# Properties

 Label 2100.2.f.h Level 2100 Weight 2 Character orbit 2100.f Analytic conductor 16.769 Analytic rank 0 Dimension 4 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.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.7685844245$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + ( 2 + \beta_{1} + \beta_{2} ) q^{7} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + ( 2 + \beta_{1} + \beta_{2} ) q^{7} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{9} + ( 1 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{11} + ( 1 - \beta_{1} - \beta_{3} ) q^{13} + ( 2 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{17} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{19} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{21} + ( 1 + 3 \beta_{1} + 3 \beta_{3} ) q^{23} + ( 3 - 2 \beta_{2} + \beta_{3} ) q^{27} + 2 \beta_{2} q^{29} + ( 1 + \beta_{1} - \beta_{3} ) q^{31} + ( 3 + \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{33} + ( 2 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{37} + ( 3 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{39} + ( -8 + 2 \beta_{1} + 2 \beta_{3} ) q^{41} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{43} + ( -1 - \beta_{1} + \beta_{3} ) q^{47} + ( 3 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{49} + ( 6 + 2 \beta_{1} + 3 \beta_{2} ) q^{51} + ( 7 + \beta_{1} + \beta_{3} ) q^{53} + ( 3 + \beta_{1} - \beta_{2} + 5 \beta_{3} ) q^{57} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{59} + ( 4 + 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{61} + ( 3 + \beta_{2} - 5 \beta_{3} ) q^{63} + ( 3 + 3 \beta_{1} - 3 \beta_{3} ) q^{67} + ( -9 - \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{69} + ( -5 - 5 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} ) q^{71} + ( 9 - \beta_{1} - \beta_{3} ) q^{73} + ( -7 + 3 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{77} + ( -6 - 2 \beta_{1} - 2 \beta_{3} ) q^{79} + ( -3 - 4 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{81} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{83} + ( 2 \beta_{2} - 4 \beta_{3} ) q^{87} + ( 8 + 2 \beta_{1} + 2 \beta_{3} ) q^{89} + ( 1 + \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{91} + ( 3 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{93} + ( -3 - 5 \beta_{1} - 5 \beta_{3} ) q^{97} + ( 9 + \beta_{1} + 4 \beta_{2} - 5 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{3} + 6q^{7} + O(q^{10})$$ $$4q + 2q^{3} + 6q^{7} + 4q^{13} + 4q^{23} + 14q^{27} + 4q^{33} + 12q^{39} - 32q^{41} + 20q^{51} + 28q^{53} + 20q^{57} + 2q^{63} - 28q^{69} + 36q^{73} - 28q^{77} - 24q^{79} + 4q^{81} - 8q^{87} + 32q^{89} - 4q^{91} + 12q^{93} - 12q^{97} + 24q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

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

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1049.1
 0.618034i − 0.618034i 1.61803i − 1.61803i
0 −0.618034 1.61803i 0 0 0 2.61803 0.381966i 0 −2.23607 + 2.00000i 0
1049.2 0 −0.618034 + 1.61803i 0 0 0 2.61803 + 0.381966i 0 −2.23607 2.00000i 0
1049.3 0 1.61803 0.618034i 0 0 0 0.381966 + 2.61803i 0 2.23607 2.00000i 0
1049.4 0 1.61803 + 0.618034i 0 0 0 0.381966 2.61803i 0 2.23607 + 2.00000i 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
105.g 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.f.h 4
3.b odd 2 1 2100.2.f.g 4
5.b even 2 1 2100.2.f.b 4
5.c odd 4 1 420.2.d.d yes 4
5.c odd 4 1 2100.2.d.g 4
7.b odd 2 1 2100.2.f.a 4
15.d odd 2 1 2100.2.f.a 4
15.e even 4 1 420.2.d.c 4
15.e even 4 1 2100.2.d.h 4
20.e even 4 1 1680.2.f.f 4
21.c even 2 1 2100.2.f.b 4
35.c odd 2 1 2100.2.f.g 4
35.f even 4 1 420.2.d.c 4
35.f even 4 1 2100.2.d.h 4
60.l odd 4 1 1680.2.f.j 4
105.g even 2 1 inner 2100.2.f.h 4
105.k odd 4 1 420.2.d.d yes 4
105.k odd 4 1 2100.2.d.g 4
140.j odd 4 1 1680.2.f.j 4
420.w even 4 1 1680.2.f.f 4

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

## Hecke kernels

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

 $$T_{11}^{4} + 28 T_{11}^{2} + 16$$ $$T_{13}^{2} - 2 T_{13} - 4$$ $$T_{23}^{2} - 2 T_{23} - 44$$ $$T_{41}^{2} + 16 T_{41} + 44$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 2 T + 2 T^{2} - 6 T^{3} + 9 T^{4}$$
$5$ 1
$7$ $$1 - 6 T + 18 T^{2} - 42 T^{3} + 49 T^{4}$$
$11$ $$1 - 16 T^{2} + 126 T^{4} - 1936 T^{6} + 14641 T^{8}$$
$13$ $$( 1 - 2 T + 22 T^{2} - 26 T^{3} + 169 T^{4} )^{2}$$
$17$ $$( 1 - 14 T^{2} + 289 T^{4} )^{2}$$
$19$ $$1 - 16 T^{2} + 286 T^{4} - 5776 T^{6} + 130321 T^{8}$$
$23$ $$( 1 - 2 T + 2 T^{2} - 46 T^{3} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 10 T + 29 T^{2} )^{2}( 1 + 10 T + 29 T^{2} )^{2}$$
$31$ $$1 - 112 T^{2} + 5038 T^{4} - 107632 T^{6} + 923521 T^{8}$$
$37$ $$( 1 - 54 T^{2} + 1369 T^{4} )^{2}$$
$41$ $$( 1 + 16 T + 126 T^{2} + 656 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$1 - 112 T^{2} + 6334 T^{4} - 207088 T^{6} + 3418801 T^{8}$$
$47$ $$1 - 176 T^{2} + 12142 T^{4} - 388784 T^{6} + 4879681 T^{8}$$
$53$ $$( 1 - 14 T + 150 T^{2} - 742 T^{3} + 2809 T^{4} )^{2}$$
$59$ $$( 1 + 38 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$1 - 52 T^{2} + 2998 T^{4} - 193492 T^{6} + 13845841 T^{8}$$
$67$ $$1 - 160 T^{2} + 13758 T^{4} - 718240 T^{6} + 20151121 T^{8}$$
$71$ $$1 - 32 T^{2} + 9838 T^{4} - 161312 T^{6} + 25411681 T^{8}$$
$73$ $$( 1 - 18 T + 222 T^{2} - 1314 T^{3} + 5329 T^{4} )^{2}$$
$79$ $$( 1 + 12 T + 174 T^{2} + 948 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$1 - 272 T^{2} + 31774 T^{4} - 1873808 T^{6} + 47458321 T^{8}$$
$89$ $$( 1 - 16 T + 222 T^{2} - 1424 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$( 1 + 6 T + 78 T^{2} + 582 T^{3} + 9409 T^{4} )^{2}$$