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Properties

Label 700.2.g.k
Level $700$
Weight $2$
Character orbit 700.g
Analytic conductor $5.590$
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] = [700,2,Mod(251,700)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(700, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("700.251");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.g (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: \(5.58952814149\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.5473632256.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 14x^{6} - 12x^{5} + 34x^{4} + 24x^{3} + 56x^{2} + 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + 1) q^{2} + ( - \beta_{5} + \beta_{2} - \beta_1 + 1) q^{3} + ( - \beta_{6} - \beta_{4} + \beta_{3}) q^{4} + (\beta_{7} - 2 \beta_{5} + \beta_{2} + \cdots + 1) q^{6}+ \cdots + ( - 2 \beta_{6} - 2 \beta_{4} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + 1) q^{2} + ( - \beta_{5} + \beta_{2} - \beta_1 + 1) q^{3} + ( - \beta_{6} - \beta_{4} + \beta_{3}) q^{4} + (\beta_{7} - 2 \beta_{5} + \beta_{2} + \cdots + 1) q^{6}+ \cdots + ( - 4 \beta_{3} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 4 q^{4} + 4 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} - 4 q^{4} + 4 q^{8} + 16 q^{9} + 12 q^{14} + 4 q^{16} + 24 q^{18} - 24 q^{21} + 20 q^{22} + 16 q^{28} + 32 q^{29} - 36 q^{32} + 24 q^{36} - 16 q^{37} - 8 q^{42} + 4 q^{44} - 16 q^{46} - 64 q^{53} + 20 q^{56} - 24 q^{57} + 40 q^{58} - 52 q^{64} - 8 q^{72} + 16 q^{74} - 24 q^{77} - 24 q^{81} + 20 q^{84} + 40 q^{86} - 12 q^{88} + 24 q^{92} - 32 q^{93} - 32 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 14x^{6} - 12x^{5} + 34x^{4} + 24x^{3} + 56x^{2} + 32x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + 10\nu^{5} + 8\nu^{4} + 50\nu^{3} + 124\nu^{2} + 304\nu + 208 ) / 216 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} - \nu^{6} + 8\nu^{5} - 56\nu^{4} + 10\nu^{3} - 82\nu^{2} - 196\nu - 148 ) / 108 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 7\nu^{6} - 28\nu^{5} + 60\nu^{4} - 90\nu^{3} + 82\nu^{2} - 28\nu + 64 ) / 36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11\nu^{7} - 46\nu^{6} + 146\nu^{5} - 80\nu^{4} + 166\nu^{3} + 404\nu^{2} + 320\nu + 176 ) / 216 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2\nu^{7} - 9\nu^{6} + 32\nu^{5} - 37\nu^{4} + 74\nu^{3} + 30\nu^{2} + 68\nu + 12 ) / 18 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 23\nu^{7} - 124\nu^{6} + 446\nu^{5} - 704\nu^{4} + 1126\nu^{3} - 448\nu^{2} + 440\nu - 496 ) / 216 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{5} + 2\beta_{4} + \beta_{3} + 2\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{7} - 2\beta_{6} - \beta_{5} + 3\beta_{4} + 8\beta_{2} - 6\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{7} - 6\beta_{6} + 10\beta_{5} - 10\beta_{4} - 4\beta_{3} + 12\beta_{2} - 20\beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -40\beta_{7} + 4\beta_{6} + 38\beta_{5} - 68\beta_{4} - 12\beta_{3} - 26\beta_{2} - 28\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -134\beta_{7} + 80\beta_{6} + 34\beta_{5} - 160\beta_{4} - 30\beta_{3} - 232\beta_{2} + 68\beta _1 + 78 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -126\beta_{7} + 268\beta_{6} - 218\beta_{5} + 42\beta_{4} - 32\beta_{3} - 732\beta_{2} + 572\beta _1 + 296 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(351\) \(477\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
2.11137 + 2.44696i
−0.404265 − 0.468519i
2.11137 − 2.44696i
−0.404265 + 0.468519i
0.556194 − 1.97874i
−0.263301 + 0.936733i
0.556194 + 1.97874i
−0.263301 − 0.936733i
−0.207107 − 1.39897i −1.47363 −1.91421 + 0.579471i 0 0.305198 + 2.06155i 2.51564 + 0.819496i 1.20711 + 2.55791i −0.828427 0
251.2 −0.207107 − 1.39897i 1.47363 −1.91421 + 0.579471i 0 −0.305198 − 2.06155i −2.51564 + 0.819496i 1.20711 + 2.55791i −0.828427 0
251.3 −0.207107 + 1.39897i −1.47363 −1.91421 − 0.579471i 0 0.305198 − 2.06155i 2.51564 − 0.819496i 1.20711 − 2.55791i −0.828427 0
251.4 −0.207107 + 1.39897i 1.47363 −1.91421 − 0.579471i 0 −0.305198 + 2.06155i −2.51564 − 0.819496i 1.20711 − 2.55791i −0.828427 0
251.5 1.20711 − 0.736813i −2.79793 0.914214 − 1.77882i 0 −3.37740 + 2.06155i 0.819496 + 2.51564i −0.207107 − 2.82083i 4.82843 0
251.6 1.20711 − 0.736813i 2.79793 0.914214 − 1.77882i 0 3.37740 − 2.06155i −0.819496 + 2.51564i −0.207107 − 2.82083i 4.82843 0
251.7 1.20711 + 0.736813i −2.79793 0.914214 + 1.77882i 0 −3.37740 − 2.06155i 0.819496 − 2.51564i −0.207107 + 2.82083i 4.82843 0
251.8 1.20711 + 0.736813i 2.79793 0.914214 + 1.77882i 0 3.37740 + 2.06155i −0.819496 − 2.51564i −0.207107 + 2.82083i 4.82843 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.8
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 700.2.g.k yes 8
4.b odd 2 1 inner 700.2.g.k yes 8
5.b even 2 1 700.2.g.i ✓ 8
5.c odd 4 2 700.2.c.k 16
7.b odd 2 1 inner 700.2.g.k yes 8
20.d odd 2 1 700.2.g.i ✓ 8
20.e even 4 2 700.2.c.k 16
28.d even 2 1 inner 700.2.g.k yes 8
35.c odd 2 1 700.2.g.i ✓ 8
35.f even 4 2 700.2.c.k 16
140.c even 2 1 700.2.g.i ✓ 8
140.j odd 4 2 700.2.c.k 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
700.2.c.k 16 5.c odd 4 2
700.2.c.k 16 20.e even 4 2
700.2.c.k 16 35.f even 4 2
700.2.c.k 16 140.j odd 4 2
700.2.g.i ✓ 8 5.b even 2 1
700.2.g.i ✓ 8 20.d odd 2 1
700.2.g.i ✓ 8 35.c odd 2 1
700.2.g.i ✓ 8 140.c even 2 1
700.2.g.k yes 8 1.a even 1 1 trivial
700.2.g.k yes 8 4.b odd 2 1 inner
700.2.g.k yes 8 7.b odd 2 1 inner
700.2.g.k yes 8 28.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(700, [\chi])\):

\( T_{3}^{4} - 10T_{3}^{2} + 17 \) Copy content Toggle raw display
\( T_{11}^{4} + 10T_{11}^{2} + 17 \) Copy content Toggle raw display
\( T_{19}^{4} - 58T_{19}^{2} + 833 \) Copy content Toggle raw display
\( T_{37}^{2} + 4T_{37} - 14 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{3} + 3 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} - 10 T^{2} + 17)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 30T^{4} + 2401 \) Copy content Toggle raw display
$11$ \( (T^{4} + 10 T^{2} + 17)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 34)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 17)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} - 58 T^{2} + 833)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 20 T^{2} + 68)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 8 T - 2)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 20 T^{2} + 68)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 4 T - 14)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 17)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 40 T^{2} + 272)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 180 T^{2} + 5508)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 16 T + 46)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 40 T^{2} + 272)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 136)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 218 T^{2} + 8993)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 116 T^{2} + 3332)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 153)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 80 T^{2} + 1088)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 10 T^{2} + 17)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 153)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 68)^{4} \) Copy content Toggle raw display
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