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Properties

Label 133.2.v.a
Level $133$
Weight $2$
Character orbit 133.v
Analytic conductor $1.062$
Analytic rank $0$
Dimension $30$
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(36,133)]
 
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([0, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("133.36");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 133 = 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 133.v (of order \(9\), 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: \(30\)
Relative dimension: \(5\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 3 q^{3} - 6 q^{4} - 3 q^{5} - 9 q^{6} - 15 q^{7} - 27 q^{8} - 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 3 q^{3} - 6 q^{4} - 3 q^{5} - 9 q^{6} - 15 q^{7} - 27 q^{8} - 15 q^{9} + 24 q^{10} - 18 q^{12} + 42 q^{15} + 18 q^{16} + 42 q^{18} - 6 q^{19} - 30 q^{20} + 6 q^{21} + 12 q^{22} + 9 q^{23} - 57 q^{24} - 15 q^{25} - 3 q^{26} - 21 q^{27} - 6 q^{28} - 24 q^{29} - 27 q^{30} + 45 q^{32} + 6 q^{33} + 54 q^{34} - 3 q^{35} - 69 q^{36} + 60 q^{37} + 9 q^{38} - 24 q^{39} + 69 q^{40} - 21 q^{41} + 9 q^{42} + 24 q^{44} + 9 q^{45} - 18 q^{46} + 3 q^{47} + 102 q^{48} - 15 q^{49} - 39 q^{50} + 9 q^{51} - 63 q^{52} + 21 q^{53} + 27 q^{54} - 51 q^{55} + 54 q^{56} - 48 q^{57} + 36 q^{58} + 12 q^{59} - 69 q^{60} + 6 q^{61} - 18 q^{62} + 3 q^{63} - 21 q^{64} - 63 q^{65} - 93 q^{66} + 27 q^{67} - 45 q^{68} + 51 q^{69} - 12 q^{70} - 42 q^{71} + 27 q^{72} + 63 q^{73} - 24 q^{74} - 24 q^{75} - 60 q^{76} + 66 q^{78} - 36 q^{79} + 75 q^{80} + 69 q^{81} - 57 q^{82} + 6 q^{83} - 18 q^{84} + 6 q^{85} - 12 q^{86} + 36 q^{88} + 24 q^{89} - 12 q^{90} + 21 q^{92} - 18 q^{93} + 6 q^{94} - 15 q^{95} + 174 q^{96} - 15 q^{97} + 69 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
36.1 −0.387603 + 2.19820i 1.80002 + 1.51040i −2.80248 − 1.02002i 0.503164 − 0.183137i −4.01785 + 3.37138i −0.500000 − 0.866025i 1.09635 − 1.89893i 0.437831 + 2.48306i 0.207544 + 1.17704i
36.2 −0.0712323 + 0.403978i 0.634840 + 0.532694i 1.72126 + 0.626488i 0.0185877 − 0.00676536i −0.260418 + 0.218517i −0.500000 − 0.866025i −0.785907 + 1.36123i −0.401686 − 2.27807i 0.00140902 + 0.00799093i
36.3 0.123631 − 0.701145i −1.31184 − 1.10077i 1.40306 + 0.510674i 1.02663 − 0.373661i −0.933982 + 0.783704i −0.500000 − 0.866025i 1.24348 − 2.15377i −0.0117003 − 0.0663559i −0.135068 − 0.766010i
36.4 0.366383 − 2.07786i −0.977821 − 0.820489i −2.30387 − 0.838542i −3.52214 + 1.28195i −2.06312 + 1.73116i −0.500000 − 0.866025i −0.476558 + 0.825422i −0.238013 − 1.34984i 1.37327 + 7.78820i
36.5 0.489766 − 2.77760i 1.65294 + 1.38698i −5.59582 − 2.03671i 1.64741 − 0.599609i 4.66203 − 3.91191i −0.500000 − 0.866025i −5.57736 + 9.66028i 0.287546 + 1.63076i −0.858628 − 4.86952i
43.1 −2.41199 + 0.877894i −0.396011 + 2.24589i 3.51492 − 2.94937i −2.98366 − 2.50359i −1.01648 − 5.76473i −0.500000 − 0.866025i −3.32195 + 5.75378i −2.06812 − 0.752734i 9.39446 + 3.41930i
43.2 −2.13117 + 0.775681i 0.550325 − 3.12105i 2.40810 − 2.02064i 0.171106 + 0.143575i 1.24810 + 7.07835i −0.500000 − 0.866025i −1.29676 + 2.24605i −6.61899 − 2.40911i −0.476023 − 0.173258i
43.3 −0.261232 + 0.0950807i 0.222356 − 1.26105i −1.47289 + 1.23590i 3.20702 + 2.69101i 0.0618145 + 0.350567i −0.500000 − 0.866025i 0.545253 − 0.944405i 1.27829 + 0.465258i −1.09364 − 0.398052i
43.4 0.138224 − 0.0503095i −0.432726 + 2.45411i −1.51551 + 1.27167i −0.799064 − 0.670494i 0.0636520 + 0.360988i −0.500000 − 0.866025i −0.292599 + 0.506796i −3.01634 − 1.09786i −0.144182 − 0.0524780i
43.5 1.84709 − 0.672286i 0.0770011 − 0.436695i 1.42769 − 1.19797i −1.03509 − 0.868545i −0.151356 − 0.858382i −0.500000 − 0.866025i −0.133948 + 0.232004i 2.63430 + 0.958808i −2.49582 − 0.908404i
85.1 −0.387603 − 2.19820i 1.80002 − 1.51040i −2.80248 + 1.02002i 0.503164 + 0.183137i −4.01785 − 3.37138i −0.500000 + 0.866025i 1.09635 + 1.89893i 0.437831 − 2.48306i 0.207544 − 1.17704i
85.2 −0.0712323 − 0.403978i 0.634840 − 0.532694i 1.72126 − 0.626488i 0.0185877 + 0.00676536i −0.260418 − 0.218517i −0.500000 + 0.866025i −0.785907 − 1.36123i −0.401686 + 2.27807i 0.00140902 − 0.00799093i
85.3 0.123631 + 0.701145i −1.31184 + 1.10077i 1.40306 − 0.510674i 1.02663 + 0.373661i −0.933982 − 0.783704i −0.500000 + 0.866025i 1.24348 + 2.15377i −0.0117003 + 0.0663559i −0.135068 + 0.766010i
85.4 0.366383 + 2.07786i −0.977821 + 0.820489i −2.30387 + 0.838542i −3.52214 − 1.28195i −2.06312 − 1.73116i −0.500000 + 0.866025i −0.476558 − 0.825422i −0.238013 + 1.34984i 1.37327 − 7.78820i
85.5 0.489766 + 2.77760i 1.65294 − 1.38698i −5.59582 + 2.03671i 1.64741 + 0.599609i 4.66203 + 3.91191i −0.500000 + 0.866025i −5.57736 − 9.66028i 0.287546 − 1.63076i −0.858628 + 4.86952i
92.1 −0.755264 − 0.633742i 0.921501 − 0.335399i −0.178501 − 1.01233i 0.247488 − 1.40357i −0.908533 − 0.330679i −0.500000 − 0.866025i −1.49267 + 2.58538i −1.56146 + 1.31022i −1.07642 + 0.903225i
92.2 −0.577739 − 0.484780i −1.43187 + 0.521157i −0.248526 − 1.40946i −0.688984 + 3.90742i 1.07989 + 0.393049i −0.500000 − 0.866025i −1.29388 + 2.24107i −0.519492 + 0.435906i 2.29229 − 1.92346i
92.3 0.413656 + 0.347098i −2.97582 + 1.08311i −0.296663 − 1.68246i 0.412785 − 2.34102i −1.60691 − 0.584868i −0.500000 − 0.866025i 1.00125 − 1.73422i 5.38425 − 4.51792i 0.983314 − 0.825098i
92.4 1.19328 + 1.00128i 1.01265 − 0.368576i 0.0740565 + 0.419995i −0.123624 + 0.701108i 1.57742 + 0.574136i −0.500000 − 0.866025i 1.22555 − 2.12272i −1.40851 + 1.18188i −0.849523 + 0.712835i
92.5 2.02420 + 1.69851i −0.845543 + 0.307752i 0.865171 + 4.90663i 0.418380 − 2.37275i −2.23427 − 0.813208i −0.500000 − 0.866025i −3.94025 + 6.82472i −1.67790 + 1.40793i 4.87702 − 4.09231i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 36.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 133.2.v.a ✓ 30
7.b odd 2 1 931.2.w.c 30
7.c even 3 1 931.2.v.c 30
7.c even 3 1 931.2.x.f 30
7.d odd 6 1 931.2.v.f 30
7.d odd 6 1 931.2.x.c 30
19.e even 9 1 inner 133.2.v.a ✓ 30
19.e even 9 1 2527.2.a.t 15
19.f odd 18 1 2527.2.a.q 15
133.u even 9 1 931.2.x.f 30
133.w even 9 1 931.2.v.c 30
133.x odd 18 1 931.2.x.c 30
133.y odd 18 1 931.2.w.c 30
133.z odd 18 1 931.2.v.f 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.2.v.a ✓ 30 1.a even 1 1 trivial
133.2.v.a ✓ 30 19.e even 9 1 inner
931.2.v.c 30 7.c even 3 1
931.2.v.c 30 133.w even 9 1
931.2.v.f 30 7.d odd 6 1
931.2.v.f 30 133.z odd 18 1
931.2.w.c 30 7.b odd 2 1
931.2.w.c 30 133.y odd 18 1
931.2.x.c 30 7.d odd 6 1
931.2.x.c 30 133.x odd 18 1
931.2.x.f 30 7.c even 3 1
931.2.x.f 30 133.u even 9 1
2527.2.a.q 15 19.f odd 18 1
2527.2.a.t 15 19.e even 9 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{30} + 3 T_{2}^{28} + 21 T_{2}^{27} - 18 T_{2}^{26} + 54 T_{2}^{25} + 416 T_{2}^{24} - 297 T_{2}^{23} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(133, [\chi])\). Copy content Toggle raw display