# Properties

 Label 1815.2.a.l Level $1815$ Weight $2$ Character orbit 1815.a Self dual yes Analytic conductor $14.493$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

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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: $$3$$ Coefficient field: 3.3.469.1 Defining polynomial: $$x^{3} - x^{2} - 5 x + 4$$ 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + q^{3} + ( 2 + \beta_{2} ) q^{4} + q^{5} -\beta_{1} q^{6} + ( -1 - \beta_{1} - \beta_{2} ) q^{7} + ( -\beta_{1} - \beta_{2} ) q^{8} + q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + q^{3} + ( 2 + \beta_{2} ) q^{4} + q^{5} -\beta_{1} q^{6} + ( -1 - \beta_{1} - \beta_{2} ) q^{7} + ( -\beta_{1} - \beta_{2} ) q^{8} + q^{9} -\beta_{1} q^{10} + ( 2 + \beta_{2} ) q^{12} + ( 2 + 2 \beta_{2} ) q^{13} + ( 4 + 2 \beta_{1} + 2 \beta_{2} ) q^{14} + q^{15} + \beta_{1} q^{16} + ( 2 - \beta_{1} + \beta_{2} ) q^{17} -\beta_{1} q^{18} + ( 1 + \beta_{1} - \beta_{2} ) q^{19} + ( 2 + \beta_{2} ) q^{20} + ( -1 - \beta_{1} - \beta_{2} ) q^{21} + ( 2 - \beta_{1} - \beta_{2} ) q^{23} + ( -\beta_{1} - \beta_{2} ) q^{24} + q^{25} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{26} + q^{27} + ( -6 - 4 \beta_{1} - 2 \beta_{2} ) q^{28} + ( -4 + 2 \beta_{1} ) q^{29} -\beta_{1} q^{30} + ( -1 - 2 \beta_{2} ) q^{31} + ( -4 + 2 \beta_{1} + \beta_{2} ) q^{32} + ( 4 - 3 \beta_{1} ) q^{34} + ( -1 - \beta_{1} - \beta_{2} ) q^{35} + ( 2 + \beta_{2} ) q^{36} + ( 1 - 3 \beta_{1} - 3 \beta_{2} ) q^{37} -4 q^{38} + ( 2 + 2 \beta_{2} ) q^{39} + ( -\beta_{1} - \beta_{2} ) q^{40} + ( 2 + 4 \beta_{1} ) q^{41} + ( 4 + 2 \beta_{1} + 2 \beta_{2} ) q^{42} -2 \beta_{1} q^{43} + q^{45} + ( 4 - \beta_{1} + 2 \beta_{2} ) q^{46} + ( 4 + 3 \beta_{1} - \beta_{2} ) q^{47} + \beta_{1} q^{48} + ( 2 + 5 \beta_{1} + 3 \beta_{2} ) q^{49} -\beta_{1} q^{50} + ( 2 - \beta_{1} + \beta_{2} ) q^{51} + ( 12 + 2 \beta_{1} + 2 \beta_{2} ) q^{52} + ( 2 - 5 \beta_{1} + \beta_{2} ) q^{53} -\beta_{1} q^{54} + ( 8 + 4 \beta_{1} + 2 \beta_{2} ) q^{56} + ( 1 + \beta_{1} - \beta_{2} ) q^{57} + ( -8 + 4 \beta_{1} - 2 \beta_{2} ) q^{58} + ( 8 - 2 \beta_{1} + 4 \beta_{2} ) q^{59} + ( 2 + \beta_{2} ) q^{60} + ( -9 + 2 \beta_{1} + 2 \beta_{2} ) q^{61} + ( 3 \beta_{1} + 2 \beta_{2} ) q^{62} + ( -1 - \beta_{1} - \beta_{2} ) q^{63} + ( -8 + \beta_{1} - 3 \beta_{2} ) q^{64} + ( 2 + 2 \beta_{2} ) q^{65} + ( 1 - 3 \beta_{1} + \beta_{2} ) q^{67} + ( 8 - 2 \beta_{1} + \beta_{2} ) q^{68} + ( 2 - \beta_{1} - \beta_{2} ) q^{69} + ( 4 + 2 \beta_{1} + 2 \beta_{2} ) q^{70} + ( 2 + 6 \beta_{1} - 2 \beta_{2} ) q^{71} + ( -\beta_{1} - \beta_{2} ) q^{72} + ( 3 - \beta_{1} + \beta_{2} ) q^{73} + ( 12 + 2 \beta_{1} + 6 \beta_{2} ) q^{74} + q^{75} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{76} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{78} + ( -5 + 4 \beta_{1} - 2 \beta_{2} ) q^{79} + \beta_{1} q^{80} + q^{81} + ( -16 - 2 \beta_{1} - 4 \beta_{2} ) q^{82} -4 q^{83} + ( -6 - 4 \beta_{1} - 2 \beta_{2} ) q^{84} + ( 2 - \beta_{1} + \beta_{2} ) q^{85} + ( 8 + 2 \beta_{2} ) q^{86} + ( -4 + 2 \beta_{1} ) q^{87} + ( 10 + 2 \beta_{1} ) q^{89} -\beta_{1} q^{90} + ( -10 - 6 \beta_{1} - 2 \beta_{2} ) q^{91} + ( -4 \beta_{1} + \beta_{2} ) q^{92} + ( -1 - 2 \beta_{2} ) q^{93} + ( -12 - 3 \beta_{1} - 2 \beta_{2} ) q^{94} + ( 1 + \beta_{1} - \beta_{2} ) q^{95} + ( -4 + 2 \beta_{1} + \beta_{2} ) q^{96} + ( -1 + 3 \beta_{1} - \beta_{2} ) q^{97} + ( -20 - 5 \beta_{1} - 8 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - q^{2} + 3q^{3} + 5q^{4} + 3q^{5} - q^{6} - 3q^{7} + 3q^{9} + O(q^{10})$$ $$3q - q^{2} + 3q^{3} + 5q^{4} + 3q^{5} - q^{6} - 3q^{7} + 3q^{9} - q^{10} + 5q^{12} + 4q^{13} + 12q^{14} + 3q^{15} + q^{16} + 4q^{17} - q^{18} + 5q^{19} + 5q^{20} - 3q^{21} + 6q^{23} + 3q^{25} - 2q^{26} + 3q^{27} - 20q^{28} - 10q^{29} - q^{30} - q^{31} - 11q^{32} + 9q^{34} - 3q^{35} + 5q^{36} + 3q^{37} - 12q^{38} + 4q^{39} + 10q^{41} + 12q^{42} - 2q^{43} + 3q^{45} + 9q^{46} + 16q^{47} + q^{48} + 8q^{49} - q^{50} + 4q^{51} + 36q^{52} - q^{54} + 26q^{56} + 5q^{57} - 18q^{58} + 18q^{59} + 5q^{60} - 27q^{61} + q^{62} - 3q^{63} - 20q^{64} + 4q^{65} - q^{67} + 21q^{68} + 6q^{69} + 12q^{70} + 14q^{71} + 7q^{73} + 32q^{74} + 3q^{75} - 6q^{76} - 2q^{78} - 9q^{79} + q^{80} + 3q^{81} - 46q^{82} - 12q^{83} - 20q^{84} + 4q^{85} + 22q^{86} - 10q^{87} + 32q^{89} - q^{90} - 34q^{91} - 5q^{92} - q^{93} - 37q^{94} + 5q^{95} - 11q^{96} + q^{97} - 57q^{98} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$

## 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
 2.39138 0.772866 −2.16425
−2.39138 1.00000 3.71871 1.00000 −2.39138 −5.11009 −4.11009 1.00000 −2.39138
1.2 −0.772866 1.00000 −1.40268 1.00000 −0.772866 1.62981 2.62981 1.00000 −0.772866
1.3 2.16425 1.00000 2.68397 1.00000 2.16425 0.480279 1.48028 1.00000 2.16425
 $$n$$: 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1815.2.a.l 3
3.b odd 2 1 5445.2.a.bc 3
5.b even 2 1 9075.2.a.ci 3
11.b odd 2 1 1815.2.a.n yes 3
33.d even 2 1 5445.2.a.ba 3
55.d odd 2 1 9075.2.a.ce 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.a.l 3 1.a even 1 1 trivial
1815.2.a.n yes 3 11.b odd 2 1
5445.2.a.ba 3 33.d even 2 1
5445.2.a.bc 3 3.b odd 2 1
9075.2.a.ce 3 55.d odd 2 1
9075.2.a.ci 3 5.b even 2 1

## Hecke kernels

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

 $$T_{2}^{3} + T_{2}^{2} - 5 T_{2} - 4$$ $$T_{7}^{3} + 3 T_{7}^{2} - 10 T_{7} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-4 - 5 T + T^{2} + T^{3}$$
$3$ $$( -1 + T )^{3}$$
$5$ $$( -1 + T )^{3}$$
$7$ $$4 - 10 T + 3 T^{2} + T^{3}$$
$11$ $$T^{3}$$
$13$ $$88 - 24 T - 4 T^{2} + T^{3}$$
$17$ $$14 - 7 T - 4 T^{2} + T^{3}$$
$19$ $$16 - 4 T - 5 T^{2} + T^{3}$$
$23$ $$34 - T - 6 T^{2} + T^{3}$$
$29$ $$-16 + 12 T + 10 T^{2} + T^{3}$$
$31$ $$-61 - 29 T + T^{2} + T^{3}$$
$37$ $$548 - 114 T - 3 T^{2} + T^{3}$$
$41$ $$392 - 52 T - 10 T^{2} + T^{3}$$
$43$ $$-32 - 20 T + 2 T^{2} + T^{3}$$
$47$ $$292 + 31 T - 16 T^{2} + T^{3}$$
$53$ $$-586 - 139 T + T^{3}$$
$59$ $$1088 - 28 T - 18 T^{2} + T^{3}$$
$61$ $$133 + 191 T + 27 T^{2} + T^{3}$$
$67$ $$-172 - 54 T + T^{2} + T^{3}$$
$71$ $$2144 - 152 T - 14 T^{2} + T^{3}$$
$73$ $$16 + 4 T - 7 T^{2} + T^{3}$$
$79$ $$83 - 85 T + 9 T^{2} + T^{3}$$
$83$ $$( 4 + T )^{3}$$
$89$ $$-968 + 320 T - 32 T^{2} + T^{3}$$
$97$ $$172 - 54 T - T^{2} + T^{3}$$
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