Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4002,2,Mod(1,4002)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4002, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4002.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4002.a (trivial) |
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: | yes |
| Analytic conductor: | \(31.9561308889\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 24x^{6} - 3x^{5} + 194x^{4} + 39x^{3} - 607x^{2} - 104x + 600 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 24x^{6} - 3x^{5} + 194x^{4} + 39x^{3} - 607x^{2} - 104x + 600 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -11\nu^{7} - 79\nu^{6} + 173\nu^{5} + 1466\nu^{4} - 180\nu^{3} - 8105\nu^{2} - 2752\nu + 12248 ) / 1048 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 9\nu^{7} + 17\nu^{6} - 213\nu^{5} - 342\nu^{4} + 1362\nu^{3} + 1701\nu^{2} - 1726\nu - 732 ) / 524 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -21\nu^{7} + 135\nu^{6} + 235\nu^{5} - 2346\nu^{4} + 228\nu^{3} + 11489\nu^{2} - 3920\nu - 15584 ) / 1048 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 57\nu^{7} - 67\nu^{6} - 1087\nu^{5} + 978\nu^{4} + 6268\nu^{3} - 3637\nu^{2} - 11368\nu + 4272 ) / 1048 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 57\nu^{7} - 67\nu^{6} - 1087\nu^{5} + 978\nu^{4} + 6268\nu^{3} - 3637\nu^{2} - 9272\nu + 4272 ) / 1048 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -20\nu^{7} + 35\nu^{6} + 386\nu^{5} - 681\nu^{4} - 2197\nu^{3} + 3949\nu^{2} + 3690\nu - 6408 ) / 262 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 103\nu^{7} - 213\nu^{6} - 2001\nu^{5} + 3422\nu^{4} + 11308\nu^{3} - 14331\nu^{2} - 17104\nu + 15552 ) / 1048 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - \beta_{4} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} - \beta_{4} + \beta_{3} - 2\beta_{2} - \beta _1 + 13 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{7} + 8\beta_{5} - 5\beta_{4} + \beta_{3} - 2\beta_{2} + \beta _1 + 3 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 11\beta_{7} - 4\beta_{6} + \beta_{5} - 16\beta_{4} + 13\beta_{3} - 26\beta_{2} - 13\beta _1 + 109 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -20\beta_{7} + 77\beta_{5} - 27\beta_{4} + 6\beta_{3} - 32\beta_{2} + 8\beta _1 + 44 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 99\beta_{7} - 72\beta_{6} + 23\beta_{5} - 188\beta_{4} + 147\beta_{3} - 286\beta_{2} - 153\beta _1 + 1017 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( -140\beta_{7} - 8\beta_{6} + 399\beta_{5} - 69\beta_{4} + 9\beta_{3} - 204\beta_{2} + 11\beta _1 + 257 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−1.00000 | 1.00000 | 1.00000 | −4.40973 | −1.00000 | −0.551242 | −1.00000 | 1.00000 | 4.40973 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | 1.00000 | 1.00000 | −2.69359 | −1.00000 | −3.87906 | −1.00000 | 1.00000 | 2.69359 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 1.00000 | 1.00000 | −1.69873 | −1.00000 | 3.41134 | −1.00000 | 1.00000 | 1.69873 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 1.00000 | 1.00000 | −0.881455 | −1.00000 | 0.253486 | −1.00000 | 1.00000 | 0.881455 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 1.00000 | 1.00000 | −0.464023 | −1.00000 | −2.35800 | −1.00000 | 1.00000 | 0.464023 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 1.00000 | 1.00000 | 1.27098 | −1.00000 | 3.55617 | −1.00000 | 1.00000 | −1.27098 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | 1.00000 | 1.00000 | 1.60812 | −1.00000 | −3.37648 | −1.00000 | 1.00000 | −1.60812 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | 1.00000 | 1.00000 | 4.26843 | −1.00000 | −3.05622 | −1.00000 | 1.00000 | −4.26843 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(23\) | \(1\) |
| \(29\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 4002.2.a.bi | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4002.2.a.bi | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4002))\):
|
\( T_{5}^{8} + 3T_{5}^{7} - 22T_{5}^{6} - 66T_{5}^{5} + 67T_{5}^{4} + 231T_{5}^{3} - 18T_{5}^{2} - 204T_{5} - 72 \)
|
|
\( T_{7}^{8} + 6T_{7}^{7} - 15T_{7}^{6} - 144T_{7}^{5} - 80T_{7}^{4} + 844T_{7}^{3} + 1400T_{7}^{2} + 224T_{7} - 160 \)
|
|
\( T_{11}^{8} + 7T_{11}^{7} - 39T_{11}^{6} - 315T_{11}^{5} + 266T_{11}^{4} + 3846T_{11}^{3} + 2376T_{11}^{2} - 8424T_{11} - 3888 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{8} \)
|
| $3$ |
\( (T - 1)^{8} \)
|
| $5$ |
\( T^{8} + 3 T^{7} + \cdots - 72 \)
|
| $7$ |
\( T^{8} + 6 T^{7} + \cdots - 160 \)
|
| $11$ |
\( T^{8} + 7 T^{7} + \cdots - 3888 \)
|
| $13$ |
\( T^{8} + 3 T^{7} + \cdots + 1264 \)
|
| $17$ |
\( T^{8} + 12 T^{7} + \cdots - 89568 \)
|
| $19$ |
\( T^{8} + 4 T^{7} + \cdots + 178784 \)
|
| $23$ |
\( (T + 1)^{8} \)
|
| $29$ |
\( (T - 1)^{8} \)
|
| $31$ |
\( T^{8} + T^{7} + \cdots - 10368 \)
|
| $37$ |
\( T^{8} + 11 T^{7} + \cdots + 18680 \)
|
| $41$ |
\( T^{8} + 17 T^{7} + \cdots + 30024 \)
|
| $43$ |
\( T^{8} + 10 T^{7} + \cdots - 222912 \)
|
| $47$ |
\( T^{8} + 26 T^{7} + \cdots + 182016 \)
|
| $53$ |
\( T^{8} + 2 T^{7} + \cdots + 3939840 \)
|
| $59$ |
\( T^{8} + 25 T^{7} + \cdots - 42129216 \)
|
| $61$ |
\( T^{8} - 3 T^{7} + \cdots - 14816 \)
|
| $67$ |
\( T^{8} - T^{7} + \cdots + 7438336 \)
|
| $71$ |
\( T^{8} + 27 T^{7} + \cdots - 6184320 \)
|
| $73$ |
\( T^{8} - 16 T^{7} + \cdots + 640384 \)
|
| $79$ |
\( T^{8} + 6 T^{7} + \cdots + 152000 \)
|
| $83$ |
\( T^{8} + 44 T^{7} + \cdots + 8418048 \)
|
| $89$ |
\( T^{8} + 52 T^{7} + \cdots - 27648 \)
|
| $97$ |
\( T^{8} + 4 T^{7} + \cdots - 2388480 \)
|
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