Properties

 Label 2352.2.a.bd Level $2352$ Weight $2$ Character orbit 2352.a Self dual yes Analytic conductor $18.781$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$2352 = 2^{4} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2352.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$18.7808145554$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1176) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} + ( -2 + \beta ) q^{5} + q^{9} +O(q^{10})$$ $$q + q^{3} + ( -2 + \beta ) q^{5} + q^{9} + ( -2 + 2 \beta ) q^{11} + \beta q^{13} + ( -2 + \beta ) q^{15} + ( 2 - 3 \beta ) q^{17} + ( 4 + 2 \beta ) q^{19} + ( -2 - 2 \beta ) q^{23} + ( 1 - 4 \beta ) q^{25} + q^{27} + 6 \beta q^{29} + ( 8 - 2 \beta ) q^{31} + ( -2 + 2 \beta ) q^{33} + ( -4 + 4 \beta ) q^{37} + \beta q^{39} + ( 2 - \beta ) q^{41} + 8 q^{43} + ( -2 + \beta ) q^{45} + ( 4 + 2 \beta ) q^{47} + ( 2 - 3 \beta ) q^{51} + ( -2 - 8 \beta ) q^{53} + ( 8 - 6 \beta ) q^{55} + ( 4 + 2 \beta ) q^{57} + ( 8 - 2 \beta ) q^{59} + ( -4 - 7 \beta ) q^{61} + ( 2 - 2 \beta ) q^{65} + 8 q^{67} + ( -2 - 2 \beta ) q^{69} + ( -2 + 2 \beta ) q^{71} + ( 4 + 5 \beta ) q^{73} + ( 1 - 4 \beta ) q^{75} + ( 8 - 4 \beta ) q^{79} + q^{81} + ( 4 + 8 \beta ) q^{83} + ( -10 + 8 \beta ) q^{85} + 6 \beta q^{87} + ( -2 + 9 \beta ) q^{89} + ( 8 - 2 \beta ) q^{93} -4 q^{95} + ( 12 - 3 \beta ) q^{97} + ( -2 + 2 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} - 4q^{5} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{3} - 4q^{5} + 2q^{9} - 4q^{11} - 4q^{15} + 4q^{17} + 8q^{19} - 4q^{23} + 2q^{25} + 2q^{27} + 16q^{31} - 4q^{33} - 8q^{37} + 4q^{41} + 16q^{43} - 4q^{45} + 8q^{47} + 4q^{51} - 4q^{53} + 16q^{55} + 8q^{57} + 16q^{59} - 8q^{61} + 4q^{65} + 16q^{67} - 4q^{69} - 4q^{71} + 8q^{73} + 2q^{75} + 16q^{79} + 2q^{81} + 8q^{83} - 20q^{85} - 4q^{89} + 16q^{93} - 8q^{95} + 24q^{97} - 4q^{99} + O(q^{100})$$

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
 −1.41421 1.41421
0 1.00000 0 −3.41421 0 0 0 1.00000 0
1.2 0 1.00000 0 −0.585786 0 0 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$7$$ $$1$$

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 2352.2.a.bd 2
3.b odd 2 1 7056.2.a.cx 2
4.b odd 2 1 1176.2.a.j 2
7.b odd 2 1 2352.2.a.bb 2
7.c even 3 2 2352.2.q.bc 4
7.d odd 6 2 2352.2.q.be 4
8.b even 2 1 9408.2.a.ds 2
8.d odd 2 1 9408.2.a.ee 2
12.b even 2 1 3528.2.a.bl 2
21.c even 2 1 7056.2.a.cg 2
28.d even 2 1 1176.2.a.o yes 2
28.f even 6 2 1176.2.q.k 4
28.g odd 6 2 1176.2.q.o 4
56.e even 2 1 9408.2.a.dg 2
56.h odd 2 1 9408.2.a.du 2
84.h odd 2 1 3528.2.a.bb 2
84.j odd 6 2 3528.2.s.bm 4
84.n even 6 2 3528.2.s.bd 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1176.2.a.j 2 4.b odd 2 1
1176.2.a.o yes 2 28.d even 2 1
1176.2.q.k 4 28.f even 6 2
1176.2.q.o 4 28.g odd 6 2
2352.2.a.bb 2 7.b odd 2 1
2352.2.a.bd 2 1.a even 1 1 trivial
2352.2.q.bc 4 7.c even 3 2
2352.2.q.be 4 7.d odd 6 2
3528.2.a.bb 2 84.h odd 2 1
3528.2.a.bl 2 12.b even 2 1
3528.2.s.bd 4 84.n even 6 2
3528.2.s.bm 4 84.j odd 6 2
7056.2.a.cg 2 21.c even 2 1
7056.2.a.cx 2 3.b odd 2 1
9408.2.a.dg 2 56.e even 2 1
9408.2.a.ds 2 8.b even 2 1
9408.2.a.du 2 56.h odd 2 1
9408.2.a.ee 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2352))$$:

 $$T_{5}^{2} + 4 T_{5} + 2$$ $$T_{11}^{2} + 4 T_{11} - 4$$ $$T_{13}^{2} - 2$$ $$T_{17}^{2} - 4 T_{17} - 14$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$2 + 4 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-4 + 4 T + T^{2}$$
$13$ $$-2 + T^{2}$$
$17$ $$-14 - 4 T + T^{2}$$
$19$ $$8 - 8 T + T^{2}$$
$23$ $$-4 + 4 T + T^{2}$$
$29$ $$-72 + T^{2}$$
$31$ $$56 - 16 T + T^{2}$$
$37$ $$-16 + 8 T + T^{2}$$
$41$ $$2 - 4 T + T^{2}$$
$43$ $$( -8 + T )^{2}$$
$47$ $$8 - 8 T + T^{2}$$
$53$ $$-124 + 4 T + T^{2}$$
$59$ $$56 - 16 T + T^{2}$$
$61$ $$-82 + 8 T + T^{2}$$
$67$ $$( -8 + T )^{2}$$
$71$ $$-4 + 4 T + T^{2}$$
$73$ $$-34 - 8 T + T^{2}$$
$79$ $$32 - 16 T + T^{2}$$
$83$ $$-112 - 8 T + T^{2}$$
$89$ $$-158 + 4 T + T^{2}$$
$97$ $$126 - 24 T + T^{2}$$