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Properties

Label 390.2.s.b
Level $390$
Weight $2$
Character orbit 390.s
Analytic conductor $3.114$
Analytic rank $0$
Dimension $24$
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(77,390)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(390, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("390.77");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.s (of order \(4\), degree \(2\), 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: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{3} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{3} - 12 q^{9} + 4 q^{10} + 4 q^{12} - 8 q^{13} - 16 q^{14} - 24 q^{16} + 52 q^{17} + 16 q^{22} - 16 q^{23} + 16 q^{25} + 40 q^{27} - 80 q^{29} + 20 q^{35} - 16 q^{38} - 32 q^{39} - 4 q^{42} + 28 q^{43} + 8 q^{48} - 20 q^{51} - 8 q^{52} + 72 q^{53} + 24 q^{55} - 64 q^{61} - 32 q^{62} - 12 q^{65} + 8 q^{66} + 52 q^{68} - 16 q^{69} - 120 q^{74} - 104 q^{75} + 32 q^{77} + 44 q^{78} + 12 q^{81} + 32 q^{82} - 32 q^{87} - 16 q^{88} + 64 q^{90} + 64 q^{91} + 16 q^{92} + 96 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
77.1 −0.707107 − 0.707107i −1.53318 − 0.805823i 1.00000i −1.53706 + 1.62402i 0.514321 + 1.65393i 0.677204 − 0.677204i 0.707107 − 0.707107i 1.70130 + 2.47095i 2.23522 − 0.0614950i
77.2 −0.707107 − 0.707107i −1.29768 − 1.14719i 1.00000i 2.23605 − 0.00995019i 0.106413 + 1.72878i −1.03162 + 1.03162i 0.707107 − 0.707107i 0.367928 + 2.97735i −1.58816 − 1.57409i
77.3 −0.707107 − 0.707107i −0.854151 + 1.50679i 1.00000i 1.11448 + 1.93854i 1.66944 − 0.461487i 3.10713 − 3.10713i 0.707107 − 0.707107i −1.54085 − 2.57406i 0.582700 − 2.15881i
77.4 −0.707107 − 0.707107i −0.128848 + 1.72725i 1.00000i −1.56114 − 1.60089i 1.31246 − 1.13024i 0.236462 − 0.236462i 0.707107 − 0.707107i −2.96680 − 0.445105i −0.0281051 + 2.23589i
77.5 −0.707107 − 0.707107i 0.174068 − 1.72328i 1.00000i 1.17831 − 1.90042i −1.34163 + 1.09546i −2.28877 + 2.28877i 0.707107 − 0.707107i −2.93940 − 0.599938i −2.17699 + 0.510611i
77.6 −0.707107 − 0.707107i 1.63979 − 0.557754i 1.00000i −2.13774 + 0.655799i −1.55390 − 0.765115i 2.12802 − 2.12802i 0.707107 − 0.707107i 2.37782 − 1.82920i 1.97533 + 1.04789i
77.7 0.707107 + 0.707107i −1.53318 − 0.805823i 1.00000i 1.53706 − 1.62402i −0.514321 − 1.65393i −0.677204 + 0.677204i −0.707107 + 0.707107i 1.70130 + 2.47095i 2.23522 − 0.0614950i
77.8 0.707107 + 0.707107i −1.29768 − 1.14719i 1.00000i −2.23605 + 0.00995019i −0.106413 − 1.72878i 1.03162 − 1.03162i −0.707107 + 0.707107i 0.367928 + 2.97735i −1.58816 − 1.57409i
77.9 0.707107 + 0.707107i −0.854151 + 1.50679i 1.00000i −1.11448 − 1.93854i −1.66944 + 0.461487i −3.10713 + 3.10713i −0.707107 + 0.707107i −1.54085 − 2.57406i 0.582700 − 2.15881i
77.10 0.707107 + 0.707107i −0.128848 + 1.72725i 1.00000i 1.56114 + 1.60089i −1.31246 + 1.13024i −0.236462 + 0.236462i −0.707107 + 0.707107i −2.96680 − 0.445105i −0.0281051 + 2.23589i
77.11 0.707107 + 0.707107i 0.174068 − 1.72328i 1.00000i −1.17831 + 1.90042i 1.34163 − 1.09546i 2.28877 − 2.28877i −0.707107 + 0.707107i −2.93940 − 0.599938i −2.17699 + 0.510611i
77.12 0.707107 + 0.707107i 1.63979 − 0.557754i 1.00000i 2.13774 − 0.655799i 1.55390 + 0.765115i −2.12802 + 2.12802i −0.707107 + 0.707107i 2.37782 − 1.82920i 1.97533 + 1.04789i
233.1 −0.707107 + 0.707107i −1.53318 + 0.805823i − 1.00000i −1.53706 − 1.62402i 0.514321 − 1.65393i 0.677204 + 0.677204i 0.707107 + 0.707107i 1.70130 − 2.47095i 2.23522 + 0.0614950i
233.2 −0.707107 + 0.707107i −1.29768 + 1.14719i − 1.00000i 2.23605 + 0.00995019i 0.106413 − 1.72878i −1.03162 − 1.03162i 0.707107 + 0.707107i 0.367928 − 2.97735i −1.58816 + 1.57409i
233.3 −0.707107 + 0.707107i −0.854151 − 1.50679i − 1.00000i 1.11448 − 1.93854i 1.66944 + 0.461487i 3.10713 + 3.10713i 0.707107 + 0.707107i −1.54085 + 2.57406i 0.582700 + 2.15881i
233.4 −0.707107 + 0.707107i −0.128848 − 1.72725i − 1.00000i −1.56114 + 1.60089i 1.31246 + 1.13024i 0.236462 + 0.236462i 0.707107 + 0.707107i −2.96680 + 0.445105i −0.0281051 − 2.23589i
233.5 −0.707107 + 0.707107i 0.174068 + 1.72328i − 1.00000i 1.17831 + 1.90042i −1.34163 − 1.09546i −2.28877 − 2.28877i 0.707107 + 0.707107i −2.93940 + 0.599938i −2.17699 − 0.510611i
233.6 −0.707107 + 0.707107i 1.63979 + 0.557754i − 1.00000i −2.13774 − 0.655799i −1.55390 + 0.765115i 2.12802 + 2.12802i 0.707107 + 0.707107i 2.37782 + 1.82920i 1.97533 − 1.04789i
233.7 0.707107 − 0.707107i −1.53318 + 0.805823i − 1.00000i 1.53706 + 1.62402i −0.514321 + 1.65393i −0.677204 − 0.677204i −0.707107 − 0.707107i 1.70130 − 2.47095i 2.23522 + 0.0614950i
233.8 0.707107 − 0.707107i −1.29768 + 1.14719i − 1.00000i −2.23605 − 0.00995019i −0.106413 + 1.72878i 1.03162 + 1.03162i −0.707107 − 0.707107i 0.367928 − 2.97735i −1.58816 + 1.57409i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 77.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
15.e even 4 1 inner
195.s even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.s.b ✓ 24
3.b odd 2 1 390.2.s.c yes 24
5.c odd 4 1 390.2.s.c yes 24
13.b even 2 1 inner 390.2.s.b ✓ 24
15.e even 4 1 inner 390.2.s.b ✓ 24
39.d odd 2 1 390.2.s.c yes 24
65.h odd 4 1 390.2.s.c yes 24
195.s even 4 1 inner 390.2.s.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.s.b ✓ 24 1.a even 1 1 trivial
390.2.s.b ✓ 24 13.b even 2 1 inner
390.2.s.b ✓ 24 15.e even 4 1 inner
390.2.s.b ✓ 24 195.s even 4 1 inner
390.2.s.c yes 24 3.b odd 2 1
390.2.s.c yes 24 5.c odd 4 1
390.2.s.c yes 24 39.d odd 2 1
390.2.s.c yes 24 65.h odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(390, [\chi])\):

\( T_{7}^{24} + 570T_{7}^{20} + 83553T_{7}^{16} + 3792536T_{7}^{12} + 18386480T_{7}^{8} + 13023616T_{7}^{4} + 160000 \) Copy content Toggle raw display
\( T_{17}^{12} - 26 T_{17}^{11} + 338 T_{17}^{10} - 2514 T_{17}^{9} + 11301 T_{17}^{8} - 26460 T_{17}^{7} + \cdots + 26419600 \) Copy content Toggle raw display