Newspace parameters
Level: | \( N \) | \(=\) | \( 384 = 2^{7} \cdot 3 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 384.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(39.6940658242\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{16} + 46 x^{14} + 1311 x^{12} + 24382 x^{10} + 338077 x^{8} + 3338772 x^{6} + 24662556 x^{4} + 120434256 x^{2} + 362673936 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
Coefficient ring index: | \( 2^{84}\cdot 3^{4} \) |
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 46 x^{14} + 1311 x^{12} + 24382 x^{10} + 338077 x^{8} + 3338772 x^{6} + 24662556 x^{4} + 120434256 x^{2} + 362673936 \) :
\(\beta_{1}\) | \(=\) | \( ( - 2266283 \nu^{14} - 66378230 \nu^{12} - 1517851305 \nu^{10} - 18296855030 \nu^{8} - 146533806035 \nu^{6} - 185743103760 \nu^{4} + \cdots + 76779634811400 ) / 728558119080 \) |
\(\beta_{2}\) | \(=\) | \( ( 4108 \nu^{14} + 76360 \nu^{12} + 1460868 \nu^{10} + 9399208 \nu^{8} + 100952428 \nu^{6} + 194090928 \nu^{4} + 9371901600 \nu^{2} + \cdots + 19234287648 ) / 752642685 \) |
\(\beta_{3}\) | \(=\) | \( ( - 556 \nu^{15} - 18699 \nu^{13} - 428444 \nu^{11} - 5765809 \nu^{9} - 58126704 \nu^{7} - 271876859 \nu^{5} - 903142338 \nu^{3} + \cdots + 4265951220 \nu ) / 3179967120 \) |
\(\beta_{4}\) | \(=\) | \( ( - 5339189 \nu^{14} - 164490066 \nu^{12} - 4390456039 \nu^{10} - 65458015394 \nu^{8} - 801533739069 \nu^{6} + \cdots - 121687906180248 ) / 769033570140 \) |
\(\beta_{5}\) | \(=\) | \( ( 57146483 \nu^{14} + 2115064406 \nu^{12} + 52703126049 \nu^{10} + 806141495414 \nu^{8} + 9307681589819 \nu^{6} + \cdots + 673855067434968 ) / 4614201420840 \) |
\(\beta_{6}\) | \(=\) | \( ( 72757 \nu^{14} + 4260382 \nu^{12} + 121527975 \nu^{10} + 2311227190 \nu^{8} + 30370027885 \nu^{6} + 289010652420 \nu^{4} + \cdots + 7044868877688 ) / 2955295530 \) |
\(\beta_{7}\) | \(=\) | \( ( 1601473 \nu^{15} + 98155168 \nu^{13} + 2885978475 \nu^{11} + 54364746640 \nu^{9} + 719902590145 \nu^{7} + 6465681409230 \nu^{5} + \cdots + 118184013684432 \nu ) / 2285023191660 \) |
\(\beta_{8}\) | \(=\) | \( ( - 31371 \nu^{14} - 1579778 \nu^{12} - 43486169 \nu^{10} - 829237674 \nu^{8} - 10759730579 \nu^{6} - 101770416284 \nu^{4} + \cdots - 2463773203656 ) / 985098510 \) |
\(\beta_{9}\) | \(=\) | \( ( 510748 \nu^{15} + 36077731 \nu^{13} + 876218028 \nu^{11} + 14129962873 \nu^{9} + 125697707008 \nu^{7} + 672547885683 \nu^{5} + \cdots - 4704024865428 \nu ) / 543774377520 \) |
\(\beta_{10}\) | \(=\) | \( ( 44399 \nu^{14} + 1891290 \nu^{12} + 45803557 \nu^{10} + 744244322 \nu^{8} + 8371524567 \nu^{6} + 68379174892 \nu^{4} + 366832939368 \nu^{2} + \cdots + 1592996663592 ) / 985098510 \) |
\(\beta_{11}\) | \(=\) | \( ( 6590197 \nu^{15} + 282332912 \nu^{13} + 7259403175 \nu^{11} + 122888600320 \nu^{9} + 1494048049205 \nu^{7} + \cdots + 276231209943888 \nu ) / 4823937849060 \) |
\(\beta_{12}\) | \(=\) | \( ( - 116582967 \nu^{15} - 6636652714 \nu^{13} - 198004463629 \nu^{11} - 3806456973654 \nu^{9} - 49210879046239 \nu^{7} + \cdots - 30\!\cdots\!60 \nu ) / 53063316339660 \) |
\(\beta_{13}\) | \(=\) | \( ( 6838460 \nu^{15} + 179593223 \nu^{13} + 3674734380 \nu^{11} + 27146644325 \nu^{9} - 2549711920 \nu^{7} - 4975055150985 \nu^{5} + \cdots - 307694922226308 \nu ) / 1631323132560 \) |
\(\beta_{14}\) | \(=\) | \( ( - 2256580781 \nu^{15} - 111612544475 \nu^{13} - 3164107444263 \nu^{11} - 58666117370813 \nu^{9} - 758281783113113 \nu^{7} + \cdots - 15\!\cdots\!88 \nu ) / 477569847056940 \) |
\(\beta_{15}\) | \(=\) | \( ( - 893983 \nu^{15} - 28765249 \nu^{13} - 726540813 \nu^{11} - 10463518723 \nu^{9} - 124685999683 \nu^{7} - 859282193763 \nu^{5} + \cdots - 7232245440804 \nu ) / 101957695785 \) |
\(\nu\) | \(=\) | \( ( 6 \beta_{15} - 16 \beta_{14} + 6 \beta_{13} + 16 \beta_{12} - 31 \beta_{11} - 6 \beta_{9} + 17 \beta_{7} - 132 \beta_{3} ) / 1536 \) |
\(\nu^{2}\) | \(=\) | \( ( 7\beta_{10} - \beta_{8} - 12\beta_{6} + 11\beta_{5} - 59\beta_{4} - 36\beta_{2} + 129\beta _1 - 8832 ) / 1536 \) |
\(\nu^{3}\) | \(=\) | \( ( -10\beta_{15} - 2\beta_{14} - 12\beta_{13} - \beta_{12} - 28\beta_{11} + 18\beta_{9} - 37\beta_{7} + 10\beta_{3} ) / 128 \) |
\(\nu^{4}\) | \(=\) | \( ( 76 \beta_{10} + 128 \beta_{8} + 276 \beta_{6} - 1207 \beta_{5} + 103 \beta_{4} + 720 \beta_{2} - 1797 \beta _1 - 97152 ) / 1536 \) |
\(\nu^{5}\) | \(=\) | \( ( 93 \beta_{15} + 5608 \beta_{14} + 2667 \beta_{13} - 2716 \beta_{12} + 29710 \beta_{11} - 2109 \beta_{9} + 3154 \beta_{7} + 175304 \beta_{3} ) / 3072 \) |
\(\nu^{6}\) | \(=\) | \( ( -555\beta_{10} - 111\beta_{8} + 3523\beta_{5} + 2925\beta_{4} + 816\beta_{2} + 2025\beta _1 + 333952 ) / 256 \) |
\(\nu^{7}\) | \(=\) | \( ( 34281 \beta_{15} - 27752 \beta_{14} - 7809 \beta_{13} + 39068 \beta_{12} - 248798 \beta_{11} - 62985 \beta_{9} + 235486 \beta_{7} - 2985688 \beta_{3} ) / 3072 \) |
\(\nu^{8}\) | \(=\) | \( ( 23246 \beta_{10} - 54302 \beta_{8} - 76164 \beta_{6} - 95783 \beta_{5} - 208873 \beta_{4} - 196200 \beta_{2} + 30507 \beta _1 - 9105792 ) / 1536 \) |
\(\nu^{9}\) | \(=\) | \( ( - 5510 \beta_{15} - 34866 \beta_{14} - 11856 \beta_{13} - 17433 \beta_{12} - 57196 \beta_{11} + 30894 \beta_{9} - 214093 \beta_{7} + 14250 \beta_{3} ) / 128 \) |
\(\nu^{10}\) | \(=\) | \( ( 142961 \beta_{10} + 1052881 \beta_{8} + 1127460 \beta_{6} - 2151179 \beta_{5} - 2358661 \beta_{4} + 185580 \beta_{2} - 2759169 \beta _1 - 58564992 ) / 1536 \) |
\(\nu^{11}\) | \(=\) | \( ( - 7730301 \beta_{15} + 8830808 \beta_{14} + 8432997 \beta_{13} + 3010012 \beta_{12} + 23747966 \beta_{11} + 5904093 \beta_{9} + 34545146 \beta_{7} + 669116968 \beta_{3} ) / 3072 \) |
\(\nu^{12}\) | \(=\) | \( ( - 57750 \beta_{10} - 11550 \beta_{8} + 7709435 \beta_{5} + 13461045 \beta_{4} + 5349720 \beta_{2} + 9319185 \beta _1 - 50199424 ) / 256 \) |
\(\nu^{13}\) | \(=\) | \( ( 110384487 \beta_{15} + 141741608 \beta_{14} - 130968543 \beta_{13} + 29702692 \beta_{12} + 274610738 \beta_{11} - 92312007 \beta_{9} + 489737654 \beta_{7} + \cdots - 9266543864 \beta_{3} ) / 3072 \) |
\(\nu^{14}\) | \(=\) | \( ( - 35312813 \beta_{10} - 236287741 \beta_{8} - 212340612 \beta_{6} - 363202879 \beta_{5} - 486159569 \beta_{4} - 4520916 \beta_{2} - 635930781 \beta _1 + 15566656128 ) / 1536 \) |
\(\nu^{15}\) | \(=\) | \( ( 19515260 \beta_{15} - 120946422 \beta_{14} + 38717958 \beta_{13} - 60473211 \beta_{12} - 50067578 \beta_{11} - 96326052 \beta_{9} - 594326477 \beta_{7} - 45014670 \beta_{3} ) / 128 \) |
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 |
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0 | −5.19615 | 0 | − | 39.0495i | 0 | 95.6459i | 0 | 27.0000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
319.2 | 0 | −5.19615 | 0 | − | 23.5183i | 0 | − | 40.3412i | 0 | 27.0000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
319.3 | 0 | −5.19615 | 0 | − | 19.1142i | 0 | − | 6.40010i | 0 | 27.0000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
319.4 | 0 | −5.19615 | 0 | − | 4.54640i | 0 | − | 61.7048i | 0 | 27.0000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
319.5 | 0 | −5.19615 | 0 | 4.54640i | 0 | 61.7048i | 0 | 27.0000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
319.6 | 0 | −5.19615 | 0 | 19.1142i | 0 | 6.40010i | 0 | 27.0000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
319.7 | 0 | −5.19615 | 0 | 23.5183i | 0 | 40.3412i | 0 | 27.0000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
319.8 | 0 | −5.19615 | 0 | 39.0495i | 0 | − | 95.6459i | 0 | 27.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
319.9 | 0 | 5.19615 | 0 | − | 39.0495i | 0 | − | 95.6459i | 0 | 27.0000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
319.10 | 0 | 5.19615 | 0 | − | 23.5183i | 0 | 40.3412i | 0 | 27.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
319.11 | 0 | 5.19615 | 0 | − | 19.1142i | 0 | 6.40010i | 0 | 27.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
319.12 | 0 | 5.19615 | 0 | − | 4.54640i | 0 | 61.7048i | 0 | 27.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
319.13 | 0 | 5.19615 | 0 | 4.54640i | 0 | − | 61.7048i | 0 | 27.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
319.14 | 0 | 5.19615 | 0 | 19.1142i | 0 | − | 6.40010i | 0 | 27.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
319.15 | 0 | 5.19615 | 0 | 23.5183i | 0 | − | 40.3412i | 0 | 27.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
319.16 | 0 | 5.19615 | 0 | 39.0495i | 0 | 95.6459i | 0 | 27.0000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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 | 384.5.b.d | ✓ | 16 |
3.b | odd | 2 | 1 | 1152.5.b.m | 16 | ||
4.b | odd | 2 | 1 | inner | 384.5.b.d | ✓ | 16 |
8.b | even | 2 | 1 | inner | 384.5.b.d | ✓ | 16 |
8.d | odd | 2 | 1 | inner | 384.5.b.d | ✓ | 16 |
12.b | even | 2 | 1 | 1152.5.b.m | 16 | ||
16.e | even | 4 | 1 | 768.5.g.h | 8 | ||
16.e | even | 4 | 1 | 768.5.g.j | 8 | ||
16.f | odd | 4 | 1 | 768.5.g.h | 8 | ||
16.f | odd | 4 | 1 | 768.5.g.j | 8 | ||
24.f | even | 2 | 1 | 1152.5.b.m | 16 | ||
24.h | odd | 2 | 1 | 1152.5.b.m | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
384.5.b.d | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
384.5.b.d | ✓ | 16 | 4.b | odd | 2 | 1 | inner |
384.5.b.d | ✓ | 16 | 8.b | even | 2 | 1 | inner |
384.5.b.d | ✓ | 16 | 8.d | odd | 2 | 1 | inner |
768.5.g.h | 8 | 16.e | even | 4 | 1 | ||
768.5.g.h | 8 | 16.f | odd | 4 | 1 | ||
768.5.g.j | 8 | 16.e | even | 4 | 1 | ||
768.5.g.j | 8 | 16.f | odd | 4 | 1 | ||
1152.5.b.m | 16 | 3.b | odd | 2 | 1 | ||
1152.5.b.m | 16 | 12.b | even | 2 | 1 | ||
1152.5.b.m | 16 | 24.f | even | 2 | 1 | ||
1152.5.b.m | 16 | 24.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 2464T_{5}^{6} + 1653120T_{5}^{4} + 341272576T_{5}^{2} + 6369316864 \)
acting on \(S_{5}^{\mathrm{new}}(384, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} \)
$3$
\( (T^{2} - 27)^{8} \)
$5$
\( (T^{8} + 2464 T^{6} + \cdots + 6369316864)^{2} \)
$7$
\( (T^{8} + 14624 T^{6} + \cdots + 2321893298176)^{2} \)
$11$
\( (T^{8} - 85184 T^{6} + \cdots + 47\!\cdots\!56)^{2} \)
$13$
\( (T^{8} + 148672 T^{6} + \cdots + 42\!\cdots\!64)^{2} \)
$17$
\( (T^{4} - 120 T^{3} + \cdots + 16654958608)^{4} \)
$19$
\( (T^{8} - 420032 T^{6} + \cdots + 45\!\cdots\!00)^{2} \)
$23$
\( (T^{8} + 503936 T^{6} + \cdots + 33\!\cdots\!00)^{2} \)
$29$
\( (T^{8} + 1506976 T^{6} + \cdots + 11\!\cdots\!84)^{2} \)
$31$
\( (T^{8} + 2826528 T^{6} + \cdots + 76\!\cdots\!24)^{2} \)
$37$
\( (T^{8} + 10909504 T^{6} + \cdots + 47\!\cdots\!04)^{2} \)
$41$
\( (T^{4} + 888 T^{3} + \cdots + 4399896700944)^{4} \)
$43$
\( (T^{8} - 3851456 T^{6} + \cdots + 21\!\cdots\!36)^{2} \)
$47$
\( (T^{8} + 15831168 T^{6} + \cdots + 16\!\cdots\!16)^{2} \)
$53$
\( (T^{8} + 38624160 T^{6} + \cdots + 73\!\cdots\!00)^{2} \)
$59$
\( (T^{8} - 23504064 T^{6} + \cdots + 14\!\cdots\!16)^{2} \)
$61$
\( (T^{8} + 58033984 T^{6} + \cdots + 28\!\cdots\!00)^{2} \)
$67$
\( (T^{8} - 85260992 T^{6} + \cdots + 73\!\cdots\!96)^{2} \)
$71$
\( (T^{8} + 39923840 T^{6} + \cdots + 41\!\cdots\!00)^{2} \)
$73$
\( (T^{4} - 8600 T^{3} + \cdots - 12\!\cdots\!76)^{4} \)
$79$
\( (T^{8} + 160497440 T^{6} + \cdots + 23\!\cdots\!04)^{2} \)
$83$
\( (T^{8} - 172743360 T^{6} + \cdots + 27\!\cdots\!76)^{2} \)
$89$
\( (T^{4} + 3912 T^{3} + \cdots - 21\!\cdots\!48)^{4} \)
$97$
\( (T^{4} - 1272 T^{3} + \cdots + 319894196086800)^{4} \)
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