Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [210,4,Mod(121,210)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(210, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("210.121");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 210.i (of order \(3\), 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: | \(12.3904011012\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 241x^{4} - 1920x^{3} + 58680x^{2} - 259200x + 1166400 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 3\cdot 7 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} + 241x^{4} - 1920x^{3} + 58680x^{2} - 259200x + 1166400 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 47\nu^{5} - 11327\nu^{4} - 46729\nu^{3} - 2757960\nu^{2} + 28841616\nu - 387098352 ) / 16659216 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 482\nu^{5} - 473\nu^{4} + 113993\nu^{3} - 402711\nu^{2} + 27755640\nu + 2342520 ) / 124944120 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 1178\nu^{5} - 168209\nu^{4} - 1109671\nu^{3} - 41243991\nu^{2} - 166673520\nu - 5241927960 ) / 124944120 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -1405\nu^{5} - 8462\nu^{4} - 274438\nu^{3} + 1116033\nu^{2} - 47188140\nu - 69300360 ) / 13882680 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 8677\nu^{5} - 8755\nu^{4} + 2109955\nu^{3} - 21190158\nu^{2} + 388799280\nu - 2227647960 ) / 41648040 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{5} - 2\beta_{4} - 4\beta_{3} + 4\beta_{2} + 10\beta _1 + 1 ) / 21 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -50\beta_{5} + 26\beta_{4} + 31\beta_{3} + 3371\beta_{2} - 88\beta _1 - 3352 ) / 21 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 167\beta_{5} + 334\beta_{4} - 37\beta_{3} + 37\beta_{2} - 278\beta _1 + 2587 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 3739\beta_{5} - 22888\beta_{4} - 15410\beta_{3} - 781792\beta_{2} + 23342\beta _1 - 3739 ) / 21 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -256990\beta_{5} - 438506\beta_{4} + 302369\beta_{3} + 7201309\beta_{2} - 166232\beta _1 - 7246688 ) / 21 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(71\) | \(127\) |
| \(\chi(n)\) | \(-1 + \beta_{2}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 121.1 |
|
1.00000 | − | 1.73205i | 1.50000 | + | 2.59808i | −2.00000 | − | 3.46410i | −2.50000 | + | 4.33013i | 6.00000 | −17.5901 | − | 5.79546i | −8.00000 | −4.50000 | + | 7.79423i | 5.00000 | + | 8.66025i | ||||||||||||||||||||||
| 121.2 | 1.00000 | − | 1.73205i | 1.50000 | + | 2.59808i | −2.00000 | − | 3.46410i | −2.50000 | + | 4.33013i | 6.00000 | −8.58712 | + | 16.4092i | −8.00000 | −4.50000 | + | 7.79423i | 5.00000 | + | 8.66025i | |||||||||||||||||||||||
| 121.3 | 1.00000 | − | 1.73205i | 1.50000 | + | 2.59808i | −2.00000 | − | 3.46410i | −2.50000 | + | 4.33013i | 6.00000 | 15.1772 | − | 10.6137i | −8.00000 | −4.50000 | + | 7.79423i | 5.00000 | + | 8.66025i | |||||||||||||||||||||||
| 151.1 | 1.00000 | + | 1.73205i | 1.50000 | − | 2.59808i | −2.00000 | + | 3.46410i | −2.50000 | − | 4.33013i | 6.00000 | −17.5901 | + | 5.79546i | −8.00000 | −4.50000 | − | 7.79423i | 5.00000 | − | 8.66025i | |||||||||||||||||||||||
| 151.2 | 1.00000 | + | 1.73205i | 1.50000 | − | 2.59808i | −2.00000 | + | 3.46410i | −2.50000 | − | 4.33013i | 6.00000 | −8.58712 | − | 16.4092i | −8.00000 | −4.50000 | − | 7.79423i | 5.00000 | − | 8.66025i | |||||||||||||||||||||||
| 151.3 | 1.00000 | + | 1.73205i | 1.50000 | − | 2.59808i | −2.00000 | + | 3.46410i | −2.50000 | − | 4.33013i | 6.00000 | 15.1772 | + | 10.6137i | −8.00000 | −4.50000 | − | 7.79423i | 5.00000 | − | 8.66025i | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 210.4.i.k | ✓ | 6 |
| 3.b | odd | 2 | 1 | 630.4.k.o | 6 | ||
| 7.c | even | 3 | 1 | inner | 210.4.i.k | ✓ | 6 |
| 7.c | even | 3 | 1 | 1470.4.a.bt | 3 | ||
| 7.d | odd | 6 | 1 | 1470.4.a.bu | 3 | ||
| 21.h | odd | 6 | 1 | 630.4.k.o | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 210.4.i.k | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 210.4.i.k | ✓ | 6 | 7.c | even | 3 | 1 | inner |
| 630.4.k.o | 6 | 3.b | odd | 2 | 1 | ||
| 630.4.k.o | 6 | 21.h | odd | 6 | 1 | ||
| 1470.4.a.bt | 3 | 7.c | even | 3 | 1 | ||
| 1470.4.a.bu | 3 | 7.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{6} + 61T_{11}^{5} + 6787T_{11}^{4} + 188706T_{11}^{3} + 20860182T_{11}^{2} + 575997156T_{11} + 35293633956 \)
acting on \(S_{4}^{\mathrm{new}}(210, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2 T + 4)^{3} \)
|
| $3$ |
\( (T^{2} - 3 T + 9)^{3} \)
|
| $5$ |
\( (T^{2} + 5 T + 25)^{3} \)
|
| $7$ |
\( T^{6} + 22 T^{5} + \cdots + 40353607 \)
|
| $11$ |
\( T^{6} + \cdots + 35293633956 \)
|
| $13$ |
\( (T^{3} - 37 T^{2} + \cdots + 191703)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 108979214400 \)
|
| $19$ |
\( T^{6} + \cdots + 746397507249 \)
|
| $23$ |
\( T^{6} + \cdots + 1411472306916 \)
|
| $29$ |
\( (T^{3} + 180 T^{2} + \cdots - 1351728)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 36523032100 \)
|
| $37$ |
\( T^{6} + \cdots + 1950751336249 \)
|
| $41$ |
\( (T^{3} - 53 T^{2} + \cdots + 18959274)^{2} \)
|
| $43$ |
\( (T^{3} + 574 T^{2} + \cdots - 5257324)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 846645883952400 \)
|
| $53$ |
\( T^{6} + \cdots + 760797174832704 \)
|
| $59$ |
\( T^{6} + \cdots + 363351609262656 \)
|
| $61$ |
\( T^{6} + \cdots + 34\!\cdots\!00 \)
|
| $67$ |
\( T^{6} + \cdots + 16\!\cdots\!00 \)
|
| $71$ |
\( (T^{3} + 34 T^{2} + \cdots - 11009124)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 880375243369924 \)
|
| $79$ |
\( T^{6} + \cdots + 51\!\cdots\!44 \)
|
| $83$ |
\( (T^{3} - 80 T^{2} + \cdots - 13768776)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 349165848560400 \)
|
| $97$ |
\( (T^{3} + 1080 T^{2} + \cdots - 23394368)^{2} \)
|
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