LMFDB - Embedded newform 2890.2.b.k.2311.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2890,2,Mod(2311,2890)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2890, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2890.2311"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2890 = 2 \cdot 5 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2890.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,4,0,4,0,0,0,4,8,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$23.0767661842$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{19})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} + 25 $ x^4 - 9*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2311.4
Root			$-2.17945 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2890.2311
Dual form			2890.2.b.k.2311.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +4.35890i q^{7} +1.00000 q^{8} +2.00000 q^{9} +1.00000i q^{10} -3.35890i q^{11} +1.00000i q^{12} +5.35890 q^{13} +4.35890i q^{14} -1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{18} +7.35890 q^{19} +1.00000i q^{20} -4.35890 q^{21} -3.35890i q^{22} -0.358899i q^{23} +1.00000i q^{24} -1.00000 q^{25} +5.35890 q^{26} +5.00000i q^{27} +4.35890i q^{28} -1.00000 q^{30} -7.35890i q^{31} +1.00000 q^{32} +3.35890 q^{33} -4.35890 q^{35} +2.00000 q^{36} +5.35890i q^{37} +7.35890 q^{38} +5.35890i q^{39} +1.00000i q^{40} -9.71780i q^{41} -4.35890 q^{42} +10.0000 q^{43} -3.35890i q^{44} +2.00000i q^{45} -0.358899i q^{46} -6.71780 q^{47} +1.00000i q^{48} -12.0000 q^{49} -1.00000 q^{50} +5.35890 q^{52} -6.71780 q^{53} +5.00000i q^{54} +3.35890 q^{55} +4.35890i q^{56} +7.35890i q^{57} +6.00000 q^{59} -1.00000 q^{60} +10.0000i q^{61} -7.35890i q^{62} +8.71780i q^{63} +1.00000 q^{64} +5.35890i q^{65} +3.35890 q^{66} -7.00000 q^{67} +0.358899 q^{69} -4.35890 q^{70} -9.35890i q^{71} +2.00000 q^{72} -1.35890i q^{73} +5.35890i q^{74} -1.00000i q^{75} +7.35890 q^{76} +14.6411 q^{77} +5.35890i q^{78} -4.64110i q^{79} +1.00000i q^{80} +1.00000 q^{81} -9.71780i q^{82} -15.7178 q^{83} -4.35890 q^{84} +10.0000 q^{86} -3.35890i q^{88} -9.00000 q^{89} +2.00000i q^{90} +23.3589i q^{91} -0.358899i q^{92} +7.35890 q^{93} -6.71780 q^{94} +7.35890i q^{95} +1.00000i q^{96} +4.64110i q^{97} -12.0000 q^{98} -6.71780i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} + 8 q^{9} + 4 q^{13} - 4 q^{15} + 4 q^{16} + 8 q^{18} + 12 q^{19} - 4 q^{25} + 4 q^{26} - 4 q^{30} + 4 q^{32} - 4 q^{33} + 8 q^{36} + 12 q^{38} + 40 q^{43} + 8 q^{47} - 48 q^{49}+ \cdots - 48 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8 + 8 * q^9 + 4 * q^13 - 4 * q^15 + 4 * q^16 + 8 * q^18 + 12 * q^19 - 4 * q^25 + 4 * q^26 - 4 * q^30 + 4 * q^32 - 4 * q^33 + 8 * q^36 + 12 * q^38 + 40 * q^43 + 8 * q^47 - 48 * q^49 - 4 * q^50 + 4 * q^52 + 8 * q^53 - 4 * q^55 + 24 * q^59 - 4 * q^60 + 4 * q^64 - 4 * q^66 - 28 * q^67 - 16 * q^69 + 8 * q^72 + 12 * q^76 + 76 * q^77 + 4 * q^81 - 28 * q^83 + 40 * q^86 - 36 * q^89 + 12 * q^93 + 8 * q^94 - 48 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2890\mathbb{Z}\right)^\times$.

$n$	$581$	$1157$
$\chi(n)$	$-1$	$1$

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000i	0.577350i	0.957427	+	0.288675i	$0.0932147\pi$
$3$	1.00000i	0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$	1.00000	0.500000
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	1.00000i	0.408248i
$7$	4.35890i	1.64751i	0.566947	+	0.823754i	$0.308125\pi$
$7$	4.35890i	1.64751i	−0.566947	+	0.823754i	$0.691875\pi$
$8$	1.00000	0.353553
$9$	2.00000	0.666667
$10$	1.00000i	0.316228i
$11$	− 3.35890i	− 1.01275i	−0.862315	−	0.506373i	$-0.830986\pi$
$11$	− 3.35890i	− 1.01275i	0.862315	−	0.506373i	$-0.169014\pi$
$12$	1.00000i	0.288675i
$13$	5.35890	1.48629	0.743146	−	0.669130i	$-0.233333\pi$
$13$	5.35890	1.48629	0.743146	+	0.669130i	$0.233333\pi$
$14$	4.35890i	1.16496i
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	2.00000	0.471405
$19$	7.35890	1.68825	0.844124	−	0.536149i	$-0.180121\pi$
$19$	7.35890	1.68825	0.844124	+	0.536149i	$0.180121\pi$
$20$	1.00000i	0.223607i
$21$	−4.35890	−0.951190
$22$	− 3.35890i	− 0.716120i
$23$	− 0.358899i	− 0.0748356i	−0.999300	−	0.0374178i	$-0.988087\pi$
$23$	− 0.358899i	− 0.0748356i	0.999300	−	0.0374178i	$-0.0119132\pi$
$24$	1.00000i	0.204124i
$25$	−1.00000	−0.200000
$26$	5.35890	1.05097
$27$	5.00000i	0.962250i
$28$	4.35890i	0.823754i
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	−1.00000	−0.182574
$31$	− 7.35890i	− 1.32170i	−0.750519	−	0.660849i	$-0.770197\pi$
$31$	− 7.35890i	− 1.32170i	0.750519	−	0.660849i	$-0.229803\pi$
$32$	1.00000	0.176777
$33$	3.35890	0.584709
$34$	0	0
$35$	−4.35890	−0.736788
$36$	2.00000	0.333333
$37$	5.35890i	0.880998i	0.897753	+	0.440499i	$0.145198\pi$
$37$	5.35890i	0.880998i	−0.897753	+	0.440499i	$0.854802\pi$
$38$	7.35890	1.19377
$39$	5.35890i	0.858111i
$40$	1.00000i	0.158114i
$41$	− 9.71780i	− 1.51767i	−0.651286	−	0.758833i	$-0.725770\pi$
$41$	− 9.71780i	− 1.51767i	0.651286	−	0.758833i	$-0.274230\pi$
$42$	−4.35890	−0.672593
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	− 3.35890i	− 0.506373i
$45$	2.00000i	0.298142i
$46$	− 0.358899i	− 0.0529168i
$47$	−6.71780	−0.979892	−0.489946	−	0.871753i	$-0.662983\pi$
$47$	−6.71780	−0.979892	−0.489946	+	0.871753i	$0.662983\pi$
$48$	1.00000i	0.144338i
$49$	−12.0000	−1.71429
$50$	−1.00000	−0.141421
$51$	0	0
$52$	5.35890	0.743146
$53$	−6.71780	−0.922761	−0.461380	−	0.887202i	$-0.652646\pi$
$53$	−6.71780	−0.922761	−0.461380	+	0.887202i	$0.652646\pi$
$54$	5.00000i	0.680414i
$55$	3.35890	0.452914
$56$	4.35890i	0.582482i
$57$	7.35890i	0.974710i
$58$	0	0
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	−1.00000	−0.129099
$61$	10.0000i	1.28037i	0.768221	+	0.640184i	$0.221142\pi$
$61$	10.0000i	1.28037i	−0.768221	+	0.640184i	$0.778858\pi$
$62$	− 7.35890i	− 0.934581i
$63$	8.71780i	1.09834i
$64$	1.00000	0.125000
$65$	5.35890i	0.664690i
$66$	3.35890	0.413452
$67$	−7.00000	−0.855186	−0.427593	−	0.903971i	$-0.640638\pi$
$67$	−7.00000	−0.855186	−0.427593	+	0.903971i	$0.640638\pi$
$68$	0	0
$69$	0.358899	0.0432064
$70$	−4.35890	−0.520988
$71$	− 9.35890i	− 1.11070i	−0.831618	−	0.555348i	$-0.812585\pi$
$71$	− 9.35890i	− 1.11070i	0.831618	−	0.555348i	$-0.187415\pi$
$72$	2.00000	0.235702
$73$	− 1.35890i	− 0.159047i	−0.996833	−	0.0795235i	$-0.974660\pi$
$73$	− 1.35890i	− 0.159047i	0.996833	−	0.0795235i	$-0.0253399\pi$
$74$	5.35890i	0.622959i
$75$	− 1.00000i	− 0.115470i
$76$	7.35890	0.844124
$77$	14.6411	1.66851
$78$	5.35890i	0.606776i
$79$	− 4.64110i	− 0.522165i	−0.965317	−	0.261082i	$-0.915921\pi$
$79$	− 4.64110i	− 0.522165i	0.965317	−	0.261082i	$-0.0840794\pi$
$80$	1.00000i	0.111803i
$81$	1.00000	0.111111
$82$	− 9.71780i	− 1.07315i
$83$	−15.7178	−1.72525	−0.862626	−	0.505842i	$-0.831182\pi$
$83$	−15.7178	−1.72525	−0.862626	+	0.505842i	$0.831182\pi$
$84$	−4.35890	−0.475595
$85$	0	0
$86$	10.0000	1.07833
$87$	0	0
$88$	− 3.35890i	− 0.358060i
$89$	−9.00000	−0.953998	−0.476999	−	0.878904i	$-0.658275\pi$
$89$	−9.00000	−0.953998	−0.476999	+	0.878904i	$0.658275\pi$
$90$	2.00000i	0.210819i
$91$	23.3589i	2.44868i
$92$	− 0.358899i	− 0.0374178i
$93$	7.35890	0.763082
$94$	−6.71780	−0.692888
$95$	7.35890i	0.755007i
$96$	1.00000i	0.102062i
$97$	4.64110i	0.471232i	0.971846	+	0.235616i	$0.0757108\pi$
$97$	4.64110i	0.471232i	−0.971846	+	0.235616i	$0.924289\pi$
$98$	−12.0000	−1.21218
$99$	− 6.71780i	− 0.675164i
$100$	−1.00000	−0.100000
$101$	−19.0767	−1.89820	−0.949101	−	0.314971i	$-0.898005\pi$
$101$	−19.0767	−1.89820	−0.949101	+	0.314971i	$0.898005\pi$
$102$	0	0
$103$	−4.35890	−0.429495	−0.214748	−	0.976670i	$-0.568893\pi$
$103$	−4.35890	−0.429495	−0.214748	+	0.976670i	$0.568893\pi$
$104$	5.35890	0.525483
$105$	− 4.35890i	− 0.425385i
$106$	−6.71780	−0.652490
$107$	0.717798i	0.0693921i	0.999398	+	0.0346961i	$0.0110463\pi$
$107$	0.717798i	0.0693921i	−0.999398	+	0.0346961i	$0.988954\pi$
$108$	5.00000i	0.481125i
$109$	10.3589i	0.992203i	0.868265	+	0.496101i	$0.165236\pi$
$109$	10.3589i	0.992203i	−0.868265	+	0.496101i	$0.834764\pi$
$110$	3.35890	0.320258
$111$	−5.35890	−0.508644
$112$	4.35890i	0.411877i
$113$	10.0767i	0.947936i	0.880542	+	0.473968i	$0.157179\pi$
$113$	10.0767i	0.947936i	−0.880542	+	0.473968i	$0.842821\pi$
$114$	7.35890i	0.689224i
$115$	0.358899	0.0334675
$116$	0	0
$117$	10.7178	0.990861
$118$	6.00000	0.552345
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	−0.282202	−0.0256547
$122$	10.0000i	0.905357i
$123$	9.71780	0.876224
$124$	− 7.35890i	− 0.660849i
$125$	− 1.00000i	− 0.0894427i
$126$	8.71780i	0.776643i
$127$	4.35890	0.386790	0.193395	−	0.981121i	$-0.438050\pi$
$127$	4.35890	0.386790	0.193395	+	0.981121i	$0.438050\pi$
$128$	1.00000	0.0883883
$129$	10.0000i	0.880451i
$130$	5.35890i	0.470007i
$131$	− 4.07670i	− 0.356183i	−0.984014	−	0.178091i	$-0.943008\pi$
$131$	− 4.07670i	− 0.356183i	0.984014	−	0.178091i	$-0.0569922\pi$
$132$	3.35890	0.292355
$133$	32.0767i	2.78140i
$134$	−7.00000	−0.604708
$135$	−5.00000	−0.430331
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0.358899	0.0305515
$139$	− 10.0000i	− 0.848189i	−0.905618	−	0.424094i	$-0.860592\pi$
$139$	− 10.0000i	− 0.848189i	0.905618	−	0.424094i	$-0.139408\pi$
$140$	−4.35890	−0.368394
$141$	− 6.71780i	− 0.565741i
$142$	− 9.35890i	− 0.785381i
$143$	− 18.0000i	− 1.50524i
$144$	2.00000	0.166667
$145$	0	0
$146$	− 1.35890i	− 0.112463i
$147$	− 12.0000i	− 0.989743i
$148$	5.35890i	0.440499i
$149$	−6.35890	−0.520941	−0.260471	−	0.965482i	$-0.583878\pi$
$149$	−6.35890	−0.520941	−0.260471	+	0.965482i	$0.583878\pi$
$150$	− 1.00000i	− 0.0816497i
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	7.35890	0.596886
$153$	0	0
$154$	14.6411	1.17981
$155$	7.35890	0.591081
$156$	5.35890i	0.429055i
$157$	−16.7178	−1.33423	−0.667113	−	0.744957i	$-0.732470\pi$
$157$	−16.7178	−1.33423	−0.667113	+	0.744957i	$0.732470\pi$
$158$	− 4.64110i	− 0.369226i
$159$	− 6.71780i	− 0.532756i
$160$	1.00000i	0.0790569i
$161$	1.56440	0.123292
$162$	1.00000	0.0785674
$163$	− 11.0000i	− 0.861586i	−0.902451	−	0.430793i	$-0.858234\pi$
$163$	− 11.0000i	− 0.861586i	0.902451	−	0.430793i	$-0.141766\pi$
$164$	− 9.71780i	− 0.758833i
$165$	3.35890i	0.261490i
$166$	−15.7178	−1.21994
$167$	12.7178i	0.984133i	0.870558	+	0.492066i	$0.163758\pi$
$167$	12.7178i	0.984133i	−0.870558	+	0.492066i	$0.836242\pi$
$168$	−4.35890	−0.336296
$169$	15.7178	1.20906
$170$	0	0
$171$	14.7178	1.12550
$172$	10.0000	0.762493
$173$	12.7178i	0.966916i	0.875367	+	0.483458i	$0.160619\pi$
$173$	12.7178i	0.966916i	−0.875367	+	0.483458i	$0.839381\pi$
$174$	0	0
$175$	− 4.35890i	− 0.329502i
$176$	− 3.35890i	− 0.253187i
$177$	6.00000i	0.450988i
$178$	−9.00000	−0.674579
$179$	19.4356	1.45268	0.726342	−	0.687333i	$-0.241219\pi$
$179$	19.4356	1.45268	0.726342	+	0.687333i	$0.241219\pi$
$180$	2.00000i	0.149071i
$181$	− 9.07670i	− 0.674666i	−0.941385	−	0.337333i	$-0.890475\pi$
$181$	− 9.07670i	− 0.674666i	0.941385	−	0.337333i	$-0.109525\pi$
$182$	23.3589i	1.73148i
$183$	−10.0000	−0.739221
$184$	− 0.358899i	− 0.0264584i
$185$	−5.35890	−0.393994
$186$	7.35890	0.539581
$187$	0	0
$188$	−6.71780	−0.489946
$189$	−21.7945	−1.58532
$190$	7.35890i	0.533871i
$191$	−8.64110	−0.625248	−0.312624	−	0.949877i	$-0.601208\pi$
$191$	−8.64110	−0.625248	−0.312624	+	0.949877i	$0.601208\pi$
$192$	1.00000i	0.0721688i
$193$	13.3589i	0.961595i	0.876832	+	0.480797i	$0.159653\pi$
$193$	13.3589i	0.961595i	−0.876832	+	0.480797i	$0.840347\pi$
$194$	4.64110i	0.333212i
$195$	−5.35890	−0.383759
$196$	−12.0000	−0.857143
$197$	− 16.0767i	− 1.14542i	−0.819759	−	0.572709i	$-0.805893\pi$
$197$	− 16.0767i	− 1.14542i	0.819759	−	0.572709i	$-0.194107\pi$
$198$	− 6.71780i	− 0.477413i
$199$	− 1.35890i	− 0.0963298i	−0.998839	−	0.0481649i	$-0.984663\pi$
$199$	− 1.35890i	− 0.0963298i	0.998839	−	0.0481649i	$-0.0153373\pi$
$200$	−1.00000	−0.0707107
$201$	− 7.00000i	− 0.493742i
$202$	−19.0767	−1.34223
$203$	0	0
$204$	0	0
$205$	9.71780	0.678720
$206$	−4.35890	−0.303699
$207$	− 0.717798i	− 0.0498904i
$208$	5.35890	0.371573
$209$	− 24.7178i	− 1.70977i
$210$	− 4.35890i	− 0.300793i
$211$	− 16.6411i	− 1.14562i	−0.819688	−	0.572810i	$-0.805853\pi$
$211$	− 16.6411i	− 1.14562i	0.819688	−	0.572810i	$-0.194147\pi$
$212$	−6.71780	−0.461380
$213$	9.35890	0.641261
$214$	0.717798i	0.0490677i
$215$	10.0000i	0.681994i
$216$	5.00000i	0.340207i
$217$	32.0767	2.17751
$218$	10.3589i	0.701593i
$219$	1.35890	0.0918259
$220$	3.35890	0.226457
$221$	0	0
$222$	−5.35890	−0.359666
$223$	4.35890	0.291893	0.145947	−	0.989292i	$-0.453377\pi$
$223$	4.35890	0.291893	0.145947	+	0.989292i	$0.453377\pi$
$224$	4.35890i	0.291241i
$225$	−2.00000	−0.133333
$226$	10.0767i	0.670292i
$227$	− 9.00000i	− 0.597351i	−0.954355	−	0.298675i	$-0.903455\pi$
$227$	− 9.00000i	− 0.597351i	0.954355	−	0.298675i	$-0.0965448\pi$
$228$	7.35890i	0.487355i
$229$	16.3589	1.08103	0.540513	−	0.841336i	$-0.318230\pi$
$229$	16.3589	1.08103	0.540513	+	0.841336i	$0.318230\pi$
$230$	0.358899	0.0236651
$231$	14.6411i	0.963314i
$232$	0	0
$233$	− 6.71780i	− 0.440098i	−0.975489	−	0.220049i	$-0.929378\pi$
$233$	− 6.71780i	− 0.440098i	0.975489	−	0.220049i	$-0.0706217\pi$
$234$	10.7178	0.700644
$235$	− 6.71780i	− 0.438221i
$236$	6.00000	0.390567
$237$	4.64110	0.301472
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	−1.00000	−0.0645497
$241$	− 12.2822i	− 0.791166i	−0.918430	−	0.395583i	$-0.870543\pi$
$241$	− 12.2822i	− 0.791166i	0.918430	−	0.395583i	$-0.129457\pi$
$242$	−0.282202	−0.0181406
$243$	16.0000i	1.02640i
$244$	10.0000i	0.640184i
$245$	− 12.0000i	− 0.766652i
$246$	9.71780	0.619584
$247$	39.4356	2.50923
$248$	− 7.35890i	− 0.467291i
$249$	− 15.7178i	− 0.996075i
$250$	− 1.00000i	− 0.0632456i
$251$	−8.64110	−0.545421	−0.272711	−	0.962096i	$-0.587920\pi$
$251$	−8.64110	−0.545421	−0.272711	+	0.962096i	$0.587920\pi$
$252$	8.71780i	0.549170i
$253$	−1.20551	−0.0757895
$254$	4.35890	0.273502
$255$	0	0
$256$	1.00000	0.0625000
$257$	28.0767	1.75138	0.875688	−	0.482877i	$-0.160408\pi$
$257$	28.0767	1.75138	0.875688	+	0.482877i	$0.160408\pi$
$258$	10.0000i	0.622573i
$259$	−23.3589	−1.45145
$260$	5.35890i	0.332345i
$261$	0	0
$262$	− 4.07670i	− 0.251859i
$263$	−6.35890	−0.392107	−0.196053	−	0.980593i	$-0.562813\pi$
$263$	−6.35890	−0.392107	−0.196053	+	0.980593i	$0.562813\pi$
$264$	3.35890	0.206726
$265$	− 6.71780i	− 0.412671i
$266$	32.0767i	1.96675i
$267$	− 9.00000i	− 0.550791i
$268$	−7.00000	−0.427593
$269$	18.3589i	1.11936i	0.828708	+	0.559681i	$0.189076\pi$
$269$	18.3589i	1.11936i	−0.828708	+	0.559681i	$0.810924\pi$
$270$	−5.00000	−0.304290
$271$	12.7945	0.777210	0.388605	−	0.921404i	$-0.372957\pi$
$271$	12.7945	0.777210	0.388605	+	0.921404i	$0.372957\pi$
$272$	0	0
$273$	−23.3589	−1.41374
$274$	−6.00000	−0.362473
$275$	3.35890i	0.202549i
$276$	0.358899	0.0216032
$277$	− 12.6411i	− 0.759530i	−0.925083	−	0.379765i	$-0.876005\pi$
$277$	− 12.6411i	− 0.759530i	0.925083	−	0.379765i	$-0.123995\pi$
$278$	− 10.0000i	− 0.599760i
$279$	− 14.7178i	− 0.881132i
$280$	−4.35890	−0.260494
$281$	−21.7178	−1.29558	−0.647788	−	0.761821i	$-0.724306\pi$
$281$	−21.7178	−1.29558	−0.647788	+	0.761821i	$0.724306\pi$
$282$	− 6.71780i	− 0.400039i
$283$	− 30.4356i	− 1.80921i	−0.426253	−	0.904604i	$-0.640167\pi$
$283$	− 30.4356i	− 1.80921i	0.426253	−	0.904604i	$-0.359833\pi$
$284$	− 9.35890i	− 0.555348i
$285$	−7.35890	−0.435904
$286$	− 18.0000i	− 1.06436i
$287$	42.3589	2.50037
$288$	2.00000	0.117851
$289$	0	0
$290$	0	0
$291$	−4.64110	−0.272066
$292$	− 1.35890i	− 0.0795235i
$293$	−10.0767	−0.588687	−0.294343	−	0.955700i	$-0.595101\pi$
$293$	−10.0767	−0.588687	−0.294343	+	0.955700i	$0.595101\pi$
$294$	− 12.0000i	− 0.699854i
$295$	6.00000i	0.349334i
$296$	5.35890i	0.311480i
$297$	16.7945	0.974515
$298$	−6.35890	−0.368361
$299$	− 1.92330i	− 0.111227i
$300$	− 1.00000i	− 0.0577350i
$301$	43.5890i	2.51243i
$302$	−2.00000	−0.115087
$303$	− 19.0767i	− 1.09593i
$304$	7.35890	0.422062
$305$	−10.0000	−0.572598
$306$	0	0
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	14.6411	0.834254
$309$	− 4.35890i	− 0.247969i
$310$	7.35890	0.417957
$311$	− 12.7178i	− 0.721160i	−0.932728	−	0.360580i	$-0.882579\pi$
$311$	− 12.7178i	− 0.721160i	0.932728	−	0.360580i	$-0.117421\pi$
$312$	5.35890i	0.303388i
$313$	− 10.6411i	− 0.601471i	−0.953708	−	0.300735i	$-0.902768\pi$
$313$	− 10.6411i	− 0.601471i	0.953708	−	0.300735i	$-0.0972321\pi$
$314$	−16.7178	−0.943440
$315$	−8.71780	−0.491192
$316$	− 4.64110i	− 0.261082i
$317$	22.0767i	1.23995i	0.784621	+	0.619975i	$0.212857\pi$
$317$	22.0767i	1.23995i	−0.784621	+	0.619975i	$0.787143\pi$
$318$	− 6.71780i	− 0.376715i
$319$	0	0
$320$	1.00000i	0.0559017i
$321$	−0.717798	−0.0400636
$322$	1.56440	0.0871808
$323$	0	0
$324$	1.00000	0.0555556
$325$	−5.35890	−0.297258
$326$	− 11.0000i	− 0.609234i
$327$	−10.3589	−0.572848
$328$	− 9.71780i	− 0.536576i
$329$	− 29.2822i	− 1.61438i
$330$	3.35890i	0.184901i
$331$	34.7178	1.90826	0.954131	−	0.299388i	$-0.0967826\pi$
$331$	34.7178	1.90826	0.954131	+	0.299388i	$0.0967826\pi$
$332$	−15.7178	−0.862626
$333$	10.7178i	0.587332i
$334$	12.7178i	0.695887i
$335$	− 7.00000i	− 0.382451i
$336$	−4.35890	−0.237797
$337$	3.92330i	0.213716i	0.994274	+	0.106858i	$0.0340790\pi$
$337$	3.92330i	0.213716i	−0.994274	+	0.106858i	$0.965921\pi$
$338$	15.7178	0.854935
$339$	−10.0767	−0.547291
$340$	0	0
$341$	−24.7178	−1.33854
$342$	14.7178	0.795847
$343$	− 21.7945i	− 1.17679i
$344$	10.0000	0.539164
$345$	0.358899i	0.0193225i
$346$	12.7178i	0.683713i
$347$	0.717798i	0.0385334i	0.999814	+	0.0192667i	$0.00613316\pi$
$347$	0.717798i	0.0385334i	−0.999814	+	0.0192667i	$0.993867\pi$
$348$	0	0
$349$	−15.7945	−0.845460	−0.422730	−	0.906256i	$-0.638928\pi$
$349$	−15.7945	−0.845460	−0.422730	+	0.906256i	$0.638928\pi$
$350$	− 4.35890i	− 0.232993i
$351$	26.7945i	1.43018i
$352$	− 3.35890i	− 0.179030i
$353$	−16.0767	−0.855676	−0.427838	−	0.903855i	$-0.640725\pi$
$353$	−16.0767	−0.855676	−0.427838	+	0.903855i	$0.640725\pi$
$354$	6.00000i	0.318896i
$355$	9.35890	0.496719
$356$	−9.00000	−0.476999
$357$	0	0
$358$	19.4356	1.02720
$359$	15.3589	0.810612	0.405306	−	0.914181i	$-0.367165\pi$
$359$	15.3589	0.810612	0.405306	+	0.914181i	$0.367165\pi$
$360$	2.00000i	0.105409i
$361$	35.1534	1.85018
$362$	− 9.07670i	− 0.477061i
$363$	− 0.282202i	− 0.0148118i
$364$	23.3589i	1.22434i
$365$	1.35890	0.0711280
$366$	−10.0000	−0.522708
$367$	17.4356i	0.910131i	0.890458	+	0.455065i	$0.150384\pi$
$367$	17.4356i	0.910131i	−0.890458	+	0.455065i	$0.849616\pi$
$368$	− 0.358899i	− 0.0187089i
$369$	− 19.4356i	− 1.01178i
$370$	−5.35890	−0.278596
$371$	− 29.2822i	− 1.52026i
$372$	7.35890	0.381541
$373$	−20.7945	−1.07670	−0.538349	−	0.842722i	$-0.680952\pi$
$373$	−20.7945	−1.07670	−0.538349	+	0.842722i	$0.680952\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	−6.71780	−0.346444
$377$	0	0
$378$	−21.7945	−1.12099
$379$	18.0767i	0.928538i	0.885694	+	0.464269i	$0.153683\pi$
$379$	18.0767i	0.928538i	−0.885694	+	0.464269i	$0.846317\pi$
$380$	7.35890i	0.377504i
$381$	4.35890i	0.223313i
$382$	−8.64110	−0.442117
$383$	−25.7945	−1.31804	−0.659019	−	0.752127i	$-0.729028\pi$
$383$	−25.7945	−1.31804	−0.659019	+	0.752127i	$0.729028\pi$
$384$	1.00000i	0.0510310i
$385$	14.6411i	0.746180i
$386$	13.3589i	0.679950i
$387$	20.0000	1.01666
$388$	4.64110i	0.235616i
$389$	−14.1534	−0.717606	−0.358803	−	0.933413i	$-0.616815\pi$
$389$	−14.1534	−0.717606	−0.358803	+	0.933413i	$0.616815\pi$
$390$	−5.35890	−0.271358
$391$	0	0
$392$	−12.0000	−0.606092
$393$	4.07670	0.205642
$394$	− 16.0767i	− 0.809932i
$395$	4.64110	0.233519
$396$	− 6.71780i	− 0.337582i
$397$	− 36.7945i	− 1.84666i	−0.384004	−	0.923331i	$-0.625455\pi$
$397$	− 36.7945i	− 1.84666i	0.384004	−	0.923331i	$-0.374545\pi$
$398$	− 1.35890i	− 0.0681154i
$399$	−32.0767	−1.60584
$400$	−1.00000	−0.0500000
$401$	28.4356i	1.42001i	0.704199	+	0.710003i	$0.251306\pi$
$401$	28.4356i	1.42001i	−0.704199	+	0.710003i	$0.748694\pi$
$402$	− 7.00000i	− 0.349128i
$403$	− 39.4356i	− 1.96443i
$404$	−19.0767	−0.949101
$405$	1.00000i	0.0496904i
$406$	0	0
$407$	18.0000	0.892227
$408$	0	0
$409$	−20.4356	−1.01048	−0.505238	−	0.862980i	$-0.668595\pi$
$409$	−20.4356	−1.01048	−0.505238	+	0.862980i	$0.668595\pi$
$410$	9.71780	0.479928
$411$	− 6.00000i	− 0.295958i
$412$	−4.35890	−0.214748
$413$	26.1534i	1.28692i
$414$	− 0.717798i	− 0.0352778i
$415$	− 15.7178i	− 0.771556i
$416$	5.35890	0.262742
$417$	10.0000	0.489702
$418$	− 24.7178i	− 1.20899i
$419$	7.43560i	0.363253i	0.983368	+	0.181626i	$0.0581361\pi$
$419$	7.43560i	0.363253i	−0.983368	+	0.181626i	$0.941864\pi$
$420$	− 4.35890i	− 0.212692i
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	− 16.6411i	− 0.810076i
$423$	−13.4356	−0.653261
$424$	−6.71780	−0.326245
$425$	0	0
$426$	9.35890	0.453440
$427$	−43.5890	−2.10942
$428$	0.717798i	0.0346961i
$429$	18.0000	0.869048
$430$	10.0000i	0.482243i
$431$	− 3.35890i	− 0.161792i	−0.996723	−	0.0808962i	$-0.974222\pi$
$431$	− 3.35890i	− 0.161792i	0.996723	−	0.0808962i	$-0.0257782\pi$
$432$	5.00000i	0.240563i
$433$	−17.3589	−0.834215	−0.417108	−	0.908857i	$-0.636956\pi$
$433$	−17.3589	−0.834215	−0.417108	+	0.908857i	$0.636956\pi$
$434$	32.0767	1.53973
$435$	0	0
$436$	10.3589i	0.496101i
$437$	− 2.64110i	− 0.126341i
$438$	1.35890	0.0649307
$439$	22.1534i	1.05732i	0.848832	+	0.528662i	$0.177306\pi$
$439$	22.1534i	1.05732i	−0.848832	+	0.528662i	$0.822694\pi$
$440$	3.35890	0.160129
$441$	−24.0000	−1.14286
$442$	0	0
$443$	17.1534	0.814982	0.407491	−	0.913209i	$-0.366404\pi$
$443$	17.1534	0.814982	0.407491	+	0.913209i	$0.366404\pi$
$444$	−5.35890	−0.254322
$445$	− 9.00000i	− 0.426641i
$446$	4.35890	0.206400
$447$	− 6.35890i	− 0.300766i
$448$	4.35890i	0.205939i
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	−2.00000	−0.0942809
$451$	−32.6411	−1.53701
$452$	10.0767i	0.473968i
$453$	− 2.00000i	− 0.0939682i
$454$	− 9.00000i	− 0.422391i
$455$	−23.3589	−1.09508
$456$	7.35890i	0.344612i
$457$	−31.5123	−1.47408	−0.737041	−	0.675848i	$-0.763778\pi$
$457$	−31.5123	−1.47408	−0.737041	+	0.675848i	$0.763778\pi$
$458$	16.3589	0.764401
$459$	0	0
$460$	0.358899	0.0167337
$461$	−25.0767	−1.16794	−0.583969	−	0.811776i	$-0.698501\pi$
$461$	−25.0767	−1.16794	−0.583969	+	0.811776i	$0.698501\pi$
$462$	14.6411i	0.681166i
$463$	−11.0767	−0.514778	−0.257389	−	0.966308i	$-0.582862\pi$
$463$	−11.0767	−0.514778	−0.257389	+	0.966308i	$0.582862\pi$
$464$	0	0
$465$	7.35890i	0.341261i
$466$	− 6.71780i	− 0.311196i
$467$	−15.7178	−0.727333	−0.363666	−	0.931529i	$-0.618475\pi$
$467$	−15.7178	−0.727333	−0.363666	+	0.931529i	$0.618475\pi$
$468$	10.7178	0.495430
$469$	− 30.5123i	− 1.40893i
$470$	− 6.71780i	− 0.309869i
$471$	− 16.7178i	− 0.770315i
$472$	6.00000	0.276172
$473$	− 33.5890i	− 1.54442i
$474$	4.64110	0.213173
$475$	−7.35890	−0.337649
$476$	0	0
$477$	−13.4356	−0.615174
$478$	−12.0000	−0.548867
$479$	20.1534i	0.920832i	0.887703	+	0.460416i	$0.152300\pi$
$479$	20.1534i	0.920832i	−0.887703	+	0.460416i	$0.847700\pi$
$480$	−1.00000	−0.0456435
$481$	28.7178i	1.30942i
$482$	− 12.2822i	− 0.559439i
$483$	1.56440i	0.0711829i
$484$	−0.282202	−0.0128274
$485$	−4.64110	−0.210742
$486$	16.0000i	0.725775i
$487$	− 38.3589i	− 1.73821i	−0.494630	−	0.869104i	$-0.664696\pi$
$487$	− 38.3589i	− 1.73821i	0.494630	−	0.869104i	$-0.335304\pi$
$488$	10.0000i	0.452679i
$489$	11.0000	0.497437
$490$	− 12.0000i	− 0.542105i
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	9.71780	0.438112
$493$	0	0
$494$	39.4356	1.77429
$495$	6.71780	0.301943
$496$	− 7.35890i	− 0.330424i
$497$	40.7945	1.82988
$498$	− 15.7178i	− 0.704331i
$499$	− 1.28220i	− 0.0573992i	−0.999588	−	0.0286996i	$-0.990863\pi$
$499$	− 1.28220i	− 0.0573992i	0.999588	−	0.0286996i	$-0.00913663\pi$
$500$	− 1.00000i	− 0.0447214i
$501$	−12.7178	−0.568189
$502$	−8.64110	−0.385671
$503$	− 13.7945i	− 0.615066i	−0.951537	−	0.307533i	$-0.900497\pi$
$503$	− 13.7945i	− 0.615066i	0.951537	−	0.307533i	$-0.0995035\pi$
$504$	8.71780i	0.388322i
$505$	− 19.0767i	− 0.848902i
$506$	−1.20551	−0.0535912
$507$	15.7178i	0.698052i
$508$	4.35890	0.193395
$509$	37.0767	1.64340	0.821698	−	0.569923i	$-0.193027\pi$
$509$	37.0767	1.64340	0.821698	+	0.569923i	$0.193027\pi$
$510$	0	0
$511$	5.92330	0.262032
$512$	1.00000	0.0441942
$513$	36.7945i	1.62452i
$514$	28.0767	1.23841
$515$	− 4.35890i	− 0.192076i
$516$	10.0000i	0.440225i
$517$	22.5644i	0.992381i
$518$	−23.3589	−1.02633
$519$	−12.7178	−0.558249
$520$	5.35890i	0.235003i
$521$	39.0000i	1.70862i	0.519763	+	0.854311i	$0.326020\pi$
$521$	39.0000i	1.70862i	−0.519763	+	0.854311i	$0.673980\pi$
$522$	0	0
$523$	−13.7178	−0.599837	−0.299919	−	0.953965i	$-0.596960\pi$
$523$	−13.7178	−0.599837	−0.299919	+	0.953965i	$0.596960\pi$
$524$	− 4.07670i	− 0.178091i
$525$	4.35890	0.190238
$526$	−6.35890	−0.277261
$527$	0	0
$528$	3.35890	0.146177
$529$	22.8712	0.994400
$530$	− 6.71780i	− 0.291802i
$531$	12.0000	0.520756
$532$	32.0767i	1.39070i
$533$	− 52.0767i	− 2.25569i
$534$	− 9.00000i	− 0.389468i
$535$	−0.717798	−0.0310331
$536$	−7.00000	−0.302354
$537$	19.4356i	0.838708i
$538$	18.3589i	0.791508i
$539$	40.3068i	1.73614i
$540$	−5.00000	−0.215166
$541$	− 21.6411i	− 0.930424i	−0.885199	−	0.465212i	$-0.845978\pi$
$541$	− 21.6411i	− 0.930424i	0.885199	−	0.465212i	$-0.154022\pi$
$542$	12.7945	0.549571
$543$	9.07670	0.389518
$544$	0	0
$545$	−10.3589	−0.443726
$546$	−23.3589	−0.999669
$547$	12.4356i	0.531708i	0.964013	+	0.265854i	$0.0856539\pi$
$547$	12.4356i	0.531708i	−0.964013	+	0.265854i	$0.914346\pi$
$548$	−6.00000	−0.256307
$549$	20.0000i	0.853579i
$550$	3.35890i	0.143224i
$551$	0	0
$552$	0.358899	0.0152758
$553$	20.2301	0.860271
$554$	− 12.6411i	− 0.537069i
$555$	− 5.35890i	− 0.227473i
$556$	− 10.0000i	− 0.424094i
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	− 14.7178i	− 0.623054i
$559$	53.5890	2.26657
$560$	−4.35890	−0.184197
$561$	0	0
$562$	−21.7178	−0.916110
$563$	24.7178	1.04173	0.520865	−	0.853639i	$-0.325609\pi$
$563$	24.7178	1.04173	0.520865	+	0.853639i	$0.325609\pi$
$564$	− 6.71780i	− 0.282870i
$565$	−10.0767	−0.423930
$566$	− 30.4356i	− 1.27930i
$567$	4.35890i	0.183057i
$568$	− 9.35890i	− 0.392691i
$569$	29.8712	1.25227	0.626133	−	0.779716i	$-0.284637\pi$
$569$	29.8712	1.25227	0.626133	+	0.779716i	$0.284637\pi$
$570$	−7.35890	−0.308230
$571$	− 36.0767i	− 1.50976i	−0.655861	−	0.754881i	$-0.727694\pi$
$571$	− 36.0767i	− 1.50976i	0.655861	−	0.754881i	$-0.272306\pi$
$572$	− 18.0000i	− 0.752618i
$573$	− 8.64110i	− 0.360987i
$574$	42.3589	1.76803
$575$	0.358899i	0.0149671i
$576$	2.00000	0.0833333
$577$	44.2301	1.84132	0.920661	−	0.390362i	$-0.127650\pi$
$577$	44.2301	1.84132	0.920661	+	0.390362i	$0.127650\pi$
$578$	0	0
$579$	−13.3589	−0.555177
$580$	0	0
$581$	− 68.5123i	− 2.84237i
$582$	−4.64110	−0.192380
$583$	22.5644i	0.934522i
$584$	− 1.35890i	− 0.0562316i
$585$	10.7178i	0.443126i
$586$	−10.0767	−0.416265
$587$	1.56440	0.0645699	0.0322849	−	0.999479i	$-0.489722\pi$
$587$	1.56440	0.0645699	0.0322849	+	0.999479i	$0.489722\pi$
$588$	− 12.0000i	− 0.494872i
$589$	− 54.1534i	− 2.23135i
$590$	6.00000i	0.247016i
$591$	16.0767	0.661307
$592$	5.35890i	0.220249i
$593$	22.7945	0.936058	0.468029	−	0.883713i	$-0.344964\pi$
$593$	22.7945	0.936058	0.468029	+	0.883713i	$0.344964\pi$
$594$	16.7945	0.689086
$595$	0	0
$596$	−6.35890	−0.260471
$597$	1.35890	0.0556160
$598$	− 1.92330i	− 0.0786497i
$599$	−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
$601$	− 19.1534i	− 0.781283i	−0.920543	−	0.390642i	$-0.872253\pi$
$601$	− 19.1534i	− 0.781283i	0.920543	−	0.390642i	$-0.127747\pi$
$602$	43.5890i	1.77655i
$603$	−14.0000	−0.570124
$604$	−2.00000	−0.0813788
$605$	− 0.282202i	− 0.0114731i
$606$	− 19.0767i	− 0.774938i
$607$	− 21.2822i	− 0.863818i	−0.901917	−	0.431909i	$-0.857840\pi$
$607$	− 21.2822i	− 0.863818i	0.901917	−	0.431909i	$-0.142160\pi$
$608$	7.35890	0.298443
$609$	0	0
$610$	−10.0000	−0.404888
$611$	−36.0000	−1.45640
$612$	0	0
$613$	5.35890	0.216444	0.108222	−	0.994127i	$-0.465484\pi$
$613$	5.35890	0.216444	0.108222	+	0.994127i	$0.465484\pi$
$614$	2.00000	0.0807134
$615$	9.71780i	0.391859i
$616$	14.6411	0.589907
$617$	− 37.4356i	− 1.50710i	−0.657390	−	0.753550i	$-0.728340\pi$
$617$	− 37.4356i	− 1.50710i	0.657390	−	0.753550i	$-0.271660\pi$
$618$	− 4.35890i	− 0.175341i
$619$	26.7945i	1.07696i	0.842638	+	0.538481i	$0.181002\pi$
$619$	26.7945i	1.07696i	−0.842638	+	0.538481i	$0.818998\pi$
$620$	7.35890	0.295540
$621$	1.79449	0.0720106
$622$	− 12.7178i	− 0.509937i
$623$	− 39.2301i	− 1.57172i
$624$	5.35890i	0.214528i
$625$	1.00000	0.0400000
$626$	− 10.6411i	− 0.425304i
$627$	24.7178	0.987134
$628$	−16.7178	−0.667113
$629$	0	0
$630$	−8.71780	−0.347325
$631$	−21.4356	−0.853338	−0.426669	−	0.904408i	$-0.640313\pi$
$631$	−21.4356	−0.853338	−0.426669	+	0.904408i	$0.640313\pi$
$632$	− 4.64110i	− 0.184613i
$633$	16.6411	0.661424
$634$	22.0767i	0.876778i
$635$	4.35890i	0.172978i
$636$	− 6.71780i	− 0.266378i
$637$	−64.3068	−2.54793
$638$	0	0
$639$	− 18.7178i	− 0.740465i
$640$	1.00000i	0.0395285i
$641$	− 41.8712i	− 1.65381i	−0.562340	−	0.826906i	$-0.690099\pi$
$641$	− 41.8712i	− 1.65381i	0.562340	−	0.826906i	$-0.309901\pi$
$642$	−0.717798	−0.0283292
$643$	37.8712i	1.49349i	0.665108	+	0.746747i	$0.268385\pi$
$643$	37.8712i	1.49349i	−0.665108	+	0.746747i	$0.731615\pi$
$644$	1.56440	0.0616462
$645$	−10.0000	−0.393750
$646$	0	0
$647$	19.0767	0.749982	0.374991	−	0.927028i	$-0.377646\pi$
$647$	19.0767	0.749982	0.374991	+	0.927028i	$0.377646\pi$
$648$	1.00000	0.0392837
$649$	− 20.1534i	− 0.791090i
$650$	−5.35890	−0.210193
$651$	32.0767i	1.25718i
$652$	− 11.0000i	− 0.430793i
$653$	25.4356i	0.995372i	0.867357	+	0.497686i	$0.165817\pi$
$653$	25.4356i	0.995372i	−0.867357	+	0.497686i	$0.834183\pi$
$654$	−10.3589	−0.405065
$655$	4.07670	0.159290
$656$	− 9.71780i	− 0.379416i
$657$	− 2.71780i	− 0.106031i
$658$	− 29.2822i	− 1.14154i
$659$	32.1534	1.25252	0.626259	−	0.779615i	$-0.284585\pi$
$659$	32.1534	1.25252	0.626259	+	0.779615i	$0.284585\pi$
$660$	3.35890i	0.130745i
$661$	−44.3589	−1.72536	−0.862681	−	0.505749i	$-0.831216\pi$
$661$	−44.3589	−1.72536	−0.862681	+	0.505749i	$0.831216\pi$
$662$	34.7178	1.34935
$663$	0	0
$664$	−15.7178	−0.609969
$665$	−32.0767	−1.24388
$666$	10.7178i	0.415306i
$667$	0	0
$668$	12.7178i	0.492066i
$669$	4.35890i	0.168525i
$670$	− 7.00000i	− 0.270434i
$671$	33.5890	1.29669
$672$	−4.35890	−0.168148
$673$	− 1.28220i	− 0.0494253i	−0.999695	−	0.0247126i	$-0.992133\pi$
$673$	− 1.28220i	− 0.0494253i	0.999695	−	0.0247126i	$-0.00786708\pi$
$674$	3.92330i	0.151120i
$675$	− 5.00000i	− 0.192450i
$676$	15.7178	0.604531
$677$	30.0000i	1.15299i	0.817099	+	0.576497i	$0.195581\pi$
$677$	30.0000i	1.15299i	−0.817099	+	0.576497i	$0.804419\pi$
$678$	−10.0767	−0.386993
$679$	−20.2301	−0.776360
$680$	0	0
$681$	9.00000	0.344881
$682$	−24.7178	−0.946493
$683$	− 16.4356i	− 0.628891i	−0.949276	−	0.314445i	$-0.898181\pi$
$683$	− 16.4356i	− 0.628891i	0.949276	−	0.314445i	$-0.101819\pi$
$684$	14.7178	0.562749
$685$	− 6.00000i	− 0.229248i
$686$	− 21.7945i	− 0.832118i
$687$	16.3589i	0.624131i
$688$	10.0000	0.381246
$689$	−36.0000	−1.37149
$690$	0.358899i	0.0136630i
$691$	35.4356i	1.34803i	0.738716	+	0.674017i	$0.235432\pi$
$691$	35.4356i	1.34803i	−0.738716	+	0.674017i	$0.764568\pi$
$692$	12.7178i	0.483458i
$693$	29.2822	1.11234
$694$	0.717798i	0.0272472i
$695$	10.0000	0.379322
$696$	0	0
$697$	0	0
$698$	−15.7945	−0.597830
$699$	6.71780	0.254090
$700$	− 4.35890i	− 0.164751i
$701$	−1.79449	−0.0677771	−0.0338886	−	0.999426i	$-0.510789\pi$
$701$	−1.79449	−0.0677771	−0.0338886	+	0.999426i	$0.510789\pi$
$702$	26.7945i	1.01129i
$703$	39.4356i	1.48734i
$704$	− 3.35890i	− 0.126593i
$705$	6.71780	0.253007
$706$	−16.0767	−0.605054
$707$	− 83.1534i	− 3.12731i
$708$	6.00000i	0.225494i
$709$	18.9233i	0.710680i	0.934737	+	0.355340i	$0.115635\pi$
$709$	18.9233i	0.710680i	−0.934737	+	0.355340i	$0.884365\pi$
$710$	9.35890	0.351233
$711$	− 9.28220i	− 0.348110i
$712$	−9.00000	−0.337289
$713$	−2.64110	−0.0989100
$714$	0	0
$715$	18.0000	0.673162
$716$	19.4356	0.726342
$717$	− 12.0000i	− 0.448148i
$718$	15.3589	0.573189
$719$	4.56440i	0.170224i	0.996371	+	0.0851118i	$0.0271247\pi$
$719$	4.56440i	0.170224i	−0.996371	+	0.0851118i	$0.972875\pi$
$720$	2.00000i	0.0745356i
$721$	− 19.0000i	− 0.707597i
$722$	35.1534	1.30827
$723$	12.2822	0.456780
$724$	− 9.07670i	− 0.337333i
$725$	0	0
$726$	− 0.282202i	− 0.0104735i
$727$	6.56440	0.243460	0.121730	−	0.992563i	$-0.461156\pi$
$727$	6.56440	0.243460	0.121730	+	0.992563i	$0.461156\pi$
$728$	23.3589i	0.865738i
$729$	−13.0000	−0.481481
$730$	1.35890	0.0502951
$731$	0	0
$732$	−10.0000	−0.369611
$733$	−6.56440	−0.242462	−0.121231	−	0.992624i	$-0.538684\pi$
$733$	−6.56440	−0.242462	−0.121231	+	0.992624i	$0.538684\pi$
$734$	17.4356i	0.643560i
$735$	12.0000	0.442627
$736$	− 0.358899i	− 0.0132292i
$737$	23.5123i	0.866086i
$738$	− 19.4356i	− 0.715434i
$739$	3.28220	0.120738	0.0603689	−	0.998176i	$-0.480772\pi$
$739$	3.28220	0.120738	0.0603689	+	0.998176i	$0.480772\pi$
$740$	−5.35890	−0.196997
$741$	39.4356i	1.44870i
$742$	− 29.2822i	− 1.07498i
$743$	− 30.3589i	− 1.11376i	−0.830593	−	0.556880i	$-0.811998\pi$
$743$	− 30.3589i	− 1.11376i	0.830593	−	0.556880i	$-0.188002\pi$
$744$	7.35890	0.269790
$745$	− 6.35890i	− 0.232972i
$746$	−20.7945	−0.761341
$747$	−31.4356	−1.15017
$748$	0	0
$749$	−3.12881	−0.114324
$750$	1.00000	0.0365148
$751$	45.4356i	1.65797i	0.559272	+	0.828984i	$0.311081\pi$
$751$	45.4356i	1.65797i	−0.559272	+	0.828984i	$0.688919\pi$
$752$	−6.71780	−0.244973
$753$	− 8.64110i	− 0.314899i
$754$	0	0
$755$	− 2.00000i	− 0.0727875i
$756$	−21.7945	−0.792658
$757$	16.0000	0.581530	0.290765	−	0.956795i	$-0.406090\pi$
$757$	16.0000	0.581530	0.290765	+	0.956795i	$0.406090\pi$
$758$	18.0767i	0.656575i
$759$	− 1.20551i	− 0.0437571i
$760$	7.35890i	0.266935i
$761$	−5.28220	−0.191480	−0.0957398	−	0.995406i	$-0.530522\pi$
$761$	−5.28220	−0.191480	−0.0957398	+	0.995406i	$0.530522\pi$
$762$	4.35890i	0.157906i
$763$	−45.1534	−1.63466
$764$	−8.64110	−0.312624
$765$	0	0
$766$	−25.7945	−0.931993
$767$	32.1534	1.16099
$768$	1.00000i	0.0360844i
$769$	29.7178	1.07165	0.535826	−	0.844329i	$-0.320000\pi$
$769$	29.7178	1.07165	0.535826	+	0.844329i	$0.320000\pi$
$770$	14.6411i	0.527629i
$771$	28.0767i	1.01116i
$772$	13.3589i	0.480797i
$773$	−12.7178	−0.457427	−0.228714	−	0.973494i	$-0.573452\pi$
$773$	−12.7178	−0.457427	−0.228714	+	0.973494i	$0.573452\pi$
$774$	20.0000	0.718885
$775$	7.35890i	0.264339i
$776$	4.64110i	0.166606i
$777$	− 23.3589i	− 0.837996i
$778$	−14.1534	−0.507424
$779$	− 71.5123i	− 2.56219i
$780$	−5.35890	−0.191879
$781$	−31.4356	−1.12485
$782$	0	0
$783$	0	0
$784$	−12.0000	−0.428571
$785$	− 16.7178i	− 0.596684i
$786$	4.07670	0.145411
$787$	− 39.1534i	− 1.39567i	−0.716260	−	0.697834i	$-0.754147\pi$
$787$	− 39.1534i	− 1.39567i	0.716260	−	0.697834i	$-0.245853\pi$
$788$	− 16.0767i	− 0.572709i
$789$	− 6.35890i	− 0.226383i
$790$	4.64110	0.165123
$791$	−43.9233	−1.56173
$792$	− 6.71780i	− 0.238707i
$793$	53.5890i	1.90300i
$794$	− 36.7945i	− 1.30579i
$795$	6.71780	0.238256
$796$	− 1.35890i	− 0.0481649i
$797$	28.0767	0.994528	0.497264	−	0.867599i	$-0.334338\pi$
$797$	28.0767	0.994528	0.497264	+	0.867599i	$0.334338\pi$
$798$	−32.0767	−1.13550
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	−18.0000	−0.635999
$802$	28.4356i	1.00410i
$803$	−4.56440	−0.161074
$804$	− 7.00000i	− 0.246871i
$805$	1.56440i	0.0551380i
$806$	− 39.4356i	− 1.38906i
$807$	−18.3589	−0.646264
$808$	−19.0767	−0.671116
$809$	23.1534i	0.814030i	0.913421	+	0.407015i	$0.133430\pi$
$809$	23.1534i	0.814030i	−0.913421	+	0.407015i	$0.866570\pi$
$810$	1.00000i	0.0351364i
$811$	− 26.0767i	− 0.915677i	−0.889035	−	0.457838i	$-0.848624\pi$
$811$	− 26.0767i	− 0.915677i	0.889035	−	0.457838i	$-0.151376\pi$
$812$	0	0
$813$	12.7945i	0.448723i
$814$	18.0000	0.630900
$815$	11.0000	0.385313
$816$	0	0
$817$	73.5890	2.57455
$818$	−20.4356	−0.714514
$819$	46.7178i	1.63245i
$820$	9.71780	0.339360
$821$	7.07670i	0.246978i	0.992346	+	0.123489i	$0.0394084\pi$
$821$	7.07670i	0.246978i	−0.992346	+	0.123489i	$0.960592\pi$
$822$	− 6.00000i	− 0.209274i
$823$	28.3589i	0.988529i	0.869312	+	0.494265i	$0.164563\pi$
$823$	28.3589i	0.988529i	−0.869312	+	0.494265i	$0.835437\pi$
$824$	−4.35890	−0.151849
$825$	−3.35890	−0.116942
$826$	26.1534i	0.909993i
$827$	− 27.0000i	− 0.938882i	−0.882964	−	0.469441i	$-0.844455\pi$
$827$	− 27.0000i	− 0.938882i	0.882964	−	0.469441i	$-0.155545\pi$
$828$	− 0.717798i	− 0.0249452i
$829$	19.6411	0.682164	0.341082	−	0.940034i	$-0.389207\pi$
$829$	19.6411	0.682164	0.341082	+	0.940034i	$0.389207\pi$
$830$	− 15.7178i	− 0.545573i
$831$	12.6411	0.438515
$832$	5.35890	0.185786
$833$	0	0
$834$	10.0000	0.346272
$835$	−12.7178	−0.440118
$836$	− 24.7178i	− 0.854883i
$837$	36.7945	1.27180
$838$	7.43560i	0.256858i
$839$	− 34.0767i	− 1.17646i	−0.808694	−	0.588229i	$-0.799825\pi$
$839$	− 34.0767i	− 1.17646i	0.808694	−	0.588229i	$-0.200175\pi$
$840$	− 4.35890i	− 0.150396i
$841$	29.0000	1.00000
$842$	8.00000	0.275698
$843$	− 21.7178i	− 0.748001i
$844$	− 16.6411i	− 0.572810i
$845$	15.7178i	0.540709i
$846$	−13.4356	−0.461925
$847$	− 1.23009i	− 0.0422664i
$848$	−6.71780	−0.230690
$849$	30.4356	1.04455
$850$	0	0
$851$	1.92330	0.0659300
$852$	9.35890	0.320631
$853$	53.3589i	1.82697i	0.406868	+	0.913487i	$0.366621\pi$
$853$	53.3589i	1.82697i	−0.406868	+	0.913487i	$0.633379\pi$
$854$	−43.5890	−1.49158
$855$	14.7178i	0.503338i
$856$	0.717798i	0.0245338i
$857$	− 0.717798i	− 0.0245195i	−0.999925	−	0.0122598i	$-0.996098\pi$
$857$	− 0.717798i	− 0.0245195i	0.999925	−	0.0122598i	$-0.00390250\pi$
$858$	18.0000	0.614510
$859$	3.28220	0.111987	0.0559936	−	0.998431i	$-0.482167\pi$
$859$	3.28220	0.111987	0.0559936	+	0.998431i	$0.482167\pi$
$860$	10.0000i	0.340997i
$861$	42.3589i	1.44359i
$862$	− 3.35890i	− 0.114405i
$863$	38.8712	1.32319	0.661595	−	0.749861i	$-0.269880\pi$
$863$	38.8712	1.32319	0.661595	+	0.749861i	$0.269880\pi$
$864$	5.00000i	0.170103i
$865$	−12.7178	−0.432418
$866$	−17.3589	−0.589879
$867$	0	0
$868$	32.0767	1.08875
$869$	−15.5890	−0.528820
$870$	0	0
$871$	−37.5123	−1.27106
$872$	10.3589i	0.350797i
$873$	9.28220i	0.314155i
$874$	− 2.64110i	− 0.0893366i
$875$	4.35890	0.147358
$876$	1.35890	0.0459129
$877$	− 25.2822i	− 0.853719i	−0.904318	−	0.426860i	$-0.859620\pi$
$877$	− 25.2822i	− 0.853719i	0.904318	−	0.426860i	$-0.140380\pi$
$878$	22.1534i	0.747641i
$879$	− 10.0767i	− 0.339879i
$880$	3.35890	0.113228
$881$	− 19.4356i	− 0.654802i	−0.944886	−	0.327401i	$-0.893827\pi$
$881$	− 19.4356i	− 0.654802i	0.944886	−	0.327401i	$-0.106173\pi$
$882$	−24.0000	−0.808122
$883$	8.71780	0.293377	0.146689	−	0.989183i	$-0.453138\pi$
$883$	8.71780	0.293377	0.146689	+	0.989183i	$0.453138\pi$
$884$	0	0
$885$	−6.00000	−0.201688
$886$	17.1534	0.576280
$887$	− 7.07670i	− 0.237612i	−0.992917	−	0.118806i	$-0.962093\pi$
$887$	− 7.07670i	− 0.237612i	0.992917	−	0.118806i	$-0.0379067\pi$
$888$	−5.35890	−0.179833
$889$	19.0000i	0.637240i
$890$	− 9.00000i	− 0.301681i
$891$	− 3.35890i	− 0.112527i
$892$	4.35890	0.145947
$893$	−49.4356	−1.65430
$894$	− 6.35890i	− 0.212673i
$895$	19.4356i	0.649660i
$896$	4.35890i	0.145621i
$897$	1.92330	0.0642172
$898$	0	0
$899$	0	0
$900$	−2.00000	−0.0666667
$901$	0	0
$902$	−32.6411	−1.08683
$903$	−43.5890	−1.45055
$904$	10.0767i	0.335146i
$905$	9.07670	0.301720
$906$	− 2.00000i	− 0.0664455i
$907$	− 22.1534i	− 0.735591i	−0.929907	−	0.367796i	$-0.880113\pi$
$907$	− 22.1534i	− 0.735591i	0.929907	−	0.367796i	$-0.119887\pi$
$908$	− 9.00000i	− 0.298675i
$909$	−38.1534	−1.26547
$910$	−23.3589	−0.774340
$911$	− 24.7178i	− 0.818937i	−0.912324	−	0.409469i	$-0.865714\pi$
$911$	− 24.7178i	− 0.818937i	0.912324	−	0.409469i	$-0.134286\pi$
$912$	7.35890i	0.243677i
$913$	52.7945i	1.74724i
$914$	−31.5123	−1.04233
$915$	− 10.0000i	− 0.330590i
$916$	16.3589	0.540513
$917$	17.7699	0.586814
$918$	0	0
$919$	−29.4356	−0.970991	−0.485495	−	0.874239i	$-0.661361\pi$
$919$	−29.4356	−0.970991	−0.485495	+	0.874239i	$0.661361\pi$
$920$	0.358899	0.0118325
$921$	2.00000i	0.0659022i
$922$	−25.0767	−0.825857
$923$	− 50.1534i	− 1.65082i
$924$	14.6411i	0.481657i
$925$	− 5.35890i	− 0.176200i
$926$	−11.0767	−0.364003
$927$	−8.71780	−0.286330
$928$	0	0
$929$	51.7178i	1.69681i	0.529351	+	0.848403i	$0.322435\pi$
$929$	51.7178i	1.69681i	−0.529351	+	0.848403i	$0.677565\pi$
$930$	7.35890i	0.241308i
$931$	−88.3068	−2.89414
$932$	− 6.71780i	− 0.220049i
$933$	12.7178	0.416362
$934$	−15.7178	−0.514302
$935$	0	0
$936$	10.7178	0.350322
$937$	−28.6411	−0.935664	−0.467832	−	0.883817i	$-0.654965\pi$
$937$	−28.6411	−0.935664	−0.467832	+	0.883817i	$0.654965\pi$
$938$	− 30.5123i	− 0.996262i
$939$	10.6411	0.347259
$940$	− 6.71780i	− 0.219110i
$941$	39.2301i	1.27886i	0.768847	+	0.639432i	$0.220831\pi$
$941$	39.2301i	1.27886i	−0.768847	+	0.639432i	$0.779169\pi$
$942$	− 16.7178i	− 0.544695i
$943$	−3.48771	−0.113575
$944$	6.00000	0.195283
$945$	− 21.7945i	− 0.708975i
$946$	− 33.5890i	− 1.09207i
$947$	3.00000i	0.0974869i	0.998811	+	0.0487435i	$0.0155217\pi$
$947$	3.00000i	0.0974869i	−0.998811	+	0.0487435i	$0.984478\pi$
$948$	4.64110	0.150736
$949$	− 7.28220i	− 0.236390i
$950$	−7.35890	−0.238754
$951$	−22.0767	−0.715886
$952$	0	0
$953$	−48.2301	−1.56233	−0.781163	−	0.624327i	$-0.785373\pi$
$953$	−48.2301	−1.56233	−0.781163	+	0.624327i	$0.785373\pi$
$954$	−13.4356	−0.434993
$955$	− 8.64110i	− 0.279619i
$956$	−12.0000	−0.388108
$957$	0	0
$958$	20.1534i	0.651127i
$959$	− 26.1534i	− 0.844537i
$960$	−1.00000	−0.0322749
$961$	−23.1534	−0.746884
$962$	28.7178i	0.925899i
$963$	1.43560i	0.0462614i
$964$	− 12.2822i	− 0.395583i
$965$	−13.3589	−0.430038
$966$	1.56440i	0.0503339i
$967$	−39.7945	−1.27970	−0.639852	−	0.768498i	$-0.721004\pi$
$967$	−39.7945	−1.27970	−0.639852	+	0.768498i	$0.721004\pi$
$968$	−0.282202	−0.00907032
$969$	0	0
$970$	−4.64110	−0.149017
$971$	−0.717798	−0.0230352	−0.0115176	−	0.999934i	$-0.503666\pi$
$971$	−0.717798	−0.0230352	−0.0115176	+	0.999934i	$0.503666\pi$
$972$	16.0000i	0.513200i
$973$	43.5890	1.39740
$974$	− 38.3589i	− 1.22910i
$975$	− 5.35890i	− 0.171622i
$976$	10.0000i	0.320092i
$977$	−3.84661	−0.123064	−0.0615319	−	0.998105i	$-0.519599\pi$
$977$	−3.84661	−0.123064	−0.0615319	+	0.998105i	$0.519599\pi$
$978$	11.0000	0.351741
$979$	30.2301i	0.966158i
$980$	− 12.0000i	− 0.383326i
$981$	20.7178i	0.661468i
$982$	12.0000	0.382935
$983$	− 16.9233i	− 0.539770i	−0.962893	−	0.269885i	$-0.913014\pi$
$983$	− 16.9233i	− 0.539770i	0.962893	−	0.269885i	$-0.0869856\pi$
$984$	9.71780	0.309792
$985$	16.0767	0.512246
$986$	0	0
$987$	29.2822	0.932063
$988$	39.4356	1.25461
$989$	− 3.58899i	− 0.114123i
$990$	6.71780	0.213506
$991$	15.4356i	0.490328i	0.969482	+	0.245164i	$0.0788418\pi$
$991$	15.4356i	0.490328i	−0.969482	+	0.245164i	$0.921158\pi$
$992$	− 7.35890i	− 0.233645i
$993$	34.7178i	1.10174i
$994$	40.7945	1.29392
$995$	1.35890	0.0430800
$996$	− 15.7178i	− 0.498037i
$997$	7.35890i	0.233059i	0.993187	+	0.116529i	$0.0371769\pi$
$997$	7.35890i	0.233059i	−0.993187	+	0.116529i	$0.962823\pi$
$998$	− 1.28220i	− 0.0405874i
$999$	−26.7945	−0.847740

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2890.2.b.k.2311.4		4
17.4	even	4		2890.2.a.s.1.1	yes	2
17.13	even	4		2890.2.a.r.1.2	✓	2
17.16	even	2	inner	2890.2.b.k.2311.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2890.2.a.r.1.2	✓	2	17.13	even	4
2890.2.a.s.1.1	yes	2	17.4	even	4
2890.2.b.k.2311.1		4	17.16	even	2	inner
2890.2.b.k.2311.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2890 = 2 \cdot 5 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2890.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +4.35890i q^{7} +1.00000 q^{8} +2.00000 q^{9} +1.00000i q^{10} -3.35890i q^{11} +1.00000i q^{12} +5.35890 q^{13} +4.35890i q^{14} -1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{18} +7.35890 q^{19} +1.00000i q^{20} -4.35890 q^{21} -3.35890i q^{22} -0.358899i q^{23} +1.00000i q^{24} -1.00000 q^{25} +5.35890 q^{26} +5.00000i q^{27} +4.35890i q^{28} -1.00000 q^{30} -7.35890i q^{31} +1.00000 q^{32} +3.35890 q^{33} -4.35890 q^{35} +2.00000 q^{36} +5.35890i q^{37} +7.35890 q^{38} +5.35890i q^{39} +1.00000i q^{40} -9.71780i q^{41} -4.35890 q^{42} +10.0000 q^{43} -3.35890i q^{44} +2.00000i q^{45} -0.358899i q^{46} -6.71780 q^{47} +1.00000i q^{48} -12.0000 q^{49} -1.00000 q^{50} +5.35890 q^{52} -6.71780 q^{53} +5.00000i q^{54} +3.35890 q^{55} +4.35890i q^{56} +7.35890i q^{57} +6.00000 q^{59} -1.00000 q^{60} +10.0000i q^{61} -7.35890i q^{62} +8.71780i q^{63} +1.00000 q^{64} +5.35890i q^{65} +3.35890 q^{66} -7.00000 q^{67} +0.358899 q^{69} -4.35890 q^{70} -9.35890i q^{71} +2.00000 q^{72} -1.35890i q^{73} +5.35890i q^{74} -1.00000i q^{75} +7.35890 q^{76} +14.6411 q^{77} +5.35890i q^{78} -4.64110i q^{79} +1.00000i q^{80} +1.00000 q^{81} -9.71780i q^{82} -15.7178 q^{83} -4.35890 q^{84} +10.0000 q^{86} -3.35890i q^{88} -9.00000 q^{89} +2.00000i q^{90} +23.3589i q^{91} -0.358899i q^{92} +7.35890 q^{93} -6.71780 q^{94} +7.35890i q^{95} +1.00000i q^{96} +4.64110i q^{97} -12.0000 q^{98} -6.71780i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} + 8 q^{9} + 4 q^{13} - 4 q^{15} + 4 q^{16} + 8 q^{18} + 12 q^{19} - 4 q^{25} + 4 q^{26} - 4 q^{30} + 4 q^{32} - 4 q^{33} + 8 q^{36} + 12 q^{38} + 40 q^{43} + 8 q^{47} - 48 q^{49}+ \cdots - 48 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8 + 8 * q^9 + 4 * q^13 - 4 * q^15 + 4 * q^16 + 8 * q^18 + 12 * q^19 - 4 * q^25 + 4 * q^26 - 4 * q^30 + 4 * q^32 - 4 * q^33 + 8 * q^36 + 12 * q^38 + 40 * q^43 + 8 * q^47 - 48 * q^49 - 4 * q^50 + 4 * q^52 + 8 * q^53 - 4 * q^55 + 24 * q^59 - 4 * q^60 + 4 * q^64 - 4 * q^66 - 28 * q^67 - 16 * q^69 + 8 * q^72 + 12 * q^76 + 76 * q^77 + 4 * q^81 - 28 * q^83 + 40 * q^86 - 36 * q^89 + 12 * q^93 + 8 * q^94 - 48 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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