LMFDB - Embedded newform 2890.2.b.p.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)

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);

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")

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 = [8,8,0,8,0,0,0,8,-16,0,0,0,24] 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:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 170)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2311.4
Root			$-0.382683 - 0.923880i$ of defining polynomial
Character	$\chi$	$=$	2890.2311
Dual form			2890.2.b.p.2311.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} -1.43355i q^{3} +1.00000 q^{4} +1.00000i q^{5} -1.43355i q^{6} +0.648847i q^{7} +1.00000 q^{8} +0.944947 q^{9} +1.00000i q^{10} +3.73124i q^{11} -1.43355i q^{12} +4.08239 q^{13} +0.648847i q^{14} +1.43355 q^{15} +1.00000 q^{16} +0.944947 q^{18} -4.28931 q^{19} +1.00000i q^{20} +0.930151 q^{21} +3.73124i q^{22} +7.27677i q^{23} -1.43355i q^{24} -1.00000 q^{25} +4.08239 q^{26} -5.65526i q^{27} +0.648847i q^{28} -1.80109i q^{29} +1.43355 q^{30} +4.97388i q^{31} +1.00000 q^{32} +5.34890 q^{33} -0.648847 q^{35} +0.944947 q^{36} +8.15640i q^{37} -4.28931 q^{38} -5.85229i q^{39} +1.00000i q^{40} +1.93694i q^{41} +0.930151 q^{42} +1.04667 q^{43} +3.73124i q^{44} +0.944947i q^{45} +7.27677i q^{46} -4.71031 q^{47} -1.43355i q^{48} +6.57900 q^{49} -1.00000 q^{50} +4.08239 q^{52} -8.72286 q^{53} -5.65526i q^{54} -3.73124 q^{55} +0.648847i q^{56} +6.14892i q^{57} -1.80109i q^{58} +10.3881 q^{59} +1.43355 q^{60} +3.54168i q^{61} +4.97388i q^{62} +0.613126i q^{63} +1.00000 q^{64} +4.08239i q^{65} +5.34890 q^{66} +3.48022 q^{67} +10.4316 q^{69} -0.648847 q^{70} -13.2729i q^{71} +0.944947 q^{72} +5.99321i q^{73} +8.15640i q^{74} +1.43355i q^{75} -4.28931 q^{76} -2.42100 q^{77} -5.85229i q^{78} +0.293157i q^{79} +1.00000i q^{80} -5.27223 q^{81} +1.93694i q^{82} -2.48406 q^{83} +0.930151 q^{84} +1.04667 q^{86} -2.58194 q^{87} +3.73124i q^{88} +3.59539 q^{89} +0.944947i q^{90} +2.64885i q^{91} +7.27677i q^{92} +7.13028 q^{93} -4.71031 q^{94} -4.28931i q^{95} -1.43355i q^{96} -19.2355i q^{97} +6.57900 q^{98} +3.52582i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 8 q^{4} + 8 q^{8} - 16 q^{9} + 24 q^{13} + 8 q^{15} + 8 q^{16} - 16 q^{18} + 24 q^{19} - 16 q^{21} - 8 q^{25} + 24 q^{26} + 8 q^{30} + 8 q^{32} + 16 q^{33} - 16 q^{36} + 24 q^{38} - 16 q^{42}+ \cdots + 24 q^{98}+O(q^{100}) $ 8 * q + 8 * q^2 + 8 * q^4 + 8 * q^8 - 16 * q^9 + 24 * q^13 + 8 * q^15 + 8 * q^16 - 16 * q^18 + 24 * q^19 - 16 * q^21 - 8 * q^25 + 24 * q^26 + 8 * q^30 + 8 * q^32 + 16 * q^33 - 16 * q^36 + 24 * q^38 - 16 * q^42 - 16 * q^43 - 8 * q^47 + 24 * q^49 - 8 * q^50 + 24 * q^52 - 8 * q^53 - 16 * q^55 + 24 * q^59 + 8 * q^60 + 8 * q^64 + 16 * q^66 - 16 * q^69 - 16 * q^72 + 24 * q^76 - 48 * q^77 - 8 * q^81 - 32 * q^83 - 16 * q^84 - 16 * q^86 + 8 * q^87 + 8 * q^89 - 56 * q^93 - 8 * q^94 + 24 * 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
$3$	− 1.43355i	− 0.827658i	−0.910355	−	0.413829i	$-0.864191\pi$
$3$	− 1.43355i	− 0.827658i	0.910355	−	0.413829i	$-0.135809\pi$
$4$	1.00000	0.500000
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	− 1.43355i	− 0.585243i
$7$	0.648847i	0.245241i	0.992454	+	0.122621i	$0.0391298\pi$
$7$	0.648847i	0.245241i	−0.992454	+	0.122621i	$0.960870\pi$
$8$	1.00000	0.353553
$9$	0.944947	0.314982
$10$	1.00000i	0.316228i
$11$	3.73124i	1.12501i	0.826794	+	0.562505i	$0.190162\pi$
$11$	3.73124i	1.12501i	−0.826794	+	0.562505i	$0.809838\pi$
$12$	− 1.43355i	− 0.413829i
$13$	4.08239	1.13225	0.566126	−	0.824319i	$-0.308442\pi$
$13$	4.08239	1.13225	0.566126	+	0.824319i	$0.308442\pi$
$14$	0.648847i	0.173412i
$15$	1.43355	0.370140
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	0.944947	0.222726
$19$	−4.28931	−0.984036	−0.492018	−	0.870585i	$-0.663741\pi$
$19$	−4.28931	−0.984036	−0.492018	+	0.870585i	$0.663741\pi$
$20$	1.00000i	0.223607i
$21$	0.930151	0.202976
$22$	3.73124i	0.795503i
$23$	7.27677i	1.51731i	0.651492	+	0.758656i	$0.274143\pi$
$23$	7.27677i	1.51731i	−0.651492	+	0.758656i	$0.725857\pi$
$24$	− 1.43355i	− 0.292621i
$25$	−1.00000	−0.200000
$26$	4.08239	0.800623
$27$	− 5.65526i	− 1.08836i
$28$	0.648847i	0.122621i
$29$	− 1.80109i	− 0.334454i	−0.985918	−	0.167227i	$-0.946519\pi$
$29$	− 1.80109i	− 0.334454i	0.985918	−	0.167227i	$-0.0534812\pi$
$30$	1.43355	0.261728
$31$	4.97388i	0.893335i	0.894700	+	0.446668i	$0.147389\pi$
$31$	4.97388i	0.893335i	−0.894700	+	0.446668i	$0.852611\pi$
$32$	1.00000	0.176777
$33$	5.34890	0.931124
$34$	0	0
$35$	−0.648847	−0.109675
$36$	0.944947	0.157491
$37$	8.15640i	1.34090i	0.741953	+	0.670452i	$0.233900\pi$
$37$	8.15640i	1.34090i	−0.741953	+	0.670452i	$0.766100\pi$
$38$	−4.28931	−0.695818
$39$	− 5.85229i	− 0.937117i
$40$	1.00000i	0.158114i
$41$	1.93694i	0.302499i	0.988496	+	0.151250i	$0.0483297\pi$
$41$	1.93694i	0.302499i	−0.988496	+	0.151250i	$0.951670\pi$
$42$	0.930151	0.143525
$43$	1.04667	0.159616	0.0798079	−	0.996810i	$-0.474569\pi$
$43$	1.04667	0.159616	0.0798079	+	0.996810i	$0.474569\pi$
$44$	3.73124i	0.562505i
$45$	0.944947i	0.140864i
$46$	7.27677i	1.07290i
$47$	−4.71031	−0.687070	−0.343535	−	0.939140i	$-0.611624\pi$
$47$	−4.71031	−0.687070	−0.343535	+	0.939140i	$0.611624\pi$
$48$	− 1.43355i	− 0.206914i
$49$	6.57900	0.939857
$50$	−1.00000	−0.141421
$51$	0	0
$52$	4.08239	0.566126
$53$	−8.72286	−1.19818	−0.599088	−	0.800683i	$-0.704470\pi$
$53$	−8.72286	−1.19818	−0.599088	+	0.800683i	$0.704470\pi$
$54$	− 5.65526i	− 0.769584i
$55$	−3.73124	−0.503120
$56$	0.648847i	0.0867058i
$57$	6.14892i	0.814445i
$58$	− 1.80109i	− 0.236494i
$59$	10.3881	1.35241	0.676207	−	0.736711i	$-0.263622\pi$
$59$	10.3881	1.35241	0.676207	+	0.736711i	$0.263622\pi$
$60$	1.43355	0.185070
$61$	3.54168i	0.453466i	0.973957	+	0.226733i	$0.0728045\pi$
$61$	3.54168i	0.453466i	−0.973957	+	0.226733i	$0.927195\pi$
$62$	4.97388i	0.631683i
$63$	0.613126i	0.0772466i
$64$	1.00000	0.125000
$65$	4.08239i	0.506358i
$66$	5.34890	0.658404
$67$	3.48022	0.425176	0.212588	−	0.977142i	$-0.431811\pi$
$67$	3.48022	0.425176	0.212588	+	0.977142i	$0.431811\pi$
$68$	0	0
$69$	10.4316	1.25581
$70$	−0.648847	−0.0775520
$71$	− 13.2729i	− 1.57521i	−0.616183	−	0.787603i	$-0.711322\pi$
$71$	− 13.2729i	− 1.57521i	0.616183	−	0.787603i	$-0.288678\pi$
$72$	0.944947	0.111363
$73$	5.99321i	0.701452i	0.936478	+	0.350726i	$0.114065\pi$
$73$	5.99321i	0.701452i	−0.936478	+	0.350726i	$0.885935\pi$
$74$	8.15640i	0.948163i
$75$	1.43355i	0.165532i
$76$	−4.28931	−0.492018
$77$	−2.42100	−0.275899
$78$	− 5.85229i	− 0.662642i
$79$	0.293157i	0.0329828i	0.999864	+	0.0164914i	$0.00524961\pi$
$79$	0.293157i	0.0329828i	−0.999864	+	0.0164914i	$0.994750\pi$
$80$	1.00000i	0.111803i
$81$	−5.27223	−0.585804
$82$	1.93694i	0.213899i
$83$	−2.48406	−0.272661	−0.136331	−	0.990663i	$-0.543531\pi$
$83$	−2.48406	−0.272661	−0.136331	+	0.990663i	$0.543531\pi$
$84$	0.930151	0.101488
$85$	0	0
$86$	1.04667	0.112865
$87$	−2.58194	−0.276813
$88$	3.73124i	0.397751i
$89$	3.59539	0.381110	0.190555	−	0.981676i	$-0.438971\pi$
$89$	3.59539	0.381110	0.190555	+	0.981676i	$0.438971\pi$
$90$	0.944947i	0.0996062i
$91$	2.64885i	0.277675i
$92$	7.27677i	0.758656i
$93$	7.13028	0.739376
$94$	−4.71031	−0.485832
$95$	− 4.28931i	− 0.440074i
$96$	− 1.43355i	− 0.146311i
$97$	− 19.2355i	− 1.95307i	−0.215366	−	0.976533i	$-0.569094\pi$
$97$	− 19.2355i	− 1.95307i	0.215366	−	0.976533i	$-0.430906\pi$
$98$	6.57900	0.664579
$99$	3.52582i	0.354359i
$100$	−1.00000	−0.100000
$101$	9.83938	0.979055	0.489527	−	0.871988i	$-0.337169\pi$
$101$	9.83938	0.979055	0.489527	+	0.871988i	$0.337169\pi$
$102$	0	0
$103$	5.96722	0.587968	0.293984	−	0.955810i	$-0.405019\pi$
$103$	5.96722	0.587968	0.293984	+	0.955810i	$0.405019\pi$
$104$	4.08239	0.400311
$105$	0.930151i	0.0907735i
$106$	−8.72286	−0.847239
$107$	− 5.43739i	− 0.525652i	−0.964843	−	0.262826i	$-0.915345\pi$
$107$	− 5.43739i	− 0.525652i	0.964843	−	0.262826i	$-0.0846546\pi$
$108$	− 5.65526i	− 0.544178i
$109$	16.3410i	1.56519i	0.622532	+	0.782594i	$0.286104\pi$
$109$	16.3410i	1.56519i	−0.622532	+	0.782594i	$0.713896\pi$
$110$	−3.73124	−0.355760
$111$	11.6926	1.10981
$112$	0.648847i	0.0613103i
$113$	2.49154i	0.234384i	0.993109	+	0.117192i	$0.0373894\pi$
$113$	2.49154i	0.234384i	−0.993109	+	0.117192i	$0.962611\pi$
$114$	6.14892i	0.575900i
$115$	−7.27677	−0.678562
$116$	− 1.80109i	− 0.167227i
$117$	3.85765	0.356639
$118$	10.3881	0.956301
$119$	0	0
$120$	1.43355	0.130864
$121$	−2.92214	−0.265649
$122$	3.54168i	0.320649i
$123$	2.77669	0.250366
$124$	4.97388i	0.446668i
$125$	− 1.00000i	− 0.0894427i
$126$	0.613126i	0.0546216i
$127$	−9.00347	−0.798929	−0.399464	−	0.916749i	$-0.630804\pi$
$127$	−9.00347	−0.798929	−0.399464	+	0.916749i	$0.630804\pi$
$128$	1.00000	0.0883883
$129$	− 1.50045i	− 0.132107i
$130$	4.08239i	0.358049i
$131$	− 15.9047i	− 1.38960i	−0.719203	−	0.694800i	$-0.755493\pi$
$131$	− 15.9047i	− 1.38960i	0.719203	−	0.694800i	$-0.244507\pi$
$132$	5.34890	0.465562
$133$	− 2.78311i	− 0.241326i
$134$	3.48022	0.300645
$135$	5.65526	0.486727
$136$	0	0
$137$	22.9252	1.95864	0.979318	−	0.202328i	$-0.0648507\pi$
$137$	22.9252	1.95864	0.979318	+	0.202328i	$0.0648507\pi$
$138$	10.4316	0.887995
$139$	11.5110i	0.976353i	0.872745	+	0.488176i	$0.162338\pi$
$139$	11.5110i	0.976353i	−0.872745	+	0.488176i	$0.837662\pi$
$140$	−0.648847	−0.0548376
$141$	6.75245i	0.568659i
$142$	− 13.2729i	− 1.11384i
$143$	15.2324i	1.27380i
$144$	0.944947	0.0787456
$145$	1.80109	0.149572
$146$	5.99321i	0.496002i
$147$	− 9.43129i	− 0.777880i
$148$	8.15640i	0.670452i
$149$	13.8251	1.13260	0.566299	−	0.824200i	$-0.308375\pi$
$149$	13.8251	1.13260	0.566299	+	0.824200i	$0.308375\pi$
$150$	1.43355i	0.117049i
$151$	18.3424	1.49268	0.746342	−	0.665563i	$-0.231809\pi$
$151$	18.3424	1.49268	0.746342	+	0.665563i	$0.231809\pi$
$152$	−4.28931	−0.347909
$153$	0	0
$154$	−2.42100	−0.195090
$155$	−4.97388	−0.399512
$156$	− 5.85229i	− 0.468559i
$157$	−17.5326	−1.39926	−0.699629	−	0.714507i	$-0.746651\pi$
$157$	−17.5326	−1.39926	−0.699629	+	0.714507i	$0.746651\pi$
$158$	0.293157i	0.0233224i
$159$	12.5046i	0.991680i
$160$	1.00000i	0.0790569i
$161$	−4.72151	−0.372107
$162$	−5.27223	−0.414226
$163$	− 8.61140i	− 0.674497i	−0.941416	−	0.337249i	$-0.890504\pi$
$163$	− 8.61140i	− 0.674497i	0.941416	−	0.337249i	$-0.109496\pi$
$164$	1.93694i	0.151250i
$165$	5.34890i	0.416411i
$166$	−2.48406	−0.192801
$167$	− 11.1026i	− 0.859143i	−0.903033	−	0.429571i	$-0.858665\pi$
$167$	− 11.1026i	− 0.859143i	0.903033	−	0.429571i	$-0.141335\pi$
$168$	0.930151	0.0717627
$169$	3.66593	0.281994
$170$	0	0
$171$	−4.05317	−0.309954
$172$	1.04667	0.0798079
$173$	− 15.3228i	− 1.16497i	−0.812842	−	0.582485i	$-0.802081\pi$
$173$	− 15.3228i	− 1.16497i	0.812842	−	0.582485i	$-0.197919\pi$
$174$	−2.58194	−0.195736
$175$	− 0.648847i	− 0.0490482i
$176$	3.73124i	0.281253i
$177$	− 14.8918i	− 1.11934i
$178$	3.59539	0.269486
$179$	−10.1899	−0.761627	−0.380813	−	0.924652i	$-0.624356\pi$
$179$	−10.1899	−0.761627	−0.380813	+	0.924652i	$0.624356\pi$
$180$	0.944947i	0.0704322i
$181$	− 13.0679i	− 0.971328i	−0.874146	−	0.485664i	$-0.838578\pi$
$181$	− 13.0679i	− 0.971328i	0.874146	−	0.485664i	$-0.161422\pi$
$182$	2.64885i	0.196346i
$183$	5.07717	0.375315
$184$	7.27677i	0.536451i
$185$	−8.15640	−0.599671
$186$	7.13028	0.522818
$187$	0	0
$188$	−4.71031	−0.343535
$189$	3.66940	0.266909
$190$	− 4.28931i	− 0.311179i
$191$	−19.7531	−1.42929	−0.714644	−	0.699489i	$-0.753411\pi$
$191$	−19.7531	−1.42929	−0.714644	+	0.699489i	$0.753411\pi$
$192$	− 1.43355i	− 0.103457i
$193$	10.3611i	0.745806i	0.927870	+	0.372903i	$0.121638\pi$
$193$	10.3611i	0.745806i	−0.927870	+	0.372903i	$0.878362\pi$
$194$	− 19.2355i	− 1.38103i
$195$	5.85229	0.419092
$196$	6.57900	0.469928
$197$	− 16.6845i	− 1.18872i	−0.804198	−	0.594361i	$-0.797405\pi$
$197$	− 16.6845i	− 1.18872i	0.804198	−	0.594361i	$-0.202595\pi$
$198$	3.52582i	0.250569i
$199$	− 9.47274i	− 0.671505i	−0.941950	−	0.335752i	$-0.891009\pi$
$199$	− 9.47274i	− 0.671505i	0.941950	−	0.335752i	$-0.108991\pi$
$200$	−1.00000	−0.0707107
$201$	− 4.98905i	− 0.351900i
$202$	9.83938	0.692296
$203$	1.16863	0.0820217
$204$	0	0
$205$	−1.93694	−0.135282
$206$	5.96722	0.415756
$207$	6.87616i	0.477926i
$208$	4.08239	0.283063
$209$	− 16.0044i	− 1.10705i
$210$	0.930151i	0.0641865i
$211$	25.1371i	1.73051i	0.501334	+	0.865254i	$0.332843\pi$
$211$	25.1371i	1.73051i	−0.501334	+	0.865254i	$0.667157\pi$
$212$	−8.72286	−0.599088
$213$	−19.0273	−1.30373
$214$	− 5.43739i	− 0.371692i
$215$	1.04667i	0.0713824i
$216$	− 5.65526i	− 0.384792i
$217$	−3.22729	−0.219082
$218$	16.3410i	1.10676i
$219$	8.59154	0.580563
$220$	−3.73124	−0.251560
$221$	0	0
$222$	11.6926	0.784754
$223$	19.0672	1.27684	0.638418	−	0.769690i	$-0.279589\pi$
$223$	19.0672	1.27684	0.638418	+	0.769690i	$0.279589\pi$
$224$	0.648847i	0.0433529i
$225$	−0.944947	−0.0629965
$226$	2.49154i	0.165735i
$227$	− 4.01217i	− 0.266297i	−0.991096	−	0.133149i	$-0.957491\pi$
$227$	− 4.01217i	− 0.266297i	0.991096	−	0.133149i	$-0.0425087\pi$
$228$	6.14892i	0.407222i
$229$	17.1455	1.13300	0.566502	−	0.824061i	$-0.308296\pi$
$229$	17.1455	1.13300	0.566502	+	0.824061i	$0.308296\pi$
$230$	−7.27677	−0.479816
$231$	3.47062i	0.228350i
$232$	− 1.80109i	− 0.118247i
$233$	− 18.9317i	− 1.24026i	−0.784499	−	0.620130i	$-0.787080\pi$
$233$	− 18.9317i	− 1.24026i	0.784499	−	0.620130i	$-0.212920\pi$
$234$	3.85765	0.252182
$235$	− 4.71031i	− 0.307267i
$236$	10.3881	0.676207
$237$	0.420255	0.0272985
$238$	0	0
$239$	−25.1215	−1.62498	−0.812488	−	0.582978i	$-0.801888\pi$
$239$	−25.1215	−1.62498	−0.812488	+	0.582978i	$0.801888\pi$
$240$	1.43355	0.0925350
$241$	− 5.98711i	− 0.385664i	−0.981232	−	0.192832i	$-0.938233\pi$
$241$	− 5.98711i	− 0.385664i	0.981232	−	0.192832i	$-0.0617672\pi$
$242$	−2.92214	−0.187843
$243$	− 9.40780i	− 0.603511i
$244$	3.54168i	0.226733i
$245$	6.57900i	0.420317i
$246$	2.77669	0.177035
$247$	−17.5107	−1.11418
$248$	4.97388i	0.315842i
$249$	3.56102i	0.225670i
$250$	− 1.00000i	− 0.0632456i
$251$	−25.4489	−1.60632	−0.803160	−	0.595763i	$-0.796850\pi$
$251$	−25.4489	−1.60632	−0.803160	+	0.595763i	$0.796850\pi$
$252$	0.613126i	0.0386233i
$253$	−27.1514	−1.70699
$254$	−9.00347	−0.564928
$255$	0	0
$256$	1.00000	0.0625000
$257$	24.8653	1.55106	0.775528	−	0.631314i	$-0.217484\pi$
$257$	24.8653	1.55106	0.775528	+	0.631314i	$0.217484\pi$
$258$	− 1.50045i	− 0.0934140i
$259$	−5.29226	−0.328845
$260$	4.08239i	0.253179i
$261$	− 1.70193i	− 0.105347i
$262$	− 15.9047i	− 0.982595i
$263$	−5.49782	−0.339010	−0.169505	−	0.985529i	$-0.554217\pi$
$263$	−5.49782	−0.339010	−0.169505	+	0.985529i	$0.554217\pi$
$264$	5.34890	0.329202
$265$	− 8.72286i	− 0.535841i
$266$	− 2.78311i	− 0.170643i
$267$	− 5.15415i	− 0.315429i
$268$	3.48022	0.212588
$269$	9.06844i	0.552913i	0.961026	+	0.276456i	$0.0891601\pi$
$269$	9.06844i	0.552913i	−0.961026	+	0.276456i	$0.910840\pi$
$270$	5.65526	0.344168
$271$	−26.0766	−1.58404	−0.792020	−	0.610496i	$-0.790970\pi$
$271$	−26.0766	−1.58404	−0.792020	+	0.610496i	$0.790970\pi$
$272$	0	0
$273$	3.79724	0.229820
$274$	22.9252	1.38496
$275$	− 3.73124i	− 0.225002i
$276$	10.4316	0.627907
$277$	2.50204i	0.150333i	0.997171	+	0.0751666i	$0.0239489\pi$
$277$	2.50204i	0.150333i	−0.997171	+	0.0751666i	$0.976051\pi$
$278$	11.5110i	0.690386i
$279$	4.70005i	0.281385i
$280$	−0.648847	−0.0387760
$281$	11.6341	0.694029	0.347015	−	0.937860i	$-0.387195\pi$
$281$	11.6341	0.694029	0.347015	+	0.937860i	$0.387195\pi$
$282$	6.75245i	0.402103i
$283$	27.1032i	1.61112i	0.592514	+	0.805560i	$0.298135\pi$
$283$	27.1032i	1.61112i	−0.592514	+	0.805560i	$0.701865\pi$
$284$	− 13.2729i	− 0.787603i
$285$	−6.14892	−0.364231
$286$	15.2324i	0.900710i
$287$	−1.25678	−0.0741852
$288$	0.944947	0.0556816
$289$	0	0
$290$	1.80109	0.105764
$291$	−27.5749	−1.61647
$292$	5.99321i	0.350726i
$293$	−19.1754	−1.12024	−0.560118	−	0.828413i	$-0.689244\pi$
$293$	−19.1754	−1.12024	−0.560118	+	0.828413i	$0.689244\pi$
$294$	− 9.43129i	− 0.550044i
$295$	10.3881i	0.604818i
$296$	8.15640i	0.474081i
$297$	21.1011	1.22441
$298$	13.8251	0.800867
$299$	29.7066i	1.71798i
$300$	1.43355i	0.0827658i
$301$	0.679129i	0.0391444i
$302$	18.3424	1.05549
$303$	− 14.1052i	− 0.810322i
$304$	−4.28931	−0.246009
$305$	−3.54168	−0.202796
$306$	0	0
$307$	8.72029	0.497693	0.248847	−	0.968543i	$-0.419949\pi$
$307$	8.72029	0.497693	0.248847	+	0.968543i	$0.419949\pi$
$308$	−2.42100	−0.137949
$309$	− 8.55428i	− 0.486636i
$310$	−4.97388	−0.282497
$311$	16.8641i	0.956275i	0.878285	+	0.478137i	$0.158688\pi$
$311$	16.8641i	0.956275i	−0.878285	+	0.478137i	$0.841312\pi$
$312$	− 5.85229i	− 0.331321i
$313$	− 7.43930i	− 0.420494i	−0.977648	−	0.210247i	$-0.932573\pi$
$313$	− 7.43930i	− 0.420494i	0.977648	−	0.210247i	$-0.0674268\pi$
$314$	−17.5326	−0.989424
$315$	−0.613126	−0.0345457
$316$	0.293157i	0.0164914i
$317$	1.29035i	0.0724730i	0.999343	+	0.0362365i	$0.0115370\pi$
$317$	1.29035i	0.0724730i	−0.999343	+	0.0362365i	$0.988463\pi$
$318$	12.5046i	0.701224i
$319$	6.72029	0.376264
$320$	1.00000i	0.0559017i
$321$	−7.79475	−0.435060
$322$	−4.72151	−0.263119
$323$	0	0
$324$	−5.27223	−0.292902
$325$	−4.08239	−0.226450
$326$	− 8.61140i	− 0.476941i
$327$	23.4256	1.29544
$328$	1.93694i	0.106950i
$329$	− 3.05627i	− 0.168498i
$330$	5.34890i	0.294447i
$331$	7.26651	0.399403	0.199702	−	0.979857i	$-0.436003\pi$
$331$	7.26651	0.399403	0.199702	+	0.979857i	$0.436003\pi$
$332$	−2.48406	−0.136331
$333$	7.70737i	0.422361i
$334$	− 11.1026i	− 0.607506i
$335$	3.48022i	0.190145i
$336$	0.930151	0.0507439
$337$	− 17.4872i	− 0.952588i	−0.879286	−	0.476294i	$-0.841980\pi$
$337$	− 17.4872i	− 0.952588i	0.879286	−	0.476294i	$-0.158020\pi$
$338$	3.66593	0.199400
$339$	3.57174	0.193990
$340$	0	0
$341$	−18.5587	−1.00501
$342$	−4.05317	−0.219171
$343$	8.81069i	0.475732i
$344$	1.04667	0.0564327
$345$	10.4316i	0.561617i
$346$	− 15.3228i	− 0.823758i
$347$	− 21.7988i	− 1.17022i	−0.810954	−	0.585110i	$-0.801051\pi$
$347$	− 21.7988i	− 1.17022i	0.810954	−	0.585110i	$-0.198949\pi$
$348$	−2.58194	−0.138407
$349$	13.2144	0.707351	0.353675	−	0.935368i	$-0.384932\pi$
$349$	13.2144	0.707351	0.353675	+	0.935368i	$0.384932\pi$
$350$	− 0.648847i	− 0.0346823i
$351$	− 23.0870i	− 1.23229i
$352$	3.73124i	0.198876i
$353$	29.5181	1.57109	0.785546	−	0.618803i	$-0.212382\pi$
$353$	29.5181	1.57109	0.785546	+	0.618803i	$0.212382\pi$
$354$	− 14.8918i	− 0.791490i
$355$	13.2729	0.704454
$356$	3.59539	0.190555
$357$	0	0
$358$	−10.1899	−0.538551
$359$	−8.21236	−0.433432	−0.216716	−	0.976235i	$-0.569535\pi$
$359$	−8.21236	−0.433432	−0.216716	+	0.976235i	$0.569535\pi$
$360$	0.944947i	0.0498031i
$361$	−0.601802	−0.0316738
$362$	− 13.0679i	− 0.686833i
$363$	4.18903i	0.219867i
$364$	2.64885i	0.138837i
$365$	−5.99321	−0.313699
$366$	5.07717	0.265388
$367$	− 10.9304i	− 0.570564i	−0.958444	−	0.285282i	$-0.907913\pi$
$367$	− 10.9304i	− 0.570564i	0.958444	−	0.285282i	$-0.0920873\pi$
$368$	7.27677i	0.379328i
$369$	1.83031i	0.0952819i
$370$	−8.15640	−0.424031
$371$	− 5.65980i	− 0.293842i
$372$	7.13028	0.369688
$373$	4.57353	0.236808	0.118404	−	0.992965i	$-0.462222\pi$
$373$	4.57353	0.236808	0.118404	+	0.992965i	$0.462222\pi$
$374$	0	0
$375$	−1.43355	−0.0740280
$376$	−4.71031	−0.242916
$377$	− 7.35275i	− 0.378686i
$378$	3.66940	0.188733
$379$	− 7.24227i	− 0.372010i	−0.982549	−	0.186005i	$-0.940446\pi$
$379$	− 7.24227i	− 0.372010i	0.982549	−	0.186005i	$-0.0595541\pi$
$380$	− 4.28931i	− 0.220037i
$381$	12.9069i	0.661240i
$382$	−19.7531	−1.01066
$383$	−35.4256	−1.81017	−0.905083	−	0.425236i	$-0.860191\pi$
$383$	−35.4256	−1.81017	−0.905083	+	0.425236i	$0.860191\pi$
$384$	− 1.43355i	− 0.0731553i
$385$	− 2.42100i	− 0.123386i
$386$	10.3611i	0.527365i
$387$	0.989049	0.0502762
$388$	− 19.2355i	− 0.976533i
$389$	−33.3188	−1.68933	−0.844666	−	0.535293i	$-0.820201\pi$
$389$	−33.3188	−1.68933	−0.844666	+	0.535293i	$0.820201\pi$
$390$	5.85229	0.296342
$391$	0	0
$392$	6.57900	0.332290
$393$	−22.8001	−1.15011
$394$	− 16.6845i	− 0.840553i
$395$	−0.293157	−0.0147504
$396$	3.52582i	0.177179i
$397$	− 13.4120i	− 0.673127i	−0.941661	−	0.336564i	$-0.890735\pi$
$397$	− 13.4120i	− 0.673127i	0.941661	−	0.336564i	$-0.109265\pi$
$398$	− 9.47274i	− 0.474825i
$399$	−3.98971	−0.199735
$400$	−1.00000	−0.0500000
$401$	34.4003i	1.71787i	0.512087	+	0.858934i	$0.328873\pi$
$401$	34.4003i	1.71787i	−0.512087	+	0.858934i	$0.671127\pi$
$402$	− 4.98905i	− 0.248831i
$403$	20.3053i	1.01148i
$404$	9.83938	0.489527
$405$	− 5.27223i	− 0.261979i
$406$	1.16863	0.0579981
$407$	−30.4335	−1.50853
$408$	0	0
$409$	37.6365	1.86101	0.930503	−	0.366283i	$-0.119370\pi$
$409$	37.6365	1.86101	0.930503	+	0.366283i	$0.119370\pi$
$410$	−1.93694	−0.0956586
$411$	− 32.8644i	− 1.62108i
$412$	5.96722	0.293984
$413$	6.74028i	0.331667i
$414$	6.87616i	0.337945i
$415$	− 2.48406i	− 0.121938i
$416$	4.08239	0.200156
$417$	16.5016	0.808086
$418$	− 16.0044i	− 0.782803i
$419$	2.02265i	0.0988129i	0.998779	+	0.0494064i	$0.0157330\pi$
$419$	2.02265i	0.0988129i	−0.998779	+	0.0494064i	$0.984267\pi$
$420$	0.930151i	0.0453867i
$421$	−34.7424	−1.69324	−0.846620	−	0.532198i	$-0.821366\pi$
$421$	−34.7424	−1.69324	−0.846620	+	0.532198i	$0.821366\pi$
$422$	25.1371i	1.22365i
$423$	−4.45100	−0.216415
$424$	−8.72286	−0.423619
$425$	0	0
$426$	−19.0273	−0.921878
$427$	−2.29801	−0.111209
$428$	− 5.43739i	− 0.262826i
$429$	21.8363	1.05427
$430$	1.04667i	0.0504750i
$431$	23.8731i	1.14993i	0.818179	+	0.574964i	$0.194984\pi$
$431$	23.8731i	1.14993i	−0.818179	+	0.574964i	$0.805016\pi$
$432$	− 5.65526i	− 0.272089i
$433$	1.15412	0.0554635	0.0277317	−	0.999615i	$-0.491172\pi$
$433$	1.15412	0.0554635	0.0277317	+	0.999615i	$0.491172\pi$
$434$	−3.22729	−0.154915
$435$	− 2.58194i	− 0.123795i
$436$	16.3410i	0.782594i
$437$	− 31.2123i	− 1.49309i
$438$	8.59154	0.410520
$439$	− 0.395124i	− 0.0188582i	−0.999956	−	0.00942912i	$-0.996999\pi$
$439$	− 0.395124i	− 0.0188582i	0.999956	−	0.00942912i	$-0.00300143\pi$
$440$	−3.73124	−0.177880
$441$	6.21681	0.296038
$442$	0	0
$443$	−17.8435	−0.847772	−0.423886	−	0.905716i	$-0.639334\pi$
$443$	−17.8435	−0.847772	−0.423886	+	0.905716i	$0.639334\pi$
$444$	11.6926	0.554905
$445$	3.59539i	0.170438i
$446$	19.0672	0.902859
$447$	− 19.8189i	− 0.937403i
$448$	0.648847i	0.0306551i
$449$	− 30.2944i	− 1.42968i	−0.699288	−	0.714840i	$-0.746500\pi$
$449$	− 30.2944i	− 1.42968i	0.699288	−	0.714840i	$-0.253500\pi$
$450$	−0.944947	−0.0445452
$451$	−7.22718	−0.340315
$452$	2.49154i	0.117192i
$453$	− 26.2947i	− 1.23543i
$454$	− 4.01217i	− 0.188300i
$455$	−2.64885	−0.124180
$456$	6.14892i	0.287950i
$457$	−35.0823	−1.64108	−0.820541	−	0.571587i	$-0.806328\pi$
$457$	−35.0823	−1.64108	−0.820541	+	0.571587i	$0.806328\pi$
$458$	17.1455	0.801154
$459$	0	0
$460$	−7.27677	−0.339281
$461$	40.2564	1.87493	0.937464	−	0.348083i	$-0.113167\pi$
$461$	40.2564	1.87493	0.937464	+	0.348083i	$0.113167\pi$
$462$	3.47062i	0.161468i
$463$	−1.74763	−0.0812191	−0.0406096	−	0.999175i	$-0.512930\pi$
$463$	−1.74763	−0.0812191	−0.0406096	+	0.999175i	$0.512930\pi$
$464$	− 1.80109i	− 0.0836134i
$465$	7.13028i	0.330659i
$466$	− 18.9317i	− 0.876996i
$467$	−17.0853	−0.790615	−0.395308	−	0.918549i	$-0.629362\pi$
$467$	−17.0853	−0.790615	−0.395308	+	0.918549i	$0.629362\pi$
$468$	3.85765	0.178320
$469$	2.25813i	0.104271i
$470$	− 4.71031i	− 0.217271i
$471$	25.1338i	1.15811i
$472$	10.3881	0.478151
$473$	3.90538i	0.179570i
$474$	0.420255	0.0193029
$475$	4.28931	0.196807
$476$	0	0
$477$	−8.24264	−0.377405
$478$	−25.1215	−1.14903
$479$	20.9115i	0.955471i	0.878504	+	0.477736i	$0.158542\pi$
$479$	20.9115i	0.955471i	−0.878504	+	0.477736i	$0.841458\pi$
$480$	1.43355	0.0654321
$481$	33.2976i	1.51824i
$482$	− 5.98711i	− 0.272706i
$483$	6.76850i	0.307977i
$484$	−2.92214	−0.132825
$485$	19.2355	0.873438
$486$	− 9.40780i	− 0.426746i
$487$	16.2520i	0.736448i	0.929737	+	0.368224i	$0.120034\pi$
$487$	16.2520i	0.736448i	−0.929737	+	0.368224i	$0.879966\pi$
$488$	3.54168i	0.160325i
$489$	−12.3448	−0.558253
$490$	6.57900i	0.297209i
$491$	16.3083	0.735985	0.367992	−	0.929829i	$-0.380045\pi$
$491$	16.3083	0.735985	0.367992	+	0.929829i	$0.380045\pi$
$492$	2.77669	0.125183
$493$	0	0
$494$	−17.5107	−0.787842
$495$	−3.52582	−0.158474
$496$	4.97388i	0.223334i
$497$	8.61209	0.386305
$498$	3.56102i	0.159573i
$499$	− 32.7000i	− 1.46385i	−0.681384	−	0.731926i	$-0.738621\pi$
$499$	− 32.7000i	− 1.46385i	0.681384	−	0.731926i	$-0.261379\pi$
$500$	− 1.00000i	− 0.0447214i
$501$	−15.9160	−0.711076
$502$	−25.4489	−1.13584
$503$	17.6314i	0.786146i	0.919507	+	0.393073i	$0.128588\pi$
$503$	17.6314i	0.786146i	−0.919507	+	0.393073i	$0.871412\pi$
$504$	0.613126i	0.0273108i
$505$	9.83938i	0.437847i
$506$	−27.1514	−1.20703
$507$	− 5.25527i	− 0.233395i
$508$	−9.00347	−0.399464
$509$	−18.7525	−0.831189	−0.415595	−	0.909550i	$-0.636426\pi$
$509$	−18.7525	−0.831189	−0.415595	+	0.909550i	$0.636426\pi$
$510$	0	0
$511$	−3.88868	−0.172025
$512$	1.00000	0.0441942
$513$	24.2572i	1.07098i
$514$	24.8653	1.09676
$515$	5.96722i	0.262947i
$516$	− 1.50045i	− 0.0660537i
$517$	− 17.5753i	− 0.772961i
$518$	−5.29226	−0.232528
$519$	−21.9659	−0.964196
$520$	4.08239i	0.179025i
$521$	− 19.8645i	− 0.870278i	−0.900363	−	0.435139i	$-0.856699\pi$
$521$	− 19.8645i	− 0.870278i	0.900363	−	0.435139i	$-0.143301\pi$
$522$	− 1.70193i	− 0.0744916i
$523$	−16.9141	−0.739601	−0.369801	−	0.929111i	$-0.620574\pi$
$523$	−16.9141	−0.739601	−0.369801	+	0.929111i	$0.620574\pi$
$524$	− 15.9047i	− 0.694800i
$525$	−0.930151	−0.0405951
$526$	−5.49782	−0.239717
$527$	0	0
$528$	5.34890	0.232781
$529$	−29.9514	−1.30223
$530$	− 8.72286i	− 0.378897i
$531$	9.81620	0.425987
$532$	− 2.78311i	− 0.120663i
$533$	7.90735i	0.342505i
$534$	− 5.15415i	− 0.223042i
$535$	5.43739	0.235079
$536$	3.48022	0.150322
$537$	14.6076i	0.630366i
$538$	9.06844i	0.390968i
$539$	24.5478i	1.05735i
$540$	5.65526	0.243364
$541$	22.1120i	0.950670i	0.879805	+	0.475335i	$0.157673\pi$
$541$	22.1120i	0.950670i	−0.879805	+	0.475335i	$0.842327\pi$
$542$	−26.0766	−1.12009
$543$	−18.7334	−0.803927
$544$	0	0
$545$	−16.3410	−0.699974
$546$	3.79724	0.162507
$547$	35.5271i	1.51903i	0.650491	+	0.759514i	$0.274563\pi$
$547$	35.5271i	1.51903i	−0.650491	+	0.759514i	$0.725437\pi$
$548$	22.9252	0.979318
$549$	3.34671i	0.142834i
$550$	− 3.73124i	− 0.159101i
$551$	7.72543i	0.329114i
$552$	10.4316	0.443997
$553$	−0.190214	−0.00808873
$554$	2.50204i	0.106302i
$555$	11.6926i	0.496322i
$556$	11.5110i	0.488176i
$557$	5.78242	0.245009	0.122504	−	0.992468i	$-0.460907\pi$
$557$	5.78242	0.245009	0.122504	+	0.992468i	$0.460907\pi$
$558$	4.70005i	0.198969i
$559$	4.27292	0.180725
$560$	−0.648847	−0.0274188
$561$	0	0
$562$	11.6341	0.490753
$563$	0.883106	0.0372185	0.0186092	−	0.999827i	$-0.494076\pi$
$563$	0.883106	0.0372185	0.0186092	+	0.999827i	$0.494076\pi$
$564$	6.75245i	0.284329i
$565$	−2.49154	−0.104820
$566$	27.1032i	1.13923i
$567$	− 3.42087i	− 0.143663i
$568$	− 13.2729i	− 0.556919i
$569$	6.06200	0.254132	0.127066	−	0.991894i	$-0.459444\pi$
$569$	6.06200	0.254132	0.127066	+	0.991894i	$0.459444\pi$
$570$	−6.14892	−0.257550
$571$	− 22.1138i	− 0.925432i	−0.886507	−	0.462716i	$-0.846875\pi$
$571$	− 22.1138i	− 0.925432i	0.886507	−	0.462716i	$-0.153125\pi$
$572$	15.2324i	0.636898i
$573$	28.3170i	1.18296i
$574$	−1.25678	−0.0524569
$575$	− 7.27677i	− 0.303462i
$576$	0.944947	0.0393728
$577$	−11.1949	−0.466051	−0.233026	−	0.972471i	$-0.574863\pi$
$577$	−11.1949	−0.466051	−0.233026	+	0.972471i	$0.574863\pi$
$578$	0	0
$579$	14.8531	0.617272
$580$	1.80109	0.0747861
$581$	− 1.61178i	− 0.0668677i
$582$	−27.5749	−1.14302
$583$	− 32.5471i	− 1.34796i
$584$	5.99321i	0.248001i
$585$	3.85765i	0.159494i
$586$	−19.1754	−0.792127
$587$	14.0669	0.580605	0.290302	−	0.956935i	$-0.406244\pi$
$587$	14.0669	0.580605	0.290302	+	0.956935i	$0.406244\pi$
$588$	− 9.43129i	− 0.388940i
$589$	− 21.3345i	− 0.879074i
$590$	10.3881i	0.427671i
$591$	−23.9180	−0.983855
$592$	8.15640i	0.335226i
$593$	−9.99549	−0.410466	−0.205233	−	0.978713i	$-0.565795\pi$
$593$	−9.99549	−0.410466	−0.205233	+	0.978713i	$0.565795\pi$
$594$	21.1011	0.865790
$595$	0	0
$596$	13.8251	0.566299
$597$	−13.5796	−0.555776
$598$	29.7066i	1.21479i
$599$	17.8658	0.729976	0.364988	−	0.931012i	$-0.381073\pi$
$599$	17.8658	0.729976	0.364988	+	0.931012i	$0.381073\pi$
$600$	1.43355i	0.0585243i
$601$	− 21.8948i	− 0.893108i	−0.894757	−	0.446554i	$-0.852651\pi$
$601$	− 21.8948i	− 0.893108i	0.894757	−	0.446554i	$-0.147349\pi$
$602$	0.679129i	0.0276792i
$603$	3.28862	0.133923
$604$	18.3424	0.746342
$605$	− 2.92214i	− 0.118802i
$606$	− 14.1052i	− 0.572984i
$607$	− 16.8314i	− 0.683164i	−0.939852	−	0.341582i	$-0.889037\pi$
$607$	− 16.8314i	− 0.683164i	0.939852	−	0.341582i	$-0.110963\pi$
$608$	−4.28931	−0.173955
$609$	− 1.67528i	− 0.0678859i
$610$	−3.54168	−0.143399
$611$	−19.2293	−0.777936
$612$	0	0
$613$	−16.8709	−0.681408	−0.340704	−	0.940171i	$-0.610665\pi$
$613$	−16.8709	−0.681408	−0.340704	+	0.940171i	$0.610665\pi$
$614$	8.72029	0.351922
$615$	2.77669i	0.111967i
$616$	−2.42100	−0.0975450
$617$	− 30.9716i	− 1.24687i	−0.781876	−	0.623434i	$-0.785737\pi$
$617$	− 30.9716i	− 1.24687i	0.781876	−	0.623434i	$-0.214263\pi$
$618$	− 8.55428i	− 0.344104i
$619$	− 9.03604i	− 0.363189i	−0.983373	−	0.181595i	$-0.941874\pi$
$619$	− 9.03604i	− 0.363189i	0.983373	−	0.181595i	$-0.0581258\pi$
$620$	−4.97388	−0.199756
$621$	41.1520	1.65137
$622$	16.8641i	0.676188i
$623$	2.33285i	0.0934639i
$624$	− 5.85229i	− 0.234279i
$625$	1.00000	0.0400000
$626$	− 7.43930i	− 0.297334i
$627$	−22.9431	−0.916259
$628$	−17.5326	−0.699629
$629$	0	0
$630$	−0.613126	−0.0244275
$631$	−24.7713	−0.986129	−0.493064	−	0.869993i	$-0.664123\pi$
$631$	−24.7713	−0.986129	−0.493064	+	0.869993i	$0.664123\pi$
$632$	0.293157i	0.0116612i
$633$	36.0351	1.43227
$634$	1.29035i	0.0512462i
$635$	− 9.00347i	− 0.357292i
$636$	12.5046i	0.495840i
$637$	26.8580	1.06415
$638$	6.72029	0.266059
$639$	− 12.5422i	− 0.496162i
$640$	1.00000i	0.0395285i
$641$	30.6326i	1.20991i	0.796258	+	0.604957i	$0.206810\pi$
$641$	30.6326i	1.20991i	−0.796258	+	0.604957i	$0.793190\pi$
$642$	−7.79475	−0.307634
$643$	− 2.38650i	− 0.0941144i	−0.998892	−	0.0470572i	$-0.985016\pi$
$643$	− 2.38650i	− 0.0941144i	0.998892	−	0.0470572i	$-0.0149843\pi$
$644$	−4.72151	−0.186053
$645$	1.50045	0.0590802
$646$	0	0
$647$	−25.0527	−0.984924	−0.492462	−	0.870334i	$-0.663903\pi$
$647$	−25.0527	−0.984924	−0.492462	+	0.870334i	$0.663903\pi$
$648$	−5.27223	−0.207113
$649$	38.7605i	1.52148i
$650$	−4.08239	−0.160125
$651$	4.62646i	0.181325i
$652$	− 8.61140i	− 0.337249i
$653$	− 15.4788i	− 0.605731i	−0.953033	−	0.302866i	$-0.902057\pi$
$653$	− 15.4788i	− 0.605731i	0.953033	−	0.302866i	$-0.0979434\pi$
$654$	23.4256	0.916015
$655$	15.9047	0.621448
$656$	1.93694i	0.0756248i
$657$	5.66327i	0.220945i
$658$	− 3.05627i	− 0.119146i
$659$	8.19841	0.319365	0.159682	−	0.987168i	$-0.448953\pi$
$659$	8.19841	0.319365	0.159682	+	0.987168i	$0.448953\pi$
$660$	5.34890i	0.208206i
$661$	45.2036	1.75822	0.879108	−	0.476622i	$-0.158139\pi$
$661$	45.2036	1.75822	0.879108	+	0.476622i	$0.158139\pi$
$662$	7.26651	0.282421
$663$	0	0
$664$	−2.48406	−0.0964003
$665$	2.78311	0.107924
$666$	7.70737i	0.298655i
$667$	13.1061	0.507470
$668$	− 11.1026i	− 0.429571i
$669$	− 27.3337i	− 1.05678i
$670$	3.48022i	0.134453i
$671$	−13.2149	−0.510154
$672$	0.930151	0.0358814
$673$	− 17.6341i	− 0.679743i	−0.940472	−	0.339871i	$-0.889616\pi$
$673$	− 17.6341i	− 0.679743i	0.940472	−	0.339871i	$-0.110384\pi$
$674$	− 17.4872i	− 0.673581i
$675$	5.65526i	0.217671i
$676$	3.66593	0.140997
$677$	− 42.1776i	− 1.62102i	−0.585728	−	0.810508i	$-0.699191\pi$
$677$	− 42.1776i	− 1.62102i	0.585728	−	0.810508i	$-0.300809\pi$
$678$	3.57174	0.137172
$679$	12.4809	0.478972
$680$	0	0
$681$	−5.75163	−0.220403
$682$	−18.5587	−0.710651
$683$	− 12.0540i	− 0.461234i	−0.973045	−	0.230617i	$-0.925926\pi$
$683$	− 12.0540i	− 0.461234i	0.973045	−	0.230617i	$-0.0740745\pi$
$684$	−4.05317	−0.154977
$685$	22.9252i	0.875928i
$686$	8.81069i	0.336394i
$687$	− 24.5788i	− 0.937739i
$688$	1.04667	0.0399040
$689$	−35.6101	−1.35664
$690$	10.4316i	0.397123i
$691$	31.4839i	1.19770i	0.800860	+	0.598852i	$0.204376\pi$
$691$	31.4839i	1.19770i	−0.800860	+	0.598852i	$0.795624\pi$
$692$	− 15.3228i	− 0.582485i
$693$	−2.28772	−0.0869033
$694$	− 21.7988i	− 0.827470i
$695$	−11.5110	−0.436638
$696$	−2.58194	−0.0978682
$697$	0	0
$698$	13.2144	0.500172
$699$	−27.1395	−1.02651
$700$	− 0.648847i	− 0.0245241i
$701$	7.86897	0.297207	0.148603	−	0.988897i	$-0.452522\pi$
$701$	7.86897	0.297207	0.148603	+	0.988897i	$0.452522\pi$
$702$	− 23.0870i	− 0.871362i
$703$	− 34.9854i	− 1.31950i
$704$	3.73124i	0.140626i
$705$	−6.75245	−0.254312
$706$	29.5181	1.11093
$707$	6.38425i	0.240104i
$708$	− 14.8918i	− 0.559668i
$709$	− 16.0385i	− 0.602338i	−0.953571	−	0.301169i	$-0.902623\pi$
$709$	− 16.0385i	− 0.602338i	0.953571	−	0.301169i	$-0.0973768\pi$
$710$	13.2729	0.498124
$711$	0.277018i	0.0103890i
$712$	3.59539	0.134743
$713$	−36.1938	−1.35547
$714$	0	0
$715$	−15.2324	−0.569659
$716$	−10.1899	−0.380813
$717$	36.0129i	1.34492i
$718$	−8.21236	−0.306482
$719$	1.13641i	0.0423810i	0.999775	+	0.0211905i	$0.00674565\pi$
$719$	1.13641i	0.0423810i	−0.999775	+	0.0211905i	$0.993254\pi$
$720$	0.944947i	0.0352161i
$721$	3.87181i	0.144194i
$722$	−0.601802	−0.0223967
$723$	−8.58280	−0.319198
$724$	− 13.0679i	− 0.485664i
$725$	1.80109i	0.0668907i
$726$	4.18903i	0.155469i
$727$	16.5261	0.612918	0.306459	−	0.951884i	$-0.400856\pi$
$727$	16.5261	0.612918	0.306459	+	0.951884i	$0.400856\pi$
$728$	2.64885i	0.0981728i
$729$	−29.3032	−1.08530
$730$	−5.99321	−0.221819
$731$	0	0
$732$	5.07717	0.187657
$733$	4.79888	0.177251	0.0886253	−	0.996065i	$-0.471753\pi$
$733$	4.79888	0.177251	0.0886253	+	0.996065i	$0.471753\pi$
$734$	− 10.9304i	− 0.403450i
$735$	9.43129	0.347878
$736$	7.27677i	0.268225i
$737$	12.9855i	0.478328i
$738$	1.83031i	0.0673745i
$739$	−48.1816	−1.77239	−0.886195	−	0.463312i	$-0.846661\pi$
$739$	−48.1816	−1.77239	−0.886195	+	0.463312i	$0.846661\pi$
$740$	−8.15640	−0.299835
$741$	25.1023i	0.922157i
$742$	− 5.65980i	− 0.207778i
$743$	− 19.7569i	− 0.724810i	−0.932021	−	0.362405i	$-0.881956\pi$
$743$	− 19.7569i	− 0.724810i	0.932021	−	0.362405i	$-0.118044\pi$
$744$	7.13028	0.261409
$745$	13.8251i	0.506513i
$746$	4.57353	0.167449
$747$	−2.34731	−0.0858835
$748$	0	0
$749$	3.52803	0.128912
$750$	−1.43355	−0.0523457
$751$	16.2921i	0.594507i	0.954799	+	0.297253i	$0.0960706\pi$
$751$	16.2921i	0.594507i	−0.954799	+	0.297253i	$0.903929\pi$
$752$	−4.71031	−0.171767
$753$	36.4822i	1.32948i
$754$	− 7.35275i	− 0.267771i
$755$	18.3424i	0.667548i
$756$	3.66940	0.133455
$757$	16.0166	0.582134	0.291067	−	0.956703i	$-0.405990\pi$
$757$	16.0166	0.582134	0.291067	+	0.956703i	$0.405990\pi$
$758$	− 7.24227i	− 0.263051i
$759$	38.9227i	1.41280i
$760$	− 4.28931i	− 0.155590i
$761$	14.4363	0.523314	0.261657	−	0.965161i	$-0.415731\pi$
$761$	14.4363	0.523314	0.261657	+	0.965161i	$0.415731\pi$
$762$	12.9069i	0.467567i
$763$	−10.6028	−0.383848
$764$	−19.7531	−0.714644
$765$	0	0
$766$	−35.4256	−1.27998
$767$	42.4083	1.53127
$768$	− 1.43355i	− 0.0517286i
$769$	−55.2022	−1.99064	−0.995321	−	0.0966274i	$-0.969194\pi$
$769$	−55.2022	−1.99064	−0.995321	+	0.0966274i	$0.969194\pi$
$770$	− 2.42100i	− 0.0872469i
$771$	− 35.6456i	− 1.28374i
$772$	10.3611i	0.372903i
$773$	−0.0978044	−0.00351778	−0.00175889	−	0.999998i	$-0.500560\pi$
$773$	−0.0978044	−0.00351778	−0.00175889	+	0.999998i	$0.500560\pi$
$774$	0.989049	0.0355506
$775$	− 4.97388i	− 0.178667i
$776$	− 19.2355i	− 0.690513i
$777$	7.58669i	0.272171i
$778$	−33.3188	−1.19454
$779$	− 8.30814i	− 0.297670i
$780$	5.85229	0.209546
$781$	49.5244	1.77212
$782$	0	0
$783$	−10.1856	−0.364004
$784$	6.57900	0.234964
$785$	− 17.5326i	− 0.625767i
$786$	−22.8001	−0.813253
$787$	− 37.4792i	− 1.33599i	−0.744167	−	0.667994i	$-0.767153\pi$
$787$	− 37.4792i	− 1.33599i	0.744167	−	0.667994i	$-0.232847\pi$
$788$	− 16.6845i	− 0.594361i
$789$	7.88138i	0.280585i
$790$	−0.293157	−0.0104301
$791$	−1.61663	−0.0574807
$792$	3.52582i	0.125285i
$793$	14.4585i	0.513438i
$794$	− 13.4120i	− 0.475973i
$795$	−12.5046	−0.443493
$796$	− 9.47274i	− 0.335752i
$797$	−26.8631	−0.951540	−0.475770	−	0.879570i	$-0.657831\pi$
$797$	−26.8631	−0.951540	−0.475770	+	0.879570i	$0.657831\pi$
$798$	−3.98971	−0.141234
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	3.39745	0.120043
$802$	34.4003i	1.21472i
$803$	−22.3621	−0.789142
$804$	− 4.98905i	− 0.175950i
$805$	− 4.72151i	− 0.166411i
$806$	20.3053i	0.715225i
$807$	13.0000	0.457623
$808$	9.83938	0.346148
$809$	− 32.0160i	− 1.12562i	−0.826586	−	0.562811i	$-0.809720\pi$
$809$	− 32.0160i	− 1.12562i	0.826586	−	0.562811i	$-0.190280\pi$
$810$	− 5.27223i	− 0.185247i
$811$	32.0305i	1.12474i	0.826885	+	0.562371i	$0.190111\pi$
$811$	32.0305i	1.12474i	−0.826885	+	0.562371i	$0.809889\pi$
$812$	1.16863	0.0410109
$813$	37.3820i	1.31104i
$814$	−30.4335	−1.06669
$815$	8.61140	0.301644
$816$	0	0
$817$	−4.48950	−0.157068
$818$	37.6365	1.31593
$819$	2.50302i	0.0874626i
$820$	−1.93694	−0.0676409
$821$	3.86563i	0.134911i	0.997722	+	0.0674557i	$0.0214881\pi$
$821$	3.86563i	0.134911i	−0.997722	+	0.0674557i	$0.978512\pi$
$822$	− 32.8644i	− 1.14628i
$823$	4.22340i	0.147218i	0.997287	+	0.0736092i	$0.0234517\pi$
$823$	4.22340i	0.147218i	−0.997287	+	0.0736092i	$0.976548\pi$
$824$	5.96722	0.207878
$825$	−5.34890	−0.186225
$826$	6.74028i	0.234524i
$827$	− 42.5037i	− 1.47800i	−0.673706	−	0.739000i	$-0.735298\pi$
$827$	− 42.5037i	− 1.47800i	0.673706	−	0.739000i	$-0.264702\pi$
$828$	6.87616i	0.238963i
$829$	10.9929	0.381798	0.190899	−	0.981610i	$-0.438860\pi$
$829$	10.9929	0.381798	0.190899	+	0.981610i	$0.438860\pi$
$830$	− 2.48406i	− 0.0862231i
$831$	3.58679	0.124424
$832$	4.08239	0.141531
$833$	0	0
$834$	16.5016	0.571403
$835$	11.1026	0.384220
$836$	− 16.0044i	− 0.553525i
$837$	28.1286	0.972266
$838$	2.02265i	0.0698712i
$839$	− 33.6345i	− 1.16119i	−0.814192	−	0.580596i	$-0.802819\pi$
$839$	− 33.6345i	− 1.16119i	0.814192	−	0.580596i	$-0.197181\pi$
$840$	0.930151i	0.0320933i
$841$	25.7561	0.888141
$842$	−34.7424	−1.19730
$843$	− 16.6779i	− 0.574419i
$844$	25.1371i	0.865254i
$845$	3.66593i	0.126112i
$846$	−4.45100	−0.153028
$847$	− 1.89602i	− 0.0651481i
$848$	−8.72286	−0.299544
$849$	38.8537	1.33346
$850$	0	0
$851$	−59.3523	−2.03457
$852$	−19.0273	−0.651866
$853$	− 10.6584i	− 0.364938i	−0.983212	−	0.182469i	$-0.941591\pi$
$853$	− 10.6584i	− 0.364938i	0.983212	−	0.182469i	$-0.0584090\pi$
$854$	−2.29801	−0.0786363
$855$	− 4.05317i	− 0.138616i
$856$	− 5.43739i	− 0.185846i
$857$	− 24.1940i	− 0.826451i	−0.910629	−	0.413226i	$-0.864402\pi$
$857$	− 24.1940i	− 0.826451i	0.910629	−	0.413226i	$-0.135598\pi$
$858$	21.8363	0.745479
$859$	19.2963	0.658381	0.329190	−	0.944264i	$-0.393224\pi$
$859$	19.2963	0.658381	0.329190	+	0.944264i	$0.393224\pi$
$860$	1.04667i	0.0356912i
$861$	1.80165i	0.0614000i
$862$	23.8731i	0.813122i
$863$	−31.7534	−1.08090	−0.540449	−	0.841377i	$-0.681746\pi$
$863$	−31.7534	−1.08090	−0.540449	+	0.841377i	$0.681746\pi$
$864$	− 5.65526i	− 0.192396i
$865$	15.3228	0.520990
$866$	1.15412	0.0392186
$867$	0	0
$868$	−3.22729	−0.109541
$869$	−1.09384	−0.0371060
$870$	− 2.58194i	− 0.0875360i
$871$	14.2076	0.481407
$872$	16.3410i	0.553378i
$873$	− 18.1765i	− 0.615182i
$874$	− 31.2123i	− 1.05577i
$875$	0.648847	0.0219350
$876$	8.59154	0.290281
$877$	− 37.0649i	− 1.25159i	−0.779986	−	0.625797i	$-0.784774\pi$
$877$	− 37.0649i	− 1.25159i	0.779986	−	0.625797i	$-0.215226\pi$
$878$	− 0.395124i	− 0.0133348i
$879$	27.4888i	0.927173i
$880$	−3.73124	−0.125780
$881$	33.0057i	1.11199i	0.831186	+	0.555995i	$0.187663\pi$
$881$	33.0057i	1.11199i	−0.831186	+	0.555995i	$0.812337\pi$
$882$	6.21681	0.209331
$883$	25.4716	0.857188	0.428594	−	0.903497i	$-0.359009\pi$
$883$	25.4716	0.857188	0.428594	+	0.903497i	$0.359009\pi$
$884$	0	0
$885$	14.8918	0.500583
$886$	−17.8435	−0.599465
$887$	− 29.2476i	− 0.982040i	−0.871148	−	0.491020i	$-0.836624\pi$
$887$	− 29.2476i	− 0.982040i	0.871148	−	0.491020i	$-0.163376\pi$
$888$	11.6926	0.392377
$889$	− 5.84187i	− 0.195930i
$890$	3.59539i	0.120518i
$891$	− 19.6720i	− 0.659035i
$892$	19.0672	0.638418
$893$	20.2040	0.676101
$894$	− 19.8189i	− 0.662844i
$895$	− 10.1899i	− 0.340610i
$896$	0.648847i	0.0216764i
$897$	42.5858	1.42190
$898$	− 30.2944i	− 1.01094i
$899$	8.95839	0.298779
$900$	−0.944947	−0.0314982
$901$	0	0
$902$	−7.22718	−0.240639
$903$	0.973563	0.0323981
$904$	2.49154i	0.0828674i
$905$	13.0679	0.434391
$906$	− 26.2947i	− 0.873582i
$907$	41.5854i	1.38082i	0.723418	+	0.690410i	$0.242570\pi$
$907$	41.5854i	1.38082i	−0.723418	+	0.690410i	$0.757430\pi$
$908$	− 4.01217i	− 0.133149i
$909$	9.29769	0.308385
$910$	−2.64885	−0.0878084
$911$	− 0.0623697i	− 0.00206640i	−0.999999	−	0.00103320i	$-0.999671\pi$
$911$	− 0.0623697i	− 0.00206640i	0.999999	−	0.00103320i	$-0.000328878\pi$
$912$	6.14892i	0.203611i
$913$	− 9.26863i	− 0.306747i
$914$	−35.0823	−1.16042
$915$	5.07717i	0.167846i
$916$	17.1455	0.566502
$917$	10.3197	0.340787
$918$	0	0
$919$	17.0376	0.562019	0.281009	−	0.959705i	$-0.409331\pi$
$919$	17.0376	0.562019	0.281009	+	0.959705i	$0.409331\pi$
$920$	−7.27677	−0.239908
$921$	− 12.5009i	− 0.411920i
$922$	40.2564	1.32577
$923$	− 54.1853i	− 1.78353i
$924$	3.47062i	0.114175i
$925$	− 8.15640i	− 0.268181i
$926$	−1.74763	−0.0574306
$927$	5.63871	0.185200
$928$	− 1.80109i	− 0.0591236i
$929$	− 6.87693i	− 0.225625i	−0.993616	−	0.112812i	$-0.964014\pi$
$929$	− 6.87693i	− 0.225625i	0.993616	−	0.112812i	$-0.0359859\pi$
$930$	7.13028i	0.233811i
$931$	−28.2194	−0.924853
$932$	− 18.9317i	− 0.620130i
$933$	24.1754	0.791468
$934$	−17.0853	−0.559049
$935$	0	0
$936$	3.85765	0.126091
$937$	19.0427	0.622098	0.311049	−	0.950394i	$-0.399320\pi$
$937$	19.0427	0.622098	0.311049	+	0.950394i	$0.399320\pi$
$938$	2.25813i	0.0737305i
$939$	−10.6646	−0.348025
$940$	− 4.71031i	− 0.153634i
$941$	2.60364i	0.0848761i	0.999099	+	0.0424381i	$0.0135125\pi$
$941$	2.60364i	0.0848761i	−0.999099	+	0.0424381i	$0.986487\pi$
$942$	25.1338i	0.818905i
$943$	−14.0947	−0.458985
$944$	10.3881	0.338104
$945$	3.66940i	0.119366i
$946$	3.90538i	0.126975i
$947$	7.42511i	0.241284i	0.992696	+	0.120642i	$0.0384952\pi$
$947$	7.42511i	0.241284i	−0.992696	+	0.120642i	$0.961505\pi$
$948$	0.420255	0.0136492
$949$	24.4666i	0.794221i
$950$	4.28931	0.139164
$951$	1.84977	0.0599829
$952$	0	0
$953$	37.5558	1.21655	0.608277	−	0.793725i	$-0.291861\pi$
$953$	37.5558	1.21655	0.608277	+	0.793725i	$0.291861\pi$
$954$	−8.24264	−0.266865
$955$	− 19.7531i	− 0.639197i
$956$	−25.1215	−0.812488
$957$	− 9.63384i	− 0.311418i
$958$	20.9115i	0.675620i
$959$	14.8750i	0.480338i
$960$	1.43355	0.0462675
$961$	6.26052	0.201952
$962$	33.2976i	1.07356i
$963$	− 5.13805i	− 0.165571i
$964$	− 5.98711i	− 0.192832i
$965$	−10.3611	−0.333535
$966$	6.76850i	0.217773i
$967$	49.1247	1.57974	0.789871	−	0.613273i	$-0.210147\pi$
$967$	49.1247	1.57974	0.789871	+	0.613273i	$0.210147\pi$
$968$	−2.92214	−0.0939213
$969$	0	0
$970$	19.2355	0.617614
$971$	−20.0813	−0.644439	−0.322220	−	0.946665i	$-0.604429\pi$
$971$	−20.0813	−0.644439	−0.322220	+	0.946665i	$0.604429\pi$
$972$	− 9.40780i	− 0.301755i
$973$	−7.46889	−0.239442
$974$	16.2520i	0.520748i
$975$	5.85229i	0.187423i
$976$	3.54168i	0.113367i
$977$	38.9009	1.24455	0.622275	−	0.782799i	$-0.286209\pi$
$977$	38.9009	1.24455	0.622275	+	0.782799i	$0.286209\pi$
$978$	−12.3448	−0.394744
$979$	13.4152i	0.428753i
$980$	6.57900i	0.210158i
$981$	15.4414i	0.493007i
$982$	16.3083	0.520420
$983$	4.42226i	0.141048i	0.997510	+	0.0705242i	$0.0224672\pi$
$983$	4.42226i	0.141048i	−0.997510	+	0.0705242i	$0.977533\pi$
$984$	2.77669	0.0885177
$985$	16.6845	0.531613
$986$	0	0
$987$	−4.38130	−0.139458
$988$	−17.5107	−0.557088
$989$	7.61639i	0.242187i
$990$	−3.52582	−0.112058
$991$	16.5120i	0.524521i	0.964997	+	0.262261i	$0.0844680\pi$
$991$	16.5120i	0.524521i	−0.964997	+	0.262261i	$0.915532\pi$
$992$	4.97388i	0.157921i
$993$	− 10.4169i	− 0.330569i
$994$	8.61209	0.273159
$995$	9.47274	0.300306
$996$	3.56102i	0.112835i
$997$	− 56.3738i	− 1.78538i	−0.450675	−	0.892688i	$-0.648817\pi$
$997$	− 56.3738i	− 1.78538i	0.450675	−	0.892688i	$-0.351183\pi$
$998$	− 32.7000i	− 1.03510i
$999$	46.1266	1.45938

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2890.2.b.p.2311.4		8
17.3	odd	16		170.2.k.a.161.1	yes	8
17.4	even	4		2890.2.a.bc.1.3		4
17.11	odd	16		170.2.k.a.151.1	✓	8
17.13	even	4		2890.2.a.bf.1.2		4
17.16	even	2	inner	2890.2.b.p.2311.5		8
85.3	even	16		850.2.o.f.399.1		8
85.28	even	16		850.2.o.c.49.2		8
85.37	even	16		850.2.o.c.399.2		8
85.54	odd	16		850.2.l.d.501.2		8
85.62	even	16		850.2.o.f.49.1		8
85.79	odd	16		850.2.l.d.151.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
170.2.k.a.151.1	✓	8	17.11	odd	16
170.2.k.a.161.1	yes	8	17.3	odd	16
850.2.l.d.151.2		8	85.79	odd	16
850.2.l.d.501.2		8	85.54	odd	16
850.2.o.c.49.2		8	85.28	even	16
850.2.o.c.399.2		8	85.37	even	16
850.2.o.f.49.1		8	85.62	even	16
850.2.o.f.399.1		8	85.3	even	16
2890.2.a.bc.1.3		4	17.4	even	4
2890.2.a.bf.1.2		4	17.13	even	4
2890.2.b.p.2311.4		8	1.1	even	1	trivial
2890.2.b.p.2311.5		8	17.16	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.43355i q^{3} +1.00000 q^{4} +1.00000i q^{5} -1.43355i q^{6} +0.648847i q^{7} +1.00000 q^{8} +0.944947 q^{9} +1.00000i q^{10} +3.73124i q^{11} -1.43355i q^{12} +4.08239 q^{13} +0.648847i q^{14} +1.43355 q^{15} +1.00000 q^{16} +0.944947 q^{18} -4.28931 q^{19} +1.00000i q^{20} +0.930151 q^{21} +3.73124i q^{22} +7.27677i q^{23} -1.43355i q^{24} -1.00000 q^{25} +4.08239 q^{26} -5.65526i q^{27} +0.648847i q^{28} -1.80109i q^{29} +1.43355 q^{30} +4.97388i q^{31} +1.00000 q^{32} +5.34890 q^{33} -0.648847 q^{35} +0.944947 q^{36} +8.15640i q^{37} -4.28931 q^{38} -5.85229i q^{39} +1.00000i q^{40} +1.93694i q^{41} +0.930151 q^{42} +1.04667 q^{43} +3.73124i q^{44} +0.944947i q^{45} +7.27677i q^{46} -4.71031 q^{47} -1.43355i q^{48} +6.57900 q^{49} -1.00000 q^{50} +4.08239 q^{52} -8.72286 q^{53} -5.65526i q^{54} -3.73124 q^{55} +0.648847i q^{56} +6.14892i q^{57} -1.80109i q^{58} +10.3881 q^{59} +1.43355 q^{60} +3.54168i q^{61} +4.97388i q^{62} +0.613126i q^{63} +1.00000 q^{64} +4.08239i q^{65} +5.34890 q^{66} +3.48022 q^{67} +10.4316 q^{69} -0.648847 q^{70} -13.2729i q^{71} +0.944947 q^{72} +5.99321i q^{73} +8.15640i q^{74} +1.43355i q^{75} -4.28931 q^{76} -2.42100 q^{77} -5.85229i q^{78} +0.293157i q^{79} +1.00000i q^{80} -5.27223 q^{81} +1.93694i q^{82} -2.48406 q^{83} +0.930151 q^{84} +1.04667 q^{86} -2.58194 q^{87} +3.73124i q^{88} +3.59539 q^{89} +0.944947i q^{90} +2.64885i q^{91} +7.27677i q^{92} +7.13028 q^{93} -4.71031 q^{94} -4.28931i q^{95} -1.43355i q^{96} -19.2355i q^{97} +6.57900 q^{98} +3.52582i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{4} + 8 q^{8} - 16 q^{9} + 24 q^{13} + 8 q^{15} + 8 q^{16} - 16 q^{18} + 24 q^{19} - 16 q^{21} - 8 q^{25} + 24 q^{26} + 8 q^{30} + 8 q^{32} + 16 q^{33} - 16 q^{36} + 24 q^{38} - 16 q^{42}+ \cdots + 24 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^4 + 8 * q^8 - 16 * q^9 + 24 * q^13 + 8 * q^15 + 8 * q^16 - 16 * q^18 + 24 * q^19 - 16 * q^21 - 8 * q^25 + 24 * q^26 + 8 * q^30 + 8 * q^32 + 16 * q^33 - 16 * q^36 + 24 * q^38 - 16 * q^42 - 16 * q^43 - 8 * q^47 + 24 * q^49 - 8 * q^50 + 24 * q^52 - 8 * q^53 - 16 * q^55 + 24 * q^59 + 8 * q^60 + 8 * q^64 + 16 * q^66 - 16 * q^69 - 16 * q^72 + 24 * q^76 - 48 * q^77 - 8 * q^81 - 32 * q^83 - 16 * q^84 - 16 * q^86 + 8 * q^87 + 8 * q^89 - 56 * q^93 - 8 * q^94 + 24 * q^98

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