# Properties

 Label 2352.4.a.bu 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: $$1$$ 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -3 q^{3} + ( 6 + \beta ) q^{5} + 9 q^{9} +O(q^{10})$$ $$q -3 q^{3} + ( 6 + \beta ) q^{5} + 9 q^{9} + ( -2 - 6 \beta ) q^{11} + ( 24 - 15 \beta ) q^{13} + ( -18 - 3 \beta ) q^{15} + ( 66 - 11 \beta ) q^{17} + ( -60 - 46 \beta ) q^{19} + ( 38 + 102 \beta ) q^{23} + ( -87 + 12 \beta ) q^{25} -27 q^{27} + ( -56 + 150 \beta ) q^{29} + ( -216 + 54 \beta ) q^{31} + ( 6 + 18 \beta ) q^{33} + ( -140 - 180 \beta ) q^{37} + ( -72 + 45 \beta ) q^{39} + ( 18 + 127 \beta ) q^{41} + ( 64 - 288 \beta ) q^{43} + ( 54 + 9 \beta ) q^{45} + ( 132 + 338 \beta ) q^{47} + ( -198 + 33 \beta ) q^{51} + ( 134 - 192 \beta ) q^{53} + ( -24 - 38 \beta ) q^{55} + ( 180 + 138 \beta ) q^{57} + ( -168 - 298 \beta ) q^{59} + ( -252 + 353 \beta ) q^{61} + ( 114 - 66 \beta ) q^{65} + ( 192 + 144 \beta ) q^{67} + ( -114 - 306 \beta ) q^{69} + ( 198 - 342 \beta ) q^{71} + ( -156 - 595 \beta ) q^{73} + ( 261 - 36 \beta ) q^{75} + ( 424 + 300 \beta ) q^{79} + 81 q^{81} + ( 324 - 80 \beta ) q^{83} + 374 q^{85} + ( 168 - 450 \beta ) q^{87} + ( -306 - 175 \beta ) q^{89} + ( 648 - 162 \beta ) q^{93} + ( -452 - 336 \beta ) q^{95} + ( -1092 + 133 \beta ) q^{97} + ( -18 - 54 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 6q^{3} + 12q^{5} + 18q^{9} + O(q^{10})$$ $$2q - 6q^{3} + 12q^{5} + 18q^{9} - 4q^{11} + 48q^{13} - 36q^{15} + 132q^{17} - 120q^{19} + 76q^{23} - 174q^{25} - 54q^{27} - 112q^{29} - 432q^{31} + 12q^{33} - 280q^{37} - 144q^{39} + 36q^{41} + 128q^{43} + 108q^{45} + 264q^{47} - 396q^{51} + 268q^{53} - 48q^{55} + 360q^{57} - 336q^{59} - 504q^{61} + 228q^{65} + 384q^{67} - 228q^{69} + 396q^{71} - 312q^{73} + 522q^{75} + 848q^{79} + 162q^{81} + 648q^{83} + 748q^{85} + 336q^{87} - 612q^{89} + 1296q^{93} - 904q^{95} - 2184q^{97} - 36q^{99} + O(q^{100})$$

## 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 4.58579 0 0 0 9.00000 0
1.2 0 −3.00000 0 7.41421 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.bu 2
4.b odd 2 1 294.4.a.o yes 2
7.b odd 2 1 2352.4.a.bw 2
12.b even 2 1 882.4.a.t 2
28.d even 2 1 294.4.a.l 2
28.f even 6 2 294.4.e.m 4
28.g odd 6 2 294.4.e.k 4
84.h odd 2 1 882.4.a.bb 2
84.j odd 6 2 882.4.g.be 4
84.n even 6 2 882.4.g.bk 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.4.a.l 2 28.d even 2 1
294.4.a.o yes 2 4.b odd 2 1
294.4.e.k 4 28.g odd 6 2
294.4.e.m 4 28.f even 6 2
882.4.a.t 2 12.b even 2 1
882.4.a.bb 2 84.h odd 2 1
882.4.g.be 4 84.j odd 6 2
882.4.g.bk 4 84.n even 6 2
2352.4.a.bu 2 1.a even 1 1 trivial
2352.4.a.bw 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2352))$$:

 $$T_{5}^{2} - 12 T_{5} + 34$$ $$T_{11}^{2} + 4 T_{11} - 68$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 3 + T )^{2}$$
$5$ $$34 - 12 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-68 + 4 T + T^{2}$$
$13$ $$126 - 48 T + T^{2}$$
$17$ $$4114 - 132 T + T^{2}$$
$19$ $$-632 + 120 T + T^{2}$$
$23$ $$-19364 - 76 T + T^{2}$$
$29$ $$-41864 + 112 T + T^{2}$$
$31$ $$40824 + 432 T + T^{2}$$
$37$ $$-45200 + 280 T + T^{2}$$
$41$ $$-31934 - 36 T + T^{2}$$
$43$ $$-161792 - 128 T + T^{2}$$
$47$ $$-211064 - 264 T + T^{2}$$
$53$ $$-55772 - 268 T + T^{2}$$
$59$ $$-149384 + 336 T + T^{2}$$
$61$ $$-185714 + 504 T + T^{2}$$
$67$ $$-4608 - 384 T + T^{2}$$
$71$ $$-194724 - 396 T + T^{2}$$
$73$ $$-683714 + 312 T + T^{2}$$
$79$ $$-224 - 848 T + T^{2}$$
$83$ $$92176 - 648 T + T^{2}$$
$89$ $$32386 + 612 T + T^{2}$$
$97$ $$1157086 + 2184 T + T^{2}$$