Properties

Label 3311.1.h.n
Level 3311
Weight 1
Character orbit 3311.h
Self dual yes
Analytic conductor 1.652
Analytic rank 0
Dimension 3
Projective image \(D_{9}\)
CM discriminant -3311
Inner twists 2

Related objects

Downloads

Learn more about

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: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Defining polynomial: \(x^{3} - 3 x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{9}\)
Projective field Galois closure of 9.1.120181251723841.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 + \beta - \beta^{2} ) q^{2} + \beta q^{3} + ( 1 - \beta ) q^{4} + ( -2 - \beta + \beta^{2} ) q^{5} + ( -1 - \beta + \beta^{2} ) q^{6} + q^{7} + ( 1 + \beta - \beta^{2} ) q^{8} + ( -1 + \beta^{2} ) q^{9} +O(q^{10})\) \( q + ( 2 + \beta - \beta^{2} ) q^{2} + \beta q^{3} + ( 1 - \beta ) q^{4} + ( -2 - \beta + \beta^{2} ) q^{5} + ( -1 - \beta + \beta^{2} ) q^{6} + q^{7} + ( 1 + \beta - \beta^{2} ) q^{8} + ( -1 + \beta^{2} ) q^{9} + ( -2 + \beta ) q^{10} + q^{11} + ( \beta - \beta^{2} ) q^{12} + q^{13} + ( 2 + \beta - \beta^{2} ) q^{14} + ( 1 + \beta - \beta^{2} ) q^{15} + ( -1 - \beta + \beta^{2} ) q^{16} + ( 2 - \beta^{2} ) q^{17} + ( -1 + \beta ) q^{18} + ( -3 - 2 \beta + 2 \beta^{2} ) q^{20} + \beta q^{21} + ( 2 + \beta - \beta^{2} ) q^{22} - q^{23} + ( -1 - 2 \beta + \beta^{2} ) q^{24} + ( 1 - \beta ) q^{25} + ( 2 + \beta - \beta^{2} ) q^{26} + ( 1 + \beta ) q^{27} + ( 1 - \beta ) q^{28} -\beta q^{29} + ( -2 \beta + \beta^{2} ) q^{30} + ( -1 + \beta ) q^{32} + \beta q^{33} + ( 3 - \beta^{2} ) q^{34} + ( -2 - \beta + \beta^{2} ) q^{35} + ( -2 - 2 \beta + \beta^{2} ) q^{36} + \beta q^{39} + ( 2 \beta - \beta^{2} ) q^{40} + ( 2 - \beta^{2} ) q^{41} + ( -1 - \beta + \beta^{2} ) q^{42} + q^{43} + ( 1 - \beta ) q^{44} + ( 1 - \beta ) q^{45} + ( -2 - \beta + \beta^{2} ) q^{46} + ( 1 + 2 \beta - \beta^{2} ) q^{48} + q^{49} + ( 3 + 2 \beta - 2 \beta^{2} ) q^{50} + ( -1 - \beta ) q^{51} + ( 1 - \beta ) q^{52} + ( -2 + \beta^{2} ) q^{53} + q^{54} + ( -2 - \beta + \beta^{2} ) q^{55} + ( 1 + \beta - \beta^{2} ) q^{56} + ( 1 + \beta - \beta^{2} ) q^{58} + ( 2 + 3 \beta - 2 \beta^{2} ) q^{60} + ( -1 + \beta^{2} ) q^{63} + ( -2 - \beta + \beta^{2} ) q^{64} + ( -2 - \beta + \beta^{2} ) q^{65} + ( -1 - \beta + \beta^{2} ) q^{66} + ( 2 + \beta - \beta^{2} ) q^{67} + ( 3 + \beta - \beta^{2} ) q^{68} -\beta q^{69} + ( -2 + \beta ) q^{70} + ( \beta - \beta^{2} ) q^{72} + ( \beta - \beta^{2} ) q^{75} + q^{77} + ( -1 - \beta + \beta^{2} ) q^{78} + ( -2 \beta + \beta^{2} ) q^{80} + ( 1 + \beta ) q^{81} + ( 3 - \beta^{2} ) q^{82} + \beta q^{83} + ( \beta - \beta^{2} ) q^{84} + ( -3 + \beta^{2} ) q^{85} + ( 2 + \beta - \beta^{2} ) q^{86} -\beta^{2} q^{87} + ( 1 + \beta - \beta^{2} ) q^{88} -2 q^{89} + ( 3 + 2 \beta - 2 \beta^{2} ) q^{90} + q^{91} + ( -1 + \beta ) q^{92} + ( -\beta + \beta^{2} ) q^{96} + ( 2 + \beta - \beta^{2} ) q^{98} + ( -1 + \beta^{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{4} + 3q^{6} + 3q^{7} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{4} + 3q^{6} + 3q^{7} - 3q^{8} + 3q^{9} - 6q^{10} + 3q^{11} - 6q^{12} + 3q^{13} - 3q^{15} + 3q^{16} - 3q^{18} + 3q^{20} - 3q^{23} + 3q^{24} + 3q^{25} + 3q^{27} + 3q^{28} + 6q^{30} - 3q^{32} + 3q^{34} - 6q^{40} + 3q^{42} + 3q^{43} + 3q^{44} + 3q^{45} - 3q^{48} + 3q^{49} - 3q^{50} - 3q^{51} + 3q^{52} + 3q^{54} - 3q^{56} - 3q^{58} - 6q^{60} + 3q^{63} + 3q^{66} + 3q^{68} - 6q^{70} - 6q^{72} - 6q^{75} + 3q^{77} + 3q^{78} + 6q^{80} + 3q^{81} + 3q^{82} - 6q^{84} - 3q^{85} - 6q^{87} - 3q^{88} - 6q^{89} - 3q^{90} + 3q^{91} - 3q^{92} + 6q^{96} + 3q^{99} + O(q^{100}) \)

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
−1.53209
1.87939
−0.347296
−1.87939 −1.53209 2.53209 1.87939 2.87939 1.00000 −2.87939 1.34730 −3.53209
3310.2 0.347296 1.87939 −0.879385 −0.347296 0.652704 1.00000 −0.652704 2.53209 −0.120615
3310.3 1.53209 −0.347296 1.34730 −1.53209 −0.532089 1.00000 0.532089 −0.879385 −2.34730
\(n\): 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.n yes 3
7.b odd 2 1 3311.1.h.l yes 3
11.b odd 2 1 3311.1.h.k 3
43.b odd 2 1 3311.1.h.m yes 3
77.b even 2 1 3311.1.h.m yes 3
301.c even 2 1 3311.1.h.k 3
473.d even 2 1 3311.1.h.l yes 3
3311.h odd 2 1 CM 3311.1.h.n yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3311.1.h.k 3 11.b odd 2 1
3311.1.h.k 3 301.c even 2 1
3311.1.h.l yes 3 7.b odd 2 1
3311.1.h.l yes 3 473.d even 2 1
3311.1.h.m yes 3 43.b odd 2 1
3311.1.h.m yes 3 77.b even 2 1
3311.1.h.n yes 3 1.a even 1 1 trivial
3311.1.h.n yes 3 3311.h odd 2 1 CM

Hecke kernels

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

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

Hecke characteristic polynomials

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