# Properties

 Label 8400.2.a.ca Level $8400$ Weight $2$ Character orbit 8400.a Self dual yes Analytic conductor $67.074$ Analytic rank $0$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 210) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{7} + q^{9} + O(q^{10})$$ $$q + q^{3} - q^{7} + q^{9} + 2 q^{11} + 2 q^{13} + 8 q^{17} + 2 q^{19} - q^{21} + q^{27} - 6 q^{29} - 6 q^{31} + 2 q^{33} + 8 q^{37} + 2 q^{39} + 6 q^{41} - 8 q^{43} - 4 q^{47} + q^{49} + 8 q^{51} - 2 q^{53} + 2 q^{57} + 8 q^{59} + 10 q^{61} - q^{63} + 12 q^{67} + 14 q^{71} - 10 q^{73} - 2 q^{77} - 4 q^{79} + q^{81} - 16 q^{83} - 6 q^{87} + 10 q^{89} - 2 q^{91} - 6 q^{93} + 10 q^{97} + 2 q^{99} + O(q^{100})$$

## 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.ca 1
4.b odd 2 1 1050.2.a.m 1
5.b even 2 1 8400.2.a.bd 1
5.c odd 4 2 1680.2.t.d 2
12.b even 2 1 3150.2.a.q 1
15.e even 4 2 5040.2.t.k 2
20.d odd 2 1 1050.2.a.g 1
20.e even 4 2 210.2.g.a 2
28.d even 2 1 7350.2.a.co 1
60.h even 2 1 3150.2.a.be 1
60.l odd 4 2 630.2.g.d 2
140.c even 2 1 7350.2.a.g 1
140.j odd 4 2 1470.2.g.e 2
140.w even 12 4 1470.2.n.g 4
140.x odd 12 4 1470.2.n.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.g.a 2 20.e even 4 2
630.2.g.d 2 60.l odd 4 2
1050.2.a.g 1 20.d odd 2 1
1050.2.a.m 1 4.b odd 2 1
1470.2.g.e 2 140.j odd 4 2
1470.2.n.c 4 140.x odd 12 4
1470.2.n.g 4 140.w even 12 4
1680.2.t.d 2 5.c odd 4 2
3150.2.a.q 1 12.b even 2 1
3150.2.a.be 1 60.h even 2 1
5040.2.t.k 2 15.e even 4 2
7350.2.a.g 1 140.c even 2 1
7350.2.a.co 1 28.d even 2 1
8400.2.a.bd 1 5.b even 2 1
8400.2.a.ca 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8400))$$:

 $$T_{11} - 2$$ $$T_{13} - 2$$ $$T_{17} - 8$$ $$T_{19} - 2$$ $$T_{23}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-1 + T$$
$5$ $$T$$
$7$ $$1 + T$$
$11$ $$-2 + T$$
$13$ $$-2 + T$$
$17$ $$-8 + T$$
$19$ $$-2 + T$$
$23$ $$T$$
$29$ $$6 + T$$
$31$ $$6 + T$$
$37$ $$-8 + T$$
$41$ $$-6 + T$$
$43$ $$8 + T$$
$47$ $$4 + T$$
$53$ $$2 + T$$
$59$ $$-8 + T$$
$61$ $$-10 + T$$
$67$ $$-12 + T$$
$71$ $$-14 + T$$
$73$ $$10 + T$$
$79$ $$4 + T$$
$83$ $$16 + T$$
$89$ $$-10 + T$$
$97$ $$-10 + T$$