# Properties

 Label 5400.2.f.v 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 216) 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.

 $$f(q)$$ $$=$$ $$q + 3 i q^{7} +O(q^{10})$$ $$q + 3 i q^{7} + 4 q^{11} + i q^{13} + 4 i q^{17} + q^{19} + 4 i q^{23} -4 q^{31} + 9 i q^{37} -8 i q^{43} + 12 i q^{47} -2 q^{49} -8 i q^{53} -4 q^{59} -5 q^{61} -11 i q^{67} + 8 q^{71} + i q^{73} + 12 i q^{77} + 5 q^{79} + 8 i q^{83} -12 q^{89} -3 q^{91} -5 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + O(q^{10})$$ $$2 q + 8 q^{11} + 2 q^{19} - 8 q^{31} - 4 q^{49} - 8 q^{59} - 10 q^{61} + 16 q^{71} + 10 q^{79} - 24 q^{89} - 6 q^{91} + O(q^{100})$$

## 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 3.00000i 0 0 0
649.2 0 0 0 0 0 3.00000i 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.v 2
3.b odd 2 1 5400.2.f.e 2
5.b even 2 1 inner 5400.2.f.v 2
5.c odd 4 1 216.2.a.d yes 1
5.c odd 4 1 5400.2.a.bp 1
15.d odd 2 1 5400.2.f.e 2
15.e even 4 1 216.2.a.a 1
15.e even 4 1 5400.2.a.bn 1
20.e even 4 1 432.2.a.h 1
40.i odd 4 1 1728.2.a.a 1
40.k even 4 1 1728.2.a.b 1
45.k odd 12 2 648.2.i.a 2
45.l even 12 2 648.2.i.h 2
60.l odd 4 1 432.2.a.a 1
120.q odd 4 1 1728.2.a.bb 1
120.w even 4 1 1728.2.a.ba 1
180.v odd 12 2 1296.2.i.q 2
180.x even 12 2 1296.2.i.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.a.a 1 15.e even 4 1
216.2.a.d yes 1 5.c odd 4 1
432.2.a.a 1 60.l odd 4 1
432.2.a.h 1 20.e even 4 1
648.2.i.a 2 45.k odd 12 2
648.2.i.h 2 45.l even 12 2
1296.2.i.a 2 180.x even 12 2
1296.2.i.q 2 180.v odd 12 2
1728.2.a.a 1 40.i odd 4 1
1728.2.a.b 1 40.k even 4 1
1728.2.a.ba 1 120.w even 4 1
1728.2.a.bb 1 120.q odd 4 1
5400.2.a.bn 1 15.e even 4 1
5400.2.a.bp 1 5.c odd 4 1
5400.2.f.e 2 3.b odd 2 1
5400.2.f.e 2 15.d odd 2 1
5400.2.f.v 2 1.a even 1 1 trivial
5400.2.f.v 2 5.b even 2 1 inner

## Hecke kernels

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

 $$T_{7}^{2} + 9$$ $$T_{11} - 4$$ $$T_{13}^{2} + 1$$ $$T_{29}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$9 + T^{2}$$
$11$ $$( -4 + T )^{2}$$
$13$ $$1 + T^{2}$$
$17$ $$16 + T^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$16 + T^{2}$$
$29$ $$T^{2}$$
$31$ $$( 4 + T )^{2}$$
$37$ $$81 + T^{2}$$
$41$ $$T^{2}$$
$43$ $$64 + T^{2}$$
$47$ $$144 + T^{2}$$
$53$ $$64 + T^{2}$$
$59$ $$( 4 + T )^{2}$$
$61$ $$( 5 + T )^{2}$$
$67$ $$121 + T^{2}$$
$71$ $$( -8 + T )^{2}$$
$73$ $$1 + T^{2}$$
$79$ $$( -5 + T )^{2}$$
$83$ $$64 + T^{2}$$
$89$ $$( 12 + T )^{2}$$
$97$ $$25 + T^{2}$$