# Properties

 Label 1840.2.m.d Level $1840$ Weight $2$ Character orbit 1840.m Analytic conductor $14.692$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1840.m (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.6924739719$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{6}, \sqrt{-14})$$ Defining polynomial: $$x^{4} + 4 x^{2} + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + q^{3} + \beta_{2} q^{5} -2 q^{9} +O(q^{10})$$ $$q + q^{3} + \beta_{2} q^{5} -2 q^{9} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{11} + \beta_{3} q^{13} + \beta_{2} q^{15} + ( \beta_{1} - \beta_{2} ) q^{17} + ( \beta_{1} - \beta_{2} ) q^{19} + ( -3 + \beta_{1} + \beta_{2} ) q^{23} + ( -2 - \beta_{3} ) q^{25} -5 q^{27} -3 q^{29} -\beta_{3} q^{31} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{33} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{37} + \beta_{3} q^{39} -3 q^{41} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{43} -2 \beta_{2} q^{45} -9 q^{47} + 7 q^{49} + ( \beta_{1} - \beta_{2} ) q^{51} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{53} + ( -6 + 2 \beta_{3} ) q^{55} + ( \beta_{1} - \beta_{2} ) q^{57} + 2 \beta_{3} q^{59} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{61} + ( -5 \beta_{1} + 2 \beta_{2} ) q^{65} + ( -3 + \beta_{1} + \beta_{2} ) q^{69} + 3 \beta_{3} q^{71} + \beta_{3} q^{73} + ( -2 - \beta_{3} ) q^{75} + ( 3 \beta_{1} - 3 \beta_{2} ) q^{79} + q^{81} + ( -\beta_{1} - \beta_{2} ) q^{83} + ( -3 + \beta_{3} ) q^{85} -3 q^{87} + ( -\beta_{1} - \beta_{2} ) q^{89} -\beta_{3} q^{93} + ( -3 + \beta_{3} ) q^{95} + ( -4 \beta_{1} + 4 \beta_{2} ) q^{97} + ( -4 \beta_{1} + 4 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{3} - 8q^{9} + O(q^{10})$$ $$4q + 4q^{3} - 8q^{9} - 12q^{23} - 8q^{25} - 20q^{27} - 12q^{29} - 12q^{41} - 36q^{47} + 28q^{49} - 24q^{55} - 12q^{69} - 8q^{75} + 4q^{81} - 12q^{85} - 12q^{87} - 12q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 4 x^{2} + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 2$$ $$\nu^{3}$$ $$=$$ $$5 \beta_{2} - 4 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$$.

 $$n$$ $$737$$ $$1151$$ $$1201$$ $$1381$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1839.1
 1.22474 − 1.87083i 1.22474 + 1.87083i −1.22474 − 1.87083i −1.22474 + 1.87083i
0 1.00000 0 −1.22474 1.87083i 0 0 0 −2.00000 0
1839.2 0 1.00000 0 −1.22474 + 1.87083i 0 0 0 −2.00000 0
1839.3 0 1.00000 0 1.22474 1.87083i 0 0 0 −2.00000 0
1839.4 0 1.00000 0 1.22474 + 1.87083i 0 0 0 −2.00000 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
20.d odd 2 1 inner
23.b odd 2 1 inner
460.g even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1840.2.m.d yes 4
4.b odd 2 1 1840.2.m.a 4
5.b even 2 1 1840.2.m.a 4
20.d odd 2 1 inner 1840.2.m.d yes 4
23.b odd 2 1 inner 1840.2.m.d yes 4
92.b even 2 1 1840.2.m.a 4
115.c odd 2 1 1840.2.m.a 4
460.g even 2 1 inner 1840.2.m.d yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1840.2.m.a 4 4.b odd 2 1
1840.2.m.a 4 5.b even 2 1
1840.2.m.a 4 92.b even 2 1
1840.2.m.a 4 115.c odd 2 1
1840.2.m.d yes 4 1.a even 1 1 trivial
1840.2.m.d yes 4 20.d odd 2 1 inner
1840.2.m.d yes 4 23.b odd 2 1 inner
1840.2.m.d yes 4 460.g even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1840, [\chi])$$:

 $$T_{3} - 1$$ $$T_{7}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -1 + T )^{4}$$
$5$ $$25 + 4 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( -24 + T^{2} )^{2}$$
$13$ $$( 21 + T^{2} )^{2}$$
$17$ $$( -6 + T^{2} )^{2}$$
$19$ $$( -6 + T^{2} )^{2}$$
$23$ $$( 23 + 6 T + T^{2} )^{2}$$
$29$ $$( 3 + T )^{4}$$
$31$ $$( 21 + T^{2} )^{2}$$
$37$ $$( -24 + T^{2} )^{2}$$
$41$ $$( 3 + T )^{4}$$
$43$ $$( 126 + T^{2} )^{2}$$
$47$ $$( 9 + T )^{4}$$
$53$ $$( -24 + T^{2} )^{2}$$
$59$ $$( 84 + T^{2} )^{2}$$
$61$ $$( 126 + T^{2} )^{2}$$
$67$ $$T^{4}$$
$71$ $$( 189 + T^{2} )^{2}$$
$73$ $$( 21 + T^{2} )^{2}$$
$79$ $$( -54 + T^{2} )^{2}$$
$83$ $$( 14 + T^{2} )^{2}$$
$89$ $$( 14 + T^{2} )^{2}$$
$97$ $$( -96 + T^{2} )^{2}$$