Properties

Label 13.4.a.a
Level 13
Weight 4
Character orbit 13.a
Self dual yes
Analytic conductor 0.767
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 13 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 13.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.767024830075\)
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

\(f(q)\) \(=\) \( q - 5q^{2} - 7q^{3} + 17q^{4} - 7q^{5} + 35q^{6} - 13q^{7} - 45q^{8} + 22q^{9} + O(q^{10}) \) \( q - 5q^{2} - 7q^{3} + 17q^{4} - 7q^{5} + 35q^{6} - 13q^{7} - 45q^{8} + 22q^{9} + 35q^{10} - 26q^{11} - 119q^{12} + 13q^{13} + 65q^{14} + 49q^{15} + 89q^{16} + 77q^{17} - 110q^{18} - 126q^{19} - 119q^{20} + 91q^{21} + 130q^{22} - 96q^{23} + 315q^{24} - 76q^{25} - 65q^{26} + 35q^{27} - 221q^{28} - 82q^{29} - 245q^{30} + 196q^{31} - 85q^{32} + 182q^{33} - 385q^{34} + 91q^{35} + 374q^{36} - 131q^{37} + 630q^{38} - 91q^{39} + 315q^{40} + 336q^{41} - 455q^{42} - 201q^{43} - 442q^{44} - 154q^{45} + 480q^{46} - 105q^{47} - 623q^{48} - 174q^{49} + 380q^{50} - 539q^{51} + 221q^{52} - 432q^{53} - 175q^{54} + 182q^{55} + 585q^{56} + 882q^{57} + 410q^{58} - 294q^{59} + 833q^{60} - 56q^{61} - 980q^{62} - 286q^{63} - 287q^{64} - 91q^{65} - 910q^{66} + 478q^{67} + 1309q^{68} + 672q^{69} - 455q^{70} + 9q^{71} - 990q^{72} + 98q^{73} + 655q^{74} + 532q^{75} - 2142q^{76} + 338q^{77} + 455q^{78} + 1304q^{79} - 623q^{80} - 839q^{81} - 1680q^{82} - 308q^{83} + 1547q^{84} - 539q^{85} + 1005q^{86} + 574q^{87} + 1170q^{88} - 1190q^{89} + 770q^{90} - 169q^{91} - 1632q^{92} - 1372q^{93} + 525q^{94} + 882q^{95} + 595q^{96} + 70q^{97} + 870q^{98} - 572q^{99} + O(q^{100}) \)

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
−5.00000 −7.00000 17.0000 −7.00000 35.0000 −13.0000 −45.0000 22.0000 35.0000
\(n\): 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 13.4.a.a 1
3.b odd 2 1 117.4.a.b 1
4.b odd 2 1 208.4.a.g 1
5.b even 2 1 325.4.a.d 1
5.c odd 4 2 325.4.b.b 2
7.b odd 2 1 637.4.a.a 1
8.b even 2 1 832.4.a.r 1
8.d odd 2 1 832.4.a.a 1
11.b odd 2 1 1573.4.a.a 1
12.b even 2 1 1872.4.a.k 1
13.b even 2 1 169.4.a.e 1
13.c even 3 2 169.4.c.e 2
13.d odd 4 2 169.4.b.a 2
13.e even 6 2 169.4.c.a 2
13.f odd 12 4 169.4.e.e 4
39.d odd 2 1 1521.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.a 1 1.a even 1 1 trivial
117.4.a.b 1 3.b odd 2 1
169.4.a.e 1 13.b even 2 1
169.4.b.a 2 13.d odd 4 2
169.4.c.a 2 13.e even 6 2
169.4.c.e 2 13.c even 3 2
169.4.e.e 4 13.f odd 12 4
208.4.a.g 1 4.b odd 2 1
325.4.a.d 1 5.b even 2 1
325.4.b.b 2 5.c odd 4 2
637.4.a.a 1 7.b odd 2 1
832.4.a.a 1 8.d odd 2 1
832.4.a.r 1 8.b even 2 1
1521.4.a.a 1 39.d odd 2 1
1573.4.a.a 1 11.b odd 2 1
1872.4.a.k 1 12.b even 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 5 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(13))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 5 T + 8 T^{2} \)
$3$ \( 1 + 7 T + 27 T^{2} \)
$5$ \( 1 + 7 T + 125 T^{2} \)
$7$ \( 1 + 13 T + 343 T^{2} \)
$11$ \( 1 + 26 T + 1331 T^{2} \)
$13$ \( 1 - 13 T \)
$17$ \( 1 - 77 T + 4913 T^{2} \)
$19$ \( 1 + 126 T + 6859 T^{2} \)
$23$ \( 1 + 96 T + 12167 T^{2} \)
$29$ \( 1 + 82 T + 24389 T^{2} \)
$31$ \( 1 - 196 T + 29791 T^{2} \)
$37$ \( 1 + 131 T + 50653 T^{2} \)
$41$ \( 1 - 336 T + 68921 T^{2} \)
$43$ \( 1 + 201 T + 79507 T^{2} \)
$47$ \( 1 + 105 T + 103823 T^{2} \)
$53$ \( 1 + 432 T + 148877 T^{2} \)
$59$ \( 1 + 294 T + 205379 T^{2} \)
$61$ \( 1 + 56 T + 226981 T^{2} \)
$67$ \( 1 - 478 T + 300763 T^{2} \)
$71$ \( 1 - 9 T + 357911 T^{2} \)
$73$ \( 1 - 98 T + 389017 T^{2} \)
$79$ \( 1 - 1304 T + 493039 T^{2} \)
$83$ \( 1 + 308 T + 571787 T^{2} \)
$89$ \( 1 + 1190 T + 704969 T^{2} \)
$97$ \( 1 - 70 T + 912673 T^{2} \)
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