Properties

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

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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: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} - 2 \beta_1 q^{3} + ( - 5 \beta_1 + 5) q^{4} + \beta_{3} q^{5} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{6} + (8 \beta_{3} - 8 \beta_{2}) q^{7} - 13 \beta_{3} q^{8} + ( - 23 \beta_1 + 23) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} - 2 \beta_1 q^{3} + ( - 5 \beta_1 + 5) q^{4} + \beta_{3} q^{5} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{6} + (8 \beta_{3} - 8 \beta_{2}) q^{7} - 13 \beta_{3} q^{8} + ( - 23 \beta_1 + 23) q^{9} - 3 \beta_1 q^{10} + 8 \beta_{2} q^{11} - 10 q^{12} - 24 q^{14} - 2 \beta_{2} q^{15} - \beta_1 q^{16} + ( - 117 \beta_1 + 117) q^{17} - 23 \beta_{3} q^{18} + ( - 66 \beta_{3} + 66 \beta_{2}) q^{19} + (5 \beta_{3} - 5 \beta_{2}) q^{20} - 16 \beta_{3} q^{21} + ( - 24 \beta_1 + 24) q^{22} - 78 \beta_1 q^{23} + 26 \beta_{2} q^{24} - 122 q^{25} - 100 q^{27} - 40 \beta_{2} q^{28} + 141 \beta_1 q^{29} + (6 \beta_1 - 6) q^{30} - 90 \beta_{3} q^{31} + ( - 105 \beta_{3} + 105 \beta_{2}) q^{32} + (16 \beta_{3} - 16 \beta_{2}) q^{33} - 117 \beta_{3} q^{34} + ( - 24 \beta_1 + 24) q^{35} - 115 \beta_1 q^{36} - 83 \beta_{2} q^{37} + 198 q^{38} - 39 q^{40} - 157 \beta_{2} q^{41} + 48 \beta_1 q^{42} + ( - 104 \beta_1 + 104) q^{43} + 40 \beta_{3} q^{44} + (23 \beta_{3} - 23 \beta_{2}) q^{45} + ( - 78 \beta_{3} + 78 \beta_{2}) q^{46} + 174 \beta_{3} q^{47} + (2 \beta_1 - 2) q^{48} + 151 \beta_1 q^{49} + 122 \beta_{2} q^{50} - 234 q^{51} + 93 q^{53} + 100 \beta_{2} q^{54} + 24 \beta_1 q^{55} + (312 \beta_1 - 312) q^{56} + 132 \beta_{3} q^{57} + (141 \beta_{3} - 141 \beta_{2}) q^{58} + (164 \beta_{3} - 164 \beta_{2}) q^{59} - 10 \beta_{3} q^{60} + (145 \beta_1 - 145) q^{61} + 270 \beta_1 q^{62} - 184 \beta_{2} q^{63} + 307 q^{64} - 48 q^{66} + 454 \beta_{2} q^{67} - 585 \beta_1 q^{68} + (156 \beta_1 - 156) q^{69} - 24 \beta_{3} q^{70} + (610 \beta_{3} - 610 \beta_{2}) q^{71} + ( - 299 \beta_{3} + 299 \beta_{2}) q^{72} + 265 \beta_{3} q^{73} + (249 \beta_1 - 249) q^{74} + 244 \beta_1 q^{75} + 330 \beta_{2} q^{76} + 192 q^{77} + 1276 q^{79} - \beta_{2} q^{80} - 421 \beta_1 q^{81} + (471 \beta_1 - 471) q^{82} + 456 \beta_{3} q^{83} + ( - 80 \beta_{3} + 80 \beta_{2}) q^{84} + (117 \beta_{3} - 117 \beta_{2}) q^{85} - 104 \beta_{3} q^{86} + ( - 282 \beta_1 + 282) q^{87} - 312 \beta_1 q^{88} - 564 \beta_{2} q^{89} - 69 q^{90} - 390 q^{92} + 180 \beta_{2} q^{93} - 522 \beta_1 q^{94} + (198 \beta_1 - 198) q^{95} + 210 \beta_{3} q^{96} + (116 \beta_{3} - 116 \beta_{2}) q^{97} + (151 \beta_{3} - 151 \beta_{2}) q^{98} + 184 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 10 q^{4} + 46 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} + 10 q^{4} + 46 q^{9} - 6 q^{10} - 40 q^{12} - 96 q^{14} - 2 q^{16} + 234 q^{17} + 48 q^{22} - 156 q^{23} - 488 q^{25} - 400 q^{27} + 282 q^{29} - 12 q^{30} + 48 q^{35} - 230 q^{36} + 792 q^{38} - 156 q^{40} + 96 q^{42} + 208 q^{43} - 4 q^{48} + 302 q^{49} - 936 q^{51} + 372 q^{53} + 48 q^{55} - 624 q^{56} - 290 q^{61} + 540 q^{62} + 1228 q^{64} - 192 q^{66} - 1170 q^{68} - 312 q^{69} - 498 q^{74} + 488 q^{75} + 768 q^{77} + 5104 q^{79} - 842 q^{81} - 942 q^{82} + 564 q^{87} - 624 q^{88} - 276 q^{90} - 1560 q^{92} - 1044 q^{94} - 396 q^{95}+O(q^{100}) \) 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
−0.866025 + 1.50000i −1.00000 + 1.73205i 2.50000 + 4.33013i 1.73205 −1.73205 3.00000i 6.92820 + 12.0000i −22.5167 11.5000 + 19.9186i −1.50000 + 2.59808i
22.2 0.866025 1.50000i −1.00000 + 1.73205i 2.50000 + 4.33013i −1.73205 1.73205 + 3.00000i −6.92820 12.0000i 22.5167 11.5000 + 19.9186i −1.50000 + 2.59808i
146.1 −0.866025 1.50000i −1.00000 1.73205i 2.50000 4.33013i 1.73205 −1.73205 + 3.00000i 6.92820 12.0000i −22.5167 11.5000 19.9186i −1.50000 2.59808i
146.2 0.866025 + 1.50000i −1.00000 1.73205i 2.50000 4.33013i −1.73205 1.73205 3.00000i −6.92820 + 12.0000i 22.5167 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.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.h 4
13.b even 2 1 inner 169.4.c.h 4
13.c even 3 1 169.4.a.i 2
13.c even 3 1 inner 169.4.c.h 4
13.d odd 4 1 13.4.e.b 2
13.d odd 4 1 169.4.e.a 2
13.e even 6 1 169.4.a.i 2
13.e even 6 1 inner 169.4.c.h 4
13.f odd 12 1 13.4.e.b 2
13.f odd 12 2 169.4.b.d 2
13.f odd 12 1 169.4.e.a 2
39.f even 4 1 117.4.q.a 2
39.h odd 6 1 1521.4.a.o 2
39.i odd 6 1 1521.4.a.o 2
39.k even 12 1 117.4.q.a 2
52.f even 4 1 208.4.w.b 2
52.l even 12 1 208.4.w.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.e.b 2 13.d odd 4 1
13.4.e.b 2 13.f odd 12 1
117.4.q.a 2 39.f even 4 1
117.4.q.a 2 39.k even 12 1
169.4.a.i 2 13.c even 3 1
169.4.a.i 2 13.e even 6 1
169.4.b.d 2 13.f odd 12 2
169.4.c.h 4 1.a even 1 1 trivial
169.4.c.h 4 13.b even 2 1 inner
169.4.c.h 4 13.c even 3 1 inner
169.4.c.h 4 13.e even 6 1 inner
169.4.e.a 2 13.d odd 4 1
169.4.e.a 2 13.f odd 12 1
208.4.w.b 2 52.f even 4 1
208.4.w.b 2 52.l even 12 1
1521.4.a.o 2 39.h odd 6 1
1521.4.a.o 2 39.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 3T_{2}^{2} + 9 \) acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$3$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 192 T^{2} + 36864 \) Copy content Toggle raw display
$11$ \( T^{4} + 192 T^{2} + 36864 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 117 T + 13689)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 13068 T^{2} + \cdots + 170772624 \) Copy content Toggle raw display
$23$ \( (T^{2} + 78 T + 6084)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 141 T + 19881)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 24300)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 20667 T^{2} + \cdots + 427124889 \) Copy content Toggle raw display
$41$ \( T^{4} + 73947 T^{2} + \cdots + 5468158809 \) Copy content Toggle raw display
$43$ \( (T^{2} - 104 T + 10816)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 90828)^{2} \) Copy content Toggle raw display
$53$ \( (T - 93)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + 80688 T^{2} + \cdots + 6510553344 \) Copy content Toggle raw display
$61$ \( (T^{2} + 145 T + 21025)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 618348 T^{2} + \cdots + 382354249104 \) Copy content Toggle raw display
$71$ \( T^{4} + 1116300 T^{2} + \cdots + 1246125690000 \) Copy content Toggle raw display
$73$ \( (T^{2} - 210675)^{2} \) Copy content Toggle raw display
$79$ \( (T - 1276)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 623808)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 954288 T^{2} + \cdots + 910665586944 \) Copy content Toggle raw display
$97$ \( T^{4} + 40368 T^{2} + \cdots + 1629575424 \) Copy content Toggle raw display
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