Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1445,2,Mod(866,1445)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1445, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1445.866");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1445 = 5 \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1445.d (of order \(2\), degree \(1\), 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: | \(11.5383830921\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} + 18x^{10} + 83x^{8} + 152x^{6} + 111x^{4} + 22x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 85) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} + 18x^{10} + 83x^{8} + 152x^{6} + 111x^{4} + 22x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{10} + 16\nu^{8} + 50\nu^{6} + 36\nu^{4} - 13\nu^{2} - 2 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{11} + 15\nu^{9} + 34\nu^{7} - 16\nu^{5} - 75\nu^{3} - 27\nu ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{10} - 18\nu^{8} - 84\nu^{6} - 166\nu^{4} - 133\nu^{2} - 18 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{11} + 19\nu^{9} + 98\nu^{7} + 188\nu^{5} + 125\nu^{3} + 17\nu ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{10} + 19\nu^{8} + 98\nu^{6} + 188\nu^{4} + 125\nu^{2} + 13 ) / 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 2\nu^{11} + 37\nu^{9} + 182\nu^{7} + 354\nu^{5} + 258\nu^{3} + 31\nu ) / 4 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -3\nu^{10} - 52\nu^{8} - 216\nu^{6} - 338\nu^{4} - 183\nu^{2} - 8 ) / 4 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 4\nu^{10} + 65\nu^{8} + 218\nu^{6} + 222\nu^{4} + 36\nu^{2} - 1 ) / 4 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( -5\nu^{11} - 87\nu^{9} - 366\nu^{7} - 592\nu^{5} - 369\nu^{3} - 61\nu ) / 4 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( -7\nu^{11} - 124\nu^{9} - 546\nu^{7} - 916\nu^{5} - 553\nu^{3} - 42\nu ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{9} + \beta_{8} + \beta_{6} - 2\beta_{2} - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{11} - \beta_{10} + 2\beta_{7} + 3\beta_{5} + 2\beta_{3} - 9\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -12\beta_{9} - 9\beta_{8} - 10\beta_{6} - 3\beta_{4} + 28\beta_{2} + 12 \)
|
| \(\nu^{5}\) | \(=\) |
\( -28\beta_{11} + 12\beta_{10} - 32\beta_{7} - 41\beta_{5} - 31\beta_{3} + 102\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 145\beta_{9} + 102\beta_{8} + 115\beta_{6} + 41\beta_{4} - 348\beta_{2} - 123 \)
|
| \(\nu^{7}\) | \(=\) |
\( 348\beta_{11} - 145\beta_{10} + 408\beta_{7} + 504\beta_{5} + 391\beta_{3} - 1222\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( -1758\beta_{9} - 1222\beta_{8} - 1378\beta_{6} - 504\beta_{4} + 4240\beta_{2} + 1447 \)
|
| \(\nu^{9}\) | \(=\) |
\( -4240\beta_{11} + 1758\beta_{10} - 4996\beta_{7} - 6122\beta_{5} - 4776\beta_{3} + 14789\beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( 21323\beta_{9} + 14789\beta_{8} + 16671\beta_{6} + 6122\beta_{4} - 51470\beta_{2} - 17458 \)
|
| \(\nu^{11}\) | \(=\) |
\( 51470\beta_{11} - 21323\beta_{10} + 60706\beta_{7} + 74263\beta_{5} + 58004\beta_{3} - 179303\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1445\mathbb{Z}\right)^\times\).
| \(n\) | \(581\) | \(1157\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 866.1 |
|
−2.51230 | − | 1.25468i | 4.31167 | − | 1.00000i | 3.15213i | 1.61485i | −5.80761 | 1.42578 | 2.51230i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 866.2 | −2.51230 | 1.25468i | 4.31167 | 1.00000i | − | 3.15213i | − | 1.61485i | −5.80761 | 1.42578 | − | 2.51230i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 866.3 | −2.07061 | − | 2.52346i | 2.28744 | − | 1.00000i | 5.22511i | 0.368961i | −0.595174 | −3.36786 | 2.07061i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 866.4 | −2.07061 | 2.52346i | 2.28744 | 1.00000i | − | 5.22511i | − | 0.368961i | −0.595174 | −3.36786 | − | 2.07061i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 866.5 | −1.12708 | − | 2.48265i | −0.729699 | 1.00000i | 2.79814i | − | 2.44256i | 3.07658 | −3.16356 | − | 1.12708i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 866.6 | −1.12708 | 2.48265i | −0.729699 | − | 1.00000i | − | 2.79814i | 2.44256i | 3.07658 | −3.16356 | 1.12708i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 866.7 | 0.677603 | − | 2.35757i | −1.54085 | − | 1.00000i | − | 1.59750i | 4.27746i | −2.39929 | −2.55814 | − | 0.677603i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 866.8 | 0.677603 | 2.35757i | −1.54085 | 1.00000i | 1.59750i | − | 4.27746i | −2.39929 | −2.55814 | 0.677603i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 866.9 | 0.783476 | − | 0.544977i | −1.38617 | − | 1.00000i | − | 0.426976i | 1.18848i | −2.65298 | 2.70300 | − | 0.783476i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 866.10 | 0.783476 | 0.544977i | −1.38617 | 1.00000i | 0.426976i | − | 1.18848i | −2.65298 | 2.70300 | 0.783476i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 866.11 | 2.24891 | − | 0.198035i | 3.05761 | 1.00000i | − | 0.445364i | 1.89231i | 2.37848 | 2.96078 | 2.24891i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 866.12 | 2.24891 | 0.198035i | 3.05761 | − | 1.00000i | 0.445364i | − | 1.89231i | 2.37848 | 2.96078 | − | 2.24891i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1445.2.d.g | 12 | |
| 17.b | even | 2 | 1 | inner | 1445.2.d.g | 12 | |
| 17.c | even | 4 | 1 | 1445.2.a.n | 6 | ||
| 17.c | even | 4 | 1 | 1445.2.a.o | 6 | ||
| 17.d | even | 8 | 2 | 85.2.e.a | ✓ | 12 | |
| 51.g | odd | 8 | 2 | 765.2.k.b | 12 | ||
| 68.g | odd | 8 | 2 | 1360.2.bt.d | 12 | ||
| 85.j | even | 4 | 1 | 7225.2.a.z | 6 | ||
| 85.j | even | 4 | 1 | 7225.2.a.bb | 6 | ||
| 85.k | odd | 8 | 2 | 425.2.j.c | 12 | ||
| 85.m | even | 8 | 2 | 425.2.e.f | 12 | ||
| 85.n | odd | 8 | 2 | 425.2.j.b | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 85.2.e.a | ✓ | 12 | 17.d | even | 8 | 2 | |
| 425.2.e.f | 12 | 85.m | even | 8 | 2 | ||
| 425.2.j.b | 12 | 85.n | odd | 8 | 2 | ||
| 425.2.j.c | 12 | 85.k | odd | 8 | 2 | ||
| 765.2.k.b | 12 | 51.g | odd | 8 | 2 | ||
| 1360.2.bt.d | 12 | 68.g | odd | 8 | 2 | ||
| 1445.2.a.n | 6 | 17.c | even | 4 | 1 | ||
| 1445.2.a.o | 6 | 17.c | even | 4 | 1 | ||
| 1445.2.d.g | 12 | 1.a | even | 1 | 1 | trivial | |
| 1445.2.d.g | 12 | 17.b | even | 2 | 1 | inner | |
| 7225.2.a.z | 6 | 85.j | even | 4 | 1 | ||
| 7225.2.a.bb | 6 | 85.j | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1445, [\chi])\):
|
\( T_{2}^{6} + 2T_{2}^{5} - 7T_{2}^{4} - 12T_{2}^{3} + 11T_{2}^{2} + 10T_{2} - 7 \)
|
|
\( T_{3}^{12} + 20T_{3}^{10} + 144T_{3}^{8} + 436T_{3}^{6} + 476T_{3}^{4} + 120T_{3}^{2} + 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{6} + 2 T^{5} - 7 T^{4} + \cdots - 7)^{2} \)
|
| $3$ |
\( T^{12} + 20 T^{10} + \cdots + 4 \)
|
| $5$ |
\( (T^{2} + 1)^{6} \)
|
| $7$ |
\( T^{12} + 32 T^{10} + \cdots + 196 \)
|
| $11$ |
\( T^{12} + 52 T^{10} + \cdots + 198916 \)
|
| $13$ |
\( (T^{6} - 58 T^{4} + \cdots - 316)^{2} \)
|
| $17$ |
\( T^{12} \)
|
| $19$ |
\( (T^{6} - 12 T^{5} + \cdots - 1472)^{2} \)
|
| $23$ |
\( T^{12} + 84 T^{10} + \cdots + 196 \)
|
| $29$ |
\( T^{12} + 172 T^{10} + \cdots + 5345344 \)
|
| $31$ |
\( T^{12} + \cdots + 1645275844 \)
|
| $37$ |
\( T^{12} + 236 T^{10} + \cdots + 3655744 \)
|
| $41$ |
\( T^{12} + \cdots + 192876544 \)
|
| $43$ |
\( (T^{6} + 8 T^{5} + \cdots - 15004)^{2} \)
|
| $47$ |
\( (T^{6} - 24 T^{5} + \cdots - 18076)^{2} \)
|
| $53$ |
\( (T^{6} - 216 T^{4} + \cdots - 132112)^{2} \)
|
| $59$ |
\( (T^{6} - 16 T^{5} + \cdots + 896)^{2} \)
|
| $61$ |
\( T^{12} + 184 T^{10} + \cdots + 984064 \)
|
| $67$ |
\( (T^{6} + 4 T^{5} - 46 T^{4} + \cdots - 92)^{2} \)
|
| $71$ |
\( T^{12} + \cdots + 29133710596 \)
|
| $73$ |
\( T^{12} + 416 T^{10} + \cdots + 541696 \)
|
| $79$ |
\( T^{12} + 424 T^{10} + \cdots + 15225604 \)
|
| $83$ |
\( (T^{6} + 20 T^{5} + \cdots + 11204)^{2} \)
|
| $89$ |
\( (T^{6} + 12 T^{5} + \cdots - 31292)^{2} \)
|
| $97$ |
\( T^{12} + \cdots + 227086559296 \)
|
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