Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1050,3,Mod(701,1050)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1050, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1050.701");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1050.e (of order \(2\), degree \(1\), 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: | \(28.6104277578\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 8 x^{15} + 34 x^{14} - 80 x^{13} + 97 x^{12} - 80 x^{11} + 498 x^{10} - 3288 x^{9} + \cdots + 43046721 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 3^{3} \) |
| Twist minimal: | no (minimal twist has level 210) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 8 x^{15} + 34 x^{14} - 80 x^{13} + 97 x^{12} - 80 x^{11} + 498 x^{10} - 3288 x^{9} + \cdots + 43046721 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 16099 \nu^{15} + 4153021 \nu^{14} - 15016145 \nu^{13} + 28476925 \nu^{12} + \cdots + 12108511707741 ) / 1562997741696 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 242026 \nu^{15} + 942563 \nu^{14} - 1277806 \nu^{13} + 701903 \nu^{12} + \cdots - 348051871161 ) / 1172248306272 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 13307 \nu^{15} + 179536 \nu^{14} - 640907 \nu^{13} + 1208056 \nu^{12} - 207224 \nu^{11} + \cdots + 532478372832 ) / 63364773312 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 563020 \nu^{15} + 3294727 \nu^{14} - 17087684 \nu^{13} + 45270475 \nu^{12} + \cdots + 18619576973379 ) / 2344496612544 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1255853 \nu^{15} - 15403237 \nu^{14} + 82119065 \nu^{13} - 140493085 \nu^{12} + \cdots - 74847290037021 ) / 4688993225088 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 141791 \nu^{15} - 1580125 \nu^{14} + 6541847 \nu^{13} - 14004913 \nu^{12} + \cdots - 6117039496449 ) / 520999247232 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 24328 \nu^{15} + 35285 \nu^{14} - 151678 \nu^{13} + 144311 \nu^{12} + 1005560 \nu^{11} + \cdots - 74835927297 ) / 86833207872 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 29989 \nu^{15} - 210221 \nu^{14} + 754963 \nu^{13} - 1055177 \nu^{12} - 61616 \nu^{11} + \cdots - 467281722393 ) / 50967317664 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 60113 \nu^{15} + 451675 \nu^{14} - 1511201 \nu^{13} + 2038843 \nu^{12} + \cdots + 966250614411 ) / 86833207872 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 608353 \nu^{15} - 2759042 \nu^{14} + 5689201 \nu^{13} - 3614582 \nu^{12} + \cdots - 534994214526 ) / 781498870848 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 33544 \nu^{15} - 152423 \nu^{14} + 301840 \nu^{13} - 248939 \nu^{12} - 144488 \nu^{11} + \cdots - 49412852739 ) / 33978211776 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 19897 \nu^{15} + 178301 \nu^{14} - 643441 \nu^{13} + 997040 \nu^{12} - 426964 \nu^{11} + \cdots + 469419709536 ) / 15841193328 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 2831687 \nu^{15} - 17128369 \nu^{14} + 46503947 \nu^{13} - 65927665 \nu^{12} + \cdots - 22619889624033 ) / 1562997741696 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 9615163 \nu^{15} + 66802685 \nu^{14} - 198110251 \nu^{13} + 258112553 \nu^{12} + \cdots + 99842531206617 ) / 4688993225088 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 819664 \nu^{15} - 3156725 \nu^{14} + 6821743 \nu^{13} - 1586774 \nu^{12} + \cdots + 302484525498 ) / 390749435424 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{14} - 2\beta_{13} + \beta_{11} - 2\beta_{10} + 2\beta_{9} - 3\beta_{3} - 4\beta_{2} + 2\beta_1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 2 \beta_{13} + 7 \beta_{11} - 6 \beta_{10} + 2 \beta_{9} + 6 \beta_{8} - 3 \beta_{4} + 6 \beta_{3} + \cdots - 3 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 3 \beta_{15} - 3 \beta_{14} + 2 \beta_{13} + 6 \beta_{12} - \beta_{11} + \beta_{9} + 6 \beta_{8} + \cdots - 9 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 36 \beta_{15} + 48 \beta_{14} - 2 \beta_{13} + 25 \beta_{11} + 24 \beta_{10} + 20 \beta_{9} - 30 \beta_{8} + \cdots + 33 ) / 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 12 \beta_{15} + 144 \beta_{14} + 184 \beta_{13} + 12 \beta_{12} - 137 \beta_{11} + 24 \beta_{10} + \cdots + 378 ) / 6 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 54 \beta_{15} - 48 \beta_{14} + 26 \beta_{13} + 8 \beta_{11} - 168 \beta_{10} - 26 \beta_{9} - 168 \beta_{8} + \cdots + 39 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 588 \beta_{15} + 18 \beta_{14} - 734 \beta_{13} + 84 \beta_{12} - 209 \beta_{11} - 30 \beta_{10} + \cdots + 720 ) / 6 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 612 \beta_{15} - 1824 \beta_{14} + 322 \beta_{13} + 2592 \beta_{12} + 1465 \beta_{11} - 498 \beta_{10} + \cdots - 2577 ) / 6 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 471 \beta_{15} + 315 \beta_{14} + 1172 \beta_{13} - 2082 \beta_{12} + 3869 \beta_{11} - 24 \beta_{10} + \cdots - 4653 ) / 3 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 21672 \beta_{15} + 5952 \beta_{14} + 2050 \beta_{13} + 11232 \beta_{12} + 34207 \beta_{11} + \cdots + 49227 ) / 6 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 25176 \beta_{15} - 18144 \beta_{14} - 76064 \beta_{13} - 23640 \beta_{12} + 67417 \beta_{11} + \cdots + 314550 ) / 6 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 76788 \beta_{15} - 72624 \beta_{14} - 177364 \beta_{13} - 62208 \beta_{12} + 58760 \beta_{11} + \cdots - 393513 ) / 3 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 360984 \beta_{15} + 274878 \beta_{14} - 51602 \beta_{13} - 58344 \beta_{12} + 52225 \beta_{11} + \cdots + 3369888 ) / 6 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 609624 \beta_{15} - 209184 \beta_{14} - 1117154 \beta_{13} + 489888 \beta_{12} - 2617049 \beta_{11} + \cdots + 2304357 ) / 6 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 2430393 \beta_{15} + 3589533 \beta_{14} - 644722 \beta_{13} + 310686 \beta_{12} + 2635775 \beta_{11} + \cdots - 5444829 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(451\) | \(701\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 701.1 |
|
− | 1.41421i | −2.85457 | + | 0.922735i | −2.00000 | 0 | 1.30494 | + | 4.03697i | −2.64575 | 2.82843i | 7.29712 | − | 5.26802i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 701.2 | − | 1.41421i | −2.18398 | − | 2.05675i | −2.00000 | 0 | −2.90869 | + | 3.08861i | −2.64575 | 2.82843i | 0.539533 | + | 8.98381i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 701.3 | − | 1.41421i | −0.650833 | + | 2.92855i | −2.00000 | 0 | 4.14160 | + | 0.920417i | 2.64575 | 2.82843i | −8.15283 | − | 3.81200i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 701.4 | − | 1.41421i | 0.812085 | + | 2.88800i | −2.00000 | 0 | 4.08424 | − | 1.14846i | 2.64575 | 2.82843i | −7.68104 | + | 4.69059i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 701.5 | − | 1.41421i | 1.51079 | − | 2.59182i | −2.00000 | 0 | −3.66538 | − | 2.13658i | 2.64575 | 2.82843i | −4.43502 | − | 7.83139i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 701.6 | − | 1.41421i | 1.86110 | + | 2.35293i | −2.00000 | 0 | 3.32755 | − | 2.63200i | −2.64575 | 2.82843i | −2.07259 | + | 8.75810i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 701.7 | − | 1.41421i | 2.53169 | + | 1.60951i | −2.00000 | 0 | 2.27619 | − | 3.58035i | −2.64575 | 2.82843i | 3.81894 | + | 8.14958i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 701.8 | − | 1.41421i | 2.97371 | − | 0.396304i | −2.00000 | 0 | −0.560459 | − | 4.20546i | 2.64575 | 2.82843i | 8.68589 | − | 2.35699i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 701.9 | 1.41421i | −2.85457 | − | 0.922735i | −2.00000 | 0 | 1.30494 | − | 4.03697i | −2.64575 | − | 2.82843i | 7.29712 | + | 5.26802i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 701.10 | 1.41421i | −2.18398 | + | 2.05675i | −2.00000 | 0 | −2.90869 | − | 3.08861i | −2.64575 | − | 2.82843i | 0.539533 | − | 8.98381i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 701.11 | 1.41421i | −0.650833 | − | 2.92855i | −2.00000 | 0 | 4.14160 | − | 0.920417i | 2.64575 | − | 2.82843i | −8.15283 | + | 3.81200i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 701.12 | 1.41421i | 0.812085 | − | 2.88800i | −2.00000 | 0 | 4.08424 | + | 1.14846i | 2.64575 | − | 2.82843i | −7.68104 | − | 4.69059i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 701.13 | 1.41421i | 1.51079 | + | 2.59182i | −2.00000 | 0 | −3.66538 | + | 2.13658i | 2.64575 | − | 2.82843i | −4.43502 | + | 7.83139i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 701.14 | 1.41421i | 1.86110 | − | 2.35293i | −2.00000 | 0 | 3.32755 | + | 2.63200i | −2.64575 | − | 2.82843i | −2.07259 | − | 8.75810i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 701.15 | 1.41421i | 2.53169 | − | 1.60951i | −2.00000 | 0 | 2.27619 | + | 3.58035i | −2.64575 | − | 2.82843i | 3.81894 | − | 8.14958i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 701.16 | 1.41421i | 2.97371 | + | 0.396304i | −2.00000 | 0 | −0.560459 | + | 4.20546i | 2.64575 | − | 2.82843i | 8.68589 | + | 2.35699i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1050.3.e.d | 16 | |
| 3.b | odd | 2 | 1 | inner | 1050.3.e.d | 16 | |
| 5.b | even | 2 | 1 | 210.3.e.a | ✓ | 16 | |
| 5.c | odd | 4 | 2 | 1050.3.c.c | 32 | ||
| 15.d | odd | 2 | 1 | 210.3.e.a | ✓ | 16 | |
| 15.e | even | 4 | 2 | 1050.3.c.c | 32 | ||
| 20.d | odd | 2 | 1 | 1680.3.l.c | 16 | ||
| 60.h | even | 2 | 1 | 1680.3.l.c | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 210.3.e.a | ✓ | 16 | 5.b | even | 2 | 1 | |
| 210.3.e.a | ✓ | 16 | 15.d | odd | 2 | 1 | |
| 1050.3.c.c | 32 | 5.c | odd | 4 | 2 | ||
| 1050.3.c.c | 32 | 15.e | even | 4 | 2 | ||
| 1050.3.e.d | 16 | 1.a | even | 1 | 1 | trivial | |
| 1050.3.e.d | 16 | 3.b | odd | 2 | 1 | inner | |
| 1680.3.l.c | 16 | 20.d | odd | 2 | 1 | ||
| 1680.3.l.c | 16 | 60.h | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1050, [\chi])\):
|
\( T_{11}^{16} + 1348 T_{11}^{14} + 718702 T_{11}^{12} + 193917500 T_{11}^{10} + 28186215185 T_{11}^{8} + \cdots + 33\!\cdots\!76 \)
|
|
\( T_{13}^{8} - 858 T_{13}^{6} + 456 T_{13}^{5} + 216597 T_{13}^{4} - 330024 T_{13}^{3} - 13327660 T_{13}^{2} + \cdots - 64451196 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2)^{8} \)
|
| $3$ |
\( T^{16} - 8 T^{15} + \cdots + 43046721 \)
|
| $5$ |
\( T^{16} \)
|
| $7$ |
\( (T^{2} - 7)^{8} \)
|
| $11$ |
\( T^{16} + \cdots + 33\!\cdots\!76 \)
|
| $13$ |
\( (T^{8} - 858 T^{6} + \cdots - 64451196)^{2} \)
|
| $17$ |
\( T^{16} + \cdots + 33\!\cdots\!96 \)
|
| $19$ |
\( (T^{8} - 24 T^{7} + \cdots + 1552481856)^{2} \)
|
| $23$ |
\( T^{16} + \cdots + 27\!\cdots\!16 \)
|
| $29$ |
\( T^{16} + \cdots + 26\!\cdots\!16 \)
|
| $31$ |
\( (T^{8} - 4004 T^{6} + \cdots + 212472592896)^{2} \)
|
| $37$ |
\( (T^{8} + 40 T^{7} + \cdots + 74680094976)^{2} \)
|
| $41$ |
\( T^{16} + \cdots + 37\!\cdots\!56 \)
|
| $43$ |
\( (T^{8} - 168 T^{7} + \cdots + 729854118144)^{2} \)
|
| $47$ |
\( T^{16} + \cdots + 47\!\cdots\!96 \)
|
| $53$ |
\( T^{16} + \cdots + 13\!\cdots\!76 \)
|
| $59$ |
\( T^{16} + \cdots + 21\!\cdots\!36 \)
|
| $61$ |
\( (T^{8} + \cdots - 28344025168896)^{2} \)
|
| $67$ |
\( (T^{8} + \cdots - 331910403784704)^{2} \)
|
| $71$ |
\( T^{16} + \cdots + 44\!\cdots\!56 \)
|
| $73$ |
\( (T^{8} + \cdots - 535924119269376)^{2} \)
|
| $79$ |
\( (T^{8} + \cdots + 148620892321344)^{2} \)
|
| $83$ |
\( T^{16} + \cdots + 53\!\cdots\!16 \)
|
| $89$ |
\( T^{16} + \cdots + 24\!\cdots\!56 \)
|
| $97$ |
\( (T^{8} + \cdots - 1371263608636)^{2} \)
|
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