Properties

Label 169.2.c.a
Level $169$
Weight $2$
Character orbit 169.c
Analytic conductor $1.349$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 169.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.34947179416\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-1 + \zeta_{12}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 1.50000i −1.00000 + 1.73205i −0.500000 0.866025i −1.73205 −1.73205 3.00000i 0 −1.73205 −0.500000 0.866025i 1.50000 2.59808i
22.2 0.866025 1.50000i −1.00000 + 1.73205i −0.500000 0.866025i 1.73205 1.73205 + 3.00000i 0 1.73205 −0.500000 0.866025i 1.50000 2.59808i
146.1 −0.866025 1.50000i −1.00000 1.73205i −0.500000 + 0.866025i −1.73205 −1.73205 + 3.00000i 0 −1.73205 −0.500000 + 0.866025i 1.50000 + 2.59808i
146.2 0.866025 + 1.50000i −1.00000 1.73205i −0.500000 + 0.866025i 1.73205 1.73205 3.00000i 0 1.73205 −0.500000 + 0.866025i 1.50000 + 2.59808i
\(n\): 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
13.c even 3 1 inner
13.e even 6 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 3 T_{2}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(169, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 9 + 3 T^{2} + T^{4} \)
$3$ \( ( 4 + 2 T + T^{2} )^{2} \)
$5$ \( ( -3 + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( T^{4} \)
$13$ \( T^{4} \)
$17$ \( ( 9 + 3 T + T^{2} )^{2} \)
$19$ \( 144 + 12 T^{2} + T^{4} \)
$23$ \( ( 36 + 6 T + T^{2} )^{2} \)
$29$ \( ( 9 + 3 T + T^{2} )^{2} \)
$31$ \( ( -12 + T^{2} )^{2} \)
$37$ \( 5625 + 75 T^{2} + T^{4} \)
$41$ \( 729 + 27 T^{2} + T^{4} \)
$43$ \( ( 64 - 8 T + T^{2} )^{2} \)
$47$ \( ( -12 + T^{2} )^{2} \)
$53$ \( ( 3 + T )^{4} \)
$59$ \( 2304 + 48 T^{2} + T^{4} \)
$61$ \( ( 1 + T + T^{2} )^{2} \)
$67$ \( 144 + 12 T^{2} + T^{4} \)
$71$ \( 144 + 12 T^{2} + T^{4} \)
$73$ \( ( -3 + T^{2} )^{2} \)
$79$ \( ( -4 + T )^{4} \)
$83$ \( ( -192 + T^{2} )^{2} \)
$89$ \( 2304 + 48 T^{2} + T^{4} \)
$97$ \( 2304 + 48 T^{2} + T^{4} \)
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