Properties

Label 670.2.e.b
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.

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

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 −3.00000 −0.500000 + 0.866025i 1.00000 1.50000 + 2.59808i −2.00000 + 3.46410i 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 −2.00000 3.46410i 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 \(S_{2}^{\mathrm{new}}(670, [\chi])\):

\( T_{3} + 3 \)
\( T_{7}^{2} + 4 T_{7} + 16 \)
\( T_{11}^{2} - 2 T_{11} + 4 \)