Newspace parameters
| Level: | \( N \) | \(=\) | \( 126 = 2 \cdot 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 126.g (of order \(3\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(39.3605132110\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{2389})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 598x^{2} + 597x + 356409 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 14) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} + 598x^{2} + 597x + 356409 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 598\nu^{2} - 598\nu + 356409 ) / 357006 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 598\nu^{2} + 714610\nu - 356409 ) / 357006 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 896 ) / 299 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 1195\beta _1 - 1195 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 299\beta_{3} - 896 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(73\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 37.1 |
|
4.00000 | − | 6.92820i | 0 | −32.0000 | − | 55.4256i | −108.377 | + | 187.715i | 0 | 726.284 | − | 544.110i | −512.000 | 0 | 867.019 | + | 1501.72i | ||||||||||||||||||||
| 37.2 | 4.00000 | − | 6.92820i | 0 | −32.0000 | − | 55.4256i | −10.6226 | + | 18.3989i | 0 | −642.284 | + | 641.104i | −512.000 | 0 | 84.9808 | + | 147.191i | |||||||||||||||||||||
| 109.1 | 4.00000 | + | 6.92820i | 0 | −32.0000 | + | 55.4256i | −108.377 | − | 187.715i | 0 | 726.284 | + | 544.110i | −512.000 | 0 | 867.019 | − | 1501.72i | |||||||||||||||||||||
| 109.2 | 4.00000 | + | 6.92820i | 0 | −32.0000 | + | 55.4256i | −10.6226 | − | 18.3989i | 0 | −642.284 | − | 641.104i | −512.000 | 0 | 84.9808 | − | 147.191i | |||||||||||||||||||||
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 | 126.8.g.e | 4 | |
| 3.b | odd | 2 | 1 | 14.8.c.a | ✓ | 4 | |
| 7.c | even | 3 | 1 | inner | 126.8.g.e | 4 | |
| 12.b | even | 2 | 1 | 112.8.i.a | 4 | ||
| 21.c | even | 2 | 1 | 98.8.c.h | 4 | ||
| 21.g | even | 6 | 1 | 98.8.a.k | 2 | ||
| 21.g | even | 6 | 1 | 98.8.c.h | 4 | ||
| 21.h | odd | 6 | 1 | 14.8.c.a | ✓ | 4 | |
| 21.h | odd | 6 | 1 | 98.8.a.h | 2 | ||
| 84.n | even | 6 | 1 | 112.8.i.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.8.c.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 14.8.c.a | ✓ | 4 | 21.h | odd | 6 | 1 | |
| 98.8.a.h | 2 | 21.h | odd | 6 | 1 | ||
| 98.8.a.k | 2 | 21.g | even | 6 | 1 | ||
| 98.8.c.h | 4 | 21.c | even | 2 | 1 | ||
| 98.8.c.h | 4 | 21.g | even | 6 | 1 | ||
| 112.8.i.a | 4 | 12.b | even | 2 | 1 | ||
| 112.8.i.a | 4 | 84.n | even | 6 | 1 | ||
| 126.8.g.e | 4 | 1.a | even | 1 | 1 | trivial | |
| 126.8.g.e | 4 | 7.c | even | 3 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 238T_{5}^{3} + 52039T_{5}^{2} + 1095990T_{5} + 21206025 \)
acting on \(S_{8}^{\mathrm{new}}(126, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 8 T + 64)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 238 T^{3} + \cdots + 21206025 \)
|
| $7$ |
\( T^{4} + \cdots + 678223072849 \)
|
| $11$ |
\( T^{4} + \cdots + 71110682271225 \)
|
| $13$ |
\( (T^{2} - 1316 T - 91342860)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 16\!\cdots\!81 \)
|
| $19$ |
\( T^{4} + \cdots + 78\!\cdots\!25 \)
|
| $23$ |
\( T^{4} + \cdots + 12\!\cdots\!69 \)
|
| $29$ |
\( (T^{2} + 172820 T + 5666758164)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 74\!\cdots\!25 \)
|
| $37$ |
\( T^{4} + \cdots + 81\!\cdots\!29 \)
|
| $41$ |
\( (T^{2} - 1156260 T + 332194626036)^{2} \)
|
| $43$ |
\( (T^{2} - 57544 T - 154524293360)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 62\!\cdots\!81 \)
|
| $53$ |
\( T^{4} + \cdots + 19\!\cdots\!61 \)
|
| $59$ |
\( T^{4} + \cdots + 73\!\cdots\!29 \)
|
| $61$ |
\( T^{4} + \cdots + 32\!\cdots\!81 \)
|
| $67$ |
\( T^{4} + \cdots + 16\!\cdots\!61 \)
|
| $71$ |
\( (T^{2} + \cdots + 2760738723264)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 28\!\cdots\!25 \)
|
| $79$ |
\( T^{4} + \cdots + 43\!\cdots\!25 \)
|
| $83$ |
\( (T^{2} + \cdots - 11352739197744)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 24\!\cdots\!21 \)
|
| $97$ |
\( (T^{2} + \cdots + 6136617007220)^{2} \)
|
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