Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [349,2,Mod(1,349)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(349, 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("349.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 349 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 349.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: | \(2.78677903054\) |
| Analytic rank: | \(1\) |
| Dimension: | \(11\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{11} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{11} - 5x^{10} - x^{9} + 35x^{8} - 24x^{7} - 80x^{6} + 66x^{5} + 77x^{4} - 56x^{3} - 31x^{2} + 15x + 4 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{10}\) 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
\( x^{11} - 5x^{10} - x^{9} + 35x^{8} - 24x^{7} - 80x^{6} + 66x^{5} + 77x^{4} - 56x^{3} - 31x^{2} + 15x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{10} + 6\nu^{9} - 4\nu^{8} - 33\nu^{7} + 49\nu^{6} + 42\nu^{5} - 85\nu^{4} - 2\nu^{3} + 33\nu^{2} - 12\nu ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{10} + 5\nu^{9} - 30\nu^{7} + 24\nu^{6} + 50\nu^{5} - 42\nu^{4} - 27\nu^{3} + 13\nu^{2} + 5\nu + 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2 \nu^{10} - 9 \nu^{9} - 6 \nu^{8} + 64 \nu^{7} - 15 \nu^{6} - 147 \nu^{5} + 36 \nu^{4} + 131 \nu^{3} + \cdots - 14 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2 \nu^{10} - 9 \nu^{9} - 6 \nu^{8} + 64 \nu^{7} - 15 \nu^{6} - 149 \nu^{5} + 40 \nu^{4} + 141 \nu^{3} + \cdots - 2 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( -\nu^{10} + 5\nu^{9} - 31\nu^{7} + 27\nu^{6} + 55\nu^{5} - 58\nu^{4} - 33\nu^{3} + 31\nu^{2} + 6\nu - 1 \)
|
| \(\beta_{8}\) | \(=\) |
\( \nu^{10} - 5\nu^{9} + 31\nu^{7} - 27\nu^{6} - 55\nu^{5} + 59\nu^{4} + 32\nu^{3} - 36\nu^{2} - 4\nu + 5 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( \nu^{10} - 2 \nu^{9} - 16 \nu^{8} + 33 \nu^{7} + 73 \nu^{6} - 144 \nu^{5} - 125 \nu^{4} + 202 \nu^{3} + \cdots - 20 ) / 2 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 2 \nu^{10} + 7 \nu^{9} + 16 \nu^{8} - 64 \nu^{7} - 47 \nu^{6} + 201 \nu^{5} + 72 \nu^{4} - 243 \nu^{3} + \cdots + 16 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{9} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{9} + \beta_{8} + 2\beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} + 6\beta_{2} + 8\beta _1 + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( -7\beta_{9} + 2\beta_{8} + 9\beta_{7} + 6\beta_{6} + 8\beta_{5} + 7\beta_{3} + 8\beta_{2} + 28\beta _1 + 7 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 2 \beta_{10} - 13 \beta_{9} + 9 \beta_{8} + 21 \beta_{7} + 9 \beta_{6} + 13 \beta_{5} + 11 \beta_{3} + \cdots + 30 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 6 \beta_{10} - 52 \beta_{9} + 21 \beta_{8} + 69 \beta_{7} + 35 \beta_{6} + 57 \beta_{5} + \beta_{4} + \cdots + 40 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 29 \beta_{10} - 121 \beta_{9} + 68 \beta_{8} + 171 \beta_{7} + 65 \beta_{6} + 118 \beta_{5} + \cdots + 138 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 84 \beta_{10} - 389 \beta_{9} + 169 \beta_{8} + 499 \beta_{7} + 206 \beta_{6} + 400 \beta_{5} + \cdots + 216 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 288 \beta_{10} - 978 \beta_{9} + 489 \beta_{8} + 1268 \beta_{7} + 427 \beta_{6} + 933 \beta_{5} + \cdots + 657 \)
|
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 |
|
−2.60070 | −0.209864 | 4.76365 | 1.72230 | 0.545795 | −2.16571 | −7.18744 | −2.95596 | −4.47919 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.42122 | 2.15404 | 3.86232 | −2.99362 | −5.21540 | 0.559596 | −4.50909 | 1.63987 | 7.24821 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.17018 | −2.21125 | 2.70967 | −2.88560 | 4.79881 | 3.08193 | −1.54011 | 1.88963 | 6.26228 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.31062 | 0.477277 | −0.282282 | −1.24067 | −0.625527 | 0.925788 | 2.99120 | −2.77221 | 1.62604 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.831463 | −3.29989 | −1.30867 | 1.20940 | 2.74374 | 3.20765 | 2.75104 | 7.88930 | −1.00557 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.734924 | 0.314167 | −1.45989 | 2.34643 | −0.230889 | −4.45589 | 2.54275 | −2.90130 | −1.72445 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.216390 | 2.05625 | −1.95318 | −4.07164 | 0.444951 | −1.74178 | −0.855428 | 1.22815 | −0.881063 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.767986 | 0.193628 | −1.41020 | −0.953573 | 0.148703 | −2.43623 | −2.61898 | −2.96251 | −0.732330 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.916892 | −2.19816 | −1.15931 | 2.45207 | −2.01547 | −1.41213 | −2.89675 | 1.83189 | 2.24828 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.25205 | −1.02718 | −0.432368 | −3.34750 | −1.28608 | 4.81118 | −3.04545 | −1.94490 | −4.19124 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.91579 | −2.24901 | 1.67025 | −1.23759 | −4.30862 | −3.37442 | −0.631741 | 2.05803 | −2.37097 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(349\) | \(1\) |
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 | 349.2.a.a | ✓ | 11 |
| 3.b | odd | 2 | 1 | 3141.2.a.b | 11 | ||
| 4.b | odd | 2 | 1 | 5584.2.a.j | 11 | ||
| 5.b | even | 2 | 1 | 8725.2.a.l | 11 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 349.2.a.a | ✓ | 11 | 1.a | even | 1 | 1 | trivial |
| 3141.2.a.b | 11 | 3.b | odd | 2 | 1 | ||
| 5584.2.a.j | 11 | 4.b | odd | 2 | 1 | ||
| 8725.2.a.l | 11 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{11} + 5 T_{2}^{10} - T_{2}^{9} - 35 T_{2}^{8} - 24 T_{2}^{7} + 80 T_{2}^{6} + 66 T_{2}^{5} + \cdots - 4 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(349))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} + 5 T^{10} + \cdots - 4 \)
|
| $3$ |
\( T^{11} + 6 T^{10} + \cdots - 1 \)
|
| $5$ |
\( T^{11} + 9 T^{10} + \cdots + 2066 \)
|
| $7$ |
\( T^{11} + 3 T^{10} + \cdots - 4808 \)
|
| $11$ |
\( T^{11} + 31 T^{10} + \cdots + 1366 \)
|
| $13$ |
\( T^{11} + 4 T^{10} + \cdots + 110828 \)
|
| $17$ |
\( T^{11} + T^{10} + \cdots - 285937 \)
|
| $19$ |
\( T^{11} + 17 T^{10} + \cdots + 191249 \)
|
| $23$ |
\( T^{11} + 24 T^{10} + \cdots + 151 \)
|
| $29$ |
\( T^{11} + 17 T^{10} + \cdots + 811361 \)
|
| $31$ |
\( T^{11} + 10 T^{10} + \cdots + 23913808 \)
|
| $37$ |
\( T^{11} + T^{10} + \cdots + 30359353 \)
|
| $41$ |
\( T^{11} + 15 T^{10} + \cdots + 2830 \)
|
| $43$ |
\( T^{11} + 5 T^{10} + \cdots + 11806 \)
|
| $47$ |
\( T^{11} - 4 T^{10} + \cdots - 4141504 \)
|
| $53$ |
\( T^{11} + \cdots + 799326040 \)
|
| $59$ |
\( T^{11} + 52 T^{10} + \cdots + 18247468 \)
|
| $61$ |
\( T^{11} - 366 T^{9} + \cdots + 76056118 \)
|
| $67$ |
\( T^{11} + 23 T^{10} + \cdots - 20223293 \)
|
| $71$ |
\( T^{11} + 30 T^{10} + \cdots - 6512458 \)
|
| $73$ |
\( T^{11} + \cdots + 4903265291 \)
|
| $79$ |
\( T^{11} - 11 T^{10} + \cdots - 21305618 \)
|
| $83$ |
\( T^{11} + \cdots + 337935589 \)
|
| $89$ |
\( T^{11} + \cdots - 13841703482 \)
|
| $97$ |
\( T^{11} + \cdots - 2578258190 \)
|
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