Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [110,4,Mod(31,110)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(110, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 6]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("110.31");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 110 = 2 \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 110.g (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: | \(6.49021010063\) |
| 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: |
\( x^{12} - 3 x^{11} + 18 x^{10} + 56 x^{9} + 1903 x^{8} + 4434 x^{7} + 95917 x^{6} + 226101 x^{5} + \cdots + 14641 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 5 \) |
| 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
\( x^{12} - 3 x^{11} + 18 x^{10} + 56 x^{9} + 1903 x^{8} + 4434 x^{7} + 95917 x^{6} + 226101 x^{5} + \cdots + 14641 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 21\!\cdots\!83 \nu^{11} + \cdots + 13\!\cdots\!28 ) / 60\!\cdots\!10 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 31\!\cdots\!94 \nu^{11} + \cdots - 19\!\cdots\!76 ) / 60\!\cdots\!10 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 42\!\cdots\!08 \nu^{11} + \cdots - 26\!\cdots\!15 ) / 66\!\cdots\!10 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 75\!\cdots\!32 \nu^{11} + \cdots + 21\!\cdots\!52 ) / 60\!\cdots\!10 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 14\!\cdots\!96 \nu^{11} + \cdots + 13\!\cdots\!43 ) / 66\!\cdots\!10 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 15\!\cdots\!92 \nu^{11} + \cdots - 30\!\cdots\!58 ) / 66\!\cdots\!10 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 70\!\cdots\!37 \nu^{11} + \cdots - 27\!\cdots\!47 ) / 66\!\cdots\!10 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 94\!\cdots\!53 \nu^{11} + \cdots - 18\!\cdots\!37 ) / 55\!\cdots\!10 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 23\!\cdots\!52 \nu^{11} + \cdots + 11\!\cdots\!29 ) / 66\!\cdots\!10 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 43\!\cdots\!97 \nu^{11} + \cdots + 18\!\cdots\!71 ) / 66\!\cdots\!10 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{11} - 2\beta_{10} - \beta_{9} + \beta_{8} - 7\beta_{7} - \beta_{5} + 29\beta_{4} + \beta_{3} + 3\beta_{2} - \beta _1 + 7 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2 \beta_{11} - 2 \beta_{10} - \beta_{9} - 46 \beta_{7} - 7 \beta_{6} + 11 \beta_{5} + 46 \beta_{4} + \cdots + 7 \)
|
| \(\nu^{4}\) | \(=\) |
\( 57 \beta_{11} - 11 \beta_{10} - 22 \beta_{9} - 35 \beta_{8} - 1387 \beta_{7} - 586 \beta_{6} + \cdots - 101 \beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( 101 \beta_{11} + 101 \beta_{10} - 122 \beta_{9} - 122 \beta_{8} - 3460 \beta_{7} - 3683 \beta_{6} + \cdots - 1311 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 967 \beta_{11} + 3346 \beta_{10} - 1412 \beta_{9} - 1934 \beta_{8} + 967 \beta_{7} - 40356 \beta_{6} + \cdots - 40236 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 17890 \beta_{11} + 20702 \beta_{10} + 2812 \beta_{9} - 10351 \beta_{8} + 264484 \beta_{7} + \cdots - 272023 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 269378 \beta_{11} + 203050 \beta_{10} + 134689 \beta_{9} - 66328 \beta_{8} + 4512937 \beta_{7} + \cdots - 2087000 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 1291384 \beta_{11} + 520694 \beta_{10} + 1041388 \beta_{9} + 249996 \beta_{8} + 27457728 \beta_{7} + \cdots + 4555715 \beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 4555715 \beta_{11} - 4555715 \beta_{10} + 8098100 \beta_{9} + 8098100 \beta_{8} + 157291920 \beta_{7} + \cdots + 119092494 \)
|
| \(\nu^{11}\) | \(=\) |
\( 34990175 \beta_{11} - 88994840 \beta_{10} + 19014490 \beta_{9} + 69980350 \beta_{8} - 34990175 \beta_{7} + \cdots + 1271885780 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/110\mathbb{Z}\right)^\times\).
| \(n\) | \(67\) | \(101\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{7}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 31.1 |
|
0.618034 | − | 1.90211i | −3.92812 | + | 2.85395i | −3.23607 | − | 2.35114i | −1.54508 | − | 4.75528i | 3.00082 | + | 9.23557i | 5.32223 | + | 3.86683i | −6.47214 | + | 4.70228i | −1.05833 | + | 3.25720i | −10.0000 | ||||||||||||||||||||||||||||||||||||||
| 31.2 | 0.618034 | − | 1.90211i | 2.90911 | − | 2.11359i | −3.23607 | − | 2.35114i | −1.54508 | − | 4.75528i | −2.22236 | − | 6.83973i | −21.1898 | − | 15.3953i | −6.47214 | + | 4.70228i | −4.34781 | + | 13.3812i | −10.0000 | |||||||||||||||||||||||||||||||||||||||
| 31.3 | 0.618034 | − | 1.90211i | 7.56410 | − | 5.49564i | −3.23607 | − | 2.35114i | −1.54508 | − | 4.75528i | −5.77846 | − | 17.7843i | 26.8848 | + | 19.5330i | −6.47214 | + | 4.70228i | 18.6701 | − | 57.4606i | −10.0000 | |||||||||||||||||||||||||||||||||||||||
| 71.1 | 0.618034 | + | 1.90211i | −3.92812 | − | 2.85395i | −3.23607 | + | 2.35114i | −1.54508 | + | 4.75528i | 3.00082 | − | 9.23557i | 5.32223 | − | 3.86683i | −6.47214 | − | 4.70228i | −1.05833 | − | 3.25720i | −10.0000 | |||||||||||||||||||||||||||||||||||||||
| 71.2 | 0.618034 | + | 1.90211i | 2.90911 | + | 2.11359i | −3.23607 | + | 2.35114i | −1.54508 | + | 4.75528i | −2.22236 | + | 6.83973i | −21.1898 | + | 15.3953i | −6.47214 | − | 4.70228i | −4.34781 | − | 13.3812i | −10.0000 | |||||||||||||||||||||||||||||||||||||||
| 71.3 | 0.618034 | + | 1.90211i | 7.56410 | + | 5.49564i | −3.23607 | + | 2.35114i | −1.54508 | + | 4.75528i | −5.77846 | + | 17.7843i | 26.8848 | − | 19.5330i | −6.47214 | − | 4.70228i | 18.6701 | + | 57.4606i | −10.0000 | |||||||||||||||||||||||||||||||||||||||
| 81.1 | −1.61803 | − | 1.17557i | −1.38070 | + | 4.24936i | 1.23607 | + | 3.80423i | 4.04508 | − | 2.93893i | 7.22944 | − | 5.25249i | −0.607198 | − | 1.86876i | 2.47214 | − | 7.60845i | 5.69276 | + | 4.13603i | −10.0000 | |||||||||||||||||||||||||||||||||||||||
| 81.2 | −1.61803 | − | 1.17557i | 0.458064 | − | 1.40977i | 1.23607 | + | 3.80423i | 4.04508 | − | 2.93893i | −2.39845 | + | 1.74258i | 4.18048 | + | 12.8662i | 2.47214 | − | 7.60845i | 20.0658 | + | 14.5787i | −10.0000 | |||||||||||||||||||||||||||||||||||||||
| 81.3 | −1.61803 | − | 1.17557i | 1.87755 | − | 5.77851i | 1.23607 | + | 3.80423i | 4.04508 | − | 2.93893i | −9.83099 | + | 7.14263i | −7.09050 | − | 21.8223i | 2.47214 | − | 7.60845i | −8.02250 | − | 5.82869i | −10.0000 | |||||||||||||||||||||||||||||||||||||||
| 91.1 | −1.61803 | + | 1.17557i | −1.38070 | − | 4.24936i | 1.23607 | − | 3.80423i | 4.04508 | + | 2.93893i | 7.22944 | + | 5.25249i | −0.607198 | + | 1.86876i | 2.47214 | + | 7.60845i | 5.69276 | − | 4.13603i | −10.0000 | |||||||||||||||||||||||||||||||||||||||
| 91.2 | −1.61803 | + | 1.17557i | 0.458064 | + | 1.40977i | 1.23607 | − | 3.80423i | 4.04508 | + | 2.93893i | −2.39845 | − | 1.74258i | 4.18048 | − | 12.8662i | 2.47214 | + | 7.60845i | 20.0658 | − | 14.5787i | −10.0000 | |||||||||||||||||||||||||||||||||||||||
| 91.3 | −1.61803 | + | 1.17557i | 1.87755 | + | 5.77851i | 1.23607 | − | 3.80423i | 4.04508 | + | 2.93893i | −9.83099 | − | 7.14263i | −7.09050 | + | 21.8223i | 2.47214 | + | 7.60845i | −8.02250 | + | 5.82869i | −10.0000 | |||||||||||||||||||||||||||||||||||||||
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 | 110.4.g.b | ✓ | 12 |
| 11.c | even | 5 | 1 | inner | 110.4.g.b | ✓ | 12 |
| 11.c | even | 5 | 1 | 1210.4.a.bf | 6 | ||
| 11.d | odd | 10 | 1 | 1210.4.a.bc | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 110.4.g.b | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 110.4.g.b | ✓ | 12 | 11.c | even | 5 | 1 | inner |
| 1210.4.a.bc | 6 | 11.d | odd | 10 | 1 | ||
| 1210.4.a.bf | 6 | 11.c | even | 5 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} - 15 T_{3}^{11} + 122 T_{3}^{10} - 476 T_{3}^{9} + 2899 T_{3}^{8} - 5934 T_{3}^{7} + \cdots + 43151761 \)
acting on \(S_{4}^{\mathrm{new}}(110, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 16)^{3} \)
|
| $3$ |
\( T^{12} - 15 T^{11} + \cdots + 43151761 \)
|
| $5$ |
\( (T^{4} - 5 T^{3} + \cdots + 625)^{3} \)
|
| $7$ |
\( T^{12} + \cdots + 12197863591936 \)
|
| $11$ |
\( T^{12} + \cdots + 55\!\cdots\!81 \)
|
| $13$ |
\( T^{12} + \cdots + 11\!\cdots\!00 \)
|
| $17$ |
\( T^{12} + \cdots + 14\!\cdots\!25 \)
|
| $19$ |
\( T^{12} + \cdots + 18\!\cdots\!25 \)
|
| $23$ |
\( (T^{6} + \cdots - 1170533509004)^{2} \)
|
| $29$ |
\( T^{12} + \cdots + 17\!\cdots\!00 \)
|
| $31$ |
\( T^{12} + \cdots + 58\!\cdots\!00 \)
|
| $37$ |
\( T^{12} + \cdots + 48\!\cdots\!36 \)
|
| $41$ |
\( T^{12} + \cdots + 56\!\cdots\!61 \)
|
| $43$ |
\( (T^{6} + \cdots - 10\!\cdots\!05)^{2} \)
|
| $47$ |
\( T^{12} + \cdots + 11\!\cdots\!00 \)
|
| $53$ |
\( T^{12} + \cdots + 43\!\cdots\!76 \)
|
| $59$ |
\( T^{12} + \cdots + 36\!\cdots\!25 \)
|
| $61$ |
\( T^{12} + \cdots + 25\!\cdots\!00 \)
|
| $67$ |
\( (T^{6} + \cdots - 55916232721561)^{2} \)
|
| $71$ |
\( T^{12} + \cdots + 34\!\cdots\!96 \)
|
| $73$ |
\( T^{12} + \cdots + 99\!\cdots\!21 \)
|
| $79$ |
\( T^{12} + \cdots + 34\!\cdots\!00 \)
|
| $83$ |
\( T^{12} + \cdots + 54\!\cdots\!25 \)
|
| $89$ |
\( (T^{6} + \cdots + 89\!\cdots\!75)^{2} \)
|
| $97$ |
\( T^{12} + \cdots + 35\!\cdots\!41 \)
|
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