# Properties

 Label 2100.2.bc.e 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: 8.0.49787136.1 Defining polynomial: $$x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16$$ x^8 + 3*x^6 + 5*x^4 + 12*x^2 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} + \beta_1 q^{7} - \beta_{3} q^{9}+O(q^{10})$$ q - b2 * q^3 + b1 * q^7 - b3 * q^9 $$q - \beta_{2} q^{3} + \beta_1 q^{7} - \beta_{3} q^{9} + ( - \beta_{7} - \beta_{3} - 1) q^{11} + ( - \beta_{6} - \beta_1) q^{13} + ( - \beta_{6} + \beta_{2}) q^{17} + (2 \beta_{5} - 3 \beta_{3}) q^{19} + ( - \beta_{7} + \beta_{5}) q^{21} + (\beta_{4} - \beta_1) q^{23} + ( - \beta_{4} - \beta_{2}) q^{27} + (\beta_{7} - \beta_{5} - 5) q^{29} + ( - 2 \beta_{7} - \beta_{3} - 1) q^{31} + ( - \beta_{4} + \beta_1) q^{33} + ( - 2 \beta_{4} + \beta_1) q^{37} + \beta_{7} q^{39} + ( - 3 \beta_{7} + 3 \beta_{5} - 3) q^{41} + ( - 3 \beta_{6} + 2 \beta_{4} + 2 \beta_{2} - 3 \beta_1) q^{43} + 6 \beta_{4} q^{47} + (7 \beta_{3} + 7) q^{49} + (\beta_{5} + \beta_{3}) q^{51} + ( - 2 \beta_{6} + 2 \beta_{2}) q^{53} + ( - 2 \beta_{6} - 3 \beta_{4} - 3 \beta_{2} - 2 \beta_1) q^{57} + ( - 3 \beta_{7} + 3 \beta_{3} + 3) q^{59} + 8 \beta_{3} q^{61} - \beta_{6} q^{63} + (\beta_{6} - 2 \beta_{2}) q^{67} + (\beta_{7} - \beta_{5} + 1) q^{69} + ( - \beta_{7} + \beta_{5} + 11) q^{71} - 5 \beta_{6} q^{73} + ( - \beta_{6} + 7 \beta_{4} + 7 \beta_{2} - \beta_1) q^{77} + (2 \beta_{5} + 3 \beta_{3}) q^{79} + ( - \beta_{3} - 1) q^{81} + ( - 3 \beta_{6} - 3 \beta_{4} - 3 \beta_{2} - 3 \beta_1) q^{83} + (\beta_{6} + 5 \beta_{2}) q^{87} + ( - 5 \beta_{5} + \beta_{3}) q^{89} - 7 \beta_{3} q^{91} + ( - \beta_{4} + 2 \beta_1) q^{93} + ( - 8 \beta_{4} - 8 \beta_{2}) q^{97} + ( - \beta_{7} + \beta_{5} - 1) q^{99}+O(q^{100})$$ q - b2 * q^3 + b1 * q^7 - b3 * q^9 + (-b7 - b3 - 1) * q^11 + (-b6 - b1) * q^13 + (-b6 + b2) * q^17 + (2*b5 - 3*b3) * q^19 + (-b7 + b5) * q^21 + (b4 - b1) * q^23 + (-b4 - b2) * q^27 + (b7 - b5 - 5) * q^29 + (-2*b7 - b3 - 1) * q^31 + (-b4 + b1) * q^33 + (-2*b4 + b1) * q^37 + b7 * q^39 + (-3*b7 + 3*b5 - 3) * q^41 + (-3*b6 + 2*b4 + 2*b2 - 3*b1) * q^43 + 6*b4 * q^47 + (7*b3 + 7) * q^49 + (b5 + b3) * q^51 + (-2*b6 + 2*b2) * q^53 + (-2*b6 - 3*b4 - 3*b2 - 2*b1) * q^57 + (-3*b7 + 3*b3 + 3) * q^59 + 8*b3 * q^61 - b6 * q^63 + (b6 - 2*b2) * q^67 + (b7 - b5 + 1) * q^69 + (-b7 + b5 + 11) * q^71 - 5*b6 * q^73 + (-b6 + 7*b4 + 7*b2 - b1) * q^77 + (2*b5 + 3*b3) * q^79 + (-b3 - 1) * q^81 + (-3*b6 - 3*b4 - 3*b2 - 3*b1) * q^83 + (b6 + 5*b2) * q^87 + (-5*b5 + b3) * q^89 - 7*b3 * q^91 + (-b4 + 2*b1) * q^93 + (-8*b4 - 8*b2) * q^97 + (-b7 + b5 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{9}+O(q^{10})$$ 8 * q + 4 * q^9 $$8 q + 4 q^{9} - 4 q^{11} + 12 q^{19} - 40 q^{29} - 4 q^{31} - 24 q^{41} + 28 q^{49} - 4 q^{51} + 12 q^{59} - 32 q^{61} + 8 q^{69} + 88 q^{71} - 12 q^{79} - 4 q^{81} - 4 q^{89} + 28 q^{91} - 8 q^{99}+O(q^{100})$$ 8 * q + 4 * q^9 - 4 * q^11 + 12 * q^19 - 40 * q^29 - 4 * q^31 - 24 * q^41 + 28 * q^49 - 4 * q^51 + 12 * q^59 - 32 * q^61 + 8 * q^69 + 88 * q^71 - 12 * q^79 - 4 * q^81 - 4 * q^89 + 28 * q^91 - 8 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{6} + 15\nu^{4} + 5\nu^{2} + 12 ) / 20$$ (v^6 + 15*v^4 + 5*v^2 + 12) / 20 $$\beta_{2}$$ $$=$$ $$( \nu^{7} - 5\nu^{5} + 5\nu^{3} + 12\nu ) / 40$$ (v^7 - 5*v^5 + 5*v^3 + 12*v) / 40 $$\beta_{3}$$ $$=$$ $$( -3\nu^{6} - 5\nu^{4} - 15\nu^{2} - 36 ) / 20$$ (-3*v^6 - 5*v^4 - 15*v^2 - 36) / 20 $$\beta_{4}$$ $$=$$ $$( -\nu^{7} - 7\nu ) / 10$$ (-v^7 - 7*v) / 10 $$\beta_{5}$$ $$=$$ $$( -\nu^{7} + 13\nu ) / 10$$ (-v^7 + 13*v) / 10 $$\beta_{6}$$ $$=$$ $$( -9\nu^{6} - 15\nu^{4} - 5\nu^{2} - 48 ) / 20$$ (-9*v^6 - 15*v^4 - 5*v^2 - 48) / 20 $$\beta_{7}$$ $$=$$ $$( 11\nu^{7} + 25\nu^{5} + 55\nu^{3} + 132\nu ) / 40$$ (11*v^7 + 25*v^5 + 55*v^3 + 132*v) / 40
 $$\nu$$ $$=$$ $$( \beta_{5} - \beta_{4} ) / 2$$ (b5 - b4) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{6} - 3\beta_{3} - 3 ) / 2$$ (b6 - 3*b3 - 3) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{7} - \beta_{5} + 5\beta_{4} + 5\beta_{2} ) / 2$$ (b7 - b5 + 5*b4 + 5*b2) / 2 $$\nu^{4}$$ $$=$$ $$( \beta_{3} + 3\beta_1 ) / 2$$ (b3 + 3*b1) / 2 $$\nu^{5}$$ $$=$$ $$( \beta_{7} - 11\beta_{2} ) / 2$$ (b7 - 11*b2) / 2 $$\nu^{6}$$ $$=$$ $$( -5\beta_{6} - 5\beta _1 - 9 ) / 2$$ (-5*b6 - 5*b1 - 9) / 2 $$\nu^{7}$$ $$=$$ $$( -7\beta_{5} - 13\beta_{4} ) / 2$$ (-7*b5 - 13*b4) / 2

## Character values

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

 $$n$$ $$701$$ $$1051$$ $$1177$$ $$1501$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$\beta_{3}$$

## 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
 1.09445 + 0.895644i −0.228425 − 1.39564i −1.09445 − 0.895644i 0.228425 + 1.39564i 1.09445 − 0.895644i −0.228425 + 1.39564i −1.09445 + 0.895644i 0.228425 − 1.39564i
0 −0.866025 0.500000i 0 0 0 −2.29129 + 1.32288i 0 0.500000 + 0.866025i 0
949.2 0 −0.866025 0.500000i 0 0 0 2.29129 1.32288i 0 0.500000 + 0.866025i 0
949.3 0 0.866025 + 0.500000i 0 0 0 −2.29129 + 1.32288i 0 0.500000 + 0.866025i 0
949.4 0 0.866025 + 0.500000i 0 0 0 2.29129 1.32288i 0 0.500000 + 0.866025i 0
1549.1 0 −0.866025 + 0.500000i 0 0 0 −2.29129 1.32288i 0 0.500000 0.866025i 0
1549.2 0 −0.866025 + 0.500000i 0 0 0 2.29129 + 1.32288i 0 0.500000 0.866025i 0
1549.3 0 0.866025 0.500000i 0 0 0 −2.29129 1.32288i 0 0.500000 0.866025i 0
1549.4 0 0.866025 0.500000i 0 0 0 2.29129 + 1.32288i 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.e 8
5.b even 2 1 inner 2100.2.bc.e 8
5.c odd 4 1 420.2.q.c 4
5.c odd 4 1 2100.2.q.h 4
7.c even 3 1 inner 2100.2.bc.e 8
15.e even 4 1 1260.2.s.f 4
20.e even 4 1 1680.2.bg.q 4
35.f even 4 1 2940.2.q.t 4
35.j even 6 1 inner 2100.2.bc.e 8
35.k even 12 1 2940.2.a.m 2
35.k even 12 1 2940.2.q.t 4
35.l odd 12 1 420.2.q.c 4
35.l odd 12 1 2100.2.q.h 4
35.l odd 12 1 2940.2.a.s 2
105.w odd 12 1 8820.2.a.bj 2
105.x even 12 1 1260.2.s.f 4
105.x even 12 1 8820.2.a.be 2
140.w even 12 1 1680.2.bg.q 4

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

## Hecke kernels

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

 $$T_{11}^{4} + 2T_{11}^{3} + 10T_{11}^{2} - 12T_{11} + 36$$ T11^4 + 2*T11^3 + 10*T11^2 - 12*T11 + 36 $$T_{13}^{2} + 7$$ T13^2 + 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{4} - T^{2} + 1)^{2}$$
$5$ $$T^{8}$$
$7$ $$(T^{4} - 7 T^{2} + 49)^{2}$$
$11$ $$(T^{4} + 2 T^{3} + 10 T^{2} - 12 T + 36)^{2}$$
$13$ $$(T^{2} + 7)^{4}$$
$17$ $$T^{8} - 16 T^{6} + 220 T^{4} + \cdots + 1296$$
$19$ $$(T^{4} - 6 T^{3} + 55 T^{2} + 114 T + 361)^{2}$$
$23$ $$T^{8} - 16 T^{6} + 220 T^{4} + \cdots + 1296$$
$29$ $$(T^{2} + 10 T + 18)^{4}$$
$31$ $$(T^{4} + 2 T^{3} + 31 T^{2} - 54 T + 729)^{2}$$
$37$ $$T^{8} - 22 T^{6} + 475 T^{4} + \cdots + 81$$
$41$ $$(T^{2} + 6 T - 54)^{4}$$
$43$ $$(T^{4} + 134 T^{2} + 3481)^{2}$$
$47$ $$(T^{4} - 36 T^{2} + 1296)^{2}$$
$53$ $$T^{8} - 64 T^{6} + 3520 T^{4} + \cdots + 331776$$
$59$ $$(T^{4} - 6 T^{3} + 90 T^{2} + 324 T + 2916)^{2}$$
$61$ $$(T^{2} + 8 T + 64)^{4}$$
$67$ $$T^{8} - 22 T^{6} + 475 T^{4} + \cdots + 81$$
$71$ $$(T^{2} - 22 T + 114)^{4}$$
$73$ $$(T^{4} - 175 T^{2} + 30625)^{2}$$
$79$ $$(T^{4} + 6 T^{3} + 55 T^{2} - 114 T + 361)^{2}$$
$83$ $$(T^{4} + 144 T^{2} + 2916)^{2}$$
$89$ $$(T^{4} + 2 T^{3} + 178 T^{2} - 348 T + 30276)^{2}$$
$97$ $$(T^{2} + 64)^{4}$$