# Properties

 Label 2100.2.d.j Level 2100 Weight 2 Character orbit 2100.d Analytic conductor 16.769 Analytic rank 0 Dimension 8 CM discriminant -35 Inner twists 8

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2100.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.7685844245$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.121550625.1 Defining polynomial: $$x^{8} - x^{7} - 4 x^{6} - 9 x^{5} + 23 x^{4} + 18 x^{3} - 16 x^{2} + 8 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{8}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 420) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} + ( -\beta_{2} + \beta_{3} ) q^{7} -\beta_{1} q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{3} + ( -\beta_{2} + \beta_{3} ) q^{7} -\beta_{1} q^{9} + \beta_{4} q^{11} + ( \beta_{2} + \beta_{3} - \beta_{6} ) q^{13} + ( -\beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{6} ) q^{17} + ( -2 - \beta_{1} - \beta_{4} ) q^{21} + ( \beta_{2} - \beta_{5} ) q^{27} + ( \beta_{1} + \beta_{4} + \beta_{7} ) q^{29} + ( 2 \beta_{2} - \beta_{3} - \beta_{6} ) q^{33} + ( -5 - \beta_{1} + 2 \beta_{4} ) q^{39} + ( \beta_{2} + \beta_{3} + 2 \beta_{5} + \beta_{6} ) q^{47} -7 q^{49} + ( -7 + 2 \beta_{1} + \beta_{7} ) q^{51} + ( -\beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} ) q^{63} + ( -\beta_{1} + 2 \beta_{4} - \beta_{7} ) q^{71} + ( -4 \beta_{2} + 4 \beta_{3} ) q^{73} + ( \beta_{2} + 4 \beta_{3} - \beta_{5} + \beta_{6} ) q^{77} + ( -2 - 5 \beta_{1} - \beta_{4} + \beta_{7} ) q^{79} + ( 4 + \beta_{4} + \beta_{7} ) q^{81} + ( 2 \beta_{3} - 2 \beta_{5} ) q^{83} + ( 4 \beta_{2} - 2 \beta_{3} + 3 \beta_{5} + \beta_{6} ) q^{87} + ( 2 - 5 \beta_{1} - \beta_{4} + \beta_{7} ) q^{91} + ( -3 \beta_{2} + 5 \beta_{3} - \beta_{6} ) q^{97} + ( -3 + \beta_{1} + 3 \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 2q^{9} + O(q^{10})$$ $$8q + 2q^{9} - 14q^{21} - 38q^{39} - 56q^{49} - 58q^{51} - 4q^{79} + 34q^{81} + 28q^{91} - 26q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} - 4 x^{6} - 9 x^{5} + 23 x^{4} + 18 x^{3} - 16 x^{2} + 8 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-83 \nu^{7} + 165 \nu^{6} + 110 \nu^{5} + 599 \nu^{4} - 2255 \nu^{3} + 1540 \nu^{2} - 1560 \nu - 2200$$$$)/816$$ $$\beta_{2}$$ $$=$$ $$($$$$-263 \nu^{7} + 309 \nu^{6} + 818 \nu^{5} + 2891 \nu^{4} - 5447 \nu^{3} - 2624 \nu^{2} - 3552 \nu - 4120$$$$)/2448$$ $$\beta_{3}$$ $$=$$ $$($$$$377 \nu^{7} - 747 \nu^{6} - 974 \nu^{5} - 2357 \nu^{4} + 11705 \nu^{3} - 1600 \nu^{2} - 6624 \nu + 4792$$$$)/2448$$ $$\beta_{4}$$ $$=$$ $$($$$$11 \nu^{7} - 2 \nu^{6} - 41 \nu^{5} - 131 \nu^{4} + 152 \nu^{3} + 293 \nu^{2} + 90 \nu + 4$$$$)/68$$ $$\beta_{5}$$ $$=$$ $$($$$$159 \nu^{7} - 253 \nu^{6} - 418 \nu^{5} - 1059 \nu^{4} + 3775 \nu^{3} - 480 \nu^{2} - 2368 \nu + 5096$$$$)/816$$ $$\beta_{6}$$ $$=$$ $$($$$$-121 \nu^{7} + 294 \nu^{6} + 43 \nu^{5} + 625 \nu^{4} - 3712 \nu^{3} + 4121 \nu^{2} + 1050 \nu - 3920$$$$)/612$$ $$\beta_{7}$$ $$=$$ $$($$$$-673 \nu^{7} + 1215 \nu^{6} + 1762 \nu^{5} + 4621 \nu^{4} - 19597 \nu^{3} + 3860 \nu^{2} + 8616 \nu - 10760$$$$)/816$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{6} + \beta_{2} - 3 \beta_{1}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{7} + \beta_{4} + 9 \beta_{3} - \beta_{1} + 6$$$$)/6$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{7} - 5 \beta_{5} + 2 \beta_{4} + \beta_{3} + 4 \beta_{2} - \beta_{1} + 20$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{7} + 9 \beta_{6} + 9 \beta_{5} + 4 \beta_{4} + 9 \beta_{3} + 27 \beta_{2} - 25 \beta_{1} - 12$$$$)/6$$ $$\nu^{5}$$ $$=$$ $$($$$$39 \beta_{7} - 44 \beta_{6} - 33 \beta_{5} + 48 \beta_{4} + 165 \beta_{3} + 22 \beta_{2} + 9 \beta_{1} + 174$$$$)/12$$ $$\nu^{6}$$ $$=$$ $$($$$$-10 \beta_{7} - 18 \beta_{5} + 20 \beta_{4} - 27 \beta_{3} + 45 \beta_{2} - 10 \beta_{1} + 81$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$117 \beta_{7} + 34 \beta_{6} + 279 \beta_{5} + 234 \beta_{4} + 279 \beta_{3} + 592 \beta_{2} - 429 \beta_{1} - 702$$$$)/12$$

## Character values

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

 $$n$$ $$701$$ $$1051$$ $$1177$$ $$1501$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1301.1
 0.553538 + 0.676408i 0.553538 − 0.676408i −1.44918 + 1.77086i −1.44918 − 1.77086i −0.862555 − 0.141174i −0.862555 + 0.141174i 2.25820 − 0.369600i 2.25820 + 0.369600i
0 −1.70466 0.306808i 0 0 0 2.64575i 0 2.81174 + 1.04601i 0
1301.2 0 −1.70466 + 0.306808i 0 0 0 2.64575i 0 2.81174 1.04601i 0
1301.3 0 −0.586627 1.62968i 0 0 0 2.64575i 0 −2.31174 + 1.91203i 0
1301.4 0 −0.586627 + 1.62968i 0 0 0 2.64575i 0 −2.31174 1.91203i 0
1301.5 0 0.586627 1.62968i 0 0 0 2.64575i 0 −2.31174 1.91203i 0
1301.6 0 0.586627 + 1.62968i 0 0 0 2.64575i 0 −2.31174 + 1.91203i 0
1301.7 0 1.70466 0.306808i 0 0 0 2.64575i 0 2.81174 1.04601i 0
1301.8 0 1.70466 + 0.306808i 0 0 0 2.64575i 0 2.81174 + 1.04601i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1301.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.c odd 2 1 CM by $$\Q(\sqrt{-35})$$
3.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
15.d odd 2 1 inner
21.c even 2 1 inner
105.g even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.d.j 8
3.b odd 2 1 inner 2100.2.d.j 8
5.b even 2 1 inner 2100.2.d.j 8
5.c odd 4 2 420.2.f.a 8
7.b odd 2 1 inner 2100.2.d.j 8
15.d odd 2 1 inner 2100.2.d.j 8
15.e even 4 2 420.2.f.a 8
20.e even 4 2 1680.2.k.d 8
21.c even 2 1 inner 2100.2.d.j 8
35.c odd 2 1 CM 2100.2.d.j 8
35.f even 4 2 420.2.f.a 8
60.l odd 4 2 1680.2.k.d 8
105.g even 2 1 inner 2100.2.d.j 8
105.k odd 4 2 420.2.f.a 8
140.j odd 4 2 1680.2.k.d 8
420.w even 4 2 1680.2.k.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.f.a 8 5.c odd 4 2
420.2.f.a 8 15.e even 4 2
420.2.f.a 8 35.f even 4 2
420.2.f.a 8 105.k odd 4 2
1680.2.k.d 8 20.e even 4 2
1680.2.k.d 8 60.l odd 4 2
1680.2.k.d 8 140.j odd 4 2
1680.2.k.d 8 420.w even 4 2
2100.2.d.j 8 1.a even 1 1 trivial
2100.2.d.j 8 3.b odd 2 1 inner
2100.2.d.j 8 5.b even 2 1 inner
2100.2.d.j 8 7.b odd 2 1 inner
2100.2.d.j 8 15.d odd 2 1 inner
2100.2.d.j 8 21.c even 2 1 inner
2100.2.d.j 8 35.c odd 2 1 CM
2100.2.d.j 8 105.g even 2 1 inner

## Hecke kernels

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

 $$T_{11}^{4} + 31 T_{11}^{2} + 4$$ $$T_{13}^{4} + 71 T_{13}^{2} + 1024$$ $$T_{17}^{4} - 97 T_{17}^{2} + 2116$$ $$T_{37}$$ $$T_{41}$$ $$T_{43}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - T^{2} - 8 T^{4} - 9 T^{6} + 81 T^{8}$$
$5$ 1
$7$ $$( 1 + 7 T^{2} )^{4}$$
$11$ $$( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} )^{2}( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} )^{2}$$
$13$ $$( 1 + 19 T^{2} + 192 T^{4} + 3211 T^{6} + 28561 T^{8} )^{2}$$
$17$ $$( 1 - 29 T^{2} + 552 T^{4} - 8381 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$( 1 - 19 T^{2} )^{8}$$
$23$ $$( 1 - 23 T^{2} )^{8}$$
$29$ $$( 1 - 9 T + 52 T^{2} - 261 T^{3} + 841 T^{4} )^{2}( 1 + 9 T + 52 T^{2} + 261 T^{3} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 31 T^{2} )^{8}$$
$37$ $$( 1 + 37 T^{2} )^{8}$$
$41$ $$( 1 + 41 T^{2} )^{8}$$
$43$ $$( 1 + 43 T^{2} )^{8}$$
$47$ $$( 1 + 31 T^{2} - 1248 T^{4} + 68479 T^{6} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 - 53 T^{2} )^{8}$$
$59$ $$( 1 + 59 T^{2} )^{8}$$
$61$ $$( 1 - 61 T^{2} )^{8}$$
$67$ $$( 1 + 67 T^{2} )^{8}$$
$71$ $$( 1 - 12 T + 71 T^{2} )^{4}( 1 + 12 T + 71 T^{2} )^{4}$$
$73$ $$( 1 - 34 T^{2} + 5329 T^{4} )^{4}$$
$79$ $$( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} )^{4}$$
$83$ $$( 1 + 86 T^{2} + 6889 T^{4} )^{4}$$
$89$ $$( 1 + 89 T^{2} )^{8}$$
$97$ $$( 1 - 149 T^{2} + 12792 T^{4} - 1401941 T^{6} + 88529281 T^{8} )^{2}$$
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