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Properties

Label 48.18.a.e.1.1
Level $48$
Weight $18$
Character 48.1
Self dual yes
Analytic conductor $87.947$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [48,18,Mod(1,48)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("48.1"); S:= CuspForms(chi, 18); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(48, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 18, names="a")
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 48.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,6561,0,-163554] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(87.9466019254\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 48.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+6561.00 q^{3} -163554. q^{5} +2.08466e7 q^{7} +4.30467e7 q^{9} -8.17372e8 q^{11} +2.99590e8 q^{13} -1.07308e9 q^{15} -4.47756e10 q^{17} -7.87487e10 q^{19} +1.36774e11 q^{21} +7.04672e11 q^{23} -7.36190e11 q^{25} +2.82430e11 q^{27} -1.63794e11 q^{29} -1.04986e12 q^{31} -5.36278e12 q^{33} -3.40954e12 q^{35} -1.98057e13 q^{37} +1.96561e12 q^{39} +1.46600e13 q^{41} -1.16039e14 q^{43} -7.04046e12 q^{45} +1.76607e14 q^{47} +2.01949e14 q^{49} -2.93773e14 q^{51} +1.52863e14 q^{53} +1.33685e14 q^{55} -5.16670e14 q^{57} +2.62797e14 q^{59} -1.35855e15 q^{61} +8.97376e14 q^{63} -4.89991e13 q^{65} -4.44864e14 q^{67} +4.62335e15 q^{69} +4.00327e15 q^{71} +9.24833e14 q^{73} -4.83014e15 q^{75} -1.70394e16 q^{77} -1.47473e16 q^{79} +1.85302e15 q^{81} -2.64230e16 q^{83} +7.32323e15 q^{85} -1.07465e15 q^{87} -3.88837e16 q^{89} +6.24542e15 q^{91} -6.88814e15 q^{93} +1.28797e16 q^{95} -2.53744e16 q^{97} -3.51852e16 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 6561.00 0.577350
\(4\) 0 0
\(5\) −163554. −0.187248 −0.0936238 − 0.995608i \(-0.529845\pi\)
−0.0936238 + 0.995608i \(0.529845\pi\)
\(6\) 0 0
\(7\) 2.08466e7 1.36679 0.683394 − 0.730050i \(-0.260503\pi\)
0.683394 + 0.730050i \(0.260503\pi\)
\(8\) 0 0
\(9\) 4.30467e7 0.333333
\(10\) 0 0
\(11\) −8.17372e8 −1.14969 −0.574847 − 0.818261i \(-0.694938\pi\)
−0.574847 + 0.818261i \(0.694938\pi\)
\(12\) 0 0
\(13\) 2.99590e8 0.101861 0.0509306 − 0.998702i \(-0.483781\pi\)
0.0509306 + 0.998702i \(0.483781\pi\)
\(14\) 0 0
\(15\) −1.07308e9 −0.108107
\(16\) 0 0
\(17\) −4.47756e10 −1.55677 −0.778387 − 0.627785i \(-0.783962\pi\)
−0.778387 + 0.627785i \(0.783962\pi\)
\(18\) 0 0
\(19\) −7.87487e10 −1.06374 −0.531872 − 0.846824i \(-0.678511\pi\)
−0.531872 + 0.846824i \(0.678511\pi\)
\(20\) 0 0
\(21\) 1.36774e11 0.789115
\(22\) 0 0
\(23\) 7.04672e11 1.87629 0.938146 − 0.346240i \(-0.112542\pi\)
0.938146 + 0.346240i \(0.112542\pi\)
\(24\) 0 0
\(25\) −7.36190e11 −0.964938
\(26\) 0 0
\(27\) 2.82430e11 0.192450
\(28\) 0 0
\(29\) −1.63794e11 −0.0608015 −0.0304008 − 0.999538i \(-0.509678\pi\)
−0.0304008 + 0.999538i \(0.509678\pi\)
\(30\) 0 0
\(31\) −1.04986e12 −0.221084 −0.110542 − 0.993871i \(-0.535259\pi\)
−0.110542 + 0.993871i \(0.535259\pi\)
\(32\) 0 0
\(33\) −5.36278e12 −0.663776
\(34\) 0 0
\(35\) −3.40954e12 −0.255928
\(36\) 0 0
\(37\) −1.98057e13 −0.926993 −0.463496 − 0.886099i \(-0.653405\pi\)
−0.463496 + 0.886099i \(0.653405\pi\)
\(38\) 0 0
\(39\) 1.96561e12 0.0588095
\(40\) 0 0
\(41\) 1.46600e13 0.286729 0.143365 − 0.989670i \(-0.454208\pi\)
0.143365 + 0.989670i \(0.454208\pi\)
\(42\) 0 0
\(43\) −1.16039e14 −1.51399 −0.756993 − 0.653424i \(-0.773332\pi\)
−0.756993 + 0.653424i \(0.773332\pi\)
\(44\) 0 0
\(45\) −7.04046e12 −0.0624158
\(46\) 0 0
\(47\) 1.76607e14 1.08187 0.540935 − 0.841064i \(-0.318070\pi\)
0.540935 + 0.841064i \(0.318070\pi\)
\(48\) 0 0
\(49\) 2.01949e14 0.868109
\(50\) 0 0
\(51\) −2.93773e14 −0.898804
\(52\) 0 0
\(53\) 1.52863e14 0.337255 0.168628 − 0.985680i \(-0.446066\pi\)
0.168628 + 0.985680i \(0.446066\pi\)
\(54\) 0 0
\(55\) 1.33685e14 0.215277
\(56\) 0 0
\(57\) −5.16670e14 −0.614153
\(58\) 0 0
\(59\) 2.62797e14 0.233012 0.116506 − 0.993190i \(-0.462831\pi\)
0.116506 + 0.993190i \(0.462831\pi\)
\(60\) 0 0
\(61\) −1.35855e15 −0.907346 −0.453673 − 0.891168i \(-0.649887\pi\)
−0.453673 + 0.891168i \(0.649887\pi\)
\(62\) 0 0
\(63\) 8.97376e14 0.455596
\(64\) 0 0
\(65\) −4.89991e13 −0.0190732
\(66\) 0 0
\(67\) −4.44864e14 −0.133842 −0.0669208 − 0.997758i \(-0.521317\pi\)
−0.0669208 + 0.997758i \(0.521317\pi\)
\(68\) 0 0
\(69\) 4.62335e15 1.08328
\(70\) 0 0
\(71\) 4.00327e15 0.735731 0.367865 − 0.929879i \(-0.380089\pi\)
0.367865 + 0.929879i \(0.380089\pi\)
\(72\) 0 0
\(73\) 9.24833e14 0.134220 0.0671102 − 0.997746i \(-0.478622\pi\)
0.0671102 + 0.997746i \(0.478622\pi\)
\(74\) 0 0
\(75\) −4.83014e15 −0.557107
\(76\) 0 0
\(77\) −1.70394e16 −1.57139
\(78\) 0 0
\(79\) −1.47473e16 −1.09366 −0.546830 − 0.837244i \(-0.684166\pi\)
−0.546830 + 0.837244i \(0.684166\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) 0 0
\(83\) −2.64230e16 −1.28771 −0.643855 − 0.765148i \(-0.722666\pi\)
−0.643855 + 0.765148i \(0.722666\pi\)
\(84\) 0 0
\(85\) 7.32323e15 0.291502
\(86\) 0 0
\(87\) −1.07465e15 −0.0351038
\(88\) 0 0
\(89\) −3.88837e16 −1.04702 −0.523508 − 0.852021i \(-0.675377\pi\)
−0.523508 + 0.852021i \(0.675377\pi\)
\(90\) 0 0
\(91\) 6.24542e15 0.139223
\(92\) 0 0
\(93\) −6.88814e15 −0.127643
\(94\) 0 0
\(95\) 1.28797e16 0.199184
\(96\) 0 0
\(97\) −2.53744e16 −0.328727 −0.164364 − 0.986400i \(-0.552557\pi\)
−0.164364 + 0.986400i \(0.552557\pi\)
\(98\) 0 0
\(99\) −3.51852e16 −0.383231
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 48.18.a.e.1.1 1
4.3 odd 2 3.18.a.a.1.1 ✓ 1
12.11 even 2 9.18.a.a.1.1 1
20.3 even 4 75.18.b.a.49.1 2
20.7 even 4 75.18.b.a.49.2 2
20.19 odd 2 75.18.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.18.a.a.1.1 ✓ 1 4.3 odd 2
9.18.a.a.1.1 1 12.11 even 2
48.18.a.e.1.1 1 1.1 even 1 trivial
75.18.a.a.1.1 1 20.19 odd 2
75.18.b.a.49.1 2 20.3 even 4
75.18.b.a.49.2 2 20.7 even 4