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Properties

Label 960.4.h.e
Level $960$
Weight $4$
Character orbit 960.h
Analytic conductor $56.642$
Analytic rank $0$
Dimension $24$
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] = [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 480)
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
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 20 q^{9} + 72 q^{11} + 72 q^{13} - 20 q^{15} + 68 q^{21} - 96 q^{23} - 600 q^{25} + 168 q^{27} - 80 q^{33} + 504 q^{37} + 456 q^{39} + 220 q^{45} - 432 q^{47} - 816 q^{49} + 1240 q^{51} + 40 q^{57} - 2184 q^{59} + 1080 q^{61} + 1720 q^{63} + 1548 q^{69} - 816 q^{71} - 1440 q^{73} - 2304 q^{81} - 2016 q^{83} + 2144 q^{87} + 112 q^{93} - 1800 q^{95} + 720 q^{97} + 3160 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1 0 −4.92143 − 1.66721i 0 − 5.00000i 0 − 15.5345i 0 21.4409 + 16.4101i 0
191.2 0 −4.92143 + 1.66721i 0 5.00000i 0 15.5345i 0 21.4409 − 16.4101i 0
191.3 0 −4.47955 − 2.63318i 0 5.00000i 0 − 30.6962i 0 13.1328 + 23.5909i 0
191.4 0 −4.47955 + 2.63318i 0 − 5.00000i 0 30.6962i 0 13.1328 − 23.5909i 0
191.5 0 −4.12770 − 3.15628i 0 5.00000i 0 3.91168i 0 7.07583 + 26.0563i 0
191.6 0 −4.12770 + 3.15628i 0 − 5.00000i 0 − 3.91168i 0 7.07583 − 26.0563i 0
191.7 0 −3.76455 − 3.58164i 0 − 5.00000i 0 30.2049i 0 1.34373 + 26.9665i 0
191.8 0 −3.76455 + 3.58164i 0 5.00000i 0 − 30.2049i 0 1.34373 − 26.9665i 0
191.9 0 −2.67023 − 4.45756i 0 − 5.00000i 0 − 28.8929i 0 −12.7397 + 23.8055i 0
191.10 0 −2.67023 + 4.45756i 0 5.00000i 0 28.8929i 0 −12.7397 − 23.8055i 0
191.11 0 −0.637007 − 5.15696i 0 5.00000i 0 26.9030i 0 −26.1884 + 6.57004i 0
191.12 0 −0.637007 + 5.15696i 0 − 5.00000i 0 − 26.9030i 0 −26.1884 − 6.57004i 0
191.13 0 0.991849 − 5.10061i 0 − 5.00000i 0 − 7.56208i 0 −25.0325 − 10.1181i 0
191.14 0 0.991849 + 5.10061i 0 5.00000i 0 7.56208i 0 −25.0325 + 10.1181i 0
191.15 0 2.28458 − 4.66698i 0 − 5.00000i 0 8.51421i 0 −16.5614 − 21.3242i 0
191.16 0 2.28458 + 4.66698i 0 5.00000i 0 − 8.51421i 0 −16.5614 + 21.3242i 0
191.17 0 3.26404 − 4.04302i 0 5.00000i 0 − 14.4221i 0 −5.69203 − 26.3932i 0
191.18 0 3.26404 + 4.04302i 0 − 5.00000i 0 14.4221i 0 −5.69203 + 26.3932i 0
191.19 0 3.78906 − 3.55571i 0 5.00000i 0 13.3637i 0 1.71392 − 26.9455i 0
191.20 0 3.78906 + 3.55571i 0 − 5.00000i 0 − 13.3637i 0 1.71392 + 26.9455i 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 960.4.h.e 24
3.b odd 2 1 960.4.h.c 24
4.b odd 2 1 960.4.h.c 24
8.b even 2 1 480.4.h.a ✓ 24
8.d odd 2 1 480.4.h.b yes 24
12.b even 2 1 inner 960.4.h.e 24
24.f even 2 1 480.4.h.a ✓ 24
24.h odd 2 1 480.4.h.b yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.4.h.a ✓ 24 8.b even 2 1
480.4.h.a ✓ 24 24.f even 2 1
480.4.h.b yes 24 8.d odd 2 1
480.4.h.b yes 24 24.h odd 2 1
960.4.h.c 24 3.b odd 2 1
960.4.h.c 24 4.b odd 2 1
960.4.h.e 24 1.a even 1 1 trivial
960.4.h.e 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 \(S_{4}^{\mathrm{new}}(960, [\chi])\):

\( T_{7}^{24} + 4524 T_{7}^{22} + 8643276 T_{7}^{20} + 9110596992 T_{7}^{18} + 5827185951024 T_{7}^{16} + \cdots + 25\!\cdots\!64 \) Copy content Toggle raw display
\( T_{11}^{12} - 36 T_{11}^{11} - 7380 T_{11}^{10} + 150976 T_{11}^{9} + 20504016 T_{11}^{8} + \cdots + 70596020436992 \) Copy content Toggle raw display