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Properties

Label 528.6.o.b
Level $528$
Weight $6$
Character orbit 528.o
Analytic conductor $84.683$
Analytic rank $0$
Dimension $40$
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] = [528,6,Mod(175,528)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(528, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("528.175");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 528.o (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: \(84.6826568613\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 88 q^{5} - 3240 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 88 q^{5} - 3240 q^{9} + 5464 q^{25} + 6840 q^{33} + 4576 q^{37} - 7128 q^{45} + 58760 q^{49} - 165336 q^{53} + 16632 q^{69} + 28136 q^{77} + 262440 q^{81} - 304608 q^{89} + 102672 q^{93} + 15664 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
175.1 0 − 9.00000i 0 −34.2910 0 120.607 0 −81.0000 0
175.2 0 9.00000i 0 −34.2910 0 120.607 0 −81.0000 0
175.3 0 − 9.00000i 0 89.8188 0 112.074 0 −81.0000 0
175.4 0 9.00000i 0 89.8188 0 112.074 0 −81.0000 0
175.5 0 − 9.00000i 0 89.8188 0 −112.074 0 −81.0000 0
175.6 0 9.00000i 0 89.8188 0 −112.074 0 −81.0000 0
175.7 0 − 9.00000i 0 48.1562 0 32.4843 0 −81.0000 0
175.8 0 9.00000i 0 48.1562 0 32.4843 0 −81.0000 0
175.9 0 − 9.00000i 0 −76.1886 0 −138.380 0 −81.0000 0
175.10 0 9.00000i 0 −76.1886 0 −138.380 0 −81.0000 0
175.11 0 − 9.00000i 0 57.6896 0 131.273 0 −81.0000 0
175.12 0 9.00000i 0 57.6896 0 131.273 0 −81.0000 0
175.13 0 − 9.00000i 0 4.87261 0 −228.204 0 −81.0000 0
175.14 0 9.00000i 0 4.87261 0 −228.204 0 −81.0000 0
175.15 0 − 9.00000i 0 1.95371 0 −158.286 0 −81.0000 0
175.16 0 9.00000i 0 1.95371 0 −158.286 0 −81.0000 0
175.17 0 − 9.00000i 0 −84.4799 0 98.5983 0 −81.0000 0
175.18 0 9.00000i 0 −84.4799 0 98.5983 0 −81.0000 0
175.19 0 − 9.00000i 0 1.95371 0 158.286 0 −81.0000 0
175.20 0 9.00000i 0 1.95371 0 158.286 0 −81.0000 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 175.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
11.b odd 2 1 inner
44.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 528.6.o.b ✓ 40
4.b odd 2 1 inner 528.6.o.b ✓ 40
11.b odd 2 1 inner 528.6.o.b ✓ 40
44.c even 2 1 inner 528.6.o.b ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
528.6.o.b ✓ 40 1.a even 1 1 trivial
528.6.o.b ✓ 40 4.b odd 2 1 inner
528.6.o.b ✓ 40 11.b odd 2 1 inner
528.6.o.b ✓ 40 44.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} - 22 T_{5}^{9} - 16066 T_{5}^{8} + 342128 T_{5}^{7} + 84406964 T_{5}^{6} + \cdots + 11\!\cdots\!28 \) acting on \(S_{6}^{\mathrm{new}}(528, [\chi])\). Copy content Toggle raw display