Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4023,2,Mod(1,4023)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4023, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("4023.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4023 = 3^{3} \cdot 149 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4023.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: | \(32.1238167332\) |
| Analytic rank: | \(1\) |
| Dimension: | \(18\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{18} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{18} - x^{17} - 23 x^{16} + 21 x^{15} + 215 x^{14} - 179 x^{13} - 1052 x^{12} + 799 x^{11} + 2884 x^{10} + \cdots - 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{17}\) 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
\( x^{18} - x^{17} - 23 x^{16} + 21 x^{15} + 215 x^{14} - 179 x^{13} - 1052 x^{12} + 799 x^{11} + 2884 x^{10} + \cdots - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 4\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 778 \nu^{17} - 5570 \nu^{16} - 14734 \nu^{15} + 120827 \nu^{14} + 112377 \nu^{13} - 1049518 \nu^{12} + \cdots - 37275 ) / 14199 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 18088 \nu^{17} + 8704 \nu^{16} + 422141 \nu^{15} - 165362 \nu^{14} - 4008125 \nu^{13} + \cdots - 249261 ) / 14199 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 18707 \nu^{17} + 13379 \nu^{16} + 434533 \nu^{15} - 266770 \nu^{14} - 4109587 \nu^{13} + \cdots - 115995 ) / 14199 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 19490 \nu^{17} + 5820 \nu^{16} + 453389 \nu^{15} - 96637 \nu^{14} - 4275680 \nu^{13} + \cdots - 214977 ) / 14199 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 19785 \nu^{17} + 12154 \nu^{16} + 459937 \nu^{15} - 238578 \nu^{14} - 4349542 \nu^{13} + \cdots - 174909 ) / 14199 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 23225 \nu^{17} + 14699 \nu^{16} + 541267 \nu^{15} - 292396 \nu^{14} - 5131831 \nu^{13} + \cdots - 135330 ) / 14199 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 23336 \nu^{17} - 19521 \nu^{16} - 539062 \nu^{15} + 393022 \nu^{14} + 5072921 \nu^{13} + \cdots + 155727 ) / 14199 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 34935 \nu^{17} + 26348 \nu^{16} + 809452 \nu^{15} - 529668 \nu^{14} - 7635320 \nu^{13} + \cdots - 282888 ) / 14199 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 38566 \nu^{17} - 27170 \nu^{16} - 897733 \nu^{15} + 544034 \nu^{14} + 8512938 \nu^{13} + \cdots + 370764 ) / 14199 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 40549 \nu^{17} + 25562 \nu^{16} + 940337 \nu^{15} - 501974 \nu^{14} - 8867499 \nu^{13} + \cdots - 385935 ) / 14199 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 43154 \nu^{17} + 29271 \nu^{16} + 1000646 \nu^{15} - 585532 \nu^{14} - 9435266 \nu^{13} + \cdots - 253041 ) / 14199 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 44972 \nu^{17} - 30217 \nu^{16} - 1049214 \nu^{15} + 603424 \nu^{14} + 9971928 \nu^{13} + \cdots + 528396 ) / 14199 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 48938 \nu^{17} - 27001 \nu^{16} - 1134422 \nu^{15} + 519304 \nu^{14} + 10681050 \nu^{13} + \cdots + 374151 ) / 14199 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 58303 \nu^{17} - 38518 \nu^{16} - 1353123 \nu^{15} + 764426 \nu^{14} + 12773723 \nu^{13} + \cdots + 441681 ) / 14199 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{16} - \beta_{12} - 2\beta_{11} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{4} + 5\beta_{2} - \beta _1 + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( - \beta_{16} - \beta_{15} + \beta_{12} - \beta_{11} - \beta_{8} - \beta_{7} + \beta_{5} - \beta_{4} + \cdots + 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{17} - 10 \beta_{16} - \beta_{15} + \beta_{14} - 10 \beta_{12} - 21 \beta_{11} - \beta_{10} + \cdots + 75 \)
|
| \(\nu^{7}\) | \(=\) |
\( 2 \beta_{17} - 13 \beta_{16} - 11 \beta_{15} + \beta_{13} + 10 \beta_{12} - 13 \beta_{11} + 2 \beta_{9} + \cdots + 23 \)
|
| \(\nu^{8}\) | \(=\) |
\( 13 \beta_{17} - 81 \beta_{16} - 14 \beta_{15} + 13 \beta_{14} - 2 \beta_{13} - 78 \beta_{12} + \cdots + 436 \)
|
| \(\nu^{9}\) | \(=\) |
\( 27 \beta_{17} - 121 \beta_{16} - 90 \beta_{15} + \beta_{14} + 12 \beta_{13} + 71 \beta_{12} - 128 \beta_{11} + \cdots + 205 \)
|
| \(\nu^{10}\) | \(=\) |
\( 120 \beta_{17} - 611 \beta_{16} - 136 \beta_{15} + 118 \beta_{14} - 32 \beta_{13} - 560 \beta_{12} + \cdots + 2671 \)
|
| \(\nu^{11}\) | \(=\) |
\( 251 \beta_{17} - 994 \beta_{16} - 667 \beta_{15} + 18 \beta_{14} + 97 \beta_{13} + 435 \beta_{12} + \cdots + 1686 \)
|
| \(\nu^{12}\) | \(=\) |
\( 970 \beta_{17} - 4452 \beta_{16} - 1141 \beta_{15} + 930 \beta_{14} - 336 \beta_{13} - 3875 \beta_{12} + \cdots + 16941 \)
|
| \(\nu^{13}\) | \(=\) |
\( 2024 \beta_{17} - 7696 \beta_{16} - 4752 \beta_{15} + 218 \beta_{14} + 658 \beta_{13} + 2414 \beta_{12} + \cdots + 13368 \)
|
| \(\nu^{14}\) | \(=\) |
\( 7349 \beta_{17} - 31816 \beta_{16} - 8900 \beta_{15} + 6837 \beta_{14} - 2938 \beta_{13} - 26319 \beta_{12} + \cdots + 110014 \)
|
| \(\nu^{15}\) | \(=\) |
\( 15292 \beta_{17} - 57725 \beta_{16} - 33312 \beta_{15} + 2226 \beta_{14} + 4008 \beta_{13} + \cdots + 103658 \)
|
| \(\nu^{16}\) | \(=\) |
\( 53751 \beta_{17} - 224802 \beta_{16} - 66717 \beta_{15} + 48403 \beta_{14} - 23270 \beta_{13} + \cdots + 726261 \)
|
| \(\nu^{17}\) | \(=\) |
\( 111935 \beta_{17} - 425063 \beta_{16} - 232089 \beta_{15} + 20648 \beta_{14} + 22383 \beta_{13} + \cdots + 790838 \)
|
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 |
|
−2.49122 | 0 | 4.20616 | 1.53980 | 0 | −1.34218 | −5.49603 | 0 | −3.83597 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.13100 | 0 | 2.54117 | −1.95284 | 0 | −3.82523 | −1.15323 | 0 | 4.16151 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.01153 | 0 | 2.04626 | 2.08172 | 0 | 1.96766 | −0.0930518 | 0 | −4.18744 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.69396 | 0 | 0.869486 | −0.287438 | 0 | −3.11450 | 1.91504 | 0 | 0.486908 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.61093 | 0 | 0.595086 | −1.42964 | 0 | 2.37809 | 2.26321 | 0 | 2.30304 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.24310 | 0 | −0.454703 | 1.28836 | 0 | −2.39093 | 3.05144 | 0 | −1.60156 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.640999 | 0 | −1.58912 | 0.119159 | 0 | 3.75386 | 2.30062 | 0 | −0.0763806 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −0.274945 | 0 | −1.92441 | 1.52999 | 0 | −1.33748 | 1.07900 | 0 | −0.420664 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −0.0953448 | 0 | −1.99091 | −2.35124 | 0 | 0.230539 | 0.380512 | 0 | 0.224178 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.345854 | 0 | −1.88039 | 3.22536 | 0 | −4.66667 | −1.34205 | 0 | 1.11550 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0.354219 | 0 | −1.87453 | −3.57164 | 0 | −0.319733 | −1.37243 | 0 | −1.26514 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1.09825 | 0 | −0.793854 | 0.317616 | 0 | 2.96417 | −3.06834 | 0 | 0.348821 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 1.15236 | 0 | −0.672070 | 0.228732 | 0 | −0.505929 | −3.07918 | 0 | 0.263581 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 1.31176 | 0 | −0.279277 | 2.49864 | 0 | −0.922712 | −2.98987 | 0 | 3.27762 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 1.96635 | 0 | 1.86653 | 1.90384 | 0 | 0.496374 | −0.262451 | 0 | 3.74362 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 2.11691 | 0 | 2.48131 | −2.92751 | 0 | −1.01950 | 1.01889 | 0 | −6.19727 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 2.19454 | 0 | 2.81598 | −2.58110 | 0 | 1.24533 | 1.79071 | 0 | −5.66432 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.18 | 2.65278 | 0 | 5.03727 | −0.631799 | 0 | −4.59116 | 8.05721 | 0 | −1.67603 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(149\) | \(-1\) |
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 | 4023.2.a.b | yes | 18 |
| 3.b | odd | 2 | 1 | 4023.2.a.a | ✓ | 18 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4023.2.a.a | ✓ | 18 | 3.b | odd | 2 | 1 | |
| 4023.2.a.b | yes | 18 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{18} - T_{2}^{17} - 23 T_{2}^{16} + 21 T_{2}^{15} + 215 T_{2}^{14} - 179 T_{2}^{13} - 1052 T_{2}^{12} + \cdots - 3 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4023))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{18} - T^{17} + \cdots - 3 \)
|
| $3$ |
\( T^{18} \)
|
| $5$ |
\( T^{18} + T^{17} + \cdots + 27 \)
|
| $7$ |
\( T^{18} + 11 T^{17} + \cdots - 1237 \)
|
| $11$ |
\( T^{18} + 2 T^{17} + \cdots - 36999 \)
|
| $13$ |
\( T^{18} + 10 T^{17} + \cdots - 1420603 \)
|
| $17$ |
\( T^{18} + 2 T^{17} + \cdots - 46146157 \)
|
| $19$ |
\( T^{18} + \cdots - 392656013 \)
|
| $23$ |
\( T^{18} - 5 T^{17} + \cdots - 23295113 \)
|
| $29$ |
\( T^{18} - 3 T^{17} + \cdots - 231767 \)
|
| $31$ |
\( T^{18} + \cdots + 85138787039 \)
|
| $37$ |
\( T^{18} + \cdots - 479320173 \)
|
| $41$ |
\( T^{18} + \cdots - 120222540309 \)
|
| $43$ |
\( T^{18} + \cdots + 180371713 \)
|
| $47$ |
\( T^{18} + \cdots + 105878216649 \)
|
| $53$ |
\( T^{18} + \cdots - 8978035007 \)
|
| $59$ |
\( T^{18} + \cdots + 123300710215 \)
|
| $61$ |
\( T^{18} + \cdots + 5643635211 \)
|
| $67$ |
\( T^{18} + \cdots - 5239279425909 \)
|
| $71$ |
\( T^{18} + \cdots - 1037320441005 \)
|
| $73$ |
\( T^{18} + \cdots - 197947420831488 \)
|
| $79$ |
\( T^{18} + \cdots + 308425107348397 \)
|
| $83$ |
\( T^{18} + \cdots - 105669675061317 \)
|
| $89$ |
\( T^{18} + \cdots - 60011236799845 \)
|
| $97$ |
\( T^{18} + \cdots + 31484583656791 \)
|
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