Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [38,8,Mod(7,38)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(38, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("38.7");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 38 = 2 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 38.c (of order \(3\), degree \(2\), minimal) |
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: | no |
| Analytic conductor: | \(11.8706309684\) |
| Analytic rank: | \(0\) |
| Dimension: | \(14\) |
| Relative dimension: | \(7\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{14} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{14} - x^{13} + 9740 x^{12} + 37173 x^{11} + 71393485 x^{10} + 196352446 x^{9} + 218671355941 x^{8} + \cdots + 66\!\cdots\!81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 3^{6} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{14} - x^{13} + 9740 x^{12} + 37173 x^{11} + 71393485 x^{10} + 196352446 x^{9} + 218671355941 x^{8} + \cdots + 66\!\cdots\!81 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 29\!\cdots\!90 \nu^{13} + \cdots + 38\!\cdots\!24 ) / 72\!\cdots\!95 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 16\!\cdots\!96 \nu^{13} + \cdots + 20\!\cdots\!72 ) / 20\!\cdots\!05 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 16\!\cdots\!92 \nu^{13} + \cdots - 20\!\cdots\!08 ) / 72\!\cdots\!95 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 20\!\cdots\!74 \nu^{13} + \cdots + 31\!\cdots\!97 ) / 79\!\cdots\!80 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 26\!\cdots\!07 \nu^{13} + \cdots - 63\!\cdots\!37 ) / 79\!\cdots\!28 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 46\!\cdots\!82 \nu^{13} + \cdots + 17\!\cdots\!49 ) / 79\!\cdots\!80 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 98\!\cdots\!84 \nu^{13} + \cdots - 29\!\cdots\!32 ) / 11\!\cdots\!60 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 15\!\cdots\!70 \nu^{13} + \cdots + 45\!\cdots\!18 ) / 11\!\cdots\!60 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 22\!\cdots\!01 \nu^{13} + \cdots + 51\!\cdots\!45 ) / 11\!\cdots\!60 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 46\!\cdots\!97 \nu^{13} + \cdots - 13\!\cdots\!96 ) / 22\!\cdots\!20 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 45\!\cdots\!94 \nu^{13} + \cdots - 30\!\cdots\!16 ) / 20\!\cdots\!05 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 67\!\cdots\!03 \nu^{13} + \cdots - 30\!\cdots\!74 ) / 22\!\cdots\!20 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{12} - \beta_{4} - 2782\beta_{3} + 3\beta_{2} + 3\beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( -6\beta_{10} - 6\beta_{9} - 8\beta_{7} + 34\beta_{6} + 8\beta_{5} - 17\beta_{4} + 4936\beta_{2} - 9337 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 632 \beta_{13} - 5616 \beta_{12} + 1544 \beta_{11} + 192 \beta_{10} - 384 \beta_{9} + 1264 \beta_{8} + \cdots - 13709153 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 84448 \beta_{13} - 86440 \beta_{12} + 37072 \beta_{11} + 92256 \beta_{10} - 46128 \beta_{9} + \cdots - 26376817 \beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 1897296 \beta_{10} + 1897296 \beta_{9} + 14418040 \beta_{7} - 8575904 \beta_{6} - 6397720 \beta_{5} + \cdots + 72769391118 \)
|
| \(\nu^{7}\) | \(=\) |
\( 649978584 \beta_{13} + 817692189 \beta_{12} - 209113944 \beta_{11} - 313126038 \beta_{10} + \cdots + 1301961499113 \)
|
| \(\nu^{8}\) | \(=\) |
\( 49174040112 \beta_{13} + 185901785440 \beta_{12} - 104815097232 \beta_{11} - 29655377280 \beta_{10} + \cdots + 3912930848352 \beta_1 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 2035526940384 \beta_{10} - 2035526940384 \beta_{9} - 1484921597216 \beta_{7} + 25423234501168 \beta_{6} + \cdots - 10\!\cdots\!28 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 344339829603440 \beta_{13} + \cdots - 23\!\cdots\!70 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 30\!\cdots\!44 \beta_{13} + \cdots - 50\!\cdots\!84 \beta_1 \)
|
| \(\nu^{12}\) | \(=\) |
\( 73\!\cdots\!40 \beta_{10} + \cdots + 13\!\cdots\!17 \)
|
| \(\nu^{13}\) | \(=\) |
\( 19\!\cdots\!84 \beta_{13} + \cdots + 60\!\cdots\!08 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/38\mathbb{Z}\right)^\times\).
| \(n\) | \(21\) |
| \(\chi(n)\) | \(-1 + \beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
4.00000 | + | 6.92820i | −36.1624 | − | 62.6352i | −32.0000 | + | 55.4256i | −36.8530 | − | 63.8313i | 289.300 | − | 501.082i | 866.283 | −512.000 | −1521.94 | + | 2636.09i | 294.824 | − | 510.650i | ||||||||||||||||||||||||||||||||||||||||||||||
| 7.2 | 4.00000 | + | 6.92820i | −24.5226 | − | 42.4744i | −32.0000 | + | 55.4256i | 81.5095 | + | 141.179i | 196.181 | − | 339.795i | −230.715 | −512.000 | −109.217 | + | 189.169i | −652.076 | + | 1129.43i | |||||||||||||||||||||||||||||||||||||||||||||||
| 7.3 | 4.00000 | + | 6.92820i | −4.67249 | − | 8.09299i | −32.0000 | + | 55.4256i | −40.1604 | − | 69.5598i | 37.3799 | − | 64.7439i | −541.879 | −512.000 | 1049.84 | − | 1818.37i | 321.283 | − | 556.478i | |||||||||||||||||||||||||||||||||||||||||||||||
| 7.4 | 4.00000 | + | 6.92820i | 7.15555 | + | 12.3938i | −32.0000 | + | 55.4256i | −222.650 | − | 385.641i | −57.2444 | + | 99.1503i | 1436.09 | −512.000 | 991.096 | − | 1716.63i | 1781.20 | − | 3085.13i | |||||||||||||||||||||||||||||||||||||||||||||||
| 7.5 | 4.00000 | + | 6.92820i | 9.55281 | + | 16.5459i | −32.0000 | + | 55.4256i | 226.101 | + | 391.619i | −76.4225 | + | 132.368i | −488.812 | −512.000 | 910.988 | − | 1577.88i | −1808.81 | + | 3132.95i | |||||||||||||||||||||||||||||||||||||||||||||||
| 7.6 | 4.00000 | + | 6.92820i | 36.4887 | + | 63.2003i | −32.0000 | + | 55.4256i | −170.433 | − | 295.199i | −291.910 | + | 505.603i | −1668.31 | −512.000 | −1569.35 | + | 2718.20i | 1363.47 | − | 2361.59i | |||||||||||||||||||||||||||||||||||||||||||||||
| 7.7 | 4.00000 | + | 6.92820i | 39.6605 | + | 68.6939i | −32.0000 | + | 55.4256i | 99.4859 | + | 172.315i | −317.284 | + | 549.551i | 1591.34 | −512.000 | −2052.40 | + | 3554.87i | −795.887 | + | 1378.52i | |||||||||||||||||||||||||||||||||||||||||||||||
| 11.1 | 4.00000 | − | 6.92820i | −36.1624 | + | 62.6352i | −32.0000 | − | 55.4256i | −36.8530 | + | 63.8313i | 289.300 | + | 501.082i | 866.283 | −512.000 | −1521.94 | − | 2636.09i | 294.824 | + | 510.650i | |||||||||||||||||||||||||||||||||||||||||||||||
| 11.2 | 4.00000 | − | 6.92820i | −24.5226 | + | 42.4744i | −32.0000 | − | 55.4256i | 81.5095 | − | 141.179i | 196.181 | + | 339.795i | −230.715 | −512.000 | −109.217 | − | 189.169i | −652.076 | − | 1129.43i | |||||||||||||||||||||||||||||||||||||||||||||||
| 11.3 | 4.00000 | − | 6.92820i | −4.67249 | + | 8.09299i | −32.0000 | − | 55.4256i | −40.1604 | + | 69.5598i | 37.3799 | + | 64.7439i | −541.879 | −512.000 | 1049.84 | + | 1818.37i | 321.283 | + | 556.478i | |||||||||||||||||||||||||||||||||||||||||||||||
| 11.4 | 4.00000 | − | 6.92820i | 7.15555 | − | 12.3938i | −32.0000 | − | 55.4256i | −222.650 | + | 385.641i | −57.2444 | − | 99.1503i | 1436.09 | −512.000 | 991.096 | + | 1716.63i | 1781.20 | + | 3085.13i | |||||||||||||||||||||||||||||||||||||||||||||||
| 11.5 | 4.00000 | − | 6.92820i | 9.55281 | − | 16.5459i | −32.0000 | − | 55.4256i | 226.101 | − | 391.619i | −76.4225 | − | 132.368i | −488.812 | −512.000 | 910.988 | + | 1577.88i | −1808.81 | − | 3132.95i | |||||||||||||||||||||||||||||||||||||||||||||||
| 11.6 | 4.00000 | − | 6.92820i | 36.4887 | − | 63.2003i | −32.0000 | − | 55.4256i | −170.433 | + | 295.199i | −291.910 | − | 505.603i | −1668.31 | −512.000 | −1569.35 | − | 2718.20i | 1363.47 | + | 2361.59i | |||||||||||||||||||||||||||||||||||||||||||||||
| 11.7 | 4.00000 | − | 6.92820i | 39.6605 | − | 68.6939i | −32.0000 | − | 55.4256i | 99.4859 | − | 172.315i | −317.284 | − | 549.551i | 1591.34 | −512.000 | −2052.40 | − | 3554.87i | −795.887 | − | 1378.52i | |||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 38.8.c.b | ✓ | 14 |
| 19.c | even | 3 | 1 | inner | 38.8.c.b | ✓ | 14 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 38.8.c.b | ✓ | 14 | 1.a | even | 1 | 1 | trivial |
| 38.8.c.b | ✓ | 14 | 19.c | even | 3 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{14} - 55 T_{3}^{13} + 11468 T_{3}^{12} - 308269 T_{3}^{11} + 75629549 T_{3}^{10} + \cdots + 27\!\cdots\!25 \)
acting on \(S_{8}^{\mathrm{new}}(38, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 8 T + 64)^{7} \)
|
| $3$ |
\( T^{14} + \cdots + 27\!\cdots\!25 \)
|
| $5$ |
\( T^{14} + \cdots + 17\!\cdots\!00 \)
|
| $7$ |
\( (T^{7} + \cdots - 20\!\cdots\!16)^{2} \)
|
| $11$ |
\( (T^{7} + \cdots + 11\!\cdots\!40)^{2} \)
|
| $13$ |
\( T^{14} + \cdots + 27\!\cdots\!44 \)
|
| $17$ |
\( T^{14} + \cdots + 41\!\cdots\!00 \)
|
| $19$ |
\( T^{14} + \cdots + 45\!\cdots\!79 \)
|
| $23$ |
\( T^{14} + \cdots + 31\!\cdots\!16 \)
|
| $29$ |
\( T^{14} + \cdots + 46\!\cdots\!00 \)
|
| $31$ |
\( (T^{7} + \cdots - 36\!\cdots\!16)^{2} \)
|
| $37$ |
\( (T^{7} + \cdots + 64\!\cdots\!52)^{2} \)
|
| $41$ |
\( T^{14} + \cdots + 30\!\cdots\!89 \)
|
| $43$ |
\( T^{14} + \cdots + 38\!\cdots\!64 \)
|
| $47$ |
\( T^{14} + \cdots + 84\!\cdots\!00 \)
|
| $53$ |
\( T^{14} + \cdots + 43\!\cdots\!00 \)
|
| $59$ |
\( T^{14} + \cdots + 27\!\cdots\!49 \)
|
| $61$ |
\( T^{14} + \cdots + 22\!\cdots\!24 \)
|
| $67$ |
\( T^{14} + \cdots + 28\!\cdots\!25 \)
|
| $71$ |
\( T^{14} + \cdots + 95\!\cdots\!36 \)
|
| $73$ |
\( T^{14} + \cdots + 13\!\cdots\!25 \)
|
| $79$ |
\( T^{14} + \cdots + 25\!\cdots\!00 \)
|
| $83$ |
\( (T^{7} + \cdots - 23\!\cdots\!32)^{2} \)
|
| $89$ |
\( T^{14} + \cdots + 53\!\cdots\!00 \)
|
| $97$ |
\( T^{14} + \cdots + 73\!\cdots\!25 \)
|
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