Properties

Label 3311.1.h.i
Level 3311
Weight 1
Character orbit 3311.h
Self dual Yes
Analytic conductor 1.652
Analytic rank 0
Dimension 2
Projective image \(D_{6}\)
CM disc. -3311
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: Yes
Analytic conductor: \(1.65240425683\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{6}\)
Projective field Galois closure of 6.0.76739047.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(+ q^{2}\) \( -\beta q^{3} \) \( -\beta q^{5} \) \( -\beta q^{6} \) \(- q^{7}\) \(- q^{8}\) \( + 2 q^{9} \) \(+O(q^{10})\) \( q\) \(+ q^{2}\) \( -\beta q^{3} \) \( -\beta q^{5} \) \( -\beta q^{6} \) \(- q^{7}\) \(- q^{8}\) \( + 2 q^{9} \) \( -\beta q^{10} \) \(- q^{11}\) \(- q^{14}\) \( + 3 q^{15} \) \(- q^{16}\) \( -\beta q^{17} \) \( + 2 q^{18} \) \( + \beta q^{21} \) \(- q^{22}\) \( -2 q^{23} \) \( + \beta q^{24} \) \( + 2 q^{25} \) \( -\beta q^{27} \) \(- q^{29}\) \( + 3 q^{30} \) \( + \beta q^{33} \) \( -\beta q^{34} \) \( + \beta q^{35} \) \( + \beta q^{40} \) \( + \beta q^{41} \) \( + \beta q^{42} \) \(+ q^{43}\) \( -2 \beta q^{45} \) \( -2 q^{46} \) \( + \beta q^{48} \) \(+ q^{49}\) \( + 2 q^{50} \) \( + 3 q^{51} \) \(- q^{53}\) \( -\beta q^{54} \) \( + \beta q^{55} \) \(+ q^{56}\) \(- q^{58}\) \( -2 q^{63} \) \(+ q^{64}\) \( + \beta q^{66} \) \(- q^{67}\) \( + 2 \beta q^{69} \) \( + \beta q^{70} \) \( -2 q^{72} \) \( -2 \beta q^{75} \) \(+ q^{77}\) \( + \beta q^{80} \) \(+ q^{81}\) \( + \beta q^{82} \) \( -\beta q^{83} \) \( + 3 q^{85} \) \(+ q^{86}\) \( + \beta q^{87} \) \(+ q^{88}\) \( -2 \beta q^{90} \) \(+ q^{98}\) \( -2 q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut -\mathstrut 2q^{8} \) \(\mathstrut +\mathstrut 4q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut -\mathstrut 2q^{8} \) \(\mathstrut +\mathstrut 4q^{9} \) \(\mathstrut -\mathstrut 2q^{11} \) \(\mathstrut -\mathstrut 2q^{14} \) \(\mathstrut +\mathstrut 6q^{15} \) \(\mathstrut -\mathstrut 2q^{16} \) \(\mathstrut +\mathstrut 4q^{18} \) \(\mathstrut -\mathstrut 2q^{22} \) \(\mathstrut -\mathstrut 4q^{23} \) \(\mathstrut +\mathstrut 4q^{25} \) \(\mathstrut -\mathstrut 2q^{29} \) \(\mathstrut +\mathstrut 6q^{30} \) \(\mathstrut +\mathstrut 2q^{43} \) \(\mathstrut -\mathstrut 4q^{46} \) \(\mathstrut +\mathstrut 2q^{49} \) \(\mathstrut +\mathstrut 4q^{50} \) \(\mathstrut +\mathstrut 6q^{51} \) \(\mathstrut -\mathstrut 2q^{53} \) \(\mathstrut +\mathstrut 2q^{56} \) \(\mathstrut -\mathstrut 2q^{58} \) \(\mathstrut -\mathstrut 4q^{63} \) \(\mathstrut +\mathstrut 2q^{64} \) \(\mathstrut -\mathstrut 2q^{67} \) \(\mathstrut -\mathstrut 4q^{72} \) \(\mathstrut +\mathstrut 2q^{77} \) \(\mathstrut +\mathstrut 2q^{81} \) \(\mathstrut +\mathstrut 6q^{85} \) \(\mathstrut +\mathstrut 2q^{86} \) \(\mathstrut +\mathstrut 2q^{88} \) \(\mathstrut +\mathstrut 2q^{98} \) \(\mathstrut -\mathstrut 4q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
1.73205
−1.73205
1.00000 −1.73205 0 −1.73205 −1.73205 −1.00000 −1.00000 2.00000 −1.73205
3310.2 1.00000 1.73205 0 1.73205 1.73205 −1.00000 −1.00000 2.00000 1.73205
\(n\): 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
3311.h Odd 1 CM by \(\Q(\sqrt{-3311}) \) yes
7.b Odd 1 yes
473.d Even 1 yes

Hecke kernels

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

\(T_{2} \) \(\mathstrut -\mathstrut 1 \)
\(T_{3}^{2} \) \(\mathstrut -\mathstrut 3 \)