Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5,11,Mod(2,5)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 11, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5.2");
S:= CuspForms(chi, 11);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5 \) |
| Weight: | \( k \) | \(=\) | \( 11 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5.c (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: | \(3.17678626337\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 1334x^{6} + 456089x^{4} + 43159076x^{2} + 31360000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 3^{2}\cdot 5^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 1334x^{6} + 456089x^{4} + 43159076x^{2} + 31360000 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{7} - 1334\nu^{5} - 450489\nu^{3} - 39423876\nu ) / 33454400 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - \nu^{7} - 35 \nu^{6} - 1789 \nu^{5} - 37625 \nu^{4} - 963064 \nu^{3} - 7002590 \nu^{2} + \cdots - 113422400 ) / 12545400 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - \nu^{7} + 35 \nu^{6} - 1789 \nu^{5} + 37625 \nu^{4} - 963064 \nu^{3} + 7002590 \nu^{2} + \cdots + 113422400 ) / 12545400 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 5 \nu^{7} - 196 \nu^{6} - 9218 \nu^{5} - 235900 \nu^{4} - 5122865 \nu^{3} - 56022904 \nu^{2} + \cdots - 756212800 ) / 10036320 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 5 \nu^{7} - 196 \nu^{6} + 9218 \nu^{5} - 235900 \nu^{4} + 5122865 \nu^{3} - 56022904 \nu^{2} + \cdots - 756212800 ) / 10036320 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{6} - 1585\nu^{4} - 719464\nu^{2} - 65776900 ) / 35844 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -29\nu^{7} - 38414\nu^{5} - 12978341\nu^{3} - 1229515316\nu ) / 573504 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - \beta_{4} + 7\beta_{3} + 7\beta_{2} - 4\beta_1 ) / 30 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -6\beta_{6} + 17\beta_{5} + 17\beta_{4} + 89\beta_{3} - 89\beta_{2} - 10058 ) / 30 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -78\beta_{7} - 629\beta_{5} + 629\beta_{4} - 4913\beta_{3} - 4913\beta_{2} + 137186\beta_1 ) / 30 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 1334\beta_{6} - 5771\beta_{5} - 5771\beta_{4} - 33727\beta_{3} + 33727\beta_{2} + 2188194 ) / 10 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 87870\beta_{7} + 515501\beta_{5} - 515501\beta_{4} + 3769457\beta_{3} + 3769457\beta_{2} - 151567154\beta_1 ) / 30 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 3101706 \beta_{6} + 15210217 \beta_{5} + 15210217 \beta_{4} + 96339589 \beta_{3} - 96339589 \beta_{2} - 5141800558 ) / 30 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 82080438 \beta_{7} - 443744629 \beta_{5} + 443744629 \beta_{4} - 3091170313 \beta_{3} + \cdots + 139543862986 \beta_1 ) / 30 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2.1 |
|
−36.6454 | − | 36.6454i | −13.8338 | + | 13.8338i | 1661.78i | −721.162 | + | 3040.65i | 1013.89 | −13891.7 | − | 13891.7i | 23371.7 | − | 23371.7i | 58666.3i | 137853. | − | 84998.7i | ||||||||||||||||||||||||||||||
| 2.2 | −4.63382 | − | 4.63382i | 70.6389 | − | 70.6389i | − | 981.055i | 1004.90 | − | 2959.02i | −654.656 | 2611.74 | + | 2611.74i | −9291.07 | + | 9291.07i | 49069.3i | −18368.1 | + | 9055.08i | ||||||||||||||||||||||||||||||
| 2.3 | 18.8402 | + | 18.8402i | −273.023 | + | 273.023i | − | 314.097i | 71.2753 | + | 3124.19i | −10287.6 | 12640.3 | + | 12640.3i | 25210.0 | − | 25210.0i | − | 90034.1i | −57517.4 | + | 60203.0i | |||||||||||||||||||||||||||||
| 2.4 | 37.4391 | + | 37.4391i | 246.218 | − | 246.218i | 1779.37i | −3025.01 | + | 784.185i | 18436.4 | −8610.35 | − | 8610.35i | −28280.5 | + | 28280.5i | − | 62197.5i | −142613. | − | 83894.5i | ||||||||||||||||||||||||||||||
| 3.1 | −36.6454 | + | 36.6454i | −13.8338 | − | 13.8338i | − | 1661.78i | −721.162 | − | 3040.65i | 1013.89 | −13891.7 | + | 13891.7i | 23371.7 | + | 23371.7i | − | 58666.3i | 137853. | + | 84998.7i | |||||||||||||||||||||||||||||
| 3.2 | −4.63382 | + | 4.63382i | 70.6389 | + | 70.6389i | 981.055i | 1004.90 | + | 2959.02i | −654.656 | 2611.74 | − | 2611.74i | −9291.07 | − | 9291.07i | − | 49069.3i | −18368.1 | − | 9055.08i | ||||||||||||||||||||||||||||||
| 3.3 | 18.8402 | − | 18.8402i | −273.023 | − | 273.023i | 314.097i | 71.2753 | − | 3124.19i | −10287.6 | 12640.3 | − | 12640.3i | 25210.0 | + | 25210.0i | 90034.1i | −57517.4 | − | 60203.0i | |||||||||||||||||||||||||||||||
| 3.4 | 37.4391 | − | 37.4391i | 246.218 | + | 246.218i | − | 1779.37i | −3025.01 | − | 784.185i | 18436.4 | −8610.35 | + | 8610.35i | −28280.5 | − | 28280.5i | 62197.5i | −142613. | + | 83894.5i | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 5.11.c.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 45.11.g.a | 8 | ||
| 4.b | odd | 2 | 1 | 80.11.p.d | 8 | ||
| 5.b | even | 2 | 1 | 25.11.c.a | 8 | ||
| 5.c | odd | 4 | 1 | inner | 5.11.c.a | ✓ | 8 |
| 5.c | odd | 4 | 1 | 25.11.c.a | 8 | ||
| 15.e | even | 4 | 1 | 45.11.g.a | 8 | ||
| 20.e | even | 4 | 1 | 80.11.p.d | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.11.c.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 5.11.c.a | ✓ | 8 | 5.c | odd | 4 | 1 | inner |
| 25.11.c.a | 8 | 5.b | even | 2 | 1 | ||
| 25.11.c.a | 8 | 5.c | odd | 4 | 1 | ||
| 45.11.g.a | 8 | 3.b | odd | 2 | 1 | ||
| 45.11.g.a | 8 | 15.e | even | 4 | 1 | ||
| 80.11.p.d | 8 | 4.b | odd | 2 | 1 | ||
| 80.11.p.d | 8 | 20.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(5, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + \cdots + 229540642816 \)
|
| $3$ |
\( T^{8} + \cdots + 69\!\cdots\!16 \)
|
| $5$ |
\( T^{8} + \cdots + 90\!\cdots\!25 \)
|
| $7$ |
\( T^{8} + \cdots + 24\!\cdots\!16 \)
|
| $11$ |
\( (T^{4} + \cdots - 15\!\cdots\!84)^{2} \)
|
| $13$ |
\( T^{8} + \cdots + 41\!\cdots\!16 \)
|
| $17$ |
\( T^{8} + \cdots + 39\!\cdots\!16 \)
|
| $19$ |
\( T^{8} + \cdots + 67\!\cdots\!00 \)
|
| $23$ |
\( T^{8} + \cdots + 48\!\cdots\!16 \)
|
| $29$ |
\( T^{8} + \cdots + 10\!\cdots\!00 \)
|
| $31$ |
\( (T^{4} + \cdots + 45\!\cdots\!16)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 93\!\cdots\!16 \)
|
| $41$ |
\( (T^{4} + \cdots + 43\!\cdots\!16)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 13\!\cdots\!16 \)
|
| $47$ |
\( T^{8} + \cdots + 32\!\cdots\!16 \)
|
| $53$ |
\( T^{8} + \cdots + 16\!\cdots\!16 \)
|
| $59$ |
\( T^{8} + \cdots + 63\!\cdots\!00 \)
|
| $61$ |
\( (T^{4} + \cdots + 25\!\cdots\!16)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 18\!\cdots\!16 \)
|
| $71$ |
\( (T^{4} + \cdots - 23\!\cdots\!84)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 18\!\cdots\!16 \)
|
| $79$ |
\( T^{8} + \cdots + 42\!\cdots\!00 \)
|
| $83$ |
\( T^{8} + \cdots + 21\!\cdots\!16 \)
|
| $89$ |
\( T^{8} + \cdots + 10\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots + 31\!\cdots\!16 \)
|
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