Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3311,2,Mod(1,3311)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3311, 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("3311.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3311 = 7 \cdot 11 \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3311.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: | \(26.4384681092\) |
| Analytic rank: | \(1\) |
| Dimension: | \(15\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{15} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{15} - 2 x^{14} - 17 x^{13} + 34 x^{12} + 109 x^{11} - 220 x^{10} - 326 x^{9} + 678 x^{8} + 448 x^{7} + \cdots + 2 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{14}\) 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
\( x^{15} - 2 x^{14} - 17 x^{13} + 34 x^{12} + 109 x^{11} - 220 x^{10} - 326 x^{9} + 678 x^{8} + 448 x^{7} + \cdots + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 419 \nu^{14} + 1306 \nu^{13} - 10284 \nu^{12} - 21139 \nu^{11} + 95449 \nu^{10} + 127878 \nu^{9} + \cdots - 21444 ) / 6221 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 527 \nu^{14} - 629 \nu^{13} - 10871 \nu^{12} + 14168 \nu^{11} + 84923 \nu^{10} - 121303 \nu^{9} + \cdots + 3421 ) / 6221 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 734 \nu^{14} + 328 \nu^{13} - 15625 \nu^{12} - 3699 \nu^{11} + 127995 \nu^{10} + 8166 \nu^{9} + \cdots - 2318 ) / 6221 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 794 \nu^{14} - 747 \nu^{13} - 12495 \nu^{12} + 9695 \nu^{11} + 71688 \nu^{10} - 40409 \nu^{9} + \cdots + 8731 ) / 6221 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 952 \nu^{14} + 2541 \nu^{13} + 14621 \nu^{12} - 41648 \nu^{11} - 79560 \nu^{10} + 256454 \nu^{9} + \cdots - 2367 ) / 6221 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 1028 \nu^{14} - 1829 \nu^{13} - 18951 \nu^{12} + 33315 \nu^{11} + 133902 \nu^{10} - 232962 \nu^{9} + \cdots - 4789 ) / 6221 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 1159 \nu^{14} + 1584 \nu^{13} + 19375 \nu^{12} - 23781 \nu^{11} - 122632 \nu^{10} + 126985 \nu^{9} + \cdots - 15291 ) / 6221 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 1255 \nu^{14} - 3304 \nu^{13} - 20588 \nu^{12} + 55165 \nu^{11} + 127100 \nu^{10} - 347788 \nu^{9} + \cdots - 6845 ) / 6221 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 1407 \nu^{14} - 1880 \nu^{13} - 23027 \nu^{12} + 32278 \nu^{11} + 136248 \nu^{10} - 213710 \nu^{9} + \cdots + 28611 ) / 6221 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 2049 \nu^{14} - 4051 \nu^{13} - 33083 \nu^{12} + 64860 \nu^{11} + 198788 \nu^{10} - 388197 \nu^{9} + \cdots + 14328 ) / 6221 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 2096 \nu^{14} + 2301 \nu^{13} + 37889 \nu^{12} - 40307 \nu^{11} - 261371 \nu^{10} + 270204 \nu^{9} + \cdots + 2212 ) / 6221 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 2323 \nu^{14} - 3776 \nu^{13} - 39526 \nu^{12} + 62157 \nu^{11} + 254569 \nu^{10} - 385030 \nu^{9} + \cdots + 1953 ) / 6221 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{12} - \beta_{10} - \beta_{6} + \beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{14} + \beta_{12} - \beta_{10} - 2\beta_{8} - 2\beta_{6} + \beta_{4} + 6\beta_{2} - \beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{14} - \beta_{13} + 8\beta_{12} - 7\beta_{10} - \beta_{8} - 8\beta_{6} + 8\beta_{2} + 18\beta _1 + 9 \)
|
| \(\nu^{6}\) | \(=\) |
\( 7 \beta_{14} - \beta_{13} + 9 \beta_{12} - 8 \beta_{10} + \beta_{9} - 17 \beta_{8} - \beta_{7} + \cdots + 80 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 9 \beta_{14} - 9 \beta_{13} + 51 \beta_{12} - 42 \beta_{10} - 11 \beta_{8} - 2 \beta_{7} - 52 \beta_{6} + \cdots + 62 \)
|
| \(\nu^{8}\) | \(=\) |
\( 38 \beta_{14} - 9 \beta_{13} + 62 \beta_{12} + \beta_{11} - 51 \beta_{10} + 9 \beta_{9} - 111 \beta_{8} + \cdots + 434 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 62 \beta_{14} - 59 \beta_{13} + 301 \beta_{12} + 2 \beta_{11} - 240 \beta_{10} - 2 \beta_{9} + \cdots + 389 \)
|
| \(\nu^{10}\) | \(=\) |
\( 187 \beta_{14} - 56 \beta_{13} + 389 \beta_{12} + 16 \beta_{11} - 300 \beta_{10} + 59 \beta_{9} + \cdots + 2370 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 392 \beta_{14} - 342 \beta_{13} + 1719 \beta_{12} + 34 \beta_{11} - 1341 \beta_{10} - 29 \beta_{9} + \cdots + 2346 \)
|
| \(\nu^{12}\) | \(=\) |
\( 864 \beta_{14} - 293 \beta_{13} + 2340 \beta_{12} + 169 \beta_{11} - 1700 \beta_{10} + 345 \beta_{9} + \cdots + 12987 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 2396 \beta_{14} - 1859 \beta_{13} + 9670 \beta_{12} + 369 \beta_{11} - 7406 \beta_{10} - 277 \beta_{9} + \cdots + 13885 \)
|
| \(\nu^{14}\) | \(=\) |
\( 3749 \beta_{14} - 1347 \beta_{13} + 13781 \beta_{12} + 1477 \beta_{11} - 9453 \beta_{10} + 1913 \beta_{9} + \cdots + 71340 \)
|
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 |
|
−2.41086 | 1.43803 | 3.81226 | 2.38378 | −3.46690 | 1.00000 | −4.36911 | −0.932064 | −5.74696 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.31662 | −0.396352 | 3.36672 | −0.986733 | 0.918196 | 1.00000 | −3.16616 | −2.84290 | 2.28588 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.85146 | −2.37307 | 1.42790 | −2.15313 | 4.39364 | 1.00000 | 1.05922 | 2.63144 | 3.98643 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.72541 | 0.104992 | 0.977033 | 0.631873 | −0.181153 | 1.00000 | 1.76504 | −2.98898 | −1.09024 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.39939 | −0.306757 | −0.0417074 | 4.21665 | 0.429273 | 1.00000 | 2.85715 | −2.90590 | −5.90073 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.936812 | 2.31934 | −1.12238 | 0.384361 | −2.17279 | 1.00000 | 2.92509 | 2.37934 | −0.360074 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.676613 | −1.26693 | −1.54219 | −3.31069 | 0.857220 | 1.00000 | 2.39670 | −1.39490 | 2.24006 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −0.146329 | 1.76139 | −1.97859 | −2.81911 | −0.257742 | 1.00000 | 0.582181 | 0.102507 | 0.412516 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.142363 | −2.69183 | −1.97973 | 2.45331 | −0.383217 | 1.00000 | −0.566566 | 4.24597 | 0.349260 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.393099 | −0.681708 | −1.84547 | −0.0411556 | −0.267979 | 1.00000 | −1.51165 | −2.53527 | −0.0161783 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.15931 | −2.37551 | −0.655993 | 1.07305 | −2.75396 | 1.00000 | −3.07913 | 2.64306 | 1.24400 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1.56134 | 2.53571 | 0.437779 | −2.05592 | 3.95910 | 1.00000 | −2.43916 | 3.42983 | −3.20998 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 1.62309 | 0.603518 | 0.634410 | 1.33524 | 0.979563 | 1.00000 | −2.21647 | −2.63577 | 2.16722 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 2.25968 | −2.39300 | 3.10613 | 1.56984 | −5.40740 | 1.00000 | 2.49951 | 2.72644 | 3.54733 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 2.32462 | −0.277832 | 3.40384 | −1.68136 | −0.645851 | 1.00000 | 3.26338 | −2.92281 | −3.90852 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(-1\) |
| \(11\) | \(1\) |
| \(43\) | \(-1\) |
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 | 3311.2.a.d | ✓ | 15 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3311.2.a.d | ✓ | 15 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3311))\):
|
\( T_{2}^{15} + 2 T_{2}^{14} - 17 T_{2}^{13} - 34 T_{2}^{12} + 109 T_{2}^{11} + 220 T_{2}^{10} - 326 T_{2}^{9} + \cdots - 2 \)
|
|
\( T_{5}^{15} - T_{5}^{14} - 33 T_{5}^{13} + 24 T_{5}^{12} + 397 T_{5}^{11} - 259 T_{5}^{10} - 2257 T_{5}^{9} + \cdots + 38 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{15} + 2 T^{14} + \cdots - 2 \)
|
| $3$ |
\( T^{15} + 4 T^{14} + \cdots + 1 \)
|
| $5$ |
\( T^{15} - T^{14} + \cdots + 38 \)
|
| $7$ |
\( (T - 1)^{15} \)
|
| $11$ |
\( (T + 1)^{15} \)
|
| $13$ |
\( T^{15} + 14 T^{14} + \cdots + 304 \)
|
| $17$ |
\( T^{15} + 9 T^{14} + \cdots + 34124 \)
|
| $19$ |
\( T^{15} + 10 T^{14} + \cdots + 17728 \)
|
| $23$ |
\( T^{15} + T^{14} + \cdots + 22966 \)
|
| $29$ |
\( T^{15} + 12 T^{14} + \cdots - 67364422 \)
|
| $31$ |
\( T^{15} + 16 T^{14} + \cdots + 22646356 \)
|
| $37$ |
\( T^{15} + \cdots - 862813454 \)
|
| $41$ |
\( T^{15} + \cdots - 2155596208 \)
|
| $43$ |
\( (T - 1)^{15} \)
|
| $47$ |
\( T^{15} + \cdots - 361046144 \)
|
| $53$ |
\( T^{15} + \cdots + 177883349942 \)
|
| $59$ |
\( T^{15} + \cdots + 32358042434 \)
|
| $61$ |
\( T^{15} + 12 T^{14} + \cdots - 6495604 \)
|
| $67$ |
\( T^{15} + \cdots - 293334124 \)
|
| $71$ |
\( T^{15} + \cdots + 21154405744 \)
|
| $73$ |
\( T^{15} + \cdots - 1431782984 \)
|
| $79$ |
\( T^{15} + \cdots - 518461424 \)
|
| $83$ |
\( T^{15} + \cdots - 2139798442 \)
|
| $89$ |
\( T^{15} + \cdots - 24626303356 \)
|
| $97$ |
\( T^{15} + \cdots + 181901769529 \)
|
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