Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [306,4,Mod(55,306)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(306, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("306.55");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 306 = 2 \cdot 3^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 306.g (of order \(4\), 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: | \(18.0545844618\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} + 2x^{6} - 114x^{5} + 3058x^{4} - 12578x^{3} + 25538x^{2} + 8814x + 1521 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 2x^{7} + 2x^{6} - 114x^{5} + 3058x^{4} - 12578x^{3} + 25538x^{2} + 8814x + 1521 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 4685 \nu^{7} + 2245 \nu^{6} - 134895 \nu^{5} - 1704897 \nu^{4} + 6165305 \nu^{3} + \cdots - 728296605 ) / 171478116 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 573587 \nu^{7} - 1343851 \nu^{6} + 1157275 \nu^{5} - 65778099 \nu^{4} + 1776591443 \nu^{3} + \cdots + 2542582029 ) / 2972287344 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 3574617 \nu^{7} + 9311953 \nu^{6} - 3031185 \nu^{5} + 414187389 \nu^{4} - 11322797793 \nu^{3} + \cdots - 16174502019 ) / 4458431016 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 13031969 \nu^{7} - 29500033 \nu^{6} + 59401281 \nu^{5} - 1434249369 \nu^{4} + 40305176801 \nu^{3} + \cdots - 9776855673 ) / 8916862032 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 719387 \nu^{7} - 1691371 \nu^{6} + 2114511 \nu^{5} - 82844019 \nu^{4} + 2228522843 \nu^{3} + \cdots - 538852275 ) / 342956232 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 10503593 \nu^{7} - 16644633 \nu^{6} + 21282045 \nu^{5} - 1217713977 \nu^{4} + \cdots + 79022574423 ) / 4458431016 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 850429 \nu^{7} + 1686369 \nu^{6} - 1693353 \nu^{5} + 96528105 \nu^{4} - 2595266053 \nu^{3} + \cdots - 7447081239 ) / 342956232 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -7\beta_{7} - 3\beta_{6} + 7\beta_{5} - 3\beta_{4} - 2\beta_{3} - 115\beta_{2} ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 65\beta_{7} + 41\beta_{6} - 53\beta_{3} + 122\beta_{2} - 53\beta _1 + 163 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( -110\beta_{7} - 60\beta_{6} - 110\beta_{5} + 60\beta_{4} + 60\beta_{2} + 2\beta _1 - 1475 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 4303\beta_{5} - 2455\beta_{4} + 2695\beta_{3} - 16601\beta_{2} - 2695\beta _1 + 16601 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 27409\beta_{7} + 15501\beta_{6} - 27409\beta_{5} + 15501\beta_{4} - 7114\beta_{3} + 313933\beta_{2} ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -283055\beta_{7} - 159119\beta_{6} + 142643\beta_{3} - 1101446\beta_{2} + 142643\beta _1 - 1260565 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/306\mathbb{Z}\right)^\times\).
| \(n\) | \(37\) | \(137\) |
| \(\chi(n)\) | \(-\beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 55.1 |
|
− | 2.00000i | 0 | −4.00000 | −13.7126 | + | 13.7126i | 0 | −6.84354 | − | 6.84354i | 8.00000i | 0 | 27.4252 | + | 27.4252i | |||||||||||||||||||||||||||||||||||
| 55.2 | − | 2.00000i | 0 | −4.00000 | −0.580334 | + | 0.580334i | 0 | 13.3735 | + | 13.3735i | 8.00000i | 0 | 1.16067 | + | 1.16067i | ||||||||||||||||||||||||||||||||||||
| 55.3 | − | 2.00000i | 0 | −4.00000 | −0.376810 | + | 0.376810i | 0 | −15.9157 | − | 15.9157i | 8.00000i | 0 | 0.753619 | + | 0.753619i | ||||||||||||||||||||||||||||||||||||
| 55.4 | − | 2.00000i | 0 | −4.00000 | 6.66975 | − | 6.66975i | 0 | 5.38568 | + | 5.38568i | 8.00000i | 0 | −13.3395 | − | 13.3395i | ||||||||||||||||||||||||||||||||||||
| 217.1 | 2.00000i | 0 | −4.00000 | −13.7126 | − | 13.7126i | 0 | −6.84354 | + | 6.84354i | − | 8.00000i | 0 | 27.4252 | − | 27.4252i | ||||||||||||||||||||||||||||||||||||
| 217.2 | 2.00000i | 0 | −4.00000 | −0.580334 | − | 0.580334i | 0 | 13.3735 | − | 13.3735i | − | 8.00000i | 0 | 1.16067 | − | 1.16067i | ||||||||||||||||||||||||||||||||||||
| 217.3 | 2.00000i | 0 | −4.00000 | −0.376810 | − | 0.376810i | 0 | −15.9157 | + | 15.9157i | − | 8.00000i | 0 | 0.753619 | − | 0.753619i | ||||||||||||||||||||||||||||||||||||
| 217.4 | 2.00000i | 0 | −4.00000 | 6.66975 | + | 6.66975i | 0 | 5.38568 | − | 5.38568i | − | 8.00000i | 0 | −13.3395 | + | 13.3395i | ||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.c | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 306.4.g.e | ✓ | 8 |
| 3.b | odd | 2 | 1 | 306.4.g.f | yes | 8 | |
| 17.c | even | 4 | 1 | inner | 306.4.g.e | ✓ | 8 |
| 51.f | odd | 4 | 1 | 306.4.g.f | yes | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 306.4.g.e | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 306.4.g.e | ✓ | 8 | 17.c | even | 4 | 1 | inner |
| 306.4.g.f | yes | 8 | 3.b | odd | 2 | 1 | |
| 306.4.g.f | yes | 8 | 51.f | odd | 4 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 16T_{5}^{7} + 128T_{5}^{6} - 2360T_{5}^{5} + 28721T_{5}^{4} + 59416T_{5}^{3} + 59168T_{5}^{2} + 27520T_{5} + 6400 \)
acting on \(S_{4}^{\mathrm{new}}(306, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 4)^{4} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + 16 T^{7} + \cdots + 6400 \)
|
| $7$ |
\( T^{8} + 8 T^{7} + \cdots + 984704400 \)
|
| $11$ |
\( T^{8} + \cdots + 33694273600 \)
|
| $13$ |
\( (T^{4} + 14 T^{3} + \cdots + 162420)^{2} \)
|
| $17$ |
\( T^{8} + \cdots + 582622237229761 \)
|
| $19$ |
\( T^{8} + \cdots + 704371600000000 \)
|
| $23$ |
\( T^{8} + \cdots + 548769824100 \)
|
| $29$ |
\( T^{8} + \cdots + 13\!\cdots\!00 \)
|
| $31$ |
\( T^{8} + \cdots + 43\!\cdots\!16 \)
|
| $37$ |
\( T^{8} + \cdots + 82\!\cdots\!00 \)
|
| $41$ |
\( T^{8} + \cdots + 15\!\cdots\!00 \)
|
| $43$ |
\( T^{8} + \cdots + 11\!\cdots\!00 \)
|
| $47$ |
\( (T^{4} + 628 T^{3} + \cdots + 388464032)^{2} \)
|
| $53$ |
\( T^{8} + \cdots + 31\!\cdots\!84 \)
|
| $59$ |
\( T^{8} + \cdots + 20\!\cdots\!00 \)
|
| $61$ |
\( T^{8} + \cdots + 47\!\cdots\!76 \)
|
| $67$ |
\( (T^{4} + 84 T^{3} + \cdots + 8186232960)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 23\!\cdots\!00 \)
|
| $73$ |
\( T^{8} + \cdots + 14\!\cdots\!00 \)
|
| $79$ |
\( T^{8} + \cdots + 14\!\cdots\!00 \)
|
| $83$ |
\( T^{8} + \cdots + 34\!\cdots\!00 \)
|
| $89$ |
\( (T^{4} + 888 T^{3} + \cdots + 154468827360)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 36\!\cdots\!00 \)
|
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