LMFDB - Embedded newform 2890.2.b.k.2311.3

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			2311.3
Root			$2.17945 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2890.2311
Dual form			2890.2.b.k.2311.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} -4.35890i q^{7} +1.00000 q^{8} +2.00000 q^{9} +1.00000i q^{10} +5.35890i q^{11} +1.00000i q^{12} -3.35890 q^{13} -4.35890i q^{14} -1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{18} -1.35890 q^{19} +1.00000i q^{20} +4.35890 q^{21} +5.35890i q^{22} +8.35890i q^{23} +1.00000i q^{24} -1.00000 q^{25} -3.35890 q^{26} +5.00000i q^{27} -4.35890i q^{28} -1.00000 q^{30} +1.35890i q^{31} +1.00000 q^{32} -5.35890 q^{33} +4.35890 q^{35} +2.00000 q^{36} -3.35890i q^{37} -1.35890 q^{38} -3.35890i q^{39} +1.00000i q^{40} +7.71780i q^{41} +4.35890 q^{42} +10.0000 q^{43} +5.35890i q^{44} +2.00000i q^{45} +8.35890i q^{46} +10.7178 q^{47} +1.00000i q^{48} -12.0000 q^{49} -1.00000 q^{50} -3.35890 q^{52} +10.7178 q^{53} +5.00000i q^{54} -5.35890 q^{55} -4.35890i q^{56} -1.35890i q^{57} +6.00000 q^{59} -1.00000 q^{60} +10.0000i q^{61} +1.35890i q^{62} -8.71780i q^{63} +1.00000 q^{64} -3.35890i q^{65} -5.35890 q^{66} -7.00000 q^{67} -8.35890 q^{69} +4.35890 q^{70} -0.641101i q^{71} +2.00000 q^{72} +7.35890i q^{73} -3.35890i q^{74} -1.00000i q^{75} -1.35890 q^{76} +23.3589 q^{77} -3.35890i q^{78} -13.3589i q^{79} +1.00000i q^{80} +1.00000 q^{81} +7.71780i q^{82} +1.71780 q^{83} +4.35890 q^{84} +10.0000 q^{86} +5.35890i q^{88} -9.00000 q^{89} +2.00000i q^{90} +14.6411i q^{91} +8.35890i q^{92} -1.35890 q^{93} +10.7178 q^{94} -1.35890i q^{95} +1.00000i q^{96} +13.3589i q^{97} -12.0000 q^{98} +10.7178i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.691875\pi$
$7$	− 4.35890i	− 1.64751i	0.566947	−	0.823754i	$-0.308125\pi$
$8$	1.00000	0.353553
$9$	2.00000	0.666667
$10$	1.00000i	0.316228i
$11$	5.35890i	1.61577i	0.589341	+	0.807884i	$0.299387\pi$
$11$	5.35890i	1.61577i	−0.589341	+	0.807884i	$0.700613\pi$
$12$	1.00000i	0.288675i
$13$	−3.35890	−0.931591	−0.465795	−	0.884892i	$-0.654232\pi$
$13$	−3.35890	−0.931591	−0.465795	+	0.884892i	$0.654232\pi$
$14$	− 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
$19$	−1.35890	−0.311753	−0.155876	−	0.987777i	$-0.549820\pi$
$19$	−1.35890	−0.311753	−0.155876	+	0.987777i	$0.549820\pi$
$20$	1.00000i	0.223607i
$21$	4.35890	0.951190
$22$	5.35890i	1.14252i
$23$	8.35890i	1.74295i	0.490439	+	0.871475i	$0.336836\pi$
$23$	8.35890i	1.74295i	−0.490439	+	0.871475i	$0.663164\pi$
$24$	1.00000i	0.204124i
$25$	−1.00000	−0.200000
$26$	−3.35890	−0.658734
$27$	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
$31$	1.35890i	0.244065i	0.992526	+	0.122033i	$0.0389413\pi$
$31$	1.35890i	0.244065i	−0.992526	+	0.122033i	$0.961059\pi$
$32$	1.00000	0.176777
$33$	−5.35890	−0.932865
$34$	0	0
$35$	4.35890	0.736788
$36$	2.00000	0.333333
$37$	− 3.35890i	− 0.552200i	−0.961129	−	0.276100i	$-0.910958\pi$
$37$	− 3.35890i	− 0.552200i	0.961129	−	0.276100i	$-0.0890420\pi$
$38$	−1.35890	−0.220443
$39$	− 3.35890i	− 0.537854i
$40$	1.00000i	0.158114i
$41$	7.71780i	1.20532i	0.797999	+	0.602659i	$0.205892\pi$
$41$	7.71780i	1.20532i	−0.797999	+	0.602659i	$0.794108\pi$
$42$	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	$0.223975\pi$
$44$	5.35890i	0.807884i
$45$	2.00000i	0.298142i
$46$	8.35890i	1.23245i
$47$	10.7178	1.56335	0.781676	−	0.623685i	$-0.214365\pi$
$47$	10.7178	1.56335	0.781676	+	0.623685i	$0.214365\pi$
$48$	1.00000i	0.144338i
$49$	−12.0000	−1.71429
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−3.35890	−0.465795
$53$	10.7178	1.47220	0.736101	−	0.676871i	$-0.236665\pi$
$53$	10.7178	1.47220	0.736101	+	0.676871i	$0.236665\pi$
$54$	5.00000i	0.680414i
$55$	−5.35890	−0.722594
$56$	− 4.35890i	− 0.582482i
$57$	− 1.35890i	− 0.179991i
$58$	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	$0.778858\pi$
$62$	1.35890i	0.172580i
$63$	− 8.71780i	− 1.09834i
$64$	1.00000	0.125000
$65$	− 3.35890i	− 0.416620i
$66$	−5.35890	−0.659635
$67$	−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
$69$	−8.35890	−1.00629
$70$	4.35890	0.520988
$71$	− 0.641101i	− 0.0760847i	−0.999276	−	0.0380423i	$-0.987888\pi$
$71$	− 0.641101i	− 0.0760847i	0.999276	−	0.0380423i	$-0.0121122\pi$
$72$	2.00000	0.235702
$73$	7.35890i	0.861294i	0.902520	+	0.430647i	$0.141715\pi$
$73$	7.35890i	0.861294i	−0.902520	+	0.430647i	$0.858285\pi$
$74$	− 3.35890i	− 0.390464i
$75$	− 1.00000i	− 0.115470i
$76$	−1.35890	−0.155876
$77$	23.3589	2.66199
$78$	− 3.35890i	− 0.380320i
$79$	− 13.3589i	− 1.50299i	−0.659737	−	0.751497i	$-0.729332\pi$
$79$	− 13.3589i	− 1.50299i	0.659737	−	0.751497i	$-0.270668\pi$
$80$	1.00000i	0.111803i
$81$	1.00000	0.111111
$82$	7.71780i	0.852288i
$83$	1.71780	0.188553	0.0942764	−	0.995546i	$-0.469946\pi$
$83$	1.71780	0.188553	0.0942764	+	0.995546i	$0.469946\pi$
$84$	4.35890	0.475595
$85$	0	0
$86$	10.0000	1.07833
$87$	0	0
$88$	5.35890i	0.571261i
$89$	−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
$91$	14.6411i	1.53480i
$92$	8.35890i	0.871475i
$93$	−1.35890	−0.140911
$94$	10.7178	1.10546
$95$	− 1.35890i	− 0.139420i
$96$	1.00000i	0.102062i
$97$	13.3589i	1.35639i	0.734882	+	0.678195i	$0.237238\pi$
$97$	13.3589i	1.35639i	−0.734882	+	0.678195i	$0.762762\pi$
$98$	−12.0000	−1.21218
$99$	10.7178i	1.07718i
$100$	−1.00000	−0.100000
$101$	7.07670	0.704158	0.352079	−	0.935970i	$-0.385475\pi$
$101$	7.07670	0.704158	0.352079	+	0.935970i	$0.385475\pi$
$102$	0	0
$103$	4.35890	0.429495	0.214748	−	0.976670i	$-0.431107\pi$
$103$	4.35890	0.429495	0.214748	+	0.976670i	$0.431107\pi$
$104$	−3.35890	−0.329367
$105$	4.35890i	0.425385i
$106$	10.7178	1.04100
$107$	− 16.7178i	− 1.61617i	−0.589066	−	0.808085i	$-0.700504\pi$
$107$	− 16.7178i	− 1.61617i	0.589066	−	0.808085i	$-0.299496\pi$
$108$	5.00000i	0.481125i
$109$	1.64110i	0.157189i	0.996907	+	0.0785945i	$0.0250432\pi$
$109$	1.64110i	0.157189i	−0.996907	+	0.0785945i	$0.974957\pi$
$110$	−5.35890	−0.510951
$111$	3.35890	0.318813
$112$	− 4.35890i	− 0.411877i
$113$	− 16.0767i	− 1.51237i	−0.654359	−	0.756184i	$-0.727061\pi$
$113$	− 16.0767i	− 1.51237i	0.654359	−	0.756184i	$-0.272939\pi$
$114$	− 1.35890i	− 0.127273i
$115$	−8.35890	−0.779471
$116$	0	0
$117$	−6.71780	−0.621061
$118$	6.00000	0.552345
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	−17.7178	−1.61071
$122$	10.0000i	0.905357i
$123$	−7.71780	−0.695890
$124$	1.35890i	0.122033i
$125$	− 1.00000i	− 0.0894427i
$126$	− 8.71780i	− 0.776643i
$127$	−4.35890	−0.386790	−0.193395	−	0.981121i	$-0.561950\pi$
$127$	−4.35890	−0.386790	−0.193395	+	0.981121i	$0.561950\pi$
$128$	1.00000	0.0883883
$129$	10.0000i	0.880451i
$130$	− 3.35890i	− 0.294595i
$131$	22.0767i	1.92885i	0.264357	+	0.964425i	$0.414840\pi$
$131$	22.0767i	1.92885i	−0.264357	+	0.964425i	$0.585160\pi$
$132$	−5.35890	−0.466432
$133$	5.92330i	0.513616i
$134$	−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	$0.582506\pi$
$138$	−8.35890	−0.711557
$139$	− 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
$141$	10.7178i	0.902601i
$142$	− 0.641101i	− 0.0538000i
$143$	− 18.0000i	− 1.50524i
$144$	2.00000	0.166667
$145$	0	0
$146$	7.35890i	0.609027i
$147$	− 12.0000i	− 0.989743i
$148$	− 3.35890i	− 0.276100i
$149$	2.35890	0.193249	0.0966243	−	0.995321i	$-0.469195\pi$
$149$	2.35890	0.193249	0.0966243	+	0.995321i	$0.469195\pi$
$150$	− 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	$0.525932\pi$
$152$	−1.35890	−0.110221
$153$	0	0
$154$	23.3589	1.88231
$155$	−1.35890	−0.109149
$156$	− 3.35890i	− 0.268927i
$157$	0.717798	0.0572865	0.0286433	−	0.999590i	$-0.490881\pi$
$157$	0.717798	0.0572865	0.0286433	+	0.999590i	$0.490881\pi$
$158$	− 13.3589i	− 1.06278i
$159$	10.7178i	0.849977i
$160$	1.00000i	0.0790569i
$161$	36.4356	2.87153
$162$	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	$-0.141766\pi$
$164$	7.71780i	0.602659i
$165$	− 5.35890i	− 0.417190i
$166$	1.71780	0.133327
$167$	− 4.71780i	− 0.365074i	−0.983199	−	0.182537i	$-0.941569\pi$
$167$	− 4.71780i	− 0.365074i	0.983199	−	0.182537i	$-0.0584310\pi$
$168$	4.35890	0.336296
$169$	−1.71780	−0.132138
$170$	0	0
$171$	−2.71780	−0.207835
$172$	10.0000	0.762493
$173$	− 4.71780i	− 0.358688i	−0.983786	−	0.179344i	$-0.942603\pi$
$173$	− 4.71780i	− 0.358688i	0.983786	−	0.179344i	$-0.0573974\pi$
$174$	0	0
$175$	4.35890i	0.329502i
$176$	5.35890i	0.403942i
$177$	6.00000i	0.450988i
$178$	−9.00000	−0.674579
$179$	−15.4356	−1.15371	−0.576855	−	0.816846i	$-0.695720\pi$
$179$	−15.4356	−1.15371	−0.576855	+	0.816846i	$0.695720\pi$
$180$	2.00000i	0.149071i
$181$	17.0767i	1.26930i	0.772799	+	0.634650i	$0.218856\pi$
$181$	17.0767i	1.26930i	−0.772799	+	0.634650i	$0.781144\pi$
$182$	14.6411i	1.08527i
$183$	−10.0000	−0.739221
$184$	8.35890i	0.616226i
$185$	3.35890	0.246951
$186$	−1.35890	−0.0996393
$187$	0	0
$188$	10.7178	0.781676
$189$	21.7945	1.58532
$190$	− 1.35890i	− 0.0985849i
$191$	−17.3589	−1.25605	−0.628023	−	0.778195i	$-0.716136\pi$
$191$	−17.3589	−1.25605	−0.628023	+	0.778195i	$0.716136\pi$
$192$	1.00000i	0.0721688i
$193$	4.64110i	0.334074i	0.985951	+	0.167037i	$0.0534199\pi$
$193$	4.64110i	0.334074i	−0.985951	+	0.167037i	$0.946580\pi$
$194$	13.3589i	0.959113i
$195$	3.35890	0.240536
$196$	−12.0000	−0.857143
$197$	10.0767i	0.717935i	0.933350	+	0.358967i	$0.116871\pi$
$197$	10.0767i	0.717935i	−0.933350	+	0.358967i	$0.883129\pi$
$198$	10.7178i	0.761681i
$199$	7.35890i	0.521659i	0.965385	+	0.260829i	$0.0839960\pi$
$199$	7.35890i	0.521659i	−0.965385	+	0.260829i	$0.916004\pi$
$200$	−1.00000	−0.0707107
$201$	− 7.00000i	− 0.493742i
$202$	7.07670	0.497915
$203$	0	0
$204$	0	0
$205$	−7.71780	−0.539034
$206$	4.35890	0.303699
$207$	16.7178i	1.16197i
$208$	−3.35890	−0.232898
$209$	− 7.28220i	− 0.503720i
$210$	4.35890i	0.300793i
$211$	− 25.3589i	− 1.74578i	−0.487918	−	0.872889i	$-0.662244\pi$
$211$	− 25.3589i	− 1.74578i	0.487918	−	0.872889i	$-0.337756\pi$
$212$	10.7178	0.736101
$213$	0.641101	0.0439275
$214$	− 16.7178i	− 1.14281i
$215$	10.0000i	0.681994i
$216$	5.00000i	0.340207i
$217$	5.92330	0.402100
$218$	1.64110i	0.111149i
$219$	−7.35890	−0.497268
$220$	−5.35890	−0.361297
$221$	0	0
$222$	3.35890	0.225435
$223$	−4.35890	−0.291893	−0.145947	−	0.989292i	$-0.546623\pi$
$223$	−4.35890	−0.291893	−0.145947	+	0.989292i	$0.546623\pi$
$224$	− 4.35890i	− 0.291241i
$225$	−2.00000	−0.133333
$226$	− 16.0767i	− 1.06941i
$227$	− 9.00000i	− 0.597351i	−0.954355	−	0.298675i	$-0.903455\pi$
$227$	− 9.00000i	− 0.597351i	0.954355	−	0.298675i	$-0.0965448\pi$
$228$	− 1.35890i	− 0.0899953i
$229$	7.64110	0.504938	0.252469	−	0.967605i	$-0.418757\pi$
$229$	7.64110	0.504938	0.252469	+	0.967605i	$0.418757\pi$
$230$	−8.35890	−0.551169
$231$	23.3589i	1.53690i
$232$	0	0
$233$	10.7178i	0.702146i	0.936348	+	0.351073i	$0.114183\pi$
$233$	10.7178i	0.702146i	−0.936348	+	0.351073i	$0.885817\pi$
$234$	−6.71780	−0.439156
$235$	10.7178i	0.699152i
$236$	6.00000	0.390567
$237$	13.3589	0.867754
$238$	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
$241$	− 29.7178i	− 1.91429i	−0.289606	−	0.957146i	$-0.593524\pi$
$241$	− 29.7178i	− 1.91429i	0.289606	−	0.957146i	$-0.406476\pi$
$242$	−17.7178	−1.13894
$243$	16.0000i	1.02640i
$244$	10.0000i	0.640184i
$245$	− 12.0000i	− 0.766652i
$246$	−7.71780	−0.492069
$247$	4.56440	0.290426
$248$	1.35890i	0.0862902i
$249$	1.71780i	0.108861i
$250$	− 1.00000i	− 0.0632456i
$251$	−17.3589	−1.09568	−0.547842	−	0.836582i	$-0.684551\pi$
$251$	−17.3589	−1.09568	−0.547842	+	0.836582i	$0.684551\pi$
$252$	− 8.71780i	− 0.549170i
$253$	−44.7945	−2.81621
$254$	−4.35890	−0.273502
$255$	0	0
$256$	1.00000	0.0625000
$257$	1.92330	0.119972	0.0599862	−	0.998199i	$-0.480894\pi$
$257$	1.92330	0.119972	0.0599862	+	0.998199i	$0.480894\pi$
$258$	10.0000i	0.622573i
$259$	−14.6411	−0.909754
$260$	− 3.35890i	− 0.208310i
$261$	0	0
$262$	22.0767i	1.36390i
$263$	2.35890	0.145456	0.0727280	−	0.997352i	$-0.476830\pi$
$263$	2.35890	0.145456	0.0727280	+	0.997352i	$0.476830\pi$
$264$	−5.35890	−0.329817
$265$	10.7178i	0.658389i
$266$	5.92330i	0.363181i
$267$	− 9.00000i	− 0.550791i
$268$	−7.00000	−0.427593
$269$	9.64110i	0.587828i	0.955832	+	0.293914i	$0.0949580\pi$
$269$	9.64110i	0.587828i	−0.955832	+	0.293914i	$0.905042\pi$
$270$	−5.00000	−0.304290
$271$	−30.7945	−1.87063	−0.935316	−	0.353813i	$-0.884885\pi$
$271$	−30.7945	−1.87063	−0.935316	+	0.353813i	$0.884885\pi$
$272$	0	0
$273$	−14.6411	−0.886120
$274$	−6.00000	−0.362473
$275$	− 5.35890i	− 0.323154i
$276$	−8.35890	−0.503147
$277$	− 21.3589i	− 1.28333i	−0.766984	−	0.641666i	$-0.778244\pi$
$277$	− 21.3589i	− 1.28333i	0.766984	−	0.641666i	$-0.221756\pi$
$278$	− 10.0000i	− 0.599760i
$279$	2.71780i	0.162710i
$280$	4.35890	0.260494
$281$	−4.28220	−0.255455	−0.127727	−	0.991809i	$-0.540768\pi$
$281$	−4.28220	−0.255455	−0.127727	+	0.991809i	$0.540768\pi$
$282$	10.7178i	0.638236i
$283$	4.43560i	0.263669i	0.991272	+	0.131834i	$0.0420867\pi$
$283$	4.43560i	0.263669i	−0.991272	+	0.131834i	$0.957913\pi$
$284$	− 0.641101i	− 0.0380423i
$285$	1.35890	0.0804942
$286$	− 18.0000i	− 1.06436i
$287$	33.6411	1.98577
$288$	2.00000	0.117851
$289$	0	0
$290$	0	0
$291$	−13.3589	−0.783113
$292$	7.35890i	0.430647i
$293$	16.0767	0.939211	0.469605	−	0.882876i	$-0.344396\pi$
$293$	16.0767	0.939211	0.469605	+	0.882876i	$0.344396\pi$
$294$	− 12.0000i	− 0.699854i
$295$	6.00000i	0.349334i
$296$	− 3.35890i	− 0.195232i
$297$	−26.7945	−1.55477
$298$	2.35890	0.136647
$299$	− 28.0767i	− 1.62372i
$300$	− 1.00000i	− 0.0577350i
$301$	− 43.5890i	− 2.51243i
$302$	−2.00000	−0.115087
$303$	7.07670i	0.406546i
$304$	−1.35890	−0.0779382
$305$	−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	$0.481823\pi$
$308$	23.3589	1.33100
$309$	4.35890i	0.247969i
$310$	−1.35890	−0.0771803
$311$	4.71780i	0.267522i	0.991014	+	0.133761i	$0.0427054\pi$
$311$	4.71780i	0.267522i	−0.991014	+	0.133761i	$0.957295\pi$
$312$	− 3.35890i	− 0.190160i
$313$	− 19.3589i	− 1.09423i	−0.837058	−	0.547115i	$-0.815726\pi$
$313$	− 19.3589i	− 1.09423i	0.837058	−	0.547115i	$-0.184274\pi$
$314$	0.717798	0.0405077
$315$	8.71780	0.491192
$316$	− 13.3589i	− 0.751497i
$317$	− 4.07670i	− 0.228970i	−0.993425	−	0.114485i	$-0.963478\pi$
$317$	− 4.07670i	− 0.228970i	0.993425	−	0.114485i	$-0.0365218\pi$
$318$	10.7178i	0.601024i
$319$	0	0
$320$	1.00000i	0.0559017i
$321$	16.7178	0.933096
$322$	36.4356	2.03048
$323$	0	0
$324$	1.00000	0.0555556
$325$	3.35890	0.186318
$326$	− 11.0000i	− 0.609234i
$327$	−1.64110	−0.0907531
$328$	7.71780i	0.426144i
$329$	− 46.7178i	− 2.57564i
$330$	− 5.35890i	− 0.294998i
$331$	17.2822	0.949916	0.474958	−	0.880009i	$-0.342463\pi$
$331$	17.2822	0.949916	0.474958	+	0.880009i	$0.342463\pi$
$332$	1.71780	0.0942764
$333$	− 6.71780i	− 0.368133i
$334$	− 4.71780i	− 0.258146i
$335$	− 7.00000i	− 0.382451i
$336$	4.35890	0.237797
$337$	30.0767i	1.63838i	0.573521	+	0.819191i	$0.305577\pi$
$337$	30.0767i	1.63838i	−0.573521	+	0.819191i	$0.694423\pi$
$338$	−1.71780	−0.0934359
$339$	16.0767	0.873166
$340$	0	0
$341$	−7.28220	−0.394353
$342$	−2.71780	−0.146962
$343$	21.7945i	1.17679i
$344$	10.0000	0.539164
$345$	− 8.35890i	− 0.450028i
$346$	− 4.71780i	− 0.253630i
$347$	− 16.7178i	− 0.897458i	−0.893668	−	0.448729i	$-0.851877\pi$
$347$	− 16.7178i	− 0.897458i	0.893668	−	0.448729i	$-0.148123\pi$
$348$	0	0
$349$	27.7945	1.48781	0.743903	−	0.668288i	$-0.232973\pi$
$349$	27.7945	1.48781	0.743903	+	0.668288i	$0.232973\pi$
$350$	4.35890i	0.232993i
$351$	− 16.7945i	− 0.896424i
$352$	5.35890i	0.285630i
$353$	10.0767	0.536328	0.268164	−	0.963373i	$-0.413583\pi$
$353$	10.0767	0.536328	0.268164	+	0.963373i	$0.413583\pi$
$354$	6.00000i	0.318896i
$355$	0.641101	0.0340261
$356$	−9.00000	−0.476999
$357$	0	0
$358$	−15.4356	−0.815797
$359$	6.64110	0.350504	0.175252	−	0.984524i	$-0.443926\pi$
$359$	6.64110	0.350504	0.175252	+	0.984524i	$0.443926\pi$
$360$	2.00000i	0.105409i
$361$	−17.1534	−0.902810
$362$	17.0767i	0.897531i
$363$	− 17.7178i	− 0.929943i
$364$	14.6411i	0.767402i
$365$	−7.35890	−0.385182
$366$	−10.0000	−0.522708
$367$	− 17.4356i	− 0.910131i	−0.890458	−	0.455065i	$-0.849616\pi$
$367$	− 17.4356i	− 0.910131i	0.890458	−	0.455065i	$-0.150384\pi$
$368$	8.35890i	0.435738i
$369$	15.4356i	0.803545i
$370$	3.35890	0.174621
$371$	− 46.7178i	− 2.42547i
$372$	−1.35890	−0.0704556
$373$	22.7945	1.18025	0.590127	−	0.807310i	$-0.299078\pi$
$373$	22.7945	1.18025	0.590127	+	0.807310i	$0.299078\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	10.7178	0.552728
$377$	0	0
$378$	21.7945	1.12099
$379$	− 8.07670i	− 0.414872i	−0.978249	−	0.207436i	$-0.933488\pi$
$379$	− 8.07670i	− 0.414872i	0.978249	−	0.207436i	$-0.0665119\pi$
$380$	− 1.35890i	− 0.0697100i
$381$	− 4.35890i	− 0.223313i
$382$	−17.3589	−0.888159
$383$	17.7945	0.909256	0.454628	−	0.890681i	$-0.349772\pi$
$383$	17.7945	0.909256	0.454628	+	0.890681i	$0.349772\pi$
$384$	1.00000i	0.0510310i
$385$	23.3589i	1.19048i
$386$	4.64110i	0.236226i
$387$	20.0000	1.01666
$388$	13.3589i	0.678195i
$389$	38.1534	1.93445	0.967227	−	0.253914i	$-0.0817179\pi$
$389$	38.1534	1.93445	0.967227	+	0.253914i	$0.0817179\pi$
$390$	3.35890	0.170084
$391$	0	0
$392$	−12.0000	−0.606092
$393$	−22.0767	−1.11362
$394$	10.0767i	0.507657i
$395$	13.3589	0.672159
$396$	10.7178i	0.538590i
$397$	6.79449i	0.341006i	0.985357	+	0.170503i	$0.0545392\pi$
$397$	6.79449i	0.341006i	−0.985357	+	0.170503i	$0.945461\pi$
$398$	7.35890i	0.368868i
$399$	−5.92330	−0.296536
$400$	−1.00000	−0.0500000
$401$	− 6.43560i	− 0.321378i	−0.987005	−	0.160689i	$-0.948628\pi$
$401$	− 6.43560i	− 0.321378i	0.987005	−	0.160689i	$-0.0513717\pi$
$402$	− 7.00000i	− 0.349128i
$403$	− 4.56440i	− 0.227369i
$404$	7.07670	0.352079
$405$	1.00000i	0.0496904i
$406$	0	0
$407$	18.0000	0.892227
$408$	0	0
$409$	14.4356	0.713794	0.356897	−	0.934144i	$-0.383835\pi$
$409$	14.4356	0.713794	0.356897	+	0.934144i	$0.383835\pi$
$410$	−7.71780	−0.381155
$411$	− 6.00000i	− 0.295958i
$412$	4.35890	0.214748
$413$	− 26.1534i	− 1.28692i
$414$	16.7178i	0.821635i
$415$	1.71780i	0.0843234i
$416$	−3.35890	−0.164684
$417$	10.0000	0.489702
$418$	− 7.28220i	− 0.356184i
$419$	− 27.4356i	− 1.34032i	−0.742218	−	0.670158i	$-0.766226\pi$
$419$	− 27.4356i	− 1.34032i	0.742218	−	0.670158i	$-0.233774\pi$
$420$	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	$0.437546\pi$
$422$	− 25.3589i	− 1.23445i
$423$	21.4356	1.04223
$424$	10.7178	0.520502
$425$	0	0
$426$	0.641101	0.0310614
$427$	43.5890	2.10942
$428$	− 16.7178i	− 0.808085i
$429$	18.0000	0.869048
$430$	10.0000i	0.482243i
$431$	5.35890i	0.258129i	0.991636	+	0.129065i	$0.0411974\pi$
$431$	5.35890i	0.258129i	−0.991636	+	0.129065i	$0.958803\pi$
$432$	5.00000i	0.240563i
$433$	−8.64110	−0.415265	−0.207632	−	0.978207i	$-0.566576\pi$
$433$	−8.64110	−0.415265	−0.207632	+	0.978207i	$0.566576\pi$
$434$	5.92330	0.284328
$435$	0	0
$436$	1.64110i	0.0785945i
$437$	− 11.3589i	− 0.543370i
$438$	−7.35890	−0.351622
$439$	− 30.1534i	− 1.43914i	−0.694418	−	0.719571i	$-0.744338\pi$
$439$	− 30.1534i	− 1.43914i	0.694418	−	0.719571i	$-0.255662\pi$
$440$	−5.35890	−0.255475
$441$	−24.0000	−1.14286
$442$	0	0
$443$	−35.1534	−1.67019	−0.835094	−	0.550107i	$-0.814587\pi$
$443$	−35.1534	−1.67019	−0.835094	+	0.550107i	$0.814587\pi$
$444$	3.35890	0.159406
$445$	− 9.00000i	− 0.426641i
$446$	−4.35890	−0.206400
$447$	2.35890i	0.111572i
$448$	− 4.35890i	− 0.205939i
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	−2.00000	−0.0942809
$451$	−41.3589	−1.94751
$452$	− 16.0767i	− 0.756184i
$453$	− 2.00000i	− 0.0939682i
$454$	− 9.00000i	− 0.422391i
$455$	−14.6411	−0.686385
$456$	− 1.35890i	− 0.0636363i
$457$	29.5123	1.38053	0.690263	−	0.723558i	$-0.257495\pi$
$457$	29.5123	1.38053	0.690263	+	0.723558i	$0.257495\pi$
$458$	7.64110	0.357045
$459$	0	0
$460$	−8.35890	−0.389736
$461$	1.07670	0.0501468	0.0250734	−	0.999686i	$-0.492018\pi$
$461$	1.07670	0.0501468	0.0250734	+	0.999686i	$0.492018\pi$
$462$	23.3589i	1.08675i
$463$	15.0767	0.700674	0.350337	−	0.936624i	$-0.386067\pi$
$463$	15.0767	0.700674	0.350337	+	0.936624i	$0.386067\pi$
$464$	0	0
$465$	− 1.35890i	− 0.0630174i
$466$	10.7178i	0.496492i
$467$	1.71780	0.0794902	0.0397451	−	0.999210i	$-0.487345\pi$
$467$	1.71780	0.0794902	0.0397451	+	0.999210i	$0.487345\pi$
$468$	−6.71780	−0.310530
$469$	30.5123i	1.40893i
$470$	10.7178i	0.494375i
$471$	0.717798i	0.0330744i
$472$	6.00000	0.276172
$473$	53.5890i	2.46402i
$474$	13.3589	0.613595
$475$	1.35890	0.0623506
$476$	0	0
$477$	21.4356	0.981469
$478$	−12.0000	−0.548867
$479$	− 32.1534i	− 1.46913i	−0.678540	−	0.734563i	$-0.737387\pi$
$479$	− 32.1534i	− 1.46913i	0.678540	−	0.734563i	$-0.262613\pi$
$480$	−1.00000	−0.0456435
$481$	11.2822i	0.514424i
$482$	− 29.7178i	− 1.35361i
$483$	36.4356i	1.65788i
$484$	−17.7178	−0.805354
$485$	−13.3589	−0.606596
$486$	16.0000i	0.725775i
$487$	− 29.6411i	− 1.34317i	−0.740929	−	0.671583i	$-0.765615\pi$
$487$	− 29.6411i	− 1.34317i	0.740929	−	0.671583i	$-0.234385\pi$
$488$	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	$0.412720\pi$
$492$	−7.71780	−0.347945
$493$	0	0
$494$	4.56440	0.205362
$495$	−10.7178	−0.481729
$496$	1.35890i	0.0610164i
$497$	−2.79449	−0.125350
$498$	1.71780i	0.0769764i
$499$	− 18.7178i	− 0.837924i	−0.908004	−	0.418962i	$-0.862394\pi$
$499$	− 18.7178i	− 0.837924i	0.908004	−	0.418962i	$-0.137606\pi$
$500$	− 1.00000i	− 0.0447214i
$501$	4.71780	0.210776
$502$	−17.3589	−0.774766
$503$	29.7945i	1.32847i	0.747523	+	0.664235i	$0.231243\pi$
$503$	29.7945i	1.32847i	−0.747523	+	0.664235i	$0.768757\pi$
$504$	− 8.71780i	− 0.388322i
$505$	7.07670i	0.314909i
$506$	−44.7945	−1.99136
$507$	− 1.71780i	− 0.0762901i
$508$	−4.35890	−0.193395
$509$	10.9233	0.484167	0.242083	−	0.970255i	$-0.422169\pi$
$509$	10.9233	0.484167	0.242083	+	0.970255i	$0.422169\pi$
$510$	0	0
$511$	32.0767	1.41899
$512$	1.00000	0.0441942
$513$	− 6.79449i	− 0.299984i
$514$	1.92330	0.0848333
$515$	4.35890i	0.192076i
$516$	10.0000i	0.440225i
$517$	57.4356i	2.52601i
$518$	−14.6411	−0.643293
$519$	4.71780	0.207088
$520$	− 3.35890i	− 0.147297i
$521$	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
$523$	3.71780	0.162568	0.0812840	−	0.996691i	$-0.474098\pi$
$523$	3.71780	0.162568	0.0812840	+	0.996691i	$0.474098\pi$
$524$	22.0767i	0.964425i
$525$	−4.35890	−0.190238
$526$	2.35890	0.102853
$527$	0	0
$528$	−5.35890	−0.233216
$529$	−46.8712	−2.03788
$530$	10.7178i	0.465551i
$531$	12.0000	0.520756
$532$	5.92330i	0.256808i
$533$	− 25.9233i	− 1.12286i
$534$	− 9.00000i	− 0.389468i
$535$	16.7178	0.722773
$536$	−7.00000	−0.302354
$537$	− 15.4356i	− 0.666095i
$538$	9.64110i	0.415657i
$539$	− 64.3068i	− 2.76989i
$540$	−5.00000	−0.215166
$541$	− 30.3589i	− 1.30523i	−0.757689	−	0.652616i	$-0.773672\pi$
$541$	− 30.3589i	− 1.30523i	0.757689	−	0.652616i	$-0.226328\pi$
$542$	−30.7945	−1.32274
$543$	−17.0767	−0.732831
$544$	0	0
$545$	−1.64110	−0.0702970
$546$	−14.6411	−0.626581
$547$	− 22.4356i	− 0.959277i	−0.877466	−	0.479638i	$-0.840768\pi$
$547$	− 22.4356i	− 0.959277i	0.877466	−	0.479638i	$-0.159232\pi$
$548$	−6.00000	−0.256307
$549$	20.0000i	0.853579i
$550$	− 5.35890i	− 0.228504i
$551$	0	0
$552$	−8.35890	−0.355778
$553$	−58.2301	−2.47620
$554$	− 21.3589i	− 0.907453i
$555$	3.35890i	0.142577i
$556$	− 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	$0.280765\pi$
$558$	2.71780i	0.115054i
$559$	−33.5890	−1.42066
$560$	4.35890	0.184197
$561$	0	0
$562$	−4.28220	−0.180634
$563$	7.28220	0.306908	0.153454	−	0.988156i	$-0.450960\pi$
$563$	7.28220	0.306908	0.153454	+	0.988156i	$0.450960\pi$
$564$	10.7178i	0.451301i
$565$	16.0767	0.676352
$566$	4.43560i	0.186442i
$567$	− 4.35890i	− 0.183057i
$568$	− 0.641101i	− 0.0269000i
$569$	−39.8712	−1.67149	−0.835744	−	0.549120i	$-0.814963\pi$
$569$	−39.8712	−1.67149	−0.835744	+	0.549120i	$0.814963\pi$
$570$	1.35890	0.0569180
$571$	− 9.92330i	− 0.415277i	−0.978206	−	0.207639i	$-0.933422\pi$
$571$	− 9.92330i	− 0.415277i	0.978206	−	0.207639i	$-0.0665778\pi$
$572$	− 18.0000i	− 0.752618i
$573$	− 17.3589i	− 0.725178i
$574$	33.6411	1.40415
$575$	− 8.35890i	− 0.348590i
$576$	2.00000	0.0833333
$577$	−34.2301	−1.42502	−0.712509	−	0.701663i	$-0.752441\pi$
$577$	−34.2301	−1.42502	−0.712509	+	0.701663i	$0.752441\pi$
$578$	0	0
$579$	−4.64110	−0.192878
$580$	0	0
$581$	− 7.48771i	− 0.310642i
$582$	−13.3589	−0.553744
$583$	57.4356i	2.37874i
$584$	7.35890i	0.304513i
$585$	− 6.71780i	− 0.277747i
$586$	16.0767	0.664122
$587$	36.4356	1.50386	0.751929	−	0.659244i	$-0.229124\pi$
$587$	36.4356	1.50386	0.751929	+	0.659244i	$0.229124\pi$
$588$	− 12.0000i	− 0.494872i
$589$	− 1.84661i	− 0.0760881i
$590$	6.00000i	0.247016i
$591$	−10.0767	−0.414500
$592$	− 3.35890i	− 0.138050i
$593$	−20.7945	−0.853928	−0.426964	−	0.904269i	$-0.640417\pi$
$593$	−20.7945	−0.853928	−0.426964	+	0.904269i	$0.640417\pi$
$594$	−26.7945	−1.09939
$595$	0	0
$596$	2.35890	0.0966243
$597$	−7.35890	−0.301180
$598$	− 28.0767i	− 1.14814i
$599$	−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
$601$	33.1534i	1.35236i	0.736739	+	0.676178i	$0.236365\pi$
$601$	33.1534i	1.35236i	−0.736739	+	0.676178i	$0.763635\pi$
$602$	− 43.5890i	− 1.77655i
$603$	−14.0000	−0.570124
$604$	−2.00000	−0.0813788
$605$	− 17.7178i	− 0.720331i
$606$	7.07670i	0.287471i
$607$	− 38.7178i	− 1.57151i	−0.618540	−	0.785753i	$-0.712275\pi$
$607$	− 38.7178i	− 1.57151i	0.618540	−	0.785753i	$-0.287725\pi$
$608$	−1.35890	−0.0551106
$609$	0	0
$610$	−10.0000	−0.404888
$611$	−36.0000	−1.45640
$612$	0	0
$613$	−3.35890	−0.135665	−0.0678323	−	0.997697i	$-0.521608\pi$
$613$	−3.35890	−0.135665	−0.0678323	+	0.997697i	$0.521608\pi$
$614$	2.00000	0.0807134
$615$	− 7.71780i	− 0.311212i
$616$	23.3589	0.941157
$617$	− 2.56440i	− 0.103239i	−0.998667	−	0.0516195i	$-0.983562\pi$
$617$	− 2.56440i	− 0.103239i	0.998667	−	0.0516195i	$-0.0164383\pi$
$618$	4.35890i	0.175341i
$619$	− 16.7945i	− 0.675028i	−0.941321	−	0.337514i	$-0.890414\pi$
$619$	− 16.7945i	− 0.675028i	0.941321	−	0.337514i	$-0.109586\pi$
$620$	−1.35890	−0.0545747
$621$	−41.7945	−1.67716
$622$	4.71780i	0.189166i
$623$	39.2301i	1.57172i
$624$	− 3.35890i	− 0.134464i
$625$	1.00000	0.0400000
$626$	− 19.3589i	− 0.773737i
$627$	7.28220	0.290823
$628$	0.717798	0.0286433
$629$	0	0
$630$	8.71780	0.347325
$631$	13.4356	0.534863	0.267431	−	0.963577i	$-0.413825\pi$
$631$	13.4356	0.534863	0.267431	+	0.963577i	$0.413825\pi$
$632$	− 13.3589i	− 0.531388i
$633$	25.3589	1.00793
$634$	− 4.07670i	− 0.161906i
$635$	− 4.35890i	− 0.172978i
$636$	10.7178i	0.424988i
$637$	40.3068	1.59701
$638$	0	0
$639$	− 1.28220i	− 0.0507231i
$640$	1.00000i	0.0395285i
$641$	27.8712i	1.10085i	0.834886	+	0.550423i	$0.185534\pi$
$641$	27.8712i	1.10085i	−0.834886	+	0.550423i	$0.814466\pi$
$642$	16.7178	0.659799
$643$	− 31.8712i	− 1.25688i	−0.777859	−	0.628439i	$-0.783694\pi$
$643$	− 31.8712i	− 1.25688i	0.777859	−	0.628439i	$-0.216306\pi$
$644$	36.4356	1.43576
$645$	−10.0000	−0.393750
$646$	0	0
$647$	−7.07670	−0.278214	−0.139107	−	0.990277i	$-0.544423\pi$
$647$	−7.07670	−0.278214	−0.139107	+	0.990277i	$0.544423\pi$
$648$	1.00000	0.0392837
$649$	32.1534i	1.26213i
$650$	3.35890	0.131747
$651$	5.92330i	0.232153i
$652$	− 11.0000i	− 0.430793i
$653$	− 9.43560i	− 0.369243i	−0.982810	−	0.184622i	$-0.940894\pi$
$653$	− 9.43560i	− 0.369243i	0.982810	−	0.184622i	$-0.0591060\pi$
$654$	−1.64110	−0.0641721
$655$	−22.0767	−0.862608
$656$	7.71780i	0.301329i
$657$	14.7178i	0.574196i
$658$	− 46.7178i	− 1.82125i
$659$	−20.1534	−0.785065	−0.392532	−	0.919738i	$-0.628401\pi$
$659$	−20.1534	−0.785065	−0.392532	+	0.919738i	$0.628401\pi$
$660$	− 5.35890i	− 0.208595i
$661$	−35.6411	−1.38628	−0.693139	−	0.720804i	$-0.743773\pi$
$661$	−35.6411	−1.38628	−0.693139	+	0.720804i	$0.743773\pi$
$662$	17.2822	0.671692
$663$	0	0
$664$	1.71780	0.0666635
$665$	−5.92330	−0.229696
$666$	− 6.71780i	− 0.260309i
$667$	0	0
$668$	− 4.71780i	− 0.182537i
$669$	− 4.35890i	− 0.168525i
$670$	− 7.00000i	− 0.270434i
$671$	−53.5890	−2.06878
$672$	4.35890	0.168148
$673$	− 18.7178i	− 0.721518i	−0.932659	−	0.360759i	$-0.882518\pi$
$673$	− 18.7178i	− 0.721518i	0.932659	−	0.360759i	$-0.117482\pi$
$674$	30.0767i	1.15851i
$675$	− 5.00000i	− 0.192450i
$676$	−1.71780	−0.0660691
$677$	30.0000i	1.15299i	0.817099	+	0.576497i	$0.195581\pi$
$677$	30.0000i	1.15299i	−0.817099	+	0.576497i	$0.804419\pi$
$678$	16.0767	0.617422
$679$	58.2301	2.23467
$680$	0	0
$681$	9.00000	0.344881
$682$	−7.28220	−0.278850
$683$	18.4356i	0.705418i	0.935733	+	0.352709i	$0.114740\pi$
$683$	18.4356i	0.705418i	−0.935733	+	0.352709i	$0.885260\pi$
$684$	−2.71780	−0.103918
$685$	− 6.00000i	− 0.229248i
$686$	21.7945i	0.832118i
$687$	7.64110i	0.291526i
$688$	10.0000	0.381246
$689$	−36.0000	−1.37149
$690$	− 8.35890i	− 0.318218i
$691$	0.564404i	0.0214709i	0.999942	+	0.0107355i	$0.00341727\pi$
$691$	0.564404i	0.0214709i	−0.999942	+	0.0107355i	$0.996583\pi$
$692$	− 4.71780i	− 0.179344i
$693$	46.7178	1.77466
$694$	− 16.7178i	− 0.634599i
$695$	10.0000	0.379322
$696$	0	0
$697$	0	0
$698$	27.7945	1.05204
$699$	−10.7178	−0.405384
$700$	4.35890i	0.164751i
$701$	41.7945	1.57856	0.789278	−	0.614036i	$-0.210455\pi$
$701$	41.7945	1.57856	0.789278	+	0.614036i	$0.210455\pi$
$702$	− 16.7945i	− 0.633867i
$703$	4.56440i	0.172150i
$704$	5.35890i	0.201971i
$705$	−10.7178	−0.403656
$706$	10.0767	0.379241
$707$	− 30.8466i	− 1.16011i
$708$	6.00000i	0.225494i
$709$	45.0767i	1.69289i	0.532475	+	0.846445i	$0.321262\pi$
$709$	45.0767i	1.69289i	−0.532475	+	0.846445i	$0.678738\pi$
$710$	0.641101	0.0240601
$711$	− 26.7178i	− 1.00200i
$712$	−9.00000	−0.337289
$713$	−11.3589	−0.425394
$714$	0	0
$715$	18.0000	0.673162
$716$	−15.4356	−0.576855
$717$	− 12.0000i	− 0.448148i
$718$	6.64110	0.247844
$719$	39.4356i	1.47070i	0.677688	+	0.735350i	$0.262982\pi$
$719$	39.4356i	1.47070i	−0.677688	+	0.735350i	$0.737018\pi$
$720$	2.00000i	0.0745356i
$721$	− 19.0000i	− 0.707597i
$722$	−17.1534	−0.638383
$723$	29.7178	1.10522
$724$	17.0767i	0.634650i
$725$	0	0
$726$	− 17.7178i	− 0.657569i
$727$	41.4356	1.53676	0.768381	−	0.639993i	$-0.221063\pi$
$727$	41.4356	1.53676	0.768381	+	0.639993i	$0.221063\pi$
$728$	14.6411i	0.542635i
$729$	−13.0000	−0.481481
$730$	−7.35890	−0.272365
$731$	0	0
$732$	−10.0000	−0.369611
$733$	−41.4356	−1.53046	−0.765229	−	0.643758i	$-0.777374\pi$
$733$	−41.4356	−1.53046	−0.765229	+	0.643758i	$0.777374\pi$
$734$	− 17.4356i	− 0.643560i
$735$	12.0000	0.442627
$736$	8.35890i	0.308113i
$737$	− 37.5123i	− 1.38178i
$738$	15.4356i	0.568192i
$739$	20.7178	0.762117	0.381058	−	0.924551i	$-0.375560\pi$
$739$	20.7178	0.762117	0.381058	+	0.924551i	$0.375560\pi$
$740$	3.35890	0.123476
$741$	4.56440i	0.167678i
$742$	− 46.7178i	− 1.71506i
$743$	− 21.6411i	− 0.793935i	−0.917833	−	0.396967i	$-0.870063\pi$
$743$	− 21.6411i	− 0.793935i	0.917833	−	0.396967i	$-0.129937\pi$
$744$	−1.35890	−0.0498197
$745$	2.35890i	0.0864234i
$746$	22.7945	0.834566
$747$	3.43560	0.125702
$748$	0	0
$749$	−72.8712	−2.66266
$750$	1.00000	0.0365148
$751$	10.5644i	0.385501i	0.981248	+	0.192750i	$0.0617407\pi$
$751$	10.5644i	0.385501i	−0.981248	+	0.192750i	$0.938259\pi$
$752$	10.7178	0.390838
$753$	− 17.3589i	− 0.632593i
$754$	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	$0.406090\pi$
$758$	− 8.07670i	− 0.293359i
$759$	− 44.7945i	− 1.62594i
$760$	− 1.35890i	− 0.0492924i
$761$	−22.7178	−0.823520	−0.411760	−	0.911292i	$-0.635086\pi$
$761$	−22.7178	−0.823520	−0.411760	+	0.911292i	$0.635086\pi$
$762$	− 4.35890i	− 0.157906i
$763$	7.15339	0.258970
$764$	−17.3589	−0.628023
$765$	0	0
$766$	17.7945	0.642941
$767$	−20.1534	−0.727697
$768$	1.00000i	0.0360844i
$769$	12.2822	0.442908	0.221454	−	0.975171i	$-0.428920\pi$
$769$	12.2822	0.442908	0.221454	+	0.975171i	$0.428920\pi$
$770$	23.3589i	0.841796i
$771$	1.92330i	0.0692661i
$772$	4.64110i	0.167037i
$773$	4.71780	0.169687	0.0848437	−	0.996394i	$-0.472961\pi$
$773$	4.71780	0.169687	0.0848437	+	0.996394i	$0.472961\pi$
$774$	20.0000	0.718885
$775$	− 1.35890i	− 0.0488131i
$776$	13.3589i	0.479557i
$777$	− 14.6411i	− 0.525247i
$778$	38.1534	1.36787
$779$	− 10.4877i	− 0.375761i
$780$	3.35890	0.120268
$781$	3.43560	0.122935
$782$	0	0
$783$	0	0
$784$	−12.0000	−0.428571
$785$	0.717798i	0.0256193i
$786$	−22.0767	−0.787450
$787$	13.1534i	0.468868i	0.972132	+	0.234434i	$0.0753236\pi$
$787$	13.1534i	0.468868i	−0.972132	+	0.234434i	$0.924676\pi$
$788$	10.0767i	0.358967i
$789$	2.35890i	0.0839790i
$790$	13.3589	0.475288
$791$	−70.0767	−2.49164
$792$	10.7178i	0.380840i
$793$	− 33.5890i	− 1.19278i
$794$	6.79449i	0.241128i
$795$	−10.7178	−0.380121
$796$	7.35890i	0.260829i
$797$	1.92330	0.0681269	0.0340634	−	0.999420i	$-0.489155\pi$
$797$	1.92330	0.0681269	0.0340634	+	0.999420i	$0.489155\pi$
$798$	−5.92330	−0.209683
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	−18.0000	−0.635999
$802$	− 6.43560i	− 0.227249i
$803$	−39.4356	−1.39165
$804$	− 7.00000i	− 0.246871i
$805$	36.4356i	1.28419i
$806$	− 4.56440i	− 0.160774i
$807$	−9.64110	−0.339383
$808$	7.07670	0.248957
$809$	− 29.1534i	− 1.02498i	−0.858694	−	0.512489i	$-0.828723\pi$
$809$	− 29.1534i	− 1.02498i	0.858694	−	0.512489i	$-0.171277\pi$
$810$	1.00000i	0.0351364i
$811$	0.0766968i	0.00269319i	0.999999	+	0.00134659i	$0.000428635\pi$
$811$	0.0766968i	0.00269319i	−0.999999	+	0.00134659i	$0.999571\pi$
$812$	0	0
$813$	− 30.7945i	− 1.08001i
$814$	18.0000	0.630900
$815$	11.0000	0.385313
$816$	0	0
$817$	−13.5890	−0.475419
$818$	14.4356	0.504729
$819$	29.2822i	1.02320i
$820$	−7.71780	−0.269517
$821$	− 19.0767i	− 0.665781i	−0.942965	−	0.332891i	$-0.891976\pi$
$821$	− 19.0767i	− 0.665781i	0.942965	−	0.332891i	$-0.108024\pi$
$822$	− 6.00000i	− 0.209274i
$823$	19.6411i	0.684646i	0.939582	+	0.342323i	$0.111214\pi$
$823$	19.6411i	0.684646i	−0.939582	+	0.342323i	$0.888786\pi$
$824$	4.35890	0.151849
$825$	5.35890	0.186573
$826$	− 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	$-0.155545\pi$
$828$	16.7178i	0.580984i
$829$	28.3589	0.984945	0.492473	−	0.870328i	$-0.336093\pi$
$829$	28.3589	0.984945	0.492473	+	0.870328i	$0.336093\pi$
$830$	1.71780i	0.0596256i
$831$	21.3589	0.740932
$832$	−3.35890	−0.116449
$833$	0	0
$834$	10.0000	0.346272
$835$	4.71780	0.163266
$836$	− 7.28220i	− 0.251860i
$837$	−6.79449	−0.234852
$838$	− 27.4356i	− 0.947747i
$839$	− 7.92330i	− 0.273543i	−0.990603	−	0.136771i	$-0.956327\pi$
$839$	− 7.92330i	− 0.273543i	0.990603	−	0.136771i	$-0.0436726\pi$
$840$	4.35890i	0.150396i
$841$	29.0000	1.00000
$842$	8.00000	0.275698
$843$	− 4.28220i	− 0.147487i
$844$	− 25.3589i	− 0.872889i
$845$	− 1.71780i	− 0.0590940i
$846$	21.4356	0.736971
$847$	77.2301i	2.65366i
$848$	10.7178	0.368051
$849$	−4.43560	−0.152229
$850$	0	0
$851$	28.0767	0.962457
$852$	0.641101	0.0219638
$853$	44.6411i	1.52848i	0.644931	+	0.764241i	$0.276886\pi$
$853$	44.6411i	1.52848i	−0.644931	+	0.764241i	$0.723114\pi$
$854$	43.5890	1.49158
$855$	− 2.71780i	− 0.0929467i
$856$	− 16.7178i	− 0.571403i
$857$	16.7178i	0.571069i	0.958368	+	0.285535i	$0.0921711\pi$
$857$	16.7178i	0.571069i	−0.958368	+	0.285535i	$0.907829\pi$
$858$	18.0000	0.614510
$859$	20.7178	0.706882	0.353441	−	0.935457i	$-0.385011\pi$
$859$	20.7178	0.706882	0.353441	+	0.935457i	$0.385011\pi$
$860$	10.0000i	0.340997i
$861$	33.6411i	1.14649i
$862$	5.35890i	0.182525i
$863$	−30.8712	−1.05087	−0.525434	−	0.850834i	$-0.676097\pi$
$863$	−30.8712	−1.05087	−0.525434	+	0.850834i	$0.676097\pi$
$864$	5.00000i	0.170103i
$865$	4.71780	0.160410
$866$	−8.64110	−0.293637
$867$	0	0
$868$	5.92330	0.201050
$869$	71.5890	2.42849
$870$	0	0
$871$	23.5123	0.796684
$872$	1.64110i	0.0555747i
$873$	26.7178i	0.904260i
$874$	− 11.3589i	− 0.384220i
$875$	−4.35890	−0.147358
$876$	−7.35890	−0.248634
$877$	− 42.7178i	− 1.44248i	−0.692687	−	0.721239i	$-0.743573\pi$
$877$	− 42.7178i	− 1.44248i	0.692687	−	0.721239i	$-0.256427\pi$
$878$	− 30.1534i	− 1.01763i
$879$	16.0767i	0.542254i
$880$	−5.35890	−0.180648
$881$	15.4356i	0.520038i	0.965603	+	0.260019i	$0.0837289\pi$
$881$	15.4356i	0.520038i	−0.965603	+	0.260019i	$0.916271\pi$
$882$	−24.0000	−0.808122
$883$	−8.71780	−0.293377	−0.146689	−	0.989183i	$-0.546862\pi$
$883$	−8.71780	−0.293377	−0.146689	+	0.989183i	$0.546862\pi$
$884$	0	0
$885$	−6.00000	−0.201688
$886$	−35.1534	−1.18100
$887$	19.0767i	0.640533i	0.947328	+	0.320266i	$0.103772\pi$
$887$	19.0767i	0.640533i	−0.947328	+	0.320266i	$0.896228\pi$
$888$	3.35890	0.112717
$889$	19.0000i	0.637240i
$890$	− 9.00000i	− 0.301681i
$891$	5.35890i	0.179530i
$892$	−4.35890	−0.145947
$893$	−14.5644	−0.487379
$894$	2.35890i	0.0788934i
$895$	− 15.4356i	− 0.515955i
$896$	− 4.35890i	− 0.145621i
$897$	28.0767	0.937454
$898$	0	0
$899$	0	0
$900$	−2.00000	−0.0666667
$901$	0	0
$902$	−41.3589	−1.37710
$903$	43.5890	1.45055
$904$	− 16.0767i	− 0.534703i
$905$	−17.0767	−0.567649
$906$	− 2.00000i	− 0.0664455i
$907$	30.1534i	1.00123i	0.865671	+	0.500614i	$0.166892\pi$
$907$	30.1534i	1.00123i	−0.865671	+	0.500614i	$0.833108\pi$
$908$	− 9.00000i	− 0.298675i
$909$	14.1534	0.469438
$910$	−14.6411	−0.485348
$911$	− 7.28220i	− 0.241270i	−0.992697	−	0.120635i	$-0.961507\pi$
$911$	− 7.28220i	− 0.241270i	0.992697	−	0.120635i	$-0.0384931\pi$
$912$	− 1.35890i	− 0.0449976i
$913$	9.20551i	0.304658i
$914$	29.5123	0.976180
$915$	− 10.0000i	− 0.330590i
$916$	7.64110	0.252469
$917$	96.2301	3.17780
$918$	0	0
$919$	5.43560	0.179304	0.0896519	−	0.995973i	$-0.471425\pi$
$919$	5.43560	0.179304	0.0896519	+	0.995973i	$0.471425\pi$
$920$	−8.35890	−0.275585
$921$	2.00000i	0.0659022i
$922$	1.07670	0.0354591
$923$	2.15339i	0.0708798i
$924$	23.3589i	0.768451i
$925$	3.35890i	0.110440i
$926$	15.0767	0.495451
$927$	8.71780	0.286330
$928$	0	0
$929$	34.2822i	1.12476i	0.826878	+	0.562381i	$0.190115\pi$
$929$	34.2822i	1.12476i	−0.826878	+	0.562381i	$0.809885\pi$
$930$	− 1.35890i	− 0.0445601i
$931$	16.3068	0.534433
$932$	10.7178i	0.351073i
$933$	−4.71780	−0.154454
$934$	1.71780	0.0562081
$935$	0	0
$936$	−6.71780	−0.219578
$937$	−37.3589	−1.22046	−0.610231	−	0.792224i	$-0.708923\pi$
$937$	−37.3589	−1.22046	−0.610231	+	0.792224i	$0.708923\pi$
$938$	30.5123i	0.996262i
$939$	19.3589	0.631754
$940$	10.7178i	0.349576i
$941$	− 39.2301i	− 1.27886i	−0.768847	−	0.639432i	$-0.779169\pi$
$941$	− 39.2301i	− 1.27886i	0.768847	−	0.639432i	$-0.220831\pi$
$942$	0.717798i	0.0233871i
$943$	−64.5123	−2.10081
$944$	6.00000	0.195283
$945$	21.7945i	0.708975i
$946$	53.5890i	1.74233i
$947$	3.00000i	0.0974869i	0.998811	+	0.0487435i	$0.0155217\pi$
$947$	3.00000i	0.0974869i	−0.998811	+	0.0487435i	$0.984478\pi$
$948$	13.3589	0.433877
$949$	− 24.7178i	− 0.802374i
$950$	1.35890	0.0440885
$951$	4.07670	0.132196
$952$	0	0
$953$	30.2301	0.979249	0.489624	−	0.871933i	$-0.337134\pi$
$953$	30.2301	0.979249	0.489624	+	0.871933i	$0.337134\pi$
$954$	21.4356	0.694003
$955$	− 17.3589i	− 0.561721i
$956$	−12.0000	−0.388108
$957$	0	0
$958$	− 32.1534i	− 1.03883i
$959$	26.1534i	0.844537i
$960$	−1.00000	−0.0322749
$961$	29.1534	0.940432
$962$	11.2822i	0.363753i
$963$	− 33.4356i	− 1.07745i
$964$	− 29.7178i	− 0.957146i
$965$	−4.64110	−0.149402
$966$	36.4356i	1.17230i
$967$	3.79449	0.122023	0.0610114	−	0.998137i	$-0.480567\pi$
$967$	3.79449	0.122023	0.0610114	+	0.998137i	$0.480567\pi$
$968$	−17.7178	−0.569472
$969$	0	0
$970$	−13.3589	−0.428928
$971$	16.7178	0.536500	0.268250	−	0.963349i	$-0.413555\pi$
$971$	16.7178	0.536500	0.268250	+	0.963349i	$0.413555\pi$
$972$	16.0000i	0.513200i
$973$	−43.5890	−1.39740
$974$	− 29.6411i	− 0.949762i
$975$	3.35890i	0.107571i
$976$	10.0000i	0.320092i
$977$	−56.1534	−1.79651	−0.898253	−	0.439478i	$-0.855163\pi$
$977$	−56.1534	−1.79651	−0.898253	+	0.439478i	$0.855163\pi$
$978$	11.0000	0.351741
$979$	− 48.2301i	− 1.54144i
$980$	− 12.0000i	− 0.383326i
$981$	3.28220i	0.104793i
$982$	12.0000	0.382935
$983$	− 43.0767i	− 1.37393i	−0.726689	−	0.686967i	$-0.758942\pi$
$983$	− 43.0767i	− 1.37393i	0.726689	−	0.686967i	$-0.241058\pi$
$984$	−7.71780	−0.246034
$985$	−10.0767	−0.321070
$986$	0	0
$987$	46.7178	1.48704
$988$	4.56440	0.145213
$989$	83.5890i	2.65798i
$990$	−10.7178	−0.340634
$991$	− 19.4356i	− 0.617392i	−0.951161	−	0.308696i	$-0.900107\pi$
$991$	− 19.4356i	− 0.617392i	0.951161	−	0.308696i	$-0.0998926\pi$
$992$	1.35890i	0.0431451i
$993$	17.2822i	0.548434i
$994$	−2.79449	−0.0886360
$995$	−7.35890	−0.233293
$996$	1.71780i	0.0544305i
$997$	− 1.35890i	− 0.0430368i	−0.999768	−	0.0215184i	$-0.993150\pi$
$997$	− 1.35890i	− 0.0430368i	0.999768	−	0.0215184i	$-0.00685004\pi$
$998$	− 18.7178i	− 0.592502i
$999$	16.7945	0.531354

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2890.2.b.k.2311.3		4
17.4	even	4		2890.2.a.s.1.2	yes	2
17.13	even	4		2890.2.a.r.1.1	✓	2
17.16	even	2	inner	2890.2.b.k.2311.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2890.2.a.r.1.1	✓	2	17.13	even	4
2890.2.a.s.1.2	yes	2	17.4	even	4
2890.2.b.k.2311.2		4	17.16	even	2	inner
2890.2.b.k.2311.3		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} -4.35890i q^{7} +1.00000 q^{8} +2.00000 q^{9} +1.00000i q^{10} +5.35890i q^{11} +1.00000i q^{12} -3.35890 q^{13} -4.35890i q^{14} -1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{18} -1.35890 q^{19} +1.00000i q^{20} +4.35890 q^{21} +5.35890i q^{22} +8.35890i q^{23} +1.00000i q^{24} -1.00000 q^{25} -3.35890 q^{26} +5.00000i q^{27} -4.35890i q^{28} -1.00000 q^{30} +1.35890i q^{31} +1.00000 q^{32} -5.35890 q^{33} +4.35890 q^{35} +2.00000 q^{36} -3.35890i q^{37} -1.35890 q^{38} -3.35890i q^{39} +1.00000i q^{40} +7.71780i q^{41} +4.35890 q^{42} +10.0000 q^{43} +5.35890i q^{44} +2.00000i q^{45} +8.35890i q^{46} +10.7178 q^{47} +1.00000i q^{48} -12.0000 q^{49} -1.00000 q^{50} -3.35890 q^{52} +10.7178 q^{53} +5.00000i q^{54} -5.35890 q^{55} -4.35890i q^{56} -1.35890i q^{57} +6.00000 q^{59} -1.00000 q^{60} +10.0000i q^{61} +1.35890i q^{62} -8.71780i q^{63} +1.00000 q^{64} -3.35890i q^{65} -5.35890 q^{66} -7.00000 q^{67} -8.35890 q^{69} +4.35890 q^{70} -0.641101i q^{71} +2.00000 q^{72} +7.35890i q^{73} -3.35890i q^{74} -1.00000i q^{75} -1.35890 q^{76} +23.3589 q^{77} -3.35890i q^{78} -13.3589i q^{79} +1.00000i q^{80} +1.00000 q^{81} +7.71780i q^{82} +1.71780 q^{83} +4.35890 q^{84} +10.0000 q^{86} +5.35890i q^{88} -9.00000 q^{89} +2.00000i q^{90} +14.6411i q^{91} +8.35890i q^{92} -1.35890 q^{93} +10.7178 q^{94} -1.35890i q^{95} +1.00000i q^{96} +13.3589i q^{97} -12.0000 q^{98} +10.7178i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} + 8 q^{9} + 4 q^{13} - 4 q^{15} + 4 q^{16} + 8 q^{18} + 12 q^{19} - 4 q^{25} + 4 q^{26} - 4 q^{30} + 4 q^{32} - 4 q^{33} + 8 q^{36} + 12 q^{38} + 40 q^{43} + 8 q^{47} - 48 q^{49}+ \cdots - 48 q^{98}+O(q^{100}) \) 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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