LMFDB - Embedded newform 552.2.f.b.277.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [552,2,Mod(277,552)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(552, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("552.277");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 552 = 2^{3} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	552.f (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$4.40774219157$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			277.4
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	552.277
Dual form			552.2.f.b.277.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +1.00000i q^{3} +2.00000 q^{4} +3.41421i q^{5} +1.41421i q^{6} -2.00000 q^{7} +2.82843 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.41421 q^{2} +1.00000i q^{3} +2.00000 q^{4} +3.41421i q^{5} +1.41421i q^{6} -2.00000 q^{7} +2.82843 q^{8} -1.00000 q^{9} +4.82843i q^{10} +4.24264i q^{11} +2.00000i q^{12} -6.82843i q^{13} -2.82843 q^{14} -3.41421 q^{15} +4.00000 q^{16} -1.17157 q^{17} -1.41421 q^{18} +2.24264i q^{19} +6.82843i q^{20} -2.00000i q^{21} +6.00000i q^{22} -1.00000 q^{23} +2.82843i q^{24} -6.65685 q^{25} -9.65685i q^{26} -1.00000i q^{27} -4.00000 q^{28} +2.82843i q^{29} -4.82843 q^{30} +10.8284 q^{31} +5.65685 q^{32} -4.24264 q^{33} -1.65685 q^{34} -6.82843i q^{35} -2.00000 q^{36} -3.07107i q^{37} +3.17157i q^{38} +6.82843 q^{39} +9.65685i q^{40} +8.48528 q^{41} -2.82843i q^{42} -1.75736i q^{43} +8.48528i q^{44} -3.41421i q^{45} -1.41421 q^{46} +6.00000 q^{47} +4.00000i q^{48} -3.00000 q^{49} -9.41421 q^{50} -1.17157i q^{51} -13.6569i q^{52} -11.4142i q^{53} -1.41421i q^{54} -14.4853 q^{55} -5.65685 q^{56} -2.24264 q^{57} +4.00000i q^{58} +5.65685i q^{59} -6.82843 q^{60} -3.75736i q^{61} +15.3137 q^{62} +2.00000 q^{63} +8.00000 q^{64} +23.3137 q^{65} -6.00000 q^{66} +9.07107i q^{67} -2.34315 q^{68} -1.00000i q^{69} -9.65685i q^{70} -11.6569 q^{71} -2.82843 q^{72} -4.34315i q^{74} -6.65685i q^{75} +4.48528i q^{76} -8.48528i q^{77} +9.65685 q^{78} +9.31371 q^{79} +13.6569i q^{80} +1.00000 q^{81} +12.0000 q^{82} -4.24264i q^{83} -4.00000i q^{84} -4.00000i q^{85} -2.48528i q^{86} -2.82843 q^{87} +12.0000i q^{88} +5.17157 q^{89} -4.82843i q^{90} +13.6569i q^{91} -2.00000 q^{92} +10.8284i q^{93} +8.48528 q^{94} -7.65685 q^{95} +5.65685i q^{96} +3.65685 q^{97} -4.24264 q^{98} -4.24264i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4} - 8 q^{7} - 4 q^{9}+O(q^{10}) $ 4 * q + 8 * q^4 - 8 * q^7 - 4 * q^9	$ 4 q + 8 q^{4} - 8 q^{7} - 4 q^{9} - 8 q^{15} + 16 q^{16} - 16 q^{17} - 4 q^{23} - 4 q^{25} - 16 q^{28} - 8 q^{30} + 32 q^{31} + 16 q^{34} - 8 q^{36} + 16 q^{39} + 24 q^{47} - 12 q^{49} - 32 q^{50} - 24 q^{55} + 8 q^{57} - 16 q^{60} + 16 q^{62} + 8 q^{63} + 32 q^{64} + 48 q^{65} - 24 q^{66} - 32 q^{68} - 24 q^{71} + 16 q^{78} - 8 q^{79} + 4 q^{81} + 48 q^{82} + 32 q^{89} - 8 q^{92} - 8 q^{95} - 8 q^{97}+O(q^{100}) $ 4 * q + 8 * q^4 - 8 * q^7 - 4 * q^9 - 8 * q^15 + 16 * q^16 - 16 * q^17 - 4 * q^23 - 4 * q^25 - 16 * q^28 - 8 * q^30 + 32 * q^31 + 16 * q^34 - 8 * q^36 + 16 * q^39 + 24 * q^47 - 12 * q^49 - 32 * q^50 - 24 * q^55 + 8 * q^57 - 16 * q^60 + 16 * q^62 + 8 * q^63 + 32 * q^64 + 48 * q^65 - 24 * q^66 - 32 * q^68 - 24 * q^71 + 16 * q^78 - 8 * q^79 + 4 * q^81 + 48 * q^82 + 32 * q^89 - 8 * q^92 - 8 * q^95 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/552\mathbb{Z}\right)^\times$.

$n$	$97$	$185$	$277$	$415$
$\chi(n)$	$1$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$	1.41421	1.00000
$2$	1.41421	1.00000
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	2.00000	1.00000
$5$	3.41421i	1.52688i	0.645877	+	0.763441i	$0.276492\pi$
$5$	3.41421i	1.52688i	−0.645877	+	0.763441i	$0.723508\pi$
$6$	1.41421i	0.577350i
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	2.82843	1.00000
$9$	−1.00000	−0.333333
$10$	4.82843i	1.52688i
$11$	4.24264i	1.27920i	0.768706	+	0.639602i	$0.220901\pi$
$11$	4.24264i	1.27920i	−0.768706	+	0.639602i	$0.779099\pi$
$12$	2.00000i	0.577350i
$13$	− 6.82843i	− 1.89386i	−0.321433	−	0.946932i	$-0.604164\pi$
$13$	− 6.82843i	− 1.89386i	0.321433	−	0.946932i	$-0.395836\pi$
$14$	−2.82843	−0.755929
$15$	−3.41421	−0.881546
$16$	4.00000	1.00000
$17$	−1.17157	−0.284148	−0.142074	−	0.989856i	$-0.545377\pi$
$17$	−1.17157	−0.284148	−0.142074	+	0.989856i	$0.545377\pi$
$18$	−1.41421	−0.333333
$19$	2.24264i	0.514497i	0.966345	+	0.257249i	$0.0828159\pi$
$19$	2.24264i	0.514497i	−0.966345	+	0.257249i	$0.917184\pi$
$20$	6.82843i	1.52688i
$21$	− 2.00000i	− 0.436436i
$22$	6.00000i	1.27920i
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	2.82843i	0.577350i
$25$	−6.65685	−1.33137
$26$	− 9.65685i	− 1.89386i
$27$	− 1.00000i	− 0.192450i
$28$	−4.00000	−0.755929
$29$	2.82843i	0.525226i	0.964901	+	0.262613i	$0.0845842\pi$
$29$	2.82843i	0.525226i	−0.964901	+	0.262613i	$0.915416\pi$
$30$	−4.82843	−0.881546
$31$	10.8284	1.94484	0.972421	−	0.233231i	$-0.0749297\pi$
$31$	10.8284	1.94484	0.972421	+	0.233231i	$0.0749297\pi$
$32$	5.65685	1.00000
$33$	−4.24264	−0.738549
$34$	−1.65685	−0.284148
$35$	− 6.82843i	− 1.15421i
$36$	−2.00000	−0.333333
$37$	− 3.07107i	− 0.504880i	−0.967612	−	0.252440i	$-0.918767\pi$
$37$	− 3.07107i	− 0.504880i	0.967612	−	0.252440i	$-0.0812331\pi$
$38$	3.17157i	0.514497i
$39$	6.82843	1.09342
$40$	9.65685i	1.52688i
$41$	8.48528	1.32518	0.662589	−	0.748983i	$-0.269458\pi$
$41$	8.48528	1.32518	0.662589	+	0.748983i	$0.269458\pi$
$42$	− 2.82843i	− 0.436436i
$43$	− 1.75736i	− 0.267995i	−0.990982	−	0.133997i	$-0.957219\pi$
$43$	− 1.75736i	− 0.267995i	0.990982	−	0.133997i	$-0.0427814\pi$
$44$	8.48528i	1.27920i
$45$	− 3.41421i	− 0.508961i
$46$	−1.41421	−0.208514
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	4.00000i	0.577350i
$49$	−3.00000	−0.428571
$50$	−9.41421	−1.33137
$51$	− 1.17157i	− 0.164053i
$52$	− 13.6569i	− 1.89386i
$53$	− 11.4142i	− 1.56786i	−0.620847	−	0.783931i	$-0.713211\pi$
$53$	− 11.4142i	− 1.56786i	0.620847	−	0.783931i	$-0.286789\pi$
$54$	− 1.41421i	− 0.192450i
$55$	−14.4853	−1.95319
$56$	−5.65685	−0.755929
$57$	−2.24264	−0.297045
$58$	4.00000i	0.525226i
$59$	5.65685i	0.736460i	0.929735	+	0.368230i	$0.120036\pi$
$59$	5.65685i	0.736460i	−0.929735	+	0.368230i	$0.879964\pi$
$60$	−6.82843	−0.881546
$61$	− 3.75736i	− 0.481081i	−0.970639	−	0.240540i	$-0.922675\pi$
$61$	− 3.75736i	− 0.481081i	0.970639	−	0.240540i	$-0.0773246\pi$
$62$	15.3137	1.94484
$63$	2.00000	0.251976
$64$	8.00000	1.00000
$65$	23.3137	2.89171
$66$	−6.00000	−0.738549
$67$	9.07107i	1.10821i	0.832448	+	0.554104i	$0.186939\pi$
$67$	9.07107i	1.10821i	−0.832448	+	0.554104i	$0.813061\pi$
$68$	−2.34315	−0.284148
$69$	− 1.00000i	− 0.120386i
$70$	− 9.65685i	− 1.15421i
$71$	−11.6569	−1.38341	−0.691707	−	0.722178i	$-0.743141\pi$
$71$	−11.6569	−1.38341	−0.691707	+	0.722178i	$0.743141\pi$
$72$	−2.82843	−0.333333
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	− 4.34315i	− 0.504880i
$75$	− 6.65685i	− 0.768667i
$76$	4.48528i	0.514497i
$77$	− 8.48528i	− 0.966988i
$78$	9.65685	1.09342
$79$	9.31371	1.04787	0.523937	−	0.851757i	$-0.324463\pi$
$79$	9.31371	1.04787	0.523937	+	0.851757i	$0.324463\pi$
$80$	13.6569i	1.52688i
$81$	1.00000	0.111111
$82$	12.0000	1.32518
$83$	− 4.24264i	− 0.465690i	−0.972514	−	0.232845i	$-0.925196\pi$
$83$	− 4.24264i	− 0.465690i	0.972514	−	0.232845i	$-0.0748035\pi$
$84$	− 4.00000i	− 0.436436i
$85$	− 4.00000i	− 0.433861i
$86$	− 2.48528i	− 0.267995i
$87$	−2.82843	−0.303239
$88$	12.0000i	1.27920i
$89$	5.17157	0.548186	0.274093	−	0.961703i	$-0.411622\pi$
$89$	5.17157	0.548186	0.274093	+	0.961703i	$0.411622\pi$
$90$	− 4.82843i	− 0.508961i
$91$	13.6569i	1.43163i
$92$	−2.00000	−0.208514
$93$	10.8284i	1.12286i
$94$	8.48528	0.875190
$95$	−7.65685	−0.785577
$96$	5.65685i	0.577350i
$97$	3.65685	0.371297	0.185649	−	0.982616i	$-0.440561\pi$
$97$	3.65685	0.371297	0.185649	+	0.982616i	$0.440561\pi$
$98$	−4.24264	−0.428571
$99$	− 4.24264i	− 0.426401i
$100$	−13.3137	−1.33137
$101$	− 10.1421i	− 1.00918i	−0.863359	−	0.504590i	$-0.831644\pi$
$101$	− 10.1421i	− 1.00918i	0.863359	−	0.504590i	$-0.168356\pi$
$102$	− 1.65685i	− 0.164053i
$103$	−19.6569	−1.93685	−0.968424	−	0.249310i	$-0.919796\pi$
$103$	−19.6569	−1.93685	−0.968424	+	0.249310i	$0.919796\pi$
$104$	− 19.3137i	− 1.89386i
$105$	6.82843	0.666386
$106$	− 16.1421i	− 1.56786i
$107$	− 2.58579i	− 0.249977i	−0.992158	−	0.124989i	$-0.960111\pi$
$107$	− 2.58579i	− 0.249977i	0.992158	−	0.124989i	$-0.0398895\pi$
$108$	− 2.00000i	− 0.192450i
$109$	− 12.7279i	− 1.21911i	−0.792742	−	0.609557i	$-0.791347\pi$
$109$	− 12.7279i	− 1.21911i	0.792742	−	0.609557i	$-0.208653\pi$
$110$	−20.4853	−1.95319
$111$	3.07107	0.291493
$112$	−8.00000	−0.755929
$113$	−18.8284	−1.77123	−0.885615	−	0.464421i	$-0.846263\pi$
$113$	−18.8284	−1.77123	−0.885615	+	0.464421i	$0.846263\pi$
$114$	−3.17157	−0.297045
$115$	− 3.41421i	− 0.318377i
$116$	5.65685i	0.525226i
$117$	6.82843i	0.631288i
$118$	8.00000i	0.736460i
$119$	2.34315	0.214796
$120$	−9.65685	−0.881546
$121$	−7.00000	−0.636364
$122$	− 5.31371i	− 0.481081i
$123$	8.48528i	0.765092i
$124$	21.6569	1.94484
$125$	− 5.65685i	− 0.505964i
$126$	2.82843	0.251976
$127$	−13.3137	−1.18140	−0.590700	−	0.806891i	$-0.701148\pi$
$127$	−13.3137	−1.18140	−0.590700	+	0.806891i	$0.701148\pi$
$128$	11.3137	1.00000
$129$	1.75736	0.154727
$130$	32.9706	2.89171
$131$	− 14.8284i	− 1.29557i	−0.761825	−	0.647783i	$-0.775696\pi$
$131$	− 14.8284i	− 1.29557i	0.761825	−	0.647783i	$-0.224304\pi$
$132$	−8.48528	−0.738549
$133$	− 4.48528i	− 0.388923i
$134$	12.8284i	1.10821i
$135$	3.41421	0.293849
$136$	−3.31371	−0.284148
$137$	14.8284	1.26688	0.633439	−	0.773793i	$-0.281643\pi$
$137$	14.8284	1.26688	0.633439	+	0.773793i	$0.281643\pi$
$138$	− 1.41421i	− 0.120386i
$139$	− 3.31371i	− 0.281065i	−0.990076	−	0.140533i	$-0.955119\pi$
$139$	− 3.31371i	− 0.281065i	0.990076	−	0.140533i	$-0.0448815\pi$
$140$	− 13.6569i	− 1.15421i
$141$	6.00000i	0.505291i
$142$	−16.4853	−1.38341
$143$	28.9706	2.42264
$144$	−4.00000	−0.333333
$145$	−9.65685	−0.801958
$146$	0	0
$147$	− 3.00000i	− 0.247436i
$148$	− 6.14214i	− 0.504880i
$149$	13.0711i	1.07082i	0.844591	+	0.535412i	$0.179844\pi$
$149$	13.0711i	1.07082i	−0.844591	+	0.535412i	$0.820156\pi$
$150$	− 9.41421i	− 0.768667i
$151$	−3.51472	−0.286024	−0.143012	−	0.989721i	$-0.545679\pi$
$151$	−3.51472	−0.286024	−0.143012	+	0.989721i	$0.545679\pi$
$152$	6.34315i	0.514497i
$153$	1.17157	0.0947161
$154$	− 12.0000i	− 0.966988i
$155$	36.9706i	2.96955i
$156$	13.6569	1.09342
$157$	− 16.7279i	− 1.33503i	−0.744595	−	0.667517i	$-0.767357\pi$
$157$	− 16.7279i	− 1.33503i	0.744595	−	0.667517i	$-0.232643\pi$
$158$	13.1716	1.04787
$159$	11.4142	0.905206
$160$	19.3137i	1.52688i
$161$	2.00000	0.157622
$162$	1.41421	0.111111
$163$	4.48528i	0.351314i	0.984451	+	0.175657i	$0.0562050\pi$
$163$	4.48528i	0.351314i	−0.984451	+	0.175657i	$0.943795\pi$
$164$	16.9706	1.32518
$165$	− 14.4853i	− 1.12768i
$166$	− 6.00000i	− 0.465690i
$167$	−7.65685	−0.592505	−0.296253	−	0.955110i	$-0.595737\pi$
$167$	−7.65685	−0.592505	−0.296253	+	0.955110i	$0.595737\pi$
$168$	− 5.65685i	− 0.436436i
$169$	−33.6274	−2.58672
$170$	− 5.65685i	− 0.433861i
$171$	− 2.24264i	− 0.171499i
$172$	− 3.51472i	− 0.267995i
$173$	− 18.8284i	− 1.43150i	−0.698357	−	0.715749i	$-0.746085\pi$
$173$	− 18.8284i	− 1.43150i	0.698357	−	0.715749i	$-0.253915\pi$
$174$	−4.00000	−0.303239
$175$	13.3137	1.00642
$176$	16.9706i	1.27920i
$177$	−5.65685	−0.425195
$178$	7.31371	0.548186
$179$	8.48528i	0.634220i	0.948389	+	0.317110i	$0.102712\pi$
$179$	8.48528i	0.634220i	−0.948389	+	0.317110i	$0.897288\pi$
$180$	− 6.82843i	− 0.508961i
$181$	12.2426i	0.909988i	0.890494	+	0.454994i	$0.150359\pi$
$181$	12.2426i	0.909988i	−0.890494	+	0.454994i	$0.849641\pi$
$182$	19.3137i	1.43163i
$183$	3.75736	0.277752
$184$	−2.82843	−0.208514
$185$	10.4853	0.770893
$186$	15.3137i	1.12286i
$187$	− 4.97056i	− 0.363484i
$188$	12.0000	0.875190
$189$	2.00000i	0.145479i
$190$	−10.8284	−0.785577
$191$	−9.65685	−0.698745	−0.349373	−	0.936984i	$-0.613605\pi$
$191$	−9.65685	−0.698745	−0.349373	+	0.936984i	$0.613605\pi$
$192$	8.00000i	0.577350i
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	5.17157	0.371297
$195$	23.3137i	1.66953i
$196$	−6.00000	−0.428571
$197$	24.0000i	1.70993i	0.518686	+	0.854965i	$0.326421\pi$
$197$	24.0000i	1.70993i	−0.518686	+	0.854965i	$0.673579\pi$
$198$	− 6.00000i	− 0.426401i
$199$	10.9706	0.777683	0.388841	−	0.921305i	$-0.372875\pi$
$199$	10.9706	0.777683	0.388841	+	0.921305i	$0.372875\pi$
$200$	−18.8284	−1.33137
$201$	−9.07107	−0.639824
$202$	− 14.3431i	− 1.00918i
$203$	− 5.65685i	− 0.397033i
$204$	− 2.34315i	− 0.164053i
$205$	28.9706i	2.02339i
$206$	−27.7990	−1.93685
$207$	1.00000	0.0695048
$208$	− 27.3137i	− 1.89386i
$209$	−9.51472	−0.658147
$210$	9.65685	0.666386
$211$	0.485281i	0.0334081i	0.999860	+	0.0167041i	$0.00531732\pi$
$211$	0.485281i	0.0334081i	−0.999860	+	0.0167041i	$0.994683\pi$
$212$	− 22.8284i	− 1.56786i
$213$	− 11.6569i	− 0.798714i
$214$	− 3.65685i	− 0.249977i
$215$	6.00000	0.409197
$216$	− 2.82843i	− 0.192450i
$217$	−21.6569	−1.47016
$218$	− 18.0000i	− 1.21911i
$219$	0	0
$220$	−28.9706	−1.95319
$221$	8.00000i	0.538138i
$222$	4.34315	0.291493
$223$	6.82843	0.457265	0.228633	−	0.973513i	$-0.426575\pi$
$223$	6.82843	0.457265	0.228633	+	0.973513i	$0.426575\pi$
$224$	−11.3137	−0.755929
$225$	6.65685	0.443790
$226$	−26.6274	−1.77123
$227$	16.2426i	1.07806i	0.842286	+	0.539031i	$0.181209\pi$
$227$	16.2426i	1.07806i	−0.842286	+	0.539031i	$0.818791\pi$
$228$	−4.48528	−0.297045
$229$	8.72792i	0.576757i	0.957516	+	0.288379i	$0.0931162\pi$
$229$	8.72792i	0.576757i	−0.957516	+	0.288379i	$0.906884\pi$
$230$	− 4.82843i	− 0.318377i
$231$	8.48528	0.558291
$232$	8.00000i	0.525226i
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	9.65685i	0.631288i
$235$	20.4853i	1.33631i
$236$	11.3137i	0.736460i
$237$	9.31371i	0.604990i
$238$	3.31371	0.214796
$239$	−5.65685	−0.365911	−0.182956	−	0.983121i	$-0.558567\pi$
$239$	−5.65685	−0.365911	−0.182956	+	0.983121i	$0.558567\pi$
$240$	−13.6569	−0.881546
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	−9.89949	−0.636364
$243$	1.00000i	0.0641500i
$244$	− 7.51472i	− 0.481081i
$245$	− 10.2426i	− 0.654378i
$246$	12.0000i	0.765092i
$247$	15.3137	0.974388
$248$	30.6274	1.94484
$249$	4.24264	0.268866
$250$	− 8.00000i	− 0.505964i
$251$	− 13.8995i	− 0.877328i	−0.898651	−	0.438664i	$-0.855452\pi$
$251$	− 13.8995i	− 0.877328i	0.898651	−	0.438664i	$-0.144548\pi$
$252$	4.00000	0.251976
$253$	− 4.24264i	− 0.266733i
$254$	−18.8284	−1.18140
$255$	4.00000	0.250490
$256$	16.0000	1.00000
$257$	6.14214	0.383136	0.191568	−	0.981479i	$-0.438643\pi$
$257$	6.14214	0.383136	0.191568	+	0.981479i	$0.438643\pi$
$258$	2.48528	0.154727
$259$	6.14214i	0.381654i
$260$	46.6274	2.89171
$261$	− 2.82843i	− 0.175075i
$262$	− 20.9706i	− 1.29557i
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	−12.0000	−0.738549
$265$	38.9706	2.39394
$266$	− 6.34315i	− 0.388923i
$267$	5.17157i	0.316495i
$268$	18.1421i	1.10821i
$269$	− 2.34315i	− 0.142864i	−0.997445	−	0.0714321i	$-0.977243\pi$
$269$	− 2.34315i	− 0.142864i	0.997445	−	0.0714321i	$-0.0227569\pi$
$270$	4.82843	0.293849
$271$	−13.3137	−0.808750	−0.404375	−	0.914593i	$-0.632511\pi$
$271$	−13.3137	−0.808750	−0.404375	+	0.914593i	$0.632511\pi$
$272$	−4.68629	−0.284148
$273$	−13.6569	−0.826550
$274$	20.9706	1.26688
$275$	− 28.2426i	− 1.70310i
$276$	− 2.00000i	− 0.120386i
$277$	22.1421i	1.33039i	0.746669	+	0.665196i	$0.231652\pi$
$277$	22.1421i	1.33039i	−0.746669	+	0.665196i	$0.768348\pi$
$278$	− 4.68629i	− 0.281065i
$279$	−10.8284	−0.648281
$280$	− 19.3137i	− 1.15421i
$281$	−11.7990	−0.703869	−0.351934	−	0.936025i	$-0.614476\pi$
$281$	−11.7990	−0.703869	−0.351934	+	0.936025i	$0.614476\pi$
$282$	8.48528i	0.505291i
$283$	− 19.8995i	− 1.18290i	−0.806341	−	0.591451i	$-0.798555\pi$
$283$	− 19.8995i	− 1.18290i	0.806341	−	0.591451i	$-0.201445\pi$
$284$	−23.3137	−1.38341
$285$	− 7.65685i	− 0.453553i
$286$	40.9706	2.42264
$287$	−16.9706	−1.00174
$288$	−5.65685	−0.333333
$289$	−15.6274	−0.919260
$290$	−13.6569	−0.801958
$291$	3.65685i	0.214369i
$292$	0	0
$293$	1.75736i	0.102666i	0.998682	+	0.0513330i	$0.0163470\pi$
$293$	1.75736i	0.102666i	−0.998682	+	0.0513330i	$0.983653\pi$
$294$	− 4.24264i	− 0.247436i
$295$	−19.3137	−1.12449
$296$	− 8.68629i	− 0.504880i
$297$	4.24264	0.246183
$298$	18.4853i	1.07082i
$299$	6.82843i	0.394898i
$300$	− 13.3137i	− 0.768667i
$301$	3.51472i	0.202585i
$302$	−4.97056	−0.286024
$303$	10.1421	0.582650
$304$	8.97056i	0.514497i
$305$	12.8284	0.734554
$306$	1.65685	0.0947161
$307$	− 12.4853i	− 0.712573i	−0.934377	−	0.356286i	$-0.884043\pi$
$307$	− 12.4853i	− 0.712573i	0.934377	−	0.356286i	$-0.115957\pi$
$308$	− 16.9706i	− 0.966988i
$309$	− 19.6569i	− 1.11824i
$310$	52.2843i	2.96955i
$311$	−21.6569	−1.22805	−0.614024	−	0.789288i	$-0.710450\pi$
$311$	−21.6569	−1.22805	−0.614024	+	0.789288i	$0.710450\pi$
$312$	19.3137	1.09342
$313$	6.97056	0.394000	0.197000	−	0.980404i	$-0.436880\pi$
$313$	6.97056	0.394000	0.197000	+	0.980404i	$0.436880\pi$
$314$	− 23.6569i	− 1.33503i
$315$	6.82843i	0.384738i
$316$	18.6274	1.04787
$317$	6.14214i	0.344977i	0.985012	+	0.172488i	$0.0551807\pi$
$317$	6.14214i	0.344977i	−0.985012	+	0.172488i	$0.944819\pi$
$318$	16.1421	0.905206
$319$	−12.0000	−0.671871
$320$	27.3137i	1.52688i
$321$	2.58579	0.144325
$322$	2.82843	0.157622
$323$	− 2.62742i	− 0.146193i
$324$	2.00000	0.111111
$325$	45.4558i	2.52144i
$326$	6.34315i	0.351314i
$327$	12.7279	0.703856
$328$	24.0000	1.32518
$329$	−12.0000	−0.661581
$330$	− 20.4853i	− 1.12768i
$331$	24.4853i	1.34583i	0.739719	+	0.672916i	$0.234959\pi$
$331$	24.4853i	1.34583i	−0.739719	+	0.672916i	$0.765041\pi$
$332$	− 8.48528i	− 0.465690i
$333$	3.07107i	0.168293i
$334$	−10.8284	−0.592505
$335$	−30.9706	−1.69210
$336$	− 8.00000i	− 0.436436i
$337$	−15.6569	−0.852883	−0.426442	−	0.904515i	$-0.640233\pi$
$337$	−15.6569	−0.852883	−0.426442	+	0.904515i	$0.640233\pi$
$338$	−47.5563	−2.58672
$339$	− 18.8284i	− 1.02262i
$340$	− 8.00000i	− 0.433861i
$341$	45.9411i	2.48785i
$342$	− 3.17157i	− 0.171499i
$343$	20.0000	1.07990
$344$	− 4.97056i	− 0.267995i
$345$	3.41421	0.183815
$346$	− 26.6274i	− 1.43150i
$347$	15.5147i	0.832874i	0.909165	+	0.416437i	$0.136721\pi$
$347$	15.5147i	0.832874i	−0.909165	+	0.416437i	$0.863279\pi$
$348$	−5.65685	−0.303239
$349$	28.9706i	1.55076i	0.631496	+	0.775379i	$0.282441\pi$
$349$	28.9706i	1.55076i	−0.631496	+	0.775379i	$0.717559\pi$
$350$	18.8284	1.00642
$351$	−6.82843	−0.364474
$352$	24.0000i	1.27920i
$353$	−3.65685	−0.194635	−0.0973174	−	0.995253i	$-0.531026\pi$
$353$	−3.65685	−0.194635	−0.0973174	+	0.995253i	$0.531026\pi$
$354$	−8.00000	−0.425195
$355$	− 39.7990i	− 2.11231i
$356$	10.3431	0.548186
$357$	2.34315i	0.124012i
$358$	12.0000i	0.634220i
$359$	−11.3137	−0.597115	−0.298557	−	0.954392i	$-0.596505\pi$
$359$	−11.3137	−0.597115	−0.298557	+	0.954392i	$0.596505\pi$
$360$	− 9.65685i	− 0.508961i
$361$	13.9706	0.735293
$362$	17.3137i	0.909988i
$363$	− 7.00000i	− 0.367405i
$364$	27.3137i	1.43163i
$365$	0	0
$366$	5.31371	0.277752
$367$	30.9706	1.61665	0.808325	−	0.588736i	$-0.200374\pi$
$367$	30.9706	1.61665	0.808325	+	0.588736i	$0.200374\pi$
$368$	−4.00000	−0.208514
$369$	−8.48528	−0.441726
$370$	14.8284	0.770893
$371$	22.8284i	1.18519i
$372$	21.6569i	1.12286i
$373$	− 36.7279i	− 1.90170i	−0.309652	−	0.950850i	$-0.600213\pi$
$373$	− 36.7279i	− 1.90170i	0.309652	−	0.950850i	$-0.399787\pi$
$374$	− 7.02944i	− 0.363484i
$375$	5.65685	0.292119
$376$	16.9706	0.875190
$377$	19.3137	0.994707
$378$	2.82843i	0.145479i
$379$	− 14.7279i	− 0.756523i	−0.925699	−	0.378261i	$-0.876522\pi$
$379$	− 14.7279i	− 0.756523i	0.925699	−	0.378261i	$-0.123478\pi$
$380$	−15.3137	−0.785577
$381$	− 13.3137i	− 0.682082i
$382$	−13.6569	−0.698745
$383$	−19.3137	−0.986884	−0.493442	−	0.869779i	$-0.664262\pi$
$383$	−19.3137	−0.986884	−0.493442	+	0.869779i	$0.664262\pi$
$384$	11.3137i	0.577350i
$385$	28.9706	1.47648
$386$	−14.1421	−0.719816
$387$	1.75736i	0.0893316i
$388$	7.31371	0.371297
$389$	14.0416i	0.711939i	0.934498	+	0.355969i	$0.115849\pi$
$389$	14.0416i	0.711939i	−0.934498	+	0.355969i	$0.884151\pi$
$390$	32.9706i	1.66953i
$391$	1.17157	0.0592490
$392$	−8.48528	−0.428571
$393$	14.8284	0.747995
$394$	33.9411i	1.70993i
$395$	31.7990i	1.59998i
$396$	− 8.48528i	− 0.426401i
$397$	0.970563i	0.0487111i	0.999703	+	0.0243556i	$0.00775339\pi$
$397$	0.970563i	0.0487111i	−0.999703	+	0.0243556i	$0.992247\pi$
$398$	15.5147	0.777683
$399$	4.48528	0.224545
$400$	−26.6274	−1.33137
$401$	−37.4558	−1.87046	−0.935228	−	0.354047i	$-0.884805\pi$
$401$	−37.4558	−1.87046	−0.935228	+	0.354047i	$0.884805\pi$
$402$	−12.8284	−0.639824
$403$	− 73.9411i	− 3.68327i
$404$	− 20.2843i	− 1.00918i
$405$	3.41421i	0.169654i
$406$	− 8.00000i	− 0.397033i
$407$	13.0294	0.645845
$408$	− 3.31371i	− 0.164053i
$409$	2.00000	0.0988936	0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000	0.0988936	0.0494468	+	0.998777i	$0.484254\pi$
$410$	40.9706i	2.02339i
$411$	14.8284i	0.731432i
$412$	−39.3137	−1.93685
$413$	− 11.3137i	− 0.556711i
$414$	1.41421	0.0695048
$415$	14.4853	0.711054
$416$	− 38.6274i	− 1.89386i
$417$	3.31371	0.162273
$418$	−13.4558	−0.658147
$419$	− 0.928932i	− 0.0453813i	−0.999743	−	0.0226907i	$-0.992777\pi$
$419$	− 0.928932i	− 0.0453813i	0.999743	−	0.0226907i	$-0.00722328\pi$
$420$	13.6569	0.666386
$421$	− 19.5563i	− 0.953118i	−0.879142	−	0.476559i	$-0.841884\pi$
$421$	− 19.5563i	− 0.953118i	0.879142	−	0.476559i	$-0.158116\pi$
$422$	0.686292i	0.0334081i
$423$	−6.00000	−0.291730
$424$	− 32.2843i	− 1.56786i
$425$	7.79899	0.378307
$426$	− 16.4853i	− 0.798714i
$427$	7.51472i	0.363663i
$428$	− 5.17157i	− 0.249977i
$429$	28.9706i	1.39871i
$430$	8.48528	0.409197
$431$	−20.2843	−0.977059	−0.488529	−	0.872547i	$-0.662467\pi$
$431$	−20.2843	−0.977059	−0.488529	+	0.872547i	$0.662467\pi$
$432$	− 4.00000i	− 0.192450i
$433$	−10.9706	−0.527212	−0.263606	−	0.964630i	$-0.584912\pi$
$433$	−10.9706	−0.527212	−0.263606	+	0.964630i	$0.584912\pi$
$434$	−30.6274	−1.47016
$435$	− 9.65685i	− 0.463011i
$436$	− 25.4558i	− 1.21911i
$437$	− 2.24264i	− 0.107280i
$438$	0	0
$439$	36.6274	1.74813	0.874066	−	0.485808i	$-0.161475\pi$
$439$	36.6274	1.74813	0.874066	+	0.485808i	$0.161475\pi$
$440$	−40.9706	−1.95319
$441$	3.00000	0.142857
$442$	11.3137i	0.538138i
$443$	16.2843i	0.773689i	0.922145	+	0.386845i	$0.126435\pi$
$443$	16.2843i	0.773689i	−0.922145	+	0.386845i	$0.873565\pi$
$444$	6.14214	0.291493
$445$	17.6569i	0.837015i
$446$	9.65685	0.457265
$447$	−13.0711	−0.618240
$448$	−16.0000	−0.755929
$449$	27.7990	1.31192	0.655958	−	0.754798i	$-0.272265\pi$
$449$	27.7990	1.31192	0.655958	+	0.754798i	$0.272265\pi$
$450$	9.41421	0.443790
$451$	36.0000i	1.69517i
$452$	−37.6569	−1.77123
$453$	− 3.51472i	− 0.165136i
$454$	22.9706i	1.07806i
$455$	−46.6274	−2.18593
$456$	−6.34315	−0.297045
$457$	27.6569	1.29373	0.646867	−	0.762603i	$-0.276079\pi$
$457$	27.6569	1.29373	0.646867	+	0.762603i	$0.276079\pi$
$458$	12.3431i	0.576757i
$459$	1.17157i	0.0546843i
$460$	− 6.82843i	− 0.318377i
$461$	− 28.2843i	− 1.31733i	−0.752436	−	0.658665i	$-0.771121\pi$
$461$	− 28.2843i	− 1.31733i	0.752436	−	0.658665i	$-0.228879\pi$
$462$	12.0000	0.558291
$463$	−25.3137	−1.17643	−0.588214	−	0.808705i	$-0.700169\pi$
$463$	−25.3137	−1.17643	−0.588214	+	0.808705i	$0.700169\pi$
$464$	11.3137i	0.525226i
$465$	−36.9706	−1.71447
$466$	25.4558	1.17922
$467$	− 35.5563i	− 1.64535i	−0.568511	−	0.822676i	$-0.692480\pi$
$467$	− 35.5563i	− 1.64535i	0.568511	−	0.822676i	$-0.307520\pi$
$468$	13.6569i	0.631288i
$469$	− 18.1421i	− 0.837726i
$470$	28.9706i	1.33631i
$471$	16.7279	0.770782
$472$	16.0000i	0.736460i
$473$	7.45584	0.342820
$474$	13.1716i	0.604990i
$475$	− 14.9289i	− 0.684986i
$476$	4.68629	0.214796
$477$	11.4142i	0.522621i
$478$	−8.00000	−0.365911
$479$	17.6569	0.806762	0.403381	−	0.915032i	$-0.367835\pi$
$479$	17.6569	0.806762	0.403381	+	0.915032i	$0.367835\pi$
$480$	−19.3137	−0.881546
$481$	−20.9706	−0.956175
$482$	14.1421	0.644157
$483$	2.00000i	0.0910032i
$484$	−14.0000	−0.636364
$485$	12.4853i	0.566927i
$486$	1.41421i	0.0641500i
$487$	−35.9411	−1.62865	−0.814324	−	0.580411i	$-0.802892\pi$
$487$	−35.9411	−1.62865	−0.814324	+	0.580411i	$0.802892\pi$
$488$	− 10.6274i	− 0.481081i
$489$	−4.48528	−0.202831
$490$	− 14.4853i	− 0.654378i
$491$	− 14.1421i	− 0.638226i	−0.947717	−	0.319113i	$-0.896615\pi$
$491$	− 14.1421i	− 0.638226i	0.947717	−	0.319113i	$-0.103385\pi$
$492$	16.9706i	0.765092i
$493$	− 3.31371i	− 0.149242i
$494$	21.6569	0.974388
$495$	14.4853	0.651065
$496$	43.3137	1.94484
$497$	23.3137	1.04576
$498$	6.00000	0.268866
$499$	6.82843i	0.305682i	0.988251	+	0.152841i	$0.0488423\pi$
$499$	6.82843i	0.305682i	−0.988251	+	0.152841i	$0.951158\pi$
$500$	− 11.3137i	− 0.505964i
$501$	− 7.65685i	− 0.342083i
$502$	− 19.6569i	− 0.877328i
$503$	1.65685	0.0738755	0.0369377	−	0.999318i	$-0.488240\pi$
$503$	1.65685	0.0738755	0.0369377	+	0.999318i	$0.488240\pi$
$504$	5.65685	0.251976
$505$	34.6274	1.54090
$506$	− 6.00000i	− 0.266733i
$507$	− 33.6274i	− 1.49345i
$508$	−26.6274	−1.18140
$509$	2.82843i	0.125368i	0.998033	+	0.0626839i	$0.0199660\pi$
$509$	2.82843i	0.125368i	−0.998033	+	0.0626839i	$0.980034\pi$
$510$	5.65685	0.250490
$511$	0	0
$512$	22.6274	1.00000
$513$	2.24264	0.0990150
$514$	8.68629	0.383136
$515$	− 67.1127i	− 2.95734i
$516$	3.51472	0.154727
$517$	25.4558i	1.11955i
$518$	8.68629i	0.381654i
$519$	18.8284	0.826476
$520$	65.9411	2.89171
$521$	−22.8284	−1.00013	−0.500066	−	0.865987i	$-0.666691\pi$
$521$	−22.8284	−1.00013	−0.500066	+	0.865987i	$0.666691\pi$
$522$	− 4.00000i	− 0.175075i
$523$	− 1.75736i	− 0.0768440i	−0.999262	−	0.0384220i	$-0.987767\pi$
$523$	− 1.75736i	− 0.0768440i	0.999262	−	0.0384220i	$-0.0122331\pi$
$524$	− 29.6569i	− 1.29557i
$525$	13.3137i	0.581058i
$526$	16.9706	0.739952
$527$	−12.6863	−0.552624
$528$	−16.9706	−0.738549
$529$	1.00000	0.0434783
$530$	55.1127	2.39394
$531$	− 5.65685i	− 0.245487i
$532$	− 8.97056i	− 0.388923i
$533$	− 57.9411i	− 2.50971i
$534$	7.31371i	0.316495i
$535$	8.82843	0.381686
$536$	25.6569i	1.10821i
$537$	−8.48528	−0.366167
$538$	− 3.31371i	− 0.142864i
$539$	− 12.7279i	− 0.548230i
$540$	6.82843	0.293849
$541$	− 36.0000i	− 1.54776i	−0.633332	−	0.773880i	$-0.718313\pi$
$541$	− 36.0000i	− 1.54776i	0.633332	−	0.773880i	$-0.281687\pi$
$542$	−18.8284	−0.808750
$543$	−12.2426	−0.525382
$544$	−6.62742	−0.284148
$545$	43.4558	1.86144
$546$	−19.3137	−0.826550
$547$	15.5147i	0.663361i	0.943392	+	0.331681i	$0.107616\pi$
$547$	15.5147i	0.663361i	−0.943392	+	0.331681i	$0.892384\pi$
$548$	29.6569	1.26688
$549$	3.75736i	0.160360i
$550$	− 39.9411i	− 1.70310i
$551$	−6.34315	−0.270227
$552$	− 2.82843i	− 0.120386i
$553$	−18.6274	−0.792118
$554$	31.3137i	1.33039i
$555$	10.4853i	0.445075i
$556$	− 6.62742i	− 0.281065i
$557$	23.8995i	1.01265i	0.862342	+	0.506327i	$0.168997\pi$
$557$	23.8995i	1.01265i	−0.862342	+	0.506327i	$0.831003\pi$
$558$	−15.3137	−0.648281
$559$	−12.0000	−0.507546
$560$	− 27.3137i	− 1.15421i
$561$	4.97056	0.209857
$562$	−16.6863	−0.703869
$563$	0.727922i	0.0306783i	0.999882	+	0.0153391i	$0.00488279\pi$
$563$	0.727922i	0.0306783i	−0.999882	+	0.0153391i	$0.995117\pi$
$564$	12.0000i	0.505291i
$565$	− 64.2843i	− 2.70446i
$566$	− 28.1421i	− 1.18290i
$567$	−2.00000	−0.0839921
$568$	−32.9706	−1.38341
$569$	26.8284	1.12471	0.562353	−	0.826897i	$-0.309896\pi$
$569$	26.8284	1.12471	0.562353	+	0.826897i	$0.309896\pi$
$570$	− 10.8284i	− 0.453553i
$571$	− 12.5858i	− 0.526699i	−0.964701	−	0.263349i	$-0.915173\pi$
$571$	− 12.5858i	− 0.526699i	0.964701	−	0.263349i	$-0.0848272\pi$
$572$	57.9411	2.42264
$573$	− 9.65685i	− 0.403421i
$574$	−24.0000	−1.00174
$575$	6.65685	0.277610
$576$	−8.00000	−0.333333
$577$	20.0000	0.832611	0.416305	−	0.909225i	$-0.363325\pi$
$577$	20.0000	0.832611	0.416305	+	0.909225i	$0.363325\pi$
$578$	−22.1005	−0.919260
$579$	− 10.0000i	− 0.415586i
$580$	−19.3137	−0.801958
$581$	8.48528i	0.352029i
$582$	5.17157i	0.214369i
$583$	48.4264	2.00562
$584$	0	0
$585$	−23.3137	−0.963903
$586$	2.48528i	0.102666i
$587$	− 15.7990i	− 0.652094i	−0.945354	−	0.326047i	$-0.894283\pi$
$587$	− 15.7990i	− 0.652094i	0.945354	−	0.326047i	$-0.105717\pi$
$588$	− 6.00000i	− 0.247436i
$589$	24.2843i	1.00062i
$590$	−27.3137	−1.12449
$591$	−24.0000	−0.987228
$592$	− 12.2843i	− 0.504880i
$593$	−22.1421	−0.909269	−0.454634	−	0.890678i	$-0.650230\pi$
$593$	−22.1421	−0.909269	−0.454634	+	0.890678i	$0.650230\pi$
$594$	6.00000	0.246183
$595$	8.00000i	0.327968i
$596$	26.1421i	1.07082i
$597$	10.9706i	0.448995i
$598$	9.65685i	0.394898i
$599$	−3.31371	−0.135394	−0.0676972	−	0.997706i	$-0.521565\pi$
$599$	−3.31371	−0.135394	−0.0676972	+	0.997706i	$0.521565\pi$
$600$	− 18.8284i	− 0.768667i
$601$	−42.6274	−1.73881	−0.869404	−	0.494101i	$-0.835497\pi$
$601$	−42.6274	−1.73881	−0.869404	+	0.494101i	$0.835497\pi$
$602$	4.97056i	0.202585i
$603$	− 9.07107i	− 0.369402i
$604$	−7.02944	−0.286024
$605$	− 23.8995i	− 0.971653i
$606$	14.3431	0.582650
$607$	20.4853	0.831472	0.415736	−	0.909485i	$-0.363524\pi$
$607$	20.4853	0.831472	0.415736	+	0.909485i	$0.363524\pi$
$608$	12.6863i	0.514497i
$609$	5.65685	0.229227
$610$	18.1421	0.734554
$611$	− 40.9706i	− 1.65749i
$612$	2.34315	0.0947161
$613$	− 11.7574i	− 0.474875i	−0.971403	−	0.237438i	$-0.923692\pi$
$613$	− 11.7574i	− 0.474875i	0.971403	−	0.237438i	$-0.0763075\pi$
$614$	− 17.6569i	− 0.712573i
$615$	−28.9706	−1.16821
$616$	− 24.0000i	− 0.966988i
$617$	−18.1421	−0.730375	−0.365187	−	0.930934i	$-0.618995\pi$
$617$	−18.1421	−0.730375	−0.365187	+	0.930934i	$0.618995\pi$
$618$	− 27.7990i	− 1.11824i
$619$	− 8.38478i	− 0.337013i	−0.985701	−	0.168506i	$-0.946106\pi$
$619$	− 8.38478i	− 0.337013i	0.985701	−	0.168506i	$-0.0538944\pi$
$620$	73.9411i	2.96955i
$621$	1.00000i	0.0401286i
$622$	−30.6274	−1.22805
$623$	−10.3431	−0.414389
$624$	27.3137	1.09342
$625$	−13.9706	−0.558823
$626$	9.85786	0.394000
$627$	− 9.51472i	− 0.379981i
$628$	− 33.4558i	− 1.33503i
$629$	3.59798i	0.143461i
$630$	9.65685i	0.384738i
$631$	26.2843	1.04636	0.523180	−	0.852222i	$-0.324745\pi$
$631$	26.2843	1.04636	0.523180	+	0.852222i	$0.324745\pi$
$632$	26.3431	1.04787
$633$	−0.485281	−0.0192882
$634$	8.68629i	0.344977i
$635$	− 45.4558i	− 1.80386i
$636$	22.8284	0.905206
$637$	20.4853i	0.811656i
$638$	−16.9706	−0.671871
$639$	11.6569	0.461138
$640$	38.6274i	1.52688i
$641$	−35.7990	−1.41398	−0.706988	−	0.707226i	$-0.749946\pi$
$641$	−35.7990	−1.41398	−0.706988	+	0.707226i	$0.749946\pi$
$642$	3.65685	0.144325
$643$	− 18.9289i	− 0.746484i	−0.927734	−	0.373242i	$-0.878246\pi$
$643$	− 18.9289i	− 0.746484i	0.927734	−	0.373242i	$-0.121754\pi$
$644$	4.00000	0.157622
$645$	6.00000i	0.236250i
$646$	− 3.71573i	− 0.146193i
$647$	19.3137	0.759300	0.379650	−	0.925130i	$-0.376044\pi$
$647$	19.3137	0.759300	0.379650	+	0.925130i	$0.376044\pi$
$648$	2.82843	0.111111
$649$	−24.0000	−0.942082
$650$	64.2843i	2.52144i
$651$	− 21.6569i	− 0.848799i
$652$	8.97056i	0.351314i
$653$	10.1421i	0.396892i	0.980112	+	0.198446i	$0.0635895\pi$
$653$	10.1421i	0.396892i	−0.980112	+	0.198446i	$0.936410\pi$
$654$	18.0000	0.703856
$655$	50.6274	1.97818
$656$	33.9411	1.32518
$657$	0	0
$658$	−16.9706	−0.661581
$659$	26.8701i	1.04671i	0.852115	+	0.523354i	$0.175320\pi$
$659$	26.8701i	1.04671i	−0.852115	+	0.523354i	$0.824680\pi$
$660$	− 28.9706i	− 1.12768i
$661$	− 21.6985i	− 0.843973i	−0.906602	−	0.421987i	$-0.861333\pi$
$661$	− 21.6985i	− 0.843973i	0.906602	−	0.421987i	$-0.138667\pi$
$662$	34.6274i	1.34583i
$663$	−8.00000	−0.310694
$664$	− 12.0000i	− 0.465690i
$665$	15.3137	0.593840
$666$	4.34315i	0.168293i
$667$	− 2.82843i	− 0.109517i
$668$	−15.3137	−0.592505
$669$	6.82843i	0.264002i
$670$	−43.7990	−1.69210
$671$	15.9411	0.615400
$672$	− 11.3137i	− 0.436436i
$673$	21.6569	0.834810	0.417405	−	0.908720i	$-0.362940\pi$
$673$	21.6569	0.834810	0.417405	+	0.908720i	$0.362940\pi$
$674$	−22.1421	−0.852883
$675$	6.65685i	0.256222i
$676$	−67.2548	−2.58672
$677$	− 3.89949i	− 0.149870i	−0.997188	−	0.0749349i	$-0.976125\pi$
$677$	− 3.89949i	− 0.149870i	0.997188	−	0.0749349i	$-0.0238749\pi$
$678$	− 26.6274i	− 1.02262i
$679$	−7.31371	−0.280674
$680$	− 11.3137i	− 0.433861i
$681$	−16.2426	−0.622419
$682$	64.9706i	2.48785i
$683$	− 43.1127i	− 1.64966i	−0.565380	−	0.824831i	$-0.691270\pi$
$683$	− 43.1127i	− 1.64966i	0.565380	−	0.824831i	$-0.308730\pi$
$684$	− 4.48528i	− 0.171499i
$685$	50.6274i	1.93437i
$686$	28.2843	1.07990
$687$	−8.72792	−0.332991
$688$	− 7.02944i	− 0.267995i
$689$	−77.9411	−2.96932
$690$	4.82843	0.183815
$691$	15.5147i	0.590208i	0.955465	+	0.295104i	$0.0953543\pi$
$691$	15.5147i	0.590208i	−0.955465	+	0.295104i	$0.904646\pi$
$692$	− 37.6569i	− 1.43150i
$693$	8.48528i	0.322329i
$694$	21.9411i	0.832874i
$695$	11.3137	0.429153
$696$	−8.00000	−0.303239
$697$	−9.94113	−0.376547
$698$	40.9706i	1.55076i
$699$	18.0000i	0.680823i
$700$	26.6274	1.00642
$701$	33.5563i	1.26741i	0.773577	+	0.633703i	$0.218466\pi$
$701$	33.5563i	1.26741i	−0.773577	+	0.633703i	$0.781534\pi$
$702$	−9.65685	−0.364474
$703$	6.88730	0.259760
$704$	33.9411i	1.27920i
$705$	−20.4853	−0.771520
$706$	−5.17157	−0.194635
$707$	20.2843i	0.762869i
$708$	−11.3137	−0.425195
$709$	− 42.5858i	− 1.59934i	−0.600438	−	0.799671i	$-0.705007\pi$
$709$	− 42.5858i	− 1.59934i	0.600438	−	0.799671i	$-0.294993\pi$
$710$	− 56.2843i	− 2.11231i
$711$	−9.31371	−0.349291
$712$	14.6274	0.548186
$713$	−10.8284	−0.405528
$714$	3.31371i	0.124012i
$715$	98.9117i	3.69909i
$716$	16.9706i	0.634220i
$717$	− 5.65685i	− 0.211259i
$718$	−16.0000	−0.597115
$719$	1.37258	0.0511887	0.0255944	−	0.999672i	$-0.491852\pi$
$719$	1.37258	0.0511887	0.0255944	+	0.999672i	$0.491852\pi$
$720$	− 13.6569i	− 0.508961i
$721$	39.3137	1.46412
$722$	19.7574	0.735293
$723$	10.0000i	0.371904i
$724$	24.4853i	0.909988i
$725$	− 18.8284i	− 0.699270i
$726$	− 9.89949i	− 0.367405i
$727$	−42.2843	−1.56824	−0.784118	−	0.620611i	$-0.786885\pi$
$727$	−42.2843	−1.56824	−0.784118	+	0.620611i	$0.786885\pi$
$728$	38.6274i	1.43163i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	2.05887i	0.0761502i
$732$	7.51472	0.277752
$733$	15.7574i	0.582011i	0.956721	+	0.291006i	$0.0939899\pi$
$733$	15.7574i	0.582011i	−0.956721	+	0.291006i	$0.906010\pi$
$734$	43.7990	1.61665
$735$	10.2426	0.377805
$736$	−5.65685	−0.208514
$737$	−38.4853	−1.41762
$738$	−12.0000	−0.441726
$739$	− 38.3431i	− 1.41048i	−0.708971	−	0.705238i	$-0.750840\pi$
$739$	− 38.3431i	− 1.41048i	0.708971	−	0.705238i	$-0.249160\pi$
$740$	20.9706	0.770893
$741$	15.3137i	0.562563i
$742$	32.2843i	1.18519i
$743$	27.5980	1.01247	0.506236	−	0.862395i	$-0.331037\pi$
$743$	27.5980	1.01247	0.506236	+	0.862395i	$0.331037\pi$
$744$	30.6274i	1.12286i
$745$	−44.6274	−1.63502
$746$	− 51.9411i	− 1.90170i
$747$	4.24264i	0.155230i
$748$	− 9.94113i	− 0.363484i
$749$	5.17157i	0.188965i
$750$	8.00000	0.292119
$751$	1.02944	0.0375647	0.0187823	−	0.999824i	$-0.494021\pi$
$751$	1.02944	0.0375647	0.0187823	+	0.999824i	$0.494021\pi$
$752$	24.0000	0.875190
$753$	13.8995	0.506526
$754$	27.3137	0.994707
$755$	− 12.0000i	− 0.436725i
$756$	4.00000i	0.145479i
$757$	9.21320i	0.334860i	0.985884	+	0.167430i	$0.0535467\pi$
$757$	9.21320i	0.334860i	−0.985884	+	0.167430i	$0.946453\pi$
$758$	− 20.8284i	− 0.756523i
$759$	4.24264	0.153998
$760$	−21.6569	−0.785577
$761$	3.65685	0.132561	0.0662804	−	0.997801i	$-0.478887\pi$
$761$	3.65685	0.132561	0.0662804	+	0.997801i	$0.478887\pi$
$762$	− 18.8284i	− 0.682082i
$763$	25.4558i	0.921563i
$764$	−19.3137	−0.698745
$765$	4.00000i	0.144620i
$766$	−27.3137	−0.986884
$767$	38.6274	1.39476
$768$	16.0000i	0.577350i
$769$	26.9706	0.972583	0.486292	−	0.873797i	$-0.338349\pi$
$769$	26.9706	0.972583	0.486292	+	0.873797i	$0.338349\pi$
$770$	40.9706	1.47648
$771$	6.14214i	0.221204i
$772$	−20.0000	−0.719816
$773$	45.0711i	1.62109i	0.585674	+	0.810547i	$0.300830\pi$
$773$	45.0711i	1.62109i	−0.585674	+	0.810547i	$0.699170\pi$
$774$	2.48528i	0.0893316i
$775$	−72.0833	−2.58931
$776$	10.3431	0.371297
$777$	−6.14214	−0.220348
$778$	19.8579i	0.711939i
$779$	19.0294i	0.681800i
$780$	46.6274i	1.66953i
$781$	− 49.4558i	− 1.76967i
$782$	1.65685	0.0592490
$783$	2.82843	0.101080
$784$	−12.0000	−0.428571
$785$	57.1127	2.03844
$786$	20.9706	0.747995
$787$	− 4.10051i	− 0.146167i	−0.997326	−	0.0730836i	$-0.976716\pi$
$787$	− 4.10051i	− 0.146167i	0.997326	−	0.0730836i	$-0.0232840\pi$
$788$	48.0000i	1.70993i
$789$	12.0000i	0.427211i
$790$	44.9706i	1.59998i
$791$	37.6569	1.33892
$792$	− 12.0000i	− 0.426401i
$793$	−25.6569	−0.911102
$794$	1.37258i	0.0487111i
$795$	38.9706i	1.38214i
$796$	21.9411	0.777683
$797$	17.3553i	0.614758i	0.951587	+	0.307379i	$0.0994519\pi$
$797$	17.3553i	0.614758i	−0.951587	+	0.307379i	$0.900548\pi$
$798$	6.34315	0.224545
$799$	−7.02944	−0.248684
$800$	−37.6569	−1.33137
$801$	−5.17157	−0.182729
$802$	−52.9706	−1.87046
$803$	0	0
$804$	−18.1421	−0.639824
$805$	6.82843i	0.240670i
$806$	− 104.569i	− 3.68327i
$807$	2.34315	0.0824826
$808$	− 28.6863i	− 1.00918i
$809$	12.3431	0.433962	0.216981	−	0.976176i	$-0.430379\pi$
$809$	12.3431	0.433962	0.216981	+	0.976176i	$0.430379\pi$
$810$	4.82843i	0.169654i
$811$	30.1421i	1.05843i	0.848487	+	0.529217i	$0.177514\pi$
$811$	30.1421i	1.05843i	−0.848487	+	0.529217i	$0.822486\pi$
$812$	− 11.3137i	− 0.397033i
$813$	− 13.3137i	− 0.466932i
$814$	18.4264	0.645845
$815$	−15.3137	−0.536416
$816$	− 4.68629i	− 0.164053i
$817$	3.94113	0.137883
$818$	2.82843	0.0988936
$819$	− 13.6569i	− 0.477209i
$820$	57.9411i	2.02339i
$821$	50.1421i	1.74997i	0.484148	+	0.874986i	$0.339130\pi$
$821$	50.1421i	1.74997i	−0.484148	+	0.874986i	$0.660870\pi$
$822$	20.9706i	0.731432i
$823$	25.4558	0.887335	0.443667	−	0.896191i	$-0.353677\pi$
$823$	25.4558	0.887335	0.443667	+	0.896191i	$0.353677\pi$
$824$	−55.5980	−1.93685
$825$	28.2426	0.983283
$826$	− 16.0000i	− 0.556711i
$827$	32.5269i	1.13107i	0.824724	+	0.565536i	$0.191331\pi$
$827$	32.5269i	1.13107i	−0.824724	+	0.565536i	$0.808669\pi$
$828$	2.00000	0.0695048
$829$	36.0000i	1.25033i	0.780492	+	0.625166i	$0.214969\pi$
$829$	36.0000i	1.25033i	−0.780492	+	0.625166i	$0.785031\pi$
$830$	20.4853	0.711054
$831$	−22.1421	−0.768102
$832$	− 54.6274i	− 1.89386i
$833$	3.51472	0.121778
$834$	4.68629	0.162273
$835$	− 26.1421i	− 0.904686i
$836$	−19.0294	−0.658147
$837$	− 10.8284i	− 0.374285i
$838$	− 1.31371i	− 0.0453813i
$839$	15.3137	0.528688	0.264344	−	0.964428i	$-0.414845\pi$
$839$	15.3137	0.528688	0.264344	+	0.964428i	$0.414845\pi$
$840$	19.3137	0.666386
$841$	21.0000	0.724138
$842$	− 27.6569i	− 0.953118i
$843$	− 11.7990i	− 0.406379i
$844$	0.970563i	0.0334081i
$845$	− 114.811i	− 3.94962i
$846$	−8.48528	−0.291730
$847$	14.0000	0.481046
$848$	− 45.6569i	− 1.56786i
$849$	19.8995	0.682949
$850$	11.0294	0.378307
$851$	3.07107i	0.105275i
$852$	− 23.3137i	− 0.798714i
$853$	− 12.2843i	− 0.420605i	−0.977636	−	0.210303i	$-0.932555\pi$
$853$	− 12.2843i	− 0.420605i	0.977636	−	0.210303i	$-0.0674450\pi$
$854$	10.6274i	0.363663i
$855$	7.65685	0.261859
$856$	− 7.31371i	− 0.249977i
$857$	21.3137	0.728062	0.364031	−	0.931387i	$-0.381400\pi$
$857$	21.3137	0.728062	0.364031	+	0.931387i	$0.381400\pi$
$858$	40.9706i	1.39871i
$859$	− 4.00000i	− 0.136478i	−0.997669	−	0.0682391i	$-0.978262\pi$
$859$	− 4.00000i	− 0.136478i	0.997669	−	0.0682391i	$-0.0217381\pi$
$860$	12.0000	0.409197
$861$	− 16.9706i	− 0.578355i
$862$	−28.6863	−0.977059
$863$	−16.9706	−0.577685	−0.288842	−	0.957377i	$-0.593270\pi$
$863$	−16.9706	−0.577685	−0.288842	+	0.957377i	$0.593270\pi$
$864$	− 5.65685i	− 0.192450i
$865$	64.2843	2.18573
$866$	−15.5147	−0.527212
$867$	− 15.6274i	− 0.530735i
$868$	−43.3137	−1.47016
$869$	39.5147i	1.34045i
$870$	− 13.6569i	− 0.463011i
$871$	61.9411	2.09879
$872$	− 36.0000i	− 1.21911i
$873$	−3.65685	−0.123766
$874$	− 3.17157i	− 0.107280i
$875$	11.3137i	0.382473i
$876$	0	0
$877$	− 8.48528i	− 0.286528i	−0.989685	−	0.143264i	$-0.954240\pi$
$877$	− 8.48528i	− 0.286528i	0.989685	−	0.143264i	$-0.0457597\pi$
$878$	51.7990	1.74813
$879$	−1.75736	−0.0592743
$880$	−57.9411	−1.95319
$881$	37.4558	1.26192	0.630960	−	0.775816i	$-0.282661\pi$
$881$	37.4558	1.26192	0.630960	+	0.775816i	$0.282661\pi$
$882$	4.24264	0.142857
$883$	8.48528i	0.285552i	0.989755	+	0.142776i	$0.0456029\pi$
$883$	8.48528i	0.285552i	−0.989755	+	0.142776i	$0.954397\pi$
$884$	16.0000i	0.538138i
$885$	− 19.3137i	− 0.649223i
$886$	23.0294i	0.773689i
$887$	15.9411	0.535251	0.267625	−	0.963523i	$-0.413761\pi$
$887$	15.9411	0.535251	0.267625	+	0.963523i	$0.413761\pi$
$888$	8.68629	0.291493
$889$	26.6274	0.893055
$890$	24.9706i	0.837015i
$891$	4.24264i	0.142134i
$892$	13.6569	0.457265
$893$	13.4558i	0.450283i
$894$	−18.4853	−0.618240
$895$	−28.9706	−0.968379
$896$	−22.6274	−0.755929
$897$	−6.82843	−0.227995
$898$	39.3137	1.31192
$899$	30.6274i	1.02148i
$900$	13.3137	0.443790
$901$	13.3726i	0.445505i
$902$	50.9117i	1.69517i
$903$	−3.51472	−0.116963
$904$	−53.2548	−1.77123
$905$	−41.7990	−1.38945
$906$	− 4.97056i	− 0.165136i
$907$	37.5563i	1.24704i	0.781808	+	0.623519i	$0.214298\pi$
$907$	37.5563i	1.24704i	−0.781808	+	0.623519i	$0.785702\pi$
$908$	32.4853i	1.07806i
$909$	10.1421i	0.336393i
$910$	−65.9411	−2.18593
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	−8.97056	−0.297045
$913$	18.0000	0.595713
$914$	39.1127	1.29373
$915$	12.8284i	0.424095i
$916$	17.4558i	0.576757i
$917$	29.6569i	0.979356i
$918$	1.65685i	0.0546843i
$919$	−27.9411	−0.921693	−0.460846	−	0.887480i	$-0.652454\pi$
$919$	−27.9411	−0.921693	−0.460846	+	0.887480i	$0.652454\pi$
$920$	− 9.65685i	− 0.318377i
$921$	12.4853	0.411404
$922$	− 40.0000i	− 1.31733i
$923$	79.5980i	2.62000i
$924$	16.9706	0.558291
$925$	20.4437i	0.672183i
$926$	−35.7990	−1.17643
$927$	19.6569	0.645616
$928$	16.0000i	0.525226i
$929$	45.1716	1.48203	0.741016	−	0.671488i	$-0.234344\pi$
$929$	45.1716	1.48203	0.741016	+	0.671488i	$0.234344\pi$
$930$	−52.2843	−1.71447
$931$	− 6.72792i	− 0.220499i
$932$	36.0000	1.17922
$933$	− 21.6569i	− 0.709014i
$934$	− 50.2843i	− 1.64535i
$935$	16.9706	0.554997
$936$	19.3137i	0.631288i
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	− 25.6569i	− 0.837726i
$939$	6.97056i	0.227476i
$940$	40.9706i	1.33631i
$941$	11.8995i	0.387912i	0.981010	+	0.193956i	$0.0621320\pi$
$941$	11.8995i	0.387912i	−0.981010	+	0.193956i	$0.937868\pi$
$942$	23.6569	0.770782
$943$	−8.48528	−0.276319
$944$	22.6274i	0.736460i
$945$	−6.82843	−0.222129
$946$	10.5442	0.342820
$947$	− 4.20101i	− 0.136514i	−0.997668	−	0.0682572i	$-0.978256\pi$
$947$	− 4.20101i	− 0.136514i	0.997668	−	0.0682572i	$-0.0217439\pi$
$948$	18.6274i	0.604990i
$949$	0	0
$950$	− 21.1127i	− 0.684986i
$951$	−6.14214	−0.199172
$952$	6.62742	0.214796
$953$	50.1421	1.62426	0.812132	−	0.583474i	$-0.198307\pi$
$953$	50.1421	1.62426	0.812132	+	0.583474i	$0.198307\pi$
$954$	16.1421i	0.522621i
$955$	− 32.9706i	− 1.06690i
$956$	−11.3137	−0.365911
$957$	− 12.0000i	− 0.387905i
$958$	24.9706	0.806762
$959$	−29.6569	−0.957670
$960$	−27.3137	−0.881546
$961$	86.2548	2.78241
$962$	−29.6569	−0.956175
$963$	2.58579i	0.0833258i
$964$	20.0000	0.644157
$965$	− 34.1421i	− 1.09907i
$966$	2.82843i	0.0910032i
$967$	−8.48528	−0.272868	−0.136434	−	0.990649i	$-0.543564\pi$
$967$	−8.48528	−0.272868	−0.136434	+	0.990649i	$0.543564\pi$
$968$	−19.7990	−0.636364
$969$	2.62742	0.0844048
$970$	17.6569i	0.566927i
$971$	30.3848i	0.975094i	0.873097	+	0.487547i	$0.162108\pi$
$971$	30.3848i	0.975094i	−0.873097	+	0.487547i	$0.837892\pi$
$972$	2.00000i	0.0641500i
$973$	6.62742i	0.212465i
$974$	−50.8284	−1.62865
$975$	−45.4558	−1.45575
$976$	− 15.0294i	− 0.481081i
$977$	−20.4853	−0.655382	−0.327691	−	0.944785i	$-0.606271\pi$
$977$	−20.4853	−0.655382	−0.327691	+	0.944785i	$0.606271\pi$
$978$	−6.34315	−0.202831
$979$	21.9411i	0.701241i
$980$	− 20.4853i	− 0.654378i
$981$	12.7279i	0.406371i
$982$	− 20.0000i	− 0.638226i
$983$	−28.9706	−0.924017	−0.462009	−	0.886875i	$-0.652871\pi$
$983$	−28.9706	−0.924017	−0.462009	+	0.886875i	$0.652871\pi$
$984$	24.0000i	0.765092i
$985$	−81.9411	−2.61086
$986$	− 4.68629i	− 0.149242i
$987$	− 12.0000i	− 0.381964i
$988$	30.6274	0.974388
$989$	1.75736i	0.0558808i
$990$	20.4853	0.651065
$991$	−42.4264	−1.34772	−0.673860	−	0.738859i	$-0.735365\pi$
$991$	−42.4264	−1.34772	−0.673860	+	0.738859i	$0.735365\pi$
$992$	61.2548	1.94484
$993$	−24.4853	−0.777017
$994$	32.9706	1.04576
$995$	37.4558i	1.18743i
$996$	8.48528	0.268866
$997$	18.6274i	0.589936i	0.955507	+	0.294968i	$0.0953090\pi$
$997$	18.6274i	0.589936i	−0.955507	+	0.294968i	$0.904691\pi$
$998$	9.65685i	0.305682i
$999$	−3.07107	−0.0971643

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.f.b.277.4	yes	4
4.3	odd	2		2208.2.f.b.1105.2		4
8.3	odd	2		2208.2.f.b.1105.3		4
8.5	even	2	inner	552.2.f.b.277.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.f.b.277.3	✓	4	8.5	even	2	inner
552.2.f.b.277.4	yes	4	1.1	even	1	trivial
2208.2.f.b.1105.2		4	4.3	odd	2
2208.2.f.b.1105.3		4	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 552 = 2^{3} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	552.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +1.00000i q^{3} +2.00000 q^{4} +3.41421i q^{5} +1.41421i q^{6} -2.00000 q^{7} +2.82843 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.41421 q^{2} +1.00000i q^{3} +2.00000 q^{4} +3.41421i q^{5} +1.41421i q^{6} -2.00000 q^{7} +2.82843 q^{8} -1.00000 q^{9} +4.82843i q^{10} +4.24264i q^{11} +2.00000i q^{12} -6.82843i q^{13} -2.82843 q^{14} -3.41421 q^{15} +4.00000 q^{16} -1.17157 q^{17} -1.41421 q^{18} +2.24264i q^{19} +6.82843i q^{20} -2.00000i q^{21} +6.00000i q^{22} -1.00000 q^{23} +2.82843i q^{24} -6.65685 q^{25} -9.65685i q^{26} -1.00000i q^{27} -4.00000 q^{28} +2.82843i q^{29} -4.82843 q^{30} +10.8284 q^{31} +5.65685 q^{32} -4.24264 q^{33} -1.65685 q^{34} -6.82843i q^{35} -2.00000 q^{36} -3.07107i q^{37} +3.17157i q^{38} +6.82843 q^{39} +9.65685i q^{40} +8.48528 q^{41} -2.82843i q^{42} -1.75736i q^{43} +8.48528i q^{44} -3.41421i q^{45} -1.41421 q^{46} +6.00000 q^{47} +4.00000i q^{48} -3.00000 q^{49} -9.41421 q^{50} -1.17157i q^{51} -13.6569i q^{52} -11.4142i q^{53} -1.41421i q^{54} -14.4853 q^{55} -5.65685 q^{56} -2.24264 q^{57} +4.00000i q^{58} +5.65685i q^{59} -6.82843 q^{60} -3.75736i q^{61} +15.3137 q^{62} +2.00000 q^{63} +8.00000 q^{64} +23.3137 q^{65} -6.00000 q^{66} +9.07107i q^{67} -2.34315 q^{68} -1.00000i q^{69} -9.65685i q^{70} -11.6569 q^{71} -2.82843 q^{72} -4.34315i q^{74} -6.65685i q^{75} +4.48528i q^{76} -8.48528i q^{77} +9.65685 q^{78} +9.31371 q^{79} +13.6569i q^{80} +1.00000 q^{81} +12.0000 q^{82} -4.24264i q^{83} -4.00000i q^{84} -4.00000i q^{85} -2.48528i q^{86} -2.82843 q^{87} +12.0000i q^{88} +5.17157 q^{89} -4.82843i q^{90} +13.6569i q^{91} -2.00000 q^{92} +10.8284i q^{93} +8.48528 q^{94} -7.65685 q^{95} +5.65685i q^{96} +3.65685 q^{97} -4.24264 q^{98} -4.24264i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4} - 8 q^{7} - 4 q^{9}+O(q^{10}) \) 4 * q + 8 * q^4 - 8 * q^7 - 4 * q^9	\( 4 q + 8 q^{4} - 8 q^{7} - 4 q^{9} - 8 q^{15} + 16 q^{16} - 16 q^{17} - 4 q^{23} - 4 q^{25} - 16 q^{28} - 8 q^{30} + 32 q^{31} + 16 q^{34} - 8 q^{36} + 16 q^{39} + 24 q^{47} - 12 q^{49} - 32 q^{50} - 24 q^{55} + 8 q^{57} - 16 q^{60} + 16 q^{62} + 8 q^{63} + 32 q^{64} + 48 q^{65} - 24 q^{66} - 32 q^{68} - 24 q^{71} + 16 q^{78} - 8 q^{79} + 4 q^{81} + 48 q^{82} + 32 q^{89} - 8 q^{92} - 8 q^{95} - 8 q^{97}+O(q^{100}) \) 4 * q + 8 * q^4 - 8 * q^7 - 4 * q^9 - 8 * q^15 + 16 * q^16 - 16 * q^17 - 4 * q^23 - 4 * q^25 - 16 * q^28 - 8 * q^30 + 32 * q^31 + 16 * q^34 - 8 * q^36 + 16 * q^39 + 24 * q^47 - 12 * q^49 - 32 * q^50 - 24 * q^55 + 8 * q^57 - 16 * q^60 + 16 * q^62 + 8 * q^63 + 32 * q^64 + 48 * q^65 - 24 * q^66 - 32 * q^68 - 24 * q^71 + 16 * q^78 - 8 * q^79 + 4 * q^81 + 48 * q^82 + 32 * q^89 - 8 * q^92 - 8 * q^95 - 8 * q^97

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