LMFDB - Embedded newform 1950.2.b.c.1351.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1950,2,Mod(1351,1950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1950, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1950.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1950.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$15.5708283941$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 78)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1950.1351
Dual form			1950.2.b.c.1351.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{12} +(3.00000 - 2.00000i) q^{13} +2.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} +1.00000i q^{18} +6.00000i q^{19} +2.00000i q^{21} -4.00000 q^{23} +1.00000i q^{24} +(2.00000 + 3.00000i) q^{26} -1.00000 q^{27} +2.00000i q^{28} -10.0000 q^{29} -10.0000i q^{31} +1.00000i q^{32} +2.00000i q^{34} -1.00000 q^{36} +8.00000i q^{37} -6.00000 q^{38} +(-3.00000 + 2.00000i) q^{39} -10.0000i q^{41} -2.00000 q^{42} -4.00000 q^{43} -4.00000i q^{46} -12.0000i q^{47} -1.00000 q^{48} +3.00000 q^{49} -2.00000 q^{51} +(-3.00000 + 2.00000i) q^{52} +6.00000 q^{53} -1.00000i q^{54} -2.00000 q^{56} -6.00000i q^{57} -10.0000i q^{58} -4.00000i q^{59} +2.00000 q^{61} +10.0000 q^{62} -2.00000i q^{63} -1.00000 q^{64} -2.00000i q^{67} -2.00000 q^{68} +4.00000 q^{69} -1.00000i q^{72} -4.00000i q^{73} -8.00000 q^{74} -6.00000i q^{76} +(-2.00000 - 3.00000i) q^{78} +1.00000 q^{81} +10.0000 q^{82} -4.00000i q^{83} -2.00000i q^{84} -4.00000i q^{86} +10.0000 q^{87} +6.00000i q^{89} +(-4.00000 - 6.00000i) q^{91} +4.00000 q^{92} +10.0000i q^{93} +12.0000 q^{94} -1.00000i q^{96} -12.0000i q^{97} +3.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9	$ 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} + 2 q^{12} + 6 q^{13} + 4 q^{14} + 2 q^{16} + 4 q^{17} - 8 q^{23} + 4 q^{26} - 2 q^{27} - 20 q^{29} - 2 q^{36} - 12 q^{38} - 6 q^{39} - 4 q^{42} - 8 q^{43} - 2 q^{48} + 6 q^{49} - 4 q^{51} - 6 q^{52} + 12 q^{53} - 4 q^{56} + 4 q^{61} + 20 q^{62} - 2 q^{64} - 4 q^{68} + 8 q^{69} - 16 q^{74} - 4 q^{78} + 2 q^{81} + 20 q^{82} + 20 q^{87} - 8 q^{91} + 8 q^{92} + 24 q^{94}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 + 2 * q^12 + 6 * q^13 + 4 * q^14 + 2 * q^16 + 4 * q^17 - 8 * q^23 + 4 * q^26 - 2 * q^27 - 20 * q^29 - 2 * q^36 - 12 * q^38 - 6 * q^39 - 4 * q^42 - 8 * q^43 - 2 * q^48 + 6 * q^49 - 4 * q^51 - 6 * q^52 + 12 * q^53 - 4 * q^56 + 4 * q^61 + 20 * q^62 - 2 * q^64 - 4 * q^68 + 8 * q^69 - 16 * q^74 - 4 * q^78 + 2 * q^81 + 20 * q^82 + 20 * q^87 - 8 * q^91 + 8 * q^92 + 24 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times$.

$n$	$301$	$1301$	$1327$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$	−1.00000			−0.500000
$5$		0			0
$5$		0			0
$6$		−	1.00000i		−	0.408248i
$7$		−	2.00000i		−	0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$		−	2.00000i		−	0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$		−	1.00000i		−	0.353553i
$9$	1.00000			0.333333
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$	1.00000			0.288675
$13$	3.00000	−	2.00000i	0.832050	−	0.554700i
$13$	3.00000	−	2.00000i	0.832050	−	0.554700i
$14$	2.00000			0.534522
$15$		0			0
$16$	1.00000			0.250000
$17$	2.00000			0.485071			0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000			0.485071			0.242536	+	0.970143i	$0.422021\pi$
$18$			1.00000i			0.235702i
$19$			6.00000i			1.37649i	0.725476	+	0.688247i	$0.241620\pi$
$19$			6.00000i			1.37649i	−0.725476	+	0.688247i	$0.758380\pi$
$20$		0			0
$21$			2.00000i			0.436436i
$22$		0			0
$23$	−4.00000			−0.834058			−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000			−0.834058			−0.417029	+	0.908893i	$0.636929\pi$
$24$			1.00000i			0.204124i
$25$		0			0
$26$	2.00000	+	3.00000i	0.392232	+	0.588348i
$27$	−1.00000			−0.192450
$28$			2.00000i			0.377964i
$29$	−10.0000			−1.85695			−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−10.0000			−1.85695			−0.928477	+	0.371391i	$0.878881\pi$
$30$		0			0
$31$		−	10.0000i		−	1.79605i	−0.439941	−	0.898027i	$-0.645001\pi$
$31$		−	10.0000i		−	1.79605i	0.439941	−	0.898027i	$-0.354999\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$			2.00000i			0.342997i
$35$		0			0
$36$	−1.00000			−0.166667
$37$			8.00000i			1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$			8.00000i			1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$	−6.00000			−0.973329
$39$	−3.00000	+	2.00000i	−0.480384	+	0.320256i
$40$		0			0
$41$		−	10.0000i		−	1.56174i	−0.624695	−	0.780869i	$-0.714777\pi$
$41$		−	10.0000i		−	1.56174i	0.624695	−	0.780869i	$-0.285223\pi$
$42$	−2.00000			−0.308607
$43$	−4.00000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$		0			0
$45$		0			0
$46$		−	4.00000i		−	0.589768i
$47$		−	12.0000i		−	1.75038i	−0.483779	−	0.875190i	$-0.660736\pi$
$47$		−	12.0000i		−	1.75038i	0.483779	−	0.875190i	$-0.339264\pi$
$48$	−1.00000			−0.144338
$49$	3.00000			0.428571
$50$		0			0
$51$	−2.00000			−0.280056
$52$	−3.00000	+	2.00000i	−0.416025	+	0.277350i
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		−	1.00000i		−	0.136083i
$55$		0			0
$56$	−2.00000			−0.267261
$57$		−	6.00000i		−	0.794719i
$58$		−	10.0000i		−	1.31306i
$59$		−	4.00000i		−	0.520756i	−0.965507	−	0.260378i	$-0.916153\pi$
$59$		−	4.00000i		−	0.520756i	0.965507	−	0.260378i	$-0.0838471\pi$
$60$		0			0
$61$	2.00000			0.256074			0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000			0.256074			0.128037	+	0.991769i	$0.459132\pi$
$62$	10.0000			1.27000
$63$		−	2.00000i		−	0.251976i
$64$	−1.00000			−0.125000
$65$		0			0
$66$		0			0
$67$		−	2.00000i		−	0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$		−	2.00000i		−	0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$	−2.00000			−0.242536
$69$	4.00000			0.481543
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		−	1.00000i		−	0.117851i
$73$		−	4.00000i		−	0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$		−	4.00000i		−	0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	−8.00000			−0.929981
$75$		0			0
$76$		−	6.00000i		−	0.688247i
$77$		0			0
$78$	−2.00000	−	3.00000i	−0.226455	−	0.339683i
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$	10.0000			1.10432
$83$		−	4.00000i		−	0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$		−	4.00000i		−	0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$		−	2.00000i		−	0.218218i
$85$		0			0
$86$		−	4.00000i		−	0.431331i
$87$	10.0000			1.07211
$88$		0			0
$89$			6.00000i			0.635999i	0.948091	+	0.317999i	$0.103011\pi$
$89$			6.00000i			0.635999i	−0.948091	+	0.317999i	$0.896989\pi$
$90$		0			0
$91$	−4.00000	−	6.00000i	−0.419314	−	0.628971i
$92$	4.00000			0.417029
$93$			10.0000i			1.03695i
$94$	12.0000			1.23771
$95$		0			0
$96$		−	1.00000i		−	0.102062i
$97$		−	12.0000i		−	1.21842i	−0.793011	−	0.609208i	$-0.791488\pi$
$97$		−	12.0000i		−	1.21842i	0.793011	−	0.609208i	$-0.208512\pi$
$98$			3.00000i			0.303046i
$99$		0			0
$100$		0			0
$101$	2.00000			0.199007			0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000			0.199007			0.0995037	+	0.995037i	$0.468274\pi$
$102$		−	2.00000i		−	0.198030i
$103$	16.0000			1.57653			0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000			1.57653			0.788263	+	0.615338i	$0.210980\pi$
$104$	−2.00000	−	3.00000i	−0.196116	−	0.294174i
$105$		0			0
$106$			6.00000i			0.582772i
$107$	−8.00000			−0.773389			−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000			−0.773389			−0.386695	+	0.922208i	$0.626383\pi$
$108$	1.00000			0.0962250
$109$		−	4.00000i		−	0.383131i	−0.981480	−	0.191565i	$-0.938644\pi$
$109$		−	4.00000i		−	0.383131i	0.981480	−	0.191565i	$-0.0613564\pi$
$110$		0			0
$111$		−	8.00000i		−	0.759326i
$112$		−	2.00000i		−	0.188982i
$113$	−14.0000			−1.31701			−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000			−1.31701			−0.658505	+	0.752577i	$0.728811\pi$
$114$	6.00000			0.561951
$115$		0			0
$116$	10.0000			0.928477
$117$	3.00000	−	2.00000i	0.277350	−	0.184900i
$118$	4.00000			0.368230
$119$		−	4.00000i		−	0.366679i
$120$		0			0
$121$	11.0000			1.00000
$122$			2.00000i			0.181071i
$123$			10.0000i			0.901670i
$124$			10.0000i			0.898027i
$125$		0			0
$126$	2.00000			0.178174
$127$	−8.00000			−0.709885			−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000			−0.709885			−0.354943	+	0.934888i	$0.615500\pi$
$128$		−	1.00000i		−	0.0883883i
$129$	4.00000			0.352180
$130$		0			0
$131$	−8.00000			−0.698963			−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000			−0.698963			−0.349482	+	0.936943i	$0.613642\pi$
$132$		0			0
$133$	12.0000			1.04053
$134$	2.00000			0.172774
$135$		0			0
$136$		−	2.00000i		−	0.171499i
$137$		−	2.00000i		−	0.170872i	−0.996344	−	0.0854358i	$-0.972772\pi$
$137$		−	2.00000i		−	0.170872i	0.996344	−	0.0854358i	$-0.0272282\pi$
$138$			4.00000i			0.340503i
$139$	−20.0000			−1.69638			−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000			−1.69638			−0.848189	+	0.529694i	$0.822307\pi$
$140$		0			0
$141$			12.0000i			1.01058i
$142$		0			0
$143$		0			0
$144$	1.00000			0.0833333
$145$		0			0
$146$	4.00000			0.331042
$147$	−3.00000			−0.247436
$148$		−	8.00000i		−	0.657596i
$149$		−	14.0000i		−	1.14692i	−0.819232	−	0.573462i	$-0.805600\pi$
$149$		−	14.0000i		−	1.14692i	0.819232	−	0.573462i	$-0.194400\pi$
$150$		0			0
$151$		−	10.0000i		−	0.813788i	−0.913475	−	0.406894i	$-0.866612\pi$
$151$		−	10.0000i		−	0.813788i	0.913475	−	0.406894i	$-0.133388\pi$
$152$	6.00000			0.486664
$153$	2.00000			0.161690
$154$		0			0
$155$		0			0
$156$	3.00000	−	2.00000i	0.240192	−	0.160128i
$157$	2.00000			0.159617			0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000			0.159617			0.0798087	+	0.996810i	$0.474569\pi$
$158$		0			0
$159$	−6.00000			−0.475831
$160$		0			0
$161$			8.00000i			0.630488i
$162$			1.00000i			0.0785674i
$163$		−	14.0000i		−	1.09656i	−0.836293	−	0.548282i	$-0.815282\pi$
$163$		−	14.0000i		−	1.09656i	0.836293	−	0.548282i	$-0.184718\pi$
$164$			10.0000i			0.780869i
$165$		0			0
$166$	4.00000			0.310460
$167$		−	12.0000i		−	0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$		−	12.0000i		−	0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$	2.00000			0.154303
$169$	5.00000	−	12.0000i	0.384615	−	0.923077i
$170$		0			0
$171$			6.00000i			0.458831i
$172$	4.00000			0.304997
$173$	6.00000			0.456172			0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000			0.456172			0.228086	+	0.973641i	$0.426753\pi$
$174$			10.0000i			0.758098i
$175$		0			0
$176$		0			0
$177$			4.00000i			0.300658i
$178$	−6.00000			−0.449719
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	22.0000			1.63525			0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000			1.63525			0.817624	+	0.575753i	$0.195291\pi$
$182$	6.00000	−	4.00000i	0.444750	−	0.296500i
$183$	−2.00000			−0.147844
$184$			4.00000i			0.294884i
$185$		0			0
$186$	−10.0000			−0.733236
$187$		0			0
$188$			12.0000i			0.875190i
$189$			2.00000i			0.145479i
$190$		0			0
$191$	12.0000			0.868290			0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000			0.868290			0.434145	+	0.900843i	$0.357051\pi$
$192$	1.00000			0.0721688
$193$			16.0000i			1.15171i	0.817554	+	0.575853i	$0.195330\pi$
$193$			16.0000i			1.15171i	−0.817554	+	0.575853i	$0.804670\pi$
$194$	12.0000			0.861550
$195$		0			0
$196$	−3.00000			−0.214286
$197$		−	22.0000i		−	1.56744i	−0.621117	−	0.783718i	$-0.713321\pi$
$197$		−	22.0000i		−	1.56744i	0.621117	−	0.783718i	$-0.286679\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$			2.00000i			0.141069i
$202$			2.00000i			0.140720i
$203$			20.0000i			1.40372i
$204$	2.00000			0.140028
$205$		0			0
$206$			16.0000i			1.11477i
$207$	−4.00000			−0.278019
$208$	3.00000	−	2.00000i	0.208013	−	0.138675i
$209$		0			0
$210$		0			0
$211$	12.0000			0.826114			0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000			0.826114			0.413057	+	0.910705i	$0.364461\pi$
$212$	−6.00000			−0.412082
$213$		0			0
$214$		−	8.00000i		−	0.546869i
$215$		0			0
$216$			1.00000i			0.0680414i
$217$	−20.0000			−1.35769
$218$	4.00000			0.270914
$219$			4.00000i			0.270295i
$220$		0			0
$221$	6.00000	−	4.00000i	0.403604	−	0.269069i
$222$	8.00000			0.536925
$223$		−	14.0000i		−	0.937509i	−0.883328	−	0.468755i	$-0.844703\pi$
$223$		−	14.0000i		−	0.937509i	0.883328	−	0.468755i	$-0.155297\pi$
$224$	2.00000			0.133631
$225$		0			0
$226$		−	14.0000i		−	0.931266i
$227$			8.00000i			0.530979i	0.964114	+	0.265489i	$0.0855335\pi$
$227$			8.00000i			0.530979i	−0.964114	+	0.265489i	$0.914466\pi$
$228$			6.00000i			0.397360i
$229$		−	4.00000i		−	0.264327i	−0.991228	−	0.132164i	$-0.957808\pi$
$229$		−	4.00000i		−	0.264327i	0.991228	−	0.132164i	$-0.0421925\pi$
$230$		0			0
$231$		0			0
$232$			10.0000i			0.656532i
$233$	6.00000			0.393073			0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000			0.393073			0.196537	+	0.980497i	$0.437031\pi$
$234$	2.00000	+	3.00000i	0.130744	+	0.196116i
$235$		0			0
$236$			4.00000i			0.260378i
$237$		0			0
$238$	4.00000			0.259281
$239$			16.0000i			1.03495i	0.855697	+	0.517477i	$0.173129\pi$
$239$			16.0000i			1.03495i	−0.855697	+	0.517477i	$0.826871\pi$
$240$		0			0
$241$		−	20.0000i		−	1.28831i	−0.764894	−	0.644157i	$-0.777208\pi$
$241$		−	20.0000i		−	1.28831i	0.764894	−	0.644157i	$-0.222792\pi$
$242$			11.0000i			0.707107i
$243$	−1.00000			−0.0641500
$244$	−2.00000			−0.128037
$245$		0			0
$246$	−10.0000			−0.637577
$247$	12.0000	+	18.0000i	0.763542	+	1.14531i
$248$	−10.0000			−0.635001
$249$			4.00000i			0.253490i
$250$		0			0
$251$	−28.0000			−1.76734			−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000			−1.76734			−0.883672	+	0.468106i	$0.844936\pi$
$252$			2.00000i			0.125988i
$253$		0			0
$254$		−	8.00000i		−	0.501965i
$255$		0			0
$256$	1.00000			0.0625000
$257$	−18.0000			−1.12281			−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000			−1.12281			−0.561405	+	0.827541i	$0.689739\pi$
$258$			4.00000i			0.249029i
$259$	16.0000			0.994192
$260$		0			0
$261$	−10.0000			−0.618984
$262$		−	8.00000i		−	0.494242i
$263$	−24.0000			−1.47990			−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000			−1.47990			−0.739952	+	0.672660i	$0.765152\pi$
$264$		0			0
$265$		0			0
$266$			12.0000i			0.735767i
$267$		−	6.00000i		−	0.367194i
$268$			2.00000i			0.122169i
$269$	−10.0000			−0.609711			−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000			−0.609711			−0.304855	+	0.952399i	$0.598608\pi$
$270$		0			0
$271$			10.0000i			0.607457i	0.952759	+	0.303728i	$0.0982315\pi$
$271$			10.0000i			0.607457i	−0.952759	+	0.303728i	$0.901768\pi$
$272$	2.00000			0.121268
$273$	4.00000	+	6.00000i	0.242091	+	0.363137i
$274$	2.00000			0.120824
$275$		0			0
$276$	−4.00000			−0.240772
$277$	2.00000			0.120168			0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000			0.120168			0.0600842	+	0.998193i	$0.480863\pi$
$278$		−	20.0000i		−	1.19952i
$279$		−	10.0000i		−	0.598684i
$280$		0			0
$281$			10.0000i			0.596550i	0.954480	+	0.298275i	$0.0964112\pi$
$281$			10.0000i			0.596550i	−0.954480	+	0.298275i	$0.903589\pi$
$282$	−12.0000			−0.714590
$283$	−4.00000			−0.237775			−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000			−0.237775			−0.118888	+	0.992908i	$0.537933\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−20.0000			−1.18056
$288$			1.00000i			0.0589256i
$289$	−13.0000			−0.764706
$290$		0			0
$291$			12.0000i			0.703452i
$292$			4.00000i			0.234082i
$293$		−	14.0000i		−	0.817889i	−0.912559	−	0.408944i	$-0.865897\pi$
$293$		−	14.0000i		−	0.817889i	0.912559	−	0.408944i	$-0.134103\pi$
$294$		−	3.00000i		−	0.174964i
$295$		0			0
$296$	8.00000			0.464991
$297$		0			0
$298$	14.0000			0.810998
$299$	−12.0000	+	8.00000i	−0.693978	+	0.462652i
$300$		0			0
$301$			8.00000i			0.461112i
$302$	10.0000			0.575435
$303$	−2.00000			−0.114897
$304$			6.00000i			0.344124i
$305$		0			0
$306$			2.00000i			0.114332i
$307$		−	2.00000i		−	0.114146i	−0.998370	−	0.0570730i	$-0.981823\pi$
$307$		−	2.00000i		−	0.114146i	0.998370	−	0.0570730i	$-0.0181768\pi$
$308$		0			0
$309$	−16.0000			−0.910208
$310$		0			0
$311$	−28.0000			−1.58773			−0.793867	−	0.608091i	$-0.791935\pi$
$311$	−28.0000			−1.58773			−0.793867	+	0.608091i	$0.791935\pi$
$312$	2.00000	+	3.00000i	0.113228	+	0.169842i
$313$	26.0000			1.46961			0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000			1.46961			0.734803	+	0.678280i	$0.237274\pi$
$314$			2.00000i			0.112867i
$315$		0			0
$316$		0			0
$317$			18.0000i			1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$			18.0000i			1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$		−	6.00000i		−	0.336463i
$319$		0			0
$320$		0			0
$321$	8.00000			0.446516
$322$	−8.00000			−0.445823
$323$			12.0000i			0.667698i
$324$	−1.00000			−0.0555556
$325$		0			0
$326$	14.0000			0.775388
$327$			4.00000i			0.221201i
$328$	−10.0000			−0.552158
$329$	−24.0000			−1.32316
$330$		0			0
$331$			10.0000i			0.549650i	0.961494	+	0.274825i	$0.0886199\pi$
$331$			10.0000i			0.549650i	−0.961494	+	0.274825i	$0.911380\pi$
$332$			4.00000i			0.219529i
$333$			8.00000i			0.438397i
$334$	12.0000			0.656611
$335$		0			0
$336$			2.00000i			0.109109i
$337$	2.00000			0.108947			0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000			0.108947			0.0544735	+	0.998515i	$0.482652\pi$
$338$	12.0000	+	5.00000i	0.652714	+	0.271964i
$339$	14.0000			0.760376
$340$		0			0
$341$		0			0
$342$	−6.00000			−0.324443
$343$		−	20.0000i		−	1.07990i
$344$			4.00000i			0.215666i
$345$		0			0
$346$			6.00000i			0.322562i
$347$	12.0000			0.644194			0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000			0.644194			0.322097	+	0.946707i	$0.395612\pi$
$348$	−10.0000			−0.536056
$349$			16.0000i			0.856460i	0.903670	+	0.428230i	$0.140863\pi$
$349$			16.0000i			0.856460i	−0.903670	+	0.428230i	$0.859137\pi$
$350$		0			0
$351$	−3.00000	+	2.00000i	−0.160128	+	0.106752i
$352$		0			0
$353$			26.0000i			1.38384i	0.721974	+	0.691920i	$0.243235\pi$
$353$			26.0000i			1.38384i	−0.721974	+	0.691920i	$0.756765\pi$
$354$	−4.00000			−0.212598
$355$		0			0
$356$		−	6.00000i		−	0.317999i
$357$			4.00000i			0.211702i
$358$		0			0
$359$		−	4.00000i		−	0.211112i	−0.994413	−	0.105556i	$-0.966338\pi$
$359$		−	4.00000i		−	0.211112i	0.994413	−	0.105556i	$-0.0336622\pi$
$360$		0			0
$361$	−17.0000			−0.894737
$362$			22.0000i			1.15629i
$363$	−11.0000			−0.577350
$364$	4.00000	+	6.00000i	0.209657	+	0.314485i
$365$		0			0
$366$		−	2.00000i		−	0.104542i
$367$	−8.00000			−0.417597			−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000			−0.417597			−0.208798	+	0.977959i	$0.566955\pi$
$368$	−4.00000			−0.208514
$369$		−	10.0000i		−	0.520579i
$370$		0			0
$371$		−	12.0000i		−	0.623009i
$372$		−	10.0000i		−	0.518476i
$373$	6.00000			0.310668			0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000			0.310668			0.155334	+	0.987862i	$0.450355\pi$
$374$		0			0
$375$		0			0
$376$	−12.0000			−0.618853
$377$	−30.0000	+	20.0000i	−1.54508	+	1.03005i
$378$	−2.00000			−0.102869
$379$		−	34.0000i		−	1.74646i	−0.487306	−	0.873231i	$-0.662020\pi$
$379$		−	34.0000i		−	1.74646i	0.487306	−	0.873231i	$-0.337980\pi$
$380$		0			0
$381$	8.00000			0.409852
$382$			12.0000i			0.613973i
$383$		−	4.00000i		−	0.204390i	−0.994764	−	0.102195i	$-0.967413\pi$
$383$		−	4.00000i		−	0.204390i	0.994764	−	0.102195i	$-0.0325866\pi$
$384$			1.00000i			0.0510310i
$385$		0			0
$386$	−16.0000			−0.814379
$387$	−4.00000			−0.203331
$388$			12.0000i			0.609208i
$389$	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
$391$	−8.00000			−0.404577
$392$		−	3.00000i		−	0.151523i
$393$	8.00000			0.403547
$394$	22.0000			1.10834
$395$		0			0
$396$		0			0
$397$			8.00000i			0.401508i	0.979642	+	0.200754i	$0.0643393\pi$
$397$			8.00000i			0.401508i	−0.979642	+	0.200754i	$0.935661\pi$
$398$		0			0
$399$	−12.0000			−0.600751
$400$		0			0
$401$			30.0000i			1.49813i	0.662497	+	0.749064i	$0.269497\pi$
$401$			30.0000i			1.49813i	−0.662497	+	0.749064i	$0.730503\pi$
$402$	−2.00000			−0.0997509
$403$	−20.0000	−	30.0000i	−0.996271	−	1.49441i
$404$	−2.00000			−0.0995037
$405$		0			0
$406$	−20.0000			−0.992583
$407$		0			0
$408$			2.00000i			0.0990148i
$409$		−	4.00000i		−	0.197787i	−0.995098	−	0.0988936i	$-0.968470\pi$
$409$		−	4.00000i		−	0.197787i	0.995098	−	0.0988936i	$-0.0315304\pi$
$410$		0			0
$411$			2.00000i			0.0986527i
$412$	−16.0000			−0.788263
$413$	−8.00000			−0.393654
$414$		−	4.00000i		−	0.196589i
$415$		0			0
$416$	2.00000	+	3.00000i	0.0980581	+	0.147087i
$417$	20.0000			0.979404
$418$		0			0
$419$	40.0000			1.95413			0.977064	−	0.212946i	$-0.0683059\pi$
$419$	40.0000			1.95413			0.977064	+	0.212946i	$0.0683059\pi$
$420$		0			0
$421$		−	20.0000i		−	0.974740i	−0.873195	−	0.487370i	$-0.837956\pi$
$421$		−	20.0000i		−	0.974740i	0.873195	−	0.487370i	$-0.162044\pi$
$422$			12.0000i			0.584151i
$423$		−	12.0000i		−	0.583460i
$424$		−	6.00000i		−	0.291386i
$425$		0			0
$426$		0			0
$427$		−	4.00000i		−	0.193574i
$428$	8.00000			0.386695
$429$		0			0
$430$		0			0
$431$			20.0000i			0.963366i	0.876346	+	0.481683i	$0.159974\pi$
$431$			20.0000i			0.963366i	−0.876346	+	0.481683i	$0.840026\pi$
$432$	−1.00000			−0.0481125
$433$	26.0000			1.24948			0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000			1.24948			0.624740	+	0.780833i	$0.285205\pi$
$434$		−	20.0000i		−	0.960031i
$435$		0			0
$436$			4.00000i			0.191565i
$437$		−	24.0000i		−	1.14808i
$438$	−4.00000			−0.191127
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	3.00000			0.142857
$442$	4.00000	+	6.00000i	0.190261	+	0.285391i
$443$	16.0000			0.760183			0.380091	−	0.924949i	$-0.375893\pi$
$443$	16.0000			0.760183			0.380091	+	0.924949i	$0.375893\pi$
$444$			8.00000i			0.379663i
$445$		0			0
$446$	14.0000			0.662919
$447$			14.0000i			0.662177i
$448$			2.00000i			0.0944911i
$449$			6.00000i			0.283158i	0.989927	+	0.141579i	$0.0452178\pi$
$449$			6.00000i			0.283158i	−0.989927	+	0.141579i	$0.954782\pi$
$450$		0			0
$451$		0			0
$452$	14.0000			0.658505
$453$			10.0000i			0.469841i
$454$	−8.00000			−0.375459
$455$		0			0
$456$	−6.00000			−0.280976
$457$			28.0000i			1.30978i	0.755722	+	0.654892i	$0.227286\pi$
$457$			28.0000i			1.30978i	−0.755722	+	0.654892i	$0.772714\pi$
$458$	4.00000			0.186908
$459$	−2.00000			−0.0933520
$460$		0			0
$461$			30.0000i			1.39724i	0.715493	+	0.698620i	$0.246202\pi$
$461$			30.0000i			1.39724i	−0.715493	+	0.698620i	$0.753798\pi$
$462$		0			0
$463$			6.00000i			0.278844i	0.990233	+	0.139422i	$0.0445244\pi$
$463$			6.00000i			0.278844i	−0.990233	+	0.139422i	$0.955476\pi$
$464$	−10.0000			−0.464238
$465$		0			0
$466$			6.00000i			0.277945i
$467$	12.0000			0.555294			0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000			0.555294			0.277647	+	0.960683i	$0.410445\pi$
$468$	−3.00000	+	2.00000i	−0.138675	+	0.0924500i
$469$	−4.00000			−0.184703
$470$		0			0
$471$	−2.00000			−0.0921551
$472$	−4.00000			−0.184115
$473$		0			0
$474$		0			0
$475$		0			0
$476$			4.00000i			0.183340i
$477$	6.00000			0.274721
$478$	−16.0000			−0.731823
$479$		−	24.0000i		−	1.09659i	−0.836286	−	0.548294i	$-0.815277\pi$
$479$		−	24.0000i		−	1.09659i	0.836286	−	0.548294i	$-0.184723\pi$
$480$		0			0
$481$	16.0000	+	24.0000i	0.729537	+	1.09431i
$482$	20.0000			0.910975
$483$		−	8.00000i		−	0.364013i
$484$	−11.0000			−0.500000
$485$		0			0
$486$		−	1.00000i		−	0.0453609i
$487$			18.0000i			0.815658i	0.913058	+	0.407829i	$0.133714\pi$
$487$			18.0000i			0.815658i	−0.913058	+	0.407829i	$0.866286\pi$
$488$		−	2.00000i		−	0.0905357i
$489$			14.0000i			0.633102i
$490$		0			0
$491$	−28.0000			−1.26362			−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000			−1.26362			−0.631811	+	0.775122i	$0.717688\pi$
$492$		−	10.0000i		−	0.450835i
$493$	−20.0000			−0.900755
$494$	−18.0000	+	12.0000i	−0.809858	+	0.539906i
$495$		0			0
$496$		−	10.0000i		−	0.449013i
$497$		0			0
$498$	−4.00000			−0.179244
$499$		−	14.0000i		−	0.626726i	−0.949633	−	0.313363i	$-0.898544\pi$
$499$		−	14.0000i		−	0.626726i	0.949633	−	0.313363i	$-0.101456\pi$
$500$		0			0
$501$			12.0000i			0.536120i
$502$		−	28.0000i		−	1.24970i
$503$	−24.0000			−1.07011			−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000			−1.07011			−0.535054	+	0.844818i	$0.679709\pi$
$504$	−2.00000			−0.0890871
$505$		0			0
$506$		0			0
$507$	−5.00000	+	12.0000i	−0.222058	+	0.532939i
$508$	8.00000			0.354943
$509$			6.00000i			0.265945i	0.991120	+	0.132973i	$0.0424523\pi$
$509$			6.00000i			0.265945i	−0.991120	+	0.132973i	$0.957548\pi$
$510$		0			0
$511$	−8.00000			−0.353899
$512$			1.00000i			0.0441942i
$513$		−	6.00000i		−	0.264906i
$514$		−	18.0000i		−	0.793946i
$515$		0			0
$516$	−4.00000			−0.176090
$517$		0			0
$518$			16.0000i			0.703000i
$519$	−6.00000			−0.263371
$520$		0			0
$521$	−18.0000			−0.788594			−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000			−0.788594			−0.394297	+	0.918983i	$0.629012\pi$
$522$		−	10.0000i		−	0.437688i
$523$	−4.00000			−0.174908			−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000			−0.174908			−0.0874539	+	0.996169i	$0.527873\pi$
$524$	8.00000			0.349482
$525$		0			0
$526$		−	24.0000i		−	1.04645i
$527$		−	20.0000i		−	0.871214i
$528$		0			0
$529$	−7.00000			−0.304348
$530$		0			0
$531$		−	4.00000i		−	0.173585i
$532$	−12.0000			−0.520266
$533$	−20.0000	−	30.0000i	−0.866296	−	1.29944i
$534$	6.00000			0.259645
$535$		0			0
$536$	−2.00000			−0.0863868
$537$		0			0
$538$		−	10.0000i		−	0.431131i
$539$		0			0
$540$		0			0
$541$			20.0000i			0.859867i	0.902861	+	0.429934i	$0.141463\pi$
$541$			20.0000i			0.859867i	−0.902861	+	0.429934i	$0.858537\pi$
$542$	−10.0000			−0.429537
$543$	−22.0000			−0.944110
$544$			2.00000i			0.0857493i
$545$		0			0
$546$	−6.00000	+	4.00000i	−0.256776	+	0.171184i
$547$	−28.0000			−1.19719			−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000			−1.19719			−0.598597	+	0.801050i	$0.704275\pi$
$548$			2.00000i			0.0854358i
$549$	2.00000			0.0853579
$550$		0			0
$551$		−	60.0000i		−	2.55609i
$552$		−	4.00000i		−	0.170251i
$553$		0			0
$554$			2.00000i			0.0849719i
$555$		0			0
$556$	20.0000			0.848189
$557$			18.0000i			0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$			18.0000i			0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$	10.0000			0.423334
$559$	−12.0000	+	8.00000i	−0.507546	+	0.338364i
$560$		0			0
$561$		0			0
$562$	−10.0000			−0.421825
$563$	16.0000			0.674320			0.337160	−	0.941447i	$-0.390534\pi$
$563$	16.0000			0.674320			0.337160	+	0.941447i	$0.390534\pi$
$564$		−	12.0000i		−	0.505291i
$565$		0			0
$566$		−	4.00000i		−	0.168133i
$567$		−	2.00000i		−	0.0839921i
$568$		0			0
$569$	10.0000			0.419222			0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000			0.419222			0.209611	+	0.977785i	$0.432780\pi$
$570$		0			0
$571$	−28.0000			−1.17176			−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000			−1.17176			−0.585882	+	0.810397i	$0.699252\pi$
$572$		0			0
$573$	−12.0000			−0.501307
$574$		−	20.0000i		−	0.834784i
$575$		0			0
$576$	−1.00000			−0.0416667
$577$			8.00000i			0.333044i	0.986038	+	0.166522i	$0.0532537\pi$
$577$			8.00000i			0.333044i	−0.986038	+	0.166522i	$0.946746\pi$
$578$		−	13.0000i		−	0.540729i
$579$		−	16.0000i		−	0.664937i
$580$		0			0
$581$	−8.00000			−0.331896
$582$	−12.0000			−0.497416
$583$		0			0
$584$	−4.00000			−0.165521
$585$		0			0
$586$	14.0000			0.578335
$587$			28.0000i			1.15568i	0.816149	+	0.577842i	$0.196105\pi$
$587$			28.0000i			1.15568i	−0.816149	+	0.577842i	$0.803895\pi$
$588$	3.00000			0.123718
$589$	60.0000			2.47226
$590$		0			0
$591$			22.0000i			0.904959i
$592$			8.00000i			0.328798i
$593$			26.0000i			1.06769i	0.845582	+	0.533846i	$0.179254\pi$
$593$			26.0000i			1.06769i	−0.845582	+	0.533846i	$0.820746\pi$
$594$		0			0
$595$		0			0
$596$			14.0000i			0.573462i
$597$		0			0
$598$	−8.00000	−	12.0000i	−0.327144	−	0.490716i
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	22.0000			0.897399			0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000			0.897399			0.448699	+	0.893683i	$0.351887\pi$
$602$	−8.00000			−0.326056
$603$		−	2.00000i		−	0.0814463i
$604$			10.0000i			0.406894i
$605$		0			0
$606$		−	2.00000i		−	0.0812444i
$607$	32.0000			1.29884			0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000			1.29884			0.649420	+	0.760430i	$0.275012\pi$
$608$	−6.00000			−0.243332
$609$		−	20.0000i		−	0.810441i
$610$		0			0
$611$	−24.0000	−	36.0000i	−0.970936	−	1.45640i
$612$	−2.00000			−0.0808452
$613$			16.0000i			0.646234i	0.946359	+	0.323117i	$0.104731\pi$
$613$			16.0000i			0.646234i	−0.946359	+	0.323117i	$0.895269\pi$
$614$	2.00000			0.0807134
$615$		0			0
$616$		0			0
$617$		−	22.0000i		−	0.885687i	−0.896599	−	0.442843i	$-0.853970\pi$
$617$		−	22.0000i		−	0.885687i	0.896599	−	0.442843i	$-0.146030\pi$
$618$		−	16.0000i		−	0.643614i
$619$			26.0000i			1.04503i	0.852631	+	0.522514i	$0.175006\pi$
$619$			26.0000i			1.04503i	−0.852631	+	0.522514i	$0.824994\pi$
$620$		0			0
$621$	4.00000			0.160514
$622$		−	28.0000i		−	1.12270i
$623$	12.0000			0.480770
$624$	−3.00000	+	2.00000i	−0.120096	+	0.0800641i
$625$		0			0
$626$			26.0000i			1.03917i
$627$		0			0
$628$	−2.00000			−0.0798087
$629$			16.0000i			0.637962i
$630$		0			0
$631$		−	10.0000i		−	0.398094i	−0.979990	−	0.199047i	$-0.936215\pi$
$631$		−	10.0000i		−	0.398094i	0.979990	−	0.199047i	$-0.0637846\pi$
$632$		0			0
$633$	−12.0000			−0.476957
$634$	−18.0000			−0.714871
$635$		0			0
$636$	6.00000			0.237915
$637$	9.00000	−	6.00000i	0.356593	−	0.237729i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−18.0000			−0.710957			−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000			−0.710957			−0.355479	+	0.934684i	$0.615682\pi$
$642$			8.00000i			0.315735i
$643$			6.00000i			0.236617i	0.992977	+	0.118308i	$0.0377472\pi$
$643$			6.00000i			0.236617i	−0.992977	+	0.118308i	$0.962253\pi$
$644$		−	8.00000i		−	0.315244i
$645$		0			0
$646$	−12.0000			−0.472134
$647$	32.0000			1.25805			0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000			1.25805			0.629025	+	0.777385i	$0.283454\pi$
$648$		−	1.00000i		−	0.0392837i
$649$		0			0
$650$		0			0
$651$	20.0000			0.783862
$652$			14.0000i			0.548282i
$653$	26.0000			1.01746			0.508729	−	0.860927i	$-0.330115\pi$
$653$	26.0000			1.01746			0.508729	+	0.860927i	$0.330115\pi$
$654$	−4.00000			−0.156412
$655$		0			0
$656$		−	10.0000i		−	0.390434i
$657$		−	4.00000i		−	0.156055i
$658$		−	24.0000i		−	0.935617i
$659$	20.0000			0.779089			0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000			0.779089			0.389545	+	0.921008i	$0.372632\pi$
$660$		0			0
$661$			40.0000i			1.55582i	0.628376	+	0.777910i	$0.283720\pi$
$661$			40.0000i			1.55582i	−0.628376	+	0.777910i	$0.716280\pi$
$662$	−10.0000			−0.388661
$663$	−6.00000	+	4.00000i	−0.233021	+	0.155347i
$664$	−4.00000			−0.155230
$665$		0			0
$666$	−8.00000			−0.309994
$667$	40.0000			1.54881
$668$			12.0000i			0.464294i
$669$			14.0000i			0.541271i
$670$		0			0
$671$		0			0
$672$	−2.00000			−0.0771517
$673$	6.00000			0.231283			0.115642	−	0.993291i	$-0.463108\pi$
$673$	6.00000			0.231283			0.115642	+	0.993291i	$0.463108\pi$
$674$			2.00000i			0.0770371i
$675$		0			0
$676$	−5.00000	+	12.0000i	−0.192308	+	0.461538i
$677$	−18.0000			−0.691796			−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000			−0.691796			−0.345898	+	0.938272i	$0.612426\pi$
$678$			14.0000i			0.537667i
$679$	−24.0000			−0.921035
$680$		0			0
$681$		−	8.00000i		−	0.306561i
$682$		0			0
$683$		−	24.0000i		−	0.918334i	−0.888350	−	0.459167i	$-0.848148\pi$
$683$		−	24.0000i		−	0.918334i	0.888350	−	0.459167i	$-0.151852\pi$
$684$		−	6.00000i		−	0.229416i
$685$		0			0
$686$	20.0000			0.763604
$687$			4.00000i			0.152610i
$688$	−4.00000			−0.152499
$689$	18.0000	−	12.0000i	0.685745	−	0.457164i
$690$		0			0
$691$			10.0000i			0.380418i	0.981744	+	0.190209i	$0.0609166\pi$
$691$			10.0000i			0.380418i	−0.981744	+	0.190209i	$0.939083\pi$
$692$	−6.00000			−0.228086
$693$		0			0
$694$			12.0000i			0.455514i
$695$		0			0
$696$		−	10.0000i		−	0.379049i
$697$		−	20.0000i		−	0.757554i
$698$	−16.0000			−0.605609
$699$	−6.00000			−0.226941
$700$		0			0
$701$	22.0000			0.830929			0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000			0.830929			0.415464	+	0.909610i	$0.363619\pi$
$702$	−2.00000	−	3.00000i	−0.0754851	−	0.113228i
$703$	−48.0000			−1.81035
$704$		0			0
$705$		0			0
$706$	−26.0000			−0.978523
$707$		−	4.00000i		−	0.150435i
$708$		−	4.00000i		−	0.150329i
$709$			36.0000i			1.35201i	0.736898	+	0.676004i	$0.236290\pi$
$709$			36.0000i			1.35201i	−0.736898	+	0.676004i	$0.763710\pi$
$710$		0			0
$711$		0			0
$712$	6.00000			0.224860
$713$			40.0000i			1.49801i
$714$	−4.00000			−0.149696
$715$		0			0
$716$		0			0
$717$		−	16.0000i		−	0.597531i
$718$	4.00000			0.149279
$719$	20.0000			0.745874			0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000			0.745874			0.372937	+	0.927857i	$0.378351\pi$
$720$		0			0
$721$		−	32.0000i		−	1.19174i
$722$		−	17.0000i		−	0.632674i
$723$			20.0000i			0.743808i
$724$	−22.0000			−0.817624
$725$		0			0
$726$		−	11.0000i		−	0.408248i
$727$	−8.00000			−0.296704			−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000			−0.296704			−0.148352	+	0.988935i	$0.547397\pi$
$728$	−6.00000	+	4.00000i	−0.222375	+	0.148250i
$729$	1.00000			0.0370370
$730$		0			0
$731$	−8.00000			−0.295891
$732$	2.00000			0.0739221
$733$		−	44.0000i		−	1.62518i	−0.582838	−	0.812589i	$-0.698058\pi$
$733$		−	44.0000i		−	1.62518i	0.582838	−	0.812589i	$-0.301942\pi$
$734$		−	8.00000i		−	0.295285i
$735$		0			0
$736$		−	4.00000i		−	0.147442i
$737$		0			0
$738$	10.0000			0.368105
$739$			26.0000i			0.956425i	0.878244	+	0.478213i	$0.158715\pi$
$739$			26.0000i			0.956425i	−0.878244	+	0.478213i	$0.841285\pi$
$740$		0			0
$741$	−12.0000	−	18.0000i	−0.440831	−	0.661247i
$742$	12.0000			0.440534
$743$			16.0000i			0.586983i	0.955962	+	0.293492i	$0.0948173\pi$
$743$			16.0000i			0.586983i	−0.955962	+	0.293492i	$0.905183\pi$
$744$	10.0000			0.366618
$745$		0			0
$746$			6.00000i			0.219676i
$747$		−	4.00000i		−	0.146352i
$748$		0			0
$749$			16.0000i			0.584627i
$750$		0			0
$751$	32.0000			1.16770			0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000			1.16770			0.583848	+	0.811863i	$0.301546\pi$
$752$		−	12.0000i		−	0.437595i
$753$	28.0000			1.02038
$754$	−20.0000	−	30.0000i	−0.728357	−	1.09254i
$755$		0			0
$756$		−	2.00000i		−	0.0727393i
$757$	22.0000			0.799604			0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000			0.799604			0.399802	+	0.916602i	$0.369079\pi$
$758$	34.0000			1.23494
$759$		0			0
$760$		0			0
$761$		−	30.0000i		−	1.08750i	−0.839248	−	0.543750i	$-0.817004\pi$
$761$		−	30.0000i		−	1.08750i	0.839248	−	0.543750i	$-0.182996\pi$
$762$			8.00000i			0.289809i
$763$	−8.00000			−0.289619
$764$	−12.0000			−0.434145
$765$		0			0
$766$	4.00000			0.144526
$767$	−8.00000	−	12.0000i	−0.288863	−	0.433295i
$768$	−1.00000			−0.0360844
$769$		−	24.0000i		−	0.865462i	−0.901523	−	0.432731i	$-0.857550\pi$
$769$		−	24.0000i		−	0.865462i	0.901523	−	0.432731i	$-0.142450\pi$
$770$		0			0
$771$	18.0000			0.648254
$772$		−	16.0000i		−	0.575853i
$773$			6.00000i			0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$			6.00000i			0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$		−	4.00000i		−	0.143777i
$775$		0			0
$776$	−12.0000			−0.430775
$777$	−16.0000			−0.573997
$778$			30.0000i			1.07555i
$779$	60.0000			2.14972
$780$		0			0
$781$		0			0
$782$		−	8.00000i		−	0.286079i
$783$	10.0000			0.357371
$784$	3.00000			0.107143
$785$		0			0
$786$			8.00000i			0.285351i
$787$			38.0000i			1.35455i	0.735728	+	0.677277i	$0.236840\pi$
$787$			38.0000i			1.35455i	−0.735728	+	0.677277i	$0.763160\pi$
$788$			22.0000i			0.783718i
$789$	24.0000			0.854423
$790$		0			0
$791$			28.0000i			0.995565i
$792$		0			0
$793$	6.00000	−	4.00000i	0.213066	−	0.142044i
$794$	−8.00000			−0.283909
$795$		0			0
$796$		0			0
$797$	2.00000			0.0708436			0.0354218	−	0.999372i	$-0.488723\pi$
$797$	2.00000			0.0708436			0.0354218	+	0.999372i	$0.488723\pi$
$798$		−	12.0000i		−	0.424795i
$799$		−	24.0000i		−	0.849059i
$800$		0			0
$801$			6.00000i			0.212000i
$802$	−30.0000			−1.05934
$803$		0			0
$804$		−	2.00000i		−	0.0705346i
$805$		0			0
$806$	30.0000	−	20.0000i	1.05670	−	0.704470i
$807$	10.0000			0.352017
$808$		−	2.00000i		−	0.0703598i
$809$	−50.0000			−1.75791			−0.878953	−	0.476908i	$-0.841757\pi$
$809$	−50.0000			−1.75791			−0.878953	+	0.476908i	$0.841757\pi$
$810$		0			0
$811$		−	10.0000i		−	0.351147i	−0.984466	−	0.175574i	$-0.943822\pi$
$811$		−	10.0000i		−	0.351147i	0.984466	−	0.175574i	$-0.0561780\pi$
$812$		−	20.0000i		−	0.701862i
$813$		−	10.0000i		−	0.350715i
$814$		0			0
$815$		0			0
$816$	−2.00000			−0.0700140
$817$		−	24.0000i		−	0.839654i
$818$	4.00000			0.139857
$819$	−4.00000	−	6.00000i	−0.139771	−	0.209657i
$820$		0			0
$821$		−	30.0000i		−	1.04701i	−0.852023	−	0.523504i	$-0.824625\pi$
$821$		−	30.0000i		−	1.04701i	0.852023	−	0.523504i	$-0.175375\pi$
$822$	−2.00000			−0.0697580
$823$	−24.0000			−0.836587			−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000			−0.836587			−0.418294	+	0.908312i	$0.637372\pi$
$824$		−	16.0000i		−	0.557386i
$825$		0			0
$826$		−	8.00000i		−	0.278356i
$827$			48.0000i			1.66912i	0.550914	+	0.834562i	$0.314279\pi$
$827$			48.0000i			1.66912i	−0.550914	+	0.834562i	$0.685721\pi$
$828$	4.00000			0.139010
$829$	−30.0000			−1.04194			−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000			−1.04194			−0.520972	+	0.853574i	$0.674430\pi$
$830$		0			0
$831$	−2.00000			−0.0693792
$832$	−3.00000	+	2.00000i	−0.104006	+	0.0693375i
$833$	6.00000			0.207888
$834$			20.0000i			0.692543i
$835$		0			0
$836$		0			0
$837$			10.0000i			0.345651i
$838$			40.0000i			1.38178i
$839$			16.0000i			0.552381i	0.961103	+	0.276191i	$0.0890721\pi$
$839$			16.0000i			0.552381i	−0.961103	+	0.276191i	$0.910928\pi$
$840$		0			0
$841$	71.0000			2.44828
$842$	20.0000			0.689246
$843$		−	10.0000i		−	0.344418i
$844$	−12.0000			−0.413057
$845$		0			0
$846$	12.0000			0.412568
$847$		−	22.0000i		−	0.755929i
$848$	6.00000			0.206041
$849$	4.00000			0.137280
$850$		0			0
$851$		−	32.0000i		−	1.09695i
$852$		0			0
$853$			56.0000i			1.91740i	0.284413	+	0.958702i	$0.408201\pi$
$853$			56.0000i			1.91740i	−0.284413	+	0.958702i	$0.591799\pi$
$854$	4.00000			0.136877
$855$		0			0
$856$			8.00000i			0.273434i
$857$	22.0000			0.751506			0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000			0.751506			0.375753	+	0.926720i	$0.377384\pi$
$858$		0			0
$859$	−20.0000			−0.682391			−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000			−0.682391			−0.341196	+	0.939992i	$0.610832\pi$
$860$		0			0
$861$	20.0000			0.681598
$862$	−20.0000			−0.681203
$863$		−	44.0000i		−	1.49778i	−0.662696	−	0.748889i	$-0.730588\pi$
$863$		−	44.0000i		−	1.49778i	0.662696	−	0.748889i	$-0.269412\pi$
$864$		−	1.00000i		−	0.0340207i
$865$		0			0
$866$			26.0000i			0.883516i
$867$	13.0000			0.441503
$868$	20.0000			0.678844
$869$		0			0
$870$		0			0
$871$	−4.00000	−	6.00000i	−0.135535	−	0.203302i
$872$	−4.00000			−0.135457
$873$		−	12.0000i		−	0.406138i
$874$	24.0000			0.811812
$875$		0			0
$876$		−	4.00000i		−	0.135147i
$877$			8.00000i			0.270141i	0.990836	+	0.135070i	$0.0431261\pi$
$877$			8.00000i			0.270141i	−0.990836	+	0.135070i	$0.956874\pi$
$878$		0			0
$879$			14.0000i			0.472208i
$880$		0			0
$881$	42.0000			1.41502			0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000			1.41502			0.707508	+	0.706705i	$0.249819\pi$
$882$			3.00000i			0.101015i
$883$	36.0000			1.21150			0.605748	−	0.795656i	$-0.292874\pi$
$883$	36.0000			1.21150			0.605748	+	0.795656i	$0.292874\pi$
$884$	−6.00000	+	4.00000i	−0.201802	+	0.134535i
$885$		0			0
$886$			16.0000i			0.537531i
$887$	12.0000			0.402921			0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000			0.402921			0.201460	+	0.979497i	$0.435431\pi$
$888$	−8.00000			−0.268462
$889$			16.0000i			0.536623i
$890$		0			0
$891$		0			0
$892$			14.0000i			0.468755i
$893$	72.0000			2.40939
$894$	−14.0000			−0.468230
$895$		0			0
$896$	−2.00000			−0.0668153
$897$	12.0000	−	8.00000i	0.400668	−	0.267112i
$898$	−6.00000			−0.200223
$899$			100.000i			3.33519i
$900$		0			0
$901$	12.0000			0.399778
$902$		0			0
$903$		−	8.00000i		−	0.266223i
$904$			14.0000i			0.465633i
$905$		0			0
$906$	−10.0000			−0.332228
$907$	−28.0000			−0.929725			−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000			−0.929725			−0.464862	+	0.885383i	$0.653896\pi$
$908$		−	8.00000i		−	0.265489i
$909$	2.00000			0.0663358
$910$		0			0
$911$	12.0000			0.397578			0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000			0.397578			0.198789	+	0.980042i	$0.436299\pi$
$912$		−	6.00000i		−	0.198680i
$913$		0			0
$914$	−28.0000			−0.926158
$915$		0			0
$916$			4.00000i			0.132164i
$917$			16.0000i			0.528367i
$918$		−	2.00000i		−	0.0660098i
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$			2.00000i			0.0659022i
$922$	−30.0000			−0.987997
$923$		0			0
$924$		0			0
$925$		0			0
$926$	−6.00000			−0.197172
$927$	16.0000			0.525509
$928$		−	10.0000i		−	0.328266i
$929$			6.00000i			0.196854i	0.995144	+	0.0984268i	$0.0313810\pi$
$929$			6.00000i			0.196854i	−0.995144	+	0.0984268i	$0.968619\pi$
$930$		0			0
$931$			18.0000i			0.589926i
$932$	−6.00000			−0.196537
$933$	28.0000			0.916679
$934$			12.0000i			0.392652i
$935$		0			0
$936$	−2.00000	−	3.00000i	−0.0653720	−	0.0980581i
$937$	2.00000			0.0653372			0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000			0.0653372			0.0326686	+	0.999466i	$0.489599\pi$
$938$		−	4.00000i		−	0.130605i
$939$	−26.0000			−0.848478
$940$		0			0
$941$			10.0000i			0.325991i	0.986627	+	0.162995i	$0.0521156\pi$
$941$			10.0000i			0.325991i	−0.986627	+	0.162995i	$0.947884\pi$
$942$		−	2.00000i		−	0.0651635i
$943$			40.0000i			1.30258i
$944$		−	4.00000i		−	0.130189i
$945$		0			0
$946$		0			0
$947$		−	52.0000i		−	1.68977i	−0.534946	−	0.844886i	$-0.679668\pi$
$947$		−	52.0000i		−	1.68977i	0.534946	−	0.844886i	$-0.320332\pi$
$948$		0			0
$949$	−8.00000	−	12.0000i	−0.259691	−	0.389536i
$950$		0			0
$951$		−	18.0000i		−	0.583690i
$952$	−4.00000			−0.129641
$953$	6.00000			0.194359			0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000			0.194359			0.0971795	+	0.995267i	$0.469018\pi$
$954$			6.00000i			0.194257i
$955$		0			0
$956$		−	16.0000i		−	0.517477i
$957$		0			0
$958$	24.0000			0.775405
$959$	−4.00000			−0.129167
$960$		0			0
$961$	−69.0000			−2.22581
$962$	−24.0000	+	16.0000i	−0.773791	+	0.515861i
$963$	−8.00000			−0.257796
$964$			20.0000i			0.644157i
$965$		0			0
$966$	8.00000			0.257396
$967$		−	22.0000i		−	0.707472i	−0.935345	−	0.353736i	$-0.884911\pi$
$967$		−	22.0000i		−	0.707472i	0.935345	−	0.353736i	$-0.115089\pi$
$968$		−	11.0000i		−	0.353553i
$969$		−	12.0000i		−	0.385496i
$970$		0			0
$971$	12.0000			0.385098			0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000			0.385098			0.192549	+	0.981287i	$0.438325\pi$
$972$	1.00000			0.0320750
$973$			40.0000i			1.28234i
$974$	−18.0000			−0.576757
$975$		0			0
$976$	2.00000			0.0640184
$977$		−	42.0000i		−	1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$		−	42.0000i		−	1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$	−14.0000			−0.447671
$979$		0			0
$980$		0			0
$981$		−	4.00000i		−	0.127710i
$982$		−	28.0000i		−	0.893516i
$983$		−	24.0000i		−	0.765481i	−0.923856	−	0.382741i	$-0.874980\pi$
$983$		−	24.0000i		−	0.765481i	0.923856	−	0.382741i	$-0.125020\pi$
$984$	10.0000			0.318788
$985$		0			0
$986$		−	20.0000i		−	0.636930i
$987$	24.0000			0.763928
$988$	−12.0000	−	18.0000i	−0.381771	−	0.572656i
$989$	16.0000			0.508770
$990$		0			0
$991$	−8.00000			−0.254128			−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000			−0.254128			−0.127064	+	0.991894i	$0.540555\pi$
$992$	10.0000			0.317500
$993$		−	10.0000i		−	0.317340i
$994$		0			0
$995$		0			0
$996$		−	4.00000i		−	0.126745i
$997$	42.0000			1.33015			0.665077	−	0.746775i	$-0.268399\pi$
$997$	42.0000			1.33015			0.665077	+	0.746775i	$0.268399\pi$
$998$	14.0000			0.443162
$999$		−	8.00000i		−	0.253109i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1950.2.b.c.1351.2		2
5.2	odd	4		1950.2.f.d.649.2		2
5.3	odd	4		1950.2.f.g.649.1		2
5.4	even	2		78.2.b.a.25.1	✓	2
13.12	even	2	inner	1950.2.b.c.1351.1		2
15.14	odd	2		234.2.b.a.181.2		2
20.19	odd	2		624.2.c.a.337.1		2
35.34	odd	2		3822.2.c.d.883.1		2
40.19	odd	2		2496.2.c.m.961.2		2
40.29	even	2		2496.2.c.f.961.2		2
60.59	even	2		1872.2.c.b.1585.2		2
65.4	even	6		1014.2.i.c.361.2		4
65.9	even	6		1014.2.i.c.361.1		4
65.12	odd	4		1950.2.f.g.649.2		2
65.19	odd	12		1014.2.e.e.991.1		2
65.24	odd	12		1014.2.e.b.529.1		2
65.29	even	6		1014.2.i.c.823.2		4
65.34	odd	4		1014.2.a.g.1.1		1
65.38	odd	4		1950.2.f.d.649.1		2
65.44	odd	4		1014.2.a.b.1.1		1
65.49	even	6		1014.2.i.c.823.1		4
65.54	odd	12		1014.2.e.e.529.1		2
65.59	odd	12		1014.2.e.b.991.1		2
65.64	even	2		78.2.b.a.25.2	yes	2
195.44	even	4		3042.2.a.n.1.1		1
195.164	even	4		3042.2.a.c.1.1		1
195.194	odd	2		234.2.b.a.181.1		2
260.99	even	4		8112.2.a.j.1.1		1
260.239	even	4		8112.2.a.g.1.1		1
260.259	odd	2		624.2.c.a.337.2		2
455.454	odd	2		3822.2.c.d.883.2		2
520.259	odd	2		2496.2.c.m.961.1		2
520.389	even	2		2496.2.c.f.961.1		2
780.779	even	2		1872.2.c.b.1585.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.2.b.a.25.1	✓	2	5.4	even	2
78.2.b.a.25.2	yes	2	65.64	even	2
234.2.b.a.181.1		2	195.194	odd	2
234.2.b.a.181.2		2	15.14	odd	2
624.2.c.a.337.1		2	20.19	odd	2
624.2.c.a.337.2		2	260.259	odd	2
1014.2.a.b.1.1		1	65.44	odd	4
1014.2.a.g.1.1		1	65.34	odd	4
1014.2.e.b.529.1		2	65.24	odd	12
1014.2.e.b.991.1		2	65.59	odd	12
1014.2.e.e.529.1		2	65.54	odd	12
1014.2.e.e.991.1		2	65.19	odd	12
1014.2.i.c.361.1		4	65.9	even	6
1014.2.i.c.361.2		4	65.4	even	6
1014.2.i.c.823.1		4	65.49	even	6
1014.2.i.c.823.2		4	65.29	even	6
1872.2.c.b.1585.1		2	780.779	even	2
1872.2.c.b.1585.2		2	60.59	even	2
1950.2.b.c.1351.1		2	13.12	even	2	inner
1950.2.b.c.1351.2		2	1.1	even	1	trivial
1950.2.f.d.649.1		2	65.38	odd	4
1950.2.f.d.649.2		2	5.2	odd	4
1950.2.f.g.649.1		2	5.3	odd	4
1950.2.f.g.649.2		2	65.12	odd	4
2496.2.c.f.961.1		2	520.389	even	2
2496.2.c.f.961.2		2	40.29	even	2
2496.2.c.m.961.1		2	520.259	odd	2
2496.2.c.m.961.2		2	40.19	odd	2
3042.2.a.c.1.1		1	195.164	even	4
3042.2.a.n.1.1		1	195.44	even	4
3822.2.c.d.883.1		2	35.34	odd	2
3822.2.c.d.883.2		2	455.454	odd	2
8112.2.a.g.1.1		1	260.239	even	4
8112.2.a.j.1.1		1	260.99	even	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:	\( N \)	\(=\)	\( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1950.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{12} +(3.00000 - 2.00000i) q^{13} +2.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} +1.00000i q^{18} +6.00000i q^{19} +2.00000i q^{21} -4.00000 q^{23} +1.00000i q^{24} +(2.00000 + 3.00000i) q^{26} -1.00000 q^{27} +2.00000i q^{28} -10.0000 q^{29} -10.0000i q^{31} +1.00000i q^{32} +2.00000i q^{34} -1.00000 q^{36} +8.00000i q^{37} -6.00000 q^{38} +(-3.00000 + 2.00000i) q^{39} -10.0000i q^{41} -2.00000 q^{42} -4.00000 q^{43} -4.00000i q^{46} -12.0000i q^{47} -1.00000 q^{48} +3.00000 q^{49} -2.00000 q^{51} +(-3.00000 + 2.00000i) q^{52} +6.00000 q^{53} -1.00000i q^{54} -2.00000 q^{56} -6.00000i q^{57} -10.0000i q^{58} -4.00000i q^{59} +2.00000 q^{61} +10.0000 q^{62} -2.00000i q^{63} -1.00000 q^{64} -2.00000i q^{67} -2.00000 q^{68} +4.00000 q^{69} -1.00000i q^{72} -4.00000i q^{73} -8.00000 q^{74} -6.00000i q^{76} +(-2.00000 - 3.00000i) q^{78} +1.00000 q^{81} +10.0000 q^{82} -4.00000i q^{83} -2.00000i q^{84} -4.00000i q^{86} +10.0000 q^{87} +6.00000i q^{89} +(-4.00000 - 6.00000i) q^{91} +4.00000 q^{92} +10.0000i q^{93} +12.0000 q^{94} -1.00000i q^{96} -12.0000i q^{97} +3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} + 2 q^{12} + 6 q^{13} + 4 q^{14} + 2 q^{16} + 4 q^{17} - 8 q^{23} + 4 q^{26} - 2 q^{27} - 20 q^{29} - 2 q^{36} - 12 q^{38} - 6 q^{39} - 4 q^{42} - 8 q^{43} - 2 q^{48} + 6 q^{49} - 4 q^{51} - 6 q^{52} + 12 q^{53} - 4 q^{56} + 4 q^{61} + 20 q^{62} - 2 q^{64} - 4 q^{68} + 8 q^{69} - 16 q^{74} - 4 q^{78} + 2 q^{81} + 20 q^{82} + 20 q^{87} - 8 q^{91} + 8 q^{92} + 24 q^{94}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 + 2 * q^12 + 6 * q^13 + 4 * q^14 + 2 * q^16 + 4 * q^17 - 8 * q^23 + 4 * q^26 - 2 * q^27 - 20 * q^29 - 2 * q^36 - 12 * q^38 - 6 * q^39 - 4 * q^42 - 8 * q^43 - 2 * q^48 + 6 * q^49 - 4 * q^51 - 6 * q^52 + 12 * q^53 - 4 * q^56 + 4 * q^61 + 20 * q^62 - 2 * q^64 - 4 * q^68 + 8 * q^69 - 16 * q^74 - 4 * q^78 + 2 * q^81 + 20 * q^82 + 20 * q^87 - 8 * q^91 + 8 * q^92 + 24 * q^94

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