# Properties

 Label 2100.2.bc.f Level $2100$ Weight $2$ Character orbit 2100.bc Analytic conductor $16.769$ Analytic rank $0$ Dimension $8$ 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.bc (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.7685844245$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 420) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} + ( - \beta_{5} + \beta_{3} - \beta_1) q^{7} + \beta_{2} q^{9}+O(q^{10})$$ q - b1 * q^3 + (-b5 + b3 - b1) * q^7 + b2 * q^9 $$q - \beta_1 q^{3} + ( - \beta_{5} + \beta_{3} - \beta_1) q^{7} + \beta_{2} q^{9} + ( - 2 \beta_{7} + \beta_{4}) q^{11} + ( - \beta_{6} + 2 \beta_{5} - \beta_{3}) q^{13} + (\beta_{6} + \beta_{5}) q^{17} - 7 \beta_{2} q^{19} + (\beta_{7} + 1) q^{21} + ( - 2 \beta_{6} + \beta_{5}) q^{23} - \beta_{3} q^{27} + (\beta_{7} - 2 \beta_{4} + 6) q^{29} + ( - 4 \beta_{7} + 2 \beta_{4} - \beta_{2} + 1) q^{31} + ( - 2 \beta_{6} + \beta_{5}) q^{33} + ( - 2 \beta_{6} + \beta_{5} - \beta_{3} + \beta_1) q^{37} + ( - 2 \beta_{7} + \beta_{4} + \beta_{2} - 1) q^{39} + (\beta_{7} - 2 \beta_{4}) q^{41} + ( - \beta_{6} + 2 \beta_{5} - \beta_{3}) q^{43} + ( - 6 \beta_{3} + 6 \beta_1) q^{47} + (2 \beta_{4} + 5 \beta_{2} - 5) q^{49} + ( - \beta_{7} - \beta_{4}) q^{51} + ( - 2 \beta_{6} - 2 \beta_{5}) q^{53} + 7 \beta_{3} q^{57} + (2 \beta_{7} - \beta_{4} + 6 \beta_{2} - 6) q^{59} + ( - 2 \beta_{7} - 2 \beta_{4} + 4 \beta_{2}) q^{61} + (\beta_{6} - \beta_{5} - \beta_1) q^{63} + ( - \beta_{6} - \beta_{5} - \beta_1) q^{67} + ( - \beta_{7} + 2 \beta_{4}) q^{69} + (3 \beta_{7} - 6 \beta_{4}) q^{71} + ( - \beta_{6} - \beta_{5} - 5 \beta_1) q^{73} + ( - \beta_{6} + 2 \beta_{5} + 6 \beta_{3} - 12 \beta_1) q^{77} + 11 \beta_{2} q^{79} + (\beta_{2} - 1) q^{81} + (\beta_{6} - 2 \beta_{5} - 6 \beta_{3}) q^{83} + (\beta_{6} + \beta_{5} - 6 \beta_1) q^{87} + ( - \beta_{7} - \beta_{4} - 6 \beta_{2}) q^{89} + ( - 2 \beta_{4} - 5 \beta_{2} + 12) q^{91} + ( - 4 \beta_{6} + 2 \beta_{5} + \beta_{3} - \beta_1) q^{93} + ( - 2 \beta_{6} + 4 \beta_{5} - 8 \beta_{3}) q^{97} + ( - \beta_{7} + 2 \beta_{4}) q^{99}+O(q^{100})$$ q - b1 * q^3 + (-b5 + b3 - b1) * q^7 + b2 * q^9 + (-2*b7 + b4) * q^11 + (-b6 + 2*b5 - b3) * q^13 + (b6 + b5) * q^17 - 7*b2 * q^19 + (b7 + 1) * q^21 + (-2*b6 + b5) * q^23 - b3 * q^27 + (b7 - 2*b4 + 6) * q^29 + (-4*b7 + 2*b4 - b2 + 1) * q^31 + (-2*b6 + b5) * q^33 + (-2*b6 + b5 - b3 + b1) * q^37 + (-2*b7 + b4 + b2 - 1) * q^39 + (b7 - 2*b4) * q^41 + (-b6 + 2*b5 - b3) * q^43 + (-6*b3 + 6*b1) * q^47 + (2*b4 + 5*b2 - 5) * q^49 + (-b7 - b4) * q^51 + (-2*b6 - 2*b5) * q^53 + 7*b3 * q^57 + (2*b7 - b4 + 6*b2 - 6) * q^59 + (-2*b7 - 2*b4 + 4*b2) * q^61 + (b6 - b5 - b1) * q^63 + (-b6 - b5 - b1) * q^67 + (-b7 + 2*b4) * q^69 + (3*b7 - 6*b4) * q^71 + (-b6 - b5 - 5*b1) * q^73 + (-b6 + 2*b5 + 6*b3 - 12*b1) * q^77 + 11*b2 * q^79 + (b2 - 1) * q^81 + (b6 - 2*b5 - 6*b3) * q^83 + (b6 + b5 - 6*b1) * q^87 + (-b7 - b4 - 6*b2) * q^89 + (-2*b4 - 5*b2 + 12) * q^91 + (-4*b6 + 2*b5 + b3 - b1) * q^93 + (-2*b6 + 4*b5 - 8*b3) * q^97 + (-b7 + 2*b4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{9}+O(q^{10})$$ 8 * q + 4 * q^9 $$8 q + 4 q^{9} - 28 q^{19} + 8 q^{21} + 48 q^{29} + 4 q^{31} - 4 q^{39} - 20 q^{49} - 24 q^{59} + 16 q^{61} + 44 q^{79} - 4 q^{81} - 24 q^{89} + 76 q^{91}+O(q^{100})$$ 8 * q + 4 * q^9 - 28 * q^19 + 8 * q^21 + 48 * q^29 + 4 * q^31 - 4 * q^39 - 20 * q^49 - 24 * q^59 + 16 * q^61 + 44 * q^79 - 4 * q^81 - 24 * q^89 + 76 * q^91

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{24}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{24}^{4}$$ v^4 $$\beta_{3}$$ $$=$$ $$\zeta_{24}^{6}$$ v^6 $$\beta_{4}$$ $$=$$ $$\zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}^{3} + 2\zeta_{24}$$ v^7 - v^5 + v^3 + 2*v $$\beta_{5}$$ $$=$$ $$-\zeta_{24}^{7} + 2\zeta_{24}^{5} + 2\zeta_{24}^{3} - \zeta_{24}$$ -v^7 + 2*v^5 + 2*v^3 - v $$\beta_{6}$$ $$=$$ $$-2\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}$$ -2*v^7 + v^5 + v^3 + v $$\beta_{7}$$ $$=$$ $$2\zeta_{24}^{7} + \zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24}$$ 2*v^7 + v^5 - v^3 + v
 $$\zeta_{24}$$ $$=$$ $$( \beta_{7} + 2\beta_{6} - \beta_{5} + \beta_{4} ) / 6$$ (b7 + 2*b6 - b5 + b4) / 6 $$\zeta_{24}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{24}^{3}$$ $$=$$ $$( -\beta_{7} - \beta_{6} + 2\beta_{5} + 2\beta_{4} ) / 6$$ (-b7 - b6 + 2*b5 + 2*b4) / 6 $$\zeta_{24}^{4}$$ $$=$$ $$\beta_{2}$$ b2 $$\zeta_{24}^{5}$$ $$=$$ $$( 2\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} ) / 6$$ (2*b7 + b6 + b5 - b4) / 6 $$\zeta_{24}^{6}$$ $$=$$ $$\beta_{3}$$ b3 $$\zeta_{24}^{7}$$ $$=$$ $$( \beta_{7} - 2\beta_{6} + \beta_{5} + \beta_{4} ) / 6$$ (b7 - 2*b6 + b5 + b4) / 6

## Character values

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

 $$n$$ $$701$$ $$1051$$ $$1177$$ $$1501$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$-\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
949.1
 0.965926 + 0.258819i −0.965926 − 0.258819i 0.258819 − 0.965926i −0.258819 + 0.965926i 0.965926 − 0.258819i −0.965926 + 0.258819i 0.258819 + 0.965926i −0.258819 − 0.965926i
0 −0.866025 0.500000i 0 0 0 −2.09077 1.62132i 0 0.500000 + 0.866025i 0
949.2 0 −0.866025 0.500000i 0 0 0 0.358719 + 2.62132i 0 0.500000 + 0.866025i 0
949.3 0 0.866025 + 0.500000i 0 0 0 −0.358719 2.62132i 0 0.500000 + 0.866025i 0
949.4 0 0.866025 + 0.500000i 0 0 0 2.09077 + 1.62132i 0 0.500000 + 0.866025i 0
1549.1 0 −0.866025 + 0.500000i 0 0 0 −2.09077 + 1.62132i 0 0.500000 0.866025i 0
1549.2 0 −0.866025 + 0.500000i 0 0 0 0.358719 2.62132i 0 0.500000 0.866025i 0
1549.3 0 0.866025 0.500000i 0 0 0 −0.358719 + 2.62132i 0 0.500000 0.866025i 0
1549.4 0 0.866025 0.500000i 0 0 0 2.09077 1.62132i 0 0.500000 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1549.4 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
7.c even 3 1 inner
35.j 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.bc.f 8
5.b even 2 1 inner 2100.2.bc.f 8
5.c odd 4 1 420.2.q.d 4
5.c odd 4 1 2100.2.q.k 4
7.c even 3 1 inner 2100.2.bc.f 8
15.e even 4 1 1260.2.s.e 4
20.e even 4 1 1680.2.bg.t 4
35.f even 4 1 2940.2.q.q 4
35.j even 6 1 inner 2100.2.bc.f 8
35.k even 12 1 2940.2.a.p 2
35.k even 12 1 2940.2.q.q 4
35.l odd 12 1 420.2.q.d 4
35.l odd 12 1 2100.2.q.k 4
35.l odd 12 1 2940.2.a.r 2
105.w odd 12 1 8820.2.a.bf 2
105.x even 12 1 1260.2.s.e 4
105.x even 12 1 8820.2.a.bk 2
140.w even 12 1 1680.2.bg.t 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.q.d 4 5.c odd 4 1
420.2.q.d 4 35.l odd 12 1
1260.2.s.e 4 15.e even 4 1
1260.2.s.e 4 105.x even 12 1
1680.2.bg.t 4 20.e even 4 1
1680.2.bg.t 4 140.w even 12 1
2100.2.q.k 4 5.c odd 4 1
2100.2.q.k 4 35.l odd 12 1
2100.2.bc.f 8 1.a even 1 1 trivial
2100.2.bc.f 8 5.b even 2 1 inner
2100.2.bc.f 8 7.c even 3 1 inner
2100.2.bc.f 8 35.j even 6 1 inner
2940.2.a.p 2 35.k even 12 1
2940.2.a.r 2 35.l odd 12 1
2940.2.q.q 4 35.f even 4 1
2940.2.q.q 4 35.k even 12 1
8820.2.a.bf 2 105.w odd 12 1
8820.2.a.bk 2 105.x even 12 1

## Hecke kernels

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

 $$T_{11}^{4} + 18T_{11}^{2} + 324$$ T11^4 + 18*T11^2 + 324 $$T_{13}^{4} + 38T_{13}^{2} + 289$$ T13^4 + 38*T13^2 + 289

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{4} - T^{2} + 1)^{2}$$
$5$ $$T^{8}$$
$7$ $$T^{8} + 10 T^{6} + 51 T^{4} + \cdots + 2401$$
$11$ $$(T^{4} + 18 T^{2} + 324)^{2}$$
$13$ $$(T^{4} + 38 T^{2} + 289)^{2}$$
$17$ $$(T^{4} - 18 T^{2} + 324)^{2}$$
$19$ $$(T^{2} + 7 T + 49)^{4}$$
$23$ $$(T^{4} - 18 T^{2} + 324)^{2}$$
$29$ $$(T^{2} - 12 T + 18)^{4}$$
$31$ $$(T^{4} - 2 T^{3} + 75 T^{2} + 142 T + 5041)^{2}$$
$37$ $$T^{8} - 38 T^{6} + 1155 T^{4} + \cdots + 83521$$
$41$ $$(T^{2} - 18)^{4}$$
$43$ $$(T^{4} + 38 T^{2} + 289)^{2}$$
$47$ $$(T^{4} - 36 T^{2} + 1296)^{2}$$
$53$ $$(T^{4} - 72 T^{2} + 5184)^{2}$$
$59$ $$(T^{4} + 12 T^{3} + 126 T^{2} + 216 T + 324)^{2}$$
$61$ $$(T^{4} - 8 T^{3} + 120 T^{2} + 448 T + 3136)^{2}$$
$67$ $$T^{8} - 38 T^{6} + 1155 T^{4} + \cdots + 83521$$
$71$ $$(T^{2} - 162)^{4}$$
$73$ $$T^{8} - 86 T^{6} + 7347 T^{4} + \cdots + 2401$$
$79$ $$(T^{2} - 11 T + 121)^{4}$$
$83$ $$(T^{4} + 108 T^{2} + 324)^{2}$$
$89$ $$(T^{4} + 12 T^{3} + 126 T^{2} + 216 T + 324)^{2}$$
$97$ $$(T^{4} + 272 T^{2} + 64)^{2}$$