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: \(x^{8} - 9 x^{6} + 65 x^{4} - 144 x^{2} + 256\)
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 9 \nu^{6} - 65 \nu^{4} + 585 \nu^{2} - 1296 \)\()/1040\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} - 181 \nu \)\()/260\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{6} + 116 \)\()/65\)
\(\beta_{5}\)\(=\)\((\)\( -29 \nu^{6} + 325 \nu^{4} - 1885 \nu^{2} + 4176 \)\()/1040\)
\(\beta_{6}\)\(=\)\((\)\( 9 \nu^{7} - 65 \nu^{5} + 585 \nu^{3} - 256 \nu \)\()/1040\)
\(\beta_{7}\)\(=\)\((\)\( 9 \nu^{7} - 65 \nu^{5} + 377 \nu^{3} - 256 \nu \)\()/832\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - \beta_{4} + 5 \beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(-4 \beta_{7} + 5 \beta_{6}\)
\(\nu^{4}\)\(=\)\(9 \beta_{5} + 29 \beta_{2}\)
\(\nu^{5}\)\(=\)\(-36 \beta_{7} + 29 \beta_{6} - 36 \beta_{3} - 29 \beta_{1}\)
\(\nu^{6}\)\(=\)\(65 \beta_{4} - 116\)
\(\nu^{7}\)\(=\)\(-260 \beta_{3} - 181 \beta_{1}\)

Character values

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

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

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 \( T_{2}^{8} - 21 T_{2}^{6} + 437 T_{2}^{4} - 84 T_{2}^{2} + 16 \) acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 - 84 T^{2} + 437 T^{4} - 21 T^{6} + T^{8} \)
$3$ \( ( 1024 - 160 T + 57 T^{2} + 5 T^{3} + T^{4} )^{2} \)
$5$ \( ( 2500 + 325 T^{2} + T^{4} )^{2} \)
$7$ \( 7311616 - 327184 T^{2} + 11937 T^{4} - 121 T^{6} + T^{8} \)
$11$ \( 610673479936 - 1607454992 T^{2} + 3449793 T^{4} - 2057 T^{6} + T^{8} \)
$13$ \( T^{8} \)
$17$ \( ( 18769 - 9590 T + 4763 T^{2} - 70 T^{3} + T^{4} )^{2} \)
$19$ \( 559724432982016 - 240204709888 T^{2} + 79424913 T^{4} - 10153 T^{6} + T^{8} \)
$23$ \( ( 394384 - 91060 T + 20397 T^{2} - 145 T^{3} + T^{4} )^{2} \)
$29$ \( ( 225330121 - 510374 T + 16167 T^{2} + 34 T^{3} + T^{4} )^{2} \)
$31$ \( ( 1413760000 + 94800 T^{2} + T^{4} )^{2} \)
$37$ \( 403490473681 - 21918521754 T^{2} + 1190028827 T^{4} - 34506 T^{6} + T^{8} \)
$41$ \( 24929818533638640241 - 739569727492362 T^{2} + 16947149963 T^{4} - 148122 T^{6} + T^{8} \)
$43$ \( ( 138485824 - 5354440 T + 195257 T^{2} - 455 T^{3} + T^{4} )^{2} \)
$47$ \( ( 6789760000 + 168400 T^{2} + T^{4} )^{2} \)
$53$ \( ( -41450 - 545 T + T^{2} )^{4} \)
$59$ \( \)\(51\!\cdots\!16\)\( - 8022545344854288 T^{2} + 101846036513 T^{4} - 352953 T^{6} + T^{8} \)
$61$ \( ( 11448786001 - 53713498 T + 359003 T^{2} + 502 T^{3} + T^{4} )^{2} \)
$67$ \( 202454052206715136 - 82431096489744 T^{2} + 33112657457 T^{4} - 183201 T^{6} + T^{8} \)
$71$ \( 3363170782998040576 - 186648328972352 T^{2} + 8524662753 T^{4} - 101777 T^{6} + T^{8} \)
$73$ \( ( 3008522500 + 232525 T^{2} + T^{4} )^{2} \)
$79$ \( ( 7600 - 240 T + T^{2} )^{4} \)
$83$ \( ( 655360000 + 118800 T^{2} + T^{4} )^{2} \)
$89$ \( \)\(45\!\cdots\!56\)\( - 11815382446935028 T^{2} + 288959278773 T^{4} - 556933 T^{6} + T^{8} \)
$97$ \( \)\(63\!\cdots\!56\)\( - 1555370046819222804 T^{2} + 3029811318677 T^{4} - 1955781 T^{6} + T^{8} \)
show more
show less