# Properties

 Label 8100.2.a.j Level $8100$ Weight $2$ Character orbit 8100.a Self dual yes Analytic conductor $64.679$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8100 = 2^{2} \cdot 3^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8100.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.6788256372$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 36) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 0 0 0 0 1.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8100.2.a.j 1
3.b odd 2 1 8100.2.a.g 1
5.b even 2 1 324.2.a.c 1
5.c odd 4 2 8100.2.d.h 2
9.c even 3 2 900.2.i.b 2
9.d odd 6 2 2700.2.i.b 2
15.d odd 2 1 324.2.a.a 1
15.e even 4 2 8100.2.d.c 2
20.d odd 2 1 1296.2.a.k 1
40.e odd 2 1 5184.2.a.f 1
40.f even 2 1 5184.2.a.e 1
45.h odd 6 2 108.2.e.a 2
45.j even 6 2 36.2.e.a 2
45.k odd 12 4 900.2.s.b 4
45.l even 12 4 2700.2.s.b 4
60.h even 2 1 1296.2.a.b 1
120.i odd 2 1 5184.2.a.ba 1
120.m even 2 1 5184.2.a.bb 1
180.n even 6 2 432.2.i.c 2
180.p odd 6 2 144.2.i.a 2
315.q odd 6 2 1764.2.i.c 2
315.r even 6 2 1764.2.i.a 2
315.u even 6 2 5292.2.l.c 2
315.v odd 6 2 5292.2.l.a 2
315.z even 6 2 5292.2.j.a 2
315.bg odd 6 2 1764.2.j.b 2
315.bn odd 6 2 1764.2.l.a 2
315.bo even 6 2 1764.2.l.c 2
315.bq even 6 2 5292.2.i.a 2
315.br odd 6 2 5292.2.i.c 2
360.z odd 6 2 576.2.i.e 2
360.bd even 6 2 1728.2.i.c 2
360.bh odd 6 2 1728.2.i.d 2
360.bk even 6 2 576.2.i.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.e.a 2 45.j even 6 2
108.2.e.a 2 45.h odd 6 2
144.2.i.a 2 180.p odd 6 2
324.2.a.a 1 15.d odd 2 1
324.2.a.c 1 5.b even 2 1
432.2.i.c 2 180.n even 6 2
576.2.i.e 2 360.z odd 6 2
576.2.i.f 2 360.bk even 6 2
900.2.i.b 2 9.c even 3 2
900.2.s.b 4 45.k odd 12 4
1296.2.a.b 1 60.h even 2 1
1296.2.a.k 1 20.d odd 2 1
1728.2.i.c 2 360.bd even 6 2
1728.2.i.d 2 360.bh odd 6 2
1764.2.i.a 2 315.r even 6 2
1764.2.i.c 2 315.q odd 6 2
1764.2.j.b 2 315.bg odd 6 2
1764.2.l.a 2 315.bn odd 6 2
1764.2.l.c 2 315.bo even 6 2
2700.2.i.b 2 9.d odd 6 2
2700.2.s.b 4 45.l even 12 4
5184.2.a.e 1 40.f even 2 1
5184.2.a.f 1 40.e odd 2 1
5184.2.a.ba 1 120.i odd 2 1
5184.2.a.bb 1 120.m even 2 1
5292.2.i.a 2 315.bq even 6 2
5292.2.i.c 2 315.br odd 6 2
5292.2.j.a 2 315.z even 6 2
5292.2.l.a 2 315.v odd 6 2
5292.2.l.c 2 315.u even 6 2
8100.2.a.g 1 3.b odd 2 1
8100.2.a.j 1 1.a even 1 1 trivial
8100.2.d.c 2 15.e even 4 2
8100.2.d.h 2 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8100))$$:

 $$T_{7} - 1$$ $$T_{11} - 3$$ $$T_{17} + 6$$

## Hecke characteristic polynomials

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