Newspace parameters
| Level: | \( N \) | \(=\) | \( 1682 = 2 \cdot 29^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1682.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(99.2412126297\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 97x^{4} - 87x^{3} + 1809x^{2} + 1890x - 1620 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 97x^{4} - 87x^{3} + 1809x^{2} + 1890x - 1620 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -77\nu^{5} + 395\nu^{4} + 5585\nu^{3} - 11565\nu^{2} - 81423\nu - 45792 ) / 58374 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} + 11\nu^{4} - 172\nu^{3} - 702\nu^{2} + 3528\nu + 3861 ) / 621 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 58\nu^{5} - 466\nu^{4} - 1006\nu^{3} + 5721\nu^{2} - 57438\nu + 175500 ) / 29187 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 53\nu^{5} - 314\nu^{4} - 3044\nu^{3} + 9645\nu^{2} + 16623\nu - 20790 ) / 9729 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 4\beta_{5} - 3\beta_{4} + 12\beta_{2} + 4\beta _1 + 36 \)
|
| \(\nu^{3}\) | \(=\) |
\( 22\beta_{5} - 3\beta_{4} + 3\beta_{3} + 90\beta_{2} + 65\beta _1 + 117 \)
|
| \(\nu^{4}\) | \(=\) |
\( 347\beta_{5} - 177\beta_{4} + 57\beta_{3} + 1236\beta_{2} + 459\beta _1 + 2421 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2775\beta_{5} - 675\beta_{4} + 510\beta_{3} + 10308\beta_{2} + 5411\beta _1 + 14904 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.00000 | −9.53664 | 4.00000 | 1.45008 | 19.0733 | −33.3969 | −8.00000 | 63.9476 | −2.90016 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | −5.97333 | 4.00000 | −9.06103 | 11.9467 | −5.48119 | −8.00000 | 8.68070 | 18.1221 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | −3.04095 | 4.00000 | 20.5863 | 6.08191 | −7.99437 | −8.00000 | −17.7526 | −41.1726 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | −2.91431 | 4.00000 | −7.56771 | 5.82861 | 17.6968 | −8.00000 | −18.5068 | 15.1354 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −2.00000 | 4.01428 | 4.00000 | −7.58102 | −8.02857 | 0.475564 | −8.00000 | −10.8855 | 15.1620 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −2.00000 | 7.45095 | 4.00000 | −7.82664 | −14.9019 | 2.70016 | −8.00000 | 28.5167 | 15.6533 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(29\) | \( -1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1682.4.a.k | ✓ | 6 |
| 29.b | even | 2 | 1 | 1682.4.a.l | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1682.4.a.k | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 1682.4.a.l | yes | 6 | 29.b | even | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 10T_{3}^{5} - 58T_{3}^{4} - 695T_{3}^{3} - 229T_{3}^{2} + 8470T_{3} + 15100 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1682))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{6} \)
|
| $3$ |
\( T^{6} + 10 T^{5} + \cdots + 15100 \)
|
| $5$ |
\( T^{6} + 10 T^{5} + \cdots + 121455 \)
|
| $7$ |
\( T^{6} + 26 T^{5} + \cdots - 33255 \)
|
| $11$ |
\( T^{6} + \cdots + 3156090039 \)
|
| $13$ |
\( T^{6} + \cdots + 1068709689 \)
|
| $17$ |
\( T^{6} + \cdots - 4813781796 \)
|
| $19$ |
\( T^{6} + \cdots + 127552104955 \)
|
| $23$ |
\( T^{6} + \cdots - 222823171989 \)
|
| $29$ |
\( T^{6} \)
|
| $31$ |
\( T^{6} + \cdots - 7738085869 \)
|
| $37$ |
\( T^{6} + \cdots - 1658282920880 \)
|
| $41$ |
\( T^{6} + \cdots - 6378535487664 \)
|
| $43$ |
\( T^{6} + \cdots - 7863899526659 \)
|
| $47$ |
\( T^{6} + \cdots - 92542526238339 \)
|
| $53$ |
\( T^{6} + \cdots - 119294451556596 \)
|
| $59$ |
\( T^{6} + \cdots - 18\!\cdots\!25 \)
|
| $61$ |
\( T^{6} + \cdots + 26\!\cdots\!95 \)
|
| $67$ |
\( T^{6} + \cdots + 26735867645905 \)
|
| $71$ |
\( T^{6} + \cdots - 17\!\cdots\!21 \)
|
| $73$ |
\( T^{6} + \cdots + 1844275431359 \)
|
| $79$ |
\( T^{6} + \cdots + 74546262563805 \)
|
| $83$ |
\( T^{6} + \cdots - 14\!\cdots\!59 \)
|
| $89$ |
\( T^{6} + \cdots + 23\!\cdots\!95 \)
|
| $97$ |
\( T^{6} + \cdots - 89\!\cdots\!64 \)
|
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