# Properties

 Label 1815.2.a.s Level $1815$ Weight $2$ Character orbit 1815.a Self dual yes Analytic conductor $14.493$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1815 = 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1815.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.4928479669$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.8112.1 Defining polynomial: $$x^{4} - 5 x^{2} + 3$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} - q^{3} + ( 2 - \beta_{3} ) q^{4} + q^{5} + \beta_{2} q^{6} + \beta_{1} q^{7} + ( \beta_{1} - 3 \beta_{2} ) q^{8} + q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{2} - q^{3} + ( 2 - \beta_{3} ) q^{4} + q^{5} + \beta_{2} q^{6} + \beta_{1} q^{7} + ( \beta_{1} - 3 \beta_{2} ) q^{8} + q^{9} -\beta_{2} q^{10} + ( -2 + \beta_{3} ) q^{12} + ( \beta_{1} + 2 \beta_{2} ) q^{13} + ( 1 - \beta_{3} ) q^{14} - q^{15} + ( 9 - 2 \beta_{3} ) q^{16} + \beta_{1} q^{17} -\beta_{2} q^{18} -2 \beta_{2} q^{19} + ( 2 - \beta_{3} ) q^{20} -\beta_{1} q^{21} + ( -2 + 2 \beta_{3} ) q^{23} + ( -\beta_{1} + 3 \beta_{2} ) q^{24} + q^{25} + ( -7 + \beta_{3} ) q^{26} - q^{27} + ( -\beta_{1} - 4 \beta_{2} ) q^{28} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{29} + \beta_{2} q^{30} + 4 q^{31} -9 \beta_{2} q^{32} + ( 1 - \beta_{3} ) q^{34} + \beta_{1} q^{35} + ( 2 - \beta_{3} ) q^{36} -2 \beta_{3} q^{37} + ( 8 - 2 \beta_{3} ) q^{38} + ( -\beta_{1} - 2 \beta_{2} ) q^{39} + ( \beta_{1} - 3 \beta_{2} ) q^{40} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{41} + ( -1 + \beta_{3} ) q^{42} + ( \beta_{1} + 4 \beta_{2} ) q^{43} + q^{45} + ( -2 \beta_{1} + 8 \beta_{2} ) q^{46} + ( -9 + 2 \beta_{3} ) q^{48} + ( 3 + 2 \beta_{3} ) q^{49} -\beta_{2} q^{50} -\beta_{1} q^{51} + ( -3 \beta_{1} + 6 \beta_{2} ) q^{52} + ( 4 + 2 \beta_{3} ) q^{53} + \beta_{2} q^{54} + ( 13 - \beta_{3} ) q^{56} + 2 \beta_{2} q^{57} + ( -10 + 4 \beta_{3} ) q^{58} + ( 2 - 2 \beta_{3} ) q^{59} + ( -2 + \beta_{3} ) q^{60} + 2 \beta_{1} q^{61} -4 \beta_{2} q^{62} + \beta_{1} q^{63} + ( 18 - 5 \beta_{3} ) q^{64} + ( \beta_{1} + 2 \beta_{2} ) q^{65} + ( -6 - 2 \beta_{3} ) q^{67} + ( -\beta_{1} - 4 \beta_{2} ) q^{68} + ( 2 - 2 \beta_{3} ) q^{69} + ( 1 - \beta_{3} ) q^{70} + ( -2 + 2 \beta_{3} ) q^{71} + ( \beta_{1} - 3 \beta_{2} ) q^{72} + ( -\beta_{1} + 2 \beta_{2} ) q^{73} + ( 2 \beta_{1} - 6 \beta_{2} ) q^{74} - q^{75} + ( 2 \beta_{1} - 10 \beta_{2} ) q^{76} + ( 7 - \beta_{3} ) q^{78} -2 \beta_{2} q^{79} + ( 9 - 2 \beta_{3} ) q^{80} + q^{81} + 6 q^{82} + ( -3 \beta_{1} - 2 \beta_{2} ) q^{83} + ( \beta_{1} + 4 \beta_{2} ) q^{84} + \beta_{1} q^{85} + ( -15 + 3 \beta_{3} ) q^{86} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{87} + 6 q^{89} -\beta_{2} q^{90} + ( 8 + 4 \beta_{3} ) q^{91} + ( -30 + 6 \beta_{3} ) q^{92} -4 q^{93} -2 \beta_{2} q^{95} + 9 \beta_{2} q^{96} + ( 12 + 2 \beta_{3} ) q^{97} + ( -2 \beta_{1} + 3 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{3} + 8q^{4} + 4q^{5} + 4q^{9} + O(q^{10})$$ $$4q - 4q^{3} + 8q^{4} + 4q^{5} + 4q^{9} - 8q^{12} + 4q^{14} - 4q^{15} + 36q^{16} + 8q^{20} - 8q^{23} + 4q^{25} - 28q^{26} - 4q^{27} + 16q^{31} + 4q^{34} + 8q^{36} + 32q^{38} - 4q^{42} + 4q^{45} - 36q^{48} + 12q^{49} + 16q^{53} + 52q^{56} - 40q^{58} + 8q^{59} - 8q^{60} + 72q^{64} - 24q^{67} + 8q^{69} + 4q^{70} - 8q^{71} - 4q^{75} + 28q^{78} + 36q^{80} + 4q^{81} + 24q^{82} - 60q^{86} + 24q^{89} + 32q^{91} - 120q^{92} - 16q^{93} + 48q^{97} + O(q^{100})$$

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −0.835000 2.07431 −2.07431 0.835000
−2.75782 −1.00000 5.60555 1.00000 2.75782 −1.67000 −9.94345 1.00000 −2.75782
1.2 −0.628052 −1.00000 −1.60555 1.00000 0.628052 4.14863 2.26447 1.00000 −0.628052
1.3 0.628052 −1.00000 −1.60555 1.00000 −0.628052 −4.14863 −2.26447 1.00000 0.628052
1.4 2.75782 −1.00000 5.60555 1.00000 −2.75782 1.67000 9.94345 1.00000 2.75782
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$5$$ $$-1$$
$$11$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b 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.a.s 4
3.b odd 2 1 5445.2.a.bo 4
5.b even 2 1 9075.2.a.dc 4
11.b odd 2 1 inner 1815.2.a.s 4
33.d even 2 1 5445.2.a.bo 4
55.d odd 2 1 9075.2.a.dc 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.a.s 4 1.a even 1 1 trivial
1815.2.a.s 4 11.b odd 2 1 inner
5445.2.a.bo 4 3.b odd 2 1
5445.2.a.bo 4 33.d even 2 1
9075.2.a.dc 4 5.b even 2 1
9075.2.a.dc 4 55.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1815))$$:

 $$T_{2}^{4} - 8 T_{2}^{2} + 3$$ $$T_{7}^{4} - 20 T_{7}^{2} + 48$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$3 - 8 T^{2} + T^{4}$$
$3$ $$( 1 + T )^{4}$$
$5$ $$( -1 + T )^{4}$$
$7$ $$48 - 20 T^{2} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$432 - 44 T^{2} + T^{4}$$
$17$ $$48 - 20 T^{2} + T^{4}$$
$19$ $$48 - 32 T^{2} + T^{4}$$
$23$ $$( -48 + 4 T + T^{2} )^{2}$$
$29$ $$3888 - 128 T^{2} + T^{4}$$
$31$ $$( -4 + T )^{4}$$
$37$ $$( -52 + T^{2} )^{2}$$
$41$ $$432 - 96 T^{2} + T^{4}$$
$43$ $$3888 - 132 T^{2} + T^{4}$$
$47$ $$T^{4}$$
$53$ $$( -36 - 8 T + T^{2} )^{2}$$
$59$ $$( -48 - 4 T + T^{2} )^{2}$$
$61$ $$768 - 80 T^{2} + T^{4}$$
$67$ $$( -16 + 12 T + T^{2} )^{2}$$
$71$ $$( -48 + 4 T + T^{2} )^{2}$$
$73$ $$432 - 60 T^{2} + T^{4}$$
$79$ $$48 - 32 T^{2} + T^{4}$$
$83$ $$48 - 188 T^{2} + T^{4}$$
$89$ $$( -6 + T )^{4}$$
$97$ $$( 92 - 24 T + T^{2} )^{2}$$