Properties

 Label 2100.2.d.b Level 2100 Weight 2 Character orbit 2100.d Analytic conductor 16.769 Analytic rank 0 Dimension 2 CM discriminant -3 Inner twists 4

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2100.d (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$16.7685844245$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 84) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 - 2 \zeta_{6} ) q^{3} + ( -1 - 2 \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})$$ $$q + ( 1 - 2 \zeta_{6} ) q^{3} + ( -1 - 2 \zeta_{6} ) q^{7} -3 q^{9} + ( -4 + 8 \zeta_{6} ) q^{13} + ( 2 - 4 \zeta_{6} ) q^{19} + ( -5 + 4 \zeta_{6} ) q^{21} + ( -3 + 6 \zeta_{6} ) q^{27} + ( -6 + 12 \zeta_{6} ) q^{31} -10 q^{37} + 12 q^{39} + 8 q^{43} + ( -3 + 8 \zeta_{6} ) q^{49} -6 q^{57} + ( -4 + 8 \zeta_{6} ) q^{61} + ( 3 + 6 \zeta_{6} ) q^{63} + 16 q^{67} + ( -8 + 16 \zeta_{6} ) q^{73} -4 q^{79} + 9 q^{81} + ( 20 - 16 \zeta_{6} ) q^{91} + 18 q^{93} + ( 8 - 16 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{7} - 6q^{9} + O(q^{10})$$ $$2q - 4q^{7} - 6q^{9} - 6q^{21} - 20q^{37} + 24q^{39} + 16q^{43} + 2q^{49} - 12q^{57} + 12q^{63} + 32q^{67} - 8q^{79} + 18q^{81} + 24q^{91} + 36q^{93} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times$$.

 $$n$$ $$701$$ $$1051$$ $$1177$$ $$1501$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1301.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.73205i 0 0 0 −2.00000 1.73205i 0 −3.00000 0
1301.2 0 1.73205i 0 0 0 −2.00000 + 1.73205i 0 −3.00000 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 $$\Q(\sqrt{-3})$$
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.d.b 2
3.b odd 2 1 CM 2100.2.d.b 2
5.b even 2 1 84.2.f.a 2
5.c odd 4 2 2100.2.f.e 4
7.b odd 2 1 inner 2100.2.d.b 2
15.d odd 2 1 84.2.f.a 2
15.e even 4 2 2100.2.f.e 4
20.d odd 2 1 336.2.k.a 2
21.c even 2 1 inner 2100.2.d.b 2
35.c odd 2 1 84.2.f.a 2
35.f even 4 2 2100.2.f.e 4
35.i odd 6 1 588.2.k.a 2
35.i odd 6 1 588.2.k.e 2
35.j even 6 1 588.2.k.a 2
35.j even 6 1 588.2.k.e 2
40.e odd 2 1 1344.2.k.a 2
40.f even 2 1 1344.2.k.b 2
45.h odd 6 1 2268.2.x.c 2
45.h odd 6 1 2268.2.x.e 2
45.j even 6 1 2268.2.x.c 2
45.j even 6 1 2268.2.x.e 2
60.h even 2 1 336.2.k.a 2
105.g even 2 1 84.2.f.a 2
105.k odd 4 2 2100.2.f.e 4
105.o odd 6 1 588.2.k.a 2
105.o odd 6 1 588.2.k.e 2
105.p even 6 1 588.2.k.a 2
105.p even 6 1 588.2.k.e 2
120.i odd 2 1 1344.2.k.b 2
120.m even 2 1 1344.2.k.a 2
140.c even 2 1 336.2.k.a 2
280.c odd 2 1 1344.2.k.b 2
280.n even 2 1 1344.2.k.a 2
315.z even 6 1 2268.2.x.c 2
315.z even 6 1 2268.2.x.e 2
315.bg odd 6 1 2268.2.x.c 2
315.bg odd 6 1 2268.2.x.e 2
420.o odd 2 1 336.2.k.a 2
840.b odd 2 1 1344.2.k.a 2
840.u even 2 1 1344.2.k.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.f.a 2 5.b even 2 1
84.2.f.a 2 15.d odd 2 1
84.2.f.a 2 35.c odd 2 1
84.2.f.a 2 105.g even 2 1
336.2.k.a 2 20.d odd 2 1
336.2.k.a 2 60.h even 2 1
336.2.k.a 2 140.c even 2 1
336.2.k.a 2 420.o odd 2 1
588.2.k.a 2 35.i odd 6 1
588.2.k.a 2 35.j even 6 1
588.2.k.a 2 105.o odd 6 1
588.2.k.a 2 105.p even 6 1
588.2.k.e 2 35.i odd 6 1
588.2.k.e 2 35.j even 6 1
588.2.k.e 2 105.o odd 6 1
588.2.k.e 2 105.p even 6 1
1344.2.k.a 2 40.e odd 2 1
1344.2.k.a 2 120.m even 2 1
1344.2.k.a 2 280.n even 2 1
1344.2.k.a 2 840.b odd 2 1
1344.2.k.b 2 40.f even 2 1
1344.2.k.b 2 120.i odd 2 1
1344.2.k.b 2 280.c odd 2 1
1344.2.k.b 2 840.u even 2 1
2100.2.d.b 2 1.a even 1 1 trivial
2100.2.d.b 2 3.b odd 2 1 CM
2100.2.d.b 2 7.b odd 2 1 inner
2100.2.d.b 2 21.c even 2 1 inner
2100.2.f.e 4 5.c odd 4 2
2100.2.f.e 4 15.e even 4 2
2100.2.f.e 4 35.f even 4 2
2100.2.f.e 4 105.k odd 4 2
2268.2.x.c 2 45.h odd 6 1
2268.2.x.c 2 45.j even 6 1
2268.2.x.c 2 315.z even 6 1
2268.2.x.c 2 315.bg odd 6 1
2268.2.x.e 2 45.h odd 6 1
2268.2.x.e 2 45.j even 6 1
2268.2.x.e 2 315.z even 6 1
2268.2.x.e 2 315.bg odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2100, [\chi])$$:

 $$T_{11}$$ $$T_{13}^{2} + 48$$ $$T_{17}$$ $$T_{37} + 10$$ $$T_{41}$$ $$T_{43} - 8$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 3 T^{2}$$
$5$ 1
$7$ $$1 + 4 T + 7 T^{2}$$
$11$ $$( 1 - 11 T^{2} )^{2}$$
$13$ $$( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )$$
$17$ $$( 1 + 17 T^{2} )^{2}$$
$19$ $$( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$
$23$ $$( 1 - 23 T^{2} )^{2}$$
$29$ $$( 1 - 29 T^{2} )^{2}$$
$31$ $$( 1 - 4 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} )$$
$37$ $$( 1 + 10 T + 37 T^{2} )^{2}$$
$41$ $$( 1 + 41 T^{2} )^{2}$$
$43$ $$( 1 - 8 T + 43 T^{2} )^{2}$$
$47$ $$( 1 + 47 T^{2} )^{2}$$
$53$ $$( 1 - 53 T^{2} )^{2}$$
$59$ $$( 1 + 59 T^{2} )^{2}$$
$61$ $$( 1 - 14 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} )$$
$67$ $$( 1 - 16 T + 67 T^{2} )^{2}$$
$71$ $$( 1 - 71 T^{2} )^{2}$$
$73$ $$( 1 - 10 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} )$$
$79$ $$( 1 + 4 T + 79 T^{2} )^{2}$$
$83$ $$( 1 + 83 T^{2} )^{2}$$
$89$ $$( 1 + 89 T^{2} )^{2}$$
$97$ $$( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} )$$