Properties

Label 114.2.i.c
Level $114$
Weight $2$
Character orbit 114.i
Analytic conductor $0.910$
Analytic rank $0$
Dimension $6$
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.i (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.910294583043\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{18} q^{2} - \zeta_{18}^{4} q^{3} + \zeta_{18}^{2} q^{4} + ( - \zeta_{18}^{5} + 2 \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} - 2 \zeta_{18} + 1) q^{5} + \zeta_{18}^{5} q^{6} + ( - 3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18}^{2} + 2 \zeta_{18} - 1) q^{7} - \zeta_{18}^{3} q^{8} + (\zeta_{18}^{5} - \zeta_{18}^{2}) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{18} q^{2} - \zeta_{18}^{4} q^{3} + \zeta_{18}^{2} q^{4} + ( - \zeta_{18}^{5} + 2 \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} - 2 \zeta_{18} + 1) q^{5} + \zeta_{18}^{5} q^{6} + ( - 3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18}^{2} + 2 \zeta_{18} - 1) q^{7} - \zeta_{18}^{3} q^{8} + (\zeta_{18}^{5} - \zeta_{18}^{2}) q^{9} + ( - 2 \zeta_{18}^{5} + \zeta_{18}^{4} + 2 \zeta_{18}^{2} - \zeta_{18} - 1) q^{10} + (2 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{2} + 2 \zeta_{18}) q^{11} + ( - \zeta_{18}^{3} + 1) q^{12} + ( - 2 \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{3} - 2 \zeta_{18} + 1) q^{13} + (3 \zeta_{18}^{5} - \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + \zeta_{18} - 3) q^{14} + ( - \zeta_{18}^{3} + 2 \zeta_{18}^{2} - \zeta_{18}) q^{15} + \zeta_{18}^{4} q^{16} + (\zeta_{18}^{5} - \zeta_{18}^{3} - \zeta_{18}^{2} + 2 \zeta_{18}) q^{17} + q^{18} + (2 \zeta_{18}^{4} + 3 \zeta_{18}) q^{19} + ( - \zeta_{18}^{5} + \zeta_{18}^{2} + \zeta_{18} - 2) q^{20} + (\zeta_{18}^{5} - \zeta_{18}^{3} - 3 \zeta_{18}^{2} + \zeta_{18} - 2) q^{21} + (3 \zeta_{18}^{5} + \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 2) q^{22} + (6 \zeta_{18}^{4} - 3 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 3 \zeta_{18} + 6) q^{23} + (\zeta_{18}^{4} - \zeta_{18}) q^{24} + ( - \zeta_{18}^{5} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{3} + \zeta_{18} - 4) q^{25} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - \zeta_{18} - 2) q^{26} + \zeta_{18}^{3} q^{27} + (\zeta_{18}^{5} - 2 \zeta_{18}^{4} - \zeta_{18}^{3} - \zeta_{18}^{2} + 3 \zeta_{18} + 3) q^{28} + (2 \zeta_{18}^{5} + \zeta_{18}^{4} - \zeta_{18}^{3} - 2 \zeta_{18}^{2}) q^{29} + (\zeta_{18}^{4} - 2 \zeta_{18}^{3} + \zeta_{18}^{2}) q^{30} + (5 \zeta_{18}^{3} + \zeta_{18}^{2} - \zeta_{18} - 5) q^{31} - \zeta_{18}^{5} q^{32} + (\zeta_{18}^{5} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{2} - 1) q^{33} + (\zeta_{18}^{4} - 2 \zeta_{18}^{2} + 1) q^{34} + (4 \zeta_{18}^{5} - 7 \zeta_{18}^{4} + 4 \zeta_{18}^{3}) q^{35} - \zeta_{18} q^{36} + (4 \zeta_{18}^{5} - 4 \zeta_{18}^{2} - 4 \zeta_{18} - 1) q^{37} + ( - 2 \zeta_{18}^{5} - 3 \zeta_{18}^{2}) q^{38} + (\zeta_{18}^{5} - 2 \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} - 2) q^{39} + ( - \zeta_{18}^{2} + 2 \zeta_{18} - 1) q^{40} + ( - 2 \zeta_{18}^{5} + 3 \zeta_{18}^{4} + \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 3) q^{41} + (\zeta_{18}^{4} + 2 \zeta_{18}^{3} - \zeta_{18}^{2} + 2 \zeta_{18} + 1) q^{42} + ( - 3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - 5 \zeta_{18}^{3} + 5 \zeta_{18}^{2} + 3 \zeta_{18} + 3) q^{43} + ( - \zeta_{18}^{4} - \zeta_{18}^{3} - 2 \zeta_{18} + 3) q^{44} + (\zeta_{18}^{5} + \zeta_{18}^{4} - 2 \zeta_{18}^{3} - \zeta_{18} + 2) q^{45} + ( - 6 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 6 \zeta_{18}) q^{46} + ( - 3 \zeta_{18}^{5} - \zeta_{18}^{4} - 3 \zeta_{18}^{3} + 3 \zeta_{18}^{2} + 4 \zeta_{18} + 4) q^{47} + ( - \zeta_{18}^{5} + \zeta_{18}^{2}) q^{48} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} - 8 \zeta_{18}^{3} - \zeta_{18}^{2} - \zeta_{18}) q^{49} + ( - 3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}^{2} + 4 \zeta_{18} - 1) q^{50} + ( - 2 \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}) q^{51} + (\zeta_{18}^{5} - 2 \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} + 2 \zeta_{18} - 1) q^{52} + (2 \zeta_{18}^{3} + 9 \zeta_{18}^{2} + 2 \zeta_{18}) q^{53} - \zeta_{18}^{4} q^{54} + (7 \zeta_{18}^{5} - 7 \zeta_{18}^{3} + \zeta_{18}^{2} - 4 \zeta_{18} + 8) q^{55} + (2 \zeta_{18}^{5} + \zeta_{18}^{4} - 3 \zeta_{18}^{2} - 3 \zeta_{18} + 1) q^{56} + ( - 5 \zeta_{18}^{5} + 2 \zeta_{18}^{2}) q^{57} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} + 2) q^{58} + (\zeta_{18}^{5} - \zeta_{18}^{3} - 5 \zeta_{18}^{2} - 6 \zeta_{18} - 4) q^{59} + ( - \zeta_{18}^{5} + 2 \zeta_{18}^{4} - \zeta_{18}^{3}) q^{60} + ( - 3 \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18}^{2} + \zeta_{18} - 3) q^{61} + ( - 5 \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} + 5 \zeta_{18}) q^{62} + ( - \zeta_{18}^{5} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - \zeta_{18} - 2) q^{63} + (\zeta_{18}^{3} - 1) q^{64} + ( - 5 \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 4 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 5 \zeta_{18}) q^{65} + ( - 3 \zeta_{18}^{4} + 2 \zeta_{18}^{3} + \zeta_{18} + 1) q^{66} + (4 \zeta_{18}^{5} - 5 \zeta_{18}^{4} + 11 \zeta_{18}^{3} - 4 \zeta_{18}^{2} - 6 \zeta_{18} - 6) q^{67} + ( - \zeta_{18}^{5} + 2 \zeta_{18}^{3} - \zeta_{18}) q^{68} + ( - 3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 6 \zeta_{18}^{2} - 3 \zeta_{18} + 2) q^{69} + (7 \zeta_{18}^{5} - 4 \zeta_{18}^{4} - 4 \zeta_{18}^{3} + 4) q^{70} + ( - 3 \zeta_{18}^{5} - 7 \zeta_{18}^{4} - 5 \zeta_{18}^{3} + 5 \zeta_{18}^{2} + 7 \zeta_{18} + 3) q^{71} + \zeta_{18}^{2} q^{72} + (\zeta_{18}^{5} + 5 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 3) q^{73} + (4 \zeta_{18}^{2} + \zeta_{18} + 4) q^{74} + ( - 4 \zeta_{18}^{5} + \zeta_{18}^{4} + 3 \zeta_{18}^{2} + 3 \zeta_{18} - 1) q^{75} + (5 \zeta_{18}^{3} - 2) q^{76} + ( - 6 \zeta_{18}^{5} + 10 \zeta_{18}^{4} - 4 \zeta_{18}^{2} - 4 \zeta_{18} - 7) q^{77} + (2 \zeta_{18}^{5} - 2 \zeta_{18}^{3} - \zeta_{18}^{2} + 2 \zeta_{18} + 1) q^{78} + (4 \zeta_{18}^{5} + 5 \zeta_{18}^{4} - \zeta_{18}^{3} - 5 \zeta_{18}^{2} + 5) q^{79} + (\zeta_{18}^{3} - 2 \zeta_{18}^{2} + \zeta_{18}) q^{80} + ( - \zeta_{18}^{4} + \zeta_{18}) q^{81} + ( - 3 \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} + 3 \zeta_{18} - 2) q^{82} + (3 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - \zeta_{18}^{3} - 4 \zeta_{18}^{2} + \zeta_{18} + 1) q^{83} + ( - \zeta_{18}^{5} - 2 \zeta_{18}^{4} + \zeta_{18}^{3} - 2 \zeta_{18}^{2} - \zeta_{18}) q^{84} + (5 \zeta_{18}^{5} - \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 5 \zeta_{18}^{2} + 3 \zeta_{18} + 3) q^{85} + (3 \zeta_{18}^{5} + 5 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 3 \zeta_{18}^{2} - 3 \zeta_{18} - 3) q^{86} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} + 2 \zeta_{18}^{3} + \zeta_{18}^{2} - \zeta_{18}) q^{87} + (\zeta_{18}^{5} + \zeta_{18}^{4} + 2 \zeta_{18}^{2} - 3 \zeta_{18}) q^{88} + ( - \zeta_{18}^{5} - 7 \zeta_{18} + 7) q^{89} + ( - \zeta_{18}^{5} + 2 \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} - 2 \zeta_{18} + 1) q^{90} + ( - 8 \zeta_{18}^{4} + 2 \zeta_{18}^{3} + 5 \zeta_{18}^{2} + 2 \zeta_{18} - 8) q^{91} + ( - 3 \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 3 \zeta_{18}^{3} + 6 \zeta_{18}^{2} - 6) q^{92} + (\zeta_{18}^{5} - \zeta_{18}^{3} + 5 \zeta_{18} + 1) q^{93} + (\zeta_{18}^{5} + 3 \zeta_{18}^{4} - 4 \zeta_{18}^{2} - 4 \zeta_{18} - 3) q^{94} + (6 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 10 \zeta_{18}^{2} + 5 \zeta_{18} + 3) q^{95} - q^{96} + (8 \zeta_{18}^{5} - 8 \zeta_{18}^{3} - \zeta_{18}^{2} - 4 \zeta_{18} + 7) q^{97} + (\zeta_{18}^{5} + 8 \zeta_{18}^{4} + 2 \zeta_{18}^{3} + \zeta_{18}^{2} - 1) q^{98} + ( - 2 \zeta_{18}^{4} + 3 \zeta_{18}^{3} + 3 \zeta_{18} - 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{5} - 3 q^{7} - 3 q^{8} - 6 q^{10} + 3 q^{12} + 9 q^{13} - 12 q^{14} - 3 q^{15} - 3 q^{17} + 6 q^{18} - 12 q^{20} - 15 q^{21} + 15 q^{22} + 27 q^{23} - 15 q^{25} - 6 q^{26} + 3 q^{27} + 15 q^{28} - 3 q^{29} - 6 q^{30} - 15 q^{31} + 3 q^{33} + 6 q^{34} + 12 q^{35} - 6 q^{37} - 12 q^{39} - 6 q^{40} - 15 q^{41} + 12 q^{42} + 3 q^{43} + 15 q^{44} + 6 q^{45} - 6 q^{46} + 15 q^{47} - 24 q^{49} - 3 q^{50} + 3 q^{51} - 9 q^{52} + 6 q^{53} + 27 q^{55} + 6 q^{56} + 12 q^{58} - 27 q^{59} - 3 q^{60} - 15 q^{61} - 3 q^{62} - 3 q^{63} - 3 q^{64} + 12 q^{65} + 12 q^{66} - 3 q^{67} + 6 q^{68} + 6 q^{69} + 12 q^{70} + 3 q^{71} + 12 q^{73} + 24 q^{74} - 6 q^{75} + 3 q^{76} - 42 q^{77} + 27 q^{79} + 3 q^{80} - 15 q^{82} + 3 q^{83} + 3 q^{84} + 12 q^{85} - 24 q^{86} + 6 q^{87} + 42 q^{89} + 3 q^{90} - 42 q^{91} - 27 q^{92} + 3 q^{93} - 18 q^{94} + 24 q^{95} - 6 q^{96} + 18 q^{97} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(77\) \(97\)
\(\chi(n)\) \(1\) \(\zeta_{18}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
−0.766044 + 0.642788i
0.939693 0.342020i
−0.173648 + 0.984808i
0.939693 + 0.342020i
−0.766044 0.642788i
−0.173648 0.984808i
0.766044 0.642788i 0.939693 + 0.342020i 0.173648 0.984808i −0.613341 3.47843i 0.939693 0.342020i −1.85844 + 3.21891i −0.500000 0.866025i 0.766044 + 0.642788i −2.70574 2.27038i
43.1 −0.939693 + 0.342020i −0.173648 + 0.984808i 0.766044 0.642788i −0.0923963 0.0775297i −0.173648 0.984808i 2.14543 + 3.71599i −0.500000 + 0.866025i −0.939693 0.342020i 0.113341 + 0.0412527i
55.1 0.173648 0.984808i −0.766044 0.642788i −0.939693 0.342020i 2.20574 0.802823i −0.766044 + 0.642788i −1.78699 3.09516i −0.500000 + 0.866025i 0.173648 + 0.984808i −0.407604 2.31164i
61.1 −0.939693 0.342020i −0.173648 0.984808i 0.766044 + 0.642788i −0.0923963 + 0.0775297i −0.173648 + 0.984808i 2.14543 3.71599i −0.500000 0.866025i −0.939693 + 0.342020i 0.113341 0.0412527i
73.1 0.766044 + 0.642788i 0.939693 0.342020i 0.173648 + 0.984808i −0.613341 + 3.47843i 0.939693 + 0.342020i −1.85844 3.21891i −0.500000 + 0.866025i 0.766044 0.642788i −2.70574 + 2.27038i
85.1 0.173648 + 0.984808i −0.766044 + 0.642788i −0.939693 + 0.342020i 2.20574 + 0.802823i −0.766044 0.642788i −1.78699 + 3.09516i −0.500000 0.866025i 0.173648 0.984808i −0.407604 + 2.31164i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 85.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 114.2.i.c 6
3.b odd 2 1 342.2.u.b 6
4.b odd 2 1 912.2.bo.d 6
19.e even 9 1 inner 114.2.i.c 6
19.e even 9 1 2166.2.a.r 3
19.f odd 18 1 2166.2.a.p 3
57.j even 18 1 6498.2.a.bu 3
57.l odd 18 1 342.2.u.b 6
57.l odd 18 1 6498.2.a.bp 3
76.l odd 18 1 912.2.bo.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.i.c 6 1.a even 1 1 trivial
114.2.i.c 6 19.e even 9 1 inner
342.2.u.b 6 3.b odd 2 1
342.2.u.b 6 57.l odd 18 1
912.2.bo.d 6 4.b odd 2 1
912.2.bo.d 6 76.l odd 18 1
2166.2.a.p 3 19.f odd 18 1
2166.2.a.r 3 19.e even 9 1
6498.2.a.bp 3 57.l odd 18 1
6498.2.a.bu 3 57.j even 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 3T_{5}^{5} + 12T_{5}^{4} - 46T_{5}^{3} + 60T_{5}^{2} + 12T_{5} + 1 \) acting on \(S_{2}^{\mathrm{new}}(114, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + T^{3} + 1 \) Copy content Toggle raw display
$3$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$5$ \( T^{6} - 3 T^{5} + 12 T^{4} - 46 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{6} + 3 T^{5} + 27 T^{4} + \cdots + 3249 \) Copy content Toggle raw display
$11$ \( T^{6} + 21 T^{4} + 74 T^{3} + \cdots + 1369 \) Copy content Toggle raw display
$13$ \( T^{6} - 9 T^{5} + 36 T^{4} - 28 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{6} + 3 T^{5} + 18 T^{4} + 24 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$19$ \( T^{6} + 107T^{3} + 6859 \) Copy content Toggle raw display
$23$ \( T^{6} - 27 T^{5} + 378 T^{4} + \cdots + 157609 \) Copy content Toggle raw display
$29$ \( T^{6} + 3 T^{5} + 12 T^{4} + 46 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{6} + 15 T^{5} + 153 T^{4} + \cdots + 11881 \) Copy content Toggle raw display
$37$ \( (T^{3} + 3 T^{2} - 45 T + 17)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 15 T^{5} + 87 T^{4} + \cdots + 2809 \) Copy content Toggle raw display
$43$ \( T^{6} - 3 T^{5} - 3 T^{4} + \cdots + 63001 \) Copy content Toggle raw display
$47$ \( T^{6} - 15 T^{5} + 51 T^{4} + \cdots + 5041 \) Copy content Toggle raw display
$53$ \( T^{6} - 6 T^{5} - 30 T^{4} + \cdots + 395641 \) Copy content Toggle raw display
$59$ \( T^{6} + 27 T^{5} + 198 T^{4} + \cdots + 81 \) Copy content Toggle raw display
$61$ \( T^{6} + 15 T^{5} + 84 T^{4} + \cdots + 289 \) Copy content Toggle raw display
$67$ \( T^{6} + 3 T^{5} + 72 T^{4} + \cdots + 7017201 \) Copy content Toggle raw display
$71$ \( T^{6} - 3 T^{5} - 87 T^{4} + \cdots + 26569 \) Copy content Toggle raw display
$73$ \( T^{6} - 12 T^{5} + 54 T^{4} + \cdots + 3249 \) Copy content Toggle raw display
$79$ \( T^{6} - 27 T^{5} + 351 T^{4} + \cdots + 32761 \) Copy content Toggle raw display
$83$ \( T^{6} - 3 T^{5} + 45 T^{4} + \cdots + 2601 \) Copy content Toggle raw display
$89$ \( T^{6} - 42 T^{5} + 756 T^{4} + \cdots + 239121 \) Copy content Toggle raw display
$97$ \( T^{6} - 18 T^{5} + 171 T^{4} + \cdots + 218089 \) Copy content Toggle raw display
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