LMFDB - Embedded newform 1764.2.b.d.1567.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,2,Mod(1567,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1764.1567");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1764.b (of order $2$, degree $1$, 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:	$14.0856109166$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1567.2
Root			$-1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1764.1567
Dual form			1764.2.b.d.1567.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.22474 + 0.707107i) q^{2} +(1.00000 - 1.73205i) q^{4} -2.82843i q^{5} +2.82843i q^{8} +O(q^{10})$	\(q+(-1.22474 + 0.707107i) q^{2} +(1.00000 - 1.73205i) q^{4} -2.82843i q^{5} +2.82843i q^{8} +(2.00000 + 3.46410i) q^{10} +5.65685i q^{11} -5.19615i q^{13} +(-2.00000 - 3.46410i) q^{16} -2.82843i q^{17} +5.00000 q^{19} +(-4.89898 - 2.82843i) q^{20} +(-4.00000 - 6.92820i) q^{22} -2.82843i q^{23} -3.00000 q^{25} +(3.67423 + 6.36396i) q^{26} -1.00000 q^{31} +(4.89898 + 2.82843i) q^{32} +(2.00000 + 3.46410i) q^{34} +5.00000 q^{37} +(-6.12372 + 3.53553i) q^{38} +8.00000 q^{40} +5.65685i q^{41} -5.19615i q^{43} +(9.79796 + 5.65685i) q^{44} +(2.00000 + 3.46410i) q^{46} +4.89898 q^{47} +(3.67423 - 2.12132i) q^{50} +(-9.00000 - 5.19615i) q^{52} -4.89898 q^{53} +16.0000 q^{55} -4.89898 q^{59} -6.92820i q^{61} +(1.22474 - 0.707107i) q^{62} -8.00000 q^{64} -14.6969 q^{65} -8.66025i q^{67} +(-4.89898 - 2.82843i) q^{68} -2.82843i q^{71} +1.73205i q^{73} +(-6.12372 + 3.53553i) q^{74} +(5.00000 - 8.66025i) q^{76} -1.73205i q^{79} +(-9.79796 + 5.65685i) q^{80} +(-4.00000 - 6.92820i) q^{82} -14.6969 q^{83} -8.00000 q^{85} +(3.67423 + 6.36396i) q^{86} -16.0000 q^{88} -11.3137i q^{89} +(-4.89898 - 2.82843i) q^{92} +(-6.00000 + 3.46410i) q^{94} -14.1421i q^{95} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4}+O(q^{10}) $ 4 * q + 4 * q^4	$ 4 q + 4 q^{4} + 8 q^{10} - 8 q^{16} + 20 q^{19} - 16 q^{22} - 12 q^{25} - 4 q^{31} + 8 q^{34} + 20 q^{37} + 32 q^{40} + 8 q^{46} - 36 q^{52} + 64 q^{55} - 32 q^{64} + 20 q^{76} - 16 q^{82} - 32 q^{85} - 64 q^{88} - 24 q^{94}+O(q^{100}) $ 4 * q + 4 * q^4 + 8 * q^10 - 8 * q^16 + 20 * q^19 - 16 * q^22 - 12 * q^25 - 4 * q^31 + 8 * q^34 + 20 * q^37 + 32 * q^40 + 8 * q^46 - 36 * q^52 + 64 * q^55 - 32 * q^64 + 20 * q^76 - 16 * q^82 - 32 * q^85 - 64 * q^88 - 24 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−1.22474	+	0.707107i	−0.866025	+	0.500000i
$2$	−1.22474	+	0.707107i	−0.866025	+	0.500000i
$3$		0			0
$3$		0			0
$4$	1.00000	−	1.73205i	0.500000	−	0.866025i
$5$		−	2.82843i		−	1.26491i	−0.774597	−	0.632456i	$-0.782047\pi$
$5$		−	2.82843i		−	1.26491i	0.774597	−	0.632456i	$-0.217953\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$			2.82843i			1.00000i
$9$		0			0
$10$	2.00000	+	3.46410i	0.632456	+	1.09545i
$11$			5.65685i			1.70561i	0.522233	+	0.852803i	$0.325099\pi$
$11$			5.65685i			1.70561i	−0.522233	+	0.852803i	$0.674901\pi$
$12$		0			0
$13$		−	5.19615i		−	1.44115i	−0.693375	−	0.720577i	$-0.743877\pi$
$13$		−	5.19615i		−	1.44115i	0.693375	−	0.720577i	$-0.256123\pi$
$14$		0			0
$15$		0			0
$16$	−2.00000	−	3.46410i	−0.500000	−	0.866025i
$17$		−	2.82843i		−	0.685994i	−0.939336	−	0.342997i	$-0.888558\pi$
$17$		−	2.82843i		−	0.685994i	0.939336	−	0.342997i	$-0.111442\pi$
$18$		0			0
$19$	5.00000			1.14708			0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000			1.14708			0.573539	+	0.819178i	$0.305570\pi$
$20$	−4.89898	−	2.82843i	−1.09545	−	0.632456i
$21$		0			0
$22$	−4.00000	−	6.92820i	−0.852803	−	1.47710i
$23$		−	2.82843i		−	0.589768i	−0.955533	−	0.294884i	$-0.904719\pi$
$23$		−	2.82843i		−	0.589768i	0.955533	−	0.294884i	$-0.0952810\pi$
$24$		0			0
$25$	−3.00000			−0.600000
$26$	3.67423	+	6.36396i	0.720577	+	1.24808i
$27$		0			0
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	−1.00000			−0.179605			−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000			−0.179605			−0.0898027	+	0.995960i	$0.528624\pi$
$32$	4.89898	+	2.82843i	0.866025	+	0.500000i
$33$		0			0
$34$	2.00000	+	3.46410i	0.342997	+	0.594089i
$35$		0			0
$36$		0			0
$37$	5.00000			0.821995			0.410997	−	0.911636i	$-0.365181\pi$
$37$	5.00000			0.821995			0.410997	+	0.911636i	$0.365181\pi$
$38$	−6.12372	+	3.53553i	−0.993399	+	0.573539i
$39$		0			0
$40$	8.00000			1.26491
$41$			5.65685i			0.883452i	0.897150	+	0.441726i	$0.145634\pi$
$41$			5.65685i			0.883452i	−0.897150	+	0.441726i	$0.854366\pi$
$42$		0			0
$43$		−	5.19615i		−	0.792406i	−0.918163	−	0.396203i	$-0.870328\pi$
$43$		−	5.19615i		−	0.792406i	0.918163	−	0.396203i	$-0.129672\pi$
$44$	9.79796	+	5.65685i	1.47710	+	0.852803i
$45$		0			0
$46$	2.00000	+	3.46410i	0.294884	+	0.510754i
$47$	4.89898			0.714590			0.357295	−	0.933992i	$-0.383699\pi$
$47$	4.89898			0.714590			0.357295	+	0.933992i	$0.383699\pi$
$48$		0			0
$49$		0			0
$50$	3.67423	−	2.12132i	0.519615	−	0.300000i
$51$		0			0
$52$	−9.00000	−	5.19615i	−1.24808	−	0.720577i
$53$	−4.89898			−0.672927			−0.336463	−	0.941697i	$-0.609231\pi$
$53$	−4.89898			−0.672927			−0.336463	+	0.941697i	$0.609231\pi$
$54$		0			0
$55$	16.0000			2.15744
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.89898			−0.637793			−0.318896	−	0.947790i	$-0.603312\pi$
$59$	−4.89898			−0.637793			−0.318896	+	0.947790i	$0.603312\pi$
$60$		0			0
$61$		−	6.92820i		−	0.887066i	−0.896258	−	0.443533i	$-0.853725\pi$
$61$		−	6.92820i		−	0.887066i	0.896258	−	0.443533i	$-0.146275\pi$
$62$	1.22474	−	0.707107i	0.155543	−	0.0898027i
$63$		0			0
$64$	−8.00000			−1.00000
$65$	−14.6969			−1.82293
$66$		0			0
$67$		−	8.66025i		−	1.05802i	−0.848616	−	0.529009i	$-0.822564\pi$
$67$		−	8.66025i		−	1.05802i	0.848616	−	0.529009i	$-0.177436\pi$
$68$	−4.89898	−	2.82843i	−0.594089	−	0.342997i
$69$		0			0
$70$		0			0
$71$		−	2.82843i		−	0.335673i	−0.985815	−	0.167836i	$-0.946322\pi$
$71$		−	2.82843i		−	0.335673i	0.985815	−	0.167836i	$-0.0536780\pi$
$72$		0			0
$73$			1.73205i			0.202721i	0.994850	+	0.101361i	$0.0323196\pi$
$73$			1.73205i			0.202721i	−0.994850	+	0.101361i	$0.967680\pi$
$74$	−6.12372	+	3.53553i	−0.711868	+	0.410997i
$75$		0			0
$76$	5.00000	−	8.66025i	0.573539	−	0.993399i
$77$		0			0
$78$		0			0
$79$		−	1.73205i		−	0.194871i	−0.995242	−	0.0974355i	$-0.968936\pi$
$79$		−	1.73205i		−	0.194871i	0.995242	−	0.0974355i	$-0.0310640\pi$
$80$	−9.79796	+	5.65685i	−1.09545	+	0.632456i
$81$		0			0
$82$	−4.00000	−	6.92820i	−0.441726	−	0.765092i
$83$	−14.6969			−1.61320			−0.806599	−	0.591099i	$-0.798694\pi$
$83$	−14.6969			−1.61320			−0.806599	+	0.591099i	$0.798694\pi$
$84$		0			0
$85$	−8.00000			−0.867722
$86$	3.67423	+	6.36396i	0.396203	+	0.686244i
$87$		0			0
$88$	−16.0000			−1.70561
$89$		−	11.3137i		−	1.19925i	−0.800281	−	0.599625i	$-0.795316\pi$
$89$		−	11.3137i		−	1.19925i	0.800281	−	0.599625i	$-0.204684\pi$
$90$		0			0
$91$		0			0
$92$	−4.89898	−	2.82843i	−0.510754	−	0.294884i
$93$		0			0
$94$	−6.00000	+	3.46410i	−0.618853	+	0.357295i
$95$		−	14.1421i		−	1.45095i
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$		0			0
$100$	−3.00000	+	5.19615i	−0.300000	+	0.519615i
$101$		−	11.3137i		−	1.12576i	−0.826540	−	0.562878i	$-0.809694\pi$
$101$		−	11.3137i		−	1.12576i	0.826540	−	0.562878i	$-0.190306\pi$
$102$		0			0
$103$	−1.00000			−0.0985329			−0.0492665	−	0.998786i	$-0.515688\pi$
$103$	−1.00000			−0.0985329			−0.0492665	+	0.998786i	$0.515688\pi$
$104$	14.6969			1.44115
$105$		0			0
$106$	6.00000	−	3.46410i	0.582772	−	0.336463i
$107$		−	2.82843i		−	0.273434i	−0.990610	−	0.136717i	$-0.956345\pi$
$107$		−	2.82843i		−	0.273434i	0.990610	−	0.136717i	$-0.0436552\pi$
$108$		0			0
$109$	5.00000			0.478913			0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000			0.478913			0.239457	+	0.970907i	$0.423031\pi$
$110$	−19.5959	+	11.3137i	−1.86840	+	1.07872i
$111$		0			0
$112$		0			0
$113$	−14.6969			−1.38257			−0.691286	−	0.722581i	$-0.742955\pi$
$113$	−14.6969			−1.38257			−0.691286	+	0.722581i	$0.742955\pi$
$114$		0			0
$115$	−8.00000			−0.746004
$116$		0			0
$117$		0			0
$118$	6.00000	−	3.46410i	0.552345	−	0.318896i
$119$		0			0
$120$		0			0
$121$	−21.0000			−1.90909
$122$	4.89898	+	8.48528i	0.443533	+	0.768221i
$123$		0			0
$124$	−1.00000	+	1.73205i	−0.0898027	+	0.155543i
$125$		−	5.65685i		−	0.505964i
$126$		0			0
$127$		−	15.5885i		−	1.38325i	−0.722256	−	0.691626i	$-0.756895\pi$
$127$		−	15.5885i		−	1.38325i	0.722256	−	0.691626i	$-0.243105\pi$
$128$	9.79796	−	5.65685i	0.866025	−	0.500000i
$129$		0			0
$130$	18.0000	−	10.3923i	1.57870	−	0.911465i
$131$	19.5959			1.71210			0.856052	−	0.516890i	$-0.172910\pi$
$131$	19.5959			1.71210			0.856052	+	0.516890i	$0.172910\pi$
$132$		0			0
$133$		0			0
$134$	6.12372	+	10.6066i	0.529009	+	0.916271i
$135$		0			0
$136$	8.00000			0.685994
$137$	9.79796			0.837096			0.418548	−	0.908195i	$-0.362539\pi$
$137$	9.79796			0.837096			0.418548	+	0.908195i	$0.362539\pi$
$138$		0			0
$139$	−13.0000			−1.10265			−0.551323	−	0.834292i	$-0.685877\pi$
$139$	−13.0000			−1.10265			−0.551323	+	0.834292i	$0.685877\pi$
$140$		0			0
$141$		0			0
$142$	2.00000	+	3.46410i	0.167836	+	0.290701i
$143$	29.3939			2.45804
$144$		0			0
$145$		0			0
$146$	−1.22474	−	2.12132i	−0.101361	−	0.175562i
$147$		0			0
$148$	5.00000	−	8.66025i	0.410997	−	0.711868i
$149$	−9.79796			−0.802680			−0.401340	−	0.915929i	$-0.631455\pi$
$149$	−9.79796			−0.802680			−0.401340	+	0.915929i	$0.631455\pi$
$150$		0			0
$151$			17.3205i			1.40952i	0.709444	+	0.704761i	$0.248946\pi$
$151$			17.3205i			1.40952i	−0.709444	+	0.704761i	$0.751054\pi$
$152$			14.1421i			1.14708i
$153$		0			0
$154$		0			0
$155$			2.82843i			0.227185i
$156$		0			0
$157$			6.92820i			0.552931i	0.961024	+	0.276465i	$0.0891631\pi$
$157$			6.92820i			0.552931i	−0.961024	+	0.276465i	$0.910837\pi$
$158$	1.22474	+	2.12132i	0.0974355	+	0.168763i
$159$		0			0
$160$	8.00000	−	13.8564i	0.632456	−	1.09545i
$161$		0			0
$162$		0			0
$163$			3.46410i			0.271329i	0.990755	+	0.135665i	$0.0433170\pi$
$163$			3.46410i			0.271329i	−0.990755	+	0.135665i	$0.956683\pi$
$164$	9.79796	+	5.65685i	0.765092	+	0.441726i
$165$		0			0
$166$	18.0000	−	10.3923i	1.39707	−	0.806599i
$167$	14.6969			1.13728			0.568642	−	0.822585i	$-0.307469\pi$
$167$	14.6969			1.13728			0.568642	+	0.822585i	$0.307469\pi$
$168$		0			0
$169$	−14.0000			−1.07692
$170$	9.79796	−	5.65685i	0.751469	−	0.433861i
$171$		0			0
$172$	−9.00000	−	5.19615i	−0.686244	−	0.396203i
$173$		−	19.7990i		−	1.50529i	−0.658427	−	0.752645i	$-0.728778\pi$
$173$		−	19.7990i		−	1.50529i	0.658427	−	0.752645i	$-0.271222\pi$
$174$		0			0
$175$		0			0
$176$	19.5959	−	11.3137i	1.47710	−	0.852803i
$177$		0			0
$178$	8.00000	+	13.8564i	0.599625	+	1.03858i
$179$		−	2.82843i		−	0.211407i	−0.994398	−	0.105703i	$-0.966291\pi$
$179$		−	2.82843i		−	0.211407i	0.994398	−	0.105703i	$-0.0337094\pi$
$180$		0			0
$181$			5.19615i			0.386227i	0.981176	+	0.193113i	$0.0618586\pi$
$181$			5.19615i			0.386227i	−0.981176	+	0.193113i	$0.938141\pi$
$182$		0			0
$183$		0			0
$184$	8.00000			0.589768
$185$		−	14.1421i		−	1.03975i
$186$		0			0
$187$	16.0000			1.17004
$188$	4.89898	−	8.48528i	0.357295	−	0.618853i
$189$		0			0
$190$	10.0000	+	17.3205i	0.725476	+	1.25656i
$191$		−	19.7990i		−	1.43260i	−0.697790	−	0.716302i	$-0.745833\pi$
$191$		−	19.7990i		−	1.43260i	0.697790	−	0.716302i	$-0.254167\pi$
$192$		0			0
$193$	−7.00000			−0.503871			−0.251936	−	0.967744i	$-0.581067\pi$
$193$	−7.00000			−0.503871			−0.251936	+	0.967744i	$0.581067\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	14.6969			1.04711			0.523557	−	0.851991i	$-0.324605\pi$
$197$	14.6969			1.04711			0.523557	+	0.851991i	$0.324605\pi$
$198$		0			0
$199$	−4.00000			−0.283552			−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000			−0.283552			−0.141776	+	0.989899i	$0.545281\pi$
$200$		−	8.48528i		−	0.600000i
$201$		0			0
$202$	8.00000	+	13.8564i	0.562878	+	0.974933i
$203$		0			0
$204$		0			0
$205$	16.0000			1.11749
$206$	1.22474	−	0.707107i	0.0853320	−	0.0492665i
$207$		0			0
$208$	−18.0000	+	10.3923i	−1.24808	+	0.720577i
$209$			28.2843i			1.95646i
$210$		0			0
$211$			10.3923i			0.715436i	0.933830	+	0.357718i	$0.116445\pi$
$211$			10.3923i			0.715436i	−0.933830	+	0.357718i	$0.883555\pi$
$212$	−4.89898	+	8.48528i	−0.336463	+	0.582772i
$213$		0			0
$214$	2.00000	+	3.46410i	0.136717	+	0.236801i
$215$	−14.6969			−1.00232
$216$		0			0
$217$		0			0
$218$	−6.12372	+	3.53553i	−0.414751	+	0.239457i
$219$		0			0
$220$	16.0000	−	27.7128i	1.07872	−	1.86840i
$221$	−14.6969			−0.988623
$222$		0			0
$223$	−4.00000			−0.267860			−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000			−0.267860			−0.133930	+	0.990991i	$0.542760\pi$
$224$		0			0
$225$		0			0
$226$	18.0000	−	10.3923i	1.19734	−	0.691286i
$227$	9.79796			0.650313			0.325157	−	0.945660i	$-0.394583\pi$
$227$	9.79796			0.650313			0.325157	+	0.945660i	$0.394583\pi$
$228$		0			0
$229$			19.0526i			1.25903i	0.776989	+	0.629514i	$0.216746\pi$
$229$			19.0526i			1.25903i	−0.776989	+	0.629514i	$0.783254\pi$
$230$	9.79796	−	5.65685i	0.646058	−	0.373002i
$231$		0			0
$232$		0			0
$233$	−24.4949			−1.60471			−0.802357	−	0.596844i	$-0.796421\pi$
$233$	−24.4949			−1.60471			−0.802357	+	0.596844i	$0.796421\pi$
$234$		0			0
$235$		−	13.8564i		−	0.903892i
$236$	−4.89898	+	8.48528i	−0.318896	+	0.552345i
$237$		0			0
$238$		0			0
$239$		−	11.3137i		−	0.731823i	−0.930650	−	0.365911i	$-0.880757\pi$
$239$		−	11.3137i		−	0.731823i	0.930650	−	0.365911i	$-0.119243\pi$
$240$		0			0
$241$		−	13.8564i		−	0.892570i	−0.894891	−	0.446285i	$-0.852747\pi$
$241$		−	13.8564i		−	0.892570i	0.894891	−	0.446285i	$-0.147253\pi$
$242$	25.7196	−	14.8492i	1.65332	−	0.954545i
$243$		0			0
$244$	−12.0000	−	6.92820i	−0.768221	−	0.443533i
$245$		0			0
$246$		0			0
$247$		−	25.9808i		−	1.65312i
$248$		−	2.82843i		−	0.179605i
$249$		0			0
$250$	4.00000	+	6.92820i	0.252982	+	0.438178i
$251$	14.6969			0.927663			0.463831	−	0.885924i	$-0.346474\pi$
$251$	14.6969			0.927663			0.463831	+	0.885924i	$0.346474\pi$
$252$		0			0
$253$	16.0000			1.00591
$254$	11.0227	+	19.0919i	0.691626	+	1.19793i
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$		−	2.82843i		−	0.176432i	−0.996101	−	0.0882162i	$-0.971883\pi$
$257$		−	2.82843i		−	0.176432i	0.996101	−	0.0882162i	$-0.0281166\pi$
$258$		0			0
$259$		0			0
$260$	−14.6969	+	25.4558i	−0.911465	+	1.57870i
$261$		0			0
$262$	−24.0000	+	13.8564i	−1.48272	+	0.856052i
$263$			5.65685i			0.348817i	0.984673	+	0.174408i	$0.0558013\pi$
$263$			5.65685i			0.348817i	−0.984673	+	0.174408i	$0.944199\pi$
$264$		0			0
$265$			13.8564i			0.851192i
$266$		0			0
$267$		0			0
$268$	−15.0000	−	8.66025i	−0.916271	−	0.529009i
$269$		−	11.3137i		−	0.689809i	−0.938638	−	0.344904i	$-0.887911\pi$
$269$		−	11.3137i		−	0.689809i	0.938638	−	0.344904i	$-0.112089\pi$
$270$		0			0
$271$	−28.0000			−1.70088			−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000			−1.70088			−0.850439	+	0.526073i	$0.823664\pi$
$272$	−9.79796	+	5.65685i	−0.594089	+	0.342997i
$273$		0			0
$274$	−12.0000	+	6.92820i	−0.724947	+	0.418548i
$275$		−	16.9706i		−	1.02336i
$276$		0			0
$277$	11.0000			0.660926			0.330463	−	0.943819i	$-0.392795\pi$
$277$	11.0000			0.660926			0.330463	+	0.943819i	$0.392795\pi$
$278$	15.9217	−	9.19239i	0.954919	−	0.551323i
$279$		0			0
$280$		0			0
$281$	−14.6969			−0.876746			−0.438373	−	0.898793i	$-0.644445\pi$
$281$	−14.6969			−0.876746			−0.438373	+	0.898793i	$0.644445\pi$
$282$		0			0
$283$	29.0000			1.72387			0.861936	−	0.507018i	$-0.169252\pi$
$283$	29.0000			1.72387			0.861936	+	0.507018i	$0.169252\pi$
$284$	−4.89898	−	2.82843i	−0.290701	−	0.167836i
$285$		0			0
$286$	−36.0000	+	20.7846i	−2.12872	+	1.22902i
$287$		0			0
$288$		0			0
$289$	9.00000			0.529412
$290$		0			0
$291$		0			0
$292$	3.00000	+	1.73205i	0.175562	+	0.101361i
$293$		−	11.3137i		−	0.660954i	−0.943814	−	0.330477i	$-0.892790\pi$
$293$		−	11.3137i		−	0.660954i	0.943814	−	0.330477i	$-0.107210\pi$
$294$		0			0
$295$			13.8564i			0.806751i
$296$			14.1421i			0.821995i
$297$		0			0
$298$	12.0000	−	6.92820i	0.695141	−	0.401340i
$299$	−14.6969			−0.849946
$300$		0			0
$301$		0			0
$302$	−12.2474	−	21.2132i	−0.704761	−	1.22068i
$303$		0			0
$304$	−10.0000	−	17.3205i	−0.573539	−	0.993399i
$305$	−19.5959			−1.12206
$306$		0			0
$307$	−1.00000			−0.0570730			−0.0285365	−	0.999593i	$-0.509085\pi$
$307$	−1.00000			−0.0570730			−0.0285365	+	0.999593i	$0.509085\pi$
$308$		0			0
$309$		0			0
$310$	−2.00000	−	3.46410i	−0.113592	−	0.196748i
$311$	−19.5959			−1.11118			−0.555591	−	0.831456i	$-0.687508\pi$
$311$	−19.5959			−1.11118			−0.555591	+	0.831456i	$0.687508\pi$
$312$		0			0
$313$		−	22.5167i		−	1.27272i	−0.771393	−	0.636358i	$-0.780440\pi$
$313$		−	22.5167i		−	1.27272i	0.771393	−	0.636358i	$-0.219560\pi$
$314$	−4.89898	−	8.48528i	−0.276465	−	0.478852i
$315$		0			0
$316$	−3.00000	−	1.73205i	−0.168763	−	0.0974355i
$317$	19.5959			1.10062			0.550308	−	0.834962i	$-0.314510\pi$
$317$	19.5959			1.10062			0.550308	+	0.834962i	$0.314510\pi$
$318$		0			0
$319$		0			0
$320$			22.6274i			1.26491i
$321$		0			0
$322$		0			0
$323$		−	14.1421i		−	0.786889i
$324$		0			0
$325$			15.5885i			0.864692i
$326$	−2.44949	−	4.24264i	−0.135665	−	0.234978i
$327$		0			0
$328$	−16.0000			−0.883452
$329$		0			0
$330$		0			0
$331$			29.4449i			1.61844i	0.587508	+	0.809218i	$0.300109\pi$
$331$			29.4449i			1.61844i	−0.587508	+	0.809218i	$0.699891\pi$
$332$	−14.6969	+	25.4558i	−0.806599	+	1.39707i
$333$		0			0
$334$	−18.0000	+	10.3923i	−0.984916	+	0.568642i
$335$	−24.4949			−1.33830
$336$		0			0
$337$	29.0000			1.57973			0.789865	−	0.613280i	$-0.210150\pi$
$337$	29.0000			1.57973			0.789865	+	0.613280i	$0.210150\pi$
$338$	17.1464	−	9.89949i	0.932643	−	0.538462i
$339$		0			0
$340$	−8.00000	+	13.8564i	−0.433861	+	0.751469i
$341$		−	5.65685i		−	0.306336i
$342$		0			0
$343$		0			0
$344$	14.6969			0.792406
$345$		0			0
$346$	14.0000	+	24.2487i	0.752645	+	1.30362i
$347$		−	2.82843i		−	0.151838i	−0.997114	−	0.0759190i	$-0.975811\pi$
$347$		−	2.82843i		−	0.151838i	0.997114	−	0.0759190i	$-0.0241890\pi$
$348$		0			0
$349$		−	20.7846i		−	1.11257i	−0.830990	−	0.556287i	$-0.812225\pi$
$349$		−	20.7846i		−	1.11257i	0.830990	−	0.556287i	$-0.187775\pi$
$350$		0			0
$351$		0			0
$352$	−16.0000	+	27.7128i	−0.852803	+	1.47710i
$353$			31.1127i			1.65596i	0.560756	+	0.827981i	$0.310510\pi$
$353$			31.1127i			1.65596i	−0.560756	+	0.827981i	$0.689490\pi$
$354$		0			0
$355$	−8.00000			−0.424596
$356$	−19.5959	−	11.3137i	−1.03858	−	0.599625i
$357$		0			0
$358$	2.00000	+	3.46410i	0.105703	+	0.183083i
$359$			22.6274i			1.19423i	0.802156	+	0.597115i	$0.203686\pi$
$359$			22.6274i			1.19423i	−0.802156	+	0.597115i	$0.796314\pi$
$360$		0			0
$361$	6.00000			0.315789
$362$	−3.67423	−	6.36396i	−0.193113	−	0.334482i
$363$		0			0
$364$		0			0
$365$	4.89898			0.256424
$366$		0			0
$367$	11.0000			0.574195			0.287098	−	0.957901i	$-0.407310\pi$
$367$	11.0000			0.574195			0.287098	+	0.957901i	$0.407310\pi$
$368$	−9.79796	+	5.65685i	−0.510754	+	0.294884i
$369$		0			0
$370$	10.0000	+	17.3205i	0.519875	+	0.900450i
$371$		0			0
$372$		0			0
$373$	−1.00000			−0.0517780			−0.0258890	−	0.999665i	$-0.508242\pi$
$373$	−1.00000			−0.0517780			−0.0258890	+	0.999665i	$0.508242\pi$
$374$	−19.5959	+	11.3137i	−1.01328	+	0.585018i
$375$		0			0
$376$			13.8564i			0.714590i
$377$		0			0
$378$		0			0
$379$			36.3731i			1.86836i	0.356803	+	0.934179i	$0.383867\pi$
$379$			36.3731i			1.86836i	−0.356803	+	0.934179i	$0.616133\pi$
$380$	−24.4949	−	14.1421i	−1.25656	−	0.725476i
$381$		0			0
$382$	14.0000	+	24.2487i	0.716302	+	1.24067i
$383$	34.2929			1.75228			0.876142	−	0.482054i	$-0.160109\pi$
$383$	34.2929			1.75228			0.876142	+	0.482054i	$0.160109\pi$
$384$		0			0
$385$		0			0
$386$	8.57321	−	4.94975i	0.436365	−	0.251936i
$387$		0			0
$388$		0			0
$389$	24.4949			1.24194			0.620970	−	0.783834i	$-0.286739\pi$
$389$	24.4949			1.24194			0.620970	+	0.783834i	$0.286739\pi$
$390$		0			0
$391$	−8.00000			−0.404577
$392$		0			0
$393$		0			0
$394$	−18.0000	+	10.3923i	−0.906827	+	0.523557i
$395$	−4.89898			−0.246494
$396$		0			0
$397$			29.4449i			1.47780i	0.673818	+	0.738898i	$0.264653\pi$
$397$			29.4449i			1.47780i	−0.673818	+	0.738898i	$0.735347\pi$
$398$	4.89898	−	2.82843i	0.245564	−	0.141776i
$399$		0			0
$400$	6.00000	+	10.3923i	0.300000	+	0.519615i
$401$	4.89898			0.244643			0.122322	−	0.992491i	$-0.460966\pi$
$401$	4.89898			0.244643			0.122322	+	0.992491i	$0.460966\pi$
$402$		0			0
$403$			5.19615i			0.258839i
$404$	−19.5959	−	11.3137i	−0.974933	−	0.562878i
$405$		0			0
$406$		0			0
$407$			28.2843i			1.40200i
$408$		0			0
$409$		−	29.4449i		−	1.45595i	−0.685601	−	0.727977i	$-0.740461\pi$
$409$		−	29.4449i		−	1.45595i	0.685601	−	0.727977i	$-0.259539\pi$
$410$	−19.5959	+	11.3137i	−0.967773	+	0.558744i
$411$		0			0
$412$	−1.00000	+	1.73205i	−0.0492665	+	0.0853320i
$413$		0			0
$414$		0			0
$415$			41.5692i			2.04055i
$416$	14.6969	−	25.4558i	0.720577	−	1.24808i
$417$		0			0
$418$	−20.0000	−	34.6410i	−0.978232	−	1.69435i
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	23.0000			1.12095			0.560476	−	0.828171i	$-0.310618\pi$
$421$	23.0000			1.12095			0.560476	+	0.828171i	$0.310618\pi$
$422$	−7.34847	−	12.7279i	−0.357718	−	0.619586i
$423$		0			0
$424$		−	13.8564i		−	0.672927i
$425$			8.48528i			0.411597i
$426$		0			0
$427$		0			0
$428$	−4.89898	−	2.82843i	−0.236801	−	0.136717i
$429$		0			0
$430$	18.0000	−	10.3923i	0.868037	−	0.501161i
$431$			14.1421i			0.681203i	0.940208	+	0.340601i	$0.110631\pi$
$431$			14.1421i			0.681203i	−0.940208	+	0.340601i	$0.889369\pi$
$432$		0			0
$433$		−	5.19615i		−	0.249711i	−0.992175	−	0.124856i	$-0.960153\pi$
$433$		−	5.19615i		−	0.249711i	0.992175	−	0.124856i	$-0.0398468\pi$
$434$		0			0
$435$		0			0
$436$	5.00000	−	8.66025i	0.239457	−	0.414751i
$437$		−	14.1421i		−	0.676510i
$438$		0			0
$439$	−4.00000			−0.190910			−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000			−0.190910			−0.0954548	+	0.995434i	$0.530431\pi$
$440$			45.2548i			2.15744i
$441$		0			0
$442$	18.0000	−	10.3923i	0.856173	−	0.494312i
$443$			31.1127i			1.47821i	0.673591	+	0.739104i	$0.264751\pi$
$443$			31.1127i			1.47821i	−0.673591	+	0.739104i	$0.735249\pi$
$444$		0			0
$445$	−32.0000			−1.51695
$446$	4.89898	−	2.82843i	0.231973	−	0.133930i
$447$		0			0
$448$		0			0
$449$	−14.6969			−0.693591			−0.346796	−	0.937941i	$-0.612730\pi$
$449$	−14.6969			−0.693591			−0.346796	+	0.937941i	$0.612730\pi$
$450$		0			0
$451$	−32.0000			−1.50682
$452$	−14.6969	+	25.4558i	−0.691286	+	1.19734i
$453$		0			0
$454$	−12.0000	+	6.92820i	−0.563188	+	0.325157i
$455$		0			0
$456$		0			0
$457$	−7.00000			−0.327446			−0.163723	−	0.986506i	$-0.552350\pi$
$457$	−7.00000			−0.327446			−0.163723	+	0.986506i	$0.552350\pi$
$458$	−13.4722	−	23.3345i	−0.629514	−	1.09035i
$459$		0			0
$460$	−8.00000	+	13.8564i	−0.373002	+	0.646058i
$461$		−	11.3137i		−	0.526932i	−0.964669	−	0.263466i	$-0.915134\pi$
$461$		−	11.3137i		−	0.526932i	0.964669	−	0.263466i	$-0.0848657\pi$
$462$		0			0
$463$			5.19615i			0.241486i	0.992684	+	0.120743i	$0.0385276\pi$
$463$			5.19615i			0.241486i	−0.992684	+	0.120743i	$0.961472\pi$
$464$		0			0
$465$		0			0
$466$	30.0000	−	17.3205i	1.38972	−	0.802357i
$467$	4.89898			0.226698			0.113349	−	0.993555i	$-0.463842\pi$
$467$	4.89898			0.226698			0.113349	+	0.993555i	$0.463842\pi$
$468$		0			0
$469$		0			0
$470$	9.79796	+	16.9706i	0.451946	+	0.782794i
$471$		0			0
$472$		−	13.8564i		−	0.637793i
$473$	29.3939			1.35153
$474$		0			0
$475$	−15.0000			−0.688247
$476$		0			0
$477$		0			0
$478$	8.00000	+	13.8564i	0.365911	+	0.633777i
$479$	−34.2929			−1.56688			−0.783440	−	0.621467i	$-0.786537\pi$
$479$	−34.2929			−1.56688			−0.783440	+	0.621467i	$0.786537\pi$
$480$		0			0
$481$		−	25.9808i		−	1.18462i
$482$	9.79796	+	16.9706i	0.446285	+	0.772988i
$483$		0			0
$484$	−21.0000	+	36.3731i	−0.954545	+	1.65332i
$485$		0			0
$486$		0			0
$487$			22.5167i			1.02033i	0.860077	+	0.510164i	$0.170415\pi$
$487$			22.5167i			1.02033i	−0.860077	+	0.510164i	$0.829585\pi$
$488$	19.5959			0.887066
$489$		0			0
$490$		0			0
$491$			5.65685i			0.255290i	0.991820	+	0.127645i	$0.0407419\pi$
$491$			5.65685i			0.255290i	−0.991820	+	0.127645i	$0.959258\pi$
$492$		0			0
$493$		0			0
$494$	18.3712	+	31.8198i	0.826558	+	1.43164i
$495$		0			0
$496$	2.00000	+	3.46410i	0.0898027	+	0.155543i
$497$		0			0
$498$		0			0
$499$		−	32.9090i		−	1.47321i	−0.676325	−	0.736604i	$-0.736428\pi$
$499$		−	32.9090i		−	1.47321i	0.676325	−	0.736604i	$-0.263572\pi$
$500$	−9.79796	−	5.65685i	−0.438178	−	0.252982i
$501$		0			0
$502$	−18.0000	+	10.3923i	−0.803379	+	0.463831i
$503$	−29.3939			−1.31061			−0.655304	−	0.755365i	$-0.727460\pi$
$503$	−29.3939			−1.31061			−0.655304	+	0.755365i	$0.727460\pi$
$504$		0			0
$505$	−32.0000			−1.42398
$506$	−19.5959	+	11.3137i	−0.871145	+	0.502956i
$507$		0			0
$508$	−27.0000	−	15.5885i	−1.19793	−	0.691626i
$509$			31.1127i			1.37905i	0.724264	+	0.689523i	$0.242180\pi$
$509$			31.1127i			1.37905i	−0.724264	+	0.689523i	$0.757820\pi$
$510$		0			0
$511$		0			0
$512$		−	22.6274i		−	1.00000i
$513$		0			0
$514$	2.00000	+	3.46410i	0.0882162	+	0.152795i
$515$			2.82843i			0.124635i
$516$		0			0
$517$			27.7128i			1.21881i
$518$		0			0
$519$		0			0
$520$		−	41.5692i		−	1.82293i
$521$		−	11.3137i		−	0.495663i	−0.968803	−	0.247831i	$-0.920282\pi$
$521$		−	11.3137i		−	0.495663i	0.968803	−	0.247831i	$-0.0797179\pi$
$522$		0			0
$523$	−31.0000			−1.35554			−0.677768	−	0.735276i	$-0.737052\pi$
$523$	−31.0000			−1.35554			−0.677768	+	0.735276i	$0.737052\pi$
$524$	19.5959	−	33.9411i	0.856052	−	1.48272i
$525$		0			0
$526$	−4.00000	−	6.92820i	−0.174408	−	0.302084i
$527$			2.82843i			0.123208i
$528$		0			0
$529$	15.0000			0.652174
$530$	−9.79796	−	16.9706i	−0.425596	−	0.737154i
$531$		0			0
$532$		0			0
$533$	29.3939			1.27319
$534$		0			0
$535$	−8.00000			−0.345870
$536$	24.4949			1.05802
$537$		0			0
$538$	8.00000	+	13.8564i	0.344904	+	0.597392i
$539$		0			0
$540$		0			0
$541$	23.0000			0.988847			0.494424	−	0.869221i	$-0.335379\pi$
$541$	23.0000			0.988847			0.494424	+	0.869221i	$0.335379\pi$
$542$	34.2929	−	19.7990i	1.47300	−	0.850439i
$543$		0			0
$544$	8.00000	−	13.8564i	0.342997	−	0.594089i
$545$		−	14.1421i		−	0.605783i
$546$		0			0
$547$		−	10.3923i		−	0.444343i	−0.975008	−	0.222171i	$-0.928686\pi$
$547$		−	10.3923i		−	0.444343i	0.975008	−	0.222171i	$-0.0713145\pi$
$548$	9.79796	−	16.9706i	0.418548	−	0.724947i
$549$		0			0
$550$	12.0000	+	20.7846i	0.511682	+	0.886259i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−13.4722	+	7.77817i	−0.572379	+	0.330463i
$555$		0			0
$556$	−13.0000	+	22.5167i	−0.551323	+	0.954919i
$557$	−19.5959			−0.830306			−0.415153	−	0.909752i	$-0.636272\pi$
$557$	−19.5959			−0.830306			−0.415153	+	0.909752i	$0.636272\pi$
$558$		0			0
$559$	−27.0000			−1.14198
$560$		0			0
$561$		0			0
$562$	18.0000	−	10.3923i	0.759284	−	0.438373i
$563$	24.4949			1.03234			0.516168	−	0.856487i	$-0.327358\pi$
$563$	24.4949			1.03234			0.516168	+	0.856487i	$0.327358\pi$
$564$		0			0
$565$			41.5692i			1.74883i
$566$	−35.5176	+	20.5061i	−1.49292	+	0.861936i
$567$		0			0
$568$	8.00000			0.335673
$569$	19.5959			0.821504			0.410752	−	0.911747i	$-0.365266\pi$
$569$	19.5959			0.821504			0.410752	+	0.911747i	$0.365266\pi$
$570$		0			0
$571$			12.1244i			0.507388i	0.967284	+	0.253694i	$0.0816457\pi$
$571$			12.1244i			0.507388i	−0.967284	+	0.253694i	$0.918354\pi$
$572$	29.3939	−	50.9117i	1.22902	−	2.12872i
$573$		0			0
$574$		0			0
$575$			8.48528i			0.353861i
$576$		0			0
$577$		−	8.66025i		−	0.360531i	−0.983618	−	0.180266i	$-0.942304\pi$
$577$		−	8.66025i		−	0.360531i	0.983618	−	0.180266i	$-0.0576957\pi$
$578$	−11.0227	+	6.36396i	−0.458484	+	0.264706i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		−	27.7128i		−	1.14775i
$584$	−4.89898			−0.202721
$585$		0			0
$586$	8.00000	+	13.8564i	0.330477	+	0.572403i
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$	−5.00000			−0.206021
$590$	−9.79796	−	16.9706i	−0.403376	−	0.698667i
$591$		0			0
$592$	−10.0000	−	17.3205i	−0.410997	−	0.711868i
$593$			39.5980i			1.62609i	0.582198	+	0.813047i	$0.302193\pi$
$593$			39.5980i			1.62609i	−0.582198	+	0.813047i	$0.697807\pi$
$594$		0			0
$595$		0			0
$596$	−9.79796	+	16.9706i	−0.401340	+	0.695141i
$597$		0			0
$598$	18.0000	−	10.3923i	0.736075	−	0.424973i
$599$			22.6274i			0.924531i	0.886742	+	0.462266i	$0.152963\pi$
$599$			22.6274i			0.924531i	−0.886742	+	0.462266i	$0.847037\pi$
$600$		0			0
$601$			25.9808i			1.05978i	0.848067	+	0.529889i	$0.177766\pi$
$601$			25.9808i			1.05978i	−0.848067	+	0.529889i	$0.822234\pi$
$602$		0			0
$603$		0			0
$604$	30.0000	+	17.3205i	1.22068	+	0.704761i
$605$			59.3970i			2.41483i
$606$		0			0
$607$	47.0000			1.90767			0.953836	−	0.300329i	$-0.0970966\pi$
$607$	47.0000			1.90767			0.953836	+	0.300329i	$0.0970966\pi$
$608$	24.4949	+	14.1421i	0.993399	+	0.573539i
$609$		0			0
$610$	24.0000	−	13.8564i	0.971732	−	0.561029i
$611$		−	25.4558i		−	1.02983i
$612$		0			0
$613$	14.0000			0.565455			0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000			0.565455			0.282727	+	0.959200i	$0.408761\pi$
$614$	1.22474	−	0.707107i	0.0494267	−	0.0285365i
$615$		0			0
$616$		0			0
$617$	29.3939			1.18335			0.591676	−	0.806176i	$-0.298466\pi$
$617$	29.3939			1.18335			0.591676	+	0.806176i	$0.298466\pi$
$618$		0			0
$619$	29.0000			1.16561			0.582804	−	0.812613i	$-0.301955\pi$
$619$	29.0000			1.16561			0.582804	+	0.812613i	$0.301955\pi$
$620$	4.89898	+	2.82843i	0.196748	+	0.113592i
$621$		0			0
$622$	24.0000	−	13.8564i	0.962312	−	0.555591i
$623$		0			0
$624$		0			0
$625$	−31.0000			−1.24000
$626$	15.9217	+	27.5772i	0.636358	+	1.10221i
$627$		0			0
$628$	12.0000	+	6.92820i	0.478852	+	0.276465i
$629$		−	14.1421i		−	0.563884i
$630$		0			0
$631$			10.3923i			0.413711i	0.978371	+	0.206856i	$0.0663230\pi$
$631$			10.3923i			0.413711i	−0.978371	+	0.206856i	$0.933677\pi$
$632$	4.89898			0.194871
$633$		0			0
$634$	−24.0000	+	13.8564i	−0.953162	+	0.550308i
$635$	−44.0908			−1.74969
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	−16.0000	−	27.7128i	−0.632456	−	1.09545i
$641$	−4.89898			−0.193498			−0.0967490	−	0.995309i	$-0.530844\pi$
$641$	−4.89898			−0.193498			−0.0967490	+	0.995309i	$0.530844\pi$
$642$		0			0
$643$	29.0000			1.14365			0.571824	−	0.820376i	$-0.306236\pi$
$643$	29.0000			1.14365			0.571824	+	0.820376i	$0.306236\pi$
$644$		0			0
$645$		0			0
$646$	10.0000	+	17.3205i	0.393445	+	0.681466i
$647$	−19.5959			−0.770395			−0.385198	−	0.922834i	$-0.625867\pi$
$647$	−19.5959			−0.770395			−0.385198	+	0.922834i	$0.625867\pi$
$648$		0			0
$649$		−	27.7128i		−	1.08782i
$650$	−11.0227	−	19.0919i	−0.432346	−	0.748845i
$651$		0			0
$652$	6.00000	+	3.46410i	0.234978	+	0.135665i
$653$	19.5959			0.766848			0.383424	−	0.923573i	$-0.374745\pi$
$653$	19.5959			0.766848			0.383424	+	0.923573i	$0.374745\pi$
$654$		0			0
$655$		−	55.4256i		−	2.16566i
$656$	19.5959	−	11.3137i	0.765092	−	0.441726i
$657$		0			0
$658$		0			0
$659$			14.1421i			0.550899i	0.961315	+	0.275450i	$0.0888267\pi$
$659$			14.1421i			0.550899i	−0.961315	+	0.275450i	$0.911173\pi$
$660$		0			0
$661$			32.9090i			1.28001i	0.768371	+	0.640005i	$0.221068\pi$
$661$			32.9090i			1.28001i	−0.768371	+	0.640005i	$0.778932\pi$
$662$	−20.8207	−	36.0624i	−0.809218	−	1.40161i
$663$		0			0
$664$		−	41.5692i		−	1.61320i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	14.6969	−	25.4558i	0.568642	−	0.984916i
$669$		0			0
$670$	30.0000	−	17.3205i	1.15900	−	0.669150i
$671$	39.1918			1.51298
$672$		0			0
$673$	−19.0000			−0.732396			−0.366198	−	0.930537i	$-0.619341\pi$
$673$	−19.0000			−0.732396			−0.366198	+	0.930537i	$0.619341\pi$
$674$	−35.5176	+	20.5061i	−1.36809	+	0.789865i
$675$		0			0
$676$	−14.0000	+	24.2487i	−0.538462	+	0.932643i
$677$			5.65685i			0.217411i	0.994074	+	0.108705i	$0.0346705\pi$
$677$			5.65685i			0.217411i	−0.994074	+	0.108705i	$0.965330\pi$
$678$		0			0
$679$		0			0
$680$		−	22.6274i		−	0.867722i
$681$		0			0
$682$	4.00000	+	6.92820i	0.153168	+	0.265295i
$683$		−	36.7696i		−	1.40695i	−0.710721	−	0.703474i	$-0.751631\pi$
$683$		−	36.7696i		−	1.40695i	0.710721	−	0.703474i	$-0.248369\pi$
$684$		0			0
$685$		−	27.7128i		−	1.05885i
$686$		0			0
$687$		0			0
$688$	−18.0000	+	10.3923i	−0.686244	+	0.396203i
$689$			25.4558i			0.969790i
$690$		0			0
$691$	5.00000			0.190209			0.0951045	−	0.995467i	$-0.469681\pi$
$691$	5.00000			0.190209			0.0951045	+	0.995467i	$0.469681\pi$
$692$	−34.2929	−	19.7990i	−1.30362	−	0.752645i
$693$		0			0
$694$	2.00000	+	3.46410i	0.0759190	+	0.131495i
$695$			36.7696i			1.39475i
$696$		0			0
$697$	16.0000			0.606043
$698$	14.6969	+	25.4558i	0.556287	+	0.963518i
$699$		0			0
$700$		0			0
$701$	−44.0908			−1.66529			−0.832644	−	0.553809i	$-0.813174\pi$
$701$	−44.0908			−1.66529			−0.832644	+	0.553809i	$0.813174\pi$
$702$		0			0
$703$	25.0000			0.942893
$704$		−	45.2548i		−	1.70561i
$705$		0			0
$706$	−22.0000	−	38.1051i	−0.827981	−	1.43411i
$707$		0			0
$708$		0			0
$709$	2.00000			0.0751116			0.0375558	−	0.999295i	$-0.488043\pi$
$709$	2.00000			0.0751116			0.0375558	+	0.999295i	$0.488043\pi$
$710$	9.79796	−	5.65685i	0.367711	−	0.212298i
$711$		0			0
$712$	32.0000			1.19925
$713$			2.82843i			0.105925i
$714$		0			0
$715$		−	83.1384i		−	3.10920i
$716$	−4.89898	−	2.82843i	−0.183083	−	0.105703i
$717$		0			0
$718$	−16.0000	−	27.7128i	−0.597115	−	1.03423i
$719$	−39.1918			−1.46161			−0.730804	−	0.682587i	$-0.760855\pi$
$719$	−39.1918			−1.46161			−0.730804	+	0.682587i	$0.760855\pi$
$720$		0			0
$721$		0			0
$722$	−7.34847	+	4.24264i	−0.273482	+	0.157895i
$723$		0			0
$724$	9.00000	+	5.19615i	0.334482	+	0.193113i
$725$		0			0
$726$		0			0
$727$	17.0000			0.630495			0.315248	−	0.949009i	$-0.397912\pi$
$727$	17.0000			0.630495			0.315248	+	0.949009i	$0.397912\pi$
$728$		0			0
$729$		0			0
$730$	−6.00000	+	3.46410i	−0.222070	+	0.128212i
$731$	−14.6969			−0.543586
$732$		0			0
$733$			8.66025i			0.319874i	0.987127	+	0.159937i	$0.0511291\pi$
$733$			8.66025i			0.319874i	−0.987127	+	0.159937i	$0.948871\pi$
$734$	−13.4722	+	7.77817i	−0.497268	+	0.287098i
$735$		0			0
$736$	8.00000	−	13.8564i	0.294884	−	0.510754i
$737$	48.9898			1.80456
$738$		0			0
$739$			32.9090i			1.21058i	0.796007	+	0.605288i	$0.206942\pi$
$739$			32.9090i			1.21058i	−0.796007	+	0.605288i	$0.793058\pi$
$740$	−24.4949	−	14.1421i	−0.900450	−	0.519875i
$741$		0			0
$742$		0			0
$743$		−	45.2548i		−	1.66024i	−0.557586	−	0.830119i	$-0.688272\pi$
$743$		−	45.2548i		−	1.66024i	0.557586	−	0.830119i	$-0.311728\pi$
$744$		0			0
$745$			27.7128i			1.01532i
$746$	1.22474	−	0.707107i	0.0448411	−	0.0258890i
$747$		0			0
$748$	16.0000	−	27.7128i	0.585018	−	1.01328i
$749$		0			0
$750$		0			0
$751$			8.66025i			0.316017i	0.987438	+	0.158009i	$0.0505074\pi$
$751$			8.66025i			0.316017i	−0.987438	+	0.158009i	$0.949493\pi$
$752$	−9.79796	−	16.9706i	−0.357295	−	0.618853i
$753$		0			0
$754$		0			0
$755$	48.9898			1.78292
$756$		0			0
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$	−25.7196	−	44.5477i	−0.934179	−	1.61805i
$759$		0			0
$760$	40.0000			1.45095
$761$		−	19.7990i		−	0.717713i	−0.933393	−	0.358856i	$-0.883167\pi$
$761$		−	19.7990i		−	0.717713i	0.933393	−	0.358856i	$-0.116833\pi$
$762$		0			0
$763$		0			0
$764$	−34.2929	−	19.7990i	−1.24067	−	0.716302i
$765$		0			0
$766$	−42.0000	+	24.2487i	−1.51752	+	0.876142i
$767$			25.4558i			0.919157i
$768$		0			0
$769$			15.5885i			0.562134i	0.959688	+	0.281067i	$0.0906883\pi$
$769$			15.5885i			0.562134i	−0.959688	+	0.281067i	$0.909312\pi$
$770$		0			0
$771$		0			0
$772$	−7.00000	+	12.1244i	−0.251936	+	0.436365i
$773$		−	11.3137i		−	0.406926i	−0.979083	−	0.203463i	$-0.934780\pi$
$773$		−	11.3137i		−	0.406926i	0.979083	−	0.203463i	$-0.0652196\pi$
$774$		0			0
$775$	3.00000			0.107763
$776$		0			0
$777$		0			0
$778$	−30.0000	+	17.3205i	−1.07555	+	0.620970i
$779$			28.2843i			1.01339i
$780$		0			0
$781$	16.0000			0.572525
$782$	9.79796	−	5.65685i	0.350374	−	0.202289i
$783$		0			0
$784$		0			0
$785$	19.5959			0.699408
$786$		0			0
$787$	−16.0000			−0.570338			−0.285169	−	0.958477i	$-0.592050\pi$
$787$	−16.0000			−0.570338			−0.285169	+	0.958477i	$0.592050\pi$
$788$	14.6969	−	25.4558i	0.523557	−	0.906827i
$789$		0			0
$790$	6.00000	−	3.46410i	0.213470	−	0.123247i
$791$		0			0
$792$		0			0
$793$	−36.0000			−1.27840
$794$	−20.8207	−	36.0624i	−0.738898	−	1.27981i
$795$		0			0
$796$	−4.00000	+	6.92820i	−0.141776	+	0.245564i
$797$			31.1127i			1.10207i	0.834483	+	0.551034i	$0.185767\pi$
$797$			31.1127i			1.10207i	−0.834483	+	0.551034i	$0.814233\pi$
$798$		0			0
$799$		−	13.8564i		−	0.490204i
$800$	−14.6969	−	8.48528i	−0.519615	−	0.300000i
$801$		0			0
$802$	−6.00000	+	3.46410i	−0.211867	+	0.122322i
$803$	−9.79796			−0.345762
$804$		0			0
$805$		0			0
$806$	−3.67423	−	6.36396i	−0.129419	−	0.224161i
$807$		0			0
$808$	32.0000			1.12576
$809$	−4.89898			−0.172239			−0.0861195	−	0.996285i	$-0.527447\pi$
$809$	−4.89898			−0.172239			−0.0861195	+	0.996285i	$0.527447\pi$
$810$		0			0
$811$	8.00000			0.280918			0.140459	−	0.990086i	$-0.455142\pi$
$811$	8.00000			0.280918			0.140459	+	0.990086i	$0.455142\pi$
$812$		0			0
$813$		0			0
$814$	−20.0000	−	34.6410i	−0.701000	−	1.21417i
$815$	9.79796			0.343208
$816$		0			0
$817$		−	25.9808i		−	0.908952i
$818$	20.8207	+	36.0624i	0.727977	+	1.26089i
$819$		0			0
$820$	16.0000	−	27.7128i	0.558744	−	0.967773i
$821$	34.2929			1.19683			0.598414	−	0.801187i	$-0.295798\pi$
$821$	34.2929			1.19683			0.598414	+	0.801187i	$0.295798\pi$
$822$		0			0
$823$		−	45.0333i		−	1.56976i	−0.619646	−	0.784881i	$-0.712724\pi$
$823$		−	45.0333i		−	1.56976i	0.619646	−	0.784881i	$-0.287276\pi$
$824$		−	2.82843i		−	0.0985329i
$825$		0			0
$826$		0			0
$827$		−	2.82843i		−	0.0983540i	−0.998790	−	0.0491770i	$-0.984340\pi$
$827$		−	2.82843i		−	0.0983540i	0.998790	−	0.0491770i	$-0.0156598\pi$
$828$		0			0
$829$			32.9090i			1.14298i	0.820611	+	0.571488i	$0.193634\pi$
$829$			32.9090i			1.14298i	−0.820611	+	0.571488i	$0.806366\pi$
$830$	−29.3939	−	50.9117i	−1.02028	−	1.76717i
$831$		0			0
$832$			41.5692i			1.44115i
$833$		0			0
$834$		0			0
$835$		−	41.5692i		−	1.43856i
$836$	48.9898	+	28.2843i	1.69435	+	0.978232i
$837$		0			0
$838$		0			0
$839$	29.3939			1.01479			0.507395	−	0.861714i	$-0.330609\pi$
$839$	29.3939			1.01479			0.507395	+	0.861714i	$0.330609\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$	−28.1691	+	16.2635i	−0.970772	+	0.560476i
$843$		0			0
$844$	18.0000	+	10.3923i	0.619586	+	0.357718i
$845$			39.5980i			1.36221i
$846$		0			0
$847$		0			0
$848$	9.79796	+	16.9706i	0.336463	+	0.582772i
$849$		0			0
$850$	−6.00000	−	10.3923i	−0.205798	−	0.356453i
$851$		−	14.1421i		−	0.484786i
$852$		0			0
$853$		−	15.5885i		−	0.533739i	−0.963733	−	0.266869i	$-0.914011\pi$
$853$		−	15.5885i		−	0.533739i	0.963733	−	0.266869i	$-0.0859892\pi$
$854$		0			0
$855$		0			0
$856$	8.00000			0.273434
$857$			5.65685i			0.193234i	0.995322	+	0.0966172i	$0.0308023\pi$
$857$			5.65685i			0.193234i	−0.995322	+	0.0966172i	$0.969198\pi$
$858$		0			0
$859$	8.00000			0.272956			0.136478	−	0.990643i	$-0.456422\pi$
$859$	8.00000			0.272956			0.136478	+	0.990643i	$0.456422\pi$
$860$	−14.6969	+	25.4558i	−0.501161	+	0.868037i
$861$		0			0
$862$	−10.0000	−	17.3205i	−0.340601	−	0.589939i
$863$		−	28.2843i		−	0.962808i	−0.876499	−	0.481404i	$-0.840127\pi$
$863$		−	28.2843i		−	0.962808i	0.876499	−	0.481404i	$-0.159873\pi$
$864$		0			0
$865$	−56.0000			−1.90406
$866$	3.67423	+	6.36396i	0.124856	+	0.216256i
$867$		0			0
$868$		0			0
$869$	9.79796			0.332373
$870$		0			0
$871$	−45.0000			−1.52477
$872$			14.1421i			0.478913i
$873$		0			0
$874$	10.0000	+	17.3205i	0.338255	+	0.585875i
$875$		0			0
$876$		0			0
$877$	38.0000			1.28317			0.641584	−	0.767052i	$-0.278277\pi$
$877$	38.0000			1.28317			0.641584	+	0.767052i	$0.278277\pi$
$878$	4.89898	−	2.82843i	0.165333	−	0.0954548i
$879$		0			0
$880$	−32.0000	−	55.4256i	−1.07872	−	1.86840i
$881$		−	53.7401i		−	1.81055i	−0.424826	−	0.905275i	$-0.639665\pi$
$881$		−	53.7401i		−	1.81055i	0.424826	−	0.905275i	$-0.360335\pi$
$882$		0			0
$883$			15.5885i			0.524593i	0.964987	+	0.262297i	$0.0844799\pi$
$883$			15.5885i			0.524593i	−0.964987	+	0.262297i	$0.915520\pi$
$884$	−14.6969	+	25.4558i	−0.494312	+	0.856173i
$885$		0			0
$886$	−22.0000	−	38.1051i	−0.739104	−	1.28017i
$887$	−39.1918			−1.31593			−0.657967	−	0.753047i	$-0.728583\pi$
$887$	−39.1918			−1.31593			−0.657967	+	0.753047i	$0.728583\pi$
$888$		0			0
$889$		0			0
$890$	39.1918	−	22.6274i	1.31371	−	0.758473i
$891$		0			0
$892$	−4.00000	+	6.92820i	−0.133930	+	0.231973i
$893$	24.4949			0.819690
$894$		0			0
$895$	−8.00000			−0.267411
$896$		0			0
$897$		0			0
$898$	18.0000	−	10.3923i	0.600668	−	0.346796i
$899$		0			0
$900$		0			0
$901$			13.8564i			0.461624i
$902$	39.1918	−	22.6274i	1.30495	−	0.753411i
$903$		0			0
$904$		−	41.5692i		−	1.38257i
$905$	14.6969			0.488543
$906$		0			0
$907$			22.5167i			0.747653i	0.927499	+	0.373827i	$0.121955\pi$
$907$			22.5167i			0.747653i	−0.927499	+	0.373827i	$0.878045\pi$
$908$	9.79796	−	16.9706i	0.325157	−	0.563188i
$909$		0			0
$910$		0			0
$911$		−	2.82843i		−	0.0937100i	−0.998902	−	0.0468550i	$-0.985080\pi$
$911$		−	2.82843i		−	0.0937100i	0.998902	−	0.0468550i	$-0.0149199\pi$
$912$		0			0
$913$		−	83.1384i		−	2.75148i
$914$	8.57321	−	4.94975i	0.283577	−	0.163723i
$915$		0			0
$916$	33.0000	+	19.0526i	1.09035	+	0.629514i
$917$		0			0
$918$		0			0
$919$			50.2295i			1.65692i	0.560050	+	0.828459i	$0.310782\pi$
$919$			50.2295i			1.65692i	−0.560050	+	0.828459i	$0.689218\pi$
$920$		−	22.6274i		−	0.746004i
$921$		0			0
$922$	8.00000	+	13.8564i	0.263466	+	0.456336i
$923$	−14.6969			−0.483756
$924$		0			0
$925$	−15.0000			−0.493197
$926$	−3.67423	−	6.36396i	−0.120743	−	0.209133i
$927$		0			0
$928$		0			0
$929$			14.1421i			0.463988i	0.972717	+	0.231994i	$0.0745250\pi$
$929$			14.1421i			0.463988i	−0.972717	+	0.231994i	$0.925475\pi$
$930$		0			0
$931$		0			0
$932$	−24.4949	+	42.4264i	−0.802357	+	1.38972i
$933$		0			0
$934$	−6.00000	+	3.46410i	−0.196326	+	0.113349i
$935$		−	45.2548i		−	1.47999i
$936$		0			0
$937$		−	36.3731i		−	1.18826i	−0.804370	−	0.594128i	$-0.797497\pi$
$937$		−	36.3731i		−	1.18826i	0.804370	−	0.594128i	$-0.202503\pi$
$938$		0			0
$939$		0			0
$940$	−24.0000	−	13.8564i	−0.782794	−	0.451946i
$941$			48.0833i			1.56747i	0.621096	+	0.783735i	$0.286688\pi$
$941$			48.0833i			1.56747i	−0.621096	+	0.783735i	$0.713312\pi$
$942$		0			0
$943$	16.0000			0.521032
$944$	9.79796	+	16.9706i	0.318896	+	0.552345i
$945$		0			0
$946$	−36.0000	+	20.7846i	−1.17046	+	0.675766i
$947$			5.65685i			0.183823i	0.995767	+	0.0919115i	$0.0292977\pi$
$947$			5.65685i			0.183823i	−0.995767	+	0.0919115i	$0.970702\pi$
$948$		0			0
$949$	9.00000			0.292152
$950$	18.3712	−	10.6066i	0.596040	−	0.344124i
$951$		0			0
$952$		0			0
$953$	44.0908			1.42824			0.714121	−	0.700022i	$-0.246827\pi$
$953$	44.0908			1.42824			0.714121	+	0.700022i	$0.246827\pi$
$954$		0			0
$955$	−56.0000			−1.81212
$956$	−19.5959	−	11.3137i	−0.633777	−	0.365911i
$957$		0			0
$958$	42.0000	−	24.2487i	1.35696	−	0.783440i
$959$		0			0
$960$		0			0
$961$	−30.0000			−0.967742
$962$	18.3712	+	31.8198i	0.592310	+	1.02591i
$963$		0			0
$964$	−24.0000	−	13.8564i	−0.772988	−	0.446285i
$965$			19.7990i			0.637352i
$966$		0			0
$967$			25.9808i			0.835485i	0.908565	+	0.417742i	$0.137179\pi$
$967$			25.9808i			0.835485i	−0.908565	+	0.417742i	$0.862821\pi$
$968$		−	59.3970i		−	1.90909i
$969$		0			0
$970$		0			0
$971$	34.2929			1.10051			0.550255	−	0.834997i	$-0.314530\pi$
$971$	34.2929			1.10051			0.550255	+	0.834997i	$0.314530\pi$
$972$		0			0
$973$		0			0
$974$	−15.9217	−	27.5772i	−0.510164	−	0.883629i
$975$		0			0
$976$	−24.0000	+	13.8564i	−0.768221	+	0.443533i
$977$	−19.5959			−0.626929			−0.313464	−	0.949600i	$-0.601490\pi$
$977$	−19.5959			−0.626929			−0.313464	+	0.949600i	$0.601490\pi$
$978$		0			0
$979$	64.0000			2.04545
$980$		0			0
$981$		0			0
$982$	−4.00000	−	6.92820i	−0.127645	−	0.221088i
$983$	24.4949			0.781266			0.390633	−	0.920547i	$-0.372256\pi$
$983$	24.4949			0.781266			0.390633	+	0.920547i	$0.372256\pi$
$984$		0			0
$985$		−	41.5692i		−	1.32451i
$986$		0			0
$987$		0			0
$988$	−45.0000	−	25.9808i	−1.43164	−	0.826558i
$989$	−14.6969			−0.467335
$990$		0			0
$991$			22.5167i			0.715265i	0.933862	+	0.357633i	$0.116416\pi$
$991$			22.5167i			0.715265i	−0.933862	+	0.357633i	$0.883584\pi$
$992$	−4.89898	−	2.82843i	−0.155543	−	0.0898027i
$993$		0			0
$994$		0			0
$995$			11.3137i			0.358669i
$996$		0			0
$997$		−	19.0526i		−	0.603401i	−0.953403	−	0.301700i	$-0.902446\pi$
$997$		−	19.0526i		−	0.603401i	0.953403	−	0.301700i	$-0.0975542\pi$
$998$	23.2702	+	40.3051i	0.736604	+	1.27584i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.2.b.d.1567.2		4
3.2	odd	2	inner	1764.2.b.d.1567.3		4
4.3	odd	2		1764.2.b.c.1567.1		4
7.4	even	3		252.2.bf.b.19.1	✓	4
7.5	odd	6		252.2.bf.c.199.2	yes	4
7.6	odd	2		1764.2.b.c.1567.2		4
12.11	even	2		1764.2.b.c.1567.4		4
21.5	even	6		252.2.bf.c.199.1	yes	4
21.11	odd	6		252.2.bf.b.19.2	yes	4
21.20	even	2		1764.2.b.c.1567.3		4
28.11	odd	6		252.2.bf.c.19.2	yes	4
28.19	even	6		252.2.bf.b.199.2	yes	4
28.27	even	2	inner	1764.2.b.d.1567.1		4
84.11	even	6		252.2.bf.c.19.1	yes	4
84.47	odd	6		252.2.bf.b.199.1	yes	4
84.83	odd	2	inner	1764.2.b.d.1567.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.bf.b.19.1	✓	4	7.4	even	3
252.2.bf.b.19.2	yes	4	21.11	odd	6
252.2.bf.b.199.1	yes	4	84.47	odd	6
252.2.bf.b.199.2	yes	4	28.19	even	6
252.2.bf.c.19.1	yes	4	84.11	even	6
252.2.bf.c.19.2	yes	4	28.11	odd	6
252.2.bf.c.199.1	yes	4	21.5	even	6
252.2.bf.c.199.2	yes	4	7.5	odd	6
1764.2.b.c.1567.1		4	4.3	odd	2
1764.2.b.c.1567.2		4	7.6	odd	2
1764.2.b.c.1567.3		4	21.20	even	2
1764.2.b.c.1567.4		4	12.11	even	2
1764.2.b.d.1567.1		4	28.27	even	2	inner
1764.2.b.d.1567.2		4	1.1	even	1	trivial
1764.2.b.d.1567.3		4	3.2	odd	2	inner
1764.2.b.d.1567.4		4	84.83	odd	2	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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1764.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.22474 + 0.707107i) q^{2} +(1.00000 - 1.73205i) q^{4} -2.82843i q^{5} +2.82843i q^{8} +O(q^{10})\)	\(q+(-1.22474 + 0.707107i) q^{2} +(1.00000 - 1.73205i) q^{4} -2.82843i q^{5} +2.82843i q^{8} +(2.00000 + 3.46410i) q^{10} +5.65685i q^{11} -5.19615i q^{13} +(-2.00000 - 3.46410i) q^{16} -2.82843i q^{17} +5.00000 q^{19} +(-4.89898 - 2.82843i) q^{20} +(-4.00000 - 6.92820i) q^{22} -2.82843i q^{23} -3.00000 q^{25} +(3.67423 + 6.36396i) q^{26} -1.00000 q^{31} +(4.89898 + 2.82843i) q^{32} +(2.00000 + 3.46410i) q^{34} +5.00000 q^{37} +(-6.12372 + 3.53553i) q^{38} +8.00000 q^{40} +5.65685i q^{41} -5.19615i q^{43} +(9.79796 + 5.65685i) q^{44} +(2.00000 + 3.46410i) q^{46} +4.89898 q^{47} +(3.67423 - 2.12132i) q^{50} +(-9.00000 - 5.19615i) q^{52} -4.89898 q^{53} +16.0000 q^{55} -4.89898 q^{59} -6.92820i q^{61} +(1.22474 - 0.707107i) q^{62} -8.00000 q^{64} -14.6969 q^{65} -8.66025i q^{67} +(-4.89898 - 2.82843i) q^{68} -2.82843i q^{71} +1.73205i q^{73} +(-6.12372 + 3.53553i) q^{74} +(5.00000 - 8.66025i) q^{76} -1.73205i q^{79} +(-9.79796 + 5.65685i) q^{80} +(-4.00000 - 6.92820i) q^{82} -14.6969 q^{83} -8.00000 q^{85} +(3.67423 + 6.36396i) q^{86} -16.0000 q^{88} -11.3137i q^{89} +(-4.89898 - 2.82843i) q^{92} +(-6.00000 + 3.46410i) q^{94} -14.1421i q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4}+O(q^{10}) \) 4 * q + 4 * q^4	\( 4 q + 4 q^{4} + 8 q^{10} - 8 q^{16} + 20 q^{19} - 16 q^{22} - 12 q^{25} - 4 q^{31} + 8 q^{34} + 20 q^{37} + 32 q^{40} + 8 q^{46} - 36 q^{52} + 64 q^{55} - 32 q^{64} + 20 q^{76} - 16 q^{82} - 32 q^{85} - 64 q^{88} - 24 q^{94}+O(q^{100}) \) 4 * q + 4 * q^4 + 8 * q^10 - 8 * q^16 + 20 * q^19 - 16 * q^22 - 12 * q^25 - 4 * q^31 + 8 * q^34 + 20 * q^37 + 32 * q^40 + 8 * q^46 - 36 * q^52 + 64 * q^55 - 32 * q^64 + 20 * q^76 - 16 * q^82 - 32 * q^85 - 64 * q^88 - 24 * q^94

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