LMFDB - Embedded newform 245.2.b.f.99.1

Error: no document with id 234107152 found in table mf_hecke_traces.

Error: table True does not exist

Error: no document with id 216729336 found in table mf_hecke_traces.

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [245,2,Mod(99,245)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("245.99"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(245, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 245 = 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	245.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,4,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$1.95633484952$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	245.99
Dual form			245.2.b.f.99.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -2.82843i q^{3} +1.00000 q^{4} +(-2.12132 + 0.707107i) q^{5} -2.82843 q^{6} -3.00000i q^{8} -5.00000 q^{9} +(0.707107 + 2.12132i) q^{10} -2.82843i q^{12} -4.24264i q^{13} +(2.00000 + 6.00000i) q^{15} -1.00000 q^{16} +4.24264i q^{17} +5.00000i q^{18} +2.82843 q^{19} +(-2.12132 + 0.707107i) q^{20} +4.00000i q^{23} -8.48528 q^{24} +(4.00000 - 3.00000i) q^{25} -4.24264 q^{26} +5.65685i q^{27} +(6.00000 - 2.00000i) q^{30} +5.65685 q^{31} -5.00000i q^{32} +4.24264 q^{34} -5.00000 q^{36} -6.00000i q^{37} -2.82843i q^{38} -12.0000 q^{39} +(2.12132 + 6.36396i) q^{40} +4.24264 q^{41} +(10.6066 - 3.53553i) q^{45} +4.00000 q^{46} +2.82843i q^{48} +(-3.00000 - 4.00000i) q^{50} +12.0000 q^{51} -4.24264i q^{52} -8.00000i q^{53} +5.65685 q^{54} -8.00000i q^{57} -8.48528 q^{59} +(2.00000 + 6.00000i) q^{60} +9.89949 q^{61} -5.65685i q^{62} -7.00000 q^{64} +(3.00000 + 9.00000i) q^{65} +12.0000i q^{67} +4.24264i q^{68} +11.3137 q^{69} +12.0000 q^{71} +15.0000i q^{72} +12.7279i q^{73} -6.00000 q^{74} +(-8.48528 - 11.3137i) q^{75} +2.82843 q^{76} +12.0000i q^{78} -12.0000 q^{79} +(2.12132 - 0.707107i) q^{80} +1.00000 q^{81} -4.24264i q^{82} +8.48528i q^{83} +(-3.00000 - 9.00000i) q^{85} +4.24264 q^{89} +(-3.53553 - 10.6066i) q^{90} +4.00000i q^{92} -16.0000i q^{93} +(-6.00000 + 2.00000i) q^{95} -14.1421 q^{96} -4.24264i q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} - 20 q^{9} + 8 q^{15} - 4 q^{16} + 16 q^{25} + 24 q^{30} - 20 q^{36} - 48 q^{39} + 16 q^{46} - 12 q^{50} + 48 q^{51} + 8 q^{60} - 28 q^{64} + 12 q^{65} + 48 q^{71} - 24 q^{74} - 48 q^{79}+ \cdots - 24 q^{95}+O(q^{100}) $ 4 * q + 4 * q^4 - 20 * q^9 + 8 * q^15 - 4 * q^16 + 16 * q^25 + 24 * q^30 - 20 * q^36 - 48 * q^39 + 16 * q^46 - 12 * q^50 + 48 * q^51 + 8 * q^60 - 28 * q^64 + 12 * q^65 + 48 * q^71 - 24 * q^74 - 48 * q^79 + 4 * q^81 - 12 * q^85 - 24 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/245\mathbb{Z}\right)^\times$.

$n$	$101$	$197$
$\chi(n)$	$1$	$-1$

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 $

$2$		−	1.00000i		−	0.707107i	−0.935414	−	0.353553i	$-0.884973\pi$
$2$		−	1.00000i		−	0.707107i	0.935414	−	0.353553i	$-0.115027\pi$
$3$		−	2.82843i		−	1.63299i	−0.577350	−	0.816497i	$-0.695913\pi$
$3$		−	2.82843i		−	1.63299i	0.577350	−	0.816497i	$-0.304087\pi$
$4$	1.00000			0.500000
$5$	−2.12132	+	0.707107i	−0.948683	+	0.316228i
$5$	−2.12132	+	0.707107i	−0.948683	+	0.316228i
$6$	−2.82843			−1.15470
$7$		0			0
$7$		0			0
$8$		−	3.00000i		−	1.06066i
$9$	−5.00000			−1.66667
$10$	0.707107	+	2.12132i	0.223607	+	0.670820i
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		−	2.82843i		−	0.816497i
$13$		−	4.24264i		−	1.17670i	−0.808608	−	0.588348i	$-0.799778\pi$
$13$		−	4.24264i		−	1.17670i	0.808608	−	0.588348i	$-0.200222\pi$
$14$		0			0
$15$	2.00000	+	6.00000i	0.516398	+	1.54919i
$16$	−1.00000			−0.250000
$17$			4.24264i			1.02899i	0.857493	+	0.514496i	$0.172021\pi$
$17$			4.24264i			1.02899i	−0.857493	+	0.514496i	$0.827979\pi$
$18$			5.00000i			1.17851i
$19$	2.82843			0.648886			0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843			0.648886			0.324443	+	0.945905i	$0.394823\pi$
$20$	−2.12132	+	0.707107i	−0.474342	+	0.158114i
$21$		0			0
$22$		0			0
$23$			4.00000i			0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$			4.00000i			0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	−8.48528			−1.73205
$25$	4.00000	−	3.00000i	0.800000	−	0.600000i
$26$	−4.24264			−0.832050
$27$			5.65685i			1.08866i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$	6.00000	−	2.00000i	1.09545	−	0.365148i
$31$	5.65685			1.01600			0.508001	−	0.861357i	$-0.330385\pi$
$31$	5.65685			1.01600			0.508001	+	0.861357i	$0.330385\pi$
$32$		−	5.00000i		−	0.883883i
$33$		0			0
$34$	4.24264			0.727607
$35$		0			0
$36$	−5.00000			−0.833333
$37$		−	6.00000i		−	0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$		−	6.00000i		−	0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$		−	2.82843i		−	0.458831i
$39$	−12.0000			−1.92154
$40$	2.12132	+	6.36396i	0.335410	+	1.00623i
$41$	4.24264			0.662589			0.331295	−	0.943527i	$-0.392515\pi$
$41$	4.24264			0.662589			0.331295	+	0.943527i	$0.392515\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	10.6066	−	3.53553i	1.58114	−	0.527046i
$46$	4.00000			0.589768
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$			2.82843i			0.408248i
$49$		0			0
$50$	−3.00000	−	4.00000i	−0.424264	−	0.565685i
$51$	12.0000			1.68034
$52$		−	4.24264i		−	0.588348i
$53$		−	8.00000i		−	1.09888i	−0.835532	−	0.549442i	$-0.814840\pi$
$53$		−	8.00000i		−	1.09888i	0.835532	−	0.549442i	$-0.185160\pi$
$54$	5.65685			0.769800
$55$		0			0
$56$		0			0
$57$		−	8.00000i		−	1.05963i
$58$		0			0
$59$	−8.48528			−1.10469			−0.552345	−	0.833616i	$-0.686267\pi$
$59$	−8.48528			−1.10469			−0.552345	+	0.833616i	$0.686267\pi$
$60$	2.00000	+	6.00000i	0.258199	+	0.774597i
$61$	9.89949			1.26750			0.633750	−	0.773538i	$-0.281515\pi$
$61$	9.89949			1.26750			0.633750	+	0.773538i	$0.281515\pi$
$62$		−	5.65685i		−	0.718421i
$63$		0			0
$64$	−7.00000			−0.875000
$65$	3.00000	+	9.00000i	0.372104	+	1.11631i
$66$		0			0
$67$			12.0000i			1.46603i	0.680211	+	0.733017i	$0.261888\pi$
$67$			12.0000i			1.46603i	−0.680211	+	0.733017i	$0.738112\pi$
$68$			4.24264i			0.514496i
$69$	11.3137			1.36201
$70$		0			0
$71$	12.0000			1.42414			0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000			1.42414			0.712069	+	0.702109i	$0.247758\pi$
$72$			15.0000i			1.76777i
$73$			12.7279i			1.48969i	0.667237	+	0.744845i	$0.267477\pi$
$73$			12.7279i			1.48969i	−0.667237	+	0.744845i	$0.732523\pi$
$74$	−6.00000			−0.697486
$75$	−8.48528	−	11.3137i	−0.979796	−	1.30639i
$76$	2.82843			0.324443
$77$		0			0
$78$			12.0000i			1.35873i
$79$	−12.0000			−1.35011			−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000			−1.35011			−0.675053	+	0.737769i	$0.735879\pi$
$80$	2.12132	−	0.707107i	0.237171	−	0.0790569i
$81$	1.00000			0.111111
$82$		−	4.24264i		−	0.468521i
$83$			8.48528i			0.931381i	0.884948	+	0.465690i	$0.154194\pi$
$83$			8.48528i			0.931381i	−0.884948	+	0.465690i	$0.845806\pi$
$84$		0			0
$85$	−3.00000	−	9.00000i	−0.325396	−	0.976187i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	4.24264			0.449719			0.224860	−	0.974391i	$-0.427808\pi$
$89$	4.24264			0.449719			0.224860	+	0.974391i	$0.427808\pi$
$90$	−3.53553	−	10.6066i	−0.372678	−	1.11803i
$91$		0			0
$92$			4.00000i			0.417029i
$93$		−	16.0000i		−	1.65912i
$94$		0			0
$95$	−6.00000	+	2.00000i	−0.615587	+	0.205196i
$96$	−14.1421			−1.44338
$97$		−	4.24264i		−	0.430775i	−0.976529	−	0.215387i	$-0.930899\pi$
$97$		−	4.24264i		−	0.430775i	0.976529	−	0.215387i	$-0.0691014\pi$
$98$		0			0
$99$		0			0
$100$	4.00000	−	3.00000i	0.400000	−	0.300000i
$101$	−4.24264			−0.422159			−0.211079	−	0.977469i	$-0.567698\pi$
$101$	−4.24264			−0.422159			−0.211079	+	0.977469i	$0.567698\pi$
$102$		−	12.0000i		−	1.18818i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$	−12.7279			−1.24808
$105$		0			0
$106$	−8.00000			−0.777029
$107$		−	4.00000i		−	0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$		−	4.00000i		−	0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$			5.65685i			0.544331i
$109$	16.0000			1.53252			0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000			1.53252			0.766261	+	0.642529i	$0.222115\pi$
$110$		0			0
$111$	−16.9706			−1.61077
$112$		0			0
$113$			8.00000i			0.752577i	0.926503	+	0.376288i	$0.122800\pi$
$113$			8.00000i			0.752577i	−0.926503	+	0.376288i	$0.877200\pi$
$114$	−8.00000			−0.749269
$115$	−2.82843	−	8.48528i	−0.263752	−	0.791257i
$116$		0			0
$117$			21.2132i			1.96116i
$118$			8.48528i			0.781133i
$119$		0			0
$120$	18.0000	−	6.00000i	1.64317	−	0.547723i
$121$	−11.0000			−1.00000
$122$		−	9.89949i		−	0.896258i
$123$		−	12.0000i		−	1.08200i
$124$	5.65685			0.508001
$125$	−6.36396	+	9.19239i	−0.569210	+	0.822192i
$126$		0			0
$127$			12.0000i			1.06483i	0.846484	+	0.532414i	$0.178715\pi$
$127$			12.0000i			1.06483i	−0.846484	+	0.532414i	$0.821285\pi$
$128$		−	3.00000i		−	0.265165i
$129$		0			0
$130$	9.00000	−	3.00000i	0.789352	−	0.263117i
$131$	−8.48528			−0.741362			−0.370681	−	0.928760i	$-0.620876\pi$
$131$	−8.48528			−0.741362			−0.370681	+	0.928760i	$0.620876\pi$
$132$		0			0
$133$		0			0
$134$	12.0000			1.03664
$135$	−4.00000	−	12.0000i	−0.344265	−	1.03280i
$136$	12.7279			1.09141
$137$			14.0000i			1.19610i	0.801459	+	0.598050i	$0.204058\pi$
$137$			14.0000i			1.19610i	−0.801459	+	0.598050i	$0.795942\pi$
$138$		−	11.3137i		−	0.963087i
$139$	−2.82843			−0.239904			−0.119952	−	0.992780i	$-0.538274\pi$
$139$	−2.82843			−0.239904			−0.119952	+	0.992780i	$0.538274\pi$
$140$		0			0
$141$		0			0
$142$		−	12.0000i		−	1.00702i
$143$		0			0
$144$	5.00000			0.416667
$145$		0			0
$146$	12.7279			1.05337
$147$		0			0
$148$		−	6.00000i		−	0.493197i
$149$	6.00000			0.491539			0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000			0.491539			0.245770	+	0.969328i	$0.420959\pi$
$150$	−11.3137	+	8.48528i	−0.923760	+	0.692820i
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		−	8.48528i		−	0.688247i
$153$		−	21.2132i		−	1.71499i
$154$		0			0
$155$	−12.0000	+	4.00000i	−0.963863	+	0.321288i
$156$	−12.0000			−0.960769
$157$		−	4.24264i		−	0.338600i	−0.985565	−	0.169300i	$-0.945849\pi$
$157$		−	4.24264i		−	0.338600i	0.985565	−	0.169300i	$-0.0541506\pi$
$158$			12.0000i			0.954669i
$159$	−22.6274			−1.79447
$160$	3.53553	+	10.6066i	0.279508	+	0.838525i
$161$		0			0
$162$		−	1.00000i		−	0.0785674i
$163$			24.0000i			1.87983i	0.341415	+	0.939913i	$0.389094\pi$
$163$			24.0000i			1.87983i	−0.341415	+	0.939913i	$0.610906\pi$
$164$	4.24264			0.331295
$165$		0			0
$166$	8.48528			0.658586
$167$		−	11.3137i		−	0.875481i	−0.899101	−	0.437741i	$-0.855779\pi$
$167$		−	11.3137i		−	0.875481i	0.899101	−	0.437741i	$-0.144221\pi$
$168$		0			0
$169$	−5.00000			−0.384615
$170$	−9.00000	+	3.00000i	−0.690268	+	0.230089i
$171$	−14.1421			−1.08148
$172$		0			0
$173$		−	4.24264i		−	0.322562i	−0.986909	−	0.161281i	$-0.948437\pi$
$173$		−	4.24264i		−	0.322562i	0.986909	−	0.161281i	$-0.0515625\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			24.0000i			1.80395i
$178$		−	4.24264i		−	0.317999i
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$	10.6066	−	3.53553i	0.790569	−	0.263523i
$181$	−18.3848			−1.36653			−0.683265	−	0.730171i	$-0.739441\pi$
$181$	−18.3848			−1.36653			−0.683265	+	0.730171i	$0.739441\pi$
$182$		0			0
$183$		−	28.0000i		−	2.06982i
$184$	12.0000			0.884652
$185$	4.24264	+	12.7279i	0.311925	+	0.935775i
$186$	−16.0000			−1.17318
$187$		0			0
$188$		0			0
$189$		0			0
$190$	2.00000	+	6.00000i	0.145095	+	0.435286i
$191$	12.0000			0.868290			0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000			0.868290			0.434145	+	0.900843i	$0.357051\pi$
$192$			19.7990i			1.42887i
$193$			24.0000i			1.72756i	0.503871	+	0.863779i	$0.331909\pi$
$193$			24.0000i			1.72756i	−0.503871	+	0.863779i	$0.668091\pi$
$194$	−4.24264			−0.304604
$195$	25.4558	−	8.48528i	1.82293	−	0.607644i
$196$		0			0
$197$			8.00000i			0.569976i	0.958531	+	0.284988i	$0.0919897\pi$
$197$			8.00000i			0.569976i	−0.958531	+	0.284988i	$0.908010\pi$
$198$		0			0
$199$	11.3137			0.802008			0.401004	−	0.916076i	$-0.368661\pi$
$199$	11.3137			0.802008			0.401004	+	0.916076i	$0.368661\pi$
$200$	−9.00000	−	12.0000i	−0.636396	−	0.848528i
$201$	33.9411			2.39402
$202$			4.24264i			0.298511i
$203$		0			0
$204$	12.0000			0.840168
$205$	−9.00000	+	3.00000i	−0.628587	+	0.209529i
$206$		0			0
$207$		−	20.0000i		−	1.39010i
$208$			4.24264i			0.294174i
$209$		0			0
$210$		0			0
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$		−	8.00000i		−	0.549442i
$213$		−	33.9411i		−	2.32561i
$214$	−4.00000			−0.273434
$215$		0			0
$216$	16.9706			1.15470
$217$		0			0
$218$		−	16.0000i		−	1.08366i
$219$	36.0000			2.43265
$220$		0			0
$221$	18.0000			1.21081
$222$			16.9706i			1.13899i
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	−20.0000	+	15.0000i	−1.33333	+	1.00000i
$226$	8.00000			0.532152
$227$		−	2.82843i		−	0.187729i	−0.995585	−	0.0938647i	$-0.970078\pi$
$227$		−	2.82843i		−	0.187729i	0.995585	−	0.0938647i	$-0.0299221\pi$
$228$		−	8.00000i		−	0.529813i
$229$	−7.07107			−0.467269			−0.233635	−	0.972324i	$-0.575062\pi$
$229$	−7.07107			−0.467269			−0.233635	+	0.972324i	$0.575062\pi$
$230$	−8.48528	+	2.82843i	−0.559503	+	0.186501i
$231$		0			0
$232$		0			0
$233$		−	14.0000i		−	0.917170i	−0.888650	−	0.458585i	$-0.848356\pi$
$233$		−	14.0000i		−	0.917170i	0.888650	−	0.458585i	$-0.151644\pi$
$234$	21.2132			1.38675
$235$		0			0
$236$	−8.48528			−0.552345
$237$			33.9411i			2.20471i
$238$		0			0
$239$	−24.0000			−1.55243			−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000			−1.55243			−0.776215	+	0.630468i	$0.782863\pi$
$240$	−2.00000	−	6.00000i	−0.129099	−	0.387298i
$241$	−9.89949			−0.637683			−0.318841	−	0.947808i	$-0.603294\pi$
$241$	−9.89949			−0.637683			−0.318841	+	0.947808i	$0.603294\pi$
$242$			11.0000i			0.707107i
$243$			14.1421i			0.907218i
$244$	9.89949			0.633750
$245$		0			0
$246$	−12.0000			−0.765092
$247$		−	12.0000i		−	0.763542i
$248$		−	16.9706i		−	1.07763i
$249$	24.0000			1.52094
$250$	9.19239	+	6.36396i	0.581378	+	0.402492i
$251$	25.4558			1.60676			0.803379	−	0.595468i	$-0.203033\pi$
$251$	25.4558			1.60676			0.803379	+	0.595468i	$0.203033\pi$
$252$		0			0
$253$		0			0
$254$	12.0000			0.752947
$255$	−25.4558	+	8.48528i	−1.59411	+	0.531369i
$256$	−17.0000			−1.06250
$257$		−	26.8701i		−	1.67611i	−0.545587	−	0.838054i	$-0.683693\pi$
$257$		−	26.8701i		−	1.67611i	0.545587	−	0.838054i	$-0.316307\pi$
$258$		0			0
$259$		0			0
$260$	3.00000	+	9.00000i	0.186052	+	0.558156i
$261$		0			0
$262$			8.48528i			0.524222i
$263$		−	16.0000i		−	0.986602i	−0.869859	−	0.493301i	$-0.835790\pi$
$263$		−	16.0000i		−	0.986602i	0.869859	−	0.493301i	$-0.164210\pi$
$264$		0			0
$265$	5.65685	+	16.9706i	0.347498	+	1.04249i
$266$		0			0
$267$		−	12.0000i		−	0.734388i
$268$			12.0000i			0.733017i
$269$	21.2132			1.29339			0.646696	−	0.762748i	$-0.276150\pi$
$269$	21.2132			1.29339			0.646696	+	0.762748i	$0.276150\pi$
$270$	−12.0000	+	4.00000i	−0.730297	+	0.243432i
$271$	−5.65685			−0.343629			−0.171815	−	0.985129i	$-0.554963\pi$
$271$	−5.65685			−0.343629			−0.171815	+	0.985129i	$0.554963\pi$
$272$		−	4.24264i		−	0.257248i
$273$		0			0
$274$	14.0000			0.845771
$275$		0			0
$276$	11.3137			0.681005
$277$		−	24.0000i		−	1.44202i	−0.692925	−	0.721010i	$-0.743678\pi$
$277$		−	24.0000i		−	1.44202i	0.692925	−	0.721010i	$-0.256322\pi$
$278$			2.82843i			0.169638i
$279$	−28.2843			−1.69334
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$			25.4558i			1.51319i	0.653882	+	0.756596i	$0.273139\pi$
$283$			25.4558i			1.51319i	−0.653882	+	0.756596i	$0.726861\pi$
$284$	12.0000			0.712069
$285$	5.65685	+	16.9706i	0.335083	+	1.00525i
$286$		0			0
$287$		0			0
$288$			25.0000i			1.47314i
$289$	−1.00000			−0.0588235
$290$		0			0
$291$	−12.0000			−0.703452
$292$			12.7279i			0.744845i
$293$		−	1.41421i		−	0.0826192i	−0.999146	−	0.0413096i	$-0.986847\pi$
$293$		−	1.41421i		−	0.0826192i	0.999146	−	0.0413096i	$-0.0131530\pi$
$294$		0			0
$295$	18.0000	−	6.00000i	1.04800	−	0.349334i
$296$	−18.0000			−1.04623
$297$		0			0
$298$		−	6.00000i		−	0.347571i
$299$	16.9706			0.981433
$300$	−8.48528	−	11.3137i	−0.489898	−	0.653197i
$301$		0			0
$302$		0			0
$303$			12.0000i			0.689382i
$304$	−2.82843			−0.162221
$305$	−21.0000	+	7.00000i	−1.20246	+	0.400819i
$306$	−21.2132			−1.21268
$307$		−	25.4558i		−	1.45284i	−0.687250	−	0.726421i	$-0.741182\pi$
$307$		−	25.4558i		−	1.45284i	0.687250	−	0.726421i	$-0.258818\pi$
$308$		0			0
$309$		0			0
$310$	4.00000	+	12.0000i	0.227185	+	0.681554i
$311$	−33.9411			−1.92462			−0.962312	−	0.271947i	$-0.912333\pi$
$311$	−33.9411			−1.92462			−0.962312	+	0.271947i	$0.912333\pi$
$312$			36.0000i			2.03810i
$313$		−	21.2132i		−	1.19904i	−0.800359	−	0.599521i	$-0.795358\pi$
$313$		−	21.2132i		−	1.19904i	0.800359	−	0.599521i	$-0.204642\pi$
$314$	−4.24264			−0.239426
$315$		0			0
$316$	−12.0000			−0.675053
$317$		−	8.00000i		−	0.449325i	−0.974437	−	0.224662i	$-0.927872\pi$
$317$		−	8.00000i		−	0.449325i	0.974437	−	0.224662i	$-0.0721279\pi$
$318$			22.6274i			1.26888i
$319$		0			0
$320$	14.8492	−	4.94975i	0.830098	−	0.276699i
$321$	−11.3137			−0.631470
$322$		0			0
$323$			12.0000i			0.667698i
$324$	1.00000			0.0555556
$325$	−12.7279	−	16.9706i	−0.706018	−	0.941357i
$326$	24.0000			1.32924
$327$		−	45.2548i		−	2.50260i
$328$		−	12.7279i		−	0.702782i
$329$		0			0
$330$		0			0
$331$	−8.00000			−0.439720			−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000			−0.439720			−0.219860	+	0.975531i	$0.570560\pi$
$332$			8.48528i			0.465690i
$333$			30.0000i			1.64399i
$334$	−11.3137			−0.619059
$335$	−8.48528	−	25.4558i	−0.463600	−	1.39080i
$336$		0			0
$337$		−	18.0000i		−	0.980522i	−0.871576	−	0.490261i	$-0.836901\pi$
$337$		−	18.0000i		−	0.980522i	0.871576	−	0.490261i	$-0.163099\pi$
$338$			5.00000i			0.271964i
$339$	22.6274			1.22895
$340$	−3.00000	−	9.00000i	−0.162698	−	0.488094i
$341$		0			0
$342$			14.1421i			0.764719i
$343$		0			0
$344$		0			0
$345$	−24.0000	+	8.00000i	−1.29212	+	0.430706i
$346$	−4.24264			−0.228086
$347$		−	8.00000i		−	0.429463i	−0.976673	−	0.214731i	$-0.931112\pi$
$347$		−	8.00000i		−	0.429463i	0.976673	−	0.214731i	$-0.0688876\pi$
$348$		0			0
$349$	−1.41421			−0.0757011			−0.0378506	−	0.999283i	$-0.512051\pi$
$349$	−1.41421			−0.0757011			−0.0378506	+	0.999283i	$0.512051\pi$
$350$		0			0
$351$	24.0000			1.28103
$352$		0			0
$353$			15.5563i			0.827981i	0.910281	+	0.413990i	$0.135865\pi$
$353$			15.5563i			0.827981i	−0.910281	+	0.413990i	$0.864135\pi$
$354$	24.0000			1.27559
$355$	−25.4558	+	8.48528i	−1.35106	+	0.450352i
$356$	4.24264			0.224860
$357$		0			0
$358$			12.0000i			0.634220i
$359$	−24.0000			−1.26667			−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000			−1.26667			−0.633336	+	0.773877i	$0.718315\pi$
$360$	−10.6066	−	31.8198i	−0.559017	−	1.67705i
$361$	−11.0000			−0.578947
$362$			18.3848i			0.966282i
$363$			31.1127i			1.63299i
$364$		0			0
$365$	−9.00000	−	27.0000i	−0.471082	−	1.41324i
$366$	−28.0000			−1.46358
$367$		−	16.9706i		−	0.885856i	−0.896557	−	0.442928i	$-0.853940\pi$
$367$		−	16.9706i		−	0.885856i	0.896557	−	0.442928i	$-0.146060\pi$
$368$		−	4.00000i		−	0.208514i
$369$	−21.2132			−1.10432
$370$	12.7279	−	4.24264i	0.661693	−	0.220564i
$371$		0			0
$372$		−	16.0000i		−	0.829561i
$373$		−	24.0000i		−	1.24267i	−0.783544	−	0.621336i	$-0.786590\pi$
$373$		−	24.0000i		−	1.24267i	0.783544	−	0.621336i	$-0.213410\pi$
$374$		0			0
$375$	26.0000	+	18.0000i	1.34263	+	0.929516i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	24.0000			1.23280			0.616399	−	0.787434i	$-0.288591\pi$
$379$	24.0000			1.23280			0.616399	+	0.787434i	$0.288591\pi$
$380$	−6.00000	+	2.00000i	−0.307794	+	0.102598i
$381$	33.9411			1.73886
$382$		−	12.0000i		−	0.613973i
$383$			33.9411i			1.73431i	0.498038	+	0.867155i	$0.334054\pi$
$383$			33.9411i			1.73431i	−0.498038	+	0.867155i	$0.665946\pi$
$384$	−8.48528			−0.433013
$385$		0			0
$386$	24.0000			1.22157
$387$		0			0
$388$		−	4.24264i		−	0.215387i
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$	−8.48528	−	25.4558i	−0.429669	−	1.28901i
$391$	−16.9706			−0.858238
$392$		0			0
$393$			24.0000i			1.21064i
$394$	8.00000			0.403034
$395$	25.4558	−	8.48528i	1.28082	−	0.426941i
$396$		0			0
$397$		−	4.24264i		−	0.212932i	−0.994316	−	0.106466i	$-0.966046\pi$
$397$		−	4.24264i		−	0.212932i	0.994316	−	0.106466i	$-0.0339535\pi$
$398$		−	11.3137i		−	0.567105i
$399$		0			0
$400$	−4.00000	+	3.00000i	−0.200000	+	0.150000i
$401$	24.0000			1.19850			0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000			1.19850			0.599251	+	0.800561i	$0.295465\pi$
$402$		−	33.9411i		−	1.69283i
$403$		−	24.0000i		−	1.19553i
$404$	−4.24264			−0.211079
$405$	−2.12132	+	0.707107i	−0.105409	+	0.0351364i
$406$		0			0
$407$		0			0
$408$		−	36.0000i		−	1.78227i
$409$	−18.3848			−0.909069			−0.454534	−	0.890729i	$-0.650194\pi$
$409$	−18.3848			−0.909069			−0.454534	+	0.890729i	$0.650194\pi$
$410$	3.00000	+	9.00000i	0.148159	+	0.444478i
$411$	39.5980			1.95322
$412$		0			0
$413$		0			0
$414$	−20.0000			−0.982946
$415$	−6.00000	−	18.0000i	−0.294528	−	0.883585i
$416$	−21.2132			−1.04006
$417$			8.00000i			0.391762i
$418$		0			0
$419$	−8.48528			−0.414533			−0.207267	−	0.978285i	$-0.566457\pi$
$419$	−8.48528			−0.414533			−0.207267	+	0.978285i	$0.566457\pi$
$420$		0			0
$421$	−6.00000			−0.292422			−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000			−0.292422			−0.146211	+	0.989253i	$0.546708\pi$
$422$		−	4.00000i		−	0.194717i
$423$		0			0
$424$	−24.0000			−1.16554
$425$	12.7279	+	16.9706i	0.617395	+	0.823193i
$426$	−33.9411			−1.64445
$427$		0			0
$428$		−	4.00000i		−	0.193347i
$429$		0			0
$430$		0			0
$431$	24.0000			1.15604			0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000			1.15604			0.578020	+	0.816023i	$0.303826\pi$
$432$		−	5.65685i		−	0.272166i
$433$			4.24264i			0.203888i	0.994790	+	0.101944i	$0.0325063\pi$
$433$			4.24264i			0.203888i	−0.994790	+	0.101944i	$0.967494\pi$
$434$		0			0
$435$		0			0
$436$	16.0000			0.766261
$437$			11.3137i			0.541208i
$438$		−	36.0000i		−	1.72015i
$439$	−28.2843			−1.34993			−0.674967	−	0.737848i	$-0.735842\pi$
$439$	−28.2843			−1.34993			−0.674967	+	0.737848i	$0.735842\pi$
$440$		0			0
$441$		0			0
$442$		−	18.0000i		−	0.856173i
$443$		−	4.00000i		−	0.190046i	−0.995475	−	0.0950229i	$-0.969708\pi$
$443$		−	4.00000i		−	0.190046i	0.995475	−	0.0950229i	$-0.0302924\pi$
$444$	−16.9706			−0.805387
$445$	−9.00000	+	3.00000i	−0.426641	+	0.142214i
$446$		0			0
$447$		−	16.9706i		−	0.802680i
$448$		0			0
$449$	18.0000			0.849473			0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000			0.849473			0.424736	+	0.905317i	$0.360367\pi$
$450$	15.0000	+	20.0000i	0.707107	+	0.942809i
$451$		0			0
$452$			8.00000i			0.376288i
$453$		0			0
$454$	−2.82843			−0.132745
$455$		0			0
$456$	−24.0000			−1.12390
$457$			24.0000i			1.12267i	0.827588	+	0.561336i	$0.189713\pi$
$457$			24.0000i			1.12267i	−0.827588	+	0.561336i	$0.810287\pi$
$458$			7.07107i			0.330409i
$459$	−24.0000			−1.12022
$460$	−2.82843	−	8.48528i	−0.131876	−	0.395628i
$461$	21.2132			0.987997			0.493999	−	0.869463i	$-0.335535\pi$
$461$	21.2132			0.987997			0.493999	+	0.869463i	$0.335535\pi$
$462$		0			0
$463$			24.0000i			1.11537i	0.830051	+	0.557687i	$0.188311\pi$
$463$			24.0000i			1.11537i	−0.830051	+	0.557687i	$0.811689\pi$
$464$		0			0
$465$	11.3137	+	33.9411i	0.524661	+	1.57398i
$466$	−14.0000			−0.648537
$467$			31.1127i			1.43972i	0.694117	+	0.719862i	$0.255795\pi$
$467$			31.1127i			1.43972i	−0.694117	+	0.719862i	$0.744205\pi$
$468$			21.2132i			0.980581i
$469$		0			0
$470$		0			0
$471$	−12.0000			−0.552931
$472$			25.4558i			1.17170i
$473$		0			0
$474$	33.9411			1.55897
$475$	11.3137	−	8.48528i	0.519109	−	0.389331i
$476$		0			0
$477$			40.0000i			1.83147i
$478$			24.0000i			1.09773i
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$	30.0000	−	10.0000i	1.36931	−	0.456435i
$481$	−25.4558			−1.16069
$482$			9.89949i			0.450910i
$483$		0			0
$484$	−11.0000			−0.500000
$485$	3.00000	+	9.00000i	0.136223	+	0.408669i
$486$	14.1421			0.641500
$487$			12.0000i			0.543772i	0.962329	+	0.271886i	$0.0876473\pi$
$487$			12.0000i			0.543772i	−0.962329	+	0.271886i	$0.912353\pi$
$488$		−	29.6985i		−	1.34439i
$489$	67.8823			3.06974
$490$		0			0
$491$	−12.0000			−0.541552			−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000			−0.541552			−0.270776	+	0.962642i	$0.587280\pi$
$492$		−	12.0000i		−	0.541002i
$493$		0			0
$494$	−12.0000			−0.539906
$495$		0			0
$496$	−5.65685			−0.254000
$497$		0			0
$498$		−	24.0000i		−	1.07547i
$499$	12.0000			0.537194			0.268597	−	0.963253i	$-0.413440\pi$
$499$	12.0000			0.537194			0.268597	+	0.963253i	$0.413440\pi$
$500$	−6.36396	+	9.19239i	−0.284605	+	0.411096i
$501$	−32.0000			−1.42965
$502$		−	25.4558i		−	1.13615i
$503$			16.9706i			0.756680i	0.925667	+	0.378340i	$0.123505\pi$
$503$			16.9706i			0.756680i	−0.925667	+	0.378340i	$0.876495\pi$
$504$		0			0
$505$	9.00000	−	3.00000i	0.400495	−	0.133498i
$506$		0			0
$507$			14.1421i			0.628074i
$508$			12.0000i			0.532414i
$509$	12.7279			0.564155			0.282078	−	0.959392i	$-0.408976\pi$
$509$	12.7279			0.564155			0.282078	+	0.959392i	$0.408976\pi$
$510$	8.48528	+	25.4558i	0.375735	+	1.12720i
$511$		0			0
$512$			11.0000i			0.486136i
$513$			16.0000i			0.706417i
$514$	−26.8701			−1.18519
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−12.0000			−0.526742
$520$	27.0000	−	9.00000i	1.18403	−	0.394676i
$521$	29.6985			1.30111			0.650557	−	0.759457i	$-0.274535\pi$
$521$	29.6985			1.30111			0.650557	+	0.759457i	$0.274535\pi$
$522$		0			0
$523$		−	8.48528i		−	0.371035i	−0.982641	−	0.185518i	$-0.940604\pi$
$523$		−	8.48528i		−	0.371035i	0.982641	−	0.185518i	$-0.0593962\pi$
$524$	−8.48528			−0.370681
$525$		0			0
$526$	−16.0000			−0.697633
$527$			24.0000i			1.04546i
$528$		0			0
$529$	7.00000			0.304348
$530$	16.9706	−	5.65685i	0.737154	−	0.245718i
$531$	42.4264			1.84115
$532$		0			0
$533$		−	18.0000i		−	0.779667i
$534$	−12.0000			−0.519291
$535$	2.82843	+	8.48528i	0.122284	+	0.366851i
$536$	36.0000			1.55496
$537$			33.9411i			1.46467i
$538$		−	21.2132i		−	0.914566i
$539$		0			0
$540$	−4.00000	−	12.0000i	−0.172133	−	0.516398i
$541$	18.0000			0.773880			0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000			0.773880			0.386940	+	0.922105i	$0.373532\pi$
$542$			5.65685i			0.242983i
$543$			52.0000i			2.23153i
$544$	21.2132			0.909509
$545$	−33.9411	+	11.3137i	−1.45388	+	0.484626i
$546$		0			0
$547$			24.0000i			1.02617i	0.858339	+	0.513083i	$0.171497\pi$
$547$			24.0000i			1.02617i	−0.858339	+	0.513083i	$0.828503\pi$
$548$			14.0000i			0.598050i
$549$	−49.4975			−2.11250
$550$		0			0
$551$		0			0
$552$		−	33.9411i		−	1.44463i
$553$		0			0
$554$	−24.0000			−1.01966
$555$	36.0000	−	12.0000i	1.52811	−	0.509372i
$556$	−2.82843			−0.119952
$557$		−	40.0000i		−	1.69485i	−0.530912	−	0.847427i	$-0.678150\pi$
$557$		−	40.0000i		−	1.69485i	0.530912	−	0.847427i	$-0.321850\pi$
$558$			28.2843i			1.19737i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	25.4558i		−	1.07284i	−0.843952	−	0.536418i	$-0.819777\pi$
$563$		−	25.4558i		−	1.07284i	0.843952	−	0.536418i	$-0.180223\pi$
$564$		0			0
$565$	−5.65685	−	16.9706i	−0.237986	−	0.713957i
$566$	25.4558			1.06999
$567$		0			0
$568$		−	36.0000i		−	1.51053i
$569$	−24.0000			−1.00613			−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000			−1.00613			−0.503066	+	0.864248i	$0.667795\pi$
$570$	16.9706	−	5.65685i	0.710819	−	0.236940i
$571$	−32.0000			−1.33916			−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000			−1.33916			−0.669579	+	0.742741i	$0.733526\pi$
$572$		0			0
$573$		−	33.9411i		−	1.41791i
$574$		0			0
$575$	12.0000	+	16.0000i	0.500435	+	0.667246i
$576$	35.0000			1.45833
$577$		−	38.1838i		−	1.58961i	−0.606864	−	0.794805i	$-0.707573\pi$
$577$		−	38.1838i		−	1.58961i	0.606864	−	0.794805i	$-0.292427\pi$
$578$			1.00000i			0.0415945i
$579$	67.8823			2.82109
$580$		0			0
$581$		0			0
$582$			12.0000i			0.497416i
$583$		0			0
$584$	38.1838			1.58006
$585$	−15.0000	−	45.0000i	−0.620174	−	1.86052i
$586$	−1.41421			−0.0584206
$587$			42.4264i			1.75113i	0.483105	+	0.875563i	$0.339509\pi$
$587$			42.4264i			1.75113i	−0.483105	+	0.875563i	$0.660491\pi$
$588$		0			0
$589$	16.0000			0.659269
$590$	−6.00000	−	18.0000i	−0.247016	−	0.741048i
$591$	22.6274			0.930768
$592$			6.00000i			0.246598i
$593$		−	7.07107i		−	0.290374i	−0.989404	−	0.145187i	$-0.953622\pi$
$593$		−	7.07107i		−	0.290374i	0.989404	−	0.145187i	$-0.0463784\pi$
$594$		0			0
$595$		0			0
$596$	6.00000			0.245770
$597$		−	32.0000i		−	1.30967i
$598$		−	16.9706i		−	0.693978i
$599$	−12.0000			−0.490307			−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000			−0.490307			−0.245153	+	0.969484i	$0.578838\pi$
$600$	−33.9411	+	25.4558i	−1.38564	+	1.03923i
$601$	−41.0122			−1.67292			−0.836461	−	0.548026i	$-0.815379\pi$
$601$	−41.0122			−1.67292			−0.836461	+	0.548026i	$0.815379\pi$
$602$		0			0
$603$		−	60.0000i		−	2.44339i
$604$		0			0
$605$	23.3345	−	7.77817i	0.948683	−	0.316228i
$606$	12.0000			0.487467
$607$			33.9411i			1.37763i	0.724938	+	0.688814i	$0.241868\pi$
$607$			33.9411i			1.37763i	−0.724938	+	0.688814i	$0.758132\pi$
$608$		−	14.1421i		−	0.573539i
$609$		0			0
$610$	7.00000	+	21.0000i	0.283422	+	0.850265i
$611$		0			0
$612$		−	21.2132i		−	0.857493i
$613$		−	6.00000i		−	0.242338i	−0.992632	−	0.121169i	$-0.961336\pi$
$613$		−	6.00000i		−	0.242338i	0.992632	−	0.121169i	$-0.0386643\pi$
$614$	−25.4558			−1.02731
$615$	8.48528	+	25.4558i	0.342160	+	1.02648i
$616$		0			0
$617$		−	38.0000i		−	1.52982i	−0.644136	−	0.764911i	$-0.722783\pi$
$617$		−	38.0000i		−	1.52982i	0.644136	−	0.764911i	$-0.277217\pi$
$618$		0			0
$619$	19.7990			0.795789			0.397894	−	0.917431i	$-0.369741\pi$
$619$	19.7990			0.795789			0.397894	+	0.917431i	$0.369741\pi$
$620$	−12.0000	+	4.00000i	−0.481932	+	0.160644i
$621$	−22.6274			−0.908007
$622$			33.9411i			1.36092i
$623$		0			0
$624$	12.0000			0.480384
$625$	7.00000	−	24.0000i	0.280000	−	0.960000i
$626$	−21.2132			−0.847850
$627$		0			0
$628$		−	4.24264i		−	0.169300i
$629$	25.4558			1.01499
$630$		0			0
$631$	12.0000			0.477712			0.238856	−	0.971055i	$-0.423228\pi$
$631$	12.0000			0.477712			0.238856	+	0.971055i	$0.423228\pi$
$632$			36.0000i			1.43200i
$633$		−	11.3137i		−	0.449680i
$634$	−8.00000			−0.317721
$635$	−8.48528	−	25.4558i	−0.336728	−	1.01018i
$636$	−22.6274			−0.897235
$637$		0			0
$638$		0			0
$639$	−60.0000			−2.37356
$640$	2.12132	+	6.36396i	0.0838525	+	0.251558i
$641$	24.0000			0.947943			0.473972	−	0.880540i	$-0.342820\pi$
$641$	24.0000			0.947943			0.473972	+	0.880540i	$0.342820\pi$
$642$			11.3137i			0.446516i
$643$			8.48528i			0.334627i	0.985904	+	0.167313i	$0.0535092\pi$
$643$			8.48528i			0.334627i	−0.985904	+	0.167313i	$0.946491\pi$
$644$		0			0
$645$		0			0
$646$	12.0000			0.472134
$647$			5.65685i			0.222394i	0.993798	+	0.111197i	$0.0354684\pi$
$647$			5.65685i			0.222394i	−0.993798	+	0.111197i	$0.964532\pi$
$648$		−	3.00000i		−	0.117851i
$649$		0			0
$650$	−16.9706	+	12.7279i	−0.665640	+	0.499230i
$651$		0			0
$652$			24.0000i			0.939913i
$653$			2.00000i			0.0782660i	0.999234	+	0.0391330i	$0.0124596\pi$
$653$			2.00000i			0.0782660i	−0.999234	+	0.0391330i	$0.987540\pi$
$654$	−45.2548			−1.76960
$655$	18.0000	−	6.00000i	0.703318	−	0.234439i
$656$	−4.24264			−0.165647
$657$		−	63.6396i		−	2.48282i
$658$		0			0
$659$	−48.0000			−1.86981			−0.934907	−	0.354892i	$-0.884518\pi$
$659$	−48.0000			−1.86981			−0.934907	+	0.354892i	$0.884518\pi$
$660$		0			0
$661$	−15.5563			−0.605072			−0.302536	−	0.953138i	$-0.597833\pi$
$661$	−15.5563			−0.605072			−0.302536	+	0.953138i	$0.597833\pi$
$662$			8.00000i			0.310929i
$663$		−	50.9117i		−	1.97725i
$664$	25.4558			0.987878
$665$		0			0
$666$	30.0000			1.16248
$667$		0			0
$668$		−	11.3137i		−	0.437741i
$669$		0			0
$670$	−25.4558	+	8.48528i	−0.983445	+	0.327815i
$671$		0			0
$672$		0			0
$673$			6.00000i			0.231283i	0.993291	+	0.115642i	$0.0368924\pi$
$673$			6.00000i			0.231283i	−0.993291	+	0.115642i	$0.963108\pi$
$674$	−18.0000			−0.693334
$675$	16.9706	+	22.6274i	0.653197	+	0.870930i
$676$	−5.00000			−0.192308
$677$		−	9.89949i		−	0.380468i	−0.981739	−	0.190234i	$-0.939075\pi$
$677$		−	9.89949i		−	0.380468i	0.981739	−	0.190234i	$-0.0609248\pi$
$678$		−	22.6274i		−	0.869001i
$679$		0			0
$680$	−27.0000	+	9.00000i	−1.03540	+	0.345134i
$681$	−8.00000			−0.306561
$682$		0			0
$683$			28.0000i			1.07139i	0.844411	+	0.535695i	$0.179950\pi$
$683$			28.0000i			1.07139i	−0.844411	+	0.535695i	$0.820050\pi$
$684$	−14.1421			−0.540738
$685$	−9.89949	−	29.6985i	−0.378240	−	1.13472i
$686$		0			0
$687$			20.0000i			0.763048i
$688$		0			0
$689$	−33.9411			−1.29305
$690$	8.00000	+	24.0000i	0.304555	+	0.913664i
$691$	2.82843			0.107598			0.0537992	−	0.998552i	$-0.482867\pi$
$691$	2.82843			0.107598			0.0537992	+	0.998552i	$0.482867\pi$
$692$		−	4.24264i		−	0.161281i
$693$		0			0
$694$	−8.00000			−0.303676
$695$	6.00000	−	2.00000i	0.227593	−	0.0758643i
$696$		0			0
$697$			18.0000i			0.681799i
$698$			1.41421i			0.0535288i
$699$	−39.5980			−1.49773
$700$		0			0
$701$	48.0000			1.81293			0.906467	−	0.422276i	$-0.138769\pi$
$701$	48.0000			1.81293			0.906467	+	0.422276i	$0.138769\pi$
$702$		−	24.0000i		−	0.905822i
$703$		−	16.9706i		−	0.640057i
$704$		0			0
$705$		0			0
$706$	15.5563			0.585471
$707$		0			0
$708$			24.0000i			0.901975i
$709$		0			0			−	1.00000i	$-0.5\pi$
$709$		0			0				1.00000i	$0.5\pi$
$710$	8.48528	+	25.4558i	0.318447	+	0.955341i
$711$	60.0000			2.25018
$712$		−	12.7279i		−	0.476999i
$713$			22.6274i			0.847403i
$714$		0			0
$715$		0			0
$716$	−12.0000			−0.448461
$717$			67.8823i			2.53511i
$718$			24.0000i			0.895672i
$719$	16.9706			0.632895			0.316448	−	0.948610i	$-0.397510\pi$
$719$	16.9706			0.632895			0.316448	+	0.948610i	$0.397510\pi$
$720$	−10.6066	+	3.53553i	−0.395285	+	0.131762i
$721$		0			0
$722$			11.0000i			0.409378i
$723$			28.0000i			1.04133i
$724$	−18.3848			−0.683265
$725$		0			0
$726$	31.1127			1.15470
$727$			33.9411i			1.25881i	0.777079	+	0.629403i	$0.216701\pi$
$727$			33.9411i			1.25881i	−0.777079	+	0.629403i	$0.783299\pi$
$728$		0			0
$729$	43.0000			1.59259
$730$	−27.0000	+	9.00000i	−0.999315	+	0.333105i
$731$		0			0
$732$		−	28.0000i		−	1.03491i
$733$			4.24264i			0.156706i	0.996926	+	0.0783528i	$0.0249660\pi$
$733$			4.24264i			0.156706i	−0.996926	+	0.0783528i	$0.975034\pi$
$734$	−16.9706			−0.626395
$735$		0			0
$736$	20.0000			0.737210
$737$		0			0
$738$			21.2132i			0.780869i
$739$	−16.0000			−0.588570			−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000			−0.588570			−0.294285	+	0.955718i	$0.595081\pi$
$740$	4.24264	+	12.7279i	0.155963	+	0.467888i
$741$	−33.9411			−1.24686
$742$		0			0
$743$			28.0000i			1.02722i	0.858024	+	0.513610i	$0.171692\pi$
$743$			28.0000i			1.02722i	−0.858024	+	0.513610i	$0.828308\pi$
$744$	−48.0000			−1.75977
$745$	−12.7279	+	4.24264i	−0.466315	+	0.155438i
$746$	−24.0000			−0.878702
$747$		−	42.4264i		−	1.55230i
$748$		0			0
$749$		0			0
$750$	18.0000	−	26.0000i	0.657267	−	0.949386i
$751$	−40.0000			−1.45962			−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000			−1.45962			−0.729810	+	0.683650i	$0.760392\pi$
$752$		0			0
$753$		−	72.0000i		−	2.62383i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			6.00000i			0.218074i	0.994038	+	0.109037i	$0.0347767\pi$
$757$			6.00000i			0.218074i	−0.994038	+	0.109037i	$0.965223\pi$
$758$		−	24.0000i		−	0.871719i
$759$		0			0
$760$	6.00000	+	18.0000i	0.217643	+	0.652929i
$761$	−29.6985			−1.07657			−0.538285	−	0.842763i	$-0.680927\pi$
$761$	−29.6985			−1.07657			−0.538285	+	0.842763i	$0.680927\pi$
$762$		−	33.9411i		−	1.22956i
$763$		0			0
$764$	12.0000			0.434145
$765$	15.0000	+	45.0000i	0.542326	+	1.62698i
$766$	33.9411			1.22634
$767$			36.0000i			1.29988i
$768$			48.0833i			1.73506i
$769$	−24.0416			−0.866963			−0.433482	−	0.901162i	$-0.642715\pi$
$769$	−24.0416			−0.866963			−0.433482	+	0.901162i	$0.642715\pi$
$770$		0			0
$771$	−76.0000			−2.73707
$772$			24.0000i			0.863779i
$773$			21.2132i			0.762986i	0.924372	+	0.381493i	$0.124590\pi$
$773$			21.2132i			0.762986i	−0.924372	+	0.381493i	$0.875410\pi$
$774$		0			0
$775$	22.6274	−	16.9706i	0.812801	−	0.609601i
$776$	−12.7279			−0.456906
$777$		0			0
$778$		0			0
$779$	12.0000			0.429945
$780$	25.4558	−	8.48528i	0.911465	−	0.303822i
$781$		0			0
$782$			16.9706i			0.606866i
$783$		0			0
$784$		0			0
$785$	3.00000	+	9.00000i	0.107075	+	0.321224i
$786$	24.0000			0.856052
$787$			8.48528i			0.302468i	0.988498	+	0.151234i	$0.0483246\pi$
$787$			8.48528i			0.302468i	−0.988498	+	0.151234i	$0.951675\pi$
$788$			8.00000i			0.284988i
$789$	−45.2548			−1.61111
$790$	−8.48528	−	25.4558i	−0.301893	−	0.905678i
$791$		0			0
$792$		0			0
$793$		−	42.0000i		−	1.49146i
$794$	−4.24264			−0.150566
$795$	48.0000	−	16.0000i	1.70238	−	0.567462i
$796$	11.3137			0.401004
$797$		−	4.24264i		−	0.150282i	−0.997173	−	0.0751410i	$-0.976059\pi$
$797$		−	4.24264i		−	0.150282i	0.997173	−	0.0751410i	$-0.0239407\pi$
$798$		0			0
$799$		0			0
$800$	−15.0000	−	20.0000i	−0.530330	−	0.707107i
$801$	−21.2132			−0.749532
$802$		−	24.0000i		−	0.847469i
$803$		0			0
$804$	33.9411			1.19701
$805$		0			0
$806$	−24.0000			−0.845364
$807$		−	60.0000i		−	2.11210i
$808$			12.7279i			0.447767i
$809$	−30.0000			−1.05474			−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000			−1.05474			−0.527372	+	0.849635i	$0.676823\pi$
$810$	0.707107	+	2.12132i	0.0248452	+	0.0745356i
$811$	14.1421			0.496598			0.248299	−	0.968683i	$-0.420129\pi$
$811$	14.1421			0.496598			0.248299	+	0.968683i	$0.420129\pi$
$812$		0			0
$813$			16.0000i			0.561144i
$814$		0			0
$815$	−16.9706	−	50.9117i	−0.594453	−	1.78336i
$816$	−12.0000			−0.420084
$817$		0			0
$818$			18.3848i			0.642809i
$819$		0			0
$820$	−9.00000	+	3.00000i	−0.314294	+	0.104765i
$821$	−6.00000			−0.209401			−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000			−0.209401			−0.104701	+	0.994504i	$0.533388\pi$
$822$		−	39.5980i		−	1.38114i
$823$		−	24.0000i		−	0.836587i	−0.908312	−	0.418294i	$-0.862628\pi$
$823$		−	24.0000i		−	0.836587i	0.908312	−	0.418294i	$-0.137372\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	52.0000i		−	1.80822i	−0.427303	−	0.904109i	$-0.640536\pi$
$827$		−	52.0000i		−	1.80822i	0.427303	−	0.904109i	$-0.359464\pi$
$828$		−	20.0000i		−	0.695048i
$829$	−24.0416			−0.835000			−0.417500	−	0.908677i	$-0.637094\pi$
$829$	−24.0416			−0.835000			−0.417500	+	0.908677i	$0.637094\pi$
$830$	−18.0000	+	6.00000i	−0.624789	+	0.208263i
$831$	−67.8823			−2.35481
$832$			29.6985i			1.02961i
$833$		0			0
$834$	8.00000			0.277017
$835$	8.00000	+	24.0000i	0.276851	+	0.830554i
$836$		0			0
$837$			32.0000i			1.10608i
$838$			8.48528i			0.293119i
$839$	−16.9706			−0.585889			−0.292944	−	0.956129i	$-0.594635\pi$
$839$	−16.9706			−0.585889			−0.292944	+	0.956129i	$0.594635\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$			6.00000i			0.206774i
$843$		0			0
$844$	4.00000			0.137686
$845$	10.6066	−	3.53553i	0.364878	−	0.121626i
$846$		0			0
$847$		0			0
$848$			8.00000i			0.274721i
$849$	72.0000			2.47103
$850$	16.9706	−	12.7279i	0.582086	−	0.436564i
$851$	24.0000			0.822709
$852$		−	33.9411i		−	1.16280i
$853$			12.7279i			0.435796i	0.975972	+	0.217898i	$0.0699200\pi$
$853$			12.7279i			0.435796i	−0.975972	+	0.217898i	$0.930080\pi$
$854$		0			0
$855$	30.0000	−	10.0000i	1.02598	−	0.341993i
$856$	−12.0000			−0.410152
$857$		−	55.1543i		−	1.88404i	−0.335562	−	0.942018i	$-0.608926\pi$
$857$		−	55.1543i		−	1.88404i	0.335562	−	0.942018i	$-0.391074\pi$
$858$		0			0
$859$	31.1127			1.06155			0.530776	−	0.847512i	$-0.321901\pi$
$859$	31.1127			1.06155			0.530776	+	0.847512i	$0.321901\pi$
$860$		0			0
$861$		0			0
$862$		−	24.0000i		−	0.817443i
$863$		−	16.0000i		−	0.544646i	−0.962206	−	0.272323i	$-0.912208\pi$
$863$		−	16.0000i		−	0.544646i	0.962206	−	0.272323i	$-0.0877920\pi$
$864$	28.2843			0.962250
$865$	3.00000	+	9.00000i	0.102003	+	0.306009i
$866$	4.24264			0.144171
$867$			2.82843i			0.0960584i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	50.9117			1.72508
$872$		−	48.0000i		−	1.62549i
$873$			21.2132i			0.717958i
$874$	11.3137			0.382692
$875$		0			0
$876$	36.0000			1.21633
$877$			30.0000i			1.01303i	0.862232	+	0.506514i	$0.169066\pi$
$877$			30.0000i			1.01303i	−0.862232	+	0.506514i	$0.830934\pi$
$878$			28.2843i			0.954548i
$879$	−4.00000			−0.134917
$880$		0			0
$881$	46.6690			1.57232			0.786160	−	0.618023i	$-0.212066\pi$
$881$	46.6690			1.57232			0.786160	+	0.618023i	$0.212066\pi$
$882$		0			0
$883$			36.0000i			1.21150i	0.795656	+	0.605748i	$0.207126\pi$
$883$			36.0000i			1.21150i	−0.795656	+	0.605748i	$0.792874\pi$
$884$	18.0000			0.605406
$885$	−16.9706	−	50.9117i	−0.570459	−	1.71138i
$886$	−4.00000			−0.134383
$887$			16.9706i			0.569816i	0.958555	+	0.284908i	$0.0919630\pi$
$887$			16.9706i			0.569816i	−0.958555	+	0.284908i	$0.908037\pi$
$888$			50.9117i			1.70848i
$889$		0			0
$890$	3.00000	+	9.00000i	0.100560	+	0.301681i
$891$		0			0
$892$		0			0
$893$		0			0
$894$	−16.9706			−0.567581
$895$	25.4558	−	8.48528i	0.850895	−	0.283632i
$896$		0			0
$897$		−	48.0000i		−	1.60267i
$898$		−	18.0000i		−	0.600668i
$899$		0			0
$900$	−20.0000	+	15.0000i	−0.666667	+	0.500000i
$901$	33.9411			1.13074
$902$		0			0
$903$		0			0
$904$	24.0000			0.798228
$905$	39.0000	−	13.0000i	1.29640	−	0.432135i
$906$		0			0
$907$			12.0000i			0.398453i	0.979953	+	0.199227i	$0.0638430\pi$
$907$			12.0000i			0.398453i	−0.979953	+	0.199227i	$0.936157\pi$
$908$		−	2.82843i		−	0.0938647i
$909$	21.2132			0.703598
$910$		0			0
$911$	24.0000			0.795155			0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000			0.795155			0.397578	+	0.917568i	$0.369851\pi$
$912$			8.00000i			0.264906i
$913$		0			0
$914$	24.0000			0.793849
$915$	19.7990	+	59.3970i	0.654534	+	1.96360i
$916$	−7.07107			−0.233635
$917$		0			0
$918$			24.0000i			0.792118i
$919$	−20.0000			−0.659739			−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000			−0.659739			−0.329870	+	0.944027i	$0.607005\pi$
$920$	−25.4558	+	8.48528i	−0.839254	+	0.279751i
$921$	−72.0000			−2.37248
$922$		−	21.2132i		−	0.698620i
$923$		−	50.9117i		−	1.67578i
$924$		0			0
$925$	−18.0000	−	24.0000i	−0.591836	−	0.789115i
$926$	24.0000			0.788689
$927$		0			0
$928$		0			0
$929$	12.7279			0.417590			0.208795	−	0.977959i	$-0.433046\pi$
$929$	12.7279			0.417590			0.208795	+	0.977959i	$0.433046\pi$
$930$	33.9411	−	11.3137i	1.11297	−	0.370991i
$931$		0			0
$932$		−	14.0000i		−	0.458585i
$933$			96.0000i			3.14290i
$934$	31.1127			1.01804
$935$		0			0
$936$	63.6396			2.08013
$937$			12.7279i			0.415803i	0.978150	+	0.207902i	$0.0666634\pi$
$937$			12.7279i			0.415803i	−0.978150	+	0.207902i	$0.933337\pi$
$938$		0			0
$939$	−60.0000			−1.95803
$940$		0			0
$941$	−21.2132			−0.691531			−0.345765	−	0.938321i	$-0.612381\pi$
$941$	−21.2132			−0.691531			−0.345765	+	0.938321i	$0.612381\pi$
$942$			12.0000i			0.390981i
$943$			16.9706i			0.552638i
$944$	8.48528			0.276172
$945$		0			0
$946$		0			0
$947$			8.00000i			0.259965i	0.991516	+	0.129983i	$0.0414921\pi$
$947$			8.00000i			0.259965i	−0.991516	+	0.129983i	$0.958508\pi$
$948$			33.9411i			1.10236i
$949$	54.0000			1.75291
$950$	−8.48528	−	11.3137i	−0.275299	−	0.367065i
$951$	−22.6274			−0.733744
$952$		0			0
$953$			32.0000i			1.03658i	0.855204	+	0.518291i	$0.173432\pi$
$953$			32.0000i			1.03658i	−0.855204	+	0.518291i	$0.826568\pi$
$954$	40.0000			1.29505
$955$	−25.4558	+	8.48528i	−0.823732	+	0.274577i
$956$	−24.0000			−0.776215
$957$		0			0
$958$		0			0
$959$		0			0
$960$	−14.0000	−	42.0000i	−0.451848	−	1.35554i
$961$	1.00000			0.0322581
$962$			25.4558i			0.820729i
$963$			20.0000i			0.644491i
$964$	−9.89949			−0.318841
$965$	−16.9706	−	50.9117i	−0.546302	−	1.63891i
$966$		0			0
$967$		−	36.0000i		−	1.15768i	−0.815440	−	0.578841i	$-0.803505\pi$
$967$		−	36.0000i		−	1.15768i	0.815440	−	0.578841i	$-0.196495\pi$
$968$			33.0000i			1.06066i
$969$	33.9411			1.09035
$970$	9.00000	−	3.00000i	0.288973	−	0.0963242i
$971$	8.48528			0.272306			0.136153	−	0.990688i	$-0.456526\pi$
$971$	8.48528			0.272306			0.136153	+	0.990688i	$0.456526\pi$
$972$			14.1421i			0.453609i
$973$		0			0
$974$	12.0000			0.384505
$975$	−48.0000	+	36.0000i	−1.53723	+	1.15292i
$976$	−9.89949			−0.316875
$977$			10.0000i			0.319928i	0.987123	+	0.159964i	$0.0511379\pi$
$977$			10.0000i			0.319928i	−0.987123	+	0.159964i	$0.948862\pi$
$978$		−	67.8823i		−	2.17064i
$979$		0			0
$980$		0			0
$981$	−80.0000			−2.55420
$982$			12.0000i			0.382935i
$983$			28.2843i			0.902128i	0.892492	+	0.451064i	$0.148955\pi$
$983$			28.2843i			0.902128i	−0.892492	+	0.451064i	$0.851045\pi$
$984$	−36.0000			−1.14764
$985$	−5.65685	−	16.9706i	−0.180242	−	0.540727i
$986$		0			0
$987$		0			0
$988$		−	12.0000i		−	0.381771i
$989$		0			0
$990$		0			0
$991$	20.0000			0.635321			0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000			0.635321			0.317660	+	0.948205i	$0.397103\pi$
$992$		−	28.2843i		−	0.898027i
$993$			22.6274i			0.718059i
$994$		0			0
$995$	−24.0000	+	8.00000i	−0.760851	+	0.253617i
$996$	24.0000			0.760469
$997$			55.1543i			1.74676i	0.487044	+	0.873378i	$0.338075\pi$
$997$			55.1543i			1.74676i	−0.487044	+	0.873378i	$0.661925\pi$
$998$		−	12.0000i		−	0.379853i
$999$	33.9411			1.07385

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.2.b.f.99.1	✓	4
3.2	odd	2		2205.2.d.k.1324.4		4
5.2	odd	4		1225.2.a.v.1.1		2
5.3	odd	4		1225.2.a.l.1.2		2
5.4	even	2	inner	245.2.b.f.99.4	yes	4
7.2	even	3		245.2.j.g.214.2		8
7.3	odd	6		245.2.j.g.79.4		8
7.4	even	3		245.2.j.g.79.3		8
7.5	odd	6		245.2.j.g.214.1		8
7.6	odd	2	inner	245.2.b.f.99.2	yes	4
15.14	odd	2		2205.2.d.k.1324.2		4
21.20	even	2		2205.2.d.k.1324.3		4
35.4	even	6		245.2.j.g.79.2		8
35.9	even	6		245.2.j.g.214.3		8
35.13	even	4		1225.2.a.l.1.1		2
35.19	odd	6		245.2.j.g.214.4		8
35.24	odd	6		245.2.j.g.79.1		8
35.27	even	4		1225.2.a.v.1.2		2
35.34	odd	2	inner	245.2.b.f.99.3	yes	4
105.104	even	2		2205.2.d.k.1324.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.b.f.99.1	✓	4	1.1	even	1	trivial
245.2.b.f.99.2	yes	4	7.6	odd	2	inner
245.2.b.f.99.3	yes	4	35.34	odd	2	inner
245.2.b.f.99.4	yes	4	5.4	even	2	inner
245.2.j.g.79.1		8	35.24	odd	6
245.2.j.g.79.2		8	35.4	even	6
245.2.j.g.79.3		8	7.4	even	3
245.2.j.g.79.4		8	7.3	odd	6
245.2.j.g.214.1		8	7.5	odd	6
245.2.j.g.214.2		8	7.2	even	3
245.2.j.g.214.3		8	35.9	even	6
245.2.j.g.214.4		8	35.19	odd	6
1225.2.a.l.1.1		2	35.13	even	4
1225.2.a.l.1.2		2	5.3	odd	4
1225.2.a.v.1.1		2	5.2	odd	4
1225.2.a.v.1.2		2	35.27	even	4
2205.2.d.k.1324.1		4	105.104	even	2
2205.2.d.k.1324.2		4	15.14	odd	2
2205.2.d.k.1324.3		4	21.20	even	2
2205.2.d.k.1324.4		4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	245.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -2.82843i q^{3} +1.00000 q^{4} +(-2.12132 + 0.707107i) q^{5} -2.82843 q^{6} -3.00000i q^{8} -5.00000 q^{9} +(0.707107 + 2.12132i) q^{10} -2.82843i q^{12} -4.24264i q^{13} +(2.00000 + 6.00000i) q^{15} -1.00000 q^{16} +4.24264i q^{17} +5.00000i q^{18} +2.82843 q^{19} +(-2.12132 + 0.707107i) q^{20} +4.00000i q^{23} -8.48528 q^{24} +(4.00000 - 3.00000i) q^{25} -4.24264 q^{26} +5.65685i q^{27} +(6.00000 - 2.00000i) q^{30} +5.65685 q^{31} -5.00000i q^{32} +4.24264 q^{34} -5.00000 q^{36} -6.00000i q^{37} -2.82843i q^{38} -12.0000 q^{39} +(2.12132 + 6.36396i) q^{40} +4.24264 q^{41} +(10.6066 - 3.53553i) q^{45} +4.00000 q^{46} +2.82843i q^{48} +(-3.00000 - 4.00000i) q^{50} +12.0000 q^{51} -4.24264i q^{52} -8.00000i q^{53} +5.65685 q^{54} -8.00000i q^{57} -8.48528 q^{59} +(2.00000 + 6.00000i) q^{60} +9.89949 q^{61} -5.65685i q^{62} -7.00000 q^{64} +(3.00000 + 9.00000i) q^{65} +12.0000i q^{67} +4.24264i q^{68} +11.3137 q^{69} +12.0000 q^{71} +15.0000i q^{72} +12.7279i q^{73} -6.00000 q^{74} +(-8.48528 - 11.3137i) q^{75} +2.82843 q^{76} +12.0000i q^{78} -12.0000 q^{79} +(2.12132 - 0.707107i) q^{80} +1.00000 q^{81} -4.24264i q^{82} +8.48528i q^{83} +(-3.00000 - 9.00000i) q^{85} +4.24264 q^{89} +(-3.53553 - 10.6066i) q^{90} +4.00000i q^{92} -16.0000i q^{93} +(-6.00000 + 2.00000i) q^{95} -14.1421 q^{96} -4.24264i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} - 20 q^{9} + 8 q^{15} - 4 q^{16} + 16 q^{25} + 24 q^{30} - 20 q^{36} - 48 q^{39} + 16 q^{46} - 12 q^{50} + 48 q^{51} + 8 q^{60} - 28 q^{64} + 12 q^{65} + 48 q^{71} - 24 q^{74} - 48 q^{79}+ \cdots - 24 q^{95}+O(q^{100}) \) 4 * q + 4 * q^4 - 20 * q^9 + 8 * q^15 - 4 * q^16 + 16 * q^25 + 24 * q^30 - 20 * q^36 - 48 * q^39 + 16 * q^46 - 12 * q^50 + 48 * q^51 + 8 * q^60 - 28 * q^64 + 12 * q^65 + 48 * q^71 - 24 * q^74 - 48 * q^79 + 4 * q^81 - 12 * q^85 - 24 * q^95

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