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: |
\( x^{6} + 364793x^{4} + 33276143056x^{2} + 15375182054400 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{2}\cdot 5^{2} \) |
| 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
\( x^{6} + 364793x^{4} + 33276143056x^{2} + 15375182054400 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} + 182397\nu^{2} + 3921120 ) / 3738724 \)
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| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} + 3556327\nu^{3} + 681924381584\nu ) / 14659985450880 \)
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| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + 303001\nu^{2} + 14670332352 ) / 120604 \)
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| \(\beta_{5}\) | \(=\) |
\( ( 52771\nu^{5} + 15939976923\nu^{3} + 1145671886510336\nu ) / 203610909040 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - 31\beta_{2} - 121608 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 3799512\beta_{3} - 182365\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( -182397\beta_{4} + 9393031\beta_{2} + 22177013256 \)
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| \(\nu^{5}\) | \(=\) |
\( 3556327\beta_{5} - 1147678338456\beta_{3} + 33374808229\beta_1 \)
|
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 |
|
− | 500.408i | − | 6561.00i | −119336. | 0 | −3.28318e6 | 8.90512e6i | − | 5.87255e6i | −4.30467e7 | 0 | |||||||||||||||||||||||||||||||||
| 49.2 | − | 352.958i | 6561.00i | 6492.31 | 0 | 2.31576e6 | − | 1.07009e7i | − | 4.85545e7i | −4.30467e7 | 0 | ||||||||||||||||||||||||||||||||||
| 49.3 | − | 105.550i | − | 6561.00i | 119931. | 0 | −692515. | − | 2.39385e7i | − | 2.64934e7i | −4.30467e7 | 0 | |||||||||||||||||||||||||||||||||
| 49.4 | 105.550i | 6561.00i | 119931. | 0 | −692515. | 2.39385e7i | 2.64934e7i | −4.30467e7 | 0 | |||||||||||||||||||||||||||||||||||||
| 49.5 | 352.958i | − | 6561.00i | 6492.31 | 0 | 2.31576e6 | 1.07009e7i | 4.85545e7i | −4.30467e7 | 0 | ||||||||||||||||||||||||||||||||||||
| 49.6 | 500.408i | 6561.00i | −119336. | 0 | −3.28318e6 | − | 8.90512e6i | 5.87255e6i | −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.e | 6 | |
| 5.b | even | 2 | 1 | inner | 75.18.b.e | 6 | |
| 5.c | odd | 4 | 1 | 15.18.a.c | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 75.18.a.d | 3 | ||
| 15.e | even | 4 | 1 | 45.18.a.d | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.18.a.c | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 45.18.a.d | 3 | 15.e | even | 4 | 1 | ||
| 75.18.a.d | 3 | 5.c | odd | 4 | 1 | ||
| 75.18.b.e | 6 | 1.a | even | 1 | 1 | trivial | |
| 75.18.b.e | 6 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 386129T_{2}^{4} + 35373491344T_{2}^{2} + 347547429605376 \)
acting on \(S_{18}^{\mathrm{new}}(75, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots + 347547429605376 \)
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| $3$ |
\( (T^{2} + 43046721)^{3} \)
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| $5$ |
\( T^{6} \)
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| $7$ |
\( T^{6} + \cdots + 52\!\cdots\!00 \)
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| $11$ |
\( (T^{3} + \cdots + 22\!\cdots\!36)^{2} \)
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| $13$ |
\( T^{6} + \cdots + 39\!\cdots\!84 \)
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| $17$ |
\( T^{6} + \cdots + 16\!\cdots\!84 \)
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| $19$ |
\( (T^{3} + \cdots + 15\!\cdots\!00)^{2} \)
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| $23$ |
\( T^{6} + \cdots + 10\!\cdots\!00 \)
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| $29$ |
\( (T^{3} + \cdots + 45\!\cdots\!00)^{2} \)
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| $31$ |
\( (T^{3} + \cdots + 58\!\cdots\!00)^{2} \)
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| $37$ |
\( T^{6} + \cdots + 70\!\cdots\!00 \)
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| $41$ |
\( (T^{3} + \cdots - 60\!\cdots\!60)^{2} \)
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| $43$ |
\( T^{6} + \cdots + 55\!\cdots\!96 \)
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| $47$ |
\( T^{6} + \cdots + 17\!\cdots\!64 \)
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| $53$ |
\( T^{6} + \cdots + 21\!\cdots\!00 \)
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| $59$ |
\( (T^{3} + \cdots - 76\!\cdots\!00)^{2} \)
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| $61$ |
\( (T^{3} + \cdots + 39\!\cdots\!68)^{2} \)
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| $67$ |
\( T^{6} + \cdots + 63\!\cdots\!24 \)
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| $71$ |
\( (T^{3} + \cdots - 71\!\cdots\!08)^{2} \)
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| $73$ |
\( T^{6} + \cdots + 56\!\cdots\!00 \)
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| $79$ |
\( (T^{3} + \cdots + 77\!\cdots\!00)^{2} \)
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| $83$ |
\( T^{6} + \cdots + 42\!\cdots\!04 \)
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| $89$ |
\( (T^{3} + \cdots + 77\!\cdots\!00)^{2} \)
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| $97$ |
\( T^{6} + \cdots + 17\!\cdots\!84 \)
|
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