Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [770,2,Mod(221,770)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(770, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("770.221");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 770.i (of order \(3\), 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: | \(6.14848095564\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Relative dimension: | \(5\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} + 12x^{8} - 6x^{7} + 113x^{6} - 43x^{5} + 381x^{4} - 75x^{3} + 982x^{2} - 217x + 49 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} + 12x^{8} - 6x^{7} + 113x^{6} - 43x^{5} + 381x^{4} - 75x^{3} + 982x^{2} - 217x + 49 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 98501 \nu^{9} + 122729 \nu^{8} - 114811 \nu^{7} + 622151 \nu^{6} - 1745919 \nu^{5} + \cdots + 87558415 ) / 383873037 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 122729 \nu^{9} + 1296823 \nu^{8} - 1213157 \nu^{7} + 12876532 \nu^{6} - 18448353 \nu^{5} + \cdots + 1924191734 ) / 383873037 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 12763717 \nu^{9} - 14388479 \nu^{8} - 180540162 \nu^{7} - 215638669 \nu^{6} + \cdots - 10157690340 ) / 5374222518 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 21436535 \nu^{9} + 39922932 \nu^{8} - 231347611 \nu^{7} + 566697487 \nu^{6} + \cdots + 16808160187 ) / 5374222518 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 12508345 \nu^{9} + 689507 \nu^{8} - 149241037 \nu^{7} + 74246393 \nu^{6} - 1409087928 \nu^{5} + \cdots + 2642146213 ) / 2687111259 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 57287134 \nu^{9} - 43187207 \nu^{8} - 636536463 \nu^{7} - 21732244 \nu^{6} + \cdots - 7072023903 ) / 5374222518 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 63494973 \nu^{9} - 29110928 \nu^{8} - 843704947 \nu^{7} - 4838781 \nu^{6} + \cdots + 6315431857 ) / 5374222518 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 62541725 \nu^{9} + 3447535 \nu^{8} - 746205185 \nu^{7} + 371231965 \nu^{6} + \cdots + 13210731065 ) / 2687111259 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{9} - 5\beta_{6} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} - \beta_{7} - \beta_{4} - \beta_{3} - 6\beta_{2} + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -10\beta_{9} + 2\beta_{8} + 3\beta_{7} + 36\beta_{6} + \beta_{5} + \beta_{4} + 10\beta_{3} + 2\beta _1 - 36 \)
|
| \(\nu^{5}\) | \(=\) |
\( 15\beta_{9} - 12\beta_{8} - 32\beta_{6} - 12\beta_{5} + 41\beta_{2} - 41\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 15\beta_{8} - 15\beta_{7} + 24\beta_{5} - 39\beta_{4} - 92\beta_{3} - 35\beta_{2} + 283 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 172 \beta_{9} + 6 \beta_{8} + 122 \beta_{7} + 395 \beta_{6} + 116 \beta_{5} + 116 \beta_{4} + \cdots - 395 \)
|
| \(\nu^{8}\) | \(=\) |
\( 832\beta_{9} - 404\beta_{8} - 226\beta_{7} - 2362\beta_{6} - 404\beta_{5} + 226\beta_{4} + 439\beta_{2} - 439\beta_1 \)
|
| \(\nu^{9}\) | \(=\) |
\( 1058\beta_{8} - 1058\beta_{7} + 130\beta_{5} - 1188\beta_{4} - 1805\beta_{3} - 2564\beta_{2} + 4345 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/770\mathbb{Z}\right)^\times\).
| \(n\) | \(211\) | \(617\) | \(661\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1 + \beta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 221.1 |
|
−0.500000 | − | 0.866025i | −1.26598 | + | 2.19274i | −0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | 2.53196 | 2.22551 | − | 1.43078i | 1.00000 | −1.70541 | − | 2.95386i | −0.500000 | + | 0.866025i | ||||||||||||||||||||||||||||||||||
| 221.2 | −0.500000 | − | 0.866025i | −1.08142 | + | 1.87308i | −0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | 2.16285 | −0.590793 | + | 2.57895i | 1.00000 | −0.838952 | − | 1.45311i | −0.500000 | + | 0.866025i | |||||||||||||||||||||||||||||||||||
| 221.3 | −0.500000 | − | 0.866025i | −0.112651 | + | 0.195118i | −0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | 0.225302 | −2.14481 | − | 1.54913i | 1.00000 | 1.47462 | + | 2.55412i | −0.500000 | + | 0.866025i | |||||||||||||||||||||||||||||||||||
| 221.4 | −0.500000 | − | 0.866025i | 0.922461 | − | 1.59775i | −0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | −1.84492 | 2.59682 | + | 0.506486i | 1.00000 | −0.201870 | − | 0.349648i | −0.500000 | + | 0.866025i | |||||||||||||||||||||||||||||||||||
| 221.5 | −0.500000 | − | 0.866025i | 1.53759 | − | 2.66319i | −0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | −3.07519 | −2.08672 | + | 1.62653i | 1.00000 | −3.22839 | − | 5.59173i | −0.500000 | + | 0.866025i | |||||||||||||||||||||||||||||||||||
| 331.1 | −0.500000 | + | 0.866025i | −1.26598 | − | 2.19274i | −0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | 2.53196 | 2.22551 | + | 1.43078i | 1.00000 | −1.70541 | + | 2.95386i | −0.500000 | − | 0.866025i | |||||||||||||||||||||||||||||||||||
| 331.2 | −0.500000 | + | 0.866025i | −1.08142 | − | 1.87308i | −0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | 2.16285 | −0.590793 | − | 2.57895i | 1.00000 | −0.838952 | + | 1.45311i | −0.500000 | − | 0.866025i | |||||||||||||||||||||||||||||||||||
| 331.3 | −0.500000 | + | 0.866025i | −0.112651 | − | 0.195118i | −0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | 0.225302 | −2.14481 | + | 1.54913i | 1.00000 | 1.47462 | − | 2.55412i | −0.500000 | − | 0.866025i | |||||||||||||||||||||||||||||||||||
| 331.4 | −0.500000 | + | 0.866025i | 0.922461 | + | 1.59775i | −0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | −1.84492 | 2.59682 | − | 0.506486i | 1.00000 | −0.201870 | + | 0.349648i | −0.500000 | − | 0.866025i | |||||||||||||||||||||||||||||||||||
| 331.5 | −0.500000 | + | 0.866025i | 1.53759 | + | 2.66319i | −0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | −3.07519 | −2.08672 | − | 1.62653i | 1.00000 | −3.22839 | + | 5.59173i | −0.500000 | − | 0.866025i | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 770.2.i.m | ✓ | 10 |
| 7.c | even | 3 | 1 | inner | 770.2.i.m | ✓ | 10 |
| 7.c | even | 3 | 1 | 5390.2.a.ci | 5 | ||
| 7.d | odd | 6 | 1 | 5390.2.a.ch | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 770.2.i.m | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 770.2.i.m | ✓ | 10 | 7.c | even | 3 | 1 | inner |
| 5390.2.a.ch | 5 | 7.d | odd | 6 | 1 | ||
| 5390.2.a.ci | 5 | 7.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(770, [\chi])\):
|
\( T_{3}^{10} + 12T_{3}^{8} + 6T_{3}^{7} + 113T_{3}^{6} + 43T_{3}^{5} + 381T_{3}^{4} + 75T_{3}^{3} + 982T_{3}^{2} + 217T_{3} + 49 \)
|
|
\( T_{13}^{5} + T_{13}^{4} - 28T_{13}^{3} - 20T_{13}^{2} + 168T_{13} + 144 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + T + 1)^{5} \)
|
| $3$ |
\( T^{10} + 12 T^{8} + \cdots + 49 \)
|
| $5$ |
\( (T^{2} + T + 1)^{5} \)
|
| $7$ |
\( T^{10} - 7 T^{8} + \cdots + 16807 \)
|
| $11$ |
\( (T^{2} + T + 1)^{5} \)
|
| $13$ |
\( (T^{5} + T^{4} - 28 T^{3} + \cdots + 144)^{2} \)
|
| $17$ |
\( T^{10} + 43 T^{8} + \cdots + 50176 \)
|
| $19$ |
\( T^{10} + 4 T^{9} + \cdots + 92416 \)
|
| $23$ |
\( T^{10} + 7 T^{9} + \cdots + 2304 \)
|
| $29$ |
\( (T^{5} - 4 T^{4} + \cdots - 317)^{2} \)
|
| $31$ |
\( T^{10} + 6 T^{9} + \cdots + 576 \)
|
| $37$ |
\( T^{10} + \cdots + 107495424 \)
|
| $41$ |
\( (T^{5} + T^{4} - 160 T^{3} + \cdots - 3584)^{2} \)
|
| $43$ |
\( (T^{5} - 13 T^{4} + \cdots - 16)^{2} \)
|
| $47$ |
\( T^{10} + 8 T^{9} + \cdots + 94633984 \)
|
| $53$ |
\( T^{10} + \cdots + 1387115536 \)
|
| $59$ |
\( T^{10} + \cdots + 289816576 \)
|
| $61$ |
\( T^{10} + 2 T^{9} + \cdots + 25331089 \)
|
| $67$ |
\( T^{10} + 5 T^{9} + \cdots + 421201 \)
|
| $71$ |
\( (T^{5} - 28 T^{4} + \cdots + 12544)^{2} \)
|
| $73$ |
\( T^{10} - T^{9} + \cdots + 29333056 \)
|
| $79$ |
\( T^{10} + \cdots + 6392322304 \)
|
| $83$ |
\( (T^{5} + 23 T^{4} + \cdots + 12672)^{2} \)
|
| $89$ |
\( T^{10} + \cdots + 176943204 \)
|
| $97$ |
\( (T^{5} + 13 T^{4} + \cdots + 120192)^{2} \)
|
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