Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [185,4,Mod(1,185)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(185, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("185.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 185 = 5 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 185.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: | \(10.9153533511\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - x^{6} - 36x^{5} + 36x^{4} + 337x^{3} - 249x^{2} - 930x + 450 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - x^{6} - 36x^{5} + 36x^{4} + 337x^{3} - 249x^{2} - 930x + 450 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} - 6\nu^{5} - 26\nu^{4} + 186\nu^{3} + 27\nu^{2} - 964\nu + 430 ) / 20 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - 4\nu^{5} - 27\nu^{4} + 120\nu^{3} + 64\nu^{2} - 546\nu + 294 ) / 6 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 7\nu^{6} - 22\nu^{5} - 192\nu^{4} + 672\nu^{3} + 559\nu^{2} - 3078\nu + 1710 ) / 30 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 19\nu^{6} - 64\nu^{5} - 534\nu^{4} + 1944\nu^{3} + 1873\nu^{2} - 8976\nu + 3030 ) / 60 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 19\nu^{6} - 64\nu^{5} - 534\nu^{4} + 1944\nu^{3} + 1813\nu^{2} - 8976\nu + 3630 ) / 60 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{6} + \beta_{5} + 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - 2\beta_{3} + 2\beta_{2} + 17\beta _1 - 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -29\beta_{6} + 25\beta_{5} + 4\beta_{4} + 2\beta_{3} - 6\beta _1 + 166 \)
|
| \(\nu^{5}\) | \(=\) |
\( 4\beta_{6} - 6\beta_{5} + 35\beta_{4} - 62\beta_{3} + 56\beta_{2} + 349\beta _1 - 100 \)
|
| \(\nu^{6}\) | \(=\) |
\( -703\beta_{6} + 587\beta_{5} + 128\beta_{4} + 52\beta_{3} - 16\beta_{2} - 260\beta _1 + 3388 \)
|
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 |
|
−4.90845 | −7.91596 | 16.0928 | −5.00000 | 38.8551 | −28.1970 | −39.7233 | 35.6625 | 24.5422 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.39497 | 5.24897 | −2.26414 | −5.00000 | −12.5711 | 3.57603 | 24.5823 | 0.551653 | 11.9748 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.36165 | −8.59174 | −2.42262 | −5.00000 | 20.2907 | 12.7008 | 24.6146 | 46.8179 | 11.8082 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.463388 | 8.32583 | −7.78527 | −5.00000 | 3.85808 | −35.3512 | −7.31470 | 42.3194 | −2.31694 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2.33681 | 2.33269 | −2.53932 | −5.00000 | 5.45105 | −4.77798 | −24.6284 | −21.5586 | −11.6840 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 3.22870 | −1.72141 | 2.42450 | −5.00000 | −5.55791 | −0.268133 | −18.0016 | −24.0367 | −16.1435 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 4.63616 | −5.67837 | 13.4940 | −5.00000 | −26.3258 | −24.6825 | 25.4711 | 5.24388 | −23.1808 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(37\) | \(-1\) |
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 | 185.4.a.b | ✓ | 7 |
| 3.b | odd | 2 | 1 | 1665.4.a.f | 7 | ||
| 5.b | even | 2 | 1 | 925.4.a.c | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 185.4.a.b | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 925.4.a.c | 7 | 5.b | even | 2 | 1 | ||
| 1665.4.a.f | 7 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} - T_{2}^{6} - 36T_{2}^{5} + 36T_{2}^{4} + 337T_{2}^{3} - 249T_{2}^{2} - 930T_{2} + 450 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(185))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - T^{6} + \cdots + 450 \)
|
| $3$ |
\( T^{7} + 8 T^{6} + \cdots - 67772 \)
|
| $5$ |
\( (T + 5)^{7} \)
|
| $7$ |
\( T^{7} + 77 T^{6} + \cdots + 1431594 \)
|
| $11$ |
\( T^{7} + \cdots + 2472209760 \)
|
| $13$ |
\( T^{7} + \cdots + 21347954944 \)
|
| $17$ |
\( T^{7} + \cdots + 1373441629056 \)
|
| $19$ |
\( T^{7} + \cdots + 214778476864 \)
|
| $23$ |
\( T^{7} + \cdots + 110025335808 \)
|
| $29$ |
\( T^{7} + \cdots + 16\!\cdots\!24 \)
|
| $31$ |
\( T^{7} + \cdots + 6896612447592 \)
|
| $37$ |
\( (T - 37)^{7} \)
|
| $41$ |
\( T^{7} + \cdots - 13\!\cdots\!20 \)
|
| $43$ |
\( T^{7} + \cdots + 10\!\cdots\!88 \)
|
| $47$ |
\( T^{7} + \cdots - 92\!\cdots\!52 \)
|
| $53$ |
\( T^{7} + \cdots + 27602580461328 \)
|
| $59$ |
\( T^{7} + \cdots + 10\!\cdots\!56 \)
|
| $61$ |
\( T^{7} + \cdots + 10\!\cdots\!80 \)
|
| $67$ |
\( T^{7} + \cdots + 39\!\cdots\!72 \)
|
| $71$ |
\( T^{7} + \cdots + 26\!\cdots\!00 \)
|
| $73$ |
\( T^{7} + \cdots + 53\!\cdots\!80 \)
|
| $79$ |
\( T^{7} + \cdots - 85\!\cdots\!80 \)
|
| $83$ |
\( T^{7} + \cdots - 82\!\cdots\!24 \)
|
| $89$ |
\( T^{7} + \cdots - 10\!\cdots\!08 \)
|
| $97$ |
\( T^{7} + \cdots + 16\!\cdots\!20 \)
|
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