Properties

Label 700.2.c.d
Level $700$
Weight $2$
Character orbit 700.c
Analytic conductor $5.590$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{7})\)
Defining polynomial: \(x^{4} - 3 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 28)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -2 \beta_{1} + \beta_{3} ) q^{7} + ( \beta_{1} + 2 \beta_{3} ) q^{8} + 3 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -2 \beta_{1} + \beta_{3} ) q^{7} + ( \beta_{1} + 2 \beta_{3} ) q^{8} + 3 q^{9} + ( -2 + 4 \beta_{2} ) q^{11} + ( -3 - \beta_{2} ) q^{14} + ( -1 + 3 \beta_{2} ) q^{16} + 3 \beta_{1} q^{18} + ( -2 \beta_{1} + 8 \beta_{3} ) q^{22} + ( 4 \beta_{1} - 2 \beta_{3} ) q^{23} + ( -3 \beta_{1} - 2 \beta_{3} ) q^{28} + 2 q^{29} + ( -\beta_{1} + 6 \beta_{3} ) q^{32} + ( 3 + 3 \beta_{2} ) q^{36} -6 \beta_{3} q^{37} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{43} + ( -10 + 6 \beta_{2} ) q^{44} + ( 6 + 2 \beta_{2} ) q^{46} + 7 q^{49} -10 \beta_{3} q^{53} + ( -1 - 5 \beta_{2} ) q^{56} + 2 \beta_{1} q^{58} + ( -6 \beta_{1} + 3 \beta_{3} ) q^{63} + ( -7 + 5 \beta_{2} ) q^{64} + ( -12 \beta_{1} + 6 \beta_{3} ) q^{67} + ( 2 - 4 \beta_{2} ) q^{71} + ( 3 \beta_{1} + 6 \beta_{3} ) q^{72} + ( 6 - 6 \beta_{2} ) q^{74} -14 \beta_{3} q^{77} + ( 6 - 12 \beta_{2} ) q^{79} + 9 q^{81} + ( -6 - 2 \beta_{2} ) q^{86} + ( -10 \beta_{1} + 12 \beta_{3} ) q^{88} + ( 6 \beta_{1} + 4 \beta_{3} ) q^{92} + 7 \beta_{1} q^{98} + ( -6 + 12 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 6q^{4} + 12q^{9} + O(q^{10}) \) \( 4q + 6q^{4} + 12q^{9} - 14q^{14} + 2q^{16} + 8q^{29} + 18q^{36} - 28q^{44} + 28q^{46} + 28q^{49} - 14q^{56} - 18q^{64} + 12q^{74} + 36q^{81} - 28q^{86} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 3 x^{2} + 4\):

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

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
−1.32288 0.500000i
−1.32288 + 0.500000i
1.32288 0.500000i
1.32288 + 0.500000i
−1.32288 0.500000i 0 1.50000 + 1.32288i 0 0 2.64575 −1.32288 2.50000i 3.00000 0
699.2 −1.32288 + 0.500000i 0 1.50000 1.32288i 0 0 2.64575 −1.32288 + 2.50000i 3.00000 0
699.3 1.32288 0.500000i 0 1.50000 1.32288i 0 0 −2.64575 1.32288 2.50000i 3.00000 0
699.4 1.32288 + 0.500000i 0 1.50000 + 1.32288i 0 0 −2.64575 1.32288 + 2.50000i 3.00000 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
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.d 4
4.b odd 2 1 inner 700.2.c.d 4
5.b even 2 1 inner 700.2.c.d 4
5.c odd 4 1 28.2.d.a 2
5.c odd 4 1 700.2.g.a 2
7.b odd 2 1 CM 700.2.c.d 4
15.e even 4 1 252.2.b.a 2
20.d odd 2 1 inner 700.2.c.d 4
20.e even 4 1 28.2.d.a 2
20.e even 4 1 700.2.g.a 2
28.d even 2 1 inner 700.2.c.d 4
35.c odd 2 1 inner 700.2.c.d 4
35.f even 4 1 28.2.d.a 2
35.f even 4 1 700.2.g.a 2
35.k even 12 2 196.2.f.b 4
35.l odd 12 2 196.2.f.b 4
40.i odd 4 1 448.2.f.b 2
40.k even 4 1 448.2.f.b 2
60.l odd 4 1 252.2.b.a 2
80.i odd 4 1 1792.2.e.b 4
80.j even 4 1 1792.2.e.b 4
80.s even 4 1 1792.2.e.b 4
80.t odd 4 1 1792.2.e.b 4
105.k odd 4 1 252.2.b.a 2
120.q odd 4 1 4032.2.b.e 2
120.w even 4 1 4032.2.b.e 2
140.c even 2 1 inner 700.2.c.d 4
140.j odd 4 1 28.2.d.a 2
140.j odd 4 1 700.2.g.a 2
140.w even 12 2 196.2.f.b 4
140.x odd 12 2 196.2.f.b 4
280.s even 4 1 448.2.f.b 2
280.y odd 4 1 448.2.f.b 2
420.w even 4 1 252.2.b.a 2
560.r even 4 1 1792.2.e.b 4
560.u odd 4 1 1792.2.e.b 4
560.bm odd 4 1 1792.2.e.b 4
560.bn even 4 1 1792.2.e.b 4
840.bm even 4 1 4032.2.b.e 2
840.bp odd 4 1 4032.2.b.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.d.a 2 5.c odd 4 1
28.2.d.a 2 20.e even 4 1
28.2.d.a 2 35.f even 4 1
28.2.d.a 2 140.j odd 4 1
196.2.f.b 4 35.k even 12 2
196.2.f.b 4 35.l odd 12 2
196.2.f.b 4 140.w even 12 2
196.2.f.b 4 140.x odd 12 2
252.2.b.a 2 15.e even 4 1
252.2.b.a 2 60.l odd 4 1
252.2.b.a 2 105.k odd 4 1
252.2.b.a 2 420.w even 4 1
448.2.f.b 2 40.i odd 4 1
448.2.f.b 2 40.k even 4 1
448.2.f.b 2 280.s even 4 1
448.2.f.b 2 280.y odd 4 1
700.2.c.d 4 1.a even 1 1 trivial
700.2.c.d 4 4.b odd 2 1 inner
700.2.c.d 4 5.b even 2 1 inner
700.2.c.d 4 7.b odd 2 1 CM
700.2.c.d 4 20.d odd 2 1 inner
700.2.c.d 4 28.d even 2 1 inner
700.2.c.d 4 35.c odd 2 1 inner
700.2.c.d 4 140.c even 2 1 inner
700.2.g.a 2 5.c odd 4 1
700.2.g.a 2 20.e even 4 1
700.2.g.a 2 35.f even 4 1
700.2.g.a 2 140.j odd 4 1
1792.2.e.b 4 80.i odd 4 1
1792.2.e.b 4 80.j even 4 1
1792.2.e.b 4 80.s even 4 1
1792.2.e.b 4 80.t odd 4 1
1792.2.e.b 4 560.r even 4 1
1792.2.e.b 4 560.u odd 4 1
1792.2.e.b 4 560.bm odd 4 1
1792.2.e.b 4 560.bn even 4 1
4032.2.b.e 2 120.q odd 4 1
4032.2.b.e 2 120.w even 4 1
4032.2.b.e 2 840.bm even 4 1
4032.2.b.e 2 840.bp odd 4 1

Hecke kernels

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

\( T_{3} \)
\( T_{11}^{2} + 28 \)
\( T_{13} \)
\( T_{19} \)
\( T_{23}^{2} - 28 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 - 3 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( -7 + T^{2} )^{2} \)
$11$ \( ( 28 + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( ( -28 + T^{2} )^{2} \)
$29$ \( ( -2 + T )^{4} \)
$31$ \( T^{4} \)
$37$ \( ( 36 + T^{2} )^{2} \)
$41$ \( T^{4} \)
$43$ \( ( -28 + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( ( 100 + T^{2} )^{2} \)
$59$ \( T^{4} \)
$61$ \( T^{4} \)
$67$ \( ( -252 + T^{2} )^{2} \)
$71$ \( ( 28 + T^{2} )^{2} \)
$73$ \( T^{4} \)
$79$ \( ( 252 + T^{2} )^{2} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
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