Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [225,2,Mod(46,225)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(225, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 6]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("225.46");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 225 = 3^{2} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 225.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: | \(1.79663404548\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | 16.0.1130304400000000000000.3 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 5x^{14} + 5x^{12} - 10x^{10} + 205x^{8} - 700x^{6} + 1250x^{4} - 1250x^{2} + 625 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 5^{2} \) |
| 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
\( x^{16} - 5x^{14} + 5x^{12} - 10x^{10} + 205x^{8} - 700x^{6} + 1250x^{4} - 1250x^{2} + 625 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 9094 \nu^{15} + 529550 \nu^{13} - 1775155 \nu^{11} - 1261485 \nu^{9} - 4566370 \nu^{7} + \cdots + 136237250 \nu ) / 54753875 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 18559 \nu^{14} + 486170 \nu^{12} - 1233230 \nu^{10} - 2183265 \nu^{8} - 1879805 \nu^{6} + \cdots + 104474000 ) / 54753875 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 85587 \nu^{15} - 683920 \nu^{13} + 729395 \nu^{11} + 734880 \nu^{9} + 21524610 \nu^{7} + \cdots - 113406125 \nu ) / 54753875 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 191309 \nu^{15} - 327040 \nu^{13} - 1605420 \nu^{11} - 1814665 \nu^{9} + 36243345 \nu^{7} + \cdots + 103147500 \nu ) / 54753875 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 216278 \nu^{15} - 1248275 \nu^{13} + 1298755 \nu^{11} - 360830 \nu^{9} + 45929540 \nu^{7} + \cdots - 120294875 \nu ) / 54753875 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 226756 \nu^{14} - 812710 \nu^{12} - 301695 \nu^{10} - 1956335 \nu^{8} + 45336155 \nu^{6} + \cdots - 85476750 ) / 54753875 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 237426 \nu^{14} + 787630 \nu^{12} + 357520 \nu^{10} + 1796010 \nu^{8} - 44976455 \nu^{6} + \cdots - 3771750 ) / 54753875 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 26057 \nu^{14} - 128005 \nu^{12} + 108960 \nu^{10} - 210920 \nu^{8} + 5037260 \nu^{6} + \cdots - 22852125 ) / 4977625 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 323013 \nu^{15} + 1471550 \nu^{13} - 371875 \nu^{11} + 1061130 \nu^{9} - 66501065 \nu^{7} + \cdots + 54880500 \nu ) / 54753875 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 330256 \nu^{15} + 1021010 \nu^{13} + 321835 \nu^{11} + 4065110 \nu^{9} - 59745955 \nu^{7} + \cdots + 81913375 \nu ) / 54753875 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 389298 \nu^{14} - 1338325 \nu^{12} - 665635 \nu^{10} - 3682905 \nu^{8} + 76510440 \nu^{6} + \cdots - 87855125 ) / 54753875 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 443034 \nu^{15} + 2060985 \nu^{13} - 997060 \nu^{11} + 2317165 \nu^{9} - 91265695 \nu^{7} + \cdots + 260525500 \nu ) / 54753875 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 40791 \nu^{14} - 196825 \nu^{12} + 194030 \nu^{10} - 383635 \nu^{8} + 8130605 \nu^{6} + \cdots - 36685125 ) / 4977625 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 597957 \nu^{15} - 2324530 \nu^{13} + 218325 \nu^{11} - 5390820 \nu^{9} + 116417835 \nu^{7} + \cdots - 244363125 \nu ) / 54753875 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 607697 \nu^{14} - 2843530 \nu^{12} + 1509785 \nu^{10} - 3468945 \nu^{8} + 126219210 \nu^{6} + \cdots - 393095875 ) / 54753875 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{14} + \beta_{12} + 2\beta_{10} - \beta_{9} + 3\beta_{5} - \beta_{4} - 3\beta_{3} + 2\beta_1 ) / 5 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{15} - \beta_{13} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{2} + 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{14} + \beta_{12} + 2\beta_{10} - \beta_{5} + \beta_{4} - \beta_{3} - 2\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{13} - 5\beta_{11} + 2\beta_{8} + \beta_{7} + 9\beta_{6} + \beta_{2} + 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( -3\beta_{12} + 2\beta_{10} - 5\beta_{9} - 5\beta_{5} - 4\beta_{4} - 5\beta_{3} + \beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 8\beta_{15} - 6\beta_{11} - 8\beta_{8} + 11\beta_{7} + 10\beta_{6} + 6\beta_{2} + 16 \)
|
| \(\nu^{7}\) | \(=\) |
\( 13\beta_{14} - 21\beta_{12} + 40\beta_{10} + 14\beta_{9} + 19\beta_{5} + 3\beta_{4} - 46\beta_{3} + 10\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( 36\beta_{15} - 29\beta_{13} - 44\beta_{11} + 11\beta_{8} - 37\beta_{7} - 18\beta_{6} + 22\beta_{2} - 48 \)
|
| \(\nu^{9}\) | \(=\) |
\( -6\beta_{14} - 16\beta_{12} + 3\beta_{10} - 39\beta_{9} - 118\beta_{5} + 56\beta_{4} - 17\beta_{3} - 127\beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( -85\beta_{15} + 140\beta_{13} - 210\beta_{11} - 140\beta_{8} - 165\beta_{7} + 320\beta_{6} - 55\beta_{2} + 35 \)
|
| \(\nu^{11}\) | \(=\) |
\( -260\beta_{14} - 340\beta_{12} - 85\beta_{10} - 60\beta_{9} + 80\beta_{5} - 175\beta_{4} - 285\beta_{3} + 225\beta_1 \)
|
| \(\nu^{12}\) | \(=\) |
\( 200\beta_{15} + 200\beta_{13} + 185\beta_{11} - 520\beta_{8} - 385\beta_{7} - 1005\beta_{6} + 120\beta_{2} - 1005 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 115 \beta_{14} - 415 \beta_{12} + 225 \beta_{10} + 835 \beta_{9} + 530 \beta_{5} + 820 \beta_{4} + \cdots - 495 \beta_1 \)
|
| \(\nu^{14}\) | \(=\) |
\( -1105\beta_{15} + 685\beta_{13} - 735\beta_{11} - 4150\beta_{7} - 1180\beta_{6} - 685\beta_{2} - 4885 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 4125 \beta_{14} + 420 \beta_{12} - 7095 \beta_{10} - 2285 \beta_{9} - 4200 \beta_{5} + \cdots - 3095 \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{11}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 46.1 |
|
−0.670386 | + | 2.06324i | 0 | −2.18949 | − | 1.59076i | 1.69588 | + | 1.45739i | 0 | −3.32440 | 1.23973 | − | 0.900718i | 0 | −4.14384 | + | 2.52199i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 46.2 | −0.167451 | + | 0.515362i | 0 | 1.38048 | + | 1.00297i | 1.16252 | − | 1.91012i | 0 | 1.08833 | −1.62484 | + | 1.18052i | 0 | 0.789737 | + | 0.918969i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 46.3 | 0.167451 | − | 0.515362i | 0 | 1.38048 | + | 1.00297i | −1.16252 | + | 1.91012i | 0 | 1.08833 | 1.62484 | − | 1.18052i | 0 | 0.789737 | + | 0.918969i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 46.4 | 0.670386 | − | 2.06324i | 0 | −2.18949 | − | 1.59076i | −1.69588 | − | 1.45739i | 0 | −3.32440 | −1.23973 | + | 0.900718i | 0 | −4.14384 | + | 2.52199i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.1 | −1.64259 | − | 1.19341i | 0 | 0.655837 | + | 2.01846i | −2.23130 | − | 0.145994i | 0 | −0.504300 | 0.0767537 | − | 0.236224i | 0 | 3.49087 | + | 2.90266i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.2 | −0.757918 | − | 0.550660i | 0 | −0.346820 | − | 1.06740i | 1.42964 | + | 1.71934i | 0 | 2.74037 | −0.903913 | + | 2.78196i | 0 | −0.136773 | − | 2.09036i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.3 | 0.757918 | + | 0.550660i | 0 | −0.346820 | − | 1.06740i | −1.42964 | − | 1.71934i | 0 | 2.74037 | 0.903913 | − | 2.78196i | 0 | −0.136773 | − | 2.09036i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.4 | 1.64259 | + | 1.19341i | 0 | 0.655837 | + | 2.01846i | 2.23130 | + | 0.145994i | 0 | −0.504300 | −0.0767537 | + | 0.236224i | 0 | 3.49087 | + | 2.90266i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 136.1 | −1.64259 | + | 1.19341i | 0 | 0.655837 | − | 2.01846i | −2.23130 | + | 0.145994i | 0 | −0.504300 | 0.0767537 | + | 0.236224i | 0 | 3.49087 | − | 2.90266i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 136.2 | −0.757918 | + | 0.550660i | 0 | −0.346820 | + | 1.06740i | 1.42964 | − | 1.71934i | 0 | 2.74037 | −0.903913 | − | 2.78196i | 0 | −0.136773 | + | 2.09036i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 136.3 | 0.757918 | − | 0.550660i | 0 | −0.346820 | + | 1.06740i | −1.42964 | + | 1.71934i | 0 | 2.74037 | 0.903913 | + | 2.78196i | 0 | −0.136773 | + | 2.09036i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 136.4 | 1.64259 | − | 1.19341i | 0 | 0.655837 | − | 2.01846i | 2.23130 | − | 0.145994i | 0 | −0.504300 | −0.0767537 | − | 0.236224i | 0 | 3.49087 | − | 2.90266i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 181.1 | −0.670386 | − | 2.06324i | 0 | −2.18949 | + | 1.59076i | 1.69588 | − | 1.45739i | 0 | −3.32440 | 1.23973 | + | 0.900718i | 0 | −4.14384 | − | 2.52199i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 181.2 | −0.167451 | − | 0.515362i | 0 | 1.38048 | − | 1.00297i | 1.16252 | + | 1.91012i | 0 | 1.08833 | −1.62484 | − | 1.18052i | 0 | 0.789737 | − | 0.918969i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 181.3 | 0.167451 | + | 0.515362i | 0 | 1.38048 | − | 1.00297i | −1.16252 | − | 1.91012i | 0 | 1.08833 | 1.62484 | + | 1.18052i | 0 | 0.789737 | − | 0.918969i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 181.4 | 0.670386 | + | 2.06324i | 0 | −2.18949 | + | 1.59076i | −1.69588 | + | 1.45739i | 0 | −3.32440 | −1.23973 | − | 0.900718i | 0 | −4.14384 | − | 2.52199i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 25.d | even | 5 | 1 | inner |
| 75.j | odd | 10 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 225.2.h.e | ✓ | 16 |
| 3.b | odd | 2 | 1 | inner | 225.2.h.e | ✓ | 16 |
| 25.d | even | 5 | 1 | inner | 225.2.h.e | ✓ | 16 |
| 25.d | even | 5 | 1 | 5625.2.a.w | 8 | ||
| 25.e | even | 10 | 1 | 5625.2.a.v | 8 | ||
| 75.h | odd | 10 | 1 | 5625.2.a.v | 8 | ||
| 75.j | odd | 10 | 1 | inner | 225.2.h.e | ✓ | 16 |
| 75.j | odd | 10 | 1 | 5625.2.a.w | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 225.2.h.e | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 225.2.h.e | ✓ | 16 | 3.b | odd | 2 | 1 | inner |
| 225.2.h.e | ✓ | 16 | 25.d | even | 5 | 1 | inner |
| 225.2.h.e | ✓ | 16 | 75.j | odd | 10 | 1 | inner |
| 5625.2.a.v | 8 | 25.e | even | 10 | 1 | ||
| 5625.2.a.v | 8 | 75.h | odd | 10 | 1 | ||
| 5625.2.a.w | 8 | 25.d | even | 5 | 1 | ||
| 5625.2.a.w | 8 | 75.j | odd | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} + 5T_{2}^{14} + 20T_{2}^{12} + 75T_{2}^{10} + 385T_{2}^{8} + 25T_{2}^{6} + 250T_{2}^{4} + 125T_{2}^{2} + 25 \)
acting on \(S_{2}^{\mathrm{new}}(225, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} + 5 T^{14} + \cdots + 25 \)
|
| $3$ |
\( T^{16} \)
|
| $5$ |
\( T^{16} - 5 T^{14} + \cdots + 390625 \)
|
| $7$ |
\( (T^{4} - 10 T^{2} + 5 T + 5)^{4} \)
|
| $11$ |
\( T^{16} + 30 T^{14} + \cdots + 15625 \)
|
| $13$ |
\( (T^{8} - 5 T^{6} + \cdots + 2025)^{2} \)
|
| $17$ |
\( T^{16} + 15 T^{14} + \cdots + 17682025 \)
|
| $19$ |
\( (T^{8} - 7 T^{7} + \cdots + 11881)^{2} \)
|
| $23$ |
\( T^{16} + \cdots + 20393268025 \)
|
| $29$ |
\( T^{16} + 155 T^{14} + \cdots + 15625 \)
|
| $31$ |
\( (T^{8} - 9 T^{7} + \cdots + 641601)^{2} \)
|
| $37$ |
\( (T^{8} - 5 T^{7} + \cdots + 9025)^{2} \)
|
| $41$ |
\( T^{16} + \cdots + 16041026265625 \)
|
| $43$ |
\( (T^{4} + 20 T^{3} + \cdots - 45)^{4} \)
|
| $47$ |
\( T^{16} + \cdots + 144120025 \)
|
| $53$ |
\( T^{16} + \cdots + 104248286142025 \)
|
| $59$ |
\( T^{16} + \cdots + 39\!\cdots\!25 \)
|
| $61$ |
\( (T^{8} - 16 T^{7} + \cdots + 101761)^{2} \)
|
| $67$ |
\( (T^{8} + 20 T^{7} + \cdots + 40896025)^{2} \)
|
| $71$ |
\( T^{16} + \cdots + 67\!\cdots\!25 \)
|
| $73$ |
\( (T^{8} - 30 T^{7} + \cdots + 117614025)^{2} \)
|
| $79$ |
\( (T^{8} - 18 T^{7} + \cdots + 28185481)^{2} \)
|
| $83$ |
\( T^{16} + 180 T^{14} + \cdots + 3258025 \)
|
| $89$ |
\( T^{16} + \cdots + 1272666015625 \)
|
| $97$ |
\( (T^{8} - 20 T^{7} + \cdots + 86397025)^{2} \)
|
| show more | |
| show less |