Properties

Label 114.2.e.a
Level $114$
Weight $2$
Character orbit 114.e
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.e (of order \(3\), degree \(2\), 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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{6} ) q^{2} + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{4} + ( 4 - 4 \zeta_{6} ) q^{5} + \zeta_{6} q^{6} -3 q^{7} + q^{8} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( -1 + \zeta_{6} ) q^{2} + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{4} + ( 4 - 4 \zeta_{6} ) q^{5} + \zeta_{6} q^{6} -3 q^{7} + q^{8} -\zeta_{6} q^{9} + 4 \zeta_{6} q^{10} + 2 q^{11} - q^{12} + 7 \zeta_{6} q^{13} + ( 3 - 3 \zeta_{6} ) q^{14} -4 \zeta_{6} q^{15} + ( -1 + \zeta_{6} ) q^{16} + q^{18} + ( -5 + 2 \zeta_{6} ) q^{19} -4 q^{20} + ( -3 + 3 \zeta_{6} ) q^{21} + ( -2 + 2 \zeta_{6} ) q^{22} + 4 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{24} -11 \zeta_{6} q^{25} -7 q^{26} - q^{27} + 3 \zeta_{6} q^{28} -4 \zeta_{6} q^{29} + 4 q^{30} + q^{31} -\zeta_{6} q^{32} + ( 2 - 2 \zeta_{6} ) q^{33} + ( -12 + 12 \zeta_{6} ) q^{35} + ( -1 + \zeta_{6} ) q^{36} + 7 q^{37} + ( 3 - 5 \zeta_{6} ) q^{38} + 7 q^{39} + ( 4 - 4 \zeta_{6} ) q^{40} + ( -4 + 4 \zeta_{6} ) q^{41} -3 \zeta_{6} q^{42} + ( -7 + 7 \zeta_{6} ) q^{43} -2 \zeta_{6} q^{44} -4 q^{45} -4 q^{46} -2 \zeta_{6} q^{47} + \zeta_{6} q^{48} + 2 q^{49} + 11 q^{50} + ( 7 - 7 \zeta_{6} ) q^{52} + 4 \zeta_{6} q^{53} + ( 1 - \zeta_{6} ) q^{54} + ( 8 - 8 \zeta_{6} ) q^{55} -3 q^{56} + ( -3 + 5 \zeta_{6} ) q^{57} + 4 q^{58} + ( 6 - 6 \zeta_{6} ) q^{59} + ( -4 + 4 \zeta_{6} ) q^{60} + \zeta_{6} q^{61} + ( -1 + \zeta_{6} ) q^{62} + 3 \zeta_{6} q^{63} + q^{64} + 28 q^{65} + 2 \zeta_{6} q^{66} -3 \zeta_{6} q^{67} + 4 q^{69} -12 \zeta_{6} q^{70} + ( -2 + 2 \zeta_{6} ) q^{71} -\zeta_{6} q^{72} + ( 3 - 3 \zeta_{6} ) q^{73} + ( -7 + 7 \zeta_{6} ) q^{74} -11 q^{75} + ( 2 + 3 \zeta_{6} ) q^{76} -6 q^{77} + ( -7 + 7 \zeta_{6} ) q^{78} + ( -5 + 5 \zeta_{6} ) q^{79} + 4 \zeta_{6} q^{80} + ( -1 + \zeta_{6} ) q^{81} -4 \zeta_{6} q^{82} -12 q^{83} + 3 q^{84} -7 \zeta_{6} q^{86} -4 q^{87} + 2 q^{88} -18 \zeta_{6} q^{89} + ( 4 - 4 \zeta_{6} ) q^{90} -21 \zeta_{6} q^{91} + ( 4 - 4 \zeta_{6} ) q^{92} + ( 1 - \zeta_{6} ) q^{93} + 2 q^{94} + ( -12 + 20 \zeta_{6} ) q^{95} - q^{96} + ( -10 + 10 \zeta_{6} ) q^{97} + ( -2 + 2 \zeta_{6} ) q^{98} -2 \zeta_{6} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} + q^{3} - q^{4} + 4q^{5} + q^{6} - 6q^{7} + 2q^{8} - q^{9} + O(q^{10}) \) \( 2q - q^{2} + q^{3} - q^{4} + 4q^{5} + q^{6} - 6q^{7} + 2q^{8} - q^{9} + 4q^{10} + 4q^{11} - 2q^{12} + 7q^{13} + 3q^{14} - 4q^{15} - q^{16} + 2q^{18} - 8q^{19} - 8q^{20} - 3q^{21} - 2q^{22} + 4q^{23} + q^{24} - 11q^{25} - 14q^{26} - 2q^{27} + 3q^{28} - 4q^{29} + 8q^{30} + 2q^{31} - q^{32} + 2q^{33} - 12q^{35} - q^{36} + 14q^{37} + q^{38} + 14q^{39} + 4q^{40} - 4q^{41} - 3q^{42} - 7q^{43} - 2q^{44} - 8q^{45} - 8q^{46} - 2q^{47} + q^{48} + 4q^{49} + 22q^{50} + 7q^{52} + 4q^{53} + q^{54} + 8q^{55} - 6q^{56} - q^{57} + 8q^{58} + 6q^{59} - 4q^{60} + q^{61} - q^{62} + 3q^{63} + 2q^{64} + 56q^{65} + 2q^{66} - 3q^{67} + 8q^{69} - 12q^{70} - 2q^{71} - q^{72} + 3q^{73} - 7q^{74} - 22q^{75} + 7q^{76} - 12q^{77} - 7q^{78} - 5q^{79} + 4q^{80} - q^{81} - 4q^{82} - 24q^{83} + 6q^{84} - 7q^{86} - 8q^{87} + 4q^{88} - 18q^{89} + 4q^{90} - 21q^{91} + 4q^{92} + q^{93} + 4q^{94} - 4q^{95} - 2q^{96} - 10q^{97} - 2q^{98} - 2q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(77\) \(97\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 2.00000 + 3.46410i 0.500000 0.866025i −3.00000 1.00000 −0.500000 + 0.866025i 2.00000 3.46410i
49.1 −0.500000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i 2.00000 3.46410i 0.500000 + 0.866025i −3.00000 1.00000 −0.500000 0.866025i 2.00000 + 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
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 114.2.e.a 2
3.b odd 2 1 342.2.g.d 2
4.b odd 2 1 912.2.q.d 2
12.b even 2 1 2736.2.s.c 2
19.c even 3 1 inner 114.2.e.a 2
19.c even 3 1 2166.2.a.f 1
19.d odd 6 1 2166.2.a.c 1
57.f even 6 1 6498.2.a.x 1
57.h odd 6 1 342.2.g.d 2
57.h odd 6 1 6498.2.a.l 1
76.g odd 6 1 912.2.q.d 2
228.m even 6 1 2736.2.s.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.e.a 2 1.a even 1 1 trivial
114.2.e.a 2 19.c even 3 1 inner
342.2.g.d 2 3.b odd 2 1
342.2.g.d 2 57.h odd 6 1
912.2.q.d 2 4.b odd 2 1
912.2.q.d 2 76.g odd 6 1
2166.2.a.c 1 19.d odd 6 1
2166.2.a.f 1 19.c even 3 1
2736.2.s.c 2 12.b even 2 1
2736.2.s.c 2 228.m even 6 1
6498.2.a.l 1 57.h odd 6 1
6498.2.a.x 1 57.f even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 4 T_{5} + 16 \) acting on \(S_{2}^{\mathrm{new}}(114, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( 16 - 4 T + T^{2} \)
$7$ \( ( 3 + T )^{2} \)
$11$ \( ( -2 + T )^{2} \)
$13$ \( 49 - 7 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 19 + 8 T + T^{2} \)
$23$ \( 16 - 4 T + T^{2} \)
$29$ \( 16 + 4 T + T^{2} \)
$31$ \( ( -1 + T )^{2} \)
$37$ \( ( -7 + T )^{2} \)
$41$ \( 16 + 4 T + T^{2} \)
$43$ \( 49 + 7 T + T^{2} \)
$47$ \( 4 + 2 T + T^{2} \)
$53$ \( 16 - 4 T + T^{2} \)
$59$ \( 36 - 6 T + T^{2} \)
$61$ \( 1 - T + T^{2} \)
$67$ \( 9 + 3 T + T^{2} \)
$71$ \( 4 + 2 T + T^{2} \)
$73$ \( 9 - 3 T + T^{2} \)
$79$ \( 25 + 5 T + T^{2} \)
$83$ \( ( 12 + T )^{2} \)
$89$ \( 324 + 18 T + T^{2} \)
$97$ \( 100 + 10 T + T^{2} \)
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