Newspace parameters
| Level: | \( N \) | \(=\) | \( 3751 = 11^{2} \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3751.t (of order \(10\), degree \(4\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(1.87199286239\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{10})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 31) |
| Projective image: | \(D_{3}\) |
| Projective field: | Galois closure of 3.1.31.1 |
| Artin image: | $C_5\times S_3$ |
| Artin field: | Galois closure of \(\mathbb{Q}[x]/(x^{15} + \cdots)\) |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3751\mathbb{Z}\right)^\times\).
| \(n\) | \(2421\) | \(2543\) |
| \(\chi(n)\) | \(-1\) | \(-\zeta_{10}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2138.1 |
|
0.809017 | + | 0.587785i | 0 | 0 | 0.809017 | − | 0.587785i | 0 | −0.309017 | − | 0.951057i | 0.309017 | − | 0.951057i | −0.809017 | − | 0.587785i | 1.00000 | ||||||||||||||||||||
| 2665.1 | 0.809017 | − | 0.587785i | 0 | 0 | 0.809017 | + | 0.587785i | 0 | −0.309017 | + | 0.951057i | 0.309017 | + | 0.951057i | −0.809017 | + | 0.587785i | 1.00000 | |||||||||||||||||||||
| 2913.1 | −0.309017 | + | 0.951057i | 0 | 0 | −0.309017 | − | 0.951057i | 0 | 0.809017 | + | 0.587785i | −0.809017 | + | 0.587785i | 0.309017 | − | 0.951057i | 1.00000 | |||||||||||||||||||||
| 3657.1 | −0.309017 | − | 0.951057i | 0 | 0 | −0.309017 | + | 0.951057i | 0 | 0.809017 | − | 0.587785i | −0.809017 | − | 0.587785i | 0.309017 | + | 0.951057i | 1.00000 | |||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 31.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-31}) \) |
| 11.c | even | 5 | 3 | inner |
| 341.t | odd | 10 | 3 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - T_{2}^{3} + T_{2}^{2} - T_{2} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(3751, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - T^{3} + T^{2} - T + 1 \)
$3$
\( T^{4} \)
$5$
\( T^{4} - T^{3} + T^{2} - T + 1 \)
$7$
\( T^{4} - T^{3} + T^{2} - T + 1 \)
$11$
\( T^{4} \)
$13$
\( T^{4} \)
$17$
\( T^{4} \)
$19$
\( T^{4} - T^{3} + T^{2} - T + 1 \)
$23$
\( T^{4} \)
$29$
\( T^{4} \)
$31$
\( T^{4} + T^{3} + T^{2} + T + 1 \)
$37$
\( T^{4} \)
$41$
\( T^{4} - T^{3} + T^{2} - T + 1 \)
$43$
\( T^{4} \)
$47$
\( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \)
$53$
\( T^{4} \)
$59$
\( T^{4} - T^{3} + T^{2} - T + 1 \)
$61$
\( T^{4} \)
$67$
\( (T - 2)^{4} \)
$71$
\( T^{4} - T^{3} + T^{2} - T + 1 \)
$73$
\( T^{4} \)
$79$
\( T^{4} \)
$83$
\( T^{4} \)
$89$
\( T^{4} \)
$97$
\( T^{4} - T^{3} + T^{2} - T + 1 \)
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