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Properties

Label 968.2.k.a
Level $968$
Weight $2$
Character orbit 968.k
Analytic conductor $7.730$
Analytic rank $0$
Dimension $8$
CM discriminant -8
Inner twists $16$

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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: \(\Z[a_1, \ldots, a_{9}]\)
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.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} - 2 \beta_{6} q^{3} - 2 \beta_{4} q^{4} - 2 \beta_{3} q^{6} - 2 \beta_1 q^{8} - \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{2} - 2 \beta_{6} q^{3} - 2 \beta_{4} q^{4} - 2 \beta_{3} q^{6} - 2 \beta_1 q^{8} - \beta_{2} q^{9} - 4 q^{12} + (4 \beta_{6} - 4 \beta_{4} + 4 \beta_{2} - 4) q^{16} - 4 \beta_{3} q^{17} + (\beta_{7} - \beta_{5} + \beta_{3} - \beta_1) q^{18} - 6 \beta_1 q^{19} + 4 \beta_{7} q^{24} - 5 \beta_{6} q^{25} + ( - 4 \beta_{6} + 4 \beta_{4} + \cdots + 4) q^{27}+ \cdots - 7 \beta_{5} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} + 4 q^{4} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{3} + 4 q^{4} - 2 q^{9} - 32 q^{12} - 8 q^{16} - 10 q^{25} + 8 q^{27} - 64 q^{34} + 4 q^{36} - 24 q^{38} - 16 q^{48} + 14 q^{49} + 12 q^{59} + 16 q^{64} + 112 q^{67} - 20 q^{75} + 22 q^{81} + 32 q^{82} + 24 q^{86} + 144 q^{89} - 20 q^{97}+O(q^{100}) \) 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 \(\left(\mathbb{Z}/968\mathbb{Z}\right)^\times\).

\(n\) \(485\) \(727\) \(849\)
\(\chi(n)\) \(-1\) \(-1\) \(-\beta_{4}\)

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 0.618034 − 1.90211i −0.618034 − 1.90211i 0 1.66251 + 2.28825i 0 2.68999 + 0.874032i −0.809017 − 0.587785i 0
403.2 0.831254 − 1.14412i 0.618034 − 1.90211i −0.618034 − 1.90211i 0 −1.66251 − 2.28825i 0 −2.68999 − 0.874032i −0.809017 − 0.587785i 0
475.1 −1.34500 − 0.437016i −1.61803 + 1.17557i 1.61803 + 1.17557i 0 2.68999 − 0.874032i 0 −1.66251 − 2.28825i 0.309017 − 0.951057i 0
475.2 1.34500 + 0.437016i −1.61803 + 1.17557i 1.61803 + 1.17557i 0 −2.68999 + 0.874032i 0 1.66251 + 2.28825i 0.309017 − 0.951057i 0
699.1 −1.34500 + 0.437016i −1.61803 − 1.17557i 1.61803 − 1.17557i 0 2.68999 + 0.874032i 0 −1.66251 + 2.28825i 0.309017 + 0.951057i 0
699.2 1.34500 − 0.437016i −1.61803 − 1.17557i 1.61803 − 1.17557i 0 −2.68999 − 0.874032i 0 1.66251 − 2.28825i 0.309017 + 0.951057i 0
723.1 −0.831254 − 1.14412i 0.618034 + 1.90211i −0.618034 + 1.90211i 0 1.66251 − 2.28825i 0 2.68999 − 0.874032i −0.809017 + 0.587785i 0
723.2 0.831254 + 1.14412i 0.618034 + 1.90211i −0.618034 + 1.90211i 0 −1.66251 + 2.28825i 0 −2.68999 + 0.874032i −0.809017 + 0.587785i 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.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
88.g even 2 1 inner
88.k even 10 3 inner
88.l odd 10 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 968.2.k.a 8
8.d odd 2 1 CM 968.2.k.a 8
11.b odd 2 1 inner 968.2.k.a 8
11.c even 5 1 88.2.g.a ✓ 2
11.c even 5 3 inner 968.2.k.a 8
11.d odd 10 1 88.2.g.a ✓ 2
11.d odd 10 3 inner 968.2.k.a 8
33.f even 10 1 792.2.h.b 2
33.h odd 10 1 792.2.h.b 2
44.g even 10 1 352.2.g.a 2
44.h odd 10 1 352.2.g.a 2
88.g even 2 1 inner 968.2.k.a 8
88.k even 10 1 88.2.g.a ✓ 2
88.k even 10 3 inner 968.2.k.a 8
88.l odd 10 1 88.2.g.a ✓ 2
88.l odd 10 3 inner 968.2.k.a 8
88.o even 10 1 352.2.g.a 2
88.p odd 10 1 352.2.g.a 2
132.n odd 10 1 3168.2.h.b 2
132.o even 10 1 3168.2.h.b 2
176.u odd 20 2 2816.2.e.d 4
176.v odd 20 2 2816.2.e.d 4
176.w even 20 2 2816.2.e.d 4
176.x even 20 2 2816.2.e.d 4
264.r odd 10 1 792.2.h.b 2
264.t odd 10 1 3168.2.h.b 2
264.u even 10 1 3168.2.h.b 2
264.w even 10 1 792.2.h.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.2.g.a ✓ 2 11.c even 5 1
88.2.g.a ✓ 2 11.d odd 10 1
88.2.g.a ✓ 2 88.k even 10 1
88.2.g.a ✓ 2 88.l odd 10 1
352.2.g.a 2 44.g even 10 1
352.2.g.a 2 44.h odd 10 1
352.2.g.a 2 88.o even 10 1
352.2.g.a 2 88.p odd 10 1
792.2.h.b 2 33.f even 10 1
792.2.h.b 2 33.h odd 10 1
792.2.h.b 2 264.r odd 10 1
792.2.h.b 2 264.w even 10 1
968.2.k.a 8 1.a even 1 1 trivial
968.2.k.a 8 8.d odd 2 1 CM
968.2.k.a 8 11.b odd 2 1 inner
968.2.k.a 8 11.c even 5 3 inner
968.2.k.a 8 11.d odd 10 3 inner
968.2.k.a 8 88.g even 2 1 inner
968.2.k.a 8 88.k even 10 3 inner
968.2.k.a 8 88.l odd 10 3 inner
2816.2.e.d 4 176.u odd 20 2
2816.2.e.d 4 176.v odd 20 2
2816.2.e.d 4 176.w even 20 2
2816.2.e.d 4 176.x even 20 2
3168.2.h.b 2 132.n odd 10 1
3168.2.h.b 2 132.o even 10 1
3168.2.h.b 2 264.t odd 10 1
3168.2.h.b 2 264.u even 10 1

Hecke kernels

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

\( T_{3}^{4} + 2T_{3}^{3} + 4T_{3}^{2} + 8T_{3} + 16 \) Copy content Toggle raw display
\( T_{17}^{8} - 32T_{17}^{6} + 1024T_{17}^{4} - 32768T_{17}^{2} + 1048576 \) 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
$3$ \( (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 16)^{2} \) 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
$17$ \( T^{8} - 32 T^{6} + \cdots + 1048576 \) Copy content Toggle raw display
$19$ \( T^{8} - 72 T^{6} + \cdots + 26873856 \) 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
$41$ \( T^{8} - 128 T^{6} + \cdots + 268435456 \) Copy content Toggle raw display
$43$ \( (T^{2} + 72)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{4} - 6 T^{3} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( (T - 14)^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 6879707136 \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} - 8 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$89$ \( (T - 18)^{8} \) Copy content Toggle raw display
$97$ \( (T^{4} + 10 T^{3} + \cdots + 10000)^{2} \) Copy content Toggle raw display
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