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Properties

Label 456.2.bm.b
Level $456$
Weight $2$
Character orbit 456.bm
Analytic conductor $3.641$
Analytic rank $0$
Dimension $60$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [456,2,Mod(41,456)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(456, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 0, 9, 13])) 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("456.41"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 456 = 2^{3} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 456.bm (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [60,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.64117833217\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(10\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 6 q^{9} + 3 q^{13} - 3 q^{15} - 6 q^{17} + 3 q^{19} + 6 q^{25} + 3 q^{27} + 6 q^{29} + 24 q^{35} + 18 q^{39} + 3 q^{41} - 21 q^{43} + 63 q^{45} - 18 q^{47} - 30 q^{49} + 33 q^{51} - 36 q^{53} + 18 q^{55}+ \cdots - 39 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 0 −1.55056 + 0.771852i 0 −3.00241 − 0.529406i 0 −0.251171 − 0.435040i 0 1.80849 − 2.39361i 0
41.2 0 −1.50282 − 0.861118i 0 −0.320800 − 0.0565656i 0 −0.0144179 − 0.0249726i 0 1.51695 + 2.58822i 0
41.3 0 −1.35104 + 1.08383i 0 3.33482 + 0.588019i 0 2.19892 + 3.80864i 0 0.650612 − 2.92860i 0
41.4 0 −0.0815079 + 1.73013i 0 −1.23805 − 0.218301i 0 −0.753669 − 1.30539i 0 −2.98671 − 0.282039i 0
41.5 0 0.144474 − 1.72601i 0 2.45783 + 0.433381i 0 −2.29577 − 3.97638i 0 −2.95825 − 0.498730i 0
41.6 0 0.264929 − 1.71167i 0 −4.07516 − 0.718561i 0 0.904376 + 1.56643i 0 −2.85963 − 0.906941i 0
41.7 0 0.505407 + 1.65667i 0 2.78114 + 0.490390i 0 −1.12097 − 1.94159i 0 −2.48913 + 1.67459i 0
41.8 0 1.20462 − 1.24454i 0 1.70062 + 0.299865i 0 1.13747 + 1.97015i 0 −0.0977739 − 2.99841i 0
41.9 0 1.57581 + 0.718900i 0 0.726176 + 0.128044i 0 1.38478 + 2.39851i 0 1.96636 + 2.26570i 0
41.10 0 1.73038 − 0.0760235i 0 −2.36417 − 0.416867i 0 −2.12924 − 3.68795i 0 2.98844 − 0.263099i 0
89.1 0 −1.55056 − 0.771852i 0 −3.00241 + 0.529406i 0 −0.251171 + 0.435040i 0 1.80849 + 2.39361i 0
89.2 0 −1.50282 + 0.861118i 0 −0.320800 + 0.0565656i 0 −0.0144179 + 0.0249726i 0 1.51695 − 2.58822i 0
89.3 0 −1.35104 − 1.08383i 0 3.33482 − 0.588019i 0 2.19892 − 3.80864i 0 0.650612 + 2.92860i 0
89.4 0 −0.0815079 − 1.73013i 0 −1.23805 + 0.218301i 0 −0.753669 + 1.30539i 0 −2.98671 + 0.282039i 0
89.5 0 0.144474 + 1.72601i 0 2.45783 − 0.433381i 0 −2.29577 + 3.97638i 0 −2.95825 + 0.498730i 0
89.6 0 0.264929 + 1.71167i 0 −4.07516 + 0.718561i 0 0.904376 − 1.56643i 0 −2.85963 + 0.906941i 0
89.7 0 0.505407 − 1.65667i 0 2.78114 − 0.490390i 0 −1.12097 + 1.94159i 0 −2.48913 − 1.67459i 0
89.8 0 1.20462 + 1.24454i 0 1.70062 − 0.299865i 0 1.13747 − 1.97015i 0 −0.0977739 + 2.99841i 0
89.9 0 1.57581 − 0.718900i 0 0.726176 − 0.128044i 0 1.38478 − 2.39851i 0 1.96636 − 2.26570i 0
89.10 0 1.73038 + 0.0760235i 0 −2.36417 + 0.416867i 0 −2.12924 + 3.68795i 0 2.98844 + 0.263099i 0
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.10
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.j even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 456.2.bm.b yes 60
3.b odd 2 1 456.2.bm.a ✓ 60
4.b odd 2 1 912.2.cc.g 60
12.b even 2 1 912.2.cc.h 60
19.f odd 18 1 456.2.bm.a ✓ 60
57.j even 18 1 inner 456.2.bm.b yes 60
76.k even 18 1 912.2.cc.h 60
228.u odd 18 1 912.2.cc.g 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.bm.a ✓ 60 3.b odd 2 1
456.2.bm.a ✓ 60 19.f odd 18 1
456.2.bm.b yes 60 1.a even 1 1 trivial
456.2.bm.b yes 60 57.j even 18 1 inner
912.2.cc.g 60 4.b odd 2 1
912.2.cc.g 60 228.u odd 18 1
912.2.cc.h 60 12.b even 2 1
912.2.cc.h 60 76.k even 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{60} - 3 T_{5}^{58} + 36 T_{5}^{57} + 3 T_{5}^{56} - 78 T_{5}^{55} - 3824 T_{5}^{54} + \cdots + 66324791296 \) acting on \(S_{2}^{\mathrm{new}}(456, [\chi])\). Copy content Toggle raw display