↑

Properties

Label 961.2.c.h
Level $961$
Weight $2$
Character orbit 961.c
Analytic conductor $7.674$
Analytic rank $0$
Dimension $8$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [961,2,Mod(439,961)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("961.439"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(961, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 961 = 31^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 961.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,12,0,12,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.67362363425\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.207360000.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 6x^{6} + 32x^{4} + 24x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 2) q^{2} - \beta_1 q^{3} + ( - 3 \beta_{2} + 3) q^{4} + ( - 2 \beta_{6} + \beta_{4} - 1) q^{5} + (\beta_{3} - \beta_1) q^{6} + \beta_{4} q^{7} + ( - 4 \beta_{2} + 5) q^{8} + (2 \beta_{6} - \beta_{4} + 1) q^{9}+ \cdots + (3 \beta_{3} - 6 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{2} + 12 q^{4} + 4 q^{7} + 24 q^{8} - 10 q^{10} + 6 q^{14} + 52 q^{16} + 10 q^{18} - 4 q^{19} - 30 q^{20} + 6 q^{28} + 60 q^{32} + 24 q^{33} + 30 q^{36} - 6 q^{38} + 72 q^{39} - 40 q^{40} - 12 q^{41}+ \cdots + 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 6x^{6} + 32x^{4} + 24x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} - 40 ) / 64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 104\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{6} + 16\nu^{4} + 96\nu^{2} + 72 ) / 64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{7} + 16\nu^{5} + 96\nu^{3} + 72\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{6} - 16\nu^{4} - 64\nu^{2} - 8 ) / 64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{7} - 16\nu^{5} - 80\nu^{3} - 8\nu ) / 32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{6} + 2\beta_{4} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} + 4\beta_{5} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -12\beta_{6} - 8\beta_{4} - 12\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -12\beta_{7} - 20\beta_{5} - 12\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 64\beta_{2} + 40 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 64\beta_{3} + 104\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/961\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1 + \beta_{4}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
439.1
1.14412 − 1.98168i
−1.14412 + 1.98168i
0.437016 − 0.756934i
−0.437016 + 0.756934i
1.14412 + 1.98168i
−1.14412 − 1.98168i
0.437016 + 0.756934i
−0.437016 − 0.756934i
0.381966 −1.14412 + 1.98168i −1.85410 1.11803 + 1.93649i −0.437016 + 0.756934i 0.500000 − 0.866025i −1.47214 −1.11803 − 1.93649i 0.427051 + 0.739674i
439.2 0.381966 1.14412 − 1.98168i −1.85410 1.11803 + 1.93649i 0.437016 − 0.756934i 0.500000 − 0.866025i −1.47214 −1.11803 − 1.93649i 0.427051 + 0.739674i
439.3 2.61803 −0.437016 + 0.756934i 4.85410 −1.11803 − 1.93649i −1.14412 + 1.98168i 0.500000 − 0.866025i 7.47214 1.11803 + 1.93649i −2.92705 − 5.06980i
439.4 2.61803 0.437016 − 0.756934i 4.85410 −1.11803 − 1.93649i 1.14412 − 1.98168i 0.500000 − 0.866025i 7.47214 1.11803 + 1.93649i −2.92705 − 5.06980i
521.1 0.381966 −1.14412 − 1.98168i −1.85410 1.11803 − 1.93649i −0.437016 − 0.756934i 0.500000 + 0.866025i −1.47214 −1.11803 + 1.93649i 0.427051 − 0.739674i
521.2 0.381966 1.14412 + 1.98168i −1.85410 1.11803 − 1.93649i 0.437016 + 0.756934i 0.500000 + 0.866025i −1.47214 −1.11803 + 1.93649i 0.427051 − 0.739674i
521.3 2.61803 −0.437016 − 0.756934i 4.85410 −1.11803 + 1.93649i −1.14412 − 1.98168i 0.500000 + 0.866025i 7.47214 1.11803 − 1.93649i −2.92705 + 5.06980i
521.4 2.61803 0.437016 + 0.756934i 4.85410 −1.11803 + 1.93649i 1.14412 + 1.98168i 0.500000 + 0.866025i 7.47214 1.11803 − 1.93649i −2.92705 + 5.06980i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 439.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.b odd 2 1 inner
31.c even 3 1 inner
31.e odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 961.2.c.h 8
31.b odd 2 1 inner 961.2.c.h 8
31.c even 3 1 961.2.a.h ✓ 4
31.c even 3 1 inner 961.2.c.h 8
31.d even 5 2 961.2.g.i 16
31.d even 5 2 961.2.g.p 16
31.e odd 6 1 961.2.a.h ✓ 4
31.e odd 6 1 inner 961.2.c.h 8
31.f odd 10 2 961.2.g.i 16
31.f odd 10 2 961.2.g.p 16
31.g even 15 2 961.2.d.h 8
31.g even 15 2 961.2.d.j 8
31.g even 15 2 961.2.g.i 16
31.g even 15 2 961.2.g.p 16
31.h odd 30 2 961.2.d.h 8
31.h odd 30 2 961.2.d.j 8
31.h odd 30 2 961.2.g.i 16
31.h odd 30 2 961.2.g.p 16
93.g even 6 1 8649.2.a.r 4
93.h odd 6 1 8649.2.a.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
961.2.a.h ✓ 4 31.c even 3 1
961.2.a.h ✓ 4 31.e odd 6 1
961.2.c.h 8 1.a even 1 1 trivial
961.2.c.h 8 31.b odd 2 1 inner
961.2.c.h 8 31.c even 3 1 inner
961.2.c.h 8 31.e odd 6 1 inner
961.2.d.h 8 31.g even 15 2
961.2.d.h 8 31.h odd 30 2
961.2.d.j 8 31.g even 15 2
961.2.d.j 8 31.h odd 30 2
961.2.g.i 16 31.d even 5 2
961.2.g.i 16 31.f odd 10 2
961.2.g.i 16 31.g even 15 2
961.2.g.i 16 31.h odd 30 2
961.2.g.p 16 31.d even 5 2
961.2.g.p 16 31.f odd 10 2
961.2.g.p 16 31.g even 15 2
961.2.g.p 16 31.h odd 30 2
8649.2.a.r 4 93.g even 6 1
8649.2.a.r 4 93.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(961, [\chi])\):

\( T_{2}^{2} - 3T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{8} + 6T_{3}^{6} + 32T_{3}^{4} + 24T_{3}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 3 T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + 6 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( (T^{4} + 5 T^{2} + 25)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 18 T^{2} + 324)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 54 T^{6} + \cdots + 104976 \) Copy content Toggle raw display
$17$ \( T^{8} + 14 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 54 T^{2} + 324)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 14 T^{2} + 4)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} + 18 T^{2} + 324)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 6 T^{3} + \cdots + 121)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 94 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$47$ \( (T^{2} + 6 T - 36)^{4} \) Copy content Toggle raw display
$53$ \( T^{8} + 216 T^{6} + \cdots + 26873856 \) Copy content Toggle raw display
$59$ \( (T^{4} + 6 T^{3} + \cdots + 5041)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 214 T^{2} + 3844)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 6 T + 36)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 6 T^{3} + \cdots + 121)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 18 T^{2} + 324)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 126 T^{6} + \cdots + 104976 \) Copy content Toggle raw display
$83$ \( (T^{4} + 10 T^{2} + 100)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 270 T^{2} + 8100)^{2} \) Copy content Toggle raw display
$97$ \( (T + 7)^{8} \) Copy content Toggle raw display
show more
show less