Newspace parameters
| Level: | \( N \) | \(=\) | \( 82 = 2 \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 82.f (of order \(10\), degree \(4\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(0.654773296574\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{10})\) |
| Coefficient field: | 8.0.346890625.1 |
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|
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| Defining polynomial: |
\( x^{8} - x^{7} + x^{5} + 9x^{4} + 31x^{3} - 50x^{2} - 11x + 121 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} + x^{5} + 9x^{4} + 31x^{3} - 50x^{2} - 11x + 121 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 9219 \nu^{7} - 23849 \nu^{6} + 41305 \nu^{5} - 46232 \nu^{4} + 166021 \nu^{3} + 73918 \nu^{2} + \cdots + 901395 ) / 182171 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 10275 \nu^{7} - 24091 \nu^{6} + 33528 \nu^{5} - 54966 \nu^{4} + 176383 \nu^{3} + 91870 \nu^{2} + \cdots + 548537 ) / 182171 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 16922 \nu^{7} - 37690 \nu^{6} + 65483 \nu^{5} - 73025 \nu^{4} + 263343 \nu^{3} + 138625 \nu^{2} + \cdots + 1248071 ) / 182171 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 17662 \nu^{7} - 61321 \nu^{6} + 89705 \nu^{5} - 121928 \nu^{4} + 349269 \nu^{3} - 97210 \nu^{2} + \cdots + 1860595 ) / 182171 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -1814\nu^{7} + 5246\nu^{6} - 9067\nu^{5} + 10173\nu^{4} - 36431\nu^{3} + 2284\nu^{2} + 115922\nu - 197758 ) / 16561 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 1888\nu^{7} - 5953\nu^{6} + 8177\nu^{5} - 10095\nu^{4} + 35087\nu^{3} - 1026\nu^{2} - 130383\nu + 169581 ) / 16561 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} - \beta_{4} + 4\beta_{2} - 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - 3\beta_{6} - 3\beta_{5} - 4\beta_{4} + 3\beta_{3} + \beta_{2} + 2\beta _1 - 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 7\beta_{7} - 2\beta_{6} - 7\beta_{5} - 6\beta_{3} + 2\beta _1 - 6 \)
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| \(\nu^{5}\) | \(=\) |
\( -6\beta_{7} + 4\beta_{5} + 4\beta_{4} - 13\beta_{3} + 13\beta_{2} - 4\beta _1 - 32 \)
|
| \(\nu^{6}\) | \(=\) |
\( -17\beta_{7} - 17\beta_{6} + 10\beta_{4} + 10\beta_{3} - 28\beta_{2} - 19\beta _1 + 11 \)
|
| \(\nu^{7}\) | \(=\) |
\( -8\beta_{6} + \beta_{5} + 88\beta_{4} - 161\beta_{2} + 88 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/82\mathbb{Z}\right)^\times\).
| \(n\) | \(47\) |
| \(\chi(n)\) | \(\beta_{2}\) |
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 |
|
−0.309017 | − | 0.951057i | 0.0787164i | −0.809017 | + | 0.587785i | 2.80592 | − | 2.03862i | 0.0748638 | − | 0.0243247i | −2.80592 | − | 0.911698i | 0.809017 | + | 0.587785i | 2.99380 | −2.80592 | − | 2.03862i | ||||||||||||||||||||||||||||
| 23.2 | −0.309017 | − | 0.951057i | 2.99897i | −0.809017 | + | 0.587785i | −1.68788 | + | 1.22632i | 2.85219 | − | 0.926732i | 1.68788 | + | 0.548427i | 0.809017 | + | 0.587785i | −5.99380 | 1.68788 | + | 1.22632i | |||||||||||||||||||||||||||||
| 25.1 | −0.309017 | + | 0.951057i | − | 2.99897i | −0.809017 | − | 0.587785i | −1.68788 | − | 1.22632i | 2.85219 | + | 0.926732i | 1.68788 | − | 0.548427i | 0.809017 | − | 0.587785i | −5.99380 | 1.68788 | − | 1.22632i | ||||||||||||||||||||||||||||
| 25.2 | −0.309017 | + | 0.951057i | − | 0.0787164i | −0.809017 | − | 0.587785i | 2.80592 | + | 2.03862i | 0.0748638 | + | 0.0243247i | −2.80592 | + | 0.911698i | 0.809017 | − | 0.587785i | 2.99380 | −2.80592 | + | 2.03862i | ||||||||||||||||||||||||||||
| 31.1 | 0.809017 | + | 0.587785i | − | 1.72671i | 0.309017 | + | 0.951057i | 0.200214 | + | 0.616196i | 1.01494 | − | 1.39694i | −0.200214 | − | 0.275571i | −0.309017 | + | 0.951057i | 0.0184624 | −0.200214 | + | 0.616196i | ||||||||||||||||||||||||||||
| 31.2 | 0.809017 | + | 0.587785i | 2.45326i | 0.309017 | + | 0.951057i | −1.31825 | − | 4.05715i | −1.44199 | + | 1.98473i | 1.31825 | + | 1.81441i | −0.309017 | + | 0.951057i | −3.01846 | 1.31825 | − | 4.05715i | |||||||||||||||||||||||||||||
| 45.1 | 0.809017 | − | 0.587785i | − | 2.45326i | 0.309017 | − | 0.951057i | −1.31825 | + | 4.05715i | −1.44199 | − | 1.98473i | 1.31825 | − | 1.81441i | −0.309017 | − | 0.951057i | −3.01846 | 1.31825 | + | 4.05715i | ||||||||||||||||||||||||||||
| 45.2 | 0.809017 | − | 0.587785i | 1.72671i | 0.309017 | − | 0.951057i | 0.200214 | − | 0.616196i | 1.01494 | + | 1.39694i | −0.200214 | + | 0.275571i | −0.309017 | − | 0.951057i | 0.0184624 | −0.200214 | − | 0.616196i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 41.f | even | 10 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 82.2.f.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 738.2.n.c | 8 | ||
| 4.b | odd | 2 | 1 | 656.2.be.b | 8 | ||
| 41.f | even | 10 | 1 | inner | 82.2.f.b | ✓ | 8 |
| 41.g | even | 20 | 2 | 3362.2.a.y | 8 | ||
| 123.l | odd | 10 | 1 | 738.2.n.c | 8 | ||
| 164.l | odd | 10 | 1 | 656.2.be.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 82.2.f.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 82.2.f.b | ✓ | 8 | 41.f | even | 10 | 1 | inner |
| 656.2.be.b | 8 | 4.b | odd | 2 | 1 | ||
| 656.2.be.b | 8 | 164.l | odd | 10 | 1 | ||
| 738.2.n.c | 8 | 3.b | odd | 2 | 1 | ||
| 738.2.n.c | 8 | 123.l | odd | 10 | 1 | ||
| 3362.2.a.y | 8 | 41.g | even | 20 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 18T_{3}^{6} + 99T_{3}^{4} + 162T_{3}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(82, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \)
|
| $3$ |
\( T^{8} + 18 T^{6} + \cdots + 1 \)
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| $5$ |
\( T^{8} + 10 T^{6} + \cdots + 400 \)
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| $7$ |
\( T^{8} - 8 T^{6} + \cdots + 16 \)
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| $11$ |
\( T^{8} + 10 T^{7} + \cdots + 22801 \)
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| $13$ |
\( T^{8} - 5 T^{7} + \cdots + 32400 \)
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| $17$ |
\( T^{8} + 15 T^{7} + \cdots + 30976 \)
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| $19$ |
\( T^{8} - 10 T^{7} + \cdots + 1 \)
|
| $23$ |
\( T^{8} + 5 T^{7} + \cdots + 400 \)
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| $29$ |
\( T^{8} + 5 T^{7} + \cdots + 1296 \)
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| $31$ |
\( T^{8} + 6 T^{7} + \cdots + 913936 \)
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| $37$ |
\( T^{8} - 19 T^{7} + \cdots + 929296 \)
|
| $41$ |
\( T^{8} + 24 T^{7} + \cdots + 2825761 \)
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| $43$ |
\( T^{8} + 7 T^{7} + \cdots + 10784656 \)
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| $47$ |
\( (T^{4} - 20 T^{3} + \cdots + 1280)^{2} \)
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| $53$ |
\( T^{8} + 5 T^{7} + \cdots + 1296 \)
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| $59$ |
\( T^{8} - 4 T^{7} + \cdots + 25563136 \)
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| $61$ |
\( T^{8} - 21 T^{7} + \cdots + 4822416 \)
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| $67$ |
\( T^{8} + 15 T^{7} + \cdots + 126736 \)
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| $71$ |
\( T^{8} - 15 T^{7} + \cdots + 48400 \)
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| $73$ |
\( (T^{4} + 21 T^{3} + \cdots - 324)^{2} \)
|
| $79$ |
\( T^{8} + 232 T^{6} + \cdots + 65536 \)
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| $83$ |
\( (T^{4} + 12 T^{3} + \cdots + 891)^{2} \)
|
| $89$ |
\( T^{8} - 25 T^{7} + \cdots + 942841 \)
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| $97$ |
\( T^{8} - 15 T^{7} + \cdots + 239909121 \)
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