Properties

Label 48.6.c.b
Level 48
Weight 6
Character orbit 48.c
Analytic conductor 7.698
Analytic rank 0
Dimension 2
CM discriminant -3
Inner twists 4

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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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 9 - 18 \zeta_{6} ) q^{3} + ( 62 - 124 \zeta_{6} ) q^{7} -243 q^{9} +O(q^{10})\) \( q + ( 9 - 18 \zeta_{6} ) q^{3} + ( 62 - 124 \zeta_{6} ) q^{7} -243 q^{9} -1202 q^{13} + ( 1618 - 3236 \zeta_{6} ) q^{19} -1674 q^{21} + 3125 q^{25} + ( -2187 + 4374 \zeta_{6} ) q^{27} + ( -1626 + 3252 \zeta_{6} ) q^{31} + 16550 q^{37} + ( -10818 + 21636 \zeta_{6} ) q^{39} + ( 13866 - 27732 \zeta_{6} ) q^{43} + 5275 q^{49} -43686 q^{57} -38626 q^{61} + ( -15066 + 30132 \zeta_{6} ) q^{63} + ( 37138 - 74276 \zeta_{6} ) q^{67} + 1450 q^{73} + ( 28125 - 56250 \zeta_{6} ) q^{75} + ( -27050 + 54100 \zeta_{6} ) q^{79} + 59049 q^{81} + ( -74524 + 149048 \zeta_{6} ) q^{91} + 43902 q^{93} + 134386 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 486q^{9} + O(q^{10}) \) \( 2q - 486q^{9} - 2404q^{13} - 3348q^{21} + 6250q^{25} + 33100q^{37} + 10550q^{49} - 87372q^{57} - 77252q^{61} + 2900q^{73} + 118098q^{81} + 87804q^{93} + 268772q^{97} + O(q^{100}) \)

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 + 0.866025i
0.500000 0.866025i
0 15.5885i 0 0 0 107.387i 0 −243.000 0
47.2 0 15.5885i 0 0 0 107.387i 0 −243.000 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
4.b odd 2 1 inner
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.b 2
3.b odd 2 1 CM 48.6.c.b 2
4.b odd 2 1 inner 48.6.c.b 2
8.b even 2 1 192.6.c.b 2
8.d odd 2 1 192.6.c.b 2
12.b even 2 1 inner 48.6.c.b 2
24.f even 2 1 192.6.c.b 2
24.h odd 2 1 192.6.c.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.6.c.b 2 1.a even 1 1 trivial
48.6.c.b 2 3.b odd 2 1 CM
48.6.c.b 2 4.b odd 2 1 inner
48.6.c.b 2 12.b even 2 1 inner
192.6.c.b 2 8.b even 2 1
192.6.c.b 2 8.d odd 2 1
192.6.c.b 2 24.f even 2 1
192.6.c.b 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 \(S_{6}^{\mathrm{new}}(48, [\chi])\):

\( T_{5} \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 243 T^{2} \)
$5$ \( ( 1 - 3125 T^{2} )^{2} \)
$7$ \( ( 1 - 236 T + 16807 T^{2} )( 1 + 236 T + 16807 T^{2} ) \)
$11$ \( ( 1 + 161051 T^{2} )^{2} \)
$13$ \( ( 1 + 1202 T + 371293 T^{2} )^{2} \)
$17$ \( ( 1 - 1419857 T^{2} )^{2} \)
$19$ \( ( 1 - 1432 T + 2476099 T^{2} )( 1 + 1432 T + 2476099 T^{2} ) \)
$23$ \( ( 1 + 6436343 T^{2} )^{2} \)
$29$ \( ( 1 - 20511149 T^{2} )^{2} \)
$31$ \( ( 1 - 10324 T + 28629151 T^{2} )( 1 + 10324 T + 28629151 T^{2} ) \)
$37$ \( ( 1 - 16550 T + 69343957 T^{2} )^{2} \)
$41$ \( ( 1 - 115856201 T^{2} )^{2} \)
$43$ \( ( 1 - 3352 T + 147008443 T^{2} )( 1 + 3352 T + 147008443 T^{2} ) \)
$47$ \( ( 1 + 229345007 T^{2} )^{2} \)
$53$ \( ( 1 - 418195493 T^{2} )^{2} \)
$59$ \( ( 1 + 714924299 T^{2} )^{2} \)
$61$ \( ( 1 + 38626 T + 844596301 T^{2} )^{2} \)
$67$ \( ( 1 - 35536 T + 1350125107 T^{2} )( 1 + 35536 T + 1350125107 T^{2} ) \)
$71$ \( ( 1 + 1804229351 T^{2} )^{2} \)
$73$ \( ( 1 - 1450 T + 2073071593 T^{2} )^{2} \)
$79$ \( ( 1 - 100564 T + 3077056399 T^{2} )( 1 + 100564 T + 3077056399 T^{2} ) \)
$83$ \( ( 1 + 3939040643 T^{2} )^{2} \)
$89$ \( ( 1 - 5584059449 T^{2} )^{2} \)
$97$ \( ( 1 - 134386 T + 8587340257 T^{2} )^{2} \)
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