# Properties

 Label 2475.2.a.u Level $2475$ Weight $2$ Character orbit 2475.a Self dual yes Analytic conductor $19.763$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2475 = 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2475.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$19.7629745003$$ Analytic rank: $$0$$ 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: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + ( 2 + \beta ) q^{4} - q^{7} + ( 4 + \beta ) q^{8} +O(q^{10})$$ $$q + \beta q^{2} + ( 2 + \beta ) q^{4} - q^{7} + ( 4 + \beta ) q^{8} + q^{11} + ( 3 - \beta ) q^{13} -\beta q^{14} + 3 \beta q^{16} + ( 4 - \beta ) q^{17} + 3 q^{19} + \beta q^{22} + ( 4 + \beta ) q^{23} + ( -4 + 2 \beta ) q^{26} + ( -2 - \beta ) q^{28} -2 \beta q^{29} + ( -3 + 3 \beta ) q^{31} + ( 4 + \beta ) q^{32} + ( -4 + 3 \beta ) q^{34} + ( -2 + 5 \beta ) q^{37} + 3 \beta q^{38} + 3 \beta q^{41} + ( 3 - 3 \beta ) q^{43} + ( 2 + \beta ) q^{44} + ( 4 + 5 \beta ) q^{46} + ( 8 - \beta ) q^{47} -6 q^{49} + 2 q^{52} + ( 4 - 2 \beta ) q^{53} + ( -4 - \beta ) q^{56} + ( -8 - 2 \beta ) q^{58} + ( -8 - \beta ) q^{59} + ( -5 - \beta ) q^{61} + 12 q^{62} + ( 4 - \beta ) q^{64} + ( -1 + 3 \beta ) q^{67} + ( 4 + \beta ) q^{68} + ( 4 - 5 \beta ) q^{71} + ( -6 - 4 \beta ) q^{73} + ( 20 + 3 \beta ) q^{74} + ( 6 + 3 \beta ) q^{76} - q^{77} + ( -8 - 3 \beta ) q^{79} + ( 12 + 3 \beta ) q^{82} -12 q^{86} + ( 4 + \beta ) q^{88} + ( 12 + 2 \beta ) q^{89} + ( -3 + \beta ) q^{91} + ( 12 + 7 \beta ) q^{92} + ( -4 + 7 \beta ) q^{94} + ( 3 - 2 \beta ) q^{97} -6 \beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 5 q^{4} - 2 q^{7} + 9 q^{8} + O(q^{10})$$ $$2 q + q^{2} + 5 q^{4} - 2 q^{7} + 9 q^{8} + 2 q^{11} + 5 q^{13} - q^{14} + 3 q^{16} + 7 q^{17} + 6 q^{19} + q^{22} + 9 q^{23} - 6 q^{26} - 5 q^{28} - 2 q^{29} - 3 q^{31} + 9 q^{32} - 5 q^{34} + q^{37} + 3 q^{38} + 3 q^{41} + 3 q^{43} + 5 q^{44} + 13 q^{46} + 15 q^{47} - 12 q^{49} + 4 q^{52} + 6 q^{53} - 9 q^{56} - 18 q^{58} - 17 q^{59} - 11 q^{61} + 24 q^{62} + 7 q^{64} + q^{67} + 9 q^{68} + 3 q^{71} - 16 q^{73} + 43 q^{74} + 15 q^{76} - 2 q^{77} - 19 q^{79} + 27 q^{82} - 24 q^{86} + 9 q^{88} + 26 q^{89} - 5 q^{91} + 31 q^{92} - q^{94} + 4 q^{97} - 6 q^{98} + O(q^{100})$$

## 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
 −1.56155 2.56155
−1.56155 0 0.438447 0 0 −1.00000 2.43845 0 0
1.2 2.56155 0 4.56155 0 0 −1.00000 6.56155 0 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

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 2475.2.a.u yes 2
3.b odd 2 1 2475.2.a.p 2
5.b even 2 1 2475.2.a.q yes 2
5.c odd 4 2 2475.2.c.j 4
15.d odd 2 1 2475.2.a.v yes 2
15.e even 4 2 2475.2.c.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2475.2.a.p 2 3.b odd 2 1
2475.2.a.q yes 2 5.b even 2 1
2475.2.a.u yes 2 1.a even 1 1 trivial
2475.2.a.v yes 2 15.d odd 2 1
2475.2.c.i 4 15.e even 4 2
2475.2.c.j 4 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2475))$$:

 $$T_{2}^{2} - T_{2} - 4$$ $$T_{7} + 1$$ $$T_{29}^{2} + 2 T_{29} - 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-4 - T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$( -1 + T )^{2}$$
$13$ $$2 - 5 T + T^{2}$$
$17$ $$8 - 7 T + T^{2}$$
$19$ $$( -3 + T )^{2}$$
$23$ $$16 - 9 T + T^{2}$$
$29$ $$-16 + 2 T + T^{2}$$
$31$ $$-36 + 3 T + T^{2}$$
$37$ $$-106 - T + T^{2}$$
$41$ $$-36 - 3 T + T^{2}$$
$43$ $$-36 - 3 T + T^{2}$$
$47$ $$52 - 15 T + T^{2}$$
$53$ $$-8 - 6 T + T^{2}$$
$59$ $$68 + 17 T + T^{2}$$
$61$ $$26 + 11 T + T^{2}$$
$67$ $$-38 - T + T^{2}$$
$71$ $$-104 - 3 T + T^{2}$$
$73$ $$-4 + 16 T + T^{2}$$
$79$ $$52 + 19 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$152 - 26 T + T^{2}$$
$97$ $$-13 - 4 T + T^{2}$$