# Properties

 Label 504.2.c.e Level 504 Weight 2 Character orbit 504.c Analytic conductor 4.024 Analytic rank 0 Dimension 8 CM no Inner twists 4

# 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: 8.0.72339481600.1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{4} -\beta_{5} q^{5} - q^{7} + \beta_{3} q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{4} -\beta_{5} q^{5} - q^{7} + \beta_{3} q^{8} + ( 1 - \beta_{2} - \beta_{7} ) q^{10} + ( 2 \beta_{1} - \beta_{5} - \beta_{6} ) q^{11} + ( 2 \beta_{2} - \beta_{4} + \beta_{7} ) q^{13} -\beta_{1} q^{14} + ( 1 + \beta_{4} ) q^{16} + ( 2 \beta_{1} + 2 \beta_{3} - \beta_{5} ) q^{17} + ( -\beta_{4} - \beta_{7} ) q^{19} + ( 2 \beta_{1} - 2 \beta_{5} ) q^{20} + ( -3 + \beta_{2} - \beta_{7} ) q^{22} + ( -2 \beta_{1} + 2 \beta_{3} - \beta_{5} - 2 \beta_{6} ) q^{23} + ( -1 - 2 \beta_{2} - \beta_{4} + \beta_{7} ) q^{25} + ( 2 \beta_{3} - 2 \beta_{6} ) q^{26} -\beta_{2} q^{28} + ( -2 \beta_{1} + 2 \beta_{5} + \beta_{6} ) q^{29} + ( 2 - 2 \beta_{2} - \beta_{4} + \beta_{7} ) q^{31} + ( -\beta_{3} + 2 \beta_{5} + 2 \beta_{6} ) q^{32} + ( 3 + \beta_{2} + 2 \beta_{4} - \beta_{7} ) q^{34} + \beta_{5} q^{35} + ( \beta_{4} + \beta_{7} ) q^{37} + ( 2 \beta_{1} + 2 \beta_{3} - 4 \beta_{5} - 2 \beta_{6} ) q^{38} + ( 2 - 2 \beta_{7} ) q^{40} + ( 2 \beta_{1} - 2 \beta_{3} + \beta_{5} + 2 \beta_{6} ) q^{41} + ( 2 \beta_{2} + 2 \beta_{7} ) q^{43} + ( -2 \beta_{1} + 2 \beta_{3} - 2 \beta_{5} ) q^{44} + ( -5 - 3 \beta_{2} + 2 \beta_{4} - \beta_{7} ) q^{46} + q^{49} + ( -\beta_{1} - 2 \beta_{3} - 2 \beta_{6} ) q^{50} + ( -6 + 2 \beta_{4} ) q^{52} + ( -2 \beta_{1} - 2 \beta_{5} + \beta_{6} ) q^{53} + ( -2 - 2 \beta_{2} - \beta_{4} + \beta_{7} ) q^{55} -\beta_{3} q^{56} + ( 2 + 2 \beta_{7} ) q^{58} + ( 4 \beta_{1} - 2 \beta_{6} ) q^{59} + ( 2 \beta_{2} - \beta_{4} + \beta_{7} ) q^{61} + ( 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{6} ) q^{62} + ( 5 + 2 \beta_{2} - \beta_{4} + 2 \beta_{7} ) q^{64} + ( 4 \beta_{1} + 2 \beta_{6} ) q^{65} + ( 2 \beta_{2} + 2 \beta_{7} ) q^{67} + ( 2 \beta_{1} + 2 \beta_{5} + 4 \beta_{6} ) q^{68} + ( -1 + \beta_{2} + \beta_{7} ) q^{70} + ( -2 \beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{71} -6 q^{73} + ( -2 \beta_{1} - 2 \beta_{3} + 4 \beta_{5} + 2 \beta_{6} ) q^{74} + ( -2 - 2 \beta_{2} + 2 \beta_{4} - 4 \beta_{7} ) q^{76} + ( -2 \beta_{1} + \beta_{5} + \beta_{6} ) q^{77} + ( 4 \beta_{1} + 2 \beta_{3} - 4 \beta_{5} ) q^{80} + ( 5 + 3 \beta_{2} - 2 \beta_{4} + \beta_{7} ) q^{82} + 2 \beta_{5} q^{83} + ( -4 \beta_{2} - \beta_{4} - 5 \beta_{7} ) q^{85} + ( -2 \beta_{1} + 4 \beta_{5} ) q^{86} + ( 4 - 4 \beta_{2} + 2 \beta_{4} - 2 \beta_{7} ) q^{88} + ( -2 \beta_{1} + 2 \beta_{3} - \beta_{5} - 2 \beta_{6} ) q^{89} + ( -2 \beta_{2} + \beta_{4} - \beta_{7} ) q^{91} + ( -6 \beta_{1} - 4 \beta_{3} + 2 \beta_{5} + 4 \beta_{6} ) q^{92} + ( -8 \beta_{1} - 4 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} ) q^{95} + ( 6 - 4 \beta_{2} - 2 \beta_{4} + 2 \beta_{7} ) q^{97} + \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 2q^{4} - 8q^{7} + O(q^{10})$$ $$8q + 2q^{4} - 8q^{7} + 8q^{10} + 10q^{16} - 20q^{22} - 16q^{25} - 2q^{28} + 8q^{31} + 32q^{34} + 20q^{40} - 40q^{46} + 8q^{49} - 44q^{52} - 24q^{55} + 12q^{58} + 38q^{64} - 8q^{70} - 48q^{73} - 8q^{76} + 40q^{82} + 32q^{88} + 32q^{97} + O(q^{100})$$

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

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

## 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
 −1.38758 − 0.273147i −1.38758 + 0.273147i −0.569745 − 1.29437i −0.569745 + 1.29437i 0.569745 − 1.29437i 0.569745 + 1.29437i 1.38758 − 0.273147i 1.38758 + 0.273147i
−1.38758 0.273147i 0 1.85078 + 0.758030i 3.66103i 0 −1.00000 −2.36106 1.55737i 0 1.00000 5.07999i
253.2 −1.38758 + 0.273147i 0 1.85078 0.758030i 3.66103i 0 −1.00000 −2.36106 + 1.55737i 0 1.00000 + 5.07999i
253.3 −0.569745 1.29437i 0 −1.35078 + 1.47492i 0.772577i 0 −1.00000 2.67869 + 0.908080i 0 1.00000 0.440172i
253.4 −0.569745 + 1.29437i 0 −1.35078 1.47492i 0.772577i 0 −1.00000 2.67869 0.908080i 0 1.00000 + 0.440172i
253.5 0.569745 1.29437i 0 −1.35078 1.47492i 0.772577i 0 −1.00000 −2.67869 + 0.908080i 0 1.00000 + 0.440172i
253.6 0.569745 + 1.29437i 0 −1.35078 + 1.47492i 0.772577i 0 −1.00000 −2.67869 0.908080i 0 1.00000 0.440172i
253.7 1.38758 0.273147i 0 1.85078 0.758030i 3.66103i 0 −1.00000 2.36106 1.55737i 0 1.00000 + 5.07999i
253.8 1.38758 + 0.273147i 0 1.85078 + 0.758030i 3.66103i 0 −1.00000 2.36106 + 1.55737i 0 1.00000 5.07999i
 $$n$$: 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
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 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.e 8
3.b odd 2 1 inner 504.2.c.e 8
4.b odd 2 1 2016.2.c.f 8
8.b even 2 1 inner 504.2.c.e 8
8.d odd 2 1 2016.2.c.f 8
12.b even 2 1 2016.2.c.f 8
24.f even 2 1 2016.2.c.f 8
24.h odd 2 1 inner 504.2.c.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.c.e 8 1.a even 1 1 trivial
504.2.c.e 8 3.b odd 2 1 inner
504.2.c.e 8 8.b even 2 1 inner
504.2.c.e 8 24.h odd 2 1 inner
2016.2.c.f 8 4.b odd 2 1
2016.2.c.f 8 8.d odd 2 1
2016.2.c.f 8 12.b even 2 1
2016.2.c.f 8 24.f even 2 1

## Hecke kernels

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

 $$T_{5}^{4} + 14 T_{5}^{2} + 8$$ $$T_{11}^{4} + 26 T_{11}^{2} + 128$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} - 2 T^{4} - 4 T^{6} + 16 T^{8}$$
$3$ 
$5$ $$( 1 - 6 T^{2} + 18 T^{4} - 150 T^{6} + 625 T^{8} )^{2}$$
$7$ $$( 1 + T )^{8}$$
$11$ $$( 1 - 18 T^{2} + 282 T^{4} - 2178 T^{6} + 14641 T^{8} )^{2}$$
$13$ $$( 1 - 8 T^{2} + 190 T^{4} - 1352 T^{6} + 28561 T^{8} )^{2}$$
$17$ $$( 1 - 6 T^{2} + 218 T^{4} - 1734 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$( 1 + 558 T^{4} + 130321 T^{8} )^{2}$$
$23$ $$( 1 + 2 T^{2} + 34 T^{4} + 1058 T^{6} + 279841 T^{8} )^{2}$$
$29$ $$( 1 - 64 T^{2} + 2542 T^{4} - 53824 T^{6} + 707281 T^{8} )^{2}$$
$31$ $$( 1 - 2 T + 22 T^{2} - 62 T^{3} + 961 T^{4} )^{4}$$
$37$ $$( 1 - 72 T^{2} + 3870 T^{4} - 98568 T^{6} + 1874161 T^{8} )^{2}$$
$41$ $$( 1 + 74 T^{2} + 3706 T^{4} + 124394 T^{6} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 - 68 T^{2} + 2230 T^{4} - 125732 T^{6} + 3418801 T^{8} )^{2}$$
$47$ $$( 1 + 47 T^{2} )^{8}$$
$53$ $$( 1 - 96 T^{2} + 7758 T^{4} - 269664 T^{6} + 7890481 T^{8} )^{2}$$
$59$ $$( 1 - 124 T^{2} + 8182 T^{4} - 431644 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 - 200 T^{2} + 17278 T^{4} - 744200 T^{6} + 13845841 T^{8} )^{2}$$
$67$ $$( 1 - 164 T^{2} + 13078 T^{4} - 736196 T^{6} + 20151121 T^{8} )^{2}$$
$71$ $$( 1 + 210 T^{2} + 20738 T^{4} + 1058610 T^{6} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 + 6 T + 73 T^{2} )^{8}$$
$79$ $$( 1 + 79 T^{2} )^{8}$$
$83$ $$( 1 - 276 T^{2} + 32166 T^{4} - 1901364 T^{6} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 + 266 T^{2} + 32506 T^{4} + 2106986 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 - 8 T + 46 T^{2} - 776 T^{3} + 9409 T^{4} )^{4}$$