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Properties

Label 740.2.bp.a
Level $740$
Weight $2$
Character orbit 740.bp
Analytic conductor $5.909$
Analytic rank $0$
Dimension $120$
Inner twists $4$

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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] = [740,2,Mod(169,740)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(740, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 9, 11])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("740.169"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 740.bp (of order \(18\), degree \(6\), 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: \(5.90892974957\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(20\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q + 3 q^{5} + 6 q^{9} + 12 q^{11} + 3 q^{15} + 6 q^{19} - 12 q^{21} - 33 q^{25} - 48 q^{35} + 24 q^{39} + 30 q^{41} - 27 q^{45} + 6 q^{49} - 3 q^{55} - 42 q^{59} + 48 q^{61} - 18 q^{65} - 108 q^{69}+ \cdots + 90 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
169.1 0 −1.11522 − 3.06403i 0 0.896315 − 2.04857i 0 −1.89421 − 0.334000i 0 −5.84644 + 4.90574i 0
169.2 0 −1.06939 − 2.93812i 0 −0.726059 + 2.11491i 0 4.51652 + 0.796384i 0 −5.19083 + 4.35562i 0
169.3 0 −0.788794 − 2.16719i 0 2.23599 + 0.0187872i 0 2.25112 + 0.396932i 0 −1.77640 + 1.49058i 0
169.4 0 −0.780576 − 2.14461i 0 0.743977 + 2.10867i 0 −3.87984 − 0.684120i 0 −1.69194 + 1.41970i 0
169.5 0 −0.671392 − 1.84463i 0 −1.87437 + 1.21932i 0 −0.102465 − 0.0180673i 0 −0.653777 + 0.548584i 0
169.6 0 −0.583105 − 1.60207i 0 −0.703709 − 2.12245i 0 −0.0792084 − 0.0139666i 0 0.0715239 − 0.0600157i 0
169.7 0 −0.410260 − 1.12718i 0 −2.23575 + 0.0375558i 0 0.632411 + 0.111511i 0 1.19591 − 1.00349i 0
169.8 0 −0.324044 − 0.890304i 0 −0.350865 − 2.20837i 0 4.31890 + 0.761538i 0 1.61050 − 1.35137i 0
169.9 0 −0.0844428 − 0.232005i 0 1.84708 − 1.26028i 0 −4.40804 − 0.777256i 0 2.25144 − 1.88918i 0
169.10 0 −0.0565853 − 0.155467i 0 1.96096 − 1.07454i 0 −0.407159 − 0.0717931i 0 2.27717 − 1.91077i 0
169.11 0 0.0565853 + 0.155467i 0 1.47519 + 1.68042i 0 0.407159 + 0.0717931i 0 2.27717 − 1.91077i 0
169.12 0 0.0844428 + 0.232005i 0 1.30464 + 1.81601i 0 4.40804 + 0.777256i 0 2.25144 − 1.88918i 0
169.13 0 0.324044 + 0.890304i 0 −1.08501 + 1.95518i 0 −4.31890 − 0.761538i 0 1.61050 − 1.35137i 0
169.14 0 0.410260 + 1.12718i 0 −2.08808 − 0.799963i 0 −0.632411 − 0.111511i 0 1.19591 − 1.00349i 0
169.15 0 0.583105 + 1.60207i 0 −1.38719 + 1.75377i 0 0.0792084 + 0.0139666i 0 0.0715239 − 0.0600157i 0
169.16 0 0.671392 + 1.84463i 0 −1.34430 − 1.78686i 0 0.102465 + 0.0180673i 0 −0.653777 + 0.548584i 0
169.17 0 0.780576 + 2.14461i 0 1.42032 − 1.72705i 0 3.87984 + 0.684120i 0 −1.69194 + 1.41970i 0
169.18 0 0.788794 + 2.16719i 0 2.10757 + 0.747099i 0 −2.25112 − 0.396932i 0 −1.77640 + 1.49058i 0
169.19 0 1.06939 + 2.93812i 0 0.0410689 − 2.23569i 0 −4.51652 − 0.796384i 0 −5.19083 + 4.35562i 0
169.20 0 1.11522 + 3.06403i 0 0.141610 + 2.23158i 0 1.89421 + 0.334000i 0 −5.84644 + 4.90574i 0
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 169.20
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
37.h even 18 1 inner
185.v even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.2.bp.a ✓ 120
5.b even 2 1 inner 740.2.bp.a ✓ 120
37.h even 18 1 inner 740.2.bp.a ✓ 120
185.v even 18 1 inner 740.2.bp.a ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.bp.a ✓ 120 1.a even 1 1 trivial
740.2.bp.a ✓ 120 5.b even 2 1 inner
740.2.bp.a ✓ 120 37.h even 18 1 inner
740.2.bp.a ✓ 120 185.v even 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(740, [\chi])\).