# Properties

 Label 1815.2.a.r Level $1815$ Weight $2$ Character orbit 1815.a Self dual yes Analytic conductor $14.493$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# 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: 4.4.5725.1 Defining polynomial: $$x^{4} - x^{3} - 8 x^{2} + 6 x + 11$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 165) 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 8 x^{2} + 6 x + 11$$:

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −0.933531 −2.48008 2.55157 1.86205
−2.55157 −1.00000 4.51049 −1.00000 2.55157 4.19499 −6.40567 1.00000 2.55157
1.2 −1.86205 −1.00000 1.46722 −1.00000 1.86205 −2.63089 0.992053 1.00000 1.86205
1.3 0.933531 −1.00000 −1.12852 −1.00000 −0.933531 2.04108 −2.92057 1.00000 −0.933531
1.4 2.48008 −1.00000 4.15081 −1.00000 −2.48008 4.39482 5.33418 1.00000 −2.48008
 $$n$$: 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.r 4
3.b odd 2 1 5445.2.a.br 4
5.b even 2 1 9075.2.a.dg 4
11.b odd 2 1 1815.2.a.v 4
11.d odd 10 2 165.2.m.b 8
33.d even 2 1 5445.2.a.bk 4
33.f even 10 2 495.2.n.b 8
55.d odd 2 1 9075.2.a.cq 4
55.h odd 10 2 825.2.n.i 8
55.l even 20 4 825.2.bx.g 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.m.b 8 11.d odd 10 2
495.2.n.b 8 33.f even 10 2
825.2.n.i 8 55.h odd 10 2
825.2.bx.g 16 55.l even 20 4
1815.2.a.r 4 1.a even 1 1 trivial
1815.2.a.v 4 11.b odd 2 1
5445.2.a.bk 4 33.d even 2 1
5445.2.a.br 4 3.b odd 2 1
9075.2.a.cq 4 55.d odd 2 1
9075.2.a.dg 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1815))$$:

 $$T_{2}^{4} + T_{2}^{3} - 8 T_{2}^{2} - 6 T_{2} + 11$$ $$T_{7}^{4} - 8 T_{7}^{3} + 8 T_{7}^{2} + 57 T_{7} - 99$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$11 - 6 T - 8 T^{2} + T^{3} + T^{4}$$
$3$ $$( 1 + T )^{4}$$
$5$ $$( 1 + T )^{4}$$
$7$ $$-99 + 57 T + 8 T^{2} - 8 T^{3} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$99 - 150 T + 76 T^{2} - 15 T^{3} + T^{4}$$
$17$ $$99 + 102 T - 20 T^{2} - 6 T^{3} + T^{4}$$
$19$ $$9 + 12 T - 10 T^{2} - T^{3} + T^{4}$$
$23$ $$341 - 29 T - 39 T^{2} + T^{3} + T^{4}$$
$29$ $$-619 - 268 T + 48 T^{2} + 17 T^{3} + T^{4}$$
$31$ $$-25 + 125 T + 35 T^{2} - 15 T^{3} + T^{4}$$
$37$ $$1151 - 21 T - 83 T^{2} + T^{3} + T^{4}$$
$41$ $$11 - 37 T + 38 T^{2} - 12 T^{3} + T^{4}$$
$43$ $$9 - 57 T + 50 T^{2} - 14 T^{3} + T^{4}$$
$47$ $$-3509 + 1081 T - 34 T^{2} - 14 T^{3} + T^{4}$$
$53$ $$1711 + 83 T - 82 T^{2} - 2 T^{3} + T^{4}$$
$59$ $$-99 - 189 T + T^{2} + 11 T^{3} + T^{4}$$
$61$ $$1111 - 106 T - 88 T^{2} + T^{3} + T^{4}$$
$67$ $$1049 + 180 T - 74 T^{2} - 5 T^{3} + T^{4}$$
$71$ $$-821 - 699 T - 145 T^{2} + 3 T^{3} + T^{4}$$
$73$ $$-151 - 3350 T + 674 T^{2} - 45 T^{3} + T^{4}$$
$79$ $$-841 - 970 T - 229 T^{2} + T^{4}$$
$83$ $$-1975 - 625 T + 5 T^{2} + 15 T^{3} + T^{4}$$
$89$ $$99 + 156 T - 55 T^{2} - 2 T^{3} + T^{4}$$
$97$ $$891 + 819 T + 232 T^{2} + 26 T^{3} + T^{4}$$