# Properties

 Label 2100.2.d.k Level 2100 Weight 2 Character orbit 2100.d 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.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.7685844245$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.14786560000.1 Defining polynomial: $$x^{8} - 6 x^{6} + 22 x^{4} - 54 x^{2} + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} + \beta_{3} - \beta_{6} ) q^{3} + ( \beta_{3} + \beta_{5} ) q^{7} + ( 2 - \beta_{2} + \beta_{5} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{1} + \beta_{3} - \beta_{6} ) q^{3} + ( \beta_{3} + \beta_{5} ) q^{7} + ( 2 - \beta_{2} + \beta_{5} ) q^{9} + ( 1 - \beta_{2} - \beta_{4} + \beta_{5} ) q^{11} + ( \beta_{3} + 2 \beta_{6} ) q^{13} + \beta_{7} q^{17} + ( -2 \beta_{3} - \beta_{6} ) q^{19} + ( -1 + \beta_{1} - \beta_{3} + \beta_{4} + \beta_{7} ) q^{21} + ( 2 - \beta_{2} - 3 \beta_{4} + 2 \beta_{5} ) q^{23} + ( \beta_{1} + 2 \beta_{3} - 3 \beta_{6} + \beta_{7} ) q^{27} + ( -1 - \beta_{2} + 3 \beta_{4} - \beta_{5} ) q^{29} + ( 3 \beta_{3} - \beta_{6} ) q^{31} + ( -\beta_{1} - \beta_{3} - 2 \beta_{6} ) q^{33} + \beta_{5} q^{37} + ( -1 + 2 \beta_{2} + 3 \beta_{4} - 2 \beta_{5} ) q^{39} + ( 2 \beta_{1} + \beta_{3} - \beta_{6} + 2 \beta_{7} ) q^{41} + ( -5 - 2 \beta_{5} ) q^{43} + ( 4 \beta_{1} + 2 \beta_{3} - 2 \beta_{6} + 3 \beta_{7} ) q^{47} + ( 3 - 2 \beta_{3} + 4 \beta_{6} ) q^{49} + ( -1 + \beta_{2} - \beta_{4} + 2 \beta_{5} ) q^{51} + ( -3 + 4 \beta_{2} + 2 \beta_{4} - 3 \beta_{5} ) q^{53} + ( 2 - \beta_{2} - 3 \beta_{4} + \beta_{5} ) q^{57} + ( -6 \beta_{1} - 3 \beta_{3} + 3 \beta_{6} - \beta_{7} ) q^{59} + ( 5 \beta_{3} - 5 \beta_{6} ) q^{61} + ( 3 + \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{63} + 3 \beta_{5} q^{67} + ( -\beta_{1} - 5 \beta_{3} - 3 \beta_{6} - \beta_{7} ) q^{69} + ( -1 + 3 \beta_{2} - \beta_{4} - \beta_{5} ) q^{71} -\beta_{3} q^{73} + ( 2 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{77} + ( -3 - 2 \beta_{5} ) q^{79} + ( -2 \beta_{2} - 2 \beta_{4} + 5 \beta_{5} ) q^{81} + ( 2 \beta_{1} + \beta_{3} - \beta_{6} + 2 \beta_{7} ) q^{83} + ( -\beta_{1} + 7 \beta_{3} + 2 \beta_{7} ) q^{87} + ( 2 \beta_{1} + \beta_{3} - \beta_{6} - 3 \beta_{7} ) q^{89} + ( -4 + 3 \beta_{3} - 2 \beta_{5} + 4 \beta_{6} ) q^{91} + ( -3 - \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{93} + ( 5 \beta_{3} - 3 \beta_{6} ) q^{97} + ( -2 - 2 \beta_{2} - 3 \beta_{4} + 2 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 12q^{9} + O(q^{10})$$ $$8q + 12q^{9} - 4q^{21} + 12q^{39} - 40q^{43} + 24q^{49} - 8q^{51} + 20q^{63} - 24q^{79} - 16q^{81} - 32q^{91} - 20q^{93} - 36q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 6 x^{6} + 22 x^{4} - 54 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 1$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + 3 \nu^{5} - 5 \nu^{3} + 9 \nu$$$$)/54$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{6} + 3 \nu^{4} - 5 \nu^{2} + 27$$$$)/18$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{6} + 6 \nu^{4} - 13 \nu^{2} + 27$$$$)/9$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} - 3 \nu^{5} + 13 \nu^{3} - 15 \nu$$$$)/18$$ $$\beta_{7}$$ $$=$$ $$($$$$-5 \nu^{7} + 21 \nu^{5} - 29 \nu^{3} + 45 \nu$$$$)/54$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + 2 \beta_{6} - \beta_{3} + \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{5} + 2 \beta_{4} + 2 \beta_{2} - 4$$ $$\nu^{5}$$ $$=$$ $$3 \beta_{7} + 3 \beta_{6} + 6 \beta_{3} - \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-3 \beta_{5} + 12 \beta_{4} - \beta_{2} - 10$$ $$\nu^{7}$$ $$=$$ $$-4 \beta_{7} + \beta_{6} + 31 \beta_{3} - \beta_{1}$$

## Character values

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

 $$n$$ $$701$$ $$1051$$ $$1177$$ $$1501$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1301.1
 −1.67601 + 0.437016i −1.67601 − 0.437016i −1.30038 + 1.14412i −1.30038 − 1.14412i 1.30038 + 1.14412i 1.30038 − 1.14412i 1.67601 + 0.437016i 1.67601 − 0.437016i
0 −1.67601 0.437016i 0 0 0 2.23607 + 1.41421i 0 2.61803 + 1.46489i 0
1301.2 0 −1.67601 + 0.437016i 0 0 0 2.23607 1.41421i 0 2.61803 1.46489i 0
1301.3 0 −1.30038 1.14412i 0 0 0 −2.23607 1.41421i 0 0.381966 + 2.97558i 0
1301.4 0 −1.30038 + 1.14412i 0 0 0 −2.23607 + 1.41421i 0 0.381966 2.97558i 0
1301.5 0 1.30038 1.14412i 0 0 0 −2.23607 1.41421i 0 0.381966 2.97558i 0
1301.6 0 1.30038 + 1.14412i 0 0 0 −2.23607 + 1.41421i 0 0.381966 + 2.97558i 0
1301.7 0 1.67601 0.437016i 0 0 0 2.23607 + 1.41421i 0 2.61803 1.46489i 0
1301.8 0 1.67601 + 0.437016i 0 0 0 2.23607 1.41421i 0 2.61803 + 1.46489i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1301.8 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 inner
7.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.d.k 8
3.b odd 2 1 inner 2100.2.d.k 8
5.b even 2 1 2100.2.d.l yes 8
5.c odd 4 2 2100.2.f.i 16
7.b odd 2 1 inner 2100.2.d.k 8
15.d odd 2 1 2100.2.d.l yes 8
15.e even 4 2 2100.2.f.i 16
21.c even 2 1 inner 2100.2.d.k 8
35.c odd 2 1 2100.2.d.l yes 8
35.f even 4 2 2100.2.f.i 16
105.g even 2 1 2100.2.d.l yes 8
105.k odd 4 2 2100.2.f.i 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.2.d.k 8 1.a even 1 1 trivial
2100.2.d.k 8 3.b odd 2 1 inner
2100.2.d.k 8 7.b odd 2 1 inner
2100.2.d.k 8 21.c even 2 1 inner
2100.2.d.l yes 8 5.b even 2 1
2100.2.d.l yes 8 15.d odd 2 1
2100.2.d.l yes 8 35.c odd 2 1
2100.2.d.l yes 8 105.g even 2 1
2100.2.f.i 16 5.c odd 4 2
2100.2.f.i 16 15.e even 4 2
2100.2.f.i 16 35.f even 4 2
2100.2.f.i 16 105.k odd 4 2

## Hecke kernels

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

 $$T_{11}^{4} + 16 T_{11}^{2} + 19$$ $$T_{13}^{4} + 36 T_{13}^{2} + 4$$ $$T_{17}^{4} - 22 T_{17}^{2} + 76$$ $$T_{37}^{2} - 5$$ $$T_{41}^{4} - 90 T_{41}^{2} + 1900$$ $$T_{43}^{2} + 10 T_{43} + 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 6 T^{2} + 22 T^{4} - 54 T^{6} + 81 T^{8}$$
$5$ 1
$7$ $$( 1 - 6 T^{2} + 49 T^{4} )^{2}$$
$11$ $$( 1 - 28 T^{2} + 393 T^{4} - 3388 T^{6} + 14641 T^{8} )^{2}$$
$13$ $$( 1 - 16 T^{2} + 82 T^{4} - 2704 T^{6} + 28561 T^{8} )^{2}$$
$17$ $$( 1 + 46 T^{2} + 1062 T^{4} + 13294 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$( 1 - 46 T^{2} + 1126 T^{4} - 16606 T^{6} + 130321 T^{8} )^{2}$$
$23$ $$( 1 - 12 T^{2} - 31 T^{4} - 6348 T^{6} + 279841 T^{8} )^{2}$$
$29$ $$( 1 - 12 T^{2} + 1313 T^{4} - 10092 T^{6} + 707281 T^{8} )^{2}$$
$31$ $$( 1 - 94 T^{2} + 4006 T^{4} - 90334 T^{6} + 923521 T^{8} )^{2}$$
$37$ $$( 1 + 69 T^{2} + 1369 T^{4} )^{4}$$
$41$ $$( 1 + 74 T^{2} + 4606 T^{4} + 124394 T^{6} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 + 10 T + 91 T^{2} + 430 T^{3} + 1849 T^{4} )^{4}$$
$47$ $$( 1 - 34 T^{2} + 1582 T^{4} - 75106 T^{6} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 - 32 T^{2} + 5374 T^{4} - 89888 T^{6} + 7890481 T^{8} )^{2}$$
$59$ $$( 1 + 76 T^{2} + 3906 T^{4} + 264556 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 - 94 T^{2} + 6526 T^{4} - 349774 T^{6} + 13845841 T^{8} )^{2}$$
$67$ $$( 1 + 89 T^{2} + 4489 T^{4} )^{4}$$
$71$ $$( 1 - 164 T^{2} + 13681 T^{4} - 826724 T^{6} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 - 144 T^{2} + 5329 T^{4} )^{4}$$
$79$ $$( 1 + 6 T + 147 T^{2} + 474 T^{3} + 6241 T^{4} )^{4}$$
$83$ $$( 1 + 242 T^{2} + 28294 T^{4} + 1667138 T^{6} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 + 116 T^{2} + 6706 T^{4} + 918836 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 - 294 T^{2} + 38222 T^{4} - 2766246 T^{6} + 88529281 T^{8} )^{2}$$