Properties

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

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

Character values

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

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

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

Hecke kernels

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

\( T_{5}^{2} + 12 \) Copy content Toggle raw display
\( T_{29} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

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