# Properties

 Label 2475.1.b.b Level $2475$ Weight $1$ Character orbit 2475.b Analytic conductor $1.235$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -55 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2475 = 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2475.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.23518590627$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 495) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.2475.1 Artin image: $\SD_{16}$ Artin field: Galois closure of 8.2.15160921875.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - q^{4} -\beta q^{7} +O(q^{10})$$ $$q + \beta q^{2} - q^{4} -\beta q^{7} - q^{11} -\beta q^{13} + 2 q^{14} - q^{16} -\beta q^{17} -\beta q^{22} + 2 q^{26} + \beta q^{28} -\beta q^{32} + 2 q^{34} -\beta q^{43} + q^{44} - q^{49} + \beta q^{52} + q^{64} + \beta q^{68} + \beta q^{73} + \beta q^{77} -\beta q^{83} + 2 q^{86} + 2 q^{89} -2 q^{91} -\beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + O(q^{10})$$ $$2 q - 2 q^{4} - 2 q^{11} + 4 q^{14} - 2 q^{16} + 4 q^{26} + 4 q^{34} + 2 q^{44} - 2 q^{49} + 2 q^{64} + 4 q^{86} + 4 q^{89} - 4 q^{91} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times$$.

 $$n$$ $$551$$ $$2026$$ $$2377$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
901.1
 − 1.41421i 1.41421i
1.41421i 0 −1.00000 0 0 1.41421i 0 0 0
901.2 1.41421i 0 −1.00000 0 0 1.41421i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
55.d odd 2 1 CM by $$\Q(\sqrt{-55})$$
5.b even 2 1 inner
11.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2475.1.b.b 2
3.b odd 2 1 2475.1.b.c 2
5.b even 2 1 inner 2475.1.b.b 2
5.c odd 4 2 495.1.h.c yes 2
11.b odd 2 1 inner 2475.1.b.b 2
15.d odd 2 1 2475.1.b.c 2
15.e even 4 2 495.1.h.b 2
33.d even 2 1 2475.1.b.c 2
55.d odd 2 1 CM 2475.1.b.b 2
55.e even 4 2 495.1.h.c yes 2
165.d even 2 1 2475.1.b.c 2
165.l odd 4 2 495.1.h.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.1.h.b 2 15.e even 4 2
495.1.h.b 2 165.l odd 4 2
495.1.h.c yes 2 5.c odd 4 2
495.1.h.c yes 2 55.e even 4 2
2475.1.b.b 2 1.a even 1 1 trivial
2475.1.b.b 2 5.b even 2 1 inner
2475.1.b.b 2 11.b odd 2 1 inner
2475.1.b.b 2 55.d odd 2 1 CM
2475.1.b.c 2 3.b odd 2 1
2475.1.b.c 2 15.d odd 2 1
2475.1.b.c 2 33.d even 2 1
2475.1.b.c 2 165.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 $$S_{1}^{\mathrm{new}}(2475, [\chi])$$:

 $$T_{2}^{2} + 2$$ $$T_{89} - 2$$

## Hecke characteristic polynomials

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