Properties

Label 48.6.c.a
Level 48
Weight 6
Character orbit 48.c
Analytic conductor 7.698
Analytic rank 1
Dimension 2
CM no
Inner twists 2

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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: \(1\)
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
0.500000 + 1.65831i
0.500000 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:

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.a 2
3.b odd 2 1 48.6.c.c yes 2
4.b odd 2 1 48.6.c.c yes 2
8.b even 2 1 192.6.c.c 2
8.d odd 2 1 192.6.c.a 2
12.b even 2 1 inner 48.6.c.a 2
24.f even 2 1 192.6.c.c 2
24.h odd 2 1 192.6.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.6.c.a 2 1.a even 1 1 trivial
48.6.c.a 2 12.b even 2 1 inner
48.6.c.c yes 2 3.b odd 2 1
48.6.c.c yes 2 4.b odd 2 1
192.6.c.a 2 8.d odd 2 1
192.6.c.a 2 24.h odd 2 1
192.6.c.c 2 8.b even 2 1
192.6.c.c 2 24.f even 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
$3$ \( 1 + 24 T + 243 T^{2} \)
$5$ \( 1 + 86 T^{2} + 9765625 T^{4} \)
$7$ \( 1 - 1538 T^{2} + 282475249 T^{4} \)
$11$ \( ( 1 + 648 T + 161051 T^{2} )^{2} \)
$13$ \( ( 1 + 242 T + 371293 T^{2} )^{2} \)
$17$ \( 1 - 2738338 T^{2} + 2015993900449 T^{4} \)
$19$ \( 1 - 3380474 T^{2} + 6131066257801 T^{4} \)
$23$ \( ( 1 + 1296 T + 6436343 T^{2} )^{2} \)
$29$ \( 1 - 37062298 T^{2} + 420707233300201 T^{4} \)
$31$ \( 1 - 45678866 T^{2} + 819628286980801 T^{4} \)
$37$ \( ( 1 + 12058 T + 69343957 T^{2} )^{2} \)
$41$ \( 1 - 12512146 T^{2} + 13422659310152401 T^{4} \)
$43$ \( 1 + 33190390 T^{2} + 21611482313284249 T^{4} \)
$47$ \( ( 1 + 12960 T + 229345007 T^{2} )^{2} \)
$53$ \( 1 - 108251530 T^{2} + 174887470365513049 T^{4} \)
$59$ \( ( 1 + 8424 T + 714924299 T^{2} )^{2} \)
$61$ \( ( 1 + 25762 T + 844596301 T^{2} )^{2} \)
$67$ \( 1 - 2596035290 T^{2} + 1822837804551761449 T^{4} \)
$71$ \( ( 1 + 55728 T + 1804229351 T^{2} )^{2} \)
$73$ \( ( 1 - 26026 T + 2073071593 T^{2} )^{2} \)
$79$ \( 1 - 5786874674 T^{2} + 9468276082626847201 T^{4} \)
$83$ \( ( 1 - 78408 T + 3939040643 T^{2} )^{2} \)
$89$ \( 1 - 4075828594 T^{2} + 31181719929966183601 T^{4} \)
$97$ \( ( 1 - 103090 T + 8587340257 T^{2} )^{2} \)
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