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Properties

Label 429.2.x.a
Level $429$
Weight $2$
Character orbit 429.x
Analytic conductor $3.426$
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] = [429,2,Mod(248,429)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("429.248");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 429 = 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 429.x (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: \(192\)
Relative dimension: \(48\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 192 q + 6 q^{3} - 48 q^{4} - 10 q^{6} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 192 q + 6 q^{3} - 48 q^{4} - 10 q^{6} - 14 q^{9} - 20 q^{12} + 6 q^{15} - 28 q^{16} - 10 q^{18} - 60 q^{19} + 4 q^{22} + 30 q^{24} - 12 q^{25} + 30 q^{27} - 20 q^{28} + 12 q^{31} - 32 q^{33} + 56 q^{34} + 44 q^{36} + 60 q^{40} - 108 q^{42} - 40 q^{45} - 28 q^{48} - 24 q^{49} + 10 q^{51} + 20 q^{52} - 20 q^{55} + 50 q^{57} - 60 q^{58} + 36 q^{60} + 20 q^{61} - 30 q^{63} + 64 q^{64} - 30 q^{66} - 120 q^{67} - 16 q^{69} + 20 q^{70} + 150 q^{72} - 20 q^{73} + 18 q^{75} + 40 q^{78} - 160 q^{79} - 46 q^{81} + 88 q^{82} - 40 q^{85} + 72 q^{88} - 80 q^{90} + 8 q^{91} - 72 q^{93} - 120 q^{94} - 140 q^{96} - 88 q^{97} + 62 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
248.1 −0.867138 − 2.66878i −1.64023 + 0.556462i −4.75241 + 3.45283i 1.12653 + 0.366031i 2.90738 + 3.89488i −1.16567 − 1.60441i 8.79543 + 6.39025i 2.38070 − 1.82545i − 3.32385i
248.2 −0.833208 − 2.56435i 1.66893 + 0.463321i −4.26363 + 3.09771i −3.14288 − 1.02118i −0.202449 − 4.66577i −1.19340 − 1.64257i 7.13336 + 5.18269i 2.57067 + 1.54650i 8.91030i
248.3 −0.828482 − 2.54980i 1.34014 − 1.09728i −4.19709 + 3.04936i 2.56502 + 0.833425i −3.90813 − 2.50803i 2.78275 + 3.83012i 6.91450 + 5.02368i 0.591966 − 2.94102i − 7.23077i
248.4 −0.759437 − 2.33731i 0.0396224 + 1.73160i −3.26823 + 2.37451i −0.541474 − 0.175936i 4.01719 − 1.40765i 0.184926 + 0.254529i 4.05551 + 2.94650i −2.99686 + 0.137220i 1.39920i
248.5 −0.751690 − 2.31346i −0.447888 − 1.67314i −3.16905 + 2.30245i 1.59703 + 0.518906i −3.53408 + 2.29386i −2.23366 − 3.07437i 3.77287 + 2.74115i −2.59879 + 1.49876i − 4.08473i
248.6 −0.742222 − 2.28432i −0.941406 − 1.45388i −3.04921 + 2.21538i −1.40487 − 0.456471i −2.62239 + 3.22957i 2.72019 + 3.74402i 3.43751 + 2.49750i −1.22751 + 2.73737i 3.54799i
248.7 −0.653756 − 2.01206i 1.33811 + 1.09976i −2.00293 + 1.45522i 0.460929 + 0.149765i 1.33797 − 3.41132i 1.17111 + 1.61190i 0.814296 + 0.591620i 0.581071 + 2.94319i − 1.02532i
248.8 −0.641276 − 1.97365i 1.56505 − 0.742034i −1.86601 + 1.35573i 2.08634 + 0.677894i −2.46814 − 2.61301i −1.98628 − 2.73388i 0.514597 + 0.373877i 1.89877 − 2.32264i − 4.55242i
248.9 −0.639563 − 1.96837i −0.957560 + 1.44329i −1.84742 + 1.34223i 0.0542705 + 0.0176336i 3.45335 + 0.961761i 0.902973 + 1.24284i 0.474744 + 0.344922i −1.16616 − 2.76407i − 0.118102i
248.10 −0.588772 − 1.81206i −1.41376 + 1.00065i −1.31886 + 0.958206i −4.08513 − 1.32734i 2.64561 + 1.97265i −0.445193 − 0.612756i −0.570025 − 0.414147i 0.997417 − 2.82934i 8.18397i
248.11 −0.585970 − 1.80343i 1.29904 − 1.14565i −1.29097 + 0.937942i −2.64888 − 0.860673i −2.82729 − 1.67141i −0.0612752 − 0.0843381i −0.620198 − 0.450600i 0.374988 − 2.97647i 5.28140i
248.12 −0.471771 − 1.45196i 0.741004 + 1.56554i −0.267592 + 0.194417i −0.890567 − 0.289363i 1.92352 − 1.81449i −2.98183 − 4.10413i −2.06170 − 1.49791i −1.90183 + 2.32014i 1.42958i
248.13 −0.403403 − 1.24155i 0.347020 − 1.69693i 0.239326 − 0.173881i −1.63352 − 0.530762i −2.24681 + 0.253706i 0.605010 + 0.832725i −2.42467 − 1.76163i −2.75915 − 1.17774i 2.24220i
248.14 −0.395367 − 1.21681i 1.57722 + 0.715804i 0.293714 − 0.213396i 3.00322 + 0.975804i 0.247420 − 2.20219i 0.904708 + 1.24522i −2.44595 − 1.77709i 1.97525 + 2.25796i − 4.04015i
248.15 −0.387859 − 1.19371i −0.747764 + 1.56232i 0.343533 − 0.249591i 4.12008 + 1.33870i 2.15498 + 0.286650i −1.94633 − 2.67890i −2.46204 − 1.78877i −1.88170 − 2.33649i − 5.43740i
248.16 −0.358717 − 1.10402i −0.371629 − 1.69171i 0.527855 − 0.383509i 2.25266 + 0.731932i −1.73437 + 1.01713i 0.626805 + 0.862723i −2.49102 − 1.80983i −2.72378 + 1.25738i − 2.74953i
248.17 −0.316587 − 0.974355i −1.11557 − 1.32495i 0.768894 − 0.558634i −1.73318 − 0.563145i −0.937801 + 1.50642i −2.49336 − 3.43181i −2.44540 − 1.77669i −0.511010 + 2.95616i 1.86702i
248.18 −0.268430 − 0.826144i −1.73136 + 0.0488003i 1.00758 − 0.732046i −1.29465 − 0.420659i 0.505067 + 1.41726i 2.39963 + 3.30281i −2.28076 − 1.65707i 2.99524 − 0.168982i 1.18249i
248.19 −0.214393 − 0.659835i −0.480417 + 1.66409i 1.22862 − 0.892642i 1.23103 + 0.399986i 1.20102 − 0.0397745i 2.53769 + 3.49283i −1.97498 − 1.43491i −2.53840 − 1.59891i − 0.898032i
248.20 −0.155057 − 0.477215i 1.66619 + 0.473072i 1.41434 − 1.02758i −2.24502 − 0.729452i −0.0325974 − 0.868487i −1.56756 − 2.15756i −1.52157 − 1.10548i 2.55241 + 1.57646i 1.18447i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 248.48
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
33.f 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.x.a ✓ 192
3.b odd 2 1 inner 429.2.x.a ✓ 192
11.d odd 10 1 inner 429.2.x.a ✓ 192
33.f even 10 1 inner 429.2.x.a ✓ 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.x.a ✓ 192 1.a even 1 1 trivial
429.2.x.a ✓ 192 3.b odd 2 1 inner
429.2.x.a ✓ 192 11.d odd 10 1 inner
429.2.x.a ✓ 192 33.f even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(429, [\chi])\).