Newspace parameters
| Level: | \( N \) | \(=\) | \( 63 = 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 63.e (of order \(3\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(19.6802566055\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
|
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| Defining polynomial: |
\( x^{8} - x^{7} + 103x^{6} - 378x^{5} + 9744x^{4} - 22680x^{3} + 149400x^{2} + 216000x + 810000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 3\cdot 7^{2} \) |
| Twist minimal: | no (minimal twist has level 7) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} + 103x^{6} - 378x^{5} + 9744x^{4} - 22680x^{3} + 149400x^{2} + 216000x + 810000 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2419 \nu^{7} + 2089856 \nu^{6} + 6596632 \nu^{5} + 191205693 \nu^{4} - 241778964 \nu^{3} + \cdots + 21568329000 ) / 3180326625 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 51287 \nu^{7} - 221882 \nu^{6} + 4986506 \nu^{5} - 35051121 \nu^{4} + 521375988 \nu^{3} + \cdots + 9362007000 ) / 25442613000 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 11627 \nu^{7} + 68647 \nu^{6} - 1542751 \nu^{5} + 10026716 \nu^{4} - 161305998 \nu^{3} + \cdots + 4975101000 ) / 605776500 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 141661 \nu^{7} + 215659 \nu^{6} + 13007813 \nu^{5} - 17965948 \nu^{4} + 1049560974 \nu^{3} + \cdots + 78343411500 ) / 4240435500 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 832833 \nu^{7} + 2650538 \nu^{6} - 59567354 \nu^{5} + 625838339 \nu^{4} + \cdots + 192094254000 ) / 8480871000 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 5145317 \nu^{7} - 23255012 \nu^{6} + 522625796 \nu^{5} - 4123329861 \nu^{4} + \cdots - 1685376396000 ) / 25442613000 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 1093321 \nu^{7} - 961699 \nu^{6} - 100392593 \nu^{5} + 138658828 \nu^{4} + \cdots - 692595526500 ) / 4240435500 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} + \beta_{5} + 5\beta_{3} - 4\beta_{2} + 4 ) / 28 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{4} - 4\beta_{3} - 104\beta_{2} - \beta _1 + 4 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -36\beta_{7} - 29\beta_{5} - 362\beta_{4} - 29\beta_{2} + 29\beta _1 + 1279 ) / 14 \)
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| \(\nu^{4}\) | \(=\) |
\( ( 12\beta_{6} + 115\beta_{5} + 442\beta_{3} + 8607\beta_{2} - 8607 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 3138\beta_{7} - 3138\beta_{6} + 38300\beta_{4} - 38300\beta_{3} - 281950\beta_{2} - 3383\beta _1 + 38300 ) / 14 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -2010\beta_{7} - 11341\beta_{5} - 48424\beta_{4} - 11341\beta_{2} + 11341\beta _1 + 750491 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 293766\beta_{6} + 432779\beta_{5} + 3984392\beta_{3} + 36494421\beta_{2} - 36494421 ) / 14 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).
| \(n\) | \(10\) | \(29\) |
| \(\chi(n)\) | \(-\beta_{2}\) | \(1\) |
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 |
|
−10.4120 | + | 18.0342i | 0 | −152.821 | − | 264.694i | 96.7016 | − | 167.492i | 0 | 612.373 | + | 669.733i | 3699.23 | 0 | 2013.72 | + | 3487.87i | ||||||||||||||||||||||||||||||||
| 37.2 | −3.69038 | + | 6.39193i | 0 | 36.7621 | + | 63.6739i | −145.409 | + | 251.857i | 0 | 358.327 | + | 833.753i | −1487.40 | 0 | −1073.23 | − | 1858.89i | |||||||||||||||||||||||||||||||||
| 37.3 | 1.59276 | − | 2.75874i | 0 | 58.9262 | + | 102.063i | −0.294487 | + | 0.510067i | 0 | 31.8040 | − | 906.935i | 783.169 | 0 | 0.938097 | + | 1.62483i | |||||||||||||||||||||||||||||||||
| 37.4 | 9.50966 | − | 16.4712i | 0 | −116.867 | − | 202.420i | 175.002 | − | 303.113i | 0 | −666.505 | + | 615.885i | −2011.00 | 0 | −3328.43 | − | 5765.00i | |||||||||||||||||||||||||||||||||
| 46.1 | −10.4120 | − | 18.0342i | 0 | −152.821 | + | 264.694i | 96.7016 | + | 167.492i | 0 | 612.373 | − | 669.733i | 3699.23 | 0 | 2013.72 | − | 3487.87i | |||||||||||||||||||||||||||||||||
| 46.2 | −3.69038 | − | 6.39193i | 0 | 36.7621 | − | 63.6739i | −145.409 | − | 251.857i | 0 | 358.327 | − | 833.753i | −1487.40 | 0 | −1073.23 | + | 1858.89i | |||||||||||||||||||||||||||||||||
| 46.3 | 1.59276 | + | 2.75874i | 0 | 58.9262 | − | 102.063i | −0.294487 | − | 0.510067i | 0 | 31.8040 | + | 906.935i | 783.169 | 0 | 0.938097 | − | 1.62483i | |||||||||||||||||||||||||||||||||
| 46.4 | 9.50966 | + | 16.4712i | 0 | −116.867 | + | 202.420i | 175.002 | + | 303.113i | 0 | −666.505 | − | 615.885i | −2011.00 | 0 | −3328.43 | + | 5765.00i | |||||||||||||||||||||||||||||||||
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 | 63.8.e.b | 8 | |
| 3.b | odd | 2 | 1 | 7.8.c.a | ✓ | 8 | |
| 7.c | even | 3 | 1 | inner | 63.8.e.b | 8 | |
| 7.c | even | 3 | 1 | 441.8.a.s | 4 | ||
| 7.d | odd | 6 | 1 | 441.8.a.t | 4 | ||
| 12.b | even | 2 | 1 | 112.8.i.c | 8 | ||
| 21.c | even | 2 | 1 | 49.8.c.g | 8 | ||
| 21.g | even | 6 | 1 | 49.8.a.e | 4 | ||
| 21.g | even | 6 | 1 | 49.8.c.g | 8 | ||
| 21.h | odd | 6 | 1 | 7.8.c.a | ✓ | 8 | |
| 21.h | odd | 6 | 1 | 49.8.a.f | 4 | ||
| 84.n | even | 6 | 1 | 112.8.i.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.8.c.a | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 7.8.c.a | ✓ | 8 | 21.h | odd | 6 | 1 | |
| 49.8.a.e | 4 | 21.g | even | 6 | 1 | ||
| 49.8.a.f | 4 | 21.h | odd | 6 | 1 | ||
| 49.8.c.g | 8 | 21.c | even | 2 | 1 | ||
| 49.8.c.g | 8 | 21.g | even | 6 | 1 | ||
| 63.8.e.b | 8 | 1.a | even | 1 | 1 | trivial | |
| 63.8.e.b | 8 | 7.c | even | 3 | 1 | inner | |
| 112.8.i.c | 8 | 12.b | even | 2 | 1 | ||
| 112.8.i.c | 8 | 84.n | even | 6 | 1 | ||
| 441.8.a.s | 4 | 7.c | even | 3 | 1 | ||
| 441.8.a.t | 4 | 7.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 6 T_{2}^{7} + 448 T_{2}^{6} + 936 T_{2}^{5} + 170656 T_{2}^{4} + 590304 T_{2}^{3} + \cdots + 86713344 \)
acting on \(S_{8}^{\mathrm{new}}(63, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 6 T^{7} + \cdots + 86713344 \)
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| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + \cdots + 134435328890625 \)
|
| $7$ |
\( T^{8} + \cdots + 45\!\cdots\!01 \)
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| $11$ |
\( T^{8} + \cdots + 53\!\cdots\!25 \)
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| $13$ |
\( (T^{4} + \cdots + 16\!\cdots\!76)^{2} \)
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| $17$ |
\( T^{8} + \cdots + 14\!\cdots\!29 \)
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| $19$ |
\( T^{8} + \cdots + 27\!\cdots\!09 \)
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| $23$ |
\( T^{8} + \cdots + 59\!\cdots\!69 \)
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| $29$ |
\( (T^{4} + \cdots - 59\!\cdots\!00)^{2} \)
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| $31$ |
\( T^{8} + \cdots + 18\!\cdots\!61 \)
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| $37$ |
\( T^{8} + \cdots + 15\!\cdots\!49 \)
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| $41$ |
\( (T^{4} + \cdots - 57\!\cdots\!00)^{2} \)
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| $43$ |
\( (T^{4} + \cdots + 32\!\cdots\!56)^{2} \)
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| $47$ |
\( T^{8} + \cdots + 11\!\cdots\!25 \)
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| $53$ |
\( T^{8} + \cdots + 19\!\cdots\!69 \)
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| $59$ |
\( T^{8} + \cdots + 26\!\cdots\!25 \)
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| $61$ |
\( T^{8} + \cdots + 32\!\cdots\!21 \)
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| $67$ |
\( T^{8} + \cdots + 39\!\cdots\!25 \)
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| $71$ |
\( (T^{4} + \cdots - 23\!\cdots\!24)^{2} \)
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| $73$ |
\( T^{8} + \cdots + 80\!\cdots\!69 \)
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| $79$ |
\( T^{8} + \cdots + 31\!\cdots\!29 \)
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| $83$ |
\( (T^{4} + \cdots - 62\!\cdots\!16)^{2} \)
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| $89$ |
\( T^{8} + \cdots + 61\!\cdots\!69 \)
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| $97$ |
\( (T^{4} + \cdots + 42\!\cdots\!36)^{2} \)
|
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