Properties

Label 384.7.b.c
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: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 4 x^{7} + 106 x^{6} - 304 x^{5} + 4359 x^{4} - 8216 x^{3} + 73366 x^{2} - 69308 x + 604693\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{10} \)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} -\beta_{1} q^{5} + \beta_{5} q^{7} + 243 q^{9} +O(q^{10})\) \( q + \beta_{4} q^{3} -\beta_{1} q^{5} + \beta_{5} q^{7} + 243 q^{9} + ( -17 \beta_{4} + \beta_{7} ) q^{11} + \beta_{3} q^{13} + \beta_{6} q^{15} + ( -818 + \beta_{2} ) q^{17} + ( -195 \beta_{4} - 5 \beta_{7} ) q^{19} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{21} + ( 2 \beta_{5} + 4 \beta_{6} ) q^{23} + ( -7079 - 2 \beta_{2} ) q^{25} + 243 \beta_{4} q^{27} + ( 47 \beta_{1} + 14 \beta_{3} ) q^{29} + ( -61 \beta_{5} + 20 \beta_{6} ) q^{31} + ( -4212 - 3 \beta_{2} ) q^{33} + ( -325 \beta_{4} + 33 \beta_{7} ) q^{35} + ( -110 \beta_{1} - 27 \beta_{3} ) q^{37} + ( 81 \beta_{5} - \beta_{6} ) q^{39} + ( -62446 + \beta_{2} ) q^{41} + ( 529 \beta_{4} - 25 \beta_{7} ) q^{43} -243 \beta_{1} q^{45} + ( 198 \beta_{5} + 32 \beta_{6} ) q^{47} + ( -51839 + 12 \beta_{2} ) q^{49} + ( -845 \beta_{4} - 81 \beta_{7} ) q^{51} + ( 949 \beta_{1} - 70 \beta_{3} ) q^{53} + ( -400 \beta_{5} - 76 \beta_{6} ) q^{55} + ( -46980 + 15 \beta_{2} ) q^{57} + ( -10620 \beta_{4} + 32 \beta_{7} ) q^{59} + ( -462 \beta_{1} - 5 \beta_{3} ) q^{61} + 243 \beta_{5} q^{63} + ( -4512 - 31 \beta_{2} ) q^{65} + ( -10568 \beta_{4} + 140 \beta_{7} ) q^{67} + ( -978 \beta_{1} + 6 \beta_{3} ) q^{69} + ( 94 \beta_{5} - 228 \beta_{6} ) q^{71} + ( -245330 - 20 \beta_{2} ) q^{73} + ( -7025 \beta_{4} + 162 \beta_{7} ) q^{75} + ( 3284 \beta_{1} + 124 \beta_{3} ) q^{77} + ( -1525 \beta_{5} - 4 \beta_{6} ) q^{79} + 59049 q^{81} + ( -13683 \beta_{4} - 309 \beta_{7} ) q^{83} + ( -4334 \beta_{1} + 400 \beta_{3} ) q^{85} + ( 1134 \beta_{5} - 61 \beta_{6} ) q^{87} + ( -211874 + 38 \beta_{2} ) q^{89} + ( -56279 \beta_{4} - 357 \beta_{7} ) q^{91} + ( -4677 \beta_{1} - 183 \beta_{3} ) q^{93} + ( 2000 \beta_{5} + 100 \beta_{6} ) q^{95} + ( 954094 - 10 \beta_{2} ) q^{97} + ( -4131 \beta_{4} + 243 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 1944 q^{9} + O(q^{10}) \) \( 8 q + 1944 q^{9} - 6544 q^{17} - 56632 q^{25} - 33696 q^{33} - 499568 q^{41} - 414712 q^{49} - 375840 q^{57} - 36096 q^{65} - 1962640 q^{73} + 472392 q^{81} - 1694992 q^{89} + 7632752 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} + 106 x^{6} - 304 x^{5} + 4359 x^{4} - 8216 x^{3} + 73366 x^{2} - 69308 x + 604693\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{6} - 6 \nu^{5} + 418 \nu^{4} - 826 \nu^{3} + 15758 \nu^{2} - 15346 \nu + 184860 \)\()/959\)
\(\beta_{2}\)\(=\)\((\)\( 72 \nu^{6} - 216 \nu^{5} + 5184 \nu^{4} - 10008 \nu^{3} + 103680 \nu^{2} - 98712 \nu - 42696 \)\()/137\)
\(\beta_{3}\)\(=\)\((\)\( 146 \nu^{6} - 438 \nu^{5} + 13252 \nu^{4} - 25774 \nu^{3} + 477116 \nu^{2} - 464302 \nu + 5226282 \)\()/959\)
\(\beta_{4}\)\(=\)\((\)\( -3078 \nu^{7} + 10773 \nu^{6} - 265689 \nu^{5} + 637290 \nu^{4} - 9124497 \nu^{3} + 13054842 \nu^{2} - 79324947 \nu + 37507653 \)\()/13442440\)
\(\beta_{5}\)\(=\)\((\)\( 6066 \nu^{7} - 21231 \nu^{6} + 1938603 \nu^{5} - 4793430 \nu^{4} + 138256659 \nu^{3} - 202602174 \nu^{2} + 4301554449 \nu - 2117169471 \)\()/23524270\)
\(\beta_{6}\)\(=\)\((\)\( 92502 \nu^{7} - 323757 \nu^{6} + 5437665 \nu^{5} - 12784770 \nu^{4} + 57721113 \nu^{3} - 73958778 \nu^{2} - 1012485933 \nu + 518150979 \)\()/4704854\)
\(\beta_{7}\)\(=\)\((\)\( -55910 \nu^{7} + 195685 \nu^{6} - 5832297 \nu^{5} + 14091530 \nu^{4} - 193915041 \nu^{3} + 276878874 \nu^{2} - 1778978627 \nu + 843807893 \)\()/2688488\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{6} + 9 \beta_{5} + 96 \beta_{4} + 432\)\()/864\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{6} + 18 \beta_{5} + 192 \beta_{4} + 12 \beta_{3} - 3 \beta_{2} - 120 \beta_{1} - 42336\)\()/1728\)
\(\nu^{3}\)\(=\)\((\)\(162 \beta_{7} - 176 \beta_{6} - 576 \beta_{5} - 30474 \beta_{4} + 36 \beta_{3} - 9 \beta_{2} - 360 \beta_{1} - 127872\)\()/3456\)
\(\nu^{4}\)\(=\)\((\)\(81 \beta_{7} - 89 \beta_{6} - 297 \beta_{5} - 15333 \beta_{4} - 264 \beta_{3} + 54 \beta_{2} + 5664 \beta_{1} + 364176\)\()/864\)
\(\nu^{5}\)\(=\)\((\)\(-12960 \beta_{7} + 2852 \beta_{6} + 2484 \beta_{5} + 1424736 \beta_{4} - 2700 \beta_{3} + 555 \beta_{2} + 57240 \beta_{1} + 3855168\)\()/3456\)
\(\nu^{6}\)\(=\)\((\)\(-19845 \beta_{7} + 4724 \beta_{6} + 5220 \beta_{5} + 2213865 \beta_{4} + 19188 \beta_{3} + 39 \beta_{2} - 581976 \beta_{1} + 7627392\)\()/1728\)
\(\nu^{7}\)\(=\)\((\)\(283311 \beta_{7} + 74398 \beta_{6} + 254286 \beta_{5} - 26602971 \beta_{4} + 71904 \beta_{3} - 840 \beta_{2} - 2137296 \beta_{1} + 19874592\)\()/1728\)

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
−1.23205 + 6.41320i
−1.23205 3.79090i
−1.23205 + 3.79090i
−1.23205 6.41320i
2.23205 6.41320i
2.23205 + 3.79090i
2.23205 3.79090i
2.23205 + 6.41320i
0 −15.5885 0 195.235i 0 277.510i 0 243.000 0
319.2 0 −15.5885 0 85.3891i 0 511.824i 0 243.000 0
319.3 0 −15.5885 0 85.3891i 0 511.824i 0 243.000 0
319.4 0 −15.5885 0 195.235i 0 277.510i 0 243.000 0
319.5 0 15.5885 0 195.235i 0 277.510i 0 243.000 0
319.6 0 15.5885 0 85.3891i 0 511.824i 0 243.000 0
319.7 0 15.5885 0 85.3891i 0 511.824i 0 243.000 0
319.8 0 15.5885 0 195.235i 0 277.510i 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.c 8
4.b odd 2 1 inner 384.7.b.c 8
8.b even 2 1 inner 384.7.b.c 8
8.d odd 2 1 inner 384.7.b.c 8
16.e even 4 2 768.7.g.g 8
16.f odd 4 2 768.7.g.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.7.b.c 8 1.a even 1 1 trivial
384.7.b.c 8 4.b odd 2 1 inner
384.7.b.c 8 8.b even 2 1 inner
384.7.b.c 8 8.d odd 2 1 inner
768.7.g.g 8 16.e even 4 2
768.7.g.g 8 16.f odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 45408 T_{5}^{2} + 277920000 \) acting on \(S_{7}^{\mathrm{new}}(384, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( -243 + T^{2} )^{4} \)
$5$ \( ( 277920000 + 45408 T^{2} + T^{4} )^{2} \)
$7$ \( ( 20174323968 + 338976 T^{2} + T^{4} )^{2} \)
$11$ \( ( 4522189383936 - 4545120 T^{2} + T^{4} )^{2} \)
$13$ \( ( 11711515004928 + 9088896 T^{2} + T^{4} )^{2} \)
$17$ \( ( -58718780 + 1636 T + T^{2} )^{4} \)
$19$ \( ( 2107360836000000 - 128143200 T^{2} + T^{4} )^{2} \)
$23$ \( ( 2996687502839808 + 180514944 T^{2} + T^{4} )^{2} \)
$29$ \( ( 123449102940268800 + 1869854304 T^{2} + T^{4} )^{2} \)
$31$ \( ( 6459874959912599808 + 5276545056 T^{2} + T^{4} )^{2} \)
$37$ \( ( 1035562157965836288 + 7121639424 T^{2} + T^{4} )^{2} \)
$41$ \( ( 3840115012 + 124892 T + T^{2} )^{4} \)
$43$ \( ( 1701874998525913344 - 2889761376 T^{2} + T^{4} )^{2} \)
$47$ \( ( 80088648191955505152 + 26657465472 T^{2} + T^{4} )^{2} \)
$53$ \( ( \)\(91\!\cdots\!00\)\( + 86629009248 T^{2} + T^{4} )^{2} \)
$59$ \( ( \)\(63\!\cdots\!00\)\( - 59428064352 T^{2} + T^{4} )^{2} \)
$61$ \( ( 19934164088443699200 + 9877596672 T^{2} + T^{4} )^{2} \)
$67$ \( ( \)\(24\!\cdots\!84\)\( - 140980616544 T^{2} + T^{4} )^{2} \)
$71$ \( ( \)\(55\!\cdots\!92\)\( + 569594613888 T^{2} + T^{4} )^{2} \)
$73$ \( ( 36431647300 + 490660 T + T^{2} )^{4} \)
$79$ \( ( \)\(11\!\cdots\!00\)\( + 790499817504 T^{2} + T^{4} )^{2} \)
$83$ \( ( \)\(27\!\cdots\!44\)\( - 509657219424 T^{2} + T^{4} )^{2} \)
$89$ \( ( -40865541500 + 423748 T + T^{2} )^{4} \)
$97$ \( ( 904356570436 - 1908188 T + T^{2} )^{4} \)
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