Properties

Label 114.2.h.d
Level $114$
Weight $2$
Character orbit 114.h
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.h (of order \(6\), 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: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 0.866025i 1.73205i −0.500000 0.866025i 3.00000 + 1.73205i −1.50000 0.866025i −2.00000 −1.00000 −3.00000 3.00000 1.73205i
107.1 0.500000 + 0.866025i 1.73205i −0.500000 + 0.866025i 3.00000 1.73205i −1.50000 + 0.866025i −2.00000 −1.00000 −3.00000 3.00000 + 1.73205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 114.2.h.d yes 2
3.b odd 2 1 114.2.h.a 2
4.b odd 2 1 912.2.bn.b 2
12.b even 2 1 912.2.bn.d 2
19.d odd 6 1 114.2.h.a 2
57.f even 6 1 inner 114.2.h.d yes 2
76.f even 6 1 912.2.bn.d 2
228.n odd 6 1 912.2.bn.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.h.a 2 3.b odd 2 1
114.2.h.a 2 19.d odd 6 1
114.2.h.d yes 2 1.a even 1 1 trivial
114.2.h.d yes 2 57.f even 6 1 inner
912.2.bn.b 2 4.b odd 2 1
912.2.bn.b 2 228.n odd 6 1
912.2.bn.d 2 12.b even 2 1
912.2.bn.d 2 76.f even 6 1

Hecke kernels

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

\( T_{5}^{2} - 6 T_{5} + 12 \)
\( T_{7} + 2 \)
\( T_{17}^{2} - 12 T_{17} + 48 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( 3 + T^{2} \)
$5$ \( 12 - 6 T + T^{2} \)
$7$ \( ( 2 + T )^{2} \)
$11$ \( 3 + T^{2} \)
$13$ \( 12 + 6 T + T^{2} \)
$17$ \( 48 - 12 T + T^{2} \)
$19$ \( 19 - T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( 36 + 6 T + T^{2} \)
$31$ \( 48 + T^{2} \)
$37$ \( 48 + T^{2} \)
$41$ \( 9 + 3 T + T^{2} \)
$43$ \( 64 + 8 T + T^{2} \)
$47$ \( 12 + 6 T + T^{2} \)
$53$ \( 36 - 6 T + T^{2} \)
$59$ \( 9 - 3 T + T^{2} \)
$61$ \( 100 + 10 T + T^{2} \)
$67$ \( 27 - 9 T + T^{2} \)
$71$ \( 36 - 6 T + T^{2} \)
$73$ \( 25 + 5 T + T^{2} \)
$79$ \( 192 + 24 T + T^{2} \)
$83$ \( 27 + T^{2} \)
$89$ \( 36 + 6 T + T^{2} \)
$97$ \( 75 - 15 T + T^{2} \)
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