Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [350,6,Mod(99,350)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(350, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("350.99");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 350 = 2 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 350.c (of order \(2\), degree \(1\), not 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: | \(56.1343369345\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} + 2x^{4} + 240x^{3} + 41209x^{2} - 33698x + 13778 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 5^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} + 2x^{4} + 240x^{3} + 41209x^{2} - 33698x + 13778 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 58\nu^{5} + 8111\nu^{4} + 11890\nu^{3} + 6960\nu^{2} - 9628\nu + 225301659 ) / 11854465 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -8299\nu^{5} + 13195\nu^{4} - 7800\nu^{3} - 2689375\nu^{2} - 342401851\nu + 139664515 ) / 140560085 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2638\nu^{5} - 39864\nu^{4} + 540790\nu^{3} + 316560\nu^{2} - 437908\nu - 938368691 ) / 11854465 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 1113559 \nu^{5} + 1763729 \nu^{4} + 342230 \nu^{3} - 221688820 \nu^{2} - 45663139341 \nu + 18626022281 ) / 983920595 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 4401981 \nu^{5} - 7208951 \nu^{4} + 47191130 \nu^{3} + 821212355 \nu^{2} + 190307361619 \nu - 77619884639 ) / 983920595 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + 4\beta_{4} - \beta_{3} - 5\beta_{2} - 4\beta _1 + 5 ) / 20 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{5} + 31\beta_{4} - 670\beta_{2} ) / 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 199\beta_{5} + 936\beta_{4} + 199\beta_{3} - 2035\beta_{2} + 936\beta _1 - 2035 ) / 20 \)
|
| \(\nu^{4}\) | \(=\) |
\( -29\beta_{3} + 1319\beta _1 - 27364 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -40149\beta_{5} - 206096\beta_{4} + 40149\beta_{3} + 737945\beta_{2} + 206096\beta _1 - 737945 ) / 20 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 99.1 |
|
− | 4.00000i | − | 19.0051i | −16.0000 | 0 | −76.0204 | 49.0000i | 64.0000i | −118.194 | 0 | ||||||||||||||||||||||||||||||||||
| 99.2 | − | 4.00000i | 2.52911i | −16.0000 | 0 | 10.1164 | 49.0000i | 64.0000i | 236.604 | 0 | ||||||||||||||||||||||||||||||||||||
| 99.3 | − | 4.00000i | 17.4760i | −16.0000 | 0 | 69.9039 | 49.0000i | 64.0000i | −62.4100 | 0 | ||||||||||||||||||||||||||||||||||||
| 99.4 | 4.00000i | − | 17.4760i | −16.0000 | 0 | 69.9039 | − | 49.0000i | − | 64.0000i | −62.4100 | 0 | ||||||||||||||||||||||||||||||||||
| 99.5 | 4.00000i | − | 2.52911i | −16.0000 | 0 | 10.1164 | − | 49.0000i | − | 64.0000i | 236.604 | 0 | ||||||||||||||||||||||||||||||||||
| 99.6 | 4.00000i | 19.0051i | −16.0000 | 0 | −76.0204 | − | 49.0000i | − | 64.0000i | −118.194 | 0 | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 350.6.c.m | 6 | |
| 5.b | even | 2 | 1 | inner | 350.6.c.m | 6 | |
| 5.c | odd | 4 | 1 | 350.6.a.v | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 350.6.a.w | yes | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 350.6.a.v | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 350.6.a.w | yes | 3 | 5.c | odd | 4 | 1 | |
| 350.6.c.m | 6 | 1.a | even | 1 | 1 | trivial | |
| 350.6.c.m | 6 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(350, [\chi])\):
|
\( T_{3}^{6} + 673T_{3}^{4} + 114576T_{3}^{2} + 705600 \)
|
|
\( T_{11}^{3} + 11T_{11}^{2} - 221125T_{11} - 40799375 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 16)^{3} \)
|
| $3$ |
\( T^{6} + 673 T^{4} + \cdots + 705600 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( (T^{2} + 2401)^{3} \)
|
| $11$ |
\( (T^{3} + 11 T^{2} + \cdots - 40799375)^{2} \)
|
| $13$ |
\( T^{6} + \cdots + 16\!\cdots\!56 \)
|
| $17$ |
\( T^{6} + \cdots + 15\!\cdots\!04 \)
|
| $19$ |
\( (T^{3} - 1811 T^{2} + \cdots + 2760714180)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 41\!\cdots\!00 \)
|
| $29$ |
\( (T^{3} + 1924 T^{2} + \cdots - 154187630694)^{2} \)
|
| $31$ |
\( (T^{3} + 6560 T^{2} + \cdots + 6751381392)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 50\!\cdots\!00 \)
|
| $41$ |
\( (T^{3} + 26981 T^{2} + \cdots - 873981504312)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 11\!\cdots\!36 \)
|
| $47$ |
\( T^{6} + \cdots + 30\!\cdots\!24 \)
|
| $53$ |
\( T^{6} + \cdots + 54\!\cdots\!84 \)
|
| $59$ |
\( (T^{3} + \cdots - 51424466453808)^{2} \)
|
| $61$ |
\( (T^{3} + \cdots - 17091016362224)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 19\!\cdots\!09 \)
|
| $71$ |
\( (T^{3} + \cdots - 88582122380350)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 17\!\cdots\!84 \)
|
| $79$ |
\( (T^{3} + \cdots + 8297801992798)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 22\!\cdots\!44 \)
|
| $89$ |
\( (T^{3} + \cdots - 560143115269220)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 68\!\cdots\!00 \)
|
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