Properties

Label 168.4.q.f
Level $168$
Weight $4$
Character orbit 168.q
Analytic conductor $9.912$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 168.q (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.91232088096\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \(x^{8} + 173 x^{6} + 9457 x^{4} + 168048 x^{2} + 746496\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 7 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + 3 \beta_{1} q^{3} + ( -1 - \beta_{1} + \beta_{6} ) q^{5} + ( 2 + \beta_{5} ) q^{7} + ( -9 - 9 \beta_{1} ) q^{9} +O(q^{10})\) \( q + 3 \beta_{1} q^{3} + ( -1 - \beta_{1} + \beta_{6} ) q^{5} + ( 2 + \beta_{5} ) q^{7} + ( -9 - 9 \beta_{1} ) q^{9} + ( 1 + 4 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{11} + ( 6 - 4 \beta_{2} - \beta_{4} - \beta_{5} ) q^{13} + ( 3 - 3 \beta_{2} ) q^{15} + ( -1 + 26 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + \beta_{4} - 3 \beta_{6} - 2 \beta_{7} ) q^{17} + ( 5 + 5 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 3 \beta_{5} + 4 \beta_{6} ) q^{19} + ( 9 \beta_{1} + 3 \beta_{3} - 3 \beta_{5} ) q^{21} + ( -26 - 26 \beta_{1} - \beta_{2} - \beta_{4} + 3 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{23} + ( 1 + 33 \beta_{1} + 2 \beta_{2} + 9 \beta_{3} - \beta_{4} - 3 \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{25} + 27 q^{27} + ( -16 + 4 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - 7 \beta_{5} - \beta_{6} - \beta_{7} ) q^{29} + ( 5 + 62 \beta_{1} - 7 \beta_{2} + 5 \beta_{3} - 5 \beta_{4} - 5 \beta_{5} + 7 \beta_{6} + 5 \beta_{7} ) q^{31} + ( -12 - 12 \beta_{1} - 3 \beta_{3} + 6 \beta_{6} - 3 \beta_{7} ) q^{33} + ( 14 - 10 \beta_{1} - 14 \beta_{2} - \beta_{3} + \beta_{5} + 21 \beta_{6} ) q^{35} + ( -143 - 143 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + 6 \beta_{5} - 12 \beta_{6} + \beta_{7} ) q^{37} + ( -3 + 15 \beta_{1} + 12 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 12 \beta_{6} - 3 \beta_{7} ) q^{39} + ( 110 + 8 \beta_{1} - 14 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} - 8 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{41} + ( -71 - 12 \beta_{1} + 15 \beta_{2} - 9 \beta_{3} + \beta_{4} + 16 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{43} + ( 9 \beta_{1} + 9 \beta_{2} - 9 \beta_{6} ) q^{45} + ( 24 + 24 \beta_{1} + \beta_{2} + 4 \beta_{3} + \beta_{4} - 3 \beta_{5} - 26 \beta_{6} + 5 \beta_{7} ) q^{47} + ( 139 + 237 \beta_{1} + 2 \beta_{3} - 7 \beta_{4} + \beta_{5} ) q^{49} + ( -66 - 66 \beta_{1} + 3 \beta_{2} + 3 \beta_{4} - 9 \beta_{5} + 6 \beta_{6} + 3 \beta_{7} ) q^{51} + ( -4 - 7 \beta_{1} - 9 \beta_{2} - 12 \beta_{3} + 4 \beta_{4} + 6 \beta_{5} + 11 \beta_{6} - 2 \beta_{7} ) q^{53} + ( 360 - 20 \beta_{1} - 18 \beta_{2} - 15 \beta_{3} - 2 \beta_{4} + 23 \beta_{5} + 5 \beta_{6} + 5 \beta_{7} ) q^{55} + ( -15 + 12 \beta_{1} - 9 \beta_{2} + 9 \beta_{3} + 3 \beta_{4} - 12 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} ) q^{57} + ( -8 + 54 \beta_{1} + 6 \beta_{2} - 20 \beta_{3} + 8 \beta_{4} + 11 \beta_{5} - 3 \beta_{6} - 5 \beta_{7} ) q^{59} + ( 68 + 68 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} + 6 \beta_{5} + 38 \beta_{6} + 4 \beta_{7} ) q^{61} + ( -18 - 27 \beta_{1} - 9 \beta_{3} ) q^{63} + ( -540 - 540 \beta_{1} + 7 \beta_{2} - 10 \beta_{3} + 7 \beta_{4} - 21 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} ) q^{65} + ( 2 + 21 \beta_{1} + 12 \beta_{2} - 26 \beta_{3} - 2 \beta_{4} + 5 \beta_{5} - 5 \beta_{6} + 9 \beta_{7} ) q^{67} + ( 75 + 12 \beta_{1} + 9 \beta_{2} + 9 \beta_{3} + 6 \beta_{4} - 9 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} ) q^{69} + ( -209 + 4 \beta_{1} + 31 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - 7 \beta_{5} - \beta_{6} - \beta_{7} ) q^{71} + ( 1 - 49 \beta_{1} + 2 \beta_{2} + 9 \beta_{3} - \beta_{4} - 3 \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{73} + ( -75 - 75 \beta_{1} + 6 \beta_{2} - 9 \beta_{3} + 6 \beta_{4} - 18 \beta_{5} + 6 \beta_{6} - 3 \beta_{7} ) q^{75} + ( 97 + 397 \beta_{1} + 63 \beta_{2} - 10 \beta_{3} + 21 \beta_{4} - \beta_{5} - 21 \beta_{6} - 7 \beta_{7} ) q^{77} + ( -444 - 444 \beta_{1} - 4 \beta_{2} + 9 \beta_{3} - 4 \beta_{4} + 12 \beta_{5} + 29 \beta_{6} + 5 \beta_{7} ) q^{79} + 81 \beta_{1} q^{81} + ( 864 + 4 \beta_{1} + 27 \beta_{2} + 3 \beta_{3} + \beta_{4} - 4 \beta_{5} - \beta_{6} - \beta_{7} ) q^{83} + ( 356 - 16 \beta_{1} - 94 \beta_{2} - 12 \beta_{3} - 2 \beta_{4} + 18 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} ) q^{85} + ( -9 - 69 \beta_{1} - 15 \beta_{2} - 21 \beta_{3} + 9 \beta_{4} + 12 \beta_{5} + 18 \beta_{6} - 6 \beta_{7} ) q^{87} + ( 94 + 94 \beta_{1} + 8 \beta_{2} - 16 \beta_{3} + 8 \beta_{4} - 24 \beta_{5} - 42 \beta_{6} - 8 \beta_{7} ) q^{89} + ( -229 + 237 \beta_{1} - 21 \beta_{2} - 5 \beta_{3} + 7 \beta_{4} + 13 \beta_{5} - 42 \beta_{6} + 14 \beta_{7} ) q^{91} + ( -186 - 186 \beta_{1} - 15 \beta_{3} - 21 \beta_{6} - 15 \beta_{7} ) q^{93} + ( 3 + 520 \beta_{1} - 74 \beta_{2} + 31 \beta_{3} - 3 \beta_{4} - 10 \beta_{5} + 67 \beta_{6} - 4 \beta_{7} ) q^{95} + ( -159 - 16 \beta_{1} + 90 \beta_{2} - 12 \beta_{3} - 9 \beta_{4} + 11 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} ) q^{97} + ( 27 - 18 \beta_{2} + 9 \beta_{4} + 9 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 12q^{3} - 4q^{5} + 18q^{7} - 36q^{9} + O(q^{10}) \) \( 8q - 12q^{3} - 4q^{5} + 18q^{7} - 36q^{9} - 14q^{11} + 44q^{13} + 24q^{15} - 96q^{17} + 26q^{19} - 36q^{21} - 96q^{23} - 110q^{25} + 216q^{27} - 152q^{29} - 238q^{31} - 42q^{33} + 152q^{35} - 562q^{37} - 66q^{39} + 856q^{41} - 516q^{43} - 36q^{45} + 80q^{47} + 156q^{49} - 288q^{51} + 2952q^{55} - 156q^{57} - 262q^{59} + 276q^{61} - 54q^{63} - 2196q^{65} - 150q^{67} + 576q^{69} - 1696q^{71} + 218q^{73} - 330q^{75} - 764q^{77} - 1762q^{79} - 324q^{81} + 6900q^{83} + 2904q^{85} + 228q^{87} + 344q^{89} - 2806q^{91} - 714q^{93} - 2004q^{95} - 1244q^{97} + 252q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 173 x^{6} + 9457 x^{4} + 168048 x^{2} + 746496\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{7} + 691 \nu^{5} + 67439 \nu^{3} + 869616 \nu - 673920 \)\()/1347840\)
\(\beta_{2}\)\(=\)\((\)\( 19 \nu^{6} + 911 \nu^{4} - 31781 \nu^{2} - 208224 \)\()/42120\)
\(\beta_{3}\)\(=\)\((\)\( 283 \nu^{7} + 768 \nu^{6} + 29087 \nu^{5} + 143232 \nu^{4} + 233803 \nu^{3} + 8185728 \nu^{2} - 19439568 \nu + 109164672 \)\()/4043520\)
\(\beta_{4}\)\(=\)\((\)\( 135 \nu^{7} - 2384 \nu^{6} + 19035 \nu^{5} - 149776 \nu^{4} + 892215 \nu^{3} + 3526576 \nu^{2} + 14802480 \nu + 104194944 \)\()/1347840\)
\(\beta_{5}\)\(=\)\((\)\( -27 \nu^{7} - 560 \nu^{6} - 3807 \nu^{5} - 62320 \nu^{4} - 178443 \nu^{3} - 1680944 \nu^{2} - 2960496 \nu - 9334656 \)\()/269568\)
\(\beta_{6}\)\(=\)\((\)\( 43 \nu^{7} + 57 \nu^{6} + 5387 \nu^{5} + 2733 \nu^{4} + 181903 \nu^{3} - 95343 \nu^{2} + 1055052 \nu - 624672 \)\()/252720\)
\(\beta_{7}\)\(=\)\((\)\( 1087 \nu^{7} - 4272 \nu^{6} + 147443 \nu^{5} - 417648 \nu^{4} + 5991727 \nu^{3} - 7751472 \nu^{2} + 78636528 \nu - 12451968 \)\()/2021760\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(3 \beta_{7} - 11 \beta_{6} - \beta_{5} - 2 \beta_{4} + 5 \beta_{3} + 5 \beta_{2} - 12 \beta_{1} - 5\)\()/28\)
\(\nu^{2}\)\(=\)\((\)\(-3 \beta_{7} - 3 \beta_{6} + \beta_{5} + 16 \beta_{4} + 9 \beta_{3} + 51 \beta_{2} + 12 \beta_{1} - 1213\)\()/28\)
\(\nu^{3}\)\(=\)\((\)\(-41 \beta_{7} + 155 \beta_{6} + 2 \beta_{5} + 39 \beta_{4} - 115 \beta_{3} - 59 \beta_{2} - 18 \beta_{1} - 11\)\()/7\)
\(\nu^{4}\)\(=\)\((\)\(191 \beta_{7} + 191 \beta_{6} - 241 \beta_{5} - 1196 \beta_{4} - 573 \beta_{3} - 4759 \beta_{2} - 764 \beta_{1} + 77321\)\()/28\)
\(\nu^{5}\)\(=\)\((\)\(11113 \beta_{7} - 42185 \beta_{6} + 393 \beta_{5} - 11506 \beta_{4} + 34911 \beta_{3} + 15143 \beta_{2} + 65700 \beta_{1} + 32457\)\()/28\)
\(\nu^{6}\)\(=\)\((\)\(-3544 \beta_{7} - 3544 \beta_{6} + 3307 \beta_{5} + 21027 \beta_{4} + 10632 \beta_{3} + 93890 \beta_{2} + 14176 \beta_{1} - 1357362\)\()/7\)
\(\nu^{7}\)\(=\)\((\)\(-110295 \beta_{7} + 442367 \beta_{6} - 8363 \beta_{5} + 118658 \beta_{4} - 364337 \beta_{3} - 157673 \beta_{2} - 1090260 \beta_{1} - 536767\)\()/4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(85\) \(113\) \(127\)
\(\chi(n)\) \(-1 - \beta_{1}\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
8.67551i
8.34231i
2.57353i
4.63878i
8.67551i
8.34231i
2.57353i
4.63878i
0 −1.50000 + 2.59808i 0 −9.47901 16.4181i 0 12.8033 + 13.3819i 0 −4.50000 7.79423i 0
25.2 0 −1.50000 + 2.59808i 0 −0.363171 0.629031i 0 −18.1420 3.72380i 0 −4.50000 7.79423i 0
25.3 0 −1.50000 + 2.59808i 0 −0.0642956 0.111363i 0 −0.866259 18.5000i 0 −4.50000 7.79423i 0
25.4 0 −1.50000 + 2.59808i 0 7.90648 + 13.6944i 0 15.2050 + 10.5739i 0 −4.50000 7.79423i 0
121.1 0 −1.50000 2.59808i 0 −9.47901 + 16.4181i 0 12.8033 13.3819i 0 −4.50000 + 7.79423i 0
121.2 0 −1.50000 2.59808i 0 −0.363171 + 0.629031i 0 −18.1420 + 3.72380i 0 −4.50000 + 7.79423i 0
121.3 0 −1.50000 2.59808i 0 −0.0642956 + 0.111363i 0 −0.866259 + 18.5000i 0 −4.50000 + 7.79423i 0
121.4 0 −1.50000 2.59808i 0 7.90648 13.6944i 0 15.2050 10.5739i 0 −4.50000 + 7.79423i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 168.4.q.f 8
3.b odd 2 1 504.4.s.j 8
4.b odd 2 1 336.4.q.m 8
7.c even 3 1 inner 168.4.q.f 8
7.c even 3 1 1176.4.a.bd 4
7.d odd 6 1 1176.4.a.ba 4
21.h odd 6 1 504.4.s.j 8
28.f even 6 1 2352.4.a.cp 4
28.g odd 6 1 336.4.q.m 8
28.g odd 6 1 2352.4.a.cm 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.q.f 8 1.a even 1 1 trivial
168.4.q.f 8 7.c even 3 1 inner
336.4.q.m 8 4.b odd 2 1
336.4.q.m 8 28.g odd 6 1
504.4.s.j 8 3.b odd 2 1
504.4.s.j 8 21.h odd 6 1
1176.4.a.ba 4 7.d odd 6 1
1176.4.a.bd 4 7.c even 3 1
2352.4.a.cm 4 28.g odd 6 1
2352.4.a.cp 4 28.f even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{8} + \cdots\) acting on \(S_{4}^{\mathrm{new}}(168, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 9 + 3 T + T^{2} )^{4} \)
$5$ \( 784 + 7168 T + 57220 T^{2} + 76256 T^{3} + 89261 T^{4} - 676 T^{5} + 313 T^{6} + 4 T^{7} + T^{8} \)
$7$ \( 13841287201 - 726364926 T + 9882516 T^{2} + 2593080 T^{3} - 128723 T^{4} + 7560 T^{5} + 84 T^{6} - 18 T^{7} + T^{8} \)
$11$ \( 33244957032336 - 391593061104 T + 35719311436 T^{2} + 204963188 T^{3} + 24291005 T^{4} + 60302 T^{5} + 5591 T^{6} + 14 T^{7} + T^{8} \)
$13$ \( ( 13795008 + 84004 T - 7423 T^{2} - 22 T^{3} + T^{4} )^{2} \)
$17$ \( 1953905916248064 + 75853776224256 T + 2312485998592 T^{2} + 33033099264 T^{3} + 413546496 T^{4} + 2058880 T^{5} + 23520 T^{6} + 96 T^{7} + T^{8} \)
$19$ \( 30110417391616 - 1419300084992 T + 145012515664 T^{2} + 3967250612 T^{3} + 190422977 T^{4} + 887414 T^{5} + 14911 T^{6} - 26 T^{7} + T^{8} \)
$23$ \( 9663676416 - 11960057856 T + 13395988480 T^{2} - 1721407488 T^{3} + 193022976 T^{4} - 1616512 T^{5} + 23520 T^{6} + 96 T^{7} + T^{8} \)
$29$ \( ( -802800 - 3696384 T - 52293 T^{2} + 76 T^{3} + T^{4} )^{2} \)
$31$ \( 27234235833160225 + 2765633698693390 T + 267850878190996 T^{2} + 1398592677692 T^{3} + 10357966421 T^{4} + 14770364 T^{5} + 135412 T^{6} + 238 T^{7} + T^{8} \)
$37$ \( 21385545645567919104 + 157703115307512960 T + 1238349439157040 T^{2} + 4641849317052 T^{3} + 24055639113 T^{4} + 77367450 T^{5} + 299539 T^{6} + 562 T^{7} + T^{8} \)
$41$ \( ( -3866949120 + 36803808 T - 41132 T^{2} - 428 T^{3} + T^{4} )^{2} \)
$43$ \( ( -2654719484 - 46672932 T - 179543 T^{2} + 258 T^{3} + T^{4} )^{2} \)
$47$ \( \)\(56\!\cdots\!24\)\( + 441086629735406592 T + 8366652681609600 T^{2} - 2395660818432 T^{3} + 90674636656 T^{4} - 10062848 T^{5} + 342648 T^{6} - 80 T^{7} + T^{8} \)
$53$ \( 25753595040000 + 15293721768000 T + 8395480482400 T^{2} + 407775320940 T^{3} + 18313600281 T^{4} + 6027320 T^{5} + 135309 T^{6} + T^{8} \)
$59$ \( 2696200518203654400 - 73352242586819520 T + 1541505278215296 T^{2} - 11493726649116 T^{3} + 66418351369 T^{4} - 161800754 T^{5} + 345195 T^{6} + 262 T^{7} + T^{8} \)
$61$ \( 97291323031404960000 - 415823608078444800 T + 6074779134836224 T^{2} + 23812464646272 T^{3} + 168331971984 T^{4} + 204566560 T^{5} + 511872 T^{6} - 276 T^{7} + T^{8} \)
$67$ \( \)\(70\!\cdots\!24\)\( + 655493426514372960 T + 16283038431222868 T^{2} - 22555081338120 T^{3} + 318750682533 T^{4} - 138028090 T^{5} + 613251 T^{6} + 150 T^{7} + T^{8} \)
$71$ \( ( 1940742864 - 46164032 T - 78520 T^{2} + 848 T^{3} + T^{4} )^{2} \)
$73$ \( 78620725698303376 + 1581530253463888 T + 40650589792684 T^{2} - 55505097884 T^{3} + 1942405933 T^{4} - 4410506 T^{5} + 79039 T^{6} - 218 T^{7} + T^{8} \)
$79$ \( \)\(86\!\cdots\!01\)\( + 11781219476000650786 T + 85234608753590332 T^{2} + 233736255591028 T^{3} + 869284996429 T^{4} + 1563690004 T^{5} + 2360908 T^{6} + 1762 T^{7} + T^{8} \)
$83$ \( ( 352673538780 - 2140010896 T + 4237149 T^{2} - 3450 T^{3} + T^{4} )^{2} \)
$89$ \( \)\(20\!\cdots\!76\)\( - 27178868651391590400 T + 299556787215937536 T^{2} - 713782053251712 T^{3} + 1571613039120 T^{4} - 791464416 T^{5} + 1267972 T^{6} - 344 T^{7} + T^{8} \)
$97$ \( ( 79407506004 - 1902403156 T - 2800699 T^{2} + 622 T^{3} + T^{4} )^{2} \)
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