Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [95,2,Mod(49,95)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(95, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("95.49");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 95 = 5 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 95.i (of order \(6\), 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: | \(0.758578819202\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(6\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | 12.0.13026266817859584.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 6x^{10} + 29x^{8} - 40x^{6} + 43x^{4} - 7x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{12} - 6x^{10} + 29x^{8} - 40x^{6} + 43x^{4} - 7x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -36\nu^{10} + 174\nu^{8} - 841\nu^{6} + 258\nu^{4} - 42\nu^{2} - 2207 ) / 1205 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 138\nu^{10} - 667\nu^{8} + 3023\nu^{6} - 989\nu^{4} + 161\nu^{2} + 2636 ) / 1205 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 138\nu^{11} - 667\nu^{9} + 3023\nu^{7} - 989\nu^{5} + 161\nu^{3} + 2636\nu ) / 1205 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 174\nu^{11} - 841\nu^{9} + 3864\nu^{7} - 1247\nu^{5} + 203\nu^{3} + 7253\nu ) / 1205 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 203\nu^{10} - 1182\nu^{8} + 5713\nu^{6} - 7279\nu^{4} + 8471\nu^{2} - 174 ) / 1205 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -203\nu^{11} + 1182\nu^{9} - 5713\nu^{7} + 7279\nu^{5} - 8471\nu^{3} + 174\nu ) / 1205 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -74\nu^{10} + 438\nu^{8} - 2117\nu^{6} + 2860\nu^{4} - 3139\nu^{2} + 511 ) / 241 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( -429\nu^{10} + 2676\nu^{8} - 12934\nu^{6} + 19342\nu^{4} - 19178\nu^{2} + 3122 ) / 1205 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( -429\nu^{11} + 2676\nu^{9} - 12934\nu^{7} + 19342\nu^{5} - 19178\nu^{3} + 3122\nu ) / 1205 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( -1002\nu^{11} + 6048\nu^{9} - 29232\nu^{7} + 40921\nu^{5} - 43344\nu^{3} + 7056\nu ) / 1205 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{8} + 2\beta_{6} + \beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{11} - \beta_{10} - 3\beta_{7} - \beta_{5} + \beta_{4} + \beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{9} + 5\beta_{8} + 7\beta_{6} - 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( 5\beta_{11} - 6\beta_{10} - 12\beta_{7} - 12\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -6\beta_{3} - 23\beta_{2} - 29 \)
|
| \(\nu^{7}\) | \(=\) |
\( 23\beta_{5} - 29\beta_{4} - 75\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( 29\beta_{9} - 104\beta_{8} - 127\beta_{6} - 29\beta_{3} - 104\beta_{2} \)
|
| \(\nu^{9}\) | \(=\) |
\( -104\beta_{11} + 133\beta_{10} + 231\beta_{7} + 104\beta_{5} - 133\beta_{4} - 104\beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( 133\beta_{9} - 468\beta_{8} - 566\beta_{6} + 566 \)
|
| \(\nu^{11}\) | \(=\) |
\( -468\beta_{11} + 601\beta_{10} + 1034\beta_{7} + 1034\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/95\mathbb{Z}\right)^\times\).
| \(n\) | \(21\) | \(77\) |
| \(\chi(n)\) | \(-\beta_{6}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
−2.12713 | − | 1.22810i | 1.35190 | + | 0.780522i | 2.01647 | + | 3.49262i | −1.45193 | + | 1.70056i | −1.91712 | − | 3.32055i | 4.50527i | − | 4.99330i | −0.281570 | − | 0.487693i | 5.17691 | − | 1.83419i | |||||||||||||||||||||||||||||||||||||||
| 49.2 | −0.747190 | − | 0.431391i | 2.66661 | + | 1.53957i | −0.627804 | − | 1.08739i | −0.0476457 | − | 2.23556i | −1.32831 | − | 2.30070i | 0.566520i | 2.80888i | 3.24054 | + | 5.61278i | −0.928799 | + | 1.69094i | |||||||||||||||||||||||||||||||||||||||||
| 49.3 | −0.408663 | − | 0.235942i | −0.900858 | − | 0.520111i | −0.888663 | − | 1.53921i | −2.19202 | − | 0.441641i | 0.245432 | + | 0.425100i | − | 1.17540i | 1.78246i | −0.958970 | − | 1.66098i | 0.791597 | + | 0.697672i | ||||||||||||||||||||||||||||||||||||||||
| 49.4 | 0.408663 | + | 0.235942i | 0.900858 | + | 0.520111i | −0.888663 | − | 1.53921i | 1.47848 | + | 1.67752i | 0.245432 | + | 0.425100i | 1.17540i | − | 1.78246i | −0.958970 | − | 1.66098i | 0.208403 | + | 1.03438i | ||||||||||||||||||||||||||||||||||||||||
| 49.5 | 0.747190 | + | 0.431391i | −2.66661 | − | 1.53957i | −0.627804 | − | 1.08739i | 1.95987 | − | 1.07652i | −1.32831 | − | 2.30070i | − | 0.566520i | − | 2.80888i | 3.24054 | + | 5.61278i | 1.92880 | + | 0.0411078i | |||||||||||||||||||||||||||||||||||||||
| 49.6 | 2.12713 | + | 1.22810i | −1.35190 | − | 0.780522i | 2.01647 | + | 3.49262i | −0.746759 | + | 2.10769i | −1.91712 | − | 3.32055i | − | 4.50527i | 4.99330i | −0.281570 | − | 0.487693i | −4.17691 | + | 3.56624i | ||||||||||||||||||||||||||||||||||||||||
| 64.1 | −2.12713 | + | 1.22810i | 1.35190 | − | 0.780522i | 2.01647 | − | 3.49262i | −1.45193 | − | 1.70056i | −1.91712 | + | 3.32055i | − | 4.50527i | 4.99330i | −0.281570 | + | 0.487693i | 5.17691 | + | 1.83419i | ||||||||||||||||||||||||||||||||||||||||
| 64.2 | −0.747190 | + | 0.431391i | 2.66661 | − | 1.53957i | −0.627804 | + | 1.08739i | −0.0476457 | + | 2.23556i | −1.32831 | + | 2.30070i | − | 0.566520i | − | 2.80888i | 3.24054 | − | 5.61278i | −0.928799 | − | 1.69094i | |||||||||||||||||||||||||||||||||||||||
| 64.3 | −0.408663 | + | 0.235942i | −0.900858 | + | 0.520111i | −0.888663 | + | 1.53921i | −2.19202 | + | 0.441641i | 0.245432 | − | 0.425100i | 1.17540i | − | 1.78246i | −0.958970 | + | 1.66098i | 0.791597 | − | 0.697672i | ||||||||||||||||||||||||||||||||||||||||
| 64.4 | 0.408663 | − | 0.235942i | 0.900858 | − | 0.520111i | −0.888663 | + | 1.53921i | 1.47848 | − | 1.67752i | 0.245432 | − | 0.425100i | − | 1.17540i | 1.78246i | −0.958970 | + | 1.66098i | 0.208403 | − | 1.03438i | ||||||||||||||||||||||||||||||||||||||||
| 64.5 | 0.747190 | − | 0.431391i | −2.66661 | + | 1.53957i | −0.627804 | + | 1.08739i | 1.95987 | + | 1.07652i | −1.32831 | + | 2.30070i | 0.566520i | 2.80888i | 3.24054 | − | 5.61278i | 1.92880 | − | 0.0411078i | |||||||||||||||||||||||||||||||||||||||||
| 64.6 | 2.12713 | − | 1.22810i | −1.35190 | + | 0.780522i | 2.01647 | − | 3.49262i | −0.746759 | − | 2.10769i | −1.91712 | + | 3.32055i | 4.50527i | − | 4.99330i | −0.281570 | + | 0.487693i | −4.17691 | − | 3.56624i | ||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 19.c | even | 3 | 1 | inner |
| 95.i | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 95.2.i.b | ✓ | 12 |
| 3.b | odd | 2 | 1 | 855.2.be.d | 12 | ||
| 5.b | even | 2 | 1 | inner | 95.2.i.b | ✓ | 12 |
| 5.c | odd | 4 | 2 | 475.2.e.g | 12 | ||
| 15.d | odd | 2 | 1 | 855.2.be.d | 12 | ||
| 19.c | even | 3 | 1 | inner | 95.2.i.b | ✓ | 12 |
| 19.c | even | 3 | 1 | 1805.2.b.f | 6 | ||
| 19.d | odd | 6 | 1 | 1805.2.b.g | 6 | ||
| 57.h | odd | 6 | 1 | 855.2.be.d | 12 | ||
| 95.h | odd | 6 | 1 | 1805.2.b.g | 6 | ||
| 95.i | even | 6 | 1 | inner | 95.2.i.b | ✓ | 12 |
| 95.i | even | 6 | 1 | 1805.2.b.f | 6 | ||
| 95.l | even | 12 | 2 | 9025.2.a.bt | 6 | ||
| 95.m | odd | 12 | 2 | 475.2.e.g | 12 | ||
| 95.m | odd | 12 | 2 | 9025.2.a.bu | 6 | ||
| 285.n | odd | 6 | 1 | 855.2.be.d | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 95.2.i.b | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 95.2.i.b | ✓ | 12 | 5.b | even | 2 | 1 | inner |
| 95.2.i.b | ✓ | 12 | 19.c | even | 3 | 1 | inner |
| 95.2.i.b | ✓ | 12 | 95.i | even | 6 | 1 | inner |
| 475.2.e.g | 12 | 5.c | odd | 4 | 2 | ||
| 475.2.e.g | 12 | 95.m | odd | 12 | 2 | ||
| 855.2.be.d | 12 | 3.b | odd | 2 | 1 | ||
| 855.2.be.d | 12 | 15.d | odd | 2 | 1 | ||
| 855.2.be.d | 12 | 57.h | odd | 6 | 1 | ||
| 855.2.be.d | 12 | 285.n | odd | 6 | 1 | ||
| 1805.2.b.f | 6 | 19.c | even | 3 | 1 | ||
| 1805.2.b.f | 6 | 95.i | even | 6 | 1 | ||
| 1805.2.b.g | 6 | 19.d | odd | 6 | 1 | ||
| 1805.2.b.g | 6 | 95.h | odd | 6 | 1 | ||
| 9025.2.a.bt | 6 | 95.l | even | 12 | 2 | ||
| 9025.2.a.bu | 6 | 95.m | odd | 12 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} - 7T_{2}^{10} + 43T_{2}^{8} - 40T_{2}^{6} + 29T_{2}^{4} - 6T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(95, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} - 7 T^{10} + \cdots + 1 \)
|
| $3$ |
\( T^{12} - 13 T^{10} + \cdots + 625 \)
|
| $5$ |
\( T^{12} + 2 T^{11} + \cdots + 15625 \)
|
| $7$ |
\( (T^{6} + 22 T^{4} + 35 T^{2} + 9)^{2} \)
|
| $11$ |
\( (T^{3} - T^{2} - 4 T + 3)^{4} \)
|
| $13$ |
\( T^{12} - 31 T^{10} + \cdots + 81 \)
|
| $17$ |
\( T^{12} - 35 T^{10} + \cdots + 6561 \)
|
| $19$ |
\( (T^{6} + 6 T^{5} + \cdots + 6859)^{2} \)
|
| $23$ |
\( T^{12} - 12 T^{10} + \cdots + 1 \)
|
| $29$ |
\( (T^{6} + 6 T^{5} + \cdots + 729)^{2} \)
|
| $31$ |
\( (T^{3} - 15 T^{2} + \cdots - 97)^{4} \)
|
| $37$ |
\( (T^{6} + 98 T^{4} + \cdots + 729)^{2} \)
|
| $41$ |
\( (T^{6} + 6 T^{5} + 77 T^{4} + \cdots + 9)^{2} \)
|
| $43$ |
\( T^{12} - 127 T^{10} + \cdots + 194481 \)
|
| $47$ |
\( T^{12} + \cdots + 47562811921 \)
|
| $53$ |
\( T^{12} + \cdots + 131079601 \)
|
| $59$ |
\( (T^{6} - 10 T^{5} + \cdots + 84681)^{2} \)
|
| $61$ |
\( (T^{6} - T^{5} + \cdots + 12769)^{2} \)
|
| $67$ |
\( T^{12} + \cdots + 207360000 \)
|
| $71$ |
\( (T^{6} - T^{5} + \cdots + 227529)^{2} \)
|
| $73$ |
\( T^{12} + \cdots + 9845600625 \)
|
| $79$ |
\( (T^{6} - 12 T^{5} + \cdots + 200704)^{2} \)
|
| $83$ |
\( (T^{6} + 459 T^{4} + \cdots + 966289)^{2} \)
|
| $89$ |
\( (T^{6} - 18 T^{5} + \cdots + 5184)^{2} \)
|
| $97$ |
\( T^{12} + \cdots + 1534548635361 \)
|
| show more | |
| show less |