# Properties

 Label 7200.2.f.d 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2400) 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.

 $$f(q)$$ $$=$$ $$q + i q^{7}+O(q^{10})$$ q + i * q^7 $$q + i q^{7} - 4 q^{11} - 3 i q^{13} - 4 i q^{17} - q^{19} - 8 q^{29} - q^{31} - 2 i q^{37} - 2 q^{41} + 11 i q^{43} + 2 i q^{47} + 6 q^{49} + 10 i q^{53} + 6 q^{59} + 11 q^{61} + 9 i q^{67} + 6 q^{71} - 14 i q^{73} - 4 i q^{77} + 16 q^{79} + 2 i q^{83} + 3 q^{91} + 11 i q^{97} +O(q^{100})$$ q + i * q^7 - 4 * q^11 - 3*i * q^13 - 4*i * q^17 - q^19 - 8 * q^29 - q^31 - 2*i * q^37 - 2 * q^41 + 11*i * q^43 + 2*i * q^47 + 6 * q^49 + 10*i * q^53 + 6 * q^59 + 11 * q^61 + 9*i * q^67 + 6 * q^71 - 14*i * q^73 - 4*i * q^77 + 16 * q^79 + 2*i * q^83 + 3 * q^91 + 11*i * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q - 8 q^{11} - 2 q^{19} - 16 q^{29} - 2 q^{31} - 4 q^{41} + 12 q^{49} + 12 q^{59} + 22 q^{61} + 12 q^{71} + 32 q^{79} + 6 q^{91}+O(q^{100})$$ 2 * q - 8 * q^11 - 2 * q^19 - 16 * q^29 - 2 * q^31 - 4 * q^41 + 12 * q^49 + 12 * q^59 + 22 * q^61 + 12 * q^71 + 32 * q^79 + 6 * 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 1.00000i 0 0 0
6049.2 0 0 0 0 0 1.00000i 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.d 2
3.b odd 2 1 2400.2.f.p 2
4.b odd 2 1 7200.2.f.z 2
5.b even 2 1 inner 7200.2.f.d 2
5.c odd 4 1 7200.2.a.q 1
5.c odd 4 1 7200.2.a.bh 1
12.b even 2 1 2400.2.f.c 2
15.d odd 2 1 2400.2.f.p 2
15.e even 4 1 2400.2.a.h yes 1
15.e even 4 1 2400.2.a.bd yes 1
20.d odd 2 1 7200.2.f.z 2
20.e even 4 1 7200.2.a.t 1
20.e even 4 1 7200.2.a.bk 1
24.f even 2 1 4800.2.f.bd 2
24.h odd 2 1 4800.2.f.g 2
60.h even 2 1 2400.2.f.c 2
60.l odd 4 1 2400.2.a.e 1
60.l odd 4 1 2400.2.a.ba yes 1
120.i odd 2 1 4800.2.f.g 2
120.m even 2 1 4800.2.f.bd 2
120.q odd 4 1 4800.2.a.y 1
120.q odd 4 1 4800.2.a.by 1
120.w even 4 1 4800.2.a.v 1
120.w even 4 1 4800.2.a.bv 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2400.2.a.e 1 60.l odd 4 1
2400.2.a.h yes 1 15.e even 4 1
2400.2.a.ba yes 1 60.l odd 4 1
2400.2.a.bd yes 1 15.e even 4 1
2400.2.f.c 2 12.b even 2 1
2400.2.f.c 2 60.h even 2 1
2400.2.f.p 2 3.b odd 2 1
2400.2.f.p 2 15.d odd 2 1
4800.2.a.v 1 120.w even 4 1
4800.2.a.y 1 120.q odd 4 1
4800.2.a.bv 1 120.w even 4 1
4800.2.a.by 1 120.q odd 4 1
4800.2.f.g 2 24.h odd 2 1
4800.2.f.g 2 120.i odd 2 1
4800.2.f.bd 2 24.f even 2 1
4800.2.f.bd 2 120.m even 2 1
7200.2.a.q 1 5.c odd 4 1
7200.2.a.t 1 20.e even 4 1
7200.2.a.bh 1 5.c odd 4 1
7200.2.a.bk 1 20.e even 4 1
7200.2.f.d 2 1.a even 1 1 trivial
7200.2.f.d 2 5.b even 2 1 inner
7200.2.f.z 2 4.b odd 2 1
7200.2.f.z 2 20.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 $$S_{2}^{\mathrm{new}}(7200, [\chi])$$:

 $$T_{7}^{2} + 1$$ T7^2 + 1 $$T_{11} + 4$$ T11 + 4 $$T_{13}^{2} + 9$$ T13^2 + 9 $$T_{17}^{2} + 16$$ T17^2 + 16 $$T_{19} + 1$$ T19 + 1 $$T_{29} + 8$$ T29 + 8 $$T_{31} + 1$$ T31 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1$$
$11$ $$(T + 4)^{2}$$
$13$ $$T^{2} + 9$$
$17$ $$T^{2} + 16$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2}$$
$29$ $$(T + 8)^{2}$$
$31$ $$(T + 1)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T + 2)^{2}$$
$43$ $$T^{2} + 121$$
$47$ $$T^{2} + 4$$
$53$ $$T^{2} + 100$$
$59$ $$(T - 6)^{2}$$
$61$ $$(T - 11)^{2}$$
$67$ $$T^{2} + 81$$
$71$ $$(T - 6)^{2}$$
$73$ $$T^{2} + 196$$
$79$ $$(T - 16)^{2}$$
$83$ $$T^{2} + 4$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 121$$