Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [147,3,Mod(116,147)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(147, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 4]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("147.116");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 147 = 3 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 147.h (of order \(6\), degree \(2\), not 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: | \(4.00545988610\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 10x^{14} + 65x^{12} - 366x^{10} + 1280x^{8} + 780x^{6} - 811x^{4} + 200x^{2} + 625 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 10x^{14} + 65x^{12} - 366x^{10} + 1280x^{8} + 780x^{6} - 811x^{4} + 200x^{2} + 625 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 757 \nu^{14} - 8520 \nu^{12} + 62480 \nu^{10} - 392752 \nu^{8} + 1573360 \nu^{6} - 2090240 \nu^{4} + \cdots + 1775000 ) / 7840350 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 61 \nu^{14} + 780 \nu^{12} - 5720 \nu^{10} + 33886 \nu^{8} - 144040 \nu^{6} + 191360 \nu^{4} + \cdots - 24950 ) / 137550 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 3494 \nu^{14} + 53085 \nu^{12} - 389290 \nu^{10} + 2272934 \nu^{8} - 9803030 \nu^{6} + \cdots - 11059375 ) / 7840350 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 8177 \nu^{15} - 116445 \nu^{13} + 853930 \nu^{11} - 4957682 \nu^{9} + 21503510 \nu^{7} + \cdots + 24259375 \nu ) / 39201750 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 1945 \nu^{14} + 19929 \nu^{12} - 132260 \nu^{10} + 754660 \nu^{8} - 2722612 \nu^{6} + \cdots - 735395 ) / 1568070 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3976 \nu^{14} - 47265 \nu^{12} + 346610 \nu^{10} - 2079526 \nu^{8} + 8728270 \nu^{6} + \cdots + 9846875 ) / 2613450 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 4217 \nu^{15} - 44355 \nu^{13} + 325270 \nu^{11} - 1982822 \nu^{9} + 8190890 \nu^{7} + \cdots + 9240625 \nu ) / 7840350 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 203 \nu^{15} + 2505 \nu^{13} - 18370 \nu^{11} + 109598 \nu^{9} - 462590 \nu^{7} + \cdots - 521875 \nu ) / 299250 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 33307 \nu^{15} + 284445 \nu^{13} - 1666730 \nu^{11} + 8883862 \nu^{9} - 23766460 \nu^{7} + \cdots - 3014525 \nu ) / 39201750 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 12749 \nu^{14} + 123720 \nu^{12} - 790690 \nu^{10} + 4409774 \nu^{8} - 14855960 \nu^{6} + \cdots - 2267050 ) / 2613450 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 10021 \nu^{14} + 102747 \nu^{12} - 681428 \nu^{10} + 3888148 \nu^{8} - 14153146 \nu^{6} + \cdots + 1173535 ) / 1568070 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 2389 \nu^{15} - 24582 \nu^{13} + 162452 \nu^{11} - 926932 \nu^{9} + 3375934 \nu^{7} + \cdots - 5285914 \nu ) / 1568070 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 64117 \nu^{15} + 825945 \nu^{13} - 6056930 \nu^{11} + 35893972 \nu^{9} - 152524510 \nu^{7} + \cdots - 172071875 \nu ) / 39201750 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 68077 \nu^{15} + 729045 \nu^{13} - 4927130 \nu^{11} + 28198882 \nu^{9} - 105869260 \nu^{7} + \cdots - 17236025 \nu ) / 39201750 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 45923 \nu^{15} + 468990 \nu^{13} - 3084250 \nu^{11} + 17471498 \nu^{9} - 62568320 \nu^{7} + \cdots - 9916600 \nu ) / 7840350 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{14} - \beta_{9} - \beta_{8} + \beta_{4} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{6} + \beta_{3} - 4\beta_{2} + 2\beta _1 + 4 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{13} - 5\beta_{8} - \beta_{7} + 2\beta_{4} \)
|
| \(\nu^{4}\) | \(=\) |
\( ( \beta_{11} - 5\beta_{10} + 20\beta_{5} + \beta_{3} - 12\beta_{2} + 20\beta _1 - 1 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -24\beta_{15} - 5\beta_{14} + 26\beta_{13} - 26\beta_{12} + 79\beta_{9} ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -10\beta_{11} - 3\beta_{10} + 3\beta_{6} + 67\beta_{5} + 25 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -160\beta_{15} + 103\beta_{14} - 120\beta_{12} + 447\beta_{9} + 447\beta_{8} + 160\beta_{7} + 103\beta_{4} ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( -123\beta_{6} + 243\beta_{3} - 802\beta_{2} - 624\beta _1 + 802 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( -129\beta_{13} + 800\beta_{8} + 372\beta_{7} + 631\beta_{4} \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 1763\beta_{11} - 1505\beta_{10} - 1340\beta_{5} + 1763\beta_{3} - 7166\beta_{2} - 1340\beta _1 - 1763 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 1598\beta_{15} - 9155\beta_{14} + 1928\beta_{13} - 1928\beta_{12} + 527\beta_{9} ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( 4495\beta_{11} - 5459\beta_{10} + 5459\beta_{6} + 5006\beta_{5} - 26320 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 11940 \beta_{15} - 46681 \beta_{14} - 29920 \beta_{12} + 66821 \beta_{9} + 66821 \beta_{8} + \cdots - 46681 \beta_{4} ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 55671\beta_{6} - 25751\beta_{3} + 178524\beta_{2} - 155362\beta _1 - 178524 ) / 2 \)
|
| \(\nu^{15}\) | \(=\) |
\( -118392\beta_{13} + 336870\beta_{8} + 92641\beta_{7} - 66857\beta_{4} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).
| \(n\) | \(50\) | \(52\) |
| \(\chi(n)\) | \(-1\) | \(-1 + \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 116.1 |
|
−2.92214 | + | 1.68710i | −1.08438 | + | 2.79716i | 3.69258 | − | 6.39574i | −4.92837 | + | 2.84540i | −1.55039 | − | 10.0031i | 0 | 11.4222i | −6.64826 | − | 6.06636i | 9.60092 | − | 16.6293i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 116.2 | −2.92214 | + | 1.68710i | 1.08438 | − | 2.79716i | 3.69258 | − | 6.39574i | 4.92837 | − | 2.84540i | 1.55039 | + | 10.0031i | 0 | 11.4222i | −6.64826 | − | 6.06636i | −9.60092 | + | 16.6293i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 116.3 | −0.679063 | + | 0.392057i | −2.07168 | − | 2.16982i | −1.69258 | + | 2.93164i | 4.02630 | − | 2.32459i | 2.25750 | + | 0.661230i | 0 | − | 5.79081i | −0.416279 | + | 8.99037i | −1.82274 | + | 3.15708i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 116.4 | −0.679063 | + | 0.392057i | 2.07168 | + | 2.16982i | −1.69258 | + | 2.93164i | −4.02630 | + | 2.32459i | −2.25750 | − | 0.661230i | 0 | − | 5.79081i | −0.416279 | + | 8.99037i | 1.82274 | − | 3.15708i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 116.5 | 0.679063 | − | 0.392057i | −2.91496 | − | 0.709216i | −1.69258 | + | 2.93164i | 4.02630 | − | 2.32459i | −2.25750 | + | 0.661230i | 0 | 5.79081i | 7.99403 | + | 4.13468i | 1.82274 | − | 3.15708i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 116.6 | 0.679063 | − | 0.392057i | 2.91496 | + | 0.709216i | −1.69258 | + | 2.93164i | −4.02630 | + | 2.32459i | 2.25750 | − | 0.661230i | 0 | 5.79081i | 7.99403 | + | 4.13468i | −1.82274 | + | 3.15708i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 116.7 | 2.92214 | − | 1.68710i | −1.88023 | + | 2.33768i | 3.69258 | − | 6.39574i | 4.92837 | − | 2.84540i | −1.55039 | + | 10.0031i | 0 | − | 11.4222i | −1.92949 | − | 8.79074i | 9.60092 | − | 16.6293i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 116.8 | 2.92214 | − | 1.68710i | 1.88023 | − | 2.33768i | 3.69258 | − | 6.39574i | −4.92837 | + | 2.84540i | 1.55039 | − | 10.0031i | 0 | − | 11.4222i | −1.92949 | − | 8.79074i | −9.60092 | + | 16.6293i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 128.1 | −2.92214 | − | 1.68710i | −1.08438 | − | 2.79716i | 3.69258 | + | 6.39574i | −4.92837 | − | 2.84540i | −1.55039 | + | 10.0031i | 0 | − | 11.4222i | −6.64826 | + | 6.06636i | 9.60092 | + | 16.6293i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 128.2 | −2.92214 | − | 1.68710i | 1.08438 | + | 2.79716i | 3.69258 | + | 6.39574i | 4.92837 | + | 2.84540i | 1.55039 | − | 10.0031i | 0 | − | 11.4222i | −6.64826 | + | 6.06636i | −9.60092 | − | 16.6293i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 128.3 | −0.679063 | − | 0.392057i | −2.07168 | + | 2.16982i | −1.69258 | − | 2.93164i | 4.02630 | + | 2.32459i | 2.25750 | − | 0.661230i | 0 | 5.79081i | −0.416279 | − | 8.99037i | −1.82274 | − | 3.15708i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 128.4 | −0.679063 | − | 0.392057i | 2.07168 | − | 2.16982i | −1.69258 | − | 2.93164i | −4.02630 | − | 2.32459i | −2.25750 | + | 0.661230i | 0 | 5.79081i | −0.416279 | − | 8.99037i | 1.82274 | + | 3.15708i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 128.5 | 0.679063 | + | 0.392057i | −2.91496 | + | 0.709216i | −1.69258 | − | 2.93164i | 4.02630 | + | 2.32459i | −2.25750 | − | 0.661230i | 0 | − | 5.79081i | 7.99403 | − | 4.13468i | 1.82274 | + | 3.15708i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 128.6 | 0.679063 | + | 0.392057i | 2.91496 | − | 0.709216i | −1.69258 | − | 2.93164i | −4.02630 | − | 2.32459i | 2.25750 | + | 0.661230i | 0 | − | 5.79081i | 7.99403 | − | 4.13468i | −1.82274 | − | 3.15708i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 128.7 | 2.92214 | + | 1.68710i | −1.88023 | − | 2.33768i | 3.69258 | + | 6.39574i | 4.92837 | + | 2.84540i | −1.55039 | − | 10.0031i | 0 | 11.4222i | −1.92949 | + | 8.79074i | 9.60092 | + | 16.6293i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 128.8 | 2.92214 | + | 1.68710i | 1.88023 | + | 2.33768i | 3.69258 | + | 6.39574i | −4.92837 | − | 2.84540i | 1.55039 | + | 10.0031i | 0 | 11.4222i | −1.92949 | + | 8.79074i | −9.60092 | − | 16.6293i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 7.c | even | 3 | 1 | inner |
| 7.d | odd | 6 | 1 | inner |
| 21.c | even | 2 | 1 | inner |
| 21.g | even | 6 | 1 | inner |
| 21.h | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 147.3.h.f | 16 | |
| 3.b | odd | 2 | 1 | inner | 147.3.h.f | 16 | |
| 7.b | odd | 2 | 1 | inner | 147.3.h.f | 16 | |
| 7.c | even | 3 | 1 | 147.3.b.g | ✓ | 8 | |
| 7.c | even | 3 | 1 | inner | 147.3.h.f | 16 | |
| 7.d | odd | 6 | 1 | 147.3.b.g | ✓ | 8 | |
| 7.d | odd | 6 | 1 | inner | 147.3.h.f | 16 | |
| 21.c | even | 2 | 1 | inner | 147.3.h.f | 16 | |
| 21.g | even | 6 | 1 | 147.3.b.g | ✓ | 8 | |
| 21.g | even | 6 | 1 | inner | 147.3.h.f | 16 | |
| 21.h | odd | 6 | 1 | 147.3.b.g | ✓ | 8 | |
| 21.h | odd | 6 | 1 | inner | 147.3.h.f | 16 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 147.3.b.g | ✓ | 8 | 7.c | even | 3 | 1 | |
| 147.3.b.g | ✓ | 8 | 7.d | odd | 6 | 1 | |
| 147.3.b.g | ✓ | 8 | 21.g | even | 6 | 1 | |
| 147.3.b.g | ✓ | 8 | 21.h | odd | 6 | 1 | |
| 147.3.h.f | 16 | 1.a | even | 1 | 1 | trivial | |
| 147.3.h.f | 16 | 3.b | odd | 2 | 1 | inner | |
| 147.3.h.f | 16 | 7.b | odd | 2 | 1 | inner | |
| 147.3.h.f | 16 | 7.c | even | 3 | 1 | inner | |
| 147.3.h.f | 16 | 7.d | odd | 6 | 1 | inner | |
| 147.3.h.f | 16 | 21.c | even | 2 | 1 | inner | |
| 147.3.h.f | 16 | 21.g | even | 6 | 1 | inner | |
| 147.3.h.f | 16 | 21.h | odd | 6 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(147, [\chi])\):
|
\( T_{2}^{8} - 12T_{2}^{6} + 137T_{2}^{4} - 84T_{2}^{2} + 49 \)
|
|
\( T_{5}^{8} - 54T_{5}^{6} + 2216T_{5}^{4} - 37800T_{5}^{2} + 490000 \)
|
|
\( T_{13}^{4} - 78T_{13}^{2} + 100 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{8} - 12 T^{6} + \cdots + 49)^{2} \)
|
| $3$ |
\( T^{16} + 2 T^{14} + \cdots + 43046721 \)
|
| $5$ |
\( (T^{8} - 54 T^{6} + \cdots + 490000)^{2} \)
|
| $7$ |
\( T^{16} \)
|
| $11$ |
\( (T^{8} - 496 T^{6} + \cdots + 2007040000)^{2} \)
|
| $13$ |
\( (T^{4} - 78 T^{2} + 100)^{4} \)
|
| $17$ |
\( (T^{8} - 328 T^{6} + \cdots + 306250000)^{2} \)
|
| $19$ |
\( (T^{8} + 622 T^{6} + \cdots + 6250000)^{2} \)
|
| $23$ |
\( (T^{8} - 1692 T^{6} + \cdots + 152000176384)^{2} \)
|
| $29$ |
\( (T^{4} + 756 T^{2} + 137200)^{4} \)
|
| $31$ |
\( (T^{8} + 1872 T^{6} + \cdots + 716392960000)^{2} \)
|
| $37$ |
\( (T^{4} + 58 T^{3} + \cdots + 336400)^{4} \)
|
| $41$ |
\( (T^{4} + 2632 T^{2} + 1680700)^{4} \)
|
| $43$ |
\( (T^{2} - 18 T - 1340)^{8} \)
|
| $47$ |
\( (T^{8} + \cdots + 49000000000000)^{2} \)
|
| $53$ |
\( (T^{8} - 1168 T^{6} + \cdots + 7840000)^{2} \)
|
| $59$ |
\( (T^{8} + \cdots + 69167498890000)^{2} \)
|
| $61$ |
\( (T^{8} + \cdots + 1252975764496)^{2} \)
|
| $67$ |
\( (T^{4} - 108 T^{3} + \cdots + 1123600)^{4} \)
|
| $71$ |
\( (T^{4} + 756 T^{2} + 137200)^{4} \)
|
| $73$ |
\( (T^{8} + \cdots + 10\!\cdots\!00)^{2} \)
|
| $79$ |
\( (T^{4} + 16 T^{3} + \cdots + 160000)^{4} \)
|
| $83$ |
\( (T^{4} + 9450 T^{2} + 21437500)^{4} \)
|
| $89$ |
\( (T^{8} + \cdots + 55149931690000)^{2} \)
|
| $97$ |
\( (T^{4} - 16500 T^{2} + 11222500)^{4} \)
|
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