Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [616,2,Mod(113,616)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(616, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 0, 0, 8]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("616.113");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 616 = 2^{3} \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 616.r (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: | \(4.91878476451\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(5\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 4 x^{19} + 21 x^{18} - 58 x^{17} + 225 x^{16} - 348 x^{15} + 1296 x^{14} - 755 x^{13} + \cdots + 10000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} - 4 x^{19} + 21 x^{18} - 58 x^{17} + 225 x^{16} - 348 x^{15} + 1296 x^{14} - 755 x^{13} + \cdots + 10000 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 25\!\cdots\!87 \nu^{19} + \cdots + 17\!\cdots\!00 ) / 87\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 33\!\cdots\!79 \nu^{19} + \cdots - 18\!\cdots\!00 ) / 87\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 15\!\cdots\!89 \nu^{19} + \cdots + 39\!\cdots\!00 ) / 21\!\cdots\!50 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 32\!\cdots\!14 \nu^{19} + \cdots + 30\!\cdots\!00 ) / 43\!\cdots\!50 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 84\!\cdots\!79 \nu^{19} + \cdots - 62\!\cdots\!00 ) / 87\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 12\!\cdots\!41 \nu^{19} + \cdots + 84\!\cdots\!00 ) / 87\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 77\!\cdots\!91 \nu^{19} + \cdots + 21\!\cdots\!00 ) / 43\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 39\!\cdots\!98 \nu^{19} + \cdots + 12\!\cdots\!00 ) / 21\!\cdots\!50 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 15\!\cdots\!89 \nu^{19} + \cdots + 41\!\cdots\!00 ) / 87\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 91\!\cdots\!69 \nu^{19} + \cdots - 13\!\cdots\!00 ) / 43\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 38\!\cdots\!87 \nu^{19} + \cdots - 37\!\cdots\!00 ) / 16\!\cdots\!50 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 48\!\cdots\!31 \nu^{19} + \cdots + 42\!\cdots\!00 ) / 17\!\cdots\!00 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 15\!\cdots\!87 \nu^{19} + \cdots + 52\!\cdots\!00 ) / 43\!\cdots\!00 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 30\!\cdots\!93 \nu^{19} + \cdots - 51\!\cdots\!00 ) / 87\!\cdots\!00 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 15\!\cdots\!64 \nu^{19} + \cdots - 52\!\cdots\!00 ) / 43\!\cdots\!00 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 95\!\cdots\!37 \nu^{19} + \cdots - 96\!\cdots\!00 ) / 17\!\cdots\!00 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 67\!\cdots\!58 \nu^{19} + \cdots + 17\!\cdots\!00 ) / 87\!\cdots\!00 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 47\!\cdots\!62 \nu^{19} + \cdots + 41\!\cdots\!00 ) / 43\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{18} + \beta_{12} + 5\beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{4} - \beta_{3} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{18} + 2 \beta_{17} + \beta_{15} + \beta_{14} - \beta_{13} - 2 \beta_{12} - \beta_{8} - \beta_{7} + \cdots - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4 \beta_{19} + 12 \beta_{17} + 12 \beta_{15} + 12 \beta_{14} - 40 \beta_{11} + 12 \beta_{10} + 8 \beta_{9} + \cdots - 32 \)
|
| \(\nu^{5}\) | \(=\) |
\( 33 \beta_{19} - 18 \beta_{18} + 14 \beta_{17} - 29 \beta_{16} + 33 \beta_{15} + 29 \beta_{14} + \cdots - 77 \)
|
| \(\nu^{6}\) | \(=\) |
\( 138 \beta_{19} - 138 \beta_{18} - 134 \beta_{16} + 63 \beta_{15} + 11 \beta_{14} + 63 \beta_{13} + \cdots - 87 \)
|
| \(\nu^{7}\) | \(=\) |
\( 238 \beta_{19} - 428 \beta_{18} - 170 \beta_{17} - 170 \beta_{16} + 238 \beta_{13} - 138 \beta_{12} + \cdots - 238 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 808 \beta_{18} - 1477 \beta_{17} - 808 \beta_{15} - 260 \beta_{14} + 773 \beta_{13} + 909 \beta_{12} + \cdots - 229 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 2901 \beta_{19} - 4266 \beta_{17} + 2005 \beta_{16} - 5150 \beta_{15} - 4266 \beta_{14} + 7643 \beta_{11} + \cdots + 9136 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 18128 \beta_{19} + 9792 \beta_{18} - 4272 \beta_{17} + 16332 \beta_{16} - 18128 \beta_{15} + \cdots + 42984 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 60317 \beta_{19} + 60317 \beta_{18} + 49633 \beta_{16} - 34406 \beta_{15} - 23582 \beta_{14} + \cdots + 79456 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 116235 \beta_{19} + 208158 \beta_{18} + 60679 \beta_{17} + 60679 \beta_{16} - 116235 \beta_{13} + \cdots + 116235 \)
|
| \(\nu^{13}\) | \(=\) |
\( 403786 \beta_{18} + 572872 \beta_{17} + 403786 \beta_{15} + 278074 \beta_{14} - 295606 \beta_{13} + \cdots - 246870 \)
|
| \(\nu^{14}\) | \(=\) |
\( 1367076 \beta_{19} + 2042737 \beta_{17} - 800204 \beta_{16} + 2393409 \beta_{15} + 2042737 \beta_{14} + \cdots - 4283224 \)
|
| \(\nu^{15}\) | \(=\) |
\( 8080442 \beta_{19} - 4716641 \beta_{18} + 3285217 \beta_{17} - 6589394 \beta_{16} + 8080442 \beta_{15} + \cdots - 17276877 \)
|
| \(\nu^{16}\) | \(=\) |
\( 27549860 \beta_{19} - 27549860 \beta_{18} - 23080828 \beta_{16} + 15999404 \beta_{15} + 10111728 \beta_{14} + \cdots - 36677924 \)
|
| \(\nu^{17}\) | \(=\) |
\( 54957194 \beta_{19} - 93250361 \beta_{18} - 38818014 \beta_{17} - 38818014 \beta_{16} + 54957194 \beta_{13} + \cdots - 54957194 \)
|
| \(\nu^{18}\) | \(=\) |
\( - 186673303 \beta_{18} - 262122886 \beta_{17} - 186673303 \beta_{15} - 124440195 \beta_{14} + \cdots + 124242485 \)
|
| \(\nu^{19}\) | \(=\) |
\( - 639316518 \beta_{19} - 869376568 \beta_{17} + 458111114 \beta_{16} - 1075896580 \beta_{15} + \cdots + 1825417254 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/616\mathbb{Z}\right)^\times\).
| \(n\) | \(57\) | \(309\) | \(353\) | \(463\) |
| \(\chi(n)\) | \(\beta_{11}\) | \(1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 113.1 |
|
0 | −1.05061 | − | 3.23344i | 0 | −0.556182 | − | 0.404090i | 0 | −0.309017 | + | 0.951057i | 0 | −6.92433 | + | 5.03082i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 113.2 | 0 | −0.645517 | − | 1.98670i | 0 | 1.70961 | + | 1.24210i | 0 | −0.309017 | + | 0.951057i | 0 | −1.10322 | + | 0.801539i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 113.3 | 0 | −0.162519 | − | 0.500181i | 0 | −1.76076 | − | 1.27927i | 0 | −0.309017 | + | 0.951057i | 0 | 2.20328 | − | 1.60078i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 113.4 | 0 | 0.384203 | + | 1.18245i | 0 | −2.71639 | − | 1.97357i | 0 | −0.309017 | + | 0.951057i | 0 | 1.17646 | − | 0.854751i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 113.5 | 0 | 0.665425 | + | 2.04797i | 0 | 2.20569 | + | 1.60253i | 0 | −0.309017 | + | 0.951057i | 0 | −1.32433 | + | 0.962184i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 169.1 | 0 | −1.05061 | + | 3.23344i | 0 | −0.556182 | + | 0.404090i | 0 | −0.309017 | − | 0.951057i | 0 | −6.92433 | − | 5.03082i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 169.2 | 0 | −0.645517 | + | 1.98670i | 0 | 1.70961 | − | 1.24210i | 0 | −0.309017 | − | 0.951057i | 0 | −1.10322 | − | 0.801539i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 169.3 | 0 | −0.162519 | + | 0.500181i | 0 | −1.76076 | + | 1.27927i | 0 | −0.309017 | − | 0.951057i | 0 | 2.20328 | + | 1.60078i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 169.4 | 0 | 0.384203 | − | 1.18245i | 0 | −2.71639 | + | 1.97357i | 0 | −0.309017 | − | 0.951057i | 0 | 1.17646 | + | 0.854751i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 169.5 | 0 | 0.665425 | − | 2.04797i | 0 | 2.20569 | − | 1.60253i | 0 | −0.309017 | − | 0.951057i | 0 | −1.32433 | − | 0.962184i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 225.1 | 0 | −2.21758 | − | 1.61116i | 0 | −0.395619 | + | 1.21759i | 0 | 0.809017 | − | 0.587785i | 0 | 1.39475 | + | 4.29259i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 225.2 | 0 | −1.67832 | − | 1.21937i | 0 | 0.914896 | − | 2.81576i | 0 | 0.809017 | − | 0.587785i | 0 | 0.402849 | + | 1.23984i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 225.3 | 0 | 0.255698 | + | 0.185775i | 0 | 0.102849 | − | 0.316537i | 0 | 0.809017 | − | 0.587785i | 0 | −0.896182 | − | 2.75816i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 225.4 | 0 | 1.46730 | + | 1.06605i | 0 | −0.699852 | + | 2.15392i | 0 | 0.809017 | − | 0.587785i | 0 | 0.0894373 | + | 0.275260i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 225.5 | 0 | 2.48193 | + | 1.80322i | 0 | 1.19576 | − | 3.68017i | 0 | 0.809017 | − | 0.587785i | 0 | 1.98128 | + | 6.09777i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 449.1 | 0 | −2.21758 | + | 1.61116i | 0 | −0.395619 | − | 1.21759i | 0 | 0.809017 | + | 0.587785i | 0 | 1.39475 | − | 4.29259i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 449.2 | 0 | −1.67832 | + | 1.21937i | 0 | 0.914896 | + | 2.81576i | 0 | 0.809017 | + | 0.587785i | 0 | 0.402849 | − | 1.23984i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 449.3 | 0 | 0.255698 | − | 0.185775i | 0 | 0.102849 | + | 0.316537i | 0 | 0.809017 | + | 0.587785i | 0 | −0.896182 | + | 2.75816i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 449.4 | 0 | 1.46730 | − | 1.06605i | 0 | −0.699852 | − | 2.15392i | 0 | 0.809017 | + | 0.587785i | 0 | 0.0894373 | − | 0.275260i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 449.5 | 0 | 2.48193 | − | 1.80322i | 0 | 1.19576 | + | 3.68017i | 0 | 0.809017 | + | 0.587785i | 0 | 1.98128 | − | 6.09777i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.c | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 616.2.r.f | ✓ | 20 |
| 11.c | even | 5 | 1 | inner | 616.2.r.f | ✓ | 20 |
| 11.c | even | 5 | 1 | 6776.2.a.bm | 10 | ||
| 11.d | odd | 10 | 1 | 6776.2.a.bn | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 616.2.r.f | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 616.2.r.f | ✓ | 20 | 11.c | even | 5 | 1 | inner |
| 6776.2.a.bm | 10 | 11.c | even | 5 | 1 | ||
| 6776.2.a.bn | 10 | 11.d | odd | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{20} + T_{3}^{19} + 11 T_{3}^{18} - 8 T_{3}^{17} + 85 T_{3}^{16} + 112 T_{3}^{15} + 1081 T_{3}^{14} + \cdots + 10000 \)
acting on \(S_{2}^{\mathrm{new}}(616, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} \)
|
| $3$ |
\( T^{20} + T^{19} + \cdots + 10000 \)
|
| $5$ |
\( T^{20} + 16 T^{18} + \cdots + 102400 \)
|
| $7$ |
\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{5} \)
|
| $11$ |
\( T^{20} + \cdots + 25937424601 \)
|
| $13$ |
\( T^{20} + \cdots + 5609410816 \)
|
| $17$ |
\( T^{20} + 39 T^{18} + \cdots + 51552400 \)
|
| $19$ |
\( T^{20} + \cdots + 2296709776 \)
|
| $23$ |
\( (T^{10} + 3 T^{9} + \cdots - 368404)^{2} \)
|
| $29$ |
\( T^{20} + \cdots + 638362648576 \)
|
| $31$ |
\( T^{20} + \cdots + 101572239616 \)
|
| $37$ |
\( T^{20} + \cdots + 5836348816 \)
|
| $41$ |
\( T^{20} + 12 T^{19} + \cdots + 20647936 \)
|
| $43$ |
\( (T^{10} + 21 T^{9} + \cdots - 398345)^{2} \)
|
| $47$ |
\( T^{20} + \cdots + 6845245696 \)
|
| $53$ |
\( T^{20} + \cdots + 17188652646400 \)
|
| $59$ |
\( T^{20} + \cdots + 23809724416 \)
|
| $61$ |
\( T^{20} + \cdots + 27\!\cdots\!00 \)
|
| $67$ |
\( (T^{10} + T^{9} + \cdots + 78764149)^{2} \)
|
| $71$ |
\( T^{20} + \cdots + 12037569030400 \)
|
| $73$ |
\( T^{20} + \cdots + 129431214182656 \)
|
| $79$ |
\( T^{20} + \cdots + 907134334096 \)
|
| $83$ |
\( T^{20} + \cdots + 15\!\cdots\!76 \)
|
| $89$ |
\( (T^{10} + 23 T^{9} + \cdots + 455935796)^{2} \)
|
| $97$ |
\( T^{20} + \cdots + 17\!\cdots\!16 \)
|
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