Properties

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

Newform invariants

Self dual: yes
Analytic conductor: \(1.65240425683\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

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

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

Twists

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ \( 1 - T^{2} + T^{4} \)
$5$ \( 1 - T^{2} + T^{4} \)
$7$ \( ( 1 - T )^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( ( 1 + T^{2} )^{2} \)
$17$ \( 1 - T^{2} + T^{4} \)
$19$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$23$ \( ( 1 + T )^{4} \)
$29$ \( ( 1 - T + T^{2} )^{2} \)
$31$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$37$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$41$ \( 1 - T^{2} + T^{4} \)
$43$ \( ( 1 + T )^{2} \)
$47$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$53$ \( ( 1 + T + T^{2} )^{2} \)
$59$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$61$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$67$ \( ( 1 + T + T^{2} )^{2} \)
$71$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$73$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$79$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$83$ \( 1 - T^{2} + T^{4} \)
$89$ \( ( 1 + T^{2} )^{2} \)
$97$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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