Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [352,2,Mod(97,352)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(352, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 0, 6]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("352.97");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 352 = 2^{5} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 352.m (of order \(5\), degree \(4\), 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: | \(2.81073415115\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 2 x^{11} + 11 x^{10} - 11 x^{9} + 39 x^{8} - 43 x^{7} + 99 x^{6} + 36 x^{5} + 431 x^{4} + \cdots + 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 2 x^{11} + 11 x^{10} - 11 x^{9} + 39 x^{8} - 43 x^{7} + 99 x^{6} + 36 x^{5} + 431 x^{4} + \cdots + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 12925652272 \nu^{11} - 689411783229 \nu^{10} + 1158393636547 \nu^{9} + \cdots - 68957262745775 ) / 206669284189945 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 171694051509 \nu^{11} - 3538073194043 \nu^{10} + 6710004699384 \nu^{9} + \cdots - 872744346462385 ) / 206669284189945 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 1093010781921 \nu^{11} - 1619787283007 \nu^{10} + 10879795136331 \nu^{9} + \cdots - 79252706903350 ) / 206669284189945 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 1093010781921 \nu^{11} + 1619787283007 \nu^{10} - 10879795136331 \nu^{9} + \cdots + 79252706903350 ) / 206669284189945 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1185075482262 \nu^{11} - 2450405323424 \nu^{10} + 12479616226722 \nu^{9} + \cdots - 68337530631950 ) / 206669284189945 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2733501225278 \nu^{11} - 4281926968294 \nu^{10} + 27618108154634 \nu^{9} + \cdots + 2642224095290 ) / 206669284189945 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 2758290509831 \nu^{11} - 5503655367390 \nu^{10} + 29651783824912 \nu^{9} + \cdots - 366378456407095 ) / 206669284189945 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 3170108276134 \nu^{11} + 5247205770347 \nu^{10} - 33251403754467 \nu^{9} + \cdots + 203977836612425 ) / 206669284189945 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 6759751764174 \nu^{11} - 11030588267832 \nu^{10} + 70986300515617 \nu^{9} + \cdots + 7410594838995 ) / 206669284189945 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 6923728370080 \nu^{11} + 13776035137694 \nu^{10} - 76812432137873 \nu^{9} + \cdots + 13\!\cdots\!10 ) / 206669284189945 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 7100336146293 \nu^{11} + 10864290641274 \nu^{10} - 70862768140539 \nu^{9} + \cdots - 7554307496840 ) / 206669284189945 \)
|
| \(\nu\) | \(=\) |
\( \beta_{4} + \beta_{3} \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{10} - 3\beta_{8} - \beta_{7} + \beta_{2} - 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{11} + \beta_{10} + \beta_{9} + \beta_{6} - 5\beta_{4} - \beta_{3} + \beta_{2} - \beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 7\beta_{11} - 2\beta_{10} + 9\beta_{8} + 14\beta_{7} + 9\beta_{6} + 2\beta_{5} - 2\beta_{4} - 2\beta_{3} - 7\beta_{2} \)
|
| \(\nu^{5}\) | \(=\) |
\( -11\beta_{10} - 11\beta_{9} + 15\beta_{7} - 15\beta_{6} + 11\beta_{5} + 11\beta_{4} - 20\beta_{2} + 19\beta _1 + 19 \)
|
| \(\nu^{6}\) | \(=\) |
\( -50\beta_{11} - 22\beta_{9} - 74\beta_{8} - 151\beta_{6} - 24\beta_{5} + 30\beta_{4} + 30\beta_{3} + 24\beta _1 + 74 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 74 \beta_{11} + 102 \beta_{10} + 74 \beta_{9} - 64 \beta_{8} - 152 \beta_{7} - 201 \beta_{5} + \cdots - 152 \)
|
| \(\nu^{8}\) | \(=\) |
\( 198 \beta_{11} + 198 \beta_{10} + 377 \beta_{9} + 485 \beta_{8} - 485 \beta_{7} + 1076 \beta_{6} + \cdots - 1076 \)
|
| \(\nu^{9}\) | \(=\) |
\( 893 \beta_{11} - 603 \beta_{10} + 1373 \beta_{8} + 700 \beta_{7} + 1373 \beta_{6} + 1453 \beta_{5} + \cdots - 664 \beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 2949 \beta_{10} - 2949 \beta_{9} + 4713 \beta_{7} - 4713 \beta_{6} + 1976 \beta_{5} + 1976 \beta_{4} + \cdots + 8097 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 7597 \beta_{11} - 4925 \beta_{9} - 11843 \beta_{8} - 18471 \beta_{6} - 6395 \beta_{5} + 11046 \beta_{4} + \cdots + 11843 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/352\mathbb{Z}\right)^\times\).
| \(n\) | \(133\) | \(287\) | \(321\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1 + \beta_{6} - \beta_{7} + \beta_{8}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 97.1 |
|
0 | −1.50933 | + | 1.09659i | 0 | −0.686361 | − | 2.11240i | 0 | 0.787585 | + | 0.572214i | 0 | 0.148512 | − | 0.457073i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 97.2 | 0 | 0.0520331 | − | 0.0378043i | 0 | 0.714760 | + | 2.19980i | 0 | −1.31923 | − | 0.958478i | 0 | −0.925773 | + | 2.84924i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 97.3 | 0 | 2.57533 | − | 1.87109i | 0 | −0.0283989 | − | 0.0874029i | 0 | 3.14968 | + | 2.28838i | 0 | 2.20431 | − | 6.78417i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 225.1 | 0 | −1.50933 | − | 1.09659i | 0 | −0.686361 | + | 2.11240i | 0 | 0.787585 | − | 0.572214i | 0 | 0.148512 | + | 0.457073i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 225.2 | 0 | 0.0520331 | + | 0.0378043i | 0 | 0.714760 | − | 2.19980i | 0 | −1.31923 | + | 0.958478i | 0 | −0.925773 | − | 2.84924i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 225.3 | 0 | 2.57533 | + | 1.87109i | 0 | −0.0283989 | + | 0.0874029i | 0 | 3.14968 | − | 2.28838i | 0 | 2.20431 | + | 6.78417i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 257.1 | 0 | −0.945302 | + | 2.90934i | 0 | −2.43185 | + | 1.76684i | 0 | 0.483582 | + | 1.48831i | 0 | −5.14361 | − | 3.73705i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 257.2 | 0 | −0.385002 | + | 1.18491i | 0 | 3.03889 | − | 2.20788i | 0 | −1.04575 | − | 3.21850i | 0 | 1.17126 | + | 0.850967i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 257.3 | 0 | 0.212270 | − | 0.653300i | 0 | −0.607036 | + | 0.441038i | 0 | 0.944137 | + | 2.90576i | 0 | 2.04531 | + | 1.48600i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 289.1 | 0 | −0.945302 | − | 2.90934i | 0 | −2.43185 | − | 1.76684i | 0 | 0.483582 | − | 1.48831i | 0 | −5.14361 | + | 3.73705i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 289.2 | 0 | −0.385002 | − | 1.18491i | 0 | 3.03889 | + | 2.20788i | 0 | −1.04575 | + | 3.21850i | 0 | 1.17126 | − | 0.850967i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 289.3 | 0 | 0.212270 | + | 0.653300i | 0 | −0.607036 | − | 0.441038i | 0 | 0.944137 | − | 2.90576i | 0 | 2.04531 | − | 1.48600i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.c | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 352.2.m.f | yes | 12 |
| 4.b | odd | 2 | 1 | 352.2.m.e | ✓ | 12 | |
| 8.b | even | 2 | 1 | 704.2.m.n | 12 | ||
| 8.d | odd | 2 | 1 | 704.2.m.m | 12 | ||
| 11.c | even | 5 | 1 | inner | 352.2.m.f | yes | 12 |
| 11.c | even | 5 | 1 | 3872.2.a.bn | 6 | ||
| 11.d | odd | 10 | 1 | 3872.2.a.bo | 6 | ||
| 44.g | even | 10 | 1 | 3872.2.a.bp | 6 | ||
| 44.h | odd | 10 | 1 | 352.2.m.e | ✓ | 12 | |
| 44.h | odd | 10 | 1 | 3872.2.a.bq | 6 | ||
| 88.k | even | 10 | 1 | 7744.2.a.dt | 6 | ||
| 88.l | odd | 10 | 1 | 704.2.m.m | 12 | ||
| 88.l | odd | 10 | 1 | 7744.2.a.du | 6 | ||
| 88.o | even | 10 | 1 | 704.2.m.n | 12 | ||
| 88.o | even | 10 | 1 | 7744.2.a.dv | 6 | ||
| 88.p | odd | 10 | 1 | 7744.2.a.dw | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 352.2.m.e | ✓ | 12 | 4.b | odd | 2 | 1 | |
| 352.2.m.e | ✓ | 12 | 44.h | odd | 10 | 1 | |
| 352.2.m.f | yes | 12 | 1.a | even | 1 | 1 | trivial |
| 352.2.m.f | yes | 12 | 11.c | even | 5 | 1 | inner |
| 704.2.m.m | 12 | 8.d | odd | 2 | 1 | ||
| 704.2.m.m | 12 | 88.l | odd | 10 | 1 | ||
| 704.2.m.n | 12 | 8.b | even | 2 | 1 | ||
| 704.2.m.n | 12 | 88.o | even | 10 | 1 | ||
| 3872.2.a.bn | 6 | 11.c | even | 5 | 1 | ||
| 3872.2.a.bo | 6 | 11.d | odd | 10 | 1 | ||
| 3872.2.a.bp | 6 | 44.g | even | 10 | 1 | ||
| 3872.2.a.bq | 6 | 44.h | odd | 10 | 1 | ||
| 7744.2.a.dt | 6 | 88.k | even | 10 | 1 | ||
| 7744.2.a.du | 6 | 88.l | odd | 10 | 1 | ||
| 7744.2.a.dv | 6 | 88.o | even | 10 | 1 | ||
| 7744.2.a.dw | 6 | 88.p | odd | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} + 5 T_{3}^{10} - 11 T_{3}^{9} + 45 T_{3}^{8} + 175 T_{3}^{7} + 451 T_{3}^{6} + 360 T_{3}^{5} + \cdots + 1 \)
acting on \(S_{2}^{\mathrm{new}}(352, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} + 5 T^{10} + \cdots + 1 \)
|
| $5$ |
\( T^{12} + T^{10} + \cdots + 16 \)
|
| $7$ |
\( T^{12} - 6 T^{11} + \cdots + 10000 \)
|
| $11$ |
\( T^{12} + 11 T^{11} + \cdots + 1771561 \)
|
| $13$ |
\( T^{12} + 2 T^{11} + \cdots + 3225616 \)
|
| $17$ |
\( T^{12} - 12 T^{11} + \cdots + 841 \)
|
| $19$ |
\( T^{12} - 5 T^{11} + \cdots + 7612081 \)
|
| $23$ |
\( (T^{6} + 6 T^{5} + \cdots + 320)^{2} \)
|
| $29$ |
\( T^{12} + 9 T^{10} + \cdots + 6990736 \)
|
| $31$ |
\( T^{12} - 16 T^{11} + \cdots + 1948816 \)
|
| $37$ |
\( T^{12} + 4 T^{11} + \cdots + 3225616 \)
|
| $41$ |
\( T^{12} + \cdots + 839782441 \)
|
| $43$ |
\( (T^{6} + 11 T^{5} + \cdots - 81776)^{2} \)
|
| $47$ |
\( T^{12} + 24 T^{11} + \cdots + 18524416 \)
|
| $53$ |
\( T^{12} + \cdots + 36796446976 \)
|
| $59$ |
\( T^{12} - 31 T^{11} + \cdots + 2825761 \)
|
| $61$ |
\( T^{12} + \cdots + 253446400 \)
|
| $67$ |
\( (T^{6} + 31 T^{5} + \cdots - 36304)^{2} \)
|
| $71$ |
\( T^{12} + \cdots + 1025280400 \)
|
| $73$ |
\( T^{12} + 8 T^{11} + \cdots + 35700625 \)
|
| $79$ |
\( T^{12} + 4 T^{11} + \cdots + 54700816 \)
|
| $83$ |
\( T^{12} + \cdots + 40080440401 \)
|
| $89$ |
\( (T^{6} + T^{5} - 265 T^{4} + \cdots + 7600)^{2} \)
|
| $97$ |
\( T^{12} - 3 T^{11} + \cdots + 121 \)
|
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