Properties

Label 116.2.c.a
Level 116
Weight 2
Character orbit 116.c
Analytic conductor 0.926
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 116 = 2^{2} \cdot 29 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 116.c (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(0.926264663447\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-7}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{3} + q^{5} -2 q^{7} -4 q^{9} +O(q^{10})\) \( q -\beta q^{3} + q^{5} -2 q^{7} -4 q^{9} -\beta q^{11} + 5 q^{13} -\beta q^{15} + 2 \beta q^{17} + 2 \beta q^{19} + 2 \beta q^{21} + 6 q^{23} -4 q^{25} + \beta q^{27} + ( -1 - 2 \beta ) q^{29} + \beta q^{31} -7 q^{33} -2 q^{35} + 4 \beta q^{37} -5 \beta q^{39} -2 \beta q^{41} + 3 \beta q^{43} -4 q^{45} -3 \beta q^{47} -3 q^{49} + 14 q^{51} + 5 q^{53} -\beta q^{55} + 14 q^{57} -14 q^{59} -2 \beta q^{61} + 8 q^{63} + 5 q^{65} -4 q^{67} -6 \beta q^{69} -8 q^{71} -4 \beta q^{73} + 4 \beta q^{75} + 2 \beta q^{77} + \beta q^{79} -5 q^{81} + 2 q^{83} + 2 \beta q^{85} + ( -14 + \beta ) q^{87} -2 \beta q^{89} -10 q^{91} + 7 q^{93} + 2 \beta q^{95} + 2 \beta q^{97} + 4 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} - 4q^{7} - 8q^{9} + O(q^{10}) \) \( 2q + 2q^{5} - 4q^{7} - 8q^{9} + 10q^{13} + 12q^{23} - 8q^{25} - 2q^{29} - 14q^{33} - 4q^{35} - 8q^{45} - 6q^{49} + 28q^{51} + 10q^{53} + 28q^{57} - 28q^{59} + 16q^{63} + 10q^{65} - 8q^{67} - 16q^{71} - 10q^{81} + 4q^{83} - 28q^{87} - 20q^{91} + 14q^{93} + O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/116\mathbb{Z}\right)^\times\).

\(n\) \(59\) \(89\)
\(\chi(n)\) \(1\) \(-1\)

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
0.500000 + 1.32288i
0.500000 1.32288i
0 2.64575i 0 1.00000 0 −2.00000 0 −4.00000 0
57.2 0 2.64575i 0 1.00000 0 −2.00000 0 −4.00000 0
\(n\): 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 \(S_{2}^{\mathrm{new}}(116, [\chi])\).