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Properties

Label 736.2.x.a
Level $736$
Weight $2$
Character orbit 736.x
Analytic conductor $5.877$
Analytic rank $0$
Dimension $220$
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] = [736,2,Mod(49,736)] 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("736.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(736, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([0, 11, 16])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 736 = 2^{5} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 736.x (of order \(22\), degree \(10\), not 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: \(5.87698958877\)
Analytic rank: \(0\)
Dimension: \(220\)
Relative dimension: \(22\) over \(\Q(\zeta_{22})\)
Twist minimal: no (minimal twist has level 184)
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 220 q + 22 q^{7} + 22 q^{15} - 18 q^{17} + 16 q^{23} - 4 q^{25} + 34 q^{31} - 30 q^{33} + 18 q^{39} - 18 q^{41} + 40 q^{47} - 28 q^{49} + 38 q^{55} - 30 q^{57} - 18 q^{63} - 38 q^{65} + 26 q^{71} - 18 q^{73}+ \cdots - 18 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 0 −3.05941 − 1.39719i 0 −1.58431 − 1.37281i 0 −2.86721 + 1.84264i 0 5.44328 + 6.28187i 0
49.2 0 −2.46787 − 1.12704i 0 −1.04195 − 0.902852i 0 2.69045 − 1.72905i 0 2.85560 + 3.29554i 0
49.3 0 −2.23415 − 1.02030i 0 0.183115 + 0.158670i 0 0.661458 − 0.425093i 0 1.98583 + 2.29177i 0
49.4 0 −2.00125 − 0.913942i 0 −2.16668 − 1.87744i 0 1.81284 − 1.16504i 0 1.20514 + 1.39081i 0
49.5 0 −1.78987 − 0.817407i 0 1.45070 + 1.25704i 0 2.66148 − 1.71043i 0 0.570905 + 0.658860i 0
49.6 0 −1.57450 − 0.719052i 0 2.68037 + 2.32255i 0 −0.314784 + 0.202299i 0 −0.00255606 − 0.00294985i 0
49.7 0 −1.31167 − 0.599018i 0 0.865287 + 0.749776i 0 −3.05894 + 1.96586i 0 −0.602936 − 0.695825i 0
49.8 0 −1.19099 − 0.543906i 0 −3.10457 − 2.69013i 0 −2.79024 + 1.79318i 0 −0.841964 − 0.971678i 0
49.9 0 −0.496871 − 0.226913i 0 1.16498 + 1.00946i 0 −0.981244 + 0.630607i 0 −1.76919 − 2.04176i 0
49.10 0 −0.302712 − 0.138244i 0 −2.42383 − 2.10026i 0 3.31871 − 2.13281i 0 −1.89206 − 2.18355i 0
49.11 0 −0.247795 − 0.113164i 0 −0.131544 − 0.113984i 0 −1.39962 + 0.899478i 0 −1.91599 − 2.21117i 0
49.12 0 0.247795 + 0.113164i 0 0.131544 + 0.113984i 0 −1.39962 + 0.899478i 0 −1.91599 − 2.21117i 0
49.13 0 0.302712 + 0.138244i 0 2.42383 + 2.10026i 0 3.31871 − 2.13281i 0 −1.89206 − 2.18355i 0
49.14 0 0.496871 + 0.226913i 0 −1.16498 − 1.00946i 0 −0.981244 + 0.630607i 0 −1.76919 − 2.04176i 0
49.15 0 1.19099 + 0.543906i 0 3.10457 + 2.69013i 0 −2.79024 + 1.79318i 0 −0.841964 − 0.971678i 0
49.16 0 1.31167 + 0.599018i 0 −0.865287 − 0.749776i 0 −3.05894 + 1.96586i 0 −0.602936 − 0.695825i 0
49.17 0 1.57450 + 0.719052i 0 −2.68037 − 2.32255i 0 −0.314784 + 0.202299i 0 −0.00255606 − 0.00294985i 0
49.18 0 1.78987 + 0.817407i 0 −1.45070 − 1.25704i 0 2.66148 − 1.71043i 0 0.570905 + 0.658860i 0
49.19 0 2.00125 + 0.913942i 0 2.16668 + 1.87744i 0 1.81284 − 1.16504i 0 1.20514 + 1.39081i 0
49.20 0 2.23415 + 1.02030i 0 −0.183115 − 0.158670i 0 0.661458 − 0.425093i 0 1.98583 + 2.29177i 0
See next 80 embeddings (of 220 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 49.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
23.c even 11 1 inner
184.p even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 736.2.x.a 220
4.b odd 2 1 184.2.p.a ✓ 220
8.b even 2 1 inner 736.2.x.a 220
8.d odd 2 1 184.2.p.a ✓ 220
23.c even 11 1 inner 736.2.x.a 220
92.g odd 22 1 184.2.p.a ✓ 220
184.k odd 22 1 184.2.p.a ✓ 220
184.p even 22 1 inner 736.2.x.a 220
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
184.2.p.a ✓ 220 4.b odd 2 1
184.2.p.a ✓ 220 8.d odd 2 1
184.2.p.a ✓ 220 92.g odd 22 1
184.2.p.a ✓ 220 184.k odd 22 1
736.2.x.a 220 1.a even 1 1 trivial
736.2.x.a 220 8.b even 2 1 inner
736.2.x.a 220 23.c even 11 1 inner
736.2.x.a 220 184.p even 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(736, [\chi])\).