Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [180,4,Mod(127,180)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(180, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("180.127");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 180.k (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: | \(10.6203438010\) |
| 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: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{14} \) |
| Twist minimal: | no (minimal twist has level 20) |
| 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 \(\left(\mathbb{Z}/180\mathbb{Z}\right)^\times\).
| \(n\) | \(37\) | \(91\) | \(101\) |
| \(\chi(n)\) | \(\beta_{1}\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 127.1 |
|
−2.70884 | + | 0.813737i | 0 | 6.67566 | − | 4.40857i | 0.435501 | − | 11.1719i | 0 | −17.7783 | + | 17.7783i | −14.4959 | + | 17.3744i | 0 | 7.91125 | + | 30.6172i | ||||||||||||||||||||||||||||||||||||||||||
| 127.2 | −1.03109 | + | 2.63379i | 0 | −5.87372 | − | 5.43134i | 10.4994 | + | 3.84216i | 0 | −1.14202 | + | 1.14202i | 20.3613 | − | 9.86997i | 0 | −20.9453 | + | 23.6917i | |||||||||||||||||||||||||||||||||||||||||||
| 127.3 | 0.510409 | + | 2.78199i | 0 | −7.47897 | + | 2.83991i | −10.9349 | + | 2.32970i | 0 | 14.4440 | − | 14.4440i | −11.7179 | − | 19.3569i | 0 | −12.0625 | − | 29.2318i | |||||||||||||||||||||||||||||||||||||||||||
| 127.4 | 0.813737 | − | 2.70884i | 0 | −6.67566 | − | 4.40857i | 0.435501 | − | 11.1719i | 0 | 17.7783 | − | 17.7783i | −17.3744 | + | 14.4959i | 0 | −29.9084 | − | 10.2707i | |||||||||||||||||||||||||||||||||||||||||||
| 127.5 | 2.63379 | − | 1.03109i | 0 | 5.87372 | − | 5.43134i | 10.4994 | + | 3.84216i | 0 | 1.14202 | − | 1.14202i | 9.86997 | − | 20.3613i | 0 | 31.6149 | − | 0.706375i | |||||||||||||||||||||||||||||||||||||||||||
| 127.6 | 2.78199 | + | 0.510409i | 0 | 7.47897 | + | 2.83991i | −10.9349 | + | 2.32970i | 0 | −14.4440 | + | 14.4440i | 19.3569 | + | 11.7179i | 0 | −31.6100 | + | 0.899920i | |||||||||||||||||||||||||||||||||||||||||||
| 163.1 | −2.70884 | − | 0.813737i | 0 | 6.67566 | + | 4.40857i | 0.435501 | + | 11.1719i | 0 | −17.7783 | − | 17.7783i | −14.4959 | − | 17.3744i | 0 | 7.91125 | − | 30.6172i | |||||||||||||||||||||||||||||||||||||||||||
| 163.2 | −1.03109 | − | 2.63379i | 0 | −5.87372 | + | 5.43134i | 10.4994 | − | 3.84216i | 0 | −1.14202 | − | 1.14202i | 20.3613 | + | 9.86997i | 0 | −20.9453 | − | 23.6917i | |||||||||||||||||||||||||||||||||||||||||||
| 163.3 | 0.510409 | − | 2.78199i | 0 | −7.47897 | − | 2.83991i | −10.9349 | − | 2.32970i | 0 | 14.4440 | + | 14.4440i | −11.7179 | + | 19.3569i | 0 | −12.0625 | + | 29.2318i | |||||||||||||||||||||||||||||||||||||||||||
| 163.4 | 0.813737 | + | 2.70884i | 0 | −6.67566 | + | 4.40857i | 0.435501 | + | 11.1719i | 0 | 17.7783 | + | 17.7783i | −17.3744 | − | 14.4959i | 0 | −29.9084 | + | 10.2707i | |||||||||||||||||||||||||||||||||||||||||||
| 163.5 | 2.63379 | + | 1.03109i | 0 | 5.87372 | + | 5.43134i | 10.4994 | − | 3.84216i | 0 | 1.14202 | + | 1.14202i | 9.86997 | + | 20.3613i | 0 | 31.6149 | + | 0.706375i | |||||||||||||||||||||||||||||||||||||||||||
| 163.6 | 2.78199 | − | 0.510409i | 0 | 7.47897 | − | 2.83991i | −10.9349 | − | 2.32970i | 0 | −14.4440 | − | 14.4440i | 19.3569 | − | 11.7179i | 0 | −31.6100 | − | 0.899920i | |||||||||||||||||||||||||||||||||||||||||||
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 | 180.4.k.e | 12 | |
| 3.b | odd | 2 | 1 | 20.4.e.b | ✓ | 12 | |
| 4.b | odd | 2 | 1 | inner | 180.4.k.e | 12 | |
| 5.c | odd | 4 | 1 | inner | 180.4.k.e | 12 | |
| 12.b | even | 2 | 1 | 20.4.e.b | ✓ | 12 | |
| 15.d | odd | 2 | 1 | 100.4.e.e | 12 | ||
| 15.e | even | 4 | 1 | 20.4.e.b | ✓ | 12 | |
| 15.e | even | 4 | 1 | 100.4.e.e | 12 | ||
| 20.e | even | 4 | 1 | inner | 180.4.k.e | 12 | |
| 24.f | even | 2 | 1 | 320.4.n.k | 12 | ||
| 24.h | odd | 2 | 1 | 320.4.n.k | 12 | ||
| 60.h | even | 2 | 1 | 100.4.e.e | 12 | ||
| 60.l | odd | 4 | 1 | 20.4.e.b | ✓ | 12 | |
| 60.l | odd | 4 | 1 | 100.4.e.e | 12 | ||
| 120.q | odd | 4 | 1 | 320.4.n.k | 12 | ||
| 120.w | even | 4 | 1 | 320.4.n.k | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 20.4.e.b | ✓ | 12 | 3.b | odd | 2 | 1 | |
| 20.4.e.b | ✓ | 12 | 12.b | even | 2 | 1 | |
| 20.4.e.b | ✓ | 12 | 15.e | even | 4 | 1 | |
| 20.4.e.b | ✓ | 12 | 60.l | odd | 4 | 1 | |
| 100.4.e.e | 12 | 15.d | odd | 2 | 1 | ||
| 100.4.e.e | 12 | 15.e | even | 4 | 1 | ||
| 100.4.e.e | 12 | 60.h | even | 2 | 1 | ||
| 100.4.e.e | 12 | 60.l | odd | 4 | 1 | ||
| 180.4.k.e | 12 | 1.a | even | 1 | 1 | trivial | |
| 180.4.k.e | 12 | 4.b | odd | 2 | 1 | inner | |
| 180.4.k.e | 12 | 5.c | odd | 4 | 1 | inner | |
| 180.4.k.e | 12 | 20.e | even | 4 | 1 | inner | |
| 320.4.n.k | 12 | 24.f | even | 2 | 1 | ||
| 320.4.n.k | 12 | 24.h | odd | 2 | 1 | ||
| 320.4.n.k | 12 | 120.q | odd | 4 | 1 | ||
| 320.4.n.k | 12 | 120.w | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(180, [\chi])\):
|
\( T_{7}^{12} + 573704T_{7}^{8} + 69574698000T_{7}^{4} + 473344000000 \)
|
|
\( T_{13}^{6} - 58T_{13}^{5} + 1682T_{13}^{4} + 54976T_{13}^{3} + 777924T_{13}^{2} + 3369240T_{13} + 7296200 \)
|
|
\( T_{17}^{6} - 166T_{17}^{5} + 13778T_{17}^{4} - 662912T_{17}^{3} + 19909444T_{17}^{2} - 347054360T_{17} + 3024864200 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} - 6 T^{11} + \cdots + 262144 \)
|
| $3$ |
\( T^{12} \)
|
| $5$ |
\( (T^{6} - 85 T^{4} + \cdots + 1953125)^{2} \)
|
| $7$ |
\( T^{12} + \cdots + 473344000000 \)
|
| $11$ |
\( (T^{6} + 3000 T^{4} + \cdots + 88064000)^{2} \)
|
| $13$ |
\( (T^{6} - 58 T^{5} + \cdots + 7296200)^{2} \)
|
| $17$ |
\( (T^{6} - 166 T^{5} + \cdots + 3024864200)^{2} \)
|
| $19$ |
\( (T^{6} - 24160 T^{4} + \cdots - 148035584000)^{2} \)
|
| $23$ |
\( T^{12} + \cdots + 43\!\cdots\!00 \)
|
| $29$ |
\( (T^{6} + 65648 T^{4} + \cdots + 234782887936)^{2} \)
|
| $31$ |
\( (T^{6} + \cdots + 4998782336000)^{2} \)
|
| $37$ |
\( (T^{6} + \cdots + 6862252857800)^{2} \)
|
| $41$ |
\( (T^{3} - 164 T^{2} + \cdots + 1791008)^{4} \)
|
| $43$ |
\( T^{12} + \cdots + 22\!\cdots\!00 \)
|
| $47$ |
\( T^{12} + \cdots + 28\!\cdots\!00 \)
|
| $53$ |
\( (T^{6} + \cdots + 273645700473800)^{2} \)
|
| $59$ |
\( (T^{6} + \cdots - 742151346176000)^{2} \)
|
| $61$ |
\( (T^{3} + 224 T^{2} + \cdots - 11698768)^{4} \)
|
| $67$ |
\( T^{12} + \cdots + 16\!\cdots\!00 \)
|
| $71$ |
\( (T^{6} + \cdots + 29\!\cdots\!00)^{2} \)
|
| $73$ |
\( (T^{6} + \cdots + 17037736128200)^{2} \)
|
| $79$ |
\( (T^{6} + \cdots - 42\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{12} + \cdots + 13\!\cdots\!00 \)
|
| $89$ |
\( (T^{6} + \cdots + 84\!\cdots\!16)^{2} \)
|
| $97$ |
\( (T^{6} + \cdots + 60\!\cdots\!00)^{2} \)
|
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