Properties

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

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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: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
−0.366025 1.36603i 1.73205 −1.73205 + 1.00000i 0 −0.633975 2.36603i 1.73205 2.00000i 2.00000 + 2.00000i 0 0
251.2 −0.366025 + 1.36603i 1.73205 −1.73205 1.00000i 0 −0.633975 + 2.36603i 1.73205 + 2.00000i 2.00000 2.00000i 0 0
251.3 1.36603 0.366025i −1.73205 1.73205 1.00000i 0 −2.36603 + 0.633975i −1.73205 + 2.00000i 2.00000 2.00000i 0 0
251.4 1.36603 + 0.366025i −1.73205 1.73205 + 1.00000i 0 −2.36603 0.633975i −1.73205 2.00000i 2.00000 + 2.00000i 0 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
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.f 4
4.b odd 2 1 700.2.g.g 4
5.b even 2 1 140.2.g.a 4
5.c odd 4 1 700.2.c.b 4
5.c odd 4 1 700.2.c.e 4
7.b odd 2 1 700.2.g.g 4
15.d odd 2 1 1260.2.c.a 4
20.d odd 2 1 140.2.g.b yes 4
20.e even 4 1 700.2.c.c 4
20.e even 4 1 700.2.c.f 4
28.d even 2 1 inner 700.2.g.f 4
35.c odd 2 1 140.2.g.b yes 4
35.f even 4 1 700.2.c.c 4
35.f even 4 1 700.2.c.f 4
35.i odd 6 1 980.2.o.b 4
35.i odd 6 1 980.2.o.d 4
35.j even 6 1 980.2.o.a 4
35.j even 6 1 980.2.o.c 4
40.e odd 2 1 2240.2.k.b 4
40.f even 2 1 2240.2.k.a 4
60.h even 2 1 1260.2.c.b 4
105.g even 2 1 1260.2.c.b 4
140.c even 2 1 140.2.g.a 4
140.j odd 4 1 700.2.c.b 4
140.j odd 4 1 700.2.c.e 4
140.p odd 6 1 980.2.o.b 4
140.p odd 6 1 980.2.o.d 4
140.s even 6 1 980.2.o.a 4
140.s even 6 1 980.2.o.c 4
280.c odd 2 1 2240.2.k.b 4
280.n even 2 1 2240.2.k.a 4
420.o odd 2 1 1260.2.c.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.g.a 4 5.b even 2 1
140.2.g.a 4 140.c even 2 1
140.2.g.b yes 4 20.d odd 2 1
140.2.g.b yes 4 35.c odd 2 1
700.2.c.b 4 5.c odd 4 1
700.2.c.b 4 140.j odd 4 1
700.2.c.c 4 20.e even 4 1
700.2.c.c 4 35.f even 4 1
700.2.c.e 4 5.c odd 4 1
700.2.c.e 4 140.j odd 4 1
700.2.c.f 4 20.e even 4 1
700.2.c.f 4 35.f even 4 1
700.2.g.f 4 1.a even 1 1 trivial
700.2.g.f 4 28.d even 2 1 inner
700.2.g.g 4 4.b odd 2 1
700.2.g.g 4 7.b odd 2 1
980.2.o.a 4 35.j even 6 1
980.2.o.a 4 140.s even 6 1
980.2.o.b 4 35.i odd 6 1
980.2.o.b 4 140.p odd 6 1
980.2.o.c 4 35.j even 6 1
980.2.o.c 4 140.s even 6 1
980.2.o.d 4 35.i odd 6 1
980.2.o.d 4 140.p odd 6 1
1260.2.c.a 4 15.d odd 2 1
1260.2.c.a 4 420.o odd 2 1
1260.2.c.b 4 60.h even 2 1
1260.2.c.b 4 105.g even 2 1
2240.2.k.a 4 40.f even 2 1
2240.2.k.a 4 280.n even 2 1
2240.2.k.b 4 40.e odd 2 1
2240.2.k.b 4 280.c odd 2 1

Hecke kernels

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

\( T_{3}^{2} - 3 \)
\( T_{11}^{4} + 14 T_{11}^{2} + 1 \)
\( T_{19} - 6 \)
\( T_{37}^{2} - 12 T_{37} + 24 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4} \)
$3$ \( ( -3 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( 49 + 2 T^{2} + T^{4} \)
$11$ \( 1 + 14 T^{2} + T^{4} \)
$13$ \( 9 + 42 T^{2} + T^{4} \)
$17$ \( 9 + 42 T^{2} + T^{4} \)
$19$ \( ( -6 + T )^{4} \)
$23$ \( 64 + 32 T^{2} + T^{4} \)
$29$ \( ( -47 - 2 T + T^{2} )^{2} \)
$31$ \( ( 6 + T )^{4} \)
$37$ \( ( 24 - 12 T + T^{2} )^{2} \)
$41$ \( ( 12 + T^{2} )^{2} \)
$43$ \( ( 4 + T^{2} )^{2} \)
$47$ \( ( -3 + T^{2} )^{2} \)
$53$ \( ( 2 + T )^{4} \)
$59$ \( ( -12 + T^{2} )^{2} \)
$61$ \( 576 + 96 T^{2} + T^{4} \)
$67$ \( ( 12 + T^{2} )^{2} \)
$71$ \( 16 + 56 T^{2} + T^{4} \)
$73$ \( 144 + 168 T^{2} + T^{4} \)
$79$ \( 1521 + 222 T^{2} + T^{4} \)
$83$ \( ( 132 + 24 T + T^{2} )^{2} \)
$89$ \( 576 + 96 T^{2} + T^{4} \)
$97$ \( 9801 + 234 T^{2} + T^{4} \)
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