Properties

Label 2011.1.b.a
Level 2011
Weight 1
Character orbit 2011.b
Self dual Yes
Analytic conductor 1.004
Analytic rank 0
Dimension 3
Projective image \(D_{7}\)
CM disc. -2011
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 2011 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 2011.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(1.0036197404\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{7}\)
Projective field Galois closure of 7.1.8132727331.1
Artin image size \(14\)
Artin image $D_7$
Artin field Galois closure of 7.1.8132727331.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(+ q^{4}\) \( -\beta_{1} q^{5} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(+ q^{4}\) \( -\beta_{1} q^{5} \) \(+ q^{9}\) \( + ( -1 + \beta_{1} - \beta_{2} ) q^{13} \) \(+ q^{16}\) \( -\beta_{1} q^{20} \) \( + \beta_{2} q^{23} \) \( + ( 1 + \beta_{2} ) q^{25} \) \( + \beta_{2} q^{31} \) \(+ q^{36}\) \( + ( -1 + \beta_{1} - \beta_{2} ) q^{41} \) \( + ( -1 + \beta_{1} - \beta_{2} ) q^{43} \) \( -\beta_{1} q^{45} \) \(+ q^{49}\) \( + ( -1 + \beta_{1} - \beta_{2} ) q^{52} \) \(+ q^{64}\) \( + ( -1 + \beta_{1} ) q^{65} \) \( + ( -1 + \beta_{1} - \beta_{2} ) q^{71} \) \( -\beta_{1} q^{80} \) \(+ q^{81}\) \( + \beta_{2} q^{83} \) \( + \beta_{2} q^{89} \) \( + \beta_{2} q^{92} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(3q \) \(\mathstrut +\mathstrut 3q^{4} \) \(\mathstrut -\mathstrut q^{5} \) \(\mathstrut +\mathstrut 3q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut +\mathstrut 3q^{4} \) \(\mathstrut -\mathstrut q^{5} \) \(\mathstrut +\mathstrut 3q^{9} \) \(\mathstrut -\mathstrut q^{13} \) \(\mathstrut +\mathstrut 3q^{16} \) \(\mathstrut -\mathstrut q^{20} \) \(\mathstrut -\mathstrut q^{23} \) \(\mathstrut +\mathstrut 2q^{25} \) \(\mathstrut -\mathstrut q^{31} \) \(\mathstrut +\mathstrut 3q^{36} \) \(\mathstrut -\mathstrut q^{41} \) \(\mathstrut -\mathstrut q^{43} \) \(\mathstrut -\mathstrut q^{45} \) \(\mathstrut +\mathstrut 3q^{49} \) \(\mathstrut -\mathstrut q^{52} \) \(\mathstrut +\mathstrut 3q^{64} \) \(\mathstrut -\mathstrut 2q^{65} \) \(\mathstrut -\mathstrut q^{71} \) \(\mathstrut -\mathstrut q^{80} \) \(\mathstrut +\mathstrut 3q^{81} \) \(\mathstrut -\mathstrut q^{83} \) \(\mathstrut -\mathstrut q^{89} \) \(\mathstrut -\mathstrut q^{92} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(2\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2011\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2010.1
1.80194
0.445042
−1.24698
0 0 1.00000 −1.80194 0 0 0 1.00000 0
2010.2 0 0 1.00000 −0.445042 0 0 0 1.00000 0
2010.3 0 0 1.00000 1.24698 0 0 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
2011.b Odd 1 CM by \(\Q(\sqrt{-2011}) \) yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(2011, [\chi])\).