# Properties

 Label 2352.4.a.bq 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: $$\Q(\sqrt{1345})$$ Defining polynomial: $$x^{2} - x - 336$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{1345})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -3 q^{3} + ( 3 - \beta ) q^{5} + 9 q^{9} +O(q^{10})$$ $$q -3 q^{3} + ( 3 - \beta ) q^{5} + 9 q^{9} + ( -33 - \beta ) q^{11} + ( 20 + \beta ) q^{13} + ( -9 + 3 \beta ) q^{15} + ( -48 + 4 \beta ) q^{17} + ( -20 - 3 \beta ) q^{19} + ( -72 - 4 \beta ) q^{23} + ( 220 - 5 \beta ) q^{25} -27 q^{27} + ( 33 + 11 \beta ) q^{29} + ( 259 + 2 \beta ) q^{31} + ( 99 + 3 \beta ) q^{33} + ( 8 - 9 \beta ) q^{37} + ( -60 - 3 \beta ) q^{39} + ( -216 + 6 \beta ) q^{41} + ( 46 + 15 \beta ) q^{43} + ( 27 - 9 \beta ) q^{45} + ( -294 + 12 \beta ) q^{47} + ( 144 - 12 \beta ) q^{51} + ( -123 + 3 \beta ) q^{53} + ( 237 + 31 \beta ) q^{55} + ( 60 + 9 \beta ) q^{57} + ( -9 + 25 \beta ) q^{59} + ( 110 + 4 \beta ) q^{61} + ( -276 - 18 \beta ) q^{65} + ( -338 - 11 \beta ) q^{67} + ( 216 + 12 \beta ) q^{69} + ( -246 + 20 \beta ) q^{71} + ( -478 + 35 \beta ) q^{73} + ( -660 + 15 \beta ) q^{75} + ( 259 + 8 \beta ) q^{79} + 81 q^{81} + ( 123 - 25 \beta ) q^{83} + ( -1488 + 56 \beta ) q^{85} + ( -99 - 33 \beta ) q^{87} + ( 366 + 42 \beta ) q^{89} + ( -777 - 6 \beta ) q^{93} + ( 948 + 14 \beta ) q^{95} + ( 959 + 35 \beta ) q^{97} + ( -297 - 9 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 6q^{3} + 5q^{5} + 18q^{9} + O(q^{10})$$ $$2q - 6q^{3} + 5q^{5} + 18q^{9} - 67q^{11} + 41q^{13} - 15q^{15} - 92q^{17} - 43q^{19} - 148q^{23} + 435q^{25} - 54q^{27} + 77q^{29} + 520q^{31} + 201q^{33} + 7q^{37} - 123q^{39} - 426q^{41} + 107q^{43} + 45q^{45} - 576q^{47} + 276q^{51} - 243q^{53} + 505q^{55} + 129q^{57} + 7q^{59} + 224q^{61} - 570q^{65} - 687q^{67} + 444q^{69} - 472q^{71} - 921q^{73} - 1305q^{75} + 526q^{79} + 162q^{81} + 221q^{83} - 2920q^{85} - 231q^{87} + 774q^{89} - 1560q^{93} + 1910q^{95} + 1953q^{97} - 603q^{99} + O(q^{100})$$

## 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
 18.8371 −17.8371
0 −3.00000 0 −15.8371 0 0 0 9.00000 0
1.2 0 −3.00000 0 20.8371 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.bq 2
4.b odd 2 1 294.4.a.n 2
7.b odd 2 1 2352.4.a.ca 2
7.c even 3 2 336.4.q.j 4
12.b even 2 1 882.4.a.v 2
28.d even 2 1 294.4.a.m 2
28.f even 6 2 294.4.e.l 4
28.g odd 6 2 42.4.e.c 4
84.h odd 2 1 882.4.a.z 2
84.j odd 6 2 882.4.g.bf 4
84.n even 6 2 126.4.g.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.c 4 28.g odd 6 2
126.4.g.g 4 84.n even 6 2
294.4.a.m 2 28.d even 2 1
294.4.a.n 2 4.b odd 2 1
294.4.e.l 4 28.f even 6 2
336.4.q.j 4 7.c even 3 2
882.4.a.v 2 12.b even 2 1
882.4.a.z 2 84.h odd 2 1
882.4.g.bf 4 84.j odd 6 2
2352.4.a.bq 2 1.a even 1 1 trivial
2352.4.a.ca 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2352))$$:

 $$T_{5}^{2} - 5 T_{5} - 330$$ $$T_{11}^{2} + 67 T_{11} + 786$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 3 + T )^{2}$$
$5$ $$-330 - 5 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$786 + 67 T + T^{2}$$
$13$ $$84 - 41 T + T^{2}$$
$17$ $$-3264 + 92 T + T^{2}$$
$19$ $$-2564 + 43 T + T^{2}$$
$23$ $$96 + 148 T + T^{2}$$
$29$ $$-39204 - 77 T + T^{2}$$
$31$ $$66255 - 520 T + T^{2}$$
$37$ $$-27224 - 7 T + T^{2}$$
$41$ $$33264 + 426 T + T^{2}$$
$43$ $$-72794 - 107 T + T^{2}$$
$47$ $$34524 + 576 T + T^{2}$$
$53$ $$11736 + 243 T + T^{2}$$
$59$ $$-210144 - 7 T + T^{2}$$
$61$ $$7164 - 224 T + T^{2}$$
$67$ $$77306 + 687 T + T^{2}$$
$71$ $$-78804 + 472 T + T^{2}$$
$73$ $$-199846 + 921 T + T^{2}$$
$79$ $$47649 - 526 T + T^{2}$$
$83$ $$-197946 - 221 T + T^{2}$$
$89$ $$-443376 - 774 T + T^{2}$$
$97$ $$541646 - 1953 T + T^{2}$$