Properties

Label 670.2.e.f
Level 670
Weight 2
Character orbit 670.e
Analytic conductor 5.350
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 670 = 2 \cdot 5 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 670.e (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(5.34997693543\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 1 - \beta_{3} ) q^{3} + ( -1 + \beta_{1} ) q^{4} - q^{5} + ( -\beta_{1} + \beta_{2} ) q^{6} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{7} + q^{8} + ( 4 - 2 \beta_{3} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 1 - \beta_{3} ) q^{3} + ( -1 + \beta_{1} ) q^{4} - q^{5} + ( -\beta_{1} + \beta_{2} ) q^{6} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{7} + q^{8} + ( 4 - 2 \beta_{3} ) q^{9} + \beta_{1} q^{10} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{12} + 4 \beta_{1} q^{13} + ( 2 - \beta_{3} ) q^{14} + ( -1 + \beta_{3} ) q^{15} -\beta_{1} q^{16} + ( -2 \beta_{1} + \beta_{2} ) q^{17} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{18} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{19} + ( 1 - \beta_{1} ) q^{20} + ( -8 + 8 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{21} + \beta_{2} q^{23} + ( 1 - \beta_{3} ) q^{24} + q^{25} + ( 4 - 4 \beta_{1} ) q^{26} + ( 13 - 3 \beta_{3} ) q^{27} + ( -2 \beta_{1} + \beta_{2} ) q^{28} + ( 6 - 6 \beta_{1} ) q^{29} + ( \beta_{1} - \beta_{2} ) q^{30} + ( 3 - 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{31} + ( -1 + \beta_{1} ) q^{32} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{34} + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{35} + ( -4 + 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{36} + ( -\beta_{1} + 2 \beta_{2} ) q^{37} + ( 2 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{38} + ( 4 \beta_{1} - 4 \beta_{2} ) q^{39} - q^{40} + ( 7 - 7 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{41} + ( 8 - 3 \beta_{3} ) q^{42} + 4 q^{43} + ( -4 + 2 \beta_{3} ) q^{45} + ( -\beta_{2} + \beta_{3} ) q^{46} + ( -3 \beta_{2} + 3 \beta_{3} ) q^{47} + ( -\beta_{1} + \beta_{2} ) q^{48} + ( -3 \beta_{1} + 4 \beta_{2} ) q^{49} -\beta_{1} q^{50} + ( -8 \beta_{1} + 3 \beta_{2} ) q^{51} -4 q^{52} -7 q^{53} + ( -13 \beta_{1} + 3 \beta_{2} ) q^{54} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{56} + ( 14 \beta_{1} - 4 \beta_{2} ) q^{57} -6 q^{58} + ( -2 + 2 \beta_{3} ) q^{59} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{60} + ( -2 \beta_{1} + \beta_{2} ) q^{61} + ( -3 - \beta_{3} ) q^{62} + ( -20 + 20 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} ) q^{63} + q^{64} -4 \beta_{1} q^{65} + ( 7 - 7 \beta_{1} + \beta_{2} + \beta_{3} ) q^{67} + ( 2 - \beta_{3} ) q^{68} + ( -6 \beta_{1} + \beta_{2} ) q^{69} + ( -2 + \beta_{3} ) q^{70} + ( 7 - 7 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{71} + ( 4 - 2 \beta_{3} ) q^{72} + ( 6 \beta_{1} - \beta_{2} ) q^{73} + ( -1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{74} + ( 1 - \beta_{3} ) q^{75} + ( -2 + 2 \beta_{3} ) q^{76} + ( 4 - 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{78} + ( -2 + 2 \beta_{1} ) q^{79} + \beta_{1} q^{80} + ( 19 - 10 \beta_{3} ) q^{81} + ( -7 - 2 \beta_{3} ) q^{82} + ( -5 \beta_{1} + 3 \beta_{2} ) q^{83} + ( -8 \beta_{1} + 3 \beta_{2} ) q^{84} + ( 2 \beta_{1} - \beta_{2} ) q^{85} -4 \beta_{1} q^{86} + ( 6 - 6 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} ) q^{87} + ( -7 - 2 \beta_{3} ) q^{89} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{90} + ( -8 + 4 \beta_{3} ) q^{91} -\beta_{3} q^{92} + ( -3 + 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{93} -3 \beta_{3} q^{94} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{95} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{96} + ( -6 \beta_{1} + 4 \beta_{2} ) q^{97} + ( -3 + 3 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} + 4q^{3} - 2q^{4} - 4q^{5} - 2q^{6} - 4q^{7} + 4q^{8} + 16q^{9} + O(q^{10}) \) \( 4q - 2q^{2} + 4q^{3} - 2q^{4} - 4q^{5} - 2q^{6} - 4q^{7} + 4q^{8} + 16q^{9} + 2q^{10} - 2q^{12} + 8q^{13} + 8q^{14} - 4q^{15} - 2q^{16} - 4q^{17} - 8q^{18} + 4q^{19} + 2q^{20} - 16q^{21} + 4q^{24} + 4q^{25} + 8q^{26} + 52q^{27} - 4q^{28} + 12q^{29} + 2q^{30} + 6q^{31} - 2q^{32} - 4q^{34} + 4q^{35} - 8q^{36} - 2q^{37} + 4q^{38} + 8q^{39} - 4q^{40} + 14q^{41} + 32q^{42} + 16q^{43} - 16q^{45} - 2q^{48} - 6q^{49} - 2q^{50} - 16q^{51} - 16q^{52} - 28q^{53} - 26q^{54} - 4q^{56} + 28q^{57} - 24q^{58} - 8q^{59} + 2q^{60} - 4q^{61} - 12q^{62} - 40q^{63} + 4q^{64} - 8q^{65} + 14q^{67} + 8q^{68} - 12q^{69} - 8q^{70} + 14q^{71} + 16q^{72} + 12q^{73} - 2q^{74} + 4q^{75} - 8q^{76} + 8q^{78} - 4q^{79} + 2q^{80} + 76q^{81} - 28q^{82} - 10q^{83} - 16q^{84} + 4q^{85} - 8q^{86} + 12q^{87} - 28q^{89} + 8q^{90} - 32q^{91} - 6q^{93} - 4q^{95} - 2q^{96} - 12q^{97} - 6q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 2 \nu \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 4 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2}\)\()/3\)
\(\nu^{2}\)\(=\)\(2 \beta_{1}\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{3} + 4 \beta_{2}\)\()/3\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/670\mathbb{Z}\right)^\times\).

\(n\) \(471\) \(537\)
\(\chi(n)\) \(-1 + \beta_{1}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
171.1
1.22474 + 0.707107i
−1.22474 0.707107i
1.22474 0.707107i
−1.22474 + 0.707107i
−0.500000 0.866025i −1.44949 −0.500000 + 0.866025i −1.00000 0.724745 + 1.25529i 0.224745 0.389270i 1.00000 −0.898979 0.500000 + 0.866025i
171.2 −0.500000 0.866025i 3.44949 −0.500000 + 0.866025i −1.00000 −1.72474 2.98735i −2.22474 + 3.85337i 1.00000 8.89898 0.500000 + 0.866025i
431.1 −0.500000 + 0.866025i −1.44949 −0.500000 0.866025i −1.00000 0.724745 1.25529i 0.224745 + 0.389270i 1.00000 −0.898979 0.500000 0.866025i
431.2 −0.500000 + 0.866025i 3.44949 −0.500000 0.866025i −1.00000 −1.72474 + 2.98735i −2.22474 3.85337i 1.00000 8.89898 0.500000 0.866025i
\(n\): 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
67.c Even 1 yes

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(670, [\chi])\):

\( T_{3}^{2} - 2 T_{3} - 5 \)
\( T_{7}^{4} + 4 T_{7}^{3} + 18 T_{7}^{2} - 8 T_{7} + 4 \)
\( T_{11} \)