Newspace parameters
Level: | \( N \) | \(=\) | \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 210.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.67685844245\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(i, \sqrt{5})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} + 3x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 2 \) |
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 \(1,\beta_1,\beta_2,\beta_3\) 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 \( x^{4} + 3x^{2} + 1 \) :
\(\beta_{1}\) | \(=\) | \( \nu^{2} + \nu + 1 \) |
\(\beta_{2}\) | \(=\) | \( \nu^{3} + 2\nu \) |
\(\beta_{3}\) | \(=\) | \( -\nu^{2} + \nu - 1 \) |
\(\nu\) | \(=\) | \( ( \beta_{3} + \beta_1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( -\beta_{3} + \beta _1 - 2 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( -\beta_{3} + \beta_{2} - \beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).
\(n\) | \(31\) | \(71\) | \(127\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
41.1 |
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− | 1.00000i | −1.61803 | + | 0.618034i | −1.00000 | 1.00000 | 0.618034 | + | 1.61803i | 2.61803 | − | 0.381966i | 1.00000i | 2.23607 | − | 2.00000i | − | 1.00000i | ||||||||||||||||||||
41.2 | − | 1.00000i | 0.618034 | − | 1.61803i | −1.00000 | 1.00000 | −1.61803 | − | 0.618034i | 0.381966 | − | 2.61803i | 1.00000i | −2.23607 | − | 2.00000i | − | 1.00000i | |||||||||||||||||||||
41.3 | 1.00000i | −1.61803 | − | 0.618034i | −1.00000 | 1.00000 | 0.618034 | − | 1.61803i | 2.61803 | + | 0.381966i | − | 1.00000i | 2.23607 | + | 2.00000i | 1.00000i | ||||||||||||||||||||||
41.4 | 1.00000i | 0.618034 | + | 1.61803i | −1.00000 | 1.00000 | −1.61803 | + | 0.618034i | 0.381966 | + | 2.61803i | − | 1.00000i | −2.23607 | + | 2.00000i | 1.00000i | ||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
21.c | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 210.2.b.a | ✓ | 4 |
3.b | odd | 2 | 1 | 210.2.b.b | yes | 4 | |
4.b | odd | 2 | 1 | 1680.2.f.i | 4 | ||
5.b | even | 2 | 1 | 1050.2.b.c | 4 | ||
5.c | odd | 4 | 1 | 1050.2.d.c | 4 | ||
5.c | odd | 4 | 1 | 1050.2.d.d | 4 | ||
7.b | odd | 2 | 1 | 210.2.b.b | yes | 4 | |
12.b | even | 2 | 1 | 1680.2.f.e | 4 | ||
15.d | odd | 2 | 1 | 1050.2.b.a | 4 | ||
15.e | even | 4 | 1 | 1050.2.d.a | 4 | ||
15.e | even | 4 | 1 | 1050.2.d.f | 4 | ||
21.c | even | 2 | 1 | inner | 210.2.b.a | ✓ | 4 |
28.d | even | 2 | 1 | 1680.2.f.e | 4 | ||
35.c | odd | 2 | 1 | 1050.2.b.a | 4 | ||
35.f | even | 4 | 1 | 1050.2.d.a | 4 | ||
35.f | even | 4 | 1 | 1050.2.d.f | 4 | ||
84.h | odd | 2 | 1 | 1680.2.f.i | 4 | ||
105.g | even | 2 | 1 | 1050.2.b.c | 4 | ||
105.k | odd | 4 | 1 | 1050.2.d.c | 4 | ||
105.k | odd | 4 | 1 | 1050.2.d.d | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
210.2.b.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
210.2.b.a | ✓ | 4 | 21.c | even | 2 | 1 | inner |
210.2.b.b | yes | 4 | 3.b | odd | 2 | 1 | |
210.2.b.b | yes | 4 | 7.b | odd | 2 | 1 | |
1050.2.b.a | 4 | 15.d | odd | 2 | 1 | ||
1050.2.b.a | 4 | 35.c | odd | 2 | 1 | ||
1050.2.b.c | 4 | 5.b | even | 2 | 1 | ||
1050.2.b.c | 4 | 105.g | even | 2 | 1 | ||
1050.2.d.a | 4 | 15.e | even | 4 | 1 | ||
1050.2.d.a | 4 | 35.f | even | 4 | 1 | ||
1050.2.d.c | 4 | 5.c | odd | 4 | 1 | ||
1050.2.d.c | 4 | 105.k | odd | 4 | 1 | ||
1050.2.d.d | 4 | 5.c | odd | 4 | 1 | ||
1050.2.d.d | 4 | 105.k | odd | 4 | 1 | ||
1050.2.d.f | 4 | 15.e | even | 4 | 1 | ||
1050.2.d.f | 4 | 35.f | even | 4 | 1 | ||
1680.2.f.e | 4 | 12.b | even | 2 | 1 | ||
1680.2.f.e | 4 | 28.d | even | 2 | 1 | ||
1680.2.f.i | 4 | 4.b | odd | 2 | 1 | ||
1680.2.f.i | 4 | 84.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{17}^{2} - 6T_{17} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(210, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 1)^{2} \)
$3$
\( T^{4} + 2 T^{3} + 2 T^{2} + 6 T + 9 \)
$5$
\( (T - 1)^{4} \)
$7$
\( T^{4} - 6 T^{3} + 18 T^{2} - 42 T + 49 \)
$11$
\( (T^{2} + 20)^{2} \)
$13$
\( T^{4} + 12T^{2} + 16 \)
$17$
\( (T^{2} - 6 T + 4)^{2} \)
$19$
\( T^{4} + 72T^{2} + 16 \)
$23$
\( (T^{2} + 16)^{2} \)
$29$
\( T^{4} + 92T^{2} + 1936 \)
$31$
\( T^{4} + 60T^{2} + 400 \)
$37$
\( (T^{2} - 6 T + 4)^{2} \)
$41$
\( (T^{2} - 4 T - 16)^{2} \)
$43$
\( (T^{2} - 8 T - 64)^{2} \)
$47$
\( (T^{2} + 4 T - 16)^{2} \)
$53$
\( T^{4} + 72T^{2} + 16 \)
$59$
\( (T^{2} - 20)^{2} \)
$61$
\( T^{4} + 60T^{2} + 400 \)
$67$
\( (T + 12)^{4} \)
$71$
\( T^{4} + 60T^{2} + 400 \)
$73$
\( T^{4} + 172T^{2} + 5776 \)
$79$
\( (T^{2} - 80)^{2} \)
$83$
\( (T^{2} + 2 T - 244)^{2} \)
$89$
\( (T^{2} + 20 T + 80)^{2} \)
$97$
\( T^{4} + 28T^{2} + 16 \)
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