Newspace parameters
Level: | \( N \) | \(=\) | \( 320 = 2^{6} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 320.e (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(8.71936845953\) |
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: | \( x^{16} + 19x^{14} + 301x^{12} + 1102x^{10} + 3238x^{8} + 1102x^{6} + 301x^{4} + 19x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
Coefficient ring index: | \( 2^{34} \) |
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 \( x^{16} + 19x^{14} + 301x^{12} + 1102x^{10} + 3238x^{8} + 1102x^{6} + 301x^{4} + 19x^{2} + 1 \) :
\(\beta_{1}\) | \(=\) | \( ( 89 \nu^{15} + 1682 \nu^{13} + 26620 \nu^{11} + 95408 \nu^{9} + 278884 \nu^{7} + 71420 \nu^{5} + 24119 \nu^{3} + 2802 \nu ) / 300 \) |
\(\beta_{2}\) | \(=\) | \( ( - 5491 \nu^{14} - 104672 \nu^{12} - 1659105 \nu^{10} - 6150577 \nu^{8} - 18098609 \nu^{6} - 6967305 \nu^{4} - 1444486 \nu^{2} - 160337 ) / 23400 \) |
\(\beta_{3}\) | \(=\) | \( ( - 484 \nu^{14} - 9043 \nu^{12} - 142780 \nu^{10} - 487388 \nu^{8} - 1399796 \nu^{6} - 45980 \nu^{4} - 2904 \nu^{2} + 7647 ) / 1950 \) |
\(\beta_{4}\) | \(=\) | \( ( - 6793 \nu^{14} - 128996 \nu^{12} - 2043195 \nu^{10} - 7461691 \nu^{8} - 21872627 \nu^{6} - 7090995 \nu^{4} - 1452298 \nu^{2} + 45169 ) / 23400 \) |
\(\beta_{5}\) | \(=\) | \( ( 60\nu^{14} + 1121\nu^{12} + 17700\nu^{10} + 60420\nu^{8} + 173642\nu^{6} + 5700\nu^{4} + 360\nu^{2} - 1729 ) / 150 \) |
\(\beta_{6}\) | \(=\) | \( ( 2182 \nu^{15} + 42080 \nu^{13} + 668520 \nu^{11} + 2590264 \nu^{9} + 7726640 \nu^{7} + 4329720 \nu^{5} + 1079482 \nu^{3} + 121760 \nu ) / 2925 \) |
\(\beta_{7}\) | \(=\) | \( ( 1007 \nu^{14} + 19074 \nu^{12} + 302005 \nu^{10} + 1092309 \nu^{8} + 3201253 \nu^{6} + 937805 \nu^{4} + 297502 \nu^{2} + 10979 ) / 1950 \) |
\(\beta_{8}\) | \(=\) | \( ( 80\nu^{14} + 1513\nu^{12} + 23950\nu^{10} + 86110\nu^{8} + 252226\nu^{6} + 68750\nu^{4} + 25230\nu^{2} + 913 ) / 150 \) |
\(\beta_{9}\) | \(=\) | \( ( 2177 \nu^{15} + 41200 \nu^{13} + 652160 \nu^{11} + 2349604 \nu^{9} + 6863340 \nu^{7} + 1846960 \nu^{5} + 400527 \nu^{3} - 67060 \nu ) / 1950 \) |
\(\beta_{10}\) | \(=\) | \( ( 379 \nu^{15} + 7182 \nu^{13} + 113715 \nu^{11} + 411883 \nu^{9} + 1205379 \nu^{7} + 353115 \nu^{5} + 84304 \nu^{3} + 1197 \nu ) / 300 \) |
\(\beta_{11}\) | \(=\) | \( ( 5949 \nu^{15} + 112740 \nu^{13} + 1785050 \nu^{11} + 6466898 \nu^{9} + 18921530 \nu^{7} + 5543050 \nu^{5} + 1271149 \nu^{3} + 18790 \nu ) / 3900 \) |
\(\beta_{12}\) | \(=\) | \( ( - 100 \nu^{14} - 1899 \nu^{12} - 30080 \nu^{10} - 109880 \nu^{8} - 322398 \nu^{6} - 105880 \nu^{4} - 25980 \nu^{2} - 999 ) / 75 \) |
\(\beta_{13}\) | \(=\) | \( ( - 381 \nu^{15} - 7218 \nu^{13} - 114285 \nu^{11} - 413597 \nu^{9} - 1211421 \nu^{7} - 354885 \nu^{5} - 100416 \nu^{3} - 1203 \nu ) / 100 \) |
\(\beta_{14}\) | \(=\) | \( ( 611 \nu^{15} + 11718 \nu^{13} + 185970 \nu^{11} + 705902 \nu^{9} + 2094906 \nu^{7} + 1012770 \nu^{5} + 264491 \nu^{3} + 29988 \nu ) / 150 \) |
\(\beta_{15}\) | \(=\) | \( ( 4443 \nu^{15} + 84180 \nu^{13} + 1332850 \nu^{11} + 4825036 \nu^{9} + 14128210 \nu^{7} + 4138850 \nu^{5} + 1104293 \nu^{3} + 14030 \nu ) / 975 \) |
\(\nu\) | \(=\) | \( ( 4\beta_{15} - \beta_{14} + 2\beta_{13} - 2\beta_{10} - 4\beta_{9} + 3\beta_{6} - 6\beta_1 ) / 32 \) |
\(\nu^{2}\) | \(=\) | \( ( 5\beta_{12} - 4\beta_{8} + 11\beta_{7} + 2\beta_{5} + 2\beta_{3} - 12\beta_{2} - 38 ) / 16 \) |
\(\nu^{3}\) | \(=\) | \( ( -7\beta_{15} - 5\beta_{13} - 4\beta_{11} + 15\beta_{10} ) / 4 \) |
\(\nu^{4}\) | \(=\) | \( ( -85\beta_{12} + 44\beta_{8} - 149\beta_{7} + 38\beta_{5} + 200\beta_{4} + 42\beta_{3} + 28\beta_{2} - 482 ) / 16 \) |
\(\nu^{5}\) | \(=\) | \( ( 124 \beta_{15} + 281 \beta_{14} + 278 \beta_{13} + 320 \beta_{11} - 998 \beta_{10} + 556 \beta_{9} - 957 \beta_{6} + 726 \beta_1 ) / 32 \) |
\(\nu^{6}\) | \(=\) | \( -75\beta_{5} - 160\beta_{4} - 85\beta_{3} + 160\beta_{2} + 874 \) |
\(\nu^{7}\) | \(=\) | \( ( 10116 \beta_{15} - 4279 \beta_{14} + 4118 \beta_{13} + 5120 \beta_{11} - 15398 \beta_{10} - 8236 \beta_{9} + 14637 \beta_{6} - 10314 \beta_1 ) / 32 \) |
\(\nu^{8}\) | \(=\) | \( ( 20075 \beta_{12} - 8836 \beta_{8} + 33029 \beta_{7} + 9158 \beta_{5} - 8200 \beta_{4} + 10438 \beta_{3} - 46748 \beta_{2} - 104642 ) / 16 \) |
\(\nu^{9}\) | \(=\) | \( ( -22593\beta_{15} - 15485\beta_{13} - 19596\beta_{11} + 58455\beta_{10} ) / 4 \) |
\(\nu^{10}\) | \(=\) | \( ( - 303835 \beta_{12} + 132716 \beta_{8} - 498491 \beta_{7} + 138722 \beta_{5} + 707200 \beta_{4} + 158318 \beta_{3} + 125132 \beta_{2} - 1577798 ) / 16 \) |
\(\nu^{11}\) | \(=\) | \( ( 429756 \beta_{15} + 979439 \beta_{14} + 935042 \beta_{13} + 1188160 \beta_{11} - 3537602 \beta_{10} + 1870084 \beta_{9} - 3354483 \beta_{6} + 2312154 \beta_1 ) / 32 \) |
\(\nu^{12}\) | \(=\) | \( -262200\beta_{5} - 549680\beta_{4} - 299330\beta_{3} + 549680\beta_{2} + 2979001 \) |
\(\nu^{13}\) | \(=\) | \( ( 34749764 \beta_{15} - 14803441 \beta_{14} + 14127362 \beta_{13} + 17968960 \beta_{11} - 53476802 \beta_{10} - 28254724 \beta_{9} + 50703123 \beta_{6} - 34913766 \beta_1 ) / 32 \) |
\(\nu^{14}\) | \(=\) | \( ( 69423845 \beta_{12} - 30257524 \beta_{8} + 113808731 \beta_{7} + 31704482 \beta_{5} - 28659200 \beta_{4} + 36196722 \beta_{3} - 161567692 \beta_{2} - 360121478 ) / 16 \) |
\(\nu^{15}\) | \(=\) | \( ( -77912207\beta_{15} - 53372525\beta_{13} - 67901204\beta_{11} + 202057575\beta_{10} ) / 4 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).
\(n\) | \(191\) | \(257\) | \(261\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
159.1 |
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0 | − | 5.13399i | 0 | −2.79662 | − | 4.14474i | 0 | −6.19337 | 0 | −17.3578 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
159.2 | 0 | − | 5.13399i | 0 | −2.79662 | + | 4.14474i | 0 | 6.19337 | 0 | −17.3578 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
159.3 | 0 | − | 5.13399i | 0 | 2.79662 | − | 4.14474i | 0 | −6.19337 | 0 | −17.3578 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
159.4 | 0 | − | 5.13399i | 0 | 2.79662 | + | 4.14474i | 0 | 6.19337 | 0 | −17.3578 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
159.5 | 0 | − | 1.90845i | 0 | −4.37937 | − | 2.41269i | 0 | −3.95502 | 0 | 5.35782 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
159.6 | 0 | − | 1.90845i | 0 | −4.37937 | + | 2.41269i | 0 | 3.95502 | 0 | 5.35782 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
159.7 | 0 | − | 1.90845i | 0 | 4.37937 | − | 2.41269i | 0 | −3.95502 | 0 | 5.35782 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
159.8 | 0 | − | 1.90845i | 0 | 4.37937 | + | 2.41269i | 0 | 3.95502 | 0 | 5.35782 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
159.9 | 0 | 1.90845i | 0 | −4.37937 | − | 2.41269i | 0 | 3.95502 | 0 | 5.35782 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
159.10 | 0 | 1.90845i | 0 | −4.37937 | + | 2.41269i | 0 | −3.95502 | 0 | 5.35782 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
159.11 | 0 | 1.90845i | 0 | 4.37937 | − | 2.41269i | 0 | 3.95502 | 0 | 5.35782 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
159.12 | 0 | 1.90845i | 0 | 4.37937 | + | 2.41269i | 0 | −3.95502 | 0 | 5.35782 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
159.13 | 0 | 5.13399i | 0 | −2.79662 | − | 4.14474i | 0 | 6.19337 | 0 | −17.3578 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
159.14 | 0 | 5.13399i | 0 | −2.79662 | + | 4.14474i | 0 | −6.19337 | 0 | −17.3578 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
159.15 | 0 | 5.13399i | 0 | 2.79662 | − | 4.14474i | 0 | 6.19337 | 0 | −17.3578 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
159.16 | 0 | 5.13399i | 0 | 2.79662 | + | 4.14474i | 0 | −6.19337 | 0 | −17.3578 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
8.b | even | 2 | 1 | inner |
8.d | odd | 2 | 1 | inner |
20.d | odd | 2 | 1 | inner |
40.e | odd | 2 | 1 | inner |
40.f | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 320.3.e.b | ✓ | 16 |
4.b | odd | 2 | 1 | inner | 320.3.e.b | ✓ | 16 |
5.b | even | 2 | 1 | inner | 320.3.e.b | ✓ | 16 |
5.c | odd | 4 | 2 | 1600.3.g.k | 16 | ||
8.b | even | 2 | 1 | inner | 320.3.e.b | ✓ | 16 |
8.d | odd | 2 | 1 | inner | 320.3.e.b | ✓ | 16 |
16.e | even | 4 | 2 | 1280.3.h.n | 16 | ||
16.f | odd | 4 | 2 | 1280.3.h.n | 16 | ||
20.d | odd | 2 | 1 | inner | 320.3.e.b | ✓ | 16 |
20.e | even | 4 | 2 | 1600.3.g.k | 16 | ||
40.e | odd | 2 | 1 | inner | 320.3.e.b | ✓ | 16 |
40.f | even | 2 | 1 | inner | 320.3.e.b | ✓ | 16 |
40.i | odd | 4 | 2 | 1600.3.g.k | 16 | ||
40.k | even | 4 | 2 | 1600.3.g.k | 16 | ||
80.k | odd | 4 | 2 | 1280.3.h.n | 16 | ||
80.q | even | 4 | 2 | 1280.3.h.n | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
320.3.e.b | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
320.3.e.b | ✓ | 16 | 4.b | odd | 2 | 1 | inner |
320.3.e.b | ✓ | 16 | 5.b | even | 2 | 1 | inner |
320.3.e.b | ✓ | 16 | 8.b | even | 2 | 1 | inner |
320.3.e.b | ✓ | 16 | 8.d | odd | 2 | 1 | inner |
320.3.e.b | ✓ | 16 | 20.d | odd | 2 | 1 | inner |
320.3.e.b | ✓ | 16 | 40.e | odd | 2 | 1 | inner |
320.3.e.b | ✓ | 16 | 40.f | even | 2 | 1 | inner |
1280.3.h.n | 16 | 16.e | even | 4 | 2 | ||
1280.3.h.n | 16 | 16.f | odd | 4 | 2 | ||
1280.3.h.n | 16 | 80.k | odd | 4 | 2 | ||
1280.3.h.n | 16 | 80.q | even | 4 | 2 | ||
1600.3.g.k | 16 | 5.c | odd | 4 | 2 | ||
1600.3.g.k | 16 | 20.e | even | 4 | 2 | ||
1600.3.g.k | 16 | 40.i | odd | 4 | 2 | ||
1600.3.g.k | 16 | 40.k | even | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 30T_{3}^{2} + 96 \)
acting on \(S_{3}^{\mathrm{new}}(320, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} \)
$3$
\( (T^{4} + 30 T^{2} + 96)^{4} \)
$5$
\( (T^{8} - 8 T^{6} + 750 T^{4} + \cdots + 390625)^{2} \)
$7$
\( (T^{4} - 54 T^{2} + 600)^{4} \)
$11$
\( (T^{4} - 440 T^{2} + 15376)^{4} \)
$13$
\( (T^{4} - 276 T^{2} + 6144)^{4} \)
$17$
\( (T^{4} + 552 T^{2} + 24576)^{4} \)
$19$
\( (T^{2} - 108)^{8} \)
$23$
\( (T^{4} - 2070 T^{2} + 786264)^{4} \)
$29$
\( (T^{4} + 944 T^{2} + 16384)^{4} \)
$31$
\( (T^{4} + 1040 T^{2} + 262144)^{4} \)
$37$
\( (T^{4} - 1836 T^{2} + 726624)^{4} \)
$41$
\( (T^{2} + 6 T - 1152)^{8} \)
$43$
\( (T^{4} + 3294 T^{2} + 1548384)^{4} \)
$47$
\( (T^{4} - 2550 T^{2} + 411864)^{4} \)
$53$
\( (T^{4} - 6324 T^{2} + 1935744)^{4} \)
$59$
\( (T^{4} - 10904 T^{2} + 2704)^{4} \)
$61$
\( (T^{4} + 6972 T^{2} + 12110400)^{4} \)
$67$
\( (T^{4} + 16686 T^{2} + 67254624)^{4} \)
$71$
\( (T^{4} + 1104 T^{2} + 230400)^{4} \)
$73$
\( (T^{4} + 9384 T^{2} + 960000)^{4} \)
$79$
\( (T^{4} + 27152 T^{2} + \cdots + 149426176)^{4} \)
$83$
\( (T^{4} + 19038 T^{2} + 89397600)^{4} \)
$89$
\( (T^{2} + 144 T + 4668)^{8} \)
$97$
\( (T^{4} + 27816 T^{2} + \cdots + 147609600)^{4} \)
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