LMFDB - Embedded newform 325.2.c.f.51.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [325,2,Mod(51,325)] 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("325.51"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	325.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,4,2,0,0,0,0,2,0,0,4,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$2.59513806569$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			51.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	325.51
Dual form			325.2.c.f.51.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} +2.00000 q^{3} +1.00000 q^{4} +2.00000i q^{6} -5.00000i q^{7} +3.00000i q^{8} +1.00000 q^{9} +3.00000i q^{11} +2.00000 q^{12} +(3.00000 - 2.00000i) q^{13} +5.00000 q^{14} -1.00000 q^{16} -5.00000 q^{17} +1.00000i q^{18} +4.00000i q^{19} -10.0000i q^{21} -3.00000 q^{22} -4.00000 q^{23} +6.00000i q^{24} +(2.00000 + 3.00000i) q^{26} -4.00000 q^{27} -5.00000i q^{28} -1.00000 q^{29} -1.00000i q^{31} +5.00000i q^{32} +6.00000i q^{33} -5.00000i q^{34} +1.00000 q^{36} +4.00000i q^{37} -4.00000 q^{38} +(6.00000 - 4.00000i) q^{39} +8.00000i q^{41} +10.0000 q^{42} +4.00000 q^{43} +3.00000i q^{44} -4.00000i q^{46} -7.00000i q^{47} -2.00000 q^{48} -18.0000 q^{49} -10.0000 q^{51} +(3.00000 - 2.00000i) q^{52} +3.00000 q^{53} -4.00000i q^{54} +15.0000 q^{56} +8.00000i q^{57} -1.00000i q^{58} -3.00000i q^{59} +1.00000 q^{61} +1.00000 q^{62} -5.00000i q^{63} -7.00000 q^{64} -6.00000 q^{66} +3.00000i q^{67} -5.00000 q^{68} -8.00000 q^{69} -8.00000i q^{71} +3.00000i q^{72} +4.00000i q^{73} -4.00000 q^{74} +4.00000i q^{76} +15.0000 q^{77} +(4.00000 + 6.00000i) q^{78} +10.0000 q^{79} -11.0000 q^{81} -8.00000 q^{82} -9.00000i q^{83} -10.0000i q^{84} +4.00000i q^{86} -2.00000 q^{87} -9.00000 q^{88} -18.0000i q^{89} +(-10.0000 - 15.0000i) q^{91} -4.00000 q^{92} -2.00000i q^{93} +7.00000 q^{94} +10.0000i q^{96} -14.0000i q^{97} -18.0000i q^{98} +3.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{3} + 2 q^{4} + 2 q^{9} + 4 q^{12} + 6 q^{13} + 10 q^{14} - 2 q^{16} - 10 q^{17} - 6 q^{22} - 8 q^{23} + 4 q^{26} - 8 q^{27} - 2 q^{29} + 2 q^{36} - 8 q^{38} + 12 q^{39} + 20 q^{42} + 8 q^{43}+ \cdots + 14 q^{94}+O(q^{100}) $ 2 * q + 4 * q^3 + 2 * q^4 + 2 * q^9 + 4 * q^12 + 6 * q^13 + 10 * q^14 - 2 * q^16 - 10 * q^17 - 6 * q^22 - 8 * q^23 + 4 * q^26 - 8 * q^27 - 2 * q^29 + 2 * q^36 - 8 * q^38 + 12 * q^39 + 20 * q^42 + 8 * q^43 - 4 * q^48 - 36 * q^49 - 20 * q^51 + 6 * q^52 + 6 * q^53 + 30 * q^56 + 2 * q^61 + 2 * q^62 - 14 * q^64 - 12 * q^66 - 10 * q^68 - 16 * q^69 - 8 * q^74 + 30 * q^77 + 8 * q^78 + 20 * q^79 - 22 * q^81 - 16 * q^82 - 4 * q^87 - 18 * q^88 - 20 * q^91 - 8 * q^92 + 14 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$.

$n$	$27$	$301$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$			1.00000i			0.707107i	0.935414	+	0.353553i	$0.115027\pi$
$2$			1.00000i			0.707107i	−0.935414	+	0.353553i	$0.884973\pi$
$3$	2.00000			1.15470			0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000			1.15470			0.577350	+	0.816497i	$0.304087\pi$
$4$	1.00000			0.500000
$5$		0			0
$5$		0			0
$6$			2.00000i			0.816497i
$7$		−	5.00000i		−	1.88982i	−0.327327	−	0.944911i	$-0.606148\pi$
$7$		−	5.00000i		−	1.88982i	0.327327	−	0.944911i	$-0.393852\pi$
$8$			3.00000i			1.06066i
$9$	1.00000			0.333333
$10$		0			0
$11$			3.00000i			0.904534i	0.891883	+	0.452267i	$0.149385\pi$
$11$			3.00000i			0.904534i	−0.891883	+	0.452267i	$0.850615\pi$
$12$	2.00000			0.577350
$13$	3.00000	−	2.00000i	0.832050	−	0.554700i
$13$	3.00000	−	2.00000i	0.832050	−	0.554700i
$14$	5.00000			1.33631
$15$		0			0
$16$	−1.00000			−0.250000
$17$	−5.00000			−1.21268			−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000			−1.21268			−0.606339	+	0.795206i	$0.707363\pi$
$18$			1.00000i			0.235702i
$19$			4.00000i			0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$			4.00000i			0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$		0			0
$21$		−	10.0000i		−	2.18218i
$22$	−3.00000			−0.639602
$23$	−4.00000			−0.834058			−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000			−0.834058			−0.417029	+	0.908893i	$0.636929\pi$
$24$			6.00000i			1.22474i
$25$		0			0
$26$	2.00000	+	3.00000i	0.392232	+	0.588348i
$27$	−4.00000			−0.769800
$28$		−	5.00000i		−	0.944911i
$29$	−1.00000			−0.185695			−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000			−0.185695			−0.0928477	+	0.995680i	$0.529597\pi$
$30$		0			0
$31$		−	1.00000i		−	0.179605i	−0.995960	−	0.0898027i	$-0.971376\pi$
$31$		−	1.00000i		−	0.179605i	0.995960	−	0.0898027i	$-0.0286236\pi$
$32$			5.00000i			0.883883i
$33$			6.00000i			1.04447i
$34$		−	5.00000i		−	0.857493i
$35$		0			0
$36$	1.00000			0.166667
$37$			4.00000i			0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$			4.00000i			0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$	−4.00000			−0.648886
$39$	6.00000	−	4.00000i	0.960769	−	0.640513i
$40$		0			0
$41$			8.00000i			1.24939i	0.780869	+	0.624695i	$0.214777\pi$
$41$			8.00000i			1.24939i	−0.780869	+	0.624695i	$0.785223\pi$
$42$	10.0000			1.54303
$43$	4.00000			0.609994			0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000			0.609994			0.304997	+	0.952353i	$0.401344\pi$
$44$			3.00000i			0.452267i
$45$		0			0
$46$		−	4.00000i		−	0.589768i
$47$		−	7.00000i		−	1.02105i	−0.859861	−	0.510527i	$-0.829450\pi$
$47$		−	7.00000i		−	1.02105i	0.859861	−	0.510527i	$-0.170550\pi$
$48$	−2.00000			−0.288675
$49$	−18.0000			−2.57143
$50$		0			0
$51$	−10.0000			−1.40028
$52$	3.00000	−	2.00000i	0.416025	−	0.277350i
$53$	3.00000			0.412082			0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000			0.412082			0.206041	+	0.978543i	$0.433942\pi$
$54$		−	4.00000i		−	0.544331i
$55$		0			0
$56$	15.0000			2.00446
$57$			8.00000i			1.05963i
$58$		−	1.00000i		−	0.131306i
$59$		−	3.00000i		−	0.390567i	−0.980747	−	0.195283i	$-0.937437\pi$
$59$		−	3.00000i		−	0.390567i	0.980747	−	0.195283i	$-0.0625627\pi$
$60$		0			0
$61$	1.00000			0.128037			0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000			0.128037			0.0640184	+	0.997949i	$0.479608\pi$
$62$	1.00000			0.127000
$63$		−	5.00000i		−	0.629941i
$64$	−7.00000			−0.875000
$65$		0			0
$66$	−6.00000			−0.738549
$67$			3.00000i			0.366508i	0.983066	+	0.183254i	$0.0586631\pi$
$67$			3.00000i			0.366508i	−0.983066	+	0.183254i	$0.941337\pi$
$68$	−5.00000			−0.606339
$69$	−8.00000			−0.963087
$70$		0			0
$71$		−	8.00000i		−	0.949425i	−0.880141	−	0.474713i	$-0.842552\pi$
$71$		−	8.00000i		−	0.949425i	0.880141	−	0.474713i	$-0.157448\pi$
$72$			3.00000i			0.353553i
$73$			4.00000i			0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$			4.00000i			0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$	−4.00000			−0.464991
$75$		0			0
$76$			4.00000i			0.458831i
$77$	15.0000			1.70941
$78$	4.00000	+	6.00000i	0.452911	+	0.679366i
$79$	10.0000			1.12509			0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000			1.12509			0.562544	+	0.826767i	$0.309823\pi$
$80$		0			0
$81$	−11.0000			−1.22222
$82$	−8.00000			−0.883452
$83$		−	9.00000i		−	0.987878i	−0.869496	−	0.493939i	$-0.835557\pi$
$83$		−	9.00000i		−	0.987878i	0.869496	−	0.493939i	$-0.164443\pi$
$84$		−	10.0000i		−	1.09109i
$85$		0			0
$86$			4.00000i			0.431331i
$87$	−2.00000			−0.214423
$88$	−9.00000			−0.959403
$89$		−	18.0000i		−	1.90800i	−0.299813	−	0.953998i	$-0.596924\pi$
$89$		−	18.0000i		−	1.90800i	0.299813	−	0.953998i	$-0.403076\pi$
$90$		0			0
$91$	−10.0000	−	15.0000i	−1.04828	−	1.57243i
$92$	−4.00000			−0.417029
$93$		−	2.00000i		−	0.207390i
$94$	7.00000			0.721995
$95$		0			0
$96$			10.0000i			1.02062i
$97$		−	14.0000i		−	1.42148i	−0.703452	−	0.710742i	$-0.748359\pi$
$97$		−	14.0000i		−	1.42148i	0.703452	−	0.710742i	$-0.251641\pi$
$98$		−	18.0000i		−	1.81827i
$99$			3.00000i			0.301511i
$100$		0			0
$101$	1.00000			0.0995037			0.0497519	−	0.998762i	$-0.484157\pi$
$101$	1.00000			0.0995037			0.0497519	+	0.998762i	$0.484157\pi$
$102$		−	10.0000i		−	0.990148i
$103$	−4.00000			−0.394132			−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000			−0.394132			−0.197066	+	0.980390i	$0.563141\pi$
$104$	6.00000	+	9.00000i	0.588348	+	0.882523i
$105$		0			0
$106$			3.00000i			0.291386i
$107$	16.0000			1.54678			0.773389	−	0.633932i	$-0.218560\pi$
$107$	16.0000			1.54678			0.773389	+	0.633932i	$0.218560\pi$
$108$	−4.00000			−0.384900
$109$			2.00000i			0.191565i	0.995402	+	0.0957826i	$0.0305354\pi$
$109$			2.00000i			0.191565i	−0.995402	+	0.0957826i	$0.969465\pi$
$110$		0			0
$111$			8.00000i			0.759326i
$112$			5.00000i			0.472456i
$113$	−10.0000			−0.940721			−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000			−0.940721			−0.470360	+	0.882474i	$0.655876\pi$
$114$	−8.00000			−0.749269
$115$		0			0
$116$	−1.00000			−0.0928477
$117$	3.00000	−	2.00000i	0.277350	−	0.184900i
$118$	3.00000			0.276172
$119$			25.0000i			2.29175i
$120$		0			0
$121$	2.00000			0.181818
$122$			1.00000i			0.0905357i
$123$			16.0000i			1.44267i
$124$		−	1.00000i		−	0.0898027i
$125$		0			0
$126$	5.00000			0.445435
$127$	−18.0000			−1.59724			−0.798621	−	0.601834i	$-0.794437\pi$
$127$	−18.0000			−1.59724			−0.798621	+	0.601834i	$0.794437\pi$
$128$			3.00000i			0.265165i
$129$	8.00000			0.704361
$130$		0			0
$131$	18.0000			1.57267			0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000			1.57267			0.786334	+	0.617802i	$0.211977\pi$
$132$			6.00000i			0.522233i
$133$	20.0000			1.73422
$134$	−3.00000			−0.259161
$135$		0			0
$136$		−	15.0000i		−	1.28624i
$137$			18.0000i			1.53784i	0.639343	+	0.768922i	$0.279207\pi$
$137$			18.0000i			1.53784i	−0.639343	+	0.768922i	$0.720793\pi$
$138$		−	8.00000i		−	0.681005i
$139$	4.00000			0.339276			0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000			0.339276			0.169638	+	0.985506i	$0.445740\pi$
$140$		0			0
$141$		−	14.0000i		−	1.17901i
$142$	8.00000			0.671345
$143$	6.00000	+	9.00000i	0.501745	+	0.752618i
$144$	−1.00000			−0.0833333
$145$		0			0
$146$	−4.00000			−0.331042
$147$	−36.0000			−2.96923
$148$			4.00000i			0.328798i
$149$		−	10.0000i		−	0.819232i	−0.912258	−	0.409616i	$-0.865663\pi$
$149$		−	10.0000i		−	0.819232i	0.912258	−	0.409616i	$-0.134337\pi$
$150$		0			0
$151$			13.0000i			1.05792i	0.848645	+	0.528962i	$0.177419\pi$
$151$			13.0000i			1.05792i	−0.848645	+	0.528962i	$0.822581\pi$
$152$	−12.0000			−0.973329
$153$	−5.00000			−0.404226
$154$			15.0000i			1.20873i
$155$		0			0
$156$	6.00000	−	4.00000i	0.480384	−	0.320256i
$157$	13.0000			1.03751			0.518756	−	0.854922i	$-0.326395\pi$
$157$	13.0000			1.03751			0.518756	+	0.854922i	$0.326395\pi$
$158$			10.0000i			0.795557i
$159$	6.00000			0.475831
$160$		0			0
$161$			20.0000i			1.57622i
$162$		−	11.0000i		−	0.864242i
$163$			12.0000i			0.939913i	0.882690	+	0.469956i	$0.155730\pi$
$163$			12.0000i			0.939913i	−0.882690	+	0.469956i	$0.844270\pi$
$164$			8.00000i			0.624695i
$165$		0			0
$166$	9.00000			0.698535
$167$			16.0000i			1.23812i	0.785345	+	0.619059i	$0.212486\pi$
$167$			16.0000i			1.23812i	−0.785345	+	0.619059i	$0.787514\pi$
$168$	30.0000			2.31455
$169$	5.00000	−	12.0000i	0.384615	−	0.923077i
$170$		0			0
$171$			4.00000i			0.305888i
$172$	4.00000			0.304997
$173$	−13.0000			−0.988372			−0.494186	−	0.869356i	$-0.664534\pi$
$173$	−13.0000			−0.988372			−0.494186	+	0.869356i	$0.664534\pi$
$174$		−	2.00000i		−	0.151620i
$175$		0			0
$176$		−	3.00000i		−	0.226134i
$177$		−	6.00000i		−	0.450988i
$178$	18.0000			1.34916
$179$	12.0000			0.896922			0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000			0.896922			0.448461	+	0.893802i	$0.351972\pi$
$180$		0			0
$181$	3.00000			0.222988			0.111494	−	0.993765i	$-0.464436\pi$
$181$	3.00000			0.222988			0.111494	+	0.993765i	$0.464436\pi$
$182$	15.0000	−	10.0000i	1.11187	−	0.741249i
$183$	2.00000			0.147844
$184$		−	12.0000i		−	0.884652i
$185$		0			0
$186$	2.00000			0.146647
$187$		−	15.0000i		−	1.09691i
$188$		−	7.00000i		−	0.510527i
$189$			20.0000i			1.45479i
$190$		0			0
$191$	10.0000			0.723575			0.361787	−	0.932261i	$-0.382167\pi$
$191$	10.0000			0.723575			0.361787	+	0.932261i	$0.382167\pi$
$192$	−14.0000			−1.01036
$193$			4.00000i			0.287926i	0.989583	+	0.143963i	$0.0459847\pi$
$193$			4.00000i			0.287926i	−0.989583	+	0.143963i	$0.954015\pi$
$194$	14.0000			1.00514
$195$		0			0
$196$	−18.0000			−1.28571
$197$		−	8.00000i		−	0.569976i	−0.958531	−	0.284988i	$-0.908010\pi$
$197$		−	8.00000i		−	0.569976i	0.958531	−	0.284988i	$-0.0919897\pi$
$198$	−3.00000			−0.213201
$199$	−6.00000			−0.425329			−0.212664	−	0.977125i	$-0.568214\pi$
$199$	−6.00000			−0.425329			−0.212664	+	0.977125i	$0.568214\pi$
$200$		0			0
$201$			6.00000i			0.423207i
$202$			1.00000i			0.0703598i
$203$			5.00000i			0.350931i
$204$	−10.0000			−0.700140
$205$		0			0
$206$		−	4.00000i		−	0.278693i
$207$	−4.00000			−0.278019
$208$	−3.00000	+	2.00000i	−0.208013	+	0.138675i
$209$	−12.0000			−0.830057
$210$		0			0
$211$	−12.0000			−0.826114			−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000			−0.826114			−0.413057	+	0.910705i	$0.635539\pi$
$212$	3.00000			0.206041
$213$		−	16.0000i		−	1.09630i
$214$			16.0000i			1.09374i
$215$		0			0
$216$		−	12.0000i		−	0.816497i
$217$	−5.00000			−0.339422
$218$	−2.00000			−0.135457
$219$			8.00000i			0.540590i
$220$		0			0
$221$	−15.0000	+	10.0000i	−1.00901	+	0.672673i
$222$	−8.00000			−0.536925
$223$		−	16.0000i		−	1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$		−	16.0000i		−	1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$	25.0000			1.67038
$225$		0			0
$226$		−	10.0000i		−	0.665190i
$227$			9.00000i			0.597351i	0.954355	+	0.298675i	$0.0965448\pi$
$227$			9.00000i			0.597351i	−0.954355	+	0.298675i	$0.903455\pi$
$228$			8.00000i			0.529813i
$229$		−	2.00000i		−	0.132164i	−0.997814	−	0.0660819i	$-0.978950\pi$
$229$		−	2.00000i		−	0.132164i	0.997814	−	0.0660819i	$-0.0210498\pi$
$230$		0			0
$231$	30.0000			1.97386
$232$		−	3.00000i		−	0.196960i
$233$	−14.0000			−0.917170			−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000			−0.917170			−0.458585	+	0.888650i	$0.651644\pi$
$234$	2.00000	+	3.00000i	0.130744	+	0.196116i
$235$		0			0
$236$		−	3.00000i		−	0.195283i
$237$	20.0000			1.29914
$238$	−25.0000			−1.62051
$239$		−	5.00000i		−	0.323423i	−0.986838	−	0.161712i	$-0.948299\pi$
$239$		−	5.00000i		−	0.323423i	0.986838	−	0.161712i	$-0.0517014\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$			2.00000i			0.128565i
$243$	−10.0000			−0.641500
$244$	1.00000			0.0640184
$245$		0			0
$246$	−16.0000			−1.02012
$247$	8.00000	+	12.0000i	0.509028	+	0.763542i
$248$	3.00000			0.190500
$249$		−	18.0000i		−	1.14070i
$250$		0			0
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$		−	5.00000i		−	0.314970i
$253$		−	12.0000i		−	0.754434i
$254$		−	18.0000i		−	1.12942i
$255$		0			0
$256$	−17.0000			−1.06250
$257$	−15.0000			−0.935674			−0.467837	−	0.883815i	$-0.654967\pi$
$257$	−15.0000			−0.935674			−0.467837	+	0.883815i	$0.654967\pi$
$258$			8.00000i			0.498058i
$259$	20.0000			1.24274
$260$		0			0
$261$	−1.00000			−0.0618984
$262$			18.0000i			1.11204i
$263$	6.00000			0.369976			0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000			0.369976			0.184988	+	0.982741i	$0.440775\pi$
$264$	−18.0000			−1.10782
$265$		0			0
$266$			20.0000i			1.22628i
$267$		−	36.0000i		−	2.20316i
$268$			3.00000i			0.183254i
$269$	−17.0000			−1.03651			−0.518254	−	0.855227i	$-0.673418\pi$
$269$	−17.0000			−1.03651			−0.518254	+	0.855227i	$0.673418\pi$
$270$		0			0
$271$			29.0000i			1.76162i	0.473466	+	0.880812i	$0.343003\pi$
$271$			29.0000i			1.76162i	−0.473466	+	0.880812i	$0.656997\pi$
$272$	5.00000			0.303170
$273$	−20.0000	−	30.0000i	−1.21046	−	1.81568i
$274$	−18.0000			−1.08742
$275$		0			0
$276$	−8.00000			−0.481543
$277$	2.00000			0.120168			0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000			0.120168			0.0600842	+	0.998193i	$0.480863\pi$
$278$			4.00000i			0.239904i
$279$		−	1.00000i		−	0.0598684i
$280$		0			0
$281$		−	18.0000i		−	1.07379i	−0.843649	−	0.536895i	$-0.819597\pi$
$281$		−	18.0000i		−	1.07379i	0.843649	−	0.536895i	$-0.180403\pi$
$282$	14.0000			0.833688
$283$	28.0000			1.66443			0.832214	−	0.554455i	$-0.187073\pi$
$283$	28.0000			1.66443			0.832214	+	0.554455i	$0.187073\pi$
$284$		−	8.00000i		−	0.474713i
$285$		0			0
$286$	−9.00000	+	6.00000i	−0.532181	+	0.354787i
$287$	40.0000			2.36113
$288$			5.00000i			0.294628i
$289$	8.00000			0.470588
$290$		0			0
$291$		−	28.0000i		−	1.64139i
$292$			4.00000i			0.234082i
$293$		−	6.00000i		−	0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$		−	6.00000i		−	0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$		−	36.0000i		−	2.09956i
$295$		0			0
$296$	−12.0000			−0.697486
$297$		−	12.0000i		−	0.696311i
$298$	10.0000			0.579284
$299$	−12.0000	+	8.00000i	−0.693978	+	0.462652i
$300$		0			0
$301$		−	20.0000i		−	1.15278i
$302$	−13.0000			−0.748066
$303$	2.00000			0.114897
$304$		−	4.00000i		−	0.229416i
$305$		0			0
$306$		−	5.00000i		−	0.285831i
$307$			8.00000i			0.456584i	0.973593	+	0.228292i	$0.0733141\pi$
$307$			8.00000i			0.456584i	−0.973593	+	0.228292i	$0.926686\pi$
$308$	15.0000			0.854704
$309$	−8.00000			−0.455104
$310$		0			0
$311$	4.00000			0.226819			0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000			0.226819			0.113410	+	0.993548i	$0.463823\pi$
$312$	12.0000	+	18.0000i	0.679366	+	1.01905i
$313$	−17.0000			−0.960897			−0.480448	−	0.877023i	$-0.659526\pi$
$313$	−17.0000			−0.960897			−0.480448	+	0.877023i	$0.659526\pi$
$314$			13.0000i			0.733632i
$315$		0			0
$316$	10.0000			0.562544
$317$			22.0000i			1.23564i	0.786318	+	0.617822i	$0.211985\pi$
$317$			22.0000i			1.23564i	−0.786318	+	0.617822i	$0.788015\pi$
$318$			6.00000i			0.336463i
$319$		−	3.00000i		−	0.167968i
$320$		0			0
$321$	32.0000			1.78607
$322$	−20.0000			−1.11456
$323$		−	20.0000i		−	1.11283i
$324$	−11.0000			−0.611111
$325$		0			0
$326$	−12.0000			−0.664619
$327$			4.00000i			0.221201i
$328$	−24.0000			−1.32518
$329$	−35.0000			−1.92961
$330$		0			0
$331$			20.0000i			1.09930i	0.835395	+	0.549650i	$0.185239\pi$
$331$			20.0000i			1.09930i	−0.835395	+	0.549650i	$0.814761\pi$
$332$		−	9.00000i		−	0.493939i
$333$			4.00000i			0.219199i
$334$	−16.0000			−0.875481
$335$		0			0
$336$			10.0000i			0.545545i
$337$	3.00000			0.163420			0.0817102	−	0.996656i	$-0.473962\pi$
$337$	3.00000			0.163420			0.0817102	+	0.996656i	$0.473962\pi$
$338$	12.0000	+	5.00000i	0.652714	+	0.271964i
$339$	−20.0000			−1.08625
$340$		0			0
$341$	3.00000			0.162459
$342$	−4.00000			−0.216295
$343$			55.0000i			2.96972i
$344$			12.0000i			0.646997i
$345$		0			0
$346$		−	13.0000i		−	0.698884i
$347$	−4.00000			−0.214731			−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000			−0.214731			−0.107366	+	0.994220i	$0.534242\pi$
$348$	−2.00000			−0.107211
$349$			2.00000i			0.107058i	0.998566	+	0.0535288i	$0.0170469\pi$
$349$			2.00000i			0.107058i	−0.998566	+	0.0535288i	$0.982953\pi$
$350$		0			0
$351$	−12.0000	+	8.00000i	−0.640513	+	0.427008i
$352$	−15.0000			−0.799503
$353$			6.00000i			0.319348i	0.987170	+	0.159674i	$0.0510443\pi$
$353$			6.00000i			0.319348i	−0.987170	+	0.159674i	$0.948956\pi$
$354$	6.00000			0.318896
$355$		0			0
$356$		−	18.0000i		−	0.953998i
$357$			50.0000i			2.64628i
$358$			12.0000i			0.634220i
$359$			23.0000i			1.21389i	0.794742	+	0.606947i	$0.207606\pi$
$359$			23.0000i			1.21389i	−0.794742	+	0.606947i	$0.792394\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$			3.00000i			0.157676i
$363$	4.00000			0.209946
$364$	−10.0000	−	15.0000i	−0.524142	−	0.786214i
$365$		0			0
$366$			2.00000i			0.104542i
$367$	18.0000			0.939592			0.469796	−	0.882775i	$-0.344327\pi$
$367$	18.0000			0.939592			0.469796	+	0.882775i	$0.344327\pi$
$368$	4.00000			0.208514
$369$			8.00000i			0.416463i
$370$		0			0
$371$		−	15.0000i		−	0.778761i
$372$		−	2.00000i		−	0.103695i
$373$	−31.0000			−1.60512			−0.802560	−	0.596572i	$-0.796529\pi$
$373$	−31.0000			−1.60512			−0.802560	+	0.596572i	$0.796529\pi$
$374$	15.0000			0.775632
$375$		0			0
$376$	21.0000			1.08299
$377$	−3.00000	+	2.00000i	−0.154508	+	0.103005i
$378$	−20.0000			−1.02869
$379$		−	27.0000i		−	1.38690i	−0.720506	−	0.693448i	$-0.756091\pi$
$379$		−	27.0000i		−	1.38690i	0.720506	−	0.693448i	$-0.243909\pi$
$380$		0			0
$381$	−36.0000			−1.84434
$382$			10.0000i			0.511645i
$383$		−	32.0000i		−	1.63512i	−0.575841	−	0.817562i	$-0.695325\pi$
$383$		−	32.0000i		−	1.63512i	0.575841	−	0.817562i	$-0.304675\pi$
$384$			6.00000i			0.306186i
$385$		0			0
$386$	−4.00000			−0.203595
$387$	4.00000			0.203331
$388$		−	14.0000i		−	0.710742i
$389$	34.0000			1.72387			0.861934	−	0.507020i	$-0.169253\pi$
$389$	34.0000			1.72387			0.861934	+	0.507020i	$0.169253\pi$
$390$		0			0
$391$	20.0000			1.01144
$392$		−	54.0000i		−	2.72741i
$393$	36.0000			1.81596
$394$	8.00000			0.403034
$395$		0			0
$396$			3.00000i			0.150756i
$397$			8.00000i			0.401508i	0.979642	+	0.200754i	$0.0643393\pi$
$397$			8.00000i			0.401508i	−0.979642	+	0.200754i	$0.935661\pi$
$398$		−	6.00000i		−	0.300753i
$399$	40.0000			2.00250
$400$		0			0
$401$		−	26.0000i		−	1.29838i	−0.760627	−	0.649189i	$-0.775108\pi$
$401$		−	26.0000i		−	1.29838i	0.760627	−	0.649189i	$-0.224892\pi$
$402$	−6.00000			−0.299253
$403$	−2.00000	−	3.00000i	−0.0996271	−	0.149441i
$404$	1.00000			0.0497519
$405$		0			0
$406$	−5.00000			−0.248146
$407$	−12.0000			−0.594818
$408$		−	30.0000i		−	1.48522i
$409$			6.00000i			0.296681i	0.988936	+	0.148340i	$0.0473931\pi$
$409$			6.00000i			0.296681i	−0.988936	+	0.148340i	$0.952607\pi$
$410$		0			0
$411$			36.0000i			1.77575i
$412$	−4.00000			−0.197066
$413$	−15.0000			−0.738102
$414$		−	4.00000i		−	0.196589i
$415$		0			0
$416$	10.0000	+	15.0000i	0.490290	+	0.735436i
$417$	8.00000			0.391762
$418$		−	12.0000i		−	0.586939i
$419$	−18.0000			−0.879358			−0.439679	−	0.898155i	$-0.644908\pi$
$419$	−18.0000			−0.879358			−0.439679	+	0.898155i	$0.644908\pi$
$420$		0			0
$421$			4.00000i			0.194948i	0.995238	+	0.0974740i	$0.0310763\pi$
$421$			4.00000i			0.194948i	−0.995238	+	0.0974740i	$0.968924\pi$
$422$		−	12.0000i		−	0.584151i
$423$		−	7.00000i		−	0.340352i
$424$			9.00000i			0.437079i
$425$		0			0
$426$	16.0000			0.775203
$427$		−	5.00000i		−	0.241967i
$428$	16.0000			0.773389
$429$	12.0000	+	18.0000i	0.579365	+	0.869048i
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	4.00000			0.192450
$433$	−14.0000			−0.672797			−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000			−0.672797			−0.336399	+	0.941720i	$0.609209\pi$
$434$		−	5.00000i		−	0.240008i
$435$		0			0
$436$			2.00000i			0.0957826i
$437$		−	16.0000i		−	0.765384i
$438$	−8.00000			−0.382255
$439$	22.0000			1.05000			0.525001	−	0.851101i	$-0.324065\pi$
$439$	22.0000			1.05000			0.525001	+	0.851101i	$0.324065\pi$
$440$		0			0
$441$	−18.0000			−0.857143
$442$	−10.0000	−	15.0000i	−0.475651	−	0.713477i
$443$	24.0000			1.14027			0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000			1.14027			0.570137	+	0.821549i	$0.306890\pi$
$444$			8.00000i			0.379663i
$445$		0			0
$446$	16.0000			0.757622
$447$		−	20.0000i		−	0.945968i
$448$			35.0000i			1.65359i
$449$		−	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
$451$	−24.0000			−1.13012
$452$	−10.0000			−0.470360
$453$			26.0000i			1.22159i
$454$	−9.00000			−0.422391
$455$		0			0
$456$	−24.0000			−1.12390
$457$		−	10.0000i		−	0.467780i	−0.972263	−	0.233890i	$-0.924854\pi$
$457$		−	10.0000i		−	0.467780i	0.972263	−	0.233890i	$-0.0751456\pi$
$458$	2.00000			0.0934539
$459$	20.0000			0.933520
$460$		0			0
$461$		−	26.0000i		−	1.21094i	−0.795868	−	0.605470i	$-0.792985\pi$
$461$		−	26.0000i		−	1.21094i	0.795868	−	0.605470i	$-0.207015\pi$
$462$			30.0000i			1.39573i
$463$		−	19.0000i		−	0.883005i	−0.897260	−	0.441502i	$-0.854446\pi$
$463$		−	19.0000i		−	0.883005i	0.897260	−	0.441502i	$-0.145554\pi$
$464$	1.00000			0.0464238
$465$		0			0
$466$		−	14.0000i		−	0.648537i
$467$	−18.0000			−0.832941			−0.416470	−	0.909149i	$-0.636733\pi$
$467$	−18.0000			−0.832941			−0.416470	+	0.909149i	$0.636733\pi$
$468$	3.00000	−	2.00000i	0.138675	−	0.0924500i
$469$	15.0000			0.692636
$470$		0			0
$471$	26.0000			1.19802
$472$	9.00000			0.414259
$473$			12.0000i			0.551761i
$474$			20.0000i			0.918630i
$475$		0			0
$476$			25.0000i			1.14587i
$477$	3.00000			0.137361
$478$	5.00000			0.228695
$479$		−	3.00000i		−	0.137073i	−0.997649	−	0.0685367i	$-0.978167\pi$
$479$		−	3.00000i		−	0.137073i	0.997649	−	0.0685367i	$-0.0218330\pi$
$480$		0			0
$481$	8.00000	+	12.0000i	0.364769	+	0.547153i
$482$		0			0
$483$			40.0000i			1.82006i
$484$	2.00000			0.0909091
$485$		0			0
$486$		−	10.0000i		−	0.453609i
$487$		−	15.0000i		−	0.679715i	−0.940477	−	0.339857i	$-0.889621\pi$
$487$		−	15.0000i		−	0.679715i	0.940477	−	0.339857i	$-0.110379\pi$
$488$			3.00000i			0.135804i
$489$			24.0000i			1.08532i
$490$		0			0
$491$	−28.0000			−1.26362			−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000			−1.26362			−0.631811	+	0.775122i	$0.717688\pi$
$492$			16.0000i			0.721336i
$493$	5.00000			0.225189
$494$	−12.0000	+	8.00000i	−0.539906	+	0.359937i
$495$		0			0
$496$			1.00000i			0.0449013i
$497$	−40.0000			−1.79425
$498$	18.0000			0.806599
$499$			19.0000i			0.850557i	0.905063	+	0.425278i	$0.139824\pi$
$499$			19.0000i			0.850557i	−0.905063	+	0.425278i	$0.860176\pi$
$500$		0			0
$501$			32.0000i			1.42965i
$502$			12.0000i			0.535586i
$503$	16.0000			0.713405			0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000			0.713405			0.356702	+	0.934218i	$0.383901\pi$
$504$	15.0000			0.668153
$505$		0			0
$506$	12.0000			0.533465
$507$	10.0000	−	24.0000i	0.444116	−	1.06588i
$508$	−18.0000			−0.798621
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$	20.0000			0.884748
$512$		−	11.0000i		−	0.486136i
$513$		−	16.0000i		−	0.706417i
$514$		−	15.0000i		−	0.661622i
$515$		0			0
$516$	8.00000			0.352180
$517$	21.0000			0.923579
$518$			20.0000i			0.878750i
$519$	−26.0000			−1.14127
$520$		0			0
$521$	10.0000			0.438108			0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000			0.438108			0.219054	+	0.975713i	$0.429703\pi$
$522$		−	1.00000i		−	0.0437688i
$523$	22.0000			0.961993			0.480996	−	0.876723i	$-0.340275\pi$
$523$	22.0000			0.961993			0.480996	+	0.876723i	$0.340275\pi$
$524$	18.0000			0.786334
$525$		0			0
$526$			6.00000i			0.261612i
$527$			5.00000i			0.217803i
$528$		−	6.00000i		−	0.261116i
$529$	−7.00000			−0.304348
$530$		0			0
$531$		−	3.00000i		−	0.130189i
$532$	20.0000			0.867110
$533$	16.0000	+	24.0000i	0.693037	+	1.03956i
$534$	36.0000			1.55787
$535$		0			0
$536$	−9.00000			−0.388741
$537$	24.0000			1.03568
$538$		−	17.0000i		−	0.732922i
$539$		−	54.0000i		−	2.32594i
$540$		0			0
$541$			38.0000i			1.63375i	0.576816	+	0.816874i	$0.304295\pi$
$541$			38.0000i			1.63375i	−0.576816	+	0.816874i	$0.695705\pi$
$542$	−29.0000			−1.24566
$543$	6.00000			0.257485
$544$		−	25.0000i		−	1.07187i
$545$		0			0
$546$	30.0000	−	20.0000i	1.28388	−	0.855921i
$547$	−38.0000			−1.62476			−0.812381	−	0.583127i	$-0.801829\pi$
$547$	−38.0000			−1.62476			−0.812381	+	0.583127i	$0.801829\pi$
$548$			18.0000i			0.768922i
$549$	1.00000			0.0426790
$550$		0			0
$551$		−	4.00000i		−	0.170406i
$552$		−	24.0000i		−	1.02151i
$553$		−	50.0000i		−	2.12622i
$554$			2.00000i			0.0849719i
$555$		0			0
$556$	4.00000			0.169638
$557$		−	24.0000i		−	1.01691i	−0.861088	−	0.508456i	$-0.830216\pi$
$557$		−	24.0000i		−	1.01691i	0.861088	−	0.508456i	$-0.169784\pi$
$558$	1.00000			0.0423334
$559$	12.0000	−	8.00000i	0.507546	−	0.338364i
$560$		0			0
$561$		−	30.0000i		−	1.26660i
$562$	18.0000			0.759284
$563$	30.0000			1.26435			0.632175	−	0.774826i	$-0.282163\pi$
$563$	30.0000			1.26435			0.632175	+	0.774826i	$0.282163\pi$
$564$		−	14.0000i		−	0.589506i
$565$		0			0
$566$			28.0000i			1.17693i
$567$			55.0000i			2.30978i
$568$	24.0000			1.00702
$569$	−33.0000			−1.38343			−0.691716	−	0.722170i	$-0.743145\pi$
$569$	−33.0000			−1.38343			−0.691716	+	0.722170i	$0.743145\pi$
$570$		0			0
$571$	2.00000			0.0836974			0.0418487	−	0.999124i	$-0.486675\pi$
$571$	2.00000			0.0836974			0.0418487	+	0.999124i	$0.486675\pi$
$572$	6.00000	+	9.00000i	0.250873	+	0.376309i
$573$	20.0000			0.835512
$574$			40.0000i			1.66957i
$575$		0			0
$576$	−7.00000			−0.291667
$577$		−	38.0000i		−	1.58196i	−0.611842	−	0.790980i	$-0.709571\pi$
$577$		−	38.0000i		−	1.58196i	0.611842	−	0.790980i	$-0.290429\pi$
$578$			8.00000i			0.332756i
$579$			8.00000i			0.332469i
$580$		0			0
$581$	−45.0000			−1.86691
$582$	28.0000			1.16064
$583$			9.00000i			0.372742i
$584$	−12.0000			−0.496564
$585$		0			0
$586$	6.00000			0.247858
$587$			25.0000i			1.03186i	0.856631	+	0.515930i	$0.172554\pi$
$587$			25.0000i			1.03186i	−0.856631	+	0.515930i	$0.827446\pi$
$588$	−36.0000			−1.48461
$589$	4.00000			0.164817
$590$		0			0
$591$		−	16.0000i		−	0.658152i
$592$		−	4.00000i		−	0.164399i
$593$			8.00000i			0.328521i	0.986417	+	0.164260i	$0.0525237\pi$
$593$			8.00000i			0.328521i	−0.986417	+	0.164260i	$0.947476\pi$
$594$	12.0000			0.492366
$595$		0			0
$596$		−	10.0000i		−	0.409616i
$597$	−12.0000			−0.491127
$598$	−8.00000	−	12.0000i	−0.327144	−	0.490716i
$599$	−14.0000			−0.572024			−0.286012	−	0.958226i	$-0.592330\pi$
$599$	−14.0000			−0.572024			−0.286012	+	0.958226i	$0.592330\pi$
$600$		0			0
$601$	29.0000			1.18293			0.591467	−	0.806329i	$-0.298549\pi$
$601$	29.0000			1.18293			0.591467	+	0.806329i	$0.298549\pi$
$602$	20.0000			0.815139
$603$			3.00000i			0.122169i
$604$			13.0000i			0.528962i
$605$		0			0
$606$			2.00000i			0.0812444i
$607$	8.00000			0.324710			0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000			0.324710			0.162355	+	0.986732i	$0.448091\pi$
$608$	−20.0000			−0.811107
$609$			10.0000i			0.405220i
$610$		0			0
$611$	−14.0000	−	21.0000i	−0.566379	−	0.849569i
$612$	−5.00000			−0.202113
$613$			10.0000i			0.403896i	0.979396	+	0.201948i	$0.0647272\pi$
$613$			10.0000i			0.403896i	−0.979396	+	0.201948i	$0.935273\pi$
$614$	−8.00000			−0.322854
$615$		0			0
$616$			45.0000i			1.81310i
$617$		−	4.00000i		−	0.161034i	−0.996753	−	0.0805170i	$-0.974343\pi$
$617$		−	4.00000i		−	0.161034i	0.996753	−	0.0805170i	$-0.0256571\pi$
$618$		−	8.00000i		−	0.321807i
$619$			28.0000i			1.12542i	0.826656	+	0.562708i	$0.190240\pi$
$619$			28.0000i			1.12542i	−0.826656	+	0.562708i	$0.809760\pi$
$620$		0			0
$621$	16.0000			0.642058
$622$			4.00000i			0.160385i
$623$	−90.0000			−3.60577
$624$	−6.00000	+	4.00000i	−0.240192	+	0.160128i
$625$		0			0
$626$		−	17.0000i		−	0.679457i
$627$	−24.0000			−0.958468
$628$	13.0000			0.518756
$629$		−	20.0000i		−	0.797452i
$630$		0			0
$631$		−	20.0000i		−	0.796187i	−0.917345	−	0.398094i	$-0.869672\pi$
$631$		−	20.0000i		−	0.796187i	0.917345	−	0.398094i	$-0.130328\pi$
$632$			30.0000i			1.19334i
$633$	−24.0000			−0.953914
$634$	−22.0000			−0.873732
$635$		0			0
$636$	6.00000			0.237915
$637$	−54.0000	+	36.0000i	−2.13956	+	1.42637i
$638$	3.00000			0.118771
$639$		−	8.00000i		−	0.316475i
$640$		0			0
$641$	−15.0000			−0.592464			−0.296232	−	0.955116i	$-0.595730\pi$
$641$	−15.0000			−0.592464			−0.296232	+	0.955116i	$0.595730\pi$
$642$			32.0000i			1.26294i
$643$			4.00000i			0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$			4.00000i			0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$			20.0000i			0.788110i
$645$		0			0
$646$	20.0000			0.786889
$647$	14.0000			0.550397			0.275198	−	0.961387i	$-0.411256\pi$
$647$	14.0000			0.550397			0.275198	+	0.961387i	$0.411256\pi$
$648$		−	33.0000i		−	1.29636i
$649$	9.00000			0.353281
$650$		0			0
$651$	−10.0000			−0.391931
$652$			12.0000i			0.469956i
$653$	−31.0000			−1.21312			−0.606562	−	0.795036i	$-0.707452\pi$
$653$	−31.0000			−1.21312			−0.606562	+	0.795036i	$0.707452\pi$
$654$	−4.00000			−0.156412
$655$		0			0
$656$		−	8.00000i		−	0.312348i
$657$			4.00000i			0.156055i
$658$		−	35.0000i		−	1.36444i
$659$	−6.00000			−0.233727			−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000			−0.233727			−0.116863	+	0.993148i	$0.537284\pi$
$660$		0			0
$661$		−	38.0000i		−	1.47803i	−0.673690	−	0.739014i	$-0.735292\pi$
$661$		−	38.0000i		−	1.47803i	0.673690	−	0.739014i	$-0.264708\pi$
$662$	−20.0000			−0.777322
$663$	−30.0000	+	20.0000i	−1.16510	+	0.776736i
$664$	27.0000			1.04780
$665$		0			0
$666$	−4.00000			−0.154997
$667$	4.00000			0.154881
$668$			16.0000i			0.619059i
$669$		−	32.0000i		−	1.23719i
$670$		0			0
$671$			3.00000i			0.115814i
$672$	50.0000			1.92879
$673$	23.0000			0.886585			0.443292	−	0.896377i	$-0.353810\pi$
$673$	23.0000			0.886585			0.443292	+	0.896377i	$0.353810\pi$
$674$			3.00000i			0.115556i
$675$		0			0
$676$	5.00000	−	12.0000i	0.192308	−	0.461538i
$677$	−6.00000			−0.230599			−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000			−0.230599			−0.115299	+	0.993331i	$0.536783\pi$
$678$		−	20.0000i		−	0.768095i
$679$	−70.0000			−2.68635
$680$		0			0
$681$			18.0000i			0.689761i
$682$			3.00000i			0.114876i
$683$			41.0000i			1.56882i	0.620242	+	0.784411i	$0.287034\pi$
$683$			41.0000i			1.56882i	−0.620242	+	0.784411i	$0.712966\pi$
$684$			4.00000i			0.152944i
$685$		0			0
$686$	−55.0000			−2.09991
$687$		−	4.00000i		−	0.152610i
$688$	−4.00000			−0.152499
$689$	9.00000	−	6.00000i	0.342873	−	0.228582i
$690$		0			0
$691$		−	17.0000i		−	0.646710i	−0.946278	−	0.323355i	$-0.895189\pi$
$691$		−	17.0000i		−	0.646710i	0.946278	−	0.323355i	$-0.104811\pi$
$692$	−13.0000			−0.494186
$693$	15.0000			0.569803
$694$		−	4.00000i		−	0.151838i
$695$		0			0
$696$		−	6.00000i		−	0.227429i
$697$		−	40.0000i		−	1.51511i
$698$	−2.00000			−0.0757011
$699$	−28.0000			−1.05906
$700$		0			0
$701$	−45.0000			−1.69963			−0.849813	−	0.527084i	$-0.823285\pi$
$701$	−45.0000			−1.69963			−0.849813	+	0.527084i	$0.823285\pi$
$702$	−8.00000	−	12.0000i	−0.301941	−	0.452911i
$703$	−16.0000			−0.603451
$704$		−	21.0000i		−	0.791467i
$705$		0			0
$706$	−6.00000			−0.225813
$707$		−	5.00000i		−	0.188044i
$708$		−	6.00000i		−	0.225494i
$709$			8.00000i			0.300446i	0.988652	+	0.150223i	$0.0479992\pi$
$709$			8.00000i			0.300446i	−0.988652	+	0.150223i	$0.952001\pi$
$710$		0			0
$711$	10.0000			0.375029
$712$	54.0000			2.02374
$713$			4.00000i			0.149801i
$714$	−50.0000			−1.87120
$715$		0			0
$716$	12.0000			0.448461
$717$		−	10.0000i		−	0.373457i
$718$	−23.0000			−0.858352
$719$	30.0000			1.11881			0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000			1.11881			0.559406	+	0.828894i	$0.311029\pi$
$720$		0			0
$721$			20.0000i			0.744839i
$722$			3.00000i			0.111648i
$723$		0			0
$724$	3.00000			0.111494
$725$		0			0
$726$			4.00000i			0.148454i
$727$	16.0000			0.593407			0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000			0.593407			0.296704	+	0.954970i	$0.404113\pi$
$728$	45.0000	−	30.0000i	1.66781	−	1.11187i
$729$	13.0000			0.481481
$730$		0			0
$731$	−20.0000			−0.739727
$732$	2.00000			0.0739221
$733$		−	42.0000i		−	1.55131i	−0.631160	−	0.775653i	$-0.717421\pi$
$733$		−	42.0000i		−	1.55131i	0.631160	−	0.775653i	$-0.282579\pi$
$734$			18.0000i			0.664392i
$735$		0			0
$736$		−	20.0000i		−	0.737210i
$737$	−9.00000			−0.331519
$738$	−8.00000			−0.294484
$739$			9.00000i			0.331070i	0.986204	+	0.165535i	$0.0529351\pi$
$739$			9.00000i			0.331070i	−0.986204	+	0.165535i	$0.947065\pi$
$740$		0			0
$741$	16.0000	+	24.0000i	0.587775	+	0.881662i
$742$	15.0000			0.550667
$743$		−	41.0000i		−	1.50414i	−0.659081	−	0.752072i	$-0.729055\pi$
$743$		−	41.0000i		−	1.50414i	0.659081	−	0.752072i	$-0.270945\pi$
$744$	6.00000			0.219971
$745$		0			0
$746$		−	31.0000i		−	1.13499i
$747$		−	9.00000i		−	0.329293i
$748$		−	15.0000i		−	0.548454i
$749$		−	80.0000i		−	2.92314i
$750$		0			0
$751$	−32.0000			−1.16770			−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000			−1.16770			−0.583848	+	0.811863i	$0.698454\pi$
$752$			7.00000i			0.255264i
$753$	24.0000			0.874609
$754$	−2.00000	−	3.00000i	−0.0728357	−	0.109254i
$755$		0			0
$756$			20.0000i			0.727393i
$757$	−5.00000			−0.181728			−0.0908640	−	0.995863i	$-0.528963\pi$
$757$	−5.00000			−0.181728			−0.0908640	+	0.995863i	$0.528963\pi$
$758$	27.0000			0.980684
$759$		−	24.0000i		−	0.871145i
$760$		0			0
$761$		−	10.0000i		−	0.362500i	−0.983437	−	0.181250i	$-0.941986\pi$
$761$		−	10.0000i		−	0.362500i	0.983437	−	0.181250i	$-0.0580143\pi$
$762$		−	36.0000i		−	1.30414i
$763$	10.0000			0.362024
$764$	10.0000			0.361787
$765$		0			0
$766$	32.0000			1.15621
$767$	−6.00000	−	9.00000i	−0.216647	−	0.324971i
$768$	−34.0000			−1.22687
$769$		−	40.0000i		−	1.44244i	−0.692708	−	0.721218i	$-0.743582\pi$
$769$		−	40.0000i		−	1.44244i	0.692708	−	0.721218i	$-0.256418\pi$
$770$		0			0
$771$	−30.0000			−1.08042
$772$			4.00000i			0.143963i
$773$			18.0000i			0.647415i	0.946157	+	0.323708i	$0.104929\pi$
$773$			18.0000i			0.647415i	−0.946157	+	0.323708i	$0.895071\pi$
$774$			4.00000i			0.143777i
$775$		0			0
$776$	42.0000			1.50771
$777$	40.0000			1.43499
$778$			34.0000i			1.21896i
$779$	−32.0000			−1.14652
$780$		0			0
$781$	24.0000			0.858788
$782$			20.0000i			0.715199i
$783$	4.00000			0.142948
$784$	18.0000			0.642857
$785$		0			0
$786$			36.0000i			1.28408i
$787$			43.0000i			1.53278i	0.642373	+	0.766392i	$0.277950\pi$
$787$			43.0000i			1.53278i	−0.642373	+	0.766392i	$0.722050\pi$
$788$		−	8.00000i		−	0.284988i
$789$	12.0000			0.427211
$790$		0			0
$791$			50.0000i			1.77780i
$792$	−9.00000			−0.319801
$793$	3.00000	−	2.00000i	0.106533	−	0.0710221i
$794$	−8.00000			−0.283909
$795$		0			0
$796$	−6.00000			−0.212664
$797$	53.0000			1.87736			0.938678	−	0.344795i	$-0.112051\pi$
$797$	53.0000			1.87736			0.938678	+	0.344795i	$0.112051\pi$
$798$			40.0000i			1.41598i
$799$			35.0000i			1.23821i
$800$		0			0
$801$		−	18.0000i		−	0.635999i
$802$	26.0000			0.918092
$803$	−12.0000			−0.423471
$804$			6.00000i			0.211604i
$805$		0			0
$806$	3.00000	−	2.00000i	0.105670	−	0.0704470i
$807$	−34.0000			−1.19686
$808$			3.00000i			0.105540i
$809$	−30.0000			−1.05474			−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000			−1.05474			−0.527372	+	0.849635i	$0.676823\pi$
$810$		0			0
$811$		−	25.0000i		−	0.877869i	−0.898519	−	0.438934i	$-0.855356\pi$
$811$		−	25.0000i		−	0.877869i	0.898519	−	0.438934i	$-0.144644\pi$
$812$			5.00000i			0.175466i
$813$			58.0000i			2.03415i
$814$		−	12.0000i		−	0.420600i
$815$		0			0
$816$	10.0000			0.350070
$817$			16.0000i			0.559769i
$818$	−6.00000			−0.209785
$819$	−10.0000	−	15.0000i	−0.349428	−	0.524142i
$820$		0			0
$821$			30.0000i			1.04701i	0.852023	+	0.523504i	$0.175375\pi$
$821$			30.0000i			1.04701i	−0.852023	+	0.523504i	$0.824625\pi$
$822$	−36.0000			−1.25564
$823$	−12.0000			−0.418294			−0.209147	−	0.977884i	$-0.567069\pi$
$823$	−12.0000			−0.418294			−0.209147	+	0.977884i	$0.567069\pi$
$824$		−	12.0000i		−	0.418040i
$825$		0			0
$826$		−	15.0000i		−	0.521917i
$827$		−	19.0000i		−	0.660695i	−0.943859	−	0.330347i	$-0.892834\pi$
$827$		−	19.0000i		−	0.660695i	0.943859	−	0.330347i	$-0.107166\pi$
$828$	−4.00000			−0.139010
$829$	−11.0000			−0.382046			−0.191023	−	0.981586i	$-0.561180\pi$
$829$	−11.0000			−0.382046			−0.191023	+	0.981586i	$0.561180\pi$
$830$		0			0
$831$	4.00000			0.138758
$832$	−21.0000	+	14.0000i	−0.728044	+	0.485363i
$833$	90.0000			3.11832
$834$			8.00000i			0.277017i
$835$		0			0
$836$	−12.0000			−0.415029
$837$			4.00000i			0.138260i
$838$		−	18.0000i		−	0.621800i
$839$			36.0000i			1.24286i	0.783470	+	0.621429i	$0.213448\pi$
$839$			36.0000i			1.24286i	−0.783470	+	0.621429i	$0.786552\pi$
$840$		0			0
$841$	−28.0000			−0.965517
$842$	−4.00000			−0.137849
$843$		−	36.0000i		−	1.23991i
$844$	−12.0000			−0.413057
$845$		0			0
$846$	7.00000			0.240665
$847$		−	10.0000i		−	0.343604i
$848$	−3.00000			−0.103020
$849$	56.0000			1.92192
$850$		0			0
$851$		−	16.0000i		−	0.548473i
$852$		−	16.0000i		−	0.548151i
$853$			54.0000i			1.84892i	0.381273	+	0.924462i	$0.375486\pi$
$853$			54.0000i			1.84892i	−0.381273	+	0.924462i	$0.624514\pi$
$854$	5.00000			0.171096
$855$		0			0
$856$			48.0000i			1.64061i
$857$	−14.0000			−0.478231			−0.239115	−	0.970991i	$-0.576857\pi$
$857$	−14.0000			−0.478231			−0.239115	+	0.970991i	$0.576857\pi$
$858$	−18.0000	+	12.0000i	−0.614510	+	0.409673i
$859$	−4.00000			−0.136478			−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000			−0.136478			−0.0682391	+	0.997669i	$0.521738\pi$
$860$		0			0
$861$	80.0000			2.72639
$862$		0			0
$863$		−	33.0000i		−	1.12333i	−0.827364	−	0.561667i	$-0.810160\pi$
$863$		−	33.0000i		−	1.12333i	0.827364	−	0.561667i	$-0.189840\pi$
$864$		−	20.0000i		−	0.680414i
$865$		0			0
$866$		−	14.0000i		−	0.475739i
$867$	16.0000			0.543388
$868$	−5.00000			−0.169711
$869$			30.0000i			1.01768i
$870$		0			0
$871$	6.00000	+	9.00000i	0.203302	+	0.304953i
$872$	−6.00000			−0.203186
$873$		−	14.0000i		−	0.473828i
$874$	16.0000			0.541208
$875$		0			0
$876$			8.00000i			0.270295i
$877$			14.0000i			0.472746i	0.971662	+	0.236373i	$0.0759588\pi$
$877$			14.0000i			0.472746i	−0.971662	+	0.236373i	$0.924041\pi$
$878$			22.0000i			0.742464i
$879$		−	12.0000i		−	0.404750i
$880$		0			0
$881$	−33.0000			−1.11180			−0.555899	−	0.831250i	$-0.687626\pi$
$881$	−33.0000			−1.11180			−0.555899	+	0.831250i	$0.687626\pi$
$882$		−	18.0000i		−	0.606092i
$883$	−28.0000			−0.942275			−0.471138	−	0.882060i	$-0.656156\pi$
$883$	−28.0000			−0.942275			−0.471138	+	0.882060i	$0.656156\pi$
$884$	−15.0000	+	10.0000i	−0.504505	+	0.336336i
$885$		0			0
$886$			24.0000i			0.806296i
$887$	22.0000			0.738688			0.369344	−	0.929293i	$-0.379582\pi$
$887$	22.0000			0.738688			0.369344	+	0.929293i	$0.379582\pi$
$888$	−24.0000			−0.805387
$889$			90.0000i			3.01850i
$890$		0			0
$891$		−	33.0000i		−	1.10554i
$892$		−	16.0000i		−	0.535720i
$893$	28.0000			0.936984
$894$	20.0000			0.668900
$895$		0			0
$896$	15.0000			0.501115
$897$	−24.0000	+	16.0000i	−0.801337	+	0.534224i
$898$	6.00000			0.200223
$899$			1.00000i			0.0333519i
$900$		0			0
$901$	−15.0000			−0.499722
$902$		−	24.0000i		−	0.799113i
$903$		−	40.0000i		−	1.33112i
$904$		−	30.0000i		−	0.997785i
$905$		0			0
$906$	−26.0000			−0.863792
$907$	20.0000			0.664089			0.332045	−	0.943264i	$-0.392262\pi$
$907$	20.0000			0.664089			0.332045	+	0.943264i	$0.392262\pi$
$908$			9.00000i			0.298675i
$909$	1.00000			0.0331679
$910$		0			0
$911$	2.00000			0.0662630			0.0331315	−	0.999451i	$-0.489452\pi$
$911$	2.00000			0.0662630			0.0331315	+	0.999451i	$0.489452\pi$
$912$		−	8.00000i		−	0.264906i
$913$	27.0000			0.893570
$914$	10.0000			0.330771
$915$		0			0
$916$		−	2.00000i		−	0.0660819i
$917$		−	90.0000i		−	2.97206i
$918$			20.0000i			0.660098i
$919$	46.0000			1.51740			0.758700	−	0.651440i	$-0.225835\pi$
$919$	46.0000			1.51740			0.758700	+	0.651440i	$0.225835\pi$
$920$		0			0
$921$			16.0000i			0.527218i
$922$	26.0000			0.856264
$923$	−16.0000	−	24.0000i	−0.526646	−	0.789970i
$924$	30.0000			0.986928
$925$		0			0
$926$	19.0000			0.624379
$927$	−4.00000			−0.131377
$928$		−	5.00000i		−	0.164133i
$929$			56.0000i			1.83730i	0.395072	+	0.918650i	$0.370720\pi$
$929$			56.0000i			1.83730i	−0.395072	+	0.918650i	$0.629280\pi$
$930$		0			0
$931$		−	72.0000i		−	2.35970i
$932$	−14.0000			−0.458585
$933$	8.00000			0.261908
$934$		−	18.0000i		−	0.588978i
$935$		0			0
$936$	6.00000	+	9.00000i	0.196116	+	0.294174i
$937$	1.00000			0.0326686			0.0163343	−	0.999867i	$-0.494800\pi$
$937$	1.00000			0.0326686			0.0163343	+	0.999867i	$0.494800\pi$
$938$			15.0000i			0.489767i
$939$	−34.0000			−1.10955
$940$		0			0
$941$		−	8.00000i		−	0.260793i	−0.991462	−	0.130396i	$-0.958375\pi$
$941$		−	8.00000i		−	0.260793i	0.991462	−	0.130396i	$-0.0416250\pi$
$942$			26.0000i			0.847126i
$943$		−	32.0000i		−	1.04206i
$944$			3.00000i			0.0976417i
$945$		0			0
$946$	−12.0000			−0.390154
$947$		−	3.00000i		−	0.0974869i	−0.998811	−	0.0487435i	$-0.984478\pi$
$947$		−	3.00000i		−	0.0974869i	0.998811	−	0.0487435i	$-0.0155217\pi$
$948$	20.0000			0.649570
$949$	8.00000	+	12.0000i	0.259691	+	0.389536i
$950$		0			0
$951$			44.0000i			1.42680i
$952$	−75.0000			−2.43076
$953$	9.00000			0.291539			0.145769	−	0.989319i	$-0.453434\pi$
$953$	9.00000			0.291539			0.145769	+	0.989319i	$0.453434\pi$
$954$			3.00000i			0.0971286i
$955$		0			0
$956$		−	5.00000i		−	0.161712i
$957$		−	6.00000i		−	0.193952i
$958$	3.00000			0.0969256
$959$	90.0000			2.90625
$960$		0			0
$961$	30.0000			0.967742
$962$	−12.0000	+	8.00000i	−0.386896	+	0.257930i
$963$	16.0000			0.515593
$964$		0			0
$965$		0			0
$966$	−40.0000			−1.28698
$967$			37.0000i			1.18984i	0.803785	+	0.594920i	$0.202816\pi$
$967$			37.0000i			1.18984i	−0.803785	+	0.594920i	$0.797184\pi$
$968$			6.00000i			0.192847i
$969$		−	40.0000i		−	1.28499i
$970$		0			0
$971$	−48.0000			−1.54039			−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000			−1.54039			−0.770197	+	0.637806i	$0.779842\pi$
$972$	−10.0000			−0.320750
$973$		−	20.0000i		−	0.641171i
$974$	15.0000			0.480631
$975$		0			0
$976$	−1.00000			−0.0320092
$977$			46.0000i			1.47167i	0.677161	+	0.735835i	$0.263210\pi$
$977$			46.0000i			1.47167i	−0.677161	+	0.735835i	$0.736790\pi$
$978$	−24.0000			−0.767435
$979$	54.0000			1.72585
$980$		0			0
$981$			2.00000i			0.0638551i
$982$		−	28.0000i		−	0.893516i
$983$			11.0000i			0.350846i	0.984493	+	0.175423i	$0.0561292\pi$
$983$			11.0000i			0.350846i	−0.984493	+	0.175423i	$0.943871\pi$
$984$	−48.0000			−1.53018
$985$		0			0
$986$			5.00000i			0.159232i
$987$	−70.0000			−2.22812
$988$	8.00000	+	12.0000i	0.254514	+	0.381771i
$989$	−16.0000			−0.508770
$990$		0			0
$991$	4.00000			0.127064			0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000			0.127064			0.0635321	+	0.997980i	$0.479763\pi$
$992$	5.00000			0.158750
$993$			40.0000i			1.26936i
$994$		−	40.0000i		−	1.26872i
$995$		0			0
$996$		−	18.0000i		−	0.570352i
$997$	−45.0000			−1.42516			−0.712582	−	0.701589i	$-0.752474\pi$
$997$	−45.0000			−1.42516			−0.712582	+	0.701589i	$0.752474\pi$
$998$	−19.0000			−0.601434
$999$		−	16.0000i		−	0.506218i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.2.c.f.51.2	yes	2
5.2	odd	4		325.2.d.b.324.1		2
5.3	odd	4		325.2.d.c.324.2		2
5.4	even	2		325.2.c.a.51.1	✓	2
13.5	odd	4		4225.2.a.n.1.1		1
13.8	odd	4		4225.2.a.f.1.1		1
13.12	even	2	inner	325.2.c.f.51.1	yes	2
65.12	odd	4		325.2.d.c.324.1		2
65.34	odd	4		4225.2.a.l.1.1		1
65.38	odd	4		325.2.d.b.324.2		2
65.44	odd	4		4225.2.a.d.1.1		1
65.64	even	2		325.2.c.a.51.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.2.c.a.51.1	✓	2	5.4	even	2
325.2.c.a.51.2	yes	2	65.64	even	2
325.2.c.f.51.1	yes	2	13.12	even	2	inner
325.2.c.f.51.2	yes	2	1.1	even	1	trivial
325.2.d.b.324.1		2	5.2	odd	4
325.2.d.b.324.2		2	65.38	odd	4
325.2.d.c.324.1		2	65.12	odd	4
325.2.d.c.324.2		2	5.3	odd	4
4225.2.a.d.1.1		1	65.44	odd	4
4225.2.a.f.1.1		1	13.8	odd	4
4225.2.a.l.1.1		1	65.34	odd	4
4225.2.a.n.1.1		1	13.5	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	325.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +2.00000 q^{3} +1.00000 q^{4} +2.00000i q^{6} -5.00000i q^{7} +3.00000i q^{8} +1.00000 q^{9} +3.00000i q^{11} +2.00000 q^{12} +(3.00000 - 2.00000i) q^{13} +5.00000 q^{14} -1.00000 q^{16} -5.00000 q^{17} +1.00000i q^{18} +4.00000i q^{19} -10.0000i q^{21} -3.00000 q^{22} -4.00000 q^{23} +6.00000i q^{24} +(2.00000 + 3.00000i) q^{26} -4.00000 q^{27} -5.00000i q^{28} -1.00000 q^{29} -1.00000i q^{31} +5.00000i q^{32} +6.00000i q^{33} -5.00000i q^{34} +1.00000 q^{36} +4.00000i q^{37} -4.00000 q^{38} +(6.00000 - 4.00000i) q^{39} +8.00000i q^{41} +10.0000 q^{42} +4.00000 q^{43} +3.00000i q^{44} -4.00000i q^{46} -7.00000i q^{47} -2.00000 q^{48} -18.0000 q^{49} -10.0000 q^{51} +(3.00000 - 2.00000i) q^{52} +3.00000 q^{53} -4.00000i q^{54} +15.0000 q^{56} +8.00000i q^{57} -1.00000i q^{58} -3.00000i q^{59} +1.00000 q^{61} +1.00000 q^{62} -5.00000i q^{63} -7.00000 q^{64} -6.00000 q^{66} +3.00000i q^{67} -5.00000 q^{68} -8.00000 q^{69} -8.00000i q^{71} +3.00000i q^{72} +4.00000i q^{73} -4.00000 q^{74} +4.00000i q^{76} +15.0000 q^{77} +(4.00000 + 6.00000i) q^{78} +10.0000 q^{79} -11.0000 q^{81} -8.00000 q^{82} -9.00000i q^{83} -10.0000i q^{84} +4.00000i q^{86} -2.00000 q^{87} -9.00000 q^{88} -18.0000i q^{89} +(-10.0000 - 15.0000i) q^{91} -4.00000 q^{92} -2.00000i q^{93} +7.00000 q^{94} +10.0000i q^{96} -14.0000i q^{97} -18.0000i q^{98} +3.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} + 2 q^{4} + 2 q^{9} + 4 q^{12} + 6 q^{13} + 10 q^{14} - 2 q^{16} - 10 q^{17} - 6 q^{22} - 8 q^{23} + 4 q^{26} - 8 q^{27} - 2 q^{29} + 2 q^{36} - 8 q^{38} + 12 q^{39} + 20 q^{42} + 8 q^{43}+ \cdots + 14 q^{94}+O(q^{100}) \) 2 * q + 4 * q^3 + 2 * q^4 + 2 * q^9 + 4 * q^12 + 6 * q^13 + 10 * q^14 - 2 * q^16 - 10 * q^17 - 6 * q^22 - 8 * q^23 + 4 * q^26 - 8 * q^27 - 2 * q^29 + 2 * q^36 - 8 * q^38 + 12 * q^39 + 20 * q^42 + 8 * q^43 - 4 * q^48 - 36 * q^49 - 20 * q^51 + 6 * q^52 + 6 * q^53 + 30 * q^56 + 2 * q^61 + 2 * q^62 - 14 * q^64 - 12 * q^66 - 10 * q^68 - 16 * q^69 - 8 * q^74 + 30 * q^77 + 8 * q^78 + 20 * q^79 - 22 * q^81 - 16 * q^82 - 4 * q^87 - 18 * q^88 - 20 * q^91 - 8 * q^92 + 14 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more