Properties

Label 48.7.g.b
Level 48
Weight 7
Character orbit 48.g
Analytic conductor 11.043
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 48.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.0425960138\)
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: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 9 - 18 \zeta_{6} ) q^{3} + 6 q^{5} + ( 116 - 232 \zeta_{6} ) q^{7} -243 q^{9} +O(q^{10})\) \( q + ( 9 - 18 \zeta_{6} ) q^{3} + 6 q^{5} + ( 116 - 232 \zeta_{6} ) q^{7} -243 q^{9} + ( 372 - 744 \zeta_{6} ) q^{11} -2654 q^{13} + ( 54 - 108 \zeta_{6} ) q^{15} -7206 q^{17} + ( 2044 - 4088 \zeta_{6} ) q^{19} -3132 q^{21} + ( 10080 - 20160 \zeta_{6} ) q^{23} -15589 q^{25} + ( -2187 + 4374 \zeta_{6} ) q^{27} + 11550 q^{29} + ( -4932 + 9864 \zeta_{6} ) q^{31} -10044 q^{33} + ( 696 - 1392 \zeta_{6} ) q^{35} + 22346 q^{37} + ( -23886 + 47772 \zeta_{6} ) q^{39} + 103626 q^{41} + ( -73548 + 147096 \zeta_{6} ) q^{43} -1458 q^{45} + ( 92856 - 185712 \zeta_{6} ) q^{47} + 77281 q^{49} + ( -64854 + 129708 \zeta_{6} ) q^{51} + 168462 q^{53} + ( 2232 - 4464 \zeta_{6} ) q^{55} -55188 q^{57} + ( 64428 - 128856 \zeta_{6} ) q^{59} -260470 q^{61} + ( -28188 + 56376 \zeta_{6} ) q^{63} -15924 q^{65} + ( -183068 + 366136 \zeta_{6} ) q^{67} -272160 q^{69} + ( 409008 - 818016 \zeta_{6} ) q^{71} -395918 q^{73} + ( -140301 + 280602 \zeta_{6} ) q^{75} -129456 q^{77} + ( 321436 - 642872 \zeta_{6} ) q^{79} + 59049 q^{81} + ( 51468 - 102936 \zeta_{6} ) q^{83} -43236 q^{85} + ( 103950 - 207900 \zeta_{6} ) q^{87} -251886 q^{89} + ( -307864 + 615728 \zeta_{6} ) q^{91} + 133164 q^{93} + ( 12264 - 24528 \zeta_{6} ) q^{95} + 517474 q^{97} + ( -90396 + 180792 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 12q^{5} - 486q^{9} + O(q^{10}) \) \( 2q + 12q^{5} - 486q^{9} - 5308q^{13} - 14412q^{17} - 6264q^{21} - 31178q^{25} + 23100q^{29} - 20088q^{33} + 44692q^{37} + 207252q^{41} - 2916q^{45} + 154562q^{49} + 336924q^{53} - 110376q^{57} - 520940q^{61} - 31848q^{65} - 544320q^{69} - 791836q^{73} - 258912q^{77} + 118098q^{81} - 86472q^{85} - 503772q^{89} + 266328q^{93} + 1034948q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
0.500000 + 0.866025i
0.500000 0.866025i
0 15.5885i 0 6.00000 0 200.918i 0 −243.000 0
31.2 0 15.5885i 0 6.00000 0 200.918i 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
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.7.g.b 2
3.b odd 2 1 144.7.g.c 2
4.b odd 2 1 inner 48.7.g.b 2
8.b even 2 1 192.7.g.b 2
8.d odd 2 1 192.7.g.b 2
12.b even 2 1 144.7.g.c 2
16.e even 4 2 768.7.b.b 4
16.f odd 4 2 768.7.b.b 4
24.f even 2 1 576.7.g.g 2
24.h odd 2 1 576.7.g.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.7.g.b 2 1.a even 1 1 trivial
48.7.g.b 2 4.b odd 2 1 inner
144.7.g.c 2 3.b odd 2 1
144.7.g.c 2 12.b even 2 1
192.7.g.b 2 8.b even 2 1
192.7.g.b 2 8.d odd 2 1
576.7.g.g 2 24.f even 2 1
576.7.g.g 2 24.h odd 2 1
768.7.b.b 4 16.e even 4 2
768.7.b.b 4 16.f odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 6 \) acting on \(S_{7}^{\mathrm{new}}(48, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 243 T^{2} \)
$5$ \( ( 1 - 6 T + 15625 T^{2} )^{2} \)
$7$ \( 1 - 194930 T^{2} + 13841287201 T^{4} \)
$11$ \( 1 - 3127970 T^{2} + 3138428376721 T^{4} \)
$13$ \( ( 1 + 2654 T + 4826809 T^{2} )^{2} \)
$17$ \( ( 1 + 7206 T + 24137569 T^{2} )^{2} \)
$19$ \( 1 - 81557954 T^{2} + 2213314919066161 T^{4} \)
$23$ \( 1 + 8747422 T^{2} + 21914624432020321 T^{4} \)
$29$ \( ( 1 - 11550 T + 594823321 T^{2} )^{2} \)
$31$ \( 1 - 1702033490 T^{2} + 787662783788549761 T^{4} \)
$37$ \( ( 1 - 22346 T + 2565726409 T^{2} )^{2} \)
$41$ \( ( 1 - 103626 T + 4750104241 T^{2} )^{2} \)
$43$ \( 1 + 3585198814 T^{2} + 39959630797262576401 T^{4} \)
$47$ \( 1 + 4308279550 T^{2} + \)\(11\!\cdots\!41\)\( T^{4} \)
$53$ \( ( 1 - 168462 T + 22164361129 T^{2} )^{2} \)
$59$ \( 1 - 71908165730 T^{2} + \)\(17\!\cdots\!81\)\( T^{4} \)
$61$ \( ( 1 + 260470 T + 51520374361 T^{2} )^{2} \)
$67$ \( 1 - 80375086466 T^{2} + \)\(81\!\cdots\!61\)\( T^{4} \)
$71$ \( 1 + 245662064350 T^{2} + \)\(16\!\cdots\!41\)\( T^{4} \)
$73$ \( ( 1 + 395918 T + 151334226289 T^{2} )^{2} \)
$79$ \( 1 - 176211604754 T^{2} + \)\(59\!\cdots\!41\)\( T^{4} \)
$83$ \( 1 - 645933881666 T^{2} + \)\(10\!\cdots\!61\)\( T^{4} \)
$89$ \( ( 1 + 251886 T + 496981290961 T^{2} )^{2} \)
$97$ \( ( 1 - 517474 T + 832972004929 T^{2} )^{2} \)
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