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Properties

Label 169.4.e.c
Level $169$
Weight $4$
Character orbit 169.e
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.e (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.97132279097\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 \zeta_{12} q^{2} + ( -2 + 2 \zeta_{12}^{2} ) q^{3} + 8 \zeta_{12}^{2} q^{4} -17 \zeta_{12}^{3} q^{5} + ( -8 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{6} + ( 20 \zeta_{12} - 20 \zeta_{12}^{3} ) q^{7} + 23 \zeta_{12}^{2} q^{9} +O(q^{10})\) \( q + 4 \zeta_{12} q^{2} + ( -2 + 2 \zeta_{12}^{2} ) q^{3} + 8 \zeta_{12}^{2} q^{4} -17 \zeta_{12}^{3} q^{5} + ( -8 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{6} + ( 20 \zeta_{12} - 20 \zeta_{12}^{3} ) q^{7} + 23 \zeta_{12}^{2} q^{9} + ( 68 - 68 \zeta_{12}^{2} ) q^{10} + 32 \zeta_{12} q^{11} -16 q^{12} + 80 q^{14} + 34 \zeta_{12} q^{15} + ( 64 - 64 \zeta_{12}^{2} ) q^{16} -13 \zeta_{12}^{2} q^{17} + 92 \zeta_{12}^{3} q^{18} + ( -30 \zeta_{12} + 30 \zeta_{12}^{3} ) q^{19} + ( 136 \zeta_{12} - 136 \zeta_{12}^{3} ) q^{20} + 40 \zeta_{12}^{3} q^{21} + 128 \zeta_{12}^{2} q^{22} + ( 78 - 78 \zeta_{12}^{2} ) q^{23} -164 q^{25} -100 q^{27} + 160 \zeta_{12} q^{28} + ( -197 + 197 \zeta_{12}^{2} ) q^{29} + 136 \zeta_{12}^{2} q^{30} + 74 \zeta_{12}^{3} q^{31} + ( 256 \zeta_{12} - 256 \zeta_{12}^{3} ) q^{32} + ( -64 \zeta_{12} + 64 \zeta_{12}^{3} ) q^{33} -52 \zeta_{12}^{3} q^{34} -340 \zeta_{12}^{2} q^{35} + ( -184 + 184 \zeta_{12}^{2} ) q^{36} + 227 \zeta_{12} q^{37} -120 q^{38} -165 \zeta_{12} q^{41} + ( -160 + 160 \zeta_{12}^{2} ) q^{42} -156 \zeta_{12}^{2} q^{43} + 256 \zeta_{12}^{3} q^{44} + ( 391 \zeta_{12} - 391 \zeta_{12}^{3} ) q^{45} + ( 312 \zeta_{12} - 312 \zeta_{12}^{3} ) q^{46} -162 \zeta_{12}^{3} q^{47} + 128 \zeta_{12}^{2} q^{48} + ( 57 - 57 \zeta_{12}^{2} ) q^{49} -656 \zeta_{12} q^{50} + 26 q^{51} + 93 q^{53} -400 \zeta_{12} q^{54} + ( 544 - 544 \zeta_{12}^{2} ) q^{55} -60 \zeta_{12}^{3} q^{57} + ( -788 \zeta_{12} + 788 \zeta_{12}^{3} ) q^{58} + ( -864 \zeta_{12} + 864 \zeta_{12}^{3} ) q^{59} + 272 \zeta_{12}^{3} q^{60} -145 \zeta_{12}^{2} q^{61} + ( -296 + 296 \zeta_{12}^{2} ) q^{62} + 460 \zeta_{12} q^{63} + 512 q^{64} -256 q^{66} + 862 \zeta_{12} q^{67} + ( 104 - 104 \zeta_{12}^{2} ) q^{68} + 156 \zeta_{12}^{2} q^{69} -1360 \zeta_{12}^{3} q^{70} + ( -654 \zeta_{12} + 654 \zeta_{12}^{3} ) q^{71} + 215 \zeta_{12}^{3} q^{73} + 908 \zeta_{12}^{2} q^{74} + ( 328 - 328 \zeta_{12}^{2} ) q^{75} -240 \zeta_{12} q^{76} + 640 q^{77} -76 q^{79} -1088 \zeta_{12} q^{80} + ( -421 + 421 \zeta_{12}^{2} ) q^{81} -660 \zeta_{12}^{2} q^{82} -628 \zeta_{12}^{3} q^{83} + ( -320 \zeta_{12} + 320 \zeta_{12}^{3} ) q^{84} + ( -221 \zeta_{12} + 221 \zeta_{12}^{3} ) q^{85} -624 \zeta_{12}^{3} q^{86} -394 \zeta_{12}^{2} q^{87} + 266 \zeta_{12} q^{89} + 1564 q^{90} + 624 q^{92} -148 \zeta_{12} q^{93} + ( 648 - 648 \zeta_{12}^{2} ) q^{94} + 510 \zeta_{12}^{2} q^{95} + 512 \zeta_{12}^{3} q^{96} + ( -238 \zeta_{12} + 238 \zeta_{12}^{3} ) q^{97} + ( 228 \zeta_{12} - 228 \zeta_{12}^{3} ) q^{98} + 736 \zeta_{12}^{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 16 q^{4} + 46 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{3} + 16 q^{4} + 46 q^{9} + 136 q^{10} - 64 q^{12} + 320 q^{14} + 128 q^{16} - 26 q^{17} + 256 q^{22} + 156 q^{23} - 656 q^{25} - 400 q^{27} - 394 q^{29} + 272 q^{30} - 680 q^{35} - 368 q^{36} - 480 q^{38} - 320 q^{42} - 312 q^{43} + 256 q^{48} + 114 q^{49} + 104 q^{51} + 372 q^{53} + 1088 q^{55} - 290 q^{61} - 592 q^{62} + 2048 q^{64} - 1024 q^{66} + 208 q^{68} + 312 q^{69} + 1816 q^{74} + 656 q^{75} + 2560 q^{77} - 304 q^{79} - 842 q^{81} - 1320 q^{82} - 788 q^{87} + 6256 q^{90} + 2496 q^{92} + 1296 q^{94} + 1020 q^{95} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(\zeta_{12}^{2}\)

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
−0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 − 0.500000i
0.866025 + 0.500000i
−3.46410 + 2.00000i −1.00000 − 1.73205i 4.00000 − 6.92820i − 17.0000i 6.92820 + 4.00000i −17.3205 − 10.0000i 0 11.5000 − 19.9186i 34.0000 + 58.8897i
23.2 3.46410 − 2.00000i −1.00000 − 1.73205i 4.00000 − 6.92820i 17.0000i −6.92820 − 4.00000i 17.3205 + 10.0000i 0 11.5000 − 19.9186i 34.0000 + 58.8897i
147.1 −3.46410 − 2.00000i −1.00000 + 1.73205i 4.00000 + 6.92820i 17.0000i 6.92820 − 4.00000i −17.3205 + 10.0000i 0 11.5000 + 19.9186i 34.0000 − 58.8897i
147.2 3.46410 + 2.00000i −1.00000 + 1.73205i 4.00000 + 6.92820i − 17.0000i −6.92820 + 4.00000i 17.3205 − 10.0000i 0 11.5000 + 19.9186i 34.0000 − 58.8897i
\(n\): 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.e.c 4
13.b even 2 1 inner 169.4.e.c 4
13.c even 3 1 169.4.b.c 2
13.c even 3 1 inner 169.4.e.c 4
13.d odd 4 1 13.4.c.a ✓ 2
13.d odd 4 1 169.4.c.d 2
13.e even 6 1 169.4.b.c 2
13.e even 6 1 inner 169.4.e.c 4
13.f odd 12 1 13.4.c.a ✓ 2
13.f odd 12 1 169.4.a.a 1
13.f odd 12 1 169.4.a.d 1
13.f odd 12 1 169.4.c.d 2
39.f even 4 1 117.4.g.c 2
39.k even 12 1 117.4.g.c 2
39.k even 12 1 1521.4.a.b 1
39.k even 12 1 1521.4.a.k 1
52.f even 4 1 208.4.i.b 2
52.l even 12 1 208.4.i.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.c.a ✓ 2 13.d odd 4 1
13.4.c.a ✓ 2 13.f odd 12 1
117.4.g.c 2 39.f even 4 1
117.4.g.c 2 39.k even 12 1
169.4.a.a 1 13.f odd 12 1
169.4.a.d 1 13.f odd 12 1
169.4.b.c 2 13.c even 3 1
169.4.b.c 2 13.e even 6 1
169.4.c.d 2 13.d odd 4 1
169.4.c.d 2 13.f odd 12 1
169.4.e.c 4 1.a even 1 1 trivial
169.4.e.c 4 13.b even 2 1 inner
169.4.e.c 4 13.c even 3 1 inner
169.4.e.c 4 13.e even 6 1 inner
208.4.i.b 2 52.f even 4 1
208.4.i.b 2 52.l even 12 1
1521.4.a.b 1 39.k even 12 1
1521.4.a.k 1 39.k even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 16 T_{2}^{2} + 256 \) acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 256 - 16 T^{2} + T^{4} \)
$3$ \( ( 4 + 2 T + T^{2} )^{2} \)
$5$ \( ( 289 + T^{2} )^{2} \)
$7$ \( 160000 - 400 T^{2} + T^{4} \)
$11$ \( 1048576 - 1024 T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( ( 169 + 13 T + T^{2} )^{2} \)
$19$ \( 810000 - 900 T^{2} + T^{4} \)
$23$ \( ( 6084 - 78 T + T^{2} )^{2} \)
$29$ \( ( 38809 + 197 T + T^{2} )^{2} \)
$31$ \( ( 5476 + T^{2} )^{2} \)
$37$ \( 2655237841 - 51529 T^{2} + T^{4} \)
$41$ \( 741200625 - 27225 T^{2} + T^{4} \)
$43$ \( ( 24336 + 156 T + T^{2} )^{2} \)
$47$ \( ( 26244 + T^{2} )^{2} \)
$53$ \( ( -93 + T )^{4} \)
$59$ \( 557256278016 - 746496 T^{2} + T^{4} \)
$61$ \( ( 21025 + 145 T + T^{2} )^{2} \)
$67$ \( 552114385936 - 743044 T^{2} + T^{4} \)
$71$ \( 182940976656 - 427716 T^{2} + T^{4} \)
$73$ \( ( 46225 + T^{2} )^{2} \)
$79$ \( ( 76 + T )^{4} \)
$83$ \( ( 394384 + T^{2} )^{2} \)
$89$ \( 5006411536 - 70756 T^{2} + T^{4} \)
$97$ \( 3208542736 - 56644 T^{2} + T^{4} \)
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