Properties

Label 169.2.a.c
Level $169$
Weight $2$
Character orbit 169.a
Self dual yes
Analytic conductor $1.349$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 169.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.34947179416\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
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.

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

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
1.80194
0.445042
−1.24698
−0.801938 −2.24698 −1.35690 −0.246980 1.80194 2.35690 2.69202 2.04892 0.198062
1.2 0.554958 0.801938 −1.69202 2.80194 0.445042 2.69202 −2.04892 −2.35690 1.55496
1.3 2.24698 −0.554958 3.04892 1.44504 −1.24698 −2.04892 2.35690 −2.69202 3.24698
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 169.2.a.c yes 3
3.b odd 2 1 1521.2.a.o 3
4.b odd 2 1 2704.2.a.ba 3
5.b even 2 1 4225.2.a.bb 3
7.b odd 2 1 8281.2.a.bj 3
13.b even 2 1 169.2.a.b 3
13.c even 3 2 169.2.c.b 6
13.d odd 4 2 169.2.b.b 6
13.e even 6 2 169.2.c.c 6
13.f odd 12 4 169.2.e.b 12
39.d odd 2 1 1521.2.a.r 3
39.f even 4 2 1521.2.b.l 6
52.b odd 2 1 2704.2.a.z 3
52.f even 4 2 2704.2.f.o 6
65.d even 2 1 4225.2.a.bg 3
91.b odd 2 1 8281.2.a.bf 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
169.2.a.b 3 13.b even 2 1
169.2.a.c yes 3 1.a even 1 1 trivial
169.2.b.b 6 13.d odd 4 2
169.2.c.b 6 13.c even 3 2
169.2.c.c 6 13.e even 6 2
169.2.e.b 12 13.f odd 12 4
1521.2.a.o 3 3.b odd 2 1
1521.2.a.r 3 39.d odd 2 1
1521.2.b.l 6 39.f even 4 2
2704.2.a.z 3 52.b odd 2 1
2704.2.a.ba 3 4.b odd 2 1
2704.2.f.o 6 52.f even 4 2
4225.2.a.bb 3 5.b even 2 1
4225.2.a.bg 3 65.d even 2 1
8281.2.a.bf 3 91.b odd 2 1
8281.2.a.bj 3 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 2 T_{2}^{2} - T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(169))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T - 2 T^{2} + T^{3} \)
$3$ \( -1 - T + 2 T^{2} + T^{3} \)
$5$ \( 1 + 3 T - 4 T^{2} + T^{3} \)
$7$ \( 13 - 4 T - 3 T^{2} + T^{3} \)
$11$ \( -13 + 19 T - 8 T^{2} + T^{3} \)
$13$ \( T^{3} \)
$17$ \( 13 - 15 T + 2 T^{2} + T^{3} \)
$19$ \( 1 - 11 T - 4 T^{2} + T^{3} \)
$23$ \( -13 - T + 5 T^{2} + T^{3} \)
$29$ \( 83 - 44 T + T^{2} + T^{3} \)
$31$ \( 167 - 36 T - 5 T^{2} + T^{3} \)
$37$ \( 29 + 41 T + 12 T^{2} + T^{3} \)
$41$ \( -49 - 49 T - 7 T^{2} + T^{3} \)
$43$ \( 13 + 40 T - 13 T^{2} + T^{3} \)
$47$ \( -167 + 101 T - 18 T^{2} + T^{3} \)
$53$ \( 337 - 86 T - T^{2} + T^{3} \)
$59$ \( -1 + 83 T - 19 T^{2} + T^{3} \)
$61$ \( 239 - 67 T - 4 T^{2} + T^{3} \)
$67$ \( -41 - 72 T - T^{2} + T^{3} \)
$71$ \( -547 + 222 T - 27 T^{2} + T^{3} \)
$73$ \( -911 - 120 T + 9 T^{2} + T^{3} \)
$79$ \( 127 - 162 T + 5 T^{2} + T^{3} \)
$83$ \( -203 - 140 T - 7 T^{2} + T^{3} \)
$89$ \( 281 - 74 T - 11 T^{2} + T^{3} \)
$97$ \( -301 - 84 T + 7 T^{2} + T^{3} \)
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