# Properties

 Label 1840.2.m.c Level $1840$ Weight $2$ Character orbit 1840.m Analytic conductor $14.692$ Analytic rank $0$ Dimension $4$ CM discriminant -20 Inner twists $8$

# 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: $$\Q(\sqrt{-2}, \sqrt{-5})$$ Defining polynomial: $$x^{4} - 4 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 2 \beta_{1} - \beta_{2} ) q^{3} + \beta_{3} q^{5} + 3 \beta_{2} q^{7} + 7 q^{9} +O(q^{10})$$ $$q + ( 2 \beta_{1} - \beta_{2} ) q^{3} + \beta_{3} q^{5} + 3 \beta_{2} q^{7} + 7 q^{9} + 5 \beta_{2} q^{15} + 6 \beta_{3} q^{21} + ( -3 \beta_{1} + \beta_{2} ) q^{23} -5 q^{25} + ( 8 \beta_{1} - 4 \beta_{2} ) q^{27} -6 q^{29} + ( -6 \beta_{1} + 3 \beta_{2} ) q^{35} + 12 q^{41} -9 \beta_{2} q^{43} + 7 \beta_{3} q^{45} + ( 6 \beta_{1} - 3 \beta_{2} ) q^{47} -11 q^{49} -6 \beta_{3} q^{61} + 21 \beta_{2} q^{63} + 3 \beta_{2} q^{67} + ( -15 - \beta_{3} ) q^{69} + ( -10 \beta_{1} + 5 \beta_{2} ) q^{75} + 19 q^{81} + 11 \beta_{2} q^{83} + ( -12 \beta_{1} + 6 \beta_{2} ) q^{87} -8 \beta_{3} q^{89} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 28q^{9} + O(q^{10})$$ $$4q + 28q^{9} - 20q^{25} - 24q^{29} + 48q^{41} - 44q^{49} - 60q^{69} + 76q^{81} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 4 x^{2} + 9$$:

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

## 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
 −1.58114 + 0.707107i −1.58114 − 0.707107i 1.58114 − 0.707107i 1.58114 + 0.707107i
0 −3.16228 0 2.23607i 0 4.24264i 0 7.00000 0
1839.2 0 −3.16228 0 2.23607i 0 4.24264i 0 7.00000 0
1839.3 0 3.16228 0 2.23607i 0 4.24264i 0 7.00000 0
1839.4 0 3.16228 0 2.23607i 0 4.24264i 0 7.00000 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 CM by $$\Q(\sqrt{-5})$$
4.b odd 2 1 inner
5.b even 2 1 inner
23.b odd 2 1 inner
92.b even 2 1 inner
115.c 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.c 4
4.b odd 2 1 inner 1840.2.m.c 4
5.b even 2 1 inner 1840.2.m.c 4
20.d odd 2 1 CM 1840.2.m.c 4
23.b odd 2 1 inner 1840.2.m.c 4
92.b even 2 1 inner 1840.2.m.c 4
115.c odd 2 1 inner 1840.2.m.c 4
460.g even 2 1 inner 1840.2.m.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1840.2.m.c 4 1.a even 1 1 trivial
1840.2.m.c 4 4.b odd 2 1 inner
1840.2.m.c 4 5.b even 2 1 inner
1840.2.m.c 4 20.d odd 2 1 CM
1840.2.m.c 4 23.b odd 2 1 inner
1840.2.m.c 4 92.b even 2 1 inner
1840.2.m.c 4 115.c odd 2 1 inner
1840.2.m.c 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 $$S_{2}^{\mathrm{new}}(1840, [\chi])$$:

 $$T_{3}^{2} - 10$$ $$T_{7}^{2} + 18$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -10 + T^{2} )^{2}$$
$5$ $$( 5 + T^{2} )^{2}$$
$7$ $$( 18 + T^{2} )^{2}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$529 - 44 T^{2} + T^{4}$$
$29$ $$( 6 + T )^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$( -12 + T )^{4}$$
$43$ $$( 162 + T^{2} )^{2}$$
$47$ $$( -90 + T^{2} )^{2}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$( 180 + T^{2} )^{2}$$
$67$ $$( 18 + T^{2} )^{2}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$( 242 + T^{2} )^{2}$$
$89$ $$( 320 + T^{2} )^{2}$$
$97$ $$T^{4}$$