LMFDB - Embedded newform 1050.3.h.b.349.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1050,3,Mod(349,1050)] 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("1050.349"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1050, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1050.h (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$28.6104277578$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.11007531417600000000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 7x^{12} + 48x^{8} - 7x^{4} + 1 $ x^16 - 7x^12 + 48x^8 - 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{28} $
Twist minimal:	no (minimal twist has level 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.6
Root			$0.596975 + 0.159959i$ of defining polynomial
Character	$\chi$	$=$	1050.349
Dual form			1050.3.h.b.349.14

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.41421i q^{2} +1.73205 q^{3} -2.00000 q^{4} -2.44949i q^{6} +(-2.02265 - 6.70141i) q^{7} +2.82843i q^{8} +3.00000 q^{9} -5.00806 q^{11} -3.46410 q^{12} -9.06929 q^{13} +(-9.47723 + 2.86045i) q^{14} +4.00000 q^{16} -19.7410 q^{17} -4.24264i q^{18} +25.6863i q^{19} +(-3.50333 - 11.6072i) q^{21} +7.08246i q^{22} +40.9686i q^{23} +4.89898i q^{24} +12.8259i q^{26} +5.19615 q^{27} +(4.04529 + 13.4028i) q^{28} +22.3071 q^{29} -15.9015i q^{31} -5.65685i q^{32} -8.67421 q^{33} +27.9179i q^{34} -6.00000 q^{36} -44.7506i q^{37} +36.3259 q^{38} -15.7085 q^{39} +27.0829i q^{41} +(-16.4150 + 4.95445i) q^{42} +30.6690i q^{43} +10.0161 q^{44} +57.9383 q^{46} -58.1430 q^{47} +6.92820 q^{48} +(-40.8178 + 27.1092i) q^{49} -34.1924 q^{51} +18.1386 q^{52} -65.6199i q^{53} -7.34847i q^{54} +(18.9545 - 5.72091i) q^{56} +44.4900i q^{57} -31.5469i q^{58} +32.5746i q^{59} +83.5349i q^{61} -22.4881 q^{62} +(-6.06794 - 20.1042i) q^{63} -8.00000 q^{64} +12.2672i q^{66} +72.0585i q^{67} +39.4819 q^{68} +70.9597i q^{69} -24.8675 q^{71} +8.48528i q^{72} +67.8725 q^{73} -63.2870 q^{74} -51.3726i q^{76} +(10.1295 + 33.5610i) q^{77} +22.2151i q^{78} -30.4452 q^{79} +9.00000 q^{81} +38.3009 q^{82} +72.4623 q^{83} +(7.00665 + 23.2144i) q^{84} +43.3725 q^{86} +38.6370 q^{87} -14.1649i q^{88} +113.756i q^{89} +(18.3440 + 60.7770i) q^{91} -81.9372i q^{92} -27.5422i q^{93} +82.2267i q^{94} -9.79796i q^{96} -103.804 q^{97} +(38.3381 + 57.7251i) q^{98} -15.0242 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{4} + 48 q^{9} - 32 q^{11} - 64 q^{14} + 64 q^{16} + 288 q^{29} - 96 q^{36} + 96 q^{39} + 64 q^{44} + 256 q^{46} + 48 q^{49} + 128 q^{56} - 128 q^{64} + 352 q^{71} + 320 q^{74} + 576 q^{79}+ \cdots - 96 q^{99}+O(q^{100}) $ 16 * q - 32 * q^4 + 48 * q^9 - 32 * q^11 - 64 * q^14 + 64 * q^16 + 288 * q^29 - 96 * q^36 + 96 * q^39 + 64 * q^44 + 256 * q^46 + 48 * q^49 + 128 * q^56 - 128 * q^64 + 352 * q^71 + 320 * q^74 + 576 * q^79 + 144 * q^81 - 128 * q^86 + 128 * q^91 - 96 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$701$
$\chi(n)$	$-1$	$-1$	$1$

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 $

$2$		−	1.41421i		−	0.707107i
$2$		−	1.41421i		−	0.707107i
$3$	1.73205			0.577350
$3$	1.73205			0.577350
$4$	−2.00000			−0.500000
$5$		0			0
$5$		0			0
$6$		−	2.44949i		−	0.408248i
$7$	−2.02265	−	6.70141i	−0.288949	−	0.957344i
$7$	−2.02265	−	6.70141i	−0.288949	−	0.957344i
$8$			2.82843i			0.353553i
$9$	3.00000			0.333333
$10$		0			0
$11$	−5.00806			−0.455278			−0.227639	−	0.973746i	$-0.573101\pi$
$11$	−5.00806			−0.455278			−0.227639	+	0.973746i	$0.573101\pi$
$12$	−3.46410			−0.288675
$13$	−9.06929			−0.697638			−0.348819	−	0.937190i	$-0.613417\pi$
$13$	−9.06929			−0.697638			−0.348819	+	0.937190i	$0.613417\pi$
$14$	−9.47723	+	2.86045i	−0.676945	+	0.204318i
$15$		0			0
$16$	4.00000			0.250000
$17$	−19.7410			−1.16123			−0.580617	−	0.814177i	$-0.697188\pi$
$17$	−19.7410			−1.16123			−0.580617	+	0.814177i	$0.697188\pi$
$18$		−	4.24264i		−	0.235702i
$19$			25.6863i			1.35191i	0.736942	+	0.675956i	$0.236269\pi$
$19$			25.6863i			1.35191i	−0.736942	+	0.675956i	$0.763731\pi$
$20$		0			0
$21$	−3.50333	−	11.6072i	−0.166825	−	0.552723i
$22$			7.08246i			0.321930i
$23$			40.9686i			1.78124i	0.454746	+	0.890621i	$0.349730\pi$
$23$			40.9686i			1.78124i	−0.454746	+	0.890621i	$0.650270\pi$
$24$			4.89898i			0.204124i
$25$		0			0
$26$			12.8259i			0.493304i
$27$	5.19615			0.192450
$28$	4.04529	+	13.4028i	0.144475	+	0.478672i
$29$	22.3071			0.769209			0.384604	−	0.923081i	$-0.374338\pi$
$29$	22.3071			0.769209			0.384604	+	0.923081i	$0.374338\pi$
$30$		0			0
$31$		−	15.9015i		−	0.512952i	−0.966551	−	0.256476i	$-0.917439\pi$
$31$		−	15.9015i		−	0.512952i	0.966551	−	0.256476i	$-0.0825614\pi$
$32$		−	5.65685i		−	0.176777i
$33$	−8.67421			−0.262855
$34$			27.9179i			0.821116i
$35$		0			0
$36$	−6.00000			−0.166667
$37$		−	44.7506i		−	1.20948i	−0.796424	−	0.604738i	$-0.793278\pi$
$37$		−	44.7506i		−	1.20948i	0.796424	−	0.604738i	$-0.206722\pi$
$38$	36.3259			0.955946
$39$	−15.7085			−0.402781
$40$		0			0
$41$			27.0829i			0.660557i	0.943883	+	0.330279i	$0.107143\pi$
$41$			27.0829i			0.660557i	−0.943883	+	0.330279i	$0.892857\pi$
$42$	−16.4150	+	4.95445i	−0.390834	+	0.117963i
$43$			30.6690i			0.713233i	0.934251	+	0.356616i	$0.116070\pi$
$43$			30.6690i			0.713233i	−0.934251	+	0.356616i	$0.883930\pi$
$44$	10.0161			0.227639
$45$		0			0
$46$	57.9383			1.25953
$47$	−58.1430			−1.23709			−0.618543	−	0.785751i	$-0.712277\pi$
$47$	−58.1430			−1.23709			−0.618543	+	0.785751i	$0.712277\pi$
$48$	6.92820			0.144338
$49$	−40.8178	+	27.1092i	−0.833016	+	0.553248i
$50$		0			0
$51$	−34.1924			−0.670438
$52$	18.1386			0.348819
$53$		−	65.6199i		−	1.23811i	−0.785347	−	0.619055i	$-0.787516\pi$
$53$		−	65.6199i		−	1.23811i	0.785347	−	0.619055i	$-0.212484\pi$
$54$		−	7.34847i		−	0.136083i
$55$		0			0
$56$	18.9545	−	5.72091i	0.338472	−	0.102159i
$57$			44.4900i			0.780526i
$58$		−	31.5469i		−	0.543913i
$59$			32.5746i			0.552112i	0.961142	+	0.276056i	$0.0890275\pi$
$59$			32.5746i			0.552112i	−0.961142	+	0.276056i	$0.910973\pi$
$60$		0			0
$61$			83.5349i			1.36943i	0.728813	+	0.684713i	$0.240072\pi$
$61$			83.5349i			1.36943i	−0.728813	+	0.684713i	$0.759928\pi$
$62$	−22.4881			−0.362712
$63$	−6.06794	−	20.1042i	−0.0963165	−	0.319115i
$64$	−8.00000			−0.125000
$65$		0			0
$66$			12.2672i			0.185866i
$67$			72.0585i			1.07550i	0.843104	+	0.537750i	$0.180726\pi$
$67$			72.0585i			1.07550i	−0.843104	+	0.537750i	$0.819274\pi$
$68$	39.4819			0.580617
$69$			70.9597i			1.02840i
$70$		0			0
$71$	−24.8675			−0.350246			−0.175123	−	0.984547i	$-0.556032\pi$
$71$	−24.8675			−0.350246			−0.175123	+	0.984547i	$0.556032\pi$
$72$			8.48528i			0.117851i
$73$	67.8725			0.929760			0.464880	−	0.885374i	$-0.346098\pi$
$73$	67.8725			0.929760			0.464880	+	0.885374i	$0.346098\pi$
$74$	−63.2870			−0.855229
$75$		0			0
$76$		−	51.3726i		−	0.675956i
$77$	10.1295	+	33.5610i	0.131552	+	0.435858i
$78$			22.2151i			0.284809i
$79$	−30.4452			−0.385383			−0.192691	−	0.981259i	$-0.561722\pi$
$79$	−30.4452			−0.385383			−0.192691	+	0.981259i	$0.561722\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$	38.3009			0.467085
$83$	72.4623			0.873040			0.436520	−	0.899695i	$-0.356211\pi$
$83$	72.4623			0.873040			0.436520	+	0.899695i	$0.356211\pi$
$84$	7.00665	+	23.2144i	0.0834125	+	0.276362i
$85$		0			0
$86$	43.3725			0.504332
$87$	38.6370			0.444103
$88$		−	14.1649i		−	0.160965i
$89$			113.756i			1.27815i	0.769143	+	0.639076i	$0.220683\pi$
$89$			113.756i			1.27815i	−0.769143	+	0.639076i	$0.779317\pi$
$90$		0			0
$91$	18.3440	+	60.7770i	0.201582	+	0.667880i
$92$		−	81.9372i		−	0.890621i
$93$		−	27.5422i		−	0.296153i
$94$			82.2267i			0.874752i
$95$		0			0
$96$		−	9.79796i		−	0.102062i
$97$	−103.804			−1.07014			−0.535072	−	0.844806i	$-0.679716\pi$
$97$	−103.804			−1.07014			−0.535072	+	0.844806i	$0.679716\pi$
$98$	38.3381	+	57.7251i	0.391206	+	0.589032i
$99$	−15.0242			−0.151759
$100$		0			0
$101$			155.379i			1.53841i	0.639003	+	0.769204i	$0.279347\pi$
$101$			155.379i			1.53841i	−0.639003	+	0.769204i	$0.720653\pi$
$102$			48.3553i			0.474071i
$103$	−65.9471			−0.640264			−0.320132	−	0.947373i	$-0.603727\pi$
$103$	−65.9471			−0.640264			−0.320132	+	0.947373i	$0.603727\pi$
$104$		−	25.6518i		−	0.246652i
$105$		0			0
$106$	−92.8005			−0.875476
$107$		−	136.269i		−	1.27354i	−0.771053	−	0.636771i	$-0.780270\pi$
$107$		−	136.269i		−	1.27354i	0.771053	−	0.636771i	$-0.219730\pi$
$108$	−10.3923			−0.0962250
$109$	−166.003			−1.52296			−0.761482	−	0.648186i	$-0.775528\pi$
$109$	−166.003			−1.52296			−0.761482	+	0.648186i	$0.775528\pi$
$110$		0			0
$111$		−	77.5104i		−	0.698292i
$112$	−8.09058	−	26.8056i	−0.0722374	−	0.239336i
$113$			25.5037i			0.225696i	0.993612	+	0.112848i	$0.0359974\pi$
$113$			25.5037i			0.225696i	−0.993612	+	0.112848i	$0.964003\pi$
$114$	62.9184			0.551915
$115$		0			0
$116$	−44.6141			−0.384604
$117$	−27.2079			−0.232546
$118$	46.0674			0.390402
$119$	39.9290	+	132.292i	0.335538	+	1.11170i
$120$		0			0
$121$	−95.9194			−0.792722
$122$	118.136			0.968330
$123$			46.9089i			0.381373i
$124$			31.8030i			0.256476i
$125$		0			0
$126$	−28.4317	+	8.58136i	−0.225648	+	0.0681060i
$127$		−	230.306i		−	1.81344i	−0.421738	−	0.906718i	$-0.638580\pi$
$127$		−	230.306i		−	1.81344i	0.421738	−	0.906718i	$-0.361420\pi$
$128$			11.3137i			0.0883883i
$129$			53.1203i			0.411785i
$130$		0			0
$131$			40.9127i			0.312311i	0.987733	+	0.156155i	$0.0499101\pi$
$131$			40.9127i			0.312311i	−0.987733	+	0.156155i	$0.950090\pi$
$132$	17.3484			0.131427
$133$	172.135	−	51.9543i	1.29424	−	0.390634i
$134$	101.906			0.760493
$135$		0			0
$136$		−	55.8359i		−	0.410558i
$137$			158.300i			1.15547i	0.816223	+	0.577737i	$0.196064\pi$
$137$			158.300i			1.15547i	−0.816223	+	0.577737i	$0.803936\pi$
$138$	100.352			0.727189
$139$		−	141.584i		−	1.01859i	−0.860592	−	0.509296i	$-0.829906\pi$
$139$		−	141.584i		−	1.01859i	0.860592	−	0.509296i	$-0.170094\pi$
$140$		0			0
$141$	−100.707			−0.714232
$142$			35.1679i			0.247661i
$143$	45.4195			0.317619
$144$	12.0000			0.0833333
$145$		0			0
$146$		−	95.9861i		−	0.657439i
$147$	−70.6985	+	46.9545i	−0.480942	+	0.319418i
$148$			89.5013i			0.604738i
$149$	121.456			0.815139			0.407569	−	0.913174i	$-0.366376\pi$
$149$	121.456			0.815139			0.407569	+	0.913174i	$0.366376\pi$
$150$		0			0
$151$	−219.093			−1.45094			−0.725472	−	0.688251i	$-0.758379\pi$
$151$	−219.093			−1.45094			−0.725472	+	0.688251i	$0.758379\pi$
$152$	−72.6519			−0.477973
$153$	−59.2229			−0.387078
$154$	47.4625	−	14.3253i	0.308198	−	0.0930215i
$155$		0			0
$156$	31.4169			0.201391
$157$	57.2815			0.364850			0.182425	−	0.983220i	$-0.441605\pi$
$157$	57.2815			0.364850			0.182425	+	0.983220i	$0.441605\pi$
$158$			43.0560i			0.272507i
$159$		−	113.657i		−	0.714824i
$160$		0			0
$161$	274.547	−	82.8649i	1.70526	−	0.514689i
$162$		−	12.7279i		−	0.0785674i
$163$			8.52297i			0.0522882i	0.999658	+	0.0261441i	$0.00832287\pi$
$163$			8.52297i			0.0522882i	−0.999658	+	0.0261441i	$0.991677\pi$
$164$		−	54.1657i		−	0.330279i
$165$		0			0
$166$		−	102.477i		−	0.617333i
$167$	258.148			1.54580			0.772899	−	0.634529i	$-0.218806\pi$
$167$	258.148			1.54580			0.772899	+	0.634529i	$0.218806\pi$
$168$	32.8301	−	9.90890i	0.195417	−	0.0589816i
$169$	−86.7480			−0.513301
$170$		0			0
$171$			77.0589i			0.450637i
$172$		−	61.3380i		−	0.356616i
$173$	−327.250			−1.89162			−0.945808	−	0.324725i	$-0.894728\pi$
$173$	−327.250			−1.89162			−0.945808	+	0.324725i	$0.894728\pi$
$174$		−	54.6409i		−	0.314028i
$175$		0			0
$176$	−20.0322			−0.113819
$177$			56.4208i			0.318762i
$178$	160.875			0.903791
$179$	167.569			0.936141			0.468071	−	0.883691i	$-0.344949\pi$
$179$	167.569			0.936141			0.468071	+	0.883691i	$0.344949\pi$
$180$		0			0
$181$			147.957i			0.817442i	0.912659	+	0.408721i	$0.134025\pi$
$181$			147.957i			0.817442i	−0.912659	+	0.408721i	$0.865975\pi$
$182$	85.9517	−	25.9423i	0.472262	−	0.142540i
$183$			144.687i			0.790638i
$184$	−115.877			−0.629764
$185$		0			0
$186$	−38.9506			−0.209412
$187$	98.8638			0.528684
$188$	116.286			0.618543
$189$	−10.5100	−	34.8216i	−0.0556083	−	0.184241i
$190$		0			0
$191$	49.4939			0.259130			0.129565	−	0.991571i	$-0.458642\pi$
$191$	49.4939			0.259130			0.129565	+	0.991571i	$0.458642\pi$
$192$	−13.8564			−0.0721688
$193$		−	172.674i		−	0.894686i	−0.894362	−	0.447343i	$-0.852370\pi$
$193$		−	172.674i		−	0.894686i	0.894362	−	0.447343i	$-0.147630\pi$
$194$			146.801i			0.756707i
$195$		0			0
$196$	81.6356	−	54.2183i	0.416508	−	0.276624i
$197$			75.6444i			0.383982i	0.981397	+	0.191991i	$0.0614944\pi$
$197$			75.6444i			0.383982i	−0.981397	+	0.191991i	$0.938506\pi$
$198$			21.2474i			0.107310i
$199$		−	147.620i		−	0.741808i	−0.928671	−	0.370904i	$-0.879048\pi$
$199$		−	147.620i		−	0.741808i	0.928671	−	0.370904i	$-0.120952\pi$
$200$		0			0
$201$			124.809i			0.620940i
$202$	219.739			1.08782
$203$	−45.1193	−	149.489i	−0.222262	−	0.736398i
$204$	68.3847			0.335219
$205$		0			0
$206$			93.2634i			0.452735i
$207$			122.906i			0.593747i
$208$	−36.2772			−0.174409
$209$		−	128.638i		−	0.615495i
$210$		0			0
$211$	156.882			0.743518			0.371759	−	0.928329i	$-0.378755\pi$
$211$	156.882			0.743518			0.371759	+	0.928329i	$0.378755\pi$
$212$			131.240i			0.619055i
$213$	−43.0717			−0.202215
$214$	−192.713			−0.900530
$215$		0			0
$216$			14.6969i			0.0680414i
$217$	−106.563	+	32.1631i	−0.491072	+	0.148217i
$218$			234.764i			1.07690i
$219$	117.559			0.536797
$220$		0			0
$221$	179.037			0.810120
$222$	−109.616			−0.493767
$223$	−60.4029			−0.270865			−0.135433	−	0.990787i	$-0.543242\pi$
$223$	−60.4029			−0.270865			−0.135433	+	0.990787i	$0.543242\pi$
$224$	−37.9089	+	11.4418i	−0.169236	+	0.0510795i
$225$		0			0
$226$	36.0677			0.159592
$227$	−313.611			−1.38155			−0.690773	−	0.723071i	$-0.742730\pi$
$227$	−313.611			−1.38155			−0.690773	+	0.723071i	$0.742730\pi$
$228$		−	88.9800i		−	0.390263i
$229$		−	110.368i		−	0.481957i	−0.970530	−	0.240979i	$-0.922532\pi$
$229$		−	110.368i		−	0.481957i	0.970530	−	0.240979i	$-0.0774684\pi$
$230$		0			0
$231$	17.5449	+	58.1294i	0.0759517	+	0.251643i
$232$			63.0939i			0.271956i
$233$		−	38.4067i		−	0.164835i	−0.996598	−	0.0824177i	$-0.973736\pi$
$233$		−	38.4067i		−	0.164835i	0.996598	−	0.0824177i	$-0.0262642\pi$
$234$			38.4777i			0.164435i
$235$		0			0
$236$		−	65.1492i		−	0.276056i
$237$	−52.7327			−0.222501
$238$	187.090	−	56.4681i	0.786091	−	0.237261i
$239$	125.129			0.523552			0.261776	−	0.965129i	$-0.415692\pi$
$239$	125.129			0.523552			0.261776	+	0.965129i	$0.415692\pi$
$240$		0			0
$241$		−	154.625i		−	0.641598i	−0.947147	−	0.320799i	$-0.896049\pi$
$241$		−	154.625i		−	0.641598i	0.947147	−	0.320799i	$-0.103951\pi$
$242$			135.650i			0.560539i
$243$	15.5885			0.0641500
$244$		−	167.070i		−	0.684713i
$245$		0			0
$246$	66.3392			0.269671
$247$		−	232.957i		−	0.943144i
$248$	44.9763			0.181356
$249$	125.508			0.504050
$250$		0			0
$251$		−	152.655i		−	0.608185i	−0.952642	−	0.304093i	$-0.901647\pi$
$251$		−	152.655i		−	0.608185i	0.952642	−	0.304093i	$-0.0983532\pi$
$252$	12.1359	+	40.2085i	0.0481582	+	0.159557i
$253$		−	205.173i		−	0.810960i
$254$	−325.702			−1.28229
$255$		0			0
$256$	16.0000			0.0625000
$257$	20.1517			0.0784113			0.0392057	−	0.999231i	$-0.487517\pi$
$257$	20.1517			0.0784113			0.0392057	+	0.999231i	$0.487517\pi$
$258$	75.1234			0.291176
$259$	−299.892	+	90.5147i	−1.15789	+	0.349478i
$260$		0			0
$261$	66.9212			0.256403
$262$	57.8593			0.220837
$263$			320.699i			1.21939i	0.792637	+	0.609693i	$0.208707\pi$
$263$			320.699i			1.21939i	−0.792637	+	0.609693i	$0.791293\pi$
$264$		−	24.5344i		−	0.0929332i
$265$		0			0
$266$	−73.4745	−	243.435i	−0.276220	−	0.915169i
$267$			197.030i			0.737942i
$268$		−	144.117i		−	0.537750i
$269$		−	39.2134i		−	0.145775i	−0.997340	−	0.0728874i	$-0.976779\pi$
$269$		−	39.2134i		−	0.145775i	0.997340	−	0.0728874i	$-0.0232214\pi$
$270$		0			0
$271$			299.071i			1.10358i	0.833982	+	0.551792i	$0.186056\pi$
$271$			299.071i			1.10358i	−0.833982	+	0.551792i	$0.813944\pi$
$272$	−78.9639			−0.290308
$273$	31.7727	+	105.269i	0.116383	+	0.385600i
$274$	223.870			0.817044
$275$		0			0
$276$		−	141.919i		−	0.514200i
$277$		−	438.966i		−	1.58472i	−0.610056	−	0.792358i	$-0.708853\pi$
$277$		−	438.966i		−	1.58472i	0.610056	−	0.792358i	$-0.291147\pi$
$278$	−200.230			−0.720253
$279$		−	47.7045i		−	0.170984i
$280$		0			0
$281$	−199.479			−0.709889			−0.354944	−	0.934887i	$-0.615500\pi$
$281$	−199.479			−0.709889			−0.354944	+	0.934887i	$0.615500\pi$
$282$			142.421i			0.505038i
$283$	−541.179			−1.91229			−0.956147	−	0.292889i	$-0.905384\pi$
$283$	−541.179			−1.91229			−0.956147	+	0.292889i	$0.905384\pi$
$284$	49.7349			0.175123
$285$		0			0
$286$		−	64.2329i		−	0.224591i
$287$	181.493	−	54.7790i	0.632381	−	0.190868i
$288$		−	16.9706i		−	0.0589256i
$289$	100.706			0.348462
$290$		0			0
$291$	−179.794			−0.617848
$292$	−135.745			−0.464880
$293$	−378.757			−1.29269			−0.646343	−	0.763047i	$-0.723702\pi$
$293$	−378.757			−1.29269			−0.646343	+	0.763047i	$0.723702\pi$
$294$	66.4036	+	99.9828i	0.225863	+	0.340078i
$295$		0			0
$296$	126.574			0.427615
$297$	−26.0226			−0.0876183
$298$		−	171.764i		−	0.576390i
$299$		−	371.556i		−	1.24266i
$300$		0			0
$301$	205.526	−	62.0326i	0.682810	−	0.206088i
$302$			309.844i			1.02597i
$303$			269.125i			0.888200i
$304$			102.745i			0.337978i
$305$		0			0
$306$			83.7538i			0.273705i
$307$	140.210			0.456710			0.228355	−	0.973578i	$-0.426665\pi$
$307$	140.210			0.456710			0.228355	+	0.973578i	$0.426665\pi$
$308$	−20.2591	−	67.1221i	−0.0657761	−	0.217929i
$309$	−114.224			−0.369656
$310$		0			0
$311$			249.886i			0.803491i	0.915751	+	0.401745i	$0.131596\pi$
$311$			249.886i			0.803491i	−0.915751	+	0.401745i	$0.868404\pi$
$312$		−	44.4303i		−	0.142405i
$313$	−517.894			−1.65461			−0.827307	−	0.561750i	$-0.810128\pi$
$313$	−517.894			−1.65461			−0.827307	+	0.561750i	$0.810128\pi$
$314$		−	81.0082i		−	0.257988i
$315$		0			0
$316$	60.8904			0.192691
$317$		−	180.043i		−	0.567959i	−0.958830	−	0.283980i	$-0.908345\pi$
$317$		−	180.043i		−	0.567959i	0.958830	−	0.283980i	$-0.0916548\pi$
$318$	−160.735			−0.505457
$319$	−111.715			−0.350204
$320$		0			0
$321$		−	236.025i		−	0.735280i
$322$	−117.189	−	388.268i	−0.363940	−	1.20580i
$323$		−	507.073i		−	1.56988i
$324$	−18.0000			−0.0555556
$325$		0			0
$326$	12.0533			0.0369733
$327$	−287.526			−0.879283
$328$	−76.6019			−0.233542
$329$	117.603	+	389.640i	0.357455	+	1.18432i
$330$		0			0
$331$	81.0283			0.244799			0.122399	−	0.992481i	$-0.460941\pi$
$331$	81.0283			0.244799			0.122399	+	0.992481i	$0.460941\pi$
$332$	−144.925			−0.436520
$333$		−	134.252i		−	0.403159i
$334$		−	365.077i		−	1.09304i
$335$		0			0
$336$	−14.0133	−	46.4287i	−0.0417063	−	0.138181i
$337$		−	372.937i		−	1.10664i	−0.832970	−	0.553318i	$-0.813361\pi$
$337$		−	372.937i		−	1.10664i	0.832970	−	0.553318i	$-0.186639\pi$
$338$			122.680i			0.362959i
$339$			44.1737i			0.130306i
$340$		0			0
$341$			79.6356i			0.233536i
$342$	108.978			0.318649
$343$	264.230	+	218.705i	0.770349	+	0.637623i
$344$	−86.7451			−0.252166
$345$		0			0
$346$			462.801i			1.33758i
$347$			415.231i			1.19663i	0.801260	+	0.598316i	$0.204163\pi$
$347$			415.231i			1.19663i	−0.801260	+	0.598316i	$0.795837\pi$
$348$	−77.2739			−0.222051
$349$		−	536.207i		−	1.53641i	−0.640203	−	0.768206i	$-0.721150\pi$
$349$		−	536.207i		−	1.53641i	0.640203	−	0.768206i	$-0.278850\pi$
$350$		0			0
$351$	−47.1254			−0.134260
$352$			28.3298i			0.0804825i
$353$	416.230			1.17912			0.589560	−	0.807724i	$-0.299301\pi$
$353$	416.230			1.17912			0.589560	+	0.807724i	$0.299301\pi$
$354$	79.7911			0.225399
$355$		0			0
$356$		−	227.511i		−	0.639076i
$357$	69.1590	+	229.137i	0.193723	+	0.641840i
$358$		−	236.979i		−	0.661952i
$359$	270.563			0.753659			0.376829	−	0.926283i	$-0.377014\pi$
$359$	270.563			0.753659			0.376829	+	0.926283i	$0.377014\pi$
$360$		0			0
$361$	−298.787			−0.827664
$362$	209.243			0.578019
$363$	−166.137			−0.457678
$364$	−36.6879	−	121.554i	−0.100791	−	0.333940i
$365$		0			0
$366$	204.618			0.559066
$367$	605.024			1.64857			0.824283	−	0.566178i	$-0.191578\pi$
$367$	605.024			1.64857			0.824283	+	0.566178i	$0.191578\pi$
$368$			163.874i			0.445311i
$369$			81.2486i			0.220186i
$370$		0			0
$371$	−439.746	+	132.726i	−1.18530	+	0.357751i
$372$			55.0844i			0.148076i
$373$			21.5124i			0.0576739i	0.999584	+	0.0288370i	$0.00918037\pi$
$373$			21.5124i			0.0576739i	−0.999584	+	0.0288370i	$0.990820\pi$
$374$		−	139.815i		−	0.373836i
$375$		0			0
$376$		−	164.453i		−	0.437376i
$377$	−202.309			−0.536629
$378$	−49.2451	+	14.8634i	−0.130278	+	0.0393210i
$379$	−449.862			−1.18697			−0.593486	−	0.804844i	$-0.702249\pi$
$379$	−449.862			−1.18697			−0.593486	+	0.804844i	$0.702249\pi$
$380$		0			0
$381$		−	398.902i		−	1.04699i
$382$		−	69.9949i		−	0.183233i
$383$	−560.756			−1.46412			−0.732058	−	0.681242i	$-0.761440\pi$
$383$	−560.756			−1.46412			−0.732058	+	0.681242i	$0.761440\pi$
$384$			19.5959i			0.0510310i
$385$		0			0
$386$	−244.199			−0.632639
$387$			92.0071i			0.237744i
$388$	207.608			0.535072
$389$	582.222			1.49671			0.748357	−	0.663296i	$-0.230843\pi$
$389$	582.222			1.49671			0.748357	+	0.663296i	$0.230843\pi$
$390$		0			0
$391$		−	808.759i		−	2.06844i
$392$	−76.6763	−	115.450i	−0.195603	−	0.294516i
$393$			70.8629i			0.180313i
$394$	106.977			0.271516
$395$		0			0
$396$	30.0483			0.0758796
$397$	−726.563			−1.83013			−0.915066	−	0.403304i	$-0.867862\pi$
$397$	−726.563			−1.83013			−0.915066	+	0.403304i	$0.867862\pi$
$398$	−208.766			−0.524538
$399$	298.146	−	89.9875i	0.747232	−	0.225533i
$400$		0			0
$401$	−32.7962			−0.0817860			−0.0408930	−	0.999164i	$-0.513020\pi$
$401$	−32.7962			−0.0817860			−0.0408930	+	0.999164i	$0.513020\pi$
$402$	176.507			0.439071
$403$			144.215i			0.357855i
$404$		−	310.758i		−	0.769204i
$405$		0			0
$406$	−211.409	+	63.8083i	−0.520712	+	0.157163i
$407$			224.114i			0.550648i
$408$		−	96.7106i		−	0.237036i
$409$			637.739i			1.55926i	0.626238	+	0.779632i	$0.284594\pi$
$409$			637.739i			1.55926i	−0.626238	+	0.779632i	$0.715406\pi$
$410$		0			0
$411$			274.184i			0.667114i
$412$	131.894			0.320132
$413$	218.296	−	65.8869i	0.528561	−	0.159532i
$414$	173.815			0.419843
$415$		0			0
$416$			51.3037i			0.123326i
$417$		−	245.231i		−	0.588084i
$418$	−181.922			−0.435221
$419$		−	344.535i		−	0.822280i	−0.911572	−	0.411140i	$-0.865131\pi$
$419$		−	344.535i		−	0.822280i	0.911572	−	0.411140i	$-0.134869\pi$
$420$		0			0
$421$	290.412			0.689814			0.344907	−	0.938637i	$-0.387910\pi$
$421$	290.412			0.689814			0.344907	+	0.938637i	$0.387910\pi$
$422$		−	221.865i		−	0.525746i
$423$	−174.429			−0.412362
$424$	185.601			0.437738
$425$		0			0
$426$			60.9126i			0.142987i
$427$	559.802	−	168.962i	1.31101	−	0.395695i
$428$			272.538i			0.636771i
$429$	78.6689			0.183377
$430$		0			0
$431$	−214.780			−0.498329			−0.249164	−	0.968461i	$-0.580156\pi$
$431$	−214.780			−0.498329			−0.249164	+	0.968461i	$0.580156\pi$
$432$	20.7846			0.0481125
$433$	425.041			0.981619			0.490809	−	0.871267i	$-0.336701\pi$
$433$	425.041			0.981619			0.490809	+	0.871267i	$0.336701\pi$
$434$	45.4855	+	150.702i	0.104805	+	0.347240i
$435$		0			0
$436$	332.006			0.761482
$437$	−1052.33			−2.40808
$438$		−	166.253i		−	0.379573i
$439$		−	204.200i		−	0.465149i	−0.972579	−	0.232574i	$-0.925285\pi$
$439$		−	204.200i		−	0.465149i	0.972579	−	0.232574i	$-0.0747149\pi$
$440$		0			0
$441$	−122.453	+	81.3275i	−0.277672	+	0.184416i
$442$		−	253.196i		−	0.572841i
$443$		−	276.367i		−	0.623854i	−0.950106	−	0.311927i	$-0.899026\pi$
$443$		−	276.367i		−	0.623854i	0.950106	−	0.311927i	$-0.100974\pi$
$444$			155.021i			0.349146i
$445$		0			0
$446$			85.4227i			0.191531i
$447$	210.367			0.470620
$448$	16.1812	+	53.6113i	0.0361187	+	0.119668i
$449$	−195.290			−0.434945			−0.217472	−	0.976066i	$-0.569781\pi$
$449$	−195.290			−0.434945			−0.217472	+	0.976066i	$0.569781\pi$
$450$		0			0
$451$		−	135.632i		−	0.300737i
$452$		−	51.0074i		−	0.112848i
$453$	−379.480			−0.837703
$454$			443.513i			0.976901i
$455$		0			0
$456$	−125.837			−0.275958
$457$			10.3428i			0.0226319i	0.999936	+	0.0113159i	$0.00360206\pi$
$457$			10.3428i			0.0226319i	−0.999936	+	0.0113159i	$0.996398\pi$
$458$	−156.084			−0.340795
$459$	−102.577			−0.223479
$460$		0			0
$461$		−	527.572i		−	1.14441i	−0.820111	−	0.572204i	$-0.806088\pi$
$461$		−	527.572i		−	1.14441i	0.820111	−	0.572204i	$-0.193912\pi$
$462$	82.2074	−	24.8122i	0.177938	−	0.0537060i
$463$		−	797.897i		−	1.72332i	−0.507486	−	0.861660i	$-0.669425\pi$
$463$		−	797.897i		−	1.72332i	0.507486	−	0.861660i	$-0.330575\pi$
$464$	89.2282			0.192302
$465$		0			0
$466$	−54.3152			−0.116556
$467$	169.651			0.363278			0.181639	−	0.983365i	$-0.441860\pi$
$467$	169.651			0.363278			0.181639	+	0.983365i	$0.441860\pi$
$468$	54.4157			0.116273
$469$	482.894	−	145.749i	1.02962	−	0.310765i
$470$		0			0
$471$	99.2144			0.210646
$472$	−92.1348			−0.195201
$473$		−	153.592i		−	0.324719i
$474$			74.5753i			0.157332i
$475$		0			0
$476$	−79.8580	−	264.585i	−0.167769	−	0.555850i
$477$		−	196.860i		−	0.412704i
$478$		−	176.959i		−	0.370207i
$479$		−	704.272i		−	1.47030i	−0.677906	−	0.735148i	$-0.737113\pi$
$479$		−	704.272i		−	1.47030i	0.677906	−	0.735148i	$-0.262887\pi$
$480$		0			0
$481$			405.857i			0.843777i
$482$	−218.673			−0.453678
$483$	475.530	−	143.526i	0.984534	−	0.297156i
$484$	191.839			0.396361
$485$		0			0
$486$		−	22.0454i		−	0.0453609i
$487$		−	66.7966i		−	0.137159i	−0.997646	−	0.0685797i	$-0.978153\pi$
$487$		−	66.7966i		−	0.137159i	0.997646	−	0.0685797i	$-0.0218467\pi$
$488$	−236.273			−0.484165
$489$			14.7622i			0.0301886i
$490$		0			0
$491$	546.122			1.11227			0.556133	−	0.831094i	$-0.312285\pi$
$491$	546.122			1.11227			0.556133	+	0.831094i	$0.312285\pi$
$492$		−	93.8178i		−	0.190687i
$493$	−440.363			−0.893231
$494$	−329.450			−0.666904
$495$		0			0
$496$		−	63.6060i		−	0.128238i
$497$	50.2981	+	166.647i	0.101203	+	0.335306i
$498$		−	177.496i		−	0.356417i
$499$	−714.060			−1.43098			−0.715491	−	0.698622i	$-0.753797\pi$
$499$	−714.060			−1.43098			−0.715491	+	0.698622i	$0.753797\pi$
$500$		0			0
$501$	447.126			0.892467
$502$	−215.886			−0.430052
$503$	878.751			1.74702			0.873510	−	0.486807i	$-0.161838\pi$
$503$	878.751			1.74702			0.873510	+	0.486807i	$0.161838\pi$
$504$	56.8634	−	17.1627i	0.112824	−	0.0340530i
$505$		0			0
$506$	−290.158			−0.573435
$507$	−150.252			−0.296355
$508$			460.613i			0.906718i
$509$			510.558i			1.00306i	0.865140	+	0.501530i	$0.167229\pi$
$509$			510.558i			1.00306i	−0.865140	+	0.501530i	$0.832771\pi$
$510$		0			0
$511$	−137.282	−	454.841i	−0.268654	−	0.890100i
$512$		−	22.6274i		−	0.0441942i
$513$			133.470i			0.260175i
$514$		−	28.4988i		−	0.0554452i
$515$		0			0
$516$		−	106.241i		−	0.205893i
$517$	291.184			0.563218
$518$	128.007	+	424.112i	0.247118	+	0.818749i
$519$	−566.813			−1.09213
$520$		0			0
$521$			17.2889i			0.0331841i	0.999862	+	0.0165921i	$0.00528166\pi$
$521$			17.2889i			0.0331841i	−0.999862	+	0.0165921i	$0.994718\pi$
$522$		−	94.6408i		−	0.181304i
$523$	−287.840			−0.550364			−0.275182	−	0.961392i	$-0.588738\pi$
$523$	−287.840			−0.550364			−0.275182	+	0.961392i	$0.588738\pi$
$524$		−	81.8254i		−	0.156155i
$525$		0			0
$526$	453.536			0.862237
$527$			313.911i			0.595657i
$528$	−34.6968			−0.0657137
$529$	−1149.42			−2.17282
$530$		0			0
$531$			97.7238i			0.184037i
$532$	−344.269	+	103.909i	−0.647122	+	0.195317i
$533$		−	245.622i		−	0.460830i
$534$	278.643			0.521804
$535$		0			0
$536$	−203.812			−0.380247
$537$	290.239			0.540481
$538$	−55.4562			−0.103078
$539$	204.418	−	135.764i	0.379254	−	0.251882i
$540$		0			0
$541$	−228.274			−0.421949			−0.210974	−	0.977492i	$-0.567664\pi$
$541$	−228.274			−0.421949			−0.210974	+	0.977492i	$0.567664\pi$
$542$	422.950			0.780351
$543$			256.269i			0.471950i
$544$			111.672i			0.205279i
$545$		0			0
$546$	148.873	−	44.9334i	0.272661	−	0.0822955i
$547$			876.996i			1.60328i	0.597805	+	0.801642i	$0.296040\pi$
$547$			876.996i			1.60328i	−0.597805	+	0.801642i	$0.703960\pi$
$548$		−	316.600i		−	0.577737i
$549$			250.605i			0.456475i
$550$		0			0
$551$			572.986i			1.03990i
$552$	−200.704			−0.363595
$553$	61.5799	+	204.026i	0.111356	+	0.368944i
$554$	−620.792			−1.12056
$555$		0			0
$556$			283.168i			0.509296i
$557$			561.224i			1.00758i	0.863825	+	0.503792i	$0.168062\pi$
$557$			561.224i			1.00758i	−0.863825	+	0.503792i	$0.831938\pi$
$558$	−67.4644			−0.120904
$559$		−	278.146i		−	0.497578i
$560$		0			0
$561$	171.237			0.305236
$562$			282.105i			0.501967i
$563$	890.386			1.58150			0.790752	−	0.612137i	$-0.209690\pi$
$563$	890.386			1.58150			0.790752	+	0.612137i	$0.209690\pi$
$564$	201.413			0.357116
$565$		0			0
$566$			765.343i			1.35220i
$567$	−18.2038	−	60.3127i	−0.0321055	−	0.106372i
$568$		−	70.3358i		−	0.123831i
$569$	−138.453			−0.243327			−0.121663	−	0.992571i	$-0.538823\pi$
$569$	−138.453			−0.243327			−0.121663	+	0.992571i	$0.538823\pi$
$570$		0			0
$571$	476.343			0.834225			0.417113	−	0.908855i	$-0.363042\pi$
$571$	476.343			0.834225			0.417113	+	0.908855i	$0.363042\pi$
$572$	−90.8390			−0.158810
$573$	85.7259			0.149609
$574$	−77.4693	−	256.670i	−0.134964	−	0.447161i
$575$		0			0
$576$	−24.0000			−0.0416667
$577$	−678.464			−1.17585			−0.587924	−	0.808916i	$-0.700055\pi$
$577$	−678.464			−1.17585			−0.587924	+	0.808916i	$0.700055\pi$
$578$		−	142.419i		−	0.246400i
$579$		−	299.081i		−	0.516547i
$580$		0			0
$581$	−146.566	−	485.600i	−0.252264	−	0.835800i
$582$			254.267i			0.436885i
$583$			328.628i			0.563684i
$584$			191.972i			0.328720i
$585$		0			0
$586$			535.643i			0.914067i
$587$	−48.9046			−0.0833128			−0.0416564	−	0.999132i	$-0.513263\pi$
$587$	−48.9046			−0.0833128			−0.0416564	+	0.999132i	$0.513263\pi$
$588$	141.397	−	93.9089i	0.240471	−	0.159709i
$589$	408.451			0.693465
$590$		0			0
$591$			131.020i			0.221692i
$592$		−	179.003i		−	0.302369i
$593$	−487.761			−0.822531			−0.411266	−	0.911516i	$-0.634913\pi$
$593$	−487.761			−0.822531			−0.411266	+	0.911516i	$0.634913\pi$
$594$			36.8015i			0.0619555i
$595$		0			0
$596$	−242.911			−0.407569
$597$		−	255.685i		−	0.428283i
$598$	−525.459			−0.878695
$599$	1083.77			1.80931			0.904653	−	0.426149i	$-0.140130\pi$
$599$	1083.77			1.80931			0.904653	+	0.426149i	$0.140130\pi$
$600$		0			0
$601$		−	274.481i		−	0.456707i	−0.973578	−	0.228353i	$-0.926666\pi$
$601$		−	274.481i		−	0.456707i	0.973578	−	0.228353i	$-0.0733342\pi$
$602$	−87.7273	−	290.657i	−0.145726	−	0.482819i
$603$			216.175i			0.358500i
$604$	438.185			0.725472
$605$		0			0
$606$	380.600			0.628052
$607$	941.550			1.55115			0.775577	−	0.631253i	$-0.217459\pi$
$607$	941.550			1.55115			0.775577	+	0.631253i	$0.217459\pi$
$608$	145.304			0.238986
$609$	−78.1489	−	258.922i	−0.128323	−	0.425159i
$610$		0			0
$611$	527.316			0.863038
$612$	118.446			0.193539
$613$			658.706i			1.07456i	0.843404	+	0.537281i	$0.180548\pi$
$613$			658.706i			1.07456i	−0.843404	+	0.537281i	$0.819452\pi$
$614$		−	198.287i		−	0.322942i
$615$		0			0
$616$	−94.9249	+	28.6506i	−0.154099	+	0.0465108i
$617$			355.965i			0.576929i	0.957491	+	0.288464i	$0.0931447\pi$
$617$			355.965i			0.576929i	−0.957491	+	0.288464i	$0.906855\pi$
$618$			161.537i			0.261387i
$619$			214.933i			0.347226i	0.984814	+	0.173613i	$0.0555442\pi$
$619$			214.933i			0.347226i	−0.984814	+	0.173613i	$0.944456\pi$
$620$		0			0
$621$			212.879i			0.342800i
$622$	353.392			0.568154
$623$	762.323	−	230.087i	1.22363	−	0.369322i
$624$	−62.8339			−0.100695
$625$		0			0
$626$			732.413i			1.16999i
$627$		−	222.808i		−	0.355356i
$628$	−114.563			−0.182425
$629$			883.421i			1.40448i
$630$		0			0
$631$	589.714			0.934570			0.467285	−	0.884107i	$-0.345232\pi$
$631$	589.714			0.934570			0.467285	+	0.884107i	$0.345232\pi$
$632$		−	86.1121i		−	0.136253i
$633$	271.728			0.429270
$634$	−254.619			−0.401608
$635$		0			0
$636$			227.314i			0.357412i
$637$	370.189	−	245.861i	0.581144	−	0.385967i
$638$			157.989i			0.247631i
$639$	−74.6024			−0.116749
$640$		0			0
$641$	−675.508			−1.05383			−0.526917	−	0.849917i	$-0.676652\pi$
$641$	−675.508			−1.05383			−0.526917	+	0.849917i	$0.676652\pi$
$642$	−333.789			−0.519921
$643$	733.747			1.14113			0.570565	−	0.821252i	$-0.306724\pi$
$643$	733.747			1.14113			0.570565	+	0.821252i	$0.306724\pi$
$644$	−549.095	+	165.730i	−0.852631	+	0.257345i
$645$		0			0
$646$	−717.109			−1.11008
$647$	645.060			0.997002			0.498501	−	0.866889i	$-0.333884\pi$
$647$	645.060			0.997002			0.498501	+	0.866889i	$0.333884\pi$
$648$			25.4558i			0.0392837i
$649$		−	163.135i		−	0.251364i
$650$		0			0
$651$	−184.572	+	55.7082i	−0.283520	+	0.0855732i
$652$		−	17.0459i		−	0.0261441i
$653$			610.744i			0.935289i	0.883917	+	0.467644i	$0.154897\pi$
$653$			610.744i			0.935289i	−0.883917	+	0.467644i	$0.845103\pi$
$654$			406.623i			0.621747i
$655$		0			0
$656$			108.331i			0.165139i
$657$	203.617			0.309920
$658$	551.035	−	166.315i	0.837439	−	0.252759i
$659$	−220.888			−0.335187			−0.167594	−	0.985856i	$-0.553600\pi$
$659$	−220.888			−0.335187			−0.167594	+	0.985856i	$0.553600\pi$
$660$		0			0
$661$		−	306.043i		−	0.463000i	−0.972835	−	0.231500i	$-0.925637\pi$
$661$		−	306.043i		−	0.463000i	0.972835	−	0.231500i	$-0.0743634\pi$
$662$		−	114.591i		−	0.173099i
$663$	310.100			0.467723
$664$			204.954i			0.308666i
$665$		0			0
$666$	−189.861			−0.285076
$667$			913.888i			1.37015i
$668$	−516.297			−0.772899
$669$	−104.621			−0.156384
$670$		0			0
$671$		−	418.348i		−	0.623469i
$672$	−65.6601	+	19.8178i	−0.0977085	+	0.0294908i
$673$			1025.63i			1.52397i	0.647594	+	0.761986i	$0.275775\pi$
$673$			1025.63i			1.52397i	−0.647594	+	0.761986i	$0.724225\pi$
$674$	−527.412			−0.782510
$675$		0			0
$676$	173.496			0.256651
$677$	622.692			0.919781			0.459891	−	0.887976i	$-0.347889\pi$
$677$	622.692			0.919781			0.459891	+	0.887976i	$0.347889\pi$
$678$	62.4711			0.0921402
$679$	209.959	+	695.634i	0.309218	+	1.02450i
$680$		0			0
$681$	−543.190			−0.797636
$682$	112.622			0.165135
$683$		−	170.676i		−	0.249891i	−0.992164	−	0.124946i	$-0.960124\pi$
$683$		−	170.676i		−	0.249891i	0.992164	−	0.124946i	$-0.0398757\pi$
$684$		−	154.118i		−	0.225319i
$685$		0			0
$686$	309.295	−	373.677i	0.450867	−	0.544719i
$687$		−	191.163i		−	0.278258i
$688$			122.676i			0.178308i
$689$			595.126i			0.863753i
$690$		0			0
$691$			1353.09i			1.95816i	0.203481	+	0.979079i	$0.434774\pi$
$691$			1353.09i			1.95816i	−0.203481	+	0.979079i	$0.565226\pi$
$692$	654.499			0.945808
$693$	30.3886	+	100.683i	0.0438508	+	0.145286i
$694$	587.226			0.846147
$695$		0			0
$696$			109.282i			0.157014i
$697$		−	534.642i		−	0.767061i
$698$	−758.312			−1.08641
$699$		−	66.5223i		−	0.0951678i
$700$		0			0
$701$	−631.426			−0.900750			−0.450375	−	0.892840i	$-0.648710\pi$
$701$	−631.426			−0.900750			−0.450375	+	0.892840i	$0.648710\pi$
$702$			66.6454i			0.0949365i
$703$	1149.48			1.63510
$704$	40.0644			0.0569097
$705$		0			0
$706$		−	588.638i		−	0.833764i
$707$	1041.26	−	314.277i	1.47279	−	0.444522i
$708$		−	112.842i		−	0.159381i
$709$	294.907			0.415947			0.207974	−	0.978134i	$-0.433313\pi$
$709$	294.907			0.415947			0.207974	+	0.978134i	$0.433313\pi$
$710$		0			0
$711$	−91.3357			−0.128461
$712$	−321.749			−0.451895
$713$	651.462			0.913692
$714$	324.049	−	97.8056i	0.453850	−	0.136983i
$715$		0			0
$716$	−335.139			−0.468071
$717$	216.730			0.302273
$718$		−	382.634i		−	0.532917i
$719$			884.788i			1.23058i	0.788301	+	0.615290i	$0.210961\pi$
$719$			884.788i			1.23058i	−0.788301	+	0.615290i	$0.789039\pi$
$720$		0			0
$721$	133.388	+	441.939i	0.185004	+	0.612953i
$722$			422.548i			0.585247i
$723$		−	267.818i		−	0.370427i
$724$		−	295.914i		−	0.408721i
$725$		0			0
$726$			234.954i			0.323627i
$727$	−361.739			−0.497578			−0.248789	−	0.968558i	$-0.580033\pi$
$727$	−361.739			−0.497578			−0.248789	+	0.968558i	$0.580033\pi$
$728$	−171.903	+	51.8846i	−0.236131	+	0.0712700i
$729$	27.0000			0.0370370
$730$		0			0
$731$		−	605.436i		−	0.828230i
$732$		−	289.374i		−	0.395319i
$733$	819.923			1.11859			0.559293	−	0.828970i	$-0.311073\pi$
$733$	819.923			1.11859			0.559293	+	0.828970i	$0.311073\pi$
$734$		−	855.633i		−	1.16571i
$735$		0			0
$736$	231.753			0.314882
$737$		−	360.873i		−	0.489651i
$738$	114.903			0.155695
$739$	802.663			1.08615			0.543074	−	0.839685i	$-0.317260\pi$
$739$	802.663			1.08615			0.543074	+	0.839685i	$0.317260\pi$
$740$		0			0
$741$		−	403.493i		−	0.544525i
$742$	187.703	+	621.894i	0.252968	+	0.838132i
$743$			1061.48i			1.42865i	0.699816	+	0.714323i	$0.253265\pi$
$743$			1061.48i			1.42865i	−0.699816	+	0.714323i	$0.746735\pi$
$744$	77.9012			0.104706
$745$		0			0
$746$	30.4231			0.0407816
$747$	217.387			0.291013
$748$	−197.728			−0.264342
$749$	−913.194	+	275.624i	−1.21922	+	0.367989i
$750$		0			0
$751$	−1044.41			−1.39069			−0.695346	−	0.718675i	$-0.744749\pi$
$751$	−1044.41			−1.39069			−0.695346	+	0.718675i	$0.744749\pi$
$752$	−232.572			−0.309272
$753$		−	264.405i		−	0.351136i
$754$			286.108i			0.379454i
$755$		0			0
$756$	21.0200	+	69.6431i	0.0278042	+	0.0921205i
$757$		−	1267.83i		−	1.67481i	−0.546581	−	0.837406i	$-0.684071\pi$
$757$		−	1267.83i		−	1.67481i	0.546581	−	0.837406i	$-0.315929\pi$
$758$			636.202i			0.839316i
$759$		−	355.370i		−	0.468208i
$760$		0			0
$761$			257.020i			0.337740i	0.985638	+	0.168870i	$0.0540118\pi$
$761$			257.020i			0.337740i	−0.985638	+	0.168870i	$0.945988\pi$
$762$	−564.133			−0.740332
$763$	335.765	+	1112.45i	0.440060	+	1.45800i
$764$	−98.9878			−0.129565
$765$		0			0
$766$			793.029i			1.03529i
$767$		−	295.428i		−	0.385174i
$768$	27.7128			0.0360844
$769$			1192.50i			1.55071i	0.631524	+	0.775357i	$0.282430\pi$
$769$			1192.50i			1.55071i	−0.631524	+	0.775357i	$0.717570\pi$
$770$		0			0
$771$	34.9038			0.0452708
$772$			345.349i			0.447343i
$773$	−1121.01			−1.45021			−0.725104	−	0.688639i	$-0.758208\pi$
$773$	−1121.01			−1.45021			−0.725104	+	0.688639i	$0.758208\pi$
$774$	130.118			0.168111
$775$		0			0
$776$		−	293.602i		−	0.378353i
$777$	−519.429	+	156.776i	−0.668506	+	0.201771i
$778$		−	823.386i		−	1.05834i
$779$	−695.659			−0.893015
$780$		0			0
$781$	124.538			0.159459
$782$	−1143.76			−1.46261
$783$	115.911			0.148034
$784$	−163.271	+	108.437i	−0.208254	+	0.138312i
$785$		0			0
$786$	100.215			0.127500
$787$	−1207.04			−1.53372			−0.766861	−	0.641813i	$-0.778183\pi$
$787$	−1207.04			−1.53372			−0.766861	+	0.641813i	$0.778183\pi$
$788$		−	151.289i		−	0.191991i
$789$			555.466i			0.704013i
$790$		0			0
$791$	170.911	−	51.5850i	0.216069	−	0.0652149i
$792$		−	42.4948i		−	0.0536550i
$793$		−	757.603i		−	0.955363i
$794$			1027.51i			1.29410i
$795$		0			0
$796$			295.240i			0.370904i
$797$	−378.901			−0.475409			−0.237704	−	0.971338i	$-0.576395\pi$
$797$	−378.901			−0.475409			−0.237704	+	0.971338i	$0.576395\pi$
$798$	−127.262	−	421.642i	−0.159476	−	0.528373i
$799$	1147.80			1.43655
$800$		0			0
$801$			341.267i			0.426051i
$802$			46.3808i			0.0578314i
$803$	−339.909			−0.423299
$804$		−	249.618i		−	0.310470i
$805$		0			0
$806$	203.951			0.253041
$807$		−	67.9196i		−	0.0841631i
$808$	−439.479			−0.543909
$809$	403.691			0.499000			0.249500	−	0.968375i	$-0.419734\pi$
$809$	403.691			0.499000			0.249500	+	0.968375i	$0.419734\pi$
$810$		0			0
$811$		−	109.713i		−	0.135281i	−0.997710	−	0.0676407i	$-0.978453\pi$
$811$		−	109.713i		−	0.135281i	0.997710	−	0.0676407i	$-0.0215471\pi$
$812$	90.2386	+	298.977i	0.111131	+	0.368199i
$813$			518.006i			0.637154i
$814$	316.945			0.389367
$815$		0			0
$816$	−136.769			−0.167610
$817$	−787.774			−0.964228
$818$	901.899			1.10257
$819$	55.0319	+	182.331i	0.0671940	+	0.222627i
$820$		0			0
$821$	426.293			0.519236			0.259618	−	0.965711i	$-0.416403\pi$
$821$	426.293			0.519236			0.259618	+	0.965711i	$0.416403\pi$
$822$	387.754			0.471721
$823$			708.932i			0.861400i	0.902495	+	0.430700i	$0.141733\pi$
$823$			708.932i			0.861400i	−0.902495	+	0.430700i	$0.858267\pi$
$824$		−	186.527i		−	0.226367i
$825$		0			0
$826$	−93.1781	−	308.717i	−0.112806	−	0.373749i
$827$		−	813.963i		−	0.984236i	−0.870528	−	0.492118i	$-0.836223\pi$
$827$		−	813.963i		−	0.984236i	0.870528	−	0.492118i	$-0.163777\pi$
$828$		−	245.811i		−	0.296874i
$829$			92.7107i			0.111834i	0.998435	+	0.0559172i	$0.0178083\pi$
$829$			92.7107i			0.111834i	−0.998435	+	0.0559172i	$0.982192\pi$
$830$		0			0
$831$		−	760.312i		−	0.914936i
$832$	72.5543			0.0872047
$833$	805.783	−	535.161i	0.967326	−	0.642450i
$834$	−346.809			−0.415838
$835$		0			0
$836$			257.277i			0.307748i
$837$		−	82.6267i		−	0.0987176i
$838$	−487.247			−0.581440
$839$			852.206i			1.01574i	0.861434	+	0.507870i	$0.169567\pi$
$839$			852.206i			1.01574i	−0.861434	+	0.507870i	$0.830433\pi$
$840$		0			0
$841$	−343.395			−0.408318
$842$		−	410.704i		−	0.487772i
$843$	−345.507			−0.409854
$844$	−313.764			−0.371759
$845$		0			0
$846$			246.680i			0.291584i
$847$	194.011	+	642.795i	0.229057	+	0.758908i
$848$		−	262.479i		−	0.309528i
$849$	−937.349			−1.10406
$850$		0			0
$851$	1833.37			2.15437
$852$	86.1434			0.101107
$853$	1472.75			1.72655			0.863275	−	0.504733i	$-0.168409\pi$
$853$	1472.75			1.72655			0.863275	+	0.504733i	$0.168409\pi$
$854$	−238.948	−	791.680i	−0.279798	−	0.927025i
$855$		0			0
$856$	385.427			0.450265
$857$	−882.824			−1.03013			−0.515066	−	0.857150i	$-0.672233\pi$
$857$	−882.824			−1.03013			−0.515066	+	0.857150i	$0.672233\pi$
$858$		−	111.255i		−	0.129667i
$859$		−	700.963i		−	0.816023i	−0.912977	−	0.408011i	$-0.866222\pi$
$859$		−	700.963i		−	0.816023i	0.912977	−	0.408011i	$-0.133778\pi$
$860$		0			0
$861$	314.356	−	94.8801i	0.365105	−	0.110198i
$862$			303.744i			0.352372i
$863$		−	1246.07i		−	1.44388i	−0.691957	−	0.721938i	$-0.743251\pi$
$863$		−	1246.07i		−	1.44388i	0.691957	−	0.721938i	$-0.256749\pi$
$864$		−	29.3939i		−	0.0340207i
$865$		0			0
$866$		−	601.099i		−	0.694109i
$867$	174.427			0.201185
$868$	213.125	−	64.3263i	0.245536	−	0.0741086i
$869$	152.471			0.175456
$870$		0			0
$871$		−	653.519i		−	0.750309i
$872$		−	469.527i		−	0.538449i
$873$	−311.412			−0.356715
$874$			1488.22i			1.70277i
$875$		0			0
$876$	−235.117			−0.268399
$877$			957.986i			1.09234i	0.837673	+	0.546172i	$0.183916\pi$
$877$			957.986i			1.09234i	−0.837673	+	0.546172i	$0.816084\pi$
$878$	−288.783			−0.328910
$879$	−656.026			−0.746333
$880$		0			0
$881$		−	865.707i		−	0.982642i	−0.870979	−	0.491321i	$-0.836514\pi$
$881$		−	865.707i		−	0.982642i	0.870979	−	0.491321i	$-0.163486\pi$
$882$	115.014	+	173.175i	0.130402	+	0.196344i
$883$		−	985.817i		−	1.11644i	−0.829693	−	0.558220i	$-0.811484\pi$
$883$		−	985.817i		−	1.11644i	0.829693	−	0.558220i	$-0.188516\pi$
$884$	−358.073			−0.405060
$885$		0			0
$886$	−390.843			−0.441132
$887$	−443.357			−0.499838			−0.249919	−	0.968267i	$-0.580404\pi$
$887$	−443.357			−0.499838			−0.249919	+	0.968267i	$0.580404\pi$
$888$	219.232			0.246883
$889$	−1543.38	+	465.828i	−1.73608	+	0.523991i
$890$		0			0
$891$	−45.0725			−0.0505864
$892$	120.806			0.135433
$893$		−	1493.48i		−	1.67243i
$894$		−	297.504i		−	0.332779i
$895$		0			0
$896$	75.8178	−	22.8836i	0.0846181	−	0.0255398i
$897$		−	643.554i		−	0.717451i
$898$			276.182i			0.307552i
$899$		−	354.716i		−	0.394567i
$900$		0			0
$901$			1295.40i			1.43774i
$902$	−191.813			−0.212653
$903$	355.981	−	107.444i	0.394220	−	0.118985i
$904$	−72.1354			−0.0797958
$905$		0			0
$906$			536.665i			0.592346i
$907$			49.0514i			0.0540810i	0.999634	+	0.0270405i	$0.00860830\pi$
$907$			49.0514i			0.0540810i	−0.999634	+	0.0270405i	$0.991392\pi$
$908$	627.222			0.690773
$909$			466.138i			0.512803i
$910$		0			0
$911$	−290.836			−0.319249			−0.159624	−	0.987178i	$-0.551028\pi$
$911$	−290.836			−0.319249			−0.159624	+	0.987178i	$0.551028\pi$
$912$			177.960i			0.195132i
$913$	−362.895			−0.397476
$914$	14.6269			0.0160032
$915$		0			0
$916$			220.736i			0.240979i
$917$	274.173	−	82.7519i	0.298989	−	0.0902420i
$918$			145.066i			0.158024i
$919$	721.872			0.785498			0.392749	−	0.919646i	$-0.371524\pi$
$919$	721.872			0.785498			0.392749	+	0.919646i	$0.371524\pi$
$920$		0			0
$921$	242.851			0.263681
$922$	−746.100			−0.809219
$923$	225.530			0.244345
$924$	−35.0897	−	116.259i	−0.0379759	−	0.125821i
$925$		0			0
$926$	−1128.40			−1.21857
$927$	−197.841			−0.213421
$928$		−	126.188i		−	0.135978i
$929$			617.161i			0.664329i	0.943222	+	0.332164i	$0.107779\pi$
$929$			617.161i			0.664329i	−0.943222	+	0.332164i	$0.892221\pi$
$930$		0			0
$931$	−696.335	−	1048.46i	−0.747943	−	1.12616i
$932$			76.8133i			0.0824177i
$933$			432.815i			0.463896i
$934$		−	239.923i		−	0.256877i
$935$		0			0
$936$		−	76.9555i		−	0.0822174i
$937$	−224.045			−0.239108			−0.119554	−	0.992828i	$-0.538147\pi$
$937$	−224.045			−0.239108			−0.119554	+	0.992828i	$0.538147\pi$
$938$	−206.120	−	682.915i	−0.219744	−	0.728054i
$939$	−897.019			−0.955292
$940$		0			0
$941$			522.371i			0.555124i	0.960708	+	0.277562i	$0.0895264\pi$
$941$			522.371i			0.555124i	−0.960708	+	0.277562i	$0.910474\pi$
$942$		−	140.310i		−	0.148949i
$943$	−1109.55			−1.17661
$944$			130.298i			0.138028i
$945$		0			0
$946$	−217.212			−0.229611
$947$		−	728.427i		−	0.769194i	−0.923084	−	0.384597i	$-0.874340\pi$
$947$		−	728.427i		−	0.769194i	0.923084	−	0.384597i	$-0.125660\pi$
$948$	105.465			0.111250
$949$	−615.555			−0.648636
$950$		0			0
$951$		−	311.844i		−	0.327911i
$952$	−374.179	+	112.936i	−0.393045	+	0.118630i
$953$			1169.38i			1.22705i	0.789674	+	0.613527i	$0.210250\pi$
$953$			1169.38i			1.22705i	−0.789674	+	0.613527i	$0.789750\pi$
$954$	−278.402			−0.291825
$955$		0			0
$956$	−250.258			−0.261776
$957$	−193.496			−0.202190
$958$	−995.991			−1.03966
$959$	1060.83	−	320.185i	1.10619	−	0.333874i
$960$		0			0
$961$	708.142			0.736880
$962$	573.968			0.596640
$963$		−	408.807i		−	0.424514i
$964$			309.250i			0.320799i
$965$		0			0
$966$	−202.977	−	672.501i	−0.210121	−	0.696170i
$967$			771.440i			0.797767i	0.917002	+	0.398883i	$0.130602\pi$
$967$			771.440i			0.797767i	−0.917002	+	0.398883i	$0.869398\pi$
$968$		−	271.301i		−	0.280270i
$969$		−	878.275i		−	0.906373i
$970$		0			0
$971$			1380.71i			1.42195i	0.703220	+	0.710973i	$0.251745\pi$
$971$			1380.71i			1.42195i	−0.703220	+	0.710973i	$0.748255\pi$
$972$	−31.1769			−0.0320750
$973$	−948.814	+	286.375i	−0.975143	+	0.294321i
$974$	−94.4647			−0.0969864
$975$		0			0
$976$			334.140i			0.342356i
$977$			974.771i			0.997718i	0.866683	+	0.498859i	$0.166248\pi$
$977$			974.771i			0.997718i	−0.866683	+	0.498859i	$0.833752\pi$
$978$	20.8769			0.0213466
$979$		−	569.694i		−	0.581915i
$980$		0			0
$981$	−498.009			−0.507655
$982$		−	772.333i		−	0.786490i
$983$	206.961			0.210540			0.105270	−	0.994444i	$-0.466429\pi$
$983$	206.961			0.210540			0.105270	+	0.994444i	$0.466429\pi$
$984$	−132.678			−0.134836
$985$		0			0
$986$			622.767i			0.631610i
$987$	203.694	+	674.877i	0.206377	+	0.683766i
$988$			465.913i			0.471572i
$989$	−1256.47			−1.27044
$990$		0			0
$991$	−413.701			−0.417459			−0.208729	−	0.977973i	$-0.566933\pi$
$991$	−413.701			−0.417459			−0.208729	+	0.977973i	$0.566933\pi$
$992$	−89.9525			−0.0906779
$993$	140.345			0.141335
$994$	235.675	−	71.1322i	0.237097	−	0.0715616i
$995$		0			0
$996$	−251.017			−0.252025
$997$	−542.511			−0.544143			−0.272072	−	0.962277i	$-0.587709\pi$
$997$	−542.511			−0.544143			−0.272072	+	0.962277i	$0.587709\pi$
$998$			1009.83i			1.01186i
$999$		−	232.531i		−	0.232764i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1050.3.h.b.349.6		16
5.2	odd	4		1050.3.f.b.601.6		8
5.3	odd	4		210.3.f.a.181.4	yes	8
5.4	even	2	inner	1050.3.h.b.349.11		16
7.6	odd	2	inner	1050.3.h.b.349.3		16
15.8	even	4		630.3.f.c.181.5		8
20.3	even	4		1680.3.s.a.1441.4		8
35.13	even	4		210.3.f.a.181.1	✓	8
35.27	even	4		1050.3.f.b.601.8		8
35.34	odd	2	inner	1050.3.h.b.349.14		16
105.83	odd	4		630.3.f.c.181.7		8
140.83	odd	4		1680.3.s.a.1441.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.f.a.181.1	✓	8	35.13	even	4
210.3.f.a.181.4	yes	8	5.3	odd	4
630.3.f.c.181.5		8	15.8	even	4
630.3.f.c.181.7		8	105.83	odd	4
1050.3.f.b.601.6		8	5.2	odd	4
1050.3.f.b.601.8		8	35.27	even	4
1050.3.h.b.349.3		16	7.6	odd	2	inner
1050.3.h.b.349.6		16	1.1	even	1	trivial
1050.3.h.b.349.11		16	5.4	even	2	inner
1050.3.h.b.349.14		16	35.34	odd	2	inner
1680.3.s.a.1441.4		8	20.3	even	4
1680.3.s.a.1441.6		8	140.83	odd	4

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:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1050.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} +1.73205 q^{3} -2.00000 q^{4} -2.44949i q^{6} +(-2.02265 - 6.70141i) q^{7} +2.82843i q^{8} +3.00000 q^{9} -5.00806 q^{11} -3.46410 q^{12} -9.06929 q^{13} +(-9.47723 + 2.86045i) q^{14} +4.00000 q^{16} -19.7410 q^{17} -4.24264i q^{18} +25.6863i q^{19} +(-3.50333 - 11.6072i) q^{21} +7.08246i q^{22} +40.9686i q^{23} +4.89898i q^{24} +12.8259i q^{26} +5.19615 q^{27} +(4.04529 + 13.4028i) q^{28} +22.3071 q^{29} -15.9015i q^{31} -5.65685i q^{32} -8.67421 q^{33} +27.9179i q^{34} -6.00000 q^{36} -44.7506i q^{37} +36.3259 q^{38} -15.7085 q^{39} +27.0829i q^{41} +(-16.4150 + 4.95445i) q^{42} +30.6690i q^{43} +10.0161 q^{44} +57.9383 q^{46} -58.1430 q^{47} +6.92820 q^{48} +(-40.8178 + 27.1092i) q^{49} -34.1924 q^{51} +18.1386 q^{52} -65.6199i q^{53} -7.34847i q^{54} +(18.9545 - 5.72091i) q^{56} +44.4900i q^{57} -31.5469i q^{58} +32.5746i q^{59} +83.5349i q^{61} -22.4881 q^{62} +(-6.06794 - 20.1042i) q^{63} -8.00000 q^{64} +12.2672i q^{66} +72.0585i q^{67} +39.4819 q^{68} +70.9597i q^{69} -24.8675 q^{71} +8.48528i q^{72} +67.8725 q^{73} -63.2870 q^{74} -51.3726i q^{76} +(10.1295 + 33.5610i) q^{77} +22.2151i q^{78} -30.4452 q^{79} +9.00000 q^{81} +38.3009 q^{82} +72.4623 q^{83} +(7.00665 + 23.2144i) q^{84} +43.3725 q^{86} +38.6370 q^{87} -14.1649i q^{88} +113.756i q^{89} +(18.3440 + 60.7770i) q^{91} -81.9372i q^{92} -27.5422i q^{93} +82.2267i q^{94} -9.79796i q^{96} -103.804 q^{97} +(38.3381 + 57.7251i) q^{98} -15.0242 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{4} + 48 q^{9} - 32 q^{11} - 64 q^{14} + 64 q^{16} + 288 q^{29} - 96 q^{36} + 96 q^{39} + 64 q^{44} + 256 q^{46} + 48 q^{49} + 128 q^{56} - 128 q^{64} + 352 q^{71} + 320 q^{74} + 576 q^{79}+ \cdots - 96 q^{99}+O(q^{100}) \) 16 * q - 32 * q^4 + 48 * q^9 - 32 * q^11 - 64 * q^14 + 64 * q^16 + 288 * q^29 - 96 * q^36 + 96 * q^39 + 64 * q^44 + 256 * q^46 + 48 * q^49 + 128 * q^56 - 128 * q^64 + 352 * q^71 + 320 * q^74 + 576 * q^79 + 144 * q^81 - 128 * q^86 + 128 * q^91 - 96 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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