Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [275,2,Mod(26,275)]
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([0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("275.26");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 275 = 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 275.h (of order \(5\), degree \(4\), 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 \(\Q(\zeta_{5})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 4 x^{15} + 10 x^{14} - 13 x^{13} + 53 x^{12} - 12 x^{11} + 136 x^{10} + 8 x^{9} + 300 x^{8} + \cdots + 16 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 5 \) |
| Twist minimal: | yes |
| 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
\( x^{16} - 4 x^{15} + 10 x^{14} - 13 x^{13} + 53 x^{12} - 12 x^{11} + 136 x^{10} + 8 x^{9} + 300 x^{8} + \cdots + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 26449018002 \nu^{15} - 236888516211 \nu^{14} + 788099178184 \nu^{13} - 1526842653922 \nu^{12} + \cdots + 6985303901728 ) / 25527087093000 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 71383474437 \nu^{15} - 289509004196 \nu^{14} + 724686737314 \nu^{13} + \cdots - 26659476408112 ) / 25527087093000 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 104750151572 \nu^{15} + 673159972921 \nu^{14} - 2076349151074 \nu^{13} + \cdots + 2568824593792 ) / 25527087093000 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 64382245392 \nu^{15} - 136843233291 \nu^{14} + 153040707434 \nu^{13} + 333330036813 \nu^{12} + \cdots + 3970336916928 ) / 12763543546500 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 160551537112 \nu^{15} + 537455996876 \nu^{14} - 932355398199 \nu^{13} + \cdots - 10162625679608 ) / 25527087093000 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 335740577504 \nu^{15} - 1576476952372 \nu^{14} + 4304252214543 \nu^{13} + \cdots + 2834825497856 ) / 25527087093000 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 169847619014 \nu^{15} - 638541728247 \nu^{14} + 1347321043353 \nu^{13} + \cdots - 8155768916224 ) / 12763543546500 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 436581493858 \nu^{15} + 1772774993434 \nu^{14} - 4602703454791 \nu^{13} + \cdots - 32620021539072 ) / 25527087093000 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 336455799554 \nu^{15} - 1405015474407 \nu^{14} + 3422183399388 \nu^{13} + \cdots - 6722455741504 ) / 12763543546500 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 784893186521 \nu^{15} + 3081039362473 \nu^{14} - 7454798472622 \nu^{13} + \cdots - 17909140090624 ) / 25527087093000 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 825910307819 \nu^{15} + 3905169355947 \nu^{14} - 10779625022808 \nu^{13} + \cdots - 50705453705536 ) / 25527087093000 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 420153483844 \nu^{15} - 1344158135822 \nu^{14} + 2796519364033 \nu^{13} + \cdots + 30173885523636 ) / 12763543546500 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 509735557264 \nu^{15} - 1869094610042 \nu^{14} + 4458813844393 \nu^{13} + \cdots + 14317509681756 ) / 12763543546500 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 1298485402504 \nu^{15} - 6312051157617 \nu^{14} + 17598840879833 \nu^{13} + \cdots + 5900772630536 ) / 25527087093000 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{10} - \beta_{8} - 2\beta_{7} + \beta_{5} - 2\beta_{4} \)
|
| \(\nu^{3}\) | \(=\) |
\( - \beta_{15} - \beta_{13} - 2 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} - 3 \beta_{8} - 2 \beta_{7} + \cdots + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( - \beta_{15} + \beta_{14} - 3 \beta_{13} + 2 \beta_{12} - 9 \beta_{11} + 3 \beta_{10} + 11 \beta_{9} + \cdots - 13 \beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( 9 \beta_{14} - 12 \beta_{13} + 9 \beta_{12} - 14 \beta_{11} + 14 \beta_{9} + 14 \beta_{6} + 42 \beta_{5} + \cdots - 11 \)
|
| \(\nu^{6}\) | \(=\) |
\( 14 \beta_{15} + 25 \beta_{14} - 25 \beta_{13} + 14 \beta_{12} - 45 \beta_{10} + 88 \beta_{8} + 45 \beta_{7} + \cdots - 45 \)
|
| \(\nu^{7}\) | \(=\) |
\( 88 \beta_{15} + 45 \beta_{14} + 163 \beta_{11} - 275 \beta_{10} - 165 \beta_{9} + 438 \beta_{8} + \cdots - 165 \)
|
| \(\nu^{8}\) | \(=\) |
\( 275 \beta_{15} + 275 \beta_{13} - 163 \beta_{12} + 919 \beta_{11} - 919 \beta_{10} - 949 \beta_{9} + \cdots - 409 \)
|
| \(\nu^{9}\) | \(=\) |
\( 534 \beta_{15} - 534 \beta_{14} + 1453 \beta_{13} - 919 \beta_{12} + 2980 \beta_{11} - 1811 \beta_{10} + \cdots + 5427 \beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 2980 \beta_{14} + 4791 \beta_{13} - 2980 \beta_{12} + 5961 \beta_{11} - 6063 \beta_{9} - 6063 \beta_{6} + \cdots + 4018 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 5961 \beta_{15} - 9845 \beta_{14} + 9845 \beta_{13} - 5961 \beta_{12} + 19818 \beta_{10} + \cdots + 20208 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 32277 \beta_{15} - 19818 \beta_{14} - 65289 \beta_{11} + 106372 \beta_{10} + 66561 \beta_{9} + \cdots + 66561 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 106372 \beta_{15} - 106372 \beta_{13} + 65289 \beta_{12} - 349879 \beta_{11} + 349879 \beta_{10} + \cdots + 137070 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 215699 \beta_{15} + 215699 \beta_{14} - 565578 \beta_{13} + 349879 \beta_{12} - 1152511 \beta_{11} + \cdots - 2127272 \beta_1 \)
|
| \(\nu^{15}\) | \(=\) |
\( 1152511 \beta_{14} - 1863133 \beta_{13} + 1152511 \beta_{12} - 2342971 \beta_{11} + 2391411 \beta_{9} + \cdots - 1482267 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(177\) |
| \(\chi(n)\) | \(-1 - \beta_{6} + \beta_{7} - \beta_{9}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 26.1 |
|
−1.85576 | − | 1.34829i | 0.0131041 | − | 0.0403304i | 1.00792 | + | 3.10207i | 0 | −0.0786950 | + | 0.0571753i | −0.352180 | − | 1.08390i | 0.894346 | − | 2.75251i | 2.42560 | + | 1.76230i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 26.2 | −0.168506 | − | 0.122427i | −0.585361 | + | 1.80155i | −0.604628 | − | 1.86085i | 0 | 0.319195 | − | 0.231909i | 0.398265 | + | 1.22573i | −0.254662 | + | 0.783769i | −0.475901 | − | 0.345762i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 26.3 | 1.06921 | + | 0.776830i | 0.626199 | − | 1.92724i | −0.0782786 | − | 0.240917i | 0 | 2.16668 | − | 1.57419i | −0.107618 | − | 0.331213i | 0.920262 | − | 2.83228i | −0.895086 | − | 0.650318i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 26.4 | 2.07308 | + | 1.50618i | −0.553942 | + | 1.70486i | 1.41105 | + | 4.34277i | 0 | −3.71620 | + | 2.69998i | −1.17454 | − | 3.61485i | −2.03208 | + | 6.25410i | −0.172643 | − | 0.125433i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 126.1 | −0.854560 | − | 2.63006i | −1.32800 | − | 0.964848i | −4.56893 | + | 3.31953i | 0 | −1.40276 | + | 4.31724i | −1.53545 | + | 1.11557i | 8.16046 | + | 5.92892i | −0.0944009 | − | 0.290536i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 126.2 | −0.516686 | − | 1.59020i | 2.51599 | + | 1.82798i | −0.643729 | + | 0.467696i | 0 | 1.60686 | − | 4.94542i | 2.49985 | − | 1.81624i | −1.62907 | − | 1.18359i | 2.06168 | + | 6.34519i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 126.3 | −0.0430779 | − | 0.132580i | −2.41558 | − | 1.75502i | 1.60231 | − | 1.16415i | 0 | −0.128623 | + | 0.395861i | 2.88786 | − | 2.09815i | −0.448926 | − | 0.326164i | 1.82787 | + | 5.62561i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 126.4 | 0.296290 | + | 0.911888i | 0.727583 | + | 0.528620i | 0.874283 | − | 0.635204i | 0 | −0.266466 | + | 0.820099i | −0.616184 | + | 0.447684i | 2.38967 | + | 1.73620i | −0.677113 | − | 2.08394i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 201.1 | −1.85576 | + | 1.34829i | 0.0131041 | + | 0.0403304i | 1.00792 | − | 3.10207i | 0 | −0.0786950 | − | 0.0571753i | −0.352180 | + | 1.08390i | 0.894346 | + | 2.75251i | 2.42560 | − | 1.76230i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 201.2 | −0.168506 | + | 0.122427i | −0.585361 | − | 1.80155i | −0.604628 | + | 1.86085i | 0 | 0.319195 | + | 0.231909i | 0.398265 | − | 1.22573i | −0.254662 | − | 0.783769i | −0.475901 | + | 0.345762i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 201.3 | 1.06921 | − | 0.776830i | 0.626199 | + | 1.92724i | −0.0782786 | + | 0.240917i | 0 | 2.16668 | + | 1.57419i | −0.107618 | + | 0.331213i | 0.920262 | + | 2.83228i | −0.895086 | + | 0.650318i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 201.4 | 2.07308 | − | 1.50618i | −0.553942 | − | 1.70486i | 1.41105 | − | 4.34277i | 0 | −3.71620 | − | 2.69998i | −1.17454 | + | 3.61485i | −2.03208 | − | 6.25410i | −0.172643 | + | 0.125433i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 251.1 | −0.854560 | + | 2.63006i | −1.32800 | + | 0.964848i | −4.56893 | − | 3.31953i | 0 | −1.40276 | − | 4.31724i | −1.53545 | − | 1.11557i | 8.16046 | − | 5.92892i | −0.0944009 | + | 0.290536i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 251.2 | −0.516686 | + | 1.59020i | 2.51599 | − | 1.82798i | −0.643729 | − | 0.467696i | 0 | 1.60686 | + | 4.94542i | 2.49985 | + | 1.81624i | −1.62907 | + | 1.18359i | 2.06168 | − | 6.34519i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 251.3 | −0.0430779 | + | 0.132580i | −2.41558 | + | 1.75502i | 1.60231 | + | 1.16415i | 0 | −0.128623 | − | 0.395861i | 2.88786 | + | 2.09815i | −0.448926 | + | 0.326164i | 1.82787 | − | 5.62561i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 251.4 | 0.296290 | − | 0.911888i | 0.727583 | − | 0.528620i | 0.874283 | + | 0.635204i | 0 | −0.266466 | − | 0.820099i | −0.616184 | − | 0.447684i | 2.38967 | − | 1.73620i | −0.677113 | + | 2.08394i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.c | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 275.2.h.c | ✓ | 16 |
| 5.b | even | 2 | 1 | 275.2.h.e | yes | 16 | |
| 5.c | odd | 4 | 2 | 275.2.z.c | 32 | ||
| 11.c | even | 5 | 1 | inner | 275.2.h.c | ✓ | 16 |
| 11.c | even | 5 | 1 | 3025.2.a.bi | 8 | ||
| 11.d | odd | 10 | 1 | 3025.2.a.bm | 8 | ||
| 55.h | odd | 10 | 1 | 3025.2.a.bj | 8 | ||
| 55.j | even | 10 | 1 | 275.2.h.e | yes | 16 | |
| 55.j | even | 10 | 1 | 3025.2.a.bn | 8 | ||
| 55.k | odd | 20 | 2 | 275.2.z.c | 32 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 275.2.h.c | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 275.2.h.c | ✓ | 16 | 11.c | even | 5 | 1 | inner |
| 275.2.h.e | yes | 16 | 5.b | even | 2 | 1 | |
| 275.2.h.e | yes | 16 | 55.j | even | 10 | 1 | |
| 275.2.z.c | 32 | 5.c | odd | 4 | 2 | ||
| 275.2.z.c | 32 | 55.k | odd | 20 | 2 | ||
| 3025.2.a.bi | 8 | 11.c | even | 5 | 1 | ||
| 3025.2.a.bj | 8 | 55.h | odd | 10 | 1 | ||
| 3025.2.a.bm | 8 | 11.d | odd | 10 | 1 | ||
| 3025.2.a.bn | 8 | 55.j | even | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} + 5 T_{2}^{14} - 12 T_{2}^{13} + 32 T_{2}^{12} + 30 T_{2}^{11} + 259 T_{2}^{10} - 193 T_{2}^{9} + \cdots + 1 \)
acting on \(S_{2}^{\mathrm{new}}(275, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} + 5 T^{14} + \cdots + 1 \)
|
| $3$ |
\( T^{16} + 2 T^{15} + \cdots + 16 \)
|
| $5$ |
\( T^{16} \)
|
| $7$ |
\( T^{16} - 4 T^{15} + \cdots + 961 \)
|
| $11$ |
\( T^{16} + \cdots + 214358881 \)
|
| $13$ |
\( T^{16} - 7 T^{15} + \cdots + 12075625 \)
|
| $17$ |
\( T^{16} - 12 T^{15} + \cdots + 192721 \)
|
| $19$ |
\( T^{16} + \cdots + 109830400 \)
|
| $23$ |
\( (T^{8} + 2 T^{7} + \cdots - 121)^{2} \)
|
| $29$ |
\( T^{16} + \cdots + 1259895025 \)
|
| $31$ |
\( T^{16} + \cdots + 1126877761 \)
|
| $37$ |
\( T^{16} + 4 T^{15} + \cdots + 19811401 \)
|
| $41$ |
\( T^{16} + \cdots + 195873515776 \)
|
| $43$ |
\( (T^{8} + 14 T^{7} + \cdots - 101)^{2} \)
|
| $47$ |
\( T^{16} + \cdots + 211111761961 \)
|
| $53$ |
\( T^{16} + \cdots + 2390818816 \)
|
| $59$ |
\( T^{16} + \cdots + 6178746025 \)
|
| $61$ |
\( T^{16} + \cdots + 11020667028121 \)
|
| $67$ |
\( (T^{8} + 7 T^{7} + \cdots + 23994961)^{2} \)
|
| $71$ |
\( T^{16} + \cdots + 923250017881 \)
|
| $73$ |
\( T^{16} + \cdots + 217828491841 \)
|
| $79$ |
\( T^{16} + \cdots + 275304843025 \)
|
| $83$ |
\( T^{16} - 28 T^{15} + \cdots + 69372241 \)
|
| $89$ |
\( (T^{8} + 11 T^{7} + \cdots - 278125)^{2} \)
|
| $97$ |
\( T^{16} + \cdots + 7330099456 \)
|
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