# Properties

 Label 2352.4.a.cm Level $2352$ Weight $4$ Character orbit 2352.a Self dual yes Analytic conductor $138.772$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - 2 x^{3} - 152 x^{2} - 177 x + 2922$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}\cdot 7$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -3 q^{3} + ( 1 + \beta_{1} ) q^{5} + 9 q^{9} +O(q^{10})$$ $$q -3 q^{3} + ( 1 + \beta_{1} ) q^{5} + 9 q^{9} + ( -3 - 2 \beta_{1} - \beta_{3} ) q^{11} + ( 6 + 4 \beta_{1} - \beta_{3} ) q^{13} + ( -3 - 3 \beta_{1} ) q^{15} + ( 25 + 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{17} + ( 7 - 3 \beta_{1} + \beta_{2} ) q^{19} + ( -23 + 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{23} + ( 28 + 4 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{25} -27 q^{27} + ( -18 - 6 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{29} + ( -57 + 7 \beta_{1} - 5 \beta_{3} ) q^{31} + ( 9 + 6 \beta_{1} + 3 \beta_{3} ) q^{33} + ( 140 - 14 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{37} + ( -18 - 12 \beta_{1} + 3 \beta_{3} ) q^{39} + ( 106 + 14 \beta_{1} - 2 \beta_{2} ) q^{41} + ( 65 + 15 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} ) q^{43} + ( 9 + 9 \beta_{1} ) q^{45} + ( 17 + 25 \beta_{1} - \beta_{2} + 5 \beta_{3} ) q^{47} + ( -75 - 9 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{51} + ( 1 - 11 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{53} + ( -370 - 18 \beta_{1} - 5 \beta_{2} - 3 \beta_{3} ) q^{55} + ( -21 + 9 \beta_{1} - 3 \beta_{2} ) q^{57} + ( -68 - 3 \beta_{1} + 3 \beta_{2} + 8 \beta_{3} ) q^{59} + ( -68 + 36 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{61} + ( 551 + 9 \beta_{1} + 7 \beta_{2} + 3 \beta_{3} ) q^{65} + ( -33 - 5 \beta_{1} + 7 \beta_{2} - 2 \beta_{3} ) q^{67} + ( 69 - 9 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{69} + ( 211 + 31 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{71} + ( -54 + 4 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{73} + ( -84 - 12 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} ) q^{75} + ( -441 - 25 \beta_{1} + 4 \beta_{2} + 5 \beta_{3} ) q^{79} + 81 q^{81} + ( -862 + 27 \beta_{1} + \beta_{2} ) q^{83} + ( 364 + 94 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{85} + ( 54 + 18 \beta_{1} + 3 \beta_{2} + 9 \beta_{3} ) q^{87} + ( -86 - 34 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} ) q^{89} + ( 171 - 21 \beta_{1} + 15 \beta_{3} ) q^{93} + ( -503 + 67 \beta_{1} - 7 \beta_{2} - 3 \beta_{3} ) q^{95} + ( -151 - 90 \beta_{1} + 4 \beta_{2} - 5 \beta_{3} ) q^{97} + ( -27 - 18 \beta_{1} - 9 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 12q^{3} + 4q^{5} + 36q^{9} + O(q^{10})$$ $$4q - 12q^{3} + 4q^{5} + 36q^{9} - 14q^{11} + 22q^{13} - 12q^{15} + 96q^{17} + 26q^{19} - 96q^{23} + 110q^{25} - 108q^{27} - 76q^{29} - 238q^{31} + 42q^{33} + 562q^{37} - 66q^{39} + 428q^{41} + 258q^{43} + 36q^{45} + 80q^{47} - 288q^{51} - 1476q^{55} - 78q^{57} - 262q^{59} - 276q^{61} + 2196q^{65} - 150q^{67} + 288q^{69} + 848q^{71} - 218q^{73} - 330q^{75} - 1762q^{79} + 324q^{81} - 3450q^{83} + 1452q^{85} + 228q^{87} - 344q^{89} + 714q^{93} - 2004q^{95} - 622q^{97} - 126q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - 152 x^{2} - 177 x + 2922$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$2 \nu^{3} + 7 \nu^{2} - 213 \nu - 1158$$$$)/105$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{3} - 98 \nu^{2} + 942 \nu + 6297$$$$)/105$$ $$\beta_{3}$$ $$=$$ $$($$$$-8 \nu^{3} + 77 \nu^{2} + 677 \nu - 3348$$$$)/35$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + 3 \beta_{2} + 9 \beta_{1} + 15$$$$)/28$$ $$\nu^{2}$$ $$=$$ $$($$$$11 \beta_{3} + 5 \beta_{2} + 127 \beta_{1} + 2153$$$$)/28$$ $$\nu^{3}$$ $$=$$ $$($$$$34 \beta_{3} + 151 \beta_{2} + 992 \beta_{1} + 5137$$$$)/14$$

## 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
 3.92665 −8.69515 −6.46638 13.2349
0 −3.00000 0 −15.8130 0 0 0 9.00000 0
1.2 0 −3.00000 0 0.128591 0 0 0 9.00000 0
1.3 0 −3.00000 0 0.726342 0 0 0 9.00000 0
1.4 0 −3.00000 0 18.9580 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.cm 4
4.b odd 2 1 1176.4.a.bd 4
7.b odd 2 1 2352.4.a.cp 4
7.c even 3 2 336.4.q.m 8
28.d even 2 1 1176.4.a.ba 4
28.g odd 6 2 168.4.q.f 8
84.n even 6 2 504.4.s.j 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.q.f 8 28.g odd 6 2
336.4.q.m 8 7.c even 3 2
504.4.s.j 8 84.n even 6 2
1176.4.a.ba 4 28.d even 2 1
1176.4.a.bd 4 4.b odd 2 1
2352.4.a.cm 4 1.a even 1 1 trivial
2352.4.a.cp 4 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2352))$$:

 $$T_{5}^{4} - 4 T_{5}^{3} - 297 T_{5}^{2} + 256 T_{5} - 28$$ $$T_{11}^{4} + 14 T_{11}^{3} - 5395 T_{11}^{2} - 67916 T_{11} + 5765844$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 3 + T )^{4}$$
$5$ $$-28 + 256 T - 297 T^{2} - 4 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$5765844 - 67916 T - 5395 T^{2} + 14 T^{3} + T^{4}$$
$13$ $$13795008 + 84004 T - 7423 T^{2} - 22 T^{3} + T^{4}$$
$17$ $$-44203008 + 1716032 T - 14304 T^{2} - 96 T^{3} + T^{4}$$
$19$ $$5487296 - 258652 T - 14235 T^{2} - 26 T^{3} + T^{4}$$
$23$ $$-98304 + 121664 T - 14304 T^{2} + 96 T^{3} + T^{4}$$
$29$ $$-802800 - 3696384 T - 52293 T^{2} + 76 T^{3} + T^{4}$$
$31$ $$-165027985 - 16758574 T - 78768 T^{2} + 238 T^{3} + T^{4}$$
$37$ $$-4624450848 + 34102020 T + 16305 T^{2} - 562 T^{3} + T^{4}$$
$41$ $$-3866949120 + 36803808 T - 41132 T^{2} - 428 T^{3} + T^{4}$$
$43$ $$-2654719484 + 46672932 T - 179543 T^{2} - 258 T^{3} + T^{4}$$
$47$ $$23866588368 + 18481344 T - 336248 T^{2} - 80 T^{3} + T^{4}$$
$53$ $$-5074800 + 3013660 T - 135309 T^{2} + T^{4}$$
$59$ $$-1642011120 + 44672196 T - 276551 T^{2} + 262 T^{3} + T^{4}$$
$61$ $$9863636400 + 42157232 T - 435696 T^{2} + 276 T^{3} + T^{4}$$
$67$ $$26529903468 + 24707720 T - 590751 T^{2} + 150 T^{3} + T^{4}$$
$71$ $$1940742864 + 46164032 T - 78520 T^{2} - 848 T^{3} + T^{4}$$
$73$ $$280393876 - 5640388 T - 31515 T^{2} + 218 T^{3} + T^{4}$$
$79$ $$-93048620201 - 126613586 T + 743736 T^{2} + 1762 T^{3} + T^{4}$$
$83$ $$352673538780 + 2140010896 T + 4237149 T^{2} + 3450 T^{3} + T^{4}$$
$89$ $$-45796564224 - 593469600 T - 1149636 T^{2} + 344 T^{3} + T^{4}$$
$97$ $$79407506004 - 1902403156 T - 2800699 T^{2} + 622 T^{3} + T^{4}$$