Newspace parameters
Level: | \( N \) | \(=\) | \( 640 = 2^{7} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 640.l (of order \(4\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(5.11042572936\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Relative dimension: | \(8\) over \(\Q(i)\) |
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: | \( x^{16} - 4 x^{15} + 4 x^{14} + 7 x^{12} - 8 x^{11} - 28 x^{10} + 28 x^{9} + 17 x^{8} + 56 x^{7} - 112 x^{6} - 64 x^{5} + 112 x^{4} + 256 x^{2} - 512 x + 256 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{12} \) |
Twist minimal: | no (minimal twist has level 80) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \( x^{16} - 4 x^{15} + 4 x^{14} + 7 x^{12} - 8 x^{11} - 28 x^{10} + 28 x^{9} + 17 x^{8} + 56 x^{7} - 112 x^{6} - 64 x^{5} + 112 x^{4} + 256 x^{2} - 512 x + 256 \) :
\(\beta_{1}\) | \(=\) | \( ( - 163 \nu^{15} + 58 \nu^{14} + 376 \nu^{13} + 568 \nu^{12} - 501 \nu^{11} - 2502 \nu^{10} - 632 \nu^{9} + 3284 \nu^{8} + 6101 \nu^{7} - 1962 \nu^{6} - 10212 \nu^{5} - 4496 \nu^{4} + \cdots - 7296 ) / 2688 \) |
\(\beta_{2}\) | \(=\) | \( ( - 9 \nu^{15} - 80 \nu^{14} + 152 \nu^{13} + 168 \nu^{12} + 65 \nu^{11} - 740 \nu^{10} - 1008 \nu^{9} + 1260 \nu^{8} + 2359 \nu^{7} + 1444 \nu^{6} - 5092 \nu^{5} - 4728 \nu^{4} + 2944 \nu^{3} + \cdots - 12288 ) / 128 \) |
\(\beta_{3}\) | \(=\) | \( ( 41 \nu^{15} + 80 \nu^{14} - 264 \nu^{13} - 312 \nu^{12} + 47 \nu^{11} + 1380 \nu^{10} + 1312 \nu^{9} - 2204 \nu^{8} - 4071 \nu^{7} - 1316 \nu^{6} + 8052 \nu^{5} + 6376 \nu^{4} - 4208 \nu^{3} + \cdots + 15488 ) / 384 \) |
\(\beta_{4}\) | \(=\) | \( ( 303 \nu^{15} - 1892 \nu^{14} + 1724 \nu^{13} + 1784 \nu^{12} + 3497 \nu^{11} - 6864 \nu^{10} - 19332 \nu^{9} + 11948 \nu^{8} + 27583 \nu^{7} + 39328 \nu^{6} - 63120 \nu^{5} + \cdots - 209536 ) / 2688 \) |
\(\beta_{5}\) | \(=\) | \( ( 397 \nu^{15} - 502 \nu^{14} - 772 \nu^{13} - 664 \nu^{12} + 2523 \nu^{11} + 5226 \nu^{10} - 3292 \nu^{9} - 9932 \nu^{8} - 10139 \nu^{7} + 16854 \nu^{6} + 24480 \nu^{5} - 4720 \nu^{4} + \cdots + 8832 ) / 2688 \) |
\(\beta_{6}\) | \(=\) | \( ( - 655 \nu^{15} + 60 \nu^{14} + 2516 \nu^{13} + 2480 \nu^{12} - 3641 \nu^{11} - 14664 \nu^{10} - 3308 \nu^{9} + 27372 \nu^{8} + 36625 \nu^{7} - 15880 \nu^{6} - 84768 \nu^{5} + \cdots - 123008 ) / 2688 \) |
\(\beta_{7}\) | \(=\) | \( ( - 715 \nu^{15} + 2724 \nu^{14} - 1240 \nu^{13} - 1648 \nu^{12} - 7037 \nu^{11} + 2592 \nu^{10} + 27040 \nu^{9} - 2292 \nu^{8} - 22475 \nu^{7} - 67264 \nu^{6} + 45252 \nu^{5} + \cdots + 227968 ) / 2688 \) |
\(\beta_{8}\) | \(=\) | \( ( 396 \nu^{15} - 1201 \nu^{14} + 256 \nu^{13} + 460 \nu^{12} + 3508 \nu^{11} + 489 \nu^{10} - 11700 \nu^{9} - 2228 \nu^{8} + 6320 \nu^{7} + 32015 \nu^{6} - 9900 \nu^{5} - 42864 \nu^{4} + \cdots - 83072 ) / 1344 \) |
\(\beta_{9}\) | \(=\) | \( ( - 137 \nu^{15} + 466 \nu^{14} - 144 \nu^{13} - 264 \nu^{12} - 1303 \nu^{11} + 106 \nu^{10} + 4672 \nu^{9} + 492 \nu^{8} - 3289 \nu^{7} - 12506 \nu^{6} + 4972 \nu^{5} + 17552 \nu^{4} + \cdots + 32000 ) / 448 \) |
\(\beta_{10}\) | \(=\) | \( ( - 935 \nu^{15} + 3056 \nu^{14} - 676 \nu^{13} - 1552 \nu^{12} - 8961 \nu^{11} - 636 \nu^{10} + 30236 \nu^{9} + 5644 \nu^{8} - 17975 \nu^{7} - 81876 \nu^{6} + 25272 \nu^{5} + \cdots + 203136 ) / 2688 \) |
\(\beta_{11}\) | \(=\) | \( ( - 1011 \nu^{15} + 2684 \nu^{14} - 164 \nu^{13} - 632 \nu^{12} - 8357 \nu^{11} - 3480 \nu^{10} + 25260 \nu^{9} + 9028 \nu^{8} - 7939 \nu^{7} - 73048 \nu^{6} + 8664 \nu^{5} + \cdots + 166912 ) / 2688 \) |
\(\beta_{12}\) | \(=\) | \( ( - 1269 \nu^{15} + 3292 \nu^{14} + 236 \nu^{13} - 664 \nu^{12} - 11155 \nu^{11} - 6240 \nu^{10} + 31596 \nu^{9} + 17228 \nu^{8} - 5477 \nu^{7} - 95888 \nu^{6} - 4752 \nu^{5} + \cdots + 176384 ) / 2688 \) |
\(\beta_{13}\) | \(=\) | \( ( - 1583 \nu^{15} + 4222 \nu^{14} - 120 \nu^{13} - 1032 \nu^{12} - 13337 \nu^{11} - 5850 \nu^{10} + 40424 \nu^{9} + 15908 \nu^{8} - 12087 \nu^{7} - 117622 \nu^{6} + 9804 \nu^{5} + \cdots + 252160 ) / 2688 \) |
\(\beta_{14}\) | \(=\) | \( ( 239 \nu^{15} - 554 \nu^{14} - 56 \nu^{13} + 16 \nu^{12} + 1833 \nu^{11} + 1254 \nu^{10} - 5096 \nu^{9} - 2764 \nu^{8} + 311 \nu^{7} + 15690 \nu^{6} + 1044 \nu^{5} - 17384 \nu^{4} + \cdots - 29952 ) / 384 \) |
\(\beta_{15}\) | \(=\) | \( ( - 356 \nu^{15} + 1000 \nu^{14} - 139 \nu^{13} - 312 \nu^{12} - 3008 \nu^{11} - 856 \nu^{10} + 9619 \nu^{9} + 2572 \nu^{8} - 4104 \nu^{7} - 27068 \nu^{6} + 5701 \nu^{5} + 34644 \nu^{4} + \cdots + 65248 ) / 224 \) |
\(\nu\) | \(=\) | \( ( - \beta_{15} - 2 \beta_{14} - \beta_{12} + \beta_{9} - \beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta _1 + 3 ) / 4 \) |
\(\nu^{2}\) | \(=\) | \( ( - \beta_{14} - \beta_{13} + \beta_{12} - 3 \beta_{11} + \beta_{10} + \beta_{9} + 2 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} + \beta_{3} - 2 \beta_{2} + 3 \beta _1 + 4 ) / 4 \) |
\(\nu^{3}\) | \(=\) | \( ( \beta_{15} - 2\beta_{13} + \beta_{12} - 2\beta_{11} + \beta_{10} - \beta_{9} - \beta_{6} + \beta_{2} + 1 ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( ( 2 \beta_{15} + \beta_{14} - 3 \beta_{13} + \beta_{12} - 3 \beta_{11} - \beta_{10} + \beta_{9} - 2 \beta_{7} - 3 \beta_{5} - 3 \beta_{3} - 2 \beta_{2} - 3 \beta _1 + 2 ) / 4 \) |
\(\nu^{5}\) | \(=\) | \( ( 3 \beta_{15} + 2 \beta_{14} - 6 \beta_{13} + 9 \beta_{12} - 8 \beta_{11} + 2 \beta_{10} - 3 \beta_{9} + 5 \beta_{8} + \beta_{7} - 4 \beta_{6} - 3 \beta_{5} - \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + 12 \beta _1 - 13 ) / 4 \) |
\(\nu^{6}\) | \(=\) | \( ( 2 \beta_{15} + 3 \beta_{14} - 2 \beta_{13} + \beta_{12} - \beta_{11} - 4 \beta_{9} - 5 \beta_{8} - 2 \beta_{7} - 2 \beta_{6} - 4 \beta_{5} + \beta_{2} - 2 \beta _1 - 6 ) / 2 \) |
\(\nu^{7}\) | \(=\) | \( ( 7 \beta_{15} + 4 \beta_{14} + 3 \beta_{12} - 6 \beta_{11} - 6 \beta_{10} - 3 \beta_{9} + 19 \beta_{8} + 9 \beta_{7} - 4 \beta_{6} + 3 \beta_{5} + \beta_{4} - 3 \beta_{3} + 5 \beta_{2} - 12 \beta _1 - 19 ) / 4 \) |
\(\nu^{8}\) | \(=\) | \( ( - 2 \beta_{15} + 13 \beta_{14} + 3 \beta_{13} + 13 \beta_{12} + 13 \beta_{11} - 17 \beta_{10} - 3 \beta_{9} - 16 \beta_{8} + 8 \beta_{7} - 14 \beta_{6} - 11 \beta_{5} + 2 \beta_{4} + 11 \beta_{3} + 14 \beta_{2} + 11 \beta _1 - 32 ) / 4 \) |
\(\nu^{9}\) | \(=\) | \( ( - 9 \beta_{15} - 6 \beta_{14} + 23 \beta_{13} - 16 \beta_{12} + 6 \beta_{11} - 7 \beta_{10} - 3 \beta_{9} - 16 \beta_{8} + 11 \beta_{7} - 2 \beta_{5} + 10 \beta_{4} - \beta_{3} - 3 \beta_{2} - 17 \beta _1 - 13 ) / 2 \) |
\(\nu^{10}\) | \(=\) | \( ( - 4 \beta_{15} - 13 \beta_{14} + 17 \beta_{13} - 3 \beta_{12} + 13 \beta_{11} - 13 \beta_{10} + 3 \beta_{9} + 36 \beta_{8} - 10 \beta_{7} + 10 \beta_{6} + 7 \beta_{5} - 30 \beta_{4} + 3 \beta_{3} + 4 \beta_{2} - 51 \beta _1 + 8 ) / 4 \) |
\(\nu^{11}\) | \(=\) | \( ( - 31 \beta_{15} + 4 \beta_{14} + 84 \beta_{13} - 43 \beta_{12} + 30 \beta_{11} - 56 \beta_{10} + 35 \beta_{9} - 41 \beta_{8} + 69 \beta_{7} - 26 \beta_{6} - \beta_{5} + 37 \beta_{4} + 9 \beta_{3} + 31 \beta_{2} - 20 \beta _1 + 23 ) / 4 \) |
\(\nu^{12}\) | \(=\) | \( ( - 46 \beta_{15} - 44 \beta_{14} + 49 \beta_{13} - 52 \beta_{12} + 26 \beta_{11} - 14 \beta_{10} + 24 \beta_{9} - 88 \beta_{8} + \beta_{7} + 33 \beta_{6} + 6 \beta_{5} + \beta_{4} - 7 \beta_{3} - 28 \beta_{2} - 47 \beta _1 + 23 ) / 2 \) |
\(\nu^{13}\) | \(=\) | \( ( - 5 \beta_{15} - 38 \beta_{14} + 76 \beta_{13} - 17 \beta_{12} - 84 \beta_{11} + 88 \beta_{10} - 3 \beta_{9} + 139 \beta_{8} - 75 \beta_{7} + 36 \beta_{6} + 55 \beta_{5} - 75 \beta_{4} - 37 \beta_{3} - 33 \beta_{2} - 6 \beta _1 + 139 ) / 4 \) |
\(\nu^{14}\) | \(=\) | \( ( - 32 \beta_{15} + 9 \beta_{14} - 23 \beta_{13} - 81 \beta_{12} + 27 \beta_{11} - 9 \beta_{10} + 95 \beta_{9} - 178 \beta_{8} + 50 \beta_{7} - 18 \beta_{6} - 61 \beta_{5} - 14 \beta_{4} - 33 \beta_{3} + 66 \beta_{2} - 131 \beta _1 + 68 ) / 4 \) |
\(\nu^{15}\) | \(=\) | \( ( - 45 \beta_{15} + 4 \beta_{14} + 82 \beta_{13} - 85 \beta_{12} - 58 \beta_{11} + 11 \beta_{10} + 77 \beta_{9} - 108 \beta_{8} + 125 \beta_{6} + 24 \beta_{5} - 4 \beta_{4} - 32 \beta_{3} - 89 \beta_{2} + 32 \beta _1 + 63 ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/640\mathbb{Z}\right)^\times\).
\(n\) | \(257\) | \(261\) | \(511\) |
\(\chi(n)\) | \(1\) | \(\beta_{8}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
161.1 |
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0 | −1.82762 | − | 1.82762i | 0 | 0.707107 | − | 0.707107i | 0 | − | 4.50961i | 0 | 3.68037i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
161.2 | 0 | −1.42313 | − | 1.42313i | 0 | −0.707107 | + | 0.707107i | 0 | − | 0.690576i | 0 | 1.05061i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
161.3 | 0 | −1.37027 | − | 1.37027i | 0 | 0.707107 | − | 0.707107i | 0 | 2.73482i | 0 | 0.755274i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
161.4 | 0 | −0.209571 | − | 0.209571i | 0 | −0.707107 | + | 0.707107i | 0 | 1.73696i | 0 | − | 2.91216i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
161.5 | 0 | 0.120009 | + | 0.120009i | 0 | 0.707107 | − | 0.707107i | 0 | 2.66881i | 0 | − | 2.97120i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
161.6 | 0 | 0.720673 | + | 0.720673i | 0 | −0.707107 | + | 0.707107i | 0 | − | 4.02840i | 0 | − | 1.96126i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
161.7 | 0 | 1.66366 | + | 1.66366i | 0 | 0.707107 | − | 0.707107i | 0 | − | 2.89402i | 0 | 2.53555i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
161.8 | 0 | 2.32624 | + | 2.32624i | 0 | −0.707107 | + | 0.707107i | 0 | 0.982011i | 0 | 7.82281i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
481.1 | 0 | −1.82762 | + | 1.82762i | 0 | 0.707107 | + | 0.707107i | 0 | 4.50961i | 0 | − | 3.68037i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
481.2 | 0 | −1.42313 | + | 1.42313i | 0 | −0.707107 | − | 0.707107i | 0 | 0.690576i | 0 | − | 1.05061i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
481.3 | 0 | −1.37027 | + | 1.37027i | 0 | 0.707107 | + | 0.707107i | 0 | − | 2.73482i | 0 | − | 0.755274i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
481.4 | 0 | −0.209571 | + | 0.209571i | 0 | −0.707107 | − | 0.707107i | 0 | − | 1.73696i | 0 | 2.91216i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
481.5 | 0 | 0.120009 | − | 0.120009i | 0 | 0.707107 | + | 0.707107i | 0 | − | 2.66881i | 0 | 2.97120i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
481.6 | 0 | 0.720673 | − | 0.720673i | 0 | −0.707107 | − | 0.707107i | 0 | 4.02840i | 0 | 1.96126i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
481.7 | 0 | 1.66366 | − | 1.66366i | 0 | 0.707107 | + | 0.707107i | 0 | 2.89402i | 0 | − | 2.53555i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
481.8 | 0 | 2.32624 | − | 2.32624i | 0 | −0.707107 | − | 0.707107i | 0 | − | 0.982011i | 0 | − | 7.82281i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
16.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 640.2.l.b | 16 | |
4.b | odd | 2 | 1 | 640.2.l.a | 16 | ||
8.b | even | 2 | 1 | 80.2.l.a | ✓ | 16 | |
8.d | odd | 2 | 1 | 320.2.l.a | 16 | ||
16.e | even | 4 | 1 | 80.2.l.a | ✓ | 16 | |
16.e | even | 4 | 1 | inner | 640.2.l.b | 16 | |
16.f | odd | 4 | 1 | 320.2.l.a | 16 | ||
16.f | odd | 4 | 1 | 640.2.l.a | 16 | ||
24.f | even | 2 | 1 | 2880.2.t.c | 16 | ||
24.h | odd | 2 | 1 | 720.2.t.c | 16 | ||
32.g | even | 8 | 1 | 5120.2.a.s | 8 | ||
32.g | even | 8 | 1 | 5120.2.a.v | 8 | ||
32.h | odd | 8 | 1 | 5120.2.a.t | 8 | ||
32.h | odd | 8 | 1 | 5120.2.a.u | 8 | ||
40.e | odd | 2 | 1 | 1600.2.l.i | 16 | ||
40.f | even | 2 | 1 | 400.2.l.h | 16 | ||
40.i | odd | 4 | 1 | 400.2.q.g | 16 | ||
40.i | odd | 4 | 1 | 400.2.q.h | 16 | ||
40.k | even | 4 | 1 | 1600.2.q.g | 16 | ||
40.k | even | 4 | 1 | 1600.2.q.h | 16 | ||
48.i | odd | 4 | 1 | 720.2.t.c | 16 | ||
48.k | even | 4 | 1 | 2880.2.t.c | 16 | ||
80.i | odd | 4 | 1 | 400.2.q.h | 16 | ||
80.j | even | 4 | 1 | 1600.2.q.h | 16 | ||
80.k | odd | 4 | 1 | 1600.2.l.i | 16 | ||
80.q | even | 4 | 1 | 400.2.l.h | 16 | ||
80.s | even | 4 | 1 | 1600.2.q.g | 16 | ||
80.t | odd | 4 | 1 | 400.2.q.g | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
80.2.l.a | ✓ | 16 | 8.b | even | 2 | 1 | |
80.2.l.a | ✓ | 16 | 16.e | even | 4 | 1 | |
320.2.l.a | 16 | 8.d | odd | 2 | 1 | ||
320.2.l.a | 16 | 16.f | odd | 4 | 1 | ||
400.2.l.h | 16 | 40.f | even | 2 | 1 | ||
400.2.l.h | 16 | 80.q | even | 4 | 1 | ||
400.2.q.g | 16 | 40.i | odd | 4 | 1 | ||
400.2.q.g | 16 | 80.t | odd | 4 | 1 | ||
400.2.q.h | 16 | 40.i | odd | 4 | 1 | ||
400.2.q.h | 16 | 80.i | odd | 4 | 1 | ||
640.2.l.a | 16 | 4.b | odd | 2 | 1 | ||
640.2.l.a | 16 | 16.f | odd | 4 | 1 | ||
640.2.l.b | 16 | 1.a | even | 1 | 1 | trivial | |
640.2.l.b | 16 | 16.e | even | 4 | 1 | inner | |
720.2.t.c | 16 | 24.h | odd | 2 | 1 | ||
720.2.t.c | 16 | 48.i | odd | 4 | 1 | ||
1600.2.l.i | 16 | 40.e | odd | 2 | 1 | ||
1600.2.l.i | 16 | 80.k | odd | 4 | 1 | ||
1600.2.q.g | 16 | 40.k | even | 4 | 1 | ||
1600.2.q.g | 16 | 80.s | even | 4 | 1 | ||
1600.2.q.h | 16 | 40.k | even | 4 | 1 | ||
1600.2.q.h | 16 | 80.j | even | 4 | 1 | ||
2880.2.t.c | 16 | 24.f | even | 2 | 1 | ||
2880.2.t.c | 16 | 48.k | even | 4 | 1 | ||
5120.2.a.s | 8 | 32.g | even | 8 | 1 | ||
5120.2.a.t | 8 | 32.h | odd | 8 | 1 | ||
5120.2.a.u | 8 | 32.h | odd | 8 | 1 | ||
5120.2.a.v | 8 | 32.g | even | 8 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{16} + 8 T_{3}^{13} + 112 T_{3}^{12} + 80 T_{3}^{11} + 32 T_{3}^{10} + 176 T_{3}^{9} + 2632 T_{3}^{8} + 2560 T_{3}^{7} + 1024 T_{3}^{6} - 2656 T_{3}^{5} + 5824 T_{3}^{4} + 1088 T_{3}^{3} + 128 T_{3}^{2} - 64 T_{3} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(640, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} \)
$3$
\( T^{16} + 8 T^{13} + 112 T^{12} + 80 T^{11} + \cdots + 16 \)
$5$
\( (T^{4} + 1)^{4} \)
$7$
\( T^{16} + 64 T^{14} + 1616 T^{12} + \cdots + 204304 \)
$11$
\( T^{16} - 8 T^{15} + 32 T^{14} + \cdots + 1290496 \)
$13$
\( T^{16} - 128 T^{13} + \cdots + 20647936 \)
$17$
\( (T^{8} - 72 T^{6} - 64 T^{5} + 1408 T^{4} + \cdots + 13888)^{2} \)
$19$
\( T^{16} - 8 T^{15} + 32 T^{14} + \cdots + 614656 \)
$23$
\( T^{16} + 128 T^{14} + 4784 T^{12} + \cdots + 1731856 \)
$29$
\( T^{16} - 16 T^{15} + \cdots + 3017085184 \)
$31$
\( (T^{8} - 96 T^{6} + 64 T^{5} + 2848 T^{4} + \cdots - 20224)^{2} \)
$37$
\( T^{16} - 16 T^{15} + 128 T^{14} + \cdots + 18939904 \)
$41$
\( T^{16} + 384 T^{14} + \cdots + 110660014336 \)
$43$
\( T^{16} + 8 T^{15} + 32 T^{14} + \cdots + 53640976 \)
$47$
\( (T^{8} + 20 T^{7} - 8 T^{6} - 2392 T^{5} + \cdots + 575044)^{2} \)
$53$
\( T^{16} + 16 T^{15} + \cdots + 383725735936 \)
$59$
\( T^{16} - 8 T^{15} + \cdots + 12227051776 \)
$61$
\( T^{16} + 16 T^{15} + \cdots + 1393986371584 \)
$67$
\( T^{16} + 40 T^{15} + \cdots + 46120451769616 \)
$71$
\( T^{16} + 640 T^{14} + \cdots + 3333516427264 \)
$73$
\( T^{16} + 560 T^{14} + \cdots + 15847788544 \)
$79$
\( (T^{8} - 8 T^{7} - 160 T^{6} + 352 T^{5} + \cdots + 4352)^{2} \)
$83$
\( T^{16} + 40 T^{15} + \cdots + 2050640656 \)
$89$
\( T^{16} + 416 T^{14} + \cdots + 684153962496 \)
$97$
\( (T^{8} - 440 T^{6} - 416 T^{5} + \cdots - 8549312)^{2} \)
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