# Properties

 Label 2352.4.a.bo 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: $$\Q(\sqrt{505})$$ Defining polynomial: $$x^{2} - x - 126$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 168) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{505})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -3 q^{3} + ( -4 - \beta ) q^{5} + 9 q^{9} +O(q^{10})$$ $$q -3 q^{3} + ( -4 - \beta ) q^{5} + 9 q^{9} -5 \beta q^{11} + ( 9 + \beta ) q^{13} + ( 12 + 3 \beta ) q^{15} + ( 4 - 4 \beta ) q^{17} + ( 55 + 7 \beta ) q^{19} + ( -76 + 4 \beta ) q^{23} + ( 17 + 9 \beta ) q^{25} -27 q^{27} + ( -50 + 5 \beta ) q^{29} + ( -179 - 2 \beta ) q^{31} + 15 \beta q^{33} + ( -27 + \beta ) q^{37} + ( -27 - 3 \beta ) q^{39} + ( 94 - 18 \beta ) q^{41} + ( -215 + 27 \beta ) q^{43} + ( -36 - 9 \beta ) q^{45} + ( 166 + 36 \beta ) q^{47} + ( -12 + 12 \beta ) q^{51} + ( 346 + 5 \beta ) q^{53} + ( 630 + 25 \beta ) q^{55} + ( -165 - 21 \beta ) q^{57} + ( -306 + 27 \beta ) q^{59} + ( 566 + 28 \beta ) q^{61} + ( -162 - 14 \beta ) q^{65} + ( -153 + 73 \beta ) q^{67} + ( 228 - 12 \beta ) q^{69} + ( -270 - 76 \beta ) q^{71} + ( 377 + 63 \beta ) q^{73} + ( -51 - 27 \beta ) q^{75} + ( 425 - 48 \beta ) q^{79} + 81 q^{81} + ( -108 - 67 \beta ) q^{83} + ( 488 + 16 \beta ) q^{85} + ( 150 - 15 \beta ) q^{87} + ( 940 - 22 \beta ) q^{89} + ( 537 + 6 \beta ) q^{93} + ( -1102 - 90 \beta ) q^{95} + ( -892 + 55 \beta ) q^{97} -45 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 6q^{3} - 9q^{5} + 18q^{9} + O(q^{10})$$ $$2q - 6q^{3} - 9q^{5} + 18q^{9} - 5q^{11} + 19q^{13} + 27q^{15} + 4q^{17} + 117q^{19} - 148q^{23} + 43q^{25} - 54q^{27} - 95q^{29} - 360q^{31} + 15q^{33} - 53q^{37} - 57q^{39} + 170q^{41} - 403q^{43} - 81q^{45} + 368q^{47} - 12q^{51} + 697q^{53} + 1285q^{55} - 351q^{57} - 585q^{59} + 1160q^{61} - 338q^{65} - 233q^{67} + 444q^{69} - 616q^{71} + 817q^{73} - 129q^{75} + 802q^{79} + 162q^{81} - 283q^{83} + 992q^{85} + 285q^{87} + 1858q^{89} + 1080q^{93} - 2294q^{95} - 1729q^{97} - 45q^{99} + O(q^{100})$$

## 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
 11.7361 −10.7361
0 −3.00000 0 −15.7361 0 0 0 9.00000 0
1.2 0 −3.00000 0 6.73610 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.bo 2
4.b odd 2 1 1176.4.a.u 2
7.b odd 2 1 2352.4.a.cc 2
7.d odd 6 2 336.4.q.g 4
28.d even 2 1 1176.4.a.r 2
28.f even 6 2 168.4.q.d 4
84.j odd 6 2 504.4.s.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.q.d 4 28.f even 6 2
336.4.q.g 4 7.d odd 6 2
504.4.s.f 4 84.j odd 6 2
1176.4.a.r 2 28.d even 2 1
1176.4.a.u 2 4.b odd 2 1
2352.4.a.bo 2 1.a even 1 1 trivial
2352.4.a.cc 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2352))$$:

 $$T_{5}^{2} + 9 T_{5} - 106$$ $$T_{11}^{2} + 5 T_{11} - 3150$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 3 + T )^{2}$$
$5$ $$-106 + 9 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-3150 + 5 T + T^{2}$$
$13$ $$-36 - 19 T + T^{2}$$
$17$ $$-2016 - 4 T + T^{2}$$
$19$ $$-2764 - 117 T + T^{2}$$
$23$ $$3456 + 148 T + T^{2}$$
$29$ $$-900 + 95 T + T^{2}$$
$31$ $$31895 + 360 T + T^{2}$$
$37$ $$576 + 53 T + T^{2}$$
$41$ $$-33680 - 170 T + T^{2}$$
$43$ $$-51434 + 403 T + T^{2}$$
$47$ $$-129764 - 368 T + T^{2}$$
$53$ $$118296 - 697 T + T^{2}$$
$59$ $$-6480 + 585 T + T^{2}$$
$61$ $$237420 - 1160 T + T^{2}$$
$67$ $$-659214 + 233 T + T^{2}$$
$71$ $$-634356 + 616 T + T^{2}$$
$73$ $$-334214 - 817 T + T^{2}$$
$79$ $$-130079 - 802 T + T^{2}$$
$83$ $$-546714 + 283 T + T^{2}$$
$89$ $$801936 - 1858 T + T^{2}$$
$97$ $$365454 + 1729 T + T^{2}$$