Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [353,2,Mod(1,353)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(353, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("353.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 353 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 353.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: | \(2.81871919135\) |
| 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: |
\( x^{14} - 4 x^{13} - 14 x^{12} + 71 x^{11} + 47 x^{10} - 452 x^{9} + 101 x^{8} + 1251 x^{7} - 740 x^{6} + \cdots - 1 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{13}\) 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
\( x^{14} - 4 x^{13} - 14 x^{12} + 71 x^{11} + 47 x^{10} - 452 x^{9} + 101 x^{8} + 1251 x^{7} - 740 x^{6} + \cdots - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - \nu^{13} + 3 \nu^{12} + 13 \nu^{11} - 42 \nu^{10} - 53 \nu^{9} + 203 \nu^{8} + 54 \nu^{7} - 401 \nu^{6} + \cdots - 29 ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - \nu^{12} + 2 \nu^{11} + 17 \nu^{10} - 33 \nu^{9} - 104 \nu^{8} + 195 \nu^{7} + 277 \nu^{6} - 500 \nu^{5} + \cdots - 1 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{13} - 7 \nu^{12} - 9 \nu^{11} + 130 \nu^{10} - 55 \nu^{9} - 855 \nu^{8} + 810 \nu^{7} + 2405 \nu^{6} + \cdots - 75 ) / 8 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{13} - 5 \nu^{12} - 13 \nu^{11} + 92 \nu^{10} + 23 \nu^{9} - 603 \nu^{8} + 280 \nu^{7} + 1703 \nu^{6} + \cdots - 29 ) / 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 3 \nu^{13} - 13 \nu^{12} - 39 \nu^{11} + 226 \nu^{10} + 99 \nu^{9} - 1409 \nu^{8} + 506 \nu^{7} + \cdots - 69 ) / 8 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 7 \nu^{13} + 27 \nu^{12} + 99 \nu^{11} - 476 \nu^{10} - 353 \nu^{9} + 3009 \nu^{8} - 524 \nu^{7} + \cdots + 139 ) / 8 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 7 \nu^{13} + 31 \nu^{12} + 91 \nu^{11} - 544 \nu^{10} - 221 \nu^{9} + 3425 \nu^{8} - 1304 \nu^{7} + \cdots + 143 ) / 8 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 3 \nu^{13} + 16 \nu^{12} + 35 \nu^{11} - 289 \nu^{10} - 6 \nu^{9} + 1863 \nu^{8} - 1209 \nu^{7} + \cdots + 100 ) / 4 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 7 \nu^{13} + 24 \nu^{12} + 105 \nu^{11} - 423 \nu^{10} - 454 \nu^{9} + 2669 \nu^{8} + 81 \nu^{7} + \cdots + 106 ) / 4 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 7 \nu^{13} - 25 \nu^{12} - 103 \nu^{11} + 442 \nu^{10} + 415 \nu^{9} - 2793 \nu^{8} + 178 \nu^{7} + \cdots - 121 ) / 4 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 5 \nu^{13} - 23 \nu^{12} - 63 \nu^{11} + 404 \nu^{10} + 121 \nu^{9} - 2545 \nu^{8} + 1172 \nu^{7} + \cdots - 145 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{9} + \beta_{8} - \beta_{4} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{10} + \beta_{9} + \beta_{7} - 2\beta_{5} + 6\beta_{2} - \beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{12} + \beta_{11} - 9\beta_{9} + 8\beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - 8\beta_{4} + \beta_{3} + 36\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{12} + \beta_{11} - 9 \beta_{10} + 8 \beta_{9} + 9 \beta_{7} - 2 \beta_{6} - 19 \beta_{5} + \cdots + 86 \)
|
| \(\nu^{7}\) | \(=\) |
\( - \beta_{13} + 11 \beta_{12} + 11 \beta_{11} - \beta_{10} - 66 \beta_{9} + 54 \beta_{8} + \cdots + 220 \beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 3 \beta_{13} + 14 \beta_{12} + 14 \beta_{11} - 65 \beta_{10} + 46 \beta_{9} + \beta_{8} + 65 \beta_{7} + \cdots + 521 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 17 \beta_{13} + 91 \beta_{12} + 92 \beta_{11} - 15 \beta_{10} - 457 \beta_{9} + 348 \beta_{8} + \cdots + 4 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 49 \beta_{13} + 133 \beta_{12} + 136 \beta_{11} - 443 \beta_{10} + 225 \beta_{9} + 12 \beta_{8} + \cdots + 3235 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 191 \beta_{13} + 681 \beta_{12} + 702 \beta_{11} - 158 \beta_{10} - 3095 \beta_{9} + 2203 \beta_{8} + \cdots + 88 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 537 \beta_{13} + 1086 \beta_{12} + 1146 \beta_{11} - 2971 \beta_{10} + 927 \beta_{9} + 94 \beta_{8} + \cdots + 20344 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 1798 \beta_{13} + 4864 \beta_{12} + 5138 \beta_{11} - 1445 \beta_{10} - 20744 \beta_{9} + 13831 \beta_{8} + \cdots + 1200 \)
|
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 |
|
−2.50114 | 2.38878 | 4.25570 | −0.957198 | −5.97467 | 1.26127 | −5.64183 | 2.70626 | 2.39409 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.50011 | −1.92577 | 4.25055 | −1.85756 | 4.81463 | 3.88270 | −5.62661 | 0.708576 | 4.64409 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.59618 | −1.17885 | 0.547793 | −2.41823 | 1.88165 | −2.99006 | 2.31799 | −1.61032 | 3.85993 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.41172 | 2.61874 | −0.00705285 | 3.04446 | −3.69692 | 5.11343 | 2.83339 | 3.85779 | −4.29792 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.766580 | 1.39011 | −1.41235 | −3.92680 | −1.06563 | 4.83756 | 2.61584 | −1.06761 | 3.01021 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.0519079 | −0.974028 | −1.99731 | 2.17556 | 0.0505598 | 2.76426 | 0.207492 | −2.05127 | −0.112929 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.0420195 | −2.61610 | −1.99823 | −4.27486 | 0.109927 | 0.908371 | 0.168004 | 3.84396 | 0.179628 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.748009 | −2.51698 | −1.44048 | 0.670244 | −1.88272 | 0.453811 | −2.57351 | 3.33518 | 0.501349 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.24663 | 1.82907 | −0.445910 | 3.16975 | 2.28018 | 0.303774 | −3.04915 | 0.345504 | 3.95152 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.61142 | 0.564492 | 0.596684 | −0.289385 | 0.909636 | 4.57842 | −2.26134 | −2.68135 | −0.466322 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.08539 | 2.60610 | 2.34887 | −2.82246 | 5.43474 | 2.17060 | 0.727521 | 3.79175 | −5.88594 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.12356 | 0.241297 | 2.50949 | 2.97619 | 0.512407 | 0.956763 | 1.08193 | −2.94178 | 6.32011 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 2.43365 | −1.22229 | 3.92266 | 2.32335 | −2.97462 | −0.374206 | 4.67908 | −1.50602 | 5.65422 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 2.62099 | −3.20458 | 4.86960 | −1.81306 | −8.39917 | 5.13330 | 7.52119 | 7.26932 | −4.75202 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(353\) | \(-1\) |
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 | 353.2.a.d | ✓ | 14 |
| 3.b | odd | 2 | 1 | 3177.2.a.h | 14 | ||
| 4.b | odd | 2 | 1 | 5648.2.a.p | 14 | ||
| 5.b | even | 2 | 1 | 8825.2.a.i | 14 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 353.2.a.d | ✓ | 14 | 1.a | even | 1 | 1 | trivial |
| 3177.2.a.h | 14 | 3.b | odd | 2 | 1 | ||
| 5648.2.a.p | 14 | 4.b | odd | 2 | 1 | ||
| 8825.2.a.i | 14 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{14} - 4 T_{2}^{13} - 14 T_{2}^{12} + 71 T_{2}^{11} + 47 T_{2}^{10} - 452 T_{2}^{9} + 101 T_{2}^{8} + \cdots - 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(353))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{14} - 4 T^{13} + \cdots - 1 \)
|
| $3$ |
\( T^{14} + 2 T^{13} + \cdots - 322 \)
|
| $5$ |
\( T^{14} + 4 T^{13} + \cdots + 10400 \)
|
| $7$ |
\( T^{14} - 29 T^{13} + \cdots + 2290 \)
|
| $11$ |
\( T^{14} + 7 T^{13} + \cdots - 57856 \)
|
| $13$ |
\( T^{14} - 2 T^{13} + \cdots + 8277472 \)
|
| $17$ |
\( T^{14} - 8 T^{13} + \cdots - 1456 \)
|
| $19$ |
\( T^{14} - 14 T^{13} + \cdots - 418368 \)
|
| $23$ |
\( T^{14} - 16 T^{13} + \cdots - 48674240 \)
|
| $29$ |
\( T^{14} + 9 T^{13} + \cdots - 35333612 \)
|
| $31$ |
\( T^{14} + \cdots - 9849648574 \)
|
| $37$ |
\( T^{14} - 8 T^{13} + \cdots - 15760288 \)
|
| $41$ |
\( T^{14} - 14 T^{13} + \cdots - 109420 \)
|
| $43$ |
\( T^{14} - 11 T^{13} + \cdots - 7146304 \)
|
| $47$ |
\( T^{14} - 14 T^{13} + \cdots - 15261760 \)
|
| $53$ |
\( T^{14} + 27 T^{13} + \cdots + 1008864 \)
|
| $59$ |
\( T^{14} + \cdots + 86637542314 \)
|
| $61$ |
\( T^{14} + \cdots - 156221060 \)
|
| $67$ |
\( T^{14} + \cdots - 3880568794 \)
|
| $71$ |
\( T^{14} + \cdots + 269110066458 \)
|
| $73$ |
\( T^{14} + \cdots - 120468679468 \)
|
| $79$ |
\( T^{14} + \cdots - 7688346874 \)
|
| $83$ |
\( T^{14} + \cdots - 838875264 \)
|
| $89$ |
\( T^{14} + \cdots - 790691221600 \)
|
| $97$ |
\( T^{14} + \cdots + 210539761328 \)
|
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