# Properties

 Label 7200.2.f.s Level $7200$ Weight $2$ Character orbit 7200.f Analytic conductor $57.492$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7200 = 2^{5} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7200.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$57.4922894553$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q+O(q^{10})$$ q $$q + \beta q^{13} + 3 \beta q^{17} + 4 q^{19} - 4 \beta q^{23} - 2 q^{29} + 4 q^{31} - 5 \beta q^{37} - 2 q^{41} - 2 \beta q^{43} + 4 \beta q^{47} + 7 q^{49} + \beta q^{53} + 8 q^{59} - 2 q^{61} + 6 \beta q^{67} - 8 q^{71} - 7 \beta q^{73} + 12 q^{79} + 2 \beta q^{83} - 14 q^{89} - \beta q^{97} +O(q^{100})$$ q + b * q^13 + 3*b * q^17 + 4 * q^19 - 4*b * q^23 - 2 * q^29 + 4 * q^31 - 5*b * q^37 - 2 * q^41 - 2*b * q^43 + 4*b * q^47 + 7 * q^49 + b * q^53 + 8 * q^59 - 2 * q^61 + 6*b * q^67 - 8 * q^71 - 7*b * q^73 + 12 * q^79 + 2*b * q^83 - 14 * q^89 - b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q + 8 q^{19} - 4 q^{29} + 8 q^{31} - 4 q^{41} + 14 q^{49} + 16 q^{59} - 4 q^{61} - 16 q^{71} + 24 q^{79} - 28 q^{89}+O(q^{100})$$ 2 * q + 8 * q^19 - 4 * q^29 + 8 * q^31 - 4 * q^41 + 14 * q^49 + 16 * q^59 - 4 * q^61 - 16 * q^71 + 24 * q^79 - 28 * q^89

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/7200\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$6401$$ $$6751$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
6049.1
 − 1.00000i 1.00000i
0 0 0 0 0 0 0 0 0
6049.2 0 0 0 0 0 0 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
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 7200.2.f.s 2
3.b odd 2 1 2400.2.f.l 2
4.b odd 2 1 7200.2.f.k 2
5.b even 2 1 inner 7200.2.f.s 2
5.c odd 4 1 1440.2.a.c 1
5.c odd 4 1 7200.2.a.z 1
12.b even 2 1 2400.2.f.g 2
15.d odd 2 1 2400.2.f.l 2
15.e even 4 1 480.2.a.d 1
15.e even 4 1 2400.2.a.z 1
20.d odd 2 1 7200.2.f.k 2
20.e even 4 1 1440.2.a.d 1
20.e even 4 1 7200.2.a.ba 1
24.f even 2 1 4800.2.f.v 2
24.h odd 2 1 4800.2.f.o 2
40.i odd 4 1 2880.2.a.ba 1
40.k even 4 1 2880.2.a.z 1
60.h even 2 1 2400.2.f.g 2
60.l odd 4 1 480.2.a.g yes 1
60.l odd 4 1 2400.2.a.i 1
120.i odd 2 1 4800.2.f.o 2
120.m even 2 1 4800.2.f.v 2
120.q odd 4 1 960.2.a.b 1
120.q odd 4 1 4800.2.a.cb 1
120.w even 4 1 960.2.a.k 1
120.w even 4 1 4800.2.a.s 1
240.z odd 4 1 3840.2.k.s 2
240.bb even 4 1 3840.2.k.n 2
240.bd odd 4 1 3840.2.k.s 2
240.bf even 4 1 3840.2.k.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.a.d 1 15.e even 4 1
480.2.a.g yes 1 60.l odd 4 1
960.2.a.b 1 120.q odd 4 1
960.2.a.k 1 120.w even 4 1
1440.2.a.c 1 5.c odd 4 1
1440.2.a.d 1 20.e even 4 1
2400.2.a.i 1 60.l odd 4 1
2400.2.a.z 1 15.e even 4 1
2400.2.f.g 2 12.b even 2 1
2400.2.f.g 2 60.h even 2 1
2400.2.f.l 2 3.b odd 2 1
2400.2.f.l 2 15.d odd 2 1
2880.2.a.z 1 40.k even 4 1
2880.2.a.ba 1 40.i odd 4 1
3840.2.k.n 2 240.bb even 4 1
3840.2.k.n 2 240.bf even 4 1
3840.2.k.s 2 240.z odd 4 1
3840.2.k.s 2 240.bd odd 4 1
4800.2.a.s 1 120.w even 4 1
4800.2.a.cb 1 120.q odd 4 1
4800.2.f.o 2 24.h odd 2 1
4800.2.f.o 2 120.i odd 2 1
4800.2.f.v 2 24.f even 2 1
4800.2.f.v 2 120.m even 2 1
7200.2.a.z 1 5.c odd 4 1
7200.2.a.ba 1 20.e even 4 1
7200.2.f.k 2 4.b odd 2 1
7200.2.f.k 2 20.d odd 2 1
7200.2.f.s 2 1.a even 1 1 trivial
7200.2.f.s 2 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(7200, [\chi])$$:

 $$T_{7}$$ T7 $$T_{11}$$ T11 $$T_{13}^{2} + 4$$ T13^2 + 4 $$T_{17}^{2} + 36$$ T17^2 + 36 $$T_{19} - 4$$ T19 - 4 $$T_{29} + 2$$ T29 + 2 $$T_{31} - 4$$ T31 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2} + 36$$
$19$ $$(T - 4)^{2}$$
$23$ $$T^{2} + 64$$
$29$ $$(T + 2)^{2}$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} + 100$$
$41$ $$(T + 2)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2} + 64$$
$53$ $$T^{2} + 4$$
$59$ $$(T - 8)^{2}$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} + 144$$
$71$ $$(T + 8)^{2}$$
$73$ $$T^{2} + 196$$
$79$ $$(T - 12)^{2}$$
$83$ $$T^{2} + 16$$
$89$ $$(T + 14)^{2}$$
$97$ $$T^{2} + 4$$