Properties

 Label 2475.2.a.f Level $2475$ Weight $2$ Character orbit 2475.a Self dual yes Analytic conductor $19.763$ Analytic rank $1$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 825) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q - 2q^{4} + q^{7} + O(q^{10})$$ $$q - 2q^{4} + q^{7} + q^{11} + q^{13} + 4q^{16} - 6q^{17} - 7q^{19} + 6q^{23} - 2q^{28} + 6q^{29} - 7q^{31} - 2q^{37} + 6q^{41} + q^{43} - 2q^{44} - 6q^{49} - 2q^{52} - 6q^{53} + 5q^{61} - 8q^{64} - 5q^{67} + 12q^{68} + 12q^{71} - 14q^{73} + 14q^{76} + q^{77} - 4q^{79} - 6q^{83} - 6q^{89} + q^{91} - 12q^{92} - 17q^{97} + O(q^{100})$$

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
 0
0 0 −2.00000 0 0 1.00000 0 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.f 1
3.b odd 2 1 825.2.a.b 1
5.b even 2 1 2475.2.a.e 1
5.c odd 4 2 2475.2.c.h 2
15.d odd 2 1 825.2.a.c yes 1
15.e even 4 2 825.2.c.b 2
33.d even 2 1 9075.2.a.i 1
165.d even 2 1 9075.2.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.a.b 1 3.b odd 2 1
825.2.a.c yes 1 15.d odd 2 1
825.2.c.b 2 15.e even 4 2
2475.2.a.e 1 5.b even 2 1
2475.2.a.f 1 1.a even 1 1 trivial
2475.2.c.h 2 5.c odd 4 2
9075.2.a.i 1 33.d even 2 1
9075.2.a.l 1 165.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2475))$$:

 $$T_{2}$$ $$T_{7} - 1$$ $$T_{29} - 6$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$-1 + T$$
$11$ $$-1 + T$$
$13$ $$-1 + T$$
$17$ $$6 + T$$
$19$ $$7 + T$$
$23$ $$-6 + T$$
$29$ $$-6 + T$$
$31$ $$7 + T$$
$37$ $$2 + T$$
$41$ $$-6 + T$$
$43$ $$-1 + T$$
$47$ $$T$$
$53$ $$6 + T$$
$59$ $$T$$
$61$ $$-5 + T$$
$67$ $$5 + T$$
$71$ $$-12 + T$$
$73$ $$14 + T$$
$79$ $$4 + T$$
$83$ $$6 + T$$
$89$ $$6 + T$$
$97$ $$17 + T$$