Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4,29,Mod(3,4)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 29, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4.3");
S:= CuspForms(chi, 29);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4 = 2^{2} \) |
| Weight: | \( k \) | \(=\) | \( 29 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4.b (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: | \(19.8673896993\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 3 x^{11} - 2473686 x^{10} + 20681698940 x^{9} - 75473901552696 x^{8} + \cdots + 18\!\cdots\!40 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{132}\cdot 3^{15}\cdot 5^{2}\cdot 7^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 3 x^{11} - 2473686 x^{10} + 20681698940 x^{9} - 75473901552696 x^{8} + \cdots + 18\!\cdots\!40 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - \nu^{11} + 509 \nu^{10} + 2216132 \nu^{9} - 21803061732 \nu^{8} + 86506250789088 \nu^{7} + \cdots + 68\!\cdots\!96 ) / 33\!\cdots\!44 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 11\!\cdots\!63 \nu^{11} + \cdots - 11\!\cdots\!52 ) / 15\!\cdots\!32 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 71\!\cdots\!81 \nu^{11} + \cdots + 28\!\cdots\!24 ) / 10\!\cdots\!88 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 88\!\cdots\!65 \nu^{11} + \cdots - 22\!\cdots\!20 ) / 15\!\cdots\!32 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 11\!\cdots\!31 \nu^{11} + \cdots - 31\!\cdots\!64 ) / 15\!\cdots\!32 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 22\!\cdots\!91 \nu^{11} + \cdots + 44\!\cdots\!72 ) / 15\!\cdots\!32 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 15\!\cdots\!09 \nu^{11} + \cdots - 35\!\cdots\!68 ) / 31\!\cdots\!64 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 87\!\cdots\!27 \nu^{11} + \cdots + 42\!\cdots\!04 ) / 15\!\cdots\!32 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 73\!\cdots\!09 \nu^{11} + \cdots + 19\!\cdots\!84 ) / 87\!\cdots\!24 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 64\!\cdots\!15 \nu^{11} + \cdots - 62\!\cdots\!24 ) / 31\!\cdots\!64 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 64\!\cdots\!65 \nu^{11} + \cdots - 13\!\cdots\!72 ) / 10\!\cdots\!88 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{5} + \beta_{4} - 3\beta_{3} + 27\beta_{2} + 13079\beta _1 + 524288 ) / 2097152 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4 \beta_{9} + 8 \beta_{8} - 56 \beta_{6} + 1755 \beta_{5} - 99 \beta_{4} + 14097 \beta_{3} + \cdots + 864617496576 ) / 2097152 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 32768 \beta_{11} - 28672 \beta_{10} + 47988 \beta_{9} - 29976 \beta_{8} + 48040 \beta_{6} + \cdots - 10\!\cdots\!24 ) / 2097152 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 4096000 \beta_{11} + 67350528 \beta_{10} - 189508108 \beta_{9} + 75081704 \beta_{8} + \cdots + 54\!\cdots\!08 ) / 2097152 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 213956198400 \beta_{11} + 44282703872 \beta_{10} + 179147076452 \beta_{9} + \cdots + 15\!\cdots\!48 ) / 2097152 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 11269749440512 \beta_{11} + \cdots - 13\!\cdots\!24 ) / 2097152 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 40\!\cdots\!68 \beta_{11} + \cdots - 96\!\cdots\!24 ) / 2097152 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 25\!\cdots\!80 \beta_{11} + \cdots + 42\!\cdots\!08 ) / 2097152 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 71\!\cdots\!52 \beta_{11} + \cdots + 96\!\cdots\!88 ) / 2097152 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 28\!\cdots\!00 \beta_{11} + \cdots - 15\!\cdots\!84 ) / 2097152 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 22\!\cdots\!16 \beta_{11} + \cdots + 11\!\cdots\!36 ) / 2097152 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3.1 |
|
−15801.7 | − | 4329.20i | 8.53819e6i | 2.30951e8 | + | 1.36817e8i | −9.02774e9 | 3.69635e10 | − | 1.34918e11i | − | 4.67071e11i | −3.05711e12 | − | 3.16178e12i | −5.00238e13 | 1.42654e14 | + | 3.90829e13i | |||||||||||||||||||||||||||||||||||||||||||
| 3.2 | −15801.7 | + | 4329.20i | − | 8.53819e6i | 2.30951e8 | − | 1.36817e8i | −9.02774e9 | 3.69635e10 | + | 1.34918e11i | 4.67071e11i | −3.05711e12 | + | 3.16178e12i | −5.00238e13 | 1.42654e14 | − | 3.90829e13i | ||||||||||||||||||||||||||||||||||||||||||||
| 3.3 | −9883.33 | − | 13067.3i | − | 2.42617e6i | −7.30750e7 | + | 2.58298e8i | −1.42980e9 | −3.17035e10 | + | 2.39786e10i | 8.18155e10i | 4.09749e12 | − | 1.59794e12i | 1.69905e13 | 1.41312e13 | + | 1.86836e13i | ||||||||||||||||||||||||||||||||||||||||||||
| 3.4 | −9883.33 | + | 13067.3i | 2.42617e6i | −7.30750e7 | − | 2.58298e8i | −1.42980e9 | −3.17035e10 | − | 2.39786e10i | − | 8.18155e10i | 4.09749e12 | + | 1.59794e12i | 1.69905e13 | 1.41312e13 | − | 1.86836e13i | ||||||||||||||||||||||||||||||||||||||||||||
| 3.5 | 1299.79 | − | 16332.4i | 5.39102e6i | −2.65057e8 | − | 4.24572e7i | 2.64454e9 | 8.80480e10 | + | 7.00717e9i | 2.77694e11i | −1.03794e12 | + | 4.27381e12i | −6.18626e12 | 3.43734e12 | − | 4.31916e13i | |||||||||||||||||||||||||||||||||||||||||||||
| 3.6 | 1299.79 | + | 16332.4i | − | 5.39102e6i | −2.65057e8 | + | 4.24572e7i | 2.64454e9 | 8.80480e10 | − | 7.00717e9i | − | 2.77694e11i | −1.03794e12 | − | 4.27381e12i | −6.18626e12 | 3.43734e12 | + | 4.31916e13i | |||||||||||||||||||||||||||||||||||||||||||
| 3.7 | 7038.52 | − | 14795.1i | − | 7.03288e6i | −1.69354e8 | − | 2.08271e8i | 2.68456e9 | −1.04052e11 | − | 4.95011e10i | − | 1.04221e12i | −4.27339e12 | + | 1.03968e12i | −2.65846e13 | 1.88954e13 | − | 3.97184e13i | |||||||||||||||||||||||||||||||||||||||||||
| 3.8 | 7038.52 | + | 14795.1i | 7.03288e6i | −1.69354e8 | + | 2.08271e8i | 2.68456e9 | −1.04052e11 | + | 4.95011e10i | 1.04221e12i | −4.27339e12 | − | 1.03968e12i | −2.65846e13 | 1.88954e13 | + | 3.97184e13i | |||||||||||||||||||||||||||||||||||||||||||||
| 3.9 | 13376.2 | − | 9461.17i | 304553.i | 8.94080e7 | − | 2.53108e8i | −8.37982e9 | 2.88143e9 | + | 4.07375e9i | 8.99064e11i | −1.19876e12 | − | 4.23152e12i | 2.27840e13 | −1.12090e14 | + | 7.92829e13i | |||||||||||||||||||||||||||||||||||||||||||||
| 3.10 | 13376.2 | + | 9461.17i | − | 304553.i | 8.94080e7 | + | 2.53108e8i | −8.37982e9 | 2.88143e9 | − | 4.07375e9i | − | 8.99064e11i | −1.19876e12 | + | 4.23152e12i | 2.27840e13 | −1.12090e14 | − | 7.92829e13i | |||||||||||||||||||||||||||||||||||||||||||
| 3.11 | 16120.5 | − | 2926.32i | 5.82821e6i | 2.51309e8 | − | 9.43478e7i | 7.30842e9 | 1.70552e10 | + | 9.39539e10i | − | 1.16512e12i | 3.77514e12 | − | 2.25635e12i | −1.10912e13 | 1.17816e14 | − | 2.13868e13i | ||||||||||||||||||||||||||||||||||||||||||||
| 3.12 | 16120.5 | + | 2926.32i | − | 5.82821e6i | 2.51309e8 | + | 9.43478e7i | 7.30842e9 | 1.70552e10 | − | 9.39539e10i | 1.16512e12i | 3.77514e12 | + | 2.25635e12i | −1.10912e13 | 1.17816e14 | + | 2.13868e13i | ||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 4.29.b.b | ✓ | 12 |
| 3.b | odd | 2 | 1 | 36.29.d.b | 12 | ||
| 4.b | odd | 2 | 1 | inner | 4.29.b.b | ✓ | 12 |
| 12.b | even | 2 | 1 | 36.29.d.b | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4.29.b.b | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 4.29.b.b | ✓ | 12 | 4.b | odd | 2 | 1 | inner |
| 36.29.d.b | 12 | 3.b | odd | 2 | 1 | ||
| 36.29.d.b | 12 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} + 191372170099968 T_{3}^{10} + \cdots + 19\!\cdots\!00 \)
acting on \(S_{29}^{\mathrm{new}}(4, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + \cdots + 37\!\cdots\!56 \)
|
| $3$ |
\( T^{12} + \cdots + 19\!\cdots\!00 \)
|
| $5$ |
\( (T^{6} + \cdots - 56\!\cdots\!00)^{2} \)
|
| $7$ |
\( T^{12} + \cdots + 13\!\cdots\!00 \)
|
| $11$ |
\( T^{12} + \cdots + 12\!\cdots\!00 \)
|
| $13$ |
\( (T^{6} + \cdots - 63\!\cdots\!00)^{2} \)
|
| $17$ |
\( (T^{6} + \cdots + 52\!\cdots\!00)^{2} \)
|
| $19$ |
\( T^{12} + \cdots + 41\!\cdots\!00 \)
|
| $23$ |
\( T^{12} + \cdots + 37\!\cdots\!00 \)
|
| $29$ |
\( (T^{6} + \cdots + 18\!\cdots\!84)^{2} \)
|
| $31$ |
\( T^{12} + \cdots + 52\!\cdots\!00 \)
|
| $37$ |
\( (T^{6} + \cdots - 99\!\cdots\!00)^{2} \)
|
| $41$ |
\( (T^{6} + \cdots - 49\!\cdots\!36)^{2} \)
|
| $43$ |
\( T^{12} + \cdots + 10\!\cdots\!00 \)
|
| $47$ |
\( T^{12} + \cdots + 55\!\cdots\!00 \)
|
| $53$ |
\( (T^{6} + \cdots + 37\!\cdots\!00)^{2} \)
|
| $59$ |
\( T^{12} + \cdots + 30\!\cdots\!00 \)
|
| $61$ |
\( (T^{6} + \cdots - 10\!\cdots\!16)^{2} \)
|
| $67$ |
\( T^{12} + \cdots + 86\!\cdots\!00 \)
|
| $71$ |
\( T^{12} + \cdots + 71\!\cdots\!00 \)
|
| $73$ |
\( (T^{6} + \cdots + 42\!\cdots\!00)^{2} \)
|
| $79$ |
\( T^{12} + \cdots + 44\!\cdots\!00 \)
|
| $83$ |
\( T^{12} + \cdots + 11\!\cdots\!00 \)
|
| $89$ |
\( (T^{6} + \cdots - 52\!\cdots\!96)^{2} \)
|
| $97$ |
\( (T^{6} + \cdots - 22\!\cdots\!00)^{2} \)
|
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