Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [420,2,Mod(13,420)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(420, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 0, 3, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("420.13");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 420.x (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: | \(3.35371688489\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 8x^{14} + 8x^{12} - 8x^{10} + 212x^{8} + 248x^{6} + 368x^{4} + 32x^{2} + 100 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{7} \) |
| 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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 8x^{14} + 8x^{12} - 8x^{10} + 212x^{8} + 248x^{6} + 368x^{4} + 32x^{2} + 100 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 1203 \nu^{14} + 8243 \nu^{12} - 15793 \nu^{10} + 159623 \nu^{8} - 497280 \nu^{6} + \cdots - 1940200 ) / 1276350 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 2969 \nu^{14} + 42129 \nu^{12} - 174534 \nu^{10} + 192824 \nu^{8} - 638870 \nu^{6} + \cdots + 2502950 ) / 1276350 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 570 \nu^{15} - 4712 \nu^{13} + 4682 \nu^{11} + 3835 \nu^{9} + 108472 \nu^{7} + 107896 \nu^{5} + \cdots - 194682 \nu ) / 170180 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5309 \nu^{14} - 52964 \nu^{12} + 133744 \nu^{10} - 152449 \nu^{8} + 925640 \nu^{6} + \cdots + 1778150 ) / 1276350 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2329 \nu^{14} - 16984 \nu^{12} + 3874 \nu^{10} + 8131 \nu^{8} + 466620 \nu^{6} + 946682 \nu^{4} + \cdots + 235200 ) / 425450 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 486 \nu^{15} - 18682 \nu^{14} - 26141 \nu^{13} + 150467 \nu^{12} + 200326 \nu^{11} + \cdots - 3176700 ) / 2552700 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 7803 \nu^{15} + 62803 \nu^{13} - 63288 \nu^{11} + 43563 \nu^{9} - 1587570 \nu^{7} + \cdots - 415970 \nu ) / 850900 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 11396 \nu^{14} - 96476 \nu^{12} + 144691 \nu^{10} - 248311 \nu^{8} + 2745380 \nu^{6} + \cdots - 406450 ) / 1276350 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 11316 \nu^{15} + 18682 \nu^{14} - 98651 \nu^{13} - 150467 \nu^{12} + 144631 \nu^{11} + \cdots + 3176700 ) / 2552700 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 14576 \nu^{14} - 105746 \nu^{12} + 19441 \nu^{10} + 57464 \nu^{8} + 2827460 \nu^{6} + \cdots + 3338750 ) / 1276350 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 697 \nu^{15} + 29713 \nu^{14} + 6718 \nu^{13} - 238253 \nu^{12} - 99458 \nu^{11} + 228628 \nu^{10} + \cdots + 293400 ) / 2552700 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 15502 \nu^{14} - 141197 \nu^{12} + 262642 \nu^{10} - 241137 \nu^{8} + 3173740 \nu^{6} + \cdots - 1844850 ) / 1276350 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 15299 \nu^{15} - 29713 \nu^{14} + 127039 \nu^{13} + 238253 \nu^{12} - 162509 \nu^{11} + \cdots - 293400 ) / 2552700 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 29273 \nu^{15} + 18682 \nu^{14} + 228943 \nu^{13} - 150467 \nu^{12} - 185753 \nu^{11} + \cdots + 3176700 ) / 2552700 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 25290 \nu^{15} + 29713 \nu^{14} - 200555 \nu^{13} - 238253 \nu^{12} + 167875 \nu^{11} + \cdots + 293400 ) / 2552700 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} - \beta_{9} - \beta_{8} - \beta_{5} - \beta_{2} - \beta _1 - 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{12} - \beta_{8} - \beta_{4} - \beta _1 + 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2 \beta_{14} + \beta_{12} - 2 \beta_{11} + 2 \beta_{10} - 6 \beta_{9} + \beta_{8} - 4 \beta_{7} + \cdots + 1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{12} - 2\beta_{10} - 6\beta_{8} + 7\beta_{5} - 5\beta_{4} - 2\beta_{2} - 6\beta _1 + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 14 \beta_{15} + 10 \beta_{14} + 6 \beta_{13} + \beta_{12} - 10 \beta_{11} + 17 \beta_{10} - 38 \beta_{9} + \cdots + 1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 27\beta_{12} - 16\beta_{10} - 30\beta_{8} + 38\beta_{5} - 29\beta_{4} - 3\beta_{2} - 30\beta _1 + 51 \)
|
| \(\nu^{7}\) | \(=\) |
\( 41 \beta_{15} + 18 \beta_{14} + 2 \beta_{13} - 5 \beta_{12} - 29 \beta_{11} + 54 \beta_{10} - 109 \beta_{9} + \cdots - 5 \)
|
| \(\nu^{8}\) | \(=\) |
\( 172\beta_{12} - 106\beta_{10} - 178\beta_{8} + 218\beta_{5} - 142\beta_{4} + 20\beta_{2} - 164\beta _1 + 258 \)
|
| \(\nu^{9}\) | \(=\) |
\( 155 \beta_{15} + 19 \beta_{14} + 25 \beta_{13} - \beta_{12} - 128 \beta_{11} + 340 \beta_{10} - 583 \beta_{9} + \cdots - 1 \)
|
| \(\nu^{10}\) | \(=\) |
\( 988\beta_{12} - 738\beta_{10} - 980\beta_{8} + 1402\beta_{5} - 670\beta_{4} + 248\beta_{2} - 920\beta _1 + 1320 \)
|
| \(\nu^{11}\) | \(=\) |
\( 768 \beta_{15} - 322 \beta_{14} + 248 \beta_{13} + 45 \beta_{12} - 610 \beta_{11} + 2148 \beta_{10} + \cdots + 45 \)
|
| \(\nu^{12}\) | \(=\) |
\( 5608 \beta_{12} - 4734 \beta_{10} - 5128 \beta_{8} + 8294 \beta_{5} - 3096 \beta_{4} + 2272 \beta_{2} + \cdots + 6878 \)
|
| \(\nu^{13}\) | \(=\) |
\( 3562 \beta_{15} - 4098 \beta_{14} + 1382 \beta_{13} + 279 \beta_{12} - 2738 \beta_{11} + 12839 \beta_{10} + \cdots + 279 \)
|
| \(\nu^{14}\) | \(=\) |
\( 31276 \beta_{12} - 28906 \beta_{10} - 26354 \beta_{8} + 47114 \beta_{5} - 12784 \beta_{4} + 17500 \beta_{2} + \cdots + 33556 \)
|
| \(\nu^{15}\) | \(=\) |
\( 13138 \beta_{15} - 35536 \beta_{14} + 8100 \beta_{13} + 2642 \beta_{12} - 10322 \beta_{11} + 74344 \beta_{10} + \cdots + 2642 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/420\mathbb{Z}\right)^\times\).
| \(n\) | \(211\) | \(241\) | \(281\) | \(337\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(-\beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 13.1 |
|
0 | −0.707107 | + | 0.707107i | 0 | −2.22219 | + | 0.248747i | 0 | 1.51930 | − | 2.16604i | 0 | − | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.2 | 0 | −0.707107 | + | 0.707107i | 0 | −0.0675488 | + | 2.23505i | 0 | −2.42508 | − | 1.05782i | 0 | − | 1.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.3 | 0 | −0.707107 | + | 0.707107i | 0 | 1.53758 | − | 1.62353i | 0 | −2.63522 | + | 0.235858i | 0 | − | 1.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.4 | 0 | −0.707107 | + | 0.707107i | 0 | 2.16637 | + | 0.553944i | 0 | 2.12678 | + | 1.57379i | 0 | − | 1.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.5 | 0 | 0.707107 | − | 0.707107i | 0 | −2.16637 | − | 0.553944i | 0 | −1.57379 | − | 2.12678i | 0 | − | 1.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.6 | 0 | 0.707107 | − | 0.707107i | 0 | −1.53758 | + | 1.62353i | 0 | −0.235858 | + | 2.63522i | 0 | − | 1.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.7 | 0 | 0.707107 | − | 0.707107i | 0 | 0.0675488 | − | 2.23505i | 0 | 1.05782 | + | 2.42508i | 0 | − | 1.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 13.8 | 0 | 0.707107 | − | 0.707107i | 0 | 2.22219 | − | 0.248747i | 0 | 2.16604 | − | 1.51930i | 0 | − | 1.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.1 | 0 | −0.707107 | − | 0.707107i | 0 | −2.22219 | − | 0.248747i | 0 | 1.51930 | + | 2.16604i | 0 | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.2 | 0 | −0.707107 | − | 0.707107i | 0 | −0.0675488 | − | 2.23505i | 0 | −2.42508 | + | 1.05782i | 0 | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.3 | 0 | −0.707107 | − | 0.707107i | 0 | 1.53758 | + | 1.62353i | 0 | −2.63522 | − | 0.235858i | 0 | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.4 | 0 | −0.707107 | − | 0.707107i | 0 | 2.16637 | − | 0.553944i | 0 | 2.12678 | − | 1.57379i | 0 | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.5 | 0 | 0.707107 | + | 0.707107i | 0 | −2.16637 | + | 0.553944i | 0 | −1.57379 | + | 2.12678i | 0 | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.6 | 0 | 0.707107 | + | 0.707107i | 0 | −1.53758 | − | 1.62353i | 0 | −0.235858 | − | 2.63522i | 0 | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.7 | 0 | 0.707107 | + | 0.707107i | 0 | 0.0675488 | + | 2.23505i | 0 | 1.05782 | − | 2.42508i | 0 | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.8 | 0 | 0.707107 | + | 0.707107i | 0 | 2.22219 | + | 0.248747i | 0 | 2.16604 | + | 1.51930i | 0 | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 35.f | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 420.2.x.a | ✓ | 16 |
| 3.b | odd | 2 | 1 | 1260.2.ba.b | 16 | ||
| 4.b | odd | 2 | 1 | 1680.2.cz.c | 16 | ||
| 5.b | even | 2 | 1 | 2100.2.x.d | 16 | ||
| 5.c | odd | 4 | 1 | inner | 420.2.x.a | ✓ | 16 |
| 5.c | odd | 4 | 1 | 2100.2.x.d | 16 | ||
| 7.b | odd | 2 | 1 | inner | 420.2.x.a | ✓ | 16 |
| 15.e | even | 4 | 1 | 1260.2.ba.b | 16 | ||
| 20.e | even | 4 | 1 | 1680.2.cz.c | 16 | ||
| 21.c | even | 2 | 1 | 1260.2.ba.b | 16 | ||
| 28.d | even | 2 | 1 | 1680.2.cz.c | 16 | ||
| 35.c | odd | 2 | 1 | 2100.2.x.d | 16 | ||
| 35.f | even | 4 | 1 | inner | 420.2.x.a | ✓ | 16 |
| 35.f | even | 4 | 1 | 2100.2.x.d | 16 | ||
| 105.k | odd | 4 | 1 | 1260.2.ba.b | 16 | ||
| 140.j | odd | 4 | 1 | 1680.2.cz.c | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 420.2.x.a | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 420.2.x.a | ✓ | 16 | 5.c | odd | 4 | 1 | inner |
| 420.2.x.a | ✓ | 16 | 7.b | odd | 2 | 1 | inner |
| 420.2.x.a | ✓ | 16 | 35.f | even | 4 | 1 | inner |
| 1260.2.ba.b | 16 | 3.b | odd | 2 | 1 | ||
| 1260.2.ba.b | 16 | 15.e | even | 4 | 1 | ||
| 1260.2.ba.b | 16 | 21.c | even | 2 | 1 | ||
| 1260.2.ba.b | 16 | 105.k | odd | 4 | 1 | ||
| 1680.2.cz.c | 16 | 4.b | odd | 2 | 1 | ||
| 1680.2.cz.c | 16 | 20.e | even | 4 | 1 | ||
| 1680.2.cz.c | 16 | 28.d | even | 2 | 1 | ||
| 1680.2.cz.c | 16 | 140.j | odd | 4 | 1 | ||
| 2100.2.x.d | 16 | 5.b | even | 2 | 1 | ||
| 2100.2.x.d | 16 | 5.c | odd | 4 | 1 | ||
| 2100.2.x.d | 16 | 35.c | odd | 2 | 1 | ||
| 2100.2.x.d | 16 | 35.f | even | 4 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(420, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} \)
|
| $3$ |
\( (T^{4} + 1)^{4} \)
|
| $5$ |
\( T^{16} - 8 T^{14} + \cdots + 390625 \)
|
| $7$ |
\( T^{16} + 32 T^{13} + \cdots + 5764801 \)
|
| $11$ |
\( (T^{4} - 4 T^{3} - 8 T^{2} + \cdots - 4)^{4} \)
|
| $13$ |
\( T^{16} + \cdots + 116985856 \)
|
| $17$ |
\( T^{16} + 2224 T^{12} + \cdots + 3748096 \)
|
| $19$ |
\( (T^{8} - 32 T^{6} + \cdots + 16)^{2} \)
|
| $23$ |
\( (T^{8} - 4 T^{7} + \cdots + 760384)^{2} \)
|
| $29$ |
\( (T^{8} + 96 T^{6} + \cdots + 30976)^{2} \)
|
| $31$ |
\( (T^{8} + 112 T^{6} + \cdots + 8464)^{2} \)
|
| $37$ |
\( (T^{8} - 8 T^{7} + \cdots + 7507600)^{2} \)
|
| $41$ |
\( (T^{8} + 128 T^{6} + \cdots + 270400)^{2} \)
|
| $43$ |
\( (T^{8} - 24 T^{7} + \cdots + 692224)^{2} \)
|
| $47$ |
\( T^{16} + 8704 T^{12} + \cdots + 1048576 \)
|
| $53$ |
\( (T^{8} + 20 T^{7} + \cdots + 1600)^{2} \)
|
| $59$ |
\( (T^{8} - 192 T^{6} + \cdots + 4096)^{2} \)
|
| $61$ |
\( (T^{8} + 176 T^{6} + \cdots + 173056)^{2} \)
|
| $67$ |
\( (T^{8} + 24 T^{7} + \cdots + 127599616)^{2} \)
|
| $71$ |
\( (T^{4} + 8 T^{3} + \cdots + 5164)^{4} \)
|
| $73$ |
\( T^{16} + \cdots + 133633600000000 \)
|
| $79$ |
\( (T^{8} + 384 T^{6} + \cdots + 262144)^{2} \)
|
| $83$ |
\( T^{16} + \cdots + 655360000 \)
|
| $89$ |
\( (T^{8} - 384 T^{6} + \cdots + 193600)^{2} \)
|
| $97$ |
\( T^{16} + \cdots + 744753706110976 \)
|
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