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: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
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| Defining polynomial: |
\( x^{6} - 175350x^{3} + 346592689x^{2} - 3264490950x + 15373811250 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{14}\cdot 3^{2}\cdot 5^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 175350x^{3} + 346592689x^{2} - 3264490950x + 15373811250 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 346592689 \nu^{5} + 1632245475 \nu^{4} + 7686905625 \nu^{3} - 30387514008075 \nu^{2} + \cdots - 56\!\cdots\!50 ) / 56\!\cdots\!00 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -78446\nu^{5} - 43641908\nu^{4} - 1460429182\nu^{3} + 6877753050\nu^{2} - 9737024196859537 ) / 16514874787813 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 8503947783851 \nu^{5} - 40048537463025 \nu^{4} + \cdots + 13\!\cdots\!50 ) / 23\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 18113704 \nu^{5} - 2554316138 \nu^{4} + 337222827368 \nu^{3} - 1588118998200 \nu^{2} - 66\!\cdots\!17 ) / 16514874787813 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 183715870589297 \nu^{5} - 865192509744675 \nu^{4} + \cdots + 31\!\cdots\!50 ) / 77\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{5} - \beta_{4} + 61\beta_{3} + 61\beta_{2} - 20\beta _1 - 20 ) / 960 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 29\beta_{5} + 8071\beta_{3} + 5954740\beta_1 ) / 480 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -18617\beta_{5} + 18617\beta_{4} + 1135637\beta_{3} - 1135637\beta_{2} - 84540340\beta _1 + 84540340 ) / 960 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -39223\beta_{4} - 9056852\beta_{2} - 6928821755 ) / 30 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 351677839 \beta_{5} + 351677839 \beta_{4} - 19726904179 \beta_{3} - 19726904179 \beta_{2} + \cdots + 2618051168780 ) / 960 \)
|
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 |
|
− | 670.613i | 6561.00i | −318650. | 0 | 4.39989e6 | − | 2.70804e7i | 1.25792e8i | −4.30467e7 | 0 | ||||||||||||||||||||||||||||||||||
| 49.2 | − | 442.567i | − | 6561.00i | −64793.8 | 0 | −2.90368e6 | − | 2.47993e7i | − | 2.93326e7i | −4.30467e7 | 0 | |||||||||||||||||||||||||||||||||
| 49.3 | − | 213.954i | 6561.00i | 85295.6 | 0 | 1.40375e6 | 7.24373e6i | − | 4.62928e7i | −4.30467e7 | 0 | |||||||||||||||||||||||||||||||||||
| 49.4 | 213.954i | − | 6561.00i | 85295.6 | 0 | 1.40375e6 | − | 7.24373e6i | 4.62928e7i | −4.30467e7 | 0 | |||||||||||||||||||||||||||||||||||
| 49.5 | 442.567i | 6561.00i | −64793.8 | 0 | −2.90368e6 | 2.47993e7i | 2.93326e7i | −4.30467e7 | 0 | |||||||||||||||||||||||||||||||||||||
| 49.6 | 670.613i | − | 6561.00i | −318650. | 0 | 4.39989e6 | 2.70804e7i | − | 1.25792e8i | −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.d | 6 | |
| 5.b | even | 2 | 1 | inner | 75.18.b.d | 6 | |
| 5.c | odd | 4 | 1 | 15.18.a.b | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 75.18.a.e | 3 | ||
| 15.e | even | 4 | 1 | 45.18.a.e | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.18.a.b | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 45.18.a.e | 3 | 15.e | even | 4 | 1 | ||
| 75.18.a.e | 3 | 5.c | odd | 4 | 1 | ||
| 75.18.b.d | 6 | 1.a | even | 1 | 1 | trivial | |
| 75.18.b.d | 6 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 691364T_{2}^{4} + 117637801984T_{2}^{2} + 4032221552050176 \)
acting on \(S_{18}^{\mathrm{new}}(75, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots + 40\!\cdots\!76 \)
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| $3$ |
\( (T^{2} + 43046721)^{3} \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + \cdots + 23\!\cdots\!00 \)
|
| $11$ |
\( (T^{3} + \cdots + 32\!\cdots\!96)^{2} \)
|
| $13$ |
\( T^{6} + \cdots + 23\!\cdots\!64 \)
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| $17$ |
\( T^{6} + \cdots + 38\!\cdots\!64 \)
|
| $19$ |
\( (T^{3} + \cdots - 30\!\cdots\!00)^{2} \)
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| $23$ |
\( T^{6} + \cdots + 60\!\cdots\!00 \)
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| $29$ |
\( (T^{3} + \cdots - 64\!\cdots\!00)^{2} \)
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| $31$ |
\( (T^{3} + \cdots + 53\!\cdots\!00)^{2} \)
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| $37$ |
\( T^{6} + \cdots + 30\!\cdots\!00 \)
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| $41$ |
\( (T^{3} + \cdots - 69\!\cdots\!80)^{2} \)
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| $43$ |
\( T^{6} + \cdots + 17\!\cdots\!36 \)
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| $47$ |
\( T^{6} + \cdots + 30\!\cdots\!64 \)
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| $53$ |
\( T^{6} + \cdots + 94\!\cdots\!00 \)
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| $59$ |
\( (T^{3} + \cdots - 38\!\cdots\!00)^{2} \)
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| $61$ |
\( (T^{3} + \cdots - 55\!\cdots\!12)^{2} \)
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| $67$ |
\( T^{6} + \cdots + 94\!\cdots\!64 \)
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| $71$ |
\( (T^{3} + \cdots + 11\!\cdots\!92)^{2} \)
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| $73$ |
\( T^{6} + \cdots + 32\!\cdots\!00 \)
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| $79$ |
\( (T^{3} + \cdots + 87\!\cdots\!00)^{2} \)
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| $83$ |
\( T^{6} + \cdots + 17\!\cdots\!04 \)
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| $89$ |
\( (T^{3} + \cdots + 28\!\cdots\!00)^{2} \)
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| $97$ |
\( T^{6} + \cdots + 13\!\cdots\!84 \)
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