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Properties

Label 390.2.bd.c
Level $390$
Weight $2$
Character orbit 390.bd
Analytic conductor $3.114$
Analytic rank $0$
Dimension $32$
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] = [390,2,Mod(7,390)]
 
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([0, 3, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("390.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.bd (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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 16 q^{4} - 4 q^{11} + 20 q^{13} + 4 q^{15} - 16 q^{16} - 20 q^{17} + 32 q^{18} - 20 q^{19} + 8 q^{21} - 4 q^{22} + 4 q^{23} + 8 q^{25} - 4 q^{26} - 24 q^{29} + 4 q^{30} + 12 q^{31} - 12 q^{33} + 16 q^{34} + 12 q^{35} + 20 q^{37} + 4 q^{38} + 20 q^{39} - 28 q^{41} + 8 q^{42} - 4 q^{43} + 4 q^{44} - 4 q^{46} - 16 q^{47} - 20 q^{49} + 28 q^{50} + 16 q^{52} - 4 q^{53} - 40 q^{55} - 12 q^{56} - 36 q^{59} - 4 q^{60} - 28 q^{61} - 36 q^{62} + 12 q^{63} - 32 q^{64} - 32 q^{65} - 36 q^{67} - 4 q^{68} + 20 q^{69} - 24 q^{70} - 4 q^{71} + 16 q^{72} - 24 q^{74} - 16 q^{76} - 20 q^{77} - 16 q^{78} + 16 q^{81} + 28 q^{82} - 40 q^{83} - 8 q^{84} + 88 q^{85} - 8 q^{86} - 16 q^{87} - 8 q^{88} + 16 q^{89} - 40 q^{91} + 20 q^{92} + 24 q^{94} - 8 q^{95} + 72 q^{97} + 72 q^{98} + 4 q^{99}+O(q^{100}) \) 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 −0.866025 − 0.500000i −0.258819 − 0.965926i 0.500000 + 0.866025i −1.51030 − 1.64894i −0.258819 + 0.965926i −2.29844 − 3.98101i − 1.00000i −0.866025 + 0.500000i 0.483485 + 2.18317i
7.2 −0.866025 − 0.500000i −0.258819 − 0.965926i 0.500000 + 0.866025i −1.27660 + 1.83584i −0.258819 + 0.965926i 0.484219 + 0.838691i − 1.00000i −0.866025 + 0.500000i 2.02348 − 0.951586i
7.3 −0.866025 − 0.500000i −0.258819 − 0.965926i 0.500000 + 0.866025i −0.0407428 − 2.23570i −0.258819 + 0.965926i 1.23545 + 2.13987i − 1.00000i −0.866025 + 0.500000i −1.08256 + 1.95654i
7.4 −0.866025 − 0.500000i −0.258819 − 0.965926i 0.500000 + 0.866025i 0.895783 + 2.04880i −0.258819 + 0.965926i −0.804897 − 1.39412i − 1.00000i −0.866025 + 0.500000i 0.248628 − 2.22220i
7.5 −0.866025 − 0.500000i 0.258819 + 0.965926i 0.500000 + 0.866025i −1.86235 + 1.23760i 0.258819 − 0.965926i −1.84360 − 3.19321i − 1.00000i −0.866025 + 0.500000i 2.23164 − 0.140614i
7.6 −0.866025 − 0.500000i 0.258819 + 0.965926i 0.500000 + 0.866025i −0.473708 − 2.18531i 0.258819 − 0.965926i 1.36023 + 2.35599i − 1.00000i −0.866025 + 0.500000i −0.682415 + 2.12939i
7.7 −0.866025 − 0.500000i 0.258819 + 0.965926i 0.500000 + 0.866025i 2.03189 + 0.933504i 0.258819 − 0.965926i 2.15872 + 3.73901i − 1.00000i −0.866025 + 0.500000i −1.29291 − 1.82438i
7.8 −0.866025 − 0.500000i 0.258819 + 0.965926i 0.500000 + 0.866025i 2.23602 + 0.0142140i 0.258819 − 0.965926i −2.02374 − 3.50522i − 1.00000i −0.866025 + 0.500000i −1.92935 − 1.13032i
37.1 0.866025 − 0.500000i −0.965926 − 0.258819i 0.500000 − 0.866025i −2.10895 − 0.743195i −0.965926 + 0.258819i 0.0343211 − 0.0594459i − 1.00000i 0.866025 + 0.500000i −2.19800 + 0.410849i
37.2 0.866025 − 0.500000i −0.965926 − 0.258819i 0.500000 − 0.866025i −1.84366 + 1.26528i −0.965926 + 0.258819i −1.86936 + 3.23783i − 1.00000i 0.866025 + 0.500000i −0.964013 + 2.01759i
37.3 0.866025 − 0.500000i −0.965926 − 0.258819i 0.500000 − 0.866025i 1.50593 − 1.65293i −0.965926 + 0.258819i 0.627513 − 1.08688i − 1.00000i 0.866025 + 0.500000i 0.477706 − 2.18444i
37.4 0.866025 − 0.500000i −0.965926 − 0.258819i 0.500000 − 0.866025i 1.92904 + 1.13085i −0.965926 + 0.258819i 0.141701 − 0.245433i − 1.00000i 0.866025 + 0.500000i 2.23602 + 0.0148223i
37.5 0.866025 − 0.500000i 0.965926 + 0.258819i 0.500000 − 0.866025i −1.95695 + 1.08182i 0.965926 − 0.258819i 2.37716 − 4.11737i − 1.00000i 0.866025 + 0.500000i −1.15386 + 1.91536i
37.6 0.866025 − 0.500000i 0.965926 + 0.258819i 0.500000 − 0.866025i −0.900968 − 2.04652i 0.965926 − 0.258819i 0.0298867 − 0.0517653i − 1.00000i 0.866025 + 0.500000i −1.80352 − 1.32186i
37.7 0.866025 − 0.500000i 0.965926 + 0.258819i 0.500000 − 0.866025i 1.30715 + 1.81421i 0.965926 − 0.258819i −0.717312 + 1.24242i − 1.00000i 0.866025 + 0.500000i 2.03913 + 0.917581i
37.8 0.866025 − 0.500000i 0.965926 + 0.258819i 0.500000 − 0.866025i 2.06841 − 0.849511i 0.965926 − 0.258819i 1.10814 − 1.91935i − 1.00000i 0.866025 + 0.500000i 1.36654 − 1.76990i
223.1 −0.866025 + 0.500000i −0.258819 + 0.965926i 0.500000 − 0.866025i −1.51030 + 1.64894i −0.258819 − 0.965926i −2.29844 + 3.98101i 1.00000i −0.866025 − 0.500000i 0.483485 − 2.18317i
223.2 −0.866025 + 0.500000i −0.258819 + 0.965926i 0.500000 − 0.866025i −1.27660 − 1.83584i −0.258819 − 0.965926i 0.484219 − 0.838691i 1.00000i −0.866025 − 0.500000i 2.02348 + 0.951586i
223.3 −0.866025 + 0.500000i −0.258819 + 0.965926i 0.500000 − 0.866025i −0.0407428 + 2.23570i −0.258819 − 0.965926i 1.23545 − 2.13987i 1.00000i −0.866025 − 0.500000i −1.08256 − 1.95654i
223.4 −0.866025 + 0.500000i −0.258819 + 0.965926i 0.500000 − 0.866025i 0.895783 − 2.04880i −0.258819 − 0.965926i −0.804897 + 1.39412i 1.00000i −0.866025 − 0.500000i 0.248628 + 2.22220i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.8
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.t 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.bd.c ✓ 32
5.c odd 4 1 390.2.bn.c yes 32
13.f odd 12 1 390.2.bn.c yes 32
65.t even 12 1 inner 390.2.bd.c ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bd.c ✓ 32 1.a even 1 1 trivial
390.2.bd.c ✓ 32 65.t even 12 1 inner
390.2.bn.c yes 32 5.c odd 4 1
390.2.bn.c yes 32 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{32} + 66 T_{7}^{30} - 24 T_{7}^{29} + 2699 T_{7}^{28} - 1516 T_{7}^{27} + 69962 T_{7}^{26} + \cdots + 65536 \) acting on \(S_{2}^{\mathrm{new}}(390, [\chi])\). Copy content Toggle raw display