Newspace parameters
| Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 100 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(558.609014683\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{8} - x^{7} + \cdots + 23\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | multiple of \( 2^{109}\cdot 3^{62}\cdot 5^{13}\cdot 7^{9}\cdot 11^{3}\cdot 13\cdot 17 \) |
| Twist minimal: | no (minimal twist has level 1) |
| 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
\( x^{8} - x^{7} + \cdots + 23\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( 72\nu - 9 \)
|
| \(\beta_{2}\) | \(=\) |
\( 5184\nu^{2} - 8600749240614624\nu - 992470782456186611960041188444 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 13\!\cdots\!23 \nu^{7} + \cdots + 73\!\cdots\!60 ) / 59\!\cdots\!92 \)
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| \(\beta_{4}\) | \(=\) |
\( ( - 17\!\cdots\!21 \nu^{7} + \cdots - 50\!\cdots\!00 ) / 74\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 25\!\cdots\!37 \nu^{7} + \cdots + 88\!\cdots\!00 ) / 39\!\cdots\!00 \)
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| \(\beta_{6}\) | \(=\) |
\( ( - 27\!\cdots\!83 \nu^{7} + \cdots - 59\!\cdots\!00 ) / 52\!\cdots\!00 \)
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| \(\beta_{7}\) | \(=\) |
\( ( - 16\!\cdots\!71 \nu^{7} + \cdots - 33\!\cdots\!80 ) / 14\!\cdots\!80 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 9 ) / 72 \)
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| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 119454850564092\beta _1 + 992470782456187687053696265272 ) / 5184 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{7} - 117 \beta_{6} + 280 \beta_{5} - 3005125991 \beta_{4} + \cdots + 11\!\cdots\!36 ) / 373248 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 10509299761779 \beta_{7} + \cdots + 21\!\cdots\!96 ) / 3359232 \)
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| \(\nu^{5}\) | \(=\) |
\( ( 21\!\cdots\!39 \beta_{7} + \cdots + 25\!\cdots\!28 ) / 15116544 \)
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| \(\nu^{6}\) | \(=\) |
\( ( - 46\!\cdots\!91 \beta_{7} + \cdots + 11\!\cdots\!36 ) / 45349632 \)
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| \(\nu^{7}\) | \(=\) |
\( ( 55\!\cdots\!77 \beta_{7} + \cdots + 61\!\cdots\!92 ) / 816293376 \)
|
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 |
|
−1.53045e15 | 0 | 1.70845e30 | 5.47796e34 | 0 | −5.03791e41 | −1.64466e45 | 0 | −8.38375e49 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.10299e15 | 0 | 5.82752e29 | −3.48045e34 | 0 | 4.01523e41 | 5.63335e43 | 0 | 3.83889e49 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −5.86024e14 | 0 | −2.90401e29 | −2.74639e34 | 0 | 3.47138e41 | 5.41619e44 | 0 | 1.60945e49 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.86466e14 | 0 | −5.51763e29 | 4.79915e34 | 0 | −6.92478e41 | 3.39630e44 | 0 | −1.37479e49 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 6.35192e14 | 0 | −2.30357e29 | −4.78100e33 | 0 | 7.17401e41 | −5.48921e44 | 0 | −3.03685e48 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 7.31556e14 | 0 | −9.86512e28 | −5.46163e34 | 0 | −9.42405e41 | −5.35848e44 | 0 | −3.99548e49 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 8.07907e14 | 0 | 1.88877e28 | 5.34820e34 | 0 | 1.04094e42 | −4.96812e44 | 0 | 4.32085e49 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.53931e15 | 0 | 1.73565e30 | 1.42424e34 | 0 | −4.25157e41 | 1.69606e45 | 0 | 2.19235e49 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
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 | 9.100.a.d | 8 | |
| 3.b | odd | 2 | 1 | 1.100.a.a | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.100.a.a | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 9.100.a.d | 8 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 208040616902520 T_{2}^{7} + \cdots + 16\!\cdots\!36 \)
acting on \(S_{100}^{\mathrm{new}}(\Gamma_0(9))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + \cdots + 16\!\cdots\!36 \)
|
| $3$ |
\( T^{8} \)
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| $5$ |
\( T^{8} + \cdots + 49\!\cdots\!00 \)
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| $7$ |
\( T^{8} + \cdots + 14\!\cdots\!16 \)
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| $11$ |
\( T^{8} + \cdots - 33\!\cdots\!84 \)
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| $13$ |
\( T^{8} + \cdots + 11\!\cdots\!16 \)
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| $17$ |
\( T^{8} + \cdots + 68\!\cdots\!76 \)
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| $19$ |
\( T^{8} + \cdots + 20\!\cdots\!00 \)
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| $23$ |
\( T^{8} + \cdots - 51\!\cdots\!84 \)
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| $29$ |
\( T^{8} + \cdots + 25\!\cdots\!00 \)
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| $31$ |
\( T^{8} + \cdots - 29\!\cdots\!64 \)
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| $37$ |
\( T^{8} + \cdots + 51\!\cdots\!96 \)
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| $41$ |
\( T^{8} + \cdots + 11\!\cdots\!96 \)
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| $43$ |
\( T^{8} + \cdots - 11\!\cdots\!84 \)
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| $47$ |
\( T^{8} + \cdots - 29\!\cdots\!44 \)
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| $53$ |
\( T^{8} + \cdots + 62\!\cdots\!16 \)
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| $59$ |
\( T^{8} + \cdots - 64\!\cdots\!00 \)
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| $61$ |
\( T^{8} + \cdots - 56\!\cdots\!84 \)
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| $67$ |
\( T^{8} + \cdots + 23\!\cdots\!76 \)
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| $71$ |
\( T^{8} + \cdots - 85\!\cdots\!24 \)
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| $73$ |
\( T^{8} + \cdots + 73\!\cdots\!16 \)
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| $79$ |
\( T^{8} + \cdots + 11\!\cdots\!00 \)
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| $83$ |
\( T^{8} + \cdots + 32\!\cdots\!16 \)
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| $89$ |
\( T^{8} + \cdots + 17\!\cdots\!00 \)
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| $97$ |
\( T^{8} + \cdots - 27\!\cdots\!44 \)
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