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: \( x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - 3 \beta_{7} + \beta_{2}) q^{3} + (\beta_{7} - \beta_{4} - 3 \beta_{2} - 4) q^{4} + (\beta_{6} + \beta_{5} - \beta_{3} - \beta_1) q^{5} + ( - 6 \beta_{6} - \beta_{5}) q^{6} + (5 \beta_{6} + 11 \beta_{5}) q^{7} + (2 \beta_{6} + 11 \beta_{5} - 2 \beta_{3} - 11 \beta_1) q^{8} + (15 \beta_{7} - 15 \beta_{4} - 10 \beta_{2} - 25) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - 3 \beta_{7} + \beta_{2}) q^{3} + (\beta_{7} - \beta_{4} - 3 \beta_{2} - 4) q^{4} + (\beta_{6} + \beta_{5} - \beta_{3} - \beta_1) q^{5} + ( - 6 \beta_{6} - \beta_{5}) q^{6} + (5 \beta_{6} + 11 \beta_{5}) q^{7} + (2 \beta_{6} + 11 \beta_{5} - 2 \beta_{3} - 11 \beta_1) q^{8} + (15 \beta_{7} - 15 \beta_{4} - 10 \beta_{2} - 25) q^{9} + (\beta_{7} - 3 \beta_{2}) q^{10} + (17 \beta_{3} - 12 \beta_1) q^{11} + (13 \beta_{4} + 28) q^{12} + ( - \beta_{4} + 44) q^{14} + (10 \beta_{3} + 7 \beta_1) q^{15} + ( - 15 \beta_{7} - 27 \beta_{2}) q^{16} + ( - 17 \beta_{7} + 17 \beta_{4} + \beta_{2} + 18) q^{17} + (30 \beta_{6} + 10 \beta_{5} - 30 \beta_{3} - 10 \beta_1) q^{18} + ( - 13 \beta_{6} + 32 \beta_{5}) q^{19} + ( - 6 \beta_{6} - 5 \beta_{5}) q^{20} + ( - 86 \beta_{6} - 41 \beta_{5} + 86 \beta_{3} + 41 \beta_1) q^{21} + ( - 46 \beta_{7} + 46 \beta_{4} - 94 \beta_{2} - 48) q^{22} + (12 \beta_{7} - 92 \beta_{2}) q^{23} + (74 \beta_{3} + 23 \beta_1) q^{24} + ( - 3 \beta_{4} + 117) q^{25} + (9 \beta_{4} + 172) q^{27} + ( - 42 \beta_{3} - 43 \beta_1) q^{28} + (96 \beta_{7} + 26 \beta_{2}) q^{29} + ( - 13 \beta_{7} + 13 \beta_{4} + 15 \beta_{2} + 28) q^{30} + (13 \beta_{6} - 34 \beta_{5} - 13 \beta_{3} + 34 \beta_1) q^{31} + ( - 46 \beta_{6} - 61 \beta_{5}) q^{32} + (4 \beta_{6} - 90 \beta_{5}) q^{33} + ( - 34 \beta_{6} - \beta_{5} + 34 \beta_{3} + \beta_1) q^{34} + (21 \beta_{7} - 21 \beta_{4} - 43 \beta_{2} - 64) q^{35} + ( - 70 \beta_{7} + 90 \beta_{2}) q^{36} + (51 \beta_{3} - 5 \beta_1) q^{37} + ( - 58 \beta_{4} + 128) q^{38} + ( - 15 \beta_{4} - 52) q^{40} + (63 \beta_{3} + 22 \beta_1) q^{41} + ( - 131 \beta_{7} + 33 \beta_{2}) q^{42} + (143 \beta_{7} - 143 \beta_{4} + 215 \beta_{2} + 72) q^{43} + (44 \beta_{6} - 2 \beta_{5} - 44 \beta_{3} + 2 \beta_1) q^{44} + ( - 55 \beta_{6} - 40 \beta_{5}) q^{45} + (24 \beta_{6} + 92 \beta_{5}) q^{46} + (139 \beta_{6} + 121 \beta_{5} - 139 \beta_{3} - 121 \beta_1) q^{47} + ( - 21 \beta_{7} + 21 \beta_{4} - 153 \beta_{2} - 132) q^{48} + (99 \beta_{7} - 142 \beta_{2}) q^{49} + ( - 6 \beta_{3} + 120 \beta_1) q^{50} + ( - 71 \beta_{4} - 276) q^{51} + ( - 30 \beta_{4} - 74) q^{53} + (18 \beta_{3} + 163 \beta_1) q^{54} + (22 \beta_{7} + 2 \beta_{2}) q^{55} + (33 \beta_{7} - 33 \beta_{4} - 491 \beta_{2} - 524) q^{56} + ( - 140 \beta_{6} + 46 \beta_{5} + 140 \beta_{3} - 46 \beta_1) q^{57} + (192 \beta_{6} - 26 \beta_{5}) q^{58} + (123 \beta_{6} + 124 \beta_{5}) q^{59} + (54 \beta_{6} + 41 \beta_{5} - 54 \beta_{3} - 41 \beta_1) q^{60} + (190 \beta_{7} - 190 \beta_{4} + 624 \beta_{2} + 434) q^{61} + (60 \beta_{7} + 196 \beta_{2}) q^{62} + ( - 455 \beta_{3} - 260 \beta_1) q^{63} + (89 \beta_{4} - 340) q^{64} + (98 \beta_{4} - 360) q^{66} + ( - 75 \beta_{3} - 232 \beta_1) q^{67} + (69 \beta_{7} - 71 \beta_{2}) q^{68} + ( - 324 \beta_{7} + 324 \beta_{4} + 236 \beta_{2} + 560) q^{69} + (42 \beta_{6} + 43 \beta_{5} - 42 \beta_{3} - 43 \beta_1) q^{70} + (25 \beta_{6} + 231 \beta_{5}) q^{71} + ( - 380 \beta_{6} - 170 \beta_{5}) q^{72} + (49 \beta_{6} - 260 \beta_{5} - 49 \beta_{3} + 260 \beta_1) q^{73} + ( - 107 \beta_{7} + 107 \beta_{4} - 127 \beta_{2} - 20) q^{74} + ( - 348 \beta_{7} + 84 \beta_{2}) q^{75} + ( - 12 \beta_{3} - 70 \beta_1) q^{76} + (386 \beta_{4} - 188) q^{77} + ( - 40 \beta_{4} - 524) q^{79} + (18 \beta_{3} + 3 \beta_1) q^{80} + ( - 120 \beta_{7} + \beta_{2}) q^{81} + ( - 104 \beta_{7} + 104 \beta_{4} - 16 \beta_{2} + 88) q^{82} + ( - 535 \beta_{6} - 182 \beta_{5} + 535 \beta_{3} + 182 \beta_1) q^{83} + (426 \beta_{6} + 295 \beta_{5}) q^{84} + (52 \beta_{6} + 35 \beta_{5}) q^{85} + (286 \beta_{6} - 215 \beta_{5} - 286 \beta_{3} + 215 \beta_1) q^{86} + ( - 306 \beta_{7} + 306 \beta_{4} + 1126 \beta_{2} + 1432) q^{87} + (458 \beta_{7} + 850 \beta_{2}) q^{88} + ( - 83 \beta_{3} + 388 \beta_1) q^{89} + ( - 70 \beta_{4} - 160) q^{90} + ( - 140 \beta_{4} - 464) q^{92} + ( - 152 \beta_{3} + 44 \beta_1) q^{93} + (157 \beta_{7} - 327 \beta_{2}) q^{94} + (6 \beta_{7} - 6 \beta_{4} - 70 \beta_{2} - 76) q^{95} + (550 \beta_{6} + 337 \beta_{5} - 550 \beta_{3} - 337 \beta_1) q^{96} + ( - 359 \beta_{6} - 508 \beta_{5}) q^{97} + (198 \beta_{6} + 142 \beta_{5}) q^{98} + (65 \beta_{6} + 390 \beta_{5} - 65 \beta_{3} - 390 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 10 q^{3} - 14 q^{4} - 70 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 10 q^{3} - 14 q^{4} - 70 q^{9} + 14 q^{10} + 172 q^{12} + 356 q^{14} + 78 q^{16} + 38 q^{17} - 284 q^{22} + 392 q^{23} + 948 q^{25} + 1340 q^{27} + 88 q^{29} + 86 q^{30} - 214 q^{35} - 500 q^{36} + 1256 q^{38} - 356 q^{40} - 394 q^{42} + 574 q^{43} - 570 q^{48} + 766 q^{49} - 1924 q^{51} - 472 q^{53} + 36 q^{55} - 2030 q^{56} + 2116 q^{61} - 664 q^{62} - 3076 q^{64} - 3272 q^{66} + 422 q^{68} + 1592 q^{69} - 294 q^{74} - 1032 q^{75} - 3048 q^{77} - 4032 q^{79} - 244 q^{81} + 144 q^{82} + 5116 q^{87} - 2484 q^{88} - 1000 q^{90} - 3152 q^{92} + 1622 q^{94} - 292 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 9\nu^{6} - 65\nu^{4} + 585\nu^{2} - 1296 ) / 1040 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 181\nu ) / 130 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 116 ) / 65 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -9\nu^{7} + 65\nu^{5} - 585\nu^{3} + 1296\nu ) / 1040 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -29\nu^{7} + 325\nu^{5} - 1885\nu^{3} + 4176\nu ) / 2080 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -29\nu^{6} + 325\nu^{4} - 1885\nu^{2} + 4176 ) / 1040 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{4} + 5\beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{6} - 5\beta_{5} - 2\beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{7} + 29\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 18\beta_{6} - 29\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 65\beta_{4} - 116 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 130\beta_{3} - 181\beta_1 \) Copy content Toggle raw display

Character values

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

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

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 \( T_{2}^{8} - 9T_{2}^{6} + 65T_{2}^{4} - 144T_{2}^{2} + 256 \) acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 9 T^{6} + 65 T^{4} - 144 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( (T^{4} + 5 T^{3} + 57 T^{2} - 160 T + 1024)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 13 T^{2} + 4)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 1069 T^{6} + \cdots + 59553569296 \) Copy content Toggle raw display
$11$ \( T^{8} - 4424 T^{6} + \cdots + 952857108736 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} - 19 T^{3} + 1499 T^{2} + \cdots + 1295044)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} - 12232 T^{6} + \cdots + 44859774689536 \) Copy content Toggle raw display
$23$ \( (T^{4} - 196 T^{3} + 29424 T^{2} + \cdots + 80856064)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 44 T^{3} + 40620 T^{2} + \cdots + 1496451856)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 13524 T^{2} + 9388096)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 22053 T^{6} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{8} - 30564 T^{6} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( (T^{4} - 287 T^{3} + 148685 T^{2} + \cdots + 4397811856)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 219061 T^{2} + \cdots + 222546724)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 118 T - 344)^{4} \) Copy content Toggle raw display
$59$ \( T^{8} - 198408 T^{6} + \cdots + 98\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( (T^{4} - 1058 T^{3} + \cdots + 15981005056)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 459816 T^{6} + \cdots + 26\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{8} - 462149 T^{6} + \cdots + 24\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( (T^{4} + 678568 T^{2} + \cdots + 55373619856)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 1008 T + 247216)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 2198436 T^{2} + \cdots + 668574416896)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 1538824 T^{6} + \cdots + 67\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{8} - 2624136 T^{6} + \cdots + 60\!\cdots\!76 \) Copy content Toggle raw display
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