Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [74,8,Mod(1,74)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(74, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("74.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 74 = 2 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 74.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: | \(23.1164918858\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 10621x^{4} + 102052x^{3} + 31004503x^{2} - 305547358x - 22608804936 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 10621x^{4} + 102052x^{3} + 31004503x^{2} - 305547358x - 22608804936 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 7747955 \nu^{5} + 503122743 \nu^{4} - 57043408454 \nu^{3} - 3094025378066 \nu^{2} + \cdots + 41\!\cdots\!94 ) / 3870993554217 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 8203492 \nu^{5} + 501227847 \nu^{4} - 57892177126 \nu^{3} - 2502670068046 \nu^{2} + \cdots + 22\!\cdots\!74 ) / 3870993554217 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 5498161 \nu^{5} - 255663367 \nu^{4} + 39239242757 \nu^{3} + 1275575322580 \nu^{2} + \cdots - 14\!\cdots\!05 ) / 1290331184739 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 27256199 \nu^{5} + 1727779758 \nu^{4} - 176835769391 \nu^{3} - 9009486066587 \nu^{2} + \cdots + 95\!\cdots\!81 ) / 3870993554217 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + 11\beta_{3} - 6\beta_{2} - 7\beta _1 + 3538 \)
|
| \(\nu^{3}\) | \(=\) |
\( 230\beta_{5} - 41\beta_{4} - 704\beta_{3} - 151\beta_{2} + 4557\beta _1 - 41815 \)
|
| \(\nu^{4}\) | \(=\) |
\( -6262\beta_{5} + 21304\beta_{4} + 91225\beta_{3} - 29206\beta_{2} - 105230\beta _1 + 16821488 \)
|
| \(\nu^{5}\) | \(=\) |
\( 1700644\beta_{5} - 1285924\beta_{4} - 6714243\beta_{3} - 1111585\beta_{2} + 24476553\beta _1 - 519414756 \)
|
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 |
|
−8.00000 | −59.4910 | 64.0000 | 326.669 | 475.928 | −633.531 | −512.000 | 1352.18 | −2613.35 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −8.00000 | −56.6850 | 64.0000 | 360.557 | 453.480 | 1603.66 | −512.000 | 1026.19 | −2884.45 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −8.00000 | −39.6523 | 64.0000 | −332.745 | 317.219 | −135.814 | −512.000 | −614.693 | 2661.96 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −8.00000 | 31.0193 | 64.0000 | −544.510 | −248.154 | −1685.44 | −512.000 | −1224.80 | 4356.08 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −8.00000 | 62.0201 | 64.0000 | 42.8650 | −496.161 | 819.178 | −512.000 | 1659.49 | −342.920 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −8.00000 | 90.7890 | 64.0000 | 133.163 | −726.312 | −948.062 | −512.000 | 6055.64 | −1065.31 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(37\) | \( +1 \) |
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 | 74.8.a.c | ✓ | 6 |
| 4.b | odd | 2 | 1 | 592.8.a.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 74.8.a.c | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 592.8.a.c | 6 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 28T_{3}^{5} - 10296T_{3}^{4} + 108368T_{3}^{3} + 30949008T_{3}^{2} - 6853572T_{3} - 23355301401 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(74))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 8)^{6} \)
|
| $3$ |
\( T^{6} + \cdots - 23355301401 \)
|
| $5$ |
\( T^{6} + \cdots + 121811045690700 \)
|
| $7$ |
\( T^{6} + \cdots + 18\!\cdots\!00 \)
|
| $11$ |
\( T^{6} + \cdots + 39\!\cdots\!43 \)
|
| $13$ |
\( T^{6} + \cdots - 29\!\cdots\!08 \)
|
| $17$ |
\( T^{6} + \cdots - 82\!\cdots\!24 \)
|
| $19$ |
\( T^{6} + \cdots - 44\!\cdots\!60 \)
|
| $23$ |
\( T^{6} + \cdots + 28\!\cdots\!64 \)
|
| $29$ |
\( T^{6} + \cdots + 10\!\cdots\!00 \)
|
| $31$ |
\( T^{6} + \cdots - 30\!\cdots\!60 \)
|
| $37$ |
\( (T + 50653)^{6} \)
|
| $41$ |
\( T^{6} + \cdots - 17\!\cdots\!07 \)
|
| $43$ |
\( T^{6} + \cdots - 26\!\cdots\!84 \)
|
| $47$ |
\( T^{6} + \cdots - 30\!\cdots\!00 \)
|
| $53$ |
\( T^{6} + \cdots - 14\!\cdots\!68 \)
|
| $59$ |
\( T^{6} + \cdots - 18\!\cdots\!32 \)
|
| $61$ |
\( T^{6} + \cdots + 36\!\cdots\!76 \)
|
| $67$ |
\( T^{6} + \cdots - 23\!\cdots\!08 \)
|
| $71$ |
\( T^{6} + \cdots - 14\!\cdots\!64 \)
|
| $73$ |
\( T^{6} + \cdots + 11\!\cdots\!49 \)
|
| $79$ |
\( T^{6} + \cdots - 22\!\cdots\!56 \)
|
| $83$ |
\( T^{6} + \cdots - 18\!\cdots\!16 \)
|
| $89$ |
\( T^{6} + \cdots + 96\!\cdots\!84 \)
|
| $97$ |
\( T^{6} + \cdots - 15\!\cdots\!64 \)
|
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