Properties

Label 114.2.h.c
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: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

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 −1.50000 0.866025i −0.500000 0.866025i −3.00000 1.73205i −1.50000 + 0.866025i 1.00000 −1.00000 1.50000 + 2.59808i −3.00000 + 1.73205i
107.1 0.500000 + 0.866025i −1.50000 + 0.866025i −0.500000 + 0.866025i −3.00000 + 1.73205i −1.50000 0.866025i 1.00000 −1.00000 1.50000 2.59808i −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.c yes 2
3.b odd 2 1 114.2.h.b 2
4.b odd 2 1 912.2.bn.c 2
12.b even 2 1 912.2.bn.f 2
19.d odd 6 1 114.2.h.b 2
57.f even 6 1 inner 114.2.h.c yes 2
76.f even 6 1 912.2.bn.f 2
228.n odd 6 1 912.2.bn.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.h.b 2 3.b odd 2 1
114.2.h.b 2 19.d odd 6 1
114.2.h.c yes 2 1.a even 1 1 trivial
114.2.h.c yes 2 57.f even 6 1 inner
912.2.bn.c 2 4.b odd 2 1
912.2.bn.c 2 228.n odd 6 1
912.2.bn.f 2 12.b even 2 1
912.2.bn.f 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 \(S_{2}^{\mathrm{new}}(114, [\chi])\):

\( T_{5}^{2} + 6T_{5} + 12 \) Copy content Toggle raw display
\( T_{7} - 1 \) Copy content Toggle raw display
\( T_{17}^{2} - 6T_{17} + 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$5$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 12 \) Copy content Toggle raw display
$13$ \( T^{2} - 9T + 27 \) Copy content Toggle raw display
$17$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$19$ \( T^{2} + 8T + 19 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$31$ \( T^{2} + 3 \) Copy content Toggle raw display
$37$ \( T^{2} + 27 \) Copy content Toggle raw display
$41$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$43$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} + 12T + 48 \) Copy content Toggle raw display
$53$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$67$ \( T^{2} - 15T + 75 \) Copy content Toggle raw display
$71$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$73$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$79$ \( T^{2} - 9T + 27 \) Copy content Toggle raw display
$83$ \( T^{2} + 108 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 12T + 48 \) Copy content Toggle raw display
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