LMFDB - Embedded newform 275.3.c.h.76.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [275,3,Mod(76,275)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("275.76"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 275 = 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	275.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,4,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$7.49320726991$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 21x^{6} + 130x^{4} + 215x^{2} + 25 $ x^8 + 21x^6 + 130x^4 + 215*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			76.8
Root			$3.27682i$ of defining polynomial
Character	$\chi$	$=$	275.76
Dual form			275.3.c.h.76.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+3.27682i q^{2} -1.25169 q^{3} -6.73752 q^{4} -4.10155i q^{6} +3.94258i q^{7} -8.97034i q^{8} -7.43328 q^{9} +(-6.83371 - 8.61977i) q^{11} +8.43328 q^{12} -6.61973i q^{13} -12.9191 q^{14} +2.44407 q^{16} -1.49050i q^{17} -24.3575i q^{18} -18.5757i q^{19} -4.93488i q^{21} +(28.2454 - 22.3928i) q^{22} +18.1115 q^{23} +11.2281i q^{24} +21.6916 q^{26} +20.5693 q^{27} -26.5632i q^{28} +24.3055i q^{29} -16.8211 q^{31} -27.8726i q^{32} +(8.55368 + 10.7893i) q^{33} +4.88410 q^{34} +50.0818 q^{36} -65.2003 q^{37} +60.8692 q^{38} +8.28584i q^{39} -67.5290i q^{41} +16.1707 q^{42} +65.6159i q^{43} +(46.0422 + 58.0759i) q^{44} +59.3480i q^{46} -81.4098 q^{47} -3.05921 q^{48} +33.4561 q^{49} +1.86564i q^{51} +44.6005i q^{52} +16.2112 q^{53} +67.4019i q^{54} +35.3663 q^{56} +23.2510i q^{57} -79.6445 q^{58} -75.7037 q^{59} -4.85215i q^{61} -55.1197i q^{62} -29.3063i q^{63} +101.110 q^{64} +(-35.3544 + 28.0288i) q^{66} -12.1006 q^{67} +10.0423i q^{68} -22.6699 q^{69} -79.6844 q^{71} +66.6790i q^{72} +42.6954i q^{73} -213.649i q^{74} +125.154i q^{76} +(33.9841 - 26.9424i) q^{77} -27.1512 q^{78} -77.6420i q^{79} +41.1531 q^{81} +221.280 q^{82} -37.1521i q^{83} +33.2489i q^{84} -215.011 q^{86} -30.4229i q^{87} +(-77.3223 + 61.3007i) q^{88} -102.707 q^{89} +26.0988 q^{91} -122.026 q^{92} +21.0548 q^{93} -266.765i q^{94} +34.8878i q^{96} -11.4069 q^{97} +109.629i q^{98} +(50.7968 + 64.0731i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{3} - 10 q^{4} + 32 q^{9} - q^{11} - 24 q^{12} + 18 q^{14} - 14 q^{16} - 35 q^{22} + 4 q^{23} + 68 q^{26} + 142 q^{27} - 42 q^{31} - 31 q^{33} - 142 q^{34} - 84 q^{36} - 104 q^{37} + 190 q^{38}+ \cdots + 368 q^{99}+O(q^{100}) $ 8 * q + 4 * q^3 - 10 * q^4 + 32 * q^9 - q^11 - 24 * q^12 + 18 * q^14 - 14 * q^16 - 35 * q^22 + 4 * q^23 + 68 * q^26 + 142 * q^27 - 42 * q^31 - 31 * q^33 - 142 * q^34 - 84 * q^36 - 104 * q^37 + 190 * q^38 + 50 * q^42 + 69 * q^44 - 64 * q^47 - 86 * q^48 - 50 * q^49 - 286 * q^53 + 166 * q^56 - 130 * q^58 - 160 * q^59 + 288 * q^64 - 315 * q^66 + 46 * q^67 - 16 * q^69 + 246 * q^71 + 80 * q^77 + 620 * q^78 + 304 * q^81 + 280 * q^82 - 282 * q^86 - 365 * q^88 - 18 * q^89 - 142 * q^91 + 26 * q^92 + 818 * q^93 - 264 * q^97 + 368 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/275\mathbb{Z}\right)^\times$.

$n$	$101$	$177$
$\chi(n)$	$-1$	$1$

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 $

$2$			3.27682i			1.63841i	0.573503	+	0.819204i	$0.305584\pi$
$2$			3.27682i			1.63841i	−0.573503	+	0.819204i	$0.694416\pi$
$3$	−1.25169			−0.417230			−0.208615	−	0.977998i	$-0.566895\pi$
$3$	−1.25169			−0.417230			−0.208615	+	0.977998i	$0.566895\pi$
$4$	−6.73752			−1.68438
$5$		0			0
$5$		0			0
$6$		−	4.10155i		−	0.683592i
$7$			3.94258i			0.563226i	0.959528	+	0.281613i	$0.0908694\pi$
$7$			3.94258i			0.563226i	−0.959528	+	0.281613i	$0.909131\pi$
$8$		−	8.97034i		−	1.12129i
$9$	−7.43328			−0.825919
$10$		0			0
$11$	−6.83371	−	8.61977i	−0.621246	−	0.783616i
$11$	−6.83371	−	8.61977i	−0.621246	−	0.783616i
$12$	8.43328			0.702773
$13$		−	6.61973i		−	0.509210i	−0.967045	−	0.254605i	$-0.918055\pi$
$13$		−	6.61973i		−	0.509210i	0.967045	−	0.254605i	$-0.0819454\pi$
$14$	−12.9191			−0.922793
$15$		0			0
$16$	2.44407			0.152754
$17$		−	1.49050i		−	0.0876766i	−0.999039	−	0.0438383i	$-0.986041\pi$
$17$		−	1.49050i		−	0.0876766i	0.999039	−	0.0438383i	$-0.0139586\pi$
$18$		−	24.3575i		−	1.35319i
$19$		−	18.5757i		−	0.977669i	−0.872377	−	0.488835i	$-0.837422\pi$
$19$		−	18.5757i		−	0.977669i	0.872377	−	0.488835i	$-0.162578\pi$
$20$		0			0
$21$		−	4.93488i		−	0.234994i
$22$	28.2454	−	22.3928i	1.28388	−	1.01785i
$23$	18.1115			0.787456			0.393728	−	0.919227i	$-0.371185\pi$
$23$	18.1115			0.787456			0.393728	+	0.919227i	$0.371185\pi$
$24$			11.2281i			0.467836i
$25$		0			0
$26$	21.6916			0.834293
$27$	20.5693			0.761828
$28$		−	26.5632i		−	0.948686i
$29$			24.3055i			0.838119i	0.907959	+	0.419060i	$0.137640\pi$
$29$			24.3055i			0.838119i	−0.907959	+	0.419060i	$0.862360\pi$
$30$		0			0
$31$	−16.8211			−0.542617			−0.271309	−	0.962492i	$-0.587456\pi$
$31$	−16.8211			−0.542617			−0.271309	+	0.962492i	$0.587456\pi$
$32$		−	27.8726i		−	0.871018i
$33$	8.55368	+	10.7893i	0.259202	+	0.326948i
$34$	4.88410			0.143650
$35$		0			0
$36$	50.0818			1.39116
$37$	−65.2003			−1.76217			−0.881086	−	0.472957i	$-0.843187\pi$
$37$	−65.2003			−1.76217			−0.881086	+	0.472957i	$0.843187\pi$
$38$	60.8692			1.60182
$39$			8.28584i			0.212457i
$40$		0			0
$41$		−	67.5290i		−	1.64705i	−0.567282	−	0.823524i	$-0.692005\pi$
$41$		−	67.5290i		−	1.64705i	0.567282	−	0.823524i	$-0.307995\pi$
$42$	16.1707			0.385017
$43$			65.6159i			1.52595i	0.646427	+	0.762976i	$0.276263\pi$
$43$			65.6159i			1.52595i	−0.646427	+	0.762976i	$0.723737\pi$
$44$	46.0422	+	58.0759i	1.04641	+	1.31991i
$45$		0			0
$46$			59.3480i			1.29017i
$47$	−81.4098			−1.73212			−0.866062	−	0.499937i	$-0.833356\pi$
$47$	−81.4098			−1.73212			−0.866062	+	0.499937i	$0.833356\pi$
$48$	−3.05921			−0.0637336
$49$	33.4561			0.682777
$50$		0			0
$51$			1.86564i			0.0365813i
$52$			44.6005i			0.857703i
$53$	16.2112			0.305872			0.152936	−	0.988236i	$-0.451127\pi$
$53$	16.2112			0.305872			0.152936	+	0.988236i	$0.451127\pi$
$54$			67.4019i			1.24818i
$55$		0			0
$56$	35.3663			0.631540
$57$			23.2510i			0.407912i
$58$	−79.6445			−1.37318
$59$	−75.7037			−1.28311			−0.641557	−	0.767075i	$-0.721711\pi$
$59$	−75.7037			−1.28311			−0.641557	+	0.767075i	$0.721711\pi$
$60$		0			0
$61$		−	4.85215i		−	0.0795434i	−0.999209	−	0.0397717i	$-0.987337\pi$
$61$		−	4.85215i		−	0.0795434i	0.999209	−	0.0397717i	$-0.0126631\pi$
$62$		−	55.1197i		−	0.889028i
$63$		−	29.3063i		−	0.465179i
$64$	101.110			1.57984
$65$		0			0
$66$	−35.3544	+	28.0288i	−0.535673	+	0.424679i
$67$	−12.1006			−0.180606			−0.0903029	−	0.995914i	$-0.528784\pi$
$67$	−12.1006			−0.180606			−0.0903029	+	0.995914i	$0.528784\pi$
$68$			10.0423i			0.147681i
$69$	−22.6699			−0.328550
$70$		0			0
$71$	−79.6844			−1.12232			−0.561158	−	0.827709i	$-0.689644\pi$
$71$	−79.6844			−1.12232			−0.561158	+	0.827709i	$0.689644\pi$
$72$			66.6790i			0.926097i
$73$			42.6954i			0.584869i	0.956286	+	0.292434i	$0.0944653\pi$
$73$			42.6954i			0.584869i	−0.956286	+	0.292434i	$0.905535\pi$
$74$		−	213.649i		−	2.88715i
$75$		0			0
$76$			125.154i			1.64677i
$77$	33.9841	−	26.9424i	0.441352	−	0.349902i
$78$	−27.1512			−0.348092
$79$		−	77.6420i		−	0.982810i	−0.870931	−	0.491405i	$-0.836484\pi$
$79$		−	77.6420i		−	0.982810i	0.870931	−	0.491405i	$-0.163516\pi$
$80$		0			0
$81$	41.1531			0.508062
$82$	221.280			2.69854
$83$		−	37.1521i		−	0.447616i	−0.974633	−	0.223808i	$-0.928151\pi$
$83$		−	37.1521i		−	0.447616i	0.974633	−	0.223808i	$-0.0718488\pi$
$84$			33.2489i			0.395820i
$85$		0			0
$86$	−215.011			−2.50013
$87$		−	30.4229i		−	0.349688i
$88$	−77.3223	+	61.3007i	−0.878662	+	0.696598i
$89$	−102.707			−1.15401			−0.577006	−	0.816740i	$-0.695779\pi$
$89$	−102.707			−1.15401			−0.577006	+	0.816740i	$0.695779\pi$
$90$		0			0
$91$	26.0988			0.286800
$92$	−122.026			−1.32637
$93$	21.0548			0.226396
$94$		−	266.765i		−	2.83792i
$95$		0			0
$96$			34.8878i			0.363415i
$97$	−11.4069			−0.117597			−0.0587985	−	0.998270i	$-0.518727\pi$
$97$	−11.4069			−0.117597			−0.0587985	+	0.998270i	$0.518727\pi$
$98$			109.629i			1.11867i
$99$	50.7968	+	64.0731i	0.513099	+	0.647203i
$100$		0			0
$101$		−	156.659i		−	1.55108i	−0.631301	−	0.775538i	$-0.717479\pi$
$101$		−	156.659i		−	1.55108i	0.631301	−	0.775538i	$-0.282521\pi$
$102$	−6.11337			−0.0599350
$103$	−105.744			−1.02664			−0.513322	−	0.858196i	$-0.671585\pi$
$103$	−105.744			−1.02664			−0.513322	+	0.858196i	$0.671585\pi$
$104$	−59.3812			−0.570973
$105$		0			0
$106$			53.1212i			0.501143i
$107$			192.014i			1.79453i	0.441497	+	0.897263i	$0.354448\pi$
$107$			192.014i			1.79453i	−0.441497	+	0.897263i	$0.645552\pi$
$108$	−138.586			−1.28321
$109$			162.478i			1.49062i	0.666716	+	0.745312i	$0.267699\pi$
$109$			162.478i			1.49062i	−0.666716	+	0.745312i	$0.732301\pi$
$110$		0			0
$111$	81.6105			0.735230
$112$			9.63594i			0.0860351i
$113$	−75.5761			−0.668815			−0.334408	−	0.942429i	$-0.608536\pi$
$113$	−75.5761			−0.668815			−0.334408	+	0.942429i	$0.608536\pi$
$114$	−76.1893			−0.668327
$115$		0			0
$116$		−	163.758i		−	1.41171i
$117$			49.2063i			0.420566i
$118$		−	248.067i		−	2.10226i
$119$	5.87642			0.0493817
$120$		0			0
$121$	−27.6009	+	117.810i	−0.228107	+	0.973636i
$122$	15.8996			0.130325
$123$			84.5252i			0.687197i
$124$	113.333			0.913973
$125$		0			0
$126$	96.0312			0.762153
$127$			125.050i			0.984647i	0.870412	+	0.492323i	$0.163852\pi$
$127$			125.050i			0.984647i	−0.870412	+	0.492323i	$0.836148\pi$
$128$			219.827i			1.71740i
$129$		−	82.1307i		−	0.636672i
$130$		0			0
$131$			10.1412i			0.0774136i	0.999251	+	0.0387068i	$0.0123238\pi$
$131$			10.1412i			0.0774136i	−0.999251	+	0.0387068i	$0.987676\pi$
$132$	−57.6305	−	72.6929i	−0.436595	−	0.550704i
$133$	73.2362			0.550648
$134$		−	39.6514i		−	0.295906i
$135$		0			0
$136$	−13.3703			−0.0983111
$137$	−70.4327			−0.514107			−0.257054	−	0.966397i	$-0.582752\pi$
$137$	−70.4327			−0.514107			−0.257054	+	0.966397i	$0.582752\pi$
$138$		−	74.2852i		−	0.538299i
$139$			203.327i			1.46279i	0.681957	+	0.731393i	$0.261129\pi$
$139$			203.327i			1.46279i	−0.681957	+	0.731393i	$0.738871\pi$
$140$		0			0
$141$	101.900			0.722693
$142$		−	261.111i		−	1.83881i
$143$	−57.0605	+	45.2373i	−0.399025	+	0.316345i
$144$	−18.1674			−0.126163
$145$		0			0
$146$	−139.905			−0.958253
$147$	−41.8766			−0.284875
$148$	439.288			2.96816
$149$		−	98.2512i		−	0.659404i	−0.944085	−	0.329702i	$-0.893052\pi$
$149$		−	98.2512i		−	0.659404i	0.944085	−	0.329702i	$-0.106948\pi$
$150$		0			0
$151$		−	2.04381i		−	0.0135351i	−0.999977	−	0.00676757i	$-0.997846\pi$
$151$		−	2.04381i		−	0.0135351i	0.999977	−	0.00676757i	$-0.00215420\pi$
$152$	−166.630			−1.09625
$153$			11.0793i			0.0724138i
$154$	88.2854	+	111.360i	0.573282	+	0.723115i
$155$		0			0
$156$		−	55.8260i		−	0.357859i
$157$	160.316			1.02112			0.510561	−	0.859841i	$-0.329438\pi$
$157$	160.316			1.02112			0.510561	+	0.859841i	$0.329438\pi$
$158$	254.418			1.61024
$159$	−20.2914			−0.127619
$160$		0			0
$161$			71.4060i			0.443515i
$162$			134.851i			0.832413i
$163$	−59.3198			−0.363925			−0.181962	−	0.983305i	$-0.558245\pi$
$163$	−59.3198			−0.363925			−0.181962	+	0.983305i	$0.558245\pi$
$164$			454.977i			2.77425i
$165$		0			0
$166$	121.741			0.733377
$167$		−	149.432i		−	0.894800i	−0.894334	−	0.447400i	$-0.852350\pi$
$167$		−	149.432i		−	0.894800i	0.894334	−	0.447400i	$-0.147650\pi$
$168$	−44.2676			−0.263497
$169$	125.179			0.740705
$170$		0			0
$171$			138.078i			0.807476i
$172$		−	442.089i		−	2.57028i
$173$		−	71.7530i		−	0.414757i	−0.978261	−	0.207379i	$-0.933507\pi$
$173$		−	71.7530i		−	0.414757i	0.978261	−	0.207379i	$-0.0664932\pi$
$174$	99.6901			0.572932
$175$		0			0
$176$	−16.7021	−	21.0673i	−0.0948980	−	0.119701i
$177$	94.7575			0.535353
$178$		−	336.552i		−	1.89074i
$179$	143.137			0.799647			0.399824	−	0.916592i	$-0.369071\pi$
$179$	143.137			0.799647			0.399824	+	0.916592i	$0.369071\pi$
$180$		0			0
$181$	305.844			1.68974			0.844872	−	0.534968i	$-0.179676\pi$
$181$	305.844			1.68974			0.844872	+	0.534968i	$0.179676\pi$
$182$			85.5210i			0.469895i
$183$			6.07338i			0.0331879i
$184$		−	162.466i		−	0.882968i
$185$		0			0
$186$			68.9928i			0.370929i
$187$	−12.8478	+	10.1857i	−0.0687047	+	0.0544687i
$188$	548.500			2.91755
$189$			81.0963i			0.429081i
$190$		0			0
$191$	−1.64065			−0.00858982			−0.00429491	−	0.999991i	$-0.501367\pi$
$191$	−1.64065			−0.00858982			−0.00429491	+	0.999991i	$0.501367\pi$
$192$	−126.558			−0.659155
$193$		−	287.478i		−	1.48952i	−0.667330	−	0.744762i	$-0.732563\pi$
$193$		−	287.478i		−	1.48952i	0.667330	−	0.744762i	$-0.267437\pi$
$194$		−	37.3784i		−	0.192672i
$195$		0			0
$196$	−225.411			−1.15006
$197$		−	109.774i		−	0.557227i	−0.960403	−	0.278614i	$-0.910125\pi$
$197$		−	109.774i		−	0.557227i	0.960403	−	0.278614i	$-0.0898749\pi$
$198$	−209.956	+	166.452i	−1.06038	+	0.840666i
$199$	38.8559			0.195256			0.0976279	−	0.995223i	$-0.468874\pi$
$199$	38.8559			0.195256			0.0976279	+	0.995223i	$0.468874\pi$
$200$		0			0
$201$	15.1462			0.0753541
$202$	513.341			2.54129
$203$	−95.8262			−0.472050
$204$		−	12.5698i		−	0.0616167i
$205$		0			0
$206$		−	346.504i		−	1.68206i
$207$	−134.628			−0.650375
$208$		−	16.1791i		−	0.0777840i
$209$	−160.118	+	126.941i	−0.766117	+	0.607373i
$210$		0			0
$211$			89.4659i			0.424009i	0.977269	+	0.212005i	$0.0679992\pi$
$211$			89.4659i			0.424009i	−0.977269	+	0.212005i	$0.932001\pi$
$212$	−109.223			−0.515205
$213$	99.7400			0.468263
$214$	−629.195			−2.94016
$215$		0			0
$216$		−	184.514i		−	0.854231i
$217$		−	66.3186i		−	0.305616i
$218$	−532.410			−2.44225
$219$		−	53.4414i		−	0.244025i
$220$		0			0
$221$	−9.86672			−0.0446458
$222$			267.423i			1.20461i
$223$	189.941			0.851752			0.425876	−	0.904782i	$-0.359966\pi$
$223$	189.941			0.851752			0.425876	+	0.904782i	$0.359966\pi$
$224$	109.890			0.490580
$225$		0			0
$226$		−	247.649i		−	1.09579i
$227$			344.741i			1.51868i	0.650691	+	0.759342i	$0.274479\pi$
$227$			344.741i			1.51868i	−0.650691	+	0.759342i	$0.725521\pi$
$228$		−	156.654i		−	0.687079i
$229$	−114.108			−0.498288			−0.249144	−	0.968466i	$-0.580149\pi$
$229$	−114.108			−0.498288			−0.249144	+	0.968466i	$0.580149\pi$
$230$		0			0
$231$	−42.5376	+	33.7235i	−0.184145	+	0.145989i
$232$	218.028			0.939777
$233$		−	16.9887i		−	0.0729129i	−0.999335	−	0.0364565i	$-0.988393\pi$
$233$		−	16.9887i		−	0.0729129i	0.999335	−	0.0364565i	$-0.0116070\pi$
$234$	−161.240			−0.689059
$235$		0			0
$236$	510.055			2.16125
$237$			97.1836i			0.410057i
$238$			19.2559i			0.0809074i
$239$		−	212.744i		−	0.890144i	−0.895495	−	0.445072i	$-0.853178\pi$
$239$		−	212.744i		−	0.890144i	0.895495	−	0.445072i	$-0.146822\pi$
$240$		0			0
$241$		−	362.416i		−	1.50380i	−0.659277	−	0.751900i	$-0.729138\pi$
$241$		−	362.416i		−	1.50380i	0.659277	−	0.751900i	$-0.270862\pi$
$242$	−386.042	−	90.4430i	−1.59521	−	0.373731i
$243$	−236.635			−0.973806
$244$			32.6914i			0.133981i
$245$		0			0
$246$	−276.974			−1.12591
$247$	−122.966			−0.497839
$248$			150.891i			0.608432i
$249$			46.5029i			0.186759i
$250$		0			0
$251$	50.5799			0.201514			0.100757	−	0.994911i	$-0.467874\pi$
$251$	50.5799			0.201514			0.100757	+	0.994911i	$0.467874\pi$
$252$			197.452i			0.783538i
$253$	−123.769	−	156.117i	−0.489204	−	0.617063i
$254$	−409.766			−1.61325
$255$		0			0
$256$	−315.894			−1.23396
$257$	429.305			1.67045			0.835224	−	0.549910i	$-0.185338\pi$
$257$	429.305			1.67045			0.835224	+	0.549910i	$0.185338\pi$
$258$	269.127			1.04313
$259$		−	257.058i		−	0.992500i
$260$		0			0
$261$		−	180.669i		−	0.692219i
$262$	−33.2308			−0.126835
$263$			157.886i			0.600329i	0.953888	+	0.300164i	$0.0970415\pi$
$263$			157.886i			0.600329i	−0.953888	+	0.300164i	$0.902958\pi$
$264$	96.7834	−	76.7294i	0.366604	−	0.290641i
$265$		0			0
$266$			239.982i			0.902186i
$267$	128.557			0.481488
$268$	81.5280			0.304209
$269$	−213.381			−0.793237			−0.396618	−	0.917984i	$-0.629816\pi$
$269$	−213.381			−0.793237			−0.396618	+	0.917984i	$0.629816\pi$
$270$		0			0
$271$			466.412i			1.72108i	0.509384	+	0.860540i	$0.329873\pi$
$271$			466.412i			1.72108i	−0.509384	+	0.860540i	$0.670127\pi$
$272$		−	3.64289i		−	0.0133930i
$273$	−32.6676			−0.119661
$274$		−	230.795i		−	0.842317i
$275$		0			0
$276$	152.739			0.553403
$277$		−	66.3436i		−	0.239508i	−0.992804	−	0.119754i	$-0.961789\pi$
$277$		−	66.3436i		−	0.239508i	0.992804	−	0.119754i	$-0.0382105\pi$
$278$	−666.265			−2.39664
$279$	125.036			0.448158
$280$		0			0
$281$			240.196i			0.854791i	0.904065	+	0.427396i	$0.140569\pi$
$281$			240.196i			0.854791i	−0.904065	+	0.427396i	$0.859431\pi$
$282$			333.907i			1.18407i
$283$			270.527i			0.955924i	0.878380	+	0.477962i	$0.158624\pi$
$283$			270.527i			0.955924i	−0.878380	+	0.477962i	$0.841376\pi$
$284$	536.875			1.89040
$285$		0			0
$286$	−148.234	−	186.977i	−0.518301	−	0.653765i
$287$	266.238			0.927659
$288$			207.185i			0.719391i
$289$	286.778			0.992313
$290$		0			0
$291$	14.2779			0.0490650
$292$		−	287.661i		−	0.985141i
$293$		−	543.407i		−	1.85463i	−0.374282	−	0.927315i	$-0.622111\pi$
$293$		−	543.407i		−	1.85463i	0.374282	−	0.927315i	$-0.377889\pi$
$294$		−	137.222i		−	0.466741i
$295$		0			0
$296$			584.869i			1.97591i
$297$	−140.565	−	177.303i	−0.473282	−	0.596980i
$298$	321.951			1.08037
$299$		−	119.893i		−	0.400980i
$300$		0			0
$301$	−258.696			−0.859455
$302$	6.69718			0.0221761
$303$			196.088i			0.647154i
$304$		−	45.4003i		−	0.149343i
$305$		0			0
$306$	−36.3049			−0.118643
$307$		−	314.156i		−	1.02331i	−0.859191	−	0.511654i	$-0.829033\pi$
$307$		−	314.156i		−	1.02331i	0.859191	−	0.511654i	$-0.170967\pi$
$308$	−228.969	+	181.525i	−0.743405	+	0.589367i
$309$	132.359			0.428346
$310$		0			0
$311$	−374.834			−1.20525			−0.602627	−	0.798023i	$-0.705879\pi$
$311$	−374.834			−1.20525			−0.602627	+	0.798023i	$0.705879\pi$
$312$	74.3268			0.238227
$313$	206.435			0.659538			0.329769	−	0.944062i	$-0.393029\pi$
$313$	206.435			0.659538			0.329769	+	0.944062i	$0.393029\pi$
$314$			525.327i			1.67301i
$315$		0			0
$316$			523.114i			1.65542i
$317$	−273.867			−0.863935			−0.431968	−	0.901889i	$-0.642181\pi$
$317$	−273.867			−0.863935			−0.431968	+	0.901889i	$0.642181\pi$
$318$		−	66.4912i		−	0.209092i
$319$	209.508	−	166.096i	0.656763	−	0.520678i
$320$		0			0
$321$		−	240.342i		−	0.748729i
$322$	−233.984			−0.726659
$323$	−27.6871			−0.0857187
$324$	−277.269			−0.855770
$325$		0			0
$326$		−	194.380i		−	0.596257i
$327$		−	203.372i		−	0.621932i
$328$	−605.758			−1.84682
$329$		−	320.965i		−	0.975576i
$330$		0			0
$331$	−441.950			−1.33520			−0.667598	−	0.744522i	$-0.732677\pi$
$331$	−441.950			−1.33520			−0.667598	+	0.744522i	$0.732677\pi$
$332$			250.313i			0.753955i
$333$	484.652			1.45541
$334$	489.660			1.46605
$335$		0			0
$336$		−	12.0612i		−	0.0358964i
$337$		−	130.963i		−	0.388613i	−0.980941	−	0.194307i	$-0.937754\pi$
$337$		−	130.963i		−	0.388613i	0.980941	−	0.194307i	$-0.0622457\pi$
$338$			410.189i			1.21358i
$339$	94.5978			0.279049
$340$		0			0
$341$	114.951	+	144.994i	0.337099	+	0.425203i
$342$	−452.457			−1.32297
$343$			325.090i			0.947783i
$344$	588.597			1.71104
$345$		0			0
$346$	235.121			0.679541
$347$		−	188.943i		−	0.544503i	−0.962226	−	0.272251i	$-0.912232\pi$
$347$		−	188.943i		−	0.544503i	0.962226	−	0.272251i	$-0.0877683\pi$
$348$			204.975i			0.589008i
$349$			500.080i			1.43289i	0.697642	+	0.716446i	$0.254233\pi$
$349$			500.080i			1.43289i	−0.697642	+	0.716446i	$0.745767\pi$
$350$		0			0
$351$		−	136.163i		−	0.387930i
$352$	−240.255	+	190.473i	−0.682544	+	0.541117i
$353$	414.398			1.17393			0.586966	−	0.809611i	$-0.300322\pi$
$353$	414.398			1.17393			0.586966	+	0.809611i	$0.300322\pi$
$354$			310.503i			0.877127i
$355$		0			0
$356$	691.991			1.94379
$357$	−7.35545			−0.0206035
$358$			469.033i			1.31015i
$359$		−	212.272i		−	0.591288i	−0.955298	−	0.295644i	$-0.904466\pi$
$359$		−	212.272i		−	0.591288i	0.955298	−	0.295644i	$-0.0955342\pi$
$360$		0			0
$361$	15.9429			0.0441633
$362$			1002.19i			2.76849i
$363$	34.5477	−	147.461i	0.0951728	−	0.406230i
$364$	−175.841			−0.483080
$365$		0			0
$366$	−19.9013			−0.0543753
$367$	−460.146			−1.25380			−0.626902	−	0.779098i	$-0.715677\pi$
$367$	−460.146			−1.25380			−0.626902	+	0.779098i	$0.715677\pi$
$368$	44.2657			0.120287
$369$			501.961i			1.36033i
$370$		0			0
$371$			63.9141i			0.172275i
$372$	−141.857			−0.381337
$373$			483.099i			1.29517i	0.761992	+	0.647586i	$0.224221\pi$
$373$			483.099i			1.29517i	−0.761992	+	0.647586i	$0.775779\pi$
$374$	−33.3765	−	42.0998i	−0.0892420	−	0.112566i
$375$		0			0
$376$			730.273i			1.94222i
$377$	160.896			0.426779
$378$	−265.738			−0.703009
$379$	262.073			0.691486			0.345743	−	0.938329i	$-0.387627\pi$
$379$	262.073			0.691486			0.345743	+	0.938329i	$0.387627\pi$
$380$		0			0
$381$		−	156.524i		−	0.410824i
$382$		−	5.37612i		−	0.0140736i
$383$	−326.405			−0.852233			−0.426116	−	0.904668i	$-0.640119\pi$
$383$	−326.405			−0.852233			−0.426116	+	0.904668i	$0.640119\pi$
$384$		−	275.155i		−	0.716550i
$385$		0			0
$386$	942.013			2.44045
$387$		−	487.741i		−	1.26031i
$388$	76.8543			0.198078
$389$	−647.719			−1.66509			−0.832544	−	0.553960i	$-0.813116\pi$
$389$	−647.719			−1.66509			−0.832544	+	0.553960i	$0.813116\pi$
$390$		0			0
$391$		−	26.9952i		−	0.0690414i
$392$		−	300.112i		−	0.765592i
$393$		−	12.6936i		−	0.0322992i
$394$	359.708			0.912965
$395$		0			0
$396$	−342.244	−	431.694i	−0.864254	−	1.09014i
$397$	−122.651			−0.308946			−0.154473	−	0.987997i	$-0.549368\pi$
$397$	−122.651			−0.308946			−0.154473	+	0.987997i	$0.549368\pi$
$398$			127.324i			0.319909i
$399$	−91.6690			−0.229747
$400$		0			0
$401$	−243.279			−0.606680			−0.303340	−	0.952882i	$-0.598102\pi$
$401$	−243.279			−0.606680			−0.303340	+	0.952882i	$0.598102\pi$
$402$			49.6312i			0.123461i
$403$			111.351i			0.276306i
$404$			1055.49i			2.61260i
$405$		0			0
$406$		−	314.005i		−	0.773411i
$407$	445.560	+	562.012i	1.09474	+	1.38086i
$408$	16.7355			0.0410183
$409$		−	424.434i		−	1.03774i	−0.854854	−	0.518868i	$-0.826353\pi$
$409$		−	424.434i		−	1.03774i	0.854854	−	0.518868i	$-0.173647\pi$
$410$		0			0
$411$	88.1598			0.214501
$412$	712.454			1.72926
$413$		−	298.468i		−	0.722683i
$414$		−	441.150i		−	1.06558i
$415$		0			0
$416$	−184.509			−0.443531
$417$		−	254.502i		−	0.610317i
$418$	−415.962	−	524.678i	−0.995125	−	1.25521i
$419$	−68.6588			−0.163864			−0.0819318	−	0.996638i	$-0.526109\pi$
$419$	−68.6588			−0.163864			−0.0819318	+	0.996638i	$0.526109\pi$
$420$		0			0
$421$	−78.1961			−0.185739			−0.0928694	−	0.995678i	$-0.529604\pi$
$421$	−78.1961			−0.185739			−0.0928694	+	0.995678i	$0.529604\pi$
$422$	−293.163			−0.694700
$423$	605.141			1.43059
$424$		−	145.420i		−	0.342972i
$425$		0			0
$426$			326.830i			0.767206i
$427$	19.1300			0.0448009
$428$		−	1293.70i		−	3.02266i
$429$	71.4220	−	56.6230i	0.166485	−	0.131988i
$430$		0			0
$431$		−	592.143i		−	1.37388i	−0.726714	−	0.686940i	$-0.758953\pi$
$431$		−	592.143i		−	1.37388i	0.726714	−	0.686940i	$-0.241047\pi$
$432$	50.2729			0.116372
$433$	92.7053			0.214100			0.107050	−	0.994254i	$-0.465860\pi$
$433$	92.7053			0.214100			0.107050	+	0.994254i	$0.465860\pi$
$434$	217.314			0.500723
$435$		0			0
$436$		−	1094.70i		−	2.51078i
$437$		−	336.434i		−	0.769871i
$438$	175.118			0.399812
$439$		−	745.450i		−	1.69806i	−0.528342	−	0.849031i	$-0.677186\pi$
$439$		−	745.450i		−	1.69806i	0.528342	−	0.849031i	$-0.322814\pi$
$440$		0			0
$441$	−248.688			−0.563919
$442$		−	32.3314i		−	0.0731480i
$443$	−583.895			−1.31805			−0.659023	−	0.752123i	$-0.729030\pi$
$443$	−583.895			−1.31805			−0.659023	+	0.752123i	$0.729030\pi$
$444$	−549.852			−1.23841
$445$		0			0
$446$			622.401i			1.39552i
$447$			122.980i			0.275123i
$448$			398.633i			0.889805i
$449$	−344.083			−0.766332			−0.383166	−	0.923679i	$-0.625166\pi$
$449$	−344.083			−0.766332			−0.383166	+	0.923679i	$0.625166\pi$
$450$		0			0
$451$	−582.084	+	461.473i	−1.29065	+	1.02322i
$452$	509.195			1.12654
$453$			2.55821i			0.00564726i
$454$	−1129.65			−2.48822
$455$		0			0
$456$	208.569			0.457389
$457$		−	409.596i		−	0.896272i	−0.893965	−	0.448136i	$-0.852088\pi$
$457$		−	409.596i		−	0.896272i	0.893965	−	0.448136i	$-0.147912\pi$
$458$		−	373.910i		−	0.816398i
$459$		−	30.6587i		−	0.0667945i
$460$		0			0
$461$		−	599.882i		−	1.30126i	−0.759394	−	0.650632i	$-0.774504\pi$
$461$		−	599.882i		−	1.30126i	0.759394	−	0.650632i	$-0.225496\pi$
$462$	−110.506	−	139.388i	−0.239190	−	0.301705i
$463$	−497.936			−1.07546			−0.537728	−	0.843119i	$-0.680717\pi$
$463$	−497.936			−1.07546			−0.537728	+	0.843119i	$0.680717\pi$
$464$			59.4042i			0.128026i
$465$		0			0
$466$	55.6689			0.119461
$467$	440.806			0.943909			0.471955	−	0.881623i	$-0.343549\pi$
$467$	440.806			0.943909			0.471955	+	0.881623i	$0.343549\pi$
$468$		−	331.528i		−	0.708393i
$469$		−	47.7076i		−	0.101722i
$470$		0			0
$471$	−200.666			−0.426043
$472$			679.088i			1.43875i
$473$	565.594	−	448.400i	1.19576	−	0.947992i
$474$	−318.453			−0.671841
$475$		0			0
$476$	−39.5925			−0.0831775
$477$	−120.503			−0.252626
$478$	697.124			1.45842
$479$			521.375i			1.08847i	0.838934	+	0.544233i	$0.183179\pi$
$479$			521.375i			1.08847i	−0.838934	+	0.544233i	$0.816821\pi$
$480$		0			0
$481$			431.609i			0.897315i
$482$	1187.57			2.46384
$483$		−	89.3780i		−	0.185048i
$484$	185.961	−	793.747i	0.384218	−	1.63997i
$485$		0			0
$486$		−	775.409i		−	1.59549i
$487$	−158.630			−0.325728			−0.162864	−	0.986648i	$-0.552073\pi$
$487$	−158.630			−0.325728			−0.162864	+	0.986648i	$0.552073\pi$
$488$	−43.5254			−0.0891914
$489$	74.2499			0.151840
$490$		0			0
$491$			558.903i			1.13829i	0.822236	+	0.569147i	$0.192727\pi$
$491$			558.903i			1.13829i	−0.822236	+	0.569147i	$0.807273\pi$
$492$		−	569.490i		−	1.15750i
$493$	36.2273			0.0734834
$494$		−	402.937i		−	0.815663i
$495$		0			0
$496$	−41.1120			−0.0828871
$497$		−	314.162i		−	0.632117i
$498$	−152.381			−0.305987
$499$	−297.691			−0.596575			−0.298288	−	0.954476i	$-0.596415\pi$
$499$	−297.691			−0.596575			−0.298288	+	0.954476i	$0.596415\pi$
$500$		0			0
$501$			187.042i			0.373337i
$502$			165.741i			0.330161i
$503$		−	169.632i		−	0.337241i	−0.985681	−	0.168620i	$-0.946069\pi$
$503$		−	169.632i		−	0.337241i	0.985681	−	0.168620i	$-0.0539312\pi$
$504$	−262.887			−0.521602
$505$		0			0
$506$	511.566	−	405.567i	1.01100	−	0.801515i
$507$	−156.685			−0.309044
$508$		−	842.527i		−	1.65852i
$509$	473.225			0.929715			0.464858	−	0.885385i	$-0.346106\pi$
$509$	473.225			0.929715			0.464858	+	0.885385i	$0.346106\pi$
$510$		0			0
$511$	−168.330			−0.329413
$512$		−	155.819i		−	0.304334i
$513$		−	382.090i		−	0.744815i
$514$			1406.75i			2.73687i
$515$		0			0
$516$			553.357i			1.07240i
$517$	556.331	+	701.734i	1.07607	+	1.35732i
$518$	842.330			1.62612
$519$			89.8124i			0.173049i
$520$		0			0
$521$	974.599			1.87063			0.935316	−	0.353813i	$-0.115115\pi$
$521$	974.599			1.87063			0.935316	+	0.353813i	$0.115115\pi$
$522$	592.020			1.13414
$523$			628.934i			1.20255i	0.799042	+	0.601275i	$0.205340\pi$
$523$			628.934i			1.20255i	−0.799042	+	0.601275i	$0.794660\pi$
$524$		−	68.3264i		−	0.130394i
$525$		0			0
$526$	−517.365			−0.983583
$527$			25.0719i			0.0475748i
$528$	20.9058	+	26.3697i	0.0395943	+	0.0499427i
$529$	−200.974			−0.379913
$530$		0			0
$531$	562.727			1.05975
$532$	−493.430			−0.927500
$533$	−447.023			−0.838693
$534$			421.259i			0.788874i
$535$		0			0
$536$			108.546i			0.202512i
$537$	−179.163			−0.333637
$538$		−	699.209i		−	1.29964i
$539$	−228.629	−	288.384i	−0.424173	−	0.535035i
$540$		0			0
$541$		−	5.72311i		−	0.0105788i	−0.999986	−	0.00528938i	$-0.998316\pi$
$541$		−	5.72311i		−	0.0105788i	0.999986	−	0.00528938i	$-0.00168367\pi$
$542$	−1528.35			−2.81983
$543$	−382.821			−0.705011
$544$	−41.5441			−0.0763679
$545$		0			0
$546$		−	107.046i		−	0.196054i
$547$		−	167.284i		−	0.305820i	−0.988240	−	0.152910i	$-0.951136\pi$
$547$		−	167.284i		−	0.305820i	0.988240	−	0.152910i	$-0.0488645\pi$
$548$	474.541			0.865952
$549$			36.0674i			0.0656965i
$550$		0			0
$551$	451.491			0.819403
$552$			203.357i			0.368400i
$553$	306.110			0.553544
$554$	217.396			0.392411
$555$		0			0
$556$		−	1369.92i		−	2.46389i
$557$			458.071i			0.822390i	0.911548	+	0.411195i	$0.134888\pi$
$557$			458.071i			0.822390i	−0.911548	+	0.411195i	$0.865112\pi$
$558$			409.720i			0.734265i
$559$	434.360			0.777030
$560$		0			0
$561$	16.0814	−	12.7493i	0.0286656	−	0.0227260i
$562$	−787.079			−1.40050
$563$			570.622i			1.01354i	0.862082	+	0.506769i	$0.169160\pi$
$563$			570.622i			1.01354i	−0.862082	+	0.506769i	$0.830840\pi$
$564$	−686.551			−1.21729
$565$		0			0
$566$	−886.465			−1.56619
$567$			162.249i			0.286154i
$568$			714.796i			1.25844i
$569$			399.293i			0.701745i	0.936423	+	0.350873i	$0.114115\pi$
$569$			399.293i			0.701745i	−0.936423	+	0.350873i	$0.885885\pi$
$570$		0			0
$571$		−	1048.75i		−	1.83668i	−0.395790	−	0.918341i	$-0.629529\pi$
$571$		−	1048.75i		−	1.83668i	0.395790	−	0.918341i	$-0.370471\pi$
$572$	384.446	−	304.787i	0.672109	−	0.532844i
$573$	2.05359			0.00358393
$574$			872.414i			1.51988i
$575$		0			0
$576$	−751.575			−1.30482
$577$	−65.8975			−0.114207			−0.0571036	−	0.998368i	$-0.518187\pi$
$577$	−65.8975			−0.114207			−0.0571036	+	0.998368i	$0.518187\pi$
$578$			939.720i			1.62581i
$579$			359.833i			0.621474i
$580$		0			0
$581$	146.475			0.252109
$582$			46.7861i			0.0803884i
$583$	−110.783	−	139.737i	−0.190022	−	0.239686i
$584$	382.992			0.655809
$585$		0			0
$586$	1780.64			3.03864
$587$	−971.299			−1.65468			−0.827342	−	0.561699i	$-0.810148\pi$
$587$	−971.299			−1.65468			−0.827342	+	0.561699i	$0.810148\pi$
$588$	282.144			0.479837
$589$			312.464i			0.530500i
$590$		0			0
$591$			137.403i			0.232492i
$592$	−159.354			−0.269179
$593$		−	381.457i		−	0.643266i	−0.946864	−	0.321633i	$-0.895768\pi$
$593$		−	381.457i		−	0.643266i	0.946864	−	0.321633i	$-0.104232\pi$
$594$	580.989	−	460.605i	0.978096	−	0.775430i
$595$		0			0
$596$			661.969i			1.11069i
$597$	−48.6355			−0.0814665
$598$	392.868			0.656969
$599$	372.420			0.621737			0.310869	−	0.950453i	$-0.399380\pi$
$599$	372.420			0.621737			0.310869	+	0.950453i	$0.399380\pi$
$600$		0			0
$601$			941.497i			1.56655i	0.621674	+	0.783276i	$0.286453\pi$
$601$			941.497i			1.56655i	−0.621674	+	0.783276i	$0.713547\pi$
$602$		−	847.699i		−	1.40814i
$603$	89.9471			0.149166
$604$			13.7702i			0.0227983i
$605$		0			0
$606$	−642.543			−1.06030
$607$			803.140i			1.32313i	0.749888	+	0.661565i	$0.230107\pi$
$607$			803.140i			1.32313i	−0.749888	+	0.661565i	$0.769893\pi$
$608$	−517.753			−0.851568
$609$	119.945			0.196953
$610$		0			0
$611$			538.911i			0.882014i
$612$		−	74.6471i		−	0.121972i
$613$			438.888i			0.715968i	0.933728	+	0.357984i	$0.116536\pi$
$613$			438.888i			0.715968i	−0.933728	+	0.357984i	$0.883464\pi$
$614$	1029.43			1.67660
$615$		0			0
$616$	−241.683	−	304.849i	−0.392342	−	0.494885i
$617$	1053.80			1.70795			0.853973	−	0.520317i	$-0.174186\pi$
$617$	1053.80			1.70795			0.853973	+	0.520317i	$0.174186\pi$
$618$			433.716i			0.701805i
$619$	113.476			0.183322			0.0916608	−	0.995790i	$-0.470782\pi$
$619$	113.476			0.183322			0.0916608	+	0.995790i	$0.470782\pi$
$620$		0			0
$621$	372.541			0.599906
$622$		−	1228.26i		−	1.97470i
$623$		−	404.931i		−	0.649969i
$624$			20.2512i			0.0324538i
$625$		0			0
$626$			676.451i			1.08059i
$627$	200.418	−	158.891i	0.319647	−	0.253414i
$628$	−1080.13			−1.71996
$629$			97.1812i			0.154501i
$630$		0			0
$631$	405.042			0.641905			0.320952	−	0.947095i	$-0.395997\pi$
$631$	405.042			0.641905			0.320952	+	0.947095i	$0.395997\pi$
$632$	−696.475			−1.10202
$633$		−	111.984i		−	0.176909i
$634$		−	897.413i		−	1.41548i
$635$		0			0
$636$	136.714			0.214959
$637$		−	221.470i		−	0.347677i
$638$	544.267	+	686.517i	0.853083	+	1.07605i
$639$	592.316			0.926942
$640$		0			0
$641$	−836.111			−1.30439			−0.652193	−	0.758053i	$-0.726151\pi$
$641$	−836.111			−1.30439			−0.652193	+	0.758053i	$0.726151\pi$
$642$	787.557			1.22672
$643$	−466.764			−0.725917			−0.362958	−	0.931805i	$-0.618233\pi$
$643$	−466.764			−0.725917			−0.362958	+	0.931805i	$0.618233\pi$
$644$		−	481.099i		−	0.747048i
$645$		0			0
$646$		−	90.7256i		−	0.140442i
$647$	−400.219			−0.618577			−0.309289	−	0.950968i	$-0.600091\pi$
$647$	−400.219			−0.618577			−0.309289	+	0.950968i	$0.600091\pi$
$648$		−	369.157i		−	0.569686i
$649$	517.337	+	652.549i	0.797130	+	1.00547i
$650$		0			0
$651$			83.0103i			0.127512i
$652$	399.668			0.612988
$653$	105.848			0.162096			0.0810478	−	0.996710i	$-0.474173\pi$
$653$	105.848			0.162096			0.0810478	+	0.996710i	$0.474173\pi$
$654$	666.412			1.01898
$655$		0			0
$656$		−	165.045i		−	0.251594i
$657$		−	317.367i		−	0.483055i
$658$	1051.74			1.59839
$659$		−	491.956i		−	0.746519i	−0.927727	−	0.373260i	$-0.878240\pi$
$659$		−	491.956i		−	0.746519i	0.927727	−	0.373260i	$-0.121760\pi$
$660$		0			0
$661$	407.084			0.615861			0.307931	−	0.951409i	$-0.400364\pi$
$661$	407.084			0.615861			0.307931	+	0.951409i	$0.400364\pi$
$662$		−	1448.19i		−	2.18759i
$663$	12.3501			0.0186275
$664$	−333.267			−0.501908
$665$		0			0
$666$			1588.12i			2.38456i
$667$			440.208i			0.659982i
$668$			1006.80i			1.50718i
$669$	−237.747			−0.355376
$670$		0			0
$671$	−41.8244	+	33.1582i	−0.0623315	+	0.0494160i
$672$	−137.548			−0.204684
$673$		−	908.695i		−	1.35022i	−0.737719	−	0.675108i	$-0.764097\pi$
$673$		−	908.695i		−	1.35022i	0.737719	−	0.675108i	$-0.235903\pi$
$674$	429.140			0.636707
$675$		0			0
$676$	−843.397			−1.24763
$677$			1064.55i			1.57245i	0.617941	+	0.786224i	$0.287967\pi$
$677$			1064.55i			1.57245i	−0.617941	+	0.786224i	$0.712033\pi$
$678$			309.979i			0.457197i
$679$		−	44.9727i		−	0.0662337i
$680$		0			0
$681$		−	431.509i		−	0.633640i
$682$	−475.119	+	376.672i	−0.696656	+	0.552305i
$683$	332.676			0.487081			0.243540	−	0.969891i	$-0.421691\pi$
$683$	332.676			0.487081			0.243540	+	0.969891i	$0.421691\pi$
$684$		−	930.305i		−	1.36010i
$685$		0			0
$686$	−1065.26			−1.55285
$687$	142.828			0.207900
$688$			160.370i			0.233096i
$689$		−	107.314i		−	0.155753i
$690$		0			0
$691$	880.234			1.27385			0.636927	−	0.770924i	$-0.280205\pi$
$691$	880.234			1.27385			0.636927	+	0.770924i	$0.280205\pi$
$692$			483.437i			0.698608i
$693$	−252.613	+	200.271i	−0.364521	+	0.288991i
$694$	619.130			0.892118
$695$		0			0
$696$	−272.903			−0.392103
$697$	−100.652			−0.144408
$698$	−1638.67			−2.34766
$699$			21.2646i			0.0304214i
$700$		0			0
$701$		−	743.435i		−	1.06054i	−0.847830	−	0.530268i	$-0.822091\pi$
$701$		−	743.435i		−	1.06054i	0.847830	−	0.530268i	$-0.177909\pi$
$702$	446.183			0.635588
$703$			1211.14i			1.72282i
$704$	−690.953	−	871.542i	−0.981468	−	1.23799i
$705$		0			0
$706$			1357.91i			1.92338i
$707$	617.639			0.873605
$708$	−638.430			−0.901738
$709$	−459.136			−0.647583			−0.323791	−	0.946128i	$-0.604958\pi$
$709$	−459.136			−0.647583			−0.323791	+	0.946128i	$0.604958\pi$
$710$		0			0
$711$			577.134i			0.811722i
$712$			921.317i			1.29399i
$713$	−304.656			−0.427287
$714$		−	24.1025i		−	0.0337569i
$715$		0			0
$716$	−964.387			−1.34691
$717$			266.290i			0.371394i
$718$	695.578			0.968771
$719$	−571.747			−0.795197			−0.397599	−	0.917559i	$-0.630156\pi$
$719$	−571.747			−0.795197			−0.397599	+	0.917559i	$0.630156\pi$
$720$		0			0
$721$		−	416.905i		−	0.578232i
$722$			52.2421i			0.0723574i
$723$			453.632i			0.627430i
$724$	−2060.63			−2.84617
$725$		0			0
$726$	483.204	+	113.207i	0.665570	+	0.155932i
$727$	−188.665			−0.259511			−0.129756	−	0.991546i	$-0.541419\pi$
$727$	−188.665			−0.259511			−0.129756	+	0.991546i	$0.541419\pi$
$728$		−	234.115i		−	0.321587i
$729$	−74.1842			−0.101762
$730$		0			0
$731$	97.8007			0.133790
$732$		−	40.9195i		−	0.0559010i
$733$		−	542.416i		−	0.739994i	−0.929033	−	0.369997i	$-0.879359\pi$
$733$		−	542.416i		−	0.739994i	0.929033	−	0.369997i	$-0.120641\pi$
$734$		−	1507.81i		−	2.05424i
$735$		0			0
$736$		−	504.814i		−	0.685888i
$737$	82.6919	+	104.304i	0.112201	+	0.141526i
$738$	−1644.83			−2.22877
$739$			637.670i			0.862882i	0.902141	+	0.431441i	$0.141995\pi$
$739$			637.670i			0.862882i	−0.902141	+	0.431441i	$0.858005\pi$
$740$		0			0
$741$	153.915			0.207713
$742$	−209.435			−0.282257
$743$		−	728.223i		−	0.980112i	−0.871691	−	0.490056i	$-0.836976\pi$
$743$		−	728.223i		−	0.980112i	0.871691	−	0.490056i	$-0.163024\pi$
$744$		−	188.869i		−	0.253856i
$745$		0			0
$746$	−1583.03			−2.12202
$747$			276.162i			0.369695i
$748$	86.5622	−	68.6260i	0.115725	−	0.0917460i
$749$	−757.031			−1.01072
$750$		0			0
$751$	33.5510			0.0446750			0.0223375	−	0.999750i	$-0.492889\pi$
$751$	33.5510			0.0446750			0.0223375	+	0.999750i	$0.492889\pi$
$752$	−198.971			−0.264589
$753$	−63.3103			−0.0840774
$754$			527.225i			0.699237i
$755$		0			0
$756$		−	546.388i		−	0.722735i
$757$	−1223.98			−1.61688			−0.808438	−	0.588581i	$-0.799687\pi$
$757$	−1223.98			−1.61688			−0.808438	+	0.588581i	$0.799687\pi$
$758$			858.765i			1.13294i
$759$	154.920	+	195.410i	0.204110	+	0.257457i
$760$		0			0
$761$		−	711.474i		−	0.934920i	−0.884014	−	0.467460i	$-0.845169\pi$
$761$		−	711.474i		−	0.934920i	0.884014	−	0.467460i	$-0.154831\pi$
$762$	512.900			0.673097
$763$	−640.582			−0.839557
$764$	11.0539			0.0144685
$765$		0			0
$766$		−	1069.57i		−	1.39630i
$767$			501.138i			0.653374i
$768$	395.401			0.514846
$769$		−	418.803i		−	0.544607i	−0.962211	−	0.272304i	$-0.912214\pi$
$769$		−	418.803i		−	0.544607i	0.962211	−	0.272304i	$-0.0877855\pi$
$770$		0			0
$771$	−537.356			−0.696960
$772$			1936.89i			2.50892i
$773$	−708.180			−0.916145			−0.458073	−	0.888915i	$-0.651460\pi$
$773$	−708.180			−0.916145			−0.458073	+	0.888915i	$0.651460\pi$
$774$	1598.24			2.06491
$775$		0			0
$776$			102.324i			0.131861i
$777$			321.756i			0.414100i
$778$		−	2122.46i		−	2.72809i
$779$	−1254.40			−1.61027
$780$		0			0
$781$	544.540	+	686.861i	0.697234	+	0.879464i
$782$	88.4583			0.113118
$783$			499.947i			0.638503i
$784$	81.7689			0.104297
$785$		0			0
$786$	41.5946			0.0529193
$787$			629.493i			0.799864i	0.916545	+	0.399932i	$0.130966\pi$
$787$			629.493i			0.799864i	−0.916545	+	0.399932i	$0.869034\pi$
$788$			739.603i			0.938582i
$789$		−	197.625i		−	0.250475i
$790$		0			0
$791$		−	297.965i		−	0.376694i
$792$	574.758	−	455.665i	0.725704	−	0.575334i
$793$	−32.1199			−0.0405043
$794$		−	401.906i		−	0.506179i
$795$		0			0
$796$	−261.792			−0.328885
$797$	−1278.65			−1.60433			−0.802165	−	0.597103i	$-0.796318\pi$
$797$	−1278.65			−1.60433			−0.802165	+	0.597103i	$0.796318\pi$
$798$		−	300.382i		−	0.376419i
$799$			121.341i			0.151867i
$800$		0			0
$801$	763.450			0.953121
$802$		−	797.179i		−	0.993989i
$803$	368.025	−	291.768i	0.458312	−	0.363347i
$804$	−102.048			−0.126925
$805$		0			0
$806$	−364.878			−0.452702
$807$	267.086			0.330962
$808$	−1405.28			−1.73921
$809$		−	516.115i		−	0.637966i	−0.947760	−	0.318983i	$-0.896659\pi$
$809$		−	516.115i		−	0.637966i	0.947760	−	0.318983i	$-0.103341\pi$
$810$		0			0
$811$			307.425i			0.379069i	0.981874	+	0.189534i	$0.0606978\pi$
$811$			307.425i			0.379069i	−0.981874	+	0.189534i	$0.939302\pi$
$812$	645.631			0.795112
$813$		−	583.803i		−	0.718085i
$814$	−1841.61	+	1460.02i	−2.26242	+	1.79363i
$815$		0			0
$816$			4.55976i			0.00558795i
$817$	1218.86			1.49188
$818$	1390.79			1.70024
$819$	−194.000			−0.236874
$820$		0			0
$821$		−	181.868i		−	0.221520i	−0.993847	−	0.110760i	$-0.964672\pi$
$821$		−	181.868i		−	0.221520i	0.993847	−	0.110760i	$-0.0353285\pi$
$822$			288.883i			0.351440i
$823$	−719.382			−0.874097			−0.437049	−	0.899438i	$-0.643976\pi$
$823$	−719.382			−0.874097			−0.437049	+	0.899438i	$0.643976\pi$
$824$			948.562i			1.15117i
$825$		0			0
$826$	978.024			1.18405
$827$		−	555.837i		−	0.672113i	−0.941842	−	0.336056i	$-0.890907\pi$
$827$		−	555.837i		−	0.672113i	0.941842	−	0.336056i	$-0.109093\pi$
$828$	907.056			1.09548
$829$	448.221			0.540677			0.270338	−	0.962765i	$-0.412864\pi$
$829$	448.221			0.540677			0.270338	+	0.962765i	$0.412864\pi$
$830$		0			0
$831$			83.0416i			0.0999297i
$832$		−	669.318i		−	0.804469i
$833$		−	49.8663i		−	0.0598635i
$834$	833.957			0.999948
$835$		0			0
$836$	1078.80	−	855.267i	1.29043	−	1.02305i
$837$	−346.000			−0.413381
$838$		−	224.982i		−	0.268475i
$839$	682.509			0.813479			0.406740	−	0.913544i	$-0.366666\pi$
$839$	682.509			0.813479			0.406740	+	0.913544i	$0.366666\pi$
$840$		0			0
$841$	250.245			0.297556
$842$		−	256.234i		−	0.304316i
$843$		−	300.651i		−	0.356644i
$844$		−	602.778i		−	0.714192i
$845$		0			0
$846$			1982.94i			2.34390i
$847$	−464.475	−	108.819i	−0.548377	−	0.128475i
$848$	39.6214			0.0467233
$849$		−	338.615i		−	0.398840i
$850$		0			0
$851$	−1180.87			−1.38763
$852$	−672.000			−0.788733
$853$			1370.47i			1.60665i	0.595544	+	0.803323i	$0.296937\pi$
$853$			1370.47i			1.60665i	−0.595544	+	0.803323i	$0.703063\pi$
$854$			62.6854i			0.0734021i
$855$		0			0
$856$	1722.43			2.01219
$857$		−	1074.81i		−	1.25416i	−0.778956	−	0.627078i	$-0.784251\pi$
$857$		−	1074.81i		−	1.25416i	0.778956	−	0.627078i	$-0.215749\pi$
$858$	185.543	+	234.037i	0.216251	+	0.272770i
$859$	969.712			1.12888			0.564442	−	0.825472i	$-0.309091\pi$
$859$	969.712			1.12888			0.564442	+	0.825472i	$0.309091\pi$
$860$		0			0
$861$	−333.247			−0.387047
$862$	1940.34			2.25098
$863$	129.003			0.149482			0.0747408	−	0.997203i	$-0.476187\pi$
$863$	129.003			0.149482			0.0747408	+	0.997203i	$0.476187\pi$
$864$		−	573.321i		−	0.663566i
$865$		0			0
$866$			303.778i			0.350783i
$867$	−358.957			−0.414022
$868$			446.823i			0.514773i
$869$	−669.256	+	530.583i	−0.770145	+	0.610567i
$870$		0			0
$871$			80.1026i			0.0919663i
$872$	1457.48			1.67142
$873$	84.7907			0.0971257
$874$	1102.43			1.26136
$875$		0			0
$876$			360.062i			0.411030i
$877$			490.300i			0.559065i	0.960136	+	0.279532i	$0.0901794\pi$
$877$			490.300i			0.559065i	−0.960136	+	0.279532i	$0.909821\pi$
$878$	2442.70			2.78212
$879$			680.176i			0.773807i
$880$		0			0
$881$	−772.952			−0.877358			−0.438679	−	0.898644i	$-0.644553\pi$
$881$	−772.952			−0.877358			−0.438679	+	0.898644i	$0.644553\pi$
$882$		−	814.905i		−	0.923929i
$883$	1723.84			1.95225			0.976124	−	0.217212i	$-0.0696963\pi$
$883$	1723.84			1.95225			0.976124	+	0.217212i	$0.0696963\pi$
$884$	66.4772			0.0752004
$885$		0			0
$886$		−	1913.31i		−	2.15950i
$887$		−	1064.55i		−	1.20017i	−0.799936	−	0.600085i	$-0.795133\pi$
$887$		−	1064.55i		−	1.20017i	0.799936	−	0.600085i	$-0.204867\pi$
$888$		−	732.074i		−	0.824408i
$889$	−493.020			−0.554578
$890$		0			0
$891$	−281.228	−	354.730i	−0.315632	−	0.398126i
$892$	−1279.73			−1.43467
$893$			1512.24i			1.69344i
$894$	−402.982			−0.450763
$895$		0			0
$896$	−866.686			−0.967283
$897$			150.069i			0.167301i
$898$		−	1127.50i		−	1.25556i
$899$		−	408.845i		−	0.454778i
$900$		0			0
$901$		−	24.1629i		−	0.0268178i
$902$	−1512.16	−	1907.38i	−1.67645	−	2.11461i
$903$	323.807			0.358590
$904$			677.943i			0.749937i
$905$		0			0
$906$	−8.38278			−0.00925252
$907$	−1115.09			−1.22943			−0.614713	−	0.788751i	$-0.710728\pi$
$907$	−1115.09			−1.22943			−0.614713	+	0.788751i	$0.710728\pi$
$908$		−	2322.70i		−	2.55804i
$909$			1164.49i			1.28106i
$910$		0			0
$911$	834.815			0.916372			0.458186	−	0.888856i	$-0.348499\pi$
$911$	834.815			0.916372			0.458186	+	0.888856i	$0.348499\pi$
$912$			56.8271i			0.0623104i
$913$	−320.243	+	253.887i	−0.350759	+	0.278080i
$914$	1342.17			1.46846
$915$		0			0
$916$	768.804			0.839305
$917$	−39.9824			−0.0436013
$918$	100.463			0.109437
$919$		−	284.457i		−	0.309529i	−0.987951	−	0.154765i	$-0.950538\pi$
$919$		−	284.457i		−	0.309529i	0.987951	−	0.154765i	$-0.0494619\pi$
$920$		0			0
$921$			393.225i			0.426955i
$922$	1965.70			2.13200
$923$			527.489i			0.571494i
$924$	286.598	−	227.213i	0.310170	−	0.245901i
$925$		0			0
$926$		−	1631.64i		−	1.76203i
$927$	786.026			0.847925
$928$	677.456			0.730017
$929$	337.429			0.363218			0.181609	−	0.983371i	$-0.441870\pi$
$929$	337.429			0.363218			0.181609	+	0.983371i	$0.441870\pi$
$930$		0			0
$931$		−	621.470i		−	0.667530i
$932$			114.462i			0.122813i
$933$	469.176			0.502868
$934$			1444.44i			1.54651i
$935$		0			0
$936$	441.397			0.471578
$937$		−	736.226i		−	0.785726i	−0.919597	−	0.392863i	$-0.871485\pi$
$937$		−	736.226i		−	0.785726i	0.919597	−	0.392863i	$-0.128515\pi$
$938$	156.329			0.166662
$939$	−258.393			−0.275179
$940$		0			0
$941$		−	433.395i		−	0.460569i	−0.973123	−	0.230284i	$-0.926034\pi$
$941$		−	433.395i		−	0.460569i	0.973123	−	0.230284i	$-0.0739656\pi$
$942$		−	657.545i		−	0.698031i
$943$		−	1223.05i		−	1.29698i
$944$	−185.025			−0.196001
$945$		0			0
$946$	1469.32	+	1853.35i	1.55320	+	1.95914i
$947$	439.058			0.463631			0.231815	−	0.972760i	$-0.425534\pi$
$947$	439.058			0.463631			0.231815	+	0.972760i	$0.425534\pi$
$948$		−	654.776i		−	0.690692i
$949$	282.632			0.297821
$950$		0			0
$951$	342.797			0.360459
$952$		−	52.7135i		−	0.0553713i
$953$			1108.12i			1.16277i	0.813630	+	0.581383i	$0.197488\pi$
$953$			1108.12i			1.16277i	−0.813630	+	0.581383i	$0.802512\pi$
$954$		−	394.865i		−	0.413904i
$955$		0			0
$956$			1433.37i			1.49934i
$957$	−262.238	+	207.901i	−0.274021	+	0.217242i
$958$	−1708.45			−1.78335
$959$		−	277.686i		−	0.289558i
$960$		0			0
$961$	−678.050			−0.705567
$962$	−1414.30			−1.47017
$963$		−	1427.29i		−	1.48213i
$964$			2441.78i			2.53297i
$965$		0			0
$966$	292.875			0.303184
$967$		−	108.838i		−	0.112553i	−0.998415	−	0.0562763i	$-0.982077\pi$
$967$		−	108.838i		−	0.112553i	0.998415	−	0.0562763i	$-0.0179228\pi$
$968$	1056.80	+	247.589i	1.09173	+	0.255774i
$969$	34.6557			0.0357644
$970$		0			0
$971$	−312.516			−0.321850			−0.160925	−	0.986967i	$-0.551448\pi$
$971$	−312.516			−0.321850			−0.160925	+	0.986967i	$0.551448\pi$
$972$	1594.33			1.64026
$973$	−801.633			−0.823878
$974$		−	519.800i		−	0.533676i
$975$		0			0
$976$		−	11.8590i		−	0.0121506i
$977$	−594.696			−0.608696			−0.304348	−	0.952561i	$-0.598439\pi$
$977$	−594.696			−0.608696			−0.304348	+	0.952561i	$0.598439\pi$
$978$			243.303i			0.248776i
$979$	701.870	+	885.312i	0.716926	+	0.904302i
$980$		0			0
$981$		−	1207.74i		−	1.23113i
$982$	−1831.42			−1.86499
$983$	554.324			0.563910			0.281955	−	0.959428i	$-0.409017\pi$
$983$	554.324			0.563910			0.281955	+	0.959428i	$0.409017\pi$
$984$	758.220			0.770549
$985$		0			0
$986$			118.710i			0.120396i
$987$			401.748i			0.407039i
$988$	828.487			0.838549
$989$			1188.40i			1.20162i
$990$		0			0
$991$	−148.655			−0.150005			−0.0750023	−	0.997183i	$-0.523896\pi$
$991$	−148.655			−0.150005			−0.0750023	+	0.997183i	$0.523896\pi$
$992$			468.848i			0.472629i
$993$	553.183			0.557083
$994$	1029.45			1.03566
$995$		0			0
$996$		−	313.314i		−	0.314572i
$997$			1023.25i			1.02633i	0.858291	+	0.513164i	$0.171527\pi$
$997$			1023.25i			1.02633i	−0.858291	+	0.513164i	$0.828473\pi$
$998$		−	975.479i		−	0.977433i
$999$	−1341.13			−1.34247

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	275.3.c.h.76.8	yes	8
5.2	odd	4		275.3.d.b.274.2		16
5.3	odd	4		275.3.d.b.274.15		16
5.4	even	2		275.3.c.g.76.1	✓	8
11.10	odd	2	inner	275.3.c.h.76.1	yes	8
55.32	even	4		275.3.d.b.274.16		16
55.43	even	4		275.3.d.b.274.1		16
55.54	odd	2		275.3.c.g.76.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.3.c.g.76.1	✓	8	5.4	even	2
275.3.c.g.76.8	yes	8	55.54	odd	2
275.3.c.h.76.1	yes	8	11.10	odd	2	inner
275.3.c.h.76.8	yes	8	1.1	even	1	trivial
275.3.d.b.274.1		16	55.43	even	4
275.3.d.b.274.2		16	5.2	odd	4
275.3.d.b.274.15		16	5.3	odd	4
275.3.d.b.274.16		16	55.32	even	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:	\( N \)	\(=\)	\( 275 = 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	275.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.27682i q^{2} -1.25169 q^{3} -6.73752 q^{4} -4.10155i q^{6} +3.94258i q^{7} -8.97034i q^{8} -7.43328 q^{9} +(-6.83371 - 8.61977i) q^{11} +8.43328 q^{12} -6.61973i q^{13} -12.9191 q^{14} +2.44407 q^{16} -1.49050i q^{17} -24.3575i q^{18} -18.5757i q^{19} -4.93488i q^{21} +(28.2454 - 22.3928i) q^{22} +18.1115 q^{23} +11.2281i q^{24} +21.6916 q^{26} +20.5693 q^{27} -26.5632i q^{28} +24.3055i q^{29} -16.8211 q^{31} -27.8726i q^{32} +(8.55368 + 10.7893i) q^{33} +4.88410 q^{34} +50.0818 q^{36} -65.2003 q^{37} +60.8692 q^{38} +8.28584i q^{39} -67.5290i q^{41} +16.1707 q^{42} +65.6159i q^{43} +(46.0422 + 58.0759i) q^{44} +59.3480i q^{46} -81.4098 q^{47} -3.05921 q^{48} +33.4561 q^{49} +1.86564i q^{51} +44.6005i q^{52} +16.2112 q^{53} +67.4019i q^{54} +35.3663 q^{56} +23.2510i q^{57} -79.6445 q^{58} -75.7037 q^{59} -4.85215i q^{61} -55.1197i q^{62} -29.3063i q^{63} +101.110 q^{64} +(-35.3544 + 28.0288i) q^{66} -12.1006 q^{67} +10.0423i q^{68} -22.6699 q^{69} -79.6844 q^{71} +66.6790i q^{72} +42.6954i q^{73} -213.649i q^{74} +125.154i q^{76} +(33.9841 - 26.9424i) q^{77} -27.1512 q^{78} -77.6420i q^{79} +41.1531 q^{81} +221.280 q^{82} -37.1521i q^{83} +33.2489i q^{84} -215.011 q^{86} -30.4229i q^{87} +(-77.3223 + 61.3007i) q^{88} -102.707 q^{89} +26.0988 q^{91} -122.026 q^{92} +21.0548 q^{93} -266.765i q^{94} +34.8878i q^{96} -11.4069 q^{97} +109.629i q^{98} +(50.7968 + 64.0731i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} - 10 q^{4} + 32 q^{9} - q^{11} - 24 q^{12} + 18 q^{14} - 14 q^{16} - 35 q^{22} + 4 q^{23} + 68 q^{26} + 142 q^{27} - 42 q^{31} - 31 q^{33} - 142 q^{34} - 84 q^{36} - 104 q^{37} + 190 q^{38}+ \cdots + 368 q^{99}+O(q^{100}) \) 8 * q + 4 * q^3 - 10 * q^4 + 32 * q^9 - q^11 - 24 * q^12 + 18 * q^14 - 14 * q^16 - 35 * q^22 + 4 * q^23 + 68 * q^26 + 142 * q^27 - 42 * q^31 - 31 * q^33 - 142 * q^34 - 84 * q^36 - 104 * q^37 + 190 * q^38 + 50 * q^42 + 69 * q^44 - 64 * q^47 - 86 * q^48 - 50 * q^49 - 286 * q^53 + 166 * q^56 - 130 * q^58 - 160 * q^59 + 288 * q^64 - 315 * q^66 + 46 * q^67 - 16 * q^69 + 246 * q^71 + 80 * q^77 + 620 * q^78 + 304 * q^81 + 280 * q^82 - 282 * q^86 - 365 * q^88 - 18 * q^89 - 142 * q^91 + 26 * q^92 + 818 * q^93 - 264 * q^97 + 368 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more