Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2075,2,Mod(1,2075)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2075, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2075.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2075 = 5^{2} \cdot 83 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2075.a (trivial) |
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: | yes |
| Analytic conductor: | \(16.5689584194\) |
| Analytic rank: | \(0\) |
| Dimension: | \(11\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{11} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{11} - 20x^{9} - x^{8} + 146x^{7} + 15x^{6} - 464x^{5} - 76x^{4} + 567x^{3} + 136x^{2} - 100x - 8 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 415) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{10}\) 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
\( x^{11} - 20x^{9} - x^{8} + 146x^{7} + 15x^{6} - 464x^{5} - 76x^{4} + 567x^{3} + 136x^{2} - 100x - 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{9} - \nu^{8} - 15\nu^{7} + 14\nu^{6} + 72\nu^{5} - 61\nu^{4} - 115\nu^{3} + 83\nu^{2} + 28\nu - 16 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{9} + \nu^{8} + 47\nu^{7} - 8\nu^{6} - 236\nu^{5} - 5\nu^{4} + 383\nu^{3} + 89\nu^{2} - 38\nu + 8 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 3 \nu^{10} - 6 \nu^{9} + 50 \nu^{8} + 101 \nu^{7} - 266 \nu^{6} - 553 \nu^{5} + 434 \nu^{4} + \cdots + 8 ) / 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 4 \nu^{10} - 7 \nu^{9} + 65 \nu^{8} + 119 \nu^{7} - 332 \nu^{6} - 656 \nu^{5} + 483 \nu^{4} + \cdots - 24 ) / 4 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 3 \nu^{10} + 9 \nu^{9} - 51 \nu^{8} - 148 \nu^{7} + 274 \nu^{6} + 789 \nu^{5} - 425 \nu^{4} - 1401 \nu^{3} + \cdots + 24 ) / 4 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 3 \nu^{10} - 5 \nu^{9} + 50 \nu^{8} + 85 \nu^{7} - 269 \nu^{6} - 469 \nu^{5} + 465 \nu^{4} + 860 \nu^{3} + \cdots + 8 ) / 2 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 5 \nu^{10} + 10 \nu^{9} - 82 \nu^{8} - 169 \nu^{7} + 424 \nu^{6} + 927 \nu^{5} - 632 \nu^{4} - 1688 \nu^{3} + \cdots + 26 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} + \beta_{6} + \beta_{5} + \beta_{3} + 8\beta_{2} + 22 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{9} + \beta_{8} - \beta_{6} + 2\beta_{5} + \beta_{4} + 11\beta_{3} + 2\beta_{2} + 26\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{10} + 10\beta_{8} + \beta_{7} + 12\beta_{6} + 11\beta_{5} + 2\beta_{4} + 10\beta_{3} + 59\beta_{2} + 131 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 2 \beta_{10} + 12 \beta_{9} + 13 \beta_{8} - 5 \beta_{7} - 11 \beta_{6} + 24 \beta_{5} + 14 \beta_{4} + \cdots + 4 \)
|
| \(\nu^{8}\) | \(=\) |
\( 15 \beta_{10} + 2 \beta_{9} + 79 \beta_{8} + 12 \beta_{7} + 109 \beta_{6} + 95 \beta_{5} + 32 \beta_{4} + \cdots + 819 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 29 \beta_{10} + 110 \beta_{9} + 123 \beta_{8} - 77 \beta_{7} - 91 \beta_{6} + 218 \beta_{5} + \cdots + 71 \)
|
| \(\nu^{10}\) | \(=\) |
\( 152 \beta_{10} + 33 \beta_{9} + 582 \beta_{8} + 97 \beta_{7} + 892 \beta_{6} + 756 \beta_{5} + \cdots + 5294 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.67874 | 1.11162 | 5.17564 | 0 | −2.97774 | −2.87466 | −8.50671 | −1.76430 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.43225 | −1.49963 | 3.91582 | 0 | 3.64748 | 4.60197 | −4.65973 | −0.751100 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.89951 | 3.30068 | 1.60815 | 0 | −6.26969 | −1.63280 | 0.744319 | 7.89449 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.89336 | −1.62347 | 1.58483 | 0 | 3.07382 | −1.52477 | 0.786074 | −0.364342 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.381602 | −1.63018 | −1.85438 | 0 | 0.622079 | 0.0761781 | 1.47084 | −0.342523 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.0747789 | 2.34579 | −1.99441 | 0 | 0.175416 | 4.39592 | −0.298697 | 2.50275 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.609799 | −0.0621862 | −1.62814 | 0 | −0.0379211 | −2.63003 | −2.21244 | −2.99613 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.47832 | −3.02546 | 0.185423 | 0 | −4.47258 | −4.22081 | −2.68252 | 6.15338 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.22601 | 3.04563 | 2.95512 | 0 | 6.77960 | 2.67381 | 2.12611 | 6.27585 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.26966 | 1.07598 | 3.15134 | 0 | 2.44211 | −0.653240 | 2.61315 | −1.84226 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.62690 | −3.03878 | 4.90061 | 0 | −7.98258 | 4.78843 | 7.61961 | 6.23419 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(83\) | \( -1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2075.2.a.k | 11 | |
| 5.b | even | 2 | 1 | 415.2.a.e | ✓ | 11 | |
| 15.d | odd | 2 | 1 | 3735.2.a.s | 11 | ||
| 20.d | odd | 2 | 1 | 6640.2.a.bi | 11 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 415.2.a.e | ✓ | 11 | 5.b | even | 2 | 1 | |
| 2075.2.a.k | 11 | 1.a | even | 1 | 1 | trivial | |
| 3735.2.a.s | 11 | 15.d | odd | 2 | 1 | ||
| 6640.2.a.bi | 11 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2075))\):
|
\( T_{2}^{11} - 20T_{2}^{9} + T_{2}^{8} + 146T_{2}^{7} - 15T_{2}^{6} - 464T_{2}^{5} + 76T_{2}^{4} + 567T_{2}^{3} - 136T_{2}^{2} - 100T_{2} + 8 \)
|
|
\( T_{3}^{11} - 27 T_{3}^{9} - 4 T_{3}^{8} + 258 T_{3}^{7} + 71 T_{3}^{6} - 1041 T_{3}^{5} - 362 T_{3}^{4} + \cdots - 64 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} - 20 T^{9} + \cdots + 8 \)
|
| $3$ |
\( T^{11} - 27 T^{9} + \cdots - 64 \)
|
| $5$ |
\( T^{11} \)
|
| $7$ |
\( T^{11} - 3 T^{10} + \cdots - 1024 \)
|
| $11$ |
\( T^{11} + T^{10} + \cdots - 26896 \)
|
| $13$ |
\( T^{11} + T^{10} + \cdots + 23296 \)
|
| $17$ |
\( T^{11} + 28 T^{10} + \cdots - 224 \)
|
| $19$ |
\( T^{11} + 2 T^{10} + \cdots - 25600 \)
|
| $23$ |
\( T^{11} + T^{10} + \cdots + 2873344 \)
|
| $29$ |
\( T^{11} - 21 T^{10} + \cdots - 16787050 \)
|
| $31$ |
\( T^{11} + \cdots + 178968640 \)
|
| $37$ |
\( T^{11} - 10 T^{10} + \cdots - 224 \)
|
| $41$ |
\( T^{11} + \cdots - 306688256 \)
|
| $43$ |
\( T^{11} + \cdots - 114398720 \)
|
| $47$ |
\( T^{11} + 8 T^{10} + \cdots - 262144 \)
|
| $53$ |
\( T^{11} + 21 T^{10} + \cdots - 2720 \)
|
| $59$ |
\( T^{11} - 5 T^{10} + \cdots - 6278092 \)
|
| $61$ |
\( T^{11} - 14 T^{10} + \cdots - 101918 \)
|
| $67$ |
\( T^{11} + \cdots + 143897920 \)
|
| $71$ |
\( T^{11} - 3 T^{10} + \cdots - 501760 \)
|
| $73$ |
\( T^{11} + 3 T^{10} + \cdots + 1549568 \)
|
| $79$ |
\( T^{11} + 20 T^{10} + \cdots + 16744448 \)
|
| $83$ |
\( (T - 1)^{11} \)
|
| $89$ |
\( T^{11} - 37 T^{10} + \cdots + 1570816 \)
|
| $97$ |
\( T^{11} + \cdots - 5595205504 \)
|
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