Newspace parameters
| Level: | \( N \) | \(=\) | \( 106 = 2 \cdot 53 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 106.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(17.0006849002\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{7} - 2x^{6} - 1315x^{5} - 3226x^{4} + 486648x^{3} + 2213838x^{2} - 51879636x - 326785428 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{7} - 2x^{6} - 1315x^{5} - 3226x^{4} + 486648x^{3} + 2213838x^{2} - 51879636x - 326785428 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 45821 \nu^{6} + 1213139 \nu^{5} - 59933276 \nu^{4} - 911441258 \nu^{3} + 12319152846 \nu^{2} + \cdots - 618474209544 ) / 42767908452 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 1027625 \nu^{6} - 43417933 \nu^{5} - 305280344 \nu^{4} + 28335474850 \nu^{3} + \cdots + 21644507976228 ) / 128303725356 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 469921 \nu^{6} + 11618737 \nu^{5} + 470082084 \nu^{4} - 8881443370 \nu^{3} + \cdots + 17487109806264 ) / 42767908452 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 953783 \nu^{6} + 10527163 \nu^{5} + 909656204 \nu^{4} - 2912370226 \nu^{3} + \cdots + 12285142043400 ) / 42767908452 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 1232044 \nu^{6} + 12649286 \nu^{5} + 1400945107 \nu^{4} - 7055412125 \nu^{3} + \cdots + 29786961229533 ) / 32075931339 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} - \beta_{5} - 2\beta_{4} - \beta_{3} + 2\beta_{2} + 7\beta _1 + 374 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{6} + 6\beta_{5} - 17\beta_{4} + 2\beta_{3} + 79\beta_{2} + 618\beta _1 + 2318 \)
|
| \(\nu^{4}\) | \(=\) |
\( 867\beta_{6} - 790\beta_{5} - 1599\beta_{4} - 612\beta_{3} + 2815\beta_{2} + 9218\beta _1 + 219555 \)
|
| \(\nu^{5}\) | \(=\) |
\( 9500\beta_{6} - 1461\beta_{5} - 23230\beta_{4} - 3422\beta_{3} + 97517\beta_{2} + 475526\beta _1 + 3083470 \)
|
| \(\nu^{6}\) | \(=\) |
\( 693218 \beta_{6} - 606427 \beta_{5} - 1276888 \beta_{4} - 401252 \beta_{3} + 3067239 \beta_{2} + \cdots + 164592754 \)
|
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 |
|
4.00000 | −28.2102 | 16.0000 | −64.7919 | −112.841 | −115.384 | 64.0000 | 552.816 | −259.168 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 4.00000 | −15.2860 | 16.0000 | −68.7893 | −61.1440 | 73.0922 | 64.0000 | −9.33853 | −275.157 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 4.00000 | −11.4847 | 16.0000 | 77.1306 | −45.9387 | 3.65511 | 64.0000 | −111.102 | 308.523 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 4.00000 | 10.4505 | 16.0000 | 73.3812 | 41.8020 | 234.374 | 64.0000 | −133.787 | 293.525 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 4.00000 | 14.4496 | 16.0000 | −30.5747 | 57.7984 | 39.9736 | 64.0000 | −34.2088 | −122.299 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 4.00000 | 23.7784 | 16.0000 | 99.5423 | 95.1135 | −212.585 | 64.0000 | 322.411 | 398.169 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 4.00000 | 25.3024 | 16.0000 | 3.10174 | 101.210 | 60.8738 | 64.0000 | 397.210 | 12.4069 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(53\) | \( +1 \) |
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 | 106.6.a.d | ✓ | 7 |
| 3.b | odd | 2 | 1 | 954.6.a.k | 7 | ||
| 4.b | odd | 2 | 1 | 848.6.a.d | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 106.6.a.d | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 848.6.a.d | 7 | 4.b | odd | 2 | 1 | ||
| 954.6.a.k | 7 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{7} - 19T_{3}^{6} - 1162T_{3}^{5} + 22276T_{3}^{4} + 331341T_{3}^{3} - 6067089T_{3}^{2} - 26335908T_{3} + 449940420 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(106))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{7} \)
|
| $3$ |
\( T^{7} - 19 T^{6} + \cdots + 449940420 \)
|
| $5$ |
\( T^{7} + \cdots + 238137495264 \)
|
| $7$ |
\( T^{7} + \cdots - 3737322217472 \)
|
| $11$ |
\( T^{7} + \cdots + 67\!\cdots\!00 \)
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| $13$ |
\( T^{7} + \cdots + 33\!\cdots\!40 \)
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| $17$ |
\( T^{7} + \cdots + 65\!\cdots\!62 \)
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| $19$ |
\( T^{7} + \cdots - 30\!\cdots\!40 \)
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| $23$ |
\( T^{7} + \cdots + 32\!\cdots\!70 \)
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| $29$ |
\( T^{7} + \cdots + 14\!\cdots\!44 \)
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| $31$ |
\( T^{7} + \cdots + 36\!\cdots\!36 \)
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| $37$ |
\( T^{7} + \cdots + 39\!\cdots\!28 \)
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| $41$ |
\( T^{7} + \cdots + 79\!\cdots\!20 \)
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| $43$ |
\( T^{7} + \cdots - 55\!\cdots\!00 \)
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| $47$ |
\( T^{7} + \cdots + 16\!\cdots\!96 \)
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| $53$ |
\( (T + 2809)^{7} \)
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| $59$ |
\( T^{7} + \cdots - 17\!\cdots\!20 \)
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| $61$ |
\( T^{7} + \cdots - 91\!\cdots\!68 \)
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| $67$ |
\( T^{7} + \cdots + 94\!\cdots\!16 \)
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| $71$ |
\( T^{7} + \cdots + 10\!\cdots\!84 \)
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| $73$ |
\( T^{7} + \cdots - 68\!\cdots\!80 \)
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| $79$ |
\( T^{7} + \cdots - 76\!\cdots\!10 \)
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| $83$ |
\( T^{7} + \cdots - 19\!\cdots\!60 \)
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| $89$ |
\( T^{7} + \cdots + 94\!\cdots\!76 \)
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| $97$ |
\( T^{7} + \cdots - 93\!\cdots\!06 \)
|
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