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Properties

Label 1260.2.ba.a
Level $1260$
Weight $2$
Character orbit 1260.ba
Analytic conductor $10.061$
Analytic rank $0$
Dimension $8$
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] = [1260,2,Mod(433,1260)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1260, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1260.433");
 
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.ba (of order \(4\), degree \(2\), 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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.11574317056.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 45x^{4} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{6} + \beta_{2}) q^{5} + ( - \beta_{7} + \beta_{4}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} + \beta_{2}) q^{5} + ( - \beta_{7} + \beta_{4}) q^{7} + ( - \beta_{5} - \beta_{4} + 1) q^{11} + \beta_1 q^{13} + ( - 2 \beta_{7} + 2 \beta_{6} - \beta_{2}) q^{17} + ( - 2 \beta_{7} + \beta_{2} - \beta_1) q^{19} + (\beta_{7} + \beta_{6} + 2 \beta_{5}) q^{23} + ( - \beta_{7} - \beta_{6} - 2 \beta_{5} + \cdots - 2) q^{25}+ \cdots + 3 \beta_{2} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{7} + 12 q^{11} - 4 q^{23} - 12 q^{25} + 14 q^{35} - 4 q^{37} - 40 q^{43} - 40 q^{53} - 20 q^{65} + 40 q^{67} - 8 q^{71} - 44 q^{77} + 48 q^{85} - 8 q^{91} + 44 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 45x^{4} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 47\nu^{3} ) / 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} - 47\nu^{2} ) / 14 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{6} + 2\nu^{5} + 2\nu^{4} + 127\nu^{2} + 94\nu + 38 ) / 28 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{6} - 2\nu^{5} + 2\nu^{4} - 127\nu^{2} - 94\nu + 38 ) / 28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7\nu^{7} + 2\nu^{5} + 315\nu^{3} + 94\nu ) / 28 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -7\nu^{7} + 2\nu^{5} - 315\nu^{3} + 94\nu ) / 28 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - 3\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{6} + 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{5} + 7\beta_{4} - 19 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7\beta_{7} + 7\beta_{6} - 47\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -47\beta_{7} - 47\beta_{6} - 47\beta_{5} + 47\beta_{4} + 127\beta_{3} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -47\beta_{7} + 47\beta_{6} - 315\beta_{2} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(281\) \(631\) \(757\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{3}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
433.1
1.83051 − 1.83051i
0.386289 − 0.386289i
−0.386289 + 0.386289i
−1.83051 + 1.83051i
1.83051 + 1.83051i
0.386289 + 0.386289i
−0.386289 − 0.386289i
−1.83051 − 1.83051i
0 0 0 −1.83051 − 1.28422i 0 −2.12393 + 1.57763i 0 0 0
433.2 0 0 0 −0.386289 + 2.20245i 0 0.0564123 − 2.64515i 0 0 0
433.3 0 0 0 0.386289 − 2.20245i 0 2.64515 − 0.0564123i 0 0 0
433.4 0 0 0 1.83051 + 1.28422i 0 −1.57763 + 2.12393i 0 0 0
937.1 0 0 0 −1.83051 + 1.28422i 0 −2.12393 − 1.57763i 0 0 0
937.2 0 0 0 −0.386289 − 2.20245i 0 0.0564123 + 2.64515i 0 0 0
937.3 0 0 0 0.386289 + 2.20245i 0 2.64515 + 0.0564123i 0 0 0
937.4 0 0 0 1.83051 − 1.28422i 0 −1.57763 − 2.12393i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 433.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
35.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1260.2.ba.a 8
3.b odd 2 1 140.2.m.a ✓ 8
5.c odd 4 1 inner 1260.2.ba.a 8
7.b odd 2 1 inner 1260.2.ba.a 8
12.b even 2 1 560.2.bj.b 8
15.d odd 2 1 700.2.m.c 8
15.e even 4 1 140.2.m.a ✓ 8
15.e even 4 1 700.2.m.c 8
21.c even 2 1 140.2.m.a ✓ 8
21.g even 6 2 980.2.v.b 16
21.h odd 6 2 980.2.v.b 16
35.f even 4 1 inner 1260.2.ba.a 8
60.l odd 4 1 560.2.bj.b 8
84.h odd 2 1 560.2.bj.b 8
105.g even 2 1 700.2.m.c 8
105.k odd 4 1 140.2.m.a ✓ 8
105.k odd 4 1 700.2.m.c 8
105.w odd 12 2 980.2.v.b 16
105.x even 12 2 980.2.v.b 16
420.w even 4 1 560.2.bj.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.m.a ✓ 8 3.b odd 2 1
140.2.m.a ✓ 8 15.e even 4 1
140.2.m.a ✓ 8 21.c even 2 1
140.2.m.a ✓ 8 105.k odd 4 1
560.2.bj.b 8 12.b even 2 1
560.2.bj.b 8 60.l odd 4 1
560.2.bj.b 8 84.h odd 2 1
560.2.bj.b 8 420.w even 4 1
700.2.m.c 8 15.d odd 2 1
700.2.m.c 8 15.e even 4 1
700.2.m.c 8 105.g even 2 1
700.2.m.c 8 105.k odd 4 1
980.2.v.b 16 21.g even 6 2
980.2.v.b 16 21.h odd 6 2
980.2.v.b 16 105.w odd 12 2
980.2.v.b 16 105.x even 12 2
1260.2.ba.a 8 1.a even 1 1 trivial
1260.2.ba.a 8 5.c odd 4 1 inner
1260.2.ba.a 8 7.b odd 2 1 inner
1260.2.ba.a 8 35.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{2} - 3T_{11} - 8 \) acting on \(S_{2}^{\mathrm{new}}(1260, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 6 T^{6} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} + 2 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( (T^{2} - 3 T - 8)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} + 45T^{4} + 4 \) Copy content Toggle raw display
$17$ \( T^{8} + 2109 T^{4} + 2500 \) Copy content Toggle raw display
$19$ \( (T^{4} - 58 T^{2} + 800)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 2 T^{3} + \cdots + 400)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 33 T^{2} + 16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 142 T^{2} + 5000)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2 T^{3} + \cdots + 400)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 58 T^{2} + 800)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 10 T + 50)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + 17325 T^{4} + 7496644 \) Copy content Toggle raw display
$53$ \( (T^{2} + 10 T + 50)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 58 T^{2} + 800)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 232 T^{2} + 12800)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 10 T + 50)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 2 T - 40)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + 420 T^{4} + 16384 \) Copy content Toggle raw display
$79$ \( (T^{4} + 45 T^{2} + 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 16500 T^{4} + 17909824 \) Copy content Toggle raw display
$89$ \( (T^{4} - 188 T^{2} + 800)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 3645 T^{4} + 26244 \) Copy content Toggle raw display
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