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Properties

Label 133.2.ba.a
Level $133$
Weight $2$
Character orbit 133.ba
Analytic conductor $1.062$
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] = [133,2,Mod(13,133)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(133, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("133.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 133 = 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 133.ba (of order \(18\), degree \(6\), 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: \(72\)
Relative dimension: \(12\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 12 q^{2} - 12 q^{4} - 6 q^{7} - 18 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 12 q^{2} - 12 q^{4} - 6 q^{7} - 18 q^{8} - 12 q^{9} - 12 q^{11} + 21 q^{14} - 18 q^{15} - 24 q^{16} - 12 q^{21} + 6 q^{23} - 18 q^{25} + 6 q^{28} - 42 q^{29} - 6 q^{30} + 60 q^{32} + 18 q^{35} - 90 q^{36} + 48 q^{39} + 81 q^{42} - 102 q^{43} - 18 q^{44} + 36 q^{46} + 18 q^{49} + 72 q^{50} - 12 q^{51} - 72 q^{53} + 108 q^{57} - 96 q^{58} - 78 q^{60} - 36 q^{63} + 30 q^{64} + 90 q^{65} - 18 q^{67} - 111 q^{70} + 48 q^{71} + 246 q^{72} + 54 q^{74} - 54 q^{77} - 78 q^{78} + 102 q^{79} + 42 q^{81} - 225 q^{84} + 24 q^{85} + 204 q^{86} - 18 q^{88} - 39 q^{91} + 30 q^{92} - 18 q^{93} - 48 q^{95} + 60 q^{98} + 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −1.72570 − 2.05661i −2.76606 − 1.00676i −0.904305 + 5.12857i 0.506166 − 0.0892508i 2.70287 + 7.42607i −2.64575 − 0.00224753i 7.45797 − 4.30586i 4.33936 + 3.64116i −1.05705 − 0.886967i
13.2 −1.72570 − 2.05661i 2.76606 + 1.00676i −0.904305 + 5.12857i −0.506166 + 0.0892508i −2.70287 − 7.42607i 1.32482 + 2.29016i 7.45797 − 4.30586i 4.33936 + 3.64116i 1.05705 + 0.886967i
13.3 −1.19814 − 1.42788i −0.392473 − 0.142848i −0.256024 + 1.45199i 3.22534 − 0.568715i 0.266265 + 0.731557i 2.59219 − 0.529651i −0.848472 + 0.489866i −2.16450 − 1.81623i −4.67646 − 3.92402i
13.4 −1.19814 − 1.42788i 0.392473 + 0.142848i −0.256024 + 1.45199i −3.22534 + 0.568715i −0.266265 − 0.731557i −0.837406 − 2.50973i −0.848472 + 0.489866i −2.16450 − 1.81623i 4.67646 + 3.92402i
13.5 −0.469155 − 0.559117i −1.71470 − 0.624101i 0.254791 − 1.44499i −1.94169 + 0.342373i 0.455517 + 1.25152i −0.910023 + 2.48432i −2.19164 + 1.26534i 0.252576 + 0.211937i 1.10238 + 0.925008i
13.6 −0.469155 − 0.559117i 1.71470 + 0.624101i 0.254791 − 1.44499i 1.94169 − 0.342373i −0.455517 − 1.25152i −1.69647 + 2.03026i −2.19164 + 1.26534i 0.252576 + 0.211937i −1.10238 − 0.925008i
13.7 0.241883 + 0.288265i −1.17737 − 0.428527i 0.322707 − 1.83016i 1.13920 − 0.200871i −0.161256 − 0.443047i −1.40826 − 2.23982i 1.25740 − 0.725963i −1.09557 − 0.919296i 0.333457 + 0.279803i
13.8 0.241883 + 0.288265i 1.17737 + 0.428527i 0.322707 − 1.83016i −1.13920 + 0.200871i 0.161256 + 0.443047i 2.64387 + 0.0996783i 1.25740 − 0.725963i −1.09557 − 0.919296i −0.333457 − 0.279803i
13.9 0.857238 + 1.02162i −2.61214 − 0.950740i 0.0384535 − 0.218080i 3.19214 − 0.562860i −1.26793 − 3.48361i 1.46597 + 2.20248i 2.56566 − 1.48129i 3.62122 + 3.03857i 3.31145 + 2.77864i
13.10 0.857238 + 1.02162i 2.61214 + 0.950740i 0.0384535 − 0.218080i −3.19214 + 0.562860i 1.26793 + 3.48361i −2.64039 − 0.168330i 2.56566 − 1.48129i 3.62122 + 3.03857i −3.31145 − 2.77864i
13.11 1.46752 + 1.74892i −0.0432520 − 0.0157424i −0.557818 + 3.16354i −1.88950 + 0.333170i −0.0359409 − 0.0987467i 2.44438 + 1.01243i −2.39703 + 1.38393i −2.29651 − 1.92700i −3.35557 − 2.81565i
13.12 1.46752 + 1.74892i 0.0432520 + 0.0157424i −0.557818 + 3.16354i 1.88950 − 0.333170i 0.0359409 + 0.0987467i −2.09898 − 1.61068i −2.39703 + 1.38393i −2.29651 − 1.92700i 3.35557 + 2.81565i
34.1 −0.783827 + 2.15355i −0.377791 − 2.14256i −2.49129 − 2.09044i 1.71380 + 2.04243i 4.91023 + 0.865806i 0.255448 + 2.63339i 2.48517 − 1.43481i −1.62876 + 0.592822i −5.74180 + 2.08984i
34.2 −0.783827 + 2.15355i 0.377791 + 2.14256i −2.49129 − 2.09044i −1.71380 − 2.04243i −4.91023 − 0.865806i −2.40831 + 1.09547i 2.48517 − 1.43481i −1.62876 + 0.592822i 5.74180 − 2.08984i
34.3 −0.712123 + 1.95654i −0.296592 − 1.68206i −1.78885 − 1.50102i −2.24591 − 2.67657i 3.50223 + 0.617538i 1.61888 − 2.09266i 0.604383 − 0.348940i 0.0777219 − 0.0282884i 6.83619 − 2.48817i
34.4 −0.712123 + 1.95654i 0.296592 + 1.68206i −1.78885 − 1.50102i 2.24591 + 2.67657i −3.50223 − 0.617538i 1.00286 − 2.44832i 0.604383 − 0.348940i 0.0777219 − 0.0282884i −6.83619 + 2.48817i
34.5 −0.188638 + 0.518279i −0.106238 − 0.602505i 1.29906 + 1.09004i −0.591581 − 0.705019i 0.332306 + 0.0585945i 1.71331 + 2.01608i −1.76529 + 1.01919i 2.46735 − 0.898043i 0.476991 − 0.173611i
34.6 −0.188638 + 0.518279i 0.106238 + 0.602505i 1.29906 + 1.09004i 0.591581 + 0.705019i −0.332306 − 0.0585945i −2.60263 − 0.475725i −1.76529 + 1.01919i 2.46735 − 0.898043i −0.476991 + 0.173611i
34.7 −0.0123531 + 0.0339398i −0.553028 − 3.13638i 1.53109 + 1.28474i 1.64398 + 1.95922i 0.113280 + 0.0199743i 0.388335 − 2.61710i −0.125076 + 0.0722125i −6.71196 + 2.44295i −0.0868037 + 0.0315940i
34.8 −0.0123531 + 0.0339398i 0.553028 + 3.13638i 1.53109 + 1.28474i −1.64398 − 1.95922i −0.113280 − 0.0199743i 2.07230 − 1.64486i −0.125076 + 0.0722125i −6.71196 + 2.44295i 0.0868037 − 0.0315940i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
19.f odd 18 1 inner
133.ba even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 133.2.ba.a ✓ 72
7.b odd 2 1 inner 133.2.ba.a ✓ 72
7.c even 3 1 931.2.bf.b 72
7.c even 3 1 931.2.bj.b 72
7.d odd 6 1 931.2.bf.b 72
7.d odd 6 1 931.2.bj.b 72
19.f odd 18 1 inner 133.2.ba.a ✓ 72
133.ba even 18 1 inner 133.2.ba.a ✓ 72
133.bb even 18 1 931.2.bj.b 72
133.bd odd 18 1 931.2.bj.b 72
133.be odd 18 1 931.2.bf.b 72
133.bf even 18 1 931.2.bf.b 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.2.ba.a ✓ 72 1.a even 1 1 trivial
133.2.ba.a ✓ 72 7.b odd 2 1 inner
133.2.ba.a ✓ 72 19.f odd 18 1 inner
133.2.ba.a ✓ 72 133.ba even 18 1 inner
931.2.bf.b 72 7.c even 3 1
931.2.bf.b 72 7.d odd 6 1
931.2.bf.b 72 133.be odd 18 1
931.2.bf.b 72 133.bf even 18 1
931.2.bj.b 72 7.c even 3 1
931.2.bj.b 72 7.d odd 6 1
931.2.bj.b 72 133.bb even 18 1
931.2.bj.b 72 133.bd odd 18 1

Hecke kernels

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