Properties

Label 48.7.e.b
Level 48
Weight 7
Character orbit 48.e
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.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.0425960138\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 6)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 12\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -21 + \beta ) q^{3} + 10 \beta q^{5} -2 q^{7} + ( 153 - 42 \beta ) q^{9} +O(q^{10})\) \( q + ( -21 + \beta ) q^{3} + 10 \beta q^{5} -2 q^{7} + ( 153 - 42 \beta ) q^{9} -2 \beta q^{11} -2950 q^{13} + ( -2880 - 210 \beta ) q^{15} -264 \beta q^{17} -5258 q^{19} + ( 42 - 2 \beta ) q^{21} -604 \beta q^{23} -13175 q^{25} + ( 8883 + 1035 \beta ) q^{27} -130 \beta q^{29} -22898 q^{31} + ( 576 + 42 \beta ) q^{33} -20 \beta q^{35} + 34058 q^{37} + ( 61950 - 2950 \beta ) q^{39} + 988 \beta q^{41} + 6406 q^{43} + ( 120960 + 1530 \beta ) q^{45} + 10600 \beta q^{47} -117645 q^{49} + ( 76032 + 5544 \beta ) q^{51} + 11346 \beta q^{53} + 5760 q^{55} + ( 110418 - 5258 \beta ) q^{57} -19258 \beta q^{59} -62566 q^{61} + ( -306 + 84 \beta ) q^{63} -29500 \beta q^{65} -438698 q^{67} + ( 173952 + 12684 \beta ) q^{69} -4020 \beta q^{71} -730510 q^{73} + ( 276675 - 13175 \beta ) q^{75} + 4 \beta q^{77} -340562 q^{79} + ( -484623 - 12852 \beta ) q^{81} + 29242 \beta q^{83} + 760320 q^{85} + ( 37440 + 2730 \beta ) q^{87} + 22788 \beta q^{89} + 5900 q^{91} + ( 480858 - 22898 \beta ) q^{93} -52580 \beta q^{95} -281086 q^{97} + ( -24192 - 306 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 42q^{3} - 4q^{7} + 306q^{9} + O(q^{10}) \) \( 2q - 42q^{3} - 4q^{7} + 306q^{9} - 5900q^{13} - 5760q^{15} - 10516q^{19} + 84q^{21} - 26350q^{25} + 17766q^{27} - 45796q^{31} + 1152q^{33} + 68116q^{37} + 123900q^{39} + 12812q^{43} + 241920q^{45} - 235290q^{49} + 152064q^{51} + 11520q^{55} + 220836q^{57} - 125132q^{61} - 612q^{63} - 877396q^{67} + 347904q^{69} - 1461020q^{73} + 553350q^{75} - 681124q^{79} - 969246q^{81} + 1520640q^{85} + 74880q^{87} + 11800q^{91} + 961716q^{93} - 562172q^{97} - 48384q^{99} + O(q^{100}) \)

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 −21.0000 16.9706i 0 169.706i 0 −2.00000 0 153.000 + 712.764i 0
17.2 0 −21.0000 + 16.9706i 0 169.706i 0 −2.00000 0 153.000 712.764i 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 48.7.e.b 2
3.b odd 2 1 inner 48.7.e.b 2
4.b odd 2 1 6.7.b.a 2
8.b even 2 1 192.7.e.f 2
8.d odd 2 1 192.7.e.c 2
12.b even 2 1 6.7.b.a 2
20.d odd 2 1 150.7.d.a 2
20.e even 4 2 150.7.b.a 4
24.f even 2 1 192.7.e.c 2
24.h odd 2 1 192.7.e.f 2
28.d even 2 1 294.7.b.a 2
36.f odd 6 2 162.7.d.b 4
36.h even 6 2 162.7.d.b 4
60.h even 2 1 150.7.d.a 2
60.l odd 4 2 150.7.b.a 4
84.h odd 2 1 294.7.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.7.b.a 2 4.b odd 2 1
6.7.b.a 2 12.b even 2 1
48.7.e.b 2 1.a even 1 1 trivial
48.7.e.b 2 3.b odd 2 1 inner
150.7.b.a 4 20.e even 4 2
150.7.b.a 4 60.l odd 4 2
150.7.d.a 2 20.d odd 2 1
150.7.d.a 2 60.h even 2 1
162.7.d.b 4 36.f odd 6 2
162.7.d.b 4 36.h even 6 2
192.7.e.c 2 8.d odd 2 1
192.7.e.c 2 24.f even 2 1
192.7.e.f 2 8.b even 2 1
192.7.e.f 2 24.h odd 2 1
294.7.b.a 2 28.d even 2 1
294.7.b.a 2 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 28800 \) acting on \(S_{7}^{\mathrm{new}}(48, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 42 T + 729 T^{2} \)
$5$ \( 1 - 2450 T^{2} + 244140625 T^{4} \)
$7$ \( ( 1 + 2 T + 117649 T^{2} )^{2} \)
$11$ \( 1 - 3541970 T^{2} + 3138428376721 T^{4} \)
$13$ \( ( 1 + 2950 T + 4826809 T^{2} )^{2} \)
$17$ \( 1 - 28202690 T^{2} + 582622237229761 T^{4} \)
$19$ \( ( 1 + 5258 T + 47045881 T^{2} )^{2} \)
$23$ \( 1 - 191004770 T^{2} + 21914624432020321 T^{4} \)
$29$ \( 1 - 1184779442 T^{2} + 353814783205469041 T^{4} \)
$31$ \( ( 1 + 22898 T + 887503681 T^{2} )^{2} \)
$37$ \( ( 1 - 34058 T + 2565726409 T^{2} )^{2} \)
$41$ \( 1 - 9219079010 T^{2} + 22563490300366186081 T^{4} \)
$43$ \( ( 1 - 6406 T + 6321363049 T^{2} )^{2} \)
$47$ \( 1 + 10801249342 T^{2} + \)\(11\!\cdots\!41\)\( T^{4} \)
$53$ \( 1 - 7253988050 T^{2} + \)\(49\!\cdots\!41\)\( T^{4} \)
$59$ \( 1 + 22449655150 T^{2} + \)\(17\!\cdots\!81\)\( T^{4} \)
$61$ \( ( 1 + 62566 T + 51520374361 T^{2} )^{2} \)
$67$ \( ( 1 + 438698 T + 90458382169 T^{2} )^{2} \)
$71$ \( 1 - 251546372642 T^{2} + \)\(16\!\cdots\!41\)\( T^{4} \)
$73$ \( ( 1 + 730510 T + 151334226289 T^{2} )^{2} \)
$79$ \( ( 1 + 340562 T + 243087455521 T^{2} )^{2} \)
$83$ \( 1 - 407613512306 T^{2} + \)\(10\!\cdots\!61\)\( T^{4} \)
$89$ \( 1 - 844406214050 T^{2} + \)\(24\!\cdots\!21\)\( T^{4} \)
$97$ \( ( 1 + 281086 T + 832972004929 T^{2} )^{2} \)
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