↑

Properties

Label 429.2.bs.a
Level $429$
Weight $2$
Character orbit 429.bs
Analytic conductor $3.426$
Analytic rank $0$
Dimension $224$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [429,2,Mod(7,429)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(429, base_ring=CyclotomicField(60))
 
chi = DirichletCharacter(H, H._module([0, 42, 55]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("429.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 429 = 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 429.bs (of order \(60\), degree \(16\), 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: \(224\)
Relative dimension: \(14\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 224 q - 28 q^{3} - 6 q^{5} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 224 q - 28 q^{3} - 6 q^{5} + 28 q^{9} - 16 q^{11} - 10 q^{13} - 36 q^{14} + 8 q^{15} - 24 q^{16} + 50 q^{20} + 10 q^{22} - 48 q^{23} - 30 q^{24} + 10 q^{26} + 56 q^{27} + 20 q^{29} + 12 q^{31} - 32 q^{33} - 104 q^{34} + 20 q^{35} + 12 q^{37} + 20 q^{39} - 80 q^{40} + 190 q^{41} + 32 q^{42} - 24 q^{44} + 4 q^{45} - 40 q^{46} - 10 q^{47} + 44 q^{48} - 24 q^{49} - 180 q^{50} - 110 q^{52} + 28 q^{53} - 62 q^{55} - 72 q^{56} + 20 q^{58} - 118 q^{59} + 88 q^{60} - 20 q^{61} + 120 q^{62} - 20 q^{66} - 156 q^{67} - 60 q^{68} + 12 q^{69} - 20 q^{70} + 64 q^{71} - 30 q^{72} + 10 q^{73} - 60 q^{74} - 12 q^{75} + 84 q^{78} + 200 q^{79} + 188 q^{80} + 28 q^{81} - 138 q^{82} - 80 q^{83} - 60 q^{84} + 40 q^{85} - 42 q^{86} - 18 q^{88} + 216 q^{89} - 88 q^{91} + 16 q^{92} + 12 q^{93} - 40 q^{94} - 300 q^{95} - 100 q^{96} + 18 q^{97} - 6 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −0.954534 + 2.48665i −0.669131 − 0.743145i −3.78598 − 3.40891i −2.07220 − 0.328204i 2.48665 − 0.954534i 3.34409 − 0.175256i 7.34411 − 3.74201i −0.104528 + 0.994522i 2.79411 − 4.83954i
7.2 −0.834037 + 2.17274i −0.669131 − 0.743145i −2.53889 − 2.28603i 0.383236 + 0.0606986i 2.17274 − 0.834037i −0.733573 + 0.0384449i 2.93716 − 1.49656i −0.104528 + 0.994522i −0.451515 + 0.782048i
7.3 −0.578603 + 1.50731i −0.669131 − 0.743145i −0.450920 − 0.406010i −2.06394 − 0.326895i 1.50731 − 0.578603i 2.81384 − 0.147467i −2.00426 + 1.02122i −0.104528 + 0.994522i 1.68693 − 2.92185i
7.4 −0.566051 + 1.47461i −0.669131 − 0.743145i −0.367780 − 0.331151i −0.523711 − 0.0829477i 1.47461 − 0.566051i −2.90169 + 0.152071i −2.11823 + 1.07929i −0.104528 + 0.994522i 0.418763 − 0.725318i
7.5 −0.501760 + 1.30713i −0.669131 − 0.743145i 0.0294667 + 0.0265320i 2.71725 + 0.430370i 1.30713 − 0.501760i −2.18905 + 0.114723i −2.54451 + 1.29649i −0.104528 + 0.994522i −1.92595 + 3.33585i
7.6 −0.185537 + 0.483339i −0.669131 − 0.743145i 1.28710 + 1.15891i 3.92281 + 0.621313i 0.483339 − 0.185537i 1.27804 − 0.0669792i −1.72154 + 0.877171i −0.104528 + 0.994522i −1.02813 + 1.78077i
7.7 −0.0378782 + 0.0986760i −0.669131 − 0.743145i 1.47799 + 1.33079i −4.11879 − 0.652352i 0.0986760 − 0.0378782i 2.16416 − 0.113419i −0.375652 + 0.191404i −0.104528 + 0.994522i 0.220383 − 0.381715i
7.8 0.0793711 − 0.206769i −0.669131 − 0.743145i 1.44984 + 1.30544i 0.996265 + 0.157793i −0.206769 + 0.0793711i 3.42689 − 0.179596i 0.779678 − 0.397266i −0.104528 + 0.994522i 0.111701 − 0.193472i
7.9 0.0815196 − 0.212366i −0.669131 − 0.743145i 1.44784 + 1.30364i −1.64085 − 0.259884i −0.212366 + 0.0815196i −3.90019 + 0.204401i 0.800237 − 0.407741i −0.104528 + 0.994522i −0.188952 + 0.327274i
7.10 0.459872 − 1.19801i −0.669131 − 0.743145i 0.262550 + 0.236401i −2.03444 − 0.322223i −1.19801 + 0.459872i −3.05544 + 0.160129i 2.69070 − 1.37098i −0.104528 + 0.994522i −1.32161 + 2.28909i
7.11 0.463345 − 1.20706i −0.669131 − 0.743145i 0.243996 + 0.219695i 0.666783 + 0.105608i −1.20706 + 0.463345i 0.661445 − 0.0346649i 2.68226 − 1.36668i −0.104528 + 0.994522i 0.436426 − 0.755911i
7.12 0.701743 − 1.82810i −0.669131 − 0.743145i −1.36323 − 1.22746i 2.43001 + 0.384875i −1.82810 + 0.701743i 1.54304 − 0.0808673i 0.288919 − 0.147212i −0.104528 + 0.994522i 2.40883 − 4.17222i
7.13 0.915452 − 2.38484i −0.669131 − 0.743145i −3.36310 − 3.02815i 1.16930 + 0.185198i −2.38484 + 0.915452i −4.20987 + 0.220630i −5.74822 + 2.92887i −0.104528 + 0.994522i 1.51210 − 2.61904i
7.14 0.957096 − 2.49332i −0.669131 − 0.743145i −3.81432 − 3.43443i −2.91905 − 0.462332i −2.49332 + 0.957096i 1.75832 − 0.0921498i −7.45456 + 3.79829i −0.104528 + 0.994522i −3.94655 + 6.83563i
19.1 −1.55941 + 1.92572i 0.978148 + 0.207912i −0.860786 − 4.04968i −3.59392 + 0.569220i −1.92572 + 1.55941i 0.829321 − 1.27704i 4.72514 + 2.40758i 0.913545 + 0.406737i 4.50825 − 7.80851i
19.2 −1.39677 + 1.72487i 0.978148 + 0.207912i −0.608384 − 2.86222i 0.162782 − 0.0257822i −1.72487 + 1.39677i −2.66615 + 4.10551i 1.83157 + 0.933234i 0.913545 + 0.406737i −0.182899 + 0.316791i
19.3 −1.22830 + 1.51683i 0.978148 + 0.207912i −0.376213 − 1.76994i 3.41571 − 0.540995i −1.51683 + 1.22830i 0.116665 − 0.179648i −0.331315 − 0.168813i 0.913545 + 0.406737i −3.37493 + 5.84554i
19.4 −1.13436 + 1.40082i 0.978148 + 0.207912i −0.259701 − 1.22179i 0.351689 − 0.0557021i −1.40082 + 1.13436i 2.23633 − 3.44365i −1.20601 − 0.614491i 0.913545 + 0.406737i −0.320915 + 0.555841i
19.5 −0.622153 + 0.768295i 0.978148 + 0.207912i 0.212621 + 1.00030i −1.97898 + 0.313439i −0.768295 + 0.622153i −2.00584 + 3.08873i −2.66253 − 1.35662i 0.913545 + 0.406737i 0.990413 − 1.71545i
19.6 −0.528774 + 0.652981i 0.978148 + 0.207912i 0.269041 + 1.26574i −1.99296 + 0.315653i −0.652981 + 0.528774i 0.784828 − 1.20853i −2.46606 − 1.25652i 0.913545 + 0.406737i 0.847708 − 1.46827i
See next 80 embeddings (of 224 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner
13.f odd 12 1 inner
143.w even 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.bs.a ✓ 224
11.d odd 10 1 inner 429.2.bs.a ✓ 224
13.f odd 12 1 inner 429.2.bs.a ✓ 224
143.w even 60 1 inner 429.2.bs.a ✓ 224
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.bs.a ✓ 224 1.a even 1 1 trivial
429.2.bs.a ✓ 224 11.d odd 10 1 inner
429.2.bs.a ✓ 224 13.f odd 12 1 inner
429.2.bs.a ✓ 224 143.w even 60 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{224} + 92 T_{2}^{220} - 100 T_{2}^{217} + 1811 T_{2}^{216} - 3350 T_{2}^{215} + \cdots + 46\!\cdots\!21 \) acting on \(S_{2}^{\mathrm{new}}(429, [\chi])\). Copy content Toggle raw display