Properties

Label 169.4.c.i
Level $169$
Weight $4$
Character orbit 169.c
Analytic conductor $9.971$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 169.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.97132279097\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta_{2} q^{2} + 7 \beta_1 q^{3} + (4 \beta_1 - 4) q^{4} + 8 \beta_{3} q^{5} + (14 \beta_{3} - 14 \beta_{2}) q^{6} + (13 \beta_{3} - 13 \beta_{2}) q^{7} - 8 \beta_{3} q^{8} + (22 \beta_1 - 22) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_{2} q^{2} + 7 \beta_1 q^{3} + (4 \beta_1 - 4) q^{4} + 8 \beta_{3} q^{5} + (14 \beta_{3} - 14 \beta_{2}) q^{6} + (13 \beta_{3} - 13 \beta_{2}) q^{7} - 8 \beta_{3} q^{8} + (22 \beta_1 - 22) q^{9} - 48 \beta_1 q^{10} + 13 \beta_{2} q^{11} - 28 q^{12} - 78 q^{14} + 56 \beta_{2} q^{15} + 80 \beta_1 q^{16} + ( - 27 \beta_1 + 27) q^{17} + 44 \beta_{3} q^{18} + (51 \beta_{3} - 51 \beta_{2}) q^{19} + ( - 32 \beta_{3} + 32 \beta_{2}) q^{20} + 91 \beta_{3} q^{21} + ( - 78 \beta_1 + 78) q^{22} + 57 \beta_1 q^{23} - 56 \beta_{2} q^{24} + 67 q^{25} + 35 q^{27} + 52 \beta_{2} q^{28} + 69 \beta_1 q^{29} + ( - 336 \beta_1 + 336) q^{30} - 42 \beta_{3} q^{31} + (96 \beta_{3} - 96 \beta_{2}) q^{32} + ( - 91 \beta_{3} + 91 \beta_{2}) q^{33} - 54 \beta_{3} q^{34} + ( - 312 \beta_1 + 312) q^{35} - 88 \beta_1 q^{36} + 23 \beta_{2} q^{37} - 306 q^{38} - 192 q^{40} - 227 \beta_{2} q^{41} - 546 \beta_1 q^{42} + (85 \beta_1 - 85) q^{43} - 52 \beta_{3} q^{44} + ( - 176 \beta_{3} + 176 \beta_{2}) q^{45} + (114 \beta_{3} - 114 \beta_{2}) q^{46} + 198 \beta_{3} q^{47} + (560 \beta_1 - 560) q^{48} - 164 \beta_1 q^{49} - 134 \beta_{2} q^{50} + 189 q^{51} + 426 q^{53} - 70 \beta_{2} q^{54} + 312 \beta_1 q^{55} + (312 \beta_1 - 312) q^{56} + 357 \beta_{3} q^{57} + (138 \beta_{3} - 138 \beta_{2}) q^{58} + ( - 11 \beta_{3} + 11 \beta_{2}) q^{59} - 224 \beta_{3} q^{60} + ( - 17 \beta_1 + 17) q^{61} + 252 \beta_1 q^{62} + 286 \beta_{2} q^{63} + 64 q^{64} + 546 q^{66} + 95 \beta_{2} q^{67} + 108 \beta_1 q^{68} + (399 \beta_1 - 399) q^{69} - 624 \beta_{3} q^{70} + ( - 337 \beta_{3} + 337 \beta_{2}) q^{71} + (176 \beta_{3} - 176 \beta_{2}) q^{72} - 580 \beta_{3} q^{73} + ( - 138 \beta_1 + 138) q^{74} + 469 \beta_1 q^{75} + 204 \beta_{2} q^{76} + 507 q^{77} - 1244 q^{79} + 640 \beta_{2} q^{80} + 839 \beta_1 q^{81} + (1362 \beta_1 - 1362) q^{82} - 246 \beta_{3} q^{83} + ( - 364 \beta_{3} + 364 \beta_{2}) q^{84} + (216 \beta_{3} - 216 \beta_{2}) q^{85} + 170 \beta_{3} q^{86} + (483 \beta_1 - 483) q^{87} - 312 \beta_1 q^{88} - 177 \beta_{2} q^{89} + 1056 q^{90} - 228 q^{92} - 294 \beta_{2} q^{93} - 1188 \beta_1 q^{94} + ( - 1224 \beta_1 + 1224) q^{95} + 672 \beta_{3} q^{96} + ( - 713 \beta_{3} + 713 \beta_{2}) q^{97} + ( - 328 \beta_{3} + 328 \beta_{2}) q^{98} - 286 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 14 q^{3} - 8 q^{4} - 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 14 q^{3} - 8 q^{4} - 44 q^{9} - 96 q^{10} - 112 q^{12} - 312 q^{14} + 160 q^{16} + 54 q^{17} + 156 q^{22} + 114 q^{23} + 268 q^{25} + 140 q^{27} + 138 q^{29} + 672 q^{30} + 624 q^{35} - 176 q^{36} - 1224 q^{38} - 768 q^{40} - 1092 q^{42} - 170 q^{43} - 1120 q^{48} - 328 q^{49} + 756 q^{51} + 1704 q^{53} + 624 q^{55} - 624 q^{56} + 34 q^{61} + 504 q^{62} + 256 q^{64} + 2184 q^{66} + 216 q^{68} - 798 q^{69} + 276 q^{74} + 938 q^{75} + 2028 q^{77} - 4976 q^{79} + 1678 q^{81} - 2724 q^{82} - 966 q^{87} - 624 q^{88} + 4224 q^{90} - 912 q^{92} - 2376 q^{94} + 2448 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-1 + \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−1.73205 + 3.00000i 3.50000 6.06218i −2.00000 3.46410i 13.8564 12.1244 + 21.0000i 11.2583 + 19.5000i −13.8564 −11.0000 19.0526i −24.0000 + 41.5692i
22.2 1.73205 3.00000i 3.50000 6.06218i −2.00000 3.46410i −13.8564 −12.1244 21.0000i −11.2583 19.5000i 13.8564 −11.0000 19.0526i −24.0000 + 41.5692i
146.1 −1.73205 3.00000i 3.50000 + 6.06218i −2.00000 + 3.46410i 13.8564 12.1244 21.0000i 11.2583 19.5000i −13.8564 −11.0000 + 19.0526i −24.0000 41.5692i
146.2 1.73205 + 3.00000i 3.50000 + 6.06218i −2.00000 + 3.46410i −13.8564 −12.1244 + 21.0000i −11.2583 + 19.5000i 13.8564 −11.0000 + 19.0526i −24.0000 41.5692i
\(n\): 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
13.c even 3 1 inner
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.c.i 4
13.b even 2 1 inner 169.4.c.i 4
13.c even 3 1 169.4.a.h 2
13.c even 3 1 inner 169.4.c.i 4
13.d odd 4 1 13.4.e.a 2
13.d odd 4 1 169.4.e.b 2
13.e even 6 1 169.4.a.h 2
13.e even 6 1 inner 169.4.c.i 4
13.f odd 12 1 13.4.e.a 2
13.f odd 12 2 169.4.b.b 2
13.f odd 12 1 169.4.e.b 2
39.f even 4 1 117.4.q.c 2
39.h odd 6 1 1521.4.a.q 2
39.i odd 6 1 1521.4.a.q 2
39.k even 12 1 117.4.q.c 2
52.f even 4 1 208.4.w.a 2
52.l even 12 1 208.4.w.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.e.a 2 13.d odd 4 1
13.4.e.a 2 13.f odd 12 1
117.4.q.c 2 39.f even 4 1
117.4.q.c 2 39.k even 12 1
169.4.a.h 2 13.c even 3 1
169.4.a.h 2 13.e even 6 1
169.4.b.b 2 13.f odd 12 2
169.4.c.i 4 1.a even 1 1 trivial
169.4.c.i 4 13.b even 2 1 inner
169.4.c.i 4 13.c even 3 1 inner
169.4.c.i 4 13.e even 6 1 inner
169.4.e.b 2 13.d odd 4 1
169.4.e.b 2 13.f odd 12 1
208.4.w.a 2 52.f even 4 1
208.4.w.a 2 52.l even 12 1
1521.4.a.q 2 39.h odd 6 1
1521.4.a.q 2 39.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 12T_{2}^{2} + 144 \) acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$3$ \( (T^{2} - 7 T + 49)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 192)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 507 T^{2} + 257049 \) Copy content Toggle raw display
$11$ \( T^{4} + 507 T^{2} + 257049 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 27 T + 729)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 7803 T^{2} + \cdots + 60886809 \) Copy content Toggle raw display
$23$ \( (T^{2} - 57 T + 3249)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 69 T + 4761)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 5292)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 1587 T^{2} + \cdots + 2518569 \) Copy content Toggle raw display
$41$ \( T^{4} + 154587 T^{2} + \cdots + 23897140569 \) Copy content Toggle raw display
$43$ \( (T^{2} + 85 T + 7225)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 117612)^{2} \) Copy content Toggle raw display
$53$ \( (T - 426)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + 363 T^{2} + 131769 \) Copy content Toggle raw display
$61$ \( (T^{2} - 17 T + 289)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 27075 T^{2} + \cdots + 733055625 \) Copy content Toggle raw display
$71$ \( T^{4} + 340707 T^{2} + \cdots + 116081259849 \) Copy content Toggle raw display
$73$ \( (T^{2} - 1009200)^{2} \) Copy content Toggle raw display
$79$ \( (T + 1244)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 181548)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 93987 T^{2} + \cdots + 8833556169 \) Copy content Toggle raw display
$97$ \( T^{4} + 1525107 T^{2} + \cdots + 2325951361449 \) Copy content Toggle raw display
show more
show less