Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [400,10,Mod(49,400)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(400, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("400.49");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 400 = 2^{4} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 400.c (of order \(2\), degree \(1\), 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: | \(206.014334466\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} + 1532x^{8} + 822940x^{6} + 190326313x^{4} + 17734867904x^{2} + 403225000000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{37}]\) |
| Coefficient ring index: | \( 2^{36}\cdot 3^{6}\cdot 5^{6} \) |
| Twist minimal: | no (minimal twist has level 200) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{10} + 1532x^{8} + 822940x^{6} + 190326313x^{4} + 17734867904x^{2} + 403225000000 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -50462\nu^{8} - 66195362\nu^{6} - 26407509708\nu^{4} - 3363263729258\nu^{2} - 69679211120375 ) / 230721301125 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -9469\nu^{9} - 12977428\nu^{7} - 5682816780\nu^{5} - 892717379077\nu^{3} - 36559996629256\nu ) / 74416775220000 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2501756 \nu^{8} + 3063659456 \nu^{6} + 1133471611704 \nu^{4} + 147191495035604 \nu^{2} + 53\!\cdots\!50 ) / 230721301125 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 1271786 \nu^{8} - 1612767986 \nu^{6} - 625038272724 \nu^{4} - 79016927757374 \nu^{2} - 14\!\cdots\!00 ) / 25635700125 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1813294 \nu^{8} + 2540833594 \nu^{6} + 1150563879996 \nu^{4} + 185829058854346 \nu^{2} + 66\!\cdots\!75 ) / 32960185875 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 11895 \nu^{9} + 1074331444 \nu^{7} + 1254966561084 \nu^{5} + 388585967009721 \nu^{3} + 28\!\cdots\!76 \nu ) / 4340978554500 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 38359889 \nu^{9} + 46825931108 \nu^{7} + 16470027749820 \nu^{5} + \cdots - 18\!\cdots\!04 \nu ) / 46\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 423216125 \nu^{9} + 542389410452 \nu^{7} + 218517302355852 \nu^{5} + \cdots + 14\!\cdots\!68 \nu ) / 468825683886000 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 2681960543 \nu^{9} - 3433024883036 \nu^{7} + \cdots + 23\!\cdots\!88 \nu ) / 669750976980000 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{9} - \beta_{8} + 566\beta_{7} + \beta_{6} - 2191\beta_{2} ) / 5760 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + 13\beta_{4} + 35\beta_{3} - 962\beta _1 - 588461 ) / 1920 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -89\beta_{9} + 197\beta_{8} - 138610\beta_{7} - 485\beta_{6} - 4704133\beta_{2} ) / 2880 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 181\beta_{5} - 9239\beta_{4} - 20233\beta_{3} + 1138078\beta _1 + 269455159 ) / 1920 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 38641\beta_{9} + 133619\beta_{8} + 154285826\beta_{7} + 739741\beta_{6} + 9637037309\beta_{2} ) / 5760 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -9811\beta_{5} + 261617\beta_{4} + 463915\beta_{3} - 38809738\beta _1 - 6157673169 ) / 80 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 6944473 \beta_{9} - 269580131 \beta_{8} - 94106819690 \beta_{7} - 512347645 \beta_{6} - 7734307939421 \beta_{2} ) / 5760 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 147509051\beta_{5} - 4267992049\beta_{4} - 6349879103\beta_{3} + 681607164218\beta _1 + 89420780644049 ) / 1920 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 376208047 \beta_{9} + 127994180707 \beta_{8} + 30165410796418 \beta_{7} + \cdots + 28\!\cdots\!17 \beta_{2} ) / 2880 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(177\) | \(351\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
0 | − | 236.819i | 0 | 0 | 0 | 5046.95i | 0 | −36400.2 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 49.2 | 0 | − | 222.159i | 0 | 0 | 0 | 7886.53i | 0 | −29671.8 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 49.3 | 0 | − | 126.650i | 0 | 0 | 0 | − | 9447.80i | 0 | 3642.88 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 49.4 | 0 | − | 61.8022i | 0 | 0 | 0 | − | 6789.55i | 0 | 15863.5 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 49.5 | 0 | − | 49.1879i | 0 | 0 | 0 | − | 1436.67i | 0 | 17263.5 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 49.6 | 0 | 49.1879i | 0 | 0 | 0 | 1436.67i | 0 | 17263.5 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 49.7 | 0 | 61.8022i | 0 | 0 | 0 | 6789.55i | 0 | 15863.5 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 49.8 | 0 | 126.650i | 0 | 0 | 0 | 9447.80i | 0 | 3642.88 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 49.9 | 0 | 222.159i | 0 | 0 | 0 | − | 7886.53i | 0 | −29671.8 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 49.10 | 0 | 236.819i | 0 | 0 | 0 | − | 5046.95i | 0 | −36400.2 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 400.10.c.u | 10 | |
| 4.b | odd | 2 | 1 | 200.10.c.h | 10 | ||
| 5.b | even | 2 | 1 | inner | 400.10.c.u | 10 | |
| 5.c | odd | 4 | 1 | 400.10.a.bd | 5 | ||
| 5.c | odd | 4 | 1 | 400.10.a.be | 5 | ||
| 20.d | odd | 2 | 1 | 200.10.c.h | 10 | ||
| 20.e | even | 4 | 1 | 200.10.a.i | ✓ | 5 | |
| 20.e | even | 4 | 1 | 200.10.a.j | yes | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 200.10.a.i | ✓ | 5 | 20.e | even | 4 | 1 | |
| 200.10.a.j | yes | 5 | 20.e | even | 4 | 1 | |
| 200.10.c.h | 10 | 4.b | odd | 2 | 1 | ||
| 200.10.c.h | 10 | 20.d | odd | 2 | 1 | ||
| 400.10.a.bd | 5 | 5.c | odd | 4 | 1 | ||
| 400.10.a.be | 5 | 5.c | odd | 4 | 1 | ||
| 400.10.c.u | 10 | 1.a | even | 1 | 1 | trivial | |
| 400.10.c.u | 10 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{10} + 127717 T_{3}^{8} + 5226345738 T_{3}^{6} + 73342008257994 T_{3}^{4} + \cdots + 41\!\cdots\!25 \)
acting on \(S_{10}^{\mathrm{new}}(400, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} \)
|
| $3$ |
\( T^{10} + \cdots + 41\!\cdots\!25 \)
|
| $5$ |
\( T^{10} \)
|
| $7$ |
\( T^{10} + \cdots + 13\!\cdots\!64 \)
|
| $11$ |
\( (T^{5} + \cdots + 38\!\cdots\!81)^{2} \)
|
| $13$ |
\( T^{10} + \cdots + 13\!\cdots\!16 \)
|
| $17$ |
\( T^{10} + \cdots + 56\!\cdots\!25 \)
|
| $19$ |
\( (T^{5} + \cdots - 97\!\cdots\!01)^{2} \)
|
| $23$ |
\( T^{10} + \cdots + 26\!\cdots\!04 \)
|
| $29$ |
\( (T^{5} + \cdots - 16\!\cdots\!88)^{2} \)
|
| $31$ |
\( (T^{5} + \cdots - 30\!\cdots\!00)^{2} \)
|
| $37$ |
\( T^{10} + \cdots + 15\!\cdots\!76 \)
|
| $41$ |
\( (T^{5} + \cdots - 26\!\cdots\!13)^{2} \)
|
| $43$ |
\( T^{10} + \cdots + 99\!\cdots\!96 \)
|
| $47$ |
\( T^{10} + \cdots + 43\!\cdots\!76 \)
|
| $53$ |
\( T^{10} + \cdots + 22\!\cdots\!56 \)
|
| $59$ |
\( (T^{5} + \cdots - 38\!\cdots\!92)^{2} \)
|
| $61$ |
\( (T^{5} + \cdots - 70\!\cdots\!00)^{2} \)
|
| $67$ |
\( T^{10} + \cdots + 31\!\cdots\!61 \)
|
| $71$ |
\( (T^{5} + \cdots + 93\!\cdots\!00)^{2} \)
|
| $73$ |
\( T^{10} + \cdots + 18\!\cdots\!29 \)
|
| $79$ |
\( (T^{5} + \cdots - 81\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{10} + \cdots + 18\!\cdots\!09 \)
|
| $89$ |
\( (T^{5} + \cdots - 49\!\cdots\!09)^{2} \)
|
| $97$ |
\( T^{10} + \cdots + 45\!\cdots\!76 \)
|
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