Properties

Label 48.3.e.b
Level 48
Weight 3
Character orbit 48.e
Analytic conductor 1.308
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(1.30790526893\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{3} + 2 \beta q^{5} + 6 q^{7} + ( -7 - 2 \beta ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta ) q^{3} + 2 \beta q^{5} + 6 q^{7} + ( -7 - 2 \beta ) q^{9} -2 \beta q^{11} + 10 q^{13} + ( -16 - 2 \beta ) q^{15} -8 \beta q^{17} -2 q^{19} + ( -6 + 6 \beta ) q^{21} + 4 \beta q^{23} -7 q^{25} + ( 23 - 5 \beta ) q^{27} + 6 \beta q^{29} + 22 q^{31} + ( 16 + 2 \beta ) q^{33} + 12 \beta q^{35} -6 q^{37} + ( -10 + 10 \beta ) q^{39} + 12 \beta q^{41} -82 q^{43} + ( 32 - 14 \beta ) q^{45} -24 \beta q^{47} -13 q^{49} + ( 64 + 8 \beta ) q^{51} -22 \beta q^{53} + 32 q^{55} + ( 2 - 2 \beta ) q^{57} -26 \beta q^{59} -86 q^{61} + ( -42 - 12 \beta ) q^{63} + 20 \beta q^{65} -2 q^{67} + ( -32 - 4 \beta ) q^{69} + 44 \beta q^{71} + 82 q^{73} + ( 7 - 7 \beta ) q^{75} -12 \beta q^{77} -10 q^{79} + ( 17 + 28 \beta ) q^{81} + 26 \beta q^{83} + 128 q^{85} + ( -48 - 6 \beta ) q^{87} -12 \beta q^{89} + 60 q^{91} + ( -22 + 22 \beta ) q^{93} -4 \beta q^{95} -94 q^{97} + ( -32 + 14 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 12q^{7} - 14q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 12q^{7} - 14q^{9} + 20q^{13} - 32q^{15} - 4q^{19} - 12q^{21} - 14q^{25} + 46q^{27} + 44q^{31} + 32q^{33} - 12q^{37} - 20q^{39} - 164q^{43} + 64q^{45} - 26q^{49} + 128q^{51} + 64q^{55} + 4q^{57} - 172q^{61} - 84q^{63} - 4q^{67} - 64q^{69} + 164q^{73} + 14q^{75} - 20q^{79} + 34q^{81} + 256q^{85} - 96q^{87} + 120q^{91} - 44q^{93} - 188q^{97} - 64q^{99} + O(q^{100}) \)

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 −1.00000 2.82843i 0 5.65685i 0 6.00000 0 −7.00000 + 5.65685i 0
17.2 0 −1.00000 + 2.82843i 0 5.65685i 0 6.00000 0 −7.00000 5.65685i 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 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.3.e.b 2
3.b odd 2 1 inner 48.3.e.b 2
4.b odd 2 1 24.3.e.a 2
5.b even 2 1 1200.3.l.n 2
5.c odd 4 2 1200.3.c.i 4
8.b even 2 1 192.3.e.d 2
8.d odd 2 1 192.3.e.c 2
9.c even 3 2 1296.3.q.e 4
9.d odd 6 2 1296.3.q.e 4
12.b even 2 1 24.3.e.a 2
15.d odd 2 1 1200.3.l.n 2
15.e even 4 2 1200.3.c.i 4
16.e even 4 2 768.3.h.c 4
16.f odd 4 2 768.3.h.d 4
20.d odd 2 1 600.3.l.b 2
20.e even 4 2 600.3.c.a 4
24.f even 2 1 192.3.e.c 2
24.h odd 2 1 192.3.e.d 2
28.d even 2 1 1176.3.d.a 2
36.f odd 6 2 648.3.m.d 4
36.h even 6 2 648.3.m.d 4
48.i odd 4 2 768.3.h.c 4
48.k even 4 2 768.3.h.d 4
60.h even 2 1 600.3.l.b 2
60.l odd 4 2 600.3.c.a 4
84.h odd 2 1 1176.3.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.3.e.a 2 4.b odd 2 1
24.3.e.a 2 12.b even 2 1
48.3.e.b 2 1.a even 1 1 trivial
48.3.e.b 2 3.b odd 2 1 inner
192.3.e.c 2 8.d odd 2 1
192.3.e.c 2 24.f even 2 1
192.3.e.d 2 8.b even 2 1
192.3.e.d 2 24.h odd 2 1
600.3.c.a 4 20.e even 4 2
600.3.c.a 4 60.l odd 4 2
600.3.l.b 2 20.d odd 2 1
600.3.l.b 2 60.h even 2 1
648.3.m.d 4 36.f odd 6 2
648.3.m.d 4 36.h even 6 2
768.3.h.c 4 16.e even 4 2
768.3.h.c 4 48.i odd 4 2
768.3.h.d 4 16.f odd 4 2
768.3.h.d 4 48.k even 4 2
1176.3.d.a 2 28.d even 2 1
1176.3.d.a 2 84.h odd 2 1
1200.3.c.i 4 5.c odd 4 2
1200.3.c.i 4 15.e even 4 2
1200.3.l.n 2 5.b even 2 1
1200.3.l.n 2 15.d odd 2 1
1296.3.q.e 4 9.c even 3 2
1296.3.q.e 4 9.d odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 32 \) acting on \(S_{3}^{\mathrm{new}}(48, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 2 T + 9 T^{2} \)
$5$ \( 1 - 18 T^{2} + 625 T^{4} \)
$7$ \( ( 1 - 6 T + 49 T^{2} )^{2} \)
$11$ \( 1 - 210 T^{2} + 14641 T^{4} \)
$13$ \( ( 1 - 10 T + 169 T^{2} )^{2} \)
$17$ \( 1 - 66 T^{2} + 83521 T^{4} \)
$19$ \( ( 1 + 2 T + 361 T^{2} )^{2} \)
$23$ \( 1 - 930 T^{2} + 279841 T^{4} \)
$29$ \( 1 - 1394 T^{2} + 707281 T^{4} \)
$31$ \( ( 1 - 22 T + 961 T^{2} )^{2} \)
$37$ \( ( 1 + 6 T + 1369 T^{2} )^{2} \)
$41$ \( 1 - 2210 T^{2} + 2825761 T^{4} \)
$43$ \( ( 1 + 82 T + 1849 T^{2} )^{2} \)
$47$ \( 1 + 190 T^{2} + 4879681 T^{4} \)
$53$ \( 1 - 1746 T^{2} + 7890481 T^{4} \)
$59$ \( 1 - 1554 T^{2} + 12117361 T^{4} \)
$61$ \( ( 1 + 86 T + 3721 T^{2} )^{2} \)
$67$ \( ( 1 + 2 T + 4489 T^{2} )^{2} \)
$71$ \( 1 + 5406 T^{2} + 25411681 T^{4} \)
$73$ \( ( 1 - 82 T + 5329 T^{2} )^{2} \)
$79$ \( ( 1 + 10 T + 6241 T^{2} )^{2} \)
$83$ \( 1 - 8370 T^{2} + 47458321 T^{4} \)
$89$ \( 1 - 14690 T^{2} + 62742241 T^{4} \)
$97$ \( ( 1 + 94 T + 9409 T^{2} )^{2} \)
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