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Properties

Label 152.4.b.b
Level $152$
Weight $4$
Character orbit 152.b
Analytic conductor $8.968$
Analytic rank $0$
Dimension $56$
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] = [152,4,Mod(75,152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(152, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("152.75");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 152 = 2^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 152.b (of order \(2\), degree \(1\), 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: \(8.96829032087\)
Analytic rank: \(0\)
Dimension: \(56\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 2 q^{4} - 14 q^{6} - 528 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 2 q^{4} - 14 q^{6} - 528 q^{9} - 40 q^{11} - 262 q^{16} - 184 q^{17} - 84 q^{19} - 12 q^{20} + 238 q^{24} - 1504 q^{25} + 378 q^{26} - 382 q^{28} + 512 q^{30} + 40 q^{35} + 1464 q^{36} + 958 q^{38} + 1030 q^{42} + 576 q^{43} + 316 q^{44} - 3664 q^{49} - 1314 q^{54} - 648 q^{57} - 1166 q^{58} + 928 q^{62} + 3746 q^{64} + 680 q^{66} + 2538 q^{68} - 432 q^{73} - 4020 q^{74} + 3968 q^{76} + 4608 q^{80} + 2296 q^{81} - 192 q^{82} + 5376 q^{83} - 1906 q^{92} - 1962 q^{96} - 6152 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
75.1 −2.82566 − 0.125191i − 3.78769i 7.96865 + 0.707493i − 18.1832i −0.474184 + 10.7027i − 24.7388i −22.4281 − 2.99673i 12.6534 −2.27637 + 51.3795i
75.2 −2.82566 + 0.125191i 3.78769i 7.96865 − 0.707493i 18.1832i −0.474184 − 10.7027i 24.7388i −22.4281 + 2.99673i 12.6534 −2.27637 − 51.3795i
75.3 −2.81728 − 0.250834i 8.04587i 7.87416 + 1.41334i − 9.63632i 2.01818 − 22.6675i 0.162175i −21.8292 − 5.95690i −37.7360 −2.41712 + 27.1482i
75.4 −2.81728 + 0.250834i − 8.04587i 7.87416 − 1.41334i 9.63632i 2.01818 + 22.6675i − 0.162175i −21.8292 + 5.95690i −37.7360 −2.41712 − 27.1482i
75.5 −2.66988 − 0.933662i − 0.255308i 6.25655 + 4.98554i − 11.8859i −0.238371 + 0.681642i 31.1204i −12.0494 − 19.1523i 26.9348 −11.0974 + 31.7339i
75.6 −2.66988 + 0.933662i 0.255308i 6.25655 − 4.98554i 11.8859i −0.238371 − 0.681642i − 31.1204i −12.0494 + 19.1523i 26.9348 −11.0974 − 31.7339i
75.7 −2.58234 − 1.15391i 4.27659i 5.33699 + 5.95958i 5.11676i 4.93480 − 11.0436i − 21.0790i −6.90512 − 21.5481i 8.71076 5.90427 − 13.2132i
75.8 −2.58234 + 1.15391i − 4.27659i 5.33699 − 5.95958i − 5.11676i 4.93480 + 11.0436i 21.0790i −6.90512 + 21.5481i 8.71076 5.90427 + 13.2132i
75.9 −2.57123 − 1.17846i − 5.03936i 5.22247 + 6.06019i 5.72550i −5.93868 + 12.9574i − 3.30417i −6.28649 − 21.7366i 1.60489 6.74727 − 14.7216i
75.10 −2.57123 + 1.17846i 5.03936i 5.22247 − 6.06019i − 5.72550i −5.93868 − 12.9574i 3.30417i −6.28649 + 21.7366i 1.60489 6.74727 + 14.7216i
75.11 −2.08379 − 1.91254i − 9.95156i 0.684366 + 7.97067i − 10.9343i −19.0328 + 20.7370i 8.59819i 13.8182 − 17.9181i −72.0335 −20.9122 + 22.7847i
75.12 −2.08379 + 1.91254i 9.95156i 0.684366 − 7.97067i 10.9343i −19.0328 − 20.7370i − 8.59819i 13.8182 + 17.9181i −72.0335 −20.9122 − 22.7847i
75.13 −2.03616 − 1.96318i 1.32918i 0.291881 + 7.99467i 12.0484i 2.60942 − 2.70643i 0.906494i 15.1006 − 16.8514i 25.2333 23.6532 − 24.5325i
75.14 −2.03616 + 1.96318i − 1.32918i 0.291881 − 7.99467i − 12.0484i 2.60942 + 2.70643i − 0.906494i 15.1006 + 16.8514i 25.2333 23.6532 + 24.5325i
75.15 −1.78669 − 2.19266i 7.28174i −1.61551 + 7.83519i − 9.90745i 15.9664 − 13.0102i − 14.2544i 20.0663 − 10.4568i −26.0237 −21.7237 + 17.7015i
75.16 −1.78669 + 2.19266i − 7.28174i −1.61551 − 7.83519i 9.90745i 15.9664 + 13.0102i 14.2544i 20.0663 + 10.4568i −26.0237 −21.7237 − 17.7015i
75.17 −1.75234 − 2.22020i 9.10949i −1.85859 + 7.78111i 8.44223i 20.2249 − 15.9629i 32.9185i 20.5325 − 9.50873i −55.9827 18.7435 − 14.7937i
75.18 −1.75234 + 2.22020i − 9.10949i −1.85859 − 7.78111i − 8.44223i 20.2249 + 15.9629i − 32.9185i 20.5325 + 9.50873i −55.9827 18.7435 + 14.7937i
75.19 −1.43264 − 2.43876i − 0.262060i −3.89506 + 6.98774i − 16.1045i −0.639101 + 0.375439i − 14.0942i 22.6216 − 0.511862i 26.9313 −39.2748 + 23.0720i
75.20 −1.43264 + 2.43876i 0.262060i −3.89506 − 6.98774i 16.1045i −0.639101 − 0.375439i 14.0942i 22.6216 + 0.511862i 26.9313 −39.2748 − 23.0720i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 75.56
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
19.b odd 2 1 inner
152.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 152.4.b.b ✓ 56
4.b odd 2 1 608.4.b.b 56
8.b even 2 1 608.4.b.b 56
8.d odd 2 1 inner 152.4.b.b ✓ 56
19.b odd 2 1 inner 152.4.b.b ✓ 56
76.d even 2 1 608.4.b.b 56
152.b even 2 1 inner 152.4.b.b ✓ 56
152.g odd 2 1 608.4.b.b 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.4.b.b ✓ 56 1.a even 1 1 trivial
152.4.b.b ✓ 56 8.d odd 2 1 inner
152.4.b.b ✓ 56 19.b odd 2 1 inner
152.4.b.b ✓ 56 152.b even 2 1 inner
608.4.b.b 56 4.b odd 2 1
608.4.b.b 56 8.b even 2 1
608.4.b.b 56 76.d even 2 1
608.4.b.b 56 152.g odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{28} + 510 T_{3}^{26} + 114211 T_{3}^{24} + 14801236 T_{3}^{22} + 1231097567 T_{3}^{20} + \cdots + 14\!\cdots\!00 \) acting on \(S_{4}^{\mathrm{new}}(152, [\chi])\). Copy content Toggle raw display