Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7,10,Mod(2,7)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7.2");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7.c (of order \(3\), degree \(2\), minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | no |
| Analytic conductor: | \(3.60525085315\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Relative dimension: | \(5\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - x^{9} + 430 x^{8} + 61 x^{7} + 146753 x^{6} + 23608 x^{5} + 16136944 x^{4} + 30575648 x^{3} + \cdots + 761760000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 3^{3}\cdot 7^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
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
\( x^{10} - x^{9} + 430 x^{8} + 61 x^{7} + 146753 x^{6} + 23608 x^{5} + 16136944 x^{4} + 30575648 x^{3} + \cdots + 761760000 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 174976544647 \nu^{9} + 543280294119 \nu^{8} - 2080323777608 \nu^{7} + 306747495137243 \nu^{6} + \cdots - 72\!\cdots\!00 ) / 49\!\cdots\!80 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 439915122044593 \nu^{9} - 359425911506973 \nu^{8} + \cdots + 45\!\cdots\!00 ) / 45\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 9108600924689 \nu^{9} + 744234087958953 \nu^{8} + \cdots + 14\!\cdots\!20 ) / 23\!\cdots\!40 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 44987322634457 \nu^{9} - 60129676436289 \nu^{8} + 230247989821048 \nu^{7} + \cdots + 26\!\cdots\!80 ) / 69\!\cdots\!20 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 10228081391315 \nu^{9} + \cdots + 13\!\cdots\!84 ) / 69\!\cdots\!72 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 24\!\cdots\!57 \nu^{9} + \cdots - 60\!\cdots\!00 ) / 15\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 20\!\cdots\!03 \nu^{9} + \cdots + 20\!\cdots\!00 ) / 31\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 10\!\cdots\!87 \nu^{9} + \cdots + 10\!\cdots\!00 ) / 53\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{8} - \beta_{7} - 687\beta_{3} - \beta_{2} + \beta_1 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -27\beta_{5} + \beta_{4} - 507\beta_{2} - 375 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 14 \beta_{9} - 169 \beta_{8} + 155 \beta_{7} - 14 \beta_{6} + 155 \beta_{5} + 169 \beta_{4} + \cdots - 87639 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 28\beta_{9} - 583\beta_{8} - 11457\beta_{7} + 99345\beta_{3} + 142643\beta_{2} - 142643\beta_1 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 12040\beta_{6} - 89029\beta_{5} - 107929\beta_{4} + 71655\beta_{2} + 49481343 ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 9450 \beta_{9} + 115386 \beta_{8} + 1949460 \beta_{7} + 9450 \beta_{6} + 1949460 \beta_{5} + \cdots + 13417998 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 4129692 \beta_{9} + 33731905 \beta_{8} - 25531489 \beta_{7} - 14657685447 \beta_{3} + \cdots + 22648273 \beta_1 ) / 4 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( -8200416\beta_{6} - 1242725823\beta_{5} + 81699181\beta_{4} - 12756581151\beta_{2} - 8505431979 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-\beta_{3}\) |
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.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2.1 |
|
−19.4382 | + | 33.6680i | 113.728 | + | 196.982i | −499.691 | − | 865.489i | −162.760 | + | 281.909i | −8842.67 | 5234.95 | − | 3598.46i | 18947.7 | −16026.5 | + | 27758.7i | −6327.54 | − | 10959.6i | ||||||||||||||||||||||||||||||||||
| 2.2 | −12.2345 | + | 21.1908i | −79.7348 | − | 138.105i | −43.3662 | − | 75.1124i | 1014.15 | − | 1756.56i | 3902.06 | −4235.51 | − | 4734.35i | −10405.9 | −2873.78 | + | 4977.54i | 24815.2 | + | 42981.2i | |||||||||||||||||||||||||||||||||||
| 2.3 | −2.74397 | + | 4.75269i | −1.70307 | − | 2.94981i | 240.941 | + | 417.323i | −828.924 | + | 1435.74i | 18.6927 | 2822.68 | + | 5690.88i | −5454.36 | 9835.70 | − | 17035.9i | −4549.08 | − | 7879.24i | |||||||||||||||||||||||||||||||||||
| 2.4 | 9.79824 | − | 16.9710i | 104.977 | + | 181.826i | 63.9892 | + | 110.832i | 983.791 | − | 1703.98i | 4114.37 | −5768.52 | + | 2660.41i | 12541.3 | −12198.9 | + | 21129.2i | −19278.8 | − | 33391.9i | |||||||||||||||||||||||||||||||||||
| 2.5 | 15.6185 | − | 27.0520i | −56.7670 | − | 98.3234i | −231.874 | − | 401.617i | −239.755 | + | 415.269i | −3546.46 | 1428.40 | − | 6189.77i | 1507.26 | 3396.51 | − | 5882.92i | 7489.23 | + | 12971.7i | |||||||||||||||||||||||||||||||||||
| 4.1 | −19.4382 | − | 33.6680i | 113.728 | − | 196.982i | −499.691 | + | 865.489i | −162.760 | − | 281.909i | −8842.67 | 5234.95 | + | 3598.46i | 18947.7 | −16026.5 | − | 27758.7i | −6327.54 | + | 10959.6i | |||||||||||||||||||||||||||||||||||
| 4.2 | −12.2345 | − | 21.1908i | −79.7348 | + | 138.105i | −43.3662 | + | 75.1124i | 1014.15 | + | 1756.56i | 3902.06 | −4235.51 | + | 4734.35i | −10405.9 | −2873.78 | − | 4977.54i | 24815.2 | − | 42981.2i | |||||||||||||||||||||||||||||||||||
| 4.3 | −2.74397 | − | 4.75269i | −1.70307 | + | 2.94981i | 240.941 | − | 417.323i | −828.924 | − | 1435.74i | 18.6927 | 2822.68 | − | 5690.88i | −5454.36 | 9835.70 | + | 17035.9i | −4549.08 | + | 7879.24i | |||||||||||||||||||||||||||||||||||
| 4.4 | 9.79824 | + | 16.9710i | 104.977 | − | 181.826i | 63.9892 | − | 110.832i | 983.791 | + | 1703.98i | 4114.37 | −5768.52 | − | 2660.41i | 12541.3 | −12198.9 | − | 21129.2i | −19278.8 | + | 33391.9i | |||||||||||||||||||||||||||||||||||
| 4.5 | 15.6185 | + | 27.0520i | −56.7670 | + | 98.3234i | −231.874 | + | 401.617i | −239.755 | − | 415.269i | −3546.46 | 1428.40 | + | 6189.77i | 1507.26 | 3396.51 | + | 5882.92i | 7489.23 | − | 12971.7i | |||||||||||||||||||||||||||||||||||
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 | 7.10.c.a | ✓ | 10 |
| 3.b | odd | 2 | 1 | 63.10.e.b | 10 | ||
| 4.b | odd | 2 | 1 | 112.10.i.c | 10 | ||
| 7.b | odd | 2 | 1 | 49.10.c.g | 10 | ||
| 7.c | even | 3 | 1 | inner | 7.10.c.a | ✓ | 10 |
| 7.c | even | 3 | 1 | 49.10.a.e | 5 | ||
| 7.d | odd | 6 | 1 | 49.10.a.f | 5 | ||
| 7.d | odd | 6 | 1 | 49.10.c.g | 10 | ||
| 21.h | odd | 6 | 1 | 63.10.e.b | 10 | ||
| 28.g | odd | 6 | 1 | 112.10.i.c | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.10.c.a | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 7.10.c.a | ✓ | 10 | 7.c | even | 3 | 1 | inner |
| 49.10.a.e | 5 | 7.c | even | 3 | 1 | ||
| 49.10.a.f | 5 | 7.d | odd | 6 | 1 | ||
| 49.10.c.g | 10 | 7.b | odd | 2 | 1 | ||
| 49.10.c.g | 10 | 7.d | odd | 6 | 1 | ||
| 63.10.e.b | 10 | 3.b | odd | 2 | 1 | ||
| 63.10.e.b | 10 | 21.h | odd | 6 | 1 | ||
| 112.10.i.c | 10 | 4.b | odd | 2 | 1 | ||
| 112.10.i.c | 10 | 28.g | odd | 6 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(7, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + \cdots + 10212166139904 \)
|
| $3$ |
\( T^{10} + \cdots + 86\!\cdots\!89 \)
|
| $5$ |
\( T^{10} + \cdots + 10\!\cdots\!25 \)
|
| $7$ |
\( T^{10} + \cdots + 10\!\cdots\!07 \)
|
| $11$ |
\( T^{10} + \cdots + 91\!\cdots\!25 \)
|
| $13$ |
\( (T^{5} + \cdots + 79\!\cdots\!92)^{2} \)
|
| $17$ |
\( T^{10} + \cdots + 29\!\cdots\!81 \)
|
| $19$ |
\( T^{10} + \cdots + 39\!\cdots\!01 \)
|
| $23$ |
\( T^{10} + \cdots + 42\!\cdots\!01 \)
|
| $29$ |
\( (T^{5} + \cdots + 73\!\cdots\!00)^{2} \)
|
| $31$ |
\( T^{10} + \cdots + 14\!\cdots\!69 \)
|
| $37$ |
\( T^{10} + \cdots + 47\!\cdots\!25 \)
|
| $41$ |
\( (T^{5} + \cdots - 20\!\cdots\!40)^{2} \)
|
| $43$ |
\( (T^{5} + \cdots + 72\!\cdots\!36)^{2} \)
|
| $47$ |
\( T^{10} + \cdots + 34\!\cdots\!25 \)
|
| $53$ |
\( T^{10} + \cdots + 47\!\cdots\!21 \)
|
| $59$ |
\( T^{10} + \cdots + 79\!\cdots\!25 \)
|
| $61$ |
\( T^{10} + \cdots + 27\!\cdots\!41 \)
|
| $67$ |
\( T^{10} + \cdots + 20\!\cdots\!25 \)
|
| $71$ |
\( (T^{5} + \cdots + 25\!\cdots\!92)^{2} \)
|
| $73$ |
\( T^{10} + \cdots + 87\!\cdots\!29 \)
|
| $79$ |
\( T^{10} + \cdots + 26\!\cdots\!21 \)
|
| $83$ |
\( (T^{5} + \cdots + 39\!\cdots\!36)^{2} \)
|
| $89$ |
\( T^{10} + \cdots + 21\!\cdots\!41 \)
|
| $97$ |
\( (T^{5} + \cdots - 84\!\cdots\!56)^{2} \)
|
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