Newspace parameters
| Level: | \( N \) | \(=\) | \( 21 = 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 21.e (of order \(3\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(6.56008553517\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Relative dimension: | \(5\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} + \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{10} + 412 x^{8} - 96 x^{7} + 133333 x^{6} - 66144 x^{5} + 15003636 x^{4} - 36459504 x^{3} + \cdots + 2149991424 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{3}\cdot 7^{3} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{10} + 412 x^{8} - 96 x^{7} + 133333 x^{6} - 66144 x^{5} + 15003636 x^{4} - 36459504 x^{3} + \cdots + 2149991424 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 19608482386725 \nu^{9} + 25091781738784 \nu^{8} + 18325249512448 \nu^{7} + \cdots - 12\!\cdots\!68 ) / 95\!\cdots\!47 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 25091781738784 \nu^{9} + \cdots - 15\!\cdots\!55 ) / 95\!\cdots\!47 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 37\!\cdots\!43 \nu^{9} + \cdots + 63\!\cdots\!20 ) / 63\!\cdots\!28 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 470544062002577 \nu^{9} + \cdots + 14\!\cdots\!40 ) / 38\!\cdots\!88 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 51\!\cdots\!30 \nu^{9} + \cdots - 42\!\cdots\!48 ) / 37\!\cdots\!96 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 44\!\cdots\!17 \nu^{9} + \cdots - 18\!\cdots\!64 ) / 21\!\cdots\!76 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 20\!\cdots\!65 \nu^{9} + \cdots + 34\!\cdots\!00 ) / 21\!\cdots\!76 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 71\!\cdots\!41 \nu^{9} + \cdots + 11\!\cdots\!88 ) / 31\!\cdots\!64 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{8} - 165\beta_{4} \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{8} - \beta_{7} - 2\beta_{6} - \beta_{5} - 3\beta_{3} + 236\beta_{2} + 29 \)
|
| \(\nu^{4}\) | \(=\) |
\( 29 \beta_{9} - 302 \beta_{8} + 2 \beta_{7} - 3 \beta_{6} - 29 \beta_{5} + 38834 \beta_{4} - 301 \beta_{3} + \cdots - 38834 \)
|
| \(\nu^{5}\) | \(=\) |
\( 412 \beta_{9} - 776 \beta_{8} + 1236 \beta_{7} - 412 \beta_{6} + 26500 \beta_{4} + 412 \beta_{3} + \cdots - 60821 \beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 364\beta_{8} + 364\beta_{7} + 728\beta_{6} + 11900\beta_{5} + 87869\beta_{3} - 127740\beta_{2} + 9993185 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 131941 \beta_{9} + 412095 \beta_{8} - 266570 \beta_{7} + 399855 \beta_{6} + 131941 \beta_{5} + \cdots + 17760607 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 3873473 \beta_{9} + 25147837 \beta_{8} - 420447 \beta_{7} + 140149 \beta_{6} - 2700590774 \beta_{4} + \cdots + 52459845 \beta_1 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 39940984 \beta_{8} - 39940984 \beta_{7} - 79881968 \beta_{6} - 40131832 \beta_{5} - 136312296 \beta_{3} + \cdots - 7673460616 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).
| \(n\) | \(8\) | \(10\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 4.1 |
|
−10.0861 | − | 17.4697i | 13.5000 | − | 23.3827i | −139.460 | + | 241.552i | 214.749 | + | 371.955i | −544.651 | −617.628 | + | 664.890i | 3044.41 | −364.500 | − | 631.333i | 4331.97 | − | 7503.19i | ||||||||||||||||||||||||||||||||||
| 4.2 | −6.64307 | − | 11.5061i | 13.5000 | − | 23.3827i | −24.2608 | + | 42.0210i | −174.171 | − | 301.672i | −358.726 | 899.874 | − | 117.341i | −1055.96 | −364.500 | − | 631.333i | −2314.06 | + | 4008.06i | |||||||||||||||||||||||||||||||||||
| 4.3 | −2.14788 | − | 3.72024i | 13.5000 | − | 23.3827i | 54.7732 | − | 94.8700i | 120.528 | + | 208.761i | −115.986 | −243.986 | − | 874.079i | −1020.44 | −364.500 | − | 631.333i | 517.760 | − | 896.787i | |||||||||||||||||||||||||||||||||||
| 4.4 | 4.67442 | + | 8.09634i | 13.5000 | − | 23.3827i | 20.2996 | − | 35.1599i | −204.050 | − | 353.424i | 252.419 | −796.201 | − | 435.438i | 1576.21 | −364.500 | − | 631.333i | 1907.63 | − | 3304.11i | |||||||||||||||||||||||||||||||||||
| 4.5 | 6.70267 | + | 11.6094i | 13.5000 | − | 23.3827i | −25.8516 | + | 44.7763i | 141.943 | + | 245.853i | 361.944 | 328.441 | + | 845.973i | 1022.78 | −364.500 | − | 631.333i | −1902.80 | + | 3295.75i | |||||||||||||||||||||||||||||||||||
| 16.1 | −10.0861 | + | 17.4697i | 13.5000 | + | 23.3827i | −139.460 | − | 241.552i | 214.749 | − | 371.955i | −544.651 | −617.628 | − | 664.890i | 3044.41 | −364.500 | + | 631.333i | 4331.97 | + | 7503.19i | |||||||||||||||||||||||||||||||||||
| 16.2 | −6.64307 | + | 11.5061i | 13.5000 | + | 23.3827i | −24.2608 | − | 42.0210i | −174.171 | + | 301.672i | −358.726 | 899.874 | + | 117.341i | −1055.96 | −364.500 | + | 631.333i | −2314.06 | − | 4008.06i | |||||||||||||||||||||||||||||||||||
| 16.3 | −2.14788 | + | 3.72024i | 13.5000 | + | 23.3827i | 54.7732 | + | 94.8700i | 120.528 | − | 208.761i | −115.986 | −243.986 | + | 874.079i | −1020.44 | −364.500 | + | 631.333i | 517.760 | + | 896.787i | |||||||||||||||||||||||||||||||||||
| 16.4 | 4.67442 | − | 8.09634i | 13.5000 | + | 23.3827i | 20.2996 | + | 35.1599i | −204.050 | + | 353.424i | 252.419 | −796.201 | + | 435.438i | 1576.21 | −364.500 | + | 631.333i | 1907.63 | + | 3304.11i | |||||||||||||||||||||||||||||||||||
| 16.5 | 6.70267 | − | 11.6094i | 13.5000 | + | 23.3827i | −25.8516 | − | 44.7763i | 141.943 | − | 245.853i | 361.944 | 328.441 | − | 845.973i | 1022.78 | −364.500 | + | 631.333i | −1902.80 | − | 3295.75i | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 21.8.e.b | ✓ | 10 |
| 3.b | odd | 2 | 1 | 63.8.e.d | 10 | ||
| 7.b | odd | 2 | 1 | 147.8.e.n | 10 | ||
| 7.c | even | 3 | 1 | inner | 21.8.e.b | ✓ | 10 |
| 7.c | even | 3 | 1 | 147.8.a.j | 5 | ||
| 7.d | odd | 6 | 1 | 147.8.a.k | 5 | ||
| 7.d | odd | 6 | 1 | 147.8.e.n | 10 | ||
| 21.g | even | 6 | 1 | 441.8.a.w | 5 | ||
| 21.h | odd | 6 | 1 | 63.8.e.d | 10 | ||
| 21.h | odd | 6 | 1 | 441.8.a.x | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 21.8.e.b | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 21.8.e.b | ✓ | 10 | 7.c | even | 3 | 1 | inner |
| 63.8.e.d | 10 | 3.b | odd | 2 | 1 | ||
| 63.8.e.d | 10 | 21.h | odd | 6 | 1 | ||
| 147.8.a.j | 5 | 7.c | even | 3 | 1 | ||
| 147.8.a.k | 5 | 7.d | odd | 6 | 1 | ||
| 147.8.e.n | 10 | 7.b | odd | 2 | 1 | ||
| 147.8.e.n | 10 | 7.d | odd | 6 | 1 | ||
| 441.8.a.w | 5 | 21.g | even | 6 | 1 | ||
| 441.8.a.x | 5 | 21.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} + 15 T_{2}^{9} + 547 T_{2}^{8} + 2142 T_{2}^{7} + 130570 T_{2}^{6} + 504660 T_{2}^{5} + \cdots + 20819026944 \)
acting on \(S_{8}^{\mathrm{new}}(21, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + \cdots + 20819026944 \)
|
| $3$ |
\( (T^{2} - 27 T + 729)^{5} \)
|
| $5$ |
\( T^{10} + \cdots + 17\!\cdots\!00 \)
|
| $7$ |
\( T^{10} + \cdots + 37\!\cdots\!43 \)
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| $11$ |
\( T^{10} + \cdots + 16\!\cdots\!00 \)
|
| $13$ |
\( (T^{5} + \cdots + 19\!\cdots\!84)^{2} \)
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| $17$ |
\( T^{10} + \cdots + 72\!\cdots\!00 \)
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| $19$ |
\( T^{10} + \cdots + 26\!\cdots\!16 \)
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| $23$ |
\( T^{10} + \cdots + 15\!\cdots\!00 \)
|
| $29$ |
\( (T^{5} + \cdots - 51\!\cdots\!88)^{2} \)
|
| $31$ |
\( T^{10} + \cdots + 59\!\cdots\!61 \)
|
| $37$ |
\( T^{10} + \cdots + 32\!\cdots\!44 \)
|
| $41$ |
\( (T^{5} + \cdots + 10\!\cdots\!00)^{2} \)
|
| $43$ |
\( (T^{5} + \cdots - 21\!\cdots\!80)^{2} \)
|
| $47$ |
\( T^{10} + \cdots + 24\!\cdots\!00 \)
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| $53$ |
\( T^{10} + \cdots + 49\!\cdots\!36 \)
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| $59$ |
\( T^{10} + \cdots + 65\!\cdots\!00 \)
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| $61$ |
\( T^{10} + \cdots + 33\!\cdots\!00 \)
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| $67$ |
\( T^{10} + \cdots + 71\!\cdots\!00 \)
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| $71$ |
\( (T^{5} + \cdots - 37\!\cdots\!12)^{2} \)
|
| $73$ |
\( T^{10} + \cdots + 23\!\cdots\!36 \)
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| $79$ |
\( T^{10} + \cdots + 53\!\cdots\!81 \)
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| $83$ |
\( (T^{5} + \cdots - 83\!\cdots\!56)^{2} \)
|
| $89$ |
\( T^{10} + \cdots + 12\!\cdots\!56 \)
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| $97$ |
\( (T^{5} + \cdots - 10\!\cdots\!28)^{2} \)
|
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