Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9295,2,Mod(1,9295)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9295, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("9295.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9295 = 5 \cdot 11 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9295.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: | \(74.2209486788\) |
| Analytic rank: | \(0\) |
| Dimension: | \(14\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{14} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{14} - x^{13} - 22 x^{12} + 21 x^{11} + 186 x^{10} - 172 x^{9} - 755 x^{8} + 690 x^{7} + 1489 x^{6} + \cdots - 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 715) |
| 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 \(1,\beta_1,\ldots,\beta_{13}\) 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
\( x^{14} - x^{13} - 22 x^{12} + 21 x^{11} + 186 x^{10} - 172 x^{9} - 755 x^{8} + 690 x^{7} + 1489 x^{6} + \cdots - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 7\nu^{2} - \nu + 6 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 101 \nu^{13} - 217 \nu^{12} - 1673 \nu^{11} + 4113 \nu^{10} + 10941 \nu^{9} - 28104 \nu^{8} + \cdots - 9307 ) / 7124 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 229 \nu^{13} - 37 \nu^{12} + 5345 \nu^{11} + 373 \nu^{10} - 48577 \nu^{9} + 2250 \nu^{8} + \cdots + 56193 ) / 7124 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 236 \nu^{13} + 357 \nu^{12} - 4544 \nu^{11} - 8605 \nu^{10} + 30432 \nu^{9} + 76053 \nu^{8} + \cdots + 46795 ) / 7124 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 266 \nu^{13} + 307 \nu^{12} + 5182 \nu^{11} - 5983 \nu^{10} - 37138 \nu^{9} + 43281 \nu^{8} + \cdots + 13561 ) / 7124 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 270 \nu^{13} - 633 \nu^{12} - 5742 \nu^{11} + 11965 \nu^{10} + 46106 \nu^{9} - 85551 \nu^{8} + \cdots - 56991 ) / 7124 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 106 \nu^{13} + 266 \nu^{12} - 2373 \nu^{11} - 5329 \nu^{10} + 20370 \nu^{9} + 39276 \nu^{8} + \cdots + 13879 ) / 1781 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 537 \nu^{13} - 131 \nu^{12} - 12845 \nu^{11} + 743 \nu^{10} + 119325 \nu^{9} + 9314 \nu^{8} + \cdots + 1407 ) / 7124 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 1189 \nu^{13} - 161 \nu^{12} + 25761 \nu^{11} + 3741 \nu^{10} - 213301 \nu^{9} - 26022 \nu^{8} + \cdots + 14237 ) / 7124 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 1887 \nu^{13} + 266 \nu^{12} - 41555 \nu^{11} - 7110 \nu^{10} + 349855 \nu^{9} + 62429 \nu^{8} + \cdots + 33470 ) / 7124 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 7\beta_{2} + \beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{12} + \beta_{10} + \beta_{9} - \beta_{8} - \beta_{6} + \beta_{4} + 9\beta_{3} + 29\beta _1 + 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -\beta_{10} - \beta_{9} + \beta_{7} - 2\beta_{6} + 10\beta_{4} + \beta_{3} + 46\beta_{2} + 13\beta _1 + 87 \)
|
| \(\nu^{7}\) | \(=\) |
\( - \beta_{13} + 11 \beta_{12} + \beta_{11} + 13 \beta_{10} + 12 \beta_{9} - 14 \beta_{8} - \beta_{7} + \cdots + 25 \)
|
| \(\nu^{8}\) | \(=\) |
\( 2 \beta_{13} + \beta_{12} - \beta_{11} - 15 \beta_{10} - 15 \beta_{9} + 2 \beta_{8} + 13 \beta_{7} + \cdots + 539 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 15 \beta_{13} + 90 \beta_{12} + 15 \beta_{11} + 120 \beta_{10} + 105 \beta_{9} - 133 \beta_{8} + \cdots + 233 \)
|
| \(\nu^{10}\) | \(=\) |
\( 33 \beta_{13} + 16 \beta_{12} - 18 \beta_{11} - 152 \beta_{10} - 153 \beta_{9} + 31 \beta_{8} + \cdots + 3457 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 156 \beta_{13} + 661 \beta_{12} + 153 \beta_{11} + 969 \beta_{10} + 814 \beta_{9} - 1093 \beta_{8} + \cdots + 1945 \)
|
| \(\nu^{12}\) | \(=\) |
\( 360 \beta_{13} + 174 \beta_{12} - 210 \beta_{11} - 1305 \beta_{10} - 1333 \beta_{9} + 317 \beta_{8} + \cdots + 22623 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 1388 \beta_{13} + 4619 \beta_{12} + 1328 \beta_{11} + 7317 \beta_{10} + 5940 \beta_{9} - 8386 \beta_{8} + \cdots + 15329 \)
|
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 |
|
−2.59375 | −1.32033 | 4.72756 | 1.00000 | 3.42462 | −1.15599 | −7.07462 | −1.25672 | −2.59375 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.33192 | 1.69928 | 3.43784 | 1.00000 | −3.96258 | −3.78700 | −3.35292 | −0.112446 | −2.33192 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.14985 | −1.08194 | 2.62184 | 1.00000 | 2.32601 | 2.75110 | −1.33685 | −1.82940 | −2.14985 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.63721 | 3.06380 | 0.680457 | 1.00000 | −5.01608 | 4.95529 | 2.16037 | 6.38686 | −1.63721 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.43626 | −3.39685 | 0.0628427 | 1.00000 | 4.87876 | −0.0323629 | 2.78226 | 8.53859 | −1.43626 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.478148 | 1.02618 | −1.77137 | 1.00000 | −0.490668 | −2.06659 | 1.80328 | −1.94695 | −0.478148 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.0139764 | −1.88547 | −1.99980 | 1.00000 | 0.0263522 | 2.48734 | 0.0559029 | 0.555014 | −0.0139764 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.541219 | 0.0767210 | −1.70708 | 1.00000 | 0.0415229 | −3.77469 | −2.00634 | −2.99411 | 0.541219 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.982324 | 2.15654 | −1.03504 | 1.00000 | 2.11842 | 2.38288 | −2.98139 | 1.65065 | 0.982324 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.18246 | 3.37047 | −0.601786 | 1.00000 | 3.98545 | −3.70742 | −3.07651 | 8.36007 | 1.18246 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.77951 | −0.468248 | 1.16664 | 1.00000 | −0.833250 | 3.52367 | −1.48296 | −2.78074 | 1.77951 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1.90209 | −1.81545 | 1.61796 | 1.00000 | −3.45316 | −2.84789 | −0.726677 | 0.295869 | 1.90209 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 2.61508 | −3.21594 | 4.83862 | 1.00000 | −8.40992 | 4.98992 | 7.42320 | 7.34225 | 2.61508 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 2.63843 | 2.79125 | 4.96133 | 1.00000 | 7.36453 | 1.28171 | 7.81327 | 4.79107 | 2.63843 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(11\) | \(1\) |
| \(13\) | \(1\) |
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 | 9295.2.a.bb | 14 | |
| 13.b | even | 2 | 1 | 9295.2.a.ba | 14 | ||
| 13.e | even | 6 | 2 | 715.2.i.e | ✓ | 28 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 715.2.i.e | ✓ | 28 | 13.e | even | 6 | 2 | |
| 9295.2.a.ba | 14 | 13.b | even | 2 | 1 | ||
| 9295.2.a.bb | 14 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9295))\):
|
\( T_{2}^{14} - T_{2}^{13} - 22 T_{2}^{12} + 21 T_{2}^{11} + 186 T_{2}^{10} - 172 T_{2}^{9} - 755 T_{2}^{8} + \cdots - 3 \)
|
|
\( T_{3}^{14} - T_{3}^{13} - 34 T_{3}^{12} + 29 T_{3}^{11} + 439 T_{3}^{10} - 282 T_{3}^{9} - 2731 T_{3}^{8} + \cdots - 208 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{14} - T^{13} + \cdots - 3 \)
|
| $3$ |
\( T^{14} - T^{13} + \cdots - 208 \)
|
| $5$ |
\( (T - 1)^{14} \)
|
| $7$ |
\( T^{14} - 5 T^{13} + \cdots - 21248 \)
|
| $11$ |
\( (T + 1)^{14} \)
|
| $13$ |
\( T^{14} \)
|
| $17$ |
\( T^{14} - 17 T^{13} + \cdots - 104721 \)
|
| $19$ |
\( T^{14} - 3 T^{13} + \cdots - 62642736 \)
|
| $23$ |
\( T^{14} - 4 T^{13} + \cdots - 3481344 \)
|
| $29$ |
\( T^{14} - 21 T^{13} + \cdots + 295488 \)
|
| $31$ |
\( T^{14} + \cdots + 191773899 \)
|
| $37$ |
\( T^{14} + \cdots - 773413376 \)
|
| $41$ |
\( T^{14} + \cdots + 2358750672 \)
|
| $43$ |
\( T^{14} - 23 T^{13} + \cdots - 2402793 \)
|
| $47$ |
\( T^{14} + 18 T^{13} + \cdots + 7006224 \)
|
| $53$ |
\( T^{14} + \cdots + 5548239792 \)
|
| $59$ |
\( T^{14} - 8 T^{13} + \cdots - 16641027 \)
|
| $61$ |
\( T^{14} + \cdots - 157744037888 \)
|
| $67$ |
\( T^{14} + \cdots - 7013396816 \)
|
| $71$ |
\( T^{14} + \cdots + 229158014208 \)
|
| $73$ |
\( T^{14} + \cdots - 1046202937 \)
|
| $79$ |
\( T^{14} + \cdots + 136084131584 \)
|
| $83$ |
\( T^{14} + \cdots + 231228009207 \)
|
| $89$ |
\( T^{14} + \cdots - 444253023 \)
|
| $97$ |
\( T^{14} + \cdots + 732173166848 \)
|
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