# Properties

 Label 2100.2.x.c Level 2100 Weight 2 Character orbit 2100.x Analytic conductor 16.769 Analytic rank 0 Dimension 16 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2100.x (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.7685844245$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(i)$$ Coefficient field: 16.0.478584585616890104119296.1 Defining polynomial: $$x^{16} - 31 x^{12} + 336 x^{8} - 19375 x^{4} + 390625$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} + ( \beta_{12} - \beta_{13} - \beta_{14} ) q^{7} -\beta_{11} q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} + ( \beta_{12} - \beta_{13} - \beta_{14} ) q^{7} -\beta_{11} q^{9} + ( -1 - 2 \beta_{3} - \beta_{4} + \beta_{6} ) q^{11} + ( -2 \beta_{1} + \beta_{8} + \beta_{9} ) q^{13} -2 \beta_{12} q^{17} + ( 3 \beta_{5} - \beta_{7} ) q^{19} + ( 1 + \beta_{3} - \beta_{6} ) q^{21} + ( \beta_{8} + 2 \beta_{9} ) q^{23} -\beta_{12} q^{27} + ( \beta_{5} - \beta_{7} + 2 \beta_{10} + 5 \beta_{11} ) q^{29} + ( -\beta_{4} + \beta_{6} ) q^{31} + ( -2 \beta_{1} + \beta_{2} + \beta_{8} + \beta_{9} ) q^{33} + ( 4 \beta_{13} - 3 \beta_{15} ) q^{37} + ( \beta_{5} - \beta_{7} + 2 \beta_{10} ) q^{39} + ( 4 \beta_{4} - 4 \beta_{6} ) q^{41} + ( 2 \beta_{8} + 3 \beta_{9} ) q^{43} -2 \beta_{12} q^{47} + ( 2 \beta_{5} + 3 \beta_{10} + 2 \beta_{11} ) q^{49} -2 q^{51} + ( 4 \beta_{8} + 4 \beta_{9} ) q^{53} + ( 3 \beta_{13} - \beta_{15} ) q^{57} + ( -2 \beta_{5} + 4 \beta_{7} ) q^{59} + ( \beta_{4} - \beta_{6} ) q^{61} + ( \beta_{1} - \beta_{2} - \beta_{8} ) q^{63} + ( -\beta_{13} + 2 \beta_{15} ) q^{67} + ( 2 \beta_{5} - \beta_{7} ) q^{69} + ( 5 - 2 \beta_{3} - \beta_{4} + \beta_{6} ) q^{71} -10 \beta_{2} q^{73} + ( -8 \beta_{12} + 2 \beta_{14} - 3 \beta_{15} ) q^{77} + ( 3 \beta_{5} - 3 \beta_{7} + 6 \beta_{10} + \beta_{11} ) q^{79} - q^{81} + ( -4 \beta_{1} - 8 \beta_{2} + 2 \beta_{8} + 2 \beta_{9} ) q^{83} + ( 5 \beta_{12} - \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{87} + 6 \beta_{7} q^{89} + ( -7 - \beta_{3} - 3 \beta_{4} - \beta_{6} ) q^{91} + ( -\beta_{8} - \beta_{9} ) q^{93} + ( -6 \beta_{12} - \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{97} + ( \beta_{5} - \beta_{7} + 2 \beta_{10} + \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q - 8q^{11} + 12q^{21} - 32q^{51} + 88q^{71} - 16q^{81} - 108q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 31 x^{12} + 336 x^{8} - 19375 x^{4} + 390625$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-71 \nu^{13} - 924 \nu^{9} - 23856 \nu^{5} + 1375625 \nu$$$$)/2362500$$ $$\beta_{3}$$ $$=$$ $$($$$$71 \nu^{12} + 924 \nu^{8} + 23856 \nu^{4} - 1375625$$$$)/472500$$ $$\beta_{4}$$ $$=$$ $$($$$$-173 \nu^{12} - 1512 \nu^{8} + 46872 \nu^{4} + 2196875$$$$)/945000$$ $$\beta_{5}$$ $$=$$ $$($$$$-81 \nu^{14} + 3136 \nu^{10} - 62216 \nu^{6} + 3529375 \nu^{2}$$$$)/7875000$$ $$\beta_{6}$$ $$=$$ $$($$$$31 \nu^{12} - 336 \nu^{8} + 10416 \nu^{4} - 495625$$$$)/105000$$ $$\beta_{7}$$ $$=$$ $$($$$$99 \nu^{14} + 56 \nu^{10} - 141736 \nu^{6} + 6875 \nu^{2}$$$$)/7875000$$ $$\beta_{8}$$ $$=$$ $$($$$$-11 \nu^{13} + 216 \nu^{9} - 6696 \nu^{5} + 246125 \nu$$$$)/135000$$ $$\beta_{9}$$ $$=$$ $$($$$$-23 \nu^{13} + 88 \nu^{9} + 2272 \nu^{5} + 335625 \nu$$$$)/225000$$ $$\beta_{10}$$ $$=$$ $$($$$$-71 \nu^{14} - 924 \nu^{10} - 23856 \nu^{6} + 1375625 \nu^{2}$$$$)/2362500$$ $$\beta_{11}$$ $$=$$ $$($$$$-31 \nu^{14} + 336 \nu^{10} - 6666 \nu^{6} + 390625 \nu^{2}$$$$)/843750$$ $$\beta_{12}$$ $$=$$ $$($$$$-\nu^{15} + 19459 \nu^{3}$$$$)/94500$$ $$\beta_{13}$$ $$=$$ $$($$$$61 \nu^{15} - 16 \nu^{11} + 40496 \nu^{7} - 1176875 \nu^{3}$$$$)/5625000$$ $$\beta_{14}$$ $$=$$ $$($$$$\nu^{15} - 31 \nu^{11} + 336 \nu^{7} - 19375 \nu^{3}$$$$)/78125$$ $$\beta_{15}$$ $$=$$ $$($$$$-221 \nu^{15} - 24 \nu^{11} + 60744 \nu^{7} + 96875 \nu^{3}$$$$)/16875000$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-2 \beta_{11} + \beta_{10} - \beta_{7} + 3 \beta_{5}$$ $$\nu^{3}$$ $$=$$ $$-4 \beta_{15} + 2 \beta_{13} + 7 \beta_{12}$$ $$\nu^{4}$$ $$=$$ $$\beta_{6} + 9 \beta_{4} + 9 \beta_{3} + 10$$ $$\nu^{5}$$ $$=$$ $$17 \beta_{9} - 11 \beta_{8} - 28 \beta_{2} + 11 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-27 \beta_{11} - 56 \beta_{7} + 28 \beta_{5}$$ $$\nu^{7}$$ $$=$$ $$\beta_{15} - \beta_{14} + 139 \beta_{13} + 140 \beta_{12}$$ $$\nu^{8}$$ $$=$$ $$-142 \beta_{6} + 279 \beta_{3} + 142$$ $$\nu^{9}$$ $$=$$ $$142 \beta_{9} + 284 \beta_{8} - 1253 \beta_{2}$$ $$\nu^{10}$$ $$=$$ $$710 \beta_{11} - 1111 \beta_{10} + 710 \beta_{5}$$ $$\nu^{11}$$ $$=$$ $$-2531 \beta_{14} + 1512 \beta_{13} - 1512 \beta_{12}$$ $$\nu^{12}$$ $$=$$ $$1512 \beta_{6} - 3024 \beta_{4} + 14167$$ $$\nu^{13}$$ $$=$$ $$-7560 \beta_{9} - 7560 \beta_{2} + 15679 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$-38918 \beta_{11} + 559 \beta_{10} - 559 \beta_{7} + 39477 \beta_{5}$$ $$\nu^{15}$$ $$=$$ $$-77836 \beta_{15} + 38918 \beta_{13} + 41713 \beta_{12}$$

## Character values

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

 $$n$$ $$701$$ $$1051$$ $$1177$$ $$1501$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{11}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1357.1
 −0.0811201 − 2.23460i −2.23460 − 0.0811201i 1.04705 + 1.97578i 1.97578 + 1.04705i −1.97578 − 1.04705i −1.04705 − 1.97578i 2.23460 + 0.0811201i 0.0811201 + 2.23460i −0.0811201 + 2.23460i −2.23460 + 0.0811201i 1.04705 − 1.97578i 1.97578 − 1.04705i −1.97578 + 1.04705i −1.04705 + 1.97578i 2.23460 − 0.0811201i 0.0811201 − 2.23460i
0 −0.707107 0.707107i 0 0 0 −2.01297 + 1.71696i 0 1.00000i 0
1357.2 0 −0.707107 0.707107i 0 0 0 −1.71696 + 2.01297i 0 1.00000i 0
1357.3 0 −0.707107 0.707107i 0 0 0 −0.884806 2.49342i 0 1.00000i 0
1357.4 0 −0.707107 0.707107i 0 0 0 2.49342 + 0.884806i 0 1.00000i 0
1357.5 0 0.707107 + 0.707107i 0 0 0 −2.49342 0.884806i 0 1.00000i 0
1357.6 0 0.707107 + 0.707107i 0 0 0 0.884806 + 2.49342i 0 1.00000i 0
1357.7 0 0.707107 + 0.707107i 0 0 0 1.71696 2.01297i 0 1.00000i 0
1357.8 0 0.707107 + 0.707107i 0 0 0 2.01297 1.71696i 0 1.00000i 0
1693.1 0 −0.707107 + 0.707107i 0 0 0 −2.01297 1.71696i 0 1.00000i 0
1693.2 0 −0.707107 + 0.707107i 0 0 0 −1.71696 2.01297i 0 1.00000i 0
1693.3 0 −0.707107 + 0.707107i 0 0 0 −0.884806 + 2.49342i 0 1.00000i 0
1693.4 0 −0.707107 + 0.707107i 0 0 0 2.49342 0.884806i 0 1.00000i 0
1693.5 0 0.707107 0.707107i 0 0 0 −2.49342 + 0.884806i 0 1.00000i 0
1693.6 0 0.707107 0.707107i 0 0 0 0.884806 2.49342i 0 1.00000i 0
1693.7 0 0.707107 0.707107i 0 0 0 1.71696 + 2.01297i 0 1.00000i 0
1693.8 0 0.707107 0.707107i 0 0 0 2.01297 + 1.71696i 0 1.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1693.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
7.b odd 2 1 inner
35.c odd 2 1 inner
35.f even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.x.c 16
5.b even 2 1 inner 2100.2.x.c 16
5.c odd 4 2 inner 2100.2.x.c 16
7.b odd 2 1 inner 2100.2.x.c 16
35.c odd 2 1 inner 2100.2.x.c 16
35.f even 4 2 inner 2100.2.x.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.2.x.c 16 1.a even 1 1 trivial
2100.2.x.c 16 5.b even 2 1 inner
2100.2.x.c 16 5.c odd 4 2 inner
2100.2.x.c 16 7.b odd 2 1 inner
2100.2.x.c 16 35.c odd 2 1 inner
2100.2.x.c 16 35.f even 4 2 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{2} + T_{11} - 14$$ acting on $$S_{2}^{\mathrm{new}}(2100, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + T^{4} )^{4}$$
$5$ 1
$7$ $$1 + 73 T^{4} + 2928 T^{8} + 175273 T^{12} + 5764801 T^{16}$$
$11$ $$( 1 + T + 8 T^{2} + 11 T^{3} + 121 T^{4} )^{8}$$
$13$ $$( 1 - 383 T^{4} + 86256 T^{8} - 10938863 T^{12} + 815730721 T^{16} )^{2}$$
$17$ $$( 1 - 16 T^{2} + 289 T^{4} )^{4}( 1 + 16 T^{2} + 289 T^{4} )^{4}$$
$19$ $$( 1 + 29 T^{2} + 576 T^{4} + 10469 T^{6} + 130321 T^{8} )^{4}$$
$23$ $$( 1 + 521 T^{4} + 16944 T^{8} + 145797161 T^{12} + 78310985281 T^{16} )^{2}$$
$29$ $$( 1 - 47 T^{2} + 1080 T^{4} - 39527 T^{6} + 707281 T^{8} )^{4}$$
$31$ $$( 1 - 113 T^{2} + 5100 T^{4} - 108593 T^{6} + 923521 T^{8} )^{4}$$
$37$ $$( 1 + 1249 T^{4} + 2134416 T^{8} + 2340827089 T^{12} + 3512479453921 T^{16} )^{2}$$
$41$ $$( 1 + 12 T^{2} - 250 T^{4} + 20172 T^{6} + 2825761 T^{8} )^{4}$$
$43$ $$( 1 + 478 T^{4} - 4140477 T^{8} + 1634186878 T^{12} + 11688200277601 T^{16} )^{2}$$
$47$ $$( 1 + 3682 T^{4} + 4879681 T^{8} )^{4}$$
$53$ $$( 1 - 3292 T^{4} + 13762470 T^{8} - 25975463452 T^{12} + 62259690411361 T^{16} )^{2}$$
$59$ $$( 1 + 42 T^{2} + 3481 T^{4} )^{8}$$
$61$ $$( 1 - 233 T^{2} + 21000 T^{4} - 866993 T^{6} + 13845841 T^{8} )^{4}$$
$67$ $$( 1 + 4247 T^{4} + 20151121 T^{8} )^{4}$$
$71$ $$( 1 - 11 T + 158 T^{2} - 781 T^{3} + 5041 T^{4} )^{8}$$
$73$ $$( 1 - 8542 T^{4} + 28398241 T^{8} )^{4}$$
$79$ $$( 1 - 59 T^{2} + 13224 T^{4} - 368219 T^{6} + 38950081 T^{8} )^{4}$$
$83$ $$( 1 + 10948 T^{4} + 66965670 T^{8} + 519573698308 T^{12} + 2252292232139041 T^{16} )^{2}$$
$89$ $$( 1 - 40 T^{2} - 2226 T^{4} - 316840 T^{6} + 62742241 T^{8} )^{4}$$
$97$ $$( 1 + 4993 T^{4} + 1167168 T^{8} + 442026700033 T^{12} + 7837433594376961 T^{16} )^{2}$$