# Properties

 Label 1800.2.k.m Level $1800$ Weight $2$ Character orbit 1800.k Analytic conductor $14.373$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1800 = 2^{3} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1800.k (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.3730723638$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 40) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( 1 + \beta_{3} ) q^{4} + ( -2 \beta_{1} + \beta_{2} ) q^{7} -2 \beta_{2} q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( 1 + \beta_{3} ) q^{4} + ( -2 \beta_{1} + \beta_{2} ) q^{7} -2 \beta_{2} q^{8} + 2 \beta_{3} q^{11} + ( 3 + \beta_{3} ) q^{14} + ( -2 + 2 \beta_{3} ) q^{16} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{17} -2 \beta_{3} q^{19} + ( 2 \beta_{1} - 4 \beta_{2} ) q^{22} + ( 2 \beta_{1} - \beta_{2} ) q^{23} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{28} + 4 q^{31} + ( 4 \beta_{1} - 4 \beta_{2} ) q^{32} + ( 6 + 2 \beta_{3} ) q^{34} -6 \beta_{2} q^{37} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{38} -3 \beta_{2} q^{43} + ( -6 + 2 \beta_{3} ) q^{44} + ( -3 - \beta_{3} ) q^{46} + ( 6 \beta_{1} - 3 \beta_{2} ) q^{47} - q^{49} + 4 \beta_{2} q^{53} + 4 \beta_{3} q^{56} -6 \beta_{3} q^{59} + 2 \beta_{3} q^{61} -4 \beta_{1} q^{62} -8 q^{64} -3 \beta_{2} q^{67} + ( -4 \beta_{1} - 4 \beta_{2} ) q^{68} + 12 q^{71} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{73} + ( -6 + 6 \beta_{3} ) q^{74} + ( 6 - 2 \beta_{3} ) q^{76} -6 \beta_{2} q^{77} + 4 q^{79} -7 \beta_{2} q^{83} + ( -3 + 3 \beta_{3} ) q^{86} + ( 8 \beta_{1} - 4 \beta_{2} ) q^{88} + 6 q^{89} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{92} + ( -9 - 3 \beta_{3} ) q^{94} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{97} + \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{4} + O(q^{10})$$ $$4 q + 4 q^{4} + 12 q^{14} - 8 q^{16} + 16 q^{31} + 24 q^{34} - 24 q^{44} - 12 q^{46} - 4 q^{49} - 32 q^{64} + 48 q^{71} - 24 q^{74} + 24 q^{76} + 16 q^{79} - 12 q^{86} + 24 q^{89} - 36 q^{94} + O(q^{100})$$

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

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$1001$$ $$1351$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
901.1
 1.22474 + 0.707107i 1.22474 − 0.707107i −1.22474 + 0.707107i −1.22474 − 0.707107i
−1.22474 0.707107i 0 1.00000 + 1.73205i 0 0 −2.44949 2.82843i 0 0
901.2 −1.22474 + 0.707107i 0 1.00000 1.73205i 0 0 −2.44949 2.82843i 0 0
901.3 1.22474 0.707107i 0 1.00000 1.73205i 0 0 2.44949 2.82843i 0 0
901.4 1.22474 + 0.707107i 0 1.00000 + 1.73205i 0 0 2.44949 2.82843i 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
5.b even 2 1 inner
8.b even 2 1 inner
40.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1800.2.k.m 4
3.b odd 2 1 200.2.d.e 4
4.b odd 2 1 7200.2.k.l 4
5.b even 2 1 inner 1800.2.k.m 4
5.c odd 4 2 360.2.d.b 4
8.b even 2 1 inner 1800.2.k.m 4
8.d odd 2 1 7200.2.k.l 4
12.b even 2 1 800.2.d.f 4
15.d odd 2 1 200.2.d.e 4
15.e even 4 2 40.2.f.a 4
20.d odd 2 1 7200.2.k.l 4
20.e even 4 2 1440.2.d.c 4
24.f even 2 1 800.2.d.f 4
24.h odd 2 1 200.2.d.e 4
40.e odd 2 1 7200.2.k.l 4
40.f even 2 1 inner 1800.2.k.m 4
40.i odd 4 2 360.2.d.b 4
40.k even 4 2 1440.2.d.c 4
48.i odd 4 2 6400.2.a.co 4
48.k even 4 2 6400.2.a.cm 4
60.h even 2 1 800.2.d.f 4
60.l odd 4 2 160.2.f.a 4
120.i odd 2 1 200.2.d.e 4
120.m even 2 1 800.2.d.f 4
120.q odd 4 2 160.2.f.a 4
120.w even 4 2 40.2.f.a 4
240.t even 4 2 6400.2.a.cm 4
240.z odd 4 2 1280.2.c.k 4
240.bb even 4 2 1280.2.c.i 4
240.bd odd 4 2 1280.2.c.k 4
240.bf even 4 2 1280.2.c.i 4
240.bm odd 4 2 6400.2.a.co 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.f.a 4 15.e even 4 2
40.2.f.a 4 120.w even 4 2
160.2.f.a 4 60.l odd 4 2
160.2.f.a 4 120.q odd 4 2
200.2.d.e 4 3.b odd 2 1
200.2.d.e 4 15.d odd 2 1
200.2.d.e 4 24.h odd 2 1
200.2.d.e 4 120.i odd 2 1
360.2.d.b 4 5.c odd 4 2
360.2.d.b 4 40.i odd 4 2
800.2.d.f 4 12.b even 2 1
800.2.d.f 4 24.f even 2 1
800.2.d.f 4 60.h even 2 1
800.2.d.f 4 120.m even 2 1
1280.2.c.i 4 240.bb even 4 2
1280.2.c.i 4 240.bf even 4 2
1280.2.c.k 4 240.z odd 4 2
1280.2.c.k 4 240.bd odd 4 2
1440.2.d.c 4 20.e even 4 2
1440.2.d.c 4 40.k even 4 2
1800.2.k.m 4 1.a even 1 1 trivial
1800.2.k.m 4 5.b even 2 1 inner
1800.2.k.m 4 8.b even 2 1 inner
1800.2.k.m 4 40.f even 2 1 inner
6400.2.a.cm 4 48.k even 4 2
6400.2.a.cm 4 240.t even 4 2
6400.2.a.co 4 48.i odd 4 2
6400.2.a.co 4 240.bm odd 4 2
7200.2.k.l 4 4.b odd 2 1
7200.2.k.l 4 8.d odd 2 1
7200.2.k.l 4 20.d odd 2 1
7200.2.k.l 4 40.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1800, [\chi])$$:

 $$T_{7}^{2} - 6$$ $$T_{11}^{2} + 12$$ $$T_{17}^{2} - 24$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 - 2 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$( -6 + T^{2} )^{2}$$
$11$ $$( 12 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$( -24 + T^{2} )^{2}$$
$19$ $$( 12 + T^{2} )^{2}$$
$23$ $$( -6 + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$( -4 + T )^{4}$$
$37$ $$( 72 + T^{2} )^{2}$$
$41$ $$T^{4}$$
$43$ $$( 18 + T^{2} )^{2}$$
$47$ $$( -54 + T^{2} )^{2}$$
$53$ $$( 32 + T^{2} )^{2}$$
$59$ $$( 108 + T^{2} )^{2}$$
$61$ $$( 12 + T^{2} )^{2}$$
$67$ $$( 18 + T^{2} )^{2}$$
$71$ $$( -12 + T )^{4}$$
$73$ $$( -24 + T^{2} )^{2}$$
$79$ $$( -4 + T )^{4}$$
$83$ $$( 98 + T^{2} )^{2}$$
$89$ $$( -6 + T )^{4}$$
$97$ $$( -24 + T^{2} )^{2}$$