Properties

Label 169.4.a.h
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: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + 2 \beta q^{2} - 7 q^{3} + 4 q^{4} + 8 \beta q^{5} - 14 \beta q^{6} - 13 \beta q^{7} - 8 \beta q^{8} + 22 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta q^{2} - 7 q^{3} + 4 q^{4} + 8 \beta q^{5} - 14 \beta q^{6} - 13 \beta q^{7} - 8 \beta q^{8} + 22 q^{9} + 48 q^{10} - 13 \beta q^{11} - 28 q^{12} - 78 q^{14} - 56 \beta q^{15} - 80 q^{16} - 27 q^{17} + 44 \beta q^{18} - 51 \beta q^{19} + 32 \beta q^{20} + 91 \beta q^{21} - 78 q^{22} - 57 q^{23} + 56 \beta q^{24} + 67 q^{25} + 35 q^{27} - 52 \beta q^{28} - 69 q^{29} - 336 q^{30} - 42 \beta q^{31} - 96 \beta q^{32} + 91 \beta q^{33} - 54 \beta q^{34} - 312 q^{35} + 88 q^{36} - 23 \beta q^{37} - 306 q^{38} - 192 q^{40} + 227 \beta q^{41} + 546 q^{42} + 85 q^{43} - 52 \beta q^{44} + 176 \beta q^{45} - 114 \beta q^{46} + 198 \beta q^{47} + 560 q^{48} + 164 q^{49} + 134 \beta q^{50} + 189 q^{51} + 426 q^{53} + 70 \beta q^{54} - 312 q^{55} + 312 q^{56} + 357 \beta q^{57} - 138 \beta q^{58} + 11 \beta q^{59} - 224 \beta q^{60} - 17 q^{61} - 252 q^{62} - 286 \beta q^{63} + 64 q^{64} + 546 q^{66} - 95 \beta q^{67} - 108 q^{68} + 399 q^{69} - 624 \beta q^{70} + 337 \beta q^{71} - 176 \beta q^{72} - 580 \beta q^{73} - 138 q^{74} - 469 q^{75} - 204 \beta q^{76} + 507 q^{77} - 1244 q^{79} - 640 \beta q^{80} - 839 q^{81} + 1362 q^{82} - 246 \beta q^{83} + 364 \beta q^{84} - 216 \beta q^{85} + 170 \beta q^{86} + 483 q^{87} + 312 q^{88} + 177 \beta q^{89} + 1056 q^{90} - 228 q^{92} + 294 \beta q^{93} + 1188 q^{94} - 1224 q^{95} + 672 \beta q^{96} + 713 \beta q^{97} + 328 \beta q^{98} - 286 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 14 q^{3} + 8 q^{4} + 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 14 q^{3} + 8 q^{4} + 44 q^{9} + 96 q^{10} - 56 q^{12} - 156 q^{14} - 160 q^{16} - 54 q^{17} - 156 q^{22} - 114 q^{23} + 134 q^{25} + 70 q^{27} - 138 q^{29} - 672 q^{30} - 624 q^{35} + 176 q^{36} - 612 q^{38} - 384 q^{40} + 1092 q^{42} + 170 q^{43} + 1120 q^{48} + 328 q^{49} + 378 q^{51} + 852 q^{53} - 624 q^{55} + 624 q^{56} - 34 q^{61} - 504 q^{62} + 128 q^{64} + 1092 q^{66} - 216 q^{68} + 798 q^{69} - 276 q^{74} - 938 q^{75} + 1014 q^{77} - 2488 q^{79} - 1678 q^{81} + 2724 q^{82} + 966 q^{87} + 624 q^{88} + 2112 q^{90} - 456 q^{92} + 2376 q^{94} - 2448 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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
−3.46410 −7.00000 4.00000 −13.8564 24.2487 22.5167 13.8564 22.0000 48.0000
1.2 3.46410 −7.00000 4.00000 13.8564 −24.2487 −22.5167 −13.8564 22.0000 48.0000
\(n\): 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 169.4.a.h 2
3.b odd 2 1 1521.4.a.q 2
13.b even 2 1 inner 169.4.a.h 2
13.c even 3 2 169.4.c.i 4
13.d odd 4 2 169.4.b.b 2
13.e even 6 2 169.4.c.i 4
13.f odd 12 2 13.4.e.a 2
13.f odd 12 2 169.4.e.b 2
39.d odd 2 1 1521.4.a.q 2
39.k even 12 2 117.4.q.c 2
52.l even 12 2 208.4.w.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.e.a 2 13.f odd 12 2
117.4.q.c 2 39.k even 12 2
169.4.a.h 2 1.a even 1 1 trivial
169.4.a.h 2 13.b even 2 1 inner
169.4.b.b 2 13.d odd 4 2
169.4.c.i 4 13.c even 3 2
169.4.c.i 4 13.e even 6 2
169.4.e.b 2 13.f odd 12 2
208.4.w.a 2 52.l even 12 2
1521.4.a.q 2 3.b odd 2 1
1521.4.a.q 2 39.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 12 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(169))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 12 \) Copy content Toggle raw display
$3$ \( (T + 7)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 192 \) Copy content Toggle raw display
$7$ \( T^{2} - 507 \) Copy content Toggle raw display
$11$ \( T^{2} - 507 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T + 27)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 7803 \) Copy content Toggle raw display
$23$ \( (T + 57)^{2} \) Copy content Toggle raw display
$29$ \( (T + 69)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 5292 \) Copy content Toggle raw display
$37$ \( T^{2} - 1587 \) Copy content Toggle raw display
$41$ \( T^{2} - 154587 \) Copy content Toggle raw display
$43$ \( (T - 85)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 117612 \) Copy content Toggle raw display
$53$ \( (T - 426)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 363 \) Copy content Toggle raw display
$61$ \( (T + 17)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 27075 \) Copy content Toggle raw display
$71$ \( T^{2} - 340707 \) Copy content Toggle raw display
$73$ \( T^{2} - 1009200 \) Copy content Toggle raw display
$79$ \( (T + 1244)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 181548 \) Copy content Toggle raw display
$89$ \( T^{2} - 93987 \) Copy content Toggle raw display
$97$ \( T^{2} - 1525107 \) Copy content Toggle raw display
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