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Properties

Label 740.2.cc.a
Level $740$
Weight $2$
Character orbit 740.cc
Analytic conductor $5.909$
Analytic rank $0$
Dimension $228$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [740,2,Mod(17,740)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(740, base_ring=CyclotomicField(36)) chi = DirichletCharacter(H, H._module([0, 9, 7])) 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("740.17"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 740.cc (of order \(36\), degree \(12\), 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: \(228\)
Relative dimension: \(19\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 228 q + 6 q^{3} - 12 q^{25} + 12 q^{27} - 36 q^{31} + 6 q^{33} + 24 q^{35} + 24 q^{37} - 72 q^{39} - 54 q^{41} - 12 q^{45} + 36 q^{49} - 6 q^{53} - 72 q^{57} - 36 q^{61} + 18 q^{65} + 42 q^{67} + 96 q^{69}+ \cdots + 72 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0 −2.63033 + 1.84178i 0 −1.73811 + 1.40676i 0 −0.139101 − 1.58993i 0 2.50044 − 6.86990i 0
17.2 0 −2.44901 + 1.71481i 0 −1.00839 − 1.99578i 0 −0.127174 − 1.45360i 0 2.03100 − 5.58011i 0
17.3 0 −2.37198 + 1.66088i 0 2.22911 − 0.176315i 0 0.449992 + 5.14343i 0 1.84172 − 5.06008i 0
17.4 0 −2.07277 + 1.45137i 0 2.16816 − 0.546899i 0 −0.322002 − 3.68050i 0 1.16384 − 3.19762i 0
17.5 0 −1.42609 + 0.998562i 0 −1.41528 − 1.73118i 0 0.136680 + 1.56226i 0 0.0105581 − 0.0290083i 0
17.6 0 −1.41358 + 0.989801i 0 1.06322 + 1.96712i 0 0.0614096 + 0.701915i 0 −0.00755175 + 0.0207483i 0
17.7 0 −0.997755 + 0.698636i 0 −1.64427 + 1.51538i 0 0.307773 + 3.51786i 0 −0.518637 + 1.42494i 0
17.8 0 −0.921592 + 0.645306i 0 −1.50338 + 1.65525i 0 −0.371685 − 4.24838i 0 −0.593148 + 1.62966i 0
17.9 0 −0.117847 + 0.0825175i 0 1.72197 − 1.42647i 0 0.248075 + 2.83551i 0 −1.01898 + 2.79963i 0
17.10 0 0.0586331 − 0.0410554i 0 1.91470 + 1.15496i 0 −0.252966 − 2.89142i 0 −1.02431 + 2.81426i 0
17.11 0 0.422741 − 0.296006i 0 −2.01901 − 0.961044i 0 0.170641 + 1.95043i 0 −0.934970 + 2.56881i 0
17.12 0 0.485165 − 0.339716i 0 −2.23102 + 0.150100i 0 −0.237390 − 2.71338i 0 −0.906082 + 2.48944i 0
17.13 0 0.856462 − 0.599701i 0 1.15330 + 1.91570i 0 0.235534 + 2.69217i 0 −0.652175 + 1.79184i 0
17.14 0 1.08931 − 0.762745i 0 −0.627642 − 2.14617i 0 −0.345090 − 3.94440i 0 −0.421238 + 1.15734i 0
17.15 0 1.17356 − 0.821732i 0 1.45438 − 1.69846i 0 −0.140025 − 1.60049i 0 −0.324073 + 0.890382i 0
17.16 0 1.79150 − 1.25442i 0 −0.433171 + 2.19371i 0 0.206298 + 2.35800i 0 0.609834 − 1.67551i 0
17.17 0 1.95651 − 1.36996i 0 0.487612 − 2.18225i 0 0.352830 + 4.03287i 0 0.925077 − 2.54163i 0
17.18 0 2.26944 − 1.58908i 0 −2.00085 + 0.998289i 0 −0.0931130 − 1.06429i 0 1.59913 − 4.39356i 0
17.19 0 2.52279 − 1.76647i 0 2.17830 + 0.504970i 0 −0.140686 − 1.60805i 0 2.21796 − 6.09379i 0
113.1 0 −3.04494 + 0.266398i 0 1.67598 + 1.48023i 0 −0.461373 + 0.989418i 0 6.24626 − 1.10138i 0
See next 80 embeddings (of 228 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.19
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.z even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.2.cc.a ✓ 228
5.c odd 4 1 740.2.ch.a yes 228
37.i odd 36 1 740.2.ch.a yes 228
185.z even 36 1 inner 740.2.cc.a ✓ 228
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.cc.a ✓ 228 1.a even 1 1 trivial
740.2.cc.a ✓ 228 185.z even 36 1 inner
740.2.ch.a yes 228 5.c odd 4 1
740.2.ch.a yes 228 37.i odd 36 1

Hecke kernels

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