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Properties

Label 148.4.g.b
Level $148$
Weight $4$
Character orbit 148.g
Analytic conductor $8.732$
Analytic rank $0$
Dimension $108$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [148,4,Mod(31,148)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(148, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("148.31");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 148 = 2^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 148.g (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.73228268085\)
Analytic rank: \(0\)
Dimension: \(108\)
Relative dimension: \(54\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 108 q + 12 q^{5} - 18 q^{6} + 964 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 108 q + 12 q^{5} - 18 q^{6} + 964 q^{9} - 76 q^{10} - 112 q^{12} - 140 q^{13} - 6 q^{14} + 424 q^{16} - 332 q^{17} - 102 q^{18} + 116 q^{20} + 70 q^{22} - 188 q^{24} + 436 q^{26} + 668 q^{29} + 720 q^{32} - 224 q^{33} + 932 q^{34} + 104 q^{37} + 1976 q^{38} + 522 q^{42} + 1024 q^{44} - 204 q^{45} + 192 q^{46} - 7068 q^{49} - 452 q^{50} - 404 q^{52} + 1136 q^{53} - 4210 q^{54} + 40 q^{56} + 392 q^{57} + 3716 q^{60} - 2772 q^{61} - 2306 q^{66} - 5544 q^{68} - 856 q^{69} - 2900 q^{70} - 1896 q^{72} - 616 q^{74} + 876 q^{76} - 6532 q^{80} + 4636 q^{81} + 322 q^{82} + 5844 q^{84} + 2116 q^{86} - 2764 q^{88} + 700 q^{89} - 2276 q^{90} + 892 q^{92} - 400 q^{93} + 566 q^{94} - 552 q^{96} - 2308 q^{97} + 3738 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 −2.82794 − 0.0524887i 0.916468 7.99449 + 0.296870i 6.00279 − 6.00279i −2.59172 − 0.0481042i − 21.9409i −22.5924 − 1.25915i −26.1601 −17.2906 + 16.6605i
31.2 −2.82679 + 0.0960817i 4.58338 7.98154 − 0.543206i −6.64956 + 6.64956i −12.9563 + 0.440379i 6.93760i −22.5100 + 2.30241i −5.99266 18.1580 − 19.4358i
31.3 −2.74628 − 0.676713i −5.15895 7.08412 + 3.71689i 8.74904 − 8.74904i 14.1679 + 3.49113i 16.4446i −16.9397 − 15.0015i −0.385282 −29.9479 + 18.1067i
31.4 −2.71520 − 0.792257i −5.96730 6.74466 + 4.30228i −13.4059 + 13.4059i 16.2024 + 4.72764i 27.2904i −14.9046 − 17.0251i 8.60872 47.0206 − 25.7788i
31.5 −2.71211 − 0.802786i −7.81503 6.71107 + 4.35449i −1.67578 + 1.67578i 21.1952 + 6.27380i − 29.3951i −14.7054 − 17.1974i 34.0747 5.89021 − 3.19961i
31.6 −2.65139 + 0.984950i 8.80359 6.05975 − 5.22298i 11.7801 − 11.7801i −23.3418 + 8.67110i 36.4468i −10.9224 + 19.8167i 50.5032 −19.6309 + 42.8366i
31.7 −2.64939 + 0.990322i −0.331335 6.03853 − 5.24750i 0.907840 − 0.907840i 0.877836 − 0.328128i 8.95565i −10.8017 + 19.8827i −26.8902 −1.50617 + 3.30427i
31.8 −2.64447 − 1.00338i 8.64038 5.98647 + 5.30681i −7.78774 + 7.78774i −22.8492 − 8.66956i − 6.34011i −10.5063 − 20.0404i 47.6561 28.4085 − 12.7804i
31.9 −2.55202 + 1.21951i −3.99536 5.02558 − 6.22444i −14.0681 + 14.0681i 10.1962 − 4.87239i − 24.7757i −5.23457 + 22.0136i −11.0371 18.7458 − 53.0584i
31.10 −2.53006 + 1.26444i −9.44883 4.80236 − 6.39823i 3.27828 − 3.27828i 23.9060 − 11.9475i 4.61333i −4.06002 + 22.2602i 62.2803 −4.14903 + 12.4395i
31.11 −2.43149 − 1.44495i 7.02897 3.82425 + 7.02674i 12.6120 − 12.6120i −17.0908 − 10.1565i − 12.8280i 0.854684 − 22.6113i 22.4064 −48.8895 + 12.4422i
31.12 −2.21228 + 1.76233i 8.51035 1.78836 − 7.79755i −0.902179 + 0.902179i −18.8273 + 14.9981i − 28.9243i 9.78553 + 20.4020i 45.4260 0.405932 − 3.58581i
31.13 −2.10869 + 1.88505i −1.59033 0.893180 − 7.94998i 15.0564 − 15.0564i 3.35351 − 2.99785i − 11.2397i 13.1027 + 18.4478i −24.4709 −3.36728 + 60.1312i
31.14 −2.09693 − 1.89813i 2.91927 0.794240 + 7.96048i 1.40681 − 1.40681i −6.12151 − 5.54115i 22.6693i 13.4445 − 18.2001i −18.4778 −5.62028 + 0.279681i
31.15 −1.89813 − 2.09693i −2.91927 −0.794240 + 7.96048i 1.40681 − 1.40681i 5.54115 + 6.12151i − 22.6693i 18.2001 − 13.4445i −18.4778 −5.62028 − 0.279681i
31.16 −1.69654 + 2.26313i 4.21404 −2.24353 − 7.67897i −5.09639 + 5.09639i −7.14928 + 9.53693i 3.48842i 21.1847 + 7.95025i −9.24185 −2.88759 − 20.1800i
31.17 −1.69525 + 2.26409i −3.41106 −2.25224 − 7.67642i −5.90425 + 5.90425i 5.78261 − 7.72297i 32.8948i 21.1982 + 7.91418i −15.3647 −3.35858 − 23.3770i
31.18 −1.44495 − 2.43149i −7.02897 −3.82425 + 7.02674i 12.6120 − 12.6120i 10.1565 + 17.0908i 12.8280i 22.6113 − 0.854684i 22.4064 −48.8895 − 12.4422i
31.19 −1.04544 + 2.62813i −6.73414 −5.81411 − 5.49510i −2.96572 + 2.96572i 7.04013 − 17.6982i − 15.2690i 20.5201 − 9.53544i 18.3486 −4.69380 − 10.8948i
31.20 −1.00338 − 2.64447i −8.64038 −5.98647 + 5.30681i −7.78774 + 7.78774i 8.66956 + 22.8492i 6.34011i 20.0404 + 10.5063i 47.6561 28.4085 + 12.7804i
See next 80 embeddings (of 108 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.54
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
37.d odd 4 1 inner
148.g even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 148.4.g.b ✓ 108
4.b odd 2 1 inner 148.4.g.b ✓ 108
37.d odd 4 1 inner 148.4.g.b ✓ 108
148.g even 4 1 inner 148.4.g.b ✓ 108
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
148.4.g.b ✓ 108 1.a even 1 1 trivial
148.4.g.b ✓ 108 4.b odd 2 1 inner
148.4.g.b ✓ 108 37.d odd 4 1 inner
148.4.g.b ✓ 108 148.g even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{54} - 970 T_{3}^{52} + 440427 T_{3}^{50} - 124433258 T_{3}^{48} + 24529983539 T_{3}^{46} + \cdots - 63\!\cdots\!48 \) acting on \(S_{4}^{\mathrm{new}}(148, [\chi])\). Copy content Toggle raw display