Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4,43,Mod(3,4)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 43, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4.3");
S:= CuspForms(chi, 43);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4 = 2^{2} \) |
| Weight: | \( k \) | \(=\) | \( 43 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4.b (of order \(2\), degree \(1\), 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: | \(44.6910828688\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 10 x^{19} + \cdots + 58\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | multiple of \( 2^{380}\cdot 3^{41}\cdot 5^{10}\cdot 7^{6} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} - 10 x^{19} + \cdots + 58\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 35\!\cdots\!75 \nu^{19} + \cdots + 44\!\cdots\!00 ) / 25\!\cdots\!00 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 35\!\cdots\!75 \nu^{19} + \cdots - 44\!\cdots\!00 ) / 32\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 31\!\cdots\!19 \nu^{19} + \cdots - 34\!\cdots\!00 ) / 25\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 71\!\cdots\!47 \nu^{19} + \cdots + 25\!\cdots\!00 ) / 35\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 53\!\cdots\!07 \nu^{19} + \cdots + 52\!\cdots\!00 ) / 62\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 40\!\cdots\!43 \nu^{19} + \cdots - 27\!\cdots\!00 ) / 25\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 30\!\cdots\!07 \nu^{19} + \cdots + 44\!\cdots\!00 ) / 15\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 27\!\cdots\!15 \nu^{19} + \cdots - 68\!\cdots\!00 ) / 25\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 60\!\cdots\!09 \nu^{19} + \cdots + 28\!\cdots\!00 ) / 17\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 17\!\cdots\!19 \nu^{19} + \cdots + 80\!\cdots\!00 ) / 27\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 43\!\cdots\!16 \nu^{19} + \cdots - 56\!\cdots\!00 ) / 15\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 99\!\cdots\!17 \nu^{19} + \cdots + 38\!\cdots\!00 ) / 26\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 33\!\cdots\!99 \nu^{19} + \cdots - 31\!\cdots\!00 ) / 25\!\cdots\!00 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 20\!\cdots\!87 \nu^{19} + \cdots + 59\!\cdots\!00 ) / 12\!\cdots\!00 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 29\!\cdots\!45 \nu^{19} + \cdots + 40\!\cdots\!00 ) / 25\!\cdots\!00 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 31\!\cdots\!17 \nu^{19} + \cdots - 19\!\cdots\!00 ) / 83\!\cdots\!00 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 20\!\cdots\!55 \nu^{19} + \cdots + 32\!\cdots\!00 ) / 25\!\cdots\!00 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 49\!\cdots\!05 \nu^{19} + \cdots - 16\!\cdots\!00 ) / 20\!\cdots\!00 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 10\!\cdots\!47 \nu^{19} + \cdots + 19\!\cdots\!00 ) / 27\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 78\beta _1 - 8 ) / 16 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{9} + 3 \beta_{7} + 15 \beta_{6} + 640 \beta_{5} - 13058 \beta_{4} + 2493609 \beta_{3} + \cdots - 14\!\cdots\!15 ) / 256 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 238808 \beta_{19} - 128096 \beta_{18} - 118986 \beta_{17} + 517042 \beta_{16} + \cdots - 18\!\cdots\!68 ) / 4096 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 633935314689637 \beta_{19} - 153032523064724 \beta_{18} - 386151583395408 \beta_{17} + \cdots + 19\!\cdots\!17 ) / 32768 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 52\!\cdots\!80 \beta_{19} + \cdots + 70\!\cdots\!66 ) / 524288 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 41\!\cdots\!05 \beta_{19} + \cdots - 78\!\cdots\!63 ) / 1048576 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 25\!\cdots\!24 \beta_{19} + \cdots - 46\!\cdots\!30 ) / 16777216 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 21\!\cdots\!15 \beta_{19} + \cdots + 34\!\cdots\!75 ) / 33554432 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 12\!\cdots\!60 \beta_{19} + \cdots + 27\!\cdots\!74 ) / 536870912 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 10\!\cdots\!60 \beta_{19} + \cdots - 15\!\cdots\!68 ) / 1073741824 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 18\!\cdots\!68 \beta_{19} + \cdots - 47\!\cdots\!74 ) / 536870912 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 16\!\cdots\!68 \beta_{19} + \cdots + 23\!\cdots\!64 ) / 1073741824 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 13\!\cdots\!68 \beta_{19} + \cdots + 39\!\cdots\!02 ) / 268435456 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 12\!\cdots\!00 \beta_{19} + \cdots - 17\!\cdots\!00 ) / 536870912 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 20\!\cdots\!04 \beta_{19} + \cdots - 66\!\cdots\!42 ) / 268435456 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( 18\!\cdots\!24 \beta_{19} + \cdots + 26\!\cdots\!24 ) / 536870912 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( 61\!\cdots\!28 \beta_{19} + \cdots + 21\!\cdots\!14 ) / 536870912 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( - 55\!\cdots\!84 \beta_{19} + \cdots - 79\!\cdots\!32 ) / 1073741824 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( - 93\!\cdots\!08 \beta_{19} + \cdots - 35\!\cdots\!78 ) / 536870912 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3.1 |
|
−2.05493e6 | − | 418685.i | 1.44540e10i | 4.04745e12 | + | 1.72074e12i | 8.07741e14 | 6.05166e15 | − | 2.97020e16i | 7.29328e17i | −7.59680e18 | − | 5.23061e18i | −9.94985e19 | −1.65985e21 | − | 3.38189e20i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.2 | −2.05493e6 | + | 418685.i | − | 1.44540e10i | 4.04745e12 | − | 1.72074e12i | 8.07741e14 | 6.05166e15 | + | 2.97020e16i | − | 7.29328e17i | −7.59680e18 | + | 5.23061e18i | −9.94985e19 | −1.65985e21 | + | 3.38189e20i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.3 | −2.05360e6 | − | 425178.i | 8.66457e9i | 4.03649e12 | + | 1.74629e12i | −5.87785e14 | 3.68399e15 | − | 1.77936e16i | − | 1.08401e18i | −7.54686e18 | − | 5.30241e18i | 3.43442e19 | 1.20707e21 | + | 2.49913e20i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.4 | −2.05360e6 | + | 425178.i | − | 8.66457e9i | 4.03649e12 | − | 1.74629e12i | −5.87785e14 | 3.68399e15 | + | 1.77936e16i | 1.08401e18i | −7.54686e18 | + | 5.30241e18i | 3.43442e19 | 1.20707e21 | − | 2.49913e20i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.5 | −1.51618e6 | − | 1.44888e6i | − | 7.92587e9i | 1.99543e11 | + | 4.39352e12i | 1.59403e13 | −1.14836e16 | + | 1.20170e16i | 6.57146e16i | 6.06313e18 | − | 6.95047e18i | 4.65995e19 | −2.41683e19 | − | 2.30956e19i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.6 | −1.51618e6 | + | 1.44888e6i | 7.92587e9i | 1.99543e11 | − | 4.39352e12i | 1.59403e13 | −1.14836e16 | − | 1.20170e16i | − | 6.57146e16i | 6.06313e18 | + | 6.95047e18i | 4.65995e19 | −2.41683e19 | + | 2.30956e19i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.7 | −766250. | − | 1.95215e6i | 1.39436e10i | −3.22377e12 | + | 2.99168e12i | −2.28561e14 | 2.72200e16 | − | 1.06843e16i | 3.10753e17i | 8.31043e18 | + | 4.00092e18i | −8.50048e19 | 1.75135e20 | + | 4.46186e20i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.8 | −766250. | + | 1.95215e6i | − | 1.39436e10i | −3.22377e12 | − | 2.99168e12i | −2.28561e14 | 2.72200e16 | + | 1.06843e16i | − | 3.10753e17i | 8.31043e18 | − | 4.00092e18i | −8.50048e19 | 1.75135e20 | − | 4.46186e20i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.9 | 85368.2 | − | 2.09541e6i | 4.36838e8i | −4.38347e12 | − | 3.57763e11i | 5.12233e14 | 9.15355e14 | + | 3.72920e13i | − | 3.44914e17i | −1.12387e18 | + | 9.15464e18i | 1.09228e20 | 4.37284e19 | − | 1.07334e21i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.10 | 85368.2 | + | 2.09541e6i | − | 4.36838e8i | −4.38347e12 | + | 3.57763e11i | 5.12233e14 | 9.15355e14 | − | 3.72920e13i | 3.44914e17i | −1.12387e18 | − | 9.15464e18i | 1.09228e20 | 4.37284e19 | + | 1.07334e21i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.11 | 232318. | − | 2.08424e6i | − | 2.00495e10i | −4.29010e12 | − | 9.68414e11i | −5.01114e14 | −4.17880e16 | − | 4.65785e15i | − | 9.07936e16i | −3.01508e18 | + | 8.71664e18i | −2.92563e20 | −1.16418e20 | + | 1.04444e21i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.12 | 232318. | + | 2.08424e6i | 2.00495e10i | −4.29010e12 | + | 9.68414e11i | −5.01114e14 | −4.17880e16 | + | 4.65785e15i | 9.07936e16i | −3.01508e18 | − | 8.71664e18i | −2.92563e20 | −1.16418e20 | − | 1.04444e21i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.13 | 1.12655e6 | − | 1.76888e6i | 5.40870e9i | −1.85982e12 | − | 3.98546e12i | −7.68266e14 | 9.56734e15 | + | 6.09318e15i | 1.45472e17i | −9.14497e18 | − | 1.20003e18i | 8.01649e19 | −8.65491e20 | + | 1.35897e21i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.14 | 1.12655e6 | + | 1.76888e6i | − | 5.40870e9i | −1.85982e12 | + | 3.98546e12i | −7.68266e14 | 9.56734e15 | − | 6.09318e15i | − | 1.45472e17i | −9.14497e18 | + | 1.20003e18i | 8.01649e19 | −8.65491e20 | − | 1.35897e21i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.15 | 1.57912e6 | − | 1.38001e6i | − | 7.51798e9i | 5.89205e11 | − | 4.35840e12i | 5.33275e14 | −1.03749e16 | − | 1.18718e16i | 7.46884e17i | −5.08420e18 | − | 7.69555e18i | 5.28989e19 | 8.42107e20 | − | 7.35924e20i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.16 | 1.57912e6 | + | 1.38001e6i | 7.51798e9i | 5.89205e11 | + | 4.35840e12i | 5.33275e14 | −1.03749e16 | + | 1.18718e16i | − | 7.46884e17i | −5.08420e18 | + | 7.69555e18i | 5.28989e19 | 8.42107e20 | + | 7.35924e20i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.17 | 1.70189e6 | − | 1.22541e6i | 1.86642e10i | 1.39480e12 | − | 4.17101e12i | 4.71964e14 | 2.28712e16 | + | 3.17644e16i | − | 6.63278e17i | −2.73739e18 | − | 8.80780e18i | −2.38933e20 | 8.03231e20 | − | 5.78348e20i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.18 | 1.70189e6 | + | 1.22541e6i | − | 1.86642e10i | 1.39480e12 | + | 4.17101e12i | 4.71964e14 | 2.28712e16 | − | 3.17644e16i | 6.63278e17i | −2.73739e18 | + | 8.80780e18i | −2.38933e20 | 8.03231e20 | + | 5.78348e20i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.19 | 2.06680e6 | − | 355528.i | − | 9.12363e9i | 4.14525e12 | − | 1.46961e12i | −1.85810e14 | −3.24370e15 | − | 1.88567e16i | − | 7.39963e17i | 8.04489e18 | − | 4.51113e18i | 2.61784e19 | −3.84032e20 | + | 6.60607e19i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.20 | 2.06680e6 | + | 355528.i | 9.12363e9i | 4.14525e12 | + | 1.46961e12i | −1.85810e14 | −3.24370e15 | + | 1.88567e16i | 7.39963e17i | 8.04489e18 | + | 4.51113e18i | 2.61784e19 | −3.84032e20 | − | 6.60607e19i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 4.43.b.a | ✓ | 20 |
| 4.b | odd | 2 | 1 | inner | 4.43.b.a | ✓ | 20 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4.43.b.a | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 4.43.b.a | ✓ | 20 | 4.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{43}^{\mathrm{new}}(4, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} + \cdots + 27\!\cdots\!76 \)
|
| $3$ |
\( T^{20} + \cdots + 70\!\cdots\!00 \)
|
| $5$ |
\( (T^{10} + \cdots - 15\!\cdots\!00)^{2} \)
|
| $7$ |
\( T^{20} + \cdots + 72\!\cdots\!00 \)
|
| $11$ |
\( T^{20} + \cdots + 33\!\cdots\!00 \)
|
| $13$ |
\( (T^{10} + \cdots - 18\!\cdots\!00)^{2} \)
|
| $17$ |
\( (T^{10} + \cdots + 11\!\cdots\!00)^{2} \)
|
| $19$ |
\( T^{20} + \cdots + 14\!\cdots\!00 \)
|
| $23$ |
\( T^{20} + \cdots + 58\!\cdots\!00 \)
|
| $29$ |
\( (T^{10} + \cdots + 19\!\cdots\!96)^{2} \)
|
| $31$ |
\( T^{20} + \cdots + 65\!\cdots\!00 \)
|
| $37$ |
\( (T^{10} + \cdots - 89\!\cdots\!00)^{2} \)
|
| $41$ |
\( (T^{10} + \cdots - 13\!\cdots\!76)^{2} \)
|
| $43$ |
\( T^{20} + \cdots + 33\!\cdots\!00 \)
|
| $47$ |
\( T^{20} + \cdots + 32\!\cdots\!00 \)
|
| $53$ |
\( (T^{10} + \cdots + 51\!\cdots\!00)^{2} \)
|
| $59$ |
\( T^{20} + \cdots + 97\!\cdots\!00 \)
|
| $61$ |
\( (T^{10} + \cdots + 33\!\cdots\!04)^{2} \)
|
| $67$ |
\( T^{20} + \cdots + 10\!\cdots\!00 \)
|
| $71$ |
\( T^{20} + \cdots + 26\!\cdots\!00 \)
|
| $73$ |
\( (T^{10} + \cdots - 19\!\cdots\!00)^{2} \)
|
| $79$ |
\( T^{20} + \cdots + 15\!\cdots\!00 \)
|
| $83$ |
\( T^{20} + \cdots + 49\!\cdots\!00 \)
|
| $89$ |
\( (T^{10} + \cdots + 94\!\cdots\!56)^{2} \)
|
| $97$ |
\( (T^{10} + \cdots + 12\!\cdots\!00)^{2} \)
|
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