Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5312,2,Mod(1,5312)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5312, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("5312.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5312 = 2^{6} \cdot 83 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5312.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: | \(42.4165335537\) |
| 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: |
\( x^{11} - x^{10} - 25 x^{9} + 24 x^{8} + 214 x^{7} - 197 x^{6} - 721 x^{5} + 620 x^{4} + 795 x^{3} + \cdots - 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 2656) |
| 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
\( x^{11} - x^{10} - 25 x^{9} + 24 x^{8} + 214 x^{7} - 197 x^{6} - 721 x^{5} + 620 x^{4} + 795 x^{3} + \cdots - 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 525 \nu^{10} + 546 \nu^{9} + 11471 \nu^{8} - 14691 \nu^{7} - 77487 \nu^{6} + 142432 \nu^{5} + \cdots - 116432 ) / 81608 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 1903 \nu^{10} - 9446 \nu^{9} + 49585 \nu^{8} + 221651 \nu^{7} - 425901 \nu^{6} - 1720088 \nu^{5} + \cdots - 707744 ) / 163216 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3325 \nu^{10} - 3458 \nu^{9} - 86251 \nu^{8} + 93043 \nu^{7} + 776379 \nu^{6} - 888468 \nu^{5} + \cdots - 1166784 ) / 163216 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 25801 \nu^{10} - 4178 \nu^{9} + 633767 \nu^{8} + 127437 \nu^{7} - 5235987 \nu^{6} + \cdots - 615072 ) / 163216 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 40039 \nu^{10} - 25278 \nu^{9} + 990561 \nu^{8} + 646291 \nu^{7} - 8261237 \nu^{6} + \cdots - 1774944 ) / 163216 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 20411 \nu^{10} + 6538 \nu^{9} + 505117 \nu^{8} - 146097 \nu^{7} - 4228273 \nu^{6} + 1151016 \nu^{5} + \cdots + 738072 ) / 81608 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 41131 \nu^{10} + 20878 \nu^{9} - 1017685 \nu^{8} - 547183 \nu^{7} + 8494225 \nu^{6} + \cdots + 1501360 ) / 163216 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 54207 \nu^{10} - 2514 \nu^{9} - 1340385 \nu^{8} + 24373 \nu^{7} + 11210869 \nu^{6} - 51376 \nu^{5} + \cdots - 954368 ) / 163216 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{10} - \beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} + 7\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{10} + \beta_{9} + \beta_{7} + 2\beta_{6} - 2\beta_{5} - \beta_{4} - \beta_{3} + 9\beta_{2} + 40 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 10 \beta_{10} - 3 \beta_{9} - 12 \beta_{8} - 14 \beta_{7} + 14 \beta_{6} - 4 \beta_{5} + 11 \beta_{4} + \cdots - 12 \)
|
| \(\nu^{6}\) | \(=\) |
\( 23 \beta_{10} + 15 \beta_{9} + 7 \beta_{8} + 16 \beta_{7} + 34 \beta_{6} - 30 \beta_{5} - 12 \beta_{4} + \cdots + 361 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 82 \beta_{10} - 61 \beta_{9} - 122 \beta_{8} - 164 \beta_{7} + 165 \beta_{6} - 74 \beta_{5} + \cdots - 114 \)
|
| \(\nu^{8}\) | \(=\) |
\( 340 \beta_{10} + 169 \beta_{9} + 145 \beta_{8} + 189 \beta_{7} + 432 \beta_{6} - 360 \beta_{5} + \cdots + 3454 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 614 \beta_{10} - 885 \beta_{9} - 1186 \beta_{8} - 1821 \beta_{7} + 1848 \beta_{6} - 1004 \beta_{5} + \cdots - 1007 \)
|
| \(\nu^{10}\) | \(=\) |
\( 4315 \beta_{10} + 1720 \beta_{9} + 2129 \beta_{8} + 2003 \beta_{7} + 4992 \beta_{6} - 4034 \beta_{5} + \cdots + 34146 \)
|
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 | −3.20333 | 0 | 0.650505 | 0 | 4.10971 | 0 | 7.26132 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.76733 | 0 | −3.72956 | 0 | −1.31520 | 0 | 4.65809 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −2.29457 | 0 | −3.14135 | 0 | 1.17805 | 0 | 2.26507 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.14143 | 0 | −0.925018 | 0 | 4.34927 | 0 | −1.69715 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −0.229020 | 0 | 2.97013 | 0 | 3.52908 | 0 | −2.94755 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −0.0986208 | 0 | 2.14614 | 0 | −3.57034 | 0 | −2.99027 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 1.21945 | 0 | −3.50043 | 0 | 5.02593 | 0 | −1.51294 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 1.25056 | 0 | −3.44083 | 0 | −2.10491 | 0 | −1.43611 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 2.15611 | 0 | 2.53671 | 0 | −0.216419 | 0 | 1.64882 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 2.83522 | 0 | −1.82883 | 0 | −4.21092 | 0 | 5.03848 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 3.27295 | 0 | 3.26253 | 0 | 3.22575 | 0 | 7.71223 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(83\) | \(-1\) |
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 | 5312.2.a.bv | 11 | |
| 4.b | odd | 2 | 1 | 5312.2.a.bu | 11 | ||
| 8.b | even | 2 | 1 | 2656.2.a.s | ✓ | 11 | |
| 8.d | odd | 2 | 1 | 2656.2.a.t | yes | 11 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2656.2.a.s | ✓ | 11 | 8.b | even | 2 | 1 | |
| 2656.2.a.t | yes | 11 | 8.d | odd | 2 | 1 | |
| 5312.2.a.bu | 11 | 4.b | odd | 2 | 1 | ||
| 5312.2.a.bv | 11 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5312))\):
|
\( T_{3}^{11} - T_{3}^{10} - 25 T_{3}^{9} + 24 T_{3}^{8} + 214 T_{3}^{7} - 197 T_{3}^{6} - 721 T_{3}^{5} + \cdots - 16 \)
|
|
\( T_{5}^{11} + 5 T_{5}^{10} - 29 T_{5}^{9} - 158 T_{5}^{8} + 300 T_{5}^{7} + 1838 T_{5}^{6} - 1276 T_{5}^{5} + \cdots - 8192 \)
|
|
\( T_{7}^{11} - 10 T_{7}^{10} - 11 T_{7}^{9} + 368 T_{7}^{8} - 531 T_{7}^{7} - 4305 T_{7}^{6} + \cdots + 10852 \)
|
|
\( T_{11}^{11} + 3 T_{11}^{10} - 77 T_{11}^{9} - 152 T_{11}^{8} + 2228 T_{11}^{7} + 1921 T_{11}^{6} + \cdots + 47312 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} \)
|
| $3$ |
\( T^{11} - T^{10} + \cdots - 16 \)
|
| $5$ |
\( T^{11} + 5 T^{10} + \cdots - 8192 \)
|
| $7$ |
\( T^{11} - 10 T^{10} + \cdots + 10852 \)
|
| $11$ |
\( T^{11} + 3 T^{10} + \cdots + 47312 \)
|
| $13$ |
\( T^{11} - T^{10} + \cdots - 917888 \)
|
| $17$ |
\( T^{11} - 6 T^{10} + \cdots - 190 \)
|
| $19$ |
\( T^{11} - 15 T^{10} + \cdots + 92416 \)
|
| $23$ |
\( T^{11} + 6 T^{10} + \cdots - 4096 \)
|
| $29$ |
\( T^{11} + 31 T^{10} + \cdots + 17493376 \)
|
| $31$ |
\( T^{11} - 14 T^{10} + \cdots - 6071372 \)
|
| $37$ |
\( T^{11} - T^{10} + \cdots - 17994472 \)
|
| $41$ |
\( T^{11} + \cdots - 133394336 \)
|
| $43$ |
\( T^{11} - 37 T^{10} + \cdots + 6303744 \)
|
| $47$ |
\( T^{11} + 2 T^{10} + \cdots + 4591616 \)
|
| $53$ |
\( T^{11} + 29 T^{10} + \cdots - 20350976 \)
|
| $59$ |
\( T^{11} + T^{10} + \cdots - 15056 \)
|
| $61$ |
\( T^{11} + 15 T^{10} + \cdots - 23813984 \)
|
| $67$ |
\( T^{11} + \cdots + 109232128 \)
|
| $71$ |
\( T^{11} - 240 T^{9} + \cdots - 4980736 \)
|
| $73$ |
\( T^{11} + \cdots + 295197696 \)
|
| $79$ |
\( T^{11} - 10 T^{10} + \cdots - 83812352 \)
|
| $83$ |
\( (T - 1)^{11} \)
|
| $89$ |
\( T^{11} + \cdots - 900909056 \)
|
| $97$ |
\( T^{11} + 18 T^{10} + \cdots - 40898560 \)
|
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