# Properties

 Label 2100.2.bo.e Level 2100 Weight 2 Character orbit 2100.bo 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.bo (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.7685844245$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( -\zeta_{12} - 2 \zeta_{12}^{3} ) q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( -\zeta_{12} - 2 \zeta_{12}^{3} ) q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{13} + ( 5 + 5 \zeta_{12}^{2} ) q^{19} + ( -5 + 4 \zeta_{12}^{2} ) q^{21} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{27} + ( -2 + \zeta_{12}^{2} ) q^{31} + 11 \zeta_{12} q^{37} -9 \zeta_{12}^{2} q^{39} -13 \zeta_{12}^{3} q^{43} + ( -8 + 5 \zeta_{12}^{2} ) q^{49} -15 \zeta_{12}^{3} q^{57} + ( -4 - 4 \zeta_{12}^{2} ) q^{61} + ( 9 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{63} + ( 5 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{67} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{73} + ( 17 - 17 \zeta_{12}^{2} ) q^{79} -9 \zeta_{12}^{2} q^{81} + ( 3 - 15 \zeta_{12}^{2} ) q^{91} + 3 \zeta_{12} q^{93} + ( -16 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 6q^{9} + O(q^{10})$$ $$4q - 6q^{9} + 30q^{19} - 12q^{21} - 6q^{31} - 18q^{39} - 22q^{49} - 24q^{61} + 34q^{79} - 18q^{81} - 18q^{91} + O(q^{100})$$

## Character values

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

 $$n$$ $$701$$ $$1051$$ $$1177$$ $$1501$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$1 - \zeta_{12}^{2}$$

## 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
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
0 −0.866025 1.50000i 0 0 0 −0.866025 2.50000i 0 −1.50000 + 2.59808i 0
1349.2 0 0.866025 + 1.50000i 0 0 0 0.866025 + 2.50000i 0 −1.50000 + 2.59808i 0
1949.1 0 −0.866025 + 1.50000i 0 0 0 −0.866025 + 2.50000i 0 −1.50000 2.59808i 0
1949.2 0 0.866025 1.50000i 0 0 0 0.866025 2.50000i 0 −1.50000 2.59808i 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.d odd 6 1 inner
15.d odd 2 1 inner
21.g even 6 1 inner
35.i odd 6 1 inner
105.p even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.bo.e 4
3.b odd 2 1 CM 2100.2.bo.e 4
5.b even 2 1 inner 2100.2.bo.e 4
5.c odd 4 1 84.2.k.c 2
5.c odd 4 1 2100.2.bi.d 2
7.d odd 6 1 inner 2100.2.bo.e 4
15.d odd 2 1 inner 2100.2.bo.e 4
15.e even 4 1 84.2.k.c 2
15.e even 4 1 2100.2.bi.d 2
20.e even 4 1 336.2.bc.a 2
21.g even 6 1 inner 2100.2.bo.e 4
35.f even 4 1 588.2.k.b 2
35.i odd 6 1 inner 2100.2.bo.e 4
35.k even 12 1 84.2.k.c 2
35.k even 12 1 588.2.f.b 2
35.k even 12 1 2100.2.bi.d 2
35.l odd 12 1 588.2.f.b 2
35.l odd 12 1 588.2.k.b 2
45.k odd 12 1 2268.2.w.e 2
45.k odd 12 1 2268.2.bm.d 2
45.l even 12 1 2268.2.w.e 2
45.l even 12 1 2268.2.bm.d 2
60.l odd 4 1 336.2.bc.a 2
105.k odd 4 1 588.2.k.b 2
105.p even 6 1 inner 2100.2.bo.e 4
105.w odd 12 1 84.2.k.c 2
105.w odd 12 1 588.2.f.b 2
105.w odd 12 1 2100.2.bi.d 2
105.x even 12 1 588.2.f.b 2
105.x even 12 1 588.2.k.b 2
140.w even 12 1 2352.2.k.b 2
140.x odd 12 1 336.2.bc.a 2
140.x odd 12 1 2352.2.k.b 2
315.bs even 12 1 2268.2.bm.d 2
315.bu odd 12 1 2268.2.bm.d 2
315.bw odd 12 1 2268.2.w.e 2
315.cg even 12 1 2268.2.w.e 2
420.bp odd 12 1 2352.2.k.b 2
420.br even 12 1 336.2.bc.a 2
420.br even 12 1 2352.2.k.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.k.c 2 5.c odd 4 1
84.2.k.c 2 15.e even 4 1
84.2.k.c 2 35.k even 12 1
84.2.k.c 2 105.w odd 12 1
336.2.bc.a 2 20.e even 4 1
336.2.bc.a 2 60.l odd 4 1
336.2.bc.a 2 140.x odd 12 1
336.2.bc.a 2 420.br even 12 1
588.2.f.b 2 35.k even 12 1
588.2.f.b 2 35.l odd 12 1
588.2.f.b 2 105.w odd 12 1
588.2.f.b 2 105.x even 12 1
588.2.k.b 2 35.f even 4 1
588.2.k.b 2 35.l odd 12 1
588.2.k.b 2 105.k odd 4 1
588.2.k.b 2 105.x even 12 1
2100.2.bi.d 2 5.c odd 4 1
2100.2.bi.d 2 15.e even 4 1
2100.2.bi.d 2 35.k even 12 1
2100.2.bi.d 2 105.w odd 12 1
2100.2.bo.e 4 1.a even 1 1 trivial
2100.2.bo.e 4 3.b odd 2 1 CM
2100.2.bo.e 4 5.b even 2 1 inner
2100.2.bo.e 4 7.d odd 6 1 inner
2100.2.bo.e 4 15.d odd 2 1 inner
2100.2.bo.e 4 21.g even 6 1 inner
2100.2.bo.e 4 35.i odd 6 1 inner
2100.2.bo.e 4 105.p even 6 1 inner
2268.2.w.e 2 45.k odd 12 1
2268.2.w.e 2 45.l even 12 1
2268.2.w.e 2 315.bw odd 12 1
2268.2.w.e 2 315.cg even 12 1
2268.2.bm.d 2 45.k odd 12 1
2268.2.bm.d 2 45.l even 12 1
2268.2.bm.d 2 315.bs even 12 1
2268.2.bm.d 2 315.bu odd 12 1
2352.2.k.b 2 140.w even 12 1
2352.2.k.b 2 140.x odd 12 1
2352.2.k.b 2 420.bp odd 12 1
2352.2.k.b 2 420.br even 12 1

## Hecke kernels

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

 $$T_{11}$$ $$T_{13}^{2} - 27$$ $$T_{19}^{2} - 15 T_{19} + 75$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 3 T^{2} + 9 T^{4}$$
$5$ 1
$7$ $$1 + 11 T^{2} + 49 T^{4}$$
$11$ $$( 1 + 11 T^{2} + 121 T^{4} )^{2}$$
$13$ $$( 1 - T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 + 17 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 - 8 T + 19 T^{2} )^{2}( 1 - 7 T + 19 T^{2} )^{2}$$
$23$ $$( 1 - 23 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 29 T^{2} )^{4}$$
$31$ $$( 1 - 4 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2}$$
$37$ $$( 1 - 73 T^{2} + 1369 T^{4} )( 1 + 26 T^{2} + 1369 T^{4} )$$
$41$ $$( 1 + 41 T^{2} )^{4}$$
$43$ $$( 1 + 83 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$( 1 + 47 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$( 1 - 53 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$( 1 - 59 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 - T + 61 T^{2} )^{2}( 1 + 13 T + 61 T^{2} )^{2}$$
$67$ $$( 1 - 13 T^{2} + 4489 T^{4} )( 1 + 122 T^{2} + 4489 T^{4} )$$
$71$ $$( 1 - 71 T^{2} )^{4}$$
$73$ $$( 1 - 97 T^{2} + 5329 T^{4} )( 1 - 46 T^{2} + 5329 T^{4} )$$
$79$ $$( 1 - 13 T + 79 T^{2} )^{2}( 1 - 4 T + 79 T^{2} )^{2}$$
$83$ $$( 1 - 83 T^{2} )^{4}$$
$89$ $$( 1 - 89 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 + 2 T^{2} + 9409 T^{4} )^{2}$$