# Properties

 Label 2475.2.a.r Level $2475$ Weight $2$ Character orbit 2475.a Self dual yes Analytic conductor $19.763$ Analytic rank $1$ 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: $$1$$ 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: no (minimal twist has level 165) 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + q^{4} -2 q^{7} -\beta q^{8} +O(q^{10})$$ $$q + \beta q^{2} + q^{4} -2 q^{7} -\beta q^{8} + q^{11} + ( -2 + 2 \beta ) q^{13} -2 \beta q^{14} -5 q^{16} + ( 2 - 2 \beta ) q^{19} + \beta q^{22} -4 \beta q^{23} + ( 6 - 2 \beta ) q^{26} -2 q^{28} -2 \beta q^{29} + ( -4 + 4 \beta ) q^{31} -3 \beta q^{32} + ( -2 - 4 \beta ) q^{37} + ( -6 + 2 \beta ) q^{38} + 2 \beta q^{41} + ( -2 - 4 \beta ) q^{43} + q^{44} -12 q^{46} + 4 \beta q^{47} -3 q^{49} + ( -2 + 2 \beta ) q^{52} + ( -6 - 4 \beta ) q^{53} + 2 \beta q^{56} -6 q^{58} -4 \beta q^{59} + 2 q^{61} + ( 12 - 4 \beta ) q^{62} + q^{64} -8 q^{67} + 8 \beta q^{71} + ( -2 - 6 \beta ) q^{73} + ( -12 - 2 \beta ) q^{74} + ( 2 - 2 \beta ) q^{76} -2 q^{77} + ( -10 - 2 \beta ) q^{79} + 6 q^{82} + ( 12 + 2 \beta ) q^{83} + ( -12 - 2 \beta ) q^{86} -\beta q^{88} + ( 6 + 4 \beta ) q^{89} + ( 4 - 4 \beta ) q^{91} -4 \beta q^{92} + 12 q^{94} + 10 q^{97} -3 \beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{4} - 4q^{7} + O(q^{10})$$ $$2q + 2q^{4} - 4q^{7} + 2q^{11} - 4q^{13} - 10q^{16} + 4q^{19} + 12q^{26} - 4q^{28} - 8q^{31} - 4q^{37} - 12q^{38} - 4q^{43} + 2q^{44} - 24q^{46} - 6q^{49} - 4q^{52} - 12q^{53} - 12q^{58} + 4q^{61} + 24q^{62} + 2q^{64} - 16q^{67} - 4q^{73} - 24q^{74} + 4q^{76} - 4q^{77} - 20q^{79} + 12q^{82} + 24q^{83} - 24q^{86} + 12q^{89} + 8q^{91} + 24q^{94} + 20q^{97} + O(q^{100})$$

## 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
 −1.73205 1.73205
−1.73205 0 1.00000 0 0 −2.00000 1.73205 0 0
1.2 1.73205 0 1.00000 0 0 −2.00000 −1.73205 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.r 2
3.b odd 2 1 825.2.a.e 2
5.b even 2 1 495.2.a.c 2
5.c odd 4 2 2475.2.c.n 4
15.d odd 2 1 165.2.a.b 2
15.e even 4 2 825.2.c.c 4
20.d odd 2 1 7920.2.a.bz 2
33.d even 2 1 9075.2.a.bh 2
55.d odd 2 1 5445.2.a.s 2
60.h even 2 1 2640.2.a.x 2
105.g even 2 1 8085.2.a.bd 2
165.d even 2 1 1815.2.a.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.a.b 2 15.d odd 2 1
495.2.a.c 2 5.b even 2 1
825.2.a.e 2 3.b odd 2 1
825.2.c.c 4 15.e even 4 2
1815.2.a.i 2 165.d even 2 1
2475.2.a.r 2 1.a even 1 1 trivial
2475.2.c.n 4 5.c odd 4 2
2640.2.a.x 2 60.h even 2 1
5445.2.a.s 2 55.d odd 2 1
7920.2.a.bz 2 20.d odd 2 1
8085.2.a.bd 2 105.g even 2 1
9075.2.a.bh 2 33.d even 2 1

## Hecke kernels

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

 $$T_{2}^{2} - 3$$ $$T_{7} + 2$$ $$T_{29}^{2} - 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-3 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$( 2 + T )^{2}$$
$11$ $$( -1 + T )^{2}$$
$13$ $$-8 + 4 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$-8 - 4 T + T^{2}$$
$23$ $$-48 + T^{2}$$
$29$ $$-12 + T^{2}$$
$31$ $$-32 + 8 T + T^{2}$$
$37$ $$-44 + 4 T + T^{2}$$
$41$ $$-12 + T^{2}$$
$43$ $$-44 + 4 T + T^{2}$$
$47$ $$-48 + T^{2}$$
$53$ $$-12 + 12 T + T^{2}$$
$59$ $$-48 + T^{2}$$
$61$ $$( -2 + T )^{2}$$
$67$ $$( 8 + T )^{2}$$
$71$ $$-192 + T^{2}$$
$73$ $$-104 + 4 T + T^{2}$$
$79$ $$88 + 20 T + T^{2}$$
$83$ $$132 - 24 T + T^{2}$$
$89$ $$-12 - 12 T + T^{2}$$
$97$ $$( -10 + T )^{2}$$