# Properties

 Label 48.5.e.b Level 48 Weight 5 Character orbit 48.e Analytic conductor 4.962 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 48.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.96175822802$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: no (minimal twist has level 6) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 6\sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 3 + \beta ) q^{3} + 2 \beta q^{5} -26 q^{7} + ( -63 + 6 \beta ) q^{9} +O(q^{10})$$ $$q + ( 3 + \beta ) q^{3} + 2 \beta q^{5} -26 q^{7} + ( -63 + 6 \beta ) q^{9} + 14 \beta q^{11} + 50 q^{13} + ( -144 + 6 \beta ) q^{15} + 24 \beta q^{17} + 358 q^{19} + ( -78 - 26 \beta ) q^{21} -44 \beta q^{23} + 337 q^{25} + ( -621 - 45 \beta ) q^{27} -170 \beta q^{29} + 742 q^{31} + ( -1008 + 42 \beta ) q^{33} -52 \beta q^{35} + 1874 q^{37} + ( 150 + 50 \beta ) q^{39} + 284 \beta q^{41} + 262 q^{43} + ( -864 - 126 \beta ) q^{45} + 200 \beta q^{47} -1725 q^{49} + ( -1728 + 72 \beta ) q^{51} -54 \beta q^{53} -2016 q^{55} + ( 1074 + 358 \beta ) q^{57} + 214 \beta q^{59} -1486 q^{61} + ( 1638 - 156 \beta ) q^{63} + 100 \beta q^{65} + 4486 q^{67} + ( 3168 - 132 \beta ) q^{69} -420 \beta q^{71} + 290 q^{73} + ( 1011 + 337 \beta ) q^{75} -364 \beta q^{77} -9818 q^{79} + ( 1377 - 756 \beta ) q^{81} -838 \beta q^{83} -3456 q^{85} + ( 12240 - 510 \beta ) q^{87} -924 \beta q^{89} -1300 q^{91} + ( 2226 + 742 \beta ) q^{93} + 716 \beta q^{95} -478 q^{97} + ( -6048 - 882 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 6q^{3} - 52q^{7} - 126q^{9} + O(q^{10})$$ $$2q + 6q^{3} - 52q^{7} - 126q^{9} + 100q^{13} - 288q^{15} + 716q^{19} - 156q^{21} + 674q^{25} - 1242q^{27} + 1484q^{31} - 2016q^{33} + 3748q^{37} + 300q^{39} + 524q^{43} - 1728q^{45} - 3450q^{49} - 3456q^{51} - 4032q^{55} + 2148q^{57} - 2972q^{61} + 3276q^{63} + 8972q^{67} + 6336q^{69} + 580q^{73} + 2022q^{75} - 19636q^{79} + 2754q^{81} - 6912q^{85} + 24480q^{87} - 2600q^{91} + 4452q^{93} - 956q^{97} - 12096q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$$.

 $$n$$ $$17$$ $$31$$ $$37$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 − 1.41421i 1.41421i
0 3.00000 8.48528i 0 16.9706i 0 −26.0000 0 −63.0000 50.9117i 0
17.2 0 3.00000 + 8.48528i 0 16.9706i 0 −26.0000 0 −63.0000 + 50.9117i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.5.e.b 2
3.b odd 2 1 inner 48.5.e.b 2
4.b odd 2 1 6.5.b.a 2
8.b even 2 1 192.5.e.c 2
8.d odd 2 1 192.5.e.d 2
12.b even 2 1 6.5.b.a 2
20.d odd 2 1 150.5.d.a 2
20.e even 4 2 150.5.b.a 4
24.f even 2 1 192.5.e.d 2
24.h odd 2 1 192.5.e.c 2
28.d even 2 1 294.5.b.a 2
36.f odd 6 2 162.5.d.a 4
36.h even 6 2 162.5.d.a 4
60.h even 2 1 150.5.d.a 2
60.l odd 4 2 150.5.b.a 4
84.h odd 2 1 294.5.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.5.b.a 2 4.b odd 2 1
6.5.b.a 2 12.b even 2 1
48.5.e.b 2 1.a even 1 1 trivial
48.5.e.b 2 3.b odd 2 1 inner
150.5.b.a 4 20.e even 4 2
150.5.b.a 4 60.l odd 4 2
150.5.d.a 2 20.d odd 2 1
150.5.d.a 2 60.h even 2 1
162.5.d.a 4 36.f odd 6 2
162.5.d.a 4 36.h even 6 2
192.5.e.c 2 8.b even 2 1
192.5.e.c 2 24.h odd 2 1
192.5.e.d 2 8.d odd 2 1
192.5.e.d 2 24.f even 2 1
294.5.b.a 2 28.d even 2 1
294.5.b.a 2 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 288$$ acting on $$S_{5}^{\mathrm{new}}(48, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 6 T + 81 T^{2}$$
$5$ $$1 - 962 T^{2} + 390625 T^{4}$$
$7$ $$( 1 + 26 T + 2401 T^{2} )^{2}$$
$11$ $$1 - 15170 T^{2} + 214358881 T^{4}$$
$13$ $$( 1 - 50 T + 28561 T^{2} )^{2}$$
$17$ $$1 - 125570 T^{2} + 6975757441 T^{4}$$
$19$ $$( 1 - 358 T + 130321 T^{2} )^{2}$$
$23$ $$1 - 420290 T^{2} + 78310985281 T^{4}$$
$29$ $$1 + 666238 T^{2} + 500246412961 T^{4}$$
$31$ $$( 1 - 742 T + 923521 T^{2} )^{2}$$
$37$ $$( 1 - 1874 T + 1874161 T^{2} )^{2}$$
$41$ $$1 + 155710 T^{2} + 7984925229121 T^{4}$$
$43$ $$( 1 - 262 T + 3418801 T^{2} )^{2}$$
$47$ $$1 - 6879362 T^{2} + 23811286661761 T^{4}$$
$53$ $$1 - 15571010 T^{2} + 62259690411361 T^{4}$$
$59$ $$1 - 20937410 T^{2} + 146830437604321 T^{4}$$
$61$ $$( 1 + 1486 T + 13845841 T^{2} )^{2}$$
$67$ $$( 1 - 4486 T + 20151121 T^{2} )^{2}$$
$71$ $$1 - 38122562 T^{2} + 645753531245761 T^{4}$$
$73$ $$( 1 - 290 T + 28398241 T^{2} )^{2}$$
$79$ $$( 1 + 9818 T + 38950081 T^{2} )^{2}$$
$83$ $$1 - 44355074 T^{2} + 2252292232139041 T^{4}$$
$89$ $$1 - 64012610 T^{2} + 3936588805702081 T^{4}$$
$97$ $$( 1 + 478 T + 88529281 T^{2} )^{2}$$