# Properties

 Label 1476.1.h.b Level $1476$ Weight $1$ Character orbit 1476.h Self dual yes Analytic conductor $0.737$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -164 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1476 = 2^{2} \cdot 3^{2} \cdot 41$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1476.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.736619958646$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 164) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.6724.1 Artin image: $D_8$ Artin field: Galois closure of 8.0.1429145856.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} -\beta q^{7} + q^{8} +O(q^{10})$$ $$q + q^{2} + q^{4} -\beta q^{7} + q^{8} + \beta q^{11} -\beta q^{14} + q^{16} + \beta q^{19} + \beta q^{22} - q^{25} -\beta q^{28} + q^{32} + \beta q^{38} - q^{41} + \beta q^{44} -\beta q^{47} + q^{49} - q^{50} -\beta q^{56} -2 q^{61} + q^{64} -\beta q^{67} -\beta q^{71} + \beta q^{76} -2 q^{77} + \beta q^{79} - q^{82} + \beta q^{88} -\beta q^{94} + q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + O(q^{10})$$ $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 2 q^{16} - 2 q^{25} + 2 q^{32} - 2 q^{41} + 2 q^{49} - 2 q^{50} - 4 q^{61} + 2 q^{64} - 4 q^{77} - 2 q^{82} + 2 q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1476\mathbb{Z}\right)^\times$$.

 $$n$$ $$739$$ $$821$$ $$1441$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
163.1
 1.41421 −1.41421
1.00000 0 1.00000 0 0 −1.41421 1.00000 0 0
163.2 1.00000 0 1.00000 0 0 1.41421 1.00000 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
164.d odd 2 1 CM by $$\Q(\sqrt{-41})$$
4.b odd 2 1 inner
41.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1476.1.h.b 2
3.b odd 2 1 164.1.d.b 2
4.b odd 2 1 inner 1476.1.h.b 2
12.b even 2 1 164.1.d.b 2
24.f even 2 1 2624.1.h.b 2
24.h odd 2 1 2624.1.h.b 2
41.b even 2 1 inner 1476.1.h.b 2
123.b odd 2 1 164.1.d.b 2
164.d odd 2 1 CM 1476.1.h.b 2
492.d even 2 1 164.1.d.b 2
984.m odd 2 1 2624.1.h.b 2
984.p even 2 1 2624.1.h.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.1.d.b 2 3.b odd 2 1
164.1.d.b 2 12.b even 2 1
164.1.d.b 2 123.b odd 2 1
164.1.d.b 2 492.d even 2 1
1476.1.h.b 2 1.a even 1 1 trivial
1476.1.h.b 2 4.b odd 2 1 inner
1476.1.h.b 2 41.b even 2 1 inner
1476.1.h.b 2 164.d odd 2 1 CM
2624.1.h.b 2 24.f even 2 1
2624.1.h.b 2 24.h odd 2 1
2624.1.h.b 2 984.m odd 2 1
2624.1.h.b 2 984.p even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}$$ acting on $$S_{1}^{\mathrm{new}}(1476, [\chi])$$.

## Hecke characteristic polynomials

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