# Properties

 Label 2100.2.bo.a Level 2100 Weight 2 Character orbit 2100.bo Analytic conductor 16.769 Analytic rank 1 Dimension 4 CM no 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.bo (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.7685844245$$ Analytic rank: $$1$$ 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: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} + ( -3 - 3 \zeta_{12}^{2} ) q^{11} -3 \zeta_{12} q^{17} + ( -2 + \zeta_{12}^{2} ) q^{19} + ( -4 - \zeta_{12}^{2} ) q^{21} + ( 3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{23} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{27} + ( -1 - \zeta_{12}^{2} ) q^{31} + 9 \zeta_{12}^{3} q^{33} + ( -7 \zeta_{12} + 7 \zeta_{12}^{3} ) q^{37} -6 q^{41} + 4 \zeta_{12}^{3} q^{43} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{47} + ( 3 - 8 \zeta_{12}^{2} ) q^{49} + ( -3 + 6 \zeta_{12}^{2} ) q^{51} + ( -3 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{53} + 3 \zeta_{12} q^{57} + ( 3 - 3 \zeta_{12}^{2} ) q^{59} + ( -14 + 7 \zeta_{12}^{2} ) q^{61} + ( 3 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{63} + 5 \zeta_{12} q^{67} + ( 9 - 9 \zeta_{12}^{2} ) q^{69} + ( 6 - 12 \zeta_{12}^{2} ) q^{71} + ( 7 \zeta_{12} - 14 \zeta_{12}^{3} ) q^{73} + ( -15 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{77} -\zeta_{12}^{2} q^{79} -9 \zeta_{12}^{2} q^{81} + 12 \zeta_{12}^{3} q^{83} + 9 \zeta_{12}^{2} q^{89} + 3 \zeta_{12}^{3} q^{93} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{97} + ( 18 - 9 \zeta_{12}^{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 6q^{9} + O(q^{10})$$ $$4q - 6q^{9} - 18q^{11} - 6q^{19} - 18q^{21} - 6q^{31} - 24q^{41} - 4q^{49} + 6q^{59} - 42q^{61} + 18q^{69} - 2q^{79} - 18q^{81} + 18q^{89} + 54q^{99} + O(q^{100})$$

## Character values

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

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

## 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 1.73205 + 2.00000i 0 −1.50000 2.59808i 0
1349.2 0 0.866025 1.50000i 0 0 0 −1.73205 2.00000i 0 −1.50000 2.59808i 0
1949.1 0 −0.866025 1.50000i 0 0 0 1.73205 2.00000i 0 −1.50000 + 2.59808i 0
1949.2 0 0.866025 + 1.50000i 0 0 0 −1.73205 + 2.00000i 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
5.b even 2 1 inner
21.g even 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.a 4
3.b odd 2 1 2100.2.bo.f 4
5.b even 2 1 inner 2100.2.bo.a 4
5.c odd 4 1 84.2.k.a 2
5.c odd 4 1 2100.2.bi.f 2
7.d odd 6 1 2100.2.bo.f 4
15.d odd 2 1 2100.2.bo.f 4
15.e even 4 1 84.2.k.b yes 2
15.e even 4 1 2100.2.bi.e 2
20.e even 4 1 336.2.bc.d 2
21.g even 6 1 inner 2100.2.bo.a 4
35.f even 4 1 588.2.k.d 2
35.i odd 6 1 2100.2.bo.f 4
35.k even 12 1 84.2.k.b yes 2
35.k even 12 1 588.2.f.a 2
35.k even 12 1 2100.2.bi.e 2
35.l odd 12 1 588.2.f.c 2
35.l odd 12 1 588.2.k.c 2
45.k odd 12 1 2268.2.w.f 2
45.k odd 12 1 2268.2.bm.a 2
45.l even 12 1 2268.2.w.a 2
45.l even 12 1 2268.2.bm.f 2
60.l odd 4 1 336.2.bc.b 2
105.k odd 4 1 588.2.k.c 2
105.p even 6 1 inner 2100.2.bo.a 4
105.w odd 12 1 84.2.k.a 2
105.w odd 12 1 588.2.f.c 2
105.w odd 12 1 2100.2.bi.f 2
105.x even 12 1 588.2.f.a 2
105.x even 12 1 588.2.k.d 2
140.w even 12 1 2352.2.k.a 2
140.x odd 12 1 336.2.bc.b 2
140.x odd 12 1 2352.2.k.d 2
315.bs even 12 1 2268.2.bm.f 2
315.bu odd 12 1 2268.2.bm.a 2
315.bw odd 12 1 2268.2.w.f 2
315.cg even 12 1 2268.2.w.a 2
420.bp odd 12 1 2352.2.k.d 2
420.br even 12 1 336.2.bc.d 2
420.br even 12 1 2352.2.k.a 2

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

## Hecke kernels

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

 $$T_{11}^{2} + 9 T_{11} + 27$$ $$T_{13}$$ $$T_{19}^{2} + 3 T_{19} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 3 T^{2} + 9 T^{4}$$
$5$ 1
$7$ $$1 + 2 T^{2} + 49 T^{4}$$
$11$ $$( 1 + 9 T + 38 T^{2} + 99 T^{3} + 121 T^{4} )^{2}$$
$13$ $$( 1 + 13 T^{2} )^{4}$$
$17$ $$1 + 25 T^{2} + 336 T^{4} + 7225 T^{6} + 83521 T^{8}$$
$19$ $$( 1 + 3 T + 22 T^{2} + 57 T^{3} + 361 T^{4} )^{2}$$
$23$ $$1 - 19 T^{2} - 168 T^{4} - 10051 T^{6} + 279841 T^{8}$$
$29$ $$( 1 - 29 T^{2} )^{4}$$
$31$ $$( 1 - 4 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2}$$
$37$ $$1 + 25 T^{2} - 744 T^{4} + 34225 T^{6} + 1874161 T^{8}$$
$41$ $$( 1 + 6 T + 41 T^{2} )^{4}$$
$43$ $$( 1 - 70 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$1 + 85 T^{2} + 5016 T^{4} + 187765 T^{6} + 4879681 T^{8}$$
$53$ $$1 - 79 T^{2} + 3432 T^{4} - 221911 T^{6} + 7890481 T^{8}$$
$59$ $$( 1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4} )^{2}$$
$61$ $$( 1 + 21 T + 208 T^{2} + 1281 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$( 1 - 13 T^{2} + 4489 T^{4} )( 1 + 122 T^{2} + 4489 T^{4} )$$
$71$ $$( 1 - 34 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$1 + T^{2} - 5328 T^{4} + 5329 T^{6} + 28398241 T^{8}$$
$79$ $$( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$( 1 - 22 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 - 9 T - 8 T^{2} - 801 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$( 1 + 146 T^{2} + 9409 T^{4} )^{2}$$