Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [155,2,Mod(92,155)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(155, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([1, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("155.92");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 155 = 5 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 155.f (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: | \(1.23768123133\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 6 x^{15} + 18 x^{14} - 42 x^{13} + 66 x^{12} - 24 x^{11} - 162 x^{10} + 612 x^{9} - 1349 x^{8} + \cdots + 6561 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{10} \) |
| 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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 6 x^{15} + 18 x^{14} - 42 x^{13} + 66 x^{12} - 24 x^{11} - 162 x^{10} + 612 x^{9} - 1349 x^{8} + \cdots + 6561 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 64 \nu^{15} + 159 \nu^{14} - 363 \nu^{13} + 798 \nu^{12} + 15 \nu^{11} - 1578 \nu^{10} + \cdots - 25515 ) / 729 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 326 \nu^{15} - 1287 \nu^{14} + 3105 \nu^{13} - 6996 \nu^{12} + 6459 \nu^{11} + 6873 \nu^{10} + \cdots - 920727 ) / 2187 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 332 \nu^{15} + 1320 \nu^{14} - 3195 \nu^{13} + 7194 \nu^{12} - 6729 \nu^{11} - 6927 \nu^{10} + \cdots + 960093 ) / 2187 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 139 \nu^{15} + 1293 \nu^{14} - 3312 \nu^{13} + 7539 \nu^{12} - 12900 \nu^{11} - 1362 \nu^{10} + \cdots + 2040471 ) / 2187 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 272 \nu^{15} - 1539 \nu^{14} + 3816 \nu^{13} - 8643 \nu^{12} + 11616 \nu^{11} + 5010 \nu^{10} + \cdots - 1791153 ) / 2187 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 103 \nu^{15} + 572 \nu^{14} - 1425 \nu^{13} + 3228 \nu^{12} - 4299 \nu^{11} - 1806 \nu^{10} + \cdots + 649539 ) / 729 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 63 \nu^{15} + 667 \nu^{14} - 1722 \nu^{13} + 3924 \nu^{12} - 6981 \nu^{11} - 330 \nu^{10} + \cdots + 1091313 ) / 729 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 820 \nu^{15} - 3399 \nu^{14} + 8244 \nu^{13} - 18591 \nu^{12} + 18345 \nu^{11} + 17256 \nu^{10} + \cdots - 2639709 ) / 2187 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 99 \nu^{15} + 491 \nu^{14} - 1210 \nu^{13} + 2733 \nu^{12} - 3306 \nu^{11} - 1923 \nu^{10} + \cdots + 496449 ) / 243 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 92 \nu^{15} + 501 \nu^{14} - 1244 \nu^{13} + 2817 \nu^{12} - 3705 \nu^{11} - 1641 \nu^{10} + \cdots + 560844 ) / 243 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 1682 \nu^{15} + 6996 \nu^{14} - 16974 \nu^{13} + 38271 \nu^{12} - 38004 \nu^{11} - 35286 \nu^{10} + \cdots + 5504679 ) / 2187 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 1439 \nu^{15} - 7155 \nu^{14} + 17622 \nu^{13} - 39837 \nu^{12} + 48327 \nu^{11} + 27681 \nu^{10} + \cdots - 7217100 ) / 2187 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 1061 \nu^{15} - 4419 \nu^{14} + 10731 \nu^{13} - 24195 \nu^{12} + 24081 \nu^{11} + 22209 \nu^{10} + \cdots - 3481704 ) / 729 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 416 \nu^{15} + 1721 \nu^{14} - 4173 \nu^{13} + 9408 \nu^{12} - 9270 \nu^{11} - 8748 \nu^{10} + \cdots + 1337715 ) / 243 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 3860 \nu^{15} + 16113 \nu^{14} - 39105 \nu^{13} + 88167 \nu^{12} - 87900 \nu^{11} + \cdots + 12715218 ) / 2187 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{11} - 2\beta_{9} - 2\beta_{8} + 2\beta_{6} - 3\beta_{5} + \beta_{3} + \beta_{2} + 2\beta _1 + 2 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{15} - \beta_{14} - 2\beta_{13} - \beta_{12} - \beta_{7} - 2\beta_{3} - 2\beta_{2} ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{15} - 3\beta_{11} - \beta_{8} - 2\beta_{6} + \beta_{5} + 2\beta_{4} - 3\beta_{2} + 2 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 3\beta_{15} - 3\beta_{14} - \beta_{12} + 2\beta_{11} - 2\beta_{9} + 4\beta_{8} + \beta_{7} - 2\beta_{4} + 22 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -4\beta_{14} + \beta_{11} - 14\beta_{9} - 20\beta_{8} + 6\beta_{7} - 19\beta_{5} - \beta_{3} - \beta_{2} + 16\beta_1 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 4 \beta_{15} - 4 \beta_{14} - 8 \beta_{13} + 7 \beta_{9} - 2 \beta_{6} - 7 \beta_{4} - 16 \beta_{3} + \cdots - 3 \beta_1 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 26 \beta_{15} + 6 \beta_{13} + 12 \beta_{12} - 53 \beta_{11} - 6 \beta_{10} + 6 \beta_{8} + 28 \beta_{6} + \cdots - 28 ) / 4 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 29 \beta_{15} - 29 \beta_{14} - 17 \beta_{12} + 60 \beta_{11} - 16 \beta_{10} - 20 \beta_{9} + 108 \beta_{8} + \cdots - 50 ) / 4 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 9 \beta_{14} - 3 \beta_{13} + 17 \beta_{11} - 3 \beta_{10} - 20 \beta_{9} - 38 \beta_{8} + 54 \beta_{7} + \cdots - 115 ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 53 \beta_{15} - 53 \beta_{14} - 94 \beta_{13} + 55 \beta_{12} + 246 \beta_{9} + 55 \beta_{7} + \cdots - 186 \beta_1 ) / 4 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 178 \beta_{15} + 168 \beta_{13} + 138 \beta_{12} - 213 \beta_{11} - 168 \beta_{10} - 70 \beta_{8} + \cdots - 616 ) / 4 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 27 \beta_{15} - 27 \beta_{14} - 91 \beta_{12} + 446 \beta_{11} - 174 \beta_{10} - 26 \beta_{9} + \cdots - 758 ) / 2 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 358 \beta_{14} - 24 \beta_{13} - 97 \beta_{11} - 24 \beta_{10} + 62 \beta_{9} + 422 \beta_{8} + \cdots - 2712 ) / 4 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 259 \beta_{15} - 259 \beta_{14} - 134 \beta_{13} - 39 \beta_{12} + 2134 \beta_{9} - 39 \beta_{7} + \cdots - 2442 \beta_1 ) / 4 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 377 \beta_{15} + 1185 \beta_{13} + 36 \beta_{12} + 349 \beta_{11} - 1185 \beta_{10} - 1638 \beta_{8} + \cdots - 1945 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/155\mathbb{Z}\right)^\times\).
| \(n\) | \(32\) | \(96\) |
| \(\chi(n)\) | \(-\beta_{6}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 92.1 |
|
−1.72585 | − | 1.72585i | −2.01058 | − | 2.01058i | 3.95712i | −2.23127 | + | 0.146426i | 6.93991i | 0.725850 | + | 0.725850i | 3.37769 | − | 3.37769i | 5.08484i | 4.10354 | + | 3.59813i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 92.2 | −1.72585 | − | 1.72585i | 2.01058 | + | 2.01058i | 3.95712i | −2.23127 | + | 0.146426i | − | 6.93991i | 0.725850 | + | 0.725850i | 3.37769 | − | 3.37769i | 5.08484i | 4.10354 | + | 3.59813i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 92.3 | −0.759725 | − | 0.759725i | −1.72499 | − | 1.72499i | − | 0.845635i | 1.60536 | − | 1.55654i | 2.62104i | −0.240275 | − | 0.240275i | −2.16190 | + | 2.16190i | 2.95118i | −2.40218 | − | 0.0370899i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 92.4 | −0.759725 | − | 0.759725i | 1.72499 | + | 1.72499i | − | 0.845635i | 1.60536 | − | 1.55654i | − | 2.62104i | −0.240275 | − | 0.240275i | −2.16190 | + | 2.16190i | 2.95118i | −2.40218 | − | 0.0370899i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 92.5 | 0.329998 | + | 0.329998i | −1.19320 | − | 1.19320i | − | 1.78220i | 1.45220 | + | 1.70032i | − | 0.787510i | −1.33000 | − | 1.33000i | 1.24812 | − | 1.24812i | − | 0.152529i | −0.0818788 | + | 1.04033i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 92.6 | 0.329998 | + | 0.329998i | 1.19320 | + | 1.19320i | − | 1.78220i | 1.45220 | + | 1.70032i | 0.787510i | −1.33000 | − | 1.33000i | 1.24812 | − | 1.24812i | − | 0.152529i | −0.0818788 | + | 1.04033i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 92.7 | 1.15558 | + | 1.15558i | −2.13501 | − | 2.13501i | 0.670719i | −1.82630 | − | 1.29021i | − | 4.93433i | −2.15558 | − | 2.15558i | 1.53609 | − | 1.53609i | 6.11651i | −0.619490 | − | 3.60136i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 92.8 | 1.15558 | + | 1.15558i | 2.13501 | + | 2.13501i | 0.670719i | −1.82630 | − | 1.29021i | 4.93433i | −2.15558 | − | 2.15558i | 1.53609 | − | 1.53609i | 6.11651i | −0.619490 | − | 3.60136i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 123.1 | −1.72585 | + | 1.72585i | −2.01058 | + | 2.01058i | − | 3.95712i | −2.23127 | − | 0.146426i | − | 6.93991i | 0.725850 | − | 0.725850i | 3.37769 | + | 3.37769i | − | 5.08484i | 4.10354 | − | 3.59813i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 123.2 | −1.72585 | + | 1.72585i | 2.01058 | − | 2.01058i | − | 3.95712i | −2.23127 | − | 0.146426i | 6.93991i | 0.725850 | − | 0.725850i | 3.37769 | + | 3.37769i | − | 5.08484i | 4.10354 | − | 3.59813i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 123.3 | −0.759725 | + | 0.759725i | −1.72499 | + | 1.72499i | 0.845635i | 1.60536 | + | 1.55654i | − | 2.62104i | −0.240275 | + | 0.240275i | −2.16190 | − | 2.16190i | − | 2.95118i | −2.40218 | + | 0.0370899i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 123.4 | −0.759725 | + | 0.759725i | 1.72499 | − | 1.72499i | 0.845635i | 1.60536 | + | 1.55654i | 2.62104i | −0.240275 | + | 0.240275i | −2.16190 | − | 2.16190i | − | 2.95118i | −2.40218 | + | 0.0370899i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 123.5 | 0.329998 | − | 0.329998i | −1.19320 | + | 1.19320i | 1.78220i | 1.45220 | − | 1.70032i | 0.787510i | −1.33000 | + | 1.33000i | 1.24812 | + | 1.24812i | 0.152529i | −0.0818788 | − | 1.04033i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 123.6 | 0.329998 | − | 0.329998i | 1.19320 | − | 1.19320i | 1.78220i | 1.45220 | − | 1.70032i | − | 0.787510i | −1.33000 | + | 1.33000i | 1.24812 | + | 1.24812i | 0.152529i | −0.0818788 | − | 1.04033i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 123.7 | 1.15558 | − | 1.15558i | −2.13501 | + | 2.13501i | − | 0.670719i | −1.82630 | + | 1.29021i | 4.93433i | −2.15558 | + | 2.15558i | 1.53609 | + | 1.53609i | − | 6.11651i | −0.619490 | + | 3.60136i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 123.8 | 1.15558 | − | 1.15558i | 2.13501 | − | 2.13501i | − | 0.670719i | −1.82630 | + | 1.29021i | − | 4.93433i | −2.15558 | + | 2.15558i | 1.53609 | + | 1.53609i | − | 6.11651i | −0.619490 | + | 3.60136i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
| 31.b | odd | 2 | 1 | inner |
| 155.f | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 155.2.f.b | ✓ | 16 |
| 5.b | even | 2 | 1 | 775.2.f.f | 16 | ||
| 5.c | odd | 4 | 1 | inner | 155.2.f.b | ✓ | 16 |
| 5.c | odd | 4 | 1 | 775.2.f.f | 16 | ||
| 31.b | odd | 2 | 1 | inner | 155.2.f.b | ✓ | 16 |
| 155.c | odd | 2 | 1 | 775.2.f.f | 16 | ||
| 155.f | even | 4 | 1 | inner | 155.2.f.b | ✓ | 16 |
| 155.f | even | 4 | 1 | 775.2.f.f | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 155.2.f.b | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 155.2.f.b | ✓ | 16 | 5.c | odd | 4 | 1 | inner |
| 155.2.f.b | ✓ | 16 | 31.b | odd | 2 | 1 | inner |
| 155.2.f.b | ✓ | 16 | 155.f | even | 4 | 1 | inner |
| 775.2.f.f | 16 | 5.b | even | 2 | 1 | ||
| 775.2.f.f | 16 | 5.c | odd | 4 | 1 | ||
| 775.2.f.f | 16 | 155.c | odd | 2 | 1 | ||
| 775.2.f.f | 16 | 155.f | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 2T_{2}^{7} + 2T_{2}^{6} - 4T_{2}^{5} + 12T_{2}^{4} + 12T_{2}^{3} + 8T_{2}^{2} - 8T_{2} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(155, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{8} + 2 T^{7} + 2 T^{6} + \cdots + 4)^{2} \)
|
| $3$ |
\( T^{16} + 192 T^{12} + \cdots + 1560001 \)
|
| $5$ |
\( (T^{8} + 2 T^{7} + \cdots + 625)^{2} \)
|
| $7$ |
\( (T^{8} + 6 T^{7} + 18 T^{6} + \cdots + 4)^{2} \)
|
| $11$ |
\( (T^{8} + 68 T^{6} + \cdots + 4996)^{2} \)
|
| $13$ |
\( T^{16} + 844 T^{12} + \cdots + 24960016 \)
|
| $17$ |
\( T^{16} + \cdots + 975000625 \)
|
| $19$ |
\( (T^{8} + 28 T^{6} + \cdots + 49)^{2} \)
|
| $23$ |
\( T^{16} + 5068 T^{12} + \cdots + 24960016 \)
|
| $29$ |
\( (T^{8} - 152 T^{6} + \cdots + 844324)^{2} \)
|
| $31$ |
\( (T^{8} + 8 T^{7} + \cdots + 923521)^{2} \)
|
| $37$ |
\( T^{16} + 5636 T^{12} + \cdots + 1560001 \)
|
| $41$ |
\( (T^{4} + 6 T^{3} + \cdots - 469)^{4} \)
|
| $43$ |
\( T^{16} + \cdots + 44555188561 \)
|
| $47$ |
\( (T^{8} + 12 T^{7} + \cdots + 1024)^{2} \)
|
| $53$ |
\( T^{16} + \cdots + 2923693034161 \)
|
| $59$ |
\( (T^{8} + 124 T^{6} + \cdots + 26569)^{2} \)
|
| $61$ |
\( (T^{8} + 304 T^{6} + \cdots + 19984)^{2} \)
|
| $67$ |
\( (T^{8} - 30 T^{7} + \cdots + 89151364)^{2} \)
|
| $71$ |
\( (T^{4} - 18 T^{3} + \cdots - 551)^{4} \)
|
| $73$ |
\( T^{16} + \cdots + 4408189985761 \)
|
| $79$ |
\( (T^{8} - 324 T^{6} + \cdots + 4201636)^{2} \)
|
| $83$ |
\( T^{16} + \cdots + 60762165310081 \)
|
| $89$ |
\( (T^{8} - 508 T^{6} + \cdots + 4996)^{2} \)
|
| $97$ |
\( (T^{8} + 36 T^{7} + \cdots + 160000)^{2} \)
|
| show more | |
| show less |