Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [105,3,Mod(31,105)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(105, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("105.31");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 105 = 3 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 105.n (of order \(6\), 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: | \(2.86104277578\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(6\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 6 x^{11} + 39 x^{10} - 140 x^{9} + 456 x^{8} - 1050 x^{7} + 1999 x^{6} - 2844 x^{5} + 2949 x^{4} + \cdots + 36 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{2}\cdot 7 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 6 x^{11} + 39 x^{10} - 140 x^{9} + 456 x^{8} - 1050 x^{7} + 1999 x^{6} - 2844 x^{5} + 2949 x^{4} + \cdots + 36 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{8} - 4\nu^{7} + 24\nu^{6} - 58\nu^{5} + 154\nu^{4} - 216\nu^{3} + 267\nu^{2} - 168\nu + 60 ) / 6 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3 \nu^{10} - 15 \nu^{9} + 101 \nu^{8} - 314 \nu^{7} + 1020 \nu^{6} - 2024 \nu^{5} + 3629 \nu^{4} + \cdots + 342 ) / 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 206 \nu^{11} - 1133 \nu^{10} + 7471 \nu^{9} - 25122 \nu^{8} + 81494 \nu^{7} - 175924 \nu^{6} + \cdots - 15420 ) / 168 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 272 \nu^{11} - 1489 \nu^{10} + 9825 \nu^{9} - 32912 \nu^{8} + 106722 \nu^{7} - 229488 \nu^{6} + \cdots - 19920 ) / 168 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 272 \nu^{11} - 1503 \nu^{10} + 9895 \nu^{9} - 33388 \nu^{8} + 108206 \nu^{7} - 234304 \nu^{6} + \cdots - 19920 ) / 168 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 87 \nu^{11} + 468 \nu^{10} - 3096 \nu^{9} + 10226 \nu^{8} - 33110 \nu^{7} + 70140 \nu^{6} + \cdots + 4542 ) / 42 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 708 \nu^{11} + 3887 \nu^{10} - 25609 \nu^{9} + 85962 \nu^{8} - 278334 \nu^{7} + 599452 \nu^{6} + \cdots + 47532 ) / 168 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 432 \nu^{11} - 2355 \nu^{10} + 15562 \nu^{9} - 51971 \nu^{8} + 168658 \nu^{7} - 361606 \nu^{6} + \cdots - 28878 ) / 84 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 852 \nu^{11} + 4735 \nu^{10} - 31123 \nu^{9} + 105458 \nu^{8} - 341502 \nu^{7} + 742336 \nu^{6} + \cdots + 65460 ) / 168 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 471 \nu^{11} - 2601 \nu^{10} + 17121 \nu^{9} - 57733 \nu^{8} + 187012 \nu^{7} - 404628 \nu^{6} + \cdots - 34290 ) / 84 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 575 \nu^{11} + 3159 \nu^{10} - 20828 \nu^{9} + 69974 \nu^{8} - 226856 \nu^{7} + 489230 \nu^{6} + \cdots + 41988 ) / 84 \)
|
| \(\nu\) | \(=\) |
\( ( 5 \beta_{11} - 28 \beta_{10} - 9 \beta_{9} + 6 \beta_{8} - 12 \beta_{7} - 3 \beta_{6} + 15 \beta_{5} + \cdots + 22 ) / 42 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 3 \beta_{11} - 28 \beta_{10} + 11 \beta_{9} + 2 \beta_{8} - 32 \beta_{7} + 13 \beta_{6} + 47 \beta_{5} + \cdots - 156 ) / 42 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 23 \beta_{11} + 56 \beta_{10} + 33 \beta_{9} - 22 \beta_{8} + 16 \beta_{7} + 11 \beta_{6} + \cdots - 76 ) / 14 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 69 \beta_{11} + 364 \beta_{10} + \beta_{9} - 122 \beta_{8} + 272 \beta_{7} - 37 \beta_{6} + \cdots + 1032 ) / 42 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 501 \beta_{11} - 952 \beta_{10} - 829 \beta_{9} + 422 \beta_{8} - 116 \beta_{7} - 197 \beta_{6} + \cdots + 2574 ) / 42 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 375 \beta_{11} - 1260 \beta_{10} - 255 \beta_{9} + 548 \beta_{8} - 690 \beta_{7} - 29 \beta_{6} + \cdots - 2102 ) / 14 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 423 \beta_{11} + 574 \beta_{10} + 907 \beta_{9} - 248 \beta_{8} - 100 \beta_{7} + 71 \beta_{6} + \cdots - 3696 ) / 6 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 12423 \beta_{11} + 34580 \beta_{10} + 12455 \beta_{9} - 16918 \beta_{8} + 15160 \beta_{7} + \cdots + 31800 ) / 42 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 3907 \beta_{11} - 112 \beta_{10} - 14405 \beta_{9} - 1382 \beta_{8} + 5508 \beta_{7} + 1685 \beta_{6} + \cdots + 80608 ) / 14 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 114897 \beta_{11} - 289352 \beta_{10} - 149297 \beta_{9} + 148522 \beta_{8} - 106924 \beta_{7} + \cdots - 73356 ) / 42 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 18675 \beta_{11} - 271684 \beta_{10} + 235577 \beta_{9} + 204326 \beta_{8} - 210020 \beta_{7} + \cdots - 2125518 ) / 42 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).
| \(n\) | \(22\) | \(31\) | \(71\) |
| \(\chi(n)\) | \(1\) | \(1 + \beta_{3}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 31.1 |
|
−1.99068 | − | 3.44796i | 1.50000 | + | 0.866025i | −5.92561 | + | 10.2635i | −1.93649 | + | 1.11803i | − | 6.89592i | 2.28451 | + | 6.61672i | 31.2586 | 1.50000 | + | 2.59808i | 7.70987 | + | 4.45130i | |||||||||||||||||||||||||||||||||||||||
| 31.2 | −1.25588 | − | 2.17524i | 1.50000 | + | 0.866025i | −1.15446 | + | 1.99958i | 1.93649 | − | 1.11803i | − | 4.35049i | 6.75110 | − | 1.85004i | −4.24760 | 1.50000 | + | 2.59808i | −4.86399 | − | 2.80823i | ||||||||||||||||||||||||||||||||||||||||
| 31.3 | 0.288563 | + | 0.499806i | 1.50000 | + | 0.866025i | 1.83346 | − | 3.17565i | 1.93649 | − | 1.11803i | 0.999611i | −1.64406 | − | 6.80420i | 4.42478 | 1.50000 | + | 2.59808i | 1.11760 | + | 0.645246i | |||||||||||||||||||||||||||||||||||||||||
| 31.4 | 0.687692 | + | 1.19112i | 1.50000 | + | 0.866025i | 1.05416 | − | 1.82586i | −1.93649 | + | 1.11803i | 2.38224i | 6.56639 | + | 2.42539i | 8.40129 | 1.50000 | + | 2.59808i | −2.66342 | − | 1.53773i | |||||||||||||||||||||||||||||||||||||||||
| 31.5 | 1.46731 | + | 2.54146i | 1.50000 | + | 0.866025i | −2.30602 | + | 3.99415i | 1.93649 | − | 1.11803i | 5.08293i | −5.41652 | + | 4.43411i | −1.79613 | 1.50000 | + | 2.59808i | 5.68288 | + | 3.28101i | |||||||||||||||||||||||||||||||||||||||||
| 31.6 | 1.80299 | + | 3.12287i | 1.50000 | + | 0.866025i | −4.50153 | + | 7.79688i | −1.93649 | + | 1.11803i | 6.24573i | 2.45857 | − | 6.55404i | −18.0409 | 1.50000 | + | 2.59808i | −6.98294 | − | 4.03160i | |||||||||||||||||||||||||||||||||||||||||
| 61.1 | −1.99068 | + | 3.44796i | 1.50000 | − | 0.866025i | −5.92561 | − | 10.2635i | −1.93649 | − | 1.11803i | 6.89592i | 2.28451 | − | 6.61672i | 31.2586 | 1.50000 | − | 2.59808i | 7.70987 | − | 4.45130i | |||||||||||||||||||||||||||||||||||||||||
| 61.2 | −1.25588 | + | 2.17524i | 1.50000 | − | 0.866025i | −1.15446 | − | 1.99958i | 1.93649 | + | 1.11803i | 4.35049i | 6.75110 | + | 1.85004i | −4.24760 | 1.50000 | − | 2.59808i | −4.86399 | + | 2.80823i | |||||||||||||||||||||||||||||||||||||||||
| 61.3 | 0.288563 | − | 0.499806i | 1.50000 | − | 0.866025i | 1.83346 | + | 3.17565i | 1.93649 | + | 1.11803i | − | 0.999611i | −1.64406 | + | 6.80420i | 4.42478 | 1.50000 | − | 2.59808i | 1.11760 | − | 0.645246i | ||||||||||||||||||||||||||||||||||||||||
| 61.4 | 0.687692 | − | 1.19112i | 1.50000 | − | 0.866025i | 1.05416 | + | 1.82586i | −1.93649 | − | 1.11803i | − | 2.38224i | 6.56639 | − | 2.42539i | 8.40129 | 1.50000 | − | 2.59808i | −2.66342 | + | 1.53773i | ||||||||||||||||||||||||||||||||||||||||
| 61.5 | 1.46731 | − | 2.54146i | 1.50000 | − | 0.866025i | −2.30602 | − | 3.99415i | 1.93649 | + | 1.11803i | − | 5.08293i | −5.41652 | − | 4.43411i | −1.79613 | 1.50000 | − | 2.59808i | 5.68288 | − | 3.28101i | ||||||||||||||||||||||||||||||||||||||||
| 61.6 | 1.80299 | − | 3.12287i | 1.50000 | − | 0.866025i | −4.50153 | − | 7.79688i | −1.93649 | − | 1.11803i | − | 6.24573i | 2.45857 | + | 6.55404i | −18.0409 | 1.50000 | − | 2.59808i | −6.98294 | + | 4.03160i | ||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 105.3.n.b | ✓ | 12 |
| 3.b | odd | 2 | 1 | 315.3.w.b | 12 | ||
| 5.b | even | 2 | 1 | 525.3.o.m | 12 | ||
| 5.c | odd | 4 | 2 | 525.3.s.j | 24 | ||
| 7.c | even | 3 | 1 | 735.3.h.b | 12 | ||
| 7.d | odd | 6 | 1 | inner | 105.3.n.b | ✓ | 12 |
| 7.d | odd | 6 | 1 | 735.3.h.b | 12 | ||
| 21.g | even | 6 | 1 | 315.3.w.b | 12 | ||
| 35.i | odd | 6 | 1 | 525.3.o.m | 12 | ||
| 35.k | even | 12 | 2 | 525.3.s.j | 24 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 105.3.n.b | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 105.3.n.b | ✓ | 12 | 7.d | odd | 6 | 1 | inner |
| 315.3.w.b | 12 | 3.b | odd | 2 | 1 | ||
| 315.3.w.b | 12 | 21.g | even | 6 | 1 | ||
| 525.3.o.m | 12 | 5.b | even | 2 | 1 | ||
| 525.3.o.m | 12 | 35.i | odd | 6 | 1 | ||
| 525.3.s.j | 24 | 5.c | odd | 4 | 2 | ||
| 525.3.s.j | 24 | 35.k | even | 12 | 2 | ||
| 735.3.h.b | 12 | 7.c | even | 3 | 1 | ||
| 735.3.h.b | 12 | 7.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} - 2 T_{2}^{11} + 25 T_{2}^{10} - 50 T_{2}^{9} + 451 T_{2}^{8} - 842 T_{2}^{7} + 3598 T_{2}^{6} + \cdots + 7056 \)
acting on \(S_{3}^{\mathrm{new}}(105, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} - 2 T^{11} + \cdots + 7056 \)
|
| $3$ |
\( (T^{2} - 3 T + 3)^{6} \)
|
| $5$ |
\( (T^{4} - 5 T^{2} + 25)^{3} \)
|
| $7$ |
\( T^{12} + \cdots + 13841287201 \)
|
| $11$ |
\( T^{12} + \cdots + 15854839056 \)
|
| $13$ |
\( T^{12} + \cdots + 732611029329 \)
|
| $17$ |
\( T^{12} + \cdots + 93022784308224 \)
|
| $19$ |
\( T^{12} + \cdots + 1794623769 \)
|
| $23$ |
\( T^{12} + \cdots + 170135605308816 \)
|
| $29$ |
\( (T^{6} + 50 T^{5} + \cdots + 32868624)^{2} \)
|
| $31$ |
\( T^{12} + \cdots + 47\!\cdots\!44 \)
|
| $37$ |
\( T^{12} + \cdots + 10\!\cdots\!41 \)
|
| $41$ |
\( T^{12} + \cdots + 25\!\cdots\!04 \)
|
| $43$ |
\( (T^{6} + 62 T^{5} + \cdots - 961914584)^{2} \)
|
| $47$ |
\( T^{12} + \cdots + 27\!\cdots\!64 \)
|
| $53$ |
\( T^{12} + \cdots + 796594176 \)
|
| $59$ |
\( T^{12} + \cdots + 48\!\cdots\!64 \)
|
| $61$ |
\( T^{12} + \cdots + 24\!\cdots\!00 \)
|
| $67$ |
\( T^{12} + \cdots + 48\!\cdots\!76 \)
|
| $71$ |
\( (T^{6} + 314 T^{5} + \cdots - 210165130176)^{2} \)
|
| $73$ |
\( T^{12} + \cdots + 21\!\cdots\!04 \)
|
| $79$ |
\( T^{12} + \cdots + 18\!\cdots\!76 \)
|
| $83$ |
\( T^{12} + \cdots + 11\!\cdots\!24 \)
|
| $89$ |
\( T^{12} + \cdots + 19\!\cdots\!44 \)
|
| $97$ |
\( T^{12} + \cdots + 38\!\cdots\!04 \)
|
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