Newspace parameters
| Level: | \( N \) | \(=\) | \( 209 = 11 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 209.j (of order \(9\), degree \(6\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(1.66887340224\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\Q(\zeta_{18})\) |
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| Defining polynomial: |
\( x^{6} - x^{3} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/209\mathbb{Z}\right)^\times\).
| \(n\) | \(78\) | \(134\) |
| \(\chi(n)\) | \(-\zeta_{18}^{5}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 23.1 |
|
1.26604 | + | 0.460802i | −0.347296 | − | 1.96962i | −0.141559 | − | 0.118782i | −0.358441 | + | 0.300767i | 0.467911 | − | 2.65366i | 1.17365 | − | 2.03282i | −1.47178 | − | 2.54920i | −0.939693 | + | 0.342020i | −0.592396 | + | 0.215615i | ||||||||||||||||||
| 100.1 | 1.26604 | − | 0.460802i | −0.347296 | + | 1.96962i | −0.141559 | + | 0.118782i | −0.358441 | − | 0.300767i | 0.467911 | + | 2.65366i | 1.17365 | + | 2.03282i | −1.47178 | + | 2.54920i | −0.939693 | − | 0.342020i | −0.592396 | − | 0.215615i | |||||||||||||||||||
| 111.1 | 0.673648 | + | 0.565258i | 1.87939 | − | 0.684040i | −0.213011 | − | 1.20805i | −0.286989 | + | 1.62760i | 1.65270 | + | 0.601535i | 0.0603074 | + | 0.104455i | 1.41875 | − | 2.45734i | 0.766044 | − | 0.642788i | −1.11334 | + | 0.934204i | |||||||||||||||||||
| 177.1 | 0.673648 | − | 0.565258i | 1.87939 | + | 0.684040i | −0.213011 | + | 1.20805i | −0.286989 | − | 1.62760i | 1.65270 | − | 0.601535i | 0.0603074 | − | 0.104455i | 1.41875 | + | 2.45734i | 0.766044 | + | 0.642788i | −1.11334 | − | 0.934204i | |||||||||||||||||||
| 188.1 | −0.439693 | + | 2.49362i | −1.53209 | − | 1.28558i | −4.14543 | − | 1.50881i | 3.64543 | − | 1.32683i | 3.87939 | − | 3.25519i | 1.76604 | + | 3.05888i | 3.05303 | − | 5.28801i | 0.173648 | + | 0.984808i | 1.70574 | + | 9.67372i | |||||||||||||||||||
| 199.1 | −0.439693 | − | 2.49362i | −1.53209 | + | 1.28558i | −4.14543 | + | 1.50881i | 3.64543 | + | 1.32683i | 3.87939 | + | 3.25519i | 1.76604 | − | 3.05888i | 3.05303 | + | 5.28801i | 0.173648 | − | 0.984808i | 1.70574 | − | 9.67372i | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.e | even | 9 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 209.2.j.a | ✓ | 6 |
| 19.e | even | 9 | 1 | inner | 209.2.j.a | ✓ | 6 |
| 19.e | even | 9 | 1 | 3971.2.a.e | 3 | ||
| 19.f | odd | 18 | 1 | 3971.2.a.f | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 209.2.j.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 209.2.j.a | ✓ | 6 | 19.e | even | 9 | 1 | inner |
| 3971.2.a.e | 3 | 19.e | even | 9 | 1 | ||
| 3971.2.a.f | 3 | 19.f | odd | 18 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 3T_{2}^{5} + 9T_{2}^{4} - 24T_{2}^{3} + 36T_{2}^{2} - 27T_{2} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(209, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 3 T^{5} + \cdots + 9 \)
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| $3$ |
\( T^{6} - 8T^{3} + 64 \)
|
| $5$ |
\( T^{6} - 6 T^{5} + \cdots + 9 \)
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| $7$ |
\( T^{6} - 6 T^{5} + \cdots + 1 \)
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| $11$ |
\( (T^{2} + T + 1)^{3} \)
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| $13$ |
\( T^{6} - 12 T^{5} + \cdots + 5041 \)
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| $17$ |
\( T^{6} + 3 T^{5} + \cdots + 12321 \)
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| $19$ |
\( T^{6} - 6 T^{5} + \cdots + 6859 \)
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| $23$ |
\( T^{6} - 9 T^{5} + \cdots + 81 \)
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| $29$ |
\( T^{6} + 6 T^{5} + \cdots + 576 \)
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| $31$ |
\( T^{6} + 3 T^{5} + \cdots + 16129 \)
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| $37$ |
\( (T^{3} + 9 T^{2} + 24 T + 19)^{2} \)
|
| $41$ |
\( T^{6} + 9 T^{5} + \cdots + 81 \)
|
| $43$ |
\( T^{6} + 12 T^{5} + \cdots + 5329 \)
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| $47$ |
\( T^{6} - 27 T^{5} + \cdots + 210681 \)
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| $53$ |
\( T^{6} - 18 T^{5} + \cdots + 263169 \)
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| $59$ |
\( T^{6} + 6 T^{5} + \cdots + 45369 \)
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| $61$ |
\( T^{6} + 21 T^{5} + \cdots + 11449 \)
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| $67$ |
\( T^{6} - 12 T^{5} + \cdots + 128881 \)
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| $71$ |
\( T^{6} + 42 T^{5} + \cdots + 3249 \)
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| $73$ |
\( T^{6} + 15 T^{5} + \cdots + 1129969 \)
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| $79$ |
\( T^{6} + 9 T^{5} + \cdots + 289 \)
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| $83$ |
\( T^{6} - 6 T^{5} + \cdots + 47961 \)
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| $89$ |
\( T^{6} - 21 T^{5} + \cdots + 12321 \)
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| $97$ |
\( T^{6} - 27 T^{5} + \cdots + 157609 \)
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