# Properties

 Label 2100.2.s.a Level 2100 Weight 2 Character orbit 2100.s 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.s (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: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 24 x^{12} + 424 x^{8} - 159 x^{4} + 16$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{9} q^{3} -\beta_{14} q^{7} + ( \beta_{7} + \beta_{11} ) q^{9} +O(q^{10})$$ $$q -\beta_{9} q^{3} -\beta_{14} q^{7} + ( \beta_{7} + \beta_{11} ) q^{9} + ( -\beta_{1} - \beta_{2} ) q^{11} + ( -2 \beta_{3} - \beta_{4} + \beta_{9} ) q^{13} + ( -\beta_{8} + 2 \beta_{13} + \beta_{15} ) q^{17} + ( 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} ) q^{19} -\beta_{1} q^{21} + ( 2 \beta_{4} - 2 \beta_{6} ) q^{23} + ( \beta_{8} + \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{27} + ( -2 \beta_{7} + 3 \beta_{10} - \beta_{12} ) q^{29} + ( -2 + 4 \beta_{1} - 4 \beta_{2} ) q^{31} + ( 4 \beta_{3} - \beta_{6} ) q^{33} + ( 2 \beta_{8} - 2 \beta_{15} ) q^{37} + ( -\beta_{7} + 2 \beta_{11} - 2 \beta_{12} ) q^{39} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{41} + ( -4 \beta_{3} + 4 \beta_{4} - 4 \beta_{9} ) q^{43} + ( \beta_{8} + \beta_{15} ) q^{47} + \beta_{11} q^{49} + ( -2 + \beta_{1} - 5 \beta_{2} - \beta_{5} ) q^{51} + ( -2 \beta_{4} + 2 \beta_{6} ) q^{53} + ( -4 \beta_{8} + 2 \beta_{13} - 4 \beta_{14} + 2 \beta_{15} ) q^{57} + ( 4 \beta_{7} - 2 \beta_{10} - 2 \beta_{12} ) q^{59} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{61} + ( \beta_{3} - \beta_{6} ) q^{63} + ( -2 \beta_{8} + 8 \beta_{14} + 2 \beta_{15} ) q^{67} + ( -2 \beta_{7} + 6 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} ) q^{69} + ( -2 \beta_{1} - 4 \beta_{2} + 2 \beta_{5} ) q^{71} -6 \beta_{3} q^{73} + ( \beta_{8} + \beta_{15} ) q^{77} + ( -\beta_{10} - 4 \beta_{11} - \beta_{12} ) q^{79} + ( 5 - 4 \beta_{2} + \beta_{5} ) q^{81} + ( 2 \beta_{4} - 2 \beta_{6} ) q^{83} + ( \beta_{8} - 3 \beta_{13} - 6 \beta_{14} + 3 \beta_{15} ) q^{87} + ( 4 \beta_{10} - 4 \beta_{12} ) q^{89} + ( 2 + \beta_{1} - \beta_{2} ) q^{91} + ( 8 \beta_{3} + 4 \beta_{6} + 2 \beta_{9} ) q^{93} + ( \beta_{8} - 2 \beta_{14} - \beta_{15} ) q^{97} + ( -\beta_{7} + 3 \beta_{10} + 2 \beta_{11} + 5 \beta_{12} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q - 4q^{21} - 4q^{51} - 16q^{61} + 92q^{81} + 40q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 24 x^{12} + 424 x^{8} - 159 x^{4} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-247 \nu^{12} - 5984 \nu^{8} - 106343 \nu^{4} + 16895$$$$)/3753$$ $$\beta_{2}$$ $$=$$ $$($$$$-83 \nu^{12} - 1999 \nu^{8} - 35446 \nu^{4} + 7585$$$$)/1251$$ $$\beta_{3}$$ $$=$$ $$($$$$-317 \nu^{13} - 7690 \nu^{9} - 136192 \nu^{5} + 16831 \nu$$$$)/7506$$ $$\beta_{4}$$ $$=$$ $$($$$$-107 \nu^{13} - 2572 \nu^{9} - 45394 \nu^{5} + 14521 \nu$$$$)/2502$$ $$\beta_{5}$$ $$=$$ $$($$$$43 \nu^{12} + 1044 \nu^{8} + 18449 \nu^{4} - 3114$$$$)/417$$ $$\beta_{6}$$ $$=$$ $$($$$$-319 \nu^{13} - 7703 \nu^{9} - 136187 \nu^{5} + 37703 \nu$$$$)/3753$$ $$\beta_{7}$$ $$=$$ $$($$$$-313 \nu^{14} - 7664 \nu^{10} - 136202 \nu^{6} - 17407 \nu^{2}$$$$)/7506$$ $$\beta_{8}$$ $$=$$ $$($$$$791 \nu^{15} + 19528 \nu^{11} + 348928 \nu^{7} + 113063 \nu^{3}$$$$)/30024$$ $$\beta_{9}$$ $$=$$ $$($$$$-815 \nu^{13} - 19684 \nu^{9} - 348868 \nu^{5} + 69847 \nu$$$$)/7506$$ $$\beta_{10}$$ $$=$$ $$($$$$-269 \nu^{14} - 6544 \nu^{10} - 116296 \nu^{6} + 2959 \nu^{2}$$$$)/5004$$ $$\beta_{11}$$ $$=$$ $$($$$$-161 \nu^{14} - 3896 \nu^{10} - 69028 \nu^{6} + 12335 \nu^{2}$$$$)/1668$$ $$\beta_{12}$$ $$=$$ $$($$$$-1457 \nu^{14} - 35116 \nu^{10} - 621232 \nu^{6} + 164479 \nu^{2}$$$$)/15012$$ $$\beta_{13}$$ $$=$$ $$($$$$1433 \nu^{15} + 34960 \nu^{11} + 621292 \nu^{7} + 10925 \nu^{3}$$$$)/15012$$ $$\beta_{14}$$ $$=$$ $$($$$$3721 \nu^{15} + 89864 \nu^{11} + 1591352 \nu^{7} - 352847 \nu^{3}$$$$)/30024$$ $$\beta_{15}$$ $$=$$ $$($$$$2201 \nu^{15} + 53296 \nu^{11} + 944632 \nu^{7} - 145975 \nu^{3}$$$$)/10008$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{6} - \beta_{4} - \beta_{3}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{12} + 3 \beta_{11} - \beta_{10} - \beta_{7}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$4 \beta_{15} - 5 \beta_{14} - 3 \beta_{13} + \beta_{8}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-7 \beta_{5} + 4 \beta_{2} - 15 \beta_{1} - 9$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-18 \beta_{9} - 7 \beta_{6} + 29 \beta_{4} + 31 \beta_{3}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$18 \beta_{12} - 11 \beta_{11} - 43 \beta_{10} + 39 \beta_{7}$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-96 \beta_{15} + 153 \beta_{14} - 7 \beta_{13} + 107 \beta_{8}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$171 \beta_{5} + 114 \beta_{2} + 153 \beta_{1} - 103$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$438 \beta_{9} - 203 \beta_{6} - 121 \beta_{4} - 595 \beta_{3}$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$520 \beta_{12} - 1065 \beta_{11} + 1421 \beta_{10} - 577 \beta_{7}$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$634 \beta_{15} - 1649 \beta_{14} + 1503 \beta_{13} - 2981 \beta_{8}$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$($$$$-1129 \beta_{5} - 4484 \beta_{2} + 2721 \beta_{1} + 6507$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$-2892 \beta_{9} + 7985 \beta_{6} - 9577 \beta_{4} + 1015 \beta_{3}$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$-20454 \beta_{12} + 30697 \beta_{11} - 16027 \beta_{10} - 2835 \beta_{7}$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$26124 \beta_{15} - 26067 \beta_{14} - 33589 \beta_{13} + 26327 \beta_{8}$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1457.1
 −1.11470 − 1.82181i −0.0465948 + 0.660512i 0.660512 − 0.0465948i 1.82181 + 1.11470i −1.82181 − 1.11470i −0.660512 + 0.0465948i 0.0465948 − 0.660512i 1.11470 + 1.82181i −1.11470 + 1.82181i −0.0465948 − 0.660512i 0.660512 + 0.0465948i 1.82181 − 1.11470i −1.82181 + 1.11470i −0.660512 − 0.0465948i 0.0465948 + 0.660512i 1.11470 − 1.82181i
0 −1.73015 + 0.0811423i 0 0 0 0.707107 + 0.707107i 0 2.98683 0.280776i 0
1457.2 0 −1.64534 0.541157i 0 0 0 −0.707107 0.707107i 0 2.41430 + 1.78078i 0
1457.3 0 −0.541157 1.64534i 0 0 0 0.707107 + 0.707107i 0 −2.41430 + 1.78078i 0
1457.4 0 −0.0811423 + 1.73015i 0 0 0 0.707107 + 0.707107i 0 −2.98683 0.280776i 0
1457.5 0 0.0811423 1.73015i 0 0 0 −0.707107 0.707107i 0 −2.98683 0.280776i 0
1457.6 0 0.541157 + 1.64534i 0 0 0 −0.707107 0.707107i 0 −2.41430 + 1.78078i 0
1457.7 0 1.64534 + 0.541157i 0 0 0 0.707107 + 0.707107i 0 2.41430 + 1.78078i 0
1457.8 0 1.73015 0.0811423i 0 0 0 −0.707107 0.707107i 0 2.98683 0.280776i 0
1793.1 0 −1.73015 0.0811423i 0 0 0 0.707107 0.707107i 0 2.98683 + 0.280776i 0
1793.2 0 −1.64534 + 0.541157i 0 0 0 −0.707107 + 0.707107i 0 2.41430 1.78078i 0
1793.3 0 −0.541157 + 1.64534i 0 0 0 0.707107 0.707107i 0 −2.41430 1.78078i 0
1793.4 0 −0.0811423 1.73015i 0 0 0 0.707107 0.707107i 0 −2.98683 + 0.280776i 0
1793.5 0 0.0811423 + 1.73015i 0 0 0 −0.707107 + 0.707107i 0 −2.98683 + 0.280776i 0
1793.6 0 0.541157 1.64534i 0 0 0 −0.707107 + 0.707107i 0 −2.41430 1.78078i 0
1793.7 0 1.64534 0.541157i 0 0 0 0.707107 0.707107i 0 2.41430 1.78078i 0
1793.8 0 1.73015 + 0.0811423i 0 0 0 −0.707107 + 0.707107i 0 2.98683 + 0.280776i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1793.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e 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.s.a 16
3.b odd 2 1 inner 2100.2.s.a 16
5.b even 2 1 inner 2100.2.s.a 16
5.c odd 4 2 inner 2100.2.s.a 16
15.d odd 2 1 inner 2100.2.s.a 16
15.e even 4 2 inner 2100.2.s.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.2.s.a 16 1.a even 1 1 trivial
2100.2.s.a 16 3.b odd 2 1 inner
2100.2.s.a 16 5.b even 2 1 inner
2100.2.s.a 16 5.c odd 4 2 inner
2100.2.s.a 16 15.d odd 2 1 inner
2100.2.s.a 16 15.e even 4 2 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{4} + 15 T_{11}^{2} + 52$$ acting on $$S_{2}^{\mathrm{new}}(2100, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 23 T^{4} + 256 T^{8} - 1863 T^{12} + 6561 T^{16}$$
$5$ 1
$7$ $$( 1 + T^{4} )^{4}$$
$11$ $$( 1 - 29 T^{2} + 448 T^{4} - 3509 T^{6} + 14641 T^{8} )^{4}$$
$13$ $$( 1 + 17 T^{4} - 44912 T^{8} + 485537 T^{12} + 815730721 T^{16} )^{2}$$
$17$ $$( 1 - 1039 T^{4} + 433824 T^{8} - 86778319 T^{12} + 6975757441 T^{16} )^{2}$$
$19$ $$( 1 - 24 T^{2} + 254 T^{4} - 8664 T^{6} + 130321 T^{8} )^{4}$$
$23$ $$( 1 + 260 T^{4} + 141382 T^{8} + 72758660 T^{12} + 78310985281 T^{16} )^{2}$$
$29$ $$( 1 + T^{2} - 1416 T^{4} + 841 T^{6} + 707281 T^{8} )^{4}$$
$31$ $$( 1 - 6 T^{2} + 961 T^{4} )^{8}$$
$37$ $$( 1 + 932 T^{4} + 3112486 T^{8} + 1746718052 T^{12} + 3512479453921 T^{16} )^{2}$$
$41$ $$( 1 - 104 T^{2} + 5998 T^{4} - 174824 T^{6} + 2825761 T^{8} )^{4}$$
$43$ $$( 1 - 4828 T^{4} + 11812006 T^{8} - 16505971228 T^{12} + 11688200277601 T^{16} )^{2}$$
$47$ $$( 1 + 6137 T^{4} + 19047856 T^{8} + 29946602297 T^{12} + 23811286661761 T^{16} )^{2}$$
$53$ $$( 1 + 740 T^{4} - 5406938 T^{8} + 5838955940 T^{12} + 62259690411361 T^{16} )^{2}$$
$59$ $$( 1 + 6350 T^{4} + 12117361 T^{8} )^{4}$$
$61$ $$( 1 + 2 T + 106 T^{2} + 122 T^{3} + 3721 T^{4} )^{8}$$
$67$ $$( 1 - 2044 T^{4} - 20281946 T^{8} - 41188891324 T^{12} + 406067677556641 T^{16} )^{2}$$
$71$ $$( 1 - 84 T^{2} + 2054 T^{4} - 423444 T^{6} + 25411681 T^{8} )^{4}$$
$73$ $$( 1 - 96 T^{2} + 5329 T^{4} )^{4}( 1 + 96 T^{2} + 5329 T^{4} )^{4}$$
$79$ $$( 1 - 283 T^{2} + 32296 T^{4} - 1766203 T^{6} + 38950081 T^{8} )^{4}$$
$83$ $$( 1 + 8420 T^{4} + 39091942 T^{8} + 399599062820 T^{12} + 2252292232139041 T^{16} )^{2}$$
$89$ $$( 1 + 116 T^{2} + 18118 T^{4} + 918836 T^{6} + 62742241 T^{8} )^{4}$$
$97$ $$( 1 + 32753 T^{4} + 439869408 T^{8} + 2899599540593 T^{12} + 7837433594376961 T^{16} )^{2}$$