# Properties

 Label 2415.2.a.o Level $2415$ Weight $2$ Character orbit 2415.a Self dual yes Analytic conductor $19.284$ Analytic rank $1$ Dimension $5$ 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: $$5$$ Coefficient field: 5.5.2508628.1 Defining polynomial: $$x^{5} - 10 x^{3} - 2 x^{2} + 23 x + 8$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} - q^{3} + ( 2 + \beta_{2} ) q^{4} - q^{5} + \beta_{1} q^{6} - q^{7} + ( -1 - \beta_{1} - \beta_{3} ) q^{8} + q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} - q^{3} + ( 2 + \beta_{2} ) q^{4} - q^{5} + \beta_{1} q^{6} - q^{7} + ( -1 - \beta_{1} - \beta_{3} ) q^{8} + q^{9} + \beta_{1} q^{10} + ( -\beta_{1} + \beta_{3} ) q^{11} + ( -2 - \beta_{2} ) q^{12} + ( \beta_{1} - \beta_{2} ) q^{13} + \beta_{1} q^{14} + q^{15} + ( 1 + \beta_{2} + \beta_{3} + \beta_{4} ) q^{16} -2 q^{17} -\beta_{1} q^{18} + ( 1 + 2 \beta_{1} - \beta_{2} - \beta_{4} ) q^{19} + ( -2 - \beta_{2} ) q^{20} + q^{21} + ( 3 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{22} - q^{23} + ( 1 + \beta_{1} + \beta_{3} ) q^{24} + q^{25} + ( -3 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{26} - q^{27} + ( -2 - \beta_{2} ) q^{28} + ( 1 - \beta_{3} ) q^{29} -\beta_{1} q^{30} + ( \beta_{1} - \beta_{2} + \beta_{4} ) q^{31} + ( -2 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{32} + ( \beta_{1} - \beta_{3} ) q^{33} + 2 \beta_{1} q^{34} + q^{35} + ( 2 + \beta_{2} ) q^{36} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{37} + ( -5 - 2 \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{38} + ( -\beta_{1} + \beta_{2} ) q^{39} + ( 1 + \beta_{1} + \beta_{3} ) q^{40} + ( -2 + \beta_{2} - \beta_{3} + \beta_{4} ) q^{41} -\beta_{1} q^{42} + ( -2 + \beta_{4} ) q^{43} + ( -\beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{44} - q^{45} + \beta_{1} q^{46} + ( -2 + \beta_{2} - \beta_{3} ) q^{47} + ( -1 - \beta_{2} - \beta_{3} - \beta_{4} ) q^{48} + q^{49} -\beta_{1} q^{50} + 2 q^{51} + ( -4 + 3 \beta_{1} - \beta_{2} - \beta_{4} ) q^{52} + ( -4 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{53} + \beta_{1} q^{54} + ( \beta_{1} - \beta_{3} ) q^{55} + ( 1 + \beta_{1} + \beta_{3} ) q^{56} + ( -1 - 2 \beta_{1} + \beta_{2} + \beta_{4} ) q^{57} + ( 1 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{58} + ( 1 + \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{59} + ( 2 + \beta_{2} ) q^{60} + ( -3 + 2 \beta_{1} + \beta_{2} - \beta_{4} ) q^{61} + ( -5 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{62} - q^{63} + ( -2 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{64} + ( -\beta_{1} + \beta_{2} ) q^{65} + ( -3 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{66} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{67} + ( -4 - 2 \beta_{2} ) q^{68} + q^{69} -\beta_{1} q^{70} + ( 3 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{71} + ( -1 - \beta_{1} - \beta_{3} ) q^{72} + ( -2 + \beta_{1} - \beta_{2} + \beta_{4} ) q^{73} + ( -6 + \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{74} - q^{75} + ( -1 + 6 \beta_{1} - 4 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{76} + ( \beta_{1} - \beta_{3} ) q^{77} + ( 3 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{78} + ( 5 + 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{79} + ( -1 - \beta_{2} - \beta_{3} - \beta_{4} ) q^{80} + q^{81} + ( -2 + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{82} + ( -5 + 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{83} + ( 2 + \beta_{2} ) q^{84} + 2 q^{85} + ( -2 + 2 \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{86} + ( -1 + \beta_{3} ) q^{87} + ( -5 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{88} + ( -1 - 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{89} + \beta_{1} q^{90} + ( -\beta_{1} + \beta_{2} ) q^{91} + ( -2 - \beta_{2} ) q^{92} + ( -\beta_{1} + \beta_{2} - \beta_{4} ) q^{93} + ( 2 \beta_{2} + \beta_{4} ) q^{94} + ( -1 - 2 \beta_{1} + \beta_{2} + \beta_{4} ) q^{95} + ( 2 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{96} + ( -3 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{97} -\beta_{1} q^{98} + ( -\beta_{1} + \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 5 q^{3} + 10 q^{4} - 5 q^{5} - 5 q^{7} - 6 q^{8} + 5 q^{9} + O(q^{10})$$ $$5 q - 5 q^{3} + 10 q^{4} - 5 q^{5} - 5 q^{7} - 6 q^{8} + 5 q^{9} + q^{11} - 10 q^{12} + 5 q^{15} + 8 q^{16} - 10 q^{17} + 3 q^{19} - 10 q^{20} + 5 q^{21} + 12 q^{22} - 5 q^{23} + 6 q^{24} + 5 q^{25} - 14 q^{26} - 5 q^{27} - 10 q^{28} + 4 q^{29} + 2 q^{31} - 12 q^{32} - q^{33} + 5 q^{35} + 10 q^{36} - 4 q^{37} - 24 q^{38} + 6 q^{40} - 9 q^{41} - 8 q^{43} + 2 q^{44} - 5 q^{45} - 11 q^{47} - 8 q^{48} + 5 q^{49} + 10 q^{51} - 22 q^{52} - 19 q^{53} - q^{55} + 6 q^{56} - 3 q^{57} + 8 q^{58} + 5 q^{59} + 10 q^{60} - 17 q^{61} - 24 q^{62} - 5 q^{63} - 8 q^{64} - 12 q^{66} - 2 q^{67} - 20 q^{68} + 5 q^{69} + 10 q^{71} - 6 q^{72} - 8 q^{73} - 34 q^{74} - 5 q^{75} - 8 q^{76} - q^{77} + 14 q^{78} + 24 q^{79} - 8 q^{80} + 5 q^{81} - 8 q^{82} - 24 q^{83} + 10 q^{84} + 10 q^{85} - 10 q^{86} - 4 q^{87} - 26 q^{88} - 4 q^{89} - 10 q^{92} - 2 q^{93} + 2 q^{94} - 3 q^{95} + 12 q^{96} - 20 q^{97} + q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 10 x^{3} - 2 x^{2} + 23 x + 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 5 \nu - 1$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - \nu^{3} - 7 \nu^{2} + 5 \nu + 8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 5 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + \beta_{3} + 7 \beta_{2} + 21$$

## 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
 2.66023 1.92367 −0.356193 −1.83189 −2.39582
−2.66023 −1.00000 5.07682 −1.00000 2.66023 −1.00000 −8.18506 1.00000 2.66023
1.2 −1.92367 −1.00000 1.70052 −1.00000 1.92367 −1.00000 0.576096 1.00000 1.92367
1.3 0.356193 −1.00000 −1.87313 −1.00000 −0.356193 −1.00000 −1.37958 1.00000 −0.356193
1.4 1.83189 −1.00000 1.35584 −1.00000 −1.83189 −1.00000 −1.18004 1.00000 −1.83189
1.5 2.39582 −1.00000 3.73994 −1.00000 −2.39582 −1.00000 4.16859 1.00000 −2.39582
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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.o 5
3.b odd 2 1 7245.2.a.bg 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2415.2.a.o 5 1.a even 1 1 trivial
7245.2.a.bg 5 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2415))$$:

 $$T_{2}^{5} - 10 T_{2}^{3} + 2 T_{2}^{2} + 23 T_{2} - 8$$ $$T_{11}^{5} - T_{11}^{4} - 24 T_{11}^{3} + 56 T_{11}^{2} - 18 T_{11} - 16$$ $$T_{13}^{5} - 18 T_{13}^{3} + 6 T_{13}^{2} + 44 T_{13} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-8 + 23 T + 2 T^{2} - 10 T^{3} + T^{5}$$
$3$ $$( 1 + T )^{5}$$
$5$ $$( 1 + T )^{5}$$
$7$ $$( 1 + T )^{5}$$
$11$ $$-16 - 18 T + 56 T^{2} - 24 T^{3} - T^{4} + T^{5}$$
$13$ $$16 + 44 T + 6 T^{2} - 18 T^{3} + T^{5}$$
$17$ $$( 2 + T )^{5}$$
$19$ $$-80 + 308 T + 228 T^{2} - 66 T^{3} - 3 T^{4} + T^{5}$$
$23$ $$( 1 + T )^{5}$$
$29$ $$-16 + 48 T + 52 T^{2} - 16 T^{3} - 4 T^{4} + T^{5}$$
$31$ $$-160 + 272 T + 118 T^{2} - 78 T^{3} - 2 T^{4} + T^{5}$$
$37$ $$-776 - 1064 T - 458 T^{2} - 60 T^{3} + 4 T^{4} + T^{5}$$
$41$ $$-16 - 80 T - 120 T^{2} - 44 T^{3} + 9 T^{4} + T^{5}$$
$43$ $$-64 + 176 T - 96 T^{2} - 24 T^{3} + 8 T^{4} + T^{5}$$
$47$ $$320 - 184 T - 108 T^{2} + 18 T^{3} + 11 T^{4} + T^{5}$$
$53$ $$3148 - 2294 T - 528 T^{2} + 64 T^{3} + 19 T^{4} + T^{5}$$
$59$ $$-7192 - 102 T + 970 T^{2} - 130 T^{3} - 5 T^{4} + T^{5}$$
$61$ $$40 + 572 T - 400 T^{2} + 18 T^{3} + 17 T^{4} + T^{5}$$
$67$ $$8192 + 3072 T - 256 T^{2} - 120 T^{3} + 2 T^{4} + T^{5}$$
$71$ $$17792 + 13840 T + 980 T^{2} - 204 T^{3} - 10 T^{4} + T^{5}$$
$73$ $$232 - 176 T - 318 T^{2} - 54 T^{3} + 8 T^{4} + T^{5}$$
$79$ $$-800 - 964 T + 410 T^{2} + 116 T^{3} - 24 T^{4} + T^{5}$$
$83$ $$-640 - 1936 T - 300 T^{2} + 120 T^{3} + 24 T^{4} + T^{5}$$
$89$ $$87200 + 23864 T - 1292 T^{2} - 324 T^{3} + 4 T^{4} + T^{5}$$
$97$ $$86864 + 1408 T - 2692 T^{2} - 84 T^{3} + 20 T^{4} + T^{5}$$