# Properties

 Label 504.4.s.j Level $504$ Weight $4$ Character orbit 504.s Analytic conductor $29.737$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 504.s (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$29.7369626429$$ 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: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{6}\cdot 7$$ Twist minimal: no (minimal twist has level 168) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{1} - \beta_{6} ) q^{5} + ( 2 + \beta_{5} ) q^{7} +O(q^{10})$$ $$q + ( 1 + \beta_{1} - \beta_{6} ) q^{5} + ( 2 + \beta_{5} ) q^{7} + ( -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} + ( 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} + ( 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} + ( 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} + ( -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} + ( -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} + ( -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} + ( 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} + ( 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} + ( 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} + ( 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} + ( -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} + ( -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} + ( -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} + ( -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} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{5} + 18q^{7} + O(q^{10})$$ $$8q + 4q^{5} + 18q^{7} + 14q^{11} + 44q^{13} + 96q^{17} + 26q^{19} + 96q^{23} - 110q^{25} + 152q^{29} - 238q^{31} - 152q^{35} - 562q^{37} - 856q^{41} - 516q^{43} - 80q^{47} + 156q^{49} + 2952q^{55} + 262q^{59} + 276q^{61} + 2196q^{65} - 150q^{67} + 1696q^{71} + 218q^{73} + 764q^{77} - 1762q^{79} - 6900q^{83} + 2904q^{85} - 344q^{89} - 2806q^{91} + 2004q^{95} - 1244q^{97} + O(q^{100})$$

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 $$\left(\mathbb{Z}/504\mathbb{Z}\right)^\times$$.

 $$n$$ $$73$$ $$127$$ $$253$$ $$281$$ $$\chi(n)$$ $$-1 - \beta_{1}$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 4.63878i − 2.57353i 8.34231i − 8.67551i − 4.63878i 2.57353i − 8.34231i 8.67551i
0 0 0 −7.90648 + 13.6944i 0 15.2050 10.5739i 0 0 0
289.2 0 0 0 0.0642956 0.111363i 0 −0.866259 + 18.5000i 0 0 0
289.3 0 0 0 0.363171 0.629031i 0 −18.1420 + 3.72380i 0 0 0
289.4 0 0 0 9.47901 16.4181i 0 12.8033 13.3819i 0 0 0
361.1 0 0 0 −7.90648 13.6944i 0 15.2050 + 10.5739i 0 0 0
361.2 0 0 0 0.0642956 + 0.111363i 0 −0.866259 18.5000i 0 0 0
361.3 0 0 0 0.363171 + 0.629031i 0 −18.1420 3.72380i 0 0 0
361.4 0 0 0 9.47901 + 16.4181i 0 12.8033 + 13.3819i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 361.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 504.4.s.j 8
3.b odd 2 1 168.4.q.f 8
7.c even 3 1 inner 504.4.s.j 8
12.b even 2 1 336.4.q.m 8
21.g even 6 1 1176.4.a.ba 4
21.h odd 6 1 168.4.q.f 8
21.h odd 6 1 1176.4.a.bd 4
84.j odd 6 1 2352.4.a.cp 4
84.n even 6 1 336.4.q.m 8
84.n even 6 1 2352.4.a.cm 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.q.f 8 3.b odd 2 1
168.4.q.f 8 21.h odd 6 1
336.4.q.m 8 12.b even 2 1
336.4.q.m 8 84.n even 6 1
504.4.s.j 8 1.a even 1 1 trivial
504.4.s.j 8 7.c even 3 1 inner
1176.4.a.ba 4 21.g even 6 1
1176.4.a.bd 4 21.h odd 6 1
2352.4.a.cm 4 84.n even 6 1
2352.4.a.cp 4 84.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} - \cdots$$ acting on $$S_{4}^{\mathrm{new}}(504, [\chi])$$.

## Hecke characteristic polynomials

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