Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [60,2,Mod(7,60)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(60, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("60.7");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 60 = 2^{2} \cdot 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 60.j (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: | \(0.479102412128\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(6\) over \(\Q(i)\) |
| Coefficient field: | 12.0.426337261060096.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 4x^{9} - 3x^{8} + 4x^{7} + 8x^{6} + 8x^{5} - 12x^{4} - 32x^{3} + 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 4x^{9} - 3x^{8} + 4x^{7} + 8x^{6} + 8x^{5} - 12x^{4} - 32x^{3} + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{10} - \nu^{9} - 4\nu^{8} + 2\nu^{7} + 3\nu^{6} - 5\nu^{5} - 24\nu^{4} - 2\nu^{3} + 8\nu^{2} + 24\nu + 48 ) / 80 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{11} + \nu^{10} + 4\nu^{9} - 2\nu^{8} - 3\nu^{7} + 5\nu^{6} + 24\nu^{5} + 2\nu^{4} - 8\nu^{3} - 24\nu^{2} - 48\nu ) / 160 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - \nu^{10} + 9 \nu^{9} + 6 \nu^{8} + 2 \nu^{7} - 17 \nu^{6} - 35 \nu^{5} + 26 \nu^{4} + 38 \nu^{3} + \cdots - 192 ) / 80 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - \nu^{11} - \nu^{10} + \nu^{9} + 2 \nu^{8} + 13 \nu^{7} - 5 \nu^{6} - 19 \nu^{5} - 22 \nu^{4} + \cdots + 88 \nu ) / 80 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{11} + 4\nu^{10} + 2\nu^{9} + \nu^{7} + 16\nu^{6} - 6\nu^{5} + 4\nu^{4} + 8\nu^{3} - 48\nu^{2} - 160\nu - 64 ) / 160 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{11} + 4\nu^{8} + 3\nu^{7} - 4\nu^{6} - 8\nu^{5} - 8\nu^{4} + 12\nu^{3} + 32\nu^{2} ) / 32 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - \nu^{11} + 9 \nu^{10} + 6 \nu^{9} + 2 \nu^{8} - 17 \nu^{7} - 35 \nu^{6} + 26 \nu^{5} + \cdots - 192 \nu ) / 160 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 3\nu^{11} + \nu^{10} - 4\nu^{8} - 5\nu^{7} + \nu^{6} + 12\nu^{5} + 8\nu^{4} + 12\nu^{3} - 36\nu^{2} - 16\nu - 64 ) / 80 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 9 \nu^{11} + 6 \nu^{10} - 2 \nu^{9} - 20 \nu^{8} - 31 \nu^{7} + 34 \nu^{6} + 46 \nu^{5} + 16 \nu^{4} + \cdots + 64 ) / 160 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - \nu^{11} - 3 \nu^{10} - \nu^{9} + 4 \nu^{8} + 7 \nu^{7} + \nu^{6} - 9 \nu^{5} - 20 \nu^{4} + \cdots + 16 ) / 40 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 11 \nu^{11} + 6 \nu^{10} + 14 \nu^{9} - 12 \nu^{8} - 53 \nu^{7} + 10 \nu^{6} + 14 \nu^{5} + 112 \nu^{4} + \cdots - 320 ) / 160 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{11} - \beta_{9} + \beta_{4} - \beta_{3} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{11} - 2\beta_{10} + \beta_{9} - 2\beta_{7} + 2\beta_{6} - 2\beta_{5} - \beta_{4} + \beta_{3} + 2\beta_{2} ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{11} - 3\beta_{9} + 2\beta_{8} + 2\beta_{5} - \beta_{4} - \beta_{3} + 2\beta _1 + 2 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -\beta_{11} - 2\beta_{10} - \beta_{9} - 2\beta_{7} - 2\beta_{6} - \beta_{4} + \beta_{3} - 2\beta_{2} - 4\beta _1 + 4 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -\beta_{11} - \beta_{9} + 2\beta_{8} - 2\beta_{5} - \beta_{4} - \beta_{3} + 8\beta_{2} - 2\beta _1 - 2 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 3 \beta_{11} - 2 \beta_{10} - 3 \beta_{9} - 4 \beta_{8} - 6 \beta_{7} + 2 \beta_{6} + 6 \beta_{5} + \cdots + 6 \beta_{2} ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -5\beta_{11} - 5\beta_{9} + 10\beta_{8} - 8\beta_{6} + 10\beta_{5} + 5\beta_{4} + \beta_{3} - 6\beta _1 + 10 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 5 \beta_{11} + 10 \beta_{10} + 5 \beta_{9} + 2 \beta_{7} + 10 \beta_{6} + 5 \beta_{4} - 5 \beta_{3} + \cdots + 4 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 9\beta_{11} - 7\beta_{9} - 10\beta_{8} + 2\beta_{5} + 9\beta_{4} + \beta_{3} + 40\beta_{2} + 10\beta _1 + 10 ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 13 \beta_{11} - 14 \beta_{10} - 13 \beta_{9} + 4 \beta_{8} + 6 \beta_{7} + 14 \beta_{6} + \cdots - 6 \beta_{2} ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 11 \beta_{11} + 29 \beta_{9} + 54 \beta_{8} - 16 \beta_{7} + 24 \beta_{6} - 18 \beta_{5} + 11 \beta_{4} + \cdots + 54 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/60\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(37\) | \(41\) |
| \(\chi(n)\) | \(-1\) | \(\beta_{8}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
−1.41127 | + | 0.0912546i | −0.707107 | − | 0.707107i | 1.98335 | − | 0.257569i | 1.32001 | − | 1.80487i | 1.06244 | + | 0.933389i | 1.86678 | − | 1.86678i | −2.77552 | + | 0.544488i | 1.00000i | −1.69819 | + | 2.66761i | ||||||||||||||||||||||||||||||||||||||
| 7.2 | −1.19252 | + | 0.760198i | 0.707107 | + | 0.707107i | 0.844199 | − | 1.81310i | 0.432320 | + | 2.19388i | −1.38078 | − | 0.305697i | −0.611393 | + | 0.611393i | 0.371591 | + | 2.80391i | 1.00000i | −2.18333 | − | 2.28759i | |||||||||||||||||||||||||||||||||||||||
| 7.3 | 0.0912546 | − | 1.41127i | 0.707107 | + | 0.707107i | −1.98335 | − | 0.257569i | 1.32001 | − | 1.80487i | 1.06244 | − | 0.933389i | −1.86678 | + | 1.86678i | −0.544488 | + | 2.77552i | 1.00000i | −2.42670 | − | 2.02759i | |||||||||||||||||||||||||||||||||||||||
| 7.4 | 0.394157 | + | 1.35818i | 0.707107 | + | 0.707107i | −1.68928 | + | 1.07067i | −1.75233 | − | 1.38900i | −0.681664 | + | 1.23909i | 2.47817 | − | 2.47817i | −2.12000 | − | 1.87233i | 1.00000i | 1.19582 | − | 2.92746i | |||||||||||||||||||||||||||||||||||||||
| 7.5 | 0.760198 | − | 1.19252i | −0.707107 | − | 0.707107i | −0.844199 | − | 1.81310i | 0.432320 | + | 2.19388i | −1.38078 | + | 0.305697i | 0.611393 | − | 0.611393i | −2.80391 | − | 0.371591i | 1.00000i | 2.94489 | + | 1.15223i | |||||||||||||||||||||||||||||||||||||||
| 7.6 | 1.35818 | + | 0.394157i | −0.707107 | − | 0.707107i | 1.68928 | + | 1.07067i | −1.75233 | − | 1.38900i | −0.681664 | − | 1.23909i | −2.47817 | + | 2.47817i | 1.87233 | + | 2.12000i | 1.00000i | −1.83249 | − | 2.57720i | |||||||||||||||||||||||||||||||||||||||
| 43.1 | −1.41127 | − | 0.0912546i | −0.707107 | + | 0.707107i | 1.98335 | + | 0.257569i | 1.32001 | + | 1.80487i | 1.06244 | − | 0.933389i | 1.86678 | + | 1.86678i | −2.77552 | − | 0.544488i | − | 1.00000i | −1.69819 | − | 2.66761i | ||||||||||||||||||||||||||||||||||||||
| 43.2 | −1.19252 | − | 0.760198i | 0.707107 | − | 0.707107i | 0.844199 | + | 1.81310i | 0.432320 | − | 2.19388i | −1.38078 | + | 0.305697i | −0.611393 | − | 0.611393i | 0.371591 | − | 2.80391i | − | 1.00000i | −2.18333 | + | 2.28759i | ||||||||||||||||||||||||||||||||||||||
| 43.3 | 0.0912546 | + | 1.41127i | 0.707107 | − | 0.707107i | −1.98335 | + | 0.257569i | 1.32001 | + | 1.80487i | 1.06244 | + | 0.933389i | −1.86678 | − | 1.86678i | −0.544488 | − | 2.77552i | − | 1.00000i | −2.42670 | + | 2.02759i | ||||||||||||||||||||||||||||||||||||||
| 43.4 | 0.394157 | − | 1.35818i | 0.707107 | − | 0.707107i | −1.68928 | − | 1.07067i | −1.75233 | + | 1.38900i | −0.681664 | − | 1.23909i | 2.47817 | + | 2.47817i | −2.12000 | + | 1.87233i | − | 1.00000i | 1.19582 | + | 2.92746i | ||||||||||||||||||||||||||||||||||||||
| 43.5 | 0.760198 | + | 1.19252i | −0.707107 | + | 0.707107i | −0.844199 | + | 1.81310i | 0.432320 | − | 2.19388i | −1.38078 | − | 0.305697i | 0.611393 | + | 0.611393i | −2.80391 | + | 0.371591i | − | 1.00000i | 2.94489 | − | 1.15223i | ||||||||||||||||||||||||||||||||||||||
| 43.6 | 1.35818 | − | 0.394157i | −0.707107 | + | 0.707107i | 1.68928 | − | 1.07067i | −1.75233 | + | 1.38900i | −0.681664 | + | 1.23909i | −2.47817 | − | 2.47817i | 1.87233 | − | 2.12000i | − | 1.00000i | −1.83249 | + | 2.57720i | ||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 5.c | odd | 4 | 1 | inner |
| 20.e | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 60.2.j.a | ✓ | 12 |
| 3.b | odd | 2 | 1 | 180.2.k.e | 12 | ||
| 4.b | odd | 2 | 1 | inner | 60.2.j.a | ✓ | 12 |
| 5.b | even | 2 | 1 | 300.2.j.d | 12 | ||
| 5.c | odd | 4 | 1 | inner | 60.2.j.a | ✓ | 12 |
| 5.c | odd | 4 | 1 | 300.2.j.d | 12 | ||
| 8.b | even | 2 | 1 | 960.2.w.g | 12 | ||
| 8.d | odd | 2 | 1 | 960.2.w.g | 12 | ||
| 12.b | even | 2 | 1 | 180.2.k.e | 12 | ||
| 15.d | odd | 2 | 1 | 900.2.k.n | 12 | ||
| 15.e | even | 4 | 1 | 180.2.k.e | 12 | ||
| 15.e | even | 4 | 1 | 900.2.k.n | 12 | ||
| 20.d | odd | 2 | 1 | 300.2.j.d | 12 | ||
| 20.e | even | 4 | 1 | inner | 60.2.j.a | ✓ | 12 |
| 20.e | even | 4 | 1 | 300.2.j.d | 12 | ||
| 40.i | odd | 4 | 1 | 960.2.w.g | 12 | ||
| 40.k | even | 4 | 1 | 960.2.w.g | 12 | ||
| 60.h | even | 2 | 1 | 900.2.k.n | 12 | ||
| 60.l | odd | 4 | 1 | 180.2.k.e | 12 | ||
| 60.l | odd | 4 | 1 | 900.2.k.n | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 60.2.j.a | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 60.2.j.a | ✓ | 12 | 4.b | odd | 2 | 1 | inner |
| 60.2.j.a | ✓ | 12 | 5.c | odd | 4 | 1 | inner |
| 60.2.j.a | ✓ | 12 | 20.e | even | 4 | 1 | inner |
| 180.2.k.e | 12 | 3.b | odd | 2 | 1 | ||
| 180.2.k.e | 12 | 12.b | even | 2 | 1 | ||
| 180.2.k.e | 12 | 15.e | even | 4 | 1 | ||
| 180.2.k.e | 12 | 60.l | odd | 4 | 1 | ||
| 300.2.j.d | 12 | 5.b | even | 2 | 1 | ||
| 300.2.j.d | 12 | 5.c | odd | 4 | 1 | ||
| 300.2.j.d | 12 | 20.d | odd | 2 | 1 | ||
| 300.2.j.d | 12 | 20.e | even | 4 | 1 | ||
| 900.2.k.n | 12 | 15.d | odd | 2 | 1 | ||
| 900.2.k.n | 12 | 15.e | even | 4 | 1 | ||
| 900.2.k.n | 12 | 60.h | even | 2 | 1 | ||
| 900.2.k.n | 12 | 60.l | odd | 4 | 1 | ||
| 960.2.w.g | 12 | 8.b | even | 2 | 1 | ||
| 960.2.w.g | 12 | 8.d | odd | 2 | 1 | ||
| 960.2.w.g | 12 | 40.i | odd | 4 | 1 | ||
| 960.2.w.g | 12 | 40.k | even | 4 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(60, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + 4 T^{9} + \cdots + 64 \)
|
| $3$ |
\( (T^{4} + 1)^{3} \)
|
| $5$ |
\( (T^{6} + 5 T^{4} + \cdots + 125)^{2} \)
|
| $7$ |
\( T^{12} + 200 T^{8} + \cdots + 4096 \)
|
| $11$ |
\( (T^{6} + 36 T^{4} + \cdots + 128)^{2} \)
|
| $13$ |
\( (T^{6} + 2 T^{5} + 2 T^{4} + \cdots + 32)^{2} \)
|
| $17$ |
\( (T^{6} + 10 T^{5} + \cdots + 800)^{2} \)
|
| $19$ |
\( (T^{6} - 40 T^{4} + \cdots - 512)^{2} \)
|
| $23$ |
\( T^{12} + 4640 T^{8} + \cdots + 65536 \)
|
| $29$ |
\( (T^{6} + 20 T^{4} + \cdots + 64)^{2} \)
|
| $31$ |
\( (T^{6} + 112 T^{4} + \cdots + 32768)^{2} \)
|
| $37$ |
\( (T^{6} - 2 T^{5} + 2 T^{4} + \cdots + 32)^{2} \)
|
| $41$ |
\( (T^{3} - 4 T^{2} - 20 T + 64)^{4} \)
|
| $43$ |
\( T^{12} + \cdots + 15352201216 \)
|
| $47$ |
\( T^{12} + 4896 T^{8} + \cdots + 40960000 \)
|
| $53$ |
\( (T^{6} - 2 T^{5} + \cdots + 128)^{2} \)
|
| $59$ |
\( (T^{6} - 100 T^{4} + \cdots - 512)^{2} \)
|
| $61$ |
\( (T^{3} + 8 T^{2} + \cdots + 176)^{4} \)
|
| $67$ |
\( T^{12} + \cdots + 15352201216 \)
|
| $71$ |
\( (T^{6} + 256 T^{4} + \cdots + 204800)^{2} \)
|
| $73$ |
\( (T^{6} - 22 T^{5} + \cdots + 55112)^{2} \)
|
| $79$ |
\( (T^{6} - 304 T^{4} + \cdots - 2048)^{2} \)
|
| $83$ |
\( T^{12} + 16672 T^{8} + \cdots + 65536 \)
|
| $89$ |
\( (T^{6} + 72 T^{4} + \cdots + 1024)^{2} \)
|
| $97$ |
\( (T^{6} + 10 T^{5} + \cdots + 35912)^{2} \)
|
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