Properties

Label 700.2.c.a
Level $700$
Weight $2$
Character orbit 700.c
Analytic conductor $5.590$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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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: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{2} + \beta_{2} q^{3} -2 \beta_{1} q^{4} + ( -\beta_{2} - \beta_{3} ) q^{6} + ( -1 - \beta_{2} ) q^{7} + ( 2 + 2 \beta_{1} ) q^{8} -3 q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{2} + \beta_{2} q^{3} -2 \beta_{1} q^{4} + ( -\beta_{2} - \beta_{3} ) q^{6} + ( -1 - \beta_{2} ) q^{7} + ( 2 + 2 \beta_{1} ) q^{8} -3 q^{9} -5 \beta_{1} q^{11} + 2 \beta_{3} q^{12} + \beta_{3} q^{13} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{14} -4 q^{16} + 2 \beta_{3} q^{17} + ( 3 - 3 \beta_{1} ) q^{18} + ( 6 - \beta_{2} ) q^{21} + ( 5 + 5 \beta_{1} ) q^{22} - q^{23} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{24} + ( \beta_{2} - \beta_{3} ) q^{26} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{28} + 5 q^{29} + 3 \beta_{3} q^{31} + ( 4 - 4 \beta_{1} ) q^{32} + 5 \beta_{3} q^{33} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{34} + 6 \beta_{1} q^{36} -3 \beta_{1} q^{37} + 6 \beta_{1} q^{39} -5 \beta_{2} q^{41} + ( -6 + 6 \beta_{1} + \beta_{2} + \beta_{3} ) q^{42} -11 q^{43} -10 q^{44} + ( 1 - \beta_{1} ) q^{46} -2 \beta_{2} q^{47} -4 \beta_{2} q^{48} + ( -5 + 2 \beta_{2} ) q^{49} + 12 \beta_{1} q^{51} -2 \beta_{2} q^{52} + 4 \beta_{1} q^{53} + ( -2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{56} + ( -5 + 5 \beta_{1} ) q^{58} + 5 \beta_{3} q^{59} + 5 \beta_{2} q^{61} + ( 3 \beta_{2} - 3 \beta_{3} ) q^{62} + ( 3 + 3 \beta_{2} ) q^{63} + 8 \beta_{1} q^{64} + ( 5 \beta_{2} - 5 \beta_{3} ) q^{66} + 3 q^{67} -4 \beta_{2} q^{68} -\beta_{2} q^{69} + 5 \beta_{1} q^{71} + ( -6 - 6 \beta_{1} ) q^{72} + \beta_{3} q^{73} + ( 3 + 3 \beta_{1} ) q^{74} + ( 5 \beta_{1} - 5 \beta_{3} ) q^{77} + ( -6 - 6 \beta_{1} ) q^{78} -9 \beta_{1} q^{79} -9 q^{81} + ( 5 \beta_{2} + 5 \beta_{3} ) q^{82} + \beta_{2} q^{83} + ( -12 \beta_{1} - 2 \beta_{3} ) q^{84} + ( 11 - 11 \beta_{1} ) q^{86} + 5 \beta_{2} q^{87} + ( 10 - 10 \beta_{1} ) q^{88} -\beta_{2} q^{89} + ( -6 \beta_{1} - \beta_{3} ) q^{91} + 2 \beta_{1} q^{92} + 18 \beta_{1} q^{93} + ( 2 \beta_{2} + 2 \beta_{3} ) q^{94} + ( 4 \beta_{2} + 4 \beta_{3} ) q^{96} -3 \beta_{3} q^{97} + ( 5 - 5 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{98} + 15 \beta_{1} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} - 4q^{7} + 8q^{8} - 12q^{9} + O(q^{10}) \) \( 4q - 4q^{2} - 4q^{7} + 8q^{8} - 12q^{9} + 4q^{14} - 16q^{16} + 12q^{18} + 24q^{21} + 20q^{22} - 4q^{23} + 20q^{29} + 16q^{32} - 24q^{42} - 44q^{43} - 40q^{44} + 4q^{46} - 20q^{49} - 8q^{56} - 20q^{58} + 12q^{63} + 12q^{67} - 24q^{72} + 12q^{74} - 24q^{78} - 36q^{81} + 44q^{86} + 40q^{88} + 20q^{98} + O(q^{100}) \)

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

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

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
1.22474 1.22474i
−1.22474 + 1.22474i
−1.22474 1.22474i
1.22474 + 1.22474i
−1.00000 1.00000i 2.44949i 2.00000i 0 −2.44949 + 2.44949i −1.00000 + 2.44949i 2.00000 2.00000i −3.00000 0
699.2 −1.00000 1.00000i 2.44949i 2.00000i 0 2.44949 2.44949i −1.00000 2.44949i 2.00000 2.00000i −3.00000 0
699.3 −1.00000 + 1.00000i 2.44949i 2.00000i 0 2.44949 + 2.44949i −1.00000 + 2.44949i 2.00000 + 2.00000i −3.00000 0
699.4 −1.00000 + 1.00000i 2.44949i 2.00000i 0 −2.44949 2.44949i −1.00000 2.44949i 2.00000 + 2.00000i −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 inner
20.d 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.a 4
4.b odd 2 1 700.2.c.g 4
5.b even 2 1 700.2.c.g 4
5.c odd 4 1 700.2.g.b 4
5.c odd 4 1 700.2.g.h yes 4
7.b odd 2 1 inner 700.2.c.a 4
20.d odd 2 1 inner 700.2.c.a 4
20.e even 4 1 700.2.g.b 4
20.e even 4 1 700.2.g.h yes 4
28.d even 2 1 700.2.c.g 4
35.c odd 2 1 700.2.c.g 4
35.f even 4 1 700.2.g.b 4
35.f even 4 1 700.2.g.h yes 4
140.c even 2 1 inner 700.2.c.a 4
140.j odd 4 1 700.2.g.b 4
140.j odd 4 1 700.2.g.h yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
700.2.c.a 4 1.a even 1 1 trivial
700.2.c.a 4 7.b odd 2 1 inner
700.2.c.a 4 20.d odd 2 1 inner
700.2.c.a 4 140.c even 2 1 inner
700.2.c.g 4 4.b odd 2 1
700.2.c.g 4 5.b even 2 1
700.2.c.g 4 28.d even 2 1
700.2.c.g 4 35.c odd 2 1
700.2.g.b 4 5.c odd 4 1
700.2.g.b 4 20.e even 4 1
700.2.g.b 4 35.f even 4 1
700.2.g.b 4 140.j odd 4 1
700.2.g.h yes 4 5.c odd 4 1
700.2.g.h yes 4 20.e even 4 1
700.2.g.h yes 4 35.f even 4 1
700.2.g.h 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 \(S_{2}^{\mathrm{new}}(700, [\chi])\):

\( T_{3}^{2} + 6 \)
\( T_{11}^{2} + 25 \)
\( T_{13}^{2} - 6 \)
\( T_{19} \)
\( T_{23} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 2 + 2 T + T^{2} )^{2} \)
$3$ \( ( 6 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( ( 7 + 2 T + T^{2} )^{2} \)
$11$ \( ( 25 + T^{2} )^{2} \)
$13$ \( ( -6 + T^{2} )^{2} \)
$17$ \( ( -24 + T^{2} )^{2} \)
$19$ \( T^{4} \)
$23$ \( ( 1 + T )^{4} \)
$29$ \( ( -5 + T )^{4} \)
$31$ \( ( -54 + T^{2} )^{2} \)
$37$ \( ( 9 + T^{2} )^{2} \)
$41$ \( ( 150 + T^{2} )^{2} \)
$43$ \( ( 11 + T )^{4} \)
$47$ \( ( 24 + T^{2} )^{2} \)
$53$ \( ( 16 + T^{2} )^{2} \)
$59$ \( ( -150 + T^{2} )^{2} \)
$61$ \( ( 150 + T^{2} )^{2} \)
$67$ \( ( -3 + T )^{4} \)
$71$ \( ( 25 + T^{2} )^{2} \)
$73$ \( ( -6 + T^{2} )^{2} \)
$79$ \( ( 81 + T^{2} )^{2} \)
$83$ \( ( 6 + T^{2} )^{2} \)
$89$ \( ( 6 + T^{2} )^{2} \)
$97$ \( ( -54 + T^{2} )^{2} \)
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