Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [125,2,Mod(26,125)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(125, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("125.26");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 125 = 5^{3} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 125.d (of order \(5\), degree \(4\), not 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: | \(0.998130025266\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} + x^{14} - 4x^{12} - 49x^{10} + 11x^{8} + 395x^{6} + 900x^{4} + 1125x^{2} + 625 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 5^{2} \) |
| Twist minimal: | no (minimal twist has level 25) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} + x^{14} - 4x^{12} - 49x^{10} + 11x^{8} + 395x^{6} + 900x^{4} + 1125x^{2} + 625 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 22849 \nu^{15} - 2422021 \nu^{13} + 6573709 \nu^{11} - 1538146 \nu^{9} + 97097069 \nu^{7} + \cdots - 35093875 \nu ) / 858900625 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 948392 \nu^{14} - 2081693 \nu^{12} - 1547103 \nu^{10} - 35207443 \nu^{8} + 136185777 \nu^{6} + \cdots + 181387875 ) / 171780125 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 122987 \nu^{14} + 97172 \nu^{12} - 701513 \nu^{10} - 5799603 \nu^{8} + 3932417 \nu^{6} + \cdots + 52412250 ) / 15616375 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 1379032 \nu^{14} - 312743 \nu^{12} - 4990978 \nu^{10} - 61900293 \nu^{8} + 94590252 \nu^{6} + \cdots + 563878625 ) / 171780125 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 281280 \nu^{14} + 69219 \nu^{12} + 939009 \nu^{10} + 12381889 \nu^{8} - 18275316 \nu^{6} + \cdots - 167096475 ) / 34356025 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 1157122 \nu^{15} + 1428203 \nu^{13} + 8678488 \nu^{11} + 43919978 \nu^{9} + \cdots + 846395125 \nu ) / 858900625 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 441862 \nu^{14} + 108648 \nu^{12} + 1544473 \nu^{10} + 20140738 \nu^{8} - 29602957 \nu^{6} + \cdots - 215410900 ) / 34356025 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 2817591 \nu^{14} + 2379174 \nu^{12} + 8884104 \nu^{10} + 118381374 \nu^{8} - 262226761 \nu^{6} + \cdots - 860602375 ) / 171780125 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 626638 \nu^{14} + 144621 \nu^{12} + 2783831 \nu^{10} + 27217946 \nu^{8} - 41997039 \nu^{6} + \cdots - 222107450 ) / 34356025 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 257689 \nu^{15} - 260409 \nu^{13} + 1184711 \nu^{11} + 13501191 \nu^{9} - 4440499 \nu^{7} + \cdots - 189940875 \nu ) / 78081875 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 3711283 \nu^{15} + 2442853 \nu^{13} - 18411087 \nu^{11} - 175485672 \nu^{9} + \cdots + 1573606875 \nu ) / 858900625 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 11296137 \nu^{15} - 2500148 \nu^{13} - 48798533 \nu^{11} - 491452273 \nu^{9} + \cdots + 4625990250 \nu ) / 858900625 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 1033909 \nu^{15} - 651386 \nu^{13} - 3517956 \nu^{11} - 44316636 \nu^{9} + 84515054 \nu^{7} + \cdots + 416073000 \nu ) / 78081875 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 1468861 \nu^{15} - 201274 \nu^{13} - 5426879 \nu^{11} - 66369199 \nu^{9} + 90901286 \nu^{7} + \cdots + 722601875 \nu ) / 78081875 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 18299323 \nu^{15} - 8550017 \nu^{13} - 66698807 \nu^{11} - 789938892 \nu^{9} + \cdots + 7417196625 \nu ) / 858900625 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{15} - 5\beta_{14} + 4\beta_{13} + 4\beta_{12} - 5\beta_{10} + 5\beta_{6} + \beta_1 ) / 5 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{9} + 2\beta_{8} + 3\beta_{7} + 2\beta_{5} + 11\beta_{4} - 4\beta_{3} - \beta_{2} + 4 ) / 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -6\beta_{15} + 14\beta_{13} - 6\beta_{12} + 5\beta_{11} - 9\beta_1 ) / 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -7\beta_{9} - 13\beta_{8} + 3\beta_{7} + 7\beta_{5} + 6\beta_{4} - 19\beta_{3} - 26\beta_{2} + 14 ) / 5 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 39\beta_{15} - 20\beta_{14} - 6\beta_{13} - 36\beta_{12} - 30\beta_{10} - 29\beta_1 ) / 5 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 23\beta_{9} - 23\beta_{8} + 63\beta_{7} + 52\beta_{5} + 156\beta_{4} + 11\beta_{3} - 11\beta_{2} + 144 ) / 5 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 26 \beta_{15} - 190 \beta_{14} + 184 \beta_{13} + 49 \beta_{12} + 190 \beta_{11} - 115 \beta_{10} + \cdots + 66 \beta_1 ) / 5 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( -57\beta_{9} - 18\beta_{8} + 363\beta_{7} - 208\beta_{5} + 381\beta_{4} - 114\beta_{3} - 96\beta_{2} + 39 ) / 5 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( -\beta_{15} + 284\beta_{13} - 286\beta_{12} + 285\beta_{11} + 285\beta_{10} + 190\beta_{6} - 479\beta_1 ) / 5 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 208\beta_{9} - 688\beta_{8} + 208\beta_{7} - 493\beta_{5} - 149\beta_{4} - 344\beta_{3} - 896\beta_{2} - 606 ) / 5 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 2344 \beta_{15} - 290 \beta_{14} - 1651 \beta_{13} - 1846 \beta_{12} + 195 \beta_{11} - 290 \beta_{10} + \cdots - 624 \beta_1 ) / 5 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 2208 \beta_{9} - 683 \beta_{8} + 2888 \beta_{7} - 683 \beta_{5} + 3956 \beta_{4} + 2891 \beta_{3} + \cdots + 1784 ) / 5 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 906 \beta_{15} - 4675 \beta_{14} + 2364 \beta_{13} + 3769 \beta_{12} + 7530 \beta_{11} + \cdots + 5166 \beta_1 ) / 5 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 393 \beta_{9} + 1012 \beta_{8} + 10778 \beta_{7} - 18118 \beta_{5} - 619 \beta_{4} + 1631 \beta_{3} + \cdots - 18511 ) / 5 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 1639\beta_{15} + 19130\beta_{14} - 12431\beta_{13} - 9336\beta_{12} + 30920\beta_{10} - 13429\beta_1 ) / 5 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/125\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(\beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 26.1 |
|
−0.644389 | − | 1.98322i | 1.77862 | − | 1.29224i | −1.89991 | + | 1.38036i | 0 | −3.70892 | − | 2.69469i | 0.992398 | 0.587785 | + | 0.427051i | 0.566541 | − | 1.74363i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 26.2 | −0.0566033 | − | 0.174207i | 1.19083 | − | 0.865190i | 1.59089 | − | 1.15585i | 0 | −0.218127 | − | 0.158479i | −3.26086 | −0.587785 | − | 0.427051i | −0.257524 | + | 0.792578i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 26.3 | 0.0566033 | + | 0.174207i | −1.19083 | + | 0.865190i | 1.59089 | − | 1.15585i | 0 | −0.218127 | − | 0.158479i | 3.26086 | 0.587785 | + | 0.427051i | −0.257524 | + | 0.792578i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 26.4 | 0.644389 | + | 1.98322i | −1.77862 | + | 1.29224i | −1.89991 | + | 1.38036i | 0 | −3.70892 | − | 2.69469i | −0.992398 | −0.587785 | − | 0.427051i | 0.566541 | − | 1.74363i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 51.1 | −1.86824 | + | 1.35736i | 0.146753 | − | 0.451659i | 1.02988 | − | 3.16963i | 0 | 0.338893 | + | 1.04301i | 3.03582 | 0.951057 | + | 2.92705i | 2.24459 | + | 1.63079i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 51.2 | −0.917186 | + | 0.666375i | −0.804303 | + | 2.47539i | −0.220859 | + | 0.679734i | 0 | −0.911842 | − | 2.80636i | 0.407162 | −0.951057 | − | 2.92705i | −3.05361 | − | 2.21858i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 51.3 | 0.917186 | − | 0.666375i | 0.804303 | − | 2.47539i | −0.220859 | + | 0.679734i | 0 | −0.911842 | − | 2.80636i | −0.407162 | 0.951057 | + | 2.92705i | −3.05361 | − | 2.21858i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 51.4 | 1.86824 | − | 1.35736i | −0.146753 | + | 0.451659i | 1.02988 | − | 3.16963i | 0 | 0.338893 | + | 1.04301i | −3.03582 | −0.951057 | − | 2.92705i | 2.24459 | + | 1.63079i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 76.1 | −1.86824 | − | 1.35736i | 0.146753 | + | 0.451659i | 1.02988 | + | 3.16963i | 0 | 0.338893 | − | 1.04301i | 3.03582 | 0.951057 | − | 2.92705i | 2.24459 | − | 1.63079i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 76.2 | −0.917186 | − | 0.666375i | −0.804303 | − | 2.47539i | −0.220859 | − | 0.679734i | 0 | −0.911842 | + | 2.80636i | 0.407162 | −0.951057 | + | 2.92705i | −3.05361 | + | 2.21858i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 76.3 | 0.917186 | + | 0.666375i | 0.804303 | + | 2.47539i | −0.220859 | − | 0.679734i | 0 | −0.911842 | + | 2.80636i | −0.407162 | 0.951057 | − | 2.92705i | −3.05361 | + | 2.21858i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 76.4 | 1.86824 | + | 1.35736i | −0.146753 | − | 0.451659i | 1.02988 | + | 3.16963i | 0 | 0.338893 | − | 1.04301i | −3.03582 | −0.951057 | + | 2.92705i | 2.24459 | − | 1.63079i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.1 | −0.644389 | + | 1.98322i | 1.77862 | + | 1.29224i | −1.89991 | − | 1.38036i | 0 | −3.70892 | + | 2.69469i | 0.992398 | 0.587785 | − | 0.427051i | 0.566541 | + | 1.74363i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.2 | −0.0566033 | + | 0.174207i | 1.19083 | + | 0.865190i | 1.59089 | + | 1.15585i | 0 | −0.218127 | + | 0.158479i | −3.26086 | −0.587785 | + | 0.427051i | −0.257524 | − | 0.792578i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.3 | 0.0566033 | − | 0.174207i | −1.19083 | − | 0.865190i | 1.59089 | + | 1.15585i | 0 | −0.218127 | + | 0.158479i | 3.26086 | 0.587785 | − | 0.427051i | −0.257524 | − | 0.792578i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.4 | 0.644389 | − | 1.98322i | −1.77862 | − | 1.29224i | −1.89991 | − | 1.38036i | 0 | −3.70892 | + | 2.69469i | −0.992398 | −0.587785 | + | 0.427051i | 0.566541 | + | 1.74363i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 25.d | even | 5 | 1 | inner |
| 25.e | even | 10 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 125.2.d.b | 16 | |
| 5.b | even | 2 | 1 | inner | 125.2.d.b | 16 | |
| 5.c | odd | 4 | 1 | 25.2.e.a | ✓ | 8 | |
| 5.c | odd | 4 | 1 | 125.2.e.b | 8 | ||
| 15.e | even | 4 | 1 | 225.2.m.a | 8 | ||
| 20.e | even | 4 | 1 | 400.2.y.c | 8 | ||
| 25.d | even | 5 | 1 | inner | 125.2.d.b | 16 | |
| 25.d | even | 5 | 1 | 625.2.a.f | 8 | ||
| 25.d | even | 5 | 2 | 625.2.d.o | 16 | ||
| 25.e | even | 10 | 1 | inner | 125.2.d.b | 16 | |
| 25.e | even | 10 | 1 | 625.2.a.f | 8 | ||
| 25.e | even | 10 | 2 | 625.2.d.o | 16 | ||
| 25.f | odd | 20 | 1 | 25.2.e.a | ✓ | 8 | |
| 25.f | odd | 20 | 1 | 125.2.e.b | 8 | ||
| 25.f | odd | 20 | 2 | 625.2.b.c | 8 | ||
| 25.f | odd | 20 | 2 | 625.2.e.a | 8 | ||
| 25.f | odd | 20 | 2 | 625.2.e.i | 8 | ||
| 75.h | odd | 10 | 1 | 5625.2.a.x | 8 | ||
| 75.j | odd | 10 | 1 | 5625.2.a.x | 8 | ||
| 75.l | even | 20 | 1 | 225.2.m.a | 8 | ||
| 100.h | odd | 10 | 1 | 10000.2.a.bj | 8 | ||
| 100.j | odd | 10 | 1 | 10000.2.a.bj | 8 | ||
| 100.l | even | 20 | 1 | 400.2.y.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 25.2.e.a | ✓ | 8 | 5.c | odd | 4 | 1 | |
| 25.2.e.a | ✓ | 8 | 25.f | odd | 20 | 1 | |
| 125.2.d.b | 16 | 1.a | even | 1 | 1 | trivial | |
| 125.2.d.b | 16 | 5.b | even | 2 | 1 | inner | |
| 125.2.d.b | 16 | 25.d | even | 5 | 1 | inner | |
| 125.2.d.b | 16 | 25.e | even | 10 | 1 | inner | |
| 125.2.e.b | 8 | 5.c | odd | 4 | 1 | ||
| 125.2.e.b | 8 | 25.f | odd | 20 | 1 | ||
| 225.2.m.a | 8 | 15.e | even | 4 | 1 | ||
| 225.2.m.a | 8 | 75.l | even | 20 | 1 | ||
| 400.2.y.c | 8 | 20.e | even | 4 | 1 | ||
| 400.2.y.c | 8 | 100.l | even | 20 | 1 | ||
| 625.2.a.f | 8 | 25.d | even | 5 | 1 | ||
| 625.2.a.f | 8 | 25.e | even | 10 | 1 | ||
| 625.2.b.c | 8 | 25.f | odd | 20 | 2 | ||
| 625.2.d.o | 16 | 25.d | even | 5 | 2 | ||
| 625.2.d.o | 16 | 25.e | even | 10 | 2 | ||
| 625.2.e.a | 8 | 25.f | odd | 20 | 2 | ||
| 625.2.e.i | 8 | 25.f | odd | 20 | 2 | ||
| 5625.2.a.x | 8 | 75.h | odd | 10 | 1 | ||
| 5625.2.a.x | 8 | 75.j | odd | 10 | 1 | ||
| 10000.2.a.bj | 8 | 100.h | odd | 10 | 1 | ||
| 10000.2.a.bj | 8 | 100.j | odd | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} + 3T_{2}^{14} + 23T_{2}^{12} + 126T_{2}^{10} + 475T_{2}^{8} - 174T_{2}^{6} + 878T_{2}^{4} + 48T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(125, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} + 3 T^{14} + \cdots + 1 \)
|
| $3$ |
\( T^{16} + 7 T^{14} + \cdots + 256 \)
|
| $5$ |
\( T^{16} \)
|
| $7$ |
\( (T^{8} - 21 T^{6} + \cdots + 16)^{2} \)
|
| $11$ |
\( (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 16)^{4} \)
|
| $13$ |
\( T^{16} + 17 T^{14} + \cdots + 1 \)
|
| $17$ |
\( T^{16} - 12 T^{14} + \cdots + 3748096 \)
|
| $19$ |
\( (T^{8} - 5 T^{7} + \cdots + 400)^{2} \)
|
| $23$ |
\( T^{16} + 27 T^{14} + \cdots + 65536 \)
|
| $29$ |
\( (T^{8} - 5 T^{7} + \cdots + 483025)^{2} \)
|
| $31$ |
\( (T^{8} + 9 T^{7} + \cdots + 1936)^{2} \)
|
| $37$ |
\( T^{16} + \cdots + 13521270961 \)
|
| $41$ |
\( (T^{8} + 4 T^{7} + \cdots + 13456)^{2} \)
|
| $43$ |
\( (T^{8} - 129 T^{6} + \cdots + 246016)^{2} \)
|
| $47$ |
\( T^{16} + \cdots + 4294967296 \)
|
| $53$ |
\( T^{16} + \cdots + 76661949773761 \)
|
| $59$ |
\( (T^{8} - 15 T^{5} + \cdots + 4080400)^{2} \)
|
| $61$ |
\( (T^{8} + 9 T^{7} + \cdots + 116281)^{2} \)
|
| $67$ |
\( T^{16} + \cdots + 60523872256 \)
|
| $71$ |
\( (T^{8} - 6 T^{7} + \cdots + 24245776)^{2} \)
|
| $73$ |
\( T^{16} + 127 T^{14} + \cdots + 1 \)
|
| $79$ |
\( (T^{8} + 15 T^{7} + \cdots + 33408400)^{2} \)
|
| $83$ |
\( T^{16} + \cdots + 9971220736 \)
|
| $89$ |
\( (T^{8} - 25 T^{7} + \cdots + 1392400)^{2} \)
|
| $97$ |
\( T^{16} + \cdots + 90\!\cdots\!61 \)
|
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