# Properties

 Label 5376.2.a.bl Level 5376 Weight 2 Character orbit 5376.a Self dual yes Analytic conductor 42.928 Analytic rank 1 Dimension 4 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$5376 = 2^{8} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 5376.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$42.9275761266$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.19664.1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{3} + ( -1 - \beta_{3} ) q^{5} + q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + ( -1 - \beta_{3} ) q^{5} + q^{7} + q^{9} + ( -2 - \beta_{2} ) q^{11} + ( -1 + \beta_{1} ) q^{13} + ( 1 + \beta_{3} ) q^{15} + ( \beta_{1} - \beta_{3} ) q^{17} + ( -1 + \beta_{2} + \beta_{3} ) q^{19} - q^{21} + ( 2 - \beta_{1} + \beta_{3} ) q^{23} + ( 4 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{25} - q^{27} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{29} + ( -1 - \beta_{2} + \beta_{3} ) q^{31} + ( 2 + \beta_{2} ) q^{33} + ( -1 - \beta_{3} ) q^{35} + ( -1 - \beta_{2} + \beta_{3} ) q^{37} + ( 1 - \beta_{1} ) q^{39} + ( \beta_{1} - \beta_{3} ) q^{41} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{43} + ( -1 - \beta_{3} ) q^{45} + ( 2 + 2 \beta_{2} + 2 \beta_{3} ) q^{47} + q^{49} + ( -\beta_{1} + \beta_{3} ) q^{51} + ( 1 - \beta_{2} - \beta_{3} ) q^{53} + ( 1 + \beta_{2} + 3 \beta_{3} ) q^{55} + ( 1 - \beta_{2} - \beta_{3} ) q^{57} -4 q^{59} + ( 1 + \beta_{1} + 2 \beta_{3} ) q^{61} + q^{63} + 4 \beta_{3} q^{65} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{67} + ( -2 + \beta_{1} - \beta_{3} ) q^{69} + ( -4 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{71} + ( 2 + 2 \beta_{1} + 2 \beta_{3} ) q^{73} + ( -4 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{75} + ( -2 - \beta_{2} ) q^{77} + ( 4 + 2 \beta_{1} - 2 \beta_{3} ) q^{79} + q^{81} + ( -8 - 2 \beta_{2} ) q^{83} + ( 7 - 2 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{85} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{87} + ( -4 - \beta_{1} - 3 \beta_{3} ) q^{89} + ( -1 + \beta_{1} ) q^{91} + ( 1 + \beta_{2} - \beta_{3} ) q^{93} + ( -6 + 2 \beta_{1} - 2 \beta_{2} ) q^{95} + ( 4 - 2 \beta_{1} - 2 \beta_{2} ) q^{97} + ( -2 - \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{3} - 2q^{5} + 4q^{7} + 4q^{9} + O(q^{10})$$ $$4q - 4q^{3} - 2q^{5} + 4q^{7} + 4q^{9} - 6q^{11} - 4q^{13} + 2q^{15} + 2q^{17} - 8q^{19} - 4q^{21} + 6q^{23} + 12q^{25} - 4q^{27} - 4q^{31} + 6q^{33} - 2q^{35} - 4q^{37} + 4q^{39} + 2q^{41} - 8q^{43} - 2q^{45} + 4q^{49} - 2q^{51} + 8q^{53} - 4q^{55} + 8q^{57} - 16q^{59} + 4q^{63} - 8q^{65} - 12q^{67} - 6q^{69} - 14q^{71} + 4q^{73} - 12q^{75} - 6q^{77} + 20q^{79} + 4q^{81} - 28q^{83} + 20q^{85} - 10q^{89} - 4q^{91} + 4q^{93} - 20q^{95} + 20q^{97} - 6q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - 5 x^{2} + 2 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu - 1$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - \nu^{2} - 6 \nu - 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 3 \nu^{2} - 2 \nu + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + \beta_{2} + 2 \beta_{1} + 7$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{3} + 3 \beta_{2} + 8 \beta_{1} + 17$$$$)/2$$

## 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
 −0.512486 0.771876 3.28139 −1.54078
0 −1.00000 0 −4.10245 0 1.00000 0 1.00000 0
1.2 0 −1.00000 0 −1.12875 0 1.00000 0 1.00000 0
1.3 0 −1.00000 0 −0.467138 0 1.00000 0 1.00000 0
1.4 0 −1.00000 0 3.69833 0 1.00000 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 5376.2.a.bl 4
4.b odd 2 1 5376.2.a.bp 4
8.b even 2 1 5376.2.a.bq 4
8.d odd 2 1 5376.2.a.bm 4
16.e even 4 2 672.2.c.b 8
16.f odd 4 2 168.2.c.b 8
48.i odd 4 2 2016.2.c.e 8
48.k even 4 2 504.2.c.f 8
112.j even 4 2 1176.2.c.c 8
112.l odd 4 2 4704.2.c.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.c.b 8 16.f odd 4 2
504.2.c.f 8 48.k even 4 2
672.2.c.b 8 16.e even 4 2
1176.2.c.c 8 112.j even 4 2
2016.2.c.e 8 48.i odd 4 2
4704.2.c.c 8 112.l odd 4 2
5376.2.a.bl 4 1.a even 1 1 trivial
5376.2.a.bm 4 8.d odd 2 1
5376.2.a.bp 4 4.b odd 2 1
5376.2.a.bq 4 8.b even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5376))$$:

 $$T_{5}^{4} + 2 T_{5}^{3} - 14 T_{5}^{2} - 24 T_{5} - 8$$ $$T_{11}^{4} + 6 T_{11}^{3} - 14 T_{11}^{2} - 136 T_{11} - 200$$ $$T_{13}^{4} + 4 T_{13}^{3} - 20 T_{13}^{2} - 80 T_{13} - 32$$ $$T_{19}^{4} + 8 T_{19}^{3} - 12 T_{19}^{2} - 96 T_{19} + 128$$ $$T_{29}^{4} - 108 T_{29}^{2} - 32 T_{29} + 2624$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + T )^{4}$$
$5$ $$1 + 2 T + 6 T^{2} + 6 T^{3} + 2 T^{4} + 30 T^{5} + 150 T^{6} + 250 T^{7} + 625 T^{8}$$
$7$ $$( 1 - T )^{4}$$
$11$ $$1 + 6 T + 30 T^{2} + 62 T^{3} + 218 T^{4} + 682 T^{5} + 3630 T^{6} + 7986 T^{7} + 14641 T^{8}$$
$13$ $$1 + 4 T + 32 T^{2} + 76 T^{3} + 462 T^{4} + 988 T^{5} + 5408 T^{6} + 8788 T^{7} + 28561 T^{8}$$
$17$ $$1 - 2 T + 38 T^{2} - 70 T^{3} + 706 T^{4} - 1190 T^{5} + 10982 T^{6} - 9826 T^{7} + 83521 T^{8}$$
$19$ $$1 + 8 T + 64 T^{2} + 360 T^{3} + 1838 T^{4} + 6840 T^{5} + 23104 T^{6} + 54872 T^{7} + 130321 T^{8}$$
$23$ $$1 - 6 T + 74 T^{2} - 334 T^{3} + 2282 T^{4} - 7682 T^{5} + 39146 T^{6} - 73002 T^{7} + 279841 T^{8}$$
$29$ $$1 + 8 T^{2} - 32 T^{3} + 1406 T^{4} - 928 T^{5} + 6728 T^{6} + 707281 T^{8}$$
$31$ $$1 + 4 T + 80 T^{2} + 244 T^{3} + 3294 T^{4} + 7564 T^{5} + 76880 T^{6} + 119164 T^{7} + 923521 T^{8}$$
$37$ $$1 + 4 T + 104 T^{2} + 316 T^{3} + 5214 T^{4} + 11692 T^{5} + 142376 T^{6} + 202612 T^{7} + 1874161 T^{8}$$
$41$ $$1 - 2 T + 134 T^{2} - 214 T^{3} + 7618 T^{4} - 8774 T^{5} + 225254 T^{6} - 137842 T^{7} + 2825761 T^{8}$$
$43$ $$1 + 8 T + 116 T^{2} + 552 T^{3} + 5526 T^{4} + 23736 T^{5} + 214484 T^{6} + 636056 T^{7} + 3418801 T^{8}$$
$47$ $$1 + 44 T^{2} + 128 T^{3} + 3302 T^{4} + 6016 T^{5} + 97196 T^{6} + 4879681 T^{8}$$
$53$ $$1 - 8 T + 200 T^{2} - 1176 T^{3} + 15710 T^{4} - 62328 T^{5} + 561800 T^{6} - 1191016 T^{7} + 7890481 T^{8}$$
$59$ $$( 1 + 4 T + 59 T^{2} )^{4}$$
$61$ $$1 + 136 T^{2} + 544 T^{3} + 8510 T^{4} + 33184 T^{5} + 506056 T^{6} + 13845841 T^{8}$$
$67$ $$1 + 12 T + 244 T^{2} + 1948 T^{3} + 23606 T^{4} + 130516 T^{5} + 1095316 T^{6} + 3609156 T^{7} + 20151121 T^{8}$$
$71$ $$1 + 14 T + 194 T^{2} + 1686 T^{3} + 14330 T^{4} + 119706 T^{5} + 977954 T^{6} + 5010754 T^{7} + 25411681 T^{8}$$
$73$ $$1 - 4 T + 92 T^{2} + 580 T^{3} + 166 T^{4} + 42340 T^{5} + 490268 T^{6} - 1556068 T^{7} + 28398241 T^{8}$$
$79$ $$1 - 20 T + 340 T^{2} - 3972 T^{3} + 38678 T^{4} - 313788 T^{5} + 2121940 T^{6} - 9860780 T^{7} + 38950081 T^{8}$$
$83$ $$1 + 28 T + 516 T^{2} + 6268 T^{3} + 64454 T^{4} + 520244 T^{5} + 3554724 T^{6} + 16010036 T^{7} + 47458321 T^{8}$$
$89$ $$1 + 10 T + 198 T^{2} + 494 T^{3} + 13122 T^{4} + 43966 T^{5} + 1568358 T^{6} + 7049690 T^{7} + 62742241 T^{8}$$
$97$ $$1 - 20 T + 300 T^{2} - 2572 T^{3} + 25574 T^{4} - 249484 T^{5} + 2822700 T^{6} - 18253460 T^{7} + 88529281 T^{8}$$