# Properties

 Label 2475.2.a.ba Level $2475$ Weight $2$ Character orbit 2475.a Self dual yes Analytic conductor $19.763$ Analytic rank $1$ 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: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( \beta_{1} + \beta_{2} ) q^{4} + ( 1 + \beta_{1} + \beta_{2} ) q^{7} + ( -1 - \beta_{2} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( \beta_{1} + \beta_{2} ) q^{4} + ( 1 + \beta_{1} + \beta_{2} ) q^{7} + ( -1 - \beta_{2} ) q^{8} - q^{11} + ( 1 - \beta_{1} - \beta_{2} ) q^{13} + ( -1 - 3 \beta_{1} - \beta_{2} ) q^{14} + ( -1 - 2 \beta_{2} ) q^{16} + ( -1 + 3 \beta_{1} + \beta_{2} ) q^{17} + ( -2 - 2 \beta_{2} ) q^{19} + \beta_{1} q^{22} -4 q^{23} + ( 1 + \beta_{1} + \beta_{2} ) q^{26} + ( 3 + 3 \beta_{1} + \beta_{2} ) q^{28} + ( -2 - 2 \beta_{1} + 2 \beta_{2} ) q^{29} + ( -2 - 2 \beta_{1} ) q^{31} + ( 3 \beta_{1} + 2 \beta_{2} ) q^{32} + ( -5 - 3 \beta_{1} - 3 \beta_{2} ) q^{34} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{37} + ( -2 + 4 \beta_{1} ) q^{38} + ( -2 - 2 \beta_{1} - 6 \beta_{2} ) q^{41} + ( 5 - 3 \beta_{1} - 3 \beta_{2} ) q^{43} + ( -\beta_{1} - \beta_{2} ) q^{44} + 4 \beta_{1} q^{46} + ( -6 + 2 \beta_{1} - 2 \beta_{2} ) q^{47} + ( -3 + 4 \beta_{1} + 2 \beta_{2} ) q^{49} + ( -3 - \beta_{1} + \beta_{2} ) q^{52} + ( -4 - 4 \beta_{1} - 2 \beta_{2} ) q^{53} + ( -3 - \beta_{1} - \beta_{2} ) q^{56} + ( 6 + 2 \beta_{1} + 2 \beta_{2} ) q^{58} + ( -4 + 4 \beta_{1} - 2 \beta_{2} ) q^{59} + ( -6 - 4 \beta_{1} ) q^{61} + ( 4 + 4 \beta_{1} + 2 \beta_{2} ) q^{62} + ( -2 - 5 \beta_{1} + \beta_{2} ) q^{64} + ( 4 - 4 \beta_{2} ) q^{67} + ( 5 + 5 \beta_{1} + \beta_{2} ) q^{68} + ( 4 + 6 \beta_{2} ) q^{71} + ( -5 + 5 \beta_{1} + \beta_{2} ) q^{73} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{74} + ( -4 - 2 \beta_{1} ) q^{76} + ( -1 - \beta_{1} - \beta_{2} ) q^{77} + ( -2 - 4 \beta_{1} + 6 \beta_{2} ) q^{79} + ( -2 + 10 \beta_{1} + 2 \beta_{2} ) q^{82} + ( 3 + \beta_{1} + 5 \beta_{2} ) q^{83} + ( 3 + \beta_{1} + 3 \beta_{2} ) q^{86} + ( 1 + \beta_{2} ) q^{88} + ( -8 + 6 \beta_{1} ) q^{89} + ( -2 - 2 \beta_{1} ) q^{91} + ( -4 \beta_{1} - 4 \beta_{2} ) q^{92} + ( -6 + 6 \beta_{1} - 2 \beta_{2} ) q^{94} + ( 4 + 4 \beta_{1} + 8 \beta_{2} ) q^{97} + ( -6 - 3 \beta_{1} - 4 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - q^{2} + q^{4} + 4q^{7} - 3q^{8} + O(q^{10})$$ $$3q - q^{2} + q^{4} + 4q^{7} - 3q^{8} - 3q^{11} + 2q^{13} - 6q^{14} - 3q^{16} - 6q^{19} + q^{22} - 12q^{23} + 4q^{26} + 12q^{28} - 8q^{29} - 8q^{31} + 3q^{32} - 18q^{34} + 4q^{37} - 2q^{38} - 8q^{41} + 12q^{43} - q^{44} + 4q^{46} - 16q^{47} - 5q^{49} - 10q^{52} - 16q^{53} - 10q^{56} + 20q^{58} - 8q^{59} - 22q^{61} + 16q^{62} - 11q^{64} + 12q^{67} + 20q^{68} + 12q^{71} - 10q^{73} + 8q^{74} - 14q^{76} - 4q^{77} - 10q^{79} + 4q^{82} + 10q^{83} + 10q^{86} + 3q^{88} - 18q^{89} - 8q^{91} - 4q^{92} - 12q^{94} + 16q^{97} - 21q^{98} + O(q^{100})$$

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
 2.17009 0.311108 −1.48119
−2.17009 0 2.70928 0 0 3.70928 −1.53919 0 0
1.2 −0.311108 0 −1.90321 0 0 −0.903212 1.21432 0 0
1.3 1.48119 0 0.193937 0 0 1.19394 −2.67513 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.ba 3
3.b odd 2 1 825.2.a.l 3
5.b even 2 1 2475.2.a.bc 3
5.c odd 4 2 495.2.c.e 6
15.d odd 2 1 825.2.a.j 3
15.e even 4 2 165.2.c.b 6
33.d even 2 1 9075.2.a.cg 3
60.l odd 4 2 2640.2.d.h 6
165.d even 2 1 9075.2.a.ch 3
165.l odd 4 2 1815.2.c.e 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.c.b 6 15.e even 4 2
495.2.c.e 6 5.c odd 4 2
825.2.a.j 3 15.d odd 2 1
825.2.a.l 3 3.b odd 2 1
1815.2.c.e 6 165.l odd 4 2
2475.2.a.ba 3 1.a even 1 1 trivial
2475.2.a.bc 3 5.b even 2 1
2640.2.d.h 6 60.l odd 4 2
9075.2.a.cg 3 33.d even 2 1
9075.2.a.ch 3 165.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2475))$$:

 $$T_{2}^{3} + T_{2}^{2} - 3 T_{2} - 1$$ $$T_{7}^{3} - 4 T_{7}^{2} + 4$$ $$T_{29}^{3} + 8 T_{29}^{2} - 16 T_{29} - 160$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 - 3 T + T^{2} + T^{3}$$
$3$ $$T^{3}$$
$5$ $$T^{3}$$
$7$ $$4 - 4 T^{2} + T^{3}$$
$11$ $$( 1 + T )^{3}$$
$13$ $$4 - 4 T - 2 T^{2} + T^{3}$$
$17$ $$-52 - 28 T + T^{3}$$
$19$ $$-40 - 4 T + 6 T^{2} + T^{3}$$
$23$ $$( 4 + T )^{3}$$
$29$ $$-160 - 16 T + 8 T^{2} + T^{3}$$
$31$ $$-16 + 8 T + 8 T^{2} + T^{3}$$
$37$ $$32 - 16 T - 4 T^{2} + T^{3}$$
$41$ $$-928 - 112 T + 8 T^{2} + T^{3}$$
$43$ $$148 - 12 T^{2} + T^{3}$$
$47$ $$32 + 48 T + 16 T^{2} + T^{3}$$
$53$ $$16 + 32 T + 16 T^{2} + T^{3}$$
$59$ $$80 - 64 T + 8 T^{2} + T^{3}$$
$61$ $$8 + 108 T + 22 T^{2} + T^{3}$$
$67$ $$64 - 16 T - 12 T^{2} + T^{3}$$
$71$ $$944 - 96 T - 12 T^{2} + T^{3}$$
$73$ $$-388 - 44 T + 10 T^{2} + T^{3}$$
$79$ $$-1720 - 212 T + 10 T^{2} + T^{3}$$
$83$ $$604 - 60 T - 10 T^{2} + T^{3}$$
$89$ $$-520 - 12 T + 18 T^{2} + T^{3}$$
$97$ $$2432 - 160 T - 16 T^{2} + T^{3}$$