# Properties

 Label 2448.2.c.j Level $2448$ Weight $2$ Character orbit 2448.c Analytic conductor $19.547$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2448 = 2^{4} \cdot 3^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2448.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.5473784148$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 51) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 4 i q^{7} +O(q^{10})$$ $$q + 4 i q^{7} + 4 i q^{11} + 2 q^{13} + ( -1 - 4 i ) q^{17} + 4 q^{19} -4 i q^{23} + 5 q^{25} + 4 i q^{31} + 8 i q^{37} + 8 i q^{41} + 4 q^{43} -8 q^{47} -9 q^{49} -6 q^{53} -12 q^{59} -8 i q^{61} -12 q^{67} + 12 i q^{71} -16 q^{77} + 4 i q^{79} + 12 q^{83} + 10 q^{89} + 8 i q^{91} + 16 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q + 4q^{13} - 2q^{17} + 8q^{19} + 10q^{25} + 8q^{43} - 16q^{47} - 18q^{49} - 12q^{53} - 24q^{59} - 24q^{67} - 32q^{77} + 24q^{83} + 20q^{89} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2448\mathbb{Z}\right)^\times$$.

 $$n$$ $$613$$ $$1361$$ $$1873$$ $$2143$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
577.1
 − 1.00000i 1.00000i
0 0 0 0 0 4.00000i 0 0 0
577.2 0 0 0 0 0 4.00000i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2448.2.c.j 2
3.b odd 2 1 816.2.c.c 2
4.b odd 2 1 153.2.d.a 2
12.b even 2 1 51.2.d.b 2
17.b even 2 1 inner 2448.2.c.j 2
24.f even 2 1 3264.2.c.e 2
24.h odd 2 1 3264.2.c.d 2
51.c odd 2 1 816.2.c.c 2
60.h even 2 1 1275.2.g.a 2
60.l odd 4 1 1275.2.d.b 2
60.l odd 4 1 1275.2.d.d 2
68.d odd 2 1 153.2.d.a 2
68.f odd 4 1 2601.2.a.i 1
68.f odd 4 1 2601.2.a.j 1
204.h even 2 1 51.2.d.b 2
204.l even 4 1 867.2.a.a 1
204.l even 4 1 867.2.a.b 1
204.p even 8 4 867.2.e.d 4
204.t odd 16 8 867.2.h.d 8
408.b odd 2 1 3264.2.c.d 2
408.h even 2 1 3264.2.c.e 2
1020.b even 2 1 1275.2.g.a 2
1020.x odd 4 1 1275.2.d.b 2
1020.x odd 4 1 1275.2.d.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.d.b 2 12.b even 2 1
51.2.d.b 2 204.h even 2 1
153.2.d.a 2 4.b odd 2 1
153.2.d.a 2 68.d odd 2 1
816.2.c.c 2 3.b odd 2 1
816.2.c.c 2 51.c odd 2 1
867.2.a.a 1 204.l even 4 1
867.2.a.b 1 204.l even 4 1
867.2.e.d 4 204.p even 8 4
867.2.h.d 8 204.t odd 16 8
1275.2.d.b 2 60.l odd 4 1
1275.2.d.b 2 1020.x odd 4 1
1275.2.d.d 2 60.l odd 4 1
1275.2.d.d 2 1020.x odd 4 1
1275.2.g.a 2 60.h even 2 1
1275.2.g.a 2 1020.b even 2 1
2448.2.c.j 2 1.a even 1 1 trivial
2448.2.c.j 2 17.b even 2 1 inner
2601.2.a.i 1 68.f odd 4 1
2601.2.a.j 1 68.f odd 4 1
3264.2.c.d 2 24.h odd 2 1
3264.2.c.d 2 408.b odd 2 1
3264.2.c.e 2 24.f even 2 1
3264.2.c.e 2 408.h even 2 1

## Hecke kernels

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

 $$T_{5}$$ $$T_{7}^{2} + 16$$ $$T_{47} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$16 + T^{2}$$
$11$ $$16 + T^{2}$$
$13$ $$( -2 + T )^{2}$$
$17$ $$17 + 2 T + T^{2}$$
$19$ $$( -4 + T )^{2}$$
$23$ $$16 + T^{2}$$
$29$ $$T^{2}$$
$31$ $$16 + T^{2}$$
$37$ $$64 + T^{2}$$
$41$ $$64 + T^{2}$$
$43$ $$( -4 + T )^{2}$$
$47$ $$( 8 + T )^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$( 12 + T )^{2}$$
$61$ $$64 + T^{2}$$
$67$ $$( 12 + T )^{2}$$
$71$ $$144 + T^{2}$$
$73$ $$T^{2}$$
$79$ $$16 + T^{2}$$
$83$ $$( -12 + T )^{2}$$
$89$ $$( -10 + T )^{2}$$
$97$ $$256 + T^{2}$$