# Properties

 Label 2100.2.q.f 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: yes 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.

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

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 −2.62132 0.358719i 0 −0.500000 0.866025i 0
1201.2 0 −0.500000 + 0.866025i 0 0 0 1.62132 + 2.09077i 0 −0.500000 0.866025i 0
1801.1 0 −0.500000 0.866025i 0 0 0 −2.62132 + 0.358719i 0 −0.500000 + 0.866025i 0
1801.2 0 −0.500000 0.866025i 0 0 0 1.62132 2.09077i 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.f 4
5.b even 2 1 2100.2.q.j yes 4
5.c odd 4 2 2100.2.bc.g 8
7.c even 3 1 inner 2100.2.q.f 4
35.j even 6 1 2100.2.q.j yes 4
35.l odd 12 2 2100.2.bc.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.2.q.f 4 1.a even 1 1 trivial
2100.2.q.f 4 7.c even 3 1 inner
2100.2.q.j yes 4 5.b even 2 1
2100.2.q.j yes 4 35.j even 6 1
2100.2.bc.g 8 5.c odd 4 2
2100.2.bc.g 8 35.l odd 12 2

## Hecke kernels

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

 $$T_{11}^{4} + 18 T_{11}^{2} + 324$$ $$T_{13} - 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + T + T^{2} )^{2}$$
$5$ 1
$7$ $$1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4}$$
$11$ $$1 - 4 T^{2} - 105 T^{4} - 484 T^{6} + 14641 T^{8}$$
$13$ $$( 1 - 5 T + 13 T^{2} )^{4}$$
$17$ $$1 - 16 T^{2} - 33 T^{4} - 4624 T^{6} + 83521 T^{8}$$
$19$ $$1 - 2 T - 17 T^{2} + 34 T^{3} + 4 T^{4} + 646 T^{5} - 6137 T^{6} - 13718 T^{7} + 130321 T^{8}$$
$23$ $$( 1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 14 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 11 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2}$$
$37$ $$( 1 + 5 T - 12 T^{2} + 185 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$( 1 + 12 T + 100 T^{2} + 492 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$( 1 + 8 T + 30 T^{2} + 344 T^{3} + 1849 T^{4} )^{2}$$
$47$ $$1 + 12 T + 32 T^{2} + 216 T^{3} + 3567 T^{4} + 10152 T^{5} + 70688 T^{6} + 1245876 T^{7} + 4879681 T^{8}$$
$53$ $$1 - 88 T^{2} + 4935 T^{4} - 247192 T^{6} + 7890481 T^{8}$$
$59$ $$1 + 12 T + 8 T^{2} + 216 T^{3} + 6519 T^{4} + 12744 T^{5} + 27848 T^{6} + 2464548 T^{7} + 12117361 T^{8}$$
$61$ $$( 1 - 14 T + 61 T^{2} )^{2}( 1 + 13 T + 61 T^{2} )^{2}$$
$67$ $$1 - 2 T + 31 T^{2} + 322 T^{3} - 4028 T^{4} + 21574 T^{5} + 139159 T^{6} - 601526 T^{7} + 20151121 T^{8}$$
$71$ $$( 1 + 70 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$1 + 10 T + T^{2} - 470 T^{3} - 2828 T^{4} - 34310 T^{5} + 5329 T^{6} + 3890170 T^{7} + 28398241 T^{8}$$
$79$ $$1 + 10 T - 65 T^{2} + 70 T^{3} + 13084 T^{4} + 5530 T^{5} - 405665 T^{6} + 4930390 T^{7} + 38950081 T^{8}$$
$83$ $$( 1 - 12 T + 184 T^{2} - 996 T^{3} + 6889 T^{4} )^{2}$$
$89$ $$1 + 24 T + 272 T^{2} + 3024 T^{3} + 33231 T^{4} + 269136 T^{5} + 2154512 T^{6} + 16919256 T^{7} + 62742241 T^{8}$$
$97$ $$( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} )^{2}$$