# Properties

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

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## 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: $$\Q(\sqrt{3}, \sqrt{7})$$ Defining polynomial: $$x^{4} - 5 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + q^{3} -\beta_{2} q^{4} - q^{5} + \beta_{3} q^{6} -2 \beta_{1} q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} + q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{2} + q^{3} -\beta_{2} q^{4} - q^{5} + \beta_{3} q^{6} -2 \beta_{1} q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} + q^{9} -\beta_{3} q^{10} -\beta_{2} q^{12} + 2 \beta_{3} q^{13} + 2 q^{14} - q^{15} + ( 1 + \beta_{2} ) q^{16} + ( -\beta_{1} - \beta_{3} ) q^{17} + \beta_{3} q^{18} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{19} + \beta_{2} q^{20} -2 \beta_{1} q^{21} + ( 3 - 2 \beta_{2} ) q^{23} + ( \beta_{1} + \beta_{3} ) q^{24} + q^{25} + ( 4 - 2 \beta_{2} ) q^{26} + q^{27} + ( 4 \beta_{1} + 2 \beta_{3} ) q^{28} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{29} -\beta_{3} q^{30} + ( -5 - 2 \beta_{2} ) q^{31} + ( -3 \beta_{1} - 4 \beta_{3} ) q^{32} + ( -1 + \beta_{2} ) q^{34} + 2 \beta_{1} q^{35} -\beta_{2} q^{36} + ( 4 + 2 \beta_{2} ) q^{37} + ( 2 - 2 \beta_{2} ) q^{38} + 2 \beta_{3} q^{39} + ( -\beta_{1} - \beta_{3} ) q^{40} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{41} + 2 q^{42} + ( 4 \beta_{1} - 2 \beta_{3} ) q^{43} - q^{45} + ( 2 \beta_{1} + 9 \beta_{3} ) q^{46} + ( 7 - 2 \beta_{2} ) q^{47} + ( 1 + \beta_{2} ) q^{48} + ( 5 + 4 \beta_{2} ) q^{49} + \beta_{3} q^{50} + ( -\beta_{1} - \beta_{3} ) q^{51} + ( 2 \beta_{1} + 6 \beta_{3} ) q^{52} + ( 5 + 4 \beta_{2} ) q^{53} + \beta_{3} q^{54} + ( -4 - 2 \beta_{2} ) q^{56} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{57} + ( 8 - 2 \beta_{2} ) q^{58} + ( 10 + 2 \beta_{2} ) q^{59} + \beta_{2} q^{60} + ( \beta_{1} - 3 \beta_{3} ) q^{61} + ( 2 \beta_{1} + \beta_{3} ) q^{62} -2 \beta_{1} q^{63} + ( -7 + 2 \beta_{2} ) q^{64} -2 \beta_{3} q^{65} + ( -8 + 2 \beta_{2} ) q^{67} + ( \beta_{1} - 2 \beta_{3} ) q^{68} + ( 3 - 2 \beta_{2} ) q^{69} -2 q^{70} + 8 q^{71} + ( \beta_{1} + \beta_{3} ) q^{72} + ( -4 \beta_{1} - 8 \beta_{3} ) q^{73} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{74} + q^{75} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{76} + ( 4 - 2 \beta_{2} ) q^{78} + ( -\beta_{1} - 3 \beta_{3} ) q^{79} + ( -1 - \beta_{2} ) q^{80} + q^{81} + ( -2 + 2 \beta_{2} ) q^{82} + ( 6 \beta_{1} - 6 \beta_{3} ) q^{83} + ( 4 \beta_{1} + 2 \beta_{3} ) q^{84} + ( \beta_{1} + \beta_{3} ) q^{85} + ( -8 + 2 \beta_{2} ) q^{86} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{87} + ( 4 + 6 \beta_{2} ) q^{89} -\beta_{3} q^{90} + 4 q^{91} + ( 10 - 5 \beta_{2} ) q^{92} + ( -5 - 2 \beta_{2} ) q^{93} + ( 2 \beta_{1} + 13 \beta_{3} ) q^{94} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{95} + ( -3 \beta_{1} - 4 \beta_{3} ) q^{96} + ( 8 - 2 \beta_{2} ) q^{97} + ( -4 \beta_{1} - 7 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{3} + 2q^{4} - 4q^{5} + 4q^{9} + O(q^{10})$$ $$4q + 4q^{3} + 2q^{4} - 4q^{5} + 4q^{9} + 2q^{12} + 8q^{14} - 4q^{15} + 2q^{16} - 2q^{20} + 16q^{23} + 4q^{25} + 20q^{26} + 4q^{27} - 16q^{31} - 6q^{34} + 2q^{36} + 12q^{37} + 12q^{38} + 8q^{42} - 4q^{45} + 32q^{47} + 2q^{48} + 12q^{49} + 12q^{53} - 12q^{56} + 36q^{58} + 36q^{59} - 2q^{60} - 32q^{64} - 36q^{67} + 16q^{69} - 8q^{70} + 32q^{71} + 4q^{75} + 20q^{78} - 2q^{80} + 4q^{81} - 12q^{82} - 36q^{86} + 4q^{89} + 16q^{91} + 50q^{92} - 16q^{93} + 36q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5 x^{2} + 1$$:

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

## 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
 0.456850 2.18890 −2.18890 −0.456850
−2.18890 1.00000 2.79129 −1.00000 −2.18890 −0.913701 −1.73205 1.00000 2.18890
1.2 −0.456850 1.00000 −1.79129 −1.00000 −0.456850 −4.37780 1.73205 1.00000 0.456850
1.3 0.456850 1.00000 −1.79129 −1.00000 0.456850 4.37780 −1.73205 1.00000 −0.456850
1.4 2.18890 1.00000 2.79129 −1.00000 2.18890 0.913701 1.73205 1.00000 −2.18890
 $$n$$: 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.t 4
3.b odd 2 1 5445.2.a.bl 4
5.b even 2 1 9075.2.a.cu 4
11.b odd 2 1 inner 1815.2.a.t 4
33.d even 2 1 5445.2.a.bl 4
55.d odd 2 1 9075.2.a.cu 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.a.t 4 1.a even 1 1 trivial
1815.2.a.t 4 11.b odd 2 1 inner
5445.2.a.bl 4 3.b odd 2 1
5445.2.a.bl 4 33.d even 2 1
9075.2.a.cu 4 5.b even 2 1
9075.2.a.cu 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1815))$$:

 $$T_{2}^{4} - 5 T_{2}^{2} + 1$$ $$T_{7}^{4} - 20 T_{7}^{2} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 5 T^{2} + T^{4}$$
$3$ $$( -1 + T )^{4}$$
$5$ $$( 1 + T )^{4}$$
$7$ $$16 - 20 T^{2} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$16 - 20 T^{2} + T^{4}$$
$17$ $$( -3 + T^{2} )^{2}$$
$19$ $$( -12 + T^{2} )^{2}$$
$23$ $$( -5 - 8 T + T^{2} )^{2}$$
$29$ $$3600 - 132 T^{2} + T^{4}$$
$31$ $$( -5 + 8 T + T^{2} )^{2}$$
$37$ $$( -12 - 6 T + T^{2} )^{2}$$
$41$ $$( -12 + T^{2} )^{2}$$
$43$ $$3600 - 132 T^{2} + T^{4}$$
$47$ $$( 43 - 16 T + T^{2} )^{2}$$
$53$ $$( -75 - 6 T + T^{2} )^{2}$$
$59$ $$( 60 - 18 T + T^{2} )^{2}$$
$61$ $$625 - 62 T^{2} + T^{4}$$
$67$ $$( 60 + 18 T + T^{2} )^{2}$$
$71$ $$( -8 + T )^{4}$$
$73$ $$6400 - 272 T^{2} + T^{4}$$
$79$ $$25 - 38 T^{2} + T^{4}$$
$83$ $$( -252 + T^{2} )^{2}$$
$89$ $$( -188 - 2 T + T^{2} )^{2}$$
$97$ $$( 60 - 18 T + T^{2} )^{2}$$
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