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Properties

Label 72.7.h.a
Level $72$
Weight $7$
Character orbit 72.h
Analytic conductor $16.564$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [72,7,Mod(53,72)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(72, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.53");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 72.h (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(16.5638940206\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 132 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 132 q^{4} + 588 q^{10} + 4536 q^{16} + 11784 q^{22} + 75000 q^{25} + 25896 q^{28} + 122496 q^{31} + 74724 q^{34} - 56880 q^{40} - 74088 q^{46} + 197448 q^{49} - 149064 q^{52} - 232704 q^{55} - 196140 q^{58} - 1269024 q^{64} - 1585752 q^{70} + 514080 q^{73} + 327984 q^{76} - 1193088 q^{79} + 2813268 q^{82} - 3176832 q^{88} - 68856 q^{94} + 2960832 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 −7.66719 − 2.28345i 0 53.5717 + 35.0153i 149.398 0 498.755 −330.790 − 390.797i 0 −1145.46 − 341.142i
53.2 −7.66719 + 2.28345i 0 53.5717 − 35.0153i 149.398 0 498.755 −330.790 + 390.797i 0 −1145.46 + 341.142i
53.3 −7.52070 − 2.72748i 0 49.1217 + 41.0251i 63.9123 0 −492.967 −257.535 − 442.515i 0 −480.665 − 174.319i
53.4 −7.52070 + 2.72748i 0 49.1217 − 41.0251i 63.9123 0 −492.967 −257.535 + 442.515i 0 −480.665 + 174.319i
53.5 −7.27356 − 3.33097i 0 41.8093 + 48.4560i −221.820 0 −192.829 −142.697 − 491.713i 0 1613.42 + 738.876i
53.6 −7.27356 + 3.33097i 0 41.8093 − 48.4560i −221.820 0 −192.829 −142.697 + 491.713i 0 1613.42 − 738.876i
53.7 −5.42619 − 5.87847i 0 −5.11284 + 63.7954i −89.1905 0 426.860 402.763 − 316.111i 0 483.965 + 524.304i
53.8 −5.42619 + 5.87847i 0 −5.11284 − 63.7954i −89.1905 0 426.860 402.763 + 316.111i 0 483.965 − 524.304i
53.9 −3.09259 − 7.37807i 0 −44.8717 + 45.6347i 45.8456 0 −32.3457 475.466 + 189.937i 0 −141.782 − 338.252i
53.10 −3.09259 + 7.37807i 0 −44.8717 − 45.6347i 45.8456 0 −32.3457 475.466 − 189.937i 0 −141.782 + 338.252i
53.11 −1.11396 − 7.92206i 0 −61.5182 + 17.6496i 163.812 0 −207.473 208.350 + 467.690i 0 −182.479 − 1297.73i
53.12 −1.11396 + 7.92206i 0 −61.5182 − 17.6496i 163.812 0 −207.473 208.350 − 467.690i 0 −182.479 + 1297.73i
53.13 1.11396 − 7.92206i 0 −61.5182 − 17.6496i −163.812 0 −207.473 −208.350 + 467.690i 0 −182.479 + 1297.73i
53.14 1.11396 + 7.92206i 0 −61.5182 + 17.6496i −163.812 0 −207.473 −208.350 − 467.690i 0 −182.479 − 1297.73i
53.15 3.09259 − 7.37807i 0 −44.8717 − 45.6347i −45.8456 0 −32.3457 −475.466 + 189.937i 0 −141.782 + 338.252i
53.16 3.09259 + 7.37807i 0 −44.8717 + 45.6347i −45.8456 0 −32.3457 −475.466 − 189.937i 0 −141.782 − 338.252i
53.17 5.42619 − 5.87847i 0 −5.11284 − 63.7954i 89.1905 0 426.860 −402.763 − 316.111i 0 483.965 − 524.304i
53.18 5.42619 + 5.87847i 0 −5.11284 + 63.7954i 89.1905 0 426.860 −402.763 + 316.111i 0 483.965 + 524.304i
53.19 7.27356 − 3.33097i 0 41.8093 − 48.4560i 221.820 0 −192.829 142.697 − 491.713i 0 1613.42 − 738.876i
53.20 7.27356 + 3.33097i 0 41.8093 + 48.4560i 221.820 0 −192.829 142.697 + 491.713i 0 1613.42 + 738.876i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.7.h.a ✓ 24
3.b odd 2 1 inner 72.7.h.a ✓ 24
4.b odd 2 1 288.7.h.a 24
8.b even 2 1 inner 72.7.h.a ✓ 24
8.d odd 2 1 288.7.h.a 24
12.b even 2 1 288.7.h.a 24
24.f even 2 1 288.7.h.a 24
24.h odd 2 1 inner 72.7.h.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.7.h.a ✓ 24 1.a even 1 1 trivial
72.7.h.a ✓ 24 3.b odd 2 1 inner
72.7.h.a ✓ 24 8.b even 2 1 inner
72.7.h.a ✓ 24 24.h odd 2 1 inner
288.7.h.a 24 4.b odd 2 1
288.7.h.a 24 8.d odd 2 1
288.7.h.a 24 12.b even 2 1
288.7.h.a 24 24.f even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(72, [\chi])\).