Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [867,2,Mod(688,867)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(867, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([0, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("867.688");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 867 = 3 \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 867.h (of order \(8\), degree \(4\), 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: | \(6.92302985525\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{8})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 8x^{14} + 32x^{12} + 296x^{10} + 1057x^{8} + 1184x^{6} + 512x^{4} - 512x^{2} + 256 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 51) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{8}]$ |
$q$-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
\( x^{16} - 8x^{14} + 32x^{12} + 296x^{10} + 1057x^{8} + 1184x^{6} + 512x^{4} - 512x^{2} + 256 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 281 \nu^{15} - 15084 \nu^{13} + 105664 \nu^{11} - 271768 \nu^{9} - 3795687 \nu^{7} + \cdots - 11028992 \nu ) / 5224576 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 405 \nu^{14} + 5580 \nu^{12} - 31872 \nu^{10} - 51240 \nu^{8} + 338891 \nu^{6} + 1469028 \nu^{4} + \cdots - 2209888 ) / 1306144 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 21\nu^{14} - 159\nu^{12} + 650\nu^{10} + 6072\nu^{8} + 26781\nu^{6} + 47241\nu^{4} + 76410\nu^{2} + 11664 ) / 46648 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 169 \nu^{14} + 755 \nu^{12} + 596 \nu^{10} - 80864 \nu^{8} - 301553 \nu^{6} - 508173 \nu^{4} + \cdots - 127440 ) / 326536 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 2241 \nu^{15} + 19928 \nu^{13} - 95248 \nu^{11} - 508200 \nu^{9} - 2302433 \nu^{7} + \cdots + 16017920 \nu ) / 5224576 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 701 \nu^{14} - 4647 \nu^{12} + 14216 \nu^{10} + 242984 \nu^{8} + 1002309 \nu^{6} + 1738129 \nu^{4} + \cdots - 221440 ) / 653072 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 1781 \nu^{14} + 14648 \nu^{12} - 56368 \nu^{10} - 554792 \nu^{8} - 1544053 \nu^{6} + \cdots + 484672 ) / 1306144 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 797 \nu^{14} + 6576 \nu^{12} - 27232 \nu^{10} - 228232 \nu^{8} - 790973 \nu^{6} - 750296 \nu^{4} + \cdots + 298368 ) / 373184 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 1333 \nu^{15} - 8383 \nu^{13} + 22124 \nu^{11} + 489720 \nu^{9} + 1980349 \nu^{7} + \cdots + 849808 \nu ) / 1306144 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 2869 \nu^{14} - 24984 \nu^{12} + 113352 \nu^{10} + 741384 \nu^{8} + 2587445 \nu^{6} + 2935120 \nu^{4} + \cdots - 14656 ) / 1306144 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 701 \nu^{15} + 4647 \nu^{13} - 14216 \nu^{11} - 242984 \nu^{9} - 1002309 \nu^{7} + \cdots + 221440 \nu ) / 653072 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 7573 \nu^{15} - 53460 \nu^{13} + 183744 \nu^{11} + 2467080 \nu^{9} + 10223829 \nu^{7} + \cdots - 7394112 \nu ) / 5224576 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 7965 \nu^{15} + 66424 \nu^{13} - 266960 \nu^{11} - 2367176 \nu^{9} - 7125181 \nu^{7} + \cdots + 6323392 \nu ) / 5224576 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 797 \nu^{15} - 6576 \nu^{13} + 27232 \nu^{11} + 228232 \nu^{9} + 790973 \nu^{7} + 750296 \nu^{5} + \cdots - 298368 \nu ) / 373184 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 14121 \nu^{15} + 114588 \nu^{13} - 474192 \nu^{11} - 4052328 \nu^{9} - 14720937 \nu^{7} + \cdots + 3003648 \nu ) / 5224576 \)
|
| \(\nu\) | \(=\) |
\( \beta_{12} - \beta_{9} + \beta_{5} \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{10} - \beta_{7} + \beta_{4} + 3\beta_{3} \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{15} - 7\beta_{13} + 2\beta_{12} - 2\beta_{11} - 9\beta_{9} + 2\beta_{5} - 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -11\beta_{10} - 4\beta_{8} + 10\beta_{6} + 9\beta_{4} + 15\beta_{3} - 9\beta_{2} - 9 \)
|
| \(\nu^{5}\) | \(=\) |
\( 22\beta_{15} + 24\beta_{14} - 24\beta_{13} - 18\beta_{12} - 39\beta_{11} - 42\beta_{9} - 24\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -81\beta_{10} - 78\beta_{8} + 81\beta_{7} + 48\beta_{6} + 81\beta_{3} - 109\beta_{2} - 157 \)
|
| \(\nu^{7}\) | \(=\) |
\( 162\beta_{15} + 349\beta_{14} + 162\beta_{13} - 456\beta_{12} - 238\beta_{11} + 238\beta_{9} - 238\beta_{5} \)
|
| \(\nu^{8}\) | \(=\) |
\( -476\beta_{8} + 1043\beta_{7} - 749\beta_{4} - 476\beta_{3} - 749\beta_{2} - 1447 \)
|
| \(\nu^{9}\) | \(=\) |
\( 2268\beta_{14} + 4354\beta_{13} - 4737\beta_{12} + 7005\beta_{9} - 3239\beta_{5} + 2268\beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( 7005\beta_{10} + 7005\beta_{7} - 4536\beta_{6} - 9861\beta_{4} - 13483\beta_{3} - 4536 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 14010 \beta_{15} + 44359 \beta_{13} - 21402 \beta_{12} + 21402 \beta_{11} + 71473 \beta_{9} + \cdots + 30349 \beta_1 \)
|
| \(\nu^{12}\) | \(=\) |
\( 92875\beta_{10} + 42804\beta_{8} - 60698\beta_{6} - 65761\beta_{4} - 135679\beta_{3} + 65761\beta_{2} + 65761 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 185750 \beta_{15} - 201440 \beta_{14} + 201440 \beta_{13} + 131522 \beta_{12} + 285095 \beta_{11} + \cdots + 201440 \beta_1 \)
|
| \(\nu^{14}\) | \(=\) |
\( 618057 \beta_{10} + 570190 \beta_{8} - 618057 \beta_{7} - 402880 \beta_{6} - 618057 \beta_{3} + \cdots + 1276605 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 1236114 \beta_{15} - 2680029 \beta_{14} - 1236114 \beta_{13} + 3642112 \beta_{12} + \cdots + 1894662 \beta_{5} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/867\mathbb{Z}\right)^\times\).
| \(n\) | \(290\) | \(292\) |
| \(\chi(n)\) | \(1\) | \(\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 688.1 |
|
−0.157113 | + | 0.157113i | −0.382683 | + | 0.923880i | 1.95063i | 0.587961 | + | 0.243542i | −0.0850290 | − | 0.205278i | 1.59687 | − | 0.661445i | −0.620696 | − | 0.620696i | −0.707107 | − | 0.707107i | −0.130640 | + | 0.0541128i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 688.2 | −0.157113 | + | 0.157113i | 0.382683 | − | 0.923880i | 1.95063i | −0.587961 | − | 0.243542i | 0.0850290 | + | 0.205278i | −1.59687 | + | 0.661445i | −0.620696 | − | 0.620696i | −0.707107 | − | 0.707107i | 0.130640 | − | 0.0541128i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 688.3 | 1.86422 | − | 1.86422i | −0.382683 | + | 0.923880i | − | 4.95063i | −2.05304 | − | 0.850396i | 1.00891 | + | 2.43572i | −2.13807 | + | 0.885616i | −5.50062 | − | 5.50062i | −0.707107 | − | 0.707107i | −5.41264 | + | 2.24199i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 688.4 | 1.86422 | − | 1.86422i | 0.382683 | − | 0.923880i | − | 4.95063i | 2.05304 | + | 0.850396i | −1.00891 | − | 2.43572i | 2.13807 | − | 0.885616i | −5.50062 | − | 5.50062i | −0.707107 | − | 0.707107i | 5.41264 | − | 2.24199i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 712.1 | −0.157113 | − | 0.157113i | −0.382683 | − | 0.923880i | − | 1.95063i | 0.587961 | − | 0.243542i | −0.0850290 | + | 0.205278i | 1.59687 | + | 0.661445i | −0.620696 | + | 0.620696i | −0.707107 | + | 0.707107i | −0.130640 | − | 0.0541128i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 712.2 | −0.157113 | − | 0.157113i | 0.382683 | + | 0.923880i | − | 1.95063i | −0.587961 | + | 0.243542i | 0.0850290 | − | 0.205278i | −1.59687 | − | 0.661445i | −0.620696 | + | 0.620696i | −0.707107 | + | 0.707107i | 0.130640 | + | 0.0541128i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 712.3 | 1.86422 | + | 1.86422i | −0.382683 | − | 0.923880i | 4.95063i | −2.05304 | + | 0.850396i | 1.00891 | − | 2.43572i | −2.13807 | − | 0.885616i | −5.50062 | + | 5.50062i | −0.707107 | + | 0.707107i | −5.41264 | − | 2.24199i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 712.4 | 1.86422 | + | 1.86422i | 0.382683 | + | 0.923880i | 4.95063i | 2.05304 | − | 0.850396i | −1.00891 | + | 2.43572i | 2.13807 | + | 0.885616i | −5.50062 | + | 5.50062i | −0.707107 | + | 0.707107i | 5.41264 | + | 2.24199i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 733.1 | −1.16830 | − | 1.16830i | −0.923880 | + | 0.382683i | 0.729840i | 1.55616 | + | 3.75690i | 1.52645 | + | 0.632278i | −0.352980 | + | 0.852170i | −1.48392 | + | 1.48392i | 0.707107 | − | 0.707107i | 2.57112 | − | 6.20723i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 733.2 | −1.16830 | − | 1.16830i | 0.923880 | − | 0.382683i | 0.729840i | −1.55616 | − | 3.75690i | −1.52645 | − | 0.632278i | 0.352980 | − | 0.852170i | −1.48392 | + | 1.48392i | 0.707107 | − | 0.707107i | −2.57112 | + | 6.20723i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 733.3 | 1.46119 | + | 1.46119i | −0.923880 | + | 0.382683i | 2.27016i | 0.133089 | + | 0.321304i | −1.90914 | − | 0.790791i | 1.65954 | − | 4.00649i | −0.394755 | + | 0.394755i | 0.707107 | − | 0.707107i | −0.275019 | + | 0.663955i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 733.4 | 1.46119 | + | 1.46119i | 0.923880 | − | 0.382683i | 2.27016i | −0.133089 | − | 0.321304i | 1.90914 | + | 0.790791i | −1.65954 | + | 4.00649i | −0.394755 | + | 0.394755i | 0.707107 | − | 0.707107i | 0.275019 | − | 0.663955i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 757.1 | −1.16830 | + | 1.16830i | −0.923880 | − | 0.382683i | − | 0.729840i | 1.55616 | − | 3.75690i | 1.52645 | − | 0.632278i | −0.352980 | − | 0.852170i | −1.48392 | − | 1.48392i | 0.707107 | + | 0.707107i | 2.57112 | + | 6.20723i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 757.2 | −1.16830 | + | 1.16830i | 0.923880 | + | 0.382683i | − | 0.729840i | −1.55616 | + | 3.75690i | −1.52645 | + | 0.632278i | 0.352980 | + | 0.852170i | −1.48392 | − | 1.48392i | 0.707107 | + | 0.707107i | −2.57112 | − | 6.20723i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 757.3 | 1.46119 | − | 1.46119i | −0.923880 | − | 0.382683i | − | 2.27016i | 0.133089 | − | 0.321304i | −1.90914 | + | 0.790791i | 1.65954 | + | 4.00649i | −0.394755 | − | 0.394755i | 0.707107 | + | 0.707107i | −0.275019 | − | 0.663955i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 757.4 | 1.46119 | − | 1.46119i | 0.923880 | + | 0.382683i | − | 2.27016i | −0.133089 | + | 0.321304i | 1.90914 | − | 0.790791i | −1.65954 | − | 4.00649i | −0.394755 | − | 0.394755i | 0.707107 | + | 0.707107i | 0.275019 | + | 0.663955i | ||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.b | even | 2 | 1 | inner |
| 17.d | even | 8 | 2 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 867.2.h.k | 16 | |
| 17.b | even | 2 | 1 | inner | 867.2.h.k | 16 | |
| 17.c | even | 4 | 2 | 867.2.h.i | 16 | ||
| 17.d | even | 8 | 2 | 867.2.h.i | 16 | ||
| 17.d | even | 8 | 2 | inner | 867.2.h.k | 16 | |
| 17.e | odd | 16 | 2 | 51.2.e.a | ✓ | 8 | |
| 17.e | odd | 16 | 1 | 867.2.a.k | 4 | ||
| 17.e | odd | 16 | 1 | 867.2.a.l | 4 | ||
| 17.e | odd | 16 | 2 | 867.2.d.f | 8 | ||
| 17.e | odd | 16 | 2 | 867.2.e.g | 8 | ||
| 51.i | even | 16 | 2 | 153.2.f.b | 8 | ||
| 51.i | even | 16 | 1 | 2601.2.a.be | 4 | ||
| 51.i | even | 16 | 1 | 2601.2.a.bf | 4 | ||
| 68.i | even | 16 | 2 | 816.2.bd.e | 8 | ||
| 204.t | odd | 16 | 2 | 2448.2.be.x | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 51.2.e.a | ✓ | 8 | 17.e | odd | 16 | 2 | |
| 153.2.f.b | 8 | 51.i | even | 16 | 2 | ||
| 816.2.bd.e | 8 | 68.i | even | 16 | 2 | ||
| 867.2.a.k | 4 | 17.e | odd | 16 | 1 | ||
| 867.2.a.l | 4 | 17.e | odd | 16 | 1 | ||
| 867.2.d.f | 8 | 17.e | odd | 16 | 2 | ||
| 867.2.e.g | 8 | 17.e | odd | 16 | 2 | ||
| 867.2.h.i | 16 | 17.c | even | 4 | 2 | ||
| 867.2.h.i | 16 | 17.d | even | 8 | 2 | ||
| 867.2.h.k | 16 | 1.a | even | 1 | 1 | trivial | |
| 867.2.h.k | 16 | 17.b | even | 2 | 1 | inner | |
| 867.2.h.k | 16 | 17.d | even | 8 | 2 | inner | |
| 2448.2.be.x | 8 | 204.t | odd | 16 | 2 | ||
| 2601.2.a.be | 4 | 51.i | even | 16 | 1 | ||
| 2601.2.a.bf | 4 | 51.i | even | 16 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(867, [\chi])\):
|
\( T_{2}^{8} - 4T_{2}^{7} + 8T_{2}^{6} + 5T_{2}^{4} - 28T_{2}^{3} + 72T_{2}^{2} + 24T_{2} + 4 \)
|
|
\( T_{5}^{16} + 16T_{5}^{14} + 128T_{5}^{12} - 1392T_{5}^{10} + 7217T_{5}^{8} - 2784T_{5}^{6} + 512T_{5}^{4} + 128T_{5}^{2} + 16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{8} - 4 T^{7} + 8 T^{6} + \cdots + 4)^{2} \)
|
| $3$ |
\( (T^{8} + 1)^{2} \)
|
| $5$ |
\( T^{16} + 16 T^{14} + \cdots + 16 \)
|
| $7$ |
\( T^{16} + 16 T^{14} + \cdots + 65536 \)
|
| $11$ |
\( T^{16} + 32 T^{14} + \cdots + 65536 \)
|
| $13$ |
\( (T^{8} + 50 T^{6} + \cdots + 4096)^{2} \)
|
| $17$ |
\( T^{16} \)
|
| $19$ |
\( (T^{8} - 80 T^{5} + \cdots + 1024)^{2} \)
|
| $23$ |
\( T^{16} + \cdots + 1146228736 \)
|
| $29$ |
\( T^{16} - 96 T^{14} + \cdots + 65536 \)
|
| $31$ |
\( T^{16} - 7168 T^{10} + \cdots + 1048576 \)
|
| $37$ |
\( T^{16} + \cdots + 355196928256 \)
|
| $41$ |
\( T^{16} + \cdots + 7703711109136 \)
|
| $43$ |
\( (T^{8} + 24 T^{7} + \cdots + 1364224)^{2} \)
|
| $47$ |
\( (T^{8} + 120 T^{6} + \cdots + 4096)^{2} \)
|
| $53$ |
\( (T^{8} + 8 T^{7} + \cdots + 16384)^{2} \)
|
| $59$ |
\( (T^{8} + 28 T^{7} + \cdots + 21827584)^{2} \)
|
| $61$ |
\( T^{16} + \cdots + 16062013696 \)
|
| $67$ |
\( (T^{4} + 4 T^{3} + \cdots + 2048)^{4} \)
|
| $71$ |
\( T^{16} - 64 T^{14} + \cdots + 1048576 \)
|
| $73$ |
\( T^{16} + \cdots + 11574317056 \)
|
| $79$ |
\( T^{16} - 192 T^{14} + \cdots + 65536 \)
|
| $83$ |
\( (T^{8} + 8 T^{7} + \cdots + 27541504)^{2} \)
|
| $89$ |
\( (T^{8} + 236 T^{6} + \cdots + 1183744)^{2} \)
|
| $97$ |
\( T^{16} + \cdots + 18339659776 \)
|
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