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Properties

Label 480.4.h.a
Level $480$
Weight $4$
Character orbit 480.h
Analytic conductor $28.321$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [480,4,Mod(191,480)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(480, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("480.191");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 480.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: \(28.3209168028\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 20 q^{9} - 72 q^{11} - 72 q^{13} - 20 q^{15} - 68 q^{21} - 96 q^{23} - 600 q^{25} - 168 q^{27} - 80 q^{33} - 504 q^{37} + 456 q^{39} - 220 q^{45} - 432 q^{47} - 816 q^{49} - 1240 q^{51} + 40 q^{57}+ \cdots - 3160 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1 0 −5.19596 − 0.0444499i 0 5.00000i 0 − 5.27888i 0 26.9960 + 0.461920i 0
191.2 0 −5.19596 + 0.0444499i 0 − 5.00000i 0 5.27888i 0 26.9960 − 0.461920i 0
191.3 0 −5.07498 − 1.11559i 0 − 5.00000i 0 − 17.6092i 0 24.5109 + 11.3232i 0
191.4 0 −5.07498 + 1.11559i 0 5.00000i 0 17.6092i 0 24.5109 − 11.3232i 0
191.5 0 −3.78906 − 3.55571i 0 5.00000i 0 − 13.3637i 0 1.71392 + 26.9455i 0
191.6 0 −3.78906 + 3.55571i 0 − 5.00000i 0 13.3637i 0 1.71392 − 26.9455i 0
191.7 0 −3.26404 − 4.04302i 0 5.00000i 0 14.4221i 0 −5.69203 + 26.3932i 0
191.8 0 −3.26404 + 4.04302i 0 − 5.00000i 0 − 14.4221i 0 −5.69203 − 26.3932i 0
191.9 0 −2.28458 − 4.66698i 0 − 5.00000i 0 − 8.51421i 0 −16.5614 + 21.3242i 0
191.10 0 −2.28458 + 4.66698i 0 5.00000i 0 8.51421i 0 −16.5614 − 21.3242i 0
191.11 0 −0.991849 − 5.10061i 0 − 5.00000i 0 7.56208i 0 −25.0325 + 10.1181i 0
191.12 0 −0.991849 + 5.10061i 0 5.00000i 0 − 7.56208i 0 −25.0325 − 10.1181i 0
191.13 0 0.637007 − 5.15696i 0 5.00000i 0 − 26.9030i 0 −26.1884 − 6.57004i 0
191.14 0 0.637007 + 5.15696i 0 − 5.00000i 0 26.9030i 0 −26.1884 + 6.57004i 0
191.15 0 2.67023 − 4.45756i 0 − 5.00000i 0 28.8929i 0 −12.7397 − 23.8055i 0
191.16 0 2.67023 + 4.45756i 0 5.00000i 0 − 28.8929i 0 −12.7397 + 23.8055i 0
191.17 0 3.76455 − 3.58164i 0 − 5.00000i 0 − 30.2049i 0 1.34373 − 26.9665i 0
191.18 0 3.76455 + 3.58164i 0 5.00000i 0 30.2049i 0 1.34373 + 26.9665i 0
191.19 0 4.12770 − 3.15628i 0 5.00000i 0 − 3.91168i 0 7.07583 − 26.0563i 0
191.20 0 4.12770 + 3.15628i 0 − 5.00000i 0 3.91168i 0 7.07583 + 26.0563i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 480.4.h.a ✓ 24
3.b odd 2 1 480.4.h.b yes 24
4.b odd 2 1 480.4.h.b yes 24
8.b even 2 1 960.4.h.e 24
8.d odd 2 1 960.4.h.c 24
12.b even 2 1 inner 480.4.h.a ✓ 24
24.f even 2 1 960.4.h.e 24
24.h odd 2 1 960.4.h.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.4.h.a ✓ 24 1.a even 1 1 trivial
480.4.h.a ✓ 24 12.b even 2 1 inner
480.4.h.b yes 24 3.b odd 2 1
480.4.h.b yes 24 4.b odd 2 1
960.4.h.c 24 8.d odd 2 1
960.4.h.c 24 24.h odd 2 1
960.4.h.e 24 8.b even 2 1
960.4.h.e 24 24.f even 2 1