# Properties

 Label 2400.2.f.n Level $2400$ Weight $2$ Character orbit 2400.f Analytic conductor $19.164$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2400 = 2^{5} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2400.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.1640964851$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 480) 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.

 $$f(q)$$ $$=$$ $$q + i q^{3} - q^{9} +O(q^{10})$$ $$q + i q^{3} - q^{9} + 4 q^{11} + 2 i q^{13} + 2 i q^{17} -8 q^{19} + 4 i q^{23} -i q^{27} + 6 q^{29} + 4 i q^{33} -2 i q^{37} -2 q^{39} -6 q^{41} + 4 i q^{43} + 12 i q^{47} + 7 q^{49} -2 q^{51} -6 i q^{53} -8 i q^{57} -12 q^{59} + 14 q^{61} + 12 i q^{67} -4 q^{69} + 2 i q^{73} + 8 q^{79} + q^{81} -4 i q^{83} + 6 i q^{87} -2 q^{89} + 14 i q^{97} -4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} + 8q^{11} - 16q^{19} + 12q^{29} - 4q^{39} - 12q^{41} + 14q^{49} - 4q^{51} - 24q^{59} + 28q^{61} - 8q^{69} + 16q^{79} + 2q^{81} - 4q^{89} - 8q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$1601$$ $$1951$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1249.1
 − 1.00000i 1.00000i
0 1.00000i 0 0 0 0 0 −1.00000 0
1249.2 0 1.00000i 0 0 0 0 0 −1.00000 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2400.2.f.n 2
3.b odd 2 1 7200.2.f.b 2
4.b odd 2 1 2400.2.f.e 2
5.b even 2 1 inner 2400.2.f.n 2
5.c odd 4 1 480.2.a.e yes 1
5.c odd 4 1 2400.2.a.j 1
8.b even 2 1 4800.2.f.j 2
8.d odd 2 1 4800.2.f.ba 2
12.b even 2 1 7200.2.f.bb 2
15.d odd 2 1 7200.2.f.b 2
15.e even 4 1 1440.2.a.j 1
15.e even 4 1 7200.2.a.u 1
20.d odd 2 1 2400.2.f.e 2
20.e even 4 1 480.2.a.b 1
20.e even 4 1 2400.2.a.y 1
40.e odd 2 1 4800.2.f.ba 2
40.f even 2 1 4800.2.f.j 2
40.i odd 4 1 960.2.a.f 1
40.i odd 4 1 4800.2.a.ca 1
40.k even 4 1 960.2.a.o 1
40.k even 4 1 4800.2.a.u 1
60.h even 2 1 7200.2.f.bb 2
60.l odd 4 1 1440.2.a.k 1
60.l odd 4 1 7200.2.a.bg 1
80.i odd 4 1 3840.2.k.k 2
80.j even 4 1 3840.2.k.p 2
80.s even 4 1 3840.2.k.p 2
80.t odd 4 1 3840.2.k.k 2
120.q odd 4 1 2880.2.a.i 1
120.w even 4 1 2880.2.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.a.b 1 20.e even 4 1
480.2.a.e yes 1 5.c odd 4 1
960.2.a.f 1 40.i odd 4 1
960.2.a.o 1 40.k even 4 1
1440.2.a.j 1 15.e even 4 1
1440.2.a.k 1 60.l odd 4 1
2400.2.a.j 1 5.c odd 4 1
2400.2.a.y 1 20.e even 4 1
2400.2.f.e 2 4.b odd 2 1
2400.2.f.e 2 20.d odd 2 1
2400.2.f.n 2 1.a even 1 1 trivial
2400.2.f.n 2 5.b even 2 1 inner
2880.2.a.i 1 120.q odd 4 1
2880.2.a.j 1 120.w even 4 1
3840.2.k.k 2 80.i odd 4 1
3840.2.k.k 2 80.t odd 4 1
3840.2.k.p 2 80.j even 4 1
3840.2.k.p 2 80.s even 4 1
4800.2.a.u 1 40.k even 4 1
4800.2.a.ca 1 40.i odd 4 1
4800.2.f.j 2 8.b even 2 1
4800.2.f.j 2 40.f even 2 1
4800.2.f.ba 2 8.d odd 2 1
4800.2.f.ba 2 40.e odd 2 1
7200.2.a.u 1 15.e even 4 1
7200.2.a.bg 1 60.l odd 4 1
7200.2.f.b 2 3.b odd 2 1
7200.2.f.b 2 15.d odd 2 1
7200.2.f.bb 2 12.b even 2 1
7200.2.f.bb 2 60.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 $$S_{2}^{\mathrm{new}}(2400, [\chi])$$:

 $$T_{7}$$ $$T_{11} - 4$$ $$T_{13}^{2} + 4$$ $$T_{19} + 8$$ $$T_{31}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$( -4 + T )^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$4 + T^{2}$$
$19$ $$( 8 + T )^{2}$$
$23$ $$16 + T^{2}$$
$29$ $$( -6 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$144 + T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$( 12 + T )^{2}$$
$61$ $$( -14 + T )^{2}$$
$67$ $$144 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$4 + T^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$16 + T^{2}$$
$89$ $$( 2 + T )^{2}$$
$97$ $$196 + T^{2}$$