Newspace parameters
Level: | \( N \) | \(=\) | \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 210.i (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(12.3904011012\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-3}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
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Defining polynomial: | \( x^{2} - x + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).
\(n\) | \(31\) | \(71\) | \(127\) |
\(\chi(n)\) | \(-\zeta_{6}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
121.1 |
|
1.00000 | − | 1.73205i | −1.50000 | − | 2.59808i | −2.00000 | − | 3.46410i | −2.50000 | + | 4.33013i | −6.00000 | 3.50000 | + | 18.1865i | −8.00000 | −4.50000 | + | 7.79423i | 5.00000 | + | 8.66025i | ||||||||||
151.1 | 1.00000 | + | 1.73205i | −1.50000 | + | 2.59808i | −2.00000 | + | 3.46410i | −2.50000 | − | 4.33013i | −6.00000 | 3.50000 | − | 18.1865i | −8.00000 | −4.50000 | − | 7.79423i | 5.00000 | − | 8.66025i | |||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 210.4.i.e | ✓ | 2 |
3.b | odd | 2 | 1 | 630.4.k.d | 2 | ||
7.c | even | 3 | 1 | inner | 210.4.i.e | ✓ | 2 |
7.c | even | 3 | 1 | 1470.4.a.l | 1 | ||
7.d | odd | 6 | 1 | 1470.4.a.b | 1 | ||
21.h | odd | 6 | 1 | 630.4.k.d | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
210.4.i.e | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
210.4.i.e | ✓ | 2 | 7.c | even | 3 | 1 | inner |
630.4.k.d | 2 | 3.b | odd | 2 | 1 | ||
630.4.k.d | 2 | 21.h | odd | 6 | 1 | ||
1470.4.a.b | 1 | 7.d | odd | 6 | 1 | ||
1470.4.a.l | 1 | 7.c | even | 3 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{2} - 32T_{11} + 1024 \)
acting on \(S_{4}^{\mathrm{new}}(210, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - 2T + 4 \)
$3$
\( T^{2} + 3T + 9 \)
$5$
\( T^{2} + 5T + 25 \)
$7$
\( T^{2} - 7T + 343 \)
$11$
\( T^{2} - 32T + 1024 \)
$13$
\( (T - 15)^{2} \)
$17$
\( T^{2} - 70T + 4900 \)
$19$
\( T^{2} + 15T + 225 \)
$23$
\( T^{2} - 42T + 1764 \)
$29$
\( (T - 90)^{2} \)
$31$
\( T^{2} - 85T + 7225 \)
$37$
\( T^{2} + 113T + 12769 \)
$41$
\( (T - 164)^{2} \)
$43$
\( (T - 169)^{2} \)
$47$
\( T^{2} + 326T + 106276 \)
$53$
\( T^{2} - 44T + 1936 \)
$59$
\( T^{2} - 782T + 611524 \)
$61$
\( T^{2} + 658T + 432964 \)
$67$
\( T^{2} + 1071 T + 1147041 \)
$71$
\( (T - 344)^{2} \)
$73$
\( T^{2} + 431T + 185761 \)
$79$
\( T^{2} + 397T + 157609 \)
$83$
\( (T - 680)^{2} \)
$89$
\( T^{2} + 1534 T + 2353156 \)
$97$
\( (T + 1234)^{2} \)
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