Properties

Label 114.2.i.a
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: \(x^{6} - x^{3} + 1\)
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.

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

Character values

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

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

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 −0.386659 2.19285i 0.939693 0.342020i 1.32635 2.29731i 0.500000 + 0.866025i 0.766044 + 0.642788i 1.70574 + 1.43128i
43.1 0.939693 0.342020i 0.173648 0.984808i 0.766044 0.642788i −0.907604 0.761570i −0.173648 0.984808i 0.733956 + 1.27125i 0.500000 0.866025i −0.939693 0.342020i −1.11334 0.405223i
55.1 −0.173648 + 0.984808i 0.766044 + 0.642788i −0.939693 0.342020i −3.20574 + 1.16679i −0.766044 + 0.642788i 2.43969 + 4.22567i 0.500000 0.866025i 0.173648 + 0.984808i −0.592396 3.35965i
61.1 0.939693 + 0.342020i 0.173648 + 0.984808i 0.766044 + 0.642788i −0.907604 + 0.761570i −0.173648 + 0.984808i 0.733956 1.27125i 0.500000 + 0.866025i −0.939693 + 0.342020i −1.11334 + 0.405223i
73.1 −0.766044 0.642788i −0.939693 + 0.342020i 0.173648 + 0.984808i −0.386659 + 2.19285i 0.939693 + 0.342020i 1.32635 + 2.29731i 0.500000 0.866025i 0.766044 0.642788i 1.70574 1.43128i
85.1 −0.173648 0.984808i 0.766044 0.642788i −0.939693 + 0.342020i −3.20574 1.16679i −0.766044 0.642788i 2.43969 4.22567i 0.500000 + 0.866025i 0.173648 0.984808i −0.592396 + 3.35965i
\(n\): 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.a 6
3.b odd 2 1 342.2.u.e 6
4.b odd 2 1 912.2.bo.a 6
19.e even 9 1 inner 114.2.i.a 6
19.e even 9 1 2166.2.a.q 3
19.f odd 18 1 2166.2.a.s 3
57.j even 18 1 6498.2.a.bm 3
57.l odd 18 1 342.2.u.e 6
57.l odd 18 1 6498.2.a.br 3
76.l odd 18 1 912.2.bo.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.i.a 6 1.a even 1 1 trivial
114.2.i.a 6 19.e even 9 1 inner
342.2.u.e 6 3.b odd 2 1
342.2.u.e 6 57.l odd 18 1
912.2.bo.a 6 4.b odd 2 1
912.2.bo.a 6 76.l odd 18 1
2166.2.a.q 3 19.e even 9 1
2166.2.a.s 3 19.f odd 18 1
6498.2.a.bm 3 57.j even 18 1
6498.2.a.br 3 57.l odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 9 T_{5}^{5} + 36 T_{5}^{4} + 90 T_{5}^{3} + 162 T_{5}^{2} + 162 T_{5} + 81 \) acting on \(S_{2}^{\mathrm{new}}(114, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{3} + T^{6} \)
$3$ \( 1 + T^{3} + T^{6} \)
$5$ \( 81 + 162 T + 162 T^{2} + 90 T^{3} + 36 T^{4} + 9 T^{5} + T^{6} \)
$7$ \( 361 - 456 T + 405 T^{2} - 178 T^{3} + 57 T^{4} - 9 T^{5} + T^{6} \)
$11$ \( 81 + 81 T + 81 T^{2} + 18 T^{3} + 9 T^{4} + T^{6} \)
$13$ \( 1369 - 1443 T + 885 T^{2} - 352 T^{3} + 96 T^{4} - 15 T^{5} + T^{6} \)
$17$ \( 81 + 81 T + 81 T^{2} + 72 T^{3} + 36 T^{4} + 9 T^{5} + T^{6} \)
$19$ \( 6859 + 4332 T + 1482 T^{2} + 385 T^{3} + 78 T^{4} + 12 T^{5} + T^{6} \)
$23$ \( 81 - 162 T + 162 T^{2} - 90 T^{3} + 36 T^{4} - 9 T^{5} + T^{6} \)
$29$ \( 29241 - 6156 T + 2106 T^{2} - 72 T^{3} - 18 T^{4} + 9 T^{5} + T^{6} \)
$31$ \( 292681 + 35706 T + 9225 T^{2} + 488 T^{3} + 147 T^{4} + 9 T^{5} + T^{6} \)
$37$ \( ( 1 + 15 T - 9 T^{2} + T^{3} )^{2} \)
$41$ \( 210681 - 111537 T + 27702 T^{2} - 4104 T^{3} + 405 T^{4} - 27 T^{5} + T^{6} \)
$43$ \( 32041 + 23091 T + 7662 T^{2} + 1592 T^{3} + 231 T^{4} + 21 T^{5} + T^{6} \)
$47$ \( 23409 - 15147 T + 6804 T^{2} - 2016 T^{3} + 333 T^{4} - 27 T^{5} + T^{6} \)
$53$ \( 81 + 324 T + 486 T^{2} + 315 T^{3} + 90 T^{4} + T^{6} \)
$59$ \( 6561 + 6561 T + 729 T^{2} - 648 T^{3} + 162 T^{4} - 9 T^{5} + T^{6} \)
$61$ \( 94249 + 12894 T + 1821 T^{2} + 161 T^{3} - 12 T^{4} + 3 T^{5} + T^{6} \)
$67$ \( 2809 - 1272 T + 5502 T^{2} + 224 T^{3} - 66 T^{4} + 3 T^{5} + T^{6} \)
$71$ \( 263169 + 13851 T + 9720 T^{2} + 1728 T^{3} - 27 T^{4} - 9 T^{5} + T^{6} \)
$73$ \( 54289 - 32853 T + 5754 T^{2} - 244 T^{3} + 78 T^{4} + 12 T^{5} + T^{6} \)
$79$ \( 2809 - 14151 T + 30144 T^{2} + 3356 T^{3} + 231 T^{4} + 21 T^{5} + T^{6} \)
$83$ \( 408321 + 92016 T + 26487 T^{2} - 18 T^{3} + 225 T^{4} + 9 T^{5} + T^{6} \)
$89$ \( 431649 - 5913 T - 4374 T^{2} + 900 T^{3} + 90 T^{4} + T^{6} \)
$97$ \( 2679769 + 751383 T + 141156 T^{2} + 17675 T^{3} + 1323 T^{4} + 54 T^{5} + T^{6} \)
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