Newspace parameters
Level: | \( N \) | \(=\) | \( 133 = 7 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 133.e (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.06201034688\) |
Analytic rank: | \(0\) |
Dimension: | \(10\) |
Relative dimension: | \(5\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{10} - x^{9} + 7x^{8} - 2x^{7} + 32x^{6} - 9x^{5} + 63x^{4} + 20x^{3} + 68x^{2} - 8x + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) 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 \( x^{10} - x^{9} + 7x^{8} - 2x^{7} + 32x^{6} - 9x^{5} + 63x^{4} + 20x^{3} + 68x^{2} - 8x + 1 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( - 1315 \nu^{9} + 10212 \nu^{8} - 29785 \nu^{7} + 62655 \nu^{6} - 116587 \nu^{5} + 246790 \nu^{4} - 580195 \nu^{3} + 353165 \nu^{2} - 41699 \nu + 94555 ) / 759446 \) |
\(\beta_{3}\) | \(=\) | \( ( - 2630 \nu^{9} + 20424 \nu^{8} - 59570 \nu^{7} + 125310 \nu^{6} - 233174 \nu^{5} + 493580 \nu^{4} - 780667 \nu^{3} + 706330 \nu^{2} - 83398 \nu + 568833 ) / 379723 \) |
\(\beta_{4}\) | \(=\) | \( ( 8897 \nu^{9} - 20580 \nu^{8} + 60025 \nu^{7} - 74507 \nu^{6} + 234955 \nu^{5} - 497350 \nu^{4} + 379465 \nu^{3} - 711725 \nu^{2} + 84035 \nu - 2277023 ) / 759446 \) |
\(\beta_{5}\) | \(=\) | \( ( - 15274 \nu^{9} + 38916 \nu^{8} - 113505 \nu^{7} + 207978 \nu^{6} - 444291 \nu^{5} + 940470 \nu^{4} - 805012 \nu^{3} + 1345845 \nu^{2} - 158907 \nu + 1130040 ) / 379723 \) |
\(\beta_{6}\) | \(=\) | \( ( 94555 \nu^{9} - 93240 \nu^{8} + 651673 \nu^{7} - 159325 \nu^{6} + 2963105 \nu^{5} - 734408 \nu^{4} + 5710175 \nu^{3} + 2471295 \nu^{2} + 6076575 \nu + 44705 ) / 759446 \) |
\(\beta_{7}\) | \(=\) | \( ( - 53119 \nu^{9} + 45493 \nu^{8} - 354193 \nu^{7} + 54788 \nu^{6} - 1690191 \nu^{5} + 276681 \nu^{4} - 3299920 \nu^{3} - 1496128 \nu^{2} - 4520934 \nu - 26801 ) / 379723 \) |
\(\beta_{8}\) | \(=\) | \( ( - 134864 \nu^{9} + 127326 \nu^{8} - 940952 \nu^{7} + 260698 \nu^{6} - 4301561 \nu^{5} + 1178430 \nu^{4} - 8421612 \nu^{3} - 2241795 \nu^{2} + \cdots + 1066270 ) / 379723 \) |
\(\beta_{9}\) | \(=\) | \( ( - 283665 \nu^{9} + 279720 \nu^{8} - 1955019 \nu^{7} + 477975 \nu^{6} - 8889315 \nu^{5} + 2203224 \nu^{4} - 17130525 \nu^{3} - 6654439 \nu^{2} + \cdots + 2144223 ) / 759446 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{9} + 3\beta_{6} - 3 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{3} - 4\beta_{2} - 1 \) |
\(\nu^{4}\) | \(=\) | \( -5\beta_{9} + \beta_{8} - 12\beta_{6} - \beta_{5} - 5\beta_{4} \) |
\(\nu^{5}\) | \(=\) | \( -\beta_{9} + \beta_{8} - 6\beta_{7} - 7\beta_{6} - 6\beta_{3} + 16\beta_{2} - 16\beta _1 + 7 \) |
\(\nu^{6}\) | \(=\) | \( 7\beta_{5} + 23\beta_{4} - 2\beta_{3} + \beta_{2} + 51 \) |
\(\nu^{7}\) | \(=\) | \( 10\beta_{9} - 9\beta_{8} + 30\beta_{7} + 40\beta_{6} + 9\beta_{5} + 10\beta_{4} + 65\beta_1 \) |
\(\nu^{8}\) | \(=\) | \( 104\beta_{9} - 39\beta_{8} + 19\beta_{7} + 223\beta_{6} + 19\beta_{3} - 11\beta_{2} + 11\beta _1 - 223 \) |
\(\nu^{9}\) | \(=\) | \( -58\beta_{5} - 69\beta_{4} + 143\beta_{3} - 269\beta_{2} - 215 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/133\mathbb{Z}\right)^\times\).
\(n\) | \(78\) | \(115\) |
\(\chi(n)\) | \(-1 + \beta_{6}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
64.1 |
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−0.983281 | − | 1.70309i | 0.130414 | + | 0.225884i | −0.933684 | + | 1.61719i | −1.22681 | − | 2.12490i | 0.256468 | − | 0.444216i | −1.00000 | −0.260829 | 1.46598 | − | 2.53916i | −2.41260 | + | 4.17875i | ||||||||||||||||||||||||||||||||||
64.2 | −0.552442 | − | 0.956858i | 1.53536 | + | 2.65933i | 0.389615 | − | 0.674832i | 0.643958 | + | 1.11537i | 1.69640 | − | 2.93825i | −1.00000 | −3.07073 | −3.21469 | + | 5.56800i | 0.711500 | − | 1.23235i | |||||||||||||||||||||||||||||||||||
64.3 | 0.0595946 | + | 0.103221i | −0.237532 | − | 0.411417i | 0.992897 | − | 1.71975i | −0.412094 | − | 0.713768i | 0.0283112 | − | 0.0490364i | −1.00000 | 0.475063 | 1.38716 | − | 2.40263i | 0.0491171 | − | 0.0850734i | |||||||||||||||||||||||||||||||||||
64.4 | 0.883493 | + | 1.53025i | −0.775497 | − | 1.34320i | −0.561119 | + | 0.971886i | 1.75378 | + | 3.03764i | 1.37029 | − | 2.37341i | −1.00000 | 1.55099 | 0.297209 | − | 0.514782i | −3.09891 | + | 5.36747i | |||||||||||||||||||||||||||||||||||
64.5 | 1.09264 | + | 1.89250i | 0.847250 | + | 1.46748i | −1.38771 | + | 2.40358i | −1.25884 | − | 2.18037i | −1.85147 | + | 3.20684i | −1.00000 | −1.69450 | 0.0643360 | − | 0.111433i | 2.75090 | − | 4.76470i | |||||||||||||||||||||||||||||||||||
106.1 | −0.983281 | + | 1.70309i | 0.130414 | − | 0.225884i | −0.933684 | − | 1.61719i | −1.22681 | + | 2.12490i | 0.256468 | + | 0.444216i | −1.00000 | −0.260829 | 1.46598 | + | 2.53916i | −2.41260 | − | 4.17875i | |||||||||||||||||||||||||||||||||||
106.2 | −0.552442 | + | 0.956858i | 1.53536 | − | 2.65933i | 0.389615 | + | 0.674832i | 0.643958 | − | 1.11537i | 1.69640 | + | 2.93825i | −1.00000 | −3.07073 | −3.21469 | − | 5.56800i | 0.711500 | + | 1.23235i | |||||||||||||||||||||||||||||||||||
106.3 | 0.0595946 | − | 0.103221i | −0.237532 | + | 0.411417i | 0.992897 | + | 1.71975i | −0.412094 | + | 0.713768i | 0.0283112 | + | 0.0490364i | −1.00000 | 0.475063 | 1.38716 | + | 2.40263i | 0.0491171 | + | 0.0850734i | |||||||||||||||||||||||||||||||||||
106.4 | 0.883493 | − | 1.53025i | −0.775497 | + | 1.34320i | −0.561119 | − | 0.971886i | 1.75378 | − | 3.03764i | 1.37029 | + | 2.37341i | −1.00000 | 1.55099 | 0.297209 | + | 0.514782i | −3.09891 | − | 5.36747i | |||||||||||||||||||||||||||||||||||
106.5 | 1.09264 | − | 1.89250i | 0.847250 | − | 1.46748i | −1.38771 | − | 2.40358i | −1.25884 | + | 2.18037i | −1.85147 | − | 3.20684i | −1.00000 | −1.69450 | 0.0643360 | + | 0.111433i | 2.75090 | + | 4.76470i | |||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 133.2.e.c | ✓ | 10 |
3.b | odd | 2 | 1 | 1197.2.k.g | 10 | ||
7.b | odd | 2 | 1 | 931.2.e.c | 10 | ||
7.c | even | 3 | 1 | 931.2.g.e | 10 | ||
7.c | even | 3 | 1 | 931.2.h.f | 10 | ||
7.d | odd | 6 | 1 | 931.2.g.f | 10 | ||
7.d | odd | 6 | 1 | 931.2.h.e | 10 | ||
19.c | even | 3 | 1 | inner | 133.2.e.c | ✓ | 10 |
19.c | even | 3 | 1 | 2527.2.a.j | 5 | ||
19.d | odd | 6 | 1 | 2527.2.a.k | 5 | ||
57.h | odd | 6 | 1 | 1197.2.k.g | 10 | ||
133.g | even | 3 | 1 | 931.2.h.f | 10 | ||
133.h | even | 3 | 1 | 931.2.g.e | 10 | ||
133.k | odd | 6 | 1 | 931.2.h.e | 10 | ||
133.m | odd | 6 | 1 | 931.2.e.c | 10 | ||
133.t | odd | 6 | 1 | 931.2.g.f | 10 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
133.2.e.c | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
133.2.e.c | ✓ | 10 | 19.c | even | 3 | 1 | inner |
931.2.e.c | 10 | 7.b | odd | 2 | 1 | ||
931.2.e.c | 10 | 133.m | odd | 6 | 1 | ||
931.2.g.e | 10 | 7.c | even | 3 | 1 | ||
931.2.g.e | 10 | 133.h | even | 3 | 1 | ||
931.2.g.f | 10 | 7.d | odd | 6 | 1 | ||
931.2.g.f | 10 | 133.t | odd | 6 | 1 | ||
931.2.h.e | 10 | 7.d | odd | 6 | 1 | ||
931.2.h.e | 10 | 133.k | odd | 6 | 1 | ||
931.2.h.f | 10 | 7.c | even | 3 | 1 | ||
931.2.h.f | 10 | 133.g | even | 3 | 1 | ||
1197.2.k.g | 10 | 3.b | odd | 2 | 1 | ||
1197.2.k.g | 10 | 57.h | odd | 6 | 1 | ||
2527.2.a.j | 5 | 19.c | even | 3 | 1 | ||
2527.2.a.k | 5 | 19.d | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} - T_{2}^{9} + 7T_{2}^{8} - 2T_{2}^{7} + 32T_{2}^{6} - 9T_{2}^{5} + 63T_{2}^{4} + 20T_{2}^{3} + 68T_{2}^{2} - 8T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(133, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{10} - T^{9} + 7 T^{8} - 2 T^{7} + 32 T^{6} + \cdots + 1 \)
$3$
\( T^{10} - 3 T^{9} + 12 T^{8} - 7 T^{7} + \cdots + 1 \)
$5$
\( T^{10} + T^{9} + 14 T^{8} + 23 T^{7} + \cdots + 529 \)
$7$
\( (T + 1)^{10} \)
$11$
\( (T^{5} - T^{4} - 12 T^{3} + 19 T^{2} + 8 T - 16)^{2} \)
$13$
\( T^{10} + 20 T^{8} - 22 T^{7} + \cdots + 11881 \)
$17$
\( T^{10} + 7 T^{9} + 84 T^{8} + \cdots + 149769 \)
$19$
\( T^{10} + 7 T^{9} + 30 T^{8} + \cdots + 2476099 \)
$23$
\( T^{10} + 2 T^{9} + 104 T^{8} + \cdots + 36036009 \)
$29$
\( T^{10} + 6 T^{9} + 120 T^{8} + \cdots + 78943225 \)
$31$
\( (T^{5} + 3 T^{4} - 80 T^{3} + 73 T^{2} + \cdots - 1280)^{2} \)
$37$
\( (T^{5} + 5 T^{4} - 90 T^{3} - 175 T^{2} + \cdots - 2924)^{2} \)
$41$
\( T^{10} - 19 T^{9} + 280 T^{8} + \cdots + 2627641 \)
$43$
\( T^{10} + 16 T^{9} + 298 T^{8} + \cdots + 727609 \)
$47$
\( T^{10} - 10 T^{9} + 149 T^{8} + \cdots + 3717184 \)
$53$
\( T^{10} - 7 T^{9} + 214 T^{8} + \cdots + 25230529 \)
$59$
\( T^{10} - 16 T^{9} + 228 T^{8} + \cdots + 3143529 \)
$61$
\( T^{10} + 27 T^{9} + 559 T^{8} + \cdots + 408321 \)
$67$
\( T^{10} + 11 T^{9} + 172 T^{8} + \cdots + 3003289 \)
$71$
\( T^{10} + 42 T^{8} - 94 T^{7} + \cdots + 109561 \)
$73$
\( T^{10} - 5 T^{9} + 291 T^{8} + \cdots + 685444761 \)
$79$
\( T^{10} - 21 T^{9} + \cdots + 5068158481 \)
$83$
\( (T^{5} + 8 T^{4} - 29 T^{3} - 167 T^{2} + \cdots + 144)^{2} \)
$89$
\( T^{10} - 22 T^{9} + \cdots + 380718144 \)
$97$
\( T^{10} - 28 T^{9} + \cdots + 201554809 \)
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