Properties

Label 169.4.b.b
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: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(\beta = \sqrt{-3}\). 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

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
3.46410i −7.00000 −4.00000 13.8564i 24.2487i 22.5167i 13.8564i 22.0000 −48.0000
168.2 3.46410i −7.00000 −4.00000 13.8564i 24.2487i 22.5167i 13.8564i 22.0000 −48.0000
\(n\): 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.b 2
13.b even 2 1 inner 169.4.b.b 2
13.c even 3 1 13.4.e.a 2
13.c even 3 1 169.4.e.b 2
13.d odd 4 2 169.4.a.h 2
13.e even 6 1 13.4.e.a 2
13.e even 6 1 169.4.e.b 2
13.f odd 12 4 169.4.c.i 4
39.f even 4 2 1521.4.a.q 2
39.h odd 6 1 117.4.q.c 2
39.i odd 6 1 117.4.q.c 2
52.i odd 6 1 208.4.w.a 2
52.j odd 6 1 208.4.w.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.e.a 2 13.c even 3 1
13.4.e.a 2 13.e even 6 1
117.4.q.c 2 39.h odd 6 1
117.4.q.c 2 39.i odd 6 1
169.4.a.h 2 13.d odd 4 2
169.4.b.b 2 1.a even 1 1 trivial
169.4.b.b 2 13.b even 2 1 inner
169.4.c.i 4 13.f odd 12 4
169.4.e.b 2 13.c even 3 1
169.4.e.b 2 13.e even 6 1
208.4.w.a 2 52.i odd 6 1
208.4.w.a 2 52.j odd 6 1
1521.4.a.q 2 39.f even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 12 \) acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\). 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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