# Properties

 Label 504.2.i.a Level 504 Weight 2 Character orbit 504.i Analytic conductor 4.024 Analytic rank 0 Dimension 8 CM discriminant -7 Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ = $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 504.i (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.157351936.1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2\cdot 3^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( \beta_{2} - \beta_{5} ) q^{7} + \beta_{3} q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( \beta_{2} - \beta_{5} ) q^{7} + \beta_{3} q^{8} + ( -\beta_{1} + \beta_{3} - \beta_{6} - \beta_{7} ) q^{11} + ( \beta_{3} + \beta_{7} ) q^{14} + \beta_{4} q^{16} + ( -2 - \beta_{2} + \beta_{4} - 2 \beta_{5} ) q^{22} + ( 3 \beta_{1} - \beta_{3} - \beta_{6} - 2 \beta_{7} ) q^{23} + 5 q^{25} + ( 4 + \beta_{4} ) q^{28} + ( -3 \beta_{1} + \beta_{3} - \beta_{6} - 2 \beta_{7} ) q^{29} + ( -\beta_{1} + 2 \beta_{6} + \beta_{7} ) q^{32} + ( 2 \beta_{2} + 2 \beta_{5} ) q^{37} + ( -4 \beta_{2} - 4 \beta_{5} ) q^{43} + ( -3 \beta_{1} - \beta_{3} + 2 \beta_{6} + 3 \beta_{7} ) q^{44} + ( -6 + 3 \beta_{2} - \beta_{4} - 2 \beta_{5} ) q^{46} + 7 q^{49} + 5 \beta_{1} q^{50} + ( -5 \beta_{1} - \beta_{3} + \beta_{6} - 2 \beta_{7} ) q^{53} + ( 3 \beta_{1} + 2 \beta_{6} + \beta_{7} ) q^{56} + ( -6 - 3 \beta_{2} + \beta_{4} - 2 \beta_{5} ) q^{58} + ( -\beta_{2} + 4 \beta_{5} ) q^{64} + ( -2 - 4 \beta_{4} ) q^{67} + ( \beta_{1} - 3 \beta_{3} - 3 \beta_{6} - 2 \beta_{7} ) q^{71} + ( 2 \beta_{3} - 2 \beta_{7} ) q^{74} + ( \beta_{1} - 3 \beta_{3} + 3 \beta_{6} + 2 \beta_{7} ) q^{77} -8 q^{79} + ( -4 \beta_{3} + 4 \beta_{7} ) q^{86} + ( 8 - 3 \beta_{2} - \beta_{4} + 4 \beta_{5} ) q^{88} + ( -5 \beta_{1} + 3 \beta_{3} - 2 \beta_{6} + \beta_{7} ) q^{92} + 7 \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 4q^{16} - 20q^{22} + 40q^{25} + 28q^{28} - 44q^{46} + 56q^{49} - 52q^{58} - 64q^{79} + 68q^{88} + O(q^{100})$$

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
125.1
 −1.28897 − 0.581861i −1.28897 + 0.581861i −0.581861 − 1.28897i −0.581861 + 1.28897i 0.581861 − 1.28897i 0.581861 + 1.28897i 1.28897 − 0.581861i 1.28897 + 0.581861i
−1.28897 0.581861i 0 1.32288 + 1.50000i 0 0 2.64575 −0.832353 2.70318i 0 0
125.2 −1.28897 + 0.581861i 0 1.32288 1.50000i 0 0 2.64575 −0.832353 + 2.70318i 0 0
125.3 −0.581861 1.28897i 0 −1.32288 + 1.50000i 0 0 −2.64575 2.70318 + 0.832353i 0 0
125.4 −0.581861 + 1.28897i 0 −1.32288 1.50000i 0 0 −2.64575 2.70318 0.832353i 0 0
125.5 0.581861 1.28897i 0 −1.32288 1.50000i 0 0 −2.64575 −2.70318 + 0.832353i 0 0
125.6 0.581861 + 1.28897i 0 −1.32288 + 1.50000i 0 0 −2.64575 −2.70318 0.832353i 0 0
125.7 1.28897 0.581861i 0 1.32288 1.50000i 0 0 2.64575 0.832353 2.70318i 0 0
125.8 1.28897 + 0.581861i 0 1.32288 + 1.50000i 0 0 2.64575 0.832353 + 2.70318i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 125.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
3.b odd 2 1 inner
8.b even 2 1 inner
21.c even 2 1 inner
24.h odd 2 1 inner
56.h odd 2 1 inner
168.i 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.i.a 8
3.b odd 2 1 inner 504.2.i.a 8
4.b odd 2 1 2016.2.i.a 8
7.b odd 2 1 CM 504.2.i.a 8
8.b even 2 1 inner 504.2.i.a 8
8.d odd 2 1 2016.2.i.a 8
12.b even 2 1 2016.2.i.a 8
21.c even 2 1 inner 504.2.i.a 8
24.f even 2 1 2016.2.i.a 8
24.h odd 2 1 inner 504.2.i.a 8
28.d even 2 1 2016.2.i.a 8
56.e even 2 1 2016.2.i.a 8
56.h odd 2 1 inner 504.2.i.a 8
84.h odd 2 1 2016.2.i.a 8
168.e odd 2 1 2016.2.i.a 8
168.i even 2 1 inner 504.2.i.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.i.a 8 1.a even 1 1 trivial
504.2.i.a 8 3.b odd 2 1 inner
504.2.i.a 8 7.b odd 2 1 CM
504.2.i.a 8 8.b even 2 1 inner
504.2.i.a 8 21.c even 2 1 inner
504.2.i.a 8 24.h odd 2 1 inner
504.2.i.a 8 56.h odd 2 1 inner
504.2.i.a 8 168.i even 2 1 inner
2016.2.i.a 8 4.b odd 2 1
2016.2.i.a 8 8.d odd 2 1
2016.2.i.a 8 12.b even 2 1
2016.2.i.a 8 24.f even 2 1
2016.2.i.a 8 28.d even 2 1
2016.2.i.a 8 56.e even 2 1
2016.2.i.a 8 84.h odd 2 1
2016.2.i.a 8 168.e odd 2 1

## Hecke kernels

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{4} + 16 T^{8}$$
$3$ 
$5$ $$( 1 - 5 T^{2} )^{8}$$
$7$ $$( 1 - 7 T^{2} )^{4}$$
$11$ $$( 1 - 206 T^{4} + 14641 T^{8} )^{2}$$
$13$ $$( 1 + 13 T^{2} )^{8}$$
$17$ $$( 1 + 17 T^{2} )^{8}$$
$19$ $$( 1 + 19 T^{2} )^{8}$$
$23$ $$( 1 - 734 T^{4} + 279841 T^{8} )^{2}$$
$29$ $$( 1 + 1234 T^{4} + 707281 T^{8} )^{2}$$
$31$ $$( 1 - 31 T^{2} )^{8}$$
$37$ $$( 1 - 38 T^{2} + 1369 T^{4} )^{4}$$
$41$ $$( 1 + 41 T^{2} )^{8}$$
$43$ $$( 1 + 58 T^{2} + 1849 T^{4} )^{4}$$
$47$ $$( 1 + 47 T^{2} )^{8}$$
$53$ $$( 1 - 5582 T^{4} + 7890481 T^{8} )^{2}$$
$59$ $$( 1 - 59 T^{2} )^{8}$$
$61$ $$( 1 + 61 T^{2} )^{8}$$
$67$ $$( 1 - 4 T + 67 T^{2} )^{4}( 1 + 4 T + 67 T^{2} )^{4}$$
$71$ $$( 1 + 2914 T^{4} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 - 73 T^{2} )^{8}$$
$79$ $$( 1 + 8 T + 79 T^{2} )^{8}$$
$83$ $$( 1 - 83 T^{2} )^{8}$$
$89$ $$( 1 + 89 T^{2} )^{8}$$
$97$ $$( 1 - 97 T^{2} )^{8}$$