Properties

 Label 8400.2.a.q 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 840) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{7} + q^{9} + O(q^{10})$$ $$q - q^{3} + q^{7} + q^{9} - 4q^{11} + 2q^{13} + 6q^{17} - 4q^{19} - q^{21} + 8q^{23} - q^{27} - 2q^{29} + 4q^{33} + 2q^{37} - 2q^{39} + 10q^{41} + 4q^{43} + q^{49} - 6q^{51} - 14q^{53} + 4q^{57} - 12q^{59} - 2q^{61} + q^{63} - 4q^{67} - 8q^{69} - 2q^{73} - 4q^{77} + 8q^{79} + q^{81} - 4q^{83} + 2q^{87} - 6q^{89} + 2q^{91} + 6q^{97} - 4q^{99} + O(q^{100})$$

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.q 1
4.b odd 2 1 4200.2.a.z 1
5.b even 2 1 1680.2.a.p 1
15.d odd 2 1 5040.2.a.j 1
20.d odd 2 1 840.2.a.f 1
20.e even 4 2 4200.2.t.q 2
40.e odd 2 1 6720.2.a.bn 1
40.f even 2 1 6720.2.a.h 1
60.h even 2 1 2520.2.a.g 1
140.c even 2 1 5880.2.a.z 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.a.f 1 20.d odd 2 1
1680.2.a.p 1 5.b even 2 1
2520.2.a.g 1 60.h even 2 1
4200.2.a.z 1 4.b odd 2 1
4200.2.t.q 2 20.e even 4 2
5040.2.a.j 1 15.d odd 2 1
5880.2.a.z 1 140.c even 2 1
6720.2.a.h 1 40.f even 2 1
6720.2.a.bn 1 40.e odd 2 1
8400.2.a.q 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8400))$$:

 $$T_{11} + 4$$ $$T_{13} - 2$$ $$T_{17} - 6$$ $$T_{19} + 4$$ $$T_{23} - 8$$

Hecke characteristic polynomials

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