Newspace parameters
| Level: | \( N \) | \(=\) | \( 45 = 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 26 \) |
| Character orbit: | \([\chi]\) | \(=\) | 45.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(178.198550979\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{8} - x^{7} - 161639393 x^{6} + 22143624127 x^{5} + \cdots + 14\!\cdots\!06 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{21}\cdot 3^{24}\cdot 5^{9} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} - 161639393 x^{6} + 22143624127 x^{5} + \cdots + 14\!\cdots\!06 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 204\nu - 40409874 \)
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| \(\beta_{3}\) | \(=\) |
\( ( - 3045811 \nu^{7} + 10319683842 \nu^{6} + 311865318487145 \nu^{5} + \cdots - 63\!\cdots\!10 ) / 13\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 3045811 \nu^{7} + 10319683842 \nu^{6} + 311865318487145 \nu^{5} + \cdots - 88\!\cdots\!10 ) / 13\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 123079621 \nu^{7} + 132193201698 \nu^{6} + \cdots + 39\!\cdots\!50 ) / 11\!\cdots\!00 \)
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| \(\beta_{6}\) | \(=\) |
\( ( - 2141337497 \nu^{7} + 3343176919350 \nu^{6} + \cdots + 43\!\cdots\!54 ) / 11\!\cdots\!00 \)
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| \(\beta_{7}\) | \(=\) |
\( ( - 55677799 \nu^{7} - 458736969654 \nu^{6} + \cdots + 77\!\cdots\!62 ) / 11\!\cdots\!00 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 204\beta _1 + 40409874 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - \beta_{3} - 639\beta_{2} + 66255711\beta _1 - 8251526319 \)
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| \(\nu^{4}\) | \(=\) |
\( 42 \beta_{7} + 4 \beta_{6} + 78 \beta_{5} + 667 \beta_{4} - 5803 \beta_{3} + 85636466 \beta_{2} + \cdots + 26\!\cdots\!39 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 186976 \beta_{7} + 128 \beta_{6} - 219488 \beta_{5} + 105714476 \beta_{4} - 172211756 \beta_{3} + \cdots - 15\!\cdots\!46 \)
|
| \(\nu^{6}\) | \(=\) |
\( 3687575768 \beta_{7} + 571749296 \beta_{6} + 10539254664 \beta_{5} + 79791646302 \beta_{4} + \cdots + 19\!\cdots\!44 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 32256558765404 \beta_{7} - 488373019288 \beta_{6} - 34318849912372 \beta_{5} + \cdots - 16\!\cdots\!29 \)
|
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 |
|
−8163.44 | 0 | 3.30873e7 | 2.44141e8 | 0 | −2.33934e9 | 3.81331e9 | 0 | −1.99303e12 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −6708.21 | 0 | 1.14457e7 | 2.44141e8 | 0 | 6.86843e10 | 1.48310e11 | 0 | −1.63775e12 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2543.03 | 0 | −2.70874e7 | 2.44141e8 | 0 | −5.02683e10 | 1.54214e11 | 0 | −6.20858e11 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −551.567 | 0 | −3.32502e7 | 2.44141e8 | 0 | 1.16324e10 | 3.68472e10 | 0 | −1.34660e11 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2129.51 | 0 | −2.90196e7 | 2.44141e8 | 0 | 4.87474e10 | −1.33252e11 | 0 | 5.19899e11 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 6089.22 | 0 | 3.52412e6 | 2.44141e8 | 0 | −8.69204e9 | −1.82861e11 | 0 | 1.48662e12 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 8291.06 | 0 | 3.51873e7 | 2.44141e8 | 0 | −6.83602e10 | 1.35379e10 | 0 | 2.02418e12 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 10201.5 | 0 | 7.05156e7 | 2.44141e8 | 0 | 3.94224e10 | 3.77058e11 | 0 | 2.49059e12 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( -1 \) |
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 | 45.26.a.h | yes | 8 |
| 3.b | odd | 2 | 1 | 45.26.a.g | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 45.26.a.g | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 45.26.a.h | yes | 8 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 8745 T_{2}^{7} - 128181570 T_{2}^{6} + 1009027599800 T_{2}^{5} + \cdots + 84\!\cdots\!00 \)
acting on \(S_{26}^{\mathrm{new}}(\Gamma_0(45))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + \cdots + 84\!\cdots\!00 \)
|
| $3$ |
\( T^{8} \)
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| $5$ |
\( (T - 244140625)^{8} \)
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| $7$ |
\( T^{8} + \cdots + 10\!\cdots\!00 \)
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| $11$ |
\( T^{8} + \cdots - 21\!\cdots\!00 \)
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| $13$ |
\( T^{8} + \cdots + 42\!\cdots\!00 \)
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| $17$ |
\( T^{8} + \cdots - 13\!\cdots\!00 \)
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| $19$ |
\( T^{8} + \cdots + 26\!\cdots\!16 \)
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| $23$ |
\( T^{8} + \cdots - 58\!\cdots\!00 \)
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| $29$ |
\( T^{8} + \cdots + 68\!\cdots\!00 \)
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| $31$ |
\( T^{8} + \cdots - 33\!\cdots\!04 \)
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| $37$ |
\( T^{8} + \cdots - 25\!\cdots\!00 \)
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| $41$ |
\( T^{8} + \cdots + 28\!\cdots\!00 \)
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| $43$ |
\( T^{8} + \cdots - 87\!\cdots\!00 \)
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| $47$ |
\( T^{8} + \cdots + 48\!\cdots\!00 \)
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| $53$ |
\( T^{8} + \cdots - 63\!\cdots\!00 \)
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| $59$ |
\( T^{8} + \cdots - 18\!\cdots\!00 \)
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| $61$ |
\( T^{8} + \cdots + 21\!\cdots\!16 \)
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| $67$ |
\( T^{8} + \cdots + 19\!\cdots\!00 \)
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| $71$ |
\( T^{8} + \cdots + 14\!\cdots\!00 \)
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| $73$ |
\( T^{8} + \cdots - 50\!\cdots\!00 \)
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| $79$ |
\( T^{8} + \cdots - 70\!\cdots\!84 \)
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| $83$ |
\( T^{8} + \cdots + 67\!\cdots\!00 \)
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| $89$ |
\( T^{8} + \cdots + 33\!\cdots\!00 \)
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| $97$ |
\( T^{8} + \cdots + 54\!\cdots\!00 \)
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