Newspace parameters
| Level: | \( N \) | \(=\) | \( 208 = 2^{4} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 208.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(159.815381556\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
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| Defining polynomial: |
\( x^{6} - x^{5} - 13546x^{4} + 130998x^{3} + 49403509x^{2} - 776207317x - 22123683244 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{17} \) |
| Twist minimal: | no (minimal twist has level 13) |
| 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
\( x^{6} - x^{5} - 13546x^{4} + 130998x^{3} + 49403509x^{2} - 776207317x - 22123683244 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 5015 \nu^{5} - 381698 \nu^{4} + 44872104 \nu^{3} + 2440298814 \nu^{2} - 98328943649 \nu - 1235017116236 ) / 1004386560 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 4549 \nu^{5} + 297830 \nu^{4} - 39656088 \nu^{3} - 3039388314 \nu^{2} + 39945266179 \nu + 6280883551844 ) / 1004386560 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 6577 \nu^{5} + 792034 \nu^{4} + 65534952 \nu^{3} - 5983758270 \nu^{2} - 55389247015 \nu + 3844299707116 ) / 1004386560 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -635\nu^{5} + 6954\nu^{4} + 7372588\nu^{3} - 11836302\nu^{2} - 20834395553\nu - 163220197772 ) / 83698880 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2591\nu^{5} + 192406\nu^{4} - 18750180\nu^{3} - 1230092514\nu^{2} + 23290422389\nu + 899339899948 ) / 62774160 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - 4\beta_{4} + 2\beta_{3} - 3\beta_{2} + 9\beta _1 + 45 ) / 256 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -29\beta_{5} + 132\beta_{4} - 74\beta_{3} - 121\beta_{2} - 453\beta _1 + 1155767 ) / 256 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 9693\beta_{5} - 24340\beta_{4} + 12378\beta_{3} - 18423\beta_{2} + 84165\beta _1 - 15011143 ) / 256 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -313101\beta_{5} + 1291428\beta_{4} - 504714\beta_{3} - 593673\beta_{2} - 4427061\beta _1 + 7204755815 ) / 256 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 76841021 \beta_{5} - 173417204 \beta_{4} + 73945210 \beta_{3} - 119713719 \beta_{2} + \cdots - 184206340935 ) / 256 \)
|
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 |
|
0 | −827.494 | 0 | 7679.45 | 0 | 6800.32 | 0 | 507599. | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −469.277 | 0 | 4406.29 | 0 | −63013.3 | 0 | 43074.3 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −468.466 | 0 | −69.9037 | 0 | 80037.3 | 0 | 42313.3 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 19.0788 | 0 | −11885.7 | 0 | 62220.3 | 0 | −176783. | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 573.204 | 0 | −3613.82 | 0 | −14401.4 | 0 | 151416. | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 696.954 | 0 | 6795.70 | 0 | −67467.4 | 0 | 308598. | 0 | |||||||||||||||||||||||||||||||||||||
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 | 208.12.a.h | 6 | |
| 4.b | odd | 2 | 1 | 13.12.a.b | ✓ | 6 | |
| 12.b | even | 2 | 1 | 117.12.a.d | 6 | ||
| 52.b | odd | 2 | 1 | 169.12.a.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 13.12.a.b | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 117.12.a.d | 6 | 12.b | even | 2 | 1 | ||
| 169.12.a.c | 6 | 52.b | odd | 2 | 1 | ||
| 208.12.a.h | 6 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 476 T_{3}^{5} - 856262 T_{3}^{4} - 361565820 T_{3}^{3} + 173969650341 T_{3}^{2} + \cdots - 13\!\cdots\!12 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(208))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
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| $3$ |
\( T^{6} + \cdots - 13\!\cdots\!12 \)
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| $5$ |
\( T^{6} + \cdots - 69\!\cdots\!00 \)
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| $7$ |
\( T^{6} + \cdots - 20\!\cdots\!96 \)
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| $11$ |
\( T^{6} + \cdots - 18\!\cdots\!92 \)
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| $13$ |
\( (T + 371293)^{6} \)
|
| $17$ |
\( T^{6} + \cdots - 14\!\cdots\!04 \)
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| $19$ |
\( T^{6} + \cdots - 20\!\cdots\!84 \)
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| $23$ |
\( T^{6} + \cdots + 21\!\cdots\!88 \)
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| $29$ |
\( T^{6} + \cdots + 82\!\cdots\!12 \)
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| $31$ |
\( T^{6} + \cdots - 54\!\cdots\!60 \)
|
| $37$ |
\( T^{6} + \cdots - 36\!\cdots\!32 \)
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| $41$ |
\( T^{6} + \cdots + 89\!\cdots\!16 \)
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| $43$ |
\( T^{6} + \cdots + 38\!\cdots\!60 \)
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| $47$ |
\( T^{6} + \cdots + 46\!\cdots\!20 \)
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| $53$ |
\( T^{6} + \cdots + 24\!\cdots\!72 \)
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| $59$ |
\( T^{6} + \cdots - 28\!\cdots\!04 \)
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| $61$ |
\( T^{6} + \cdots + 32\!\cdots\!92 \)
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| $67$ |
\( T^{6} + \cdots - 12\!\cdots\!84 \)
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| $71$ |
\( T^{6} + \cdots - 32\!\cdots\!16 \)
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| $73$ |
\( T^{6} + \cdots - 73\!\cdots\!68 \)
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| $79$ |
\( T^{6} + \cdots - 25\!\cdots\!96 \)
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| $83$ |
\( T^{6} + \cdots - 50\!\cdots\!72 \)
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| $89$ |
\( T^{6} + \cdots + 28\!\cdots\!00 \)
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| $97$ |
\( T^{6} + \cdots - 13\!\cdots\!00 \)
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