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Properties

Label 700.2.c.k
Level $700$
Weight $2$
Character orbit 700.c
Analytic conductor $5.590$
Analytic rank $0$
Dimension $16$
CM no
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(699,700)]
 
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, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("700.699");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.c (of order \(2\), degree \(1\), 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: \(5.58952814149\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: 16.0.29960650073923649536.7
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 7x^{12} + 40x^{8} - 112x^{4} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} + ( - \beta_{9} + \beta_{8} + \cdots - \beta_{5}) q^{3}+ \cdots + (2 \beta_{4} + 2 \beta_{2} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{2} + ( - \beta_{9} + \beta_{8} + \cdots - \beta_{5}) q^{3}+ \cdots + ( - 4 \beta_{3} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4} - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{4} - 32 q^{9} - 24 q^{14} + 8 q^{16} - 48 q^{21} - 64 q^{29} + 48 q^{36} - 8 q^{44} - 32 q^{46} + 40 q^{56} + 104 q^{64} - 32 q^{74} - 48 q^{81} - 40 q^{84} + 80 q^{86}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 7x^{12} + 40x^{8} - 112x^{4} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{10} - 3\nu^{6} + 12\nu^{2} ) / 16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{15} - 4 \nu^{13} + 2 \nu^{12} - 3 \nu^{11} + 28 \nu^{9} + 10 \nu^{8} + 28 \nu^{7} - 96 \nu^{5} + \cdots + 128 ) / 256 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{15} + 2 \nu^{13} - 8 \nu^{12} + 11 \nu^{11} + 10 \nu^{9} + 24 \nu^{8} - 20 \nu^{7} + \cdots + 512 ) / 512 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{15} - \nu^{13} - \nu^{12} + 3\nu^{11} + 3\nu^{9} - 5\nu^{8} - 12\nu^{7} - 28\nu^{5} + 28\nu^{4} + 16\nu^{3} - 64 ) / 128 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3 \nu^{15} + 2 \nu^{13} - 12 \nu^{12} - 17 \nu^{11} - 22 \nu^{9} + 4 \nu^{8} + 76 \nu^{7} + 168 \nu^{5} + \cdots - 256 ) / 512 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2 \nu^{15} + \nu^{14} + 2 \nu^{13} + 6 \nu^{11} + 5 \nu^{10} - 6 \nu^{9} - 24 \nu^{7} + \cdots + 192 \nu^{2} ) / 256 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - \nu^{15} + \nu^{14} - 4 \nu^{13} + 3 \nu^{11} + 5 \nu^{10} + 28 \nu^{9} - 28 \nu^{7} + \cdots + 256 \nu ) / 256 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 4 \nu^{15} + 3 \nu^{14} + 2 \nu^{13} - 2 \nu^{12} - 44 \nu^{11} - 33 \nu^{10} + 10 \nu^{9} + \cdots + 384 ) / 512 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5 \nu^{15} + 3 \nu^{14} + 8 \nu^{13} + 2 \nu^{12} - 39 \nu^{11} - 33 \nu^{10} + 8 \nu^{9} - 22 \nu^{8} + \cdots - 384 ) / 512 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - \nu^{15} - 5 \nu^{14} - 4 \nu^{13} - 6 \nu^{12} - 21 \nu^{11} + 23 \nu^{10} + 12 \nu^{9} + \cdots + 128 ) / 512 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2 \nu^{15} + 3 \nu^{14} - 4 \nu^{13} + 6 \nu^{11} - 17 \nu^{10} + 28 \nu^{9} - 24 \nu^{7} + \cdots + 256 \nu ) / 256 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( \nu^{15} + 2\nu^{13} - 18\nu^{12} - 3\nu^{11} - 6\nu^{9} + 102\nu^{8} + 28\nu^{7} + 56\nu^{5} - 328\nu^{4} + 640 ) / 256 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 5 \nu^{15} - 5 \nu^{14} + 8 \nu^{13} - 6 \nu^{12} + 39 \nu^{11} + 23 \nu^{10} + 8 \nu^{9} + \cdots + 128 ) / 512 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( \nu^{15} + 8 \nu^{14} + 2 \nu^{13} - 11 \nu^{11} - 24 \nu^{10} + 10 \nu^{9} + 20 \nu^{7} + \cdots + 384 \nu ) / 512 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 5 \nu^{15} + 6 \nu^{14} - 10 \nu^{13} + 23 \nu^{11} - 34 \nu^{10} + 46 \nu^{9} - 68 \nu^{7} + \cdots + 128 \nu ) / 512 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{14} + \beta_{10} - \beta_{8} - \beta_{4} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{15} - \beta_{14} - \beta_{13} + \beta_{11} - \beta_{10} + \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{15} + 2 \beta_{14} + \beta_{13} - \beta_{9} - 2 \beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} + \cdots + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{12} + 2\beta_{9} - 2\beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - 3\beta_{3} - \beta_{2} + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3 \beta_{15} - \beta_{14} + \beta_{13} + 2 \beta_{10} + \beta_{9} - 2 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + \cdots + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 4 \beta_{15} - 3 \beta_{14} - \beta_{13} + 7 \beta_{11} - \beta_{10} - \beta_{7} - \beta_{6} + \cdots + 3 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - \beta_{15} - 2 \beta_{14} - 3 \beta_{13} - 2 \beta_{10} + 3 \beta_{9} - 2 \beta_{8} - 10 \beta_{7} + \cdots + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 7\beta_{12} + 2\beta_{9} - 2\beta_{8} - \beta_{7} + \beta_{6} - 11\beta_{5} - \beta_{4} - 13\beta_{3} + \beta_{2} - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 9 \beta_{15} - 15 \beta_{14} + 11 \beta_{13} - 6 \beta_{10} + 11 \beta_{9} + 6 \beta_{8} + 14 \beta_{7} + \cdots + 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 12 \beta_{15} + 3 \beta_{14} + 9 \beta_{13} + 9 \beta_{11} + 9 \beta_{10} - 15 \beta_{7} + \cdots + 53 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 9 \beta_{15} - 30 \beta_{14} - 21 \beta_{13} - 22 \beta_{10} + 21 \beta_{9} - 22 \beta_{8} - 6 \beta_{7} + \cdots - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 7 \beta_{12} - 18 \beta_{9} + 18 \beta_{8} - 31 \beta_{7} + 31 \beta_{6} - 37 \beta_{5} - 31 \beta_{4} + \cdots - 40 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 25 \beta_{15} - 17 \beta_{14} + 37 \beta_{13} - 42 \beta_{10} + 37 \beta_{9} + 42 \beta_{8} + \cdots - 25 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 84 \beta_{15} + 93 \beta_{14} - 9 \beta_{13} - 41 \beta_{11} - 9 \beta_{10} - 17 \beta_{7} + \cdots + 139 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 9 \beta_{15} - 34 \beta_{14} - 43 \beta_{13} - 74 \beta_{10} + 43 \beta_{9} - 74 \beta_{8} + 134 \beta_{7} + \cdots + 9 ) / 2 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
699.1
−1.32968 − 0.481610i
1.38588 − 0.281691i
1.38588 + 0.281691i
−1.32968 + 0.481610i
0.281691 + 1.38588i
1.32968 − 0.481610i
1.32968 + 0.481610i
0.281691 − 1.38588i
0.481610 − 1.32968i
−1.38588 − 0.281691i
−1.38588 + 0.281691i
0.481610 + 1.32968i
−0.281691 + 1.38588i
−0.481610 − 1.32968i
−0.481610 + 1.32968i
−0.281691 − 1.38588i
−1.39897 − 0.207107i − 1.47363i 1.91421 + 0.579471i 0 −0.305198 + 2.06155i 0.819496 − 2.51564i −2.55791 − 1.20711i 0.828427 0
699.2 −1.39897 − 0.207107i 1.47363i 1.91421 + 0.579471i 0 0.305198 − 2.06155i 0.819496 + 2.51564i −2.55791 − 1.20711i 0.828427 0
699.3 −1.39897 + 0.207107i − 1.47363i 1.91421 − 0.579471i 0 0.305198 + 2.06155i 0.819496 − 2.51564i −2.55791 + 1.20711i 0.828427 0
699.4 −1.39897 + 0.207107i 1.47363i 1.91421 − 0.579471i 0 −0.305198 − 2.06155i 0.819496 + 2.51564i −2.55791 + 1.20711i 0.828427 0
699.5 −0.736813 − 1.20711i − 2.79793i −0.914214 + 1.77882i 0 −3.37740 + 2.06155i 2.51564 − 0.819496i 2.82083 − 0.207107i −4.82843 0
699.6 −0.736813 − 1.20711i 2.79793i −0.914214 + 1.77882i 0 3.37740 − 2.06155i 2.51564 + 0.819496i 2.82083 − 0.207107i −4.82843 0
699.7 −0.736813 + 1.20711i − 2.79793i −0.914214 − 1.77882i 0 3.37740 + 2.06155i 2.51564 − 0.819496i 2.82083 + 0.207107i −4.82843 0
699.8 −0.736813 + 1.20711i 2.79793i −0.914214 − 1.77882i 0 −3.37740 − 2.06155i 2.51564 + 0.819496i 2.82083 + 0.207107i −4.82843 0
699.9 0.736813 − 1.20711i − 2.79793i −0.914214 − 1.77882i 0 −3.37740 − 2.06155i −2.51564 − 0.819496i −2.82083 − 0.207107i −4.82843 0
699.10 0.736813 − 1.20711i 2.79793i −0.914214 − 1.77882i 0 3.37740 + 2.06155i −2.51564 + 0.819496i −2.82083 − 0.207107i −4.82843 0
699.11 0.736813 + 1.20711i − 2.79793i −0.914214 + 1.77882i 0 3.37740 − 2.06155i −2.51564 − 0.819496i −2.82083 + 0.207107i −4.82843 0
699.12 0.736813 + 1.20711i 2.79793i −0.914214 + 1.77882i 0 −3.37740 + 2.06155i −2.51564 + 0.819496i −2.82083 + 0.207107i −4.82843 0
699.13 1.39897 − 0.207107i − 1.47363i 1.91421 − 0.579471i 0 −0.305198 − 2.06155i −0.819496 − 2.51564i 2.55791 − 1.20711i 0.828427 0
699.14 1.39897 − 0.207107i 1.47363i 1.91421 − 0.579471i 0 0.305198 + 2.06155i −0.819496 + 2.51564i 2.55791 − 1.20711i 0.828427 0
699.15 1.39897 + 0.207107i − 1.47363i 1.91421 + 0.579471i 0 0.305198 − 2.06155i −0.819496 − 2.51564i 2.55791 + 1.20711i 0.828427 0
699.16 1.39897 + 0.207107i 1.47363i 1.91421 + 0.579471i 0 −0.305198 + 2.06155i −0.819496 + 2.51564i 2.55791 + 1.20711i 0.828427 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 699.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
20.d odd 2 1 inner
28.d even 2 1 inner
35.c odd 2 1 inner
140.c 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.c.k 16
4.b odd 2 1 inner 700.2.c.k 16
5.b even 2 1 inner 700.2.c.k 16
5.c odd 4 1 700.2.g.i ✓ 8
5.c odd 4 1 700.2.g.k yes 8
7.b odd 2 1 inner 700.2.c.k 16
20.d odd 2 1 inner 700.2.c.k 16
20.e even 4 1 700.2.g.i ✓ 8
20.e even 4 1 700.2.g.k yes 8
28.d even 2 1 inner 700.2.c.k 16
35.c odd 2 1 inner 700.2.c.k 16
35.f even 4 1 700.2.g.i ✓ 8
35.f even 4 1 700.2.g.k yes 8
140.c even 2 1 inner 700.2.c.k 16
140.j odd 4 1 700.2.g.i ✓ 8
140.j odd 4 1 700.2.g.k yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
700.2.c.k 16 1.a even 1 1 trivial
700.2.c.k 16 4.b odd 2 1 inner
700.2.c.k 16 5.b even 2 1 inner
700.2.c.k 16 7.b odd 2 1 inner
700.2.c.k 16 20.d odd 2 1 inner
700.2.c.k 16 28.d even 2 1 inner
700.2.c.k 16 35.c odd 2 1 inner
700.2.c.k 16 140.c even 2 1 inner
700.2.g.i ✓ 8 5.c odd 4 1
700.2.g.i ✓ 8 20.e even 4 1
700.2.g.i ✓ 8 35.f even 4 1
700.2.g.i ✓ 8 140.j odd 4 1
700.2.g.k yes 8 5.c odd 4 1
700.2.g.k yes 8 20.e even 4 1
700.2.g.k yes 8 35.f even 4 1
700.2.g.k yes 8 140.j odd 4 1

Hecke kernels

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

\( T_{3}^{4} + 10T_{3}^{2} + 17 \) Copy content Toggle raw display
\( T_{11}^{4} + 10T_{11}^{2} + 17 \) Copy content Toggle raw display
\( T_{13}^{2} - 34 \) Copy content Toggle raw display
\( T_{19}^{4} - 58T_{19}^{2} + 833 \) Copy content Toggle raw display
\( T_{23}^{4} - 20T_{23}^{2} + 68 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 2 T^{6} + T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} + 10 T^{2} + 17)^{4} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - 30 T^{4} + 2401)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 10 T^{2} + 17)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 34)^{8} \) Copy content Toggle raw display
$17$ \( (T^{2} - 17)^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} - 58 T^{2} + 833)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 20 T^{2} + 68)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 8 T - 2)^{8} \) Copy content Toggle raw display
$31$ \( (T^{4} - 20 T^{2} + 68)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 44 T^{2} + 196)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 17)^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} - 40 T^{2} + 272)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 180 T^{2} + 5508)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} + 164 T^{2} + 2116)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 40 T^{2} + 272)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 136)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} - 218 T^{2} + 8993)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 116 T^{2} + 3332)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 153)^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} + 80 T^{2} + 1088)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 10 T^{2} + 17)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 153)^{8} \) Copy content Toggle raw display
$97$ \( (T^{2} - 68)^{8} \) Copy content Toggle raw display
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