Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [400,2,Mod(81,400)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(400, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 0, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("400.81");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 400 = 2^{4} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 400.u (of order \(5\), degree \(4\), 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: | \(3.19401608085\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 3 x^{11} + 13 x^{10} - 24 x^{9} + 93 x^{8} - 6 x^{7} + 342 x^{6} + 786 x^{5} + 1473 x^{4} + \cdots + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 100) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 3 x^{11} + 13 x^{10} - 24 x^{9} + 93 x^{8} - 6 x^{7} + 342 x^{6} + 786 x^{5} + 1473 x^{4} + \cdots + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 260325772 \nu^{11} - 576792708 \nu^{10} + 3020952123 \nu^{9} - 4900209927 \nu^{8} + \cdots + 112273499910 ) / 438461548785 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3457672033 \nu^{11} - 11783661403 \nu^{10} + 48283397793 \nu^{9} - 97783045044 \nu^{8} + \cdots + 178985069048 ) / 1753846195140 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5369632561 \nu^{11} - 15433734165 \nu^{10} + 67292419131 \nu^{9} - 119576878026 \nu^{8} + \cdots + 391154012184 ) / 1753846195140 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 22373133631 \nu^{11} - 74034744959 \nu^{10} + 314418060009 \nu^{9} - 633522002730 \nu^{8} + \cdots - 273874062464 ) / 3507692390280 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 29615632063 \nu^{11} + 87105581309 \nu^{10} - 382317412959 \nu^{9} + 696924743820 \nu^{8} + \cdots - 5593310307856 ) / 3507692390280 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 99321102659 \nu^{11} - 361617020679 \nu^{10} + 1484218111509 \nu^{9} - 3218484118998 \nu^{8} + \cdots - 17878470653712 ) / 8769230975700 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 48894251523 \nu^{11} - 157422019691 \nu^{10} + 666492738129 \nu^{9} - 1308046874814 \nu^{8} + \cdots + 2898411243400 ) / 3507692390280 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 167923324379 \nu^{11} + 478101391109 \nu^{10} - 2090376253539 \nu^{9} + \cdots - 37003359759088 ) / 8769230975700 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 542891618887 \nu^{11} - 1717504490237 \nu^{10} + 7294073305527 \nu^{9} + \cdots + 77595685775224 ) / 17538461951400 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 572643183207 \nu^{11} - 1777970044487 \nu^{10} + 7631251682697 \nu^{9} + \cdots + 26266096925344 ) / 17538461951400 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{10} - 4\beta_{8} + \beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} + \beta _1 - 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{11} + 2\beta_{10} + \beta_{9} - 6\beta_{8} - \beta_{7} + 4\beta_{6} + 5\beta_{5} + 7\beta_{4} + \beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 8 \beta_{11} + 8 \beta_{10} + 8 \beta_{9} - 2 \beta_{8} - 16 \beta_{7} + 24 \beta_{6} + 10 \beta_{5} + \cdots - 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 3 \beta_{11} + 3 \beta_{10} - 12 \beta_{9} - 12 \beta_{7} + 29 \beta_{6} + 29 \beta_{5} - 69 \beta_{4} + \cdots - 47 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 75 \beta_{11} - 66 \beta_{10} - 132 \beta_{9} + 54 \beta_{8} + 57 \beta_{7} + 66 \beta_{6} + \cdots + 12 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 256 \beta_{11} - 178 \beta_{10} - 128 \beta_{9} + 844 \beta_{8} + 78 \beta_{7} - 220 \beta_{6} + \cdots - 220 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 753 \beta_{11} - 368 \beta_{10} + 385 \beta_{9} + 3112 \beta_{8} + 184 \beta_{7} - 2744 \beta_{6} + \cdots - 1852 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 652 \beta_{11} - 2009 \beta_{10} + 705 \beta_{9} + 4658 \beta_{8} + 1304 \beta_{7} - 7165 \beta_{6} + \cdots + 2009 \)
|
| \(\nu^{10}\) | \(=\) |
\( 5100 \beta_{11} - 5100 \beta_{10} + 2700 \beta_{9} + 2700 \beta_{7} - 7110 \beta_{6} - 7110 \beta_{5} + \cdots + 25394 \)
|
| \(\nu^{11}\) | \(=\) |
\( 22335 \beta_{11} + 7815 \beta_{10} + 15630 \beta_{9} - 53280 \beta_{8} + 6705 \beta_{7} - 7815 \beta_{6} + \cdots + 45465 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(177\) | \(351\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{8}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 81.1 |
|
0 | −2.19573 | + | 1.59529i | 0 | 0.548020 | + | 2.16787i | 0 | 4.40288 | 0 | 1.34923 | − | 4.15250i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 81.2 | 0 | −0.559818 | + | 0.406731i | 0 | −0.463031 | − | 2.18760i | 0 | −3.25686 | 0 | −0.779086 | + | 2.39778i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 81.3 | 0 | 2.25555 | − | 1.63875i | 0 | −2.20302 | + | 0.383000i | 0 | 2.70809 | 0 | 1.47494 | − | 4.53940i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 161.1 | 0 | −1.02360 | + | 3.15031i | 0 | −1.34720 | + | 1.78467i | 0 | −1.69984 | 0 | −6.44966 | − | 4.68596i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 161.2 | 0 | 0.0461234 | − | 0.141953i | 0 | 0.383646 | − | 2.20291i | 0 | −3.58696 | 0 | 2.40903 | + | 1.75026i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 161.3 | 0 | 0.477475 | − | 1.46952i | 0 | 1.08159 | + | 1.95708i | 0 | 2.43270 | 0 | 0.495552 | + | 0.360039i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 241.1 | 0 | −1.02360 | − | 3.15031i | 0 | −1.34720 | − | 1.78467i | 0 | −1.69984 | 0 | −6.44966 | + | 4.68596i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 241.2 | 0 | 0.0461234 | + | 0.141953i | 0 | 0.383646 | + | 2.20291i | 0 | −3.58696 | 0 | 2.40903 | − | 1.75026i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 241.3 | 0 | 0.477475 | + | 1.46952i | 0 | 1.08159 | − | 1.95708i | 0 | 2.43270 | 0 | 0.495552 | − | 0.360039i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 321.1 | 0 | −2.19573 | − | 1.59529i | 0 | 0.548020 | − | 2.16787i | 0 | 4.40288 | 0 | 1.34923 | + | 4.15250i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 321.2 | 0 | −0.559818 | − | 0.406731i | 0 | −0.463031 | + | 2.18760i | 0 | −3.25686 | 0 | −0.779086 | − | 2.39778i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 321.3 | 0 | 2.25555 | + | 1.63875i | 0 | −2.20302 | − | 0.383000i | 0 | 2.70809 | 0 | 1.47494 | + | 4.53940i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 25.d | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 400.2.u.f | 12 | |
| 4.b | odd | 2 | 1 | 100.2.g.a | ✓ | 12 | |
| 12.b | even | 2 | 1 | 900.2.n.c | 12 | ||
| 20.d | odd | 2 | 1 | 500.2.g.a | 12 | ||
| 20.e | even | 4 | 2 | 500.2.i.b | 24 | ||
| 25.d | even | 5 | 1 | inner | 400.2.u.f | 12 | |
| 25.d | even | 5 | 1 | 10000.2.a.bc | 6 | ||
| 25.e | even | 10 | 1 | 10000.2.a.bd | 6 | ||
| 100.h | odd | 10 | 1 | 500.2.g.a | 12 | ||
| 100.h | odd | 10 | 1 | 2500.2.a.c | 6 | ||
| 100.j | odd | 10 | 1 | 100.2.g.a | ✓ | 12 | |
| 100.j | odd | 10 | 1 | 2500.2.a.d | 6 | ||
| 100.l | even | 20 | 2 | 500.2.i.b | 24 | ||
| 100.l | even | 20 | 2 | 2500.2.c.c | 12 | ||
| 300.n | even | 10 | 1 | 900.2.n.c | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 100.2.g.a | ✓ | 12 | 4.b | odd | 2 | 1 | |
| 100.2.g.a | ✓ | 12 | 100.j | odd | 10 | 1 | |
| 400.2.u.f | 12 | 1.a | even | 1 | 1 | trivial | |
| 400.2.u.f | 12 | 25.d | even | 5 | 1 | inner | |
| 500.2.g.a | 12 | 20.d | odd | 2 | 1 | ||
| 500.2.g.a | 12 | 100.h | odd | 10 | 1 | ||
| 500.2.i.b | 24 | 20.e | even | 4 | 2 | ||
| 500.2.i.b | 24 | 100.l | even | 20 | 2 | ||
| 900.2.n.c | 12 | 12.b | even | 2 | 1 | ||
| 900.2.n.c | 12 | 300.n | even | 10 | 1 | ||
| 2500.2.a.c | 6 | 100.h | odd | 10 | 1 | ||
| 2500.2.a.d | 6 | 100.j | odd | 10 | 1 | ||
| 2500.2.c.c | 12 | 100.l | even | 20 | 2 | ||
| 10000.2.a.bc | 6 | 25.d | even | 5 | 1 | ||
| 10000.2.a.bd | 6 | 25.e | even | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} + 2 T_{3}^{11} + 8 T_{3}^{10} - 4 T_{3}^{9} + 23 T_{3}^{8} + 124 T_{3}^{7} + 637 T_{3}^{6} + \cdots + 16 \)
acting on \(S_{2}^{\mathrm{new}}(400, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} + 2 T^{11} + \cdots + 16 \)
|
| $5$ |
\( T^{12} + 4 T^{11} + \cdots + 15625 \)
|
| $7$ |
\( (T^{6} - T^{5} - 29 T^{4} + \cdots - 576)^{2} \)
|
| $11$ |
\( T^{12} - 5 T^{11} + \cdots + 160000 \)
|
| $13$ |
\( T^{12} + 2 T^{11} + \cdots + 32761 \)
|
| $17$ |
\( T^{12} - T^{11} + \cdots + 1296 \)
|
| $19$ |
\( T^{12} - 8 T^{11} + \cdots + 10471696 \)
|
| $23$ |
\( T^{12} - 6 T^{11} + \cdots + 4096 \)
|
| $29$ |
\( T^{12} + 18 T^{11} + \cdots + 1042441 \)
|
| $31$ |
\( T^{12} + \cdots + 102495376 \)
|
| $37$ |
\( T^{12} - 13 T^{11} + \cdots + 2128681 \)
|
| $41$ |
\( T^{12} + 23 T^{11} + \cdots + 41422096 \)
|
| $43$ |
\( (T^{6} + 25 T^{5} + \cdots + 6400)^{2} \)
|
| $47$ |
\( T^{12} + T^{11} + \cdots + 256 \)
|
| $53$ |
\( T^{12} - 21 T^{11} + \cdots + 1745041 \)
|
| $59$ |
\( T^{12} + \cdots + 130051216 \)
|
| $61$ |
\( T^{12} + \cdots + 1661296081 \)
|
| $67$ |
\( T^{12} + \cdots + 58285547776 \)
|
| $71$ |
\( T^{12} + \cdots + 5547866256 \)
|
| $73$ |
\( T^{12} + \cdots + 4336354201 \)
|
| $79$ |
\( T^{12} + \cdots + 56261942416 \)
|
| $83$ |
\( T^{12} - 46 T^{11} + \cdots + 36048016 \)
|
| $89$ |
\( T^{12} + 2 T^{11} + \cdots + 63744256 \)
|
| $97$ |
\( T^{12} + \cdots + 121807282081 \)
|
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