LMFDB - Embedded newform 945.2.cs.a.419.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [945,2,Mod(104,945)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(945, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([11, 9, 9]))

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

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

chi := DirichletCharacter("945.104");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 945 = 3^{3} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	945.cs (of order $18$, degree $6$, 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:	$7.54586299101$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$4$ over $\Q(\zeta_{18})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{18}]$

Embedding invariants

Embedding label			419.2
Character	$\chi$	$=$	945.419
Dual form			945.2.cs.a.839.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.306808 + 1.70466i) q^{3} +(1.87939 - 0.684040i) q^{4} +(1.43732 - 1.71293i) q^{5} +(2.48619 + 0.904900i) q^{7} +(-2.81174 - 1.04601i) q^{9} +O(q^{10})$	\(q+(-0.306808 + 1.70466i) q^{3} +(1.87939 - 0.684040i) q^{4} +(1.43732 - 1.71293i) q^{5} +(2.48619 + 0.904900i) q^{7} +(-2.81174 - 1.04601i) q^{9} +(-2.55941 - 3.05019i) q^{11} +(0.589446 + 3.41358i) q^{12} +(1.10071 - 6.24241i) q^{13} +(2.47898 + 2.97568i) q^{15} +(3.06418 - 2.57115i) q^{16} +(4.17071 + 2.40796i) q^{17} +(1.52956 - 4.20243i) q^{20} +(-2.30533 + 3.96049i) q^{21} +(-0.868241 - 4.92404i) q^{25} +(2.64575 - 4.47214i) q^{27} +5.29150 q^{28} +(-10.5889 + 1.86711i) q^{29} +(5.98479 - 3.42711i) q^{33} +(5.12348 - 2.95804i) q^{35} +(-5.99985 - 0.0425087i) q^{36} +(10.3035 + 3.79155i) q^{39} +(-6.89658 - 3.98174i) q^{44} +(-5.83309 + 3.31286i) q^{45} +(-4.67425 + 12.8424i) q^{47} +(3.44283 + 6.01223i) q^{48} +(5.36231 + 4.49951i) q^{49} +(-5.38436 + 6.37086i) q^{51} +(-2.20141 - 12.4848i) q^{52} -8.90344 q^{55} +(6.69444 + 3.89672i) q^{60} +(-6.04399 - 5.14492i) q^{63} +(4.00000 - 6.92820i) q^{64} +(-9.11074 - 10.8578i) q^{65} +(9.48551 + 1.67255i) q^{68} +(6.07461 + 3.50718i) q^{71} +(7.26673 + 12.5863i) q^{73} +(8.66020 + 0.0306782i) q^{75} +(-3.60308 - 9.89937i) q^{77} +(3.00497 + 17.0421i) q^{79} -8.94427i q^{80} +(6.81174 + 5.88220i) q^{81} +(13.8663 - 2.44500i) q^{83} +(-1.62348 + 9.02022i) q^{84} +(10.1193 - 3.68312i) q^{85} +(0.0659721 - 18.6234i) q^{87} +(8.38533 - 14.5238i) q^{91} +(-11.5742 + 9.71188i) q^{97} +(4.00588 + 11.2535i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 6 q^{9}+O(q^{10}) $ 24 * q - 6 * q^9	$ 24 q - 6 q^{9} - 18 q^{11} - 24 q^{39} + 96 q^{64} - 180 q^{65} - 12 q^{79} + 102 q^{81} + 84 q^{84} + 60 q^{85} + 228 q^{99}+O(q^{100}) $ 24 * q - 6 * q^9 - 18 * q^11 - 24 * q^39 + 96 * q^64 - 180 * q^65 - 12 * q^79 + 102 * q^81 + 84 * q^84 + 60 * q^85 + 228 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/945\mathbb{Z}\right)^\times$.

$n$	$136$	$596$	$757$
$\chi(n)$	$-1$	$e\left(\frac{17}{18}\right)$	$-1$

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		0.984808	−	0.173648i	$-0.0555556\pi$
$2$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$3$	−0.306808	+	1.70466i	−0.177136	+	0.984186i
$3$	−0.306808	+	1.70466i	−0.177136	+	0.984186i
$4$	1.87939	−	0.684040i	0.939693	−	0.342020i
$5$	1.43732	−	1.71293i	0.642788	−	0.766044i
$5$	1.43732	−	1.71293i	0.642788	−	0.766044i
$6$		0			0
$7$	2.48619	+	0.904900i	0.939693	+	0.342020i
$7$	2.48619	+	0.904900i	0.939693	+	0.342020i
$8$		0			0
$9$	−2.81174	−	1.04601i	−0.937246	−	0.348669i
$10$		0			0
$11$	−2.55941	−	3.05019i	−0.771692	−	0.919667i	0.226834	−	0.973933i	$-0.427162\pi$
$11$	−2.55941	−	3.05019i	−0.771692	−	0.919667i	−0.998526	+	0.0542666i	$0.982718\pi$
$12$	0.589446	+	3.41358i	0.170158	+	0.985417i
$13$	1.10071	−	6.24241i	0.305281	−	1.73133i	−0.316898	−	0.948459i	$-0.602641\pi$
$13$	1.10071	−	6.24241i	0.305281	−	1.73133i	0.622179	−	0.782875i	$-0.286247\pi$
$14$		0			0
$15$	2.47898	+	2.97568i	0.640070	+	0.768317i
$16$	3.06418	−	2.57115i	0.766044	−	0.642788i
$17$	4.17071	+	2.40796i	1.01154	+	0.584016i	0.911644	−	0.410982i	$-0.134814\pi$
$17$	4.17071	+	2.40796i	1.01154	+	0.584016i	0.0999013	+	0.994997i	$0.468147\pi$
$18$		0			0
$19$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$19$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$20$	1.52956	−	4.20243i	0.342020	−	0.939693i
$21$	−2.30533	+	3.96049i	−0.503065	+	0.864249i
$22$		0			0
$23$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$23$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$24$		0			0
$25$	−0.868241	−	4.92404i	−0.173648	−	0.984808i
$26$		0			0
$27$	2.64575	−	4.47214i	0.509175	−	0.860663i
$28$	5.29150			1.00000
$29$	−10.5889	+	1.86711i	−1.96631	+	0.346714i	−0.973104	+	0.230366i	$0.926008\pi$
$29$	−10.5889	+	1.86711i	−1.96631	+	0.346714i	−0.993208	+	0.116348i	$0.962881\pi$
$30$		0			0
$31$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$31$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$32$		0			0
$33$	5.98479	−	3.42711i	1.04182	−	0.596583i
$34$		0			0
$35$	5.12348	−	2.95804i	0.866025	−	0.500000i
$36$	−5.99985	−	0.0425087i	−0.999975	−	0.00708479i
$37$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$37$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$38$		0			0
$39$	10.3035	+	3.79155i	1.64988	+	0.607134i
$40$		0			0
$41$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$41$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$42$		0			0
$43$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$43$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$44$	−6.89658	−	3.98174i	−1.03970	−	0.600270i
$45$	−5.83309	+	3.31286i	−0.869546	+	0.493852i
$46$		0			0
$47$	−4.67425	+	12.8424i	−0.681810	+	1.87326i	−0.265086	+	0.964225i	$0.585400\pi$
$47$	−4.67425	+	12.8424i	−0.681810	+	1.87326i	−0.416724	+	0.909033i	$0.636822\pi$
$48$	3.44283	+	6.01223i	0.496929	+	0.867791i
$49$	5.36231	+	4.49951i	0.766044	+	0.642788i
$50$		0			0
$51$	−5.38436	+	6.37086i	−0.753961	+	0.892099i
$52$	−2.20141	−	12.4848i	−0.305281	−	1.73133i
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$	−8.90344			−1.20054
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$59$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$60$	6.69444	+	3.89672i	0.864249	+	0.503065i
$61$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$61$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$62$		0			0
$63$	−6.04399	−	5.14492i	−0.761471	−	0.648199i
$64$	4.00000	−	6.92820i	0.500000	−	0.866025i
$65$	−9.11074	−	10.8578i	−1.13005	−	1.34674i
$66$		0			0
$67$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$67$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$68$	9.48551	+	1.67255i	1.15029	+	0.202827i
$69$		0			0
$70$		0			0
$71$	6.07461	+	3.50718i	0.720923	+	0.416225i	0.815092	−	0.579331i	$-0.196686\pi$
$71$	6.07461	+	3.50718i	0.720923	+	0.416225i	−0.0941692	+	0.995556i	$0.530019\pi$
$72$		0			0
$73$	7.26673	+	12.5863i	0.850506	+	1.47312i	0.880752	+	0.473577i	$0.157038\pi$
$73$	7.26673	+	12.5863i	0.850506	+	1.47312i	−0.0302463	+	0.999542i	$0.509629\pi$
$74$		0			0
$75$	8.66020	+	0.0306782i	0.999994	+	0.00354242i
$76$		0			0
$77$	−3.60308	−	9.89937i	−0.410609	−	1.12814i
$78$		0			0
$79$	3.00497	+	17.0421i	0.338086	+	1.91738i	0.394340	+	0.918964i	$0.370973\pi$
$79$	3.00497	+	17.0421i	0.338086	+	1.91738i	−0.0562544	+	0.998416i	$0.517916\pi$
$80$		−	8.94427i		−	1.00000i
$81$	6.81174	+	5.88220i	0.756860	+	0.653577i
$82$		0			0
$83$	13.8663	−	2.44500i	1.52202	−	0.268373i	0.650794	−	0.759254i	$-0.274436\pi$
$83$	13.8663	−	2.44500i	1.52202	−	0.268373i	0.871227	+	0.490881i	$0.163325\pi$
$84$	−1.62348	+	9.02022i	−0.177136	+	0.984186i
$85$	10.1193	−	3.68312i	1.09759	−	0.399490i
$86$		0			0
$87$	0.0659721	−	18.6234i	0.00707295	−	1.99663i
$88$		0			0
$89$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$		0			0
$91$	8.38533	−	14.5238i	0.879021	−	1.52251i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−11.5742	+	9.71188i	−1.17518	+	0.986092i	−0.175180	+	0.984536i	$0.556051\pi$
$97$	−11.5742	+	9.71188i	−1.17518	+	0.986092i	−0.999999	+	0.00155612i	$0.999505\pi$
$98$		0			0
$99$	4.00588	+	11.2535i	0.402606	+	1.13102i
$100$	−5.00000	−	8.66025i	−0.500000	−	0.866025i
$101$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$101$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$102$		0			0
$103$	12.3376	+	10.3524i	1.21566	+	1.02006i	0.999040	+	0.0438001i	$0.0139465\pi$
$103$	12.3376	+	10.3524i	1.21566	+	1.02006i	0.216616	+	0.976257i	$0.430498\pi$
$104$		0			0
$105$	3.47053	+	9.64134i	0.338689	+	0.940898i
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	1.91326	−	10.2147i	0.184104	−	0.982907i
$109$	−2.98186			−0.285610			−0.142805	−	0.989751i	$-0.545612\pi$
$109$	−2.98186			−0.285610			−0.142805	+	0.989751i	$0.545612\pi$
$110$		0			0
$111$		0			0
$112$	9.94477	−	3.61960i	0.939693	−	0.342020i
$113$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$113$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$114$		0			0
$115$		0			0
$116$	−18.6235	+	10.7523i	−1.72915	+	0.998323i
$117$	−9.62451	+	16.4007i	−0.889786	+	1.51624i
$118$		0			0
$119$	8.19022	+	9.76072i	0.750796	+	0.894764i
$120$		0			0
$121$	−0.842932	+	4.78050i	−0.0766302	+	0.434591i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−9.68246	−	5.59017i	−0.866025	−	0.500000i
$126$		0			0
$127$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$127$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$131$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$132$	8.90344	−	10.5347i	0.774945	−	0.916927i
$133$		0			0
$134$		0			0
$135$	−3.85766	−	10.9599i	−0.332015	−	0.943274i
$136$		0			0
$137$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$137$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$138$		0			0
$139$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$139$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$140$	7.60557	−	9.06396i	0.642788	−	0.766044i
$141$	−20.4579	−	11.9082i	−1.72286	−	1.00285i
$142$		0			0
$143$	−21.8577	+	12.6196i	−1.82783	+	1.05530i
$144$	−11.3051	+	4.02425i	−0.942092	+	0.335354i
$145$	−12.0214	+	20.8217i	−0.998323	+	1.72915i
$146$		0			0
$147$	−9.31534	+	7.76044i	−0.768317	+	0.640070i
$148$		0			0
$149$	−21.6506	−	3.81759i	−1.77369	−	0.312749i	−0.811343	−	0.584571i	$-0.801263\pi$
$149$	−21.6506	−	3.81759i	−1.77369	−	0.312749i	−0.962348	+	0.271821i	$0.912374\pi$
$150$		0			0
$151$	15.6508	−	13.1326i	1.27364	−	1.06871i	0.279554	−	0.960130i	$-0.409814\pi$
$151$	15.6508	−	13.1326i	1.27364	−	1.06871i	0.994088	−	0.108582i	$-0.0346309\pi$
$152$		0			0
$153$	−9.20819	−	11.1331i	−0.744438	−	0.900061i
$154$		0			0
$155$		0			0
$156$	21.9578	+	0.0777842i	1.75803	+	0.00622772i
$157$	7.30588	+	6.13036i	0.583072	+	0.489256i	0.885954	−	0.463772i	$-0.153504\pi$
$157$	7.30588	+	6.13036i	0.583072	+	0.489256i	−0.302882	+	0.953028i	$0.597949\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$	2.73165	−	15.1773i	0.212658	−	1.18155i
$166$		0			0
$167$	−2.27900	+	2.71600i	−0.176354	+	0.210171i	−0.846979	−	0.531626i	$-0.821581\pi$
$167$	−2.27900	+	2.71600i	−0.176354	+	0.210171i	0.670625	+	0.741796i	$0.266026\pi$
$168$		0			0
$169$	−25.5402	−	9.29586i	−1.96463	−	0.715066i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−9.66201	−	11.5147i	−0.734589	−	0.875449i	0.261372	−	0.965238i	$-0.415825\pi$
$173$	−9.66201	−	11.5147i	−0.734589	−	0.875449i	−0.995961	+	0.0897890i	$0.971381\pi$
$174$		0			0
$175$	2.29715	−	13.0278i	0.173648	−	0.984808i
$176$	−15.6850	−	2.76569i	−1.18230	−	0.208472i
$177$		0			0
$178$		0			0
$179$	5.51047	+	3.18147i	0.411872	+	0.237794i	0.691594	−	0.722287i	$-0.256909\pi$
$179$	5.51047	+	3.18147i	0.411872	+	0.237794i	−0.279722	+	0.960081i	$0.590242\pi$
$180$	−8.69650	+	10.2162i	−0.648199	+	0.761471i
$181$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$181$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−3.32983	−	18.8844i	−0.243501	−	1.38096i
$188$			27.3332i			1.99348i
$189$	10.6247	−	8.72445i	0.772832	−	0.634611i
$190$		0			0
$191$	5.82620	−	1.02732i	0.421569	−	0.0743340i	0.0411606	−	0.999153i	$-0.486894\pi$
$191$	5.82620	−	1.02732i	0.421569	−	0.0743340i	0.380409	+	0.924819i	$0.375783\pi$
$192$	10.5830	+	8.94427i	0.763763	+	0.645497i
$193$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$193$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$194$		0			0
$195$	21.3040	−	12.1995i	1.52561	−	0.873623i
$196$	13.1557	+	4.78828i	0.939693	+	0.342020i
$197$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$197$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$198$		0			0
$199$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$199$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−28.0156	−	4.93991i	−1.96631	−	0.346714i
$204$	−5.76136	+	15.6564i	−0.403376	+	1.09617i
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−12.6774	−	21.9579i	−0.879021	−	1.52251i
$209$		0			0
$210$		0			0
$211$	−21.2066	−	17.7944i	−1.45992	−	1.22502i	−0.924918	−	0.380166i	$-0.875867\pi$
$211$	−21.2066	−	17.7944i	−1.45992	−	1.22502i	−0.535001	−	0.844851i	$-0.679689\pi$
$212$		0			0
$213$	−7.84228	+	9.27911i	−0.537344	+	0.635794i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−23.6849	+	8.52572i	−1.60048	+	0.576114i
$220$	−16.7330	+	6.09031i	−1.12814	+	0.410609i
$221$	19.6222	−	23.3848i	1.31993	−	1.57303i
$222$		0			0
$223$	−27.8548	−	10.1383i	−1.86530	−	0.678912i	−0.974449	−	0.224607i	$-0.927890\pi$
$223$	−27.8548	−	10.1383i	−1.86530	−	0.678912i	−0.890846	−	0.454305i	$-0.849888\pi$
$224$		0			0
$225$	−2.70931	+	14.7533i	−0.180621	+	0.983553i
$226$		0			0
$227$	11.0341	+	13.1500i	0.732362	+	0.872795i	0.995769	−	0.0918895i	$-0.0292907\pi$
$227$	11.0341	+	13.1500i	0.732362	+	0.872795i	−0.263407	+	0.964685i	$0.584846\pi$
$228$		0			0
$229$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$229$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$230$		0			0
$231$	17.9805	−	3.10482i	1.18303	−	0.204282i
$232$		0			0
$233$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$233$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$234$		0			0
$235$	15.2797	+	26.4653i	0.996740	+	1.72640i
$236$		0			0
$237$	−29.9729	−	0.106177i	−1.94695	−	0.00689694i
$238$		0			0
$239$	2.39276	+	6.57406i	0.154775	+	0.425241i	0.992710	−	0.120530i	$-0.0384593\pi$
$239$	2.39276	+	6.57406i	0.154775	+	0.425241i	−0.837935	+	0.545770i	$0.816237\pi$
$240$	15.2470	+	2.74417i	0.984186	+	0.177136i
$241$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$241$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$242$		0			0
$243$	−12.1170	+	9.80700i	−0.777309	+	0.629119i
$244$		0			0
$245$	15.4147	−	2.71802i	0.984808	−	0.173648i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−0.0863910	+	24.3874i	−0.00547481	+	1.54549i
$250$		0			0
$251$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$251$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$252$	−14.8783	−	5.53495i	−0.937246	−	0.348669i
$253$		0			0
$254$		0			0
$255$	3.17379	+	18.3800i	0.198750	+	1.15100i
$256$	2.77837	−	15.7569i	0.173648	−	0.984808i
$257$	−23.4716	−	4.13868i	−1.46412	−	0.258164i	−0.615907	−	0.787819i	$-0.711210\pi$
$257$	−23.4716	−	4.13868i	−1.46412	−	0.258164i	−0.848213	+	0.529655i	$0.822322\pi$
$258$		0			0
$259$		0			0
$260$	−24.5497	−	14.1738i	−1.52251	−	0.879021i
$261$	31.7263	+	5.82626i	1.96381	+	0.360636i
$262$		0			0
$263$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$263$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	18.9710	−	3.34510i	1.15029	−	0.202827i
$273$	22.1855	+	18.7502i	1.34273	+	1.13481i
$274$		0			0
$275$	−12.7971	+	15.2509i	−0.771692	+	0.919667i
$276$		0			0
$277$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$277$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−3.68148	−	4.38742i	−0.219619	−	0.261731i	0.644974	−	0.764204i	$-0.276868\pi$
$281$	−3.68148	−	4.38742i	−0.219619	−	0.261731i	−0.864593	+	0.502473i	$0.832424\pi$
$282$		0			0
$283$	−2.42376	+	13.7458i	−0.144078	+	0.817106i	0.824025	+	0.566553i	$0.191723\pi$
$283$	−2.42376	+	13.7458i	−0.144078	+	0.817106i	−0.968103	+	0.250553i	$0.919388\pi$
$284$	13.8156	+	2.43606i	0.819804	+	0.144553i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	3.09653	+	5.36335i	0.182149	+	0.315491i
$290$		0			0
$291$	−13.0044	−	22.7097i	−0.762333	−	1.33127i
$292$	22.2665	+	18.6838i	1.30305	+	1.09339i
$293$	2.84671	+	7.82127i	0.166306	+	0.456923i	0.994651	−	0.103296i	$-0.0329390\pi$
$293$	2.84671	+	7.82127i	0.166306	+	0.456923i	−0.828344	+	0.560220i	$0.810717\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−20.4124	+	3.37600i	−1.18445	+	0.195895i
$298$		0			0
$299$		0			0
$300$	16.2968	−	5.86627i	0.940898	−	0.338689i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	0.316851	−	0.548803i	0.0180837	−	0.0313218i	−0.856842	−	0.515579i	$-0.827577\pi$
$307$	0.316851	−	0.548803i	0.0180837	−	0.0313218i	0.874926	+	0.484257i	$0.160910\pi$
$308$	−13.5431	−	16.1401i	−0.771692	−	0.919667i
$309$	−21.4327	+	17.8552i	−1.21926	+	1.01574i
$310$		0			0
$311$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$311$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$312$		0			0
$313$	24.4981	−	20.5564i	1.38472	−	1.16192i	0.417288	−	0.908774i	$-0.362981\pi$
$313$	24.4981	−	20.5564i	1.38472	−	1.16192i	0.967429	−	0.253141i	$-0.0814637\pi$
$314$		0			0
$315$	−17.5000	+	2.95804i	−0.986013	+	0.166667i
$316$	17.3050	+	29.9731i	0.973480	+	1.68612i
$317$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$317$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$318$		0			0
$319$	32.7965	+	27.5195i	1.83625	+	1.54080i
$320$	−6.11824	−	16.8097i	−0.342020	−	0.939693i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	16.8255	+	6.39541i	0.934752	+	0.355301i
$325$	−31.6936			−1.75804
$326$		0			0
$327$	0.914858	−	5.08306i	0.0505918	−	0.281094i
$328$		0			0
$329$	−23.2422	+	27.6990i	−1.28138	+	1.52709i
$330$		0			0
$331$	−4.34417	−	1.58115i	−0.238777	−	0.0869078i	0.219860	−	0.975531i	$-0.429440\pi$
$331$	−4.34417	−	1.58115i	−0.238777	−	0.0869078i	−0.458637	+	0.888624i	$0.651662\pi$
$332$	24.3876	−	14.0802i	1.33844	−	0.772750i
$333$		0			0
$334$		0			0
$335$		0			0
$336$	3.11906	+	18.0630i	0.170158	+	0.985417i
$337$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$337$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$338$		0			0
$339$		0			0
$340$	16.4986	−	13.8440i	0.894764	−	0.750796i
$341$		0			0
$342$		0			0
$343$	9.26013	+	16.0390i	0.500000	+	0.866025i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$347$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$348$	−12.6151	−	35.0456i	−0.676243	−	1.87864i
$349$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$349$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$350$		0			0
$351$	−25.0047	−	21.4384i	−1.33465	−	1.14430i
$352$		0			0
$353$	34.6219	−	6.10478i	1.84274	−	0.324925i	0.860054	−	0.510204i	$-0.170430\pi$
$353$	34.6219	−	6.10478i	1.84274	−	0.324925i	0.982686	+	0.185279i	$0.0593189\pi$
$354$		0			0
$355$	14.7387	−	5.36444i	0.782247	−	0.284715i
$356$		0			0
$357$	−19.1515	+	10.9669i	−1.01361	+	0.580429i
$358$		0			0
$359$	21.8765	−	12.6304i	1.15460	−	0.666608i	0.204595	−	0.978847i	$-0.434412\pi$
$359$	21.8765	−	12.6304i	1.15460	−	0.666608i	0.950004	+	0.312239i	$0.101079\pi$
$360$		0			0
$361$	−9.50000	+	16.4545i	−0.500000	+	0.866025i
$362$		0			0
$363$	−7.89052	−	2.90361i	−0.414145	−	0.152400i
$364$	5.82439	−	33.0318i	0.305281	−	1.73133i
$365$	32.0041	+	5.64318i	1.67517	+	0.295378i
$366$		0			0
$367$	27.7672	−	23.2994i	1.44943	−	1.21622i	0.516443	−	0.856322i	$-0.327256\pi$
$367$	27.7672	−	23.2994i	1.44943	−	1.21622i	0.932992	−	0.359898i	$-0.117189\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$373$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$374$		0			0
$375$	12.5000	−	14.7902i	0.645497	−	0.763763i
$376$		0			0
$377$			68.1555i			3.51019i
$378$		0			0
$379$	−37.7356			−1.93835			−0.969173	−	0.246380i	$-0.920759\pi$
$379$	−37.7356			−1.93835			−0.969173	+	0.246380i	$0.920759\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	18.1202	−	21.5948i	0.925900	−	1.10344i	−0.0684880	−	0.997652i	$-0.521817\pi$
$383$	18.1202	−	21.5948i	0.925900	−	1.10344i	0.994388	−	0.105793i	$-0.0337381\pi$
$384$		0			0
$385$	−22.1357	−	8.05673i	−1.12814	−	0.410609i
$386$		0			0
$387$		0			0
$388$	−15.1090	+	26.1696i	−0.767044	+	1.32856i
$389$	13.2008	+	15.7321i	0.669305	+	0.797647i	0.988689	−	0.149979i	$-0.0479205\pi$
$389$	13.2008	+	15.7321i	0.669305	+	0.797647i	−0.319384	+	0.947625i	$0.603476\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	33.5109	+	19.3475i	1.68612	+	0.973480i
$396$	15.2264	+	18.4095i	0.765157	+	0.925111i
$397$	−0.115521	−	0.200088i	−0.00579782	−	0.0100421i	0.863112	−	0.505013i	$-0.168512\pi$
$397$	−0.115521	−	0.200088i	−0.00579782	−	0.0100421i	−0.868910	+	0.494971i	$0.835179\pi$
$398$		0			0
$399$		0			0
$400$	−15.3209	−	12.8558i	−0.766044	−	0.642788i
$401$	−10.1171	−	27.7965i	−0.505223	−	1.38809i	−0.886113	−	0.463469i	$-0.846604\pi$
$401$	−10.1171	−	27.7965i	−0.505223	−	1.38809i	0.380889	−	0.924621i	$-0.375618\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	19.8664	−	3.21343i	0.987169	−	0.159677i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$409$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$410$		0			0
$411$		0			0
$412$	30.2685	+	11.0168i	1.49122	+	0.542761i
$413$		0			0
$414$		0			0
$415$	15.7421	−	27.2662i	0.772750	−	1.33844i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$419$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$420$	13.1175	+	15.7458i	0.640070	+	0.768317i
$421$	8.46757	−	7.10513i	0.412684	−	0.346283i	−0.412688	−	0.910872i	$-0.635410\pi$
$421$	8.46757	−	7.10513i	0.412684	−	0.346283i	0.825372	+	0.564590i	$0.190966\pi$
$422$		0			0
$423$	26.5760	−	31.2202i	1.29217	−	1.51798i
$424$		0			0
$425$	8.23570	−	22.6274i	0.399490	−	1.09759i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−14.8059	−	41.1318i	−0.714838	−	1.98586i
$430$		0			0
$431$		−	29.7955i		−	1.43520i	−0.696456	−	0.717599i	$-0.745241\pi$
$431$		−	29.7955i		−	1.43520i	0.696456	−	0.717599i	$-0.254759\pi$
$432$	−3.39148	−	20.5060i	−0.163173	−	0.986598i
$433$	−40.1484			−1.92941			−0.964703	−	0.263339i	$-0.915176\pi$
$433$	−40.1484			−1.92941			−0.964703	+	0.263339i	$0.915176\pi$
$434$		0			0
$435$	−31.8056	−	26.8807i	−1.52496	−	1.28883i
$436$	−5.60406	+	2.03971i	−0.268386	+	0.0976845i
$437$		0			0
$438$		0			0
$439$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$439$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$440$		0			0
$441$	−10.3709	−	18.2605i	−0.493852	−	0.869546i
$442$		0			0
$443$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$443$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	13.1503	−	35.7357i	0.621988	−	1.69024i
$448$	16.2141	−	13.6052i	0.766044	−	0.642788i
$449$	−24.6822	−	14.2503i	−1.16482	−	0.672511i	−0.212368	−	0.977190i	$-0.568118\pi$
$449$	−24.6822	−	14.2503i	−1.16482	−	0.672511i	−0.952455	+	0.304679i	$0.901451\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	17.5848	+	30.7084i	0.826204	+	1.44281i
$454$		0			0
$455$	−12.8259	−	35.2388i	−0.601286	−	1.65202i
$456$		0			0
$457$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$457$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$458$		0			0
$459$	21.8034	−	12.2811i	1.01769	−	0.573233i
$460$		0			0
$461$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$461$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$462$		0			0
$463$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$463$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$464$	−27.6457	+	32.9469i	−1.28342	+	1.52952i
$465$		0			0
$466$		0			0
$467$	−36.9254	+	21.3189i	−1.70871	+	0.986521i	−0.772536	+	0.634970i	$0.781012\pi$
$467$	−36.9254	+	21.3189i	−1.70871	+	0.986521i	−0.936169	+	0.351551i	$0.885654\pi$
$468$	−6.86943	+	37.4068i	−0.317539	+	1.72913i
$469$		0			0
$470$		0			0
$471$	−12.6917	+	10.5732i	−0.584802	+	0.487187i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$	22.0693	+	12.7417i	1.01154	+	0.584016i
$477$		0			0
$478$		0			0
$479$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$479$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	1.68586	+	9.56101i	0.0766302	+	0.434591i
$485$			33.7848i			1.53409i
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−0.930698	+	1.10916i	−0.0420018	+	0.0500558i	−0.786636	−	0.617417i	$-0.788179\pi$
$491$	−0.930698	+	1.10916i	−0.0420018	+	0.0500558i	0.744635	+	0.667472i	$0.232624\pi$
$492$		0			0
$493$	−48.6592	−	17.7105i	−2.19150	−	0.797641i
$494$		0			0
$495$	25.0341	+	9.31306i	1.12520	+	0.418591i
$496$		0			0
$497$	11.9290	+	14.2164i	0.535089	+	0.637694i
$498$		0			0
$499$	−7.74430	+	43.9201i	−0.346683	+	1.96613i	−0.114816	+	0.993387i	$0.536628\pi$
$499$	−7.74430	+	43.9201i	−0.346683	+	1.96613i	−0.231867	+	0.972748i	$0.574483\pi$
$500$	−22.0210	−	3.88289i	−0.984808	−	0.173648i
$501$	−3.93065	−	4.71821i	−0.175608	−	0.210794i
$502$		0			0
$503$	1.39864	+	0.807506i	0.0623624	+	0.0360049i	0.530857	−	0.847461i	$-0.321870\pi$
$503$	1.39864	+	0.807506i	0.0623624	+	0.0360049i	−0.468495	+	0.883466i	$0.655203\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	23.6822	−	40.6853i	1.05176	−	1.80690i
$508$		0			0
$509$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$509$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$510$		0			0
$511$	6.67710	+	37.8677i	0.295378	+	1.67517i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	35.4660	−	6.25361i	1.56282	−	0.275567i
$516$		0			0
$517$	51.1351	−	18.6117i	2.24892	−	0.818540i
$518$		0			0
$519$	22.5931	−	12.9376i	0.991727	−	0.567899i
$520$		0			0
$521$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$521$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$522$		0			0
$523$	5.56344	−	9.63616i	0.243272	−	0.421360i	−0.718372	−	0.695659i	$-0.755112\pi$
$523$	5.56344	−	9.63616i	0.243272	−	0.421360i	0.961644	+	0.274299i	$0.0884458\pi$
$524$		0			0
$525$	21.5032	+	7.91289i	0.938475	+	0.345347i
$526$		0			0
$527$		0			0
$528$	9.52684	−	25.8891i	0.414603	−	1.12668i
$529$	−17.6190	+	14.7841i	−0.766044	+	0.642788i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−7.11399	+	8.41738i	−0.306991	+	0.363237i
$538$		0			0
$539$		−	27.8722i		−	1.20054i
$540$	−14.7470	−	17.9590i	−0.634611	−	0.772832i
$541$	−44.3483			−1.90668			−0.953340	−	0.301898i	$-0.902380\pi$
$541$	−44.3483			−1.90668			−0.953340	+	0.301898i	$0.902380\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−4.28588	+	5.10771i	−0.183587	+	0.218790i
$546$		0			0
$547$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$547$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−7.95041	+	45.0890i	−0.338086	+	1.91738i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$557$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$558$		0			0
$559$		0			0
$560$	8.09367	−	22.2372i	0.342020	−	0.939693i
$561$	33.2131	+	0.117655i	1.40226	+	0.00496742i
$562$		0			0
$563$	−16.2301	−	44.5919i	−0.684019	−	1.87933i	−0.349503	−	0.936935i	$-0.613650\pi$
$563$	−16.2301	−	44.5919i	−0.684019	−	1.87933i	−0.334515	−	0.942390i	$-0.608573\pi$
$564$	−46.5939	−	8.38605i	−1.96196	−	0.353116i
$565$		0			0
$566$		0			0
$567$	11.6125	+	20.7882i	0.487679	+	0.873023i
$568$		0			0
$569$	45.8197	−	8.07924i	1.92086	−	0.338699i	0.922032	−	0.387113i	$-0.126528\pi$
$569$	45.8197	−	8.07924i	1.92086	−	0.338699i	0.998827	+	0.0484135i	$0.0154165\pi$
$570$		0			0
$571$	−44.4758	+	16.1879i	−1.86125	+	0.677441i	−0.883231	+	0.468938i	$0.844637\pi$
$571$	−44.4758	+	16.1879i	−1.86125	+	0.677441i	−0.978022	+	0.208502i	$0.933141\pi$
$572$	−32.4468	+	38.6686i	−1.35667	+	1.61681i
$573$	−0.0362990	+	10.2469i	−0.00151641	+	0.428070i
$574$		0			0
$575$		0			0
$576$	−18.4939	+	15.2963i	−0.770579	+	0.637344i
$577$	−2.39408	+	4.14668i	−0.0996670	+	0.172628i	−0.911547	−	0.411196i	$-0.865111\pi$
$577$	−2.39408	+	4.14668i	−0.0996670	+	0.172628i	0.811880	+	0.583825i	$0.198444\pi$
$578$		0			0
$579$		0			0
$580$	−8.34998	+	47.3551i	−0.346714	+	1.96631i
$581$	36.6867	+	6.46886i	1.52202	+	0.268373i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	14.2597	+	40.0591i	0.589567	+	1.65624i
$586$		0			0
$587$	12.5764	−	34.5535i	0.519085	−	1.42617i	−0.352445	−	0.935833i	$-0.614650\pi$
$587$	12.5764	−	34.5535i	0.519085	−	1.42617i	0.871530	−	0.490342i	$-0.163128\pi$
$588$	−12.1987	+	20.9569i	−0.503065	+	0.864249i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	41.8642i		−	1.71916i	−0.511003	−	0.859579i	$-0.670726\pi$
$593$		−	41.8642i		−	1.71916i	0.511003	−	0.859579i	$-0.329274\pi$
$594$		0			0
$595$	28.4914			1.16803
$596$	−43.3013	+	7.63519i	−1.77369	+	0.312749i
$597$		0			0
$598$		0			0
$599$	21.3861	−	25.4870i	0.873814	−	1.04137i	−0.124975	−	0.992160i	$-0.539885\pi$
$599$	21.3861	−	25.4870i	0.873814	−	1.04137i	0.998788	−	0.0492108i	$-0.0156706\pi$
$600$		0			0
$601$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$601$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$602$		0			0
$603$		0			0
$604$	20.4306	−	35.3869i	0.831310	−	1.43987i
$605$	6.97710	+	8.31498i	0.283659	+	0.338052i
$606$		0			0
$607$	6.93157	−	39.3109i	0.281344	−	1.59558i	−0.436717	−	0.899599i	$-0.643859\pi$
$607$	6.93157	−	39.3109i	0.281344	−	1.59558i	0.718061	−	0.695980i	$-0.245030\pi$
$608$		0			0
$609$	17.0163	−	46.2416i	0.689535	−	1.87380i
$610$		0			0
$611$	75.0227	+	43.3143i	3.03509	+	1.75231i
$612$	−24.9213	−	14.6247i	−1.00738	−	0.591168i
$613$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$613$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$617$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$618$		0			0
$619$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$619$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	41.3204	−	14.8738i	1.65414	−	0.595430i
$625$	−23.4923	+	8.55050i	−0.939693	+	0.342020i
$626$		0			0
$627$		0			0
$628$	17.9240	+	6.52379i	0.715244	+	0.260328i
$629$		0			0
$630$		0			0
$631$	−12.2979	+	21.3005i	−0.489570	+	0.847961i	−0.999928	−	0.0120014i	$-0.996180\pi$
$631$	−12.2979	+	21.3005i	−0.489570	+	0.847961i	0.510357	+	0.859962i	$0.329513\pi$
$632$		0			0
$633$	36.8398	−	30.6905i	1.46425	−	1.21984i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	33.9902	−	28.5211i	1.34674	−	1.13005i
$638$		0			0
$639$	−13.4117	−	16.2153i	−0.530557	−	0.641469i
$640$		0			0
$641$	−13.4252	+	36.8855i	−0.530265	+	1.45689i	0.328491	+	0.944507i	$0.393460\pi$
$641$	−13.4252	+	36.8855i	−0.530265	+	1.45689i	−0.858756	+	0.512384i	$0.828762\pi$
$642$		0			0
$643$	32.8024	+	27.5245i	1.29360	+	1.08546i	0.991213	+	0.132273i	$0.0422275\pi$
$643$	32.8024	+	27.5245i	1.29360	+	1.08546i	0.302386	+	0.953186i	$0.402217\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			50.1875i			1.97307i	0.163543	+	0.986536i	$0.447708\pi$
$647$			50.1875i			1.97307i	−0.163543	+	0.986536i	$0.552292\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$653$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−7.26673	−	42.9905i	−0.283502	−	1.67722i
$658$		0			0
$659$	31.6238	+	37.6878i	1.23189	+	1.46811i	0.835009	+	0.550236i	$0.185462\pi$
$659$	31.6238	+	37.6878i	1.23189	+	1.46811i	0.396878	+	0.917871i	$0.370093\pi$
$660$	−5.24810	−	30.3926i	−0.204282	−	1.18303i
$661$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$661$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$662$		0			0
$663$	33.8429	+	40.6238i	1.31435	+	1.57770i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−2.42526	+	6.66334i	−0.0938360	+	0.257812i
$669$	25.8285	−	44.3725i	0.998587	−	1.71554i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$673$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$674$		0			0
$675$	−24.3181	−	9.14489i	−0.936005	−	0.351987i
$676$	−54.3586			−2.09071
$677$	43.8233	−	7.72722i	1.68426	−	0.296981i	0.752109	−	0.659039i	$-0.229037\pi$
$677$	43.8233	−	7.72722i	1.68426	−	0.296981i	0.932156	+	0.362058i	$0.117926\pi$
$678$		0			0
$679$	−37.5639	+	13.6721i	−1.44157	+	0.524689i
$680$		0			0
$681$	−25.8016	+	14.7750i	−0.988721	+	0.566178i
$682$		0			0
$683$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$683$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$691$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$692$	−26.0352	−	15.0314i	−0.989709	−	0.571409i
$693$	−0.223908	+	31.6033i	−0.00850557	+	1.20051i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−4.59430	−	26.0556i	−0.173648	−	0.984808i
$701$			25.3074i			0.955849i	0.878401	+	0.477924i	$0.158611\pi$
$701$			25.3074i			0.955849i	−0.878401	+	0.477924i	$0.841389\pi$
$702$		0			0
$703$		0			0
$704$	−31.3700	+	5.53137i	−1.18230	+	0.208472i
$705$	−49.8023	+	17.9270i	−1.87566	+	0.675170i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	42.8606	+	15.6000i	1.60966	+	0.585869i	0.981373	−	0.192110i	$-0.0615331\pi$
$709$	42.8606	+	15.6000i	1.60966	+	0.585869i	0.628290	+	0.777980i	$0.283755\pi$
$710$		0			0
$711$	9.37691	−	51.0610i	0.351662	−	1.91494i
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−9.80007	+	55.5790i	−0.366502	+	2.07854i
$716$	12.5326	+	2.20983i	0.468363	+	0.0825851i
$717$	−11.9407	+	2.06187i	−0.445932	+	0.0770021i
$718$		0			0
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$	−9.35577	+	25.1489i	−0.348669	+	0.937246i
$721$	21.3056	+	36.9024i	0.793463	+	1.37432i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	18.3875	+	50.5191i	0.682893	+	1.87623i
$726$		0			0
$727$	−2.32382	−	13.1790i	−0.0861857	−	0.488783i	−0.997094	−	0.0761754i	$-0.975729\pi$
$727$	−2.32382	−	13.1790i	−0.0861857	−	0.488783i	0.910909	−	0.412608i	$-0.135382\pi$
$728$		0			0
$729$	−13.0000	−	23.6643i	−0.481481	−	0.876456i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	7.24147	−	2.63568i	0.267470	−	0.0973511i	−0.204804	−	0.978803i	$-0.565656\pi$
$733$	7.24147	−	2.63568i	0.267470	−	0.0973511i	0.472274	+	0.881452i	$0.343433\pi$
$734$		0			0
$735$	−0.0960380	+	27.1107i	−0.00354242	+	0.999994i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	25.8056	−	44.6967i	0.949276	−	1.64419i	0.202321	−	0.979319i	$-0.435152\pi$
$739$	25.8056	−	44.6967i	0.949276	−	1.64419i	0.746955	−	0.664875i	$-0.231515\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$743$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$744$		0			0
$745$	−37.6581	+	31.5989i	−1.37969	+	1.15769i
$746$		0			0
$747$	−41.5458	−	7.62953i	−1.52008	−	0.279150i
$748$	−19.1757	−	33.2133i	−0.701134	−	1.21440i
$749$		0			0
$750$		0			0
$751$	−30.3440	−	25.4616i	−1.10727	−	0.929108i	−0.109376	−	0.994000i	$-0.534885\pi$
$751$	−30.3440	−	25.4616i	−1.10727	−	0.929108i	−0.997892	+	0.0648920i	$0.979330\pi$
$752$	18.6970	+	51.3696i	0.681810	+	1.87326i
$753$		0			0
$754$		0			0
$755$		−	45.6843i		−	1.66262i
$756$	14.0000	−	23.6643i	0.509175	−	0.860663i
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$761$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$762$		0			0
$763$	−7.41348	−	2.69829i	−0.268386	−	0.0976845i
$764$	10.2470	−	5.91608i	0.370722	−	0.214036i
$765$	−32.3053	−	0.228882i	−1.16800	−	0.00827525i
$766$		0			0
$767$		0			0
$768$	26.0078	+	9.57053i	0.938475	+	0.345347i
$769$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$769$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$770$		0			0
$771$	14.2563	−	38.7414i	0.513430	−	1.39524i
$772$		0			0
$773$	29.4456	+	17.0004i	1.05908	+	0.611463i	0.925179	−	0.379531i	$-0.123915\pi$
$773$	29.4456	+	17.0004i	1.05908	+	0.611463i	0.133906	+	0.990994i	$0.457248\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$	31.6936	−	37.5003i	1.13481	−	1.34273i
$781$	−4.84988	−	27.5050i	−0.173542	−	0.984207i
$782$		0			0
$783$	−19.6657	+	52.2950i	−0.702794	+	1.86887i
$784$	28.0000			1.00000
$785$	21.0017	−	3.70317i	0.749583	−	0.132172i
$786$		0			0
$787$	−28.9365	+	10.5320i	−1.03147	+	0.375426i	−0.801642	−	0.597804i	$-0.796040\pi$
$787$	−28.9365	+	10.5320i	−1.03147	+	0.375426i	−0.229832	+	0.973230i	$0.573818\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	17.2577	+	3.04299i	0.611298	+	0.107788i	0.470722	−	0.882282i	$-0.343993\pi$
$797$	17.2577	+	3.04299i	0.611298	+	0.107788i	0.140576	+	0.990070i	$0.455105\pi$
$798$		0			0
$799$	−50.4189	+	42.3065i	−1.78369	+	1.49670i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	19.7922	−	54.3785i	0.698450	−	1.91898i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			52.2539i			1.83715i	0.395249	+	0.918574i	$0.370658\pi$
$809$			52.2539i			1.83715i	−0.395249	+	0.918574i	$0.629342\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	−56.0313	+	9.87983i	−1.96631	+	0.346714i
$813$		0			0
$814$		0			0
$815$		0			0
$816$	−0.118195	+	33.3654i	−0.00413765	+	1.16802i
$817$		0			0
$818$		0			0
$819$	−38.7694	+	32.0660i	−1.35471	+	1.12048i
$820$		0			0
$821$	−36.7426	−	43.7881i	−1.28233	−	1.52822i	−0.695592	−	0.718437i	$-0.744858\pi$
$821$	−36.7426	−	43.7881i	−1.28233	−	1.52822i	−0.586734	−	0.809780i	$-0.699586\pi$
$822$		0			0
$823$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$823$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$824$		0			0
$825$	−22.0715	−	26.4938i	−0.768429	−	0.922395i
$826$		0			0
$827$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$827$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$828$		0			0
$829$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$829$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$830$		0			0
$831$		0			0
$832$	−38.8459	−	32.5956i	−1.34674	−	1.13005i
$833$	11.5300	+	31.6784i	0.399490	+	1.09759i
$834$		0			0
$835$	1.37667	+	7.80751i	0.0476418	+	0.270190i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$839$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$840$		0			0
$841$	81.3880	−	29.6228i	2.80648	−	1.02148i
$842$		0			0
$843$	8.60857	−	4.92958i	0.296495	−	0.169784i
$844$	−52.0274	−	18.9364i	−1.79086	−	0.651818i
$845$	−52.6325	+	30.3874i	−1.81061	+	1.04536i
$846$		0			0
$847$	−6.42157	+	11.1225i	−0.220648	+	0.382173i
$848$		0			0
$849$	−22.6884	−	8.34903i	−0.778663	−	0.286538i
$850$		0			0
$851$		0			0
$852$	−8.39138	+	22.8035i	−0.287484	+	0.781234i
$853$	−42.9160	+	36.0108i	−1.46942	+	1.23299i	−0.552741	+	0.833353i	$0.686418\pi$
$853$	−42.9160	+	36.0108i	−1.46942	+	1.23299i	−0.916675	+	0.399633i	$0.869138\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−19.5244	+	53.6429i	−0.666941	+	1.83241i	−0.124678	+	0.992197i	$0.539790\pi$
$857$	−19.5244	+	53.6429i	−0.666941	+	1.83241i	−0.542263	+	0.840209i	$0.682432\pi$
$858$		0			0
$859$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$859$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$	−33.6113			−1.14282
$866$		0			0
$867$	−10.0927	+	3.63302i	−0.342767	+	0.123384i
$868$		0			0
$869$	44.2905	−	52.7834i	1.50245	−	1.79055i
$870$		0			0
$871$		0			0
$872$		0			0
$873$	42.7022	−	15.2006i	1.44525	−	0.514462i
$874$		0			0
$875$	−19.0139	−	22.6599i	−0.642788	−	0.766044i
$876$	−38.6812	+	32.2246i	−1.30692	+	1.08877i
$877$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$877$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$878$		0			0
$879$	−14.2060	+	2.45304i	−0.479156	+	0.0827391i
$880$	−27.2817	+	22.8921i	−0.919667	+	0.771692i
$881$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$881$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$882$		0			0
$883$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$883$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$884$	20.8815	−	57.3715i	0.702321	−	1.92961i
$885$		0			0
$886$		0			0
$887$	−7.98776	−	21.9462i	−0.268203	−	0.736881i	−0.998551	−	0.0538062i	$-0.982865\pi$
$887$	−7.98776	−	21.9462i	−0.268203	−	0.736881i	0.730349	−	0.683074i	$-0.239358\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	0.507763	−	35.8321i	0.0170107	−	1.20042i
$892$	−59.2849			−1.98501
$893$		0			0
$894$		0			0
$895$	13.3699	−	4.86625i	0.446907	−	0.162661i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	5.00000	+	29.5804i	0.166667	+	0.986013i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$907$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$908$	29.7325	+	17.1661i	0.986709	+	0.569677i
$909$		0			0
$910$		0			0
$911$	−12.8621	+	35.3384i	−0.426141	+	1.17081i	0.521995	+	0.852949i	$0.325188\pi$
$911$	−12.8621	+	35.3384i	−0.426141	+	1.17081i	−0.948136	+	0.317865i	$0.897034\pi$
$912$		0			0
$913$	−42.9472	−	36.0370i	−1.42135	−	1.19265i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−60.6113			−1.99938			−0.999691	−	0.0248659i	$-0.992084\pi$
$919$	−60.6113			−1.99938			−0.999691	+	0.0248659i	$0.992084\pi$
$920$		0			0
$921$	0.838310	+	0.708501i	0.0276232	+	0.0233459i
$922$		0			0
$923$	28.5796	−	34.0598i	0.940709	−	1.12109i
$924$	31.6685	−	18.1346i	1.04182	−	0.596583i
$925$		0			0
$926$		0			0
$927$	−23.8613	−	42.0136i	−0.783707	−	1.37991i
$928$		0			0
$929$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$929$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−37.1336	−	21.4391i	−1.21440	−	0.701134i
$936$		0			0
$937$	8.58659	+	14.8724i	0.280512	+	0.485860i	0.971511	−	0.236995i	$-0.0761625\pi$
$937$	8.58659	+	14.8724i	0.280512	+	0.485860i	−0.690999	+	0.722856i	$0.742829\pi$
$938$		0			0
$939$	27.5254	+	48.0679i	0.898259	+	1.56864i
$940$	46.8198	+	39.2865i	1.52709	+	1.28138i
$941$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$941$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	0.326685	−	30.7391i	0.0106271	−	0.999944i
$946$		0			0
$947$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$947$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$948$	−56.4032	+	20.3031i	−1.83189	+	0.659414i
$949$	86.5677	−	31.5081i	2.81011	−	1.02279i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$953$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$954$		0			0
$955$	6.61438	−	11.4564i	0.214036	−	0.370722i
$956$	8.99385	+	10.7184i	0.290882	+	0.346659i
$957$	−56.9736	+	47.4636i	−1.84170	+	1.53428i
$958$		0			0
$959$		0			0
$960$	30.5320	−	5.27217i	0.985417	−	0.170158i
$961$	23.7474	−	19.9264i	0.766044	−	0.642788i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$967$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	−16.0642	+	26.7197i	−0.515260	+	0.857034i
$973$		0			0
$974$		0			0
$975$	9.72384	−	54.0268i	0.311412	−	1.73024i
$976$		0			0
$977$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$977$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$978$		0			0
$979$		0			0
$980$	27.1109	−	15.6525i	0.866025	−	0.500000i
$981$	8.38421	+	3.11905i	0.267687	+	0.0995835i
$982$		0			0
$983$	38.4158	+	45.7822i	1.22528	+	1.46023i	0.844498	+	0.535559i	$0.179899\pi$
$983$	38.4158	+	45.7822i	1.22528	+	1.46023i	0.380778	+	0.924667i	$0.375656\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−40.0865	−	48.1183i	−1.27597	−	1.53162i
$988$		0			0
$989$		0			0
$990$		0			0
$991$	21.9935	+	38.0938i	0.698645	+	1.21009i	0.968936	+	0.247310i	$0.0795467\pi$
$991$	21.9935	+	38.0938i	0.698645	+	1.21009i	−0.270291	+	0.962779i	$0.587120\pi$
$992$		0			0
$993$	4.02815	−	6.92023i	0.127829	−	0.219607i
$994$		0			0
$995$		0			0
$996$	16.5196	+	45.8925i	0.523444	+	1.45416i
$997$	3.21601	+	18.2389i	0.101852	+	0.577632i	0.992431	+	0.122803i	$0.0391883\pi$
$997$	3.21601	+	18.2389i	0.101852	+	0.577632i	−0.890579	+	0.454829i	$0.849701\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	945.2.cs.a.419.2	✓	24
5.4	even	2	inner	945.2.cs.a.419.3	yes	24
7.6	odd	2	inner	945.2.cs.a.419.3	yes	24
27.2	odd	18	inner	945.2.cs.a.839.2	yes	24
35.34	odd	2	CM	945.2.cs.a.419.2	✓	24
135.29	odd	18	inner	945.2.cs.a.839.3	yes	24
189.83	even	18	inner	945.2.cs.a.839.3	yes	24
945.839	even	18	inner	945.2.cs.a.839.2	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
945.2.cs.a.419.2	✓	24	1.1	even	1	trivial
945.2.cs.a.419.2	✓	24	35.34	odd	2	CM
945.2.cs.a.419.3	yes	24	5.4	even	2	inner
945.2.cs.a.419.3	yes	24	7.6	odd	2	inner
945.2.cs.a.839.2	yes	24	27.2	odd	18	inner
945.2.cs.a.839.2	yes	24	945.839	even	18	inner
945.2.cs.a.839.3	yes	24	135.29	odd	18	inner
945.2.cs.a.839.3	yes	24	189.83	even	18	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	945.cs (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.306808 + 1.70466i) q^{3} +(1.87939 - 0.684040i) q^{4} +(1.43732 - 1.71293i) q^{5} +(2.48619 + 0.904900i) q^{7} +(-2.81174 - 1.04601i) q^{9} +O(q^{10})\)	\(q+(-0.306808 + 1.70466i) q^{3} +(1.87939 - 0.684040i) q^{4} +(1.43732 - 1.71293i) q^{5} +(2.48619 + 0.904900i) q^{7} +(-2.81174 - 1.04601i) q^{9} +(-2.55941 - 3.05019i) q^{11} +(0.589446 + 3.41358i) q^{12} +(1.10071 - 6.24241i) q^{13} +(2.47898 + 2.97568i) q^{15} +(3.06418 - 2.57115i) q^{16} +(4.17071 + 2.40796i) q^{17} +(1.52956 - 4.20243i) q^{20} +(-2.30533 + 3.96049i) q^{21} +(-0.868241 - 4.92404i) q^{25} +(2.64575 - 4.47214i) q^{27} +5.29150 q^{28} +(-10.5889 + 1.86711i) q^{29} +(5.98479 - 3.42711i) q^{33} +(5.12348 - 2.95804i) q^{35} +(-5.99985 - 0.0425087i) q^{36} +(10.3035 + 3.79155i) q^{39} +(-6.89658 - 3.98174i) q^{44} +(-5.83309 + 3.31286i) q^{45} +(-4.67425 + 12.8424i) q^{47} +(3.44283 + 6.01223i) q^{48} +(5.36231 + 4.49951i) q^{49} +(-5.38436 + 6.37086i) q^{51} +(-2.20141 - 12.4848i) q^{52} -8.90344 q^{55} +(6.69444 + 3.89672i) q^{60} +(-6.04399 - 5.14492i) q^{63} +(4.00000 - 6.92820i) q^{64} +(-9.11074 - 10.8578i) q^{65} +(9.48551 + 1.67255i) q^{68} +(6.07461 + 3.50718i) q^{71} +(7.26673 + 12.5863i) q^{73} +(8.66020 + 0.0306782i) q^{75} +(-3.60308 - 9.89937i) q^{77} +(3.00497 + 17.0421i) q^{79} -8.94427i q^{80} +(6.81174 + 5.88220i) q^{81} +(13.8663 - 2.44500i) q^{83} +(-1.62348 + 9.02022i) q^{84} +(10.1193 - 3.68312i) q^{85} +(0.0659721 - 18.6234i) q^{87} +(8.38533 - 14.5238i) q^{91} +(-11.5742 + 9.71188i) q^{97} +(4.00588 + 11.2535i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 6 q^{9}+O(q^{10}) \) 24 * q - 6 * q^9	\( 24 q - 6 q^{9} - 18 q^{11} - 24 q^{39} + 96 q^{64} - 180 q^{65} - 12 q^{79} + 102 q^{81} + 84 q^{84} + 60 q^{85} + 228 q^{99}+O(q^{100}) \) 24 * q - 6 * q^9 - 18 * q^11 - 24 * q^39 + 96 * q^64 - 180 * q^65 - 12 * q^79 + 102 * q^81 + 84 * q^84 + 60 * q^85 + 228 * q^99

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