Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [968,2,Mod(485,968)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(968, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("968.485");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 968 = 2^{3} \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 968.c (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: | \(7.72951891566\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - x^{18} - 2 x^{16} - 2 x^{15} - 4 x^{14} - 4 x^{13} + 12 x^{12} + 16 x^{11} + 32 x^{9} + \cdots + 1024 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 88) |
| 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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} - x^{18} - 2 x^{16} - 2 x^{15} - 4 x^{14} - 4 x^{13} + 12 x^{12} + 16 x^{11} + 32 x^{9} + \cdots + 1024 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} + \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( -\nu^{2} + \nu \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 3 \nu^{19} + 10 \nu^{18} + 39 \nu^{17} + 30 \nu^{16} + 18 \nu^{15} - 86 \nu^{14} - 128 \nu^{13} + \cdots - 2560 ) / 12288 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{18} - \nu^{16} - 2 \nu^{14} - 2 \nu^{13} - 4 \nu^{12} - 4 \nu^{11} + 12 \nu^{10} + 16 \nu^{9} + \cdots - 256 ) / 256 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - \nu^{19} - 10 \nu^{18} - 15 \nu^{17} - 6 \nu^{16} - 22 \nu^{15} - 26 \nu^{14} + 80 \nu^{13} + \cdots - 512 ) / 3072 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3 \nu^{19} + 2 \nu^{18} + 9 \nu^{17} + 6 \nu^{16} + 30 \nu^{15} + 14 \nu^{14} + 8 \nu^{13} + \cdots + 2560 ) / 3072 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{19} + 2 \nu^{18} - \nu^{17} - 2 \nu^{16} + 6 \nu^{15} - 6 \nu^{14} + 20 \nu^{12} + 4 \nu^{11} + \cdots + 512 ) / 1024 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( \nu^{19} + 2 \nu^{18} + 3 \nu^{17} + 6 \nu^{16} + 10 \nu^{15} + 18 \nu^{14} - 4 \nu^{12} - 28 \nu^{11} + \cdots - 512 ) / 1024 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 7 \nu^{19} + 10 \nu^{18} - 21 \nu^{17} - 18 \nu^{16} + 26 \nu^{15} + 66 \nu^{14} + 16 \nu^{13} + \cdots + 3584 ) / 6144 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 5 \nu^{19} - 6 \nu^{18} - \nu^{17} - 18 \nu^{16} - 30 \nu^{15} - 6 \nu^{14} - 52 \nu^{12} + \cdots - 2560 ) / 4096 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 5 \nu^{19} - 10 \nu^{18} + 17 \nu^{17} + 2 \nu^{16} + 46 \nu^{15} + 70 \nu^{14} + 32 \nu^{13} + \cdots + 6656 ) / 4096 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 15 \nu^{19} - 46 \nu^{18} - 45 \nu^{17} - 42 \nu^{16} + 42 \nu^{15} + 50 \nu^{14} + 128 \nu^{13} + \cdots + 24064 ) / 12288 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 5 \nu^{19} + 6 \nu^{18} - 3 \nu^{17} - 6 \nu^{16} + 10 \nu^{15} - 2 \nu^{14} + 48 \nu^{13} + \cdots + 4608 ) / 3072 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 23 \nu^{19} - 30 \nu^{18} - 21 \nu^{17} + 6 \nu^{16} + 10 \nu^{15} + 34 \nu^{14} + 192 \nu^{13} + \cdots + 7680 ) / 12288 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( \nu^{19} - \nu^{17} - 2 \nu^{15} - 2 \nu^{14} - 4 \nu^{13} - 4 \nu^{12} + 12 \nu^{11} + 16 \nu^{10} + \cdots - 256 \nu ) / 512 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 27 \nu^{19} - 70 \nu^{18} - 81 \nu^{17} - 18 \nu^{16} + 114 \nu^{15} + 314 \nu^{14} + \cdots + 30208 ) / 12288 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( \nu^{19} + \nu^{15} - 2 \nu^{14} - 6 \nu^{13} - 6 \nu^{12} - 12 \nu^{11} + 4 \nu^{10} + 12 \nu^{9} + \cdots - 384 ) / 384 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 47 \nu^{19} - 78 \nu^{18} - 45 \nu^{17} - 42 \nu^{16} + 10 \nu^{15} + 274 \nu^{14} + 384 \nu^{13} + \cdots + 19968 ) / 12288 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 37 \nu^{19} + 34 \nu^{18} + 15 \nu^{17} + 54 \nu^{16} - 14 \nu^{15} - 166 \nu^{14} - 368 \nu^{13} + \cdots - 27136 ) / 6144 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 2 \beta_{18} + 2 \beta_{16} - 2 \beta_{15} + 2 \beta_{12} - 2 \beta_{11} + 2 \beta_{10} + \cdots - \beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2 \beta_{18} + 2 \beta_{17} - 2 \beta_{14} + 2 \beta_{13} - 2 \beta_{11} - 2 \beta_{7} + 2 \beta_{6} + \cdots + \beta_1 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2 \beta_{18} - 2 \beta_{17} - 2 \beta_{14} - 2 \beta_{13} - 2 \beta_{11} + 2 \beta_{7} + 2 \beta_{6} + \cdots + \beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 2 \beta_{18} + 2 \beta_{17} - 4 \beta_{15} + 2 \beta_{14} + 2 \beta_{13} + 4 \beta_{12} - 2 \beta_{11} + \cdots + 4 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 4 \beta_{19} + 2 \beta_{18} - 2 \beta_{17} + 4 \beta_{15} + 10 \beta_{14} + 2 \beta_{13} + 2 \beta_{11} + \cdots + 8 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 4 \beta_{19} - 6 \beta_{18} - 2 \beta_{17} - 4 \beta_{15} - 6 \beta_{14} - 6 \beta_{13} + 8 \beta_{12} + \cdots - 8 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 4 \beta_{19} + 2 \beta_{18} + 6 \beta_{17} + 4 \beta_{15} + 10 \beta_{14} - 6 \beta_{13} - 8 \beta_{12} + \cdots + 8 ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 4 \beta_{19} + 10 \beta_{18} + 6 \beta_{17} + 8 \beta_{16} + 12 \beta_{15} - 14 \beta_{14} + \cdots + \beta_1 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 4 \beta_{19} + 10 \beta_{18} - 10 \beta_{17} - 8 \beta_{16} + 28 \beta_{15} + 2 \beta_{14} + 18 \beta_{13} + \cdots - 32 ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 20 \beta_{19} - 6 \beta_{18} + 6 \beta_{17} - 8 \beta_{16} + 12 \beta_{15} + 34 \beta_{14} + \cdots - 32 ) / 2 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 4 \beta_{19} - 22 \beta_{18} - 10 \beta_{17} + 24 \beta_{16} - 4 \beta_{15} + 18 \beta_{14} + 34 \beta_{13} + \cdots + 16 ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 36 \beta_{19} + 58 \beta_{18} + 22 \beta_{17} - 56 \beta_{16} + 92 \beta_{15} - 14 \beta_{14} + \cdots + \beta_1 ) / 2 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 52 \beta_{19} + 42 \beta_{18} - 10 \beta_{17} - 8 \beta_{16} - 4 \beta_{15} - 14 \beta_{14} - 30 \beta_{13} + \cdots - 80 ) / 2 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( 12 \beta_{19} - 70 \beta_{18} - 42 \beta_{17} + 8 \beta_{16} + 156 \beta_{15} + 242 \beta_{14} + \cdots + 128 ) / 2 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( - 124 \beta_{19} + 42 \beta_{18} + 22 \beta_{17} - 104 \beta_{16} + 188 \beta_{15} - 14 \beta_{14} + \cdots - 336 ) / 2 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( - 132 \beta_{19} - 38 \beta_{18} - 170 \beta_{17} - 88 \beta_{16} + 220 \beta_{15} - 206 \beta_{14} + \cdots - 160 ) / 2 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( - 204 \beta_{19} - 118 \beta_{18} + 278 \beta_{17} + 56 \beta_{16} + 444 \beta_{15} - 142 \beta_{14} + \cdots - 176 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/968\mathbb{Z}\right)^\times\).
| \(n\) | \(485\) | \(727\) | \(849\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 485.1 |
|
−1.40720 | − | 0.140637i | 2.55485i | 1.96044 | + | 0.395809i | 2.69541i | 0.359306 | − | 3.59519i | 3.94908 | −2.70308 | − | 0.832694i | −3.52726 | 0.379074 | − | 3.79299i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 485.2 | −1.40720 | + | 0.140637i | − | 2.55485i | 1.96044 | − | 0.395809i | − | 2.69541i | 0.359306 | + | 3.59519i | 3.94908 | −2.70308 | + | 0.832694i | −3.52726 | 0.379074 | + | 3.79299i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 485.3 | −1.30820 | − | 0.537231i | 1.28633i | 1.42276 | + | 1.40561i | 1.58193i | 0.691057 | − | 1.68277i | −1.86825 | −1.10612 | − | 2.60317i | 1.34535 | 0.849862 | − | 2.06948i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 485.4 | −1.30820 | + | 0.537231i | − | 1.28633i | 1.42276 | − | 1.40561i | − | 1.58193i | 0.691057 | + | 1.68277i | −1.86825 | −1.10612 | + | 2.60317i | 1.34535 | 0.849862 | + | 2.06948i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 485.5 | −1.07012 | − | 0.924577i | 0.321222i | 0.290315 | + | 1.97882i | − | 1.28733i | 0.296994 | − | 0.343746i | −3.30669 | 1.51890 | − | 2.38599i | 2.89682 | −1.19023 | + | 1.37759i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 485.6 | −1.07012 | + | 0.924577i | − | 0.321222i | 0.290315 | − | 1.97882i | 1.28733i | 0.296994 | + | 0.343746i | −3.30669 | 1.51890 | + | 2.38599i | 2.89682 | −1.19023 | − | 1.37759i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 485.7 | −0.549610 | − | 1.30305i | 0.612207i | −1.39586 | + | 1.43233i | − | 1.46030i | 0.797733 | − | 0.336475i | 4.42813 | 2.63357 | + | 1.03164i | 2.62520 | −1.90283 | + | 0.802592i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 485.8 | −0.549610 | + | 1.30305i | − | 0.612207i | −1.39586 | − | 1.43233i | 1.46030i | 0.797733 | + | 0.336475i | 4.42813 | 2.63357 | − | 1.03164i | 2.62520 | −1.90283 | − | 0.802592i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 485.9 | −0.343983 | − | 1.37174i | − | 2.85679i | −1.76335 | + | 0.943713i | 2.60791i | −3.91878 | + | 0.982689i | 0.196362 | 1.90109 | + | 2.09424i | −5.16127 | 3.57737 | − | 0.897076i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 485.10 | −0.343983 | + | 1.37174i | 2.85679i | −1.76335 | − | 0.943713i | − | 2.60791i | −3.91878 | − | 0.982689i | 0.196362 | 1.90109 | − | 2.09424i | −5.16127 | 3.57737 | + | 0.897076i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 485.11 | 0.324509 | − | 1.37648i | 0.540427i | −1.78939 | − | 0.893360i | − | 2.90090i | 0.743887 | + | 0.175374i | 2.60451 | −1.81036 | + | 2.17315i | 2.70794 | −3.99302 | − | 0.941367i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 485.12 | 0.324509 | + | 1.37648i | − | 0.540427i | −1.78939 | + | 0.893360i | 2.90090i | 0.743887 | − | 0.175374i | 2.60451 | −1.81036 | − | 2.17315i | 2.70794 | −3.99302 | + | 0.941367i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 485.13 | 0.651763 | − | 1.25507i | 2.44793i | −1.15041 | − | 1.63602i | − | 0.200474i | 3.07232 | + | 1.59547i | −2.34615 | −2.80312 | + | 0.377551i | −2.99235 | −0.251609 | − | 0.130662i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 485.14 | 0.651763 | + | 1.25507i | − | 2.44793i | −1.15041 | + | 1.63602i | 0.200474i | 3.07232 | − | 1.59547i | −2.34615 | −2.80312 | − | 0.377551i | −2.99235 | −0.251609 | + | 0.130662i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 485.15 | 0.926370 | − | 1.06857i | − | 2.75783i | −0.283677 | − | 1.97978i | − | 2.90574i | −2.94693 | − | 2.55477i | 2.36352 | −2.37832 | − | 1.53088i | −4.60564 | −3.10498 | − | 2.69179i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 485.16 | 0.926370 | + | 1.06857i | 2.75783i | −0.283677 | + | 1.97978i | 2.90574i | −2.94693 | + | 2.55477i | 2.36352 | −2.37832 | + | 1.53088i | −4.60564 | −3.10498 | + | 2.69179i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 485.17 | 1.37876 | − | 0.314658i | 1.87996i | 1.80198 | − | 0.867678i | − | 0.493106i | 0.591545 | + | 2.59203i | 0.174770 | 2.21148 | − | 1.76333i | −0.534260 | −0.155160 | − | 0.679877i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 485.18 | 1.37876 | + | 0.314658i | − | 1.87996i | 1.80198 | + | 0.867678i | 0.493106i | 0.591545 | − | 2.59203i | 0.174770 | 2.21148 | + | 1.76333i | −0.534260 | −0.155160 | + | 0.679877i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 485.19 | 1.39771 | − | 0.215430i | − | 0.868641i | 1.90718 | − | 0.602216i | 3.67418i | −0.187131 | − | 1.21411i | −1.19528 | 2.53595 | − | 1.25259i | 2.24546 | 0.791527 | + | 5.13543i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 485.20 | 1.39771 | + | 0.215430i | 0.868641i | 1.90718 | + | 0.602216i | − | 3.67418i | −0.187131 | + | 1.21411i | −1.19528 | 2.53595 | + | 1.25259i | 2.24546 | 0.791527 | − | 5.13543i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 968.2.c.h | 20 | |
| 4.b | odd | 2 | 1 | 3872.2.c.h | 20 | ||
| 8.b | even | 2 | 1 | inner | 968.2.c.h | 20 | |
| 8.d | odd | 2 | 1 | 3872.2.c.h | 20 | ||
| 11.b | odd | 2 | 1 | 968.2.c.i | 20 | ||
| 11.c | even | 5 | 2 | 88.2.o.a | ✓ | 40 | |
| 11.c | even | 5 | 2 | 968.2.o.j | 40 | ||
| 11.d | odd | 10 | 2 | 968.2.o.d | 40 | ||
| 11.d | odd | 10 | 2 | 968.2.o.i | 40 | ||
| 33.h | odd | 10 | 2 | 792.2.br.b | 40 | ||
| 44.c | even | 2 | 1 | 3872.2.c.i | 20 | ||
| 44.h | odd | 10 | 2 | 352.2.w.a | 40 | ||
| 88.b | odd | 2 | 1 | 968.2.c.i | 20 | ||
| 88.g | even | 2 | 1 | 3872.2.c.i | 20 | ||
| 88.l | odd | 10 | 2 | 352.2.w.a | 40 | ||
| 88.o | even | 10 | 2 | 88.2.o.a | ✓ | 40 | |
| 88.o | even | 10 | 2 | 968.2.o.j | 40 | ||
| 88.p | odd | 10 | 2 | 968.2.o.d | 40 | ||
| 88.p | odd | 10 | 2 | 968.2.o.i | 40 | ||
| 264.t | odd | 10 | 2 | 792.2.br.b | 40 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 88.2.o.a | ✓ | 40 | 11.c | even | 5 | 2 | |
| 88.2.o.a | ✓ | 40 | 88.o | even | 10 | 2 | |
| 352.2.w.a | 40 | 44.h | odd | 10 | 2 | ||
| 352.2.w.a | 40 | 88.l | odd | 10 | 2 | ||
| 792.2.br.b | 40 | 33.h | odd | 10 | 2 | ||
| 792.2.br.b | 40 | 264.t | odd | 10 | 2 | ||
| 968.2.c.h | 20 | 1.a | even | 1 | 1 | trivial | |
| 968.2.c.h | 20 | 8.b | even | 2 | 1 | inner | |
| 968.2.c.i | 20 | 11.b | odd | 2 | 1 | ||
| 968.2.c.i | 20 | 88.b | odd | 2 | 1 | ||
| 968.2.o.d | 40 | 11.d | odd | 10 | 2 | ||
| 968.2.o.d | 40 | 88.p | odd | 10 | 2 | ||
| 968.2.o.i | 40 | 11.d | odd | 10 | 2 | ||
| 968.2.o.i | 40 | 88.p | odd | 10 | 2 | ||
| 968.2.o.j | 40 | 11.c | even | 5 | 2 | ||
| 968.2.o.j | 40 | 88.o | even | 10 | 2 | ||
| 3872.2.c.h | 20 | 4.b | odd | 2 | 1 | ||
| 3872.2.c.h | 20 | 8.d | odd | 2 | 1 | ||
| 3872.2.c.i | 20 | 44.c | even | 2 | 1 | ||
| 3872.2.c.i | 20 | 88.g | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(968, [\chi])\):
|
\( T_{3}^{20} + 35 T_{3}^{18} + 503 T_{3}^{16} + 3822 T_{3}^{14} + 16493 T_{3}^{12} + 40565 T_{3}^{10} + \cdots + 121 \)
|
|
\( T_{5}^{20} + 51 T_{5}^{18} + 1082 T_{5}^{16} + 12427 T_{5}^{14} + 84082 T_{5}^{12} + 341655 T_{5}^{10} + \cdots + 4096 \)
|
|
\( T_{7}^{10} - 5 T_{7}^{9} - 22 T_{7}^{8} + 111 T_{7}^{7} + 176 T_{7}^{6} - 775 T_{7}^{5} - 701 T_{7}^{4} + \cdots + 64 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} - T^{18} + \cdots + 1024 \)
|
| $3$ |
\( T^{20} + 35 T^{18} + \cdots + 121 \)
|
| $5$ |
\( T^{20} + 51 T^{18} + \cdots + 4096 \)
|
| $7$ |
\( (T^{10} - 5 T^{9} + \cdots + 64)^{2} \)
|
| $11$ |
\( T^{20} \)
|
| $13$ |
\( T^{20} + 119 T^{18} + \cdots + 5308416 \)
|
| $17$ |
\( (T^{10} - T^{9} + \cdots - 34029)^{2} \)
|
| $19$ |
\( T^{20} + 147 T^{18} + \cdots + 31036041 \)
|
| $23$ |
\( (T^{10} + 2 T^{9} + \cdots - 451584)^{2} \)
|
| $29$ |
\( T^{20} + \cdots + 527896576 \)
|
| $31$ |
\( (T^{10} - T^{9} + \cdots - 590144)^{2} \)
|
| $37$ |
\( T^{20} + \cdots + 478554034176 \)
|
| $41$ |
\( (T^{10} + T^{9} + \cdots + 685431)^{2} \)
|
| $43$ |
\( T^{20} + \cdots + 258242846976 \)
|
| $47$ |
\( (T^{10} - T^{9} + \cdots + 251136)^{2} \)
|
| $53$ |
\( T^{20} + \cdots + 4813029376 \)
|
| $59$ |
\( T^{20} + \cdots + 91511695081 \)
|
| $61$ |
\( T^{20} + \cdots + 47\!\cdots\!96 \)
|
| $67$ |
\( T^{20} + \cdots + 22416209899776 \)
|
| $71$ |
\( (T^{10} + 17 T^{9} + \cdots - 2283264)^{2} \)
|
| $73$ |
\( (T^{10} - T^{9} + \cdots + 4889699)^{2} \)
|
| $79$ |
\( (T^{10} + 29 T^{9} + \cdots + 36844544)^{2} \)
|
| $83$ |
\( T^{20} + \cdots + 17926799881 \)
|
| $89$ |
\( (T^{10} + 4 T^{9} + \cdots - 6007233744)^{2} \)
|
| $97$ |
\( (T^{10} - 5 T^{9} + \cdots - 140501)^{2} \)
|
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