Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [945,2,Mod(316,945)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(945, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("945.316");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 945 = 3^{3} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 945.i (of order \(3\), degree \(2\), 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: | \(7.54586299101\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 2 x^{15} + 8 x^{14} - 10 x^{13} + 40 x^{12} - 45 x^{11} + 159 x^{10} - 180 x^{9} + 576 x^{8} + \cdots + 6561 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 3^{8} \) |
| Twist minimal: | no (minimal twist has level 315) |
| 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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 2 x^{15} + 8 x^{14} - 10 x^{13} + 40 x^{12} - 45 x^{11} + 159 x^{10} - 180 x^{9} + 576 x^{8} + \cdots + 6561 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{15} + 54 \nu^{14} + 13 \nu^{13} + 285 \nu^{12} + 92 \nu^{11} + 1376 \nu^{10} + 564 \nu^{9} + \cdots + 105705 ) / 5832 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - \nu^{15} + 26 \nu^{14} - 5 \nu^{13} + 145 \nu^{12} - 16 \nu^{11} + 666 \nu^{10} + 108 \nu^{9} + \cdots + 50301 ) / 2916 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 331 \nu^{15} - 384 \nu^{14} - 1549 \nu^{13} - 3459 \nu^{12} - 7034 \nu^{11} - 20030 \nu^{10} + \cdots - 1245861 ) / 763992 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 23 \nu^{15} - 98 \nu^{14} - 139 \nu^{13} - 571 \nu^{12} - 668 \nu^{11} - 2808 \nu^{10} + \cdots - 168399 ) / 17496 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 705 \nu^{15} + 2000 \nu^{14} - 1861 \nu^{13} + 10303 \nu^{12} - 7064 \nu^{11} + 54794 \nu^{10} + \cdots + 4917105 ) / 381996 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 35 \nu^{15} + 118 \nu^{14} - 151 \nu^{13} + 581 \nu^{12} - 836 \nu^{11} + 2892 \nu^{10} + \cdots + 269001 ) / 17496 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 55 \nu^{15} - 82 \nu^{14} - 299 \nu^{13} - 527 \nu^{12} - 1684 \nu^{11} - 2748 \nu^{10} + \cdots - 212139 ) / 17496 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 67 \nu^{15} - 206 \nu^{14} + 311 \nu^{13} - 1021 \nu^{12} + 1636 \nu^{11} - 4932 \nu^{10} + \cdots - 513945 ) / 17496 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 8852 \nu^{15} + 15445 \nu^{14} - 45301 \nu^{13} + 88880 \nu^{12} - 235523 \nu^{11} + \cdots + 37601091 ) / 2291976 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 1055 \nu^{15} + 1459 \nu^{14} - 4932 \nu^{13} + 6837 \nu^{12} - 25203 \nu^{11} + 32645 \nu^{10} + \cdots + 3174552 ) / 254664 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 9662 \nu^{15} - 17977 \nu^{14} + 46603 \nu^{13} - 104798 \nu^{12} + 193703 \nu^{11} + \cdots - 46493433 ) / 2291976 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 11173 \nu^{15} - 25886 \nu^{14} + 58457 \nu^{13} - 132427 \nu^{12} + 268228 \nu^{11} + \cdots - 58438827 ) / 2291976 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 12086 \nu^{15} + 23761 \nu^{14} - 60991 \nu^{13} + 125582 \nu^{12} - 286595 \nu^{11} + \cdots + 58198257 ) / 2291976 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 7093 \nu^{15} + 6524 \nu^{14} - 38333 \nu^{13} + 28813 \nu^{12} - 177868 \nu^{11} + \cdots + 17043291 ) / 1145988 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 14266 \nu^{15} - 28955 \nu^{14} + 77993 \nu^{13} - 140554 \nu^{12} + 382837 \nu^{11} + \cdots - 63250227 ) / 2291976 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{14} + \beta_{12} - \beta_{4} + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{15} - \beta_{12} - 2\beta_{10} + \beta_{9} + \beta_{7} + \beta_{6} + \beta_{4} - \beta_{2} + 2\beta _1 - 2 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 2 \beta_{15} - 2 \beta_{14} - 5 \beta_{10} + \beta_{9} + \beta_{7} + \beta_{6} + 3 \beta_{5} + \cdots - 4 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 4 \beta_{13} - \beta_{12} + 5 \beta_{11} + 2 \beta_{10} - \beta_{9} - 2 \beta_{8} - 5 \beta_{7} + \cdots - 15 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( \beta_{15} - \beta_{14} + 7 \beta_{13} + 3 \beta_{12} + 5 \beta_{11} + 9 \beta_{10} + 2 \beta_{9} + \cdots - 11 ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 4 \beta_{15} - \beta_{14} + 6 \beta_{12} - 9 \beta_{11} + 2 \beta_{10} - 3 \beta_{9} + 3 \beta_{8} + \cdots - 18 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 13 \beta_{15} + 5 \beta_{14} - 40 \beta_{13} - 19 \beta_{12} - 20 \beta_{11} + 17 \beta_{10} - 7 \beta_{8} + \cdots + 47 ) / 3 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - \beta_{15} + 26 \beta_{14} - 44 \beta_{13} + 35 \beta_{12} - 28 \beta_{11} + 31 \beta_{10} + \cdots + 128 ) / 3 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 24 \beta_{15} + 18 \beta_{14} + 60 \beta_{13} - 18 \beta_{12} + 3 \beta_{11} - 24 \beta_{10} - 77 \beta_{9} + \cdots + 2 ) / 3 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 59 \beta_{15} - 109 \beta_{14} - 87 \beta_{13} - 173 \beta_{12} + 9 \beta_{11} - 137 \beta_{10} + \cdots + 224 ) / 3 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 162 \beta_{15} - 8 \beta_{14} + 86 \beta_{13} + 35 \beta_{12} + 79 \beta_{11} - 38 \beta_{10} + \cdots - 57 ) / 3 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 290 \beta_{15} + 11 \beta_{14} + 708 \beta_{13} + 288 \beta_{12} + 192 \beta_{11} + 293 \beta_{10} + \cdots - 499 ) / 3 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 23 \beta_{15} - 602 \beta_{14} - 34 \beta_{13} + 669 \beta_{12} - 158 \beta_{11} - 90 \beta_{10} + \cdots - 428 ) / 3 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 52 \beta_{15} + 600 \beta_{14} - 840 \beta_{13} - 413 \beta_{12} + 294 \beta_{11} - 580 \beta_{10} + \cdots - 2833 ) / 3 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 272 \beta_{15} + 2018 \beta_{14} - 1260 \beta_{13} - 720 \beta_{12} + 1080 \beta_{11} - 580 \beta_{10} + \cdots + 1777 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/945\mathbb{Z}\right)^\times\).
| \(n\) | \(136\) | \(596\) | \(757\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{3}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 316.1 |
|
−1.38900 | + | 2.40581i | 0 | −2.85862 | − | 4.95128i | 0.500000 | + | 0.866025i | 0 | 0.500000 | − | 0.866025i | 10.3265 | 0 | −2.77799 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 316.2 | −1.09035 | + | 1.88855i | 0 | −1.37775 | − | 2.38633i | 0.500000 | + | 0.866025i | 0 | 0.500000 | − | 0.866025i | 1.64751 | 0 | −2.18071 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 316.3 | −0.627726 | + | 1.08725i | 0 | 0.211920 | + | 0.367057i | 0.500000 | + | 0.866025i | 0 | 0.500000 | − | 0.866025i | −3.04302 | 0 | −1.25545 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 316.4 | −0.441371 | + | 0.764477i | 0 | 0.610383 | + | 1.05721i | 0.500000 | + | 0.866025i | 0 | 0.500000 | − | 0.866025i | −2.84311 | 0 | −0.882742 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 316.5 | 0.219523 | − | 0.380224i | 0 | 0.903620 | + | 1.56512i | 0.500000 | + | 0.866025i | 0 | 0.500000 | − | 0.866025i | 1.67155 | 0 | 0.439045 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 316.6 | 0.522039 | − | 0.904198i | 0 | 0.454951 | + | 0.787999i | 0.500000 | + | 0.866025i | 0 | 0.500000 | − | 0.866025i | 3.03816 | 0 | 1.04408 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 316.7 | 0.978244 | − | 1.69437i | 0 | −0.913922 | − | 1.58296i | 0.500000 | + | 0.866025i | 0 | 0.500000 | − | 0.866025i | 0.336819 | 0 | 1.95649 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 316.8 | 1.32864 | − | 2.30128i | 0 | −2.53058 | − | 4.38310i | 0.500000 | + | 0.866025i | 0 | 0.500000 | − | 0.866025i | −8.13440 | 0 | 2.65729 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 631.1 | −1.38900 | − | 2.40581i | 0 | −2.85862 | + | 4.95128i | 0.500000 | − | 0.866025i | 0 | 0.500000 | + | 0.866025i | 10.3265 | 0 | −2.77799 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 631.2 | −1.09035 | − | 1.88855i | 0 | −1.37775 | + | 2.38633i | 0.500000 | − | 0.866025i | 0 | 0.500000 | + | 0.866025i | 1.64751 | 0 | −2.18071 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 631.3 | −0.627726 | − | 1.08725i | 0 | 0.211920 | − | 0.367057i | 0.500000 | − | 0.866025i | 0 | 0.500000 | + | 0.866025i | −3.04302 | 0 | −1.25545 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 631.4 | −0.441371 | − | 0.764477i | 0 | 0.610383 | − | 1.05721i | 0.500000 | − | 0.866025i | 0 | 0.500000 | + | 0.866025i | −2.84311 | 0 | −0.882742 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 631.5 | 0.219523 | + | 0.380224i | 0 | 0.903620 | − | 1.56512i | 0.500000 | − | 0.866025i | 0 | 0.500000 | + | 0.866025i | 1.67155 | 0 | 0.439045 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 631.6 | 0.522039 | + | 0.904198i | 0 | 0.454951 | − | 0.787999i | 0.500000 | − | 0.866025i | 0 | 0.500000 | + | 0.866025i | 3.03816 | 0 | 1.04408 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 631.7 | 0.978244 | + | 1.69437i | 0 | −0.913922 | + | 1.58296i | 0.500000 | − | 0.866025i | 0 | 0.500000 | + | 0.866025i | 0.336819 | 0 | 1.95649 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 631.8 | 1.32864 | + | 2.30128i | 0 | −2.53058 | + | 4.38310i | 0.500000 | − | 0.866025i | 0 | 0.500000 | + | 0.866025i | −8.13440 | 0 | 2.65729 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 945.2.i.f | 16 | |
| 3.b | odd | 2 | 1 | 315.2.i.f | ✓ | 16 | |
| 9.c | even | 3 | 1 | inner | 945.2.i.f | 16 | |
| 9.c | even | 3 | 1 | 2835.2.a.y | 8 | ||
| 9.d | odd | 6 | 1 | 315.2.i.f | ✓ | 16 | |
| 9.d | odd | 6 | 1 | 2835.2.a.x | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 315.2.i.f | ✓ | 16 | 3.b | odd | 2 | 1 | |
| 315.2.i.f | ✓ | 16 | 9.d | odd | 6 | 1 | |
| 945.2.i.f | 16 | 1.a | even | 1 | 1 | trivial | |
| 945.2.i.f | 16 | 9.c | even | 3 | 1 | inner | |
| 2835.2.a.x | 8 | 9.d | odd | 6 | 1 | ||
| 2835.2.a.y | 8 | 9.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} + T_{2}^{15} + 14 T_{2}^{14} + 9 T_{2}^{13} + 131 T_{2}^{12} + 77 T_{2}^{11} + 616 T_{2}^{10} + \cdots + 256 \)
acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} + T^{15} + \cdots + 256 \)
|
| $3$ |
\( T^{16} \)
|
| $5$ |
\( (T^{2} - T + 1)^{8} \)
|
| $7$ |
\( (T^{2} - T + 1)^{8} \)
|
| $11$ |
\( T^{16} + \cdots + 241367296 \)
|
| $13$ |
\( T^{16} + 5 T^{15} + \cdots + 3118756 \)
|
| $17$ |
\( (T^{8} + 4 T^{7} + \cdots - 13682)^{2} \)
|
| $19$ |
\( (T^{8} - 3 T^{7} + \cdots + 96256)^{2} \)
|
| $23$ |
\( T^{16} + \cdots + 358875136 \)
|
| $29$ |
\( T^{16} + \cdots + 181104718096 \)
|
| $31$ |
\( T^{16} + \cdots + 6879707136 \)
|
| $37$ |
\( (T^{8} - 21 T^{7} + \cdots - 82944)^{2} \)
|
| $41$ |
\( T^{16} + \cdots + 836829184 \)
|
| $43$ |
\( T^{16} + \cdots + 345330171904 \)
|
| $47$ |
\( T^{16} + 11 T^{15} + \cdots + 4096 \)
|
| $53$ |
\( (T^{8} - 8 T^{7} + \cdots - 165056)^{2} \)
|
| $59$ |
\( T^{16} + \cdots + 63471748096 \)
|
| $61$ |
\( T^{16} + \cdots + 27518828544 \)
|
| $67$ |
\( T^{16} + 16 T^{15} + \cdots + 5456896 \)
|
| $71$ |
\( (T^{8} - 5 T^{7} + \cdots - 1249607)^{2} \)
|
| $73$ |
\( (T^{8} - 10 T^{7} + \cdots + 124561)^{2} \)
|
| $79$ |
\( T^{16} + 27 T^{15} + \cdots + 16384 \)
|
| $83$ |
\( T^{16} + \cdots + 6791950062769 \)
|
| $89$ |
\( (T^{8} + 27 T^{7} + \cdots - 6305216)^{2} \)
|
| $97$ |
\( T^{16} + \cdots + 4365349992964 \)
|
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