Properties

Label 48.7.g.a
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + ( 9 - 18 \zeta_{6} ) q^{3} -90 q^{5} + ( -108 + 216 \zeta_{6} ) q^{7} -243 q^{9} +O(q^{10})\) \( q + ( 9 - 18 \zeta_{6} ) q^{3} -90 q^{5} + ( -108 + 216 \zeta_{6} ) q^{7} -243 q^{9} + ( -972 + 1944 \zeta_{6} ) q^{11} + 1762 q^{13} + ( -810 + 1620 \zeta_{6} ) q^{15} -1638 q^{17} + ( -7236 + 14472 \zeta_{6} ) q^{19} + 2916 q^{21} + ( -7776 + 15552 \zeta_{6} ) q^{23} -7525 q^{25} + ( -2187 + 4374 \zeta_{6} ) q^{27} -16002 q^{29} + ( 9180 - 18360 \zeta_{6} ) q^{31} + 26244 q^{33} + ( 9720 - 19440 \zeta_{6} ) q^{35} + 61130 q^{37} + ( 15858 - 31716 \zeta_{6} ) q^{39} -98550 q^{41} + ( 28404 - 56808 \zeta_{6} ) q^{43} + 21870 q^{45} + ( 103032 - 206064 \zeta_{6} ) q^{47} + 82657 q^{49} + ( -14742 + 29484 \zeta_{6} ) q^{51} -275346 q^{53} + ( 87480 - 174960 \zeta_{6} ) q^{55} + 195372 q^{57} + ( -146772 + 293544 \zeta_{6} ) q^{59} + 106634 q^{61} + ( 26244 - 52488 \zeta_{6} ) q^{63} -158580 q^{65} + ( -318492 + 636984 \zeta_{6} ) q^{67} + 209952 q^{69} + ( -42768 + 85536 \zeta_{6} ) q^{71} + 100978 q^{73} + ( -67725 + 135450 \zeta_{6} ) q^{75} -314928 q^{77} + ( 44604 - 89208 \zeta_{6} ) q^{79} + 59049 q^{81} + ( -372276 + 744552 \zeta_{6} ) q^{83} + 147420 q^{85} + ( -144018 + 288036 \zeta_{6} ) q^{87} -819054 q^{89} + ( -190296 + 380592 \zeta_{6} ) q^{91} -247860 q^{93} + ( 651240 - 1302480 \zeta_{6} ) q^{95} + 557026 q^{97} + ( 236196 - 472392 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 180q^{5} - 486q^{9} + O(q^{10}) \) \( 2q - 180q^{5} - 486q^{9} + 3524q^{13} - 3276q^{17} + 5832q^{21} - 15050q^{25} - 32004q^{29} + 52488q^{33} + 122260q^{37} - 197100q^{41} + 43740q^{45} + 165314q^{49} - 550692q^{53} + 390744q^{57} + 213268q^{61} - 317160q^{65} + 419904q^{69} + 201956q^{73} - 629856q^{77} + 118098q^{81} + 294840q^{85} - 1638108q^{89} - 495720q^{93} + 1114052q^{97} + O(q^{100}) \)

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 −90.0000 0 187.061i 0 −243.000 0
31.2 0 15.5885i 0 −90.0000 0 187.061i 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.a 2
3.b odd 2 1 144.7.g.e 2
4.b odd 2 1 inner 48.7.g.a 2
8.b even 2 1 192.7.g.c 2
8.d odd 2 1 192.7.g.c 2
12.b even 2 1 144.7.g.e 2
16.e even 4 2 768.7.b.c 4
16.f odd 4 2 768.7.b.c 4
24.f even 2 1 576.7.g.e 2
24.h odd 2 1 576.7.g.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.7.g.a 2 1.a even 1 1 trivial
48.7.g.a 2 4.b odd 2 1 inner
144.7.g.e 2 3.b odd 2 1
144.7.g.e 2 12.b even 2 1
192.7.g.c 2 8.b even 2 1
192.7.g.c 2 8.d odd 2 1
576.7.g.e 2 24.f even 2 1
576.7.g.e 2 24.h odd 2 1
768.7.b.c 4 16.e even 4 2
768.7.b.c 4 16.f odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 90 \) acting on \(S_{7}^{\mathrm{new}}(48, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 243 T^{2} \)
$5$ \( ( 1 + 90 T + 15625 T^{2} )^{2} \)
$7$ \( 1 - 200306 T^{2} + 13841287201 T^{4} \)
$11$ \( 1 - 708770 T^{2} + 3138428376721 T^{4} \)
$13$ \( ( 1 - 1762 T + 4826809 T^{2} )^{2} \)
$17$ \( ( 1 + 1638 T + 24137569 T^{2} )^{2} \)
$19$ \( 1 + 62987326 T^{2} + 2213314919066161 T^{4} \)
$23$ \( 1 - 114673250 T^{2} + 21914624432020321 T^{4} \)
$29$ \( ( 1 + 16002 T + 594823321 T^{2} )^{2} \)
$31$ \( 1 - 1522190162 T^{2} + 787662783788549761 T^{4} \)
$37$ \( ( 1 - 61130 T + 2565726409 T^{2} )^{2} \)
$41$ \( ( 1 + 98550 T + 4750104241 T^{2} )^{2} \)
$43$ \( 1 - 10222364450 T^{2} + 39959630797262576401 T^{4} \)
$47$ \( 1 + 10288348414 T^{2} + \)\(11\!\cdots\!41\)\( T^{4} \)
$53$ \( ( 1 + 275346 T + 22164361129 T^{2} )^{2} \)
$59$ \( 1 - 19735007330 T^{2} + \)\(17\!\cdots\!81\)\( T^{4} \)
$61$ \( ( 1 - 106634 T + 51520374361 T^{2} )^{2} \)
$67$ \( 1 + 123394697854 T^{2} + \)\(81\!\cdots\!61\)\( T^{4} \)
$71$ \( 1 - 250713262370 T^{2} + \)\(16\!\cdots\!41\)\( T^{4} \)
$73$ \( ( 1 - 100978 T + 151334226289 T^{2} )^{2} \)
$79$ \( 1 - 480206360594 T^{2} + \)\(59\!\cdots\!41\)\( T^{4} \)
$83$ \( 1 - 238112486210 T^{2} + \)\(10\!\cdots\!61\)\( T^{4} \)
$89$ \( ( 1 + 819054 T + 496981290961 T^{2} )^{2} \)
$97$ \( ( 1 - 557026 T + 832972004929 T^{2} )^{2} \)
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