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 $2$

Related objects

Downloads

Learn more

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: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{2} + ( - \zeta_{6} + 1) q^{3} - \zeta_{6} q^{4} + ( - 4 \zeta_{6} + 4) q^{5} + \zeta_{6} q^{6} - 3 q^{7} + q^{8} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{2} + ( - \zeta_{6} + 1) q^{3} - \zeta_{6} q^{4} + ( - 4 \zeta_{6} + 4) q^{5} + \zeta_{6} q^{6} - 3 q^{7} + q^{8} - \zeta_{6} q^{9} + 4 \zeta_{6} q^{10} + 2 q^{11} - q^{12} + 7 \zeta_{6} q^{13} + ( - 3 \zeta_{6} + 3) q^{14} - 4 \zeta_{6} q^{15} + (\zeta_{6} - 1) q^{16} + q^{18} + (2 \zeta_{6} - 5) q^{19} - 4 q^{20} + (3 \zeta_{6} - 3) q^{21} + (2 \zeta_{6} - 2) q^{22} + 4 \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{24} - 11 \zeta_{6} q^{25} - 7 q^{26} - q^{27} + 3 \zeta_{6} q^{28} - 4 \zeta_{6} q^{29} + 4 q^{30} + q^{31} - \zeta_{6} q^{32} + ( - 2 \zeta_{6} + 2) q^{33} + (12 \zeta_{6} - 12) q^{35} + (\zeta_{6} - 1) q^{36} + 7 q^{37} + ( - 5 \zeta_{6} + 3) q^{38} + 7 q^{39} + ( - 4 \zeta_{6} + 4) q^{40} + (4 \zeta_{6} - 4) q^{41} - 3 \zeta_{6} q^{42} + (7 \zeta_{6} - 7) q^{43} - 2 \zeta_{6} q^{44} - 4 q^{45} - 4 q^{46} - 2 \zeta_{6} q^{47} + \zeta_{6} q^{48} + 2 q^{49} + 11 q^{50} + ( - 7 \zeta_{6} + 7) q^{52} + 4 \zeta_{6} q^{53} + ( - \zeta_{6} + 1) q^{54} + ( - 8 \zeta_{6} + 8) q^{55} - 3 q^{56} + (5 \zeta_{6} - 3) q^{57} + 4 q^{58} + ( - 6 \zeta_{6} + 6) q^{59} + (4 \zeta_{6} - 4) q^{60} + \zeta_{6} q^{61} + (\zeta_{6} - 1) q^{62} + 3 \zeta_{6} q^{63} + q^{64} + 28 q^{65} + 2 \zeta_{6} q^{66} - 3 \zeta_{6} q^{67} + 4 q^{69} - 12 \zeta_{6} q^{70} + (2 \zeta_{6} - 2) q^{71} - \zeta_{6} q^{72} + ( - 3 \zeta_{6} + 3) q^{73} + (7 \zeta_{6} - 7) q^{74} - 11 q^{75} + (3 \zeta_{6} + 2) q^{76} - 6 q^{77} + (7 \zeta_{6} - 7) q^{78} + (5 \zeta_{6} - 5) q^{79} + 4 \zeta_{6} q^{80} + (\zeta_{6} - 1) q^{81} - 4 \zeta_{6} q^{82} - 12 q^{83} + 3 q^{84} - 7 \zeta_{6} q^{86} - 4 q^{87} + 2 q^{88} - 18 \zeta_{6} q^{89} + ( - 4 \zeta_{6} + 4) q^{90} - 21 \zeta_{6} q^{91} + ( - 4 \zeta_{6} + 4) q^{92} + ( - \zeta_{6} + 1) q^{93} + 2 q^{94} + (20 \zeta_{6} - 12) q^{95} - q^{96} + (10 \zeta_{6} - 10) q^{97} + (2 \zeta_{6} - 2) q^{98} - 2 \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + q^{3} - q^{4} + 4 q^{5} + q^{6} - 6 q^{7} + 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + q^{3} - q^{4} + 4 q^{5} + q^{6} - 6 q^{7} + 2 q^{8} - q^{9} + 4 q^{10} + 4 q^{11} - 2 q^{12} + 7 q^{13} + 3 q^{14} - 4 q^{15} - q^{16} + 2 q^{18} - 8 q^{19} - 8 q^{20} - 3 q^{21} - 2 q^{22} + 4 q^{23} + q^{24} - 11 q^{25} - 14 q^{26} - 2 q^{27} + 3 q^{28} - 4 q^{29} + 8 q^{30} + 2 q^{31} - q^{32} + 2 q^{33} - 12 q^{35} - q^{36} + 14 q^{37} + q^{38} + 14 q^{39} + 4 q^{40} - 4 q^{41} - 3 q^{42} - 7 q^{43} - 2 q^{44} - 8 q^{45} - 8 q^{46} - 2 q^{47} + q^{48} + 4 q^{49} + 22 q^{50} + 7 q^{52} + 4 q^{53} + q^{54} + 8 q^{55} - 6 q^{56} - q^{57} + 8 q^{58} + 6 q^{59} - 4 q^{60} + q^{61} - q^{62} + 3 q^{63} + 2 q^{64} + 56 q^{65} + 2 q^{66} - 3 q^{67} + 8 q^{69} - 12 q^{70} - 2 q^{71} - q^{72} + 3 q^{73} - 7 q^{74} - 22 q^{75} + 7 q^{76} - 12 q^{77} - 7 q^{78} - 5 q^{79} + 4 q^{80} - q^{81} - 4 q^{82} - 24 q^{83} + 6 q^{84} - 7 q^{86} - 8 q^{87} + 4 q^{88} - 18 q^{89} + 4 q^{90} - 21 q^{91} + 4 q^{92} + q^{93} + 4 q^{94} - 4 q^{95} - 2 q^{96} - 10 q^{97} - 2 q^{98} - 2 q^{99}+O(q^{100}) \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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 \( T_{5}^{2} - 4T_{5} + 16 \) acting on \(S_{2}^{\mathrm{new}}(114, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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