Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [77,2,Mod(15,77)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(77, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("77.15");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 77 = 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 77.f (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: | \(0.614848095564\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 3 x^{15} + 14 x^{14} - 32 x^{13} + 86 x^{12} - 145 x^{11} + 245 x^{10} - 245 x^{9} + 640 x^{8} + \cdots + 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 3 x^{15} + 14 x^{14} - 32 x^{13} + 86 x^{12} - 145 x^{11} + 245 x^{10} - 245 x^{9} + 640 x^{8} + \cdots + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 33722118880975 \nu^{15} - 215936836813390 \nu^{14} + 154302737507927 \nu^{13} + \cdots + 88\!\cdots\!50 ) / 45\!\cdots\!60 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 686277383720070 \nu^{15} + \cdots - 15\!\cdots\!35 ) / 45\!\cdots\!60 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 874936745197272 \nu^{15} - 514706296938285 \nu^{14} + \cdots - 92\!\cdots\!35 ) / 45\!\cdots\!60 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 15\!\cdots\!39 \nu^{15} + \cdots + 95\!\cdots\!50 ) / 45\!\cdots\!60 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 16\!\cdots\!90 \nu^{15} + \cdots - 18\!\cdots\!25 ) / 45\!\cdots\!60 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 16\!\cdots\!90 \nu^{15} + \cdots + 18\!\cdots\!25 ) / 45\!\cdots\!60 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 35\!\cdots\!14 \nu^{15} + \cdots + 82\!\cdots\!05 ) / 45\!\cdots\!60 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 38\!\cdots\!02 \nu^{15} + \cdots + 22\!\cdots\!25 ) / 45\!\cdots\!60 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 41\!\cdots\!60 \nu^{15} + \cdots - 29\!\cdots\!75 ) / 45\!\cdots\!60 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 73\!\cdots\!73 \nu^{15} + \cdots - 84\!\cdots\!55 ) / 45\!\cdots\!60 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 54\!\cdots\!19 \nu^{15} + \cdots + 86\!\cdots\!50 ) / 22\!\cdots\!80 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 11\!\cdots\!43 \nu^{15} + \cdots + 59\!\cdots\!75 ) / 45\!\cdots\!60 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 16\!\cdots\!01 \nu^{15} + \cdots + 16\!\cdots\!55 ) / 45\!\cdots\!60 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 16\!\cdots\!14 \nu^{15} + \cdots - 10\!\cdots\!35 ) / 45\!\cdots\!60 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 31\!\cdots\!64 \nu^{15} + \cdots + 34\!\cdots\!95 ) / 45\!\cdots\!60 \)
|
| \(\nu\) | \(=\) |
\( \beta_{6} + \beta_{5} \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{11} + 2\beta_{10} - 2\beta_{8} + 3\beta_{7} + \beta_{2} - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{15} + \beta_{13} + \beta_{12} - \beta_{10} + \beta_{8} - 5\beta_{6} - \beta_{5} - \beta_{2} + \beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( - \beta_{15} - \beta_{14} - \beta_{11} - 8 \beta_{10} - 6 \beta_{9} - 15 \beta_{7} + \beta_{5} + \cdots + 1 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 7 \beta_{14} - 7 \beta_{13} - 7 \beta_{12} + 7 \beta_{11} - 8 \beta_{8} + 8 \beta_{7} + 10 \beta_{6} + \cdots - 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( 7 \beta_{15} + \beta_{14} - 7 \beta_{13} + 11 \beta_{12} + 54 \beta_{10} + 35 \beta_{9} + 40 \beta_{8} + \cdots + 2 \beta_1 \)
|
| \(\nu^{7}\) | \(=\) |
\( 42 \beta_{15} + 54 \beta_{14} + 13 \beta_{12} - 45 \beta_{11} + 12 \beta_{10} - 13 \beta_{9} + 30 \beta_{8} + \cdots + 28 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 9 \beta_{15} + 38 \beta_{13} - 210 \beta_{12} + 89 \beta_{11} - 234 \beta_{10} - 89 \beta_{9} + \cdots - 98 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 346 \beta_{15} - 346 \beta_{14} + 248 \beta_{13} + 117 \beta_{11} + 38 \beta_{10} + 290 \beta_{9} + \cdots - 94 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 56 \beta_{14} - 56 \beta_{13} + 1290 \beta_{12} - 1290 \beta_{11} + 1992 \beta_{8} - 1992 \beta_{7} + \cdots + 1406 \)
|
| \(\nu^{11}\) | \(=\) |
\( 2174 \beta_{15} + 1477 \beta_{14} - 2174 \beta_{13} - 913 \beta_{12} - 447 \beta_{10} + \cdots - 3461 \beta_1 \)
|
| \(\nu^{12}\) | \(=\) |
\( 284 \beta_{15} + 848 \beta_{14} - 4374 \beta_{12} + 8050 \beta_{11} + 8579 \beta_{10} + 4374 \beta_{9} + \cdots - 8669 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 8898 \beta_{15} + 13556 \beta_{13} + 12425 \beta_{12} - 6647 \beta_{11} + 2237 \beta_{10} + \cdots + 5394 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 3382 \beta_{15} - 3382 \beta_{14} + 1131 \beta_{13} - 29041 \beta_{11} - 85364 \beta_{10} + \cdots + 31232 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 54132 \beta_{14} - 54132 \beta_{13} - 82100 \beta_{12} + 82100 \beta_{11} - 120596 \beta_{8} + \cdots - 67213 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/77\mathbb{Z}\right)^\times\).
| \(n\) | \(45\) | \(57\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{7} - \beta_{8} + \beta_{10}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 15.1 |
|
−1.60551 | − | 1.16647i | 0.861043 | − | 2.65002i | 0.598967 | + | 1.84343i | 0.0217822 | − | 0.0158257i | −4.47357 | + | 3.25024i | 0.309017 | + | 0.951057i | −0.0378378 | + | 0.116453i | −3.85415 | − | 2.80020i | −0.0534317 | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 15.2 | −0.901622 | − | 0.655067i | −0.883423 | + | 2.71890i | −0.234224 | − | 0.720867i | −2.79603 | + | 2.03143i | 2.57757 | − | 1.87272i | 0.309017 | + | 0.951057i | −0.949813 | + | 2.92322i | −4.18492 | − | 3.04052i | 3.85168 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 15.3 | −0.183009 | − | 0.132964i | −0.0677147 | + | 0.208405i | −0.602221 | − | 1.85345i | 2.01892 | − | 1.46683i | 0.0401026 | − | 0.0291363i | 0.309017 | + | 0.951057i | −0.276036 | + | 0.849550i | 2.38820 | + | 1.73513i | −0.564516 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 15.4 | 1.38112 | + | 1.00344i | 0.708129 | − | 2.17940i | 0.282562 | + | 0.869638i | −3.28976 | + | 2.39015i | 3.16491 | − | 2.29944i | 0.309017 | + | 0.951057i | 0.572703 | − | 1.76260i | −1.82128 | − | 1.32323i | −6.94194 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 36.1 | −1.60551 | + | 1.16647i | 0.861043 | + | 2.65002i | 0.598967 | − | 1.84343i | 0.0217822 | + | 0.0158257i | −4.47357 | − | 3.25024i | 0.309017 | − | 0.951057i | −0.0378378 | − | 0.116453i | −3.85415 | + | 2.80020i | −0.0534317 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 36.2 | −0.901622 | + | 0.655067i | −0.883423 | − | 2.71890i | −0.234224 | + | 0.720867i | −2.79603 | − | 2.03143i | 2.57757 | + | 1.87272i | 0.309017 | − | 0.951057i | −0.949813 | − | 2.92322i | −4.18492 | + | 3.04052i | 3.85168 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 36.3 | −0.183009 | + | 0.132964i | −0.0677147 | − | 0.208405i | −0.602221 | + | 1.85345i | 2.01892 | + | 1.46683i | 0.0401026 | + | 0.0291363i | 0.309017 | − | 0.951057i | −0.276036 | − | 0.849550i | 2.38820 | − | 1.73513i | −0.564516 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 36.4 | 1.38112 | − | 1.00344i | 0.708129 | + | 2.17940i | 0.282562 | − | 0.869638i | −3.28976 | − | 2.39015i | 3.16491 | + | 2.29944i | 0.309017 | − | 0.951057i | 0.572703 | + | 1.76260i | −1.82128 | + | 1.32323i | −6.94194 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.1 | −0.751051 | + | 2.31150i | −1.16030 | + | 0.843005i | −3.16091 | − | 2.29654i | 0.388938 | + | 1.19703i | −1.07716 | − | 3.31516i | −0.809017 | − | 0.587785i | 3.74989 | − | 2.72445i | −0.291419 | + | 0.896896i | −3.05904 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.2 | −0.435488 | + | 1.34029i | 1.75021 | − | 1.27160i | 0.0112975 | + | 0.00820814i | −0.565930 | − | 1.74175i | 0.942126 | + | 2.89957i | −0.809017 | − | 0.587785i | −2.29616 | + | 1.66826i | 0.519216 | − | 1.59798i | 2.58091 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.3 | 0.206962 | − | 0.636964i | −2.54013 | + | 1.84551i | 1.25514 | + | 0.911915i | 0.662464 | + | 2.03885i | 0.649815 | + | 1.99992i | −0.809017 | − | 0.587785i | 1.92429 | − | 1.39808i | 2.11929 | − | 6.52251i | 1.43578 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.4 | 0.788594 | − | 2.42704i | 0.332181 | − | 0.241344i | −3.65062 | − | 2.65233i | 1.05961 | + | 3.26115i | −0.323795 | − | 0.996539i | −0.809017 | − | 0.587785i | −5.18703 | + | 3.76860i | −0.874954 | + | 2.69283i | 8.75055 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 71.1 | −0.751051 | − | 2.31150i | −1.16030 | − | 0.843005i | −3.16091 | + | 2.29654i | 0.388938 | − | 1.19703i | −1.07716 | + | 3.31516i | −0.809017 | + | 0.587785i | 3.74989 | + | 2.72445i | −0.291419 | − | 0.896896i | −3.05904 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 71.2 | −0.435488 | − | 1.34029i | 1.75021 | + | 1.27160i | 0.0112975 | − | 0.00820814i | −0.565930 | + | 1.74175i | 0.942126 | − | 2.89957i | −0.809017 | + | 0.587785i | −2.29616 | − | 1.66826i | 0.519216 | + | 1.59798i | 2.58091 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 71.3 | 0.206962 | + | 0.636964i | −2.54013 | − | 1.84551i | 1.25514 | − | 0.911915i | 0.662464 | − | 2.03885i | 0.649815 | − | 1.99992i | −0.809017 | + | 0.587785i | 1.92429 | + | 1.39808i | 2.11929 | + | 6.52251i | 1.43578 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 71.4 | 0.788594 | + | 2.42704i | 0.332181 | + | 0.241344i | −3.65062 | + | 2.65233i | 1.05961 | − | 3.26115i | −0.323795 | + | 0.996539i | −0.809017 | + | 0.587785i | −5.18703 | − | 3.76860i | −0.874954 | − | 2.69283i | 8.75055 | |||||||||||||||||||||||||||||||||||||||||||||||||||
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 | 77.2.f.b | ✓ | 16 |
| 3.b | odd | 2 | 1 | 693.2.m.i | 16 | ||
| 7.b | odd | 2 | 1 | 539.2.f.e | 16 | ||
| 7.c | even | 3 | 2 | 539.2.q.g | 32 | ||
| 7.d | odd | 6 | 2 | 539.2.q.f | 32 | ||
| 11.b | odd | 2 | 1 | 847.2.f.x | 16 | ||
| 11.c | even | 5 | 1 | inner | 77.2.f.b | ✓ | 16 |
| 11.c | even | 5 | 1 | 847.2.a.p | 8 | ||
| 11.c | even | 5 | 2 | 847.2.f.w | 16 | ||
| 11.d | odd | 10 | 1 | 847.2.a.o | 8 | ||
| 11.d | odd | 10 | 2 | 847.2.f.v | 16 | ||
| 11.d | odd | 10 | 1 | 847.2.f.x | 16 | ||
| 33.f | even | 10 | 1 | 7623.2.a.cw | 8 | ||
| 33.h | odd | 10 | 1 | 693.2.m.i | 16 | ||
| 33.h | odd | 10 | 1 | 7623.2.a.ct | 8 | ||
| 77.j | odd | 10 | 1 | 539.2.f.e | 16 | ||
| 77.j | odd | 10 | 1 | 5929.2.a.bt | 8 | ||
| 77.l | even | 10 | 1 | 5929.2.a.bs | 8 | ||
| 77.m | even | 15 | 2 | 539.2.q.g | 32 | ||
| 77.p | odd | 30 | 2 | 539.2.q.f | 32 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 77.2.f.b | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 77.2.f.b | ✓ | 16 | 11.c | even | 5 | 1 | inner |
| 539.2.f.e | 16 | 7.b | odd | 2 | 1 | ||
| 539.2.f.e | 16 | 77.j | odd | 10 | 1 | ||
| 539.2.q.f | 32 | 7.d | odd | 6 | 2 | ||
| 539.2.q.f | 32 | 77.p | odd | 30 | 2 | ||
| 539.2.q.g | 32 | 7.c | even | 3 | 2 | ||
| 539.2.q.g | 32 | 77.m | even | 15 | 2 | ||
| 693.2.m.i | 16 | 3.b | odd | 2 | 1 | ||
| 693.2.m.i | 16 | 33.h | odd | 10 | 1 | ||
| 847.2.a.o | 8 | 11.d | odd | 10 | 1 | ||
| 847.2.a.p | 8 | 11.c | even | 5 | 1 | ||
| 847.2.f.v | 16 | 11.d | odd | 10 | 2 | ||
| 847.2.f.w | 16 | 11.c | even | 5 | 2 | ||
| 847.2.f.x | 16 | 11.b | odd | 2 | 1 | ||
| 847.2.f.x | 16 | 11.d | odd | 10 | 1 | ||
| 5929.2.a.bs | 8 | 77.l | even | 10 | 1 | ||
| 5929.2.a.bt | 8 | 77.j | odd | 10 | 1 | ||
| 7623.2.a.ct | 8 | 33.h | odd | 10 | 1 | ||
| 7623.2.a.cw | 8 | 33.f | even | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} + 3 T_{2}^{15} + 14 T_{2}^{14} + 32 T_{2}^{13} + 86 T_{2}^{12} + 145 T_{2}^{11} + 245 T_{2}^{10} + \cdots + 25 \)
acting on \(S_{2}^{\mathrm{new}}(77, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} + 3 T^{15} + \cdots + 25 \)
|
| $3$ |
\( T^{16} + 2 T^{15} + \cdots + 256 \)
|
| $5$ |
\( T^{16} + 5 T^{15} + \cdots + 256 \)
|
| $7$ |
\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{4} \)
|
| $11$ |
\( T^{16} + \cdots + 214358881 \)
|
| $13$ |
\( T^{16} + \cdots + 188897536 \)
|
| $17$ |
\( T^{16} + 5 T^{15} + \cdots + 2611456 \)
|
| $19$ |
\( T^{16} - 19 T^{15} + \cdots + 62726400 \)
|
| $23$ |
\( (T^{8} - 16 T^{7} + \cdots + 859)^{2} \)
|
| $29$ |
\( T^{16} - 3 T^{15} + \cdots + 245025 \)
|
| $31$ |
\( T^{16} + \cdots + 2629638400 \)
|
| $37$ |
\( T^{16} - 4 T^{15} + \cdots + 212521 \)
|
| $41$ |
\( T^{16} + 10 T^{15} + \cdots + 13424896 \)
|
| $43$ |
\( (T^{8} + 4 T^{7} + \cdots - 971)^{2} \)
|
| $47$ |
\( T^{16} + \cdots + 5345713926400 \)
|
| $53$ |
\( T^{16} + \cdots + 310840815961 \)
|
| $59$ |
\( T^{16} + \cdots + 187142400 \)
|
| $61$ |
\( T^{16} + \cdots + 253446400 \)
|
| $67$ |
\( (T^{8} + 19 T^{7} + \cdots - 27395)^{2} \)
|
| $71$ |
\( T^{16} + \cdots + 379119841 \)
|
| $73$ |
\( T^{16} + \cdots + 105069332736 \)
|
| $79$ |
\( T^{16} + \cdots + 15858514175625 \)
|
| $83$ |
\( T^{16} + \cdots + 756470016 \)
|
| $89$ |
\( (T^{8} - 37 T^{7} + \cdots + 952400)^{2} \)
|
| $97$ |
\( T^{16} + \cdots + 4647025244416 \)
|
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