Properties

 Label 9600.2.a.bk Level $9600$ Weight $2$ Character orbit 9600.a Self dual yes Analytic conductor $76.656$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$9600 = 2^{7} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9600.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$76.6563859404$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 384) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} - 2q^{7} + q^{9} + O(q^{10})$$ $$q + q^{3} - 2q^{7} + q^{9} + 4q^{11} - 2q^{13} + 2q^{17} + 8q^{19} - 2q^{21} - 4q^{23} + q^{27} - 6q^{31} + 4q^{33} + 2q^{37} - 2q^{39} + 6q^{41} - 4q^{47} - 3q^{49} + 2q^{51} + 8q^{57} - 4q^{59} + 14q^{61} - 2q^{63} - 4q^{67} - 4q^{69} + 12q^{71} + 10q^{73} - 8q^{77} + 10q^{79} + q^{81} + 12q^{83} - 14q^{89} + 4q^{91} - 6q^{93} - 10q^{97} + 4q^{99} + O(q^{100})$$

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 1.00000 0 0 0 −2.00000 0 1.00000 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 9600.2.a.bk 1
4.b odd 2 1 9600.2.a.t 1
5.b even 2 1 384.2.a.a 1
8.b even 2 1 9600.2.a.e 1
8.d odd 2 1 9600.2.a.bz 1
15.d odd 2 1 1152.2.a.t 1
20.d odd 2 1 384.2.a.e yes 1
40.e odd 2 1 384.2.a.d yes 1
40.f even 2 1 384.2.a.h yes 1
60.h even 2 1 1152.2.a.s 1
80.k odd 4 2 768.2.d.f 2
80.q even 4 2 768.2.d.c 2
120.i odd 2 1 1152.2.a.b 1
120.m even 2 1 1152.2.a.a 1
240.t even 4 2 2304.2.d.o 2
240.bm odd 4 2 2304.2.d.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.2.a.a 1 5.b even 2 1
384.2.a.d yes 1 40.e odd 2 1
384.2.a.e yes 1 20.d odd 2 1
384.2.a.h yes 1 40.f even 2 1
768.2.d.c 2 80.q even 4 2
768.2.d.f 2 80.k odd 4 2
1152.2.a.a 1 120.m even 2 1
1152.2.a.b 1 120.i odd 2 1
1152.2.a.s 1 60.h even 2 1
1152.2.a.t 1 15.d odd 2 1
2304.2.d.f 2 240.bm odd 4 2
2304.2.d.o 2 240.t even 4 2
9600.2.a.e 1 8.b even 2 1
9600.2.a.t 1 4.b odd 2 1
9600.2.a.bk 1 1.a even 1 1 trivial
9600.2.a.bz 1 8.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9600))$$:

 $$T_{7} + 2$$ $$T_{11} - 4$$ $$T_{13} + 2$$ $$T_{17} - 2$$ $$T_{29}$$ $$T_{31} + 6$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-1 + T$$
$5$ $$T$$
$7$ $$2 + T$$
$11$ $$-4 + T$$
$13$ $$2 + T$$
$17$ $$-2 + T$$
$19$ $$-8 + T$$
$23$ $$4 + T$$
$29$ $$T$$
$31$ $$6 + T$$
$37$ $$-2 + T$$
$41$ $$-6 + T$$
$43$ $$T$$
$47$ $$4 + T$$
$53$ $$T$$
$59$ $$4 + T$$
$61$ $$-14 + T$$
$67$ $$4 + T$$
$71$ $$-12 + T$$
$73$ $$-10 + T$$
$79$ $$-10 + T$$
$83$ $$-12 + T$$
$89$ $$14 + T$$
$97$ $$10 + T$$