# Properties

 Label 7200.2.f.ba 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.

 $$f(q)$$ $$=$$ $$q + 2 \beta q^{7}+O(q^{10})$$ q + 2*b * q^7 $$q + 2 \beta q^{7} + 4 q^{11} - 3 \beta q^{13} - \beta q^{17} + 4 q^{19} + 10 q^{29} + 4 q^{31} - 5 \beta q^{37} - 2 q^{41} - 2 \beta q^{43} + 4 \beta q^{47} - 9 q^{49} + \beta q^{53} - 12 q^{59} - 10 q^{61} - 6 \beta q^{67} - 5 \beta q^{73} + 8 \beta q^{77} - 4 q^{79} - 2 \beta q^{83} - 6 q^{89} + 24 q^{91} - 7 \beta q^{97} +O(q^{100})$$ q + 2*b * q^7 + 4 * q^11 - 3*b * q^13 - b * q^17 + 4 * q^19 + 10 * q^29 + 4 * q^31 - 5*b * q^37 - 2 * q^41 - 2*b * q^43 + 4*b * q^47 - 9 * q^49 + b * q^53 - 12 * q^59 - 10 * q^61 - 6*b * q^67 - 5*b * q^73 + 8*b * q^77 - 4 * q^79 - 2*b * q^83 - 6 * q^89 + 24 * q^91 - 7*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q + 8 q^{11} + 8 q^{19} + 20 q^{29} + 8 q^{31} - 4 q^{41} - 18 q^{49} - 24 q^{59} - 20 q^{61} - 8 q^{79} - 12 q^{89} + 48 q^{91}+O(q^{100})$$ 2 * q + 8 * q^11 + 8 * q^19 + 20 * q^29 + 8 * q^31 - 4 * q^41 - 18 * q^49 - 24 * q^59 - 20 * q^61 - 8 * q^79 - 12 * q^89 + 48 * q^91

## 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 4.00000i 0 0 0
6049.2 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
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.ba 2
3.b odd 2 1 2400.2.f.d 2
4.b odd 2 1 7200.2.f.c 2
5.b even 2 1 inner 7200.2.f.ba 2
5.c odd 4 1 1440.2.a.n 1
5.c odd 4 1 7200.2.a.d 1
12.b even 2 1 2400.2.f.o 2
15.d odd 2 1 2400.2.f.d 2
15.e even 4 1 480.2.a.f yes 1
15.e even 4 1 2400.2.a.a 1
20.d odd 2 1 7200.2.f.c 2
20.e even 4 1 1440.2.a.g 1
20.e even 4 1 7200.2.a.bw 1
24.f even 2 1 4800.2.f.h 2
24.h odd 2 1 4800.2.f.bb 2
40.i odd 4 1 2880.2.a.p 1
40.k even 4 1 2880.2.a.c 1
60.h even 2 1 2400.2.f.o 2
60.l odd 4 1 480.2.a.a 1
60.l odd 4 1 2400.2.a.bh 1
120.i odd 2 1 4800.2.f.bb 2
120.m even 2 1 4800.2.f.h 2
120.q odd 4 1 960.2.a.m 1
120.q odd 4 1 4800.2.a.bg 1
120.w even 4 1 960.2.a.h 1
120.w even 4 1 4800.2.a.bo 1
240.z odd 4 1 3840.2.k.bb 2
240.bb even 4 1 3840.2.k.c 2
240.bd odd 4 1 3840.2.k.bb 2
240.bf even 4 1 3840.2.k.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.a.a 1 60.l odd 4 1
480.2.a.f yes 1 15.e even 4 1
960.2.a.h 1 120.w even 4 1
960.2.a.m 1 120.q odd 4 1
1440.2.a.g 1 20.e even 4 1
1440.2.a.n 1 5.c odd 4 1
2400.2.a.a 1 15.e even 4 1
2400.2.a.bh 1 60.l odd 4 1
2400.2.f.d 2 3.b odd 2 1
2400.2.f.d 2 15.d odd 2 1
2400.2.f.o 2 12.b even 2 1
2400.2.f.o 2 60.h even 2 1
2880.2.a.c 1 40.k even 4 1
2880.2.a.p 1 40.i odd 4 1
3840.2.k.c 2 240.bb even 4 1
3840.2.k.c 2 240.bf even 4 1
3840.2.k.bb 2 240.z odd 4 1
3840.2.k.bb 2 240.bd odd 4 1
4800.2.a.bg 1 120.q odd 4 1
4800.2.a.bo 1 120.w even 4 1
4800.2.f.h 2 24.f even 2 1
4800.2.f.h 2 120.m even 2 1
4800.2.f.bb 2 24.h odd 2 1
4800.2.f.bb 2 120.i odd 2 1
7200.2.a.d 1 5.c odd 4 1
7200.2.a.bw 1 20.e even 4 1
7200.2.f.c 2 4.b odd 2 1
7200.2.f.c 2 20.d odd 2 1
7200.2.f.ba 2 1.a even 1 1 trivial
7200.2.f.ba 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 $$S_{2}^{\mathrm{new}}(7200, [\chi])$$:

 $$T_{7}^{2} + 16$$ T7^2 + 16 $$T_{11} - 4$$ T11 - 4 $$T_{13}^{2} + 36$$ T13^2 + 36 $$T_{17}^{2} + 4$$ T17^2 + 4 $$T_{19} - 4$$ T19 - 4 $$T_{29} - 10$$ T29 - 10 $$T_{31} - 4$$ T31 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 16$$
$11$ $$(T - 4)^{2}$$
$13$ $$T^{2} + 36$$
$17$ $$T^{2} + 4$$
$19$ $$(T - 4)^{2}$$
$23$ $$T^{2}$$
$29$ $$(T - 10)^{2}$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} + 100$$
$41$ $$(T + 2)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2} + 64$$
$53$ $$T^{2} + 4$$
$59$ $$(T + 12)^{2}$$
$61$ $$(T + 10)^{2}$$
$67$ $$T^{2} + 144$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$(T + 4)^{2}$$
$83$ $$T^{2} + 16$$
$89$ $$(T + 6)^{2}$$
$97$ $$T^{2} + 196$$