Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [12,19,Mod(5,12)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(12, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 19, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("12.5");
S:= CuspForms(chi, 19);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 12 = 2^{2} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 19 \) |
| Character orbit: | \([\chi]\) | \(=\) | 12.c (of order \(2\), degree \(1\), 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: | \(24.6463365252\) |
| 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: |
\( x^{6} + 132762x^{4} + 1042140330x^{2} + 1430023595000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{32}\cdot 3^{19}\cdot 5^{2} \) |
| 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
\( x^{6} + 132762x^{4} + 1042140330x^{2} + 1430023595000 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} - 140\nu^{4} + 152362\nu^{3} - 21330680\nu^{2} + 4028435530\nu - 218699406800 ) / 14586075 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2494\nu^{5} - 560\nu^{4} + 322837228\nu^{3} - 85322720\nu^{2} + 1562923520620\nu - 874797627200 ) / 14586075 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 90352 \nu^{5} + 12040 \nu^{4} + 12337371424 \nu^{3} + 1834438480 \nu^{2} + 115582517582560 \nu + 18808148984800 ) / 14586075 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 229\nu^{5} + 35980\nu^{4} + 34890898\nu^{3} + 5481984760\nu^{2} + 922511736370\nu + 56205747547600 ) / 4862025 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 22\nu^{5} + 588112\nu^{4} + 3351964\nu^{3} + 75203213344\nu^{2} + 88625581660\nu + 281335063656640 ) / 2917215 \)
|
| \(\nu\) | \(=\) |
\( ( 58\beta_{4} - 3\beta_{3} + 75\beta_{2} + 44160\beta_1 ) / 7464960 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -189\beta_{5} + 2737\beta_{4} - 1859529\beta _1 - 41294292480 ) / 933120 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2678018\beta_{4} + 222663\beta_{3} - 6519135\beta_{2} - 2019526320\beta_1 ) / 3732480 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 14398209\beta_{5} - 175167797\beta_{4} + 118756473549\beta _1 + 2417009100871680 ) / 466560 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 109514633382\beta_{4} - 9294242237\beta_{3} + 280734038165\beta_{2} + 82513541352480\beta_1 ) / 1244160 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) | \(7\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 5.1 |
|
0 | −14327.9 | − | 13495.6i | 0 | 480487.i | 0 | −7.51877e6 | 0 | 2.31580e7 | + | 3.86728e8i | 0 | ||||||||||||||||||||||||||||||||
| 5.2 | 0 | −14327.9 | + | 13495.6i | 0 | − | 480487.i | 0 | −7.51877e6 | 0 | 2.31580e7 | − | 3.86728e8i | 0 | ||||||||||||||||||||||||||||||||
| 5.3 | 0 | 9860.33 | − | 17035.1i | 0 | − | 2.47556e6i | 0 | 6.41846e7 | 0 | −1.92968e8 | − | 3.35943e8i | 0 | ||||||||||||||||||||||||||||||||
| 5.4 | 0 | 9860.33 | + | 17035.1i | 0 | 2.47556e6i | 0 | 6.41846e7 | 0 | −1.92968e8 | + | 3.35943e8i | 0 | |||||||||||||||||||||||||||||||||
| 5.5 | 0 | 16434.6 | − | 10831.7i | 0 | 2.83969e6i | 0 | −5.11537e7 | 0 | 1.52771e8 | − | 3.56028e8i | 0 | |||||||||||||||||||||||||||||||||
| 5.6 | 0 | 16434.6 | + | 10831.7i | 0 | − | 2.83969e6i | 0 | −5.11537e7 | 0 | 1.52771e8 | + | 3.56028e8i | 0 | ||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 12.19.c.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | inner | 12.19.c.a | ✓ | 6 |
| 4.b | odd | 2 | 1 | 48.19.e.c | 6 | ||
| 12.b | even | 2 | 1 | 48.19.e.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 12.19.c.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 12.19.c.a | ✓ | 6 | 3.b | odd | 2 | 1 | inner |
| 48.19.e.c | 6 | 4.b | odd | 2 | 1 | ||
| 48.19.e.c | 6 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{19}^{\mathrm{new}}(12, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + \cdots + 58\!\cdots\!69 \)
|
| $5$ |
\( T^{6} + \cdots + 11\!\cdots\!00 \)
|
| $7$ |
\( (T^{3} + \cdots - 24\!\cdots\!52)^{2} \)
|
| $11$ |
\( T^{6} + \cdots + 72\!\cdots\!00 \)
|
| $13$ |
\( (T^{3} + \cdots - 20\!\cdots\!28)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 29\!\cdots\!00 \)
|
| $19$ |
\( (T^{3} + \cdots + 10\!\cdots\!04)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 31\!\cdots\!00 \)
|
| $29$ |
\( T^{6} + \cdots + 73\!\cdots\!00 \)
|
| $31$ |
\( (T^{3} + \cdots - 22\!\cdots\!84)^{2} \)
|
| $37$ |
\( (T^{3} + \cdots - 93\!\cdots\!84)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 12\!\cdots\!00 \)
|
| $43$ |
\( (T^{3} + \cdots + 61\!\cdots\!92)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 14\!\cdots\!00 \)
|
| $53$ |
\( T^{6} + \cdots + 18\!\cdots\!00 \)
|
| $59$ |
\( T^{6} + \cdots + 27\!\cdots\!00 \)
|
| $61$ |
\( (T^{3} + \cdots + 75\!\cdots\!56)^{2} \)
|
| $67$ |
\( (T^{3} + \cdots + 26\!\cdots\!08)^{2} \)
|
| $71$ |
\( T^{6} + \cdots + 18\!\cdots\!00 \)
|
| $73$ |
\( (T^{3} + \cdots + 50\!\cdots\!64)^{2} \)
|
| $79$ |
\( (T^{3} + \cdots - 48\!\cdots\!52)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 80\!\cdots\!00 \)
|
| $89$ |
\( T^{6} + \cdots + 86\!\cdots\!00 \)
|
| $97$ |
\( (T^{3} + \cdots + 17\!\cdots\!88)^{2} \)
|
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