Newspace parameters
| Level: | \( N \) | \(=\) | \( 275 = 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 275.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(44.1055504486\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.30597006400.1 |
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|
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| Defining polynomial: |
\( x^{6} + 61x^{4} + 980x^{2} + 1600 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 55) |
| 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
\( x^{6} + 61x^{4} + 980x^{2} + 1600 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 3\nu^{5} + 63\nu^{3} - 380\nu ) / 400 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} + 21\nu^{2} - 160 ) / 20 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -7\nu^{5} - 147\nu^{3} + 2220\nu ) / 400 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3\nu^{5} + 113\nu^{3} + 670\nu ) / 100 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{4} + 67\nu^{2} + 190 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( ( 3\beta_{3} + 7\beta_1 ) / 10 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} - 8\beta_{2} - 102 ) / 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 20\beta_{4} - 63\beta_{3} - 227\beta_1 ) / 10 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -21\beta_{5} + 268\beta_{2} + 2942 ) / 5 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -420\beta_{4} + 1703\beta_{3} + 6987\beta_1 ) / 10 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(177\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 199.1 |
|
− | 8.45512i | 26.7755i | −39.4891 | 0 | 226.390 | 22.2927i | 63.3212i | −473.926 | 0 | |||||||||||||||||||||||||||||||||||
| 199.2 | − | 6.40437i | − | 15.0652i | −9.01590 | 0 | −96.4830 | 122.399i | − | 147.199i | 16.0398 | 0 | ||||||||||||||||||||||||||||||||||
| 199.3 | − | 4.94924i | − | 5.84068i | 7.50499 | 0 | −28.9069 | − | 1.89401i | − | 195.520i | 208.886 | 0 | |||||||||||||||||||||||||||||||||
| 199.4 | 4.94924i | 5.84068i | 7.50499 | 0 | −28.9069 | 1.89401i | 195.520i | 208.886 | 0 | |||||||||||||||||||||||||||||||||||||
| 199.5 | 6.40437i | 15.0652i | −9.01590 | 0 | −96.4830 | − | 122.399i | 147.199i | 16.0398 | 0 | ||||||||||||||||||||||||||||||||||||
| 199.6 | 8.45512i | − | 26.7755i | −39.4891 | 0 | 226.390 | − | 22.2927i | − | 63.3212i | −473.926 | 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 | 275.6.b.c | 6 | |
| 5.b | even | 2 | 1 | inner | 275.6.b.c | 6 | |
| 5.c | odd | 4 | 1 | 55.6.a.a | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 275.6.a.c | 3 | ||
| 15.e | even | 4 | 1 | 495.6.a.f | 3 | ||
| 20.e | even | 4 | 1 | 880.6.a.l | 3 | ||
| 55.e | even | 4 | 1 | 605.6.a.b | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 55.6.a.a | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 275.6.a.c | 3 | 5.c | odd | 4 | 1 | ||
| 275.6.b.c | 6 | 1.a | even | 1 | 1 | trivial | |
| 275.6.b.c | 6 | 5.b | even | 2 | 1 | inner | |
| 495.6.a.f | 3 | 15.e | even | 4 | 1 | ||
| 605.6.a.b | 3 | 55.e | even | 4 | 1 | ||
| 880.6.a.l | 3 | 20.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 137T_{2}^{4} + 5688T_{2}^{2} + 71824 \)
acting on \(S_{6}^{\mathrm{new}}(275, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 137 T^{4} + \cdots + 71824 \)
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| $3$ |
\( T^{6} + 978 T^{4} + \cdots + 5550736 \)
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| $5$ |
\( T^{6} \)
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| $7$ |
\( T^{6} + 15482 T^{4} + \cdots + 26708224 \)
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| $11$ |
\( (T - 121)^{6} \)
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| $13$ |
\( T^{6} + \cdots + 79\!\cdots\!36 \)
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| $17$ |
\( T^{6} + \cdots + 21\!\cdots\!04 \)
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| $19$ |
\( (T^{3} - 10 T^{2} + \cdots - 4578226300)^{2} \)
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| $23$ |
\( T^{6} + \cdots + 66\!\cdots\!76 \)
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| $29$ |
\( (T^{3} + 1970 T^{2} + \cdots - 146901534950)^{2} \)
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| $31$ |
\( (T^{3} - 6596 T^{2} + \cdots + 96581326832)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 23\!\cdots\!64 \)
|
| $41$ |
\( (T^{3} + 7214 T^{2} + \cdots - 888835182328)^{2} \)
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| $43$ |
\( T^{6} + \cdots + 98\!\cdots\!56 \)
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| $47$ |
\( T^{6} + \cdots + 76\!\cdots\!64 \)
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| $53$ |
\( T^{6} + \cdots + 16\!\cdots\!96 \)
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| $59$ |
\( (T^{3} + \cdots + 28840231836400)^{2} \)
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| $61$ |
\( (T^{3} + \cdots - 10687427094938)^{2} \)
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| $67$ |
\( T^{6} + \cdots + 22\!\cdots\!84 \)
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| $71$ |
\( (T^{3} + \cdots - 41054108151208)^{2} \)
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| $73$ |
\( T^{6} + \cdots + 23\!\cdots\!16 \)
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| $79$ |
\( (T^{3} + \cdots - 36194232363200)^{2} \)
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| $83$ |
\( T^{6} + \cdots + 29\!\cdots\!56 \)
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| $89$ |
\( (T^{3} + \cdots - 572172535345750)^{2} \)
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| $97$ |
\( T^{6} + \cdots + 77\!\cdots\!04 \)
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