# Properties

 Label 8100.2.d.l Level $8100$ Weight $2$ Character orbit 8100.d Analytic conductor $64.679$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$8100 = 2^{2} \cdot 3^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8100.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$64.6788256372$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 1620) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{7} +O(q^{10})$$ $$q + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{7} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{11} + ( 2 - 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{13} + ( -1 + 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{17} + ( 1 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{19} + ( -2 + 4 \zeta_{12}^{2} ) q^{23} + ( -6 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{29} + ( -1 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{31} + ( 3 - 6 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{37} + ( 6 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{41} + ( -1 + 2 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{43} + ( 1 - 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{47} + ( 3 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{49} + ( -1 + 2 \zeta_{12}^{2} + 9 \zeta_{12}^{3} ) q^{53} + ( -6 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{59} -4 q^{61} + ( -6 + 12 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{67} + ( 6 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{71} + ( -3 + 6 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{73} + ( -1 + 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{77} + ( 4 - 12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{79} + ( 7 - 14 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{83} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{89} + ( -8 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{91} + ( -1 + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + O(q^{10})$$ $$4 q + 4 q^{19} - 24 q^{29} - 4 q^{31} + 24 q^{41} + 12 q^{49} - 24 q^{59} - 16 q^{61} + 24 q^{71} + 16 q^{79} - 32 q^{91} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/8100\mathbb{Z}\right)^\times$$.

 $$n$$ $$4051$$ $$6401$$ $$7777$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
649.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
0 0 0 0 0 2.73205i 0 0 0
649.2 0 0 0 0 0 0.732051i 0 0 0
649.3 0 0 0 0 0 0.732051i 0 0 0
649.4 0 0 0 0 0 2.73205i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8100.2.d.l 4
3.b odd 2 1 8100.2.d.m 4
5.b even 2 1 inner 8100.2.d.l 4
5.c odd 4 1 1620.2.a.h yes 2
5.c odd 4 1 8100.2.a.s 2
15.d odd 2 1 8100.2.d.m 4
15.e even 4 1 1620.2.a.g 2
15.e even 4 1 8100.2.a.t 2
20.e even 4 1 6480.2.a.bp 2
45.k odd 12 2 1620.2.i.m 4
45.l even 12 2 1620.2.i.n 4
60.l odd 4 1 6480.2.a.bh 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.2.a.g 2 15.e even 4 1
1620.2.a.h yes 2 5.c odd 4 1
1620.2.i.m 4 45.k odd 12 2
1620.2.i.n 4 45.l even 12 2
6480.2.a.bh 2 60.l odd 4 1
6480.2.a.bp 2 20.e even 4 1
8100.2.a.s 2 5.c odd 4 1
8100.2.a.t 2 15.e even 4 1
8100.2.d.l 4 1.a even 1 1 trivial
8100.2.d.l 4 5.b even 2 1 inner
8100.2.d.m 4 3.b odd 2 1
8100.2.d.m 4 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(8100, [\chi])$$:

 $$T_{7}^{4} + 8 T_{7}^{2} + 4$$ $$T_{11}^{2} - 3$$ $$T_{29}^{2} + 12 T_{29} + 33$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$4 + 8 T^{2} + T^{4}$$
$11$ $$( -3 + T^{2} )^{2}$$
$13$ $$64 + 32 T^{2} + T^{4}$$
$17$ $$36 + 24 T^{2} + T^{4}$$
$19$ $$( -11 - 2 T + T^{2} )^{2}$$
$23$ $$( 12 + T^{2} )^{2}$$
$29$ $$( 33 + 12 T + T^{2} )^{2}$$
$31$ $$( -47 + 2 T + T^{2} )^{2}$$
$37$ $$676 + 56 T^{2} + T^{4}$$
$41$ $$( 9 - 12 T + T^{2} )^{2}$$
$43$ $$484 + 56 T^{2} + T^{4}$$
$47$ $$36 + 24 T^{2} + T^{4}$$
$53$ $$6084 + 168 T^{2} + T^{4}$$
$59$ $$( 33 + 12 T + T^{2} )^{2}$$
$61$ $$( 4 + T )^{4}$$
$67$ $$8464 + 248 T^{2} + T^{4}$$
$71$ $$( 9 - 12 T + T^{2} )^{2}$$
$73$ $$4 + 104 T^{2} + T^{4}$$
$79$ $$( -92 - 8 T + T^{2} )^{2}$$
$83$ $$19044 + 312 T^{2} + T^{4}$$
$89$ $$( -27 + T^{2} )^{2}$$
$97$ $$4 + 8 T^{2} + T^{4}$$
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