Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [240,4,Mod(17,240)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(240, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 0, 2, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("240.17");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 240 = 2^{4} \cdot 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 240.v (of order \(4\), degree \(2\), not 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: | \(14.1604584014\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(6\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} + 1577x^{8} + 284056x^{4} + 810000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 30) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} + 1577x^{8} + 284056x^{4} + 810000 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -83\nu^{9} - 150271\nu^{5} - 41313548\nu ) / 26668320 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2729 \nu^{10} - 32070 \nu^{9} + 12150 \nu^{8} + 4619533 \nu^{6} - 48423390 \nu^{5} + \cdots - 1808202600 ) / 1600099200 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -53\nu^{10} - 84481\nu^{6} - 17028668\nu^{2} ) / 28573200 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 1484 \nu^{11} - 1245 \nu^{10} + 61650 \nu^{8} + 2365468 \nu^{7} - 2254065 \nu^{6} + \cdots + 9374344200 ) / 800049600 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 37339\nu^{11} + 59248103\nu^{7} + 10977117484\nu^{3} ) / 12000744000 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 81859 \nu^{11} + 40935 \nu^{10} + 518400 \nu^{9} + 546750 \nu^{8} - 130212143 \nu^{7} + \cdots - 57367629000 ) / 24001488000 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 81859 \nu^{11} + 78285 \nu^{10} + 518400 \nu^{9} + 182250 \nu^{8} + 130212143 \nu^{7} + \cdots - 27123039000 ) / 24001488000 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 1484 \nu^{11} - 21493 \nu^{10} - 12150 \nu^{8} - 2365468 \nu^{7} - 33450761 \nu^{6} + \cdots + 1008153000 ) / 800049600 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 213257 \nu^{11} + 40935 \nu^{10} - 182250 \nu^{8} + 333727789 \nu^{7} + 69292995 \nu^{6} + \cdots + 27123039000 ) / 24001488000 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 83532 \nu^{11} + 13645 \nu^{10} + 12450 \nu^{9} - 182250 \nu^{8} - 130991964 \nu^{7} + \cdots + 19122543000 ) / 8000496000 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 257777 \nu^{11} + 481050 \nu^{9} + 364500 \nu^{8} + 404691829 \nu^{7} + 726350850 \nu^{5} + \cdots - 30244590000 ) / 24001488000 \)
|
| \(\nu\) | \(=\) |
\( ( 3\beta_{11} + 3\beta_{10} - 2\beta_{7} + \beta_{6} + 3\beta_{5} + 2\beta_{3} + 2\beta_{2} + \beta _1 - 1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 5 \beta_{11} - 5 \beta_{10} - 10 \beta_{9} + 6 \beta_{8} - 4 \beta_{7} - 5 \beta_{6} - 3 \beta_{5} + \cdots + 1 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 31 \beta_{11} - 31 \beta_{10} - 50 \beta_{9} + 56 \beta_{7} - 87 \beta_{6} - 225 \beta_{5} - 56 \beta_{3} + \cdots + 25 ) / 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 191 \beta_{11} + 191 \beta_{10} + 382 \beta_{9} - 29 \beta_{6} + 81 \beta_{5} + 162 \beta_{4} + \cdots - 3345 ) / 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 2943 \beta_{11} - 2943 \beta_{10} + 1796 \beta_{7} - 1147 \beta_{6} - 2943 \beta_{5} - 1796 \beta_{3} + \cdots + 1147 ) / 6 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 7043 \beta_{11} + 7043 \beta_{10} + 14086 \beta_{9} - 4890 \beta_{8} + 9196 \beta_{7} + 7043 \beta_{6} + \cdots + 2153 ) / 6 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 42595 \beta_{11} + 42595 \beta_{10} + 39794 \beta_{9} - 62492 \beta_{7} + 105087 \beta_{6} + \cdots - 19897 ) / 6 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 260123 \beta_{11} - 260123 \beta_{10} - 520246 \beta_{9} + 95345 \beta_{6} - 82389 \beta_{5} + \cdots + 4097757 ) / 6 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 3835023 \beta_{11} + 3835023 \beta_{10} - 2256140 \beta_{7} + 1578883 \beta_{6} + 3835023 \beta_{5} + \cdots - 1578883 ) / 6 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 9619931 \beta_{11} - 9619931 \beta_{10} - 19239862 \beta_{9} + 5866794 \beta_{8} - 13373068 \beta_{7} + \cdots - 3753137 ) / 6 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 58474579 \beta_{11} - 58474579 \beta_{10} - 48444338 \beta_{9} + 82696748 \beta_{7} - 141171327 \beta_{6} + \cdots + 24222169 ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(97\) | \(161\) | \(181\) |
| \(\chi(n)\) | \(1\) | \(\beta_{3}\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 17.1 |
|
0 | −4.51886 | + | 2.56513i | 0 | −10.0777 | + | 4.84155i | 0 | −10.4050 | − | 10.4050i | 0 | 13.8402 | − | 23.1830i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 17.2 | 0 | −3.51870 | − | 3.82345i | 0 | −5.59622 | + | 9.67896i | 0 | −8.77386 | − | 8.77386i | 0 | −2.23754 | + | 26.9071i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 17.3 | 0 | −2.56513 | + | 4.51886i | 0 | 10.0777 | − | 4.84155i | 0 | −10.4050 | − | 10.4050i | 0 | −13.8402 | − | 23.1830i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 17.4 | 0 | −2.01170 | − | 4.79094i | 0 | 10.8454 | + | 2.71609i | 0 | 16.1789 | + | 16.1789i | 0 | −18.9062 | + | 19.2758i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 17.5 | 0 | 3.82345 | + | 3.51870i | 0 | 5.59622 | − | 9.67896i | 0 | −8.77386 | − | 8.77386i | 0 | 2.23754 | + | 26.9071i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 17.6 | 0 | 4.79094 | + | 2.01170i | 0 | −10.8454 | − | 2.71609i | 0 | 16.1789 | + | 16.1789i | 0 | 18.9062 | + | 19.2758i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 113.1 | 0 | −4.51886 | − | 2.56513i | 0 | −10.0777 | − | 4.84155i | 0 | −10.4050 | + | 10.4050i | 0 | 13.8402 | + | 23.1830i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 113.2 | 0 | −3.51870 | + | 3.82345i | 0 | −5.59622 | − | 9.67896i | 0 | −8.77386 | + | 8.77386i | 0 | −2.23754 | − | 26.9071i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 113.3 | 0 | −2.56513 | − | 4.51886i | 0 | 10.0777 | + | 4.84155i | 0 | −10.4050 | + | 10.4050i | 0 | −13.8402 | + | 23.1830i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 113.4 | 0 | −2.01170 | + | 4.79094i | 0 | 10.8454 | − | 2.71609i | 0 | 16.1789 | − | 16.1789i | 0 | −18.9062 | − | 19.2758i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 113.5 | 0 | 3.82345 | − | 3.51870i | 0 | 5.59622 | + | 9.67896i | 0 | −8.77386 | + | 8.77386i | 0 | 2.23754 | − | 26.9071i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 113.6 | 0 | 4.79094 | − | 2.01170i | 0 | −10.8454 | + | 2.71609i | 0 | 16.1789 | − | 16.1789i | 0 | 18.9062 | − | 19.2758i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 5.c | odd | 4 | 1 | inner |
| 15.e | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 240.4.v.d | 12 | |
| 3.b | odd | 2 | 1 | inner | 240.4.v.d | 12 | |
| 4.b | odd | 2 | 1 | 30.4.e.a | ✓ | 12 | |
| 5.c | odd | 4 | 1 | inner | 240.4.v.d | 12 | |
| 12.b | even | 2 | 1 | 30.4.e.a | ✓ | 12 | |
| 15.e | even | 4 | 1 | inner | 240.4.v.d | 12 | |
| 20.d | odd | 2 | 1 | 150.4.e.c | 12 | ||
| 20.e | even | 4 | 1 | 30.4.e.a | ✓ | 12 | |
| 20.e | even | 4 | 1 | 150.4.e.c | 12 | ||
| 60.h | even | 2 | 1 | 150.4.e.c | 12 | ||
| 60.l | odd | 4 | 1 | 30.4.e.a | ✓ | 12 | |
| 60.l | odd | 4 | 1 | 150.4.e.c | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 30.4.e.a | ✓ | 12 | 4.b | odd | 2 | 1 | |
| 30.4.e.a | ✓ | 12 | 12.b | even | 2 | 1 | |
| 30.4.e.a | ✓ | 12 | 20.e | even | 4 | 1 | |
| 30.4.e.a | ✓ | 12 | 60.l | odd | 4 | 1 | |
| 150.4.e.c | 12 | 20.d | odd | 2 | 1 | ||
| 150.4.e.c | 12 | 20.e | even | 4 | 1 | ||
| 150.4.e.c | 12 | 60.h | even | 2 | 1 | ||
| 150.4.e.c | 12 | 60.l | odd | 4 | 1 | ||
| 240.4.v.d | 12 | 1.a | even | 1 | 1 | trivial | |
| 240.4.v.d | 12 | 3.b | odd | 2 | 1 | inner | |
| 240.4.v.d | 12 | 5.c | odd | 4 | 1 | inner | |
| 240.4.v.d | 12 | 15.e | even | 4 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{6} + 6T_{7}^{5} + 18T_{7}^{4} + 3280T_{7}^{3} + 191844T_{7}^{2} + 2587704T_{7} + 17452232 \)
acting on \(S_{4}^{\mathrm{new}}(240, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} + \cdots + 387420489 \)
|
| $5$ |
\( T^{12} + \cdots + 3814697265625 \)
|
| $7$ |
\( (T^{6} + 6 T^{5} + \cdots + 17452232)^{2} \)
|
| $11$ |
\( (T^{6} + 2826 T^{4} + \cdots + 154457888)^{2} \)
|
| $13$ |
\( (T^{6} + 60 T^{5} + \cdots + 64800000000)^{2} \)
|
| $17$ |
\( T^{12} + \cdots + 21\!\cdots\!56 \)
|
| $19$ |
\( (T^{6} + 21288 T^{4} + \cdots + 12078889216)^{2} \)
|
| $23$ |
\( T^{12} + \cdots + 24\!\cdots\!16 \)
|
| $29$ |
\( (T^{6} - 82170 T^{4} + \cdots - 8528180000)^{2} \)
|
| $31$ |
\( (T^{3} - 126 T^{2} + \cdots - 39488)^{4} \)
|
| $37$ |
\( (T^{6} + \cdots + 33414920101152)^{2} \)
|
| $41$ |
\( (T^{6} + 176736 T^{4} + \cdots + 165740576768)^{2} \)
|
| $43$ |
\( (T^{6} + \cdots + 40492872313632)^{2} \)
|
| $47$ |
\( T^{12} + \cdots + 10\!\cdots\!00 \)
|
| $53$ |
\( T^{12} + \cdots + 53\!\cdots\!76 \)
|
| $59$ |
\( (T^{6} + \cdots - 13\!\cdots\!92)^{2} \)
|
| $61$ |
\( (T^{3} - 462 T^{2} + \cdots + 123755256)^{4} \)
|
| $67$ |
\( (T^{6} + \cdots + 122924426333312)^{2} \)
|
| $71$ |
\( (T^{6} + \cdots + 5009245520000)^{2} \)
|
| $73$ |
\( (T^{6} + \cdots + 46\!\cdots\!00)^{2} \)
|
| $79$ |
\( (T^{6} + \cdots + 23\!\cdots\!96)^{2} \)
|
| $83$ |
\( T^{12} + \cdots + 52\!\cdots\!16 \)
|
| $89$ |
\( (T^{6} + \cdots - 53\!\cdots\!72)^{2} \)
|
| $97$ |
\( (T^{6} + \cdots + 13\!\cdots\!28)^{2} \)
|
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