Newspace parameters
Level: | \( N \) | \(=\) | \( 5796 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 5796.k (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(46.2812930115\) |
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: | \( x^{16} - 2x^{14} + 68x^{12} - 46x^{10} + 4950x^{8} - 2254x^{6} + 163268x^{4} - 235298x^{2} + 5764801 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
Coefficient ring index: | \( 2^{6} \) |
Twist minimal: | no (minimal twist has level 644) |
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 \( x^{16} - 2x^{14} + 68x^{12} - 46x^{10} + 4950x^{8} - 2254x^{6} + 163268x^{4} - 235298x^{2} + 5764801 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{15} - 849 \nu^{13} - 4069 \nu^{11} - 62787 \nu^{9} - 201333 \nu^{7} - 3304819 \nu^{5} - 7782327 \nu^{3} - 107279081 \nu ) / 45177216 \) |
\(\beta_{3}\) | \(=\) | \( ( - 22 \nu^{14} - 495 \nu^{12} + 4384 \nu^{10} + 72405 \nu^{8} + 242430 \nu^{6} + 3396043 \nu^{4} + 31981320 \nu^{2} + 197532671 ) / 56471520 \) |
\(\beta_{4}\) | \(=\) | \( ( 19 \nu^{14} + 60 \nu^{12} + 3497 \nu^{10} + 988 \nu^{8} + 135161 \nu^{6} + 331828 \nu^{4} + 5001283 \nu^{2} - 470596 ) / 7529536 \) |
\(\beta_{5}\) | \(=\) | \( ( - 1087 \nu^{14} + 4575 \nu^{12} - 37901 \nu^{10} + 249285 \nu^{8} - 950805 \nu^{6} + 22222333 \nu^{4} + 10336305 \nu^{2} + 846955151 ) / 225886080 \) |
\(\beta_{6}\) | \(=\) | \( ( - 19 \nu^{14} - 452 \nu^{12} - 2713 \nu^{10} - 27644 \nu^{8} - 117129 \nu^{6} - 1331036 \nu^{4} - 6000099 \nu^{2} - 35294700 ) / 3764768 \) |
\(\beta_{7}\) | \(=\) | \( ( - 2889 \nu^{14} - 935 \nu^{12} - 89387 \nu^{10} + 312675 \nu^{8} - 4683075 \nu^{6} - 318549 \nu^{4} - 26567065 \nu^{2} + 1410964457 ) / 451772160 \) |
\(\beta_{8}\) | \(=\) | \( ( \nu^{15} - 2\nu^{13} + 68\nu^{11} - 46\nu^{9} + 4950\nu^{7} - 2254\nu^{5} + 163268\nu^{3} - 235298\nu ) / 823543 \) |
\(\beta_{9}\) | \(=\) | \( ( - 769 \nu^{15} + 20697 \nu^{13} + 137485 \nu^{11} + 1705539 \nu^{9} + 3410709 \nu^{7} + 75371947 \nu^{5} + 364911183 \nu^{3} + 3093815753 \nu ) / 632481024 \) |
\(\beta_{10}\) | \(=\) | \( ( 841 \nu^{14} + 1699 \nu^{12} + 52827 \nu^{10} - 166527 \nu^{8} + 2170659 \nu^{6} - 1388023 \nu^{4} + 104513129 \nu^{2} - 603892317 ) / 90354432 \) |
\(\beta_{11}\) | \(=\) | \( ( 1207 \nu^{15} - 11871 \nu^{13} + 2549 \nu^{11} - 619365 \nu^{9} + 774525 \nu^{7} - 20886397 \nu^{5} - 116076345 \nu^{3} - 1242255791 \nu ) / 632481024 \) |
\(\beta_{12}\) | \(=\) | \( ( - 1559 \nu^{15} - 6535 \nu^{13} - 113117 \nu^{11} + 291675 \nu^{9} - 4951245 \nu^{7} - 7641109 \nu^{5} - 153507935 \nu^{3} + 1086253217 \nu ) / 790601280 \) |
\(\beta_{13}\) | \(=\) | \( ( 1153 \nu^{14} - 1277 \nu^{12} + 21123 \nu^{10} - 225567 \nu^{8} + 2728395 \nu^{6} - 11082967 \nu^{4} + 19517729 \nu^{2} - 773306877 ) / 90354432 \) |
\(\beta_{14}\) | \(=\) | \( ( - 1251 \nu^{15} + 885 \nu^{13} - 79433 \nu^{11} + 295735 \nu^{9} - 4190065 \nu^{7} + 14587839 \nu^{5} - 65775395 \nu^{3} + 738482773 \nu ) / 527067520 \) |
\(\beta_{15}\) | \(=\) | \( ( 4507 \nu^{15} + 14555 \nu^{13} + 218521 \nu^{11} + 182865 \nu^{9} + 11979225 \nu^{7} + 24501617 \nu^{5} + 348325075 \nu^{3} - 1142254141 \nu ) / 790601280 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{10} + \beta_{7} - \beta_{4} + \beta_{3} \) |
\(\nu^{3}\) | \(=\) | \( \beta_{15} + 4\beta_{14} - \beta_{12} - \beta_{11} + \beta_{9} + 4\beta_{8} + 4\beta_{2} + \beta_1 \) |
\(\nu^{4}\) | \(=\) | \( -3\beta_{13} - \beta_{10} - 16\beta_{7} + 2\beta_{6} + 10\beta_{5} + 2\beta_{4} + 4\beta_{3} - 15 \) |
\(\nu^{5}\) | \(=\) | \( -7\beta_{15} + 28\beta_{14} - 29\beta_{12} + 17\beta_{11} - \beta_{9} + 13\beta_{8} - 12\beta_{2} + 4\beta_1 \) |
\(\nu^{6}\) | \(=\) | \( 51\beta_{13} - 32\beta_{10} + 43\beta_{7} + 44\beta_{5} + 51\beta_{4} - 13\beta_{3} - 28 \) |
\(\nu^{7}\) | \(=\) | \( -68\beta_{15} - 72\beta_{14} + 8\beta_{12} - 100\beta_{11} - 140\beta_{9} + 213\beta_{8} - 248\beta_{2} - 48\beta_1 \) |
\(\nu^{8}\) | \(=\) | \( 72\beta_{13} - 184\beta_{10} + 296\beta_{7} - 136\beta_{6} - 468\beta_{5} - 80\beta_{4} + 128\beta_{3} - 1511 \) |
\(\nu^{9}\) | \(=\) | \( 484 \beta_{15} - 856 \beta_{14} + 1928 \beta_{12} + 12 \beta_{11} + 324 \beta_{9} - 512 \beta_{8} + 168 \beta_{2} - 2059 \beta_1 \) |
\(\nu^{10}\) | \(=\) | \( - 1600 \beta_{13} - 395 \beta_{10} - 1011 \beta_{7} + 16 \beta_{6} - 2292 \beta_{5} + 2443 \beta_{4} - 1323 \beta_{3} + 348 \) |
\(\nu^{11}\) | \(=\) | \( - 2767 \beta_{15} - 8468 \beta_{14} - 2333 \beta_{12} + 6039 \beta_{11} + 6201 \beta_{9} - 10700 \beta_{8} + 3708 \beta_{2} - 1651 \beta_1 \) |
\(\nu^{12}\) | \(=\) | \( 1897 \beta_{13} + 10251 \beta_{10} + 18064 \beta_{7} - 5422 \beta_{6} + 3774 \beta_{5} - 9086 \beta_{4} - 24692 \beta_{3} + 49153 \) |
\(\nu^{13}\) | \(=\) | \( 12137 \beta_{15} - 24396 \beta_{14} - 10913 \beta_{12} - 62935 \beta_{11} - 25673 \beta_{9} - 48927 \beta_{8} - 14228 \beta_{2} + 20916 \beta_1 \) |
\(\nu^{14}\) | \(=\) | \( - 25657 \beta_{13} + 31776 \beta_{10} - 176041 \beta_{7} - 13680 \beta_{6} - 53384 \beta_{5} - 155001 \beta_{4} + 74215 \beta_{3} + 345224 \) |
\(\nu^{15}\) | \(=\) | \( 392248 \beta_{15} + 254096 \beta_{14} + 283808 \beta_{12} + 160616 \beta_{11} + 69368 \beta_{9} - 248383 \beta_{8} + 274608 \beta_{2} + 378032 \beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5796\mathbb{Z}\right)^\times\).
\(n\) | \(829\) | \(1289\) | \(2899\) | \(4789\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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5473.1 |
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0 | 0 | 0 | −3.38856 | 0 | −2.38903 | − | 1.13690i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
5473.2 | 0 | 0 | 0 | −3.38856 | 0 | −2.38903 | + | 1.13690i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
5473.3 | 0 | 0 | 0 | −2.66041 | 0 | 1.54356 | − | 2.14882i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
5473.4 | 0 | 0 | 0 | −2.66041 | 0 | 1.54356 | + | 2.14882i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
5473.5 | 0 | 0 | 0 | −1.77749 | 0 | 0.986926 | − | 2.45479i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
5473.6 | 0 | 0 | 0 | −1.77749 | 0 | 0.986926 | + | 2.45479i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
5473.7 | 0 | 0 | 0 | −0.529536 | 0 | 2.33151 | − | 1.25062i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
5473.8 | 0 | 0 | 0 | −0.529536 | 0 | 2.33151 | + | 1.25062i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
5473.9 | 0 | 0 | 0 | 0.529536 | 0 | −2.33151 | − | 1.25062i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
5473.10 | 0 | 0 | 0 | 0.529536 | 0 | −2.33151 | + | 1.25062i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
5473.11 | 0 | 0 | 0 | 1.77749 | 0 | −0.986926 | − | 2.45479i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
5473.12 | 0 | 0 | 0 | 1.77749 | 0 | −0.986926 | + | 2.45479i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
5473.13 | 0 | 0 | 0 | 2.66041 | 0 | −1.54356 | − | 2.14882i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
5473.14 | 0 | 0 | 0 | 2.66041 | 0 | −1.54356 | + | 2.14882i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
5473.15 | 0 | 0 | 0 | 3.38856 | 0 | 2.38903 | − | 1.13690i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
5473.16 | 0 | 0 | 0 | 3.38856 | 0 | 2.38903 | + | 1.13690i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
23.b | odd | 2 | 1 | inner |
161.c | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 5796.2.k.b | 16 | |
3.b | odd | 2 | 1 | 644.2.d.a | ✓ | 16 | |
7.b | odd | 2 | 1 | inner | 5796.2.k.b | 16 | |
12.b | even | 2 | 1 | 2576.2.f.g | 16 | ||
21.c | even | 2 | 1 | 644.2.d.a | ✓ | 16 | |
23.b | odd | 2 | 1 | inner | 5796.2.k.b | 16 | |
69.c | even | 2 | 1 | 644.2.d.a | ✓ | 16 | |
84.h | odd | 2 | 1 | 2576.2.f.g | 16 | ||
161.c | even | 2 | 1 | inner | 5796.2.k.b | 16 | |
276.h | odd | 2 | 1 | 2576.2.f.g | 16 | ||
483.c | odd | 2 | 1 | 644.2.d.a | ✓ | 16 | |
1932.b | even | 2 | 1 | 2576.2.f.g | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
644.2.d.a | ✓ | 16 | 3.b | odd | 2 | 1 | |
644.2.d.a | ✓ | 16 | 21.c | even | 2 | 1 | |
644.2.d.a | ✓ | 16 | 69.c | even | 2 | 1 | |
644.2.d.a | ✓ | 16 | 483.c | odd | 2 | 1 | |
2576.2.f.g | 16 | 12.b | even | 2 | 1 | ||
2576.2.f.g | 16 | 84.h | odd | 2 | 1 | ||
2576.2.f.g | 16 | 276.h | odd | 2 | 1 | ||
2576.2.f.g | 16 | 1932.b | even | 2 | 1 | ||
5796.2.k.b | 16 | 1.a | even | 1 | 1 | trivial | |
5796.2.k.b | 16 | 7.b | odd | 2 | 1 | inner | |
5796.2.k.b | 16 | 23.b | odd | 2 | 1 | inner | |
5796.2.k.b | 16 | 161.c | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} - 22T_{5}^{6} + 146T_{5}^{4} - 296T_{5}^{2} + 72 \)
acting on \(S_{2}^{\mathrm{new}}(5796, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} \)
$3$
\( T^{16} \)
$5$
\( (T^{8} - 22 T^{6} + 146 T^{4} - 296 T^{2} + \cdots + 72)^{2} \)
$7$
\( T^{16} - 2 T^{14} + 68 T^{12} + \cdots + 5764801 \)
$11$
\( (T^{8} + 54 T^{6} + 568 T^{4} + 1624 T^{2} + \cdots + 144)^{2} \)
$13$
\( (T^{8} + 40 T^{6} + 507 T^{4} + 2152 T^{2} + \cdots + 2592)^{2} \)
$17$
\( (T^{8} - 60 T^{6} + 966 T^{4} - 4724 T^{2} + \cdots + 648)^{2} \)
$19$
\( (T^{8} - 78 T^{6} + 1980 T^{4} + \cdots + 28800)^{2} \)
$23$
\( (T^{8} - 8 T^{7} + 8 T^{6} - 8 T^{5} + \cdots + 279841)^{2} \)
$29$
\( (T^{4} - 93 T^{2} + 34 T + 1854)^{4} \)
$31$
\( (T^{8} + 114 T^{6} + 2425 T^{4} + \cdots + 450)^{2} \)
$37$
\( (T^{8} + 84 T^{6} + 1252 T^{4} + \cdots + 9216)^{2} \)
$41$
\( (T^{8} + 264 T^{6} + 24587 T^{4} + \cdots + 12781568)^{2} \)
$43$
\( (T^{8} + 176 T^{6} + 7296 T^{4} + \cdots + 36864)^{2} \)
$47$
\( (T^{8} + 258 T^{6} + 17585 T^{4} + \cdots + 55778)^{2} \)
$53$
\( (T^{8} + 310 T^{6} + 35716 T^{4} + \cdots + 34105600)^{2} \)
$59$
\( (T^{8} + 154 T^{6} + 8494 T^{4} + \cdots + 1620000)^{2} \)
$61$
\( (T^{8} - 102 T^{6} + 3702 T^{4} + \cdots + 307328)^{2} \)
$67$
\( (T^{8} + 412 T^{6} + 51368 T^{4} + \cdots + 36144144)^{2} \)
$71$
\( (T^{4} - 2 T^{3} - 123 T^{2} + 468 T + 1080)^{4} \)
$73$
\( (T^{8} + 240 T^{6} + 11659 T^{4} + \cdots + 968832)^{2} \)
$79$
\( (T^{8} + 502 T^{6} + 86040 T^{4} + \cdots + 114318864)^{2} \)
$83$
\( (T^{8} - 250 T^{6} + 6716 T^{4} + \cdots + 28800)^{2} \)
$89$
\( (T^{8} - 482 T^{6} + 74038 T^{4} + \cdots + 165888)^{2} \)
$97$
\( (T^{8} - 342 T^{6} + 26466 T^{4} + \cdots + 19208)^{2} \)
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