# Properties

 Label 2700.3.b.d Level 2700 Weight 3 Character orbit 2700.b Analytic conductor 73.570 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: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 2700.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$73.5696713773$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ 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.

 $$f(q)$$ $$=$$ $$q + 11 i q^{7} +O(q^{10})$$ $$q + 11 i q^{7} -23 i q^{13} + 37 q^{19} -46 q^{31} -73 i q^{37} + 22 i q^{43} -72 q^{49} + 47 q^{61} -13 i q^{67} -143 i q^{73} -11 q^{79} + 253 q^{91} -169 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q + 74q^{19} - 92q^{31} - 144q^{49} + 94q^{61} - 22q^{79} + 506q^{91} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1349.1
 − 1.00000i 1.00000i
0 0 0 0 0 11.0000i 0 0 0
1349.2 0 0 0 0 0 11.0000i 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 2700.3.b.d 2
3.b odd 2 1 CM 2700.3.b.d 2
5.b even 2 1 inner 2700.3.b.d 2
5.c odd 4 1 108.3.c.a 1
5.c odd 4 1 2700.3.g.b 1
15.d odd 2 1 inner 2700.3.b.d 2
15.e even 4 1 108.3.c.a 1
15.e even 4 1 2700.3.g.b 1
20.e even 4 1 432.3.e.a 1
40.i odd 4 1 1728.3.e.c 1
40.k even 4 1 1728.3.e.b 1
45.k odd 12 2 324.3.g.a 2
45.l even 12 2 324.3.g.a 2
60.l odd 4 1 432.3.e.a 1
120.q odd 4 1 1728.3.e.b 1
120.w even 4 1 1728.3.e.c 1
180.v odd 12 2 1296.3.q.c 2
180.x even 12 2 1296.3.q.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.3.c.a 1 5.c odd 4 1
108.3.c.a 1 15.e even 4 1
324.3.g.a 2 45.k odd 12 2
324.3.g.a 2 45.l even 12 2
432.3.e.a 1 20.e even 4 1
432.3.e.a 1 60.l odd 4 1
1296.3.q.c 2 180.v odd 12 2
1296.3.q.c 2 180.x even 12 2
1728.3.e.b 1 40.k even 4 1
1728.3.e.b 1 120.q odd 4 1
1728.3.e.c 1 40.i odd 4 1
1728.3.e.c 1 120.w even 4 1
2700.3.b.d 2 1.a even 1 1 trivial
2700.3.b.d 2 3.b odd 2 1 CM
2700.3.b.d 2 5.b even 2 1 inner
2700.3.b.d 2 15.d odd 2 1 inner
2700.3.g.b 1 5.c odd 4 1
2700.3.g.b 1 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(2700, [\chi])$$:

 $$T_{7}^{2} + 121$$ $$T_{11}$$ $$T_{13}^{2} + 529$$ $$T_{17}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ 1
$7$ $$1 + 23 T^{2} + 2401 T^{4}$$
$11$ $$( 1 - 11 T )^{2}( 1 + 11 T )^{2}$$
$13$ $$1 + 191 T^{2} + 28561 T^{4}$$
$17$ $$( 1 + 289 T^{2} )^{2}$$
$19$ $$( 1 - 37 T + 361 T^{2} )^{2}$$
$23$ $$( 1 + 529 T^{2} )^{2}$$
$29$ $$( 1 - 29 T )^{2}( 1 + 29 T )^{2}$$
$31$ $$( 1 + 46 T + 961 T^{2} )^{2}$$
$37$ $$1 + 2591 T^{2} + 1874161 T^{4}$$
$41$ $$( 1 - 41 T )^{2}( 1 + 41 T )^{2}$$
$43$ $$1 - 3214 T^{2} + 3418801 T^{4}$$
$47$ $$( 1 + 2209 T^{2} )^{2}$$
$53$ $$( 1 + 2809 T^{2} )^{2}$$
$59$ $$( 1 - 59 T )^{2}( 1 + 59 T )^{2}$$
$61$ $$( 1 - 47 T + 3721 T^{2} )^{2}$$
$67$ $$1 - 8809 T^{2} + 20151121 T^{4}$$
$71$ $$( 1 - 71 T )^{2}( 1 + 71 T )^{2}$$
$73$ $$1 + 9791 T^{2} + 28398241 T^{4}$$
$79$ $$( 1 + 11 T + 6241 T^{2} )^{2}$$
$83$ $$( 1 + 6889 T^{2} )^{2}$$
$89$ $$( 1 - 89 T )^{2}( 1 + 89 T )^{2}$$
$97$ $$1 + 9743 T^{2} + 88529281 T^{4}$$