Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [120,4,Mod(61,120)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(120, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("120.61");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 120 = 2^{3} \cdot 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 120.k (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: | \(7.08022920069\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.55839580416.4 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} + 7x^{6} - 6x^{5} + 18x^{4} - 24x^{3} + 112x^{2} - 128x + 256 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 2x^{7} + 7x^{6} - 6x^{5} + 18x^{4} - 24x^{3} + 112x^{2} - 128x + 256 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 23\nu^{7} + 70\nu^{6} + 201\nu^{5} - 126\nu^{4} + 342\nu^{3} + 672\nu^{2} + 80\nu + 1216 ) / 2880 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -29\nu^{7} - 10\nu^{6} - 3\nu^{5} + 18\nu^{4} - 306\nu^{3} + 624\nu^{2} - 1040\nu - 1408 ) / 2880 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 8\nu^{7} - 5\nu^{6} - 24\nu^{5} + 9\nu^{4} + 72\nu^{3} + 402\nu^{2} + 380\nu + 1096 ) / 360 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -67\nu^{7} + 250\nu^{6} - 429\nu^{5} + 414\nu^{4} - 1278\nu^{3} + 2832\nu^{2} - 4240\nu + 9856 ) / 2880 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 67\nu^{7} - 250\nu^{6} + 429\nu^{5} - 414\nu^{4} + 1278\nu^{3} - 2832\nu^{2} + 10000\nu - 12736 ) / 2880 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -13\nu^{7} + 34\nu^{6} - 51\nu^{5} + 54\nu^{4} - 18\nu^{3} + 312\nu^{2} - 928\nu + 1504 ) / 288 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 137\nu^{7} + 370\nu^{6} + 39\nu^{5} + 1926\nu^{4} + 1818\nu^{3} + 3408\nu^{2} + 5360\nu + 27904 ) / 2880 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{4} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 3\beta_{2} + \beta _1 - 2 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{6} + \beta_{5} - 3\beta_{4} + \beta_{3} - 2\beta_{2} - 5 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 3\beta_{7} - \beta_{6} - 2\beta_{5} - 3\beta_{4} - 2\beta_{3} + 11\beta_{2} - 7\beta _1 - 8 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2\beta_{7} - 3\beta_{6} - \beta_{5} - 3\beta_{4} - 5\beta_{3} + 26\beta_{2} + 12\beta _1 + 25 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -\beta_{7} + 3\beta_{6} + 10\beta_{5} + 21\beta_{4} - 8\beta_{3} - 43\beta_{2} + 23\beta _1 - 40 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 2\beta_{7} - 33\beta_{6} - 51\beta_{5} - 13\beta_{4} + 13\beta_{3} - 94\beta_{2} + 8\beta _1 - 117 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/120\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(41\) | \(61\) | \(97\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 61.1 |
|
−1.77053 | − | 2.20573i | 3.00000i | −1.73045 | + | 7.81060i | 5.00000i | 6.61718 | − | 5.31159i | −18.2117 | 20.2919 | − | 10.0120i | −9.00000 | 11.0286 | − | 8.85264i | ||||||||||||||||||||||||||||||||
| 61.2 | −1.77053 | + | 2.20573i | − | 3.00000i | −1.73045 | − | 7.81060i | − | 5.00000i | 6.61718 | + | 5.31159i | −18.2117 | 20.2919 | + | 10.0120i | −9.00000 | 11.0286 | + | 8.85264i | |||||||||||||||||||||||||||||||
| 61.3 | −0.246076 | − | 2.81770i | − | 3.00000i | −7.87889 | + | 1.38674i | − | 5.00000i | −8.45311 | + | 0.738229i | −19.1119 | 5.84623 | + | 21.8591i | −9.00000 | −14.0885 | + | 1.23038i | |||||||||||||||||||||||||||||||
| 61.4 | −0.246076 | + | 2.81770i | 3.00000i | −7.87889 | − | 1.38674i | 5.00000i | −8.45311 | − | 0.738229i | −19.1119 | 5.84623 | − | 21.8591i | −9.00000 | −14.0885 | − | 1.23038i | |||||||||||||||||||||||||||||||||
| 61.5 | 0.446531 | − | 2.79296i | 3.00000i | −7.60122 | − | 2.49428i | 5.00000i | 8.37887 | + | 1.33959i | −3.73490 | −10.3606 | + | 20.1161i | −9.00000 | 13.9648 | + | 2.23265i | |||||||||||||||||||||||||||||||||
| 61.6 | 0.446531 | + | 2.79296i | − | 3.00000i | −7.60122 | + | 2.49428i | − | 5.00000i | 8.37887 | − | 1.33959i | −3.73490 | −10.3606 | − | 20.1161i | −9.00000 | 13.9648 | − | 2.23265i | |||||||||||||||||||||||||||||||
| 61.7 | 2.57007 | − | 1.18098i | − | 3.00000i | 5.21057 | − | 6.07042i | − | 5.00000i | −3.54294 | − | 7.71022i | 1.05849 | 6.22250 | − | 21.7550i | −9.00000 | −5.90490 | − | 12.8504i | |||||||||||||||||||||||||||||||
| 61.8 | 2.57007 | + | 1.18098i | 3.00000i | 5.21057 | + | 6.07042i | 5.00000i | −3.54294 | + | 7.71022i | 1.05849 | 6.22250 | + | 21.7550i | −9.00000 | −5.90490 | + | 12.8504i | |||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 120.4.k.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 360.4.k.b | 8 | ||
| 4.b | odd | 2 | 1 | 480.4.k.b | 8 | ||
| 8.b | even | 2 | 1 | inner | 120.4.k.b | ✓ | 8 |
| 8.d | odd | 2 | 1 | 480.4.k.b | 8 | ||
| 12.b | even | 2 | 1 | 1440.4.k.b | 8 | ||
| 24.f | even | 2 | 1 | 1440.4.k.b | 8 | ||
| 24.h | odd | 2 | 1 | 360.4.k.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 120.4.k.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 120.4.k.b | ✓ | 8 | 8.b | even | 2 | 1 | inner |
| 360.4.k.b | 8 | 3.b | odd | 2 | 1 | ||
| 360.4.k.b | 8 | 24.h | odd | 2 | 1 | ||
| 480.4.k.b | 8 | 4.b | odd | 2 | 1 | ||
| 480.4.k.b | 8 | 8.d | odd | 2 | 1 | ||
| 1440.4.k.b | 8 | 12.b | even | 2 | 1 | ||
| 1440.4.k.b | 8 | 24.f | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} + 40T_{7}^{3} + 444T_{7}^{2} + 784T_{7} - 1376 \)
acting on \(S_{4}^{\mathrm{new}}(120, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 2 T^{7} + \cdots + 4096 \)
|
| $3$ |
\( (T^{2} + 9)^{4} \)
|
| $5$ |
\( (T^{2} + 25)^{4} \)
|
| $7$ |
\( (T^{4} + 40 T^{3} + \cdots - 1376)^{2} \)
|
| $11$ |
\( T^{8} + \cdots + 62236278784 \)
|
| $13$ |
\( T^{8} + \cdots + 38353305600 \)
|
| $17$ |
\( (T^{4} + 108 T^{3} + \cdots - 61632)^{2} \)
|
| $19$ |
\( T^{8} + \cdots + 31548801716224 \)
|
| $23$ |
\( (T^{4} + 16 T^{3} + \cdots + 7277888)^{2} \)
|
| $29$ |
\( T^{8} + \cdots + 29\!\cdots\!64 \)
|
| $31$ |
\( (T^{4} + 68 T^{3} + \cdots - 140627840)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 82\!\cdots\!84 \)
|
| $41$ |
\( (T^{4} - 88 T^{3} + \cdots - 3143107120)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 70\!\cdots\!16 \)
|
| $47$ |
\( (T^{4} + 424 T^{3} + \cdots - 10676481472)^{2} \)
|
| $53$ |
\( T^{8} + \cdots + 21\!\cdots\!00 \)
|
| $59$ |
\( T^{8} + \cdots + 11\!\cdots\!44 \)
|
| $61$ |
\( T^{8} + \cdots + 33\!\cdots\!64 \)
|
| $67$ |
\( T^{8} + \cdots + 90\!\cdots\!64 \)
|
| $71$ |
\( (T^{4} - 1544 T^{3} + \cdots - 11209493248)^{2} \)
|
| $73$ |
\( (T^{4} + 248 T^{3} + \cdots + 112584267280)^{2} \)
|
| $79$ |
\( (T^{4} + 1028 T^{3} + \cdots - 443106379136)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 47\!\cdots\!00 \)
|
| $89$ |
\( (T^{4} - 1736 T^{3} + \cdots - 68323863856)^{2} \)
|
| $97$ |
\( (T^{4} + \cdots - 1040188142960)^{2} \)
|
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