Newspace parameters
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 18 \) |
| Character orbit: | \([\chi]\) | \(=\) | 75.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(137.416565508\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{8} + 963373x^{6} + 304121840676x^{4} + 35407691713753600x^{2} + 1297706673500416000000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{14}\cdot 3^{8}\cdot 5^{4} \) |
| Twist minimal: | no (minimal twist has level 15) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{8} + 963373x^{6} + 304121840676x^{4} + 35407691713753600x^{2} + 1297706673500416000000 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{6} + 25559667\nu^{4} + 12507705423804\nu^{2} + 938105819924384000 ) / 690731897492160 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{6} + 76679001\nu^{4} + 63105779141492\nu^{2} + 8976259969312101120 ) / 25582662870080 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 13103197 \nu^{6} - 10452596790681 \nu^{4} + \cdots - 14\!\cdots\!20 ) / 25\!\cdots\!20 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 13020677 \nu^{7} + 12561780511521 \nu^{5} + \cdots + 23\!\cdots\!00 \nu ) / 12\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 98569506029 \nu^{7} + \cdots - 12\!\cdots\!00 \nu ) / 38\!\cdots\!00 \)
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| \(\beta_{7}\) | \(=\) |
\( ( 10648742879101 \nu^{7} + \cdots - 32\!\cdots\!00 \nu ) / 28\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 81\beta_{2} - 240864 \)
|
| \(\nu^{3}\) | \(=\) |
\( -74\beta_{7} - 189\beta_{6} - 19614080\beta_{5} - 331955\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -74\beta_{4} - 481497\beta_{3} + 65207651\beta_{2} + 79977506624 \)
|
| \(\nu^{5}\) | \(=\) |
\( 35644838\beta_{7} + 117080197\beta_{6} + 15756295738816\beta_{5} + 125957538719\beta_1 \)
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| \(\nu^{6}\) | \(=\) |
\( -1891415358\beta_{4} + 200802442305\beta_{3} - 37170191408067\beta_{2} - 30351702475008448 \)
|
| \(\nu^{7}\) | \(=\) |
\( -14500011812550\beta_{7} - 62157374976717\beta_{6} - 8973910318821252288\beta_{5} - 50406071934091207\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).
| \(n\) | \(26\) | \(52\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
− | 654.567i | − | 6561.00i | −297386. | 0 | −4.29461e6 | − | 359681.i | 1.08863e8i | −4.30467e7 | 0 | |||||||||||||||||||||||||||||||||||||||
| 49.2 | − | 577.922i | 6561.00i | −202922. | 0 | 3.79175e6 | 2.79117e7i | 4.15238e7i | −4.30467e7 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 49.3 | − | 367.218i | 6561.00i | −3777.06 | 0 | 2.40932e6 | − | 1.91248e7i | − | 4.67450e7i | −4.30467e7 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 49.4 | − | 257.574i | − | 6561.00i | 64727.8 | 0 | −1.68994e6 | − | 8.43659e6i | − | 5.04329e7i | −4.30467e7 | 0 | |||||||||||||||||||||||||||||||||||||||
| 49.5 | 257.574i | 6561.00i | 64727.8 | 0 | −1.68994e6 | 8.43659e6i | 5.04329e7i | −4.30467e7 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 49.6 | 367.218i | − | 6561.00i | −3777.06 | 0 | 2.40932e6 | 1.91248e7i | 4.67450e7i | −4.30467e7 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 49.7 | 577.922i | − | 6561.00i | −202922. | 0 | 3.79175e6 | − | 2.79117e7i | − | 4.15238e7i | −4.30467e7 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 49.8 | 654.567i | 6561.00i | −297386. | 0 | −4.29461e6 | 359681.i | − | 1.08863e8i | −4.30467e7 | 0 | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 75.18.b.f | 8 | |
| 5.b | even | 2 | 1 | inner | 75.18.b.f | 8 | |
| 5.c | odd | 4 | 1 | 15.18.a.d | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 75.18.a.f | 4 | ||
| 15.e | even | 4 | 1 | 45.18.a.f | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.18.a.d | ✓ | 4 | 5.c | odd | 4 | 1 | |
| 45.18.a.f | 4 | 15.e | even | 4 | 1 | ||
| 75.18.a.f | 4 | 5.c | odd | 4 | 1 | ||
| 75.18.b.f | 8 | 1.a | even | 1 | 1 | trivial | |
| 75.18.b.f | 8 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 963645T_{2}^{6} + 305448923268T_{2}^{4} + 35612456938055680T_{2}^{2} + 1280257661232912531456 \)
acting on \(S_{18}^{\mathrm{new}}(75, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + \cdots + 12\!\cdots\!56 \)
|
| $3$ |
\( (T^{2} + 43046721)^{4} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + \cdots + 26\!\cdots\!00 \)
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| $11$ |
\( (T^{4} + \cdots + 10\!\cdots\!84)^{2} \)
|
| $13$ |
\( T^{8} + \cdots + 17\!\cdots\!96 \)
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| $17$ |
\( T^{8} + \cdots + 66\!\cdots\!96 \)
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| $19$ |
\( (T^{4} + \cdots - 60\!\cdots\!00)^{2} \)
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| $23$ |
\( T^{8} + \cdots + 12\!\cdots\!00 \)
|
| $29$ |
\( (T^{4} + \cdots + 16\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{4} + \cdots + 41\!\cdots\!00)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 99\!\cdots\!00 \)
|
| $41$ |
\( (T^{4} + \cdots - 79\!\cdots\!00)^{2} \)
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| $43$ |
\( T^{8} + \cdots + 18\!\cdots\!96 \)
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| $47$ |
\( T^{8} + \cdots + 19\!\cdots\!16 \)
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| $53$ |
\( T^{8} + \cdots + 21\!\cdots\!00 \)
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| $59$ |
\( (T^{4} + \cdots - 16\!\cdots\!00)^{2} \)
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| $61$ |
\( (T^{4} + \cdots - 43\!\cdots\!44)^{2} \)
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| $67$ |
\( T^{8} + \cdots + 10\!\cdots\!96 \)
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| $71$ |
\( (T^{4} + \cdots - 72\!\cdots\!64)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 61\!\cdots\!00 \)
|
| $79$ |
\( (T^{4} + \cdots - 54\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 22\!\cdots\!76 \)
|
| $89$ |
\( (T^{4} + \cdots + 26\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 63\!\cdots\!16 \)
|
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