Properties

Label 3311.1.h.f
Level 3311
Weight 1
Character orbit 3311.h
Self dual Yes
Analytic conductor 1.652
Analytic rank 0
Dimension 1
Projective image \(D_{2}\)
CM/RM disc. -7, -3311, 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: Yes
Analytic conductor: \(1.65240425683\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{-7}, \sqrt{473})\)
Artin image size \(8\)
Artin image $D_4$
Artin field Galois closure of 4.0.23177.1

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{2} + 3q^{4} + q^{7} + 4q^{8} - q^{9} + O(q^{10}) \) \( q + 2q^{2} + 3q^{4} + q^{7} + 4q^{8} - q^{9} - q^{11} + 2q^{14} + 5q^{16} - 2q^{18} - 2q^{22} - 2q^{23} - q^{25} + 3q^{28} - 2q^{29} + 6q^{32} - 3q^{36} - q^{43} - 3q^{44} - 4q^{46} + q^{49} - 2q^{50} + 2q^{53} + 4q^{56} - 4q^{58} - q^{63} + 7q^{64} + 2q^{67} - 4q^{72} - q^{77} + q^{81} - 2q^{86} - 4q^{88} - 6q^{92} + 2q^{98} + q^{99} + O(q^{100}) \)

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
2.00000 0 3.00000 0 0 1.00000 4.00000 −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
7.b Odd 1 CM by \(\Q(\sqrt{-7}) \) yes
473.d Even 1 RM by \(\Q(\sqrt{473}) \) 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 \(S_{1}^{\mathrm{new}}(3311, [\chi])\):

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