Properties

Label 968.2.c.b
Level $968$
Weight $2$
Character orbit 968.c
Analytic conductor $7.730$
Analytic rank $0$
Dimension $4$
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] = [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: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + ( - \beta_{2} - 1) q^{4} + ( - 2 \beta_{2} - 1) q^{5} + (2 \beta_{3} - 2 \beta_1) q^{7} + (\beta_{3} - 2 \beta_1) q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + ( - \beta_{2} - 1) q^{4} + ( - 2 \beta_{2} - 1) q^{5} + (2 \beta_{3} - 2 \beta_1) q^{7} + (\beta_{3} - 2 \beta_1) q^{8} + 3 q^{9} + (\beta_{3} - 4 \beta_1) q^{10} + ( - \beta_{3} - \beta_1) q^{13} + (2 \beta_{2} - 2) q^{14} + (\beta_{2} - 3) q^{16} + (\beta_{3} - \beta_1) q^{17} - 3 \beta_{3} q^{18} + (2 \beta_{3} + 2 \beta_1) q^{19} + (\beta_{2} - 7) q^{20} + 8 q^{23} - 10 q^{25} + ( - \beta_{2} - 3) q^{26} + (2 \beta_{3} + 4 \beta_1) q^{28} + ( - 3 \beta_{3} - 3 \beta_1) q^{29} - 6 q^{31} + (3 \beta_{3} + 2 \beta_1) q^{32} + (\beta_{2} - 1) q^{34} + (6 \beta_{3} + 6 \beta_1) q^{35} + ( - 3 \beta_{2} - 3) q^{36} + (2 \beta_{2} + 1) q^{37} + (2 \beta_{2} + 6) q^{38} + (7 \beta_{3} + 2 \beta_1) q^{40} + (\beta_{3} - \beta_1) q^{41} + (2 \beta_{3} + 2 \beta_1) q^{43} + ( - 6 \beta_{2} - 3) q^{45} - 8 \beta_{3} q^{46} + 4 q^{47} + 5 q^{49} + 10 \beta_{3} q^{50} + (3 \beta_{3} - 2 \beta_1) q^{52} + ( - 2 \beta_{2} - 1) q^{53} + (2 \beta_{2} + 10) q^{56} + ( - 3 \beta_{2} - 9) q^{58} + ( - 4 \beta_{2} - 2) q^{59} + ( - 2 \beta_{3} - 2 \beta_1) q^{61} + 6 \beta_{3} q^{62} + (6 \beta_{3} - 6 \beta_1) q^{63} + (3 \beta_{2} + 7) q^{64} + (5 \beta_{3} - 5 \beta_1) q^{65} + ( - 4 \beta_{2} - 2) q^{67} + (\beta_{3} + 2 \beta_1) q^{68} + (6 \beta_{2} + 18) q^{70} - 2 q^{71} + (3 \beta_{3} - 6 \beta_1) q^{72} + ( - 4 \beta_{3} + 4 \beta_1) q^{73} + ( - \beta_{3} + 4 \beta_1) q^{74} + ( - 6 \beta_{3} + 4 \beta_1) q^{76} + (7 \beta_{2} + 11) q^{80} + 9 q^{81} + (\beta_{2} - 1) q^{82} + ( - 6 \beta_{3} - 6 \beta_1) q^{83} + (3 \beta_{3} + 3 \beta_1) q^{85} + (2 \beta_{2} + 6) q^{86} - 7 q^{89} + (3 \beta_{3} - 12 \beta_1) q^{90} + (4 \beta_{2} + 2) q^{91} + ( - 8 \beta_{2} - 8) q^{92} - 4 \beta_{3} q^{94} + ( - 10 \beta_{3} + 10 \beta_1) q^{95} - 9 q^{97} - 5 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{4} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{4} + 12 q^{9} - 12 q^{14} - 14 q^{16} - 30 q^{20} + 32 q^{23} - 40 q^{25} - 10 q^{26} - 24 q^{31} - 6 q^{34} - 6 q^{36} + 20 q^{38} + 16 q^{47} + 20 q^{49} + 36 q^{56} - 30 q^{58} + 22 q^{64} + 60 q^{70} - 8 q^{71} + 30 q^{80} + 36 q^{81} - 6 q^{82} + 20 q^{86} - 28 q^{89} - 16 q^{92} - 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

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

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
−0.866025 + 1.11803i
−0.866025 1.11803i
0.866025 + 1.11803i
0.866025 1.11803i
−0.866025 1.11803i 0 −0.500000 + 1.93649i 3.87298i 0 3.46410 2.59808 1.11803i 3.00000 4.33013 3.35410i
485.2 −0.866025 + 1.11803i 0 −0.500000 1.93649i 3.87298i 0 3.46410 2.59808 + 1.11803i 3.00000 4.33013 + 3.35410i
485.3 0.866025 1.11803i 0 −0.500000 1.93649i 3.87298i 0 −3.46410 −2.59808 1.11803i 3.00000 −4.33013 3.35410i
485.4 0.866025 + 1.11803i 0 −0.500000 + 1.93649i 3.87298i 0 −3.46410 −2.59808 + 1.11803i 3.00000 −4.33013 + 3.35410i
\(n\): e.g. 2-40 or 990-1000
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.b 4
4.b odd 2 1 3872.2.c.b 4
8.b even 2 1 inner 968.2.c.b 4
8.d odd 2 1 3872.2.c.b 4
11.b odd 2 1 inner 968.2.c.b 4
11.c even 5 4 968.2.o.b 16
11.d odd 10 4 968.2.o.b 16
44.c even 2 1 3872.2.c.b 4
88.b odd 2 1 inner 968.2.c.b 4
88.g even 2 1 3872.2.c.b 4
88.o even 10 4 968.2.o.b 16
88.p odd 10 4 968.2.o.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
968.2.c.b 4 1.a even 1 1 trivial
968.2.c.b 4 8.b even 2 1 inner
968.2.c.b 4 11.b odd 2 1 inner
968.2.c.b 4 88.b odd 2 1 inner
968.2.o.b 16 11.c even 5 4
968.2.o.b 16 11.d odd 10 4
968.2.o.b 16 88.o even 10 4
968.2.o.b 16 88.p odd 10 4
3872.2.c.b 4 4.b odd 2 1
3872.2.c.b 4 8.d odd 2 1
3872.2.c.b 4 44.c even 2 1
3872.2.c.b 4 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 \(S_{2}^{\mathrm{new}}(968, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5}^{2} + 15 \) Copy content Toggle raw display
\( T_{7}^{2} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 15)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$23$ \( (T - 8)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 45)^{2} \) Copy content Toggle raw display
$31$ \( (T + 6)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 15)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$47$ \( (T - 4)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 15)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 60)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 60)^{2} \) Copy content Toggle raw display
$71$ \( (T + 2)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 180)^{2} \) Copy content Toggle raw display
$89$ \( (T + 7)^{4} \) Copy content Toggle raw display
$97$ \( (T + 9)^{4} \) Copy content Toggle raw display
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