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Properties

Label 968.2.c.c
Level $968$
Weight $2$
Character orbit 968.c
Analytic conductor $7.730$
Analytic rank $0$
Dimension $8$
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(485,968)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(968, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("968.485");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 968 = 2^{3} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 968.c (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: \(7.72951891566\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1212153856.10
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 10x^{4} - 16x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{3} - \beta_{2}) q^{3} + (\beta_{3} + 1) q^{4} + (\beta_{7} - \beta_{3} - \beta_{2}) q^{5} + (\beta_{5} - \beta_{4}) q^{6} + ( - \beta_{5} - 2 \beta_{4} - 2 \beta_1) q^{7} + ( - \beta_{6} - \beta_{4} - \beta_1) q^{8} + (2 \beta_{7} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{3} - \beta_{2}) q^{3} + (\beta_{3} + 1) q^{4} + (\beta_{7} - \beta_{3} - \beta_{2}) q^{5} + (\beta_{5} - \beta_{4}) q^{6} + ( - \beta_{5} - 2 \beta_{4} - 2 \beta_1) q^{7} + ( - \beta_{6} - \beta_{4} - \beta_1) q^{8} + (2 \beta_{7} - 2) q^{9} + (2 \beta_{6} - \beta_1) q^{10} + (3 \beta_{7} - \beta_{2} - 2) q^{12} + ( - \beta_{6} - 2 \beta_{5}) q^{13} + (3 \beta_{7} - \beta_{3} + \beta_{2} + 1) q^{14} + ( - 3 \beta_{7} - 1) q^{15} + (\beta_{7} + \beta_{3} + \beta_{2} - 1) q^{16} + ( - \beta_{6} - 2 \beta_{4} + 2 \beta_1) q^{17} + ( - 2 \beta_{5} - 2 \beta_{4}) q^{18} + ( - 2 \beta_{6} + \beta_{5}) q^{19} + (2 \beta_{7} - \beta_{3} - 2 \beta_{2} + 3) q^{20} + (3 \beta_{6} - 2 \beta_{5}) q^{21} + ( - 3 \beta_{7} + 1) q^{23} + (\beta_{6} - 2 \beta_{5} + \cdots - \beta_1) q^{24}+ \cdots + ( - 2 \beta_{5} - 2 \beta_{4} - 7 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{4} - 16 q^{9} - 16 q^{12} + 8 q^{14} - 8 q^{15} - 8 q^{16} + 24 q^{20} + 8 q^{23} - 16 q^{25} + 8 q^{26} - 40 q^{31} - 40 q^{34} - 24 q^{38} + 40 q^{42} - 16 q^{47} + 8 q^{48} + 40 q^{49} + 48 q^{56} - 32 q^{60} - 16 q^{64} + 8 q^{70} + 72 q^{71} + 88 q^{78} + 56 q^{80} - 24 q^{81} - 56 q^{82} - 32 q^{86} - 40 q^{89} - 16 q^{92} - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{6} + 10x^{4} - 16x^{2} + 16 \) : Copy content Toggle raw display

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

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{4} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + \beta_{3} + \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{6} + 2\beta_{5} + 2\beta_{4} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{3} + 4\beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2\beta_{6} + 4\beta_{5} - 2\beta_{4} - 2\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(485\) \(727\) \(849\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
485.1
1.36145 + 0.382683i
1.36145 − 0.382683i
1.07072 + 0.923880i
1.07072 − 0.923880i
−1.07072 + 0.923880i
−1.07072 − 0.923880i
−1.36145 + 0.382683i
−1.36145 − 0.382683i
−1.36145 − 0.382683i − 1.47363i 1.70711 + 1.04201i − 3.55765i −0.563932 + 2.00627i −3.85077 −1.92538 − 2.07193i 0.828427 −1.36145 + 4.84357i
485.2 −1.36145 + 0.382683i 1.47363i 1.70711 − 1.04201i 3.55765i −0.563932 − 2.00627i −3.85077 −1.92538 + 2.07193i 0.828427 −1.36145 − 4.84357i
485.3 −1.07072 − 0.923880i 2.79793i 0.292893 + 1.97844i − 1.15894i 2.58495 − 2.99581i 3.02846 1.51423 − 2.38896i −4.82843 −1.07072 + 1.24090i
485.4 −1.07072 + 0.923880i − 2.79793i 0.292893 − 1.97844i 1.15894i 2.58495 + 2.99581i 3.02846 1.51423 + 2.38896i −4.82843 −1.07072 − 1.24090i
485.5 1.07072 − 0.923880i − 2.79793i 0.292893 − 1.97844i 1.15894i −2.58495 − 2.99581i −3.02846 −1.51423 − 2.38896i −4.82843 1.07072 + 1.24090i
485.6 1.07072 + 0.923880i 2.79793i 0.292893 + 1.97844i − 1.15894i −2.58495 + 2.99581i −3.02846 −1.51423 + 2.38896i −4.82843 1.07072 − 1.24090i
485.7 1.36145 − 0.382683i 1.47363i 1.70711 − 1.04201i 3.55765i 0.563932 + 2.00627i 3.85077 1.92538 − 2.07193i 0.828427 1.36145 + 4.84357i
485.8 1.36145 + 0.382683i − 1.47363i 1.70711 + 1.04201i − 3.55765i 0.563932 − 2.00627i 3.85077 1.92538 + 2.07193i 0.828427 1.36145 − 4.84357i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 485.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
11.b odd 2 1 inner
88.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 968.2.c.c ✓ 8
4.b odd 2 1 3872.2.c.c 8
8.b even 2 1 inner 968.2.c.c ✓ 8
8.d odd 2 1 3872.2.c.c 8
11.b odd 2 1 inner 968.2.c.c ✓ 8
11.c even 5 4 968.2.o.c 32
11.d odd 10 4 968.2.o.c 32
44.c even 2 1 3872.2.c.c 8
88.b odd 2 1 inner 968.2.c.c ✓ 8
88.g even 2 1 3872.2.c.c 8
88.o even 10 4 968.2.o.c 32
88.p odd 10 4 968.2.o.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
968.2.c.c ✓ 8 1.a even 1 1 trivial
968.2.c.c ✓ 8 8.b even 2 1 inner
968.2.c.c ✓ 8 11.b odd 2 1 inner
968.2.c.c ✓ 8 88.b odd 2 1 inner
968.2.o.c 32 11.c even 5 4
968.2.o.c 32 11.d odd 10 4
968.2.o.c 32 88.o even 10 4
968.2.o.c 32 88.p odd 10 4
3872.2.c.c 8 4.b odd 2 1
3872.2.c.c 8 8.d odd 2 1
3872.2.c.c 8 44.c even 2 1
3872.2.c.c 8 88.g even 2 1

Hecke kernels

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

\( T_{3}^{4} + 10T_{3}^{2} + 17 \) Copy content Toggle raw display
\( T_{5}^{4} + 14T_{5}^{2} + 17 \) Copy content Toggle raw display
\( T_{7}^{4} - 24T_{7}^{2} + 136 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 4 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( (T^{4} + 10 T^{2} + 17)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 14 T^{2} + 17)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 24 T^{2} + 136)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} + 40 T^{2} + 392)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 56 T^{2} + 136)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 40 T^{2} + 392)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 2 T - 17)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 64 T^{2} + 512)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 10 T + 23)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 46 T^{2} + 17)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 88 T^{2} + 136)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 64 T^{2} + 512)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 4 T - 4)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} + 80 T^{2} + 1088)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 58 T^{2} + 833)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 32 T^{2} + 128)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 10 T^{2} + 17)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 18 T + 79)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} - 88 T^{2} + 136)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 224 T^{2} + 2176)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 160 T^{2} + 128)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 10 T - 7)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 6 T - 191)^{4} \) Copy content Toggle raw display
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