Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5,23,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, 23, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5.2");
S:= CuspForms(chi, 23);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5 \) |
| Weight: | \( k \) | \(=\) | \( 23 \) |
| 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: | \(15.3353717421\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(10\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} + 28868605 x^{18} + 349376652422760 x^{16} + \cdots + 46\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{61}\cdot 3^{18}\cdot 5^{41} \) |
| 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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} + 28868605 x^{18} + 349376652422760 x^{16} + \cdots + 46\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 18\!\cdots\!29 \nu^{18} + \cdots - 55\!\cdots\!00 ) / 10\!\cdots\!00 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 18\!\cdots\!29 \nu^{18} + \cdots + 55\!\cdots\!00 ) / 10\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 20\!\cdots\!29 \nu^{18} + \cdots + 26\!\cdots\!00 ) / 40\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 12\!\cdots\!59 \nu^{19} + \cdots - 90\!\cdots\!00 \nu ) / 10\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 10\!\cdots\!71 \nu^{19} + \cdots - 35\!\cdots\!00 ) / 10\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 10\!\cdots\!71 \nu^{19} + \cdots + 35\!\cdots\!00 ) / 10\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 14\!\cdots\!61 \nu^{19} + \cdots - 69\!\cdots\!00 \nu ) / 13\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 12\!\cdots\!59 \nu^{19} + \cdots + 16\!\cdots\!00 ) / 27\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 13\!\cdots\!63 \nu^{19} + \cdots + 87\!\cdots\!00 ) / 27\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 13\!\cdots\!63 \nu^{19} + \cdots - 50\!\cdots\!00 ) / 27\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 12\!\cdots\!06 \nu^{19} + \cdots - 88\!\cdots\!00 ) / 11\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 46\!\cdots\!43 \nu^{19} + \cdots + 80\!\cdots\!00 ) / 27\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 18\!\cdots\!31 \nu^{19} + \cdots + 21\!\cdots\!00 ) / 54\!\cdots\!00 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 68\!\cdots\!54 \nu^{19} + \cdots + 83\!\cdots\!00 ) / 13\!\cdots\!00 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 22\!\cdots\!67 \nu^{19} + \cdots + 16\!\cdots\!00 ) / 27\!\cdots\!00 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 41\!\cdots\!22 \nu^{19} + \cdots + 15\!\cdots\!00 ) / 45\!\cdots\!00 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 64\!\cdots\!87 \nu^{19} + \cdots - 22\!\cdots\!00 ) / 30\!\cdots\!00 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 26\!\cdots\!91 \nu^{19} + \cdots + 39\!\cdots\!00 ) / 90\!\cdots\!00 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 60\!\cdots\!41 \nu^{19} + \cdots - 62\!\cdots\!00 ) / 33\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{6} + \beta_{5} + \beta_{3} - 160\beta_{2} + 160\beta _1 - 5773881 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{17} - \beta_{16} - 6 \beta_{15} + \beta_{14} + \beta_{13} - 7 \beta_{9} + 21 \beta_{8} + \cdots + 6125 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 162 \beta_{18} - 33 \beta_{17} + 1429 \beta_{16} + 33 \beta_{14} + 1429 \beta_{13} + \cdots + 53859342964725 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 225350 \beta_{19} + 570892 \beta_{18} - 4128864 \beta_{17} + 4879508 \beta_{16} + 33465804 \beta_{15} + \cdots - 34966354950 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 2948325090 \beta_{18} - 25496247 \beta_{17} - 8145093501 \beta_{16} + 25496247 \beta_{14} + \cdots - 14\!\cdots\!96 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 4999648225400 \beta_{19} - 14298960497392 \beta_{18} + 57763842359571 \beta_{17} + \cdots + 49\!\cdots\!79 ) / 4 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 60\!\cdots\!42 \beta_{18} + \cdots + 16\!\cdots\!99 ) / 4 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 20\!\cdots\!26 \beta_{19} + \cdots - 15\!\cdots\!99 ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 23\!\cdots\!64 \beta_{18} + \cdots - 49\!\cdots\!33 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 29\!\cdots\!80 \beta_{19} + \cdots + 18\!\cdots\!85 ) / 4 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 34\!\cdots\!74 \beta_{18} + \cdots + 61\!\cdots\!17 ) / 4 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 10\!\cdots\!94 \beta_{19} + \cdots - 53\!\cdots\!44 ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 11\!\cdots\!98 \beta_{18} + \cdots - 19\!\cdots\!18 ) / 2 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 13\!\cdots\!60 \beta_{19} + \cdots + 63\!\cdots\!75 ) / 4 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( - 15\!\cdots\!90 \beta_{18} + \cdots + 23\!\cdots\!67 ) / 4 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( 45\!\cdots\!30 \beta_{19} + \cdots - 19\!\cdots\!17 ) / 2 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( 50\!\cdots\!16 \beta_{18} + \cdots - 75\!\cdots\!07 ) / 2 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( - 59\!\cdots\!32 \beta_{19} + \cdots + 23\!\cdots\!73 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(\beta_{4}\) |
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 |
|
−2641.96 | − | 2641.96i | −75652.0 | + | 75652.0i | 9.76564e6i | 3.47036e7 | + | 3.43489e7i | 3.99740e8 | −1.89429e9 | − | 1.89429e9i | 1.47193e10 | − | 1.47193e10i | 1.99346e10i | −9.37131e8 | − | 1.82434e11i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2.2 | −1984.08 | − | 1984.08i | 46177.6 | − | 46177.6i | 3.67882e6i | −3.77431e6 | − | 4.86820e7i | −1.83240e8 | 2.65570e9 | + | 2.65570e9i | −1.02276e9 | + | 1.02276e9i | 2.71163e10i | −8.91004e10 | + | 1.04077e11i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2.3 | −1597.67 | − | 1597.67i | 220666. | − | 220666.i | 910798.i | −1.47171e7 | + | 4.65574e7i | −7.05104e8 | −3.15935e8 | − | 3.15935e8i | −5.24596e9 | + | 5.24596e9i | − | 6.60062e10i | 9.78965e10 | − | 5.08702e10i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2.4 | −1278.37 | − | 1278.37i | −104300. | + | 104300.i | − | 925861.i | −4.70788e7 | − | 1.29528e7i | 2.66667e8 | −1.87237e9 | − | 1.87237e9i | −6.54545e9 | + | 6.54545e9i | 9.62412e9i | 4.36254e10 | + | 7.67424e10i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2.5 | −555.007 | − | 555.007i | −210102. | + | 210102.i | − | 3.57824e6i | 4.01261e7 | + | 2.78223e7i | 2.33216e8 | 2.22187e9 | + | 2.22187e9i | −4.31382e9 | + | 4.31382e9i | − | 5.69048e10i | −6.82867e9 | − | 3.77119e10i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2.6 | 115.234 | + | 115.234i | 90525.7 | − | 90525.7i | − | 4.16775e6i | 4.36045e7 | − | 2.19734e7i | 2.08632e7 | −1.04929e9 | − | 1.04929e9i | 9.63589e8 | − | 9.63589e8i | 1.49913e10i | 7.55678e9 | + | 2.49263e9i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2.7 | 726.475 | + | 726.475i | 16088.6 | − | 16088.6i | − | 3.13877e6i | −3.46623e7 | + | 3.43906e7i | 2.33759e7 | 6.19067e8 | + | 6.19067e8i | 5.32729e9 | − | 5.32729e9i | 3.08634e10i | −5.01652e10 | − | 1.97348e8i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2.8 | 1720.64 | + | 1720.64i | −173446. | + | 173446.i | 1.72688e6i | −1.41390e7 | − | 4.67362e7i | −5.96877e8 | −5.23318e8 | − | 5.23318e8i | 4.24555e9 | − | 4.24555e9i | − | 2.87863e10i | 5.60880e10 | − | 1.04744e11i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2.9 | 2045.93 | + | 2045.93i | 221671. | − | 221671.i | 4.17735e6i | −3.01365e7 | − | 3.84185e7i | 9.07047e8 | 1.14631e9 | + | 1.14631e9i | 3.46918e7 | − | 3.46918e7i | − | 6.68951e10i | 1.69444e10 | − | 1.40259e11i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2.10 | 2423.81 | + | 2423.81i | −14678.7 | + | 14678.7i | 7.55542e6i | 3.06960e7 | + | 3.79729e7i | −7.11567e7 | −3.06524e8 | − | 3.06524e8i | −8.14672e9 | + | 8.14672e9i | 3.09501e10i | −1.76377e10 | + | 1.66441e11i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.1 | −2641.96 | + | 2641.96i | −75652.0 | − | 75652.0i | − | 9.76564e6i | 3.47036e7 | − | 3.43489e7i | 3.99740e8 | −1.89429e9 | + | 1.89429e9i | 1.47193e10 | + | 1.47193e10i | − | 1.99346e10i | −9.37131e8 | + | 1.82434e11i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.2 | −1984.08 | + | 1984.08i | 46177.6 | + | 46177.6i | − | 3.67882e6i | −3.77431e6 | + | 4.86820e7i | −1.83240e8 | 2.65570e9 | − | 2.65570e9i | −1.02276e9 | − | 1.02276e9i | − | 2.71163e10i | −8.91004e10 | − | 1.04077e11i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.3 | −1597.67 | + | 1597.67i | 220666. | + | 220666.i | − | 910798.i | −1.47171e7 | − | 4.65574e7i | −7.05104e8 | −3.15935e8 | + | 3.15935e8i | −5.24596e9 | − | 5.24596e9i | 6.60062e10i | 9.78965e10 | + | 5.08702e10i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.4 | −1278.37 | + | 1278.37i | −104300. | − | 104300.i | 925861.i | −4.70788e7 | + | 1.29528e7i | 2.66667e8 | −1.87237e9 | + | 1.87237e9i | −6.54545e9 | − | 6.54545e9i | − | 9.62412e9i | 4.36254e10 | − | 7.67424e10i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.5 | −555.007 | + | 555.007i | −210102. | − | 210102.i | 3.57824e6i | 4.01261e7 | − | 2.78223e7i | 2.33216e8 | 2.22187e9 | − | 2.22187e9i | −4.31382e9 | − | 4.31382e9i | 5.69048e10i | −6.82867e9 | + | 3.77119e10i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.6 | 115.234 | − | 115.234i | 90525.7 | + | 90525.7i | 4.16775e6i | 4.36045e7 | + | 2.19734e7i | 2.08632e7 | −1.04929e9 | + | 1.04929e9i | 9.63589e8 | + | 9.63589e8i | − | 1.49913e10i | 7.55678e9 | − | 2.49263e9i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.7 | 726.475 | − | 726.475i | 16088.6 | + | 16088.6i | 3.13877e6i | −3.46623e7 | − | 3.43906e7i | 2.33759e7 | 6.19067e8 | − | 6.19067e8i | 5.32729e9 | + | 5.32729e9i | − | 3.08634e10i | −5.01652e10 | + | 1.97348e8i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.8 | 1720.64 | − | 1720.64i | −173446. | − | 173446.i | − | 1.72688e6i | −1.41390e7 | + | 4.67362e7i | −5.96877e8 | −5.23318e8 | + | 5.23318e8i | 4.24555e9 | + | 4.24555e9i | 2.87863e10i | 5.60880e10 | + | 1.04744e11i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.9 | 2045.93 | − | 2045.93i | 221671. | + | 221671.i | − | 4.17735e6i | −3.01365e7 | + | 3.84185e7i | 9.07047e8 | 1.14631e9 | − | 1.14631e9i | 3.46918e7 | + | 3.46918e7i | 6.68951e10i | 1.69444e10 | + | 1.40259e11i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.10 | 2423.81 | − | 2423.81i | −14678.7 | − | 14678.7i | − | 7.55542e6i | 3.06960e7 | − | 3.79729e7i | −7.11567e7 | −3.06524e8 | + | 3.06524e8i | −8.14672e9 | − | 8.14672e9i | − | 3.09501e10i | −1.76377e10 | − | 1.66441e11i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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 | 5.23.c.a | ✓ | 20 |
| 5.b | even | 2 | 1 | 25.23.c.b | 20 | ||
| 5.c | odd | 4 | 1 | inner | 5.23.c.a | ✓ | 20 |
| 5.c | odd | 4 | 1 | 25.23.c.b | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.23.c.a | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 5.23.c.a | ✓ | 20 | 5.c | odd | 4 | 1 | inner |
| 25.23.c.b | 20 | 5.b | even | 2 | 1 | ||
| 25.23.c.b | 20 | 5.c | odd | 4 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{23}^{\mathrm{new}}(5, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} + \cdots + 18\!\cdots\!24 \)
|
| $3$ |
\( T^{20} + \cdots + 19\!\cdots\!24 \)
|
| $5$ |
\( T^{20} + \cdots + 59\!\cdots\!25 \)
|
| $7$ |
\( T^{20} + \cdots + 63\!\cdots\!24 \)
|
| $11$ |
\( (T^{10} + \cdots + 46\!\cdots\!24)^{2} \)
|
| $13$ |
\( T^{20} + \cdots + 55\!\cdots\!24 \)
|
| $17$ |
\( T^{20} + \cdots + 45\!\cdots\!24 \)
|
| $19$ |
\( T^{20} + \cdots + 82\!\cdots\!00 \)
|
| $23$ |
\( T^{20} + \cdots + 92\!\cdots\!24 \)
|
| $29$ |
\( T^{20} + \cdots + 11\!\cdots\!00 \)
|
| $31$ |
\( (T^{10} + \cdots + 42\!\cdots\!24)^{2} \)
|
| $37$ |
\( T^{20} + \cdots + 15\!\cdots\!24 \)
|
| $41$ |
\( (T^{10} + \cdots + 12\!\cdots\!24)^{2} \)
|
| $43$ |
\( T^{20} + \cdots + 87\!\cdots\!24 \)
|
| $47$ |
\( T^{20} + \cdots + 34\!\cdots\!24 \)
|
| $53$ |
\( T^{20} + \cdots + 36\!\cdots\!24 \)
|
| $59$ |
\( T^{20} + \cdots + 14\!\cdots\!00 \)
|
| $61$ |
\( (T^{10} + \cdots - 97\!\cdots\!76)^{2} \)
|
| $67$ |
\( T^{20} + \cdots + 57\!\cdots\!24 \)
|
| $71$ |
\( (T^{10} + \cdots + 15\!\cdots\!24)^{2} \)
|
| $73$ |
\( T^{20} + \cdots + 21\!\cdots\!24 \)
|
| $79$ |
\( T^{20} + \cdots + 13\!\cdots\!00 \)
|
| $83$ |
\( T^{20} + \cdots + 28\!\cdots\!24 \)
|
| $89$ |
\( T^{20} + \cdots + 21\!\cdots\!00 \)
|
| $97$ |
\( T^{20} + \cdots + 10\!\cdots\!24 \)
|
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