# Properties

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

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## 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{505})$$ Defining polynomial: $$x^{4} - x^{3} + 127 x^{2} + 126 x + 15876$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} + 4 \beta_{2} ) q^{5} + ( 9 - 17 \beta_{2} - \beta_{3} ) q^{7} +O(q^{10})$$ $$q + ( \beta_{1} + 4 \beta_{2} ) q^{5} + ( 9 - 17 \beta_{2} - \beta_{3} ) q^{7} + ( 5 - 5 \beta_{1} - 5 \beta_{3} ) q^{11} + ( -10 + \beta_{3} ) q^{13} + ( -4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{17} + ( -7 \beta_{1} - 55 \beta_{2} ) q^{19} + ( -4 \beta_{1} + 76 \beta_{2} ) q^{23} + ( -26 + 9 \beta_{1} + 17 \beta_{2} + 9 \beta_{3} ) q^{25} + ( 45 + 5 \beta_{3} ) q^{29} + ( 181 - 2 \beta_{1} - 179 \beta_{2} - 2 \beta_{3} ) q^{31} + ( 85 - 4 \beta_{1} + 90 \beta_{2} - 17 \beta_{3} ) q^{35} + ( -\beta_{1} + 27 \beta_{2} ) q^{37} + ( 76 + 18 \beta_{3} ) q^{41} + ( 188 + 27 \beta_{3} ) q^{43} + ( 36 \beta_{1} + 166 \beta_{2} ) q^{47} + ( -82 - 34 \beta_{1} + 17 \beta_{2} - 17 \beta_{3} ) q^{49} + ( 351 - 5 \beta_{1} - 346 \beta_{2} - 5 \beta_{3} ) q^{53} + ( 655 - 25 \beta_{3} ) q^{55} + ( -279 - 27 \beta_{1} + 306 \beta_{2} - 27 \beta_{3} ) q^{59} + ( 28 \beta_{1} + 566 \beta_{2} ) q^{61} + ( -14 \beta_{1} - 162 \beta_{2} ) q^{65} + ( -80 - 73 \beta_{1} + 153 \beta_{2} - 73 \beta_{3} ) q^{67} + ( -346 + 76 \beta_{3} ) q^{71} + ( 440 - 63 \beta_{1} - 377 \beta_{2} - 63 \beta_{3} ) q^{73} + ( 590 - 45 \beta_{1} - 630 \beta_{2} + 40 \beta_{3} ) q^{77} + ( -48 \beta_{1} + 425 \beta_{2} ) q^{79} + ( 175 - 67 \beta_{3} ) q^{83} + ( 504 - 16 \beta_{3} ) q^{85} + ( 22 \beta_{1} - 940 \beta_{2} ) q^{89} + ( -216 + 17 \beta_{1} + 153 \beta_{2} + 18 \beta_{3} ) q^{91} + ( 1192 - 90 \beta_{1} - 1102 \beta_{2} - 90 \beta_{3} ) q^{95} + ( 837 + 55 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 9q^{5} + O(q^{10})$$ $$4q + 9q^{5} + 5q^{11} - 38q^{13} - 4q^{17} - 117q^{19} + 148q^{23} - 43q^{25} + 190q^{29} + 360q^{31} + 482q^{35} + 53q^{37} + 340q^{41} + 806q^{43} + 368q^{47} - 362q^{49} + 697q^{53} + 2570q^{55} - 585q^{59} + 1160q^{61} - 338q^{65} - 233q^{67} - 1232q^{71} + 817q^{73} + 1135q^{77} + 802q^{79} + 566q^{83} + 1984q^{85} - 1858q^{89} - 505q^{91} + 2294q^{95} + 3458q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 127 x^{2} + 126 x + 15876$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 127 \nu^{2} - 127 \nu + 15876$$$$)/16002$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 253$$$$)/127$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 126 \beta_{2} + \beta_{1} - 127$$ $$\nu^{3}$$ $$=$$ $$127 \beta_{3} - 253$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/504\mathbb{Z}\right)^\times$$.

 $$n$$ $$73$$ $$127$$ $$253$$ $$281$$ $$\chi(n)$$ $$-\beta_{2}$$ $$1$$ $$1$$ $$1$$

## 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
 −5.36805 + 9.29774i 5.86805 − 10.1638i −5.36805 − 9.29774i 5.86805 + 10.1638i
0 0 0 −3.36805 + 5.83364i 0 −11.2361 + 14.7224i 0 0 0
289.2 0 0 0 7.86805 13.6279i 0 11.2361 + 14.7224i 0 0 0
361.1 0 0 0 −3.36805 5.83364i 0 −11.2361 14.7224i 0 0 0
361.2 0 0 0 7.86805 + 13.6279i 0 11.2361 14.7224i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 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.f 4
3.b odd 2 1 168.4.q.d 4
7.c even 3 1 inner 504.4.s.f 4
12.b even 2 1 336.4.q.g 4
21.g even 6 1 1176.4.a.u 2
21.h odd 6 1 168.4.q.d 4
21.h odd 6 1 1176.4.a.r 2
84.j odd 6 1 2352.4.a.bo 2
84.n even 6 1 336.4.q.g 4
84.n even 6 1 2352.4.a.cc 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.q.d 4 3.b odd 2 1
168.4.q.d 4 21.h odd 6 1
336.4.q.g 4 12.b even 2 1
336.4.q.g 4 84.n even 6 1
504.4.s.f 4 1.a even 1 1 trivial
504.4.s.f 4 7.c even 3 1 inner
1176.4.a.r 2 21.h odd 6 1
1176.4.a.u 2 21.g even 6 1
2352.4.a.bo 2 84.j odd 6 1
2352.4.a.cc 2 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 9 T_{5}^{3} + 187 T_{5}^{2} + 954 T_{5} + 11236$$ acting on $$S_{4}^{\mathrm{new}}(504, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$11236 + 954 T + 187 T^{2} - 9 T^{3} + T^{4}$$
$7$ $$117649 + 181 T^{2} + T^{4}$$
$11$ $$9922500 + 15750 T + 3175 T^{2} - 5 T^{3} + T^{4}$$
$13$ $$( -36 + 19 T + T^{2} )^{2}$$
$17$ $$4064256 - 8064 T + 2032 T^{2} + 4 T^{3} + T^{4}$$
$19$ $$7639696 - 323388 T + 16453 T^{2} + 117 T^{3} + T^{4}$$
$23$ $$11943936 - 511488 T + 18448 T^{2} - 148 T^{3} + T^{4}$$
$29$ $$( -900 - 95 T + T^{2} )^{2}$$
$31$ $$1017291025 - 11482200 T + 97705 T^{2} - 360 T^{3} + T^{4}$$
$37$ $$331776 - 30528 T + 2233 T^{2} - 53 T^{3} + T^{4}$$
$41$ $$( -33680 - 170 T + T^{2} )^{2}$$
$43$ $$( -51434 - 403 T + T^{2} )^{2}$$
$47$ $$16838695696 + 47753152 T + 265188 T^{2} - 368 T^{3} + T^{4}$$
$53$ $$13993943616 - 82452312 T + 367513 T^{2} - 697 T^{3} + T^{4}$$
$59$ $$41990400 - 3790800 T + 348705 T^{2} + 585 T^{3} + T^{4}$$
$61$ $$56368256400 - 275407200 T + 1108180 T^{2} - 1160 T^{3} + T^{4}$$
$67$ $$434563097796 - 153596862 T + 713503 T^{2} + 233 T^{3} + T^{4}$$
$71$ $$( -634356 + 616 T + T^{2} )^{2}$$
$73$ $$111698997796 + 273052838 T + 1001703 T^{2} - 817 T^{3} + T^{4}$$
$79$ $$16920546241 + 104323358 T + 773283 T^{2} - 802 T^{3} + T^{4}$$
$83$ $$( -546714 - 283 T + T^{2} )^{2}$$
$89$ $$643101348096 + 1489997088 T + 2650228 T^{2} + 1858 T^{3} + T^{4}$$
$97$ $$( 365454 - 1729 T + T^{2} )^{2}$$
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