Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [280,6,Mod(81,280)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(280, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("280.81");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 280 = 2^{3} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 280.q (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: | \(44.9074695476\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(10\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 2 x^{19} - 3021 x^{18} + 6594 x^{17} + 3763051 x^{16} - 7374124 x^{15} - 2500362614 x^{14} + \cdots + 53\!\cdots\!81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{32}\cdot 7^{3} \) |
| 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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} - 2 x^{19} - 3021 x^{18} + 6594 x^{17} + 3763051 x^{16} - 7374124 x^{15} - 2500362614 x^{14} + \cdots + 53\!\cdots\!81 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 81\!\cdots\!90 \nu^{19} + \cdots + 21\!\cdots\!36 ) / 95\!\cdots\!12 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 81\!\cdots\!90 \nu^{19} + \cdots - 14\!\cdots\!48 ) / 47\!\cdots\!56 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 13\!\cdots\!96 \nu^{19} + \cdots - 42\!\cdots\!19 ) / 95\!\cdots\!12 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 13\!\cdots\!86 \nu^{19} + \cdots - 21\!\cdots\!83 ) / 95\!\cdots\!12 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 14\!\cdots\!89 \nu^{19} + \cdots - 21\!\cdots\!58 ) / 54\!\cdots\!84 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 36\!\cdots\!16 \nu^{19} + \cdots + 22\!\cdots\!93 ) / 54\!\cdots\!84 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 46\!\cdots\!32 \nu^{19} + \cdots + 50\!\cdots\!68 ) / 60\!\cdots\!76 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 71\!\cdots\!40 \nu^{19} + \cdots + 80\!\cdots\!38 ) / 54\!\cdots\!84 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 25\!\cdots\!10 \nu^{19} + \cdots + 63\!\cdots\!71 ) / 18\!\cdots\!28 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 10\!\cdots\!81 \nu^{19} + \cdots + 98\!\cdots\!03 ) / 54\!\cdots\!84 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 11\!\cdots\!60 \nu^{19} + \cdots + 99\!\cdots\!04 ) / 54\!\cdots\!84 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 18\!\cdots\!83 \nu^{19} + \cdots - 11\!\cdots\!32 ) / 54\!\cdots\!84 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 62\!\cdots\!57 \nu^{19} + \cdots - 37\!\cdots\!61 ) / 18\!\cdots\!28 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 40\!\cdots\!46 \nu^{19} + \cdots + 16\!\cdots\!86 ) / 95\!\cdots\!12 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 26\!\cdots\!73 \nu^{19} + \cdots - 70\!\cdots\!38 ) / 54\!\cdots\!84 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 27\!\cdots\!52 \nu^{19} + \cdots + 17\!\cdots\!79 ) / 54\!\cdots\!84 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 33\!\cdots\!43 \nu^{19} + \cdots + 18\!\cdots\!47 ) / 54\!\cdots\!84 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 14\!\cdots\!18 \nu^{19} + \cdots - 19\!\cdots\!77 ) / 18\!\cdots\!28 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 52\!\cdots\!11 \nu^{19} + \cdots + 41\!\cdots\!61 ) / 54\!\cdots\!84 \)
|
| \(\nu\) | \(=\) |
\( \beta_{4} - \beta_{3} - \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} + \beta_{2} + \beta _1 + 302 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{19} + \beta_{18} + \beta_{15} + 3 \beta_{14} - 3 \beta_{12} + \beta_{10} - 6 \beta_{9} + \cdots - 331 \)
|
| \(\nu^{4}\) | \(=\) |
\( - \beta_{19} - 12 \beta_{18} + 19 \beta_{17} + 4 \beta_{16} - 8 \beta_{15} - 18 \beta_{14} + \cdots + 159245 \)
|
| \(\nu^{5}\) | \(=\) |
\( 1170 \beta_{19} + 1210 \beta_{18} - 103 \beta_{17} + 5 \beta_{16} + 1160 \beta_{15} + 3255 \beta_{14} + \cdots - 735230 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 8971 \beta_{19} - 15274 \beta_{18} + 12090 \beta_{17} + 7065 \beta_{16} - 8134 \beta_{15} + \cdots + 96119410 \)
|
| \(\nu^{7}\) | \(=\) |
\( 1078844 \beta_{19} + 1062445 \beta_{18} + 33887 \beta_{17} - 38017 \beta_{16} + 979992 \beta_{15} + \cdots - 883814186 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 13045895 \beta_{19} - 15330662 \beta_{18} + 3390563 \beta_{17} + 8577678 \beta_{16} + \cdots + 63083513140 \)
|
| \(\nu^{9}\) | \(=\) |
\( 900376314 \beta_{19} + 849348804 \beta_{18} + 158907189 \beta_{17} - 79497621 \beta_{16} + \cdots - 866081512030 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 13970923419 \beta_{19} - 14154666798 \beta_{18} - 2367957030 \beta_{17} + 8799718425 \beta_{16} + \cdots + 43935719232264 \)
|
| \(\nu^{11}\) | \(=\) |
\( 717472471363 \beta_{19} + 658117282378 \beta_{18} + 228326102349 \beta_{17} - 107471653047 \beta_{16} + \cdots - 774625996195913 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 13166351664626 \beta_{19} - 12467617589848 \beta_{18} - 5479659511292 \beta_{17} + \cdots + 31\!\cdots\!53 \)
|
| \(\nu^{13}\) | \(=\) |
\( 559941349017056 \beta_{19} + 505136155284152 \beta_{18} + 255197621619896 \beta_{17} + \cdots - 66\!\cdots\!26 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 11\!\cdots\!86 \beta_{19} + \cdots + 23\!\cdots\!68 \)
|
| \(\nu^{15}\) | \(=\) |
\( 43\!\cdots\!77 \beta_{19} + \cdots - 54\!\cdots\!77 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 98\!\cdots\!49 \beta_{19} + \cdots + 18\!\cdots\!95 \)
|
| \(\nu^{17}\) | \(=\) |
\( 33\!\cdots\!66 \beta_{19} + \cdots - 44\!\cdots\!46 \)
|
| \(\nu^{18}\) | \(=\) |
\( - 80\!\cdots\!45 \beta_{19} + \cdots + 13\!\cdots\!62 \)
|
| \(\nu^{19}\) | \(=\) |
\( 25\!\cdots\!72 \beta_{19} + \cdots - 35\!\cdots\!08 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/280\mathbb{Z}\right)^\times\).
| \(n\) | \(57\) | \(71\) | \(141\) | \(241\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-1 - \beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 81.1 |
|
0 | −13.2839 | + | 23.0084i | 0 | −12.5000 | − | 21.6506i | 0 | 59.4418 | − | 115.211i | 0 | −231.425 | − | 400.840i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 81.2 | 0 | −9.65029 | + | 16.7148i | 0 | −12.5000 | − | 21.6506i | 0 | −106.729 | + | 73.5928i | 0 | −64.7563 | − | 112.161i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 81.3 | 0 | −6.11444 | + | 10.5905i | 0 | −12.5000 | − | 21.6506i | 0 | 114.561 | + | 60.6851i | 0 | 46.7272 | + | 80.9339i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 81.4 | 0 | −4.01228 | + | 6.94947i | 0 | −12.5000 | − | 21.6506i | 0 | −106.704 | − | 73.6285i | 0 | 89.3032 | + | 154.678i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 81.5 | 0 | 0.377212 | − | 0.653350i | 0 | −12.5000 | − | 21.6506i | 0 | −11.4635 | − | 129.134i | 0 | 121.215 | + | 209.951i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 81.6 | 0 | 1.14588 | − | 1.98473i | 0 | −12.5000 | − | 21.6506i | 0 | 118.402 | + | 52.8023i | 0 | 118.874 | + | 205.896i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 81.7 | 0 | 4.00560 | − | 6.93791i | 0 | −12.5000 | − | 21.6506i | 0 | −92.4067 | + | 90.9286i | 0 | 89.4103 | + | 154.863i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 81.8 | 0 | 10.8854 | − | 18.8540i | 0 | −12.5000 | − | 21.6506i | 0 | 102.778 | − | 79.0166i | 0 | −115.483 | − | 200.023i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 81.9 | 0 | 11.5492 | − | 20.0038i | 0 | −12.5000 | − | 21.6506i | 0 | 10.7115 | + | 129.199i | 0 | −145.268 | − | 251.611i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 81.10 | 0 | 13.0977 | − | 22.6858i | 0 | −12.5000 | − | 21.6506i | 0 | −20.5908 | − | 127.996i | 0 | −221.598 | − | 383.819i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 121.1 | 0 | −13.2839 | − | 23.0084i | 0 | −12.5000 | + | 21.6506i | 0 | 59.4418 | + | 115.211i | 0 | −231.425 | + | 400.840i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 121.2 | 0 | −9.65029 | − | 16.7148i | 0 | −12.5000 | + | 21.6506i | 0 | −106.729 | − | 73.5928i | 0 | −64.7563 | + | 112.161i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 121.3 | 0 | −6.11444 | − | 10.5905i | 0 | −12.5000 | + | 21.6506i | 0 | 114.561 | − | 60.6851i | 0 | 46.7272 | − | 80.9339i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 121.4 | 0 | −4.01228 | − | 6.94947i | 0 | −12.5000 | + | 21.6506i | 0 | −106.704 | + | 73.6285i | 0 | 89.3032 | − | 154.678i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 121.5 | 0 | 0.377212 | + | 0.653350i | 0 | −12.5000 | + | 21.6506i | 0 | −11.4635 | + | 129.134i | 0 | 121.215 | − | 209.951i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 121.6 | 0 | 1.14588 | + | 1.98473i | 0 | −12.5000 | + | 21.6506i | 0 | 118.402 | − | 52.8023i | 0 | 118.874 | − | 205.896i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 121.7 | 0 | 4.00560 | + | 6.93791i | 0 | −12.5000 | + | 21.6506i | 0 | −92.4067 | − | 90.9286i | 0 | 89.4103 | − | 154.863i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 121.8 | 0 | 10.8854 | + | 18.8540i | 0 | −12.5000 | + | 21.6506i | 0 | 102.778 | + | 79.0166i | 0 | −115.483 | + | 200.023i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 121.9 | 0 | 11.5492 | + | 20.0038i | 0 | −12.5000 | + | 21.6506i | 0 | 10.7115 | − | 129.199i | 0 | −145.268 | + | 251.611i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 121.10 | 0 | 13.0977 | + | 22.6858i | 0 | −12.5000 | + | 21.6506i | 0 | −20.5908 | + | 127.996i | 0 | −221.598 | + | 383.819i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 280.6.q.c | ✓ | 20 |
| 7.c | even | 3 | 1 | inner | 280.6.q.c | ✓ | 20 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 280.6.q.c | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 280.6.q.c | ✓ | 20 | 7.c | even | 3 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{20} - 16 T_{3}^{19} + 1656 T_{3}^{18} - 16560 T_{3}^{17} + 1640530 T_{3}^{16} + \cdots + 84\!\cdots\!16 \)
acting on \(S_{6}^{\mathrm{new}}(280, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} \)
|
| $3$ |
\( T^{20} + \cdots + 84\!\cdots\!16 \)
|
| $5$ |
\( (T^{2} + 25 T + 625)^{10} \)
|
| $7$ |
\( T^{20} + \cdots + 17\!\cdots\!49 \)
|
| $11$ |
\( T^{20} + \cdots + 88\!\cdots\!96 \)
|
| $13$ |
\( (T^{10} + \cdots + 66\!\cdots\!48)^{2} \)
|
| $17$ |
\( T^{20} + \cdots + 13\!\cdots\!00 \)
|
| $19$ |
\( T^{20} + \cdots + 31\!\cdots\!00 \)
|
| $23$ |
\( T^{20} + \cdots + 30\!\cdots\!61 \)
|
| $29$ |
\( (T^{10} + \cdots - 25\!\cdots\!64)^{2} \)
|
| $31$ |
\( T^{20} + \cdots + 52\!\cdots\!00 \)
|
| $37$ |
\( T^{20} + \cdots + 89\!\cdots\!00 \)
|
| $41$ |
\( (T^{10} + \cdots + 31\!\cdots\!45)^{2} \)
|
| $43$ |
\( (T^{10} + \cdots + 87\!\cdots\!96)^{2} \)
|
| $47$ |
\( T^{20} + \cdots + 18\!\cdots\!00 \)
|
| $53$ |
\( T^{20} + \cdots + 59\!\cdots\!44 \)
|
| $59$ |
\( T^{20} + \cdots + 13\!\cdots\!00 \)
|
| $61$ |
\( T^{20} + \cdots + 76\!\cdots\!00 \)
|
| $67$ |
\( T^{20} + \cdots + 21\!\cdots\!44 \)
|
| $71$ |
\( (T^{10} + \cdots - 97\!\cdots\!40)^{2} \)
|
| $73$ |
\( T^{20} + \cdots + 55\!\cdots\!36 \)
|
| $79$ |
\( T^{20} + \cdots + 16\!\cdots\!84 \)
|
| $83$ |
\( (T^{10} + \cdots + 14\!\cdots\!96)^{2} \)
|
| $89$ |
\( T^{20} + \cdots + 10\!\cdots\!44 \)
|
| $97$ |
\( (T^{10} + \cdots + 70\!\cdots\!96)^{2} \)
|
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