# Properties

 Label 5400.2.f.f Level $5400$ Weight $2$ Character orbit 5400.f Analytic conductor $43.119$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5400 = 2^{3} \cdot 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5400.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$43.1192170915$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1080) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 i q^{7} +O(q^{10})$$ $$q + 2 i q^{7} -4 q^{11} + 2 i q^{13} -5 i q^{17} + 5 q^{19} + i q^{23} -2 q^{29} + 7 q^{31} -6 i q^{37} -4 i q^{43} -4 i q^{47} + 3 q^{49} + 9 i q^{53} + 14 q^{59} -11 q^{61} + 14 i q^{67} + 12 i q^{73} -8 i q^{77} + 3 q^{79} -i q^{83} -4 q^{91} + 16 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q - 8q^{11} + 10q^{19} - 4q^{29} + 14q^{31} + 6q^{49} + 28q^{59} - 22q^{61} + 6q^{79} - 8q^{91} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5400\mathbb{Z}\right)^\times$$.

 $$n$$ $$1001$$ $$1351$$ $$2377$$ $$2701$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
649.1
 − 1.00000i 1.00000i
0 0 0 0 0 2.00000i 0 0 0
649.2 0 0 0 0 0 2.00000i 0 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5400.2.f.f 2
3.b odd 2 1 5400.2.f.x 2
5.b even 2 1 inner 5400.2.f.f 2
5.c odd 4 1 1080.2.a.e 1
5.c odd 4 1 5400.2.a.j 1
15.d odd 2 1 5400.2.f.x 2
15.e even 4 1 1080.2.a.l yes 1
15.e even 4 1 5400.2.a.q 1
20.e even 4 1 2160.2.a.e 1
40.i odd 4 1 8640.2.a.cd 1
40.k even 4 1 8640.2.a.bi 1
45.k odd 12 2 3240.2.q.p 2
45.l even 12 2 3240.2.q.b 2
60.l odd 4 1 2160.2.a.m 1
120.q odd 4 1 8640.2.a.k 1
120.w even 4 1 8640.2.a.t 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.a.e 1 5.c odd 4 1
1080.2.a.l yes 1 15.e even 4 1
2160.2.a.e 1 20.e even 4 1
2160.2.a.m 1 60.l odd 4 1
3240.2.q.b 2 45.l even 12 2
3240.2.q.p 2 45.k odd 12 2
5400.2.a.j 1 5.c odd 4 1
5400.2.a.q 1 15.e even 4 1
5400.2.f.f 2 1.a even 1 1 trivial
5400.2.f.f 2 5.b even 2 1 inner
5400.2.f.x 2 3.b odd 2 1
5400.2.f.x 2 15.d odd 2 1
8640.2.a.k 1 120.q odd 4 1
8640.2.a.t 1 120.w even 4 1
8640.2.a.bi 1 40.k even 4 1
8640.2.a.cd 1 40.i odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(5400, [\chi])$$:

 $$T_{7}^{2} + 4$$ $$T_{11} + 4$$ $$T_{13}^{2} + 4$$ $$T_{29} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$4 + T^{2}$$
$11$ $$( 4 + T )^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$25 + T^{2}$$
$19$ $$( -5 + T )^{2}$$
$23$ $$1 + T^{2}$$
$29$ $$( 2 + T )^{2}$$
$31$ $$( -7 + T )^{2}$$
$37$ $$36 + T^{2}$$
$41$ $$T^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$16 + T^{2}$$
$53$ $$81 + T^{2}$$
$59$ $$( -14 + T )^{2}$$
$61$ $$( 11 + T )^{2}$$
$67$ $$196 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$144 + T^{2}$$
$79$ $$( -3 + T )^{2}$$
$83$ $$1 + T^{2}$$
$89$ $$T^{2}$$
$97$ $$256 + T^{2}$$