Newspace parameters
Level: | \( N \) | \(=\) | \( 30 = 2 \cdot 3 \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 30.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(4.81151459439\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(i, \sqrt{1249})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} + 625x^{2} + 97344 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2\cdot 5 \) |
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 \(1,\beta_1,\beta_2,\beta_3\) 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 \( x^{4} + 625x^{2} + 97344 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{3} + 313\nu ) / 312 \) |
\(\beta_{2}\) | \(=\) | \( ( -\nu^{3} + 312\nu^{2} - 1249\nu + 97656 ) / 312 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{3} + 936\nu^{2} + 625\nu + 292656 ) / 312 \) |
\(\nu\) | \(=\) | \( ( \beta_{3} - 3\beta_{2} - 4\beta _1 + 1 ) / 10 \) |
\(\nu^{2}\) | \(=\) | \( ( 3\beta_{3} + \beta_{2} - 2\beta _1 - 3127 ) / 10 \) |
\(\nu^{3}\) | \(=\) | \( ( -313\beta_{3} + 939\beta_{2} + 4372\beta _1 - 313 ) / 10 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/30\mathbb{Z}\right)^\times\).
\(n\) | \(7\) | \(11\) |
\(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 |
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− | 4.00000i | 9.00000i | −16.0000 | −16.1706 | + | 53.5118i | 36.0000 | 119.706i | 64.0000i | −81.0000 | 214.047 | + | 64.6824i | |||||||||||||||||||||||||
19.2 | − | 4.00000i | 9.00000i | −16.0000 | 19.1706 | − | 52.5118i | 36.0000 | − | 233.706i | 64.0000i | −81.0000 | −210.047 | − | 76.6824i | |||||||||||||||||||||||||
19.3 | 4.00000i | − | 9.00000i | −16.0000 | −16.1706 | − | 53.5118i | 36.0000 | − | 119.706i | − | 64.0000i | −81.0000 | 214.047 | − | 64.6824i | ||||||||||||||||||||||||
19.4 | 4.00000i | − | 9.00000i | −16.0000 | 19.1706 | + | 52.5118i | 36.0000 | 233.706i | − | 64.0000i | −81.0000 | −210.047 | + | 76.6824i | |||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 30.6.c.b | ✓ | 4 |
3.b | odd | 2 | 1 | 90.6.c.c | 4 | ||
4.b | odd | 2 | 1 | 240.6.f.b | 4 | ||
5.b | even | 2 | 1 | inner | 30.6.c.b | ✓ | 4 |
5.c | odd | 4 | 1 | 150.6.a.n | 2 | ||
5.c | odd | 4 | 1 | 150.6.a.o | 2 | ||
12.b | even | 2 | 1 | 720.6.f.i | 4 | ||
15.d | odd | 2 | 1 | 90.6.c.c | 4 | ||
15.e | even | 4 | 1 | 450.6.a.bb | 2 | ||
15.e | even | 4 | 1 | 450.6.a.bc | 2 | ||
20.d | odd | 2 | 1 | 240.6.f.b | 4 | ||
60.h | even | 2 | 1 | 720.6.f.i | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
30.6.c.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
30.6.c.b | ✓ | 4 | 5.b | even | 2 | 1 | inner |
90.6.c.c | 4 | 3.b | odd | 2 | 1 | ||
90.6.c.c | 4 | 15.d | odd | 2 | 1 | ||
150.6.a.n | 2 | 5.c | odd | 4 | 1 | ||
150.6.a.o | 2 | 5.c | odd | 4 | 1 | ||
240.6.f.b | 4 | 4.b | odd | 2 | 1 | ||
240.6.f.b | 4 | 20.d | odd | 2 | 1 | ||
450.6.a.bb | 2 | 15.e | even | 4 | 1 | ||
450.6.a.bc | 2 | 15.e | even | 4 | 1 | ||
720.6.f.i | 4 | 12.b | even | 2 | 1 | ||
720.6.f.i | 4 | 60.h | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} + 68948T_{7}^{2} + 782656576 \)
acting on \(S_{6}^{\mathrm{new}}(30, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 16)^{2} \)
$3$
\( (T^{2} + 81)^{2} \)
$5$
\( T^{4} - 6 T^{3} + 5010 T^{2} + \cdots + 9765625 \)
$7$
\( T^{4} + 68948 T^{2} + \cdots + 782656576 \)
$11$
\( (T^{2} - 174 T - 23656)^{2} \)
$13$
\( T^{4} + 768132 T^{2} + \cdots + 31678304256 \)
$17$
\( T^{4} + 1708148 T^{2} + \cdots + 85278016576 \)
$19$
\( (T^{2} - 120 T - 4492800)^{2} \)
$23$
\( T^{4} + 18462752 T^{2} + \cdots + 56918507776 \)
$29$
\( (T^{2} - 2430 T - 21286800)^{2} \)
$31$
\( (T^{2} - 11684 T + 33004864)^{2} \)
$37$
\( T^{4} + 177178148 T^{2} + \cdots + 43\!\cdots\!76 \)
$41$
\( (T^{2} - 24984 T + 119953964)^{2} \)
$43$
\( T^{4} + 487889312 T^{2} + \cdots + 23\!\cdots\!36 \)
$47$
\( T^{4} + 90906128 T^{2} + \cdots + 18\!\cdots\!96 \)
$53$
\( T^{4} + 863264052 T^{2} + \cdots + 11\!\cdots\!76 \)
$59$
\( (T^{2} - 23730 T - 927897400)^{2} \)
$61$
\( (T^{2} - 57124 T + 528018244)^{2} \)
$67$
\( T^{4} + 1195938128 T^{2} + \cdots + 18\!\cdots\!96 \)
$71$
\( (T^{2} + 11076 T - 2668294656)^{2} \)
$73$
\( T^{4} + 4520143952 T^{2} + \cdots + 27\!\cdots\!76 \)
$79$
\( (T^{2} + 14220 T - 1173592800)^{2} \)
$83$
\( T^{4} + 12944582432 T^{2} + \cdots + 26\!\cdots\!56 \)
$89$
\( (T^{2} - 60420 T - 336355900)^{2} \)
$97$
\( T^{4} + 14673446528 T^{2} + \cdots + 53\!\cdots\!96 \)
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