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Properties

Label 390.2.bh.c
Level $390$
Weight $2$
Character orbit 390.bh
Analytic conductor $3.114$
Analytic rank $0$
Dimension $40$
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] = [390,2,Mod(11,390)]
 
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, 0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("390.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.bh (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: \(40\)
Relative dimension: \(10\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 8 q^{6} - 8 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 8 q^{6} - 8 q^{7} + 8 q^{9} + 8 q^{13} + 4 q^{15} + 20 q^{16} - 8 q^{18} - 16 q^{19} + 4 q^{21} - 8 q^{22} + 8 q^{24} - 48 q^{27} + 8 q^{28} + 36 q^{31} + 12 q^{33} + 16 q^{34} - 12 q^{36} - 64 q^{37} - 60 q^{39} + 40 q^{40} - 12 q^{42} + 8 q^{45} - 32 q^{46} + 48 q^{49} + 24 q^{52} - 28 q^{54} + 8 q^{55} - 76 q^{57} - 28 q^{58} + 4 q^{60} - 40 q^{61} + 80 q^{63} + 40 q^{66} - 28 q^{67} - 36 q^{69} + 16 q^{70} + 16 q^{72} + 8 q^{73} - 16 q^{76} - 28 q^{78} + 40 q^{79} + 12 q^{81} + 20 q^{84} + 44 q^{85} + 40 q^{87} + 12 q^{88} + 120 q^{91} - 8 q^{93} - 16 q^{94} + 4 q^{96} - 160 q^{97} - 96 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −0.965926 − 0.258819i −1.63891 + 0.560324i 0.866025 + 0.500000i −0.707107 + 0.707107i 1.72809 − 0.117049i 1.23887 + 4.62352i −0.707107 − 0.707107i 2.37208 − 1.83664i 0.866025 − 0.500000i
11.2 −0.965926 − 0.258819i −1.39472 − 1.02701i 0.866025 + 0.500000i −0.707107 + 0.707107i 1.08139 + 1.35299i −0.213554 − 0.796995i −0.707107 − 0.707107i 0.890511 + 2.86478i 0.866025 − 0.500000i
11.3 −0.965926 − 0.258819i −0.654636 + 1.60357i 0.866025 + 0.500000i −0.707107 + 0.707107i 1.04737 − 1.37950i −0.839013 − 3.13124i −0.707107 − 0.707107i −2.14290 − 2.09952i 0.866025 − 0.500000i
11.4 −0.965926 − 0.258819i 0.888751 − 1.48665i 0.866025 + 0.500000i −0.707107 + 0.707107i −1.24324 + 1.20597i 0.431103 + 1.60890i −0.707107 − 0.707107i −1.42024 − 2.64252i 0.866025 − 0.500000i
11.5 −0.965926 − 0.258819i 1.12649 + 1.31568i 0.866025 + 0.500000i −0.707107 + 0.707107i −0.747582 − 1.56241i 0.114645 + 0.427862i −0.707107 − 0.707107i −0.462042 + 2.96421i 0.866025 − 0.500000i
11.6 0.965926 + 0.258819i −1.73185 + 0.0263575i 0.866025 + 0.500000i 0.707107 − 0.707107i −1.67966 − 0.422776i 0.431103 + 1.60890i 0.707107 + 0.707107i 2.99861 − 0.0912946i 0.866025 − 0.500000i
11.7 0.965926 + 0.258819i −0.192053 − 1.72137i 0.866025 + 0.500000i 0.707107 − 0.707107i 0.260015 − 1.71242i −0.213554 − 0.796995i 0.707107 + 0.707107i −2.92623 + 0.661187i 0.866025 − 0.500000i
11.8 0.965926 + 0.258819i 0.576170 + 1.63341i 0.866025 + 0.500000i 0.707107 − 0.707107i 0.133780 + 1.72688i 0.114645 + 0.427862i 0.707107 + 0.707107i −2.33606 + 1.88224i 0.866025 − 0.500000i
11.9 0.965926 + 0.258819i 1.30471 − 1.13918i 0.866025 + 0.500000i 0.707107 − 0.707107i 1.55510 − 0.762678i 1.23887 + 4.62352i 0.707107 + 0.707107i 0.404543 − 2.97260i 0.866025 − 0.500000i
11.10 0.965926 + 0.258819i 1.71605 + 0.234856i 0.866025 + 0.500000i 0.707107 − 0.707107i 1.59680 + 0.671001i −0.839013 − 3.13124i 0.707107 + 0.707107i 2.88969 + 0.806051i 0.866025 − 0.500000i
41.1 −0.258819 + 0.965926i −1.69507 − 0.356024i −0.866025 − 0.500000i 0.707107 + 0.707107i 0.782608 − 1.54516i −2.45106 + 0.656759i 0.707107 − 0.707107i 2.74649 + 1.20697i −0.866025 + 0.500000i
41.2 −0.258819 + 0.965926i −1.47760 + 0.903718i −0.866025 − 0.500000i 0.707107 + 0.707107i −0.490495 − 1.66115i 4.16341 − 1.11558i 0.707107 − 0.707107i 1.36659 − 2.67066i −0.866025 + 0.500000i
41.3 −0.258819 + 0.965926i 0.569090 − 1.63589i −0.866025 − 0.500000i 0.707107 + 0.707107i 1.43286 + 0.973098i −0.894079 + 0.239568i 0.707107 − 0.707107i −2.35227 − 1.86194i −0.866025 + 0.500000i
41.4 −0.258819 + 0.965926i 1.34057 + 1.09675i −0.866025 − 0.500000i 0.707107 + 0.707107i −1.40635 + 1.01103i −4.69691 + 1.25853i 0.707107 − 0.707107i 0.594264 + 2.94055i −0.866025 + 0.500000i
41.5 −0.258819 + 0.965926i 1.71129 − 0.267377i −0.866025 − 0.500000i 0.707107 + 0.707107i −0.184648 + 1.72218i 1.14659 − 0.307228i 0.707107 − 0.707107i 2.85702 − 0.915118i −0.866025 + 0.500000i
41.6 0.258819 − 0.965926i −1.70127 − 0.325099i −0.866025 − 0.500000i −0.707107 − 0.707107i −0.754342 + 1.55916i −0.894079 + 0.239568i −0.707107 + 0.707107i 2.78862 + 1.10616i −0.866025 + 0.500000i
41.7 0.258819 − 0.965926i −1.08720 + 1.34833i −0.866025 − 0.500000i −0.707107 − 0.707107i 1.02100 + 1.39913i 1.14659 − 0.307228i −0.707107 + 0.707107i −0.635994 − 2.93181i −0.866025 + 0.500000i
41.8 0.258819 − 0.965926i 0.279531 + 1.70935i −0.866025 − 0.500000i −0.707107 − 0.707107i 1.72345 + 0.172405i −4.69691 + 1.25853i −0.707107 + 0.707107i −2.84373 + 0.955629i −0.866025 + 0.500000i
41.9 0.258819 − 0.965926i 0.539207 − 1.64598i −0.866025 − 0.500000i −0.707107 − 0.707107i −1.45034 − 0.946846i −2.45106 + 0.656759i −0.707107 + 0.707107i −2.41851 − 1.77505i −0.866025 + 0.500000i
41.10 0.258819 − 0.965926i 1.52144 − 0.827777i −0.866025 − 0.500000i −0.707107 − 0.707107i −0.405794 − 1.68384i 4.16341 − 1.11558i −0.707107 + 0.707107i 1.62957 − 2.51883i −0.866025 + 0.500000i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.10
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.f odd 12 1 inner
39.k 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.bh.c ✓ 40
3.b odd 2 1 inner 390.2.bh.c ✓ 40
13.f odd 12 1 inner 390.2.bh.c ✓ 40
39.k even 12 1 inner 390.2.bh.c ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bh.c ✓ 40 1.a even 1 1 trivial
390.2.bh.c ✓ 40 3.b odd 2 1 inner
390.2.bh.c ✓ 40 13.f odd 12 1 inner
390.2.bh.c ✓ 40 39.k even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{20} + 4 T_{7}^{19} - 4 T_{7}^{18} + 24 T_{7}^{17} - 166 T_{7}^{16} - 2368 T_{7}^{15} + \cdots + 304704 \) acting on \(S_{2}^{\mathrm{new}}(390, [\chi])\). Copy content Toggle raw display