# Properties

 Label 1815.4.a.j Level $1815$ Weight $4$ Character orbit 1815.a Self dual yes Analytic conductor $107.088$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1815 = 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1815.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$107.088466660$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}\cdot 3$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 12\sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 q^{3} -8 q^{4} -5 q^{5} + \beta q^{7} + 9 q^{9} +O(q^{10})$$ $$q + 3 q^{3} -8 q^{4} -5 q^{5} + \beta q^{7} + 9 q^{9} -24 q^{12} + \beta q^{13} -15 q^{15} + 64 q^{16} -2 \beta q^{17} -3 \beta q^{19} + 40 q^{20} + 3 \beta q^{21} -72 q^{23} + 25 q^{25} + 27 q^{27} -8 \beta q^{28} -11 \beta q^{29} -20 q^{31} -5 \beta q^{35} -72 q^{36} + 214 q^{37} + 3 \beta q^{39} -\beta q^{41} + 9 \beta q^{43} -45 q^{45} -264 q^{47} + 192 q^{48} + 377 q^{49} -6 \beta q^{51} -8 \beta q^{52} + 78 q^{53} -9 \beta q^{57} + 480 q^{59} + 120 q^{60} + 4 \beta q^{61} + 9 \beta q^{63} -512 q^{64} -5 \beta q^{65} -524 q^{67} + 16 \beta q^{68} -216 q^{69} + 492 q^{71} + 35 \beta q^{73} + 75 q^{75} + 24 \beta q^{76} -11 \beta q^{79} -320 q^{80} + 81 q^{81} -22 \beta q^{83} -24 \beta q^{84} + 10 \beta q^{85} -33 \beta q^{87} -1206 q^{89} + 720 q^{91} + 576 q^{92} -60 q^{93} + 15 \beta q^{95} -1186 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 6q^{3} - 16q^{4} - 10q^{5} + 18q^{9} + O(q^{10})$$ $$2q + 6q^{3} - 16q^{4} - 10q^{5} + 18q^{9} - 48q^{12} - 30q^{15} + 128q^{16} + 80q^{20} - 144q^{23} + 50q^{25} + 54q^{27} - 40q^{31} - 144q^{36} + 428q^{37} - 90q^{45} - 528q^{47} + 384q^{48} + 754q^{49} + 156q^{53} + 960q^{59} + 240q^{60} - 1024q^{64} - 1048q^{67} - 432q^{69} + 984q^{71} + 150q^{75} - 640q^{80} + 162q^{81} - 2412q^{89} + 1440q^{91} + 1152q^{92} - 120q^{93} - 2372q^{97} + O(q^{100})$$

## 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
 −0.618034 1.61803
0 3.00000 −8.00000 −5.00000 0 −26.8328 0 9.00000 0
1.2 0 3.00000 −8.00000 −5.00000 0 26.8328 0 9.00000 0
 $$n$$: 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 1815.4.a.j 2
11.b odd 2 1 inner 1815.4.a.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.4.a.j 2 1.a even 1 1 trivial
1815.4.a.j 2 11.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1815))$$:

 $$T_{2}$$ $$T_{7}^{2} - 720$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -3 + T )^{2}$$
$5$ $$( 5 + T )^{2}$$
$7$ $$-720 + T^{2}$$
$11$ $$T^{2}$$
$13$ $$-720 + T^{2}$$
$17$ $$-2880 + T^{2}$$
$19$ $$-6480 + T^{2}$$
$23$ $$( 72 + T )^{2}$$
$29$ $$-87120 + T^{2}$$
$31$ $$( 20 + T )^{2}$$
$37$ $$( -214 + T )^{2}$$
$41$ $$-720 + T^{2}$$
$43$ $$-58320 + T^{2}$$
$47$ $$( 264 + T )^{2}$$
$53$ $$( -78 + T )^{2}$$
$59$ $$( -480 + T )^{2}$$
$61$ $$-11520 + T^{2}$$
$67$ $$( 524 + T )^{2}$$
$71$ $$( -492 + T )^{2}$$
$73$ $$-882000 + T^{2}$$
$79$ $$-87120 + T^{2}$$
$83$ $$-348480 + T^{2}$$
$89$ $$( 1206 + T )^{2}$$
$97$ $$( 1186 + T )^{2}$$