Newspace parameters
Level: | \( N \) | \(=\) | \( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4140.i (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(33.0580664368\) |
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: | \( x^{16} + 62x^{14} + 1303x^{12} + 12842x^{10} + 65359x^{8} + 170834x^{6} + 207293x^{4} + 91366x^{2} + 9604 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{23}]\) |
Coefficient ring index: | \( 2^{5} \) |
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 \( x^{16} + 62x^{14} + 1303x^{12} + 12842x^{10} + 65359x^{8} + 170834x^{6} + 207293x^{4} + 91366x^{2} + 9604 \) :
\(\beta_{1}\) | \(=\) | \( ( 124491880 \nu^{14} + 5531647309 \nu^{12} + 37688713508 \nu^{10} - 618906405792 \nu^{8} - 8576884945810 \nu^{6} + \cdots + 5081910751692 ) / 3306361853624 \) |
\(\beta_{2}\) | \(=\) | \( ( - 1099532275 \nu^{14} - 65729686391 \nu^{12} - 1281439225700 \nu^{10} - 10996984617426 \nu^{8} - 43285810004227 \nu^{6} + \cdots - 1200536614236 ) / 3306361853624 \) |
\(\beta_{3}\) | \(=\) | \( ( - 1705812243 \nu^{14} - 95894237026 \nu^{12} - 1666130880928 \nu^{10} - 12170363351342 \nu^{8} - 39670880077709 \nu^{6} + \cdots - 14661309337056 ) / 3306361853624 \) |
\(\beta_{4}\) | \(=\) | \( ( 1953768963 \nu^{14} + 112819382586 \nu^{12} + 2068772711892 \nu^{10} + 16453894067170 \nu^{8} + 60303704828537 \nu^{6} + \cdots - 1780511035288 ) / 3306361853624 \) |
\(\beta_{5}\) | \(=\) | \( ( 2166758938 \nu^{14} + 125161237541 \nu^{12} + 2294781870908 \nu^{10} + 18185274766792 \nu^{8} + 65524796371428 \nu^{6} + \cdots - 5607240199924 ) / 3306361853624 \) |
\(\beta_{6}\) | \(=\) | \( ( 2498489531 \nu^{14} + 151124599275 \nu^{12} + 3033785819160 \nu^{10} + 27878689600934 \nu^{8} + 127544892361671 \nu^{6} + \cdots + 48526335472260 ) / 3306361853624 \) |
\(\beta_{7}\) | \(=\) | \( ( 3048577016 \nu^{14} + 180777077505 \nu^{12} + 3498253609372 \nu^{10} + 30510475697496 \nu^{8} + 131178809583478 \nu^{6} + \cdots + 52453166710812 ) / 3306361853624 \) |
\(\beta_{8}\) | \(=\) | \( ( - 14724018943 \nu^{15} - 1093972349025 \nu^{13} - 29349714587204 \nu^{11} - 364715172446186 \nu^{9} + \cdots - 19\!\cdots\!52 \nu ) / 162011730827576 \) |
\(\beta_{9}\) | \(=\) | \( ( - 15363299351 \nu^{15} - 937621248815 \nu^{13} - 19163955374068 \nu^{11} - 181475071162522 \nu^{9} + \cdots - 547253380343924 \nu ) / 162011730827576 \) |
\(\beta_{10}\) | \(=\) | \( ( - 16590392249 \nu^{15} - 991263848858 \nu^{13} - 19272328078224 \nu^{11} - 163184490956034 \nu^{9} + \cdots + 18\!\cdots\!88 \nu ) / 162011730827576 \) |
\(\beta_{11}\) | \(=\) | \( ( - 101161287 \nu^{15} - 6326074969 \nu^{13} - 134946581276 \nu^{11} - 1356493811342 \nu^{9} - 7051373895279 \nu^{7} + \cdots - 5858744937772 \nu ) / 839439019832 \) |
\(\beta_{12}\) | \(=\) | \( ( 53520056427 \nu^{15} + 3044902710698 \nu^{13} + 54206097656324 \nu^{11} + 411579426630098 \nu^{9} + \cdots + 55390175420520 \nu ) / 162011730827576 \) |
\(\beta_{13}\) | \(=\) | \( ( - 12538314823 \nu^{15} - 765686789820 \nu^{13} - 15657214632828 \nu^{11} - 148299700666470 \nu^{9} + \cdots - 501097207085760 \nu ) / 23144532975368 \) |
\(\beta_{14}\) | \(=\) | \( ( - 92337525767 \nu^{15} - 5452625870940 \nu^{13} - 104585770514544 \nu^{11} - 897509336354710 \nu^{9} + \cdots - 31\!\cdots\!64 \nu ) / 162011730827576 \) |
\(\beta_{15}\) | \(=\) | \( ( 147338037348 \nu^{15} + 8962991653737 \nu^{13} + 181689302841464 \nu^{11} + \cdots + 16\!\cdots\!96 \nu ) / 162011730827576 \) |
\(\nu\) | \(=\) | \( ( -\beta_{13} - \beta_{12} + \beta_{11} + \beta_{8} ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( -\beta_{7} + \beta_{6} - 2\beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta _1 - 8 \) |
\(\nu^{3}\) | \(=\) | \( -3\beta_{15} - 3\beta_{14} + 15\beta_{13} + 12\beta_{12} - 25\beta_{11} - 5\beta_{10} - 5\beta_{9} - 13\beta_{8} \) |
\(\nu^{4}\) | \(=\) | \( 31\beta_{7} - 28\beta_{6} + 75\beta_{5} - 56\beta_{4} + 25\beta_{3} + 29\beta_{2} - 25\beta _1 + 176 \) |
\(\nu^{5}\) | \(=\) | \( 103 \beta_{15} + 110 \beta_{14} - 485 \beta_{13} - 348 \beta_{12} + 889 \beta_{11} + 188 \beta_{10} + 191 \beta_{9} + 377 \beta_{8} \) |
\(\nu^{6}\) | \(=\) | \( -945\beta_{7} + 842\beta_{6} - 2462\beta_{5} + 1980\beta_{4} - 698\beta_{3} - 868\beta_{2} + 789\beta _1 - 5098 \) |
\(\nu^{7}\) | \(=\) | \( - 3263 \beta_{15} - 3637 \beta_{14} + 15605 \beta_{13} + 10734 \beta_{12} - 29123 \beta_{11} - 6111 \beta_{10} - 6431 \beta_{9} - 11633 \beta_{8} \) |
\(\nu^{8}\) | \(=\) | \( 29697 \beta_{7} - 26434 \beta_{6} + 79015 \beta_{5} - 64918 \beta_{4} + 21133 \beta_{3} + 26957 \beta_{2} - 25473 \beta _1 + 158532 \) |
\(\nu^{9}\) | \(=\) | \( 103411 \beta_{15} + 117348 \beta_{14} - 500033 \beta_{13} - 338856 \beta_{12} + 936583 \beta_{11} + 195286 \beta_{10} + 209433 \beta_{9} + 367341 \beta_{8} \) |
\(\nu^{10}\) | \(=\) | \( - 944115 \beta_{7} + 840704 \beta_{6} - 2526354 \beta_{5} + 2089968 \beta_{4} - 662368 \beta_{3} - 851820 \beta_{2} + 818717 \beta _1 - 5024736 \) |
\(\nu^{11}\) | \(=\) | \( - 3292133 \beta_{15} - 3760505 \beta_{14} + 15994667 \beta_{13} + 10783900 \beta_{12} - 29982539 \beta_{11} - 6233921 \beta_{10} - 6739253 \beta_{9} - 11691485 \beta_{8} \) |
\(\nu^{12}\) | \(=\) | \( 30122223 \beta_{7} - 26830090 \beta_{6} + 80730559 \beta_{5} - 66939762 \beta_{4} + 21025123 \beta_{3} + 27106515 \beta_{2} - 26223785 \beta _1 + 160149968 \) |
\(\nu^{13}\) | \(=\) | \( 105047815 \beta_{15} + 120270062 \beta_{14} - 511290731 \beta_{13} - 344125986 \beta_{12} + 958615535 \beta_{11} + 199094092 \beta_{10} + 215896467 \beta_{9} + 373095275 \beta_{8} \) |
\(\nu^{14}\) | \(=\) | \( - 962087017 \beta_{7} + 857039202 \beta_{6} - 2579652506 \beta_{5} + 2140659196 \beta_{4} - 670326578 \beta_{3} - 864880484 \beta_{2} + 838709617 \beta _1 - 5113220882 \) |
\(\nu^{15}\) | \(=\) | \( - 3355026899 \beta_{15} - 3844249297 \beta_{14} + 16340259777 \beta_{13} + 10991421558 \beta_{12} - 30637798611 \beta_{11} - 6360566579 \beta_{10} + \cdots - 11916729213 \beta_{8} \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times\).
\(n\) | \(461\) | \(1657\) | \(2071\) | \(3961\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1241.1 |
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0 | 0 | 0 | −1.00000 | 0 | − | 4.85908i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1241.2 | 0 | 0 | 0 | −1.00000 | 0 | − | 4.35549i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1241.3 | 0 | 0 | 0 | −1.00000 | 0 | − | 4.17184i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1241.4 | 0 | 0 | 0 | −1.00000 | 0 | − | 3.38743i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1241.5 | 0 | 0 | 0 | −1.00000 | 0 | − | 1.73994i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1241.6 | 0 | 0 | 0 | −1.00000 | 0 | − | 0.826970i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1241.7 | 0 | 0 | 0 | −1.00000 | 0 | − | 0.807745i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1241.8 | 0 | 0 | 0 | −1.00000 | 0 | − | 0.420015i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1241.9 | 0 | 0 | 0 | −1.00000 | 0 | 0.420015i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1241.10 | 0 | 0 | 0 | −1.00000 | 0 | 0.807745i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1241.11 | 0 | 0 | 0 | −1.00000 | 0 | 0.826970i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1241.12 | 0 | 0 | 0 | −1.00000 | 0 | 1.73994i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1241.13 | 0 | 0 | 0 | −1.00000 | 0 | 3.38743i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1241.14 | 0 | 0 | 0 | −1.00000 | 0 | 4.17184i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1241.15 | 0 | 0 | 0 | −1.00000 | 0 | 4.35549i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1241.16 | 0 | 0 | 0 | −1.00000 | 0 | 4.85908i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
69.c | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 4140.2.i.a | ✓ | 16 |
3.b | odd | 2 | 1 | 4140.2.i.b | yes | 16 | |
23.b | odd | 2 | 1 | 4140.2.i.b | yes | 16 | |
69.c | even | 2 | 1 | inner | 4140.2.i.a | ✓ | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4140.2.i.a | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
4140.2.i.a | ✓ | 16 | 69.c | even | 2 | 1 | inner |
4140.2.i.b | yes | 16 | 3.b | odd | 2 | 1 | |
4140.2.i.b | yes | 16 | 23.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{8} - 56T_{11}^{6} - 16T_{11}^{5} + 930T_{11}^{4} + 636T_{11}^{3} - 4164T_{11}^{2} - 5024T_{11} - 256 \)
acting on \(S_{2}^{\mathrm{new}}(4140, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} \)
$3$
\( T^{16} \)
$5$
\( (T + 1)^{16} \)
$7$
\( T^{16} + 76 T^{14} + 2207 T^{12} + \cdots + 21316 \)
$11$
\( (T^{8} - 56 T^{6} - 16 T^{5} + 930 T^{4} + \cdots - 256)^{2} \)
$13$
\( (T^{8} - 52 T^{6} - 36 T^{5} + 662 T^{4} + \cdots + 208)^{2} \)
$17$
\( (T^{8} - 73 T^{6} + 48 T^{5} + 1377 T^{4} + \cdots + 448)^{2} \)
$19$
\( T^{16} + 152 T^{14} + \cdots + 14500864 \)
$23$
\( T^{16} + 8 T^{15} + \cdots + 78310985281 \)
$29$
\( T^{16} + 188 T^{14} + \cdots + 11237272036 \)
$31$
\( (T^{8} + 4 T^{7} - 113 T^{6} + \cdots - 372356)^{2} \)
$37$
\( T^{16} + 196 T^{14} + \cdots + 13410244 \)
$41$
\( T^{16} + 324 T^{14} + \cdots + 14273284 \)
$43$
\( T^{16} + 272 T^{14} + \cdots + 14546289664 \)
$47$
\( T^{16} + 384 T^{14} + \cdots + 805651456 \)
$53$
\( (T^{8} - 2 T^{7} - 185 T^{6} + \cdots - 110404)^{2} \)
$59$
\( T^{16} + 420 T^{14} + \cdots + 62188996 \)
$61$
\( T^{16} + 744 T^{14} + \cdots + 3487272395776 \)
$67$
\( T^{16} + 412 T^{14} + \cdots + 201024896164 \)
$71$
\( T^{16} + 548 T^{14} + \cdots + 69650599396 \)
$73$
\( (T^{8} - 4 T^{7} - 396 T^{6} + \cdots + 33260432)^{2} \)
$79$
\( T^{16} + 640 T^{14} + \cdots + 1073741824 \)
$83$
\( (T^{8} - 10 T^{7} - 353 T^{6} + \cdots - 104384)^{2} \)
$89$
\( (T^{8} - 16 T^{7} - 336 T^{6} + \cdots - 541184)^{2} \)
$97$
\( T^{16} + 792 T^{14} + \cdots + 4196794737664 \)
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