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Properties

Label 148.5.b.c
Level $148$
Weight $5$
Character orbit 148.b
Analytic conductor $15.299$
Analytic rank $0$
Dimension $72$
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] = [148,5,Mod(147,148)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(148, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("148.147");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 148 = 2^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 148.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: \(15.2987545364\)
Analytic rank: \(0\)
Dimension: \(72\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 28 q^{4} - 2056 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 28 q^{4} - 2056 q^{9} + 30 q^{10} - 70 q^{12} - 204 q^{16} + 32 q^{21} - 10344 q^{25} - 2034 q^{26} + 560 q^{28} + 4032 q^{30} + 320 q^{33} - 3460 q^{34} - 1178 q^{36} - 2824 q^{37} - 7596 q^{38} - 1578 q^{40} - 1440 q^{41} - 4734 q^{44} + 2134 q^{46} - 4770 q^{48} - 18584 q^{49} + 1104 q^{53} - 12534 q^{58} + 19974 q^{62} + 11324 q^{64} + 2208 q^{65} + 12012 q^{70} + 27424 q^{73} + 13560 q^{74} + 192 q^{77} + 28874 q^{78} + 29480 q^{81} + 21916 q^{84} + 6368 q^{85} + 73512 q^{86} + 4868 q^{90}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
147.1 −3.99438 − 0.211899i 15.7739i 15.9102 + 1.69282i 22.2504i 3.34247 − 63.0068i 26.8039i −63.1927 − 10.1331i −167.814 4.71484 − 88.8765i
147.2 −3.99438 + 0.211899i − 15.7739i 15.9102 − 1.69282i − 22.2504i 3.34247 + 63.0068i − 26.8039i −63.1927 + 10.1331i −167.814 4.71484 + 88.8765i
147.3 −3.93144 − 0.737388i 11.9349i 14.9125 + 5.79800i − 34.3876i 8.80066 − 46.9214i 35.6639i −54.3524 − 33.7908i −61.4420 −25.3570 + 135.193i
147.4 −3.93144 + 0.737388i − 11.9349i 14.9125 − 5.79800i 34.3876i 8.80066 + 46.9214i − 35.6639i −54.3524 + 33.7908i −61.4420 −25.3570 − 135.193i
147.5 −3.92227 − 0.784717i 8.77180i 14.7684 + 6.15574i 36.1272i 6.88337 − 34.4054i − 71.2058i −53.0953 − 35.7335i 4.05559 28.3496 − 141.701i
147.6 −3.92227 + 0.784717i − 8.77180i 14.7684 − 6.15574i − 36.1272i 6.88337 + 34.4054i 71.2058i −53.0953 + 35.7335i 4.05559 28.3496 + 141.701i
147.7 −3.79647 − 1.25970i 0.122792i 12.8263 + 9.56480i − 40.3385i 0.154681 − 0.466176i − 89.7174i −36.6460 − 52.4697i 80.9849 −50.8144 + 153.144i
147.8 −3.79647 + 1.25970i − 0.122792i 12.8263 − 9.56480i 40.3385i 0.154681 + 0.466176i 89.7174i −36.6460 + 52.4697i 80.9849 −50.8144 − 153.144i
147.9 −3.61618 − 1.70974i − 8.59806i 10.1536 + 12.3655i − 5.96155i −14.7005 + 31.0922i 72.0900i −15.5754 − 62.0758i 7.07329 −10.1927 + 21.5581i
147.10 −3.61618 + 1.70974i 8.59806i 10.1536 − 12.3655i 5.96155i −14.7005 − 31.0922i − 72.0900i −15.5754 + 62.0758i 7.07329 −10.1927 − 21.5581i
147.11 −3.60727 − 1.72847i − 10.4528i 10.0248 + 12.4701i 17.8272i −18.0673 + 37.7059i − 29.8017i −14.6078 − 62.3106i −28.2600 30.8137 − 64.3074i
147.12 −3.60727 + 1.72847i 10.4528i 10.0248 − 12.4701i − 17.8272i −18.0673 − 37.7059i 29.8017i −14.6078 + 62.3106i −28.2600 30.8137 + 64.3074i
147.13 −3.36916 − 2.15611i 3.21057i 6.70242 + 14.5285i 29.5269i 6.92233 − 10.8169i 36.0051i 8.74350 − 63.3999i 70.6922 63.6630 − 99.4806i
147.14 −3.36916 + 2.15611i − 3.21057i 6.70242 − 14.5285i − 29.5269i 6.92233 + 10.8169i − 36.0051i 8.74350 + 63.3999i 70.6922 63.6630 + 99.4806i
147.15 −3.07052 − 2.56358i 8.01516i 2.85613 + 15.7430i − 12.0398i 20.5475 − 24.6107i 17.3644i 31.5887 − 55.6611i 16.7572 −30.8649 + 36.9682i
147.16 −3.07052 + 2.56358i − 8.01516i 2.85613 − 15.7430i 12.0398i 20.5475 + 24.6107i − 17.3644i 31.5887 + 55.6611i 16.7572 −30.8649 − 36.9682i
147.17 −2.72385 − 2.92928i 16.3376i −1.16132 + 15.9578i − 5.83052i 47.8575 − 44.5012i − 57.2375i 49.9081 − 40.0648i −185.919 −17.0792 + 15.8815i
147.18 −2.72385 + 2.92928i − 16.3376i −1.16132 − 15.9578i 5.83052i 47.8575 + 44.5012i 57.2375i 49.9081 + 40.0648i −185.919 −17.0792 − 15.8815i
147.19 −2.63645 − 3.00817i − 16.6992i −2.09821 + 15.8618i − 27.7702i −50.2342 + 44.0268i − 8.64995i 53.2470 − 35.5072i −197.865 −83.5376 + 73.2149i
147.20 −2.63645 + 3.00817i 16.6992i −2.09821 − 15.8618i 27.7702i −50.2342 − 44.0268i 8.64995i 53.2470 + 35.5072i −197.865 −83.5376 − 73.2149i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 147.72
Significant digits:
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Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 148.5.b.c ✓ 72
4.b odd 2 1 inner 148.5.b.c ✓ 72
37.b even 2 1 inner 148.5.b.c ✓ 72
148.b odd 2 1 inner 148.5.b.c ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
148.5.b.c ✓ 72 1.a even 1 1 trivial
148.5.b.c ✓ 72 4.b odd 2 1 inner
148.5.b.c ✓ 72 37.b even 2 1 inner
148.5.b.c ✓ 72 148.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(148, [\chi])\):

\( T_{3}^{36} + 1972 T_{3}^{34} + 1760158 T_{3}^{32} + 941738792 T_{3}^{30} + 337174088169 T_{3}^{28} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
\( T_{19}^{36} - 2446304 T_{19}^{34} + 2658834798304 T_{19}^{32} + \cdots + 12\!\cdots\!04 \) Copy content Toggle raw display