# Properties

 Label 2352.4.a.cq 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: 4.4.136768.1 Defining polynomial: $$x^{4} - 2 x^{3} - 23 x^{2} + 18 x + 119$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}\cdot 7$$ Twist minimal: no (minimal twist has level 588) 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.

 $$f(q)$$ $$=$$ $$q + 3 q^{3} -\beta_{1} q^{5} + 9 q^{9} +O(q^{10})$$ $$q + 3 q^{3} -\beta_{1} q^{5} + 9 q^{9} + ( -3 \beta_{1} + \beta_{3} ) q^{11} + 3 \beta_{3} q^{13} -3 \beta_{1} q^{15} + ( -12 - \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{17} + ( 48 - 6 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} ) q^{19} + ( -48 - 3 \beta_{1} - 6 \beta_{2} - \beta_{3} ) q^{23} + ( 81 - 6 \beta_{2} + 6 \beta_{3} ) q^{25} + 27 q^{27} + ( 24 - 9 \beta_{2} + \beta_{3} ) q^{29} + ( 12 - 6 \beta_{1} + 9 \beta_{2} - 3 \beta_{3} ) q^{31} + ( -9 \beta_{1} + 3 \beta_{3} ) q^{33} + ( 64 - 6 \beta_{1} + 24 \beta_{2} - 6 \beta_{3} ) q^{37} + 9 \beta_{3} q^{39} + ( -252 - \beta_{1} + 15 \beta_{2} - 3 \beta_{3} ) q^{41} + ( 28 + 6 \beta_{1} + 12 \beta_{2} + 6 \beta_{3} ) q^{43} -9 \beta_{1} q^{45} + ( 216 + 10 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{47} + ( -36 - 3 \beta_{1} - 9 \beta_{2} - 9 \beta_{3} ) q^{51} + ( -162 - 24 \beta_{1} - 18 \beta_{2} - 10 \beta_{3} ) q^{53} + ( 588 - 6 \beta_{1} + 5 \beta_{2} + 21 \beta_{3} ) q^{55} + ( 144 - 18 \beta_{1} - 15 \beta_{2} + 9 \beta_{3} ) q^{57} + ( 84 + 10 \beta_{1} + 15 \beta_{2} - 21 \beta_{3} ) q^{59} + ( -240 + 30 \beta_{1} - 14 \beta_{2} + 15 \beta_{3} ) q^{61} + ( -90 - 18 \beta_{1} + 69 \beta_{2} + 9 \beta_{3} ) q^{65} + ( -180 + 6 \beta_{1} + 48 \beta_{2} - 30 \beta_{3} ) q^{67} + ( -144 - 9 \beta_{1} - 18 \beta_{2} - 3 \beta_{3} ) q^{69} + ( 336 - 3 \beta_{1} + 30 \beta_{2} - 9 \beta_{3} ) q^{71} + ( -168 + 30 \beta_{1} + 49 \beta_{2} - 18 \beta_{3} ) q^{73} + ( 243 - 18 \beta_{2} + 18 \beta_{3} ) q^{75} + ( 496 - 36 \beta_{1} - 96 \beta_{2} + 24 \beta_{3} ) q^{79} + 81 q^{81} + ( 780 + 8 \beta_{1} - 18 \beta_{2} - 18 \beta_{3} ) q^{83} + ( 170 + 30 \beta_{1} - 66 \beta_{2} - 24 \beta_{3} ) q^{85} + ( 72 - 27 \beta_{2} + 3 \beta_{3} ) q^{87} + ( -540 - 11 \beta_{1} - 69 \beta_{2} - 15 \beta_{3} ) q^{89} + ( 36 - 18 \beta_{1} + 27 \beta_{2} - 9 \beta_{3} ) q^{93} + ( 936 - 66 \beta_{1} + 48 \beta_{2} + 10 \beta_{3} ) q^{95} + ( -504 + 18 \beta_{1} - 49 \beta_{2} + 48 \beta_{3} ) q^{97} + ( -27 \beta_{1} + 9 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 12q^{3} + 36q^{9} + O(q^{10})$$ $$4q + 12q^{3} + 36q^{9} - 48q^{17} + 192q^{19} - 192q^{23} + 324q^{25} + 108q^{27} + 96q^{29} + 48q^{31} + 256q^{37} - 1008q^{41} + 112q^{43} + 864q^{47} - 144q^{51} - 648q^{53} + 2352q^{55} + 576q^{57} + 336q^{59} - 960q^{61} - 360q^{65} - 720q^{67} - 576q^{69} + 1344q^{71} - 672q^{73} + 972q^{75} + 1984q^{79} + 324q^{81} + 3120q^{83} + 680q^{85} + 288q^{87} - 2160q^{89} + 144q^{93} + 3744q^{95} - 2016q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - 23 x^{2} + 18 x + 119$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} - 3 \nu^{2} + 25 \nu + 2$$$$)/9$$ $$\beta_{2}$$ $$=$$ $$($$$$-7 \nu^{3} + 21 \nu^{2} + 77 \nu - 140$$$$)/9$$ $$\beta_{3}$$ $$=$$ $$-\nu^{3} + 7 \nu^{2} + 7 \nu - 68$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 7 \beta_{1} + 14$$$$)/28$$ $$\nu^{2}$$ $$=$$ $$($$$$7 \beta_{3} - 8 \beta_{2} + 7 \beta_{1} + 350$$$$)/28$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{3} - 7 \beta_{2} + 14 \beta_{1} + 92$$$$)/4$$

## 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
 4.93153 2.89590 −2.51732 −3.31012
0 3.00000 0 −19.1403 0 0 0 9.00000 0
1.2 0 3.00000 0 −8.16940 0 0 0 9.00000 0
1.3 0 3.00000 0 10.6550 0 0 0 9.00000 0
1.4 0 3.00000 0 16.6547 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.cq 4
4.b odd 2 1 588.4.a.j 4
7.b odd 2 1 2352.4.a.cl 4
12.b even 2 1 1764.4.a.bc 4
28.d even 2 1 588.4.a.k yes 4
28.f even 6 2 588.4.i.k 8
28.g odd 6 2 588.4.i.l 8
84.h odd 2 1 1764.4.a.ba 4
84.j odd 6 2 1764.4.k.bd 8
84.n even 6 2 1764.4.k.bb 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.4.a.j 4 4.b odd 2 1
588.4.a.k yes 4 28.d even 2 1
588.4.i.k 8 28.f even 6 2
588.4.i.l 8 28.g odd 6 2
1764.4.a.ba 4 84.h odd 2 1
1764.4.a.bc 4 12.b even 2 1
1764.4.k.bb 8 84.n even 6 2
1764.4.k.bd 8 84.j odd 6 2
2352.4.a.cl 4 7.b odd 2 1
2352.4.a.cq 4 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2352))$$:

 $$T_{5}^{4} - 412 T_{5}^{2} + 576 T_{5} + 27748$$ $$T_{11}^{4} - 4136 T_{11}^{2} - 82944 T_{11} + 733072$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -3 + T )^{4}$$
$5$ $$27748 + 576 T - 412 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$733072 - 82944 T - 4136 T^{2} + T^{4}$$
$13$ $$10008036 + 31104 T - 7092 T^{2} + T^{4}$$
$17$ $$-4865084 - 590496 T - 9772 T^{2} + 48 T^{3} + T^{4}$$
$19$ $$-41971136 + 1549824 T - 3280 T^{2} - 192 T^{3} + T^{4}$$
$23$ $$-40554608 - 908544 T + 3928 T^{2} + 192 T^{3} + T^{4}$$
$29$ $$38719552 + 648960 T - 11696 T^{2} - 96 T^{3} + T^{4}$$
$31$ $$-189895104 + 5705856 T - 43632 T^{2} - 48 T^{3} + T^{4}$$
$37$ $$-1479272192 + 41160704 T - 135840 T^{2} - 256 T^{3} + T^{4}$$
$41$ $$1611829828 + 41616864 T + 334100 T^{2} + 1008 T^{3} + T^{4}$$
$43$ $$-789373952 + 14991872 T - 71040 T^{2} - 112 T^{3} + T^{4}$$
$47$ $$-1168478144 - 13356288 T + 226928 T^{2} - 864 T^{3} + T^{4}$$
$53$ $$-20504773616 - 174760416 T - 208616 T^{2} + 648 T^{3} + T^{4}$$
$59$ $$18986185792 + 47665536 T - 287152 T^{2} - 336 T^{3} + T^{4}$$
$61$ $$-106656271196 - 499716480 T - 330196 T^{2} + 960 T^{3} + T^{4}$$
$67$ $$14336621568 - 330683904 T - 669312 T^{2} + 720 T^{3} + T^{4}$$
$71$ $$-17989567344 + 15945984 T + 460440 T^{2} - 1344 T^{3} + T^{4}$$
$73$ $$-90986816444 - 459082560 T - 467428 T^{2} + 672 T^{3} + T^{4}$$
$79$ $$-1013049875456 + 2268354560 T - 246720 T^{2} - 1984 T^{3} + T^{4}$$
$83$ $$74256064768 - 1203548928 T + 3243872 T^{2} - 3120 T^{3} + T^{4}$$
$89$ $$-311467391228 - 994684320 T + 523124 T^{2} + 2160 T^{3} + T^{4}$$
$97$ $$-580580611196 - 2679572160 T - 752260 T^{2} + 2016 T^{3} + T^{4}$$