Properties

Label 3311.1.h.e
Level 3311
Weight 1
Character orbit 3311.h
Self dual Yes
Analytic conductor 1.652
Analytic rank 0
Dimension 1
Projective image \(D_{3}\)
CM disc. -3311
Inner twists 2

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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: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.3311.1
Artin image size \(12\)
Artin image $D_6$
Artin field Galois closure of 6.0.471397003.1

$q$-expansion

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut +\mathstrut q^{3} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut +\mathstrut q^{6} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut -\mathstrut q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut +\mathstrut q^{3} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut +\mathstrut q^{6} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut -\mathstrut q^{8} \) \(\mathstrut +\mathstrut q^{10} \) \(\mathstrut +\mathstrut q^{11} \) \(\mathstrut +\mathstrut 2q^{13} \) \(\mathstrut -\mathstrut q^{14} \) \(\mathstrut +\mathstrut q^{15} \) \(\mathstrut -\mathstrut q^{16} \) \(\mathstrut -\mathstrut q^{17} \) \(\mathstrut -\mathstrut q^{21} \) \(\mathstrut +\mathstrut q^{22} \) \(\mathstrut +\mathstrut 2q^{23} \) \(\mathstrut -\mathstrut q^{24} \) \(\mathstrut +\mathstrut 2q^{26} \) \(\mathstrut -\mathstrut q^{27} \) \(\mathstrut +\mathstrut q^{29} \) \(\mathstrut +\mathstrut q^{30} \) \(\mathstrut +\mathstrut q^{33} \) \(\mathstrut -\mathstrut q^{34} \) \(\mathstrut -\mathstrut q^{35} \) \(\mathstrut +\mathstrut 2q^{39} \) \(\mathstrut -\mathstrut q^{40} \) \(\mathstrut -\mathstrut q^{41} \) \(\mathstrut -\mathstrut q^{42} \) \(\mathstrut -\mathstrut q^{43} \) \(\mathstrut +\mathstrut 2q^{46} \) \(\mathstrut -\mathstrut q^{48} \) \(\mathstrut +\mathstrut q^{49} \) \(\mathstrut -\mathstrut q^{51} \) \(\mathstrut -\mathstrut q^{53} \) \(\mathstrut -\mathstrut q^{54} \) \(\mathstrut +\mathstrut q^{55} \) \(\mathstrut +\mathstrut q^{56} \) \(\mathstrut +\mathstrut q^{58} \) \(\mathstrut +\mathstrut q^{64} \) \(\mathstrut +\mathstrut 2q^{65} \) \(\mathstrut +\mathstrut q^{66} \) \(\mathstrut -\mathstrut q^{67} \) \(\mathstrut +\mathstrut 2q^{69} \) \(\mathstrut -\mathstrut q^{70} \) \(\mathstrut -\mathstrut q^{77} \) \(\mathstrut +\mathstrut 2q^{78} \) \(\mathstrut -\mathstrut q^{80} \) \(\mathstrut -\mathstrut q^{81} \) \(\mathstrut -\mathstrut q^{82} \) \(\mathstrut -\mathstrut q^{83} \) \(\mathstrut -\mathstrut q^{85} \) \(\mathstrut -\mathstrut q^{86} \) \(\mathstrut +\mathstrut q^{87} \) \(\mathstrut -\mathstrut q^{88} \) \(\mathstrut -\mathstrut 2q^{89} \) \(\mathstrut -\mathstrut 2q^{91} \) \(\mathstrut +\mathstrut q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
1.00000 1.00000 0 1.00000 1.00000 −1.00000 −1.00000 0 1.00000
\(n\): 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

Hecke kernels

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

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