# Properties

 Label 2415.2.a.j Level $2415$ Weight $2$ Character orbit 2415.a Self dual yes Analytic conductor $19.284$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2415 = 3 \cdot 5 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2415.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$19.2838720881$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + q^{3} + q^{4} - q^{5} + \beta q^{6} - q^{7} -\beta q^{8} + q^{9} +O(q^{10})$$ $$q + \beta q^{2} + q^{3} + q^{4} - q^{5} + \beta q^{6} - q^{7} -\beta q^{8} + q^{9} -\beta q^{10} + ( 1 + \beta ) q^{11} + q^{12} + ( -3 - \beta ) q^{13} -\beta q^{14} - q^{15} -5 q^{16} + ( -4 - 2 \beta ) q^{17} + \beta q^{18} -2 \beta q^{19} - q^{20} - q^{21} + ( 3 + \beta ) q^{22} + q^{23} -\beta q^{24} + q^{25} + ( -3 - 3 \beta ) q^{26} + q^{27} - q^{28} -\beta q^{30} + ( -3 + \beta ) q^{31} -3 \beta q^{32} + ( 1 + \beta ) q^{33} + ( -6 - 4 \beta ) q^{34} + q^{35} + q^{36} + ( -3 + \beta ) q^{37} -6 q^{38} + ( -3 - \beta ) q^{39} + \beta q^{40} -2 \beta q^{41} -\beta q^{42} + 4 \beta q^{43} + ( 1 + \beta ) q^{44} - q^{45} + \beta q^{46} + ( 2 - 6 \beta ) q^{47} -5 q^{48} + q^{49} + \beta q^{50} + ( -4 - 2 \beta ) q^{51} + ( -3 - \beta ) q^{52} + ( -1 + \beta ) q^{53} + \beta q^{54} + ( -1 - \beta ) q^{55} + \beta q^{56} -2 \beta q^{57} + ( -3 + 3 \beta ) q^{59} - q^{60} + 4 q^{61} + ( 3 - 3 \beta ) q^{62} - q^{63} + q^{64} + ( 3 + \beta ) q^{65} + ( 3 + \beta ) q^{66} + ( -8 - 4 \beta ) q^{67} + ( -4 - 2 \beta ) q^{68} + q^{69} + \beta q^{70} -6 q^{71} -\beta q^{72} + ( -5 + 5 \beta ) q^{73} + ( 3 - 3 \beta ) q^{74} + q^{75} -2 \beta q^{76} + ( -1 - \beta ) q^{77} + ( -3 - 3 \beta ) q^{78} + ( 5 + 3 \beta ) q^{79} + 5 q^{80} + q^{81} -6 q^{82} -2 q^{83} - q^{84} + ( 4 + 2 \beta ) q^{85} + 12 q^{86} + ( -3 - \beta ) q^{88} + ( 10 - 2 \beta ) q^{89} -\beta q^{90} + ( 3 + \beta ) q^{91} + q^{92} + ( -3 + \beta ) q^{93} + ( -18 + 2 \beta ) q^{94} + 2 \beta q^{95} -3 \beta q^{96} + ( -4 + 4 \beta ) q^{97} + \beta q^{98} + ( 1 + \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{9} + O(q^{10})$$ $$2 q + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{12} - 6 q^{13} - 2 q^{15} - 10 q^{16} - 8 q^{17} - 2 q^{20} - 2 q^{21} + 6 q^{22} + 2 q^{23} + 2 q^{25} - 6 q^{26} + 2 q^{27} - 2 q^{28} - 6 q^{31} + 2 q^{33} - 12 q^{34} + 2 q^{35} + 2 q^{36} - 6 q^{37} - 12 q^{38} - 6 q^{39} + 2 q^{44} - 2 q^{45} + 4 q^{47} - 10 q^{48} + 2 q^{49} - 8 q^{51} - 6 q^{52} - 2 q^{53} - 2 q^{55} - 6 q^{59} - 2 q^{60} + 8 q^{61} + 6 q^{62} - 2 q^{63} + 2 q^{64} + 6 q^{65} + 6 q^{66} - 16 q^{67} - 8 q^{68} + 2 q^{69} - 12 q^{71} - 10 q^{73} + 6 q^{74} + 2 q^{75} - 2 q^{77} - 6 q^{78} + 10 q^{79} + 10 q^{80} + 2 q^{81} - 12 q^{82} - 4 q^{83} - 2 q^{84} + 8 q^{85} + 24 q^{86} - 6 q^{88} + 20 q^{89} + 6 q^{91} + 2 q^{92} - 6 q^{93} - 36 q^{94} - 8 q^{97} + 2 q^{99} + O(q^{100})$$

## 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
 −1.73205 1.73205
−1.73205 1.00000 1.00000 −1.00000 −1.73205 −1.00000 1.73205 1.00000 1.73205
1.2 1.73205 1.00000 1.00000 −1.00000 1.73205 −1.00000 −1.73205 1.00000 −1.73205
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$1$$
$$7$$ $$1$$
$$23$$ $$-1$$

## 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 2415.2.a.j 2
3.b odd 2 1 7245.2.a.w 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2415.2.a.j 2 1.a even 1 1 trivial
7245.2.a.w 2 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2415))$$:

 $$T_{2}^{2} - 3$$ $$T_{11}^{2} - 2 T_{11} - 2$$ $$T_{13}^{2} + 6 T_{13} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-3 + T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$-2 - 2 T + T^{2}$$
$13$ $$6 + 6 T + T^{2}$$
$17$ $$4 + 8 T + T^{2}$$
$19$ $$-12 + T^{2}$$
$23$ $$( -1 + T )^{2}$$
$29$ $$T^{2}$$
$31$ $$6 + 6 T + T^{2}$$
$37$ $$6 + 6 T + T^{2}$$
$41$ $$-12 + T^{2}$$
$43$ $$-48 + T^{2}$$
$47$ $$-104 - 4 T + T^{2}$$
$53$ $$-2 + 2 T + T^{2}$$
$59$ $$-18 + 6 T + T^{2}$$
$61$ $$( -4 + T )^{2}$$
$67$ $$16 + 16 T + T^{2}$$
$71$ $$( 6 + T )^{2}$$
$73$ $$-50 + 10 T + T^{2}$$
$79$ $$-2 - 10 T + T^{2}$$
$83$ $$( 2 + T )^{2}$$
$89$ $$88 - 20 T + T^{2}$$
$97$ $$-32 + 8 T + T^{2}$$