LMFDB - Embedded newform 1950.2.b.c.1351.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1950,2,Mod(1351,1950)] 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("1950.1351"); S:= CuspForms(chi, 2); N := Newforms(S);

sage: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")

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:traces = [2,0,-2,-2,0,0,0,0,2,0,0,2,6,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

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

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	$0.123376\pi$
$7$			2.00000i			0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$			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	$-0.758380\pi$
$19$		−	6.00000i		−	1.37649i	0.725476	−	0.688247i	$-0.241620\pi$
$20$		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	$0.354999\pi$
$31$			10.0000i			1.79605i	−0.439941	+	0.898027i	$0.645001\pi$
$32$		−	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	$-0.771573\pi$
$37$		−	8.00000i		−	1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$	−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	$0.285223\pi$
$41$			10.0000i			1.56174i	−0.624695	+	0.780869i	$0.714777\pi$
$42$	−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	$0.339264\pi$
$47$			12.0000i			1.75038i	−0.483779	+	0.875190i	$0.660736\pi$
$48$	−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	$0.0838471\pi$
$59$			4.00000i			0.520756i	−0.965507	+	0.260378i	$0.916153\pi$
$60$		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	$0.0389851\pi$
$67$			2.00000i			0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$	−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	$0.0752085\pi$
$73$			4.00000i			0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$	−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	$0.0704519\pi$
$83$			4.00000i			0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$			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	$-0.896989\pi$
$89$		−	6.00000i		−	0.635999i	0.948091	−	0.317999i	$-0.103011\pi$
$90$		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	$0.208512\pi$
$97$			12.0000i			1.21842i	−0.793011	+	0.609208i	$0.791488\pi$
$98$		−	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	$0.0613564\pi$
$109$			4.00000i			0.383131i	−0.981480	+	0.191565i	$0.938644\pi$
$110$		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	$0.0272282\pi$
$137$			2.00000i			0.170872i	−0.996344	+	0.0854358i	$0.972772\pi$
$138$		−	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	$0.194400\pi$
$149$			14.0000i			1.14692i	−0.819232	+	0.573462i	$0.805600\pi$
$150$		0			0
$151$			10.0000i			0.813788i	0.913475	+	0.406894i	$0.133388\pi$
$151$			10.0000i			0.813788i	−0.913475	+	0.406894i	$0.866612\pi$
$152$	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	$0.184718\pi$
$163$			14.0000i			1.09656i	−0.836293	+	0.548282i	$0.815282\pi$
$164$		−	10.0000i		−	0.780869i
$165$		0			0
$166$	4.00000			0.310460
$167$			12.0000i			0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$			12.0000i			0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$	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	$-0.804670\pi$
$193$		−	16.0000i		−	1.15171i	0.817554	−	0.575853i	$-0.195330\pi$
$194$	12.0000			0.861550
$195$		0			0
$196$	−3.00000			−0.214286
$197$			22.0000i			1.56744i	0.621117	+	0.783718i	$0.286679\pi$
$197$			22.0000i			1.56744i	−0.621117	+	0.783718i	$0.713321\pi$
$198$		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	$0.155297\pi$
$223$			14.0000i			0.937509i	−0.883328	+	0.468755i	$0.844703\pi$
$224$	2.00000			0.133631
$225$		0			0
$226$			14.0000i			0.931266i
$227$		−	8.00000i		−	0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$		−	8.00000i		−	0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$		−	6.00000i		−	0.397360i
$229$			4.00000i			0.264327i	0.991228	+	0.132164i	$0.0421925\pi$
$229$			4.00000i			0.264327i	−0.991228	+	0.132164i	$0.957808\pi$
$230$		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	$-0.826871\pi$
$239$		−	16.0000i		−	1.03495i	0.855697	−	0.517477i	$-0.173129\pi$
$240$		0			0
$241$			20.0000i			1.28831i	0.764894	+	0.644157i	$0.222792\pi$
$241$			20.0000i			1.28831i	−0.764894	+	0.644157i	$0.777208\pi$
$242$		−	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	$-0.901768\pi$
$271$		−	10.0000i		−	0.607457i	0.952759	−	0.303728i	$-0.0982315\pi$
$272$	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	$-0.903589\pi$
$281$		−	10.0000i		−	0.596550i	0.954480	−	0.298275i	$-0.0964112\pi$
$282$	−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	$0.134103\pi$
$293$			14.0000i			0.817889i	−0.912559	+	0.408944i	$0.865897\pi$
$294$			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	$0.0181768\pi$
$307$			2.00000i			0.114146i	−0.998370	+	0.0570730i	$0.981823\pi$
$308$		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	$-0.831312\pi$
$317$		−	18.0000i		−	1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$			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	$-0.911380\pi$
$331$		−	10.0000i		−	0.549650i	0.961494	−	0.274825i	$-0.0886199\pi$
$332$		−	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	$-0.859137\pi$
$349$		−	16.0000i		−	0.856460i	0.903670	−	0.428230i	$-0.140863\pi$
$350$		0			0
$351$	−3.00000	−	2.00000i	−0.160128	−	0.106752i
$352$		0			0
$353$		−	26.0000i		−	1.38384i	−0.721974	−	0.691920i	$-0.756765\pi$
$353$		−	26.0000i		−	1.38384i	0.721974	−	0.691920i	$-0.243235\pi$
$354$	−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	$0.0336622\pi$
$359$			4.00000i			0.211112i	−0.994413	+	0.105556i	$0.966338\pi$
$360$		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	$0.337980\pi$
$379$			34.0000i			1.74646i	−0.487306	+	0.873231i	$0.662020\pi$
$380$		0			0
$381$	8.00000			0.409852
$382$		−	12.0000i		−	0.613973i
$383$			4.00000i			0.204390i	0.994764	+	0.102195i	$0.0325866\pi$
$383$			4.00000i			0.204390i	−0.994764	+	0.102195i	$0.967413\pi$
$384$		−	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	$-0.935661\pi$
$397$		−	8.00000i		−	0.401508i	0.979642	−	0.200754i	$-0.0643393\pi$
$398$		0			0
$399$	−12.0000			−0.600751
$400$		0			0
$401$		−	30.0000i		−	1.49813i	−0.662497	−	0.749064i	$-0.730503\pi$
$401$		−	30.0000i		−	1.49813i	0.662497	−	0.749064i	$-0.269497\pi$
$402$	−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	$0.0315304\pi$
$409$			4.00000i			0.197787i	−0.995098	+	0.0988936i	$0.968470\pi$
$410$		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	$0.162044\pi$
$421$			20.0000i			0.974740i	−0.873195	+	0.487370i	$0.837956\pi$
$422$		−	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	$-0.840026\pi$
$431$		−	20.0000i		−	0.963366i	0.876346	−	0.481683i	$-0.159974\pi$
$432$	−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	$-0.954782\pi$
$449$		−	6.00000i		−	0.283158i	0.989927	−	0.141579i	$-0.0452178\pi$
$450$		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	$-0.772714\pi$
$457$		−	28.0000i		−	1.30978i	0.755722	−	0.654892i	$-0.227286\pi$
$458$	4.00000			0.186908
$459$	−2.00000			−0.0933520
$460$		0			0
$461$		−	30.0000i		−	1.39724i	−0.715493	−	0.698620i	$-0.753798\pi$
$461$		−	30.0000i		−	1.39724i	0.715493	−	0.698620i	$-0.246202\pi$
$462$		0			0
$463$		−	6.00000i		−	0.278844i	−0.990233	−	0.139422i	$-0.955476\pi$
$463$		−	6.00000i		−	0.278844i	0.990233	−	0.139422i	$-0.0445244\pi$
$464$	−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	$0.184723\pi$
$479$			24.0000i			1.09659i	−0.836286	+	0.548294i	$0.815277\pi$
$480$		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	$-0.866286\pi$
$487$		−	18.0000i		−	0.815658i	0.913058	−	0.407829i	$-0.133714\pi$
$488$			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	$0.101456\pi$
$499$			14.0000i			0.626726i	−0.949633	+	0.313363i	$0.898544\pi$
$500$		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	$-0.957548\pi$
$509$		−	6.00000i		−	0.265945i	0.991120	−	0.132973i	$-0.0424523\pi$
$510$		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	$-0.858537\pi$
$541$		−	20.0000i		−	0.859867i	0.902861	−	0.429934i	$-0.141463\pi$
$542$	−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	$-0.875462\pi$
$557$		−	18.0000i		−	0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$	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	$-0.946746\pi$
$577$		−	8.00000i		−	0.333044i	0.986038	−	0.166522i	$-0.0532537\pi$
$578$			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	$-0.803895\pi$
$587$		−	28.0000i		−	1.15568i	0.816149	−	0.577842i	$-0.196105\pi$
$588$	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	$-0.820746\pi$
$593$		−	26.0000i		−	1.06769i	0.845582	−	0.533846i	$-0.179254\pi$
$594$		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	$-0.895269\pi$
$613$		−	16.0000i		−	0.646234i	0.946359	−	0.323117i	$-0.104731\pi$
$614$	2.00000			0.0807134
$615$		0			0
$616$		0			0
$617$			22.0000i			0.885687i	0.896599	+	0.442843i	$0.146030\pi$
$617$			22.0000i			0.885687i	−0.896599	+	0.442843i	$0.853970\pi$
$618$			16.0000i			0.643614i
$619$		−	26.0000i		−	1.04503i	−0.852631	−	0.522514i	$-0.824994\pi$
$619$		−	26.0000i		−	1.04503i	0.852631	−	0.522514i	$-0.175006\pi$
$620$		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	$0.0637846\pi$
$631$			10.0000i			0.398094i	−0.979990	+	0.199047i	$0.936215\pi$
$632$		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	$-0.962253\pi$
$643$		−	6.00000i		−	0.236617i	0.992977	−	0.118308i	$-0.0377472\pi$
$644$			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	$-0.716280\pi$
$661$		−	40.0000i		−	1.55582i	0.628376	−	0.777910i	$-0.283720\pi$
$662$	−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	$0.151852\pi$
$683$			24.0000i			0.918334i	−0.888350	+	0.459167i	$0.848148\pi$
$684$			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	$-0.939083\pi$
$691$		−	10.0000i		−	0.380418i	0.981744	−	0.190209i	$-0.0609166\pi$
$692$	−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	$-0.763710\pi$
$709$		−	36.0000i		−	1.35201i	0.736898	−	0.676004i	$-0.236290\pi$
$710$		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	$0.301942\pi$
$733$			44.0000i			1.62518i	−0.582838	+	0.812589i	$0.698058\pi$
$734$			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	$-0.841285\pi$
$739$		−	26.0000i		−	0.956425i	0.878244	−	0.478213i	$-0.158715\pi$
$740$		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	$-0.905183\pi$
$743$		−	16.0000i		−	0.586983i	0.955962	−	0.293492i	$-0.0948173\pi$
$744$	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	$0.182996\pi$
$761$			30.0000i			1.08750i	−0.839248	+	0.543750i	$0.817004\pi$
$762$		−	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	$0.142450\pi$
$769$			24.0000i			0.865462i	−0.901523	+	0.432731i	$0.857550\pi$
$770$		0			0
$771$	18.0000			0.648254
$772$			16.0000i			0.575853i
$773$		−	6.00000i		−	0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$		−	6.00000i		−	0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$			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	$-0.763160\pi$
$787$		−	38.0000i		−	1.35455i	0.735728	−	0.677277i	$-0.236840\pi$
$788$		−	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	$0.0561780\pi$
$811$			10.0000i			0.351147i	−0.984466	+	0.175574i	$0.943822\pi$
$812$			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	$0.175375\pi$
$821$			30.0000i			1.04701i	−0.852023	+	0.523504i	$0.824625\pi$
$822$	−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	$-0.685721\pi$
$827$		−	48.0000i		−	1.66912i	0.550914	−	0.834562i	$-0.314279\pi$
$828$	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	$-0.910928\pi$
$839$		−	16.0000i		−	0.552381i	0.961103	−	0.276191i	$-0.0890721\pi$
$840$		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	$-0.591799\pi$
$853$		−	56.0000i		−	1.91740i	0.284413	−	0.958702i	$-0.408201\pi$
$854$	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	$0.269412\pi$
$863$			44.0000i			1.49778i	−0.662696	+	0.748889i	$0.730588\pi$
$864$			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	$-0.956874\pi$
$877$		−	8.00000i		−	0.270141i	0.990836	−	0.135070i	$-0.0431261\pi$
$878$		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	$-0.968619\pi$
$929$		−	6.00000i		−	0.196854i	0.995144	−	0.0984268i	$-0.0313810\pi$
$930$		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	$-0.947884\pi$
$941$		−	10.0000i		−	0.325991i	0.986627	−	0.162995i	$-0.0521156\pi$
$942$			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	$0.320332\pi$
$947$			52.0000i			1.68977i	−0.534946	+	0.844886i	$0.679668\pi$
$948$		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	$0.115089\pi$
$967$			22.0000i			0.707472i	−0.935345	+	0.353736i	$0.884911\pi$
$968$			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	$0.234500\pi$
$977$			42.0000i			1.34370i	−0.740688	+	0.671850i	$0.765500\pi$
$978$	−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	$0.125020\pi$
$983$			24.0000i			0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$	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.1351.1		2
5.2	odd	4		1950.2.f.g.649.2		2
5.3	odd	4		1950.2.f.d.649.1		2
5.4	even	2		78.2.b.a.25.2	yes	2
13.12	even	2	inner	1950.2.b.c.1351.2		2
15.14	odd	2		234.2.b.a.181.1		2
20.19	odd	2		624.2.c.a.337.2		2
35.34	odd	2		3822.2.c.d.883.2		2
40.19	odd	2		2496.2.c.m.961.1		2
40.29	even	2		2496.2.c.f.961.1		2
60.59	even	2		1872.2.c.b.1585.1		2
65.4	even	6		1014.2.i.c.361.1		4
65.9	even	6		1014.2.i.c.361.2		4
65.12	odd	4		1950.2.f.d.649.2		2
65.19	odd	12		1014.2.e.b.991.1		2
65.24	odd	12		1014.2.e.e.529.1		2
65.29	even	6		1014.2.i.c.823.1		4
65.34	odd	4		1014.2.a.b.1.1		1
65.38	odd	4		1950.2.f.g.649.1		2
65.44	odd	4		1014.2.a.g.1.1		1
65.49	even	6		1014.2.i.c.823.2		4
65.54	odd	12		1014.2.e.b.529.1		2
65.59	odd	12		1014.2.e.e.991.1		2
65.64	even	2		78.2.b.a.25.1	✓	2
195.44	even	4		3042.2.a.c.1.1		1
195.164	even	4		3042.2.a.n.1.1		1
195.194	odd	2		234.2.b.a.181.2		2
260.99	even	4		8112.2.a.g.1.1		1
260.239	even	4		8112.2.a.j.1.1		1
260.259	odd	2		624.2.c.a.337.1		2
455.454	odd	2		3822.2.c.d.883.1		2
520.259	odd	2		2496.2.c.m.961.2		2
520.389	even	2		2496.2.c.f.961.2		2
780.779	even	2		1872.2.c.b.1585.2		2

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

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)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more