# Properties

 Label 2100.2.f.e Level 2100 Weight 2 Character orbit 2100.f Analytic conductor 16.769 Analytic rank 0 Dimension 4 CM discriminant -3 Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2100.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.7685844245$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ 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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( -2 \zeta_{12} - \zeta_{12}^{3} ) q^{7} + 3 q^{9} +O(q^{10})$$ $$q + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( -2 \zeta_{12} - \zeta_{12}^{3} ) q^{7} + 3 q^{9} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{13} + ( 2 - 4 \zeta_{12}^{2} ) q^{19} + ( -1 - 4 \zeta_{12}^{2} ) q^{21} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{27} + ( 6 - 12 \zeta_{12}^{2} ) q^{31} -10 \zeta_{12}^{3} q^{37} -12 q^{39} -8 \zeta_{12}^{3} q^{43} + ( -5 + 8 \zeta_{12}^{2} ) q^{49} -6 \zeta_{12}^{3} q^{57} + ( 4 - 8 \zeta_{12}^{2} ) q^{61} + ( -6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{63} + 16 \zeta_{12}^{3} q^{67} + ( -16 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{73} + 4 q^{79} + 9 q^{81} + ( 4 + 16 \zeta_{12}^{2} ) q^{91} -18 \zeta_{12}^{3} q^{93} + ( -16 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 12q^{9} + O(q^{10})$$ $$4q + 12q^{9} - 12q^{21} - 48q^{39} - 4q^{49} + 16q^{79} + 36q^{81} + 48q^{91} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1049.1
 −0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i
0 −1.73205 0 0 0 1.73205 2.00000i 0 3.00000 0
1049.2 0 −1.73205 0 0 0 1.73205 + 2.00000i 0 3.00000 0
1049.3 0 1.73205 0 0 0 −1.73205 2.00000i 0 3.00000 0
1049.4 0 1.73205 0 0 0 −1.73205 + 2.00000i 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 $$\Q(\sqrt{-3})$$
5.b even 2 1 inner
7.b odd 2 1 inner
15.d odd 2 1 inner
21.c even 2 1 inner
35.c odd 2 1 inner
105.g 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.f.e 4
3.b odd 2 1 CM 2100.2.f.e 4
5.b even 2 1 inner 2100.2.f.e 4
5.c odd 4 1 84.2.f.a 2
5.c odd 4 1 2100.2.d.b 2
7.b odd 2 1 inner 2100.2.f.e 4
15.d odd 2 1 inner 2100.2.f.e 4
15.e even 4 1 84.2.f.a 2
15.e even 4 1 2100.2.d.b 2
20.e even 4 1 336.2.k.a 2
21.c even 2 1 inner 2100.2.f.e 4
35.c odd 2 1 inner 2100.2.f.e 4
35.f even 4 1 84.2.f.a 2
35.f even 4 1 2100.2.d.b 2
35.k even 12 1 588.2.k.a 2
35.k even 12 1 588.2.k.e 2
35.l odd 12 1 588.2.k.a 2
35.l odd 12 1 588.2.k.e 2
40.i odd 4 1 1344.2.k.b 2
40.k even 4 1 1344.2.k.a 2
45.k odd 12 1 2268.2.x.c 2
45.k odd 12 1 2268.2.x.e 2
45.l even 12 1 2268.2.x.c 2
45.l even 12 1 2268.2.x.e 2
60.l odd 4 1 336.2.k.a 2
105.g even 2 1 inner 2100.2.f.e 4
105.k odd 4 1 84.2.f.a 2
105.k odd 4 1 2100.2.d.b 2
105.w odd 12 1 588.2.k.a 2
105.w odd 12 1 588.2.k.e 2
105.x even 12 1 588.2.k.a 2
105.x even 12 1 588.2.k.e 2
120.q odd 4 1 1344.2.k.a 2
120.w even 4 1 1344.2.k.b 2
140.j odd 4 1 336.2.k.a 2
280.s even 4 1 1344.2.k.b 2
280.y odd 4 1 1344.2.k.a 2
315.cb even 12 1 2268.2.x.c 2
315.cb even 12 1 2268.2.x.e 2
315.cf odd 12 1 2268.2.x.c 2
315.cf odd 12 1 2268.2.x.e 2
420.w even 4 1 336.2.k.a 2
840.bm even 4 1 1344.2.k.a 2
840.bp odd 4 1 1344.2.k.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.f.a 2 5.c odd 4 1
84.2.f.a 2 15.e even 4 1
84.2.f.a 2 35.f even 4 1
84.2.f.a 2 105.k odd 4 1
336.2.k.a 2 20.e even 4 1
336.2.k.a 2 60.l odd 4 1
336.2.k.a 2 140.j odd 4 1
336.2.k.a 2 420.w even 4 1
588.2.k.a 2 35.k even 12 1
588.2.k.a 2 35.l odd 12 1
588.2.k.a 2 105.w odd 12 1
588.2.k.a 2 105.x even 12 1
588.2.k.e 2 35.k even 12 1
588.2.k.e 2 35.l odd 12 1
588.2.k.e 2 105.w odd 12 1
588.2.k.e 2 105.x even 12 1
1344.2.k.a 2 40.k even 4 1
1344.2.k.a 2 120.q odd 4 1
1344.2.k.a 2 280.y odd 4 1
1344.2.k.a 2 840.bm even 4 1
1344.2.k.b 2 40.i odd 4 1
1344.2.k.b 2 120.w even 4 1
1344.2.k.b 2 280.s even 4 1
1344.2.k.b 2 840.bp odd 4 1
2100.2.d.b 2 5.c odd 4 1
2100.2.d.b 2 15.e even 4 1
2100.2.d.b 2 35.f even 4 1
2100.2.d.b 2 105.k odd 4 1
2100.2.f.e 4 1.a even 1 1 trivial
2100.2.f.e 4 3.b odd 2 1 CM
2100.2.f.e 4 5.b even 2 1 inner
2100.2.f.e 4 7.b odd 2 1 inner
2100.2.f.e 4 15.d odd 2 1 inner
2100.2.f.e 4 21.c even 2 1 inner
2100.2.f.e 4 35.c odd 2 1 inner
2100.2.f.e 4 105.g even 2 1 inner
2268.2.x.c 2 45.k odd 12 1
2268.2.x.c 2 45.l even 12 1
2268.2.x.c 2 315.cb even 12 1
2268.2.x.c 2 315.cf odd 12 1
2268.2.x.e 2 45.k odd 12 1
2268.2.x.e 2 45.l even 12 1
2268.2.x.e 2 315.cb even 12 1
2268.2.x.e 2 315.cf odd 12 1

## Hecke kernels

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

 $$T_{11}$$ $$T_{13}^{2} - 48$$ $$T_{23}$$ $$T_{41}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - 3 T^{2} )^{2}$$
$5$ 1
$7$ $$1 + 2 T^{2} + 49 T^{4}$$
$11$ $$( 1 - 11 T^{2} )^{4}$$
$13$ $$( 1 - 22 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 - 17 T^{2} )^{4}$$
$19$ $$( 1 - 8 T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2}$$
$23$ $$( 1 + 23 T^{2} )^{4}$$
$29$ $$( 1 - 29 T^{2} )^{4}$$
$31$ $$( 1 - 4 T + 31 T^{2} )^{2}( 1 + 4 T + 31 T^{2} )^{2}$$
$37$ $$( 1 + 26 T^{2} + 1369 T^{4} )^{2}$$
$41$ $$( 1 + 41 T^{2} )^{4}$$
$43$ $$( 1 - 22 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$( 1 - 47 T^{2} )^{4}$$
$53$ $$( 1 + 53 T^{2} )^{4}$$
$59$ $$( 1 + 59 T^{2} )^{4}$$
$61$ $$( 1 - 14 T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2}$$
$67$ $$( 1 + 122 T^{2} + 4489 T^{4} )^{2}$$
$71$ $$( 1 - 71 T^{2} )^{4}$$
$73$ $$( 1 - 46 T^{2} + 5329 T^{4} )^{2}$$
$79$ $$( 1 - 4 T + 79 T^{2} )^{4}$$
$83$ $$( 1 - 83 T^{2} )^{4}$$
$89$ $$( 1 + 89 T^{2} )^{4}$$
$97$ $$( 1 + 2 T^{2} + 9409 T^{4} )^{2}$$