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Properties

Label 968.2.k.c
Level $968$
Weight $2$
Character orbit 968.k
Analytic conductor $7.730$
Analytic rank $0$
Dimension $8$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [968,2,Mod(403,968)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(968, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 5, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("968.403");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 968 = 2^{3} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 968.k (of order \(10\), degree \(4\), 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: \(7.72951891566\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: 8.0.64000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 88)
Sato-Tate group: $\mathrm{U}(1)[D_{10}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + (\beta_{6} + \beta_{5} + \cdots + \beta_{2}) q^{3}+ \cdots + (4 \beta_{6} + 2 \beta_{5} - 3 \beta_{4} + \cdots - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + (\beta_{6} + \beta_{5} + \cdots + \beta_{2}) q^{3}+ \cdots + 7 \beta_{5} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{3} + 4 q^{4} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6 q^{3} + 4 q^{4} - 12 q^{9} + 8 q^{12} - 8 q^{16} - 30 q^{17} - 10 q^{25} - 42 q^{27} + 16 q^{34} + 4 q^{36} + 36 q^{38} - 30 q^{41} + 24 q^{48} + 14 q^{49} + 10 q^{51} + 50 q^{57} + 12 q^{59} + 16 q^{64} - 28 q^{67} - 60 q^{68} + 80 q^{72} + 10 q^{73} - 20 q^{75} - 48 q^{81} + 32 q^{82} + 24 q^{86} - 36 q^{89} + 80 q^{96} - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 8 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{7} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/968\mathbb{Z}\right)^\times\).

\(n\) \(485\) \(727\) \(849\)
\(\chi(n)\) \(-1\) \(-1\) \(\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
403.1
−1.34500 + 0.437016i
1.34500 − 0.437016i
0.831254 − 1.14412i
−0.831254 + 1.14412i
0.831254 + 1.14412i
−0.831254 − 1.14412i
−1.34500 − 0.437016i
1.34500 + 0.437016i
−0.831254 + 1.14412i −0.640271 + 1.97055i −0.618034 − 1.90211i 0 −1.72232 − 2.37058i 0 2.68999 + 0.874032i −1.04607 − 0.760017i 0
403.2 0.831254 − 1.14412i 1.02224 − 3.14612i −0.618034 − 1.90211i 0 −2.74981 − 3.78479i 0 −2.68999 − 0.874032i −6.42606 − 4.66881i 0
475.1 −1.34500 − 0.437016i −0.0359800 + 0.0261410i 1.61803 + 1.17557i 0 0.0598171 − 0.0194357i 0 −1.66251 − 2.28825i −0.926440 + 2.85129i 0
475.2 1.34500 + 0.437016i 2.65401 − 1.92825i 1.61803 + 1.17557i 0 4.41232 − 1.43365i 0 1.66251 + 2.28825i 2.39858 − 7.38206i 0
699.1 −1.34500 + 0.437016i −0.0359800 − 0.0261410i 1.61803 − 1.17557i 0 0.0598171 + 0.0194357i 0 −1.66251 + 2.28825i −0.926440 − 2.85129i 0
699.2 1.34500 − 0.437016i 2.65401 + 1.92825i 1.61803 − 1.17557i 0 4.41232 + 1.43365i 0 1.66251 − 2.28825i 2.39858 + 7.38206i 0
723.1 −0.831254 − 1.14412i −0.640271 − 1.97055i −0.618034 + 1.90211i 0 −1.72232 + 2.37058i 0 2.68999 − 0.874032i −1.04607 + 0.760017i 0
723.2 0.831254 + 1.14412i 1.02224 + 3.14612i −0.618034 + 1.90211i 0 −2.74981 + 3.78479i 0 −2.68999 + 0.874032i −6.42606 + 4.66881i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 403.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
11.d odd 10 1 inner
88.k even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 968.2.k.c 8
8.d odd 2 1 CM 968.2.k.c 8
11.b odd 2 1 968.2.k.d 8
11.c even 5 1 88.2.k.a ✓ 8
11.c even 5 1 968.2.g.a 8
11.c even 5 1 968.2.k.b 8
11.c even 5 1 968.2.k.d 8
11.d odd 10 1 88.2.k.a ✓ 8
11.d odd 10 1 968.2.g.a 8
11.d odd 10 1 968.2.k.b 8
11.d odd 10 1 inner 968.2.k.c 8
33.f even 10 1 792.2.bp.a 8
33.h odd 10 1 792.2.bp.a 8
44.g even 10 1 352.2.s.a 8
44.g even 10 1 3872.2.g.b 8
44.h odd 10 1 352.2.s.a 8
44.h odd 10 1 3872.2.g.b 8
88.g even 2 1 968.2.k.d 8
88.k even 10 1 88.2.k.a ✓ 8
88.k even 10 1 968.2.g.a 8
88.k even 10 1 968.2.k.b 8
88.k even 10 1 inner 968.2.k.c 8
88.l odd 10 1 88.2.k.a ✓ 8
88.l odd 10 1 968.2.g.a 8
88.l odd 10 1 968.2.k.b 8
88.l odd 10 1 968.2.k.d 8
88.o even 10 1 352.2.s.a 8
88.o even 10 1 3872.2.g.b 8
88.p odd 10 1 352.2.s.a 8
88.p odd 10 1 3872.2.g.b 8
264.r odd 10 1 792.2.bp.a 8
264.w even 10 1 792.2.bp.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.2.k.a ✓ 8 11.c even 5 1
88.2.k.a ✓ 8 11.d odd 10 1
88.2.k.a ✓ 8 88.k even 10 1
88.2.k.a ✓ 8 88.l odd 10 1
352.2.s.a 8 44.g even 10 1
352.2.s.a 8 44.h odd 10 1
352.2.s.a 8 88.o even 10 1
352.2.s.a 8 88.p odd 10 1
792.2.bp.a 8 33.f even 10 1
792.2.bp.a 8 33.h odd 10 1
792.2.bp.a 8 264.r odd 10 1
792.2.bp.a 8 264.w even 10 1
968.2.g.a 8 11.c even 5 1
968.2.g.a 8 11.d odd 10 1
968.2.g.a 8 88.k even 10 1
968.2.g.a 8 88.l odd 10 1
968.2.k.b 8 11.c even 5 1
968.2.k.b 8 11.d odd 10 1
968.2.k.b 8 88.k even 10 1
968.2.k.b 8 88.l odd 10 1
968.2.k.c 8 1.a even 1 1 trivial
968.2.k.c 8 8.d odd 2 1 CM
968.2.k.c 8 11.d odd 10 1 inner
968.2.k.c 8 88.k even 10 1 inner
968.2.k.d 8 11.b odd 2 1
968.2.k.d 8 11.c even 5 1
968.2.k.d 8 88.g even 2 1
968.2.k.d 8 88.l odd 10 1
3872.2.g.b 8 44.g even 10 1
3872.2.g.b 8 44.h odd 10 1
3872.2.g.b 8 88.o even 10 1
3872.2.g.b 8 88.p odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(968, [\chi])\):

\( T_{3}^{8} - 6T_{3}^{7} + 27T_{3}^{6} - 68T_{3}^{5} + 150T_{3}^{4} - 182T_{3}^{3} + 492T_{3}^{2} + 36T_{3} + 1 \) Copy content Toggle raw display
\( T_{17}^{8} + 30 T_{17}^{7} + 413 T_{17}^{6} + 3240 T_{17}^{5} + 14514 T_{17}^{4} + 31470 T_{17}^{3} + \cdots + 63001 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 2 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{8} - 6 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} + 30 T^{7} + \cdots + 63001 \) Copy content Toggle raw display
$19$ \( T^{8} - 72 T^{6} + \cdots + 22201 \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} + 30 T^{7} + \cdots + 2105401 \) Copy content Toggle raw display
$43$ \( T^{8} + 358 T^{6} + \cdots + 1104601 \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} - 12 T^{7} + \cdots + 65302561 \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} + 14 T^{3} + \cdots - 4799)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} - 10 T^{7} + \cdots + 19971961 \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} - 8 T^{6} + \cdots + 972753721 \) Copy content Toggle raw display
$89$ \( (T^{4} + 18 T^{3} + \cdots + 401)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 20 T^{7} + \cdots + 73017025 \) Copy content Toggle raw display
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