Properties

Label 670.2.e.d
Level 670
Weight 2
Character orbit 670.e
Analytic conductor 5.350
Analytic rank 0
Dimension 2
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{2} + q^{3} -\zeta_{6} q^{4} - q^{5} + ( 1 - \zeta_{6} ) q^{6} -2 \zeta_{6} q^{7} - q^{8} -2 q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{2} + q^{3} -\zeta_{6} q^{4} - q^{5} + ( 1 - \zeta_{6} ) q^{6} -2 \zeta_{6} q^{7} - q^{8} -2 q^{9} + ( -1 + \zeta_{6} ) q^{10} -6 \zeta_{6} q^{11} -\zeta_{6} q^{12} + ( -2 + 2 \zeta_{6} ) q^{13} -2 q^{14} - q^{15} + ( -1 + \zeta_{6} ) q^{16} + ( -6 + 6 \zeta_{6} ) q^{17} + ( -2 + 2 \zeta_{6} ) q^{18} + ( 4 - 4 \zeta_{6} ) q^{19} + \zeta_{6} q^{20} -2 \zeta_{6} q^{21} -6 q^{22} + ( 6 - 6 \zeta_{6} ) q^{23} - q^{24} + q^{25} + 2 \zeta_{6} q^{26} -5 q^{27} + ( -2 + 2 \zeta_{6} ) q^{28} + 6 \zeta_{6} q^{29} + ( -1 + \zeta_{6} ) q^{30} -5 \zeta_{6} q^{31} + \zeta_{6} q^{32} -6 \zeta_{6} q^{33} + 6 \zeta_{6} q^{34} + 2 \zeta_{6} q^{35} + 2 \zeta_{6} q^{36} + ( 7 - 7 \zeta_{6} ) q^{37} -4 \zeta_{6} q^{38} + ( -2 + 2 \zeta_{6} ) q^{39} + q^{40} -3 \zeta_{6} q^{41} -2 q^{42} + 8 q^{43} + ( -6 + 6 \zeta_{6} ) q^{44} + 2 q^{45} -6 \zeta_{6} q^{46} -6 \zeta_{6} q^{47} + ( -1 + \zeta_{6} ) q^{48} + ( 3 - 3 \zeta_{6} ) q^{49} + ( 1 - \zeta_{6} ) q^{50} + ( -6 + 6 \zeta_{6} ) q^{51} + 2 q^{52} -3 q^{53} + ( -5 + 5 \zeta_{6} ) q^{54} + 6 \zeta_{6} q^{55} + 2 \zeta_{6} q^{56} + ( 4 - 4 \zeta_{6} ) q^{57} + 6 q^{58} -6 q^{59} + \zeta_{6} q^{60} + ( -8 + 8 \zeta_{6} ) q^{61} -5 q^{62} + 4 \zeta_{6} q^{63} + q^{64} + ( 2 - 2 \zeta_{6} ) q^{65} -6 q^{66} + ( -2 + 9 \zeta_{6} ) q^{67} + 6 q^{68} + ( 6 - 6 \zeta_{6} ) q^{69} + 2 q^{70} + 15 \zeta_{6} q^{71} + 2 q^{72} + ( 4 - 4 \zeta_{6} ) q^{73} -7 \zeta_{6} q^{74} + q^{75} -4 q^{76} + ( -12 + 12 \zeta_{6} ) q^{77} + 2 \zeta_{6} q^{78} -8 \zeta_{6} q^{79} + ( 1 - \zeta_{6} ) q^{80} + q^{81} -3 q^{82} + ( 15 - 15 \zeta_{6} ) q^{83} + ( -2 + 2 \zeta_{6} ) q^{84} + ( 6 - 6 \zeta_{6} ) q^{85} + ( 8 - 8 \zeta_{6} ) q^{86} + 6 \zeta_{6} q^{87} + 6 \zeta_{6} q^{88} + 9 q^{89} + ( 2 - 2 \zeta_{6} ) q^{90} + 4 q^{91} -6 q^{92} -5 \zeta_{6} q^{93} -6 q^{94} + ( -4 + 4 \zeta_{6} ) q^{95} + \zeta_{6} q^{96} + ( 10 - 10 \zeta_{6} ) q^{97} -3 \zeta_{6} q^{98} + 12 \zeta_{6} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + 2q^{3} - q^{4} - 2q^{5} + q^{6} - 2q^{7} - 2q^{8} - 4q^{9} + O(q^{10}) \) \( 2q + q^{2} + 2q^{3} - q^{4} - 2q^{5} + q^{6} - 2q^{7} - 2q^{8} - 4q^{9} - q^{10} - 6q^{11} - q^{12} - 2q^{13} - 4q^{14} - 2q^{15} - q^{16} - 6q^{17} - 2q^{18} + 4q^{19} + q^{20} - 2q^{21} - 12q^{22} + 6q^{23} - 2q^{24} + 2q^{25} + 2q^{26} - 10q^{27} - 2q^{28} + 6q^{29} - q^{30} - 5q^{31} + q^{32} - 6q^{33} + 6q^{34} + 2q^{35} + 2q^{36} + 7q^{37} - 4q^{38} - 2q^{39} + 2q^{40} - 3q^{41} - 4q^{42} + 16q^{43} - 6q^{44} + 4q^{45} - 6q^{46} - 6q^{47} - q^{48} + 3q^{49} + q^{50} - 6q^{51} + 4q^{52} - 6q^{53} - 5q^{54} + 6q^{55} + 2q^{56} + 4q^{57} + 12q^{58} - 12q^{59} + q^{60} - 8q^{61} - 10q^{62} + 4q^{63} + 2q^{64} + 2q^{65} - 12q^{66} + 5q^{67} + 12q^{68} + 6q^{69} + 4q^{70} + 15q^{71} + 4q^{72} + 4q^{73} - 7q^{74} + 2q^{75} - 8q^{76} - 12q^{77} + 2q^{78} - 8q^{79} + q^{80} + 2q^{81} - 6q^{82} + 15q^{83} - 2q^{84} + 6q^{85} + 8q^{86} + 6q^{87} + 6q^{88} + 18q^{89} + 2q^{90} + 8q^{91} - 12q^{92} - 5q^{93} - 12q^{94} - 4q^{95} + q^{96} + 10q^{97} - 3q^{98} + 12q^{99} + O(q^{100}) \)

Character Values

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

\(n\) \(471\) \(537\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i −1.00000 0.500000 + 0.866025i −1.00000 + 1.73205i −1.00000 −2.00000 −0.500000 0.866025i
431.1 0.500000 0.866025i 1.00000 −0.500000 0.866025i −1.00000 0.500000 0.866025i −1.00000 1.73205i −1.00000 −2.00000 −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 \(S_{2}^{\mathrm{new}}(670, [\chi])\):

\( T_{3} - 1 \)
\( T_{7}^{2} + 2 T_{7} + 4 \)
\( T_{11}^{2} + 6 T_{11} + 36 \)