Newspace parameters
| Level: | \( N \) | \(=\) | \( 35 = 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 35.e (of order \(3\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(5.61343369345\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{16} - 3 x^{15} + 234 x^{14} - 399 x^{13} + 37070 x^{12} - 54845 x^{11} + 2990279 x^{10} + \cdots + 1025569391616 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{10}\cdot 7^{5} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 3 x^{15} + 234 x^{14} - 399 x^{13} + 37070 x^{12} - 54845 x^{11} + 2990279 x^{10} + \cdots + 1025569391616 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 62\!\cdots\!88 \nu^{15} + \cdots + 19\!\cdots\!28 ) / 55\!\cdots\!60 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 73\!\cdots\!45 \nu^{15} + \cdots - 42\!\cdots\!44 ) / 37\!\cdots\!80 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 88\!\cdots\!13 \nu^{15} + \cdots - 19\!\cdots\!96 ) / 37\!\cdots\!80 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 27\!\cdots\!44 \nu^{15} + \cdots + 58\!\cdots\!84 ) / 69\!\cdots\!20 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 19\!\cdots\!57 \nu^{15} + \cdots - 49\!\cdots\!16 ) / 14\!\cdots\!60 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 18\!\cdots\!64 \nu^{15} + \cdots - 10\!\cdots\!04 ) / 73\!\cdots\!80 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 74\!\cdots\!43 \nu^{15} + \cdots + 55\!\cdots\!68 ) / 14\!\cdots\!60 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 20\!\cdots\!59 \nu^{15} + \cdots + 48\!\cdots\!00 ) / 36\!\cdots\!40 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 33\!\cdots\!10 \nu^{15} + \cdots - 19\!\cdots\!84 ) / 55\!\cdots\!60 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 32\!\cdots\!87 \nu^{15} + \cdots + 16\!\cdots\!48 ) / 48\!\cdots\!20 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 17\!\cdots\!01 \nu^{15} + \cdots + 26\!\cdots\!28 ) / 24\!\cdots\!60 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 60\!\cdots\!82 \nu^{15} + \cdots - 18\!\cdots\!64 ) / 73\!\cdots\!80 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 27\!\cdots\!29 \nu^{15} + \cdots + 65\!\cdots\!36 ) / 14\!\cdots\!60 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 29\!\cdots\!11 \nu^{15} + \cdots - 22\!\cdots\!12 ) / 73\!\cdots\!80 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{10} - \beta_{4} - \beta_{3} - 57\beta_{2} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{14} - \beta_{9} + \beta_{8} - \beta_{4} - 92\beta_{3} - 21 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 2 \beta_{15} + 2 \beta_{14} - 2 \beta_{13} + 2 \beta_{12} - 4 \beta_{11} - 129 \beta_{10} + \cdots - 5229 \)
|
| \(\nu^{5}\) | \(=\) |
\( 22 \beta_{15} + 155 \beta_{14} - 159 \beta_{13} - 2 \beta_{12} + 131 \beta_{11} - 105 \beta_{10} + \cdots + 10 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 688 \beta_{15} - 996 \beta_{14} - 268 \beta_{13} - 204 \beta_{12} - 688 \beta_{11} + 432 \beta_{9} + \cdots + 552541 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 2232 \beta_{15} + 2232 \beta_{14} + 23521 \beta_{13} - 8176 \beta_{12} - 12201 \beta_{11} + \cdots + 411289 \)
|
| \(\nu^{8}\) | \(=\) |
\( 124230 \beta_{15} + 101928 \beta_{14} + 100336 \beta_{13} - 66946 \beta_{12} + 246908 \beta_{11} + \cdots + 28642 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 175036 \beta_{15} - 2767577 \beta_{14} - 358498 \beta_{13} + 1204452 \beta_{12} - 175036 \beta_{11} + \cdots - 35376707 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 2888284 \beta_{15} + 2888284 \beta_{14} - 10587796 \beta_{13} + 17971632 \beta_{12} + \cdots - 7116610289 \)
|
| \(\nu^{11}\) | \(=\) |
\( 67626368 \beta_{15} + 288747941 \beta_{14} - 341172461 \beta_{13} + 34229664 \beta_{12} + \cdots + 50928016 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 1565408900 \beta_{15} - 1933414574 \beta_{14} - 287129634 \beta_{13} - 1256245056 \beta_{12} + \cdots + 829210958651 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 6818243834 \beta_{15} + 6818243834 \beta_{14} + 49107672801 \beta_{13} - 28259486578 \beta_{12} + \cdots + 79918583721 \)
|
| \(\nu^{14}\) | \(=\) |
\( 223047133372 \beta_{15} + 202996821360 \beta_{14} + 222318207876 \beta_{13} - 165868204068 \beta_{12} + \cdots + 28589464652 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 132894014880 \beta_{15} - 5001147059825 \beta_{14} - 882819017416 \beta_{13} + 2982124636728 \beta_{12} + \cdots + 9509382356655 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).
| \(n\) | \(22\) | \(31\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 11.1 |
|
−5.48128 | − | 9.49385i | 11.8224 | − | 20.4771i | −44.0888 | + | 76.3640i | −12.5000 | − | 21.6506i | −259.208 | −128.275 | + | 18.7779i | 615.850 | −158.040 | − | 273.733i | −137.032 | + | 237.346i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.2 | −4.82595 | − | 8.35879i | −11.4297 | + | 19.7969i | −30.5796 | + | 52.9654i | −12.5000 | − | 21.6506i | 220.637 | 107.309 | − | 72.7443i | 281.441 | −139.778 | − | 242.102i | −120.649 | + | 208.970i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.3 | −2.45778 | − | 4.25701i | −4.95859 | + | 8.58852i | 3.91860 | − | 6.78721i | −12.5000 | − | 21.6506i | 48.7485 | −109.337 | + | 69.6590i | −195.822 | 72.3249 | + | 125.270i | −61.4446 | + | 106.425i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.4 | −1.55686 | − | 2.69657i | 10.3031 | − | 17.8456i | 11.1524 | − | 19.3164i | −12.5000 | − | 21.6506i | −64.1623 | 98.0088 | + | 84.8603i | −169.090 | −90.8096 | − | 157.287i | −38.9216 | + | 67.4141i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.5 | 0.634553 | + | 1.09908i | −3.50500 | + | 6.07083i | 15.1947 | − | 26.3180i | −12.5000 | − | 21.6506i | −8.89643 | 69.6963 | − | 109.313i | 79.1787 | 96.9300 | + | 167.888i | 15.8638 | − | 27.4770i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.6 | 3.31717 | + | 5.74550i | −15.2893 | + | 26.4818i | −6.00720 | + | 10.4048i | −12.5000 | − | 21.6506i | −202.868 | 54.7457 | + | 117.516i | 132.591 | −346.023 | − | 599.329i | 82.9292 | − | 143.638i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.7 | 3.38502 | + | 5.86303i | 11.7457 | − | 20.3441i | −6.91677 | + | 11.9802i | −12.5000 | − | 21.6506i | 159.037 | 23.1361 | − | 127.561i | 122.988 | −154.421 | − | 267.465i | 84.6256 | − | 146.576i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.8 | 5.48513 | + | 9.50052i | 2.31134 | − | 4.00336i | −44.1733 | + | 76.5104i | −12.5000 | − | 21.6506i | 50.7120 | −36.2843 | + | 124.461i | −618.137 | 110.815 | + | 191.938i | 137.128 | − | 237.513i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 16.1 | −5.48128 | + | 9.49385i | 11.8224 | + | 20.4771i | −44.0888 | − | 76.3640i | −12.5000 | + | 21.6506i | −259.208 | −128.275 | − | 18.7779i | 615.850 | −158.040 | + | 273.733i | −137.032 | − | 237.346i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 16.2 | −4.82595 | + | 8.35879i | −11.4297 | − | 19.7969i | −30.5796 | − | 52.9654i | −12.5000 | + | 21.6506i | 220.637 | 107.309 | + | 72.7443i | 281.441 | −139.778 | + | 242.102i | −120.649 | − | 208.970i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 16.3 | −2.45778 | + | 4.25701i | −4.95859 | − | 8.58852i | 3.91860 | + | 6.78721i | −12.5000 | + | 21.6506i | 48.7485 | −109.337 | − | 69.6590i | −195.822 | 72.3249 | − | 125.270i | −61.4446 | − | 106.425i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 16.4 | −1.55686 | + | 2.69657i | 10.3031 | + | 17.8456i | 11.1524 | + | 19.3164i | −12.5000 | + | 21.6506i | −64.1623 | 98.0088 | − | 84.8603i | −169.090 | −90.8096 | + | 157.287i | −38.9216 | − | 67.4141i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 16.5 | 0.634553 | − | 1.09908i | −3.50500 | − | 6.07083i | 15.1947 | + | 26.3180i | −12.5000 | + | 21.6506i | −8.89643 | 69.6963 | + | 109.313i | 79.1787 | 96.9300 | − | 167.888i | 15.8638 | + | 27.4770i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 16.6 | 3.31717 | − | 5.74550i | −15.2893 | − | 26.4818i | −6.00720 | − | 10.4048i | −12.5000 | + | 21.6506i | −202.868 | 54.7457 | − | 117.516i | 132.591 | −346.023 | + | 599.329i | 82.9292 | + | 143.638i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 16.7 | 3.38502 | − | 5.86303i | 11.7457 | + | 20.3441i | −6.91677 | − | 11.9802i | −12.5000 | + | 21.6506i | 159.037 | 23.1361 | + | 127.561i | 122.988 | −154.421 | + | 267.465i | 84.6256 | + | 146.576i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 16.8 | 5.48513 | − | 9.50052i | 2.31134 | + | 4.00336i | −44.1733 | − | 76.5104i | −12.5000 | + | 21.6506i | 50.7120 | −36.2843 | − | 124.461i | −618.137 | 110.815 | − | 191.938i | 137.128 | + | 237.513i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
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 | 35.6.e.b | ✓ | 16 |
| 7.c | even | 3 | 1 | inner | 35.6.e.b | ✓ | 16 |
| 7.c | even | 3 | 1 | 245.6.a.j | 8 | ||
| 7.d | odd | 6 | 1 | 245.6.a.k | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 35.6.e.b | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 35.6.e.b | ✓ | 16 | 7.c | even | 3 | 1 | inner |
| 245.6.a.j | 8 | 7.c | even | 3 | 1 | ||
| 245.6.a.k | 8 | 7.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} + 3 T_{2}^{15} + 234 T_{2}^{14} + 399 T_{2}^{13} + 37070 T_{2}^{12} + 54845 T_{2}^{11} + \cdots + 1025569391616 \)
acting on \(S_{6}^{\mathrm{new}}(35, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} + \cdots + 1025569391616 \)
|
| $3$ |
\( T^{16} + \cdots + 66\!\cdots\!56 \)
|
| $5$ |
\( (T^{2} + 25 T + 625)^{8} \)
|
| $7$ |
\( T^{16} + \cdots + 63\!\cdots\!01 \)
|
| $11$ |
\( T^{16} + \cdots + 16\!\cdots\!00 \)
|
| $13$ |
\( (T^{8} + \cdots - 52\!\cdots\!44)^{2} \)
|
| $17$ |
\( T^{16} + \cdots + 31\!\cdots\!36 \)
|
| $19$ |
\( T^{16} + \cdots + 22\!\cdots\!24 \)
|
| $23$ |
\( T^{16} + \cdots + 54\!\cdots\!61 \)
|
| $29$ |
\( (T^{8} + \cdots + 54\!\cdots\!00)^{2} \)
|
| $31$ |
\( T^{16} + \cdots + 18\!\cdots\!56 \)
|
| $37$ |
\( T^{16} + \cdots + 12\!\cdots\!00 \)
|
| $41$ |
\( (T^{8} + \cdots - 44\!\cdots\!75)^{2} \)
|
| $43$ |
\( (T^{8} + \cdots - 19\!\cdots\!12)^{2} \)
|
| $47$ |
\( T^{16} + \cdots + 20\!\cdots\!00 \)
|
| $53$ |
\( T^{16} + \cdots + 34\!\cdots\!04 \)
|
| $59$ |
\( T^{16} + \cdots + 12\!\cdots\!00 \)
|
| $61$ |
\( T^{16} + \cdots + 69\!\cdots\!24 \)
|
| $67$ |
\( T^{16} + \cdots + 85\!\cdots\!00 \)
|
| $71$ |
\( (T^{8} + \cdots - 11\!\cdots\!24)^{2} \)
|
| $73$ |
\( T^{16} + \cdots + 47\!\cdots\!76 \)
|
| $79$ |
\( T^{16} + \cdots + 31\!\cdots\!96 \)
|
| $83$ |
\( (T^{8} + \cdots + 20\!\cdots\!56)^{2} \)
|
| $89$ |
\( T^{16} + \cdots + 10\!\cdots\!44 \)
|
| $97$ |
\( (T^{8} + \cdots + 10\!\cdots\!16)^{2} \)
|
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