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Properties

Label 390.2.bl.a
Level $390$
Weight $2$
Character orbit 390.bl
Analytic conductor $3.114$
Analytic rank $0$
Dimension $112$
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] = [390,2,Mod(107,390)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(390, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("390.107");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.bl (of order \(12\), 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: \(3.11416567883\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(28\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 112 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q + 24 q^{13} + 56 q^{16} - 16 q^{18} - 16 q^{21} + 24 q^{27} + 12 q^{30} - 80 q^{31} + 52 q^{33} - 8 q^{36} - 8 q^{37} - 16 q^{40} + 12 q^{42} - 8 q^{43} + 20 q^{45} - 8 q^{46} - 80 q^{51} + 24 q^{52} + 8 q^{55} - 96 q^{57} + 8 q^{58} - 32 q^{60} - 16 q^{61} - 16 q^{63} - 32 q^{66} + 40 q^{67} - 32 q^{70} - 8 q^{72} - 80 q^{73} - 64 q^{75} + 16 q^{76} - 36 q^{78} + 8 q^{81} - 16 q^{82} + 96 q^{85} - 8 q^{87} + 24 q^{90} - 32 q^{91} - 68 q^{93} - 40 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1 −0.965926 − 0.258819i −1.61085 + 0.636532i 0.866025 + 0.500000i −2.18408 − 0.479382i 1.72070 − 0.197925i 2.00808 − 0.538065i −0.707107 − 0.707107i 2.18965 − 2.05071i 1.98558 + 1.02833i
107.2 −0.965926 − 0.258819i −1.58374 − 0.701255i 0.866025 + 0.500000i 2.23306 + 0.115957i 1.34828 + 1.08726i 1.62020 − 0.434132i −0.707107 − 0.707107i 2.01648 + 2.22122i −2.12696 − 0.689964i
107.3 −0.965926 − 0.258819i −1.50410 + 0.858883i 0.866025 + 0.500000i 2.00792 − 0.984002i 1.67515 − 0.440328i −4.26423 + 1.14260i −0.707107 − 0.707107i 1.52464 − 2.58369i −2.19418 + 0.430786i
107.4 −0.965926 − 0.258819i −1.15100 − 1.29429i 0.866025 + 0.500000i −1.49285 − 1.66475i 0.776796 + 1.54809i −3.88695 + 1.04151i −0.707107 − 0.707107i −0.350385 + 2.97947i 1.01112 + 1.99440i
107.5 −0.965926 − 0.258819i −1.02467 + 1.39645i 0.866025 + 0.500000i 0.994998 + 2.00249i 1.35118 − 1.08366i 3.13148 − 0.839078i −0.707107 − 0.707107i −0.900122 − 2.86178i −0.442811 − 2.19178i
107.6 −0.965926 − 0.258819i −0.673711 − 1.59565i 0.866025 + 0.500000i −1.48648 + 1.67044i 0.237769 + 1.71565i 2.53048 − 0.678039i −0.707107 − 0.707107i −2.09223 + 2.15002i 1.86817 − 1.22879i
107.7 −0.965926 − 0.258819i −0.355858 + 1.69510i 0.866025 + 0.500000i −0.476388 − 2.18473i 0.782457 − 1.54524i −0.0200355 + 0.00536850i −0.707107 − 0.707107i −2.74673 − 1.20643i −0.105295 + 2.23359i
107.8 −0.965926 − 0.258819i 0.346763 + 1.69698i 0.866025 + 0.500000i 0.0369297 + 2.23576i 0.104264 − 1.72891i −2.83139 + 0.758670i −0.707107 − 0.707107i −2.75951 + 1.17690i 0.542987 − 2.16914i
107.9 −0.965926 − 0.258819i 0.619891 − 1.61732i 0.866025 + 0.500000i 0.791415 − 2.09133i −1.01736 + 1.40178i 4.90302 − 1.31376i −0.707107 − 0.707107i −2.23147 − 2.00513i −1.30572 + 1.81524i
107.10 −0.965926 − 0.258819i 0.990853 − 1.42064i 0.866025 + 0.500000i 2.23167 + 0.140175i −1.32478 + 1.11578i −2.92435 + 0.783577i −0.707107 − 0.707107i −1.03642 − 2.81529i −2.11935 − 0.712997i
107.11 −0.965926 − 0.258819i 1.21097 − 1.23836i 0.866025 + 0.500000i −2.06252 − 0.863726i −1.49022 + 0.882742i −0.883897 + 0.236840i −0.707107 − 0.707107i −0.0670792 − 2.99925i 1.76869 + 1.36811i
107.12 −0.965926 − 0.258819i 1.33215 + 1.10697i 0.866025 + 0.500000i 0.299177 − 2.21596i −1.00025 − 1.41404i −0.0934063 + 0.0250281i −0.707107 − 0.707107i 0.549229 + 2.94930i −0.862516 + 2.06302i
107.13 −0.965926 − 0.258819i 1.67139 + 0.454371i 0.866025 + 0.500000i 1.91429 + 1.15563i −1.49684 − 0.871476i 3.01549 − 0.807999i −0.707107 − 0.707107i 2.58709 + 1.51886i −1.54997 − 1.61171i
107.14 −0.965926 − 0.258819i 1.73191 + 0.0222363i 0.866025 + 0.500000i −1.39293 + 1.74921i −1.66714 − 0.469729i −2.30449 + 0.617486i −0.707107 − 0.707107i 2.99901 + 0.0770225i 1.79820 − 1.32909i
107.15 0.965926 + 0.258819i −1.72219 − 0.184567i 0.866025 + 0.500000i −2.23306 − 0.115957i −1.61574 − 0.624013i 1.62020 − 0.434132i 0.707107 + 0.707107i 2.93187 + 0.635717i −2.12696 − 0.689964i
107.16 0.965926 + 0.258819i −1.64394 + 0.545389i 0.866025 + 0.500000i 1.49285 + 1.66475i −1.72908 + 0.101321i −3.88695 + 1.04151i 0.707107 + 0.707107i 2.40510 − 1.79318i 1.01112 + 1.99440i
107.17 0.965926 + 0.258819i −1.38128 + 1.04502i 0.866025 + 0.500000i 1.48648 − 1.67044i −1.60468 + 0.651913i 2.53048 − 0.678039i 0.707107 + 0.707107i 0.815858 − 2.88693i 1.86817 − 1.22879i
107.18 0.965926 + 0.258819i −1.07677 − 1.35668i 0.866025 + 0.500000i 2.18408 + 0.479382i −0.688944 − 1.58914i 2.00808 − 0.538065i 0.707107 + 0.707107i −0.681142 + 2.92165i 1.98558 + 1.02833i
107.19 0.965926 + 0.258819i −0.873148 − 1.49587i 0.866025 + 0.500000i −2.00792 + 0.984002i −0.456238 − 1.67088i −4.26423 + 1.14260i 0.707107 + 0.707107i −1.47523 + 2.61222i −2.19418 + 0.430786i
107.20 0.965926 + 0.258819i −0.271821 + 1.71059i 0.866025 + 0.500000i −0.791415 + 2.09133i −0.705292 + 1.58195i 4.90302 − 1.31376i 0.707107 + 0.707107i −2.85223 − 0.929947i −1.30572 + 1.81524i
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
13.c even 3 1 inner
15.e even 4 1 inner
39.i odd 6 1 inner
65.q odd 12 1 inner
195.bl even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.bl.a ✓ 112
3.b odd 2 1 inner 390.2.bl.a ✓ 112
5.c odd 4 1 inner 390.2.bl.a ✓ 112
13.c even 3 1 inner 390.2.bl.a ✓ 112
15.e even 4 1 inner 390.2.bl.a ✓ 112
39.i odd 6 1 inner 390.2.bl.a ✓ 112
65.q odd 12 1 inner 390.2.bl.a ✓ 112
195.bl even 12 1 inner 390.2.bl.a ✓ 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bl.a ✓ 112 1.a even 1 1 trivial
390.2.bl.a ✓ 112 3.b odd 2 1 inner
390.2.bl.a ✓ 112 5.c odd 4 1 inner
390.2.bl.a ✓ 112 13.c even 3 1 inner
390.2.bl.a ✓ 112 15.e even 4 1 inner
390.2.bl.a ✓ 112 39.i odd 6 1 inner
390.2.bl.a ✓ 112 65.q odd 12 1 inner
390.2.bl.a ✓ 112 195.bl even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(390, [\chi])\).