# Properties

 Label 2499.2.a.o Level $2499$ Weight $2$ Character orbit 2499.a Self dual yes Analytic conductor $19.955$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$2499 = 3 \cdot 7^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2499.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$19.9546154651$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 51) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 2.56155 −1.56155
−2.56155 1.00000 4.56155 −3.56155 −2.56155 0 −6.56155 1.00000 9.12311
1.2 1.56155 1.00000 0.438447 0.561553 1.56155 0 −2.43845 1.00000 0.876894
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$-1$$
$$17$$ $$1$$

## 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 2499.2.a.o 2
3.b odd 2 1 7497.2.a.v 2
7.b odd 2 1 51.2.a.b 2
21.c even 2 1 153.2.a.e 2
28.d even 2 1 816.2.a.m 2
35.c odd 2 1 1275.2.a.n 2
35.f even 4 2 1275.2.b.d 4
56.e even 2 1 3264.2.a.bg 2
56.h odd 2 1 3264.2.a.bl 2
77.b even 2 1 6171.2.a.p 2
84.h odd 2 1 2448.2.a.v 2
91.b odd 2 1 8619.2.a.q 2
105.g even 2 1 3825.2.a.s 2
119.d odd 2 1 867.2.a.f 2
119.f odd 4 2 867.2.d.c 4
119.l odd 8 4 867.2.e.f 8
119.p even 16 8 867.2.h.j 16
168.e odd 2 1 9792.2.a.cz 2
168.i even 2 1 9792.2.a.cy 2
357.c even 2 1 2601.2.a.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.a.b 2 7.b odd 2 1
153.2.a.e 2 21.c even 2 1
816.2.a.m 2 28.d even 2 1
867.2.a.f 2 119.d odd 2 1
867.2.d.c 4 119.f odd 4 2
867.2.e.f 8 119.l odd 8 4
867.2.h.j 16 119.p even 16 8
1275.2.a.n 2 35.c odd 2 1
1275.2.b.d 4 35.f even 4 2
2448.2.a.v 2 84.h odd 2 1
2499.2.a.o 2 1.a even 1 1 trivial
2601.2.a.t 2 357.c even 2 1
3264.2.a.bg 2 56.e even 2 1
3264.2.a.bl 2 56.h odd 2 1
3825.2.a.s 2 105.g even 2 1
6171.2.a.p 2 77.b even 2 1
7497.2.a.v 2 3.b odd 2 1
8619.2.a.q 2 91.b odd 2 1
9792.2.a.cy 2 168.i even 2 1
9792.2.a.cz 2 168.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2499))$$:

 $$T_{2}^{2} + T_{2} - 4$$ $$T_{5}^{2} + 3 T_{5} - 2$$ $$T_{11}^{2} + T_{11} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-4 + T + T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$-2 + 3 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-4 + T + T^{2}$$
$13$ $$2 + 5 T + T^{2}$$
$17$ $$( 1 + T )^{2}$$
$19$ $$-36 + 3 T + T^{2}$$
$23$ $$16 + 9 T + T^{2}$$
$29$ $$-68 + T^{2}$$
$31$ $$-16 - 2 T + T^{2}$$
$37$ $$-16 + 2 T + T^{2}$$
$41$ $$-2 - 3 T + T^{2}$$
$43$ $$-36 + 3 T + T^{2}$$
$47$ $$32 - 14 T + T^{2}$$
$53$ $$-52 - 8 T + T^{2}$$
$59$ $$-8 + 6 T + T^{2}$$
$61$ $$8 + 10 T + T^{2}$$
$67$ $$( -4 + T )^{2}$$
$71$ $$-64 - 4 T + T^{2}$$
$73$ $$-52 - 8 T + T^{2}$$
$79$ $$-144 - 6 T + T^{2}$$
$83$ $$8 - 10 T + T^{2}$$
$89$ $$-8 + 6 T + T^{2}$$
$97$ $$32 - 14 T + T^{2}$$
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