LMFDB - Embedded newform 675.2.f.b.107.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,2,Mod(107,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1]))

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

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

chi := DirichletCharacter("675.107");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	675.f (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:	$5.38990213644$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			107.1
Root			$-1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	675.107
Dual form			675.2.f.b.593.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.22474 + 1.22474i) q^{2} -1.00000i q^{4} +(1.22474 + 1.22474i) q^{7} +(-1.22474 - 1.22474i) q^{8} +O(q^{10})$	\(q+(-1.22474 + 1.22474i) q^{2} -1.00000i q^{4} +(1.22474 + 1.22474i) q^{7} +(-1.22474 - 1.22474i) q^{8} -6.00000i q^{11} -3.00000 q^{14} +5.00000 q^{16} +(-2.44949 + 2.44949i) q^{17} -7.00000i q^{19} +(7.34847 + 7.34847i) q^{22} +(-4.89898 - 4.89898i) q^{23} +(1.22474 - 1.22474i) q^{28} +6.00000 q^{29} -5.00000 q^{31} +(-3.67423 + 3.67423i) q^{32} -6.00000i q^{34} +(-1.22474 - 1.22474i) q^{37} +(8.57321 + 8.57321i) q^{38} +6.00000i q^{41} +(6.12372 - 6.12372i) q^{43} -6.00000 q^{44} +12.0000 q^{46} +(7.34847 - 7.34847i) q^{47} -4.00000i q^{49} +(-2.44949 - 2.44949i) q^{53} -3.00000i q^{56} +(-7.34847 + 7.34847i) q^{58} -1.00000 q^{61} +(6.12372 - 6.12372i) q^{62} +1.00000i q^{64} +(7.34847 + 7.34847i) q^{67} +(2.44949 + 2.44949i) q^{68} +12.0000i q^{71} +(6.12372 - 6.12372i) q^{73} +3.00000 q^{74} -7.00000 q^{76} +(7.34847 - 7.34847i) q^{77} +1.00000i q^{79} +(-7.34847 - 7.34847i) q^{82} +(2.44949 + 2.44949i) q^{83} +15.0000i q^{86} +(-7.34847 + 7.34847i) q^{88} +12.0000 q^{89} +(-4.89898 + 4.89898i) q^{92} +18.0000i q^{94} +(-8.57321 - 8.57321i) q^{97} +(4.89898 + 4.89898i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 12 q^{14} + 20 q^{16} + 24 q^{29} - 20 q^{31} - 24 q^{44} + 48 q^{46} - 4 q^{61} + 12 q^{74} - 28 q^{76} + 48 q^{89}+O(q^{100}) $ 4 * q - 12 * q^14 + 20 * q^16 + 24 * q^29 - 20 * q^31 - 24 * q^44 + 48 * q^46 - 4 * q^61 + 12 * q^74 - 28 * q^76 + 48 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$

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	−0.866025	+	0.866025i	−0.992030	−	0.126004i	$-0.959785\pi$
$2$	−1.22474	+	1.22474i	−0.866025	+	0.866025i	0.126004	+	0.992030i	$0.459785\pi$
$3$		0			0
$3$		0			0
$4$		−	1.00000i		−	0.500000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.22474	+	1.22474i	0.462910	+	0.462910i	0.899608	−	0.436698i	$-0.143852\pi$
$7$	1.22474	+	1.22474i	0.462910	+	0.462910i	−0.436698	+	0.899608i	$0.643852\pi$
$8$	−1.22474	−	1.22474i	−0.433013	−	0.433013i
$9$		0			0
$10$		0			0
$11$		−	6.00000i		−	1.80907i	−0.426401	−	0.904534i	$-0.640219\pi$
$11$		−	6.00000i		−	1.80907i	0.426401	−	0.904534i	$-0.359781\pi$
$12$		0			0
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$	−3.00000			−0.801784
$15$		0			0
$16$	5.00000			1.25000
$17$	−2.44949	+	2.44949i	−0.594089	+	0.594089i	−0.938733	−	0.344645i	$-0.887999\pi$
$17$	−2.44949	+	2.44949i	−0.594089	+	0.594089i	0.344645	+	0.938733i	$0.387999\pi$
$18$		0			0
$19$		−	7.00000i		−	1.60591i	−0.596040	−	0.802955i	$-0.703260\pi$
$19$		−	7.00000i		−	1.60591i	0.596040	−	0.802955i	$-0.296740\pi$
$20$		0			0
$21$		0			0
$22$	7.34847	+	7.34847i	1.56670	+	1.56670i
$23$	−4.89898	−	4.89898i	−1.02151	−	1.02151i	−0.999764	−	0.0217443i	$-0.993078\pi$
$23$	−4.89898	−	4.89898i	−1.02151	−	1.02151i	−0.0217443	−	0.999764i	$-0.506922\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$	1.22474	−	1.22474i	0.231455	−	0.231455i
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$	−5.00000			−0.898027			−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000			−0.898027			−0.449013	+	0.893525i	$0.648224\pi$
$32$	−3.67423	+	3.67423i	−0.649519	+	0.649519i
$33$		0			0
$34$		−	6.00000i		−	1.02899i
$35$		0			0
$36$		0			0
$37$	−1.22474	−	1.22474i	−0.201347	−	0.201347i	0.599230	−	0.800577i	$-0.295473\pi$
$37$	−1.22474	−	1.22474i	−0.201347	−	0.201347i	−0.800577	+	0.599230i	$0.795473\pi$
$38$	8.57321	+	8.57321i	1.39076	+	1.39076i
$39$		0			0
$40$		0			0
$41$			6.00000i			0.937043i	0.883452	+	0.468521i	$0.155213\pi$
$41$			6.00000i			0.937043i	−0.883452	+	0.468521i	$0.844787\pi$
$42$		0			0
$43$	6.12372	−	6.12372i	0.933859	−	0.933859i	−0.0640852	−	0.997944i	$-0.520413\pi$
$43$	6.12372	−	6.12372i	0.933859	−	0.933859i	0.997944	+	0.0640852i	$0.0204129\pi$
$44$	−6.00000			−0.904534
$45$		0			0
$46$	12.0000			1.76930
$47$	7.34847	−	7.34847i	1.07188	−	1.07188i	0.0746766	−	0.997208i	$-0.476208\pi$
$47$	7.34847	−	7.34847i	1.07188	−	1.07188i	0.997208	−	0.0746766i	$-0.0237924\pi$
$48$		0			0
$49$		−	4.00000i		−	0.571429i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.44949	−	2.44949i	−0.336463	−	0.336463i	0.518571	−	0.855034i	$-0.326464\pi$
$53$	−2.44949	−	2.44949i	−0.336463	−	0.336463i	−0.855034	+	0.518571i	$0.826464\pi$
$54$		0			0
$55$		0			0
$56$		−	3.00000i		−	0.400892i
$57$		0			0
$58$	−7.34847	+	7.34847i	−0.964901	+	0.964901i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	−1.00000			−0.128037			−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000			−0.128037			−0.0640184	+	0.997949i	$0.520392\pi$
$62$	6.12372	−	6.12372i	0.777714	−	0.777714i
$63$		0			0
$64$			1.00000i			0.125000i
$65$		0			0
$66$		0			0
$67$	7.34847	+	7.34847i	0.897758	+	0.897758i	0.995238	−	0.0974792i	$-0.0310779\pi$
$67$	7.34847	+	7.34847i	0.897758	+	0.897758i	−0.0974792	+	0.995238i	$0.531078\pi$
$68$	2.44949	+	2.44949i	0.297044	+	0.297044i
$69$		0			0
$70$		0			0
$71$			12.0000i			1.42414i	0.702109	+	0.712069i	$0.252242\pi$
$71$			12.0000i			1.42414i	−0.702109	+	0.712069i	$0.747758\pi$
$72$		0			0
$73$	6.12372	−	6.12372i	0.716728	−	0.716728i	−0.251206	−	0.967934i	$-0.580827\pi$
$73$	6.12372	−	6.12372i	0.716728	−	0.716728i	0.967934	+	0.251206i	$0.0808271\pi$
$74$	3.00000			0.348743
$75$		0			0
$76$	−7.00000			−0.802955
$77$	7.34847	−	7.34847i	0.837436	−	0.837436i
$78$		0			0
$79$			1.00000i			0.112509i	0.998416	+	0.0562544i	$0.0179158\pi$
$79$			1.00000i			0.112509i	−0.998416	+	0.0562544i	$0.982084\pi$
$80$		0			0
$81$		0			0
$82$	−7.34847	−	7.34847i	−0.811503	−	0.811503i
$83$	2.44949	+	2.44949i	0.268866	+	0.268866i	0.828643	−	0.559777i	$-0.189113\pi$
$83$	2.44949	+	2.44949i	0.268866	+	0.268866i	−0.559777	+	0.828643i	$0.689113\pi$
$84$		0			0
$85$		0			0
$86$			15.0000i			1.61749i
$87$		0			0
$88$	−7.34847	+	7.34847i	−0.783349	+	0.783349i
$89$	12.0000			1.27200			0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000			1.27200			0.635999	+	0.771690i	$0.280588\pi$
$90$		0			0
$91$		0			0
$92$	−4.89898	+	4.89898i	−0.510754	+	0.510754i
$93$		0			0
$94$			18.0000i			1.85656i
$95$		0			0
$96$		0			0
$97$	−8.57321	−	8.57321i	−0.870478	−	0.870478i	0.122046	−	0.992524i	$-0.461054\pi$
$97$	−8.57321	−	8.57321i	−0.870478	−	0.870478i	−0.992524	+	0.122046i	$0.961054\pi$
$98$	4.89898	+	4.89898i	0.494872	+	0.494872i
$99$		0			0
$100$		0			0
$101$		−	6.00000i		−	0.597022i	−0.954406	−	0.298511i	$-0.903510\pi$
$101$		−	6.00000i		−	0.597022i	0.954406	−	0.298511i	$-0.0964900\pi$
$102$		0			0
$103$	−6.12372	+	6.12372i	−0.603388	+	0.603388i	−0.941210	−	0.337822i	$-0.890310\pi$
$103$	−6.12372	+	6.12372i	−0.603388	+	0.603388i	0.337822	+	0.941210i	$0.390310\pi$
$104$		0			0
$105$		0			0
$106$	6.00000			0.582772
$107$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$107$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$108$		0			0
$109$			7.00000i			0.670478i	0.942133	+	0.335239i	$0.108817\pi$
$109$			7.00000i			0.670478i	−0.942133	+	0.335239i	$0.891183\pi$
$110$		0			0
$111$		0			0
$112$	6.12372	+	6.12372i	0.578638	+	0.578638i
$113$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$113$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$114$		0			0
$115$		0			0
$116$		−	6.00000i		−	0.557086i
$117$		0			0
$118$		0			0
$119$	−6.00000			−0.550019
$120$		0			0
$121$	−25.0000			−2.27273
$122$	1.22474	−	1.22474i	0.110883	−	0.110883i
$123$		0			0
$124$			5.00000i			0.449013i
$125$		0			0
$126$		0			0
$127$	−7.34847	−	7.34847i	−0.652071	−	0.652071i	0.301420	−	0.953491i	$-0.402539\pi$
$127$	−7.34847	−	7.34847i	−0.652071	−	0.652071i	−0.953491	+	0.301420i	$0.902539\pi$
$128$	−8.57321	−	8.57321i	−0.757772	−	0.757772i
$129$		0			0
$130$		0			0
$131$		−	18.0000i		−	1.57267i	−0.617802	−	0.786334i	$-0.711977\pi$
$131$		−	18.0000i		−	1.57267i	0.617802	−	0.786334i	$-0.288023\pi$
$132$		0			0
$133$	8.57321	−	8.57321i	0.743392	−	0.743392i
$134$	−18.0000			−1.55496
$135$		0			0
$136$	6.00000			0.514496
$137$	−2.44949	+	2.44949i	−0.209274	+	0.209274i	−0.803959	−	0.594685i	$-0.797277\pi$
$137$	−2.44949	+	2.44949i	−0.209274	+	0.209274i	0.594685	+	0.803959i	$0.297277\pi$
$138$		0			0
$139$		−	7.00000i		−	0.593732i	−0.954919	−	0.296866i	$-0.904058\pi$
$139$		−	7.00000i		−	0.593732i	0.954919	−	0.296866i	$-0.0959415\pi$
$140$		0			0
$141$		0			0
$142$	−14.6969	−	14.6969i	−1.23334	−	1.23334i
$143$		0			0
$144$		0			0
$145$		0			0
$146$			15.0000i			1.24141i
$147$		0			0
$148$	−1.22474	+	1.22474i	−0.100673	+	0.100673i
$149$	−6.00000			−0.491539			−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000			−0.491539			−0.245770	+	0.969328i	$0.579041\pi$
$150$		0			0
$151$	−5.00000			−0.406894			−0.203447	−	0.979086i	$-0.565214\pi$
$151$	−5.00000			−0.406894			−0.203447	+	0.979086i	$0.565214\pi$
$152$	−8.57321	+	8.57321i	−0.695379	+	0.695379i
$153$		0			0
$154$			18.0000i			1.45048i
$155$		0			0
$156$		0			0
$157$	−13.4722	−	13.4722i	−1.07520	−	1.07520i	−0.996933	−	0.0782656i	$-0.975062\pi$
$157$	−13.4722	−	13.4722i	−1.07520	−	1.07520i	−0.0782656	−	0.996933i	$-0.524938\pi$
$158$	−1.22474	−	1.22474i	−0.0974355	−	0.0974355i
$159$		0			0
$160$		0			0
$161$		−	12.0000i		−	0.945732i
$162$		0			0
$163$	7.34847	−	7.34847i	0.575577	−	0.575577i	−0.358105	−	0.933681i	$-0.616577\pi$
$163$	7.34847	−	7.34847i	0.575577	−	0.575577i	0.933681	+	0.358105i	$0.116577\pi$
$164$	6.00000			0.468521
$165$		0			0
$166$	−6.00000			−0.465690
$167$	−12.2474	+	12.2474i	−0.947736	+	0.947736i	−0.998700	−	0.0509644i	$-0.983770\pi$
$167$	−12.2474	+	12.2474i	−0.947736	+	0.947736i	0.0509644	+	0.998700i	$0.483770\pi$
$168$		0			0
$169$			13.0000i			1.00000i
$170$		0			0
$171$		0			0
$172$	−6.12372	−	6.12372i	−0.466930	−	0.466930i
$173$	−9.79796	−	9.79796i	−0.744925	−	0.744925i	0.228596	−	0.973521i	$-0.426586\pi$
$173$	−9.79796	−	9.79796i	−0.744925	−	0.744925i	−0.973521	+	0.228596i	$0.926586\pi$
$174$		0			0
$175$		0			0
$176$		−	30.0000i		−	2.26134i
$177$		0			0
$178$	−14.6969	+	14.6969i	−1.10158	+	1.10158i
$179$	24.0000			1.79384			0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000			1.79384			0.896922	+	0.442189i	$0.145798\pi$
$180$		0			0
$181$	−14.0000			−1.04061			−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000			−1.04061			−0.520306	+	0.853980i	$0.674182\pi$
$182$		0			0
$183$		0			0
$184$			12.0000i			0.884652i
$185$		0			0
$186$		0			0
$187$	14.6969	+	14.6969i	1.07475	+	1.07475i
$188$	−7.34847	−	7.34847i	−0.535942	−	0.535942i
$189$		0			0
$190$		0			0
$191$			18.0000i			1.30243i	0.758891	+	0.651217i	$0.225741\pi$
$191$			18.0000i			1.30243i	−0.758891	+	0.651217i	$0.774259\pi$
$192$		0			0
$193$	−1.22474	+	1.22474i	−0.0881591	+	0.0881591i	−0.749811	−	0.661652i	$-0.769856\pi$
$193$	−1.22474	+	1.22474i	−0.0881591	+	0.0881591i	0.661652	+	0.749811i	$0.269856\pi$
$194$	21.0000			1.50771
$195$		0			0
$196$	−4.00000			−0.285714
$197$	−12.2474	+	12.2474i	−0.872595	+	0.872595i	−0.992755	−	0.120160i	$-0.961659\pi$
$197$	−12.2474	+	12.2474i	−0.872595	+	0.872595i	0.120160	+	0.992755i	$0.461659\pi$
$198$		0			0
$199$		−	20.0000i		−	1.41776i	−0.705328	−	0.708881i	$-0.749200\pi$
$199$		−	20.0000i		−	1.41776i	0.705328	−	0.708881i	$-0.250800\pi$
$200$		0			0
$201$		0			0
$202$	7.34847	+	7.34847i	0.517036	+	0.517036i
$203$	7.34847	+	7.34847i	0.515761	+	0.515761i
$204$		0			0
$205$		0			0
$206$		−	15.0000i		−	1.04510i
$207$		0			0
$208$		0			0
$209$	−42.0000			−2.90520
$210$		0			0
$211$	8.00000			0.550743			0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000			0.550743			0.275371	+	0.961338i	$0.411199\pi$
$212$	−2.44949	+	2.44949i	−0.168232	+	0.168232i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−6.12372	−	6.12372i	−0.415705	−	0.415705i
$218$	−8.57321	−	8.57321i	−0.580651	−	0.580651i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	8.57321	−	8.57321i	0.574105	−	0.574105i	−0.359168	−	0.933273i	$-0.616940\pi$
$223$	8.57321	−	8.57321i	0.574105	−	0.574105i	0.933273	+	0.359168i	$0.116940\pi$
$224$	−9.00000			−0.601338
$225$		0			0
$226$		0			0
$227$	−4.89898	+	4.89898i	−0.325157	+	0.325157i	−0.850741	−	0.525585i	$-0.823847\pi$
$227$	−4.89898	+	4.89898i	−0.325157	+	0.325157i	0.525585	+	0.850741i	$0.323847\pi$
$228$		0			0
$229$			5.00000i			0.330409i	0.986259	+	0.165205i	$0.0528285\pi$
$229$			5.00000i			0.330409i	−0.986259	+	0.165205i	$0.947172\pi$
$230$		0			0
$231$		0			0
$232$	−7.34847	−	7.34847i	−0.482451	−	0.482451i
$233$	7.34847	+	7.34847i	0.481414	+	0.481414i	0.905583	−	0.424169i	$-0.139434\pi$
$233$	7.34847	+	7.34847i	0.481414	+	0.481414i	−0.424169	+	0.905583i	$0.639434\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$	7.34847	−	7.34847i	0.476331	−	0.476331i
$239$	−6.00000			−0.388108			−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000			−0.388108			−0.194054	+	0.980991i	$0.562164\pi$
$240$		0			0
$241$	2.00000			0.128831			0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000			0.128831			0.0644157	+	0.997923i	$0.479482\pi$
$242$	30.6186	−	30.6186i	1.96824	−	1.96824i
$243$		0			0
$244$			1.00000i			0.0640184i
$245$		0			0
$246$		0			0
$247$		0			0
$248$	6.12372	+	6.12372i	0.388857	+	0.388857i
$249$		0			0
$250$		0			0
$251$			6.00000i			0.378717i	0.981908	+	0.189358i	$0.0606408\pi$
$251$			6.00000i			0.378717i	−0.981908	+	0.189358i	$0.939359\pi$
$252$		0			0
$253$	−29.3939	+	29.3939i	−1.84798	+	1.84798i
$254$	18.0000			1.12942
$255$		0			0
$256$	19.0000			1.18750
$257$	12.2474	−	12.2474i	0.763975	−	0.763975i	−0.213063	−	0.977038i	$-0.568344\pi$
$257$	12.2474	−	12.2474i	0.763975	−	0.763975i	0.977038	+	0.213063i	$0.0683441\pi$
$258$		0			0
$259$		−	3.00000i		−	0.186411i
$260$		0			0
$261$		0			0
$262$	22.0454	+	22.0454i	1.36197	+	1.36197i
$263$	−7.34847	−	7.34847i	−0.453126	−	0.453126i	0.443265	−	0.896391i	$-0.353820\pi$
$263$	−7.34847	−	7.34847i	−0.453126	−	0.453126i	−0.896391	+	0.443265i	$0.853820\pi$
$264$		0			0
$265$		0			0
$266$			21.0000i			1.28759i
$267$		0			0
$268$	7.34847	−	7.34847i	0.448879	−	0.448879i
$269$	−12.0000			−0.731653			−0.365826	−	0.930683i	$-0.619214\pi$
$269$	−12.0000			−0.731653			−0.365826	+	0.930683i	$0.619214\pi$
$270$		0			0
$271$	11.0000			0.668202			0.334101	−	0.942537i	$-0.391567\pi$
$271$	11.0000			0.668202			0.334101	+	0.942537i	$0.391567\pi$
$272$	−12.2474	+	12.2474i	−0.742611	+	0.742611i
$273$		0			0
$274$		−	6.00000i		−	0.362473i
$275$		0			0
$276$		0			0
$277$	6.12372	+	6.12372i	0.367939	+	0.367939i	0.866725	−	0.498786i	$-0.166221\pi$
$277$	6.12372	+	6.12372i	0.367939	+	0.367939i	−0.498786	+	0.866725i	$0.666221\pi$
$278$	8.57321	+	8.57321i	0.514187	+	0.514187i
$279$		0			0
$280$		0			0
$281$			6.00000i			0.357930i	0.983855	+	0.178965i	$0.0572749\pi$
$281$			6.00000i			0.357930i	−0.983855	+	0.178965i	$0.942725\pi$
$282$		0			0
$283$	1.22474	−	1.22474i	0.0728035	−	0.0728035i	−0.669767	−	0.742571i	$-0.733606\pi$
$283$	1.22474	−	1.22474i	0.0728035	−	0.0728035i	0.742571	+	0.669767i	$0.233606\pi$
$284$	12.0000			0.712069
$285$		0			0
$286$		0			0
$287$	−7.34847	+	7.34847i	−0.433766	+	0.433766i
$288$		0			0
$289$			5.00000i			0.294118i
$290$		0			0
$291$		0			0
$292$	−6.12372	−	6.12372i	−0.358364	−	0.358364i
$293$	19.5959	+	19.5959i	1.14481	+	1.14481i	0.987559	+	0.157246i	$0.0502617\pi$
$293$	19.5959	+	19.5959i	1.14481	+	1.14481i	0.157246	+	0.987559i	$0.449738\pi$
$294$		0			0
$295$		0			0
$296$			3.00000i			0.174371i
$297$		0			0
$298$	7.34847	−	7.34847i	0.425685	−	0.425685i
$299$		0			0
$300$		0			0
$301$	15.0000			0.864586
$302$	6.12372	−	6.12372i	0.352381	−	0.352381i
$303$		0			0
$304$		−	35.0000i		−	2.00739i
$305$		0			0
$306$		0			0
$307$	1.22474	+	1.22474i	0.0698999	+	0.0698999i	0.741192	−	0.671293i	$-0.234261\pi$
$307$	1.22474	+	1.22474i	0.0698999	+	0.0698999i	−0.671293	+	0.741192i	$0.734261\pi$
$308$	−7.34847	−	7.34847i	−0.418718	−	0.418718i
$309$		0			0
$310$		0			0
$311$			6.00000i			0.340229i	0.985424	+	0.170114i	$0.0544137\pi$
$311$			6.00000i			0.340229i	−0.985424	+	0.170114i	$0.945586\pi$
$312$		0			0
$313$	14.6969	−	14.6969i	0.830720	−	0.830720i	−0.156895	−	0.987615i	$-0.550148\pi$
$313$	14.6969	−	14.6969i	0.830720	−	0.830720i	0.987615	+	0.156895i	$0.0501485\pi$
$314$	33.0000			1.86230
$315$		0			0
$316$	1.00000			0.0562544
$317$	17.1464	−	17.1464i	0.963039	−	0.963039i	−0.0363015	−	0.999341i	$-0.511558\pi$
$317$	17.1464	−	17.1464i	0.963039	−	0.963039i	0.999341	+	0.0363015i	$0.0115577\pi$
$318$		0			0
$319$		−	36.0000i		−	2.01561i
$320$		0			0
$321$		0			0
$322$	14.6969	+	14.6969i	0.819028	+	0.819028i
$323$	17.1464	+	17.1464i	0.954053	+	0.954053i
$324$		0			0
$325$		0			0
$326$			18.0000i			0.996928i
$327$		0			0
$328$	7.34847	−	7.34847i	0.405751	−	0.405751i
$329$	18.0000			0.992372
$330$		0			0
$331$	5.00000			0.274825			0.137412	−	0.990514i	$-0.456121\pi$
$331$	5.00000			0.274825			0.137412	+	0.990514i	$0.456121\pi$
$332$	2.44949	−	2.44949i	0.134433	−	0.134433i
$333$		0			0
$334$		−	30.0000i		−	1.64153i
$335$		0			0
$336$		0			0
$337$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$337$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$338$	−15.9217	−	15.9217i	−0.866025	−	0.866025i
$339$		0			0
$340$		0			0
$341$			30.0000i			1.62459i
$342$		0			0
$343$	13.4722	−	13.4722i	0.727430	−	0.727430i
$344$	−15.0000			−0.808746
$345$		0			0
$346$	24.0000			1.29025
$347$	14.6969	−	14.6969i	0.788973	−	0.788973i	−0.192353	−	0.981326i	$-0.561612\pi$
$347$	14.6969	−	14.6969i	0.788973	−	0.788973i	0.981326	+	0.192353i	$0.0616118\pi$
$348$		0			0
$349$			35.0000i			1.87351i	0.349990	+	0.936754i	$0.386185\pi$
$349$			35.0000i			1.87351i	−0.349990	+	0.936754i	$0.613815\pi$
$350$		0			0
$351$		0			0
$352$	22.0454	+	22.0454i	1.17502	+	1.17502i
$353$	14.6969	+	14.6969i	0.782239	+	0.782239i	0.980208	−	0.197969i	$-0.0634346\pi$
$353$	14.6969	+	14.6969i	0.782239	+	0.782239i	−0.197969	+	0.980208i	$0.563435\pi$
$354$		0			0
$355$		0			0
$356$		−	12.0000i		−	0.635999i
$357$		0			0
$358$	−29.3939	+	29.3939i	−1.55351	+	1.55351i
$359$	−18.0000			−0.950004			−0.475002	−	0.879985i	$-0.657553\pi$
$359$	−18.0000			−0.950004			−0.475002	+	0.879985i	$0.657553\pi$
$360$		0			0
$361$	−30.0000			−1.57895
$362$	17.1464	−	17.1464i	0.901196	−	0.901196i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	22.0454	+	22.0454i	1.15076	+	1.15076i	0.986400	+	0.164361i	$0.0525561\pi$
$367$	22.0454	+	22.0454i	1.15076	+	1.15076i	0.164361	+	0.986400i	$0.447444\pi$
$368$	−24.4949	−	24.4949i	−1.27688	−	1.27688i
$369$		0			0
$370$		0			0
$371$		−	6.00000i		−	0.311504i
$372$		0			0
$373$	1.22474	−	1.22474i	0.0634149	−	0.0634149i	−0.674688	−	0.738103i	$-0.735722\pi$
$373$	1.22474	−	1.22474i	0.0634149	−	0.0634149i	0.738103	+	0.674688i	$0.235722\pi$
$374$	−36.0000			−1.86152
$375$		0			0
$376$	−18.0000			−0.928279
$377$		0			0
$378$		0			0
$379$		−	4.00000i		−	0.205466i	−0.994709	−	0.102733i	$-0.967241\pi$
$379$		−	4.00000i		−	0.205466i	0.994709	−	0.102733i	$-0.0327588\pi$
$380$		0			0
$381$		0			0
$382$	−22.0454	−	22.0454i	−1.12794	−	1.12794i
$383$	−2.44949	−	2.44949i	−0.125163	−	0.125163i	0.641750	−	0.766914i	$-0.278208\pi$
$383$	−2.44949	−	2.44949i	−0.125163	−	0.125163i	−0.766914	+	0.641750i	$0.778208\pi$
$384$		0			0
$385$		0			0
$386$		−	3.00000i		−	0.152696i
$387$		0			0
$388$	−8.57321	+	8.57321i	−0.435239	+	0.435239i
$389$	30.0000			1.52106			0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000			1.52106			0.760530	+	0.649303i	$0.224939\pi$
$390$		0			0
$391$	24.0000			1.21373
$392$	−4.89898	+	4.89898i	−0.247436	+	0.247436i
$393$		0			0
$394$		−	30.0000i		−	1.51138i
$395$		0			0
$396$		0			0
$397$	13.4722	+	13.4722i	0.676150	+	0.676150i	0.959127	−	0.282977i	$-0.0913219\pi$
$397$	13.4722	+	13.4722i	0.676150	+	0.676150i	−0.282977	+	0.959127i	$0.591322\pi$
$398$	24.4949	+	24.4949i	1.22782	+	1.22782i
$399$		0			0
$400$		0			0
$401$		−	12.0000i		−	0.599251i	−0.954057	−	0.299626i	$-0.903138\pi$
$401$		−	12.0000i		−	0.599251i	0.954057	−	0.299626i	$-0.0968618\pi$
$402$		0			0
$403$		0			0
$404$	−6.00000			−0.298511
$405$		0			0
$406$	−18.0000			−0.893325
$407$	−7.34847	+	7.34847i	−0.364250	+	0.364250i
$408$		0			0
$409$		−	10.0000i		−	0.494468i	−0.968956	−	0.247234i	$-0.920478\pi$
$409$		−	10.0000i		−	0.494468i	0.968956	−	0.247234i	$-0.0795217\pi$
$410$		0			0
$411$		0			0
$412$	6.12372	+	6.12372i	0.301694	+	0.301694i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	51.4393	−	51.4393i	2.51598	−	2.51598i
$419$	−24.0000			−1.17248			−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000			−1.17248			−0.586238	+	0.810139i	$0.699392\pi$
$420$		0			0
$421$	19.0000			0.926003			0.463002	−	0.886357i	$-0.346772\pi$
$421$	19.0000			0.926003			0.463002	+	0.886357i	$0.346772\pi$
$422$	−9.79796	+	9.79796i	−0.476957	+	0.476957i
$423$		0			0
$424$			6.00000i			0.291386i
$425$		0			0
$426$		0			0
$427$	−1.22474	−	1.22474i	−0.0592696	−	0.0592696i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	30.0000i		−	1.44505i	−0.691345	−	0.722525i	$-0.742982\pi$
$431$		−	30.0000i		−	1.44505i	0.691345	−	0.722525i	$-0.257018\pi$
$432$		0			0
$433$	−13.4722	+	13.4722i	−0.647432	+	0.647432i	−0.952372	−	0.304939i	$-0.901364\pi$
$433$	−13.4722	+	13.4722i	−0.647432	+	0.647432i	0.304939	+	0.952372i	$0.401364\pi$
$434$	15.0000			0.720023
$435$		0			0
$436$	7.00000			0.335239
$437$	−34.2929	+	34.2929i	−1.64045	+	1.64045i
$438$		0			0
$439$		−	17.0000i		−	0.811366i	−0.914014	−	0.405683i	$-0.867034\pi$
$439$		−	17.0000i		−	0.811366i	0.914014	−	0.405683i	$-0.132966\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	4.89898	+	4.89898i	0.232758	+	0.232758i	0.813843	−	0.581085i	$-0.197372\pi$
$443$	4.89898	+	4.89898i	0.232758	+	0.232758i	−0.581085	+	0.813843i	$0.697372\pi$
$444$		0			0
$445$		0			0
$446$			21.0000i			0.994379i
$447$		0			0
$448$	−1.22474	+	1.22474i	−0.0578638	+	0.0578638i
$449$	6.00000			0.283158			0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000			0.283158			0.141579	+	0.989927i	$0.454782\pi$
$450$		0			0
$451$	36.0000			1.69517
$452$		0			0
$453$		0			0
$454$		−	12.0000i		−	0.563188i
$455$		0			0
$456$		0			0
$457$	−14.6969	−	14.6969i	−0.687494	−	0.687494i	0.274184	−	0.961677i	$-0.411592\pi$
$457$	−14.6969	−	14.6969i	−0.687494	−	0.687494i	−0.961677	+	0.274184i	$0.911592\pi$
$458$	−6.12372	−	6.12372i	−0.286143	−	0.286143i
$459$		0			0
$460$		0			0
$461$		−	24.0000i		−	1.11779i	−0.829238	−	0.558896i	$-0.811225\pi$
$461$		−	24.0000i		−	1.11779i	0.829238	−	0.558896i	$-0.188775\pi$
$462$		0			0
$463$	−23.2702	+	23.2702i	−1.08146	+	1.08146i	−0.0850817	+	0.996374i	$0.527115\pi$
$463$	−23.2702	+	23.2702i	−1.08146	+	1.08146i	−0.996374	+	0.0850817i	$0.972885\pi$
$464$	30.0000			1.39272
$465$		0			0
$466$	−18.0000			−0.833834
$467$	−4.89898	+	4.89898i	−0.226698	+	0.226698i	−0.811312	−	0.584614i	$-0.801246\pi$
$467$	−4.89898	+	4.89898i	−0.226698	+	0.226698i	0.584614	+	0.811312i	$0.301246\pi$
$468$		0			0
$469$			18.0000i			0.831163i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−36.7423	−	36.7423i	−1.68941	−	1.68941i
$474$		0			0
$475$		0			0
$476$			6.00000i			0.275010i
$477$		0			0
$478$	7.34847	−	7.34847i	0.336111	−	0.336111i
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$	−2.44949	+	2.44949i	−0.111571	+	0.111571i
$483$		0			0
$484$			25.0000i			1.13636i
$485$		0			0
$486$		0			0
$487$	22.0454	+	22.0454i	0.998973	+	0.998973i	0.999999	−	0.00102669i	$-0.000326807\pi$
$487$	22.0454	+	22.0454i	0.998973	+	0.998973i	−0.00102669	+	0.999999i	$0.500327\pi$
$488$	1.22474	+	1.22474i	0.0554416	+	0.0554416i
$489$		0			0
$490$		0			0
$491$			30.0000i			1.35388i	0.736038	+	0.676941i	$0.236695\pi$
$491$			30.0000i			1.35388i	−0.736038	+	0.676941i	$0.763305\pi$
$492$		0			0
$493$	−14.6969	+	14.6969i	−0.661917	+	0.661917i
$494$		0			0
$495$		0			0
$496$	−25.0000			−1.12253
$497$	−14.6969	+	14.6969i	−0.659248	+	0.659248i
$498$		0			0
$499$			13.0000i			0.581960i	0.956729	+	0.290980i	$0.0939813\pi$
$499$			13.0000i			0.581960i	−0.956729	+	0.290980i	$0.906019\pi$
$500$		0			0
$501$		0			0
$502$	−7.34847	−	7.34847i	−0.327978	−	0.327978i
$503$	−12.2474	−	12.2474i	−0.546087	−	0.546087i	0.379220	−	0.925307i	$-0.376192\pi$
$503$	−12.2474	−	12.2474i	−0.546087	−	0.546087i	−0.925307	+	0.379220i	$0.876192\pi$
$504$		0			0
$505$		0			0
$506$		−	72.0000i		−	3.20079i
$507$		0			0
$508$	−7.34847	+	7.34847i	−0.326036	+	0.326036i
$509$	−24.0000			−1.06378			−0.531891	−	0.846813i	$-0.678518\pi$
$509$	−24.0000			−1.06378			−0.531891	+	0.846813i	$0.678518\pi$
$510$		0			0
$511$	15.0000			0.663561
$512$	−6.12372	+	6.12372i	−0.270633	+	0.270633i
$513$		0			0
$514$			30.0000i			1.32324i
$515$		0			0
$516$		0			0
$517$	−44.0908	−	44.0908i	−1.93911	−	1.93911i
$518$	3.67423	+	3.67423i	0.161437	+	0.161437i
$519$		0			0
$520$		0			0
$521$			12.0000i			0.525730i	0.964833	+	0.262865i	$0.0846673\pi$
$521$			12.0000i			0.525730i	−0.964833	+	0.262865i	$0.915333\pi$
$522$		0			0
$523$	−6.12372	+	6.12372i	−0.267772	+	0.267772i	−0.828202	−	0.560430i	$-0.810636\pi$
$523$	−6.12372	+	6.12372i	−0.267772	+	0.267772i	0.560430	+	0.828202i	$0.310636\pi$
$524$	−18.0000			−0.786334
$525$		0			0
$526$	18.0000			0.784837
$527$	12.2474	−	12.2474i	0.533507	−	0.533507i
$528$		0			0
$529$			25.0000i			1.08696i
$530$		0			0
$531$		0			0
$532$	−8.57321	−	8.57321i	−0.371696	−	0.371696i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		−	18.0000i		−	0.777482i
$537$		0			0
$538$	14.6969	−	14.6969i	0.633630	−	0.633630i
$539$	−24.0000			−1.03375
$540$		0			0
$541$	−19.0000			−0.816874			−0.408437	−	0.912787i	$-0.633926\pi$
$541$	−19.0000			−0.816874			−0.408437	+	0.912787i	$0.633926\pi$
$542$	−13.4722	+	13.4722i	−0.578680	+	0.578680i
$543$		0			0
$544$		−	18.0000i		−	0.771744i
$545$		0			0
$546$		0			0
$547$	8.57321	+	8.57321i	0.366564	+	0.366564i	0.866223	−	0.499658i	$-0.166541\pi$
$547$	8.57321	+	8.57321i	0.366564	+	0.366564i	−0.499658	+	0.866223i	$0.666541\pi$
$548$	2.44949	+	2.44949i	0.104637	+	0.104637i
$549$		0			0
$550$		0			0
$551$		−	42.0000i		−	1.78926i
$552$		0			0
$553$	−1.22474	+	1.22474i	−0.0520814	+	0.0520814i
$554$	−15.0000			−0.637289
$555$		0			0
$556$	−7.00000			−0.296866
$557$	7.34847	−	7.34847i	0.311365	−	0.311365i	−0.534073	−	0.845438i	$-0.679339\pi$
$557$	7.34847	−	7.34847i	0.311365	−	0.311365i	0.845438	+	0.534073i	$0.179339\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−7.34847	−	7.34847i	−0.309976	−	0.309976i
$563$	−14.6969	−	14.6969i	−0.619402	−	0.619402i	0.325976	−	0.945378i	$-0.394307\pi$
$563$	−14.6969	−	14.6969i	−0.619402	−	0.619402i	−0.945378	+	0.325976i	$0.894307\pi$
$564$		0			0
$565$		0			0
$566$			3.00000i			0.126099i
$567$		0			0
$568$	14.6969	−	14.6969i	0.616670	−	0.616670i
$569$	36.0000			1.50920			0.754599	−	0.656186i	$-0.227831\pi$
$569$	36.0000			1.50920			0.754599	+	0.656186i	$0.227831\pi$
$570$		0			0
$571$	5.00000			0.209243			0.104622	−	0.994512i	$-0.466637\pi$
$571$	5.00000			0.209243			0.104622	+	0.994512i	$0.466637\pi$
$572$		0			0
$573$		0			0
$574$		−	18.0000i		−	0.751305i
$575$		0			0
$576$		0			0
$577$	−1.22474	−	1.22474i	−0.0509868	−	0.0509868i	0.681154	−	0.732140i	$-0.261479\pi$
$577$	−1.22474	−	1.22474i	−0.0509868	−	0.0509868i	−0.732140	+	0.681154i	$0.761479\pi$
$578$	−6.12372	−	6.12372i	−0.254713	−	0.254713i
$579$		0			0
$580$		0			0
$581$			6.00000i			0.248922i
$582$		0			0
$583$	−14.6969	+	14.6969i	−0.608685	+	0.608685i
$584$	−15.0000			−0.620704
$585$		0			0
$586$	−48.0000			−1.98286
$587$	12.2474	−	12.2474i	0.505506	−	0.505506i	−0.407638	−	0.913144i	$-0.633647\pi$
$587$	12.2474	−	12.2474i	0.505506	−	0.505506i	0.913144	+	0.407638i	$0.133647\pi$
$588$		0			0
$589$			35.0000i			1.44215i
$590$		0			0
$591$		0			0
$592$	−6.12372	−	6.12372i	−0.251684	−	0.251684i
$593$	−19.5959	−	19.5959i	−0.804708	−	0.804708i	0.179119	−	0.983827i	$-0.442675\pi$
$593$	−19.5959	−	19.5959i	−0.804708	−	0.804708i	−0.983827	+	0.179119i	$0.942675\pi$
$594$		0			0
$595$		0			0
$596$			6.00000i			0.245770i
$597$		0			0
$598$		0			0
$599$	36.0000			1.47092			0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000			1.47092			0.735460	+	0.677568i	$0.236966\pi$
$600$		0			0
$601$	−35.0000			−1.42768			−0.713840	−	0.700309i	$-0.753046\pi$
$601$	−35.0000			−1.42768			−0.713840	+	0.700309i	$0.753046\pi$
$602$	−18.3712	+	18.3712i	−0.748753	+	0.748753i
$603$		0			0
$604$			5.00000i			0.203447i
$605$		0			0
$606$		0			0
$607$	6.12372	+	6.12372i	0.248554	+	0.248554i	0.820377	−	0.571823i	$-0.193764\pi$
$607$	6.12372	+	6.12372i	0.248554	+	0.248554i	−0.571823	+	0.820377i	$0.693764\pi$
$608$	25.7196	+	25.7196i	1.04307	+	1.04307i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−13.4722	+	13.4722i	−0.544137	+	0.544137i	−0.924739	−	0.380602i	$-0.875717\pi$
$613$	−13.4722	+	13.4722i	−0.544137	+	0.544137i	0.380602	+	0.924739i	$0.375717\pi$
$614$	−3.00000			−0.121070
$615$		0			0
$616$	−18.0000			−0.725241
$617$	−9.79796	+	9.79796i	−0.394451	+	0.394451i	−0.876270	−	0.481820i	$-0.839976\pi$
$617$	−9.79796	+	9.79796i	−0.394451	+	0.394451i	0.481820	+	0.876270i	$0.339976\pi$
$618$		0			0
$619$			31.0000i			1.24600i	0.782224	+	0.622998i	$0.214085\pi$
$619$			31.0000i			1.24600i	−0.782224	+	0.622998i	$0.785915\pi$
$620$		0			0
$621$		0			0
$622$	−7.34847	−	7.34847i	−0.294647	−	0.294647i
$623$	14.6969	+	14.6969i	0.588820	+	0.588820i
$624$		0			0
$625$		0			0
$626$			36.0000i			1.43885i
$627$		0			0
$628$	−13.4722	+	13.4722i	−0.537599	+	0.537599i
$629$	6.00000			0.239236
$630$		0			0
$631$	32.0000			1.27390			0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000			1.27390			0.636950	+	0.770905i	$0.280196\pi$
$632$	1.22474	−	1.22474i	0.0487177	−	0.0487177i
$633$		0			0
$634$			42.0000i			1.66803i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	44.0908	+	44.0908i	1.74557	+	1.74557i
$639$		0			0
$640$		0			0
$641$			24.0000i			0.947943i	0.880540	+	0.473972i	$0.157180\pi$
$641$			24.0000i			0.947943i	−0.880540	+	0.473972i	$0.842820\pi$
$642$		0			0
$643$	7.34847	−	7.34847i	0.289795	−	0.289795i	−0.547204	−	0.836999i	$-0.684308\pi$
$643$	7.34847	−	7.34847i	0.289795	−	0.289795i	0.836999	+	0.547204i	$0.184308\pi$
$644$	−12.0000			−0.472866
$645$		0			0
$646$	−42.0000			−1.65247
$647$	24.4949	−	24.4949i	0.962994	−	0.962994i	−0.0363455	−	0.999339i	$-0.511572\pi$
$647$	24.4949	−	24.4949i	0.962994	−	0.962994i	0.999339	+	0.0363455i	$0.0115717\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−7.34847	−	7.34847i	−0.287788	−	0.287788i
$653$	4.89898	+	4.89898i	0.191712	+	0.191712i	0.796435	−	0.604724i	$-0.206716\pi$
$653$	4.89898	+	4.89898i	0.191712	+	0.191712i	−0.604724	+	0.796435i	$0.706716\pi$
$654$		0			0
$655$		0			0
$656$			30.0000i			1.17130i
$657$		0			0
$658$	−22.0454	+	22.0454i	−0.859419	+	0.859419i
$659$	6.00000			0.233727			0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000			0.233727			0.116863	+	0.993148i	$0.462716\pi$
$660$		0			0
$661$	25.0000			0.972387			0.486194	−	0.873851i	$-0.338385\pi$
$661$	25.0000			0.972387			0.486194	+	0.873851i	$0.338385\pi$
$662$	−6.12372	+	6.12372i	−0.238005	+	0.238005i
$663$		0			0
$664$		−	6.00000i		−	0.232845i
$665$		0			0
$666$		0			0
$667$	−29.3939	−	29.3939i	−1.13814	−	1.13814i
$668$	12.2474	+	12.2474i	0.473868	+	0.473868i
$669$		0			0
$670$		0			0
$671$			6.00000i			0.231627i
$672$		0			0
$673$	28.1691	−	28.1691i	1.08584	−	1.08584i	0.0898884	−	0.995952i	$-0.471349\pi$
$673$	28.1691	−	28.1691i	1.08584	−	1.08584i	0.995952	−	0.0898884i	$-0.0286510\pi$
$674$		0			0
$675$		0			0
$676$	13.0000			0.500000
$677$	7.34847	−	7.34847i	0.282425	−	0.282425i	−0.551651	−	0.834075i	$-0.686002\pi$
$677$	7.34847	−	7.34847i	0.282425	−	0.282425i	0.834075	+	0.551651i	$0.186002\pi$
$678$		0			0
$679$		−	21.0000i		−	0.805906i
$680$		0			0
$681$		0			0
$682$	−36.7423	−	36.7423i	−1.40694	−	1.40694i
$683$	−31.8434	−	31.8434i	−1.21845	−	1.21845i	−0.968174	−	0.250279i	$-0.919478\pi$
$683$	−31.8434	−	31.8434i	−1.21845	−	1.21845i	−0.250279	−	0.968174i	$-0.580522\pi$
$684$		0			0
$685$		0			0
$686$			33.0000i			1.25995i
$687$		0			0
$688$	30.6186	−	30.6186i	1.16732	−	1.16732i
$689$		0			0
$690$		0			0
$691$	−20.0000			−0.760836			−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000			−0.760836			−0.380418	+	0.924815i	$0.624220\pi$
$692$	−9.79796	+	9.79796i	−0.372463	+	0.372463i
$693$		0			0
$694$			36.0000i			1.36654i
$695$		0			0
$696$		0			0
$697$	−14.6969	−	14.6969i	−0.556686	−	0.556686i
$698$	−42.8661	−	42.8661i	−1.62250	−	1.62250i
$699$		0			0
$700$		0			0
$701$		−	24.0000i		−	0.906467i	−0.891392	−	0.453234i	$-0.850270\pi$
$701$		−	24.0000i		−	0.906467i	0.891392	−	0.453234i	$-0.149730\pi$
$702$		0			0
$703$	−8.57321	+	8.57321i	−0.323345	+	0.323345i
$704$	6.00000			0.226134
$705$		0			0
$706$	−36.0000			−1.35488
$707$	7.34847	−	7.34847i	0.276368	−	0.276368i
$708$		0			0
$709$			5.00000i			0.187779i	0.995583	+	0.0938895i	$0.0299300\pi$
$709$			5.00000i			0.187779i	−0.995583	+	0.0938895i	$0.970070\pi$
$710$		0			0
$711$		0			0
$712$	−14.6969	−	14.6969i	−0.550791	−	0.550791i
$713$	24.4949	+	24.4949i	0.917341	+	0.917341i
$714$		0			0
$715$		0			0
$716$		−	24.0000i		−	0.896922i
$717$		0			0
$718$	22.0454	−	22.0454i	0.822727	−	0.822727i
$719$	30.0000			1.11881			0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000			1.11881			0.559406	+	0.828894i	$0.311029\pi$
$720$		0			0
$721$	−15.0000			−0.558629
$722$	36.7423	−	36.7423i	1.36741	−	1.36741i
$723$		0			0
$724$			14.0000i			0.520306i
$725$		0			0
$726$		0			0
$727$	23.2702	+	23.2702i	0.863042	+	0.863042i	0.991690	−	0.128648i	$-0.0410638\pi$
$727$	23.2702	+	23.2702i	0.863042	+	0.863042i	−0.128648	+	0.991690i	$0.541064\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			30.0000i			1.10959i
$732$		0			0
$733$	−29.3939	+	29.3939i	−1.08569	+	1.08569i	−0.0897206	+	0.995967i	$0.528597\pi$
$733$	−29.3939	+	29.3939i	−1.08569	+	1.08569i	−0.995967	+	0.0897206i	$0.971403\pi$
$734$	−54.0000			−1.99318
$735$		0			0
$736$	36.0000			1.32698
$737$	44.0908	−	44.0908i	1.62411	−	1.62411i
$738$		0			0
$739$		−	40.0000i		−	1.47142i	−0.677295	−	0.735712i	$-0.736848\pi$
$739$		−	40.0000i		−	1.47142i	0.677295	−	0.735712i	$-0.263152\pi$
$740$		0			0
$741$		0			0
$742$	7.34847	+	7.34847i	0.269771	+	0.269771i
$743$	29.3939	+	29.3939i	1.07836	+	1.07836i	0.996657	+	0.0816998i	$0.0260349\pi$
$743$	29.3939	+	29.3939i	1.07836	+	1.07836i	0.0816998	+	0.996657i	$0.473965\pi$
$744$		0			0
$745$		0			0
$746$			3.00000i			0.109838i
$747$		0			0
$748$	14.6969	−	14.6969i	0.537373	−	0.537373i
$749$		0			0
$750$		0			0
$751$	−23.0000			−0.839282			−0.419641	−	0.907690i	$-0.637844\pi$
$751$	−23.0000			−0.839282			−0.419641	+	0.907690i	$0.637844\pi$
$752$	36.7423	−	36.7423i	1.33986	−	1.33986i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	8.57321	+	8.57321i	0.311599	+	0.311599i	0.845529	−	0.533930i	$-0.179285\pi$
$757$	8.57321	+	8.57321i	0.311599	+	0.311599i	−0.533930	+	0.845529i	$0.679285\pi$
$758$	4.89898	+	4.89898i	0.177939	+	0.177939i
$759$		0			0
$760$		0			0
$761$		−	18.0000i		−	0.652499i	−0.945284	−	0.326250i	$-0.894215\pi$
$761$		−	18.0000i		−	0.652499i	0.945284	−	0.326250i	$-0.105785\pi$
$762$		0			0
$763$	−8.57321	+	8.57321i	−0.310371	+	0.310371i
$764$	18.0000			0.651217
$765$		0			0
$766$	6.00000			0.216789
$767$		0			0
$768$		0			0
$769$			10.0000i			0.360609i	0.983611	+	0.180305i	$0.0577084\pi$
$769$			10.0000i			0.360609i	−0.983611	+	0.180305i	$0.942292\pi$
$770$		0			0
$771$		0			0
$772$	1.22474	+	1.22474i	0.0440795	+	0.0440795i
$773$	17.1464	+	17.1464i	0.616714	+	0.616714i	0.944687	−	0.327973i	$-0.106365\pi$
$773$	17.1464	+	17.1464i	0.616714	+	0.616714i	−0.327973	+	0.944687i	$0.606365\pi$
$774$		0			0
$775$		0			0
$776$			21.0000i			0.753856i
$777$		0			0
$778$	−36.7423	+	36.7423i	−1.31728	+	1.31728i
$779$	42.0000			1.50481
$780$		0			0
$781$	72.0000			2.57636
$782$	−29.3939	+	29.3939i	−1.05112	+	1.05112i
$783$		0			0
$784$		−	20.0000i		−	0.714286i
$785$		0			0
$786$		0			0
$787$	37.9671	+	37.9671i	1.35338	+	1.35338i	0.881849	+	0.471531i	$0.156299\pi$
$787$	37.9671	+	37.9671i	1.35338	+	1.35338i	0.471531	+	0.881849i	$0.343701\pi$
$788$	12.2474	+	12.2474i	0.436297	+	0.436297i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	−33.0000			−1.17113
$795$		0			0
$796$	−20.0000			−0.708881
$797$	9.79796	−	9.79796i	0.347062	−	0.347062i	−0.511952	−	0.859014i	$-0.671078\pi$
$797$	9.79796	−	9.79796i	0.347062	−	0.347062i	0.859014	+	0.511952i	$0.171078\pi$
$798$		0			0
$799$			36.0000i			1.27359i
$800$		0			0
$801$		0			0
$802$	14.6969	+	14.6969i	0.518967	+	0.518967i
$803$	−36.7423	−	36.7423i	−1.29661	−	1.29661i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−7.34847	+	7.34847i	−0.258518	+	0.258518i
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$	1.00000			0.0351147			0.0175574	−	0.999846i	$-0.494411\pi$
$811$	1.00000			0.0351147			0.0175574	+	0.999846i	$0.494411\pi$
$812$	7.34847	−	7.34847i	0.257881	−	0.257881i
$813$		0			0
$814$		−	18.0000i		−	0.630900i
$815$		0			0
$816$		0			0
$817$	−42.8661	−	42.8661i	−1.49969	−	1.49969i
$818$	12.2474	+	12.2474i	0.428222	+	0.428222i
$819$		0			0
$820$		0			0
$821$			30.0000i			1.04701i	0.852023	+	0.523504i	$0.175375\pi$
$821$			30.0000i			1.04701i	−0.852023	+	0.523504i	$0.824625\pi$
$822$		0			0
$823$	7.34847	−	7.34847i	0.256152	−	0.256152i	−0.567335	−	0.823487i	$-0.692026\pi$
$823$	7.34847	−	7.34847i	0.256152	−	0.256152i	0.823487	+	0.567335i	$0.192026\pi$
$824$	15.0000			0.522550
$825$		0			0
$826$		0			0
$827$	−17.1464	+	17.1464i	−0.596240	+	0.596240i	−0.939310	−	0.343070i	$-0.888533\pi$
$827$	−17.1464	+	17.1464i	−0.596240	+	0.596240i	0.343070	+	0.939310i	$0.388533\pi$
$828$		0			0
$829$			53.0000i			1.84077i	0.391018	+	0.920383i	$0.372123\pi$
$829$			53.0000i			1.84077i	−0.391018	+	0.920383i	$0.627877\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	9.79796	+	9.79796i	0.339479	+	0.339479i
$834$		0			0
$835$		0			0
$836$			42.0000i			1.45260i
$837$		0			0
$838$	29.3939	−	29.3939i	1.01539	−	1.01539i
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$	−23.2702	+	23.2702i	−0.801942	+	0.801942i
$843$		0			0
$844$		−	8.00000i		−	0.275371i
$845$		0			0
$846$		0			0
$847$	−30.6186	−	30.6186i	−1.05207	−	1.05207i
$848$	−12.2474	−	12.2474i	−0.420579	−	0.420579i
$849$		0			0
$850$		0			0
$851$			12.0000i			0.411355i
$852$		0			0
$853$	−14.6969	+	14.6969i	−0.503214	+	0.503214i	−0.912435	−	0.409221i	$-0.865800\pi$
$853$	−14.6969	+	14.6969i	−0.503214	+	0.503214i	0.409221	+	0.912435i	$0.365800\pi$
$854$	3.00000			0.102658
$855$		0			0
$856$		0			0
$857$	−24.4949	+	24.4949i	−0.836730	+	0.836730i	−0.988427	−	0.151697i	$-0.951526\pi$
$857$	−24.4949	+	24.4949i	−0.836730	+	0.836730i	0.151697	+	0.988427i	$0.451526\pi$
$858$		0			0
$859$		−	5.00000i		−	0.170598i	−0.996355	−	0.0852989i	$-0.972815\pi$
$859$		−	5.00000i		−	0.170598i	0.996355	−	0.0852989i	$-0.0271845\pi$
$860$		0			0
$861$		0			0
$862$	36.7423	+	36.7423i	1.25145	+	1.25145i
$863$	12.2474	+	12.2474i	0.416908	+	0.416908i	0.884137	−	0.467229i	$-0.154747\pi$
$863$	12.2474	+	12.2474i	0.416908	+	0.416908i	−0.467229	+	0.884137i	$0.654747\pi$
$864$		0			0
$865$		0			0
$866$		−	33.0000i		−	1.12139i
$867$		0			0
$868$	−6.12372	+	6.12372i	−0.207853	+	0.207853i
$869$	6.00000			0.203536
$870$		0			0
$871$		0			0
$872$	8.57321	−	8.57321i	0.290326	−	0.290326i
$873$		0			0
$874$		−	84.0000i		−	2.84134i
$875$		0			0
$876$		0			0
$877$	30.6186	+	30.6186i	1.03392	+	1.03392i	0.999404	+	0.0345131i	$0.0109881\pi$
$877$	30.6186	+	30.6186i	1.03392	+	1.03392i	0.0345131	+	0.999404i	$0.489012\pi$
$878$	20.8207	+	20.8207i	0.702663	+	0.702663i
$879$		0			0
$880$		0			0
$881$		−	18.0000i		−	0.606435i	−0.952921	−	0.303218i	$-0.901939\pi$
$881$		−	18.0000i		−	0.606435i	0.952921	−	0.303218i	$-0.0980609\pi$
$882$		0			0
$883$	1.22474	−	1.22474i	0.0412159	−	0.0412159i	−0.686198	−	0.727414i	$-0.740722\pi$
$883$	1.22474	−	1.22474i	0.0412159	−	0.0412159i	0.727414	+	0.686198i	$0.240722\pi$
$884$		0			0
$885$		0			0
$886$	−12.0000			−0.403148
$887$	−2.44949	+	2.44949i	−0.0822458	+	0.0822458i	−0.747033	−	0.664787i	$-0.768522\pi$
$887$	−2.44949	+	2.44949i	−0.0822458	+	0.0822458i	0.664787	+	0.747033i	$0.268522\pi$
$888$		0			0
$889$		−	18.0000i		−	0.603701i
$890$		0			0
$891$		0			0
$892$	−8.57321	−	8.57321i	−0.287052	−	0.287052i
$893$	−51.4393	−	51.4393i	−1.72135	−	1.72135i
$894$		0			0
$895$		0			0
$896$		−	21.0000i		−	0.701561i
$897$		0			0
$898$	−7.34847	+	7.34847i	−0.245222	+	0.245222i
$899$	−30.0000			−1.00056
$900$		0			0
$901$	12.0000			0.399778
$902$	−44.0908	+	44.0908i	−1.46806	+	1.46806i
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	28.1691	+	28.1691i	0.935341	+	0.935341i	0.998033	−	0.0626922i	$-0.0199687\pi$
$907$	28.1691	+	28.1691i	0.935341	+	0.935341i	−0.0626922	+	0.998033i	$0.519969\pi$
$908$	4.89898	+	4.89898i	0.162578	+	0.162578i
$909$		0			0
$910$		0			0
$911$			18.0000i			0.596367i	0.954509	+	0.298183i	$0.0963807\pi$
$911$			18.0000i			0.596367i	−0.954509	+	0.298183i	$0.903619\pi$
$912$		0			0
$913$	14.6969	−	14.6969i	0.486398	−	0.486398i
$914$	36.0000			1.19077
$915$		0			0
$916$	5.00000			0.165205
$917$	22.0454	−	22.0454i	0.728003	−	0.728003i
$918$		0			0
$919$		−	1.00000i		−	0.0329870i	−0.999864	−	0.0164935i	$-0.994750\pi$
$919$		−	1.00000i		−	0.0329870i	0.999864	−	0.0164935i	$-0.00525028\pi$
$920$		0			0
$921$		0			0
$922$	29.3939	+	29.3939i	0.968036	+	0.968036i
$923$		0			0
$924$		0			0
$925$		0			0
$926$		−	57.0000i		−	1.87314i
$927$		0			0
$928$	−22.0454	+	22.0454i	−0.723676	+	0.723676i
$929$	−24.0000			−0.787414			−0.393707	−	0.919236i	$-0.628808\pi$
$929$	−24.0000			−0.787414			−0.393707	+	0.919236i	$0.628808\pi$
$930$		0			0
$931$	−28.0000			−0.917663
$932$	7.34847	−	7.34847i	0.240707	−	0.240707i
$933$		0			0
$934$		−	12.0000i		−	0.392652i
$935$		0			0
$936$		0			0
$937$	30.6186	+	30.6186i	1.00027	+	1.00027i	1.00000		0.000266809i	$8.49279e-5\pi$
$937$	30.6186	+	30.6186i	1.00027	+	1.00027i	0.000266809		1.00000i	$0.499915\pi$
$938$	−22.0454	−	22.0454i	−0.719808	−	0.719808i
$939$		0			0
$940$		0			0
$941$		−	30.0000i		−	0.977972i	−0.872292	−	0.488986i	$-0.837367\pi$
$941$		−	30.0000i		−	0.977972i	0.872292	−	0.488986i	$-0.162633\pi$
$942$		0			0
$943$	29.3939	−	29.3939i	0.957196	−	0.957196i
$944$		0			0
$945$		0			0
$946$	90.0000			2.92615
$947$	−31.8434	+	31.8434i	−1.03477	+	1.03477i	−0.0353971	+	0.999373i	$0.511270\pi$
$947$	−31.8434	+	31.8434i	−1.03477	+	1.03477i	−0.999373	+	0.0353971i	$0.988730\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$	7.34847	+	7.34847i	0.238165	+	0.238165i
$953$	−7.34847	−	7.34847i	−0.238040	−	0.238040i	0.577998	−	0.816038i	$-0.303834\pi$
$953$	−7.34847	−	7.34847i	−0.238040	−	0.238040i	−0.816038	+	0.577998i	$0.803834\pi$
$954$		0			0
$955$		0			0
$956$			6.00000i			0.194054i
$957$		0			0
$958$		0			0
$959$	−6.00000			−0.193750
$960$		0			0
$961$	−6.00000			−0.193548
$962$		0			0
$963$		0			0
$964$		−	2.00000i		−	0.0644157i
$965$		0			0
$966$		0			0
$967$	13.4722	+	13.4722i	0.433237	+	0.433237i	0.889728	−	0.456491i	$-0.150894\pi$
$967$	13.4722	+	13.4722i	0.433237	+	0.433237i	−0.456491	+	0.889728i	$0.650894\pi$
$968$	30.6186	+	30.6186i	0.984120	+	0.984120i
$969$		0			0
$970$		0			0
$971$		−	36.0000i		−	1.15529i	−0.816286	−	0.577647i	$-0.803971\pi$
$971$		−	36.0000i		−	1.15529i	0.816286	−	0.577647i	$-0.196029\pi$
$972$		0			0
$973$	8.57321	−	8.57321i	0.274845	−	0.274845i
$974$	−54.0000			−1.73027
$975$		0			0
$976$	−5.00000			−0.160046
$977$	14.6969	−	14.6969i	0.470197	−	0.470197i	−0.431782	−	0.901978i	$-0.642115\pi$
$977$	14.6969	−	14.6969i	0.470197	−	0.470197i	0.901978	+	0.431782i	$0.142115\pi$
$978$		0			0
$979$		−	72.0000i		−	2.30113i
$980$		0			0
$981$		0			0
$982$	−36.7423	−	36.7423i	−1.17250	−	1.17250i
$983$	2.44949	+	2.44949i	0.0781266	+	0.0781266i	0.745090	−	0.666964i	$-0.232406\pi$
$983$	2.44949	+	2.44949i	0.0781266	+	0.0781266i	−0.666964	+	0.745090i	$0.732406\pi$
$984$		0			0
$985$		0			0
$986$		−	36.0000i		−	1.14647i
$987$		0			0
$988$		0			0
$989$	−60.0000			−1.90789
$990$		0			0
$991$	47.0000			1.49300			0.746502	−	0.665383i	$-0.231732\pi$
$991$	47.0000			1.49300			0.746502	+	0.665383i	$0.231732\pi$
$992$	18.3712	−	18.3712i	0.583285	−	0.583285i
$993$		0			0
$994$		−	36.0000i		−	1.14185i
$995$		0			0
$996$		0			0
$997$	−29.3939	−	29.3939i	−0.930913	−	0.930913i	0.0668496	−	0.997763i	$-0.478705\pi$
$997$	−29.3939	−	29.3939i	−0.930913	−	0.930913i	−0.997763	+	0.0668496i	$0.978705\pi$
$998$	−15.9217	−	15.9217i	−0.503992	−	0.503992i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.2.f.b.107.1	✓	4
3.2	odd	2		675.2.f.e.107.2	yes	4
5.2	odd	4		675.2.f.e.593.1	yes	4
5.3	odd	4		675.2.f.e.593.2	yes	4
5.4	even	2	inner	675.2.f.b.107.2	yes	4
15.2	even	4	inner	675.2.f.b.593.2	yes	4
15.8	even	4	inner	675.2.f.b.593.1	yes	4
15.14	odd	2		675.2.f.e.107.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
675.2.f.b.107.1	✓	4	1.1	even	1	trivial
675.2.f.b.107.2	yes	4	5.4	even	2	inner
675.2.f.b.593.1	yes	4	15.8	even	4	inner
675.2.f.b.593.2	yes	4	15.2	even	4	inner
675.2.f.e.107.1	yes	4	15.14	odd	2
675.2.f.e.107.2	yes	4	3.2	odd	2
675.2.f.e.593.1	yes	4	5.2	odd	4
675.2.f.e.593.2	yes	4	5.3	odd	4

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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	675.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.22474 + 1.22474i) q^{2} -1.00000i q^{4} +(1.22474 + 1.22474i) q^{7} +(-1.22474 - 1.22474i) q^{8} +O(q^{10})\)	\(q+(-1.22474 + 1.22474i) q^{2} -1.00000i q^{4} +(1.22474 + 1.22474i) q^{7} +(-1.22474 - 1.22474i) q^{8} -6.00000i q^{11} -3.00000 q^{14} +5.00000 q^{16} +(-2.44949 + 2.44949i) q^{17} -7.00000i q^{19} +(7.34847 + 7.34847i) q^{22} +(-4.89898 - 4.89898i) q^{23} +(1.22474 - 1.22474i) q^{28} +6.00000 q^{29} -5.00000 q^{31} +(-3.67423 + 3.67423i) q^{32} -6.00000i q^{34} +(-1.22474 - 1.22474i) q^{37} +(8.57321 + 8.57321i) q^{38} +6.00000i q^{41} +(6.12372 - 6.12372i) q^{43} -6.00000 q^{44} +12.0000 q^{46} +(7.34847 - 7.34847i) q^{47} -4.00000i q^{49} +(-2.44949 - 2.44949i) q^{53} -3.00000i q^{56} +(-7.34847 + 7.34847i) q^{58} -1.00000 q^{61} +(6.12372 - 6.12372i) q^{62} +1.00000i q^{64} +(7.34847 + 7.34847i) q^{67} +(2.44949 + 2.44949i) q^{68} +12.0000i q^{71} +(6.12372 - 6.12372i) q^{73} +3.00000 q^{74} -7.00000 q^{76} +(7.34847 - 7.34847i) q^{77} +1.00000i q^{79} +(-7.34847 - 7.34847i) q^{82} +(2.44949 + 2.44949i) q^{83} +15.0000i q^{86} +(-7.34847 + 7.34847i) q^{88} +12.0000 q^{89} +(-4.89898 + 4.89898i) q^{92} +18.0000i q^{94} +(-8.57321 - 8.57321i) q^{97} +(4.89898 + 4.89898i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 12 q^{14} + 20 q^{16} + 24 q^{29} - 20 q^{31} - 24 q^{44} + 48 q^{46} - 4 q^{61} + 12 q^{74} - 28 q^{76} + 48 q^{89}+O(q^{100}) \) 4 * q - 12 * q^14 + 20 * q^16 + 24 * q^29 - 20 * q^31 - 24 * q^44 + 48 * q^46 - 4 * q^61 + 12 * q^74 - 28 * q^76 + 48 * q^89

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