Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [240,2,Mod(43,240)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(240, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 1, 0, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("240.43");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 240 = 2^{4} \cdot 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 240.bc (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: | \(1.91640964851\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(10\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 2 x^{19} + 3 x^{18} - 6 x^{17} + 2 x^{16} + 4 x^{14} + 20 x^{13} - 24 x^{12} + 40 x^{11} + \cdots + 1024 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| 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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} - 2 x^{19} + 3 x^{18} - 6 x^{17} + 2 x^{16} + 4 x^{14} + 20 x^{13} - 24 x^{12} + 40 x^{11} + \cdots + 1024 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 3 \nu^{19} - 6 \nu^{18} + 35 \nu^{17} - 34 \nu^{16} + 30 \nu^{15} - 88 \nu^{14} - 36 \nu^{13} + \cdots - 11776 ) / 2560 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 15 \nu^{19} + 2 \nu^{18} - 63 \nu^{17} - 2 \nu^{16} - 102 \nu^{15} + 152 \nu^{14} + 260 \nu^{13} + \cdots + 18944 ) / 2560 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 17 \nu^{19} + 8 \nu^{18} + 17 \nu^{17} + 32 \nu^{16} + 58 \nu^{15} - 60 \nu^{14} - 84 \nu^{13} + \cdots - 5120 ) / 2560 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2 \nu^{19} + \nu^{18} - 8 \nu^{17} - \nu^{16} - 2 \nu^{15} + 14 \nu^{14} + 44 \nu^{13} + 20 \nu^{12} + \cdots + 3328 ) / 320 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 7 \nu^{19} - 8 \nu^{18} - 7 \nu^{17} - 32 \nu^{16} + 22 \nu^{15} + 20 \nu^{14} + 144 \nu^{13} + \cdots + 7680 ) / 1280 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 23 \nu^{19} - 8 \nu^{18} + 21 \nu^{17} - 72 \nu^{16} - 106 \nu^{15} - 36 \nu^{14} - 4 \nu^{13} + \cdots - 10752 ) / 2560 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 11 \nu^{19} + 5 \nu^{18} + 3 \nu^{17} - 25 \nu^{16} - 68 \nu^{15} - 26 \nu^{14} + 12 \nu^{13} + \cdots - 5632 ) / 1280 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 13 \nu^{19} - 36 \nu^{18} + 23 \nu^{17} - 84 \nu^{16} + 2 \nu^{15} + 116 \nu^{14} + 116 \nu^{13} + \cdots - 17408 ) / 2560 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 29 \nu^{19} - 30 \nu^{18} + 7 \nu^{17} - 50 \nu^{16} - 62 \nu^{15} + 96 \nu^{14} + 108 \nu^{13} + \cdots + 5632 ) / 2560 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 5 \nu^{19} - 44 \nu^{18} + 31 \nu^{17} + 4 \nu^{16} + 74 \nu^{15} + 116 \nu^{14} - 100 \nu^{13} + \cdots - 9728 ) / 2560 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 2 \nu^{19} + 5 \nu^{18} + 2 \nu^{17} + 3 \nu^{16} - 26 \nu^{14} - 16 \nu^{13} - 28 \nu^{12} + \cdots - 512 ) / 256 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 4 \nu^{19} + 25 \nu^{18} - 58 \nu^{17} - 5 \nu^{16} - 102 \nu^{15} + 26 \nu^{14} + 228 \nu^{13} + \cdots + 15872 ) / 1280 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 3 \nu^{18} + 6 \nu^{17} - 3 \nu^{16} + 8 \nu^{15} - 4 \nu^{14} - 14 \nu^{13} - 16 \nu^{12} + \cdots - 1792 ) / 128 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 15 \nu^{19} + 64 \nu^{18} - 61 \nu^{17} + 56 \nu^{16} - 94 \nu^{15} - 116 \nu^{14} + 60 \nu^{13} + \cdots + 19968 ) / 2560 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 33 \nu^{19} - 64 \nu^{18} + 35 \nu^{17} - 56 \nu^{16} + 10 \nu^{15} + 188 \nu^{14} + 36 \nu^{13} + \cdots - 3584 ) / 2560 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 13 \nu^{19} + 34 \nu^{18} - 35 \nu^{17} + 46 \nu^{16} - 30 \nu^{15} - 8 \nu^{14} + 4 \nu^{13} + \cdots + 8704 ) / 1280 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 43 \nu^{19} - 60 \nu^{18} + 69 \nu^{17} - 140 \nu^{16} - 54 \nu^{15} + 12 \nu^{14} + 116 \nu^{13} + \cdots - 19456 ) / 2560 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} - \beta_{3} \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{17} - \beta_{13} + \beta_{11} + \beta_{9} - \beta_{8} - \beta_{4} - \beta_{2} + 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( - \beta_{19} + \beta_{17} - 2 \beta_{16} - \beta_{15} + \beta_{14} - \beta_{10} + \beta_{8} + \beta_{5} + \cdots - 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( - \beta_{19} - 2 \beta_{16} + \beta_{15} - 2 \beta_{12} - \beta_{10} + \beta_{9} + \beta_{7} - \beta_{6} + \cdots - 1 \)
|
| \(\nu^{7}\) | \(=\) |
\( \beta_{19} + 2 \beta_{18} - \beta_{17} + \beta_{15} + \beta_{14} - 2 \beta_{13} + 2 \beta_{11} + \cdots - 5 \)
|
| \(\nu^{8}\) | \(=\) |
\( \beta_{19} + 2 \beta_{18} + 2 \beta_{17} - 4 \beta_{16} + \beta_{15} + 2 \beta_{14} - 2 \beta_{13} + \cdots + 5 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 5 \beta_{19} + 2 \beta_{18} + 3 \beta_{17} - 4 \beta_{16} + 3 \beta_{15} + 3 \beta_{14} + 4 \beta_{13} + \cdots - 7 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 3 \beta_{19} + 2 \beta_{18} - 4 \beta_{16} + 9 \beta_{15} + 6 \beta_{14} - 3 \beta_{10} - 7 \beta_{9} + \cdots + 5 \)
|
| \(\nu^{11}\) | \(=\) |
\( 9 \beta_{19} + 10 \beta_{18} + 9 \beta_{17} + 9 \beta_{15} + \beta_{14} - 8 \beta_{13} - 12 \beta_{12} + \cdots - 1 \)
|
| \(\nu^{12}\) | \(=\) |
\( 11 \beta_{19} + 14 \beta_{18} - 4 \beta_{17} - 4 \beta_{16} + 7 \beta_{15} + 2 \beta_{14} + 4 \beta_{13} + \cdots + 11 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 5 \beta_{19} - 2 \beta_{18} + 11 \beta_{17} + 8 \beta_{16} + 11 \beta_{15} + 19 \beta_{14} + 16 \beta_{13} + \cdots + 5 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 7 \beta_{19} + 26 \beta_{18} + 16 \beta_{17} + 20 \beta_{16} + 45 \beta_{15} - 10 \beta_{14} + \cdots + 97 \)
|
| \(\nu^{15}\) | \(=\) |
\( 21 \beta_{19} + 26 \beta_{18} - 11 \beta_{17} - 16 \beta_{16} + 5 \beta_{15} - 3 \beta_{14} + 20 \beta_{12} + \cdots + 3 \)
|
| \(\nu^{16}\) | \(=\) |
\( 63 \beta_{19} - 18 \beta_{18} - 16 \beta_{17} + 4 \beta_{16} - 45 \beta_{15} - 22 \beta_{14} - 32 \beta_{13} + \cdots + 55 \)
|
| \(\nu^{17}\) | \(=\) |
\( 3 \beta_{19} + 54 \beta_{18} + 35 \beta_{17} + 80 \beta_{16} - 29 \beta_{15} - 45 \beta_{14} + 40 \beta_{13} + \cdots + 21 \)
|
| \(\nu^{18}\) | \(=\) |
\( - 31 \beta_{19} + 66 \beta_{18} - 32 \beta_{17} - 20 \beta_{16} + 77 \beta_{15} - 18 \beta_{14} + \cdots + 265 \)
|
| \(\nu^{19}\) | \(=\) |
\( 21 \beta_{19} - 22 \beta_{18} + 61 \beta_{17} - 75 \beta_{15} - 115 \beta_{14} - 136 \beta_{13} + \cdots - 173 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(97\) | \(161\) | \(181\) |
| \(\chi(n)\) | \(-1\) | \(-\beta_{5}\) | \(1\) | \(\beta_{5}\) |
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 |
|
−1.41400 | + | 0.0245121i | − | 1.00000i | 1.99880 | − | 0.0693203i | 1.24985 | − | 1.85415i | 0.0245121 | + | 1.41400i | 1.96536 | − | 1.96536i | −2.82460 | + | 0.147014i | −1.00000 | −1.72184 | + | 2.65241i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.2 | −1.09334 | + | 0.897004i | − | 1.00000i | 0.390769 | − | 1.96145i | 2.23165 | + | 0.140415i | 0.897004 | + | 1.09334i | −2.83167 | + | 2.83167i | 1.33219 | + | 2.49505i | −1.00000 | −2.56590 | + | 1.84828i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.3 | −0.843121 | − | 1.13541i | − | 1.00000i | −0.578296 | + | 1.91457i | 0.927338 | + | 2.03471i | −1.13541 | + | 0.843121i | 1.72177 | − | 1.72177i | 2.66139 | − | 0.957612i | −1.00000 | 1.52837 | − | 2.76841i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.4 | −0.257862 | − | 1.39051i | − | 1.00000i | −1.86701 | + | 0.717118i | −2.09611 | + | 0.778677i | −1.39051 | + | 0.257862i | −2.44055 | + | 2.44055i | 1.47859 | + | 2.41118i | −1.00000 | 1.62326 | + | 2.71386i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.5 | 0.0912451 | + | 1.41127i | − | 1.00000i | −1.98335 | + | 0.257542i | −0.453294 | − | 2.18964i | 1.41127 | − | 0.0912451i | 1.25143 | − | 1.25143i | −0.544432 | − | 2.77553i | −1.00000 | 3.04881 | − | 0.839513i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.6 | 0.149805 | − | 1.40626i | − | 1.00000i | −1.95512 | − | 0.421330i | 1.92841 | − | 1.13192i | −1.40626 | − | 0.149805i | 1.21767 | − | 1.21767i | −0.885385 | + | 2.68628i | −1.00000 | −1.30289 | − | 2.88140i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.7 | 0.356677 | + | 1.36850i | − | 1.00000i | −1.74556 | + | 0.976222i | −1.09619 | + | 1.94894i | 1.36850 | − | 0.356677i | −2.09269 | + | 2.09269i | −1.95856 | − | 2.04060i | −1.00000 | −3.05810 | − | 0.804985i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.8 | 1.28179 | + | 0.597511i | − | 1.00000i | 1.28596 | + | 1.53177i | 0.454390 | + | 2.18941i | 0.597511 | − | 1.28179i | 0.328507 | − | 0.328507i | 0.733083 | + | 2.73177i | −1.00000 | −0.725767 | + | 3.07787i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.9 | 1.32130 | − | 0.504160i | − | 1.00000i | 1.49164 | − | 1.33229i | 2.23072 | − | 0.154491i | −0.504160 | − | 1.32130i | −1.27936 | + | 1.27936i | 1.29922 | − | 2.51238i | −1.00000 | 2.86956 | − | 1.32877i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.10 | 1.40751 | + | 0.137540i | − | 1.00000i | 1.96217 | + | 0.387177i | −1.37678 | − | 1.76195i | 0.137540 | − | 1.40751i | 0.159531 | − | 0.159531i | 2.70851 | + | 0.814832i | −1.00000 | −1.69549 | − | 2.66933i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 67.1 | −1.41400 | − | 0.0245121i | 1.00000i | 1.99880 | + | 0.0693203i | 1.24985 | + | 1.85415i | 0.0245121 | − | 1.41400i | 1.96536 | + | 1.96536i | −2.82460 | − | 0.147014i | −1.00000 | −1.72184 | − | 2.65241i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 67.2 | −1.09334 | − | 0.897004i | 1.00000i | 0.390769 | + | 1.96145i | 2.23165 | − | 0.140415i | 0.897004 | − | 1.09334i | −2.83167 | − | 2.83167i | 1.33219 | − | 2.49505i | −1.00000 | −2.56590 | − | 1.84828i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 67.3 | −0.843121 | + | 1.13541i | 1.00000i | −0.578296 | − | 1.91457i | 0.927338 | − | 2.03471i | −1.13541 | − | 0.843121i | 1.72177 | + | 1.72177i | 2.66139 | + | 0.957612i | −1.00000 | 1.52837 | + | 2.76841i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 67.4 | −0.257862 | + | 1.39051i | 1.00000i | −1.86701 | − | 0.717118i | −2.09611 | − | 0.778677i | −1.39051 | − | 0.257862i | −2.44055 | − | 2.44055i | 1.47859 | − | 2.41118i | −1.00000 | 1.62326 | − | 2.71386i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 67.5 | 0.0912451 | − | 1.41127i | 1.00000i | −1.98335 | − | 0.257542i | −0.453294 | + | 2.18964i | 1.41127 | + | 0.0912451i | 1.25143 | + | 1.25143i | −0.544432 | + | 2.77553i | −1.00000 | 3.04881 | + | 0.839513i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 67.6 | 0.149805 | + | 1.40626i | 1.00000i | −1.95512 | + | 0.421330i | 1.92841 | + | 1.13192i | −1.40626 | + | 0.149805i | 1.21767 | + | 1.21767i | −0.885385 | − | 2.68628i | −1.00000 | −1.30289 | + | 2.88140i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 67.7 | 0.356677 | − | 1.36850i | 1.00000i | −1.74556 | − | 0.976222i | −1.09619 | − | 1.94894i | 1.36850 | + | 0.356677i | −2.09269 | − | 2.09269i | −1.95856 | + | 2.04060i | −1.00000 | −3.05810 | + | 0.804985i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 67.8 | 1.28179 | − | 0.597511i | 1.00000i | 1.28596 | − | 1.53177i | 0.454390 | − | 2.18941i | 0.597511 | + | 1.28179i | 0.328507 | + | 0.328507i | 0.733083 | − | 2.73177i | −1.00000 | −0.725767 | − | 3.07787i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 67.9 | 1.32130 | + | 0.504160i | 1.00000i | 1.49164 | + | 1.33229i | 2.23072 | + | 0.154491i | −0.504160 | + | 1.32130i | −1.27936 | − | 1.27936i | 1.29922 | + | 2.51238i | −1.00000 | 2.86956 | + | 1.32877i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 67.10 | 1.40751 | − | 0.137540i | 1.00000i | 1.96217 | − | 0.387177i | −1.37678 | + | 1.76195i | 0.137540 | + | 1.40751i | 0.159531 | + | 0.159531i | 2.70851 | − | 0.814832i | −1.00000 | −1.69549 | + | 2.66933i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 80.j | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 240.2.bc.f | yes | 20 |
| 3.b | odd | 2 | 1 | 720.2.bd.h | 20 | ||
| 4.b | odd | 2 | 1 | 960.2.bc.f | 20 | ||
| 5.c | odd | 4 | 1 | 240.2.y.f | ✓ | 20 | |
| 8.b | even | 2 | 1 | 1920.2.bc.k | 20 | ||
| 8.d | odd | 2 | 1 | 1920.2.bc.l | 20 | ||
| 15.e | even | 4 | 1 | 720.2.z.h | 20 | ||
| 16.e | even | 4 | 1 | 960.2.y.f | 20 | ||
| 16.e | even | 4 | 1 | 1920.2.y.k | 20 | ||
| 16.f | odd | 4 | 1 | 240.2.y.f | ✓ | 20 | |
| 16.f | odd | 4 | 1 | 1920.2.y.l | 20 | ||
| 20.e | even | 4 | 1 | 960.2.y.f | 20 | ||
| 40.i | odd | 4 | 1 | 1920.2.y.l | 20 | ||
| 40.k | even | 4 | 1 | 1920.2.y.k | 20 | ||
| 48.k | even | 4 | 1 | 720.2.z.h | 20 | ||
| 80.i | odd | 4 | 1 | 1920.2.bc.l | 20 | ||
| 80.j | even | 4 | 1 | inner | 240.2.bc.f | yes | 20 |
| 80.s | even | 4 | 1 | 1920.2.bc.k | 20 | ||
| 80.t | odd | 4 | 1 | 960.2.bc.f | 20 | ||
| 240.bd | odd | 4 | 1 | 720.2.bd.h | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 240.2.y.f | ✓ | 20 | 5.c | odd | 4 | 1 | |
| 240.2.y.f | ✓ | 20 | 16.f | odd | 4 | 1 | |
| 240.2.bc.f | yes | 20 | 1.a | even | 1 | 1 | trivial |
| 240.2.bc.f | yes | 20 | 80.j | even | 4 | 1 | inner |
| 720.2.z.h | 20 | 15.e | even | 4 | 1 | ||
| 720.2.z.h | 20 | 48.k | even | 4 | 1 | ||
| 720.2.bd.h | 20 | 3.b | odd | 2 | 1 | ||
| 720.2.bd.h | 20 | 240.bd | odd | 4 | 1 | ||
| 960.2.y.f | 20 | 16.e | even | 4 | 1 | ||
| 960.2.y.f | 20 | 20.e | even | 4 | 1 | ||
| 960.2.bc.f | 20 | 4.b | odd | 2 | 1 | ||
| 960.2.bc.f | 20 | 80.t | odd | 4 | 1 | ||
| 1920.2.y.k | 20 | 16.e | even | 4 | 1 | ||
| 1920.2.y.k | 20 | 40.k | even | 4 | 1 | ||
| 1920.2.y.l | 20 | 16.f | odd | 4 | 1 | ||
| 1920.2.y.l | 20 | 40.i | odd | 4 | 1 | ||
| 1920.2.bc.k | 20 | 8.b | even | 2 | 1 | ||
| 1920.2.bc.k | 20 | 80.s | even | 4 | 1 | ||
| 1920.2.bc.l | 20 | 8.d | odd | 2 | 1 | ||
| 1920.2.bc.l | 20 | 80.i | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(240, [\chi])\):
|
\( T_{7}^{20} + 4 T_{7}^{19} + 8 T_{7}^{18} - 32 T_{7}^{17} + 140 T_{7}^{16} + 448 T_{7}^{15} + \cdots + 25600 \)
|
|
\( T_{11}^{20} - 8 T_{11}^{19} + 32 T_{11}^{18} - 24 T_{11}^{17} + 1164 T_{11}^{16} - 8656 T_{11}^{15} + \cdots + 6390784 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} - 2 T^{19} + \cdots + 1024 \)
|
| $3$ |
\( (T^{2} + 1)^{10} \)
|
| $5$ |
\( T^{20} - 8 T^{19} + \cdots + 9765625 \)
|
| $7$ |
\( T^{20} + 4 T^{19} + \cdots + 25600 \)
|
| $11$ |
\( T^{20} - 8 T^{19} + \cdots + 6390784 \)
|
| $13$ |
\( (T^{10} - 4 T^{9} + \cdots - 36608)^{2} \)
|
| $17$ |
\( T^{20} + \cdots + 37171840000 \)
|
| $19$ |
\( T^{20} + 16 T^{19} + \cdots + 6553600 \)
|
| $23$ |
\( T^{20} + \cdots + 8971878400 \)
|
| $29$ |
\( T^{20} + \cdots + 8641718502400 \)
|
| $31$ |
\( T^{20} + 232 T^{18} + \cdots + 98406400 \)
|
| $37$ |
\( (T^{10} - 12 T^{9} + \cdots - 491264)^{2} \)
|
| $41$ |
\( T^{20} + \cdots + 1717986918400 \)
|
| $43$ |
\( (T^{10} + 4 T^{9} + \cdots + 29696)^{2} \)
|
| $47$ |
\( T^{20} + \cdots + 163840000 \)
|
| $53$ |
\( T^{20} + \cdots + 20\!\cdots\!00 \)
|
| $59$ |
\( T^{20} + \cdots + 1603768960000 \)
|
| $61$ |
\( T^{20} + \cdots + 55960453571584 \)
|
| $67$ |
\( (T^{10} + 8 T^{9} + \cdots + 56394752)^{2} \)
|
| $71$ |
\( (T^{10} - 312 T^{8} + \cdots + 147865600)^{2} \)
|
| $73$ |
\( T^{20} + \cdots + 631014400 \)
|
| $79$ |
\( (T^{10} + 24 T^{9} + \cdots + 222791680)^{2} \)
|
| $83$ |
\( T^{20} + \cdots + 10\!\cdots\!04 \)
|
| $89$ |
\( (T^{10} + 20 T^{9} + \cdots - 7429120)^{2} \)
|
| $97$ |
\( T^{20} + \cdots + 1895578240000 \)
|
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