# Properties

 Label 1800.2.k.s Level $1800$ Weight $2$ Character orbit 1800.k Analytic conductor $14.373$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Coefficient field: 8.0.212336640000.29 Defining polynomial: $$x^{8} + 2 x^{4} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{2} -\beta_{5} q^{4} + ( 1 + \beta_{2} - \beta_{5} ) q^{7} + \beta_{4} q^{8} +O(q^{10})$$ $$q + \beta_{6} q^{2} -\beta_{5} q^{4} + ( 1 + \beta_{2} - \beta_{5} ) q^{7} + \beta_{4} q^{8} + ( \beta_{3} + \beta_{4} ) q^{11} -\beta_{7} q^{13} + ( -\beta_{1} + \beta_{4} + \beta_{6} ) q^{14} + ( -1 - \beta_{7} ) q^{16} + ( \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{6} ) q^{17} + ( \beta_{2} + \beta_{5} + \beta_{7} ) q^{19} + ( -1 - 2 \beta_{2} - \beta_{7} ) q^{22} + ( -\beta_{3} + \beta_{4} ) q^{23} + ( 2 \beta_{3} - \beta_{6} ) q^{26} + ( 3 - \beta_{5} - \beta_{7} ) q^{28} + ( -\beta_{1} + \beta_{3} + \beta_{4} - 2 \beta_{6} ) q^{29} + ( 1 - \beta_{2} + \beta_{5} ) q^{31} + ( 2 \beta_{3} - 2 \beta_{6} ) q^{32} + ( -3 - 2 \beta_{2} + 2 \beta_{5} + \beta_{7} ) q^{34} + ( 2 \beta_{2} + 2 \beta_{5} ) q^{37} + ( -\beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{6} ) q^{38} + ( -\beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{6} ) q^{41} + ( \beta_{2} + \beta_{5} - \beta_{7} ) q^{43} + ( 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{6} ) q^{44} + ( -1 + 2 \beta_{2} - \beta_{7} ) q^{46} + ( -2 \beta_{1} + 4 \beta_{6} ) q^{47} + ( 2 \beta_{2} - 2 \beta_{5} ) q^{49} + ( -4 \beta_{2} + \beta_{5} ) q^{52} + ( -\beta_{1} - \beta_{3} - \beta_{4} - 2 \beta_{6} ) q^{53} + ( 2 \beta_{3} + \beta_{4} + 2 \beta_{6} ) q^{56} + ( 3 - 2 \beta_{2} + 2 \beta_{5} - \beta_{7} ) q^{58} + ( 2 \beta_{1} + 4 \beta_{6} ) q^{59} + ( -2 \beta_{2} - 2 \beta_{5} - \beta_{7} ) q^{61} + ( \beta_{1} - \beta_{4} + \beta_{6} ) q^{62} + ( -4 \beta_{2} + 2 \beta_{5} ) q^{64} + ( 3 \beta_{2} + 3 \beta_{5} - \beta_{7} ) q^{67} + ( 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} ) q^{68} + ( 2 \beta_{1} + 3 \beta_{3} - 3 \beta_{4} - 4 \beta_{6} ) q^{71} + ( 2 + 2 \beta_{2} - 2 \beta_{5} ) q^{73} + ( -2 \beta_{1} - 2 \beta_{4} ) q^{74} + ( 5 + 4 \beta_{2} - \beta_{5} + \beta_{7} ) q^{76} + ( \beta_{1} + 3 \beta_{3} + 3 \beta_{4} + 2 \beta_{6} ) q^{77} + ( -2 + 2 \beta_{2} - 2 \beta_{5} ) q^{79} + ( 5 - 2 \beta_{2} - 2 \beta_{5} + \beta_{7} ) q^{82} + ( 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{6} ) q^{83} + ( -\beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{6} ) q^{86} + ( -8 - 4 \beta_{2} + 2 \beta_{5} ) q^{88} + ( -2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{6} ) q^{89} + ( -3 \beta_{2} - 3 \beta_{5} - \beta_{7} ) q^{91} + ( -2 \beta_{1} + 2 \beta_{3} - 2 \beta_{6} ) q^{92} + ( 8 - 4 \beta_{5} ) q^{94} + ( 1 + 6 \beta_{2} - 6 \beta_{5} ) q^{97} + ( -2 \beta_{1} + 2 \beta_{4} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{7} + O(q^{10})$$ $$8 q + 8 q^{7} - 8 q^{16} - 8 q^{22} + 24 q^{28} + 8 q^{31} - 24 q^{34} - 8 q^{46} + 24 q^{58} + 16 q^{73} + 40 q^{76} - 16 q^{79} + 40 q^{82} - 64 q^{88} + 64 q^{94} + 8 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 2 x^{4} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{5} + 2 \nu$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{6} + 2 \nu^{2}$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} + 2 \nu^{3}$$$$)/8$$ $$\beta_{7}$$ $$=$$ $$\nu^{4} + 1$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{2}$$ $$\nu^{3}$$ $$=$$ $$\beta_{3}$$ $$\nu^{4}$$ $$=$$ $$\beta_{7} - 1$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{4} - \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$4 \beta_{5} - 2 \beta_{2}$$ $$\nu^{7}$$ $$=$$ $$8 \beta_{6} - 2 \beta_{3}$$

## 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
 1.26979 − 0.622597i 1.26979 + 0.622597i 0.622597 − 1.26979i 0.622597 + 1.26979i −0.622597 − 1.26979i −0.622597 + 1.26979i −1.26979 − 0.622597i −1.26979 + 0.622597i
−1.26979 0.622597i 0 1.22474 + 1.58114i 0 0 3.44949 −0.570759 2.77024i 0 0
901.2 −1.26979 + 0.622597i 0 1.22474 1.58114i 0 0 3.44949 −0.570759 + 2.77024i 0 0
901.3 −0.622597 1.26979i 0 −1.22474 + 1.58114i 0 0 −1.44949 2.77024 + 0.570759i 0 0
901.4 −0.622597 + 1.26979i 0 −1.22474 1.58114i 0 0 −1.44949 2.77024 0.570759i 0 0
901.5 0.622597 1.26979i 0 −1.22474 1.58114i 0 0 −1.44949 −2.77024 + 0.570759i 0 0
901.6 0.622597 + 1.26979i 0 −1.22474 + 1.58114i 0 0 −1.44949 −2.77024 0.570759i 0 0
901.7 1.26979 0.622597i 0 1.22474 1.58114i 0 0 3.44949 0.570759 2.77024i 0 0
901.8 1.26979 + 0.622597i 0 1.22474 + 1.58114i 0 0 3.44949 0.570759 + 2.77024i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 901.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h 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.s yes 8
3.b odd 2 1 inner 1800.2.k.s yes 8
4.b odd 2 1 7200.2.k.q 8
5.b even 2 1 1800.2.k.r 8
5.c odd 4 2 1800.2.d.u 16
8.b even 2 1 inner 1800.2.k.s yes 8
8.d odd 2 1 7200.2.k.q 8
12.b even 2 1 7200.2.k.q 8
15.d odd 2 1 1800.2.k.r 8
15.e even 4 2 1800.2.d.u 16
20.d odd 2 1 7200.2.k.t 8
20.e even 4 2 7200.2.d.u 16
24.f even 2 1 7200.2.k.q 8
24.h odd 2 1 inner 1800.2.k.s yes 8
40.e odd 2 1 7200.2.k.t 8
40.f even 2 1 1800.2.k.r 8
40.i odd 4 2 1800.2.d.u 16
40.k even 4 2 7200.2.d.u 16
60.h even 2 1 7200.2.k.t 8
60.l odd 4 2 7200.2.d.u 16
120.i odd 2 1 1800.2.k.r 8
120.m even 2 1 7200.2.k.t 8
120.q odd 4 2 7200.2.d.u 16
120.w even 4 2 1800.2.d.u 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1800.2.d.u 16 5.c odd 4 2
1800.2.d.u 16 15.e even 4 2
1800.2.d.u 16 40.i odd 4 2
1800.2.d.u 16 120.w even 4 2
1800.2.k.r 8 5.b even 2 1
1800.2.k.r 8 15.d odd 2 1
1800.2.k.r 8 40.f even 2 1
1800.2.k.r 8 120.i odd 2 1
1800.2.k.s yes 8 1.a even 1 1 trivial
1800.2.k.s yes 8 3.b odd 2 1 inner
1800.2.k.s yes 8 8.b even 2 1 inner
1800.2.k.s yes 8 24.h odd 2 1 inner
7200.2.d.u 16 20.e even 4 2
7200.2.d.u 16 40.k even 4 2
7200.2.d.u 16 60.l odd 4 2
7200.2.d.u 16 120.q odd 4 2
7200.2.k.q 8 4.b odd 2 1
7200.2.k.q 8 8.d odd 2 1
7200.2.k.q 8 12.b even 2 1
7200.2.k.q 8 24.f even 2 1
7200.2.k.t 8 20.d odd 2 1
7200.2.k.t 8 40.e odd 2 1
7200.2.k.t 8 60.h even 2 1
7200.2.k.t 8 120.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 $$S_{2}^{\mathrm{new}}(1800, [\chi])$$:

 $$T_{7}^{2} - 2 T_{7} - 5$$ $$T_{11}^{4} + 32 T_{11}^{2} + 40$$ $$T_{17}^{4} - 48 T_{17}^{2} + 360$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + 2 T^{4} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$( -5 - 2 T + T^{2} )^{4}$$
$11$ $$( 40 + 32 T^{2} + T^{4} )^{2}$$
$13$ $$( 15 + T^{2} )^{4}$$
$17$ $$( 360 - 48 T^{2} + T^{4} )^{2}$$
$19$ $$( 25 + 50 T^{2} + T^{4} )^{2}$$
$23$ $$( 40 - 32 T^{2} + T^{4} )^{2}$$
$29$ $$( 360 + 48 T^{2} + T^{4} )^{2}$$
$31$ $$( -5 - 2 T + T^{2} )^{4}$$
$37$ $$( 40 + T^{2} )^{4}$$
$41$ $$( 1000 - 80 T^{2} + T^{4} )^{2}$$
$43$ $$( 25 + 50 T^{2} + T^{4} )^{2}$$
$47$ $$( 2560 - 128 T^{2} + T^{4} )^{2}$$
$53$ $$( 1000 + 80 T^{2} + T^{4} )^{2}$$
$59$ $$( 2560 + 128 T^{2} + T^{4} )^{2}$$
$61$ $$( 625 + 110 T^{2} + T^{4} )^{2}$$
$67$ $$( 5625 + 210 T^{2} + T^{4} )^{2}$$
$71$ $$( 25000 - 320 T^{2} + T^{4} )^{2}$$
$73$ $$( -20 - 4 T + T^{2} )^{4}$$
$79$ $$( -20 + 4 T + T^{2} )^{4}$$
$83$ $$( 5760 + 192 T^{2} + T^{4} )^{2}$$
$89$ $$( 16000 - 320 T^{2} + T^{4} )^{2}$$
$97$ $$( -215 - 2 T + T^{2} )^{4}$$