Newspace parameters
Level: | \( N \) | \(=\) | \( 100 = 2^{2} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 100.d (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(10.3369963084\) |
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: | \( x^{16} - 5x^{14} + 21x^{12} + 35x^{10} - 199x^{8} + 560x^{6} + 5376x^{4} - 20480x^{2} + 65536 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
Coefficient ring index: | \( 2^{37}\cdot 5^{8} \) |
Twist minimal: | no (minimal twist has level 20) |
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 \( x^{16} - 5x^{14} + 21x^{12} + 35x^{10} - 199x^{8} + 560x^{6} + 5376x^{4} - 20480x^{2} + 65536 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{15} - 5\nu^{13} + 21\nu^{11} + 35\nu^{9} - 199\nu^{7} + 560\nu^{5} + 5376\nu^{3} + 45056\nu ) / 8192 \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{15} - 5\nu^{13} + 21\nu^{11} + 35\nu^{9} - 199\nu^{7} + 560\nu^{5} + 5376\nu^{3} - 20480\nu ) / 8192 \) |
\(\beta_{3}\) | \(=\) | \( ( -23\nu^{14} - 13\nu^{12} - 99\nu^{10} - 2213\nu^{8} - 1183\nu^{6} + 15920\nu^{4} - 128000\nu^{2} - 26624 ) / 15360 \) |
\(\beta_{4}\) | \(=\) | \( ( \nu^{14} - 5\nu^{12} + 21\nu^{10} + 35\nu^{8} - 199\nu^{6} + 560\nu^{4} + 5376\nu^{2} - 18432 ) / 1024 \) |
\(\beta_{5}\) | \(=\) | \( ( \nu^{14} + 59\nu^{12} - 43\nu^{10} + 99\nu^{8} + 3321\nu^{6} + 880\nu^{4} - 42496\nu^{2} + 207872 ) / 3072 \) |
\(\beta_{6}\) | \(=\) | \( ( 7\nu^{14} + 117\nu^{12} - 229\nu^{10} + 1517\nu^{8} + 16087\nu^{6} - 600\nu^{4} + 21760\nu^{2} + 810496 ) / 7680 \) |
\(\beta_{7}\) | \(=\) | \( ( -13\nu^{14} + 45\nu^{12} - 45\nu^{10} - 491\nu^{8} + 479\nu^{6} + 9372\nu^{4} - 33856\nu^{2} + 81664 ) / 3840 \) |
\(\beta_{8}\) | \(=\) | \( ( 47 \nu^{15} - 475 \nu^{13} + 395 \nu^{11} + 1469 \nu^{9} - 18521 \nu^{7} - 41888 \nu^{5} + 347904 \nu^{3} - 1458176 \nu ) / 122880 \) |
\(\beta_{9}\) | \(=\) | \( ( - 33 \nu^{14} + 357 \nu^{12} - 629 \nu^{10} - 2243 \nu^{8} + 18407 \nu^{6} + 11920 \nu^{4} - 175360 \nu^{2} + 1135616 ) / 15360 \) |
\(\beta_{10}\) | \(=\) | \( ( - 13 \nu^{15} + 113 \nu^{13} - 257 \nu^{11} - 727 \nu^{9} + 9643 \nu^{7} + 8512 \nu^{5} - 126720 \nu^{3} + 503808 \nu ) / 24576 \) |
\(\beta_{11}\) | \(=\) | \( ( -9\nu^{15} + 39\nu^{13} - 63\nu^{11} + 103\nu^{9} + 2573\nu^{7} + 6682\nu^{5} + 14144\nu^{3} + 219136\nu ) / 15360 \) |
\(\beta_{12}\) | \(=\) | \( ( - 45 \nu^{15} + 1041 \nu^{13} - 1697 \nu^{11} - 9271 \nu^{9} + 50059 \nu^{7} + 2624 \nu^{5} - 733952 \nu^{3} + 2543616 \nu ) / 122880 \) |
\(\beta_{13}\) | \(=\) | \( ( 61\nu^{14} - 81\nu^{12} - 223\nu^{10} + 4663\nu^{8} + 1973\nu^{6} - 21808\nu^{4} + 251264\nu^{2} + 124416 ) / 7680 \) |
\(\beta_{14}\) | \(=\) | \( ( 49 \nu^{15} + 275 \nu^{13} + 605 \nu^{11} + 3803 \nu^{9} + 35713 \nu^{7} - 55176 \nu^{5} + 185088 \nu^{3} + 1128448 \nu ) / 61440 \) |
\(\beta_{15}\) | \(=\) | \( ( 107 \nu^{15} - 331 \nu^{13} + 267 \nu^{11} + 6685 \nu^{9} - 11785 \nu^{7} - 49092 \nu^{5} + 436096 \nu^{3} - 745472 \nu ) / 30720 \) |
\(\nu\) | \(=\) | \( ( -\beta_{2} + \beta_1 ) / 8 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{9} - \beta_{5} + \beta_{4} - \beta_{3} + 10 ) / 16 \) |
\(\nu^{3}\) | \(=\) | \( ( \beta_{12} + 2\beta_{11} - \beta_{10} + 2\beta_{8} + 2\beta_{2} ) / 8 \) |
\(\nu^{4}\) | \(=\) | \( ( 2\beta_{13} + 4\beta_{7} + \beta_{5} + 8\beta_{4} + 7\beta_{3} - 29 ) / 16 \) |
\(\nu^{5}\) | \(=\) | \( ( \beta_{15} - 3\beta_{14} + \beta_{12} + 4\beta_{11} + 4\beta_{10} - 3\beta_{8} + 42\beta_{2} - 2\beta_1 ) / 8 \) |
\(\nu^{6}\) | \(=\) | \( ( -4\beta_{13} - \beta_{9} - 8\beta_{7} + 12\beta_{6} - 4\beta_{5} + 5\beta_{4} + 8\beta_{3} - 583 ) / 16 \) |
\(\nu^{7}\) | \(=\) | \( ( -2\beta_{15} + 6\beta_{14} - 3\beta_{12} + 5\beta_{11} + 26\beta_{10} + 34\beta_{8} + 64\beta_{2} - 25\beta_1 ) / 8 \) |
\(\nu^{8}\) | \(=\) | \( ( 3\beta_{13} - 51\beta_{9} + 36\beta_{7} + 12\beta_{6} + 18\beta_{5} - 52\beta_{4} - 15\beta_{3} - 490 ) / 8 \) |
\(\nu^{9}\) | \(=\) | \( ( 21 \beta_{15} + 9 \beta_{14} - 99 \beta_{12} - 21 \beta_{11} + 21 \beta_{10} - 201 \beta_{8} - 215 \beta_{2} - 120 \beta_1 ) / 8 \) |
\(\nu^{10}\) | \(=\) | \( ( -90\beta_{13} - 57\beta_{9} - 30\beta_{7} + 30\beta_{6} + 39\beta_{5} + 135\beta_{4} - 213\beta_{3} + 2566 ) / 8 \) |
\(\nu^{11}\) | \(=\) | \( ( - 144 \beta_{15} + 216 \beta_{14} - 204 \beta_{12} - 336 \beta_{11} - 204 \beta_{10} - 384 \beta_{8} + 397 \beta_{2} + 395 \beta_1 ) / 8 \) |
\(\nu^{12}\) | \(=\) | \( ( 24 \beta_{13} + 791 \beta_{9} + 288 \beta_{7} - 624 \beta_{6} + 277 \beta_{5} + 551 \beta_{4} - 1619 \beta_{3} - 10774 ) / 16 \) |
\(\nu^{13}\) | \(=\) | \( ( 132 \beta_{15} + 84 \beta_{14} + 803 \beta_{12} + 310 \beta_{11} - 2039 \beta_{10} - 2654 \beta_{8} + 490 \beta_{2} - 1428 \beta_1 ) / 8 \) |
\(\nu^{14}\) | \(=\) | \( ( 1774 \beta_{13} + 4344 \beta_{9} - 3652 \beta_{7} - 2832 \beta_{6} + 2507 \beta_{5} + 8248 \beta_{4} + 4949 \beta_{3} + 14033 ) / 16 \) |
\(\nu^{15}\) | \(=\) | \( ( 1991 \beta_{15} - 1557 \beta_{14} + 5231 \beta_{12} - 2656 \beta_{11} + 1664 \beta_{10} - 477 \beta_{8} + 25158 \beta_{2} + 5390 \beta_1 ) / 8 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/100\mathbb{Z}\right)^\times\).
\(n\) | \(51\) | \(77\) |
\(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
99.1 |
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−3.78898 | − | 1.28203i | 8.14153 | 12.7128 | + | 9.71518i | 0 | −30.8481 | − | 10.4377i | −63.6032 | −35.7134 | − | 53.1089i | −14.7155 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
99.2 | −3.78898 | + | 1.28203i | 8.14153 | 12.7128 | − | 9.71518i | 0 | −30.8481 | + | 10.4377i | −63.6032 | −35.7134 | + | 53.1089i | −14.7155 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
99.3 | −3.43282 | − | 2.05323i | −15.5779 | 7.56853 | + | 14.0967i | 0 | 53.4761 | + | 31.9849i | 37.6230 | 2.96235 | − | 63.9314i | 161.671 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
99.4 | −3.43282 | + | 2.05323i | −15.5779 | 7.56853 | − | 14.0967i | 0 | 53.4761 | − | 31.9849i | 37.6230 | 2.96235 | + | 63.9314i | 161.671 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
99.5 | −3.17330 | − | 2.43519i | 3.20523 | 4.13968 | + | 15.4552i | 0 | −10.1712 | − | 7.80536i | 30.6227 | 24.4999 | − | 59.1249i | −70.7265 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
99.6 | −3.17330 | + | 2.43519i | 3.20523 | 4.13968 | − | 15.4552i | 0 | −10.1712 | + | 7.80536i | 30.6227 | 24.4999 | + | 59.1249i | −70.7265 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
99.7 | −0.888538 | − | 3.90006i | −12.9912 | −14.4210 | + | 6.93071i | 0 | 11.5432 | + | 50.6665i | −78.0345 | 39.8438 | + | 50.0846i | 87.7712 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
99.8 | −0.888538 | + | 3.90006i | −12.9912 | −14.4210 | − | 6.93071i | 0 | 11.5432 | − | 50.6665i | −78.0345 | 39.8438 | − | 50.0846i | 87.7712 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
99.9 | 0.888538 | − | 3.90006i | 12.9912 | −14.4210 | − | 6.93071i | 0 | 11.5432 | − | 50.6665i | 78.0345 | −39.8438 | + | 50.0846i | 87.7712 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
99.10 | 0.888538 | + | 3.90006i | 12.9912 | −14.4210 | + | 6.93071i | 0 | 11.5432 | + | 50.6665i | 78.0345 | −39.8438 | − | 50.0846i | 87.7712 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
99.11 | 3.17330 | − | 2.43519i | −3.20523 | 4.13968 | − | 15.4552i | 0 | −10.1712 | + | 7.80536i | −30.6227 | −24.4999 | − | 59.1249i | −70.7265 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
99.12 | 3.17330 | + | 2.43519i | −3.20523 | 4.13968 | + | 15.4552i | 0 | −10.1712 | − | 7.80536i | −30.6227 | −24.4999 | + | 59.1249i | −70.7265 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
99.13 | 3.43282 | − | 2.05323i | 15.5779 | 7.56853 | − | 14.0967i | 0 | 53.4761 | − | 31.9849i | −37.6230 | −2.96235 | − | 63.9314i | 161.671 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
99.14 | 3.43282 | + | 2.05323i | 15.5779 | 7.56853 | + | 14.0967i | 0 | 53.4761 | + | 31.9849i | −37.6230 | −2.96235 | + | 63.9314i | 161.671 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
99.15 | 3.78898 | − | 1.28203i | −8.14153 | 12.7128 | − | 9.71518i | 0 | −30.8481 | + | 10.4377i | 63.6032 | 35.7134 | − | 53.1089i | −14.7155 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
99.16 | 3.78898 | + | 1.28203i | −8.14153 | 12.7128 | + | 9.71518i | 0 | −30.8481 | − | 10.4377i | 63.6032 | 35.7134 | + | 53.1089i | −14.7155 | 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 | 100.5.d.c | 16 | |
4.b | odd | 2 | 1 | inner | 100.5.d.c | 16 | |
5.b | even | 2 | 1 | inner | 100.5.d.c | 16 | |
5.c | odd | 4 | 1 | 20.5.b.a | ✓ | 8 | |
5.c | odd | 4 | 1 | 100.5.b.c | 8 | ||
15.e | even | 4 | 1 | 180.5.c.a | 8 | ||
20.d | odd | 2 | 1 | inner | 100.5.d.c | 16 | |
20.e | even | 4 | 1 | 20.5.b.a | ✓ | 8 | |
20.e | even | 4 | 1 | 100.5.b.c | 8 | ||
40.i | odd | 4 | 1 | 320.5.b.d | 8 | ||
40.k | even | 4 | 1 | 320.5.b.d | 8 | ||
60.l | odd | 4 | 1 | 180.5.c.a | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
20.5.b.a | ✓ | 8 | 5.c | odd | 4 | 1 | |
20.5.b.a | ✓ | 8 | 20.e | even | 4 | 1 | |
100.5.b.c | 8 | 5.c | odd | 4 | 1 | ||
100.5.b.c | 8 | 20.e | even | 4 | 1 | ||
100.5.d.c | 16 | 1.a | even | 1 | 1 | trivial | |
100.5.d.c | 16 | 4.b | odd | 2 | 1 | inner | |
100.5.d.c | 16 | 5.b | even | 2 | 1 | inner | |
100.5.d.c | 16 | 20.d | odd | 2 | 1 | inner | |
180.5.c.a | 8 | 15.e | even | 4 | 1 | ||
180.5.c.a | 8 | 60.l | odd | 4 | 1 | ||
320.5.b.d | 8 | 40.i | odd | 4 | 1 | ||
320.5.b.d | 8 | 40.k | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 488T_{3}^{6} + 73136T_{3}^{4} - 3415680T_{3}^{2} + 27889920 \)
acting on \(S_{5}^{\mathrm{new}}(100, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} - 20 T^{14} + \cdots + 4294967296 \)
$3$
\( (T^{8} - 488 T^{6} + 73136 T^{4} + \cdots + 27889920)^{2} \)
$5$
\( T^{16} \)
$7$
\( (T^{8} - 12488 T^{6} + \cdots + 32698357408000)^{2} \)
$11$
\( (T^{8} + 39200 T^{6} + \cdots + 204994355200000)^{2} \)
$13$
\( (T^{8} + 104208 T^{6} + \cdots + 64249843360000)^{2} \)
$17$
\( (T^{8} + 353168 T^{6} + \cdots + 47\!\cdots\!00)^{2} \)
$19$
\( (T^{8} + 311680 T^{6} + \cdots + 11\!\cdots\!00)^{2} \)
$23$
\( (T^{8} - 1352968 T^{6} + \cdots + 15\!\cdots\!20)^{2} \)
$29$
\( (T^{4} + 600 T^{3} + \cdots - 98595968624)^{4} \)
$31$
\( (T^{8} + 5480480 T^{6} + \cdots + 18\!\cdots\!00)^{2} \)
$37$
\( (T^{8} + 12321168 T^{6} + \cdots + 39\!\cdots\!00)^{2} \)
$41$
\( (T^{4} - 2448 T^{3} + \cdots + 1586334915856)^{4} \)
$43$
\( (T^{8} - 8441128 T^{6} + \cdots + 13\!\cdots\!00)^{2} \)
$47$
\( (T^{8} - 19967368 T^{6} + \cdots + 43\!\cdots\!20)^{2} \)
$53$
\( (T^{8} + 31106448 T^{6} + \cdots + 23\!\cdots\!00)^{2} \)
$59$
\( (T^{8} + 39075840 T^{6} + \cdots + 35\!\cdots\!00)^{2} \)
$61$
\( (T^{4} - 3968 T^{3} + \cdots + 17262940540816)^{4} \)
$67$
\( (T^{8} - 87547688 T^{6} + \cdots + 10\!\cdots\!20)^{2} \)
$71$
\( (T^{8} + 109932320 T^{6} + \cdots + 16\!\cdots\!00)^{2} \)
$73$
\( (T^{8} + 47333648 T^{6} + \cdots + 15\!\cdots\!00)^{2} \)
$79$
\( (T^{8} + 57226880 T^{6} + \cdots + 26\!\cdots\!00)^{2} \)
$83$
\( (T^{8} - 63696808 T^{6} + \cdots + 13\!\cdots\!20)^{2} \)
$89$
\( (T^{4} + 11880 T^{3} + \cdots + 26555598339856)^{4} \)
$97$
\( (T^{8} + 432840528 T^{6} + \cdots + 15\!\cdots\!00)^{2} \)
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