Properties

Label 37.2.a.a
Level $37$
Weight $2$
Character orbit 37.a
Self dual yes
Analytic conductor $0.295$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 37.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.295446487479\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q - 2q^{2} - 3q^{3} + 2q^{4} - 2q^{5} + 6q^{6} - q^{7} + 6q^{9} + O(q^{10}) \) \( q - 2q^{2} - 3q^{3} + 2q^{4} - 2q^{5} + 6q^{6} - q^{7} + 6q^{9} + 4q^{10} - 5q^{11} - 6q^{12} - 2q^{13} + 2q^{14} + 6q^{15} - 4q^{16} - 12q^{18} - 4q^{20} + 3q^{21} + 10q^{22} + 2q^{23} - q^{25} + 4q^{26} - 9q^{27} - 2q^{28} + 6q^{29} - 12q^{30} - 4q^{31} + 8q^{32} + 15q^{33} + 2q^{35} + 12q^{36} - q^{37} + 6q^{39} - 9q^{41} - 6q^{42} + 2q^{43} - 10q^{44} - 12q^{45} - 4q^{46} - 9q^{47} + 12q^{48} - 6q^{49} + 2q^{50} - 4q^{52} + q^{53} + 18q^{54} + 10q^{55} - 12q^{58} + 8q^{59} + 12q^{60} - 8q^{61} + 8q^{62} - 6q^{63} - 8q^{64} + 4q^{65} - 30q^{66} + 8q^{67} - 6q^{69} - 4q^{70} + 9q^{71} - q^{73} + 2q^{74} + 3q^{75} + 5q^{77} - 12q^{78} + 4q^{79} + 8q^{80} + 9q^{81} + 18q^{82} - 15q^{83} + 6q^{84} - 4q^{86} - 18q^{87} + 4q^{89} + 24q^{90} + 2q^{91} + 4q^{92} + 12q^{93} + 18q^{94} - 24q^{96} + 4q^{97} + 12q^{98} - 30q^{99} + O(q^{100}) \)

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
−2.00000 −3.00000 2.00000 −2.00000 6.00000 −1.00000 0 6.00000 4.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(37\) \(1\)

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 37.2.a.a 1
3.b odd 2 1 333.2.a.d 1
4.b odd 2 1 592.2.a.e 1
5.b even 2 1 925.2.a.e 1
5.c odd 4 2 925.2.b.b 2
7.b odd 2 1 1813.2.a.a 1
8.b even 2 1 2368.2.a.q 1
8.d odd 2 1 2368.2.a.b 1
11.b odd 2 1 4477.2.a.b 1
12.b even 2 1 5328.2.a.r 1
13.b even 2 1 6253.2.a.c 1
15.d odd 2 1 8325.2.a.e 1
37.b even 2 1 1369.2.a.e 1
37.d odd 4 2 1369.2.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.2.a.a 1 1.a even 1 1 trivial
333.2.a.d 1 3.b odd 2 1
592.2.a.e 1 4.b odd 2 1
925.2.a.e 1 5.b even 2 1
925.2.b.b 2 5.c odd 4 2
1369.2.a.e 1 37.b even 2 1
1369.2.b.c 2 37.d odd 4 2
1813.2.a.a 1 7.b odd 2 1
2368.2.a.b 1 8.d odd 2 1
2368.2.a.q 1 8.b even 2 1
4477.2.a.b 1 11.b odd 2 1
5328.2.a.r 1 12.b even 2 1
6253.2.a.c 1 13.b even 2 1
8325.2.a.e 1 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(37))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 2 T^{2} \)
$3$ \( 1 + 3 T + 3 T^{2} \)
$5$ \( 1 + 2 T + 5 T^{2} \)
$7$ \( 1 + T + 7 T^{2} \)
$11$ \( 1 + 5 T + 11 T^{2} \)
$13$ \( 1 + 2 T + 13 T^{2} \)
$17$ \( 1 + 17 T^{2} \)
$19$ \( 1 + 19 T^{2} \)
$23$ \( 1 - 2 T + 23 T^{2} \)
$29$ \( 1 - 6 T + 29 T^{2} \)
$31$ \( 1 + 4 T + 31 T^{2} \)
$37$ \( 1 + T \)
$41$ \( 1 + 9 T + 41 T^{2} \)
$43$ \( 1 - 2 T + 43 T^{2} \)
$47$ \( 1 + 9 T + 47 T^{2} \)
$53$ \( 1 - T + 53 T^{2} \)
$59$ \( 1 - 8 T + 59 T^{2} \)
$61$ \( 1 + 8 T + 61 T^{2} \)
$67$ \( 1 - 8 T + 67 T^{2} \)
$71$ \( 1 - 9 T + 71 T^{2} \)
$73$ \( 1 + T + 73 T^{2} \)
$79$ \( 1 - 4 T + 79 T^{2} \)
$83$ \( 1 + 15 T + 83 T^{2} \)
$89$ \( 1 - 4 T + 89 T^{2} \)
$97$ \( 1 - 4 T + 97 T^{2} \)
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