LMFDB - Embedded newform 1104.2.m.c.689.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1104,2,Mod(689,1104)] 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("1104.689"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1104, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1104 = 2^{4} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1104.m (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,4,0,0,0,0,0,0,0,0,0,-8,0,0,0,0,0,0,0,0,0,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	no
Analytic conductor:	$8.81548438315$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.40960000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 552)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			689.6
Root			$-0.437016 - 0.437016i$ of defining polynomial
Character	$\chi$	$=$	1104.689
Dual form			1104.2.m.c.689.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.61803 - 0.618034i) q^{3} +0.874032 q^{5} +1.41421i q^{7} +(2.23607 - 2.00000i) q^{9} +5.11667 q^{11} -3.23607 q^{13} +(1.41421 - 0.540182i) q^{15} +4.91034 q^{17} +8.27895i q^{19} +(0.874032 + 2.28825i) q^{21} +(-4.24264 - 2.23607i) q^{23} -4.23607 q^{25} +(2.38197 - 4.61803i) q^{27} -5.23607i q^{29} +6.00000 q^{31} +(8.27895 - 3.16228i) q^{33} +1.23607i q^{35} -10.7735i q^{37} +(-5.23607 + 2.00000i) q^{39} +0.472136i q^{41} +10.0270i q^{43} +(1.95440 - 1.74806i) q^{45} +11.7082i q^{47} +5.00000 q^{49} +(7.94510 - 3.03476i) q^{51} -0.874032 q^{53} +4.47214 q^{55} +(5.11667 + 13.3956i) q^{57} -8.00000i q^{59} +1.20788i q^{61} +(2.82843 + 3.16228i) q^{63} -2.82843 q^{65} -6.53089i q^{67} +(-8.24670 - 0.995938i) q^{69} -12.1803i q^{71} -7.23607 q^{73} +(-6.85410 + 2.61803i) q^{75} +7.23607i q^{77} -6.65841i q^{79} +(1.00000 - 8.94427i) q^{81} -7.94510 q^{83} +4.29180 q^{85} +(-3.23607 - 8.47214i) q^{87} -0.333851 q^{89} -4.57649i q^{91} +(9.70820 - 3.70820i) q^{93} +7.23607i q^{95} +2.41577i q^{97} +(11.4412 - 10.2333i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{3} - 8 q^{13} - 16 q^{25} + 28 q^{27} + 48 q^{31} - 24 q^{39} + 40 q^{49} - 20 q^{69} - 40 q^{73} - 28 q^{75} + 8 q^{81} + 88 q^{85} - 8 q^{87} + 24 q^{93}+O(q^{100}) $ 8 * q + 4 * q^3 - 8 * q^13 - 16 * q^25 + 28 * q^27 + 48 * q^31 - 24 * q^39 + 40 * q^49 - 20 * q^69 - 40 * q^73 - 28 * q^75 + 8 * q^81 + 88 * q^85 - 8 * q^87 + 24 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1104\mathbb{Z}\right)^\times$.

$n$	$97$	$277$	$415$	$737$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$	1.61803	−	0.618034i	0.934172	−	0.356822i
$3$	1.61803	−	0.618034i	0.934172	−	0.356822i
$4$		0			0
$5$	0.874032			0.390879			0.195440	−	0.980716i	$-0.437387\pi$
$5$	0.874032			0.390879			0.195440	+	0.980716i	$0.437387\pi$
$6$		0			0
$7$			1.41421i			0.534522i	0.963624	+	0.267261i	$0.0861187\pi$
$7$			1.41421i			0.534522i	−0.963624	+	0.267261i	$0.913881\pi$
$8$		0			0
$9$	2.23607	−	2.00000i	0.745356	−	0.666667i
$10$		0			0
$11$	5.11667			1.54273			0.771367	−	0.636390i	$-0.219573\pi$
$11$	5.11667			1.54273			0.771367	+	0.636390i	$0.219573\pi$
$12$		0			0
$13$	−3.23607			−0.897524			−0.448762	−	0.893651i	$-0.648135\pi$
$13$	−3.23607			−0.897524			−0.448762	+	0.893651i	$0.648135\pi$
$14$		0			0
$15$	1.41421	−	0.540182i	0.365148	−	0.139474i
$16$		0			0
$17$	4.91034			1.19093			0.595466	−	0.803380i	$-0.296967\pi$
$17$	4.91034			1.19093			0.595466	+	0.803380i	$0.296967\pi$
$18$		0			0
$19$			8.27895i			1.89932i	0.313280	+	0.949661i	$0.398572\pi$
$19$			8.27895i			1.89932i	−0.313280	+	0.949661i	$0.601428\pi$
$20$		0			0
$21$	0.874032	+	2.28825i	0.190729	+	0.499336i
$22$		0			0
$23$	−4.24264	−	2.23607i	−0.884652	−	0.466252i
$23$	−4.24264	−	2.23607i	−0.884652	−	0.466252i
$24$		0			0
$25$	−4.23607			−0.847214
$26$		0			0
$27$	2.38197	−	4.61803i	0.458410	−	0.888741i
$28$		0			0
$29$		−	5.23607i		−	0.972313i	−0.873872	−	0.486157i	$-0.838398\pi$
$29$		−	5.23607i		−	0.972313i	0.873872	−	0.486157i	$-0.161602\pi$
$30$		0			0
$31$	6.00000			1.07763			0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000			1.07763			0.538816	+	0.842424i	$0.318872\pi$
$32$		0			0
$33$	8.27895	−	3.16228i	1.44118	−	0.550482i
$34$		0			0
$35$			1.23607i			0.208934i
$36$		0			0
$37$		−	10.7735i		−	1.77116i	−0.464490	−	0.885578i	$-0.653762\pi$
$37$		−	10.7735i		−	1.77116i	0.464490	−	0.885578i	$-0.346238\pi$
$38$		0			0
$39$	−5.23607	+	2.00000i	−0.838442	+	0.320256i
$40$		0			0
$41$			0.472136i			0.0737352i	0.999320	+	0.0368676i	$0.0117380\pi$
$41$			0.472136i			0.0737352i	−0.999320	+	0.0368676i	$0.988262\pi$
$42$		0			0
$43$			10.0270i			1.52911i	0.644561	+	0.764553i	$0.277040\pi$
$43$			10.0270i			1.52911i	−0.644561	+	0.764553i	$0.722960\pi$
$44$		0			0
$45$	1.95440	−	1.74806i	0.291344	−	0.260586i
$46$		0			0
$47$			11.7082i			1.70782i	0.520423	+	0.853909i	$0.325774\pi$
$47$			11.7082i			1.70782i	−0.520423	+	0.853909i	$0.674226\pi$
$48$		0			0
$49$	5.00000			0.714286
$50$		0			0
$51$	7.94510	−	3.03476i	1.11254	−	0.424951i
$52$		0			0
$53$	−0.874032			−0.120058			−0.0600288	−	0.998197i	$-0.519119\pi$
$53$	−0.874032			−0.120058			−0.0600288	+	0.998197i	$0.519119\pi$
$54$		0			0
$55$	4.47214			0.603023
$56$		0			0
$57$	5.11667	+	13.3956i	0.677720	+	1.77429i
$58$		0			0
$59$		−	8.00000i		−	1.04151i	−0.853706	−	0.520756i	$-0.825650\pi$
$59$		−	8.00000i		−	1.04151i	0.853706	−	0.520756i	$-0.174350\pi$
$60$		0			0
$61$			1.20788i			0.154654i	0.997006	+	0.0773268i	$0.0246385\pi$
$61$			1.20788i			0.154654i	−0.997006	+	0.0773268i	$0.975362\pi$
$62$		0			0
$63$	2.82843	+	3.16228i	0.356348	+	0.398410i
$64$		0			0
$65$	−2.82843			−0.350823
$66$		0			0
$67$		−	6.53089i		−	0.797875i	−0.916978	−	0.398937i	$-0.869379\pi$
$67$		−	6.53089i		−	0.797875i	0.916978	−	0.398937i	$-0.130621\pi$
$68$		0			0
$69$	−8.24670	−	0.995938i	−0.992786	−	0.119897i
$70$		0			0
$71$		−	12.1803i		−	1.44554i	−0.691088	−	0.722770i	$-0.742869\pi$
$71$		−	12.1803i		−	1.44554i	0.691088	−	0.722770i	$-0.257131\pi$
$72$		0			0
$73$	−7.23607			−0.846918			−0.423459	−	0.905915i	$-0.639184\pi$
$73$	−7.23607			−0.846918			−0.423459	+	0.905915i	$0.639184\pi$
$74$		0			0
$75$	−6.85410	+	2.61803i	−0.791444	+	0.302305i
$76$		0			0
$77$			7.23607i			0.824626i
$78$		0			0
$79$		−	6.65841i		−	0.749129i	−0.927201	−	0.374565i	$-0.877792\pi$
$79$		−	6.65841i		−	0.749129i	0.927201	−	0.374565i	$-0.122208\pi$
$80$		0			0
$81$	1.00000	−	8.94427i	0.111111	−	0.993808i
$82$		0			0
$83$	−7.94510			−0.872088			−0.436044	−	0.899925i	$-0.643621\pi$
$83$	−7.94510			−0.872088			−0.436044	+	0.899925i	$0.643621\pi$
$84$		0			0
$85$	4.29180			0.465511
$86$		0			0
$87$	−3.23607	−	8.47214i	−0.346943	−	0.908308i
$88$		0			0
$89$	−0.333851			−0.0353881			−0.0176940	−	0.999843i	$-0.505632\pi$
$89$	−0.333851			−0.0353881			−0.0176940	+	0.999843i	$0.505632\pi$
$90$		0			0
$91$		−	4.57649i		−	0.479747i
$92$		0			0
$93$	9.70820	−	3.70820i	1.00669	−	0.384523i
$94$		0			0
$95$			7.23607i			0.742405i
$96$		0			0
$97$			2.41577i			0.245284i	0.992451	+	0.122642i	$0.0391367\pi$
$97$			2.41577i			0.245284i	−0.992451	+	0.122642i	$0.960863\pi$
$98$		0			0
$99$	11.4412	−	10.2333i	1.14989	−	1.02849i
$100$		0			0
$101$			8.47214i			0.843009i	0.906826	+	0.421505i	$0.138498\pi$
$101$			8.47214i			0.843009i	−0.906826	+	0.421505i	$0.861502\pi$
$102$		0			0
$103$			7.07107i			0.696733i	0.937358	+	0.348367i	$0.113264\pi$
$103$			7.07107i			0.696733i	−0.937358	+	0.348367i	$0.886736\pi$
$104$		0			0
$105$	0.763932	+	2.00000i	0.0745521	+	0.195180i
$106$		0			0
$107$	9.28050			0.897180			0.448590	−	0.893738i	$-0.351926\pi$
$107$	9.28050			0.897180			0.448590	+	0.893738i	$0.351926\pi$
$108$		0			0
$109$			4.03631i			0.386608i	0.981139	+	0.193304i	$0.0619204\pi$
$109$			4.03631i			0.386608i	−0.981139	+	0.193304i	$0.938080\pi$
$110$		0			0
$111$	−6.65841	−	17.4319i	−0.631988	−	1.65457i
$112$		0			0
$113$	−9.89949			−0.931266			−0.465633	−	0.884978i	$-0.654173\pi$
$113$	−9.89949			−0.931266			−0.465633	+	0.884978i	$0.654173\pi$
$114$		0			0
$115$	−3.70820	−	1.95440i	−0.345792	−	0.182248i
$116$		0			0
$117$	−7.23607	+	6.47214i	−0.668975	+	0.598349i
$118$		0			0
$119$			6.94427i			0.636580i
$120$		0			0
$121$	15.1803			1.38003
$122$		0			0
$123$	0.291796	+	0.763932i	0.0263104	+	0.0688814i
$124$		0			0
$125$	−8.07262			−0.722037
$126$		0			0
$127$	−12.0000			−1.06483			−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000			−1.06483			−0.532414	+	0.846484i	$0.678715\pi$
$128$		0			0
$129$	6.19704	+	16.2241i	0.545619	+	1.42845i
$130$		0			0
$131$		−	2.76393i		−	0.241486i	−0.992684	−	0.120743i	$-0.961472\pi$
$131$		−	2.76393i		−	0.241486i	0.992684	−	0.120743i	$-0.0385277\pi$
$132$		0			0
$133$	−11.7082			−1.01523
$134$		0			0
$135$	2.08191	−	4.03631i	0.179183	−	0.347390i
$136$		0			0
$137$	−17.3044			−1.47842			−0.739208	−	0.673477i	$-0.764800\pi$
$137$	−17.3044			−1.47842			−0.739208	+	0.673477i	$0.764800\pi$
$138$		0			0
$139$	−6.47214			−0.548959			−0.274480	−	0.961593i	$-0.588506\pi$
$139$	−6.47214			−0.548959			−0.274480	+	0.961593i	$0.588506\pi$
$140$		0			0
$141$	7.23607	+	18.9443i	0.609387	+	1.59540i
$142$		0			0
$143$	−16.5579			−1.38464
$144$		0			0
$145$		−	4.57649i		−	0.380057i
$146$		0			0
$147$	8.09017	−	3.09017i	0.667266	−	0.254873i
$148$		0			0
$149$	9.35931			0.766745			0.383372	−	0.923594i	$-0.374763\pi$
$149$	9.35931			0.766745			0.383372	+	0.923594i	$0.374763\pi$
$150$		0			0
$151$	−1.05573			−0.0859139			−0.0429570	−	0.999077i	$-0.513678\pi$
$151$	−1.05573			−0.0859139			−0.0429570	+	0.999077i	$0.513678\pi$
$152$		0			0
$153$	10.9799	−	9.82068i	0.887669	−	0.793955i
$154$		0			0
$155$	5.24419			0.421224
$156$		0			0
$157$			0.952843i			0.0760452i	0.999277	+	0.0380226i	$0.0121059\pi$
$157$			0.952843i			0.0760452i	−0.999277	+	0.0380226i	$0.987894\pi$
$158$		0			0
$159$	−1.41421	+	0.540182i	−0.112154	+	0.0428392i
$160$		0			0
$161$	3.16228	−	6.00000i	0.249222	−	0.472866i
$162$		0			0
$163$	12.9443			1.01387			0.506937	−	0.861983i	$-0.330778\pi$
$163$	12.9443			1.01387			0.506937	+	0.861983i	$0.330778\pi$
$164$		0			0
$165$	7.23607	−	2.76393i	0.563327	−	0.215172i
$166$		0			0
$167$			17.1246i			1.32514i	0.748999	+	0.662571i	$0.230535\pi$
$167$			17.1246i			1.32514i	−0.748999	+	0.662571i	$0.769465\pi$
$168$		0			0
$169$	−2.52786			−0.194451
$170$		0			0
$171$	16.5579	+	18.5123i	1.26621	+	1.41567i
$172$		0			0
$173$		−	10.1803i		−	0.773997i	−0.922080	−	0.386998i	$-0.873512\pi$
$173$		−	10.1803i		−	0.773997i	0.922080	−	0.386998i	$-0.126488\pi$
$174$		0			0
$175$		−	5.99070i		−	0.452855i
$176$		0			0
$177$	−4.94427	−	12.9443i	−0.371634	−	0.972951i
$178$		0			0
$179$		−	8.94427i		−	0.668526i	−0.942480	−	0.334263i	$-0.891513\pi$
$179$		−	8.94427i		−	0.668526i	0.942480	−	0.334263i	$-0.108487\pi$
$180$		0			0
$181$			5.78437i			0.429949i	0.976620	+	0.214975i	$0.0689669\pi$
$181$			5.78437i			0.429949i	−0.976620	+	0.214975i	$0.931033\pi$
$182$		0			0
$183$	0.746512	+	1.95440i	0.0551838	+	0.144473i
$184$		0			0
$185$		−	9.41641i		−	0.692308i
$186$		0			0
$187$	25.1246			1.83729
$188$		0			0
$189$	6.53089	+	3.36861i	0.475052	+	0.245030i
$190$		0			0
$191$	−25.0432			−1.81206			−0.906031	−	0.423212i	$-0.860902\pi$
$191$	−25.0432			−1.81206			−0.906031	+	0.423212i	$0.860902\pi$
$192$		0			0
$193$	3.41641			0.245918			0.122959	−	0.992412i	$-0.460762\pi$
$193$	3.41641			0.245918			0.122959	+	0.992412i	$0.460762\pi$
$194$		0			0
$195$	−4.57649	+	1.74806i	−0.327729	+	0.125181i
$196$		0			0
$197$		−	7.52786i		−	0.536338i	−0.963372	−	0.268169i	$-0.913581\pi$
$197$		−	7.52786i		−	0.536338i	0.963372	−	0.268169i	$-0.0864186\pi$
$198$		0			0
$199$			10.9799i			0.778341i	0.921166	+	0.389171i	$0.127238\pi$
$199$			10.9799i			0.778341i	−0.921166	+	0.389171i	$0.872762\pi$
$200$		0			0
$201$	−4.03631	−	10.5672i	−0.284699	−	0.745353i
$202$		0			0
$203$	7.40492			0.519723
$204$		0			0
$205$			0.412662i			0.0288216i
$206$		0			0
$207$	−13.9590	+	3.48528i	−0.970215	+	0.242244i
$208$		0			0
$209$			42.3607i			2.93015i
$210$		0			0
$211$	−14.6525			−1.00872			−0.504359	−	0.863494i	$-0.668271\pi$
$211$	−14.6525			−1.00872			−0.504359	+	0.863494i	$0.668271\pi$
$212$		0			0
$213$	−7.52786	−	19.7082i	−0.515801	−	1.35038i
$214$		0			0
$215$			8.76393i			0.597695i
$216$		0			0
$217$			8.48528i			0.576018i
$218$		0			0
$219$	−11.7082	+	4.47214i	−0.791167	+	0.302199i
$220$		0			0
$221$	−15.8902			−1.06889
$222$		0			0
$223$	−18.9443			−1.26860			−0.634301	−	0.773086i	$-0.718712\pi$
$223$	−18.9443			−1.26860			−0.634301	+	0.773086i	$0.718712\pi$
$224$		0			0
$225$	−9.47214	+	8.47214i	−0.631476	+	0.564809i
$226$		0			0
$227$	−1.20788			−0.0801700			−0.0400850	−	0.999196i	$-0.512763\pi$
$227$	−1.20788			−0.0801700			−0.0400850	+	0.999196i	$0.512763\pi$
$228$		0			0
$229$			9.02546i			0.596419i	0.954500	+	0.298210i	$0.0963895\pi$
$229$			9.02546i			0.596419i	−0.954500	+	0.298210i	$0.903611\pi$
$230$		0			0
$231$	4.47214	+	11.7082i	0.294245	+	0.770343i
$232$		0			0
$233$			21.8885i			1.43397i	0.697091	+	0.716983i	$0.254478\pi$
$233$			21.8885i			1.43397i	−0.697091	+	0.716983i	$0.745522\pi$
$234$		0			0
$235$			10.2333i			0.667550i
$236$		0			0
$237$	−4.11512	−	10.7735i	−0.267306	−	0.699816i
$238$		0			0
$239$		−	2.00000i		−	0.129369i	−0.997906	−	0.0646846i	$-0.979396\pi$
$239$		−	2.00000i		−	0.129369i	0.997906	−	0.0646846i	$-0.0206041\pi$
$240$		0			0
$241$		−	22.2148i		−	1.43098i	−0.698624	−	0.715489i	$-0.746204\pi$
$241$		−	22.2148i		−	1.43098i	0.698624	−	0.715489i	$-0.253796\pi$
$242$		0			0
$243$	−3.90983	−	15.0902i	−0.250816	−	0.968035i
$244$		0			0
$245$	4.37016			0.279199
$246$		0			0
$247$		−	26.7912i		−	1.70469i
$248$		0			0
$249$	−12.8554	+	4.91034i	−0.814681	+	0.311180i
$250$		0			0
$251$	2.70091			0.170480			0.0852399	−	0.996360i	$-0.472834\pi$
$251$	2.70091			0.170480			0.0852399	+	0.996360i	$0.472834\pi$
$252$		0			0
$253$	−21.7082	−	11.4412i	−1.36478	−	0.719304i
$254$		0			0
$255$	6.94427	−	2.65248i	0.434867	−	0.166104i
$256$		0			0
$257$		−	15.5279i		−	0.968602i	−0.874902	−	0.484301i	$-0.839074\pi$
$257$		−	15.5279i		−	0.968602i	0.874902	−	0.484301i	$-0.160926\pi$
$258$		0			0
$259$	15.2361			0.946723
$260$		0			0
$261$	−10.4721	−	11.7082i	−0.648209	−	0.724720i
$262$		0			0
$263$	−30.2874			−1.86760			−0.933800	−	0.357796i	$-0.883528\pi$
$263$	−30.2874			−1.86760			−0.933800	+	0.357796i	$0.883528\pi$
$264$		0			0
$265$	−0.763932			−0.0469280
$266$		0			0
$267$	−0.540182	+	0.206331i	−0.0330586	+	0.0126273i
$268$		0			0
$269$			7.52786i			0.458982i	0.973311	+	0.229491i	$0.0737061\pi$
$269$			7.52786i			0.458982i	−0.973311	+	0.229491i	$0.926294\pi$
$270$		0			0
$271$	4.00000			0.242983			0.121491	−	0.992592i	$-0.461232\pi$
$271$	4.00000			0.242983			0.121491	+	0.992592i	$0.461232\pi$
$272$		0			0
$273$	−2.82843	−	7.40492i	−0.171184	−	0.448166i
$274$		0			0
$275$	−21.6746			−1.30703
$276$		0			0
$277$	22.3607			1.34352			0.671762	−	0.740767i	$-0.265538\pi$
$277$	22.3607			1.34352			0.671762	+	0.740767i	$0.265538\pi$
$278$		0			0
$279$	13.4164	−	12.0000i	0.803219	−	0.718421i
$280$		0			0
$281$	−27.5378			−1.64276			−0.821382	−	0.570378i	$-0.806797\pi$
$281$	−27.5378			−1.64276			−0.821382	+	0.570378i	$0.806797\pi$
$282$		0			0
$283$			9.61435i			0.571514i	0.958302	+	0.285757i	$0.0922450\pi$
$283$			9.61435i			0.571514i	−0.958302	+	0.285757i	$0.907755\pi$
$284$		0			0
$285$	4.47214	+	11.7082i	0.264906	+	0.693534i
$286$		0			0
$287$	−0.667701			−0.0394131
$288$		0			0
$289$	7.11146			0.418321
$290$		0			0
$291$	1.49302	+	3.90879i	0.0875227	+	0.229137i
$292$		0			0
$293$	−4.78282			−0.279415			−0.139708	−	0.990193i	$-0.544616\pi$
$293$	−4.78282			−0.279415			−0.139708	+	0.990193i	$0.544616\pi$
$294$		0			0
$295$		−	6.99226i		−	0.407105i
$296$		0			0
$297$	12.1877	−	23.6290i	0.707204	−	1.37109i
$298$		0			0
$299$	13.7295	+	7.23607i	0.793996	+	0.418473i
$300$		0			0
$301$	−14.1803			−0.817341
$302$		0			0
$303$	5.23607	+	13.7082i	0.300804	+	0.787516i
$304$		0			0
$305$			1.05573i			0.0604508i
$306$		0			0
$307$	−20.3607			−1.16205			−0.581023	−	0.813887i	$-0.697347\pi$
$307$	−20.3607			−1.16205			−0.581023	+	0.813887i	$0.697347\pi$
$308$		0			0
$309$	4.37016	+	11.4412i	0.248610	+	0.650869i
$310$		0			0
$311$			20.0000i			1.13410i	0.823685	+	0.567048i	$0.191915\pi$
$311$			20.0000i			1.13410i	−0.823685	+	0.567048i	$0.808085\pi$
$312$		0			0
$313$		−	17.8933i		−	1.01139i	−0.862713	−	0.505695i	$-0.831236\pi$
$313$		−	17.8933i		−	1.01139i	0.862713	−	0.505695i	$-0.168764\pi$
$314$		0			0
$315$	2.47214	+	2.76393i	0.139289	+	0.155730i
$316$		0			0
$317$			3.70820i			0.208273i	0.994563	+	0.104137i	$0.0332080\pi$
$317$			3.70820i			0.208273i	−0.994563	+	0.104137i	$0.966792\pi$
$318$		0			0
$319$		−	26.7912i		−	1.50002i
$320$		0			0
$321$	15.0162	−	5.73567i	0.838121	−	0.320134i
$322$		0			0
$323$			40.6525i			2.26196i
$324$		0			0
$325$	13.7082			0.760394
$326$		0			0
$327$	2.49458	+	6.53089i	0.137950	+	0.361159i
$328$		0			0
$329$	−16.5579			−0.912867
$330$		0			0
$331$	14.6525			0.805373			0.402686	−	0.915338i	$-0.368077\pi$
$331$	14.6525			0.805373			0.402686	+	0.915338i	$0.368077\pi$
$332$		0			0
$333$	−21.5471	−	24.0903i	−1.18077	−	1.32014i
$334$		0			0
$335$		−	5.70820i		−	0.311872i
$336$		0			0
$337$			16.9706i			0.924445i	0.886764	+	0.462223i	$0.152948\pi$
$337$			16.9706i			0.924445i	−0.886764	+	0.462223i	$0.847052\pi$
$338$		0			0
$339$	−16.0177	+	6.11822i	−0.869963	+	0.332296i
$340$		0			0
$341$	30.7000			1.66250
$342$		0			0
$343$			16.9706i			0.916324i
$344$		0			0
$345$	−7.20788	−	0.870482i	−0.388059	−	0.0468652i
$346$		0			0
$347$		−	26.8328i		−	1.44046i	−0.693735	−	0.720231i	$-0.744036\pi$
$347$		−	26.8328i		−	1.44046i	0.693735	−	0.720231i	$-0.255964\pi$
$348$		0			0
$349$	13.7082			0.733783			0.366892	−	0.930264i	$-0.380422\pi$
$349$	13.7082			0.733783			0.366892	+	0.930264i	$0.380422\pi$
$350$		0			0
$351$	−7.70820	+	14.9443i	−0.411433	+	0.797666i
$352$		0			0
$353$		−	18.4721i		−	0.983173i	−0.870829	−	0.491586i	$-0.836417\pi$
$353$		−	18.4721i		−	0.983173i	0.870829	−	0.491586i	$-0.163583\pi$
$354$		0			0
$355$		−	10.6460i		−	0.565031i
$356$		0			0
$357$	4.29180	+	11.2361i	0.227146	+	0.594676i
$358$		0			0
$359$	22.8825			1.20769			0.603845	−	0.797102i	$-0.293635\pi$
$359$	22.8825			1.20769			0.603845	+	0.797102i	$0.293635\pi$
$360$		0			0
$361$	−49.5410			−2.60742
$362$		0			0
$363$	24.5623	−	9.38197i	1.28919	−	0.492426i
$364$		0			0
$365$	−6.32456			−0.331042
$366$		0			0
$367$			33.0370i			1.72452i	0.506469	+	0.862258i	$0.330951\pi$
$367$			33.0370i			1.72452i	−0.506469	+	0.862258i	$0.669049\pi$
$368$		0			0
$369$	0.944272	+	1.05573i	0.0491568	+	0.0549590i
$370$		0			0
$371$		−	1.23607i		−	0.0641735i
$372$		0			0
$373$			11.1862i			0.579199i	0.957148	+	0.289599i	$0.0935221\pi$
$373$			11.1862i			0.579199i	−0.957148	+	0.289599i	$0.906478\pi$
$374$		0			0
$375$	−13.0618	+	4.98915i	−0.674507	+	0.257639i
$376$		0			0
$377$			16.9443i			0.872674i
$378$		0			0
$379$		−	22.0084i		−	1.13050i	−0.824921	−	0.565248i	$-0.808780\pi$
$379$		−	22.0084i		−	1.13050i	0.824921	−	0.565248i	$-0.191220\pi$
$380$		0			0
$381$	−19.4164	+	7.41641i	−0.994733	+	0.379954i
$382$		0			0
$383$	−28.1266			−1.43720			−0.718602	−	0.695421i	$-0.755218\pi$
$383$	−28.1266			−1.43720			−0.718602	+	0.695421i	$0.755218\pi$
$384$		0			0
$385$			6.32456i			0.322329i
$386$		0			0
$387$	20.0540	+	22.4211i	1.01940	+	1.13973i
$388$		0			0
$389$	32.6544			1.65565			0.827823	−	0.560990i	$-0.189579\pi$
$389$	32.6544			1.65565			0.827823	+	0.560990i	$0.189579\pi$
$390$		0			0
$391$	−20.8328	−	10.9799i	−1.05356	−	0.555275i
$392$		0			0
$393$	−1.70820	−	4.47214i	−0.0861675	−	0.225589i
$394$		0			0
$395$		−	5.81966i		−	0.292819i
$396$		0			0
$397$	31.3050			1.57115			0.785575	−	0.618766i	$-0.212367\pi$
$397$	31.3050			1.57115			0.785575	+	0.618766i	$0.212367\pi$
$398$		0			0
$399$	−18.9443	+	7.23607i	−0.948400	+	0.362257i
$400$		0			0
$401$	−9.48683			−0.473750			−0.236875	−	0.971540i	$-0.576123\pi$
$401$	−9.48683			−0.473750			−0.236875	+	0.971540i	$0.576123\pi$
$402$		0			0
$403$	−19.4164			−0.967200
$404$		0			0
$405$	0.874032	−	7.81758i	0.0434310	−	0.388459i
$406$		0			0
$407$		−	55.1246i		−	2.73243i
$408$		0			0
$409$	2.47214			0.122239			0.0611196	−	0.998130i	$-0.480533\pi$
$409$	2.47214			0.122239			0.0611196	+	0.998130i	$0.480533\pi$
$410$		0			0
$411$	−27.9991	+	10.6947i	−1.38110	+	0.527531i
$412$		0			0
$413$	11.3137			0.556711
$414$		0			0
$415$	−6.94427			−0.340881
$416$		0			0
$417$	−10.4721	+	4.00000i	−0.512823	+	0.195881i
$418$		0			0
$419$	7.11978			0.347824			0.173912	−	0.984761i	$-0.444359\pi$
$419$	7.11978			0.347824			0.173912	+	0.984761i	$0.444359\pi$
$420$		0			0
$421$			4.29135i			0.209148i	0.994517	+	0.104574i	$0.0333478\pi$
$421$			4.29135i			0.209148i	−0.994517	+	0.104574i	$0.966652\pi$
$422$		0			0
$423$	23.4164	+	26.1803i	1.13854	+	1.27293i
$424$		0			0
$425$	−20.8005			−1.00897
$426$		0			0
$427$	−1.70820			−0.0826658
$428$		0			0
$429$	−26.7912	+	10.2333i	−1.29349	+	0.494071i
$430$		0			0
$431$	−2.41577			−0.116363			−0.0581817	−	0.998306i	$-0.518530\pi$
$431$	−2.41577			−0.116363			−0.0581817	+	0.998306i	$0.518530\pi$
$432$		0			0
$433$		−	7.40492i		−	0.355858i	−0.984043	−	0.177929i	$-0.943060\pi$
$433$		−	7.40492i		−	0.355858i	0.984043	−	0.177929i	$-0.0569397\pi$
$434$		0			0
$435$	−2.82843	−	7.40492i	−0.135613	−	0.355039i
$436$		0			0
$437$	18.5123	−	35.1246i	0.885563	−	1.68024i
$438$		0			0
$439$	3.41641			0.163056			0.0815281	−	0.996671i	$-0.474020\pi$
$439$	3.41641			0.163056			0.0815281	+	0.996671i	$0.474020\pi$
$440$		0			0
$441$	11.1803	−	10.0000i	0.532397	−	0.476190i
$442$		0			0
$443$			19.1246i			0.908638i	0.890839	+	0.454319i	$0.150117\pi$
$443$			19.1246i			0.908638i	−0.890839	+	0.454319i	$0.849883\pi$
$444$		0			0
$445$	−0.291796			−0.0138325
$446$		0			0
$447$	15.1437	−	5.78437i	0.716272	−	0.273591i
$448$		0			0
$449$		−	28.8328i		−	1.36070i	−0.732885	−	0.680352i	$-0.761827\pi$
$449$		−	28.8328i		−	1.36070i	0.732885	−	0.680352i	$-0.238173\pi$
$450$		0			0
$451$			2.41577i			0.113754i
$452$		0			0
$453$	−1.70820	+	0.652476i	−0.0802584	+	0.0306560i
$454$		0			0
$455$		−	4.00000i		−	0.187523i
$456$		0			0
$457$			8.07262i			0.377621i	0.982014	+	0.188811i	$0.0604632\pi$
$457$			8.07262i			0.377621i	−0.982014	+	0.188811i	$0.939537\pi$
$458$		0			0
$459$	11.6963	−	22.6761i	0.545935	−	1.05843i
$460$		0			0
$461$		−	27.8885i		−	1.29890i	−0.760405	−	0.649449i	$-0.774999\pi$
$461$		−	27.8885i		−	1.29890i	0.760405	−	0.649449i	$-0.225001\pi$
$462$		0			0
$463$	12.3607			0.574450			0.287225	−	0.957863i	$-0.407267\pi$
$463$	12.3607			0.574450			0.287225	+	0.957863i	$0.407267\pi$
$464$		0			0
$465$	8.48528	−	3.24109i	0.393496	−	0.150302i
$466$		0			0
$467$	23.8353			1.10297			0.551483	−	0.834186i	$-0.314062\pi$
$467$	23.8353			1.10297			0.551483	+	0.834186i	$0.314062\pi$
$468$		0			0
$469$	9.23607			0.426482
$470$		0			0
$471$	0.588890	+	1.54173i	0.0271346	+	0.0710393i
$472$		0			0
$473$			51.3050i			2.35900i
$474$		0			0
$475$		−	35.0702i		−	1.60913i
$476$		0			0
$477$	−1.95440	+	1.74806i	−0.0894856	+	0.0800384i
$478$		0			0
$479$	1.33540			0.0610161			0.0305080	−	0.999535i	$-0.490287\pi$
$479$	1.33540			0.0610161			0.0305080	+	0.999535i	$0.490287\pi$
$480$		0			0
$481$			34.8639i			1.58966i
$482$		0			0
$483$	1.40847	−	11.6626i	0.0640875	−	0.530667i
$484$		0			0
$485$			2.11146i			0.0958763i
$486$		0			0
$487$	20.9443			0.949076			0.474538	−	0.880235i	$-0.342615\pi$
$487$	20.9443			0.949076			0.474538	+	0.880235i	$0.342615\pi$
$488$		0			0
$489$	20.9443	−	8.00000i	0.947133	−	0.361773i
$490$		0			0
$491$		−	21.5279i		−	0.971539i	−0.874087	−	0.485769i	$-0.838539\pi$
$491$		−	21.5279i		−	0.971539i	0.874087	−	0.485769i	$-0.161461\pi$
$492$		0			0
$493$		−	25.7109i		−	1.15796i
$494$		0			0
$495$	10.0000	−	8.94427i	0.449467	−	0.402015i
$496$		0			0
$497$	17.2256			0.772674
$498$		0			0
$499$	−22.4721			−1.00599			−0.502995	−	0.864289i	$-0.667769\pi$
$499$	−22.4721			−1.00599			−0.502995	+	0.864289i	$0.667769\pi$
$500$		0			0
$501$	10.5836	+	27.7082i	0.472840	+	1.23791i
$502$		0			0
$503$	21.5471			0.960736			0.480368	−	0.877067i	$-0.340503\pi$
$503$	21.5471			0.960736			0.480368	+	0.877067i	$0.340503\pi$
$504$		0			0
$505$			7.40492i			0.329515i
$506$		0			0
$507$	−4.09017	+	1.56231i	−0.181651	+	0.0693844i
$508$		0			0
$509$			9.59675i			0.425368i	0.977121	+	0.212684i	$0.0682205\pi$
$509$			9.59675i			0.425368i	−0.977121	+	0.212684i	$0.931779\pi$
$510$		0			0
$511$		−	10.2333i		−	0.452697i
$512$		0			0
$513$	38.2325	+	19.7202i	1.68800	+	0.870667i
$514$		0			0
$515$			6.18034i			0.272338i
$516$		0			0
$517$			59.9070i			2.63471i
$518$		0			0
$519$	−6.29180	−	16.4721i	−0.276179	−	0.723047i
$520$		0			0
$521$	5.57804			0.244378			0.122189	−	0.992507i	$-0.461009\pi$
$521$	5.57804			0.244378			0.122189	+	0.992507i	$0.461009\pi$
$522$		0			0
$523$			26.3299i			1.15133i	0.817687	+	0.575663i	$0.195256\pi$
$523$			26.3299i			1.15133i	−0.817687	+	0.575663i	$0.804744\pi$
$524$		0			0
$525$	−3.70246	−	9.69316i	−0.161589	−	0.423044i
$526$		0			0
$527$	29.4621			1.28339
$528$		0			0
$529$	13.0000	+	18.9737i	0.565217	+	0.824942i
$530$		0			0
$531$	−16.0000	−	17.8885i	−0.694341	−	0.776297i
$532$		0			0
$533$		−	1.52786i		−	0.0661791i
$534$		0			0
$535$	8.11146			0.350689
$536$		0			0
$537$	−5.52786	−	14.4721i	−0.238545	−	0.624519i
$538$		0			0
$539$	25.5834			1.10195
$540$		0			0
$541$	−7.23607			−0.311103			−0.155551	−	0.987828i	$-0.549715\pi$
$541$	−7.23607			−0.311103			−0.155551	+	0.987828i	$0.549715\pi$
$542$		0			0
$543$	3.57494	+	9.35931i	0.153415	+	0.401647i
$544$		0			0
$545$			3.52786i			0.151117i
$546$		0			0
$547$	−3.81966			−0.163317			−0.0816584	−	0.996660i	$-0.526022\pi$
$547$	−3.81966			−0.163317			−0.0816584	+	0.996660i	$0.526022\pi$
$548$		0			0
$549$	2.41577	+	2.70091i	0.103102	+	0.115272i
$550$		0			0
$551$	43.3491			1.84674
$552$		0			0
$553$	9.41641			0.400426
$554$		0			0
$555$	−5.81966	−	15.2361i	−0.247031	−	0.646735i
$556$		0			0
$557$	15.4288			0.653740			0.326870	−	0.945069i	$-0.394006\pi$
$557$	15.4288			0.653740			0.326870	+	0.945069i	$0.394006\pi$
$558$		0			0
$559$		−	32.4481i		−	1.37241i
$560$		0			0
$561$	40.6525	−	15.5279i	1.71635	−	0.655587i
$562$		0			0
$563$	−4.03631			−0.170110			−0.0850551	−	0.996376i	$-0.527107\pi$
$563$	−4.03631			−0.170110			−0.0850551	+	0.996376i	$0.527107\pi$
$564$		0			0
$565$	−8.65248			−0.364012
$566$		0			0
$567$	12.6491	+	1.41421i	0.531213	+	0.0593914i
$568$		0			0
$569$	−15.3987			−0.645548			−0.322774	−	0.946476i	$-0.604615\pi$
$569$	−15.3987			−0.645548			−0.322774	+	0.946476i	$0.604615\pi$
$570$		0			0
$571$			6.78593i			0.283982i	0.989868	+	0.141991i	$0.0453504\pi$
$571$			6.78593i			0.283982i	−0.989868	+	0.141991i	$0.954650\pi$
$572$		0			0
$573$	−40.5207	+	15.4775i	−1.69278	+	0.646583i
$574$		0			0
$575$	17.9721	+	9.47214i	0.749489	+	0.395015i
$576$		0			0
$577$	11.3475			0.472404			0.236202	−	0.971704i	$-0.424097\pi$
$577$	11.3475			0.472404			0.236202	+	0.971704i	$0.424097\pi$
$578$		0			0
$579$	5.52786	−	2.11146i	0.229730	−	0.0877491i
$580$		0			0
$581$		−	11.2361i		−	0.466151i
$582$		0			0
$583$	−4.47214			−0.185217
$584$		0			0
$585$	−6.32456	+	5.65685i	−0.261488	+	0.233882i
$586$		0			0
$587$		−	7.12461i		−	0.294064i	−0.989132	−	0.147032i	$-0.953028\pi$
$587$		−	7.12461i		−	0.294064i	0.989132	−	0.147032i	$-0.0469721\pi$
$588$		0			0
$589$			49.6737i			2.04677i
$590$		0			0
$591$	−4.65248	−	12.1803i	−0.191377	−	0.501032i
$592$		0			0
$593$		−	21.0557i		−	0.864655i	−0.901717	−	0.432328i	$-0.857692\pi$
$593$		−	21.0557i		−	0.864655i	0.901717	−	0.432328i	$-0.142308\pi$
$594$		0			0
$595$			6.06952i			0.248826i
$596$		0			0
$597$	6.78593	+	17.7658i	0.277729	+	0.727105i
$598$		0			0
$599$			15.5279i			0.634451i	0.948350	+	0.317226i	$0.102751\pi$
$599$			15.5279i			0.634451i	−0.948350	+	0.317226i	$0.897249\pi$
$600$		0			0
$601$	15.7082			0.640751			0.320376	−	0.947291i	$-0.396191\pi$
$601$	15.7082			0.640751			0.320376	+	0.947291i	$0.396191\pi$
$602$		0			0
$603$	−13.0618	−	14.6035i	−0.531917	−	0.594701i
$604$		0			0
$605$	13.2681			0.539425
$606$		0			0
$607$	−10.0000			−0.405887			−0.202944	−	0.979190i	$-0.565051\pi$
$607$	−10.0000			−0.405887			−0.202944	+	0.979190i	$0.565051\pi$
$608$		0			0
$609$	11.9814	−	4.57649i	0.485511	−	0.185449i
$610$		0			0
$611$		−	37.8885i		−	1.53281i
$612$		0			0
$613$		−	33.4009i		−	1.34905i	−0.738251	−	0.674526i	$-0.764348\pi$
$613$		−	33.4009i		−	1.34905i	0.738251	−	0.674526i	$-0.235652\pi$
$614$		0			0
$615$	0.255039	+	0.667701i	0.0102842	+	0.0269243i
$616$		0			0
$617$	34.5300			1.39013			0.695063	−	0.718949i	$-0.255376\pi$
$617$	34.5300			1.39013			0.695063	+	0.718949i	$0.255376\pi$
$618$		0			0
$619$			11.1074i			0.446443i	0.974768	+	0.223222i	$0.0716574\pi$
$619$			11.1074i			0.446443i	−0.974768	+	0.223222i	$0.928343\pi$
$620$		0			0
$621$	−20.4321	+	14.2664i	−0.819910	+	0.572492i
$622$		0			0
$623$		−	0.472136i		−	0.0189157i
$624$		0			0
$625$	14.1246			0.564984
$626$		0			0
$627$	26.1803	+	68.5410i	1.04554	+	2.73726i
$628$		0			0
$629$		−	52.9017i		−	2.10933i
$630$		0			0
$631$			31.7016i			1.26202i	0.775775	+	0.631010i	$0.217359\pi$
$631$			31.7016i			1.26202i	−0.775775	+	0.631010i	$0.782641\pi$
$632$		0			0
$633$	−23.7082	+	9.05573i	−0.942317	+	0.359933i
$634$		0			0
$635$	−10.4884			−0.416219
$636$		0			0
$637$	−16.1803			−0.641088
$638$		0			0
$639$	−24.3607	−	27.2361i	−0.963694	−	1.07744i
$640$		0			0
$641$	28.4605			1.12412			0.562061	−	0.827096i	$-0.310009\pi$
$641$	28.4605			1.12412			0.562061	+	0.827096i	$0.310009\pi$
$642$		0			0
$643$		−	11.7751i		−	0.464364i	−0.972672	−	0.232182i	$-0.925414\pi$
$643$		−	11.7751i		−	0.464364i	0.972672	−	0.232182i	$-0.0745865\pi$
$644$		0			0
$645$	5.41641	+	14.1803i	0.213271	+	0.558350i
$646$		0			0
$647$		−	20.9443i		−	0.823404i	−0.911318	−	0.411702i	$-0.864934\pi$
$647$		−	20.9443i		−	0.823404i	0.911318	−	0.411702i	$-0.135066\pi$
$648$		0			0
$649$		−	40.9334i		−	1.60678i
$650$		0			0
$651$	5.24419	+	13.7295i	0.205536	+	0.538100i
$652$		0			0
$653$			34.0000i			1.33052i	0.746611	+	0.665261i	$0.231680\pi$
$653$			34.0000i			1.33052i	−0.746611	+	0.665261i	$0.768320\pi$
$654$		0			0
$655$		−	2.41577i		−	0.0943918i
$656$		0			0
$657$	−16.1803	+	14.4721i	−0.631255	+	0.564612i
$658$		0			0
$659$	12.7766			0.497707			0.248853	−	0.968541i	$-0.419946\pi$
$659$	12.7766			0.497707			0.248853	+	0.968541i	$0.419946\pi$
$660$		0			0
$661$		−	31.2402i		−	1.21510i	−0.794280	−	0.607552i	$-0.792152\pi$
$661$		−	31.2402i		−	1.21510i	0.794280	−	0.607552i	$-0.207848\pi$
$662$		0			0
$663$	−25.7109	+	9.82068i	−0.998528	+	0.381404i
$664$		0			0
$665$	−10.2333			−0.396832
$666$		0			0
$667$	−11.7082	+	22.2148i	−0.453343	+	0.860159i
$668$		0			0
$669$	−30.6525	+	11.7082i	−1.18509	+	0.452665i
$670$		0			0
$671$			6.18034i			0.238589i
$672$		0			0
$673$	24.6525			0.950283			0.475142	−	0.879909i	$-0.342397\pi$
$673$	24.6525			0.950283			0.475142	+	0.879909i	$0.342397\pi$
$674$		0			0
$675$	−10.0902	+	19.5623i	−0.388371	+	0.752954i
$676$		0			0
$677$	25.5046			0.980220			0.490110	−	0.871661i	$-0.336957\pi$
$677$	25.5046			0.980220			0.490110	+	0.871661i	$0.336957\pi$
$678$		0			0
$679$	−3.41641			−0.131110
$680$		0			0
$681$	−1.95440	+	0.746512i	−0.0748926	+	0.0286064i
$682$		0			0
$683$		−	33.0132i		−	1.26321i	−0.775289	−	0.631607i	$-0.782396\pi$
$683$		−	33.0132i		−	1.26321i	0.775289	−	0.631607i	$-0.217604\pi$
$684$		0			0
$685$	−15.1246			−0.577882
$686$		0			0
$687$	5.57804	+	14.6035i	0.212816	+	0.557158i
$688$		0			0
$689$	2.82843			0.107754
$690$		0			0
$691$	27.2361			1.03611			0.518054	−	0.855348i	$-0.326656\pi$
$691$	27.2361			1.03611			0.518054	+	0.855348i	$0.326656\pi$
$692$		0			0
$693$	14.4721	+	16.1803i	0.549751	+	0.614640i
$694$		0			0
$695$	−5.65685			−0.214577
$696$		0			0
$697$			2.31835i			0.0878137i
$698$		0			0
$699$	13.5279	+	35.4164i	0.511671	+	1.33957i
$700$		0			0
$701$	32.3994			1.22371			0.611854	−	0.790971i	$-0.290424\pi$
$701$	32.3994			1.22371			0.611854	+	0.790971i	$0.290424\pi$
$702$		0			0
$703$	89.1935			3.36400
$704$		0			0
$705$	6.32456	+	16.5579i	0.238197	+	0.623607i
$706$		0			0
$707$	−11.9814			−0.450607
$708$		0			0
$709$			23.6777i			0.889234i	0.895721	+	0.444617i	$0.146660\pi$
$709$			23.6777i			0.889234i	−0.895721	+	0.444617i	$0.853340\pi$
$710$		0			0
$711$	−13.3168	−	14.8886i	−0.499419	−	0.558368i
$712$		0			0
$713$	−25.4558	−	13.4164i	−0.953329	−	0.502448i
$714$		0			0
$715$	−14.4721			−0.541227
$716$		0			0
$717$	−1.23607	−	3.23607i	−0.0461618	−	0.120853i
$718$		0			0
$719$		−	28.0000i		−	1.04422i	−0.852877	−	0.522112i	$-0.825144\pi$
$719$		−	28.0000i		−	1.04422i	0.852877	−	0.522112i	$-0.174856\pi$
$720$		0			0
$721$	−10.0000			−0.372419
$722$		0			0
$723$	−13.7295	−	35.9442i	−0.510605	−	1.33678i
$724$		0			0
$725$			22.1803i			0.823757i
$726$		0			0
$727$			16.0664i			0.595871i	0.954586	+	0.297935i	$0.0962980\pi$
$727$			16.0664i			0.595871i	−0.954586	+	0.297935i	$0.903702\pi$
$728$		0			0
$729$	−15.6525	−	22.0000i	−0.579721	−	0.814815i
$730$		0			0
$731$			49.2361i			1.82106i
$732$		0			0
$733$		−	37.8198i		−	1.39691i	−0.715656	−	0.698453i	$-0.753872\pi$
$733$		−	37.8198i		−	1.39691i	0.715656	−	0.698453i	$-0.246128\pi$
$734$		0			0
$735$	7.07107	−	2.70091i	0.260820	−	0.0996245i
$736$		0			0
$737$		−	33.4164i		−	1.23091i
$738$		0			0
$739$	45.4853			1.67320			0.836602	−	0.547812i	$-0.184539\pi$
$739$	45.4853			1.67320			0.836602	+	0.547812i	$0.184539\pi$
$740$		0			0
$741$	−16.5579	−	43.3491i	−0.608270	−	1.59247i
$742$		0			0
$743$	28.2843			1.03765			0.518825	−	0.854881i	$-0.326370\pi$
$743$	28.2843			1.03765			0.518825	+	0.854881i	$0.326370\pi$
$744$		0			0
$745$	8.18034			0.299704
$746$		0			0
$747$	−17.7658	+	15.8902i	−0.650016	+	0.581392i
$748$		0			0
$749$			13.1246i			0.479563i
$750$		0			0
$751$		−	7.89639i		−	0.288143i	−0.989567	−	0.144072i	$-0.953980\pi$
$751$		−	7.89639i		−	0.288143i	0.989567	−	0.144072i	$-0.0460196\pi$
$752$		0			0
$753$	4.37016	−	1.66925i	0.159257	−	0.0608309i
$754$		0			0
$755$	−0.922740			−0.0335820
$756$		0			0
$757$		−	49.2911i		−	1.79152i	−0.444541	−	0.895759i	$-0.646633\pi$
$757$		−	49.2911i		−	1.79152i	0.444541	−	0.895759i	$-0.353367\pi$
$758$		0			0
$759$	−42.1957	−	5.09589i	−1.53161	−	0.184969i
$760$		0			0
$761$			7.05573i			0.255770i	0.991789	+	0.127885i	$0.0408188\pi$
$761$			7.05573i			0.255770i	−0.991789	+	0.127885i	$0.959181\pi$
$762$		0			0
$763$	−5.70820			−0.206651
$764$		0			0
$765$	9.59675	−	8.58359i	0.346971	−	0.310340i
$766$		0			0
$767$			25.8885i			0.934781i
$768$		0			0
$769$		−	11.7264i		−	0.422864i	−0.977393	−	0.211432i	$-0.932187\pi$
$769$		−	11.7264i		−	0.422864i	0.977393	−	0.211432i	$-0.0678126\pi$
$770$		0			0
$771$	−9.59675	−	25.1246i	−0.345618	−	0.904841i
$772$		0			0
$773$	18.5123			0.665841			0.332921	−	0.942955i	$-0.391966\pi$
$773$	18.5123			0.665841			0.332921	+	0.942955i	$0.391966\pi$
$774$		0			0
$775$	−25.4164			−0.912984
$776$		0			0
$777$	24.6525	−	9.41641i	0.884403	−	0.337812i
$778$		0			0
$779$	−3.90879			−0.140047
$780$		0			0
$781$		−	62.3228i		−	2.23009i
$782$		0			0
$783$	−24.1803	−	12.4721i	−0.864135	−	0.445718i
$784$		0			0
$785$			0.832816i			0.0297245i
$786$		0			0
$787$		−	16.3516i		−	0.582871i	−0.956591	−	0.291435i	$-0.905867\pi$
$787$		−	16.3516i		−	0.582871i	0.956591	−	0.291435i	$-0.0941328\pi$
$788$		0			0
$789$	−49.0060	+	18.7186i	−1.74466	+	0.666401i
$790$		0			0
$791$		−	14.0000i		−	0.497783i
$792$		0			0
$793$		−	3.90879i		−	0.138805i
$794$		0			0
$795$	−1.23607	+	0.472136i	−0.0438388	+	0.0167449i
$796$		0			0
$797$	−12.4428			−0.440746			−0.220373	−	0.975416i	$-0.570727\pi$
$797$	−12.4428			−0.440746			−0.220373	+	0.975416i	$0.570727\pi$
$798$		0			0
$799$			57.4913i			2.03390i
$800$		0			0
$801$	−0.746512	+	0.667701i	−0.0263767	+	0.0235921i
$802$		0			0
$803$	−37.0246			−1.30657
$804$		0			0
$805$	2.76393	−	5.24419i	0.0974158	−	0.184833i
$806$		0			0
$807$	4.65248	+	12.1803i	0.163775	+	0.428768i
$808$		0			0
$809$		−	8.00000i		−	0.281265i	−0.990062	−	0.140633i	$-0.955086\pi$
$809$		−	8.00000i		−	0.281265i	0.990062	−	0.140633i	$-0.0449136\pi$
$810$		0			0
$811$	14.6525			0.514518			0.257259	−	0.966342i	$-0.417181\pi$
$811$	14.6525			0.514518			0.257259	+	0.966342i	$0.417181\pi$
$812$		0			0
$813$	6.47214	−	2.47214i	0.226988	−	0.0867016i
$814$		0			0
$815$	11.3137			0.396302
$816$		0			0
$817$	−83.0132			−2.90426
$818$		0			0
$819$	−9.15298	−	10.2333i	−0.319831	−	0.357582i
$820$		0			0
$821$			27.8885i			0.973317i	0.873592	+	0.486658i	$0.161784\pi$
$821$			27.8885i			0.973317i	−0.873592	+	0.486658i	$0.838216\pi$
$822$		0			0
$823$	−28.4721			−0.992476			−0.496238	−	0.868186i	$-0.665286\pi$
$823$	−28.4721			−0.992476			−0.496238	+	0.868186i	$0.665286\pi$
$824$		0			0
$825$	−35.0702	+	13.3956i	−1.22099	+	0.466376i
$826$		0			0
$827$	−27.1738			−0.944926			−0.472463	−	0.881351i	$-0.656635\pi$
$827$	−27.1738			−0.944926			−0.472463	+	0.881351i	$0.656635\pi$
$828$		0			0
$829$	−15.8197			−0.549440			−0.274720	−	0.961524i	$-0.588585\pi$
$829$	−15.8197			−0.549440			−0.274720	+	0.961524i	$0.588585\pi$
$830$		0			0
$831$	36.1803	−	13.8197i	1.25508	−	0.479399i
$832$		0			0
$833$	24.5517			0.850666
$834$		0			0
$835$			14.9675i			0.517970i
$836$		0			0
$837$	14.2918	−	27.7082i	0.493997	−	0.957736i
$838$		0			0
$839$	7.81758			0.269893			0.134946	−	0.990853i	$-0.456914\pi$
$839$	7.81758			0.269893			0.134946	+	0.990853i	$0.456914\pi$
$840$		0			0
$841$	1.58359			0.0546066
$842$		0			0
$843$	−44.5570	+	17.0193i	−1.53463	+	0.586175i
$844$		0			0
$845$	−2.20943			−0.0760068
$846$		0			0
$847$			21.4682i			0.737658i
$848$		0			0
$849$	5.94200	+	15.5563i	0.203929	+	0.533893i
$850$		0			0
$851$	−24.0903	+	45.7082i	−0.825806	+	1.56686i
$852$		0			0
$853$	−38.7214			−1.32579			−0.662897	−	0.748711i	$-0.730673\pi$
$853$	−38.7214			−1.32579			−0.662897	+	0.748711i	$0.730673\pi$
$854$		0			0
$855$	14.4721	+	16.1803i	0.494937	+	0.553356i
$856$		0			0
$857$			15.4164i			0.526614i	0.964712	+	0.263307i	$0.0848133\pi$
$857$			15.4164i			0.526614i	−0.964712	+	0.263307i	$0.915187\pi$
$858$		0			0
$859$	−47.5967			−1.62398			−0.811990	−	0.583671i	$-0.801616\pi$
$859$	−47.5967			−1.62398			−0.811990	+	0.583671i	$0.801616\pi$
$860$		0			0
$861$	−1.08036	+	0.412662i	−0.0368187	+	0.0140635i
$862$		0			0
$863$			2.11146i			0.0718748i	0.999354	+	0.0359374i	$0.0114417\pi$
$863$			2.11146i			0.0718748i	−0.999354	+	0.0359374i	$0.988558\pi$
$864$		0			0
$865$		−	8.89794i		−	0.302539i
$866$		0			0
$867$	11.5066	−	4.39512i	0.390784	−	0.149266i
$868$		0			0
$869$		−	34.0689i		−	1.15571i
$870$		0			0
$871$			21.1344i			0.716112i
$872$		0			0
$873$	4.83153	+	5.40182i	0.163523	+	0.182824i
$874$		0			0
$875$		−	11.4164i		−	0.385945i
$876$		0			0
$877$	21.4164			0.723181			0.361590	−	0.932337i	$-0.382234\pi$
$877$	21.4164			0.723181			0.361590	+	0.932337i	$0.382234\pi$
$878$		0			0
$879$	−7.73877	+	2.95595i	−0.261022	+	0.0997016i
$880$		0			0
$881$	21.2132			0.714691			0.357345	−	0.933972i	$-0.383682\pi$
$881$	21.2132			0.714691			0.357345	+	0.933972i	$0.383682\pi$
$882$		0			0
$883$	15.5967			0.524872			0.262436	−	0.964949i	$-0.415474\pi$
$883$	15.5967			0.524872			0.262436	+	0.964949i	$0.415474\pi$
$884$		0			0
$885$	−4.32145	−	11.3137i	−0.145264	−	0.380306i
$886$		0			0
$887$		−	15.1246i		−	0.507835i	−0.967226	−	0.253917i	$-0.918281\pi$
$887$		−	15.1246i		−	0.507835i	0.967226	−	0.253917i	$-0.0817191\pi$
$888$		0			0
$889$		−	16.9706i		−	0.569174i
$890$		0			0
$891$	5.11667	−	45.7649i	0.171415	−	1.53318i
$892$		0			0
$893$	−96.9316			−3.24369
$894$		0			0
$895$		−	7.81758i		−	0.261313i
$896$		0			0
$897$	26.6869	+	3.22292i	0.891049	+	0.107610i
$898$		0			0
$899$		−	31.4164i		−	1.04780i
$900$		0			0
$901$	−4.29180			−0.142980
$902$		0			0
$903$	−22.9443	+	8.76393i	−0.763538	+	0.291645i
$904$		0			0
$905$			5.05573i			0.168058i
$906$		0			0
$907$			5.60815i			0.186215i	0.995656	+	0.0931077i	$0.0296801\pi$
$907$			5.60815i			0.186215i	−0.995656	+	0.0931077i	$0.970320\pi$
$908$		0			0
$909$	16.9443	+	18.9443i	0.562006	+	0.628342i
$910$		0			0
$911$	10.2333			0.339046			0.169523	−	0.985526i	$-0.445777\pi$
$911$	10.2333			0.339046			0.169523	+	0.985526i	$0.445777\pi$
$912$		0			0
$913$	−40.6525			−1.34540
$914$		0			0
$915$	0.652476	+	1.70820i	0.0215702	+	0.0564715i
$916$		0			0
$917$	3.90879			0.129080
$918$		0			0
$919$		−	30.8763i		−	1.01851i	−0.860615	−	0.509257i	$-0.829920\pi$
$919$		−	30.8763i		−	1.01851i	0.860615	−	0.509257i	$-0.170080\pi$
$920$		0			0
$921$	−32.9443	+	12.5836i	−1.08555	+	0.414643i
$922$		0			0
$923$			39.4164i			1.29741i
$924$		0			0
$925$			45.6374i			1.50055i
$926$		0			0
$927$	14.1421	+	15.8114i	0.464489	+	0.519314i
$928$		0			0
$929$			34.3607i			1.12734i	0.826001	+	0.563669i	$0.190611\pi$
$929$			34.3607i			1.12734i	−0.826001	+	0.563669i	$0.809389\pi$
$930$		0			0
$931$			41.3948i			1.35666i
$932$		0			0
$933$	12.3607	+	32.3607i	0.404670	+	1.05944i
$934$		0			0
$935$	21.9597			0.718160
$936$		0			0
$937$			48.7510i			1.59262i	0.604886	+	0.796312i	$0.293219\pi$
$937$			48.7510i			1.59262i	−0.604886	+	0.796312i	$0.706781\pi$
$938$		0			0
$939$	−11.0587	−	28.9520i	−0.360886	−	0.944812i
$940$		0			0
$941$	−1.03165			−0.0336310			−0.0168155	−	0.999859i	$-0.505353\pi$
$941$	−1.03165			−0.0336310			−0.0168155	+	0.999859i	$0.505353\pi$
$942$		0			0
$943$	1.05573	−	2.00310i	0.0343792	−	0.0652300i
$944$		0			0
$945$	5.70820	+	2.94427i	0.185688	+	0.0957772i
$946$		0			0
$947$			22.2492i			0.723003i	0.932372	+	0.361501i	$0.117736\pi$
$947$			22.2492i			0.723003i	−0.932372	+	0.361501i	$0.882264\pi$
$948$		0			0
$949$	23.4164			0.760129
$950$		0			0
$951$	2.29180	+	6.00000i	0.0743166	+	0.194563i
$952$		0			0
$953$	7.32611			0.237316			0.118658	−	0.992935i	$-0.462141\pi$
$953$	7.32611			0.237316			0.118658	+	0.992935i	$0.462141\pi$
$954$		0			0
$955$	−21.8885			−0.708297
$956$		0			0
$957$	−16.5579	−	43.3491i	−0.535241	−	1.40128i
$958$		0			0
$959$		−	24.4721i		−	0.790246i
$960$		0			0
$961$	5.00000			0.161290
$962$		0			0
$963$	20.7518	−	18.5610i	0.668718	−	0.598120i
$964$		0			0
$965$	2.98605			0.0961243
$966$		0			0
$967$	12.1115			0.389478			0.194739	−	0.980855i	$-0.437614\pi$
$967$	12.1115			0.389478			0.194739	+	0.980855i	$0.437614\pi$
$968$		0			0
$969$	25.1246	+	65.7771i	0.807119	+	2.11306i
$970$		0			0
$971$	31.6529			1.01579			0.507895	−	0.861419i	$-0.330424\pi$
$971$	31.6529			1.01579			0.507895	+	0.861419i	$0.330424\pi$
$972$		0			0
$973$		−	9.15298i		−	0.293431i
$974$		0			0
$975$	22.1803	−	8.47214i	0.710339	−	0.271325i
$976$		0			0
$977$	−33.1946			−1.06199			−0.530995	−	0.847375i	$-0.678182\pi$
$977$	−33.1946			−1.06199			−0.530995	+	0.847375i	$0.678182\pi$
$978$		0			0
$979$	−1.70820			−0.0545944
$980$		0			0
$981$	8.07262	+	9.02546i	0.257739	+	0.288161i
$982$		0			0
$983$	−12.2364			−0.390282			−0.195141	−	0.980775i	$-0.562516\pi$
$983$	−12.2364			−0.390282			−0.195141	+	0.980775i	$0.562516\pi$
$984$		0			0
$985$		−	6.57959i		−	0.209643i
$986$		0			0
$987$	−26.7912	+	10.2333i	−0.852775	+	0.325731i
$988$		0			0
$989$	22.4211	−	42.5410i	0.712949	−	1.35273i
$990$		0			0
$991$	−29.4164			−0.934443			−0.467221	−	0.884140i	$-0.654745\pi$
$991$	−29.4164			−0.934443			−0.467221	+	0.884140i	$0.654745\pi$
$992$		0			0
$993$	23.7082	−	9.05573i	0.752357	−	0.287375i
$994$		0			0
$995$			9.59675i			0.304237i
$996$		0			0
$997$	−19.5967			−0.620635			−0.310318	−	0.950633i	$-0.600435\pi$
$997$	−19.5967			−0.620635			−0.310318	+	0.950633i	$0.600435\pi$
$998$		0			0
$999$	−49.7525	−	25.6622i	−1.57410	−	0.811915i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1104.2.m.c.689.6		8
3.2	odd	2	inner	1104.2.m.c.689.7		8
4.3	odd	2		552.2.m.a.137.4	yes	8
12.11	even	2		552.2.m.a.137.1	✓	8
23.22	odd	2	inner	1104.2.m.c.689.5		8
69.68	even	2	inner	1104.2.m.c.689.8		8
92.91	even	2		552.2.m.a.137.3	yes	8
276.275	odd	2		552.2.m.a.137.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.m.a.137.1	✓	8	12.11	even	2
552.2.m.a.137.2	yes	8	276.275	odd	2
552.2.m.a.137.3	yes	8	92.91	even	2
552.2.m.a.137.4	yes	8	4.3	odd	2
1104.2.m.c.689.5		8	23.22	odd	2	inner
1104.2.m.c.689.6		8	1.1	even	1	trivial
1104.2.m.c.689.7		8	3.2	odd	2	inner
1104.2.m.c.689.8		8	69.68	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:	\( N \)	\(=\)	\( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1104.m (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.61803 - 0.618034i) q^{3} +0.874032 q^{5} +1.41421i q^{7} +(2.23607 - 2.00000i) q^{9} +5.11667 q^{11} -3.23607 q^{13} +(1.41421 - 0.540182i) q^{15} +4.91034 q^{17} +8.27895i q^{19} +(0.874032 + 2.28825i) q^{21} +(-4.24264 - 2.23607i) q^{23} -4.23607 q^{25} +(2.38197 - 4.61803i) q^{27} -5.23607i q^{29} +6.00000 q^{31} +(8.27895 - 3.16228i) q^{33} +1.23607i q^{35} -10.7735i q^{37} +(-5.23607 + 2.00000i) q^{39} +0.472136i q^{41} +10.0270i q^{43} +(1.95440 - 1.74806i) q^{45} +11.7082i q^{47} +5.00000 q^{49} +(7.94510 - 3.03476i) q^{51} -0.874032 q^{53} +4.47214 q^{55} +(5.11667 + 13.3956i) q^{57} -8.00000i q^{59} +1.20788i q^{61} +(2.82843 + 3.16228i) q^{63} -2.82843 q^{65} -6.53089i q^{67} +(-8.24670 - 0.995938i) q^{69} -12.1803i q^{71} -7.23607 q^{73} +(-6.85410 + 2.61803i) q^{75} +7.23607i q^{77} -6.65841i q^{79} +(1.00000 - 8.94427i) q^{81} -7.94510 q^{83} +4.29180 q^{85} +(-3.23607 - 8.47214i) q^{87} -0.333851 q^{89} -4.57649i q^{91} +(9.70820 - 3.70820i) q^{93} +7.23607i q^{95} +2.41577i q^{97} +(11.4412 - 10.2333i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} - 8 q^{13} - 16 q^{25} + 28 q^{27} + 48 q^{31} - 24 q^{39} + 40 q^{49} - 20 q^{69} - 40 q^{73} - 28 q^{75} + 8 q^{81} + 88 q^{85} - 8 q^{87} + 24 q^{93}+O(q^{100}) \) 8 * q + 4 * q^3 - 8 * q^13 - 16 * q^25 + 28 * q^27 + 48 * q^31 - 24 * q^39 + 40 * q^49 - 20 * q^69 - 40 * q^73 - 28 * q^75 + 8 * q^81 + 88 * q^85 - 8 * q^87 + 24 * q^93

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