Properties

Label 700.2.g.d
Level $700$
Weight $2$
Character orbit 700.g
Analytic conductor $5.590$
Analytic rank $0$
Dimension $4$
CM discriminant -20
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.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.58952814149\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
Defining polynomial: \(x^{4} - 4 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( 2 \beta_{1} - \beta_{2} ) q^{3} -2 q^{4} + 2 \beta_{3} q^{6} + ( \beta_{1} - 2 \beta_{2} ) q^{7} -2 \beta_{2} q^{8} + 7 q^{9} +O(q^{10})\) \( q + \beta_{2} q^{2} + ( 2 \beta_{1} - \beta_{2} ) q^{3} -2 q^{4} + 2 \beta_{3} q^{6} + ( \beta_{1} - 2 \beta_{2} ) q^{7} -2 \beta_{2} q^{8} + 7 q^{9} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{12} + ( 3 + \beta_{3} ) q^{14} + 4 q^{16} + 7 \beta_{2} q^{18} + ( 5 - 3 \beta_{3} ) q^{21} + \beta_{2} q^{23} -4 \beta_{3} q^{24} + ( 8 \beta_{1} - 4 \beta_{2} ) q^{27} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{28} -6 q^{29} + 4 \beta_{2} q^{32} -14 q^{36} + 2 \beta_{3} q^{41} + ( 6 \beta_{1} + 2 \beta_{2} ) q^{42} + 9 \beta_{2} q^{43} -2 q^{46} + ( -6 \beta_{1} + 3 \beta_{2} ) q^{47} + ( 8 \beta_{1} - 4 \beta_{2} ) q^{48} + ( -2 - 3 \beta_{3} ) q^{49} + 8 \beta_{3} q^{54} + ( -6 - 2 \beta_{3} ) q^{56} -6 \beta_{2} q^{58} -6 \beta_{3} q^{61} + ( 7 \beta_{1} - 14 \beta_{2} ) q^{63} -8 q^{64} + 3 \beta_{2} q^{67} + 2 \beta_{3} q^{69} -14 \beta_{2} q^{72} + 19 q^{81} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{82} + ( -6 \beta_{1} + 3 \beta_{2} ) q^{83} + ( -10 + 6 \beta_{3} ) q^{84} -18 q^{86} + ( -12 \beta_{1} + 6 \beta_{2} ) q^{87} -8 \beta_{3} q^{89} -2 \beta_{2} q^{92} -6 \beta_{3} q^{94} + 8 \beta_{3} q^{96} + ( 6 \beta_{1} - 5 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{4} + 28q^{9} + O(q^{10}) \) \( 4q - 8q^{4} + 28q^{9} + 12q^{14} + 16q^{16} + 20q^{21} - 24q^{29} - 56q^{36} - 8q^{46} - 8q^{49} - 24q^{56} - 32q^{64} + 76q^{81} - 40q^{84} - 72q^{86} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 4 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - \nu \)\()/3\)
\(\beta_{3}\)\(=\)\( \nu^{2} - 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 2\)
\(\nu^{3}\)\(=\)\(3 \beta_{2} + \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
−1.58114 0.707107i
1.58114 0.707107i
−1.58114 + 0.707107i
1.58114 + 0.707107i
1.41421i −3.16228 −2.00000 0 4.47214i −1.58114 + 2.12132i 2.82843i 7.00000 0
251.2 1.41421i 3.16228 −2.00000 0 4.47214i 1.58114 + 2.12132i 2.82843i 7.00000 0
251.3 1.41421i −3.16228 −2.00000 0 4.47214i −1.58114 2.12132i 2.82843i 7.00000 0
251.4 1.41421i 3.16228 −2.00000 0 4.47214i 1.58114 2.12132i 2.82843i 7.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
7.b 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.g.d 4
4.b odd 2 1 inner 700.2.g.d 4
5.b even 2 1 inner 700.2.g.d 4
5.c odd 4 2 140.2.c.a 4
7.b odd 2 1 inner 700.2.g.d 4
20.d odd 2 1 CM 700.2.g.d 4
20.e even 4 2 140.2.c.a 4
28.d even 2 1 inner 700.2.g.d 4
35.c odd 2 1 inner 700.2.g.d 4
35.f even 4 2 140.2.c.a 4
35.k even 12 4 980.2.s.b 8
35.l odd 12 4 980.2.s.b 8
40.i odd 4 2 2240.2.e.a 4
40.k even 4 2 2240.2.e.a 4
140.c even 2 1 inner 700.2.g.d 4
140.j odd 4 2 140.2.c.a 4
140.w even 12 4 980.2.s.b 8
140.x odd 12 4 980.2.s.b 8
280.s even 4 2 2240.2.e.a 4
280.y odd 4 2 2240.2.e.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.c.a 4 5.c odd 4 2
140.2.c.a 4 20.e even 4 2
140.2.c.a 4 35.f even 4 2
140.2.c.a 4 140.j odd 4 2
700.2.g.d 4 1.a even 1 1 trivial
700.2.g.d 4 4.b odd 2 1 inner
700.2.g.d 4 5.b even 2 1 inner
700.2.g.d 4 7.b odd 2 1 inner
700.2.g.d 4 20.d odd 2 1 CM
700.2.g.d 4 28.d even 2 1 inner
700.2.g.d 4 35.c odd 2 1 inner
700.2.g.d 4 140.c even 2 1 inner
980.2.s.b 8 35.k even 12 4
980.2.s.b 8 35.l odd 12 4
980.2.s.b 8 140.w even 12 4
980.2.s.b 8 140.x odd 12 4
2240.2.e.a 4 40.i odd 4 2
2240.2.e.a 4 40.k even 4 2
2240.2.e.a 4 280.s even 4 2
2240.2.e.a 4 280.y odd 4 2

Hecke kernels

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

\( T_{3}^{2} - 10 \)
\( T_{11} \)
\( T_{19} \)
\( T_{37} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 2 + T^{2} )^{2} \)
$3$ \( ( -10 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( 49 + 4 T^{2} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( ( 2 + T^{2} )^{2} \)
$29$ \( ( 6 + T )^{4} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( ( 20 + T^{2} )^{2} \)
$43$ \( ( 162 + T^{2} )^{2} \)
$47$ \( ( -90 + T^{2} )^{2} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( 180 + T^{2} )^{2} \)
$67$ \( ( 18 + T^{2} )^{2} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( T^{4} \)
$83$ \( ( -90 + T^{2} )^{2} \)
$89$ \( ( 320 + T^{2} )^{2} \)
$97$ \( T^{4} \)
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