.492350i	$-0.163862\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−17.1464			−1.32683			−0.663415	−	0.748251i	$-0.730894\pi$
$167$	−17.1464			−1.32683			−0.663415	+	0.748251i	$0.730894\pi$
$168$		0			0
$169$	10.0000			0.769231
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$	2.00000	−	1.73205i	0.151186	−	0.130931i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	11.0227	−	6.36396i	0.823876	−	0.475665i	−0.0278755	−	0.999611i	$-0.508874\pi$
$179$	11.0227	−	6.36396i	0.823876	−	0.475665i	0.851751	+	0.523947i	$0.175541\pi$
$180$		0			0
$181$			5.19615i			0.386227i	0.981176	+	0.193113i	$0.0618586\pi$
$181$			5.19615i			0.386227i	−0.981176	+	0.193113i	$0.938141\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−6.12372	−	10.6066i	−0.450225	−	0.779813i
$186$		0			0
$187$	−18.0000	−	10.3923i	−1.31629	−	0.759961i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−3.67423	−	2.12132i	−0.265858	−	0.153493i	0.361146	−	0.932509i	$-0.382386\pi$
$191$	−3.67423	−	2.12132i	−0.265858	−	0.153493i	−0.627004	+	0.779016i	$0.715719\pi$
$192$		0			0
$193$	−9.50000	−	16.4545i	−0.683825	−	1.18442i	−0.973805	−	0.227387i	$-0.926982\pi$
$193$	−9.50000	−	16.4545i	−0.683825	−	1.18442i	0.289980	−	0.957033i	$-0.406351\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			8.48528i			0.604551i	0.953221	+	0.302276i	$0.0977463\pi$
$197$			8.48528i			0.604551i	−0.953221	+	0.302276i	$0.902254\pi$
$198$		0			0
$199$	−12.0000	+	6.92820i	−0.850657	+	0.491127i	−0.860873	−	0.508821i	$-0.830082\pi$
$199$	−12.0000	+	6.92820i	−0.850657	+	0.491127i	0.0102152	+	0.999948i	$0.496748\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−22.0454	+	4.24264i	−1.54728	+	0.297775i
$204$		0			0
$205$	−15.0000	+	25.9808i	−1.04765	+	1.81458i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	22.0454			1.52491
$210$		0			0
$211$	−2.00000			−0.137686			−0.0688428	−	0.997628i	$-0.521931\pi$
$211$	−2.00000			−0.137686			−0.0688428	+	0.997628i	$0.521931\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−13.4722	+	23.3345i	−0.918796	+	1.59140i
$216$		0			0
$217$	−3.00000	−	3.46410i	−0.203653	−	0.235159i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−7.34847	+	4.24264i	−0.494312	+	0.285391i
$222$		0			0
$223$		−	20.7846i		−	1.39184i	−0.718119	−	0.695920i	$-0.754997\pi$
$223$		−	20.7846i		−	1.39184i	0.718119	−	0.695920i	$-0.245003\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	11.0227	+	19.0919i	0.731603	+	1.26717i	0.956198	+	0.292721i	$0.0945606\pi$
$227$	11.0227	+	19.0919i	0.731603	+	1.26717i	−0.224595	+	0.974452i	$0.572106\pi$
$228$		0			0
$229$	1.50000	+	0.866025i	0.0991228	+	0.0572286i	0.548742	−	0.835992i	$-0.315107\pi$
$229$	1.50000	+	0.866025i	0.0991228	+	0.0572286i	−0.449619	+	0.893220i	$0.648440\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−3.67423	−	2.12132i	−0.240707	−	0.138972i	0.374795	−	0.927108i	$-0.377713\pi$
$233$	−3.67423	−	2.12132i	−0.240707	−	0.138972i	−0.615502	+	0.788136i	$0.711047\pi$
$234$		0			0
$235$	−3.00000	−	5.19615i	−0.195698	−	0.338960i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	4.24264i		−	0.274434i	−0.990541	−	0.137217i	$-0.956184\pi$
$239$		−	4.24264i		−	0.274434i	0.990541	−	0.137217i	$-0.0438157\pi$
$240$		0			0
$241$	12.0000	−	6.92820i	0.772988	−	0.446285i	−0.0609515	−	0.998141i	$-0.519414\pi$
$241$	12.0000	−	6.92820i	0.772988	−	0.446285i	0.833939	+	0.551856i	$0.186080\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.44949	+	16.9706i	−0.156492	+	1.08421i
$246$		0			0
$247$	4.50000	−	7.79423i	0.286328	−	0.495935i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−9.79796			−0.618442			−0.309221	−	0.950990i	$-0.600068\pi$
$251$	−9.79796			−0.618442			−0.309221	+	0.950990i	$0.600068\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3.67423	+	6.36396i	−0.229192	+	0.396973i	−0.957569	−	0.288204i	$-0.906942\pi$
$257$	−3.67423	+	6.36396i	−0.229192	+	0.396973i	0.728377	+	0.685177i	$0.240275\pi$
$258$		0			0
$259$	12.5000	+	4.33013i	0.776712	+	0.269061i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−22.0454	+	12.7279i	−1.35938	+	0.784837i	−0.989540	−	0.144259i	$-0.953920\pi$
$263$	−22.0454	+	12.7279i	−1.35938	+	0.784837i	−0.369838	+	0.929096i	$0.620587\pi$
$264$		0			0
$265$			20.7846i			1.27679i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	6.12372	+	10.6066i	0.373370	+	0.646696i	0.990082	−	0.140493i	$-0.0448688\pi$
$269$	6.12372	+	10.6066i	0.373370	+	0.646696i	−0.616712	+	0.787189i	$0.711536\pi$
$270$		0			0
$271$	−24.0000	−	13.8564i	−1.45790	−	0.841717i	−0.458988	−	0.888442i	$-0.651788\pi$
$271$	−24.0000	−	13.8564i	−1.45790	−	0.841717i	−0.998908	+	0.0467255i	$0.985121\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3.67423	−	2.12132i	−0.221565	−	0.127920i
$276$		0			0
$277$	−5.50000	−	9.52628i	−0.330463	−	0.572379i	0.652140	−	0.758099i	$-0.273872\pi$
$277$	−5.50000	−	9.52628i	−0.330463	−	0.572379i	−0.982603	+	0.185720i	$0.940538\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	16.9706i		−	1.01238i	−0.862422	−	0.506189i	$-0.831054\pi$
$281$		−	16.9706i		−	1.01238i	0.862422	−	0.506189i	$-0.168946\pi$
$282$		0			0
$283$	−7.50000	+	4.33013i	−0.445829	+	0.257399i	−0.706067	−	0.708145i	$-0.749532\pi$
$283$	−7.50000	+	4.33013i	−0.445829	+	0.257399i	0.260238	+	0.965544i	$0.416199\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−6.12372	−	31.8198i	−0.361472	−	1.87826i
$288$		0			0
$289$	−3.50000	+	6.06218i	−0.205882	+	0.356599i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	24.4949			1.43101			0.715504	−	0.698609i	$-0.246197\pi$
$293$	24.4949			1.43101			0.715504	+	0.698609i	$0.246197\pi$
$294$		0			0
$295$	12.0000			0.698667
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−5.50000	−	28.5788i	−0.317015	−	1.64726i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−7.34847	+	4.24264i	−0.420772	+	0.242933i
$306$		0			0
$307$			19.0526i			1.08739i	0.839284	+	0.543693i	$0.182975\pi$
$307$			19.0526i			1.08739i	−0.839284	+	0.543693i	$0.817025\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	8.57321	+	14.8492i	0.486142	+	0.842023i	0.999873	−	0.0159282i	$-0.00507031\pi$
$311$	8.57321	+	14.8492i	0.486142	+	0.842023i	−0.513731	+	0.857951i	$0.671737\pi$
$312$		0			0
$313$	−13.5000	−	7.79423i	−0.763065	−	0.440556i	0.0673300	−	0.997731i	$-0.478552\pi$
$313$	−13.5000	−	7.79423i	−0.763065	−	0.440556i	−0.830395	+	0.557175i	$0.811885\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	7.34847	+	4.24264i	0.412731	+	0.238290i	0.691963	−	0.721933i	$-0.256746\pi$
$317$	7.34847	+	4.24264i	0.412731	+	0.238290i	−0.279231	+	0.960224i	$0.590080\pi$
$318$		0			0
$319$	18.0000	+	31.1769i	1.00781	+	1.74557i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	25.4558i		−	1.41640i
$324$		0			0
$325$	−1.50000	+	0.866025i	−0.0832050	+	0.0480384i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	6.12372	+	2.12132i	0.337612	+	0.116952i
$330$		0			0
$331$	2.50000	−	4.33013i	0.137412	−	0.238005i	−0.789104	−	0.614260i	$-0.789455\pi$
$331$	2.50000	−	4.33013i	0.137412	−	0.238005i	0.926516	+	0.376254i	$0.122788\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	17.1464			0.936809
$336$		0			0
$337$	7.00000			0.381314			0.190657	−	0.981657i	$-0.438938\pi$
$337$	7.00000			0.381314			0.190657	+	0.981657i	$0.438938\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−3.67423	+	6.36396i	−0.198971	+	0.344628i
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	7.34847	−	4.24264i	0.394486	−	0.227757i	−0.289616	−	0.957143i	$-0.593528\pi$
$347$	7.34847	−	4.24264i	0.394486	−	0.227757i	0.684102	+	0.729386i	$0.260194\pi$
$348$		0			0
$349$			24.2487i			1.29800i	0.760787	+	0.649002i	$0.224813\pi$
$349$			24.2487i			1.29800i	−0.760787	+	0.649002i	$0.775187\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−13.4722	−	23.3345i	−0.717053	−	1.24197i	−0.962163	−	0.272476i	$-0.912157\pi$
$353$	−13.4722	−	23.3345i	−0.717053	−	1.24197i	0.245110	−	0.969495i	$-0.421176\pi$
$354$		0			0
$355$	27.0000	+	15.5885i	1.43301	+	0.827349i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	14.6969	+	8.48528i	0.775675	+	0.447836i	0.834895	−	0.550409i	$-0.185528\pi$
$359$	14.6969	+	8.48528i	0.775675	+	0.447836i	−0.0592205	+	0.998245i	$0.518862\pi$
$360$		0			0
$361$	4.00000	+	6.92820i	0.210526	+	0.364642i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	38.1838i		−	1.99863i
$366$		0			0
$367$	22.5000	−	12.9904i	1.17449	−	0.678092i	0.219757	−	0.975555i	$-0.429474\pi$
$367$	22.5000	−	12.9904i	1.17449	−	0.678092i	0.954734	+	0.297462i	$0.0961403\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−14.6969	−	16.9706i	−0.763027	−	0.881068i
$372$		0			0
$373$	3.50000	−	6.06218i	0.181223	−	0.313888i	−0.761074	−	0.648665i	$-0.775328\pi$
$373$	3.50000	−	6.06218i	0.181223	−	0.313888i	0.942297	+	0.334777i	$0.108661\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	14.6969			0.756931
$378$		0			0
$379$	−5.00000			−0.256833			−0.128416	−	0.991720i	$-0.540989\pi$
$379$	−5.00000			−0.256833			−0.128416	+	0.991720i	$0.540989\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−4.89898	+	8.48528i	−0.250326	+	0.433578i	−0.963616	−	0.267292i	$-0.913871\pi$
$383$	−4.89898	+	8.48528i	−0.250326	+	0.433578i	0.713289	+	0.700870i	$0.247205\pi$
$384$		0			0
$385$	27.0000	−	5.19615i	1.37605	−	0.264820i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3.67423	−	2.12132i	0.186291	−	0.107555i	−0.403954	−	0.914779i	$-0.632364\pi$
$389$	3.67423	−	2.12132i	0.186291	−	0.107555i	0.590245	+	0.807224i	$0.299031\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	13.4722	+	23.3345i	0.677860	+	1.17409i
$396$		0			0
$397$	7.50000	+	4.33013i	0.376414	+	0.217323i	0.676257	−	0.736666i	$-0.263601\pi$
$397$	7.50000	+	4.33013i	0.376414	+	0.217323i	−0.299843	+	0.953989i	$0.596934\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−7.34847	−	4.24264i	−0.366965	−	0.211867i	0.305167	−	0.952299i	$-0.401288\pi$
$401$	−7.34847	−	4.24264i	−0.366965	−	0.211867i	−0.672132	+	0.740432i	$0.734621\pi$
$402$		0			0
$403$	1.50000	+	2.59808i	0.0747203	+	0.129419i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	21.2132i		−	1.05150i
$408$		0			0
$409$	−7.50000	+	4.33013i	−0.370851	+	0.214111i	−0.673830	−	0.738886i	$-0.735352\pi$
$409$	−7.50000	+	4.33013i	−0.370851	+	0.214111i	0.302979	+	0.952997i	$0.402019\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−9.79796	+	8.48528i	−0.482126	+	0.417533i
$414$		0			0
$415$	−15.0000	+	25.9808i	−0.736321	+	1.27535i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	26.9444			1.31632			0.658160	−	0.752878i	$-0.271335\pi$
$419$	26.9444			1.31632			0.658160	+	0.752878i	$0.271335\pi$
$420$		0			0
$421$	35.0000			1.70580			0.852898	−	0.522078i	$-0.174843\pi$
$421$	35.0000			1.70580			0.852898	+	0.522078i	$0.174843\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2.44949	+	4.24264i	−0.118818	+	0.205798i
$426$		0			0
$427$	3.00000	−	8.66025i	0.145180	−	0.419099i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	18.3712	−	10.6066i	0.884908	−	0.510902i	0.0126347	−	0.999920i	$-0.495978\pi$
$431$	18.3712	−	10.6066i	0.884908	−	0.510902i	0.872274	+	0.489018i	$0.162645\pi$
$432$		0			0
$433$			12.1244i			0.582659i	0.956623	+	0.291330i	$0.0940977\pi$
$433$			12.1244i			0.582659i	−0.956623	+	0.291330i	$0.905902\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−36.0000	−	20.7846i	−1.71819	−	0.991995i	−0.922231	−	0.386640i	$-0.873635\pi$
$439$	−36.0000	−	20.7846i	−1.71819	−	0.991995i	−0.795956	−	0.605355i	$-0.793031\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$		0			0
$445$	−18.0000	−	31.1769i	−0.853282	−	1.47793i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	12.7279i		−	0.600668i	−0.953834	−	0.300334i	$-0.902902\pi$
$449$		−	12.7279i		−	0.600668i	0.953834	−	0.300334i	$-0.0970981\pi$
$450$		0			0
$451$	−45.0000	+	25.9808i	−2.11897	+	1.22339i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	3.67423	−	10.6066i	0.172251	−	0.497245i
$456$		0			0
$457$	−6.50000	+	11.2583i	−0.304057	+	0.526642i	−0.977051	−	0.213006i	$-0.931675\pi$
$457$	−6.50000	+	11.2583i	−0.304057	+	0.526642i	0.672994	+	0.739648i	$0.265008\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−4.89898			−0.228168			−0.114084	−	0.993471i	$-0.536393\pi$
$461$	−4.89898			−0.228168			−0.114084	+	0.993471i	$0.536393\pi$
$462$		0			0
$463$	29.0000			1.34774			0.673872	−	0.738848i	$-0.264630\pi$
$463$	29.0000			1.34774			0.673872	+	0.738848i	$0.264630\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	3.67423	−	6.36396i	0.170023	−	0.294489i	−0.768404	−	0.639965i	$-0.778949\pi$
$467$	3.67423	−	6.36396i	0.170023	−	0.294489i	0.938428	+	0.345476i	$0.112282\pi$
$468$		0			0
$469$	−14.0000	+	12.1244i	−0.646460	+	0.559851i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−40.4166	+	23.3345i	−1.85836	+	1.07292i
$474$		0			0
$475$		−	5.19615i		−	0.238416i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	2.44949	+	4.24264i	0.111920	+	0.193851i	0.916544	−	0.399933i	$-0.130967\pi$
$479$	2.44949	+	4.24264i	0.111920	+	0.193851i	−0.804624	+	0.593784i	$0.797633\pi$
$480$		0			0
$481$	−7.50000	−	4.33013i	−0.341971	−	0.197437i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−7.34847	−	4.24264i	−0.333677	−	0.192648i
$486$		0			0
$487$	−3.50000	−	6.06218i	−0.158600	−	0.274703i	0.775764	−	0.631023i	$-0.217365\pi$
$487$	−3.50000	−	6.06218i	−0.158600	−	0.274703i	−0.934364	+	0.356320i	$0.884031\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			33.9411i			1.53174i	0.642995	+	0.765871i	$0.277692\pi$
$491$			33.9411i			1.53174i	−0.642995	+	0.765871i	$0.722308\pi$
$492$		0			0
$493$	36.0000	−	20.7846i	1.62136	−	0.936092i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−33.0681	+	6.36396i	−1.48331	+	0.285463i
$498$		0			0
$499$	17.5000	−	30.3109i	0.783408	−	1.35690i	−0.146538	−	0.989205i	$-0.546813\pi$
$499$	17.5000	−	30.3109i	0.783408	−	1.35690i	0.929946	−	0.367697i	$-0.119854\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	2.44949			0.109217			0.0546087	−	0.998508i	$-0.482609\pi$
$503$	2.44949			0.109217			0.0546087	+	0.998508i	$0.482609\pi$
$504$		0			0
$505$	−18.0000			−0.800989
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−8.57321	+	14.8492i	−0.380001	+	0.658181i	−0.991062	−	0.133402i	$-0.957410\pi$
$509$	−8.57321	+	14.8492i	−0.380001	+	0.658181i	0.611061	+	0.791584i	$0.290743\pi$
$510$		0			0
$511$	27.0000	+	31.1769i	1.19441	+	1.37919i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	18.3712	−	10.6066i	0.809531	−	0.467383i
$516$		0			0
$517$		−	10.3923i		−	0.457053i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	2.44949	+	4.24264i	0.107314	+	0.185873i	0.914681	−	0.404176i	$-0.132442\pi$
$521$	2.44949	+	4.24264i	0.107314	+	0.185873i	−0.807367	+	0.590049i	$0.799108\pi$
$522$		0			0
$523$	−7.50000	−	4.33013i	−0.327952	−	0.189343i	0.326979	−	0.945031i	$-0.393969\pi$
$523$	−7.50000	−	4.33013i	−0.327952	−	0.189343i	−0.654932	+	0.755688i	$0.727303\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	7.34847	+	4.24264i	0.320104	+	0.184812i
$528$		0			0
$529$	−11.5000	−	19.9186i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			21.2132i			0.918846i
$534$		0			0
$535$	−18.0000	+	10.3923i	−0.778208	+	0.449299i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−18.3712	+	23.3345i	−0.791302	+	1.00509i
$540$		0			0
$541$	9.50000	−	16.4545i	0.408437	−	0.707433i	−0.586278	−	0.810110i	$-0.699407\pi$
$541$	9.50000	−	16.4545i	0.408437	−	0.707433i	0.994715	+	0.102677i	$0.0327407\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	12.2474			0.524623
$546$		0			0
$547$	2.00000			0.0855138			0.0427569	−	0.999086i	$-0.486386\pi$
$547$	2.00000			0.0855138			0.0427569	+	0.999086i	$0.486386\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−22.0454	+	38.1838i	−0.939166	+	1.62668i
$552$		0			0
$553$	−27.5000	−	9.52628i	−1.16942	−	0.405099i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−18.3712	+	10.6066i	−0.778412	+	0.449416i	−0.835867	−	0.548932i	$-0.815035\pi$
$557$	−18.3712	+	10.6066i	−0.778412	+	0.449416i	0.0574555	+	0.998348i	$0.481701\pi$
$558$		0			0
$559$			19.0526i			0.805837i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−3.67423	−	6.36396i	−0.154851	−	0.268209i	0.778154	−	0.628073i	$-0.216156\pi$
$563$	−3.67423	−	6.36396i	−0.154851	−	0.268209i	−0.933005	+	0.359864i	$0.882823\pi$
$564$		0			0
$565$	−9.00000	−	5.19615i	−0.378633	−	0.218604i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	18.3712	+	10.6066i	0.770160	+	0.444652i	0.832932	−	0.553376i	$-0.186661\pi$
$569$	18.3712	+	10.6066i	0.770160	+	0.444652i	−0.0627719	+	0.998028i	$0.519994\pi$
$570$		0			0
$571$	3.50000	+	6.06218i	0.146470	+	0.253694i	0.929921	−	0.367760i	$-0.119875\pi$
$571$	3.50000	+	6.06218i	0.146470	+	0.253694i	−0.783450	+	0.621455i	$0.786542\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−34.5000	+	19.9186i	−1.43625	+	0.829222i	−0.997587	−	0.0694283i	$-0.977883\pi$
$577$	−34.5000	+	19.9186i	−1.43625	+	0.829222i	−0.438667	+	0.898650i	$0.644549\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−6.12372	−	31.8198i	−0.254055	−	1.32011i
$582$		0			0
$583$	−18.0000	+	31.1769i	−0.745484	+	1.29122i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−14.6969			−0.606608			−0.303304	−	0.952894i	$-0.598090\pi$
$587$	−14.6969			−0.606608			−0.303304	+	0.952894i	$0.598090\pi$
$588$		0			0
$589$	−9.00000			−0.370839
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.22474	+	2.12132i	−0.0502942	+	0.0871122i	−0.890077	−	0.455811i	$-0.849349\pi$
$593$	−1.22474	+	2.12132i	−0.0502942	+	0.0871122i	0.839782	+	0.542923i	$0.182683\pi$
$594$		0			0
$595$	−6.00000	−	31.1769i	−0.245976	−	1.27813i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	14.6969	−	8.48528i	0.600501	−	0.346699i	−0.168738	−	0.985661i	$-0.553969\pi$
$599$	14.6969	−	8.48528i	0.600501	−	0.346699i	0.769238	+	0.638962i	$0.220636\pi$
$600$		0			0
$601$		−	43.3013i		−	1.76630i	−0.469095	−	0.883148i	$-0.655420\pi$
$601$		−	43.3013i		−	1.76630i	0.469095	−	0.883148i	$-0.344580\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−8.57321	−	14.8492i	−0.348551	−	0.603708i
$606$		0			0
$607$	22.5000	+	12.9904i	0.913247	+	0.527263i	0.881474	−	0.472232i	$-0.156551\pi$
$607$	22.5000	+	12.9904i	0.913247	+	0.527263i	0.0317724	+	0.999495i	$0.489885\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−3.67423	−	2.12132i	−0.148644	−	0.0858194i
$612$		0			0
$613$	−4.00000	−	6.92820i	−0.161558	−	0.279827i	0.773869	−	0.633345i	$-0.218319\pi$
$613$	−4.00000	−	6.92820i	−0.161558	−	0.279827i	−0.935428	+	0.353518i	$0.884985\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			21.2132i			0.854011i	0.904249	+	0.427006i	$0.140432\pi$
$617$			21.2132i			0.854011i	−0.904249	+	0.427006i	$0.859568\pi$
$618$		0			0
$619$	7.50000	−	4.33013i	0.301450	−	0.174042i	−0.341644	−	0.939829i	$-0.610984\pi$
$619$	7.50000	−	4.33013i	0.301450	−	0.174042i	0.643094	+	0.765787i	$0.277650\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	36.7423	+	12.7279i	1.47205	+	0.509933i
$624$		0			0
$625$	14.5000	−	25.1147i	0.580000	−	1.00459i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−24.4949			−0.976676
$630$		0			0
$631$	−22.0000			−0.875806			−0.437903	−	0.899022i	$-0.644279\pi$
$631$	−22.0000			−0.875806			−0.437903	+	0.899022i	$0.644279\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	6.12372	−	10.6066i	0.243013	−	0.420910i
$636$		0			0
$637$	4.50000	+	11.2583i	0.178296	+	0.446071i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	7.34847	−	4.24264i	0.290247	−	0.167574i	−0.347806	−	0.937566i	$-0.613073\pi$
$641$	7.34847	−	4.24264i	0.290247	−	0.167574i	0.638053	+	0.769992i	$0.279740\pi$
$642$		0			0
$643$			32.9090i			1.29780i	0.760872	+	0.648901i	$0.224771\pi$
$643$			32.9090i			1.29780i	−0.760872	+	0.648901i	$0.775229\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−11.0227	−	19.0919i	−0.433347	−	0.750579i	0.563812	−	0.825903i	$-0.309334\pi$
$647$	−11.0227	−	19.0919i	−0.433347	−	0.750579i	−0.997159	+	0.0753238i	$0.976001\pi$
$648$		0			0
$649$	18.0000	+	10.3923i	0.706562	+	0.407934i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−18.3712	−	10.6066i	−0.718920	−	0.415068i	0.0954353	−	0.995436i	$-0.469576\pi$
$653$	−18.3712	−	10.6066i	−0.718920	−	0.415068i	−0.814355	+	0.580367i	$0.802909\pi$
$654$		0			0
$655$	−9.00000	−	15.5885i	−0.351659	−	0.609091i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	16.9706i		−	0.661079i	−0.943792	−	0.330540i	$-0.892769\pi$
$659$		−	16.9706i		−	0.661079i	0.943792	−	0.330540i	$-0.107231\pi$
$660$		0			0
$661$	4.50000	−	2.59808i	0.175030	−	0.101053i	−0.409926	−	0.912119i	$-0.634445\pi$
$661$	4.50000	−	2.59808i	0.175030	−	0.101053i	0.584955	+	0.811065i	$0.301112\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	22.0454	+	25.4558i	0.854884	+	0.987135i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−14.6969			−0.567369
$672$		0			0
$673$	19.0000			0.732396			0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000			0.732396			0.366198	+	0.930537i	$0.380659\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	14.6969	−	25.4558i	0.564849	−	0.978348i	−0.432214	−	0.901771i	$-0.642268\pi$
$677$	14.6969	−	25.4558i	0.564849	−	0.978348i	0.997064	−	0.0765767i	$-0.0243990\pi$
$678$		0			0
$679$	9.00000	−	1.73205i	0.345388	−	0.0664700i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−7.34847	+	4.24264i	−0.281181	+	0.162340i	−0.633958	−	0.773367i	$-0.718571\pi$
$683$	−7.34847	+	4.24264i	−0.281181	+	0.162340i	0.352777	+	0.935708i	$0.385238\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	7.34847	+	12.7279i	0.279954	+	0.484895i
$690$		0			0
$691$	16.5000	+	9.52628i	0.627690	+	0.362397i	0.779857	−	0.625958i	$-0.215292\pi$
$691$	16.5000	+	9.52628i	0.627690	+	0.362397i	−0.152167	+	0.988355i	$0.548625\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	3.67423	+	2.12132i	0.139372	+	0.0804663i
$696$		0			0
$697$	30.0000	+	51.9615i	1.13633	+	1.96818i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	33.9411i		−	1.28194i	−0.767567	−	0.640969i	$-0.778533\pi$
$701$		−	33.9411i		−	1.28194i	0.767567	−	0.640969i	$-0.221467\pi$
$702$		0			0
$703$	22.5000	−	12.9904i	0.848604	−	0.489942i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	14.6969	−	12.7279i	0.552735	−	0.478683i
$708$		0			0
$709$	−8.00000	+	13.8564i	−0.300446	+	0.520388i	−0.976237	−	0.216705i	$-0.930469\pi$
$709$	−8.00000	+	13.8564i	−0.300446	+	0.520388i	0.675791	+	0.737093i	$0.263802\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−18.0000			−0.673162
$716$		0			0
$717$		0			0
$718$		0			0
$719$	13.4722	−	23.3345i	0.502428	−	0.870231i	−0.497568	−	0.867425i	$-0.665774\pi$
$719$	13.4722	−	23.3345i	0.502428	−	0.870231i	0.999996	−	0.00280593i	$-0.000893157\pi$
$720$		0			0
$721$	−7.50000	+	21.6506i	−0.279315	+	0.806312i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	7.34847	−	4.24264i	0.272915	−	0.157568i
$726$		0			0
$727$		−	39.8372i		−	1.47748i	−0.673991	−	0.738739i	$-0.735421\pi$
$727$		−	39.8372i		−	1.47748i	0.673991	−	0.738739i	$-0.264579\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	26.9444	+	46.6690i	0.996574	+	1.72612i
$732$		0			0
$733$	7.50000	+	4.33013i	0.277019	+	0.159937i	0.632073	−	0.774909i	$-0.282204\pi$
$733$	7.50000	+	4.33013i	0.277019	+	0.159937i	−0.355054	+	0.934846i	$0.615538\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	25.7196	+	14.8492i	0.947395	+	0.546979i
$738$		0			0
$739$	9.50000	+	16.4545i	0.349463	+	0.605288i	0.986154	−	0.165831i	$-0.0530307\pi$
$739$	9.50000	+	16.4545i	0.349463	+	0.605288i	−0.636691	+	0.771119i	$0.719697\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			12.7279i			0.466942i	0.972364	+	0.233471i	$0.0750084\pi$
$743$			12.7279i			0.466942i	−0.972364	+	0.233471i	$0.924992\pi$
$744$		0			0
$745$	36.0000	−	20.7846i	1.31894	−	0.761489i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	7.34847	−	21.2132i	0.268507	−	0.775114i
$750$		0			0
$751$	2.50000	−	4.33013i	0.0912263	−	0.158009i	−0.816801	−	0.576919i	$-0.804255\pi$
$751$	2.50000	−	4.33013i	0.0912263	−	0.158009i	0.908027	+	0.418911i	$0.137588\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	4.89898			0.178292
$756$		0			0
$757$	−14.0000			−0.508839			−0.254419	−	0.967094i	$-0.581884\pi$
$757$	−14.0000			−0.508839			−0.254419	+	0.967094i	$0.581884\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	17.1464	−	29.6985i	0.621558	−	1.07657i	−0.367638	−	0.929969i	$-0.619834\pi$
$761$	17.1464	−	29.6985i	0.621558	−	1.07657i	0.989196	−	0.146600i	$-0.0468331\pi$
$762$		0			0
$763$	−10.0000	+	8.66025i	−0.362024	+	0.313522i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	7.34847	−	4.24264i	0.265338	−	0.153193i
$768$		0			0
$769$			15.5885i			0.562134i	0.959688	+	0.281067i	$0.0906883\pi$
$769$			15.5885i			0.562134i	−0.959688	+	0.281067i	$0.909312\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−20.8207	−	36.0624i	−0.748867	−	1.29708i	−0.948366	−	0.317178i	$-0.897265\pi$
$773$	−20.8207	−	36.0624i	−0.748867	−	1.29708i	0.199499	−	0.979898i	$-0.436069\pi$
$774$		0			0
$775$	1.50000	+	0.866025i	0.0538816	+	0.0311086i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−55.1135	−	31.8198i	−1.97465	−	1.14006i
$780$		0			0
$781$	27.0000	+	46.7654i	0.966136	+	1.67340i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			25.4558i			0.908558i
$786$		0			0
$787$	15.0000	−	8.66025i	0.534692	−	0.308705i	−0.208233	−	0.978079i	$-0.566771\pi$
$787$	15.0000	−	8.66025i	0.534692	−	0.308705i	0.742925	+	0.669375i	$0.233438\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	11.0227	−	2.12132i	0.391922	−	0.0754255i
$792$		0			0
$793$	−3.00000	+	5.19615i	−0.106533	+	0.184521i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$	−12.0000			−0.424529
$800$		0			0
$801$		0			0
$802$		0			0
$803$	33.0681	−	57.2756i	1.16695	−	2.02121i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	18.3712	−	10.6066i	0.645896	−	0.372908i	−0.140986	−	0.990012i	$-0.545027\pi$
$809$	18.3712	−	10.6066i	0.645896	−	0.372908i	0.786882	+	0.617103i	$0.211694\pi$
$810$		0			0
$811$		−	31.1769i		−	1.09477i	−0.836881	−	0.547385i	$-0.815623\pi$
$811$		−	31.1769i		−	1.09477i	0.836881	−	0.547385i	$-0.184377\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−26.9444	−	46.6690i	−0.943821	−	1.63475i
$816$		0			0
$817$	−49.5000	−	28.5788i	−1.73179	−	0.999847i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−18.3712	−	10.6066i	−0.641158	−	0.370173i	0.143902	−	0.989592i	$-0.454035\pi$
$821$	−18.3712	−	10.6066i	−0.641158	−	0.370173i	−0.785061	+	0.619419i	$0.787368\pi$
$822$		0			0
$823$	−13.0000	−	22.5167i	−0.453152	−	0.784881i	0.545428	−	0.838157i	$-0.316367\pi$
$823$	−13.0000	−	22.5167i	−0.453152	−	0.784881i	−0.998580	+	0.0532760i	$0.983034\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			29.6985i			1.03272i	0.856372	+	0.516359i	$0.172713\pi$
$827$			29.6985i			1.03272i	−0.856372	+	0.516359i	$0.827287\pi$
$828$		0			0
$829$	31.5000	−	18.1865i	1.09404	−	0.631644i	0.159391	−	0.987216i	$-0.449047\pi$
$829$	31.5000	−	18.1865i	1.09404	−	0.631644i	0.934649	+	0.355571i	$0.115714\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	26.9444	+	21.2132i	0.933568	+	0.734994i
$834$		0			0
$835$	−21.0000	+	36.3731i	−0.726735	+	1.25874i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−53.8888			−1.86045			−0.930224	−	0.366993i	$-0.880387\pi$
$839$	−53.8888			−1.86045			−0.930224	+	0.366993i	$0.880387\pi$
$840$		0			0
$841$	−43.0000			−1.48276
$842$		0			0
$843$		0			0
$844$		0			0
$845$	12.2474	−	21.2132i	0.421325	−	0.729756i
$846$		0			0
$847$	17.5000	+	6.06218i	0.601307	+	0.208299i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$			25.9808i			0.889564i	0.895639	+	0.444782i	$0.146719\pi$
$853$			25.9808i			0.889564i	−0.895639	+	0.444782i	$0.853281\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	19.5959	+	33.9411i	0.669384	+	1.15941i	0.978077	+	0.208245i	$0.0667751\pi$
$857$	19.5959	+	33.9411i	0.669384	+	1.15941i	−0.308693	+	0.951162i	$0.599892\pi$
$858$		0			0
$859$	39.0000	+	22.5167i	1.33066	+	0.768259i	0.985401	−	0.170248i	$-0.0544569\pi$
$859$	39.0000	+	22.5167i	1.33066	+	0.768259i	0.345262	+	0.938506i	$0.387790\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	33.0681	+	19.0919i	1.12565	+	0.649895i	0.942838	−	0.333252i	$-0.108146\pi$
$863$	33.0681	+	19.0919i	1.12565	+	0.649895i	0.182814	+	0.983148i	$0.441479\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$			46.6690i			1.58314i
$870$		0			0
$871$	10.5000	−	6.06218i	0.355779	−	0.205409i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	4.89898	+	25.4558i	0.165616	+	0.860565i
$876$		0			0
$877$	14.0000	−	24.2487i	0.472746	−	0.818821i	−0.526767	−	0.850010i	$-0.676596\pi$
$877$	14.0000	−	24.2487i	0.472746	−	0.818821i	0.999514	+	0.0311889i	$0.00992933\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−29.3939			−0.990305			−0.495152	−	0.868806i	$-0.664888\pi$
$881$	−29.3939			−0.990305			−0.495152	+	0.868806i	$0.664888\pi$
$882$		0			0
$883$	1.00000			0.0336527			0.0168263	−	0.999858i	$-0.494644\pi$
$883$	1.00000			0.0336527			0.0168263	+	0.999858i	$0.494644\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−28.1691	+	48.7904i	−0.945827	+	1.63822i	−0.191740	+	0.981446i	$0.561413\pi$
$887$	−28.1691	+	48.7904i	−0.945827	+	1.63822i	−0.754087	+	0.656774i	$0.771920\pi$
$888$		0			0
$889$	2.50000	+	12.9904i	0.0838473	+	0.435683i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	11.0227	−	6.36396i	0.368861	−	0.212962i
$894$		0			0
$895$		−	31.1769i		−	1.04213i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−7.34847	−	12.7279i	−0.245085	−	0.424500i
$900$		0			0
$901$	36.0000	+	20.7846i	1.19933	+	0.692436i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	11.0227	+	6.36396i	0.366407	+	0.211545i
$906$		0			0
$907$	−3.50000	−	6.06218i	−0.116216	−	0.201291i	0.802049	−	0.597258i	$-0.203743\pi$
$907$	−3.50000	−	6.06218i	−0.116216	−	0.201291i	−0.918265	+	0.395966i	$0.870410\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	8.48528i		−	0.281130i	−0.990071	−	0.140565i	$-0.955108\pi$
$911$		−	8.48528i		−	0.281130i	0.990071	−	0.140565i	$-0.0448919\pi$
$912$		0			0
$913$	−45.0000	+	25.9808i	−1.48928	+	0.859838i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	18.3712	+	6.36396i	0.606670	+	0.210157i
$918$		0			0
$919$	20.5000	−	35.5070i	0.676233	−	1.17127i	−0.299874	−	0.953979i	$-0.596945\pi$
$919$	20.5000	−	35.5070i	0.676233	−	1.17127i	0.976107	−	0.217291i	$-0.0697219\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	22.0454			0.725633
$924$		0			0
$925$	−5.00000			−0.164399
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−13.4722	+	23.3345i	−0.442008	+	0.765581i	−0.997838	−	0.0657150i	$-0.979067\pi$
$929$	−13.4722	+	23.3345i	−0.442008	+	0.765581i	0.555830	+	0.831296i	$0.312401\pi$
$930$		0			0
$931$	−36.0000	−	5.19615i	−1.17985	−	0.170297i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−44.0908	+	25.4558i	−1.44192	+	0.832495i
$936$		0			0
$937$		−	19.0526i		−	0.622420i	−0.950341	−	0.311210i	$-0.899266\pi$
$937$		−	19.0526i		−	0.622420i	0.950341	−	0.311210i	$-0.100734\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	9.79796	+	16.9706i	0.319404	+	0.553225i	0.980364	−	0.197197i	$-0.0631839\pi$
$941$	9.79796	+	16.9706i	0.319404	+	0.553225i	−0.660960	+	0.750422i	$0.729851\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−47.7650	−	27.5772i	−1.55216	−	0.896137i	−0.997966	−	0.0637469i	$-0.979695\pi$
$947$	−47.7650	−	27.5772i	−1.55216	−	0.896137i	−0.554189	−	0.832391i	$-0.686972\pi$
$948$		0			0
$949$	−13.5000	−	23.3827i	−0.438229	−	0.759034i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			33.9411i			1.09946i	0.835342	+	0.549730i	$0.185270\pi$
$953$			33.9411i			1.09946i	−0.835342	+	0.549730i	$0.814730\pi$
$954$		0			0
$955$	−9.00000	+	5.19615i	−0.291233	+	0.168144i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−14.0000	+	24.2487i	−0.451613	+	0.782216i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−46.5403			−1.49819
$966$		0			0
$967$	−43.0000			−1.38279			−0.691393	−	0.722478i	$-0.743003\pi$
$967$	−43.0000			−1.38279			−0.691393	+	0.722478i	$0.743003\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−4.89898	+	8.48528i	−0.157216	+	0.272306i	−0.933864	−	0.357629i	$-0.883585\pi$
$971$	−4.89898	+	8.48528i	−0.157216	+	0.272306i	0.776648	+	0.629935i	$0.216918\pi$
$972$		0			0
$973$	−4.50000	+	0.866025i	−0.144263	+	0.0277635i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	33.0681	−	19.0919i	1.05794	−	0.610803i	0.133080	−	0.991105i	$-0.457513\pi$
$977$	33.0681	−	19.0919i	1.05794	−	0.610803i	0.924862	+	0.380302i	$0.124180\pi$
$978$		0			0
$979$		−	62.3538i		−	1.99284i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−26.9444	−	46.6690i	−0.859392	−	1.48851i	−0.872509	−	0.488597i	$-0.837509\pi$
$983$	−26.9444	−	46.6690i	−0.859392	−	1.48851i	0.0131168	−	0.999914i	$-0.495825\pi$
$984$		0			0
$985$	18.0000	+	10.3923i	0.573528	+	0.331126i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−6.50000	−	11.2583i	−0.206479	−	0.357633i	0.744124	−	0.668042i	$-0.232867\pi$
$991$	−6.50000	−	11.2583i	−0.206479	−	0.357633i	−0.950603	+	0.310409i	$0.899534\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			33.9411i			1.07601i
$996$		0			0
$997$	−43.5000	+	25.1147i	−1.37766	+	0.795392i	−0.991877	−	0.127198i	$-0.959401\pi$
$997$	−43.5000	+	25.1147i	−1.37766	+	0.795392i	−0.385782	+	0.922590i	$0.626068\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.2.t.a.89.2	yes	4
3.2	odd	2	inner	252.2.t.a.89.1	yes	4
4.3	odd	2		1008.2.bt.a.593.2		4
5.2	odd	4		6300.2.dd.a.1349.2		8
5.3	odd	4		6300.2.dd.a.1349.4		8
5.4	even	2		6300.2.ch.a.1601.2		4
7.2	even	3		1764.2.f.a.881.2		4
7.3	odd	6	inner	252.2.t.a.17.1	✓	4
7.4	even	3		1764.2.t.a.521.2		4
7.5	odd	6		1764.2.f.a.881.4		4
7.6	odd	2		1764.2.t.a.1097.1		4
9.2	odd	6		2268.2.w.h.1349.1		4
9.4	even	3		2268.2.bm.g.593.1		4
9.5	odd	6		2268.2.bm.g.593.2		4
9.7	even	3		2268.2.w.h.1349.2		4
12.11	even	2		1008.2.bt.a.593.1		4
15.2	even	4		6300.2.dd.a.1349.1		8
15.8	even	4		6300.2.dd.a.1349.3		8
15.14	odd	2		6300.2.ch.a.1601.1		4
21.2	odd	6		1764.2.f.a.881.3		4
21.5	even	6		1764.2.f.a.881.1		4
21.11	odd	6		1764.2.t.a.521.1		4
21.17	even	6	inner	252.2.t.a.17.2	yes	4
21.20	even	2		1764.2.t.a.1097.2		4
28.3	even	6		1008.2.bt.a.17.1		4
28.19	even	6		7056.2.k.a.881.3		4
28.23	odd	6		7056.2.k.a.881.1		4
35.3	even	12		6300.2.dd.a.4049.1		8
35.17	even	12		6300.2.dd.a.4049.3		8
35.24	odd	6		6300.2.ch.a.4301.1		4
63.31	odd	6		2268.2.w.h.269.1		4
63.38	even	6		2268.2.bm.g.1025.1		4
63.52	odd	6		2268.2.bm.g.1025.2		4
63.59	even	6		2268.2.w.h.269.2		4
84.23	even	6		7056.2.k.a.881.4		4
84.47	odd	6		7056.2.k.a.881.2		4
84.59	odd	6		1008.2.bt.a.17.2		4
105.17	odd	12		6300.2.dd.a.4049.4		8
105.38	odd	12		6300.2.dd.a.4049.2		8
105.59	even	6		6300.2.ch.a.4301.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.t.a.17.1	✓	4	7.3	odd	6	inner
252.2.t.a.17.2	yes	4	21.17	even	6	inner
252.2.t.a.89.1	yes	4	3.2	odd	2	inner
252.2.t.a.89.2	yes	4	1.1	even	1	trivial
1008.2.bt.a.17.1		4	28.3	even	6
1008.2.bt.a.17.2		4	84.59	odd	6
1008.2.bt.a.593.1		4	12.11	even	2
1008.2.bt.a.593.2		4	4.3	odd	2
1764.2.f.a.881.1		4	21.5	even	6
1764.2.f.a.881.2		4	7.2	even	3
1764.2.f.a.881.3		4	21.2	odd	6
1764.2.f.a.881.4		4	7.5	odd	6
1764.2.t.a.521.1		4	21.11	odd	6
1764.2.t.a.521.2		4	7.4	even	3
1764.2.t.a.1097.1		4	7.6	odd	2
1764.2.t.a.1097.2		4	21.20	even	2
2268.2.w.h.269.1		4	63.31	odd	6
2268.2.w.h.269.2		4	63.59	even	6
2268.2.w.h.1349.1		4	9.2	odd	6
2268.2.w.h.1349.2		4	9.7	even	3
2268.2.bm.g.593.1		4	9.4	even	3
2268.2.bm.g.593.2		4	9.5	odd	6
2268.2.bm.g.1025.1		4	63.38	even	6
2268.2.bm.g.1025.2		4	63.52	odd	6
6300.2.ch.a.1601.1		4	15.14	odd	2
6300.2.ch.a.1601.2		4	5.4	even	2
6300.2.ch.a.4301.1		4	35.24	odd	6
6300.2.ch.a.4301.2		4	105.59	even	6
6300.2.dd.a.1349.1		8	15.2	even	4
6300.2.dd.a.1349.2		8	5.2	odd	4
6300.2.dd.a.1349.3		8	15.8	even	4
6300.2.dd.a.1349.4		8	5.3	odd	4
6300.2.dd.a.4049.1		8	35.3	even	12
6300.2.dd.a.4049.2		8	105.38	odd	12
6300.2.dd.a.4049.3		8	35.17	even	12
6300.2.dd.a.4049.4		8	105.17	odd	12
7056.2.k.a.881.1		4	28.23	odd	6
7056.2.k.a.881.2		4	84.47	odd	6
7056.2.k.a.881.3		4	28.19	even	6
7056.2.k.a.881.4		4	84.23	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	252.t (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.22474 - 2.12132i) q^{5} +(0.500000 + 2.59808i) q^{7} +O(q^{10})\)	\(q+(1.22474 - 2.12132i) q^{5} +(0.500000 + 2.59808i) q^{7} +(3.67423 - 2.12132i) q^{11} -1.73205i q^{13} +(-2.44949 - 4.24264i) q^{17} +(4.50000 + 2.59808i) q^{19} +(-0.500000 - 0.866025i) q^{25} +8.48528i q^{29} +(-1.50000 + 0.866025i) q^{31} +(6.12372 + 2.12132i) q^{35} +(2.50000 - 4.33013i) q^{37} -12.2474 q^{41} -11.0000 q^{43} +(1.22474 - 2.12132i) q^{47} +(-6.50000 + 2.59808i) q^{49} +(-7.34847 + 4.24264i) q^{53} -10.3923i q^{55} +(2.44949 + 4.24264i) q^{59} +(-3.00000 - 1.73205i) q^{61} +(-3.67423 - 2.12132i) q^{65} +(3.50000 + 6.06218i) q^{67} +12.7279i q^{71} +(13.5000 - 7.79423i) q^{73} +(7.34847 + 8.48528i) q^{77} +(-5.50000 + 9.52628i) q^{79} -12.2474 q^{83} -12.0000 q^{85} +(7.34847 - 12.7279i) q^{89} +(4.50000 - 0.866025i) q^{91} +(11.0227 - 6.36396i) q^{95} -3.46410i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{7}+O(q^{10}) \) 4 * q + 2 * q^7	\( 4 q + 2 q^{7} + 18 q^{19} - 2 q^{25} - 6 q^{31} + 10 q^{37} - 44 q^{43} - 26 q^{49} - 12 q^{61} + 14 q^{67} + 54 q^{73} - 22 q^{79} - 48 q^{85} + 18 q^{91}+O(q^{100}) \) 4 * q + 2 * q^7 + 18 * q^19 - 2 * q^25 - 6 * q^31 + 10 * q^37 - 44 * q^43 - 26 * q^49 - 12 * q^61 + 14 * q^67 + 54 * q^73 - 22 * q^79 - 48 * q^85 + 18 * q^91

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