# Properties

 Label 2100.2.q.m Level 2100 Weight 2 Character orbit 2100.q Analytic conductor 16.769 Analytic rank 0 Dimension 8 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: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.17819046144.3 Defining polynomial: $$x^{8} - 2 x^{7} + 10 x^{6} + 8 x^{5} + 38 x^{4} - 4 x^{3} + 16 x^{2} + 4 x + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{4} q^{3} + ( \beta_{1} - \beta_{6} ) q^{7} + ( -1 - \beta_{4} ) q^{9} +O(q^{10})$$ $$q -\beta_{4} q^{3} + ( \beta_{1} - \beta_{6} ) q^{7} + ( -1 - \beta_{4} ) q^{9} + ( \beta_{4} + \beta_{5} - \beta_{6} ) q^{11} + ( \beta_{3} + \beta_{6} + \beta_{7} ) q^{13} + ( \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{17} + ( 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{5} ) q^{19} + ( \beta_{1} + \beta_{2} - \beta_{3} - \beta_{6} ) q^{21} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{23} - q^{27} + ( -2 + \beta_{3} + 2 \beta_{6} + 2 \beta_{7} ) q^{29} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{31} + ( 1 + \beta_{2} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{33} + ( 2 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{6} + \beta_{7} ) q^{37} + ( \beta_{1} - \beta_{2} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{39} + ( 3 - \beta_{2} + \beta_{5} - \beta_{7} ) q^{41} + ( -4 - 3 \beta_{2} + 3 \beta_{5} - 3 \beta_{7} ) q^{43} + ( -1 - 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - \beta_{4} + 3 \beta_{6} - 3 \beta_{7} ) q^{47} + ( 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{5} - \beta_{6} - \beta_{7} ) q^{49} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{51} + ( 2 \beta_{1} - 4 \beta_{4} ) q^{53} + ( -2 \beta_{3} + \beta_{6} + \beta_{7} ) q^{57} + ( 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{59} + ( -5 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 5 \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{61} + ( \beta_{2} - \beta_{3} ) q^{63} + ( -3 \beta_{1} - \beta_{2} - 6 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{67} + ( 1 + \beta_{3} + \beta_{6} + \beta_{7} ) q^{69} + ( -2 - 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} ) q^{71} + ( 2 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} ) q^{73} + ( -3 - \beta_{1} + 3 \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{7} ) q^{77} + ( 4 + 4 \beta_{1} + 5 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} + \beta_{5} - 4 \beta_{6} + 4 \beta_{7} ) q^{79} + \beta_{4} q^{81} + ( 3 - \beta_{3} - 3 \beta_{6} - 3 \beta_{7} ) q^{83} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{87} + ( 5 + 6 \beta_{1} - \beta_{2} - 6 \beta_{3} + 5 \beta_{4} + \beta_{6} - \beta_{7} ) q^{89} + ( -5 + 2 \beta_{1} - \beta_{2} - 4 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{91} + ( 1 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{93} + ( 4 + \beta_{2} - 3 \beta_{3} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{97} + ( 1 + \beta_{2} - \beta_{5} + \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{3} + 2q^{7} - 4q^{9} + O(q^{10})$$ $$8q + 4q^{3} + 2q^{7} - 4q^{9} - 4q^{11} + 4q^{13} - 2q^{17} - 4q^{19} - 2q^{21} + 6q^{23} - 8q^{27} - 12q^{29} + 4q^{33} + 4q^{37} + 2q^{39} + 24q^{41} - 32q^{43} + 2q^{47} - 4q^{49} + 2q^{51} + 20q^{53} - 8q^{57} + 14q^{59} - 16q^{61} - 4q^{63} + 18q^{67} + 12q^{69} - 28q^{71} - 8q^{73} - 10q^{77} + 8q^{79} - 4q^{81} + 20q^{83} - 6q^{87} + 8q^{89} - 20q^{91} + 20q^{97} + 8q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{7} + 10 x^{6} + 8 x^{5} + 38 x^{4} - 4 x^{3} + 16 x^{2} + 4 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$7 \nu^{7} + 78 \nu^{6} - 137 \nu^{5} + 1009 \nu^{4} + 820 \nu^{3} + 2992 \nu^{2} - 786 \nu + 712$$$$)/450$$ $$\beta_{3}$$ $$=$$ $$($$$$-9 \nu^{7} + 14 \nu^{6} - 81 \nu^{5} - 108 \nu^{4} - 390 \nu^{3} - 54 \nu^{2} - 18 \nu - 44$$$$)/150$$ $$\beta_{4}$$ $$=$$ $$($$$$11 \nu^{7} - 31 \nu^{6} + 124 \nu^{5} + 7 \nu^{4} + 310 \nu^{3} - 434 \nu^{2} + 122 \nu - 124$$$$)/150$$ $$\beta_{5}$$ $$=$$ $$($$$$-52 \nu^{7} + 117 \nu^{6} - 493 \nu^{5} - 424 \nu^{4} - 1270 \nu^{3} + 788 \nu^{2} + 696 \nu - 682$$$$)/450$$ $$\beta_{6}$$ $$=$$ $$($$$$-58 \nu^{7} + 168 \nu^{6} - 697 \nu^{5} + 29 \nu^{4} - 1780 \nu^{3} + 1502 \nu^{2} - 1716 \nu - 478$$$$)/450$$ $$\beta_{7}$$ $$=$$ $$($$$$4 \nu^{7} - 9 \nu^{6} + 43 \nu^{5} + 19 \nu^{4} + 154 \nu^{3} - 50 \nu^{2} + 72 \nu - 20$$$$)/18$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{5} - 3 \beta_{4} - 2 \beta_{3} - \beta_{2} + 2 \beta_{1} - 3$$ $$\nu^{3}$$ $$=$$ $$-3 \beta_{7} - 2 \beta_{6} + \beta_{5} - 9 \beta_{3} - \beta_{2} - 5$$ $$\nu^{4}$$ $$=$$ $$-10 \beta_{7} - 12 \beta_{6} + 12 \beta_{5} + 26 \beta_{4} + 10 \beta_{2} - 28 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$10 \beta_{7} - 10 \beta_{6} + 30 \beta_{5} + 76 \beta_{4} + 104 \beta_{3} + 40 \beta_{2} - 104 \beta_{1} + 76$$ $$\nu^{6}$$ $$=$$ $$144 \beta_{7} + 114 \beta_{6} - 30 \beta_{5} + 354 \beta_{3} + 30 \beta_{2} + 292$$ $$\nu^{7}$$ $$=$$ $$384 \beta_{7} + 498 \beta_{6} - 498 \beta_{5} - 978 \beta_{4} - 384 \beta_{2} + 1258 \beta_{1}$$

## Character values

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

 $$n$$ $$701$$ $$1051$$ $$1177$$ $$1501$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1 - \beta_{4}$$

## 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
 −0.868255 + 1.50386i 1.75161 − 3.03388i −0.234240 + 0.405716i 0.350883 − 0.607748i −0.868255 − 1.50386i 1.75161 + 3.03388i −0.234240 − 0.405716i 0.350883 + 0.607748i
0 0.500000 0.866025i 0 0 0 −2.61250 0.418148i 0 −0.500000 0.866025i 0
1201.2 0 0.500000 0.866025i 0 0 0 0.618546 2.57243i 0 −0.500000 0.866025i 0
1201.3 0 0.500000 0.866025i 0 0 0 0.687541 + 2.55486i 0 −0.500000 0.866025i 0
1201.4 0 0.500000 0.866025i 0 0 0 2.30641 1.29633i 0 −0.500000 0.866025i 0
1801.1 0 0.500000 + 0.866025i 0 0 0 −2.61250 + 0.418148i 0 −0.500000 + 0.866025i 0
1801.2 0 0.500000 + 0.866025i 0 0 0 0.618546 + 2.57243i 0 −0.500000 + 0.866025i 0
1801.3 0 0.500000 + 0.866025i 0 0 0 0.687541 2.55486i 0 −0.500000 + 0.866025i 0
1801.4 0 0.500000 + 0.866025i 0 0 0 2.30641 + 1.29633i 0 −0.500000 + 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1801.4 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.m 8
5.b even 2 1 2100.2.q.l 8
5.c odd 4 2 420.2.bb.a 16
7.c even 3 1 inner 2100.2.q.m 8
15.e even 4 2 1260.2.bm.c 16
20.e even 4 2 1680.2.di.e 16
35.f even 4 2 2940.2.bb.i 16
35.j even 6 1 2100.2.q.l 8
35.k even 12 2 2940.2.k.g 8
35.k even 12 2 2940.2.bb.i 16
35.l odd 12 2 420.2.bb.a 16
35.l odd 12 2 2940.2.k.f 8
105.x even 12 2 1260.2.bm.c 16
140.w even 12 2 1680.2.di.e 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.bb.a 16 5.c odd 4 2
420.2.bb.a 16 35.l odd 12 2
1260.2.bm.c 16 15.e even 4 2
1260.2.bm.c 16 105.x even 12 2
1680.2.di.e 16 20.e even 4 2
1680.2.di.e 16 140.w even 12 2
2100.2.q.l 8 5.b even 2 1
2100.2.q.l 8 35.j even 6 1
2100.2.q.m 8 1.a even 1 1 trivial
2100.2.q.m 8 7.c even 3 1 inner
2940.2.k.f 8 35.l odd 12 2
2940.2.k.g 8 35.k even 12 2
2940.2.bb.i 16 35.f even 4 2
2940.2.bb.i 16 35.k even 12 2

## Hecke kernels

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

 $$T_{11}^{8} + \cdots$$ $$T_{13}^{4} - 2 T_{13}^{3} - 24 T_{13}^{2} + 75$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - T + T^{2} )^{4}$$
$5$ 1
$7$ $$1 - 2 T + 4 T^{2} + 22 T^{3} - 83 T^{4} + 154 T^{5} + 196 T^{6} - 686 T^{7} + 2401 T^{8}$$
$11$ $$1 + 4 T - 22 T^{2} - 68 T^{3} + 396 T^{4} + 526 T^{5} - 6648 T^{6} - 1364 T^{7} + 89791 T^{8} - 15004 T^{9} - 804408 T^{10} + 700106 T^{11} + 5797836 T^{12} - 10951468 T^{13} - 38974342 T^{14} + 77948684 T^{15} + 214358881 T^{16}$$
$13$ $$( 1 - 2 T + 28 T^{2} - 78 T^{3} + 465 T^{4} - 1014 T^{5} + 4732 T^{6} - 4394 T^{7} + 28561 T^{8} )^{2}$$
$17$ $$1 + 2 T - 40 T^{2} - 144 T^{3} + 788 T^{4} + 3412 T^{5} - 5360 T^{6} - 32022 T^{7} + 6583 T^{8} - 544374 T^{9} - 1549040 T^{10} + 16763156 T^{11} + 65814548 T^{12} - 204459408 T^{13} - 965502760 T^{14} + 820677346 T^{15} + 6975757441 T^{16}$$
$19$ $$1 + 4 T - 18 T^{2} + 112 T^{3} + 697 T^{4} - 2652 T^{5} + 9806 T^{6} + 65960 T^{7} - 161964 T^{8} + 1253240 T^{9} + 3539966 T^{10} - 18190068 T^{11} + 90833737 T^{12} + 277323088 T^{13} - 846825858 T^{14} + 3575486956 T^{15} + 16983563041 T^{16}$$
$23$ $$1 - 6 T - 44 T^{2} + 272 T^{3} + 1468 T^{4} - 6616 T^{5} - 39952 T^{6} + 50582 T^{7} + 1119911 T^{8} + 1163386 T^{9} - 21134608 T^{10} - 80496872 T^{11} + 410806588 T^{12} + 1750685296 T^{13} - 6513579116 T^{14} - 20428952682 T^{15} + 78310985281 T^{16}$$
$29$ $$( 1 + 6 T + 50 T^{2} + 300 T^{3} + 2316 T^{4} + 8700 T^{5} + 42050 T^{6} + 146334 T^{7} + 707281 T^{8} )^{2}$$
$31$ $$1 - 22 T^{2} + 400 T^{3} - 59 T^{4} - 10600 T^{5} + 70338 T^{6} + 164200 T^{7} - 2333796 T^{8} + 5090200 T^{9} + 67594818 T^{10} - 315784600 T^{11} - 54487739 T^{12} + 11451660400 T^{13} - 19525080982 T^{14} + 852891037441 T^{16}$$
$37$ $$1 - 4 T - 108 T^{2} + 292 T^{3} + 7187 T^{4} - 10530 T^{5} - 383576 T^{6} + 128186 T^{7} + 16685028 T^{8} + 4742882 T^{9} - 525115544 T^{10} - 533376090 T^{11} + 13469595107 T^{12} + 20248435444 T^{13} - 277098452172 T^{14} - 379727508532 T^{15} + 3512479453921 T^{16}$$
$41$ $$( 1 - 12 T + 206 T^{2} - 1514 T^{3} + 13536 T^{4} - 62074 T^{5} + 346286 T^{6} - 827052 T^{7} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 + 16 T + 160 T^{2} + 1402 T^{3} + 10561 T^{4} + 60286 T^{5} + 295840 T^{6} + 1272112 T^{7} + 3418801 T^{8} )^{2}$$
$47$ $$1 - 2 T - 64 T^{2} - 84 T^{3} + 1982 T^{4} + 7370 T^{5} + 122536 T^{6} - 409218 T^{7} - 7587185 T^{8} - 19233246 T^{9} + 270682024 T^{10} + 765175510 T^{11} + 9671527742 T^{12} - 19264980588 T^{13} - 689869781056 T^{14} - 1013246240926 T^{15} + 23811286661761 T^{16}$$
$53$ $$1 - 20 T + 68 T^{2} + 200 T^{3} + 12154 T^{4} - 129900 T^{5} + 85936 T^{6} - 2744060 T^{7} + 61991603 T^{8} - 145435180 T^{9} + 241394224 T^{10} - 19339122300 T^{11} + 95900906074 T^{12} + 83639098600 T^{13} + 1507176556772 T^{14} - 23494222796740 T^{15} + 62259690411361 T^{16}$$
$59$ $$1 - 14 T + 98 T^{2} - 2364 T^{3} + 23888 T^{4} - 127648 T^{5} + 2057308 T^{6} - 17027622 T^{7} + 74320339 T^{8} - 1004629698 T^{9} + 7161489148 T^{10} - 26216218592 T^{11} + 289459519568 T^{12} - 1690081042836 T^{13} + 4133692296818 T^{14} - 34841120787466 T^{15} + 146830437604321 T^{16}$$
$61$ $$1 + 16 T + 48 T^{2} - 408 T^{3} - 2562 T^{4} - 7524 T^{5} - 100912 T^{6} - 1615000 T^{7} - 17451933 T^{8} - 98515000 T^{9} - 375493552 T^{10} - 1707805044 T^{11} - 35473044642 T^{12} - 344595290808 T^{13} + 2472977969328 T^{14} + 50283885376336 T^{15} + 191707312997281 T^{16}$$
$67$ $$1 - 18 T + 20 T^{2} + 1396 T^{3} - 1157 T^{4} - 130618 T^{5} + 724920 T^{6} + 842608 T^{7} - 21438852 T^{8} + 56454736 T^{9} + 3254165880 T^{10} - 39285061534 T^{11} - 23314846997 T^{12} + 1884774649372 T^{13} + 1809167643380 T^{14} - 109092808895814 T^{15} + 406067677556641 T^{16}$$
$71$ $$( 1 + 14 T + 158 T^{2} + 444 T^{3} + 3552 T^{4} + 31524 T^{5} + 796478 T^{6} + 5010754 T^{7} + 25411681 T^{8} )^{2}$$
$73$ $$1 + 8 T - 144 T^{2} - 1148 T^{3} + 10427 T^{4} + 49662 T^{5} - 1117916 T^{6} - 363682 T^{7} + 112617972 T^{8} - 26548786 T^{9} - 5957374364 T^{10} + 19319362254 T^{11} + 296108458907 T^{12} - 2379886188764 T^{13} - 21792128585616 T^{14} + 88379188152776 T^{15} + 806460091894081 T^{16}$$
$79$ $$1 - 8 T - 18 T^{2} - 632 T^{3} + 6121 T^{4} + 6180 T^{5} + 974510 T^{6} - 7889020 T^{7} - 17559900 T^{8} - 623232580 T^{9} + 6081916910 T^{10} + 3046981020 T^{11} + 238413445801 T^{12} - 1944699644168 T^{13} - 4375574199378 T^{14} - 153631271889272 T^{15} + 1517108809906561 T^{16}$$
$83$ $$( 1 - 10 T + 200 T^{2} - 1808 T^{3} + 24704 T^{4} - 150064 T^{5} + 1377800 T^{6} - 5717870 T^{7} + 47458321 T^{8} )^{2}$$
$89$ $$1 - 8 T + 2 T^{2} - 1724 T^{3} + 19584 T^{4} - 80534 T^{5} + 2467704 T^{6} - 22993064 T^{7} + 80738407 T^{8} - 2046382696 T^{9} + 19546683384 T^{10} - 56773973446 T^{11} + 1228744047744 T^{12} - 9626918490076 T^{13} + 993962581922 T^{14} - 353850679164232 T^{15} + 3936588805702081 T^{16}$$
$97$ $$( 1 - 10 T + 340 T^{2} - 2590 T^{3} + 47986 T^{4} - 251230 T^{5} + 3199060 T^{6} - 9126730 T^{7} + 88529281 T^{8} )^{2}$$