Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [20,4,Mod(3,20)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(20, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 3]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("20.3");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 20 = 2^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 20.e (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: | \(1.18003820011\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(6\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 7x^{10} + 44x^{8} - 156x^{6} + 704x^{4} - 1792x^{2} + 4096 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{14} \) |
| 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
\( x^{12} - 7x^{10} + 44x^{8} - 156x^{6} + 704x^{4} - 1792x^{2} + 4096 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{11} - 8\nu^{9} + 27\nu^{7} - 128\nu^{5} + 412\nu^{3} - 1504\nu ) / 640 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5 \nu^{11} + 32 \nu^{10} - 35 \nu^{9} - 256 \nu^{8} + 220 \nu^{7} + 864 \nu^{6} - 780 \nu^{5} + \cdots - 48128 ) / 5120 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - \nu^{11} + 26 \nu^{10} + 23 \nu^{9} - 198 \nu^{8} - 92 \nu^{7} + 872 \nu^{6} + 668 \nu^{5} + \cdots - 34304 ) / 2560 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{11} + 26 \nu^{10} - 23 \nu^{9} - 198 \nu^{8} + 92 \nu^{7} + 872 \nu^{6} - 668 \nu^{5} + \cdots - 34304 ) / 2560 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 13 \nu^{11} - 32 \nu^{10} - 99 \nu^{9} + 256 \nu^{8} + 436 \nu^{7} - 864 \nu^{6} - 1804 \nu^{5} + \cdots + 48128 ) / 5120 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 5 \nu^{11} - 34 \nu^{10} - 55 \nu^{9} + 62 \nu^{8} + 200 \nu^{7} - 648 \nu^{6} - 1180 \nu^{5} + \cdots - 1024 ) / 2560 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 5 \nu^{11} + 34 \nu^{10} - 55 \nu^{9} - 62 \nu^{8} + 200 \nu^{7} + 648 \nu^{6} - 1180 \nu^{5} + \cdots + 1024 ) / 2560 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 13\nu^{10} - 99\nu^{8} + 436\nu^{6} - 1804\nu^{4} + 6816\nu^{2} - 19072 ) / 640 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 45 \nu^{11} - 12 \nu^{10} - 115 \nu^{9} - 44 \nu^{8} + 900 \nu^{7} - 144 \nu^{6} - 1740 \nu^{5} + \cdots - 28672 ) / 5120 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 45 \nu^{11} + 12 \nu^{10} - 115 \nu^{9} + 44 \nu^{8} + 900 \nu^{7} + 144 \nu^{6} - 1740 \nu^{5} + \cdots + 28672 ) / 5120 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( -3\nu^{11} + 7\nu^{9} - 50\nu^{7} + 156\nu^{5} - 824\nu^{3} + 448\nu ) / 256 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} - 2\beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{8} - \beta_{4} - \beta_{3} + 3 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{11} - \beta_{10} - \beta_{9} + 3\beta_{5} - 2\beta_{4} + 2\beta_{3} + 3\beta_{2} - 2\beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 3\beta_{10} - 3\beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta _1 - 11 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 3 \beta_{11} + \beta_{10} + \beta_{9} - 5 \beta_{7} - 5 \beta_{6} + 5 \beta_{5} - 3 \beta_{4} + \cdots + 14 \beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 7 \beta_{10} - 7 \beta_{9} - \beta_{8} - 3 \beta_{7} + 3 \beta_{6} + 19 \beta_{5} + 15 \beta_{4} + \cdots - 61 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 25 \beta_{11} + 19 \beta_{10} + 19 \beta_{9} + 3 \beta_{7} + 3 \beta_{6} + 15 \beta_{5} + \cdots - 74 \beta_1 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 35 \beta_{10} + 35 \beta_{9} + 9 \beta_{8} + 23 \beta_{7} - 23 \beta_{6} + 65 \beta_{5} + 9 \beta_{4} + \cdots - 339 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 49 \beta_{11} - 3 \beta_{10} - 3 \beta_{9} - 47 \beta_{7} - 47 \beta_{6} + 33 \beta_{5} + \cdots - 358 \beta_1 ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 85 \beta_{10} + 85 \beta_{9} - 185 \beta_{8} + 137 \beta_{7} - 137 \beta_{6} - 281 \beta_{5} + \cdots - 701 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 271 \beta_{11} + 3 \beta_{10} + 3 \beta_{9} - 345 \beta_{7} - 345 \beta_{6} - 737 \beta_{5} + \cdots + 1526 \beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).
| \(n\) | \(11\) | \(17\) |
| \(\chi(n)\) | \(-1\) | \(-\beta_{1}\) |
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 |
|
−2.78199 | + | 0.510409i | 4.02923 | − | 4.02923i | 7.47897 | − | 2.83991i | 10.9349 | + | 2.32970i | −9.15273 | + | 13.2658i | −14.4440 | − | 14.4440i | −19.3569 | + | 11.7179i | − | 5.46937i | −31.6100 | − | 0.899920i | |||||||||||||||||||||||||||||||||||||
| 3.2 | −2.63379 | − | 1.03109i | −5.55970 | + | 5.55970i | 5.87372 | + | 5.43134i | −10.4994 | + | 3.84216i | 20.3756 | − | 8.91056i | 1.14202 | + | 1.14202i | −9.86997 | − | 20.3613i | − | 34.8205i | 31.6149 | + | 0.706375i | ||||||||||||||||||||||||||||||||||||||
| 3.3 | −0.813737 | − | 2.70884i | 2.61822 | − | 2.61822i | −6.67566 | + | 4.40857i | −0.435501 | − | 11.1719i | −9.22289 | − | 4.96181i | 17.7783 | + | 17.7783i | 17.3744 | + | 14.4959i | 13.2899i | −29.9084 | + | 10.2707i | |||||||||||||||||||||||||||||||||||||||
| 3.4 | −0.510409 | + | 2.78199i | −4.02923 | + | 4.02923i | −7.47897 | − | 2.83991i | 10.9349 | + | 2.32970i | −9.15273 | − | 13.2658i | 14.4440 | + | 14.4440i | 11.7179 | − | 19.3569i | − | 5.46937i | −12.0625 | + | 29.2318i | ||||||||||||||||||||||||||||||||||||||
| 3.5 | 1.03109 | + | 2.63379i | 5.55970 | − | 5.55970i | −5.87372 | + | 5.43134i | −10.4994 | + | 3.84216i | 20.3756 | + | 8.91056i | −1.14202 | − | 1.14202i | −20.3613 | − | 9.86997i | − | 34.8205i | −20.9453 | − | 23.6917i | ||||||||||||||||||||||||||||||||||||||
| 3.6 | 2.70884 | + | 0.813737i | −2.61822 | + | 2.61822i | 6.67566 | + | 4.40857i | −0.435501 | − | 11.1719i | −9.22289 | + | 4.96181i | −17.7783 | − | 17.7783i | 14.4959 | + | 17.3744i | 13.2899i | 7.91125 | − | 30.6172i | |||||||||||||||||||||||||||||||||||||||
| 7.1 | −2.78199 | − | 0.510409i | 4.02923 | + | 4.02923i | 7.47897 | + | 2.83991i | 10.9349 | − | 2.32970i | −9.15273 | − | 13.2658i | −14.4440 | + | 14.4440i | −19.3569 | − | 11.7179i | 5.46937i | −31.6100 | + | 0.899920i | |||||||||||||||||||||||||||||||||||||||
| 7.2 | −2.63379 | + | 1.03109i | −5.55970 | − | 5.55970i | 5.87372 | − | 5.43134i | −10.4994 | − | 3.84216i | 20.3756 | + | 8.91056i | 1.14202 | − | 1.14202i | −9.86997 | + | 20.3613i | 34.8205i | 31.6149 | − | 0.706375i | |||||||||||||||||||||||||||||||||||||||
| 7.3 | −0.813737 | + | 2.70884i | 2.61822 | + | 2.61822i | −6.67566 | − | 4.40857i | −0.435501 | + | 11.1719i | −9.22289 | + | 4.96181i | 17.7783 | − | 17.7783i | 17.3744 | − | 14.4959i | − | 13.2899i | −29.9084 | − | 10.2707i | ||||||||||||||||||||||||||||||||||||||
| 7.4 | −0.510409 | − | 2.78199i | −4.02923 | − | 4.02923i | −7.47897 | + | 2.83991i | 10.9349 | − | 2.32970i | −9.15273 | + | 13.2658i | 14.4440 | − | 14.4440i | 11.7179 | + | 19.3569i | 5.46937i | −12.0625 | − | 29.2318i | |||||||||||||||||||||||||||||||||||||||
| 7.5 | 1.03109 | − | 2.63379i | 5.55970 | + | 5.55970i | −5.87372 | − | 5.43134i | −10.4994 | − | 3.84216i | 20.3756 | − | 8.91056i | −1.14202 | + | 1.14202i | −20.3613 | + | 9.86997i | 34.8205i | −20.9453 | + | 23.6917i | |||||||||||||||||||||||||||||||||||||||
| 7.6 | 2.70884 | − | 0.813737i | −2.61822 | − | 2.61822i | 6.67566 | − | 4.40857i | −0.435501 | + | 11.1719i | −9.22289 | − | 4.96181i | −17.7783 | + | 17.7783i | 14.4959 | − | 17.3744i | − | 13.2899i | 7.91125 | + | 30.6172i | ||||||||||||||||||||||||||||||||||||||
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 | 20.4.e.b | ✓ | 12 |
| 3.b | odd | 2 | 1 | 180.4.k.e | 12 | ||
| 4.b | odd | 2 | 1 | inner | 20.4.e.b | ✓ | 12 |
| 5.b | even | 2 | 1 | 100.4.e.e | 12 | ||
| 5.c | odd | 4 | 1 | inner | 20.4.e.b | ✓ | 12 |
| 5.c | odd | 4 | 1 | 100.4.e.e | 12 | ||
| 8.b | even | 2 | 1 | 320.4.n.k | 12 | ||
| 8.d | odd | 2 | 1 | 320.4.n.k | 12 | ||
| 12.b | even | 2 | 1 | 180.4.k.e | 12 | ||
| 15.e | even | 4 | 1 | 180.4.k.e | 12 | ||
| 20.d | odd | 2 | 1 | 100.4.e.e | 12 | ||
| 20.e | even | 4 | 1 | inner | 20.4.e.b | ✓ | 12 |
| 20.e | even | 4 | 1 | 100.4.e.e | 12 | ||
| 40.i | odd | 4 | 1 | 320.4.n.k | 12 | ||
| 40.k | even | 4 | 1 | 320.4.n.k | 12 | ||
| 60.l | odd | 4 | 1 | 180.4.k.e | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 20.4.e.b | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 20.4.e.b | ✓ | 12 | 4.b | odd | 2 | 1 | inner |
| 20.4.e.b | ✓ | 12 | 5.c | odd | 4 | 1 | inner |
| 20.4.e.b | ✓ | 12 | 20.e | even | 4 | 1 | inner |
| 100.4.e.e | 12 | 5.b | even | 2 | 1 | ||
| 100.4.e.e | 12 | 5.c | odd | 4 | 1 | ||
| 100.4.e.e | 12 | 20.d | odd | 2 | 1 | ||
| 100.4.e.e | 12 | 20.e | even | 4 | 1 | ||
| 180.4.k.e | 12 | 3.b | odd | 2 | 1 | ||
| 180.4.k.e | 12 | 12.b | even | 2 | 1 | ||
| 180.4.k.e | 12 | 15.e | even | 4 | 1 | ||
| 180.4.k.e | 12 | 60.l | odd | 4 | 1 | ||
| 320.4.n.k | 12 | 8.b | even | 2 | 1 | ||
| 320.4.n.k | 12 | 8.d | odd | 2 | 1 | ||
| 320.4.n.k | 12 | 40.i | odd | 4 | 1 | ||
| 320.4.n.k | 12 | 40.k | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} + 5064T_{3}^{8} + 4945680T_{3}^{4} + 757350400 \)
acting on \(S_{4}^{\mathrm{new}}(20, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + 6 T^{11} + \cdots + 262144 \)
|
| $3$ |
\( T^{12} + \cdots + 757350400 \)
|
| $5$ |
\( (T^{6} - 85 T^{4} + \cdots + 1953125)^{2} \)
|
| $7$ |
\( T^{12} + \cdots + 473344000000 \)
|
| $11$ |
\( (T^{6} + 3000 T^{4} + \cdots + 88064000)^{2} \)
|
| $13$ |
\( (T^{6} - 58 T^{5} + \cdots + 7296200)^{2} \)
|
| $17$ |
\( (T^{6} + 166 T^{5} + \cdots + 3024864200)^{2} \)
|
| $19$ |
\( (T^{6} - 24160 T^{4} + \cdots - 148035584000)^{2} \)
|
| $23$ |
\( T^{12} + \cdots + 43\!\cdots\!00 \)
|
| $29$ |
\( (T^{6} + 65648 T^{4} + \cdots + 234782887936)^{2} \)
|
| $31$ |
\( (T^{6} + \cdots + 4998782336000)^{2} \)
|
| $37$ |
\( (T^{6} + \cdots + 6862252857800)^{2} \)
|
| $41$ |
\( (T^{3} + 164 T^{2} + \cdots - 1791008)^{4} \)
|
| $43$ |
\( T^{12} + \cdots + 22\!\cdots\!00 \)
|
| $47$ |
\( T^{12} + \cdots + 28\!\cdots\!00 \)
|
| $53$ |
\( (T^{6} + \cdots + 273645700473800)^{2} \)
|
| $59$ |
\( (T^{6} + \cdots - 742151346176000)^{2} \)
|
| $61$ |
\( (T^{3} + 224 T^{2} + \cdots - 11698768)^{4} \)
|
| $67$ |
\( T^{12} + \cdots + 16\!\cdots\!00 \)
|
| $71$ |
\( (T^{6} + \cdots + 29\!\cdots\!00)^{2} \)
|
| $73$ |
\( (T^{6} + \cdots + 17037736128200)^{2} \)
|
| $79$ |
\( (T^{6} + \cdots - 42\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{12} + \cdots + 13\!\cdots\!00 \)
|
| $89$ |
\( (T^{6} + \cdots + 84\!\cdots\!16)^{2} \)
|
| $97$ |
\( (T^{6} + \cdots + 60\!\cdots\!00)^{2} \)
|
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