# Properties

 Label 1815.2.c.c Level $1815$ Weight $2$ Character orbit 1815.c Analytic conductor $14.493$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1815 = 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1815.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.4928479669$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} -\beta_{1} q^{3} -3 q^{4} + ( -1 - 2 \beta_{1} ) q^{5} -\beta_{3} q^{6} + \beta_{2} q^{8} - q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{2} -\beta_{1} q^{3} -3 q^{4} + ( -1 - 2 \beta_{1} ) q^{5} -\beta_{3} q^{6} + \beta_{2} q^{8} - q^{9} + ( \beta_{2} - 2 \beta_{3} ) q^{10} + 3 \beta_{1} q^{12} -2 \beta_{2} q^{13} + ( -2 + \beta_{1} ) q^{15} - q^{16} + 2 \beta_{2} q^{17} + \beta_{2} q^{18} + ( 3 + 6 \beta_{1} ) q^{20} + 4 \beta_{1} q^{23} + \beta_{3} q^{24} + ( -3 + 4 \beta_{1} ) q^{25} -10 q^{26} + \beta_{1} q^{27} -4 \beta_{3} q^{29} + ( 2 \beta_{2} + \beta_{3} ) q^{30} + 3 \beta_{2} q^{32} + 10 q^{34} + 3 q^{36} -8 \beta_{1} q^{37} -2 \beta_{3} q^{39} + ( -\beta_{2} + 2 \beta_{3} ) q^{40} -4 \beta_{3} q^{41} + 4 \beta_{2} q^{43} + ( 1 + 2 \beta_{1} ) q^{45} + 4 \beta_{3} q^{46} -12 \beta_{1} q^{47} + \beta_{1} q^{48} + 7 q^{49} + ( 3 \beta_{2} + 4 \beta_{3} ) q^{50} + 2 \beta_{3} q^{51} + 6 \beta_{2} q^{52} + 4 \beta_{1} q^{53} + \beta_{3} q^{54} + 20 \beta_{1} q^{58} + ( 6 - 3 \beta_{1} ) q^{60} + 13 q^{64} + ( 2 \beta_{2} - 4 \beta_{3} ) q^{65} -12 \beta_{1} q^{67} -6 \beta_{2} q^{68} + 4 q^{69} + 12 q^{71} -\beta_{2} q^{72} + 6 \beta_{2} q^{73} -8 \beta_{3} q^{74} + ( 4 + 3 \beta_{1} ) q^{75} + 10 \beta_{1} q^{78} + ( 1 + 2 \beta_{1} ) q^{80} + q^{81} + 20 \beta_{1} q^{82} + ( -2 \beta_{2} + 4 \beta_{3} ) q^{85} + 20 q^{86} + 4 \beta_{2} q^{87} -6 q^{89} + ( -\beta_{2} + 2 \beta_{3} ) q^{90} -12 \beta_{1} q^{92} -12 \beta_{3} q^{94} + 3 \beta_{3} q^{96} -8 \beta_{1} q^{97} -7 \beta_{2} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 12q^{4} - 4q^{5} - 4q^{9} + O(q^{10})$$ $$4q - 12q^{4} - 4q^{5} - 4q^{9} - 8q^{15} - 4q^{16} + 12q^{20} - 12q^{25} - 40q^{26} + 40q^{34} + 12q^{36} + 4q^{45} + 28q^{49} + 24q^{60} + 52q^{64} + 16q^{69} + 48q^{71} + 16q^{75} + 4q^{80} + 4q^{81} + 80q^{86} - 24q^{89} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1815\mathbb{Z}\right)^\times$$.

 $$n$$ $$727$$ $$1211$$ $$1696$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
364.1
 0.618034i 1.61803i − 1.61803i − 0.618034i
2.23607i 1.00000i −3.00000 −1.00000 2.00000i −2.23607 0 2.23607i −1.00000 −4.47214 + 2.23607i
364.2 2.23607i 1.00000i −3.00000 −1.00000 + 2.00000i 2.23607 0 2.23607i −1.00000 4.47214 + 2.23607i
364.3 2.23607i 1.00000i −3.00000 −1.00000 2.00000i 2.23607 0 2.23607i −1.00000 4.47214 2.23607i
364.4 2.23607i 1.00000i −3.00000 −1.00000 + 2.00000i −2.23607 0 2.23607i −1.00000 −4.47214 2.23607i
 $$n$$: 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
5.b even 2 1 inner
11.b odd 2 1 inner
55.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1815.2.c.c 4
5.b even 2 1 inner 1815.2.c.c 4
5.c odd 4 1 9075.2.a.bj 2
5.c odd 4 1 9075.2.a.bq 2
11.b odd 2 1 inner 1815.2.c.c 4
55.d odd 2 1 inner 1815.2.c.c 4
55.e even 4 1 9075.2.a.bj 2
55.e even 4 1 9075.2.a.bq 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.c.c 4 1.a even 1 1 trivial
1815.2.c.c 4 5.b even 2 1 inner
1815.2.c.c 4 11.b odd 2 1 inner
1815.2.c.c 4 55.d odd 2 1 inner
9075.2.a.bj 2 5.c odd 4 1
9075.2.a.bj 2 55.e even 4 1
9075.2.a.bq 2 5.c odd 4 1
9075.2.a.bq 2 55.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1815, [\chi])$$:

 $$T_{2}^{2} + 5$$ $$T_{19}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 5 + T^{2} )^{2}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$( 5 + 2 T + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$( 20 + T^{2} )^{2}$$
$17$ $$( 20 + T^{2} )^{2}$$
$19$ $$T^{4}$$
$23$ $$( 16 + T^{2} )^{2}$$
$29$ $$( -80 + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$( 64 + T^{2} )^{2}$$
$41$ $$( -80 + T^{2} )^{2}$$
$43$ $$( 80 + T^{2} )^{2}$$
$47$ $$( 144 + T^{2} )^{2}$$
$53$ $$( 16 + T^{2} )^{2}$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$( 144 + T^{2} )^{2}$$
$71$ $$( -12 + T )^{4}$$
$73$ $$( 180 + T^{2} )^{2}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$( 6 + T )^{4}$$
$97$ $$( 64 + T^{2} )^{2}$$