Properties

Label 700.2.g.j
Level $700$
Weight $2$
Character orbit 700.g
Analytic conductor $5.590$
Analytic rank $0$
Dimension $8$
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.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.58952814149\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.342102016.5
Defining polynomial: \(x^{8} + x^{6} + 4 x^{4} + 4 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( \beta_{5} - \beta_{7} ) q^{3} + ( -\beta_{1} + \beta_{2} - \beta_{4} ) q^{4} + ( \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{6} + ( \beta_{2} - \beta_{5} + \beta_{7} ) q^{7} + ( -2 + \beta_{1} + \beta_{5} - \beta_{6} ) q^{8} + ( 3 + 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( \beta_{5} - \beta_{7} ) q^{3} + ( -\beta_{1} + \beta_{2} - \beta_{4} ) q^{4} + ( \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{6} + ( \beta_{2} - \beta_{5} + \beta_{7} ) q^{7} + ( -2 + \beta_{1} + \beta_{5} - \beta_{6} ) q^{8} + ( 3 + 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{9} + ( 2 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{11} + ( \beta_{3} - \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{12} + ( -\beta_{6} - \beta_{7} ) q^{13} + ( -1 - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} ) q^{14} + ( -2 - 3 \beta_{1} - \beta_{5} + \beta_{6} ) q^{16} + ( -\beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{17} + ( 4 + \beta_{1} + 2 \beta_{2} - 2 \beta_{4} ) q^{18} + ( \beta_{2} + \beta_{4} + \beta_{5} - \beta_{7} ) q^{19} + ( -5 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{21} + ( -4 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} ) q^{22} + ( -4 \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{23} + ( \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{24} + ( -\beta_{3} + \beta_{4} ) q^{26} + ( 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{27} + ( -2 - \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{28} -2 q^{29} + ( 2 \beta_{2} + 2 \beta_{4} ) q^{31} + ( 2 + \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + \beta_{5} - \beta_{6} ) q^{32} + ( 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} ) q^{33} + ( 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} ) q^{34} + ( -4 + 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{36} -2 q^{37} + ( \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{38} + ( -2 \beta_{2} + 2 \beta_{4} ) q^{39} + ( -\beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{41} + ( -2 - 4 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{42} + ( 4 \beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{43} + ( -4 - 2 \beta_{1} + 2 \beta_{5} - 2 \beta_{6} ) q^{44} + ( 6 + 4 \beta_{1} - 3 \beta_{2} + 3 \beta_{4} + \beta_{5} - \beta_{6} ) q^{46} + ( 3 \beta_{2} + 3 \beta_{4} ) q^{47} + ( -2 \beta_{2} - 3 \beta_{3} + \beta_{4} - 4 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{48} + ( 4 - \beta_{1} - \beta_{3} - 3 \beta_{6} - 2 \beta_{7} ) q^{49} + ( -4 \beta_{1} - 2 \beta_{5} + 2 \beta_{6} ) q^{51} + ( \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{52} -2 q^{53} + ( 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} + 4 \beta_{6} ) q^{54} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{56} + ( 8 + 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{57} -2 \beta_{1} q^{58} + ( 3 \beta_{2} + 3 \beta_{4} - \beta_{5} + \beta_{7} ) q^{59} + ( \beta_{6} + \beta_{7} ) q^{61} + ( -2 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{62} + ( 2 \beta_{1} - 5 \beta_{4} - 2 \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{63} + ( 2 + \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} ) q^{64} + ( 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} + 8 \beta_{7} ) q^{66} + ( 8 \beta_{1} - \beta_{2} + \beta_{4} + 4 \beta_{5} - 4 \beta_{6} ) q^{67} + ( 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} ) q^{68} + ( -3 \beta_{2} - 6 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - 5 \beta_{6} - 2 \beta_{7} ) q^{69} + ( 2 \beta_{1} - 3 \beta_{2} + 3 \beta_{4} + \beta_{5} - \beta_{6} ) q^{71} + ( -10 - 7 \beta_{1} + 4 \beta_{2} - 4 \beta_{4} + \beta_{5} - \beta_{6} ) q^{72} + ( -3 \beta_{2} - 6 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} ) q^{73} -2 \beta_{1} q^{74} + ( 2 \beta_{2} + 3 \beta_{3} - \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{76} + ( -4 - 2 \beta_{1} - \beta_{2} - 4 \beta_{3} + \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{77} + ( 4 - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{78} + ( -2 \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{79} + ( 7 + 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{81} + ( 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} ) q^{82} + ( \beta_{5} - \beta_{7} ) q^{83} + ( 2 + 2 \beta_{1} - 6 \beta_{2} - \beta_{3} + 5 \beta_{4} - 4 \beta_{5} + 2 \beta_{7} ) q^{84} + ( -2 - 4 \beta_{1} + \beta_{2} - \beta_{4} - 3 \beta_{5} + 3 \beta_{6} ) q^{86} + ( -2 \beta_{5} + 2 \beta_{7} ) q^{87} + ( -4 - 2 \beta_{1} - 4 \beta_{2} + 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{88} + ( 6 \beta_{6} + 6 \beta_{7} ) q^{89} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} + \beta_{6} - \beta_{7} ) q^{91} + ( 4 + 2 \beta_{1} - 4 \beta_{5} + 4 \beta_{6} ) q^{92} + ( 4 + 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{93} + ( -3 \beta_{2} - 3 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} ) q^{94} + ( -2 \beta_{2} - 3 \beta_{3} + \beta_{4} - 6 \beta_{6} - 6 \beta_{7} ) q^{96} + ( \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{97} + ( -2 + 5 \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{98} + ( 2 \beta_{1} + 9 \beta_{2} - 9 \beta_{4} + \beta_{5} - \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 2q^{2} + 2q^{4} - 14q^{8} + 16q^{9} + O(q^{10}) \) \( 8q - 2q^{2} + 2q^{4} - 14q^{8} + 16q^{9} - 6q^{14} - 14q^{16} + 30q^{18} - 36q^{21} - 28q^{22} - 14q^{28} - 16q^{29} + 18q^{32} - 30q^{36} - 16q^{37} - 8q^{42} - 20q^{44} + 44q^{46} + 36q^{49} - 16q^{53} + 2q^{56} + 48q^{57} + 4q^{58} + 2q^{64} - 62q^{72} + 4q^{74} - 24q^{77} + 24q^{78} + 48q^{81} + 8q^{84} - 20q^{86} - 36q^{88} + 12q^{92} + 16q^{93} - 26q^{98} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{6} - 3 \nu^{4} - 2 \nu^{2} - 8 \)\()/8\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} - \nu^{6} - \nu^{5} - 3 \nu^{4} + 6 \nu^{3} - 10 \nu^{2} - 8 \nu - 8 \)\()/16\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + \nu^{6} - \nu^{5} + 3 \nu^{4} + 6 \nu^{3} + 10 \nu^{2} + 24 \nu + 8 \)\()/16\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} + \nu^{6} - \nu^{5} + 3 \nu^{4} + 6 \nu^{3} + 10 \nu^{2} - 8 \nu + 8 \)\()/16\)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{6} + 4 \nu^{5} + \nu^{4} + 4 \nu^{3} + 6 \nu^{2} + 8 \)\()/16\)
\(\beta_{6}\)\(=\)\((\)\( -3 \nu^{6} + 4 \nu^{5} - \nu^{4} + 4 \nu^{3} - 6 \nu^{2} - 8 \)\()/16\)
\(\beta_{7}\)\(=\)\((\)\( -2 \nu^{7} + 3 \nu^{6} - 2 \nu^{5} + \nu^{4} + 6 \nu^{2} + 8 \)\()/16\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{4} + \beta_{3}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{4} - \beta_{2} + \beta_{1}\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{7} + 2 \beta_{6} + \beta_{4} + \beta_{3} + 2 \beta_{2}\)\()/2\)
\(\nu^{4}\)\(=\)\(\beta_{6} - \beta_{5} - 3 \beta_{1} - 2\)
\(\nu^{5}\)\(=\)\((\)\(-2 \beta_{7} + 2 \beta_{6} + 4 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2}\)\()/2\)
\(\nu^{6}\)\(=\)\(-3 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} + 2 \beta_{2} - \beta_{1} - 2\)
\(\nu^{7}\)\(=\)\((\)\(-14 \beta_{7} - 10 \beta_{6} + 4 \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2}\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
−0.599676 + 1.28078i
0.599676 1.28078i
−0.599676 1.28078i
0.599676 + 1.28078i
1.17915 + 0.780776i
−1.17915 0.780776i
1.17915 0.780776i
−1.17915 + 0.780776i
−1.28078 0.599676i −0.936426 1.28078 + 1.53610i 0 1.19935 + 0.561553i 2.60399 + 0.468213i −0.719224 2.73546i −2.12311 0
251.2 −1.28078 0.599676i 0.936426 1.28078 + 1.53610i 0 −1.19935 0.561553i −2.60399 + 0.468213i −0.719224 2.73546i −2.12311 0
251.3 −1.28078 + 0.599676i −0.936426 1.28078 1.53610i 0 1.19935 0.561553i 2.60399 0.468213i −0.719224 + 2.73546i −2.12311 0
251.4 −1.28078 + 0.599676i 0.936426 1.28078 1.53610i 0 −1.19935 + 0.561553i −2.60399 0.468213i −0.719224 + 2.73546i −2.12311 0
251.5 0.780776 1.17915i −3.02045 −0.780776 1.84130i 0 −2.35829 + 3.56155i 2.17238 1.51022i −2.78078 0.516994i 6.12311 0
251.6 0.780776 1.17915i 3.02045 −0.780776 1.84130i 0 2.35829 3.56155i −2.17238 1.51022i −2.78078 0.516994i 6.12311 0
251.7 0.780776 + 1.17915i −3.02045 −0.780776 + 1.84130i 0 −2.35829 3.56155i 2.17238 + 1.51022i −2.78078 + 0.516994i 6.12311 0
251.8 0.780776 + 1.17915i 3.02045 −0.780776 + 1.84130i 0 2.35829 + 3.56155i −2.17238 + 1.51022i −2.78078 + 0.516994i 6.12311 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
28.d 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.j 8
4.b odd 2 1 inner 700.2.g.j 8
5.b even 2 1 140.2.g.c 8
5.c odd 4 1 700.2.c.i 8
5.c odd 4 1 700.2.c.j 8
7.b odd 2 1 inner 700.2.g.j 8
15.d odd 2 1 1260.2.c.c 8
20.d odd 2 1 140.2.g.c 8
20.e even 4 1 700.2.c.i 8
20.e even 4 1 700.2.c.j 8
28.d even 2 1 inner 700.2.g.j 8
35.c odd 2 1 140.2.g.c 8
35.f even 4 1 700.2.c.i 8
35.f even 4 1 700.2.c.j 8
35.i odd 6 2 980.2.o.e 16
35.j even 6 2 980.2.o.e 16
40.e odd 2 1 2240.2.k.e 8
40.f even 2 1 2240.2.k.e 8
60.h even 2 1 1260.2.c.c 8
105.g even 2 1 1260.2.c.c 8
140.c even 2 1 140.2.g.c 8
140.j odd 4 1 700.2.c.i 8
140.j odd 4 1 700.2.c.j 8
140.p odd 6 2 980.2.o.e 16
140.s even 6 2 980.2.o.e 16
280.c odd 2 1 2240.2.k.e 8
280.n even 2 1 2240.2.k.e 8
420.o odd 2 1 1260.2.c.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.g.c 8 5.b even 2 1
140.2.g.c 8 20.d odd 2 1
140.2.g.c 8 35.c odd 2 1
140.2.g.c 8 140.c even 2 1
700.2.c.i 8 5.c odd 4 1
700.2.c.i 8 20.e even 4 1
700.2.c.i 8 35.f even 4 1
700.2.c.i 8 140.j odd 4 1
700.2.c.j 8 5.c odd 4 1
700.2.c.j 8 20.e even 4 1
700.2.c.j 8 35.f even 4 1
700.2.c.j 8 140.j odd 4 1
700.2.g.j 8 1.a even 1 1 trivial
700.2.g.j 8 4.b odd 2 1 inner
700.2.g.j 8 7.b odd 2 1 inner
700.2.g.j 8 28.d even 2 1 inner
980.2.o.e 16 35.i odd 6 2
980.2.o.e 16 35.j even 6 2
980.2.o.e 16 140.p odd 6 2
980.2.o.e 16 140.s even 6 2
1260.2.c.c 8 15.d odd 2 1
1260.2.c.c 8 60.h even 2 1
1260.2.c.c 8 105.g even 2 1
1260.2.c.c 8 420.o odd 2 1
2240.2.k.e 8 40.e odd 2 1
2240.2.k.e 8 40.f even 2 1
2240.2.k.e 8 280.c odd 2 1
2240.2.k.e 8 280.n even 2 1

Hecke kernels

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

\( T_{3}^{4} - 10 T_{3}^{2} + 8 \)
\( T_{11}^{4} + 28 T_{11}^{2} + 128 \)
\( T_{19}^{4} - 28 T_{19}^{2} + 128 \)
\( T_{37} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 + 2 T + T^{3} + T^{4} )^{2} \)
$3$ \( ( 8 - 10 T^{2} + T^{4} )^{2} \)
$5$ \( T^{8} \)
$7$ \( 2401 - 882 T^{2} + 162 T^{4} - 18 T^{6} + T^{8} \)
$11$ \( ( 128 + 28 T^{2} + T^{4} )^{2} \)
$13$ \( ( 4 + T^{2} )^{4} \)
$17$ \( ( 64 + 52 T^{2} + T^{4} )^{2} \)
$19$ \( ( 128 - 28 T^{2} + T^{4} )^{2} \)
$23$ \( ( 1352 + 74 T^{2} + T^{4} )^{2} \)
$29$ \( ( 2 + T )^{8} \)
$31$ \( ( 512 - 56 T^{2} + T^{4} )^{2} \)
$37$ \( ( 2 + T )^{8} \)
$41$ \( ( 64 + 52 T^{2} + T^{4} )^{2} \)
$43$ \( ( 8 + 58 T^{2} + T^{4} )^{2} \)
$47$ \( ( 2592 - 126 T^{2} + T^{4} )^{2} \)
$53$ \( ( 2 + T )^{8} \)
$59$ \( ( 512 - 124 T^{2} + T^{4} )^{2} \)
$61$ \( ( 4 + T^{2} )^{4} \)
$67$ \( ( 2888 + 218 T^{2} + T^{4} )^{2} \)
$71$ \( ( 2048 + 92 T^{2} + T^{4} )^{2} \)
$73$ \( ( 20736 + 324 T^{2} + T^{4} )^{2} \)
$79$ \( ( 32 + 20 T^{2} + T^{4} )^{2} \)
$83$ \( ( 8 - 10 T^{2} + T^{4} )^{2} \)
$89$ \( ( 144 + T^{2} )^{4} \)
$97$ \( ( 64 + 52 T^{2} + T^{4} )^{2} \)
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