Properties

Label 384.7.b.d
Level $384$
Weight $7$
Character orbit 384.b
Analytic conductor $88.341$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: 8.0.1731891456.1
Defining polynomial: \(x^{8} - 9 x^{6} + 65 x^{4} - 144 x^{2} + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + 5 \beta_{3} q^{5} + ( 3 \beta_{2} + \beta_{4} ) q^{7} + 243 q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + 5 \beta_{3} q^{5} + ( 3 \beta_{2} + \beta_{4} ) q^{7} + 243 q^{9} + ( 60 \beta_{1} + \beta_{6} ) q^{11} + ( 360 \beta_{3} + \beta_{7} ) q^{13} -5 \beta_{2} q^{15} + ( 558 + \beta_{5} ) q^{17} + ( 12 \beta_{1} - 3 \beta_{6} ) q^{19} + ( -729 \beta_{3} + 3 \beta_{7} ) q^{21} + ( 114 \beta_{2} - 30 \beta_{4} ) q^{23} + 15225 q^{25} -243 \beta_{1} q^{27} + ( -3163 \beta_{3} - 18 \beta_{7} ) q^{29} + ( -123 \beta_{2} + 67 \beta_{4} ) q^{31} + ( -14580 + 3 \beta_{5} ) q^{33} + ( -240 \beta_{1} + 5 \beta_{6} ) q^{35} + ( -13266 \beta_{3} + 5 \beta_{7} ) q^{37} + ( -360 \beta_{2} + 81 \beta_{4} ) q^{39} + ( -12366 - 3 \beta_{5} ) q^{41} + ( 3996 \beta_{1} - 27 \beta_{6} ) q^{43} + 1215 \beta_{3} q^{45} + ( 1266 \beta_{2} - 282 \beta_{4} ) q^{47} + ( -34847 - 18 \beta_{5} ) q^{49} + ( -558 \beta_{1} + 81 \beta_{6} ) q^{51} + ( -16025 \beta_{3} + 90 \beta_{7} ) q^{53} + ( 300 \beta_{2} - 80 \beta_{4} ) q^{55} + ( -2916 - 9 \beta_{5} ) q^{57} + ( -7524 \beta_{1} + 60 \beta_{6} ) q^{59} + ( -5058 \beta_{3} - 37 \beta_{7} ) q^{61} + ( 729 \beta_{2} + 243 \beta_{4} ) q^{63} + ( -28800 + 5 \beta_{5} ) q^{65} + ( 2316 \beta_{1} - 336 \beta_{6} ) q^{67} + ( -27702 \beta_{3} - 90 \beta_{7} ) q^{69} + ( 3918 \beta_{2} + 606 \beta_{4} ) q^{71} + ( 9614 + 54 \beta_{5} ) q^{73} -15225 \beta_{1} q^{75} + ( -73764 \beta_{3} - 36 \beta_{7} ) q^{77} + ( 8757 \beta_{2} + 75 \beta_{4} ) q^{79} + 59049 q^{81} + ( 8652 \beta_{1} + 335 \beta_{6} ) q^{83} + ( 2790 \beta_{3} - 80 \beta_{7} ) q^{85} + ( 3163 \beta_{2} - 1458 \beta_{4} ) q^{87} + ( 829854 - 10 \beta_{5} ) q^{89} + ( 21888 \beta_{1} + 117 \beta_{6} ) q^{91} + ( 29889 \beta_{3} + 201 \beta_{7} ) q^{93} + ( 60 \beta_{2} + 240 \beta_{4} ) q^{95} + ( -616210 + 90 \beta_{5} ) q^{97} + ( 14580 \beta_{1} + 243 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 1944 q^{9} + O(q^{10}) \) \( 8 q + 1944 q^{9} + 4464 q^{17} + 121800 q^{25} - 116640 q^{33} - 98928 q^{41} - 278776 q^{49} - 23328 q^{57} - 230400 q^{65} + 76912 q^{73} + 472392 q^{81} + 6638832 q^{89} - 4929680 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 9 x^{6} + 65 x^{4} - 144 x^{2} + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 9 \nu^{7} - 225 \nu^{5} + 1305 \nu^{3} - 4896 \nu \)\()/320\)
\(\beta_{2}\)\(=\)\((\)\( -81 \nu^{6} + 585 \nu^{4} - 5265 \nu^{2} + 6984 \)\()/130\)
\(\beta_{3}\)\(=\)\((\)\( 9 \nu^{7} - 65 \nu^{5} + 377 \nu^{3} - 256 \nu \)\()/208\)
\(\beta_{4}\)\(=\)\( 6 \nu^{6} - 54 \nu^{4} + 294 \nu^{2} - 432 \)
\(\beta_{5}\)\(=\)\((\)\( -3456 \nu^{6} - 513216 \)\()/65\)
\(\beta_{6}\)\(=\)\((\)\( 51 \nu^{7} - 507 \nu^{5} + 3939 \nu^{3} - 15456 \nu \)\()/13\)
\(\beta_{7}\)\(=\)\((\)\( 243 \nu^{7} - 1755 \nu^{5} + 13635 \nu^{3} - 6912 \nu \)\()/20\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(4 \beta_{7} - 9 \beta_{6} - 432 \beta_{3} + 192 \beta_{1}\)\()/6912\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} - 36 \beta_{4} - 432 \beta_{2} + 15552\)\()/6912\)
\(\nu^{3}\)\(=\)\((\)\(5 \beta_{7} - 1404 \beta_{3}\)\()/864\)
\(\nu^{4}\)\(=\)\((\)\(-3 \beta_{5} - 108 \beta_{4} - 784 \beta_{2} - 28224\)\()/2304\)
\(\nu^{5}\)\(=\)\((\)\(116 \beta_{7} + 261 \beta_{6} - 43632 \beta_{3} - 19392 \beta_{1}\)\()/6912\)
\(\nu^{6}\)\(=\)\((\)\(-65 \beta_{5} - 513216\)\()/3456\)
\(\nu^{7}\)\(=\)\((\)\(-724 \beta_{7} + 1629 \beta_{6} + 302832 \beta_{3} - 134592 \beta_{1}\)\()/6912\)

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
2.21837 + 1.28078i
−1.35234 0.780776i
−1.35234 + 0.780776i
2.21837 1.28078i
1.35234 0.780776i
−2.21837 + 1.28078i
−2.21837 1.28078i
1.35234 + 0.780776i
0 −15.5885 0 20.0000i 0 529.850i 0 243.000 0
319.2 0 −15.5885 0 20.0000i 0 155.727i 0 243.000 0
319.3 0 −15.5885 0 20.0000i 0 155.727i 0 243.000 0
319.4 0 −15.5885 0 20.0000i 0 529.850i 0 243.000 0
319.5 0 15.5885 0 20.0000i 0 155.727i 0 243.000 0
319.6 0 15.5885 0 20.0000i 0 529.850i 0 243.000 0
319.7 0 15.5885 0 20.0000i 0 529.850i 0 243.000 0
319.8 0 15.5885 0 20.0000i 0 155.727i 0 243.000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 319.8
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.d 8
4.b odd 2 1 inner 384.7.b.d 8
8.b even 2 1 inner 384.7.b.d 8
8.d odd 2 1 inner 384.7.b.d 8
16.e even 4 1 768.7.g.c 4
16.e even 4 1 768.7.g.e 4
16.f odd 4 1 768.7.g.c 4
16.f odd 4 1 768.7.g.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.7.b.d 8 1.a even 1 1 trivial
384.7.b.d 8 4.b odd 2 1 inner
384.7.b.d 8 8.b even 2 1 inner
384.7.b.d 8 8.d odd 2 1 inner
768.7.g.c 4 16.e even 4 1
768.7.g.c 4 16.f odd 4 1
768.7.g.e 4 16.e even 4 1
768.7.g.e 4 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 400 \) acting on \(S_{7}^{\mathrm{new}}(384, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( -243 + T^{2} )^{4} \)
$5$ \( ( 400 + T^{2} )^{4} \)
$7$ \( ( 6808230144 + 304992 T^{2} + T^{4} )^{2} \)
$11$ \( ( 1010555709696 - 5509728 T^{2} + T^{4} )^{2} \)
$13$ \( ( 1207818584064 + 10492416 T^{2} + T^{4} )^{2} \)
$17$ \( ( -50450364 - 1116 T + T^{2} )^{4} \)
$19$ \( ( 285122947021056 - 33911136 T^{2} + T^{4} )^{2} \)
$23$ \( ( 3049817413423104 + 312564096 T^{2} + T^{4} )^{2} \)
$29$ \( ( 753166899505164544 + 2375996192 T^{2} + T^{4} )^{2} \)
$31$ \( ( 219636481734441216 + 1172594016 T^{2} + T^{4} )^{2} \)
$37$ \( ( 7488283910542626816 + 5790206592 T^{2} + T^{4} )^{2} \)
$41$ \( ( -303937596 + 24732 T + T^{2} )^{4} \)
$43$ \( ( 6298399499403757824 - 10501589088 T^{2} + T^{4} )^{2} \)
$47$ \( ( 9689976869755981824 + 31151806848 T^{2} + T^{4} )^{2} \)
$53$ \( ( \)\(46\!\cdots\!00\)\( + 59613869600 T^{2} + T^{4} )^{2} \)
$59$ \( ( 48834094621847226624 - 41049200736 T^{2} + T^{4} )^{2} \)
$61$ \( ( 15476092643424374784 + 9505268352 T^{2} + T^{4} )^{2} \)
$67$ \( ( \)\(44\!\cdots\!96\)\( - 427110244704 T^{2} + T^{4} )^{2} \)
$71$ \( ( \)\(27\!\cdots\!24\)\( + 205670627712 T^{2} + T^{4} )^{2} \)
$73$ \( ( -147928769852 - 19228 T + T^{2} )^{4} \)
$79$ \( ( \)\(88\!\cdots\!44\)\( + 597624861024 T^{2} + T^{4} )^{2} \)
$83$ \( ( \)\(37\!\cdots\!84\)\( - 458360917344 T^{2} + T^{4} )^{2} \)
$89$ \( ( 683581488516 - 1659708 T + T^{2} )^{4} \)
$97$ \( ( -31455232700 + 1232420 T + T^{2} )^{4} \)
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