Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [464,2,Mod(191,464)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(464, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 0, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("464.191");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 464 = 2^{4} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 464.k (of order \(4\), 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: | \(3.70505865379\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(10\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 6 x^{17} - 6 x^{16} - 2 x^{15} + 18 x^{14} + 42 x^{13} + 9 x^{12} - 30 x^{11} - 142 x^{10} + \cdots + 1024 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{17} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{20} - 6 x^{17} - 6 x^{16} - 2 x^{15} + 18 x^{14} + 42 x^{13} + 9 x^{12} - 30 x^{11} - 142 x^{10} + \cdots + 1024 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - \nu^{18} + 6 \nu^{15} + 6 \nu^{14} + 2 \nu^{13} - 18 \nu^{12} - 42 \nu^{11} - 9 \nu^{10} + \cdots + 768 \nu ) / 256 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 33 \nu^{19} - 4 \nu^{18} + 124 \nu^{17} + 310 \nu^{16} + 78 \nu^{15} - 782 \nu^{14} - 1682 \nu^{13} + \cdots - 32768 ) / 3072 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 5 \nu^{19} - 62 \nu^{18} - 48 \nu^{17} + 70 \nu^{16} + 386 \nu^{15} + 510 \nu^{14} - 110 \nu^{13} + \cdots - 6656 ) / 1536 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 129 \nu^{19} - 50 \nu^{18} - 52 \nu^{17} + 782 \nu^{16} + 1506 \nu^{15} + 2054 \nu^{14} + \cdots + 184832 ) / 3072 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 18 \nu^{19} + 29 \nu^{18} + 32 \nu^{17} - 72 \nu^{16} - 234 \nu^{15} - 354 \nu^{14} - 78 \nu^{13} + \cdots - 14592 ) / 512 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 37 \nu^{19} - 82 \nu^{18} - 124 \nu^{17} + 134 \nu^{16} + 666 \nu^{15} + 1150 \nu^{14} + \cdots + 35328 ) / 1024 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 53 \nu^{19} - 62 \nu^{18} + 4 \nu^{17} + 342 \nu^{16} + 674 \nu^{15} + 486 \nu^{14} - 678 \nu^{13} + \cdots + 24064 ) / 1024 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 36 \nu^{19} - 51 \nu^{18} - 84 \nu^{17} + 84 \nu^{16} + 346 \nu^{15} + 642 \nu^{14} + 366 \nu^{13} + \cdots + 26368 ) / 512 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 106 \nu^{19} - 173 \nu^{18} - 272 \nu^{17} + 264 \nu^{16} + 1114 \nu^{15} + 2002 \nu^{14} + \cdots + 71424 ) / 1536 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 135 \nu^{19} - 169 \nu^{18} - 248 \nu^{17} + 478 \nu^{16} + 1536 \nu^{15} + 2500 \nu^{14} + \cdots + 124672 ) / 1536 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 103 \nu^{19} + 144 \nu^{18} + 204 \nu^{17} - 282 \nu^{16} - 954 \nu^{15} - 1590 \nu^{14} + \cdots - 67584 ) / 1024 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 103 \nu^{19} - 150 \nu^{18} - 236 \nu^{17} + 258 \nu^{16} + 990 \nu^{15} + 1786 \nu^{14} + \cdots + 72192 ) / 1024 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 25 \nu^{19} - 42 \nu^{18} - 66 \nu^{17} + 62 \nu^{16} + 282 \nu^{15} + 506 \nu^{14} + 270 \nu^{13} + \cdots + 18432 ) / 256 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 185 \nu^{19} + 245 \nu^{18} + 288 \nu^{17} - 778 \nu^{16} - 2132 \nu^{15} - 2976 \nu^{14} + \cdots - 133888 ) / 1536 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 137 \nu^{19} - 166 \nu^{18} - 252 \nu^{17} + 446 \nu^{16} + 1354 \nu^{15} + 2302 \nu^{14} + \cdots + 115200 ) / 1024 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 143 \nu^{19} + 194 \nu^{18} + 244 \nu^{17} - 578 \nu^{16} - 1686 \nu^{15} - 2498 \nu^{14} + \cdots - 112128 ) / 1024 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 242 \nu^{19} - 297 \nu^{18} - 380 \nu^{17} + 968 \nu^{16} + 2642 \nu^{15} + 3922 \nu^{14} + \cdots + 196352 ) / 1536 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 207 \nu^{19} - 316 \nu^{18} - 468 \nu^{17} + 554 \nu^{16} + 2130 \nu^{15} + 3646 \nu^{14} + \cdots + 141312 ) / 1024 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 853 \nu^{19} - 1114 \nu^{18} - 1428 \nu^{17} + 3158 \nu^{16} + 9034 \nu^{15} + 13278 \nu^{14} + \cdots + 599552 ) / 3072 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{18} - 2\beta_{11} + \beta_{6} + \beta_{5} + \beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{19} - 2\beta_{17} - \beta_{13} - \beta_{11} + \beta_{4} + \beta_{3} - \beta_{2} - 2\beta_1 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{18} - 2\beta_{15} - 2\beta_{13} - 2\beta_{10} + 4\beta_{8} + \beta_{6} - \beta_{5} + 2\beta_{4} - \beta _1 + 4 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - \beta_{19} + 2 \beta_{16} + 2 \beta_{15} + \beta_{13} - \beta_{11} + 2 \beta_{7} + 2 \beta_{6} + \cdots + 4 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2 \beta_{19} - 3 \beta_{18} - 2 \beta_{17} + 2 \beta_{14} - 4 \beta_{11} + 2 \beta_{9} - 2 \beta_{8} + \cdots + 2 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 7 \beta_{19} - 4 \beta_{18} - 6 \beta_{17} - 2 \beta_{15} + 6 \beta_{14} - \beta_{13} + 4 \beta_{12} + \cdots + 2 \beta_1 ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 7 \beta_{18} - 2 \beta_{17} + 6 \beta_{16} - 2 \beta_{15} - 6 \beta_{13} + 6 \beta_{12} + 2 \beta_{10} + \cdots - 8 ) / 4 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 3 \beta_{19} + 6 \beta_{16} + 4 \beta_{15} + 4 \beta_{14} - 9 \beta_{13} + 9 \beta_{11} + 4 \beta_{10} + \cdots + 32 ) / 4 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 10 \beta_{19} - 13 \beta_{18} - 10 \beta_{16} + 30 \beta_{14} + 10 \beta_{12} - 16 \beta_{11} + \cdots + 10 ) / 4 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 11 \beta_{19} - 12 \beta_{18} - 34 \beta_{17} - 10 \beta_{14} - 7 \beta_{13} + 36 \beta_{12} + \cdots + 6 \beta_1 ) / 4 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 13 \beta_{18} + 24 \beta_{17} + 16 \beta_{16} - 34 \beta_{15} - 58 \beta_{13} + 16 \beta_{12} + \cdots + 20 ) / 4 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - \beta_{19} - 6 \beta_{16} + 10 \beta_{15} + 44 \beta_{14} - 7 \beta_{13} + 7 \beta_{11} + 44 \beta_{10} + \cdots + 140 ) / 4 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 38 \beta_{19} - 39 \beta_{18} - 34 \beta_{17} - 48 \beta_{16} - 30 \beta_{14} + 48 \beta_{12} + \cdots + 42 ) / 4 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 27 \beta_{19} - 100 \beta_{18} - 6 \beta_{17} - 42 \beta_{15} + 30 \beta_{14} - 53 \beta_{13} + \cdots - 86 \beta_1 ) / 4 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 27 \beta_{18} - 18 \beta_{17} + 54 \beta_{16} - 162 \beta_{15} + 18 \beta_{13} + 54 \beta_{12} + \cdots + 80 ) / 4 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( - 81 \beta_{19} + 54 \beta_{16} + 84 \beta_{15} - 60 \beta_{14} - 53 \beta_{13} + 53 \beta_{11} + \cdots + 96 ) / 4 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( - 38 \beta_{19} - 129 \beta_{18} + 56 \beta_{17} - 138 \beta_{16} + 390 \beta_{14} + 138 \beta_{12} + \cdots - 102 ) / 4 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( 223 \beta_{19} - 476 \beta_{18} - 402 \beta_{17} - 160 \beta_{15} - 2 \beta_{14} + 269 \beta_{13} + \cdots + 14 \beta_1 ) / 4 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( 169 \beta_{18} + 384 \beta_{17} + 472 \beta_{16} - 322 \beta_{15} - 146 \beta_{13} + 472 \beta_{12} + \cdots - 916 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/464\mathbb{Z}\right)^\times\).
| \(n\) | \(117\) | \(175\) | \(321\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(\beta_{8}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 191.1 |
|
0 | −1.92975 | − | 1.92975i | 0 | 2.39709i | 0 | − | 1.01973i | 0 | 4.44790i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 191.2 | 0 | −1.52694 | − | 1.52694i | 0 | − | 1.92027i | 0 | − | 3.76421i | 0 | 1.66310i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 191.3 | 0 | −1.47958 | − | 1.47958i | 0 | 3.06966i | 0 | 0.851201i | 0 | 1.37831i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 191.4 | 0 | −1.30508 | − | 1.30508i | 0 | − | 3.62458i | 0 | 1.31987i | 0 | 0.406445i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 191.5 | 0 | −0.228309 | − | 0.228309i | 0 | 0.0781024i | 0 | 3.21315i | 0 | − | 2.89575i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 191.6 | 0 | 0.228309 | + | 0.228309i | 0 | 0.0781024i | 0 | − | 3.21315i | 0 | − | 2.89575i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 191.7 | 0 | 1.30508 | + | 1.30508i | 0 | − | 3.62458i | 0 | − | 1.31987i | 0 | 0.406445i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 191.8 | 0 | 1.47958 | + | 1.47958i | 0 | 3.06966i | 0 | − | 0.851201i | 0 | 1.37831i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 191.9 | 0 | 1.52694 | + | 1.52694i | 0 | − | 1.92027i | 0 | 3.76421i | 0 | 1.66310i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 191.10 | 0 | 1.92975 | + | 1.92975i | 0 | 2.39709i | 0 | 1.01973i | 0 | 4.44790i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 447.1 | 0 | −1.92975 | + | 1.92975i | 0 | − | 2.39709i | 0 | 1.01973i | 0 | − | 4.44790i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 447.2 | 0 | −1.52694 | + | 1.52694i | 0 | 1.92027i | 0 | 3.76421i | 0 | − | 1.66310i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 447.3 | 0 | −1.47958 | + | 1.47958i | 0 | − | 3.06966i | 0 | − | 0.851201i | 0 | − | 1.37831i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 447.4 | 0 | −1.30508 | + | 1.30508i | 0 | 3.62458i | 0 | − | 1.31987i | 0 | − | 0.406445i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 447.5 | 0 | −0.228309 | + | 0.228309i | 0 | − | 0.0781024i | 0 | − | 3.21315i | 0 | 2.89575i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 447.6 | 0 | 0.228309 | − | 0.228309i | 0 | − | 0.0781024i | 0 | 3.21315i | 0 | 2.89575i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 447.7 | 0 | 1.30508 | − | 1.30508i | 0 | 3.62458i | 0 | 1.31987i | 0 | − | 0.406445i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 447.8 | 0 | 1.47958 | − | 1.47958i | 0 | − | 3.06966i | 0 | 0.851201i | 0 | − | 1.37831i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 447.9 | 0 | 1.52694 | − | 1.52694i | 0 | 1.92027i | 0 | − | 3.76421i | 0 | − | 1.66310i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 447.10 | 0 | 1.92975 | − | 1.92975i | 0 | − | 2.39709i | 0 | − | 1.01973i | 0 | − | 4.44790i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 29.c | odd | 4 | 1 | inner |
| 116.e | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 464.2.k.c | ✓ | 20 |
| 4.b | odd | 2 | 1 | inner | 464.2.k.c | ✓ | 20 |
| 29.c | odd | 4 | 1 | inner | 464.2.k.c | ✓ | 20 |
| 116.e | even | 4 | 1 | inner | 464.2.k.c | ✓ | 20 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 464.2.k.c | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 464.2.k.c | ✓ | 20 | 4.b | odd | 2 | 1 | inner |
| 464.2.k.c | ✓ | 20 | 29.c | odd | 4 | 1 | inner |
| 464.2.k.c | ✓ | 20 | 116.e | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{20} + 108T_{3}^{16} + 3806T_{3}^{12} + 54336T_{3}^{8} + 268897T_{3}^{4} + 2916 \)
acting on \(S_{2}^{\mathrm{new}}(464, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} \)
|
| $3$ |
\( T^{20} + 108 T^{16} + \cdots + 2916 \)
|
| $5$ |
\( (T^{10} + 32 T^{8} + \cdots + 16)^{2} \)
|
| $7$ |
\( (T^{10} + 28 T^{8} + \cdots + 192)^{2} \)
|
| $11$ |
\( T^{20} + 1116 T^{16} + \cdots + 36 \)
|
| $13$ |
\( (T^{10} + 84 T^{8} + \cdots + 104976)^{2} \)
|
| $17$ |
\( (T^{10} + 2 T^{9} + \cdots + 288)^{2} \)
|
| $19$ |
\( T^{20} + 2172 T^{16} + \cdots + 9216 \)
|
| $23$ |
\( (T^{10} + 112 T^{8} + \cdots + 3195072)^{2} \)
|
| $29$ |
\( (T^{10} + 2 T^{9} + \cdots + 20511149)^{2} \)
|
| $31$ |
\( T^{20} + \cdots + 372790840356 \)
|
| $37$ |
\( (T^{10} + 10 T^{9} + \cdots + 41472)^{2} \)
|
| $41$ |
\( (T^{10} + 2 T^{9} + \cdots + 16450848)^{2} \)
|
| $43$ |
\( T^{20} + \cdots + 19\!\cdots\!16 \)
|
| $47$ |
\( T^{20} + \cdots + 652077796642596 \)
|
| $53$ |
\( (T^{5} - 12 T^{4} + \cdots + 9302)^{4} \)
|
| $59$ |
\( (T^{10} + 304 T^{8} + \cdots + 46287552)^{2} \)
|
| $61$ |
\( (T^{10} - 2 T^{9} + \cdots + 1316050208)^{2} \)
|
| $67$ |
\( (T^{10} - 488 T^{8} + \cdots - 875999232)^{2} \)
|
| $71$ |
\( (T^{10} - 356 T^{8} + \cdots - 1625088)^{2} \)
|
| $73$ |
\( (T^{10} + 10 T^{9} + \cdots + 2580992)^{2} \)
|
| $79$ |
\( T^{20} + \cdots + 18\!\cdots\!56 \)
|
| $83$ |
\( (T^{10} + 272 T^{8} + \cdots + 1728)^{2} \)
|
| $89$ |
\( (T^{10} - 18 T^{9} + \cdots + 40443955232)^{2} \)
|
| $97$ |
\( (T^{10} + 18 T^{9} + \cdots + 2562848)^{2} \)
|
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