# Properties

 Label 2400.2.a.b Level $2400$ Weight $2$ Character orbit 2400.a Self dual yes Analytic conductor $19.164$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 480) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} - 4 q^{7} + q^{9}+O(q^{10})$$ q - q^3 - 4 * q^7 + q^9 $$q - q^{3} - 4 q^{7} + q^{9} + 2 q^{13} + 6 q^{17} + 4 q^{21} - 4 q^{23} - q^{27} - 2 q^{29} + 8 q^{31} - 6 q^{37} - 2 q^{39} - 6 q^{41} + 12 q^{43} - 12 q^{47} + 9 q^{49} - 6 q^{51} + 10 q^{53} - 8 q^{59} - 10 q^{61} - 4 q^{63} - 12 q^{67} + 4 q^{69} - 8 q^{71} - 10 q^{73} - 16 q^{79} + q^{81} + 12 q^{83} + 2 q^{87} - 6 q^{89} - 8 q^{91} - 8 q^{93} - 18 q^{97}+O(q^{100})$$ q - q^3 - 4 * q^7 + q^9 + 2 * q^13 + 6 * q^17 + 4 * q^21 - 4 * q^23 - q^27 - 2 * q^29 + 8 * q^31 - 6 * q^37 - 2 * q^39 - 6 * q^41 + 12 * q^43 - 12 * q^47 + 9 * q^49 - 6 * q^51 + 10 * q^53 - 8 * q^59 - 10 * q^61 - 4 * q^63 - 12 * q^67 + 4 * q^69 - 8 * q^71 - 10 * q^73 - 16 * q^79 + q^81 + 12 * q^83 + 2 * q^87 - 6 * q^89 - 8 * q^91 - 8 * q^93 - 18 * q^97

## 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.b 1
3.b odd 2 1 7200.2.a.b 1
4.b odd 2 1 2400.2.a.bg 1
5.b even 2 1 480.2.a.h yes 1
5.c odd 4 2 2400.2.f.j 2
8.b even 2 1 4800.2.a.bm 1
8.d odd 2 1 4800.2.a.bi 1
12.b even 2 1 7200.2.a.bz 1
15.d odd 2 1 1440.2.a.f 1
15.e even 4 2 7200.2.f.p 2
20.d odd 2 1 480.2.a.c 1
20.e even 4 2 2400.2.f.i 2
40.e odd 2 1 960.2.a.i 1
40.f even 2 1 960.2.a.d 1
40.i odd 4 2 4800.2.f.s 2
40.k even 4 2 4800.2.f.r 2
60.h even 2 1 1440.2.a.a 1
60.l odd 4 2 7200.2.f.o 2
80.k odd 4 2 3840.2.k.w 2
80.q even 4 2 3840.2.k.d 2
120.i odd 2 1 2880.2.a.bh 1
120.m even 2 1 2880.2.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.a.c 1 20.d odd 2 1
480.2.a.h yes 1 5.b even 2 1
960.2.a.d 1 40.f even 2 1
960.2.a.i 1 40.e odd 2 1
1440.2.a.a 1 60.h even 2 1
1440.2.a.f 1 15.d odd 2 1
2400.2.a.b 1 1.a even 1 1 trivial
2400.2.a.bg 1 4.b odd 2 1
2400.2.f.i 2 20.e even 4 2
2400.2.f.j 2 5.c odd 4 2
2880.2.a.s 1 120.m even 2 1
2880.2.a.bh 1 120.i odd 2 1
3840.2.k.d 2 80.q even 4 2
3840.2.k.w 2 80.k odd 4 2
4800.2.a.bi 1 8.d odd 2 1
4800.2.a.bm 1 8.b even 2 1
4800.2.f.r 2 40.k even 4 2
4800.2.f.s 2 40.i odd 4 2
7200.2.a.b 1 3.b odd 2 1
7200.2.a.bz 1 12.b even 2 1
7200.2.f.o 2 60.l odd 4 2
7200.2.f.p 2 15.e even 4 2

## Hecke kernels

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

 $$T_{7} + 4$$ T7 + 4 $$T_{11}$$ T11 $$T_{13} - 2$$ T13 - 2 $$T_{19}$$ T19

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T$$
$7$ $$T + 4$$
$11$ $$T$$
$13$ $$T - 2$$
$17$ $$T - 6$$
$19$ $$T$$
$23$ $$T + 4$$
$29$ $$T + 2$$
$31$ $$T - 8$$
$37$ $$T + 6$$
$41$ $$T + 6$$
$43$ $$T - 12$$
$47$ $$T + 12$$
$53$ $$T - 10$$
$59$ $$T + 8$$
$61$ $$T + 10$$
$67$ $$T + 12$$
$71$ $$T + 8$$
$73$ $$T + 10$$
$79$ $$T + 16$$
$83$ $$T - 12$$
$89$ $$T + 6$$
$97$ $$T + 18$$