Properties

Label 384.7.b.a
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 18 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{3} + 196 \zeta_{12}^{3} q^{5} + ( 156 - 312 \zeta_{12}^{2} ) q^{7} + 243 q^{9} +O(q^{10})\) \( q + ( 18 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{3} + 196 \zeta_{12}^{3} q^{5} + ( 156 - 312 \zeta_{12}^{2} ) q^{7} + 243 q^{9} + ( 1992 \zeta_{12} - 996 \zeta_{12}^{3} ) q^{11} + 2880 \zeta_{12}^{3} q^{13} + ( -1764 + 3528 \zeta_{12}^{2} ) q^{15} -2898 q^{17} + ( 11304 \zeta_{12} - 5652 \zeta_{12}^{3} ) q^{19} -4212 \zeta_{12}^{3} q^{21} + ( 936 - 1872 \zeta_{12}^{2} ) q^{23} -22791 q^{25} + ( 4374 \zeta_{12} - 2187 \zeta_{12}^{3} ) q^{27} + 14596 \zeta_{12}^{3} q^{29} + ( 3972 - 7944 \zeta_{12}^{2} ) q^{31} + 26892 q^{33} + ( 61152 \zeta_{12} - 30576 \zeta_{12}^{3} ) q^{35} -12168 \zeta_{12}^{3} q^{37} + ( -25920 + 51840 \zeta_{12}^{2} ) q^{39} + 18738 q^{41} + ( 156168 \zeta_{12} - 78084 \zeta_{12}^{3} ) q^{43} + 47628 \zeta_{12}^{3} q^{45} + ( 87336 - 174672 \zeta_{12}^{2} ) q^{47} + 44641 q^{49} + ( -52164 \zeta_{12} + 26082 \zeta_{12}^{3} ) q^{51} + 144236 \zeta_{12}^{3} q^{53} + ( -195216 + 390432 \zeta_{12}^{2} ) q^{55} + 152604 q^{57} + ( -385272 \zeta_{12} + 192636 \zeta_{12}^{3} ) q^{59} + 333432 \zeta_{12}^{3} q^{61} + ( 37908 - 75816 \zeta_{12}^{2} ) q^{63} -564480 q^{65} + ( -368856 \zeta_{12} + 184428 \zeta_{12}^{3} ) q^{67} -25272 \zeta_{12}^{3} q^{69} + ( -242856 + 485712 \zeta_{12}^{2} ) q^{71} + 628238 q^{73} + ( -410238 \zeta_{12} + 205119 \zeta_{12}^{3} ) q^{75} -466128 \zeta_{12}^{3} q^{77} + ( 168516 - 337032 \zeta_{12}^{2} ) q^{79} + 59049 q^{81} + ( -412248 \zeta_{12} + 206124 \zeta_{12}^{3} ) q^{83} -568008 \zeta_{12}^{3} q^{85} + ( -131364 + 262728 \zeta_{12}^{2} ) q^{87} -1375074 q^{89} + ( 898560 \zeta_{12} - 449280 \zeta_{12}^{3} ) q^{91} -107244 \zeta_{12}^{3} q^{93} + ( -1107792 + 2215584 \zeta_{12}^{2} ) q^{95} + 489710 q^{97} + ( 484056 \zeta_{12} - 242028 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 972q^{9} + O(q^{10}) \) \( 4q + 972q^{9} - 11592q^{17} - 91164q^{25} + 107568q^{33} + 74952q^{41} + 178564q^{49} + 610416q^{57} - 2257920q^{65} + 2512952q^{73} + 236196q^{81} - 5500296q^{89} + 1958840q^{97} + O(q^{100}) \)

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 0 196.000i 0 270.200i 0 243.000 0
319.2 0 −15.5885 0 196.000i 0 270.200i 0 243.000 0
319.3 0 15.5885 0 196.000i 0 270.200i 0 243.000 0
319.4 0 15.5885 0 196.000i 0 270.200i 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.a 4
4.b odd 2 1 inner 384.7.b.a 4
8.b even 2 1 inner 384.7.b.a 4
8.d odd 2 1 inner 384.7.b.a 4
16.e even 4 1 768.7.g.a 2
16.e even 4 1 768.7.g.b 2
16.f odd 4 1 768.7.g.a 2
16.f odd 4 1 768.7.g.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.7.b.a 4 1.a even 1 1 trivial
384.7.b.a 4 4.b odd 2 1 inner
384.7.b.a 4 8.b even 2 1 inner
384.7.b.a 4 8.d odd 2 1 inner
768.7.g.a 2 16.e even 4 1
768.7.g.a 2 16.f odd 4 1
768.7.g.b 2 16.e even 4 1
768.7.g.b 2 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 38416 \) acting on \(S_{7}^{\mathrm{new}}(384, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -243 + T^{2} )^{2} \)
$5$ \( ( 38416 + T^{2} )^{2} \)
$7$ \( ( 73008 + T^{2} )^{2} \)
$11$ \( ( -2976048 + T^{2} )^{2} \)
$13$ \( ( 8294400 + T^{2} )^{2} \)
$17$ \( ( 2898 + T )^{4} \)
$19$ \( ( -95835312 + T^{2} )^{2} \)
$23$ \( ( 2628288 + T^{2} )^{2} \)
$29$ \( ( 213043216 + T^{2} )^{2} \)
$31$ \( ( 47330352 + T^{2} )^{2} \)
$37$ \( ( 148060224 + T^{2} )^{2} \)
$41$ \( ( -18738 + T )^{4} \)
$43$ \( ( -18291333168 + T^{2} )^{2} \)
$47$ \( ( 22882730688 + T^{2} )^{2} \)
$53$ \( ( 20804023696 + T^{2} )^{2} \)
$59$ \( ( -111325885488 + T^{2} )^{2} \)
$61$ \( ( 111176898624 + T^{2} )^{2} \)
$67$ \( ( -102041061552 + T^{2} )^{2} \)
$71$ \( ( 176937110208 + T^{2} )^{2} \)
$73$ \( ( -628238 + T )^{4} \)
$79$ \( ( 85192926768 + T^{2} )^{2} \)
$83$ \( ( -127461310128 + T^{2} )^{2} \)
$89$ \( ( 1375074 + T )^{4} \)
$97$ \( ( -489710 + T )^{4} \)
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