Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [490,2,Mod(97,490)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(490, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([1, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("490.97");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 490 = 2 \cdot 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 490.g (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: | \(3.91266969904\) |
| 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: |
\( x^{16} + 10x^{14} + 61x^{12} + 266x^{10} + 852x^{8} + 1438x^{6} + 1933x^{4} + 3038x^{2} + 2401 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{8} \) |
| Twist minimal: | no (minimal twist has level 70) |
| 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
\( x^{16} + 10x^{14} + 61x^{12} + 266x^{10} + 852x^{8} + 1438x^{6} + 1933x^{4} + 3038x^{2} + 2401 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 525491 \nu^{14} - 4061221 \nu^{12} - 20248352 \nu^{10} - 70335258 \nu^{8} - 168482130 \nu^{6} + \cdots - 16581649 ) / 785969800 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 34297 \nu^{14} + 364187 \nu^{12} + 2237304 \nu^{10} + 9493246 \nu^{8} + 30357550 \nu^{6} + \cdots + 53786663 ) / 34172600 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 88213 \nu^{15} + 1057753 \nu^{13} + 8202336 \nu^{11} + 41107094 \nu^{9} + 157143590 \nu^{7} + \cdots + 1691266857 \nu ) / 785969800 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 623003 \nu^{14} - 5366258 \nu^{12} - 30938216 \nu^{10} - 125675214 \nu^{8} + \cdots - 1076511772 ) / 196492450 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 4174573 \nu^{15} - 307671 \nu^{14} - 41450603 \nu^{13} - 4646621 \nu^{12} + \cdots - 8000085009 ) / 11003577200 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 4174573 \nu^{15} - 307671 \nu^{14} + 41450603 \nu^{13} - 4646621 \nu^{12} + 238976156 \nu^{11} + \cdots - 8000085009 ) / 11003577200 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 1276339 \nu^{15} - 1594698 \nu^{14} - 12140159 \nu^{13} - 13006098 \nu^{12} + \cdots - 269649107 ) / 2750894300 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 1276339 \nu^{15} - 1594698 \nu^{14} + 12140159 \nu^{13} - 13006098 \nu^{12} + 75497868 \nu^{11} + \cdots - 269649107 ) / 2750894300 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 26590 \nu^{15} - 257521 \nu^{13} - 1484692 \nu^{11} - 6079906 \nu^{9} - 17945290 \nu^{7} + \cdots - 42231189 \nu ) / 50016260 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 124363 \nu^{15} - 151501 \nu^{14} - 1257693 \nu^{13} - 1812391 \nu^{12} - 7185176 \nu^{11} + \cdots - 459236869 ) / 239208200 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 124363 \nu^{15} + 151501 \nu^{14} - 1257693 \nu^{13} + 1812391 \nu^{12} - 7185176 \nu^{11} + \cdots + 459236869 ) / 239208200 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 154017 \nu^{15} + 1553547 \nu^{13} + 9597064 \nu^{11} + 41588666 \nu^{9} + 133651610 \nu^{7} + \cdots + 238292243 \nu ) / 239208200 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 357305 \nu^{15} - 472647 \nu^{14} - 2868675 \nu^{13} - 3965353 \nu^{12} - 15737980 \nu^{11} + \cdots - 781884621 ) / 200065040 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 357305 \nu^{15} + 472647 \nu^{14} - 2868675 \nu^{13} + 3965353 \nu^{12} - 15737980 \nu^{11} + \cdots + 781884621 ) / 200065040 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 104513 \nu^{15} + 887273 \nu^{13} + 5023936 \nu^{11} + 20069294 \nu^{9} + 58201830 \nu^{7} + \cdots + 147858697 \nu ) / 34172600 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{12} - \beta_{9} + \beta_{3} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{14} + \beta_{13} - \beta_{6} - \beta_{5} - 2\beta_{4} - \beta_{2} + \beta _1 - 3 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{15} + \beta_{14} + \beta_{13} + 2\beta_{12} + 8\beta_{9} - 2\beta_{8} + 2\beta_{7} + 3\beta_{6} - 3\beta_{5} ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 5 \beta_{14} - 5 \beta_{13} - 2 \beta_{11} + 2 \beta_{10} - 2 \beta_{8} - 2 \beta_{7} - 5 \beta_{6} + \cdots - 3 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 22 \beta_{15} - 14 \beta_{14} - 14 \beta_{13} + 19 \beta_{12} - 3 \beta_{11} - 3 \beta_{10} + \cdots - 3 \beta_{3} ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -7\beta_{14} + 7\beta_{13} + 14\beta_{8} + 14\beta_{7} + 5\beta_{6} + 5\beta_{5} - 11\beta_{4} - 22\beta _1 + 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 94 \beta_{15} + 64 \beta_{14} + 64 \beta_{13} - 87 \beta_{12} - 2 \beta_{11} - 2 \beta_{10} + \cdots - 7 \beta_{3} ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 75 \beta_{14} - 75 \beta_{13} + 62 \beta_{11} - 62 \beta_{10} - 62 \beta_{8} - 62 \beta_{7} + 75 \beta_{6} + \cdots + 25 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 112 \beta_{15} - 75 \beta_{14} - 75 \beta_{13} - 112 \beta_{12} + 658 \beta_{9} + \cdots - 471 \beta_{5} ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 337 \beta_{14} + 337 \beta_{13} - 198 \beta_{11} + 198 \beta_{10} - 198 \beta_{8} - 198 \beta_{7} + \cdots + 759 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 456 \beta_{15} - 332 \beta_{14} - 332 \beta_{13} + 1215 \beta_{12} + 535 \beta_{11} + \cdots + 759 \beta_{3} ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 166 \beta_{14} + 166 \beta_{13} + 332 \beta_{8} + 332 \beta_{7} - 2530 \beta_{6} - 2530 \beta_{5} + \cdots - 3575 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 2032 \beta_{15} + 1452 \beta_{14} + 1452 \beta_{13} + 1771 \beta_{12} - 2696 \beta_{11} + \cdots - 3803 \beta_{3} ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 10647 \beta_{14} - 10647 \beta_{13} - 1244 \beta_{11} + 1244 \beta_{10} + 1244 \beta_{8} + 1244 \beta_{7} + \cdots + 16801 ) / 2 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 15030 \beta_{15} - 10647 \beta_{14} - 10647 \beta_{13} - 15030 \beta_{12} - 30632 \beta_{9} + \cdots + 21653 \beta_{5} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/490\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(197\) |
| \(\chi(n)\) | \(-1\) | \(-\beta_{9}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 97.1 |
|
−0.707107 | − | 0.707107i | −2.05532 | − | 2.05532i | 1.00000i | −0.830578 | + | 2.07609i | 2.90667i | 0 | 0.707107 | − | 0.707107i | 5.44871i | 2.05532 | − | 0.880708i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.2 | −0.707107 | − | 0.707107i | −0.830578 | − | 0.830578i | 1.00000i | −2.05532 | − | 0.880708i | 1.17462i | 0 | 0.707107 | − | 0.707107i | − | 1.62028i | 0.830578 | + | 2.07609i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.3 | −0.707107 | − | 0.707107i | 0.830578 | + | 0.830578i | 1.00000i | 2.05532 | + | 0.880708i | − | 1.17462i | 0 | 0.707107 | − | 0.707107i | − | 1.62028i | −0.830578 | − | 2.07609i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.4 | −0.707107 | − | 0.707107i | 2.05532 | + | 2.05532i | 1.00000i | 0.830578 | − | 2.07609i | − | 2.90667i | 0 | 0.707107 | − | 0.707107i | 5.44871i | −2.05532 | + | 0.880708i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.5 | 0.707107 | + | 0.707107i | −1.42962 | − | 1.42962i | 1.00000i | −0.204875 | − | 2.22666i | − | 2.02179i | 0 | −0.707107 | + | 0.707107i | 1.08763i | 1.42962 | − | 1.71936i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.6 | 0.707107 | + | 0.707107i | −0.204875 | − | 0.204875i | 1.00000i | −1.42962 | − | 1.71936i | − | 0.289737i | 0 | −0.707107 | + | 0.707107i | − | 2.91605i | 0.204875 | − | 2.22666i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.7 | 0.707107 | + | 0.707107i | 0.204875 | + | 0.204875i | 1.00000i | 1.42962 | + | 1.71936i | 0.289737i | 0 | −0.707107 | + | 0.707107i | − | 2.91605i | −0.204875 | + | 2.22666i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.8 | 0.707107 | + | 0.707107i | 1.42962 | + | 1.42962i | 1.00000i | 0.204875 | + | 2.22666i | 2.02179i | 0 | −0.707107 | + | 0.707107i | 1.08763i | −1.42962 | + | 1.71936i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 293.1 | −0.707107 | + | 0.707107i | −2.05532 | + | 2.05532i | − | 1.00000i | −0.830578 | − | 2.07609i | − | 2.90667i | 0 | 0.707107 | + | 0.707107i | − | 5.44871i | 2.05532 | + | 0.880708i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 293.2 | −0.707107 | + | 0.707107i | −0.830578 | + | 0.830578i | − | 1.00000i | −2.05532 | + | 0.880708i | − | 1.17462i | 0 | 0.707107 | + | 0.707107i | 1.62028i | 0.830578 | − | 2.07609i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 293.3 | −0.707107 | + | 0.707107i | 0.830578 | − | 0.830578i | − | 1.00000i | 2.05532 | − | 0.880708i | 1.17462i | 0 | 0.707107 | + | 0.707107i | 1.62028i | −0.830578 | + | 2.07609i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 293.4 | −0.707107 | + | 0.707107i | 2.05532 | − | 2.05532i | − | 1.00000i | 0.830578 | + | 2.07609i | 2.90667i | 0 | 0.707107 | + | 0.707107i | − | 5.44871i | −2.05532 | − | 0.880708i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 293.5 | 0.707107 | − | 0.707107i | −1.42962 | + | 1.42962i | − | 1.00000i | −0.204875 | + | 2.22666i | 2.02179i | 0 | −0.707107 | − | 0.707107i | − | 1.08763i | 1.42962 | + | 1.71936i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 293.6 | 0.707107 | − | 0.707107i | −0.204875 | + | 0.204875i | − | 1.00000i | −1.42962 | + | 1.71936i | 0.289737i | 0 | −0.707107 | − | 0.707107i | 2.91605i | 0.204875 | + | 2.22666i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 293.7 | 0.707107 | − | 0.707107i | 0.204875 | − | 0.204875i | − | 1.00000i | 1.42962 | − | 1.71936i | − | 0.289737i | 0 | −0.707107 | − | 0.707107i | 2.91605i | −0.204875 | − | 2.22666i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 293.8 | 0.707107 | − | 0.707107i | 1.42962 | − | 1.42962i | − | 1.00000i | 0.204875 | − | 2.22666i | − | 2.02179i | 0 | −0.707107 | − | 0.707107i | − | 1.08763i | −1.42962 | − | 1.71936i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
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 | 490.2.g.c | 16 | |
| 5.c | odd | 4 | 1 | inner | 490.2.g.c | 16 | |
| 7.b | odd | 2 | 1 | inner | 490.2.g.c | 16 | |
| 7.c | even | 3 | 1 | 70.2.k.a | ✓ | 16 | |
| 7.c | even | 3 | 1 | 490.2.l.c | 16 | ||
| 7.d | odd | 6 | 1 | 70.2.k.a | ✓ | 16 | |
| 7.d | odd | 6 | 1 | 490.2.l.c | 16 | ||
| 21.g | even | 6 | 1 | 630.2.bv.c | 16 | ||
| 21.h | odd | 6 | 1 | 630.2.bv.c | 16 | ||
| 28.f | even | 6 | 1 | 560.2.ci.c | 16 | ||
| 28.g | odd | 6 | 1 | 560.2.ci.c | 16 | ||
| 35.f | even | 4 | 1 | inner | 490.2.g.c | 16 | |
| 35.i | odd | 6 | 1 | 350.2.o.c | 16 | ||
| 35.j | even | 6 | 1 | 350.2.o.c | 16 | ||
| 35.k | even | 12 | 1 | 70.2.k.a | ✓ | 16 | |
| 35.k | even | 12 | 1 | 350.2.o.c | 16 | ||
| 35.k | even | 12 | 1 | 490.2.l.c | 16 | ||
| 35.l | odd | 12 | 1 | 70.2.k.a | ✓ | 16 | |
| 35.l | odd | 12 | 1 | 350.2.o.c | 16 | ||
| 35.l | odd | 12 | 1 | 490.2.l.c | 16 | ||
| 105.w | odd | 12 | 1 | 630.2.bv.c | 16 | ||
| 105.x | even | 12 | 1 | 630.2.bv.c | 16 | ||
| 140.w | even | 12 | 1 | 560.2.ci.c | 16 | ||
| 140.x | odd | 12 | 1 | 560.2.ci.c | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 70.2.k.a | ✓ | 16 | 7.c | even | 3 | 1 | |
| 70.2.k.a | ✓ | 16 | 7.d | odd | 6 | 1 | |
| 70.2.k.a | ✓ | 16 | 35.k | even | 12 | 1 | |
| 70.2.k.a | ✓ | 16 | 35.l | odd | 12 | 1 | |
| 350.2.o.c | 16 | 35.i | odd | 6 | 1 | ||
| 350.2.o.c | 16 | 35.j | even | 6 | 1 | ||
| 350.2.o.c | 16 | 35.k | even | 12 | 1 | ||
| 350.2.o.c | 16 | 35.l | odd | 12 | 1 | ||
| 490.2.g.c | 16 | 1.a | even | 1 | 1 | trivial | |
| 490.2.g.c | 16 | 5.c | odd | 4 | 1 | inner | |
| 490.2.g.c | 16 | 7.b | odd | 2 | 1 | inner | |
| 490.2.g.c | 16 | 35.f | even | 4 | 1 | inner | |
| 490.2.l.c | 16 | 7.c | even | 3 | 1 | ||
| 490.2.l.c | 16 | 7.d | odd | 6 | 1 | ||
| 490.2.l.c | 16 | 35.k | even | 12 | 1 | ||
| 490.2.l.c | 16 | 35.l | odd | 12 | 1 | ||
| 560.2.ci.c | 16 | 28.f | even | 6 | 1 | ||
| 560.2.ci.c | 16 | 28.g | odd | 6 | 1 | ||
| 560.2.ci.c | 16 | 140.w | even | 12 | 1 | ||
| 560.2.ci.c | 16 | 140.x | odd | 12 | 1 | ||
| 630.2.bv.c | 16 | 21.g | even | 6 | 1 | ||
| 630.2.bv.c | 16 | 21.h | odd | 6 | 1 | ||
| 630.2.bv.c | 16 | 105.w | odd | 12 | 1 | ||
| 630.2.bv.c | 16 | 105.x | even | 12 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{16} + 90T_{3}^{12} + 1361T_{3}^{8} + 2280T_{3}^{4} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(490, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 1)^{4} \)
|
| $3$ |
\( T^{16} + 90 T^{12} + \cdots + 16 \)
|
| $5$ |
\( T^{16} + 12 T^{14} + \cdots + 390625 \)
|
| $7$ |
\( T^{16} \)
|
| $11$ |
\( (T^{4} - 6 T^{3} - 13 T^{2} + \cdots - 62)^{4} \)
|
| $13$ |
\( T^{16} + 90 T^{12} + \cdots + 16 \)
|
| $17$ |
\( T^{16} + 2664 T^{12} + \cdots + 9834496 \)
|
| $19$ |
\( (T^{8} - 62 T^{6} + \cdots + 10000)^{2} \)
|
| $23$ |
\( (T^{8} - 4 T^{7} + \cdots + 16129)^{2} \)
|
| $29$ |
\( (T^{8} + 162 T^{6} + \cdots + 329476)^{2} \)
|
| $31$ |
\( (T^{8} + 80 T^{6} + \cdots + 16)^{2} \)
|
| $37$ |
\( (T^{8} + 4 T^{7} + \cdots + 256)^{2} \)
|
| $41$ |
\( (T^{8} + 140 T^{6} + \cdots + 18769)^{2} \)
|
| $43$ |
\( (T^{8} + 4 T^{7} + \cdots + 784)^{2} \)
|
| $47$ |
\( T^{16} + 3378 T^{12} + \cdots + 9834496 \)
|
| $53$ |
\( (T^{8} - 28 T^{7} + \cdots + 204304)^{2} \)
|
| $59$ |
\( (T^{8} - 152 T^{6} + \cdots + 16384)^{2} \)
|
| $61$ |
\( (T^{8} + 110 T^{6} + \cdots + 148996)^{2} \)
|
| $67$ |
\( (T^{8} + 32 T^{7} + \cdots + 58564)^{2} \)
|
| $71$ |
\( (T^{4} - 4 T^{3} + \cdots - 4424)^{4} \)
|
| $73$ |
\( T^{16} + \cdots + 10\!\cdots\!16 \)
|
| $79$ |
\( (T^{8} + 288 T^{6} + \cdots + 7840000)^{2} \)
|
| $83$ |
\( T^{16} + \cdots + 38\!\cdots\!96 \)
|
| $89$ |
\( (T^{8} - 190 T^{6} + \cdots + 99856)^{2} \)
|
| $97$ |
\( (T^{8} + 8136 T^{4} + 3111696)^{2} \)
|
| show more | |
| show less |