Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [115,4,Mod(1,115)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(115, 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("115.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 115 = 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 115.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: | \(6.78521965066\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{8} - 2x^{7} - 49x^{6} + 31x^{5} + 750x^{4} + 249x^{3} - 2892x^{2} - 620x + 2400 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 2x^{7} - 49x^{6} + 31x^{5} + 750x^{4} + 249x^{3} - 2892x^{2} - 620x + 2400 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -21\nu^{7} + 372\nu^{6} - 611\nu^{5} - 10501\nu^{4} + 19940\nu^{3} + 88331\nu^{2} - 77178\nu - 104240 ) / 3680 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -43\nu^{7} + 236\nu^{6} + 1027\nu^{5} - 3803\nu^{4} - 10340\nu^{3} + 373\nu^{2} + 31226\nu + 17520 ) / 3680 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 57\nu^{7} - 484\nu^{6} - 1233\nu^{5} + 15097\nu^{4} + 6860\nu^{3} - 135927\nu^{2} - 25774\nu + 211440 ) / 3680 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 79\nu^{7} - 348\nu^{6} - 2871\nu^{5} + 8399\nu^{4} + 33460\nu^{3} - 36929\nu^{2} - 71618\nu + 30800 ) / 3680 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 233\nu^{7} - 1236\nu^{6} - 6977\nu^{5} + 29593\nu^{4} + 63260\nu^{3} - 146183\nu^{2} - 44766\nu + 109520 ) / 3680 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + 3\beta_{2} + 23\beta _1 + 20 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{7} - 9\beta_{6} + 5\beta_{5} - 2\beta_{4} + 6\beta_{3} + 30\beta_{2} + 87\beta _1 + 268 \)
|
| \(\nu^{5}\) | \(=\) |
\( 14\beta_{7} - 79\beta_{6} + 45\beta_{5} - 36\beta_{4} + 54\beta_{3} + 133\beta_{2} + 652\beta _1 + 956 \)
|
| \(\nu^{6}\) | \(=\) |
\( 128\beta_{7} - 548\beta_{6} + 266\beta_{5} - 133\beta_{4} + 353\beta_{3} + 951\beta_{2} + 3189\beta _1 + 7538 \)
|
| \(\nu^{7}\) | \(=\) |
\( 860 \beta_{7} - 3858 \beta_{6} + 1852 \beta_{5} - 1258 \beta_{4} + 2456 \beta_{3} + 5030 \beta_{2} + \cdots + 36204 \)
|
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 |
|
−5.03341 | −4.21852 | 17.3352 | 5.00000 | 21.2335 | −19.9088 | −46.9881 | −9.20409 | −25.1671 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −3.85207 | 8.45088 | 6.83843 | 5.00000 | −32.5534 | 16.9190 | 4.47444 | 44.4174 | −19.2603 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.769202 | −0.0181662 | −7.40833 | 5.00000 | 0.0139735 | 10.0918 | 11.8521 | −26.9997 | −3.84601 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.0457645 | −10.2193 | −7.99791 | 5.00000 | −0.467681 | −33.8594 | −0.732137 | 77.4338 | 0.228823 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2.16661 | 8.59146 | −3.30580 | 5.00000 | 18.6143 | 9.86011 | −24.4953 | 46.8131 | 10.8331 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 4.05234 | −8.94300 | 8.42146 | 5.00000 | −36.2401 | 32.7645 | 1.70790 | 52.9772 | 20.2617 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 4.57019 | 4.96142 | 12.8867 | 5.00000 | 22.6746 | −18.4259 | 22.3330 | −2.38431 | 22.8510 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 4.81977 | 1.39521 | 15.2302 | 5.00000 | 6.72460 | 13.5587 | 34.8481 | −25.0534 | 24.0989 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( -1 \) |
| \(23\) | \( -1 \) |
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 | 115.4.a.f | ✓ | 8 |
| 3.b | odd | 2 | 1 | 1035.4.a.r | 8 | ||
| 4.b | odd | 2 | 1 | 1840.4.a.v | 8 | ||
| 5.b | even | 2 | 1 | 575.4.a.n | 8 | ||
| 5.c | odd | 4 | 2 | 575.4.b.k | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 115.4.a.f | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 575.4.a.n | 8 | 5.b | even | 2 | 1 | ||
| 575.4.b.k | 16 | 5.c | odd | 4 | 2 | ||
| 1035.4.a.r | 8 | 3.b | odd | 2 | 1 | ||
| 1840.4.a.v | 8 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 6T_{2}^{7} - 35T_{2}^{6} + 249T_{2}^{5} + 170T_{2}^{4} - 2565T_{2}^{3} + 1916T_{2}^{2} + 2802T_{2} - 132 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(115))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 6 T^{7} + \cdots - 132 \)
|
| $3$ |
\( T^{8} - 187 T^{6} + \cdots + 3520 \)
|
| $5$ |
\( (T - 5)^{8} \)
|
| $7$ |
\( T^{8} + \cdots - 9289584128 \)
|
| $11$ |
\( T^{8} + \cdots - 11345758080 \)
|
| $13$ |
\( T^{8} + \cdots - 5251923133176 \)
|
| $17$ |
\( T^{8} + \cdots + 52834929586560 \)
|
| $19$ |
\( T^{8} + \cdots - 54772847726720 \)
|
| $23$ |
\( (T - 23)^{8} \)
|
| $29$ |
\( T^{8} + \cdots + 12\!\cdots\!00 \)
|
| $31$ |
\( T^{8} + \cdots + 74\!\cdots\!00 \)
|
| $37$ |
\( T^{8} + \cdots - 15\!\cdots\!64 \)
|
| $41$ |
\( T^{8} + \cdots - 18\!\cdots\!22 \)
|
| $43$ |
\( T^{8} + \cdots + 10\!\cdots\!00 \)
|
| $47$ |
\( T^{8} + \cdots - 30\!\cdots\!80 \)
|
| $53$ |
\( T^{8} + \cdots - 62\!\cdots\!56 \)
|
| $59$ |
\( T^{8} + \cdots + 10\!\cdots\!00 \)
|
| $61$ |
\( T^{8} + \cdots - 94\!\cdots\!32 \)
|
| $67$ |
\( T^{8} + \cdots - 27\!\cdots\!80 \)
|
| $71$ |
\( T^{8} + \cdots + 30\!\cdots\!20 \)
|
| $73$ |
\( T^{8} + \cdots - 59\!\cdots\!16 \)
|
| $79$ |
\( T^{8} + \cdots + 11\!\cdots\!40 \)
|
| $83$ |
\( T^{8} + \cdots + 84\!\cdots\!48 \)
|
| $89$ |
\( T^{8} + \cdots + 13\!\cdots\!20 \)
|
| $97$ |
\( T^{8} + \cdots + 12\!\cdots\!44 \)
|
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