Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2175,4,Mod(1,2175)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2175, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("2175.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2175 = 3 \cdot 5^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2175.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: | \(128.329154262\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 3 x^{15} - 92 x^{14} + 239 x^{13} + 3416 x^{12} - 7461 x^{11} - 65355 x^{10} + 115826 x^{9} + \cdots - 891088 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 3 x^{15} - 92 x^{14} + 239 x^{13} + 3416 x^{12} - 7461 x^{11} - 65355 x^{10} + 115826 x^{9} + \cdots - 891088 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 19\nu + 7 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 75\!\cdots\!71 \nu^{15} + \cdots + 10\!\cdots\!16 ) / 38\!\cdots\!60 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 49\!\cdots\!79 \nu^{15} + \cdots - 95\!\cdots\!04 ) / 15\!\cdots\!40 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 49\!\cdots\!79 \nu^{15} + \cdots - 94\!\cdots\!24 ) / 15\!\cdots\!40 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 57\!\cdots\!17 \nu^{15} + \cdots + 10\!\cdots\!32 ) / 15\!\cdots\!40 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 87\!\cdots\!21 \nu^{15} + \cdots + 17\!\cdots\!56 ) / 15\!\cdots\!40 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 24\!\cdots\!40 \nu^{15} + \cdots + 40\!\cdots\!36 ) / 38\!\cdots\!96 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 64\!\cdots\!07 \nu^{15} + \cdots - 11\!\cdots\!12 ) / 77\!\cdots\!20 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 97\!\cdots\!33 \nu^{15} + \cdots - 16\!\cdots\!88 ) / 77\!\cdots\!20 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 11\!\cdots\!91 \nu^{15} + \cdots + 19\!\cdots\!96 ) / 77\!\cdots\!20 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 11\!\cdots\!03 \nu^{15} + \cdots - 20\!\cdots\!08 ) / 77\!\cdots\!20 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 33\!\cdots\!91 \nu^{15} + \cdots + 60\!\cdots\!96 ) / 15\!\cdots\!40 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 55\!\cdots\!81 \nu^{15} + \cdots + 99\!\cdots\!56 ) / 15\!\cdots\!40 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 20\beta _1 + 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} - \beta_{5} + 2\beta_{3} + 27\beta_{2} + 35\beta _1 + 233 \)
|
| \(\nu^{5}\) | \(=\) |
\( - \beta_{15} + 2 \beta_{14} - \beta_{12} - 2 \beta_{11} + 2 \beta_{10} - \beta_{9} + \beta_{8} + \cdots + 231 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 4 \beta_{15} - 2 \beta_{14} - 6 \beta_{13} - 3 \beta_{12} - 7 \beta_{11} + \beta_{10} + \beta_{9} + \cdots + 5221 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 61 \beta_{15} + 94 \beta_{14} - 12 \beta_{13} - 58 \beta_{12} - 127 \beta_{11} + 105 \beta_{10} + \cdots + 8341 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 243 \beta_{15} - 82 \beta_{14} - 320 \beta_{13} - 230 \beta_{12} - 488 \beta_{11} + 114 \beta_{10} + \cdots + 127456 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 2605 \beta_{15} + 3262 \beta_{14} - 764 \beta_{13} - 2455 \beta_{12} - 5535 \beta_{11} + \cdots + 277339 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 10575 \beta_{15} - 2152 \beta_{14} - 12254 \beta_{13} - 11197 \beta_{12} - 23141 \beta_{11} + \cdots + 3296347 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 96349 \beta_{15} + 101188 \beta_{14} - 33842 \beta_{13} - 91445 \beta_{12} - 208426 \beta_{11} + \cdots + 8897540 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 404233 \beta_{15} - 38158 \beta_{14} - 415796 \beta_{13} - 450371 \beta_{12} - 931600 \beta_{11} + \cdots + 88851805 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 3314995 \beta_{15} + 2981458 \beta_{14} - 1297704 \beta_{13} - 3184681 \beta_{12} - 7301092 \beta_{11} + \cdots + 280930931 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 14448930 \beta_{15} - 103100 \beta_{14} - 13361028 \beta_{13} - 16403098 \beta_{12} + \cdots + 2470535187 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 109564961 \beta_{15} + 85725756 \beta_{14} - 46225438 \beta_{13} - 106614312 \beta_{12} + \cdots + 8809490893 \)
|
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 |
|
−5.58145 | −3.00000 | 23.1526 | 0 | 16.7443 | 10.0377 | −84.5733 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.87062 | −3.00000 | 15.7230 | 0 | 14.6119 | −16.1001 | −37.6156 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −4.67314 | −3.00000 | 13.8382 | 0 | 14.0194 | −23.4063 | −27.2828 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −3.58918 | −3.00000 | 4.88219 | 0 | 10.7675 | 30.7036 | 11.1904 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −3.08631 | −3.00000 | 1.52531 | 0 | 9.25893 | 25.9780 | 19.9829 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −2.66029 | −3.00000 | −0.922855 | 0 | 7.98087 | −18.8651 | 23.7374 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.515365 | −3.00000 | −7.73440 | 0 | 1.54609 | 15.3973 | 8.10895 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −0.501041 | −3.00000 | −7.74896 | 0 | 1.50312 | −17.4547 | 7.89087 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −0.450617 | −3.00000 | −7.79694 | 0 | 1.35185 | −0.788922 | 7.11838 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.18577 | −3.00000 | −6.59395 | 0 | −3.55731 | −7.22246 | −17.3051 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.86231 | −3.00000 | −4.53181 | 0 | −5.58692 | 13.3897 | −23.3381 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.99493 | −3.00000 | 0.969579 | 0 | −8.98478 | 4.43075 | −21.0556 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 3.73424 | −3.00000 | 5.94451 | 0 | −11.2027 | −11.3963 | −7.67567 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 3.83202 | −3.00000 | 6.68436 | 0 | −11.4961 | −36.2405 | −5.04155 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 4.35146 | −3.00000 | 10.9352 | 0 | −13.0544 | 29.5878 | 12.7724 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 4.96729 | −3.00000 | 16.6740 | 0 | −14.9019 | 13.9493 | 43.0864 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( +1 \) |
| \(29\) | \( +1 \) |
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 | 2175.4.a.u | ✓ | 16 |
| 5.b | even | 2 | 1 | 2175.4.a.v | yes | 16 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2175.4.a.u | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 2175.4.a.v | yes | 16 | 5.b | even | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2175))\):
|
\( T_{2}^{16} + 3 T_{2}^{15} - 92 T_{2}^{14} - 239 T_{2}^{13} + 3416 T_{2}^{12} + 7461 T_{2}^{11} + \cdots - 891088 \)
|
|
\( T_{7}^{16} - 12 T_{7}^{15} - 3022 T_{7}^{14} + 35587 T_{7}^{13} + 3434893 T_{7}^{12} + \cdots + 88\!\cdots\!64 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} + 3 T^{15} + \cdots - 891088 \)
|
| $3$ |
\( (T + 3)^{16} \)
|
| $5$ |
\( T^{16} \)
|
| $7$ |
\( T^{16} + \cdots + 88\!\cdots\!64 \)
|
| $11$ |
\( T^{16} + \cdots + 59\!\cdots\!88 \)
|
| $13$ |
\( T^{16} + \cdots + 10\!\cdots\!76 \)
|
| $17$ |
\( T^{16} + \cdots + 19\!\cdots\!16 \)
|
| $19$ |
\( T^{16} + \cdots - 26\!\cdots\!60 \)
|
| $23$ |
\( T^{16} + \cdots + 19\!\cdots\!76 \)
|
| $29$ |
\( (T + 29)^{16} \)
|
| $31$ |
\( T^{16} + \cdots - 10\!\cdots\!00 \)
|
| $37$ |
\( T^{16} + \cdots + 10\!\cdots\!28 \)
|
| $41$ |
\( T^{16} + \cdots + 71\!\cdots\!84 \)
|
| $43$ |
\( T^{16} + \cdots - 27\!\cdots\!16 \)
|
| $47$ |
\( T^{16} + \cdots + 30\!\cdots\!52 \)
|
| $53$ |
\( T^{16} + \cdots + 28\!\cdots\!00 \)
|
| $59$ |
\( T^{16} + \cdots + 82\!\cdots\!40 \)
|
| $61$ |
\( T^{16} + \cdots - 16\!\cdots\!00 \)
|
| $67$ |
\( T^{16} + \cdots - 71\!\cdots\!16 \)
|
| $71$ |
\( T^{16} + \cdots + 47\!\cdots\!92 \)
|
| $73$ |
\( T^{16} + \cdots + 18\!\cdots\!12 \)
|
| $79$ |
\( T^{16} + \cdots - 59\!\cdots\!40 \)
|
| $83$ |
\( T^{16} + \cdots - 63\!\cdots\!96 \)
|
| $89$ |
\( T^{16} + \cdots + 51\!\cdots\!00 \)
|
| $97$ |
\( T^{16} + \cdots + 46\!\cdots\!04 \)
|
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