# Properties

 Label 2475.2.a.be Level $2475$ Weight $2$ Character orbit 2475.a Self dual yes Analytic conductor $19.763$ Analytic rank $0$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3 x + 1$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{2} ) q^{2} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} + ( -3 + \beta_{1} + \beta_{2} ) q^{7} + ( 4 - 3 \beta_{1} ) q^{8} +O(q^{10})$$ $$q + ( 1 + \beta_{2} ) q^{2} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} + ( -3 + \beta_{1} + \beta_{2} ) q^{7} + ( 4 - 3 \beta_{1} ) q^{8} + q^{11} + ( -1 - 3 \beta_{1} + \beta_{2} ) q^{13} + ( -1 + \beta_{1} - 3 \beta_{2} ) q^{14} + ( 3 - 4 \beta_{1} + 2 \beta_{2} ) q^{16} + ( 1 + \beta_{1} + 3 \beta_{2} ) q^{17} + ( -2 + 4 \beta_{1} + 2 \beta_{2} ) q^{19} + ( 1 + \beta_{2} ) q^{22} + 4 q^{23} + ( 5 - 7 \beta_{1} - \beta_{2} ) q^{26} + ( -5 + 3 \beta_{1} - 3 \beta_{2} ) q^{28} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{29} + ( 2 + 2 \beta_{1} ) q^{31} + ( 5 - 4 \beta_{1} + 3 \beta_{2} ) q^{32} + ( 9 - \beta_{1} + \beta_{2} ) q^{34} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{37} + ( 6 \beta_{1} - 2 \beta_{2} ) q^{38} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{41} + ( -3 + \beta_{1} + \beta_{2} ) q^{43} + ( 2 - \beta_{1} + \beta_{2} ) q^{44} + ( 4 + 4 \beta_{2} ) q^{46} + ( 2 + 2 \beta_{1} - 2 \beta_{2} ) q^{47} + ( 5 - 4 \beta_{1} - 6 \beta_{2} ) q^{49} + ( 11 - 7 \beta_{1} + 3 \beta_{2} ) q^{52} + ( -4 + 4 \beta_{1} + 2 \beta_{2} ) q^{53} + ( -15 + 7 \beta_{1} + \beta_{2} ) q^{56} + ( 6 + 2 \beta_{1} + 2 \beta_{2} ) q^{58} + ( 4 - 4 \beta_{1} + 2 \beta_{2} ) q^{59} + ( 2 - 4 \beta_{1} ) q^{61} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{62} + ( 12 - 3 \beta_{1} + \beta_{2} ) q^{64} + ( -4 - 4 \beta_{2} ) q^{67} + ( 11 - 5 \beta_{1} + 3 \beta_{2} ) q^{68} + ( 4 + 2 \beta_{2} ) q^{71} + ( -7 + 3 \beta_{1} + 3 \beta_{2} ) q^{73} + ( 2 + 2 \beta_{1} - 2 \beta_{2} ) q^{74} + ( -8 + 6 \beta_{1} - 4 \beta_{2} ) q^{76} + ( -3 + \beta_{1} + \beta_{2} ) q^{77} + ( -2 + 2 \beta_{2} ) q^{79} + ( 6 + 2 \beta_{1} + 2 \beta_{2} ) q^{82} + ( -1 + \beta_{1} + \beta_{2} ) q^{83} + ( -1 + \beta_{1} - 3 \beta_{2} ) q^{86} + ( 4 - 3 \beta_{1} ) q^{88} -2 \beta_{1} q^{89} + ( 2 + 2 \beta_{1} - 8 \beta_{2} ) q^{91} + ( 8 - 4 \beta_{1} + 4 \beta_{2} ) q^{92} + ( -6 + 6 \beta_{1} + 2 \beta_{2} ) q^{94} + ( 4 - 4 \beta_{1} ) q^{97} + ( -9 - 2 \beta_{1} + 5 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{2} + 5q^{4} - 8q^{7} + 9q^{8} + O(q^{10})$$ $$3q + 3q^{2} + 5q^{4} - 8q^{7} + 9q^{8} + 3q^{11} - 6q^{13} - 2q^{14} + 5q^{16} + 4q^{17} - 2q^{19} + 3q^{22} + 12q^{23} + 8q^{26} - 12q^{28} + 8q^{29} + 8q^{31} + 11q^{32} + 26q^{34} - 4q^{37} + 6q^{38} + 8q^{41} - 8q^{43} + 5q^{44} + 12q^{46} + 8q^{47} + 11q^{49} + 26q^{52} - 8q^{53} - 38q^{56} + 20q^{58} + 8q^{59} + 2q^{61} + 4q^{62} + 33q^{64} - 12q^{67} + 28q^{68} + 12q^{71} - 18q^{73} + 8q^{74} - 18q^{76} - 8q^{77} - 6q^{79} + 20q^{82} - 2q^{83} - 2q^{86} + 9q^{88} - 2q^{89} + 8q^{91} + 20q^{92} - 12q^{94} + 8q^{97} - 29q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 2$$

## 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
 0.311108 2.17009 −1.48119
−1.21432 0 −0.525428 0 0 −4.90321 3.06668 0 0
1.2 1.53919 0 0.369102 0 0 −0.290725 −2.51026 0 0
1.3 2.67513 0 5.15633 0 0 −2.80606 8.44358 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.be 3
3.b odd 2 1 825.2.a.h 3
5.b even 2 1 2475.2.a.y 3
5.c odd 4 2 495.2.c.d 6
15.d odd 2 1 825.2.a.n 3
15.e even 4 2 165.2.c.a 6
33.d even 2 1 9075.2.a.ck 3
60.l odd 4 2 2640.2.d.i 6
165.d even 2 1 9075.2.a.cc 3
165.l odd 4 2 1815.2.c.d 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.c.a 6 15.e even 4 2
495.2.c.d 6 5.c odd 4 2
825.2.a.h 3 3.b odd 2 1
825.2.a.n 3 15.d odd 2 1
1815.2.c.d 6 165.l odd 4 2
2475.2.a.y 3 5.b even 2 1
2475.2.a.be 3 1.a even 1 1 trivial
2640.2.d.i 6 60.l odd 4 2
9075.2.a.cc 3 165.d even 2 1
9075.2.a.ck 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2475))$$:

 $$T_{2}^{3} - 3 T_{2}^{2} - T_{2} + 5$$ $$T_{7}^{3} + 8 T_{7}^{2} + 16 T_{7} + 4$$ $$T_{29}^{3} - 8 T_{29}^{2} + 32$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$5 - T - 3 T^{2} + T^{3}$$
$3$ $$T^{3}$$
$5$ $$T^{3}$$
$7$ $$4 + 16 T + 8 T^{2} + T^{3}$$
$11$ $$( -1 + T )^{3}$$
$13$ $$-148 - 28 T + 6 T^{2} + T^{3}$$
$17$ $$116 - 28 T - 4 T^{2} + T^{3}$$
$19$ $$-184 - 52 T + 2 T^{2} + T^{3}$$
$23$ $$( -4 + T )^{3}$$
$29$ $$32 - 8 T^{2} + T^{3}$$
$31$ $$16 + 8 T - 8 T^{2} + T^{3}$$
$37$ $$-32 - 16 T + 4 T^{2} + T^{3}$$
$41$ $$32 - 8 T^{2} + T^{3}$$
$43$ $$4 + 16 T + 8 T^{2} + T^{3}$$
$47$ $$160 - 16 T - 8 T^{2} + T^{3}$$
$53$ $$-272 - 32 T + 8 T^{2} + T^{3}$$
$59$ $$-80 - 64 T - 8 T^{2} + T^{3}$$
$61$ $$40 - 52 T - 2 T^{2} + T^{3}$$
$67$ $$-320 - 16 T + 12 T^{2} + T^{3}$$
$71$ $$16 + 32 T - 12 T^{2} + T^{3}$$
$73$ $$-92 + 60 T + 18 T^{2} + T^{3}$$
$79$ $$-8 - 4 T + 6 T^{2} + T^{3}$$
$83$ $$-4 - 4 T + 2 T^{2} + T^{3}$$
$89$ $$-8 - 12 T + 2 T^{2} + T^{3}$$
$97$ $$128 - 32 T - 8 T^{2} + T^{3}$$