# Properties

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

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{-7})$$ Defining polynomial: $$x^{2} - x + 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 200) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{-7})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta ) q^{2} + ( -1 - \beta ) q^{4} + 4 q^{7} + ( 3 - \beta ) q^{8} +O(q^{10})$$ $$q + ( -1 + \beta ) q^{2} + ( -1 - \beta ) q^{4} + 4 q^{7} + ( 3 - \beta ) q^{8} + ( 1 - 2 \beta ) q^{11} + ( -4 + 4 \beta ) q^{14} + ( -1 + 3 \beta ) q^{16} + 3 q^{17} + ( -1 + 2 \beta ) q^{19} + ( 3 + \beta ) q^{22} -4 q^{23} + ( -4 - 4 \beta ) q^{28} + 4 q^{31} + ( -5 - \beta ) q^{32} + ( -3 + 3 \beta ) q^{34} + ( 4 - 8 \beta ) q^{37} + ( -3 - \beta ) q^{38} + 5 q^{41} + ( 2 - 4 \beta ) q^{43} + ( -5 + 3 \beta ) q^{44} + ( 4 - 4 \beta ) q^{46} + 8 q^{47} + 9 q^{49} + ( 4 - 8 \beta ) q^{53} + ( 12 - 4 \beta ) q^{56} + ( 2 - 4 \beta ) q^{59} + ( -4 + 8 \beta ) q^{61} + ( -4 + 4 \beta ) q^{62} + ( 7 - 5 \beta ) q^{64} + ( -3 + 6 \beta ) q^{67} + ( -3 - 3 \beta ) q^{68} -8 q^{71} -7 q^{73} + ( 12 + 4 \beta ) q^{74} + ( 5 - 3 \beta ) q^{76} + ( 4 - 8 \beta ) q^{77} + 4 q^{79} + ( -5 + 5 \beta ) q^{82} + ( 3 - 6 \beta ) q^{83} + ( 6 + 2 \beta ) q^{86} + ( -1 - 5 \beta ) q^{88} + q^{89} + ( 4 + 4 \beta ) q^{92} + ( -8 + 8 \beta ) q^{94} -2 q^{97} + ( -9 + 9 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - 3 q^{4} + 8 q^{7} + 5 q^{8} + O(q^{10})$$ $$2 q - q^{2} - 3 q^{4} + 8 q^{7} + 5 q^{8} - 4 q^{14} + q^{16} + 6 q^{17} + 7 q^{22} - 8 q^{23} - 12 q^{28} + 8 q^{31} - 11 q^{32} - 3 q^{34} - 7 q^{38} + 10 q^{41} - 7 q^{44} + 4 q^{46} + 16 q^{47} + 18 q^{49} + 20 q^{56} - 4 q^{62} + 9 q^{64} - 9 q^{68} - 16 q^{71} - 14 q^{73} + 28 q^{74} + 7 q^{76} + 8 q^{79} - 5 q^{82} + 14 q^{86} - 7 q^{88} + 2 q^{89} + 12 q^{92} - 8 q^{94} - 4 q^{97} - 9 q^{98} + O(q^{100})$$

## 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
 0.5 − 1.32288i 0.5 + 1.32288i
−0.500000 1.32288i 0 −1.50000 + 1.32288i 0 0 4.00000 2.50000 + 1.32288i 0 0
901.2 −0.500000 + 1.32288i 0 −1.50000 1.32288i 0 0 4.00000 2.50000 1.32288i 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
8.b 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.d 2
3.b odd 2 1 200.2.d.c yes 2
4.b odd 2 1 7200.2.k.b 2
5.b even 2 1 1800.2.k.f 2
5.c odd 4 2 1800.2.d.m 4
8.b even 2 1 inner 1800.2.k.d 2
8.d odd 2 1 7200.2.k.b 2
12.b even 2 1 800.2.d.a 2
15.d odd 2 1 200.2.d.b 2
15.e even 4 2 200.2.f.d 4
20.d odd 2 1 7200.2.k.i 2
20.e even 4 2 7200.2.d.m 4
24.f even 2 1 800.2.d.a 2
24.h odd 2 1 200.2.d.c yes 2
40.e odd 2 1 7200.2.k.i 2
40.f even 2 1 1800.2.k.f 2
40.i odd 4 2 1800.2.d.m 4
40.k even 4 2 7200.2.d.m 4
48.i odd 4 2 6400.2.a.bg 2
48.k even 4 2 6400.2.a.cb 2
60.h even 2 1 800.2.d.d 2
60.l odd 4 2 800.2.f.d 4
120.i odd 2 1 200.2.d.b 2
120.m even 2 1 800.2.d.d 2
120.q odd 4 2 800.2.f.d 4
120.w even 4 2 200.2.f.d 4
240.t even 4 2 6400.2.a.bh 2
240.bm odd 4 2 6400.2.a.cc 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.d.b 2 15.d odd 2 1
200.2.d.b 2 120.i odd 2 1
200.2.d.c yes 2 3.b odd 2 1
200.2.d.c yes 2 24.h odd 2 1
200.2.f.d 4 15.e even 4 2
200.2.f.d 4 120.w even 4 2
800.2.d.a 2 12.b even 2 1
800.2.d.a 2 24.f even 2 1
800.2.d.d 2 60.h even 2 1
800.2.d.d 2 120.m even 2 1
800.2.f.d 4 60.l odd 4 2
800.2.f.d 4 120.q odd 4 2
1800.2.d.m 4 5.c odd 4 2
1800.2.d.m 4 40.i odd 4 2
1800.2.k.d 2 1.a even 1 1 trivial
1800.2.k.d 2 8.b even 2 1 inner
1800.2.k.f 2 5.b even 2 1
1800.2.k.f 2 40.f even 2 1
6400.2.a.bg 2 48.i odd 4 2
6400.2.a.bh 2 240.t even 4 2
6400.2.a.cb 2 48.k even 4 2
6400.2.a.cc 2 240.bm odd 4 2
7200.2.d.m 4 20.e even 4 2
7200.2.d.m 4 40.k even 4 2
7200.2.k.b 2 4.b odd 2 1
7200.2.k.b 2 8.d odd 2 1
7200.2.k.i 2 20.d odd 2 1
7200.2.k.i 2 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 $$S_{2}^{\mathrm{new}}(1800, [\chi])$$:

 $$T_{7} - 4$$ $$T_{11}^{2} + 7$$ $$T_{17} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$( -4 + T )^{2}$$
$11$ $$7 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$( -3 + T )^{2}$$
$19$ $$7 + T^{2}$$
$23$ $$( 4 + T )^{2}$$
$29$ $$T^{2}$$
$31$ $$( -4 + T )^{2}$$
$37$ $$112 + T^{2}$$
$41$ $$( -5 + T )^{2}$$
$43$ $$28 + T^{2}$$
$47$ $$( -8 + T )^{2}$$
$53$ $$112 + T^{2}$$
$59$ $$28 + T^{2}$$
$61$ $$112 + T^{2}$$
$67$ $$63 + T^{2}$$
$71$ $$( 8 + T )^{2}$$
$73$ $$( 7 + T )^{2}$$
$79$ $$( -4 + T )^{2}$$
$83$ $$63 + T^{2}$$
$89$ $$( -1 + T )^{2}$$
$97$ $$( 2 + T )^{2}$$