LMFDB - Embedded newform 1728.2.r.e.1441.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,2,Mod(289,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 4]))

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

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

chi := DirichletCharacter("1728.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1728.r (of order $6$, degree $2$, 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:	$13.7981494693$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 $ x^12 - 16x^8 - 24x^7 + 96x^5 + 304x^4 + 384x^3 + 288x^2 + 144*x + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 576)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1441.6
Root			$-0.180407 + 0.673288i$ of defining polynomial
Character	$\chi$	$=$	1728.1441
Dual form			1728.2.r.e.289.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.50000 - 0.866025i) q^{5} +(2.17731 + 3.77121i) q^{7} +O(q^{10})$	$q+(-1.50000 - 0.866025i) q^{5} +(2.17731 + 3.77121i) q^{7} +(-5.05328 + 2.91751i) q^{11} +(-3.48133 - 2.00994i) q^{13} -1.70739 q^{17} -2.00000i q^{19} +(3.32123 - 5.75253i) q^{23} +(-1.00000 - 1.73205i) q^{25} +(4.06108 - 2.34467i) q^{29} +(2.43071 - 4.21012i) q^{31} -7.54241i q^{35} -5.75194i q^{37} +(-0.646305 + 1.11943i) q^{41} +(3.06782 - 1.77121i) q^{43} +(-5.30669 - 9.19145i) q^{47} +(-5.98133 + 10.3600i) q^{49} +0.506816i q^{53} +10.1066 q^{55} +(1.62152 + 0.936184i) q^{59} +(1.06108 - 0.612617i) q^{61} +(3.48133 + 6.02983i) q^{65} +(1.36811 + 0.789879i) q^{67} +9.88549 q^{71} -7.96265 q^{73} +(-22.0051 - 12.7046i) q^{77} +(1.03339 + 1.78988i) q^{79} +(1.33577 - 0.771205i) q^{83} +(2.56108 + 1.47864i) q^{85} -3.96265 q^{89} -17.5051i q^{91} +(-1.73205 + 3.00000i) q^{95} +(-3.06108 - 5.30195i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 18 q^{5}+O(q^{10}) $ 12 * q - 18 * q^5	$ 12 q - 18 q^{5} + 6 q^{13} - 12 q^{25} + 18 q^{29} - 18 q^{41} - 24 q^{49} - 18 q^{61} - 6 q^{65} - 90 q^{77} + 48 q^{89} - 6 q^{97}+O(q^{100}) $ 12 * q - 18 * q^5 + 6 * q^13 - 12 * q^25 + 18 * q^29 - 18 * q^41 - 24 * q^49 - 18 * q^61 - 6 * q^65 - 90 * q^77 + 48 * q^89 - 6 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	−1.50000	−	0.866025i	−0.670820	−	0.387298i	0.125567	−	0.992085i	$-0.459925\pi$
$5$	−1.50000	−	0.866025i	−0.670820	−	0.387298i	−0.796387	+	0.604787i	$0.793258\pi$
$6$		0			0
$7$	2.17731	+	3.77121i	0.822944	+	1.42538i	0.903480	+	0.428630i	$0.141004\pi$
$7$	2.17731	+	3.77121i	0.822944	+	1.42538i	−0.0805357	+	0.996752i	$0.525663\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−5.05328	+	2.91751i	−1.52362	+	0.879663i	−0.524011	+	0.851712i	$0.675565\pi$
$11$	−5.05328	+	2.91751i	−1.52362	+	0.879663i	−0.999609	+	0.0279511i	$0.991102\pi$
$12$		0			0
$13$	−3.48133	−	2.00994i	−0.965546	−	0.557458i	−0.0676707	−	0.997708i	$-0.521557\pi$
$13$	−3.48133	−	2.00994i	−0.965546	−	0.557458i	−0.897876	+	0.440249i	$0.854890\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.70739			−0.414103			−0.207051	−	0.978330i	$-0.566387\pi$
$17$	−1.70739			−0.414103			−0.207051	+	0.978330i	$0.566387\pi$
$18$		0			0
$19$		−	2.00000i		−	0.458831i	−0.973329	−	0.229416i	$-0.926318\pi$
$19$		−	2.00000i		−	0.458831i	0.973329	−	0.229416i	$-0.0736815\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.32123	−	5.75253i	0.692523	−	1.19949i	−0.278485	−	0.960441i	$-0.589832\pi$
$23$	3.32123	−	5.75253i	0.692523	−	1.19949i	0.971008	−	0.239045i	$-0.0768344\pi$
$24$		0			0
$25$	−1.00000	−	1.73205i	−0.200000	−	0.346410i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.06108	−	2.34467i	0.754124	−	0.435394i	−0.0730578	−	0.997328i	$-0.523276\pi$
$29$	4.06108	−	2.34467i	0.754124	−	0.435394i	0.827182	+	0.561934i	$0.189942\pi$
$30$		0			0
$31$	2.43071	−	4.21012i	0.436569	−	0.756160i	−0.560853	−	0.827915i	$-0.689527\pi$
$31$	2.43071	−	4.21012i	0.436569	−	0.756160i	0.997422	+	0.0717553i	$0.0228601\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	7.54241i		−	1.27490i
$36$		0			0
$37$		−	5.75194i		−	0.945613i	−0.881166	−	0.472807i	$-0.843241\pi$
$37$		−	5.75194i		−	0.945613i	0.881166	−	0.472807i	$-0.156759\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.646305	+	1.11943i	−0.100936	+	0.174826i	−0.912071	−	0.410033i	$-0.865517\pi$
$41$	−0.646305	+	1.11943i	−0.100936	+	0.174826i	0.811135	+	0.584860i	$0.198850\pi$
$42$		0			0
$43$	3.06782	−	1.77121i	0.467838	−	0.270106i	−0.247496	−	0.968889i	$-0.579608\pi$
$43$	3.06782	−	1.77121i	0.467838	−	0.270106i	0.715334	+	0.698783i	$0.246274\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5.30669	−	9.19145i	−0.774060	−	1.34071i	−0.935322	−	0.353799i	$-0.884890\pi$
$47$	−5.30669	−	9.19145i	−0.774060	−	1.34071i	0.161262	−	0.986912i	$-0.448444\pi$
$48$		0			0
$49$	−5.98133	+	10.3600i	−0.854475	+	1.47999i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			0.506816i			0.0696166i	0.999394	+	0.0348083i	$0.0110821\pi$
$53$			0.506816i			0.0696166i	−0.999394	+	0.0348083i	$0.988918\pi$
$54$		0			0
$55$	10.1066			1.36277
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.62152	+	0.936184i	0.211104	+	0.121881i	0.601824	−	0.798629i	$-0.294441\pi$
$59$	1.62152	+	0.936184i	0.211104	+	0.121881i	−0.390721	+	0.920509i	$0.627774\pi$
$60$		0			0
$61$	1.06108	−	0.612617i	0.135858	−	0.0784376i	−0.430531	−	0.902576i	$-0.641674\pi$
$61$	1.06108	−	0.612617i	0.135858	−	0.0784376i	0.566388	+	0.824138i	$0.308340\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.48133	+	6.02983i	0.431805	+	0.747909i
$66$		0			0
$67$	1.36811	+	0.789879i	0.167141	+	0.0964990i	0.581237	−	0.813734i	$-0.302569\pi$
$67$	1.36811	+	0.789879i	0.167141	+	0.0964990i	−0.414096	+	0.910233i	$0.635902\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	9.88549			1.17319			0.586596	−	0.809880i	$-0.300468\pi$
$71$	9.88549			1.17319			0.586596	+	0.809880i	$0.300468\pi$
$72$		0			0
$73$	−7.96265			−0.931958			−0.465979	−	0.884796i	$-0.654298\pi$
$73$	−7.96265			−0.931958			−0.465979	+	0.884796i	$0.654298\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−22.0051	−	12.7046i	−2.50771	−	1.44783i
$78$		0			0
$79$	1.03339	+	1.78988i	0.116265	+	0.201377i	0.918285	−	0.395921i	$-0.129574\pi$
$79$	1.03339	+	1.78988i	0.116265	+	0.201377i	−0.802020	+	0.597298i	$0.796241\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1.33577	−	0.771205i	0.146619	−	0.0846508i	−0.424896	−	0.905242i	$-0.639689\pi$
$83$	1.33577	−	0.771205i	0.146619	−	0.0846508i	0.571515	+	0.820592i	$0.306356\pi$
$84$		0			0
$85$	2.56108	+	1.47864i	0.277789	+	0.160381i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.96265			−0.420040			−0.210020	−	0.977697i	$-0.567353\pi$
$89$	−3.96265			−0.420040			−0.210020	+	0.977697i	$0.567353\pi$
$90$		0			0
$91$		−	17.5051i		−	1.83503i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.73205	+	3.00000i	−0.177705	+	0.307794i
$96$		0			0
$97$	−3.06108	−	5.30195i	−0.310806	−	0.538332i	0.667731	−	0.744403i	$-0.267266\pi$
$97$	−3.06108	−	5.30195i	−0.310806	−	0.538332i	−0.978537	+	0.206071i	$0.933932\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−10.0051	+	5.77643i	−0.995541	+	0.574776i	−0.906926	−	0.421290i	$-0.861577\pi$
$101$	−10.0051	+	5.77643i	−0.995541	+	0.574776i	−0.0886151	+	0.996066i	$0.528244\pi$
$102$		0			0
$103$	0.110533	−	0.191448i	0.0108911	−	0.0188639i	−0.860528	−	0.509402i	$-0.829867\pi$
$103$	0.110533	−	0.191448i	0.0108911	−	0.0188639i	0.871420	+	0.490538i	$0.163200\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	0.329957i		−	0.0318982i	−0.999873	−	0.0159491i	$-0.994923\pi$
$107$		−	0.329957i		−	0.0318982i	0.999873	−	0.0159491i	$-0.00507697\pi$
$108$		0			0
$109$		−	17.9253i		−	1.71693i	−0.512873	−	0.858465i	$-0.671419\pi$
$109$		−	17.9253i		−	1.71693i	0.512873	−	0.858465i	$-0.328581\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−9.04241	+	15.6619i	−0.850638	+	1.47335i	0.0299946	+	0.999550i	$0.490451\pi$
$113$	−9.04241	+	15.6619i	−0.850638	+	1.47335i	−0.880633	+	0.473799i	$0.842882\pi$
$114$		0			0
$115$	−9.96368	+	5.75253i	−0.929118	+	0.536426i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−3.71751	−	6.43892i	−0.340784	−	0.590254i
$120$		0			0
$121$	11.5237	−	19.9597i	1.04761	−	1.81452i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			12.1244i			1.08444i
$126$		0			0
$127$	4.57568			0.406026			0.203013	−	0.979176i	$-0.434927\pi$
$127$	4.57568			0.406026			0.203013	+	0.979176i	$0.434927\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−11.7281	−	6.77121i	−1.02469	−	0.591603i	−0.109228	−	0.994017i	$-0.534838\pi$
$131$	−11.7281	−	6.77121i	−1.02469	−	0.591603i	−0.915458	+	0.402414i	$0.868171\pi$
$132$		0			0
$133$	7.54241	−	4.35461i	0.654010	−	0.377593i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−11.1887	−	19.3794i	−0.955917	−	1.65570i	−0.732257	−	0.681029i	$-0.761533\pi$
$137$	−11.1887	−	19.3794i	−0.955917	−	1.65570i	−0.223660	−	0.974667i	$-0.571800\pi$
$138$		0			0
$139$	−9.99602	−	5.77121i	−0.847851	−	0.489507i	0.0120739	−	0.999927i	$-0.496157\pi$
$139$	−9.99602	−	5.77121i	−0.847851	−	0.489507i	−0.859925	+	0.510420i	$0.829490\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	23.4561			1.96150
$144$		0			0
$145$	−8.12217			−0.674509
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.93892	−	1.11943i	−0.158842	−	0.0917076i	0.418472	−	0.908230i	$-0.362566\pi$
$149$	−1.93892	−	1.11943i	−0.158842	−	0.0917076i	−0.577314	+	0.816522i	$0.695899\pi$
$150$		0			0
$151$	−8.51738	−	14.7525i	−0.693134	−	1.20054i	−0.970806	−	0.239868i	$-0.922896\pi$
$151$	−8.51738	−	14.7525i	−0.693134	−	1.20054i	0.277671	−	0.960676i	$-0.410437\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−7.29214	+	4.21012i	−0.585719	+	0.338165i
$156$		0			0
$157$	0.0424108	+	0.0244859i	0.00338475	+	0.00195419i	0.501691	−	0.865047i	$-0.332711\pi$
$157$	0.0424108	+	0.0244859i	0.00338475	+	0.00195419i	−0.498307	+	0.867001i	$0.666045\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	28.9253			2.27963
$162$		0			0
$163$			19.0848i			1.49484i	0.664353	+	0.747419i	$0.268707\pi$
$163$			19.0848i			1.49484i	−0.664353	+	0.747419i	$0.731293\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−9.99602	+	17.3136i	−0.773515	+	1.33977i	0.162110	+	0.986773i	$0.448170\pi$
$167$	−9.99602	+	17.3136i	−0.773515	+	1.33977i	−0.935625	+	0.352995i	$0.885163\pi$
$168$		0			0
$169$	1.57976	+	2.73622i	0.121520	+	0.210478i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	10.5000	−	6.06218i	0.798300	−	0.460899i	−0.0445762	−	0.999006i	$-0.514194\pi$
$173$	10.5000	−	6.06218i	0.798300	−	0.460899i	0.842876	+	0.538107i	$0.180860\pi$
$174$		0			0
$175$	4.35461	−	7.54241i	0.329178	−	0.570153i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			6.00000i			0.448461i	0.974536	+	0.224231i	$0.0719869\pi$
$179$			6.00000i			0.448461i	−0.974536	+	0.224231i	$0.928013\pi$
$180$		0			0
$181$			5.75194i			0.427538i	0.976884	+	0.213769i	$0.0685740\pi$
$181$			5.75194i			0.427538i	−0.976884	+	0.213769i	$0.931426\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−4.98133	+	8.62791i	−0.366234	+	0.634337i
$186$		0			0
$187$	8.62791	−	4.98133i	0.630935	−	0.364271i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	0.110533	+	0.191448i	0.00799786	+	0.0138527i	0.869997	−	0.493058i	$-0.164121\pi$
$191$	0.110533	+	0.191448i	0.00799786	+	0.0138527i	−0.861999	+	0.506910i	$0.830788\pi$
$192$		0			0
$193$	−9.02374	+	15.6296i	−0.649543	+	1.12504i	0.333689	+	0.942683i	$0.391706\pi$
$193$	−9.02374	+	15.6296i	−0.649543	+	1.12504i	−0.983232	+	0.182358i	$0.941627\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	9.82080i		−	0.699703i	−0.936805	−	0.349852i	$-0.886232\pi$
$197$		−	9.82080i		−	0.699703i	0.936805	−	0.349852i	$-0.113768\pi$
$198$		0			0
$199$	−0.604760			−0.0428703			−0.0214352	−	0.999770i	$-0.506824\pi$
$199$	−0.604760			−0.0428703			−0.0214352	+	0.999770i	$0.506824\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	17.6844	+	10.2101i	1.24120	+	0.716610i
$204$		0			0
$205$	1.93892	−	1.11943i	0.135420	−	0.0781846i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	5.83502	+	10.1066i	0.403617	+	0.699085i
$210$		0			0
$211$	0.363941	+	0.210121i	0.0250547	+	0.0144654i	0.512475	−	0.858702i	$-0.328729\pi$
$211$	0.363941	+	0.210121i	0.0250547	+	0.0144654i	−0.487420	+	0.873167i	$0.662062\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−6.13564			−0.418447
$216$		0			0
$217$	21.1696			1.43709
$218$		0			0
$219$		0			0
$220$		0			0
$221$	5.94398	+	3.43176i	0.399835	+	0.230845i
$222$		0			0
$223$	11.3933	+	19.7339i	0.762955	+	1.32148i	0.941321	+	0.337513i	$0.109586\pi$
$223$	11.3933	+	19.7339i	0.762955	+	1.32148i	−0.178366	+	0.983964i	$0.557081\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	10.2494	−	5.91751i	0.680279	−	0.392759i	−0.119681	−	0.992812i	$-0.538187\pi$
$227$	10.2494	−	5.91751i	0.680279	−	0.392759i	0.799960	+	0.600053i	$0.204854\pi$
$228$		0			0
$229$	−13.5848	−	7.84320i	−0.897710	−	0.518293i	−0.0212537	−	0.999774i	$-0.506766\pi$
$229$	−13.5848	−	7.84320i	−0.897710	−	0.518293i	−0.876457	+	0.481481i	$0.840099\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	0.829557			0.0543461			0.0271731	−	0.999631i	$-0.491349\pi$
$233$	0.829557			0.0543461			0.0271731	+	0.999631i	$0.491349\pi$
$234$		0			0
$235$			18.3829i			1.19917i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−3.60698	+	6.24747i	−0.233316	+	0.404115i	−0.958782	−	0.284143i	$-0.908291\pi$
$239$	−3.60698	+	6.24747i	−0.233316	+	0.404115i	0.725466	+	0.688258i	$0.241624\pi$
$240$		0			0
$241$	10.4627	+	18.1218i	0.673959	+	1.16733i	0.976772	+	0.214281i	$0.0687408\pi$
$241$	10.4627	+	18.1218i	0.673959	+	1.16733i	−0.302813	+	0.953050i	$0.597926\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	17.9440	−	10.3600i	1.14640	−	0.661874i
$246$		0			0
$247$	−4.01989	+	6.96265i	−0.255779	+	0.443023i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	20.2553i		−	1.27850i	−0.768999	−	0.639250i	$-0.779245\pi$
$251$		−	20.2553i		−	1.27850i	0.768999	−	0.639250i	$-0.220755\pi$
$252$		0			0
$253$			38.7588i			2.43675i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3.37237	+	5.84111i	−0.210363	+	0.364359i	−0.951828	−	0.306632i	$-0.900798\pi$
$257$	−3.37237	+	5.84111i	−0.210363	+	0.364359i	0.741465	+	0.670991i	$0.234131\pi$
$258$		0			0
$259$	21.6917	−	12.5237i	1.34786	−	0.778187i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−13.2067	−	22.8747i	−0.814361	−	1.41051i	−0.909786	−	0.415078i	$-0.863754\pi$
$263$	−13.2067	−	22.8747i	−0.814361	−	1.41051i	0.0954250	−	0.995437i	$-0.469579\pi$
$264$		0			0
$265$	0.438916	−	0.760225i	0.0269624	−	0.0467002i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			23.1704i			1.41272i	0.707851	+	0.706362i	$0.249665\pi$
$269$			23.1704i			1.41272i	−0.707851	+	0.706362i	$0.750335\pi$
$270$		0			0
$271$	−17.7626			−1.07900			−0.539502	−	0.841984i	$-0.681387\pi$
$271$	−17.7626			−1.07900			−0.539502	+	0.841984i	$0.681387\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	10.1066	+	5.83502i	0.609448	+	0.351865i
$276$		0			0
$277$	−9.04241	+	5.22064i	−0.543306	+	0.313678i	−0.746418	−	0.665478i	$-0.768228\pi$
$277$	−9.04241	+	5.22064i	−0.543306	+	0.313678i	0.203112	+	0.979156i	$0.434894\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−13.4440	−	23.2857i	−0.802001	−	1.38911i	−0.918297	−	0.395891i	$-0.870436\pi$
$281$	−13.4440	−	23.2857i	−0.802001	−	1.38911i	0.116297	−	0.993215i	$-0.462898\pi$
$282$		0			0
$283$	16.1317	+	9.31362i	0.958927	+	0.553637i	0.895843	−	0.444371i	$-0.146573\pi$
$283$	16.1317	+	9.31362i	0.958927	+	0.553637i	0.0630847	+	0.998008i	$0.479906\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−5.62882			−0.332259
$288$		0			0
$289$	−14.0848			−0.828519
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.00506	+	0.580274i	0.0587165	+	0.0339000i	0.529071	−	0.848578i	$-0.322541\pi$
$293$	1.00506	+	0.580274i	0.0587165	+	0.0339000i	−0.470354	+	0.882478i	$0.655874\pi$
$294$		0			0
$295$	−1.62152	−	2.80855i	−0.0944084	−	0.163520i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−23.1245	+	13.3510i	−1.33733	+	0.772106i
$300$		0			0
$301$	13.3592	+	7.71291i	0.770009	+	0.444565i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2.12217			−0.121515
$306$		0			0
$307$		−	19.0848i		−	1.08923i	−0.838687	−	0.544614i	$-0.816676\pi$
$307$		−	19.0848i		−	1.08923i	0.838687	−	0.544614i	$-0.183324\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−5.56009	+	9.63036i	−0.315284	+	0.546088i	−0.979498	−	0.201455i	$-0.935433\pi$
$311$	−5.56009	+	9.63036i	−0.315284	+	0.546088i	0.664214	+	0.747543i	$0.268766\pi$
$312$		0			0
$313$	−13.4627	−	23.3180i	−0.760954	−	1.31801i	−0.942359	−	0.334603i	$-0.891398\pi$
$313$	−13.4627	−	23.3180i	−0.760954	−	1.31801i	0.181405	−	0.983408i	$-0.441935\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	19.4440	−	11.2260i	1.09208	−	0.630514i	0.157953	−	0.987447i	$-0.449511\pi$
$317$	19.4440	−	11.2260i	1.09208	−	0.630514i	0.934130	+	0.356932i	$0.116177\pi$
$318$		0			0
$319$	−13.6812	+	23.6965i	−0.765999	+	1.32675i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			3.41478i			0.190003i
$324$		0			0
$325$			8.03978i			0.445967i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	23.1086	−	40.0252i	1.27402	−	2.20666i
$330$		0			0
$331$	−19.5634	+	11.2949i	−1.07530	+	0.620826i	−0.929625	−	0.368506i	$-0.879869\pi$
$331$	−19.5634	+	11.2949i	−1.07530	+	0.620826i	−0.145677	+	0.989332i	$0.546536\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1.36811	−	2.36964i	−0.0747478	−	0.129467i
$336$		0			0
$337$	7.04241	−	12.1978i	0.383625	−	0.664457i	−0.607953	−	0.793973i	$-0.708009\pi$
$337$	7.04241	−	12.1978i	0.383625	−	0.664457i	0.991577	+	0.129516i	$0.0413423\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			28.3665i			1.53613i
$342$		0			0
$343$	−21.6104			−1.16685
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6.05745	+	3.49727i	0.325181	+	0.187743i	0.653699	−	0.756754i	$-0.273216\pi$
$347$	6.05745	+	3.49727i	0.325181	+	0.187743i	−0.328519	+	0.944497i	$0.606549\pi$
$348$		0			0
$349$	−8.16458	+	4.71382i	−0.437040	+	0.252325i	−0.702341	−	0.711840i	$-0.747862\pi$
$349$	−8.16458	+	4.71382i	−0.437040	+	0.252325i	0.265301	+	0.964166i	$0.414529\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−3.81128	−	6.60134i	−0.202854	−	0.351354i	0.746593	−	0.665281i	$-0.231688\pi$
$353$	−3.81128	−	6.60134i	−0.202854	−	0.351354i	−0.949447	+	0.313928i	$0.898355\pi$
$354$		0			0
$355$	−14.8282	−	8.56108i	−0.787001	−	0.454375i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	10.8344			0.571820			0.285910	−	0.958257i	$-0.407704\pi$
$359$	10.8344			0.571820			0.285910	+	0.958257i	$0.407704\pi$
$360$		0			0
$361$	15.0000			0.789474
$362$		0			0
$363$		0			0
$364$		0			0
$365$	11.9440	+	6.89586i	0.625176	+	0.360946i
$366$		0			0
$367$	−11.4747	−	19.8747i	−0.598973	−	1.03745i	−0.992973	−	0.118341i	$-0.962243\pi$
$367$	−11.4747	−	19.8747i	−0.598973	−	1.03745i	0.394000	−	0.919110i	$-0.371091\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1.91131	+	1.10349i	−0.0992302	+	0.0572906i
$372$		0			0
$373$	−0.0984305	−	0.0568289i	−0.00509654	−	0.00294249i	0.497450	−	0.867493i	$-0.334270\pi$
$373$	−0.0984305	−	0.0568289i	−0.00509654	−	0.00294249i	−0.502546	+	0.864550i	$0.667603\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−18.8506			−0.970856
$378$		0			0
$379$			16.2070i			0.832497i	0.909251	+	0.416249i	$0.136655\pi$
$379$			16.2070i			0.832497i	−0.909251	+	0.416249i	$0.863345\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−7.03874	+	12.1914i	−0.359663	+	0.622954i	−0.987904	−	0.155064i	$-0.950442\pi$
$383$	−7.03874	+	12.1914i	−0.359663	+	0.622954i	0.628242	+	0.778018i	$0.283775\pi$
$384$		0			0
$385$	22.0051	+	38.1139i	1.12148	+	1.94246i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	25.0051	−	14.4367i	1.26781	−	0.731969i	0.293234	−	0.956041i	$-0.405268\pi$
$389$	25.0051	−	14.4367i	1.26781	−	0.731969i	0.974573	+	0.224072i	$0.0719350\pi$
$390$		0			0
$391$	−5.67063	+	9.82181i	−0.286776	+	0.496710i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	3.57976i		−	0.180117i
$396$		0			0
$397$		−	2.85934i		−	0.143506i	−0.997422	−	0.0717531i	$-0.977141\pi$
$397$		−	2.85934i		−	0.143506i	0.997422	−	0.0717531i	$-0.0228594\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	10.7739	−	18.6610i	0.538025	−	0.931886i	−0.460986	−	0.887408i	$-0.652504\pi$
$401$	10.7739	−	18.6610i	0.538025	−	0.931886i	0.999010	−	0.0444786i	$-0.0141626\pi$
$402$		0			0
$403$	−16.9242	+	9.77121i	−0.843056	+	0.486738i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	16.7813	+	29.0661i	0.831821	+	1.44076i
$408$		0			0
$409$	10.4627	−	18.1218i	0.517345	−	0.896068i	−0.482452	−	0.875922i	$-0.660254\pi$
$409$	10.4627	−	18.1218i	0.517345	−	0.896068i	0.999797	−	0.0201454i	$-0.00641290\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			8.15344i			0.401204i
$414$		0			0
$415$	−2.67153			−0.131140
$416$		0			0
$417$		0			0
$418$		0			0
$419$	10.9678	+	6.33229i	0.535814	+	0.309353i	0.743381	−	0.668868i	$-0.233221\pi$
$419$	10.9678	+	6.33229i	0.535814	+	0.309353i	−0.207567	+	0.978221i	$0.566554\pi$
$420$		0			0
$421$	−23.1086	+	13.3417i	−1.12624	+	0.650236i	−0.942987	−	0.332829i	$-0.891997\pi$
$421$	−23.1086	+	13.3417i	−1.12624	+	0.650236i	−0.183255	+	0.983065i	$0.558663\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.70739	+	2.95729i	0.0828205	+	0.143449i
$426$		0			0
$427$	4.62061	+	2.66771i	0.223607	+	0.129100i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−23.6772			−1.14049			−0.570246	−	0.821474i	$-0.693152\pi$
$431$	−23.6772			−1.14049			−0.570246	+	0.821474i	$0.693152\pi$
$432$		0			0
$433$	7.96265			0.382661			0.191330	−	0.981526i	$-0.438720\pi$
$433$	7.96265			0.382661			0.191330	+	0.981526i	$0.438720\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−11.5051	−	6.64245i	−0.550362	−	0.317752i
$438$		0			0
$439$	5.33903	+	9.24747i	0.254818	+	0.441358i	0.964846	−	0.262816i	$-0.0846511\pi$
$439$	5.33903	+	9.24747i	0.254818	+	0.441358i	−0.710028	+	0.704173i	$0.751318\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	11.1566	−	6.44125i	0.530065	−	0.306033i	−0.210978	−	0.977491i	$-0.567665\pi$
$443$	11.1566	−	6.44125i	0.530065	−	0.306033i	0.741043	+	0.671458i	$0.234332\pi$
$444$		0			0
$445$	5.94398	+	3.43176i	0.281772	+	0.162681i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.547875			0.0258558			0.0129279	−	0.999916i	$-0.495885\pi$
$449$	0.547875			0.0258558			0.0129279	+	0.999916i	$0.495885\pi$
$450$		0			0
$451$		−	7.54241i		−	0.355158i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−15.1598	+	26.2576i	−0.710704	+	1.23098i
$456$		0			0
$457$	−9.90157	−	17.1500i	−0.463176	−	0.802244i	0.535941	−	0.844255i	$-0.319957\pi$
$457$	−9.90157	−	17.1500i	−0.463176	−	0.802244i	−0.999117	+	0.0420111i	$0.986624\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−12.1833	+	7.03400i	−0.567431	+	0.327606i	−0.756122	−	0.654430i	$-0.772909\pi$
$461$	−12.1833	+	7.03400i	−0.567431	+	0.327606i	0.188692	+	0.982036i	$0.439575\pi$
$462$		0			0
$463$	6.22954	−	10.7899i	0.289511	−	0.501448i	−0.684182	−	0.729311i	$-0.739841\pi$
$463$	6.22954	−	10.7899i	0.289511	−	0.501448i	0.973693	+	0.227863i	$0.0731739\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			29.3401i			1.35770i	0.734278	+	0.678849i	$0.237521\pi$
$467$			29.3401i			1.35770i	−0.734278	+	0.678849i	$0.762479\pi$
$468$		0			0
$469$			6.87923i			0.317653i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−10.3350	+	17.9008i	−0.475205	+	0.823079i
$474$		0			0
$475$	−3.46410	+	2.00000i	−0.158944	+	0.0917663i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	9.96368	+	17.2576i	0.455252	+	0.788520i	0.998703	−	0.0509214i	$-0.0162158\pi$
$479$	9.96368	+	17.2576i	0.455252	+	0.788520i	−0.543451	+	0.839441i	$0.682882\pi$
$480$		0			0
$481$	−11.5611	+	20.0244i	−0.527140	+	0.913033i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			10.6039i			0.481499i
$486$		0			0
$487$	7.46828			0.338420			0.169210	−	0.985580i	$-0.445878\pi$
$487$	7.46828			0.338420			0.169210	+	0.985580i	$0.445878\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	30.2186	+	17.4467i	1.36375	+	0.787359i	0.990120	−	0.140220i	$-0.0447811\pi$
$491$	30.2186	+	17.4467i	1.36375	+	0.787359i	0.373626	+	0.927580i	$0.378114\pi$
$492$		0			0
$493$	−6.93385	+	4.00326i	−0.312285	+	0.180298i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	21.5237	+	37.2802i	0.965472	+	1.67225i
$498$		0			0
$499$	32.4480	+	18.7339i	1.45257	+	0.838643i	0.998627	−	0.0523870i	$-0.0166830\pi$
$499$	32.4480	+	18.7339i	1.45257	+	0.838643i	0.453945	+	0.891030i	$0.350016\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	10.8991			0.485968			0.242984	−	0.970030i	$-0.421874\pi$
$503$	10.8991			0.485968			0.242984	+	0.970030i	$0.421874\pi$
$504$		0			0
$505$	20.0101			0.890439
$506$		0			0
$507$		0			0
$508$		0			0
$509$	4.06108	+	2.34467i	0.180004	+	0.103926i	0.587295	−	0.809373i	$-0.300193\pi$
$509$	4.06108	+	2.34467i	0.180004	+	0.103926i	−0.407290	+	0.913299i	$0.633526\pi$
$510$		0			0
$511$	−17.3371	−	30.0288i	−0.766950	−	1.32840i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−0.331598	+	0.191448i	−0.0146119	+	0.00843621i
$516$		0			0
$517$	53.6323	+	30.9646i	2.35875	+	1.36182i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	26.4249			1.15770			0.578848	−	0.815435i	$-0.303502\pi$
$521$	26.4249			1.15770			0.578848	+	0.815435i	$0.303502\pi$
$522$		0			0
$523$			13.8880i			0.607278i	0.952787	+	0.303639i	$0.0982017\pi$
$523$			13.8880i			0.607278i	−0.952787	+	0.303639i	$0.901798\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−4.15018	+	7.18832i	−0.180785	+	0.313128i
$528$		0			0
$529$	−10.5611	−	18.2923i	−0.459178	−	0.795319i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	4.50000	−	2.59808i	0.194917	−	0.112535i
$534$		0			0
$535$	−0.285751	+	0.494936i	−0.0123541	+	0.0213979i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	69.8023i		−	3.00660i
$540$		0			0
$541$		−	25.0161i		−	1.07553i	−0.843096	−	0.537763i	$-0.819269\pi$
$541$		−	25.0161i		−	1.07553i	0.843096	−	0.537763i	$-0.180731\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−15.5237	+	26.8879i	−0.664964	+	1.15175i
$546$		0			0
$547$	12.4236	−	7.17277i	0.531195	−	0.306686i	−0.210308	−	0.977635i	$-0.567447\pi$
$547$	12.4236	−	7.17277i	0.531195	−	0.306686i	0.741503	+	0.670950i	$0.234113\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−4.68934	−	8.12217i	−0.199772	−	0.346016i
$552$		0			0
$553$	−4.50000	+	7.79423i	−0.191359	+	0.331444i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	32.9912i		−	1.39788i	−0.715179	−	0.698941i	$-0.753655\pi$
$557$		−	32.9912i		−	1.39788i	0.715179	−	0.698941i	$-0.246345\pi$
$558$		0			0
$559$	−14.2401			−0.602292
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−21.6459	−	12.4973i	−0.912266	−	0.526697i	−0.0311065	−	0.999516i	$-0.509903\pi$
$563$	−21.6459	−	12.4973i	−0.912266	−	0.526697i	−0.881160	+	0.472819i	$0.843236\pi$
$564$		0			0
$565$	27.1272	−	15.6619i	1.14125	−	0.658902i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−2.79261	−	4.83694i	−0.117072	−	0.202775i	0.801534	−	0.597949i	$-0.204018\pi$
$569$	−2.79261	−	4.83694i	−0.117072	−	0.202775i	−0.918606	+	0.395174i	$0.870684\pi$
$570$		0			0
$571$	4.83221	+	2.78988i	0.202222	+	0.116753i	0.597691	−	0.801726i	$-0.296085\pi$
$571$	4.83221	+	2.78988i	0.202222	+	0.116753i	−0.395470	+	0.918479i	$0.629418\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−13.2849			−0.554019
$576$		0			0
$577$	−9.04748			−0.376651			−0.188326	−	0.982107i	$-0.560306\pi$
$577$	−9.04748			−0.376651			−0.188326	+	0.982107i	$0.560306\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	5.81675	+	3.35830i	0.241319	+	0.139326i
$582$		0			0
$583$	−1.47864	−	2.56108i	−0.0612391	−	0.106069i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	6.24617	−	3.60623i	0.257807	−	0.148845i	−0.365527	−	0.930801i	$-0.619111\pi$
$587$	6.24617	−	3.60623i	0.257807	−	0.148845i	0.623334	+	0.781956i	$0.285778\pi$
$588$		0			0
$589$	−8.42024	−	4.86143i	−0.346950	−	0.200312i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	25.0475			1.02858			0.514288	−	0.857617i	$-0.328056\pi$
$593$	25.0475			1.02858			0.514288	+	0.857617i	$0.328056\pi$
$594$		0			0
$595$			12.8778i			0.527940i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	20.1026	−	34.8187i	0.821369	−	1.42265i	−0.0832941	−	0.996525i	$-0.526544\pi$
$599$	20.1026	−	34.8187i	0.821369	−	1.42265i	0.904663	−	0.426128i	$-0.140123\pi$
$600$		0			0
$601$	5.48133	+	9.49394i	0.223588	+	0.387266i	0.955895	−	0.293709i	$-0.0948897\pi$
$601$	5.48133	+	9.49394i	0.223588	+	0.387266i	−0.732307	+	0.680975i	$0.761556\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−34.5712	+	19.9597i	−1.40552	+	0.811477i
$606$		0			0
$607$	14.2370	−	24.6592i	0.577861	−	1.00088i	−0.417863	−	0.908510i	$-0.637221\pi$
$607$	14.2370	−	24.6592i	0.577861	−	1.00088i	0.995724	−	0.0923747i	$-0.0294458\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			42.6646i			1.72602i
$612$		0			0
$613$		−	14.3632i		−	0.580125i	−0.957008	−	0.290063i	$-0.906324\pi$
$613$		−	14.3632i		−	0.580125i	0.957008	−	0.290063i	$-0.0936761\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.316348	+	0.547931i	−0.0127357	+	0.0220589i	−0.872323	−	0.488930i	$-0.837387\pi$
$617$	−0.316348	+	0.547931i	−0.0127357	+	0.0220589i	0.859587	+	0.510989i	$0.170721\pi$
$618$		0			0
$619$	−28.9516	+	16.7152i	−1.16366	+	0.671840i	−0.952179	−	0.305542i	$-0.901162\pi$
$619$	−28.9516	+	16.7152i	−1.16366	+	0.671840i	−0.211482	+	0.977382i	$0.567829\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−8.62791	−	14.9440i	−0.345670	−	0.598718i
$624$		0			0
$625$	5.50000	−	9.52628i	0.220000	−	0.381051i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			9.82080i			0.391581i
$630$		0			0
$631$	7.59765			0.302458			0.151229	−	0.988499i	$-0.451677\pi$
$631$	7.59765			0.302458			0.151229	+	0.988499i	$0.451677\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−6.86352	−	3.96265i	−0.272370	−	0.157253i
$636$		0			0
$637$	41.6459	−	24.0443i	1.65007	−	0.952669i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−11.8774	−	20.5723i	−0.469130	−	0.812558i	0.530247	−	0.847843i	$-0.322099\pi$
$641$	−11.8774	−	20.5723i	−0.469130	−	0.812558i	−0.999377	+	0.0352856i	$0.988766\pi$
$642$		0			0
$643$	−12.4236	−	7.17277i	−0.489939	−	0.282867i	0.234610	−	0.972090i	$-0.424619\pi$
$643$	−12.4236	−	7.17277i	−0.489939	−	0.282867i	−0.724549	+	0.689223i	$0.757952\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	28.5781			1.12352			0.561761	−	0.827299i	$-0.310124\pi$
$647$	28.5781			1.12352			0.561761	+	0.827299i	$0.310124\pi$
$648$		0			0
$649$	−10.9253			−0.428856
$650$		0			0
$651$		0			0
$652$		0			0
$653$	14.7443	+	8.51265i	0.576990	+	0.333126i	0.759936	−	0.649997i	$-0.225230\pi$
$653$	14.7443	+	8.51265i	0.576990	+	0.333126i	−0.182946	+	0.983123i	$0.558563\pi$
$654$		0			0
$655$	11.7281	+	20.3136i	0.458254	+	0.793719i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	10.8209	−	6.24747i	0.421524	−	0.243367i	−0.274205	−	0.961671i	$-0.588415\pi$
$659$	10.8209	−	6.24747i	0.421524	−	0.243367i	0.695729	+	0.718304i	$0.255081\pi$
$660$		0			0
$661$	−30.1272	−	17.3940i	−1.17181	−	0.676547i	−0.217707	−	0.976014i	$-0.569858\pi$
$661$	−30.1272	−	17.3940i	−1.17181	−	0.676547i	−0.954107	+	0.299467i	$0.903191\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−15.0848			−0.584964
$666$		0			0
$667$		−	31.1487i		−	1.20608i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−3.57463	+	6.19145i	−0.137997	+	0.239018i
$672$		0			0
$673$	−13.0237	−	22.5578i	−0.502028	−	0.869538i	−0.999997	−	0.00234354i	$-0.999254\pi$
$673$	−13.0237	−	22.5578i	−0.502028	−	0.869538i	0.497969	−	0.867195i	$-0.334079\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−28.8829	+	16.6755i	−1.11006	+	0.640893i	−0.938845	−	0.344340i	$-0.888103\pi$
$677$	−28.8829	+	16.6755i	−1.11006	+	0.640893i	−0.171215	+	0.985234i	$0.554769\pi$
$678$		0			0
$679$	13.3298	−	23.0880i	0.511552	−	0.886034i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			26.4249i			1.01112i	0.862791	+	0.505560i	$0.168714\pi$
$683$			26.4249i			1.01112i	−0.862791	+	0.505560i	$0.831286\pi$
$684$		0			0
$685$			38.7588i			1.48090i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.01867	−	1.76439i	0.0388084	−	0.0672181i
$690$		0			0
$691$	−17.6521	+	10.1914i	−0.671518	+	0.387701i	−0.796651	−	0.604439i	$-0.793397\pi$
$691$	−17.6521	+	10.1914i	−0.671518	+	0.387701i	0.125134	+	0.992140i	$0.460064\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	9.99602	+	17.3136i	0.379171	+	0.656743i
$696$		0			0
$697$	1.10349	−	1.91131i	0.0417978	−	0.0723960i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	49.8696i		−	1.88355i	−0.336247	−	0.941774i	$-0.609158\pi$
$701$		−	49.8696i		−	1.88355i	0.336247	−	0.941774i	$-0.390842\pi$
$702$		0			0
$703$	−11.5039			−0.433877
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−43.5682	−	25.1541i	−1.63855	−	0.946017i
$708$		0			0
$709$	−8.57469	+	4.95060i	−0.322029	+	0.185924i	−0.652297	−	0.757964i	$-0.726194\pi$
$709$	−8.57469	+	4.95060i	−0.322029	+	0.185924i	0.330267	+	0.943887i	$0.392861\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−16.1459	−	27.9655i	−0.604669	−	1.04732i
$714$		0			0
$715$	−35.1842	−	20.3136i	−1.31582	−	0.759686i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	25.5562			0.953084			0.476542	−	0.879152i	$-0.341890\pi$
$719$	25.5562			0.953084			0.476542	+	0.879152i	$0.341890\pi$
$720$		0			0
$721$	0.962653			0.0358511
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−8.12217	−	4.68934i	−0.301650	−	0.174158i
$726$		0			0
$727$	20.0702	+	34.7627i	0.744364	+	1.28928i	0.950491	+	0.310751i	$0.100580\pi$
$727$	20.0702	+	34.7627i	0.744364	+	1.28928i	−0.206128	+	0.978525i	$0.566086\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−5.23796	+	3.02414i	−0.193733	+	0.111852i
$732$		0			0
$733$	−40.9965	−	23.6694i	−1.51424	−	0.874247i	−0.999861	−	0.0166852i	$-0.994689\pi$
$733$	−40.9965	−	23.6694i	−1.51424	−	0.874247i	−0.514380	−	0.857562i	$-0.671978\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−9.21792			−0.339546
$738$		0			0
$739$		−	37.9627i		−	1.39648i	−0.715864	−	0.698239i	$-0.753967\pi$
$739$		−	37.9627i		−	1.39648i	0.715864	−	0.698239i	$-0.246033\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	7.25980	−	12.5743i	0.266336	−	0.461308i	−0.701577	−	0.712594i	$-0.747520\pi$
$743$	7.25980	−	12.5743i	0.266336	−	0.461308i	0.967913	+	0.251286i	$0.0808535\pi$
$744$		0			0
$745$	1.93892	+	3.35830i	0.0710364	+	0.123039i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.24434	−	0.718418i	0.0454671	−	0.0262504i
$750$		0			0
$751$	−18.3216	+	31.7339i	−0.668563	+	1.15798i	0.309743	+	0.950820i	$0.399757\pi$
$751$	−18.3216	+	31.7339i	−0.668563	+	1.15798i	−0.978306	+	0.207165i	$0.933576\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			29.5051i			1.07380i
$756$		0			0
$757$		−	2.02727i		−	0.0736822i	−0.999321	−	0.0368411i	$-0.988270\pi$
$757$		−	2.02727i		−	0.0736822i	0.999321	−	0.0368411i	$-0.0117295\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−8.60349	+	14.9017i	−0.311876	+	0.540186i	−0.978769	−	0.204968i	$-0.934291\pi$
$761$	−8.60349	+	14.9017i	−0.311876	+	0.540186i	0.666892	+	0.745154i	$0.267624\pi$
$762$		0			0
$763$	67.5999	−	39.0288i	2.44728	−	1.41294i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−3.76336	−	6.51832i	−0.135887	−	0.235363i
$768$		0			0
$769$	−13.4253	+	23.2533i	−0.484129	+	0.838536i	−0.999834	−	0.0182304i	$-0.994197\pi$
$769$	−13.4253	+	23.2533i	−0.484129	+	0.838536i	0.515705	+	0.856766i	$0.327530\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			4.41305i			0.158726i	0.996846	+	0.0793632i	$0.0252887\pi$
$773$			4.41305i			0.158726i	−0.996846	+	0.0793632i	$0.974711\pi$
$774$		0			0
$775$	−9.72286			−0.349255
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2.23887	+	1.29261i	0.0802157	+	0.0463126i
$780$		0			0
$781$	−49.9541	+	28.8410i	−1.78750	+	1.03201i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−0.0424108	−	0.0734576i	−0.00151371	−	0.00262182i
$786$		0			0
$787$	−29.7765	−	17.1914i	−1.06142	−	0.612809i	−0.135593	−	0.990765i	$-0.543294\pi$
$787$	−29.7765	−	17.1914i	−1.06142	−	0.612809i	−0.925824	+	0.377956i	$0.876627\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−78.7524			−2.80011
$792$		0			0
$793$	−4.92531			−0.174903
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−19.8269	−	11.4471i	−0.702304	−	0.405475i	0.105901	−	0.994377i	$-0.466227\pi$
$797$	−19.8269	−	11.4471i	−0.702304	−	0.405475i	−0.808205	+	0.588901i	$0.799561\pi$
$798$		0			0
$799$	9.06058	+	15.6934i	0.320540	+	0.555192i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	40.2375	−	23.2311i	1.41995	−	0.819809i
$804$		0			0
$805$	−43.3880	−	25.0501i	−1.52922	−	0.882898i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−34.7922			−1.22323			−0.611614	−	0.791156i	$-0.709480\pi$
$809$	−34.7922			−1.22323			−0.611614	+	0.791156i	$0.709480\pi$
$810$		0			0
$811$			11.6810i			0.410174i	0.978744	+	0.205087i	$0.0657478\pi$
$811$			11.6810i			0.410174i	−0.978744	+	0.205087i	$0.934252\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	16.5279	−	28.6272i	0.578948	−	1.00277i
$816$		0			0
$817$	−3.54241	−	6.13564i	−0.123933	−	0.214659i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−13.9389	+	8.04764i	−0.486472	+	0.280864i	−0.723109	−	0.690733i	$-0.757288\pi$
$821$	−13.9389	+	8.04764i	−0.486472	+	0.280864i	0.236638	+	0.971598i	$0.423954\pi$
$822$		0			0
$823$	6.40163	−	11.0880i	0.223147	−	0.386502i	−0.732615	−	0.680643i	$-0.761700\pi$
$823$	6.40163	−	11.0880i	0.223147	−	0.386502i	0.955762	+	0.294142i	$0.0950337\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	25.9253i		−	0.901511i	−0.892647	−	0.450756i	$-0.851155\pi$
$827$		−	25.9253i		−	0.901511i	0.892647	−	0.450756i	$-0.148845\pi$
$828$		0			0
$829$		−	49.0708i		−	1.70430i	−0.523299	−	0.852149i	$-0.675299\pi$
$829$		−	49.0708i		−	1.70430i	0.523299	−	0.852149i	$-0.324701\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	10.2125	−	17.6885i	0.353841	−	0.612870i
$834$		0			0
$835$	29.9881	−	17.3136i	1.03778	−	0.599162i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−15.1275	−	26.2016i	−0.522259	−	0.904579i	−0.999665	−	0.0258959i	$-0.991756\pi$
$839$	−15.1275	−	26.2016i	−0.522259	−	0.904579i	0.477406	−	0.878683i	$-0.341577\pi$
$840$		0			0
$841$	−3.50506	+	6.07095i	−0.120864	+	0.209343i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	5.47244i		−	0.188258i
$846$		0			0
$847$	100.363			3.44851
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−33.0882	−	19.1035i	−1.13425	−	0.654859i
$852$		0			0
$853$	25.9389	−	14.9758i	0.888132	−	0.512763i	0.0148007	−	0.999890i	$-0.495289\pi$
$853$	25.9389	−	14.9758i	0.888132	−	0.512763i	0.873331	+	0.487127i	$0.161955\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	11.9016	+	20.6141i	0.406550	+	0.704165i	0.994501	−	0.104732i	$-0.0333984\pi$
$857$	11.9016	+	20.6141i	0.406550	+	0.704165i	−0.587951	+	0.808897i	$0.700065\pi$
$858$		0			0
$859$	28.5000	+	16.4545i	0.972406	+	0.561419i	0.899969	−	0.435954i	$-0.143589\pi$
$859$	28.5000	+	16.4545i	0.972406	+	0.561419i	0.0724371	+	0.997373i	$0.476922\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	36.3906			1.23875			0.619375	−	0.785095i	$-0.287386\pi$
$863$	36.3906			1.23875			0.619375	+	0.785095i	$0.287386\pi$
$864$		0			0
$865$	−21.0000			−0.714021
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−10.4440	−	6.02983i	−0.354288	−	0.204548i
$870$		0			0
$871$	−3.17523	−	5.49965i	−0.107588	−	0.186349i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−45.7234	+	26.3984i	−1.54573	+	0.892430i
$876$		0			0
$877$	42.3529	+	24.4525i	1.43016	+	0.825701i	0.997132	−	0.0756818i	$-0.0241133\pi$
$877$	42.3529	+	24.4525i	1.43016	+	0.825701i	0.433024	+	0.901383i	$0.357447\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−39.9627			−1.34638			−0.673188	−	0.739471i	$-0.735076\pi$
$881$	−39.9627			−1.34638			−0.673188	+	0.739471i	$0.735076\pi$
$882$		0			0
$883$			10.0000i			0.336527i	0.985742	+	0.168263i	$0.0538159\pi$
$883$			10.0000i			0.336527i	−0.985742	+	0.168263i	$0.946184\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	22.5949	−	39.1354i	0.758661	−	1.31404i	−0.184873	−	0.982762i	$-0.559187\pi$
$887$	22.5949	−	39.1354i	0.758661	−	1.31404i	0.943534	−	0.331277i	$-0.107479\pi$
$888$		0			0
$889$	9.96265	+	17.2558i	0.334137	+	0.578742i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−18.3829	+	10.6134i	−0.615160	+	0.355163i
$894$		0			0
$895$	5.19615	−	9.00000i	0.173688	−	0.300837i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	22.7969i		−	0.760318i
$900$		0			0
$901$		−	0.865333i		−	0.0288284i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	4.98133	−	8.62791i	0.165585	−	0.286801i
$906$		0			0
$907$	−34.9726	+	20.1914i	−1.16125	+	0.670446i	−0.951602	−	0.307332i	$-0.900564\pi$
$907$	−34.9726	+	20.1914i	−1.16125	+	0.670446i	−0.209644	+	0.977778i	$0.567231\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−4.26071	−	7.37976i	−0.141164	−	0.244503i	0.786771	−	0.617244i	$-0.211751\pi$
$911$	−4.26071	−	7.37976i	−0.141164	−	0.244503i	−0.927935	+	0.372742i	$0.878418\pi$
$912$		0			0
$913$	−4.50000	+	7.79423i	−0.148928	+	0.257951i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	58.9720i		−	1.94743i
$918$		0			0
$919$	−2.00834			−0.0662490			−0.0331245	−	0.999451i	$-0.510546\pi$
$919$	−2.00834			−0.0662490			−0.0331245	+	0.999451i	$0.510546\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−34.4146	−	19.8693i	−1.13277	−	0.654006i
$924$		0			0
$925$	−9.96265	+	5.75194i	−0.327570	+	0.189123i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	14.9864	+	25.9572i	0.491688	+	0.851628i	0.999954	−	0.00957195i	$-0.00304689\pi$
$929$	14.9864	+	25.9572i	0.491688	+	0.851628i	−0.508267	+	0.861200i	$0.669714\pi$
$930$		0			0
$931$	20.7199	+	11.9627i	0.679068	+	0.392060i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−17.2558			−0.564326
$936$		0			0
$937$	−45.0475			−1.47164			−0.735818	−	0.677179i	$-0.763202\pi$
$937$	−45.0475			−1.47164			−0.735818	+	0.677179i	$0.763202\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	46.3880	+	26.7821i	1.51220	+	0.873072i	0.999898	+	0.0142686i	$0.00454200\pi$
$941$	46.3880	+	26.7821i	1.51220	+	0.873072i	0.512306	+	0.858803i	$0.328791\pi$
$942$		0			0
$943$	4.29305	+	7.43578i	0.139801	+	0.242142i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	20.6417	−	11.9175i	0.670766	−	0.387267i	−0.125601	−	0.992081i	$-0.540086\pi$
$947$	20.6417	−	11.9175i	0.670766	−	0.387267i	0.796367	+	0.604814i	$0.206752\pi$
$948$		0			0
$949$	27.7206	+	16.0045i	0.899849	+	0.519528i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	32.4732			1.05191			0.525955	−	0.850513i	$-0.323708\pi$
$953$	32.4732			1.05191			0.525955	+	0.850513i	$0.323708\pi$
$954$		0			0
$955$		−	0.382896i		−	0.0123902i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	48.7225	−	84.3899i	1.57333	−	2.72509i
$960$		0			0
$961$	3.68325	+	6.37958i	0.118815	+	0.205793i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	27.0712	−	15.6296i	0.871453	−	0.503134i
$966$		0			0
$967$	−4.83221	+	8.36964i	−0.155393	+	0.269149i	−0.933202	−	0.359352i	$-0.882998\pi$
$967$	−4.83221	+	8.36964i	−0.155393	+	0.269149i	0.777809	+	0.628501i	$0.216331\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			49.8133i			1.59858i	0.600943	+	0.799292i	$0.294792\pi$
$971$			49.8133i			1.59858i	−0.600943	+	0.799292i	$0.705208\pi$
$972$		0			0
$973$		−	50.2627i		−	1.61135i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−3.59029	+	6.21856i	−0.114863	+	0.198949i	−0.917725	−	0.397216i	$-0.869976\pi$
$977$	−3.59029	+	6.21856i	−0.114863	+	0.198949i	0.802862	+	0.596165i	$0.203310\pi$
$978$		0			0
$979$	20.0244	−	11.5611i	0.639982	−	0.369494i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−2.38174	−	4.12530i	−0.0759658	−	0.131577i	0.825540	−	0.564344i	$-0.190871\pi$
$983$	−2.38174	−	4.12530i	−0.0759658	−	0.131577i	−0.901506	+	0.432767i	$0.857537\pi$
$984$		0			0
$985$	−8.50506	+	14.7312i	−0.270994	+	0.469375i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	23.5303i		−	0.748220i
$990$		0			0
$991$	28.5781			0.907815			0.453907	−	0.891049i	$-0.350030\pi$
$991$	28.5781			0.907815			0.453907	+	0.891049i	$0.350030\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	0.907140	+	0.523738i	0.0287583	+	0.0166036i
$996$		0			0
$997$	32.2494	−	18.6192i	1.02135	−	0.589676i	0.106854	−	0.994275i	$-0.465922\pi$
$997$	32.2494	−	18.6192i	1.02135	−	0.589676i	0.914494	+	0.404599i	$0.132589\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.2.r.e.1441.6		12
3.2	odd	2		576.2.r.f.481.4	yes	12
4.3	odd	2	inner	1728.2.r.e.1441.1		12
8.3	odd	2		1728.2.r.f.1441.1		12
8.5	even	2		1728.2.r.f.1441.6		12
9.2	odd	6		576.2.r.e.97.3	✓	12
9.4	even	3		5184.2.d.r.2593.7		12
9.5	odd	6		5184.2.d.q.2593.1		12
9.7	even	3		1728.2.r.f.289.6		12
12.11	even	2		576.2.r.f.481.3	yes	12
24.5	odd	2		576.2.r.e.481.3	yes	12
24.11	even	2		576.2.r.e.481.4	yes	12
36.7	odd	6		1728.2.r.f.289.1		12
36.11	even	6		576.2.r.e.97.4	yes	12
36.23	even	6		5184.2.d.q.2593.6		12
36.31	odd	6		5184.2.d.r.2593.12		12
72.5	odd	6		5184.2.d.q.2593.7		12
72.11	even	6		576.2.r.f.97.3	yes	12
72.13	even	6		5184.2.d.r.2593.1		12
72.29	odd	6		576.2.r.f.97.4	yes	12
72.43	odd	6	inner	1728.2.r.e.289.1		12
72.59	even	6		5184.2.d.q.2593.12		12
72.61	even	6	inner	1728.2.r.e.289.6		12
72.67	odd	6		5184.2.d.r.2593.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.r.e.97.3	✓	12	9.2	odd	6
576.2.r.e.97.4	yes	12	36.11	even	6
576.2.r.e.481.3	yes	12	24.5	odd	2
576.2.r.e.481.4	yes	12	24.11	even	2
576.2.r.f.97.3	yes	12	72.11	even	6
576.2.r.f.97.4	yes	12	72.29	odd	6
576.2.r.f.481.3	yes	12	12.11	even	2
576.2.r.f.481.4	yes	12	3.2	odd	2
1728.2.r.e.289.1		12	72.43	odd	6	inner
1728.2.r.e.289.6		12	72.61	even	6	inner
1728.2.r.e.1441.1		12	4.3	odd	2	inner
1728.2.r.e.1441.6		12	1.1	even	1	trivial
1728.2.r.f.289.1		12	36.7	odd	6
1728.2.r.f.289.6		12	9.7	even	3
1728.2.r.f.1441.1		12	8.3	odd	2
1728.2.r.f.1441.6		12	8.5	even	2
5184.2.d.q.2593.1		12	9.5	odd	6
5184.2.d.q.2593.6		12	36.23	even	6
5184.2.d.q.2593.7		12	72.5	odd	6
5184.2.d.q.2593.12		12	72.59	even	6
5184.2.d.r.2593.1		12	72.13	even	6
5184.2.d.r.2593.6		12	72.67	odd	6
5184.2.d.r.2593.7		12	9.4	even	3
5184.2.d.r.2593.12		12	36.31	odd	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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1728.r (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.50000 - 0.866025i) q^{5} +(2.17731 + 3.77121i) q^{7} +O(q^{10})\)	\(q+(-1.50000 - 0.866025i) q^{5} +(2.17731 + 3.77121i) q^{7} +(-5.05328 + 2.91751i) q^{11} +(-3.48133 - 2.00994i) q^{13} -1.70739 q^{17} -2.00000i q^{19} +(3.32123 - 5.75253i) q^{23} +(-1.00000 - 1.73205i) q^{25} +(4.06108 - 2.34467i) q^{29} +(2.43071 - 4.21012i) q^{31} -7.54241i q^{35} -5.75194i q^{37} +(-0.646305 + 1.11943i) q^{41} +(3.06782 - 1.77121i) q^{43} +(-5.30669 - 9.19145i) q^{47} +(-5.98133 + 10.3600i) q^{49} +0.506816i q^{53} +10.1066 q^{55} +(1.62152 + 0.936184i) q^{59} +(1.06108 - 0.612617i) q^{61} +(3.48133 + 6.02983i) q^{65} +(1.36811 + 0.789879i) q^{67} +9.88549 q^{71} -7.96265 q^{73} +(-22.0051 - 12.7046i) q^{77} +(1.03339 + 1.78988i) q^{79} +(1.33577 - 0.771205i) q^{83} +(2.56108 + 1.47864i) q^{85} -3.96265 q^{89} -17.5051i q^{91} +(-1.73205 + 3.00000i) q^{95} +(-3.06108 - 5.30195i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 18 q^{5}+O(q^{10}) \) 12 * q - 18 * q^5	\( 12 q - 18 q^{5} + 6 q^{13} - 12 q^{25} + 18 q^{29} - 18 q^{41} - 24 q^{49} - 18 q^{61} - 6 q^{65} - 90 q^{77} + 48 q^{89} - 6 q^{97}+O(q^{100}) \) 12 * q - 18 * q^5 + 6 * q^13 - 12 * q^25 + 18 * q^29 - 18 * q^41 - 24 * q^49 - 18 * q^61 - 6 * q^65 - 90 * q^77 + 48 * q^89 - 6 * q^97

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