Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [325,2,Mod(18,325)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(325, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([3, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("325.18");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 325 = 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 325.f (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: | \(2.59513806569\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | 8.0.619810816.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 65) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 64\nu^{7} + 16\nu^{6} + 4\nu^{5} - 127\nu^{4} + 944\nu^{3} - 276\nu^{2} + 378\nu + 63 ) / 319 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -63\nu^{7} + 64\nu^{6} + 16\nu^{5} + 130\nu^{4} - 1009\nu^{3} + 1448\nu^{2} - 402\nu - 67 ) / 319 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -67\nu^{7} + 63\nu^{6} - 64\nu^{5} + 118\nu^{4} - 1068\nu^{3} + 1545\nu^{2} - 1263\nu + 268 ) / 319 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 83\nu^{7} - 59\nu^{6} + 65\nu^{5} - 70\nu^{4} + 1304\nu^{3} - 1614\nu^{2} + 1198\nu + 306 ) / 319 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -172\nu^{7} - 43\nu^{6} + 69\nu^{5} + 441\nu^{4} - 2218\nu^{3} + 662\nu^{2} + 619\nu - 269 ) / 319 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -196\nu^{7} - 49\nu^{6} - 92\nu^{5} + 369\nu^{4} - 2572\nu^{3} + 1244\nu^{2} - 1038\nu - 173 ) / 319 \)
|
| \(\beta_{7}\) | \(=\) |
\( \nu^{7} - 2\nu^{4} + 14\nu^{3} - 8\nu^{2} + \nu + 2 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{7} - \beta_{5} + \beta_{4} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 2\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{7} + 5\beta_{5} - 5\beta_{4} - 2\beta_{2} + 3\beta _1 + 2 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{7} - \beta_{5} + 5\beta_{4} + 4\beta_{3} - 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 11\beta_{7} + 2\beta_{6} + 9\beta_{5} - 11\beta_{4} - 12\beta_{3} + 12\beta_{2} - 11\beta _1 + 12 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -15\beta_{6} + 16\beta_{5} - 16\beta_{4} + 16\beta_{3} - 28\beta_{2} + 7\beta_1 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -43\beta_{7} + 16\beta_{6} - 89\beta_{5} + 105\beta_{4} + 60\beta_{2} - 43\beta _1 - 60 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/325\mathbb{Z}\right)^\times\).
| \(n\) | \(27\) | \(301\) |
| \(\chi(n)\) | \(-\beta_{2}\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 18.1 |
|
− | 2.31627i | 0.240275 | + | 0.240275i | −3.36509 | 0 | 0.556540 | − | 0.556540i | −3.95872 | 3.16190i | − | 2.88454i | 0 | ||||||||||||||||||||||||||||||||||||
| 18.2 | − | 1.57942i | −0.725850 | − | 0.725850i | −0.494582 | 0 | −1.14643 | + | 1.14643i | 4.24997 | − | 2.37769i | − | 1.94628i | 0 | ||||||||||||||||||||||||||||||||||||
| 18.3 | − | 0.134632i | 2.15558 | + | 2.15558i | 1.98187 | 0 | 0.290209 | − | 0.290209i | −1.90970 | − | 0.536087i | 6.29303i | 0 | |||||||||||||||||||||||||||||||||||||
| 18.4 | 2.03032i | 1.33000 | + | 1.33000i | −2.12221 | 0 | −2.70032 | + | 2.70032i | 1.61845 | − | 0.248119i | 0.537789i | 0 | ||||||||||||||||||||||||||||||||||||||
| 307.1 | − | 2.03032i | 1.33000 | − | 1.33000i | −2.12221 | 0 | −2.70032 | − | 2.70032i | 1.61845 | 0.248119i | − | 0.537789i | 0 | |||||||||||||||||||||||||||||||||||||
| 307.2 | 0.134632i | 2.15558 | − | 2.15558i | 1.98187 | 0 | 0.290209 | + | 0.290209i | −1.90970 | 0.536087i | − | 6.29303i | 0 | ||||||||||||||||||||||||||||||||||||||
| 307.3 | 1.57942i | −0.725850 | + | 0.725850i | −0.494582 | 0 | −1.14643 | − | 1.14643i | 4.24997 | 2.37769i | 1.94628i | 0 | |||||||||||||||||||||||||||||||||||||||
| 307.4 | 2.31627i | 0.240275 | − | 0.240275i | −3.36509 | 0 | 0.556540 | + | 0.556540i | −3.95872 | − | 3.16190i | 2.88454i | 0 | ||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 65.f | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 325.2.f.b | 8 | |
| 5.b | even | 2 | 1 | 65.2.f.b | ✓ | 8 | |
| 5.c | odd | 4 | 1 | 65.2.k.b | yes | 8 | |
| 5.c | odd | 4 | 1 | 325.2.k.b | 8 | ||
| 13.d | odd | 4 | 1 | 325.2.k.b | 8 | ||
| 15.d | odd | 2 | 1 | 585.2.n.e | 8 | ||
| 15.e | even | 4 | 1 | 585.2.w.e | 8 | ||
| 20.d | odd | 2 | 1 | 1040.2.cd.n | 8 | ||
| 20.e | even | 4 | 1 | 1040.2.bg.n | 8 | ||
| 65.d | even | 2 | 1 | 845.2.f.b | 8 | ||
| 65.f | even | 4 | 1 | inner | 325.2.f.b | 8 | |
| 65.f | even | 4 | 1 | 845.2.f.b | 8 | ||
| 65.g | odd | 4 | 1 | 65.2.k.b | yes | 8 | |
| 65.g | odd | 4 | 1 | 845.2.k.b | 8 | ||
| 65.h | odd | 4 | 1 | 845.2.k.b | 8 | ||
| 65.k | even | 4 | 1 | 65.2.f.b | ✓ | 8 | |
| 65.l | even | 6 | 2 | 845.2.t.d | 16 | ||
| 65.n | even | 6 | 2 | 845.2.t.c | 16 | ||
| 65.o | even | 12 | 2 | 845.2.t.c | 16 | ||
| 65.q | odd | 12 | 2 | 845.2.o.d | 16 | ||
| 65.r | odd | 12 | 2 | 845.2.o.c | 16 | ||
| 65.s | odd | 12 | 2 | 845.2.o.c | 16 | ||
| 65.s | odd | 12 | 2 | 845.2.o.d | 16 | ||
| 65.t | even | 12 | 2 | 845.2.t.d | 16 | ||
| 195.j | odd | 4 | 1 | 585.2.n.e | 8 | ||
| 195.n | even | 4 | 1 | 585.2.w.e | 8 | ||
| 260.s | odd | 4 | 1 | 1040.2.cd.n | 8 | ||
| 260.u | even | 4 | 1 | 1040.2.bg.n | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 65.2.f.b | ✓ | 8 | 5.b | even | 2 | 1 | |
| 65.2.f.b | ✓ | 8 | 65.k | even | 4 | 1 | |
| 65.2.k.b | yes | 8 | 5.c | odd | 4 | 1 | |
| 65.2.k.b | yes | 8 | 65.g | odd | 4 | 1 | |
| 325.2.f.b | 8 | 1.a | even | 1 | 1 | trivial | |
| 325.2.f.b | 8 | 65.f | even | 4 | 1 | inner | |
| 325.2.k.b | 8 | 5.c | odd | 4 | 1 | ||
| 325.2.k.b | 8 | 13.d | odd | 4 | 1 | ||
| 585.2.n.e | 8 | 15.d | odd | 2 | 1 | ||
| 585.2.n.e | 8 | 195.j | odd | 4 | 1 | ||
| 585.2.w.e | 8 | 15.e | even | 4 | 1 | ||
| 585.2.w.e | 8 | 195.n | even | 4 | 1 | ||
| 845.2.f.b | 8 | 65.d | even | 2 | 1 | ||
| 845.2.f.b | 8 | 65.f | even | 4 | 1 | ||
| 845.2.k.b | 8 | 65.g | odd | 4 | 1 | ||
| 845.2.k.b | 8 | 65.h | odd | 4 | 1 | ||
| 845.2.o.c | 16 | 65.r | odd | 12 | 2 | ||
| 845.2.o.c | 16 | 65.s | odd | 12 | 2 | ||
| 845.2.o.d | 16 | 65.q | odd | 12 | 2 | ||
| 845.2.o.d | 16 | 65.s | odd | 12 | 2 | ||
| 845.2.t.c | 16 | 65.n | even | 6 | 2 | ||
| 845.2.t.c | 16 | 65.o | even | 12 | 2 | ||
| 845.2.t.d | 16 | 65.l | even | 6 | 2 | ||
| 845.2.t.d | 16 | 65.t | even | 12 | 2 | ||
| 1040.2.bg.n | 8 | 20.e | even | 4 | 1 | ||
| 1040.2.bg.n | 8 | 260.u | even | 4 | 1 | ||
| 1040.2.cd.n | 8 | 20.d | odd | 2 | 1 | ||
| 1040.2.cd.n | 8 | 260.s | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 12T_{2}^{6} + 46T_{2}^{4} + 56T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(325, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 12 T^{6} + \cdots + 1 \)
|
| $3$ |
\( T^{8} - 6 T^{7} + \cdots + 4 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( (T^{4} - 20 T^{2} + \cdots + 52)^{2} \)
|
| $11$ |
\( T^{8} - 6 T^{7} + \cdots + 4 \)
|
| $13$ |
\( T^{8} + 14 T^{7} + \cdots + 28561 \)
|
| $17$ |
\( T^{8} + 16 T^{7} + \cdots + 13456 \)
|
| $19$ |
\( T^{8} + 14 T^{7} + \cdots + 100 \)
|
| $23$ |
\( T^{8} - 14 T^{7} + \cdots + 40804 \)
|
| $29$ |
\( T^{8} + 44 T^{6} + \cdots + 10000 \)
|
| $31$ |
\( T^{8} - 2 T^{7} + \cdots + 16900 \)
|
| $37$ |
\( (T^{4} - 22 T^{3} + \cdots + 580)^{2} \)
|
| $41$ |
\( T^{8} - 16 T^{7} + \cdots + 13456 \)
|
| $43$ |
\( T^{8} - 6 T^{7} + \cdots + 8836 \)
|
| $47$ |
\( (T^{4} + 8 T^{3} + \cdots - 164)^{2} \)
|
| $53$ |
\( T^{8} - 24 T^{7} + \cdots + 19600 \)
|
| $59$ |
\( T^{8} + 22 T^{7} + \cdots + 119716 \)
|
| $61$ |
\( (T^{4} - 10 T^{3} + \cdots - 3628)^{2} \)
|
| $67$ |
\( T^{8} + 100 T^{6} + \cdots + 21904 \)
|
| $71$ |
\( T^{8} + 10 T^{7} + \cdots + 1223236 \)
|
| $73$ |
\( T^{8} + 116 T^{6} + \cdots + 547600 \)
|
| $79$ |
\( T^{8} + 292 T^{6} + \cdots + 13719616 \)
|
| $83$ |
\( (T^{4} + 24 T^{3} + \cdots - 7372)^{2} \)
|
| $89$ |
\( T^{8} - 28 T^{7} + \cdots + 1795600 \)
|
| $97$ |
\( T^{8} + 292 T^{6} + \cdots + 13719616 \)
|
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