Newspace parameters
| Level: | \( N \) | \(=\) | \( 114 = 2 \cdot 3 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 114.e (of order \(3\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(6.72621774065\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.6967728.1 |
| Defining polynomial: |
\( x^{6} - x^{5} + 8x^{4} + 5x^{3} + 50x^{2} - 7x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} + 8x^{4} + 5x^{3} + 50x^{2} - 7x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 10\nu^{5} - 80\nu^{4} + 116\nu^{3} - 500\nu^{2} + 70\nu - 2525 ) / 131 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -52\nu^{5} + 23\nu^{4} - 184\nu^{3} - 544\nu^{2} - 1936\nu + 161 ) / 393 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -56\nu^{5} + 55\nu^{4} - 440\nu^{3} - 344\nu^{2} - 2750\nu - 8 ) / 393 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -58\nu^{5} + 71\nu^{4} - 568\nu^{3} - 244\nu^{2} - 1978\nu - 289 ) / 393 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -1036\nu^{5} + 821\nu^{4} - 8140\nu^{3} - 6364\nu^{2} - 52840\nu - 148 ) / 393 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{4} - 3\beta_{3} + \beta_{2} + 1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{5} + \beta_{4} - 60\beta_{3} + 2\beta_{2} - 3\beta _1 - 58 ) / 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -5\beta_{4} + 5\beta_{2} - \beta _1 - 25 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -6\beta_{5} - 10\beta_{4} + 126\beta_{3} - 5\beta_{2} - 5 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -21\beta_{5} - 62\beta_{4} + 537\beta_{3} - 124\beta_{2} + 21\beta _1 + 413 ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/114\mathbb{Z}\right)^\times\).
| \(n\) | \(77\) | \(97\) |
| \(\chi(n)\) | \(1\) | \(\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
1.00000 | + | 1.73205i | 1.50000 | + | 2.59808i | −2.00000 | + | 3.46410i | −10.1010 | − | 17.4954i | −3.00000 | + | 5.19615i | −22.8872 | −8.00000 | −4.50000 | + | 7.79423i | 20.2020 | − | 34.9909i | ||||||||||||||||||||||
| 7.2 | 1.00000 | + | 1.73205i | 1.50000 | + | 2.59808i | −2.00000 | + | 3.46410i | 2.60679 | + | 4.51510i | −3.00000 | + | 5.19615i | 31.4905 | −8.00000 | −4.50000 | + | 7.79423i | −5.21359 | + | 9.03020i | |||||||||||||||||||||||
| 7.3 | 1.00000 | + | 1.73205i | 1.50000 | + | 2.59808i | −2.00000 | + | 3.46410i | 6.49420 | + | 11.2483i | −3.00000 | + | 5.19615i | −25.6033 | −8.00000 | −4.50000 | + | 7.79423i | −12.9884 | + | 22.4966i | |||||||||||||||||||||||
| 49.1 | 1.00000 | − | 1.73205i | 1.50000 | − | 2.59808i | −2.00000 | − | 3.46410i | −10.1010 | + | 17.4954i | −3.00000 | − | 5.19615i | −22.8872 | −8.00000 | −4.50000 | − | 7.79423i | 20.2020 | + | 34.9909i | |||||||||||||||||||||||
| 49.2 | 1.00000 | − | 1.73205i | 1.50000 | − | 2.59808i | −2.00000 | − | 3.46410i | 2.60679 | − | 4.51510i | −3.00000 | − | 5.19615i | 31.4905 | −8.00000 | −4.50000 | − | 7.79423i | −5.21359 | − | 9.03020i | |||||||||||||||||||||||
| 49.3 | 1.00000 | − | 1.73205i | 1.50000 | − | 2.59808i | −2.00000 | − | 3.46410i | 6.49420 | − | 11.2483i | −3.00000 | − | 5.19615i | −25.6033 | −8.00000 | −4.50000 | − | 7.79423i | −12.9884 | − | 22.4966i | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 114.4.e.e | ✓ | 6 |
| 3.b | odd | 2 | 1 | 342.4.g.g | 6 | ||
| 19.c | even | 3 | 1 | inner | 114.4.e.e | ✓ | 6 |
| 19.c | even | 3 | 1 | 2166.4.a.s | 3 | ||
| 19.d | odd | 6 | 1 | 2166.4.a.w | 3 | ||
| 57.h | odd | 6 | 1 | 342.4.g.g | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 114.4.e.e | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 114.4.e.e | ✓ | 6 | 19.c | even | 3 | 1 | inner |
| 342.4.g.g | 6 | 3.b | odd | 2 | 1 | ||
| 342.4.g.g | 6 | 57.h | odd | 6 | 1 | ||
| 2166.4.a.s | 3 | 19.c | even | 3 | 1 | ||
| 2166.4.a.w | 3 | 19.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} + 2T_{5}^{5} + 304T_{5}^{4} - 3336T_{5}^{3} + 87264T_{5}^{2} - 410400T_{5} + 1871424 \)
acting on \(S_{4}^{\mathrm{new}}(114, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} - 2 T + 4)^{3} \)
$3$
\( (T^{2} - 3 T + 9)^{3} \)
$5$
\( T^{6} + 2 T^{5} + 304 T^{4} + \cdots + 1871424 \)
$7$
\( (T^{3} + 17 T^{2} - 941 T - 18453)^{2} \)
$11$
\( (T^{3} + 52 T^{2} - 888 T - 32688)^{2} \)
$13$
\( T^{6} + 75 T^{5} + \cdots + 174424849 \)
$17$
\( T^{6} - 48 T^{5} + \cdots + 107495424 \)
$19$
\( T^{6} - 104 T^{5} + \cdots + 322687697779 \)
$23$
\( T^{6} - 238 T^{5} + \cdots + 37266922552896 \)
$29$
\( T^{6} - 8 T^{5} + \cdots + 93900570624 \)
$31$
\( (T^{3} - 107 T^{2} - 14005 T + 1435247)^{2} \)
$37$
\( (T^{3} - 305 T^{2} - 125173 T + 28900349)^{2} \)
$41$
\( T^{6} + 16 T^{5} + \cdots + 49106682003456 \)
$43$
\( T^{6} - 331 T^{5} + \cdots + 9744392803201 \)
$47$
\( T^{6} + \cdots + 407898773822016 \)
$53$
\( T^{6} - 118 T^{5} + \cdots + 16\!\cdots\!76 \)
$59$
\( T^{6} + 936 T^{5} + \cdots + 10\!\cdots\!44 \)
$61$
\( T^{6} - 399 T^{5} + \cdots + 61\!\cdots\!69 \)
$67$
\( T^{6} + \cdots + 762088088919249 \)
$71$
\( T^{6} + \cdots + 288657109767744 \)
$73$
\( T^{6} + 91 T^{5} + \cdots + 27\!\cdots\!21 \)
$79$
\( T^{6} - 321 T^{5} + \cdots + 40\!\cdots\!29 \)
$83$
\( (T^{3} + 2148 T^{2} + 1271664 T + 211415616)^{2} \)
$89$
\( T^{6} + \cdots + 121838150433024 \)
$97$
\( T^{6} + 1382 T^{5} + \cdots + 62\!\cdots\!44 \)
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