Properties

Label 58.2.b.a
Level 58
Weight 2
Character orbit 58.b
Analytic conductor 0.463
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 58 = 2 \cdot 29 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 58.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(0.463132331723\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{2} + i q^{3} - q^{4} + q^{5} - q^{6} -2 q^{7} -i q^{8} + 2 q^{9} +O(q^{10})\) \( q + i q^{2} + i q^{3} - q^{4} + q^{5} - q^{6} -2 q^{7} -i q^{8} + 2 q^{9} + i q^{10} -5 i q^{11} -i q^{12} - q^{13} -2 i q^{14} + i q^{15} + q^{16} -2 i q^{17} + 2 i q^{18} + 4 i q^{19} - q^{20} -2 i q^{21} + 5 q^{22} -6 q^{23} + q^{24} -4 q^{25} -i q^{26} + 5 i q^{27} + 2 q^{28} + ( 5 + 2 i ) q^{29} - q^{30} + 5 i q^{31} + i q^{32} + 5 q^{33} + 2 q^{34} -2 q^{35} -2 q^{36} + 8 i q^{37} -4 q^{38} -i q^{39} -i q^{40} -10 i q^{41} + 2 q^{42} -9 i q^{43} + 5 i q^{44} + 2 q^{45} -6 i q^{46} + 3 i q^{47} + i q^{48} -3 q^{49} -4 i q^{50} + 2 q^{51} + q^{52} - q^{53} -5 q^{54} -5 i q^{55} + 2 i q^{56} -4 q^{57} + ( -2 + 5 i ) q^{58} + 10 q^{59} -i q^{60} -10 i q^{61} -5 q^{62} -4 q^{63} - q^{64} - q^{65} + 5 i q^{66} + 8 q^{67} + 2 i q^{68} -6 i q^{69} -2 i q^{70} -8 q^{71} -2 i q^{72} + 16 i q^{73} -8 q^{74} -4 i q^{75} -4 i q^{76} + 10 i q^{77} + q^{78} -i q^{79} + q^{80} + q^{81} + 10 q^{82} + 14 q^{83} + 2 i q^{84} -2 i q^{85} + 9 q^{86} + ( -2 + 5 i ) q^{87} -5 q^{88} + 14 i q^{89} + 2 i q^{90} + 2 q^{91} + 6 q^{92} -5 q^{93} -3 q^{94} + 4 i q^{95} - q^{96} -2 i q^{97} -3 i q^{98} -10 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 2q^{5} - 2q^{6} - 4q^{7} + 4q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 2q^{5} - 2q^{6} - 4q^{7} + 4q^{9} - 2q^{13} + 2q^{16} - 2q^{20} + 10q^{22} - 12q^{23} + 2q^{24} - 8q^{25} + 4q^{28} + 10q^{29} - 2q^{30} + 10q^{33} + 4q^{34} - 4q^{35} - 4q^{36} - 8q^{38} + 4q^{42} + 4q^{45} - 6q^{49} + 4q^{51} + 2q^{52} - 2q^{53} - 10q^{54} - 8q^{57} - 4q^{58} + 20q^{59} - 10q^{62} - 8q^{63} - 2q^{64} - 2q^{65} + 16q^{67} - 16q^{71} - 16q^{74} + 2q^{78} + 2q^{80} + 2q^{81} + 20q^{82} + 28q^{83} + 18q^{86} - 4q^{87} - 10q^{88} + 4q^{91} + 12q^{92} - 10q^{93} - 6q^{94} - 2q^{96} + O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/58\mathbb{Z}\right)^\times\).

\(n\) \(31\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
57.1
1.00000i
1.00000i
1.00000i 1.00000i −1.00000 1.00000 −1.00000 −2.00000 1.00000i 2.00000 1.00000i
57.2 1.00000i 1.00000i −1.00000 1.00000 −1.00000 −2.00000 1.00000i 2.00000 1.00000i
\(n\): 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
29.b Even 1 yes

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(58, [\chi])\).