Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1248,2,Mod(287,1248)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1248, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1248.287");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1248 = 2^{5} \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1248.d (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: | \(9.96533017226\) |
| 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: |
\( x^{20} - 4 x^{17} + 18 x^{16} + 8 x^{14} - 8 x^{13} + 241 x^{12} - 44 x^{11} - 112 x^{10} - 132 x^{9} + \cdots + 59049 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{23}\cdot 3^{2} \) |
| 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
\( x^{20} - 4 x^{17} + 18 x^{16} + 8 x^{14} - 8 x^{13} + 241 x^{12} - 44 x^{11} - 112 x^{10} - 132 x^{9} + \cdots + 59049 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 8159 \nu^{19} - 33927 \nu^{18} - 580797 \nu^{17} - 528599 \nu^{16} - 994557 \nu^{15} + \cdots + 1967965389 ) / 452236608 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 13 \nu^{19} - 960 \nu^{18} + 297 \nu^{17} + 677 \nu^{16} + 2049 \nu^{15} - 10692 \nu^{14} + \cdots + 2263545 ) / 629856 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 5069 \nu^{19} + 186150 \nu^{18} - 75519 \nu^{17} - 367957 \nu^{16} - 538977 \nu^{15} + \cdots - 1084828545 ) / 226118304 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3427 \nu^{19} + 10773 \nu^{18} - 17721 \nu^{17} + 193085 \nu^{16} + 275607 \nu^{15} + \cdots + 1483842321 ) / 150745536 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 13859 \nu^{19} + 70893 \nu^{18} - 143667 \nu^{17} + 99085 \nu^{16} + 523575 \nu^{15} + \cdots + 1431091881 ) / 226118304 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 16682 \nu^{19} - 78003 \nu^{18} + 168516 \nu^{17} - 358895 \nu^{16} - 181026 \nu^{15} + \cdots - 3213151335 ) / 226118304 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 55979 \nu^{19} + 103266 \nu^{18} + 107775 \nu^{17} - 338656 \nu^{16} - 1426923 \nu^{15} + \cdots - 3770869140 ) / 452236608 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 56783 \nu^{19} - 213714 \nu^{18} + 169983 \nu^{17} - 588484 \nu^{16} - 1033695 \nu^{15} + \cdots - 4315300920 ) / 452236608 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 18533 \nu^{19} + 85695 \nu^{18} - 38367 \nu^{17} + 192284 \nu^{16} - 268053 \nu^{15} + \cdots + 1146810312 ) / 113059152 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 43340 \nu^{19} + 108759 \nu^{18} - 158616 \nu^{17} + 282157 \nu^{16} + 1071168 \nu^{15} + \cdots + 2395795077 ) / 226118304 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 46841 \nu^{19} - 434160 \nu^{18} + 89955 \nu^{17} + 9272 \nu^{16} - 387513 \nu^{15} + \cdots - 2619019980 ) / 226118304 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 47653 \nu^{19} - 32784 \nu^{18} + 37413 \nu^{17} - 62794 \nu^{16} + 829725 \nu^{15} + \cdots + 94124106 ) / 226118304 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 105721 \nu^{19} - 26730 \nu^{18} + 179307 \nu^{17} + 679472 \nu^{16} + 2605977 \nu^{15} + \cdots + 2490529356 ) / 452236608 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 88 \nu^{19} + 360 \nu^{17} - 487 \nu^{16} + 612 \nu^{15} - 1440 \nu^{14} + 4808 \nu^{13} + \cdots - 5924583 ) / 314928 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 42375 \nu^{19} + 55610 \nu^{18} + 217515 \nu^{17} + 118776 \nu^{16} - 532967 \nu^{15} + \cdots - 797791356 ) / 150745536 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 2935 \nu^{19} + 576 \nu^{18} - 2343 \nu^{17} + 3775 \nu^{16} - 36315 \nu^{15} + 24924 \nu^{14} + \cdots + 439587 ) / 8374752 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 43795 \nu^{19} + 10281 \nu^{18} + 161559 \nu^{17} + 73552 \nu^{16} - 558003 \nu^{15} + \cdots - 463810212 ) / 113059152 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 23657 \nu^{19} + 34751 \nu^{18} - 14112 \nu^{17} - 59303 \nu^{16} + 273565 \nu^{15} + \cdots - 938223 ) / 37686384 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 52381 \nu^{19} + 21187 \nu^{18} - 68829 \nu^{17} - 158792 \nu^{16} - 794677 \nu^{15} + \cdots - 560703060 ) / 75372768 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{8} - \beta_{7} - \beta_{3} - \beta_{2} ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{13} + 2\beta_{10} - 2\beta_{7} - 2\beta_{5} - \beta_{4} - \beta_1 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{18} + \beta_{17} + \beta_{16} - 2 \beta_{15} - 3 \beta_{13} - 2 \beta_{12} - \beta_{11} + \cdots + 2 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{17} - 2\beta_{15} + \beta_{12} - \beta_{3} - \beta_{2} - 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 2 \beta_{19} + 6 \beta_{18} + \beta_{16} - 10 \beta_{13} + \beta_{12} + 5 \beta_{11} - 2 \beta_{10} + \cdots - 4 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -4\beta_{19} - 4\beta_{14} - 20\beta_{10} - 8\beta_{8} + 24\beta_{5} - 11\beta_{4} - 3\beta_1 ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 16 \beta_{19} + 8 \beta_{18} + 8 \beta_{16} + 16 \beta_{15} + 8 \beta_{14} + 32 \beta_{13} + 12 \beta_{12} + \cdots - 102 ) / 4 \)
|
| \(\nu^{8}\) | \(=\) |
\( - \beta_{18} - 10 \beta_{17} - 5 \beta_{16} + 10 \beta_{15} - 9 \beta_{12} - 2 \beta_{11} + 4 \beta_{9} + \cdots - 25 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 16 \beta_{19} - 25 \beta_{18} + 19 \beta_{17} + 71 \beta_{16} - 22 \beta_{15} - 8 \beta_{14} + \cdots + 40 ) / 4 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 36 \beta_{19} + 104 \beta_{14} + 186 \beta_{13} - 62 \beta_{10} + 96 \beta_{8} + 142 \beta_{7} + \cdots + 127 \beta_1 ) / 4 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 94 \beta_{19} - 42 \beta_{18} - 256 \beta_{17} - 19 \beta_{16} + 176 \beta_{15} - 168 \beta_{14} + \cdots + 754 ) / 4 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 26 \beta_{18} - 38 \beta_{17} + 18 \beta_{16} + 24 \beta_{15} - 44 \beta_{12} - 36 \beta_{11} + \cdots + 241 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 592 \beta_{19} - 372 \beta_{18} - 256 \beta_{17} - 724 \beta_{16} + 16 \beta_{15} + 248 \beta_{14} + \cdots + 220 ) / 4 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 992 \beta_{19} - 1192 \beta_{14} - 2726 \beta_{13} + 1702 \beta_{10} + 928 \beta_{8} + \cdots - 1207 \beta_1 ) / 4 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 480 \beta_{19} - 325 \beta_{18} + 2415 \beta_{17} - 357 \beta_{16} - 2862 \beta_{15} + 984 \beta_{14} + \cdots + 3770 ) / 4 \)
|
| \(\nu^{16}\) | \(=\) |
\( 384 \beta_{18} + 50 \beta_{17} - 12 \beta_{16} + 78 \beta_{15} + 535 \beta_{12} + 754 \beta_{11} + \cdots - 446 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( - 4126 \beta_{19} + 2302 \beta_{18} + 3584 \beta_{17} - 4541 \beta_{16} + 2352 \beta_{15} - 4760 \beta_{14} + \cdots - 5288 ) / 4 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( 11252 \beta_{19} + 8364 \beta_{14} + 10384 \beta_{13} - 2284 \beta_{10} - 17832 \beta_{8} + \cdots + 3307 \beta_1 ) / 4 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( - 12080 \beta_{19} - 308 \beta_{18} + 3584 \beta_{17} - 2068 \beta_{16} + 5344 \beta_{15} + \cdots - 75014 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1248\mathbb{Z}\right)^\times\).
| \(n\) | \(703\) | \(769\) | \(833\) | \(1093\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 287.1 |
|
0 | −1.72174 | − | 0.188664i | 0 | − | 0.867664i | 0 | − | 1.51615i | 0 | 2.92881 | + | 0.649664i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 287.2 | 0 | −1.72174 | + | 0.188664i | 0 | 0.867664i | 0 | 1.51615i | 0 | 2.92881 | − | 0.649664i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 287.3 | 0 | −1.62531 | − | 0.598646i | 0 | 3.43429i | 0 | − | 5.05054i | 0 | 2.28325 | + | 1.94597i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 287.4 | 0 | −1.62531 | + | 0.598646i | 0 | − | 3.43429i | 0 | 5.05054i | 0 | 2.28325 | − | 1.94597i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 287.5 | 0 | −0.736734 | − | 1.56755i | 0 | − | 2.65543i | 0 | 4.30836i | 0 | −1.91445 | + | 2.30974i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 287.6 | 0 | −0.736734 | + | 1.56755i | 0 | 2.65543i | 0 | − | 4.30836i | 0 | −1.91445 | − | 2.30974i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 287.7 | 0 | −0.461900 | − | 1.66933i | 0 | 1.99193i | 0 | − | 1.52766i | 0 | −2.57330 | + | 1.54212i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 287.8 | 0 | −0.461900 | + | 1.66933i | 0 | − | 1.99193i | 0 | 1.52766i | 0 | −2.57330 | − | 1.54212i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 287.9 | 0 | −0.371272 | − | 1.69179i | 0 | − | 3.23012i | 0 | − | 2.30164i | 0 | −2.72431 | + | 1.25623i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 287.10 | 0 | −0.371272 | + | 1.69179i | 0 | 3.23012i | 0 | 2.30164i | 0 | −2.72431 | − | 1.25623i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 287.11 | 0 | 0.371272 | − | 1.69179i | 0 | 3.23012i | 0 | − | 2.30164i | 0 | −2.72431 | − | 1.25623i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 287.12 | 0 | 0.371272 | + | 1.69179i | 0 | − | 3.23012i | 0 | 2.30164i | 0 | −2.72431 | + | 1.25623i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 287.13 | 0 | 0.461900 | − | 1.66933i | 0 | − | 1.99193i | 0 | − | 1.52766i | 0 | −2.57330 | − | 1.54212i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 287.14 | 0 | 0.461900 | + | 1.66933i | 0 | 1.99193i | 0 | 1.52766i | 0 | −2.57330 | + | 1.54212i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 287.15 | 0 | 0.736734 | − | 1.56755i | 0 | 2.65543i | 0 | 4.30836i | 0 | −1.91445 | − | 2.30974i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 287.16 | 0 | 0.736734 | + | 1.56755i | 0 | − | 2.65543i | 0 | − | 4.30836i | 0 | −1.91445 | + | 2.30974i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 287.17 | 0 | 1.62531 | − | 0.598646i | 0 | − | 3.43429i | 0 | − | 5.05054i | 0 | 2.28325 | − | 1.94597i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 287.18 | 0 | 1.62531 | + | 0.598646i | 0 | 3.43429i | 0 | 5.05054i | 0 | 2.28325 | + | 1.94597i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 287.19 | 0 | 1.72174 | − | 0.188664i | 0 | 0.867664i | 0 | − | 1.51615i | 0 | 2.92881 | − | 0.649664i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 287.20 | 0 | 1.72174 | + | 0.188664i | 0 | − | 0.867664i | 0 | 1.51615i | 0 | 2.92881 | + | 0.649664i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 4.b | odd | 2 | 1 | inner |
| 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 | 1248.2.d.d | ✓ | 20 |
| 3.b | odd | 2 | 1 | inner | 1248.2.d.d | ✓ | 20 |
| 4.b | odd | 2 | 1 | inner | 1248.2.d.d | ✓ | 20 |
| 8.b | even | 2 | 1 | 2496.2.d.q | 20 | ||
| 8.d | odd | 2 | 1 | 2496.2.d.q | 20 | ||
| 12.b | even | 2 | 1 | inner | 1248.2.d.d | ✓ | 20 |
| 24.f | even | 2 | 1 | 2496.2.d.q | 20 | ||
| 24.h | odd | 2 | 1 | 2496.2.d.q | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1248.2.d.d | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 1248.2.d.d | ✓ | 20 | 3.b | odd | 2 | 1 | inner |
| 1248.2.d.d | ✓ | 20 | 4.b | odd | 2 | 1 | inner |
| 1248.2.d.d | ✓ | 20 | 12.b | even | 2 | 1 | inner |
| 2496.2.d.q | 20 | 8.b | even | 2 | 1 | ||
| 2496.2.d.q | 20 | 8.d | odd | 2 | 1 | ||
| 2496.2.d.q | 20 | 24.f | even | 2 | 1 | ||
| 2496.2.d.q | 20 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{10} + 34T_{5}^{8} + 421T_{5}^{6} + 2276T_{5}^{4} + 4932T_{5}^{2} + 2592 \)
acting on \(S_{2}^{\mathrm{new}}(1248, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} \)
|
| $3$ |
\( T^{20} + 4 T^{18} + \cdots + 59049 \)
|
| $5$ |
\( (T^{10} + 34 T^{8} + \cdots + 2592)^{2} \)
|
| $7$ |
\( (T^{10} + 54 T^{8} + \cdots + 13456)^{2} \)
|
| $11$ |
\( (T^{2} - 8)^{10} \)
|
| $13$ |
\( (T - 1)^{20} \)
|
| $17$ |
\( (T^{10} + 102 T^{8} + \cdots + 32768)^{2} \)
|
| $19$ |
\( (T^{10} + 80 T^{8} + \cdots + 256)^{2} \)
|
| $23$ |
\( (T^{10} - 96 T^{8} + \cdots - 8192)^{2} \)
|
| $29$ |
\( (T^{2} + 8)^{10} \)
|
| $31$ |
\( (T^{10} + 192 T^{8} + \cdots + 20736)^{2} \)
|
| $37$ |
\( (T^{5} - 4 T^{4} + \cdots + 6824)^{4} \)
|
| $41$ |
\( (T^{10} + 236 T^{8} + \cdots + 158277632)^{2} \)
|
| $43$ |
\( (T^{10} + 290 T^{8} + \cdots + 32993536)^{2} \)
|
| $47$ |
\( (T^{10} - 218 T^{8} + \cdots - 16222208)^{2} \)
|
| $53$ |
\( (T^{10} + 336 T^{8} + \cdots + 2654208)^{2} \)
|
| $59$ |
\( (T^{2} - 8)^{10} \)
|
| $61$ |
\( (T^{5} + 14 T^{4} + \cdots + 91136)^{4} \)
|
| $67$ |
\( (T^{10} + 304 T^{8} + \cdots + 28047616)^{2} \)
|
| $71$ |
\( (T^{10} - 650 T^{8} + \cdots - 9590571008)^{2} \)
|
| $73$ |
\( (T^{5} - 22 T^{4} + \cdots + 35136)^{4} \)
|
| $79$ |
\( (T^{10} + 500 T^{8} + \cdots + 1230045184)^{2} \)
|
| $83$ |
\( (T^{10} - 472 T^{8} + \cdots - 10616832)^{2} \)
|
| $89$ |
\( (T^{10} + 572 T^{8} + \cdots + 4806705152)^{2} \)
|
| $97$ |
\( (T^{5} + 18 T^{4} + \cdots + 12608)^{4} \)
|
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