Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [308,2,Mod(43,308)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(308, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 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("308.43");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 308 = 2^{2} \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 308.d (of order \(2\), degree \(1\), 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: | \(2.45939238226\) |
| Analytic rank: | \(0\) |
| Dimension: | \(18\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{18} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{18} - x^{17} - 2 x^{15} + 3 x^{14} - 3 x^{13} + 4 x^{12} - 14 x^{11} + 12 x^{10} + 16 x^{9} + \cdots + 512 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{7} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{17}\) 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
\( x^{18} - x^{17} - 2 x^{15} + 3 x^{14} - 3 x^{13} + 4 x^{12} - 14 x^{11} + 12 x^{10} + 16 x^{9} + \cdots + 512 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( -\nu^{3} + \nu^{2} \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - \nu^{16} + \nu^{15} - 4 \nu^{14} - 2 \nu^{13} + 5 \nu^{12} - 5 \nu^{11} - 16 \nu^{10} + 2 \nu^{9} + \cdots - 128 ) / 256 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{17} - \nu^{16} + 4 \nu^{15} + 2 \nu^{14} - 5 \nu^{13} + 5 \nu^{12} + 16 \nu^{11} - 2 \nu^{10} + \cdots + 128 \nu ) / 512 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{17} + \nu^{16} - 6 \nu^{15} - 6 \nu^{14} + 7 \nu^{13} + 11 \nu^{12} - 30 \nu^{11} - 50 \nu^{10} + \cdots - 768 ) / 512 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{17} - \nu^{16} + 2 \nu^{14} + 7 \nu^{13} - 3 \nu^{12} + 4 \nu^{11} - 18 \nu^{10} + 24 \nu^{9} + \cdots - 128 \nu ) / 256 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - \nu^{16} + \nu^{15} - 2 \nu^{14} - 3 \nu^{12} + 3 \nu^{11} - 2 \nu^{10} + 8 \nu^{9} - 4 \nu^{8} + \cdots + 64 \nu ) / 128 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( \nu^{17} + \nu^{16} - 2 \nu^{15} - 2 \nu^{14} - \nu^{13} + 3 \nu^{12} - 2 \nu^{11} - 6 \nu^{10} + \cdots - 256 ) / 256 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( \nu^{17} + 3 \nu^{16} - 6 \nu^{14} - 5 \nu^{13} + \nu^{12} + 36 \nu^{11} - 42 \nu^{10} - 92 \nu^{9} + \cdots - 1536 ) / 512 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 3 \nu^{17} - \nu^{16} - 6 \nu^{14} - \nu^{13} + 21 \nu^{12} + 20 \nu^{11} + 22 \nu^{10} + \cdots + 512 ) / 512 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( \nu^{17} + \nu^{16} + 2 \nu^{15} - 6 \nu^{14} - \nu^{13} - 5 \nu^{12} + 10 \nu^{11} - 18 \nu^{10} + \cdots - 768 ) / 256 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( \nu^{17} - \nu^{16} - 2 \nu^{14} + 3 \nu^{13} - 3 \nu^{12} + 4 \nu^{11} - 14 \nu^{10} + 12 \nu^{9} + \cdots - 256 ) / 256 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( \nu^{17} + 2 \nu^{16} + \nu^{15} - 2 \nu^{14} + 9 \nu^{13} - 2 \nu^{12} - 17 \nu^{11} - 34 \nu^{10} + \cdots - 640 ) / 256 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( \nu^{17} + 2 \nu^{16} + \nu^{15} - 6 \nu^{14} - 3 \nu^{13} - 10 \nu^{12} + 15 \nu^{11} + 2 \nu^{10} + \cdots - 640 ) / 256 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 2 \nu^{17} + 3 \nu^{16} - \nu^{15} - 4 \nu^{14} + 9 \nu^{12} - 3 \nu^{11} - 40 \nu^{10} - 18 \nu^{9} + \cdots - 384 ) / 256 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 2 \nu^{17} - \nu^{16} + 3 \nu^{15} + 4 \nu^{13} + 5 \nu^{12} - 7 \nu^{11} - 28 \nu^{10} - 6 \nu^{9} + \cdots - 384 ) / 256 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 7 \nu^{17} + 5 \nu^{16} - 2 \nu^{15} + 10 \nu^{14} - 25 \nu^{13} + 39 \nu^{12} - 18 \nu^{11} + \cdots + 1792 ) / 512 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{14} - \beta_{11} - \beta_{8} + \beta_{5} + \beta_{4} + \beta_{2} ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{14} - \beta_{11} - \beta_{8} + \beta_{5} + \beta_{4} - \beta_{2} ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( \beta_{14} - 2 \beta_{12} - \beta_{11} + \beta_{8} + 2 \beta_{6} - \beta_{5} - \beta_{4} + 2 \beta_{3} + \cdots - 2 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2\beta_{16} + \beta_{14} - 2\beta_{13} + \beta_{11} - 3\beta_{8} - 2\beta_{7} + \beta_{5} - 3\beta_{4} + \beta_{2} + 2 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 2 \beta_{16} - \beta_{14} - 2 \beta_{13} - 6 \beta_{12} + 3 \beta_{11} + \beta_{8} + 2 \beta_{7} + \cdots - \beta_{2} ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 4 \beta_{15} - \beta_{14} - 4 \beta_{12} + 5 \beta_{11} + 5 \beta_{8} - 4 \beta_{7} + 4 \beta_{6} + \cdots + 8 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 4 \beta_{17} + 4 \beta_{16} - 4 \beta_{15} + \beta_{14} + 6 \beta_{12} + 3 \beta_{11} + 5 \beta_{8} + \cdots + 6 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 4 \beta_{17} - 6 \beta_{16} + 4 \beta_{15} + 5 \beta_{14} + 2 \beta_{13} + 12 \beta_{12} - 3 \beta_{11} + \cdots - 22 ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 4 \beta_{17} + 2 \beta_{16} - 4 \beta_{15} + 11 \beta_{14} + 2 \beta_{13} - 10 \beta_{12} - 9 \beta_{11} + \cdots - 16 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 4 \beta_{17} - 5 \beta_{14} - 4 \beta_{13} + 8 \beta_{12} + 9 \beta_{11} + 8 \beta_{10} + 8 \beta_{9} + \cdots - 2 \beta_1 ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 12 \beta_{16} + 8 \beta_{15} - 19 \beta_{14} - 4 \beta_{13} + 2 \beta_{12} + 15 \beta_{11} + 24 \beta_{10} + \cdots - 34 ) / 2 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 8 \beta_{17} + 18 \beta_{16} - 8 \beta_{15} - 15 \beta_{14} + 22 \beta_{13} - 8 \beta_{12} + \cdots + 26 ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 8 \beta_{17} - 14 \beta_{16} - 24 \beta_{15} - 65 \beta_{14} + 6 \beta_{13} - 14 \beta_{12} + 51 \beta_{11} + \cdots - 16 ) / 2 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 24 \beta_{17} + 16 \beta_{16} - 44 \beta_{15} + 15 \beta_{14} + 88 \beta_{13} + 4 \beta_{12} + 5 \beta_{11} + \cdots + 40 ) / 2 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( 44 \beta_{17} - 60 \beta_{16} + 100 \beta_{15} + 57 \beta_{14} + 88 \beta_{13} + 158 \beta_{12} + \cdots - 74 ) / 2 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( - 100 \beta_{17} - 22 \beta_{16} + 156 \beta_{15} + 29 \beta_{14} - 6 \beta_{13} - 76 \beta_{12} + \cdots - 182 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/308\mathbb{Z}\right)^\times\).
| \(n\) | \(45\) | \(57\) | \(155\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 43.1 |
|
−1.39867 | − | 0.209120i | 2.69604i | 1.91254 | + | 0.584978i | −3.01989 | 0.563795 | − | 3.77086i | 1.00000 | −2.55267 | − | 1.21814i | −4.26863 | 4.22382 | + | 0.631519i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.2 | −1.39867 | + | 0.209120i | − | 2.69604i | 1.91254 | − | 0.584978i | −3.01989 | 0.563795 | + | 3.77086i | 1.00000 | −2.55267 | + | 1.21814i | −4.26863 | 4.22382 | − | 0.631519i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.3 | −1.35077 | − | 0.418821i | − | 1.57463i | 1.64918 | + | 1.13146i | 1.58188 | −0.659487 | + | 2.12696i | 1.00000 | −1.75378 | − | 2.21906i | 0.520549 | −2.13676 | − | 0.662524i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.4 | −1.35077 | + | 0.418821i | 1.57463i | 1.64918 | − | 1.13146i | 1.58188 | −0.659487 | − | 2.12696i | 1.00000 | −1.75378 | + | 2.21906i | 0.520549 | −2.13676 | + | 0.662524i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.5 | −0.916371 | − | 1.07716i | 2.19521i | −0.320527 | + | 1.97415i | 3.16783 | 2.36458 | − | 2.01163i | 1.00000 | 2.42019 | − | 1.46380i | −1.81894 | −2.90291 | − | 3.41224i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.6 | −0.916371 | + | 1.07716i | − | 2.19521i | −0.320527 | − | 1.97415i | 3.16783 | 2.36458 | + | 2.01163i | 1.00000 | 2.42019 | + | 1.46380i | −1.81894 | −2.90291 | + | 3.41224i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.7 | −0.730056 | − | 1.21121i | 0.700517i | −0.934037 | + | 1.76850i | −3.85785 | 0.848470 | − | 0.511417i | 1.00000 | 2.82391 | − | 0.159790i | 2.50928 | 2.81644 | + | 4.67264i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.8 | −0.730056 | + | 1.21121i | − | 0.700517i | −0.934037 | − | 1.76850i | −3.85785 | 0.848470 | + | 0.511417i | 1.00000 | 2.82391 | + | 0.159790i | 2.50928 | 2.81644 | − | 4.67264i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.9 | 0.166850 | − | 1.40434i | − | 3.22189i | −1.94432 | − | 0.468627i | −1.36931 | −4.52462 | − | 0.537572i | 1.00000 | −0.982520 | + | 2.65229i | −7.38059 | −0.228469 | + | 1.92297i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.10 | 0.166850 | + | 1.40434i | 3.22189i | −1.94432 | + | 0.468627i | −1.36931 | −4.52462 | + | 0.537572i | 1.00000 | −0.982520 | − | 2.65229i | −7.38059 | −0.228469 | − | 1.92297i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.11 | 0.308387 | − | 1.38018i | − | 0.280638i | −1.80980 | − | 0.851259i | 2.41173 | −0.387331 | − | 0.0865450i | 1.00000 | −1.73301 | + | 2.23533i | 2.92124 | 0.743745 | − | 3.32862i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.12 | 0.308387 | + | 1.38018i | 0.280638i | −1.80980 | + | 0.851259i | 2.41173 | −0.387331 | + | 0.0865450i | 1.00000 | −1.73301 | − | 2.23533i | 2.92124 | 0.743745 | + | 3.32862i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.13 | 0.963054 | − | 1.03563i | 3.04909i | −0.145053 | − | 1.99473i | 2.80597 | 3.15773 | + | 2.93644i | 1.00000 | −2.20550 | − | 1.77082i | −6.29696 | 2.70230 | − | 2.90594i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.14 | 0.963054 | + | 1.03563i | − | 3.04909i | −0.145053 | + | 1.99473i | 2.80597 | 3.15773 | − | 2.93644i | 1.00000 | −2.20550 | + | 1.77082i | −6.29696 | 2.70230 | + | 2.90594i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.15 | 1.11440 | − | 0.870694i | − | 1.74833i | 0.483785 | − | 1.94061i | −0.404609 | −1.52226 | − | 1.94835i | 1.00000 | −1.15054 | − | 2.58384i | −0.0566695 | −0.450897 | + | 0.352290i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.16 | 1.11440 | + | 0.870694i | 1.74833i | 0.483785 | + | 1.94061i | −0.404609 | −1.52226 | + | 1.94835i | 1.00000 | −1.15054 | + | 2.58384i | −0.0566695 | −0.450897 | − | 0.352290i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.17 | 1.34317 | − | 0.442587i | 0.359546i | 1.60823 | − | 1.18894i | −1.31575 | 0.159130 | + | 0.482933i | 1.00000 | 1.63393 | − | 2.30874i | 2.87073 | −1.76728 | + | 0.582334i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.18 | 1.34317 | + | 0.442587i | − | 0.359546i | 1.60823 | + | 1.18894i | −1.31575 | 0.159130 | − | 0.482933i | 1.00000 | 1.63393 | + | 2.30874i | 2.87073 | −1.76728 | − | 0.582334i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 44.c | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 308.2.d.a | ✓ | 18 |
| 4.b | odd | 2 | 1 | 308.2.d.b | yes | 18 | |
| 11.b | odd | 2 | 1 | 308.2.d.b | yes | 18 | |
| 44.c | even | 2 | 1 | inner | 308.2.d.a | ✓ | 18 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 308.2.d.a | ✓ | 18 | 1.a | even | 1 | 1 | trivial |
| 308.2.d.a | ✓ | 18 | 44.c | even | 2 | 1 | inner |
| 308.2.d.b | yes | 18 | 4.b | odd | 2 | 1 | |
| 308.2.d.b | yes | 18 | 11.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{19}^{9} - 4 T_{19}^{8} - 84 T_{19}^{7} + 264 T_{19}^{6} + 2204 T_{19}^{5} - 6160 T_{19}^{4} + \cdots - 12288 \)
acting on \(S_{2}^{\mathrm{new}}(308, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{18} + T^{17} + \cdots + 512 \)
|
| $3$ |
\( T^{18} + 38 T^{16} + \cdots + 128 \)
|
| $5$ |
\( (T^{9} - 27 T^{7} + \cdots + 288)^{2} \)
|
| $7$ |
\( (T - 1)^{18} \)
|
| $11$ |
\( T^{18} + \cdots + 2357947691 \)
|
| $13$ |
\( T^{18} + \cdots + 214990848 \)
|
| $17$ |
\( T^{18} + 124 T^{16} + \cdots + 131072 \)
|
| $19$ |
\( (T^{9} - 4 T^{8} + \cdots - 12288)^{2} \)
|
| $23$ |
\( T^{18} + \cdots + 70904741888 \)
|
| $29$ |
\( T^{18} + \cdots + 3699376128 \)
|
| $31$ |
\( T^{18} + \cdots + 1831819392 \)
|
| $37$ |
\( (T^{9} + 4 T^{8} + \cdots - 277504)^{2} \)
|
| $41$ |
\( T^{18} + \cdots + 2717908992 \)
|
| $43$ |
\( (T^{9} + 20 T^{8} + \cdots - 68608)^{2} \)
|
| $47$ |
\( T^{18} + \cdots + 492095490048 \)
|
| $53$ |
\( (T^{9} - 2 T^{8} + \cdots + 8051712)^{2} \)
|
| $59$ |
\( T^{18} + \cdots + 68765298647168 \)
|
| $61$ |
\( T^{18} + \cdots + 35044924096512 \)
|
| $67$ |
\( T^{18} + \cdots + 125753131008 \)
|
| $71$ |
\( T^{18} + \cdots + 185585370693632 \)
|
| $73$ |
\( T^{18} + \cdots + 8321633353728 \)
|
| $79$ |
\( (T^{9} + 8 T^{8} + \cdots + 9453568)^{2} \)
|
| $83$ |
\( (T^{9} + 12 T^{8} + \cdots - 1161216)^{2} \)
|
| $89$ |
\( (T^{9} - 4 T^{8} + \cdots - 624000)^{2} \)
|
| $97$ |
\( (T^{9} - 4 T^{8} + \cdots - 29749888)^{2} \)
|
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