LMFDB - Embedded newform 5070.2.b.a.1351.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1351,5070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5070.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5070.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$40.4841538248$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	5070.1351
Dual form			5070.2.b.a.1351.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} -3.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} -3.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000i q^{11} +1.00000 q^{12} -3.00000 q^{14} +1.00000i q^{15} +1.00000 q^{16} -1.00000i q^{18} +5.00000i q^{19} +1.00000i q^{20} +3.00000i q^{21} +1.00000 q^{22} +4.00000 q^{23} -1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} +3.00000i q^{28} +1.00000 q^{30} -10.0000i q^{31} -1.00000i q^{32} -1.00000i q^{33} -3.00000 q^{35} -1.00000 q^{36} -1.00000i q^{37} +5.00000 q^{38} +1.00000 q^{40} -6.00000i q^{41} +3.00000 q^{42} +2.00000 q^{43} -1.00000i q^{44} -1.00000i q^{45} -4.00000i q^{46} -9.00000i q^{47} -1.00000 q^{48} -2.00000 q^{49} +1.00000i q^{50} -13.0000 q^{53} +1.00000i q^{54} +1.00000 q^{55} +3.00000 q^{56} -5.00000i q^{57} +4.00000i q^{59} -1.00000i q^{60} -2.00000 q^{61} -10.0000 q^{62} -3.00000i q^{63} -1.00000 q^{64} -1.00000 q^{66} +12.0000i q^{67} -4.00000 q^{69} +3.00000i q^{70} +2.00000i q^{71} +1.00000i q^{72} -16.0000i q^{73} -1.00000 q^{74} +1.00000 q^{75} -5.00000i q^{76} +3.00000 q^{77} -10.0000 q^{79} -1.00000i q^{80} +1.00000 q^{81} -6.00000 q^{82} -12.0000i q^{83} -3.00000i q^{84} -2.00000i q^{86} -1.00000 q^{88} +1.00000i q^{89} -1.00000 q^{90} -4.00000 q^{92} +10.0000i q^{93} -9.00000 q^{94} +5.00000 q^{95} +1.00000i q^{96} -12.0000i q^{97} +2.00000i q^{98} +1.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9	$ 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} - 2 q^{10} + 2 q^{12} - 6 q^{14} + 2 q^{16} + 2 q^{22} + 8 q^{23} - 2 q^{25} - 2 q^{27} + 2 q^{30} - 6 q^{35} - 2 q^{36} + 10 q^{38} + 2 q^{40} + 6 q^{42} + 4 q^{43} - 2 q^{48} - 4 q^{49} - 26 q^{53} + 2 q^{55} + 6 q^{56} - 4 q^{61} - 20 q^{62} - 2 q^{64} - 2 q^{66} - 8 q^{69} - 2 q^{74} + 2 q^{75} + 6 q^{77} - 20 q^{79} + 2 q^{81} - 12 q^{82} - 2 q^{88} - 2 q^{90} - 8 q^{92} - 18 q^{94} + 10 q^{95}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 - 2 * q^10 + 2 * q^12 - 6 * q^14 + 2 * q^16 + 2 * q^22 + 8 * q^23 - 2 * q^25 - 2 * q^27 + 2 * q^30 - 6 * q^35 - 2 * q^36 + 10 * q^38 + 2 * q^40 + 6 * q^42 + 4 * q^43 - 2 * q^48 - 4 * q^49 - 26 * q^53 + 2 * q^55 + 6 * q^56 - 4 * q^61 - 20 * q^62 - 2 * q^64 - 2 * q^66 - 8 * q^69 - 2 * q^74 + 2 * q^75 + 6 * q^77 - 20 * q^79 + 2 * q^81 - 12 * q^82 - 2 * q^88 - 2 * q^90 - 8 * q^92 - 18 * q^94 + 10 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times$.

$n$	$1691$	$1861$	$4057$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.00000	−0.500000
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	1.00000i	0.408248i
$7$	− 3.00000i	− 1.13389i	−0.823754	−	0.566947i	$-0.808125\pi$
$7$	− 3.00000i	− 1.13389i	0.823754	−	0.566947i	$-0.191875\pi$
$8$	1.00000i	0.353553i
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	1.00000i	0.301511i	0.988571	+	0.150756i	$0.0481707\pi$
$11$	1.00000i	0.301511i	−0.988571	+	0.150756i	$0.951829\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	−3.00000	−0.801784
$15$	1.00000i	0.258199i
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	− 1.00000i	− 0.235702i
$19$	5.00000i	1.14708i	0.819178	+	0.573539i	$0.194430\pi$
$19$	5.00000i	1.14708i	−0.819178	+	0.573539i	$0.805570\pi$
$20$	1.00000i	0.223607i
$21$	3.00000i	0.654654i
$22$	1.00000	0.213201
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	− 1.00000i	− 0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	3.00000i	0.566947i
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	1.00000	0.182574
$31$	− 10.0000i	− 1.79605i	−0.439941	−	0.898027i	$-0.645001\pi$
$31$	− 10.0000i	− 1.79605i	0.439941	−	0.898027i	$-0.354999\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 1.00000i	− 0.174078i
$34$	0	0
$35$	−3.00000	−0.507093
$36$	−1.00000	−0.166667
$37$	− 1.00000i	− 0.164399i	−0.996616	−	0.0821995i	$-0.973806\pi$
$37$	− 1.00000i	− 0.164399i	0.996616	−	0.0821995i	$-0.0261945\pi$
$38$	5.00000	0.811107
$39$	0	0
$40$	1.00000	0.158114
$41$	− 6.00000i	− 0.937043i	−0.883452	−	0.468521i	$-0.844787\pi$
$41$	− 6.00000i	− 0.937043i	0.883452	−	0.468521i	$-0.155213\pi$
$42$	3.00000	0.462910
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	− 1.00000i	− 0.150756i
$45$	− 1.00000i	− 0.149071i
$46$	− 4.00000i	− 0.589768i
$47$	− 9.00000i	− 1.31278i	−0.754420	−	0.656392i	$-0.772082\pi$
$47$	− 9.00000i	− 1.31278i	0.754420	−	0.656392i	$-0.227918\pi$
$48$	−1.00000	−0.144338
$49$	−2.00000	−0.285714
$50$	1.00000i	0.141421i
$51$	0	0
$52$	0	0
$53$	−13.0000	−1.78569	−0.892844	−	0.450367i	$-0.851293\pi$
$53$	−13.0000	−1.78569	−0.892844	+	0.450367i	$0.851293\pi$
$54$	1.00000i	0.136083i
$55$	1.00000	0.134840
$56$	3.00000	0.400892
$57$	− 5.00000i	− 0.662266i
$58$	0	0
$59$	4.00000i	0.520756i	0.965507	+	0.260378i	$0.0838471\pi$
$59$	4.00000i	0.520756i	−0.965507	+	0.260378i	$0.916153\pi$
$60$	− 1.00000i	− 0.129099i
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−10.0000	−1.27000
$63$	− 3.00000i	− 0.377964i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−1.00000	−0.123091
$67$	12.0000i	1.46603i	0.680211	+	0.733017i	$0.261888\pi$
$67$	12.0000i	1.46603i	−0.680211	+	0.733017i	$0.738112\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	3.00000i	0.358569i
$71$	2.00000i	0.237356i	0.992933	+	0.118678i	$0.0378657\pi$
$71$	2.00000i	0.237356i	−0.992933	+	0.118678i	$0.962134\pi$
$72$	1.00000i	0.117851i
$73$	− 16.0000i	− 1.87266i	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	− 16.0000i	− 1.87266i	0.351123	−	0.936329i	$-0.385800\pi$
$74$	−1.00000	−0.116248
$75$	1.00000	0.115470
$76$	− 5.00000i	− 0.573539i
$77$	3.00000	0.341882
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	− 1.00000i	− 0.111803i
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	− 3.00000i	− 0.327327i
$85$	0	0
$86$	− 2.00000i	− 0.215666i
$87$	0	0
$88$	−1.00000	−0.106600
$89$	1.00000i	0.106000i	0.998595	+	0.0529999i	$0.0168783\pi$
$89$	1.00000i	0.106000i	−0.998595	+	0.0529999i	$0.983122\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	−4.00000	−0.417029
$93$	10.0000i	1.03695i
$94$	−9.00000	−0.928279
$95$	5.00000	0.512989
$96$	1.00000i	0.102062i
$97$	− 12.0000i	− 1.21842i	−0.793011	−	0.609208i	$-0.791488\pi$
$97$	− 12.0000i	− 1.21842i	0.793011	−	0.609208i	$-0.208512\pi$
$98$	2.00000i	0.202031i
$99$	1.00000i	0.100504i
$100$	1.00000	0.100000
$101$	−4.00000	−0.398015	−0.199007	−	0.979998i	$-0.563772\pi$
$101$	−4.00000	−0.398015	−0.199007	+	0.979998i	$0.563772\pi$
$102$	0	0
$103$	9.00000	0.886796	0.443398	−	0.896325i	$-0.353773\pi$
$103$	9.00000	0.886796	0.443398	+	0.896325i	$0.353773\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	13.0000i	1.26267i
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	1.00000	0.0962250
$109$	10.0000i	0.957826i	0.877862	+	0.478913i	$0.158969\pi$
$109$	10.0000i	0.957826i	−0.877862	+	0.478913i	$0.841031\pi$
$110$	− 1.00000i	− 0.0953463i
$111$	1.00000i	0.0949158i
$112$	− 3.00000i	− 0.283473i
$113$	16.0000	1.50515	0.752577	−	0.658505i	$-0.228811\pi$
$113$	16.0000	1.50515	0.752577	+	0.658505i	$0.228811\pi$
$114$	−5.00000	−0.468293
$115$	− 4.00000i	− 0.373002i
$116$	0	0
$117$	0	0
$118$	4.00000	0.368230
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	10.0000	0.909091
$122$	2.00000i	0.181071i
$123$	6.00000i	0.541002i
$124$	10.0000i	0.898027i
$125$	1.00000i	0.0894427i
$126$	−3.00000	−0.267261
$127$	−5.00000	−0.443678	−0.221839	−	0.975083i	$-0.571206\pi$
$127$	−5.00000	−0.443678	−0.221839	+	0.975083i	$0.571206\pi$
$128$	1.00000i	0.0883883i
$129$	−2.00000	−0.176090
$130$	0	0
$131$	−15.0000	−1.31056	−0.655278	−	0.755388i	$-0.727449\pi$
$131$	−15.0000	−1.31056	−0.655278	+	0.755388i	$0.727449\pi$
$132$	1.00000i	0.0870388i
$133$	15.0000	1.30066
$134$	12.0000	1.03664
$135$	1.00000i	0.0860663i
$136$	0	0
$137$	16.0000i	1.36697i	0.729964	+	0.683486i	$0.239537\pi$
$137$	16.0000i	1.36697i	−0.729964	+	0.683486i	$0.760463\pi$
$138$	4.00000i	0.340503i
$139$	−9.00000	−0.763370	−0.381685	−	0.924292i	$-0.624656\pi$
$139$	−9.00000	−0.763370	−0.381685	+	0.924292i	$0.624656\pi$
$140$	3.00000	0.253546
$141$	9.00000i	0.757937i
$142$	2.00000	0.167836
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	−16.0000	−1.32417
$147$	2.00000	0.164957
$148$	1.00000i	0.0821995i
$149$	6.00000i	0.491539i	0.969328	+	0.245770i	$0.0790407\pi$
$149$	6.00000i	0.491539i	−0.969328	+	0.245770i	$0.920959\pi$
$150$	− 1.00000i	− 0.0816497i
$151$	− 6.00000i	− 0.488273i	−0.969741	−	0.244137i	$-0.921495\pi$
$151$	− 6.00000i	− 0.488273i	0.969741	−	0.244137i	$-0.0785045\pi$
$152$	−5.00000	−0.405554
$153$	0	0
$154$	− 3.00000i	− 0.241747i
$155$	−10.0000	−0.803219
$156$	0	0
$157$	1.00000	0.0798087	0.0399043	−	0.999204i	$-0.487295\pi$
$157$	1.00000	0.0798087	0.0399043	+	0.999204i	$0.487295\pi$
$158$	10.0000i	0.795557i
$159$	13.0000	1.03097
$160$	−1.00000	−0.0790569
$161$	− 12.0000i	− 0.945732i
$162$	− 1.00000i	− 0.0785674i
$163$	− 20.0000i	− 1.56652i	−0.621694	−	0.783260i	$-0.713555\pi$
$163$	− 20.0000i	− 1.56652i	0.621694	−	0.783260i	$-0.286445\pi$
$164$	6.00000i	0.468521i
$165$	−1.00000	−0.0778499
$166$	−12.0000	−0.931381
$167$	− 13.0000i	− 1.00597i	−0.864295	−	0.502985i	$-0.832235\pi$
$167$	− 13.0000i	− 1.00597i	0.864295	−	0.502985i	$-0.167765\pi$
$168$	−3.00000	−0.231455
$169$	0	0
$170$	0	0
$171$	5.00000i	0.382360i
$172$	−2.00000	−0.152499
$173$	9.00000	0.684257	0.342129	−	0.939653i	$-0.388852\pi$
$173$	9.00000	0.684257	0.342129	+	0.939653i	$0.388852\pi$
$174$	0	0
$175$	3.00000i	0.226779i
$176$	1.00000i	0.0753778i
$177$	− 4.00000i	− 0.300658i
$178$	1.00000	0.0749532
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	1.00000i	0.0745356i
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	4.00000i	0.294884i
$185$	−1.00000	−0.0735215
$186$	10.0000	0.733236
$187$	0	0
$188$	9.00000i	0.656392i
$189$	3.00000i	0.218218i
$190$	− 5.00000i	− 0.362738i
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	1.00000	0.0721688
$193$	16.0000i	1.15171i	0.817554	+	0.575853i	$0.195330\pi$
$193$	16.0000i	1.15171i	−0.817554	+	0.575853i	$0.804670\pi$
$194$	−12.0000	−0.861550
$195$	0	0
$196$	2.00000	0.142857
$197$	15.0000i	1.06871i	0.845262	+	0.534353i	$0.179445\pi$
$197$	15.0000i	1.06871i	−0.845262	+	0.534353i	$0.820555\pi$
$198$	1.00000	0.0710669
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	− 1.00000i	− 0.0707107i
$201$	− 12.0000i	− 0.846415i
$202$	4.00000i	0.281439i
$203$	0	0
$204$	0	0
$205$	−6.00000	−0.419058
$206$	− 9.00000i	− 0.627060i
$207$	4.00000	0.278019
$208$	0	0
$209$	−5.00000	−0.345857
$210$	− 3.00000i	− 0.207020i
$211$	−15.0000	−1.03264	−0.516321	−	0.856395i	$-0.672699\pi$
$211$	−15.0000	−1.03264	−0.516321	+	0.856395i	$0.672699\pi$
$212$	13.0000	0.892844
$213$	− 2.00000i	− 0.137038i
$214$	6.00000i	0.410152i
$215$	− 2.00000i	− 0.136399i
$216$	− 1.00000i	− 0.0680414i
$217$	−30.0000	−2.03653
$218$	10.0000	0.677285
$219$	16.0000i	1.08118i
$220$	−1.00000	−0.0674200
$221$	0	0
$222$	1.00000	0.0671156
$223$	− 11.0000i	− 0.736614i	−0.929704	−	0.368307i	$-0.879937\pi$
$223$	− 11.0000i	− 0.736614i	0.929704	−	0.368307i	$-0.120063\pi$
$224$	−3.00000	−0.200446
$225$	−1.00000	−0.0666667
$226$	− 16.0000i	− 1.06430i
$227$	20.0000i	1.32745i	0.747978	+	0.663723i	$0.231025\pi$
$227$	20.0000i	1.32745i	−0.747978	+	0.663723i	$0.768975\pi$
$228$	5.00000i	0.331133i
$229$	− 10.0000i	− 0.660819i	−0.943838	−	0.330409i	$-0.892813\pi$
$229$	− 10.0000i	− 0.660819i	0.943838	−	0.330409i	$-0.107187\pi$
$230$	−4.00000	−0.263752
$231$	−3.00000	−0.197386
$232$	0	0
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	−9.00000	−0.587095
$236$	− 4.00000i	− 0.260378i
$237$	10.0000	0.649570
$238$	0	0
$239$	2.00000i	0.129369i	0.997906	+	0.0646846i	$0.0206041\pi$
$239$	2.00000i	0.129369i	−0.997906	+	0.0646846i	$0.979396\pi$
$240$	1.00000i	0.0645497i
$241$	15.0000i	0.966235i	0.875556	+	0.483117i	$0.160496\pi$
$241$	15.0000i	0.966235i	−0.875556	+	0.483117i	$0.839504\pi$
$242$	− 10.0000i	− 0.642824i
$243$	−1.00000	−0.0641500
$244$	2.00000	0.128037
$245$	2.00000i	0.127775i
$246$	6.00000	0.382546
$247$	0	0
$248$	10.0000	0.635001
$249$	12.0000i	0.760469i
$250$	1.00000	0.0632456
$251$	−11.0000	−0.694314	−0.347157	−	0.937807i	$-0.612853\pi$
$251$	−11.0000	−0.694314	−0.347157	+	0.937807i	$0.612853\pi$
$252$	3.00000i	0.188982i
$253$	4.00000i	0.251478i
$254$	5.00000i	0.313728i
$255$	0	0
$256$	1.00000	0.0625000
$257$	24.0000	1.49708	0.748539	−	0.663090i	$-0.230755\pi$
$257$	24.0000	1.49708	0.748539	+	0.663090i	$0.230755\pi$
$258$	2.00000i	0.124515i
$259$	−3.00000	−0.186411
$260$	0	0
$261$	0	0
$262$	15.0000i	0.926703i
$263$	3.00000	0.184988	0.0924940	−	0.995713i	$-0.470516\pi$
$263$	3.00000	0.184988	0.0924940	+	0.995713i	$0.470516\pi$
$264$	1.00000	0.0615457
$265$	13.0000i	0.798584i
$266$	− 15.0000i	− 0.919709i
$267$	− 1.00000i	− 0.0611990i
$268$	− 12.0000i	− 0.733017i
$269$	−20.0000	−1.21942	−0.609711	−	0.792624i	$-0.708714\pi$
$269$	−20.0000	−1.21942	−0.609711	+	0.792624i	$0.708714\pi$
$270$	1.00000	0.0608581
$271$	− 24.0000i	− 1.45790i	−0.684569	−	0.728948i	$-0.740010\pi$
$271$	− 24.0000i	− 1.45790i	0.684569	−	0.728948i	$-0.259990\pi$
$272$	0	0
$273$	0	0
$274$	16.0000	0.966595
$275$	− 1.00000i	− 0.0603023i
$276$	4.00000	0.240772
$277$	−23.0000	−1.38194	−0.690968	−	0.722885i	$-0.742815\pi$
$277$	−23.0000	−1.38194	−0.690968	+	0.722885i	$0.742815\pi$
$278$	9.00000i	0.539784i
$279$	− 10.0000i	− 0.598684i
$280$	− 3.00000i	− 0.179284i
$281$	10.0000i	0.596550i	0.954480	+	0.298275i	$0.0964112\pi$
$281$	10.0000i	0.596550i	−0.954480	+	0.298275i	$0.903589\pi$
$282$	9.00000	0.535942
$283$	−2.00000	−0.118888	−0.0594438	−	0.998232i	$-0.518933\pi$
$283$	−2.00000	−0.118888	−0.0594438	+	0.998232i	$0.518933\pi$
$284$	− 2.00000i	− 0.118678i
$285$	−5.00000	−0.296174
$286$	0	0
$287$	−18.0000	−1.06251
$288$	− 1.00000i	− 0.0589256i
$289$	−17.0000	−1.00000
$290$	0	0
$291$	12.0000i	0.703452i
$292$	16.0000i	0.936329i
$293$	13.0000i	0.759468i	0.925096	+	0.379734i	$0.123985\pi$
$293$	13.0000i	0.759468i	−0.925096	+	0.379734i	$0.876015\pi$
$294$	− 2.00000i	− 0.116642i
$295$	4.00000	0.232889
$296$	1.00000	0.0581238
$297$	− 1.00000i	− 0.0580259i
$298$	6.00000	0.347571
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	− 6.00000i	− 0.345834i
$302$	−6.00000	−0.345261
$303$	4.00000	0.229794
$304$	5.00000i	0.286770i
$305$	2.00000i	0.114520i
$306$	0	0
$307$	18.0000i	1.02731i	0.857996	+	0.513657i	$0.171710\pi$
$307$	18.0000i	1.02731i	−0.857996	+	0.513657i	$0.828290\pi$
$308$	−3.00000	−0.170941
$309$	−9.00000	−0.511992
$310$	10.0000i	0.567962i
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	−34.0000	−1.92179	−0.960897	−	0.276907i	$-0.910691\pi$
$313$	−34.0000	−1.92179	−0.960897	+	0.276907i	$0.910691\pi$
$314$	− 1.00000i	− 0.0564333i
$315$	−3.00000	−0.169031
$316$	10.0000	0.562544
$317$	− 19.0000i	− 1.06715i	−0.845754	−	0.533573i	$-0.820849\pi$
$317$	− 19.0000i	− 1.06715i	0.845754	−	0.533573i	$-0.179151\pi$
$318$	− 13.0000i	− 0.729004i
$319$	0	0
$320$	1.00000i	0.0559017i
$321$	6.00000	0.334887
$322$	−12.0000	−0.668734
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−20.0000	−1.10770
$327$	− 10.0000i	− 0.553001i
$328$	6.00000	0.331295
$329$	−27.0000	−1.48856
$330$	1.00000i	0.0550482i
$331$	− 28.0000i	− 1.53902i	−0.638635	−	0.769510i	$-0.720501\pi$
$331$	− 28.0000i	− 1.53902i	0.638635	−	0.769510i	$-0.279499\pi$
$332$	12.0000i	0.658586i
$333$	− 1.00000i	− 0.0547997i
$334$	−13.0000	−0.711328
$335$	12.0000	0.655630
$336$	3.00000i	0.163663i
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	0	0
$339$	−16.0000	−0.869001
$340$	0	0
$341$	10.0000	0.541530
$342$	5.00000	0.270369
$343$	− 15.0000i	− 0.809924i
$344$	2.00000i	0.107833i
$345$	4.00000i	0.215353i
$346$	− 9.00000i	− 0.483843i
$347$	−28.0000	−1.50312	−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000	−1.50312	−0.751559	+	0.659665i	$0.770698\pi$
$348$	0	0
$349$	36.0000i	1.92704i	0.267644	+	0.963518i	$0.413755\pi$
$349$	36.0000i	1.92704i	−0.267644	+	0.963518i	$0.586245\pi$
$350$	3.00000	0.160357
$351$	0	0
$352$	1.00000	0.0533002
$353$	36.0000i	1.91609i	0.286623	+	0.958043i	$0.407467\pi$
$353$	36.0000i	1.91609i	−0.286623	+	0.958043i	$0.592533\pi$
$354$	−4.00000	−0.212598
$355$	2.00000	0.106149
$356$	− 1.00000i	− 0.0529999i
$357$	0	0
$358$	12.0000i	0.634220i
$359$	34.0000i	1.79445i	0.441572	+	0.897226i	$0.354421\pi$
$359$	34.0000i	1.79445i	−0.441572	+	0.897226i	$0.645579\pi$
$360$	1.00000	0.0527046
$361$	−6.00000	−0.315789
$362$	6.00000i	0.315353i
$363$	−10.0000	−0.524864
$364$	0	0
$365$	−16.0000	−0.837478
$366$	− 2.00000i	− 0.104542i
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	4.00000	0.208514
$369$	− 6.00000i	− 0.312348i
$370$	1.00000i	0.0519875i
$371$	39.0000i	2.02478i
$372$	− 10.0000i	− 0.518476i
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	0	0
$375$	− 1.00000i	− 0.0516398i
$376$	9.00000	0.464140
$377$	0	0
$378$	3.00000	0.154303
$379$	1.00000i	0.0513665i	0.999670	+	0.0256833i	$0.00817614\pi$
$379$	1.00000i	0.0513665i	−0.999670	+	0.0256833i	$0.991824\pi$
$380$	−5.00000	−0.256495
$381$	5.00000	0.256158
$382$	18.0000i	0.920960i
$383$	28.0000i	1.43073i	0.698749	+	0.715367i	$0.253740\pi$
$383$	28.0000i	1.43073i	−0.698749	+	0.715367i	$0.746260\pi$
$384$	− 1.00000i	− 0.0510310i
$385$	− 3.00000i	− 0.152894i
$386$	16.0000	0.814379
$387$	2.00000	0.101666
$388$	12.0000i	0.609208i
$389$	16.0000	0.811232	0.405616	−	0.914044i	$-0.367057\pi$
$389$	16.0000	0.811232	0.405616	+	0.914044i	$0.367057\pi$
$390$	0	0
$391$	0	0
$392$	− 2.00000i	− 0.101015i
$393$	15.0000	0.756650
$394$	15.0000	0.755689
$395$	10.0000i	0.503155i
$396$	− 1.00000i	− 0.0502519i
$397$	− 33.0000i	− 1.65622i	−0.560564	−	0.828111i	$-0.689416\pi$
$397$	− 33.0000i	− 1.65622i	0.560564	−	0.828111i	$-0.310584\pi$
$398$	− 2.00000i	− 0.100251i
$399$	−15.0000	−0.750939
$400$	−1.00000	−0.0500000
$401$	25.0000i	1.24844i	0.781248	+	0.624220i	$0.214583\pi$
$401$	25.0000i	1.24844i	−0.781248	+	0.624220i	$0.785417\pi$
$402$	−12.0000	−0.598506
$403$	0	0
$404$	4.00000	0.199007
$405$	− 1.00000i	− 0.0496904i
$406$	0	0
$407$	1.00000	0.0495682
$408$	0	0
$409$	− 17.0000i	− 0.840596i	−0.907386	−	0.420298i	$-0.861926\pi$
$409$	− 17.0000i	− 0.840596i	0.907386	−	0.420298i	$-0.138074\pi$
$410$	6.00000i	0.296319i
$411$	− 16.0000i	− 0.789222i
$412$	−9.00000	−0.443398
$413$	12.0000	0.590481
$414$	− 4.00000i	− 0.196589i
$415$	−12.0000	−0.589057
$416$	0	0
$417$	9.00000	0.440732
$418$	5.00000i	0.244558i
$419$	−28.0000	−1.36789	−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000	−1.36789	−0.683945	+	0.729534i	$0.739737\pi$
$420$	−3.00000	−0.146385
$421$	− 20.0000i	− 0.974740i	−0.873195	−	0.487370i	$-0.837956\pi$
$421$	− 20.0000i	− 0.974740i	0.873195	−	0.487370i	$-0.162044\pi$
$422$	15.0000i	0.730189i
$423$	− 9.00000i	− 0.437595i
$424$	− 13.0000i	− 0.631336i
$425$	0	0
$426$	−2.00000	−0.0969003
$427$	6.00000i	0.290360i
$428$	6.00000	0.290021
$429$	0	0
$430$	−2.00000	−0.0964486
$431$	− 36.0000i	− 1.73406i	−0.498257	−	0.867029i	$-0.666026\pi$
$431$	− 36.0000i	− 1.73406i	0.498257	−	0.867029i	$-0.333974\pi$
$432$	−1.00000	−0.0481125
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	30.0000i	1.44005i
$435$	0	0
$436$	− 10.0000i	− 0.478913i
$437$	20.0000i	0.956730i
$438$	16.0000	0.764510
$439$	10.0000	0.477274	0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000	0.477274	0.238637	+	0.971109i	$0.423299\pi$
$440$	1.00000i	0.0476731i
$441$	−2.00000	−0.0952381
$442$	0	0
$443$	−18.0000	−0.855206	−0.427603	−	0.903967i	$-0.640642\pi$
$443$	−18.0000	−0.855206	−0.427603	+	0.903967i	$0.640642\pi$
$444$	− 1.00000i	− 0.0474579i
$445$	1.00000	0.0474045
$446$	−11.0000	−0.520865
$447$	− 6.00000i	− 0.283790i
$448$	3.00000i	0.141737i
$449$	− 15.0000i	− 0.707894i	−0.935266	−	0.353947i	$-0.884839\pi$
$449$	− 15.0000i	− 0.707894i	0.935266	−	0.353947i	$-0.115161\pi$
$450$	1.00000i	0.0471405i
$451$	6.00000	0.282529
$452$	−16.0000	−0.752577
$453$	6.00000i	0.281905i
$454$	20.0000	0.938647
$455$	0	0
$456$	5.00000	0.234146
$457$	− 22.0000i	− 1.02912i	−0.857455	−	0.514558i	$-0.827956\pi$
$457$	− 22.0000i	− 1.02912i	0.857455	−	0.514558i	$-0.172044\pi$
$458$	−10.0000	−0.467269
$459$	0	0
$460$	4.00000i	0.186501i
$461$	12.0000i	0.558896i	0.960161	+	0.279448i	$0.0901514\pi$
$461$	12.0000i	0.558896i	−0.960161	+	0.279448i	$0.909849\pi$
$462$	3.00000i	0.139573i
$463$	− 16.0000i	− 0.743583i	−0.928316	−	0.371792i	$-0.878744\pi$
$463$	− 16.0000i	− 0.743583i	0.928316	−	0.371792i	$-0.121256\pi$
$464$	0	0
$465$	10.0000	0.463739
$466$	10.0000i	0.463241i
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	0	0
$469$	36.0000	1.66233
$470$	9.00000i	0.415139i
$471$	−1.00000	−0.0460776
$472$	−4.00000	−0.184115
$473$	2.00000i	0.0919601i
$474$	− 10.0000i	− 0.459315i
$475$	− 5.00000i	− 0.229416i
$476$	0	0
$477$	−13.0000	−0.595229
$478$	2.00000	0.0914779
$479$	− 8.00000i	− 0.365529i	−0.983157	−	0.182765i	$-0.941495\pi$
$479$	− 8.00000i	− 0.365529i	0.983157	−	0.182765i	$-0.0585046\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	15.0000	0.683231
$483$	12.0000i	0.546019i
$484$	−10.0000	−0.454545
$485$	−12.0000	−0.544892
$486$	1.00000i	0.0453609i
$487$	− 35.0000i	− 1.58600i	−0.609221	−	0.793001i	$-0.708518\pi$
$487$	− 35.0000i	− 1.58600i	0.609221	−	0.793001i	$-0.291482\pi$
$488$	− 2.00000i	− 0.0905357i
$489$	20.0000i	0.904431i
$490$	2.00000	0.0903508
$491$	−25.0000	−1.12823	−0.564117	−	0.825695i	$-0.690783\pi$
$491$	−25.0000	−1.12823	−0.564117	+	0.825695i	$0.690783\pi$
$492$	− 6.00000i	− 0.270501i
$493$	0	0
$494$	0	0
$495$	1.00000	0.0449467
$496$	− 10.0000i	− 0.449013i
$497$	6.00000	0.269137
$498$	12.0000	0.537733
$499$	− 20.0000i	− 0.895323i	−0.894203	−	0.447661i	$-0.852257\pi$
$499$	− 20.0000i	− 0.895323i	0.894203	−	0.447661i	$-0.147743\pi$
$500$	− 1.00000i	− 0.0447214i
$501$	13.0000i	0.580797i
$502$	11.0000i	0.490954i
$503$	−1.00000	−0.0445878	−0.0222939	−	0.999751i	$-0.507097\pi$
$503$	−1.00000	−0.0445878	−0.0222939	+	0.999751i	$0.507097\pi$
$504$	3.00000	0.133631
$505$	4.00000i	0.177998i
$506$	4.00000	0.177822
$507$	0	0
$508$	5.00000	0.221839
$509$	− 18.0000i	− 0.797836i	−0.916987	−	0.398918i	$-0.869386\pi$
$509$	− 18.0000i	− 0.797836i	0.916987	−	0.398918i	$-0.130614\pi$
$510$	0	0
$511$	−48.0000	−2.12339
$512$	− 1.00000i	− 0.0441942i
$513$	− 5.00000i	− 0.220755i
$514$	− 24.0000i	− 1.05859i
$515$	− 9.00000i	− 0.396587i
$516$	2.00000	0.0880451
$517$	9.00000	0.395820
$518$	3.00000i	0.131812i
$519$	−9.00000	−0.395056
$520$	0	0
$521$	33.0000	1.44576	0.722878	−	0.690976i	$-0.242819\pi$
$521$	33.0000	1.44576	0.722878	+	0.690976i	$0.242819\pi$
$522$	0	0
$523$	6.00000	0.262362	0.131181	−	0.991358i	$-0.458123\pi$
$523$	6.00000	0.262362	0.131181	+	0.991358i	$0.458123\pi$
$524$	15.0000	0.655278
$525$	− 3.00000i	− 0.130931i
$526$	− 3.00000i	− 0.130806i
$527$	0	0
$528$	− 1.00000i	− 0.0435194i
$529$	−7.00000	−0.304348
$530$	13.0000	0.564684
$531$	4.00000i	0.173585i
$532$	−15.0000	−0.650332
$533$	0	0
$534$	−1.00000	−0.0432742
$535$	6.00000i	0.259403i
$536$	−12.0000	−0.518321
$537$	12.0000	0.517838
$538$	20.0000i	0.862261i
$539$	− 2.00000i	− 0.0861461i
$540$	− 1.00000i	− 0.0430331i
$541$	− 2.00000i	− 0.0859867i	−0.999075	−	0.0429934i	$-0.986311\pi$
$541$	− 2.00000i	− 0.0859867i	0.999075	−	0.0429934i	$-0.0136894\pi$
$542$	−24.0000	−1.03089
$543$	6.00000	0.257485
$544$	0	0
$545$	10.0000	0.428353
$546$	0	0
$547$	34.0000	1.45374	0.726868	−	0.686778i	$-0.240975\pi$
$547$	34.0000	1.45374	0.726868	+	0.686778i	$0.240975\pi$
$548$	− 16.0000i	− 0.683486i
$549$	−2.00000	−0.0853579
$550$	−1.00000	−0.0426401
$551$	0	0
$552$	− 4.00000i	− 0.170251i
$553$	30.0000i	1.27573i
$554$	23.0000i	0.977176i
$555$	1.00000	0.0424476
$556$	9.00000	0.381685
$557$	− 3.00000i	− 0.127114i	−0.997978	−	0.0635570i	$-0.979756\pi$
$557$	− 3.00000i	− 0.127114i	0.997978	−	0.0635570i	$-0.0202445\pi$
$558$	−10.0000	−0.423334
$559$	0	0
$560$	−3.00000	−0.126773
$561$	0	0
$562$	10.0000	0.421825
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	− 9.00000i	− 0.378968i
$565$	− 16.0000i	− 0.673125i
$566$	2.00000i	0.0840663i
$567$	− 3.00000i	− 0.125988i
$568$	−2.00000	−0.0839181
$569$	11.0000	0.461144	0.230572	−	0.973055i	$-0.425940\pi$
$569$	11.0000	0.461144	0.230572	+	0.973055i	$0.425940\pi$
$570$	5.00000i	0.209427i
$571$	−7.00000	−0.292941	−0.146470	−	0.989215i	$-0.546791\pi$
$571$	−7.00000	−0.292941	−0.146470	+	0.989215i	$0.546791\pi$
$572$	0	0
$573$	18.0000	0.751961
$574$	18.0000i	0.751305i
$575$	−4.00000	−0.166812
$576$	−1.00000	−0.0416667
$577$	− 18.0000i	− 0.749350i	−0.927156	−	0.374675i	$-0.877754\pi$
$577$	− 18.0000i	− 0.749350i	0.927156	−	0.374675i	$-0.122246\pi$
$578$	17.0000i	0.707107i
$579$	− 16.0000i	− 0.664937i
$580$	0	0
$581$	−36.0000	−1.49353
$582$	12.0000	0.497416
$583$	− 13.0000i	− 0.538405i
$584$	16.0000	0.662085
$585$	0	0
$586$	13.0000	0.537025
$587$	18.0000i	0.742940i	0.928445	+	0.371470i	$0.121146\pi$
$587$	18.0000i	0.742940i	−0.928445	+	0.371470i	$0.878854\pi$
$588$	−2.00000	−0.0824786
$589$	50.0000	2.06021
$590$	− 4.00000i	− 0.164677i
$591$	− 15.0000i	− 0.617018i
$592$	− 1.00000i	− 0.0410997i
$593$	28.0000i	1.14982i	0.818216	+	0.574911i	$0.194963\pi$
$593$	28.0000i	1.14982i	−0.818216	+	0.574911i	$0.805037\pi$
$594$	−1.00000	−0.0410305
$595$	0	0
$596$	− 6.00000i	− 0.245770i
$597$	−2.00000	−0.0818546
$598$	0	0
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	1.00000i	0.0408248i
$601$	−27.0000	−1.10135	−0.550676	−	0.834719i	$-0.685630\pi$
$601$	−27.0000	−1.10135	−0.550676	+	0.834719i	$0.685630\pi$
$602$	−6.00000	−0.244542
$603$	12.0000i	0.488678i
$604$	6.00000i	0.244137i
$605$	− 10.0000i	− 0.406558i
$606$	− 4.00000i	− 0.162489i
$607$	37.0000	1.50178	0.750892	−	0.660425i	$-0.229624\pi$
$607$	37.0000	1.50178	0.750892	+	0.660425i	$0.229624\pi$
$608$	5.00000	0.202777
$609$	0	0
$610$	2.00000	0.0809776
$611$	0	0
$612$	0	0
$613$	− 23.0000i	− 0.928961i	−0.885583	−	0.464481i	$-0.846241\pi$
$613$	− 23.0000i	− 0.928961i	0.885583	−	0.464481i	$-0.153759\pi$
$614$	18.0000	0.726421
$615$	6.00000	0.241943
$616$	3.00000i	0.120873i
$617$	6.00000i	0.241551i	0.992680	+	0.120775i	$0.0385381\pi$
$617$	6.00000i	0.241551i	−0.992680	+	0.120775i	$0.961462\pi$
$618$	9.00000i	0.362033i
$619$	− 31.0000i	− 1.24600i	−0.782224	−	0.622998i	$-0.785915\pi$
$619$	− 31.0000i	− 1.24600i	0.782224	−	0.622998i	$-0.214085\pi$
$620$	10.0000	0.401610
$621$	−4.00000	−0.160514
$622$	− 12.0000i	− 0.481156i
$623$	3.00000	0.120192
$624$	0	0
$625$	1.00000	0.0400000
$626$	34.0000i	1.35891i
$627$	5.00000	0.199681
$628$	−1.00000	−0.0399043
$629$	0	0
$630$	3.00000i	0.119523i
$631$	− 4.00000i	− 0.159237i	−0.996825	−	0.0796187i	$-0.974630\pi$
$631$	− 4.00000i	− 0.159237i	0.996825	−	0.0796187i	$-0.0253703\pi$
$632$	− 10.0000i	− 0.397779i
$633$	15.0000	0.596196
$634$	−19.0000	−0.754586
$635$	5.00000i	0.198419i
$636$	−13.0000	−0.515484
$637$	0	0
$638$	0	0
$639$	2.00000i	0.0791188i
$640$	1.00000	0.0395285
$641$	9.00000	0.355479	0.177739	−	0.984078i	$-0.443122\pi$
$641$	9.00000	0.355479	0.177739	+	0.984078i	$0.443122\pi$
$642$	− 6.00000i	− 0.236801i
$643$	44.0000i	1.73519i	0.497271	+	0.867595i	$0.334335\pi$
$643$	44.0000i	1.73519i	−0.497271	+	0.867595i	$0.665665\pi$
$644$	12.0000i	0.472866i
$645$	2.00000i	0.0787499i
$646$	0	0
$647$	−9.00000	−0.353827	−0.176913	−	0.984226i	$-0.556611\pi$
$647$	−9.00000	−0.353827	−0.176913	+	0.984226i	$0.556611\pi$
$648$	1.00000i	0.0392837i
$649$	−4.00000	−0.157014
$650$	0	0
$651$	30.0000	1.17579
$652$	20.0000i	0.783260i
$653$	41.0000	1.60445	0.802227	−	0.597019i	$-0.203648\pi$
$653$	41.0000	1.60445	0.802227	+	0.597019i	$0.203648\pi$
$654$	−10.0000	−0.391031
$655$	15.0000i	0.586098i
$656$	− 6.00000i	− 0.234261i
$657$	− 16.0000i	− 0.624219i
$658$	27.0000i	1.05257i
$659$	4.00000	0.155818	0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000	0.155818	0.0779089	+	0.996960i	$0.475176\pi$
$660$	1.00000	0.0389249
$661$	14.0000i	0.544537i	0.962221	+	0.272268i	$0.0877739\pi$
$661$	14.0000i	0.544537i	−0.962221	+	0.272268i	$0.912226\pi$
$662$	−28.0000	−1.08825
$663$	0	0
$664$	12.0000	0.465690
$665$	− 15.0000i	− 0.581675i
$666$	−1.00000	−0.0387492
$667$	0	0
$668$	13.0000i	0.502985i
$669$	11.0000i	0.425285i
$670$	− 12.0000i	− 0.463600i
$671$	− 2.00000i	− 0.0772091i
$672$	3.00000	0.115728
$673$	32.0000	1.23351	0.616755	−	0.787155i	$-0.288447\pi$
$673$	32.0000	1.23351	0.616755	+	0.787155i	$0.288447\pi$
$674$	2.00000i	0.0770371i
$675$	1.00000	0.0384900
$676$	0	0
$677$	−26.0000	−0.999261	−0.499631	−	0.866239i	$-0.666531\pi$
$677$	−26.0000	−0.999261	−0.499631	+	0.866239i	$0.666531\pi$
$678$	16.0000i	0.614476i
$679$	−36.0000	−1.38155
$680$	0	0
$681$	− 20.0000i	− 0.766402i
$682$	− 10.0000i	− 0.382920i
$683$	− 26.0000i	− 0.994862i	−0.867503	−	0.497431i	$-0.834277\pi$
$683$	− 26.0000i	− 0.994862i	0.867503	−	0.497431i	$-0.165723\pi$
$684$	− 5.00000i	− 0.191180i
$685$	16.0000	0.611329
$686$	−15.0000	−0.572703
$687$	10.0000i	0.381524i
$688$	2.00000	0.0762493
$689$	0	0
$690$	4.00000	0.152277
$691$	− 33.0000i	− 1.25538i	−0.778464	−	0.627690i	$-0.784001\pi$
$691$	− 33.0000i	− 1.25538i	0.778464	−	0.627690i	$-0.215999\pi$
$692$	−9.00000	−0.342129
$693$	3.00000	0.113961
$694$	28.0000i	1.06287i
$695$	9.00000i	0.341389i
$696$	0	0
$697$	0	0
$698$	36.0000	1.36262
$699$	10.0000	0.378235
$700$	− 3.00000i	− 0.113389i
$701$	−4.00000	−0.151078	−0.0755390	−	0.997143i	$-0.524068\pi$
$701$	−4.00000	−0.151078	−0.0755390	+	0.997143i	$0.524068\pi$
$702$	0	0
$703$	5.00000	0.188579
$704$	− 1.00000i	− 0.0376889i
$705$	9.00000	0.338960
$706$	36.0000	1.35488
$707$	12.0000i	0.451306i
$708$	4.00000i	0.150329i
$709$	4.00000i	0.150223i	0.997175	+	0.0751116i	$0.0239313\pi$
$709$	4.00000i	0.150223i	−0.997175	+	0.0751116i	$0.976069\pi$
$710$	− 2.00000i	− 0.0750587i
$711$	−10.0000	−0.375029
$712$	−1.00000	−0.0374766
$713$	− 40.0000i	− 1.49801i
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	− 2.00000i	− 0.0746914i
$718$	34.0000	1.26887
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	− 1.00000i	− 0.0372678i
$721$	− 27.0000i	− 1.00553i
$722$	6.00000i	0.223297i
$723$	− 15.0000i	− 0.557856i
$724$	6.00000	0.222988
$725$	0	0
$726$	10.0000i	0.371135i
$727$	37.0000	1.37225	0.686127	−	0.727482i	$-0.259309\pi$
$727$	37.0000	1.37225	0.686127	+	0.727482i	$0.259309\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	16.0000i	0.592187i
$731$	0	0
$732$	−2.00000	−0.0739221
$733$	5.00000i	0.184679i	0.995728	+	0.0923396i	$0.0294345\pi$
$733$	5.00000i	0.184679i	−0.995728	+	0.0923396i	$0.970565\pi$
$734$	0	0
$735$	− 2.00000i	− 0.0737711i
$736$	− 4.00000i	− 0.147442i
$737$	−12.0000	−0.442026
$738$	−6.00000	−0.220863
$739$	35.0000i	1.28750i	0.765238	+	0.643748i	$0.222621\pi$
$739$	35.0000i	1.28750i	−0.765238	+	0.643748i	$0.777379\pi$
$740$	1.00000	0.0367607
$741$	0	0
$742$	39.0000	1.43174
$743$	− 24.0000i	− 0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$	− 24.0000i	− 0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$	−10.0000	−0.366618
$745$	6.00000	0.219823
$746$	− 10.0000i	− 0.366126i
$747$	− 12.0000i	− 0.439057i
$748$	0	0
$749$	18.0000i	0.657706i
$750$	−1.00000	−0.0365148
$751$	−44.0000	−1.60558	−0.802791	−	0.596260i	$-0.796653\pi$
$751$	−44.0000	−1.60558	−0.802791	+	0.596260i	$0.796653\pi$
$752$	− 9.00000i	− 0.328196i
$753$	11.0000	0.400862
$754$	0	0
$755$	−6.00000	−0.218362
$756$	− 3.00000i	− 0.109109i
$757$	39.0000	1.41748	0.708740	−	0.705470i	$-0.249264\pi$
$757$	39.0000	1.41748	0.708740	+	0.705470i	$0.249264\pi$
$758$	1.00000	0.0363216
$759$	− 4.00000i	− 0.145191i
$760$	5.00000i	0.181369i
$761$	45.0000i	1.63125i	0.578582	+	0.815624i	$0.303606\pi$
$761$	45.0000i	1.63125i	−0.578582	+	0.815624i	$0.696394\pi$
$762$	− 5.00000i	− 0.181131i
$763$	30.0000	1.08607
$764$	18.0000	0.651217
$765$	0	0
$766$	28.0000	1.01168
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	26.0000i	0.937584i	0.883309	+	0.468792i	$0.155311\pi$
$769$	26.0000i	0.937584i	−0.883309	+	0.468792i	$0.844689\pi$
$770$	−3.00000	−0.108112
$771$	−24.0000	−0.864339
$772$	− 16.0000i	− 0.575853i
$773$	37.0000i	1.33080i	0.746488	+	0.665399i	$0.231738\pi$
$773$	37.0000i	1.33080i	−0.746488	+	0.665399i	$0.768262\pi$
$774$	− 2.00000i	− 0.0718885i
$775$	10.0000i	0.359211i
$776$	12.0000	0.430775
$777$	3.00000	0.107624
$778$	− 16.0000i	− 0.573628i
$779$	30.0000	1.07486
$780$	0	0
$781$	−2.00000	−0.0715656
$782$	0	0
$783$	0	0
$784$	−2.00000	−0.0714286
$785$	− 1.00000i	− 0.0356915i
$786$	− 15.0000i	− 0.535032i
$787$	16.0000i	0.570338i	0.958477	+	0.285169i	$0.0920498\pi$
$787$	16.0000i	0.570338i	−0.958477	+	0.285169i	$0.907950\pi$
$788$	− 15.0000i	− 0.534353i
$789$	−3.00000	−0.106803
$790$	10.0000	0.355784
$791$	− 48.0000i	− 1.70668i
$792$	−1.00000	−0.0355335
$793$	0	0
$794$	−33.0000	−1.17113
$795$	− 13.0000i	− 0.461062i
$796$	−2.00000	−0.0708881
$797$	14.0000	0.495905	0.247953	−	0.968772i	$-0.420242\pi$
$797$	14.0000	0.495905	0.247953	+	0.968772i	$0.420242\pi$
$798$	15.0000i	0.530994i
$799$	0	0
$800$	1.00000i	0.0353553i
$801$	1.00000i	0.0353333i
$802$	25.0000	0.882781
$803$	16.0000	0.564628
$804$	12.0000i	0.423207i
$805$	−12.0000	−0.422944
$806$	0	0
$807$	20.0000	0.704033
$808$	− 4.00000i	− 0.140720i
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	−1.00000	−0.0351364
$811$	− 33.0000i	− 1.15879i	−0.815048	−	0.579393i	$-0.803290\pi$
$811$	− 33.0000i	− 1.15879i	0.815048	−	0.579393i	$-0.196710\pi$
$812$	0	0
$813$	24.0000i	0.841717i
$814$	− 1.00000i	− 0.0350500i
$815$	−20.0000	−0.700569
$816$	0	0
$817$	10.0000i	0.349856i
$818$	−17.0000	−0.594391
$819$	0	0
$820$	6.00000	0.209529
$821$	− 18.0000i	− 0.628204i	−0.949389	−	0.314102i	$-0.898297\pi$
$821$	− 18.0000i	− 0.628204i	0.949389	−	0.314102i	$-0.101703\pi$
$822$	−16.0000	−0.558064
$823$	5.00000	0.174289	0.0871445	−	0.996196i	$-0.472226\pi$
$823$	5.00000	0.174289	0.0871445	+	0.996196i	$0.472226\pi$
$824$	9.00000i	0.313530i
$825$	1.00000i	0.0348155i
$826$	− 12.0000i	− 0.417533i
$827$	− 30.0000i	− 1.04320i	−0.853189	−	0.521601i	$-0.825335\pi$
$827$	− 30.0000i	− 1.04320i	0.853189	−	0.521601i	$-0.174665\pi$
$828$	−4.00000	−0.139010
$829$	44.0000	1.52818	0.764092	−	0.645108i	$-0.223188\pi$
$829$	44.0000	1.52818	0.764092	+	0.645108i	$0.223188\pi$
$830$	12.0000i	0.416526i
$831$	23.0000	0.797861
$832$	0	0
$833$	0	0
$834$	− 9.00000i	− 0.311645i
$835$	−13.0000	−0.449884
$836$	5.00000	0.172929
$837$	10.0000i	0.345651i
$838$	28.0000i	0.967244i
$839$	− 54.0000i	− 1.86429i	−0.362089	−	0.932144i	$-0.617936\pi$
$839$	− 54.0000i	− 1.86429i	0.362089	−	0.932144i	$-0.382064\pi$
$840$	3.00000i	0.103510i
$841$	−29.0000	−1.00000
$842$	−20.0000	−0.689246
$843$	− 10.0000i	− 0.344418i
$844$	15.0000	0.516321
$845$	0	0
$846$	−9.00000	−0.309426
$847$	− 30.0000i	− 1.03081i
$848$	−13.0000	−0.446422
$849$	2.00000	0.0686398
$850$	0	0
$851$	− 4.00000i	− 0.137118i
$852$	2.00000i	0.0685189i
$853$	− 14.0000i	− 0.479351i	−0.970853	−	0.239675i	$-0.922959\pi$
$853$	− 14.0000i	− 0.479351i	0.970853	−	0.239675i	$-0.0770410\pi$
$854$	6.00000	0.205316
$855$	5.00000	0.170996
$856$	− 6.00000i	− 0.205076i
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	19.0000	0.648272	0.324136	−	0.946011i	$-0.394927\pi$
$859$	19.0000	0.648272	0.324136	+	0.946011i	$0.394927\pi$
$860$	2.00000i	0.0681994i
$861$	18.0000	0.613438
$862$	−36.0000	−1.22616
$863$	48.0000i	1.63394i	0.576681	+	0.816970i	$0.304348\pi$
$863$	48.0000i	1.63394i	−0.576681	+	0.816970i	$0.695652\pi$
$864$	1.00000i	0.0340207i
$865$	− 9.00000i	− 0.306009i
$866$	16.0000i	0.543702i
$867$	17.0000	0.577350
$868$	30.0000	1.01827
$869$	− 10.0000i	− 0.339227i
$870$	0	0
$871$	0	0
$872$	−10.0000	−0.338643
$873$	− 12.0000i	− 0.406138i
$874$	20.0000	0.676510
$875$	3.00000	0.101419
$876$	− 16.0000i	− 0.540590i
$877$	− 38.0000i	− 1.28317i	−0.767052	−	0.641584i	$-0.778277\pi$
$877$	− 38.0000i	− 1.28317i	0.767052	−	0.641584i	$-0.221723\pi$
$878$	− 10.0000i	− 0.337484i
$879$	− 13.0000i	− 0.438479i
$880$	1.00000	0.0337100
$881$	−5.00000	−0.168454	−0.0842271	−	0.996447i	$-0.526842\pi$
$881$	−5.00000	−0.168454	−0.0842271	+	0.996447i	$0.526842\pi$
$882$	2.00000i	0.0673435i
$883$	−42.0000	−1.41341	−0.706706	−	0.707507i	$-0.749820\pi$
$883$	−42.0000	−1.41341	−0.706706	+	0.707507i	$0.749820\pi$
$884$	0	0
$885$	−4.00000	−0.134459
$886$	18.0000i	0.604722i
$887$	−13.0000	−0.436497	−0.218249	−	0.975893i	$-0.570034\pi$
$887$	−13.0000	−0.436497	−0.218249	+	0.975893i	$0.570034\pi$
$888$	−1.00000	−0.0335578
$889$	15.0000i	0.503084i
$890$	− 1.00000i	− 0.0335201i
$891$	1.00000i	0.0335013i
$892$	11.0000i	0.368307i
$893$	45.0000	1.50587
$894$	−6.00000	−0.200670
$895$	12.0000i	0.401116i
$896$	3.00000	0.100223
$897$	0	0
$898$	−15.0000	−0.500556
$899$	0	0
$900$	1.00000	0.0333333
$901$	0	0
$902$	− 6.00000i	− 0.199778i
$903$	6.00000i	0.199667i
$904$	16.0000i	0.532152i
$905$	6.00000i	0.199447i
$906$	6.00000	0.199337
$907$	−42.0000	−1.39459	−0.697294	−	0.716786i	$-0.745613\pi$
$907$	−42.0000	−1.39459	−0.697294	+	0.716786i	$0.745613\pi$
$908$	− 20.0000i	− 0.663723i
$909$	−4.00000	−0.132672
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	− 5.00000i	− 0.165567i
$913$	12.0000	0.397142
$914$	−22.0000	−0.727695
$915$	− 2.00000i	− 0.0661180i
$916$	10.0000i	0.330409i
$917$	45.0000i	1.48603i
$918$	0	0
$919$	34.0000	1.12156	0.560778	−	0.827966i	$-0.310502\pi$
$919$	34.0000	1.12156	0.560778	+	0.827966i	$0.310502\pi$
$920$	4.00000	0.131876
$921$	− 18.0000i	− 0.593120i
$922$	12.0000	0.395199
$923$	0	0
$924$	3.00000	0.0986928
$925$	1.00000i	0.0328798i
$926$	−16.0000	−0.525793
$927$	9.00000	0.295599
$928$	0	0
$929$	− 34.0000i	− 1.11550i	−0.830008	−	0.557752i	$-0.811664\pi$
$929$	− 34.0000i	− 1.11550i	0.830008	−	0.557752i	$-0.188336\pi$
$930$	− 10.0000i	− 0.327913i
$931$	− 10.0000i	− 0.327737i
$932$	10.0000	0.327561
$933$	−12.0000	−0.392862
$934$	− 36.0000i	− 1.17796i
$935$	0	0
$936$	0	0
$937$	−10.0000	−0.326686	−0.163343	−	0.986569i	$-0.552228\pi$
$937$	−10.0000	−0.326686	−0.163343	+	0.986569i	$0.552228\pi$
$938$	− 36.0000i	− 1.17544i
$939$	34.0000	1.10955
$940$	9.00000	0.293548
$941$	− 24.0000i	− 0.782378i	−0.920310	−	0.391189i	$-0.872064\pi$
$941$	− 24.0000i	− 0.782378i	0.920310	−	0.391189i	$-0.127936\pi$
$942$	1.00000i	0.0325818i
$943$	− 24.0000i	− 0.781548i
$944$	4.00000i	0.130189i
$945$	3.00000	0.0975900
$946$	2.00000	0.0650256
$947$	− 38.0000i	− 1.23483i	−0.786636	−	0.617417i	$-0.788179\pi$
$947$	− 38.0000i	− 1.23483i	0.786636	−	0.617417i	$-0.211821\pi$
$948$	−10.0000	−0.324785
$949$	0	0
$950$	−5.00000	−0.162221
$951$	19.0000i	0.616117i
$952$	0	0
$953$	−22.0000	−0.712650	−0.356325	−	0.934362i	$-0.615970\pi$
$953$	−22.0000	−0.712650	−0.356325	+	0.934362i	$0.615970\pi$
$954$	13.0000i	0.420891i
$955$	18.0000i	0.582466i
$956$	− 2.00000i	− 0.0646846i
$957$	0	0
$958$	−8.00000	−0.258468
$959$	48.0000	1.55000
$960$	− 1.00000i	− 0.0322749i
$961$	−69.0000	−2.22581
$962$	0	0
$963$	−6.00000	−0.193347
$964$	− 15.0000i	− 0.483117i
$965$	16.0000	0.515058
$966$	12.0000	0.386094
$967$	− 7.00000i	− 0.225105i	−0.993646	−	0.112552i	$-0.964097\pi$
$967$	− 7.00000i	− 0.225105i	0.993646	−	0.112552i	$-0.0359026\pi$
$968$	10.0000i	0.321412i
$969$	0	0
$970$	12.0000i	0.385297i
$971$	27.0000	0.866471	0.433236	−	0.901281i	$-0.357372\pi$
$971$	27.0000	0.866471	0.433236	+	0.901281i	$0.357372\pi$
$972$	1.00000	0.0320750
$973$	27.0000i	0.865580i
$974$	−35.0000	−1.12147
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	− 48.0000i	− 1.53566i	−0.640656	−	0.767828i	$-0.721338\pi$
$977$	− 48.0000i	− 1.53566i	0.640656	−	0.767828i	$-0.278662\pi$
$978$	20.0000	0.639529
$979$	−1.00000	−0.0319601
$980$	− 2.00000i	− 0.0638877i
$981$	10.0000i	0.319275i
$982$	25.0000i	0.797782i
$983$	53.0000i	1.69044i	0.534421	+	0.845219i	$0.320530\pi$
$983$	53.0000i	1.69044i	−0.534421	+	0.845219i	$0.679470\pi$
$984$	−6.00000	−0.191273
$985$	15.0000	0.477940
$986$	0	0
$987$	27.0000	0.859419
$988$	0	0
$989$	8.00000	0.254385
$990$	− 1.00000i	− 0.0317821i
$991$	38.0000	1.20711	0.603555	−	0.797321i	$-0.293750\pi$
$991$	38.0000	1.20711	0.603555	+	0.797321i	$0.293750\pi$
$992$	−10.0000	−0.317500
$993$	28.0000i	0.888553i
$994$	− 6.00000i	− 0.190308i
$995$	− 2.00000i	− 0.0634043i
$996$	− 12.0000i	− 0.380235i
$997$	−7.00000	−0.221692	−0.110846	−	0.993838i	$-0.535356\pi$
$997$	−7.00000	−0.221692	−0.110846	+	0.993838i	$0.535356\pi$
$998$	−20.0000	−0.633089
$999$	1.00000i	0.0316386i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.b.a.1351.1		2
13.5	odd	4		5070.2.a.c.1.1		1
13.7	odd	12		390.2.i.b.211.1	yes	2
13.8	odd	4		5070.2.a.q.1.1		1
13.11	odd	12		390.2.i.b.61.1	✓	2
13.12	even	2	inner	5070.2.b.a.1351.2		2
39.11	even	12		1170.2.i.j.451.1		2
39.20	even	12		1170.2.i.j.991.1		2
65.7	even	12		1950.2.z.i.1849.2		4
65.24	odd	12		1950.2.i.o.451.1		2
65.33	even	12		1950.2.z.i.1849.1		4
65.37	even	12		1950.2.z.i.1699.1		4
65.59	odd	12		1950.2.i.o.601.1		2
65.63	even	12		1950.2.z.i.1699.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.i.b.61.1	✓	2	13.11	odd	12
390.2.i.b.211.1	yes	2	13.7	odd	12
1170.2.i.j.451.1		2	39.11	even	12
1170.2.i.j.991.1		2	39.20	even	12
1950.2.i.o.451.1		2	65.24	odd	12
1950.2.i.o.601.1		2	65.59	odd	12
1950.2.z.i.1699.1		4	65.37	even	12
1950.2.z.i.1699.2		4	65.63	even	12
1950.2.z.i.1849.1		4	65.33	even	12
1950.2.z.i.1849.2		4	65.7	even	12
5070.2.a.c.1.1		1	13.5	odd	4
5070.2.a.q.1.1		1	13.8	odd	4
5070.2.b.a.1351.1		2	1.1	even	1	trivial
5070.2.b.a.1351.2		2	13.12	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} -3.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} -3.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000i q^{11} +1.00000 q^{12} -3.00000 q^{14} +1.00000i q^{15} +1.00000 q^{16} -1.00000i q^{18} +5.00000i q^{19} +1.00000i q^{20} +3.00000i q^{21} +1.00000 q^{22} +4.00000 q^{23} -1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} +3.00000i q^{28} +1.00000 q^{30} -10.0000i q^{31} -1.00000i q^{32} -1.00000i q^{33} -3.00000 q^{35} -1.00000 q^{36} -1.00000i q^{37} +5.00000 q^{38} +1.00000 q^{40} -6.00000i q^{41} +3.00000 q^{42} +2.00000 q^{43} -1.00000i q^{44} -1.00000i q^{45} -4.00000i q^{46} -9.00000i q^{47} -1.00000 q^{48} -2.00000 q^{49} +1.00000i q^{50} -13.0000 q^{53} +1.00000i q^{54} +1.00000 q^{55} +3.00000 q^{56} -5.00000i q^{57} +4.00000i q^{59} -1.00000i q^{60} -2.00000 q^{61} -10.0000 q^{62} -3.00000i q^{63} -1.00000 q^{64} -1.00000 q^{66} +12.0000i q^{67} -4.00000 q^{69} +3.00000i q^{70} +2.00000i q^{71} +1.00000i q^{72} -16.0000i q^{73} -1.00000 q^{74} +1.00000 q^{75} -5.00000i q^{76} +3.00000 q^{77} -10.0000 q^{79} -1.00000i q^{80} +1.00000 q^{81} -6.00000 q^{82} -12.0000i q^{83} -3.00000i q^{84} -2.00000i q^{86} -1.00000 q^{88} +1.00000i q^{89} -1.00000 q^{90} -4.00000 q^{92} +10.0000i q^{93} -9.00000 q^{94} +5.00000 q^{95} +1.00000i q^{96} -12.0000i q^{97} +2.00000i q^{98} +1.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} - 2 q^{10} + 2 q^{12} - 6 q^{14} + 2 q^{16} + 2 q^{22} + 8 q^{23} - 2 q^{25} - 2 q^{27} + 2 q^{30} - 6 q^{35} - 2 q^{36} + 10 q^{38} + 2 q^{40} + 6 q^{42} + 4 q^{43} - 2 q^{48} - 4 q^{49} - 26 q^{53} + 2 q^{55} + 6 q^{56} - 4 q^{61} - 20 q^{62} - 2 q^{64} - 2 q^{66} - 8 q^{69} - 2 q^{74} + 2 q^{75} + 6 q^{77} - 20 q^{79} + 2 q^{81} - 12 q^{82} - 2 q^{88} - 2 q^{90} - 8 q^{92} - 18 q^{94} + 10 q^{95}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 - 2 * q^10 + 2 * q^12 - 6 * q^14 + 2 * q^16 + 2 * q^22 + 8 * q^23 - 2 * q^25 - 2 * q^27 + 2 * q^30 - 6 * q^35 - 2 * q^36 + 10 * q^38 + 2 * q^40 + 6 * q^42 + 4 * q^43 - 2 * q^48 - 4 * q^49 - 26 * q^53 + 2 * q^55 + 6 * q^56 - 4 * q^61 - 20 * q^62 - 2 * q^64 - 2 * q^66 - 8 * q^69 - 2 * q^74 + 2 * q^75 + 6 * q^77 - 20 * q^79 + 2 * q^81 - 12 * q^82 - 2 * q^88 - 2 * q^90 - 8 * q^92 - 18 * q^94 + 10 * q^95

Introduction

L-functions

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