Newspace parameters
Level: | \( N \) | \(=\) | \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 720.o (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(115.476350265\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - 4x^{7} + 370x^{6} - 1096x^{5} + 36273x^{4} - 70724x^{3} + 465952x^{2} - 430772x + 1070929 \) |
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 \( x^{8} - 4x^{7} + 370x^{6} - 1096x^{5} + 36273x^{4} - 70724x^{3} + 465952x^{2} - 430772x + 1070929 \) :
\(\beta_{1}\) | \(=\) | \( ( 2\nu^{6} - 6\nu^{5} + 667\nu^{4} - 1324\nu^{3} + 56279\nu^{2} - 55618\nu + 345414 ) / 14803 \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{6} - 3\nu^{5} + 277\nu^{4} - 549\nu^{3} + 3901\nu^{2} - 3627\nu - 1150184 ) / 6893 \) |
\(\beta_{3}\) | \(=\) | \( ( -3\nu^{6} + 9\nu^{5} - 1197\nu^{4} + 2379\nu^{3} - 120771\nu^{2} + 119583\nu - 755886 ) / 7991 \) |
\(\beta_{4}\) | \(=\) | \( ( 702 \nu^{7} - 2457 \nu^{6} + 250651 \nu^{5} - 620485 \nu^{4} + 22976045 \nu^{3} - 33844811 \nu^{2} + 171147235 \nu - 79953440 ) / 108364853 \) |
\(\beta_{5}\) | \(=\) | \( ( - 2226 \nu^{7} + 7791 \nu^{6} - 827585 \nu^{5} + 2049485 \nu^{4} - 82855295 \nu^{3} + 122237353 \nu^{2} - 1418157135 \nu + 688773806 ) / 125626511 \) |
\(\beta_{6}\) | \(=\) | \( ( - 15696 \nu^{7} + 54936 \nu^{6} - 5968704 \nu^{5} + 14784420 \nu^{4} - 624864240 \nu^{3} + 922539408 \nu^{2} - 10764578064 \nu + 5229023970 ) / 232717963 \) |
\(\beta_{7}\) | \(=\) | \( ( - 16 \nu^{7} + 56 \nu^{6} - 5892 \nu^{5} + 14590 \nu^{4} - 557352 \nu^{3} + 821466 \nu^{2} - 4167556 \nu + 1947352 ) / 15721 \) |
\(\nu\) | \(=\) | \( ( \beta_{6} - 6\beta_{5} - 6\beta_{4} + 6 ) / 12 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{6} - 6\beta_{5} - 6\beta_{4} + 2\beta_{3} - 6\beta_{2} + 12\beta _1 - 1086 ) / 12 \) |
\(\nu^{3}\) | \(=\) | \( ( 9\beta_{7} - 214\beta_{6} + 891\beta_{5} + 1623\beta_{4} + 3\beta_{3} - 9\beta_{2} + 18\beta _1 - 1632 ) / 12 \) |
\(\nu^{4}\) | \(=\) | \( ( 6\beta_{7} - 143\beta_{6} + 596\beta_{5} + 1084\beta_{4} - 284\beta_{3} + 364\beta_{2} - 1180\beta _1 + 61410 ) / 4 \) |
\(\nu^{5}\) | \(=\) | \( ( - 2700 \beta_{7} + 37950 \beta_{6} - 155781 \beta_{5} - 455169 \beta_{4} - 2135 \beta_{3} + 2745 \beta_{2} - 8880 \beta _1 + 463296 ) / 12 \) |
\(\nu^{6}\) | \(=\) | \( ( - 8145 \beta_{7} + 114923 \beta_{6} - 471816 \beta_{5} - 1373640 \beta_{4} + 223444 \beta_{3} - 193068 \beta_{2} + 917010 \beta _1 - 32477334 ) / 12 \) |
\(\nu^{7}\) | \(=\) | \( ( 656880 \beta_{7} - 6718599 \beta_{6} + 27562674 \beta_{5} + 110492052 \beta_{4} + 789530 \beta_{3} - 685356 \beta_{2} + 3240636 \beta _1 - 115294110 ) / 12 \) |
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 | −55.4527 | − | 7.07107i | 0 | −168.196 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
719.2 | 0 | 0 | 0 | −55.4527 | − | 7.07107i | 0 | 168.196 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
719.3 | 0 | 0 | 0 | −55.4527 | + | 7.07107i | 0 | −168.196 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
719.4 | 0 | 0 | 0 | −55.4527 | + | 7.07107i | 0 | 168.196 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
719.5 | 0 | 0 | 0 | 55.4527 | − | 7.07107i | 0 | −168.196 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
719.6 | 0 | 0 | 0 | 55.4527 | − | 7.07107i | 0 | 168.196 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
719.7 | 0 | 0 | 0 | 55.4527 | + | 7.07107i | 0 | −168.196 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
719.8 | 0 | 0 | 0 | 55.4527 | + | 7.07107i | 0 | 168.196 | 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 | 720.6.o.c | ✓ | 8 |
3.b | odd | 2 | 1 | inner | 720.6.o.c | ✓ | 8 |
4.b | odd | 2 | 1 | inner | 720.6.o.c | ✓ | 8 |
5.b | even | 2 | 1 | inner | 720.6.o.c | ✓ | 8 |
12.b | even | 2 | 1 | inner | 720.6.o.c | ✓ | 8 |
15.d | odd | 2 | 1 | inner | 720.6.o.c | ✓ | 8 |
20.d | odd | 2 | 1 | inner | 720.6.o.c | ✓ | 8 |
60.h | even | 2 | 1 | inner | 720.6.o.c | ✓ | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
720.6.o.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
720.6.o.c | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
720.6.o.c | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
720.6.o.c | ✓ | 8 | 5.b | even | 2 | 1 | inner |
720.6.o.c | ✓ | 8 | 12.b | even | 2 | 1 | inner |
720.6.o.c | ✓ | 8 | 15.d | odd | 2 | 1 | inner |
720.6.o.c | ✓ | 8 | 20.d | odd | 2 | 1 | inner |
720.6.o.c | ✓ | 8 | 60.h | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{2} - 28290 \)
acting on \(S_{6}^{\mathrm{new}}(720, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( T^{8} \)
$5$
\( (T^{4} - 6050 T^{2} + 9765625)^{2} \)
$7$
\( (T^{2} - 28290)^{4} \)
$11$
\( (T^{2} - 101430)^{4} \)
$13$
\( (T^{2} + 29766)^{4} \)
$17$
\( (T^{2} - 238128)^{4} \)
$19$
\( (T^{2} + 2798640)^{4} \)
$23$
\( (T^{2} + 6846180)^{4} \)
$29$
\( (T^{2} + 18702728)^{4} \)
$31$
\( (T^{2} + 40576140)^{4} \)
$37$
\( (T^{2} + 38771814)^{4} \)
$41$
\( (T^{2} + 171162002)^{4} \)
$43$
\( (T^{2} - 82493640)^{4} \)
$47$
\( (T^{2} + 109538880)^{4} \)
$53$
\( (T^{2} - 55544832)^{4} \)
$59$
\( (T^{2} - 72385830)^{4} \)
$61$
\( (T - 4510)^{8} \)
$67$
\( (T^{2} - 219077760)^{4} \)
$71$
\( (T^{2} - 1767316320)^{4} \)
$73$
\( (T^{2} + 1120273176)^{4} \)
$79$
\( (T^{2} + 11670829740)^{4} \)
$83$
\( (T^{2} + 842136720)^{4} \)
$89$
\( (T^{2} + 315657938)^{4} \)
$97$
\( (T^{2} + 6353374104)^{4} \)
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