# Properties

 Label 2352.4.a.cf Level $2352$ Weight $4$ Character orbit 2352.a Self dual yes Analytic conductor $138.772$ 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: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2352.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$138.772492334$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 147) 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.

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + ( 10 + 7 \beta ) q^{5} + 9 q^{9} +O(q^{10})$$ $$q + 3 q^{3} + ( 10 + 7 \beta ) q^{5} + 9 q^{9} + ( 10 - 24 \beta ) q^{11} + ( 52 + 25 \beta ) q^{13} + ( 30 + 21 \beta ) q^{15} + ( 58 - 45 \beta ) q^{17} + ( -96 + 22 \beta ) q^{19} + ( -14 + 28 \beta ) q^{23} + ( 73 + 140 \beta ) q^{25} + 27 q^{27} + ( 148 - 62 \beta ) q^{29} + ( 52 + 50 \beta ) q^{31} + ( 30 - 72 \beta ) q^{33} + ( -124 + 48 \beta ) q^{37} + ( 156 + 75 \beta ) q^{39} + ( 10 - 219 \beta ) q^{41} + ( 360 + 100 \beta ) q^{43} + ( 90 + 63 \beta ) q^{45} + ( 48 - 250 \beta ) q^{47} + ( 174 - 135 \beta ) q^{51} + ( 134 - 360 \beta ) q^{53} + ( -236 - 170 \beta ) q^{55} + ( -288 + 66 \beta ) q^{57} + ( 308 + 226 \beta ) q^{59} + ( 8 - 3 \beta ) q^{61} + ( 870 + 614 \beta ) q^{65} + ( 72 + 524 \beta ) q^{67} + ( -42 + 84 \beta ) q^{69} + ( -494 + 232 \beta ) q^{71} + ( 52 + 401 \beta ) q^{73} + ( 219 + 420 \beta ) q^{75} + ( 472 - 236 \beta ) q^{79} + 81 q^{81} + ( -508 - 80 \beta ) q^{83} + ( -50 - 44 \beta ) q^{85} + ( 444 - 186 \beta ) q^{87} + ( -194 + 339 \beta ) q^{89} + ( 156 + 150 \beta ) q^{93} + ( -652 - 452 \beta ) q^{95} + ( 244 - 599 \beta ) q^{97} + ( 90 - 216 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 6q^{3} + 20q^{5} + 18q^{9} + O(q^{10})$$ $$2q + 6q^{3} + 20q^{5} + 18q^{9} + 20q^{11} + 104q^{13} + 60q^{15} + 116q^{17} - 192q^{19} - 28q^{23} + 146q^{25} + 54q^{27} + 296q^{29} + 104q^{31} + 60q^{33} - 248q^{37} + 312q^{39} + 20q^{41} + 720q^{43} + 180q^{45} + 96q^{47} + 348q^{51} + 268q^{53} - 472q^{55} - 576q^{57} + 616q^{59} + 16q^{61} + 1740q^{65} + 144q^{67} - 84q^{69} - 988q^{71} + 104q^{73} + 438q^{75} + 944q^{79} + 162q^{81} - 1016q^{83} - 100q^{85} + 888q^{87} - 388q^{89} + 312q^{93} - 1304q^{95} + 488q^{97} + 180q^{99} + O(q^{100})$$

## 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 0.100505 0 0 0 9.00000 0
1.2 0 3.00000 0 19.8995 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.cf 2
4.b odd 2 1 147.4.a.j 2
7.b odd 2 1 2352.4.a.bl 2
12.b even 2 1 441.4.a.n 2
28.d even 2 1 147.4.a.k yes 2
28.f even 6 2 147.4.e.j 4
28.g odd 6 2 147.4.e.k 4
84.h odd 2 1 441.4.a.o 2
84.j odd 6 2 441.4.e.u 4
84.n even 6 2 441.4.e.v 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.4.a.j 2 4.b odd 2 1
147.4.a.k yes 2 28.d even 2 1
147.4.e.j 4 28.f even 6 2
147.4.e.k 4 28.g odd 6 2
441.4.a.n 2 12.b even 2 1
441.4.a.o 2 84.h odd 2 1
441.4.e.u 4 84.j odd 6 2
441.4.e.v 4 84.n even 6 2
2352.4.a.bl 2 7.b odd 2 1
2352.4.a.cf 2 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{5}^{2} - 20 T_{5} + 2$$ $$T_{11}^{2} - 20 T_{11} - 1052$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -3 + T )^{2}$$
$5$ $$2 - 20 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-1052 - 20 T + T^{2}$$
$13$ $$1454 - 104 T + T^{2}$$
$17$ $$-686 - 116 T + T^{2}$$
$19$ $$8248 + 192 T + T^{2}$$
$23$ $$-1372 + 28 T + T^{2}$$
$29$ $$14216 - 296 T + T^{2}$$
$31$ $$-2296 - 104 T + T^{2}$$
$37$ $$10768 + 248 T + T^{2}$$
$41$ $$-95822 - 20 T + T^{2}$$
$43$ $$109600 - 720 T + T^{2}$$
$47$ $$-122696 - 96 T + T^{2}$$
$53$ $$-241244 - 268 T + T^{2}$$
$59$ $$-7288 - 616 T + T^{2}$$
$61$ $$46 - 16 T + T^{2}$$
$67$ $$-543968 - 144 T + T^{2}$$
$71$ $$136388 + 988 T + T^{2}$$
$73$ $$-318898 - 104 T + T^{2}$$
$79$ $$111392 - 944 T + T^{2}$$
$83$ $$245264 + 1016 T + T^{2}$$
$89$ $$-192206 + 388 T + T^{2}$$
$97$ $$-658066 - 488 T + T^{2}$$