Properties

Label 670.2.e.e
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: \(\Q(\sqrt{-3}, \sqrt{193})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
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 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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 49 x^{2} + 48 x + 2304\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 49 \nu^{2} - 49 \nu + 2304 \)\()/2352\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 97 \)\()/49\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 48 \beta_{2} + \beta_{1} - 49\)
\(\nu^{3}\)\(=\)\(49 \beta_{3} - 97\)

Character Values

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

\(n\) \(471\) \(537\)
\(\chi(n)\) \(-1 + \beta_{2}\) \(1\)

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
−3.22311 5.58259i
3.72311 + 6.44862i
−3.22311 + 5.58259i
3.72311 6.44862i
−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
171.2 −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
431.2 −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 \(S_{2}^{\mathrm{new}}(670, [\chi])\):

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