Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1078,2,Mod(901,1078)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1078, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([5, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1078.901");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1078 = 2 \cdot 7^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1078.i (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: | \(8.60787333789\) |
| 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: |
\( x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 34 x^{12} + 18 x^{11} - 72 x^{10} + 132 x^{9} - 93 x^{8} + \cdots + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 154) |
| 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
\( x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 34 x^{12} + 18 x^{11} - 72 x^{10} + 132 x^{9} - 93 x^{8} + \cdots + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 80029143512 \nu^{15} + 385788744870 \nu^{14} - 820783926284 \nu^{13} + 848040618120 \nu^{12} + \cdots + 33432180594 ) / 3707507912227 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 115452774644 \nu^{15} + 886173093256 \nu^{14} - 3342656190846 \nu^{13} + \cdots - 1622048373702 ) / 3707507912227 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 229876192869 \nu^{15} - 1434169055319 \nu^{14} + 4481995762636 \nu^{13} + \cdots + 7097366139528 ) / 3707507912227 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 234407101203 \nu^{15} - 1463062841541 \nu^{14} + 4568245377402 \nu^{13} + \cdots - 5702240414038 ) / 3707507912227 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 323659513794 \nu^{15} + 2126436132890 \nu^{14} - 7021940502570 \nu^{13} + \cdots - 2750714974023 ) / 3707507912227 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 394538957168 \nu^{15} + 2231811901383 \nu^{14} - 6199106801734 \nu^{13} + \cdots - 1505816495286 ) / 3707507912227 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 401980321088 \nu^{15} - 2507531613006 \nu^{14} + 7838812215514 \nu^{13} + \cdots + 6420259797322 ) / 3707507912227 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 403027374657 \nu^{15} + 2521501247748 \nu^{14} - 7885517844470 \nu^{13} + \cdots + 4012225523467 ) / 3707507912227 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 556158552240 \nu^{15} - 3247720931080 \nu^{14} + 9404820278882 \nu^{13} + \cdots + 2653688275785 ) / 3707507912227 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 785074438 \nu^{15} + 4907305158 \nu^{14} - 15343416116 \nu^{13} + 31997236762 \nu^{12} + \cdots - 1370541375 ) / 1973128213 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 1616321538 \nu^{15} + 9938916556 \nu^{14} - 30594542475 \nu^{13} + 62871582510 \nu^{12} + \cdots - 10233974547 ) / 3810388399 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 2432602484212 \nu^{15} - 15210362430228 \nu^{14} + 47566230428388 \nu^{13} + \cdots + 3660982645558 ) / 3707507912227 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 2667083154356 \nu^{15} + 16670596841871 \nu^{14} - 52123769377554 \nu^{13} + \cdots - 5233663502032 ) / 3707507912227 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 2737320095452 \nu^{15} - 16635640723011 \nu^{14} + 50529646182960 \nu^{13} + \cdots + 16350688060416 ) / 3707507912227 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 2891771622084 \nu^{15} + 17710086389784 \nu^{14} - 54266401205842 \nu^{13} + \cdots - 17956801097484 ) / 3707507912227 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{15} - \beta_{14} + \beta_{6} - 3\beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 3 \beta_{15} - 4 \beta_{14} - \beta_{13} + \beta_{12} - 2 \beta_{11} + 4 \beta_{10} + \beta_{9} + \cdots - 2 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 5\beta_{12} + 9\beta_{10} + 2\beta_{8} - \beta_{7} + 5\beta_{4} + 5\beta_{3} - 1 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 8 \beta_{15} + 8 \beta_{14} + 11 \beta_{13} + 19 \beta_{12} + 28 \beta_{11} + 14 \beta_{10} + \cdots + 6 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -26\beta_{15} + \beta_{14} + 46\beta_{11} - 8\beta_{9} + \beta_{6} + 16\beta_{5} - 26\beta_{2} - 18\beta_1 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 96 \beta_{15} - 47 \beta_{14} - 96 \beta_{13} - 47 \beta_{12} + 82 \beta_{11} + 82 \beta_{10} + \cdots - 113 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( -138\beta_{13} + 10\beta_{12} + 245\beta_{10} + 50\beta_{8} - 100\beta_{7} - 10\beta_{4} + 138\beta_{3} - 195 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 223 \beta_{15} + 501 \beta_{14} - 278 \beta_{13} + 278 \beta_{12} + 456 \beta_{11} + 912 \beta_{10} + \cdots - 456 ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 70 \beta_{15} + 739 \beta_{14} + 1320 \beta_{11} - 576 \beta_{9} - 669 \beta_{6} + 288 \beta_{5} + \cdots + 962 \beta_1 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 1533 \beta_{15} + 1533 \beta_{14} - 1125 \beta_{13} - 2658 \beta_{12} + 4980 \beta_{11} - 2490 \beta_{10} + \cdots - 817 ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( -3545\beta_{13} - 3545\beta_{12} - 1603\beta_{8} - 1603\beta_{7} - 3973\beta_{4} + 428\beta_{3} - 3923 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 14219 \beta_{15} + 5865 \beta_{14} - 14219 \beta_{13} - 5865 \beta_{12} - 13502 \beta_{11} + \cdots - 17794 ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( 18939\beta_{15} + 18939\beta_{14} - 8782\beta_{9} - 21398\beta_{6} - 8782\beta_{5} + 2459\beta_{2} + 39879\beta_1 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 31097 \beta_{15} + 76382 \beta_{14} + 45285 \beta_{13} - 45285 \beta_{12} + 73006 \beta_{11} + \cdots + 73006 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1078\mathbb{Z}\right)^\times\).
| \(n\) | \(199\) | \(981\) |
| \(\chi(n)\) | \(1 - \beta_{10}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 901.1 |
|
−0.866025 | − | 0.500000i | −2.24749 | + | 1.29759i | 0.500000 | + | 0.866025i | 3.06486 | + | 1.76950i | 2.59518 | 0 | − | 1.00000i | 1.86747 | − | 3.23456i | −1.76950 | − | 3.06486i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 901.2 | −0.866025 | − | 0.500000i | −1.35034 | + | 0.779618i | 0.500000 | + | 0.866025i | −0.882559 | − | 0.509546i | 1.55924 | 0 | − | 1.00000i | −0.284392 | + | 0.492581i | 0.509546 | + | 0.882559i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 901.3 | −0.866025 | − | 0.500000i | 0.889740 | − | 0.513691i | 0.500000 | + | 0.866025i | −1.09005 | − | 0.629341i | −1.02738 | 0 | − | 1.00000i | −0.972242 | + | 1.68397i | 0.629341 | + | 1.09005i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 901.4 | −0.866025 | − | 0.500000i | 2.70809 | − | 1.56352i | 0.500000 | + | 0.866025i | 1.90775 | + | 1.10144i | −3.12703 | 0 | − | 1.00000i | 3.38916 | − | 5.87020i | −1.10144 | − | 1.90775i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 901.5 | 0.866025 | + | 0.500000i | −2.24749 | + | 1.29759i | 0.500000 | + | 0.866025i | 3.06486 | + | 1.76950i | −2.59518 | 0 | 1.00000i | 1.86747 | − | 3.23456i | 1.76950 | + | 3.06486i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 901.6 | 0.866025 | + | 0.500000i | −1.35034 | + | 0.779618i | 0.500000 | + | 0.866025i | −0.882559 | − | 0.509546i | −1.55924 | 0 | 1.00000i | −0.284392 | + | 0.492581i | −0.509546 | − | 0.882559i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 901.7 | 0.866025 | + | 0.500000i | 0.889740 | − | 0.513691i | 0.500000 | + | 0.866025i | −1.09005 | − | 0.629341i | 1.02738 | 0 | 1.00000i | −0.972242 | + | 1.68397i | −0.629341 | − | 1.09005i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 901.8 | 0.866025 | + | 0.500000i | 2.70809 | − | 1.56352i | 0.500000 | + | 0.866025i | 1.90775 | + | 1.10144i | 3.12703 | 0 | 1.00000i | 3.38916 | − | 5.87020i | 1.10144 | + | 1.90775i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1011.1 | −0.866025 | + | 0.500000i | −2.24749 | − | 1.29759i | 0.500000 | − | 0.866025i | 3.06486 | − | 1.76950i | 2.59518 | 0 | 1.00000i | 1.86747 | + | 3.23456i | −1.76950 | + | 3.06486i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1011.2 | −0.866025 | + | 0.500000i | −1.35034 | − | 0.779618i | 0.500000 | − | 0.866025i | −0.882559 | + | 0.509546i | 1.55924 | 0 | 1.00000i | −0.284392 | − | 0.492581i | 0.509546 | − | 0.882559i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1011.3 | −0.866025 | + | 0.500000i | 0.889740 | + | 0.513691i | 0.500000 | − | 0.866025i | −1.09005 | + | 0.629341i | −1.02738 | 0 | 1.00000i | −0.972242 | − | 1.68397i | 0.629341 | − | 1.09005i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1011.4 | −0.866025 | + | 0.500000i | 2.70809 | + | 1.56352i | 0.500000 | − | 0.866025i | 1.90775 | − | 1.10144i | −3.12703 | 0 | 1.00000i | 3.38916 | + | 5.87020i | −1.10144 | + | 1.90775i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1011.5 | 0.866025 | − | 0.500000i | −2.24749 | − | 1.29759i | 0.500000 | − | 0.866025i | 3.06486 | − | 1.76950i | −2.59518 | 0 | − | 1.00000i | 1.86747 | + | 3.23456i | 1.76950 | − | 3.06486i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1011.6 | 0.866025 | − | 0.500000i | −1.35034 | − | 0.779618i | 0.500000 | − | 0.866025i | −0.882559 | + | 0.509546i | −1.55924 | 0 | − | 1.00000i | −0.284392 | − | 0.492581i | −0.509546 | + | 0.882559i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1011.7 | 0.866025 | − | 0.500000i | 0.889740 | + | 0.513691i | 0.500000 | − | 0.866025i | −1.09005 | + | 0.629341i | 1.02738 | 0 | − | 1.00000i | −0.972242 | − | 1.68397i | −0.629341 | + | 1.09005i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1011.8 | 0.866025 | − | 0.500000i | 2.70809 | + | 1.56352i | 0.500000 | − | 0.866025i | 1.90775 | − | 1.10144i | 3.12703 | 0 | − | 1.00000i | 3.38916 | + | 5.87020i | 1.10144 | − | 1.90775i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.d | odd | 6 | 1 | inner |
| 11.b | odd | 2 | 1 | inner |
| 77.i | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1078.2.i.c | 16 | |
| 7.b | odd | 2 | 1 | 154.2.i.a | ✓ | 16 | |
| 7.c | even | 3 | 1 | 154.2.i.a | ✓ | 16 | |
| 7.c | even | 3 | 1 | 1078.2.c.b | 16 | ||
| 7.d | odd | 6 | 1 | 1078.2.c.b | 16 | ||
| 7.d | odd | 6 | 1 | inner | 1078.2.i.c | 16 | |
| 11.b | odd | 2 | 1 | inner | 1078.2.i.c | 16 | |
| 21.c | even | 2 | 1 | 1386.2.bk.c | 16 | ||
| 21.h | odd | 6 | 1 | 1386.2.bk.c | 16 | ||
| 28.d | even | 2 | 1 | 1232.2.bn.b | 16 | ||
| 28.g | odd | 6 | 1 | 1232.2.bn.b | 16 | ||
| 77.b | even | 2 | 1 | 154.2.i.a | ✓ | 16 | |
| 77.h | odd | 6 | 1 | 154.2.i.a | ✓ | 16 | |
| 77.h | odd | 6 | 1 | 1078.2.c.b | 16 | ||
| 77.i | even | 6 | 1 | 1078.2.c.b | 16 | ||
| 77.i | even | 6 | 1 | inner | 1078.2.i.c | 16 | |
| 231.h | odd | 2 | 1 | 1386.2.bk.c | 16 | ||
| 231.l | even | 6 | 1 | 1386.2.bk.c | 16 | ||
| 308.g | odd | 2 | 1 | 1232.2.bn.b | 16 | ||
| 308.n | even | 6 | 1 | 1232.2.bn.b | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 154.2.i.a | ✓ | 16 | 7.b | odd | 2 | 1 | |
| 154.2.i.a | ✓ | 16 | 7.c | even | 3 | 1 | |
| 154.2.i.a | ✓ | 16 | 77.b | even | 2 | 1 | |
| 154.2.i.a | ✓ | 16 | 77.h | odd | 6 | 1 | |
| 1078.2.c.b | 16 | 7.c | even | 3 | 1 | ||
| 1078.2.c.b | 16 | 7.d | odd | 6 | 1 | ||
| 1078.2.c.b | 16 | 77.h | odd | 6 | 1 | ||
| 1078.2.c.b | 16 | 77.i | even | 6 | 1 | ||
| 1078.2.i.c | 16 | 1.a | even | 1 | 1 | trivial | |
| 1078.2.i.c | 16 | 7.d | odd | 6 | 1 | inner | |
| 1078.2.i.c | 16 | 11.b | odd | 2 | 1 | inner | |
| 1078.2.i.c | 16 | 77.i | even | 6 | 1 | inner | |
| 1232.2.bn.b | 16 | 28.d | even | 2 | 1 | ||
| 1232.2.bn.b | 16 | 28.g | odd | 6 | 1 | ||
| 1232.2.bn.b | 16 | 308.g | odd | 2 | 1 | ||
| 1232.2.bn.b | 16 | 308.n | even | 6 | 1 | ||
| 1386.2.bk.c | 16 | 21.c | even | 2 | 1 | ||
| 1386.2.bk.c | 16 | 21.h | odd | 6 | 1 | ||
| 1386.2.bk.c | 16 | 231.h | odd | 2 | 1 | ||
| 1386.2.bk.c | 16 | 231.l | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 10T_{3}^{6} + 87T_{3}^{4} + 60T_{3}^{3} - 118T_{3}^{2} - 78T_{3} + 169 \)
acting on \(S_{2}^{\mathrm{new}}(1078, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - T^{2} + 1)^{4} \)
|
| $3$ |
\( (T^{8} - 10 T^{6} + \cdots + 169)^{2} \)
|
| $5$ |
\( (T^{8} - 6 T^{7} + \cdots + 100)^{2} \)
|
| $7$ |
\( T^{16} \)
|
| $11$ |
\( T^{16} + \cdots + 214358881 \)
|
| $13$ |
\( (T^{8} - 44 T^{6} + \cdots + 25)^{2} \)
|
| $17$ |
\( T^{16} + \cdots + 6146560000 \)
|
| $19$ |
\( T^{16} + 80 T^{14} + \cdots + 16 \)
|
| $23$ |
\( (T^{8} - 8 T^{7} + \cdots + 196)^{2} \)
|
| $29$ |
\( (T^{8} + 36 T^{6} + 366 T^{4} + \cdots + 9)^{2} \)
|
| $31$ |
\( (T^{8} - 6 T^{7} + \cdots + 3136)^{2} \)
|
| $37$ |
\( (T^{8} + 8 T^{7} + 52 T^{6} + \cdots + 4)^{2} \)
|
| $41$ |
\( (T^{8} - 200 T^{6} + \cdots + 1674436)^{2} \)
|
| $43$ |
\( (T^{8} + 244 T^{6} + \cdots + 1201216)^{2} \)
|
| $47$ |
\( (T^{8} + 12 T^{7} + \cdots + 4096)^{2} \)
|
| $53$ |
\( (T^{8} + 14 T^{7} + \cdots + 1170724)^{2} \)
|
| $59$ |
\( (T^{8} + 30 T^{7} + \cdots + 2356225)^{2} \)
|
| $61$ |
\( T^{16} + \cdots + 1706808989601 \)
|
| $67$ |
\( (T^{8} - 6 T^{7} + \cdots + 173889)^{2} \)
|
| $71$ |
\( (T^{4} - 2 T^{3} + \cdots + 3262)^{4} \)
|
| $73$ |
\( T^{16} + \cdots + 15704099856 \)
|
| $79$ |
\( T^{16} + \cdots + 11\!\cdots\!41 \)
|
| $83$ |
\( (T^{8} - 260 T^{6} + \cdots + 107584)^{2} \)
|
| $89$ |
\( (T^{8} + 48 T^{7} + \cdots + 49617936)^{2} \)
|
| $97$ |
\( (T^{8} + 644 T^{6} + \cdots + 496532089)^{2} \)
|
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