# Properties

 Label 8400.2.a.d 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

# Learn more about

## 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 1050) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} - q^{7} + q^{9} + O(q^{10})$$ $$q - q^{3} - q^{7} + q^{9} - 2q^{11} - q^{13} + 3q^{17} + q^{21} + q^{23} - q^{27} - 5q^{29} - 7q^{31} + 2q^{33} - 2q^{37} + q^{39} + 7q^{41} + 11q^{43} - 8q^{47} + q^{49} - 3q^{51} - q^{53} + 5q^{59} - 3q^{61} - q^{63} + 12q^{67} - q^{69} - 12q^{71} - 6q^{73} + 2q^{77} - 10q^{79} + q^{81} + 11q^{83} + 5q^{87} - 10q^{89} + q^{91} + 7q^{93} - 2q^{97} - 2q^{99} + O(q^{100})$$

## 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

## 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.d 1
4.b odd 2 1 1050.2.a.r yes 1
5.b even 2 1 8400.2.a.ck 1
12.b even 2 1 3150.2.a.n 1
20.d odd 2 1 1050.2.a.b 1
20.e even 4 2 1050.2.g.i 2
28.d even 2 1 7350.2.a.cb 1
60.h even 2 1 3150.2.a.y 1
60.l odd 4 2 3150.2.g.h 2
140.c even 2 1 7350.2.a.bi 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.2.a.b 1 20.d odd 2 1
1050.2.a.r yes 1 4.b odd 2 1
1050.2.g.i 2 20.e even 4 2
3150.2.a.n 1 12.b even 2 1
3150.2.a.y 1 60.h even 2 1
3150.2.g.h 2 60.l odd 4 2
7350.2.a.bi 1 140.c even 2 1
7350.2.a.cb 1 28.d even 2 1
8400.2.a.d 1 1.a even 1 1 trivial
8400.2.a.ck 1 5.b even 2 1

## Atkin-Lehner signs

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

## Hecke kernels

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

 $$T_{11} + 2$$ $$T_{13} + 1$$ $$T_{17} - 3$$ $$T_{19}$$ $$T_{23} - 1$$

## Hecke Characteristic Polynomials

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