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Properties

Label 354.3.b.a
Level $354$
Weight $3$
Character orbit 354.b
Analytic conductor $9.646$
Analytic rank $0$
Dimension $40$
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] = [354,3,Mod(119,354)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(354, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("354.119");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 354 = 2 \cdot 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 354.b (of order \(2\), degree \(1\), 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: \(9.64580135835\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 80 q^{4} + 8 q^{6} + 8 q^{7} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 80 q^{4} + 8 q^{6} + 8 q^{7} - 24 q^{9} - 16 q^{10} + 34 q^{15} + 160 q^{16} + 16 q^{18} + 24 q^{19} - 18 q^{21} - 16 q^{22} - 16 q^{24} - 216 q^{25} - 30 q^{27} - 16 q^{28} - 64 q^{30} + 96 q^{31} + 76 q^{33} + 80 q^{34} + 48 q^{36} - 200 q^{37} - 28 q^{39} + 32 q^{40} + 48 q^{42} - 104 q^{43} + 58 q^{45} + 32 q^{46} + 288 q^{49} - 176 q^{51} - 40 q^{54} + 360 q^{55} + 214 q^{57} - 128 q^{58} - 68 q^{60} - 32 q^{61} - 132 q^{63} - 320 q^{64} - 112 q^{66} - 344 q^{67} + 88 q^{69} + 192 q^{70} - 32 q^{72} + 40 q^{73} + 28 q^{75} - 48 q^{76} + 96 q^{78} + 32 q^{79} + 336 q^{81} - 80 q^{82} + 36 q^{84} + 168 q^{85} - 162 q^{87} + 32 q^{88} + 112 q^{90} + 88 q^{91} - 316 q^{93} - 400 q^{94} + 32 q^{96} - 184 q^{97} - 148 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
119.1 − 1.41421i −2.99974 + 0.0395926i −2.00000 8.98648i 0.0559924 + 4.24227i 1.46160 2.82843i 8.99686 − 0.237535i 12.7088
119.2 − 1.41421i −2.88248 + 0.831458i −2.00000 − 4.87722i 1.17586 + 4.07644i 5.25551 2.82843i 7.61735 − 4.79332i −6.89743
119.3 − 1.41421i −2.71554 − 1.27509i −2.00000 1.78565i −1.80326 + 3.84035i −11.5483 2.82843i 5.74827 + 6.92513i 2.52529
119.4 − 1.41421i −2.67506 − 1.35797i −2.00000 − 1.30422i −1.92046 + 3.78310i 6.99884 2.82843i 5.31185 + 7.26528i −1.84445
119.5 − 1.41421i −2.37497 + 1.83290i −2.00000 − 0.546180i 2.59211 + 3.35872i 0.308312 2.82843i 2.28097 − 8.70616i −0.772415
119.6 − 1.41421i −1.48935 + 2.60419i −2.00000 4.30678i 3.68289 + 2.10626i −7.55737 2.82843i −4.56365 − 7.75713i 6.09070
119.7 − 1.41421i −1.23071 − 2.73594i −2.00000 − 5.41438i −3.86920 + 1.74048i 0.567729 2.82843i −5.97072 + 6.73428i −7.65709
119.8 − 1.41421i −0.865873 − 2.87233i −2.00000 4.52476i −4.06208 + 1.22453i 8.90636 2.82843i −7.50053 + 4.97414i 6.39898
119.9 − 1.41421i −0.495685 + 2.95877i −2.00000 − 8.96910i 4.18433 + 0.701004i −9.25455 2.82843i −8.50859 − 2.93323i −12.6842
119.10 − 1.41421i −0.0610517 − 2.99938i −2.00000 − 5.31992i −4.24176 + 0.0863401i −5.79959 2.82843i −8.99255 + 0.366234i −7.52350
119.11 − 1.41421i −0.0106251 + 2.99998i −2.00000 − 7.50817i 4.24261 + 0.0150262i 12.9231 2.82843i −8.99977 − 0.0637503i −10.6182
119.12 − 1.41421i 0.306491 + 2.98430i −2.00000 9.44498i 4.22044 − 0.433443i 10.5753 2.82843i −8.81213 + 1.82932i 13.3572
119.13 − 1.41421i 1.10775 − 2.78799i −2.00000 5.86001i −3.94282 − 1.56659i −9.12163 2.82843i −6.54579 − 6.17678i 8.28731
119.14 − 1.41421i 1.45977 + 2.62089i −2.00000 1.67024i 3.70650 − 2.06443i −2.10379 2.82843i −4.73812 + 7.65181i 2.36208
119.15 − 1.41421i 1.89028 + 2.32956i −2.00000 − 1.71502i 3.29449 − 2.67326i −3.39354 2.82843i −1.85367 + 8.80704i −2.42540
119.16 − 1.41421i 1.91195 − 2.31181i −2.00000 3.94988i −3.26939 − 2.70390i 4.74743 2.82843i −1.68893 − 8.84011i 5.58597
119.17 − 1.41421i 2.49176 − 1.67067i −2.00000 − 7.29367i −2.36269 − 3.52388i 6.28110 2.82843i 3.41771 − 8.32582i −10.3148
119.18 − 1.41421i 2.68322 + 1.34177i −2.00000 − 2.14890i 1.89755 − 3.79464i 6.97234 2.82843i 5.39932 + 7.20051i −3.03900
119.19 − 1.41421i 2.95715 + 0.505227i −2.00000 − 6.09077i 0.714499 − 4.18204i −12.9107 2.82843i 8.48949 + 2.98806i −8.61364
119.20 − 1.41421i 2.99271 − 0.209029i −2.00000 5.00191i −0.295611 − 4.23233i 0.691870 2.82843i 8.91261 − 1.25112i 7.07377
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 119.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 354.3.b.a ✓ 40
3.b odd 2 1 inner 354.3.b.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
354.3.b.a ✓ 40 1.a even 1 1 trivial
354.3.b.a ✓ 40 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(354, [\chi])\).