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Properties

Label 340.2.bi.a
Level $340$
Weight $2$
Character orbit 340.bi
Analytic conductor $2.715$
Analytic rank $0$
Dimension $72$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [340,2,Mod(37,340)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("340.37"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(340, base_ring=CyclotomicField(16)) chi = DirichletCharacter(H, H._module([0, 4, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 340 = 2^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 340.bi (of order \(16\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.71491366872\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(9\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 24 q^{15} + 8 q^{25} + 48 q^{27} - 32 q^{31} - 16 q^{33} - 32 q^{37} + 32 q^{39} - 40 q^{41} + 32 q^{53} - 16 q^{55} + 72 q^{57} - 112 q^{59} - 48 q^{63} + 32 q^{67} - 16 q^{71} - 96 q^{73} + 24 q^{75}+ \cdots - 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 0 −2.25094 + 1.50403i 0 1.65287 − 1.50600i 0 −3.49200 + 0.694602i 0 1.65656 − 3.99930i 0
37.2 0 −2.02401 + 1.35240i 0 −0.127990 + 2.23240i 0 3.90900 − 0.777549i 0 1.11958 − 2.70291i 0
37.3 0 −1.87906 + 1.25555i 0 −2.23182 − 0.137737i 0 −0.142860 + 0.0284166i 0 0.806426 − 1.94689i 0
37.4 0 −0.770194 + 0.514627i 0 2.21225 − 0.325531i 0 2.08800 − 0.415329i 0 −0.819692 + 1.97891i 0
37.5 0 0.428130 − 0.286067i 0 −2.08921 + 0.797005i 0 −3.17790 + 0.632123i 0 −1.04659 + 2.52669i 0
37.6 0 0.522949 − 0.349423i 0 1.38657 + 1.75426i 0 −0.175059 + 0.0348214i 0 −0.996671 + 2.40618i 0
37.7 0 1.25876 − 0.841077i 0 −1.69071 − 1.46339i 0 4.99604 − 0.993774i 0 −0.270982 + 0.654209i 0
37.8 0 1.90862 − 1.27530i 0 1.36653 − 1.76991i 0 −1.56805 + 0.311904i 0 0.868385 − 2.09647i 0
37.9 0 2.80575 − 1.87474i 0 −0.478489 + 2.18427i 0 0.283598 − 0.0564111i 0 3.20952 − 7.74847i 0
97.1 0 −0.632787 + 3.18123i 0 −1.60222 − 1.55977i 0 −2.10838 − 1.40877i 0 −6.94819 − 2.87803i 0
97.2 0 −0.377259 + 1.89661i 0 2.15286 − 0.604326i 0 0.404555 + 0.270315i 0 −0.683156 − 0.282973i 0
97.3 0 −0.324086 + 1.62929i 0 0.769891 + 2.09935i 0 0.118962 + 0.0794881i 0 0.222085 + 0.0919905i 0
97.4 0 −0.164959 + 0.829303i 0 −0.975356 − 2.01213i 0 1.39194 + 0.930065i 0 2.11111 + 0.874449i 0
97.5 0 0.0566376 − 0.284737i 0 −2.17471 + 0.520242i 0 −3.86798 − 2.58450i 0 2.69377 + 1.11580i 0
97.6 0 0.192727 − 0.968904i 0 −1.91554 + 1.15356i 0 2.02209 + 1.35112i 0 1.87001 + 0.774583i 0
97.7 0 0.296491 − 1.49056i 0 2.22866 + 0.181908i 0 3.53511 + 2.36209i 0 0.637769 + 0.264172i 0
97.8 0 0.426888 − 2.14611i 0 1.55259 + 1.60919i 0 −2.85665 − 1.90876i 0 −1.65192 − 0.684248i 0
97.9 0 0.526346 − 2.64612i 0 −0.0361654 − 2.23578i 0 −0.595058 − 0.397605i 0 −3.95327 − 1.63750i 0
113.1 0 −1.76939 + 2.64808i 0 −1.59649 + 1.56564i 0 −0.489690 + 2.46184i 0 −2.73352 − 6.59930i 0
113.2 0 −1.68652 + 2.52405i 0 0.688405 − 2.12746i 0 0.678287 − 3.40998i 0 −2.37845 − 5.74210i 0
See all 72 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 37.9
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
85.r even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 340.2.bi.a yes 72
5.c odd 4 1 340.2.bd.a ✓ 72
17.e odd 16 1 340.2.bd.a ✓ 72
85.r even 16 1 inner 340.2.bi.a yes 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
340.2.bd.a ✓ 72 5.c odd 4 1
340.2.bd.a ✓ 72 17.e odd 16 1
340.2.bi.a yes 72 1.a even 1 1 trivial
340.2.bi.a yes 72 85.r even 16 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(340, [\chi])\).