Properties

Label 169.2.b.a
Level $169$
Weight $2$
Character orbit 169.b
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.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.34947179416\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
168.1
0.500000 + 0.866025i
0.500000 0.866025i
1.73205i 2.00000 −1.00000 1.73205i 3.46410i 0 1.73205i 1.00000 3.00000
168.2 1.73205i 2.00000 −1.00000 1.73205i 3.46410i 0 1.73205i 1.00000 3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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

Hecke kernels

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

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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