Newspace parameters
Level: | \( N \) | \(=\) | \( 201 = 3 \cdot 67 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 201.f (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.60499308063\) |
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: | \(\Z[a_1, \ldots, a_{13}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{U}(1)[D_{6}]$ |
$q$-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 \(\left(\mathbb{Z}/201\mathbb{Z}\right)^\times\).
\(n\) | \(68\) | \(136\) |
\(\chi(n)\) | \(-1\) | \(\zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
38.1 |
|
0 | − | 1.73205i | 1.00000 | + | 1.73205i | 0 | 0 | 3.00000 | − | 1.73205i | 0 | −3.00000 | 0 | |||||||||||||||||||
164.1 | 0 | 1.73205i | 1.00000 | − | 1.73205i | 0 | 0 | 3.00000 | + | 1.73205i | 0 | −3.00000 | 0 | |||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-3}) \) |
67.d | odd | 6 | 1 | inner |
201.f | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 201.2.f.a | ✓ | 2 |
3.b | odd | 2 | 1 | CM | 201.2.f.a | ✓ | 2 |
67.d | odd | 6 | 1 | inner | 201.2.f.a | ✓ | 2 |
201.f | even | 6 | 1 | inner | 201.2.f.a | ✓ | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
201.2.f.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
201.2.f.a | ✓ | 2 | 3.b | odd | 2 | 1 | CM |
201.2.f.a | ✓ | 2 | 67.d | odd | 6 | 1 | inner |
201.2.f.a | ✓ | 2 | 201.f | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} \)
acting on \(S_{2}^{\mathrm{new}}(201, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} + 3 \)
$5$
\( T^{2} \)
$7$
\( T^{2} - 6T + 12 \)
$11$
\( T^{2} \)
$13$
\( T^{2} - 3T + 3 \)
$17$
\( T^{2} \)
$19$
\( T^{2} - 8T + 64 \)
$23$
\( T^{2} \)
$29$
\( T^{2} \)
$31$
\( T^{2} + 15T + 75 \)
$37$
\( T^{2} + 10T + 100 \)
$41$
\( T^{2} \)
$43$
\( T^{2} + 147 \)
$47$
\( T^{2} \)
$53$
\( T^{2} \)
$59$
\( T^{2} \)
$61$
\( T^{2} + 15T + 75 \)
$67$
\( T^{2} + 5T + 67 \)
$71$
\( T^{2} \)
$73$
\( T^{2} - 17T + 289 \)
$79$
\( T^{2} - 9T + 27 \)
$83$
\( T^{2} \)
$89$
\( T^{2} \)
$97$
\( T^{2} + 9T + 27 \)
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