# Properties

 Label 2448.2.a.o Level $2448$ Weight $2$ Character orbit 2448.a Self dual yes Analytic conductor $19.547$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2448 = 2^{4} \cdot 3^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2448.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$19.5473784148$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 17) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2 q^{5} - 4 q^{7}+O(q^{10})$$ q + 2 * q^5 - 4 * q^7 $$q + 2 q^{5} - 4 q^{7} - 2 q^{13} - q^{17} + 4 q^{19} + 4 q^{23} - q^{25} - 6 q^{29} - 4 q^{31} - 8 q^{35} - 2 q^{37} + 6 q^{41} - 4 q^{43} + 9 q^{49} - 6 q^{53} - 12 q^{59} - 10 q^{61} - 4 q^{65} - 4 q^{67} - 4 q^{71} - 6 q^{73} - 12 q^{79} - 4 q^{83} - 2 q^{85} - 10 q^{89} + 8 q^{91} + 8 q^{95} + 2 q^{97}+O(q^{100})$$ q + 2 * q^5 - 4 * q^7 - 2 * q^13 - q^17 + 4 * q^19 + 4 * q^23 - q^25 - 6 * q^29 - 4 * q^31 - 8 * q^35 - 2 * q^37 + 6 * q^41 - 4 * q^43 + 9 * q^49 - 6 * q^53 - 12 * q^59 - 10 * q^61 - 4 * q^65 - 4 * q^67 - 4 * q^71 - 6 * q^73 - 12 * q^79 - 4 * q^83 - 2 * q^85 - 10 * q^89 + 8 * q^91 + 8 * q^95 + 2 * 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 0 0 2.00000 0 −4.00000 0 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$17$$ $$1$$

## 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 2448.2.a.o 1
3.b odd 2 1 272.2.a.b 1
4.b odd 2 1 153.2.a.c 1
8.b even 2 1 9792.2.a.i 1
8.d odd 2 1 9792.2.a.n 1
12.b even 2 1 17.2.a.a 1
15.d odd 2 1 6800.2.a.n 1
20.d odd 2 1 3825.2.a.d 1
24.f even 2 1 1088.2.a.i 1
24.h odd 2 1 1088.2.a.h 1
28.d even 2 1 7497.2.a.l 1
51.c odd 2 1 4624.2.a.d 1
60.h even 2 1 425.2.a.d 1
60.l odd 4 2 425.2.b.b 2
68.d odd 2 1 2601.2.a.g 1
84.h odd 2 1 833.2.a.a 1
84.j odd 6 2 833.2.e.a 2
84.n even 6 2 833.2.e.b 2
132.d odd 2 1 2057.2.a.e 1
156.h even 2 1 2873.2.a.c 1
204.h even 2 1 289.2.a.a 1
204.l even 4 2 289.2.b.a 2
204.p even 8 4 289.2.c.a 4
204.t odd 16 8 289.2.d.d 8
228.b odd 2 1 6137.2.a.b 1
276.h odd 2 1 8993.2.a.a 1
1020.b even 2 1 7225.2.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.2.a.a 1 12.b even 2 1
153.2.a.c 1 4.b odd 2 1
272.2.a.b 1 3.b odd 2 1
289.2.a.a 1 204.h even 2 1
289.2.b.a 2 204.l even 4 2
289.2.c.a 4 204.p even 8 4
289.2.d.d 8 204.t odd 16 8
425.2.a.d 1 60.h even 2 1
425.2.b.b 2 60.l odd 4 2
833.2.a.a 1 84.h odd 2 1
833.2.e.a 2 84.j odd 6 2
833.2.e.b 2 84.n even 6 2
1088.2.a.h 1 24.h odd 2 1
1088.2.a.i 1 24.f even 2 1
2057.2.a.e 1 132.d odd 2 1
2448.2.a.o 1 1.a even 1 1 trivial
2601.2.a.g 1 68.d odd 2 1
2873.2.a.c 1 156.h even 2 1
3825.2.a.d 1 20.d odd 2 1
4624.2.a.d 1 51.c odd 2 1
6137.2.a.b 1 228.b odd 2 1
6800.2.a.n 1 15.d odd 2 1
7225.2.a.g 1 1020.b even 2 1
7497.2.a.l 1 28.d even 2 1
8993.2.a.a 1 276.h odd 2 1
9792.2.a.i 1 8.b even 2 1
9792.2.a.n 1 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2448))$$:

 $$T_{5} - 2$$ T5 - 2 $$T_{7} + 4$$ T7 + 4 $$T_{11}$$ T11 $$T_{19} - 4$$ T19 - 4 $$T_{23} - 4$$ T23 - 4

## Hecke characteristic polynomials

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