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Properties

Label 429.2.y.b
Level $429$
Weight $2$
Character orbit 429.y
Analytic conductor $3.426$
Analytic rank $0$
Dimension $176$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [429,2,Mod(116,429)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(429, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 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("429.116");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 429 = 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 429.y (of order \(10\), degree \(4\), 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: \(3.42558224671\)
Analytic rank: \(0\)
Dimension: \(176\)
Relative dimension: \(44\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 176 q - 6 q^{3} + 20 q^{4} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 176 q - 6 q^{3} + 20 q^{4} + 6 q^{9} + 12 q^{12} - 10 q^{13} + 36 q^{16} - 96 q^{22} - 20 q^{25} + 6 q^{27} - 100 q^{30} - 116 q^{36} + 50 q^{39} + 60 q^{40} + 52 q^{42} + 32 q^{48} - 88 q^{49} - 90 q^{51} + 210 q^{52} - 20 q^{55} - 92 q^{64} - 62 q^{66} + 96 q^{69} + 166 q^{75} - 112 q^{78} - 60 q^{79} + 70 q^{81} - 144 q^{82} - 148 q^{88} + 100 q^{90} + 62 q^{91} - 180 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1 −2.50188 + 0.812912i −0.234485 + 1.71611i 3.98057 − 2.89205i 0.944274 − 2.90618i −0.808388 − 4.48411i −2.69270 + 1.95636i −4.51544 + 6.21497i −2.89003 − 0.804801i 8.03853i
116.2 −2.50188 + 0.812912i 1.55965 − 0.753314i 3.98057 − 2.89205i 0.944274 − 2.90618i −3.28969 + 3.15256i 2.69270 − 1.95636i −4.51544 + 6.21497i 1.86504 − 2.34982i 8.03853i
116.3 −2.12303 + 0.689813i −1.71440 + 0.246677i 2.41336 − 1.75341i −0.370527 + 1.14036i 3.46954 − 1.70631i −1.00158 + 0.727691i −1.28990 + 1.77539i 2.87830 − 0.845805i − 2.67661i
116.4 −2.12303 + 0.689813i −0.295173 − 1.70671i 2.41336 − 1.75341i −0.370527 + 1.14036i 1.80397 + 3.41978i 1.00158 − 0.727691i −1.28990 + 1.77539i −2.82575 + 1.00755i − 2.67661i
116.5 −2.06588 + 0.671246i −1.48049 + 0.898973i 2.19926 − 1.59786i 0.803773 − 2.47376i 2.45508 − 2.85094i 3.30516 − 2.40134i −0.917289 + 1.26254i 1.38370 − 2.66184i 5.65002i
116.6 −2.06588 + 0.671246i 0.397478 − 1.68583i 2.19926 − 1.59786i 0.803773 − 2.47376i 0.310462 + 3.74952i −3.30516 + 2.40134i −0.917289 + 1.26254i −2.68402 − 1.34016i 5.65002i
116.7 −2.03433 + 0.660995i −0.778983 + 1.54699i 2.08356 − 1.51379i −0.766534 + 2.35915i 0.562157 − 3.66200i −2.59152 + 1.88285i −0.723470 + 0.995771i −1.78637 − 2.41016i − 5.30597i
116.8 −2.03433 + 0.660995i 1.23056 − 1.21890i 2.08356 − 1.51379i −0.766534 + 2.35915i −1.69767 + 3.29305i 2.59152 − 1.88285i −0.723470 + 0.995771i 0.0285456 − 2.99986i − 5.30597i
116.9 −1.39922 + 0.454633i 0.606125 + 1.62253i 0.133079 − 0.0966876i 0.724806 − 2.23072i −1.58576 − 1.99471i −0.197187 + 0.143265i 1.58728 − 2.18470i −2.26523 + 1.96691i 3.45079i
116.10 −1.39922 + 0.454633i 1.73042 + 0.0750685i 0.133079 − 0.0966876i 0.724806 − 2.23072i −2.45536 + 0.681670i 0.197187 − 0.143265i 1.58728 − 2.18470i 2.98873 + 0.259801i 3.45079i
116.11 −1.39675 + 0.453832i −0.128417 + 1.72728i 0.126919 − 0.0922120i −0.121706 + 0.374573i −0.604530 − 2.47087i 2.86530 − 2.08177i 1.59106 − 2.18990i −2.96702 − 0.443626i − 0.578420i
116.12 −1.39675 + 0.453832i 1.60306 − 0.655892i 0.126919 − 0.0922120i −0.121706 + 0.374573i −1.94141 + 1.64364i −2.86530 + 2.08177i 1.59106 − 2.18990i 2.13961 − 2.10287i − 0.578420i
116.13 −1.15602 + 0.375612i −1.73161 − 0.0391064i −0.422745 + 0.307142i −1.20121 + 3.69693i 2.01646 − 0.605206i 3.46009 − 2.51391i 1.80225 − 2.48058i 2.99694 + 0.135434i − 4.72490i
116.14 −1.15602 + 0.375612i −0.572289 − 1.63477i −0.422745 + 0.307142i −1.20121 + 3.69693i 1.27562 + 1.67487i −3.46009 + 2.51391i 1.80225 − 2.48058i −2.34497 + 1.87113i − 4.72490i
116.15 −0.743149 + 0.241464i −1.46911 + 0.917452i −1.12407 + 0.816683i 0.877069 − 2.69934i 0.870236 − 1.03654i −1.52431 + 1.10747i 1.55673 − 2.14266i 1.31656 − 2.69567i 2.21779i
116.16 −0.743149 + 0.241464i 0.418569 − 1.68071i −1.12407 + 0.816683i 0.877069 − 2.69934i 0.0947722 + 1.35009i 1.52431 − 1.10747i 1.55673 − 2.14266i −2.64960 − 1.40699i 2.21779i
116.17 −0.687043 + 0.223234i −1.63884 − 0.560531i −1.19584 + 0.868828i −0.0885211 + 0.272440i 1.25108 + 0.0192637i −1.47481 + 1.07151i 1.47687 − 2.03274i 2.37161 + 1.83724i − 0.206939i
116.18 −0.687043 + 0.223234i −1.03953 − 1.38542i −1.19584 + 0.868828i −0.0885211 + 0.272440i 1.02347 + 0.719784i 1.47481 − 1.07151i 1.47687 − 2.03274i −0.838768 + 2.88036i − 0.206939i
116.19 −0.473784 + 0.153942i 0.663585 + 1.59989i −1.41726 + 1.02970i 0.269770 − 0.830268i −0.560687 − 0.655850i −3.14053 + 2.28173i 1.09859 − 1.51208i −2.11931 + 2.12333i 0.434897i
116.20 −0.473784 + 0.153942i 1.72665 + 0.136713i −1.41726 + 1.02970i 0.269770 − 0.830268i −0.839104 + 0.201031i 3.14053 − 2.28173i 1.09859 − 1.51208i 2.96262 + 0.472112i 0.434897i
See next 80 embeddings (of 176 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 116.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.d odd 10 1 inner
13.b even 2 1 inner
33.f even 10 1 inner
39.d odd 2 1 inner
143.l odd 10 1 inner
429.y even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.y.b ✓ 176
3.b odd 2 1 inner 429.2.y.b ✓ 176
11.d odd 10 1 inner 429.2.y.b ✓ 176
13.b even 2 1 inner 429.2.y.b ✓ 176
33.f even 10 1 inner 429.2.y.b ✓ 176
39.d odd 2 1 inner 429.2.y.b ✓ 176
143.l odd 10 1 inner 429.2.y.b ✓ 176
429.y even 10 1 inner 429.2.y.b ✓ 176
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.y.b ✓ 176 1.a even 1 1 trivial
429.2.y.b ✓ 176 3.b odd 2 1 inner
429.2.y.b ✓ 176 11.d odd 10 1 inner
429.2.y.b ✓ 176 13.b even 2 1 inner
429.2.y.b ✓ 176 33.f even 10 1 inner
429.2.y.b ✓ 176 39.d odd 2 1 inner
429.2.y.b ✓ 176 143.l odd 10 1 inner
429.2.y.b ✓ 176 429.y even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{88} - 27 T_{2}^{86} + 434 T_{2}^{84} - 5386 T_{2}^{82} + 57065 T_{2}^{80} - 520748 T_{2}^{78} + \cdots + 14641 \) acting on \(S_{2}^{\mathrm{new}}(429, [\chi])\). Copy content Toggle raw display