↑

Properties

Label 390.2.bh.b
Level $390$
Weight $2$
Character orbit 390.bh
Analytic conductor $3.114$
Analytic rank $0$
Dimension $32$
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] = [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: \(32\)
Relative dimension: \(8\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 8 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 8 q^{6} + 4 q^{9} + 16 q^{13} + 4 q^{15} + 16 q^{16} + 24 q^{18} + 32 q^{19} + 36 q^{21} + 8 q^{22} - 8 q^{24} + 12 q^{30} - 24 q^{31} - 28 q^{33} - 8 q^{34} - 12 q^{36} + 16 q^{37} - 4 q^{39} - 32 q^{40} - 12 q^{42} + 48 q^{43} - 24 q^{45} + 24 q^{46} - 48 q^{49} - 16 q^{52} - 20 q^{54} + 8 q^{55} + 4 q^{57} - 24 q^{58} + 4 q^{60} + 32 q^{61} - 48 q^{63} - 48 q^{66} - 64 q^{67} - 36 q^{69} - 40 q^{73} - 12 q^{75} + 32 q^{76} + 4 q^{78} - 32 q^{79} + 16 q^{81} + 12 q^{84} + 40 q^{85} - 72 q^{87} - 48 q^{88} - 144 q^{91} + 56 q^{94} - 4 q^{96} + 112 q^{97} + 72 q^{99}+O(q^{100}) \) 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 −0.658470 − 1.60200i 0.866025 + 0.500000i 0.707107 − 0.707107i 0.221404 + 1.71784i 1.27100 + 4.74345i −0.707107 − 0.707107i −2.13283 + 2.10974i −0.866025 + 0.500000i
11.2 −0.965926 − 0.258819i −0.350069 + 1.69631i 0.866025 + 0.500000i 0.707107 − 0.707107i 0.777177 − 1.54790i −0.400431 − 1.49443i −0.707107 − 0.707107i −2.75490 − 1.18765i −0.866025 + 0.500000i
11.3 −0.965926 − 0.258819i 0.309872 − 1.70411i 0.866025 + 0.500000i 0.707107 − 0.707107i −0.740368 + 1.56584i −1.03566 − 3.86514i −0.707107 − 0.707107i −2.80796 − 1.05611i −0.866025 + 0.500000i
11.4 −0.965926 − 0.258819i 1.50568 − 0.856120i 0.866025 + 0.500000i 0.707107 − 0.707107i −1.67595 + 0.437251i 0.165090 + 0.616125i −0.707107 − 0.707107i 1.53412 − 2.57808i −0.866025 + 0.500000i
11.5 0.965926 + 0.258819i −1.63074 − 0.583697i 0.866025 + 0.500000i −0.707107 + 0.707107i −1.42410 − 0.985873i −1.03566 − 3.86514i 0.707107 + 0.707107i 2.31860 + 1.90371i −0.866025 + 0.500000i
11.6 0.965926 + 0.258819i −1.49426 + 0.875893i 0.866025 + 0.500000i −0.707107 + 0.707107i −1.67004 + 0.459305i 0.165090 + 0.616125i 0.707107 + 0.707107i 1.46562 − 2.61762i −0.866025 + 0.500000i
11.7 0.965926 + 0.258819i −1.05814 − 1.37125i 0.866025 + 0.500000i −0.707107 + 0.707107i −0.667179 − 1.59840i 1.27100 + 4.74345i 0.707107 + 0.707107i −0.760676 + 2.90196i −0.866025 + 0.500000i
11.8 0.965926 + 0.258819i 1.64408 + 0.544984i 0.866025 + 0.500000i −0.707107 + 0.707107i 1.44701 + 0.951933i −0.400431 − 1.49443i 0.707107 + 0.707107i 2.40599 + 1.79199i −0.866025 + 0.500000i
41.1 −0.258819 + 0.965926i −1.63346 + 0.576040i −0.866025 − 0.500000i −0.707107 − 0.707107i −0.133643 − 1.72689i 1.16909 − 0.313256i 0.707107 − 0.707107i 2.33635 − 1.88187i 0.866025 − 0.500000i
41.2 −0.258819 + 0.965926i −0.385873 + 1.68852i −0.866025 − 0.500000i −0.707107 − 0.707107i −1.53111 − 0.809746i 0.0857661 − 0.0229809i 0.707107 − 0.707107i −2.70220 − 1.30311i 0.866025 − 0.500000i
41.3 −0.258819 + 0.965926i 0.973754 − 1.43241i −0.866025 − 0.500000i −0.707107 − 0.707107i 1.13158 + 1.31131i −4.16034 + 1.11476i 0.707107 − 0.707107i −1.10361 − 2.78963i 0.866025 − 0.500000i
41.4 −0.258819 + 0.965926i 1.46331 + 0.926670i −0.866025 − 0.500000i −0.707107 − 0.707107i −1.27383 + 1.17361i 2.90548 − 0.778522i 0.707107 − 0.707107i 1.28257 + 2.71201i 0.866025 − 0.500000i
41.5 0.258819 − 0.965926i −1.72738 + 0.127090i −0.866025 − 0.500000i 0.707107 + 0.707107i −0.324320 + 1.70142i −4.16034 + 1.11476i −0.707107 + 0.707107i 2.96770 − 0.439065i 0.866025 − 0.500000i
41.6 0.258819 − 0.965926i 0.0708632 + 1.73060i −0.866025 − 0.500000i 0.707107 + 0.707107i 1.68997 + 0.379464i 2.90548 − 0.778522i −0.707107 + 0.707107i −2.98996 + 0.245272i 0.866025 − 0.500000i
41.7 0.258819 − 0.965926i 1.31559 − 1.12659i −0.866025 − 0.500000i 0.707107 + 0.707107i −0.747705 − 1.56235i 1.16909 − 0.313256i −0.707107 + 0.707107i 0.461572 − 2.96428i 0.866025 − 0.500000i
41.8 0.258819 − 0.965926i 1.65524 + 0.510084i −0.866025 − 0.500000i 0.707107 + 0.707107i 0.921111 − 1.46682i 0.0857661 − 0.0229809i −0.707107 + 0.707107i 2.47963 + 1.68862i 0.866025 − 0.500000i
71.1 −0.965926 + 0.258819i −0.658470 + 1.60200i 0.866025 − 0.500000i 0.707107 + 0.707107i 0.221404 − 1.71784i 1.27100 − 4.74345i −0.707107 + 0.707107i −2.13283 − 2.10974i −0.866025 − 0.500000i
71.2 −0.965926 + 0.258819i −0.350069 − 1.69631i 0.866025 − 0.500000i 0.707107 + 0.707107i 0.777177 + 1.54790i −0.400431 + 1.49443i −0.707107 + 0.707107i −2.75490 + 1.18765i −0.866025 − 0.500000i
71.3 −0.965926 + 0.258819i 0.309872 + 1.70411i 0.866025 − 0.500000i 0.707107 + 0.707107i −0.740368 − 1.56584i −1.03566 + 3.86514i −0.707107 + 0.707107i −2.80796 + 1.05611i −0.866025 − 0.500000i
71.4 −0.965926 + 0.258819i 1.50568 + 0.856120i 0.866025 − 0.500000i 0.707107 + 0.707107i −1.67595 − 0.437251i 0.165090 − 0.616125i −0.707107 + 0.707107i 1.53412 + 2.57808i −0.866025 − 0.500000i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.8
Significant digits:
Format:

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.b ✓ 32
3.b odd 2 1 inner 390.2.bh.b ✓ 32
13.f odd 12 1 inner 390.2.bh.b ✓ 32
39.k even 12 1 inner 390.2.bh.b ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bh.b ✓ 32 1.a even 1 1 trivial
390.2.bh.b ✓ 32 3.b odd 2 1 inner
390.2.bh.b ✓ 32 13.f odd 12 1 inner
390.2.bh.b ✓ 32 39.k even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} + 12 T_{7}^{14} + 48 T_{7}^{13} - 246 T_{7}^{12} - 2376 T_{7}^{10} - 6120 T_{7}^{9} + \cdots + 729 \) acting on \(S_{2}^{\mathrm{new}}(390, [\chi])\). Copy content Toggle raw display