Properties

Label 3311.1.h.c
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 discriminant -3311
Inner twists 2

Related objects

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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 \(D_{3}\)
Projective field Galois closure of 3.1.3311.1
Artin image $D_6$
Artin field Galois closure of 6.0.76739047.1

$q$-expansion

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
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.c yes 1
7.b odd 2 1 3311.1.h.b 1
11.b odd 2 1 3311.1.h.e yes 1
43.b odd 2 1 3311.1.h.d yes 1
77.b even 2 1 3311.1.h.d yes 1
301.c even 2 1 3311.1.h.e yes 1
473.d even 2 1 3311.1.h.b 1
3311.h odd 2 1 CM 3311.1.h.c yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3311.1.h.b 1 7.b odd 2 1
3311.1.h.b 1 473.d even 2 1
3311.1.h.c yes 1 1.a even 1 1 trivial
3311.1.h.c yes 1 3311.h odd 2 1 CM
3311.1.h.d yes 1 43.b odd 2 1
3311.1.h.d yes 1 77.b even 2 1
3311.1.h.e yes 1 11.b odd 2 1
3311.1.h.e 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 \(S_{1}^{\mathrm{new}}(3311, [\chi])\):

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

Hecke characteristic polynomials

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