Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [363,3,Mod(245,363)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(363, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 8]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("363.245");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 363 = 3 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 363.h (of order \(10\), degree \(4\), not 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: | \(9.89103359628\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{10})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 12x^{14} + 180x^{12} - 2562x^{10} + 25179x^{8} - 96398x^{6} + 239275x^{4} - 536393x^{2} + 1771561 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 33) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 12x^{14} + 180x^{12} - 2562x^{10} + 25179x^{8} - 96398x^{6} + 239275x^{4} - 536393x^{2} + 1771561 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 127407047787804 \nu^{14} - 322126035973969 \nu^{12} + \cdots + 37\!\cdots\!56 ) / 21\!\cdots\!19 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 234959048766210 \nu^{14} + \cdots - 77\!\cdots\!99 ) / 21\!\cdots\!19 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 127407047787804 \nu^{15} + 322126035973969 \nu^{13} + \cdots - 37\!\cdots\!56 \nu ) / 21\!\cdots\!19 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 234959048766210 \nu^{15} + \cdots - 77\!\cdots\!99 \nu ) / 21\!\cdots\!19 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 44\!\cdots\!44 \nu^{14} + \cdots + 20\!\cdots\!70 ) / 21\!\cdots\!19 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 29\!\cdots\!01 \nu^{15} + \cdots + 38\!\cdots\!99 ) / 47\!\cdots\!18 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 29\!\cdots\!01 \nu^{15} + \cdots - 38\!\cdots\!99 ) / 47\!\cdots\!18 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 55\!\cdots\!28 \nu^{15} + \cdots - 91\!\cdots\!65 ) / 47\!\cdots\!18 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 55\!\cdots\!28 \nu^{15} + \cdots - 91\!\cdots\!65 ) / 47\!\cdots\!18 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 30\!\cdots\!44 \nu^{15} + \cdots - 55\!\cdots\!46 \nu ) / 23\!\cdots\!09 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 72\!\cdots\!54 \nu^{15} + \cdots + 25\!\cdots\!82 ) / 47\!\cdots\!18 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 72\!\cdots\!54 \nu^{15} + \cdots - 25\!\cdots\!82 ) / 47\!\cdots\!18 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 99\!\cdots\!41 \nu^{15} + \cdots - 84\!\cdots\!42 ) / 47\!\cdots\!18 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 94\!\cdots\!95 \nu^{15} + \cdots - 74\!\cdots\!23 ) / 47\!\cdots\!18 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{13} + \beta_{12} - \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{3} - 8\beta_{2} - 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{10} - \beta_{9} + 2\beta_{8} + 2\beta_{7} - \beta_{5} + 10\beta_{4} - \beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{15} - \beta_{14} - 15 \beta_{13} + 15 \beta_{12} + \beta_{9} + \beta_{8} + \beta_{7} + \cdots + \beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 8 \beta_{15} - 8 \beta_{14} - 41 \beta_{13} - 41 \beta_{12} - 85 \beta_{11} + 11 \beta_{10} + \cdots - 96 \beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 190 \beta_{15} + 190 \beta_{14} + 220 \beta_{13} - 220 \beta_{12} + 220 \beta_{10} + 30 \beta_{9} + \cdots + 378 \)
|
| \(\nu^{7}\) | \(=\) |
\( 603 \beta_{15} + 603 \beta_{14} + 1143 \beta_{13} + 1143 \beta_{12} + 1073 \beta_{11} + 540 \beta_{10} + \cdots + 721 \beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 2341 \beta_{10} - 2341 \beta_{9} + 588 \beta_{8} - 588 \beta_{7} + 6969 \beta_{6} - 12506 \beta_{3} + \cdots - 6969 \)
|
| \(\nu^{9}\) | \(=\) |
\( 6381 \beta_{15} + 6381 \beta_{14} + 12816 \beta_{13} + 12816 \beta_{12} + 12827 \beta_{11} + \cdots + 6446 \beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( 28893 \beta_{15} - 28893 \beta_{14} - 38546 \beta_{13} + 38546 \beta_{12} - 28893 \beta_{10} + \cdots - 80060 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 48156 \beta_{15} - 48156 \beta_{14} - 134901 \beta_{13} - 134901 \beta_{12} - 211927 \beta_{11} + \cdots - 205308 \beta_1 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 144620 \beta_{15} + 144620 \beta_{14} + 359360 \beta_{13} - 359360 \beta_{12} + 503980 \beta_{10} + \cdots + 1043432 \)
|
| \(\nu^{13}\) | \(=\) |
\( 219120 \beta_{10} - 219120 \beta_{9} - 1152580 \beta_{8} - 1152580 \beta_{7} + 464952 \beta_{5} + \cdots + 464952 \beta_1 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 2055772 \beta_{15} + 2055772 \beta_{14} + 4504525 \beta_{13} - 4504525 \beta_{12} + \cdots - 2055772 \beta_1 \)
|
| \(\nu^{15}\) | \(=\) |
\( 9398824 \beta_{15} + 9398824 \beta_{14} + 24575190 \beta_{13} + 24575190 \beta_{12} + \cdots + 32314430 \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).
| \(n\) | \(122\) | \(244\) |
| \(\chi(n)\) | \(-1\) | \(-\beta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 245.1 |
|
−1.79877 | − | 2.47580i | 2.88971 | − | 0.805978i | −1.65793 | + | 5.10257i | −3.90250 | + | 5.37133i | −7.19336 | − | 5.70456i | 0.946512 | − | 2.91306i | 3.97326 | − | 1.29099i | 7.70080 | − | 4.65808i | 20.3180 | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 245.2 | −1.17804 | − | 1.62143i | −0.157363 | + | 2.99587i | −0.00519352 | + | 0.0159840i | 3.22364 | − | 4.43696i | 5.04297 | − | 3.27409i | −1.72766 | + | 5.31721i | −7.59238 | + | 2.46692i | −8.95047 | − | 0.942880i | −10.9918 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 245.3 | 1.17804 | + | 1.62143i | 1.88824 | + | 2.33121i | −0.00519352 | + | 0.0159840i | −3.22364 | + | 4.43696i | −1.55548 | + | 5.80790i | −1.72766 | + | 5.31721i | 7.59238 | − | 2.46692i | −1.86912 | + | 8.80377i | −10.9918 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 245.4 | 1.79877 | + | 2.47580i | −2.81156 | + | 1.04648i | −1.65793 | + | 5.10257i | 3.90250 | − | 5.37133i | −7.64823 | − | 5.07849i | 0.946512 | − | 2.91306i | −3.97326 | + | 1.29099i | 6.80977 | − | 5.88447i | 20.3180 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 251.1 | −3.41170 | − | 1.10853i | 2.93308 | − | 0.630124i | 7.17480 | + | 5.21280i | 1.98428 | − | 0.644731i | −10.7053 | − | 1.10160i | 7.17034 | + | 5.20956i | −10.2655 | − | 14.1293i | 8.20589 | − | 3.69641i | −7.48447 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 251.2 | −1.57700 | − | 0.512399i | −0.753215 | + | 2.90391i | −1.01168 | − | 0.735029i | −0.664316 | + | 0.215849i | 2.67578 | − | 4.19352i | 2.11081 | + | 1.53360i | 5.11736 | + | 7.04345i | −7.86534 | − | 4.37453i | 1.15823 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 251.3 | 1.57700 | + | 0.512399i | −2.99453 | − | 0.181006i | −1.01168 | − | 0.735029i | 0.664316 | − | 0.215849i | −4.62964 | − | 1.81985i | 2.11081 | + | 1.53360i | −5.11736 | − | 7.04345i | 8.93447 | + | 1.08406i | 1.15823 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 251.4 | 3.41170 | + | 1.10853i | 1.50565 | − | 2.59480i | 7.17480 | + | 5.21280i | −1.98428 | + | 0.644731i | 8.01326 | − | 7.18363i | 7.17034 | + | 5.20956i | 10.2655 | + | 14.1293i | −4.46601 | − | 7.81376i | −7.48447 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 269.1 | −3.41170 | + | 1.10853i | 2.93308 | + | 0.630124i | 7.17480 | − | 5.21280i | 1.98428 | + | 0.644731i | −10.7053 | + | 1.10160i | 7.17034 | − | 5.20956i | −10.2655 | + | 14.1293i | 8.20589 | + | 3.69641i | −7.48447 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 269.2 | −1.57700 | + | 0.512399i | −0.753215 | − | 2.90391i | −1.01168 | + | 0.735029i | −0.664316 | − | 0.215849i | 2.67578 | + | 4.19352i | 2.11081 | − | 1.53360i | 5.11736 | − | 7.04345i | −7.86534 | + | 4.37453i | 1.15823 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 269.3 | 1.57700 | − | 0.512399i | −2.99453 | + | 0.181006i | −1.01168 | + | 0.735029i | 0.664316 | + | 0.215849i | −4.62964 | + | 1.81985i | 2.11081 | − | 1.53360i | −5.11736 | + | 7.04345i | 8.93447 | − | 1.08406i | 1.15823 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 269.4 | 3.41170 | − | 1.10853i | 1.50565 | + | 2.59480i | 7.17480 | − | 5.21280i | −1.98428 | − | 0.644731i | 8.01326 | + | 7.18363i | 7.17034 | − | 5.20956i | 10.2655 | − | 14.1293i | −4.46601 | + | 7.81376i | −7.48447 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 323.1 | −1.79877 | + | 2.47580i | 2.88971 | + | 0.805978i | −1.65793 | − | 5.10257i | −3.90250 | − | 5.37133i | −7.19336 | + | 5.70456i | 0.946512 | + | 2.91306i | 3.97326 | + | 1.29099i | 7.70080 | + | 4.65808i | 20.3180 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 323.2 | −1.17804 | + | 1.62143i | −0.157363 | − | 2.99587i | −0.00519352 | − | 0.0159840i | 3.22364 | + | 4.43696i | 5.04297 | + | 3.27409i | −1.72766 | − | 5.31721i | −7.59238 | − | 2.46692i | −8.95047 | + | 0.942880i | −10.9918 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 323.3 | 1.17804 | − | 1.62143i | 1.88824 | − | 2.33121i | −0.00519352 | − | 0.0159840i | −3.22364 | − | 4.43696i | −1.55548 | − | 5.80790i | −1.72766 | − | 5.31721i | 7.59238 | + | 2.46692i | −1.86912 | − | 8.80377i | −10.9918 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 323.4 | 1.79877 | − | 2.47580i | −2.81156 | − | 1.04648i | −1.65793 | − | 5.10257i | 3.90250 | + | 5.37133i | −7.64823 | + | 5.07849i | 0.946512 | + | 2.91306i | −3.97326 | − | 1.29099i | 6.80977 | + | 5.88447i | 20.3180 | |||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 11.c | even | 5 | 1 | inner |
| 33.h | odd | 10 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 363.3.h.n | 16 | |
| 3.b | odd | 2 | 1 | inner | 363.3.h.n | 16 | |
| 11.b | odd | 2 | 1 | 363.3.h.o | 16 | ||
| 11.c | even | 5 | 1 | 363.3.b.l | 8 | ||
| 11.c | even | 5 | 2 | 363.3.h.j | 16 | ||
| 11.c | even | 5 | 1 | inner | 363.3.h.n | 16 | |
| 11.d | odd | 10 | 2 | 33.3.h.b | ✓ | 16 | |
| 11.d | odd | 10 | 1 | 363.3.b.m | 8 | ||
| 11.d | odd | 10 | 1 | 363.3.h.o | 16 | ||
| 33.d | even | 2 | 1 | 363.3.h.o | 16 | ||
| 33.f | even | 10 | 2 | 33.3.h.b | ✓ | 16 | |
| 33.f | even | 10 | 1 | 363.3.b.m | 8 | ||
| 33.f | even | 10 | 1 | 363.3.h.o | 16 | ||
| 33.h | odd | 10 | 1 | 363.3.b.l | 8 | ||
| 33.h | odd | 10 | 2 | 363.3.h.j | 16 | ||
| 33.h | odd | 10 | 1 | inner | 363.3.h.n | 16 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 33.3.h.b | ✓ | 16 | 11.d | odd | 10 | 2 | |
| 33.3.h.b | ✓ | 16 | 33.f | even | 10 | 2 | |
| 363.3.b.l | 8 | 11.c | even | 5 | 1 | ||
| 363.3.b.l | 8 | 33.h | odd | 10 | 1 | ||
| 363.3.b.m | 8 | 11.d | odd | 10 | 1 | ||
| 363.3.b.m | 8 | 33.f | even | 10 | 1 | ||
| 363.3.h.j | 16 | 11.c | even | 5 | 2 | ||
| 363.3.h.j | 16 | 33.h | odd | 10 | 2 | ||
| 363.3.h.n | 16 | 1.a | even | 1 | 1 | trivial | |
| 363.3.h.n | 16 | 3.b | odd | 2 | 1 | inner | |
| 363.3.h.n | 16 | 11.c | even | 5 | 1 | inner | |
| 363.3.h.n | 16 | 33.h | odd | 10 | 1 | inner | |
| 363.3.h.o | 16 | 11.b | odd | 2 | 1 | ||
| 363.3.h.o | 16 | 11.d | odd | 10 | 1 | ||
| 363.3.h.o | 16 | 33.d | even | 2 | 1 | ||
| 363.3.h.o | 16 | 33.f | even | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(363, [\chi])\):
|
\( T_{2}^{16} - 17 T_{2}^{14} + 175 T_{2}^{12} - 1372 T_{2}^{10} + 18829 T_{2}^{8} - 48408 T_{2}^{6} + \cdots + 1771561 \)
|
|
\( T_{5}^{16} + 38 T_{5}^{14} + 3020 T_{5}^{12} + 35613 T_{5}^{10} + 1364209 T_{5}^{8} - 12321408 T_{5}^{6} + \cdots + 7929856 \)
|
|
\( T_{7}^{8} - 17T_{7}^{7} + 151T_{7}^{6} - 861T_{7}^{5} + 5629T_{7}^{4} - 23148T_{7}^{3} + 72504T_{7}^{2} - 140184T_{7} + 156816 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} - 17 T^{14} + \cdots + 1771561 \)
|
| $3$ |
\( T^{16} - 5 T^{15} + \cdots + 43046721 \)
|
| $5$ |
\( T^{16} + 38 T^{14} + \cdots + 7929856 \)
|
| $7$ |
\( (T^{8} - 17 T^{7} + \cdots + 156816)^{2} \)
|
| $11$ |
\( T^{16} \)
|
| $13$ |
\( (T^{8} - T^{7} + \cdots + 19749136)^{2} \)
|
| $17$ |
\( T^{16} + \cdots + 69002444446441 \)
|
| $19$ |
\( (T^{8} + 33 T^{7} + \cdots + 4224870001)^{2} \)
|
| $23$ |
\( (T^{8} + 1848 T^{6} + \cdots + 25255373616)^{2} \)
|
| $29$ |
\( T^{16} + \cdots + 116101021696 \)
|
| $31$ |
\( (T^{8} + 63 T^{7} + \cdots + 9448617616)^{2} \)
|
| $37$ |
\( (T^{8} + 115 T^{7} + \cdots + 487254257296)^{2} \)
|
| $41$ |
\( T^{16} + \cdots + 15\!\cdots\!81 \)
|
| $43$ |
\( (T^{4} - 39 T^{3} + \cdots - 684409)^{4} \)
|
| $47$ |
\( T^{16} + \cdots + 21\!\cdots\!96 \)
|
| $53$ |
\( T^{16} + \cdots + 30\!\cdots\!36 \)
|
| $59$ |
\( T^{16} + \cdots + 26\!\cdots\!01 \)
|
| $61$ |
\( (T^{8} + \cdots + 1393523586576)^{2} \)
|
| $67$ |
\( (T^{4} - 92 T^{3} + \cdots - 11977619)^{4} \)
|
| $71$ |
\( T^{16} + \cdots + 27\!\cdots\!96 \)
|
| $73$ |
\( (T^{8} + \cdots + 5458559358736)^{2} \)
|
| $79$ |
\( (T^{8} + \cdots + 743504776736656)^{2} \)
|
| $83$ |
\( T^{16} + \cdots + 21\!\cdots\!81 \)
|
| $89$ |
\( (T^{8} + \cdots + 3248250664131)^{2} \)
|
| $97$ |
\( (T^{8} + \cdots + 9061117408561)^{2} \)
|
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