Newspace parameters
Level: | \( N \) | \(=\) | \( 155 = 5 \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 155.f (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.23768123133\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Relative dimension: | \(6\) over \(\Q(i)\) |
Coefficient field: | 12.0.1931902335935778816.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{12} - 15x^{6} + 64 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2\cdot 5 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{U}(1)[D_{4}]$ |
$q$-expansion
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 \( x^{12} - 15x^{6} + 64 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{9} - 7\nu^{3} ) / 8 \) |
\(\beta_{2}\) | \(=\) | \( ( -\nu^{10} + 15\nu^{4} - 16\nu ) / 16 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{8} - 7\nu^{2} - 4\nu ) / 4 \) |
\(\beta_{4}\) | \(=\) | \( ( -\nu^{11} + 16\nu^{7} + 15\nu^{5} - 112\nu ) / 32 \) |
\(\beta_{5}\) | \(=\) | \( ( \nu^{10} + 4\nu^{8} + \nu^{4} - 28\nu^{2} + 16\nu ) / 16 \) |
\(\beta_{6}\) | \(=\) | \( ( \nu^{11} - 4\nu^{10} - 15\nu^{5} + 28\nu^{4} ) / 32 \) |
\(\beta_{7}\) | \(=\) | \( ( -\nu^{8} + 11\nu^{2} ) / 4 \) |
\(\beta_{8}\) | \(=\) | \( ( 3\nu^{11} - 13\nu^{5} ) / 32 \) |
\(\beta_{9}\) | \(=\) | \( ( -\nu^{11} + 8\nu^{8} + 15\nu^{5} + 32\nu^{3} - 56\nu^{2} + 32\nu ) / 32 \) |
\(\beta_{10}\) | \(=\) | \( ( -\nu^{11} - 4\nu^{10} - 16\nu^{7} + 15\nu^{5} + 28\nu^{4} + 144\nu ) / 32 \) |
\(\beta_{11}\) | \(=\) | \( ( -\nu^{11} + 8\nu^{8} + 32\nu^{6} + 15\nu^{5} - 56\nu^{2} + 32\nu - 256 ) / 32 \) |
\(\nu\) | \(=\) | \( ( \beta_{10} + 2\beta_{6} + 3\beta_{5} + \beta_{4} - 3\beta_{3} - 3\beta_{2} ) / 10 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{10} + 10\beta_{7} + 2\beta_{6} + 3\beta_{5} + \beta_{4} + 7\beta_{3} - 3\beta_{2} ) / 10 \) |
\(\nu^{3}\) | \(=\) | \( ( -\beta_{10} + 2\beta_{9} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( ( -\beta_{10} - 2\beta_{6} + 7\beta_{5} - \beta_{4} - 7\beta_{3} + 13\beta_{2} ) / 10 \) |
\(\nu^{5}\) | \(=\) | \( ( 9\beta_{10} + 10\beta_{8} - 12\beta_{6} - 3\beta_{5} + 9\beta_{4} + 3\beta_{3} + 3\beta_{2} ) / 10 \) |
\(\nu^{6}\) | \(=\) | \( ( 2\beta_{11} - \beta_{10} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 16 ) / 2 \) |
\(\nu^{7}\) | \(=\) | \( ( \beta_{10} + 22\beta_{6} + 23\beta_{5} + 21\beta_{4} - 23\beta_{3} - 23\beta_{2} ) / 10 \) |
\(\nu^{8}\) | \(=\) | \( ( 11\beta_{10} + 70\beta_{7} + 22\beta_{6} + 33\beta_{5} + 11\beta_{4} + 77\beta_{3} - 33\beta_{2} ) / 10 \) |
\(\nu^{9}\) | \(=\) | \( ( -7\beta_{10} + 14\beta_{9} - 7\beta_{5} - 7\beta_{4} - 7\beta_{3} + 7\beta_{2} + 16\beta_1 ) / 2 \) |
\(\nu^{10}\) | \(=\) | \( ( -31\beta_{10} - 62\beta_{6} + 57\beta_{5} - 31\beta_{4} - 57\beta_{3} + 83\beta_{2} ) / 10 \) |
\(\nu^{11}\) | \(=\) | \( ( 39\beta_{10} + 150\beta_{8} - 52\beta_{6} - 13\beta_{5} + 39\beta_{4} + 13\beta_{3} + 13\beta_{2} ) / 10 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/155\mathbb{Z}\right)^\times\).
\(n\) | \(32\) | \(96\) |
\(\chi(n)\) | \(\beta_{1}\) | \(-1\) |
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.
Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
92.1 |
|
−1.89782 | − | 1.89782i | 0 | 5.20343i | 2.23507 | + | 0.0667460i | 0 | −3.43053 | − | 3.43053i | 6.07952 | − | 6.07952i | − | 3.00000i | −4.11509 | − | 4.36843i | |||||||||||||||||||||||||||||||||||||||||||
92.2 | −1.32801 | − | 1.32801i | 0 | 1.52721i | 1.17534 | + | 1.90226i | 0 | 3.71783 | + | 3.71783i | −0.627865 | + | 0.627865i | − | 3.00000i | 0.965351 | − | 4.08707i | ||||||||||||||||||||||||||||||||||||||||||||
92.3 | −0.631100 | − | 0.631100i | 0 | − | 1.20343i | −2.23507 | + | 0.0667460i | 0 | −1.49382 | − | 1.49382i | −2.02168 | + | 2.02168i | − | 3.00000i | 1.45268 | + | 1.36843i | |||||||||||||||||||||||||||||||||||||||||||
92.4 | 0.402360 | + | 0.402360i | 0 | − | 1.67621i | −1.05973 | − | 1.96900i | 0 | 3.00895 | + | 3.00895i | 1.47916 | − | 1.47916i | − | 3.00000i | 0.365854 | − | 1.21864i | |||||||||||||||||||||||||||||||||||||||||||
92.5 | 1.49546 | + | 1.49546i | 0 | 2.47279i | −1.17534 | + | 1.90226i | 0 | 0.421578 | + | 0.421578i | −0.707033 | + | 0.707033i | − | 3.00000i | −4.60241 | + | 1.08707i | ||||||||||||||||||||||||||||||||||||||||||||
92.6 | 1.95911 | + | 1.95911i | 0 | 5.67621i | 1.05973 | − | 1.96900i | 0 | −2.22401 | − | 2.22401i | −7.20210 | + | 7.20210i | − | 3.00000i | 5.93362 | − | 1.78136i | ||||||||||||||||||||||||||||||||||||||||||||
123.1 | −1.89782 | + | 1.89782i | 0 | − | 5.20343i | 2.23507 | − | 0.0667460i | 0 | −3.43053 | + | 3.43053i | 6.07952 | + | 6.07952i | 3.00000i | −4.11509 | + | 4.36843i | ||||||||||||||||||||||||||||||||||||||||||||
123.2 | −1.32801 | + | 1.32801i | 0 | − | 1.52721i | 1.17534 | − | 1.90226i | 0 | 3.71783 | − | 3.71783i | −0.627865 | − | 0.627865i | 3.00000i | 0.965351 | + | 4.08707i | ||||||||||||||||||||||||||||||||||||||||||||
123.3 | −0.631100 | + | 0.631100i | 0 | 1.20343i | −2.23507 | − | 0.0667460i | 0 | −1.49382 | + | 1.49382i | −2.02168 | − | 2.02168i | 3.00000i | 1.45268 | − | 1.36843i | |||||||||||||||||||||||||||||||||||||||||||||
123.4 | 0.402360 | − | 0.402360i | 0 | 1.67621i | −1.05973 | + | 1.96900i | 0 | 3.00895 | − | 3.00895i | 1.47916 | + | 1.47916i | 3.00000i | 0.365854 | + | 1.21864i | |||||||||||||||||||||||||||||||||||||||||||||
123.5 | 1.49546 | − | 1.49546i | 0 | − | 2.47279i | −1.17534 | − | 1.90226i | 0 | 0.421578 | − | 0.421578i | −0.707033 | − | 0.707033i | 3.00000i | −4.60241 | − | 1.08707i | ||||||||||||||||||||||||||||||||||||||||||||
123.6 | 1.95911 | − | 1.95911i | 0 | − | 5.67621i | 1.05973 | + | 1.96900i | 0 | −2.22401 | + | 2.22401i | −7.20210 | − | 7.20210i | 3.00000i | 5.93362 | + | 1.78136i | ||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
31.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-31}) \) |
5.c | odd | 4 | 1 | inner |
155.f | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 155.2.f.a | ✓ | 12 |
5.b | even | 2 | 1 | 775.2.f.e | 12 | ||
5.c | odd | 4 | 1 | inner | 155.2.f.a | ✓ | 12 |
5.c | odd | 4 | 1 | 775.2.f.e | 12 | ||
31.b | odd | 2 | 1 | CM | 155.2.f.a | ✓ | 12 |
155.c | odd | 2 | 1 | 775.2.f.e | 12 | ||
155.f | even | 4 | 1 | inner | 155.2.f.a | ✓ | 12 |
155.f | even | 4 | 1 | 775.2.f.e | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
155.2.f.a | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
155.2.f.a | ✓ | 12 | 5.c | odd | 4 | 1 | inner |
155.2.f.a | ✓ | 12 | 31.b | odd | 2 | 1 | CM |
155.2.f.a | ✓ | 12 | 155.f | even | 4 | 1 | inner |
775.2.f.e | 12 | 5.b | even | 2 | 1 | ||
775.2.f.e | 12 | 5.c | odd | 4 | 1 | ||
775.2.f.e | 12 | 155.c | odd | 2 | 1 | ||
775.2.f.e | 12 | 155.f | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} + 2T_{2}^{9} + 72T_{2}^{8} + 12T_{2}^{7} + 2T_{2}^{6} + 72T_{2}^{5} + 936T_{2}^{4} + 402T_{2}^{3} + 72T_{2}^{2} - 180T_{2} + 225 \)
acting on \(S_{2}^{\mathrm{new}}(155, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} + 2 T^{9} + 72 T^{8} + 12 T^{7} + \cdots + 225 \)
$3$
\( T^{12} \)
$5$
\( T^{12} - 246 T^{6} + 15625 \)
$7$
\( T^{12} + 32 T^{9} + 882 T^{8} + \cdots + 184900 \)
$11$
\( T^{12} \)
$13$
\( T^{12} \)
$17$
\( T^{12} \)
$19$
\( (T^{6} + 114 T^{4} + 3249 T^{2} + \cdots + 24336)^{2} \)
$23$
\( T^{12} \)
$29$
\( T^{12} \)
$31$
\( (T^{2} - 31)^{6} \)
$37$
\( T^{12} \)
$41$
\( (T^{3} - 123 T + 278)^{4} \)
$43$
\( T^{12} \)
$47$
\( (T^{4} - 16 T^{3} + 128 T^{2} + 480 T + 900)^{3} \)
$53$
\( T^{12} \)
$59$
\( (T^{6} + 354 T^{4} + 31329 T^{2} + \cdots + 273916)^{2} \)
$61$
\( T^{12} \)
$67$
\( (T^{4} + 24 T^{3} + 288 T^{2} + 240 T + 100)^{3} \)
$71$
\( (T^{6} - 426 T^{4} + 45369 T^{2} + \cdots - 431644)^{2} \)
$73$
\( T^{12} \)
$79$
\( T^{12} \)
$83$
\( T^{12} \)
$89$
\( T^{12} \)
$97$
\( T^{12} + 3812 T^{9} + \cdots + 3267020100100 \)
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