# Properties

 Label 495.1.h.b Level $495$ Weight $1$ Character orbit 495.h Self dual yes Analytic conductor $0.247$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -55 Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$495 = 3^{2} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 495.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.247037181253$$ 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: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.2475.1 Artin image: $D_8$ Artin field: Galois closure of 8.2.606436875.2

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} + q^{4} - q^{5} -\beta q^{7} +O(q^{10})$$ $$q -\beta q^{2} + q^{4} - q^{5} -\beta q^{7} + \beta q^{10} + q^{11} + \beta q^{13} + 2 q^{14} - q^{16} + \beta q^{17} - q^{20} -\beta q^{22} + q^{25} -2 q^{26} -\beta q^{28} + \beta q^{32} -2 q^{34} + \beta q^{35} + \beta q^{43} + q^{44} + q^{49} -\beta q^{50} + \beta q^{52} - q^{55} - q^{64} -\beta q^{65} + \beta q^{68} -2 q^{70} -\beta q^{73} -\beta q^{77} + q^{80} -\beta q^{83} -\beta q^{85} -2 q^{86} + 2 q^{89} -2 q^{91} -\beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} - 2 q^{5} + O(q^{10})$$ $$2 q + 2 q^{4} - 2 q^{5} + 2 q^{11} + 4 q^{14} - 2 q^{16} - 2 q^{20} + 2 q^{25} - 4 q^{26} - 4 q^{34} + 2 q^{44} + 2 q^{49} - 2 q^{55} - 2 q^{64} - 4 q^{70} + 2 q^{80} - 4 q^{86} + 4 q^{89} - 4 q^{91} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/495\mathbb{Z}\right)^\times$$.

 $$n$$ $$46$$ $$56$$ $$397$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 1.41421 −1.41421
−1.41421 0 1.00000 −1.00000 0 −1.41421 0 0 1.41421
109.2 1.41421 0 1.00000 −1.00000 0 1.41421 0 0 −1.41421
 $$n$$: 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 495.1.h.b 2
3.b odd 2 1 495.1.h.c yes 2
5.b even 2 1 inner 495.1.h.b 2
5.c odd 4 2 2475.1.b.c 2
11.b odd 2 1 inner 495.1.h.b 2
15.d odd 2 1 495.1.h.c yes 2
15.e even 4 2 2475.1.b.b 2
33.d even 2 1 495.1.h.c yes 2
55.d odd 2 1 CM 495.1.h.b 2
55.e even 4 2 2475.1.b.c 2
165.d even 2 1 495.1.h.c yes 2
165.l odd 4 2 2475.1.b.b 2

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(495, [\chi])$$:

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

## Hecke characteristic polynomials

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