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: \( x^{8} - 172x^{6} + 13179x^{4} - 522628x^{2} + 8755681 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + (\beta_{5} - 9) q^{3} - \beta_{6} q^{5} + (\beta_{6} + \beta_{3}) q^{7} + (3 \beta_{7} - 21 \beta_{5} - 15 \beta_{4} - 207) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} - 9) q^{3} - \beta_{6} q^{5} + (\beta_{6} + \beta_{3}) q^{7} + (3 \beta_{7} - 21 \beta_{5} - 15 \beta_{4} - 207) q^{9} + ( - 13 \beta_{5} - 13 \beta_{4} - 206) q^{11} - \beta_1 q^{13} + (9 \beta_{6} - 5 \beta_{3} + \beta_{2} + \beta_1) q^{15} + (22 \beta_{7} + 30 \beta_{5} - 30 \beta_{4}) q^{17} + (4 \beta_{7} + 51 \beta_{5} - 51 \beta_{4}) q^{19} + (27 \beta_{6} - 4 \beta_{3} - 4 \beta_{2} - \beta_1) q^{21} + (6 \beta_{2} - 2 \beta_1) q^{23} + (20 \beta_{5} + 20 \beta_{4} - 25) q^{25} + ( - 108 \beta_{7} - 306 \beta_{5} - 189 \beta_{4} - 2673) q^{27} + (99 \beta_{6} - 88 \beta_{3}) q^{29} + (125 \beta_{6} - 23 \beta_{3}) q^{31} + ( - 39 \beta_{7} - 167 \beta_{5} + 195 \beta_{4} - 2826) q^{33} + ( - 1580 \beta_{5} - 1580 \beta_{4} - 17040) q^{35} + ( - 24 \beta_{2} - 19 \beta_1) q^{37} + ( - 252 \beta_{6} + 66 \beta_{3} - 3 \beta_{2} + 21 \beta_1) q^{39} + ( - 280 \beta_{7} - 468 \beta_{5} + 468 \beta_{4}) q^{41} + ( - 572 \beta_{7} + 561 \beta_{5} - 561 \beta_{4}) q^{43} + (207 \beta_{6} + 180 \beta_{3} + 12 \beta_{2} - 18 \beta_1) q^{45} + ( - 48 \beta_{2} - 40 \beta_1) q^{47} + (3284 \beta_{5} + 3284 \beta_{4} + 13151) q^{49} + ( - 240 \beta_{7} - 2214 \beta_{5} - 4146 \beta_{4} + \cdots - 34416) q^{51}+ \cdots + (84 \beta_{7} + 3741 \beta_{5} + 9057 \beta_{4} + 211122) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 72 q^{3} - 1656 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 72 q^{3} - 1656 q^{9} - 1648 q^{11} - 200 q^{25} - 21384 q^{27} - 22608 q^{33} - 136320 q^{35} + 105208 q^{49} - 275328 q^{51} - 391104 q^{57} - 836624 q^{59} - 1964944 q^{73} + 59400 q^{75} + 166536 q^{81} + 587024 q^{83} - 1477232 q^{97} + 1688976 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( 4304\nu^{6} - 469248\nu^{4} + 18250656\nu^{2} - 251351264 ) / 97081 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10144\nu^{6} - 1244544\nu^{4} + 74403840\nu^{2} - 1830375232 ) / 97081 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -31800\nu^{7} + 5753664\nu^{5} - 475691952\nu^{3} + 27076321320\nu ) / 287262679 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 37990 \nu^{7} - 47344 \nu^{6} - 4705618 \nu^{5} + 9433292 \nu^{4} + 279940646 \nu^{3} - 593741104 \nu^{2} - 6688318664 \nu + 16218154722 ) / 287262679 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 37990 \nu^{7} - 47344 \nu^{6} + 4705618 \nu^{5} + 9433292 \nu^{4} - 279940646 \nu^{3} - 593741104 \nu^{2} + 6688318664 \nu + 16218154722 ) / 287262679 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -293636\nu^{7} + 47925144\nu^{5} - 2781189072\nu^{3} + 71694324452\nu ) / 861788037 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -240320\nu^{7} + 26137616\nu^{5} - 1288141264\nu^{3} + 26931123856\nu ) / 287262679 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} - 4\beta_{5} + 4\beta_{4} + 2\beta_{3} ) / 96 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 48\beta_{5} + 48\beta_{4} + \beta_{2} - 2\beta _1 + 8256 ) / 192 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 127\beta_{7} + 90\beta_{6} - 544\beta_{5} + 544\beta_{4} + 63\beta_{3} ) / 96 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2718\beta_{5} + 2718\beta_{4} + 23\beta_{2} - 34\beta _1 + 38712 ) / 48 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5056\beta_{7} + 11970\beta_{6} - 31372\beta_{5} + 31372\beta_{4} - 95\beta_{3} ) / 96 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 163632\beta_{5} + 163632\beta_{4} + 965\beta_{2} - 336\beta _1 - 1152256 ) / 32 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -133523\beta_{7} + 819468\beta_{6} - 944428\beta_{5} + 944428\beta_{4} - 123892\beta_{3} ) / 96 \) Copy content Toggle raw display

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 \(S_{7}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{4} - 31200T_{5}^{2} + 243072000 \) Copy content Toggle raw display
\( T_{11}^{2} + 412T_{11} - 79244 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 36 T^{3} + 1062 T^{2} + \cdots + 531441)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - 31200 T^{2} + \cdots + 243072000)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 261600 T^{2} + \cdots + 9343687680)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 412 T - 79244)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 12372480 T^{2} + \cdots + 1562505707520)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 31653504 T^{2} + \cdots + 231275569987584)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 11092896 T^{2} + \cdots + 6011206401024)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 590284800 T^{2} + \cdots + 87\!\cdots\!20)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 1995222240 T^{2} + \cdots + 51\!\cdots\!80)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 589774560 T^{2} + \cdots + 33\!\cdots\!80)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 7027776000 T^{2} + \cdots + 45\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 5457148416 T^{2} + \cdots + 62\!\cdots\!44)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 18530946720 T^{2} + \cdots + 34\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 29138780160 T^{2} + \cdots + 93\!\cdots\!20)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 36473745120 T^{2} + \cdots + 10\!\cdots\!80)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 209156 T - 29855566796)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 108994152960 T^{2} + \cdots + 28\!\cdots\!20)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 189436899744 T^{2} + \cdots + 60\!\cdots\!04)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 258245314560 T^{2} + \cdots + 16\!\cdots\!20)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 491236 T + 49489033924)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 1037364956640 T^{2} + \cdots + 26\!\cdots\!80)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 146756 T + 1115288884)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 1658159600256 T^{2} + \cdots + 25\!\cdots\!64)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 369308 T - 1772532994364)^{4} \) Copy content Toggle raw display
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