# Properties

 Label 2100.2.q.h 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: 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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{2} ) q^{3} + ( -\beta_{1} - \beta_{3} ) q^{7} + \beta_{2} q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{2} ) q^{3} + ( -\beta_{1} - \beta_{3} ) q^{7} + \beta_{2} q^{9} + ( -1 + \beta_{1} - \beta_{2} ) q^{11} -\beta_{3} q^{13} + ( 1 - \beta_{1} + \beta_{2} ) q^{17} + ( 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{19} -\beta_{3} q^{21} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{23} - q^{27} + ( 5 - \beta_{3} ) q^{29} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{31} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{33} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{37} + \beta_{1} q^{39} + ( -3 - 3 \beta_{3} ) q^{41} + ( -2 - 3 \beta_{3} ) q^{43} -6 \beta_{2} q^{47} + ( -7 - 7 \beta_{2} ) q^{49} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{51} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{53} + ( -3 + 2 \beta_{3} ) q^{57} + ( -3 - 3 \beta_{1} - 3 \beta_{2} ) q^{59} + 8 \beta_{2} q^{61} + \beta_{1} q^{63} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{67} + ( -1 - \beta_{3} ) q^{69} + ( 11 - \beta_{3} ) q^{71} + 5 \beta_{1} q^{73} + ( 7 + \beta_{3} ) q^{77} + ( 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{79} + ( -1 - \beta_{2} ) q^{81} + ( 3 - 3 \beta_{3} ) q^{83} + ( 5 + \beta_{1} + 5 \beta_{2} ) q^{87} + ( -5 \beta_{1} - \beta_{2} - 5 \beta_{3} ) q^{89} -7 \beta_{2} q^{91} + ( 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{93} -8 q^{97} + ( 1 + \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{3} - 2q^{9} + O(q^{10})$$ $$4q + 2q^{3} - 2q^{9} - 2q^{11} + 2q^{17} - 6q^{19} - 2q^{23} - 4q^{27} + 20q^{29} - 2q^{31} + 2q^{33} - 4q^{37} - 12q^{41} - 8q^{43} + 12q^{47} - 14q^{49} - 2q^{51} - 4q^{53} - 12q^{57} - 6q^{59} - 16q^{61} - 4q^{67} - 4q^{69} + 44q^{71} + 28q^{77} + 6q^{79} - 2q^{81} + 12q^{83} + 10q^{87} + 2q^{89} + 14q^{91} + 2q^{93} - 32q^{97} + 4q^{99} + O(q^{100})$$

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.h 4
5.b even 2 1 420.2.q.c 4
5.c odd 4 2 2100.2.bc.e 8
7.c even 3 1 inner 2100.2.q.h 4
15.d odd 2 1 1260.2.s.f 4
20.d odd 2 1 1680.2.bg.q 4
35.c odd 2 1 2940.2.q.t 4
35.i odd 6 1 2940.2.a.m 2
35.i odd 6 1 2940.2.q.t 4
35.j even 6 1 420.2.q.c 4
35.j even 6 1 2940.2.a.s 2
35.l odd 12 2 2100.2.bc.e 8
105.o odd 6 1 1260.2.s.f 4
105.o odd 6 1 8820.2.a.be 2
105.p even 6 1 8820.2.a.bj 2
140.p odd 6 1 1680.2.bg.q 4

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

## Hecke kernels

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

 $$T_{11}^{4} + 2 T_{11}^{3} + 10 T_{11}^{2} - 12 T_{11} + 36$$ $$T_{13}^{2} - 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - T + T^{2} )^{2}$$
$5$ 1
$7$ $$1 + 7 T^{2} + 49 T^{4}$$
$11$ $$1 + 2 T - 12 T^{2} - 12 T^{3} + 91 T^{4} - 132 T^{5} - 1452 T^{6} + 2662 T^{7} + 14641 T^{8}$$
$13$ $$( 1 + 19 T^{2} + 169 T^{4} )^{2}$$
$17$ $$1 - 2 T - 24 T^{2} + 12 T^{3} + 427 T^{4} + 204 T^{5} - 6936 T^{6} - 9826 T^{7} + 83521 T^{8}$$
$19$ $$1 + 6 T + 17 T^{2} - 114 T^{3} - 684 T^{4} - 2166 T^{5} + 6137 T^{6} + 41154 T^{7} + 130321 T^{8}$$
$23$ $$1 + 2 T - 36 T^{2} - 12 T^{3} + 979 T^{4} - 276 T^{5} - 19044 T^{6} + 24334 T^{7} + 279841 T^{8}$$
$29$ $$( 1 - 10 T + 76 T^{2} - 290 T^{3} + 841 T^{4} )^{2}$$
$31$ $$1 + 2 T - 31 T^{2} - 54 T^{3} + 140 T^{4} - 1674 T^{5} - 29791 T^{6} + 59582 T^{7} + 923521 T^{8}$$
$37$ $$1 + 4 T - 55 T^{2} - 12 T^{3} + 3080 T^{4} - 444 T^{5} - 75295 T^{6} + 202612 T^{7} + 1874161 T^{8}$$
$41$ $$( 1 + 6 T + 28 T^{2} + 246 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$( 1 + 4 T + 27 T^{2} + 172 T^{3} + 1849 T^{4} )^{2}$$
$47$ $$( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} )^{2}$$
$53$ $$1 + 4 T - 66 T^{2} - 96 T^{3} + 3067 T^{4} - 5088 T^{5} - 185394 T^{6} + 595508 T^{7} + 7890481 T^{8}$$
$59$ $$1 + 6 T - 28 T^{2} - 324 T^{3} - 1509 T^{4} - 19116 T^{5} - 97468 T^{6} + 1232274 T^{7} + 12117361 T^{8}$$
$61$ $$( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$1 + 4 T - 115 T^{2} - 12 T^{3} + 11600 T^{4} - 804 T^{5} - 516235 T^{6} + 1203052 T^{7} + 20151121 T^{8}$$
$71$ $$( 1 - 22 T + 256 T^{2} - 1562 T^{3} + 5041 T^{4} )^{2}$$
$73$ $$1 + 29 T^{2} - 4488 T^{4} + 154541 T^{6} + 28398241 T^{8}$$
$79$ $$1 - 6 T - 103 T^{2} + 114 T^{3} + 10236 T^{4} + 9006 T^{5} - 642823 T^{6} - 2958234 T^{7} + 38950081 T^{8}$$
$83$ $$( 1 - 6 T + 112 T^{2} - 498 T^{3} + 6889 T^{4} )^{2}$$
$89$ $$1 - 2 T + 348 T^{3} - 8261 T^{4} + 30972 T^{5} - 1409938 T^{7} + 62742241 T^{8}$$
$97$ $$( 1 + 8 T + 97 T^{2} )^{4}$$
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