Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6032,2,Mod(1,6032)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6032, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 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("6032.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6032 = 2^{4} \cdot 13 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6032.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: | \(48.1657624992\) |
| Analytic rank: | \(0\) |
| Dimension: | \(13\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{13} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{13} - 4 x^{12} - 22 x^{11} + 96 x^{10} + 159 x^{9} - 827 x^{8} - 362 x^{7} + 3029 x^{6} - 142 x^{5} + \cdots + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 3016) |
| 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 \(1,\beta_1,\ldots,\beta_{12}\) 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
\( x^{13} - 4 x^{12} - 22 x^{11} + 96 x^{10} + 159 x^{9} - 827 x^{8} - 362 x^{7} + 3029 x^{6} - 142 x^{5} + \cdots + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2367 \nu^{12} - 10978 \nu^{11} - 51546 \nu^{10} + 275072 \nu^{9} + 363465 \nu^{8} - 2514391 \nu^{7} + \cdots + 744768 ) / 56872 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2470 \nu^{12} - 8161 \nu^{11} - 59988 \nu^{10} + 200172 \nu^{9} + 515082 \nu^{8} - 1786081 \nu^{7} + \cdots + 527848 ) / 56872 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 6995 \nu^{12} - 32106 \nu^{11} - 145536 \nu^{10} + 783032 \nu^{9} + 916693 \nu^{8} - 6888979 \nu^{7} + \cdots + 1582928 ) / 113744 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 4738 \nu^{12} - 19707 \nu^{11} - 103972 \nu^{10} + 478768 \nu^{9} + 746898 \nu^{8} - 4196767 \nu^{7} + \cdots + 1339208 ) / 56872 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 4841 \nu^{12} + 16890 \nu^{11} + 112414 \nu^{10} - 403868 \nu^{9} - 898515 \nu^{8} + \cdots - 724184 ) / 56872 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 3551 \nu^{12} - 13520 \nu^{11} - 77429 \nu^{10} + 318378 \nu^{9} + 544139 \nu^{8} - 2664327 \nu^{7} + \cdots + 351916 ) / 28436 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 8417 \nu^{12} - 32349 \nu^{11} - 188234 \nu^{10} + 773776 \nu^{9} + 1403947 \nu^{8} - 6625222 \nu^{7} + \cdots + 1345080 ) / 56872 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 10575 \nu^{12} + 45316 \nu^{11} + 222236 \nu^{10} - 1081564 \nu^{9} - 1442093 \nu^{8} + \cdots - 1655392 ) / 56872 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 6406 \nu^{12} - 27371 \nu^{11} - 136809 \nu^{10} + 658084 \nu^{9} + 926640 \nu^{8} - 5679929 \nu^{7} + \cdots + 1268504 ) / 28436 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 15926 \nu^{12} + 63373 \nu^{11} + 347324 \nu^{10} - 1512948 \nu^{9} - 2455250 \nu^{8} + \cdots - 2862032 ) / 56872 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - \beta_{4} + \beta_{3} + 6\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{10} + 2\beta_{9} + 2\beta_{8} + 3\beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} + 9\beta_{2} - \beta _1 + 42 \)
|
| \(\nu^{5}\) | \(=\) |
\( 13 \beta_{10} + 13 \beta_{9} + 14 \beta_{8} + 15 \beta_{7} + \beta_{6} - 2 \beta_{5} - 12 \beta_{4} + \cdots + 36 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 2 \beta_{12} + 30 \beta_{10} + 31 \beta_{9} + 29 \beta_{8} + 48 \beta_{7} + 10 \beta_{6} - 2 \beta_{5} + \cdots + 389 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 5 \beta_{12} - 4 \beta_{11} + 141 \beta_{10} + 147 \beta_{9} + 155 \beta_{8} + 184 \beta_{7} + \cdots + 514 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 55 \beta_{12} - 10 \beta_{11} + 359 \beta_{10} + 384 \beta_{9} + 327 \beta_{8} + 606 \beta_{7} + \cdots + 3788 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 148 \beta_{12} - 110 \beta_{11} + 1469 \beta_{10} + 1609 \beta_{9} + 1586 \beta_{8} + 2128 \beta_{7} + \cdots + 6587 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 1014 \beta_{12} - 296 \beta_{11} + 4018 \beta_{10} + 4439 \beta_{9} + 3400 \beta_{8} + 7110 \beta_{7} + \cdots + 38125 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 2891 \beta_{12} - 2028 \beta_{11} + 15207 \beta_{10} + 17444 \beta_{9} + 15658 \beta_{8} + \cdots + 79661 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 15778 \beta_{12} - 5782 \beta_{11} + 43817 \beta_{10} + 49936 \beta_{9} + 34068 \beta_{8} + \cdots + 392822 \)
|
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 |
|
0 | −2.90994 | 0 | −1.66472 | 0 | −2.42557 | 0 | 5.46773 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.69571 | 0 | 2.65571 | 0 | −3.33796 | 0 | 4.26683 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −2.47527 | 0 | 2.58551 | 0 | 4.52168 | 0 | 3.12695 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.993990 | 0 | 3.24551 | 0 | −4.74513 | 0 | −2.01198 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −0.863805 | 0 | −2.53485 | 0 | −1.34341 | 0 | −2.25384 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −0.0989896 | 0 | −2.29988 | 0 | 3.37309 | 0 | −2.99020 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | −0.0772773 | 0 | 0.908464 | 0 | −0.210501 | 0 | −2.99403 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 1.32867 | 0 | −1.97424 | 0 | 1.77375 | 0 | −1.23465 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 1.82663 | 0 | 2.84424 | 0 | 3.25048 | 0 | 0.336589 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 2.07047 | 0 | −1.28770 | 0 | −4.21640 | 0 | 1.28687 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 2.30310 | 0 | 4.32944 | 0 | −4.01657 | 0 | 2.30428 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0 | 3.23329 | 0 | 1.87064 | 0 | 2.94834 | 0 | 7.45419 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 0 | 3.35280 | 0 | −3.67813 | 0 | −1.57180 | 0 | 8.24127 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(13\) | \(1\) |
| \(29\) | \(-1\) |
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 | 6032.2.a.be | 13 | |
| 4.b | odd | 2 | 1 | 3016.2.a.k | ✓ | 13 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3016.2.a.k | ✓ | 13 | 4.b | odd | 2 | 1 | |
| 6032.2.a.be | 13 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6032))\):
|
\( T_{3}^{13} - 4 T_{3}^{12} - 22 T_{3}^{11} + 96 T_{3}^{10} + 159 T_{3}^{9} - 827 T_{3}^{8} - 362 T_{3}^{7} + \cdots + 16 \)
|
|
\( T_{5}^{13} - 5 T_{5}^{12} - 32 T_{5}^{11} + 172 T_{5}^{10} + 380 T_{5}^{9} - 2249 T_{5}^{8} + \cdots - 42320 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{13} \)
|
| $3$ |
\( T^{13} - 4 T^{12} + \cdots + 16 \)
|
| $5$ |
\( T^{13} - 5 T^{12} + \cdots - 42320 \)
|
| $7$ |
\( T^{13} + 6 T^{12} + \cdots - 74980 \)
|
| $11$ |
\( T^{13} + 6 T^{12} + \cdots + 1760 \)
|
| $13$ |
\( (T + 1)^{13} \)
|
| $17$ |
\( T^{13} - 4 T^{12} + \cdots - 121664 \)
|
| $19$ |
\( T^{13} - 3 T^{12} + \cdots - 243616 \)
|
| $23$ |
\( T^{13} + \cdots + 201615872 \)
|
| $29$ |
\( (T - 1)^{13} \)
|
| $31$ |
\( T^{13} + 21 T^{12} + \cdots - 84201472 \)
|
| $37$ |
\( T^{13} + \cdots - 140526656 \)
|
| $41$ |
\( T^{13} + \cdots + 198900256 \)
|
| $43$ |
\( T^{13} + \cdots + 486485696 \)
|
| $47$ |
\( T^{13} + \cdots + 120796736 \)
|
| $53$ |
\( T^{13} - 17 T^{12} + \cdots - 1769472 \)
|
| $59$ |
\( T^{13} + \cdots + 37004856032 \)
|
| $61$ |
\( T^{13} + \cdots + 30760677344 \)
|
| $67$ |
\( T^{13} + \cdots - 107429440 \)
|
| $71$ |
\( T^{13} + \cdots - 101955328 \)
|
| $73$ |
\( T^{13} + \cdots + 720663040 \)
|
| $79$ |
\( T^{13} + \cdots + 66893495000 \)
|
| $83$ |
\( T^{13} + \cdots - 239643232 \)
|
| $89$ |
\( T^{13} + \cdots - 466249408 \)
|
| $97$ |
\( T^{13} + \cdots + 3330563648 \)
|
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