# Properties

 Label 2700.2.d.g Level $2700$ Weight $2$ Character orbit 2700.d Analytic conductor $21.560$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2700 = 2^{2} \cdot 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2700.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$21.5596085457$$ 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: $$1$$ Twist minimal: no (minimal twist has level 108) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 5 i q^{7}+O(q^{10})$$ q + 5*i * q^7 $$q + 5 i q^{7} + 7 i q^{13} + q^{19} - 4 q^{31} - i q^{37} - 8 i q^{43} - 18 q^{49} - 13 q^{61} + 11 i q^{67} - 17 i q^{73} + 13 q^{79} - 35 q^{91} + 5 i q^{97} +O(q^{100})$$ q + 5*i * q^7 + 7*i * q^13 + q^19 - 4 * q^31 - i * q^37 - 8*i * q^43 - 18 * q^49 - 13 * q^61 + 11*i * q^67 - 17*i * q^73 + 13 * q^79 - 35 * q^91 + 5*i * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q + 2 q^{19} - 8 q^{31} - 36 q^{49} - 26 q^{61} + 26 q^{79} - 70 q^{91}+O(q^{100})$$ 2 * q + 2 * q^19 - 8 * q^31 - 36 * q^49 - 26 * q^61 + 26 * q^79 - 70 * q^91

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times$$.

 $$n$$ $$1001$$ $$1351$$ $$2377$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
649.1
 − 1.00000i 1.00000i
0 0 0 0 0 5.00000i 0 0 0
649.2 0 0 0 0 0 5.00000i 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2700.2.d.g 2
3.b odd 2 1 CM 2700.2.d.g 2
5.b even 2 1 inner 2700.2.d.g 2
5.c odd 4 1 108.2.a.a 1
5.c odd 4 1 2700.2.a.b 1
15.d odd 2 1 inner 2700.2.d.g 2
15.e even 4 1 108.2.a.a 1
15.e even 4 1 2700.2.a.b 1
20.e even 4 1 432.2.a.d 1
35.f even 4 1 5292.2.a.j 1
40.i odd 4 1 1728.2.a.p 1
40.k even 4 1 1728.2.a.m 1
45.k odd 12 2 324.2.e.b 2
45.l even 12 2 324.2.e.b 2
60.l odd 4 1 432.2.a.d 1
105.k odd 4 1 5292.2.a.j 1
120.q odd 4 1 1728.2.a.m 1
120.w even 4 1 1728.2.a.p 1
180.v odd 12 2 1296.2.i.j 2
180.x even 12 2 1296.2.i.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.2.a.a 1 5.c odd 4 1
108.2.a.a 1 15.e even 4 1
324.2.e.b 2 45.k odd 12 2
324.2.e.b 2 45.l even 12 2
432.2.a.d 1 20.e even 4 1
432.2.a.d 1 60.l odd 4 1
1296.2.i.j 2 180.v odd 12 2
1296.2.i.j 2 180.x even 12 2
1728.2.a.m 1 40.k even 4 1
1728.2.a.m 1 120.q odd 4 1
1728.2.a.p 1 40.i odd 4 1
1728.2.a.p 1 120.w even 4 1
2700.2.a.b 1 5.c odd 4 1
2700.2.a.b 1 15.e even 4 1
2700.2.d.g 2 1.a even 1 1 trivial
2700.2.d.g 2 3.b odd 2 1 CM
2700.2.d.g 2 5.b even 2 1 inner
2700.2.d.g 2 15.d odd 2 1 inner
5292.2.a.j 1 35.f even 4 1
5292.2.a.j 1 105.k odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2700, [\chi])$$:

 $$T_{7}^{2} + 25$$ T7^2 + 25 $$T_{11}$$ T11 $$T_{13}^{2} + 49$$ T13^2 + 49 $$T_{29}$$ T29

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 25$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 49$$
$17$ $$T^{2}$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} + 1$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 64$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$(T + 13)^{2}$$
$67$ $$T^{2} + 121$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 289$$
$79$ $$(T - 13)^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 25$$