Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [95,4,Mod(11,95)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(95, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("95.11");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 95 = 5 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 95.e (of order \(3\), degree \(2\), 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: | \(5.60518145055\) |
| Analytic rank: | \(0\) |
| Dimension: | \(18\) |
| Relative dimension: | \(9\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{18} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{18} - 3 x^{17} + 64 x^{16} - 83 x^{15} + 2369 x^{14} - 2209 x^{13} + 52787 x^{12} - 15807 x^{11} + \cdots + 156250000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) 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
\( x^{18} - 3 x^{17} + 64 x^{16} - 83 x^{15} + 2369 x^{14} - 2209 x^{13} + 52787 x^{12} - 15807 x^{11} + \cdots + 156250000 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 34\!\cdots\!95 \nu^{17} + \cdots + 30\!\cdots\!00 ) / 32\!\cdots\!39 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 13\!\cdots\!55 \nu^{17} + \cdots + 46\!\cdots\!07 ) / 32\!\cdots\!39 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 24\!\cdots\!37 \nu^{17} + \cdots - 31\!\cdots\!00 ) / 40\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 13\!\cdots\!43 \nu^{17} + \cdots + 22\!\cdots\!00 ) / 39\!\cdots\!80 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 29\!\cdots\!81 \nu^{17} + \cdots - 12\!\cdots\!00 ) / 40\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 83\!\cdots\!37 \nu^{17} + \cdots + 69\!\cdots\!00 ) / 49\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 68\!\cdots\!89 \nu^{17} + \cdots - 24\!\cdots\!20 ) / 39\!\cdots\!80 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 44\!\cdots\!33 \nu^{17} + \cdots + 68\!\cdots\!00 ) / 24\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 77\!\cdots\!43 \nu^{17} + \cdots + 23\!\cdots\!00 ) / 39\!\cdots\!80 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 13\!\cdots\!53 \nu^{17} + \cdots - 12\!\cdots\!00 ) / 49\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 82\!\cdots\!43 \nu^{17} + \cdots - 21\!\cdots\!60 ) / 19\!\cdots\!40 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 67\!\cdots\!09 \nu^{17} + \cdots - 63\!\cdots\!00 ) / 12\!\cdots\!00 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 47\!\cdots\!71 \nu^{17} + \cdots + 14\!\cdots\!60 ) / 39\!\cdots\!80 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 29\!\cdots\!61 \nu^{17} + \cdots - 99\!\cdots\!20 ) / 19\!\cdots\!40 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 37\!\cdots\!69 \nu^{17} + \cdots + 38\!\cdots\!00 ) / 24\!\cdots\!00 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 24\!\cdots\!48 \nu^{17} + \cdots - 27\!\cdots\!00 ) / 12\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + 13\beta_{4} + \beta_{3} - \beta_{2} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{12} + 2\beta_{10} + \beta_{3} - 21\beta_{2} - 9 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{13} - 3\beta_{11} + \beta_{9} - 2\beta_{7} - 28\beta_{6} - 265\beta_{4} - 44\beta _1 - 265 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 2 \beta_{16} + 2 \beta_{14} + 36 \beta_{13} - 36 \beta_{12} - 83 \beta_{11} - 83 \beta_{10} + \cdots - 519 \beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 4 \beta_{15} + 6 \beta_{14} - 113 \beta_{12} - 244 \beta_{10} + 64 \beta_{8} + 92 \beta_{5} + \cdots + 6340 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 8 \beta_{17} + 130 \beta_{16} - 1147 \beta_{13} + 3003 \beta_{11} - 295 \beta_{9} + 472 \beta_{7} + \cdots + 16381 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 252 \beta_{17} + 476 \beta_{16} + 252 \beta_{15} - 476 \beta_{14} - 4587 \beta_{13} + \cdots + 54124 \beta_1 \)
|
| \(\nu^{9}\) | \(=\) |
\( 700 \beta_{15} - 5778 \beta_{14} + 36169 \beta_{12} + 104594 \beta_{10} - 14359 \beta_{8} + \cdots - 578391 \)
|
| \(\nu^{10}\) | \(=\) |
\( 10856 \beta_{17} - 24340 \beta_{16} + 166606 \beta_{13} - 486688 \beta_{11} + 112287 \beta_{9} + \cdots - 4738475 \)
|
| \(\nu^{11}\) | \(=\) |
\( 37824 \beta_{17} - 221946 \beta_{16} - 37824 \beta_{15} + 221946 \beta_{14} + 1151139 \beta_{13} + \cdots - 12325457 \beta_1 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 406068 \beta_{15} + 1031914 \beta_{14} - 5777373 \beta_{12} - 18132635 \beta_{10} + 4090195 \beta_{8} + \cdots + 141639062 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 1657760 \beta_{17} + 7960612 \beta_{16} - 37058866 \beta_{13} + 120920536 \beta_{11} + \cdots + 669826090 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 14263464 \beta_{17} + 39655230 \beta_{16} + 14263464 \beta_{15} - 39655230 \beta_{14} + \cdots + 2018571725 \beta_1 \)
|
| \(\nu^{15}\) | \(=\) |
\( 65046996 \beta_{15} - 275648254 \beta_{14} + 1204401670 \beta_{12} + 4059380662 \beta_{10} + \cdots - 22428479932 \)
|
| \(\nu^{16}\) | \(=\) |
\( 486249512 \beta_{17} - 1440287230 \beta_{16} + 6583155344 \beta_{13} - 22285338986 \beta_{11} + \cdots - 139687389821 \)
|
| \(\nu^{17}\) | \(=\) |
\( 2394324948 \beta_{17} - 9365554010 \beta_{16} - 2394324948 \beta_{15} + 9365554010 \beta_{14} + \cdots - 399419240547 \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/95\mathbb{Z}\right)^\times\).
| \(n\) | \(21\) | \(77\) |
| \(\chi(n)\) | \(\beta_{4}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 11.1 |
|
−2.37881 | + | 4.12022i | 3.90669 | − | 6.76659i | −7.31750 | − | 12.6743i | −2.50000 | + | 4.33013i | 18.5866 | + | 32.1929i | −9.75335 | 31.5668 | −17.0245 | − | 29.4873i | −11.8941 | − | 20.6011i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.2 | −1.78693 | + | 3.09506i | −2.87389 | + | 4.97773i | −2.38626 | − | 4.13313i | −2.50000 | + | 4.33013i | −10.2709 | − | 17.7897i | −12.7820 | −11.5346 | −3.01852 | − | 5.22822i | −8.93467 | − | 15.4753i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.3 | −1.35387 | + | 2.34498i | 2.16510 | − | 3.75006i | 0.334056 | + | 0.578603i | −2.50000 | + | 4.33013i | 5.86254 | + | 10.1542i | 9.08880 | −23.4710 | 4.12469 | + | 7.14417i | −6.76936 | − | 11.7249i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.4 | −0.0856000 | + | 0.148263i | 0.0597169 | − | 0.103433i | 3.98535 | + | 6.90282i | −2.50000 | + | 4.33013i | 0.0102235 | + | 0.0177077i | −29.8703 | −2.73418 | 13.4929 | + | 23.3703i | −0.428000 | − | 0.741317i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.5 | −0.0489495 | + | 0.0847830i | −3.79409 | + | 6.57155i | 3.99521 | + | 6.91990i | −2.50000 | + | 4.33013i | −0.371437 | − | 0.643348i | 18.5564 | −1.56545 | −15.2902 | − | 26.4834i | −0.244747 | − | 0.423915i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.6 | 1.55500 | − | 2.69333i | −1.90691 | + | 3.30286i | −0.836019 | − | 1.44803i | −2.50000 | + | 4.33013i | 5.93046 | + | 10.2719i | 5.61900 | 19.6799 | 6.22742 | + | 10.7862i | 7.77498 | + | 13.4667i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.7 | 1.74804 | − | 3.02770i | 4.62018 | − | 8.00239i | −2.11129 | − | 3.65686i | −2.50000 | + | 4.33013i | −16.1525 | − | 27.9770i | −29.1065 | 13.2062 | −29.1921 | − | 50.5622i | 8.74020 | + | 15.1385i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.8 | 2.59053 | − | 4.48694i | 2.66870 | − | 4.62232i | −9.42174 | − | 16.3189i | −2.50000 | + | 4.33013i | −13.8267 | − | 23.9485i | 30.6349 | −56.1808 | −0.743882 | − | 1.28844i | 12.9527 | + | 22.4347i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.9 | 2.76060 | − | 4.78150i | −3.84550 | + | 6.66060i | −11.2418 | − | 19.4714i | −2.50000 | + | 4.33013i | 21.2318 | + | 36.7745i | −27.3870 | −79.9668 | −16.0758 | − | 27.8440i | 13.8030 | + | 23.9075i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 26.1 | −2.37881 | − | 4.12022i | 3.90669 | + | 6.76659i | −7.31750 | + | 12.6743i | −2.50000 | − | 4.33013i | 18.5866 | − | 32.1929i | −9.75335 | 31.5668 | −17.0245 | + | 29.4873i | −11.8941 | + | 20.6011i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 26.2 | −1.78693 | − | 3.09506i | −2.87389 | − | 4.97773i | −2.38626 | + | 4.13313i | −2.50000 | − | 4.33013i | −10.2709 | + | 17.7897i | −12.7820 | −11.5346 | −3.01852 | + | 5.22822i | −8.93467 | + | 15.4753i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 26.3 | −1.35387 | − | 2.34498i | 2.16510 | + | 3.75006i | 0.334056 | − | 0.578603i | −2.50000 | − | 4.33013i | 5.86254 | − | 10.1542i | 9.08880 | −23.4710 | 4.12469 | − | 7.14417i | −6.76936 | + | 11.7249i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 26.4 | −0.0856000 | − | 0.148263i | 0.0597169 | + | 0.103433i | 3.98535 | − | 6.90282i | −2.50000 | − | 4.33013i | 0.0102235 | − | 0.0177077i | −29.8703 | −2.73418 | 13.4929 | − | 23.3703i | −0.428000 | + | 0.741317i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 26.5 | −0.0489495 | − | 0.0847830i | −3.79409 | − | 6.57155i | 3.99521 | − | 6.91990i | −2.50000 | − | 4.33013i | −0.371437 | + | 0.643348i | 18.5564 | −1.56545 | −15.2902 | + | 26.4834i | −0.244747 | + | 0.423915i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 26.6 | 1.55500 | + | 2.69333i | −1.90691 | − | 3.30286i | −0.836019 | + | 1.44803i | −2.50000 | − | 4.33013i | 5.93046 | − | 10.2719i | 5.61900 | 19.6799 | 6.22742 | − | 10.7862i | 7.77498 | − | 13.4667i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 26.7 | 1.74804 | + | 3.02770i | 4.62018 | + | 8.00239i | −2.11129 | + | 3.65686i | −2.50000 | − | 4.33013i | −16.1525 | + | 27.9770i | −29.1065 | 13.2062 | −29.1921 | + | 50.5622i | 8.74020 | − | 15.1385i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 26.8 | 2.59053 | + | 4.48694i | 2.66870 | + | 4.62232i | −9.42174 | + | 16.3189i | −2.50000 | − | 4.33013i | −13.8267 | + | 23.9485i | 30.6349 | −56.1808 | −0.743882 | + | 1.28844i | 12.9527 | − | 22.4347i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 26.9 | 2.76060 | + | 4.78150i | −3.84550 | − | 6.66060i | −11.2418 | + | 19.4714i | −2.50000 | − | 4.33013i | 21.2318 | − | 36.7745i | −27.3870 | −79.9668 | −16.0758 | + | 27.8440i | 13.8030 | − | 23.9075i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 95.4.e.b | ✓ | 18 |
| 19.c | even | 3 | 1 | inner | 95.4.e.b | ✓ | 18 |
| 19.c | even | 3 | 1 | 1805.4.a.n | 9 | ||
| 19.d | odd | 6 | 1 | 1805.4.a.o | 9 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 95.4.e.b | ✓ | 18 | 1.a | even | 1 | 1 | trivial |
| 95.4.e.b | ✓ | 18 | 19.c | even | 3 | 1 | inner |
| 1805.4.a.n | 9 | 19.c | even | 3 | 1 | ||
| 1805.4.a.o | 9 | 19.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{18} - 6 T_{2}^{17} + 79 T_{2}^{16} - 264 T_{2}^{15} + 2840 T_{2}^{14} - 7757 T_{2}^{13} + \cdots + 57600 \)
acting on \(S_{4}^{\mathrm{new}}(95, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{18} - 6 T^{17} + \cdots + 57600 \)
|
| $3$ |
\( T^{18} + \cdots + 65004601600 \)
|
| $5$ |
\( (T^{2} + 5 T + 25)^{9} \)
|
| $7$ |
\( (T^{9} + 45 T^{8} + \cdots + 86179484384)^{2} \)
|
| $11$ |
\( (T^{9} + \cdots - 1459864914849)^{2} \)
|
| $13$ |
\( T^{18} + \cdots + 22\!\cdots\!00 \)
|
| $17$ |
\( T^{18} + \cdots + 20\!\cdots\!44 \)
|
| $19$ |
\( T^{18} + \cdots + 33\!\cdots\!39 \)
|
| $23$ |
\( T^{18} + \cdots + 42\!\cdots\!00 \)
|
| $29$ |
\( T^{18} + \cdots + 79\!\cdots\!24 \)
|
| $31$ |
\( (T^{9} + \cdots - 12\!\cdots\!48)^{2} \)
|
| $37$ |
\( (T^{9} + \cdots - 18\!\cdots\!76)^{2} \)
|
| $41$ |
\( T^{18} + \cdots + 49\!\cdots\!36 \)
|
| $43$ |
\( T^{18} + \cdots + 60\!\cdots\!16 \)
|
| $47$ |
\( T^{18} + \cdots + 24\!\cdots\!84 \)
|
| $53$ |
\( T^{18} + \cdots + 14\!\cdots\!24 \)
|
| $59$ |
\( T^{18} + \cdots + 52\!\cdots\!00 \)
|
| $61$ |
\( T^{18} + \cdots + 10\!\cdots\!00 \)
|
| $67$ |
\( T^{18} + \cdots + 20\!\cdots\!00 \)
|
| $71$ |
\( T^{18} + \cdots + 18\!\cdots\!64 \)
|
| $73$ |
\( T^{18} + \cdots + 11\!\cdots\!00 \)
|
| $79$ |
\( T^{18} + \cdots + 13\!\cdots\!00 \)
|
| $83$ |
\( (T^{9} + \cdots + 20\!\cdots\!64)^{2} \)
|
| $89$ |
\( T^{18} + \cdots + 28\!\cdots\!00 \)
|
| $97$ |
\( T^{18} + \cdots + 74\!\cdots\!36 \)
|
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