Properties

Label 3311.1.h.o
Level 3311
Weight 1
Character orbit 3311.h
Self dual yes
Analytic conductor 1.652
Analytic rank 0
Dimension 6
Projective image \(D_{18}\)
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: \(6\)
Coefficient field: \(\Q(\zeta_{36})^+\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{18}\)
Projective field Galois closure of \(\mathbb{Q}[x]/(x^{18} - \cdots)\)

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a root \(\beta\) of the polynomial \(x^{6} - 6 x^{4} + 9 x^{2} - 3\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 + 4 \beta^{2} - \beta^{4} ) q^{2} -\beta q^{3} + ( -3 + 5 \beta^{2} - \beta^{4} ) q^{4} + ( 5 \beta - 5 \beta^{3} + \beta^{5} ) q^{5} + ( 2 \beta - 4 \beta^{3} + \beta^{5} ) q^{6} - q^{7} + ( -1 + 4 \beta^{2} - \beta^{4} ) q^{8} + ( -1 + \beta^{2} ) q^{9} +O(q^{10})\) \( q + ( -2 + 4 \beta^{2} - \beta^{4} ) q^{2} -\beta q^{3} + ( -3 + 5 \beta^{2} - \beta^{4} ) q^{4} + ( 5 \beta - 5 \beta^{3} + \beta^{5} ) q^{5} + ( 2 \beta - 4 \beta^{3} + \beta^{5} ) q^{6} - q^{7} + ( -1 + 4 \beta^{2} - \beta^{4} ) q^{8} + ( -1 + \beta^{2} ) q^{9} -\beta q^{10} - q^{11} + ( 3 \beta - 5 \beta^{3} + \beta^{5} ) q^{12} + ( 3 \beta - \beta^{3} ) q^{13} + ( 2 - 4 \beta^{2} + \beta^{4} ) q^{14} + ( -3 + 4 \beta^{2} - \beta^{4} ) q^{15} + ( -1 + 4 \beta^{2} - \beta^{4} ) q^{16} + ( -4 \beta + 5 \beta^{3} - \beta^{5} ) q^{17} + ( -1 + 3 \beta^{2} - \beta^{4} ) q^{18} + ( -3 \beta + \beta^{3} ) q^{20} + \beta q^{21} + ( 2 - 4 \beta^{2} + \beta^{4} ) q^{22} + q^{23} + ( \beta - 4 \beta^{3} + \beta^{5} ) q^{24} + ( 5 - 5 \beta^{2} + \beta^{4} ) q^{25} + ( -3 \beta + 5 \beta^{3} - \beta^{5} ) q^{26} + ( 2 \beta - \beta^{3} ) q^{27} + ( 3 - 5 \beta^{2} + \beta^{4} ) q^{28} + ( -4 + 5 \beta^{2} - \beta^{4} ) q^{29} + \beta^{2} q^{30} + ( -3 + 5 \beta^{2} - \beta^{4} ) q^{32} + \beta q^{33} + ( -\beta + 4 \beta^{3} - \beta^{5} ) q^{34} + ( -5 \beta + 5 \beta^{3} - \beta^{5} ) q^{35} + \beta^{2} q^{36} + ( -3 \beta^{2} + \beta^{4} ) q^{39} + ( 4 \beta - 5 \beta^{3} + \beta^{5} ) q^{40} + ( 4 \beta - 5 \beta^{3} + \beta^{5} ) q^{41} + ( -2 \beta + 4 \beta^{3} - \beta^{5} ) q^{42} + q^{43} + ( 3 - 5 \beta^{2} + \beta^{4} ) q^{44} + ( -2 \beta + \beta^{3} ) q^{45} + ( -2 + 4 \beta^{2} - \beta^{4} ) q^{46} + ( \beta - 4 \beta^{3} + \beta^{5} ) q^{48} + q^{49} - q^{50} + ( 3 - 5 \beta^{2} + \beta^{4} ) q^{51} + ( -6 \beta + 9 \beta^{3} - 2 \beta^{5} ) q^{52} + ( 2 - \beta^{2} ) q^{53} + ( -\beta + \beta^{3} ) q^{54} + ( -5 \beta + 5 \beta^{3} - \beta^{5} ) q^{55} + ( 1 - 4 \beta^{2} + \beta^{4} ) q^{56} + ( -1 + 4 \beta^{2} - \beta^{4} ) q^{58} + ( 3 \beta^{2} - \beta^{4} ) q^{60} + ( 1 - \beta^{2} ) q^{63} + ( -2 + 4 \beta^{2} - \beta^{4} ) q^{64} + ( 6 - 6 \beta^{2} + \beta^{4} ) q^{65} + ( -2 \beta + 4 \beta^{3} - \beta^{5} ) q^{66} + ( 2 - 4 \beta^{2} + \beta^{4} ) q^{67} + ( 4 \beta^{3} - \beta^{5} ) q^{68} -\beta q^{69} + \beta q^{70} + ( -2 + 4 \beta^{2} - \beta^{4} ) q^{72} + ( -5 \beta + 5 \beta^{3} - \beta^{5} ) q^{75} + q^{77} + ( 3 - 6 \beta^{2} + \beta^{4} ) q^{78} + ( 4 \beta - 5 \beta^{3} + \beta^{5} ) q^{80} + ( 1 - 3 \beta^{2} + \beta^{4} ) q^{81} + ( \beta - 4 \beta^{3} + \beta^{5} ) q^{82} -\beta q^{83} + ( -3 \beta + 5 \beta^{3} - \beta^{5} ) q^{84} + ( -3 + \beta^{2} ) q^{85} + ( -2 + 4 \beta^{2} - \beta^{4} ) q^{86} + ( 4 \beta - 5 \beta^{3} + \beta^{5} ) q^{87} + ( 1 - 4 \beta^{2} + \beta^{4} ) q^{88} + ( \beta - \beta^{3} ) q^{90} + ( -3 \beta + \beta^{3} ) q^{91} + ( -3 + 5 \beta^{2} - \beta^{4} ) q^{92} + ( 3 \beta - 5 \beta^{3} + \beta^{5} ) q^{96} + ( -2 + 4 \beta^{2} - \beta^{4} ) q^{98} + ( 1 - \beta^{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{4} - 6q^{7} + 6q^{8} + 6q^{9} + O(q^{10}) \) \( 6q + 6q^{4} - 6q^{7} + 6q^{8} + 6q^{9} - 6q^{11} - 6q^{15} + 6q^{16} - 6q^{18} + 6q^{23} + 6q^{25} - 6q^{28} + 12q^{30} + 6q^{32} + 12q^{36} + 6q^{43} - 6q^{44} + 6q^{49} - 6q^{50} - 6q^{51} - 6q^{56} + 6q^{58} - 6q^{63} + 6q^{77} - 18q^{78} + 6q^{81} - 6q^{85} - 6q^{88} + 6q^{92} - 6q^{99} + O(q^{100}) \)

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
1.96962
−1.96962
0.684040
−0.684040
1.28558
−1.28558
−1.53209 −1.96962 1.34730 1.28558 3.01763 −1.00000 −0.532089 2.87939 −1.96962
3310.2 −1.53209 1.96962 1.34730 −1.28558 −3.01763 −1.00000 −0.532089 2.87939 1.96962
3310.3 −0.347296 −0.684040 −0.879385 1.96962 0.237565 −1.00000 0.652704 −0.532089 −0.684040
3310.4 −0.347296 0.684040 −0.879385 −1.96962 −0.237565 −1.00000 0.652704 −0.532089 0.684040
3310.5 1.87939 −1.28558 2.53209 −0.684040 −2.41609 −1.00000 2.87939 0.652704 −1.28558
3310.6 1.87939 1.28558 2.53209 0.684040 2.41609 −1.00000 2.87939 0.652704 1.28558
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3310.6
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.o 6
7.b odd 2 1 inner 3311.1.h.o 6
11.b odd 2 1 3311.1.h.p yes 6
43.b odd 2 1 3311.1.h.p yes 6
77.b even 2 1 3311.1.h.p yes 6
301.c even 2 1 3311.1.h.p yes 6
473.d even 2 1 inner 3311.1.h.o 6
3311.h odd 2 1 CM 3311.1.h.o 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3311.1.h.o 6 1.a even 1 1 trivial
3311.1.h.o 6 7.b odd 2 1 inner
3311.1.h.o 6 473.d even 2 1 inner
3311.1.h.o 6 3311.h odd 2 1 CM
3311.1.h.p yes 6 11.b odd 2 1
3311.1.h.p yes 6 43.b odd 2 1
3311.1.h.p yes 6 77.b even 2 1
3311.1.h.p yes 6 301.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 \(S_{1}^{\mathrm{new}}(3311, [\chi])\):

\( T_{2}^{3} - 3 T_{2} - 1 \)
\( T_{3}^{6} - 6 T_{3}^{4} + 9 T_{3}^{2} - 3 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{3} + T^{6} )^{2} \)
$3$ \( 1 - T^{6} + T^{12} \)
$5$ \( 1 - T^{6} + T^{12} \)
$7$ \( ( 1 + T )^{6} \)
$11$ \( ( 1 + T )^{6} \)
$13$ \( ( 1 - T^{2} + T^{4} )^{3} \)
$17$ \( 1 - T^{6} + T^{12} \)
$19$ \( ( 1 - T )^{6}( 1 + T )^{6} \)
$23$ \( ( 1 - T + T^{2} )^{6} \)
$29$ \( ( 1 + T^{3} + T^{6} )^{2} \)
$31$ \( ( 1 - T )^{6}( 1 + T )^{6} \)
$37$ \( ( 1 - T )^{6}( 1 + T )^{6} \)
$41$ \( 1 - T^{6} + T^{12} \)
$43$ \( ( 1 - T )^{6} \)
$47$ \( ( 1 - T )^{6}( 1 + T )^{6} \)
$53$ \( ( 1 + T^{3} + T^{6} )^{2} \)
$59$ \( ( 1 - T )^{6}( 1 + T )^{6} \)
$61$ \( ( 1 - T )^{6}( 1 + T )^{6} \)
$67$ \( ( 1 + T^{3} + T^{6} )^{2} \)
$71$ \( ( 1 - T )^{6}( 1 + T )^{6} \)
$73$ \( ( 1 - T )^{6}( 1 + T )^{6} \)
$79$ \( ( 1 - T )^{6}( 1 + T )^{6} \)
$83$ \( 1 - T^{6} + T^{12} \)
$89$ \( ( 1 + T^{2} )^{6} \)
$97$ \( ( 1 - T )^{6}( 1 + T )^{6} \)
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