Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2014,2,Mod(1,2014)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2014, 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("2014.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2014 = 2 \cdot 19 \cdot 53 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2014.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: | \(16.0818709671\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 4 x^{11} - 25 x^{10} + 117 x^{9} + 160 x^{8} - 1121 x^{7} + 167 x^{6} + 3834 x^{5} - 2950 x^{4} + \cdots + 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 4 x^{11} - 25 x^{10} + 117 x^{9} + 160 x^{8} - 1121 x^{7} + 167 x^{6} + 3834 x^{5} - 2950 x^{4} + \cdots + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 771 \nu^{11} + 18454 \nu^{10} - 53905 \nu^{9} - 513543 \nu^{8} + 1136936 \nu^{7} + 4722451 \nu^{6} + \cdots + 3226952 ) / 679656 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3325 \nu^{11} + 19457 \nu^{10} - 128817 \nu^{9} - 567676 \nu^{8} + 1912297 \nu^{7} + 5577411 \nu^{6} + \cdots + 5519376 ) / 679656 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 36211 \nu^{11} + 117140 \nu^{10} + 963851 \nu^{9} - 3509783 \nu^{8} - 7596536 \nu^{7} + \cdots + 9752096 ) / 4077936 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 40799 \nu^{11} - 38032 \nu^{10} - 1210723 \nu^{9} + 1218559 \nu^{8} + 12215896 \nu^{7} + \cdots - 3365632 ) / 4077936 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 5043 \nu^{11} + 4564 \nu^{10} - 161955 \nu^{9} - 119185 \nu^{8} + 1870584 \nu^{7} + 1063125 \nu^{6} + \cdots + 1694960 ) / 453104 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 13460 \nu^{11} - 15604 \nu^{10} - 383023 \nu^{9} + 466084 \nu^{8} + 3631291 \nu^{7} + \cdots - 2064388 ) / 1019484 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 75781 \nu^{11} - 328130 \nu^{10} - 1970009 \nu^{9} + 9594911 \nu^{8} + 14379326 \nu^{7} + \cdots - 39959168 ) / 4077936 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 44161 \nu^{11} - 129386 \nu^{10} - 1205417 \nu^{9} + 3838847 \nu^{8} + 10234586 \nu^{7} + \cdots + 1993312 ) / 2038968 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 185045 \nu^{11} - 494386 \nu^{10} - 5041801 \nu^{9} + 14624839 \nu^{8} + 42044542 \nu^{7} + \cdots - 33437632 ) / 4077936 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 185045 \nu^{11} + 494386 \nu^{10} + 5041801 \nu^{9} - 14624839 \nu^{8} - 42044542 \nu^{7} + \cdots + 57905248 ) / 4077936 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{11} - \beta_{10} + 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{11} + \beta_{10} - \beta_{6} + 2\beta_{5} + \beta_{4} + 10\beta _1 - 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 12 \beta_{11} - 12 \beta_{10} + 2 \beta_{9} - \beta_{8} - \beta_{7} + 2 \beta_{6} - \beta_{5} + \cdots + 58 \)
|
| \(\nu^{5}\) | \(=\) |
\( 18 \beta_{11} + 17 \beta_{10} + \beta_{8} + \beta_{7} - 12 \beta_{6} + 30 \beta_{5} + 16 \beta_{4} + \cdots - 52 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 135 \beta_{11} - 135 \beta_{10} + 31 \beta_{9} - 14 \beta_{8} - 12 \beta_{7} + 32 \beta_{6} + \cdots + 617 \)
|
| \(\nu^{7}\) | \(=\) |
\( 252 \beta_{11} + 229 \beta_{10} + 4 \beta_{9} + 21 \beta_{8} + 28 \beta_{7} - 134 \beta_{6} + 371 \beta_{5} + \cdots - 745 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 1517 \beta_{11} - 1514 \beta_{10} + 403 \beta_{9} - 177 \beta_{8} - 119 \beta_{7} + 407 \beta_{6} + \cdots + 6822 \)
|
| \(\nu^{9}\) | \(=\) |
\( 3238 \beta_{11} + 2871 \beta_{10} + 95 \beta_{9} + 290 \beta_{8} + 535 \beta_{7} - 1545 \beta_{6} + \cdots - 9859 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 17235 \beta_{11} - 17106 \beta_{10} + 4973 \beta_{9} - 2282 \beta_{8} - 1111 \beta_{7} + 4817 \beta_{6} + \cdots + 76935 \)
|
| \(\nu^{11}\) | \(=\) |
\( 40048 \beta_{11} + 34917 \beta_{10} + 1511 \beta_{9} + 3460 \beta_{8} + 8621 \beta_{7} - 18310 \beta_{6} + \cdots - 125270 \)
|
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 |
|
−1.00000 | −3.35521 | 1.00000 | −2.54740 | 3.35521 | 2.23317 | −1.00000 | 8.25746 | 2.54740 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −3.04122 | 1.00000 | −0.231635 | 3.04122 | −4.77041 | −1.00000 | 6.24903 | 0.231635 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −2.60226 | 1.00000 | 4.39278 | 2.60226 | −2.56821 | −1.00000 | 3.77174 | −4.39278 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | −2.48949 | 1.00000 | 4.16304 | 2.48949 | 3.72590 | −1.00000 | 3.19757 | −4.16304 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | −1.53996 | 1.00000 | −0.447921 | 1.53996 | 0.432806 | −1.00000 | −0.628522 | 0.447921 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | −0.857229 | 1.00000 | −0.579785 | 0.857229 | 2.08076 | −1.00000 | −2.26516 | 0.579785 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | −0.314519 | 1.00000 | 2.75757 | 0.314519 | 4.81916 | −1.00000 | −2.90108 | −2.75757 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | −0.0684718 | 1.00000 | −0.598023 | 0.0684718 | −4.46189 | −1.00000 | −2.99531 | 0.598023 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −1.00000 | 1.53837 | 1.00000 | −4.08602 | −1.53837 | −3.71240 | −1.00000 | −0.633414 | 4.08602 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −1.00000 | 1.89638 | 1.00000 | 3.41012 | −1.89638 | −0.727086 | −1.00000 | 0.596266 | −3.41012 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | −1.00000 | 3.38298 | 1.00000 | 2.88192 | −3.38298 | −0.823652 | −1.00000 | 8.44457 | −2.88192 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | −1.00000 | 3.45063 | 1.00000 | −3.11465 | −3.45063 | 1.77185 | −1.00000 | 8.90685 | 3.11465 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(19\) | \(-1\) |
| \(53\) | \(1\) |
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 | 2014.2.a.l | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2014.2.a.l | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2014))\):
|
\( T_{3}^{12} + 4 T_{3}^{11} - 25 T_{3}^{10} - 117 T_{3}^{9} + 160 T_{3}^{8} + 1121 T_{3}^{7} + 167 T_{3}^{6} + \cdots + 64 \)
|
|
\( T_{7}^{12} + 2 T_{7}^{11} - 55 T_{7}^{10} - 87 T_{7}^{9} + 1079 T_{7}^{8} + 1120 T_{7}^{7} - 9310 T_{7}^{6} + \cdots + 7776 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{12} \)
|
| $3$ |
\( T^{12} + 4 T^{11} + \cdots + 64 \)
|
| $5$ |
\( T^{12} - 6 T^{11} + \cdots - 578 \)
|
| $7$ |
\( T^{12} + 2 T^{11} + \cdots + 7776 \)
|
| $11$ |
\( T^{12} - 18 T^{11} + \cdots + 50016 \)
|
| $13$ |
\( T^{12} + 9 T^{11} + \cdots - 7027100 \)
|
| $17$ |
\( T^{12} - 5 T^{11} + \cdots + 517558 \)
|
| $19$ |
\( (T - 1)^{12} \)
|
| $23$ |
\( T^{12} + \cdots + 350577952 \)
|
| $29$ |
\( T^{12} - 31 T^{11} + \cdots - 13713804 \)
|
| $31$ |
\( T^{12} + \cdots + 143567904 \)
|
| $37$ |
\( T^{12} + \cdots - 154655896 \)
|
| $41$ |
\( T^{12} - 26 T^{11} + \cdots - 2919852 \)
|
| $43$ |
\( T^{12} - 7 T^{11} + \cdots + 591616 \)
|
| $47$ |
\( T^{12} + \cdots - 84467625552 \)
|
| $53$ |
\( (T + 1)^{12} \)
|
| $59$ |
\( T^{12} + \cdots + 390321072 \)
|
| $61$ |
\( T^{12} + \cdots + 8331327608 \)
|
| $67$ |
\( T^{12} + \cdots + 412474112 \)
|
| $71$ |
\( T^{12} + \cdots + 686491200 \)
|
| $73$ |
\( T^{12} - 35 T^{11} + \cdots + 4894464 \)
|
| $79$ |
\( T^{12} + \cdots - 60180152656 \)
|
| $83$ |
\( T^{12} + \cdots + 3250687464 \)
|
| $89$ |
\( T^{12} + T^{11} + \cdots - 708496 \)
|
| $97$ |
\( T^{12} + 10 T^{11} + \cdots + 97621136 \)
|
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