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: \(x^{2} - x + 1\)
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 a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 - 4 \zeta_{6} ) q^{2} -7 q^{3} -4 q^{4} + ( 8 - 16 \zeta_{6} ) q^{5} + ( -14 + 28 \zeta_{6} ) q^{6} + ( 13 - 26 \zeta_{6} ) q^{7} + ( 8 - 16 \zeta_{6} ) q^{8} + 22 q^{9} +O(q^{10})\) \( q + ( 2 - 4 \zeta_{6} ) q^{2} -7 q^{3} -4 q^{4} + ( 8 - 16 \zeta_{6} ) q^{5} + ( -14 + 28 \zeta_{6} ) q^{6} + ( 13 - 26 \zeta_{6} ) q^{7} + ( 8 - 16 \zeta_{6} ) q^{8} + 22 q^{9} -48 q^{10} + ( 13 - 26 \zeta_{6} ) q^{11} + 28 q^{12} -78 q^{14} + ( -56 + 112 \zeta_{6} ) q^{15} -80 q^{16} + 27 q^{17} + ( 44 - 88 \zeta_{6} ) q^{18} + ( -51 + 102 \zeta_{6} ) q^{19} + ( -32 + 64 \zeta_{6} ) q^{20} + ( -91 + 182 \zeta_{6} ) q^{21} -78 q^{22} + 57 q^{23} + ( -56 + 112 \zeta_{6} ) q^{24} -67 q^{25} + 35 q^{27} + ( -52 + 104 \zeta_{6} ) q^{28} -69 q^{29} + 336 q^{30} + ( -42 + 84 \zeta_{6} ) q^{31} + ( -96 + 192 \zeta_{6} ) q^{32} + ( -91 + 182 \zeta_{6} ) q^{33} + ( 54 - 108 \zeta_{6} ) q^{34} -312 q^{35} -88 q^{36} + ( 23 - 46 \zeta_{6} ) q^{37} + 306 q^{38} -192 q^{40} + ( 227 - 454 \zeta_{6} ) q^{41} + 546 q^{42} -85 q^{43} + ( -52 + 104 \zeta_{6} ) q^{44} + ( 176 - 352 \zeta_{6} ) q^{45} + ( 114 - 228 \zeta_{6} ) q^{46} + ( -198 + 396 \zeta_{6} ) q^{47} + 560 q^{48} -164 q^{49} + ( -134 + 268 \zeta_{6} ) q^{50} -189 q^{51} + 426 q^{53} + ( 70 - 140 \zeta_{6} ) q^{54} -312 q^{55} -312 q^{56} + ( 357 - 714 \zeta_{6} ) q^{57} + ( -138 + 276 \zeta_{6} ) q^{58} + ( -11 + 22 \zeta_{6} ) q^{59} + ( 224 - 448 \zeta_{6} ) q^{60} -17 q^{61} + 252 q^{62} + ( 286 - 572 \zeta_{6} ) q^{63} -64 q^{64} + 546 q^{66} + ( -95 + 190 \zeta_{6} ) q^{67} -108 q^{68} -399 q^{69} + ( -624 + 1248 \zeta_{6} ) q^{70} + ( 337 - 674 \zeta_{6} ) q^{71} + ( 176 - 352 \zeta_{6} ) q^{72} + ( 580 - 1160 \zeta_{6} ) q^{73} -138 q^{74} + 469 q^{75} + ( 204 - 408 \zeta_{6} ) q^{76} -507 q^{77} -1244 q^{79} + ( -640 + 1280 \zeta_{6} ) q^{80} -839 q^{81} -1362 q^{82} + ( -246 + 492 \zeta_{6} ) q^{83} + ( 364 - 728 \zeta_{6} ) q^{84} + ( 216 - 432 \zeta_{6} ) q^{85} + ( -170 + 340 \zeta_{6} ) q^{86} + 483 q^{87} -312 q^{88} + ( -177 + 354 \zeta_{6} ) q^{89} -1056 q^{90} -228 q^{92} + ( 294 - 588 \zeta_{6} ) q^{93} + 1188 q^{94} + 1224 q^{95} + ( 672 - 1344 \zeta_{6} ) q^{96} + ( 713 - 1426 \zeta_{6} ) q^{97} + ( -328 + 656 \zeta_{6} ) q^{98} + ( 286 - 572 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 14 q^{3} - 8 q^{4} + 44 q^{9} + O(q^{10}) \) \( 2 q - 14 q^{3} - 8 q^{4} + 44 q^{9} - 96 q^{10} + 56 q^{12} - 156 q^{14} - 160 q^{16} + 54 q^{17} - 156 q^{22} + 114 q^{23} - 134 q^{25} + 70 q^{27} - 138 q^{29} + 672 q^{30} - 624 q^{35} - 176 q^{36} + 612 q^{38} - 384 q^{40} + 1092 q^{42} - 170 q^{43} + 1120 q^{48} - 328 q^{49} - 378 q^{51} + 852 q^{53} - 624 q^{55} - 624 q^{56} - 34 q^{61} + 504 q^{62} - 128 q^{64} + 1092 q^{66} - 216 q^{68} - 798 q^{69} - 276 q^{74} + 938 q^{75} - 1014 q^{77} - 2488 q^{79} - 1678 q^{81} - 2724 q^{82} + 966 q^{87} - 624 q^{88} - 2112 q^{90} - 456 q^{92} + 2376 q^{94} + 2448 q^{95} + O(q^{100}) \)

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 \(S_{4}^{\mathrm{new}}(169, [\chi])\).

Hecke characteristic polynomials

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