Properties

Label 384.6.c.b
Level $384$
Weight $6$
Character orbit 384.c
Analytic conductor $61.587$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 384.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(61.5873868082\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial: \( x^{20} + 306 x^{18} + 37827 x^{16} + 2442168 x^{14} + 88368509 x^{12} + 1774000974 x^{10} + 18093172325 x^{8} + 74958811500 x^{6} + 79355888475 x^{4} + \cdots + 2870280625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{88}\cdot 3^{14}\cdot 41^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} - \beta_1 q^{5} + \beta_{9} q^{7} + (\beta_{11} + \beta_{6}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{3} - \beta_1 q^{5} + \beta_{9} q^{7} + (\beta_{11} + \beta_{6}) q^{9} + (\beta_{5} - \beta_{4} - 47) q^{11} + ( - \beta_{7} - 3 \beta_{5}) q^{13} + ( - \beta_{13} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_1 + 42) q^{15} + ( - \beta_{19} - \beta_{13} - 2 \beta_{9} - \beta_{6}) q^{17} + (\beta_{19} + 2 \beta_{11} - \beta_{10} - \beta_{8} + 5 \beta_{6} - 4 \beta_{5} + 6 \beta_1 - 1) q^{19} + ( - \beta_{18} - \beta_{11} - \beta_{10} + 4 \beta_{9} - \beta_{8} - \beta_{7} + 2 \beta_{4} + \beta_1 + 81) q^{21} + (\beta_{18} + \beta_{16} - \beta_{15} - \beta_{14} - \beta_{13} + \beta_{12} - 5 \beta_{11} + \beta_{10} + \beta_{9} + \cdots + 15) q^{23}+ \cdots + ( - 6 \beta_{19} - 44 \beta_{18} - 3 \beta_{17} - 13 \beta_{16} - 35 \beta_{15} + \cdots - 18919) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 2 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 2 q^{3} - 948 q^{11} + 852 q^{15} + 1640 q^{21} + 328 q^{23} - 12500 q^{25} - 2030 q^{27} + 2836 q^{33} + 7184 q^{35} + 15056 q^{37} - 12980 q^{39} + 11800 q^{45} + 36640 q^{47} - 33388 q^{49} - 1936 q^{51} + 15404 q^{57} - 62908 q^{59} + 73264 q^{61} + 23608 q^{63} - 84024 q^{69} + 34888 q^{71} + 52568 q^{73} - 115698 q^{75} + 55444 q^{81} + 225172 q^{83} - 30112 q^{85} - 225700 q^{87} - 148016 q^{93} + 418616 q^{95} + 7600 q^{97} - 378260 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 306 x^{18} + 37827 x^{16} + 2442168 x^{14} + 88368509 x^{12} + 1774000974 x^{10} + 18093172325 x^{8} + 74958811500 x^{6} + 79355888475 x^{4} + \cdots + 2870280625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 87\!\cdots\!54 \nu^{19} + \cdots + 43\!\cdots\!50 \nu ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 12\!\cdots\!64 \nu^{19} + \cdots + 61\!\cdots\!00 ) / 11\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 45\!\cdots\!26 \nu^{19} + \cdots - 11\!\cdots\!25 ) / 94\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 45\!\cdots\!26 \nu^{19} + \cdots + 15\!\cdots\!75 ) / 94\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 45\!\cdots\!26 \nu^{19} + \cdots + 13\!\cdots\!25 ) / 94\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 46\!\cdots\!64 \nu^{19} + \cdots + 31\!\cdots\!00 \nu ) / 50\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13\!\cdots\!78 \nu^{19} + \cdots + 32\!\cdots\!25 ) / 94\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 22\!\cdots\!74 \nu^{19} + \cdots - 31\!\cdots\!75 ) / 94\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 61\!\cdots\!54 \nu^{19} + \cdots - 73\!\cdots\!25 \nu ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 45\!\cdots\!26 \nu^{19} + \cdots - 13\!\cdots\!25 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 29\!\cdots\!96 \nu^{19} + \cdots - 76\!\cdots\!75 ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 16\!\cdots\!26 \nu^{19} + \cdots - 42\!\cdots\!75 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 16\!\cdots\!64 \nu^{19} + \cdots + 13\!\cdots\!50 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 11\!\cdots\!62 \nu^{19} + \cdots - 74\!\cdots\!75 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 44\!\cdots\!06 \nu^{19} + \cdots + 34\!\cdots\!25 ) / 94\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 53\!\cdots\!98 \nu^{19} + \cdots + 18\!\cdots\!25 ) / 94\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 35\!\cdots\!18 \nu^{19} + \cdots + 86\!\cdots\!50 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 15\!\cdots\!18 \nu^{19} + \cdots + 22\!\cdots\!75 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 51\!\cdots\!88 \nu^{19} + \cdots - 13\!\cdots\!50 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( - 158 \beta_{19} - 32 \beta_{18} - 190 \beta_{17} + 222 \beta_{16} - 148 \beta_{15} - 18 \beta_{14} + 36 \beta_{13} + 74 \beta_{12} + 24 \beta_{11} - 596 \beta_{10} + 1948 \beta_{9} + 36 \beta_{8} - 190 \beta_{7} + 907 \beta_{6} + \cdots - 290 ) / 94464 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 22 \beta_{19} + 536 \beta_{18} + 22 \beta_{17} + 558 \beta_{16} + 487 \beta_{15} - 66 \beta_{14} - 492 \beta_{13} + 1423 \beta_{12} + 5571 \beta_{11} - 1016 \beta_{10} + 536 \beta_{9} + 2806 \beta_{7} + \cdots - 2886677 ) / 94464 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 12588 \beta_{19} - 60 \beta_{18} + 7608 \beta_{17} - 7548 \beta_{16} + 7574 \beta_{15} - 700 \beta_{14} - 1716 \beta_{13} + 26 \beta_{12} + 86 \beta_{11} + 67332 \beta_{10} - 138228 \beta_{9} - 6636 \beta_{8} + \cdots + 52090 ) / 94464 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 11516 \beta_{19} - 67004 \beta_{18} + 11516 \beta_{17} - 55488 \beta_{16} - 93969 \beta_{15} + 26788 \beta_{14} + 90036 \beta_{13} - 95367 \beta_{12} - 468545 \beta_{11} + \cdots + 169463777 ) / 94464 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1157992 \beta_{19} + 142616 \beta_{18} - 333464 \beta_{17} + 190848 \beta_{16} - 555190 \beta_{15} + 143464 \beta_{14} - 11244 \beta_{13} - 364342 \beta_{12} + 574294 \beta_{11} + \cdots - 4975110 ) / 94464 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 9063 \beta_{19} + 40098 \beta_{18} - 9063 \beta_{17} + 31035 \beta_{16} + 72948 \beta_{15} - 21114 \beta_{14} - 58224 \beta_{13} + 37302 \beta_{12} + 264006 \beta_{11} - 70594 \beta_{10} + \cdots - 72004531 ) / 576 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 109740394 \beta_{19} - 18581960 \beta_{18} + 15180842 \beta_{17} + 3401118 \beta_{16} + 45757774 \beta_{15} - 12501430 \beta_{14} + 11813652 \beta_{13} + 49158892 \beta_{12} + \cdots + 455565084 ) / 94464 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 150155950 \beta_{19} - 619978528 \beta_{18} + 150155950 \beta_{17} - 469822578 \beta_{16} - 1291363435 \beta_{15} + 365593198 \beta_{14} + \cdots + 916127584893 ) / 94464 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 10412254104 \beta_{19} + 1923147732 \beta_{18} - 677615484 \beta_{17} - 1245532248 \beta_{16} - 3986014962 \beta_{15} + 944260320 \beta_{14} + \cdots - 41783882666 ) / 94464 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 14308089724 \beta_{19} + 57863278012 \beta_{18} - 14308089724 \beta_{17} + 43555188288 \beta_{16} + 129672026001 \beta_{15} - 35944057184 \beta_{14} + \cdots - 76164955966193 ) / 94464 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 982668597896 \beta_{19} - 185692209688 \beta_{18} + 26830889248 \beta_{17} + 158861320440 \beta_{16} + 357064421880 \beta_{15} - 69994249204 \beta_{14} + \cdots + 3851636553484 ) / 94464 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 4085546226 \beta_{19} - 16416458988 \beta_{18} + 4085546226 \beta_{17} - 12330912762 \beta_{16} - 38308667920 \beta_{15} + 10452870287 \beta_{14} + \cdots + 20156931540638 ) / 288 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 92227714270454 \beta_{19} + 17490873542560 \beta_{18} - 633412128262 \beta_{17} - 16857461414298 \beta_{16} - 32453856638544 \beta_{15} + \cdots - 356237369188494 ) / 94464 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 124793470842386 \beta_{19} + 500555902335512 \beta_{18} - 124793470842386 \beta_{17} + 375762431493126 \beta_{16} + \cdots - 58\!\cdots\!29 ) / 94464 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 86\!\cdots\!96 \beta_{19} + \cdots + 33\!\cdots\!78 ) / 94464 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 11\!\cdots\!60 \beta_{19} + \cdots + 53\!\cdots\!49 ) / 94464 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 80\!\cdots\!16 \beta_{19} + \cdots - 30\!\cdots\!14 ) / 94464 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 65\!\cdots\!59 \beta_{19} + \cdots - 29\!\cdots\!47 ) / 576 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 74\!\cdots\!58 \beta_{19} + \cdots + 28\!\cdots\!48 ) / 94464 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
383.1
7.62048i
7.62048i
0.671758i
0.671758i
0.406991i
0.406991i
9.62962i
9.62962i
5.94193i
5.94193i
6.14819i
6.14819i
7.32004i
7.32004i
4.52328i
4.52328i
2.58939i
2.58939i
0.852576i
0.852576i
0 −15.4378 2.16198i 0 81.6092i 0 91.7503i 0 233.652 + 66.7525i 0
383.2 0 −15.4378 + 2.16198i 0 81.6092i 0 91.7503i 0 233.652 66.7525i 0
383.3 0 −15.1145 3.81453i 0 40.5648i 0 81.4905i 0 213.899 + 115.310i 0
383.4 0 −15.1145 + 3.81453i 0 40.5648i 0 81.4905i 0 213.899 115.310i 0
383.5 0 −10.1204 11.8566i 0 19.3214i 0 82.7816i 0 −38.1561 + 239.986i 0
383.6 0 −10.1204 + 11.8566i 0 19.3214i 0 82.7816i 0 −38.1561 239.986i 0
383.7 0 −5.16061 14.7095i 0 48.6693i 0 167.754i 0 −189.736 + 151.820i 0
383.8 0 −5.16061 + 14.7095i 0 48.6693i 0 167.754i 0 −189.736 151.820i 0
383.9 0 −2.25525 15.4245i 0 86.6842i 0 110.468i 0 −232.828 + 69.5720i 0
383.10 0 −2.25525 + 15.4245i 0 86.6842i 0 110.468i 0 −232.828 69.5720i 0
383.11 0 −1.67629 15.4981i 0 6.76651i 0 132.568i 0 −237.380 + 51.9584i 0
383.12 0 −1.67629 + 15.4981i 0 6.76651i 0 132.568i 0 −237.380 51.9584i 0
383.13 0 10.3565 11.6509i 0 68.3281i 0 220.652i 0 −28.4850 241.325i 0
383.14 0 10.3565 + 11.6509i 0 68.3281i 0 220.652i 0 −28.4850 + 241.325i 0
383.15 0 10.5848 11.4438i 0 101.915i 0 13.9460i 0 −18.9231 242.262i 0
383.16 0 10.5848 + 11.4438i 0 101.915i 0 13.9460i 0 −18.9231 + 242.262i 0
383.17 0 12.3458 9.51744i 0 26.7752i 0 70.6872i 0 61.8366 235.001i 0
383.18 0 12.3458 + 9.51744i 0 26.7752i 0 70.6872i 0 61.8366 + 235.001i 0
383.19 0 15.4777 1.85457i 0 55.8580i 0 225.953i 0 236.121 57.4091i 0
383.20 0 15.4777 + 1.85457i 0 55.8580i 0 225.953i 0 236.121 + 57.4091i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 383.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.6.c.b yes 20
3.b odd 2 1 384.6.c.c yes 20
4.b odd 2 1 384.6.c.c yes 20
8.b even 2 1 384.6.c.d yes 20
8.d odd 2 1 384.6.c.a 20
12.b even 2 1 inner 384.6.c.b yes 20
24.f even 2 1 384.6.c.d yes 20
24.h odd 2 1 384.6.c.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.6.c.a 20 8.d odd 2 1
384.6.c.a 20 24.h odd 2 1
384.6.c.b yes 20 1.a even 1 1 trivial
384.6.c.b yes 20 12.b even 2 1 inner
384.6.c.c yes 20 3.b odd 2 1
384.6.c.c yes 20 4.b odd 2 1
384.6.c.d yes 20 8.b even 2 1
384.6.c.d yes 20 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 \(S_{6}^{\mathrm{new}}(384, [\chi])\):

\( T_{11}^{10} + 474 T_{11}^{9} - 742804 T_{11}^{8} - 368139504 T_{11}^{7} + 129628916560 T_{11}^{6} + 76348018199520 T_{11}^{5} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
\( T_{13}^{10} - 1985572 T_{13}^{8} + 389121024 T_{13}^{7} + 1179051645856 T_{13}^{6} - 377750230069248 T_{13}^{5} + \cdots + 30\!\cdots\!80 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + 2 T^{19} + \cdots + 71\!\cdots\!49 \) Copy content Toggle raw display
$5$ \( T^{20} + 37500 T^{18} + \cdots + 36\!\cdots\!16 \) Copy content Toggle raw display
$7$ \( T^{20} + 184764 T^{18} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{10} + 474 T^{9} + \cdots + 11\!\cdots\!16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} - 1985572 T^{8} + \cdots + 30\!\cdots\!80)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + 15994608 T^{18} + \cdots + 45\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{20} + 26387276 T^{18} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{10} - 164 T^{9} + \cdots + 90\!\cdots\!44)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + 219109596 T^{18} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{20} + 320247692 T^{18} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{10} - 7528 T^{9} + \cdots + 13\!\cdots\!60)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + 1324231920 T^{18} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( T^{20} + 1477477740 T^{18} + \cdots + 79\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{10} - 18320 T^{9} + \cdots + 30\!\cdots\!20)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + 4334244732 T^{18} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{10} + 31454 T^{9} + \cdots - 30\!\cdots\!36)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} - 36632 T^{9} + \cdots + 17\!\cdots\!08)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + 16158911196 T^{18} + \cdots + 28\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{10} - 17444 T^{9} + \cdots - 13\!\cdots\!84)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} - 26284 T^{9} + \cdots - 16\!\cdots\!40)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + 37377699244 T^{18} + \cdots + 91\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( (T^{10} - 112586 T^{9} + \cdots - 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + 58375922848 T^{18} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{10} - 3800 T^{9} + \cdots + 14\!\cdots\!68)^{2} \) Copy content Toggle raw display
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