# Properties

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

# Related objects

## 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 324) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 i q^{7} +O(q^{10})$$ $$q + 2 i q^{7} + 6 q^{11} -5 i q^{13} + 3 i q^{17} -2 q^{19} + 6 i q^{23} + 3 q^{29} -4 q^{31} + 5 i q^{37} + 6 q^{41} + 10 i q^{43} + 3 q^{49} -6 i q^{53} -12 q^{59} + 5 q^{61} + 2 i q^{67} -6 q^{71} + i q^{73} + 12 i q^{77} + 10 q^{79} -3 q^{89} + 10 q^{91} -10 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + O(q^{10})$$ $$2 q + 12 q^{11} - 4 q^{19} + 6 q^{29} - 8 q^{31} + 12 q^{41} + 6 q^{49} - 24 q^{59} + 10 q^{61} - 12 q^{71} + 20 q^{79} - 6 q^{89} + 20 q^{91} + O(q^{100})$$

## 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
 − 1.00000i 1.00000i
0 0 0 0 0 2.00000i 0 0 0
649.2 0 0 0 0 0 2.00000i 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.j 2
3.b odd 2 1 8100.2.d.a 2
5.b even 2 1 inner 8100.2.d.j 2
5.c odd 4 1 324.2.a.b 1
5.c odd 4 1 8100.2.a.f 1
15.d odd 2 1 8100.2.d.a 2
15.e even 4 1 324.2.a.d yes 1
15.e even 4 1 8100.2.a.a 1
20.e even 4 1 1296.2.a.a 1
40.i odd 4 1 5184.2.a.bc 1
40.k even 4 1 5184.2.a.z 1
45.k odd 12 2 324.2.e.d 2
45.l even 12 2 324.2.e.a 2
60.l odd 4 1 1296.2.a.j 1
120.q odd 4 1 5184.2.a.d 1
120.w even 4 1 5184.2.a.g 1
180.v odd 12 2 1296.2.i.d 2
180.x even 12 2 1296.2.i.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
324.2.a.b 1 5.c odd 4 1
324.2.a.d yes 1 15.e even 4 1
324.2.e.a 2 45.l even 12 2
324.2.e.d 2 45.k odd 12 2
1296.2.a.a 1 20.e even 4 1
1296.2.a.j 1 60.l odd 4 1
1296.2.i.d 2 180.v odd 12 2
1296.2.i.p 2 180.x even 12 2
5184.2.a.d 1 120.q odd 4 1
5184.2.a.g 1 120.w even 4 1
5184.2.a.z 1 40.k even 4 1
5184.2.a.bc 1 40.i odd 4 1
8100.2.a.a 1 15.e even 4 1
8100.2.a.f 1 5.c odd 4 1
8100.2.d.a 2 3.b odd 2 1
8100.2.d.a 2 15.d odd 2 1
8100.2.d.j 2 1.a even 1 1 trivial
8100.2.d.j 2 5.b 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 $$S_{2}^{\mathrm{new}}(8100, [\chi])$$:

 $$T_{7}^{2} + 4$$ $$T_{11} - 6$$ $$T_{29} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$4 + T^{2}$$
$11$ $$( -6 + T )^{2}$$
$13$ $$25 + T^{2}$$
$17$ $$9 + T^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$36 + T^{2}$$
$29$ $$( -3 + T )^{2}$$
$31$ $$( 4 + T )^{2}$$
$37$ $$25 + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$100 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$( 12 + T )^{2}$$
$61$ $$( -5 + T )^{2}$$
$67$ $$4 + T^{2}$$
$71$ $$( 6 + T )^{2}$$
$73$ $$1 + T^{2}$$
$79$ $$( -10 + T )^{2}$$
$83$ $$T^{2}$$
$89$ $$( 3 + T )^{2}$$
$97$ $$100 + T^{2}$$