# Properties

 Label 2352.4.a.bn Level $2352$ Weight $4$ Character orbit 2352.a Self dual yes Analytic conductor $138.772$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$138.772492334$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$7$$ Twist minimal: no (minimal twist has level 294) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 7\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -3 q^{3} + ( -6 + 7 \beta ) q^{5} + 9 q^{9} +O(q^{10})$$ $$q -3 q^{3} + ( -6 + 7 \beta ) q^{5} + 9 q^{9} + ( 2 + 42 \beta ) q^{11} + ( -24 - 21 \beta ) q^{13} + ( 18 - 21 \beta ) q^{15} + ( -42 + 7 \beta ) q^{17} + ( -36 - 14 \beta ) q^{19} + ( 154 - 42 \beta ) q^{23} + ( 9 - 84 \beta ) q^{25} -27 q^{27} + ( -40 + 126 \beta ) q^{29} + ( 192 - 42 \beta ) q^{31} + ( -6 - 126 \beta ) q^{33} + ( 268 - 84 \beta ) q^{37} + ( 72 + 63 \beta ) q^{39} + ( -378 - 35 \beta ) q^{41} + ( -200 - 168 \beta ) q^{43} + ( -54 + 63 \beta ) q^{45} + ( 156 - 70 \beta ) q^{47} + ( 126 - 21 \beta ) q^{51} + ( -26 - 168 \beta ) q^{53} + ( 576 - 238 \beta ) q^{55} + ( 108 + 42 \beta ) q^{57} + ( 432 + 14 \beta ) q^{59} + ( -708 + 91 \beta ) q^{61} + ( -150 - 42 \beta ) q^{65} + ( -72 - 168 \beta ) q^{67} + ( -462 + 126 \beta ) q^{69} + ( 762 + 210 \beta ) q^{71} + ( -372 - 581 \beta ) q^{73} + ( -27 + 252 \beta ) q^{75} + ( -488 - 588 \beta ) q^{79} + 81 q^{81} + ( -156 + 952 \beta ) q^{83} + ( 350 - 336 \beta ) q^{85} + ( 120 - 378 \beta ) q^{87} + ( -54 + 203 \beta ) q^{89} + ( -576 + 126 \beta ) q^{93} + ( 20 - 168 \beta ) q^{95} + ( 372 + 875 \beta ) q^{97} + ( 18 + 378 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 6q^{3} - 12q^{5} + 18q^{9} + O(q^{10})$$ $$2q - 6q^{3} - 12q^{5} + 18q^{9} + 4q^{11} - 48q^{13} + 36q^{15} - 84q^{17} - 72q^{19} + 308q^{23} + 18q^{25} - 54q^{27} - 80q^{29} + 384q^{31} - 12q^{33} + 536q^{37} + 144q^{39} - 756q^{41} - 400q^{43} - 108q^{45} + 312q^{47} + 252q^{51} - 52q^{53} + 1152q^{55} + 216q^{57} + 864q^{59} - 1416q^{61} - 300q^{65} - 144q^{67} - 924q^{69} + 1524q^{71} - 744q^{73} - 54q^{75} - 976q^{79} + 162q^{81} - 312q^{83} + 700q^{85} + 240q^{87} - 108q^{89} - 1152q^{93} + 40q^{95} + 744q^{97} + 36q^{99} + O(q^{100})$$

## 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 −3.00000 0 −15.8995 0 0 0 9.00000 0
1.2 0 −3.00000 0 3.89949 0 0 0 9.00000 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 2352.4.a.bn 2
4.b odd 2 1 294.4.a.k yes 2
7.b odd 2 1 2352.4.a.cd 2
12.b even 2 1 882.4.a.bi 2
28.d even 2 1 294.4.a.j 2
28.f even 6 2 294.4.e.o 4
28.g odd 6 2 294.4.e.n 4
84.h odd 2 1 882.4.a.bc 2
84.j odd 6 2 882.4.g.bd 4
84.n even 6 2 882.4.g.y 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.4.a.j 2 28.d even 2 1
294.4.a.k yes 2 4.b odd 2 1
294.4.e.n 4 28.g odd 6 2
294.4.e.o 4 28.f even 6 2
882.4.a.bc 2 84.h odd 2 1
882.4.a.bi 2 12.b even 2 1
882.4.g.y 4 84.n even 6 2
882.4.g.bd 4 84.j odd 6 2
2352.4.a.bn 2 1.a even 1 1 trivial
2352.4.a.cd 2 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2352))$$:

 $$T_{5}^{2} + 12 T_{5} - 62$$ $$T_{11}^{2} - 4 T_{11} - 3524$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 3 + T )^{2}$$
$5$ $$-62 + 12 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-3524 - 4 T + T^{2}$$
$13$ $$-306 + 48 T + T^{2}$$
$17$ $$1666 + 84 T + T^{2}$$
$19$ $$904 + 72 T + T^{2}$$
$23$ $$20188 - 308 T + T^{2}$$
$29$ $$-30152 + 80 T + T^{2}$$
$31$ $$33336 - 384 T + T^{2}$$
$37$ $$57712 - 536 T + T^{2}$$
$41$ $$140434 + 756 T + T^{2}$$
$43$ $$-16448 + 400 T + T^{2}$$
$47$ $$14536 - 312 T + T^{2}$$
$53$ $$-55772 + 52 T + T^{2}$$
$59$ $$186232 - 864 T + T^{2}$$
$61$ $$484702 + 1416 T + T^{2}$$
$67$ $$-51264 + 144 T + T^{2}$$
$71$ $$492444 - 1524 T + T^{2}$$
$73$ $$-536738 + 744 T + T^{2}$$
$79$ $$-453344 + 976 T + T^{2}$$
$83$ $$-1788272 + 312 T + T^{2}$$
$89$ $$-79502 + 108 T + T^{2}$$
$97$ $$-1392866 - 744 T + T^{2}$$