# Properties

 Label 8400.2.a.cb 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 1050) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{7} + q^{9} + O(q^{10})$$ $$q + q^{3} - q^{7} + q^{9} + 2q^{11} + 7q^{13} - 7q^{17} - 8q^{19} - q^{21} + 5q^{23} + q^{27} + 9q^{29} - q^{31} + 2q^{33} - 2q^{37} + 7q^{39} + 11q^{41} - 3q^{43} - 4q^{47} + q^{49} - 7q^{51} + 3q^{53} - 8q^{57} - 7q^{59} - 5q^{61} - q^{63} + 12q^{67} + 5q^{69} + 4q^{71} + 10q^{73} - 2q^{77} + 6q^{79} + q^{81} + 9q^{83} + 9q^{87} - 10q^{89} - 7q^{91} - q^{93} + 10q^{97} + 2q^{99} + O(q^{100})$$

## 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.cb 1
4.b odd 2 1 1050.2.a.n yes 1
5.b even 2 1 8400.2.a.bb 1
12.b even 2 1 3150.2.a.r 1
20.d odd 2 1 1050.2.a.f 1
20.e even 4 2 1050.2.g.b 2
28.d even 2 1 7350.2.a.cm 1
60.h even 2 1 3150.2.a.bd 1
60.l odd 4 2 3150.2.g.p 2
140.c even 2 1 7350.2.a.i 1

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

## 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8400))$$:

 $$T_{11} - 2$$ $$T_{13} - 7$$ $$T_{17} + 7$$ $$T_{19} + 8$$ $$T_{23} - 5$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 - T$$
$5$ 
$7$ $$1 + T$$
$11$ $$1 - 2 T + 11 T^{2}$$
$13$ $$1 - 7 T + 13 T^{2}$$
$17$ $$1 + 7 T + 17 T^{2}$$
$19$ $$1 + 8 T + 19 T^{2}$$
$23$ $$1 - 5 T + 23 T^{2}$$
$29$ $$1 - 9 T + 29 T^{2}$$
$31$ $$1 + T + 31 T^{2}$$
$37$ $$1 + 2 T + 37 T^{2}$$
$41$ $$1 - 11 T + 41 T^{2}$$
$43$ $$1 + 3 T + 43 T^{2}$$
$47$ $$1 + 4 T + 47 T^{2}$$
$53$ $$1 - 3 T + 53 T^{2}$$
$59$ $$1 + 7 T + 59 T^{2}$$
$61$ $$1 + 5 T + 61 T^{2}$$
$67$ $$1 - 12 T + 67 T^{2}$$
$71$ $$1 - 4 T + 71 T^{2}$$
$73$ $$1 - 10 T + 73 T^{2}$$
$79$ $$1 - 6 T + 79 T^{2}$$
$83$ $$1 - 9 T + 83 T^{2}$$
$89$ $$1 + 10 T + 89 T^{2}$$
$97$ $$1 - 10 T + 97 T^{2}$$