Properties

Label 1862.2.a.f
Level 1862
Weight 2
Character orbit 1862.a
Self dual yes
Analytic conductor 14.868
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 1862 = 2 \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1862.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.8681448564\)
Analytic rank: \(0\)
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

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + 4q^{5} + q^{6} + q^{8} - 2q^{9} + O(q^{10}) \) \( q + q^{2} + q^{3} + q^{4} + 4q^{5} + q^{6} + q^{8} - 2q^{9} + 4q^{10} + 2q^{11} + q^{12} + q^{13} + 4q^{15} + q^{16} - 3q^{17} - 2q^{18} + q^{19} + 4q^{20} + 2q^{22} - q^{23} + q^{24} + 11q^{25} + q^{26} - 5q^{27} - 5q^{29} + 4q^{30} + 8q^{31} + q^{32} + 2q^{33} - 3q^{34} - 2q^{36} - 2q^{37} + q^{38} + q^{39} + 4q^{40} + 8q^{41} + 4q^{43} + 2q^{44} - 8q^{45} - q^{46} - 8q^{47} + q^{48} + 11q^{50} - 3q^{51} + q^{52} - q^{53} - 5q^{54} + 8q^{55} + q^{57} - 5q^{58} - 15q^{59} + 4q^{60} - 2q^{61} + 8q^{62} + q^{64} + 4q^{65} + 2q^{66} + 3q^{67} - 3q^{68} - q^{69} + 2q^{71} - 2q^{72} - 9q^{73} - 2q^{74} + 11q^{75} + q^{76} + q^{78} - 10q^{79} + 4q^{80} + q^{81} + 8q^{82} + 6q^{83} - 12q^{85} + 4q^{86} - 5q^{87} + 2q^{88} - 8q^{90} - q^{92} + 8q^{93} - 8q^{94} + 4q^{95} + q^{96} + 2q^{97} - 4q^{99} + O(q^{100}) \)

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 4.00000 1.00000 0 1.00000 −2.00000 4.00000
\(n\): 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 1862.2.a.f 1
7.b odd 2 1 38.2.a.b 1
21.c even 2 1 342.2.a.d 1
28.d even 2 1 304.2.a.d 1
35.c odd 2 1 950.2.a.b 1
35.f even 4 2 950.2.b.c 2
56.e even 2 1 1216.2.a.g 1
56.h odd 2 1 1216.2.a.n 1
77.b even 2 1 4598.2.a.a 1
84.h odd 2 1 2736.2.a.w 1
91.b odd 2 1 6422.2.a.b 1
105.g even 2 1 8550.2.a.u 1
133.c even 2 1 722.2.a.b 1
133.m odd 6 2 722.2.c.d 2
133.p even 6 2 722.2.c.f 2
133.y odd 18 6 722.2.e.c 6
133.ba even 18 6 722.2.e.d 6
140.c even 2 1 7600.2.a.h 1
399.h odd 2 1 6498.2.a.y 1
532.b odd 2 1 5776.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.a.b 1 7.b odd 2 1
304.2.a.d 1 28.d even 2 1
342.2.a.d 1 21.c even 2 1
722.2.a.b 1 133.c even 2 1
722.2.c.d 2 133.m odd 6 2
722.2.c.f 2 133.p even 6 2
722.2.e.c 6 133.y odd 18 6
722.2.e.d 6 133.ba even 18 6
950.2.a.b 1 35.c odd 2 1
950.2.b.c 2 35.f even 4 2
1216.2.a.g 1 56.e even 2 1
1216.2.a.n 1 56.h odd 2 1
1862.2.a.f 1 1.a even 1 1 trivial
2736.2.a.w 1 84.h odd 2 1
4598.2.a.a 1 77.b even 2 1
5776.2.a.d 1 532.b odd 2 1
6422.2.a.b 1 91.b odd 2 1
6498.2.a.y 1 399.h odd 2 1
7600.2.a.h 1 140.c even 2 1
8550.2.a.u 1 105.g even 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)
\(19\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1862))\):

\( T_{3} - 1 \)
\( T_{5} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T \)
$3$ \( 1 - T + 3 T^{2} \)
$5$ \( 1 - 4 T + 5 T^{2} \)
$7$ 1
$11$ \( 1 - 2 T + 11 T^{2} \)
$13$ \( 1 - T + 13 T^{2} \)
$17$ \( 1 + 3 T + 17 T^{2} \)
$19$ \( 1 - T \)
$23$ \( 1 + T + 23 T^{2} \)
$29$ \( 1 + 5 T + 29 T^{2} \)
$31$ \( 1 - 8 T + 31 T^{2} \)
$37$ \( 1 + 2 T + 37 T^{2} \)
$41$ \( 1 - 8 T + 41 T^{2} \)
$43$ \( 1 - 4 T + 43 T^{2} \)
$47$ \( 1 + 8 T + 47 T^{2} \)
$53$ \( 1 + T + 53 T^{2} \)
$59$ \( 1 + 15 T + 59 T^{2} \)
$61$ \( 1 + 2 T + 61 T^{2} \)
$67$ \( 1 - 3 T + 67 T^{2} \)
$71$ \( 1 - 2 T + 71 T^{2} \)
$73$ \( 1 + 9 T + 73 T^{2} \)
$79$ \( 1 + 10 T + 79 T^{2} \)
$83$ \( 1 - 6 T + 83 T^{2} \)
$89$ \( 1 + 89 T^{2} \)
$97$ \( 1 - 2 T + 97 T^{2} \)
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