LMFDB - Embedded newform 325.2.d.c.324.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [325,2,Mod(324,325)] 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("325.324"); 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([1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	325.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == 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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			324.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	325.324
Dual form			325.2.d.c.324.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

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

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 $

$2$	1.00000			0.707107			0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000			0.707107			0.353553	+	0.935414i	$0.384973\pi$
$3$			2.00000i			1.15470i	0.816497	+	0.577350i	$0.195913\pi$
$3$			2.00000i			1.15470i	−0.816497	+	0.577350i	$0.804087\pi$
$4$	−1.00000			−0.500000
$5$		0			0
$5$		0			0
$6$			2.00000i			0.816497i
$7$	−5.00000			−1.88982			−0.944911	−	0.327327i	$-0.893852\pi$
$7$	−5.00000			−1.88982			−0.944911	+	0.327327i	$0.893852\pi$
$8$	−3.00000			−1.06066
$9$	−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	$0.850615\pi$
$12$		−	2.00000i		−	0.577350i
$13$	2.00000	+	3.00000i	0.554700	+	0.832050i
$13$	2.00000	+	3.00000i	0.554700	+	0.832050i
$14$	−5.00000			−1.33631
$15$		0			0
$16$	−1.00000			−0.250000
$17$			5.00000i			1.21268i	0.795206	+	0.606339i	$0.207363\pi$
$17$			5.00000i			1.21268i	−0.795206	+	0.606339i	$0.792637\pi$
$18$	−1.00000			−0.235702
$19$		−	4.00000i		−	0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$		−	4.00000i		−	0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$		0			0
$21$		−	10.0000i		−	2.18218i
$22$			3.00000i			0.639602i
$23$		−	4.00000i		−	0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$		−	4.00000i		−	0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$		−	6.00000i		−	1.22474i
$25$		0			0
$26$	2.00000	+	3.00000i	0.392232	+	0.588348i
$27$			4.00000i			0.769800i
$28$	5.00000			0.944911
$29$	1.00000			0.185695			0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000			0.185695			0.0928477	+	0.995680i	$0.470403\pi$
$30$		0			0
$31$		−	1.00000i		−	0.179605i	−0.995960	−	0.0898027i	$-0.971376\pi$
$31$		−	1.00000i		−	0.179605i	0.995960	−	0.0898027i	$-0.0286236\pi$
$32$	5.00000			0.883883
$33$	−6.00000			−1.04447
$34$			5.00000i			0.857493i
$35$		0			0
$36$	1.00000			0.166667
$37$	4.00000			0.657596			0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000			0.657596			0.328798	+	0.944400i	$0.393356\pi$
$38$		−	4.00000i		−	0.648886i
$39$	−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	$0.785223\pi$
$42$		−	10.0000i		−	1.54303i
$43$			4.00000i			0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$			4.00000i			0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$		−	3.00000i		−	0.452267i
$45$		0			0
$46$		−	4.00000i		−	0.589768i
$47$	−7.00000			−1.02105			−0.510527	−	0.859861i	$-0.670550\pi$
$47$	−7.00000			−1.02105			−0.510527	+	0.859861i	$0.670550\pi$
$48$		−	2.00000i		−	0.288675i
$49$	18.0000			2.57143
$50$		0			0
$51$	−10.0000			−1.40028
$52$	−2.00000	−	3.00000i	−0.277350	−	0.416025i
$53$			3.00000i			0.412082i	0.978543	+	0.206041i	$0.0660580\pi$
$53$			3.00000i			0.412082i	−0.978543	+	0.206041i	$0.933942\pi$
$54$			4.00000i			0.544331i
$55$		0			0
$56$	15.0000			2.00446
$57$	8.00000			1.05963
$58$	1.00000			0.131306
$59$			3.00000i			0.390567i	0.980747	+	0.195283i	$0.0625627\pi$
$59$			3.00000i			0.390567i	−0.980747	+	0.195283i	$0.937437\pi$
$60$		0			0
$61$	1.00000			0.128037			0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000			0.128037			0.0640184	+	0.997949i	$0.479608\pi$
$62$		−	1.00000i		−	0.127000i
$63$	5.00000			0.629941
$64$	7.00000			0.875000
$65$		0			0
$66$	−6.00000			−0.738549
$67$	3.00000			0.366508			0.183254	−	0.983066i	$-0.441337\pi$
$67$	3.00000			0.366508			0.183254	+	0.983066i	$0.441337\pi$
$68$		−	5.00000i		−	0.606339i
$69$	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	$-0.157448\pi$
$72$	3.00000			0.353553
$73$	−4.00000			−0.468165			−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000			−0.468165			−0.234082	+	0.972217i	$0.575209\pi$
$74$	4.00000			0.464991
$75$		0			0
$76$			4.00000i			0.458831i
$77$		−	15.0000i		−	1.70941i
$78$	−6.00000	+	4.00000i	−0.679366	+	0.452911i
$79$	−10.0000			−1.12509			−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000			−1.12509			−0.562544	+	0.826767i	$0.690177\pi$
$80$		0			0
$81$	−11.0000			−1.22222
$82$			8.00000i			0.883452i
$83$	9.00000			0.987878			0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000			0.987878			0.493939	+	0.869496i	$0.335557\pi$
$84$			10.0000i			1.09109i
$85$		0			0
$86$			4.00000i			0.431331i
$87$			2.00000i			0.214423i
$88$		−	9.00000i		−	0.959403i
$89$			18.0000i			1.90800i	0.299813	+	0.953998i	$0.403076\pi$
$89$			18.0000i			1.90800i	−0.299813	+	0.953998i	$0.596924\pi$
$90$		0			0
$91$	−10.0000	−	15.0000i	−1.04828	−	1.57243i
$92$			4.00000i			0.417029i
$93$	2.00000			0.207390
$94$	−7.00000			−0.721995
$95$		0			0
$96$			10.0000i			1.02062i
$97$	−14.0000			−1.42148			−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000			−1.42148			−0.710742	+	0.703452i	$0.751641\pi$
$98$	18.0000			1.81827
$99$		−	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	$0.484157\pi$
$102$	−10.0000			−0.990148
$103$		−	4.00000i		−	0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$		−	4.00000i		−	0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	−6.00000	−	9.00000i	−0.588348	−	0.882523i
$105$		0			0
$106$			3.00000i			0.291386i
$107$		−	16.0000i		−	1.54678i	−0.633932	−	0.773389i	$-0.718560\pi$
$107$		−	16.0000i		−	1.54678i	0.633932	−	0.773389i	$-0.281440\pi$
$108$		−	4.00000i		−	0.384900i
$109$		−	2.00000i		−	0.191565i	−0.995402	−	0.0957826i	$-0.969465\pi$
$109$		−	2.00000i		−	0.191565i	0.995402	−	0.0957826i	$-0.0305354\pi$
$110$		0			0
$111$			8.00000i			0.759326i
$112$	5.00000			0.472456
$113$		−	10.0000i		−	0.940721i	−0.882474	−	0.470360i	$-0.844124\pi$
$113$		−	10.0000i		−	0.940721i	0.882474	−	0.470360i	$-0.155876\pi$
$114$	8.00000			0.749269
$115$		0			0
$116$	−1.00000			−0.0928477
$117$	−2.00000	−	3.00000i	−0.184900	−	0.277350i
$118$			3.00000i			0.276172i
$119$		−	25.0000i		−	2.29175i
$120$		0			0
$121$	2.00000			0.181818
$122$	1.00000			0.0905357
$123$	−16.0000			−1.44267
$124$			1.00000i			0.0898027i
$125$		0			0
$126$	5.00000			0.445435
$127$			18.0000i			1.59724i	0.601834	+	0.798621i	$0.294437\pi$
$127$			18.0000i			1.59724i	−0.601834	+	0.798621i	$0.705563\pi$
$128$	−3.00000			−0.265165
$129$	−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	$0.211977\pi$
$132$	6.00000			0.522233
$133$			20.0000i			1.73422i
$134$	3.00000			0.259161
$135$		0			0
$136$		−	15.0000i		−	1.28624i
$137$	18.0000			1.53784			0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000			1.53784			0.768922	+	0.639343i	$0.220793\pi$
$138$	8.00000			0.681005
$139$	−4.00000			−0.339276			−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000			−0.339276			−0.169638	+	0.985506i	$0.554260\pi$
$140$		0			0
$141$		−	14.0000i		−	1.17901i
$142$		−	8.00000i		−	0.671345i
$143$	−9.00000	+	6.00000i	−0.752618	+	0.501745i
$144$	1.00000			0.0833333
$145$		0			0
$146$	−4.00000			−0.331042
$147$			36.0000i			2.96923i
$148$	−4.00000			−0.328798
$149$			10.0000i			0.819232i	0.912258	+	0.409616i	$0.134337\pi$
$149$			10.0000i			0.819232i	−0.912258	+	0.409616i	$0.865663\pi$
$150$		0			0
$151$			13.0000i			1.05792i	0.848645	+	0.528962i	$0.177419\pi$
$151$			13.0000i			1.05792i	−0.848645	+	0.528962i	$0.822581\pi$
$152$			12.0000i			0.973329i
$153$		−	5.00000i		−	0.404226i
$154$		−	15.0000i		−	1.20873i
$155$		0			0
$156$	6.00000	−	4.00000i	0.480384	−	0.320256i
$157$		−	13.0000i		−	1.03751i	−0.854922	−	0.518756i	$-0.826395\pi$
$157$		−	13.0000i		−	1.03751i	0.854922	−	0.518756i	$-0.173605\pi$
$158$	−10.0000			−0.795557
$159$	−6.00000			−0.475831
$160$		0			0
$161$			20.0000i			1.57622i
$162$	−11.0000			−0.864242
$163$	−12.0000			−0.939913			−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000			−0.939913			−0.469956	+	0.882690i	$0.655730\pi$
$164$		−	8.00000i		−	0.624695i
$165$		0			0
$166$	9.00000			0.698535
$167$	16.0000			1.23812			0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000			1.23812			0.619059	+	0.785345i	$0.287514\pi$
$168$			30.0000i			2.31455i
$169$	−5.00000	+	12.0000i	−0.384615	+	0.923077i
$170$		0			0
$171$			4.00000i			0.305888i
$172$		−	4.00000i		−	0.304997i
$173$		−	13.0000i		−	0.988372i	−0.869356	−	0.494186i	$-0.835466\pi$
$173$		−	13.0000i		−	0.988372i	0.869356	−	0.494186i	$-0.164534\pi$
$174$			2.00000i			0.151620i
$175$		0			0
$176$		−	3.00000i		−	0.226134i
$177$	−6.00000			−0.450988
$178$			18.0000i			1.34916i
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$	3.00000			0.222988			0.111494	−	0.993765i	$-0.464436\pi$
$181$	3.00000			0.222988			0.111494	+	0.993765i	$0.464436\pi$
$182$	−10.0000	−	15.0000i	−0.741249	−	1.11187i
$183$			2.00000i			0.147844i
$184$			12.0000i			0.884652i
$185$		0			0
$186$	2.00000			0.146647
$187$	−15.0000			−1.09691
$188$	7.00000			0.510527
$189$		−	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	$0.382167\pi$
$192$			14.0000i			1.01036i
$193$	−4.00000			−0.287926			−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000			−0.287926			−0.143963	+	0.989583i	$0.545985\pi$
$194$	−14.0000			−1.00514
$195$		0			0
$196$	−18.0000			−1.28571
$197$	−8.00000			−0.569976			−0.284988	−	0.958531i	$-0.591990\pi$
$197$	−8.00000			−0.569976			−0.284988	+	0.958531i	$0.591990\pi$
$198$		−	3.00000i		−	0.213201i
$199$	6.00000			0.425329			0.212664	−	0.977125i	$-0.431786\pi$
$199$	6.00000			0.425329			0.212664	+	0.977125i	$0.431786\pi$
$200$		0			0
$201$			6.00000i			0.423207i
$202$	1.00000			0.0703598
$203$	−5.00000			−0.350931
$204$	10.0000			0.700140
$205$		0			0
$206$		−	4.00000i		−	0.278693i
$207$			4.00000i			0.278019i
$208$	−2.00000	−	3.00000i	−0.138675	−	0.208013i
$209$	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	$0.635539\pi$
$212$		−	3.00000i		−	0.206041i
$213$	16.0000			1.09630
$214$		−	16.0000i		−	1.09374i
$215$		0			0
$216$		−	12.0000i		−	0.816497i
$217$			5.00000i			0.339422i
$218$		−	2.00000i		−	0.135457i
$219$		−	8.00000i		−	0.540590i
$220$		0			0
$221$	−15.0000	+	10.0000i	−1.00901	+	0.672673i
$222$			8.00000i			0.536925i
$223$	16.0000			1.07144			0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000			1.07144			0.535720	+	0.844396i	$0.320040\pi$
$224$	−25.0000			−1.67038
$225$		0			0
$226$		−	10.0000i		−	0.665190i
$227$	9.00000			0.597351			0.298675	−	0.954355i	$-0.403455\pi$
$227$	9.00000			0.597351			0.298675	+	0.954355i	$0.403455\pi$
$228$	−8.00000			−0.529813
$229$			2.00000i			0.132164i	0.997814	+	0.0660819i	$0.0210498\pi$
$229$			2.00000i			0.132164i	−0.997814	+	0.0660819i	$0.978950\pi$
$230$		0			0
$231$	30.0000			1.97386
$232$	−3.00000			−0.196960
$233$		−	14.0000i		−	0.917170i	−0.888650	−	0.458585i	$-0.848356\pi$
$233$		−	14.0000i		−	0.917170i	0.888650	−	0.458585i	$-0.151644\pi$
$234$	−2.00000	−	3.00000i	−0.130744	−	0.196116i
$235$		0			0
$236$		−	3.00000i		−	0.195283i
$237$		−	20.0000i		−	1.29914i
$238$		−	25.0000i		−	1.62051i
$239$			5.00000i			0.323423i	0.986838	+	0.161712i	$0.0517014\pi$
$239$			5.00000i			0.323423i	−0.986838	+	0.161712i	$0.948299\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	2.00000			0.128565
$243$		−	10.0000i		−	0.641500i
$244$	−1.00000			−0.0640184
$245$		0			0
$246$	−16.0000			−1.02012
$247$	12.0000	−	8.00000i	0.763542	−	0.509028i
$248$			3.00000i			0.190500i
$249$			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	$0.376365\pi$
$252$	−5.00000			−0.314970
$253$	12.0000			0.754434
$254$			18.0000i			1.12942i
$255$		0			0
$256$	−17.0000			−1.06250
$257$			15.0000i			0.935674i	0.883815	+	0.467837i	$0.154967\pi$
$257$			15.0000i			0.935674i	−0.883815	+	0.467837i	$0.845033\pi$
$258$	−8.00000			−0.498058
$259$	−20.0000			−1.24274
$260$		0			0
$261$	−1.00000			−0.0618984
$262$	18.0000			1.11204
$263$			6.00000i			0.369976i	0.982741	+	0.184988i	$0.0592246\pi$
$263$			6.00000i			0.369976i	−0.982741	+	0.184988i	$0.940775\pi$
$264$	18.0000			1.10782
$265$		0			0
$266$			20.0000i			1.22628i
$267$	−36.0000			−2.20316
$268$	−3.00000			−0.183254
$269$	17.0000			1.03651			0.518254	−	0.855227i	$-0.326582\pi$
$269$	17.0000			1.03651			0.518254	+	0.855227i	$0.326582\pi$
$270$		0			0
$271$			29.0000i			1.76162i	0.473466	+	0.880812i	$0.343003\pi$
$271$			29.0000i			1.76162i	−0.473466	+	0.880812i	$0.656997\pi$
$272$		−	5.00000i		−	0.303170i
$273$	30.0000	−	20.0000i	1.81568	−	1.21046i
$274$	18.0000			1.08742
$275$		0			0
$276$	−8.00000			−0.481543
$277$		−	2.00000i		−	0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$		−	2.00000i		−	0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$	−4.00000			−0.239904
$279$			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	$-0.180403\pi$
$282$		−	14.0000i		−	0.833688i
$283$			28.0000i			1.66443i	0.554455	+	0.832214i	$0.312927\pi$
$283$			28.0000i			1.66443i	−0.554455	+	0.832214i	$0.687073\pi$
$284$			8.00000i			0.474713i
$285$		0			0
$286$	−9.00000	+	6.00000i	−0.532181	+	0.354787i
$287$		−	40.0000i		−	2.36113i
$288$	−5.00000			−0.294628
$289$	−8.00000			−0.470588
$290$		0			0
$291$		−	28.0000i		−	1.64139i
$292$	4.00000			0.234082
$293$	6.00000			0.350524			0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000			0.350524			0.175262	+	0.984522i	$0.443923\pi$
$294$			36.0000i			2.09956i
$295$		0			0
$296$	−12.0000			−0.697486
$297$	−12.0000			−0.696311
$298$			10.0000i			0.579284i
$299$	12.0000	−	8.00000i	0.693978	−	0.462652i
$300$		0			0
$301$		−	20.0000i		−	1.15278i
$302$			13.0000i			0.748066i
$303$			2.00000i			0.114897i
$304$			4.00000i			0.229416i
$305$		0			0
$306$		−	5.00000i		−	0.285831i
$307$	8.00000			0.456584			0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000			0.456584			0.228292	+	0.973593i	$0.426686\pi$
$308$			15.0000i			0.854704i
$309$	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	$0.463823\pi$
$312$	18.0000	−	12.0000i	1.01905	−	0.679366i
$313$		−	17.0000i		−	0.960897i	−0.877023	−	0.480448i	$-0.840474\pi$
$313$		−	17.0000i		−	0.960897i	0.877023	−	0.480448i	$-0.159526\pi$
$314$		−	13.0000i		−	0.733632i
$315$		0			0
$316$	10.0000			0.562544
$317$	22.0000			1.23564			0.617822	−	0.786318i	$-0.288015\pi$
$317$	22.0000			1.23564			0.617822	+	0.786318i	$0.288015\pi$
$318$	−6.00000			−0.336463
$319$			3.00000i			0.167968i
$320$		0			0
$321$	32.0000			1.78607
$322$			20.0000i			1.11456i
$323$	20.0000			1.11283
$324$	11.0000			0.611111
$325$		0			0
$326$	−12.0000			−0.664619
$327$	4.00000			0.221201
$328$		−	24.0000i		−	1.32518i
$329$	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	$0.814761\pi$
$332$	−9.00000			−0.493939
$333$	−4.00000			−0.219199
$334$	16.0000			0.875481
$335$		0			0
$336$			10.0000i			0.545545i
$337$		−	3.00000i		−	0.163420i	−0.996656	−	0.0817102i	$-0.973962\pi$
$337$		−	3.00000i		−	0.163420i	0.996656	−	0.0817102i	$-0.0260382\pi$
$338$	−5.00000	+	12.0000i	−0.271964	+	0.652714i
$339$	20.0000			1.08625
$340$		0			0
$341$	3.00000			0.162459
$342$			4.00000i			0.216295i
$343$	−55.0000			−2.96972
$344$		−	12.0000i		−	0.646997i
$345$		0			0
$346$		−	13.0000i		−	0.698884i
$347$			4.00000i			0.214731i	0.994220	+	0.107366i	$0.0342415\pi$
$347$			4.00000i			0.214731i	−0.994220	+	0.107366i	$0.965758\pi$
$348$		−	2.00000i		−	0.107211i
$349$		−	2.00000i		−	0.107058i	−0.998566	−	0.0535288i	$-0.982953\pi$
$349$		−	2.00000i		−	0.107058i	0.998566	−	0.0535288i	$-0.0170469\pi$
$350$		0			0
$351$	−12.0000	+	8.00000i	−0.640513	+	0.427008i
$352$			15.0000i			0.799503i
$353$	−6.00000			−0.319348			−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000			−0.319348			−0.159674	+	0.987170i	$0.551044\pi$
$354$	−6.00000			−0.318896
$355$		0			0
$356$		−	18.0000i		−	0.953998i
$357$	50.0000			2.64628
$358$	−12.0000			−0.634220
$359$		−	23.0000i		−	1.21389i	−0.794742	−	0.606947i	$-0.792394\pi$
$359$		−	23.0000i		−	1.21389i	0.794742	−	0.606947i	$-0.207606\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$	3.00000			0.157676
$363$			4.00000i			0.209946i
$364$	10.0000	+	15.0000i	0.524142	+	0.786214i
$365$		0			0
$366$			2.00000i			0.104542i
$367$		−	18.0000i		−	0.939592i	−0.882775	−	0.469796i	$-0.844327\pi$
$367$		−	18.0000i		−	0.939592i	0.882775	−	0.469796i	$-0.155673\pi$
$368$			4.00000i			0.208514i
$369$		−	8.00000i		−	0.416463i
$370$		0			0
$371$		−	15.0000i		−	0.778761i
$372$	−2.00000			−0.103695
$373$		−	31.0000i		−	1.60512i	−0.596572	−	0.802560i	$-0.703471\pi$
$373$		−	31.0000i		−	1.60512i	0.596572	−	0.802560i	$-0.296529\pi$
$374$	−15.0000			−0.775632
$375$		0			0
$376$	21.0000			1.08299
$377$	2.00000	+	3.00000i	0.103005	+	0.154508i
$378$		−	20.0000i		−	1.02869i
$379$			27.0000i			1.38690i	0.720506	+	0.693448i	$0.243909\pi$
$379$			27.0000i			1.38690i	−0.720506	+	0.693448i	$0.756091\pi$
$380$		0			0
$381$	−36.0000			−1.84434
$382$	10.0000			0.511645
$383$	32.0000			1.63512			0.817562	−	0.575841i	$-0.195325\pi$
$383$	32.0000			1.63512			0.817562	+	0.575841i	$0.195325\pi$
$384$		−	6.00000i		−	0.306186i
$385$		0			0
$386$	−4.00000			−0.203595
$387$		−	4.00000i		−	0.203331i
$388$	14.0000			0.710742
$389$	−34.0000			−1.72387			−0.861934	−	0.507020i	$-0.830747\pi$
$389$	−34.0000			−1.72387			−0.861934	+	0.507020i	$0.830747\pi$
$390$		0			0
$391$	20.0000			1.01144
$392$	−54.0000			−2.72741
$393$			36.0000i			1.81596i
$394$	−8.00000			−0.403034
$395$		0			0
$396$			3.00000i			0.150756i
$397$	8.00000			0.401508			0.200754	−	0.979642i	$-0.435661\pi$
$397$	8.00000			0.401508			0.200754	+	0.979642i	$0.435661\pi$
$398$	6.00000			0.300753
$399$	−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	$-0.224892\pi$
$402$			6.00000i			0.299253i
$403$	3.00000	−	2.00000i	0.149441	−	0.0996271i
$404$	−1.00000			−0.0497519
$405$		0			0
$406$	−5.00000			−0.248146
$407$			12.0000i			0.594818i
$408$	30.0000			1.48522
$409$		−	6.00000i		−	0.296681i	−0.988936	−	0.148340i	$-0.952607\pi$
$409$		−	6.00000i		−	0.296681i	0.988936	−	0.148340i	$-0.0473931\pi$
$410$		0			0
$411$			36.0000i			1.77575i
$412$			4.00000i			0.197066i
$413$		−	15.0000i		−	0.738102i
$414$			4.00000i			0.196589i
$415$		0			0
$416$	10.0000	+	15.0000i	0.490290	+	0.735436i
$417$		−	8.00000i		−	0.391762i
$418$	12.0000			0.586939
$419$	18.0000			0.879358			0.439679	−	0.898155i	$-0.355092\pi$
$419$	18.0000			0.879358			0.439679	+	0.898155i	$0.355092\pi$
$420$		0			0
$421$			4.00000i			0.194948i	0.995238	+	0.0974740i	$0.0310763\pi$
$421$			4.00000i			0.194948i	−0.995238	+	0.0974740i	$0.968924\pi$
$422$	−12.0000			−0.584151
$423$	7.00000			0.340352
$424$		−	9.00000i		−	0.437079i
$425$		0			0
$426$	16.0000			0.775203
$427$	−5.00000			−0.241967
$428$			16.0000i			0.773389i
$429$	−12.0000	−	18.0000i	−0.579365	−	0.869048i
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		−	4.00000i		−	0.192450i
$433$		−	14.0000i		−	0.672797i	−0.941720	−	0.336399i	$-0.890791\pi$
$433$		−	14.0000i		−	0.672797i	0.941720	−	0.336399i	$-0.109209\pi$
$434$			5.00000i			0.240008i
$435$		0			0
$436$			2.00000i			0.0957826i
$437$	−16.0000			−0.765384
$438$		−	8.00000i		−	0.382255i
$439$	−22.0000			−1.05000			−0.525001	−	0.851101i	$-0.675935\pi$
$439$	−22.0000			−1.05000			−0.525001	+	0.851101i	$0.675935\pi$
$440$		0			0
$441$	−18.0000			−0.857143
$442$	−15.0000	+	10.0000i	−0.713477	+	0.475651i
$443$			24.0000i			1.14027i	0.821549	+	0.570137i	$0.193110\pi$
$443$			24.0000i			1.14027i	−0.821549	+	0.570137i	$0.806890\pi$
$444$		−	8.00000i		−	0.379663i
$445$		0			0
$446$	16.0000			0.757622
$447$	−20.0000			−0.945968
$448$	−35.0000			−1.65359
$449$			6.00000i			0.283158i	0.989927	+	0.141579i	$0.0452178\pi$
$449$			6.00000i			0.283158i	−0.989927	+	0.141579i	$0.954782\pi$
$450$		0			0
$451$	−24.0000			−1.13012
$452$			10.0000i			0.470360i
$453$	−26.0000			−1.22159
$454$	9.00000			0.422391
$455$		0			0
$456$	−24.0000			−1.12390
$457$	−10.0000			−0.467780			−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000			−0.467780			−0.233890	+	0.972263i	$0.575146\pi$
$458$			2.00000i			0.0934539i
$459$	−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	$-0.207015\pi$
$462$	30.0000			1.39573
$463$	19.0000			0.883005			0.441502	−	0.897260i	$-0.354446\pi$
$463$	19.0000			0.883005			0.441502	+	0.897260i	$0.354446\pi$
$464$	−1.00000			−0.0464238
$465$		0			0
$466$		−	14.0000i		−	0.648537i
$467$			18.0000i			0.832941i	0.909149	+	0.416470i	$0.136733\pi$
$467$			18.0000i			0.832941i	−0.909149	+	0.416470i	$0.863267\pi$
$468$	2.00000	+	3.00000i	0.0924500	+	0.138675i
$469$	−15.0000			−0.692636
$470$		0			0
$471$	26.0000			1.19802
$472$		−	9.00000i		−	0.414259i
$473$	−12.0000			−0.551761
$474$		−	20.0000i		−	0.918630i
$475$		0			0
$476$			25.0000i			1.14587i
$477$		−	3.00000i		−	0.137361i
$478$			5.00000i			0.228695i
$479$			3.00000i			0.137073i	0.997649	+	0.0685367i	$0.0218330\pi$
$479$			3.00000i			0.137073i	−0.997649	+	0.0685367i	$0.978167\pi$
$480$		0			0
$481$	8.00000	+	12.0000i	0.364769	+	0.547153i
$482$		0			0
$483$	−40.0000			−1.82006
$484$	−2.00000			−0.0909091
$485$		0			0
$486$		−	10.0000i		−	0.453609i
$487$	−15.0000			−0.679715			−0.339857	−	0.940477i	$-0.610379\pi$
$487$	−15.0000			−0.679715			−0.339857	+	0.940477i	$0.610379\pi$
$488$	−3.00000			−0.135804
$489$		−	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	$0.717688\pi$
$492$	16.0000			0.721336
$493$			5.00000i			0.225189i
$494$	12.0000	−	8.00000i	0.539906	−	0.359937i
$495$		0			0
$496$			1.00000i			0.0449013i
$497$			40.0000i			1.79425i
$498$			18.0000i			0.806599i
$499$		−	19.0000i		−	0.850557i	−0.905063	−	0.425278i	$-0.860176\pi$
$499$		−	19.0000i		−	0.850557i	0.905063	−	0.425278i	$-0.139824\pi$
$500$		0			0
$501$			32.0000i			1.42965i
$502$	12.0000			0.535586
$503$			16.0000i			0.713405i	0.934218	+	0.356702i	$0.116099\pi$
$503$			16.0000i			0.713405i	−0.934218	+	0.356702i	$0.883901\pi$
$504$	−15.0000			−0.668153
$505$		0			0
$506$	12.0000			0.533465
$507$	−24.0000	−	10.0000i	−1.06588	−	0.444116i
$508$		−	18.0000i		−	0.798621i
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$	20.0000			0.884748
$512$	−11.0000			−0.486136
$513$	16.0000			0.706417
$514$			15.0000i			0.661622i
$515$		0			0
$516$	8.00000			0.352180
$517$		−	21.0000i		−	0.923579i
$518$	−20.0000			−0.878750
$519$	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	$0.429703\pi$
$522$	−1.00000			−0.0437688
$523$			22.0000i			0.961993i	0.876723	+	0.480996i	$0.159725\pi$
$523$			22.0000i			0.961993i	−0.876723	+	0.480996i	$0.840275\pi$
$524$	−18.0000			−0.786334
$525$		0			0
$526$			6.00000i			0.261612i
$527$	5.00000			0.217803
$528$	6.00000			0.261116
$529$	7.00000			0.304348
$530$		0			0
$531$		−	3.00000i		−	0.130189i
$532$		−	20.0000i		−	0.867110i
$533$	−24.0000	+	16.0000i	−1.03956	+	0.693037i
$534$	−36.0000			−1.55787
$535$		0			0
$536$	−9.00000			−0.388741
$537$		−	24.0000i		−	1.03568i
$538$	17.0000			0.732922
$539$			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	$0.695705\pi$
$542$			29.0000i			1.24566i
$543$			6.00000i			0.257485i
$544$			25.0000i			1.07187i
$545$		0			0
$546$	30.0000	−	20.0000i	1.28388	−	0.855921i
$547$			38.0000i			1.62476i	0.583127	+	0.812381i	$0.301829\pi$
$547$			38.0000i			1.62476i	−0.583127	+	0.812381i	$0.698171\pi$
$548$	−18.0000			−0.768922
$549$	−1.00000			−0.0426790
$550$		0			0
$551$		−	4.00000i		−	0.170406i
$552$	−24.0000			−1.02151
$553$	50.0000			2.12622
$554$		−	2.00000i		−	0.0849719i
$555$		0			0
$556$	4.00000			0.169638
$557$	−24.0000			−1.01691			−0.508456	−	0.861088i	$-0.669784\pi$
$557$	−24.0000			−1.01691			−0.508456	+	0.861088i	$0.669784\pi$
$558$			1.00000i			0.0423334i
$559$	−12.0000	+	8.00000i	−0.507546	+	0.338364i
$560$		0			0
$561$		−	30.0000i		−	1.26660i
$562$		−	18.0000i		−	0.759284i
$563$			30.0000i			1.26435i	0.774826	+	0.632175i	$0.217837\pi$
$563$			30.0000i			1.26435i	−0.774826	+	0.632175i	$0.782163\pi$
$564$			14.0000i			0.589506i
$565$		0			0
$566$			28.0000i			1.17693i
$567$	55.0000			2.30978
$568$			24.0000i			1.00702i
$569$	33.0000			1.38343			0.691716	−	0.722170i	$-0.256855\pi$
$569$	33.0000			1.38343			0.691716	+	0.722170i	$0.256855\pi$
$570$		0			0
$571$	2.00000			0.0836974			0.0418487	−	0.999124i	$-0.486675\pi$
$571$	2.00000			0.0836974			0.0418487	+	0.999124i	$0.486675\pi$
$572$	9.00000	−	6.00000i	0.376309	−	0.250873i
$573$			20.0000i			0.835512i
$574$		−	40.0000i		−	1.66957i
$575$		0			0
$576$	−7.00000			−0.291667
$577$	−38.0000			−1.58196			−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000			−1.58196			−0.790980	+	0.611842i	$0.790429\pi$
$578$	−8.00000			−0.332756
$579$		−	8.00000i		−	0.332469i
$580$		0			0
$581$	−45.0000			−1.86691
$582$		−	28.0000i		−	1.16064i
$583$	−9.00000			−0.372742
$584$	12.0000			0.496564
$585$		0			0
$586$	6.00000			0.247858
$587$	25.0000			1.03186			0.515930	−	0.856631i	$-0.327446\pi$
$587$	25.0000			1.03186			0.515930	+	0.856631i	$0.327446\pi$
$588$		−	36.0000i		−	1.48461i
$589$	−4.00000			−0.164817
$590$		0			0
$591$		−	16.0000i		−	0.658152i
$592$	−4.00000			−0.164399
$593$	−8.00000			−0.328521			−0.164260	−	0.986417i	$-0.552524\pi$
$593$	−8.00000			−0.328521			−0.164260	+	0.986417i	$0.552524\pi$
$594$	−12.0000			−0.492366
$595$		0			0
$596$		−	10.0000i		−	0.409616i
$597$			12.0000i			0.491127i
$598$	12.0000	−	8.00000i	0.490716	−	0.327144i
$599$	14.0000			0.572024			0.286012	−	0.958226i	$-0.407670\pi$
$599$	14.0000			0.572024			0.286012	+	0.958226i	$0.407670\pi$
$600$		0			0
$601$	29.0000			1.18293			0.591467	−	0.806329i	$-0.298549\pi$
$601$	29.0000			1.18293			0.591467	+	0.806329i	$0.298549\pi$
$602$		−	20.0000i		−	0.815139i
$603$	−3.00000			−0.122169
$604$		−	13.0000i		−	0.528962i
$605$		0			0
$606$			2.00000i			0.0812444i
$607$		−	8.00000i		−	0.324710i	−0.986732	−	0.162355i	$-0.948091\pi$
$607$		−	8.00000i		−	0.324710i	0.986732	−	0.162355i	$-0.0519090\pi$
$608$		−	20.0000i		−	0.811107i
$609$		−	10.0000i		−	0.405220i
$610$		0			0
$611$	−14.0000	−	21.0000i	−0.566379	−	0.849569i
$612$			5.00000i			0.202113i
$613$	−10.0000			−0.403896			−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000			−0.403896			−0.201948	+	0.979396i	$0.564727\pi$
$614$	8.00000			0.322854
$615$		0			0
$616$			45.0000i			1.81310i
$617$	−4.00000			−0.161034			−0.0805170	−	0.996753i	$-0.525657\pi$
$617$	−4.00000			−0.161034			−0.0805170	+	0.996753i	$0.525657\pi$
$618$	8.00000			0.321807
$619$		−	28.0000i		−	1.12542i	−0.826656	−	0.562708i	$-0.809760\pi$
$619$		−	28.0000i		−	1.12542i	0.826656	−	0.562708i	$-0.190240\pi$
$620$		0			0
$621$	16.0000			0.642058
$622$	4.00000			0.160385
$623$		−	90.0000i		−	3.60577i
$624$	6.00000	−	4.00000i	0.240192	−	0.160128i
$625$		0			0
$626$		−	17.0000i		−	0.679457i
$627$			24.0000i			0.958468i
$628$			13.0000i			0.518756i
$629$			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	$-0.130328\pi$
$632$	30.0000			1.19334
$633$		−	24.0000i		−	0.953914i
$634$	22.0000			0.873732
$635$		0			0
$636$	6.00000			0.237915
$637$	36.0000	+	54.0000i	1.42637	+	2.13956i
$638$			3.00000i			0.118771i
$639$			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	$0.595730\pi$
$642$	32.0000			1.26294
$643$	−4.00000			−0.157745			−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000			−0.157745			−0.0788723	+	0.996885i	$0.525132\pi$
$644$		−	20.0000i		−	0.788110i
$645$		0			0
$646$	20.0000			0.786889
$647$		−	14.0000i		−	0.550397i	−0.961387	−	0.275198i	$-0.911256\pi$
$647$		−	14.0000i		−	0.550397i	0.961387	−	0.275198i	$-0.0887435\pi$
$648$	33.0000			1.29636
$649$	−9.00000			−0.353281
$650$		0			0
$651$	−10.0000			−0.391931
$652$	12.0000			0.469956
$653$		−	31.0000i		−	1.21312i	−0.795036	−	0.606562i	$-0.792548\pi$
$653$		−	31.0000i		−	1.21312i	0.795036	−	0.606562i	$-0.207452\pi$
$654$	4.00000			0.156412
$655$		0			0
$656$		−	8.00000i		−	0.312348i
$657$	4.00000			0.156055
$658$	35.0000			1.36444
$659$	6.00000			0.233727			0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000			0.233727			0.116863	+	0.993148i	$0.462716\pi$
$660$		0			0
$661$		−	38.0000i		−	1.47803i	−0.673690	−	0.739014i	$-0.735292\pi$
$661$		−	38.0000i		−	1.47803i	0.673690	−	0.739014i	$-0.264708\pi$
$662$			20.0000i			0.777322i
$663$	−20.0000	−	30.0000i	−0.776736	−	1.16510i
$664$	−27.0000			−1.04780
$665$		0			0
$666$	−4.00000			−0.154997
$667$		−	4.00000i		−	0.154881i
$668$	−16.0000			−0.619059
$669$			32.0000i			1.23719i
$670$		0			0
$671$			3.00000i			0.115814i
$672$		−	50.0000i		−	1.92879i
$673$			23.0000i			0.886585i	0.896377	+	0.443292i	$0.146190\pi$
$673$			23.0000i			0.886585i	−0.896377	+	0.443292i	$0.853810\pi$
$674$		−	3.00000i		−	0.115556i
$675$		0			0
$676$	5.00000	−	12.0000i	0.192308	−	0.461538i
$677$			6.00000i			0.230599i	0.993331	+	0.115299i	$0.0367827\pi$
$677$			6.00000i			0.230599i	−0.993331	+	0.115299i	$0.963217\pi$
$678$	20.0000			0.768095
$679$	70.0000			2.68635
$680$		0			0
$681$			18.0000i			0.689761i
$682$	3.00000			0.114876
$683$	−41.0000			−1.56882			−0.784411	−	0.620242i	$-0.787034\pi$
$683$	−41.0000			−1.56882			−0.784411	+	0.620242i	$0.787034\pi$
$684$		−	4.00000i		−	0.152944i
$685$		0			0
$686$	−55.0000			−2.09991
$687$	−4.00000			−0.152610
$688$		−	4.00000i		−	0.152499i
$689$	−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	$-0.104811\pi$
$692$			13.0000i			0.494186i
$693$			15.0000i			0.569803i
$694$			4.00000i			0.151838i
$695$		0			0
$696$		−	6.00000i		−	0.227429i
$697$	−40.0000			−1.51511
$698$		−	2.00000i		−	0.0757011i
$699$	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	$0.823285\pi$
$702$	−12.0000	+	8.00000i	−0.452911	+	0.301941i
$703$		−	16.0000i		−	0.603451i
$704$			21.0000i			0.791467i
$705$		0			0
$706$	−6.00000			−0.225813
$707$	−5.00000			−0.188044
$708$	6.00000			0.225494
$709$		−	8.00000i		−	0.300446i	−0.988652	−	0.150223i	$-0.952001\pi$
$709$		−	8.00000i		−	0.300446i	0.988652	−	0.150223i	$-0.0479992\pi$
$710$		0			0
$711$	10.0000			0.375029
$712$		−	54.0000i		−	2.02374i
$713$	−4.00000			−0.149801
$714$	50.0000			1.87120
$715$		0			0
$716$	12.0000			0.448461
$717$	−10.0000			−0.373457
$718$		−	23.0000i		−	0.858352i
$719$	−30.0000			−1.11881			−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000			−1.11881			−0.559406	+	0.828894i	$0.688971\pi$
$720$		0			0
$721$			20.0000i			0.744839i
$722$	3.00000			0.111648
$723$		0			0
$724$	−3.00000			−0.111494
$725$		0			0
$726$			4.00000i			0.148454i
$727$		−	16.0000i		−	0.593407i	−0.954970	−	0.296704i	$-0.904113\pi$
$727$		−	16.0000i		−	0.593407i	0.954970	−	0.296704i	$-0.0958873\pi$
$728$	30.0000	+	45.0000i	1.11187	+	1.66781i
$729$	−13.0000			−0.481481
$730$		0			0
$731$	−20.0000			−0.739727
$732$		−	2.00000i		−	0.0739221i
$733$	42.0000			1.55131			0.775653	−	0.631160i	$-0.217421\pi$
$733$	42.0000			1.55131			0.775653	+	0.631160i	$0.217421\pi$
$734$		−	18.0000i		−	0.664392i
$735$		0			0
$736$		−	20.0000i		−	0.737210i
$737$			9.00000i			0.331519i
$738$		−	8.00000i		−	0.294484i
$739$		−	9.00000i		−	0.331070i	−0.986204	−	0.165535i	$-0.947065\pi$
$739$		−	9.00000i		−	0.331070i	0.986204	−	0.165535i	$-0.0529351\pi$
$740$		0			0
$741$	16.0000	+	24.0000i	0.587775	+	0.881662i
$742$		−	15.0000i		−	0.550667i
$743$	41.0000			1.50414			0.752072	−	0.659081i	$-0.229055\pi$
$743$	41.0000			1.50414			0.752072	+	0.659081i	$0.229055\pi$
$744$	−6.00000			−0.219971
$745$		0			0
$746$		−	31.0000i		−	1.13499i
$747$	−9.00000			−0.329293
$748$	15.0000			0.548454
$749$			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	$0.698454\pi$
$752$	7.00000			0.255264
$753$			24.0000i			0.874609i
$754$	2.00000	+	3.00000i	0.0728357	+	0.109254i
$755$		0			0
$756$			20.0000i			0.727393i
$757$			5.00000i			0.181728i	0.995863	+	0.0908640i	$0.0289629\pi$
$757$			5.00000i			0.181728i	−0.995863	+	0.0908640i	$0.971037\pi$
$758$			27.0000i			0.980684i
$759$			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	$-0.0580143\pi$
$762$	−36.0000			−1.30414
$763$			10.0000i			0.362024i
$764$	−10.0000			−0.361787
$765$		0			0
$766$	32.0000			1.15621
$767$	−9.00000	+	6.00000i	−0.324971	+	0.216647i
$768$		−	34.0000i		−	1.22687i
$769$			40.0000i			1.44244i	0.692708	+	0.721218i	$0.256418\pi$
$769$			40.0000i			1.44244i	−0.692708	+	0.721218i	$0.743582\pi$
$770$		0			0
$771$	−30.0000			−1.08042
$772$	4.00000			0.143963
$773$	−18.0000			−0.647415			−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000			−0.647415			−0.323708	+	0.946157i	$0.604929\pi$
$774$		−	4.00000i		−	0.143777i
$775$		0			0
$776$	42.0000			1.50771
$777$		−	40.0000i		−	1.43499i
$778$	−34.0000			−1.21896
$779$	32.0000			1.14652
$780$		0			0
$781$	24.0000			0.858788
$782$	20.0000			0.715199
$783$			4.00000i			0.142948i
$784$	−18.0000			−0.642857
$785$		0			0
$786$			36.0000i			1.28408i
$787$	43.0000			1.53278			0.766392	−	0.642373i	$-0.222050\pi$
$787$	43.0000			1.53278			0.766392	+	0.642373i	$0.222050\pi$
$788$	8.00000			0.284988
$789$	−12.0000			−0.427211
$790$		0			0
$791$			50.0000i			1.77780i
$792$			9.00000i			0.319801i
$793$	2.00000	+	3.00000i	0.0710221	+	0.106533i
$794$	8.00000			0.283909
$795$		0			0
$796$	−6.00000			−0.212664
$797$		−	53.0000i		−	1.87736i	−0.344795	−	0.938678i	$-0.612051\pi$
$797$		−	53.0000i		−	1.87736i	0.344795	−	0.938678i	$-0.387949\pi$
$798$	−40.0000			−1.41598
$799$		−	35.0000i		−	1.23821i
$800$		0			0
$801$		−	18.0000i		−	0.635999i
$802$		−	26.0000i		−	0.918092i
$803$		−	12.0000i		−	0.423471i
$804$		−	6.00000i		−	0.211604i
$805$		0			0
$806$	3.00000	−	2.00000i	0.105670	−	0.0704470i
$807$			34.0000i			1.19686i
$808$	−3.00000			−0.105540
$809$	30.0000			1.05474			0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000			1.05474			0.527372	+	0.849635i	$0.323177\pi$
$810$		0			0
$811$		−	25.0000i		−	0.877869i	−0.898519	−	0.438934i	$-0.855356\pi$
$811$		−	25.0000i		−	0.877869i	0.898519	−	0.438934i	$-0.144644\pi$
$812$	5.00000			0.175466
$813$	−58.0000			−2.03415
$814$			12.0000i			0.420600i
$815$		0			0
$816$	10.0000			0.350070
$817$	16.0000			0.559769
$818$		−	6.00000i		−	0.209785i
$819$	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	$0.824625\pi$
$822$			36.0000i			1.25564i
$823$		−	12.0000i		−	0.418294i	−0.977884	−	0.209147i	$-0.932931\pi$
$823$		−	12.0000i		−	0.418294i	0.977884	−	0.209147i	$-0.0670687\pi$
$824$			12.0000i			0.418040i
$825$		0			0
$826$		−	15.0000i		−	0.521917i
$827$	−19.0000			−0.660695			−0.330347	−	0.943859i	$-0.607166\pi$
$827$	−19.0000			−0.660695			−0.330347	+	0.943859i	$0.607166\pi$
$828$		−	4.00000i		−	0.139010i
$829$	11.0000			0.382046			0.191023	−	0.981586i	$-0.438820\pi$
$829$	11.0000			0.382046			0.191023	+	0.981586i	$0.438820\pi$
$830$		0			0
$831$	4.00000			0.138758
$832$	14.0000	+	21.0000i	0.485363	+	0.728044i
$833$			90.0000i			3.11832i
$834$		−	8.00000i		−	0.277017i
$835$		0			0
$836$	−12.0000			−0.415029
$837$	4.00000			0.138260
$838$	18.0000			0.621800
$839$		−	36.0000i		−	1.24286i	−0.783470	−	0.621429i	$-0.786552\pi$
$839$		−	36.0000i		−	1.24286i	0.783470	−	0.621429i	$-0.213448\pi$
$840$		0			0
$841$	−28.0000			−0.965517
$842$			4.00000i			0.137849i
$843$	36.0000			1.23991
$844$	12.0000			0.413057
$845$		0			0
$846$	7.00000			0.240665
$847$	−10.0000			−0.343604
$848$		−	3.00000i		−	0.103020i
$849$	−56.0000			−1.92192
$850$		0			0
$851$		−	16.0000i		−	0.548473i
$852$	−16.0000			−0.548151
$853$	−54.0000			−1.84892			−0.924462	−	0.381273i	$-0.875486\pi$
$853$	−54.0000			−1.84892			−0.924462	+	0.381273i	$0.875486\pi$
$854$	−5.00000			−0.171096
$855$		0			0
$856$			48.0000i			1.64061i
$857$			14.0000i			0.478231i	0.970991	+	0.239115i	$0.0768574\pi$
$857$			14.0000i			0.478231i	−0.970991	+	0.239115i	$0.923143\pi$
$858$	−12.0000	−	18.0000i	−0.409673	−	0.614510i
$859$	4.00000			0.136478			0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000			0.136478			0.0682391	+	0.997669i	$0.478262\pi$
$860$		0			0
$861$	80.0000			2.72639
$862$		0			0
$863$	33.0000			1.12333			0.561667	−	0.827364i	$-0.310160\pi$
$863$	33.0000			1.12333			0.561667	+	0.827364i	$0.310160\pi$
$864$			20.0000i			0.680414i
$865$		0			0
$866$		−	14.0000i		−	0.475739i
$867$		−	16.0000i		−	0.543388i
$868$		−	5.00000i		−	0.169711i
$869$		−	30.0000i		−	1.01768i
$870$		0			0
$871$	6.00000	+	9.00000i	0.203302	+	0.304953i
$872$			6.00000i			0.203186i
$873$	14.0000			0.473828
$874$	−16.0000			−0.541208
$875$		0			0
$876$			8.00000i			0.270295i
$877$	14.0000			0.472746			0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000			0.472746			0.236373	+	0.971662i	$0.424041\pi$
$878$	−22.0000			−0.742464
$879$			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	$0.687626\pi$
$882$	−18.0000			−0.606092
$883$		−	28.0000i		−	0.942275i	−0.882060	−	0.471138i	$-0.843844\pi$
$883$		−	28.0000i		−	0.942275i	0.882060	−	0.471138i	$-0.156156\pi$
$884$	15.0000	−	10.0000i	0.504505	−	0.336336i
$885$		0			0
$886$			24.0000i			0.806296i
$887$		−	22.0000i		−	0.738688i	−0.929293	−	0.369344i	$-0.879582\pi$
$887$		−	22.0000i		−	0.738688i	0.929293	−	0.369344i	$-0.120418\pi$
$888$		−	24.0000i		−	0.805387i
$889$		−	90.0000i		−	3.01850i
$890$		0			0
$891$		−	33.0000i		−	1.10554i
$892$	−16.0000			−0.535720
$893$			28.0000i			0.936984i
$894$	−20.0000			−0.668900
$895$		0			0
$896$	15.0000			0.501115
$897$	16.0000	+	24.0000i	0.534224	+	0.801337i
$898$			6.00000i			0.200223i
$899$		−	1.00000i		−	0.0333519i
$900$		0			0
$901$	−15.0000			−0.499722
$902$	−24.0000			−0.799113
$903$	40.0000			1.33112
$904$			30.0000i			0.997785i
$905$		0			0
$906$	−26.0000			−0.863792
$907$		−	20.0000i		−	0.664089i	−0.943264	−	0.332045i	$-0.892262\pi$
$907$		−	20.0000i		−	0.664089i	0.943264	−	0.332045i	$-0.107738\pi$
$908$	−9.00000			−0.298675
$909$	−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	$0.489452\pi$
$912$	−8.00000			−0.264906
$913$			27.0000i			0.893570i
$914$	−10.0000			−0.330771
$915$		0			0
$916$		−	2.00000i		−	0.0660819i
$917$	−90.0000			−2.97206
$918$	−20.0000			−0.660098
$919$	−46.0000			−1.51740			−0.758700	−	0.651440i	$-0.774165\pi$
$919$	−46.0000			−1.51740			−0.758700	+	0.651440i	$0.774165\pi$
$920$		0			0
$921$			16.0000i			0.527218i
$922$		−	26.0000i		−	0.856264i
$923$	24.0000	−	16.0000i	0.789970	−	0.526646i
$924$	−30.0000			−0.986928
$925$		0			0
$926$	19.0000			0.624379
$927$			4.00000i			0.131377i
$928$	5.00000			0.164133
$929$		−	56.0000i		−	1.83730i	−0.395072	−	0.918650i	$-0.629280\pi$
$929$		−	56.0000i		−	1.83730i	0.395072	−	0.918650i	$-0.370720\pi$
$930$		0			0
$931$		−	72.0000i		−	2.35970i
$932$			14.0000i			0.458585i
$933$			8.00000i			0.261908i
$934$			18.0000i			0.588978i
$935$		0			0
$936$	6.00000	+	9.00000i	0.196116	+	0.294174i
$937$		−	1.00000i		−	0.0326686i	−0.999867	−	0.0163343i	$-0.994800\pi$
$937$		−	1.00000i		−	0.0326686i	0.999867	−	0.0163343i	$-0.00519960\pi$
$938$	−15.0000			−0.489767
$939$	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	$-0.0416250\pi$
$942$	26.0000			0.847126
$943$	32.0000			1.04206
$944$		−	3.00000i		−	0.0976417i
$945$		0			0
$946$	−12.0000			−0.390154
$947$	−3.00000			−0.0974869			−0.0487435	−	0.998811i	$-0.515522\pi$
$947$	−3.00000			−0.0974869			−0.0487435	+	0.998811i	$0.515522\pi$
$948$			20.0000i			0.649570i
$949$	−8.00000	−	12.0000i	−0.259691	−	0.389536i
$950$		0			0
$951$			44.0000i			1.42680i
$952$			75.0000i			2.43076i
$953$			9.00000i			0.291539i	0.989319	+	0.145769i	$0.0465657\pi$
$953$			9.00000i			0.291539i	−0.989319	+	0.145769i	$0.953434\pi$
$954$		−	3.00000i		−	0.0971286i
$955$		0			0
$956$		−	5.00000i		−	0.161712i
$957$	−6.00000			−0.193952
$958$			3.00000i			0.0969256i
$959$	−90.0000			−2.90625
$960$		0			0
$961$	30.0000			0.967742
$962$	8.00000	+	12.0000i	0.257930	+	0.386896i
$963$			16.0000i			0.515593i
$964$		0			0
$965$		0			0
$966$	−40.0000			−1.28698
$967$	37.0000			1.18984			0.594920	−	0.803785i	$-0.297184\pi$
$967$	37.0000			1.18984			0.594920	+	0.803785i	$0.297184\pi$
$968$	−6.00000			−0.192847
$969$			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	$0.779842\pi$
$972$			10.0000i			0.320750i
$973$	20.0000			0.641171
$974$	−15.0000			−0.480631
$975$		0			0
$976$	−1.00000			−0.0320092
$977$	46.0000			1.47167			0.735835	−	0.677161i	$-0.236790\pi$
$977$	46.0000			1.47167			0.735835	+	0.677161i	$0.236790\pi$
$978$		−	24.0000i		−	0.767435i
$979$	−54.0000			−1.72585
$980$		0			0
$981$			2.00000i			0.0638551i
$982$	−28.0000			−0.893516
$983$	−11.0000			−0.350846			−0.175423	−	0.984493i	$-0.556129\pi$
$983$	−11.0000			−0.350846			−0.175423	+	0.984493i	$0.556129\pi$
$984$	48.0000			1.53018
$985$		0			0
$986$			5.00000i			0.159232i
$987$			70.0000i			2.22812i
$988$	−12.0000	+	8.00000i	−0.381771	+	0.254514i
$989$	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	$0.479763\pi$
$992$		−	5.00000i		−	0.158750i
$993$	−40.0000			−1.26936
$994$			40.0000i			1.26872i
$995$		0			0
$996$		−	18.0000i		−	0.570352i
$997$			45.0000i			1.42516i	0.701589	+	0.712582i	$0.252474\pi$
$997$			45.0000i			1.42516i	−0.701589	+	0.712582i	$0.747526\pi$
$998$		−	19.0000i		−	0.601434i
$999$			16.0000i			0.506218i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.2.c.a.51.1	✓	2	5.3	odd	4
325.2.c.a.51.2	yes	2	65.38	odd	4
325.2.c.f.51.1	yes	2	65.12	odd	4
325.2.c.f.51.2	yes	2	5.2	odd	4
325.2.d.b.324.1		2	5.4	even	2
325.2.d.b.324.2		2	13.12	even	2
325.2.d.c.324.1		2	65.64	even	2	inner
325.2.d.c.324.2		2	1.1	even	1	trivial
4225.2.a.d.1.1		1	65.18	even	4
4225.2.a.f.1.1		1	65.47	even	4
4225.2.a.l.1.1		1	65.8	even	4
4225.2.a.n.1.1		1	65.57	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:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	325.d (of order \(2\), degree \(1\), not minimal)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more