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: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{10} - 2 x^{9} - 124193 x^{8} - 182380 x^{7} + 5434866119 x^{6} + 59076086702 x^{5} + \cdots + 20\!\cdots\!04 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{10}\cdot 3\cdot 13^{6} \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 2 x^{9} - 124193 x^{8} - 182380 x^{7} + 5434866119 x^{6} + 59076086702 x^{5} + \cdots + 20\!\cdots\!04 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 13\!\cdots\!57 \nu^{9} + \cdots + 10\!\cdots\!92 ) / 16\!\cdots\!56 \)
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| \(\beta_{3}\) | \(=\) |
\( ( 17\!\cdots\!77 \nu^{9} + \cdots + 58\!\cdots\!68 ) / 97\!\cdots\!44 \)
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| \(\beta_{4}\) | \(=\) |
\( ( 19\!\cdots\!31 \nu^{9} + \cdots + 13\!\cdots\!88 ) / 78\!\cdots\!52 \)
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| \(\beta_{5}\) | \(=\) |
\( ( 47\!\cdots\!75 \nu^{9} + \cdots - 32\!\cdots\!44 ) / 78\!\cdots\!52 \)
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| \(\beta_{6}\) | \(=\) |
\( ( - 83\!\cdots\!93 \nu^{9} + \cdots - 15\!\cdots\!88 ) / 97\!\cdots\!44 \)
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| \(\beta_{7}\) | \(=\) |
\( ( 68\!\cdots\!13 \nu^{9} + \cdots + 40\!\cdots\!28 ) / 78\!\cdots\!52 \)
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| \(\beta_{8}\) | \(=\) |
\( ( 10\!\cdots\!25 \nu^{9} + \cdots + 37\!\cdots\!60 ) / 12\!\cdots\!68 \)
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| \(\beta_{9}\) | \(=\) |
\( ( - 17\!\cdots\!27 \nu^{9} + \cdots + 18\!\cdots\!32 ) / 78\!\cdots\!52 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + 2\beta_{5} - \beta_{4} - 2\beta_{3} + 3\beta_{2} + 5\beta _1 + 24838 \)
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| \(\nu^{3}\) | \(=\) |
\( 11 \beta_{9} + 106 \beta_{8} - 132 \beta_{7} + 52 \beta_{6} - 105 \beta_{5} + 347 \beta_{4} + \cdots + 121990 \)
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| \(\nu^{4}\) | \(=\) |
\( 11917 \beta_{9} + 6569 \beta_{8} + 17268 \beta_{7} + 71280 \beta_{6} + 93845 \beta_{5} + \cdots + 911246754 \)
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| \(\nu^{5}\) | \(=\) |
\( 498568 \beta_{9} + 2722403 \beta_{8} - 8130432 \beta_{7} + 1106771 \beta_{6} - 5650398 \beta_{5} + \cdots - 8518391914 \)
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| \(\nu^{6}\) | \(=\) |
\( 822775058 \beta_{9} + 386152714 \beta_{8} + 1083402768 \beta_{7} + 4220311979 \beta_{6} + \cdots + 37713409743242 \)
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| \(\nu^{7}\) | \(=\) |
\( 1578207537 \beta_{9} - 39648991236 \beta_{8} - 408155284116 \beta_{7} - 60285746286 \beta_{6} + \cdots - 11\!\cdots\!42 \)
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| \(\nu^{8}\) | \(=\) |
\( 46087993369659 \beta_{9} + 17043313583679 \beta_{8} + 58818259346292 \beta_{7} + 231528796948330 \beta_{6} + \cdots + 16\!\cdots\!26 \)
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| \(\nu^{9}\) | \(=\) |
\( - 15\!\cdots\!78 \beta_{9} + \cdots - 87\!\cdots\!18 \)
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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 | −228.520 | 256.000 | −1660.64 | 3656.31 | 737.048 | −4096.00 | 32538.2 | 26570.2 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −16.0000 | −208.242 | 256.000 | 1666.10 | 3331.87 | 11980.1 | −4096.00 | 23681.6 | −26657.6 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −16.0000 | −183.269 | 256.000 | −518.346 | 2932.31 | −8697.72 | −4096.00 | 13904.6 | 8293.53 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −16.0000 | −153.089 | 256.000 | 1938.97 | 2449.42 | 8817.58 | −4096.00 | 3753.21 | −31023.5 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −16.0000 | −7.07825 | 256.000 | −302.293 | 113.252 | −6519.27 | −4096.00 | −19632.9 | 4836.69 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −16.0000 | 23.8286 | 256.000 | 683.525 | −381.258 | −4980.66 | −4096.00 | −19115.2 | −10936.4 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −16.0000 | 92.1245 | 256.000 | −641.917 | −1473.99 | 7255.53 | −4096.00 | −11196.1 | 10270.7 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −16.0000 | 113.854 | 256.000 | −2120.34 | −1821.66 | 2716.31 | −4096.00 | −6720.36 | 33925.4 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −16.0000 | 172.829 | 256.000 | 1469.33 | −2765.26 | 4966.19 | −4096.00 | 10186.8 | −23509.2 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −16.0000 | 215.562 | 256.000 | 2573.60 | −3449.00 | −4899.07 | −4096.00 | 26784.1 | −41177.7 | |||||||||||||||||||||||||||||||||||||||||||||||||
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.m | 10 | |
| 13.b | even | 2 | 1 | 338.10.a.n | 10 | ||
| 13.f | odd | 12 | 2 | 26.10.e.a | ✓ | 20 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 26.10.e.a | ✓ | 20 | 13.f | odd | 12 | 2 | |
| 338.10.a.m | 10 | 1.a | even | 1 | 1 | trivial | |
| 338.10.a.n | 10 | 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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\( T_{3}^{10} + 162 T_{3}^{9} - 112385 T_{3}^{8} - 15204372 T_{3}^{7} + 4579527943 T_{3}^{6} + \cdots - 87\!\cdots\!28 \)
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\( T_{5}^{10} - 3088 T_{5}^{9} - 7137486 T_{5}^{8} + 25886451056 T_{5}^{7} + 11664060879169 T_{5}^{6} + \cdots - 29\!\cdots\!00 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 16)^{10} \)
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| $3$ |
\( T^{10} + \cdots - 87\!\cdots\!28 \)
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| $5$ |
\( T^{10} + \cdots - 29\!\cdots\!00 \)
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| $7$ |
\( T^{10} + \cdots + 10\!\cdots\!44 \)
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| $11$ |
\( T^{10} + \cdots - 33\!\cdots\!32 \)
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| $13$ |
\( T^{10} \)
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| $17$ |
\( T^{10} + \cdots + 47\!\cdots\!37 \)
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| $19$ |
\( T^{10} + \cdots + 61\!\cdots\!56 \)
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| $23$ |
\( T^{10} + \cdots + 14\!\cdots\!28 \)
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| $29$ |
\( T^{10} + \cdots + 10\!\cdots\!53 \)
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| $31$ |
\( T^{10} + \cdots + 46\!\cdots\!28 \)
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| $37$ |
\( T^{10} + \cdots + 16\!\cdots\!41 \)
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| $41$ |
\( T^{10} + \cdots - 19\!\cdots\!71 \)
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| $43$ |
\( T^{10} + \cdots + 32\!\cdots\!52 \)
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| $47$ |
\( T^{10} + \cdots + 35\!\cdots\!76 \)
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| $53$ |
\( T^{10} + \cdots - 22\!\cdots\!64 \)
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| $59$ |
\( T^{10} + \cdots - 73\!\cdots\!64 \)
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| $61$ |
\( T^{10} + \cdots + 22\!\cdots\!89 \)
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| $67$ |
\( T^{10} + \cdots - 69\!\cdots\!00 \)
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| $71$ |
\( T^{10} + \cdots + 52\!\cdots\!56 \)
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| $73$ |
\( T^{10} + \cdots + 61\!\cdots\!16 \)
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| $79$ |
\( T^{10} + \cdots + 10\!\cdots\!12 \)
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| $83$ |
\( T^{10} + \cdots + 24\!\cdots\!52 \)
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| $89$ |
\( T^{10} + \cdots + 59\!\cdots\!68 \)
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| $97$ |
\( T^{10} + \cdots + 35\!\cdots\!08 \)
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