Properties

Label 169.4.b.d
Level 169
Weight 4
Character orbit 169.b
Analytic conductor 9.971
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 169.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.97132279097\)
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.

\(f(q)\) \(=\) \( q + ( 1 - 2 \zeta_{6} ) q^{2} + 2 q^{3} + 5 q^{4} + ( 1 - 2 \zeta_{6} ) q^{5} + ( 2 - 4 \zeta_{6} ) q^{6} + ( 8 - 16 \zeta_{6} ) q^{7} + ( 13 - 26 \zeta_{6} ) q^{8} -23 q^{9} +O(q^{10})\) \( q + ( 1 - 2 \zeta_{6} ) q^{2} + 2 q^{3} + 5 q^{4} + ( 1 - 2 \zeta_{6} ) q^{5} + ( 2 - 4 \zeta_{6} ) q^{6} + ( 8 - 16 \zeta_{6} ) q^{7} + ( 13 - 26 \zeta_{6} ) q^{8} -23 q^{9} -3 q^{10} + ( 8 - 16 \zeta_{6} ) q^{11} + 10 q^{12} -24 q^{14} + ( 2 - 4 \zeta_{6} ) q^{15} + q^{16} + 117 q^{17} + ( -23 + 46 \zeta_{6} ) q^{18} + ( 66 - 132 \zeta_{6} ) q^{19} + ( 5 - 10 \zeta_{6} ) q^{20} + ( 16 - 32 \zeta_{6} ) q^{21} -24 q^{22} -78 q^{23} + ( 26 - 52 \zeta_{6} ) q^{24} + 122 q^{25} -100 q^{27} + ( 40 - 80 \zeta_{6} ) q^{28} -141 q^{29} -6 q^{30} + ( -90 + 180 \zeta_{6} ) q^{31} + ( 105 - 210 \zeta_{6} ) q^{32} + ( 16 - 32 \zeta_{6} ) q^{33} + ( 117 - 234 \zeta_{6} ) q^{34} -24 q^{35} -115 q^{36} + ( -83 + 166 \zeta_{6} ) q^{37} -198 q^{38} -39 q^{40} + ( 157 - 314 \zeta_{6} ) q^{41} -48 q^{42} + 104 q^{43} + ( 40 - 80 \zeta_{6} ) q^{44} + ( -23 + 46 \zeta_{6} ) q^{45} + ( -78 + 156 \zeta_{6} ) q^{46} + ( -174 + 348 \zeta_{6} ) q^{47} + 2 q^{48} + 151 q^{49} + ( 122 - 244 \zeta_{6} ) q^{50} + 234 q^{51} + 93 q^{53} + ( -100 + 200 \zeta_{6} ) q^{54} -24 q^{55} -312 q^{56} + ( 132 - 264 \zeta_{6} ) q^{57} + ( -141 + 282 \zeta_{6} ) q^{58} + ( 164 - 328 \zeta_{6} ) q^{59} + ( 10 - 20 \zeta_{6} ) q^{60} + 145 q^{61} + 270 q^{62} + ( -184 + 368 \zeta_{6} ) q^{63} -307 q^{64} -48 q^{66} + ( -454 + 908 \zeta_{6} ) q^{67} + 585 q^{68} -156 q^{69} + ( -24 + 48 \zeta_{6} ) q^{70} + ( -610 + 1220 \zeta_{6} ) q^{71} + ( -299 + 598 \zeta_{6} ) q^{72} + ( -265 + 530 \zeta_{6} ) q^{73} + 249 q^{74} + 244 q^{75} + ( 330 - 660 \zeta_{6} ) q^{76} -192 q^{77} + 1276 q^{79} + ( 1 - 2 \zeta_{6} ) q^{80} + 421 q^{81} -471 q^{82} + ( 456 - 912 \zeta_{6} ) q^{83} + ( 80 - 160 \zeta_{6} ) q^{84} + ( 117 - 234 \zeta_{6} ) q^{85} + ( 104 - 208 \zeta_{6} ) q^{86} -282 q^{87} -312 q^{88} + ( -564 + 1128 \zeta_{6} ) q^{89} + 69 q^{90} -390 q^{92} + ( -180 + 360 \zeta_{6} ) q^{93} + 522 q^{94} -198 q^{95} + ( 210 - 420 \zeta_{6} ) q^{96} + ( -116 + 232 \zeta_{6} ) q^{97} + ( 151 - 302 \zeta_{6} ) q^{98} + ( -184 + 368 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{3} + 10q^{4} - 46q^{9} + O(q^{10}) \) \( 2q + 4q^{3} + 10q^{4} - 46q^{9} - 6q^{10} + 20q^{12} - 48q^{14} + 2q^{16} + 234q^{17} - 48q^{22} - 156q^{23} + 244q^{25} - 200q^{27} - 282q^{29} - 12q^{30} - 48q^{35} - 230q^{36} - 396q^{38} - 78q^{40} - 96q^{42} + 208q^{43} + 4q^{48} + 302q^{49} + 468q^{51} + 186q^{53} - 48q^{55} - 624q^{56} + 290q^{61} + 540q^{62} - 614q^{64} - 96q^{66} + 1170q^{68} - 312q^{69} + 498q^{74} + 488q^{75} - 384q^{77} + 2552q^{79} + 842q^{81} - 942q^{82} - 564q^{87} - 624q^{88} + 138q^{90} - 780q^{92} + 1044q^{94} - 396q^{95} + O(q^{100}) \)

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 5.00000 1.73205i 3.46410i 13.8564i 22.5167i −23.0000 −3.00000
168.2 1.73205i 2.00000 5.00000 1.73205i 3.46410i 13.8564i 22.5167i −23.0000 −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 169.4.b.d 2
13.b even 2 1 inner 169.4.b.d 2
13.c even 3 1 13.4.e.b 2
13.c even 3 1 169.4.e.a 2
13.d odd 4 2 169.4.a.i 2
13.e even 6 1 13.4.e.b 2
13.e even 6 1 169.4.e.a 2
13.f odd 12 4 169.4.c.h 4
39.f even 4 2 1521.4.a.o 2
39.h odd 6 1 117.4.q.a 2
39.i odd 6 1 117.4.q.a 2
52.i odd 6 1 208.4.w.b 2
52.j odd 6 1 208.4.w.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.e.b 2 13.c even 3 1
13.4.e.b 2 13.e even 6 1
117.4.q.a 2 39.h odd 6 1
117.4.q.a 2 39.i odd 6 1
169.4.a.i 2 13.d odd 4 2
169.4.b.d 2 1.a even 1 1 trivial
169.4.b.d 2 13.b even 2 1 inner
169.4.c.h 4 13.f odd 12 4
169.4.e.a 2 13.c even 3 1
169.4.e.a 2 13.e even 6 1
208.4.w.b 2 52.i odd 6 1
208.4.w.b 2 52.j odd 6 1
1521.4.a.o 2 39.f even 4 2

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 13 T^{2} + 64 T^{4} \)
$3$ \( ( 1 - 2 T + 27 T^{2} )^{2} \)
$5$ \( 1 - 247 T^{2} + 15625 T^{4} \)
$7$ \( 1 - 494 T^{2} + 117649 T^{4} \)
$11$ \( 1 - 2470 T^{2} + 1771561 T^{4} \)
$13$ 1
$17$ \( ( 1 - 117 T + 4913 T^{2} )^{2} \)
$19$ \( 1 - 650 T^{2} + 47045881 T^{4} \)
$23$ \( ( 1 + 78 T + 12167 T^{2} )^{2} \)
$29$ \( ( 1 + 141 T + 24389 T^{2} )^{2} \)
$31$ \( ( 1 - 308 T + 29791 T^{2} )( 1 + 308 T + 29791 T^{2} ) \)
$37$ \( 1 - 80639 T^{2} + 2565726409 T^{4} \)
$41$ \( 1 - 63895 T^{2} + 4750104241 T^{4} \)
$43$ \( ( 1 - 104 T + 79507 T^{2} )^{2} \)
$47$ \( 1 - 116818 T^{2} + 10779215329 T^{4} \)
$53$ \( ( 1 - 93 T + 148877 T^{2} )^{2} \)
$59$ \( 1 - 330070 T^{2} + 42180533641 T^{4} \)
$61$ \( ( 1 - 145 T + 226981 T^{2} )^{2} \)
$67$ \( 1 + 16822 T^{2} + 90458382169 T^{4} \)
$71$ \( 1 + 400478 T^{2} + 128100283921 T^{4} \)
$73$ \( 1 - 567359 T^{2} + 151334226289 T^{4} \)
$79$ \( ( 1 - 1276 T + 493039 T^{2} )^{2} \)
$83$ \( 1 - 519766 T^{2} + 326940373369 T^{4} \)
$89$ \( 1 - 455650 T^{2} + 496981290961 T^{4} \)
$97$ \( 1 - 1784978 T^{2} + 832972004929 T^{4} \)
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