Properties

 Label 2700.2.i.b Level $2700$ Weight $2$ Character orbit 2700.i Analytic conductor $21.560$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

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

Newform invariants

 Self dual: no Analytic conductor: $$21.5596085457$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 36) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{7} + ( 3 - 3 \zeta_{6} ) q^{11} -\zeta_{6} q^{13} + 6 q^{17} -4 q^{19} + 3 \zeta_{6} q^{23} + ( 3 - 3 \zeta_{6} ) q^{29} -5 \zeta_{6} q^{31} -2 q^{37} + 3 \zeta_{6} q^{41} + ( -1 + \zeta_{6} ) q^{43} + ( 9 - 9 \zeta_{6} ) q^{47} + 6 \zeta_{6} q^{49} -6 q^{53} -3 \zeta_{6} q^{59} + ( 13 - 13 \zeta_{6} ) q^{61} -7 \zeta_{6} q^{67} + 12 q^{71} + 10 q^{73} + 3 \zeta_{6} q^{77} + ( -11 + 11 \zeta_{6} ) q^{79} + ( 9 - 9 \zeta_{6} ) q^{83} -6 q^{89} + q^{91} + ( 11 - 11 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{7} + O(q^{10})$$ $$2q - q^{7} + 3q^{11} - q^{13} + 12q^{17} - 8q^{19} + 3q^{23} + 3q^{29} - 5q^{31} - 4q^{37} + 3q^{41} - q^{43} + 9q^{47} + 6q^{49} - 12q^{53} - 3q^{59} + 13q^{61} - 7q^{67} + 24q^{71} + 20q^{73} + 3q^{77} - 11q^{79} + 9q^{83} - 12q^{89} + 2q^{91} + 11q^{97} + O(q^{100})$$

Character values

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

 $$n$$ $$1001$$ $$1351$$ $$2377$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
901.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 0 0 −0.500000 0.866025i 0 0 0
1801.1 0 0 0 0 0 −0.500000 + 0.866025i 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
9.c even 3 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2700.2.i.b 2
3.b odd 2 1 900.2.i.b 2
5.b even 2 1 108.2.e.a 2
5.c odd 4 2 2700.2.s.b 4
9.c even 3 1 inner 2700.2.i.b 2
9.c even 3 1 8100.2.a.g 1
9.d odd 6 1 900.2.i.b 2
9.d odd 6 1 8100.2.a.j 1
15.d odd 2 1 36.2.e.a 2
15.e even 4 2 900.2.s.b 4
20.d odd 2 1 432.2.i.c 2
35.c odd 2 1 5292.2.j.a 2
35.i odd 6 1 5292.2.i.a 2
35.i odd 6 1 5292.2.l.c 2
35.j even 6 1 5292.2.i.c 2
35.j even 6 1 5292.2.l.a 2
40.e odd 2 1 1728.2.i.c 2
40.f even 2 1 1728.2.i.d 2
45.h odd 6 1 36.2.e.a 2
45.h odd 6 1 324.2.a.c 1
45.j even 6 1 108.2.e.a 2
45.j even 6 1 324.2.a.a 1
45.k odd 12 2 2700.2.s.b 4
45.k odd 12 2 8100.2.d.c 2
45.l even 12 2 900.2.s.b 4
45.l even 12 2 8100.2.d.h 2
60.h even 2 1 144.2.i.a 2
105.g even 2 1 1764.2.j.b 2
105.o odd 6 1 1764.2.i.a 2
105.o odd 6 1 1764.2.l.c 2
105.p even 6 1 1764.2.i.c 2
105.p even 6 1 1764.2.l.a 2
120.i odd 2 1 576.2.i.f 2
120.m even 2 1 576.2.i.e 2
180.n even 6 1 144.2.i.a 2
180.n even 6 1 1296.2.a.k 1
180.p odd 6 1 432.2.i.c 2
180.p odd 6 1 1296.2.a.b 1
315.q odd 6 1 5292.2.l.c 2
315.r even 6 1 5292.2.l.a 2
315.u even 6 1 1764.2.i.c 2
315.v odd 6 1 1764.2.i.a 2
315.z even 6 1 1764.2.j.b 2
315.bg odd 6 1 5292.2.j.a 2
315.bn odd 6 1 5292.2.i.a 2
315.bo even 6 1 5292.2.i.c 2
315.bq even 6 1 1764.2.l.a 2
315.br odd 6 1 1764.2.l.c 2
360.z odd 6 1 1728.2.i.c 2
360.z odd 6 1 5184.2.a.bb 1
360.bd even 6 1 576.2.i.e 2
360.bd even 6 1 5184.2.a.f 1
360.bh odd 6 1 576.2.i.f 2
360.bh odd 6 1 5184.2.a.e 1
360.bk even 6 1 1728.2.i.d 2
360.bk even 6 1 5184.2.a.ba 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.e.a 2 15.d odd 2 1
36.2.e.a 2 45.h odd 6 1
108.2.e.a 2 5.b even 2 1
108.2.e.a 2 45.j even 6 1
144.2.i.a 2 60.h even 2 1
144.2.i.a 2 180.n even 6 1
324.2.a.a 1 45.j even 6 1
324.2.a.c 1 45.h odd 6 1
432.2.i.c 2 20.d odd 2 1
432.2.i.c 2 180.p odd 6 1
576.2.i.e 2 120.m even 2 1
576.2.i.e 2 360.bd even 6 1
576.2.i.f 2 120.i odd 2 1
576.2.i.f 2 360.bh odd 6 1
900.2.i.b 2 3.b odd 2 1
900.2.i.b 2 9.d odd 6 1
900.2.s.b 4 15.e even 4 2
900.2.s.b 4 45.l even 12 2
1296.2.a.b 1 180.p odd 6 1
1296.2.a.k 1 180.n even 6 1
1728.2.i.c 2 40.e odd 2 1
1728.2.i.c 2 360.z odd 6 1
1728.2.i.d 2 40.f even 2 1
1728.2.i.d 2 360.bk even 6 1
1764.2.i.a 2 105.o odd 6 1
1764.2.i.a 2 315.v odd 6 1
1764.2.i.c 2 105.p even 6 1
1764.2.i.c 2 315.u even 6 1
1764.2.j.b 2 105.g even 2 1
1764.2.j.b 2 315.z even 6 1
1764.2.l.a 2 105.p even 6 1
1764.2.l.a 2 315.bq even 6 1
1764.2.l.c 2 105.o odd 6 1
1764.2.l.c 2 315.br odd 6 1
2700.2.i.b 2 1.a even 1 1 trivial
2700.2.i.b 2 9.c even 3 1 inner
2700.2.s.b 4 5.c odd 4 2
2700.2.s.b 4 45.k odd 12 2
5184.2.a.e 1 360.bh odd 6 1
5184.2.a.f 1 360.bd even 6 1
5184.2.a.ba 1 360.bk even 6 1
5184.2.a.bb 1 360.z odd 6 1
5292.2.i.a 2 35.i odd 6 1
5292.2.i.a 2 315.bn odd 6 1
5292.2.i.c 2 35.j even 6 1
5292.2.i.c 2 315.bo even 6 1
5292.2.j.a 2 35.c odd 2 1
5292.2.j.a 2 315.bg odd 6 1
5292.2.l.a 2 35.j even 6 1
5292.2.l.a 2 315.r even 6 1
5292.2.l.c 2 35.i odd 6 1
5292.2.l.c 2 315.q odd 6 1
8100.2.a.g 1 9.c even 3 1
8100.2.a.j 1 9.d odd 6 1
8100.2.d.c 2 45.k odd 12 2
8100.2.d.h 2 45.l even 12 2

Hecke kernels

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

 $$T_{7}^{2} + T_{7} + 1$$ $$T_{11}^{2} - 3 T_{11} + 9$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$1 + T + T^{2}$$
$11$ $$9 - 3 T + T^{2}$$
$13$ $$1 + T + T^{2}$$
$17$ $$( -6 + T )^{2}$$
$19$ $$( 4 + T )^{2}$$
$23$ $$9 - 3 T + T^{2}$$
$29$ $$9 - 3 T + T^{2}$$
$31$ $$25 + 5 T + T^{2}$$
$37$ $$( 2 + T )^{2}$$
$41$ $$9 - 3 T + T^{2}$$
$43$ $$1 + T + T^{2}$$
$47$ $$81 - 9 T + T^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$9 + 3 T + T^{2}$$
$61$ $$169 - 13 T + T^{2}$$
$67$ $$49 + 7 T + T^{2}$$
$71$ $$( -12 + T )^{2}$$
$73$ $$( -10 + T )^{2}$$
$79$ $$121 + 11 T + T^{2}$$
$83$ $$81 - 9 T + T^{2}$$
$89$ $$( 6 + T )^{2}$$
$97$ $$121 - 11 T + T^{2}$$