Properties

Label 384.7.b.b
Level $384$
Weight $7$
Character orbit 384.b
Analytic conductor $88.341$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 384.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(88.3407681100\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{5} \)
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 18 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{3} + ( 60 - 120 \zeta_{12}^{2} ) q^{5} -108 \zeta_{12}^{3} q^{7} + 243 q^{9} +O(q^{10})\) \( q + ( 18 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{3} + ( 60 - 120 \zeta_{12}^{2} ) q^{5} -108 \zeta_{12}^{3} q^{7} + 243 q^{9} + ( 456 \zeta_{12} - 228 \zeta_{12}^{3} ) q^{11} + ( -1272 + 2544 \zeta_{12}^{2} ) q^{13} -1620 \zeta_{12}^{3} q^{15} + 974 q^{17} + ( 1128 \zeta_{12} - 564 \zeta_{12}^{3} ) q^{19} + ( 972 - 1944 \zeta_{12}^{2} ) q^{21} + 20952 \zeta_{12}^{3} q^{23} + 4825 q^{25} + ( 4374 \zeta_{12} - 2187 \zeta_{12}^{3} ) q^{27} + ( 5484 - 10968 \zeta_{12}^{2} ) q^{29} + 15660 \zeta_{12}^{3} q^{31} + 6156 q^{33} + ( -12960 \zeta_{12} + 6480 \zeta_{12}^{3} ) q^{35} + ( -29712 + 59424 \zeta_{12}^{2} ) q^{37} + 34344 \zeta_{12}^{3} q^{39} + 33298 q^{41} + ( 18888 \zeta_{12} - 9444 \zeta_{12}^{3} ) q^{43} + ( 14580 - 29160 \zeta_{12}^{2} ) q^{45} -73224 \zeta_{12}^{3} q^{47} + 105985 q^{49} + ( 17532 \zeta_{12} - 8766 \zeta_{12}^{3} ) q^{51} + ( 95076 - 190152 \zeta_{12}^{2} ) q^{53} -41040 \zeta_{12}^{3} q^{55} + 15228 q^{57} + ( 86664 \zeta_{12} - 43332 \zeta_{12}^{3} ) q^{59} + ( -2784 + 5568 \zeta_{12}^{2} ) q^{61} -26244 \zeta_{12}^{3} q^{63} + 228960 q^{65} + ( 301992 \zeta_{12} - 150996 \zeta_{12}^{3} ) q^{67} + ( -188568 + 377136 \zeta_{12}^{2} ) q^{69} + 165672 \zeta_{12}^{3} q^{71} -113618 q^{73} + ( 86850 \zeta_{12} - 43425 \zeta_{12}^{3} ) q^{75} + ( 24624 - 49248 \zeta_{12}^{2} ) q^{77} -658260 \zeta_{12}^{3} q^{79} + 59049 q^{81} + ( 665640 \zeta_{12} - 332820 \zeta_{12}^{3} ) q^{83} + ( 58440 - 116880 \zeta_{12}^{2} ) q^{85} -148068 \zeta_{12}^{3} q^{87} -464290 q^{89} + ( 274752 \zeta_{12} - 137376 \zeta_{12}^{3} ) q^{91} + ( -140940 + 281880 \zeta_{12}^{2} ) q^{93} -101520 \zeta_{12}^{3} q^{95} + 51694 q^{97} + ( 110808 \zeta_{12} - 55404 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 972q^{9} + O(q^{10}) \) \( 4q + 972q^{9} + 3896q^{17} + 19300q^{25} + 24624q^{33} + 133192q^{41} + 423940q^{49} + 60912q^{57} + 915840q^{65} - 454472q^{73} + 236196q^{81} - 1857160q^{89} + 206776q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
319.1
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
0 −15.5885 0 103.923i 0 108.000i 0 243.000 0
319.2 0 −15.5885 0 103.923i 0 108.000i 0 243.000 0
319.3 0 15.5885 0 103.923i 0 108.000i 0 243.000 0
319.4 0 15.5885 0 103.923i 0 108.000i 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
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.7.b.b 4
4.b odd 2 1 inner 384.7.b.b 4
8.b even 2 1 inner 384.7.b.b 4
8.d odd 2 1 inner 384.7.b.b 4
16.e even 4 2 768.7.g.d 4
16.f odd 4 2 768.7.g.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.7.b.b 4 1.a even 1 1 trivial
384.7.b.b 4 4.b odd 2 1 inner
384.7.b.b 4 8.b even 2 1 inner
384.7.b.b 4 8.d odd 2 1 inner
768.7.g.d 4 16.e even 4 2
768.7.g.d 4 16.f odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 10800 \) acting on \(S_{7}^{\mathrm{new}}(384, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -243 + T^{2} )^{2} \)
$5$ \( ( 10800 + T^{2} )^{2} \)
$7$ \( ( 11664 + T^{2} )^{2} \)
$11$ \( ( -155952 + T^{2} )^{2} \)
$13$ \( ( 4853952 + T^{2} )^{2} \)
$17$ \( ( -974 + T )^{4} \)
$19$ \( ( -954288 + T^{2} )^{2} \)
$23$ \( ( 438986304 + T^{2} )^{2} \)
$29$ \( ( 90222768 + T^{2} )^{2} \)
$31$ \( ( 245235600 + T^{2} )^{2} \)
$37$ \( ( 2648408832 + T^{2} )^{2} \)
$41$ \( ( -33298 + T )^{4} \)
$43$ \( ( -267567408 + T^{2} )^{2} \)
$47$ \( ( 5361754176 + T^{2} )^{2} \)
$53$ \( ( 27118337328 + T^{2} )^{2} \)
$59$ \( ( -5632986672 + T^{2} )^{2} \)
$61$ \( ( 23251968 + T^{2} )^{2} \)
$67$ \( ( -68399376048 + T^{2} )^{2} \)
$71$ \( ( 27447211584 + T^{2} )^{2} \)
$73$ \( ( 113618 + T )^{4} \)
$79$ \( ( 433306227600 + T^{2} )^{2} \)
$83$ \( ( -332307457200 + T^{2} )^{2} \)
$89$ \( ( 464290 + T )^{4} \)
$97$ \( ( -51694 + T )^{4} \)
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