Properties

 Label 8400.2.a.bj 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

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{7} + q^{9} + O(q^{10})$$ $$q - q^{3} + q^{7} + q^{9} + 6q^{11} - 2q^{13} + 4q^{17} + 6q^{19} - q^{21} - q^{27} - 2q^{29} + 10q^{31} - 6q^{33} - 4q^{37} + 2q^{39} + 2q^{41} + 4q^{43} + q^{49} - 4q^{51} + 6q^{53} - 6q^{57} + 8q^{59} - 2q^{61} + q^{63} + 16q^{67} - 10q^{71} - 6q^{73} + 6q^{77} - 4q^{79} + q^{81} - 8q^{83} + 2q^{87} + 6q^{89} - 2q^{91} - 10q^{93} - 2q^{97} + 6q^{99} + O(q^{100})$$

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.bj 1
4.b odd 2 1 525.2.a.b 1
5.b even 2 1 8400.2.a.ch 1
5.c odd 4 2 1680.2.t.f 2
12.b even 2 1 1575.2.a.i 1
15.e even 4 2 5040.2.t.e 2
20.d odd 2 1 525.2.a.c 1
20.e even 4 2 105.2.d.a 2
28.d even 2 1 3675.2.a.d 1
60.h even 2 1 1575.2.a.e 1
60.l odd 4 2 315.2.d.c 2
140.c even 2 1 3675.2.a.l 1
140.j odd 4 2 735.2.d.a 2
140.w even 12 4 735.2.q.a 4
140.x odd 12 4 735.2.q.b 4
420.w even 4 2 2205.2.d.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.d.a 2 20.e even 4 2
315.2.d.c 2 60.l odd 4 2
525.2.a.b 1 4.b odd 2 1
525.2.a.c 1 20.d odd 2 1
735.2.d.a 2 140.j odd 4 2
735.2.q.a 4 140.w even 12 4
735.2.q.b 4 140.x odd 12 4
1575.2.a.e 1 60.h even 2 1
1575.2.a.i 1 12.b even 2 1
1680.2.t.f 2 5.c odd 4 2
2205.2.d.f 2 420.w even 4 2
3675.2.a.d 1 28.d even 2 1
3675.2.a.l 1 140.c even 2 1
5040.2.t.e 2 15.e even 4 2
8400.2.a.bj 1 1.a even 1 1 trivial
8400.2.a.ch 1 5.b even 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

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