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Properties

Label 608.6.b.b
Level $608$
Weight $6$
Character orbit 608.b
Analytic conductor $97.513$
Analytic rank $0$
Dimension $96$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [608,6,Mod(303,608)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(608, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("608.303");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 608 = 2^{5} \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 608.b (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: \(97.5133624463\)
Analytic rank: \(0\)
Dimension: \(96\)
Twist minimal: no (minimal twist has level 152)
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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 6168 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q - 6168 q^{9} - 944 q^{11} - 3832 q^{17} - 5240 q^{19} - 62504 q^{25} - 7720 q^{35} - 45096 q^{43} - 210840 q^{49} - 36336 q^{57} - 4336 q^{73} - 20624 q^{81} - 52152 q^{83} + 752768 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
303.1 0 − 28.5036i 0 − 107.944i 0 87.6251i 0 −569.454 0
303.2 0 − 28.5036i 0 107.944i 0 − 87.6251i 0 −569.454 0
303.3 0 − 27.1983i 0 − 36.9812i 0 150.607i 0 −496.748 0
303.4 0 − 27.1983i 0 36.9812i 0 − 150.607i 0 −496.748 0
303.5 0 − 26.7990i 0 − 21.0026i 0 − 145.610i 0 −475.186 0
303.6 0 − 26.7990i 0 21.0026i 0 145.610i 0 −475.186 0
303.7 0 − 26.7852i 0 − 46.5237i 0 − 208.957i 0 −474.446 0
303.8 0 − 26.7852i 0 46.5237i 0 208.957i 0 −474.446 0
303.9 0 − 25.5985i 0 − 73.7438i 0 50.3974i 0 −412.283 0
303.10 0 − 25.5985i 0 73.7438i 0 − 50.3974i 0 −412.283 0
303.11 0 − 23.2069i 0 − 79.1121i 0 − 208.766i 0 −295.559 0
303.12 0 − 23.2069i 0 79.1121i 0 208.766i 0 −295.559 0
303.13 0 − 22.8051i 0 − 24.0756i 0 157.637i 0 −277.072 0
303.14 0 − 22.8051i 0 24.0756i 0 − 157.637i 0 −277.072 0
303.15 0 − 20.4507i 0 − 36.1821i 0 − 13.6727i 0 −175.233 0
303.16 0 − 20.4507i 0 36.1821i 0 13.6727i 0 −175.233 0
303.17 0 − 19.3062i 0 − 63.3569i 0 60.7576i 0 −129.731 0
303.18 0 − 19.3062i 0 63.3569i 0 − 60.7576i 0 −129.731 0
303.19 0 − 19.2136i 0 − 61.7747i 0 34.9784i 0 −126.164 0
303.20 0 − 19.2136i 0 61.7747i 0 − 34.9784i 0 −126.164 0
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 303.96
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
19.b odd 2 1 inner
152.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 608.6.b.b 96
4.b odd 2 1 152.6.b.b ✓ 96
8.b even 2 1 152.6.b.b ✓ 96
8.d odd 2 1 inner 608.6.b.b 96
19.b odd 2 1 inner 608.6.b.b 96
76.d even 2 1 152.6.b.b ✓ 96
152.b even 2 1 inner 608.6.b.b 96
152.g odd 2 1 152.6.b.b ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.6.b.b ✓ 96 4.b odd 2 1
152.6.b.b ✓ 96 8.b even 2 1
152.6.b.b ✓ 96 76.d even 2 1
152.6.b.b ✓ 96 152.g odd 2 1
608.6.b.b 96 1.a even 1 1 trivial
608.6.b.b 96 8.d odd 2 1 inner
608.6.b.b 96 19.b odd 2 1 inner
608.6.b.b 96 152.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} + 7374 T_{3}^{46} + 25211281 T_{3}^{44} + 53082818500 T_{3}^{42} + 77113215958634 T_{3}^{40} + \cdots + 11\!\cdots\!84 \) acting on \(S_{6}^{\mathrm{new}}(608, [\chi])\). Copy content Toggle raw display