Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [105,2,Mod(13,105)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(105, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("105.13");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 105 = 3 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 105.m (of order \(4\), 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: | \(0.838429221223\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 4x^{14} + 6x^{12} - 12x^{10} + 33x^{8} - 48x^{6} + 96x^{4} - 256x^{2} + 256 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{7} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{16} - 4x^{14} + 6x^{12} - 12x^{10} + 33x^{8} - 48x^{6} + 96x^{4} - 256x^{2} + 256 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{14} - 21\nu^{12} + 18\nu^{10} - 18\nu^{8} + 189\nu^{6} - 81\nu^{4} - 72\nu^{2} - 1072 ) / 480 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{14} - 6\nu^{10} - 12\nu^{8} - 81\nu^{6} + 108\nu^{4} - 48\nu^{2} + 448 ) / 384 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 17\nu^{14} - 72\nu^{12} + 6\nu^{10} - 36\nu^{8} + 513\nu^{6} - 1332\nu^{4} + 3696\nu^{2} - 4544 ) / 3840 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{15} - 3\nu^{13} + 24\nu^{11} + 6\nu^{9} - 78\nu^{7} + 117\nu^{5} - 216\nu^{3} - 16\nu ) / 960 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -29\nu^{15} + 144\nu^{13} + 18\nu^{11} + 132\nu^{9} - 621\nu^{7} + 204\nu^{5} - 1392\nu^{3} + 4928\nu ) / 7680 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -43\nu^{14} + 48\nu^{12} + 126\nu^{10} - 36\nu^{8} - 27\nu^{6} - 492\nu^{4} - 144\nu^{2} + 1216 ) / 3840 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -13\nu^{14} + 18\nu^{12} - 54\nu^{10} + 144\nu^{8} - 117\nu^{6} + 558\nu^{4} - 624\nu^{2} + 736 ) / 960 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( \nu^{15} - 4\nu^{13} + 6\nu^{11} - 12\nu^{9} + 33\nu^{7} - 48\nu^{5} + 96\nu^{3} - 128\nu ) / 128 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( -43\nu^{14} + 108\nu^{12} - 114\nu^{10} + 324\nu^{8} - 747\nu^{6} + 528\nu^{4} - 3024\nu^{2} + 6016 ) / 1920 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 149\nu^{15} - 264\nu^{13} + 222\nu^{11} - 1332\nu^{9} + 2181\nu^{7} - 1764\nu^{5} + 8112\nu^{3} - 14528\nu ) / 7680 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( -11\nu^{14} + 24\nu^{12} - 18\nu^{10} + 108\nu^{8} - 219\nu^{6} + 156\nu^{4} - 864\nu^{2} + 1472 ) / 192 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 105 \nu^{15} + 274 \nu^{14} - 360 \nu^{13} - 384 \nu^{12} + 150 \nu^{11} + 492 \nu^{10} + \cdots - 28288 ) / 7680 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 56 \nu^{15} + 137 \nu^{14} - 96 \nu^{13} - 192 \nu^{12} + 48 \nu^{11} + 246 \nu^{10} - 288 \nu^{9} + \cdots - 14144 ) / 3840 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 94 \nu^{15} - 137 \nu^{14} + 204 \nu^{13} + 192 \nu^{12} - 132 \nu^{11} - 246 \nu^{10} + \cdots + 14144 ) / 3840 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 109\nu^{15} - 224\nu^{13} + 142\nu^{11} - 932\nu^{9} + 1661\nu^{7} - 1244\nu^{5} + 8432\nu^{3} - 13888\nu ) / 2560 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{14} + \beta_{13} + \beta_{8} - \beta_{4} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{9} + \beta_{7} + \beta_{6} + \beta_{3} - \beta _1 + 1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{15} + \beta_{14} + \beta_{13} - 4\beta_{10} + \beta_{8} + 2\beta_{5} - \beta_{4} ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -3\beta_{9} + 3\beta_{7} + \beta_{6} - 3\beta_{3} - \beta _1 + 1 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 3 \beta_{14} + 3 \beta_{13} + 4 \beta_{12} + 2 \beta_{11} - \beta_{8} + 2 \beta_{7} + 4 \beta_{5} + \cdots + 2 \beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -2\beta_{11} + 3\beta_{9} + 3\beta_{7} + \beta_{6} - 3\beta_{3} - 6\beta_{2} + \beta _1 + 9 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 6 \beta_{15} + \beta_{14} + 5 \beta_{13} + 8 \beta_{12} + 6 \beta_{11} + 4 \beta_{10} + \cdots + 6 \beta_1 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 6\beta_{11} - 15\beta_{9} + 5\beta_{7} - 3\beta_{6} + 5\beta_{3} - 10\beta_{2} - 3\beta _1 + 9 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 12 \beta_{15} - 21 \beta_{14} + 27 \beta_{13} - 8 \beta_{12} + 20 \beta_{11} - 20 \beta_{10} + \cdots + 20 \beta_1 ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 12\beta_{11} - 45\beta_{9} + \beta_{7} + 21\beta_{6} - 19\beta_{3} - 12\beta_{2} - 9\beta _1 + 13 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 14 \beta_{15} - 15 \beta_{14} + 25 \beta_{13} + 20 \beta_{11} - 4 \beta_{10} + 25 \beta_{8} + \cdots + 20 \beta_1 ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( -12\beta_{11} + 13\beta_{9} + 7\beta_{7} + 17\beta_{6} - 43\beta_{3} - 60\beta_{2} - 37\beta _1 - 23 ) / 2 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 24 \beta_{15} - 37 \beta_{14} + 11 \beta_{13} + 12 \beta_{12} + 30 \beta_{11} + 24 \beta_{10} + \cdots + 30 \beta_1 ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 18\beta_{11} - 69\beta_{9} - 33\beta_{7} - 111\beta_{6} - 75\beta_{3} - 90\beta_{2} - 51\beta _1 + 41 ) / 2 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 150 \beta_{15} - 167 \beta_{14} + 229 \beta_{13} - 120 \beta_{12} + 138 \beta_{11} + 132 \beta_{10} + \cdots + 138 \beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).
| \(n\) | \(22\) | \(31\) | \(71\) |
| \(\chi(n)\) | \(\beta_{9}\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 13.1 |
|
−1.86147 | + | 1.86147i | −0.707107 | + | 0.707107i | − | 4.93012i | −1.50619 | − | 1.65269i | − | 2.63251i | −2.20563 | − | 1.46123i | 5.45433 | + | 5.45433i | − | 1.00000i | 5.88016 | + | 0.272713i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.2 | −1.86147 | + | 1.86147i | 0.707107 | − | 0.707107i | − | 4.93012i | 1.50619 | + | 1.65269i | 2.63251i | 1.46123 | + | 2.20563i | 5.45433 | + | 5.45433i | − | 1.00000i | −5.88016 | − | 0.272713i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.3 | −0.167056 | + | 0.167056i | −0.707107 | + | 0.707107i | 1.94418i | −2.23450 | + | 0.0836010i | − | 0.236253i | −0.0627175 | + | 2.64501i | −0.658899 | − | 0.658899i | − | 1.00000i | 0.359321 | − | 0.387253i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.4 | −0.167056 | + | 0.167056i | 0.707107 | − | 0.707107i | 1.94418i | 2.23450 | − | 0.0836010i | 0.236253i | −2.64501 | + | 0.0627175i | −0.658899 | − | 0.658899i | − | 1.00000i | −0.359321 | + | 0.387253i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.5 | 0.540143 | − | 0.540143i | −0.707107 | + | 0.707107i | 1.41649i | 1.03649 | + | 1.98133i | 0.763878i | 0.614060 | − | 2.57351i | 1.84539 | + | 1.84539i | − | 1.00000i | 1.63006 | + | 0.510348i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.6 | 0.540143 | − | 0.540143i | 0.707107 | − | 0.707107i | 1.41649i | −1.03649 | − | 1.98133i | − | 0.763878i | 2.57351 | − | 0.614060i | 1.84539 | + | 1.84539i | − | 1.00000i | −1.63006 | − | 0.510348i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.7 | 1.48838 | − | 1.48838i | −0.707107 | + | 0.707107i | − | 2.43055i | 1.28999 | − | 1.82645i | 2.10489i | −1.75993 | + | 1.97552i | −0.640825 | − | 0.640825i | − | 1.00000i | −0.798469 | − | 4.63845i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.8 | 1.48838 | − | 1.48838i | 0.707107 | − | 0.707107i | − | 2.43055i | −1.28999 | + | 1.82645i | − | 2.10489i | −1.97552 | + | 1.75993i | −0.640825 | − | 0.640825i | − | 1.00000i | 0.798469 | + | 4.63845i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.1 | −1.86147 | − | 1.86147i | −0.707107 | − | 0.707107i | 4.93012i | −1.50619 | + | 1.65269i | 2.63251i | −2.20563 | + | 1.46123i | 5.45433 | − | 5.45433i | 1.00000i | 5.88016 | − | 0.272713i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.2 | −1.86147 | − | 1.86147i | 0.707107 | + | 0.707107i | 4.93012i | 1.50619 | − | 1.65269i | − | 2.63251i | 1.46123 | − | 2.20563i | 5.45433 | − | 5.45433i | 1.00000i | −5.88016 | + | 0.272713i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.3 | −0.167056 | − | 0.167056i | −0.707107 | − | 0.707107i | − | 1.94418i | −2.23450 | − | 0.0836010i | 0.236253i | −0.0627175 | − | 2.64501i | −0.658899 | + | 0.658899i | 1.00000i | 0.359321 | + | 0.387253i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.4 | −0.167056 | − | 0.167056i | 0.707107 | + | 0.707107i | − | 1.94418i | 2.23450 | + | 0.0836010i | − | 0.236253i | −2.64501 | − | 0.0627175i | −0.658899 | + | 0.658899i | 1.00000i | −0.359321 | − | 0.387253i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.5 | 0.540143 | + | 0.540143i | −0.707107 | − | 0.707107i | − | 1.41649i | 1.03649 | − | 1.98133i | − | 0.763878i | 0.614060 | + | 2.57351i | 1.84539 | − | 1.84539i | 1.00000i | 1.63006 | − | 0.510348i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.6 | 0.540143 | + | 0.540143i | 0.707107 | + | 0.707107i | − | 1.41649i | −1.03649 | + | 1.98133i | 0.763878i | 2.57351 | + | 0.614060i | 1.84539 | − | 1.84539i | 1.00000i | −1.63006 | + | 0.510348i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.7 | 1.48838 | + | 1.48838i | −0.707107 | − | 0.707107i | 2.43055i | 1.28999 | + | 1.82645i | − | 2.10489i | −1.75993 | − | 1.97552i | −0.640825 | + | 0.640825i | 1.00000i | −0.798469 | + | 4.63845i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.8 | 1.48838 | + | 1.48838i | 0.707107 | + | 0.707107i | 2.43055i | −1.28999 | − | 1.82645i | 2.10489i | −1.97552 | − | 1.75993i | −0.640825 | + | 0.640825i | 1.00000i | 0.798469 | − | 4.63845i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 35.f | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 105.2.m.a | ✓ | 16 |
| 3.b | odd | 2 | 1 | 315.2.p.e | 16 | ||
| 4.b | odd | 2 | 1 | 1680.2.cz.d | 16 | ||
| 5.b | even | 2 | 1 | 525.2.m.b | 16 | ||
| 5.c | odd | 4 | 1 | inner | 105.2.m.a | ✓ | 16 |
| 5.c | odd | 4 | 1 | 525.2.m.b | 16 | ||
| 7.b | odd | 2 | 1 | inner | 105.2.m.a | ✓ | 16 |
| 7.c | even | 3 | 2 | 735.2.v.a | 32 | ||
| 7.d | odd | 6 | 2 | 735.2.v.a | 32 | ||
| 15.e | even | 4 | 1 | 315.2.p.e | 16 | ||
| 20.e | even | 4 | 1 | 1680.2.cz.d | 16 | ||
| 21.c | even | 2 | 1 | 315.2.p.e | 16 | ||
| 28.d | even | 2 | 1 | 1680.2.cz.d | 16 | ||
| 35.c | odd | 2 | 1 | 525.2.m.b | 16 | ||
| 35.f | even | 4 | 1 | inner | 105.2.m.a | ✓ | 16 |
| 35.f | even | 4 | 1 | 525.2.m.b | 16 | ||
| 35.k | even | 12 | 2 | 735.2.v.a | 32 | ||
| 35.l | odd | 12 | 2 | 735.2.v.a | 32 | ||
| 105.k | odd | 4 | 1 | 315.2.p.e | 16 | ||
| 140.j | odd | 4 | 1 | 1680.2.cz.d | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 105.2.m.a | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 105.2.m.a | ✓ | 16 | 5.c | odd | 4 | 1 | inner |
| 105.2.m.a | ✓ | 16 | 7.b | odd | 2 | 1 | inner |
| 105.2.m.a | ✓ | 16 | 35.f | even | 4 | 1 | inner |
| 315.2.p.e | 16 | 3.b | odd | 2 | 1 | ||
| 315.2.p.e | 16 | 15.e | even | 4 | 1 | ||
| 315.2.p.e | 16 | 21.c | even | 2 | 1 | ||
| 315.2.p.e | 16 | 105.k | odd | 4 | 1 | ||
| 525.2.m.b | 16 | 5.b | even | 2 | 1 | ||
| 525.2.m.b | 16 | 5.c | odd | 4 | 1 | ||
| 525.2.m.b | 16 | 35.c | odd | 2 | 1 | ||
| 525.2.m.b | 16 | 35.f | even | 4 | 1 | ||
| 735.2.v.a | 32 | 7.c | even | 3 | 2 | ||
| 735.2.v.a | 32 | 7.d | odd | 6 | 2 | ||
| 735.2.v.a | 32 | 35.k | even | 12 | 2 | ||
| 735.2.v.a | 32 | 35.l | odd | 12 | 2 | ||
| 1680.2.cz.d | 16 | 4.b | odd | 2 | 1 | ||
| 1680.2.cz.d | 16 | 20.e | even | 4 | 1 | ||
| 1680.2.cz.d | 16 | 28.d | even | 2 | 1 | ||
| 1680.2.cz.d | 16 | 140.j | odd | 4 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(105, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{8} - 4 T^{5} + 34 T^{4} + \cdots + 1)^{2} \)
|
| $3$ |
\( (T^{4} + 1)^{4} \)
|
| $5$ |
\( T^{16} + 28 T^{12} + \cdots + 390625 \)
|
| $7$ |
\( T^{16} + 8 T^{15} + \cdots + 5764801 \)
|
| $11$ |
\( (T^{4} + 4 T^{3} - 12 T^{2} + \cdots - 60)^{4} \)
|
| $13$ |
\( T^{16} + 736 T^{12} + \cdots + 4096 \)
|
| $17$ |
\( T^{16} + \cdots + 100000000 \)
|
| $19$ |
\( (T^{8} - 104 T^{6} + \cdots + 144)^{2} \)
|
| $23$ |
\( (T^{8} + 20 T^{7} + \cdots + 64)^{2} \)
|
| $29$ |
\( (T^{8} + 48 T^{6} + \cdots + 256)^{2} \)
|
| $31$ |
\( (T^{8} + 120 T^{6} + \cdots + 274576)^{2} \)
|
| $37$ |
\( (T^{8} - 16 T^{7} + \cdots + 144)^{2} \)
|
| $41$ |
\( (T^{8} + 184 T^{6} + \cdots + 129600)^{2} \)
|
| $43$ |
\( (T^{8} + 8 T^{7} + \cdots + 4096)^{2} \)
|
| $47$ |
\( T^{16} + 25216 T^{12} + \cdots + 1048576 \)
|
| $53$ |
\( (T^{8} - 12 T^{7} + \cdots + 129600)^{2} \)
|
| $59$ |
\( (T^{8} - 160 T^{6} + \cdots + 1183744)^{2} \)
|
| $61$ |
\( (T^{8} + 288 T^{6} + \cdots + 640000)^{2} \)
|
| $67$ |
\( (T^{8} + 16 T^{7} + \cdots + 1024)^{2} \)
|
| $71$ |
\( (T^{4} - 16 T^{3} + \cdots - 1132)^{4} \)
|
| $73$ |
\( T^{16} + \cdots + 169459576016896 \)
|
| $79$ |
\( (T^{8} + 320 T^{6} + \cdots + 18939904)^{2} \)
|
| $83$ |
\( T^{16} + \cdots + 45137758519296 \)
|
| $89$ |
\( (T^{8} - 136 T^{6} + \cdots + 107584)^{2} \)
|
| $97$ |
\( T^{16} + \cdots + 60\!\cdots\!56 \)
|
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