Properties

Label 48.22.a.e.1.1
Level $48$
Weight $22$
Character 48.1
Self dual yes
Analytic conductor $134.149$
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] = [48,22,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, 22); 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, 22, names="a")
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 48.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,59049,0,3109950] 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: \(134.149125258\)
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)
 
\(f(q)\) \(=\) \(q+59049.0 q^{3} +3.10995e6 q^{5} -3.63304e8 q^{7} +3.48678e9 q^{9} -1.45818e10 q^{11} +1.13351e11 q^{13} +1.83639e11 q^{15} -8.58939e12 q^{17} +2.92029e13 q^{19} -2.14527e13 q^{21} +1.55899e14 q^{23} -4.67165e14 q^{25} +2.05891e14 q^{27} +2.40079e15 q^{29} -2.23982e15 q^{31} -8.61043e14 q^{33} -1.12986e15 q^{35} -3.07851e16 q^{37} +6.69325e15 q^{39} -1.03208e17 q^{41} +1.65557e17 q^{43} +1.08437e16 q^{45} +6.65872e16 q^{47} -4.26556e17 q^{49} -5.07195e17 q^{51} +4.35423e17 q^{53} -4.53488e16 q^{55} +1.72440e18 q^{57} -5.53437e18 q^{59} -7.17621e18 q^{61} -1.26676e18 q^{63} +3.52515e17 q^{65} +1.57554e19 q^{67} +9.20569e18 q^{69} -2.64579e19 q^{71} +1.34712e19 q^{73} -2.75856e19 q^{75} +5.29764e18 q^{77} +1.68861e19 q^{79} +1.21577e19 q^{81} +1.70688e20 q^{83} -2.67126e19 q^{85} +1.41764e20 q^{87} -3.12592e20 q^{89} -4.11808e19 q^{91} -1.32259e20 q^{93} +9.08197e19 q^{95} +9.49015e20 q^{97} -5.08437e19 q^{99} +O(q^{100})\)

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
\(3\) 59049.0 0.577350
\(4\) 0 0
\(5\) 3.10995e6 0.142419 0.0712096 − 0.997461i \(-0.477314\pi\)
0.0712096 + 0.997461i \(0.477314\pi\)
\(6\) 0 0
\(7\) −3.63304e8 −0.486117 −0.243058 − 0.970012i \(-0.578151\pi\)
−0.243058 + 0.970012i \(0.578151\pi\)
\(8\) 0 0
\(9\) 3.48678e9 0.333333
\(10\) 0 0
\(11\) −1.45818e10 −0.169508 −0.0847538 − 0.996402i \(-0.527010\pi\)
−0.0847538 + 0.996402i \(0.527010\pi\)
\(12\) 0 0
\(13\) 1.13351e11 0.228044 0.114022 − 0.993478i \(-0.463627\pi\)
0.114022 + 0.993478i \(0.463627\pi\)
\(14\) 0 0
\(15\) 1.83639e11 0.0822257
\(16\) 0 0
\(17\) −8.58939e12 −1.03335 −0.516676 − 0.856181i \(-0.672831\pi\)
−0.516676 + 0.856181i \(0.672831\pi\)
\(18\) 0 0
\(19\) 2.92029e13 1.09273 0.546366 − 0.837546i \(-0.316011\pi\)
0.546366 + 0.837546i \(0.316011\pi\)
\(20\) 0 0
\(21\) −2.14527e13 −0.280660
\(22\) 0 0
\(23\) 1.55899e14 0.784696 0.392348 − 0.919817i \(-0.371663\pi\)
0.392348 + 0.919817i \(0.371663\pi\)
\(24\) 0 0
\(25\) −4.67165e14 −0.979717
\(26\) 0 0
\(27\) 2.05891e14 0.192450
\(28\) 0 0
\(29\) 2.40079e15 1.05968 0.529840 − 0.848097i \(-0.322252\pi\)
0.529840 + 0.848097i \(0.322252\pi\)
\(30\) 0 0
\(31\) −2.23982e15 −0.490812 −0.245406 − 0.969420i \(-0.578921\pi\)
−0.245406 + 0.969420i \(0.578921\pi\)
\(32\) 0 0
\(33\) −8.61043e14 −0.0978652
\(34\) 0 0
\(35\) −1.12986e15 −0.0692323
\(36\) 0 0
\(37\) −3.07851e16 −1.05250 −0.526250 − 0.850330i \(-0.676402\pi\)
−0.526250 + 0.850330i \(0.676402\pi\)
\(38\) 0 0
\(39\) 6.69325e15 0.131661
\(40\) 0 0
\(41\) −1.03208e17 −1.20083 −0.600414 − 0.799689i \(-0.704998\pi\)
−0.600414 + 0.799689i \(0.704998\pi\)
\(42\) 0 0
\(43\) 1.65557e17 1.16823 0.584117 − 0.811670i \(-0.301441\pi\)
0.584117 + 0.811670i \(0.301441\pi\)
\(44\) 0 0
\(45\) 1.08437e16 0.0474730
\(46\) 0 0
\(47\) 6.65872e16 0.184656 0.0923280 − 0.995729i \(-0.470569\pi\)
0.0923280 + 0.995729i \(0.470569\pi\)
\(48\) 0 0
\(49\) −4.26556e17 −0.763690
\(50\) 0 0
\(51\) −5.07195e17 −0.596607
\(52\) 0 0
\(53\) 4.35423e17 0.341991 0.170995 − 0.985272i \(-0.445302\pi\)
0.170995 + 0.985272i \(0.445302\pi\)
\(54\) 0 0
\(55\) −4.53488e16 −0.0241411
\(56\) 0 0
\(57\) 1.72440e18 0.630889
\(58\) 0 0
\(59\) −5.53437e18 −1.40968 −0.704842 − 0.709364i \(-0.748982\pi\)
−0.704842 + 0.709364i \(0.748982\pi\)
\(60\) 0 0
\(61\) −7.17621e18 −1.28805 −0.644023 − 0.765006i \(-0.722736\pi\)
−0.644023 + 0.765006i \(0.722736\pi\)
\(62\) 0 0
\(63\) −1.26676e18 −0.162039
\(64\) 0 0
\(65\) 3.52515e17 0.0324779
\(66\) 0 0
\(67\) 1.57554e19 1.05595 0.527977 − 0.849258i \(-0.322951\pi\)
0.527977 + 0.849258i \(0.322951\pi\)
\(68\) 0 0
\(69\) 9.20569e18 0.453045
\(70\) 0 0
\(71\) −2.64579e19 −0.964588 −0.482294 − 0.876009i \(-0.660196\pi\)
−0.482294 + 0.876009i \(0.660196\pi\)
\(72\) 0 0
\(73\) 1.34712e19 0.366875 0.183437 − 0.983031i \(-0.441278\pi\)
0.183437 + 0.983031i \(0.441278\pi\)
\(74\) 0 0
\(75\) −2.75856e19 −0.565640
\(76\) 0 0
\(77\) 5.29764e18 0.0824005
\(78\) 0 0
\(79\) 1.68861e19 0.200653 0.100326 − 0.994955i \(-0.468011\pi\)
0.100326 + 0.994955i \(0.468011\pi\)
\(80\) 0 0
\(81\) 1.21577e19 0.111111
\(82\) 0 0
\(83\) 1.70688e20 1.20749 0.603744 − 0.797178i \(-0.293675\pi\)
0.603744 + 0.797178i \(0.293675\pi\)
\(84\) 0 0
\(85\) −2.67126e19 −0.147169
\(86\) 0 0
\(87\) 1.41764e20 0.611807
\(88\) 0 0
\(89\) −3.12592e20 −1.06263 −0.531317 − 0.847173i \(-0.678303\pi\)
−0.531317 + 0.847173i \(0.678303\pi\)
\(90\) 0 0
\(91\) −4.11808e19 −0.110856
\(92\) 0 0
\(93\) −1.32259e20 −0.283371
\(94\) 0 0
\(95\) 9.08197e19 0.155626
\(96\) 0 0
\(97\) 9.49015e20 1.30668 0.653341 − 0.757064i \(-0.273367\pi\)
0.653341 + 0.757064i \(0.273367\pi\)
\(98\) 0 0
\(99\) −5.08437e19 −0.0565025
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 48.22.a.e.1.1 1
4.3 odd 2 3.22.a.a.1.1 ✓ 1
12.11 even 2 9.22.a.d.1.1 1
20.3 even 4 75.22.b.a.49.2 2
20.7 even 4 75.22.b.a.49.1 2
20.19 odd 2 75.22.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.22.a.a.1.1 ✓ 1 4.3 odd 2
9.22.a.d.1.1 1 12.11 even 2
48.22.a.e.1.1 1 1.1 even 1 trivial
75.22.a.c.1.1 1 20.19 odd 2
75.22.b.a.49.1 2 20.7 even 4
75.22.b.a.49.2 2 20.3 even 4