LMFDB - Embedded newform 3654.2.f.b.755.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3654,2,Mod(755,3654)] 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("3654.755"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3654, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3654 = 2 \cdot 3^{2} \cdot 7 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3654.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [40,0,0,-40,0,0,-4,0,0,0,0,0,0,0,0,40,0,0,0,0,0,8,0,0,64] 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:	$29.1773368986$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			755.7
Character	$\chi$	$=$	3654.755
Dual form			3654.2.f.b.755.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -1.00000 q^{4} -2.09396 q^{5} +(-2.34749 - 1.22036i) q^{7} +1.00000i q^{8} +2.09396i q^{10} +5.08562i q^{11} +5.72887i q^{13} +(-1.22036 + 2.34749i) q^{14} +1.00000 q^{16} +0.0838140 q^{17} +2.68374i q^{19} +2.09396 q^{20} +5.08562 q^{22} -5.04508i q^{23} -0.615315 q^{25} +5.72887 q^{26} +(2.34749 + 1.22036i) q^{28} -1.00000i q^{29} -2.19960i q^{31} -1.00000i q^{32} -0.0838140i q^{34} +(4.91557 + 2.55538i) q^{35} -1.46824 q^{37} +2.68374 q^{38} -2.09396i q^{40} +6.62193 q^{41} +7.01012 q^{43} -5.08562i q^{44} -5.04508 q^{46} -10.0266 q^{47} +(4.02146 + 5.72956i) q^{49} +0.615315i q^{50} -5.72887i q^{52} -3.12887i q^{53} -10.6491i q^{55} +(1.22036 - 2.34749i) q^{56} -1.00000 q^{58} -6.23070 q^{59} -0.169107i q^{61} -2.19960 q^{62} -1.00000 q^{64} -11.9960i q^{65} -0.665504 q^{67} -0.0838140 q^{68} +(2.55538 - 4.91557i) q^{70} -14.8331i q^{71} -3.15651i q^{73} +1.46824i q^{74} -2.68374i q^{76} +(6.20628 - 11.9385i) q^{77} +6.53737 q^{79} -2.09396 q^{80} -6.62193i q^{82} -7.54921 q^{83} -0.175503 q^{85} -7.01012i q^{86} -5.08562 q^{88} -13.0060 q^{89} +(6.99127 - 13.4485i) q^{91} +5.04508i q^{92} +10.0266i q^{94} -5.61966i q^{95} +7.01498i q^{97} +(5.72956 - 4.02146i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 40 q^{4} - 4 q^{7} + 40 q^{16} + 8 q^{22} + 64 q^{25} + 4 q^{28} - 48 q^{37} - 8 q^{43} + 8 q^{46} + 36 q^{49} - 40 q^{58} - 40 q^{64} + 24 q^{67} + 28 q^{70} - 144 q^{79} - 24 q^{85} - 8 q^{88}+O(q^{100}) $ 40 * q - 40 * q^4 - 4 * q^7 + 40 * q^16 + 8 * q^22 + 64 * q^25 + 4 * q^28 - 48 * q^37 - 8 * q^43 + 8 * q^46 + 36 * q^49 - 40 * q^58 - 40 * q^64 + 24 * q^67 + 28 * q^70 - 144 * q^79 - 24 * q^85 - 8 * q^88

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3654\mathbb{Z}\right)^\times$.

$n$	$379$	$407$	$2089$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	−2.09396			−0.936449			−0.468225	−	0.883610i	$-0.655106\pi$
$5$	−2.09396			−0.936449			−0.468225	+	0.883610i	$0.655106\pi$
$6$		0			0
$7$	−2.34749	−	1.22036i	−0.887269	−	0.461252i
$7$	−2.34749	−	1.22036i	−0.887269	−	0.461252i
$8$			1.00000i			0.353553i
$9$		0			0
$10$			2.09396i			0.662170i
$11$			5.08562i			1.53337i	0.642022	+	0.766686i	$0.278096\pi$
$11$			5.08562i			1.53337i	−0.642022	+	0.766686i	$0.721904\pi$
$12$		0			0
$13$			5.72887i			1.58890i	0.607328	+	0.794451i	$0.292241\pi$
$13$			5.72887i			1.58890i	−0.607328	+	0.794451i	$0.707759\pi$
$14$	−1.22036	+	2.34749i	−0.326154	+	0.627394i
$15$		0			0
$16$	1.00000			0.250000
$17$	0.0838140			0.0203279			0.0101639	−	0.999948i	$-0.496765\pi$
$17$	0.0838140			0.0203279			0.0101639	+	0.999948i	$0.496765\pi$
$18$		0			0
$19$			2.68374i			0.615692i	0.951436	+	0.307846i	$0.0996082\pi$
$19$			2.68374i			0.615692i	−0.951436	+	0.307846i	$0.900392\pi$
$20$	2.09396			0.468225
$21$		0			0
$22$	5.08562			1.08426
$23$		−	5.04508i		−	1.05197i	−0.850493	−	0.525986i	$-0.823696\pi$
$23$		−	5.04508i		−	1.05197i	0.850493	−	0.525986i	$-0.176304\pi$
$24$		0			0
$25$	−0.615315			−0.123063
$26$	5.72887			1.12352
$27$		0			0
$28$	2.34749	+	1.22036i	0.443635	+	0.230626i
$29$		−	1.00000i		−	0.185695i
$29$		−	1.00000i		−	0.185695i
$30$		0			0
$31$		−	2.19960i		−	0.395060i	−0.980297	−	0.197530i	$-0.936708\pi$
$31$		−	2.19960i		−	0.395060i	0.980297	−	0.197530i	$-0.0632919\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$		−	0.0838140i		−	0.0143740i
$35$	4.91557	+	2.55538i	0.830883	+	0.431939i
$36$		0			0
$37$	−1.46824			−0.241377			−0.120689	−	0.992690i	$-0.538510\pi$
$37$	−1.46824			−0.241377			−0.120689	+	0.992690i	$0.538510\pi$
$38$	2.68374			0.435360
$39$		0			0
$40$		−	2.09396i		−	0.331085i
$41$	6.62193			1.03417			0.517086	−	0.855934i	$-0.327017\pi$
$41$	6.62193			1.03417			0.517086	+	0.855934i	$0.327017\pi$
$42$		0			0
$43$	7.01012			1.06903			0.534517	−	0.845158i	$-0.320494\pi$
$43$	7.01012			1.06903			0.534517	+	0.845158i	$0.320494\pi$
$44$		−	5.08562i		−	0.766686i
$45$		0			0
$46$	−5.04508			−0.743856
$47$	−10.0266			−1.46253			−0.731266	−	0.682093i	$-0.761070\pi$
$47$	−10.0266			−1.46253			−0.731266	+	0.682093i	$0.761070\pi$
$48$		0			0
$49$	4.02146	+	5.72956i	0.574494	+	0.818509i
$50$			0.615315i			0.0870187i
$51$		0			0
$52$		−	5.72887i		−	0.794451i
$53$		−	3.12887i		−	0.429783i	−0.976638	−	0.214892i	$-0.931060\pi$
$53$		−	3.12887i		−	0.429783i	0.976638	−	0.214892i	$-0.0689398\pi$
$54$		0			0
$55$		−	10.6491i		−	1.43593i
$56$	1.22036	−	2.34749i	0.163077	−	0.313697i
$57$		0			0
$58$	−1.00000			−0.131306
$59$	−6.23070			−0.811169			−0.405584	−	0.914058i	$-0.632932\pi$
$59$	−6.23070			−0.811169			−0.405584	+	0.914058i	$0.632932\pi$
$60$		0			0
$61$		−	0.169107i		−	0.0216519i	−0.999941	−	0.0108260i	$-0.996554\pi$
$61$		−	0.169107i		−	0.0216519i	0.999941	−	0.0108260i	$-0.00344607\pi$
$62$	−2.19960			−0.279349
$63$		0			0
$64$	−1.00000			−0.125000
$65$		−	11.9960i		−	1.48793i
$66$		0			0
$67$	−0.665504			−0.0813043			−0.0406521	−	0.999173i	$-0.512944\pi$
$67$	−0.665504			−0.0813043			−0.0406521	+	0.999173i	$0.512944\pi$
$68$	−0.0838140			−0.0101639
$69$		0			0
$70$	2.55538	−	4.91557i	0.305427	−	0.587523i
$71$		−	14.8331i		−	1.76036i	−0.474638	−	0.880181i	$-0.657421\pi$
$71$		−	14.8331i		−	1.76036i	0.474638	−	0.880181i	$-0.342579\pi$
$72$		0			0
$73$		−	3.15651i		−	0.369441i	−0.982791	−	0.184721i	$-0.940862\pi$
$73$		−	3.15651i		−	0.369441i	0.982791	−	0.184721i	$-0.0591380\pi$
$74$			1.46824i			0.170679i
$75$		0			0
$76$		−	2.68374i		−	0.307846i
$77$	6.20628	−	11.9385i	0.707271	−	1.36051i
$78$		0			0
$79$	6.53737			0.735512			0.367756	−	0.929922i	$-0.380126\pi$
$79$	6.53737			0.735512			0.367756	+	0.929922i	$0.380126\pi$
$80$	−2.09396			−0.234112
$81$		0			0
$82$		−	6.62193i		−	0.731269i
$83$	−7.54921			−0.828634			−0.414317	−	0.910133i	$-0.635980\pi$
$83$	−7.54921			−0.828634			−0.414317	+	0.910133i	$0.635980\pi$
$84$		0			0
$85$	−0.175503			−0.0190360
$86$		−	7.01012i		−	0.755921i
$87$		0			0
$88$	−5.08562			−0.542129
$89$	−13.0060			−1.37863			−0.689317	−	0.724460i	$-0.742089\pi$
$89$	−13.0060			−1.37863			−0.689317	+	0.724460i	$0.742089\pi$
$90$		0			0
$91$	6.99127	−	13.4485i	0.732884	−	1.40978i
$92$			5.04508i			0.525986i
$93$		0			0
$94$			10.0266i			1.03417i
$95$		−	5.61966i		−	0.576565i
$96$		0			0
$97$			7.01498i			0.712263i	0.934436	+	0.356132i	$0.115905\pi$
$97$			7.01498i			0.712263i	−0.934436	+	0.356132i	$0.884095\pi$
$98$	5.72956	−	4.02146i	0.578773	−	0.406228i
$99$		0			0
$100$	0.615315			0.0615315
$101$	−11.7521			−1.16938			−0.584689	−	0.811258i	$-0.698783\pi$
$101$	−11.7521			−1.16938			−0.584689	+	0.811258i	$0.698783\pi$
$102$		0			0
$103$			16.9944i			1.67450i	0.546817	+	0.837252i	$0.315839\pi$
$103$			16.9944i			1.67450i	−0.546817	+	0.837252i	$0.684161\pi$
$104$	−5.72887			−0.561762
$105$		0			0
$106$	−3.12887			−0.303903
$107$		−	7.80211i		−	0.754258i	−0.926161	−	0.377129i	$-0.876911\pi$
$107$		−	7.80211i		−	0.754258i	0.926161	−	0.377129i	$-0.123089\pi$
$108$		0			0
$109$	−0.524959			−0.0502819			−0.0251410	−	0.999684i	$-0.508003\pi$
$109$	−0.524959			−0.0502819			−0.0251410	+	0.999684i	$0.508003\pi$
$110$	−10.6491			−1.01535
$111$		0			0
$112$	−2.34749	−	1.22036i	−0.221817	−	0.115313i
$113$		−	9.71468i		−	0.913880i	−0.889497	−	0.456940i	$-0.848945\pi$
$113$		−	9.71468i		−	0.913880i	0.889497	−	0.456940i	$-0.151055\pi$
$114$		0			0
$115$			10.5642i			0.985118i
$116$			1.00000i			0.0928477i
$117$		0			0
$118$			6.23070i			0.573583i
$119$	−0.196753	−	0.102283i	−0.0180363	−	0.00937627i
$120$		0			0
$121$	−14.8635			−1.35123
$122$	−0.169107			−0.0153102
$123$		0			0
$124$			2.19960i			0.197530i
$125$	11.7583			1.05169
$126$		0			0
$127$	6.11874			0.542950			0.271475	−	0.962445i	$-0.412489\pi$
$127$	6.11874			0.542950			0.271475	+	0.962445i	$0.412489\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	−11.9960			−1.05212
$131$	−1.78939			−0.156339			−0.0781697	−	0.996940i	$-0.524908\pi$
$131$	−1.78939			−0.156339			−0.0781697	+	0.996940i	$0.524908\pi$
$132$		0			0
$133$	3.27512	−	6.30007i	0.283989	−	0.546285i
$134$			0.665504i			0.0574908i
$135$		0			0
$136$			0.0838140i			0.00718699i
$137$			9.58464i			0.818871i	0.912339	+	0.409436i	$0.134274\pi$
$137$			9.58464i			0.818871i	−0.912339	+	0.409436i	$0.865726\pi$
$138$		0			0
$139$			3.98318i			0.337849i	0.985629	+	0.168924i	$0.0540293\pi$
$139$			3.98318i			0.337849i	−0.985629	+	0.168924i	$0.945971\pi$
$140$	−4.91557	−	2.55538i	−0.415441	−	0.215969i
$141$		0			0
$142$	−14.8331			−1.24476
$143$	−29.1348			−2.43638
$144$		0			0
$145$			2.09396i			0.173894i
$146$	−3.15651			−0.261235
$147$		0			0
$148$	1.46824			0.120689
$149$		−	6.42874i		−	0.526663i	−0.964705	−	0.263332i	$-0.915179\pi$
$149$		−	6.42874i		−	0.526663i	0.964705	−	0.263332i	$-0.0848213\pi$
$150$		0			0
$151$	−12.1904			−0.992041			−0.496020	−	0.868311i	$-0.665206\pi$
$151$	−12.1904			−0.992041			−0.496020	+	0.868311i	$0.665206\pi$
$152$	−2.68374			−0.217680
$153$		0			0
$154$	−11.9385	−	6.20628i	−0.962029	−	0.500116i
$155$			4.60588i			0.369953i
$156$		0			0
$157$		−	8.65041i		−	0.690378i	−0.938533	−	0.345189i	$-0.887815\pi$
$157$		−	8.65041i		−	0.690378i	0.938533	−	0.345189i	$-0.112185\pi$
$158$		−	6.53737i		−	0.520085i
$159$		0			0
$160$			2.09396i			0.165542i
$161$	−6.15680	+	11.8433i	−0.485224	+	0.933382i
$162$		0			0
$163$	22.7342			1.78068			0.890341	−	0.455295i	$-0.150466\pi$
$163$	22.7342			1.78068			0.890341	+	0.455295i	$0.150466\pi$
$164$	−6.62193			−0.517086
$165$		0			0
$166$			7.54921i			0.585933i
$167$	−8.25057			−0.638448			−0.319224	−	0.947679i	$-0.603422\pi$
$167$	−8.25057			−0.638448			−0.319224	+	0.947679i	$0.603422\pi$
$168$		0			0
$169$	−19.8199			−1.52461
$170$			0.175503i			0.0134605i
$171$		0			0
$172$	−7.01012			−0.534517
$173$	23.0659			1.75367			0.876834	−	0.480793i	$-0.159651\pi$
$173$	23.0659			1.75367			0.876834	+	0.480793i	$0.159651\pi$
$174$		0			0
$175$	1.44445	+	0.750905i	0.109190	+	0.0567631i
$176$			5.08562i			0.383343i
$177$		0			0
$178$			13.0060i			0.974842i
$179$			12.2767i			0.917602i	0.888539	+	0.458801i	$0.151721\pi$
$179$			12.2767i			0.917602i	−0.888539	+	0.458801i	$0.848279\pi$
$180$		0			0
$181$			16.2132i			1.20512i	0.798075	+	0.602558i	$0.205852\pi$
$181$			16.2132i			1.20512i	−0.798075	+	0.602558i	$0.794148\pi$
$182$	−13.4485	−	6.99127i	−0.996868	−	0.518227i
$183$		0			0
$184$	5.04508			0.371928
$185$	3.07444			0.226037
$186$		0			0
$187$			0.426246i			0.0311702i
$188$	10.0266			0.731266
$189$		0			0
$190$	−5.61966			−0.407693
$191$		−	3.14319i		−	0.227433i	−0.993513	−	0.113717i	$-0.963724\pi$
$191$		−	3.14319i		−	0.227433i	0.993513	−	0.113717i	$-0.0362756\pi$
$192$		0			0
$193$	−0.241791			−0.0174045			−0.00870225	−	0.999962i	$-0.502770\pi$
$193$	−0.241791			−0.0174045			−0.00870225	+	0.999962i	$0.502770\pi$
$194$	7.01498			0.503646
$195$		0			0
$196$	−4.02146	−	5.72956i	−0.287247	−	0.409255i
$197$		−	24.6873i		−	1.75890i	−0.475996	−	0.879448i	$-0.657912\pi$
$197$		−	24.6873i		−	1.75890i	0.475996	−	0.879448i	$-0.342088\pi$
$198$		0			0
$199$		−	24.5655i		−	1.74140i	−0.491813	−	0.870701i	$-0.663665\pi$
$199$		−	24.5655i		−	1.74140i	0.491813	−	0.870701i	$-0.336335\pi$
$200$		−	0.615315i		−	0.0435094i
$201$		0			0
$202$			11.7521i			0.826875i
$203$	−1.22036	+	2.34749i	−0.0856523	+	0.164762i
$204$		0			0
$205$	−13.8661			−0.968449
$206$	16.9944			1.18405
$207$		0			0
$208$			5.72887i			0.397225i
$209$	−13.6485			−0.944086
$210$		0			0
$211$	−6.52086			−0.448914			−0.224457	−	0.974484i	$-0.572061\pi$
$211$	−6.52086			−0.448914			−0.224457	+	0.974484i	$0.572061\pi$
$212$			3.12887i			0.214892i
$213$		0			0
$214$	−7.80211			−0.533341
$215$	−14.6789			−1.00110
$216$		0			0
$217$	−2.68430	+	5.16354i	−0.182222	+	0.350524i
$218$			0.524959i			0.0355547i
$219$		0			0
$220$			10.6491i			0.717963i
$221$			0.480159i			0.0322990i
$222$		0			0
$223$		−	27.8778i		−	1.86683i	−0.358794	−	0.933417i	$-0.616812\pi$
$223$		−	27.8778i		−	1.86683i	0.358794	−	0.933417i	$-0.383188\pi$
$224$	−1.22036	+	2.34749i	−0.0815386	+	0.156849i
$225$		0			0
$226$	−9.71468			−0.646211
$227$	−10.3089			−0.684227			−0.342114	−	0.939659i	$-0.611143\pi$
$227$	−10.3089			−0.684227			−0.342114	+	0.939659i	$0.611143\pi$
$228$		0			0
$229$			3.04938i			0.201509i	0.994911	+	0.100754i	$0.0321256\pi$
$229$			3.04938i			0.201509i	−0.994911	+	0.100754i	$0.967874\pi$
$230$	10.5642			0.696584
$231$		0			0
$232$	1.00000			0.0656532
$233$		−	21.3292i		−	1.39732i	−0.715453	−	0.698661i	$-0.753780\pi$
$233$		−	21.3292i		−	1.39732i	0.715453	−	0.698661i	$-0.246220\pi$
$234$		0			0
$235$	20.9954			1.36959
$236$	6.23070			0.405584
$237$		0			0
$238$	−0.102283	+	0.196753i	−0.00663002	+	0.0127536i
$239$		−	16.6158i		−	1.07478i	−0.843333	−	0.537392i	$-0.819410\pi$
$239$		−	16.6158i		−	1.07478i	0.843333	−	0.537392i	$-0.180590\pi$
$240$		0			0
$241$		−	4.60540i		−	0.296660i	−0.988938	−	0.148330i	$-0.952610\pi$
$241$		−	4.60540i		−	0.296660i	0.988938	−	0.148330i	$-0.0473898\pi$
$242$			14.8635i			0.955465i
$243$		0			0
$244$			0.169107i			0.0108260i
$245$	−8.42078	−	11.9975i	−0.537984	−	0.766492i
$246$		0			0
$247$	−15.3748			−0.978275
$248$	2.19960			0.139675
$249$		0			0
$250$		−	11.7583i		−	0.743658i
$251$	14.0707			0.888136			0.444068	−	0.895993i	$-0.353535\pi$
$251$	14.0707			0.888136			0.444068	+	0.895993i	$0.353535\pi$
$252$		0			0
$253$	25.6574			1.61306
$254$		−	6.11874i		−	0.383924i
$255$		0			0
$256$	1.00000			0.0625000
$257$	−9.15106			−0.570827			−0.285414	−	0.958404i	$-0.592131\pi$
$257$	−9.15106			−0.570827			−0.285414	+	0.958404i	$0.592131\pi$
$258$		0			0
$259$	3.44668	+	1.79178i	0.214166	+	0.111336i
$260$			11.9960i			0.743963i
$261$		0			0
$262$			1.78939i			0.110549i
$263$		−	27.3278i		−	1.68510i	−0.538615	−	0.842552i	$-0.681052\pi$
$263$		−	27.3278i		−	1.68510i	0.538615	−	0.842552i	$-0.318948\pi$
$264$		0			0
$265$			6.55174i			0.402470i
$266$	−6.30007	−	3.27512i	−0.386282	−	0.200811i
$267$		0			0
$268$	0.665504			0.0406521
$269$	−12.9376			−0.788820			−0.394410	−	0.918935i	$-0.629051\pi$
$269$	−12.9376			−0.788820			−0.394410	+	0.918935i	$0.629051\pi$
$270$		0			0
$271$		−	20.4152i		−	1.24013i	−0.784549	−	0.620067i	$-0.787105\pi$
$271$		−	20.4152i		−	1.24013i	0.784549	−	0.620067i	$-0.212895\pi$
$272$	0.0838140			0.00508197
$273$		0			0
$274$	9.58464			0.579029
$275$		−	3.12926i		−	0.188702i
$276$		0			0
$277$	−18.2251			−1.09504			−0.547520	−	0.836792i	$-0.684428\pi$
$277$	−18.2251			−1.09504			−0.547520	+	0.836792i	$0.684428\pi$
$278$	3.98318			0.238895
$279$		0			0
$280$	−2.55538	+	4.91557i	−0.152713	+	0.293761i
$281$			18.3462i			1.09444i	0.836988	+	0.547221i	$0.184314\pi$
$281$			18.3462i			1.09444i	−0.836988	+	0.547221i	$0.815686\pi$
$282$		0			0
$283$		−	24.9977i		−	1.48596i	−0.669313	−	0.742981i	$-0.733411\pi$
$283$		−	24.9977i		−	1.48596i	0.669313	−	0.742981i	$-0.266589\pi$
$284$			14.8331i			0.880181i
$285$		0			0
$286$			29.1348i			1.72278i
$287$	−15.5449	−	8.08112i	−0.917588	−	0.477013i
$288$		0			0
$289$	−16.9930			−0.999587
$290$	2.09396			0.122962
$291$		0			0
$292$			3.15651i			0.184721i
$293$	27.2313			1.59087			0.795436	−	0.606038i	$-0.207242\pi$
$293$	27.2313			1.59087			0.795436	+	0.606038i	$0.207242\pi$
$294$		0			0
$295$	13.0469			0.759618
$296$		−	1.46824i		−	0.0853397i
$297$		0			0
$298$	−6.42874			−0.372407
$299$	28.9026			1.67148
$300$		0			0
$301$	−16.4562	−	8.55485i	−0.948521	−	0.493094i
$302$			12.1904i			0.701479i
$303$		0			0
$304$			2.68374i			0.153923i
$305$			0.354103i			0.0202759i
$306$		0			0
$307$			11.1820i			0.638190i	0.947723	+	0.319095i	$0.103379\pi$
$307$			11.1820i			0.638190i	−0.947723	+	0.319095i	$0.896621\pi$
$308$	−6.20628	+	11.9385i	−0.353635	+	0.680257i
$309$		0			0
$310$	4.60588			0.261596
$311$	23.5365			1.33463			0.667317	−	0.744773i	$-0.267442\pi$
$311$	23.5365			1.33463			0.667317	+	0.744773i	$0.267442\pi$
$312$		0			0
$313$		−	23.8054i		−	1.34556i	−0.739841	−	0.672782i	$-0.765099\pi$
$313$		−	23.8054i		−	1.34556i	0.739841	−	0.672782i	$-0.234901\pi$
$314$	−8.65041			−0.488171
$315$		0			0
$316$	−6.53737			−0.367756
$317$		−	22.6738i		−	1.27349i	−0.771075	−	0.636745i	$-0.780281\pi$
$317$		−	22.6738i		−	1.27349i	0.771075	−	0.636745i	$-0.219719\pi$
$318$		0			0
$319$	5.08562			0.284740
$320$	2.09396			0.117056
$321$		0			0
$322$	11.8433	+	6.15680i	0.660001	+	0.343105i
$323$			0.224935i			0.0125157i
$324$		0			0
$325$		−	3.52506i		−	0.195535i
$326$		−	22.7342i		−	1.25913i
$327$		0			0
$328$			6.62193i			0.365635i
$329$	23.5374	+	12.2360i	1.29766	+	0.674595i
$330$		0			0
$331$	−18.0202			−0.990480			−0.495240	−	0.868756i	$-0.664920\pi$
$331$	−18.0202			−0.990480			−0.495240	+	0.868756i	$0.664920\pi$
$332$	7.54921			0.414317
$333$		0			0
$334$			8.25057i			0.451451i
$335$	1.39354			0.0761373
$336$		0			0
$337$	15.8472			0.863250			0.431625	−	0.902053i	$-0.357940\pi$
$337$	15.8472			0.863250			0.431625	+	0.902053i	$0.357940\pi$
$338$			19.8199i			1.07806i
$339$		0			0
$340$	0.175503			0.00951801
$341$	11.1863			0.605774
$342$		0			0
$343$	−2.44823	−	18.3577i	−0.132192	−	0.991224i
$344$			7.01012i			0.377960i
$345$		0			0
$346$		−	23.0659i		−	1.24003i
$347$		−	10.6587i		−	0.572187i	−0.958202	−	0.286093i	$-0.907643\pi$
$347$		−	10.6587i		−	0.572187i	0.958202	−	0.286093i	$-0.0923568\pi$
$348$		0			0
$349$			28.9116i			1.54760i	0.633427	+	0.773802i	$0.281648\pi$
$349$			28.9116i			1.54760i	−0.633427	+	0.773802i	$0.718352\pi$
$350$	0.750905	−	1.44445i	0.0401375	−	0.0772091i
$351$		0			0
$352$	5.08562			0.271065
$353$	−21.2889			−1.13309			−0.566546	−	0.824030i	$-0.691721\pi$
$353$	−21.2889			−1.13309			−0.566546	+	0.824030i	$0.691721\pi$
$354$		0			0
$355$			31.0599i			1.64849i
$356$	13.0060			0.689317
$357$		0			0
$358$	12.2767			0.648843
$359$		−	10.5541i		−	0.557024i	−0.960433	−	0.278512i	$-0.910159\pi$
$359$		−	10.5541i		−	0.557024i	0.960433	−	0.278512i	$-0.0898411\pi$
$360$		0			0
$361$	11.7975			0.620923
$362$	16.2132			0.852146
$363$		0			0
$364$	−6.99127	+	13.4485i	−0.366442	+	0.704892i
$365$			6.60961i			0.345963i
$366$		0			0
$367$			30.1107i			1.57176i	0.618376	+	0.785882i	$0.287791\pi$
$367$			30.1107i			1.57176i	−0.618376	+	0.785882i	$0.712209\pi$
$368$		−	5.04508i		−	0.262993i
$369$		0			0
$370$		−	3.07444i		−	0.159833i
$371$	−3.81834	+	7.34500i	−0.198238	+	0.381334i
$372$		0			0
$373$	−0.225433			−0.0116725			−0.00583625	−	0.999983i	$-0.501858\pi$
$373$	−0.225433			−0.0116725			−0.00583625	+	0.999983i	$0.501858\pi$
$374$	0.426246			0.0220407
$375$		0			0
$376$		−	10.0266i		−	0.517083i
$377$	5.72887			0.295052
$378$		0			0
$379$	21.9143			1.12566			0.562831	−	0.826572i	$-0.309712\pi$
$379$	21.9143			1.12566			0.562831	+	0.826572i	$0.309712\pi$
$380$			5.61966i			0.288282i
$381$		0			0
$382$	−3.14319			−0.160820
$383$	−13.2298			−0.676013			−0.338006	−	0.941144i	$-0.609753\pi$
$383$	−13.2298			−0.676013			−0.338006	+	0.941144i	$0.609753\pi$
$384$		0			0
$385$	−12.9957	+	24.9987i	−0.662323	+	1.27405i
$386$			0.241791i			0.0123068i
$387$		0			0
$388$		−	7.01498i		−	0.356132i
$389$			20.3750i			1.03305i	0.856272	+	0.516526i	$0.172775\pi$
$389$			20.3750i			1.03305i	−0.856272	+	0.516526i	$0.827225\pi$
$390$		0			0
$391$		−	0.422848i		−	0.0213843i
$392$	−5.72956	+	4.02146i	−0.289387	+	0.203114i
$393$		0			0
$394$	−24.6873			−1.24373
$395$	−13.6890			−0.688769
$396$		0			0
$397$			16.6678i			0.836531i	0.908325	+	0.418265i	$0.137362\pi$
$397$			16.6678i			0.836531i	−0.908325	+	0.418265i	$0.862638\pi$
$398$	−24.5655			−1.23136
$399$		0			0
$400$	−0.615315			−0.0307658
$401$		−	15.7493i		−	0.786485i	−0.919435	−	0.393242i	$-0.871353\pi$
$401$		−	15.7493i		−	0.786485i	0.919435	−	0.393242i	$-0.128647\pi$
$402$		0			0
$403$	12.6012			0.627711
$404$	11.7521			0.584689
$405$		0			0
$406$	2.34749	+	1.22036i	0.116504	+	0.0605653i
$407$		−	7.46691i		−	0.370121i
$408$		0			0
$409$		−	17.0312i		−	0.842139i	−0.907028	−	0.421069i	$-0.861655\pi$
$409$		−	17.0312i		−	0.842139i	0.907028	−	0.421069i	$-0.138345\pi$
$410$			13.8661i			0.684797i
$411$		0			0
$412$		−	16.9944i		−	0.837252i
$413$	14.6265	+	7.60369i	0.719725	+	0.374153i
$414$		0			0
$415$	15.8078			0.775974
$416$	5.72887			0.280881
$417$		0			0
$418$			13.6485i			0.667570i
$419$	−11.5262			−0.563093			−0.281546	−	0.959548i	$-0.590847\pi$
$419$	−11.5262			−0.563093			−0.281546	+	0.959548i	$0.590847\pi$
$420$		0			0
$421$	−21.5417			−1.04988			−0.524939	−	0.851140i	$-0.675912\pi$
$421$	−21.5417			−1.04988			−0.524939	+	0.851140i	$0.675912\pi$
$422$			6.52086i			0.317430i
$423$		0			0
$424$	3.12887			0.151951
$425$	−0.0515720			−0.00250161
$426$		0			0
$427$	−0.206371	+	0.396977i	−0.00998698	+	0.0192111i
$428$			7.80211i			0.377129i
$429$		0			0
$430$			14.6789i			0.707881i
$431$		−	14.9330i		−	0.719298i	−0.933088	−	0.359649i	$-0.882896\pi$
$431$		−	14.9330i		−	0.719298i	0.933088	−	0.359649i	$-0.117104\pi$
$432$		0			0
$433$		−	16.3522i		−	0.785837i	−0.919573	−	0.392918i	$-0.871465\pi$
$433$		−	16.3522i		−	0.785837i	0.919573	−	0.392918i	$-0.128535\pi$
$434$	5.16354	+	2.68430i	0.247858	+	0.128850i
$435$		0			0
$436$	0.524959			0.0251410
$437$	13.5397			0.647691
$438$		0			0
$439$			20.9177i			0.998348i	0.866502	+	0.499174i	$0.166363\pi$
$439$			20.9177i			0.998348i	−0.866502	+	0.499174i	$0.833637\pi$
$440$	10.6491			0.507676
$441$		0			0
$442$	0.480159			0.0228388
$443$			17.2290i			0.818576i	0.912405	+	0.409288i	$0.134223\pi$
$443$			17.2290i			0.818576i	−0.912405	+	0.409288i	$0.865777\pi$
$444$		0			0
$445$	27.2341			1.29102
$446$	−27.8778			−1.32005
$447$		0			0
$448$	2.34749	+	1.22036i	0.110909	+	0.0576565i
$449$		−	19.5036i		−	0.920434i	−0.887807	−	0.460217i	$-0.847772\pi$
$449$		−	19.5036i		−	0.920434i	0.887807	−	0.460217i	$-0.152228\pi$
$450$		0			0
$451$			33.6766i			1.58577i
$452$			9.71468i			0.456940i
$453$		0			0
$454$			10.3089i			0.483822i
$455$	−14.6395	+	28.1606i	−0.686308	+	1.32019i
$456$		0			0
$457$	−22.5735			−1.05594			−0.527972	−	0.849262i	$-0.677047\pi$
$457$	−22.5735			−1.05594			−0.527972	+	0.849262i	$0.677047\pi$
$458$	3.04938			0.142488
$459$		0			0
$460$		−	10.5642i		−	0.492559i
$461$	0.0153958			0.000717054			0.000358527	−	1.00000i	$-0.499886\pi$
$461$	0.0153958			0.000717054			0.000358527		1.00000i	$0.499886\pi$
$462$		0			0
$463$	12.9800			0.603230			0.301615	−	0.953430i	$-0.402474\pi$
$463$	12.9800			0.603230			0.301615	+	0.953430i	$0.402474\pi$
$464$		−	1.00000i		−	0.0464238i
$465$		0			0
$466$	−21.3292			−0.988056
$467$	15.2541			0.705875			0.352937	−	0.935647i	$-0.385183\pi$
$467$	15.2541			0.705875			0.352937	+	0.935647i	$0.385183\pi$
$468$		0			0
$469$	1.56227	+	0.812153i	0.0721388	+	0.0375017i
$470$		−	20.9954i		−	0.968444i
$471$		0			0
$472$		−	6.23070i		−	0.286791i
$473$			35.6508i			1.63923i
$474$		0			0
$475$		−	1.65135i		−	0.0757690i
$476$	0.196753	+	0.102283i	0.00901815	+	0.00468813i
$477$		0			0
$478$	−16.6158			−0.759987
$479$	18.0263			0.823643			0.411821	−	0.911265i	$-0.364893\pi$
$479$	18.0263			0.823643			0.411821	+	0.911265i	$0.364893\pi$
$480$		0			0
$481$		−	8.41135i		−	0.383524i
$482$	−4.60540			−0.209770
$483$		0			0
$484$	14.8635			0.675616
$485$		−	14.6891i		−	0.666999i
$486$		0			0
$487$	−7.06154			−0.319989			−0.159995	−	0.987118i	$-0.551148\pi$
$487$	−7.06154			−0.319989			−0.159995	+	0.987118i	$0.551148\pi$
$488$	0.169107			0.00765510
$489$		0			0
$490$	−11.9975	+	8.42078i	−0.541992	+	0.380412i
$491$			23.8411i			1.07593i	0.842966	+	0.537967i	$0.180807\pi$
$491$			23.8411i			1.07593i	−0.842966	+	0.537967i	$0.819193\pi$
$492$		0			0
$493$		−	0.0838140i		−	0.00377479i
$494$			15.3748i			0.691745i
$495$		0			0
$496$		−	2.19960i		−	0.0987649i
$497$	−18.1017	+	34.8205i	−0.811970	+	1.56191i
$498$		0			0
$499$	−38.1415			−1.70745			−0.853723	−	0.520727i	$-0.825661\pi$
$499$	−38.1415			−1.70745			−0.853723	+	0.520727i	$0.825661\pi$
$500$	−11.7583			−0.525846
$501$		0			0
$502$		−	14.0707i		−	0.628007i
$503$	−21.7081			−0.967915			−0.483958	−	0.875091i	$-0.660801\pi$
$503$	−21.7081			−0.967915			−0.483958	+	0.875091i	$0.660801\pi$
$504$		0			0
$505$	24.6085			1.09506
$506$		−	25.6574i		−	1.14061i
$507$		0			0
$508$	−6.11874			−0.271475
$509$	−0.959344			−0.0425222			−0.0212611	−	0.999774i	$-0.506768\pi$
$509$	−0.959344			−0.0425222			−0.0212611	+	0.999774i	$0.506768\pi$
$510$		0			0
$511$	−3.85207	+	7.40988i	−0.170405	+	0.327794i
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$			9.15106i			0.403636i
$515$		−	35.5856i		−	1.56809i
$516$		0			0
$517$		−	50.9915i		−	2.24261i
$518$	1.79178	−	3.44668i	0.0787261	−	0.151439i
$519$		0			0
$520$	11.9960			0.526061
$521$	40.1191			1.75765			0.878826	−	0.477143i	$-0.158328\pi$
$521$	40.1191			1.75765			0.878826	+	0.477143i	$0.158328\pi$
$522$		0			0
$523$		−	30.6470i		−	1.34010i	−0.742316	−	0.670050i	$-0.766272\pi$
$523$		−	30.6470i		−	1.34010i	0.742316	−	0.670050i	$-0.233728\pi$
$524$	1.78939			0.0781697
$525$		0			0
$526$	−27.3278			−1.19155
$527$		−	0.184357i		−	0.00803072i
$528$		0			0
$529$	−2.45282			−0.106644
$530$	6.55174			0.284589
$531$		0			0
$532$	−3.27512	+	6.30007i	−0.141995	+	0.273142i
$533$			37.9361i			1.64320i
$534$		0			0
$535$			16.3373i			0.706324i
$536$		−	0.665504i		−	0.0287454i
$537$		0			0
$538$			12.9376i			0.557780i
$539$	−29.1384	+	20.4516i	−1.25508	+	0.880913i
$540$		0			0
$541$	14.3509			0.616993			0.308497	−	0.951225i	$-0.400174\pi$
$541$	14.3509			0.616993			0.308497	+	0.951225i	$0.400174\pi$
$542$	−20.4152			−0.876907
$543$		0			0
$544$		−	0.0838140i		−	0.00359349i
$545$	1.09924			0.0470864
$546$		0			0
$547$	41.7345			1.78444			0.892219	−	0.451602i	$-0.149147\pi$
$547$	41.7345			1.78444			0.892219	+	0.451602i	$0.149147\pi$
$548$		−	9.58464i		−	0.409436i
$549$		0			0
$550$	−3.12926			−0.133432
$551$	2.68374			0.114331
$552$		0			0
$553$	−15.3464	−	7.97793i	−0.652597	−	0.339256i
$554$			18.2251i			0.774311i
$555$		0			0
$556$		−	3.98318i		−	0.168924i
$557$		−	2.51524i		−	0.106574i	−0.998579	−	0.0532870i	$-0.983030\pi$
$557$		−	2.51524i		−	0.106574i	0.998579	−	0.0532870i	$-0.0169698\pi$
$558$		0			0
$559$			40.1600i			1.69859i
$560$	4.91557	+	2.55538i	0.207721	+	0.107985i
$561$		0			0
$562$	18.3462			0.773887
$563$	39.8123			1.67789			0.838944	−	0.544217i	$-0.183173\pi$
$563$	39.8123			1.67789			0.838944	+	0.544217i	$0.183173\pi$
$564$		0			0
$565$			20.3422i			0.855802i
$566$	−24.9977			−1.05073
$567$		0			0
$568$	14.8331			0.622382
$569$			24.6357i			1.03278i	0.856353	+	0.516391i	$0.172725\pi$
$569$			24.6357i			1.03278i	−0.856353	+	0.516391i	$0.827275\pi$
$570$		0			0
$571$	−8.43767			−0.353106			−0.176553	−	0.984291i	$-0.556495\pi$
$571$	−8.43767			−0.353106			−0.176553	+	0.984291i	$0.556495\pi$
$572$	29.1348			1.21819
$573$		0			0
$574$	−8.08112	+	15.5449i	−0.337299	+	0.648833i
$575$			3.10431i			0.129459i
$576$		0			0
$577$			1.19465i			0.0497338i	0.999691	+	0.0248669i	$0.00791620\pi$
$577$			1.19465i			0.0497338i	−0.999691	+	0.0248669i	$0.992084\pi$
$578$			16.9930i			0.706815i
$579$		0			0
$580$		−	2.09396i		−	0.0869471i
$581$	17.7217	+	9.21274i	0.735221	+	0.382209i
$582$		0			0
$583$	15.9122			0.659018
$584$	3.15651			0.130617
$585$		0			0
$586$		−	27.2313i		−	1.12492i
$587$	−0.594115			−0.0245217			−0.0122609	−	0.999925i	$-0.503903\pi$
$587$	−0.594115			−0.0245217			−0.0122609	+	0.999925i	$0.503903\pi$
$588$		0			0
$589$	5.90315			0.243235
$590$		−	13.0469i		−	0.537131i
$591$		0			0
$592$	−1.46824			−0.0603443
$593$	−30.0394			−1.23357			−0.616786	−	0.787131i	$-0.711566\pi$
$593$	−30.0394			−1.23357			−0.616786	+	0.787131i	$0.711566\pi$
$594$		0			0
$595$	0.411993	+	0.214177i	0.0168901	+	0.00878040i
$596$			6.42874i			0.263332i
$597$		0			0
$598$		−	28.9026i		−	1.18191i
$599$		−	30.3348i		−	1.23944i	−0.784821	−	0.619722i	$-0.787245\pi$
$599$		−	30.3348i		−	1.23944i	0.784821	−	0.619722i	$-0.212755\pi$
$600$		0			0
$601$			6.04673i			0.246651i	0.992366	+	0.123326i	$0.0393560\pi$
$601$			6.04673i			0.246651i	−0.992366	+	0.123326i	$0.960644\pi$
$602$	−8.55485	+	16.4562i	−0.348670	+	0.670705i
$603$		0			0
$604$	12.1904			0.496020
$605$	31.1237			1.26536
$606$		0			0
$607$			20.4214i			0.828879i	0.910077	+	0.414439i	$0.136022\pi$
$607$			20.4214i			0.828879i	−0.910077	+	0.414439i	$0.863978\pi$
$608$	2.68374			0.108840
$609$		0			0
$610$	0.354103			0.0143372
$611$		−	57.4411i		−	2.32382i
$612$		0			0
$613$	20.4858			0.827412			0.413706	−	0.910410i	$-0.364234\pi$
$613$	20.4858			0.827412			0.413706	+	0.910410i	$0.364234\pi$
$614$	11.1820			0.451268
$615$		0			0
$616$	11.9385	+	6.20628i	0.481014	+	0.250058i
$617$			16.0855i			0.647577i	0.946129	+	0.323789i	$0.104957\pi$
$617$			16.0855i			0.647577i	−0.946129	+	0.323789i	$0.895043\pi$
$618$		0			0
$619$		−	11.6153i		−	0.466860i	−0.972374	−	0.233430i	$-0.925005\pi$
$619$		−	11.6153i		−	0.466860i	0.972374	−	0.233430i	$-0.0749949\pi$
$620$		−	4.60588i		−	0.184977i
$621$		0			0
$622$		−	23.5365i		−	0.943729i
$623$	30.5315	+	15.8720i	1.22322	+	0.635897i
$624$		0			0
$625$	−21.5448			−0.861792
$626$	−23.8054			−0.951457
$627$		0			0
$628$			8.65041i			0.345189i
$629$	−0.123059			−0.00490668
$630$		0			0
$631$	32.9932			1.31344			0.656719	−	0.754135i	$-0.271944\pi$
$631$	32.9932			1.31344			0.656719	+	0.754135i	$0.271944\pi$
$632$			6.53737i			0.260043i
$633$		0			0
$634$	−22.6738			−0.900493
$635$	−12.8124			−0.508445
$636$		0			0
$637$	−32.8239	+	23.0384i	−1.30053	+	0.912814i
$638$		−	5.08562i		−	0.201342i
$639$		0			0
$640$		−	2.09396i		−	0.0827712i
$641$			12.4544i			0.491919i	0.969280	+	0.245960i	$0.0791030\pi$
$641$			12.4544i			0.491919i	−0.969280	+	0.245960i	$0.920897\pi$
$642$		0			0
$643$		−	38.4148i		−	1.51493i	−0.652874	−	0.757466i	$-0.726437\pi$
$643$		−	38.4148i		−	1.51493i	0.652874	−	0.757466i	$-0.273563\pi$
$644$	6.15680	−	11.8433i	0.242612	−	0.466691i
$645$		0			0
$646$	0.224935			0.00884995
$647$	−8.14883			−0.320363			−0.160182	−	0.987088i	$-0.551208\pi$
$647$	−8.14883			−0.320363			−0.160182	+	0.987088i	$0.551208\pi$
$648$		0			0
$649$		−	31.6870i		−	1.24382i
$650$	−3.52506			−0.138264
$651$		0			0
$652$	−22.7342			−0.890341
$653$			40.6424i			1.59046i	0.606309	+	0.795229i	$0.292650\pi$
$653$			40.6424i			1.59046i	−0.606309	+	0.795229i	$0.707350\pi$
$654$		0			0
$655$	3.74691			0.146404
$656$	6.62193			0.258543
$657$		0			0
$658$	12.2360	−	23.5374i	0.477011	−	0.917584i
$659$			18.1503i			0.707034i	0.935428	+	0.353517i	$0.115014\pi$
$659$			18.1503i			0.707034i	−0.935428	+	0.353517i	$0.884986\pi$
$660$		0			0
$661$		−	23.4847i		−	0.913448i	−0.889608	−	0.456724i	$-0.849023\pi$
$661$		−	23.4847i		−	0.913448i	0.889608	−	0.456724i	$-0.150977\pi$
$662$			18.0202i			0.700375i
$663$		0			0
$664$		−	7.54921i		−	0.292966i
$665$	−6.85799	+	13.1921i	−0.265941	+	0.511568i
$666$		0			0
$667$	−5.04508			−0.195346
$668$	8.25057			0.319224
$669$		0			0
$670$		−	1.39354i		−	0.0538372i
$671$	0.860013			0.0332004
$672$		0			0
$673$	15.8132			0.609553			0.304776	−	0.952424i	$-0.401418\pi$
$673$	15.8132			0.609553			0.304776	+	0.952424i	$0.401418\pi$
$674$		−	15.8472i		−	0.610410i
$675$		0			0
$676$	19.8199			0.762305
$677$	−26.9488			−1.03573			−0.517864	−	0.855463i	$-0.673273\pi$
$677$	−26.9488			−1.03573			−0.517864	+	0.855463i	$0.673273\pi$
$678$		0			0
$679$	8.56079	−	16.4676i	0.328533	−	0.631970i
$680$		−	0.175503i		−	0.00673025i
$681$		0			0
$682$		−	11.1863i		−	0.428347i
$683$			15.4122i			0.589731i	0.955539	+	0.294865i	$0.0952748\pi$
$683$			15.4122i			0.589731i	−0.955539	+	0.294865i	$0.904725\pi$
$684$		0			0
$685$		−	20.0699i		−	0.766831i
$686$	−18.3577	+	2.44823i	−0.700901	+	0.0934737i
$687$		0			0
$688$	7.01012			0.267258
$689$	17.9249			0.682883
$690$		0			0
$691$		−	13.3947i		−	0.509559i	−0.966999	−	0.254780i	$-0.917997\pi$
$691$		−	13.3947i		−	0.509559i	0.966999	−	0.254780i	$-0.0820029\pi$
$692$	−23.0659			−0.876834
$693$		0			0
$694$	−10.6587			−0.404597
$695$		−	8.34063i		−	0.316378i
$696$		0			0
$697$	0.555010			0.0210225
$698$	28.9116			1.09432
$699$		0			0
$700$	−1.44445	−	0.750905i	−0.0545950	−	0.0283815i
$701$		−	10.7595i		−	0.406382i	−0.979139	−	0.203191i	$-0.934869\pi$
$701$		−	10.7595i		−	0.406382i	0.979139	−	0.203191i	$-0.0651312\pi$
$702$		0			0
$703$		−	3.94037i		−	0.148614i
$704$		−	5.08562i		−	0.191672i
$705$		0			0
$706$			21.2889i			0.801218i
$707$	27.5880	+	14.3418i	1.03755	+	0.539377i
$708$		0			0
$709$	−30.3384			−1.13938			−0.569691	−	0.821859i	$-0.692937\pi$
$709$	−30.3384			−1.13938			−0.569691	+	0.821859i	$0.692937\pi$
$710$	31.0599			1.16566
$711$		0			0
$712$		−	13.0060i		−	0.487421i
$713$	−11.0971			−0.415592
$714$		0			0
$715$	61.0073			2.28154
$716$		−	12.2767i		−	0.458801i
$717$		0			0
$718$	−10.5541			−0.393875
$719$	−17.5915			−0.656052			−0.328026	−	0.944669i	$-0.606383\pi$
$719$	−17.5915			−0.656052			−0.328026	+	0.944669i	$0.606383\pi$
$720$		0			0
$721$	20.7392	−	39.8942i	0.772368	−	1.48574i
$722$		−	11.7975i		−	0.439059i
$723$		0			0
$724$		−	16.2132i		−	0.602558i
$725$			0.615315i			0.0228522i
$726$		0			0
$727$		−	22.6997i		−	0.841885i	−0.907087	−	0.420942i	$-0.861699\pi$
$727$		−	22.6997i		−	0.841885i	0.907087	−	0.420942i	$-0.138301\pi$
$728$	13.4485	+	6.99127i	0.498434	+	0.259114i
$729$		0			0
$730$	6.60961			0.244633
$731$	0.587546			0.0217312
$732$		0			0
$733$		−	33.0984i		−	1.22252i	−0.791431	−	0.611259i	$-0.790664\pi$
$733$		−	33.0984i		−	1.22252i	0.791431	−	0.611259i	$-0.209336\pi$
$734$	30.1107			1.11141
$735$		0			0
$736$	−5.04508			−0.185964
$737$		−	3.38450i		−	0.124670i
$738$		0			0
$739$	−28.2258			−1.03830			−0.519151	−	0.854683i	$-0.673752\pi$
$739$	−28.2258			−1.03830			−0.519151	+	0.854683i	$0.673752\pi$
$740$	−3.07444			−0.113019
$741$		0			0
$742$	7.34500	+	3.81834i	0.269644	+	0.140176i
$743$			6.40900i			0.235123i	0.993066	+	0.117562i	$0.0375078\pi$
$743$			6.40900i			0.235123i	−0.993066	+	0.117562i	$0.962492\pi$
$744$		0			0
$745$			13.4616i			0.493193i
$746$			0.225433i			0.00825370i
$747$		0			0
$748$		−	0.426246i		−	0.0155851i
$749$	−9.52136	+	18.3154i	−0.347903	+	0.669230i
$750$		0			0
$751$	−23.8444			−0.870094			−0.435047	−	0.900408i	$-0.643268\pi$
$751$	−23.8444			−0.870094			−0.435047	+	0.900408i	$0.643268\pi$
$752$	−10.0266			−0.365633
$753$		0			0
$754$		−	5.72887i		−	0.208633i
$755$	25.5263			0.928996
$756$		0			0
$757$	30.2068			1.09789			0.548943	−	0.835860i	$-0.315030\pi$
$757$	30.2068			1.09789			0.548943	+	0.835860i	$0.315030\pi$
$758$		−	21.9143i		−	0.795964i
$759$		0			0
$760$	5.61966			0.203846
$761$	−18.5750			−0.673343			−0.336672	−	0.941622i	$-0.609301\pi$
$761$	−18.5750			−0.673343			−0.336672	+	0.941622i	$0.609301\pi$
$762$		0			0
$763$	1.23234	+	0.640637i	0.0446136	+	0.0231926i
$764$			3.14319i			0.113717i
$765$		0			0
$766$			13.2298i			0.478013i
$767$		−	35.6949i		−	1.28887i
$768$		0			0
$769$			42.7354i			1.54108i	0.637392	+	0.770539i	$0.280013\pi$
$769$			42.7354i			1.54108i	−0.637392	+	0.770539i	$0.719987\pi$
$770$	24.9987	+	12.9957i	0.900891	+	0.468333i
$771$		0			0
$772$	0.241791			0.00870225
$773$	−12.7089			−0.457108			−0.228554	−	0.973531i	$-0.573400\pi$
$773$	−12.7089			−0.457108			−0.228554	+	0.973531i	$0.573400\pi$
$774$		0			0
$775$			1.35345i			0.0486173i
$776$	−7.01498			−0.251823
$777$		0			0
$778$	20.3750			0.730477
$779$			17.7715i			0.636731i
$780$		0			0
$781$	75.4354			2.69929
$782$	−0.422848			−0.0151210
$783$		0			0
$784$	4.02146	+	5.72956i	0.143623	+	0.204627i
$785$			18.1137i			0.646504i
$786$		0			0
$787$		−	6.13579i		−	0.218717i	−0.994002	−	0.109359i	$-0.965120\pi$
$787$		−	6.13579i		−	0.218717i	0.994002	−	0.109359i	$-0.0348797\pi$
$788$			24.6873i			0.879448i
$789$		0			0
$790$			13.6890i			0.487034i
$791$	−11.8554	+	22.8052i	−0.421529	+	0.810858i
$792$		0			0
$793$	0.968790			0.0344027
$794$	16.6678			0.591517
$795$		0			0
$796$			24.5655i			0.870701i
$797$	18.9255			0.670374			0.335187	−	0.942152i	$-0.391200\pi$
$797$	18.9255			0.670374			0.335187	+	0.942152i	$0.391200\pi$
$798$		0			0
$799$	−0.840370			−0.0297302
$800$			0.615315i			0.0217547i
$801$		0			0
$802$	−15.7493			−0.556129
$803$	16.0528			0.566491
$804$		0			0
$805$	12.8921	−	24.7994i	0.454387	−	0.874065i
$806$		−	12.6012i		−	0.443859i
$807$		0			0
$808$		−	11.7521i		−	0.413437i
$809$			37.7996i			1.32896i	0.747305	+	0.664481i	$0.231347\pi$
$809$			37.7996i			1.32896i	−0.747305	+	0.664481i	$0.768653\pi$
$810$		0			0
$811$		−	18.9933i		−	0.666946i	−0.942760	−	0.333473i	$-0.891779\pi$
$811$		−	18.9933i		−	0.666946i	0.942760	−	0.333473i	$-0.108221\pi$
$812$	1.22036	−	2.34749i	0.0428262	−	0.0823809i
$813$		0			0
$814$	−7.46691			−0.261715
$815$	−47.6046			−1.66752
$816$		0			0
$817$			18.8133i			0.658196i
$818$	−17.0312			−0.595482
$819$		0			0
$820$	13.8661			0.484224
$821$		−	1.93324i		−	0.0674705i	−0.999431	−	0.0337353i	$-0.989260\pi$
$821$		−	1.93324i		−	0.0674705i	0.999431	−	0.0337353i	$-0.0107403\pi$
$822$		0			0
$823$	21.0869			0.735044			0.367522	−	0.930015i	$-0.380206\pi$
$823$	21.0869			0.735044			0.367522	+	0.930015i	$0.380206\pi$
$824$	−16.9944			−0.592027
$825$		0			0
$826$	7.60369	−	14.6265i	0.264566	−	0.508922i
$827$		−	27.1054i		−	0.942547i	−0.881987	−	0.471273i	$-0.843794\pi$
$827$		−	27.1054i		−	0.942547i	0.881987	−	0.471273i	$-0.156206\pi$
$828$		0			0
$829$			2.17373i			0.0754966i	0.999287	+	0.0377483i	$0.0120185\pi$
$829$			2.17373i			0.0754966i	−0.999287	+	0.0377483i	$0.987981\pi$
$830$		−	15.8078i		−	0.548696i
$831$		0			0
$832$		−	5.72887i		−	0.198613i
$833$	0.337054	+	0.480218i	0.0116782	+	0.0166386i
$834$		0			0
$835$	17.2764			0.597874
$836$	13.6485			0.472043
$837$		0			0
$838$			11.5262i			0.398167i
$839$	−39.7166			−1.37117			−0.685585	−	0.727993i	$-0.740453\pi$
$839$	−39.7166			−1.37117			−0.685585	+	0.727993i	$0.740453\pi$
$840$		0			0
$841$	−1.00000			−0.0344828
$842$			21.5417i			0.742376i
$843$		0			0
$844$	6.52086			0.224457
$845$	41.5022			1.42772
$846$		0			0
$847$	34.8921	+	18.1388i	1.19891	+	0.623258i
$848$		−	3.12887i		−	0.107446i
$849$		0			0
$850$			0.0515720i			0.00176891i
$851$			7.40738i			0.253922i
$852$		0			0
$853$			44.6701i			1.52947i	0.644342	+	0.764737i	$0.277131\pi$
$853$			44.6701i			1.52947i	−0.644342	+	0.764737i	$0.722869\pi$
$854$	0.396977	+	0.206371i	0.0135843	+	0.00706186i
$855$		0			0
$856$	7.80211			0.266671
$857$	0.487280			0.0166452			0.00832259	−	0.999965i	$-0.497351\pi$
$857$	0.487280			0.0166452			0.00832259	+	0.999965i	$0.497351\pi$
$858$		0			0
$859$			9.16916i			0.312848i	0.987690	+	0.156424i	$0.0499966\pi$
$859$			9.16916i			0.312848i	−0.987690	+	0.156424i	$0.950003\pi$
$860$	14.6789			0.500548
$861$		0			0
$862$	−14.9330			−0.508621
$863$		−	32.0533i		−	1.09111i	−0.838076	−	0.545553i	$-0.816320\pi$
$863$		−	32.0533i		−	1.09111i	0.838076	−	0.545553i	$-0.183680\pi$
$864$		0			0
$865$	−48.2992			−1.64222
$866$	−16.3522			−0.555671
$867$		0			0
$868$	2.68430	−	5.16354i	0.0911110	−	0.175262i
$869$			33.2466i			1.12781i
$870$		0			0
$871$		−	3.81258i		−	0.129184i
$872$		−	0.524959i		−	0.0177773i
$873$		0			0
$874$		−	13.5397i		−	0.457987i
$875$	−27.6025	−	14.3493i	−0.933133	−	0.485095i
$876$		0			0
$877$	−5.69375			−0.192264			−0.0961321	−	0.995369i	$-0.530647\pi$
$877$	−5.69375			−0.192264			−0.0961321	+	0.995369i	$0.530647\pi$
$878$	20.9177			0.705938
$879$		0			0
$880$		−	10.6491i		−	0.358981i
$881$	8.40319			0.283111			0.141555	−	0.989930i	$-0.454790\pi$
$881$	8.40319			0.283111			0.141555	+	0.989930i	$0.454790\pi$
$882$		0			0
$883$	42.9878			1.44665			0.723327	−	0.690506i	$-0.242612\pi$
$883$	42.9878			1.44665			0.723327	+	0.690506i	$0.242612\pi$
$884$		−	0.480159i		−	0.0161495i
$885$		0			0
$886$	17.2290			0.578820
$887$	0.0859168			0.00288480			0.00144240	−	0.999999i	$-0.499541\pi$
$887$	0.0859168			0.00288480			0.00144240	+	0.999999i	$0.499541\pi$
$888$		0			0
$889$	−14.3637	−	7.46705i	−0.481743	−	0.250437i
$890$		−	27.2341i		−	0.912890i
$891$		0			0
$892$			27.8778i			0.933417i
$893$		−	26.9088i		−	0.900469i
$894$		0			0
$895$		−	25.7069i		−	0.859288i
$896$	1.22036	−	2.34749i	0.0407693	−	0.0784243i
$897$		0			0
$898$	−19.5036			−0.650845
$899$	−2.19960			−0.0733607
$900$		0			0
$901$		−	0.262243i		−	0.00873658i
$902$	33.6766			1.12131
$903$		0			0
$904$	9.71468			0.323105
$905$		−	33.9498i		−	1.12853i
$906$		0			0
$907$	−46.7163			−1.55119			−0.775594	−	0.631232i	$-0.782549\pi$
$907$	−46.7163			−1.55119			−0.775594	+	0.631232i	$0.782549\pi$
$908$	10.3089			0.342114
$909$		0			0
$910$	28.1606	+	14.6395i	0.933516	+	0.485293i
$911$			49.3108i			1.63374i	0.576821	+	0.816870i	$0.304293\pi$
$911$			49.3108i			1.63374i	−0.576821	+	0.816870i	$0.695707\pi$
$912$		0			0
$913$		−	38.3924i		−	1.27060i
$914$			22.5735i			0.746665i
$915$		0			0
$916$		−	3.04938i		−	0.100754i
$917$	4.20057	+	2.18369i	0.138715	+	0.0721118i
$918$		0			0
$919$	0.0433033			0.00142844			0.000714221	−	1.00000i	$-0.499773\pi$
$919$	0.0433033			0.00142844			0.000714221		1.00000i	$0.499773\pi$
$920$	−10.5642			−0.348292
$921$		0			0
$922$		−	0.0153958i		−	0.000507034i
$923$	84.9767			2.79704
$924$		0			0
$925$	0.903430			0.0297046
$926$		−	12.9800i		−	0.426548i
$927$		0			0
$928$	−1.00000			−0.0328266
$929$	−49.6966			−1.63049			−0.815246	−	0.579114i	$-0.803399\pi$
$929$	−49.6966			−1.63049			−0.815246	+	0.579114i	$0.803399\pi$
$930$		0			0
$931$	−15.3767	+	10.7925i	−0.503950	+	0.353711i
$932$			21.3292i			0.698661i
$933$		0			0
$934$		−	15.2541i		−	0.499129i
$935$		−	0.892544i		−	0.0291893i
$936$		0			0
$937$		−	2.11204i		−	0.0689974i	−0.999405	−	0.0344987i	$-0.989017\pi$
$937$		−	2.11204i		−	0.0689974i	0.999405	−	0.0344987i	$-0.0109835\pi$
$938$	0.812153	−	1.56227i	0.0265177	−	0.0510098i
$939$		0			0
$940$	−20.9954			−0.684793
$941$	−50.9898			−1.66222			−0.831110	−	0.556108i	$-0.812294\pi$
$941$	−50.9898			−1.66222			−0.831110	+	0.556108i	$0.812294\pi$
$942$		0			0
$943$		−	33.4081i		−	1.08792i
$944$	−6.23070			−0.202792
$945$		0			0
$946$	35.6508			1.15911
$947$		−	19.8857i		−	0.646197i	−0.946365	−	0.323098i	$-0.895276\pi$
$947$		−	19.8857i		−	0.646197i	0.946365	−	0.323098i	$-0.104724\pi$
$948$		0			0
$949$	18.0832			0.587006
$950$	−1.65135			−0.0535768
$951$		0			0
$952$	0.102283	−	0.196753i	0.00331501	−	0.00637680i
$953$			30.1302i			0.976012i	0.872840	+	0.488006i	$0.162276\pi$
$953$			30.1302i			0.976012i	−0.872840	+	0.488006i	$0.837724\pi$
$954$		0			0
$955$			6.58173i			0.212980i
$956$			16.6158i			0.537392i
$957$		0			0
$958$		−	18.0263i		−	0.582403i
$959$	11.6967	−	22.4999i	0.377706	−	0.726559i
$960$		0			0
$961$	26.1618			0.843928
$962$	−8.41135			−0.271193
$963$		0			0
$964$			4.60540i			0.148330i
$965$	0.506302			0.0162984
$966$		0			0
$967$	−27.0829			−0.870927			−0.435463	−	0.900206i	$-0.643415\pi$
$967$	−27.0829			−0.870927			−0.435463	+	0.900206i	$0.643415\pi$
$968$		−	14.8635i		−	0.477733i
$969$		0			0
$970$	−14.6891			−0.471639
$971$	−25.6204			−0.822199			−0.411100	−	0.911590i	$-0.634855\pi$
$971$	−25.6204			−0.822199			−0.411100	+	0.911590i	$0.634855\pi$
$972$		0			0
$973$	4.86090	−	9.35048i	0.155833	−	0.299763i
$974$			7.06154i			0.226266i
$975$		0			0
$976$		−	0.169107i		−	0.00541298i
$977$			5.98940i			0.191618i	0.995400	+	0.0958090i	$0.0305438\pi$
$977$			5.98940i			0.191618i	−0.995400	+	0.0958090i	$0.969456\pi$
$978$		0			0
$979$		−	66.1436i		−	2.11396i
$980$	8.42078	+	11.9975i	0.268992	+	0.383246i
$981$		0			0
$982$	23.8411			0.760800
$983$	14.2604			0.454835			0.227417	−	0.973797i	$-0.426972\pi$
$983$	14.2604			0.454835			0.227417	+	0.973797i	$0.426972\pi$
$984$		0			0
$985$			51.6943i			1.64712i
$986$	−0.0838140			−0.00266918
$987$		0			0
$988$	15.3748			0.489137
$989$		−	35.3666i		−	1.12459i
$990$		0			0
$991$	16.5383			0.525356			0.262678	−	0.964884i	$-0.415394\pi$
$991$	16.5383			0.525356			0.262678	+	0.964884i	$0.415394\pi$
$992$	−2.19960			−0.0698373
$993$		0			0
$994$	34.8205	+	18.1017i	1.10444	+	0.574149i
$995$			51.4393i			1.63073i
$996$		0			0
$997$			30.4252i			0.963575i	0.876288	+	0.481787i	$0.160012\pi$
$997$			30.4252i			0.963575i	−0.876288	+	0.481787i	$0.839988\pi$
$998$			38.1415i			1.20735i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3654.2.f.b.755.7	✓	40
3.2	odd	2	inner	3654.2.f.b.755.34	yes	40
7.6	odd	2	inner	3654.2.f.b.755.33	yes	40
21.20	even	2	inner	3654.2.f.b.755.8	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3654.2.f.b.755.7	✓	40	1.1	even	1	trivial
3654.2.f.b.755.8	yes	40	21.20	even	2	inner
3654.2.f.b.755.33	yes	40	7.6	odd	2	inner
3654.2.f.b.755.34	yes	40	3.2	odd	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:	\( N \)	\(=\)	\( 3654 = 2 \cdot 3^{2} \cdot 7 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3654.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -2.09396 q^{5} +(-2.34749 - 1.22036i) q^{7} +1.00000i q^{8} +2.09396i q^{10} +5.08562i q^{11} +5.72887i q^{13} +(-1.22036 + 2.34749i) q^{14} +1.00000 q^{16} +0.0838140 q^{17} +2.68374i q^{19} +2.09396 q^{20} +5.08562 q^{22} -5.04508i q^{23} -0.615315 q^{25} +5.72887 q^{26} +(2.34749 + 1.22036i) q^{28} -1.00000i q^{29} -2.19960i q^{31} -1.00000i q^{32} -0.0838140i q^{34} +(4.91557 + 2.55538i) q^{35} -1.46824 q^{37} +2.68374 q^{38} -2.09396i q^{40} +6.62193 q^{41} +7.01012 q^{43} -5.08562i q^{44} -5.04508 q^{46} -10.0266 q^{47} +(4.02146 + 5.72956i) q^{49} +0.615315i q^{50} -5.72887i q^{52} -3.12887i q^{53} -10.6491i q^{55} +(1.22036 - 2.34749i) q^{56} -1.00000 q^{58} -6.23070 q^{59} -0.169107i q^{61} -2.19960 q^{62} -1.00000 q^{64} -11.9960i q^{65} -0.665504 q^{67} -0.0838140 q^{68} +(2.55538 - 4.91557i) q^{70} -14.8331i q^{71} -3.15651i q^{73} +1.46824i q^{74} -2.68374i q^{76} +(6.20628 - 11.9385i) q^{77} +6.53737 q^{79} -2.09396 q^{80} -6.62193i q^{82} -7.54921 q^{83} -0.175503 q^{85} -7.01012i q^{86} -5.08562 q^{88} -13.0060 q^{89} +(6.99127 - 13.4485i) q^{91} +5.04508i q^{92} +10.0266i q^{94} -5.61966i q^{95} +7.01498i q^{97} +(5.72956 - 4.02146i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 40 q^{4} - 4 q^{7} + 40 q^{16} + 8 q^{22} + 64 q^{25} + 4 q^{28} - 48 q^{37} - 8 q^{43} + 8 q^{46} + 36 q^{49} - 40 q^{58} - 40 q^{64} + 24 q^{67} + 28 q^{70} - 144 q^{79} - 24 q^{85} - 8 q^{88}+O(q^{100}) \) 40 * q - 40 * q^4 - 4 * q^7 + 40 * q^16 + 8 * q^22 + 64 * q^25 + 4 * q^28 - 48 * q^37 - 8 * q^43 + 8 * q^46 + 36 * q^49 - 40 * q^58 - 40 * q^64 + 24 * q^67 + 28 * q^70 - 144 * q^79 - 24 * q^85 - 8 * q^88

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