Newspace parameters
Level: | \( N \) | \(=\) | \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 720.o (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(42.4813752041\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | 8.0.928445276160000.218 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - 4x^{7} + 130x^{6} - 376x^{5} + 5673x^{4} - 10724x^{3} + 96952x^{2} - 91652x + 530329 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
Coefficient ring index: | \( 2^{8}\cdot 3^{2} \) |
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 \(1,\beta_1,\ldots,\beta_{7}\) 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 \( x^{8} - 4x^{7} + 130x^{6} - 376x^{5} + 5673x^{4} - 10724x^{3} + 96952x^{2} - 91652x + 530329 \) :
\(\beta_{1}\) | \(=\) | \( ( 2\nu^{6} - 6\nu^{5} + 307\nu^{4} - 604\nu^{3} + 12719\nu^{2} - 12418\nu + 130554 ) / 12317 \) |
\(\beta_{2}\) | \(=\) | \( ( -\nu^{6} + 3\nu^{5} - 97\nu^{4} + 189\nu^{3} - 2461\nu^{2} + 2367\nu - 13636 ) / 113 \) |
\(\beta_{3}\) | \(=\) | \( ( 3\nu^{6} - 9\nu^{5} + 297\nu^{4} - 579\nu^{3} + 8451\nu^{2} - 8163\nu + 66666 ) / 109 \) |
\(\beta_{4}\) | \(=\) | \( ( - 174 \nu^{7} + 609 \nu^{6} - 16615 \nu^{5} + 40015 \nu^{4} - 402745 \nu^{3} + 564407 \nu^{2} - 1771845 \nu + 793174 ) / 1242709 \) |
\(\beta_{5}\) | \(=\) | \( ( - 222 \nu^{7} + 777 \nu^{6} - 22771 \nu^{5} + 54985 \nu^{4} - 678965 \nu^{3} + 963851 \nu^{2} - 5724175 \nu + 2703260 ) / 1288313 \) |
\(\beta_{6}\) | \(=\) | \( ( 27216 \nu^{7} - 95256 \nu^{6} + 3627264 \nu^{5} - 8830020 \nu^{4} + 170982000 \nu^{3} - 247690608 \nu^{2} + 4227412944 \nu - 2072716770 ) / 140426117 \) |
\(\beta_{7}\) | \(=\) | \( ( -16\nu^{7} + 56\nu^{6} - 2052\nu^{5} + 4990\nu^{4} - 76872\nu^{3} + 110346\nu^{2} - 777796\nu + 370672 ) / 11401 \) |
\(\nu\) | \(=\) | \( ( \beta_{6} + 6\beta_{5} - 6\beta_{4} + 6 ) / 12 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{6} + 6\beta_{5} - 6\beta_{4} + 2\beta_{3} + 6\beta_{2} - 12\beta _1 - 366 ) / 12 \) |
\(\nu^{3}\) | \(=\) | \( ( 9\beta_{7} - 34\beta_{6} - 543\beta_{5} + 531\beta_{4} + 3\beta_{3} + 9\beta_{2} - 18\beta _1 - 552 ) / 12 \) |
\(\nu^{4}\) | \(=\) | \( ( 6\beta_{7} - 23\beta_{6} - 364\beta_{5} + 356\beta_{4} - 44\beta_{3} - 124\beta_{2} + 700\beta _1 + 4530 ) / 4 \) |
\(\nu^{5}\) | \(=\) | \( ( - 900 \beta_{7} + 1350 \beta_{6} + 32169 \beta_{5} - 28701 \beta_{4} - 335 \beta_{3} - 945 \beta_{2} + 5280 \beta _1 + 34896 ) / 12 \) |
\(\nu^{6}\) | \(=\) | \( ( - 2745 \beta_{7} + 4223 \beta_{6} + 99240 \beta_{5} - 88776 \beta_{4} + 7444 \beta_{3} + 18828 \beta_{2} - 161730 \beta _1 - 566574 ) / 12 \) |
\(\nu^{7}\) | \(=\) | \( ( 59640 \beta_{7} - 58239 \beta_{6} - 1760352 \beta_{5} + 1402374 \beta_{4} + 27230 \beta_{3} + 69216 \beta_{2} - 584556 \beta _1 - 2105790 ) / 12 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).
\(n\) | \(181\) | \(271\) | \(577\) | \(641\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
719.1 |
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0 | 0 | 0 | −8.66025 | − | 7.07107i | 0 | −26.2679 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
719.2 | 0 | 0 | 0 | −8.66025 | − | 7.07107i | 0 | 26.2679 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
719.3 | 0 | 0 | 0 | −8.66025 | + | 7.07107i | 0 | −26.2679 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
719.4 | 0 | 0 | 0 | −8.66025 | + | 7.07107i | 0 | 26.2679 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
719.5 | 0 | 0 | 0 | 8.66025 | − | 7.07107i | 0 | −26.2679 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
719.6 | 0 | 0 | 0 | 8.66025 | − | 7.07107i | 0 | 26.2679 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
719.7 | 0 | 0 | 0 | 8.66025 | + | 7.07107i | 0 | −26.2679 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
719.8 | 0 | 0 | 0 | 8.66025 | + | 7.07107i | 0 | 26.2679 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
4.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
12.b | even | 2 | 1 | inner |
15.d | odd | 2 | 1 | inner |
20.d | odd | 2 | 1 | inner |
60.h | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 720.4.o.b | ✓ | 8 |
3.b | odd | 2 | 1 | inner | 720.4.o.b | ✓ | 8 |
4.b | odd | 2 | 1 | inner | 720.4.o.b | ✓ | 8 |
5.b | even | 2 | 1 | inner | 720.4.o.b | ✓ | 8 |
12.b | even | 2 | 1 | inner | 720.4.o.b | ✓ | 8 |
15.d | odd | 2 | 1 | inner | 720.4.o.b | ✓ | 8 |
20.d | odd | 2 | 1 | inner | 720.4.o.b | ✓ | 8 |
60.h | even | 2 | 1 | inner | 720.4.o.b | ✓ | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
720.4.o.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
720.4.o.b | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
720.4.o.b | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
720.4.o.b | ✓ | 8 | 5.b | even | 2 | 1 | inner |
720.4.o.b | ✓ | 8 | 12.b | even | 2 | 1 | inner |
720.4.o.b | ✓ | 8 | 15.d | odd | 2 | 1 | inner |
720.4.o.b | ✓ | 8 | 20.d | odd | 2 | 1 | inner |
720.4.o.b | ✓ | 8 | 60.h | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{2} - 690 \)
acting on \(S_{4}^{\mathrm{new}}(720, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( T^{8} \)
$5$
\( (T^{4} - 50 T^{2} + 15625)^{2} \)
$7$
\( (T^{2} - 690)^{4} \)
$11$
\( (T^{2} - 2070)^{4} \)
$13$
\( (T^{2} + 1014)^{4} \)
$17$
\( (T^{2} - 2352)^{4} \)
$19$
\( (T^{2} + 16560)^{4} \)
$23$
\( (T^{2} + 1380)^{4} \)
$29$
\( (T^{2} + 2888)^{4} \)
$31$
\( (T^{2} + 37260)^{4} \)
$37$
\( (T^{2} + 20886)^{4} \)
$41$
\( (T^{2} + 23762)^{4} \)
$43$
\( (T^{2} - 24840)^{4} \)
$47$
\( (T^{2} + 353280)^{4} \)
$53$
\( (T^{2} - 442368)^{4} \)
$59$
\( (T^{2} - 250470)^{4} \)
$61$
\( (T - 550)^{8} \)
$67$
\( (T^{2} - 706560)^{4} \)
$71$
\( (T^{2} - 298080)^{4} \)
$73$
\( (T^{2} + 351384)^{4} \)
$79$
\( (T^{2} + 202860)^{4} \)
$83$
\( (T^{2} + 667920)^{4} \)
$89$
\( (T^{2} + 103058)^{4} \)
$97$
\( (T^{2} + 3594456)^{4} \)
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