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Properties

Label 180.9.c.b
Level $180$
Weight $9$
Character orbit 180.c
Analytic conductor $73.328$
Analytic rank $0$
Dimension $32$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [180,9,Mod(91,180)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(180, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 9, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("180.91"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 180.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(73.3281498110\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 610 q^{4} - 8750 q^{10} - 51392 q^{13} + 11986 q^{16} - 758068 q^{22} + 2500000 q^{25} + 976324 q^{28} - 6117428 q^{34} + 5152064 q^{37} - 96250 q^{40} - 10391752 q^{46} - 11002976 q^{49} + 13976584 q^{52}+ \cdots + 80579520 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1 −15.9001 − 1.78503i 0 249.627 + 56.7644i −279.508 0 1266.61i −3867.78 − 1348.15i 0 4444.22 + 498.931i
91.2 −15.9001 + 1.78503i 0 249.627 − 56.7644i −279.508 0 − 1266.61i −3867.78 + 1348.15i 0 4444.22 − 498.931i
91.3 −15.8926 − 1.85034i 0 249.152 + 58.8137i 279.508 0 − 4368.38i −3850.87 − 1395.72i 0 −4442.13 − 517.186i
91.4 −15.8926 + 1.85034i 0 249.152 − 58.8137i 279.508 0 4368.38i −3850.87 + 1395.72i 0 −4442.13 + 517.186i
91.5 −14.6346 − 6.46742i 0 172.345 + 189.297i 279.508 0 1279.81i −1297.94 − 3884.91i 0 −4090.50 − 1807.70i
91.6 −14.6346 + 6.46742i 0 172.345 − 189.297i 279.508 0 − 1279.81i −1297.94 + 3884.91i 0 −4090.50 + 1807.70i
91.7 −12.7138 − 9.71382i 0 67.2836 + 247.000i −279.508 0 − 3004.85i 1543.88 − 3793.90i 0 3553.63 + 2715.09i
91.8 −12.7138 + 9.71382i 0 67.2836 − 247.000i −279.508 0 3004.85i 1543.88 + 3793.90i 0 3553.63 − 2715.09i
91.9 −9.51821 − 12.8609i 0 −74.8072 + 244.826i 279.508 0 − 2236.25i 3860.72 − 1368.22i 0 −2660.42 − 3594.74i
91.10 −9.51821 + 12.8609i 0 −74.8072 − 244.826i 279.508 0 2236.25i 3860.72 + 1368.22i 0 −2660.42 + 3594.74i
91.11 −8.96713 − 13.2511i 0 −95.1813 + 237.648i −279.508 0 3400.42i 4002.59 − 869.766i 0 2506.39 + 3703.78i
91.12 −8.96713 + 13.2511i 0 −95.1813 − 237.648i −279.508 0 − 3400.42i 4002.59 + 869.766i 0 2506.39 − 3703.78i
91.13 −6.78369 − 14.4907i 0 −163.963 + 196.601i 279.508 0 782.759i 3961.17 + 1042.27i 0 −1896.10 − 4050.29i
91.14 −6.78369 + 14.4907i 0 −163.963 − 196.601i 279.508 0 − 782.759i 3961.17 − 1042.27i 0 −1896.10 + 4050.29i
91.15 −1.42186 − 15.9367i 0 −251.957 + 45.3194i −279.508 0 581.971i 1080.49 + 3950.92i 0 397.421 + 4454.44i
91.16 −1.42186 + 15.9367i 0 −251.957 − 45.3194i −279.508 0 − 581.971i 1080.49 − 3950.92i 0 397.421 − 4454.44i
91.17 1.42186 − 15.9367i 0 −251.957 − 45.3194i 279.508 0 − 581.971i −1080.49 + 3950.92i 0 397.421 − 4454.44i
91.18 1.42186 + 15.9367i 0 −251.957 + 45.3194i 279.508 0 581.971i −1080.49 − 3950.92i 0 397.421 + 4454.44i
91.19 6.78369 − 14.4907i 0 −163.963 − 196.601i −279.508 0 − 782.759i −3961.17 + 1042.27i 0 −1896.10 + 4050.29i
91.20 6.78369 + 14.4907i 0 −163.963 + 196.601i −279.508 0 782.759i −3961.17 − 1042.27i 0 −1896.10 − 4050.29i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.32
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.9.c.b ✓ 32
3.b odd 2 1 inner 180.9.c.b ✓ 32
4.b odd 2 1 inner 180.9.c.b ✓ 32
12.b even 2 1 inner 180.9.c.b ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.9.c.b ✓ 32 1.a even 1 1 trivial
180.9.c.b ✓ 32 3.b odd 2 1 inner
180.9.c.b ✓ 32 4.b odd 2 1 inner
180.9.c.b ✓ 32 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} + 48869152 T_{7}^{14} + 889031371271360 T_{7}^{12} + \cdots + 54\!\cdots\!00 \) acting on \(S_{9}^{\mathrm{new}}(180, [\chi])\). Copy content Toggle raw display