Properties

Label 114.2.b.c
Level $114$
Weight $2$
Character orbit 114.b
Analytic conductor $0.910$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 114 = 2 \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 114.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.910294583043\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + q^{2} + ( -1 - \zeta_{6} ) q^{3} + q^{4} + ( 2 - 4 \zeta_{6} ) q^{5} + ( -1 - \zeta_{6} ) q^{6} + q^{7} + q^{8} + 3 \zeta_{6} q^{9} +O(q^{10})\) \( q + q^{2} + ( -1 - \zeta_{6} ) q^{3} + q^{4} + ( 2 - 4 \zeta_{6} ) q^{5} + ( -1 - \zeta_{6} ) q^{6} + q^{7} + q^{8} + 3 \zeta_{6} q^{9} + ( 2 - 4 \zeta_{6} ) q^{10} + ( -2 + 4 \zeta_{6} ) q^{11} + ( -1 - \zeta_{6} ) q^{12} + ( -1 + 2 \zeta_{6} ) q^{13} + q^{14} + ( -6 + 6 \zeta_{6} ) q^{15} + q^{16} + ( 1 - 2 \zeta_{6} ) q^{17} + 3 \zeta_{6} q^{18} + ( -3 - 2 \zeta_{6} ) q^{19} + ( 2 - 4 \zeta_{6} ) q^{20} + ( -1 - \zeta_{6} ) q^{21} + ( -2 + 4 \zeta_{6} ) q^{22} + ( -3 + 6 \zeta_{6} ) q^{23} + ( -1 - \zeta_{6} ) q^{24} -7 q^{25} + ( -1 + 2 \zeta_{6} ) q^{26} + ( 3 - 6 \zeta_{6} ) q^{27} + q^{28} + 9 q^{29} + ( -6 + 6 \zeta_{6} ) q^{30} + ( -6 + 12 \zeta_{6} ) q^{31} + q^{32} + ( 6 - 6 \zeta_{6} ) q^{33} + ( 1 - 2 \zeta_{6} ) q^{34} + ( 2 - 4 \zeta_{6} ) q^{35} + 3 \zeta_{6} q^{36} + ( 4 - 8 \zeta_{6} ) q^{37} + ( -3 - 2 \zeta_{6} ) q^{38} + ( 3 - 3 \zeta_{6} ) q^{39} + ( 2 - 4 \zeta_{6} ) q^{40} + ( -1 - \zeta_{6} ) q^{42} + 2 q^{43} + ( -2 + 4 \zeta_{6} ) q^{44} + ( 12 - 6 \zeta_{6} ) q^{45} + ( -3 + 6 \zeta_{6} ) q^{46} + ( 2 - 4 \zeta_{6} ) q^{47} + ( -1 - \zeta_{6} ) q^{48} -6 q^{49} -7 q^{50} + ( -3 + 3 \zeta_{6} ) q^{51} + ( -1 + 2 \zeta_{6} ) q^{52} -9 q^{53} + ( 3 - 6 \zeta_{6} ) q^{54} + 12 q^{55} + q^{56} + ( 1 + 7 \zeta_{6} ) q^{57} + 9 q^{58} -3 q^{59} + ( -6 + 6 \zeta_{6} ) q^{60} -8 q^{61} + ( -6 + 12 \zeta_{6} ) q^{62} + 3 \zeta_{6} q^{63} + q^{64} + 6 q^{65} + ( 6 - 6 \zeta_{6} ) q^{66} + ( 5 - 10 \zeta_{6} ) q^{67} + ( 1 - 2 \zeta_{6} ) q^{68} + ( 9 - 9 \zeta_{6} ) q^{69} + ( 2 - 4 \zeta_{6} ) q^{70} -12 q^{71} + 3 \zeta_{6} q^{72} + 11 q^{73} + ( 4 - 8 \zeta_{6} ) q^{74} + ( 7 + 7 \zeta_{6} ) q^{75} + ( -3 - 2 \zeta_{6} ) q^{76} + ( -2 + 4 \zeta_{6} ) q^{77} + ( 3 - 3 \zeta_{6} ) q^{78} + ( 4 - 8 \zeta_{6} ) q^{79} + ( 2 - 4 \zeta_{6} ) q^{80} + ( -9 + 9 \zeta_{6} ) q^{81} + ( 6 - 12 \zeta_{6} ) q^{83} + ( -1 - \zeta_{6} ) q^{84} -6 q^{85} + 2 q^{86} + ( -9 - 9 \zeta_{6} ) q^{87} + ( -2 + 4 \zeta_{6} ) q^{88} + 6 q^{89} + ( 12 - 6 \zeta_{6} ) q^{90} + ( -1 + 2 \zeta_{6} ) q^{91} + ( -3 + 6 \zeta_{6} ) q^{92} + ( 18 - 18 \zeta_{6} ) q^{93} + ( 2 - 4 \zeta_{6} ) q^{94} + ( -14 + 16 \zeta_{6} ) q^{95} + ( -1 - \zeta_{6} ) q^{96} + ( -8 + 16 \zeta_{6} ) q^{97} -6 q^{98} + ( -12 + 6 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 3q^{3} + 2q^{4} - 3q^{6} + 2q^{7} + 2q^{8} + 3q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 3q^{3} + 2q^{4} - 3q^{6} + 2q^{7} + 2q^{8} + 3q^{9} - 3q^{12} + 2q^{14} - 6q^{15} + 2q^{16} + 3q^{18} - 8q^{19} - 3q^{21} - 3q^{24} - 14q^{25} + 2q^{28} + 18q^{29} - 6q^{30} + 2q^{32} + 6q^{33} + 3q^{36} - 8q^{38} + 3q^{39} - 3q^{42} + 4q^{43} + 18q^{45} - 3q^{48} - 12q^{49} - 14q^{50} - 3q^{51} - 18q^{53} + 24q^{55} + 2q^{56} + 9q^{57} + 18q^{58} - 6q^{59} - 6q^{60} - 16q^{61} + 3q^{63} + 2q^{64} + 12q^{65} + 6q^{66} + 9q^{69} - 24q^{71} + 3q^{72} + 22q^{73} + 21q^{75} - 8q^{76} + 3q^{78} - 9q^{81} - 3q^{84} - 12q^{85} + 4q^{86} - 27q^{87} + 12q^{89} + 18q^{90} + 18q^{93} - 12q^{95} - 3q^{96} - 12q^{98} - 18q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/114\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(97\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1
0.500000 + 0.866025i
0.500000 0.866025i
1.00000 −1.50000 0.866025i 1.00000 3.46410i −1.50000 0.866025i 1.00000 1.00000 1.50000 + 2.59808i 3.46410i
113.2 1.00000 −1.50000 + 0.866025i 1.00000 3.46410i −1.50000 + 0.866025i 1.00000 1.00000 1.50000 2.59808i 3.46410i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 114.2.b.c yes 2
3.b odd 2 1 114.2.b.b 2
4.b odd 2 1 912.2.f.e 2
12.b even 2 1 912.2.f.a 2
19.b odd 2 1 114.2.b.b 2
57.d even 2 1 inner 114.2.b.c yes 2
76.d even 2 1 912.2.f.a 2
228.b odd 2 1 912.2.f.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.b.b 2 3.b odd 2 1
114.2.b.b 2 19.b odd 2 1
114.2.b.c yes 2 1.a even 1 1 trivial
114.2.b.c yes 2 57.d even 2 1 inner
912.2.f.a 2 12.b even 2 1
912.2.f.a 2 76.d even 2 1
912.2.f.e 2 4.b odd 2 1
912.2.f.e 2 228.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(114, [\chi])\):

\( T_{5}^{2} + 12 \)
\( T_{29} - 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( 3 + 3 T + T^{2} \)
$5$ \( 12 + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( 12 + T^{2} \)
$13$ \( 3 + T^{2} \)
$17$ \( 3 + T^{2} \)
$19$ \( 19 + 8 T + T^{2} \)
$23$ \( 27 + T^{2} \)
$29$ \( ( -9 + T )^{2} \)
$31$ \( 108 + T^{2} \)
$37$ \( 48 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -2 + T )^{2} \)
$47$ \( 12 + T^{2} \)
$53$ \( ( 9 + T )^{2} \)
$59$ \( ( 3 + T )^{2} \)
$61$ \( ( 8 + T )^{2} \)
$67$ \( 75 + T^{2} \)
$71$ \( ( 12 + T )^{2} \)
$73$ \( ( -11 + T )^{2} \)
$79$ \( 48 + T^{2} \)
$83$ \( 108 + T^{2} \)
$89$ \( ( -6 + T )^{2} \)
$97$ \( 192 + T^{2} \)
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