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Properties

Label 72.8.f.a
Level $72$
Weight $8$
Character orbit 72.f
Analytic conductor $22.492$
Analytic rank $0$
Dimension $28$
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] = [72,8,Mod(35,72)]
 
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([1, 1, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.35");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 72.f (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: \(22.4917218349\)
Analytic rank: \(0\)
Dimension: \(28\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 52 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 52 q^{4} + 10092 q^{10} - 1928 q^{16} - 121168 q^{19} + 59576 q^{22} + 437500 q^{25} + 46872 q^{28} - 114748 q^{34} + 1054752 q^{40} + 1505696 q^{43} - 476184 q^{46} - 2272076 q^{49} + 1468392 q^{52} + 3054996 q^{58} - 4186016 q^{64} - 776272 q^{67} + 3238872 q^{70} - 2534128 q^{73} - 21642832 q^{76} + 10334372 q^{82} + 10834016 q^{88} - 3406992 q^{91} - 22555944 q^{94} - 26311456 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1 −11.3087 − 0.338231i 0 127.771 + 7.64988i 93.0830 0 − 1025.06i −1442.33 − 129.726i 0 −1052.64 − 31.4836i
35.2 −11.3087 + 0.338231i 0 127.771 − 7.64988i 93.0830 0 1025.06i −1442.33 + 129.726i 0 −1052.64 + 31.4836i
35.3 −10.7674 − 3.47333i 0 103.872 + 74.7971i −527.660 0 − 1312.48i −858.632 − 1166.15i 0 5681.50 + 1832.74i
35.4 −10.7674 + 3.47333i 0 103.872 − 74.7971i −527.660 0 1312.48i −858.632 + 1166.15i 0 5681.50 − 1832.74i
35.5 −9.08380 − 6.74423i 0 37.0308 + 122.526i 168.010 0 1488.58i 489.966 − 1362.75i 0 −1526.17 − 1133.10i
35.6 −9.08380 + 6.74423i 0 37.0308 − 122.526i 168.010 0 − 1488.58i 489.966 + 1362.75i 0 −1526.17 + 1133.10i
35.7 −8.14946 − 7.84769i 0 4.82744 + 127.909i 294.266 0 − 328.633i 964.449 − 1080.27i 0 −2398.11 − 2309.31i
35.8 −8.14946 + 7.84769i 0 4.82744 − 127.909i 294.266 0 328.633i 964.449 + 1080.27i 0 −2398.11 + 2309.31i
35.9 −7.18397 − 8.74017i 0 −24.7812 + 125.578i −166.398 0 − 750.222i 1275.60 − 685.559i 0 1195.40 + 1454.35i
35.10 −7.18397 + 8.74017i 0 −24.7812 − 125.578i −166.398 0 750.222i 1275.60 + 685.559i 0 1195.40 − 1454.35i
35.11 −3.16822 − 10.8610i 0 −107.925 + 68.8204i −238.250 0 737.398i 1089.39 + 954.138i 0 754.827 + 2587.64i
35.12 −3.16822 + 10.8610i 0 −107.925 − 68.8204i −238.250 0 − 737.398i 1089.39 − 954.138i 0 754.827 − 2587.64i
35.13 −0.319783 − 11.3092i 0 −127.795 + 7.23297i 412.177 0 − 359.164i 122.666 + 1442.95i 0 −131.807 − 4661.38i
35.14 −0.319783 + 11.3092i 0 −127.795 − 7.23297i 412.177 0 359.164i 122.666 − 1442.95i 0 −131.807 + 4661.38i
35.15 0.319783 − 11.3092i 0 −127.795 − 7.23297i −412.177 0 359.164i −122.666 + 1442.95i 0 −131.807 + 4661.38i
35.16 0.319783 + 11.3092i 0 −127.795 + 7.23297i −412.177 0 − 359.164i −122.666 − 1442.95i 0 −131.807 − 4661.38i
35.17 3.16822 − 10.8610i 0 −107.925 − 68.8204i 238.250 0 − 737.398i −1089.39 + 954.138i 0 754.827 − 2587.64i
35.18 3.16822 + 10.8610i 0 −107.925 + 68.8204i 238.250 0 737.398i −1089.39 − 954.138i 0 754.827 + 2587.64i
35.19 7.18397 − 8.74017i 0 −24.7812 − 125.578i 166.398 0 750.222i −1275.60 − 685.559i 0 1195.40 − 1454.35i
35.20 7.18397 + 8.74017i 0 −24.7812 + 125.578i 166.398 0 − 750.222i −1275.60 + 685.559i 0 1195.40 + 1454.35i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.28
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.8.f.a ✓ 28
3.b odd 2 1 inner 72.8.f.a ✓ 28
4.b odd 2 1 288.8.f.a 28
8.b even 2 1 288.8.f.a 28
8.d odd 2 1 inner 72.8.f.a ✓ 28
12.b even 2 1 288.8.f.a 28
24.f even 2 1 inner 72.8.f.a ✓ 28
24.h odd 2 1 288.8.f.a 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.8.f.a ✓ 28 1.a even 1 1 trivial
72.8.f.a ✓ 28 3.b odd 2 1 inner
72.8.f.a ✓ 28 8.d odd 2 1 inner
72.8.f.a ✓ 28 24.f even 2 1 inner
288.8.f.a 28 4.b odd 2 1
288.8.f.a 28 8.b even 2 1
288.8.f.a 28 12.b even 2 1
288.8.f.a 28 24.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(72, [\chi])\).