Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2057,4,Mod(1,2057)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2057, 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("2057.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2057 = 11^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2057.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: | \(121.366928882\) |
| Analytic rank: | \(1\) |
| Dimension: | \(19\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{19} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{19} - 3 x^{18} - 115 x^{17} + 311 x^{16} + 5535 x^{15} - 13091 x^{14} - 145551 x^{13} + \cdots - 201424256 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(1,\beta_1,\ldots,\beta_{18}\) 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
\( x^{19} - 3 x^{18} - 115 x^{17} + 311 x^{16} + 5535 x^{15} - 13091 x^{14} - 145551 x^{13} + \cdots - 201424256 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 13 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 81\!\cdots\!95 \nu^{18} + \cdots + 77\!\cdots\!48 ) / 44\!\cdots\!36 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 20\!\cdots\!59 \nu^{18} + \cdots + 12\!\cdots\!92 ) / 89\!\cdots\!72 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 71\!\cdots\!05 \nu^{18} + \cdots + 15\!\cdots\!48 ) / 22\!\cdots\!68 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 28\!\cdots\!13 \nu^{18} + \cdots - 45\!\cdots\!56 ) / 89\!\cdots\!72 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 28\!\cdots\!13 \nu^{18} + \cdots + 45\!\cdots\!40 ) / 89\!\cdots\!72 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 31\!\cdots\!91 \nu^{18} + \cdots - 30\!\cdots\!28 ) / 89\!\cdots\!72 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 16\!\cdots\!37 \nu^{18} + \cdots - 13\!\cdots\!00 ) / 44\!\cdots\!36 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 22\!\cdots\!19 \nu^{18} + \cdots + 34\!\cdots\!32 ) / 44\!\cdots\!36 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 45\!\cdots\!51 \nu^{18} + \cdots - 57\!\cdots\!92 ) / 89\!\cdots\!72 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 57\!\cdots\!25 \nu^{18} + \cdots - 71\!\cdots\!32 ) / 89\!\cdots\!72 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 31\!\cdots\!53 \nu^{18} + \cdots - 31\!\cdots\!20 ) / 44\!\cdots\!36 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 10\!\cdots\!93 \nu^{18} + \cdots + 53\!\cdots\!92 ) / 89\!\cdots\!72 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 12\!\cdots\!75 \nu^{18} + \cdots - 11\!\cdots\!00 ) / 89\!\cdots\!72 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 13\!\cdots\!69 \nu^{18} + \cdots + 15\!\cdots\!44 ) / 81\!\cdots\!52 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 10\!\cdots\!05 \nu^{18} + \cdots - 15\!\cdots\!16 ) / 40\!\cdots\!76 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 12\!\cdots\!61 \nu^{18} + \cdots - 13\!\cdots\!00 ) / 44\!\cdots\!36 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 13 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{6} + 20\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{15} + \beta_{11} + 2\beta_{8} + 2\beta_{7} - 3\beta_{6} + \beta_{4} + 29\beta_{2} + 2\beta _1 + 266 \)
|
| \(\nu^{5}\) | \(=\) |
\( - \beta_{18} - 2 \beta_{17} - 4 \beta_{16} - \beta_{15} - \beta_{14} + 2 \beta_{13} - \beta_{12} + \cdots + 105 \)
|
| \(\nu^{6}\) | \(=\) |
\( 7 \beta_{18} - 2 \beta_{17} + 4 \beta_{16} + 43 \beta_{15} + 5 \beta_{14} - 5 \beta_{13} + 2 \beta_{12} + \cdots + 6329 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 32 \beta_{18} - 130 \beta_{17} - 234 \beta_{16} - 55 \beta_{15} - 37 \beta_{14} + 89 \beta_{13} + \cdots + 3642 \)
|
| \(\nu^{8}\) | \(=\) |
\( 399 \beta_{18} - 182 \beta_{17} + 150 \beta_{16} + 1459 \beta_{15} + 318 \beta_{14} - 241 \beta_{13} + \cdots + 162085 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 657 \beta_{18} - 5665 \beta_{17} - 9589 \beta_{16} - 2151 \beta_{15} - 941 \beta_{14} + 2911 \beta_{13} + \cdots + 124115 \)
|
| \(\nu^{10}\) | \(=\) |
\( 16425 \beta_{18} - 10064 \beta_{17} + 2944 \beta_{16} + 45791 \beta_{15} + 13879 \beta_{14} + \cdots + 4344084 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 6476 \beta_{18} - 212174 \beta_{17} - 342610 \beta_{16} - 72872 \beta_{15} - 18215 \beta_{14} + \cdots + 4143043 \)
|
| \(\nu^{12}\) | \(=\) |
\( 597940 \beta_{18} - 448940 \beta_{17} - 2330 \beta_{16} + 1389592 \beta_{15} + 521537 \beta_{14} + \cdots + 120213131 \)
|
| \(\nu^{13}\) | \(=\) |
\( 242268 \beta_{18} - 7383293 \beta_{17} - 11455651 \beta_{16} - 2283851 \beta_{15} - 192338 \beta_{14} + \cdots + 136410154 \)
|
| \(\nu^{14}\) | \(=\) |
\( 20490660 \beta_{18} - 17838048 \beta_{17} - 3475590 \beta_{16} + 41510389 \beta_{15} + 18177529 \beta_{14} + \cdots + 3406520521 \)
|
| \(\nu^{15}\) | \(=\) |
\( 20420286 \beta_{18} - 246657170 \beta_{17} - 369566644 \beta_{16} - 68187206 \beta_{15} + \cdots + 4459433059 \)
|
| \(\nu^{16}\) | \(=\) |
\( 678625325 \beta_{18} - 661033807 \beta_{17} - 216035137 \beta_{16} + 1231150962 \beta_{15} + \cdots + 98276844384 \)
|
| \(\nu^{17}\) | \(=\) |
\( 994084550 \beta_{18} - 8040008157 \beta_{17} - 11679921931 \beta_{16} - 1967973039 \beta_{15} + \cdots + 145349299353 \)
|
| \(\nu^{18}\) | \(=\) |
\( 22007875795 \beta_{18} - 23418298977 \beta_{17} - 9946140955 \beta_{16} + 36414534856 \beta_{15} + \cdots + 2874230761457 \)
|
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 |
|
−5.59523 | 0.895140 | 23.3066 | −1.92876 | −5.00851 | −34.6154 | −85.6437 | −26.1987 | 10.7918 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.97347 | 6.68324 | 16.7354 | 14.9902 | −33.2389 | 21.2163 | −43.4455 | 17.6657 | −74.5533 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −4.81185 | −3.66014 | 15.1539 | −7.39744 | 17.6120 | 23.4523 | −34.4232 | −13.6034 | 35.5954 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −4.08515 | 9.66310 | 8.68847 | 4.51036 | −39.4752 | −32.8387 | −2.81252 | 66.3755 | −18.4255 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −3.75359 | −4.92617 | 6.08945 | 16.0638 | 18.4908 | −14.0219 | 7.17142 | −2.73287 | −60.2968 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −2.81634 | 7.24695 | −0.0682022 | −13.5104 | −20.4099 | 6.32650 | 22.7228 | 25.5183 | 38.0500 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −2.61591 | −1.84520 | −1.15700 | −2.49111 | 4.82688 | −12.8578 | 23.9539 | −23.5952 | 6.51653 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −2.34538 | −9.82681 | −2.49920 | −3.14065 | 23.0476 | 4.23265 | 24.6246 | 69.5662 | 7.36601 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −0.935743 | 2.92967 | −7.12438 | −18.0632 | −2.74142 | −28.9989 | 14.1525 | −18.4170 | 16.9025 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.607806 | −6.88207 | −7.63057 | −11.1811 | −4.18297 | 2.22231 | −9.50036 | 20.3629 | −6.79597 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.19175 | −0.503439 | −6.57973 | 9.91730 | −0.599973 | 18.4265 | −17.3754 | −26.7465 | 11.8189 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1.25139 | 7.03672 | −6.43402 | −12.5431 | 8.80569 | 26.4598 | −18.0626 | 22.5155 | −15.6963 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 2.34217 | −3.46564 | −2.51423 | −7.63566 | −8.11714 | −18.8236 | −24.6261 | −14.9893 | −17.8840 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 2.47116 | 8.89819 | −1.89337 | 5.19254 | 21.9889 | −5.56575 | −24.4481 | 52.1778 | 12.8316 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 2.58074 | −10.0740 | −1.33980 | 18.7836 | −25.9983 | −18.2613 | −24.1036 | 74.4851 | 48.4756 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 4.18105 | −4.69460 | 9.48121 | 11.9577 | −19.6284 | 23.5043 | 6.19301 | −4.96072 | 49.9957 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 4.42068 | 3.81504 | 11.5424 | −2.56198 | 16.8651 | −0.718501 | 15.6597 | −12.4455 | −11.3257 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.18 | 4.50857 | 4.38405 | 12.3272 | 6.95541 | 19.7658 | −32.0757 | 19.5094 | −7.78011 | 31.3589 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.19 | 5.37736 | −1.67405 | 20.9160 | −18.9174 | −9.00195 | 13.9369 | 69.4537 | −24.1976 | −101.725 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \(-1\) |
| \(17\) | \(1\) |
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 | 2057.4.a.k | ✓ | 19 |
| 11.b | odd | 2 | 1 | 2057.4.a.l | yes | 19 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2057.4.a.k | ✓ | 19 | 1.a | even | 1 | 1 | trivial |
| 2057.4.a.l | yes | 19 | 11.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2057))\):
|
\( T_{2}^{19} + 3 T_{2}^{18} - 115 T_{2}^{17} - 311 T_{2}^{16} + 5535 T_{2}^{15} + 13091 T_{2}^{14} + \cdots + 201424256 \)
|
|
\( T_{3}^{19} - 4 T_{3}^{18} - 335 T_{3}^{17} + 1302 T_{3}^{16} + 44795 T_{3}^{15} - 160848 T_{3}^{14} + \cdots - 399479748096 \)
|
|
\( T_{5}^{19} + 11 T_{5}^{18} - 1170 T_{5}^{17} - 12638 T_{5}^{16} + 543168 T_{5}^{15} + \cdots + 12\!\cdots\!88 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{19} + \cdots + 201424256 \)
|
| $3$ |
\( T^{19} + \cdots - 399479748096 \)
|
| $5$ |
\( T^{19} + \cdots + 12\!\cdots\!88 \)
|
| $7$ |
\( T^{19} + \cdots - 12\!\cdots\!60 \)
|
| $11$ |
\( T^{19} \)
|
| $13$ |
\( T^{19} + \cdots - 90\!\cdots\!48 \)
|
| $17$ |
\( (T + 17)^{19} \)
|
| $19$ |
\( T^{19} + \cdots + 19\!\cdots\!00 \)
|
| $23$ |
\( T^{19} + \cdots - 55\!\cdots\!60 \)
|
| $29$ |
\( T^{19} + \cdots - 96\!\cdots\!40 \)
|
| $31$ |
\( T^{19} + \cdots + 23\!\cdots\!80 \)
|
| $37$ |
\( T^{19} + \cdots + 17\!\cdots\!04 \)
|
| $41$ |
\( T^{19} + \cdots + 57\!\cdots\!00 \)
|
| $43$ |
\( T^{19} + \cdots - 57\!\cdots\!71 \)
|
| $47$ |
\( T^{19} + \cdots - 34\!\cdots\!82 \)
|
| $53$ |
\( T^{19} + \cdots - 38\!\cdots\!08 \)
|
| $59$ |
\( T^{19} + \cdots + 35\!\cdots\!48 \)
|
| $61$ |
\( T^{19} + \cdots + 33\!\cdots\!24 \)
|
| $67$ |
\( T^{19} + \cdots + 65\!\cdots\!00 \)
|
| $71$ |
\( T^{19} + \cdots - 19\!\cdots\!00 \)
|
| $73$ |
\( T^{19} + \cdots - 13\!\cdots\!24 \)
|
| $79$ |
\( T^{19} + \cdots + 20\!\cdots\!00 \)
|
| $83$ |
\( T^{19} + \cdots - 63\!\cdots\!85 \)
|
| $89$ |
\( T^{19} + \cdots + 46\!\cdots\!38 \)
|
| $97$ |
\( T^{19} + \cdots + 49\!\cdots\!56 \)
|
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