# Properties

 Label 1800.2.k.l Level $1800$ Weight $2$ Character orbit 1800.k Analytic conductor $14.373$ Analytic rank $0$ Dimension $4$ CM discriminant -15 Inner twists $8$

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## 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: $$\Q(\sqrt{3}, \sqrt{-5})$$ Defining polynomial: $$x^{4} + x^{2} + 4$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 360) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a root $$\beta$$ of the polynomial $$x^{4} + x^{2} + 4$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} + \beta^{2} q^{4} -\beta^{3} q^{8} +O(q^{10})$$ $$q -\beta q^{2} + \beta^{2} q^{4} -\beta^{3} q^{8} + ( -4 - \beta^{2} ) q^{16} + ( 2 \beta - 2 \beta^{3} ) q^{17} + ( 2 + 4 \beta^{2} ) q^{19} + ( -\beta + \beta^{3} ) q^{23} -8 q^{31} + ( 4 \beta + \beta^{3} ) q^{32} + ( -8 - 4 \beta^{2} ) q^{34} + ( -2 \beta - 4 \beta^{3} ) q^{38} + ( 4 + 2 \beta^{2} ) q^{46} + ( 3 \beta - 3 \beta^{3} ) q^{47} -7 q^{49} + ( -3 \beta - \beta^{3} ) q^{53} + ( 4 + 8 \beta^{2} ) q^{61} + 8 \beta q^{62} + ( 4 - 3 \beta^{2} ) q^{64} + ( 8 \beta + 4 \beta^{3} ) q^{68} + ( -16 - 2 \beta^{2} ) q^{76} + 16 q^{79} + ( 12 \beta + 4 \beta^{3} ) q^{83} + ( -4 \beta - 2 \beta^{3} ) q^{92} + ( -12 - 6 \beta^{2} ) q^{94} + 7 \beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{4} + O(q^{10})$$ $$4 q - 2 q^{4} - 14 q^{16} - 32 q^{31} - 24 q^{34} + 12 q^{46} - 28 q^{49} + 22 q^{64} - 60 q^{76} + 64 q^{79} - 36 q^{94} + O(q^{100})$$

## 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
 0.866025 + 1.11803i 0.866025 − 1.11803i −0.866025 + 1.11803i −0.866025 − 1.11803i
−0.866025 1.11803i 0 −0.500000 + 1.93649i 0 0 0 2.59808 1.11803i 0 0
901.2 −0.866025 + 1.11803i 0 −0.500000 1.93649i 0 0 0 2.59808 + 1.11803i 0 0
901.3 0.866025 1.11803i 0 −0.500000 1.93649i 0 0 0 −2.59808 1.11803i 0 0
901.4 0.866025 + 1.11803i 0 −0.500000 + 1.93649i 0 0 0 −2.59808 + 1.11803i 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
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
3.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner
40.f even 2 1 inner
120.i odd 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.l 4
3.b odd 2 1 inner 1800.2.k.l 4
4.b odd 2 1 7200.2.k.n 4
5.b even 2 1 inner 1800.2.k.l 4
5.c odd 4 2 360.2.d.c 4
8.b even 2 1 inner 1800.2.k.l 4
8.d odd 2 1 7200.2.k.n 4
12.b even 2 1 7200.2.k.n 4
15.d odd 2 1 CM 1800.2.k.l 4
15.e even 4 2 360.2.d.c 4
20.d odd 2 1 7200.2.k.n 4
20.e even 4 2 1440.2.d.d 4
24.f even 2 1 7200.2.k.n 4
24.h odd 2 1 inner 1800.2.k.l 4
40.e odd 2 1 7200.2.k.n 4
40.f even 2 1 inner 1800.2.k.l 4
40.i odd 4 2 360.2.d.c 4
40.k even 4 2 1440.2.d.d 4
60.h even 2 1 7200.2.k.n 4
60.l odd 4 2 1440.2.d.d 4
120.i odd 2 1 inner 1800.2.k.l 4
120.m even 2 1 7200.2.k.n 4
120.q odd 4 2 1440.2.d.d 4
120.w even 4 2 360.2.d.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.d.c 4 5.c odd 4 2
360.2.d.c 4 15.e even 4 2
360.2.d.c 4 40.i odd 4 2
360.2.d.c 4 120.w even 4 2
1440.2.d.d 4 20.e even 4 2
1440.2.d.d 4 40.k even 4 2
1440.2.d.d 4 60.l odd 4 2
1440.2.d.d 4 120.q odd 4 2
1800.2.k.l 4 1.a even 1 1 trivial
1800.2.k.l 4 3.b odd 2 1 inner
1800.2.k.l 4 5.b even 2 1 inner
1800.2.k.l 4 8.b even 2 1 inner
1800.2.k.l 4 15.d odd 2 1 CM
1800.2.k.l 4 24.h odd 2 1 inner
1800.2.k.l 4 40.f even 2 1 inner
1800.2.k.l 4 120.i odd 2 1 inner
7200.2.k.n 4 4.b odd 2 1
7200.2.k.n 4 8.d odd 2 1
7200.2.k.n 4 12.b even 2 1
7200.2.k.n 4 20.d odd 2 1
7200.2.k.n 4 24.f even 2 1
7200.2.k.n 4 40.e odd 2 1
7200.2.k.n 4 60.h even 2 1
7200.2.k.n 4 120.m even 2 1

## Hecke kernels

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

 $$T_{7}$$ $$T_{11}$$ $$T_{17}^{2} - 48$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$( -48 + T^{2} )^{2}$$
$19$ $$( 60 + T^{2} )^{2}$$
$23$ $$( -12 + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$( 8 + T )^{4}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$( -108 + T^{2} )^{2}$$
$53$ $$( 20 + T^{2} )^{2}$$
$59$ $$T^{4}$$
$61$ $$( 240 + T^{2} )^{2}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$( -16 + T )^{4}$$
$83$ $$( 320 + T^{2} )^{2}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$
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