Properties

 Label 8100.2.a.f 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 324) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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 −2.00000 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.f 1
3.b odd 2 1 8100.2.a.a 1
5.b even 2 1 324.2.a.b 1
5.c odd 4 2 8100.2.d.j 2
15.d odd 2 1 324.2.a.d yes 1
15.e even 4 2 8100.2.d.a 2
20.d odd 2 1 1296.2.a.a 1
40.e odd 2 1 5184.2.a.z 1
40.f even 2 1 5184.2.a.bc 1
45.h odd 6 2 324.2.e.a 2
45.j even 6 2 324.2.e.d 2
60.h even 2 1 1296.2.a.j 1
120.i odd 2 1 5184.2.a.g 1
120.m even 2 1 5184.2.a.d 1
180.n even 6 2 1296.2.i.d 2
180.p odd 6 2 1296.2.i.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
324.2.a.b 1 5.b even 2 1
324.2.a.d yes 1 15.d odd 2 1
324.2.e.a 2 45.h odd 6 2
324.2.e.d 2 45.j even 6 2
1296.2.a.a 1 20.d odd 2 1
1296.2.a.j 1 60.h even 2 1
1296.2.i.d 2 180.n even 6 2
1296.2.i.p 2 180.p odd 6 2
5184.2.a.d 1 120.m even 2 1
5184.2.a.g 1 120.i odd 2 1
5184.2.a.z 1 40.e odd 2 1
5184.2.a.bc 1 40.f even 2 1
8100.2.a.a 1 3.b odd 2 1
8100.2.a.f 1 1.a even 1 1 trivial
8100.2.d.a 2 15.e even 4 2
8100.2.d.j 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8100))$$:

 $$T_{7} + 2$$ $$T_{11} - 6$$ $$T_{17} + 3$$

Hecke characteristic polynomials

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