# Properties

 Label 2400.2.a.q Level $2400$ Weight $2$ Character orbit 2400.a Self dual yes Analytic conductor $19.164$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2400 = 2^{5} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2400.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$19.1640964851$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 96) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} + 4q^{7} + q^{9} + O(q^{10})$$ $$q - q^{3} + 4q^{7} + q^{9} + 4q^{11} + 2q^{13} + 6q^{17} - 4q^{19} - 4q^{21} - q^{27} + 2q^{29} + 4q^{31} - 4q^{33} + 2q^{37} - 2q^{39} + 2q^{41} - 4q^{43} - 8q^{47} + 9q^{49} - 6q^{51} - 10q^{53} + 4q^{57} - 4q^{59} + 6q^{61} + 4q^{63} - 4q^{67} - 16q^{71} + 6q^{73} + 16q^{77} + 4q^{79} + q^{81} - 12q^{83} - 2q^{87} + 10q^{89} + 8q^{91} - 4q^{93} + 14q^{97} + 4q^{99} + O(q^{100})$$

## 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 4.00000 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$1$$

## 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 2400.2.a.q 1
3.b odd 2 1 7200.2.a.bx 1
4.b odd 2 1 2400.2.a.r 1
5.b even 2 1 96.2.a.b yes 1
5.c odd 4 2 2400.2.f.r 2
8.b even 2 1 4800.2.a.co 1
8.d odd 2 1 4800.2.a.f 1
12.b even 2 1 7200.2.a.e 1
15.d odd 2 1 288.2.a.b 1
15.e even 4 2 7200.2.f.f 2
20.d odd 2 1 96.2.a.a 1
20.e even 4 2 2400.2.f.a 2
35.c odd 2 1 4704.2.a.e 1
40.e odd 2 1 192.2.a.c 1
40.f even 2 1 192.2.a.a 1
40.i odd 4 2 4800.2.f.e 2
40.k even 4 2 4800.2.f.bh 2
45.h odd 6 2 2592.2.i.w 2
45.j even 6 2 2592.2.i.h 2
60.h even 2 1 288.2.a.c 1
60.l odd 4 2 7200.2.f.x 2
80.k odd 4 2 768.2.d.a 2
80.q even 4 2 768.2.d.h 2
120.i odd 2 1 576.2.a.g 1
120.m even 2 1 576.2.a.h 1
140.c even 2 1 4704.2.a.t 1
180.n even 6 2 2592.2.i.q 2
180.p odd 6 2 2592.2.i.b 2
240.t even 4 2 2304.2.d.c 2
240.bm odd 4 2 2304.2.d.s 2
280.c odd 2 1 9408.2.a.ct 1
280.n even 2 1 9408.2.a.bj 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.2.a.a 1 20.d odd 2 1
96.2.a.b yes 1 5.b even 2 1
192.2.a.a 1 40.f even 2 1
192.2.a.c 1 40.e odd 2 1
288.2.a.b 1 15.d odd 2 1
288.2.a.c 1 60.h even 2 1
576.2.a.g 1 120.i odd 2 1
576.2.a.h 1 120.m even 2 1
768.2.d.a 2 80.k odd 4 2
768.2.d.h 2 80.q even 4 2
2304.2.d.c 2 240.t even 4 2
2304.2.d.s 2 240.bm odd 4 2
2400.2.a.q 1 1.a even 1 1 trivial
2400.2.a.r 1 4.b odd 2 1
2400.2.f.a 2 20.e even 4 2
2400.2.f.r 2 5.c odd 4 2
2592.2.i.b 2 180.p odd 6 2
2592.2.i.h 2 45.j even 6 2
2592.2.i.q 2 180.n even 6 2
2592.2.i.w 2 45.h odd 6 2
4704.2.a.e 1 35.c odd 2 1
4704.2.a.t 1 140.c even 2 1
4800.2.a.f 1 8.d odd 2 1
4800.2.a.co 1 8.b even 2 1
4800.2.f.e 2 40.i odd 4 2
4800.2.f.bh 2 40.k even 4 2
7200.2.a.e 1 12.b even 2 1
7200.2.a.bx 1 3.b odd 2 1
7200.2.f.f 2 15.e even 4 2
7200.2.f.x 2 60.l odd 4 2
9408.2.a.bj 1 280.n even 2 1
9408.2.a.ct 1 280.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2400))$$:

 $$T_{7} - 4$$ $$T_{11} - 4$$ $$T_{13} - 2$$ $$T_{19} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + T$$
$5$ 1
$7$ $$1 - 4 T + 7 T^{2}$$
$11$ $$1 - 4 T + 11 T^{2}$$
$13$ $$1 - 2 T + 13 T^{2}$$
$17$ $$1 - 6 T + 17 T^{2}$$
$19$ $$1 + 4 T + 19 T^{2}$$
$23$ $$1 + 23 T^{2}$$
$29$ $$1 - 2 T + 29 T^{2}$$
$31$ $$1 - 4 T + 31 T^{2}$$
$37$ $$1 - 2 T + 37 T^{2}$$
$41$ $$1 - 2 T + 41 T^{2}$$
$43$ $$1 + 4 T + 43 T^{2}$$
$47$ $$1 + 8 T + 47 T^{2}$$
$53$ $$1 + 10 T + 53 T^{2}$$
$59$ $$1 + 4 T + 59 T^{2}$$
$61$ $$1 - 6 T + 61 T^{2}$$
$67$ $$1 + 4 T + 67 T^{2}$$
$71$ $$1 + 16 T + 71 T^{2}$$
$73$ $$1 - 6 T + 73 T^{2}$$
$79$ $$1 - 4 T + 79 T^{2}$$
$83$ $$1 + 12 T + 83 T^{2}$$
$89$ $$1 - 10 T + 89 T^{2}$$
$97$ $$1 - 14 T + 97 T^{2}$$