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Properties

Label 384.7.l.a
Level $384$
Weight $7$
Character orbit 384.l
Analytic conductor $88.341$
Analytic rank $0$
Dimension $48$
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] = [384,7,Mod(31,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.31");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 384.l (of order \(4\), degree \(2\), not 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: \(88.3407681100\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 2720 q^{11} - 3936 q^{19} - 26240 q^{23} - 66400 q^{29} + 162336 q^{35} + 7200 q^{37} + 340704 q^{43} + 806736 q^{49} + 80352 q^{51} - 443680 q^{53} - 232704 q^{55} - 886144 q^{59} + 326496 q^{61} - 372832 q^{65} - 962112 q^{67} - 541728 q^{69} - 534016 q^{71} - 1073088 q^{75} + 932960 q^{77} - 2834352 q^{81} - 2497760 q^{83} + 372000 q^{85} + 775008 q^{91} - 660960 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 0 −11.0227 − 11.0227i 0 −122.335 − 122.335i 0 −392.292 0 243.000i 0
31.2 0 −11.0227 − 11.0227i 0 −107.837 − 107.837i 0 −5.36994 0 243.000i 0
31.3 0 −11.0227 − 11.0227i 0 −52.2146 − 52.2146i 0 282.592 0 243.000i 0
31.4 0 −11.0227 − 11.0227i 0 15.5658 + 15.5658i 0 −10.2991 0 243.000i 0
31.5 0 −11.0227 − 11.0227i 0 −29.4894 − 29.4894i 0 −461.206 0 243.000i 0
31.6 0 −11.0227 − 11.0227i 0 55.7840 + 55.7840i 0 −496.753 0 243.000i 0
31.7 0 −11.0227 − 11.0227i 0 −45.7781 − 45.7781i 0 565.544 0 243.000i 0
31.8 0 −11.0227 − 11.0227i 0 59.6079 + 59.6079i 0 661.000 0 243.000i 0
31.9 0 −11.0227 − 11.0227i 0 95.7535 + 95.7535i 0 −338.697 0 243.000i 0
31.10 0 −11.0227 − 11.0227i 0 98.7574 + 98.7574i 0 218.381 0 243.000i 0
31.11 0 −11.0227 − 11.0227i 0 −128.299 − 128.299i 0 −76.4178 0 243.000i 0
31.12 0 −11.0227 − 11.0227i 0 160.484 + 160.484i 0 53.5181 0 243.000i 0
31.13 0 11.0227 + 11.0227i 0 −156.277 − 156.277i 0 −23.3623 0 243.000i 0
31.14 0 11.0227 + 11.0227i 0 −145.472 − 145.472i 0 −535.877 0 243.000i 0
31.15 0 11.0227 + 11.0227i 0 −132.791 − 132.791i 0 560.492 0 243.000i 0
31.16 0 11.0227 + 11.0227i 0 99.4317 + 99.4317i 0 −407.926 0 243.000i 0
31.17 0 11.0227 + 11.0227i 0 −47.0845 − 47.0845i 0 −400.703 0 243.000i 0
31.18 0 11.0227 + 11.0227i 0 9.26689 + 9.26689i 0 320.373 0 243.000i 0
31.19 0 11.0227 + 11.0227i 0 −38.1198 − 38.1198i 0 −34.3159 0 243.000i 0
31.20 0 11.0227 + 11.0227i 0 26.6395 + 26.6395i 0 403.840 0 243.000i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.7.l.a 48
4.b odd 2 1 384.7.l.b 48
8.b even 2 1 192.7.l.a 48
8.d odd 2 1 48.7.l.a ✓ 48
16.e even 4 1 48.7.l.a ✓ 48
16.e even 4 1 384.7.l.b 48
16.f odd 4 1 192.7.l.a 48
16.f odd 4 1 inner 384.7.l.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.7.l.a ✓ 48 8.d odd 2 1
48.7.l.a ✓ 48 16.e even 4 1
192.7.l.a 48 8.b even 2 1
192.7.l.a 48 16.f odd 4 1
384.7.l.a 48 1.a even 1 1 trivial
384.7.l.a 48 16.f odd 4 1 inner
384.7.l.b 48 4.b odd 2 1
384.7.l.b 48 16.e even 4 1