Properties

Label 384.7.h.e
Level $384$
Weight $7$
Character orbit 384.h
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.h (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: \(x^{8} - 172 x^{6} + 13179 x^{4} - 522628 x^{2} + 8755681\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{6} \)
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.

\(f(q)\) \(=\) \( q + ( -9 + \beta_{5} ) q^{3} -\beta_{6} q^{5} + ( \beta_{3} + \beta_{6} ) q^{7} + ( -207 - 15 \beta_{4} - 21 \beta_{5} + 3 \beta_{7} ) q^{9} +O(q^{10})\) \( q + ( -9 + \beta_{5} ) q^{3} -\beta_{6} q^{5} + ( \beta_{3} + \beta_{6} ) q^{7} + ( -207 - 15 \beta_{4} - 21 \beta_{5} + 3 \beta_{7} ) q^{9} + ( -206 - 13 \beta_{4} - 13 \beta_{5} ) q^{11} -\beta_{1} q^{13} + ( \beta_{1} + \beta_{2} - 5 \beta_{3} + 9 \beta_{6} ) q^{15} + ( -30 \beta_{4} + 30 \beta_{5} + 22 \beta_{7} ) q^{17} + ( -51 \beta_{4} + 51 \beta_{5} + 4 \beta_{7} ) q^{19} + ( -\beta_{1} - 4 \beta_{2} - 4 \beta_{3} + 27 \beta_{6} ) q^{21} + ( -2 \beta_{1} + 6 \beta_{2} ) q^{23} + ( -25 + 20 \beta_{4} + 20 \beta_{5} ) q^{25} + ( -2673 - 189 \beta_{4} - 306 \beta_{5} - 108 \beta_{7} ) q^{27} + ( -88 \beta_{3} + 99 \beta_{6} ) q^{29} + ( -23 \beta_{3} + 125 \beta_{6} ) q^{31} + ( -2826 + 195 \beta_{4} - 167 \beta_{5} - 39 \beta_{7} ) q^{33} + ( -17040 - 1580 \beta_{4} - 1580 \beta_{5} ) q^{35} + ( -19 \beta_{1} - 24 \beta_{2} ) q^{37} + ( 21 \beta_{1} - 3 \beta_{2} + 66 \beta_{3} - 252 \beta_{6} ) q^{39} + ( 468 \beta_{4} - 468 \beta_{5} - 280 \beta_{7} ) q^{41} + ( -561 \beta_{4} + 561 \beta_{5} - 572 \beta_{7} ) q^{43} + ( -18 \beta_{1} + 12 \beta_{2} + 180 \beta_{3} + 207 \beta_{6} ) q^{45} + ( -40 \beta_{1} - 48 \beta_{2} ) q^{47} + ( 13151 + 3284 \beta_{4} + 3284 \beta_{5} ) q^{49} + ( -34416 - 4146 \beta_{4} - 2214 \beta_{5} - 240 \beta_{7} ) q^{51} + ( -272 \beta_{3} - 773 \beta_{6} ) q^{53} + ( 130 \beta_{3} + 206 \beta_{6} ) q^{55} + ( -48888 - 1437 \beta_{4} - 1359 \beta_{5} + 93 \beta_{7} ) q^{57} + ( -104578 - 7527 \beta_{4} - 7527 \beta_{5} ) q^{59} + ( 95 \beta_{1} + 72 \beta_{2} ) q^{61} + ( -54 \beta_{1} - 30 \beta_{2} - 387 \beta_{3} - 1503 \beta_{6} ) q^{63} + ( 4560 \beta_{4} - 4560 \beta_{5} + 1500 \beta_{7} ) q^{65} + ( -789 \beta_{4} + 789 \beta_{5} - 1856 \beta_{7} ) q^{67} + ( 114 \beta_{1} + 12 \beta_{2} + 1068 \beta_{3} + 792 \beta_{6} ) q^{69} + ( 150 \beta_{1} + 102 \beta_{2} ) q^{71} + ( -245618 - 3880 \beta_{4} - 3880 \beta_{5} ) q^{73} + ( 7425 - 300 \beta_{4} - 85 \beta_{5} + 60 \beta_{7} ) q^{75} + ( -336 \beta_{3} - 1142 \beta_{6} ) q^{77} + ( -43 \beta_{3} + 5769 \beta_{6} ) q^{79} + ( 20817 + 22734 \beta_{4} + 7074 \beta_{5} + 702 \beta_{7} ) q^{81} + ( 73378 + 2435 \beta_{4} + 2435 \beta_{5} ) q^{83} + ( -28 \beta_{1} + 192 \beta_{2} ) q^{85} + ( -99 \beta_{1} + 165 \beta_{2} + 1287 \beta_{3} - 4059 \beta_{6} ) q^{87} + ( 12810 \beta_{4} - 12810 \beta_{5} - 4086 \beta_{7} ) q^{89} + ( 4368 \beta_{4} - 4368 \beta_{5} + 3036 \beta_{7} ) q^{91} + ( -125 \beta_{1} - 56 \beta_{2} + 832 \beta_{3} - 1953 \beta_{6} ) q^{93} + ( 86 \beta_{1} + 126 \beta_{2} ) q^{95} + ( -184654 + 50092 \beta_{4} + 50092 \beta_{5} ) q^{97} + ( 211122 + 9057 \beta_{4} + 3741 \beta_{5} + 84 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 72q^{3} - 1656q^{9} + O(q^{10}) \) \( 8q - 72q^{3} - 1656q^{9} - 1648q^{11} - 200q^{25} - 21384q^{27} - 22608q^{33} - 136320q^{35} + 105208q^{49} - 275328q^{51} - 391104q^{57} - 836624q^{59} - 1964944q^{73} + 59400q^{75} + 166536q^{81} + 587024q^{83} - 1477232q^{97} + 1688976q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 172 x^{6} + 13179 x^{4} - 522628 x^{2} + 8755681\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 4304 \nu^{6} - 469248 \nu^{4} + 18250656 \nu^{2} - 251351264 \)\()/97081\)
\(\beta_{2}\)\(=\)\((\)\( 10144 \nu^{6} - 1244544 \nu^{4} + 74403840 \nu^{2} - 1830375232 \)\()/97081\)
\(\beta_{3}\)\(=\)\((\)\( -31800 \nu^{7} + 5753664 \nu^{5} - 475691952 \nu^{3} + 27076321320 \nu \)\()/ 287262679 \)
\(\beta_{4}\)\(=\)\((\)\( 37990 \nu^{7} - 47344 \nu^{6} - 4705618 \nu^{5} + 9433292 \nu^{4} + 279940646 \nu^{3} - 593741104 \nu^{2} - 6688318664 \nu + 16218154722 \)\()/ 287262679 \)
\(\beta_{5}\)\(=\)\((\)\( -37990 \nu^{7} - 47344 \nu^{6} + 4705618 \nu^{5} + 9433292 \nu^{4} - 279940646 \nu^{3} - 593741104 \nu^{2} + 6688318664 \nu + 16218154722 \)\()/ 287262679 \)
\(\beta_{6}\)\(=\)\((\)\( -293636 \nu^{7} + 47925144 \nu^{5} - 2781189072 \nu^{3} + 71694324452 \nu \)\()/ 861788037 \)
\(\beta_{7}\)\(=\)\((\)\( -240320 \nu^{7} + 26137616 \nu^{5} - 1288141264 \nu^{3} + 26931123856 \nu \)\()/ 287262679 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} - 4 \beta_{5} + 4 \beta_{4} + 2 \beta_{3}\)\()/96\)
\(\nu^{2}\)\(=\)\((\)\(48 \beta_{5} + 48 \beta_{4} + \beta_{2} - 2 \beta_{1} + 8256\)\()/192\)
\(\nu^{3}\)\(=\)\((\)\(127 \beta_{7} + 90 \beta_{6} - 544 \beta_{5} + 544 \beta_{4} + 63 \beta_{3}\)\()/96\)
\(\nu^{4}\)\(=\)\((\)\(2718 \beta_{5} + 2718 \beta_{4} + 23 \beta_{2} - 34 \beta_{1} + 38712\)\()/48\)
\(\nu^{5}\)\(=\)\((\)\(5056 \beta_{7} + 11970 \beta_{6} - 31372 \beta_{5} + 31372 \beta_{4} - 95 \beta_{3}\)\()/96\)
\(\nu^{6}\)\(=\)\((\)\(163632 \beta_{5} + 163632 \beta_{4} + 965 \beta_{2} - 336 \beta_{1} - 1152256\)\()/32\)
\(\nu^{7}\)\(=\)\((\)\(-133523 \beta_{7} + 819468 \beta_{6} - 944428 \beta_{5} + 944428 \beta_{4} - 123892 \beta_{3}\)\()/96\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
−6.86097 + 3.28347i
6.86097 + 3.28347i
−6.86097 3.28347i
6.86097 3.28347i
7.10120 + 0.847848i
−7.10120 + 0.847848i
7.10120 0.847848i
−7.10120 0.847848i
0 −22.4164 15.0501i 0 −122.733 0 −206.594 0 275.991 + 674.737i 0
65.2 0 −22.4164 15.0501i 0 122.733 0 206.594 0 275.991 + 674.737i 0
65.3 0 −22.4164 + 15.0501i 0 −122.733 0 −206.594 0 275.991 674.737i 0
65.4 0 −22.4164 + 15.0501i 0 122.733 0 206.594 0 275.991 674.737i 0
65.5 0 4.41641 26.6364i 0 −127.030 0 467.888 0 −689.991 235.274i 0
65.6 0 4.41641 26.6364i 0 127.030 0 −467.888 0 −689.991 235.274i 0
65.7 0 4.41641 + 26.6364i 0 −127.030 0 467.888 0 −689.991 + 235.274i 0
65.8 0 4.41641 + 26.6364i 0 127.030 0 −467.888 0 −689.991 + 235.274i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
12.b even 2 1 inner
24.h 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.h.e 8
3.b odd 2 1 384.7.h.f yes 8
4.b odd 2 1 384.7.h.f yes 8
8.b even 2 1 384.7.h.f yes 8
8.d odd 2 1 inner 384.7.h.e 8
12.b even 2 1 inner 384.7.h.e 8
24.f even 2 1 384.7.h.f yes 8
24.h odd 2 1 inner 384.7.h.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.7.h.e 8 1.a even 1 1 trivial
384.7.h.e 8 8.d odd 2 1 inner
384.7.h.e 8 12.b even 2 1 inner
384.7.h.e 8 24.h odd 2 1 inner
384.7.h.f yes 8 3.b odd 2 1
384.7.h.f yes 8 4.b odd 2 1
384.7.h.f yes 8 8.b even 2 1
384.7.h.f yes 8 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{4} - 31200 T_{5}^{2} + 243072000 \)
\( T_{11}^{2} + 412 T_{11} - 79244 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 531441 + 26244 T + 1062 T^{2} + 36 T^{3} + T^{4} )^{2} \)
$5$ \( ( 243072000 - 31200 T^{2} + T^{4} )^{2} \)
$7$ \( ( 9343687680 - 261600 T^{2} + T^{4} )^{2} \)
$11$ \( ( -79244 + 412 T + T^{2} )^{4} \)
$13$ \( ( 1562505707520 + 12372480 T^{2} + T^{4} )^{2} \)
$17$ \( ( 231275569987584 + 31653504 T^{2} + T^{4} )^{2} \)
$19$ \( ( 6011206401024 + 11092896 T^{2} + T^{4} )^{2} \)
$23$ \( ( 87025317886033920 + 590284800 T^{2} + T^{4} )^{2} \)
$29$ \( ( 513352398894382080 - 1995222240 T^{2} + T^{4} )^{2} \)
$31$ \( ( 33938340358855680 - 589774560 T^{2} + T^{4} )^{2} \)
$37$ \( ( 4542243154403328000 + 7027776000 T^{2} + T^{4} )^{2} \)
$41$ \( ( 6253706256515334144 + 5457148416 T^{2} + T^{4} )^{2} \)
$43$ \( ( 34005633871914777600 + 18530946720 T^{2} + T^{4} )^{2} \)
$47$ \( ( 93702742277326110720 + 29138780160 T^{2} + T^{4} )^{2} \)
$53$ \( ( 1022061919954222080 - 36473745120 T^{2} + T^{4} )^{2} \)
$59$ \( ( -29855566796 + 209156 T + T^{2} )^{4} \)
$61$ \( ( \)\(28\!\cdots\!20\)\( + 108994152960 T^{2} + T^{4} )^{2} \)
$67$ \( ( \)\(60\!\cdots\!04\)\( + 189436899744 T^{2} + T^{4} )^{2} \)
$71$ \( ( \)\(16\!\cdots\!20\)\( + 258245314560 T^{2} + T^{4} )^{2} \)
$73$ \( ( 49489033924 + 491236 T + T^{2} )^{4} \)
$79$ \( ( \)\(26\!\cdots\!80\)\( - 1037364956640 T^{2} + T^{4} )^{2} \)
$83$ \( ( 1115288884 - 146756 T + T^{2} )^{4} \)
$89$ \( ( \)\(25\!\cdots\!64\)\( + 1658159600256 T^{2} + T^{4} )^{2} \)
$97$ \( ( -1772532994364 + 369308 T + T^{2} )^{4} \)
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