Newspace parameters
| Level: | \( N \) | \(=\) | \( 446 = 2 \cdot 223 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 446.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(3.56132793015\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{7} - x^{6} - 14x^{5} + 12x^{4} + 50x^{3} - 36x^{2} - 38x + 18 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - x^{6} - 14x^{5} + 12x^{4} + 50x^{3} - 36x^{2} - 38x + 18 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( ( -6\nu^{6} + 37\nu^{5} + 92\nu^{4} - 388\nu^{3} - 287\nu^{2} + 703\nu - 98 ) / 239 \)
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| \(\beta_{3}\) | \(=\) |
\( ( 6\nu^{6} - 37\nu^{5} - 92\nu^{4} + 388\nu^{3} + 526\nu^{2} - 703\nu - 858 ) / 239 \)
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| \(\beta_{4}\) | \(=\) |
\( ( 11\nu^{6} - 28\nu^{5} - 89\nu^{4} + 313\nu^{3} - 151\nu^{2} - 771\nu + 817 ) / 239 \)
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| \(\beta_{5}\) | \(=\) |
\( ( -32\nu^{6} + 38\nu^{5} + 411\nu^{4} - 476\nu^{3} - 1212\nu^{2} + 1439\nu + 513 ) / 239 \)
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| \(\beta_{6}\) | \(=\) |
\( ( -47\nu^{6} + 11\nu^{5} + 641\nu^{4} - 12\nu^{3} - 2049\nu^{2} - 269\nu + 985 ) / 239 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{6} - 2\beta_{5} - \beta_{4} - \beta_{3} + 7\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 4\beta_{4} + 10\beta_{3} + 12\beta_{2} + \beta _1 + 25 \)
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| \(\nu^{5}\) | \(=\) |
\( 10\beta_{6} - 22\beta_{5} - 12\beta_{4} - 13\beta_{3} + 4\beta_{2} + 55\beta _1 + 2 \)
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| \(\nu^{6}\) | \(=\) |
\( -3\beta_{6} + 9\beta_{5} + 52\beta_{4} + 90\beta_{3} + 121\beta_{2} + 19\beta _1 + 188 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
1.00000 | −2.90898 | 1.00000 | −3.52361 | −2.90898 | −0.552246 | 1.00000 | 5.46214 | −3.52361 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.01250 | 1.00000 | 1.52490 | −2.01250 | 0.908231 | 1.00000 | 1.05016 | 1.52490 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −0.922829 | 1.00000 | 0.437167 | −0.922829 | 3.40247 | 1.00000 | −2.14839 | 0.437167 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 0.409721 | 1.00000 | 4.32652 | 0.409721 | 2.79833 | 1.00000 | −2.83213 | 4.32652 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 1.26499 | 1.00000 | 1.88711 | 1.26499 | −3.39932 | 1.00000 | −1.39980 | 1.88711 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 2.08198 | 1.00000 | −3.23274 | 2.08198 | 4.23794 | 1.00000 | 1.33463 | −3.23274 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 3.08762 | 1.00000 | 0.580657 | 3.08762 | −1.39541 | 1.00000 | 6.53337 | 0.580657 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(223\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 446.2.a.e | ✓ | 7 |
| 3.b | odd | 2 | 1 | 4014.2.a.w | 7 | ||
| 4.b | odd | 2 | 1 | 3568.2.a.l | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 446.2.a.e | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 3568.2.a.l | 7 | 4.b | odd | 2 | 1 | ||
| 4014.2.a.w | 7 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(446))\):
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\( T_{3}^{7} - T_{3}^{6} - 14T_{3}^{5} + 12T_{3}^{4} + 50T_{3}^{3} - 36T_{3}^{2} - 38T_{3} + 18 \)
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\( T_{5}^{7} - 2T_{5}^{6} - 22T_{5}^{5} + 42T_{5}^{4} + 92T_{5}^{3} - 256T_{5}^{2} + 174T_{5} - 36 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{7} \)
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| $3$ |
\( T^{7} - T^{6} + \cdots + 18 \)
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| $5$ |
\( T^{7} - 2 T^{6} + \cdots - 36 \)
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| $7$ |
\( T^{7} - 6 T^{6} + \cdots + 96 \)
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| $11$ |
\( T^{7} - 9 T^{6} + \cdots + 3494 \)
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| $13$ |
\( T^{7} + 2 T^{6} + \cdots + 7056 \)
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| $17$ |
\( T^{7} + 7 T^{6} + \cdots + 1212 \)
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| $19$ |
\( T^{7} + 2 T^{6} + \cdots - 2304 \)
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| $23$ |
\( T^{7} - 15 T^{6} + \cdots - 256 \)
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| $29$ |
\( T^{7} - 9 T^{6} + \cdots + 12096 \)
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| $31$ |
\( T^{7} + 2 T^{6} + \cdots + 10888 \)
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| $37$ |
\( T^{7} - 5 T^{6} + \cdots - 114624 \)
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| $41$ |
\( T^{7} + 33 T^{6} + \cdots - 45248 \)
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| $43$ |
\( T^{7} - 20 T^{6} + \cdots - 55296 \)
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| $47$ |
\( T^{7} + 2 T^{6} + \cdots - 536856 \)
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| $53$ |
\( T^{7} + 13 T^{6} + \cdots - 395712 \)
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| $59$ |
\( T^{7} - 9 T^{6} + \cdots + 346162 \)
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| $61$ |
\( T^{7} - 8 T^{6} + \cdots + 14636 \)
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| $67$ |
\( T^{7} - 29 T^{6} + \cdots - 1778034 \)
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| $71$ |
\( T^{7} - 424 T^{5} + \cdots - 4460544 \)
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| $73$ |
\( T^{7} + 37 T^{6} + \cdots - 206064 \)
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| $79$ |
\( T^{7} - 32 T^{6} + \cdots - 371712 \)
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| $83$ |
\( T^{7} + 6 T^{6} + \cdots - 263424 \)
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| $89$ |
\( T^{7} + 17 T^{6} + \cdots + 186084 \)
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| $97$ |
\( T^{7} - 12 T^{6} + \cdots + 5015808 \)
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