Properties

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

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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: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.73205
1.73205
−1.73205 2.00000 1.00000 1.73205 −3.46410 0 1.73205 1.00000 −3.00000
1.2 1.73205 2.00000 1.00000 −1.73205 3.46410 0 −1.73205 1.00000 −3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.2.a.a 2
3.b odd 2 1 1521.2.a.k 2
4.b odd 2 1 2704.2.a.o 2
5.b even 2 1 4225.2.a.v 2
7.b odd 2 1 8281.2.a.q 2
13.b even 2 1 inner 169.2.a.a 2
13.c even 3 2 169.2.c.a 4
13.d odd 4 2 169.2.b.a 2
13.e even 6 2 169.2.c.a 4
13.f odd 12 2 13.2.e.a 2
13.f odd 12 2 169.2.e.a 2
39.d odd 2 1 1521.2.a.k 2
39.f even 4 2 1521.2.b.a 2
39.k even 12 2 117.2.q.c 2
52.b odd 2 1 2704.2.a.o 2
52.f even 4 2 2704.2.f.b 2
52.l even 12 2 208.2.w.b 2
65.d even 2 1 4225.2.a.v 2
65.o even 12 2 325.2.m.a 4
65.s odd 12 2 325.2.n.a 2
65.t even 12 2 325.2.m.a 4
91.b odd 2 1 8281.2.a.q 2
91.w even 12 2 637.2.u.b 2
91.x odd 12 2 637.2.k.a 2
91.ba even 12 2 637.2.k.c 2
91.bc even 12 2 637.2.q.a 2
91.bd odd 12 2 637.2.u.c 2
104.u even 12 2 832.2.w.a 2
104.x odd 12 2 832.2.w.d 2
156.v odd 12 2 1872.2.by.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.2.e.a 2 13.f odd 12 2
117.2.q.c 2 39.k even 12 2
169.2.a.a 2 1.a even 1 1 trivial
169.2.a.a 2 13.b even 2 1 inner
169.2.b.a 2 13.d odd 4 2
169.2.c.a 4 13.c even 3 2
169.2.c.a 4 13.e even 6 2
169.2.e.a 2 13.f odd 12 2
208.2.w.b 2 52.l even 12 2
325.2.m.a 4 65.o even 12 2
325.2.m.a 4 65.t even 12 2
325.2.n.a 2 65.s odd 12 2
637.2.k.a 2 91.x odd 12 2
637.2.k.c 2 91.ba even 12 2
637.2.q.a 2 91.bc even 12 2
637.2.u.b 2 91.w even 12 2
637.2.u.c 2 91.bd odd 12 2
832.2.w.a 2 104.u even 12 2
832.2.w.d 2 104.x odd 12 2
1521.2.a.k 2 3.b odd 2 1
1521.2.a.k 2 39.d odd 2 1
1521.2.b.a 2 39.f even 4 2
1872.2.by.d 2 156.v odd 12 2
2704.2.a.o 2 4.b odd 2 1
2704.2.a.o 2 52.b odd 2 1
2704.2.f.b 2 52.f even 4 2
4225.2.a.v 2 5.b even 2 1
4225.2.a.v 2 65.d even 2 1
8281.2.a.q 2 7.b odd 2 1
8281.2.a.q 2 91.b odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -3 + T^{2} \)
$3$ \( ( -2 + T )^{2} \)
$5$ \( -3 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( T^{2} \)
$17$ \( ( -3 + T )^{2} \)
$19$ \( -12 + T^{2} \)
$23$ \( ( -6 + T )^{2} \)
$29$ \( ( -3 + T )^{2} \)
$31$ \( -12 + T^{2} \)
$37$ \( -75 + T^{2} \)
$41$ \( -27 + T^{2} \)
$43$ \( ( 8 + T )^{2} \)
$47$ \( -12 + T^{2} \)
$53$ \( ( 3 + T )^{2} \)
$59$ \( -48 + T^{2} \)
$61$ \( ( -1 + T )^{2} \)
$67$ \( -12 + T^{2} \)
$71$ \( -12 + T^{2} \)
$73$ \( -3 + T^{2} \)
$79$ \( ( -4 + T )^{2} \)
$83$ \( -192 + T^{2} \)
$89$ \( -48 + T^{2} \)
$97$ \( -48 + T^{2} \)
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