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Properties

Label 840.2.bk.c
Level $840$
Weight $2$
Character orbit 840.bk
Analytic conductor $6.707$
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] = [840,2,Mod(113,840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(840, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("840.113");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 840.bk (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: \(6.70743376979\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 4 q^{3} + 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 4 q^{3} + 8 q^{5} - 8 q^{15} - 40 q^{17} - 4 q^{21} - 8 q^{23} + 16 q^{25} + 32 q^{27} - 16 q^{29} + 16 q^{31} + 28 q^{33} + 8 q^{35} + 32 q^{37} - 24 q^{39} + 8 q^{43} - 36 q^{45} - 16 q^{47} - 60 q^{51} - 8 q^{55} - 4 q^{57} + 16 q^{59} + 16 q^{61} - 8 q^{63} + 24 q^{65} - 40 q^{67} - 40 q^{69} - 44 q^{75} - 8 q^{77} - 12 q^{81} - 40 q^{83} + 88 q^{85} + 60 q^{87} + 80 q^{89} - 8 q^{91} + 52 q^{93} + 80 q^{95} + 8 q^{97} - 48 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1 0 −1.67715 + 0.432628i 0 −1.93337 − 1.12342i 0 −0.707107 + 0.707107i 0 2.62567 − 1.45117i 0
113.2 0 −1.55202 + 0.768924i 0 2.06693 + 0.853124i 0 0.707107 − 0.707107i 0 1.81751 − 2.38677i 0
113.3 0 −1.48199 − 0.896493i 0 1.59667 − 1.56545i 0 0.707107 − 0.707107i 0 1.39260 + 2.65719i 0
113.4 0 −1.33301 + 1.10593i 0 1.75093 − 1.39077i 0 −0.707107 + 0.707107i 0 0.553828 − 2.94844i 0
113.5 0 −1.15334 + 1.29221i 0 −1.37371 − 1.76435i 0 0.707107 − 0.707107i 0 −0.339591 − 2.98072i 0
113.6 0 −1.03705 − 1.38727i 0 −1.15525 − 1.91452i 0 −0.707107 + 0.707107i 0 −0.849049 + 2.87735i 0
113.7 0 −0.976806 − 1.43033i 0 −0.229967 + 2.22421i 0 0.707107 − 0.707107i 0 −1.09170 + 2.79431i 0
113.8 0 −0.494796 − 1.65987i 0 2.22953 + 0.170883i 0 −0.707107 + 0.707107i 0 −2.51035 + 1.64260i 0
113.9 0 −0.103971 + 1.72893i 0 −2.18714 + 0.465195i 0 −0.707107 + 0.707107i 0 −2.97838 − 0.359515i 0
113.10 0 0.0984437 + 1.72925i 0 −1.88563 + 1.20183i 0 0.707107 − 0.707107i 0 −2.98062 + 0.340468i 0
113.11 0 0.496012 − 1.65951i 0 2.08357 − 0.811622i 0 0.707107 − 0.707107i 0 −2.50795 − 1.64627i 0
113.12 0 0.891384 + 1.48507i 0 2.22204 + 0.250066i 0 −0.707107 + 0.707107i 0 −1.41087 + 2.64754i 0
113.13 0 1.27235 + 1.17522i 0 1.70772 + 1.44350i 0 0.707107 − 0.707107i 0 0.237733 + 2.99057i 0
113.14 0 1.59025 − 0.686369i 0 −0.551373 − 2.16702i 0 0.707107 − 0.707107i 0 2.05780 − 2.18300i 0
113.15 0 1.73082 + 0.0653143i 0 0.614711 + 2.14991i 0 −0.707107 + 0.707107i 0 2.99147 + 0.226095i 0
113.16 0 1.73088 − 0.0636215i 0 −0.955669 − 2.02156i 0 −0.707107 + 0.707107i 0 2.99190 − 0.220243i 0
617.1 0 −1.67715 − 0.432628i 0 −1.93337 + 1.12342i 0 −0.707107 − 0.707107i 0 2.62567 + 1.45117i 0
617.2 0 −1.55202 − 0.768924i 0 2.06693 − 0.853124i 0 0.707107 + 0.707107i 0 1.81751 + 2.38677i 0
617.3 0 −1.48199 + 0.896493i 0 1.59667 + 1.56545i 0 0.707107 + 0.707107i 0 1.39260 − 2.65719i 0
617.4 0 −1.33301 − 1.10593i 0 1.75093 + 1.39077i 0 −0.707107 − 0.707107i 0 0.553828 + 2.94844i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 840.2.bk.c ✓ 32
3.b odd 2 1 840.2.bk.d yes 32
5.c odd 4 1 840.2.bk.d yes 32
15.e even 4 1 inner 840.2.bk.c ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.bk.c ✓ 32 1.a even 1 1 trivial
840.2.bk.c ✓ 32 15.e even 4 1 inner
840.2.bk.d yes 32 3.b odd 2 1
840.2.bk.d yes 32 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\):

\( T_{11}^{32} + 220 T_{11}^{30} + 21670 T_{11}^{28} + 1265740 T_{11}^{26} + 48963857 T_{11}^{24} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
\( T_{17}^{32} + 40 T_{17}^{31} + 800 T_{17}^{30} + 10240 T_{17}^{29} + 93250 T_{17}^{28} + \cdots + 35759251849216 \) Copy content Toggle raw display