Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [720,2,Mod(163,720)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(720, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 3, 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("720.163");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 720.z (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: | \(5.74922894553\) |
| 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: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | no (minimal twist has level 240) |
| 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
\( x^{20} - 2 x^{19} + 3 x^{18} - 6 x^{17} + 2 x^{16} + 4 x^{14} + 20 x^{13} - 24 x^{12} + 40 x^{11} + \cdots + 1024 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 3 \nu^{19} + 6 \nu^{18} - 35 \nu^{17} + 34 \nu^{16} - 30 \nu^{15} + 88 \nu^{14} + 36 \nu^{13} + \cdots + 11776 ) / 2560 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - \nu^{19} + 2 \nu^{18} - 3 \nu^{17} + 6 \nu^{16} - 2 \nu^{15} - 4 \nu^{13} - 20 \nu^{12} + \cdots + 1024 ) / 512 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - \nu^{19} + \nu^{18} - \nu^{17} + 3 \nu^{16} + 4 \nu^{15} - 2 \nu^{14} - 4 \nu^{13} - 24 \nu^{12} + \cdots + 256 ) / 256 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 7 \nu^{19} + 2 \nu^{18} - 17 \nu^{17} - 22 \nu^{16} - 8 \nu^{15} - 20 \nu^{14} + 164 \nu^{13} + \cdots + 7680 ) / 1280 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 15 \nu^{19} - 2 \nu^{18} + 63 \nu^{17} + 2 \nu^{16} + 102 \nu^{15} - 152 \nu^{14} - 260 \nu^{13} + \cdots - 18944 ) / 2560 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 17 \nu^{19} + 8 \nu^{18} + 17 \nu^{17} + 32 \nu^{16} + 58 \nu^{15} - 60 \nu^{14} - 84 \nu^{13} + \cdots - 5120 ) / 2560 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 23 \nu^{19} - 8 \nu^{18} + 21 \nu^{17} - 72 \nu^{16} - 106 \nu^{15} - 36 \nu^{14} - 4 \nu^{13} + \cdots - 10752 ) / 2560 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 5 \nu^{19} - 32 \nu^{18} + 53 \nu^{17} - 8 \nu^{16} + 122 \nu^{15} - 12 \nu^{14} - 140 \nu^{13} + \cdots - 8704 ) / 2560 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 33 \nu^{19} - 6 \nu^{18} + 23 \nu^{17} - 114 \nu^{16} - 138 \nu^{15} - 104 \nu^{14} + 196 \nu^{13} + \cdots - 4608 ) / 2560 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 29 \nu^{19} - 30 \nu^{18} + 7 \nu^{17} - 50 \nu^{16} - 62 \nu^{15} + 96 \nu^{14} + 108 \nu^{13} + \cdots + 5632 ) / 2560 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 29 \nu^{19} - 38 \nu^{18} - 21 \nu^{17} - 82 \nu^{16} + 6 \nu^{15} + 168 \nu^{14} + 348 \nu^{13} + \cdots + 9216 ) / 2560 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 2 \nu^{19} + 5 \nu^{18} + 2 \nu^{17} + 3 \nu^{16} - 26 \nu^{14} - 16 \nu^{13} - 28 \nu^{12} + \cdots - 512 ) / 256 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 5 \nu^{19} + 27 \nu^{18} - 23 \nu^{17} + 13 \nu^{16} - 42 \nu^{15} - 58 \nu^{14} + 40 \nu^{13} + \cdots + 7424 ) / 1280 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 3 \nu^{18} - 6 \nu^{17} + 3 \nu^{16} - 8 \nu^{15} + 4 \nu^{14} + 14 \nu^{13} + 16 \nu^{12} + \cdots + 1792 ) / 128 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 13 \nu^{19} - 28 \nu^{18} + 21 \nu^{17} - 12 \nu^{16} + 24 \nu^{15} + 44 \nu^{14} - 44 \nu^{13} + \cdots - 1792 ) / 1280 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 13 \nu^{19} - 34 \nu^{18} + 35 \nu^{17} - 46 \nu^{16} + 30 \nu^{15} + 8 \nu^{14} - 4 \nu^{13} + \cdots - 8704 ) / 1280 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 13 \nu^{19} + 34 \nu^{18} - 35 \nu^{17} + 46 \nu^{16} - 30 \nu^{15} - 8 \nu^{14} + 4 \nu^{13} + \cdots + 8704 ) / 1280 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 17 \nu^{19} - 32 \nu^{18} + 39 \nu^{17} - 48 \nu^{16} + 6 \nu^{15} + 36 \nu^{14} - 36 \nu^{13} + \cdots - 12288 ) / 1280 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 17 \nu^{19} + 47 \nu^{18} - 39 \nu^{17} + 73 \nu^{16} - 26 \nu^{15} - 66 \nu^{14} - 44 \nu^{13} + \cdots + 14848 ) / 1280 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{17} + \beta_{16} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{14} - \beta_{12} + \beta_{10} + \beta_{7} + 2\beta_{6} + \beta_{5} + 2\beta_{3} - \beta_1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2 \beta_{18} + \beta_{14} + 2 \beta_{13} - \beta_{12} - \beta_{10} + 2 \beta_{8} + \beta_{7} + \cdots - \beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2 \beta_{18} + 2 \beta_{17} - \beta_{14} - \beta_{12} - 2 \beta_{11} - \beta_{10} - \beta_{7} - 2 \beta_{6} + \cdots + 4 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2 \beta_{19} - 2 \beta_{17} + 4 \beta_{16} - 2 \beta_{15} + 3 \beta_{14} + \beta_{12} + 2 \beta_{11} + \cdots - 4 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 2 \beta_{19} + 2 \beta_{18} - 2 \beta_{17} + \beta_{14} + 4 \beta_{13} - 3 \beta_{12} + \beta_{10} + \cdots - 2 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 2 \beta_{19} + 4 \beta_{18} - 2 \beta_{16} - 2 \beta_{15} - \beta_{14} + 4 \beta_{13} - 7 \beta_{12} + \cdots - 8 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 6 \beta_{19} + 6 \beta_{18} - 6 \beta_{17} + 4 \beta_{16} - 4 \beta_{15} + 3 \beta_{14} - \beta_{12} + \cdots + 2 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 6 \beta_{19} - 4 \beta_{18} - 8 \beta_{17} + 6 \beta_{16} - 6 \beta_{15} - 7 \beta_{14} + 15 \beta_{12} + \cdots - 20 ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 6 \beta_{19} - 14 \beta_{18} - 18 \beta_{17} - 4 \beta_{16} - 12 \beta_{15} - 11 \beta_{14} - 8 \beta_{13} + \cdots + 2 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 18 \beta_{19} + 4 \beta_{18} - 4 \beta_{17} + 14 \beta_{16} - 2 \beta_{15} + 3 \beta_{14} + 8 \beta_{13} + \cdots - 20 ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 10 \beta_{19} + 6 \beta_{18} + 10 \beta_{17} - 12 \beta_{16} - 4 \beta_{15} - 13 \beta_{14} - 40 \beta_{13} + \cdots + 30 ) / 2 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 10 \beta_{19} - 28 \beta_{18} - 20 \beta_{17} + 34 \beta_{16} - 38 \beta_{15} - 11 \beta_{14} + \cdots - 44 ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 18 \beta_{19} - 54 \beta_{18} + 22 \beta_{17} - 36 \beta_{16} + 20 \beta_{15} - 43 \beta_{14} + \cdots + 106 ) / 2 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 26 \beta_{19} + 12 \beta_{18} + 60 \beta_{17} + 6 \beta_{16} + 6 \beta_{15} + 27 \beta_{14} - 80 \beta_{13} + \cdots - 4 ) / 2 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( 34 \beta_{19} + 118 \beta_{18} + 106 \beta_{17} + 100 \beta_{16} + 44 \beta_{15} + 131 \beta_{14} + \cdots + 54 ) / 2 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( - 106 \beta_{19} + 4 \beta_{18} + 148 \beta_{17} - 86 \beta_{16} + 90 \beta_{15} + 45 \beta_{14} + \cdots + 4 ) / 2 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( - 162 \beta_{19} + 42 \beta_{18} + 246 \beta_{17} - 148 \beta_{16} + 36 \beta_{15} - 43 \beta_{14} + \cdots + 282 ) / 2 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( 186 \beta_{19} + 236 \beta_{18} + 236 \beta_{17} + 358 \beta_{16} + 230 \beta_{15} + 411 \beta_{14} + \cdots - 884 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).
| \(n\) | \(181\) | \(271\) | \(577\) | \(641\) |
| \(\chi(n)\) | \(-\beta_{6}\) | \(-1\) | \(-\beta_{6}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 163.1 |
|
−1.40626 | + | 0.149805i | 0 | 1.95512 | − | 0.421330i | 1.13192 | + | 1.92841i | 0 | 1.21767 | − | 1.21767i | −2.68628 | + | 0.885385i | 0 | −1.88066 | − | 2.54227i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.2 | −1.39051 | − | 0.257862i | 0 | 1.86701 | + | 0.717118i | −0.778677 | − | 2.09611i | 0 | −2.44055 | + | 2.44055i | −2.41118 | − | 1.47859i | 0 | 0.542249 | + | 3.11544i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.3 | −1.13541 | − | 0.843121i | 0 | 0.578296 | + | 1.91457i | −2.03471 | + | 0.927338i | 0 | 1.72177 | − | 1.72177i | 0.957612 | − | 2.66139i | 0 | 3.09208 | + | 0.662601i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.4 | −0.504160 | + | 1.32130i | 0 | −1.49164 | − | 1.33229i | 0.154491 | + | 2.23072i | 0 | −1.27936 | + | 1.27936i | 2.51238 | − | 1.29922i | 0 | −3.02534 | − | 0.920515i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.5 | 0.0245121 | − | 1.41400i | 0 | −1.99880 | − | 0.0693203i | 1.85415 | + | 1.24985i | 0 | 1.96536 | − | 1.96536i | −0.147014 | + | 2.82460i | 0 | 1.81274 | − | 2.59113i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.6 | 0.137540 | + | 1.40751i | 0 | −1.96217 | + | 0.387177i | 1.76195 | − | 1.37678i | 0 | 0.159531 | − | 0.159531i | −0.814832 | − | 2.70851i | 0 | 2.18017 | + | 2.29061i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.7 | 0.597511 | + | 1.28179i | 0 | −1.28596 | + | 1.53177i | −2.18941 | + | 0.454390i | 0 | 0.328507 | − | 0.328507i | −2.73177 | − | 0.733083i | 0 | −1.89063 | − | 2.53486i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.8 | 0.897004 | − | 1.09334i | 0 | −0.390769 | − | 1.96145i | −0.140415 | + | 2.23165i | 0 | −2.83167 | + | 2.83167i | −2.49505 | − | 1.33219i | 0 | 2.31400 | + | 2.15532i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.9 | 1.36850 | + | 0.356677i | 0 | 1.74556 | + | 0.976222i | −1.94894 | − | 1.09619i | 0 | −2.09269 | + | 2.09269i | 2.04060 | + | 1.95856i | 0 | −2.27614 | − | 2.19527i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.10 | 1.41127 | + | 0.0912451i | 0 | 1.98335 | + | 0.257542i | 2.18964 | − | 0.453294i | 0 | 1.25143 | − | 1.25143i | 2.77553 | + | 0.544432i | 0 | 3.13153 | − | 0.439925i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 667.1 | −1.40626 | − | 0.149805i | 0 | 1.95512 | + | 0.421330i | 1.13192 | − | 1.92841i | 0 | 1.21767 | + | 1.21767i | −2.68628 | − | 0.885385i | 0 | −1.88066 | + | 2.54227i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 667.2 | −1.39051 | + | 0.257862i | 0 | 1.86701 | − | 0.717118i | −0.778677 | + | 2.09611i | 0 | −2.44055 | − | 2.44055i | −2.41118 | + | 1.47859i | 0 | 0.542249 | − | 3.11544i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 667.3 | −1.13541 | + | 0.843121i | 0 | 0.578296 | − | 1.91457i | −2.03471 | − | 0.927338i | 0 | 1.72177 | + | 1.72177i | 0.957612 | + | 2.66139i | 0 | 3.09208 | − | 0.662601i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 667.4 | −0.504160 | − | 1.32130i | 0 | −1.49164 | + | 1.33229i | 0.154491 | − | 2.23072i | 0 | −1.27936 | − | 1.27936i | 2.51238 | + | 1.29922i | 0 | −3.02534 | + | 0.920515i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 667.5 | 0.0245121 | + | 1.41400i | 0 | −1.99880 | + | 0.0693203i | 1.85415 | − | 1.24985i | 0 | 1.96536 | + | 1.96536i | −0.147014 | − | 2.82460i | 0 | 1.81274 | + | 2.59113i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 667.6 | 0.137540 | − | 1.40751i | 0 | −1.96217 | − | 0.387177i | 1.76195 | + | 1.37678i | 0 | 0.159531 | + | 0.159531i | −0.814832 | + | 2.70851i | 0 | 2.18017 | − | 2.29061i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 667.7 | 0.597511 | − | 1.28179i | 0 | −1.28596 | − | 1.53177i | −2.18941 | − | 0.454390i | 0 | 0.328507 | + | 0.328507i | −2.73177 | + | 0.733083i | 0 | −1.89063 | + | 2.53486i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 667.8 | 0.897004 | + | 1.09334i | 0 | −0.390769 | + | 1.96145i | −0.140415 | − | 2.23165i | 0 | −2.83167 | − | 2.83167i | −2.49505 | + | 1.33219i | 0 | 2.31400 | − | 2.15532i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 667.9 | 1.36850 | − | 0.356677i | 0 | 1.74556 | − | 0.976222i | −1.94894 | + | 1.09619i | 0 | −2.09269 | − | 2.09269i | 2.04060 | − | 1.95856i | 0 | −2.27614 | + | 2.19527i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 667.10 | 1.41127 | − | 0.0912451i | 0 | 1.98335 | − | 0.257542i | 2.18964 | + | 0.453294i | 0 | 1.25143 | + | 1.25143i | 2.77553 | − | 0.544432i | 0 | 3.13153 | + | 0.439925i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 80.s | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 720.2.z.h | 20 | |
| 3.b | odd | 2 | 1 | 240.2.y.f | ✓ | 20 | |
| 5.c | odd | 4 | 1 | 720.2.bd.h | 20 | ||
| 12.b | even | 2 | 1 | 960.2.y.f | 20 | ||
| 15.e | even | 4 | 1 | 240.2.bc.f | yes | 20 | |
| 16.f | odd | 4 | 1 | 720.2.bd.h | 20 | ||
| 24.f | even | 2 | 1 | 1920.2.y.k | 20 | ||
| 24.h | odd | 2 | 1 | 1920.2.y.l | 20 | ||
| 48.i | odd | 4 | 1 | 960.2.bc.f | 20 | ||
| 48.i | odd | 4 | 1 | 1920.2.bc.l | 20 | ||
| 48.k | even | 4 | 1 | 240.2.bc.f | yes | 20 | |
| 48.k | even | 4 | 1 | 1920.2.bc.k | 20 | ||
| 60.l | odd | 4 | 1 | 960.2.bc.f | 20 | ||
| 80.s | even | 4 | 1 | inner | 720.2.z.h | 20 | |
| 120.q | odd | 4 | 1 | 1920.2.bc.l | 20 | ||
| 120.w | even | 4 | 1 | 1920.2.bc.k | 20 | ||
| 240.z | odd | 4 | 1 | 240.2.y.f | ✓ | 20 | |
| 240.bb | even | 4 | 1 | 960.2.y.f | 20 | ||
| 240.bd | odd | 4 | 1 | 1920.2.y.l | 20 | ||
| 240.bf | even | 4 | 1 | 1920.2.y.k | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 240.2.y.f | ✓ | 20 | 3.b | odd | 2 | 1 | |
| 240.2.y.f | ✓ | 20 | 240.z | odd | 4 | 1 | |
| 240.2.bc.f | yes | 20 | 15.e | even | 4 | 1 | |
| 240.2.bc.f | yes | 20 | 48.k | even | 4 | 1 | |
| 720.2.z.h | 20 | 1.a | even | 1 | 1 | trivial | |
| 720.2.z.h | 20 | 80.s | even | 4 | 1 | inner | |
| 720.2.bd.h | 20 | 5.c | odd | 4 | 1 | ||
| 720.2.bd.h | 20 | 16.f | odd | 4 | 1 | ||
| 960.2.y.f | 20 | 12.b | even | 2 | 1 | ||
| 960.2.y.f | 20 | 240.bb | even | 4 | 1 | ||
| 960.2.bc.f | 20 | 48.i | odd | 4 | 1 | ||
| 960.2.bc.f | 20 | 60.l | odd | 4 | 1 | ||
| 1920.2.y.k | 20 | 24.f | even | 2 | 1 | ||
| 1920.2.y.k | 20 | 240.bf | even | 4 | 1 | ||
| 1920.2.y.l | 20 | 24.h | odd | 2 | 1 | ||
| 1920.2.y.l | 20 | 240.bd | odd | 4 | 1 | ||
| 1920.2.bc.k | 20 | 48.k | even | 4 | 1 | ||
| 1920.2.bc.k | 20 | 120.w | even | 4 | 1 | ||
| 1920.2.bc.l | 20 | 48.i | odd | 4 | 1 | ||
| 1920.2.bc.l | 20 | 120.q | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(720, [\chi])\):
|
\( T_{7}^{20} + 4 T_{7}^{19} + 8 T_{7}^{18} - 32 T_{7}^{17} + 140 T_{7}^{16} + 448 T_{7}^{15} + \cdots + 25600 \)
|
|
\( T_{11}^{20} + 8 T_{11}^{19} + 32 T_{11}^{18} + 24 T_{11}^{17} + 1164 T_{11}^{16} + 8656 T_{11}^{15} + \cdots + 6390784 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} - T^{18} + \cdots + 1024 \)
|
| $3$ |
\( T^{20} \)
|
| $5$ |
\( T^{20} - 2 T^{18} + \cdots + 9765625 \)
|
| $7$ |
\( T^{20} + 4 T^{19} + \cdots + 25600 \)
|
| $11$ |
\( T^{20} + 8 T^{19} + \cdots + 6390784 \)
|
| $13$ |
\( T^{20} + \cdots + 1340145664 \)
|
| $17$ |
\( T^{20} + \cdots + 37171840000 \)
|
| $19$ |
\( T^{20} - 16 T^{19} + \cdots + 6553600 \)
|
| $23$ |
\( T^{20} + \cdots + 8971878400 \)
|
| $29$ |
\( T^{20} + \cdots + 8641718502400 \)
|
| $31$ |
\( T^{20} + 232 T^{18} + \cdots + 98406400 \)
|
| $37$ |
\( T^{20} + \cdots + 241340317696 \)
|
| $41$ |
\( T^{20} + \cdots + 1717986918400 \)
|
| $43$ |
\( T^{20} + \cdots + 881852416 \)
|
| $47$ |
\( T^{20} + \cdots + 163840000 \)
|
| $53$ |
\( (T^{10} + 4 T^{9} + \cdots - 454907840)^{2} \)
|
| $59$ |
\( T^{20} + \cdots + 1603768960000 \)
|
| $61$ |
\( T^{20} + \cdots + 55960453571584 \)
|
| $67$ |
\( T^{20} + \cdots + 31\!\cdots\!04 \)
|
| $71$ |
\( (T^{10} - 312 T^{8} + \cdots + 147865600)^{2} \)
|
| $73$ |
\( T^{20} + \cdots + 631014400 \)
|
| $79$ |
\( (T^{10} - 24 T^{9} + \cdots + 222791680)^{2} \)
|
| $83$ |
\( (T^{10} - 12 T^{9} + \cdots - 32616448)^{2} \)
|
| $89$ |
\( (T^{10} + 20 T^{9} + \cdots - 7429120)^{2} \)
|
| $97$ |
\( T^{20} + \cdots + 1895578240000 \)
|
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