Properties

Label 70.2.a.a
Level $70$
Weight $2$
Character orbit 70.a
Self dual yes
Analytic conductor $0.559$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 70 = 2 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 70.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.558952814149\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - q^{5} - q^{7} + q^{8} - 3q^{9} + O(q^{10}) \) \( q + q^{2} + q^{4} - q^{5} - q^{7} + q^{8} - 3q^{9} - q^{10} + 4q^{11} - 6q^{13} - q^{14} + q^{16} + 2q^{17} - 3q^{18} - q^{20} + 4q^{22} + q^{25} - 6q^{26} - q^{28} + 6q^{29} + 8q^{31} + q^{32} + 2q^{34} + q^{35} - 3q^{36} - 10q^{37} - q^{40} + 2q^{41} + 4q^{43} + 4q^{44} + 3q^{45} + 8q^{47} + q^{49} + q^{50} - 6q^{52} - 2q^{53} - 4q^{55} - q^{56} + 6q^{58} - 8q^{59} - 14q^{61} + 8q^{62} + 3q^{63} + q^{64} + 6q^{65} - 12q^{67} + 2q^{68} + q^{70} - 16q^{71} - 3q^{72} + 2q^{73} - 10q^{74} - 4q^{77} - 8q^{79} - q^{80} + 9q^{81} + 2q^{82} + 8q^{83} - 2q^{85} + 4q^{86} + 4q^{88} + 10q^{89} + 3q^{90} + 6q^{91} + 8q^{94} + 2q^{97} + q^{98} - 12q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
1.00000 0 1.00000 −1.00000 0 −1.00000 1.00000 −3.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(7\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 70.2.a.a 1
3.b odd 2 1 630.2.a.d 1
4.b odd 2 1 560.2.a.d 1
5.b even 2 1 350.2.a.b 1
5.c odd 4 2 350.2.c.b 2
7.b odd 2 1 490.2.a.h 1
7.c even 3 2 490.2.e.d 2
7.d odd 6 2 490.2.e.c 2
8.b even 2 1 2240.2.a.n 1
8.d odd 2 1 2240.2.a.q 1
11.b odd 2 1 8470.2.a.j 1
12.b even 2 1 5040.2.a.bm 1
15.d odd 2 1 3150.2.a.bj 1
15.e even 4 2 3150.2.g.c 2
20.d odd 2 1 2800.2.a.m 1
20.e even 4 2 2800.2.g.n 2
21.c even 2 1 4410.2.a.b 1
28.d even 2 1 3920.2.a.t 1
35.c odd 2 1 2450.2.a.l 1
35.f even 4 2 2450.2.c.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.a.a 1 1.a even 1 1 trivial
350.2.a.b 1 5.b even 2 1
350.2.c.b 2 5.c odd 4 2
490.2.a.h 1 7.b odd 2 1
490.2.e.c 2 7.d odd 6 2
490.2.e.d 2 7.c even 3 2
560.2.a.d 1 4.b odd 2 1
630.2.a.d 1 3.b odd 2 1
2240.2.a.n 1 8.b even 2 1
2240.2.a.q 1 8.d odd 2 1
2450.2.a.l 1 35.c odd 2 1
2450.2.c.k 2 35.f even 4 2
2800.2.a.m 1 20.d odd 2 1
2800.2.g.n 2 20.e even 4 2
3150.2.a.bj 1 15.d odd 2 1
3150.2.g.c 2 15.e even 4 2
3920.2.a.t 1 28.d even 2 1
4410.2.a.b 1 21.c even 2 1
5040.2.a.bm 1 12.b even 2 1
8470.2.a.j 1 11.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(\Gamma_0(70))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( T \)
$5$ \( 1 + T \)
$7$ \( 1 + T \)
$11$ \( -4 + T \)
$13$ \( 6 + T \)
$17$ \( -2 + T \)
$19$ \( T \)
$23$ \( T \)
$29$ \( -6 + T \)
$31$ \( -8 + T \)
$37$ \( 10 + T \)
$41$ \( -2 + T \)
$43$ \( -4 + T \)
$47$ \( -8 + T \)
$53$ \( 2 + T \)
$59$ \( 8 + T \)
$61$ \( 14 + T \)
$67$ \( 12 + T \)
$71$ \( 16 + T \)
$73$ \( -2 + T \)
$79$ \( 8 + T \)
$83$ \( -8 + T \)
$89$ \( -10 + T \)
$97$ \( -2 + T \)
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