Newspace parameters
| Level: | \( N \) | \(=\) | \( 208 = 2^{4} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 208.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(107.127453922\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
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| Defining polynomial: |
\( x^{7} - x^{6} - 112810 x^{5} + 1645934 x^{4} + 3493976849 x^{3} - 83049726457 x^{2} + \cdots + 864293655586764 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{19}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 104) |
| 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
\( x^{7} - x^{6} - 112810 x^{5} + 1645934 x^{4} + 3493976849 x^{3} - 83049726457 x^{2} + \cdots + 864293655586764 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 15795768817 \nu^{6} + 7343349530000 \nu^{5} + \cdots - 40\!\cdots\!92 ) / 28\!\cdots\!20 \)
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| \(\beta_{3}\) | \(=\) |
\( ( - 211182971761 \nu^{6} - 18750683527060 \nu^{5} + \cdots + 23\!\cdots\!44 ) / 14\!\cdots\!60 \)
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| \(\beta_{4}\) | \(=\) |
\( ( 13594023070 \nu^{6} + 1534496734189 \nu^{5} + \cdots - 45\!\cdots\!40 ) / 72\!\cdots\!18 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 290869471067 \nu^{6} + 28034206523120 \nu^{5} + \cdots - 74\!\cdots\!48 ) / 72\!\cdots\!80 \)
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| \(\beta_{6}\) | \(=\) |
\( ( 1346076674651 \nu^{6} + 75787353512360 \nu^{5} + \cdots - 22\!\cdots\!24 ) / 28\!\cdots\!20 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + 3\beta_{3} + 5\beta_{2} - 19\beta _1 + 32236 \)
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| \(\nu^{3}\) | \(=\) |
\( -30\beta_{6} + 31\beta_{5} - 89\beta_{4} - 123\beta_{3} - 48\beta_{2} + 50852\beta _1 - 664328 \)
|
| \(\nu^{4}\) | \(=\) |
\( 60533 \beta_{6} + 2992 \beta_{5} + 13665 \beta_{4} + 208312 \beta_{3} + 279175 \beta_{2} + \cdots + 1638969460 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 4117404 \beta_{6} + 2106530 \beta_{5} - 7401118 \beta_{4} - 17649132 \beta_{3} + 21430620 \beta_{2} + \cdots - 67110000604 \)
|
| \(\nu^{6}\) | \(=\) |
\( 3672138117 \beta_{6} + 348731636 \beta_{5} + 1507906206 \beta_{4} + 13441220573 \beta_{3} + \cdots + 92010955632088 \)
|
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 | −264.020 | 0 | −498.097 | 0 | −8282.02 | 0 | 50023.3 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −204.282 | 0 | 2155.51 | 0 | 6106.83 | 0 | 22048.1 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −100.057 | 0 | −2398.21 | 0 | 1546.37 | 0 | −9671.67 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 44.9663 | 0 | 193.817 | 0 | 763.904 | 0 | −17661.0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 69.8667 | 0 | −170.421 | 0 | 12012.7 | 0 | −14801.7 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 208.821 | 0 | −1446.79 | 0 | −8051.09 | 0 | 23923.4 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 231.704 | 0 | 1363.19 | 0 | −3005.73 | 0 | 34003.6 | 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 | 208.10.a.l | 7 | |
| 4.b | odd | 2 | 1 | 104.10.a.c | ✓ | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 104.10.a.c | ✓ | 7 | 4.b | odd | 2 | 1 | |
| 208.10.a.l | 7 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{7} + 13 T_{3}^{6} - 112738 T_{3}^{5} + 518054 T_{3}^{4} + 3502632321 T_{3}^{3} + \cdots + 820307149606656 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(208))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
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| $3$ |
\( T^{7} + \cdots + 820307149606656 \)
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| $5$ |
\( T^{7} + \cdots - 16\!\cdots\!00 \)
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| $7$ |
\( T^{7} + \cdots + 17\!\cdots\!00 \)
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| $11$ |
\( T^{7} + \cdots + 25\!\cdots\!32 \)
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| $13$ |
\( (T + 28561)^{7} \)
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| $17$ |
\( T^{7} + \cdots + 59\!\cdots\!20 \)
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| $19$ |
\( T^{7} + \cdots - 14\!\cdots\!72 \)
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| $23$ |
\( T^{7} + \cdots - 51\!\cdots\!32 \)
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| $29$ |
\( T^{7} + \cdots + 26\!\cdots\!92 \)
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| $31$ |
\( T^{7} + \cdots - 32\!\cdots\!60 \)
|
| $37$ |
\( T^{7} + \cdots + 34\!\cdots\!32 \)
|
| $41$ |
\( T^{7} + \cdots - 35\!\cdots\!32 \)
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| $43$ |
\( T^{7} + \cdots + 14\!\cdots\!48 \)
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| $47$ |
\( T^{7} + \cdots + 19\!\cdots\!44 \)
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| $53$ |
\( T^{7} + \cdots + 52\!\cdots\!48 \)
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| $59$ |
\( T^{7} + \cdots + 28\!\cdots\!12 \)
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| $61$ |
\( T^{7} + \cdots + 99\!\cdots\!00 \)
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| $67$ |
\( T^{7} + \cdots + 78\!\cdots\!96 \)
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| $71$ |
\( T^{7} + \cdots - 70\!\cdots\!44 \)
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| $73$ |
\( T^{7} + \cdots - 50\!\cdots\!36 \)
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| $79$ |
\( T^{7} + \cdots - 87\!\cdots\!52 \)
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| $83$ |
\( T^{7} + \cdots - 41\!\cdots\!48 \)
|
| $89$ |
\( T^{7} + \cdots + 17\!\cdots\!00 \)
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| $97$ |
\( T^{7} + \cdots - 25\!\cdots\!00 \)
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