Properties

Label 726.2.a.i
Level 726
Weight 2
Character orbit 726.a
Self dual yes
Analytic conductor 5.797
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 726 = 2 \cdot 3 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 726.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.79713918674\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 66)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + q^{6} - 2q^{7} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} + q^{3} + q^{4} + q^{6} - 2q^{7} + q^{8} + q^{9} + q^{12} + 4q^{13} - 2q^{14} + q^{16} + 6q^{17} + q^{18} + 4q^{19} - 2q^{21} + 6q^{23} + q^{24} - 5q^{25} + 4q^{26} + q^{27} - 2q^{28} - 6q^{29} + 8q^{31} + q^{32} + 6q^{34} + q^{36} - 10q^{37} + 4q^{38} + 4q^{39} - 6q^{41} - 2q^{42} - 8q^{43} + 6q^{46} - 6q^{47} + q^{48} - 3q^{49} - 5q^{50} + 6q^{51} + 4q^{52} + q^{54} - 2q^{56} + 4q^{57} - 6q^{58} - 8q^{61} + 8q^{62} - 2q^{63} + q^{64} - 4q^{67} + 6q^{68} + 6q^{69} + 6q^{71} + q^{72} - 2q^{73} - 10q^{74} - 5q^{75} + 4q^{76} + 4q^{78} - 14q^{79} + q^{81} - 6q^{82} + 12q^{83} - 2q^{84} - 8q^{86} - 6q^{87} - 6q^{89} - 8q^{91} + 6q^{92} + 8q^{93} - 6q^{94} + q^{96} + 14q^{97} - 3q^{98} + O(q^{100}) \)

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 −2.00000 1.00000 1.00000 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 726.2.a.i 1
3.b odd 2 1 2178.2.a.b 1
4.b odd 2 1 5808.2.a.l 1
11.b odd 2 1 66.2.a.a 1
11.c even 5 4 726.2.e.b 4
11.d odd 10 4 726.2.e.k 4
33.d even 2 1 198.2.a.e 1
44.c even 2 1 528.2.a.d 1
55.d odd 2 1 1650.2.a.m 1
55.e even 4 2 1650.2.c.d 2
77.b even 2 1 3234.2.a.d 1
88.b odd 2 1 2112.2.a.i 1
88.g even 2 1 2112.2.a.v 1
99.g even 6 2 1782.2.e.f 2
99.h odd 6 2 1782.2.e.s 2
132.d odd 2 1 1584.2.a.h 1
165.d even 2 1 4950.2.a.g 1
165.l odd 4 2 4950.2.c.r 2
231.h odd 2 1 9702.2.a.bu 1
264.m even 2 1 6336.2.a.bj 1
264.p odd 2 1 6336.2.a.bf 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.a.a 1 11.b odd 2 1
198.2.a.e 1 33.d even 2 1
528.2.a.d 1 44.c even 2 1
726.2.a.i 1 1.a even 1 1 trivial
726.2.e.b 4 11.c even 5 4
726.2.e.k 4 11.d odd 10 4
1584.2.a.h 1 132.d odd 2 1
1650.2.a.m 1 55.d odd 2 1
1650.2.c.d 2 55.e even 4 2
1782.2.e.f 2 99.g even 6 2
1782.2.e.s 2 99.h odd 6 2
2112.2.a.i 1 88.b odd 2 1
2112.2.a.v 1 88.g even 2 1
2178.2.a.b 1 3.b odd 2 1
3234.2.a.d 1 77.b even 2 1
4950.2.a.g 1 165.d even 2 1
4950.2.c.r 2 165.l odd 4 2
5808.2.a.l 1 4.b odd 2 1
6336.2.a.bf 1 264.p odd 2 1
6336.2.a.bj 1 264.m even 2 1
9702.2.a.bu 1 231.h odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(11\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(726))\):

\( T_{5} \)
\( T_{7} + 2 \)
\( T_{13} - 4 \)

Hecke Characteristic Polynomials

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