Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [48,3,Mod(5,48)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(48, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 1, 2]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("48.5");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 48 = 2^{4} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 48.i (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: | \(1.30790526893\) |
| 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: |
\( x^{20} - 2 x^{18} + 6 x^{16} - 24 x^{14} - 24 x^{12} + 1216 x^{10} - 384 x^{8} - 6144 x^{6} + \cdots + 1048576 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{13} \) |
| 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
\( x^{20} - 2 x^{18} + 6 x^{16} - 24 x^{14} - 24 x^{12} + 1216 x^{10} - 384 x^{8} - 6144 x^{6} + \cdots + 1048576 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 33 \nu^{18} + 42 \nu^{16} - 1090 \nu^{14} + 528 \nu^{12} + 11816 \nu^{10} - 6496 \nu^{8} + \cdots + 16842752 ) / 10158080 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 25 \nu^{19} + 68 \nu^{18} - 2270 \nu^{17} - 1448 \nu^{16} + 7850 \nu^{15} - 10920 \nu^{14} + \cdots + 120061952 ) / 40632320 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 25 \nu^{19} - 68 \nu^{18} - 2270 \nu^{17} + 1448 \nu^{16} + 7850 \nu^{15} + 10920 \nu^{14} + \cdots - 120061952 ) / 40632320 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 4 \nu^{19} + 112 \nu^{18} - 101 \nu^{17} - 542 \nu^{16} - 410 \nu^{15} + 980 \nu^{14} + \cdots - 18481152 ) / 10158080 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 4 \nu^{19} + 112 \nu^{18} + 101 \nu^{17} - 542 \nu^{16} + 410 \nu^{15} + 980 \nu^{14} + \cdots - 18481152 ) / 10158080 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 4 \nu^{19} - 149 \nu^{18} + 101 \nu^{17} - 836 \nu^{16} + 410 \nu^{15} + 790 \nu^{14} + \cdots - 63307776 ) / 10158080 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 4 \nu^{19} + 145 \nu^{18} + 101 \nu^{17} - 500 \nu^{16} + 410 \nu^{15} - 110 \nu^{14} + \cdots - 1638400 ) / 10158080 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 79 \nu^{19} - 160 \nu^{18} + 34 \nu^{17} - 2040 \nu^{16} + 2030 \nu^{15} + 2960 \nu^{14} + \cdots - 82575360 ) / 40632320 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 51 \nu^{19} - 448 \nu^{18} - 1606 \nu^{17} - 32 \nu^{16} + 2450 \nu^{15} + 12480 \nu^{14} + \cdots - 70254592 ) / 40632320 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 33 \nu^{19} - 42 \nu^{17} + 1090 \nu^{15} - 528 \nu^{13} - 11816 \nu^{11} + 6496 \nu^{9} + \cdots - 6684672 \nu ) / 10158080 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 33 \nu^{19} + 42 \nu^{17} - 1090 \nu^{15} + 528 \nu^{13} + 11816 \nu^{11} - 6496 \nu^{9} + \cdots + 27000832 \nu ) / 10158080 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 79 \nu^{19} + 592 \nu^{18} + 34 \nu^{17} - 3112 \nu^{16} + 2030 \nu^{15} + 2320 \nu^{14} + \cdots - 96468992 ) / 40632320 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 51 \nu^{19} - 448 \nu^{18} + 1606 \nu^{17} - 32 \nu^{16} - 2450 \nu^{15} + 12480 \nu^{14} + \cdots - 70254592 ) / 40632320 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 79 \nu^{19} + 592 \nu^{18} - 34 \nu^{17} - 3112 \nu^{16} - 2030 \nu^{15} + 2320 \nu^{14} + \cdots - 96468992 ) / 40632320 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( \nu^{19} - 2 \nu^{17} + 6 \nu^{15} - 24 \nu^{13} - 24 \nu^{11} + 1216 \nu^{9} - 384 \nu^{7} + \cdots - 131072 \nu ) / 262144 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 211 \nu^{19} - 468 \nu^{18} + 334 \nu^{17} + 1008 \nu^{16} - 2470 \nu^{15} + 2520 \nu^{14} + \cdots - 122683392 ) / 40632320 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 211 \nu^{19} + 468 \nu^{18} + 334 \nu^{17} - 1008 \nu^{16} - 2470 \nu^{15} - 2520 \nu^{14} + \cdots + 122683392 ) / 40632320 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 257 \nu^{19} - 1042 \nu^{17} + 870 \nu^{15} + 11272 \nu^{13} - 14616 \nu^{11} + \cdots - 9961472 \nu ) / 40632320 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 157 \nu^{19} - 140 \nu^{18} + 982 \nu^{17} + 240 \nu^{16} - 1110 \nu^{15} + 2600 \nu^{14} + \cdots + 14745600 ) / 10158080 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{11} + \beta_{10} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} - \beta_{5} - \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{19} + 2\beta_{18} + \beta_{16} - \beta_{13} - \beta_{11} + \beta_{7} - \beta_{5} + \beta_{4} \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{17} - \beta_{16} - \beta_{14} - \beta_{13} - 2\beta_{12} - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{4} - 7\beta _1 - 1 \)
|
| \(\nu^{5}\) | \(=\) |
\( 6 \beta_{18} - \beta_{17} - \beta_{16} - 6 \beta_{15} + 4 \beta_{14} - \beta_{13} - 4 \beta_{12} + \cdots - 2 \beta_{2} \)
|
| \(\nu^{6}\) | \(=\) |
\( - 2 \beta_{17} + 2 \beta_{16} - 10 \beta_{14} - 4 \beta_{12} - 6 \beta_{8} + 6 \beta_{7} + 2 \beta_{6} + \cdots + 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( 10 \beta_{19} + 28 \beta_{18} - 12 \beta_{17} - 2 \beta_{16} + 8 \beta_{15} + 4 \beta_{14} + \cdots + 8 \beta_{2} \)
|
| \(\nu^{8}\) | \(=\) |
\( - 2 \beta_{17} + 2 \beta_{16} - 10 \beta_{14} + 2 \beta_{13} + 44 \beta_{12} + 2 \beta_{9} - 54 \beta_{8} + \cdots + 6 \)
|
| \(\nu^{9}\) | \(=\) |
\( 4 \beta_{19} - 44 \beta_{18} - 2 \beta_{17} + 2 \beta_{16} + 180 \beta_{15} - 24 \beta_{14} + \cdots - 12 \beta_{2} \)
|
| \(\nu^{10}\) | \(=\) |
\( 80 \beta_{17} - 80 \beta_{16} - 64 \beta_{14} + 76 \beta_{13} + 16 \beta_{12} + 76 \beta_{9} + \cdots - 660 \)
|
| \(\nu^{11}\) | \(=\) |
\( 68 \beta_{19} + 128 \beta_{18} - 116 \beta_{17} - 48 \beta_{16} + 296 \beta_{15} - 200 \beta_{14} + \cdots - 56 \beta_{2} \)
|
| \(\nu^{12}\) | \(=\) |
\( - 44 \beta_{17} + 44 \beta_{16} - 76 \beta_{14} + 52 \beta_{13} - 312 \beta_{12} + 52 \beta_{9} + \cdots - 1348 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 1120 \beta_{19} - 328 \beta_{18} + 236 \beta_{17} - 884 \beta_{16} - 1720 \beta_{15} + \cdots - 536 \beta_{2} \)
|
| \(\nu^{14}\) | \(=\) |
\( - 344 \beta_{17} + 344 \beta_{16} - 56 \beta_{14} + 1424 \beta_{13} + 1648 \beta_{12} + 1424 \beta_{9} + \cdots + 7840 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 1928 \beta_{19} - 4240 \beta_{18} - 1928 \beta_{16} + 6592 \beta_{15} - 5808 \beta_{14} + \cdots + 576 \beta_{2} \)
|
| \(\nu^{16}\) | \(=\) |
\( - 3928 \beta_{17} + 3928 \beta_{16} + 2152 \beta_{14} - 72 \beta_{13} + 6608 \beta_{12} - 72 \beta_{9} + \cdots - 17656 \)
|
| \(\nu^{17}\) | \(=\) |
\( - 1424 \beta_{19} - 7568 \beta_{18} + 9960 \beta_{17} + 8536 \beta_{16} + 6512 \beta_{15} + \cdots - 7760 \beta_{2} \)
|
| \(\nu^{18}\) | \(=\) |
\( 2880 \beta_{17} - 2880 \beta_{16} + 3456 \beta_{14} - 14192 \beta_{13} - 5568 \beta_{12} - 14192 \beta_{9} + \cdots - 11376 \)
|
| \(\nu^{19}\) | \(=\) |
\( - 42128 \beta_{19} + 57472 \beta_{18} + 14480 \beta_{17} - 27648 \beta_{16} - 51232 \beta_{15} + \cdots - 27808 \beta_{2} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).
| \(n\) | \(17\) | \(31\) | \(37\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(-\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 5.1 |
|
−1.96139 | + | 0.391068i | −2.99548 | + | 0.164573i | 3.69413 | − | 1.53408i | 3.61305 | − | 3.61305i | 5.81096 | − | 1.49423i | − | 12.2792i | −6.64572 | + | 4.45358i | 8.94583 | − | 0.985948i | −5.67366 | + | 8.49955i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.2 | −1.85381 | + | 0.750590i | 1.50491 | + | 2.59524i | 2.87323 | − | 2.78290i | −2.59897 | + | 2.59897i | −4.73777 | − | 3.68151i | 7.30027i | −3.23761 | + | 7.31559i | −4.47050 | + | 7.81118i | 2.86723 | − | 6.76875i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.3 | −1.28499 | − | 1.53258i | −2.06336 | + | 2.17774i | −0.697601 | + | 3.93870i | −3.17955 | + | 3.17955i | 5.98896 | + | 0.363879i | 6.03979i | 6.93278 | − | 3.99206i | −0.485128 | − | 8.98692i | 8.95859 | + | 0.787223i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.4 | −1.21144 | + | 1.59136i | 1.14944 | − | 2.77106i | −1.06484 | − | 3.85566i | 4.80434 | − | 4.80434i | 3.01728 | + | 5.18614i | 7.36187i | 7.42573 | + | 2.97634i | −6.35757 | − | 6.37035i | 1.82527 | + | 13.4656i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.5 | −0.312316 | − | 1.97546i | −1.18505 | − | 2.75602i | −3.80492 | + | 1.23394i | −0.00985921 | + | 0.00985921i | −5.07432 | + | 3.20176i | − | 6.42277i | 3.62594 | + | 7.13110i | −6.19134 | + | 6.53203i | 0.0225557 | + | 0.0163973i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.6 | 0.312316 | + | 1.97546i | 2.75602 | + | 1.18505i | −3.80492 | + | 1.23394i | 0.00985921 | − | 0.00985921i | −1.48026 | + | 5.81454i | − | 6.42277i | −3.62594 | − | 7.13110i | 6.19134 | + | 6.53203i | 0.0225557 | + | 0.0163973i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.7 | 1.21144 | − | 1.59136i | 2.77106 | − | 1.14944i | −1.06484 | − | 3.85566i | −4.80434 | + | 4.80434i | 1.52779 | − | 5.80223i | 7.36187i | −7.42573 | − | 2.97634i | 6.35757 | − | 6.37035i | 1.82527 | + | 13.4656i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.8 | 1.28499 | + | 1.53258i | −2.17774 | + | 2.06336i | −0.697601 | + | 3.93870i | 3.17955 | − | 3.17955i | −5.96063 | − | 0.686173i | 6.03979i | −6.93278 | + | 3.99206i | 0.485128 | − | 8.98692i | 8.95859 | + | 0.787223i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.9 | 1.85381 | − | 0.750590i | −2.59524 | − | 1.50491i | 2.87323 | − | 2.78290i | 2.59897 | − | 2.59897i | −5.94065 | − | 0.841858i | 7.30027i | 3.23761 | − | 7.31559i | 4.47050 | + | 7.81118i | 2.86723 | − | 6.76875i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.10 | 1.96139 | − | 0.391068i | −0.164573 | + | 2.99548i | 3.69413 | − | 1.53408i | −3.61305 | + | 3.61305i | 0.848646 | + | 5.93968i | − | 12.2792i | 6.64572 | − | 4.45358i | −8.94583 | − | 0.985948i | −5.67366 | + | 8.49955i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 29.1 | −1.96139 | − | 0.391068i | −2.99548 | − | 0.164573i | 3.69413 | + | 1.53408i | 3.61305 | + | 3.61305i | 5.81096 | + | 1.49423i | 12.2792i | −6.64572 | − | 4.45358i | 8.94583 | + | 0.985948i | −5.67366 | − | 8.49955i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 29.2 | −1.85381 | − | 0.750590i | 1.50491 | − | 2.59524i | 2.87323 | + | 2.78290i | −2.59897 | − | 2.59897i | −4.73777 | + | 3.68151i | − | 7.30027i | −3.23761 | − | 7.31559i | −4.47050 | − | 7.81118i | 2.86723 | + | 6.76875i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 29.3 | −1.28499 | + | 1.53258i | −2.06336 | − | 2.17774i | −0.697601 | − | 3.93870i | −3.17955 | − | 3.17955i | 5.98896 | − | 0.363879i | − | 6.03979i | 6.93278 | + | 3.99206i | −0.485128 | + | 8.98692i | 8.95859 | − | 0.787223i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 29.4 | −1.21144 | − | 1.59136i | 1.14944 | + | 2.77106i | −1.06484 | + | 3.85566i | 4.80434 | + | 4.80434i | 3.01728 | − | 5.18614i | − | 7.36187i | 7.42573 | − | 2.97634i | −6.35757 | + | 6.37035i | 1.82527 | − | 13.4656i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 29.5 | −0.312316 | + | 1.97546i | −1.18505 | + | 2.75602i | −3.80492 | − | 1.23394i | −0.00985921 | − | 0.00985921i | −5.07432 | − | 3.20176i | 6.42277i | 3.62594 | − | 7.13110i | −6.19134 | − | 6.53203i | 0.0225557 | − | 0.0163973i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 29.6 | 0.312316 | − | 1.97546i | 2.75602 | − | 1.18505i | −3.80492 | − | 1.23394i | 0.00985921 | + | 0.00985921i | −1.48026 | − | 5.81454i | 6.42277i | −3.62594 | + | 7.13110i | 6.19134 | − | 6.53203i | 0.0225557 | − | 0.0163973i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 29.7 | 1.21144 | + | 1.59136i | 2.77106 | + | 1.14944i | −1.06484 | + | 3.85566i | −4.80434 | − | 4.80434i | 1.52779 | + | 5.80223i | − | 7.36187i | −7.42573 | + | 2.97634i | 6.35757 | + | 6.37035i | 1.82527 | − | 13.4656i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 29.8 | 1.28499 | − | 1.53258i | −2.17774 | − | 2.06336i | −0.697601 | − | 3.93870i | 3.17955 | + | 3.17955i | −5.96063 | + | 0.686173i | − | 6.03979i | −6.93278 | − | 3.99206i | 0.485128 | + | 8.98692i | 8.95859 | − | 0.787223i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 29.9 | 1.85381 | + | 0.750590i | −2.59524 | + | 1.50491i | 2.87323 | + | 2.78290i | 2.59897 | + | 2.59897i | −5.94065 | + | 0.841858i | − | 7.30027i | 3.23761 | + | 7.31559i | 4.47050 | − | 7.81118i | 2.86723 | + | 6.76875i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 29.10 | 1.96139 | + | 0.391068i | −0.164573 | − | 2.99548i | 3.69413 | + | 1.53408i | −3.61305 | − | 3.61305i | 0.848646 | − | 5.93968i | 12.2792i | 6.64572 | + | 4.45358i | −8.94583 | + | 0.985948i | −5.67366 | − | 8.49955i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 16.e | even | 4 | 1 | inner |
| 48.i | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 48.3.i.b | ✓ | 20 |
| 3.b | odd | 2 | 1 | inner | 48.3.i.b | ✓ | 20 |
| 4.b | odd | 2 | 1 | 192.3.i.b | 20 | ||
| 8.b | even | 2 | 1 | 384.3.i.d | 20 | ||
| 8.d | odd | 2 | 1 | 384.3.i.c | 20 | ||
| 12.b | even | 2 | 1 | 192.3.i.b | 20 | ||
| 16.e | even | 4 | 1 | inner | 48.3.i.b | ✓ | 20 |
| 16.e | even | 4 | 1 | 384.3.i.d | 20 | ||
| 16.f | odd | 4 | 1 | 192.3.i.b | 20 | ||
| 16.f | odd | 4 | 1 | 384.3.i.c | 20 | ||
| 24.f | even | 2 | 1 | 384.3.i.c | 20 | ||
| 24.h | odd | 2 | 1 | 384.3.i.d | 20 | ||
| 48.i | odd | 4 | 1 | inner | 48.3.i.b | ✓ | 20 |
| 48.i | odd | 4 | 1 | 384.3.i.d | 20 | ||
| 48.k | even | 4 | 1 | 192.3.i.b | 20 | ||
| 48.k | even | 4 | 1 | 384.3.i.c | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 48.3.i.b | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 48.3.i.b | ✓ | 20 | 3.b | odd | 2 | 1 | inner |
| 48.3.i.b | ✓ | 20 | 16.e | even | 4 | 1 | inner |
| 48.3.i.b | ✓ | 20 | 48.i | odd | 4 | 1 | inner |
| 192.3.i.b | 20 | 4.b | odd | 2 | 1 | ||
| 192.3.i.b | 20 | 12.b | even | 2 | 1 | ||
| 192.3.i.b | 20 | 16.f | odd | 4 | 1 | ||
| 192.3.i.b | 20 | 48.k | even | 4 | 1 | ||
| 384.3.i.c | 20 | 8.d | odd | 2 | 1 | ||
| 384.3.i.c | 20 | 16.f | odd | 4 | 1 | ||
| 384.3.i.c | 20 | 24.f | even | 2 | 1 | ||
| 384.3.i.c | 20 | 48.k | even | 4 | 1 | ||
| 384.3.i.d | 20 | 8.b | even | 2 | 1 | ||
| 384.3.i.d | 20 | 16.e | even | 4 | 1 | ||
| 384.3.i.d | 20 | 24.h | odd | 2 | 1 | ||
| 384.3.i.d | 20 | 48.i | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{20} + 3404T_{5}^{16} + 3190384T_{5}^{12} + 1068787520T_{5}^{8} + 108375444480T_{5}^{4} + 4096 \)
acting on \(S_{3}^{\mathrm{new}}(48, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} - 2 T^{18} + \cdots + 1048576 \)
|
| $3$ |
\( T^{20} + \cdots + 3486784401 \)
|
| $5$ |
\( T^{20} + 3404 T^{16} + \cdots + 4096 \)
|
| $7$ |
\( (T^{10} + 336 T^{8} + \cdots + 655360000)^{2} \)
|
| $11$ |
\( T^{20} + \cdots + 22\!\cdots\!00 \)
|
| $13$ |
\( (T^{10} - 46 T^{9} + \cdots + 33620000)^{2} \)
|
| $17$ |
\( (T^{10} + 952 T^{8} + \cdots + 15510536192)^{2} \)
|
| $19$ |
\( (T^{10} + 26 T^{9} + \cdots + 23975244288)^{2} \)
|
| $23$ |
\( (T^{10} + \cdots - 2157878476800)^{2} \)
|
| $29$ |
\( T^{20} + \cdots + 14\!\cdots\!00 \)
|
| $31$ |
\( (T^{5} + 20 T^{4} + \cdots + 6473680)^{4} \)
|
| $37$ |
\( (T^{10} + \cdots + 93878430976800)^{2} \)
|
| $41$ |
\( (T^{10} + \cdots - 89172136396800)^{2} \)
|
| $43$ |
\( (T^{10} + \cdots + 398518394892800)^{2} \)
|
| $47$ |
\( (T^{10} + \cdots + 2199023255552)^{2} \)
|
| $53$ |
\( T^{20} + \cdots + 51\!\cdots\!00 \)
|
| $59$ |
\( T^{20} + \cdots + 55\!\cdots\!00 \)
|
| $61$ |
\( (T^{10} + \cdots + 79\!\cdots\!68)^{2} \)
|
| $67$ |
\( (T^{10} + \cdots + 14\!\cdots\!00)^{2} \)
|
| $71$ |
\( (T^{10} + \cdots - 63\!\cdots\!00)^{2} \)
|
| $73$ |
\( (T^{10} + \cdots + 900192010240000)^{2} \)
|
| $79$ |
\( (T^{5} - 96 T^{4} + \cdots - 147403248)^{4} \)
|
| $83$ |
\( T^{20} + \cdots + 53\!\cdots\!00 \)
|
| $89$ |
\( (T^{10} + \cdots - 15\!\cdots\!00)^{2} \)
|
| $97$ |
\( (T^{5} - 118 T^{4} + \cdots - 2657552000)^{4} \)
|
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