Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [28,3,Mod(11,28)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(28, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 4]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("28.11");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 28 = 2^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 28.g (of order \(6\), 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: | \(0.762944740209\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(6\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 3 x^{11} - 4 x^{10} + 3 x^{9} + 86 x^{8} - 163 x^{7} + 155 x^{6} - 166 x^{5} + 164 x^{4} + \cdots + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{8} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 3 x^{11} - 4 x^{10} + 3 x^{9} + 86 x^{8} - 163 x^{7} + 155 x^{6} - 166 x^{5} + 164 x^{4} + \cdots + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 548052 \nu^{11} + 1422116 \nu^{10} - 6191612 \nu^{9} - 21007396 \nu^{8} + 23867696 \nu^{7} + \cdots - 9243685 ) / 5128417 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 2016555 \nu^{11} + 4608465 \nu^{10} + 11867940 \nu^{9} + 1459025 \nu^{8} - 175954938 \nu^{7} + \cdots + 13204764 ) / 10256834 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 6080681 \nu^{11} + 174798 \nu^{10} - 64448111 \nu^{9} - 87898803 \nu^{8} + 490503169 \nu^{7} + \cdots + 199145960 ) / 20513668 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 8092731 \nu^{11} + 13062376 \nu^{10} + 58450763 \nu^{9} + 38422753 \nu^{8} - 688812905 \nu^{7} + \cdots - 38518016 ) / 20513668 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 8864783 \nu^{11} - 20455402 \nu^{10} - 48834261 \nu^{9} - 8785933 \nu^{8} + 751559539 \nu^{7} + \cdots - 70375800 ) / 20513668 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 2781796 \nu^{11} + 6284192 \nu^{10} + 15510516 \nu^{9} + 3410783 \nu^{8} - 234558069 \nu^{7} + \cdots + 15160294 ) / 5128417 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 11588911 \nu^{11} - 2778978 \nu^{10} - 108332469 \nu^{9} - 167711837 \nu^{8} + 897253275 \nu^{7} + \cdots + 51100076 ) / 20513668 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 7577433 \nu^{11} - 23003531 \nu^{10} - 36322258 \nu^{9} + 36406939 \nu^{8} + 694698148 \nu^{7} + \cdots - 10748502 ) / 10256834 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 18854063 \nu^{11} + 54731322 \nu^{10} + 82312181 \nu^{9} - 56986907 \nu^{8} - 1629791971 \nu^{7} + \cdots + 162375652 ) / 20513668 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 13906733 \nu^{11} - 32333536 \nu^{10} - 77826224 \nu^{9} - 8906454 \nu^{8} + 1190477049 \nu^{7} + \cdots - 51147512 ) / 10256834 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 6254493 \nu^{11} + 16328866 \nu^{10} + 30974575 \nu^{9} - 6021165 \nu^{8} - 537499493 \nu^{7} + \cdots + 44275732 ) / 2930524 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{11} + 2\beta_{10} + 2\beta_{6} + 3\beta_{5} + 2\beta_{4} + 3 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{9} + 2\beta_{8} + 2\beta_{7} - 11\beta_{5} + 2\beta_{4} - \beta_{3} - 6\beta_{2} + 2\beta_1 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 9\beta_{11} + 18\beta_{10} + 2\beta_{8} + 6\beta_{6} - 9\beta_{3} + 6\beta_{2} + 4\beta _1 + 25 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -\beta_{11} + 38\beta_{10} - 14\beta_{9} + 18\beta_{7} + 54\beta_{6} - 75\beta_{5} + 38\beta_{4} - 75 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -52\beta_{9} + 38\beta_{8} + 38\beta_{7} - 293\beta_{5} - 122\beta_{4} - 81\beta_{3} + 2\beta_{2} + 52\beta_1 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 71\beta_{11} + 462\beta_{10} - 122\beta_{8} + 482\beta_{6} - 71\beta_{3} + 482\beta_{2} - 70\beta _1 - 379 ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 651 \beta_{11} - 622 \beta_{10} - 532 \beta_{9} + 462 \beta_{7} + 462 \beta_{6} - 2987 \beta_{5} + \cdots - 2987 ) / 4 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 90 \beta_{9} - 622 \beta_{8} - 622 \beta_{7} + 391 \beta_{5} - 4734 \beta_{4} - 1283 \beta_{3} + \cdots - 90 \beta_1 ) / 4 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 4513 \beta_{11} - 802 \beta_{10} - 4734 \beta_{8} + 7942 \beta_{6} + 4513 \beta_{3} + 7942 \beta_{2} + \cdots - 26985 ) / 4 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 15935 \beta_{11} - 42942 \beta_{10} - 4022 \beta_{9} - 802 \beta_{7} - 26154 \beta_{6} - 23505 \beta_{5} + \cdots - 23505 ) / 4 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 38920 \beta_{9} - 42942 \beta_{8} - 42942 \beta_{7} + 216715 \beta_{5} - 35802 \beta_{4} + \cdots - 38920 \beta_1 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/28\mathbb{Z}\right)^\times\).
| \(n\) | \(15\) | \(17\) |
| \(\chi(n)\) | \(-1\) | \(-1 - \beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 11.1 |
|
−1.98615 | + | 0.234945i | 1.86796 | + | 1.07847i | 3.88960 | − | 0.933271i | 3.25304 | + | 5.63443i | −3.96343 | − | 1.70313i | −2.39669 | − | 6.57692i | −7.50608 | + | 2.76746i | −2.17382 | − | 3.76517i | −7.78481 | − | 10.4265i | ||||||||||||||||||||||||||||||||||||
| 11.2 | −1.51615 | − | 1.30434i | −3.95004 | − | 2.28056i | 0.597396 | + | 3.95514i | −2.62655 | − | 4.54932i | 3.01422 | + | 8.60985i | 5.86799 | − | 3.81663i | 4.25310 | − | 6.77577i | 5.90188 | + | 10.2224i | −1.95163 | + | 10.3234i | |||||||||||||||||||||||||||||||||||||
| 11.3 | −0.371518 | − | 1.96519i | 3.95004 | + | 2.28056i | −3.72395 | + | 1.46021i | −2.62655 | − | 4.54932i | 3.01422 | − | 8.60985i | −5.86799 | + | 3.81663i | 4.25310 | + | 6.77577i | 5.90188 | + | 10.2224i | −7.96447 | + | 6.85183i | |||||||||||||||||||||||||||||||||||||
| 11.4 | −0.104798 | + | 1.99725i | 1.63031 | + | 0.941260i | −3.97803 | − | 0.418616i | −1.12649 | − | 1.95113i | −2.05079 | + | 3.15750i | 6.84270 | + | 1.47562i | 1.25297 | − | 7.90127i | −2.72806 | − | 4.72514i | 4.01495 | − | 2.04540i | |||||||||||||||||||||||||||||||||||||
| 11.5 | 1.19654 | − | 1.60259i | −1.86796 | − | 1.07847i | −1.13656 | − | 3.83513i | 3.25304 | + | 5.63443i | −3.96343 | + | 1.70313i | 2.39669 | + | 6.57692i | −7.50608 | − | 2.76746i | −2.17382 | − | 3.76517i | 12.9221 | + | 1.52857i | |||||||||||||||||||||||||||||||||||||
| 11.6 | 1.78207 | + | 0.907869i | −1.63031 | − | 0.941260i | 2.35155 | + | 3.23577i | −1.12649 | − | 1.95113i | −2.05079 | − | 3.15750i | −6.84270 | − | 1.47562i | 1.25297 | + | 7.90127i | −2.72806 | − | 4.72514i | −0.236107 | − | 4.49975i | |||||||||||||||||||||||||||||||||||||
| 23.1 | −1.98615 | − | 0.234945i | 1.86796 | − | 1.07847i | 3.88960 | + | 0.933271i | 3.25304 | − | 5.63443i | −3.96343 | + | 1.70313i | −2.39669 | + | 6.57692i | −7.50608 | − | 2.76746i | −2.17382 | + | 3.76517i | −7.78481 | + | 10.4265i | |||||||||||||||||||||||||||||||||||||
| 23.2 | −1.51615 | + | 1.30434i | −3.95004 | + | 2.28056i | 0.597396 | − | 3.95514i | −2.62655 | + | 4.54932i | 3.01422 | − | 8.60985i | 5.86799 | + | 3.81663i | 4.25310 | + | 6.77577i | 5.90188 | − | 10.2224i | −1.95163 | − | 10.3234i | |||||||||||||||||||||||||||||||||||||
| 23.3 | −0.371518 | + | 1.96519i | 3.95004 | − | 2.28056i | −3.72395 | − | 1.46021i | −2.62655 | + | 4.54932i | 3.01422 | + | 8.60985i | −5.86799 | − | 3.81663i | 4.25310 | − | 6.77577i | 5.90188 | − | 10.2224i | −7.96447 | − | 6.85183i | |||||||||||||||||||||||||||||||||||||
| 23.4 | −0.104798 | − | 1.99725i | 1.63031 | − | 0.941260i | −3.97803 | + | 0.418616i | −1.12649 | + | 1.95113i | −2.05079 | − | 3.15750i | 6.84270 | − | 1.47562i | 1.25297 | + | 7.90127i | −2.72806 | + | 4.72514i | 4.01495 | + | 2.04540i | |||||||||||||||||||||||||||||||||||||
| 23.5 | 1.19654 | + | 1.60259i | −1.86796 | + | 1.07847i | −1.13656 | + | 3.83513i | 3.25304 | − | 5.63443i | −3.96343 | − | 1.70313i | 2.39669 | − | 6.57692i | −7.50608 | + | 2.76746i | −2.17382 | + | 3.76517i | 12.9221 | − | 1.52857i | |||||||||||||||||||||||||||||||||||||
| 23.6 | 1.78207 | − | 0.907869i | −1.63031 | + | 0.941260i | 2.35155 | − | 3.23577i | −1.12649 | + | 1.95113i | −2.05079 | + | 3.15750i | −6.84270 | + | 1.47562i | 1.25297 | − | 7.90127i | −2.72806 | + | 4.72514i | −0.236107 | + | 4.49975i | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 7.c | even | 3 | 1 | inner |
| 28.g | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 28.3.g.a | ✓ | 12 |
| 3.b | odd | 2 | 1 | 252.3.y.c | 12 | ||
| 4.b | odd | 2 | 1 | inner | 28.3.g.a | ✓ | 12 |
| 7.b | odd | 2 | 1 | 196.3.g.i | 12 | ||
| 7.c | even | 3 | 1 | inner | 28.3.g.a | ✓ | 12 |
| 7.c | even | 3 | 1 | 196.3.c.i | 6 | ||
| 7.d | odd | 6 | 1 | 196.3.c.h | 6 | ||
| 7.d | odd | 6 | 1 | 196.3.g.i | 12 | ||
| 8.b | even | 2 | 1 | 448.3.r.h | 12 | ||
| 8.d | odd | 2 | 1 | 448.3.r.h | 12 | ||
| 12.b | even | 2 | 1 | 252.3.y.c | 12 | ||
| 21.h | odd | 6 | 1 | 252.3.y.c | 12 | ||
| 28.d | even | 2 | 1 | 196.3.g.i | 12 | ||
| 28.f | even | 6 | 1 | 196.3.c.h | 6 | ||
| 28.f | even | 6 | 1 | 196.3.g.i | 12 | ||
| 28.g | odd | 6 | 1 | inner | 28.3.g.a | ✓ | 12 |
| 28.g | odd | 6 | 1 | 196.3.c.i | 6 | ||
| 56.k | odd | 6 | 1 | 448.3.r.h | 12 | ||
| 56.p | even | 6 | 1 | 448.3.r.h | 12 | ||
| 84.n | even | 6 | 1 | 252.3.y.c | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 28.3.g.a | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 28.3.g.a | ✓ | 12 | 4.b | odd | 2 | 1 | inner |
| 28.3.g.a | ✓ | 12 | 7.c | even | 3 | 1 | inner |
| 28.3.g.a | ✓ | 12 | 28.g | odd | 6 | 1 | inner |
| 196.3.c.h | 6 | 7.d | odd | 6 | 1 | ||
| 196.3.c.h | 6 | 28.f | even | 6 | 1 | ||
| 196.3.c.i | 6 | 7.c | even | 3 | 1 | ||
| 196.3.c.i | 6 | 28.g | odd | 6 | 1 | ||
| 196.3.g.i | 12 | 7.b | odd | 2 | 1 | ||
| 196.3.g.i | 12 | 7.d | odd | 6 | 1 | ||
| 196.3.g.i | 12 | 28.d | even | 2 | 1 | ||
| 196.3.g.i | 12 | 28.f | even | 6 | 1 | ||
| 252.3.y.c | 12 | 3.b | odd | 2 | 1 | ||
| 252.3.y.c | 12 | 12.b | even | 2 | 1 | ||
| 252.3.y.c | 12 | 21.h | odd | 6 | 1 | ||
| 252.3.y.c | 12 | 84.n | even | 6 | 1 | ||
| 448.3.r.h | 12 | 8.b | even | 2 | 1 | ||
| 448.3.r.h | 12 | 8.d | odd | 2 | 1 | ||
| 448.3.r.h | 12 | 56.k | odd | 6 | 1 | ||
| 448.3.r.h | 12 | 56.p | even | 6 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(28, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + 2 T^{11} + \cdots + 4096 \)
|
| $3$ |
\( T^{12} - 29 T^{10} + \cdots + 117649 \)
|
| $5$ |
\( (T^{6} + T^{5} + 38 T^{4} + \cdots + 5929)^{2} \)
|
| $7$ |
\( T^{12} + \cdots + 13841287201 \)
|
| $11$ |
\( T^{12} - 101 T^{10} + \cdots + 30625 \)
|
| $13$ |
\( (T^{3} + 6 T^{2} + \cdots - 280)^{4} \)
|
| $17$ |
\( (T^{6} + T^{5} + \cdots + 17935225)^{2} \)
|
| $19$ |
\( T^{12} + \cdots + 28722900390625 \)
|
| $23$ |
\( T^{12} + \cdots + 234433177489 \)
|
| $29$ |
\( (T^{3} - 18 T^{2} + \cdots - 4408)^{4} \)
|
| $31$ |
\( T^{12} + \cdots + 15\!\cdots\!25 \)
|
| $37$ |
\( (T^{6} - 43 T^{5} + \cdots + 1926771025)^{2} \)
|
| $41$ |
\( (T^{3} - 2 T^{2} + \cdots + 44296)^{4} \)
|
| $43$ |
\( (T^{6} + 6976 T^{4} + \cdots + 1101463552)^{2} \)
|
| $47$ |
\( T^{12} + \cdots + 43\!\cdots\!89 \)
|
| $53$ |
\( (T^{6} + 37 T^{5} + \cdots + 417180625)^{2} \)
|
| $59$ |
\( T^{12} + \cdots + 24\!\cdots\!25 \)
|
| $61$ |
\( (T^{6} - 43 T^{5} + \cdots + 5716418449)^{2} \)
|
| $67$ |
\( T^{12} + \cdots + 49\!\cdots\!09 \)
|
| $71$ |
\( (T^{6} + 11648 T^{4} + \cdots + 6884147200)^{2} \)
|
| $73$ |
\( (T^{6} + 117 T^{5} + \cdots + 1997643025)^{2} \)
|
| $79$ |
\( T^{12} + \cdots + 36\!\cdots\!25 \)
|
| $83$ |
\( (T^{6} + 2944 T^{4} + \cdots + 275365888)^{2} \)
|
| $89$ |
\( (T^{6} - 3 T^{5} + \cdots + 52287361)^{2} \)
|
| $97$ |
\( (T^{3} - 186 T^{2} + \cdots + 994280)^{4} \)
|
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