Properties

Label 3311.1.h.a
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 discs -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 \(2\), degree \(1\), minimal)

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: \( 1 \)
Twist minimal: yes
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{-7}, \sqrt{473})\)
Artin image $D_4$
Artin field Galois closure of 4.0.23177.2

$q$-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 Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
473.d even 2 1 RM by \(\Q(\sqrt{473}) \)
3311.h odd 2 1 CM by \(\Q(\sqrt{-3311}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3311.1.h.a 1
7.b odd 2 1 CM 3311.1.h.a 1
11.b odd 2 1 3311.1.h.f yes 1
43.b odd 2 1 3311.1.h.f yes 1
77.b even 2 1 3311.1.h.f yes 1
301.c even 2 1 3311.1.h.f yes 1
473.d even 2 1 RM 3311.1.h.a 1
3311.h odd 2 1 CM 3311.1.h.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3311.1.h.a 1 1.a even 1 1 trivial
3311.1.h.a 1 7.b odd 2 1 CM
3311.1.h.a 1 473.d even 2 1 RM
3311.1.h.a 1 3311.h odd 2 1 CM
3311.1.h.f yes 1 11.b odd 2 1
3311.1.h.f yes 1 43.b odd 2 1
3311.1.h.f yes 1 77.b even 2 1
3311.1.h.f yes 1 301.c even 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( 1 + T^{2} \)
$7$ \( 1 + T \)
$11$ \( 1 + T \)
$13$ \( 1 + T^{2} \)
$17$ \( 1 + T^{2} \)
$19$ \( ( 1 - T )( 1 + T ) \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( ( 1 - T )^{2} \)
$31$ \( ( 1 - T )( 1 + T ) \)
$37$ \( ( 1 - T )( 1 + T ) \)
$41$ \( 1 + T^{2} \)
$43$ \( 1 - T \)
$47$ \( ( 1 - T )( 1 + T ) \)
$53$ \( ( 1 - T )^{2} \)
$59$ \( ( 1 - T )( 1 + T ) \)
$61$ \( ( 1 - T )( 1 + T ) \)
$67$ \( ( 1 - T )^{2} \)
$71$ \( ( 1 - T )( 1 + T ) \)
$73$ \( ( 1 - T )( 1 + T ) \)
$79$ \( ( 1 - T )( 1 + T ) \)
$83$ \( 1 + T^{2} \)
$89$ \( 1 + T^{2} \)
$97$ \( ( 1 - T )( 1 + T ) \)
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