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Properties

Label 700.2.be.b
Level $700$
Weight $2$
Character orbit 700.be
Analytic conductor $5.590$
Analytic rank $0$
Dimension $8$
CM discriminant -20
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [700,2,Mod(107,700)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(700, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 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("700.107");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.be (of order \(12\), degree \(4\), 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: \(5.58952814149\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: 8.0.12960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{7} - \beta_{4} + 1) q^{2} + (\beta_{7} + 2 \beta_{6} - \beta_{3} - 1) q^{3} - 2 \beta_1 q^{4} + (\beta_{7} + \beta_{6} - \beta_{5} + \cdots - 2) q^{6}+ \cdots + ( - 5 \beta_{7} - \beta_{6} + \cdots - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} - \beta_{4} + 1) q^{2} + (\beta_{7} + 2 \beta_{6} - \beta_{3} - 1) q^{3} - 2 \beta_1 q^{4} + (\beta_{7} + \beta_{6} - \beta_{5} + \cdots - 2) q^{6}+ \cdots + ( - 5 \beta_{4} - 6 \beta_{3} - \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 2 q^{3} - 8 q^{6} - 6 q^{7} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} - 2 q^{3} - 8 q^{6} - 6 q^{7} + 16 q^{8} - 4 q^{12} + 16 q^{16} + 20 q^{18} - 36 q^{21} - 2 q^{23} - 68 q^{27} + 12 q^{28} - 16 q^{32} + 80 q^{36} - 48 q^{41} - 24 q^{42} - 36 q^{43} + 4 q^{46} + 28 q^{47} - 16 q^{48} - 24 q^{56} - 12 q^{58} - 16 q^{61} + 60 q^{63} + 6 q^{67} + 40 q^{72} + 64 q^{81} - 24 q^{82} + 44 q^{83} - 36 q^{86} + 54 q^{87} + 8 q^{92} - 16 q^{96} - 8 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 13\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} - 4\nu^{4} + 8\nu^{2} + 4\nu - 3 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + 4\nu^{4} - 8\nu^{2} + 4\nu + 3 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{6} + 8\nu^{4} - 24\nu^{2} + 9 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{7} - \nu^{6} - 8\nu^{5} + 24\nu^{3} - \nu - 13 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{7} - 3\nu^{6} + 8\nu^{5} + 8\nu^{4} - 24\nu^{3} - 16\nu^{2} + 9\nu + 1 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{7} + 16\nu^{5} - 40\nu^{3} + 15\nu ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + \beta_{5} - 2\beta_{4} - \beta_{2} + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} - \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4\beta_{4} + 3\beta_{3} - 3\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6\beta_{7} - 5\beta_{6} + 5\beta_{5} - 5\beta_{2} + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -4\beta_{6} - 4\beta_{5} + 4\beta_{3} - 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -13\beta_{3} - 13\beta_{2} + 16\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(477\)
\(\chi(n)\) \(-1 + \beta_{4}\) \(-1\) \(-\beta_{1} + \beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
0.535233 − 0.309017i
−1.40126 + 0.809017i
−0.535233 − 0.309017i
1.40126 + 0.809017i
−0.535233 + 0.309017i
1.40126 − 0.809017i
0.535233 + 0.309017i
−1.40126 − 0.809017i
−0.366025 − 1.36603i −0.525792 + 1.96228i −1.73205 + 1.00000i 0 2.87298 −2.45827 + 0.978225i 2.00000 + 2.00000i −0.976025 − 0.563508i 0
107.2 −0.366025 − 1.36603i 0.891818 − 3.32831i −1.73205 + 1.00000i 0 −4.87298 −1.63981 − 2.07630i 2.00000 + 2.00000i −7.68423 − 4.43649i 0
207.1 1.36603 + 0.366025i −3.32831 + 0.891818i 1.73205 + 1.00000i 0 −4.87298 2.07630 + 1.63981i 2.00000 + 2.00000i 7.68423 − 4.43649i 0
207.2 1.36603 + 0.366025i 1.96228 − 0.525792i 1.73205 + 1.00000i 0 2.87298 −0.978225 + 2.45827i 2.00000 + 2.00000i 0.976025 − 0.563508i 0
443.1 1.36603 − 0.366025i −3.32831 − 0.891818i 1.73205 − 1.00000i 0 −4.87298 2.07630 − 1.63981i 2.00000 − 2.00000i 7.68423 + 4.43649i 0
443.2 1.36603 − 0.366025i 1.96228 + 0.525792i 1.73205 − 1.00000i 0 2.87298 −0.978225 − 2.45827i 2.00000 − 2.00000i 0.976025 + 0.563508i 0
543.1 −0.366025 + 1.36603i −0.525792 − 1.96228i −1.73205 − 1.00000i 0 2.87298 −2.45827 − 0.978225i 2.00000 − 2.00000i −0.976025 + 0.563508i 0
543.2 −0.366025 + 1.36603i 0.891818 + 3.32831i −1.73205 − 1.00000i 0 −4.87298 −1.63981 + 2.07630i 2.00000 − 2.00000i −7.68423 + 4.43649i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
5.c odd 4 1 inner
7.c even 3 1 inner
20.e even 4 1 inner
35.l odd 12 1 inner
140.p odd 6 1 inner
140.w even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.2.be.b yes 8
4.b odd 2 1 700.2.be.a ✓ 8
5.b even 2 1 700.2.be.a ✓ 8
5.c odd 4 1 700.2.be.a ✓ 8
5.c odd 4 1 inner 700.2.be.b yes 8
7.c even 3 1 inner 700.2.be.b yes 8
20.d odd 2 1 CM 700.2.be.b yes 8
20.e even 4 1 700.2.be.a ✓ 8
20.e even 4 1 inner 700.2.be.b yes 8
28.g odd 6 1 700.2.be.a ✓ 8
35.j even 6 1 700.2.be.a ✓ 8
35.l odd 12 1 700.2.be.a ✓ 8
35.l odd 12 1 inner 700.2.be.b yes 8
140.p odd 6 1 inner 700.2.be.b yes 8
140.w even 12 1 700.2.be.a ✓ 8
140.w even 12 1 inner 700.2.be.b yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
700.2.be.a ✓ 8 4.b odd 2 1
700.2.be.a ✓ 8 5.b even 2 1
700.2.be.a ✓ 8 5.c odd 4 1
700.2.be.a ✓ 8 20.e even 4 1
700.2.be.a ✓ 8 28.g odd 6 1
700.2.be.a ✓ 8 35.j even 6 1
700.2.be.a ✓ 8 35.l odd 12 1
700.2.be.a ✓ 8 140.w even 12 1
700.2.be.b yes 8 1.a even 1 1 trivial
700.2.be.b yes 8 5.c odd 4 1 inner
700.2.be.b yes 8 7.c even 3 1 inner
700.2.be.b yes 8 20.d odd 2 1 CM
700.2.be.b yes 8 20.e even 4 1 inner
700.2.be.b yes 8 35.l odd 12 1 inner
700.2.be.b yes 8 140.p odd 6 1 inner
700.2.be.b yes 8 140.w even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 2T_{3}^{7} + 2T_{3}^{6} + 32T_{3}^{5} - 17T_{3}^{4} - 224T_{3}^{3} + 98T_{3}^{2} - 686T_{3} + 2401 \) acting on \(S_{2}^{\mathrm{new}}(700, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + 2 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 6 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} + 2 T^{7} + \cdots + 20151121 \) Copy content Toggle raw display
$29$ \( (T^{4} + 138 T^{2} + 2601)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( (T^{2} + 12 T + 21)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 18 T^{3} + \cdots + 1089)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 14 T^{3} + \cdots + 9604)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T^{4} + 8 T^{3} + \cdots + 14161)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 1121513121 \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} - 22 T^{3} + \cdots + 49)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 2847396321 \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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