Newspace parameters
Level: | \( N \) | \(=\) | \( 147 = 3 \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 147.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(23.5764215125\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
Defining polynomial: |
\( x^{16} - 171 x^{14} + 21495 x^{12} - 1128902 x^{10} + 42970860 x^{8} - 655075344 x^{6} + 7244325760 x^{4} - 29387167488 x^{2} + 90230547456 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
Coefficient ring index: | \( 2^{12}\cdot 3^{8}\cdot 7^{4} \) |
Twist minimal: | no (minimal twist has level 21) |
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
\( x^{16} - 171 x^{14} + 21495 x^{12} - 1128902 x^{10} + 42970860 x^{8} - 655075344 x^{6} + 7244325760 x^{4} - 29387167488 x^{2} + 90230547456 \)
:
\(\beta_{1}\) | \(=\) |
\( ( - 12\!\cdots\!65 \nu^{14} + \cdots + 21\!\cdots\!64 ) / 64\!\cdots\!16 \)
|
\(\beta_{2}\) | \(=\) |
\( ( - 32290359027 \nu^{14} + 5080229838581 \nu^{12} - 625594991538609 \nu^{10} + \cdots - 55\!\cdots\!48 ) / 14\!\cdots\!64 \)
|
\(\beta_{3}\) | \(=\) |
\( ( - 13230933495217 \nu^{15} + \cdots + 93\!\cdots\!12 \nu ) / 25\!\cdots\!32 \)
|
\(\beta_{4}\) | \(=\) |
\( ( 13230933495217 \nu^{14} + \cdots - 22\!\cdots\!28 ) / 18\!\cdots\!88 \)
|
\(\beta_{5}\) | \(=\) |
\( ( 42\!\cdots\!10 \nu^{15} + \cdots - 48\!\cdots\!96 ) / 32\!\cdots\!08 \)
|
\(\beta_{6}\) | \(=\) |
\( ( - 250787622883333 \nu^{14} + \cdots - 74\!\cdots\!16 ) / 14\!\cdots\!28 \)
|
\(\beta_{7}\) | \(=\) |
\( ( 37\!\cdots\!85 \nu^{15} + \cdots - 40\!\cdots\!92 ) / 25\!\cdots\!64 \)
|
\(\beta_{8}\) | \(=\) |
\( ( 25\!\cdots\!69 \nu^{15} + \cdots - 86\!\cdots\!68 \nu ) / 12\!\cdots\!32 \)
|
\(\beta_{9}\) | \(=\) |
\( ( - 16\!\cdots\!51 \nu^{14} + \cdots + 26\!\cdots\!92 ) / 80\!\cdots\!52 \)
|
\(\beta_{10}\) | \(=\) |
\( ( 21\!\cdots\!43 \nu^{15} + \cdots - 97\!\cdots\!92 ) / 64\!\cdots\!16 \)
|
\(\beta_{11}\) | \(=\) |
\( ( - 464108966034257 \nu^{14} + \cdots - 59\!\cdots\!44 ) / 14\!\cdots\!28 \)
|
\(\beta_{12}\) | \(=\) |
\( ( 10\!\cdots\!45 \nu^{15} + \cdots + 40\!\cdots\!92 ) / 25\!\cdots\!64 \)
|
\(\beta_{13}\) | \(=\) |
\( ( - 16\!\cdots\!29 \nu^{15} + \cdots + 51\!\cdots\!80 ) / 21\!\cdots\!72 \)
|
\(\beta_{14}\) | \(=\) |
\( ( 32\!\cdots\!39 \nu^{15} + \cdots + 33\!\cdots\!76 ) / 86\!\cdots\!88 \)
|
\(\beta_{15}\) | \(=\) |
\( ( 61\!\cdots\!77 \nu^{15} + \cdots + 33\!\cdots\!76 ) / 86\!\cdots\!88 \)
|
\(\nu\) | \(=\) |
\( ( -\beta_{12} + \beta_{10} - \beta_{7} - \beta_{5} - 7\beta_{3} ) / 14 \)
|
\(\nu^{2}\) | \(=\) |
\( ( 7\beta_{12} - 7\beta_{7} - 14\beta_{5} + 40\beta_{4} + 7\beta_{2} + 7\beta _1 + 301 ) / 14 \)
|
\(\nu^{3}\) | \(=\) |
\( \beta_{15} - 3\beta_{14} - \beta_{11} - 80\beta_{3} - \beta_{2} - 1 \)
|
\(\nu^{4}\) | \(=\) |
\( ( 651 \beta_{12} - 14 \beta_{11} - 182 \beta_{9} - 651 \beta_{7} + 112 \beta_{6} - 1302 \beta_{5} + 3182 \beta_{4} - 763 \beta_{2} + 1435 \beta _1 - 24255 ) / 14 \)
|
\(\nu^{5}\) | \(=\) |
\( ( 973 \beta_{15} - 2499 \beta_{14} + 476 \beta_{13} + 3000 \beta_{12} - 1001 \beta_{11} - 11260 \beta_{10} - 49 \beta_{9} + 2919 \beta_{8} + 3098 \beta_{7} + 238 \beta_{6} + 11064 \beta_{5} - 53774 \beta_{3} + \cdots - 525 ) / 14 \)
|
\(\nu^{6}\) | \(=\) |
\( -566\beta_{11} + 2524\beta_{6} - 11445\beta_{2} - 333005 \)
|
\(\nu^{7}\) | \(=\) |
\( ( - 111489 \beta_{15} + 271719 \beta_{14} - 78596 \beta_{13} + 279656 \beta_{12} + 119413 \beta_{11} - 1174928 \beta_{10} + 37779 \beta_{9} + 334467 \beta_{8} + 204098 \beta_{7} + \cdots + 40817 ) / 14 \)
|
\(\nu^{8}\) | \(=\) |
\( ( - 6320615 \beta_{12} - 569058 \beta_{11} + 2600178 \beta_{9} + 6320615 \beta_{7} + 2153676 \beta_{6} + 12641230 \beta_{5} - 30796250 \beta_{4} - 8474291 \beta_{2} + \cdots - 238072499 ) / 14 \)
|
\(\nu^{9}\) | \(=\) |
\( - 1744655 \beta_{15} + 4165881 \beta_{14} - 1393260 \beta_{13} + 1907243 \beta_{11} - 696630 \beta_{6} + 82169878 \beta_{3} + 513983 \beta_{2} + 513983 \)
|
\(\nu^{10}\) | \(=\) |
\( ( - 658696647 \beta_{12} + 67988914 \beta_{11} + 280792162 \beta_{9} + 658696647 \beta_{7} - 242379452 \beta_{6} + 1317393294 \beta_{5} - 3217585738 \beta_{4} + \cdots + 24936699795 ) / 14 \)
|
\(\nu^{11}\) | \(=\) |
\( ( - 1317846089 \beta_{15} + 3119998287 \beta_{14} - 1105495636 \beta_{13} - 2936793576 \beta_{12} + 1453823917 \beta_{11} + 13197864536 \beta_{10} + \cdots + 348328281 ) / 14 \)
|
\(\nu^{12}\) | \(=\) |
\( 1085940862\beta_{11} - 3779909252\beta_{6} + 13721538933\beta_{2} + 377261441413 \)
|
\(\nu^{13}\) | \(=\) |
\( ( 141367916025 \beta_{15} - 333469461087 \beta_{14} + 121040631124 \beta_{13} - 310243081576 \beta_{12} - 156571088093 \beta_{11} + 1406702237368 \beta_{10} + \cdots - 35530456969 ) / 14 \)
|
\(\nu^{14}\) | \(=\) |
\( ( 7397466827527 \beta_{12} + 827711044818 \beta_{11} - 3221809636290 \beta_{9} - 7397466827527 \beta_{7} - 2852471385372 \beta_{6} + \cdots + 281013171864019 ) / 14 \)
|
\(\nu^{15}\) | \(=\) |
\( 2161024276975 \beta_{15} - 5089578052089 \beta_{14} + 1866472518732 \beta_{13} - 2397513146923 \beta_{11} + 933236259366 \beta_{6} - 98739956883782 \beta_{3} + \cdots - 531040628191 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).
\(n\) | \(50\) | \(52\) |
\(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
146.1 |
|
− | 10.3347i | −12.5210 | − | 9.28579i | −74.8050 | −31.7408 | −95.9654 | + | 129.400i | 0 | 442.375i | 70.5484 | + | 232.534i | 328.031i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
146.2 | − | 10.3347i | 12.5210 | + | 9.28579i | −74.8050 | 31.7408 | 95.9654 | − | 129.400i | 0 | 442.375i | 70.5484 | + | 232.534i | − | 328.031i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
146.3 | − | 6.80349i | −12.4320 | + | 9.40448i | −14.2874 | −14.9141 | 63.9832 | + | 84.5813i | 0 | − | 120.507i | 66.1117 | − | 233.834i | 101.468i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
146.4 | − | 6.80349i | 12.4320 | − | 9.40448i | −14.2874 | 14.9141 | −63.9832 | − | 84.5813i | 0 | − | 120.507i | 66.1117 | − | 233.834i | − | 101.468i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
146.5 | − | 3.65505i | −15.5635 | + | 0.882561i | 18.6406 | 74.3244 | 3.22580 | + | 56.8851i | 0 | − | 185.094i | 241.442 | − | 27.4714i | − | 271.659i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
146.6 | − | 3.65505i | 15.5635 | − | 0.882561i | 18.6406 | −74.3244 | −3.22580 | − | 56.8851i | 0 | − | 185.094i | 241.442 | − | 27.4714i | 271.659i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
146.7 | − | 2.13264i | −9.16236 | + | 12.6115i | 27.4518 | −95.0525 | 26.8959 | + | 19.5400i | 0 | − | 126.790i | −75.1022 | − | 231.103i | 202.713i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
146.8 | − | 2.13264i | 9.16236 | − | 12.6115i | 27.4518 | 95.0525 | −26.8959 | − | 19.5400i | 0 | − | 126.790i | −75.1022 | − | 231.103i | − | 202.713i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
146.9 | 2.13264i | −9.16236 | − | 12.6115i | 27.4518 | −95.0525 | 26.8959 | − | 19.5400i | 0 | 126.790i | −75.1022 | + | 231.103i | − | 202.713i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
146.10 | 2.13264i | 9.16236 | + | 12.6115i | 27.4518 | 95.0525 | −26.8959 | + | 19.5400i | 0 | 126.790i | −75.1022 | + | 231.103i | 202.713i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
146.11 | 3.65505i | −15.5635 | − | 0.882561i | 18.6406 | 74.3244 | 3.22580 | − | 56.8851i | 0 | 185.094i | 241.442 | + | 27.4714i | 271.659i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
146.12 | 3.65505i | 15.5635 | + | 0.882561i | 18.6406 | −74.3244 | −3.22580 | + | 56.8851i | 0 | 185.094i | 241.442 | + | 27.4714i | − | 271.659i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
146.13 | 6.80349i | −12.4320 | − | 9.40448i | −14.2874 | −14.9141 | 63.9832 | − | 84.5813i | 0 | 120.507i | 66.1117 | + | 233.834i | − | 101.468i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
146.14 | 6.80349i | 12.4320 | + | 9.40448i | −14.2874 | 14.9141 | −63.9832 | + | 84.5813i | 0 | 120.507i | 66.1117 | + | 233.834i | 101.468i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
146.15 | 10.3347i | −12.5210 | + | 9.28579i | −74.8050 | −31.7408 | −95.9654 | − | 129.400i | 0 | − | 442.375i | 70.5484 | − | 232.534i | − | 328.031i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
146.16 | 10.3347i | 12.5210 | − | 9.28579i | −74.8050 | 31.7408 | 95.9654 | + | 129.400i | 0 | − | 442.375i | 70.5484 | − | 232.534i | 328.031i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.b | odd | 2 | 1 | inner |
21.c | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 147.6.c.c | 16 | |
3.b | odd | 2 | 1 | inner | 147.6.c.c | 16 | |
7.b | odd | 2 | 1 | inner | 147.6.c.c | 16 | |
7.c | even | 3 | 1 | 21.6.g.c | ✓ | 16 | |
7.d | odd | 6 | 1 | 21.6.g.c | ✓ | 16 | |
21.c | even | 2 | 1 | inner | 147.6.c.c | 16 | |
21.g | even | 6 | 1 | 21.6.g.c | ✓ | 16 | |
21.h | odd | 6 | 1 | 21.6.g.c | ✓ | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
21.6.g.c | ✓ | 16 | 7.c | even | 3 | 1 | |
21.6.g.c | ✓ | 16 | 7.d | odd | 6 | 1 | |
21.6.g.c | ✓ | 16 | 21.g | even | 6 | 1 | |
21.6.g.c | ✓ | 16 | 21.h | odd | 6 | 1 | |
147.6.c.c | 16 | 1.a | even | 1 | 1 | trivial | |
147.6.c.c | 16 | 3.b | odd | 2 | 1 | inner | |
147.6.c.c | 16 | 7.b | odd | 2 | 1 | inner | |
147.6.c.c | 16 | 21.c | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 171T_{2}^{6} + 7746T_{2}^{4} + 97832T_{2}^{2} + 300384 \)
acting on \(S_{6}^{\mathrm{new}}(147, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{8} + 171 T^{6} + 7746 T^{4} + \cdots + 300384)^{2} \)
$3$
\( T^{16} - 606 T^{14} + \cdots + 12\!\cdots\!01 \)
$5$
\( (T^{8} - 15789 T^{6} + \cdots + 11184619211136)^{2} \)
$7$
\( T^{16} \)
$11$
\( (T^{8} + 233511 T^{6} + \cdots + 50\!\cdots\!44)^{2} \)
$13$
\( (T^{8} + 825711 T^{6} + \cdots + 39\!\cdots\!16)^{2} \)
$17$
\( (T^{8} - 2274750 T^{6} + \cdots + 79\!\cdots\!44)^{2} \)
$19$
\( (T^{8} + 16853013 T^{6} + \cdots + 29\!\cdots\!24)^{2} \)
$23$
\( (T^{8} + 34314906 T^{6} + \cdots + 25\!\cdots\!96)^{2} \)
$29$
\( (T^{8} + 95665689 T^{6} + \cdots + 48\!\cdots\!04)^{2} \)
$31$
\( (T^{8} + 84970740 T^{6} + \cdots + 62\!\cdots\!61)^{2} \)
$37$
\( (T^{4} - 5233 T^{3} + \cdots + 244571248924934)^{4} \)
$41$
\( (T^{8} - 509738712 T^{6} + \cdots + 13\!\cdots\!44)^{2} \)
$43$
\( (T^{4} - 27763 T^{3} + \cdots - 52\!\cdots\!64)^{4} \)
$47$
\( (T^{8} - 175724658 T^{6} + \cdots + 20\!\cdots\!96)^{2} \)
$53$
\( (T^{8} + 1823558175 T^{6} + \cdots + 33\!\cdots\!84)^{2} \)
$59$
\( (T^{8} - 2009137293 T^{6} + \cdots + 16\!\cdots\!96)^{2} \)
$61$
\( (T^{8} + 2425473822 T^{6} + \cdots + 12\!\cdots\!76)^{2} \)
$67$
\( (T^{4} - 51101 T^{3} + \cdots - 49\!\cdots\!86)^{4} \)
$71$
\( (T^{8} + 6392338344 T^{6} + \cdots + 25\!\cdots\!96)^{2} \)
$73$
\( (T^{8} + 5517500409 T^{6} + \cdots + 10\!\cdots\!36)^{2} \)
$79$
\( (T^{4} - 125654 T^{3} + \cdots - 28\!\cdots\!53)^{4} \)
$83$
\( (T^{8} - 19036730139 T^{6} + \cdots + 56\!\cdots\!04)^{2} \)
$89$
\( (T^{8} - 20057888862 T^{6} + \cdots + 56\!\cdots\!96)^{2} \)
$97$
\( (T^{8} + 34813643163 T^{6} + \cdots + 60\!\cdots\!24)^{2} \)
show more
show less