Properties

Label 48.7.g.c
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.

\(f(q)\) \(=\) \( q + ( 9 - 18 \zeta_{6} ) q^{3} + 150 q^{5} + ( -188 + 376 \zeta_{6} ) q^{7} -243 q^{9} +O(q^{10})\) \( q + ( 9 - 18 \zeta_{6} ) q^{3} + 150 q^{5} + ( -188 + 376 \zeta_{6} ) q^{7} -243 q^{9} + ( 852 - 1704 \zeta_{6} ) q^{11} + 3394 q^{13} + ( 1350 - 2700 \zeta_{6} ) q^{15} + 5178 q^{17} + ( 3932 - 7864 \zeta_{6} ) q^{19} + 5076 q^{21} + ( -2304 + 4608 \zeta_{6} ) q^{23} + 6875 q^{25} + ( -2187 + 4374 \zeta_{6} ) q^{27} + 32142 q^{29} + ( 18828 - 37656 \zeta_{6} ) q^{31} -23004 q^{33} + ( -28200 + 56400 \zeta_{6} ) q^{35} -76150 q^{37} + ( 30546 - 61092 \zeta_{6} ) q^{39} -70038 q^{41} + ( -58284 + 116568 \zeta_{6} ) q^{43} -36450 q^{45} + ( -87528 + 175056 \zeta_{6} ) q^{47} + 11617 q^{49} + ( 46602 - 93204 \zeta_{6} ) q^{51} + 66942 q^{53} + ( 127800 - 255600 \zeta_{6} ) q^{55} -106164 q^{57} + ( -225492 + 450984 \zeta_{6} ) q^{59} -257014 q^{61} + ( 45684 - 91368 \zeta_{6} ) q^{63} + 509100 q^{65} + ( 185444 - 370888 \zeta_{6} ) q^{67} + 62208 q^{69} + ( 198096 - 396192 \zeta_{6} ) q^{71} + 243442 q^{73} + ( 61875 - 123750 \zeta_{6} ) q^{75} + 480528 q^{77} + ( 274220 - 548440 \zeta_{6} ) q^{79} + 59049 q^{81} + ( -597012 + 1194024 \zeta_{6} ) q^{83} + 776700 q^{85} + ( 289278 - 578556 \zeta_{6} ) q^{87} -686766 q^{89} + ( -638072 + 1276144 \zeta_{6} ) q^{91} -508356 q^{93} + ( 589800 - 1179600 \zeta_{6} ) q^{95} -942686 q^{97} + ( -207036 + 414072 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 300q^{5} - 486q^{9} + O(q^{10}) \) \( 2q + 300q^{5} - 486q^{9} + 6788q^{13} + 10356q^{17} + 10152q^{21} + 13750q^{25} + 64284q^{29} - 46008q^{33} - 152300q^{37} - 140076q^{41} - 72900q^{45} + 23234q^{49} + 133884q^{53} - 212328q^{57} - 514028q^{61} + 1018200q^{65} + 124416q^{69} + 486884q^{73} + 961056q^{77} + 118098q^{81} + 1553400q^{85} - 1373532q^{89} - 1016712q^{93} - 1885372q^{97} + O(q^{100}) \)

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 0 150.000 0 325.626i 0 −243.000 0
31.2 0 15.5885i 0 150.000 0 325.626i 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.c 2
3.b odd 2 1 144.7.g.b 2
4.b odd 2 1 inner 48.7.g.c 2
8.b even 2 1 192.7.g.a 2
8.d odd 2 1 192.7.g.a 2
12.b even 2 1 144.7.g.b 2
16.e even 4 2 768.7.b.d 4
16.f odd 4 2 768.7.b.d 4
24.f even 2 1 576.7.g.j 2
24.h odd 2 1 576.7.g.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.7.g.c 2 1.a even 1 1 trivial
48.7.g.c 2 4.b odd 2 1 inner
144.7.g.b 2 3.b odd 2 1
144.7.g.b 2 12.b even 2 1
192.7.g.a 2 8.b even 2 1
192.7.g.a 2 8.d odd 2 1
576.7.g.j 2 24.f even 2 1
576.7.g.j 2 24.h odd 2 1
768.7.b.d 4 16.e even 4 2
768.7.b.d 4 16.f odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 150 \) acting on \(S_{7}^{\mathrm{new}}(48, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 243 T^{2} \)
$5$ \( ( 1 - 150 T + 15625 T^{2} )^{2} \)
$7$ \( 1 - 129266 T^{2} + 13841287201 T^{4} \)
$11$ \( 1 - 1365410 T^{2} + 3138428376721 T^{4} \)
$13$ \( ( 1 - 3394 T + 4826809 T^{2} )^{2} \)
$17$ \( ( 1 - 5178 T + 24137569 T^{2} )^{2} \)
$19$ \( 1 - 47709890 T^{2} + 2213314919066161 T^{4} \)
$23$ \( 1 - 280146530 T^{2} + 21914624432020321 T^{4} \)
$29$ \( ( 1 - 32142 T + 594823321 T^{2} )^{2} \)
$31$ \( 1 - 711526610 T^{2} + 787662783788549761 T^{4} \)
$37$ \( ( 1 + 76150 T + 2565726409 T^{2} )^{2} \)
$41$ \( ( 1 + 70038 T + 4750104241 T^{2} )^{2} \)
$43$ \( 1 - 2451652130 T^{2} + 39959630797262576401 T^{4} \)
$47$ \( 1 + 1425021694 T^{2} + \)\(11\!\cdots\!41\)\( T^{4} \)
$53$ \( ( 1 - 66942 T + 22164361129 T^{2} )^{2} \)
$59$ \( 1 + 68178858910 T^{2} + \)\(17\!\cdots\!81\)\( T^{4} \)
$61$ \( ( 1 + 257014 T + 51520374361 T^{2} )^{2} \)
$67$ \( 1 - 77748332930 T^{2} + \)\(81\!\cdots\!61\)\( T^{4} \)
$71$ \( 1 - 138474492194 T^{2} + \)\(16\!\cdots\!41\)\( T^{4} \)
$73$ \( ( 1 - 243442 T + 151334226289 T^{2} )^{2} \)
$79$ \( 1 - 260585085842 T^{2} + \)\(59\!\cdots\!41\)\( T^{4} \)
$83$ \( 1 + 415389237694 T^{2} + \)\(10\!\cdots\!61\)\( T^{4} \)
$89$ \( ( 1 + 686766 T + 496981290961 T^{2} )^{2} \)
$97$ \( ( 1 + 942686 T + 832972004929 T^{2} )^{2} \)
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