# Properties

 Label 8400.2.a.co 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 105) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{7} + q^{9} + O(q^{10})$$ $$q + q^{3} + q^{7} + q^{9} + 6q^{13} - 2q^{17} + 8q^{19} + q^{21} + 8q^{23} + q^{27} - 2q^{29} - 4q^{31} + 2q^{37} + 6q^{39} - 6q^{41} + 4q^{43} + 8q^{47} + q^{49} - 2q^{51} - 10q^{53} + 8q^{57} - 4q^{59} - 2q^{61} + q^{63} + 4q^{67} + 8q^{69} + 12q^{71} + 2q^{73} - 8q^{79} + q^{81} - 4q^{83} - 2q^{87} - 6q^{89} + 6q^{91} - 4q^{93} + 18q^{97} + O(q^{100})$$

## 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.co 1
4.b odd 2 1 525.2.a.a 1
5.b even 2 1 1680.2.a.f 1
12.b even 2 1 1575.2.a.h 1
15.d odd 2 1 5040.2.a.d 1
20.d odd 2 1 105.2.a.a 1
20.e even 4 2 525.2.d.b 2
28.d even 2 1 3675.2.a.f 1
40.e odd 2 1 6720.2.a.p 1
40.f even 2 1 6720.2.a.bk 1
60.h even 2 1 315.2.a.a 1
60.l odd 4 2 1575.2.d.b 2
140.c even 2 1 735.2.a.f 1
140.p odd 6 2 735.2.i.a 2
140.s even 6 2 735.2.i.b 2
420.o odd 2 1 2205.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.a.a 1 20.d odd 2 1
315.2.a.a 1 60.h even 2 1
525.2.a.a 1 4.b odd 2 1
525.2.d.b 2 20.e even 4 2
735.2.a.f 1 140.c even 2 1
735.2.i.a 2 140.p odd 6 2
735.2.i.b 2 140.s even 6 2
1575.2.a.h 1 12.b even 2 1
1575.2.d.b 2 60.l odd 4 2
1680.2.a.f 1 5.b even 2 1
2205.2.a.b 1 420.o odd 2 1
3675.2.a.f 1 28.d even 2 1
5040.2.a.d 1 15.d odd 2 1
6720.2.a.p 1 40.e odd 2 1
6720.2.a.bk 1 40.f even 2 1
8400.2.a.co 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8400))$$:

 $$T_{11}$$ $$T_{13} - 6$$ $$T_{17} + 2$$ $$T_{19} - 8$$ $$T_{23} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-1 + T$$
$5$ $$T$$
$7$ $$-1 + T$$
$11$ $$T$$
$13$ $$-6 + T$$
$17$ $$2 + T$$
$19$ $$-8 + T$$
$23$ $$-8 + T$$
$29$ $$2 + T$$
$31$ $$4 + T$$
$37$ $$-2 + T$$
$41$ $$6 + T$$
$43$ $$-4 + T$$
$47$ $$-8 + T$$
$53$ $$10 + T$$
$59$ $$4 + T$$
$61$ $$2 + T$$
$67$ $$-4 + T$$
$71$ $$-12 + T$$
$73$ $$-2 + T$$
$79$ $$8 + T$$
$83$ $$4 + T$$
$89$ $$6 + T$$
$97$ $$-18 + T$$