↑

Properties

Label 960.4.h.d
Level $960$
Weight $4$
Character orbit 960.h
Analytic conductor $56.642$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [960,4,Mod(191,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, 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("960.191");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 960.h (of order \(2\), degree \(1\), 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: \(56.6418336055\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 60)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 20 q^{9} - 72 q^{13} + 68 q^{21} - 600 q^{25} + 848 q^{33} - 504 q^{37} + 220 q^{45} - 2256 q^{49} + 1416 q^{57} - 1992 q^{61} + 1548 q^{69} - 2304 q^{73} + 3840 q^{81} - 240 q^{85} + 4384 q^{93} - 2448 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1 0 −5.18580 − 0.327893i 0 5.00000i 0 − 26.3120i 0 26.7850 + 3.40077i 0
191.2 0 −5.18580 + 0.327893i 0 − 5.00000i 0 26.3120i 0 26.7850 − 3.40077i 0
191.3 0 −5.00938 − 1.38062i 0 − 5.00000i 0 0.228949i 0 23.1878 + 13.8321i 0
191.4 0 −5.00938 + 1.38062i 0 5.00000i 0 − 0.228949i 0 23.1878 − 13.8321i 0
191.5 0 −4.56698 − 2.47844i 0 5.00000i 0 20.9745i 0 14.7146 + 22.6380i 0
191.6 0 −4.56698 + 2.47844i 0 − 5.00000i 0 − 20.9745i 0 14.7146 − 22.6380i 0
191.7 0 −2.09257 − 4.75617i 0 − 5.00000i 0 − 32.2690i 0 −18.2423 + 19.9053i 0
191.8 0 −2.09257 + 4.75617i 0 5.00000i 0 32.2690i 0 −18.2423 − 19.9053i 0
191.9 0 −1.95519 − 4.81427i 0 − 5.00000i 0 20.3642i 0 −19.3544 + 18.8257i 0
191.10 0 −1.95519 + 4.81427i 0 5.00000i 0 − 20.3642i 0 −19.3544 − 18.8257i 0
191.11 0 −1.56674 − 4.95432i 0 5.00000i 0 5.80602i 0 −22.0907 + 15.5243i 0
191.12 0 −1.56674 + 4.95432i 0 − 5.00000i 0 − 5.80602i 0 −22.0907 − 15.5243i 0
191.13 0 1.56674 − 4.95432i 0 − 5.00000i 0 5.80602i 0 −22.0907 − 15.5243i 0
191.14 0 1.56674 + 4.95432i 0 5.00000i 0 − 5.80602i 0 −22.0907 + 15.5243i 0
191.15 0 1.95519 − 4.81427i 0 5.00000i 0 20.3642i 0 −19.3544 − 18.8257i 0
191.16 0 1.95519 + 4.81427i 0 − 5.00000i 0 − 20.3642i 0 −19.3544 + 18.8257i 0
191.17 0 2.09257 − 4.75617i 0 5.00000i 0 − 32.2690i 0 −18.2423 − 19.9053i 0
191.18 0 2.09257 + 4.75617i 0 − 5.00000i 0 32.2690i 0 −18.2423 + 19.9053i 0
191.19 0 4.56698 − 2.47844i 0 − 5.00000i 0 20.9745i 0 14.7146 − 22.6380i 0
191.20 0 4.56698 + 2.47844i 0 5.00000i 0 − 20.9745i 0 14.7146 + 22.6380i 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
3.b odd 2 1 inner
4.b odd 2 1 inner
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 960.4.h.d 24
3.b odd 2 1 inner 960.4.h.d 24
4.b odd 2 1 inner 960.4.h.d 24
8.b even 2 1 60.4.e.a ✓ 24
8.d odd 2 1 60.4.e.a ✓ 24
12.b even 2 1 inner 960.4.h.d 24
24.f even 2 1 60.4.e.a ✓ 24
24.h odd 2 1 60.4.e.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.4.e.a ✓ 24 8.b even 2 1
60.4.e.a ✓ 24 8.d odd 2 1
60.4.e.a ✓ 24 24.f even 2 1
60.4.e.a ✓ 24 24.h odd 2 1
960.4.h.d 24 1.a even 1 1 trivial
960.4.h.d 24 3.b odd 2 1 inner
960.4.h.d 24 4.b odd 2 1 inner
960.4.h.d 24 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(960, [\chi])\):

\( T_{7}^{12} + 2622 T_{7}^{10} + 2472324 T_{7}^{8} + 1012910536 T_{7}^{6} + 163004962848 T_{7}^{4} + \cdots + 232398348800 \) Copy content Toggle raw display
\( T_{11}^{12} - 8028 T_{11}^{10} + 24000528 T_{11}^{8} - 34550751296 T_{11}^{6} + 25234949571072 T_{11}^{4} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display