Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3283,2,Mod(1,3283)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3283, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("3283.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3283 = 7^{2} \cdot 67 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3283.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: | \(26.2148869836\) |
| 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: |
\( x^{10} - 4x^{9} - 12x^{8} + 40x^{7} + 54x^{6} - 104x^{5} - 68x^{4} + 88x^{3} + 13x^{2} - 14x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3 \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 4x^{9} - 12x^{8} + 40x^{7} + 54x^{6} - 104x^{5} - 68x^{4} + 88x^{3} + 13x^{2} - 14x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 1185 \nu^{9} + 1577 \nu^{8} - 37772 \nu^{7} - 30342 \nu^{6} + 280081 \nu^{5} + 218156 \nu^{4} + \cdots - 67523 ) / 43883 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 4588 \nu^{9} + 23594 \nu^{8} + 36554 \nu^{7} - 259029 \nu^{6} - 49685 \nu^{5} + 849613 \nu^{4} + \cdots + 170406 ) / 43883 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 5059 \nu^{9} - 22745 \nu^{8} - 47901 \nu^{7} + 212307 \nu^{6} + 188338 \nu^{5} - 522824 \nu^{4} + \cdots - 140363 ) / 43883 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 7330 \nu^{9} - 21167 \nu^{8} - 122178 \nu^{7} + 203929 \nu^{6} + 722990 \nu^{5} - 370702 \nu^{4} + \cdots + 62632 ) / 43883 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 12993 \nu^{9} + 49811 \nu^{8} + 160965 \nu^{7} - 480982 \nu^{6} - 741610 \nu^{5} + 1127766 \nu^{4} + \cdots + 58118 ) / 43883 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 19001 \nu^{9} - 59665 \nu^{8} - 281295 \nu^{7} + 522565 \nu^{6} + 1516682 \nu^{5} - 730907 \nu^{4} + \cdots + 105877 ) / 43883 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 23294 \nu^{9} - 91058 \nu^{8} - 284375 \nu^{7} + 891356 \nu^{6} + 1302409 \nu^{5} - 2162721 \nu^{4} + \cdots - 159186 ) / 43883 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 24522 \nu^{9} + 89683 \nu^{8} + 320481 \nu^{7} - 856469 \nu^{6} - 1546141 \nu^{5} + 1858363 \nu^{4} + \cdots + 61478 ) / 43883 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 4455 \nu^{9} - 16846 \nu^{8} - 57015 \nu^{7} + 166368 \nu^{6} + 270290 \nu^{5} - 407695 \nu^{4} + \cdots - 22058 ) / 6269 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{9} - 3\beta_{8} - 3\beta_{7} - \beta_{6} + 2\beta_{5} + 2\beta_{3} + 3 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{9} - 9\beta_{8} - 3\beta_{7} - 5\beta_{6} + 4\beta_{5} + 4\beta_{3} + 6\beta _1 + 27 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 5 \beta_{9} - 48 \beta_{8} - 27 \beta_{7} - 26 \beta_{6} + 19 \beta_{5} - 3 \beta_{4} + 19 \beta_{3} + \cdots + 66 ) / 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( \beta_{9} - 61 \beta_{8} - 21 \beta_{7} - 35 \beta_{6} + 30 \beta_{5} - 6 \beta_{4} + 20 \beta_{3} + \cdots + 111 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 68 \beta_{9} - 807 \beta_{8} - 324 \beta_{7} - 473 \beta_{6} + 385 \beta_{5} - 105 \beta_{4} + \cdots + 1203 ) / 6 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 113 \beta_{9} - 1665 \beta_{8} - 549 \beta_{7} - 989 \beta_{6} + 877 \beta_{5} - 258 \beta_{4} + \cdots + 2634 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 1285 \beta_{9} - 14259 \beta_{8} - 4821 \beta_{7} - 8569 \beta_{6} + 7532 \beta_{5} - 2472 \beta_{4} + \cdots + 21261 ) / 6 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 1781 \beta_{9} - 20055 \beta_{8} - 6283 \beta_{7} - 12121 \beta_{6} + 11008 \beta_{5} - 3734 \beta_{4} + \cdots + 30177 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 24811 \beta_{9} - 256452 \beta_{8} - 79911 \beta_{7} - 155854 \beta_{6} + 141803 \beta_{5} + \cdots + 378354 ) / 6 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.33920 | −2.73347 | 3.47186 | −0.826056 | 6.39413 | 0 | −3.44297 | 4.47186 | 1.93231 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.33920 | 2.73347 | 3.47186 | 0.826056 | −6.39413 | 0 | −3.44297 | 4.47186 | −1.93231 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.18852 | −1.84732 | −0.587409 | −3.87396 | 2.19559 | 0 | 3.07520 | 0.412591 | 4.60430 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.18852 | 1.84732 | −0.587409 | 3.87396 | −2.19559 | 0 | 3.07520 | 0.412591 | −4.60430 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.314634 | −1.44879 | −1.90101 | 1.91526 | 0.455840 | 0 | 1.22739 | −0.901005 | −0.602607 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.314634 | 1.44879 | −1.90101 | −1.91526 | −0.455840 | 0 | 1.22739 | −0.901005 | 0.602607 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.03886 | −2.48132 | 2.15696 | −2.05518 | −5.05908 | 0 | 0.320023 | 3.15696 | −4.19023 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.03886 | 2.48132 | 2.15696 | 2.05518 | 5.05908 | 0 | 0.320023 | 3.15696 | 4.19023 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.80350 | −3.14000 | 5.85959 | 1.19083 | −8.80298 | 0 | 10.8204 | 6.85959 | 3.33848 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.80350 | 3.14000 | 5.85959 | −1.19083 | 8.80298 | 0 | 10.8204 | 6.85959 | −3.33848 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \( -1 \) |
| \(67\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3283.2.a.v | ✓ | 10 |
| 7.b | odd | 2 | 1 | inner | 3283.2.a.v | ✓ | 10 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3283.2.a.v | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 3283.2.a.v | ✓ | 10 | 7.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3283))\):
|
\( T_{2}^{5} - T_{2}^{4} - 9T_{2}^{3} + 4T_{2}^{2} + 18T_{2} + 5 \)
|
|
\( T_{3}^{10} - 29T_{3}^{8} + 317T_{3}^{6} - 1616T_{3}^{4} + 3792T_{3}^{2} - 3249 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{5} - T^{4} - 9 T^{3} + \cdots + 5)^{2} \)
|
| $3$ |
\( T^{10} - 29 T^{8} + \cdots - 3249 \)
|
| $5$ |
\( T^{10} - 25 T^{8} + \cdots - 225 \)
|
| $7$ |
\( T^{10} \)
|
| $11$ |
\( (T^{5} - 6 T^{4} - 5 T^{3} + \cdots - 20)^{2} \)
|
| $13$ |
\( T^{10} - 105 T^{8} + \cdots - 31329 \)
|
| $17$ |
\( T^{10} - 53 T^{8} + \cdots - 11664 \)
|
| $19$ |
\( T^{10} - 55 T^{8} + \cdots - 576 \)
|
| $23$ |
\( (T^{5} - 2 T^{4} + \cdots - 280)^{2} \)
|
| $29$ |
\( (T^{5} - 10 T^{4} + \cdots - 409)^{2} \)
|
| $31$ |
\( T^{10} - 121 T^{8} + \cdots - 3426201 \)
|
| $37$ |
\( (T^{5} - 5 T^{4} + \cdots + 909)^{2} \)
|
| $41$ |
\( T^{10} - 253 T^{8} + \cdots - 71216721 \)
|
| $43$ |
\( (T^{5} - T^{4} - 112 T^{3} + \cdots - 36)^{2} \)
|
| $47$ |
\( T^{10} - 96 T^{8} + \cdots - 197136 \)
|
| $53$ |
\( (T^{5} - 24 T^{4} + \cdots + 1304)^{2} \)
|
| $59$ |
\( T^{10} + \cdots - 432473616 \)
|
| $61$ |
\( T^{10} - 182 T^{8} + \cdots - 2471184 \)
|
| $67$ |
\( (T + 1)^{10} \)
|
| $71$ |
\( (T^{5} + 9 T^{4} + \cdots + 1861)^{2} \)
|
| $73$ |
\( T^{10} - 163 T^{8} + \cdots - 144 \)
|
| $79$ |
\( (T^{5} - 32 T^{4} + \cdots - 3008)^{2} \)
|
| $83$ |
\( T^{10} + \cdots - 807469056 \)
|
| $89$ |
\( T^{10} + \cdots - 142277184 \)
|
| $97$ |
\( T^{10} + \cdots - 179024400 \)
|
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