# Properties

 Label 2624.1.h.c Level $2624$ Weight $1$ Character orbit 2624.h Self dual yes Analytic conductor $1.310$ Analytic rank $0$ Dimension $4$ Projective image $D_{8}$ 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: $$4$$ Coefficient field: $$\Q(\zeta_{16})^+$$ Defining polynomial: $$x^{4} - 4 x^{2} + 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 656) Projective image: $$D_{8}$$ Projective field: Galois closure of 8.0.17643776.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{3} -\beta_{2} q^{5} -\beta_{1} q^{7} + ( 1 - \beta_{2} ) q^{9} +O(q^{10})$$ $$q -\beta_{3} q^{3} -\beta_{2} q^{5} -\beta_{1} q^{7} + ( 1 - \beta_{2} ) q^{9} -\beta_{1} q^{11} + ( \beta_{1} - \beta_{3} ) q^{15} + \beta_{3} q^{19} + \beta_{2} q^{21} + q^{25} + ( \beta_{1} - \beta_{3} ) q^{27} + \beta_{2} q^{33} + ( \beta_{1} + \beta_{3} ) q^{35} + \beta_{2} q^{37} - q^{41} + ( 2 - \beta_{2} ) q^{45} -\beta_{3} q^{47} + ( 1 + \beta_{2} ) q^{49} + ( \beta_{1} + \beta_{3} ) q^{55} + ( -2 + \beta_{2} ) q^{57} + \beta_{3} q^{63} -\beta_{1} q^{67} + \beta_{3} q^{71} -\beta_{2} q^{73} -\beta_{3} q^{75} + ( 2 + \beta_{2} ) q^{77} -\beta_{3} q^{79} + ( 1 - \beta_{2} ) q^{81} + ( -\beta_{1} + \beta_{3} ) q^{95} + \beta_{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{9} + O(q^{10})$$ $$4 q + 4 q^{9} + 4 q^{25} - 4 q^{41} + 8 q^{45} + 4 q^{49} - 8 q^{57} + 8 q^{77} + 4 q^{81} + O(q^{100})$$

## 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
 −0.765367 1.84776 −1.84776 0.765367
0 −1.84776 0 1.41421 0 0.765367 0 2.41421 0
2623.2 0 −0.765367 0 −1.41421 0 −1.84776 0 −0.414214 0
2623.3 0 0.765367 0 −1.41421 0 1.84776 0 −0.414214 0
2623.4 0 1.84776 0 1.41421 0 −0.765367 0 2.41421 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.c 4
4.b odd 2 1 inner 2624.1.h.c 4
8.b even 2 1 656.1.h.a 4
8.d odd 2 1 656.1.h.a 4
41.b even 2 1 inner 2624.1.h.c 4
164.d odd 2 1 CM 2624.1.h.c 4
328.c odd 2 1 656.1.h.a 4
328.g even 2 1 656.1.h.a 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 4 T_{3}^{2} + 2$$ acting on $$S_{1}^{\mathrm{new}}(2624, [\chi])$$.

## Hecke characteristic polynomials

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