Properties

Label 3311.1.h.g
Level 3311
Weight 1
Character orbit 3311.h
Analytic conductor 1.652
Analytic rank 0
Dimension 2
Projective image \(D_{6}\)
RM disc. 473
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 3311 = 7 \cdot 11 \cdot 43 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 3311.h (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(1.65240425683\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{6}\)
Projective field Galois closure of 6.0.76739047.3

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q\) \(- q^{2}\) \( + \zeta_{6}^{2} q^{7} \) \(+ q^{8}\) \(- q^{9}\) \(+O(q^{10})\) \( q\) \(- q^{2}\) \( + \zeta_{6}^{2} q^{7} \) \(+ q^{8}\) \(- q^{9}\) \(- q^{11}\) \( -\zeta_{6}^{2} q^{14} \) \(- q^{16}\) \(+ q^{18}\) \( + ( \zeta_{6} + \zeta_{6}^{2} ) q^{19} \) \(+ q^{22}\) \(+ q^{23}\) \(- q^{25}\) \(+ q^{29}\) \( + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{31} \) \( + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{38} \) \(- q^{43}\) \(- q^{46}\) \( + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{47} \) \( -\zeta_{6} q^{49} \) \(+ q^{50}\) \(- q^{53}\) \( + \zeta_{6}^{2} q^{56} \) \(- q^{58}\) \( + ( \zeta_{6} + \zeta_{6}^{2} ) q^{59} \) \( + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{61} \) \( + ( \zeta_{6} + \zeta_{6}^{2} ) q^{62} \) \( -\zeta_{6}^{2} q^{63} \) \(+ q^{64}\) \(- q^{67}\) \(- q^{72}\) \( -\zeta_{6}^{2} q^{77} \) \(+ q^{81}\) \(+ q^{86}\) \(- q^{88}\) \( + ( \zeta_{6} + \zeta_{6}^{2} ) q^{94} \) \( + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{97} \) \( + \zeta_{6} q^{98} \) \(+ q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut +\mathstrut 2q^{8} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut +\mathstrut 2q^{8} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut -\mathstrut 2q^{11} \) \(\mathstrut +\mathstrut q^{14} \) \(\mathstrut -\mathstrut 2q^{16} \) \(\mathstrut +\mathstrut 2q^{18} \) \(\mathstrut +\mathstrut 2q^{22} \) \(\mathstrut +\mathstrut 2q^{23} \) \(\mathstrut -\mathstrut 2q^{25} \) \(\mathstrut +\mathstrut 2q^{29} \) \(\mathstrut -\mathstrut 2q^{43} \) \(\mathstrut -\mathstrut 2q^{46} \) \(\mathstrut -\mathstrut q^{49} \) \(\mathstrut +\mathstrut 2q^{50} \) \(\mathstrut -\mathstrut 2q^{53} \) \(\mathstrut -\mathstrut q^{56} \) \(\mathstrut -\mathstrut 2q^{58} \) \(\mathstrut +\mathstrut q^{63} \) \(\mathstrut +\mathstrut 2q^{64} \) \(\mathstrut -\mathstrut 2q^{67} \) \(\mathstrut -\mathstrut 2q^{72} \) \(\mathstrut +\mathstrut q^{77} \) \(\mathstrut +\mathstrut 2q^{81} \) \(\mathstrut +\mathstrut 2q^{86} \) \(\mathstrut -\mathstrut 2q^{88} \) \(\mathstrut +\mathstrut q^{98} \) \(\mathstrut +\mathstrut 2q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3311\mathbb{Z}\right)^\times\).

\(n\) \(904\) \(1893\) \(2927\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3310.1
0.500000 0.866025i
0.500000 + 0.866025i
−1.00000 0 0 0 0 −0.500000 0.866025i 1.00000 −1.00000 0
3310.2 −1.00000 0 0 0 0 −0.500000 + 0.866025i 1.00000 −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
473.d Even 1 RM by \(\Q(\sqrt{473}) \) yes
7.b Odd 1 yes
3311.h Odd 1 yes

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3311, [\chi])\):

\(T_{2} \) \(\mathstrut +\mathstrut 1 \)
\(T_{3} \)