Properties

Label 78.2.a.a
Level $78$
Weight $2$
Character orbit 78.a
Self dual yes
Analytic conductor $0.623$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 78 = 2 \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 78.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.622833135766\)
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

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + 2 q^{5} + q^{6} + 4 q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} + q^{4} + 2 q^{5} + q^{6} + 4 q^{7} - q^{8} + q^{9} - 2 q^{10} - 4 q^{11} - q^{12} + q^{13} - 4 q^{14} - 2 q^{15} + q^{16} + 2 q^{17} - q^{18} - 8 q^{19} + 2 q^{20} - 4 q^{21} + 4 q^{22} + q^{24} - q^{25} - q^{26} - q^{27} + 4 q^{28} + 6 q^{29} + 2 q^{30} - 4 q^{31} - q^{32} + 4 q^{33} - 2 q^{34} + 8 q^{35} + q^{36} - 2 q^{37} + 8 q^{38} - q^{39} - 2 q^{40} - 10 q^{41} + 4 q^{42} + 4 q^{43} - 4 q^{44} + 2 q^{45} + 8 q^{47} - q^{48} + 9 q^{49} + q^{50} - 2 q^{51} + q^{52} - 10 q^{53} + q^{54} - 8 q^{55} - 4 q^{56} + 8 q^{57} - 6 q^{58} + 4 q^{59} - 2 q^{60} - 2 q^{61} + 4 q^{62} + 4 q^{63} + q^{64} + 2 q^{65} - 4 q^{66} - 16 q^{67} + 2 q^{68} - 8 q^{70} - 8 q^{71} - q^{72} + 2 q^{73} + 2 q^{74} + q^{75} - 8 q^{76} - 16 q^{77} + q^{78} + 8 q^{79} + 2 q^{80} + q^{81} + 10 q^{82} + 12 q^{83} - 4 q^{84} + 4 q^{85} - 4 q^{86} - 6 q^{87} + 4 q^{88} + 14 q^{89} - 2 q^{90} + 4 q^{91} + 4 q^{93} - 8 q^{94} - 16 q^{95} + q^{96} + 10 q^{97} - 9 q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −1.00000 1.00000 2.00000 1.00000 4.00000 −1.00000 1.00000 −2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(13\) \(-1\)

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 78.2.a.a 1
3.b odd 2 1 234.2.a.c 1
4.b odd 2 1 624.2.a.h 1
5.b even 2 1 1950.2.a.w 1
5.c odd 4 2 1950.2.e.i 2
7.b odd 2 1 3822.2.a.j 1
8.b even 2 1 2496.2.a.t 1
8.d odd 2 1 2496.2.a.b 1
9.c even 3 2 2106.2.e.q 2
9.d odd 6 2 2106.2.e.j 2
11.b odd 2 1 9438.2.a.t 1
12.b even 2 1 1872.2.a.c 1
13.b even 2 1 1014.2.a.d 1
13.c even 3 2 1014.2.e.f 2
13.d odd 4 2 1014.2.b.b 2
13.e even 6 2 1014.2.e.c 2
13.f odd 12 4 1014.2.i.d 4
15.d odd 2 1 5850.2.a.d 1
15.e even 4 2 5850.2.e.bb 2
24.f even 2 1 7488.2.a.bk 1
24.h odd 2 1 7488.2.a.bz 1
39.d odd 2 1 3042.2.a.f 1
39.f even 4 2 3042.2.b.g 2
52.b odd 2 1 8112.2.a.v 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.a.a 1 1.a even 1 1 trivial
234.2.a.c 1 3.b odd 2 1
624.2.a.h 1 4.b odd 2 1
1014.2.a.d 1 13.b even 2 1
1014.2.b.b 2 13.d odd 4 2
1014.2.e.c 2 13.e even 6 2
1014.2.e.f 2 13.c even 3 2
1014.2.i.d 4 13.f odd 12 4
1872.2.a.c 1 12.b even 2 1
1950.2.a.w 1 5.b even 2 1
1950.2.e.i 2 5.c odd 4 2
2106.2.e.j 2 9.d odd 6 2
2106.2.e.q 2 9.c even 3 2
2496.2.a.b 1 8.d odd 2 1
2496.2.a.t 1 8.b even 2 1
3042.2.a.f 1 39.d odd 2 1
3042.2.b.g 2 39.f even 4 2
3822.2.a.j 1 7.b odd 2 1
5850.2.a.d 1 15.d odd 2 1
5850.2.e.bb 2 15.e even 4 2
7488.2.a.bk 1 24.f even 2 1
7488.2.a.bz 1 24.h odd 2 1
8112.2.a.v 1 52.b odd 2 1
9438.2.a.t 1 11.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(\Gamma_0(78))\).

Hecke characteristic polynomials

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