Properties

Label 722.2.a.e
Level 722
Weight 2
Character orbit 722.a
Self dual yes
Analytic conductor 5.765
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 722 = 2 \cdot 19^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 722.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.76519902594\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - q^{6} - q^{7} + q^{8} - 2q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} + q^{4} - q^{6} - q^{7} + q^{8} - 2q^{9} - 6q^{11} - q^{12} - 5q^{13} - q^{14} + q^{16} + 3q^{17} - 2q^{18} + q^{21} - 6q^{22} + 3q^{23} - q^{24} - 5q^{25} - 5q^{26} + 5q^{27} - q^{28} - 9q^{29} + 4q^{31} + q^{32} + 6q^{33} + 3q^{34} - 2q^{36} - 2q^{37} + 5q^{39} + q^{42} + 8q^{43} - 6q^{44} + 3q^{46} - q^{48} - 6q^{49} - 5q^{50} - 3q^{51} - 5q^{52} + 3q^{53} + 5q^{54} - q^{56} - 9q^{58} - 9q^{59} - 10q^{61} + 4q^{62} + 2q^{63} + q^{64} + 6q^{66} - 5q^{67} + 3q^{68} - 3q^{69} + 6q^{71} - 2q^{72} - 7q^{73} - 2q^{74} + 5q^{75} + 6q^{77} + 5q^{78} + 10q^{79} + q^{81} - 6q^{83} + q^{84} + 8q^{86} + 9q^{87} - 6q^{88} + 12q^{89} + 5q^{91} + 3q^{92} - 4q^{93} - q^{96} + 10q^{97} - 6q^{98} + 12q^{99} + O(q^{100}) \)

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 −1.00000 1.00000 0 −1.00000 −1.00000 1.00000 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 722.2.a.e 1
3.b odd 2 1 6498.2.a.f 1
4.b odd 2 1 5776.2.a.m 1
19.b odd 2 1 38.2.a.a 1
19.c even 3 2 722.2.c.c 2
19.d odd 6 2 722.2.c.e 2
19.e even 9 6 722.2.e.e 6
19.f odd 18 6 722.2.e.f 6
57.d even 2 1 342.2.a.e 1
76.d even 2 1 304.2.a.c 1
95.d odd 2 1 950.2.a.d 1
95.g even 4 2 950.2.b.b 2
133.c even 2 1 1862.2.a.b 1
152.b even 2 1 1216.2.a.m 1
152.g odd 2 1 1216.2.a.e 1
209.d even 2 1 4598.2.a.p 1
228.b odd 2 1 2736.2.a.n 1
247.d odd 2 1 6422.2.a.h 1
285.b even 2 1 8550.2.a.m 1
380.d even 2 1 7600.2.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.a.a 1 19.b odd 2 1
304.2.a.c 1 76.d even 2 1
342.2.a.e 1 57.d even 2 1
722.2.a.e 1 1.a even 1 1 trivial
722.2.c.c 2 19.c even 3 2
722.2.c.e 2 19.d odd 6 2
722.2.e.e 6 19.e even 9 6
722.2.e.f 6 19.f odd 18 6
950.2.a.d 1 95.d odd 2 1
950.2.b.b 2 95.g even 4 2
1216.2.a.e 1 152.g odd 2 1
1216.2.a.m 1 152.b even 2 1
1862.2.a.b 1 133.c even 2 1
2736.2.a.n 1 228.b odd 2 1
4598.2.a.p 1 209.d even 2 1
5776.2.a.m 1 4.b odd 2 1
6422.2.a.h 1 247.d odd 2 1
6498.2.a.f 1 3.b odd 2 1
7600.2.a.n 1 380.d even 2 1
8550.2.a.m 1 285.b even 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(19\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(722))\):

\( T_{3} + 1 \)
\( T_{5} \)
\( T_{7} + 1 \)
\( T_{13} + 5 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T \)
$3$ \( 1 + T + 3 T^{2} \)
$5$ \( 1 + 5 T^{2} \)
$7$ \( 1 + T + 7 T^{2} \)
$11$ \( 1 + 6 T + 11 T^{2} \)
$13$ \( 1 + 5 T + 13 T^{2} \)
$17$ \( 1 - 3 T + 17 T^{2} \)
$19$ 1
$23$ \( 1 - 3 T + 23 T^{2} \)
$29$ \( 1 + 9 T + 29 T^{2} \)
$31$ \( 1 - 4 T + 31 T^{2} \)
$37$ \( 1 + 2 T + 37 T^{2} \)
$41$ \( 1 + 41 T^{2} \)
$43$ \( 1 - 8 T + 43 T^{2} \)
$47$ \( 1 + 47 T^{2} \)
$53$ \( 1 - 3 T + 53 T^{2} \)
$59$ \( 1 + 9 T + 59 T^{2} \)
$61$ \( 1 + 10 T + 61 T^{2} \)
$67$ \( 1 + 5 T + 67 T^{2} \)
$71$ \( 1 - 6 T + 71 T^{2} \)
$73$ \( 1 + 7 T + 73 T^{2} \)
$79$ \( 1 - 10 T + 79 T^{2} \)
$83$ \( 1 + 6 T + 83 T^{2} \)
$89$ \( 1 - 12 T + 89 T^{2} \)
$97$ \( 1 - 10 T + 97 T^{2} \)
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