Properties

Label 3311.1.h.p
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 disc. -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 \(2\) and degree \(1\))

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: \( 1 \)
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 basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of \(\nu = \zeta_{36} + \zeta_{36}^{-1}\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 3 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 5 \nu^{2} + 4 \)
\(\beta_{5}\)\(=\)\( \nu^{5} - 5 \nu^{3} + 4 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(2\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(3\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4}\mathstrut +\mathstrut \) \(5\) \(\beta_{2}\mathstrut +\mathstrut \) \(6\)
\(\nu^{5}\)\(=\)\(\beta_{5}\mathstrut +\mathstrut \) \(5\) \(\beta_{3}\mathstrut +\mathstrut \) \(11\) \(\beta_{1}\)

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
1.28558
−1.28558
0.684040
−0.684040
1.96962
−1.96962
−1.87939 −1.28558 2.53209 −0.684040 2.41609 1.00000 −2.87939 0.652704 1.28558
3310.2 −1.87939 1.28558 2.53209 0.684040 −2.41609 1.00000 −2.87939 0.652704 −1.28558
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 1.53209 −1.96962 1.34730 1.28558 −3.01763 1.00000 0.532089 2.87939 1.96962
3310.6 1.53209 1.96962 1.34730 −1.28558 3.01763 1.00000 0.532089 2.87939 −1.96962
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3310.6
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
7.b Odd 1 yes
473.d Even 1 yes

Hecke kernels

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

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