Properties

Label 169.4.e.g
Level $169$
Weight $4$
Character orbit 169.e
Analytic conductor $9.971$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

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: \(\Z[a_1, \ldots, a_{7}]\)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{7} + \beta_{6} - 2 \beta_{3} - \beta_1) q^{2} + ( - 3 \beta_{5} + 3 \beta_{4} - 4 \beta_{2} - 1) q^{3} + ( - 5 \beta_{5} - 5 \beta_{2}) q^{4} + (10 \beta_{7} + 5 \beta_{6}) q^{5} + (14 \beta_{3} + 10 \beta_1) q^{6} + (7 \beta_{3} + \beta_1) q^{7} + ( - 4 \beta_{7} + 7 \beta_{6}) q^{8} + (15 \beta_{5} + 25 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{7} + \beta_{6} - 2 \beta_{3} - \beta_1) q^{2} + ( - 3 \beta_{5} + 3 \beta_{4} - 4 \beta_{2} - 1) q^{3} + ( - 5 \beta_{5} - 5 \beta_{2}) q^{4} + (10 \beta_{7} + 5 \beta_{6}) q^{5} + (14 \beta_{3} + 10 \beta_1) q^{6} + (7 \beta_{3} + \beta_1) q^{7} + ( - 4 \beta_{7} + 7 \beta_{6}) q^{8} + (15 \beta_{5} + 25 \beta_{2}) q^{9} + ( - 5 \beta_{5} + 5 \beta_{4} - 5 \beta_{2}) q^{10} + ( - \beta_{7} + 15 \beta_{6} - \beta_{3} - 15 \beta_1) q^{11} + (20 \beta_{4} - 60) q^{12} + (10 \beta_{4} - 18) q^{14} + (50 \beta_{7} + 10 \beta_{6} + 50 \beta_{3} - 10 \beta_1) q^{15} + (15 \beta_{5} - 15 \beta_{4} - 21 \beta_{2} - 36) q^{16} + (16 \beta_{5} - 27 \beta_{2}) q^{17} + (80 \beta_{7} - 55 \beta_{6}) q^{18} + (73 \beta_{3} - 5 \beta_1) q^{19} + (100 \beta_{3} - 25 \beta_1) q^{20} + (19 \beta_{7} - 25 \beta_{6}) q^{21} + ( - 46 \beta_{5} - 108 \beta_{2}) q^{22} + ( - 33 \beta_{5} + 33 \beta_{4} + 56 \beta_{2} + 89) q^{23} + (88 \beta_{7} - 40 \beta_{6} + 88 \beta_{3} + 40 \beta_1) q^{24} + ( - 75 \beta_{4} - 75) q^{25} + ( - 9 \beta_{4} + 163) q^{27} + (20 \beta_{7} - 40 \beta_{6} + 20 \beta_{3} + 40 \beta_1) q^{28} + ( - 60 \beta_{5} + 60 \beta_{4} - 47 \beta_{2} + 13) q^{29} + (20 \beta_{5} + 80 \beta_{2}) q^{30} + (120 \beta_{7} + 100 \beta_{6}) q^{31} + ( - 20 \beta_{3} - 65 \beta_1) q^{32} + (181 \beta_{3} + 63 \beta_1) q^{33} + ( - 22 \beta_{7} - 5 \beta_{6}) q^{34} + (30 \beta_{5} - 20 \beta_{2}) q^{35} + (125 \beta_{5} - 125 \beta_{4} + 425 \beta_{2} + 300) q^{36} + (73 \beta_{7} - 44 \beta_{6} + 73 \beta_{3} + 44 \beta_1) q^{37} + (58 \beta_{4} - 126) q^{38} + ( - 15 \beta_{4} - 100) q^{40} + (259 \beta_{7} - 20 \beta_{6} + 259 \beta_{3} + 20 \beta_1) q^{41} + (94 \beta_{5} - 94 \beta_{4} + 232 \beta_{2} + 138) q^{42} + ( - 97 \beta_{5} - 276 \beta_{2}) q^{43} + ( - 300 \beta_{7} + 80 \beta_{6}) q^{44} + ( - 200 \beta_{3} + 25 \beta_1) q^{45} + ( - 46 \beta_{3} + 10 \beta_1) q^{46} + (100 \beta_{7} + 140 \beta_{6}) q^{47} + (48 \beta_{5} - 96 \beta_{2}) q^{48} + (15 \beta_{5} - 15 \beta_{4} - 275 \beta_{2} - 290) q^{49} + ( - 150 \beta_{7} + 150 \beta_{6} - 150 \beta_{3} - 150 \beta_1) q^{50} + (65 \beta_{4} + 149) q^{51} + ( - 165 \beta_{4} + 190) q^{53} + ( - 362 \beta_{7} + 190 \beta_{6} - 362 \beta_{3} - 190 \beta_1) q^{54} + (70 \beta_{5} - 70 \beta_{4} - 220 \beta_{2} - 290) q^{55} + (60 \beta_{5} + 116 \beta_{2}) q^{56} + (13 \beta_{7} - 199 \beta_{6}) q^{57} + (214 \beta_{3} + 167 \beta_1) q^{58} + ( - 377 \beta_{3} - 55 \beta_1) q^{59} + ( - 200 \beta_{7} - 200 \beta_{6}) q^{60} + ( - 200 \beta_{5} + 151 \beta_{2}) q^{61} + ( - 180 \beta_{5} + 180 \beta_{4} - 340 \beta_{2} - 160) q^{62} + ( - 130 \beta_{7} + 130 \beta_{6} - 130 \beta_{3} - 130 \beta_1) q^{63} + ( - 95 \beta_{4} + 588) q^{64} + (370 \beta_{4} - 614) q^{66} + ( - 283 \beta_{7} - 91 \beta_{6} - 283 \beta_{3} + 91 \beta_1) q^{67} + ( - 135 \beta_{5} + 135 \beta_{4} + 185 \beta_{2} + 320) q^{68} + ( - 135 \beta_{5} + 172 \beta_{2}) q^{69} + (20 \beta_{7} - 40 \beta_{6}) q^{70} + ( - 11 \beta_{3} - 105 \beta_1) q^{71} + ( - 460 \beta_{3} - 235 \beta_1) q^{72} + (250 \beta_{7} - 85 \beta_{6}) q^{73} + (205 \beta_{5} + 527 \beta_{2}) q^{74} + ( - 75 \beta_{5} + 75 \beta_{4} - 900 \beta_{2} - 825) q^{75} + ( - 100 \beta_{7} - 340 \beta_{6} - 100 \beta_{3} + 340 \beta_1) q^{76} + (121 \beta_{4} - 67) q^{77} + (40 \beta_{4} + 140) q^{79} + ( - 660 \beta_{7} - 255 \beta_{6} - 660 \beta_{3} + 255 \beta_1) q^{80} + ( - 120 \beta_{5} + 120 \beta_{4} - 121 \beta_{2} - 1) q^{81} + (319 \beta_{5} + 917 \beta_{2}) q^{82} + ( - 180 \beta_{7} - 100 \beta_{6}) q^{83} + ( - 500 \beta_{3} - 220 \beta_1) q^{84} + ( - 750 \beta_{3} + 295 \beta_1) q^{85} + ( - 746 \beta_{7} + 470 \beta_{6}) q^{86} + (201 \beta_{5} + 908 \beta_{2}) q^{87} + ( - 172 \beta_{5} + 172 \beta_{4} - 596 \beta_{2} - 424) q^{88} + ( - 523 \beta_{7} - 125 \beta_{6} - 523 \beta_{3} + 125 \beta_1) q^{89} + ( - 125 \beta_{4} + 300) q^{90} + ( - 280 \beta_{4} - 660) q^{92} + (1080 \beta_{7} - 40 \beta_{6} + 1080 \beta_{3} + 40 \beta_1) q^{93} + ( - 320 \beta_{5} + 320 \beta_{4} - 680 \beta_{2} - 360) q^{94} + (390 \beta_{5} - 440 \beta_{2}) q^{95} + ( - 800 \beta_{7} + 320 \beta_{6}) q^{96} + ( - 27 \beta_{3} + 469 \beta_1) q^{97} + (520 \beta_{3} + 245 \beta_1) q^{98} + (910 \beta_{7} - 390 \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 10 q^{3} + 10 q^{4} - 70 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 10 q^{3} + 10 q^{4} - 70 q^{9} - 10 q^{10} - 560 q^{12} - 184 q^{14} - 114 q^{16} + 140 q^{17} + 340 q^{22} + 290 q^{23} - 300 q^{25} + 1340 q^{27} - 68 q^{29} - 280 q^{30} + 140 q^{35} + 1450 q^{36} - 1240 q^{38} - 740 q^{40} + 740 q^{42} + 910 q^{43} + 480 q^{48} - 1130 q^{49} + 932 q^{51} + 2180 q^{53} - 1020 q^{55} - 344 q^{56} - 1004 q^{61} - 1000 q^{62} + 5084 q^{64} - 6392 q^{66} + 1010 q^{68} - 958 q^{69} - 1698 q^{74} - 3450 q^{75} - 1020 q^{77} + 960 q^{79} - 244 q^{81} - 3030 q^{82} - 3230 q^{87} - 2040 q^{88} + 2900 q^{90} - 4160 q^{92} - 2080 q^{94} + 2540 q^{95}+O(q^{100}) \) 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
\(\beta_{3}\)\(=\) \( ( -\nu^{7} - 181\nu ) / 260 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 116 ) / 65 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -29\nu^{6} + 325\nu^{4} - 1885\nu^{2} + 4176 ) / 1040 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9\nu^{7} - 65\nu^{5} + 585\nu^{3} - 256\nu ) / 1040 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9\nu^{7} - 65\nu^{5} + 377\nu^{3} - 256\nu ) / 832 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{4} + 5\beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{7} + 5\beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{5} + 29\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -36\beta_{7} + 29\beta_{6} - 36\beta_{3} - 29\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 65\beta_{4} - 116 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -260\beta_{3} - 181\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{2}\)

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
−3.95042 + 2.28078i −4.34233 7.52113i 6.40388 11.0918i 2.80776i 34.3081 + 19.8078i 8.28055 + 4.78078i 21.9309i −24.2116 + 41.9358i −6.40388 11.0918i
23.2 −0.379706 + 0.219224i 1.84233 + 3.19101i −3.90388 + 6.76172i 17.8078i −1.39909 0.807764i 4.70983 + 2.71922i 6.93087i 6.71165 11.6249i 3.90388 + 6.76172i
23.3 0.379706 0.219224i 1.84233 + 3.19101i −3.90388 + 6.76172i 17.8078i 1.39909 + 0.807764i −4.70983 2.71922i 6.93087i 6.71165 11.6249i 3.90388 + 6.76172i
23.4 3.95042 2.28078i −4.34233 7.52113i 6.40388 11.0918i 2.80776i −34.3081 19.8078i −8.28055 4.78078i 21.9309i −24.2116 + 41.9358i −6.40388 11.0918i
147.1 −3.95042 2.28078i −4.34233 + 7.52113i 6.40388 + 11.0918i 2.80776i 34.3081 19.8078i 8.28055 4.78078i 21.9309i −24.2116 41.9358i −6.40388 + 11.0918i
147.2 −0.379706 0.219224i 1.84233 3.19101i −3.90388 6.76172i 17.8078i −1.39909 + 0.807764i 4.70983 2.71922i 6.93087i 6.71165 + 11.6249i 3.90388 6.76172i
147.3 0.379706 + 0.219224i 1.84233 3.19101i −3.90388 6.76172i 17.8078i 1.39909 0.807764i −4.70983 + 2.71922i 6.93087i 6.71165 + 11.6249i 3.90388 6.76172i
147.4 3.95042 + 2.28078i −4.34233 + 7.52113i 6.40388 + 11.0918i 2.80776i −34.3081 + 19.8078i −8.28055 + 4.78078i 21.9309i −24.2116 41.9358i −6.40388 + 11.0918i
\(n\): 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.g 8
13.b even 2 1 inner 169.4.e.g 8
13.c even 3 1 169.4.b.e 4
13.c even 3 1 inner 169.4.e.g 8
13.d odd 4 1 13.4.c.b 4
13.d odd 4 1 169.4.c.f 4
13.e even 6 1 169.4.b.e 4
13.e even 6 1 inner 169.4.e.g 8
13.f odd 12 1 13.4.c.b 4
13.f odd 12 1 169.4.a.f 2
13.f odd 12 1 169.4.a.j 2
13.f odd 12 1 169.4.c.f 4
39.f even 4 1 117.4.g.d 4
39.k even 12 1 117.4.g.d 4
39.k even 12 1 1521.4.a.l 2
39.k even 12 1 1521.4.a.t 2
52.f even 4 1 208.4.i.e 4
52.l even 12 1 208.4.i.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.c.b 4 13.d odd 4 1
13.4.c.b 4 13.f odd 12 1
117.4.g.d 4 39.f even 4 1
117.4.g.d 4 39.k even 12 1
169.4.a.f 2 13.f odd 12 1
169.4.a.j 2 13.f odd 12 1
169.4.b.e 4 13.c even 3 1
169.4.b.e 4 13.e even 6 1
169.4.c.f 4 13.d odd 4 1
169.4.c.f 4 13.f odd 12 1
169.4.e.g 8 1.a even 1 1 trivial
169.4.e.g 8 13.b even 2 1 inner
169.4.e.g 8 13.c even 3 1 inner
169.4.e.g 8 13.e even 6 1 inner
208.4.i.e 4 52.f even 4 1
208.4.i.e 4 52.l even 12 1
1521.4.a.l 2 39.k even 12 1
1521.4.a.t 2 39.k even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 21T_{2}^{6} + 437T_{2}^{4} - 84T_{2}^{2} + 16 \) acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 21 T^{6} + 437 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( (T^{4} + 5 T^{3} + 57 T^{2} - 160 T + 1024)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 325 T^{2} + 2500)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 121 T^{6} + 11937 T^{4} + \cdots + 7311616 \) Copy content Toggle raw display
$11$ \( T^{8} - 2057 T^{6} + \cdots + 610673479936 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} - 70 T^{3} + 4763 T^{2} + \cdots + 18769)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 559724432982016 \) Copy content Toggle raw display
$23$ \( (T^{4} - 145 T^{3} + 20397 T^{2} + \cdots + 394384)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 34 T^{3} + 16167 T^{2} + \cdots + 225330121)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 94800 T^{2} + \cdots + 1413760000)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 34506 T^{6} + \cdots + 403490473681 \) Copy content Toggle raw display
$41$ \( T^{8} - 148122 T^{6} + \cdots + 24\!\cdots\!41 \) Copy content Toggle raw display
$43$ \( (T^{4} - 455 T^{3} + 195257 T^{2} + \cdots + 138485824)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 168400 T^{2} + \cdots + 6789760000)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 545 T - 41450)^{4} \) Copy content Toggle raw display
$59$ \( T^{8} - 352953 T^{6} + \cdots + 51\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{4} + 502 T^{3} + \cdots + 11448786001)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 183201 T^{6} + \cdots + 20\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{8} - 101777 T^{6} + \cdots + 33\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( (T^{4} + 232525 T^{2} + \cdots + 3008522500)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 240 T + 7600)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 118800 T^{2} + \cdots + 655360000)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 556933 T^{6} + \cdots + 45\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{8} - 1955781 T^{6} + \cdots + 63\!\cdots\!56 \) Copy content Toggle raw display
show more
show less