Newspace parameters
Level: | \( N \) | \(=\) | \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 300.f (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(8.17440793081\) |
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: | \( x^{16} + 5x^{14} + 12x^{12} + 25x^{10} + 53x^{8} + 100x^{6} + 192x^{4} + 320x^{2} + 256 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
Coefficient ring index: | \( 2^{24} \) |
Twist minimal: | no (minimal twist has level 60) |
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 \( x^{16} + 5x^{14} + 12x^{12} + 25x^{10} + 53x^{8} + 100x^{6} + 192x^{4} + 320x^{2} + 256 \) :
\(\beta_{1}\) | \(=\) | \( ( -\nu^{14} - \nu^{12} - 8\nu^{10} - 25\nu^{8} - 17\nu^{6} - 96\nu^{4} - 160\nu^{2} - 128 ) / 64 \) |
\(\beta_{2}\) | \(=\) | \( ( 5\nu^{14} - 3\nu^{12} + 16\nu^{10} + 5\nu^{8} - 67\nu^{6} + 24\nu^{4} + 248\nu^{2} - 336 ) / 304 \) |
\(\beta_{3}\) | \(=\) | \( ( -17\nu^{14} - 81\nu^{12} - 176\nu^{10} - 321\nu^{8} - 593\nu^{6} - 1176\nu^{4} - 2120\nu^{2} - 3296 ) / 304 \) |
\(\beta_{4}\) | \(=\) | \( ( -5\nu^{14} - 16\nu^{12} - 35\nu^{10} - 81\nu^{8} - 180\nu^{6} - 271\nu^{4} - 552\nu^{2} - 804 ) / 76 \) |
\(\beta_{5}\) | \(=\) | \( ( 165\nu^{14} + 357\nu^{12} + 680\nu^{10} + 1533\nu^{8} + 2805\nu^{6} + 4896\nu^{4} + 8640\nu^{2} + 9280 ) / 2432 \) |
\(\beta_{6}\) | \(=\) | \( ( 41\nu^{15} + 97\nu^{13} + 192\nu^{11} + 193\nu^{9} + 241\nu^{7} - 1384\nu^{5} + 1760\nu^{3} + 832\nu ) / 2432 \) |
\(\beta_{7}\) | \(=\) | \( ( 267\nu^{14} + 843\nu^{12} + 1432\nu^{10} + 3763\nu^{8} + 6971\nu^{6} + 11040\nu^{4} + 25920\nu^{2} + 32704 ) / 2432 \) |
\(\beta_{8}\) | \(=\) | \( ( 71\nu^{15} + 231\nu^{13} + 440\nu^{11} + 1135\nu^{9} + 1815\nu^{7} + 3168\nu^{5} + 7200\nu^{3} + 9152\nu ) / 2432 \) |
\(\beta_{9}\) | \(=\) | \( ( 15\nu^{15} + 47\nu^{13} + 88\nu^{11} + 215\nu^{9} + 479\nu^{7} + 704\nu^{5} + 1536\nu^{3} + 1984\nu ) / 512 \) |
\(\beta_{10}\) | \(=\) | \( ( - 407 \nu^{14} - 1367 \nu^{12} - 2488 \nu^{10} - 5727 \nu^{8} - 11783 \nu^{6} - 20832 \nu^{4} - 43200 \nu^{2} - 56128 ) / 2432 \) |
\(\beta_{11}\) | \(=\) | \( ( - 353 \nu^{15} - 1217 \nu^{13} - 2984 \nu^{11} - 4761 \nu^{9} - 10257 \nu^{7} - 19904 \nu^{5} - 38272 \nu^{3} - 53312 \nu ) / 9728 \) |
\(\beta_{12}\) | \(=\) | \( ( -103\nu^{15} - 303\nu^{13} - 512\nu^{11} - 1167\nu^{9} - 2207\nu^{7} - 4264\nu^{5} - 8544\nu^{3} - 9920\nu ) / 2432 \) |
\(\beta_{13}\) | \(=\) | \( ( - 449 \nu^{15} - 1889 \nu^{13} - 3048 \nu^{11} - 7289 \nu^{9} - 15537 \nu^{7} - 27904 \nu^{5} - 62976 \nu^{3} - 99392 \nu ) / 9728 \) |
\(\beta_{14}\) | \(=\) | \( ( 527 \nu^{15} + 1295 \nu^{13} + 2872 \nu^{11} + 5847 \nu^{9} + 10175 \nu^{7} + 21408 \nu^{5} + 54016 \nu^{3} + 48064 \nu ) / 9728 \) |
\(\beta_{15}\) | \(=\) | \( ( 427 \nu^{15} + 1051 \nu^{13} + 2248 \nu^{11} + 5139 \nu^{9} + 9995 \nu^{7} + 19408 \nu^{5} + 39936 \nu^{3} + 56000 \nu ) / 4864 \) |
\(\nu\) | \(=\) | \( ( 2\beta_{15} - 2\beta_{14} - \beta_{13} + \beta_{12} - 3\beta_{9} + \beta_{6} ) / 8 \) |
\(\nu^{2}\) | \(=\) | \( ( -\beta_{10} + \beta_{7} - 4\beta_{5} + \beta_{3} + 4\beta_{2} + \beta _1 - 4 ) / 8 \) |
\(\nu^{3}\) | \(=\) | \( ( -\beta_{15} + 3\beta_{14} - \beta_{13} + 2\beta_{12} + \beta_{11} + \beta_{9} - \beta_{8} ) / 4 \) |
\(\nu^{4}\) | \(=\) | \( ( -5\beta_{10} - 7\beta_{7} + 4\beta_{5} + 4\beta_{4} + \beta_{3} - 6\beta_{2} - 5\beta_1 ) / 8 \) |
\(\nu^{5}\) | \(=\) | \( ( \beta_{13} - 7\beta_{12} - 2\beta_{11} - 5\beta_{9} - 2\beta_{8} - 7\beta_{6} ) / 8 \) |
\(\nu^{6}\) | \(=\) | \( ( -2\beta_{7} - 6\beta_{4} + 2\beta_{3} - \beta_{2} + 3\beta _1 - 10 ) / 4 \) |
\(\nu^{7}\) | \(=\) | \( ( -2\beta_{15} - 2\beta_{14} + 3\beta_{13} + 5\beta_{12} + 41\beta_{9} - 20\beta_{8} + \beta_{6} ) / 8 \) |
\(\nu^{8}\) | \(=\) | \( ( 21\beta_{10} + 27\beta_{7} + 4\beta_{5} - 5\beta_{3} - 20\beta_{2} - 21\beta _1 - 12 ) / 8 \) |
\(\nu^{9}\) | \(=\) | \( ( \beta_{15} - 3\beta_{14} + 5\beta_{13} + 10\beta_{12} + 7\beta_{11} - 5\beta_{9} + 41\beta_{8} - 4\beta_{6} ) / 4 \) |
\(\nu^{10}\) | \(=\) | \( ( 21\beta_{10} - 9\beta_{7} - 4\beta_{5} - 36\beta_{4} - 33\beta_{3} + 54\beta_{2} + 21\beta _1 - 16 ) / 8 \) |
\(\nu^{11}\) | \(=\) | \( ( 23\beta_{13} + 47\beta_{12} - 78\beta_{11} - 19\beta_{9} + 18\beta_{8} + 15\beta_{6} ) / 8 \) |
\(\nu^{12}\) | \(=\) | \( ( -12\beta_{10} + 14\beta_{7} + 54\beta_{4} - 14\beta_{3} - 31\beta_{2} + 17\beta _1 - 46 ) / 4 \) |
\(\nu^{13}\) | \(=\) | \( ( -94\beta_{15} - 30\beta_{14} - 67\beta_{13} - 213\beta_{12} - 73\beta_{9} - 12\beta_{8} + 15\beta_{6} ) / 8 \) |
\(\nu^{14}\) | \(=\) | \( ( -29\beta_{10} - 51\beta_{7} + 188\beta_{5} + 93\beta_{3} + 100\beta_{2} + 29\beta _1 + 476 ) / 8 \) |
\(\nu^{15}\) | \(=\) | \( ( 135 \beta_{15} - 53 \beta_{14} + 63 \beta_{13} - 134 \beta_{12} + 73 \beta_{11} - 63 \beta_{9} - 153 \beta_{8} + 72 \beta_{6} ) / 4 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).
\(n\) | \(101\) | \(151\) | \(277\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
199.1 |
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−1.95141 | − | 0.438172i | 1.73205 | 3.61601 | + | 1.71011i | 0 | −3.37994 | − | 0.758935i | −6.33166 | −6.30701 | − | 4.92155i | 3.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
199.2 | −1.95141 | + | 0.438172i | 1.73205 | 3.61601 | − | 1.71011i | 0 | −3.37994 | + | 0.758935i | −6.33166 | −6.30701 | + | 4.92155i | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
199.3 | −1.08539 | − | 1.67986i | −1.73205 | −1.64388 | + | 3.64660i | 0 | 1.87994 | + | 2.90961i | 0.596540 | 7.91002 | − | 1.19648i | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
199.4 | −1.08539 | + | 1.67986i | −1.73205 | −1.64388 | − | 3.64660i | 0 | 1.87994 | − | 2.90961i | 0.596540 | 7.91002 | + | 1.19648i | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
199.5 | −0.696577 | − | 1.87477i | 1.73205 | −3.02956 | + | 2.61185i | 0 | −1.20651 | − | 3.24721i | 5.46770 | 7.00695 | + | 3.86039i | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
199.6 | −0.696577 | + | 1.87477i | 1.73205 | −3.02956 | − | 2.61185i | 0 | −1.20651 | + | 3.24721i | 5.46770 | 7.00695 | − | 3.86039i | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
199.7 | −0.169449 | − | 1.99281i | 1.73205 | −3.94257 | + | 0.675358i | 0 | −0.293494 | − | 3.45165i | −12.3959 | 2.01392 | + | 7.74236i | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
199.8 | −0.169449 | + | 1.99281i | 1.73205 | −3.94257 | − | 0.675358i | 0 | −0.293494 | + | 3.45165i | −12.3959 | 2.01392 | − | 7.74236i | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
199.9 | 0.169449 | − | 1.99281i | −1.73205 | −3.94257 | − | 0.675358i | 0 | −0.293494 | + | 3.45165i | 12.3959 | −2.01392 | + | 7.74236i | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
199.10 | 0.169449 | + | 1.99281i | −1.73205 | −3.94257 | + | 0.675358i | 0 | −0.293494 | − | 3.45165i | 12.3959 | −2.01392 | − | 7.74236i | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
199.11 | 0.696577 | − | 1.87477i | −1.73205 | −3.02956 | − | 2.61185i | 0 | −1.20651 | + | 3.24721i | −5.46770 | −7.00695 | + | 3.86039i | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
199.12 | 0.696577 | + | 1.87477i | −1.73205 | −3.02956 | + | 2.61185i | 0 | −1.20651 | − | 3.24721i | −5.46770 | −7.00695 | − | 3.86039i | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
199.13 | 1.08539 | − | 1.67986i | 1.73205 | −1.64388 | − | 3.64660i | 0 | 1.87994 | − | 2.90961i | −0.596540 | −7.91002 | − | 1.19648i | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
199.14 | 1.08539 | + | 1.67986i | 1.73205 | −1.64388 | + | 3.64660i | 0 | 1.87994 | + | 2.90961i | −0.596540 | −7.91002 | + | 1.19648i | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
199.15 | 1.95141 | − | 0.438172i | −1.73205 | 3.61601 | − | 1.71011i | 0 | −3.37994 | + | 0.758935i | 6.33166 | 6.30701 | − | 4.92155i | 3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
199.16 | 1.95141 | + | 0.438172i | −1.73205 | 3.61601 | + | 1.71011i | 0 | −3.37994 | − | 0.758935i | 6.33166 | 6.30701 | + | 4.92155i | 3.00000 | 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 |
20.d | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 300.3.f.b | 16 | |
3.b | odd | 2 | 1 | 900.3.f.f | 16 | ||
4.b | odd | 2 | 1 | inner | 300.3.f.b | 16 | |
5.b | even | 2 | 1 | inner | 300.3.f.b | 16 | |
5.c | odd | 4 | 1 | 60.3.c.a | ✓ | 8 | |
5.c | odd | 4 | 1 | 300.3.c.d | 8 | ||
12.b | even | 2 | 1 | 900.3.f.f | 16 | ||
15.d | odd | 2 | 1 | 900.3.f.f | 16 | ||
15.e | even | 4 | 1 | 180.3.c.b | 8 | ||
15.e | even | 4 | 1 | 900.3.c.u | 8 | ||
20.d | odd | 2 | 1 | inner | 300.3.f.b | 16 | |
20.e | even | 4 | 1 | 60.3.c.a | ✓ | 8 | |
20.e | even | 4 | 1 | 300.3.c.d | 8 | ||
40.i | odd | 4 | 1 | 960.3.e.c | 8 | ||
40.k | even | 4 | 1 | 960.3.e.c | 8 | ||
60.h | even | 2 | 1 | 900.3.f.f | 16 | ||
60.l | odd | 4 | 1 | 180.3.c.b | 8 | ||
60.l | odd | 4 | 1 | 900.3.c.u | 8 | ||
120.q | odd | 4 | 1 | 2880.3.e.j | 8 | ||
120.w | even | 4 | 1 | 2880.3.e.j | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
60.3.c.a | ✓ | 8 | 5.c | odd | 4 | 1 | |
60.3.c.a | ✓ | 8 | 20.e | even | 4 | 1 | |
180.3.c.b | 8 | 15.e | even | 4 | 1 | ||
180.3.c.b | 8 | 60.l | odd | 4 | 1 | ||
300.3.c.d | 8 | 5.c | odd | 4 | 1 | ||
300.3.c.d | 8 | 20.e | even | 4 | 1 | ||
300.3.f.b | 16 | 1.a | even | 1 | 1 | trivial | |
300.3.f.b | 16 | 4.b | odd | 2 | 1 | inner | |
300.3.f.b | 16 | 5.b | even | 2 | 1 | inner | |
300.3.f.b | 16 | 20.d | odd | 2 | 1 | inner | |
900.3.c.u | 8 | 15.e | even | 4 | 1 | ||
900.3.c.u | 8 | 60.l | odd | 4 | 1 | ||
900.3.f.f | 16 | 3.b | odd | 2 | 1 | ||
900.3.f.f | 16 | 12.b | even | 2 | 1 | ||
900.3.f.f | 16 | 15.d | odd | 2 | 1 | ||
900.3.f.f | 16 | 60.h | even | 2 | 1 | ||
960.3.e.c | 8 | 40.i | odd | 4 | 1 | ||
960.3.e.c | 8 | 40.k | even | 4 | 1 | ||
2880.3.e.j | 8 | 120.q | odd | 4 | 1 | ||
2880.3.e.j | 8 | 120.w | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} - 224T_{7}^{6} + 12032T_{7}^{4} - 188416T_{7}^{2} + 65536 \)
acting on \(S_{3}^{\mathrm{new}}(300, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} + 10 T^{14} + 33 T^{12} + \cdots + 65536 \)
$3$
\( (T^{2} - 3)^{8} \)
$5$
\( T^{16} \)
$7$
\( (T^{8} - 224 T^{6} + 12032 T^{4} + \cdots + 65536)^{2} \)
$11$
\( (T^{4} + 208 T^{2} + 10496)^{4} \)
$13$
\( (T^{8} + 1008 T^{6} + 290528 T^{4} + \cdots + 155351296)^{2} \)
$17$
\( (T^{8} + 848 T^{6} + 162144 T^{4} + \cdots + 77721856)^{2} \)
$19$
\( (T^{8} + 1696 T^{6} + \cdots + 6544162816)^{2} \)
$23$
\( (T^{8} - 3616 T^{6} + \cdots + 101419319296)^{2} \)
$29$
\( (T^{4} + 32 T^{3} - 2152 T^{2} + \cdots + 1334416)^{4} \)
$31$
\( (T^{8} + 5408 T^{6} + \cdots + 59895709696)^{2} \)
$37$
\( (T^{8} + 6192 T^{6} + \cdots + 59919206656)^{2} \)
$41$
\( (T^{4} + 8 T^{3} - 1800 T^{2} + \cdots + 87184)^{4} \)
$43$
\( (T^{8} - 10816 T^{6} + \cdots + 33624411406336)^{2} \)
$47$
\( (T^{8} - 8032 T^{6} + \cdots + 1056981385216)^{2} \)
$53$
\( (T^{8} + 11472 T^{6} + \cdots + 228545188096)^{2} \)
$59$
\( (T^{8} + 4896 T^{6} + \cdots + 173909016576)^{2} \)
$61$
\( (T^{4} + 88 T^{3} - 2536 T^{2} + \cdots - 2142704)^{4} \)
$67$
\( (T^{8} - 16064 T^{6} + \cdots + 281086590976)^{2} \)
$71$
\( (T^{8} + 13952 T^{6} + \cdots + 16079971680256)^{2} \)
$73$
\( (T^{8} + 17552 T^{6} + \cdots + 24622079140096)^{2} \)
$79$
\( (T^{8} + 41888 T^{6} + \cdots + 31\!\cdots\!36)^{2} \)
$83$
\( (T^{8} - 36928 T^{6} + \cdots + 42\!\cdots\!36)^{2} \)
$89$
\( (T^{4} + 40 T^{3} - 20584 T^{2} + \cdots + 70652944)^{4} \)
$97$
\( (T^{8} + 34896 T^{6} + \cdots + 35\!\cdots\!76)^{2} \)
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