Properties

Label 169.4.e.f
Level $169$
Weight $4$
Character orbit 169.e
Analytic conductor $9.971$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.1731891456.1
Defining polynomial: \(x^{8} - 9 x^{6} + 65 x^{4} - 144 x^{2} + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( \beta_{2} - 3 \beta_{7} ) q^{3} + ( -4 - 3 \beta_{2} - \beta_{4} + \beta_{7} ) q^{4} + ( -\beta_{1} - \beta_{3} + \beta_{5} + \beta_{6} ) q^{5} + ( -\beta_{5} - 6 \beta_{6} ) q^{6} + ( 11 \beta_{5} + 5 \beta_{6} ) q^{7} + ( -11 \beta_{1} - 2 \beta_{3} + 11 \beta_{5} + 2 \beta_{6} ) q^{8} + ( -25 - 10 \beta_{2} - 15 \beta_{4} + 15 \beta_{7} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( \beta_{2} - 3 \beta_{7} ) q^{3} + ( -4 - 3 \beta_{2} - \beta_{4} + \beta_{7} ) q^{4} + ( -\beta_{1} - \beta_{3} + \beta_{5} + \beta_{6} ) q^{5} + ( -\beta_{5} - 6 \beta_{6} ) q^{6} + ( 11 \beta_{5} + 5 \beta_{6} ) q^{7} + ( -11 \beta_{1} - 2 \beta_{3} + 11 \beta_{5} + 2 \beta_{6} ) q^{8} + ( -25 - 10 \beta_{2} - 15 \beta_{4} + 15 \beta_{7} ) q^{9} + ( -3 \beta_{2} + \beta_{7} ) q^{10} + ( -12 \beta_{1} + 17 \beta_{3} ) q^{11} + ( 28 + 13 \beta_{4} ) q^{12} + ( 44 - \beta_{4} ) q^{14} + ( 7 \beta_{1} + 10 \beta_{3} ) q^{15} + ( -27 \beta_{2} - 15 \beta_{7} ) q^{16} + ( 18 + \beta_{2} + 17 \beta_{4} - 17 \beta_{7} ) q^{17} + ( -10 \beta_{1} - 30 \beta_{3} + 10 \beta_{5} + 30 \beta_{6} ) q^{18} + ( 32 \beta_{5} - 13 \beta_{6} ) q^{19} + ( -5 \beta_{5} - 6 \beta_{6} ) q^{20} + ( 41 \beta_{1} + 86 \beta_{3} - 41 \beta_{5} - 86 \beta_{6} ) q^{21} + ( -48 - 94 \beta_{2} + 46 \beta_{4} - 46 \beta_{7} ) q^{22} + ( -92 \beta_{2} + 12 \beta_{7} ) q^{23} + ( 23 \beta_{1} + 74 \beta_{3} ) q^{24} + ( 117 - 3 \beta_{4} ) q^{25} + ( 172 + 9 \beta_{4} ) q^{27} + ( -43 \beta_{1} - 42 \beta_{3} ) q^{28} + ( 26 \beta_{2} + 96 \beta_{7} ) q^{29} + ( 28 + 15 \beta_{2} + 13 \beta_{4} - 13 \beta_{7} ) q^{30} + ( 34 \beta_{1} - 13 \beta_{3} - 34 \beta_{5} + 13 \beta_{6} ) q^{31} + ( -61 \beta_{5} - 46 \beta_{6} ) q^{32} + ( -90 \beta_{5} + 4 \beta_{6} ) q^{33} + ( \beta_{1} + 34 \beta_{3} - \beta_{5} - 34 \beta_{6} ) q^{34} + ( -64 - 43 \beta_{2} - 21 \beta_{4} + 21 \beta_{7} ) q^{35} + ( 90 \beta_{2} - 70 \beta_{7} ) q^{36} + ( -5 \beta_{1} + 51 \beta_{3} ) q^{37} + ( 128 - 58 \beta_{4} ) q^{38} + ( -52 - 15 \beta_{4} ) q^{40} + ( 22 \beta_{1} + 63 \beta_{3} ) q^{41} + ( 33 \beta_{2} - 131 \beta_{7} ) q^{42} + ( 72 + 215 \beta_{2} - 143 \beta_{4} + 143 \beta_{7} ) q^{43} + ( 2 \beta_{1} - 44 \beta_{3} - 2 \beta_{5} + 44 \beta_{6} ) q^{44} + ( -40 \beta_{5} - 55 \beta_{6} ) q^{45} + ( 92 \beta_{5} + 24 \beta_{6} ) q^{46} + ( -121 \beta_{1} - 139 \beta_{3} + 121 \beta_{5} + 139 \beta_{6} ) q^{47} + ( -132 - 153 \beta_{2} + 21 \beta_{4} - 21 \beta_{7} ) q^{48} + ( -142 \beta_{2} + 99 \beta_{7} ) q^{49} + ( 120 \beta_{1} - 6 \beta_{3} ) q^{50} + ( -276 - 71 \beta_{4} ) q^{51} + ( -74 - 30 \beta_{4} ) q^{53} + ( 163 \beta_{1} + 18 \beta_{3} ) q^{54} + ( 2 \beta_{2} + 22 \beta_{7} ) q^{55} + ( -524 - 491 \beta_{2} - 33 \beta_{4} + 33 \beta_{7} ) q^{56} + ( -46 \beta_{1} + 140 \beta_{3} + 46 \beta_{5} - 140 \beta_{6} ) q^{57} + ( -26 \beta_{5} + 192 \beta_{6} ) q^{58} + ( 124 \beta_{5} + 123 \beta_{6} ) q^{59} + ( -41 \beta_{1} - 54 \beta_{3} + 41 \beta_{5} + 54 \beta_{6} ) q^{60} + ( 434 + 624 \beta_{2} - 190 \beta_{4} + 190 \beta_{7} ) q^{61} + ( 196 \beta_{2} + 60 \beta_{7} ) q^{62} + ( -260 \beta_{1} - 455 \beta_{3} ) q^{63} + ( -340 + 89 \beta_{4} ) q^{64} + ( -360 + 98 \beta_{4} ) q^{66} + ( -232 \beta_{1} - 75 \beta_{3} ) q^{67} + ( -71 \beta_{2} + 69 \beta_{7} ) q^{68} + ( 560 + 236 \beta_{2} + 324 \beta_{4} - 324 \beta_{7} ) q^{69} + ( -43 \beta_{1} - 42 \beta_{3} + 43 \beta_{5} + 42 \beta_{6} ) q^{70} + ( 231 \beta_{5} + 25 \beta_{6} ) q^{71} + ( -170 \beta_{5} - 380 \beta_{6} ) q^{72} + ( 260 \beta_{1} - 49 \beta_{3} - 260 \beta_{5} + 49 \beta_{6} ) q^{73} + ( -20 - 127 \beta_{2} + 107 \beta_{4} - 107 \beta_{7} ) q^{74} + ( 84 \beta_{2} - 348 \beta_{7} ) q^{75} + ( -70 \beta_{1} - 12 \beta_{3} ) q^{76} + ( -188 + 386 \beta_{4} ) q^{77} + ( -524 - 40 \beta_{4} ) q^{79} + ( 3 \beta_{1} + 18 \beta_{3} ) q^{80} + ( \beta_{2} - 120 \beta_{7} ) q^{81} + ( 88 - 16 \beta_{2} + 104 \beta_{4} - 104 \beta_{7} ) q^{82} + ( 182 \beta_{1} + 535 \beta_{3} - 182 \beta_{5} - 535 \beta_{6} ) q^{83} + ( 295 \beta_{5} + 426 \beta_{6} ) q^{84} + ( 35 \beta_{5} + 52 \beta_{6} ) q^{85} + ( 215 \beta_{1} - 286 \beta_{3} - 215 \beta_{5} + 286 \beta_{6} ) q^{86} + ( 1432 + 1126 \beta_{2} + 306 \beta_{4} - 306 \beta_{7} ) q^{87} + ( 850 \beta_{2} + 458 \beta_{7} ) q^{88} + ( 388 \beta_{1} - 83 \beta_{3} ) q^{89} + ( -160 - 70 \beta_{4} ) q^{90} + ( -464 - 140 \beta_{4} ) q^{92} + ( 44 \beta_{1} - 152 \beta_{3} ) q^{93} + ( -327 \beta_{2} + 157 \beta_{7} ) q^{94} + ( -76 - 70 \beta_{2} - 6 \beta_{4} + 6 \beta_{7} ) q^{95} + ( -337 \beta_{1} - 550 \beta_{3} + 337 \beta_{5} + 550 \beta_{6} ) q^{96} + ( -508 \beta_{5} - 359 \beta_{6} ) q^{97} + ( 142 \beta_{5} + 198 \beta_{6} ) q^{98} + ( -390 \beta_{1} - 65 \beta_{3} + 390 \beta_{5} + 65 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 10 q^{3} - 14 q^{4} - 70 q^{9} + O(q^{10}) \) \( 8 q - 10 q^{3} - 14 q^{4} - 70 q^{9} + 14 q^{10} + 172 q^{12} + 356 q^{14} + 78 q^{16} + 38 q^{17} - 284 q^{22} + 392 q^{23} + 948 q^{25} + 1340 q^{27} + 88 q^{29} + 86 q^{30} - 214 q^{35} - 500 q^{36} + 1256 q^{38} - 356 q^{40} - 394 q^{42} + 574 q^{43} - 570 q^{48} + 766 q^{49} - 1924 q^{51} - 472 q^{53} + 36 q^{55} - 2030 q^{56} + 2116 q^{61} - 664 q^{62} - 3076 q^{64} - 3272 q^{66} + 422 q^{68} + 1592 q^{69} - 294 q^{74} - 1032 q^{75} - 3048 q^{77} - 4032 q^{79} - 244 q^{81} + 144 q^{82} + 5116 q^{87} - 2484 q^{88} - 1000 q^{90} - 3152 q^{92} + 1622 q^{94} - 292 q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 9 x^{6} + 65 x^{4} - 144 x^{2} + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 9 \nu^{6} - 65 \nu^{4} + 585 \nu^{2} - 1296 \)\()/1040\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + 181 \nu \)\()/130\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{6} + 116 \)\()/65\)
\(\beta_{5}\)\(=\)\((\)\( -9 \nu^{7} + 65 \nu^{5} - 585 \nu^{3} + 1296 \nu \)\()/1040\)
\(\beta_{6}\)\(=\)\((\)\( -29 \nu^{7} + 325 \nu^{5} - 1885 \nu^{3} + 4176 \nu \)\()/2080\)
\(\beta_{7}\)\(=\)\((\)\( -29 \nu^{6} + 325 \nu^{4} - 1885 \nu^{2} + 4176 \)\()/1040\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} - \beta_{4} + 5 \beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(2 \beta_{6} - 5 \beta_{5} - 2 \beta_{3} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(9 \beta_{7} + 29 \beta_{2}\)
\(\nu^{5}\)\(=\)\(18 \beta_{6} - 29 \beta_{5}\)
\(\nu^{6}\)\(=\)\(65 \beta_{4} - 116\)
\(\nu^{7}\)\(=\)\(130 \beta_{3} - 181 \beta_{1}\)

Character values

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

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

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
−2.21837 + 1.28078i
−1.35234 + 0.780776i
1.35234 0.780776i
2.21837 1.28078i
−2.21837 1.28078i
−1.35234 0.780776i
1.35234 + 0.780776i
2.21837 + 1.28078i
−2.21837 + 1.28078i 1.84233 + 3.19101i −0.719224 + 1.24573i 0.561553i −8.17394 4.71922i −15.7418 9.08854i 24.1771i 6.71165 11.6249i 0.719224 + 1.24573i
23.2 −1.35234 + 0.780776i −4.34233 7.52113i −2.78078 + 4.81645i 3.56155i 11.7446 + 6.78078i −23.5360 13.5885i 21.1771i −24.2116 + 41.9358i 2.78078 + 4.81645i
23.3 1.35234 0.780776i −4.34233 7.52113i −2.78078 + 4.81645i 3.56155i −11.7446 6.78078i 23.5360 + 13.5885i 21.1771i −24.2116 + 41.9358i 2.78078 + 4.81645i
23.4 2.21837 1.28078i 1.84233 + 3.19101i −0.719224 + 1.24573i 0.561553i 8.17394 + 4.71922i 15.7418 + 9.08854i 24.1771i 6.71165 11.6249i 0.719224 + 1.24573i
147.1 −2.21837 1.28078i 1.84233 3.19101i −0.719224 1.24573i 0.561553i −8.17394 + 4.71922i −15.7418 + 9.08854i 24.1771i 6.71165 + 11.6249i 0.719224 1.24573i
147.2 −1.35234 0.780776i −4.34233 + 7.52113i −2.78078 4.81645i 3.56155i 11.7446 6.78078i −23.5360 + 13.5885i 21.1771i −24.2116 41.9358i 2.78078 4.81645i
147.3 1.35234 + 0.780776i −4.34233 + 7.52113i −2.78078 4.81645i 3.56155i −11.7446 + 6.78078i 23.5360 13.5885i 21.1771i −24.2116 41.9358i 2.78078 4.81645i
147.4 2.21837 + 1.28078i 1.84233 3.19101i −0.719224 1.24573i 0.561553i 8.17394 4.71922i 15.7418 9.08854i 24.1771i 6.71165 + 11.6249i 0.719224 1.24573i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 147.4
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.f 8
13.b even 2 1 inner 169.4.e.f 8
13.c even 3 1 169.4.b.f 4
13.c even 3 1 inner 169.4.e.f 8
13.d odd 4 1 169.4.c.g 4
13.d odd 4 1 169.4.c.j 4
13.e even 6 1 169.4.b.f 4
13.e even 6 1 inner 169.4.e.f 8
13.f odd 12 1 13.4.a.b 2
13.f odd 12 1 169.4.a.g 2
13.f odd 12 1 169.4.c.g 4
13.f odd 12 1 169.4.c.j 4
39.k even 12 1 117.4.a.d 2
39.k even 12 1 1521.4.a.r 2
52.l even 12 1 208.4.a.h 2
65.o even 12 1 325.4.b.e 4
65.s odd 12 1 325.4.a.f 2
65.t even 12 1 325.4.b.e 4
91.bc even 12 1 637.4.a.b 2
104.u even 12 1 832.4.a.z 2
104.x odd 12 1 832.4.a.s 2
143.o even 12 1 1573.4.a.b 2
156.v odd 12 1 1872.4.a.bb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.b 2 13.f odd 12 1
117.4.a.d 2 39.k even 12 1
169.4.a.g 2 13.f odd 12 1
169.4.b.f 4 13.c even 3 1
169.4.b.f 4 13.e even 6 1
169.4.c.g 4 13.d odd 4 1
169.4.c.g 4 13.f odd 12 1
169.4.c.j 4 13.d odd 4 1
169.4.c.j 4 13.f odd 12 1
169.4.e.f 8 1.a even 1 1 trivial
169.4.e.f 8 13.b even 2 1 inner
169.4.e.f 8 13.c even 3 1 inner
169.4.e.f 8 13.e even 6 1 inner
208.4.a.h 2 52.l even 12 1
325.4.a.f 2 65.s odd 12 1
325.4.b.e 4 65.o even 12 1
325.4.b.e 4 65.t even 12 1
637.4.a.b 2 91.bc even 12 1
832.4.a.s 2 104.x odd 12 1
832.4.a.z 2 104.u even 12 1
1521.4.a.r 2 39.k even 12 1
1573.4.a.b 2 143.o even 12 1
1872.4.a.bb 2 156.v odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 9 T_{2}^{6} + 65 T_{2}^{4} - 144 T_{2}^{2} + 256 \) acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 256 - 144 T^{2} + 65 T^{4} - 9 T^{6} + T^{8} \)
$3$ \( ( 1024 - 160 T + 57 T^{2} + 5 T^{3} + T^{4} )^{2} \)
$5$ \( ( 4 + 13 T^{2} + T^{4} )^{2} \)
$7$ \( 59553569296 - 260874484 T^{2} + 898725 T^{4} - 1069 T^{6} + T^{8} \)
$11$ \( 952857108736 - 4318461056 T^{2} + 18595632 T^{4} - 4424 T^{6} + T^{8} \)
$13$ \( T^{8} \)
$17$ \( ( 1295044 + 21622 T + 1499 T^{2} - 19 T^{3} + T^{4} )^{2} \)
$19$ \( 44859774689536 - 81926804608 T^{2} + 142924080 T^{4} - 12232 T^{6} + T^{8} \)
$23$ \( ( 80856064 - 1762432 T + 29424 T^{2} - 196 T^{3} + T^{4} )^{2} \)
$29$ \( ( 1496451856 + 1702096 T + 40620 T^{2} - 44 T^{3} + T^{4} )^{2} \)
$31$ \( ( 9388096 + 13524 T^{2} + T^{4} )^{2} \)
$37$ \( 13675570759219216 - 2578935069588 T^{2} + 369392213 T^{4} - 22053 T^{6} + T^{8} \)
$41$ \( 15556132052402176 - 3812071182336 T^{2} + 809433872 T^{4} - 30564 T^{6} + T^{8} \)
$43$ \( ( 4397811856 + 19032692 T + 148685 T^{2} - 287 T^{3} + T^{4} )^{2} \)
$47$ \( ( 222546724 + 219061 T^{2} + T^{4} )^{2} \)
$53$ \( ( -344 + 118 T + T^{2} )^{4} \)
$59$ \( 983560255459492096 - 196770353066112 T^{2} + 38373988400 T^{4} - 198408 T^{6} + T^{8} \)
$61$ \( ( 15981005056 - 133748128 T + 992948 T^{2} - 1058 T^{3} + T^{4} )^{2} \)
$67$ \( \)\(26\!\cdots\!56\)\( - 23818440850544256 T^{2} + 159630814640 T^{4} - 459816 T^{6} + T^{8} \)
$71$ \( \)\(24\!\cdots\!96\)\( - 22878030010057364 T^{2} + 164078118165 T^{4} - 462149 T^{6} + T^{8} \)
$73$ \( ( 55373619856 + 678568 T^{2} + T^{4} )^{2} \)
$79$ \( ( 247216 + 1008 T + T^{2} )^{4} \)
$83$ \( ( 668574416896 + 2198436 T^{2} + T^{4} )^{2} \)
$89$ \( \)\(67\!\cdots\!36\)\( - 400580946862665856 T^{2} + 2107663018032 T^{4} - 1538824 T^{6} + T^{8} \)
$97$ \( \)\(60\!\cdots\!76\)\( - 2038953510815070336 T^{2} + 6109089807920 T^{4} - 2624136 T^{6} + T^{8} \)
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