Properties

Label 670.2.e.a
Level 670
Weight 2
Character orbit 670.e
Analytic conductor 5.350
Analytic rank 1
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: \(1\)
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.

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

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 −3.00000 −0.500000 + 0.866025i −1.00000 1.50000 + 2.59808i 1.00000 1.73205i 1.00000 6.00000 0.500000 + 0.866025i
431.1 −0.500000 + 0.866025i −3.00000 −0.500000 0.866025i −1.00000 1.50000 2.59808i 1.00000 + 1.73205i 1.00000 6.00000 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 \(S_{2}^{\mathrm{new}}(670, [\chi])\):

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