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Properties

Label 133.2.h.a
Level $133$
Weight $2$
Character orbit 133.h
Analytic conductor $1.062$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [133,2,Mod(11,133)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(133, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("133.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 133 = 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 133.h (of order \(3\), 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: \(1.06201034688\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{2} + 3 q^{3} + 22 q^{4} - 6 q^{6} - 2 q^{7} - 18 q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{2} + 3 q^{3} + 22 q^{4} - 6 q^{6} - 2 q^{7} - 18 q^{8} - 9 q^{9} - 32 q^{10} - q^{11} - 2 q^{12} + 6 q^{13} + 2 q^{14} - 9 q^{15} + 18 q^{16} + 8 q^{17} + 5 q^{18} + 2 q^{19} + 12 q^{21} - 2 q^{22} - 9 q^{23} - 8 q^{24} + 28 q^{25} + q^{26} - 18 q^{27} + 13 q^{28} - 2 q^{29} - 9 q^{30} + 11 q^{31} - 48 q^{32} - 6 q^{33} + 6 q^{34} - 31 q^{35} - 7 q^{36} - 14 q^{37} + 12 q^{38} - 10 q^{39} - 84 q^{40} + 20 q^{41} - 30 q^{42} + 2 q^{43} + 2 q^{44} - 12 q^{45} - 6 q^{46} + 39 q^{48} + 18 q^{49} + 22 q^{50} + 21 q^{51} + 11 q^{52} - 14 q^{53} + 86 q^{54} + 9 q^{55} - 21 q^{56} + 21 q^{57} + 35 q^{58} + 42 q^{59} - 6 q^{60} + 6 q^{61} - 19 q^{62} - 12 q^{63} - 2 q^{64} - 27 q^{65} - 6 q^{66} + 28 q^{67} + 51 q^{68} - 34 q^{69} + 42 q^{70} + q^{71} + 18 q^{72} - 21 q^{73} - 25 q^{74} + 31 q^{75} - 70 q^{76} - 20 q^{77} + 57 q^{78} + 10 q^{79} - 26 q^{80} + 28 q^{81} - 12 q^{82} + 10 q^{83} + 129 q^{84} - 27 q^{85} + 18 q^{86} + 53 q^{87} - 36 q^{88} - q^{89} + 27 q^{90} + 36 q^{91} - 72 q^{92} - 68 q^{93} + 12 q^{94} - 5 q^{95} - 94 q^{96} + 31 q^{97} - 65 q^{98} + 43 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −2.72364 1.24563 + 2.15750i 5.41824 1.22998 −3.39266 − 5.87626i 2.19933 − 1.47070i −9.31007 −1.60320 + 2.77683i −3.35002
11.2 −2.29215 −1.07625 − 1.86413i 3.25394 3.17431 2.46693 + 4.27285i −2.63669 − 0.218793i −2.87420 −0.816647 + 1.41447i −7.27599
11.3 −2.12002 −0.147810 − 0.256015i 2.49447 −1.61932 0.313361 + 0.542756i −0.177013 + 2.63982i −1.04829 1.45630 − 2.52239i 3.43298
11.4 −1.18985 −0.0886809 − 0.153600i −0.584258 0.114185 0.105517 + 0.182761i 1.73984 − 1.99323i 3.07488 1.48427 − 2.57083i −0.135863
11.5 −0.867953 1.27972 + 2.21653i −1.24666 3.56436 −1.11073 − 1.92385i −2.63921 − 0.185998i 2.81795 −1.77535 + 3.07499i −3.09369
11.6 −0.286706 1.53242 + 2.65422i −1.91780 −3.58001 −0.439353 − 0.760981i 2.10209 + 1.60661i 1.12326 −3.19660 + 5.53667i 1.02641
11.7 0.0743621 −0.698298 − 1.20949i −1.99447 −2.46272 −0.0519269 − 0.0899401i −2.54166 + 0.734810i −0.297037 0.524760 − 0.908910i −0.183133
11.8 1.09126 −1.15977 − 2.00878i −0.809151 2.88898 −1.26561 − 2.19210i −0.319979 − 2.62633i −3.06551 −1.19014 + 2.06138i 3.15263
11.9 1.11673 0.464784 + 0.805029i −0.752913 1.22815 0.519038 + 0.899001i 2.58326 + 0.571654i −3.07426 1.06795 − 1.84975i 1.37151
11.10 1.59493 0.741885 + 1.28498i 0.543815 0.620112 1.18326 + 2.04946i −2.33703 + 1.24028i −2.32252 0.399213 − 0.691457i 0.989038
11.11 2.28098 −1.23198 − 2.13385i 3.20287 −1.05456 −2.81012 − 4.86728i 1.23585 + 2.33938i 2.74373 −1.53555 + 2.65966i −2.40543
11.12 2.32205 0.638361 + 1.10567i 3.39191 −4.10346 1.48231 + 2.56743i −0.208777 − 2.63750i 3.23209 0.684991 − 1.18644i −9.52844
121.1 −2.72364 1.24563 − 2.15750i 5.41824 1.22998 −3.39266 + 5.87626i 2.19933 + 1.47070i −9.31007 −1.60320 − 2.77683i −3.35002
121.2 −2.29215 −1.07625 + 1.86413i 3.25394 3.17431 2.46693 − 4.27285i −2.63669 + 0.218793i −2.87420 −0.816647 − 1.41447i −7.27599
121.3 −2.12002 −0.147810 + 0.256015i 2.49447 −1.61932 0.313361 − 0.542756i −0.177013 − 2.63982i −1.04829 1.45630 + 2.52239i 3.43298
121.4 −1.18985 −0.0886809 + 0.153600i −0.584258 0.114185 0.105517 − 0.182761i 1.73984 + 1.99323i 3.07488 1.48427 + 2.57083i −0.135863
121.5 −0.867953 1.27972 − 2.21653i −1.24666 3.56436 −1.11073 + 1.92385i −2.63921 + 0.185998i 2.81795 −1.77535 − 3.07499i −3.09369
121.6 −0.286706 1.53242 − 2.65422i −1.91780 −3.58001 −0.439353 + 0.760981i 2.10209 − 1.60661i 1.12326 −3.19660 − 5.53667i 1.02641
121.7 0.0743621 −0.698298 + 1.20949i −1.99447 −2.46272 −0.0519269 + 0.0899401i −2.54166 − 0.734810i −0.297037 0.524760 + 0.908910i −0.183133
121.8 1.09126 −1.15977 + 2.00878i −0.809151 2.88898 −1.26561 + 2.19210i −0.319979 + 2.62633i −3.06551 −1.19014 − 2.06138i 3.15263
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
133.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 133.2.h.a yes 24
7.b odd 2 1 931.2.h.h 24
7.c even 3 1 133.2.g.a ✓ 24
7.c even 3 1 931.2.e.f 24
7.d odd 6 1 931.2.e.e 24
7.d odd 6 1 931.2.g.h 24
19.c even 3 1 133.2.g.a ✓ 24
133.g even 3 1 931.2.e.f 24
133.h even 3 1 inner 133.2.h.a yes 24
133.k odd 6 1 931.2.e.e 24
133.m odd 6 1 931.2.g.h 24
133.t odd 6 1 931.2.h.h 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.2.g.a ✓ 24 7.c even 3 1
133.2.g.a ✓ 24 19.c even 3 1
133.2.h.a yes 24 1.a even 1 1 trivial
133.2.h.a yes 24 133.h even 3 1 inner
931.2.e.e 24 7.d odd 6 1
931.2.e.e 24 133.k odd 6 1
931.2.e.f 24 7.c even 3 1
931.2.e.f 24 133.g even 3 1
931.2.g.h 24 7.d odd 6 1
931.2.g.h 24 133.m odd 6 1
931.2.h.h 24 7.b odd 2 1
931.2.h.h 24 133.t odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(133, [\chi])\).