LMFDB - Embedded newform 483.3.f.a.22.18

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [483,3,Mod(22,483)] 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("483.22"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(483, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 483 = 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	483.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$13.1607967686$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			22.18
Character	$\chi$	$=$	483.22
Dual form			483.3.f.a.22.17

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.984102 q^{2} +1.73205 q^{3} -3.03154 q^{4} +2.95865i q^{5} -1.70451 q^{6} -2.64575i q^{7} +6.91976 q^{8} +3.00000 q^{9} -2.91162i q^{10} -1.64161i q^{11} -5.25079 q^{12} -20.6451 q^{13} +2.60369i q^{14} +5.12454i q^{15} +5.31643 q^{16} -1.80914i q^{17} -2.95231 q^{18} +30.5833i q^{19} -8.96928i q^{20} -4.58258i q^{21} +1.61551i q^{22} +(-4.61202 - 22.5328i) q^{23} +11.9854 q^{24} +16.2464 q^{25} +20.3169 q^{26} +5.19615 q^{27} +8.02071i q^{28} -23.0792 q^{29} -5.04307i q^{30} -46.3849 q^{31} -32.9109 q^{32} -2.84335i q^{33} +1.78038i q^{34} +7.82786 q^{35} -9.09463 q^{36} -61.7024i q^{37} -30.0971i q^{38} -35.7583 q^{39} +20.4732i q^{40} -46.0087 q^{41} +4.50972i q^{42} -47.1312i q^{43} +4.97660i q^{44} +8.87596i q^{45} +(4.53870 + 22.1746i) q^{46} -77.1309 q^{47} +9.20832 q^{48} -7.00000 q^{49} -15.9881 q^{50} -3.13352i q^{51} +62.5864 q^{52} +75.2969i q^{53} -5.11354 q^{54} +4.85694 q^{55} -18.3080i q^{56} +52.9719i q^{57} +22.7123 q^{58} -59.3394 q^{59} -15.5353i q^{60} +13.6900i q^{61} +45.6475 q^{62} -7.93725i q^{63} +11.1220 q^{64} -61.0816i q^{65} +2.79814i q^{66} +24.4518i q^{67} +5.48449i q^{68} +(-7.98826 - 39.0280i) q^{69} -7.70341 q^{70} -24.4667 q^{71} +20.7593 q^{72} -5.03409 q^{73} +60.7214i q^{74} +28.1395 q^{75} -92.7147i q^{76} -4.34328 q^{77} +35.1898 q^{78} +120.358i q^{79} +15.7295i q^{80} +9.00000 q^{81} +45.2773 q^{82} -84.7183i q^{83} +13.8923i q^{84} +5.35262 q^{85} +46.3819i q^{86} -39.9744 q^{87} -11.3595i q^{88} -62.8049i q^{89} -8.73485i q^{90} +54.6217i q^{91} +(13.9815 + 68.3093i) q^{92} -80.3410 q^{93} +75.9047 q^{94} -90.4855 q^{95} -57.0034 q^{96} +49.7308i q^{97} +6.88871 q^{98} -4.92482i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 4 q^{2} + 116 q^{4} - 24 q^{6} - 4 q^{8} + 144 q^{9} + 16 q^{13} + 324 q^{16} + 12 q^{18} - 4 q^{23} - 24 q^{24} - 176 q^{25} + 136 q^{26} - 128 q^{29} - 8 q^{31} - 252 q^{32} - 56 q^{35} + 348 q^{36}+ \cdots - 28 q^{98}+O(q^{100}) $ 48 * q + 4 * q^2 + 116 * q^4 - 24 * q^6 - 4 * q^8 + 144 * q^9 + 16 * q^13 + 324 * q^16 + 12 * q^18 - 4 * q^23 - 24 * q^24 - 176 * q^25 + 136 * q^26 - 128 * q^29 - 8 * q^31 - 252 * q^32 - 56 * q^35 + 348 * q^36 + 96 * q^39 - 24 * q^41 - 148 * q^46 - 408 * q^47 - 96 * q^48 - 336 * q^49 + 236 * q^50 - 32 * q^52 - 72 * q^54 - 24 * q^55 - 56 * q^58 + 136 * q^59 + 184 * q^62 + 716 * q^64 - 48 * q^69 - 112 * q^70 + 48 * q^71 - 12 * q^72 - 224 * q^73 - 48 * q^75 + 224 * q^77 - 96 * q^78 + 432 * q^81 + 640 * q^82 - 424 * q^85 + 312 * q^87 + 1060 * q^92 + 192 * q^93 + 216 * q^94 + 624 * q^95 + 48 * q^96 - 28 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/483\mathbb{Z}\right)^\times$.

$n$	$323$	$346$	$442$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	−0.984102			−0.492051			−0.246025	−	0.969263i	$-0.579125\pi$
$2$	−0.984102			−0.492051			−0.246025	+	0.969263i	$0.579125\pi$
$3$	1.73205			0.577350
$3$	1.73205			0.577350
$4$	−3.03154			−0.757886
$5$			2.95865i			0.591731i	0.955230	+	0.295865i	$0.0956079\pi$
$5$			2.95865i			0.591731i	−0.955230	+	0.295865i	$0.904392\pi$
$6$	−1.70451			−0.284086
$7$		−	2.64575i		−	0.377964i
$7$		−	2.64575i		−	0.377964i
$8$	6.91976			0.864969
$9$	3.00000			0.333333
$10$		−	2.91162i		−	0.291162i
$11$		−	1.64161i		−	0.149237i	−0.997212	−	0.0746185i	$-0.976226\pi$
$11$		−	1.64161i		−	0.149237i	0.997212	−	0.0746185i	$-0.0237739\pi$
$12$	−5.25079			−0.437566
$13$	−20.6451			−1.58808			−0.794041	−	0.607864i	$-0.792027\pi$
$13$	−20.6451			−1.58808			−0.794041	+	0.607864i	$0.792027\pi$
$14$			2.60369i			0.185978i
$15$			5.12454i			0.341636i
$16$	5.31643			0.332277
$17$		−	1.80914i		−	0.106420i	−0.998583	−	0.0532100i	$-0.983055\pi$
$17$		−	1.80914i		−	0.106420i	0.998583	−	0.0532100i	$-0.0169453\pi$
$18$	−2.95231			−0.164017
$19$			30.5833i			1.60965i	0.593513	+	0.804825i	$0.297741\pi$
$19$			30.5833i			1.60965i	−0.593513	+	0.804825i	$0.702259\pi$
$20$		−	8.96928i		−	0.448464i
$21$		−	4.58258i		−	0.218218i
$22$			1.61551i			0.0734322i
$23$	−4.61202	−	22.5328i	−0.200523	−	0.979689i
$23$	−4.61202	−	22.5328i	−0.200523	−	0.979689i
$24$	11.9854			0.499390
$25$	16.2464			0.649855
$26$	20.3169			0.781417
$27$	5.19615			0.192450
$28$			8.02071i			0.286454i
$29$	−23.0792			−0.795835			−0.397918	−	0.917421i	$-0.630267\pi$
$29$	−23.0792			−0.795835			−0.397918	+	0.917421i	$0.630267\pi$
$30$		−	5.04307i		−	0.168102i
$31$	−46.3849			−1.49629			−0.748144	−	0.663537i	$-0.769055\pi$
$31$	−46.3849			−1.49629			−0.748144	+	0.663537i	$0.769055\pi$
$32$	−32.9109			−1.02847
$33$		−	2.84335i		−	0.0861620i
$34$			1.78038i			0.0523641i
$35$	7.82786			0.223653
$36$	−9.09463			−0.252629
$37$		−	61.7024i		−	1.66763i	−0.552042	−	0.833816i	$-0.686151\pi$
$37$		−	61.7024i		−	1.66763i	0.552042	−	0.833816i	$-0.313849\pi$
$38$		−	30.0971i		−	0.792029i
$39$	−35.7583			−0.916880
$40$			20.4732i			0.511829i
$41$	−46.0087			−1.12216			−0.561082	−	0.827760i	$-0.689615\pi$
$41$	−46.0087			−1.12216			−0.561082	+	0.827760i	$0.689615\pi$
$42$			4.50972i			0.107374i
$43$		−	47.1312i		−	1.09608i	−0.836454	−	0.548038i	$-0.815375\pi$
$43$		−	47.1312i		−	1.09608i	0.836454	−	0.548038i	$-0.184625\pi$
$44$			4.97660i			0.113105i
$45$			8.87596i			0.197244i
$46$	4.53870	+	22.1746i	0.0986674	+	0.482057i
$47$	−77.1309			−1.64108			−0.820541	−	0.571587i	$-0.806328\pi$
$47$	−77.1309			−1.64108			−0.820541	+	0.571587i	$0.806328\pi$
$48$	9.20832			0.191840
$49$	−7.00000			−0.142857
$50$	−15.9881			−0.319762
$51$		−	3.13352i		−	0.0614417i
$52$	62.5864			1.20359
$53$			75.2969i			1.42070i	0.703851	+	0.710348i	$0.251462\pi$
$53$			75.2969i			1.42070i	−0.703851	+	0.710348i	$0.748538\pi$
$54$	−5.11354			−0.0946953
$55$	4.85694			0.0883081
$56$		−	18.3080i		−	0.326928i
$57$			52.9719i			0.929331i
$58$	22.7123			0.391592
$59$	−59.3394			−1.00575			−0.502876	−	0.864359i	$-0.667725\pi$
$59$	−59.3394			−1.00575			−0.502876	+	0.864359i	$0.667725\pi$
$60$		−	15.5353i		−	0.258921i
$61$			13.6900i			0.224426i	0.993684	+	0.112213i	$0.0357939\pi$
$61$			13.6900i			0.224426i	−0.993684	+	0.112213i	$0.964206\pi$
$62$	45.6475			0.736250
$63$		−	7.93725i		−	0.125988i
$64$	11.1220			0.173781
$65$		−	61.0816i		−	0.939717i
$66$			2.79814i			0.0423961i
$67$			24.4518i			0.364953i	0.983210	+	0.182476i	$0.0584113\pi$
$67$			24.4518i			0.364953i	−0.983210	+	0.182476i	$0.941589\pi$
$68$			5.48449i			0.0806543i
$69$	−7.98826	−	39.0280i	−0.115772	−	0.565624i
$70$	−7.70341			−0.110049
$71$	−24.4667			−0.344601			−0.172301	−	0.985044i	$-0.555120\pi$
$71$	−24.4667			−0.344601			−0.172301	+	0.985044i	$0.555120\pi$
$72$	20.7593			0.288323
$73$	−5.03409			−0.0689602			−0.0344801	−	0.999405i	$-0.510978\pi$
$73$	−5.03409			−0.0689602			−0.0344801	+	0.999405i	$0.510978\pi$
$74$			60.7214i			0.820560i
$75$	28.1395			0.375194
$76$		−	92.7147i		−	1.21993i
$77$	−4.34328			−0.0564063
$78$	35.1898			0.451152
$79$			120.358i			1.52352i	0.647862	+	0.761758i	$0.275663\pi$
$79$			120.358i			1.52352i	−0.647862	+	0.761758i	$0.724337\pi$
$80$			15.7295i			0.196618i
$81$	9.00000			0.111111
$82$	45.2773			0.552162
$83$		−	84.7183i		−	1.02070i	−0.859966	−	0.510351i	$-0.829515\pi$
$83$		−	84.7183i		−	1.02070i	0.859966	−	0.510351i	$-0.170485\pi$
$84$			13.8923i			0.165384i
$85$	5.35262			0.0629720
$86$			46.3819i			0.539325i
$87$	−39.9744			−0.459476
$88$		−	11.3595i		−	0.129085i
$89$		−	62.8049i		−	0.705673i	−0.935685	−	0.352837i	$-0.885217\pi$
$89$		−	62.8049i		−	0.705673i	0.935685	−	0.352837i	$-0.114783\pi$
$90$		−	8.73485i		−	0.0970539i
$91$			54.6217i			0.600239i
$92$	13.9815	+	68.3093i	0.151973	+	0.742492i
$93$	−80.3410			−0.863882
$94$	75.9047			0.807496
$95$	−90.4855			−0.952479
$96$	−57.0034			−0.593785
$97$			49.7308i			0.512689i	0.966586	+	0.256344i	$0.0825181\pi$
$97$			49.7308i			0.512689i	−0.966586	+	0.256344i	$0.917482\pi$
$98$	6.88871			0.0702930
$99$		−	4.92482i		−	0.0497456i
$100$	−49.2516			−0.492516
$101$	−94.1838			−0.932513			−0.466257	−	0.884650i	$-0.654398\pi$
$101$	−94.1838			−0.932513			−0.466257	+	0.884650i	$0.654398\pi$
$102$			3.08371i			0.0302324i
$103$		−	124.143i		−	1.20527i	−0.798015	−	0.602637i	$-0.794117\pi$
$103$		−	124.143i		−	1.20527i	0.798015	−	0.602637i	$-0.205883\pi$
$104$	−142.859			−1.37364
$105$	13.5582			0.129126
$106$		−	74.0998i		−	0.699055i
$107$		−	209.537i		−	1.95829i	−0.203156	−	0.979146i	$-0.565120\pi$
$107$		−	209.537i		−	1.95829i	0.203156	−	0.979146i	$-0.434880\pi$
$108$	−15.7524			−0.145855
$109$			98.4084i			0.902829i	0.892314	+	0.451415i	$0.149080\pi$
$109$			98.4084i			0.902829i	−0.892314	+	0.451415i	$0.850920\pi$
$110$	−4.77973			−0.0434521
$111$		−	106.872i		−	0.962808i
$112$		−	14.0659i		−	0.125589i
$113$			163.527i			1.44714i	0.690251	+	0.723570i	$0.257500\pi$
$113$			163.527i			1.44714i	−0.690251	+	0.723570i	$0.742500\pi$
$114$		−	52.1297i		−	0.457278i
$115$	66.6669	−	13.6454i	0.579712	−	0.118655i
$116$	69.9657			0.603152
$117$	−61.9352			−0.529361
$118$	58.3960			0.494881
$119$	−4.78654			−0.0402230
$120$			35.4605i			0.295505i
$121$	118.305			0.977728
$122$		−	13.4723i		−	0.110429i
$123$	−79.6894			−0.647882
$124$	140.618			1.13402
$125$			122.034i			0.976270i
$126$			7.81107i			0.0619926i
$127$	132.335			1.04201			0.521003	−	0.853555i	$-0.325558\pi$
$127$	132.335			1.04201			0.521003	+	0.853555i	$0.325558\pi$
$128$	120.699			0.942957
$129$		−	81.6337i		−	0.632819i
$130$			60.1105i			0.462389i
$131$	112.519			0.858923			0.429462	−	0.903085i	$-0.358703\pi$
$131$	112.519			0.858923			0.429462	+	0.903085i	$0.358703\pi$
$132$			8.61972i			0.0653009i
$133$	80.9159			0.608390
$134$		−	24.0631i		−	0.179575i
$135$			15.3736i			0.113879i
$136$		−	12.5188i		−	0.0920501i
$137$			88.6187i			0.646851i	0.946254	+	0.323426i	$0.104835\pi$
$137$			88.6187i			0.646851i	−0.946254	+	0.323426i	$0.895165\pi$
$138$	7.86126	+	38.4076i	0.0569656	+	0.278316i
$139$	158.393			1.13952			0.569760	−	0.821811i	$-0.307036\pi$
$139$	158.393			1.13952			0.569760	+	0.821811i	$0.307036\pi$
$140$	−23.7305			−0.169504
$141$	−133.595			−0.947480
$142$	24.0777			0.169561
$143$			33.8911i			0.237001i
$144$	15.9493			0.110759
$145$		−	68.2834i		−	0.470920i
$146$	4.95406			0.0339319
$147$	−12.1244			−0.0824786
$148$			187.053i			1.26387i
$149$		−	241.254i		−	1.61915i	−0.587015	−	0.809576i	$-0.699697\pi$
$149$		−	241.254i		−	1.61915i	0.587015	−	0.809576i	$-0.300303\pi$
$150$	−27.6922			−0.184615
$151$	−39.1314			−0.259148			−0.129574	−	0.991570i	$-0.541361\pi$
$151$	−39.1314			−0.259148			−0.129574	+	0.991570i	$0.541361\pi$
$152$			211.629i			1.39230i
$153$		−	5.42742i		−	0.0354734i
$154$	4.27423			0.0277548
$155$		−	137.237i		−	0.885399i
$156$	108.403			0.694890
$157$			33.6563i			0.214371i	0.994239	+	0.107186i	$0.0341839\pi$
$157$			33.6563i			0.214371i	−0.994239	+	0.107186i	$0.965816\pi$
$158$		−	118.444i		−	0.749647i
$159$			130.418i			0.820239i
$160$		−	97.3720i		−	0.608575i
$161$	−59.6163	+	12.2023i	−0.370288	+	0.0757905i
$162$	−8.85692			−0.0546723
$163$	24.0778			0.147717			0.0738583	−	0.997269i	$-0.476469\pi$
$163$	24.0778			0.147717			0.0738583	+	0.997269i	$0.476469\pi$
$164$	139.477			0.850472
$165$	8.41247			0.0509847
$166$			83.3715i			0.502238i
$167$	−118.883			−0.711876			−0.355938	−	0.934510i	$-0.615839\pi$
$167$	−118.883			−0.711876			−0.355938	+	0.934510i	$0.615839\pi$
$168$		−	31.7103i		−	0.188752i
$169$	257.219			1.52201
$170$	−5.26752			−0.0309854
$171$			91.7500i			0.536550i
$172$			142.880i			0.830700i
$173$	−74.7281			−0.431955			−0.215977	−	0.976398i	$-0.569294\pi$
$173$	−74.7281			−0.431955			−0.215977	+	0.976398i	$0.569294\pi$
$174$	39.3389			0.226085
$175$		−	42.9839i		−	0.245622i
$176$		−	8.72748i		−	0.0495880i
$177$	−102.779			−0.580671
$178$			61.8065i			0.347227i
$179$	277.617			1.55093			0.775466	−	0.631390i	$-0.217515\pi$
$179$	277.617			1.55093			0.775466	+	0.631390i	$0.217515\pi$
$180$		−	26.9079i		−	0.149488i
$181$			53.3247i			0.294611i	0.989091	+	0.147306i	$0.0470601\pi$
$181$			53.3247i			0.294611i	−0.989091	+	0.147306i	$0.952940\pi$
$182$		−	53.7533i		−	0.295348i
$183$			23.7117i			0.129572i
$184$	−31.9141	−	155.922i	−0.173446	−	0.847401i
$185$	182.556			0.986789
$186$	79.0638			0.425074
$187$	−2.96990			−0.0158818
$188$	233.826			1.24375
$189$		−	13.7477i		−	0.0727393i
$190$	89.0469			0.468668
$191$			262.168i			1.37261i	0.727316	+	0.686303i	$0.240768\pi$
$191$			262.168i			1.37261i	−0.727316	+	0.686303i	$0.759232\pi$
$192$	19.2639			0.100333
$193$	64.1974			0.332629			0.166314	−	0.986073i	$-0.446813\pi$
$193$	64.1974			0.332629			0.166314	+	0.986073i	$0.446813\pi$
$194$		−	48.9402i		−	0.252269i
$195$		−	105.796i		−	0.542546i
$196$	21.2208			0.108269
$197$	−328.623			−1.66814			−0.834068	−	0.551662i	$-0.813994\pi$
$197$	−328.623			−1.66814			−0.834068	+	0.551662i	$0.813994\pi$
$198$			4.84652i			0.0244774i
$199$			354.709i			1.78246i	0.453556	+	0.891228i	$0.350155\pi$
$199$			354.709i			1.78246i	−0.453556	+	0.891228i	$0.649845\pi$
$200$	112.421			0.562105
$201$			42.3518i			0.210706i
$202$	92.6865			0.458844
$203$			61.0619i			0.300797i
$204$			9.49941i			0.0465658i
$205$		−	136.124i		−	0.664019i
$206$			122.170i			0.593056i
$207$	−13.8361	−	67.5985i	−0.0668409	−	0.326563i
$208$	−109.758			−0.527683
$209$	50.2058			0.240219
$210$	−13.3427			−0.0635367
$211$	−285.944			−1.35519			−0.677593	−	0.735437i	$-0.736977\pi$
$211$	−285.944			−1.35519			−0.677593	+	0.735437i	$0.736977\pi$
$212$		−	228.266i		−	1.07673i
$213$	−42.3775			−0.198955
$214$			206.206i			0.963580i
$215$	139.445			0.648581
$216$	35.9561			0.166463
$217$			122.723i			0.565544i
$218$		−	96.8439i		−	0.444238i
$219$	−8.71931			−0.0398142
$220$	−14.7240			−0.0669274
$221$			37.3498i			0.169004i
$222$			105.173i			0.473751i
$223$	396.446			1.77779			0.888893	−	0.458115i	$-0.151475\pi$
$223$	396.446			1.77779			0.888893	+	0.458115i	$0.151475\pi$
$224$			87.0741i			0.388724i
$225$	48.7391			0.216618
$226$		−	160.927i		−	0.712067i
$227$		−	168.217i		−	0.741042i	−0.928824	−	0.370521i	$-0.879179\pi$
$227$		−	168.217i		−	0.741042i	0.928824	−	0.370521i	$-0.120821\pi$
$228$		−	160.587i		−	0.704327i
$229$		−	182.998i		−	0.799118i	−0.916708	−	0.399559i	$-0.869163\pi$
$229$		−	182.998i		−	0.799118i	0.916708	−	0.399559i	$-0.130837\pi$
$230$	−65.6070	+	13.4284i	−0.285248	+	0.0583845i
$231$	−7.52278			−0.0325662
$232$	−159.703			−0.688373
$233$	146.908			0.630505			0.315253	−	0.949008i	$-0.397911\pi$
$233$	146.908			0.630505			0.315253	+	0.949008i	$0.397911\pi$
$234$	60.9506			0.260472
$235$		−	228.203i		−	0.971079i
$236$	179.890			0.762245
$237$			208.466i			0.879602i
$238$	4.71044			0.0197918
$239$	279.831			1.17084			0.585421	−	0.810729i	$-0.300929\pi$
$239$	279.831			1.17084			0.585421	+	0.810729i	$0.300929\pi$
$240$			27.2442i			0.113518i
$241$			274.634i			1.13956i	0.821797	+	0.569780i	$0.192972\pi$
$241$			274.634i			1.13956i	−0.821797	+	0.569780i	$0.807028\pi$
$242$	−116.424			−0.481092
$243$	15.5885			0.0641500
$244$		−	41.5018i		−	0.170089i
$245$		−	20.7106i		−	0.0845329i
$246$	78.4225			0.318791
$247$		−	631.395i		−	2.55626i
$248$	−320.972			−1.29424
$249$		−	146.736i		−	0.589303i
$250$		−	120.094i		−	0.480374i
$251$		−	176.637i		−	0.703731i	−0.936051	−	0.351866i	$-0.885547\pi$
$251$		−	176.637i		−	0.703731i	0.936051	−	0.351866i	$-0.114453\pi$
$252$			24.0621i			0.0954846i
$253$	−36.9901	+	7.57113i	−0.146206	+	0.0299254i
$254$	−130.231			−0.512720
$255$	9.27101			0.0363569
$256$	−163.268			−0.637764
$257$	−322.584			−1.25519			−0.627595	−	0.778540i	$-0.715961\pi$
$257$	−322.584			−1.25519			−0.627595	+	0.778540i	$0.715961\pi$
$258$			80.3359i			0.311379i
$259$	−163.249			−0.630306
$260$			185.171i			0.712198i
$261$	−69.2377			−0.265278
$262$	−110.730			−0.422634
$263$			177.102i			0.673393i	0.941613	+	0.336696i	$0.109310\pi$
$263$			177.102i			0.673393i	−0.941613	+	0.336696i	$0.890690\pi$
$264$		−	19.6753i		−	0.0745275i
$265$	−222.777			−0.840669
$266$	−79.6295			−0.299359
$267$		−	108.781i		−	0.407421i
$268$		−	74.1268i		−	0.276593i
$269$	−55.4512			−0.206138			−0.103069	−	0.994674i	$-0.532866\pi$
$269$	−55.4512			−0.206138			−0.103069	+	0.994674i	$0.532866\pi$
$270$		−	15.1292i		−	0.0560341i
$271$	−180.846			−0.667327			−0.333663	−	0.942692i	$-0.608285\pi$
$271$	−180.846			−0.667327			−0.333663	+	0.942692i	$0.608285\pi$
$272$		−	9.61817i		−	0.0353609i
$273$			94.6076i			0.346548i
$274$		−	87.2098i		−	0.318284i
$275$		−	26.6701i		−	0.0969824i
$276$	24.2167	+	118.315i	0.0877418	+	0.428678i
$277$	−152.629			−0.551007			−0.275503	−	0.961300i	$-0.588845\pi$
$277$	−152.629			−0.551007			−0.275503	+	0.961300i	$0.588845\pi$
$278$	−155.875			−0.560702
$279$	−139.155			−0.498763
$280$	54.1669			0.193453
$281$			211.984i			0.754391i	0.926134	+	0.377196i	$0.123112\pi$
$281$			211.984i			0.754391i	−0.926134	+	0.377196i	$0.876888\pi$
$282$	131.471			0.466208
$283$			496.918i			1.75589i	0.478757	+	0.877947i	$0.341087\pi$
$283$			496.918i			1.75589i	−0.478757	+	0.877947i	$0.658913\pi$
$284$	74.1718			0.261168
$285$	−156.725			−0.549914
$286$		−	33.3523i		−	0.116616i
$287$			121.728i			0.424138i
$288$	−98.7328			−0.342822
$289$	285.727			0.988675
$290$			67.1978i			0.231717i
$291$			86.1363i			0.296001i
$292$	15.2611			0.0522640
$293$		−	584.701i		−	1.99557i	−0.0665528	−	0.997783i	$-0.521200\pi$
$293$		−	584.701i		−	1.99557i	0.0665528	−	0.997783i	$-0.478800\pi$
$294$	11.9316			0.0405837
$295$		−	175.565i		−	0.595134i
$296$		−	426.965i		−	1.44245i
$297$		−	8.53004i		−	0.0287207i
$298$			237.418i			0.796706i
$299$	95.2155	+	465.192i	0.318447	+	1.55583i
$300$	−85.3063			−0.284354
$301$	−124.698			−0.414278
$302$	38.5093			0.127514
$303$	−163.131			−0.538387
$304$			162.594i			0.534849i
$305$	−40.5039			−0.132800
$306$			5.34114i			0.0174547i
$307$	−536.733			−1.74832			−0.874158	−	0.485641i	$-0.838586\pi$
$307$	−536.733			−1.74832			−0.874158	+	0.485641i	$0.838586\pi$
$308$	13.1668			0.0427495
$309$		−	215.022i		−	0.695866i
$310$			135.055i			0.435662i
$311$	−171.080			−0.550096			−0.275048	−	0.961431i	$-0.588694\pi$
$311$	−171.080			−0.550096			−0.275048	+	0.961431i	$0.588694\pi$
$312$	−247.439			−0.793073
$313$		−	132.047i		−	0.421875i	−0.977500	−	0.210937i	$-0.932348\pi$
$313$		−	132.047i		−	0.421875i	0.977500	−	0.210937i	$-0.0676516\pi$
$314$		−	33.1212i		−	0.105482i
$315$	23.4836			0.0745510
$316$		−	364.870i		−	1.15465i
$317$	41.3975			0.130591			0.0652957	−	0.997866i	$-0.479201\pi$
$317$	41.3975			0.130591			0.0652957	+	0.997866i	$0.479201\pi$
$318$		−	128.345i		−	0.403599i
$319$			37.8870i			0.118768i
$320$			32.9061i			0.102832i
$321$		−	362.929i		−	1.13062i
$322$	58.6685	−	12.0083i	0.182200	−	0.0372928i
$323$	55.3296			0.171299
$324$	−27.2839			−0.0842095
$325$	−335.408			−1.03202
$326$	−23.6950			−0.0726841
$327$			170.448i			0.521249i
$328$	−318.369			−0.970637
$329$			204.069i			0.620271i
$330$	−8.27873			−0.0250871
$331$	−529.880			−1.60085			−0.800423	−	0.599436i	$-0.795392\pi$
$331$	−529.880			−1.60085			−0.800423	+	0.599436i	$0.795392\pi$
$332$			256.827i			0.773576i
$333$		−	185.107i		−	0.555877i
$334$	116.993			0.350279
$335$	−72.3445			−0.215954
$336$		−	24.3629i		−	0.0725087i
$337$			107.305i			0.318412i	0.987245	+	0.159206i	$0.0508933\pi$
$337$			107.305i			0.318412i	−0.987245	+	0.159206i	$0.949107\pi$
$338$	−253.130			−0.748904
$339$			283.237i			0.835507i
$340$	−16.2267			−0.0477256
$341$			76.1458i			0.223301i
$342$		−	90.2914i		−	0.264010i
$343$			18.5203i			0.0539949i
$344$		−	326.137i		−	0.948072i
$345$	115.470	−	23.6345i	0.334697	−	0.0685057i
$346$	73.5401			0.212544
$347$	−205.554			−0.592373			−0.296187	−	0.955130i	$-0.595715\pi$
$347$	−205.554			−0.592373			−0.296187	+	0.955130i	$0.595715\pi$
$348$	121.184			0.348230
$349$	257.695			0.738382			0.369191	−	0.929354i	$-0.379635\pi$
$349$	257.695			0.738382			0.369191	+	0.929354i	$0.379635\pi$
$350$			42.3005i			0.120859i
$351$	−107.275			−0.305627
$352$			54.0268i			0.153485i
$353$	−219.135			−0.620779			−0.310390	−	0.950609i	$-0.600460\pi$
$353$	−219.135			−0.620779			−0.310390	+	0.950609i	$0.600460\pi$
$354$	101.145			0.285720
$355$		−	72.3884i		−	0.203911i
$356$			190.396i			0.534820i
$357$	−8.29053			−0.0232228
$358$	−273.203			−0.763137
$359$		−	188.537i		−	0.525173i	−0.964908	−	0.262587i	$-0.915424\pi$
$359$		−	188.537i		−	0.525173i	0.964908	−	0.262587i	$-0.0845756\pi$
$360$			61.4195i			0.170610i
$361$	−574.340			−1.59097
$362$		−	52.4769i		−	0.144964i
$363$	204.910			0.564492
$364$		−	165.588i		−	0.454912i
$365$		−	14.8941i		−	0.0408059i
$366$		−	23.3348i		−	0.0637562i
$367$			262.250i			0.714577i	0.933994	+	0.357288i	$0.116299\pi$
$367$			262.250i			0.714577i	−0.933994	+	0.357288i	$0.883701\pi$
$368$	−24.5195	−	119.794i	−0.0666290	−	0.325528i
$369$	−138.026			−0.374055
$370$	−179.654			−0.485550
$371$	199.217			0.536973
$372$	243.557			0.654724
$373$			65.3990i			0.175333i	0.996150	+	0.0876663i	$0.0279409\pi$
$373$			65.3990i			0.175333i	−0.996150	+	0.0876663i	$0.972059\pi$
$374$	2.92268			0.00781466
$375$			211.369i			0.563649i
$376$	−533.727			−1.41949
$377$	476.472			1.26385
$378$			13.5292i			0.0357914i
$379$		−	296.488i		−	0.782291i	−0.920329	−	0.391145i	$-0.872079\pi$
$379$		−	296.488i		−	0.782291i	0.920329	−	0.391145i	$-0.127921\pi$
$380$	274.311			0.721870
$381$	229.210			0.601602
$382$		−	258.000i		−	0.675392i
$383$			479.187i			1.25114i	0.780168	+	0.625570i	$0.215134\pi$
$383$			479.187i			1.25114i	−0.780168	+	0.625570i	$0.784866\pi$
$384$	209.056			0.544417
$385$		−	12.8503i		−	0.0333773i
$386$	−63.1767			−0.163670
$387$		−	141.394i		−	0.365358i
$388$		−	150.761i		−	0.388560i
$389$		−	101.204i		−	0.260164i	−0.991503	−	0.130082i	$-0.958476\pi$
$389$		−	101.204i		−	0.260164i	0.991503	−	0.130082i	$-0.0415241\pi$
$390$			104.114i			0.266960i
$391$	−40.7651	+	8.34380i	−0.104259	+	0.0213396i
$392$	−48.4383			−0.123567
$393$	194.889			0.495900
$394$	323.398			0.820808
$395$	−356.097			−0.901510
$396$			14.9298i			0.0377015i
$397$	376.793			0.949102			0.474551	−	0.880228i	$-0.342611\pi$
$397$	376.793			0.949102			0.474551	+	0.880228i	$0.342611\pi$
$398$		−	349.069i		−	0.877059i
$399$	140.150			0.351254
$400$	86.3727			0.215932
$401$		−	5.99307i		−	0.0149453i	−0.999972	−	0.00747265i	$-0.997621\pi$
$401$		−	5.99307i		−	0.0149453i	0.999972	−	0.00747265i	$-0.00237864\pi$
$402$		−	41.6785i		−	0.103678i
$403$	957.620			2.37623
$404$	285.522			0.706738
$405$			26.6279i			0.0657478i
$406$		−	60.0911i		−	0.148008i
$407$	−101.291			−0.248872
$408$		−	21.6832i		−	0.0531452i
$409$	−392.937			−0.960726			−0.480363	−	0.877070i	$-0.659495\pi$
$409$	−392.937			−0.960726			−0.480363	+	0.877070i	$0.659495\pi$
$410$			133.960i			0.326731i
$411$			153.492i			0.373460i
$412$			376.346i			0.913460i
$413$			156.997i			0.380139i
$414$	13.6161	+	66.5239i	0.0328891	+	0.160686i
$415$	250.652			0.603981
$416$	679.448			1.63329
$417$	274.345			0.657902
$418$	−49.4076			−0.118200
$419$			337.328i			0.805080i	0.915402	+	0.402540i	$0.131873\pi$
$419$			337.328i			0.805080i	−0.915402	+	0.402540i	$0.868127\pi$
$420$	−41.1024			−0.0978629
$421$		−	715.484i		−	1.69949i	−0.527197	−	0.849743i	$-0.676757\pi$
$421$		−	715.484i		−	1.69949i	0.527197	−	0.849743i	$-0.323243\pi$
$422$	281.398			0.666821
$423$	−231.393			−0.547028
$424$			521.036i			1.22886i
$425$		−	29.3920i		−	0.0691576i
$426$	41.7038			0.0978962
$427$	36.2203			0.0848250
$428$			635.221i			1.48416i
$429$			58.7011i			0.136832i
$430$	−137.228			−0.319135
$431$		−	795.766i		−	1.84632i	−0.384410	−	0.923162i	$-0.625595\pi$
$431$		−	795.766i		−	1.84632i	0.384410	−	0.923162i	$-0.374405\pi$
$432$	27.6250			0.0639467
$433$		−	217.649i		−	0.502654i	−0.967902	−	0.251327i	$-0.919133\pi$
$433$		−	217.649i		−	0.502654i	0.967902	−	0.251327i	$-0.0808669\pi$
$434$		−	120.772i		−	0.278276i
$435$		−	118.270i		−	0.271886i
$436$		−	298.329i		−	0.684241i
$437$	689.130	−	141.051i	1.57696	−	0.322771i
$438$	8.58069			0.0195906
$439$	431.228			0.982297			0.491148	−	0.871076i	$-0.336577\pi$
$439$	431.228			0.982297			0.491148	+	0.871076i	$0.336577\pi$
$440$	33.6089			0.0763838
$441$	−21.0000			−0.0476190
$442$		−	36.7561i		−	0.0831585i
$443$	−239.193			−0.539940			−0.269970	−	0.962869i	$-0.587014\pi$
$443$	−239.193			−0.539940			−0.269970	+	0.962869i	$0.587014\pi$
$444$			323.986i			0.729698i
$445$	185.818			0.417569
$446$	−390.143			−0.874761
$447$		−	417.864i		−	0.934818i
$448$		−	29.4260i		−	0.0656831i
$449$	−97.7033			−0.217602			−0.108801	−	0.994064i	$-0.534701\pi$
$449$	−97.7033			−0.217602			−0.108801	+	0.994064i	$0.534701\pi$
$450$	−47.9643			−0.106587
$451$			75.5282i			0.167468i
$452$		−	495.739i		−	1.09677i
$453$	−67.7776			−0.149619
$454$			165.542i			0.364630i
$455$	−161.607			−0.355180
$456$			366.553i			0.803843i
$457$		−	196.469i		−	0.429911i	−0.976624	−	0.214956i	$-0.931039\pi$
$457$		−	196.469i		−	0.429911i	0.976624	−	0.214956i	$-0.0689607\pi$
$458$			180.089i			0.393207i
$459$		−	9.40057i		−	0.0204806i
$460$	−202.104	+	41.3665i	−0.439355	+	0.0899273i
$461$	−217.422			−0.471632			−0.235816	−	0.971798i	$-0.575776\pi$
$461$	−217.422			−0.471632			−0.235816	+	0.971798i	$0.575776\pi$
$462$	7.40319			0.0160242
$463$	838.394			1.81079			0.905393	−	0.424574i	$-0.139576\pi$
$463$	838.394			1.81079			0.905393	+	0.424574i	$0.139576\pi$
$464$	−122.699			−0.264438
$465$		−	237.701i		−	0.511185i
$466$	−144.572			−0.310241
$467$			392.675i			0.840845i	0.907328	+	0.420423i	$0.138118\pi$
$467$			392.675i			0.840845i	−0.907328	+	0.420423i	$0.861882\pi$
$468$	187.759			0.401195
$469$	64.6935			0.137939
$470$			224.576i			0.477820i
$471$			58.2944i			0.123767i
$472$	−410.614			−0.869945
$473$	−77.3709			−0.163575
$474$		−	205.151i		−	0.432809i
$475$			496.868i			1.04604i
$476$	14.5106			0.0304844
$477$			225.891i			0.473565i
$478$	−275.383			−0.576114
$479$		−	285.575i		−	0.596190i	−0.954536	−	0.298095i	$-0.903649\pi$
$479$		−	285.575i		−	0.596190i	0.954536	−	0.298095i	$-0.0963512\pi$
$480$		−	168.653i		−	0.351361i
$481$			1273.85i			2.64834i
$482$		−	270.268i		−	0.560721i
$483$	−103.258	+	21.1349i	−0.213786	+	0.0437576i
$484$	−358.647			−0.741006
$485$	−147.136			−0.303374
$486$	−15.3406			−0.0315651
$487$	199.314			0.409269			0.204635	−	0.978838i	$-0.434399\pi$
$487$	199.314			0.409269			0.204635	+	0.978838i	$0.434399\pi$
$488$			94.7313i			0.194122i
$489$	41.7040			0.0852842
$490$			20.3813i			0.0415945i
$491$	−178.193			−0.362918			−0.181459	−	0.983398i	$-0.558082\pi$
$491$	−178.193			−0.362918			−0.181459	+	0.983398i	$0.558082\pi$
$492$	241.582			0.491020
$493$			41.7536i			0.0846928i
$494$			621.357i			1.25781i
$495$	14.5708			0.0294360
$496$	−246.602			−0.497182
$497$			64.7327i			0.130247i
$498$			144.404i			0.289967i
$499$	623.881			1.25026			0.625131	−	0.780520i	$-0.285046\pi$
$499$	623.881			1.25026			0.625131	+	0.780520i	$0.285046\pi$
$500$		−	369.950i		−	0.739901i
$501$	−205.912			−0.411002
$502$			173.828i			0.346272i
$503$		−	259.558i		−	0.516021i	−0.966142	−	0.258010i	$-0.916933\pi$
$503$		−	259.558i		−	0.516021i	0.966142	−	0.258010i	$-0.0830669\pi$
$504$		−	54.9239i		−	0.108976i
$505$		−	278.657i		−	0.551796i
$506$	36.4020	−	7.45076i	0.0719407	−	0.0147248i
$507$	445.516			0.878730
$508$	−401.178			−0.789721
$509$	−590.340			−1.15980			−0.579902	−	0.814686i	$-0.696909\pi$
$509$	−590.340			−1.15980			−0.579902	+	0.814686i	$0.696909\pi$
$510$	−9.12362			−0.0178894
$511$			13.3190i			0.0260645i
$512$	−322.122			−0.629145
$513$			158.916i			0.309777i
$514$	317.456			0.617618
$515$	367.297			0.713198
$516$			247.476i			0.479605i
$517$			126.619i			0.244910i
$518$	160.654			0.310143
$519$	−129.433			−0.249389
$520$		−	422.670i		−	0.812826i
$521$		−	597.731i		−	1.14728i	−0.819109	−	0.573638i	$-0.805532\pi$
$521$		−	597.731i		−	1.14728i	0.819109	−	0.573638i	$-0.194468\pi$
$522$	68.1369			0.130531
$523$		−	976.357i		−	1.86684i	−0.358786	−	0.933420i	$-0.616809\pi$
$523$		−	976.357i		−	1.86684i	0.358786	−	0.933420i	$-0.383191\pi$
$524$	−341.106			−0.650966
$525$		−	74.4502i		−	0.141810i
$526$		−	174.287i		−	0.331344i
$527$			83.9169i			0.159235i
$528$		−	15.1164i		−	0.0286296i
$529$	−486.458	+	207.844i	−0.919581	+	0.392900i
$530$	219.236			0.413652
$531$	−178.018			−0.335251
$532$	−245.300			−0.461090
$533$	949.853			1.78209
$534$			107.052i			0.200472i
$535$	619.948			1.15878
$536$			169.201i			0.315673i
$537$	480.846			0.895431
$538$	54.5696			0.101430
$539$			11.4912i			0.0213196i
$540$		−	46.6058i		−	0.0863070i
$541$	−978.363			−1.80843			−0.904217	−	0.427074i	$-0.859545\pi$
$541$	−978.363			−1.80843			−0.904217	+	0.427074i	$0.859545\pi$
$542$	177.970			0.328359
$543$			92.3610i			0.170094i
$544$			59.5405i			0.109449i
$545$	−291.156			−0.534232
$546$		−	93.1035i		−	0.170519i
$547$	682.278			1.24731			0.623654	−	0.781700i	$-0.285647\pi$
$547$	682.278			1.24731			0.623654	+	0.781700i	$0.285647\pi$
$548$		−	268.651i		−	0.490240i
$549$			41.0699i			0.0748086i
$550$			26.2461i			0.0477203i
$551$		−	705.840i		−	1.28102i
$552$	−55.2768	−	270.064i	−0.100139	−	0.489247i
$553$	318.437			0.575835
$554$	150.202			0.271123
$555$	316.196			0.569723
$556$	−480.176			−0.863626
$557$			896.298i			1.60915i	0.593850	+	0.804576i	$0.297607\pi$
$557$			896.298i			1.60915i	−0.593850	+	0.804576i	$0.702393\pi$
$558$	136.942			0.245417
$559$			973.028i			1.74066i
$560$	41.6163			0.0743147
$561$	−5.14401			−0.00916936
$562$		−	208.614i		−	0.371199i
$563$			311.938i			0.554064i	0.960861	+	0.277032i	$0.0893508\pi$
$563$			311.938i			0.554064i	−0.960861	+	0.277032i	$0.910649\pi$
$564$	404.998			0.718081
$565$	−483.819			−0.856317
$566$		−	489.018i		−	0.863990i
$567$		−	23.8118i		−	0.0419961i
$568$	−169.303			−0.298069
$569$			685.321i			1.20443i	0.798334	+	0.602216i	$0.205715\pi$
$569$			685.321i			1.20443i	−0.798334	+	0.602216i	$0.794285\pi$
$570$	154.234			0.270586
$571$			284.733i			0.498656i	0.968419	+	0.249328i	$0.0802097\pi$
$571$			284.733i			0.498656i	−0.968419	+	0.249328i	$0.919790\pi$
$572$		−	102.742i		−	0.179619i
$573$			454.088i			0.792475i
$574$		−	119.792i		−	0.208698i
$575$	−74.9286	−	366.077i	−0.130311	−	0.636656i
$576$	33.3660			0.0579271
$577$	−427.106			−0.740219			−0.370109	−	0.928988i	$-0.620680\pi$
$577$	−427.106			−0.740219			−0.370109	+	0.928988i	$0.620680\pi$
$578$	−281.185			−0.486478
$579$	111.193			0.192043
$580$			207.004i			0.356904i
$581$	−224.144			−0.385789
$582$		−	84.7669i		−	0.145648i
$583$	123.608			0.212020
$584$	−34.8347			−0.0596485
$585$		−	183.245i		−	0.313239i
$586$			575.405i			0.981920i
$587$	936.975			1.59621			0.798105	−	0.602519i	$-0.205836\pi$
$587$	936.975			1.59621			0.798105	+	0.602519i	$0.205836\pi$
$588$	36.7555			0.0625094
$589$		−	1418.61i		−	2.40850i
$590$			172.773i			0.292836i
$591$	−569.191			−0.963099
$592$		−	328.036i		−	0.554115i
$593$	967.284			1.63117			0.815585	−	0.578637i	$-0.196415\pi$
$593$	967.284			1.63117			0.815585	+	0.578637i	$0.196415\pi$
$594$			8.39443i			0.0141320i
$595$		−	14.1617i		−	0.0238012i
$596$			731.371i			1.22713i
$597$			614.373i			1.02910i
$598$	−93.7018	−	457.797i	−0.156692	−	0.765546i
$599$	−695.902			−1.16177			−0.580887	−	0.813985i	$-0.697294\pi$
$599$	−695.902			−1.16177			−0.580887	+	0.813985i	$0.697294\pi$
$600$	194.719			0.324531
$601$	−482.402			−0.802665			−0.401333	−	0.915932i	$-0.631453\pi$
$601$	−482.402			−0.802665			−0.401333	+	0.915932i	$0.631453\pi$
$602$	122.715			0.203846
$603$			73.3555i			0.121651i
$604$	118.629			0.196405
$605$			350.024i			0.578552i
$606$	160.538			0.264914
$607$	−534.976			−0.881344			−0.440672	−	0.897668i	$-0.645260\pi$
$607$	−534.976			−0.881344			−0.440672	+	0.897668i	$0.645260\pi$
$608$		−	1006.53i		−	1.65547i
$609$			105.762i			0.173666i
$610$	39.8600			0.0653442
$611$	1592.37			2.60617
$612$			16.4535i			0.0268848i
$613$			7.13100i			0.0116329i	0.999983	+	0.00581647i	$0.00185145\pi$
$613$			7.13100i			0.0116329i	−0.999983	+	0.00581647i	$0.998149\pi$
$614$	528.200			0.860261
$615$		−	235.773i		−	0.383371i
$616$	−30.0544			−0.0487897
$617$		−	675.408i		−	1.09466i	−0.836916	−	0.547332i	$-0.815643\pi$
$617$		−	675.408i		−	1.09466i	0.836916	−	0.547332i	$-0.184357\pi$
$618$			211.604i			0.342401i
$619$			201.503i			0.325529i	0.986665	+	0.162765i	$0.0520411\pi$
$619$			201.503i			0.325529i	−0.986665	+	0.162765i	$0.947959\pi$
$620$			416.039i			0.671031i
$621$	−23.9648	−	117.084i	−0.0385906	−	0.188541i
$622$	168.360			0.270675
$623$	−166.166			−0.266719
$624$	−190.106			−0.304658
$625$	45.1040			0.0721665
$626$			129.947i			0.207584i
$627$	86.9590			0.138691
$628$		−	102.031i		−	0.162469i
$629$	−111.628			−0.177470
$630$	−23.1102			−0.0366829
$631$		−	804.427i		−	1.27485i	−0.770514	−	0.637423i	$-0.780001\pi$
$631$		−	804.427i		−	1.27485i	0.770514	−	0.637423i	$-0.219999\pi$
$632$			832.846i			1.31779i
$633$	−495.270			−0.782418
$634$	−40.7394			−0.0642577
$635$			391.532i			0.616587i
$636$		−	395.368i		−	0.621648i
$637$	144.515			0.226869
$638$		−	37.2847i		−	0.0584399i
$639$	−73.4000			−0.114867
$640$			357.105i			0.557977i
$641$		−	184.130i		−	0.287255i	−0.989632	−	0.143627i	$-0.954123\pi$
$641$		−	184.130i		−	0.287255i	0.989632	−	0.143627i	$-0.0458767\pi$
$642$			357.159i			0.556323i
$643$			778.840i			1.21126i	0.795747	+	0.605630i	$0.207079\pi$
$643$			778.840i			1.21126i	−0.795747	+	0.605630i	$0.792921\pi$
$644$	180.729	−	36.9917i	0.280636	−	0.0574405i
$645$	241.526			0.374459
$646$	−54.4499			−0.0842878
$647$	143.648			0.222022			0.111011	−	0.993819i	$-0.464591\pi$
$647$	143.648			0.222022			0.111011	+	0.993819i	$0.464591\pi$
$648$	62.2778			0.0961077
$649$			97.4119i			0.150095i
$650$	330.075			0.507808
$651$			212.562i			0.326517i
$652$	−72.9929			−0.111952
$653$	−533.097			−0.816382			−0.408191	−	0.912897i	$-0.633840\pi$
$653$	−533.097			−0.816382			−0.408191	+	0.912897i	$0.633840\pi$
$654$		−	167.739i		−	0.256481i
$655$			332.904i			0.508251i
$656$	−244.602			−0.372869
$657$	−15.1023			−0.0229867
$658$		−	200.825i		−	0.305205i
$659$		−	129.534i		−	0.196562i	−0.995159	−	0.0982809i	$-0.968666\pi$
$659$		−	129.534i		−	0.196562i	0.995159	−	0.0982809i	$-0.0313344\pi$
$660$	−25.5028			−0.0386406
$661$		−	613.199i		−	0.927684i	−0.885918	−	0.463842i	$-0.846470\pi$
$661$		−	613.199i		−	0.927684i	0.885918	−	0.463842i	$-0.153530\pi$
$662$	521.456			0.787697
$663$			64.6918i			0.0975744i
$664$		−	586.230i		−	0.882877i
$665$			239.402i			0.360003i
$666$			182.164i			0.273520i
$667$	106.442	+	520.041i	0.159583	+	0.779671i
$668$	360.400			0.539520
$669$	686.665			1.02641
$670$	71.1944			0.106260
$671$	22.4736			0.0334926
$672$			150.817i			0.224430i
$673$	−1086.59			−1.61455			−0.807275	−	0.590175i	$-0.799059\pi$
$673$	−1086.59			−1.61455			−0.807275	+	0.590175i	$0.799059\pi$
$674$		−	105.599i		−	0.156675i
$675$	84.4186			0.125065
$676$	−779.770			−1.15351
$677$		−	591.555i		−	0.873788i	−0.899513	−	0.436894i	$-0.856078\pi$
$677$		−	591.555i		−	0.873788i	0.899513	−	0.436894i	$-0.143922\pi$
$678$		−	278.734i		−	0.411112i
$679$	131.575			0.193778
$680$	37.0388			0.0544689
$681$		−	291.360i		−	0.427841i
$682$		−	74.9352i		−	0.109876i
$683$	−213.493			−0.312581			−0.156290	−	0.987711i	$-0.549954\pi$
$683$	−213.493			−0.312581			−0.156290	+	0.987711i	$0.549954\pi$
$684$		−	278.144i		−	0.406643i
$685$	−262.192			−0.382762
$686$		−	18.2258i		−	0.0265683i
$687$		−	316.962i		−	0.461371i
$688$		−	250.570i		−	0.364200i
$689$		−	1554.51i		−	2.25618i
$690$	−113.635	+	23.2587i	−0.164688	+	0.0337083i
$691$	1199.99			1.73660			0.868298	−	0.496044i	$-0.165214\pi$
$691$	1199.99			1.73660			0.868298	+	0.496044i	$0.165214\pi$
$692$	226.542			0.327372
$693$	−13.0298			−0.0188021
$694$	202.286			0.291478
$695$			468.630i			0.674288i
$696$	−276.613			−0.397432
$697$			83.2363i			0.119421i
$698$	−253.598			−0.363321
$699$	254.452			0.364022
$700$			130.307i			0.186154i
$701$			974.622i			1.39033i	0.718850	+	0.695165i	$0.244669\pi$
$701$			974.622i			1.39033i	−0.718850	+	0.695165i	$0.755331\pi$
$702$	105.569			0.150384
$703$	1887.06			2.68430
$704$		−	18.2579i		−	0.0259346i
$705$		−	395.260i		−	0.560653i
$706$	215.651			0.305455
$707$			249.187i			0.352457i
$708$	311.578			0.440083
$709$		−	688.631i		−	0.971271i	−0.874162	−	0.485635i	$-0.838588\pi$
$709$		−	688.631i		−	0.971271i	0.874162	−	0.485635i	$-0.161412\pi$
$710$			71.2376i			0.100335i
$711$			361.073i			0.507838i
$712$		−	434.595i		−	0.610386i
$713$	213.928	+	1045.18i	0.300040	+	1.46590i
$714$	8.15872			0.0114268
$715$	−100.272			−0.140240
$716$	−841.607			−1.17543
$717$	484.682			0.675986
$718$			185.540i			0.258412i
$719$	57.7261			0.0802866			0.0401433	−	0.999194i	$-0.487219\pi$
$719$	57.7261			0.0802866			0.0401433	+	0.999194i	$0.487219\pi$
$720$			47.1884i			0.0655394i
$721$	−328.452			−0.455551
$722$	565.210			0.782839
$723$			475.680i			0.657925i
$724$		−	161.656i		−	0.223282i
$725$	−374.954			−0.517178
$726$	−201.653			−0.277759
$727$			734.127i			1.00980i	0.863177	+	0.504902i	$0.168471\pi$
$727$			734.127i			1.00980i	−0.863177	+	0.504902i	$0.831529\pi$
$728$			377.969i			0.519188i
$729$	27.0000			0.0370370
$730$			14.6574i			0.0200786i
$731$	−85.2671			−0.116644
$732$		−	71.8832i		−	0.0982010i
$733$			692.573i			0.944847i	0.881372	+	0.472423i	$0.156621\pi$
$733$			692.573i			0.944847i	−0.881372	+	0.472423i	$0.843379\pi$
$734$		−	258.080i		−	0.351608i
$735$		−	35.8718i		−	0.0488051i
$736$	151.786	+	741.577i	0.206231	+	1.00758i
$737$	40.1403			0.0544645
$738$	135.832			0.184054
$739$	639.157			0.864895			0.432447	−	0.901659i	$-0.357650\pi$
$739$	639.157			0.864895			0.432447	+	0.901659i	$0.357650\pi$
$740$	−553.426			−0.747873
$741$		−	1093.61i		−	1.47585i
$742$	−196.050			−0.264218
$743$			745.887i			1.00389i	0.864901	+	0.501943i	$0.167381\pi$
$743$			745.887i			1.00389i	−0.864901	+	0.501943i	$0.832619\pi$
$744$	−555.940			−0.747232
$745$	713.786			0.958102
$746$		−	64.3593i		−	0.0862725i
$747$		−	254.155i		−	0.340234i
$748$	9.00337			0.0120366
$749$	−554.384			−0.740165
$750$		−	208.008i		−	0.277344i
$751$		−	1.32171i		−	0.00175993i	−1.00000		0.000879967i	$-0.999720\pi$
$751$		−	1.32171i		−	0.00175993i	1.00000		0.000879967i	$-0.000280102\pi$
$752$	−410.061			−0.545294
$753$		−	305.943i		−	0.406299i
$754$	−468.897			−0.621880
$755$		−	115.776i		−	0.153346i
$756$			41.6768i			0.0551281i
$757$			464.771i			0.613964i	0.951715	+	0.306982i	$0.0993192\pi$
$757$			464.771i			0.613964i	−0.951715	+	0.306982i	$0.900681\pi$
$758$			291.775i			0.384927i
$759$	−64.0687	+	13.1136i	−0.0844120	+	0.0172774i
$760$	−626.137			−0.823865
$761$	−1046.21			−1.37479			−0.687394	−	0.726285i	$-0.741245\pi$
$761$	−1046.21			−1.37479			−0.687394	+	0.726285i	$0.741245\pi$
$762$	−225.566			−0.296019
$763$	260.364			0.341237
$764$		−	794.773i		−	1.04028i
$765$	16.0579			0.0209907
$766$		−	471.569i		−	0.615625i
$767$	1225.07			1.59722
$768$	−282.788			−0.368213
$769$			1061.61i			1.38051i	0.723566	+	0.690255i	$0.242502\pi$
$769$			1061.61i			1.38051i	−0.723566	+	0.690255i	$0.757498\pi$
$770$			12.6460i			0.0164233i
$771$	−558.732			−0.724685
$772$	−194.617			−0.252095
$773$		−	384.077i		−	0.496866i	−0.968649	−	0.248433i	$-0.920084\pi$
$773$		−	384.077i		−	0.496866i	0.968649	−	0.248433i	$-0.0799155\pi$
$774$			139.146i			0.179775i
$775$	−753.587			−0.972370
$776$			344.125i			0.443460i
$777$	−282.756			−0.363907
$778$			99.5950i			0.128014i
$779$		−	1407.10i		−	1.80629i
$780$			320.726i			0.411188i
$781$			40.1646i			0.0514272i
$782$	40.1170	−	8.21115i	0.0513005	−	0.0105002i
$783$	−119.923			−0.153159
$784$	−37.2150			−0.0474681
$785$	−99.5773			−0.126850
$786$	−191.790			−0.244008
$787$			493.710i			0.627332i	0.949533	+	0.313666i	$0.101557\pi$
$787$			493.710i			0.627332i	−0.949533	+	0.313666i	$0.898443\pi$
$788$	996.234			1.26426
$789$			306.750i			0.388783i
$790$	350.435			0.443589
$791$	432.651			0.546968
$792$		−	34.0785i		−	0.0430285i
$793$		−	282.631i		−	0.356407i
$794$	−370.803			−0.467006
$795$	−385.862			−0.485361
$796$		−	1075.31i		−	1.35090i
$797$			57.2706i			0.0718577i	0.999354	+	0.0359288i	$0.0114390\pi$
$797$			57.2706i			0.0718577i	−0.999354	+	0.0359288i	$0.988561\pi$
$798$	−137.922			−0.172835
$799$			139.541i			0.174644i
$800$	−534.683			−0.668354
$801$		−	188.415i		−	0.235224i
$802$			5.89779i			0.00735385i
$803$			8.26400i			0.0102914i
$804$		−	128.391i		−	0.159691i
$805$	−36.1023	−	176.384i	−0.0448475	−	0.219111i
$806$	−942.396			−1.16923
$807$	−96.0442			−0.119014
$808$	−651.729			−0.806595
$809$	−557.511			−0.689136			−0.344568	−	0.938761i	$-0.611975\pi$
$809$	−557.511			−0.689136			−0.344568	+	0.938761i	$0.611975\pi$
$810$		−	26.2045i		−	0.0323513i
$811$	181.946			0.224348			0.112174	−	0.993689i	$-0.464219\pi$
$811$	181.946			0.224348			0.112174	+	0.993689i	$0.464219\pi$
$812$		−	185.112i		−	0.227970i
$813$	−313.234			−0.385281
$814$	99.6807			0.122458
$815$			71.2379i			0.0874084i
$816$		−	16.6592i		−	0.0204156i
$817$	1441.43			1.76430
$818$	386.690			0.472726
$819$			163.865i			0.200080i
$820$			412.665i			0.503250i
$821$	−1108.28			−1.34992			−0.674959	−	0.737855i	$-0.735839\pi$
$821$	−1108.28			−1.34992			−0.674959	+	0.737855i	$0.735839\pi$
$822$		−	151.052i		−	0.183761i
$823$	639.531			0.777073			0.388537	−	0.921433i	$-0.372981\pi$
$823$	639.531			0.777073			0.388537	+	0.921433i	$0.372981\pi$
$824$		−	859.041i		−	1.04253i
$825$		−	46.1941i		−	0.0559928i
$826$		−	154.501i		−	0.187048i
$827$		−	1328.39i		−	1.60628i	−0.595791	−	0.803140i	$-0.703161\pi$
$827$		−	1328.39i		−	1.60628i	0.595791	−	0.803140i	$-0.296839\pi$
$828$	41.9446	+	204.928i	0.0506578	+	0.247497i
$829$	−585.865			−0.706712			−0.353356	−	0.935489i	$-0.614960\pi$
$829$	−585.865			−0.706712			−0.353356	+	0.935489i	$0.614960\pi$
$830$	−246.667			−0.297189
$831$	−264.361			−0.318124
$832$	−229.614			−0.275979
$833$			12.6640i			0.0152029i
$834$	−269.984			−0.323721
$835$		−	351.734i		−	0.421239i
$836$	−152.201			−0.182059
$837$	−241.023			−0.287961
$838$		−	331.966i		−	0.396140i
$839$			1423.22i			1.69633i	0.529734	+	0.848164i	$0.322292\pi$
$839$			1423.22i			1.69633i	−0.529734	+	0.848164i	$0.677708\pi$
$840$	93.8198			0.111690
$841$	−308.349			−0.366646
$842$			704.109i			0.836234i
$843$			367.167i			0.435548i
$844$	866.853			1.02708
$845$			761.021i			0.900617i
$846$	227.714			0.269165
$847$		−	313.006i		−	0.369547i
$848$			400.310i			0.472064i
$849$			860.688i			1.01377i
$850$			28.9247i			0.0340291i
$851$	−1390.33	+	284.573i	−1.63376	+	0.334398i
$852$	128.469			0.150786
$853$	−178.396			−0.209140			−0.104570	−	0.994518i	$-0.533347\pi$
$853$	−178.396			−0.209140			−0.104570	+	0.994518i	$0.533347\pi$
$854$	−35.6444			−0.0417382
$855$	−271.456			−0.317493
$856$		−	1449.95i		−	1.69386i
$857$	−15.2764			−0.0178254			−0.00891271	−	0.999960i	$-0.502837\pi$
$857$	−15.2764			−0.0178254			−0.00891271	+	0.999960i	$0.502837\pi$
$858$		−	57.7678i		−	0.0673285i
$859$	−460.564			−0.536163			−0.268081	−	0.963396i	$-0.586390\pi$
$859$	−460.564			−0.536163			−0.268081	+	0.963396i	$0.586390\pi$
$860$	−422.733			−0.491551
$861$			210.838i			0.244876i
$862$			783.115i			0.908486i
$863$	472.358			0.547345			0.273672	−	0.961823i	$-0.411762\pi$
$863$	472.358			0.547345			0.273672	+	0.961823i	$0.411762\pi$
$864$	−171.010			−0.197928
$865$		−	221.095i		−	0.255601i
$866$			214.189i			0.247331i
$867$	494.894			0.570812
$868$		−	372.040i		−	0.428617i
$869$	197.580			0.227365
$870$			116.390i			0.133782i
$871$		−	504.810i		−	0.579575i
$872$			680.962i			0.780920i
$873$			149.192i			0.170896i
$874$	−678.174	+	138.809i	−0.775943	+	0.158820i
$875$	322.871			0.368995
$876$	26.4330			0.0301746
$877$	−561.186			−0.639893			−0.319947	−	0.947436i	$-0.603665\pi$
$877$	−561.186			−0.639893			−0.319947	+	0.947436i	$0.603665\pi$
$878$	−424.372			−0.483340
$879$		−	1012.73i		−	1.15214i
$880$	25.8216			0.0293427
$881$		−	219.031i		−	0.248617i	−0.992244	−	0.124308i	$-0.960329\pi$
$881$		−	219.031i		−	0.248617i	0.992244	−	0.124308i	$-0.0396712\pi$
$882$	20.6661			0.0234310
$883$	335.195			0.379609			0.189805	−	0.981822i	$-0.439215\pi$
$883$	335.195			0.379609			0.189805	+	0.981822i	$0.439215\pi$
$884$		−	113.228i		−	0.128086i
$885$		−	304.087i		−	0.343601i
$886$	235.391			0.265678
$887$	−911.215			−1.02730			−0.513650	−	0.858000i	$-0.671707\pi$
$887$	−911.215			−1.02730			−0.513650	+	0.858000i	$0.671707\pi$
$888$		−	739.526i		−	0.832799i
$889$		−	350.125i		−	0.393841i
$890$	−182.864			−0.205465
$891$		−	14.7745i		−	0.0165819i
$892$	−1201.84			−1.34736
$893$		−	2358.92i		−	2.64157i
$894$			411.220i			0.459978i
$895$			821.371i			0.917733i
$896$		−	319.338i		−	0.356404i
$897$	164.918	+	805.737i	0.183855	+	0.898257i
$898$	96.1500			0.107071
$899$	1070.53			1.19080
$900$	−147.755			−0.164172
$901$	136.223			0.151191
$902$		−	74.3274i		−	0.0824029i
$903$	−215.982			−0.239183
$904$			1131.57i			1.25173i
$905$	−157.769			−0.174331
$906$	66.7001			0.0736204
$907$		−	727.272i		−	0.801843i	−0.916112	−	0.400922i	$-0.868690\pi$
$907$		−	727.272i		−	0.801843i	0.916112	−	0.400922i	$-0.131310\pi$
$908$			509.956i			0.561625i
$909$	−282.551			−0.310838
$910$	159.037			0.174766
$911$			160.214i			0.175866i	0.996126	+	0.0879329i	$0.0280261\pi$
$911$			160.214i			0.175866i	−0.996126	+	0.0879329i	$0.971974\pi$
$912$			281.621i			0.308795i
$913$	−139.074			−0.152327
$914$			193.346i			0.211538i
$915$	−70.1548			−0.0766719
$916$			554.766i			0.605640i
$917$		−	297.697i		−	0.324642i
$918$			9.25112i			0.0100775i
$919$			885.855i			0.963933i	0.876190	+	0.481967i	$0.160077\pi$
$919$			885.855i			0.963933i	−0.876190	+	0.481967i	$0.839923\pi$
$920$	461.318	−	94.4226i	0.501433	−	0.102633i
$921$	−929.649			−1.00939
$922$	213.966			0.232067
$923$	505.116			0.547255
$924$	22.8056			0.0246814
$925$		−	1002.44i		−	1.08372i
$926$	−825.065			−0.890999
$927$		−	372.430i		−	0.401758i
$928$	759.559			0.818490
$929$	152.772			0.164448			0.0822241	−	0.996614i	$-0.473798\pi$
$929$	152.772			0.164448			0.0822241	+	0.996614i	$0.473798\pi$
$930$			233.922i			0.251529i
$931$		−	214.083i		−	0.229950i
$932$	−445.357			−0.477851
$933$	−296.319			−0.317598
$934$		−	386.432i		−	0.413739i
$935$		−	8.78689i		−	0.00939775i
$936$	−428.577			−0.457881
$937$		−	584.541i		−	0.623843i	−0.950108	−	0.311921i	$-0.899027\pi$
$937$		−	584.541i		−	0.623843i	0.950108	−	0.311921i	$-0.100973\pi$
$938$	−63.6650			−0.0678731
$939$		−	228.712i		−	0.243569i
$940$			691.809i			0.735967i
$941$			591.924i			0.629037i	0.949251	+	0.314518i	$0.101843\pi$
$941$			591.924i			0.629037i	−0.949251	+	0.314518i	$0.898157\pi$
$942$		−	57.3676i		−	0.0608998i
$943$	212.193	+	1036.71i	0.225019	+	1.09937i
$944$	−315.474			−0.334188
$945$	40.6747			0.0430421
$946$	76.1409			0.0804872
$947$	1307.91			1.38111			0.690553	−	0.723281i	$-0.257367\pi$
$947$	1307.91			1.38111			0.690553	+	0.723281i	$0.257367\pi$
$948$		−	631.973i		−	0.666638i
$949$	103.929			0.109514
$950$		−	488.969i		−	0.514704i
$951$	71.7026			0.0753970
$952$	−33.1217			−0.0347917
$953$			1276.91i			1.33989i	0.742412	+	0.669943i	$0.233682\pi$
$953$			1276.91i			1.33989i	−0.742412	+	0.669943i	$0.766318\pi$
$954$		−	222.299i		−	0.233018i
$955$	−775.663			−0.812213
$956$	−848.321			−0.887365
$957$			65.6222i			0.0685708i
$958$			281.035i			0.293356i
$959$	234.463			0.244487
$960$			56.9951i			0.0593699i
$961$	1190.56			1.23888
$962$		−	1253.60i		−	1.30312i
$963$		−	628.612i		−	0.652764i
$964$		−	832.564i		−	0.863656i
$965$			189.938i			0.196827i
$966$	101.617	−	20.7989i	0.105193	−	0.0215310i
$967$	223.374			0.230997			0.115498	−	0.993308i	$-0.463154\pi$
$967$	223.374			0.230997			0.115498	+	0.993308i	$0.463154\pi$
$968$	818.643			0.845705
$969$	95.8336			0.0988995
$970$	144.797			0.149275
$971$		−	1668.29i		−	1.71811i	−0.511879	−	0.859057i	$-0.671050\pi$
$971$		−	1668.29i		−	1.71811i	0.511879	−	0.859057i	$-0.328950\pi$
$972$	−47.2571			−0.0486184
$973$		−	419.069i		−	0.430698i
$974$	−196.145			−0.201381
$975$	−580.943			−0.595839
$976$			72.7818i			0.0745715i
$977$		−	494.183i		−	0.505817i	−0.967490	−	0.252909i	$-0.918613\pi$
$977$		−	494.183i		−	0.505817i	0.967490	−	0.252909i	$-0.0813872\pi$
$978$	−41.0410			−0.0419642
$979$	−103.101			−0.105313
$980$			62.7850i			0.0640663i
$981$			295.225i			0.300943i
$982$	175.360			0.178574
$983$		−	1237.71i		−	1.25912i	−0.776953	−	0.629559i	$-0.783236\pi$
$983$		−	1237.71i		−	1.25912i	0.776953	−	0.629559i	$-0.216764\pi$
$984$	−551.431			−0.560398
$985$		−	972.280i		−	0.987087i
$986$		−	41.0898i		−	0.0416732i
$987$			353.458i			0.358114i
$988$			1914.10i			1.93735i
$989$	−1062.00	+	217.370i	−1.07381	+	0.219788i
$990$	−14.3392			−0.0144840
$991$	643.200			0.649042			0.324521	−	0.945879i	$-0.394797\pi$
$991$	643.200			0.649042			0.324521	+	0.945879i	$0.394797\pi$
$992$	1526.57			1.53888
$993$	−917.779			−0.924248
$994$		−	63.7036i		−	0.0640881i
$995$	−1049.46			−1.05473
$996$			444.838i			0.446624i
$997$	210.864			0.211499			0.105749	−	0.994393i	$-0.466276\pi$
$997$	210.864			0.211499			0.105749	+	0.994393i	$0.466276\pi$
$998$	−613.962			−0.615193
$999$		−	320.615i		−	0.320936i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.3.f.a.22.18	yes	48
23.22	odd	2	inner	483.3.f.a.22.17	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.3.f.a.22.17	✓	48	23.22	odd	2	inner
483.3.f.a.22.18	yes	48	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 483 = 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	483.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.984102 q^{2} +1.73205 q^{3} -3.03154 q^{4} +2.95865i q^{5} -1.70451 q^{6} -2.64575i q^{7} +6.91976 q^{8} +3.00000 q^{9} -2.91162i q^{10} -1.64161i q^{11} -5.25079 q^{12} -20.6451 q^{13} +2.60369i q^{14} +5.12454i q^{15} +5.31643 q^{16} -1.80914i q^{17} -2.95231 q^{18} +30.5833i q^{19} -8.96928i q^{20} -4.58258i q^{21} +1.61551i q^{22} +(-4.61202 - 22.5328i) q^{23} +11.9854 q^{24} +16.2464 q^{25} +20.3169 q^{26} +5.19615 q^{27} +8.02071i q^{28} -23.0792 q^{29} -5.04307i q^{30} -46.3849 q^{31} -32.9109 q^{32} -2.84335i q^{33} +1.78038i q^{34} +7.82786 q^{35} -9.09463 q^{36} -61.7024i q^{37} -30.0971i q^{38} -35.7583 q^{39} +20.4732i q^{40} -46.0087 q^{41} +4.50972i q^{42} -47.1312i q^{43} +4.97660i q^{44} +8.87596i q^{45} +(4.53870 + 22.1746i) q^{46} -77.1309 q^{47} +9.20832 q^{48} -7.00000 q^{49} -15.9881 q^{50} -3.13352i q^{51} +62.5864 q^{52} +75.2969i q^{53} -5.11354 q^{54} +4.85694 q^{55} -18.3080i q^{56} +52.9719i q^{57} +22.7123 q^{58} -59.3394 q^{59} -15.5353i q^{60} +13.6900i q^{61} +45.6475 q^{62} -7.93725i q^{63} +11.1220 q^{64} -61.0816i q^{65} +2.79814i q^{66} +24.4518i q^{67} +5.48449i q^{68} +(-7.98826 - 39.0280i) q^{69} -7.70341 q^{70} -24.4667 q^{71} +20.7593 q^{72} -5.03409 q^{73} +60.7214i q^{74} +28.1395 q^{75} -92.7147i q^{76} -4.34328 q^{77} +35.1898 q^{78} +120.358i q^{79} +15.7295i q^{80} +9.00000 q^{81} +45.2773 q^{82} -84.7183i q^{83} +13.8923i q^{84} +5.35262 q^{85} +46.3819i q^{86} -39.9744 q^{87} -11.3595i q^{88} -62.8049i q^{89} -8.73485i q^{90} +54.6217i q^{91} +(13.9815 + 68.3093i) q^{92} -80.3410 q^{93} +75.9047 q^{94} -90.4855 q^{95} -57.0034 q^{96} +49.7308i q^{97} +6.88871 q^{98} -4.92482i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 4 q^{2} + 116 q^{4} - 24 q^{6} - 4 q^{8} + 144 q^{9} + 16 q^{13} + 324 q^{16} + 12 q^{18} - 4 q^{23} - 24 q^{24} - 176 q^{25} + 136 q^{26} - 128 q^{29} - 8 q^{31} - 252 q^{32} - 56 q^{35} + 348 q^{36}+ \cdots - 28 q^{98}+O(q^{100}) \) 48 * q + 4 * q^2 + 116 * q^4 - 24 * q^6 - 4 * q^8 + 144 * q^9 + 16 * q^13 + 324 * q^16 + 12 * q^18 - 4 * q^23 - 24 * q^24 - 176 * q^25 + 136 * q^26 - 128 * q^29 - 8 * q^31 - 252 * q^32 - 56 * q^35 + 348 * q^36 + 96 * q^39 - 24 * q^41 - 148 * q^46 - 408 * q^47 - 96 * q^48 - 336 * q^49 + 236 * q^50 - 32 * q^52 - 72 * q^54 - 24 * q^55 - 56 * q^58 + 136 * q^59 + 184 * q^62 + 716 * q^64 - 48 * q^69 - 112 * q^70 + 48 * q^71 - 12 * q^72 - 224 * q^73 - 48 * q^75 + 224 * q^77 - 96 * q^78 + 432 * q^81 + 640 * q^82 - 424 * q^85 + 312 * q^87 + 1060 * q^92 + 192 * q^93 + 216 * q^94 + 624 * q^95 + 48 * q^96 - 28 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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