# Properties

 Label 2624.1.h.b Level $2624$ Weight $1$ Character orbit 2624.h Self dual yes Analytic conductor $1.310$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -164 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2624 = 2^{6} \cdot 41$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2624.h (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.30954659315$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 8.0.4516806656.3

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta q^{3} -\beta q^{7} + q^{9} +O(q^{10})$$ $$q -\beta q^{3} -\beta q^{7} + q^{9} + \beta q^{11} -\beta q^{19} + 2 q^{21} - q^{25} -2 q^{33} + q^{41} + \beta q^{47} + q^{49} + 2 q^{57} + 2 q^{61} -\beta q^{63} + \beta q^{67} + \beta q^{71} + \beta q^{75} -2 q^{77} + \beta q^{79} - q^{81} + \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{9} + O(q^{10})$$ $$2q + 2q^{9} + 4q^{21} - 2q^{25} - 4q^{33} + 2q^{41} + 2q^{49} + 4q^{57} + 4q^{61} - 4q^{77} - 2q^{81} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2624\mathbb{Z}\right)^\times$$.

 $$n$$ $$129$$ $$575$$ $$1477$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2623.1
 1.41421 −1.41421
0 −1.41421 0 0 0 −1.41421 0 1.00000 0
2623.2 0 1.41421 0 0 0 1.41421 0 1.00000 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 2624.1.h.b 2
4.b odd 2 1 inner 2624.1.h.b 2
8.b even 2 1 164.1.d.b 2
8.d odd 2 1 164.1.d.b 2
24.f even 2 1 1476.1.h.b 2
24.h odd 2 1 1476.1.h.b 2
41.b even 2 1 inner 2624.1.h.b 2
164.d odd 2 1 CM 2624.1.h.b 2
328.c odd 2 1 164.1.d.b 2
328.g even 2 1 164.1.d.b 2
984.m odd 2 1 1476.1.h.b 2
984.p even 2 1 1476.1.h.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.1.d.b 2 8.b even 2 1
164.1.d.b 2 8.d odd 2 1
164.1.d.b 2 328.c odd 2 1
164.1.d.b 2 328.g even 2 1
1476.1.h.b 2 24.f even 2 1
1476.1.h.b 2 24.h odd 2 1
1476.1.h.b 2 984.m odd 2 1
1476.1.h.b 2 984.p even 2 1
2624.1.h.b 2 1.a even 1 1 trivial
2624.1.h.b 2 4.b odd 2 1 inner
2624.1.h.b 2 41.b even 2 1 inner
2624.1.h.b 2 164.d odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 2$$ acting on $$S_{1}^{\mathrm{new}}(2624, [\chi])$$.

## Hecke characteristic polynomials

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