Properties

Label 61.2.f.a
Level 61
Weight 2
Character orbit 61.f
Analytic conductor 0.487
Analytic rank 1
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 61 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 61.f (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.48708745233\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

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

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/61\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1
0.500000 0.866025i
0.500000 + 0.866025i
−1.50000 + 0.866025i −2.00000 0.500000 0.866025i −1.50000 2.59808i 3.00000 1.73205i −3.00000 + 1.73205i 1.73205i 1.00000 4.50000 + 2.59808i
48.1 −1.50000 0.866025i −2.00000 0.500000 + 0.866025i −1.50000 + 2.59808i 3.00000 + 1.73205i −3.00000 1.73205i 1.73205i 1.00000 4.50000 2.59808i
\(n\): 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
61.f Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2}^{2} + 3 T_{2} + 3 \) acting on \(S_{2}^{\mathrm{new}}(61, [\chi])\).