Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [275,2,Mod(49,275)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(275, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("275.49");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 275 = 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 275.z (of order \(10\), 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: | \(2.19588605559\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{10})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 11x^{14} + 56x^{12} - 141x^{10} + 551x^{8} - 1245x^{6} + 1400x^{4} + 125x^{2} + 625 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 55) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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
\( x^{16} - 11x^{14} + 56x^{12} - 141x^{10} + 551x^{8} - 1245x^{6} + 1400x^{4} + 125x^{2} + 625 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 8794653 \nu^{14} + 57157787 \nu^{12} - 985758052 \nu^{10} + 5165013322 \nu^{8} + \cdots + 94458988125 ) / 119433312625 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 13082829 \nu^{15} + 600984896 \nu^{13} - 6998480441 \nu^{11} + 35723314026 \nu^{9} + \cdots + 616297150000 \nu ) / 597166563125 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 17059629 \nu^{14} - 232150169 \nu^{12} + 1788142999 \nu^{10} - 7589325464 \nu^{8} + \cdots - 21411461000 ) / 119433312625 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 18633578 \nu^{14} + 148646863 \nu^{12} - 445792298 \nu^{10} + 786001903 \nu^{8} + \cdots + 102394604625 ) / 119433312625 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 51768912 \nu^{14} - 610042682 \nu^{12} + 3408703022 \nu^{10} - 9963171417 \nu^{8} + \cdots - 35649764250 ) / 119433312625 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 57056094 \nu^{15} + 886773831 \nu^{13} - 11927270701 \nu^{11} + 61548380636 \nu^{9} + \cdots + 1088592090625 \nu ) / 597166563125 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 17295639 \nu^{14} - 197587039 \nu^{12} + 1005852609 \nu^{10} - 2467403474 \nu^{8} + \cdots + 13909049025 ) / 23886662525 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 88867804 \nu^{14} + 1145491714 \nu^{12} - 6759969294 \nu^{10} + 20728132534 \nu^{8} + \cdots - 26514235375 ) / 119433312625 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 148979203 \nu^{14} - 1546223773 \nu^{12} + 7513598583 \nu^{10} - 17986740138 \nu^{8} + \cdots - 1635353625 ) / 119433312625 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 171291688 \nu^{15} + 1798910423 \nu^{13} - 8431583683 \nu^{11} + 15211413013 \nu^{9} + \cdots - 729830779750 \nu ) / 597166563125 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 383405571 \nu^{15} + 4576502156 \nu^{13} - 26037419701 \nu^{11} + 78919316986 \nu^{9} + \cdots + 275540651500 \nu ) / 597166563125 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 17295639 \nu^{15} - 197587039 \nu^{13} + 1005852609 \nu^{11} - 2467403474 \nu^{9} + \cdots + 13909049025 \nu ) / 23886662525 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 6275 \nu^{15} - 71216 \nu^{13} + 364716 \nu^{11} - 901036 \nu^{9} + 3271196 \nu^{7} + \cdots + 4902600 \nu ) / 6308875 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 29408352 \nu^{15} - 308164018 \nu^{13} + 1490441603 \nu^{11} - 3427035093 \nu^{9} + \cdots + 25670893800 \nu ) / 23886662525 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 29408352 \nu^{15} - 308164018 \nu^{13} + 1490441603 \nu^{11} - 3427035093 \nu^{9} + \cdots + 1784231275 \nu ) / 23886662525 \)
|
| \(\nu\) | \(=\) |
\( -\beta_{15} + \beta_{14} \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{7} + 2\beta_{5} - \beta_{3} + 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{15} + 4\beta_{14} + \beta_{13} - 4\beta_{12} + 2\beta_{11} + \beta_{10} - 3\beta_{6} + 3\beta_{2} \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{9} - 7\beta_{8} - 7\beta_{7} + 6\beta_{5} - 2\beta_{4} - 7\beta_{3} + 6\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8\beta_{14} + 12\beta_{13} - 20\beta_{12} + 11\beta_{11} - 11\beta_{2} \)
|
| \(\nu^{6}\) | \(=\) |
\( -43\beta_{9} - 43\beta_{8} + 36\beta_{5} - 24\beta_{4} - 19\beta_{3} + 68\beta _1 - 36 \)
|
| \(\nu^{7}\) | \(=\) |
\( 55\beta_{15} - 55\beta_{12} + 81\beta_{11} - 81\beta_{10} + 56\beta_{6} - 204\beta_{2} \)
|
| \(\nu^{8}\) | \(=\) |
\( -259\beta_{9} - 136\beta_{8} + 167\beta_{7} + 167\beta_{5} + 388\beta _1 - 388 \)
|
| \(\nu^{9}\) | \(=\) |
\( 647\beta_{15} - 357\beta_{14} - 382\beta_{13} + 382\beta_{11} - 913\beta_{10} + 357\beta_{6} - 1270\beta_{2} \)
|
| \(\nu^{10}\) | \(=\) |
\( -888\beta_{9} + 1361\beta_{7} + 672\beta_{4} + 888\beta_{3} + 1361\beta _1 - 2298 \)
|
| \(\nu^{11}\) | \(=\) |
\( 3858\beta_{15} - 3858\beta_{14} - 3336\beta_{13} + 2249\beta_{12} - 5568\beta_{10} + 2249\beta_{6} - 5585\beta_{2} \)
|
| \(\nu^{12}\) | \(=\) |
\( 5585\beta_{8} + 8356\beta_{7} - 5467\beta_{5} + 5585\beta_{4} + 9426\beta_{3} - 8356 \)
|
| \(\nu^{13}\) | \(=\) |
\( 13941 \beta_{15} - 23249 \beta_{14} - 20596 \beta_{13} + 23249 \beta_{12} - 13267 \beta_{11} + \cdots - 9308 \beta_{2} \)
|
| \(\nu^{14}\) | \(=\) |
\( 34537\beta_{9} + 57112\beta_{8} + 32557\beta_{7} - 51131\beta_{5} + 34537\beta_{4} + 57112\beta_{3} - 51131\beta_1 \)
|
| \(\nu^{15}\) | \(=\) |
\( -85668\beta_{14} - 79687\beta_{13} + 140800\beta_{12} - 126186\beta_{11} + 126186\beta_{2} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(177\) |
| \(\chi(n)\) | \(-\beta_{7}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
−2.14748 | + | 0.697759i | −0.456994 | + | 0.628998i | 2.50678 | − | 1.82128i | 0 | 0.542497 | − | 1.66963i | −0.0728678 | − | 0.100294i | −1.45801 | + | 2.00678i | 0.740256 | + | 2.27827i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.2 | −1.78421 | + | 0.579725i | −1.04478 | + | 1.43801i | 1.22929 | − | 0.893133i | 0 | 1.03045 | − | 3.17141i | 2.50279 | + | 3.44479i | 0.529862 | − | 0.729292i | −0.0492728 | − | 0.151646i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.3 | 1.78421 | − | 0.579725i | 1.04478 | − | 1.43801i | 1.22929 | − | 0.893133i | 0 | 1.03045 | − | 3.17141i | −2.50279 | − | 3.44479i | −0.529862 | + | 0.729292i | −0.0492728 | − | 0.151646i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.4 | 2.14748 | − | 0.697759i | 0.456994 | − | 0.628998i | 2.50678 | − | 1.82128i | 0 | 0.542497 | − | 1.66963i | 0.0728678 | + | 0.100294i | 1.45801 | − | 2.00678i | 0.740256 | + | 2.27827i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 124.1 | −1.48388 | − | 2.04238i | −2.34600 | + | 0.762262i | −1.35140 | + | 4.15918i | 0 | 5.03801 | + | 3.66033i | 1.99105 | + | 0.646930i | 5.69802 | − | 1.85140i | 2.49563 | − | 1.81318i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 124.2 | −0.0549637 | − | 0.0756511i | 1.39494 | − | 0.453245i | 0.615332 | − | 1.89380i | 0 | −0.110960 | − | 0.0806171i | 4.30308 | + | 1.39815i | −0.354955 | + | 0.115332i | −0.686611 | + | 0.498852i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 124.3 | 0.0549637 | + | 0.0756511i | −1.39494 | + | 0.453245i | 0.615332 | − | 1.89380i | 0 | −0.110960 | − | 0.0806171i | −4.30308 | − | 1.39815i | 0.354955 | − | 0.115332i | −0.686611 | + | 0.498852i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 124.4 | 1.48388 | + | 2.04238i | 2.34600 | − | 0.762262i | −1.35140 | + | 4.15918i | 0 | 5.03801 | + | 3.66033i | −1.99105 | − | 0.646930i | −5.69802 | + | 1.85140i | 2.49563 | − | 1.81318i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 174.1 | −2.14748 | − | 0.697759i | −0.456994 | − | 0.628998i | 2.50678 | + | 1.82128i | 0 | 0.542497 | + | 1.66963i | −0.0728678 | + | 0.100294i | −1.45801 | − | 2.00678i | 0.740256 | − | 2.27827i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 174.2 | −1.78421 | − | 0.579725i | −1.04478 | − | 1.43801i | 1.22929 | + | 0.893133i | 0 | 1.03045 | + | 3.17141i | 2.50279 | − | 3.44479i | 0.529862 | + | 0.729292i | −0.0492728 | + | 0.151646i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 174.3 | 1.78421 | + | 0.579725i | 1.04478 | + | 1.43801i | 1.22929 | + | 0.893133i | 0 | 1.03045 | + | 3.17141i | −2.50279 | + | 3.44479i | −0.529862 | − | 0.729292i | −0.0492728 | + | 0.151646i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 174.4 | 2.14748 | + | 0.697759i | 0.456994 | + | 0.628998i | 2.50678 | + | 1.82128i | 0 | 0.542497 | + | 1.66963i | 0.0728678 | − | 0.100294i | 1.45801 | + | 2.00678i | 0.740256 | − | 2.27827i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 224.1 | −1.48388 | + | 2.04238i | −2.34600 | − | 0.762262i | −1.35140 | − | 4.15918i | 0 | 5.03801 | − | 3.66033i | 1.99105 | − | 0.646930i | 5.69802 | + | 1.85140i | 2.49563 | + | 1.81318i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 224.2 | −0.0549637 | + | 0.0756511i | 1.39494 | + | 0.453245i | 0.615332 | + | 1.89380i | 0 | −0.110960 | + | 0.0806171i | 4.30308 | − | 1.39815i | −0.354955 | − | 0.115332i | −0.686611 | − | 0.498852i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 224.3 | 0.0549637 | − | 0.0756511i | −1.39494 | − | 0.453245i | 0.615332 | + | 1.89380i | 0 | −0.110960 | + | 0.0806171i | −4.30308 | + | 1.39815i | 0.354955 | + | 0.115332i | −0.686611 | − | 0.498852i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 224.4 | 1.48388 | − | 2.04238i | 2.34600 | + | 0.762262i | −1.35140 | − | 4.15918i | 0 | 5.03801 | − | 3.66033i | −1.99105 | + | 0.646930i | −5.69802 | − | 1.85140i | 2.49563 | + | 1.81318i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 11.c | even | 5 | 1 | inner |
| 55.j | even | 10 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 275.2.z.b | 16 | |
| 5.b | even | 2 | 1 | inner | 275.2.z.b | 16 | |
| 5.c | odd | 4 | 1 | 55.2.g.a | ✓ | 8 | |
| 5.c | odd | 4 | 1 | 275.2.h.b | 8 | ||
| 11.c | even | 5 | 1 | inner | 275.2.z.b | 16 | |
| 15.e | even | 4 | 1 | 495.2.n.f | 8 | ||
| 20.e | even | 4 | 1 | 880.2.bo.e | 8 | ||
| 55.e | even | 4 | 1 | 605.2.g.n | 8 | ||
| 55.j | even | 10 | 1 | inner | 275.2.z.b | 16 | |
| 55.k | odd | 20 | 1 | 55.2.g.a | ✓ | 8 | |
| 55.k | odd | 20 | 1 | 275.2.h.b | 8 | ||
| 55.k | odd | 20 | 1 | 605.2.a.l | 4 | ||
| 55.k | odd | 20 | 2 | 605.2.g.j | 8 | ||
| 55.k | odd | 20 | 1 | 3025.2.a.v | 4 | ||
| 55.l | even | 20 | 1 | 605.2.a.i | 4 | ||
| 55.l | even | 20 | 2 | 605.2.g.g | 8 | ||
| 55.l | even | 20 | 1 | 605.2.g.n | 8 | ||
| 55.l | even | 20 | 1 | 3025.2.a.be | 4 | ||
| 165.u | odd | 20 | 1 | 5445.2.a.bu | 4 | ||
| 165.v | even | 20 | 1 | 495.2.n.f | 8 | ||
| 165.v | even | 20 | 1 | 5445.2.a.bg | 4 | ||
| 220.v | even | 20 | 1 | 880.2.bo.e | 8 | ||
| 220.v | even | 20 | 1 | 9680.2.a.cs | 4 | ||
| 220.w | odd | 20 | 1 | 9680.2.a.cv | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 55.2.g.a | ✓ | 8 | 5.c | odd | 4 | 1 | |
| 55.2.g.a | ✓ | 8 | 55.k | odd | 20 | 1 | |
| 275.2.h.b | 8 | 5.c | odd | 4 | 1 | ||
| 275.2.h.b | 8 | 55.k | odd | 20 | 1 | ||
| 275.2.z.b | 16 | 1.a | even | 1 | 1 | trivial | |
| 275.2.z.b | 16 | 5.b | even | 2 | 1 | inner | |
| 275.2.z.b | 16 | 11.c | even | 5 | 1 | inner | |
| 275.2.z.b | 16 | 55.j | even | 10 | 1 | inner | |
| 495.2.n.f | 8 | 15.e | even | 4 | 1 | ||
| 495.2.n.f | 8 | 165.v | even | 20 | 1 | ||
| 605.2.a.i | 4 | 55.l | even | 20 | 1 | ||
| 605.2.a.l | 4 | 55.k | odd | 20 | 1 | ||
| 605.2.g.g | 8 | 55.l | even | 20 | 2 | ||
| 605.2.g.j | 8 | 55.k | odd | 20 | 2 | ||
| 605.2.g.n | 8 | 55.e | even | 4 | 1 | ||
| 605.2.g.n | 8 | 55.l | even | 20 | 1 | ||
| 880.2.bo.e | 8 | 20.e | even | 4 | 1 | ||
| 880.2.bo.e | 8 | 220.v | even | 20 | 1 | ||
| 3025.2.a.v | 4 | 55.k | odd | 20 | 1 | ||
| 3025.2.a.be | 4 | 55.l | even | 20 | 1 | ||
| 5445.2.a.bg | 4 | 165.v | even | 20 | 1 | ||
| 5445.2.a.bu | 4 | 165.u | odd | 20 | 1 | ||
| 9680.2.a.cs | 4 | 220.v | even | 20 | 1 | ||
| 9680.2.a.cv | 4 | 220.w | odd | 20 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} - 10T_{2}^{14} + 71T_{2}^{12} - 480T_{2}^{10} + 2801T_{2}^{8} - 8880T_{2}^{6} + 13031T_{2}^{4} + 70T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(275, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} - 10 T^{14} + \cdots + 1 \)
|
| $3$ |
\( T^{16} - 11 T^{14} + \cdots + 625 \)
|
| $5$ |
\( T^{16} \)
|
| $7$ |
\( T^{16} - 29 T^{14} + \cdots + 625 \)
|
| $11$ |
\( (T^{8} + 5 T^{7} + \cdots + 14641)^{2} \)
|
| $13$ |
\( T^{16} - 50 T^{14} + \cdots + 14641 \)
|
| $17$ |
\( T^{16} + 5 T^{14} + \cdots + 14641 \)
|
| $19$ |
\( (T^{8} - T^{7} + 31 T^{6} + \cdots + 625)^{2} \)
|
| $23$ |
\( (T^{8} + 189 T^{6} + \cdots + 2785561)^{2} \)
|
| $29$ |
\( (T^{8} + 19 T^{7} + \cdots + 3025)^{2} \)
|
| $31$ |
\( (T^{8} - 6 T^{7} + \cdots + 10201)^{2} \)
|
| $37$ |
\( T^{16} + \cdots + 519885601 \)
|
| $41$ |
\( (T^{8} + 4 T^{7} + \cdots + 249001)^{2} \)
|
| $43$ |
\( (T^{8} + 199 T^{6} + \cdots + 3481)^{2} \)
|
| $47$ |
\( T^{16} - 158 T^{14} + \cdots + 25411681 \)
|
| $53$ |
\( T^{16} - 65 T^{14} + \cdots + 1 \)
|
| $59$ |
\( (T^{8} - 19 T^{7} + \cdots + 9150625)^{2} \)
|
| $61$ |
\( (T^{8} + 2 T^{7} + \cdots + 3025)^{2} \)
|
| $67$ |
\( (T^{8} + 165 T^{6} + \cdots + 10201)^{2} \)
|
| $71$ |
\( (T^{8} - 40 T^{7} + \cdots + 60824401)^{2} \)
|
| $73$ |
\( T^{16} + \cdots + 22898045041 \)
|
| $79$ |
\( (T^{8} + 17 T^{7} + \cdots + 4644025)^{2} \)
|
| $83$ |
\( T^{16} + 83 T^{14} + \cdots + 707281 \)
|
| $89$ |
\( (T^{4} - 150 T^{2} + \cdots + 725)^{4} \)
|
| $97$ |
\( T^{16} - 214 T^{14} + \cdots + 390625 \)
|
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