# Properties

 Label 504.2.c.f Level 504 Weight 2 Character orbit 504.c Analytic conductor 4.024 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: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 504.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.386672896.3 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: no (minimal twist has level 168) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{2} -\beta_{2} q^{4} + ( -\beta_{2} + \beta_{6} ) q^{5} + q^{7} + ( 1 - \beta_{3} - \beta_{4} - \beta_{6} ) q^{8} +O(q^{10})$$ $$q -\beta_{3} q^{2} -\beta_{2} q^{4} + ( -\beta_{2} + \beta_{6} ) q^{5} + q^{7} + ( 1 - \beta_{3} - \beta_{4} - \beta_{6} ) q^{8} + ( -1 - \beta_{1} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{10} + ( \beta_{3} + \beta_{5} + \beta_{7} ) q^{11} + ( \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{13} -\beta_{3} q^{14} + ( -1 + \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{16} + ( -2 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{17} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{7} ) q^{19} + ( -2 + 2 \beta_{1} ) q^{20} + ( 1 + \beta_{1} - \beta_{4} + \beta_{6} + \beta_{7} ) q^{22} + ( \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{23} + ( -1 + 2 \beta_{1} - \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{25} + ( 4 + 2 \beta_{2} ) q^{26} -\beta_{2} q^{28} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{7} ) q^{29} + ( 2 + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{31} + ( 3 + \beta_{3} + \beta_{4} - \beta_{6} ) q^{32} + ( 1 + \beta_{1} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{34} + ( -\beta_{2} + \beta_{6} ) q^{35} + ( -\beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{37} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{38} + ( 2 + 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{40} + ( 2 + \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{41} + ( 2 \beta_{6} + 2 \beta_{7} ) q^{43} + ( -4 + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 4 \beta_{7} ) q^{44} + ( -1 - \beta_{1} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{46} + ( -2 \beta_{1} + 4 \beta_{3} - 2 \beta_{4} ) q^{47} + q^{49} + ( 3 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} + 2 \beta_{7} ) q^{50} + ( -2 - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} ) q^{52} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{7} ) q^{53} + ( -2 - 2 \beta_{1} - \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{55} + ( 1 - \beta_{3} - \beta_{4} - \beta_{6} ) q^{56} + ( 6 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{58} + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{59} + ( \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{6} ) q^{61} + ( -2 - 2 \beta_{1} + 2 \beta_{6} ) q^{62} + ( 5 + \beta_{1} - 5 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{64} + ( 4 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{65} + ( -2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{7} ) q^{67} + ( 2 - 2 \beta_{5} + 2 \beta_{6} ) q^{68} + ( -1 - \beta_{1} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{70} + ( 4 - \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{71} + ( -2 + 2 \beta_{2} + 4 \beta_{4} + 2 \beta_{6} ) q^{73} + ( -2 \beta_{4} + 2 \beta_{7} ) q^{74} + ( -2 - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} - 4 \beta_{7} ) q^{76} + ( \beta_{3} + \beta_{5} + \beta_{7} ) q^{77} + ( -8 - 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{79} + ( 2 + 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{80} + ( -1 - \beta_{1} + \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{82} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{6} - \beta_{7} ) q^{83} + ( 2 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{85} + ( -4 + 2 \beta_{3} + 4 \beta_{7} ) q^{86} + ( 2 + 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{88} + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{89} + ( \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{91} + ( -2 + 2 \beta_{2} + 2 \beta_{5} - 2 \beta_{6} ) q^{92} + ( -8 + 4 \beta_{2} ) q^{94} + ( -4 + 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} ) q^{95} + ( 2 - 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{97} -\beta_{3} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 2q^{4} + 8q^{7} + 6q^{8} + O(q^{10})$$ $$8q + 2q^{4} + 8q^{7} + 6q^{8} - 4q^{10} - 6q^{16} - 4q^{17} - 24q^{20} - 12q^{23} - 24q^{25} + 28q^{26} + 2q^{28} + 8q^{31} + 30q^{32} - 4q^{34} - 12q^{38} + 28q^{40} + 4q^{41} - 16q^{44} + 4q^{46} + 8q^{49} + 20q^{50} - 12q^{52} - 8q^{55} + 6q^{56} + 44q^{58} - 12q^{62} + 26q^{64} + 16q^{65} + 16q^{68} - 4q^{70} + 28q^{71} - 8q^{73} - 4q^{74} - 24q^{76} - 40q^{79} + 4q^{80} + 4q^{82} - 24q^{86} + 4q^{88} - 20q^{89} - 20q^{92} - 72q^{94} - 40q^{95} + 40q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{6} - 2 x^{5} + 2 x^{4} - 4 x^{3} - 4 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{6} - \nu^{4} + 2 \nu^{3} + 2 \nu^{2} - 8$$$$)/4$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} - \nu^{4} - 2 \nu^{3} + 2 \nu^{2} - 4 \nu - 4$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} - \nu^{5} - 2 \nu^{4} + 2 \nu^{3} - 4 \nu^{2} - 4 \nu$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{6} + \nu^{4} - 2 \nu^{3} - 2 \nu^{2} + 8 \nu + 8$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + 2 \nu^{6} - \nu^{5} + 4 \nu^{4} - 2 \nu^{3} - 8 \nu^{2} - 12 \nu$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{6} + 2 \nu^{5} - \nu^{4} - 2 \nu^{2} - 4 \nu - 12$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{7} + \nu^{6} + \nu^{5} + \nu^{4} + 2 \nu^{3} - 2 \nu^{2} - 8 \nu - 8$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} + \beta_{2}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{4} - 2 \beta_{2} + \beta_{1} + 2$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{7} - \beta_{6} + \beta_{5} - 3 \beta_{3} - \beta_{2} - 2$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_{1} + 6$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-\beta_{7} + \beta_{6} + 3 \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} + 6 \beta_{1} + 10$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$8 \beta_{7} - 4 \beta_{6} - 4 \beta_{5} + 7 \beta_{4} + 4 \beta_{3} + 6 \beta_{2} + \beta_{1} - 2$$$$)/2$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
253.1
 −1.19503 + 0.756243i −1.19503 − 0.756243i −0.835949 + 1.14070i −0.835949 − 1.14070i 0.621372 + 1.27039i 0.621372 − 1.27039i 1.40961 + 0.114062i 1.40961 − 0.114062i
−1.19503 0.756243i 0 0.856193 + 1.80747i 4.10245i 0 1.00000 0.343707 2.80747i 0 3.10245 4.90255i
253.2 −1.19503 + 0.756243i 0 0.856193 1.80747i 4.10245i 0 1.00000 0.343707 + 2.80747i 0 3.10245 + 4.90255i
253.3 −0.835949 1.14070i 0 −0.602380 + 1.90713i 0.467138i 0 1.00000 2.67901 0.907128i 0 −0.532862 + 0.390503i
253.4 −0.835949 + 1.14070i 0 −0.602380 1.90713i 0.467138i 0 1.00000 2.67901 + 0.907128i 0 −0.532862 0.390503i
253.5 0.621372 1.27039i 0 −1.22779 1.57877i 3.69833i 0 1.00000 −2.76858 + 0.578773i 0 −4.69833 2.29804i
253.6 0.621372 + 1.27039i 0 −1.22779 + 1.57877i 3.69833i 0 1.00000 −2.76858 0.578773i 0 −4.69833 + 2.29804i
253.7 1.40961 0.114062i 0 1.97398 0.321565i 1.12875i 0 1.00000 2.74586 0.678435i 0 0.128747 + 1.59109i
253.8 1.40961 + 0.114062i 0 1.97398 + 0.321565i 1.12875i 0 1.00000 2.74586 + 0.678435i 0 0.128747 1.59109i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 253.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.c.f 8
3.b odd 2 1 168.2.c.b 8
4.b odd 2 1 2016.2.c.e 8
8.b even 2 1 inner 504.2.c.f 8
8.d odd 2 1 2016.2.c.e 8
12.b even 2 1 672.2.c.b 8
21.c even 2 1 1176.2.c.c 8
24.f even 2 1 672.2.c.b 8
24.h odd 2 1 168.2.c.b 8
48.i odd 4 1 5376.2.a.bm 4
48.i odd 4 1 5376.2.a.bp 4
48.k even 4 1 5376.2.a.bl 4
48.k even 4 1 5376.2.a.bq 4
84.h odd 2 1 4704.2.c.c 8
168.e odd 2 1 4704.2.c.c 8
168.i even 2 1 1176.2.c.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.c.b 8 3.b odd 2 1
168.2.c.b 8 24.h odd 2 1
504.2.c.f 8 1.a even 1 1 trivial
504.2.c.f 8 8.b even 2 1 inner
672.2.c.b 8 12.b even 2 1
672.2.c.b 8 24.f even 2 1
1176.2.c.c 8 21.c even 2 1
1176.2.c.c 8 168.i even 2 1
2016.2.c.e 8 4.b odd 2 1
2016.2.c.e 8 8.d odd 2 1
4704.2.c.c 8 84.h odd 2 1
4704.2.c.c 8 168.e odd 2 1
5376.2.a.bl 4 48.k even 4 1
5376.2.a.bm 4 48.i odd 4 1
5376.2.a.bp 4 48.i odd 4 1
5376.2.a.bq 4 48.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(504, [\chi])$$:

 $$T_{5}^{8} + 32 T_{5}^{6} + 276 T_{5}^{4} + 352 T_{5}^{2} + 64$$ $$T_{11}^{8} + 64 T_{11}^{6} + 1428 T_{11}^{4} + 12896 T_{11}^{2} + 40000$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} - 2 T^{3} + 2 T^{4} - 4 T^{5} - 4 T^{6} + 16 T^{8}$$
$3$ 
$5$ $$1 - 8 T^{2} + 16 T^{4} - 168 T^{6} + 1694 T^{8} - 4200 T^{10} + 10000 T^{12} - 125000 T^{14} + 390625 T^{16}$$
$7$ $$( 1 - T )^{8}$$
$11$ $$1 - 24 T^{2} + 592 T^{4} - 8312 T^{6} + 114206 T^{8} - 1005752 T^{10} + 8667472 T^{12} - 42517464 T^{14} + 214358881 T^{16}$$
$13$ $$1 - 48 T^{2} + 1340 T^{4} - 26704 T^{6} + 396198 T^{8} - 4512976 T^{10} + 38271740 T^{12} - 231686832 T^{14} + 815730721 T^{16}$$
$17$ $$( 1 + 2 T + 38 T^{2} + 70 T^{3} + 706 T^{4} + 1190 T^{5} + 10982 T^{6} + 9826 T^{7} + 83521 T^{8} )^{2}$$
$19$ $$1 - 64 T^{2} + 2012 T^{4} - 42432 T^{6} + 793446 T^{8} - 15317952 T^{10} + 262205852 T^{12} - 3010936384 T^{14} + 16983563041 T^{16}$$
$23$ $$( 1 + 6 T + 74 T^{2} + 334 T^{3} + 2282 T^{4} + 7682 T^{5} + 39146 T^{6} + 73002 T^{7} + 279841 T^{8} )^{2}$$
$29$ $$1 - 16 T^{2} + 2876 T^{4} - 34928 T^{6} + 3439654 T^{8} - 29374448 T^{10} + 2034140156 T^{12} - 9517173136 T^{14} + 500246412961 T^{16}$$
$31$ $$( 1 - 4 T + 80 T^{2} - 244 T^{3} + 3294 T^{4} - 7564 T^{5} + 76880 T^{6} - 119164 T^{7} + 923521 T^{8} )^{2}$$
$37$ $$1 - 192 T^{2} + 18716 T^{4} - 1175872 T^{6} + 51538086 T^{8} - 1609768768 T^{10} + 35076797276 T^{12} - 492619470528 T^{14} + 3512479453921 T^{16}$$
$41$ $$( 1 - 2 T + 134 T^{2} - 214 T^{3} + 7618 T^{4} - 8774 T^{5} + 225254 T^{6} - 137842 T^{7} + 2825761 T^{8} )^{2}$$
$43$ $$1 - 168 T^{2} + 15676 T^{4} - 1026520 T^{6} + 50753126 T^{8} - 1898035480 T^{10} + 53593124476 T^{12} - 1061988992232 T^{14} + 11688200277601 T^{16}$$
$47$ $$( 1 + 44 T^{2} + 128 T^{3} + 3302 T^{4} + 6016 T^{5} + 97196 T^{6} + 4879681 T^{8} )^{2}$$
$53$ $$1 - 336 T^{2} + 52604 T^{4} - 5027376 T^{6} + 321653350 T^{8} - 14121899184 T^{10} + 415070862524 T^{12} - 7447225339344 T^{14} + 62259690411361 T^{16}$$
$59$ $$( 1 - 102 T^{2} + 3481 T^{4} )^{4}$$
$61$ $$1 - 272 T^{2} + 35516 T^{4} - 3030896 T^{6} + 201654822 T^{8} - 11277964016 T^{10} + 491748888956 T^{12} - 14013541826192 T^{14} + 191707312997281 T^{16}$$
$67$ $$1 - 344 T^{2} + 59996 T^{4} - 6783272 T^{6} + 536949606 T^{8} - 30450108008 T^{10} + 1208986655516 T^{12} - 31117683466136 T^{14} + 406067677556641 T^{16}$$
$71$ $$( 1 - 14 T + 194 T^{2} - 1686 T^{3} + 14330 T^{4} - 119706 T^{5} + 977954 T^{6} - 5010754 T^{7} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 + 4 T + 92 T^{2} - 580 T^{3} + 166 T^{4} - 42340 T^{5} + 490268 T^{6} + 1556068 T^{7} + 28398241 T^{8} )^{2}$$
$79$ $$( 1 + 20 T + 340 T^{2} + 3972 T^{3} + 38678 T^{4} + 313788 T^{5} + 2121940 T^{6} + 9860780 T^{7} + 38950081 T^{8} )^{2}$$
$83$ $$1 - 248 T^{2} + 44156 T^{4} - 5204488 T^{6} + 499369126 T^{8} - 35853717832 T^{10} + 2095569622076 T^{12} - 81081212595512 T^{14} + 2252292232139041 T^{16}$$
$89$ $$( 1 + 10 T + 198 T^{2} + 494 T^{3} + 13122 T^{4} + 43966 T^{5} + 1568358 T^{6} + 7049690 T^{7} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 - 20 T + 300 T^{2} - 2572 T^{3} + 25574 T^{4} - 249484 T^{5} + 2822700 T^{6} - 18253460 T^{7} + 88529281 T^{8} )^{2}$$