Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3360,2,Mod(2591,3360)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3360, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1, 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("3360.2591");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3360.ba (of order \(2\), degree \(1\), 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: | \(26.8297350792\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 4 x^{19} + 8 x^{18} - 20 x^{17} + 50 x^{16} - 92 x^{15} + 144 x^{14} - 244 x^{13} + \cdots + 59049 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{15} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} - 4 x^{19} + 8 x^{18} - 20 x^{17} + 50 x^{16} - 92 x^{15} + 144 x^{14} - 244 x^{13} + \cdots + 59049 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 3065 \nu^{19} - 4748 \nu^{18} + 24339 \nu^{17} - 14083 \nu^{16} + 15787 \nu^{15} + \cdots + 74775717 ) / 33014952 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 4 \nu^{19} - 13 \nu^{18} + 20 \nu^{17} - 56 \nu^{16} + 140 \nu^{15} - 218 \nu^{14} + 300 \nu^{13} + \cdots - 157464 ) / 19683 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 1046 \nu^{19} + 4016 \nu^{18} - 7894 \nu^{17} + 15751 \nu^{16} - 41938 \nu^{15} + \cdots + 50644359 ) / 2913084 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 14087 \nu^{19} - 29093 \nu^{18} + 54229 \nu^{17} - 179431 \nu^{16} + 360067 \nu^{15} + \cdots - 453201075 ) / 5826168 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 15587 \nu^{19} - 35645 \nu^{18} + 68833 \nu^{17} - 206197 \nu^{16} + 432103 \nu^{15} + \cdots - 554489793 ) / 5826168 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 10591 \nu^{19} + 23833 \nu^{18} - 42761 \nu^{17} + 135986 \nu^{16} - 287963 \nu^{15} + \cdots + 359687142 ) / 2676888 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 293635 \nu^{19} + 660817 \nu^{18} - 1211543 \nu^{17} + 3827339 \nu^{16} - 8045015 \nu^{15} + \cdots + 10175343363 ) / 66029904 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 305785 \nu^{19} - 688141 \nu^{18} + 1212575 \nu^{17} - 3888257 \nu^{16} + 8345501 \nu^{15} + \cdots - 10111609809 ) / 66029904 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 102563 \nu^{19} - 231171 \nu^{18} + 423719 \nu^{17} - 1335231 \nu^{16} + 2829207 \nu^{15} + \cdots - 3603872007 ) / 22009968 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 968387 \nu^{19} + 2225687 \nu^{18} - 3924853 \nu^{17} + 12426469 \nu^{16} + \cdots + 32540388309 ) / 198089712 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 549287 \nu^{19} - 1274081 \nu^{18} + 2313757 \nu^{17} - 7172269 \nu^{16} + 15447271 \nu^{15} + \cdots - 19060761321 ) / 99044856 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 663485 \nu^{19} + 1581179 \nu^{18} - 2792950 \nu^{17} + 8748385 \nu^{16} - 18956245 \nu^{15} + \cdots + 22939493301 ) / 99044856 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 531505 \nu^{19} + 1205809 \nu^{18} - 2179559 \nu^{17} + 6904502 \nu^{16} - 14734325 \nu^{15} + \cdots + 18199807218 ) / 49522428 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 282421 \nu^{19} - 639424 \nu^{18} + 1147703 \nu^{17} - 3678638 \nu^{16} + 7835453 \nu^{15} + \cdots - 9713481768 ) / 24761214 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 2291315 \nu^{19} + 5299301 \nu^{18} - 9366223 \nu^{17} + 29478187 \nu^{16} + \cdots + 78213648195 ) / 198089712 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 766795 \nu^{19} + 1739871 \nu^{18} - 3059047 \nu^{17} + 9863967 \nu^{16} - 20921895 \nu^{15} + \cdots + 25041020967 ) / 66029904 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 397 \nu^{19} + 894 \nu^{18} - 1593 \nu^{17} + 5183 \nu^{16} - 10945 \nu^{15} + 17338 \nu^{14} + \cdots + 13509099 ) / 33048 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 10769 \nu^{19} - 24802 \nu^{18} + 44829 \nu^{17} - 140687 \nu^{16} + 301253 \nu^{15} + \cdots - 370112571 ) / 892296 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{14} - 2\beta_{12} - \beta_{4} ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2 \beta_{19} - 2 \beta_{18} + 2 \beta_{17} - 2 \beta_{15} + 3 \beta_{14} + 2 \beta_{13} - 6 \beta_{11} + \cdots + 6 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{19} - \beta_{15} - \beta_{14} + \beta_{11} - 2 \beta_{10} - \beta_{9} - 2 \beta_{8} - \beta_{7} + \cdots + 1 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 16 \beta_{19} - 12 \beta_{18} + 10 \beta_{17} + 6 \beta_{16} - 24 \beta_{15} + 11 \beta_{14} + \cdots + 2 \beta_1 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{19} + 3 \beta_{18} + \beta_{17} - \beta_{16} + 3 \beta_{15} + 2 \beta_{14} + 2 \beta_{12} + \cdots + 5 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 40 \beta_{19} - 4 \beta_{18} + 20 \beta_{17} + 28 \beta_{16} - 20 \beta_{15} - 7 \beta_{14} - 14 \beta_{13} + \cdots + 80 ) / 4 \)
|
| \(\nu^{8}\) | \(=\) |
\( 2 \beta_{19} - 3 \beta_{18} + 2 \beta_{17} - 10 \beta_{16} - 7 \beta_{15} - 3 \beta_{14} - 12 \beta_{12} + \cdots - 21 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 18 \beta_{19} + 62 \beta_{18} + 22 \beta_{17} - 8 \beta_{16} + 70 \beta_{15} + 57 \beta_{14} - 26 \beta_{13} + \cdots - 98 ) / 4 \)
|
| \(\nu^{10}\) | \(=\) |
\( 40 \beta_{19} - 24 \beta_{18} + 38 \beta_{17} + 30 \beta_{16} - 28 \beta_{15} + 28 \beta_{14} + \cdots + 18 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 116 \beta_{19} + 36 \beta_{18} - 24 \beta_{17} - 164 \beta_{16} - 36 \beta_{15} - 43 \beta_{14} + \cdots + 752 ) / 4 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 6 \beta_{19} + 38 \beta_{18} + 74 \beta_{17} + 58 \beta_{16} + 42 \beta_{15} + 34 \beta_{14} + \cdots - 137 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 828 \beta_{19} + 524 \beta_{18} + 16 \beta_{17} + 324 \beta_{16} + 1348 \beta_{15} - 511 \beta_{14} + \cdots + 1872 ) / 4 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 320 \beta_{19} + 16 \beta_{18} - 70 \beta_{17} - 158 \beta_{16} + 164 \beta_{15} - 92 \beta_{14} + \cdots + 1070 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 2622 \beta_{19} + 1838 \beta_{18} - 474 \beta_{17} - 1468 \beta_{16} + 2582 \beta_{15} - 3171 \beta_{14} + \cdots - 6710 ) / 4 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 223 \beta_{19} - 130 \beta_{18} + 510 \beta_{17} + 682 \beta_{16} - 37 \beta_{15} + 447 \beta_{14} + \cdots + 2195 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( - 3780 \beta_{19} + 4848 \beta_{18} - 2538 \beta_{17} - 5258 \beta_{16} + 3140 \beta_{15} + \cdots + 20464 ) / 4 \)
|
| \(\nu^{18}\) | \(=\) |
\( 251 \beta_{19} + 1913 \beta_{18} + 373 \beta_{17} + 427 \beta_{16} + 21 \beta_{15} - 5082 \beta_{14} + \cdots + 6105 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( - 5180 \beta_{19} + 4520 \beta_{18} + 548 \beta_{17} - 3216 \beta_{16} + 976 \beta_{15} - 1433 \beta_{14} + \cdots + 2752 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3360\mathbb{Z}\right)^\times\).
| \(n\) | \(421\) | \(1121\) | \(1471\) | \(1921\) | \(2017\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2591.1 |
|
0 | −1.70000 | − | 0.331674i | 0 | 1.00000i | 0 | − | 1.00000i | 0 | 2.77998 | + | 1.12769i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2591.2 | 0 | −1.70000 | + | 0.331674i | 0 | − | 1.00000i | 0 | 1.00000i | 0 | 2.77998 | − | 1.12769i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2591.3 | 0 | −1.61485 | − | 0.626300i | 0 | − | 1.00000i | 0 | 1.00000i | 0 | 2.21550 | + | 2.02277i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2591.4 | 0 | −1.61485 | + | 0.626300i | 0 | 1.00000i | 0 | − | 1.00000i | 0 | 2.21550 | − | 2.02277i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2591.5 | 0 | −1.49219 | − | 0.879407i | 0 | 1.00000i | 0 | − | 1.00000i | 0 | 1.45329 | + | 2.62449i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2591.6 | 0 | −1.49219 | + | 0.879407i | 0 | − | 1.00000i | 0 | 1.00000i | 0 | 1.45329 | − | 2.62449i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2591.7 | 0 | −0.814731 | − | 1.52847i | 0 | 1.00000i | 0 | − | 1.00000i | 0 | −1.67243 | + | 2.49058i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2591.8 | 0 | −0.814731 | + | 1.52847i | 0 | − | 1.00000i | 0 | 1.00000i | 0 | −1.67243 | − | 2.49058i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2591.9 | 0 | −0.774110 | − | 1.54944i | 0 | − | 1.00000i | 0 | 1.00000i | 0 | −1.80151 | + | 2.39887i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2591.10 | 0 | −0.774110 | + | 1.54944i | 0 | 1.00000i | 0 | − | 1.00000i | 0 | −1.80151 | − | 2.39887i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2591.11 | 0 | −0.705354 | − | 1.58192i | 0 | − | 1.00000i | 0 | 1.00000i | 0 | −2.00495 | + | 2.23163i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2591.12 | 0 | −0.705354 | + | 1.58192i | 0 | 1.00000i | 0 | − | 1.00000i | 0 | −2.00495 | − | 2.23163i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2591.13 | 0 | 0.474600 | − | 1.66576i | 0 | − | 1.00000i | 0 | 1.00000i | 0 | −2.54951 | − | 1.58114i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2591.14 | 0 | 0.474600 | + | 1.66576i | 0 | 1.00000i | 0 | − | 1.00000i | 0 | −2.54951 | + | 1.58114i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2591.15 | 0 | 1.22276 | − | 1.22673i | 0 | 1.00000i | 0 | − | 1.00000i | 0 | −0.00973701 | − | 2.99998i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2591.16 | 0 | 1.22276 | + | 1.22673i | 0 | − | 1.00000i | 0 | 1.00000i | 0 | −0.00973701 | + | 2.99998i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2591.17 | 0 | 1.67479 | − | 0.441676i | 0 | − | 1.00000i | 0 | 1.00000i | 0 | 2.60984 | − | 1.47943i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2591.18 | 0 | 1.67479 | + | 0.441676i | 0 | 1.00000i | 0 | − | 1.00000i | 0 | 2.60984 | + | 1.47943i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2591.19 | 0 | 1.72909 | − | 0.101185i | 0 | − | 1.00000i | 0 | 1.00000i | 0 | 2.97952 | − | 0.349915i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2591.20 | 0 | 1.72909 | + | 0.101185i | 0 | 1.00000i | 0 | − | 1.00000i | 0 | 2.97952 | + | 0.349915i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 12.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3360.2.ba.c | ✓ | 20 |
| 3.b | odd | 2 | 1 | 3360.2.ba.d | yes | 20 | |
| 4.b | odd | 2 | 1 | 3360.2.ba.d | yes | 20 | |
| 12.b | even | 2 | 1 | inner | 3360.2.ba.c | ✓ | 20 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3360.2.ba.c | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 3360.2.ba.c | ✓ | 20 | 12.b | even | 2 | 1 | inner |
| 3360.2.ba.d | yes | 20 | 3.b | odd | 2 | 1 | |
| 3360.2.ba.d | yes | 20 | 4.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3360, [\chi])\):
|
\( T_{11}^{10} + 4 T_{11}^{9} - 58 T_{11}^{8} - 204 T_{11}^{7} + 897 T_{11}^{6} + 1984 T_{11}^{5} + \cdots - 576 \)
|
|
\( T_{23}^{10} - 4 T_{23}^{9} - 92 T_{23}^{8} + 496 T_{23}^{7} + 1184 T_{23}^{6} - 8960 T_{23}^{5} + \cdots - 2048 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} \)
|
| $3$ |
\( T^{20} + 4 T^{19} + \cdots + 59049 \)
|
| $5$ |
\( (T^{2} + 1)^{10} \)
|
| $7$ |
\( (T^{2} + 1)^{10} \)
|
| $11$ |
\( (T^{10} + 4 T^{9} + \cdots - 576)^{2} \)
|
| $13$ |
\( (T^{10} - 4 T^{9} + \cdots + 38336)^{2} \)
|
| $17$ |
\( T^{20} + \cdots + 67949891584 \)
|
| $19$ |
\( T^{20} + 160 T^{18} + \cdots + 4194304 \)
|
| $23$ |
\( (T^{10} - 4 T^{9} + \cdots - 2048)^{2} \)
|
| $29$ |
\( T^{20} + \cdots + 12131700736 \)
|
| $31$ |
\( T^{20} + 312 T^{18} + \cdots + 84934656 \)
|
| $37$ |
\( (T^{10} - 8 T^{9} + \cdots + 14336)^{2} \)
|
| $41$ |
\( T^{20} + \cdots + 82796609536 \)
|
| $43$ |
\( T^{20} + \cdots + 109650720587776 \)
|
| $47$ |
\( (T^{10} - 166 T^{8} + \cdots - 170432)^{2} \)
|
| $53$ |
\( T^{20} + \cdots + 59\!\cdots\!24 \)
|
| $59$ |
\( (T^{10} + 32 T^{9} + \cdots - 1082331136)^{2} \)
|
| $61$ |
\( (T^{10} + 32 T^{9} + \cdots + 5874688)^{2} \)
|
| $67$ |
\( T^{20} + \cdots + 143735929176064 \)
|
| $71$ |
\( (T^{10} + 20 T^{9} + \cdots + 123715584)^{2} \)
|
| $73$ |
\( (T^{10} - 4 T^{9} + \cdots + 278528)^{2} \)
|
| $79$ |
\( T^{20} + \cdots + 39\!\cdots\!36 \)
|
| $83$ |
\( (T^{10} - 12 T^{9} + \cdots + 81019904)^{2} \)
|
| $89$ |
\( T^{20} + \cdots + 20979624443904 \)
|
| $97$ |
\( (T^{10} - 44 T^{9} + \cdots - 25700288)^{2} \)
|
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