# Properties

 Label 48.6.c.c Level 48 Weight 6 Character orbit 48.c Analytic conductor 7.698 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 48.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.69842335102$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-11})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 3\sqrt{-11}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 12 + \beta ) q^{3} -8 \beta q^{5} -18 \beta q^{7} + ( 45 + 24 \beta ) q^{9} +O(q^{10})$$ $$q + ( 12 + \beta ) q^{3} -8 \beta q^{5} -18 \beta q^{7} + ( 45 + 24 \beta ) q^{9} + 648 q^{11} -242 q^{13} + ( 792 - 96 \beta ) q^{15} + 32 \beta q^{17} -126 \beta q^{19} + ( 1782 - 216 \beta ) q^{21} + 1296 q^{23} -3211 q^{25} + ( -1836 + 333 \beta ) q^{27} + 200 \beta q^{29} + 342 \beta q^{31} + ( 7776 + 648 \beta ) q^{33} -14256 q^{35} -12058 q^{37} + ( -2904 - 242 \beta ) q^{39} -1488 \beta q^{41} + 1818 \beta q^{43} + ( 19008 - 360 \beta ) q^{45} + 12960 q^{47} -15269 q^{49} + ( -3168 + 384 \beta ) q^{51} + 2712 \beta q^{53} -5184 \beta q^{55} + ( 12474 - 1512 \beta ) q^{57} + 8424 q^{59} -25762 q^{61} + ( 42768 - 810 \beta ) q^{63} + 1936 \beta q^{65} + 1026 \beta q^{67} + ( 15552 + 1296 \beta ) q^{69} + 55728 q^{71} + 26026 q^{73} + ( -38532 - 3211 \beta ) q^{75} -11664 \beta q^{77} + 1926 \beta q^{79} + ( -54999 + 2160 \beta ) q^{81} -78408 q^{83} + 25344 q^{85} + ( -19800 + 2400 \beta ) q^{87} + 8464 \beta q^{89} + 4356 \beta q^{91} + ( -33858 + 4104 \beta ) q^{93} -99792 q^{95} + 103090 q^{97} + ( 29160 + 15552 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 24q^{3} + 90q^{9} + O(q^{10})$$ $$2q + 24q^{3} + 90q^{9} + 1296q^{11} - 484q^{13} + 1584q^{15} + 3564q^{21} + 2592q^{23} - 6422q^{25} - 3672q^{27} + 15552q^{33} - 28512q^{35} - 24116q^{37} - 5808q^{39} + 38016q^{45} + 25920q^{47} - 30538q^{49} - 6336q^{51} + 24948q^{57} + 16848q^{59} - 51524q^{61} + 85536q^{63} + 31104q^{69} + 111456q^{71} + 52052q^{73} - 77064q^{75} - 109998q^{81} - 156816q^{83} + 50688q^{85} - 39600q^{87} - 67716q^{93} - 199584q^{95} + 206180q^{97} + 58320q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1
 0.5 − 1.65831i 0.5 + 1.65831i
0 12.0000 9.94987i 0 79.5990i 0 179.098i 0 45.0000 238.797i 0
47.2 0 12.0000 + 9.94987i 0 79.5990i 0 179.098i 0 45.0000 + 238.797i 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
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.6.c.c yes 2
3.b odd 2 1 48.6.c.a 2
4.b odd 2 1 48.6.c.a 2
8.b even 2 1 192.6.c.a 2
8.d odd 2 1 192.6.c.c 2
12.b even 2 1 inner 48.6.c.c yes 2
24.f even 2 1 192.6.c.a 2
24.h odd 2 1 192.6.c.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.6.c.a 2 3.b odd 2 1
48.6.c.a 2 4.b odd 2 1
48.6.c.c yes 2 1.a even 1 1 trivial
48.6.c.c yes 2 12.b even 2 1 inner
192.6.c.a 2 8.b even 2 1
192.6.c.a 2 24.f even 2 1
192.6.c.c 2 8.d odd 2 1
192.6.c.c 2 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(48, [\chi])$$:

 $$T_{5}^{2} + 6336$$ $$T_{11} - 648$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 24 T + 243 T^{2}$$
$5$ $$1 + 86 T^{2} + 9765625 T^{4}$$
$7$ $$1 - 1538 T^{2} + 282475249 T^{4}$$
$11$ $$( 1 - 648 T + 161051 T^{2} )^{2}$$
$13$ $$( 1 + 242 T + 371293 T^{2} )^{2}$$
$17$ $$1 - 2738338 T^{2} + 2015993900449 T^{4}$$
$19$ $$1 - 3380474 T^{2} + 6131066257801 T^{4}$$
$23$ $$( 1 - 1296 T + 6436343 T^{2} )^{2}$$
$29$ $$1 - 37062298 T^{2} + 420707233300201 T^{4}$$
$31$ $$1 - 45678866 T^{2} + 819628286980801 T^{4}$$
$37$ $$( 1 + 12058 T + 69343957 T^{2} )^{2}$$
$41$ $$1 - 12512146 T^{2} + 13422659310152401 T^{4}$$
$43$ $$1 + 33190390 T^{2} + 21611482313284249 T^{4}$$
$47$ $$( 1 - 12960 T + 229345007 T^{2} )^{2}$$
$53$ $$1 - 108251530 T^{2} + 174887470365513049 T^{4}$$
$59$ $$( 1 - 8424 T + 714924299 T^{2} )^{2}$$
$61$ $$( 1 + 25762 T + 844596301 T^{2} )^{2}$$
$67$ $$1 - 2596035290 T^{2} + 1822837804551761449 T^{4}$$
$71$ $$( 1 - 55728 T + 1804229351 T^{2} )^{2}$$
$73$ $$( 1 - 26026 T + 2073071593 T^{2} )^{2}$$
$79$ $$1 - 5786874674 T^{2} + 9468276082626847201 T^{4}$$
$83$ $$( 1 + 78408 T + 3939040643 T^{2} )^{2}$$
$89$ $$1 - 4075828594 T^{2} + 31181719929966183601 T^{4}$$
$97$ $$( 1 - 103090 T + 8587340257 T^{2} )^{2}$$