# Properties

 Label 2100.2.bi.e Level 2100 Weight 2 Character orbit 2100.bi Analytic conductor 16.769 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2100.bi (of order $$6$$, degree $$2$$, 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: $$\Z[a_1, \ldots, a_{11}]$$ 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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$701$$ $$1051$$ $$1177$$ $$1501$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.73205i 0 0 0 −2.00000 1.73205i 0 −3.00000 0
1601.1 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
21.g 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.bi.e 2
3.b odd 2 1 2100.2.bi.f 2
5.b even 2 1 84.2.k.b yes 2
5.c odd 4 2 2100.2.bo.f 4
7.d odd 6 1 2100.2.bi.f 2
15.d odd 2 1 84.2.k.a 2
15.e even 4 2 2100.2.bo.a 4
20.d odd 2 1 336.2.bc.b 2
21.g even 6 1 inner 2100.2.bi.e 2
35.c odd 2 1 588.2.k.c 2
35.i odd 6 1 84.2.k.a 2
35.i odd 6 1 588.2.f.c 2
35.j even 6 1 588.2.f.a 2
35.j even 6 1 588.2.k.d 2
35.k even 12 2 2100.2.bo.a 4
45.h odd 6 1 2268.2.w.f 2
45.h odd 6 1 2268.2.bm.a 2
45.j even 6 1 2268.2.w.a 2
45.j even 6 1 2268.2.bm.f 2
60.h even 2 1 336.2.bc.d 2
105.g even 2 1 588.2.k.d 2
105.o odd 6 1 588.2.f.c 2
105.o odd 6 1 588.2.k.c 2
105.p even 6 1 84.2.k.b yes 2
105.p even 6 1 588.2.f.a 2
105.w odd 12 2 2100.2.bo.f 4
140.p odd 6 1 2352.2.k.d 2
140.s even 6 1 336.2.bc.d 2
140.s even 6 1 2352.2.k.a 2
315.q odd 6 1 2268.2.bm.a 2
315.u even 6 1 2268.2.w.a 2
315.bn odd 6 1 2268.2.w.f 2
315.bq even 6 1 2268.2.bm.f 2
420.ba even 6 1 2352.2.k.a 2
420.be odd 6 1 336.2.bc.b 2
420.be odd 6 1 2352.2.k.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.k.a 2 15.d odd 2 1
84.2.k.a 2 35.i odd 6 1
84.2.k.b yes 2 5.b even 2 1
84.2.k.b yes 2 105.p even 6 1
336.2.bc.b 2 20.d odd 2 1
336.2.bc.b 2 420.be odd 6 1
336.2.bc.d 2 60.h even 2 1
336.2.bc.d 2 140.s even 6 1
588.2.f.a 2 35.j even 6 1
588.2.f.a 2 105.p even 6 1
588.2.f.c 2 35.i odd 6 1
588.2.f.c 2 105.o odd 6 1
588.2.k.c 2 35.c odd 2 1
588.2.k.c 2 105.o odd 6 1
588.2.k.d 2 35.j even 6 1
588.2.k.d 2 105.g even 2 1
2100.2.bi.e 2 1.a even 1 1 trivial
2100.2.bi.e 2 21.g even 6 1 inner
2100.2.bi.f 2 3.b odd 2 1
2100.2.bi.f 2 7.d odd 6 1
2100.2.bo.a 4 15.e even 4 2
2100.2.bo.a 4 35.k even 12 2
2100.2.bo.f 4 5.c odd 4 2
2100.2.bo.f 4 105.w odd 12 2
2268.2.w.a 2 45.j even 6 1
2268.2.w.a 2 315.u even 6 1
2268.2.w.f 2 45.h odd 6 1
2268.2.w.f 2 315.bn odd 6 1
2268.2.bm.a 2 45.h odd 6 1
2268.2.bm.a 2 315.q odd 6 1
2268.2.bm.f 2 45.j even 6 1
2268.2.bm.f 2 315.bq even 6 1
2352.2.k.a 2 140.s even 6 1
2352.2.k.a 2 420.ba even 6 1
2352.2.k.d 2 140.p odd 6 1
2352.2.k.d 2 420.be odd 6 1

## Hecke kernels

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

 $$T_{11}^{2} - 9 T_{11} + 27$$ $$T_{13}$$ $$T_{19}^{2} - 3 T_{19} + 3$$ $$T_{37}^{2} - 7 T_{37} + 49$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 3 T^{2}$$
$5$ 1
$7$ $$1 + 4 T + 7 T^{2}$$
$11$ $$1 - 9 T + 38 T^{2} - 99 T^{3} + 121 T^{4}$$
$13$ $$( 1 - 13 T^{2} )^{2}$$
$17$ $$1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4}$$
$19$ $$1 - 3 T + 22 T^{2} - 57 T^{3} + 361 T^{4}$$
$23$ $$1 - 9 T + 50 T^{2} - 207 T^{3} + 529 T^{4}$$
$29$ $$( 1 - 29 T^{2} )^{2}$$
$31$ $$( 1 - 4 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} )$$
$37$ $$1 - 7 T + 12 T^{2} - 259 T^{3} + 1369 T^{4}$$
$41$ $$( 1 - 6 T + 41 T^{2} )^{2}$$
$43$ $$( 1 + 4 T + 43 T^{2} )^{2}$$
$47$ $$1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4}$$
$53$ $$1 - 9 T + 80 T^{2} - 477 T^{3} + 2809 T^{4}$$
$59$ $$1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4}$$
$61$ $$1 + 21 T + 208 T^{2} + 1281 T^{3} + 3721 T^{4}$$
$67$ $$( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} )$$
$71$ $$1 - 34 T^{2} + 5041 T^{4}$$
$73$ $$1 - 21 T + 220 T^{2} - 1533 T^{3} + 5329 T^{4}$$
$79$ $$1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4}$$
$83$ $$( 1 - 12 T + 83 T^{2} )^{2}$$
$89$ $$1 - 9 T - 8 T^{2} - 801 T^{3} + 7921 T^{4}$$
$97$ $$1 - 146 T^{2} + 9409 T^{4}$$