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Properties

Label 1400.2.bh.g
Level $1400$
Weight $2$
Character orbit 1400.bh
Analytic conductor $11.179$
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] = [1400,2,Mod(249,1400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1400, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1400.249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1400.bh (of order \(6\), degree \(2\), not 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: \(11.1790562830\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + ( - \beta_{4} + \beta_{3} - \beta_{2}) q^{7} - \beta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + ( - \beta_{4} + \beta_{3} - \beta_{2}) q^{7} - \beta_{6} q^{9} + ( - \beta_{5} + 2 \beta_1) q^{11} + ( - \beta_{7} + \beta_{4} + \cdots - \beta_{2}) q^{13}+ \cdots + ( - 2 \beta_{6} - 2 \beta_{5} + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{11} - 20 q^{21} + 40 q^{29} + 8 q^{31} - 8 q^{39} + 24 q^{41} - 20 q^{49} + 24 q^{51} - 16 q^{59} - 4 q^{61} - 72 q^{69} - 96 q^{71} - 16 q^{79} + 4 q^{81} - 44 q^{89} - 8 q^{91} + 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} - \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{6} - \zeta_{24}^{2} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 2\zeta_{24}^{7} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -2\zeta_{24}^{7} - 2\zeta_{24}^{5} + 2\zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{24}^{7} - \zeta_{24}^{6} + \zeta_{24}^{2} + \zeta_{24} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{7} + \beta_{5} + \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{4} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{6} + \beta_{4} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( -\beta_{7} + \beta_{4} + \beta_{3} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{7} + \beta_{5} - \beta_{3} ) / 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(701\) \(801\) \(1177\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{1}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
249.1
−0.965926 + 0.258819i
−0.258819 − 0.965926i
0.965926 − 0.258819i
0.258819 + 0.965926i
−0.965926 − 0.258819i
−0.258819 + 0.965926i
0.965926 + 0.258819i
0.258819 − 0.965926i
0 −2.09077 − 1.20711i 0 0 0 0.358719 − 2.62132i 0 1.41421 + 2.44949i 0
249.2 0 −0.358719 − 0.207107i 0 0 0 2.09077 − 1.62132i 0 −1.41421 − 2.44949i 0
249.3 0 0.358719 + 0.207107i 0 0 0 −2.09077 + 1.62132i 0 −1.41421 − 2.44949i 0
249.4 0 2.09077 + 1.20711i 0 0 0 −0.358719 + 2.62132i 0 1.41421 + 2.44949i 0
849.1 0 −2.09077 + 1.20711i 0 0 0 0.358719 + 2.62132i 0 1.41421 − 2.44949i 0
849.2 0 −0.358719 + 0.207107i 0 0 0 2.09077 + 1.62132i 0 −1.41421 + 2.44949i 0
849.3 0 0.358719 − 0.207107i 0 0 0 −2.09077 − 1.62132i 0 −1.41421 + 2.44949i 0
849.4 0 2.09077 − 1.20711i 0 0 0 −0.358719 − 2.62132i 0 1.41421 − 2.44949i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 249.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.2.bh.g 8
5.b even 2 1 inner 1400.2.bh.g 8
5.c odd 4 1 280.2.q.d ✓ 4
5.c odd 4 1 1400.2.q.h 4
7.c even 3 1 inner 1400.2.bh.g 8
15.e even 4 1 2520.2.bi.k 4
20.e even 4 1 560.2.q.j 4
35.f even 4 1 1960.2.q.q 4
35.j even 6 1 inner 1400.2.bh.g 8
35.k even 12 1 1960.2.a.t 2
35.k even 12 1 1960.2.q.q 4
35.k even 12 1 9800.2.a.br 2
35.l odd 12 1 280.2.q.d ✓ 4
35.l odd 12 1 1400.2.q.h 4
35.l odd 12 1 1960.2.a.p 2
35.l odd 12 1 9800.2.a.bz 2
105.x even 12 1 2520.2.bi.k 4
140.w even 12 1 560.2.q.j 4
140.w even 12 1 3920.2.a.bz 2
140.x odd 12 1 3920.2.a.bp 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.d ✓ 4 5.c odd 4 1
280.2.q.d ✓ 4 35.l odd 12 1
560.2.q.j 4 20.e even 4 1
560.2.q.j 4 140.w even 12 1
1400.2.q.h 4 5.c odd 4 1
1400.2.q.h 4 35.l odd 12 1
1400.2.bh.g 8 1.a even 1 1 trivial
1400.2.bh.g 8 5.b even 2 1 inner
1400.2.bh.g 8 7.c even 3 1 inner
1400.2.bh.g 8 35.j even 6 1 inner
1960.2.a.p 2 35.l odd 12 1
1960.2.a.t 2 35.k even 12 1
1960.2.q.q 4 35.f even 4 1
1960.2.q.q 4 35.k even 12 1
2520.2.bi.k 4 15.e even 4 1
2520.2.bi.k 4 105.x even 12 1
3920.2.a.bp 2 140.x odd 12 1
3920.2.a.bz 2 140.w even 12 1
9800.2.a.br 2 35.k even 12 1
9800.2.a.bz 2 35.l odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1400, [\chi])\):

\( T_{3}^{8} - 6T_{3}^{6} + 35T_{3}^{4} - 6T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{4} - 4T_{11}^{3} + 20T_{11}^{2} + 16T_{11} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 6 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 10 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( (T^{4} - 4 T^{3} + 20 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} - 72 T^{6} + \cdots + 614656 \) Copy content Toggle raw display
$19$ \( (T^{4} + 32 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 102 T^{6} + \cdots + 4879681 \) Copy content Toggle raw display
$29$ \( (T^{2} - 10 T + 17)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 4 T^{3} + 20 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 32 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 6 T + 1)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 214 T^{2} + 7921)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 136 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$53$ \( (T^{4} - 32 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 4 T + 16)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 2 T^{3} + \cdots + 961)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 214 T^{6} + \cdots + 62742241 \) Copy content Toggle raw display
$71$ \( (T + 12)^{8} \) Copy content Toggle raw display
$73$ \( T^{8} - 72 T^{6} + \cdots + 614656 \) Copy content Toggle raw display
$79$ \( (T^{2} + 4 T + 16)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 198 T^{2} + 3969)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 22 T^{3} + \cdots + 7921)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 36)^{4} \) Copy content Toggle raw display
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