# Properties

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

# Learn more about

## 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: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{3} ) q^{4} + \beta_{2} q^{5} + q^{7} + 2 \beta_{2} q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{3} ) q^{4} + \beta_{2} q^{5} + q^{7} + 2 \beta_{2} q^{8} + ( -1 + \beta_{3} ) q^{10} + \beta_{2} q^{11} + \beta_{1} q^{14} + ( -2 + 2 \beta_{3} ) q^{16} + ( -2 \beta_{1} + \beta_{2} ) q^{17} -2 \beta_{3} q^{19} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{20} + ( -1 + \beta_{3} ) q^{22} + ( 2 \beta_{1} - \beta_{2} ) q^{23} + 3 q^{25} + ( 1 + \beta_{3} ) q^{28} + 4 \beta_{2} q^{29} + 2 q^{31} + ( -4 \beta_{1} + 4 \beta_{2} ) q^{32} + ( -3 - \beta_{3} ) q^{34} + \beta_{2} q^{35} -2 \beta_{3} q^{37} + ( 2 \beta_{1} - 4 \beta_{2} ) q^{38} -4 q^{40} + ( -6 \beta_{1} + 3 \beta_{2} ) q^{41} -6 \beta_{3} q^{43} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{44} + ( 3 + \beta_{3} ) q^{46} + ( 8 \beta_{1} - 4 \beta_{2} ) q^{47} + q^{49} + 3 \beta_{1} q^{50} + 4 \beta_{2} q^{53} -2 q^{55} + 2 \beta_{2} q^{56} + ( -4 + 4 \beta_{3} ) q^{58} -8 \beta_{2} q^{59} -8 \beta_{3} q^{61} + 2 \beta_{1} q^{62} -8 q^{64} + 2 \beta_{3} q^{67} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{68} + ( -1 + \beta_{3} ) q^{70} + ( 6 \beta_{1} - 3 \beta_{2} ) q^{71} + 2 q^{73} + ( 2 \beta_{1} - 4 \beta_{2} ) q^{74} + ( 6 - 2 \beta_{3} ) q^{76} + \beta_{2} q^{77} + 8 q^{79} -4 \beta_{1} q^{80} + ( -9 - 3 \beta_{3} ) q^{82} -2 \beta_{2} q^{83} -2 \beta_{3} q^{85} + ( 6 \beta_{1} - 12 \beta_{2} ) q^{86} -4 q^{88} + ( -10 \beta_{1} + 5 \beta_{2} ) q^{89} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{92} + ( 12 + 4 \beta_{3} ) q^{94} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{95} -10 q^{97} + \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{4} + 4q^{7} + O(q^{10})$$ $$4q + 4q^{4} + 4q^{7} - 4q^{10} - 8q^{16} - 4q^{22} + 12q^{25} + 4q^{28} + 8q^{31} - 12q^{34} - 16q^{40} + 12q^{46} + 4q^{49} - 8q^{55} - 16q^{58} - 32q^{64} - 4q^{70} + 8q^{73} + 24q^{76} + 32q^{79} - 36q^{82} - 16q^{88} + 48q^{94} - 40q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{3}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} - 1$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 1$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{2}$$

## 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
 −1.22474 − 0.707107i −1.22474 + 0.707107i 1.22474 − 0.707107i 1.22474 + 0.707107i
−1.22474 0.707107i 0 1.00000 + 1.73205i 1.41421i 0 1.00000 2.82843i 0 −1.00000 + 1.73205i
253.2 −1.22474 + 0.707107i 0 1.00000 1.73205i 1.41421i 0 1.00000 2.82843i 0 −1.00000 1.73205i
253.3 1.22474 0.707107i 0 1.00000 1.73205i 1.41421i 0 1.00000 2.82843i 0 −1.00000 1.73205i
253.4 1.22474 + 0.707107i 0 1.00000 + 1.73205i 1.41421i 0 1.00000 2.82843i 0 −1.00000 + 1.73205i
 $$n$$: 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
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.c 4
3.b odd 2 1 inner 504.2.c.c 4
4.b odd 2 1 2016.2.c.b 4
8.b even 2 1 inner 504.2.c.c 4
8.d odd 2 1 2016.2.c.b 4
12.b even 2 1 2016.2.c.b 4
24.f even 2 1 2016.2.c.b 4
24.h odd 2 1 inner 504.2.c.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.c.c 4 1.a even 1 1 trivial
504.2.c.c 4 3.b odd 2 1 inner
504.2.c.c 4 8.b even 2 1 inner
504.2.c.c 4 24.h odd 2 1 inner
2016.2.c.b 4 4.b odd 2 1
2016.2.c.b 4 8.d odd 2 1
2016.2.c.b 4 12.b even 2 1
2016.2.c.b 4 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 $$S_{2}^{\mathrm{new}}(504, [\chi])$$:

 $$T_{5}^{2} + 2$$ $$T_{11}^{2} + 2$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T^{2} + 4 T^{4}$$
$3$ 
$5$ $$( 1 - 8 T^{2} + 25 T^{4} )^{2}$$
$7$ $$( 1 - T )^{4}$$
$11$ $$( 1 - 20 T^{2} + 121 T^{4} )^{2}$$
$13$ $$( 1 - 13 T^{2} )^{4}$$
$17$ $$( 1 + 28 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 - 8 T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2}$$
$23$ $$( 1 + 40 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 26 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 2 T + 31 T^{2} )^{4}$$
$37$ $$( 1 - 62 T^{2} + 1369 T^{4} )^{2}$$
$41$ $$( 1 + 28 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 - 8 T + 43 T^{2} )^{2}( 1 + 8 T + 43 T^{2} )^{2}$$
$47$ $$( 1 - 2 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$( 1 - 74 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$( 1 + 10 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 + 70 T^{2} + 3721 T^{4} )^{2}$$
$67$ $$( 1 - 16 T + 67 T^{2} )^{2}( 1 + 16 T + 67 T^{2} )^{2}$$
$71$ $$( 1 + 88 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 - 2 T + 73 T^{2} )^{4}$$
$79$ $$( 1 - 8 T + 79 T^{2} )^{4}$$
$83$ $$( 1 - 18 T + 83 T^{2} )^{2}( 1 + 18 T + 83 T^{2} )^{2}$$
$89$ $$( 1 + 28 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 + 10 T + 97 T^{2} )^{4}$$
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