# Properties

 Label 2100.2.q.i Level 2100 Weight 2 Character orbit 2100.q Analytic conductor 16.769 Analytic rank 0 Dimension 4 CM no Inner twists 2

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## 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: $$\Q(\sqrt{-3}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 7 x^{2} + 49$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7 x^{2} + 49$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/7$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$7 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$7 \beta_{3}$$

## Character values

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

 $$n$$ $$701$$ $$1051$$ $$1177$$ $$1501$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$\beta_{2}$$

## 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
 −1.32288 + 2.29129i 1.32288 − 2.29129i −1.32288 − 2.29129i 1.32288 + 2.29129i
0 0.500000 0.866025i 0 0 0 −1.32288 + 2.29129i 0 −0.500000 0.866025i 0
1201.2 0 0.500000 0.866025i 0 0 0 1.32288 2.29129i 0 −0.500000 0.866025i 0
1801.1 0 0.500000 + 0.866025i 0 0 0 −1.32288 2.29129i 0 −0.500000 + 0.866025i 0
1801.2 0 0.500000 + 0.866025i 0 0 0 1.32288 + 2.29129i 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.i yes 4
5.b even 2 1 2100.2.q.g 4
5.c odd 4 2 2100.2.bc.h 8
7.c even 3 1 inner 2100.2.q.i yes 4
35.j even 6 1 2100.2.q.g 4
35.l odd 12 2 2100.2.bc.h 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.2.q.g 4 5.b even 2 1
2100.2.q.g 4 35.j even 6 1
2100.2.q.i yes 4 1.a even 1 1 trivial
2100.2.q.i yes 4 7.c even 3 1 inner
2100.2.bc.h 8 5.c odd 4 2
2100.2.bc.h 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 $$S_{2}^{\mathrm{new}}(2100, [\chi])$$:

 $$T_{11}^{4} - 6 T_{11}^{3} + 34 T_{11}^{2} - 12 T_{11} + 4$$ $$T_{13} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - T + T^{2} )^{2}$$
$5$ 1
$7$ $$1 + 7 T^{2} + 49 T^{4}$$
$11$ $$1 - 6 T + 12 T^{2} - 12 T^{3} + 59 T^{4} - 132 T^{5} + 1452 T^{6} - 7986 T^{7} + 14641 T^{8}$$
$13$ $$( 1 - T + 13 T^{2} )^{4}$$
$17$ $$1 + 2 T - 24 T^{2} - 12 T^{3} + 427 T^{4} - 204 T^{5} - 6936 T^{6} + 9826 T^{7} + 83521 T^{8}$$
$19$ $$1 - 4 T - 19 T^{2} + 12 T^{3} + 560 T^{4} + 228 T^{5} - 6859 T^{6} - 27436 T^{7} + 130321 T^{8}$$
$23$ $$( 1 - 2 T - 19 T^{2} - 46 T^{3} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 4 T + 34 T^{2} - 116 T^{3} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 7 T + 31 T^{2} )^{2}( 1 + 11 T + 31 T^{2} )^{2}$$
$37$ $$1 + 6 T - 19 T^{2} - 114 T^{3} + 324 T^{4} - 4218 T^{5} - 26011 T^{6} + 303918 T^{7} + 1874161 T^{8}$$
$41$ $$( 1 + 6 T + 84 T^{2} + 246 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$( 1 + 4 T + 62 T^{2} + 172 T^{3} + 1849 T^{4} )^{2}$$
$47$ $$1 - 10 T + 44 T^{2} + 380 T^{3} - 3773 T^{4} + 17860 T^{5} + 97196 T^{6} - 1038230 T^{7} + 4879681 T^{8}$$
$53$ $$1 + 10 T - 24 T^{2} + 180 T^{3} + 7267 T^{4} + 9540 T^{5} - 67416 T^{6} + 1488770 T^{7} + 7890481 T^{8}$$
$59$ $$1 + 14 T + 92 T^{2} - 196 T^{3} - 4229 T^{4} - 11564 T^{5} + 320252 T^{6} + 2875306 T^{7} + 12117361 T^{8}$$
$61$ $$1 - 2 T - 7 T^{2} + 222 T^{3} - 3844 T^{4} + 13542 T^{5} - 26047 T^{6} - 453962 T^{7} + 13845841 T^{8}$$
$67$ $$1 - 71 T^{2} + 552 T^{4} - 318719 T^{6} + 20151121 T^{8}$$
$71$ $$( 1 + 4 T + 118 T^{2} + 284 T^{3} + 5041 T^{4} )^{2}$$
$73$ $$1 - 2 T - 115 T^{2} + 54 T^{3} + 8540 T^{4} + 3942 T^{5} - 612835 T^{6} - 778034 T^{7} + 28398241 T^{8}$$
$79$ $$1 - 8 T - 103 T^{2} - 72 T^{3} + 16592 T^{4} - 5688 T^{5} - 642823 T^{6} - 3944312 T^{7} + 38950081 T^{8}$$
$83$ $$( 1 - 2 T + 160 T^{2} - 166 T^{3} + 6889 T^{4} )^{2}$$
$89$ $$1 - 2 T - 112 T^{2} + 124 T^{3} + 5179 T^{4} + 11036 T^{5} - 887152 T^{6} - 1409938 T^{7} + 62742241 T^{8}$$
$97$ $$( 1 - 18 T + 247 T^{2} - 1746 T^{3} + 9409 T^{4} )^{2}$$
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