Properties

Label 76.4.i.a
Level $76$
Weight $4$
Character orbit 76.i
Analytic conductor $4.484$
Analytic rank $0$
Dimension $30$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [76,4,Mod(5,76)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(76, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 16]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("76.5");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 76.i (of order \(9\), degree \(6\), 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: \(4.48414516044\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(5\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 3 q^{3} + 6 q^{7} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 3 q^{3} + 6 q^{7} + 15 q^{9} + 42 q^{11} - 42 q^{13} + 78 q^{15} + 30 q^{17} + 282 q^{19} + 198 q^{21} - 300 q^{23} - 276 q^{25} + 219 q^{27} + 216 q^{29} + 30 q^{31} - 597 q^{33} - 636 q^{35} + 60 q^{37} - 2172 q^{39} - 63 q^{41} - 246 q^{43} - 882 q^{45} + 762 q^{47} - 525 q^{49} + 2613 q^{51} + 882 q^{53} + 1350 q^{55} + 924 q^{57} + 2085 q^{59} + 1530 q^{61} + 2424 q^{63} + 1530 q^{65} - 3609 q^{67} + 756 q^{69} - 4962 q^{71} - 2394 q^{73} - 3516 q^{77} - 630 q^{79} - 3723 q^{81} - 2382 q^{83} + 3228 q^{85} - 1110 q^{87} + 2196 q^{89} + 6036 q^{91} + 5010 q^{93} + 6204 q^{95} + 6459 q^{97} + 6189 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 0 −1.53481 + 8.70432i 0 10.3935 + 8.72119i 0 −7.14447 12.3746i 0 −48.0379 17.4843i 0
5.2 0 −0.580234 + 3.29067i 0 −11.9065 9.99077i 0 −13.7659 23.8433i 0 14.8799 + 5.41582i 0
5.3 0 −0.267608 + 1.51768i 0 −1.63306 1.37030i 0 13.0616 + 22.6234i 0 23.1400 + 8.42226i 0
5.4 0 0.629037 3.56745i 0 12.9592 + 10.8741i 0 −4.22583 7.31935i 0 13.0407 + 4.74643i 0
5.5 0 1.42726 8.09439i 0 −6.74894 5.66304i 0 0.266189 + 0.461053i 0 −38.1104 13.8710i 0
9.1 0 −7.47864 + 6.27533i 0 −3.19346 1.16233i 0 4.69206 8.12689i 0 11.8619 67.2720i 0
9.2 0 −1.65540 + 1.38904i 0 8.99353 + 3.27338i 0 −0.0466734 + 0.0808407i 0 −3.87760 + 21.9910i 0
9.3 0 −0.487209 + 0.408817i 0 −11.6536 4.24157i 0 −11.7723 + 20.3903i 0 −4.61826 + 26.1915i 0
9.4 0 3.64834 3.06132i 0 −10.2365 3.72578i 0 16.4303 28.4581i 0 −0.749805 + 4.25235i 0
9.5 0 6.23895 5.23510i 0 12.3313 + 4.48821i 0 −4.34015 + 7.51736i 0 6.82973 38.7333i 0
17.1 0 −7.47864 6.27533i 0 −3.19346 + 1.16233i 0 4.69206 + 8.12689i 0 11.8619 + 67.2720i 0
17.2 0 −1.65540 1.38904i 0 8.99353 3.27338i 0 −0.0466734 0.0808407i 0 −3.87760 21.9910i 0
17.3 0 −0.487209 0.408817i 0 −11.6536 + 4.24157i 0 −11.7723 20.3903i 0 −4.61826 26.1915i 0
17.4 0 3.64834 + 3.06132i 0 −10.2365 + 3.72578i 0 16.4303 + 28.4581i 0 −0.749805 4.25235i 0
17.5 0 6.23895 + 5.23510i 0 12.3313 4.48821i 0 −4.34015 7.51736i 0 6.82973 + 38.7333i 0
25.1 0 −8.30252 3.02187i 0 2.68735 + 15.2407i 0 13.2377 22.9284i 0 39.1169 + 32.8230i 0
25.2 0 −3.29897 1.20073i 0 −0.0631039 0.357880i 0 −13.5658 + 23.4967i 0 −11.2417 9.43292i 0
25.3 0 −1.97311 0.718152i 0 −1.01111 5.73427i 0 2.37147 4.10751i 0 −17.3058 14.5213i 0
25.4 0 5.87446 + 2.13813i 0 −3.07163 17.4201i 0 12.3773 21.4380i 0 9.25452 + 7.76546i 0
25.5 0 6.26044 + 2.27861i 0 2.15308 + 12.2107i 0 −4.57544 + 7.92489i 0 13.3179 + 11.1750i 0
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.4.i.a 30
19.e even 9 1 inner 76.4.i.a 30
19.e even 9 1 1444.4.a.j 15
19.f odd 18 1 1444.4.a.k 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.4.i.a 30 1.a even 1 1 trivial
76.4.i.a 30 19.e even 9 1 inner
1444.4.a.j 15 19.e even 9 1
1444.4.a.k 15 19.f odd 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(76, [\chi])\).