LMFDB - Embedded newform 5070.2.b.g.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.g.1351.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000i q^{11} +1.00000 q^{12} +2.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} -2.00000 q^{17} -1.00000i q^{18} -6.00000i q^{19} -1.00000i q^{20} -2.00000i q^{21} +1.00000 q^{22} +3.00000 q^{23} -1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} -2.00000i q^{28} -1.00000 q^{29} -1.00000 q^{30} +3.00000i q^{31} -1.00000i q^{32} -1.00000i q^{33} +2.00000i q^{34} -2.00000 q^{35} -1.00000 q^{36} -5.00000i q^{37} -6.00000 q^{38} -1.00000 q^{40} -10.0000i q^{41} -2.00000 q^{42} -5.00000 q^{43} -1.00000i q^{44} +1.00000i q^{45} -3.00000i q^{46} +3.00000i q^{47} -1.00000 q^{48} +3.00000 q^{49} +1.00000i q^{50} +2.00000 q^{51} +14.0000 q^{53} +1.00000i q^{54} -1.00000 q^{55} -2.00000 q^{56} +6.00000i q^{57} +1.00000i q^{58} -5.00000i q^{59} +1.00000i q^{60} -10.0000 q^{61} +3.00000 q^{62} +2.00000i q^{63} -1.00000 q^{64} -1.00000 q^{66} +2.00000 q^{68} -3.00000 q^{69} +2.00000i q^{70} -4.00000i q^{71} +1.00000i q^{72} -2.00000i q^{73} -5.00000 q^{74} +1.00000 q^{75} +6.00000i q^{76} -2.00000 q^{77} +5.00000 q^{79} +1.00000i q^{80} +1.00000 q^{81} -10.0000 q^{82} +6.00000i q^{83} +2.00000i q^{84} -2.00000i q^{85} +5.00000i q^{86} +1.00000 q^{87} -1.00000 q^{88} +10.0000i q^{89} +1.00000 q^{90} -3.00000 q^{92} -3.00000i q^{93} +3.00000 q^{94} +6.00000 q^{95} +1.00000i q^{96} +10.0000i q^{97} -3.00000i q^{98} +1.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 4 q^{14} + 2 q^{16} - 4 q^{17} + 2 q^{22} + 6 q^{23} - 2 q^{25} - 2 q^{27} - 2 q^{29} - 2 q^{30} - 4 q^{35} - 2 q^{36} - 12 q^{38} - 2 q^{40} - 4 q^{42} - 10 q^{43} - 2 q^{48} + 6 q^{49} + 4 q^{51} + 28 q^{53} - 2 q^{55} - 4 q^{56} - 20 q^{61} + 6 q^{62} - 2 q^{64} - 2 q^{66} + 4 q^{68} - 6 q^{69} - 10 q^{74} + 2 q^{75} - 4 q^{77} + 10 q^{79} + 2 q^{81} - 20 q^{82} + 2 q^{87} - 2 q^{88} + 2 q^{90} - 6 q^{92} + 6 q^{94} + 12 q^{95}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 + 2 * q^10 + 2 * q^12 + 4 * q^14 + 2 * q^16 - 4 * q^17 + 2 * q^22 + 6 * q^23 - 2 * q^25 - 2 * q^27 - 2 * q^29 - 2 * q^30 - 4 * q^35 - 2 * q^36 - 12 * q^38 - 2 * q^40 - 4 * q^42 - 10 * q^43 - 2 * q^48 + 6 * q^49 + 4 * q^51 + 28 * q^53 - 2 * q^55 - 4 * q^56 - 20 * q^61 + 6 * q^62 - 2 * q^64 - 2 * q^66 + 4 * q^68 - 6 * q^69 - 10 * q^74 + 2 * q^75 - 4 * q^77 + 10 * q^79 + 2 * q^81 - 20 * q^82 + 2 * q^87 - 2 * q^88 + 2 * q^90 - 6 * q^92 + 6 * q^94 + 12 * 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
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	1.00000i	0.353553i
$9$	1.00000	0.333333
$10$	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
$14$	2.00000	0.534522
$15$	− 1.00000i	− 0.258199i
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	− 1.00000i	− 0.235702i
$19$	− 6.00000i	− 1.37649i	−0.725476	−	0.688247i	$-0.758380\pi$
$19$	− 6.00000i	− 1.37649i	0.725476	−	0.688247i	$-0.241620\pi$
$20$	− 1.00000i	− 0.223607i
$21$	− 2.00000i	− 0.436436i
$22$	1.00000	0.213201
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	− 1.00000i	− 0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	− 2.00000i	− 0.377964i
$29$	−1.00000	−0.185695	−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000	−0.185695	−0.0928477	+	0.995680i	$0.529597\pi$
$30$	−1.00000	−0.182574
$31$	3.00000i	0.538816i	0.963026	+	0.269408i	$0.0868280\pi$
$31$	3.00000i	0.538816i	−0.963026	+	0.269408i	$0.913172\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 1.00000i	− 0.174078i
$34$	2.00000i	0.342997i
$35$	−2.00000	−0.338062
$36$	−1.00000	−0.166667
$37$	− 5.00000i	− 0.821995i	−0.911636	−	0.410997i	$-0.865181\pi$
$37$	− 5.00000i	− 0.821995i	0.911636	−	0.410997i	$-0.134819\pi$
$38$	−6.00000	−0.973329
$39$	0	0
$40$	−1.00000	−0.158114
$41$	− 10.0000i	− 1.56174i	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	− 10.0000i	− 1.56174i	0.624695	−	0.780869i	$-0.285223\pi$
$42$	−2.00000	−0.308607
$43$	−5.00000	−0.762493	−0.381246	−	0.924473i	$-0.624505\pi$
$43$	−5.00000	−0.762493	−0.381246	+	0.924473i	$0.624505\pi$
$44$	− 1.00000i	− 0.150756i
$45$	1.00000i	0.149071i
$46$	− 3.00000i	− 0.442326i
$47$	3.00000i	0.437595i	0.975770	+	0.218797i	$0.0702134\pi$
$47$	3.00000i	0.437595i	−0.975770	+	0.218797i	$0.929787\pi$
$48$	−1.00000	−0.144338
$49$	3.00000	0.428571
$50$	1.00000i	0.141421i
$51$	2.00000	0.280056
$52$	0	0
$53$	14.0000	1.92305	0.961524	−	0.274721i	$-0.0885855\pi$
$53$	14.0000	1.92305	0.961524	+	0.274721i	$0.0885855\pi$
$54$	1.00000i	0.136083i
$55$	−1.00000	−0.134840
$56$	−2.00000	−0.267261
$57$	6.00000i	0.794719i
$58$	1.00000i	0.131306i
$59$	− 5.00000i	− 0.650945i	−0.945552	−	0.325472i	$-0.894477\pi$
$59$	− 5.00000i	− 0.650945i	0.945552	−	0.325472i	$-0.105523\pi$
$60$	1.00000i	0.129099i
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	3.00000	0.381000
$63$	2.00000i	0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−1.00000	−0.123091
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	2.00000	0.242536
$69$	−3.00000	−0.361158
$70$	2.00000i	0.239046i
$71$	− 4.00000i	− 0.474713i	−0.971423	−	0.237356i	$-0.923719\pi$
$71$	− 4.00000i	− 0.474713i	0.971423	−	0.237356i	$-0.0762809\pi$
$72$	1.00000i	0.117851i
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	−5.00000	−0.581238
$75$	1.00000	0.115470
$76$	6.00000i	0.688247i
$77$	−2.00000	−0.227921
$78$	0	0
$79$	5.00000	0.562544	0.281272	−	0.959628i	$-0.409244\pi$
$79$	5.00000	0.562544	0.281272	+	0.959628i	$0.409244\pi$
$80$	1.00000i	0.111803i
$81$	1.00000	0.111111
$82$	−10.0000	−1.10432
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	2.00000i	0.218218i
$85$	− 2.00000i	− 0.216930i
$86$	5.00000i	0.539164i
$87$	1.00000	0.107211
$88$	−1.00000	−0.106600
$89$	10.0000i	1.06000i	0.847998	+	0.529999i	$0.177808\pi$
$89$	10.0000i	1.06000i	−0.847998	+	0.529999i	$0.822192\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	−3.00000	−0.312772
$93$	− 3.00000i	− 0.311086i
$94$	3.00000	0.309426
$95$	6.00000	0.615587
$96$	1.00000i	0.102062i
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	− 3.00000i	− 0.303046i
$99$	1.00000i	0.100504i
$100$	1.00000	0.100000
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	− 2.00000i	− 0.198030i
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	− 14.0000i	− 1.35980i
$107$	−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
$109$	6.00000i	0.574696i	0.957826	+	0.287348i	$0.0927736\pi$
$109$	6.00000i	0.574696i	−0.957826	+	0.287348i	$0.907226\pi$
$110$	1.00000i	0.0953463i
$111$	5.00000i	0.474579i
$112$	2.00000i	0.188982i
$113$	17.0000	1.59923	0.799613	−	0.600516i	$-0.205038\pi$
$113$	17.0000	1.59923	0.799613	+	0.600516i	$0.205038\pi$
$114$	6.00000	0.561951
$115$	3.00000i	0.279751i
$116$	1.00000	0.0928477
$117$	0	0
$118$	−5.00000	−0.460287
$119$	− 4.00000i	− 0.366679i
$120$	1.00000	0.0912871
$121$	10.0000	0.909091
$122$	10.0000i	0.905357i
$123$	10.0000i	0.901670i
$124$	− 3.00000i	− 0.269408i
$125$	− 1.00000i	− 0.0894427i
$126$	2.00000	0.178174
$127$	14.0000	1.24230	0.621150	−	0.783692i	$-0.286666\pi$
$127$	14.0000	1.24230	0.621150	+	0.783692i	$0.286666\pi$
$128$	1.00000i	0.0883883i
$129$	5.00000	0.440225
$130$	0	0
$131$	13.0000	1.13582	0.567908	−	0.823092i	$-0.307753\pi$
$131$	13.0000	1.13582	0.567908	+	0.823092i	$0.307753\pi$
$132$	1.00000i	0.0870388i
$133$	12.0000	1.04053
$134$	0	0
$135$	− 1.00000i	− 0.0860663i
$136$	− 2.00000i	− 0.171499i
$137$	− 9.00000i	− 0.768922i	−0.923141	−	0.384461i	$-0.874387\pi$
$137$	− 9.00000i	− 0.768922i	0.923141	−	0.384461i	$-0.125613\pi$
$138$	3.00000i	0.255377i
$139$	10.0000	0.848189	0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000	0.848189	0.424094	+	0.905618i	$0.360592\pi$
$140$	2.00000	0.169031
$141$	− 3.00000i	− 0.252646i
$142$	−4.00000	−0.335673
$143$	0	0
$144$	1.00000	0.0833333
$145$	− 1.00000i	− 0.0830455i
$146$	−2.00000	−0.165521
$147$	−3.00000	−0.247436
$148$	5.00000i	0.410997i
$149$	− 11.0000i	− 0.901155i	−0.892737	−	0.450578i	$-0.851218\pi$
$149$	− 11.0000i	− 0.901155i	0.892737	−	0.450578i	$-0.148782\pi$
$150$	− 1.00000i	− 0.0816497i
$151$	24.0000i	1.95309i	0.215308	+	0.976546i	$0.430924\pi$
$151$	24.0000i	1.95309i	−0.215308	+	0.976546i	$0.569076\pi$
$152$	6.00000	0.486664
$153$	−2.00000	−0.161690
$154$	2.00000i	0.161165i
$155$	−3.00000	−0.240966
$156$	0	0
$157$	25.0000	1.99522	0.997609	−	0.0691164i	$-0.0220180\pi$
$157$	25.0000	1.99522	0.997609	+	0.0691164i	$0.0220180\pi$
$158$	− 5.00000i	− 0.397779i
$159$	−14.0000	−1.11027
$160$	1.00000	0.0790569
$161$	6.00000i	0.472866i
$162$	− 1.00000i	− 0.0785674i
$163$	17.0000i	1.33154i	0.746156	+	0.665771i	$0.231897\pi$
$163$	17.0000i	1.33154i	−0.746156	+	0.665771i	$0.768103\pi$
$164$	10.0000i	0.780869i
$165$	1.00000	0.0778499
$166$	6.00000	0.465690
$167$	− 7.00000i	− 0.541676i	−0.962625	−	0.270838i	$-0.912699\pi$
$167$	− 7.00000i	− 0.541676i	0.962625	−	0.270838i	$-0.0873008\pi$
$168$	2.00000	0.154303
$169$	0	0
$170$	−2.00000	−0.153393
$171$	− 6.00000i	− 0.458831i
$172$	5.00000	0.381246
$173$	−4.00000	−0.304114	−0.152057	−	0.988372i	$-0.548590\pi$
$173$	−4.00000	−0.304114	−0.152057	+	0.988372i	$0.548590\pi$
$174$	− 1.00000i	− 0.0758098i
$175$	− 2.00000i	− 0.151186i
$176$	1.00000i	0.0753778i
$177$	5.00000i	0.375823i
$178$	10.0000	0.749532
$179$	7.00000	0.523205	0.261602	−	0.965176i	$-0.415749\pi$
$179$	7.00000	0.523205	0.261602	+	0.965176i	$0.415749\pi$
$180$	− 1.00000i	− 0.0745356i
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	0	0
$183$	10.0000	0.739221
$184$	3.00000i	0.221163i
$185$	5.00000	0.367607
$186$	−3.00000	−0.219971
$187$	− 2.00000i	− 0.146254i
$188$	− 3.00000i	− 0.218797i
$189$	− 2.00000i	− 0.145479i
$190$	− 6.00000i	− 0.435286i
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	1.00000	0.0721688
$193$	− 4.00000i	− 0.287926i	−0.989583	−	0.143963i	$-0.954015\pi$
$193$	− 4.00000i	− 0.287926i	0.989583	−	0.143963i	$-0.0459847\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	−3.00000	−0.214286
$197$	12.0000i	0.854965i	0.904024	+	0.427482i	$0.140599\pi$
$197$	12.0000i	0.854965i	−0.904024	+	0.427482i	$0.859401\pi$
$198$	1.00000	0.0710669
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	− 1.00000i	− 0.0707107i
$201$	0	0
$202$	14.0000i	0.985037i
$203$	− 2.00000i	− 0.140372i
$204$	−2.00000	−0.140028
$205$	10.0000	0.698430
$206$	− 6.00000i	− 0.418040i
$207$	3.00000	0.208514
$208$	0	0
$209$	6.00000	0.415029
$210$	− 2.00000i	− 0.138013i
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	−14.0000	−0.961524
$213$	4.00000i	0.274075i
$214$	6.00000i	0.410152i
$215$	− 5.00000i	− 0.340997i
$216$	− 1.00000i	− 0.0680414i
$217$	−6.00000	−0.407307
$218$	6.00000	0.406371
$219$	2.00000i	0.135147i
$220$	1.00000	0.0674200
$221$	0	0
$222$	5.00000	0.335578
$223$	− 10.0000i	− 0.669650i	−0.942280	−	0.334825i	$-0.891323\pi$
$223$	− 10.0000i	− 0.669650i	0.942280	−	0.334825i	$-0.108677\pi$
$224$	2.00000	0.133631
$225$	−1.00000	−0.0666667
$226$	− 17.0000i	− 1.13082i
$227$	− 20.0000i	− 1.32745i	−0.747978	−	0.663723i	$-0.768975\pi$
$227$	− 20.0000i	− 1.32745i	0.747978	−	0.663723i	$-0.231025\pi$
$228$	− 6.00000i	− 0.397360i
$229$	22.0000i	1.45380i	0.686743	+	0.726900i	$0.259040\pi$
$229$	22.0000i	1.45380i	−0.686743	+	0.726900i	$0.740960\pi$
$230$	3.00000	0.197814
$231$	2.00000	0.131590
$232$	− 1.00000i	− 0.0656532i
$233$	−3.00000	−0.196537	−0.0982683	−	0.995160i	$-0.531330\pi$
$233$	−3.00000	−0.196537	−0.0982683	+	0.995160i	$0.531330\pi$
$234$	0	0
$235$	−3.00000	−0.195698
$236$	5.00000i	0.325472i
$237$	−5.00000	−0.324785
$238$	−4.00000	−0.259281
$239$	8.00000i	0.517477i	0.965947	+	0.258738i	$0.0833068\pi$
$239$	8.00000i	0.517477i	−0.965947	+	0.258738i	$0.916693\pi$
$240$	− 1.00000i	− 0.0645497i
$241$	− 7.00000i	− 0.450910i	−0.974254	−	0.225455i	$-0.927613\pi$
$241$	− 7.00000i	− 0.450910i	0.974254	−	0.225455i	$-0.0723868\pi$
$242$	− 10.0000i	− 0.642824i
$243$	−1.00000	−0.0641500
$244$	10.0000	0.640184
$245$	3.00000i	0.191663i
$246$	10.0000	0.637577
$247$	0	0
$248$	−3.00000	−0.190500
$249$	− 6.00000i	− 0.380235i
$250$	−1.00000	−0.0632456
$251$	−3.00000	−0.189358	−0.0946792	−	0.995508i	$-0.530183\pi$
$251$	−3.00000	−0.189358	−0.0946792	+	0.995508i	$0.530183\pi$
$252$	− 2.00000i	− 0.125988i
$253$	3.00000i	0.188608i
$254$	− 14.0000i	− 0.878438i
$255$	2.00000i	0.125245i
$256$	1.00000	0.0625000
$257$	−21.0000	−1.30994	−0.654972	−	0.755653i	$-0.727320\pi$
$257$	−21.0000	−1.30994	−0.654972	+	0.755653i	$0.727320\pi$
$258$	− 5.00000i	− 0.311286i
$259$	10.0000	0.621370
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	− 13.0000i	− 0.803143i
$263$	23.0000	1.41824	0.709120	−	0.705087i	$-0.249092\pi$
$263$	23.0000	1.41824	0.709120	+	0.705087i	$0.249092\pi$
$264$	1.00000	0.0615457
$265$	14.0000i	0.860013i
$266$	− 12.0000i	− 0.735767i
$267$	− 10.0000i	− 0.611990i
$268$	0	0
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	−1.00000	−0.0608581
$271$	− 29.0000i	− 1.76162i	−0.473466	−	0.880812i	$-0.656997\pi$
$271$	− 29.0000i	− 1.76162i	0.473466	−	0.880812i	$-0.343003\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	−9.00000	−0.543710
$275$	− 1.00000i	− 0.0603023i
$276$	3.00000	0.180579
$277$	19.0000	1.14160	0.570800	−	0.821089i	$-0.306633\pi$
$277$	19.0000	1.14160	0.570800	+	0.821089i	$0.306633\pi$
$278$	− 10.0000i	− 0.599760i
$279$	3.00000i	0.179605i
$280$	− 2.00000i	− 0.119523i
$281$	− 30.0000i	− 1.78965i	−0.446417	−	0.894825i	$-0.647300\pi$
$281$	− 30.0000i	− 1.78965i	0.446417	−	0.894825i	$-0.352700\pi$
$282$	−3.00000	−0.178647
$283$	13.0000	0.772770	0.386385	−	0.922338i	$-0.373724\pi$
$283$	13.0000	0.772770	0.386385	+	0.922338i	$0.373724\pi$
$284$	4.00000i	0.237356i
$285$	−6.00000	−0.355409
$286$	0	0
$287$	20.0000	1.18056
$288$	− 1.00000i	− 0.0589256i
$289$	−13.0000	−0.764706
$290$	−1.00000	−0.0587220
$291$	− 10.0000i	− 0.586210i
$292$	2.00000i	0.117041i
$293$	− 14.0000i	− 0.817889i	−0.912559	−	0.408944i	$-0.865897\pi$
$293$	− 14.0000i	− 0.817889i	0.912559	−	0.408944i	$-0.134103\pi$
$294$	3.00000i	0.174964i
$295$	5.00000	0.291111
$296$	5.00000	0.290619
$297$	− 1.00000i	− 0.0580259i
$298$	−11.0000	−0.637213
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	− 10.0000i	− 0.576390i
$302$	24.0000	1.38104
$303$	14.0000	0.804279
$304$	− 6.00000i	− 0.344124i
$305$	− 10.0000i	− 0.572598i
$306$	2.00000i	0.114332i
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	2.00000	0.113961
$309$	−6.00000	−0.341328
$310$	3.00000i	0.170389i
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−12.0000	−0.678280	−0.339140	−	0.940736i	$-0.610136\pi$
$313$	−12.0000	−0.678280	−0.339140	+	0.940736i	$0.610136\pi$
$314$	− 25.0000i	− 1.41083i
$315$	−2.00000	−0.112687
$316$	−5.00000	−0.281272
$317$	− 12.0000i	− 0.673987i	−0.941507	−	0.336994i	$-0.890590\pi$
$317$	− 12.0000i	− 0.673987i	0.941507	−	0.336994i	$-0.109410\pi$
$318$	14.0000i	0.785081i
$319$	− 1.00000i	− 0.0559893i
$320$	− 1.00000i	− 0.0559017i
$321$	6.00000	0.334887
$322$	6.00000	0.334367
$323$	12.0000i	0.667698i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	17.0000	0.941543
$327$	− 6.00000i	− 0.331801i
$328$	10.0000	0.552158
$329$	−6.00000	−0.330791
$330$	− 1.00000i	− 0.0550482i
$331$	− 4.00000i	− 0.219860i	−0.993939	−	0.109930i	$-0.964937\pi$
$331$	− 4.00000i	− 0.219860i	0.993939	−	0.109930i	$-0.0350627\pi$
$332$	− 6.00000i	− 0.329293i
$333$	− 5.00000i	− 0.273998i
$334$	−7.00000	−0.383023
$335$	0	0
$336$	− 2.00000i	− 0.109109i
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	0	0
$339$	−17.0000	−0.923313
$340$	2.00000i	0.108465i
$341$	−3.00000	−0.162459
$342$	−6.00000	−0.324443
$343$	20.0000i	1.07990i
$344$	− 5.00000i	− 0.269582i
$345$	− 3.00000i	− 0.161515i
$346$	4.00000i	0.215041i
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	−1.00000	−0.0536056
$349$	8.00000i	0.428230i	0.976808	+	0.214115i	$0.0686868\pi$
$349$	8.00000i	0.428230i	−0.976808	+	0.214115i	$0.931313\pi$
$350$	−2.00000	−0.106904
$351$	0	0
$352$	1.00000	0.0533002
$353$	14.0000i	0.745145i	0.928003	+	0.372572i	$0.121524\pi$
$353$	14.0000i	0.745145i	−0.928003	+	0.372572i	$0.878476\pi$
$354$	5.00000	0.265747
$355$	4.00000	0.212298
$356$	− 10.0000i	− 0.529999i
$357$	4.00000i	0.211702i
$358$	− 7.00000i	− 0.369961i
$359$	12.0000i	0.633336i	0.948536	+	0.316668i	$0.102564\pi$
$359$	12.0000i	0.633336i	−0.948536	+	0.316668i	$0.897436\pi$
$360$	−1.00000	−0.0527046
$361$	−17.0000	−0.894737
$362$	16.0000i	0.840941i
$363$	−10.0000	−0.524864
$364$	0	0
$365$	2.00000	0.104685
$366$	− 10.0000i	− 0.522708i
$367$	36.0000	1.87918	0.939592	−	0.342296i	$-0.111204\pi$
$367$	36.0000	1.87918	0.939592	+	0.342296i	$0.111204\pi$
$368$	3.00000	0.156386
$369$	− 10.0000i	− 0.520579i
$370$	− 5.00000i	− 0.259938i
$371$	28.0000i	1.45369i
$372$	3.00000i	0.155543i
$373$	37.0000	1.91579	0.957894	−	0.287123i	$-0.0926989\pi$
$373$	37.0000	1.91579	0.957894	+	0.287123i	$0.0926989\pi$
$374$	−2.00000	−0.103418
$375$	1.00000i	0.0516398i
$376$	−3.00000	−0.154713
$377$	0	0
$378$	−2.00000	−0.102869
$379$	30.0000i	1.54100i	0.637442	+	0.770498i	$0.279993\pi$
$379$	30.0000i	1.54100i	−0.637442	+	0.770498i	$0.720007\pi$
$380$	−6.00000	−0.307794
$381$	−14.0000	−0.717242
$382$	− 24.0000i	− 1.22795i
$383$	27.0000i	1.37964i	0.723983	+	0.689818i	$0.242309\pi$
$383$	27.0000i	1.37964i	−0.723983	+	0.689818i	$0.757691\pi$
$384$	− 1.00000i	− 0.0510310i
$385$	− 2.00000i	− 0.101929i
$386$	−4.00000	−0.203595
$387$	−5.00000	−0.254164
$388$	− 10.0000i	− 0.507673i
$389$	1.00000	0.0507020	0.0253510	−	0.999679i	$-0.491930\pi$
$389$	1.00000	0.0507020	0.0253510	+	0.999679i	$0.491930\pi$
$390$	0	0
$391$	−6.00000	−0.303433
$392$	3.00000i	0.151523i
$393$	−13.0000	−0.655763
$394$	12.0000	0.604551
$395$	5.00000i	0.251577i
$396$	− 1.00000i	− 0.0502519i
$397$	− 13.0000i	− 0.652451i	−0.945292	−	0.326226i	$-0.894223\pi$
$397$	− 13.0000i	− 0.652451i	0.945292	−	0.326226i	$-0.105777\pi$
$398$	0	0
$399$	−12.0000	−0.600751
$400$	−1.00000	−0.0500000
$401$	− 12.0000i	− 0.599251i	−0.954057	−	0.299626i	$-0.903138\pi$
$401$	− 12.0000i	− 0.599251i	0.954057	−	0.299626i	$-0.0968618\pi$
$402$	0	0
$403$	0	0
$404$	14.0000	0.696526
$405$	1.00000i	0.0496904i
$406$	−2.00000	−0.0992583
$407$	5.00000	0.247841
$408$	2.00000i	0.0990148i
$409$	26.0000i	1.28562i	0.766027	+	0.642809i	$0.222231\pi$
$409$	26.0000i	1.28562i	−0.766027	+	0.642809i	$0.777769\pi$
$410$	− 10.0000i	− 0.493865i
$411$	9.00000i	0.443937i
$412$	−6.00000	−0.295599
$413$	10.0000	0.492068
$414$	− 3.00000i	− 0.147442i
$415$	−6.00000	−0.294528
$416$	0	0
$417$	−10.0000	−0.489702
$418$	− 6.00000i	− 0.293470i
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	−2.00000	−0.0975900
$421$	− 28.0000i	− 1.36464i	−0.731055	−	0.682318i	$-0.760972\pi$
$421$	− 28.0000i	− 1.36464i	0.731055	−	0.682318i	$-0.239028\pi$
$422$	8.00000i	0.389434i
$423$	3.00000i	0.145865i
$424$	14.0000i	0.679900i
$425$	2.00000	0.0970143
$426$	4.00000	0.193801
$427$	− 20.0000i	− 0.967868i
$428$	6.00000	0.290021
$429$	0	0
$430$	−5.00000	−0.241121
$431$	− 12.0000i	− 0.578020i	−0.957326	−	0.289010i	$-0.906674\pi$
$431$	− 12.0000i	− 0.578020i	0.957326	−	0.289010i	$-0.0933260\pi$
$432$	−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	$0.625611\pi$
$434$	6.00000i	0.288009i
$435$	1.00000i	0.0479463i
$436$	− 6.00000i	− 0.287348i
$437$	− 18.0000i	− 0.861057i
$438$	2.00000	0.0955637
$439$	−28.0000	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$440$	− 1.00000i	− 0.0476731i
$441$	3.00000	0.142857
$442$	0	0
$443$	16.0000	0.760183	0.380091	−	0.924949i	$-0.375893\pi$
$443$	16.0000	0.760183	0.380091	+	0.924949i	$0.375893\pi$
$444$	− 5.00000i	− 0.237289i
$445$	−10.0000	−0.474045
$446$	−10.0000	−0.473514
$447$	11.0000i	0.520282i
$448$	− 2.00000i	− 0.0944911i
$449$	− 36.0000i	− 1.69895i	−0.527633	−	0.849473i	$-0.676920\pi$
$449$	− 36.0000i	− 1.69895i	0.527633	−	0.849473i	$-0.323080\pi$
$450$	1.00000i	0.0471405i
$451$	10.0000	0.470882
$452$	−17.0000	−0.799613
$453$	− 24.0000i	− 1.12762i
$454$	−20.0000	−0.938647
$455$	0	0
$456$	−6.00000	−0.280976
$457$	− 14.0000i	− 0.654892i	−0.944870	−	0.327446i	$-0.893812\pi$
$457$	− 14.0000i	− 0.654892i	0.944870	−	0.327446i	$-0.106188\pi$
$458$	22.0000	1.02799
$459$	2.00000	0.0933520
$460$	− 3.00000i	− 0.139876i
$461$	− 27.0000i	− 1.25752i	−0.777601	−	0.628758i	$-0.783564\pi$
$461$	− 27.0000i	− 1.25752i	0.777601	−	0.628758i	$-0.216436\pi$
$462$	− 2.00000i	− 0.0930484i
$463$	10.0000i	0.464739i	0.972628	+	0.232370i	$0.0746479\pi$
$463$	10.0000i	0.464739i	−0.972628	+	0.232370i	$0.925352\pi$
$464$	−1.00000	−0.0464238
$465$	3.00000	0.139122
$466$	3.00000i	0.138972i
$467$	2.00000	0.0925490	0.0462745	−	0.998929i	$-0.485265\pi$
$467$	2.00000	0.0925490	0.0462745	+	0.998929i	$0.485265\pi$
$468$	0	0
$469$	0	0
$470$	3.00000i	0.138380i
$471$	−25.0000	−1.15194
$472$	5.00000	0.230144
$473$	− 5.00000i	− 0.229900i
$474$	5.00000i	0.229658i
$475$	6.00000i	0.275299i
$476$	4.00000i	0.183340i
$477$	14.0000	0.641016
$478$	8.00000	0.365911
$479$	− 4.00000i	− 0.182765i	−0.995816	−	0.0913823i	$-0.970871\pi$
$479$	− 4.00000i	− 0.182765i	0.995816	−	0.0913823i	$-0.0291285\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	−7.00000	−0.318841
$483$	− 6.00000i	− 0.273009i
$484$	−10.0000	−0.454545
$485$	−10.0000	−0.454077
$486$	1.00000i	0.0453609i
$487$	− 2.00000i	− 0.0906287i	−0.998973	−	0.0453143i	$-0.985571\pi$
$487$	− 2.00000i	− 0.0906287i	0.998973	−	0.0453143i	$-0.0144289\pi$
$488$	− 10.0000i	− 0.452679i
$489$	− 17.0000i	− 0.768767i
$490$	3.00000	0.135526
$491$	−24.0000	−1.08310	−0.541552	−	0.840667i	$-0.682163\pi$
$491$	−24.0000	−1.08310	−0.541552	+	0.840667i	$0.682163\pi$
$492$	− 10.0000i	− 0.450835i
$493$	2.00000	0.0900755
$494$	0	0
$495$	−1.00000	−0.0449467
$496$	3.00000i	0.134704i
$497$	8.00000	0.358849
$498$	−6.00000	−0.268866
$499$	40.0000i	1.79065i	0.445418	+	0.895323i	$0.353055\pi$
$499$	40.0000i	1.79065i	−0.445418	+	0.895323i	$0.646945\pi$
$500$	1.00000i	0.0447214i
$501$	7.00000i	0.312737i
$502$	3.00000i	0.133897i
$503$	40.0000	1.78351	0.891756	−	0.452517i	$-0.149474\pi$
$503$	40.0000	1.78351	0.891756	+	0.452517i	$0.149474\pi$
$504$	−2.00000	−0.0890871
$505$	− 14.0000i	− 0.622992i
$506$	3.00000	0.133366
$507$	0	0
$508$	−14.0000	−0.621150
$509$	9.00000i	0.398918i	0.979906	+	0.199459i	$0.0639185\pi$
$509$	9.00000i	0.398918i	−0.979906	+	0.199459i	$0.936082\pi$
$510$	2.00000	0.0885615
$511$	4.00000	0.176950
$512$	− 1.00000i	− 0.0441942i
$513$	6.00000i	0.264906i
$514$	21.0000i	0.926270i
$515$	6.00000i	0.264392i
$516$	−5.00000	−0.220113
$517$	−3.00000	−0.131940
$518$	− 10.0000i	− 0.439375i
$519$	4.00000	0.175581
$520$	0	0
$521$	22.0000	0.963837	0.481919	−	0.876216i	$-0.339940\pi$
$521$	22.0000	0.963837	0.481919	+	0.876216i	$0.339940\pi$
$522$	1.00000i	0.0437688i
$523$	33.0000	1.44299	0.721495	−	0.692420i	$-0.243455\pi$
$523$	33.0000	1.44299	0.721495	+	0.692420i	$0.243455\pi$
$524$	−13.0000	−0.567908
$525$	2.00000i	0.0872872i
$526$	− 23.0000i	− 1.00285i
$527$	− 6.00000i	− 0.261364i
$528$	− 1.00000i	− 0.0435194i
$529$	−14.0000	−0.608696
$530$	14.0000	0.608121
$531$	− 5.00000i	− 0.216982i
$532$	−12.0000	−0.520266
$533$	0	0
$534$	−10.0000	−0.432742
$535$	− 6.00000i	− 0.259403i
$536$	0	0
$537$	−7.00000	−0.302072
$538$	− 18.0000i	− 0.776035i
$539$	3.00000i	0.129219i
$540$	1.00000i	0.0430331i
$541$	− 8.00000i	− 0.343947i	−0.985102	−	0.171973i	$-0.944986\pi$
$541$	− 8.00000i	− 0.343947i	0.985102	−	0.171973i	$-0.0550143\pi$
$542$	−29.0000	−1.24566
$543$	16.0000	0.686626
$544$	2.00000i	0.0857493i
$545$	−6.00000	−0.257012
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	9.00000i	0.384461i
$549$	−10.0000	−0.426790
$550$	−1.00000	−0.0426401
$551$	6.00000i	0.255609i
$552$	− 3.00000i	− 0.127688i
$553$	10.0000i	0.425243i
$554$	− 19.0000i	− 0.807233i
$555$	−5.00000	−0.212238
$556$	−10.0000	−0.424094
$557$	− 34.0000i	− 1.44063i	−0.693649	−	0.720313i	$-0.743998\pi$
$557$	− 34.0000i	− 1.44063i	0.693649	−	0.720313i	$-0.256002\pi$
$558$	3.00000	0.127000
$559$	0	0
$560$	−2.00000	−0.0845154
$561$	2.00000i	0.0844401i
$562$	−30.0000	−1.26547
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	3.00000i	0.126323i
$565$	17.0000i	0.715195i
$566$	− 13.0000i	− 0.546431i
$567$	2.00000i	0.0839921i
$568$	4.00000	0.167836
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	6.00000i	0.251312i
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	0	0
$573$	−24.0000	−1.00261
$574$	− 20.0000i	− 0.834784i
$575$	−3.00000	−0.125109
$576$	−1.00000	−0.0416667
$577$	18.0000i	0.749350i	0.927156	+	0.374675i	$0.122246\pi$
$577$	18.0000i	0.749350i	−0.927156	+	0.374675i	$0.877754\pi$
$578$	13.0000i	0.540729i
$579$	4.00000i	0.166234i
$580$	1.00000i	0.0415227i
$581$	−12.0000	−0.497844
$582$	−10.0000	−0.414513
$583$	14.0000i	0.579821i
$584$	2.00000	0.0827606
$585$	0	0
$586$	−14.0000	−0.578335
$587$	18.0000i	0.742940i	0.928445	+	0.371470i	$0.121146\pi$
$587$	18.0000i	0.742940i	−0.928445	+	0.371470i	$0.878854\pi$
$588$	3.00000	0.123718
$589$	18.0000	0.741677
$590$	− 5.00000i	− 0.205847i
$591$	− 12.0000i	− 0.493614i
$592$	− 5.00000i	− 0.205499i
$593$	− 35.0000i	− 1.43728i	−0.695383	−	0.718639i	$-0.744765\pi$
$593$	− 35.0000i	− 1.43728i	0.695383	−	0.718639i	$-0.255235\pi$
$594$	−1.00000	−0.0410305
$595$	4.00000	0.163984
$596$	11.0000i	0.450578i
$597$	0	0
$598$	0	0
$599$	30.0000	1.22577	0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000	1.22577	0.612883	+	0.790173i	$0.290010\pi$
$600$	1.00000i	0.0408248i
$601$	−11.0000	−0.448699	−0.224350	−	0.974509i	$-0.572026\pi$
$601$	−11.0000	−0.448699	−0.224350	+	0.974509i	$0.572026\pi$
$602$	−10.0000	−0.407570
$603$	0	0
$604$	− 24.0000i	− 0.976546i
$605$	10.0000i	0.406558i
$606$	− 14.0000i	− 0.568711i
$607$	−10.0000	−0.405887	−0.202944	−	0.979190i	$-0.565051\pi$
$607$	−10.0000	−0.405887	−0.202944	+	0.979190i	$0.565051\pi$
$608$	−6.00000	−0.243332
$609$	2.00000i	0.0810441i
$610$	−10.0000	−0.404888
$611$	0	0
$612$	2.00000	0.0808452
$613$	37.0000i	1.49442i	0.664590	+	0.747208i	$0.268606\pi$
$613$	37.0000i	1.49442i	−0.664590	+	0.747208i	$0.731394\pi$
$614$	0	0
$615$	−10.0000	−0.403239
$616$	− 2.00000i	− 0.0805823i
$617$	− 3.00000i	− 0.120775i	−0.998175	−	0.0603877i	$-0.980766\pi$
$617$	− 3.00000i	− 0.120775i	0.998175	−	0.0603877i	$-0.0192337\pi$
$618$	6.00000i	0.241355i
$619$	− 10.0000i	− 0.401934i	−0.979598	−	0.200967i	$-0.935592\pi$
$619$	− 10.0000i	− 0.401934i	0.979598	−	0.200967i	$-0.0644084\pi$
$620$	3.00000	0.120483
$621$	−3.00000	−0.120386
$622$	0	0
$623$	−20.0000	−0.801283
$624$	0	0
$625$	1.00000	0.0400000
$626$	12.0000i	0.479616i
$627$	−6.00000	−0.239617
$628$	−25.0000	−0.997609
$629$	10.0000i	0.398726i
$630$	2.00000i	0.0796819i
$631$	48.0000i	1.91085i	0.295234	+	0.955425i	$0.404602\pi$
$631$	48.0000i	1.91085i	−0.295234	+	0.955425i	$0.595398\pi$
$632$	5.00000i	0.198889i
$633$	8.00000	0.317971
$634$	−12.0000	−0.476581
$635$	14.0000i	0.555573i
$636$	14.0000	0.555136
$637$	0	0
$638$	−1.00000	−0.0395904
$639$	− 4.00000i	− 0.158238i
$640$	−1.00000	−0.0395285
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	− 6.00000i	− 0.236801i
$643$	40.0000i	1.57745i	0.614749	+	0.788723i	$0.289257\pi$
$643$	40.0000i	1.57745i	−0.614749	+	0.788723i	$0.710743\pi$
$644$	− 6.00000i	− 0.236433i
$645$	5.00000i	0.196875i
$646$	12.0000	0.472134
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	1.00000i	0.0392837i
$649$	5.00000	0.196267
$650$	0	0
$651$	6.00000	0.235159
$652$	− 17.0000i	− 0.665771i
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	−6.00000	−0.234619
$655$	13.0000i	0.507952i
$656$	− 10.0000i	− 0.390434i
$657$	− 2.00000i	− 0.0780274i
$658$	6.00000i	0.233904i
$659$	−13.0000	−0.506408	−0.253204	−	0.967413i	$-0.581484\pi$
$659$	−13.0000	−0.506408	−0.253204	+	0.967413i	$0.581484\pi$
$660$	−1.00000	−0.0389249
$661$	− 12.0000i	− 0.466746i	−0.972387	−	0.233373i	$-0.925024\pi$
$661$	− 12.0000i	− 0.466746i	0.972387	−	0.233373i	$-0.0749763\pi$
$662$	−4.00000	−0.155464
$663$	0	0
$664$	−6.00000	−0.232845
$665$	12.0000i	0.465340i
$666$	−5.00000	−0.193746
$667$	−3.00000	−0.116160
$668$	7.00000i	0.270838i
$669$	10.0000i	0.386622i
$670$	0	0
$671$	− 10.0000i	− 0.386046i
$672$	−2.00000	−0.0771517
$673$	16.0000	0.616755	0.308377	−	0.951264i	$-0.400214\pi$
$673$	16.0000	0.616755	0.308377	+	0.951264i	$0.400214\pi$
$674$	− 22.0000i	− 0.847408i
$675$	1.00000	0.0384900
$676$	0	0
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	17.0000i	0.652881i
$679$	−20.0000	−0.767530
$680$	2.00000	0.0766965
$681$	20.0000i	0.766402i
$682$	3.00000i	0.114876i
$683$	− 44.0000i	− 1.68361i	−0.539779	−	0.841807i	$-0.681492\pi$
$683$	− 44.0000i	− 1.68361i	0.539779	−	0.841807i	$-0.318508\pi$
$684$	6.00000i	0.229416i
$685$	9.00000	0.343872
$686$	20.0000	0.763604
$687$	− 22.0000i	− 0.839352i
$688$	−5.00000	−0.190623
$689$	0	0
$690$	−3.00000	−0.114208
$691$	14.0000i	0.532585i	0.963892	+	0.266293i	$0.0857987\pi$
$691$	14.0000i	0.532585i	−0.963892	+	0.266293i	$0.914201\pi$
$692$	4.00000	0.152057
$693$	−2.00000	−0.0759737
$694$	18.0000i	0.683271i
$695$	10.0000i	0.379322i
$696$	1.00000i	0.0379049i
$697$	20.0000i	0.757554i
$698$	8.00000	0.302804
$699$	3.00000	0.113470
$700$	2.00000i	0.0755929i
$701$	−23.0000	−0.868698	−0.434349	−	0.900745i	$-0.643022\pi$
$701$	−23.0000	−0.868698	−0.434349	+	0.900745i	$0.643022\pi$
$702$	0	0
$703$	−30.0000	−1.13147
$704$	− 1.00000i	− 0.0376889i
$705$	3.00000	0.112987
$706$	14.0000	0.526897
$707$	− 28.0000i	− 1.05305i
$708$	− 5.00000i	− 0.187912i
$709$	− 16.0000i	− 0.600893i	−0.953799	−	0.300446i	$-0.902864\pi$
$709$	− 16.0000i	− 0.600893i	0.953799	−	0.300446i	$-0.0971356\pi$
$710$	− 4.00000i	− 0.150117i
$711$	5.00000	0.187515
$712$	−10.0000	−0.374766
$713$	9.00000i	0.337053i
$714$	4.00000	0.149696
$715$	0	0
$716$	−7.00000	−0.261602
$717$	− 8.00000i	− 0.298765i
$718$	12.0000	0.447836
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	1.00000i	0.0372678i
$721$	12.0000i	0.446903i
$722$	17.0000i	0.632674i
$723$	7.00000i	0.260333i
$724$	16.0000	0.594635
$725$	1.00000	0.0371391
$726$	10.0000i	0.371135i
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	− 2.00000i	− 0.0740233i
$731$	10.0000	0.369863
$732$	−10.0000	−0.369611
$733$	− 14.0000i	− 0.517102i	−0.965998	−	0.258551i	$-0.916755\pi$
$733$	− 14.0000i	− 0.517102i	0.965998	−	0.258551i	$-0.0832450\pi$
$734$	− 36.0000i	− 1.32878i
$735$	− 3.00000i	− 0.110657i
$736$	− 3.00000i	− 0.110581i
$737$	0	0
$738$	−10.0000	−0.368105
$739$	24.0000i	0.882854i	0.897297	+	0.441427i	$0.145528\pi$
$739$	24.0000i	0.882854i	−0.897297	+	0.441427i	$0.854472\pi$
$740$	−5.00000	−0.183804
$741$	0	0
$742$	28.0000	1.02791
$743$	15.0000i	0.550297i	0.961402	+	0.275148i	$0.0887270\pi$
$743$	15.0000i	0.550297i	−0.961402	+	0.275148i	$0.911273\pi$
$744$	3.00000	0.109985
$745$	11.0000	0.403009
$746$	− 37.0000i	− 1.35467i
$747$	6.00000i	0.219529i
$748$	2.00000i	0.0731272i
$749$	− 12.0000i	− 0.438470i
$750$	1.00000	0.0365148
$751$	−41.0000	−1.49611	−0.748056	−	0.663636i	$-0.769012\pi$
$751$	−41.0000	−1.49611	−0.748056	+	0.663636i	$0.769012\pi$
$752$	3.00000i	0.109399i
$753$	3.00000	0.109326
$754$	0	0
$755$	−24.0000	−0.873449
$756$	2.00000i	0.0727393i
$757$	−54.0000	−1.96266	−0.981332	−	0.192323i	$-0.938398\pi$
$757$	−54.0000	−1.96266	−0.981332	+	0.192323i	$0.938398\pi$
$758$	30.0000	1.08965
$759$	− 3.00000i	− 0.108893i
$760$	6.00000i	0.217643i
$761$	− 20.0000i	− 0.724999i	−0.931984	−	0.362500i	$-0.881923\pi$
$761$	− 20.0000i	− 0.724999i	0.931984	−	0.362500i	$-0.118077\pi$
$762$	14.0000i	0.507166i
$763$	−12.0000	−0.434429
$764$	−24.0000	−0.868290
$765$	− 2.00000i	− 0.0723102i
$766$	27.0000	0.975550
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	29.0000i	1.04577i	0.852404	+	0.522883i	$0.175144\pi$
$769$	29.0000i	1.04577i	−0.852404	+	0.522883i	$0.824856\pi$
$770$	−2.00000	−0.0720750
$771$	21.0000	0.756297
$772$	4.00000i	0.143963i
$773$	16.0000i	0.575480i	0.957709	+	0.287740i	$0.0929039\pi$
$773$	16.0000i	0.575480i	−0.957709	+	0.287740i	$0.907096\pi$
$774$	5.00000i	0.179721i
$775$	− 3.00000i	− 0.107763i
$776$	−10.0000	−0.358979
$777$	−10.0000	−0.358748
$778$	− 1.00000i	− 0.0358517i
$779$	−60.0000	−2.14972
$780$	0	0
$781$	4.00000	0.143131
$782$	6.00000i	0.214560i
$783$	1.00000	0.0357371
$784$	3.00000	0.107143
$785$	25.0000i	0.892288i
$786$	13.0000i	0.463695i
$787$	25.0000i	0.891154i	0.895244	+	0.445577i	$0.147001\pi$
$787$	25.0000i	0.891154i	−0.895244	+	0.445577i	$0.852999\pi$
$788$	− 12.0000i	− 0.427482i
$789$	−23.0000	−0.818822
$790$	5.00000	0.177892
$791$	34.0000i	1.20890i
$792$	−1.00000	−0.0355335
$793$	0	0
$794$	−13.0000	−0.461353
$795$	− 14.0000i	− 0.496529i
$796$	0	0
$797$	52.0000	1.84193	0.920967	−	0.389640i	$-0.127401\pi$
$797$	52.0000	1.84193	0.920967	+	0.389640i	$0.127401\pi$
$798$	12.0000i	0.424795i
$799$	− 6.00000i	− 0.212265i
$800$	1.00000i	0.0353553i
$801$	10.0000i	0.353333i
$802$	−12.0000	−0.423735
$803$	2.00000	0.0705785
$804$	0	0
$805$	−6.00000	−0.211472
$806$	0	0
$807$	−18.0000	−0.633630
$808$	− 14.0000i	− 0.492518i
$809$	2.00000	0.0703163	0.0351581	−	0.999382i	$-0.488807\pi$
$809$	2.00000	0.0703163	0.0351581	+	0.999382i	$0.488807\pi$
$810$	1.00000	0.0351364
$811$	− 52.0000i	− 1.82597i	−0.407997	−	0.912983i	$-0.633772\pi$
$811$	− 52.0000i	− 1.82597i	0.407997	−	0.912983i	$-0.366228\pi$
$812$	2.00000i	0.0701862i
$813$	29.0000i	1.01707i
$814$	− 5.00000i	− 0.175250i
$815$	−17.0000	−0.595484
$816$	2.00000	0.0700140
$817$	30.0000i	1.04957i
$818$	26.0000	0.909069
$819$	0	0
$820$	−10.0000	−0.349215
$821$	− 23.0000i	− 0.802706i	−0.915924	−	0.401353i	$-0.868540\pi$
$821$	− 23.0000i	− 0.802706i	0.915924	−	0.401353i	$-0.131460\pi$
$822$	9.00000	0.313911
$823$	22.0000	0.766872	0.383436	−	0.923567i	$-0.374741\pi$
$823$	22.0000	0.766872	0.383436	+	0.923567i	$0.374741\pi$
$824$	6.00000i	0.209020i
$825$	1.00000i	0.0348155i
$826$	− 10.0000i	− 0.347945i
$827$	− 18.0000i	− 0.625921i	−0.949766	−	0.312961i	$-0.898679\pi$
$827$	− 18.0000i	− 0.625921i	0.949766	−	0.312961i	$-0.101321\pi$
$828$	−3.00000	−0.104257
$829$	−26.0000	−0.903017	−0.451509	−	0.892267i	$-0.649114\pi$
$829$	−26.0000	−0.903017	−0.451509	+	0.892267i	$0.649114\pi$
$830$	6.00000i	0.208263i
$831$	−19.0000	−0.659103
$832$	0	0
$833$	−6.00000	−0.207888
$834$	10.0000i	0.346272i
$835$	7.00000	0.242245
$836$	−6.00000	−0.207514
$837$	− 3.00000i	− 0.103695i
$838$	− 4.00000i	− 0.138178i
$839$	38.0000i	1.31191i	0.754802	+	0.655953i	$0.227733\pi$
$839$	38.0000i	1.31191i	−0.754802	+	0.655953i	$0.772267\pi$
$840$	2.00000i	0.0690066i
$841$	−28.0000	−0.965517
$842$	−28.0000	−0.964944
$843$	30.0000i	1.03325i
$844$	8.00000	0.275371
$845$	0	0
$846$	3.00000	0.103142
$847$	20.0000i	0.687208i
$848$	14.0000	0.480762
$849$	−13.0000	−0.446159
$850$	− 2.00000i	− 0.0685994i
$851$	− 15.0000i	− 0.514193i
$852$	− 4.00000i	− 0.137038i
$853$	7.00000i	0.239675i	0.992793	+	0.119838i	$0.0382374\pi$
$853$	7.00000i	0.239675i	−0.992793	+	0.119838i	$0.961763\pi$
$854$	−20.0000	−0.684386
$855$	6.00000	0.205196
$856$	− 6.00000i	− 0.205076i
$857$	35.0000	1.19558	0.597789	−	0.801654i	$-0.296046\pi$
$857$	35.0000	1.19558	0.597789	+	0.801654i	$0.296046\pi$
$858$	0	0
$859$	18.0000	0.614152	0.307076	−	0.951685i	$-0.400649\pi$
$859$	18.0000	0.614152	0.307076	+	0.951685i	$0.400649\pi$
$860$	5.00000i	0.170499i
$861$	−20.0000	−0.681598
$862$	−12.0000	−0.408722
$863$	51.0000i	1.73606i	0.496512	+	0.868030i	$0.334614\pi$
$863$	51.0000i	1.73606i	−0.496512	+	0.868030i	$0.665386\pi$
$864$	1.00000i	0.0340207i
$865$	− 4.00000i	− 0.136004i
$866$	16.0000i	0.543702i
$867$	13.0000	0.441503
$868$	6.00000	0.203653
$869$	5.00000i	0.169613i
$870$	1.00000	0.0339032
$871$	0	0
$872$	−6.00000	−0.203186
$873$	10.0000i	0.338449i
$874$	−18.0000	−0.608859
$875$	2.00000	0.0676123
$876$	− 2.00000i	− 0.0675737i
$877$	− 13.0000i	− 0.438979i	−0.975615	−	0.219489i	$-0.929561\pi$
$877$	− 13.0000i	− 0.438979i	0.975615	−	0.219489i	$-0.0704391\pi$
$878$	28.0000i	0.944954i
$879$	14.0000i	0.472208i
$880$	−1.00000	−0.0337100
$881$	−54.0000	−1.81931	−0.909653	−	0.415369i	$-0.863653\pi$
$881$	−54.0000	−1.81931	−0.909653	+	0.415369i	$0.863653\pi$
$882$	− 3.00000i	− 0.101015i
$883$	1.00000	0.0336527	0.0168263	−	0.999858i	$-0.494644\pi$
$883$	1.00000	0.0336527	0.0168263	+	0.999858i	$0.494644\pi$
$884$	0	0
$885$	−5.00000	−0.168073
$886$	− 16.0000i	− 0.537531i
$887$	1.00000	0.0335767	0.0167884	−	0.999859i	$-0.494656\pi$
$887$	1.00000	0.0335767	0.0167884	+	0.999859i	$0.494656\pi$
$888$	−5.00000	−0.167789
$889$	28.0000i	0.939090i
$890$	10.0000i	0.335201i
$891$	1.00000i	0.0335013i
$892$	10.0000i	0.334825i
$893$	18.0000	0.602347
$894$	11.0000	0.367895
$895$	7.00000i	0.233984i
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	−36.0000	−1.20134
$899$	− 3.00000i	− 0.100056i
$900$	1.00000	0.0333333
$901$	−28.0000	−0.932815
$902$	− 10.0000i	− 0.332964i
$903$	10.0000i	0.332779i
$904$	17.0000i	0.565412i
$905$	− 16.0000i	− 0.531858i
$906$	−24.0000	−0.797347
$907$	53.0000	1.75984	0.879918	−	0.475125i	$-0.157597\pi$
$907$	53.0000	1.75984	0.879918	+	0.475125i	$0.157597\pi$
$908$	20.0000i	0.663723i
$909$	−14.0000	−0.464351
$910$	0	0
$911$	34.0000	1.12647	0.563235	−	0.826297i	$-0.309557\pi$
$911$	34.0000	1.12647	0.563235	+	0.826297i	$0.309557\pi$
$912$	6.00000i	0.198680i
$913$	−6.00000	−0.198571
$914$	−14.0000	−0.463079
$915$	10.0000i	0.330590i
$916$	− 22.0000i	− 0.726900i
$917$	26.0000i	0.858596i
$918$	− 2.00000i	− 0.0660098i
$919$	−56.0000	−1.84727	−0.923635	−	0.383274i	$-0.874797\pi$
$919$	−56.0000	−1.84727	−0.923635	+	0.383274i	$0.874797\pi$
$920$	−3.00000	−0.0989071
$921$	0	0
$922$	−27.0000	−0.889198
$923$	0	0
$924$	−2.00000	−0.0657952
$925$	5.00000i	0.164399i
$926$	10.0000	0.328620
$927$	6.00000	0.197066
$928$	1.00000i	0.0328266i
$929$	− 24.0000i	− 0.787414i	−0.919236	−	0.393707i	$-0.871192\pi$
$929$	− 24.0000i	− 0.787414i	0.919236	−	0.393707i	$-0.128808\pi$
$930$	− 3.00000i	− 0.0983739i
$931$	− 18.0000i	− 0.589926i
$932$	3.00000	0.0982683
$933$	0	0
$934$	− 2.00000i	− 0.0654420i
$935$	2.00000	0.0654070
$936$	0	0
$937$	−18.0000	−0.588034	−0.294017	−	0.955800i	$-0.594992\pi$
$937$	−18.0000	−0.588034	−0.294017	+	0.955800i	$0.594992\pi$
$938$	0	0
$939$	12.0000	0.391605
$940$	3.00000	0.0978492
$941$	− 26.0000i	− 0.847576i	−0.905761	−	0.423788i	$-0.860700\pi$
$941$	− 26.0000i	− 0.847576i	0.905761	−	0.423788i	$-0.139300\pi$
$942$	25.0000i	0.814544i
$943$	− 30.0000i	− 0.976934i
$944$	− 5.00000i	− 0.162736i
$945$	2.00000	0.0650600
$946$	−5.00000	−0.162564
$947$	52.0000i	1.68977i	0.534946	+	0.844886i	$0.320332\pi$
$947$	52.0000i	1.68977i	−0.534946	+	0.844886i	$0.679668\pi$
$948$	5.00000	0.162392
$949$	0	0
$950$	6.00000	0.194666
$951$	12.0000i	0.389127i
$952$	4.00000	0.129641
$953$	−15.0000	−0.485898	−0.242949	−	0.970039i	$-0.578115\pi$
$953$	−15.0000	−0.485898	−0.242949	+	0.970039i	$0.578115\pi$
$954$	− 14.0000i	− 0.453267i
$955$	24.0000i	0.776622i
$956$	− 8.00000i	− 0.258738i
$957$	1.00000i	0.0323254i
$958$	−4.00000	−0.129234
$959$	18.0000	0.581250
$960$	1.00000i	0.0322749i
$961$	22.0000	0.709677
$962$	0	0
$963$	−6.00000	−0.193347
$964$	7.00000i	0.225455i
$965$	4.00000	0.128765
$966$	−6.00000	−0.193047
$967$	− 16.0000i	− 0.514525i	−0.966342	−	0.257263i	$-0.917179\pi$
$967$	− 16.0000i	− 0.514525i	0.966342	−	0.257263i	$-0.0828206\pi$
$968$	10.0000i	0.321412i
$969$	− 12.0000i	− 0.385496i
$970$	10.0000i	0.321081i
$971$	−40.0000	−1.28366	−0.641831	−	0.766846i	$-0.721825\pi$
$971$	−40.0000	−1.28366	−0.641831	+	0.766846i	$0.721825\pi$
$972$	1.00000	0.0320750
$973$	20.0000i	0.641171i
$974$	−2.00000	−0.0640841
$975$	0	0
$976$	−10.0000	−0.320092
$977$	39.0000i	1.24772i	0.781536	+	0.623860i	$0.214437\pi$
$977$	39.0000i	1.24772i	−0.781536	+	0.623860i	$0.785563\pi$
$978$	−17.0000	−0.543600
$979$	−10.0000	−0.319601
$980$	− 3.00000i	− 0.0958315i
$981$	6.00000i	0.191565i
$982$	24.0000i	0.765871i
$983$	− 25.0000i	− 0.797376i	−0.917087	−	0.398688i	$-0.869466\pi$
$983$	− 25.0000i	− 0.797376i	0.917087	−	0.398688i	$-0.130534\pi$
$984$	−10.0000	−0.318788
$985$	−12.0000	−0.382352
$986$	− 2.00000i	− 0.0636930i
$987$	6.00000	0.190982
$988$	0	0
$989$	−15.0000	−0.476972
$990$	1.00000i	0.0317821i
$991$	39.0000	1.23888	0.619438	−	0.785046i	$-0.287361\pi$
$991$	39.0000	1.23888	0.619438	+	0.785046i	$0.287361\pi$
$992$	3.00000	0.0952501
$993$	4.00000i	0.126936i
$994$	− 8.00000i	− 0.253745i
$995$	0	0
$996$	6.00000i	0.190117i
$997$	−2.00000	−0.0633406	−0.0316703	−	0.999498i	$-0.510083\pi$
$997$	−2.00000	−0.0633406	−0.0316703	+	0.999498i	$0.510083\pi$
$998$	40.0000	1.26618
$999$	5.00000i	0.158193i

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000i q^{11} +1.00000 q^{12} +2.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} -2.00000 q^{17} -1.00000i q^{18} -6.00000i q^{19} -1.00000i q^{20} -2.00000i q^{21} +1.00000 q^{22} +3.00000 q^{23} -1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} -2.00000i q^{28} -1.00000 q^{29} -1.00000 q^{30} +3.00000i q^{31} -1.00000i q^{32} -1.00000i q^{33} +2.00000i q^{34} -2.00000 q^{35} -1.00000 q^{36} -5.00000i q^{37} -6.00000 q^{38} -1.00000 q^{40} -10.0000i q^{41} -2.00000 q^{42} -5.00000 q^{43} -1.00000i q^{44} +1.00000i q^{45} -3.00000i q^{46} +3.00000i q^{47} -1.00000 q^{48} +3.00000 q^{49} +1.00000i q^{50} +2.00000 q^{51} +14.0000 q^{53} +1.00000i q^{54} -1.00000 q^{55} -2.00000 q^{56} +6.00000i q^{57} +1.00000i q^{58} -5.00000i q^{59} +1.00000i q^{60} -10.0000 q^{61} +3.00000 q^{62} +2.00000i q^{63} -1.00000 q^{64} -1.00000 q^{66} +2.00000 q^{68} -3.00000 q^{69} +2.00000i q^{70} -4.00000i q^{71} +1.00000i q^{72} -2.00000i q^{73} -5.00000 q^{74} +1.00000 q^{75} +6.00000i q^{76} -2.00000 q^{77} +5.00000 q^{79} +1.00000i q^{80} +1.00000 q^{81} -10.0000 q^{82} +6.00000i q^{83} +2.00000i q^{84} -2.00000i q^{85} +5.00000i q^{86} +1.00000 q^{87} -1.00000 q^{88} +10.0000i q^{89} +1.00000 q^{90} -3.00000 q^{92} -3.00000i q^{93} +3.00000 q^{94} +6.00000 q^{95} +1.00000i q^{96} +10.0000i q^{97} -3.00000i q^{98} +1.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} + 2 q^{10} + 2 q^{12} + 4 q^{14} + 2 q^{16} - 4 q^{17} + 2 q^{22} + 6 q^{23} - 2 q^{25} - 2 q^{27} - 2 q^{29} - 2 q^{30} - 4 q^{35} - 2 q^{36} - 12 q^{38} - 2 q^{40} - 4 q^{42} - 10 q^{43} - 2 q^{48} + 6 q^{49} + 4 q^{51} + 28 q^{53} - 2 q^{55} - 4 q^{56} - 20 q^{61} + 6 q^{62} - 2 q^{64} - 2 q^{66} + 4 q^{68} - 6 q^{69} - 10 q^{74} + 2 q^{75} - 4 q^{77} + 10 q^{79} + 2 q^{81} - 20 q^{82} + 2 q^{87} - 2 q^{88} + 2 q^{90} - 6 q^{92} + 6 q^{94} + 12 q^{95}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 + 2 * q^10 + 2 * q^12 + 4 * q^14 + 2 * q^16 - 4 * q^17 + 2 * q^22 + 6 * q^23 - 2 * q^25 - 2 * q^27 - 2 * q^29 - 2 * q^30 - 4 * q^35 - 2 * q^36 - 12 * q^38 - 2 * q^40 - 4 * q^42 - 10 * q^43 - 2 * q^48 + 6 * q^49 + 4 * q^51 + 28 * q^53 - 2 * q^55 - 4 * q^56 - 20 * q^61 + 6 * q^62 - 2 * q^64 - 2 * q^66 + 4 * q^68 - 6 * q^69 - 10 * q^74 + 2 * q^75 - 4 * q^77 + 10 * q^79 + 2 * q^81 - 20 * q^82 + 2 * q^87 - 2 * q^88 + 2 * q^90 - 6 * q^92 + 6 * q^94 + 12 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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