# Properties

 Label 1800.2.k.j Level $1800$ Weight $2$ Character orbit 1800.k Analytic conductor $14.373$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ 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 primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{2} + 2 \zeta_{12} q^{4} + ( 1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{7} + ( -2 - 2 \zeta_{12}^{3} ) q^{8} +O(q^{10})$$ $$q + ( -\zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{2} + 2 \zeta_{12} q^{4} + ( 1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{7} + ( -2 - 2 \zeta_{12}^{3} ) q^{8} + 2 \zeta_{12}^{3} q^{11} + ( -2 + 4 \zeta_{12}^{2} ) q^{13} + ( 2 - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{14} + 4 \zeta_{12}^{2} q^{16} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{17} + ( 2 - 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{19} + ( 2 \zeta_{12} - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{22} + ( -1 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{23} + ( 4 - 2 \zeta_{12} - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{26} + ( -2 + 2 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{28} + ( 4 - 8 \zeta_{12}^{2} ) q^{29} + ( -2 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{31} + ( 4 - 4 \zeta_{12} - 4 \zeta_{12}^{2} ) q^{32} + ( -4 - 2 \zeta_{12} + 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{34} + 2 \zeta_{12}^{3} q^{37} + ( -4 + 6 \zeta_{12} - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{38} + ( 2 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{41} + ( 1 - 2 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{43} + ( -4 + 4 \zeta_{12}^{2} ) q^{44} + ( 6 + 4 \zeta_{12} - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{46} + ( 5 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{47} + ( -3 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{49} + ( -4 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{52} + ( -2 + 4 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{53} + ( -4 + 4 \zeta_{12} + 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{56} + ( -8 + 4 \zeta_{12} + 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{58} + ( -2 + 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{59} + ( 4 - 8 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{61} + ( 4 + 4 \zeta_{12} ) q^{62} + 8 \zeta_{12}^{3} q^{64} + ( 1 - 2 \zeta_{12}^{2} - 9 \zeta_{12}^{3} ) q^{67} + ( 4 + 4 \zeta_{12}^{2} ) q^{68} + ( -2 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{71} + ( -4 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{73} + ( 2 \zeta_{12} - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{74} + ( -8 + 4 \zeta_{12} + 8 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{76} + ( 2 - 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{77} + ( -8 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{79} + ( 4 - 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{82} + ( 1 - 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{83} + ( -2 + 8 \zeta_{12} - 6 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{86} + ( 4 - 4 \zeta_{12}^{3} ) q^{88} + ( -2 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{89} + ( -2 + 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{91} + ( -6 - 2 \zeta_{12} - 6 \zeta_{12}^{2} ) q^{92} + ( 2 - 4 \zeta_{12} - 6 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{94} + ( 4 + 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{97} + ( 4 + 5 \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} + 4q^{7} - 8q^{8} + O(q^{10})$$ $$4q - 2q^{2} + 4q^{7} - 8q^{8} + 4q^{14} + 8q^{16} - 4q^{22} - 4q^{23} + 12q^{26} - 12q^{28} - 8q^{31} + 8q^{32} - 12q^{34} - 20q^{38} + 8q^{41} - 8q^{44} + 20q^{46} + 20q^{47} - 12q^{49} - 8q^{56} - 24q^{58} + 16q^{62} + 24q^{68} - 8q^{71} - 16q^{73} - 4q^{74} - 16q^{76} - 32q^{79} + 8q^{82} - 20q^{86} + 16q^{88} - 8q^{89} - 36q^{92} - 4q^{94} + 16q^{97} + 18q^{98} + O(q^{100})$$

## 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
 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i −0.866025 + 0.500000i
−1.36603 0.366025i 0 1.73205 + 1.00000i 0 0 −0.732051 −2.00000 2.00000i 0 0
901.2 −1.36603 + 0.366025i 0 1.73205 1.00000i 0 0 −0.732051 −2.00000 + 2.00000i 0 0
901.3 0.366025 1.36603i 0 −1.73205 1.00000i 0 0 2.73205 −2.00000 + 2.00000i 0 0
901.4 0.366025 + 1.36603i 0 −1.73205 + 1.00000i 0 0 2.73205 −2.00000 2.00000i 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.j 4
3.b odd 2 1 200.2.d.f 4
4.b odd 2 1 7200.2.k.j 4
5.b even 2 1 360.2.k.e 4
5.c odd 4 1 1800.2.d.l 4
5.c odd 4 1 1800.2.d.p 4
8.b even 2 1 inner 1800.2.k.j 4
8.d odd 2 1 7200.2.k.j 4
12.b even 2 1 800.2.d.e 4
15.d odd 2 1 40.2.d.a 4
15.e even 4 1 200.2.f.c 4
15.e even 4 1 200.2.f.e 4
20.d odd 2 1 1440.2.k.e 4
20.e even 4 1 7200.2.d.n 4
20.e even 4 1 7200.2.d.o 4
24.f even 2 1 800.2.d.e 4
24.h odd 2 1 200.2.d.f 4
40.e odd 2 1 1440.2.k.e 4
40.f even 2 1 360.2.k.e 4
40.i odd 4 1 1800.2.d.l 4
40.i odd 4 1 1800.2.d.p 4
40.k even 4 1 7200.2.d.n 4
40.k even 4 1 7200.2.d.o 4
48.i odd 4 1 6400.2.a.z 2
48.i odd 4 1 6400.2.a.ce 2
48.k even 4 1 6400.2.a.be 2
48.k even 4 1 6400.2.a.cj 2
60.h even 2 1 160.2.d.a 4
60.l odd 4 1 800.2.f.c 4
60.l odd 4 1 800.2.f.e 4
120.i odd 2 1 40.2.d.a 4
120.m even 2 1 160.2.d.a 4
120.q odd 4 1 800.2.f.c 4
120.q odd 4 1 800.2.f.e 4
120.w even 4 1 200.2.f.c 4
120.w even 4 1 200.2.f.e 4
240.t even 4 1 1280.2.a.d 2
240.t even 4 1 1280.2.a.n 2
240.bm odd 4 1 1280.2.a.a 2
240.bm odd 4 1 1280.2.a.o 2

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

## Hecke kernels

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

 $$T_{7}^{2} - 2 T_{7} - 2$$ $$T_{11}^{2} + 4$$ $$T_{17}^{2} - 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 4 T + 2 T^{2} + 2 T^{3} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$( -2 - 2 T + T^{2} )^{2}$$
$11$ $$( 4 + T^{2} )^{2}$$
$13$ $$( 12 + T^{2} )^{2}$$
$17$ $$( -12 + T^{2} )^{2}$$
$19$ $$16 + 56 T^{2} + T^{4}$$
$23$ $$( -26 + 2 T + T^{2} )^{2}$$
$29$ $$( 48 + T^{2} )^{2}$$
$31$ $$( -8 + 4 T + T^{2} )^{2}$$
$37$ $$( 4 + T^{2} )^{2}$$
$41$ $$( -8 - 4 T + T^{2} )^{2}$$
$43$ $$2116 + 104 T^{2} + T^{4}$$
$47$ $$( 22 - 10 T + T^{2} )^{2}$$
$53$ $$2704 + 152 T^{2} + T^{4}$$
$59$ $$16 + 56 T^{2} + T^{4}$$
$61$ $$1936 + 104 T^{2} + T^{4}$$
$67$ $$6084 + 168 T^{2} + T^{4}$$
$71$ $$( -8 + 4 T + T^{2} )^{2}$$
$73$ $$( 4 + 8 T + T^{2} )^{2}$$
$79$ $$( 16 + 16 T + T^{2} )^{2}$$
$83$ $$36 + 24 T^{2} + T^{4}$$
$89$ $$( -44 + 4 T + T^{2} )^{2}$$
$97$ $$( -92 - 8 T + T^{2} )^{2}$$