Properties

Label 148.2.q.b
Level $148$
Weight $2$
Character orbit 148.q
Analytic conductor $1.182$
Analytic rank $0$
Dimension $192$
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] = [148,2,Mod(15,148)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(148, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([18, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("148.15");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 148 = 2^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 148.q (of order \(36\), degree \(12\), 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: \(1.18178594991\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(16\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 192 q - 12 q^{2} - 18 q^{4} - 24 q^{5} - 12 q^{6} - 30 q^{8} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 192 q - 12 q^{2} - 18 q^{4} - 24 q^{5} - 12 q^{6} - 30 q^{8} - 24 q^{9} - 6 q^{10} - 12 q^{12} - 24 q^{13} - 12 q^{14} + 18 q^{16} - 42 q^{18} - 24 q^{21} - 12 q^{25} + 24 q^{26} - 12 q^{28} - 24 q^{29} - 72 q^{30} - 42 q^{32} + 12 q^{33} + 6 q^{34} - 48 q^{37} + 60 q^{38} - 36 q^{40} - 48 q^{41} - 12 q^{42} - 18 q^{44} - 24 q^{45} - 72 q^{46} - 18 q^{48} - 96 q^{49} - 66 q^{50} + 18 q^{52} - 24 q^{53} + 48 q^{54} - 12 q^{56} - 24 q^{57} - 30 q^{58} + 144 q^{60} + 48 q^{61} + 48 q^{62} + 108 q^{64} - 36 q^{65} + 240 q^{66} - 18 q^{68} - 24 q^{69} + 78 q^{70} + 78 q^{72} + 72 q^{74} + 126 q^{76} + 108 q^{77} + 198 q^{78} + 120 q^{80} + 12 q^{81} + 120 q^{82} + 30 q^{84} - 144 q^{85} + 30 q^{86} + 96 q^{88} - 120 q^{89} - 42 q^{90} + 114 q^{92} - 144 q^{93} - 24 q^{94} - 96 q^{96} - 24 q^{97} - 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1 −1.35843 + 0.393269i 0.312666 0.113801i 1.69068 1.06846i −2.13888 3.05464i −0.379981 + 0.277553i −2.14447 0.378127i −1.87648 + 2.11632i −2.21332 + 1.85720i 4.10682 + 3.30837i
15.2 −1.35454 0.406466i −1.17609 + 0.428063i 1.66957 + 1.10115i 0.506128 + 0.722826i 1.76706 0.101787i −3.36523 0.593381i −1.81392 2.17018i −1.09817 + 0.921478i −0.391767 1.18482i
15.3 −1.28548 + 0.589529i 2.82129 1.02687i 1.30491 1.51565i 1.88251 + 2.68851i −3.02134 + 2.98325i −2.77218 0.488810i −0.783913 + 2.71762i 4.60710 3.86581i −4.00488 2.34622i
15.4 −1.25586 0.650244i 1.92957 0.702306i 1.15437 + 1.63323i −0.635590 0.907717i −2.87994 0.372694i 2.95389 + 0.520851i −0.387722 2.80173i 0.931877 0.781938i 0.207974 + 1.55325i
15.5 −0.987028 + 1.01281i −0.343248 + 0.124932i −0.0515514 1.99934i 0.568635 + 0.812094i 0.212263 0.470955i 2.90658 + 0.512509i 2.07582 + 1.92119i −2.19592 + 1.84260i −1.38375 0.225643i
15.6 −0.954588 1.04344i −2.65144 + 0.965046i −0.177523 + 1.99211i −0.859834 1.22797i 3.53800 + 1.84539i 3.28436 + 0.579122i 2.24810 1.71641i 3.80070 3.18917i −0.460523 + 2.06939i
15.7 −0.185722 1.40197i 2.65144 0.965046i −1.93101 + 0.520752i −0.859834 1.22797i −1.84539 3.53800i −3.28436 0.579122i 1.08871 + 2.61050i 3.80070 3.18917i −1.56188 + 1.43352i
15.8 −0.106823 + 1.41017i −1.93332 + 0.703672i −1.97718 0.301279i −1.75433 2.50545i −0.785775 2.80149i 1.00092 + 0.176489i 0.636063 2.75598i 0.944446 0.792484i 3.72052 2.20627i
15.9 0.254373 + 1.39115i 1.68945 0.614911i −1.87059 + 0.707741i 0.733711 + 1.04785i 1.28518 + 2.19386i 0.596743 + 0.105222i −1.46040 2.42224i 0.178005 0.149364i −1.27108 + 1.28724i
15.10 0.309135 1.38001i −1.92957 + 0.702306i −1.80887 0.853220i −0.635590 0.907717i 0.372694 + 2.87994i −2.95389 0.520851i −1.73664 + 2.23251i 0.931877 0.781938i −1.44914 + 0.596516i
15.11 0.559312 1.29891i 1.17609 0.428063i −1.37434 1.45299i 0.506128 + 0.722826i 0.101787 1.76706i 3.36523 + 0.593381i −2.65599 + 0.972471i −1.09817 + 0.921478i 1.22197 0.253130i
15.12 0.902174 + 1.08907i −1.68945 + 0.614911i −0.372164 + 1.96507i 0.733711 + 1.04785i −2.19386 1.28518i −0.596743 0.105222i −2.47586 + 1.36752i 0.178005 0.149364i −0.479249 + 1.74441i
15.13 1.14892 + 0.824611i 1.93332 0.703672i 0.640035 + 1.89482i −1.75433 2.50545i 2.80149 + 0.785775i −1.00092 0.176489i −0.827143 + 2.70478i 0.944446 0.792484i 0.0504299 4.32520i
15.14 1.17445 0.787832i −0.312666 + 0.113801i 0.758642 1.85053i −2.13888 3.05464i −0.277553 + 0.379981i 2.14447 + 0.378127i −0.566923 2.77103i −2.21332 + 1.85720i −4.91854 1.90243i
15.15 1.27790 0.605792i −2.82129 + 1.02687i 1.26603 1.54828i 1.88251 + 2.68851i −2.98325 + 3.02134i 2.77218 + 0.488810i 0.679923 2.74549i 4.60710 3.86581i 4.03433 + 2.29522i
15.16 1.41030 0.105088i 0.343248 0.124932i 1.97791 0.296413i 0.568635 + 0.812094i 0.470955 0.212263i −2.90658 0.512509i 2.75831 0.625887i −2.19592 + 1.84260i 0.887289 + 1.08554i
19.1 −1.41220 0.0753572i 1.79147 1.50323i 1.98864 + 0.212840i −1.25294 2.68694i −2.64321 + 1.98786i 0.306442 0.841942i −2.79233 0.450432i 0.428749 2.43156i 1.56693 + 3.88892i
19.2 −1.37766 + 0.319439i −1.79147 + 1.50323i 1.79592 0.880160i −1.25294 2.68694i 1.98786 2.64321i −0.306442 + 0.841942i −2.19301 + 1.78625i 0.428749 2.43156i 2.58444 + 3.30146i
19.3 −1.35715 0.397685i 0.898867 0.754239i 1.68369 + 1.07943i 1.24600 + 2.67206i −1.51984 + 0.666147i −1.12931 + 3.10275i −1.85575 2.13453i −0.281859 + 1.59850i −0.628371 4.12190i
19.4 −1.26747 + 0.627309i −0.898867 + 0.754239i 1.21297 1.59019i 1.24600 + 2.67206i 0.666147 1.51984i 1.12931 3.10275i −0.539858 + 2.77643i −0.281859 + 1.59850i −3.25548 2.60513i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
37.i odd 36 1 inner
148.q even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 148.2.q.b 192
4.b odd 2 1 inner 148.2.q.b 192
37.i odd 36 1 inner 148.2.q.b 192
148.q even 36 1 inner 148.2.q.b 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
148.2.q.b 192 1.a even 1 1 trivial
148.2.q.b 192 4.b odd 2 1 inner
148.2.q.b 192 37.i odd 36 1 inner
148.2.q.b 192 148.q even 36 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{192} + 12 T_{3}^{190} + 42 T_{3}^{188} + 3174 T_{3}^{186} + 35427 T_{3}^{184} + 79668 T_{3}^{182} + 5671575 T_{3}^{180} + 62327286 T_{3}^{178} + 103869771 T_{3}^{176} + 6648514522 T_{3}^{174} + \cdots + 10\!\cdots\!84 \) acting on \(S_{2}^{\mathrm{new}}(148, [\chi])\). Copy content Toggle raw display