# Properties

 Label 8400.2.a.cd 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 420) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{7} + q^{9} + O(q^{10})$$ $$q + q^{3} - q^{7} + q^{9} + 4q^{11} - 6q^{13} + 2q^{17} - 6q^{19} - q^{21} - 2q^{23} + q^{27} + 6q^{29} + 2q^{31} + 4q^{33} - 4q^{37} - 6q^{39} + 8q^{41} + 4q^{43} - 4q^{47} + q^{49} + 2q^{51} + 6q^{53} - 6q^{57} - 4q^{59} + 14q^{61} - q^{63} - 4q^{67} - 2q^{69} - 10q^{73} - 4q^{77} + q^{81} + 16q^{83} + 6q^{87} + 8q^{89} + 6q^{91} + 2q^{93} + 10q^{97} + 4q^{99} + O(q^{100})$$

## 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.cd 1
4.b odd 2 1 2100.2.a.e 1
5.b even 2 1 8400.2.a.bh 1
5.c odd 4 2 1680.2.t.a 2
12.b even 2 1 6300.2.a.bc 1
15.e even 4 2 5040.2.t.o 2
20.d odd 2 1 2100.2.a.j 1
20.e even 4 2 420.2.k.a 2
60.h even 2 1 6300.2.a.n 1
60.l odd 4 2 1260.2.k.d 2
140.j odd 4 2 2940.2.k.d 2
140.w even 12 4 2940.2.bb.h 4
140.x odd 12 4 2940.2.bb.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.k.a 2 20.e even 4 2
1260.2.k.d 2 60.l odd 4 2
1680.2.t.a 2 5.c odd 4 2
2100.2.a.e 1 4.b odd 2 1
2100.2.a.j 1 20.d odd 2 1
2940.2.k.d 2 140.j odd 4 2
2940.2.bb.c 4 140.x odd 12 4
2940.2.bb.h 4 140.w even 12 4
5040.2.t.o 2 15.e even 4 2
6300.2.a.n 1 60.h even 2 1
6300.2.a.bc 1 12.b even 2 1
8400.2.a.bh 1 5.b even 2 1
8400.2.a.cd 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8400))$$:

 $$T_{11} - 4$$ $$T_{13} + 6$$ $$T_{17} - 2$$ $$T_{19} + 6$$ $$T_{23} + 2$$

## Hecke characteristic polynomials

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