Properties

Label 700.2.c.h
Level $700$
Weight $2$
Character orbit 700.c
Analytic conductor $5.590$
Analytic rank $0$
Dimension $8$
CM discriminant -7
Inner twists $8$

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Newspace parameters

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

Self dual: no
Analytic conductor: \(5.58952814149\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.49787136.1
Defining polynomial: \(x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -1 + \beta_{2} ) q^{4} + ( \beta_{1} - \beta_{7} ) q^{7} + ( -\beta_{1} + \beta_{3} ) q^{8} + 3 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -1 + \beta_{2} ) q^{4} + ( \beta_{1} - \beta_{7} ) q^{7} + ( -\beta_{1} + \beta_{3} ) q^{8} + 3 q^{9} + ( 1 - 2 \beta_{6} ) q^{11} + ( 1 + \beta_{2} + \beta_{6} ) q^{14} + ( -\beta_{2} + \beta_{4} ) q^{16} + 3 \beta_{1} q^{18} + ( \beta_{1} + 2 \beta_{5} - 2 \beta_{7} ) q^{22} + ( \beta_{1} + 4 \beta_{5} - \beta_{7} ) q^{23} + ( \beta_{1} + \beta_{3} - \beta_{5} + \beta_{7} ) q^{28} + ( -1 + 4 \beta_{2} - 2 \beta_{4} - 2 \beta_{6} ) q^{29} + ( \beta_{1} - \beta_{3} + 3 \beta_{5} + \beta_{7} ) q^{32} + ( -3 + 3 \beta_{2} ) q^{36} + ( 5 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} + \beta_{7} ) q^{37} + ( -\beta_{1} + 6 \beta_{5} + \beta_{7} ) q^{43} + ( 1 - \beta_{2} + 4 \beta_{6} ) q^{44} + ( -3 - 3 \beta_{2} + 5 \beta_{6} ) q^{46} + 7 q^{49} + ( 2 \beta_{1} - 4 \beta_{3} + 2 \beta_{7} ) q^{53} + ( -3 + 2 \beta_{2} + \beta_{4} - 2 \beta_{6} ) q^{56} + ( -3 \beta_{1} + 4 \beta_{3} - 4 \beta_{5} - 4 \beta_{7} ) q^{58} + ( 3 \beta_{1} - 3 \beta_{7} ) q^{63} + ( -5 - 2 \beta_{2} - \beta_{4} + 2 \beta_{6} ) q^{64} + ( -3 \beta_{1} - 2 \beta_{5} + 3 \beta_{7} ) q^{67} + ( 3 - 4 \beta_{2} - 2 \beta_{4} - 4 \beta_{6} ) q^{71} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{72} + ( -11 + 3 \beta_{2} + 2 \beta_{4} + \beta_{6} ) q^{74} + ( -7 \beta_{1} + 2 \beta_{3} - 2 \beta_{5} - 3 \beta_{7} ) q^{77} + ( 1 - 4 \beta_{2} - 2 \beta_{4} ) q^{79} + 9 q^{81} + ( -7 - 7 \beta_{2} + 5 \beta_{6} ) q^{86} + ( \beta_{1} - \beta_{3} - 4 \beta_{5} + 4 \beta_{7} ) q^{88} + ( -3 \beta_{1} - 3 \beta_{3} - 5 \beta_{5} + 5 \beta_{7} ) q^{92} + 7 \beta_{1} q^{98} + ( 3 - 6 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 6q^{4} + 24q^{9} + O(q^{10}) \) \( 8q - 6q^{4} + 24q^{9} + 14q^{14} - 2q^{16} - 8q^{29} - 18q^{36} + 22q^{44} - 10q^{46} + 56q^{49} - 28q^{56} - 36q^{64} - 78q^{74} + 72q^{81} - 50q^{86} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 1 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} + \nu^{2} + 1 \)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} + \nu^{5} - \nu^{3} - 8 \nu \)\()/8\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{6} + \nu^{4} + 3 \nu^{2} + 8 \)\()/2\)
\(\beta_{7}\)\(=\)\((\)\( 3 \nu^{7} + 5 \nu^{5} + 11 \nu^{3} + 24 \nu \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 1\)
\(\nu^{3}\)\(=\)\(\beta_{3} - \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4} - \beta_{2}\)
\(\nu^{5}\)\(=\)\(\beta_{7} + 3 \beta_{5} - \beta_{3} + \beta_{1}\)
\(\nu^{6}\)\(=\)\(2 \beta_{6} - \beta_{4} - 2 \beta_{2} - 5\)
\(\nu^{7}\)\(=\)\(\beta_{7} - 5 \beta_{5} - 2 \beta_{3} - 6 \beta_{1}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
699.1
−1.09445 0.895644i
−1.09445 + 0.895644i
−0.228425 1.39564i
−0.228425 + 1.39564i
0.228425 1.39564i
0.228425 + 1.39564i
1.09445 0.895644i
1.09445 + 0.895644i
−1.09445 0.895644i 0 0.395644 + 1.96048i 0 0 −2.64575 1.32288 2.50000i 3.00000 0
699.2 −1.09445 + 0.895644i 0 0.395644 1.96048i 0 0 −2.64575 1.32288 + 2.50000i 3.00000 0
699.3 −0.228425 1.39564i 0 −1.89564 + 0.637600i 0 0 −2.64575 1.32288 + 2.50000i 3.00000 0
699.4 −0.228425 + 1.39564i 0 −1.89564 0.637600i 0 0 −2.64575 1.32288 2.50000i 3.00000 0
699.5 0.228425 1.39564i 0 −1.89564 0.637600i 0 0 2.64575 −1.32288 + 2.50000i 3.00000 0
699.6 0.228425 + 1.39564i 0 −1.89564 + 0.637600i 0 0 2.64575 −1.32288 2.50000i 3.00000 0
699.7 1.09445 0.895644i 0 0.395644 1.96048i 0 0 2.64575 −1.32288 2.50000i 3.00000 0
699.8 1.09445 + 0.895644i 0 0.395644 + 1.96048i 0 0 2.64575 −1.32288 + 2.50000i 3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 699.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
4.b odd 2 1 inner
5.b even 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.h 8
4.b odd 2 1 inner 700.2.c.h 8
5.b even 2 1 inner 700.2.c.h 8
5.c odd 4 1 700.2.g.c 4
5.c odd 4 1 700.2.g.e yes 4
7.b odd 2 1 CM 700.2.c.h 8
20.d odd 2 1 inner 700.2.c.h 8
20.e even 4 1 700.2.g.c 4
20.e even 4 1 700.2.g.e yes 4
28.d even 2 1 inner 700.2.c.h 8
35.c odd 2 1 inner 700.2.c.h 8
35.f even 4 1 700.2.g.c 4
35.f even 4 1 700.2.g.e yes 4
140.c even 2 1 inner 700.2.c.h 8
140.j odd 4 1 700.2.g.c 4
140.j odd 4 1 700.2.g.e yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
700.2.c.h 8 1.a even 1 1 trivial
700.2.c.h 8 4.b odd 2 1 inner
700.2.c.h 8 5.b even 2 1 inner
700.2.c.h 8 7.b odd 2 1 CM
700.2.c.h 8 20.d odd 2 1 inner
700.2.c.h 8 28.d even 2 1 inner
700.2.c.h 8 35.c odd 2 1 inner
700.2.c.h 8 140.c even 2 1 inner
700.2.g.c 4 5.c odd 4 1
700.2.g.c 4 20.e even 4 1
700.2.g.c 4 35.f even 4 1
700.2.g.c 4 140.j odd 4 1
700.2.g.e yes 4 5.c odd 4 1
700.2.g.e yes 4 20.e even 4 1
700.2.g.e yes 4 35.f even 4 1
700.2.g.e yes 4 140.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 \(S_{2}^{\mathrm{new}}(700, [\chi])\):

\( T_{3} \)
\( T_{11}^{4} + 38 T_{11}^{2} + 25 \)
\( T_{13} \)
\( T_{19} \)
\( T_{23}^{4} - 110 T_{23}^{2} + 1681 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 + 12 T^{2} + 5 T^{4} + 3 T^{6} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( T^{8} \)
$7$ \( ( -7 + T^{2} )^{4} \)
$11$ \( ( 25 + 38 T^{2} + T^{4} )^{2} \)
$13$ \( T^{8} \)
$17$ \( T^{8} \)
$19$ \( T^{8} \)
$23$ \( ( 1681 - 110 T^{2} + T^{4} )^{2} \)
$29$ \( ( -83 + 2 T + T^{2} )^{4} \)
$31$ \( T^{8} \)
$37$ \( ( 5625 + 186 T^{2} + T^{4} )^{2} \)
$41$ \( T^{8} \)
$43$ \( ( 10201 - 230 T^{2} + T^{4} )^{2} \)
$47$ \( T^{8} \)
$53$ \( ( 100 + T^{2} )^{4} \)
$59$ \( T^{8} \)
$61$ \( T^{8} \)
$67$ \( ( 2601 - 150 T^{2} + T^{4} )^{2} \)
$71$ \( ( 34225 + 398 T^{2} + T^{4} )^{2} \)
$73$ \( T^{8} \)
$79$ \( ( 225 + 222 T^{2} + T^{4} )^{2} \)
$83$ \( T^{8} \)
$89$ \( T^{8} \)
$97$ \( T^{8} \)
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