↑

Properties

Label 1260.2.c.c
Level $1260$
Weight $2$
Character orbit 1260.c
Analytic conductor $10.061$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1260,2,Mod(811,1260)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1260, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1260.811");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1260.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: \(10.0611506547\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.342102016.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{6} + 4x^{4} + 4x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 140)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + ( - \beta_{5} - \beta_{4} - \beta_{3}) q^{4} + \beta_{2} q^{5} + (\beta_{7} - \beta_{6} + \beta_{3} - \beta_1) q^{7} + ( - \beta_{7} - \beta_{6} + \beta_{4} - 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + ( - \beta_{5} - \beta_{4} - \beta_{3}) q^{4} + \beta_{2} q^{5} + (\beta_{7} - \beta_{6} + \beta_{3} - \beta_1) q^{7} + ( - \beta_{7} - \beta_{6} + \beta_{4} - 2) q^{8} + \beta_1 q^{10} + (\beta_{7} + \beta_{6} + \cdots + \beta_{3}) q^{11}+ \cdots + (\beta_{7} - \beta_{6} + \beta_{5} + \cdots - 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} + 2 q^{4} - 14 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} + 2 q^{4} - 14 q^{8} + 6 q^{14} - 14 q^{16} + 28 q^{22} - 8 q^{25} + 14 q^{28} + 16 q^{29} + 18 q^{32} + 16 q^{37} + 20 q^{44} + 44 q^{46} + 36 q^{49} + 2 q^{50} - 16 q^{53} - 2 q^{56} - 4 q^{58} + 2 q^{64} + 16 q^{65} + 4 q^{70} - 4 q^{74} - 24 q^{77} + 24 q^{85} + 20 q^{86} + 36 q^{88} + 12 q^{92} - 26 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + \nu^{5} + 2\nu^{3} ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + \nu^{6} - \nu^{5} + 3\nu^{4} + 6\nu^{3} + 10\nu^{2} + 8\nu + 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} - 3\nu^{4} - 2\nu^{2} - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + \nu^{6} + \nu^{5} + 3\nu^{4} - 6\nu^{3} + 10\nu^{2} - 8\nu + 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 3\nu^{6} + 3\nu^{5} - \nu^{4} + 2\nu^{3} - 6\nu^{2} - 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} - 3\nu^{6} - 3\nu^{5} - \nu^{4} - 2\nu^{3} - 6\nu^{2} - 8 ) / 16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} + \beta_{3} + 2\beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + \beta_{6} - 3\beta_{4} - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{7} + 2\beta_{6} + \beta_{5} - \beta_{3} + 2\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -3\beta_{7} - 3\beta_{6} - 2\beta_{5} - \beta_{4} - 2\beta_{3} - 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -2\beta_{7} + 2\beta_{6} - \beta_{5} + \beta_{3} - 10\beta_{2} - \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(631\) \(757\) \(1081\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
811.1
−0.599676 + 1.28078i
0.599676 − 1.28078i
0.599676 + 1.28078i
−0.599676 − 1.28078i
−1.17915 − 0.780776i
1.17915 + 0.780776i
1.17915 − 0.780776i
−1.17915 + 0.780776i
−1.28078 − 0.599676i 0 1.28078 + 1.53610i − 1.00000i 0 2.60399 − 0.468213i −0.719224 − 2.73546i 0 −0.599676 + 1.28078i
811.2 −1.28078 − 0.599676i 0 1.28078 + 1.53610i 1.00000i 0 −2.60399 − 0.468213i −0.719224 − 2.73546i 0 0.599676 − 1.28078i
811.3 −1.28078 + 0.599676i 0 1.28078 − 1.53610i − 1.00000i 0 −2.60399 + 0.468213i −0.719224 + 2.73546i 0 0.599676 + 1.28078i
811.4 −1.28078 + 0.599676i 0 1.28078 − 1.53610i 1.00000i 0 2.60399 + 0.468213i −0.719224 + 2.73546i 0 −0.599676 − 1.28078i
811.5 0.780776 − 1.17915i 0 −0.780776 − 1.84130i − 1.00000i 0 −2.17238 + 1.51022i −2.78078 − 0.516994i 0 −1.17915 − 0.780776i
811.6 0.780776 − 1.17915i 0 −0.780776 − 1.84130i 1.00000i 0 2.17238 + 1.51022i −2.78078 − 0.516994i 0 1.17915 + 0.780776i
811.7 0.780776 + 1.17915i 0 −0.780776 + 1.84130i − 1.00000i 0 2.17238 − 1.51022i −2.78078 + 0.516994i 0 1.17915 − 0.780776i
811.8 0.780776 + 1.17915i 0 −0.780776 + 1.84130i 1.00000i 0 −2.17238 − 1.51022i −2.78078 + 0.516994i 0 −1.17915 + 0.780776i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 811.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1260.2.c.c 8
3.b odd 2 1 140.2.g.c ✓ 8
4.b odd 2 1 inner 1260.2.c.c 8
7.b odd 2 1 inner 1260.2.c.c 8
12.b even 2 1 140.2.g.c ✓ 8
15.d odd 2 1 700.2.g.j 8
15.e even 4 1 700.2.c.i 8
15.e even 4 1 700.2.c.j 8
21.c even 2 1 140.2.g.c ✓ 8
21.g even 6 2 980.2.o.e 16
21.h odd 6 2 980.2.o.e 16
24.f even 2 1 2240.2.k.e 8
24.h odd 2 1 2240.2.k.e 8
28.d even 2 1 inner 1260.2.c.c 8
60.h even 2 1 700.2.g.j 8
60.l odd 4 1 700.2.c.i 8
60.l odd 4 1 700.2.c.j 8
84.h odd 2 1 140.2.g.c ✓ 8
84.j odd 6 2 980.2.o.e 16
84.n even 6 2 980.2.o.e 16
105.g even 2 1 700.2.g.j 8
105.k odd 4 1 700.2.c.i 8
105.k odd 4 1 700.2.c.j 8
168.e odd 2 1 2240.2.k.e 8
168.i even 2 1 2240.2.k.e 8
420.o odd 2 1 700.2.g.j 8
420.w even 4 1 700.2.c.i 8
420.w even 4 1 700.2.c.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.g.c ✓ 8 3.b odd 2 1
140.2.g.c ✓ 8 12.b even 2 1
140.2.g.c ✓ 8 21.c even 2 1
140.2.g.c ✓ 8 84.h odd 2 1
700.2.c.i 8 15.e even 4 1
700.2.c.i 8 60.l odd 4 1
700.2.c.i 8 105.k odd 4 1
700.2.c.i 8 420.w even 4 1
700.2.c.j 8 15.e even 4 1
700.2.c.j 8 60.l odd 4 1
700.2.c.j 8 105.k odd 4 1
700.2.c.j 8 420.w even 4 1
700.2.g.j 8 15.d odd 2 1
700.2.g.j 8 60.h even 2 1
700.2.g.j 8 105.g even 2 1
700.2.g.j 8 420.o odd 2 1
980.2.o.e 16 21.g even 6 2
980.2.o.e 16 21.h odd 6 2
980.2.o.e 16 84.j odd 6 2
980.2.o.e 16 84.n even 6 2
1260.2.c.c 8 1.a even 1 1 trivial
1260.2.c.c 8 4.b odd 2 1 inner
1260.2.c.c 8 7.b odd 2 1 inner
1260.2.c.c 8 28.d even 2 1 inner
2240.2.k.e 8 24.f even 2 1
2240.2.k.e 8 24.h odd 2 1
2240.2.k.e 8 168.e odd 2 1
2240.2.k.e 8 168.i even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1260, [\chi])\):

\( T_{11}^{4} + 28T_{11}^{2} + 128 \) Copy content Toggle raw display
\( T_{19}^{4} - 28T_{19}^{2} + 128 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + T^{3} + 2 T + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} - 18 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( (T^{4} + 28 T^{2} + 128)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 52 T^{2} + 64)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 28 T^{2} + 128)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 74 T^{2} + 1352)^{2} \) Copy content Toggle raw display
$29$ \( (T - 2)^{8} \) Copy content Toggle raw display
$31$ \( (T^{4} - 56 T^{2} + 512)^{2} \) Copy content Toggle raw display
$37$ \( (T - 2)^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} + 52 T^{2} + 64)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 58 T^{2} + 8)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 126 T^{2} + 2592)^{2} \) Copy content Toggle raw display
$53$ \( (T + 2)^{8} \) Copy content Toggle raw display
$59$ \( (T^{4} - 124 T^{2} + 512)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 218 T^{2} + 2888)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 92 T^{2} + 2048)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 324 T^{2} + 20736)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 20 T^{2} + 32)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 10 T^{2} + 8)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 144)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 52 T^{2} + 64)^{2} \) Copy content Toggle raw display
show more
show less