LMFDB - Embedded newform 225.3.g.f.82.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,3,Mod(82,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(4))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("225.82");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	225.g (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$6.13080594811$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			82.1
Root			$-1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	225.82
Dual form			225.3.g.f.118.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.22474 - 1.22474i) q^{2} -1.00000i q^{4} +(7.34847 + 7.34847i) q^{7} +(-6.12372 + 6.12372i) q^{8} +18.0000 q^{11} +(-7.34847 + 7.34847i) q^{13} -18.0000i q^{14} +11.0000 q^{16} +(4.89898 + 4.89898i) q^{17} -10.0000i q^{19} +(-22.0454 - 22.0454i) q^{22} +(19.5959 - 19.5959i) q^{23} +18.0000 q^{26} +(7.34847 - 7.34847i) q^{28} +22.0000 q^{31} +(11.0227 + 11.0227i) q^{32} -12.0000i q^{34} +(7.34847 + 7.34847i) q^{37} +(-12.2474 + 12.2474i) q^{38} +18.0000 q^{41} +(29.3939 - 29.3939i) q^{43} -18.0000i q^{44} -48.0000 q^{46} +(-44.0908 - 44.0908i) q^{47} +59.0000i q^{49} +(7.34847 + 7.34847i) q^{52} +(-4.89898 + 4.89898i) q^{53} -90.0000 q^{56} +90.0000i q^{59} +2.00000 q^{61} +(-26.9444 - 26.9444i) q^{62} -71.0000i q^{64} +(44.0908 + 44.0908i) q^{67} +(4.89898 - 4.89898i) q^{68} -72.0000 q^{71} +(-44.0908 + 44.0908i) q^{73} -18.0000i q^{74} -10.0000 q^{76} +(132.272 + 132.272i) q^{77} +70.0000i q^{79} +(-22.0454 - 22.0454i) q^{82} +(-53.8888 + 53.8888i) q^{83} -72.0000 q^{86} +(-110.227 + 110.227i) q^{88} +90.0000i q^{89} -108.000 q^{91} +(-19.5959 - 19.5959i) q^{92} +108.000i q^{94} +(-102.879 - 102.879i) q^{97} +(72.2599 - 72.2599i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 72 q^{11} + 44 q^{16} + 72 q^{26} + 88 q^{31} + 72 q^{41} - 192 q^{46} - 360 q^{56} + 8 q^{61} - 288 q^{71} - 40 q^{76} - 288 q^{86} - 432 q^{91}+O(q^{100}) $ 4 * q + 72 * q^11 + 44 * q^16 + 72 * q^26 + 88 * q^31 + 72 * q^41 - 192 * q^46 - 360 * q^56 + 8 * q^61 - 288 * q^71 - 40 * q^76 - 288 * q^86 - 432 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$

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 $

$2$	−1.22474	−	1.22474i	−0.612372	−	0.612372i	0.331191	−	0.943564i	$-0.392549\pi$
$2$	−1.22474	−	1.22474i	−0.612372	−	0.612372i	−0.943564	+	0.331191i	$0.892549\pi$
$3$		0			0
$3$		0			0
$4$		−	1.00000i		−	0.250000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	7.34847	+	7.34847i	1.04978	+	1.04978i	0.998694	+	0.0510871i	$0.0162686\pi$
$7$	7.34847	+	7.34847i	1.04978	+	1.04978i	0.0510871	+	0.998694i	$0.483731\pi$
$8$	−6.12372	+	6.12372i	−0.765466	+	0.765466i
$9$		0			0
$10$		0			0
$11$	18.0000			1.63636			0.818182	−	0.574960i	$-0.194982\pi$
$11$	18.0000			1.63636			0.818182	+	0.574960i	$0.194982\pi$
$12$		0			0
$13$	−7.34847	+	7.34847i	−0.565267	+	0.565267i	−0.930799	−	0.365532i	$-0.880887\pi$
$13$	−7.34847	+	7.34847i	−0.565267	+	0.565267i	0.365532	+	0.930799i	$0.380887\pi$
$14$		−	18.0000i		−	1.28571i
$15$		0			0
$16$	11.0000			0.687500
$17$	4.89898	+	4.89898i	0.288175	+	0.288175i	0.836358	−	0.548183i	$-0.184680\pi$
$17$	4.89898	+	4.89898i	0.288175	+	0.288175i	−0.548183	+	0.836358i	$0.684680\pi$
$18$		0			0
$19$		−	10.0000i		−	0.526316i	−0.964753	−	0.263158i	$-0.915236\pi$
$19$		−	10.0000i		−	0.526316i	0.964753	−	0.263158i	$-0.0847640\pi$
$20$		0			0
$21$		0			0
$22$	−22.0454	−	22.0454i	−1.00206	−	1.00206i
$23$	19.5959	−	19.5959i	0.851996	−	0.851996i	−0.138382	−	0.990379i	$-0.544190\pi$
$23$	19.5959	−	19.5959i	0.851996	−	0.851996i	0.990379	+	0.138382i	$0.0441903\pi$
$24$		0			0
$25$		0			0
$26$	18.0000			0.692308
$27$		0			0
$28$	7.34847	−	7.34847i	0.262445	−	0.262445i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	22.0000			0.709677			0.354839	−	0.934928i	$-0.384536\pi$
$31$	22.0000			0.709677			0.354839	+	0.934928i	$0.384536\pi$
$32$	11.0227	+	11.0227i	0.344459	+	0.344459i
$33$		0			0
$34$		−	12.0000i		−	0.352941i
$35$		0			0
$36$		0			0
$37$	7.34847	+	7.34847i	0.198607	+	0.198607i	0.799403	−	0.600795i	$-0.205149\pi$
$37$	7.34847	+	7.34847i	0.198607	+	0.198607i	−0.600795	+	0.799403i	$0.705149\pi$
$38$	−12.2474	+	12.2474i	−0.322301	+	0.322301i
$39$		0			0
$40$		0			0
$41$	18.0000			0.439024			0.219512	−	0.975610i	$-0.429553\pi$
$41$	18.0000			0.439024			0.219512	+	0.975610i	$0.429553\pi$
$42$		0			0
$43$	29.3939	−	29.3939i	0.683579	−	0.683579i	−0.277226	−	0.960805i	$-0.589415\pi$
$43$	29.3939	−	29.3939i	0.683579	−	0.683579i	0.960805	+	0.277226i	$0.0894151\pi$
$44$		−	18.0000i		−	0.409091i
$45$		0			0
$46$	−48.0000			−1.04348
$47$	−44.0908	−	44.0908i	−0.938102	−	0.938102i	0.0600905	−	0.998193i	$-0.480861\pi$
$47$	−44.0908	−	44.0908i	−0.938102	−	0.938102i	−0.998193	+	0.0600905i	$0.980861\pi$
$48$		0			0
$49$			59.0000i			1.20408i
$50$		0			0
$51$		0			0
$52$	7.34847	+	7.34847i	0.141317	+	0.141317i
$53$	−4.89898	+	4.89898i	−0.0924336	+	0.0924336i	−0.751812	−	0.659378i	$-0.770820\pi$
$53$	−4.89898	+	4.89898i	−0.0924336	+	0.0924336i	0.659378	+	0.751812i	$0.270820\pi$
$54$		0			0
$55$		0			0
$56$	−90.0000			−1.60714
$57$		0			0
$58$		0			0
$59$			90.0000i			1.52542i	0.646738	+	0.762712i	$0.276133\pi$
$59$			90.0000i			1.52542i	−0.646738	+	0.762712i	$0.723867\pi$
$60$		0			0
$61$	2.00000			0.0327869			0.0163934	−	0.999866i	$-0.494782\pi$
$61$	2.00000			0.0327869			0.0163934	+	0.999866i	$0.494782\pi$
$62$	−26.9444	−	26.9444i	−0.434587	−	0.434587i
$63$		0			0
$64$		−	71.0000i		−	1.10938i
$65$		0			0
$66$		0			0
$67$	44.0908	+	44.0908i	0.658072	+	0.658072i	0.954924	−	0.296852i	$-0.0959367\pi$
$67$	44.0908	+	44.0908i	0.658072	+	0.658072i	−0.296852	+	0.954924i	$0.595937\pi$
$68$	4.89898	−	4.89898i	0.0720438	−	0.0720438i
$69$		0			0
$70$		0			0
$71$	−72.0000			−1.01408			−0.507042	−	0.861921i	$-0.669261\pi$
$71$	−72.0000			−1.01408			−0.507042	+	0.861921i	$0.669261\pi$
$72$		0			0
$73$	−44.0908	+	44.0908i	−0.603984	+	0.603984i	−0.941367	−	0.337384i	$-0.890458\pi$
$73$	−44.0908	+	44.0908i	−0.603984	+	0.603984i	0.337384	+	0.941367i	$0.390458\pi$
$74$		−	18.0000i		−	0.243243i
$75$		0			0
$76$	−10.0000			−0.131579
$77$	132.272	+	132.272i	1.71782	+	1.71782i
$78$		0			0
$79$			70.0000i			0.886076i	0.896503	+	0.443038i	$0.146099\pi$
$79$			70.0000i			0.886076i	−0.896503	+	0.443038i	$0.853901\pi$
$80$		0			0
$81$		0			0
$82$	−22.0454	−	22.0454i	−0.268846	−	0.268846i
$83$	−53.8888	+	53.8888i	−0.649262	+	0.649262i	−0.952815	−	0.303552i	$-0.901827\pi$
$83$	−53.8888	+	53.8888i	−0.649262	+	0.649262i	0.303552	+	0.952815i	$0.401827\pi$
$84$		0			0
$85$		0			0
$86$	−72.0000			−0.837209
$87$		0			0
$88$	−110.227	+	110.227i	−1.25258	+	1.25258i
$89$			90.0000i			1.01124i	0.862757	+	0.505618i	$0.168735\pi$
$89$			90.0000i			1.01124i	−0.862757	+	0.505618i	$0.831265\pi$
$90$		0			0
$91$	−108.000			−1.18681
$92$	−19.5959	−	19.5959i	−0.212999	−	0.212999i
$93$		0			0
$94$			108.000i			1.14894i
$95$		0			0
$96$		0			0
$97$	−102.879	−	102.879i	−1.06060	−	1.06060i	−0.998041	−	0.0625628i	$-0.980073\pi$
$97$	−102.879	−	102.879i	−1.06060	−	1.06060i	−0.0625628	−	0.998041i	$-0.519927\pi$
$98$	72.2599	−	72.2599i	0.737346	−	0.737346i
$99$		0			0
$100$		0			0
$101$	108.000			1.06931			0.534653	−	0.845071i	$-0.320442\pi$
$101$	108.000			1.06931			0.534653	+	0.845071i	$0.320442\pi$
$102$		0			0
$103$	66.1362	−	66.1362i	0.642099	−	0.642099i	−0.308972	−	0.951071i	$-0.599985\pi$
$103$	66.1362	−	66.1362i	0.642099	−	0.642099i	0.951071	+	0.308972i	$0.0999849\pi$
$104$		−	90.0000i		−	0.865385i
$105$		0			0
$106$	12.0000			0.113208
$107$	−44.0908	−	44.0908i	−0.412064	−	0.412064i	0.470393	−	0.882457i	$-0.344112\pi$
$107$	−44.0908	−	44.0908i	−0.412064	−	0.412064i	−0.882457	+	0.470393i	$0.844112\pi$
$108$		0			0
$109$		−	170.000i		−	1.55963i	−0.626008	−	0.779817i	$-0.715312\pi$
$109$		−	170.000i		−	1.55963i	0.626008	−	0.779817i	$-0.284688\pi$
$110$		0			0
$111$		0			0
$112$	80.8332	+	80.8332i	0.721725	+	0.721725i
$113$	44.0908	−	44.0908i	0.390184	−	0.390184i	−0.484569	−	0.874753i	$-0.661024\pi$
$113$	44.0908	−	44.0908i	0.390184	−	0.390184i	0.874753	+	0.484569i	$0.161024\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	110.227	−	110.227i	0.934127	−	0.934127i
$119$			72.0000i			0.605042i
$120$		0			0
$121$	203.000			1.67769
$122$	−2.44949	−	2.44949i	−0.0200778	−	0.0200778i
$123$		0			0
$124$		−	22.0000i		−	0.177419i
$125$		0			0
$126$		0			0
$127$	7.34847	+	7.34847i	0.0578620	+	0.0578620i	0.735446	−	0.677584i	$-0.236973\pi$
$127$	7.34847	+	7.34847i	0.0578620	+	0.0578620i	−0.677584	+	0.735446i	$0.736973\pi$
$128$	−42.8661	+	42.8661i	−0.334891	+	0.334891i
$129$		0			0
$130$		0			0
$131$	18.0000			0.137405			0.0687023	−	0.997637i	$-0.478114\pi$
$131$	18.0000			0.137405			0.0687023	+	0.997637i	$0.478114\pi$
$132$		0			0
$133$	73.4847	−	73.4847i	0.552516	−	0.552516i
$134$		−	108.000i		−	0.805970i
$135$		0			0
$136$	−60.0000			−0.441176
$137$	−142.070	−	142.070i	−1.03701	−	1.03701i	−0.999288	−	0.0377220i	$-0.987990\pi$
$137$	−142.070	−	142.070i	−1.03701	−	1.03701i	−0.0377220	−	0.999288i	$-0.512010\pi$
$138$		0			0
$139$			170.000i			1.22302i	0.791236	+	0.611511i	$0.209438\pi$
$139$			170.000i			1.22302i	−0.791236	+	0.611511i	$0.790562\pi$
$140$		0			0
$141$		0			0
$142$	88.1816	+	88.1816i	0.620997	+	0.620997i
$143$	−132.272	+	132.272i	−0.924982	+	0.924982i
$144$		0			0
$145$		0			0
$146$	108.000			0.739726
$147$		0			0
$148$	7.34847	−	7.34847i	0.0496518	−	0.0496518i
$149$		−	180.000i		−	1.20805i	−0.796964	−	0.604027i	$-0.793562\pi$
$149$		−	180.000i		−	1.20805i	0.796964	−	0.604027i	$-0.206438\pi$
$150$		0			0
$151$	22.0000			0.145695			0.0728477	−	0.997343i	$-0.476791\pi$
$151$	22.0000			0.145695			0.0728477	+	0.997343i	$0.476791\pi$
$152$	61.2372	+	61.2372i	0.402877	+	0.402877i
$153$		0			0
$154$		−	324.000i		−	2.10390i
$155$		0			0
$156$		0			0
$157$	−139.621	−	139.621i	−0.889305	−	0.889305i	0.105151	−	0.994456i	$-0.466467\pi$
$157$	−139.621	−	139.621i	−0.889305	−	0.889305i	−0.994456	+	0.105151i	$0.966467\pi$
$158$	85.7321	−	85.7321i	0.542608	−	0.542608i
$159$		0			0
$160$		0			0
$161$	288.000			1.78882
$162$		0			0
$163$	−117.576	+	117.576i	−0.721322	+	0.721322i	−0.968875	−	0.247552i	$-0.920374\pi$
$163$	−117.576	+	117.576i	−0.721322	+	0.721322i	0.247552	+	0.968875i	$0.420374\pi$
$164$		−	18.0000i		−	0.109756i
$165$		0			0
$166$	132.000			0.795181
$167$	−19.5959	−	19.5959i	−0.117341	−	0.117341i	0.645998	−	0.763339i	$-0.276441\pi$
$167$	−19.5959	−	19.5959i	−0.117341	−	0.117341i	−0.763339	+	0.645998i	$0.776441\pi$
$168$		0			0
$169$			61.0000i			0.360947i
$170$		0			0
$171$		0			0
$172$	−29.3939	−	29.3939i	−0.170895	−	0.170895i
$173$	142.070	−	142.070i	0.821216	−	0.821216i	−0.165066	−	0.986282i	$-0.552784\pi$
$173$	142.070	−	142.070i	0.821216	−	0.821216i	0.986282	+	0.165066i	$0.0527838\pi$
$174$		0			0
$175$		0			0
$176$	198.000			1.12500
$177$		0			0
$178$	110.227	−	110.227i	0.619253	−	0.619253i
$179$		−	90.0000i		−	0.502793i	−0.967884	−	0.251397i	$-0.919110\pi$
$179$		−	90.0000i		−	0.502793i	0.967884	−	0.251397i	$-0.0808899\pi$
$180$		0			0
$181$	−98.0000			−0.541436			−0.270718	−	0.962659i	$-0.587261\pi$
$181$	−98.0000			−0.541436			−0.270718	+	0.962659i	$0.587261\pi$
$182$	132.272	+	132.272i	0.726772	+	0.726772i
$183$		0			0
$184$			240.000i			1.30435i
$185$		0			0
$186$		0			0
$187$	88.1816	+	88.1816i	0.471560	+	0.471560i
$188$	−44.0908	+	44.0908i	−0.234526	+	0.234526i
$189$		0			0
$190$		0			0
$191$	−252.000			−1.31937			−0.659686	−	0.751541i	$-0.729311\pi$
$191$	−252.000			−1.31937			−0.659686	+	0.751541i	$0.729311\pi$
$192$		0			0
$193$	−264.545	+	264.545i	−1.37070	+	1.37070i	−0.511292	+	0.859407i	$0.670833\pi$
$193$	−264.545	+	264.545i	−1.37070	+	1.37070i	−0.859407	+	0.511292i	$0.829167\pi$
$194$			252.000i			1.29897i
$195$		0			0
$196$	59.0000			0.301020
$197$	127.373	+	127.373i	0.646566	+	0.646566i	0.952161	−	0.305596i	$-0.0988556\pi$
$197$	127.373	+	127.373i	0.646566	+	0.646566i	−0.305596	+	0.952161i	$0.598856\pi$
$198$		0			0
$199$		−	290.000i		−	1.45729i	−0.684893	−	0.728643i	$-0.740151\pi$
$199$		−	290.000i		−	1.45729i	0.684893	−	0.728643i	$-0.259849\pi$
$200$		0			0
$201$		0			0
$202$	−132.272	−	132.272i	−0.654814	−	0.654814i
$203$		0			0
$204$		0			0
$205$		0			0
$206$	−162.000			−0.786408
$207$		0			0
$208$	−80.8332	+	80.8332i	−0.388621	+	0.388621i
$209$		−	180.000i		−	0.861244i
$210$		0			0
$211$	122.000			0.578199			0.289100	−	0.957299i	$-0.406644\pi$
$211$	122.000			0.578199			0.289100	+	0.957299i	$0.406644\pi$
$212$	4.89898	+	4.89898i	0.0231084	+	0.0231084i
$213$		0			0
$214$			108.000i			0.504673i
$215$		0			0
$216$		0			0
$217$	161.666	+	161.666i	0.745006	+	0.745006i
$218$	−208.207	+	208.207i	−0.955076	+	0.955076i
$219$		0			0
$220$		0			0
$221$	−72.0000			−0.325792
$222$		0			0
$223$	−80.8332	+	80.8332i	−0.362481	+	0.362481i	−0.864725	−	0.502245i	$-0.832508\pi$
$223$	−80.8332	+	80.8332i	−0.362481	+	0.362481i	0.502245	+	0.864725i	$0.332508\pi$
$224$			162.000i			0.723214i
$225$		0			0
$226$	−108.000			−0.477876
$227$	53.8888	+	53.8888i	0.237395	+	0.237395i	0.815771	−	0.578375i	$-0.196313\pi$
$227$	53.8888	+	53.8888i	0.237395	+	0.237395i	−0.578375	+	0.815771i	$0.696313\pi$
$228$		0			0
$229$			50.0000i			0.218341i	0.994023	+	0.109170i	$0.0348194\pi$
$229$			50.0000i			0.218341i	−0.994023	+	0.109170i	$0.965181\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	44.0908	−	44.0908i	0.189231	−	0.189231i	−0.606133	−	0.795364i	$-0.707280\pi$
$233$	44.0908	−	44.0908i	0.189231	−	0.189231i	0.795364	+	0.606133i	$0.207280\pi$
$234$		0			0
$235$		0			0
$236$	90.0000			0.381356
$237$		0			0
$238$	88.1816	−	88.1816i	0.370511	−	0.370511i
$239$			180.000i			0.753138i	0.926389	+	0.376569i	$0.122896\pi$
$239$			180.000i			0.753138i	−0.926389	+	0.376569i	$0.877104\pi$
$240$		0			0
$241$	−178.000			−0.738589			−0.369295	−	0.929312i	$-0.620401\pi$
$241$	−178.000			−0.738589			−0.369295	+	0.929312i	$0.620401\pi$
$242$	−248.623	−	248.623i	−1.02737	−	1.02737i
$243$		0			0
$244$		−	2.00000i		−	0.00819672i
$245$		0			0
$246$		0			0
$247$	73.4847	+	73.4847i	0.297509	+	0.297509i
$248$	−134.722	+	134.722i	−0.543234	+	0.543234i
$249$		0			0
$250$		0			0
$251$	−342.000			−1.36255			−0.681275	−	0.732028i	$-0.738574\pi$
$251$	−342.000			−1.36255			−0.681275	+	0.732028i	$0.738574\pi$
$252$		0			0
$253$	352.727	−	352.727i	1.39418	−	1.39418i
$254$		−	18.0000i		−	0.0708661i
$255$		0			0
$256$	−179.000			−0.699219
$257$	−289.040	−	289.040i	−1.12467	−	1.12467i	−0.991030	−	0.133638i	$-0.957334\pi$
$257$	−289.040	−	289.040i	−1.12467	−	1.12467i	−0.133638	−	0.991030i	$-0.542666\pi$
$258$		0			0
$259$			108.000i			0.416988i
$260$		0			0
$261$		0			0
$262$	−22.0454	−	22.0454i	−0.0841428	−	0.0841428i
$263$	264.545	−	264.545i	1.00587	−	1.00587i	0.00589147	−	0.999983i	$-0.498125\pi$
$263$	264.545	−	264.545i	1.00587	−	1.00587i	0.999983	−	0.00589147i	$-0.00187532\pi$
$264$		0			0
$265$		0			0
$266$	−180.000			−0.676692
$267$		0			0
$268$	44.0908	−	44.0908i	0.164518	−	0.164518i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	−478.000			−1.76384			−0.881919	−	0.471401i	$-0.843748\pi$
$271$	−478.000			−1.76384			−0.881919	+	0.471401i	$0.843748\pi$
$272$	53.8888	+	53.8888i	0.198120	+	0.198120i
$273$		0			0
$274$			348.000i			1.27007i
$275$		0			0
$276$		0			0
$277$	80.8332	+	80.8332i	0.291816	+	0.291816i	0.837798	−	0.545981i	$-0.183843\pi$
$277$	80.8332	+	80.8332i	0.291816	+	0.291816i	−0.545981	+	0.837798i	$0.683843\pi$
$278$	208.207	−	208.207i	0.748945	−	0.748945i
$279$		0			0
$280$		0			0
$281$	−162.000			−0.576512			−0.288256	−	0.957553i	$-0.593075\pi$
$281$	−162.000			−0.576512			−0.288256	+	0.957553i	$0.593075\pi$
$282$		0			0
$283$	249.848	−	249.848i	0.882855	−	0.882855i	−0.110969	−	0.993824i	$-0.535395\pi$
$283$	249.848	−	249.848i	0.882855	−	0.882855i	0.993824	+	0.110969i	$0.0353954\pi$
$284$			72.0000i			0.253521i
$285$		0			0
$286$	324.000			1.13287
$287$	132.272	+	132.272i	0.460880	+	0.460880i
$288$		0			0
$289$		−	241.000i		−	0.833910i
$290$		0			0
$291$		0			0
$292$	44.0908	+	44.0908i	0.150996	+	0.150996i
$293$	−372.322	+	372.322i	−1.27073	+	1.27073i	−0.325017	+	0.945708i	$0.605370\pi$
$293$	−372.322	+	372.322i	−1.27073	+	1.27073i	−0.945708	+	0.325017i	$0.894630\pi$
$294$		0			0
$295$		0			0
$296$	−90.0000			−0.304054
$297$		0			0
$298$	−220.454	+	220.454i	−0.739779	+	0.739779i
$299$			288.000i			0.963211i
$300$		0			0
$301$	432.000			1.43522
$302$	−26.9444	−	26.9444i	−0.0892198	−	0.0892198i
$303$		0			0
$304$		−	110.000i		−	0.361842i
$305$		0			0
$306$		0			0
$307$	−396.817	−	396.817i	−1.29256	−	1.29256i	−0.933195	−	0.359369i	$-0.882992\pi$
$307$	−396.817	−	396.817i	−1.29256	−	1.29256i	−0.359369	−	0.933195i	$-0.617008\pi$
$308$	132.272	−	132.272i	0.429456	−	0.429456i
$309$		0			0
$310$		0			0
$311$	−252.000			−0.810289			−0.405145	−	0.914253i	$-0.632779\pi$
$311$	−252.000			−0.810289			−0.405145	+	0.914253i	$0.632779\pi$
$312$		0			0
$313$	−117.576	+	117.576i	−0.375641	+	0.375641i	−0.869527	−	0.493886i	$-0.835576\pi$
$313$	−117.576	+	117.576i	−0.375641	+	0.375641i	0.493886	+	0.869527i	$0.335576\pi$
$314$			342.000i			1.08917i
$315$		0			0
$316$	70.0000			0.221519
$317$	274.343	+	274.343i	0.865435	+	0.865435i	0.991963	−	0.126528i	$-0.0403834\pi$
$317$	274.343	+	274.343i	0.865435	+	0.865435i	−0.126528	+	0.991963i	$0.540383\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$	−352.727	−	352.727i	−1.09542	−	1.09542i
$323$	48.9898	−	48.9898i	0.151671	−	0.151671i
$324$		0			0
$325$		0			0
$326$	288.000			0.883436
$327$		0			0
$328$	−110.227	+	110.227i	−0.336058	+	0.336058i
$329$		−	648.000i		−	1.96960i
$330$		0			0
$331$	−418.000			−1.26284			−0.631420	−	0.775441i	$-0.717528\pi$
$331$	−418.000			−1.26284			−0.631420	+	0.775441i	$0.717528\pi$
$332$	53.8888	+	53.8888i	0.162316	+	0.162316i
$333$		0			0
$334$			48.0000i			0.143713i
$335$		0			0
$336$		0			0
$337$	191.060	+	191.060i	0.566944	+	0.566944i	0.931271	−	0.364327i	$-0.118701\pi$
$337$	191.060	+	191.060i	0.566944	+	0.566944i	−0.364327	+	0.931271i	$0.618701\pi$
$338$	74.7094	−	74.7094i	0.221034	−	0.221034i
$339$		0			0
$340$		0			0
$341$	396.000			1.16129
$342$		0			0
$343$	−73.4847	+	73.4847i	−0.214241	+	0.214241i
$344$			360.000i			1.04651i
$345$		0			0
$346$	−348.000			−1.00578
$347$	−44.0908	−	44.0908i	−0.127063	−	0.127063i	0.640716	−	0.767778i	$-0.278638\pi$
$347$	−44.0908	−	44.0908i	−0.127063	−	0.127063i	−0.767778	+	0.640716i	$0.778638\pi$
$348$		0			0
$349$		−	70.0000i		−	0.200573i	−0.994959	−	0.100287i	$-0.968024\pi$
$349$		−	70.0000i		−	0.200573i	0.994959	−	0.100287i	$-0.0319759\pi$
$350$		0			0
$351$		0			0
$352$	198.409	+	198.409i	0.563661	+	0.563661i
$353$	44.0908	−	44.0908i	0.124903	−	0.124903i	−0.641892	−	0.766795i	$-0.721850\pi$
$353$	44.0908	−	44.0908i	0.124903	−	0.124903i	0.766795	+	0.641892i	$0.221850\pi$
$354$		0			0
$355$		0			0
$356$	90.0000			0.252809
$357$		0			0
$358$	−110.227	+	110.227i	−0.307897	+	0.307897i
$359$		−	540.000i		−	1.50418i	−0.659061	−	0.752089i	$-0.729046\pi$
$359$		−	540.000i		−	1.50418i	0.659061	−	0.752089i	$-0.270954\pi$
$360$		0			0
$361$	261.000			0.722992
$362$	120.025	+	120.025i	0.331561	+	0.331561i
$363$		0			0
$364$			108.000i			0.296703i
$365$		0			0
$366$		0			0
$367$	−66.1362	−	66.1362i	−0.180208	−	0.180208i	0.611239	−	0.791446i	$-0.290672\pi$
$367$	−66.1362	−	66.1362i	−0.180208	−	0.180208i	−0.791446	+	0.611239i	$0.790672\pi$
$368$	215.555	−	215.555i	0.585748	−	0.585748i
$369$		0			0
$370$		0			0
$371$	−72.0000			−0.194070
$372$		0			0
$373$	213.106	−	213.106i	0.571329	−	0.571329i	−0.361171	−	0.932500i	$-0.617623\pi$
$373$	213.106	−	213.106i	0.571329	−	0.571329i	0.932500	+	0.361171i	$0.117623\pi$
$374$		−	216.000i		−	0.577540i
$375$		0			0
$376$	540.000			1.43617
$377$		0			0
$378$		0			0
$379$			170.000i			0.448549i	0.974526	+	0.224274i	$0.0720012\pi$
$379$			170.000i			0.448549i	−0.974526	+	0.224274i	$0.927999\pi$
$380$		0			0
$381$		0			0
$382$	308.636	+	308.636i	0.807947	+	0.807947i
$383$	−151.868	+	151.868i	−0.396523	+	0.396523i	−0.877005	−	0.480482i	$-0.840462\pi$
$383$	−151.868	+	151.868i	−0.396523	+	0.396523i	0.480482	+	0.877005i	$0.340462\pi$
$384$		0			0
$385$		0			0
$386$	648.000			1.67876
$387$		0			0
$388$	−102.879	+	102.879i	−0.265151	+	0.265151i
$389$		−	360.000i		−	0.925450i	−0.886502	−	0.462725i	$-0.846872\pi$
$389$		−	360.000i		−	0.925450i	0.886502	−	0.462725i	$-0.153128\pi$
$390$		0			0
$391$	192.000			0.491049
$392$	−361.300	−	361.300i	−0.921683	−	0.921683i
$393$		0			0
$394$		−	312.000i		−	0.791878i
$395$		0			0
$396$		0			0
$397$	301.287	+	301.287i	0.758910	+	0.758910i	0.976124	−	0.217214i	$-0.0696970\pi$
$397$	301.287	+	301.287i	0.758910	+	0.758910i	−0.217214	+	0.976124i	$0.569697\pi$
$398$	−355.176	+	355.176i	−0.892402	+	0.892402i
$399$		0			0
$400$		0			0
$401$	558.000			1.39152			0.695761	−	0.718274i	$-0.255067\pi$
$401$	558.000			1.39152			0.695761	+	0.718274i	$0.255067\pi$
$402$		0			0
$403$	−161.666	+	161.666i	−0.401157	+	0.401157i
$404$		−	108.000i		−	0.267327i
$405$		0			0
$406$		0			0
$407$	132.272	+	132.272i	0.324994	+	0.324994i
$408$		0			0
$409$		−	670.000i		−	1.63814i	−0.573692	−	0.819071i	$-0.694489\pi$
$409$		−	670.000i		−	1.63814i	0.573692	−	0.819071i	$-0.305511\pi$
$410$		0			0
$411$		0			0
$412$	−66.1362	−	66.1362i	−0.160525	−	0.160525i
$413$	−661.362	+	661.362i	−1.60136	+	1.60136i
$414$		0			0
$415$		0			0
$416$	−162.000			−0.389423
$417$		0			0
$418$	−220.454	+	220.454i	−0.527402	+	0.527402i
$419$			630.000i			1.50358i	0.659403	+	0.751790i	$0.270809\pi$
$419$			630.000i			1.50358i	−0.659403	+	0.751790i	$0.729191\pi$
$420$		0			0
$421$	142.000			0.337292			0.168646	−	0.985677i	$-0.446061\pi$
$421$	142.000			0.337292			0.168646	+	0.985677i	$0.446061\pi$
$422$	−149.419	−	149.419i	−0.354073	−	0.354073i
$423$		0			0
$424$		−	60.0000i		−	0.141509i
$425$		0			0
$426$		0			0
$427$	14.6969	+	14.6969i	0.0344191	+	0.0344191i
$428$	−44.0908	+	44.0908i	−0.103016	+	0.103016i
$429$		0			0
$430$		0			0
$431$	−612.000			−1.41995			−0.709977	−	0.704225i	$-0.751295\pi$
$431$	−612.000			−1.41995			−0.709977	+	0.704225i	$0.751295\pi$
$432$		0			0
$433$	102.879	−	102.879i	0.237595	−	0.237595i	−0.578259	−	0.815853i	$-0.696268\pi$
$433$	102.879	−	102.879i	0.237595	−	0.237595i	0.815853	+	0.578259i	$0.196268\pi$
$434$		−	396.000i		−	0.912442i
$435$		0			0
$436$	−170.000			−0.389908
$437$	−195.959	−	195.959i	−0.448419	−	0.448419i
$438$		0			0
$439$			430.000i			0.979499i	0.871863	+	0.489749i	$0.162912\pi$
$439$			430.000i			0.979499i	−0.871863	+	0.489749i	$0.837088\pi$
$440$		0			0
$441$		0			0
$442$	88.1816	+	88.1816i	0.199506	+	0.199506i
$443$	582.979	−	582.979i	1.31598	−	1.31598i	0.399049	−	0.916930i	$-0.369340\pi$
$443$	582.979	−	582.979i	1.31598	−	1.31598i	0.916930	−	0.399049i	$-0.130660\pi$
$444$		0			0
$445$		0			0
$446$	198.000			0.443946
$447$		0			0
$448$	521.741	−	521.741i	1.16460	−	1.16460i
$449$			90.0000i			0.200445i	0.994965	+	0.100223i	$0.0319555\pi$
$449$			90.0000i			0.200445i	−0.994965	+	0.100223i	$0.968044\pi$
$450$		0			0
$451$	324.000			0.718404
$452$	−44.0908	−	44.0908i	−0.0975461	−	0.0975461i
$453$		0			0
$454$		−	132.000i		−	0.290749i
$455$		0			0
$456$		0			0
$457$	191.060	+	191.060i	0.418075	+	0.418075i	0.884540	−	0.466465i	$-0.154473\pi$
$457$	191.060	+	191.060i	0.418075	+	0.418075i	−0.466465	+	0.884540i	$0.654473\pi$
$458$	61.2372	−	61.2372i	0.133706	−	0.133706i
$459$		0			0
$460$		0			0
$461$	828.000			1.79610			0.898048	−	0.439898i	$-0.144985\pi$
$461$	828.000			1.79610			0.898048	+	0.439898i	$0.144985\pi$
$462$		0			0
$463$	−7.34847	+	7.34847i	−0.0158714	+	0.0158714i	−0.714998	−	0.699127i	$-0.753572\pi$
$463$	−7.34847	+	7.34847i	−0.0158714	+	0.0158714i	0.699127	+	0.714998i	$0.253572\pi$
$464$		0			0
$465$		0			0
$466$	−108.000			−0.231760
$467$	347.828	+	347.828i	0.744813	+	0.744813i	0.973500	−	0.228687i	$-0.0734433\pi$
$467$	347.828	+	347.828i	0.744813	+	0.744813i	−0.228687	+	0.973500i	$0.573443\pi$
$468$		0			0
$469$			648.000i			1.38166i
$470$		0			0
$471$		0			0
$472$	−551.135	−	551.135i	−1.16766	−	1.16766i
$473$	529.090	−	529.090i	1.11858	−	1.11858i
$474$		0			0
$475$		0			0
$476$	72.0000			0.151261
$477$		0			0
$478$	220.454	−	220.454i	0.461201	−	0.461201i
$479$		−	360.000i		−	0.751566i	−0.926708	−	0.375783i	$-0.877374\pi$
$479$		−	360.000i		−	0.751566i	0.926708	−	0.375783i	$-0.122626\pi$
$480$		0			0
$481$	−108.000			−0.224532
$482$	218.005	+	218.005i	0.452292	+	0.452292i
$483$		0			0
$484$		−	203.000i		−	0.419421i
$485$		0			0
$486$		0			0
$487$	−580.529	−	580.529i	−1.19205	−	1.19205i	−0.976490	−	0.215561i	$-0.930842\pi$
$487$	−580.529	−	580.529i	−1.19205	−	1.19205i	−0.215561	−	0.976490i	$-0.569158\pi$
$488$	−12.2474	+	12.2474i	−0.0250972	+	0.0250972i
$489$		0			0
$490$		0			0
$491$	18.0000			0.0366599			0.0183299	−	0.999832i	$-0.494165\pi$
$491$	18.0000			0.0366599			0.0183299	+	0.999832i	$0.494165\pi$
$492$		0			0
$493$		0			0
$494$		−	180.000i		−	0.364372i
$495$		0			0
$496$	242.000			0.487903
$497$	−529.090	−	529.090i	−1.06457	−	1.06457i
$498$		0			0
$499$		−	590.000i		−	1.18236i	−0.806538	−	0.591182i	$-0.798661\pi$
$499$		−	590.000i		−	1.18236i	0.806538	−	0.591182i	$-0.201339\pi$
$500$		0			0
$501$		0			0
$502$	418.863	+	418.863i	0.834388	+	0.834388i
$503$	93.0806	−	93.0806i	0.185051	−	0.185051i	−0.608502	−	0.793553i	$-0.708229\pi$
$503$	93.0806	−	93.0806i	0.185051	−	0.185051i	0.793553	+	0.608502i	$0.208229\pi$
$504$		0			0
$505$		0			0
$506$	−864.000			−1.70751
$507$		0			0
$508$	7.34847	−	7.34847i	0.0144655	−	0.0144655i
$509$			540.000i			1.06090i	0.847715	+	0.530452i	$0.177978\pi$
$509$			540.000i			1.06090i	−0.847715	+	0.530452i	$0.822022\pi$
$510$		0			0
$511$	−648.000			−1.26810
$512$	390.694	+	390.694i	0.763073	+	0.763073i
$513$		0			0
$514$			708.000i			1.37743i
$515$		0			0
$516$		0			0
$517$	−793.635	−	793.635i	−1.53508	−	1.53508i
$518$	132.272	−	132.272i	0.255352	−	0.255352i
$519$		0			0
$520$		0			0
$521$	−342.000			−0.656430			−0.328215	−	0.944603i	$-0.606447\pi$
$521$	−342.000			−0.656430			−0.328215	+	0.944603i	$0.606447\pi$
$522$		0			0
$523$	−264.545	+	264.545i	−0.505822	+	0.505822i	−0.913241	−	0.407419i	$-0.866429\pi$
$523$	−264.545	+	264.545i	−0.505822	+	0.505822i	0.407419	+	0.913241i	$0.366429\pi$
$524$		−	18.0000i		−	0.0343511i
$525$		0			0
$526$	−648.000			−1.23194
$527$	107.778	+	107.778i	0.204511	+	0.204511i
$528$		0			0
$529$		−	239.000i		−	0.451796i
$530$		0			0
$531$		0			0
$532$	−73.4847	−	73.4847i	−0.138129	−	0.138129i
$533$	−132.272	+	132.272i	−0.248166	+	0.248166i
$534$		0			0
$535$		0			0
$536$	−540.000			−1.00746
$537$		0			0
$538$		0			0
$539$			1062.00i			1.97032i
$540$		0			0
$541$	2.00000			0.00369686			0.00184843	−	0.999998i	$-0.499412\pi$
$541$	2.00000			0.00369686			0.00184843	+	0.999998i	$0.499412\pi$
$542$	585.428	+	585.428i	1.08013	+	1.08013i
$543$		0			0
$544$			108.000i			0.198529i
$545$		0			0
$546$		0			0
$547$	264.545	+	264.545i	0.483629	+	0.483629i	0.906288	−	0.422660i	$-0.138903\pi$
$547$	264.545	+	264.545i	0.483629	+	0.483629i	−0.422660	+	0.906288i	$0.638903\pi$
$548$	−142.070	+	142.070i	−0.259253	+	0.259253i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−514.393	+	514.393i	−0.930186	+	0.930186i
$554$		−	198.000i		−	0.357401i
$555$		0			0
$556$	170.000			0.305755
$557$	−484.999	−	484.999i	−0.870734	−	0.870734i	0.121818	−	0.992552i	$-0.461128\pi$
$557$	−484.999	−	484.999i	−0.870734	−	0.870734i	−0.992552	+	0.121818i	$0.961128\pi$
$558$		0			0
$559$			432.000i			0.772809i
$560$		0			0
$561$		0			0
$562$	198.409	+	198.409i	0.353040	+	0.353040i
$563$	−396.817	+	396.817i	−0.704827	+	0.704827i	−0.965443	−	0.260616i	$-0.916074\pi$
$563$	−396.817	+	396.817i	−0.704827	+	0.704827i	0.260616	+	0.965443i	$0.416074\pi$
$564$		0			0
$565$		0			0
$566$	−612.000			−1.08127
$567$		0			0
$568$	440.908	−	440.908i	0.776247	−	0.776247i
$569$			630.000i			1.10721i	0.832781	+	0.553603i	$0.186747\pi$
$569$			630.000i			1.10721i	−0.832781	+	0.553603i	$0.813253\pi$
$570$		0			0
$571$	302.000			0.528897			0.264448	−	0.964400i	$-0.414810\pi$
$571$	302.000			0.528897			0.264448	+	0.964400i	$0.414810\pi$
$572$	132.272	+	132.272i	0.231246	+	0.231246i
$573$		0			0
$574$		−	324.000i		−	0.564460i
$575$		0			0
$576$		0			0
$577$	484.999	+	484.999i	0.840553	+	0.840553i	0.988931	−	0.148378i	$-0.0474052\pi$
$577$	484.999	+	484.999i	0.840553	+	0.840553i	−0.148378	+	0.988931i	$0.547405\pi$
$578$	−295.164	+	295.164i	−0.510664	+	0.510664i
$579$		0			0
$580$		0			0
$581$	−792.000			−1.36317
$582$		0			0
$583$	−88.1816	+	88.1816i	−0.151255	+	0.151255i
$584$		−	540.000i		−	0.924658i
$585$		0			0
$586$	912.000			1.55631
$587$	−582.979	−	582.979i	−0.993149	−	0.993149i	0.00682753	−	0.999977i	$-0.497827\pi$
$587$	−582.979	−	582.979i	−0.993149	−	0.993149i	−0.999977	+	0.00682753i	$0.997827\pi$
$588$		0			0
$589$		−	220.000i		−	0.373514i
$590$		0			0
$591$		0			0
$592$	80.8332	+	80.8332i	0.136543	+	0.136543i
$593$	533.989	−	533.989i	0.900487	−	0.900487i	−0.0949912	−	0.995478i	$-0.530282\pi$
$593$	533.989	−	533.989i	0.900487	−	0.900487i	0.995478	+	0.0949912i	$0.0302823\pi$
$594$		0			0
$595$		0			0
$596$	−180.000			−0.302013
$597$		0			0
$598$	352.727	−	352.727i	0.589844	−	0.589844i
$599$		−	540.000i		−	0.901503i	−0.892650	−	0.450751i	$-0.851156\pi$
$599$		−	540.000i		−	0.901503i	0.892650	−	0.450751i	$-0.148844\pi$
$600$		0			0
$601$	−758.000			−1.26123			−0.630616	−	0.776095i	$-0.717198\pi$
$601$	−758.000			−1.26123			−0.630616	+	0.776095i	$0.717198\pi$
$602$	−529.090	−	529.090i	−0.878887	−	0.878887i
$603$		0			0
$604$		−	22.0000i		−	0.0364238i
$605$		0			0
$606$		0			0
$607$	668.711	+	668.711i	1.10167	+	1.10167i	0.994210	+	0.107455i	$0.0342702\pi$
$607$	668.711	+	668.711i	1.10167	+	1.10167i	0.107455	+	0.994210i	$0.465730\pi$
$608$	110.227	−	110.227i	0.181294	−	0.181294i
$609$		0			0
$610$		0			0
$611$	648.000			1.06056
$612$		0			0
$613$	66.1362	−	66.1362i	0.107889	−	0.107889i	−0.651101	−	0.758991i	$-0.725693\pi$
$613$	66.1362	−	66.1362i	0.107889	−	0.107889i	0.758991	+	0.651101i	$0.225693\pi$
$614$			972.000i			1.58306i
$615$		0			0
$616$	−1620.00			−2.62987
$617$	592.777	+	592.777i	0.960740	+	0.960740i	0.999258	−	0.0385180i	$-0.0122637\pi$
$617$	592.777	+	592.777i	0.960740	+	0.960740i	−0.0385180	+	0.999258i	$0.512264\pi$
$618$		0			0
$619$			1030.00i			1.66397i	0.554795	+	0.831987i	$0.312797\pi$
$619$			1030.00i			1.66397i	−0.554795	+	0.831987i	$0.687203\pi$
$620$		0			0
$621$		0			0
$622$	308.636	+	308.636i	0.496199	+	0.496199i
$623$	−661.362	+	661.362i	−1.06158	+	1.06158i
$624$		0			0
$625$		0			0
$626$	288.000			0.460064
$627$		0			0
$628$	−139.621	+	139.621i	−0.222326	+	0.222326i
$629$			72.0000i			0.114467i
$630$		0			0
$631$	242.000			0.383518			0.191759	−	0.981442i	$-0.438581\pi$
$631$	242.000			0.383518			0.191759	+	0.981442i	$0.438581\pi$
$632$	−428.661	−	428.661i	−0.678261	−	0.678261i
$633$		0			0
$634$		−	672.000i		−	1.05994i
$635$		0			0
$636$		0			0
$637$	−433.560	−	433.560i	−0.680627	−	0.680627i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−162.000			−0.252730			−0.126365	−	0.991984i	$-0.540331\pi$
$641$	−162.000			−0.252730			−0.126365	+	0.991984i	$0.540331\pi$
$642$		0			0
$643$	−264.545	+	264.545i	−0.411423	+	0.411423i	−0.882234	−	0.470811i	$-0.843961\pi$
$643$	−264.545	+	264.545i	−0.411423	+	0.411423i	0.470811	+	0.882234i	$0.343961\pi$
$644$		−	288.000i		−	0.447205i
$645$		0			0
$646$	−120.000			−0.185759
$647$	−93.0806	−	93.0806i	−0.143865	−	0.143865i	0.631506	−	0.775371i	$-0.282437\pi$
$647$	−93.0806	−	93.0806i	−0.143865	−	0.143865i	−0.775371	+	0.631506i	$0.782437\pi$
$648$		0			0
$649$			1620.00i			2.49615i
$650$		0			0
$651$		0			0
$652$	117.576	+	117.576i	0.180331	+	0.180331i
$653$	68.5857	−	68.5857i	0.105032	−	0.105032i	−0.652638	−	0.757670i	$-0.726338\pi$
$653$	68.5857	−	68.5857i	0.105032	−	0.105032i	0.757670	+	0.652638i	$0.226338\pi$
$654$		0			0
$655$		0			0
$656$	198.000			0.301829
$657$		0			0
$658$	−793.635	+	793.635i	−1.20613	+	1.20613i
$659$			630.000i			0.955994i	0.878361	+	0.477997i	$0.158637\pi$
$659$			630.000i			0.955994i	−0.878361	+	0.477997i	$0.841363\pi$
$660$		0			0
$661$	622.000			0.940998			0.470499	−	0.882400i	$-0.344074\pi$
$661$	622.000			0.940998			0.470499	+	0.882400i	$0.344074\pi$
$662$	511.943	+	511.943i	0.773328	+	0.773328i
$663$		0			0
$664$		−	660.000i		−	0.993976i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−19.5959	+	19.5959i	−0.0293352	+	0.0293352i
$669$		0			0
$670$		0			0
$671$	36.0000			0.0536513
$672$		0			0
$673$	102.879	−	102.879i	0.152866	−	0.152866i	−0.626531	−	0.779397i	$-0.715526\pi$
$673$	102.879	−	102.879i	0.152866	−	0.152866i	0.779397	+	0.626531i	$0.215526\pi$
$674$		−	468.000i		−	0.694362i
$675$		0			0
$676$	61.0000			0.0902367
$677$	176.363	+	176.363i	0.260507	+	0.260507i	0.825260	−	0.564753i	$-0.191029\pi$
$677$	176.363	+	176.363i	0.260507	+	0.260507i	−0.564753	+	0.825260i	$0.691029\pi$
$678$		0			0
$679$		−	1512.00i		−	2.22680i
$680$		0			0
$681$		0			0
$682$	−484.999	−	484.999i	−0.711142	−	0.711142i
$683$	−445.807	+	445.807i	−0.652719	+	0.652719i	−0.953647	−	0.300928i	$-0.902704\pi$
$683$	−445.807	+	445.807i	−0.652719	+	0.652719i	0.300928	+	0.953647i	$0.402704\pi$
$684$		0			0
$685$		0			0
$686$	180.000			0.262391
$687$		0			0
$688$	323.333	−	323.333i	0.469960	−	0.469960i
$689$		−	72.0000i		−	0.104499i
$690$		0			0
$691$	682.000			0.986975			0.493488	−	0.869753i	$-0.335722\pi$
$691$	682.000			0.986975			0.493488	+	0.869753i	$0.335722\pi$
$692$	−142.070	−	142.070i	−0.205304	−	0.205304i
$693$		0			0
$694$			108.000i			0.155620i
$695$		0			0
$696$		0			0
$697$	88.1816	+	88.1816i	0.126516	+	0.126516i
$698$	−85.7321	+	85.7321i	−0.122825	+	0.122825i
$699$		0			0
$700$		0			0
$701$	468.000			0.667618			0.333809	−	0.942641i	$-0.391666\pi$
$701$	468.000			0.667618			0.333809	+	0.942641i	$0.391666\pi$
$702$		0			0
$703$	73.4847	−	73.4847i	0.104530	−	0.104530i
$704$		−	1278.00i		−	1.81534i
$705$		0			0
$706$	−108.000			−0.152975
$707$	793.635	+	793.635i	1.12254	+	1.12254i
$708$		0			0
$709$		−	310.000i		−	0.437236i	−0.975811	−	0.218618i	$-0.929845\pi$
$709$		−	310.000i		−	0.437236i	0.975811	−	0.218618i	$-0.0701548\pi$
$710$		0			0
$711$		0			0
$712$	−551.135	−	551.135i	−0.774066	−	0.774066i
$713$	431.110	−	431.110i	0.604643	−	0.604643i
$714$		0			0
$715$		0			0
$716$	−90.0000			−0.125698
$717$		0			0
$718$	−661.362	+	661.362i	−0.921117	+	0.921117i
$719$			180.000i			0.250348i	0.992135	+	0.125174i	$0.0399489\pi$
$719$			180.000i			0.250348i	−0.992135	+	0.125174i	$0.960051\pi$
$720$		0			0
$721$	972.000			1.34813
$722$	−319.658	−	319.658i	−0.442740	−	0.442740i
$723$		0			0
$724$			98.0000i			0.135359i
$725$		0			0
$726$		0			0
$727$	−507.044	−	507.044i	−0.697448	−	0.697448i	0.266412	−	0.963859i	$-0.414162\pi$
$727$	−507.044	−	507.044i	−0.697448	−	0.697448i	−0.963859	+	0.266412i	$0.914162\pi$
$728$	661.362	−	661.362i	0.908465	−	0.908465i
$729$		0			0
$730$		0			0
$731$	288.000			0.393981
$732$		0			0
$733$	800.983	−	800.983i	1.09275	−	1.09275i	0.0975121	−	0.995234i	$-0.468912\pi$
$733$	800.983	−	800.983i	1.09275	−	1.09275i	0.995234	−	0.0975121i	$-0.0310885\pi$
$734$			162.000i			0.220708i
$735$		0			0
$736$	432.000			0.586957
$737$	793.635	+	793.635i	1.07684	+	1.07684i
$738$		0			0
$739$			350.000i			0.473613i	0.971557	+	0.236806i	$0.0761007\pi$
$739$			350.000i			0.473613i	−0.971557	+	0.236806i	$0.923899\pi$
$740$		0			0
$741$		0			0
$742$	88.1816	+	88.1816i	0.118843	+	0.118843i
$743$	925.907	−	925.907i	1.24617	−	1.24617i	0.288778	−	0.957396i	$-0.406751\pi$
$743$	925.907	−	925.907i	1.24617	−	1.24617i	0.957396	−	0.288778i	$-0.0932488\pi$
$744$		0			0
$745$		0			0
$746$	−522.000			−0.699732
$747$		0			0
$748$	88.1816	−	88.1816i	0.117890	−	0.117890i
$749$		−	648.000i		−	0.865154i
$750$		0			0
$751$	−338.000			−0.450067			−0.225033	−	0.974351i	$-0.572249\pi$
$751$	−338.000			−0.450067			−0.225033	+	0.974351i	$0.572249\pi$
$752$	−484.999	−	484.999i	−0.644945	−	0.644945i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−286.590	−	286.590i	−0.378587	−	0.378587i	0.492005	−	0.870592i	$-0.336264\pi$
$757$	−286.590	−	286.590i	−0.378587	−	0.378587i	−0.870592	+	0.492005i	$0.836264\pi$
$758$	208.207	−	208.207i	0.274679	−	0.274679i
$759$		0			0
$760$		0			0
$761$	1278.00			1.67937			0.839685	−	0.543074i	$-0.182740\pi$
$761$	1278.00			1.67937			0.839685	+	0.543074i	$0.182740\pi$
$762$		0			0
$763$	1249.24	−	1249.24i	1.63727	−	1.63727i
$764$			252.000i			0.329843i
$765$		0			0
$766$	372.000			0.485640
$767$	−661.362	−	661.362i	−0.862271	−	0.862271i
$768$		0			0
$769$		−	590.000i		−	0.767230i	−0.923493	−	0.383615i	$-0.874679\pi$
$769$		−	590.000i		−	0.767230i	0.923493	−	0.383615i	$-0.125321\pi$
$770$		0			0
$771$		0			0
$772$	264.545	+	264.545i	0.342675	+	0.342675i
$773$	−127.373	+	127.373i	−0.164778	+	0.164778i	−0.784680	−	0.619902i	$-0.787173\pi$
$773$	−127.373	+	127.373i	−0.164778	+	0.164778i	0.619902	+	0.784680i	$0.287173\pi$
$774$		0			0
$775$		0			0
$776$	1260.00			1.62371
$777$		0			0
$778$	−440.908	+	440.908i	−0.566720	+	0.566720i
$779$		−	180.000i		−	0.231065i
$780$		0			0
$781$	−1296.00			−1.65941
$782$	−235.151	−	235.151i	−0.300705	−	0.300705i
$783$		0			0
$784$			649.000i			0.827806i
$785$		0			0
$786$		0			0
$787$	264.545	+	264.545i	0.336143	+	0.336143i	0.854914	−	0.518770i	$-0.173610\pi$
$787$	264.545	+	264.545i	0.336143	+	0.336143i	−0.518770	+	0.854914i	$0.673610\pi$
$788$	127.373	−	127.373i	0.161641	−	0.161641i
$789$		0			0
$790$		0			0
$791$	648.000			0.819216
$792$		0			0
$793$	−14.6969	+	14.6969i	−0.0185333	+	0.0185333i
$794$		−	738.000i		−	0.929471i
$795$		0			0
$796$	−290.000			−0.364322
$797$	200.858	+	200.858i	0.252018	+	0.252018i	0.821797	−	0.569780i	$-0.192971\pi$
$797$	200.858	+	200.858i	0.252018	+	0.252018i	−0.569780	+	0.821797i	$0.692971\pi$
$798$		0			0
$799$		−	432.000i		−	0.540676i
$800$		0			0
$801$		0			0
$802$	−683.408	−	683.408i	−0.852129	−	0.852129i
$803$	−793.635	+	793.635i	−0.988337	+	0.988337i
$804$		0			0
$805$		0			0
$806$	396.000			0.491315
$807$		0			0
$808$	−661.362	+	661.362i	−0.818518	+	0.818518i
$809$		−	630.000i		−	0.778739i	−0.921082	−	0.389370i	$-0.872693\pi$
$809$		−	630.000i		−	0.778739i	0.921082	−	0.389370i	$-0.127307\pi$
$810$		0			0
$811$	−218.000			−0.268804			−0.134402	−	0.990927i	$-0.542911\pi$
$811$	−218.000			−0.268804			−0.134402	+	0.990927i	$0.542911\pi$
$812$		0			0
$813$		0			0
$814$		−	324.000i		−	0.398034i
$815$		0			0
$816$		0			0
$817$	−293.939	−	293.939i	−0.359778	−	0.359778i
$818$	−820.579	+	820.579i	−1.00315	+	1.00315i
$819$		0			0
$820$		0			0
$821$	−432.000			−0.526188			−0.263094	−	0.964770i	$-0.584743\pi$
$821$	−432.000			−0.526188			−0.263094	+	0.964770i	$0.584743\pi$
$822$		0			0
$823$	360.075	−	360.075i	0.437515	−	0.437515i	−0.453660	−	0.891175i	$-0.649882\pi$
$823$	360.075	−	360.075i	0.437515	−	0.437515i	0.891175	+	0.453660i	$0.149882\pi$
$824$			810.000i			0.983010i
$825$		0			0
$826$	1620.00			1.96126
$827$	−876.917	−	876.917i	−1.06036	−	1.06036i	−0.998057	−	0.0623022i	$-0.980156\pi$
$827$	−876.917	−	876.917i	−1.06036	−	1.06036i	−0.0623022	−	0.998057i	$-0.519844\pi$
$828$		0			0
$829$		−	70.0000i		−	0.0844391i	−0.999108	−	0.0422195i	$-0.986557\pi$
$829$		−	70.0000i		−	0.0844391i	0.999108	−	0.0422195i	$-0.0134429\pi$
$830$		0			0
$831$		0			0
$832$	521.741	+	521.741i	0.627093	+	0.627093i
$833$	−289.040	+	289.040i	−0.346987	+	0.346987i
$834$		0			0
$835$		0			0
$836$	−180.000			−0.215311
$837$		0			0
$838$	771.589	−	771.589i	0.920751	−	0.920751i
$839$		−	540.000i		−	0.643623i	−0.946804	−	0.321812i	$-0.895708\pi$
$839$		−	540.000i		−	0.643623i	0.946804	−	0.321812i	$-0.104292\pi$
$840$		0			0
$841$	841.000			1.00000
$842$	−173.914	−	173.914i	−0.206548	−	0.206548i
$843$		0			0
$844$		−	122.000i		−	0.144550i
$845$		0			0
$846$		0			0
$847$	1491.74	+	1491.74i	1.76120	+	1.76120i
$848$	−53.8888	+	53.8888i	−0.0635481	+	0.0635481i
$849$		0			0
$850$		0			0
$851$	288.000			0.338425
$852$		0			0
$853$	−815.680	+	815.680i	−0.956249	+	0.956249i	−0.999082	−	0.0428336i	$-0.986361\pi$
$853$	−815.680	+	815.680i	−0.956249	+	0.956249i	0.0428336	+	0.999082i	$0.486361\pi$
$854$		−	36.0000i		−	0.0421546i
$855$		0			0
$856$	540.000			0.630841
$857$	298.838	+	298.838i	0.348702	+	0.348702i	0.859626	−	0.510924i	$-0.170697\pi$
$857$	298.838	+	298.838i	0.348702	+	0.348702i	−0.510924	+	0.859626i	$0.670697\pi$
$858$		0			0
$859$			310.000i			0.360885i	0.983586	+	0.180442i	$0.0577529\pi$
$859$			310.000i			0.360885i	−0.983586	+	0.180442i	$0.942247\pi$
$860$		0			0
$861$		0			0
$862$	749.544	+	749.544i	0.869540	+	0.869540i
$863$	−225.353	+	225.353i	−0.261128	+	0.261128i	−0.825512	−	0.564385i	$-0.809114\pi$
$863$	−225.353	+	225.353i	−0.261128	+	0.261128i	0.564385	+	0.825512i	$0.309114\pi$
$864$		0			0
$865$		0			0
$866$	−252.000			−0.290993
$867$		0			0
$868$	161.666	−	161.666i	0.186252	−	0.186252i
$869$			1260.00i			1.44994i
$870$		0			0
$871$	−648.000			−0.743972
$872$	1041.03	+	1041.03i	1.19385	+	1.19385i
$873$		0			0
$874$			480.000i			0.549199i
$875$		0			0
$876$		0			0
$877$	−213.106	−	213.106i	−0.242994	−	0.242994i	0.575094	−	0.818088i	$-0.304966\pi$
$877$	−213.106	−	213.106i	−0.242994	−	0.242994i	−0.818088	+	0.575094i	$0.804966\pi$
$878$	526.640	−	526.640i	0.599818	−	0.599818i
$879$		0			0
$880$		0			0
$881$	−1062.00			−1.20545			−0.602724	−	0.797950i	$-0.705918\pi$
$881$	−1062.00			−1.20545			−0.602724	+	0.797950i	$0.705918\pi$
$882$		0			0
$883$	−117.576	+	117.576i	−0.133155	+	0.133155i	−0.770543	−	0.637388i	$-0.780015\pi$
$883$	−117.576	+	117.576i	−0.133155	+	0.133155i	0.637388	+	0.770543i	$0.280015\pi$
$884$			72.0000i			0.0814480i
$885$		0			0
$886$	−1428.00			−1.61174
$887$	−509.494	−	509.494i	−0.574401	−	0.574401i	0.358954	−	0.933355i	$-0.383133\pi$
$887$	−509.494	−	509.494i	−0.574401	−	0.574401i	−0.933355	+	0.358954i	$0.883133\pi$
$888$		0			0
$889$			108.000i			0.121485i
$890$		0			0
$891$		0			0
$892$	80.8332	+	80.8332i	0.0906201	+	0.0906201i
$893$	−440.908	+	440.908i	−0.493738	+	0.493738i
$894$		0			0
$895$		0			0
$896$	−630.000			−0.703125
$897$		0			0
$898$	110.227	−	110.227i	0.122747	−	0.122747i
$899$		0			0
$900$		0			0
$901$	−48.0000			−0.0532741
$902$	−396.817	−	396.817i	−0.439931	−	0.439931i
$903$		0			0
$904$			540.000i			0.597345i
$905$		0			0
$906$		0			0
$907$	411.514	+	411.514i	0.453709	+	0.453709i	0.896584	−	0.442874i	$-0.146041\pi$
$907$	411.514	+	411.514i	0.453709	+	0.453709i	−0.442874	+	0.896584i	$0.646041\pi$
$908$	53.8888	−	53.8888i	0.0593489	−	0.0593489i
$909$		0			0
$910$		0			0
$911$	−792.000			−0.869374			−0.434687	−	0.900582i	$-0.643141\pi$
$911$	−792.000			−0.869374			−0.434687	+	0.900582i	$0.643141\pi$
$912$		0			0
$913$	−969.998	+	969.998i	−1.06243	+	1.06243i
$914$		−	468.000i		−	0.512035i
$915$		0			0
$916$	50.0000			0.0545852
$917$	132.272	+	132.272i	0.144245	+	0.144245i
$918$		0			0
$919$		−	430.000i		−	0.467900i	−0.972249	−	0.233950i	$-0.924835\pi$
$919$		−	430.000i		−	0.467900i	0.972249	−	0.233950i	$-0.0751652\pi$
$920$		0			0
$921$		0			0
$922$	−1014.09	−	1014.09i	−1.09988	−	1.09988i
$923$	529.090	−	529.090i	0.573228	−	0.573228i
$924$		0			0
$925$		0			0
$926$	18.0000			0.0194384
$927$		0			0
$928$		0			0
$929$			1530.00i			1.64693i	0.567365	+	0.823466i	$0.307963\pi$
$929$			1530.00i			1.64693i	−0.567365	+	0.823466i	$0.692037\pi$
$930$		0			0
$931$	590.000			0.633727
$932$	−44.0908	−	44.0908i	−0.0473077	−	0.0473077i
$933$		0			0
$934$		−	852.000i		−	0.912206i
$935$		0			0
$936$		0			0
$937$	−323.333	−	323.333i	−0.345072	−	0.345072i	0.513198	−	0.858270i	$-0.328461\pi$
$937$	−323.333	−	323.333i	−0.345072	−	0.345072i	−0.858270	+	0.513198i	$0.828461\pi$
$938$	793.635	−	793.635i	0.846092	−	0.846092i
$939$		0			0
$940$		0			0
$941$	−1152.00			−1.22423			−0.612115	−	0.790769i	$-0.709681\pi$
$941$	−1152.00			−1.22423			−0.612115	+	0.790769i	$0.709681\pi$
$942$		0			0
$943$	352.727	−	352.727i	0.374047	−	0.374047i
$944$			990.000i			1.04873i
$945$		0			0
$946$	−1296.00			−1.36998
$947$	739.746	+	739.746i	0.781147	+	0.781147i	0.980024	−	0.198878i	$-0.0637296\pi$
$947$	739.746	+	739.746i	0.781147	+	0.781147i	−0.198878	+	0.980024i	$0.563730\pi$
$948$		0			0
$949$		−	648.000i		−	0.682824i
$950$		0			0
$951$		0			0
$952$	−440.908	−	440.908i	−0.463139	−	0.463139i
$953$	44.0908	−	44.0908i	0.0462653	−	0.0462653i	−0.683596	−	0.729861i	$-0.739585\pi$
$953$	44.0908	−	44.0908i	0.0462653	−	0.0462653i	0.729861	+	0.683596i	$0.239585\pi$
$954$		0			0
$955$		0			0
$956$	180.000			0.188285
$957$		0			0
$958$	−440.908	+	440.908i	−0.460238	+	0.460238i
$959$		−	2088.00i		−	2.17727i
$960$		0			0
$961$	−477.000			−0.496358
$962$	132.272	+	132.272i	0.137497	+	0.137497i
$963$		0			0
$964$			178.000i			0.184647i
$965$		0			0
$966$		0			0
$967$	−727.498	−	727.498i	−0.752325	−	0.752325i	0.222588	−	0.974913i	$-0.428550\pi$
$967$	−727.498	−	727.498i	−0.752325	−	0.752325i	−0.974913	+	0.222588i	$0.928550\pi$
$968$	−1243.12	+	1243.12i	−1.28421	+	1.28421i
$969$		0			0
$970$		0			0
$971$	1278.00			1.31617			0.658084	−	0.752944i	$-0.271367\pi$
$971$	1278.00			1.31617			0.658084	+	0.752944i	$0.271367\pi$
$972$		0			0
$973$	−1249.24	+	1249.24i	−1.28391	+	1.28391i
$974$			1422.00i			1.45996i
$975$		0			0
$976$	22.0000			0.0225410
$977$	396.817	+	396.817i	0.406159	+	0.406159i	0.880397	−	0.474238i	$-0.157276\pi$
$977$	396.817	+	396.817i	0.406159	+	0.406159i	−0.474238	+	0.880397i	$0.657276\pi$
$978$		0			0
$979$			1620.00i			1.65475i
$980$		0			0
$981$		0			0
$982$	−22.0454	−	22.0454i	−0.0224495	−	0.0224495i
$983$	827.928	−	827.928i	0.842246	−	0.842246i	−0.146905	−	0.989151i	$-0.546931\pi$
$983$	827.928	−	827.928i	0.842246	−	0.842246i	0.989151	+	0.146905i	$0.0469311\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	73.4847	−	73.4847i	0.0743772	−	0.0743772i
$989$		−	1152.00i		−	1.16481i
$990$		0			0
$991$	−118.000			−0.119072			−0.0595358	−	0.998226i	$-0.518962\pi$
$991$	−118.000			−0.119072			−0.0595358	+	0.998226i	$0.518962\pi$
$992$	242.499	+	242.499i	0.244455	+	0.244455i
$993$		0			0
$994$			1296.00i			1.30382i
$995$		0			0
$996$		0			0
$997$	815.680	+	815.680i	0.818134	+	0.818134i	0.985838	−	0.167703i	$-0.0536350\pi$
$997$	815.680	+	815.680i	0.818134	+	0.818134i	−0.167703	+	0.985838i	$0.553635\pi$
$998$	−722.599	+	722.599i	−0.724048	+	0.724048i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.3.g.f.82.1		4
3.2	odd	2		75.3.f.a.7.2	yes	4
5.2	odd	4	inner	225.3.g.f.118.2		4
5.3	odd	4	inner	225.3.g.f.118.1		4
5.4	even	2	inner	225.3.g.f.82.2		4
12.11	even	2		1200.3.bg.j.1057.1		4
15.2	even	4		75.3.f.a.43.1	yes	4
15.8	even	4		75.3.f.a.43.2	yes	4
15.14	odd	2		75.3.f.a.7.1	✓	4
60.23	odd	4		1200.3.bg.j.193.1		4
60.47	odd	4		1200.3.bg.j.193.2		4
60.59	even	2		1200.3.bg.j.1057.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.3.f.a.7.1	✓	4	15.14	odd	2
75.3.f.a.7.2	yes	4	3.2	odd	2
75.3.f.a.43.1	yes	4	15.2	even	4
75.3.f.a.43.2	yes	4	15.8	even	4
225.3.g.f.82.1		4	1.1	even	1	trivial
225.3.g.f.82.2		4	5.4	even	2	inner
225.3.g.f.118.1		4	5.3	odd	4	inner
225.3.g.f.118.2		4	5.2	odd	4	inner
1200.3.bg.j.193.1		4	60.23	odd	4
1200.3.bg.j.193.2		4	60.47	odd	4
1200.3.bg.j.1057.1		4	12.11	even	2
1200.3.bg.j.1057.2		4	60.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	225.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.22474 - 1.22474i) q^{2} -1.00000i q^{4} +(7.34847 + 7.34847i) q^{7} +(-6.12372 + 6.12372i) q^{8} +18.0000 q^{11} +(-7.34847 + 7.34847i) q^{13} -18.0000i q^{14} +11.0000 q^{16} +(4.89898 + 4.89898i) q^{17} -10.0000i q^{19} +(-22.0454 - 22.0454i) q^{22} +(19.5959 - 19.5959i) q^{23} +18.0000 q^{26} +(7.34847 - 7.34847i) q^{28} +22.0000 q^{31} +(11.0227 + 11.0227i) q^{32} -12.0000i q^{34} +(7.34847 + 7.34847i) q^{37} +(-12.2474 + 12.2474i) q^{38} +18.0000 q^{41} +(29.3939 - 29.3939i) q^{43} -18.0000i q^{44} -48.0000 q^{46} +(-44.0908 - 44.0908i) q^{47} +59.0000i q^{49} +(7.34847 + 7.34847i) q^{52} +(-4.89898 + 4.89898i) q^{53} -90.0000 q^{56} +90.0000i q^{59} +2.00000 q^{61} +(-26.9444 - 26.9444i) q^{62} -71.0000i q^{64} +(44.0908 + 44.0908i) q^{67} +(4.89898 - 4.89898i) q^{68} -72.0000 q^{71} +(-44.0908 + 44.0908i) q^{73} -18.0000i q^{74} -10.0000 q^{76} +(132.272 + 132.272i) q^{77} +70.0000i q^{79} +(-22.0454 - 22.0454i) q^{82} +(-53.8888 + 53.8888i) q^{83} -72.0000 q^{86} +(-110.227 + 110.227i) q^{88} +90.0000i q^{89} -108.000 q^{91} +(-19.5959 - 19.5959i) q^{92} +108.000i q^{94} +(-102.879 - 102.879i) q^{97} +(72.2599 - 72.2599i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 72 q^{11} + 44 q^{16} + 72 q^{26} + 88 q^{31} + 72 q^{41} - 192 q^{46} - 360 q^{56} + 8 q^{61} - 288 q^{71} - 40 q^{76} - 288 q^{86} - 432 q^{91}+O(q^{100}) \) 4 * q + 72 * q^11 + 44 * q^16 + 72 * q^26 + 88 * q^31 + 72 * q^41 - 192 * q^46 - 360 * q^56 + 8 * q^61 - 288 * q^71 - 40 * q^76 - 288 * q^86 - 432 * q^91

Introduction

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