Properties

Label 169.4.b.c
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: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 i q^{2} + 2 q^{3} - 8 q^{4} + 17 i q^{5} + 8 i q^{6} - 20 i q^{7} - 23 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 4 i q^{2} + 2 q^{3} - 8 q^{4} + 17 i q^{5} + 8 i q^{6} - 20 i q^{7} - 23 q^{9} - 68 q^{10} + 32 i q^{11} - 16 q^{12} + 80 q^{14} + 34 i q^{15} - 64 q^{16} + 13 q^{17} - 92 i q^{18} + 30 i q^{19} - 136 i q^{20} - 40 i q^{21} - 128 q^{22} - 78 q^{23} - 164 q^{25} - 100 q^{27} + 160 i q^{28} + 197 q^{29} - 136 q^{30} - 74 i q^{31} - 256 i q^{32} + 64 i q^{33} + 52 i q^{34} + 340 q^{35} + 184 q^{36} + 227 i q^{37} - 120 q^{38} - 165 i q^{41} + 160 q^{42} + 156 q^{43} - 256 i q^{44} - 391 i q^{45} - 312 i q^{46} + 162 i q^{47} - 128 q^{48} - 57 q^{49} - 656 i q^{50} + 26 q^{51} + 93 q^{53} - 400 i q^{54} - 544 q^{55} + 60 i q^{57} + 788 i q^{58} + 864 i q^{59} - 272 i q^{60} + 145 q^{61} + 296 q^{62} + 460 i q^{63} + 512 q^{64} - 256 q^{66} + 862 i q^{67} - 104 q^{68} - 156 q^{69} + 1360 i q^{70} + 654 i q^{71} - 215 i q^{73} - 908 q^{74} - 328 q^{75} - 240 i q^{76} + 640 q^{77} - 76 q^{79} - 1088 i q^{80} + 421 q^{81} + 660 q^{82} + 628 i q^{83} + 320 i q^{84} + 221 i q^{85} + 624 i q^{86} + 394 q^{87} + 266 i q^{89} + 1564 q^{90} + 624 q^{92} - 148 i q^{93} - 648 q^{94} - 510 q^{95} - 512 i q^{96} + 238 i q^{97} - 228 i q^{98} - 736 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{3} - 16 q^{4} - 46 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{3} - 16 q^{4} - 46 q^{9} - 136 q^{10} - 32 q^{12} + 160 q^{14} - 128 q^{16} + 26 q^{17} - 256 q^{22} - 156 q^{23} - 328 q^{25} - 200 q^{27} + 394 q^{29} - 272 q^{30} + 680 q^{35} + 368 q^{36} - 240 q^{38} + 320 q^{42} + 312 q^{43} - 256 q^{48} - 114 q^{49} + 52 q^{51} + 186 q^{53} - 1088 q^{55} + 290 q^{61} + 592 q^{62} + 1024 q^{64} - 512 q^{66} - 208 q^{68} - 312 q^{69} - 1816 q^{74} - 656 q^{75} + 1280 q^{77} - 152 q^{79} + 842 q^{81} + 1320 q^{82} + 788 q^{87} + 3128 q^{90} + 1248 q^{92} - 1296 q^{94} - 1020 q^{95}+O(q^{100}) \) 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
1.00000i
1.00000i
4.00000i 2.00000 −8.00000 17.0000i 8.00000i 20.0000i 0 −23.0000 −68.0000
168.2 4.00000i 2.00000 −8.00000 17.0000i 8.00000i 20.0000i 0 −23.0000 −68.0000
\(n\): 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.c 2
13.b even 2 1 inner 169.4.b.c 2
13.c even 3 2 169.4.e.c 4
13.d odd 4 1 169.4.a.a 1
13.d odd 4 1 169.4.a.d 1
13.e even 6 2 169.4.e.c 4
13.f odd 12 2 13.4.c.a 2
13.f odd 12 2 169.4.c.d 2
39.f even 4 1 1521.4.a.b 1
39.f even 4 1 1521.4.a.k 1
39.k even 12 2 117.4.g.c 2
52.l even 12 2 208.4.i.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.c.a 2 13.f odd 12 2
117.4.g.c 2 39.k even 12 2
169.4.a.a 1 13.d odd 4 1
169.4.a.d 1 13.d odd 4 1
169.4.b.c 2 1.a even 1 1 trivial
169.4.b.c 2 13.b even 2 1 inner
169.4.c.d 2 13.f odd 12 2
169.4.e.c 4 13.c even 3 2
169.4.e.c 4 13.e even 6 2
208.4.i.b 2 52.l even 12 2
1521.4.a.b 1 39.f even 4 1
1521.4.a.k 1 39.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 16 \) acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 16 \) Copy content Toggle raw display
$3$ \( (T - 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 289 \) Copy content Toggle raw display
$7$ \( T^{2} + 400 \) Copy content Toggle raw display
$11$ \( T^{2} + 1024 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T - 13)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 900 \) Copy content Toggle raw display
$23$ \( (T + 78)^{2} \) Copy content Toggle raw display
$29$ \( (T - 197)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 5476 \) Copy content Toggle raw display
$37$ \( T^{2} + 51529 \) Copy content Toggle raw display
$41$ \( T^{2} + 27225 \) Copy content Toggle raw display
$43$ \( (T - 156)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 26244 \) Copy content Toggle raw display
$53$ \( (T - 93)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 746496 \) Copy content Toggle raw display
$61$ \( (T - 145)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 743044 \) Copy content Toggle raw display
$71$ \( T^{2} + 427716 \) Copy content Toggle raw display
$73$ \( T^{2} + 46225 \) Copy content Toggle raw display
$79$ \( (T + 76)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 394384 \) Copy content Toggle raw display
$89$ \( T^{2} + 70756 \) Copy content Toggle raw display
$97$ \( T^{2} + 56644 \) Copy content Toggle raw display
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