Properties

 Label 5400.2.f.b 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.

 $$f(q)$$ $$=$$ $$q + 3 i q^{7} +O(q^{10})$$ $$q + 3 i q^{7} -5 q^{11} -4 i q^{13} + 8 i q^{17} -2 q^{19} + 2 i q^{23} + 6 q^{29} -7 q^{31} -6 i q^{37} + 6 q^{41} + 2 i q^{43} -6 i q^{47} -2 q^{49} + 5 i q^{53} -4 q^{59} -8 q^{61} -10 i q^{67} + 8 q^{71} -i q^{73} -15 i q^{77} -16 q^{79} -11 i q^{83} + 6 q^{89} + 12 q^{91} -i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q - 10q^{11} - 4q^{19} + 12q^{29} - 14q^{31} + 12q^{41} - 4q^{49} - 8q^{59} - 16q^{61} + 16q^{71} - 32q^{79} + 12q^{89} + 24q^{91} + O(q^{100})$$

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.b 2
3.b odd 2 1 5400.2.f.z 2
5.b even 2 1 inner 5400.2.f.b 2
5.c odd 4 1 216.2.a.c yes 1
5.c odd 4 1 5400.2.a.e 1
15.d odd 2 1 5400.2.f.z 2
15.e even 4 1 216.2.a.b 1
15.e even 4 1 5400.2.a.h 1
20.e even 4 1 432.2.a.f 1
40.i odd 4 1 1728.2.a.l 1
40.k even 4 1 1728.2.a.i 1
45.k odd 12 2 648.2.i.c 2
45.l even 12 2 648.2.i.e 2
60.l odd 4 1 432.2.a.c 1
120.q odd 4 1 1728.2.a.r 1
120.w even 4 1 1728.2.a.s 1
180.v odd 12 2 1296.2.i.l 2
180.x even 12 2 1296.2.i.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.a.b 1 15.e even 4 1
216.2.a.c yes 1 5.c odd 4 1
432.2.a.c 1 60.l odd 4 1
432.2.a.f 1 20.e even 4 1
648.2.i.c 2 45.k odd 12 2
648.2.i.e 2 45.l even 12 2
1296.2.i.g 2 180.x even 12 2
1296.2.i.l 2 180.v odd 12 2
1728.2.a.i 1 40.k even 4 1
1728.2.a.l 1 40.i odd 4 1
1728.2.a.r 1 120.q odd 4 1
1728.2.a.s 1 120.w even 4 1
5400.2.a.e 1 5.c odd 4 1
5400.2.a.h 1 15.e even 4 1
5400.2.f.b 2 1.a even 1 1 trivial
5400.2.f.b 2 5.b even 2 1 inner
5400.2.f.z 2 3.b odd 2 1
5400.2.f.z 2 15.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 $$S_{2}^{\mathrm{new}}(5400, [\chi])$$:

 $$T_{7}^{2} + 9$$ $$T_{11} + 5$$ $$T_{13}^{2} + 16$$ $$T_{29} - 6$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$9 + T^{2}$$
$11$ $$( 5 + T )^{2}$$
$13$ $$16 + T^{2}$$
$17$ $$64 + T^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$4 + T^{2}$$
$29$ $$( -6 + T )^{2}$$
$31$ $$( 7 + T )^{2}$$
$37$ $$36 + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$4 + T^{2}$$
$47$ $$36 + T^{2}$$
$53$ $$25 + T^{2}$$
$59$ $$( 4 + T )^{2}$$
$61$ $$( 8 + T )^{2}$$
$67$ $$100 + T^{2}$$
$71$ $$( -8 + T )^{2}$$
$73$ $$1 + T^{2}$$
$79$ $$( 16 + T )^{2}$$
$83$ $$121 + T^{2}$$
$89$ $$( -6 + T )^{2}$$
$97$ $$1 + T^{2}$$