# Properties

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

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## 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: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q - i q^{3} + i q^{7} - q^{9} +O(q^{10})$$ q - i * q^3 + i * q^7 - q^9 $$q - i q^{3} + i q^{7} - q^{9} + i q^{13} + 3 q^{19} + q^{21} - 4 i q^{23} + i q^{27} - 4 q^{29} + 7 q^{31} + 6 i q^{37} + q^{39} + 6 q^{41} - 9 i q^{43} + 6 i q^{47} + 6 q^{49} + 2 i q^{53} - 3 i q^{57} + 10 q^{59} - q^{61} - i q^{63} - 3 i q^{67} - 4 q^{69} + 14 q^{71} + 10 i q^{73} + 8 q^{79} + q^{81} - 18 i q^{83} + 4 i q^{87} - q^{91} - 7 i q^{93} + 3 i q^{97} +O(q^{100})$$ q - i * q^3 + i * q^7 - q^9 + i * q^13 + 3 * q^19 + q^21 - 4*i * q^23 + i * q^27 - 4 * q^29 + 7 * q^31 + 6*i * q^37 + q^39 + 6 * q^41 - 9*i * q^43 + 6*i * q^47 + 6 * q^49 + 2*i * q^53 - 3*i * q^57 + 10 * q^59 - q^61 - i * q^63 - 3*i * q^67 - 4 * q^69 + 14 * q^71 + 10*i * q^73 + 8 * q^79 + q^81 - 18*i * q^83 + 4*i * q^87 - q^91 - 7*i * q^93 + 3*i * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} + 6 q^{19} + 2 q^{21} - 8 q^{29} + 14 q^{31} + 2 q^{39} + 12 q^{41} + 12 q^{49} + 20 q^{59} - 2 q^{61} - 8 q^{69} + 28 q^{71} + 16 q^{79} + 2 q^{81} - 2 q^{91}+O(q^{100})$$ 2 * q - 2 * q^9 + 6 * q^19 + 2 * q^21 - 8 * q^29 + 14 * q^31 + 2 * q^39 + 12 * q^41 + 12 * q^49 + 20 * q^59 - 2 * q^61 - 8 * q^69 + 28 * q^71 + 16 * q^79 + 2 * q^81 - 2 * q^91

## 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 1.00000i 0 −1.00000 0
1249.2 0 1.00000i 0 0 0 1.00000i 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.k 2
3.b odd 2 1 7200.2.f.r 2
4.b odd 2 1 2400.2.f.h 2
5.b even 2 1 inner 2400.2.f.k 2
5.c odd 4 1 2400.2.a.g yes 1
5.c odd 4 1 2400.2.a.bb yes 1
8.b even 2 1 4800.2.f.q 2
8.d odd 2 1 4800.2.f.t 2
12.b even 2 1 7200.2.f.l 2
15.d odd 2 1 7200.2.f.r 2
15.e even 4 1 7200.2.a.s 1
15.e even 4 1 7200.2.a.bi 1
20.d odd 2 1 2400.2.f.h 2
20.e even 4 1 2400.2.a.f 1
20.e even 4 1 2400.2.a.bc yes 1
40.e odd 2 1 4800.2.f.t 2
40.f even 2 1 4800.2.f.q 2
40.i odd 4 1 4800.2.a.x 1
40.i odd 4 1 4800.2.a.bw 1
40.k even 4 1 4800.2.a.w 1
40.k even 4 1 4800.2.a.bx 1
60.h even 2 1 7200.2.f.l 2
60.l odd 4 1 7200.2.a.r 1
60.l odd 4 1 7200.2.a.bj 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2400.2.a.f 1 20.e even 4 1
2400.2.a.g yes 1 5.c odd 4 1
2400.2.a.bb yes 1 5.c odd 4 1
2400.2.a.bc yes 1 20.e even 4 1
2400.2.f.h 2 4.b odd 2 1
2400.2.f.h 2 20.d odd 2 1
2400.2.f.k 2 1.a even 1 1 trivial
2400.2.f.k 2 5.b even 2 1 inner
4800.2.a.w 1 40.k even 4 1
4800.2.a.x 1 40.i odd 4 1
4800.2.a.bw 1 40.i odd 4 1
4800.2.a.bx 1 40.k even 4 1
4800.2.f.q 2 8.b even 2 1
4800.2.f.q 2 40.f even 2 1
4800.2.f.t 2 8.d odd 2 1
4800.2.f.t 2 40.e odd 2 1
7200.2.a.r 1 60.l odd 4 1
7200.2.a.s 1 15.e even 4 1
7200.2.a.bi 1 15.e even 4 1
7200.2.a.bj 1 60.l odd 4 1
7200.2.f.l 2 12.b even 2 1
7200.2.f.l 2 60.h even 2 1
7200.2.f.r 2 3.b odd 2 1
7200.2.f.r 2 15.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 $$S_{2}^{\mathrm{new}}(2400, [\chi])$$:

 $$T_{7}^{2} + 1$$ T7^2 + 1 $$T_{11}$$ T11 $$T_{13}^{2} + 1$$ T13^2 + 1 $$T_{19} - 3$$ T19 - 3 $$T_{31} - 7$$ T31 - 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 1$$
$17$ $$T^{2}$$
$19$ $$(T - 3)^{2}$$
$23$ $$T^{2} + 16$$
$29$ $$(T + 4)^{2}$$
$31$ $$(T - 7)^{2}$$
$37$ $$T^{2} + 36$$
$41$ $$(T - 6)^{2}$$
$43$ $$T^{2} + 81$$
$47$ $$T^{2} + 36$$
$53$ $$T^{2} + 4$$
$59$ $$(T - 10)^{2}$$
$61$ $$(T + 1)^{2}$$
$67$ $$T^{2} + 9$$
$71$ $$(T - 14)^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 324$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 9$$
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