Newspace parameters
Level: | \( N \) | \(=\) | \( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 3120.r (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(24.9133254306\) |
Analytic rank: | \(0\) |
Dimension: | \(10\) |
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: | \( x^{10} - 4x^{9} + 8x^{8} + 6x^{7} + 25x^{6} - 68x^{5} + 90x^{4} - 10x^{3} + 2 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
Coefficient ring index: | \( 2^{3} \) |
Twist minimal: | no (minimal twist has level 1560) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) 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 \( x^{10} - 4x^{9} + 8x^{8} + 6x^{7} + 25x^{6} - 68x^{5} + 90x^{4} - 10x^{3} + 2 \) :
\(\beta_{1}\) | \(=\) | \( ( 20289 \nu^{9} - 27913 \nu^{8} + 301505 \nu^{7} - 843151 \nu^{6} + 3442683 \nu^{5} + 2255638 \nu^{4} + 7091017 \nu^{3} - 19441646 \nu^{2} + 17822350 \nu - 2847572 ) / 4427135 \) |
\(\beta_{2}\) | \(=\) | \( ( - 44413 \nu^{9} + 600458 \nu^{8} - 1891760 \nu^{7} + 2417367 \nu^{6} + 2815661 \nu^{5} + 14459192 \nu^{4} - 29717829 \nu^{3} + 22592138 \nu^{2} + \cdots + 2033447 ) / 4427135 \) |
\(\beta_{3}\) | \(=\) | \( ( - 23486 \nu^{9} + 28951 \nu^{8} + 69457 \nu^{7} - 654191 \nu^{6} - 994108 \nu^{5} - 46474 \nu^{4} + 2199641 \nu^{3} - 5823214 \nu^{2} + 331634 \nu + 17151 ) / 885427 \) |
\(\beta_{4}\) | \(=\) | \( ( 251948 \nu^{9} - 1742428 \nu^{8} + 4805420 \nu^{7} - 3653532 \nu^{6} + 431409 \nu^{5} - 36258467 \nu^{4} + 71006404 \nu^{3} - 53366378 \nu^{2} + \cdots - 2182552 ) / 4427135 \) |
\(\beta_{5}\) | \(=\) | \( ( - 64993 \nu^{9} + 257345 \nu^{8} - 513275 \nu^{7} - 406958 \nu^{6} - 1643522 \nu^{5} + 4313381 \nu^{4} - 6058074 \nu^{3} + 331634 \nu^{2} + 902578 \nu - 838455 ) / 885427 \) |
\(\beta_{6}\) | \(=\) | \( ( - 637662 \nu^{9} + 2237951 \nu^{8} - 4150070 \nu^{7} - 5409667 \nu^{6} - 19316427 \nu^{5} + 32262199 \nu^{4} - 46694286 \nu^{3} - 10027296 \nu^{2} + \cdots - 7344086 ) / 4427135 \) |
\(\beta_{7}\) | \(=\) | \( ( 1059601 \nu^{9} - 4076561 \nu^{8} + 7731595 \nu^{7} + 7902081 \nu^{6} + 27343613 \nu^{5} - 69897989 \nu^{4} + 81137268 \nu^{3} + 8123614 \nu^{2} + \cdots - 8585234 ) / 4427135 \) |
\(\beta_{8}\) | \(=\) | \( ( - 1262419 \nu^{9} + 4347082 \nu^{8} - 7328505 \nu^{7} - 13031214 \nu^{6} - 36014989 \nu^{5} + 67988193 \nu^{4} - 66731682 \nu^{3} - 45813532 \nu^{2} + \cdots + 2931208 ) / 4427135 \) |
\(\beta_{9}\) | \(=\) | \( ( - 1332935 \nu^{9} + 5449273 \nu^{8} - 11054205 \nu^{7} - 7144335 \nu^{6} - 32542477 \nu^{5} + 94405962 \nu^{4} - 121874305 \nu^{3} + \cdots - 1842018 ) / 4427135 \) |
\(\nu\) | \(=\) | \( ( \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( -\beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} + 3\beta_{4} - 8\beta_{3} + 2\beta_{2} + \beta_1 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( -2\beta_{9} + 3\beta_{8} - 3\beta_{7} + 3\beta_{6} - 10\beta_{5} + 7\beta_{4} - 14\beta_{3} + 10\beta_{2} - 14 ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( ( -10\beta_{9} + 15\beta_{8} - 5\beta_{7} - 5\beta_{6} - 36\beta_{5} - 15\beta_{4} - 10\beta _1 - 80 ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( 35 \beta_{8} + 35 \beta_{7} - 41 \beta_{6} - 108 \beta_{5} - 149 \beta_{4} + 176 \beta_{3} - 108 \beta_{2} - 36 \beta _1 - 176 ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( ( 108 \beta_{9} - 37 \beta_{8} + 185 \beta_{7} - 185 \beta_{6} - 511 \beta_{4} + 910 \beta_{3} - 474 \beta_{2} - 108 \beta_1 ) / 2 \) |
\(\nu^{7}\) | \(=\) | \( ( 474 \beta_{9} - 511 \beta_{8} + 511 \beta_{7} - 401 \beta_{6} + 1238 \beta_{5} - 837 \beta_{4} + 2148 \beta_{3} - 1238 \beta_{2} + 2148 ) / 2 \) |
\(\nu^{8}\) | \(=\) | \( ( 1238 \beta_{9} - 2223 \beta_{8} + 363 \beta_{7} + 363 \beta_{6} + 5830 \beta_{5} + 2223 \beta_{4} + 1238 \beta _1 + 10706 ) / 2 \) |
\(\nu^{9}\) | \(=\) | \( ( - 4699 \beta_{8} - 4699 \beta_{7} + 6193 \beta_{6} + 14568 \beta_{5} + 20761 \beta_{4} - 25862 \beta_{3} + 14568 \beta_{2} + 5830 \beta _1 + 25862 ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3120\mathbb{Z}\right)^\times\).
\(n\) | \(1951\) | \(2081\) | \(2341\) | \(2497\) | \(2641\) |
\(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
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2209.1 |
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0 | − | 1.00000i | 0 | −1.34354 | + | 1.78743i | 0 | −3.74118 | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
2209.2 | 0 | − | 1.00000i | 0 | −0.852717 | − | 2.06709i | 0 | 2.76012 | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
2209.3 | 0 | − | 1.00000i | 0 | 1.26930 | + | 1.84089i | 0 | −0.793851 | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
2209.4 | 0 | − | 1.00000i | 0 | 1.69315 | − | 1.46056i | 0 | 0.420177 | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
2209.5 | 0 | − | 1.00000i | 0 | 2.23380 | − | 0.100662i | 0 | −4.64527 | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
2209.6 | 0 | 1.00000i | 0 | −1.34354 | − | 1.78743i | 0 | −3.74118 | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
2209.7 | 0 | 1.00000i | 0 | −0.852717 | + | 2.06709i | 0 | 2.76012 | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
2209.8 | 0 | 1.00000i | 0 | 1.26930 | − | 1.84089i | 0 | −0.793851 | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
2209.9 | 0 | 1.00000i | 0 | 1.69315 | + | 1.46056i | 0 | 0.420177 | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
2209.10 | 0 | 1.00000i | 0 | 2.23380 | + | 0.100662i | 0 | −4.64527 | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
65.d | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 3120.2.r.l | 10 | |
4.b | odd | 2 | 1 | 1560.2.r.h | yes | 10 | |
5.b | even | 2 | 1 | 3120.2.r.k | 10 | ||
13.b | even | 2 | 1 | 3120.2.r.k | 10 | ||
20.d | odd | 2 | 1 | 1560.2.r.g | ✓ | 10 | |
52.b | odd | 2 | 1 | 1560.2.r.g | ✓ | 10 | |
65.d | even | 2 | 1 | inner | 3120.2.r.l | 10 | |
260.g | odd | 2 | 1 | 1560.2.r.h | yes | 10 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1560.2.r.g | ✓ | 10 | 20.d | odd | 2 | 1 | |
1560.2.r.g | ✓ | 10 | 52.b | odd | 2 | 1 | |
1560.2.r.h | yes | 10 | 4.b | odd | 2 | 1 | |
1560.2.r.h | yes | 10 | 260.g | odd | 2 | 1 | |
3120.2.r.k | 10 | 5.b | even | 2 | 1 | ||
3120.2.r.k | 10 | 13.b | even | 2 | 1 | ||
3120.2.r.l | 10 | 1.a | even | 1 | 1 | trivial | |
3120.2.r.l | 10 | 65.d | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{5} + 6T_{7}^{4} - 4T_{7}^{3} - 52T_{7}^{2} - 16T_{7} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(3120, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{10} \)
$3$
\( (T^{2} + 1)^{5} \)
$5$
\( T^{10} - 6 T^{9} + 19 T^{8} + \cdots + 3125 \)
$7$
\( (T^{5} + 6 T^{4} - 4 T^{3} - 52 T^{2} + \cdots + 16)^{2} \)
$11$
\( T^{10} + 72 T^{8} + 1572 T^{6} + \cdots + 30976 \)
$13$
\( T^{10} + 8 T^{9} + 49 T^{8} + \cdots + 371293 \)
$17$
\( T^{10} + 88 T^{8} + 2800 T^{6} + \cdots + 565504 \)
$19$
\( T^{10} + 104 T^{8} + 2768 T^{6} + \cdots + 256 \)
$23$
\( T^{10} + 124 T^{8} + 4640 T^{6} + \cdots + 4096 \)
$29$
\( (T^{5} - 6 T^{4} - 48 T^{3} + 460 T^{2} + \cdots + 944)^{2} \)
$31$
\( T^{10} + 164 T^{8} + \cdots + 16516096 \)
$37$
\( (T^{5} + 12 T^{4} - 136 T^{3} + \cdots + 39296)^{2} \)
$41$
\( T^{10} + 192 T^{8} + 11396 T^{6} + \cdots + 3041536 \)
$43$
\( T^{10} + 216 T^{8} + 11024 T^{6} + \cdots + 4096 \)
$47$
\( (T^{5} + 4 T^{4} - 22 T^{3} - 16 T^{2} + \cdots - 64)^{2} \)
$53$
\( T^{10} + 148 T^{8} + 7456 T^{6} + \cdots + 123904 \)
$59$
\( T^{10} + 504 T^{8} + \cdots + 479785216 \)
$61$
\( (T^{5} + 18 T^{4} - 68 T^{3} - 2088 T^{2} + \cdots + 44800)^{2} \)
$67$
\( (T^{5} - 18 T^{4} - 104 T^{3} + \cdots - 14176)^{2} \)
$71$
\( T^{10} + 300 T^{8} + \cdots + 142468096 \)
$73$
\( (T^{5} - 20 T^{4} - 36 T^{3} + 2212 T^{2} + \cdots - 27712)^{2} \)
$79$
\( (T^{5} - 4 T^{4} - 156 T^{3} + 816 T^{2} + \cdots - 12352)^{2} \)
$83$
\( (T^{5} - 20 T^{4} - 22 T^{3} + 2160 T^{2} + \cdots - 4000)^{2} \)
$89$
\( T^{10} + 656 T^{8} + \cdots + 6209440000 \)
$97$
\( (T^{5} - 32 T^{4} + 148 T^{3} + \cdots - 94912)^{2} \)
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