# Properties

 Label 8400.2.a.bs Level $8400$ Weight $2$ Character orbit 8400.a Self dual yes Analytic conductor $67.074$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8400.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$67.0743376979$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 840) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{7} + q^{9} + O(q^{10})$$ $$q + q^{3} - q^{7} + q^{9} - 2q^{11} - 2q^{13} + 6q^{19} - q^{21} + q^{27} - 6q^{29} - 10q^{31} - 2q^{33} - 2q^{39} + 6q^{41} + 8q^{43} + 12q^{47} + q^{49} - 6q^{53} + 6q^{57} - 6q^{61} - q^{63} - 4q^{67} - 6q^{71} - 14q^{73} + 2q^{77} - 4q^{79} + q^{81} - 6q^{87} - 6q^{89} + 2q^{91} - 10q^{93} - 2q^{97} - 2q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 1.00000 0 0 0 −1.00000 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8400.2.a.bs 1
4.b odd 2 1 4200.2.a.j 1
5.b even 2 1 8400.2.a.v 1
5.c odd 4 2 1680.2.t.b 2
15.e even 4 2 5040.2.t.l 2
20.d odd 2 1 4200.2.a.w 1
20.e even 4 2 840.2.t.b 2
60.l odd 4 2 2520.2.t.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.t.b 2 20.e even 4 2
1680.2.t.b 2 5.c odd 4 2
2520.2.t.e 2 60.l odd 4 2
4200.2.a.j 1 4.b odd 2 1
4200.2.a.w 1 20.d odd 2 1
5040.2.t.l 2 15.e even 4 2
8400.2.a.v 1 5.b even 2 1
8400.2.a.bs 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8400))$$:

 $$T_{11} + 2$$ $$T_{13} + 2$$ $$T_{17}$$ $$T_{19} - 6$$ $$T_{23}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-1 + T$$
$5$ $$T$$
$7$ $$1 + T$$
$11$ $$2 + T$$
$13$ $$2 + T$$
$17$ $$T$$
$19$ $$-6 + T$$
$23$ $$T$$
$29$ $$6 + T$$
$31$ $$10 + T$$
$37$ $$T$$
$41$ $$-6 + T$$
$43$ $$-8 + T$$
$47$ $$-12 + T$$
$53$ $$6 + T$$
$59$ $$T$$
$61$ $$6 + T$$
$67$ $$4 + T$$
$71$ $$6 + T$$
$73$ $$14 + T$$
$79$ $$4 + T$$
$83$ $$T$$
$89$ $$6 + T$$
$97$ $$2 + T$$