Properties

Label 1339.2.a.c
Level $1339$
Weight $2$
Character orbit 1339.a
Self dual yes
Analytic conductor $10.692$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1339 = 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1339.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.6919688306\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( -\beta_{1} - \beta_{2} ) q^{3} + ( \beta_{1} + \beta_{2} ) q^{4} + ( -2 - \beta_{1} ) q^{5} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{6} + ( 1 - \beta_{1} - \beta_{2} ) q^{7} + ( -1 - \beta_{2} ) q^{8} + 2 \beta_{1} q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( -\beta_{1} - \beta_{2} ) q^{3} + ( \beta_{1} + \beta_{2} ) q^{4} + ( -2 - \beta_{1} ) q^{5} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{6} + ( 1 - \beta_{1} - \beta_{2} ) q^{7} + ( -1 - \beta_{2} ) q^{8} + 2 \beta_{1} q^{9} + ( 2 + 3 \beta_{1} + \beta_{2} ) q^{10} + ( -3 - \beta_{1} + \beta_{2} ) q^{11} + ( -3 - 2 \beta_{1} ) q^{12} + q^{13} + ( 1 + \beta_{1} + \beta_{2} ) q^{14} + ( 1 + 4 \beta_{1} + 3 \beta_{2} ) q^{15} + ( -1 - 2 \beta_{2} ) q^{16} + ( 1 - 2 \beta_{1} - 2 \beta_{2} ) q^{17} + ( -4 - 2 \beta_{1} - 2 \beta_{2} ) q^{18} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{19} + ( -1 - 4 \beta_{1} - 3 \beta_{2} ) q^{20} + ( 3 + \beta_{1} - \beta_{2} ) q^{21} + ( 3 + 3 \beta_{1} + \beta_{2} ) q^{22} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{23} + ( 2 + \beta_{1} ) q^{24} + ( 1 + 5 \beta_{1} + \beta_{2} ) q^{25} -\beta_{1} q^{26} + ( -2 - \beta_{1} + \beta_{2} ) q^{27} + ( -3 - \beta_{1} + \beta_{2} ) q^{28} + ( -6 + \beta_{1} + \beta_{2} ) q^{29} + ( -5 - 8 \beta_{1} - 4 \beta_{2} ) q^{30} + ( -4 + 4 \beta_{1} + 2 \beta_{2} ) q^{31} + ( 3 \beta_{1} + 2 \beta_{2} ) q^{32} + ( -1 + 5 \beta_{1} + 5 \beta_{2} ) q^{33} + ( 2 + 3 \beta_{1} + 2 \beta_{2} ) q^{34} + ( -1 + 3 \beta_{1} + 3 \beta_{2} ) q^{35} + ( 2 + 4 \beta_{1} + 2 \beta_{2} ) q^{36} + ( 3 \beta_{1} + 2 \beta_{2} ) q^{37} + ( 5 + 2 \beta_{1} + 2 \beta_{2} ) q^{38} + ( -\beta_{1} - \beta_{2} ) q^{39} + ( 1 + 2 \beta_{1} + 2 \beta_{2} ) q^{40} + ( -2 - 2 \beta_{1} + 2 \beta_{2} ) q^{41} + ( -3 - 3 \beta_{1} - \beta_{2} ) q^{42} + ( 2 + 2 \beta_{1} ) q^{43} + ( 1 - 5 \beta_{1} - 5 \beta_{2} ) q^{44} + ( -4 - 6 \beta_{1} - 2 \beta_{2} ) q^{45} + ( -10 - 2 \beta_{1} - 4 \beta_{2} ) q^{46} + ( -3 + 5 \beta_{1} + \beta_{2} ) q^{47} + ( 4 + \beta_{1} - \beta_{2} ) q^{48} + ( -3 - 2 \beta_{2} ) q^{49} + ( -9 - 7 \beta_{1} - 5 \beta_{2} ) q^{50} + ( 6 + 3 \beta_{1} - \beta_{2} ) q^{51} + ( \beta_{1} + \beta_{2} ) q^{52} + ( -5 + 3 \beta_{1} + \beta_{2} ) q^{53} + ( 3 + 2 \beta_{1} + \beta_{2} ) q^{54} + ( 9 + 5 \beta_{1} - \beta_{2} ) q^{55} + ( 1 + \beta_{1} - \beta_{2} ) q^{56} + ( 5 \beta_{1} + 4 \beta_{2} ) q^{57} + ( -1 + 4 \beta_{1} - \beta_{2} ) q^{58} + ( -2 - 5 \beta_{1} + 2 \beta_{2} ) q^{59} + ( 10 + 9 \beta_{1} + 2 \beta_{2} ) q^{60} + ( 2 + 3 \beta_{1} + 3 \beta_{2} ) q^{61} + ( -6 - 2 \beta_{1} - 4 \beta_{2} ) q^{62} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{63} + ( -2 - 5 \beta_{1} + \beta_{2} ) q^{64} + ( -2 - \beta_{1} ) q^{65} + ( -5 - 9 \beta_{1} - 5 \beta_{2} ) q^{66} + ( 4 - 4 \beta_{1} ) q^{67} + ( -6 - 3 \beta_{1} + \beta_{2} ) q^{68} + ( -8 \beta_{1} - 6 \beta_{2} ) q^{69} + ( -3 - 5 \beta_{1} - 3 \beta_{2} ) q^{70} + ( 2 + 4 \beta_{1} - 4 \beta_{2} ) q^{71} + ( 2 - 4 \beta_{1} ) q^{72} + ( 6 - 9 \beta_{1} - 4 \beta_{2} ) q^{73} + ( -4 - 5 \beta_{1} - 3 \beta_{2} ) q^{74} + ( -7 - 11 \beta_{1} - 5 \beta_{2} ) q^{75} + ( -5 \beta_{1} - 4 \beta_{2} ) q^{76} + ( -4 + 4 \beta_{1} + 6 \beta_{2} ) q^{77} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{78} + 6 \beta_{1} q^{79} + ( 3 \beta_{1} + 4 \beta_{2} ) q^{80} + ( -1 - 2 \beta_{1} + 4 \beta_{2} ) q^{81} + ( 6 + 2 \beta_{1} + 2 \beta_{2} ) q^{82} + ( -4 + \beta_{1} + 4 \beta_{2} ) q^{83} + ( -1 + 5 \beta_{1} + 5 \beta_{2} ) q^{84} + ( 7 \beta_{1} + 6 \beta_{2} ) q^{85} + ( -4 - 4 \beta_{1} - 2 \beta_{2} ) q^{86} + ( -3 + 4 \beta_{1} + 6 \beta_{2} ) q^{87} + ( -1 + 3 \beta_{1} + 3 \beta_{2} ) q^{88} + ( -\beta_{1} + 6 \beta_{2} ) q^{89} + ( 10 + 12 \beta_{1} + 6 \beta_{2} ) q^{90} + ( 1 - \beta_{1} - \beta_{2} ) q^{91} + ( 8 \beta_{1} + 6 \beta_{2} ) q^{92} + ( -8 - 4 \beta_{1} + 2 \beta_{2} ) q^{93} + ( -9 - 3 \beta_{1} - 5 \beta_{2} ) q^{94} + ( 7 + 6 \beta_{1} ) q^{95} + ( -7 - 6 \beta_{1} - \beta_{2} ) q^{96} + ( 8 - 2 \beta_{1} + 2 \beta_{2} ) q^{97} + ( -2 + 5 \beta_{1} ) q^{98} + ( -6 - 6 \beta_{1} - 2 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - q^{2} - q^{3} + q^{4} - 7q^{5} + 5q^{6} + 2q^{7} - 3q^{8} + 2q^{9} + O(q^{10}) \) \( 3q - q^{2} - q^{3} + q^{4} - 7q^{5} + 5q^{6} + 2q^{7} - 3q^{8} + 2q^{9} + 9q^{10} - 10q^{11} - 11q^{12} + 3q^{13} + 4q^{14} + 7q^{15} - 3q^{16} + q^{17} - 14q^{18} - 5q^{19} - 7q^{20} + 10q^{21} + 12q^{22} + 4q^{23} + 7q^{24} + 8q^{25} - q^{26} - 7q^{27} - 10q^{28} - 17q^{29} - 23q^{30} - 8q^{31} + 3q^{32} + 2q^{33} + 9q^{34} + 10q^{36} + 3q^{37} + 17q^{38} - q^{39} + 5q^{40} - 8q^{41} - 12q^{42} + 8q^{43} - 2q^{44} - 18q^{45} - 32q^{46} - 4q^{47} + 13q^{48} - 9q^{49} - 34q^{50} + 21q^{51} + q^{52} - 12q^{53} + 11q^{54} + 32q^{55} + 4q^{56} + 5q^{57} + q^{58} - 11q^{59} + 39q^{60} + 9q^{61} - 20q^{62} - 8q^{63} - 11q^{64} - 7q^{65} - 24q^{66} + 8q^{67} - 21q^{68} - 8q^{69} - 14q^{70} + 10q^{71} + 2q^{72} + 9q^{73} - 17q^{74} - 32q^{75} - 5q^{76} - 8q^{77} + 5q^{78} + 6q^{79} + 3q^{80} - 5q^{81} + 20q^{82} - 11q^{83} + 2q^{84} + 7q^{85} - 16q^{86} - 5q^{87} - q^{89} + 42q^{90} + 2q^{91} + 8q^{92} - 28q^{93} - 30q^{94} + 27q^{95} - 27q^{96} + 22q^{97} - q^{98} - 24q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 3 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)

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
2.17009
0.311108
−1.48119
−2.17009 −2.70928 2.70928 −4.17009 5.87936 −1.70928 −1.53919 4.34017 9.04945
1.2 −0.311108 1.90321 −1.90321 −2.31111 −0.592104 2.90321 1.21432 0.622216 0.719004
1.3 1.48119 −0.193937 0.193937 −0.518806 −0.287258 0.806063 −2.67513 −2.96239 −0.768452
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(13\) \(-1\)
\(103\) \(-1\)

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 1339.2.a.c 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1339.2.a.c 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1339))\):

\( T_{2}^{3} + T_{2}^{2} - 3 T_{2} - 1 \)
\( T_{3}^{3} + T_{3}^{2} - 5 T_{3} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + 3 T^{2} + 3 T^{3} + 6 T^{4} + 4 T^{5} + 8 T^{6} \)
$3$ \( 1 + T + 4 T^{2} + 5 T^{3} + 12 T^{4} + 9 T^{5} + 27 T^{6} \)
$5$ \( 1 + 7 T + 28 T^{2} + 75 T^{3} + 140 T^{4} + 175 T^{5} + 125 T^{6} \)
$7$ \( 1 - 2 T + 17 T^{2} - 24 T^{3} + 119 T^{4} - 98 T^{5} + 343 T^{6} \)
$11$ \( 1 + 10 T + 57 T^{2} + 216 T^{3} + 627 T^{4} + 1210 T^{5} + 1331 T^{6} \)
$13$ \( ( 1 - T )^{3} \)
$17$ \( 1 - T + 30 T^{2} - 21 T^{3} + 510 T^{4} - 289 T^{5} + 4913 T^{6} \)
$19$ \( 1 + 5 T + 44 T^{2} + 123 T^{3} + 836 T^{4} + 1805 T^{5} + 6859 T^{6} \)
$23$ \( 1 - 4 T - 11 T^{2} + 216 T^{3} - 253 T^{4} - 2116 T^{5} + 12167 T^{6} \)
$29$ \( 1 + 17 T + 178 T^{2} + 1137 T^{3} + 5162 T^{4} + 14297 T^{5} + 24389 T^{6} \)
$31$ \( 1 + 8 T + 61 T^{2} + 224 T^{3} + 1891 T^{4} + 7688 T^{5} + 29791 T^{6} \)
$37$ \( 1 - 3 T + 80 T^{2} - 251 T^{3} + 2960 T^{4} - 4107 T^{5} + 50653 T^{6} \)
$41$ \( 1 + 8 T + 107 T^{2} + 496 T^{3} + 4387 T^{4} + 13448 T^{5} + 68921 T^{6} \)
$43$ \( 1 - 8 T + 137 T^{2} - 672 T^{3} + 5891 T^{4} - 14792 T^{5} + 79507 T^{6} \)
$47$ \( 1 + 4 T + 69 T^{2} + 108 T^{3} + 3243 T^{4} + 8836 T^{5} + 103823 T^{6} \)
$53$ \( 1 + 12 T + 179 T^{2} + 1172 T^{3} + 9487 T^{4} + 33708 T^{5} + 148877 T^{6} \)
$59$ \( 1 + 11 T + 98 T^{2} + 475 T^{3} + 5782 T^{4} + 38291 T^{5} + 205379 T^{6} \)
$61$ \( 1 - 9 T + 162 T^{2} - 1001 T^{3} + 9882 T^{4} - 33489 T^{5} + 226981 T^{6} \)
$67$ \( 1 - 8 T + 169 T^{2} - 944 T^{3} + 11323 T^{4} - 35912 T^{5} + 300763 T^{6} \)
$71$ \( 1 - 10 T + 97 T^{2} - 324 T^{3} + 6887 T^{4} - 50410 T^{5} + 357911 T^{6} \)
$73$ \( 1 - 9 T - 16 T^{2} + 1075 T^{3} - 1168 T^{4} - 47961 T^{5} + 389017 T^{6} \)
$79$ \( 1 - 6 T + 129 T^{2} - 732 T^{3} + 10191 T^{4} - 37446 T^{5} + 493039 T^{6} \)
$83$ \( 1 + 11 T + 230 T^{2} + 1831 T^{3} + 19090 T^{4} + 75779 T^{5} + 571787 T^{6} \)
$89$ \( 1 + T + 108 T^{2} + 345 T^{3} + 9612 T^{4} + 7921 T^{5} + 704969 T^{6} \)
$97$ \( 1 - 22 T + 415 T^{2} - 4468 T^{3} + 40255 T^{4} - 206998 T^{5} + 912673 T^{6} \)
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