# Properties

 Label 2100.2.d.f Level 2100 Weight 2 Character orbit 2100.d Analytic conductor 16.769 Analytic rank 0 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.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.7685844245$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{5})$$ Defining polynomial: $$x^{4} + 6 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 420) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{1} ) q^{3} + ( \beta_{1} + \beta_{3} ) q^{7} + ( -1 + 2 \beta_{1} ) q^{9} +O(q^{10})$$ $$q + ( -1 - \beta_{1} ) q^{3} + ( \beta_{1} + \beta_{3} ) q^{7} + ( -1 + 2 \beta_{1} ) q^{9} -2 \beta_{1} q^{11} + 2 \beta_{1} q^{13} -4 q^{17} + 2 \beta_{2} q^{19} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{21} -2 \beta_{2} q^{23} + ( 5 - \beta_{1} ) q^{27} + 4 \beta_{1} q^{29} + 2 \beta_{2} q^{31} + ( -4 + 2 \beta_{1} ) q^{33} -4 \beta_{3} q^{37} + ( 4 - 2 \beta_{1} ) q^{39} -2 \beta_{3} q^{41} -2 \beta_{3} q^{43} -6 q^{47} + ( 3 + 2 \beta_{2} ) q^{49} + ( 4 + 4 \beta_{1} ) q^{51} -2 \beta_{2} q^{53} + ( -2 \beta_{2} + 4 \beta_{3} ) q^{57} -4 \beta_{3} q^{59} + ( -4 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{63} -2 \beta_{3} q^{67} + ( 2 \beta_{2} - 4 \beta_{3} ) q^{69} + 10 \beta_{1} q^{71} -6 \beta_{1} q^{73} + ( 4 - 2 \beta_{2} ) q^{77} + 12 q^{79} + ( -7 - 4 \beta_{1} ) q^{81} + 10 q^{83} + ( 8 - 4 \beta_{1} ) q^{87} -2 \beta_{3} q^{89} + ( -4 + 2 \beta_{2} ) q^{91} + ( -2 \beta_{2} + 4 \beta_{3} ) q^{93} + 6 \beta_{1} q^{97} + ( 8 + 2 \beta_{1} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{3} - 4q^{9} + O(q^{10})$$ $$4q - 4q^{3} - 4q^{9} - 16q^{17} + 8q^{21} + 20q^{27} - 16q^{33} + 16q^{39} - 24q^{47} + 12q^{49} + 16q^{51} - 16q^{63} + 16q^{77} + 48q^{79} - 28q^{81} + 40q^{83} + 32q^{87} - 16q^{91} + 32q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 8 \nu$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 3$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{2} + 4 \beta_{1}$$

## 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
 − 2.28825i 0.874032i 2.28825i − 0.874032i
0 −1.00000 1.41421i 0 0 0 −2.23607 + 1.41421i 0 −1.00000 + 2.82843i 0
1301.2 0 −1.00000 1.41421i 0 0 0 2.23607 + 1.41421i 0 −1.00000 + 2.82843i 0
1301.3 0 −1.00000 + 1.41421i 0 0 0 −2.23607 1.41421i 0 −1.00000 2.82843i 0
1301.4 0 −1.00000 + 1.41421i 0 0 0 2.23607 1.41421i 0 −1.00000 2.82843i 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
15.d odd 2 1 inner
21.c even 2 1 inner
35.c odd 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.f 4
3.b odd 2 1 2100.2.d.i 4
5.b even 2 1 2100.2.d.i 4
5.c odd 4 2 420.2.f.b 8
7.b odd 2 1 2100.2.d.i 4
15.d odd 2 1 inner 2100.2.d.f 4
15.e even 4 2 420.2.f.b 8
20.e even 4 2 1680.2.k.g 8
21.c even 2 1 inner 2100.2.d.f 4
35.c odd 2 1 inner 2100.2.d.f 4
35.f even 4 2 420.2.f.b 8
60.l odd 4 2 1680.2.k.g 8
105.g even 2 1 2100.2.d.i 4
105.k odd 4 2 420.2.f.b 8
140.j odd 4 2 1680.2.k.g 8
420.w even 4 2 1680.2.k.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.f.b 8 5.c odd 4 2
420.2.f.b 8 15.e even 4 2
420.2.f.b 8 35.f even 4 2
420.2.f.b 8 105.k odd 4 2
1680.2.k.g 8 20.e even 4 2
1680.2.k.g 8 60.l odd 4 2
1680.2.k.g 8 140.j odd 4 2
1680.2.k.g 8 420.w even 4 2
2100.2.d.f 4 1.a even 1 1 trivial
2100.2.d.f 4 15.d odd 2 1 inner
2100.2.d.f 4 21.c even 2 1 inner
2100.2.d.f 4 35.c odd 2 1 inner
2100.2.d.i 4 3.b odd 2 1
2100.2.d.i 4 5.b even 2 1
2100.2.d.i 4 7.b odd 2 1
2100.2.d.i 4 105.g even 2 1

## Hecke kernels

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

 $$T_{11}^{2} + 8$$ $$T_{13}^{2} + 8$$ $$T_{17} + 4$$ $$T_{37}^{2} - 80$$ $$T_{41}^{2} - 20$$ $$T_{43}^{2} - 20$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + 2 T + 3 T^{2} )^{2}$$
$5$ 1
$7$ $$1 - 6 T^{2} + 49 T^{4}$$
$11$ $$( 1 - 6 T + 11 T^{2} )^{2}( 1 + 6 T + 11 T^{2} )^{2}$$
$13$ $$( 1 - 18 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 + 4 T + 17 T^{2} )^{4}$$
$19$ $$( 1 - 6 T + 19 T^{2} )^{2}( 1 + 6 T + 19 T^{2} )^{2}$$
$23$ $$( 1 - 6 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 26 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 22 T^{2} + 961 T^{4} )^{2}$$
$37$ $$( 1 - 6 T^{2} + 1369 T^{4} )^{2}$$
$41$ $$( 1 + 62 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 + 66 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$( 1 + 6 T + 47 T^{2} )^{4}$$
$53$ $$( 1 - 66 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$( 1 + 38 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 - 61 T^{2} )^{4}$$
$67$ $$( 1 + 114 T^{2} + 4489 T^{4} )^{2}$$
$71$ $$( 1 + 58 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 - 74 T^{2} + 5329 T^{4} )^{2}$$
$79$ $$( 1 - 12 T + 79 T^{2} )^{4}$$
$83$ $$( 1 - 10 T + 83 T^{2} )^{4}$$
$89$ $$( 1 + 158 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 122 T^{2} + 9409 T^{4} )^{2}$$