Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [348,2,Mod(59,348)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(348, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("348.59");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 348 = 2^{2} \cdot 3 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 348.c (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: | \(2.77879399034\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 3x^{12} - 44x^{8} - 243x^{4} + 6561 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 3x^{12} - 44x^{8} - 243x^{4} + 6561 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{15} + 171 \nu^{13} - 432 \nu^{12} + 78 \nu^{11} + 1674 \nu^{9} + 1296 \nu^{8} + 6274 \nu^{7} + \cdots - 69984 ) / 489888 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{15} + 171 \nu^{13} + 432 \nu^{12} + 78 \nu^{11} + 1674 \nu^{9} - 1296 \nu^{8} + 6274 \nu^{7} + \cdots + 69984 ) / 489888 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - \nu^{15} - 30 \nu^{14} + 171 \nu^{13} - 78 \nu^{11} + 1548 \nu^{10} + 1674 \nu^{9} + \cdots - 142155 \nu ) / 489888 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{15} - 30 \nu^{14} - 171 \nu^{13} + 78 \nu^{11} + 1548 \nu^{10} - 1674 \nu^{9} + \cdots + 142155 \nu ) / 489888 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 19\nu^{13} + 45\nu^{12} + 186\nu^{9} - 378\nu^{8} - 1322\nu^{5} + 2394\nu^{4} - 15795\nu + 8019 ) / 27216 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 8 \nu^{15} + 9 \nu^{14} + 63 \nu^{13} + 108 \nu^{12} + 348 \nu^{11} - 270 \nu^{10} + \cdots + 17496 ) / 244944 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 7 \nu^{15} - 9 \nu^{14} - 108 \nu^{13} - 108 \nu^{12} + 426 \nu^{11} + 270 \nu^{10} + \cdots - 17496 ) / 244944 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 5\nu^{15} - 258\nu^{11} - 1678\nu^{7} - 21141\nu^{3} ) / 122472 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( -11\nu^{14} + 57\nu^{13} + 438\nu^{10} + 558\nu^{9} - 2918\nu^{6} - 3966\nu^{5} + 24219\nu^{2} - 47385\nu ) / 81648 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( -17\nu^{13} - 30\nu^{9} + 262\nu^{5} + 11097\nu ) / 13608 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 5 \nu^{15} + 5 \nu^{14} - 129 \nu^{13} + 99 \nu^{12} - 258 \nu^{11} - 258 \nu^{10} - 342 \nu^{9} + \cdots + 9477 ) / 163296 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 5 \nu^{15} - 5 \nu^{14} - 129 \nu^{13} - 99 \nu^{12} - 258 \nu^{11} + 258 \nu^{10} - 342 \nu^{9} + \cdots + 153819 ) / 163296 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( -17\nu^{14} - 30\nu^{10} + 262\nu^{6} - 2511\nu^{2} ) / 40824 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( -17\nu^{15} - 30\nu^{11} + 262\nu^{7} - 2511\nu^{3} ) / 40824 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{10} - \beta_{8} + \beta_{7} + \beta_{5} - 2\beta_{4} - \beta_{3} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{13} + \beta_{12} - 3\beta_{9} - \beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{13} + \beta_{12} + 3\beta_{6} + 2\beta_{5} - \beta_{4} - 5\beta_{3} + 2\beta_{2} + 1 \)
|
| \(\nu^{5}\) | \(=\) |
\( 4 \beta_{13} + 4 \beta_{12} - 9 \beta_{11} + 4 \beta_{8} + 4 \beta_{7} + 3 \beta_{5} - 3 \beta_{4} + \cdots - 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{14} - 4\beta_{10} - 10\beta_{8} + 10\beta_{7} + 15\beta_{5} - \beta_{4} - 3\beta_{3} + 7\beta_{2} \)
|
| \(\nu^{7}\) | \(=\) |
\( \beta_{15} - 2 \beta_{13} - 2 \beta_{12} + 12 \beta_{9} + 2 \beta_{8} + 2 \beta_{7} + 23 \beta_{5} + \cdots + 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( -33\beta_{13} + 33\beta_{12} - 13\beta_{6} + 10\beta_{5} + 23\beta_{4} - 4\beta_{3} + 17\beta_{2} + 36 \)
|
| \(\nu^{9}\) | \(=\) |
\( 27 \beta_{13} + 27 \beta_{12} + 39 \beta_{11} + 27 \beta_{8} + 27 \beta_{7} - 69 \beta_{5} + 69 \beta_{4} + \cdots - 27 \)
|
| \(\nu^{10}\) | \(=\) |
\( -51\beta_{14} + 95\beta_{10} - 11\beta_{8} + 11\beta_{7} + 134\beta_{5} + 17\beta_{4} - 53\beta_{3} - 42\beta_{2} \)
|
| \(\nu^{11}\) | \(=\) |
\( - 51 \beta_{15} - 70 \beta_{13} - 70 \beta_{12} - 285 \beta_{9} + 70 \beta_{8} + 70 \beta_{7} + \cdots + 70 \)
|
| \(\nu^{12}\) | \(=\) |
\( -224\beta_{13} + 224\beta_{12} + 336\beta_{6} + 280\beta_{5} - 56\beta_{4} - 70\beta_{3} - 266\beta_{2} + 71 \)
|
| \(\nu^{13}\) | \(=\) |
\( 14 \beta_{13} + 14 \beta_{12} - 1008 \beta_{11} + 14 \beta_{8} + 14 \beta_{7} + 168 \beta_{5} + \cdots - 14 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 2296 \beta_{14} - 377 \beta_{10} + 13 \beta_{8} - 13 \beta_{7} - 153 \beta_{5} + 250 \beta_{4} + \cdots + 182 \beta_{2} \)
|
| \(\nu^{15}\) | \(=\) |
\( - 2296 \beta_{15} - 55 \beta_{13} - 55 \beta_{12} + 1131 \beta_{9} + 55 \beta_{8} + 55 \beta_{7} + \cdots + 55 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/348\mathbb{Z}\right)^\times\).
| \(n\) | \(175\) | \(205\) | \(233\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 59.1 |
|
−1.17915 | − | 0.780776i | −0.209152 | + | 1.71938i | 0.780776 | + | 1.84130i | − | 1.49324i | 1.58907 | − | 1.86410i | − | 2.68490i | 0.516994 | − | 2.78078i | −2.91251 | − | 0.719224i | −1.16589 | + | 1.76075i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 59.2 | −1.17915 | − | 0.780776i | 1.71938 | − | 0.209152i | 0.780776 | + | 1.84130i | 3.05480i | −2.19070 | − | 1.09583i | 0.326603i | 0.516994 | − | 2.78078i | 2.91251 | − | 0.719224i | 2.38511 | − | 3.60205i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 59.3 | −1.17915 | + | 0.780776i | −0.209152 | − | 1.71938i | 0.780776 | − | 1.84130i | 1.49324i | 1.58907 | + | 1.86410i | 2.68490i | 0.516994 | + | 2.78078i | −2.91251 | + | 0.719224i | −1.16589 | − | 1.76075i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 59.4 | −1.17915 | + | 0.780776i | 1.71938 | + | 0.209152i | 0.780776 | − | 1.84130i | − | 3.05480i | −2.19070 | + | 1.09583i | − | 0.326603i | 0.516994 | + | 2.78078i | 2.91251 | + | 0.719224i | 2.38511 | + | 3.60205i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 59.5 | −0.599676 | − | 1.28078i | −1.43627 | − | 0.968056i | −1.28078 | + | 1.53610i | 2.72259i | −0.378566 | + | 2.42006i | − | 2.47973i | 2.73546 | + | 0.719224i | 1.12574 | + | 2.78078i | 3.48703 | − | 1.63267i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 59.6 | −0.599676 | − | 1.28078i | 0.968056 | + | 1.43627i | −1.28078 | + | 1.53610i | − | 0.161040i | 1.25902 | − | 2.10116i | 3.67908i | 2.73546 | + | 0.719224i | −1.12574 | + | 2.78078i | −0.206257 | + | 0.0965721i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 59.7 | −0.599676 | + | 1.28078i | −1.43627 | + | 0.968056i | −1.28078 | − | 1.53610i | − | 2.72259i | −0.378566 | − | 2.42006i | 2.47973i | 2.73546 | − | 0.719224i | 1.12574 | − | 2.78078i | 3.48703 | + | 1.63267i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 59.8 | −0.599676 | + | 1.28078i | 0.968056 | − | 1.43627i | −1.28078 | − | 1.53610i | 0.161040i | 1.25902 | + | 2.10116i | − | 3.67908i | 2.73546 | − | 0.719224i | −1.12574 | − | 2.78078i | −0.206257 | − | 0.0965721i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 59.9 | 0.599676 | − | 1.28078i | −0.968056 | − | 1.43627i | −1.28078 | − | 1.53610i | − | 0.161040i | −2.42006 | + | 0.378566i | − | 3.67908i | −2.73546 | + | 0.719224i | −1.12574 | + | 2.78078i | −0.206257 | − | 0.0965721i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 59.10 | 0.599676 | − | 1.28078i | 1.43627 | + | 0.968056i | −1.28078 | − | 1.53610i | 2.72259i | 2.10116 | − | 1.25902i | 2.47973i | −2.73546 | + | 0.719224i | 1.12574 | + | 2.78078i | 3.48703 | + | 1.63267i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 59.11 | 0.599676 | + | 1.28078i | −0.968056 | + | 1.43627i | −1.28078 | + | 1.53610i | 0.161040i | −2.42006 | − | 0.378566i | 3.67908i | −2.73546 | − | 0.719224i | −1.12574 | − | 2.78078i | −0.206257 | + | 0.0965721i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 59.12 | 0.599676 | + | 1.28078i | 1.43627 | − | 0.968056i | −1.28078 | + | 1.53610i | − | 2.72259i | 2.10116 | + | 1.25902i | − | 2.47973i | −2.73546 | − | 0.719224i | 1.12574 | − | 2.78078i | 3.48703 | − | 1.63267i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 59.13 | 1.17915 | − | 0.780776i | −1.71938 | + | 0.209152i | 0.780776 | − | 1.84130i | 3.05480i | −1.86410 | + | 1.58907i | − | 0.326603i | −0.516994 | − | 2.78078i | 2.91251 | − | 0.719224i | 2.38511 | + | 3.60205i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 59.14 | 1.17915 | − | 0.780776i | 0.209152 | − | 1.71938i | 0.780776 | − | 1.84130i | − | 1.49324i | −1.09583 | − | 2.19070i | 2.68490i | −0.516994 | − | 2.78078i | −2.91251 | − | 0.719224i | −1.16589 | − | 1.76075i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 59.15 | 1.17915 | + | 0.780776i | −1.71938 | − | 0.209152i | 0.780776 | + | 1.84130i | − | 3.05480i | −1.86410 | − | 1.58907i | 0.326603i | −0.516994 | + | 2.78078i | 2.91251 | + | 0.719224i | 2.38511 | − | 3.60205i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 59.16 | 1.17915 | + | 0.780776i | 0.209152 | + | 1.71938i | 0.780776 | + | 1.84130i | 1.49324i | −1.09583 | + | 2.19070i | − | 2.68490i | −0.516994 | + | 2.78078i | −2.91251 | + | 0.719224i | −1.16589 | + | 1.76075i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 4.b | odd | 2 | 1 | inner |
| 12.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 348.2.c.a | ✓ | 16 |
| 3.b | odd | 2 | 1 | inner | 348.2.c.a | ✓ | 16 |
| 4.b | odd | 2 | 1 | inner | 348.2.c.a | ✓ | 16 |
| 12.b | even | 2 | 1 | inner | 348.2.c.a | ✓ | 16 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 348.2.c.a | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 348.2.c.a | ✓ | 16 | 3.b | odd | 2 | 1 | inner |
| 348.2.c.a | ✓ | 16 | 4.b | odd | 2 | 1 | inner |
| 348.2.c.a | ✓ | 16 | 12.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 19T_{5}^{6} + 107T_{5}^{4} + 157T_{5}^{2} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(348, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{8} + T^{6} + 4 T^{4} + \cdots + 16)^{2} \)
|
| $3$ |
\( T^{16} - 3 T^{12} + \cdots + 6561 \)
|
| $5$ |
\( (T^{8} + 19 T^{6} + 107 T^{4} + \cdots + 4)^{2} \)
|
| $7$ |
\( (T^{8} + 27 T^{6} + \cdots + 64)^{2} \)
|
| $11$ |
\( (T^{8} - 56 T^{6} + \cdots + 256)^{2} \)
|
| $13$ |
\( (T^{4} - 6 T^{3} + \cdots - 288)^{4} \)
|
| $17$ |
\( (T^{8} + 77 T^{6} + \cdots + 1024)^{2} \)
|
| $19$ |
\( (T^{8} + 67 T^{6} + \cdots + 5184)^{2} \)
|
| $23$ |
\( (T^{4} - 56 T^{2} + 512)^{4} \)
|
| $29$ |
\( (T^{2} + 1)^{8} \)
|
| $31$ |
\( (T^{8} + 144 T^{6} + \cdots + 440896)^{2} \)
|
| $37$ |
\( (T^{4} - 11 T^{3} + \cdots - 256)^{4} \)
|
| $41$ |
\( (T^{8} + 189 T^{6} + \cdots + 153664)^{2} \)
|
| $43$ |
\( (T^{8} + 91 T^{6} + \cdots + 82944)^{2} \)
|
| $47$ |
\( (T^{8} - 207 T^{6} + \cdots + 4)^{2} \)
|
| $53$ |
\( (T^{8} + 338 T^{6} + \cdots + 1056784)^{2} \)
|
| $59$ |
\( (T^{8} - 235 T^{6} + \cdots + 4096)^{2} \)
|
| $61$ |
\( (T^{4} - 14 T^{3} + \cdots - 1152)^{4} \)
|
| $67$ |
\( (T^{8} + 284 T^{6} + \cdots + 331776)^{2} \)
|
| $71$ |
\( (T^{8} - 220 T^{6} + \cdots + 1024)^{2} \)
|
| $73$ |
\( (T^{2} + 6 T - 8)^{8} \)
|
| $79$ |
\( (T^{8} + 156 T^{6} + \cdots + 16)^{2} \)
|
| $83$ |
\( (T^{8} - 552 T^{6} + \cdots + 142468096)^{2} \)
|
| $89$ |
\( (T^{4} + 112 T^{2} + 688)^{4} \)
|
| $97$ |
\( (T + 14)^{16} \)
|
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