Properties

Label 3311.1.h.m
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 disc. -3311
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 3311 = 7 \cdot 11 \cdot 43 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 3311.h (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(1.65240425683\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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.

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

Basis of coefficient ring in terms of \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(2\)

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
−0.347296
1.87939
−1.53209
−1.53209 0.347296 1.34730 1.53209 −0.532089 −1.00000 −0.532089 −0.879385 −2.34730
3310.2 −0.347296 −1.87939 −0.879385 0.347296 0.652704 −1.00000 0.652704 2.53209 −0.120615
3310.3 1.87939 1.53209 2.53209 −1.87939 2.87939 −1.00000 2.87939 1.34730 −3.53209
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3311.h Odd 1 CM by \(\Q(\sqrt{-3311}) \) yes

Hecke kernels

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

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