Properties

Label 169.4.e.a
Level $169$
Weight $4$
Character orbit 169.e
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.e (of order \(6\), degree \(2\), 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: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 - \zeta_{6} ) q^{2} + ( -2 + 2 \zeta_{6} ) q^{3} -5 \zeta_{6} q^{4} + ( -1 + 2 \zeta_{6} ) q^{5} + ( 4 - 2 \zeta_{6} ) q^{6} + ( 16 - 8 \zeta_{6} ) q^{7} + ( -13 + 26 \zeta_{6} ) q^{8} + 23 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( -1 - \zeta_{6} ) q^{2} + ( -2 + 2 \zeta_{6} ) q^{3} -5 \zeta_{6} q^{4} + ( -1 + 2 \zeta_{6} ) q^{5} + ( 4 - 2 \zeta_{6} ) q^{6} + ( 16 - 8 \zeta_{6} ) q^{7} + ( -13 + 26 \zeta_{6} ) q^{8} + 23 \zeta_{6} q^{9} + ( 3 - 3 \zeta_{6} ) q^{10} + ( -8 - 8 \zeta_{6} ) q^{11} + 10 q^{12} -24 q^{14} + ( -2 - 2 \zeta_{6} ) q^{15} + ( -1 + \zeta_{6} ) q^{16} -117 \zeta_{6} q^{17} + ( 23 - 46 \zeta_{6} ) q^{18} + ( 132 - 66 \zeta_{6} ) q^{19} + ( 10 - 5 \zeta_{6} ) q^{20} + ( -16 + 32 \zeta_{6} ) q^{21} + 24 \zeta_{6} q^{22} + ( 78 - 78 \zeta_{6} ) q^{23} + ( -26 - 26 \zeta_{6} ) q^{24} + 122 q^{25} -100 q^{27} + ( -40 - 40 \zeta_{6} ) q^{28} + ( 141 - 141 \zeta_{6} ) q^{29} + 6 \zeta_{6} q^{30} + ( 90 - 180 \zeta_{6} ) q^{31} + ( 210 - 105 \zeta_{6} ) q^{32} + ( 32 - 16 \zeta_{6} ) q^{33} + ( -117 + 234 \zeta_{6} ) q^{34} + 24 \zeta_{6} q^{35} + ( 115 - 115 \zeta_{6} ) q^{36} + ( 83 + 83 \zeta_{6} ) q^{37} -198 q^{38} -39 q^{40} + ( -157 - 157 \zeta_{6} ) q^{41} + ( 48 - 48 \zeta_{6} ) q^{42} -104 \zeta_{6} q^{43} + ( -40 + 80 \zeta_{6} ) q^{44} + ( -46 + 23 \zeta_{6} ) q^{45} + ( -156 + 78 \zeta_{6} ) q^{46} + ( 174 - 348 \zeta_{6} ) q^{47} -2 \zeta_{6} q^{48} + ( -151 + 151 \zeta_{6} ) q^{49} + ( -122 - 122 \zeta_{6} ) q^{50} + 234 q^{51} + 93 q^{53} + ( 100 + 100 \zeta_{6} ) q^{54} + ( 24 - 24 \zeta_{6} ) q^{55} + 312 \zeta_{6} q^{56} + ( -132 + 264 \zeta_{6} ) q^{57} + ( -282 + 141 \zeta_{6} ) q^{58} + ( 328 - 164 \zeta_{6} ) q^{59} + ( -10 + 20 \zeta_{6} ) q^{60} -145 \zeta_{6} q^{61} + ( -270 + 270 \zeta_{6} ) q^{62} + ( 184 + 184 \zeta_{6} ) q^{63} -307 q^{64} -48 q^{66} + ( 454 + 454 \zeta_{6} ) q^{67} + ( -585 + 585 \zeta_{6} ) q^{68} + 156 \zeta_{6} q^{69} + ( 24 - 48 \zeta_{6} ) q^{70} + ( -1220 + 610 \zeta_{6} ) q^{71} + ( -598 + 299 \zeta_{6} ) q^{72} + ( 265 - 530 \zeta_{6} ) q^{73} -249 \zeta_{6} q^{74} + ( -244 + 244 \zeta_{6} ) q^{75} + ( -330 - 330 \zeta_{6} ) q^{76} -192 q^{77} + 1276 q^{79} + ( -1 - \zeta_{6} ) q^{80} + ( -421 + 421 \zeta_{6} ) q^{81} + 471 \zeta_{6} q^{82} + ( -456 + 912 \zeta_{6} ) q^{83} + ( 160 - 80 \zeta_{6} ) q^{84} + ( 234 - 117 \zeta_{6} ) q^{85} + ( -104 + 208 \zeta_{6} ) q^{86} + 282 \zeta_{6} q^{87} + ( 312 - 312 \zeta_{6} ) q^{88} + ( 564 + 564 \zeta_{6} ) q^{89} + 69 q^{90} -390 q^{92} + ( 180 + 180 \zeta_{6} ) q^{93} + ( -522 + 522 \zeta_{6} ) q^{94} + 198 \zeta_{6} q^{95} + ( -210 + 420 \zeta_{6} ) q^{96} + ( -232 + 116 \zeta_{6} ) q^{97} + ( 302 - 151 \zeta_{6} ) q^{98} + ( 184 - 368 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} - 2 q^{3} - 5 q^{4} + 6 q^{6} + 24 q^{7} + 23 q^{9} + O(q^{10}) \) \( 2 q - 3 q^{2} - 2 q^{3} - 5 q^{4} + 6 q^{6} + 24 q^{7} + 23 q^{9} + 3 q^{10} - 24 q^{11} + 20 q^{12} - 48 q^{14} - 6 q^{15} - q^{16} - 117 q^{17} + 198 q^{19} + 15 q^{20} + 24 q^{22} + 78 q^{23} - 78 q^{24} + 244 q^{25} - 200 q^{27} - 120 q^{28} + 141 q^{29} + 6 q^{30} + 315 q^{32} + 48 q^{33} + 24 q^{35} + 115 q^{36} + 249 q^{37} - 396 q^{38} - 78 q^{40} - 471 q^{41} + 48 q^{42} - 104 q^{43} - 69 q^{45} - 234 q^{46} - 2 q^{48} - 151 q^{49} - 366 q^{50} + 468 q^{51} + 186 q^{53} + 300 q^{54} + 24 q^{55} + 312 q^{56} - 423 q^{58} + 492 q^{59} - 145 q^{61} - 270 q^{62} + 552 q^{63} - 614 q^{64} - 96 q^{66} + 1362 q^{67} - 585 q^{68} + 156 q^{69} - 1830 q^{71} - 897 q^{72} - 249 q^{74} - 244 q^{75} - 990 q^{76} - 384 q^{77} + 2552 q^{79} - 3 q^{80} - 421 q^{81} + 471 q^{82} + 240 q^{84} + 351 q^{85} + 282 q^{87} + 312 q^{88} + 1692 q^{89} + 138 q^{90} - 780 q^{92} + 540 q^{93} - 522 q^{94} + 198 q^{95} - 348 q^{97} + 453 q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1
0.500000 0.866025i
0.500000 + 0.866025i
−1.50000 + 0.866025i −1.00000 1.73205i −2.50000 + 4.33013i 1.73205i 3.00000 + 1.73205i 12.0000 + 6.92820i 22.5167i 11.5000 19.9186i 1.50000 + 2.59808i
147.1 −1.50000 0.866025i −1.00000 + 1.73205i −2.50000 4.33013i 1.73205i 3.00000 1.73205i 12.0000 6.92820i 22.5167i 11.5000 + 19.9186i 1.50000 2.59808i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.4.e.a 2
13.b even 2 1 13.4.e.b 2
13.c even 3 1 13.4.e.b 2
13.c even 3 1 169.4.b.d 2
13.d odd 4 2 169.4.c.h 4
13.e even 6 1 169.4.b.d 2
13.e even 6 1 inner 169.4.e.a 2
13.f odd 12 2 169.4.a.i 2
13.f odd 12 2 169.4.c.h 4
39.d odd 2 1 117.4.q.a 2
39.i odd 6 1 117.4.q.a 2
39.k even 12 2 1521.4.a.o 2
52.b odd 2 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.b even 2 1
13.4.e.b 2 13.c even 3 1
117.4.q.a 2 39.d odd 2 1
117.4.q.a 2 39.i odd 6 1
169.4.a.i 2 13.f odd 12 2
169.4.b.d 2 13.c even 3 1
169.4.b.d 2 13.e even 6 1
169.4.c.h 4 13.d odd 4 2
169.4.c.h 4 13.f odd 12 2
169.4.e.a 2 1.a even 1 1 trivial
169.4.e.a 2 13.e even 6 1 inner
208.4.w.b 2 52.b odd 2 1
208.4.w.b 2 52.j odd 6 1
1521.4.a.o 2 39.k even 12 2

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 3 + 3 T + T^{2} \)
$3$ \( 4 + 2 T + T^{2} \)
$5$ \( 3 + T^{2} \)
$7$ \( 192 - 24 T + T^{2} \)
$11$ \( 192 + 24 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( 13689 + 117 T + T^{2} \)
$19$ \( 13068 - 198 T + T^{2} \)
$23$ \( 6084 - 78 T + T^{2} \)
$29$ \( 19881 - 141 T + T^{2} \)
$31$ \( 24300 + T^{2} \)
$37$ \( 20667 - 249 T + T^{2} \)
$41$ \( 73947 + 471 T + T^{2} \)
$43$ \( 10816 + 104 T + T^{2} \)
$47$ \( 90828 + T^{2} \)
$53$ \( ( -93 + T )^{2} \)
$59$ \( 80688 - 492 T + T^{2} \)
$61$ \( 21025 + 145 T + T^{2} \)
$67$ \( 618348 - 1362 T + T^{2} \)
$71$ \( 1116300 + 1830 T + T^{2} \)
$73$ \( 210675 + T^{2} \)
$79$ \( ( -1276 + T )^{2} \)
$83$ \( 623808 + T^{2} \)
$89$ \( 954288 - 1692 T + T^{2} \)
$97$ \( 40368 + 348 T + T^{2} \)
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