Newspace parameters
| Level: | \( N \) | \(=\) | \( 121 = 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 121.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(37.7985880836\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{6} - 742x^{4} + 146056x^{2} - 6022080 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3 \) |
| 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
\( x^{6} - 742x^{4} + 146056x^{2} - 6022080 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 422\nu^{3} + 10200\nu ) / 408 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 422\nu^{2} + 18360 ) / 408 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{4} + 626\nu^{2} - 68748 ) / 204 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} + 626\nu^{3} - 81396\nu ) / 204 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + 2\beta_{3} + 247 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 2\beta_{2} + 349\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 422\beta_{4} + 1252\beta_{3} + 85874 \)
|
| \(\nu^{5}\) | \(=\) |
\( 422\beta_{5} + 1252\beta_{2} + 137078\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−21.0469 | 64.7680 | 314.971 | 316.405 | −1363.16 | 703.693 | −3935.14 | 2007.89 | −6659.34 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −15.5920 | −64.5932 | 115.109 | −389.261 | 1007.13 | 1439.06 | 200.996 | 1985.28 | 6069.34 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −7.47800 | −8.17483 | −72.0796 | 192.856 | 61.1314 | −1553.40 | 1496.19 | −2120.17 | −1442.17 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 7.47800 | −8.17483 | −72.0796 | 192.856 | −61.1314 | 1553.40 | −1496.19 | −2120.17 | 1442.17 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 15.5920 | −64.5932 | 115.109 | −389.261 | −1007.13 | −1439.06 | −200.996 | 1985.28 | −6069.34 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 21.0469 | 64.7680 | 314.971 | 316.405 | 1363.16 | −703.693 | 3935.14 | 2007.89 | 6659.34 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 121.8.a.f | ✓ | 6 |
| 11.b | odd | 2 | 1 | inner | 121.8.a.f | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 121.8.a.f | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 121.8.a.f | ✓ | 6 | 11.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 742T_{2}^{4} + 146056T_{2}^{2} - 6022080 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(121))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 742 T^{4} + \cdots - 6022080 \)
|
| $3$ |
\( (T^{3} + 8 T^{2} + \cdots - 34200)^{2} \)
|
| $5$ |
\( (T^{3} - 120 T^{2} + \cdots + 23752950)^{2} \)
|
| $7$ |
\( T^{6} + \cdots - 24\!\cdots\!80 \)
|
| $11$ |
\( T^{6} \)
|
| $13$ |
\( T^{6} + \cdots - 65\!\cdots\!80 \)
|
| $17$ |
\( T^{6} + \cdots - 12\!\cdots\!00 \)
|
| $19$ |
\( T^{6} + \cdots - 31\!\cdots\!00 \)
|
| $23$ |
\( (T^{3} + \cdots - 27592496030700)^{2} \)
|
| $29$ |
\( T^{6} + \cdots - 43\!\cdots\!00 \)
|
| $31$ |
\( (T^{3} + \cdots - 19\!\cdots\!00)^{2} \)
|
| $37$ |
\( (T^{3} + \cdots - 58\!\cdots\!50)^{2} \)
|
| $41$ |
\( T^{6} + \cdots - 19\!\cdots\!00 \)
|
| $43$ |
\( T^{6} + \cdots - 52\!\cdots\!20 \)
|
| $47$ |
\( (T^{3} + \cdots + 24\!\cdots\!00)^{2} \)
|
| $53$ |
\( (T^{3} + \cdots - 25\!\cdots\!00)^{2} \)
|
| $59$ |
\( (T^{3} + \cdots - 34\!\cdots\!60)^{2} \)
|
| $61$ |
\( T^{6} + \cdots - 80\!\cdots\!00 \)
|
| $67$ |
\( (T^{3} + \cdots - 63\!\cdots\!00)^{2} \)
|
| $71$ |
\( (T^{3} + \cdots + 22\!\cdots\!32)^{2} \)
|
| $73$ |
\( T^{6} + \cdots - 44\!\cdots\!80 \)
|
| $79$ |
\( T^{6} + \cdots - 43\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots - 66\!\cdots\!20 \)
|
| $89$ |
\( (T^{3} + \cdots - 23\!\cdots\!90)^{2} \)
|
| $97$ |
\( (T^{3} + \cdots + 85\!\cdots\!50)^{2} \)
|
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