Newspace parameters
| Level: | \( N \) | \(=\) | \( 338 = 2 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 338.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(174.082112623\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
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| Defining polynomial: |
\( x^{6} - 3x^{5} - 10375x^{4} - 44865x^{3} + 25702990x^{2} + 68900300x - 16723086000 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{4}\cdot 13 \) |
| Twist minimal: | no (minimal twist has level 26) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 3x^{5} - 10375x^{4} - 44865x^{3} + 25702990x^{2} + 68900300x - 16723086000 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3\nu - 1 \)
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| \(\beta_{2}\) | \(=\) |
\( ( 17\nu^{5} + 11954\nu^{4} - 689565\nu^{3} - 86988930\nu^{2} + 2266527380\nu + 91027300800 ) / 38058000 \)
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| \(\beta_{3}\) | \(=\) |
\( ( 281\nu^{5} - 26278\nu^{4} - 2331345\nu^{3} + 190223010\nu^{2} + 4420505840\nu - 225775497600 ) / 9514500 \)
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| \(\beta_{4}\) | \(=\) |
\( ( 281\nu^{5} - 26278\nu^{4} - 2331345\nu^{3} + 275853510\nu^{2} + 3535657340\nu - 521600331600 ) / 9514500 \)
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| \(\beta_{5}\) | \(=\) |
\( ( 4949\nu^{5} + 47338\nu^{4} - 42691905\nu^{3} - 835555210\nu^{2} + 53042051860\nu + 939959779600 ) / 12686000 \)
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| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 3 \)
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| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} - \beta_{3} + 31\beta _1 + 31123 ) / 9 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 6\beta_{5} + 25\beta_{4} - 94\beta_{3} - 678\beta_{2} + 16558\beta _1 + 333592 ) / 9 \)
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| \(\nu^{4}\) | \(=\) |
\( ( 243\beta_{5} + 8285\beta_{4} - 11462\beta_{3} - 2169\beta_{2} + 453239\beta _1 + 169844366 ) / 9 \)
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| \(\nu^{5}\) | \(=\) |
\( ( 72504\beta_{5} + 305245\beta_{4} - 870076\beta_{3} - 5827932\beta_{2} + 111580762\beta _1 + 4766176198 ) / 9 \)
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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 |
|
−16.0000 | −255.762 | 256.000 | −1476.40 | 4092.20 | 8627.38 | −4096.00 | 45731.4 | 23622.4 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −16.0000 | −117.679 | 256.000 | −279.538 | 1882.87 | −7662.99 | −4096.00 | −5834.58 | 4472.61 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −16.0000 | −78.0204 | 256.000 | 1338.85 | 1248.33 | 132.180 | −4096.00 | −13595.8 | −21421.6 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −16.0000 | 115.652 | 256.000 | −1193.59 | −1850.43 | 11008.8 | −4096.00 | −6307.60 | 19097.5 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −16.0000 | 174.344 | 256.000 | −2153.96 | −2789.50 | −12255.2 | −4096.00 | 10712.7 | 34463.3 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −16.0000 | 242.466 | 256.000 | 2071.64 | −3879.46 | −323.249 | −4096.00 | 39107.0 | −33146.3 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(13\) | \( +1 \) |
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 | 338.10.a.k | 6 | |
| 13.b | even | 2 | 1 | 338.10.a.l | 6 | ||
| 13.e | even | 6 | 2 | 26.10.c.b | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 26.10.c.b | ✓ | 12 | 13.e | even | 6 | 2 | |
| 338.10.a.k | 6 | 1.a | even | 1 | 1 | trivial | |
| 338.10.a.l | 6 | 13.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(338))\):
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\( T_{3}^{6} - 81T_{3}^{5} - 90675T_{3}^{4} + 6766605T_{3}^{3} + 1901830590T_{3}^{2} - 77125089600T_{3} - 11480361984000 \)
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\( T_{5}^{6} + 1693 T_{5}^{5} - 5770035 T_{5}^{4} - 10171826233 T_{5}^{3} + 5531891086994 T_{5}^{2} + \cdots + 29\!\cdots\!00 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 16)^{6} \)
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| $3$ |
\( T^{6} + \cdots - 11480361984000 \)
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| $5$ |
\( T^{6} + \cdots + 29\!\cdots\!00 \)
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| $7$ |
\( T^{6} + \cdots - 38\!\cdots\!40 \)
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| $11$ |
\( T^{6} + \cdots - 11\!\cdots\!44 \)
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| $13$ |
\( T^{6} \)
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| $17$ |
\( T^{6} + \cdots + 18\!\cdots\!69 \)
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| $19$ |
\( T^{6} + \cdots + 31\!\cdots\!00 \)
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| $23$ |
\( T^{6} + \cdots + 17\!\cdots\!24 \)
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| $29$ |
\( T^{6} + \cdots - 68\!\cdots\!49 \)
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| $31$ |
\( T^{6} + \cdots + 11\!\cdots\!00 \)
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| $37$ |
\( T^{6} + \cdots + 60\!\cdots\!63 \)
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| $41$ |
\( T^{6} + \cdots + 60\!\cdots\!25 \)
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| $43$ |
\( T^{6} + \cdots + 21\!\cdots\!32 \)
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| $47$ |
\( T^{6} + \cdots + 42\!\cdots\!64 \)
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| $53$ |
\( T^{6} + \cdots - 58\!\cdots\!96 \)
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| $59$ |
\( T^{6} + \cdots + 17\!\cdots\!04 \)
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| $61$ |
\( T^{6} + \cdots - 24\!\cdots\!65 \)
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| $67$ |
\( T^{6} + \cdots + 20\!\cdots\!16 \)
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| $71$ |
\( T^{6} + \cdots - 46\!\cdots\!36 \)
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| $73$ |
\( T^{6} + \cdots + 27\!\cdots\!80 \)
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| $79$ |
\( T^{6} + \cdots - 14\!\cdots\!20 \)
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| $83$ |
\( T^{6} + \cdots - 97\!\cdots\!00 \)
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| $89$ |
\( T^{6} + \cdots - 18\!\cdots\!84 \)
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| $97$ |
\( T^{6} + \cdots + 79\!\cdots\!96 \)
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