Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [361,6,Mod(1,361)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(361, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("361.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 361 = 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 361.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: | \(57.8985589525\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 7 x^{15} - 381 x^{14} + 2556 x^{13} + 56941 x^{12} - 359421 x^{11} - 4285777 x^{10} + \cdots - 78052705280 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{10}\cdot 19^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 7 x^{15} - 381 x^{14} + 2556 x^{13} + 56941 x^{12} - 359421 x^{11} - 4285777 x^{10} + \cdots - 78052705280 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 51 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 21\!\cdots\!23 \nu^{15} + \cdots + 10\!\cdots\!16 ) / 15\!\cdots\!24 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 79\!\cdots\!55 \nu^{15} + \cdots + 68\!\cdots\!56 ) / 33\!\cdots\!28 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 20\!\cdots\!83 \nu^{15} + \cdots - 10\!\cdots\!96 ) / 32\!\cdots\!64 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 60\!\cdots\!37 \nu^{15} + \cdots - 21\!\cdots\!24 ) / 92\!\cdots\!48 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 12\!\cdots\!77 \nu^{15} + \cdots + 81\!\cdots\!76 ) / 17\!\cdots\!84 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 69\!\cdots\!17 \nu^{15} + \cdots - 10\!\cdots\!00 ) / 92\!\cdots\!48 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 32\!\cdots\!57 \nu^{15} + \cdots - 12\!\cdots\!20 ) / 27\!\cdots\!44 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 90\!\cdots\!43 \nu^{15} + \cdots + 13\!\cdots\!60 ) / 50\!\cdots\!08 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 45\!\cdots\!37 \nu^{15} + \cdots - 27\!\cdots\!92 ) / 16\!\cdots\!32 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 18\!\cdots\!35 \nu^{15} + \cdots - 20\!\cdots\!80 ) / 55\!\cdots\!88 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 12\!\cdots\!33 \nu^{15} + \cdots - 34\!\cdots\!56 ) / 34\!\cdots\!68 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 26\!\cdots\!13 \nu^{15} + \cdots - 12\!\cdots\!56 ) / 55\!\cdots\!88 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 38\!\cdots\!51 \nu^{15} + \cdots + 10\!\cdots\!08 ) / 62\!\cdots\!76 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 51 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{13} + \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + 4\beta_{6} + \beta_{5} - 2\beta_{4} + 84\beta _1 + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{15} - \beta_{14} - \beta_{13} - 5 \beta_{12} - \beta_{11} + 9 \beta_{10} - 7 \beta_{8} + \cdots + 4293 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 15 \beta_{15} + 9 \beta_{14} + 139 \beta_{13} - 12 \beta_{12} + 133 \beta_{11} - 104 \beta_{10} + \cdots - 118 \)
|
| \(\nu^{6}\) | \(=\) |
\( 151 \beta_{15} - 171 \beta_{14} - 276 \beta_{13} - 922 \beta_{12} - 214 \beta_{11} + 1293 \beta_{10} + \cdots + 422589 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 2564 \beta_{15} + 1638 \beta_{14} + 17576 \beta_{13} - 2031 \beta_{12} + 16026 \beta_{11} + \cdots - 154028 \)
|
| \(\nu^{8}\) | \(=\) |
\( 16550 \beta_{15} - 25312 \beta_{14} - 52786 \beta_{13} - 137490 \beta_{12} - 32126 \beta_{11} + \cdots + 44235179 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 333110 \beta_{15} + 232872 \beta_{14} + 2160127 \beta_{13} - 262650 \beta_{12} + 1915811 \beta_{11} + \cdots - 35368760 \)
|
| \(\nu^{10}\) | \(=\) |
\( 1566295 \beta_{15} - 3591917 \beta_{14} - 8616671 \beta_{13} - 18933483 \beta_{12} - 4305643 \beta_{11} + \cdots + 4768326461 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 39699589 \beta_{15} + 30806121 \beta_{14} + 261554457 \beta_{13} - 30471666 \beta_{12} + \cdots - 5972076930 \)
|
| \(\nu^{12}\) | \(=\) |
\( 133621389 \beta_{15} - 494003759 \beta_{14} - 1288137026 \beta_{13} - 2496563648 \beta_{12} + \cdots + 522757747045 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 4570365002 \beta_{15} + 3959024814 \beta_{14} + 31422303758 \beta_{13} - 3277139565 \beta_{12} + \cdots - 883936501560 \)
|
| \(\nu^{14}\) | \(=\) |
\( 10116876980 \beta_{15} - 66118122836 \beta_{14} - 182149013408 \beta_{13} - 320137370920 \beta_{12} + \cdots + 57965479655891 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 517691117388 \beta_{15} + 501470327976 \beta_{14} + 3760532123173 \beta_{13} - 326493231912 \beta_{12} + \cdots - 122046825234756 \)
|
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 |
|
−10.9400 | −17.6324 | 87.6826 | 71.8268 | 192.898 | −90.3231 | −609.166 | 67.9010 | −785.783 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −10.2124 | −8.67397 | 72.2934 | 37.2239 | 88.5822 | 128.851 | −411.493 | −167.762 | −380.146 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −6.93607 | 22.4941 | 16.1091 | −10.9029 | −156.021 | −255.766 | 110.220 | 262.986 | 75.6236 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −6.58829 | 28.5345 | 11.4056 | 76.6745 | −187.994 | −61.3593 | 135.682 | 571.218 | −505.154 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −5.73991 | −0.943791 | 0.946609 | −50.1544 | 5.41728 | 36.9062 | 178.244 | −242.109 | 287.882 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −3.71840 | 3.15947 | −18.1735 | −30.7496 | −11.7482 | 187.170 | 186.565 | −233.018 | 114.339 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −3.11632 | −26.1504 | −22.2886 | 22.5136 | 81.4930 | −157.965 | 169.180 | 440.845 | −70.1596 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.140227 | −27.0853 | −31.9803 | 42.3956 | −3.79808 | 218.239 | −8.97174 | 490.614 | 5.94499 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.26082 | 2.22509 | −26.8887 | 32.4805 | 5.03053 | −114.973 | −133.137 | −238.049 | 73.4326 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 3.63239 | 16.7281 | −18.8057 | 78.6998 | 60.7630 | −5.30898 | −184.546 | 36.8292 | 285.869 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 4.63983 | −7.54219 | −10.4719 | −52.9999 | −34.9945 | −50.3181 | −197.063 | −186.115 | −245.911 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 5.15549 | 26.1612 | −5.42092 | −85.4800 | 134.874 | 179.827 | −192.923 | 441.406 | −440.691 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 8.10664 | −17.2130 | 33.7177 | −80.5566 | −139.540 | −171.666 | 13.9247 | 53.2886 | −653.044 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 9.66398 | −23.1467 | 61.3925 | 85.2837 | −223.690 | 186.947 | 284.048 | 292.771 | 824.179 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 9.98694 | 7.57182 | 67.7390 | 95.7922 | 75.6193 | 103.834 | 356.924 | −185.668 | 956.671 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 10.6650 | 22.5136 | 81.7431 | −73.0473 | 240.108 | −18.0952 | 530.512 | 263.862 | −779.053 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(19\) | \( -1 \) |
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 | 361.6.a.j | yes | 16 |
| 19.b | odd | 2 | 1 | 361.6.a.i | ✓ | 16 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 361.6.a.i | ✓ | 16 | 19.b | odd | 2 | 1 | |
| 361.6.a.j | yes | 16 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} - 7 T_{2}^{15} - 381 T_{2}^{14} + 2556 T_{2}^{13} + 56941 T_{2}^{12} - 359421 T_{2}^{11} + \cdots - 78052705280 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(361))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} + \cdots - 78052705280 \)
|
| $3$ |
\( T^{16} + \cdots + 10\!\cdots\!24 \)
|
| $5$ |
\( T^{16} + \cdots - 18\!\cdots\!36 \)
|
| $7$ |
\( T^{16} + \cdots - 14\!\cdots\!96 \)
|
| $11$ |
\( T^{16} + \cdots + 88\!\cdots\!36 \)
|
| $13$ |
\( T^{16} + \cdots - 31\!\cdots\!25 \)
|
| $17$ |
\( T^{16} + \cdots + 48\!\cdots\!36 \)
|
| $19$ |
\( T^{16} \)
|
| $23$ |
\( T^{16} + \cdots - 54\!\cdots\!80 \)
|
| $29$ |
\( T^{16} + \cdots + 12\!\cdots\!55 \)
|
| $31$ |
\( T^{16} + \cdots + 19\!\cdots\!80 \)
|
| $37$ |
\( T^{16} + \cdots - 72\!\cdots\!31 \)
|
| $41$ |
\( T^{16} + \cdots + 57\!\cdots\!75 \)
|
| $43$ |
\( T^{16} + \cdots - 69\!\cdots\!64 \)
|
| $47$ |
\( T^{16} + \cdots - 18\!\cdots\!16 \)
|
| $53$ |
\( T^{16} + \cdots - 22\!\cdots\!61 \)
|
| $59$ |
\( T^{16} + \cdots - 38\!\cdots\!80 \)
|
| $61$ |
\( T^{16} + \cdots - 78\!\cdots\!31 \)
|
| $67$ |
\( T^{16} + \cdots - 29\!\cdots\!44 \)
|
| $71$ |
\( T^{16} + \cdots + 40\!\cdots\!36 \)
|
| $73$ |
\( T^{16} + \cdots + 95\!\cdots\!21 \)
|
| $79$ |
\( T^{16} + \cdots + 57\!\cdots\!20 \)
|
| $83$ |
\( T^{16} + \cdots - 18\!\cdots\!80 \)
|
| $89$ |
\( T^{16} + \cdots - 33\!\cdots\!75 \)
|
| $97$ |
\( T^{16} + \cdots + 29\!\cdots\!69 \)
|
| show more | |
| show less |