# Properties

 Label 2100.2.bc.h Level 2100 Weight 2 Character orbit 2100.bc Analytic conductor 16.769 Analytic rank 0 Dimension 8 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2100.bc (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.7685844245$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.49787136.1 Defining polynomial: $$x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{6} + 15 \nu^{4} + 5 \nu^{2} + 12$$$$)/20$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} - 5 \nu^{5} + 5 \nu^{3} + 12 \nu$$$$)/40$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{6} - 5 \nu^{4} - 15 \nu^{2} - 36$$$$)/20$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} - 7 \nu$$$$)/10$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} + 13 \nu$$$$)/10$$ $$\beta_{6}$$ $$=$$ $$($$$$-9 \nu^{6} - 15 \nu^{4} - 5 \nu^{2} - 48$$$$)/20$$ $$\beta_{7}$$ $$=$$ $$($$$$11 \nu^{7} + 25 \nu^{5} + 55 \nu^{3} + 132 \nu$$$$)/40$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} - \beta_{4}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} - 3 \beta_{3} - 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{7} - \beta_{5} + 5 \beta_{4} + 5 \beta_{2}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{3} + 3 \beta_{1}$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$\beta_{7} - 11 \beta_{2}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-5 \beta_{6} - 5 \beta_{1} - 9$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-7 \beta_{5} - 13 \beta_{4}$$$$)/2$$

## Character values

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

 $$n$$ $$701$$ $$1051$$ $$1177$$ $$1501$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$\beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
949.1
 1.09445 + 0.895644i −0.228425 − 1.39564i −1.09445 − 0.895644i 0.228425 + 1.39564i 1.09445 − 0.895644i −0.228425 + 1.39564i −1.09445 + 0.895644i 0.228425 − 1.39564i
0 −0.866025 0.500000i 0 0 0 −2.29129 1.32288i 0 0.500000 + 0.866025i 0
949.2 0 −0.866025 0.500000i 0 0 0 2.29129 + 1.32288i 0 0.500000 + 0.866025i 0
949.3 0 0.866025 + 0.500000i 0 0 0 −2.29129 1.32288i 0 0.500000 + 0.866025i 0
949.4 0 0.866025 + 0.500000i 0 0 0 2.29129 + 1.32288i 0 0.500000 + 0.866025i 0
1549.1 0 −0.866025 + 0.500000i 0 0 0 −2.29129 + 1.32288i 0 0.500000 0.866025i 0
1549.2 0 −0.866025 + 0.500000i 0 0 0 2.29129 1.32288i 0 0.500000 0.866025i 0
1549.3 0 0.866025 0.500000i 0 0 0 −2.29129 + 1.32288i 0 0.500000 0.866025i 0
1549.4 0 0.866025 0.500000i 0 0 0 2.29129 1.32288i 0 0.500000 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1549.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.bc.h 8
5.b even 2 1 inner 2100.2.bc.h 8
5.c odd 4 1 2100.2.q.g 4
5.c odd 4 1 2100.2.q.i yes 4
7.c even 3 1 inner 2100.2.bc.h 8
35.j even 6 1 inner 2100.2.bc.h 8
35.l odd 12 1 2100.2.q.g 4
35.l odd 12 1 2100.2.q.i yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.2.q.g 4 5.c odd 4 1
2100.2.q.g 4 35.l odd 12 1
2100.2.q.i yes 4 5.c odd 4 1
2100.2.q.i yes 4 35.l odd 12 1
2100.2.bc.h 8 1.a even 1 1 trivial
2100.2.bc.h 8 5.b even 2 1 inner
2100.2.bc.h 8 7.c even 3 1 inner
2100.2.bc.h 8 35.j even 6 1 inner

## Hecke kernels

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

 $$T_{11}^{4} - 6 T_{11}^{3} + 34 T_{11}^{2} - 12 T_{11} + 4$$ $$T_{13}^{2} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - T^{2} + T^{4} )^{2}$$
$5$ 1
$7$ $$( 1 - 7 T^{2} + 49 T^{4} )^{2}$$
$11$ $$( 1 - 6 T + 12 T^{2} - 12 T^{3} + 59 T^{4} - 132 T^{5} + 1452 T^{6} - 7986 T^{7} + 14641 T^{8} )^{2}$$
$13$ $$( 1 - 25 T^{2} + 169 T^{4} )^{4}$$
$17$ $$1 + 52 T^{2} + 1478 T^{4} + 33696 T^{6} + 638099 T^{8} + 9738144 T^{10} + 123444038 T^{12} + 1255153588 T^{14} + 6975757441 T^{16}$$
$19$ $$( 1 + 4 T - 19 T^{2} - 12 T^{3} + 560 T^{4} - 228 T^{5} - 6859 T^{6} + 27436 T^{7} + 130321 T^{8} )^{2}$$
$23$ $$( 1 + 42 T^{2} + 1235 T^{4} + 22218 T^{6} + 279841 T^{8} )^{2}$$
$29$ $$( 1 + 4 T + 34 T^{2} + 116 T^{3} + 841 T^{4} )^{4}$$
$31$ $$( 1 - 7 T + 31 T^{2} )^{4}( 1 + 11 T + 31 T^{2} )^{4}$$
$37$ $$1 + 74 T^{2} + 2377 T^{4} + 26714 T^{6} + 232996 T^{8} + 36571466 T^{10} + 4454880697 T^{12} + 189863754266 T^{14} + 3512479453921 T^{16}$$
$41$ $$( 1 + 6 T + 84 T^{2} + 246 T^{3} + 1681 T^{4} )^{4}$$
$43$ $$( 1 - 108 T^{2} + 6166 T^{4} - 199692 T^{6} + 3418801 T^{8} )^{2}$$
$47$ $$1 + 12 T^{2} + 1990 T^{4} - 75168 T^{6} - 1790061 T^{8} - 166046112 T^{10} + 9710565190 T^{12} + 129350583948 T^{14} + 23811286661761 T^{16}$$
$53$ $$1 + 148 T^{2} + 11510 T^{4} + 706848 T^{6} + 38616419 T^{8} + 1985536032 T^{10} + 90819436310 T^{12} + 3280325447092 T^{14} + 62259690411361 T^{16}$$
$59$ $$( 1 - 14 T + 92 T^{2} + 196 T^{3} - 4229 T^{4} + 11564 T^{5} + 320252 T^{6} - 2875306 T^{7} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 - 2 T - 7 T^{2} + 222 T^{3} - 3844 T^{4} + 13542 T^{5} - 26047 T^{6} - 453962 T^{7} + 13845841 T^{8} )^{2}$$
$67$ $$( 1 + 71 T^{2} + 552 T^{4} + 318719 T^{6} + 20151121 T^{8} )^{2}$$
$71$ $$( 1 + 4 T + 118 T^{2} + 284 T^{3} + 5041 T^{4} )^{4}$$
$73$ $$1 + 234 T^{2} + 30521 T^{4} + 3177018 T^{6} + 267142260 T^{8} + 16930328922 T^{10} + 866742713561 T^{12} + 35412208951626 T^{14} + 806460091894081 T^{16}$$
$79$ $$( 1 + 8 T - 103 T^{2} + 72 T^{3} + 16592 T^{4} + 5688 T^{5} - 642823 T^{6} + 3944312 T^{7} + 38950081 T^{8} )^{2}$$
$83$ $$( 1 - 316 T^{2} + 38714 T^{4} - 2176924 T^{6} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 + 2 T - 112 T^{2} - 124 T^{3} + 5179 T^{4} - 11036 T^{5} - 887152 T^{6} + 1409938 T^{7} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 - 170 T^{2} + 16971 T^{4} - 1599530 T^{6} + 88529281 T^{8} )^{2}$$