Properties

Label 630.2.i.b
Level 630
Weight 2
Character orbit 630.i
Analytic conductor 5.031
Analytic rank 1
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 630.i (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(5.03057532734\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

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

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/630\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
0.500000 0.866025i
0.500000 + 0.866025i
−1.00000 −1.50000 0.866025i 1.00000 0.500000 + 0.866025i 1.50000 + 0.866025i −2.00000 1.73205i −1.00000 1.50000 + 2.59808i −0.500000 0.866025i
151.1 −1.00000 −1.50000 + 0.866025i 1.00000 0.500000 0.866025i 1.50000 0.866025i −2.00000 + 1.73205i −1.00000 1.50000 2.59808i −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
63.h Even 1 yes

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(630, [\chi])\):

\( T_{11}^{2} - 4 T_{11} + 16 \)
\( T_{13}^{2} + 4 T_{13} + 16 \)