Properties

Label 169.4.a.i
Level 169
Weight 4
Character orbit 169.a
Self dual yes
Analytic conductor 9.971
Analytic rank 1
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.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.97132279097\)
Analytic rank: \(1\)
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.

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

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 −5.00000 −1.73205 −3.46410 13.8564 22.5167 −23.0000 3.00000
1.2 1.73205 2.00000 −5.00000 1.73205 3.46410 −13.8564 −22.5167 −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.a.i 2
3.b odd 2 1 1521.4.a.o 2
13.b even 2 1 inner 169.4.a.i 2
13.c even 3 2 169.4.c.h 4
13.d odd 4 2 169.4.b.d 2
13.e even 6 2 169.4.c.h 4
13.f odd 12 2 13.4.e.b 2
13.f odd 12 2 169.4.e.a 2
39.d odd 2 1 1521.4.a.o 2
39.k even 12 2 117.4.q.a 2
52.l even 12 2 208.4.w.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.e.b 2 13.f odd 12 2
117.4.q.a 2 39.k even 12 2
169.4.a.i 2 1.a even 1 1 trivial
169.4.a.i 2 13.b even 2 1 inner
169.4.b.d 2 13.d odd 4 2
169.4.c.h 4 13.c even 3 2
169.4.c.h 4 13.e even 6 2
169.4.e.a 2 13.f odd 12 2
208.4.w.b 2 52.l even 12 2
1521.4.a.o 2 3.b odd 2 1
1521.4.a.o 2 39.d odd 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 3 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(169))\).

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
$17$ \( ( 1 + 117 T + 4913 T^{2} )^{2} \)
$19$ \( 1 + 650 T^{2} + 47045881 T^{4} \)
$23$ \( ( 1 - 78 T + 12167 T^{2} )^{2} \)
$29$ \( ( 1 + 141 T + 24389 T^{2} )^{2} \)
$31$ \( 1 + 35282 T^{2} + 887503681 T^{4} \)
$37$ \( 1 + 80639 T^{2} + 2565726409 T^{4} \)
$41$ \( 1 + 63895 T^{2} + 4750104241 T^{4} \)
$43$ \( ( 1 + 104 T + 79507 T^{2} )^{2} \)
$47$ \( 1 + 116818 T^{2} + 10779215329 T^{4} \)
$53$ \( ( 1 - 93 T + 148877 T^{2} )^{2} \)
$59$ \( 1 + 330070 T^{2} + 42180533641 T^{4} \)
$61$ \( ( 1 - 145 T + 226981 T^{2} )^{2} \)
$67$ \( 1 - 16822 T^{2} + 90458382169 T^{4} \)
$71$ \( 1 - 400478 T^{2} + 128100283921 T^{4} \)
$73$ \( 1 + 567359 T^{2} + 151334226289 T^{4} \)
$79$ \( ( 1 - 1276 T + 493039 T^{2} )^{2} \)
$83$ \( 1 + 519766 T^{2} + 326940373369 T^{4} \)
$89$ \( 1 + 455650 T^{2} + 496981290961 T^{4} \)
$97$ \( 1 + 1784978 T^{2} + 832972004929 T^{4} \)
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