# Properties

 Label 2100.2.q.k Level 2100 Weight 2 Character orbit 2100.q 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.q (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

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

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{1} ) q^{3} + ( -\beta_{1} - \beta_{3} ) q^{7} + \beta_{1} q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{1} ) q^{3} + ( -\beta_{1} - \beta_{3} ) q^{7} + \beta_{1} q^{9} + ( \beta_{2} + \beta_{3} ) q^{11} + ( 1 + \beta_{2} - 2 \beta_{3} ) q^{13} + ( \beta_{2} + \beta_{3} ) q^{17} -7 \beta_{1} q^{19} + ( 1 - \beta_{2} ) q^{21} + ( 2 \beta_{2} - \beta_{3} ) q^{23} - q^{27} + ( -6 - \beta_{2} + 2 \beta_{3} ) q^{29} + ( 1 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{31} + ( 2 \beta_{2} - \beta_{3} ) q^{33} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{37} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{39} + ( \beta_{2} - 2 \beta_{3} ) q^{41} + ( 1 + \beta_{2} - 2 \beta_{3} ) q^{43} + 6 \beta_{1} q^{47} + ( 5 + 5 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{49} + ( 2 \beta_{2} - \beta_{3} ) q^{51} + ( 2 \beta_{2} + 2 \beta_{3} ) q^{53} + 7 q^{57} + ( 6 + 6 \beta_{1} + \beta_{2} + \beta_{3} ) q^{59} + ( -4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{61} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{63} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{67} + ( \beta_{2} - 2 \beta_{3} ) q^{69} + ( 3 \beta_{2} - 6 \beta_{3} ) q^{71} + ( 5 + 5 \beta_{1} + \beta_{2} + \beta_{3} ) q^{73} + ( -6 - 12 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{77} + 11 \beta_{1} q^{79} + ( -1 - \beta_{1} ) q^{81} + ( 6 - \beta_{2} + 2 \beta_{3} ) q^{83} + ( -6 - 6 \beta_{1} + \beta_{2} + \beta_{3} ) q^{87} + ( -6 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{89} + ( 12 + 5 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{91} + ( \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{93} + ( -8 - 2 \beta_{2} + 4 \beta_{3} ) q^{97} + ( \beta_{2} - 2 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{3} + 2q^{7} - 2q^{9} + O(q^{10})$$ $$4q + 2q^{3} + 2q^{7} - 2q^{9} + 4q^{13} + 14q^{19} + 4q^{21} - 4q^{27} - 24q^{29} + 2q^{31} - 2q^{37} + 2q^{39} + 4q^{43} - 12q^{47} + 10q^{49} + 28q^{57} + 12q^{59} + 8q^{61} + 2q^{63} - 2q^{67} + 10q^{73} - 22q^{79} - 2q^{81} + 24q^{83} - 12q^{87} + 12q^{89} + 38q^{91} - 2q^{93} - 32q^{97} + O(q^{100})$$

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

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

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1201.1
 0.707107 − 1.22474i −0.707107 + 1.22474i 0.707107 + 1.22474i −0.707107 − 1.22474i
0 0.500000 0.866025i 0 0 0 −1.62132 + 2.09077i 0 −0.500000 0.866025i 0
1201.2 0 0.500000 0.866025i 0 0 0 2.62132 0.358719i 0 −0.500000 0.866025i 0
1801.1 0 0.500000 + 0.866025i 0 0 0 −1.62132 2.09077i 0 −0.500000 + 0.866025i 0
1801.2 0 0.500000 + 0.866025i 0 0 0 2.62132 + 0.358719i 0 −0.500000 + 0.866025i 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
7.c even 3 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{11}^{4} + 18 T_{11}^{2} + 324$$ $$T_{13}^{2} - 2 T_{13} - 17$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - T + T^{2} )^{2}$$
$5$ 1
$7$ $$1 - 2 T - 3 T^{2} - 14 T^{3} + 49 T^{4}$$
$11$ $$1 - 4 T^{2} - 105 T^{4} - 484 T^{6} + 14641 T^{8}$$
$13$ $$( 1 - 2 T + 9 T^{2} - 26 T^{3} + 169 T^{4} )^{2}$$
$17$ $$1 - 16 T^{2} - 33 T^{4} - 4624 T^{6} + 83521 T^{8}$$
$19$ $$( 1 - 8 T + 19 T^{2} )^{2}( 1 + T + 19 T^{2} )^{2}$$
$23$ $$1 - 28 T^{2} + 255 T^{4} - 14812 T^{6} + 279841 T^{8}$$
$29$ $$( 1 + 12 T + 76 T^{2} + 348 T^{3} + 841 T^{4} )^{2}$$
$31$ $$1 - 2 T + 13 T^{2} + 142 T^{3} - 1004 T^{4} + 4402 T^{5} + 12493 T^{6} - 59582 T^{7} + 923521 T^{8}$$
$37$ $$1 + 2 T - 53 T^{2} - 34 T^{3} + 1732 T^{4} - 1258 T^{5} - 72557 T^{6} + 101306 T^{7} + 1874161 T^{8}$$
$41$ $$( 1 + 64 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 - 2 T + 69 T^{2} - 86 T^{3} + 1849 T^{4} )^{2}$$
$47$ $$( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} )^{2}$$
$53$ $$1 - 34 T^{2} - 1653 T^{4} - 95506 T^{6} + 7890481 T^{8}$$
$59$ $$1 - 12 T + 8 T^{2} - 216 T^{3} + 6519 T^{4} - 12744 T^{5} + 27848 T^{6} - 2464548 T^{7} + 12117361 T^{8}$$
$61$ $$1 - 8 T - 2 T^{2} + 448 T^{3} - 3269 T^{4} + 27328 T^{5} - 7442 T^{6} - 1815848 T^{7} + 13845841 T^{8}$$
$67$ $$1 + 2 T - 113 T^{2} - 34 T^{3} + 8932 T^{4} - 2278 T^{5} - 507257 T^{6} + 601526 T^{7} + 20151121 T^{8}$$
$71$ $$( 1 - 20 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$1 - 10 T - 53 T^{2} - 70 T^{3} + 10780 T^{4} - 5110 T^{5} - 282437 T^{6} - 3890170 T^{7} + 28398241 T^{8}$$
$79$ $$( 1 + 11 T + 42 T^{2} + 869 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$( 1 - 12 T + 184 T^{2} - 996 T^{3} + 6889 T^{4} )^{2}$$
$89$ $$1 - 12 T - 52 T^{2} - 216 T^{3} + 17679 T^{4} - 19224 T^{5} - 411892 T^{6} - 8459628 T^{7} + 62742241 T^{8}$$
$97$ $$( 1 + 16 T + 186 T^{2} + 1552 T^{3} + 9409 T^{4} )^{2}$$