Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [825,2,Mod(626,825)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(825, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("825.626");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 825 = 3 \cdot 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 825.f (of order \(2\), degree \(1\), 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: | \(6.58765816676\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| 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 165) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).
| \(n\) | \(376\) | \(551\) | \(727\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 626.1 |
|
−1.87228 | −0.146426 | − | 1.72585i | 1.50542 | 0 | 0.274150 | + | 3.23127i | 1.49458i | 0.925994 | −2.95712 | + | 0.505418i | 0 | ||||||||||||||||||||||||||||||||||||
| 626.2 | −1.87228 | −0.146426 | + | 1.72585i | 1.50542 | 0 | 0.274150 | − | 3.23127i | − | 1.49458i | 0.925994 | −2.95712 | − | 0.505418i | 0 | ||||||||||||||||||||||||||||||||||||
| 626.3 | −1.37033 | −1.70032 | − | 0.329998i | −0.122207 | 0 | 2.33000 | + | 0.452205i | − | 3.12221i | 2.90812 | 2.78220 | + | 1.12221i | 0 | ||||||||||||||||||||||||||||||||||||
| 626.4 | −1.37033 | −1.70032 | + | 0.329998i | −0.122207 | 0 | 2.33000 | − | 0.452205i | 3.12221i | 2.90812 | 2.78220 | − | 1.12221i | 0 | |||||||||||||||||||||||||||||||||||||
| 626.5 | 0.796815 | 1.55654 | − | 0.759725i | −1.36509 | 0 | 1.24027 | − | 0.605361i | 4.36509i | −2.68135 | 1.84564 | − | 2.36509i | 0 | |||||||||||||||||||||||||||||||||||||
| 626.6 | 0.796815 | 1.55654 | + | 0.759725i | −1.36509 | 0 | 1.24027 | + | 0.605361i | − | 4.36509i | −2.68135 | 1.84564 | + | 2.36509i | 0 | ||||||||||||||||||||||||||||||||||||
| 626.7 | 2.44579 | 1.29021 | − | 1.15558i | 3.98187 | 0 | 3.15558 | − | 2.82630i | 0.981874i | 4.84724 | 0.329281 | − | 2.98187i | 0 | |||||||||||||||||||||||||||||||||||||
| 626.8 | 2.44579 | 1.29021 | + | 1.15558i | 3.98187 | 0 | 3.15558 | + | 2.82630i | − | 0.981874i | 4.84724 | 0.329281 | + | 2.98187i | 0 | ||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 33.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 825.2.f.d | 8 | |
| 3.b | odd | 2 | 1 | 825.2.f.c | 8 | ||
| 5.b | even | 2 | 1 | 165.2.f.b | yes | 8 | |
| 5.c | odd | 4 | 1 | 825.2.d.c | 8 | ||
| 5.c | odd | 4 | 1 | 825.2.d.e | 8 | ||
| 11.b | odd | 2 | 1 | 825.2.f.c | 8 | ||
| 15.d | odd | 2 | 1 | 165.2.f.a | ✓ | 8 | |
| 15.e | even | 4 | 1 | 825.2.d.b | 8 | ||
| 15.e | even | 4 | 1 | 825.2.d.d | 8 | ||
| 20.d | odd | 2 | 1 | 2640.2.f.c | 8 | ||
| 33.d | even | 2 | 1 | inner | 825.2.f.d | 8 | |
| 55.d | odd | 2 | 1 | 165.2.f.a | ✓ | 8 | |
| 55.e | even | 4 | 1 | 825.2.d.b | 8 | ||
| 55.e | even | 4 | 1 | 825.2.d.d | 8 | ||
| 60.h | even | 2 | 1 | 2640.2.f.d | 8 | ||
| 165.d | even | 2 | 1 | 165.2.f.b | yes | 8 | |
| 165.l | odd | 4 | 1 | 825.2.d.c | 8 | ||
| 165.l | odd | 4 | 1 | 825.2.d.e | 8 | ||
| 220.g | even | 2 | 1 | 2640.2.f.d | 8 | ||
| 660.g | odd | 2 | 1 | 2640.2.f.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 165.2.f.a | ✓ | 8 | 15.d | odd | 2 | 1 | |
| 165.2.f.a | ✓ | 8 | 55.d | odd | 2 | 1 | |
| 165.2.f.b | yes | 8 | 5.b | even | 2 | 1 | |
| 165.2.f.b | yes | 8 | 165.d | even | 2 | 1 | |
| 825.2.d.b | 8 | 15.e | even | 4 | 1 | ||
| 825.2.d.b | 8 | 55.e | even | 4 | 1 | ||
| 825.2.d.c | 8 | 5.c | odd | 4 | 1 | ||
| 825.2.d.c | 8 | 165.l | odd | 4 | 1 | ||
| 825.2.d.d | 8 | 15.e | even | 4 | 1 | ||
| 825.2.d.d | 8 | 55.e | even | 4 | 1 | ||
| 825.2.d.e | 8 | 5.c | odd | 4 | 1 | ||
| 825.2.d.e | 8 | 165.l | odd | 4 | 1 | ||
| 825.2.f.c | 8 | 3.b | odd | 2 | 1 | ||
| 825.2.f.c | 8 | 11.b | odd | 2 | 1 | ||
| 825.2.f.d | 8 | 1.a | even | 1 | 1 | trivial | |
| 825.2.f.d | 8 | 33.d | even | 2 | 1 | inner | |
| 2640.2.f.c | 8 | 20.d | odd | 2 | 1 | ||
| 2640.2.f.c | 8 | 660.g | odd | 2 | 1 | ||
| 2640.2.f.d | 8 | 60.h | even | 2 | 1 | ||
| 2640.2.f.d | 8 | 220.g | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(825, [\chi])\):
|
\( T_{2}^{4} - 6T_{2}^{2} - 2T_{2} + 5 \)
|
|
\( T_{23}^{8} + 100T_{23}^{6} + 3120T_{23}^{4} + 31232T_{23}^{2} + 87616 \)
|
|
\( T_{37}^{4} + 4T_{37}^{3} - 16T_{37}^{2} - 64T_{37} - 16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - 6 T^{2} - 2 T + 5)^{2} \)
|
| $3$ |
\( T^{8} - 2 T^{7} + \cdots + 81 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + 32 T^{6} + \cdots + 400 \)
|
| $11$ |
\( T^{8} - 12 T^{7} + \cdots + 14641 \)
|
| $13$ |
\( T^{8} + 56 T^{6} + \cdots + 19600 \)
|
| $17$ |
\( (T^{4} + 2 T^{3} + \cdots + 740)^{2} \)
|
| $19$ |
\( T^{8} + 68 T^{6} + \cdots + 1600 \)
|
| $23$ |
\( T^{8} + 100 T^{6} + \cdots + 87616 \)
|
| $29$ |
\( (T^{4} + 10 T^{3} + \cdots + 40)^{2} \)
|
| $31$ |
\( (T^{4} - 24 T^{2} + \cdots + 80)^{2} \)
|
| $37$ |
\( (T^{4} + 4 T^{3} - 16 T^{2} + \cdots - 16)^{2} \)
|
| $41$ |
\( (T^{4} - 6 T^{3} + \cdots + 280)^{2} \)
|
| $43$ |
\( T^{8} + 208 T^{6} + \cdots + 2131600 \)
|
| $47$ |
\( T^{8} + 148 T^{6} + \cdots + 3136 \)
|
| $53$ |
\( T^{8} + 128 T^{6} + \cdots + 43264 \)
|
| $59$ |
\( T^{8} + 368 T^{6} + \cdots + 32809984 \)
|
| $61$ |
\( T^{8} + 256 T^{6} + \cdots + 640000 \)
|
| $67$ |
\( (T^{4} + 22 T^{3} + \cdots + 632)^{2} \)
|
| $71$ |
\( T^{8} + 80 T^{6} + \cdots + 25600 \)
|
| $73$ |
\( T^{8} + 424 T^{6} + \cdots + 4080400 \)
|
| $79$ |
\( T^{8} + 340 T^{6} + \cdots + 270400 \)
|
| $83$ |
\( (T^{4} + 18 T^{3} + \cdots - 20)^{2} \)
|
| $89$ |
\( T^{8} + 496 T^{6} + \cdots + 15366400 \)
|
| $97$ |
\( (T^{4} - 12 T^{3} + \cdots - 1168)^{2} \)
|
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