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Properties

Label 114.2.b.b.113.1
Level $114$
Weight $2$
Character 114.113
Analytic conductor $0.910$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [114,2,Mod(113,114)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("114.113"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(114, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 114 = 2 \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 114.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.910294583043\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
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: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 113.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 114.113
Dual form 114.2.b.b.113.2

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +(1.50000 - 0.866025i) q^{3} +1.00000 q^{4} +3.46410i q^{5} +(-1.50000 + 0.866025i) q^{6} +1.00000 q^{7} -1.00000 q^{8} +(1.50000 - 2.59808i) q^{9} -3.46410i q^{10} -3.46410i q^{11} +(1.50000 - 0.866025i) q^{12} +1.73205i q^{13} -1.00000 q^{14} +(3.00000 + 5.19615i) q^{15} +1.00000 q^{16} +1.73205i q^{17} +(-1.50000 + 2.59808i) q^{18} +(-4.00000 - 1.73205i) q^{19} +3.46410i q^{20} +(1.50000 - 0.866025i) q^{21} +3.46410i q^{22} -5.19615i q^{23} +(-1.50000 + 0.866025i) q^{24} -7.00000 q^{25} -1.73205i q^{26} -5.19615i q^{27} +1.00000 q^{28} -9.00000 q^{29} +(-3.00000 - 5.19615i) q^{30} +10.3923i q^{31} -1.00000 q^{32} +(-3.00000 - 5.19615i) q^{33} -1.73205i q^{34} +3.46410i q^{35} +(1.50000 - 2.59808i) q^{36} -6.92820i q^{37} +(4.00000 + 1.73205i) q^{38} +(1.50000 + 2.59808i) q^{39} -3.46410i q^{40} +(-1.50000 + 0.866025i) q^{42} +2.00000 q^{43} -3.46410i q^{44} +(9.00000 + 5.19615i) q^{45} +5.19615i q^{46} +3.46410i q^{47} +(1.50000 - 0.866025i) q^{48} -6.00000 q^{49} +7.00000 q^{50} +(1.50000 + 2.59808i) q^{51} +1.73205i q^{52} +9.00000 q^{53} +5.19615i q^{54} +12.0000 q^{55} -1.00000 q^{56} +(-7.50000 + 0.866025i) q^{57} +9.00000 q^{58} +3.00000 q^{59} +(3.00000 + 5.19615i) q^{60} -8.00000 q^{61} -10.3923i q^{62} +(1.50000 - 2.59808i) q^{63} +1.00000 q^{64} -6.00000 q^{65} +(3.00000 + 5.19615i) q^{66} -8.66025i q^{67} +1.73205i q^{68} +(-4.50000 - 7.79423i) q^{69} -3.46410i q^{70} +12.0000 q^{71} +(-1.50000 + 2.59808i) q^{72} +11.0000 q^{73} +6.92820i q^{74} +(-10.5000 + 6.06218i) q^{75} +(-4.00000 - 1.73205i) q^{76} -3.46410i q^{77} +(-1.50000 - 2.59808i) q^{78} -6.92820i q^{79} +3.46410i q^{80} +(-4.50000 - 7.79423i) q^{81} +10.3923i q^{83} +(1.50000 - 0.866025i) q^{84} -6.00000 q^{85} -2.00000 q^{86} +(-13.5000 + 7.79423i) q^{87} +3.46410i q^{88} -6.00000 q^{89} +(-9.00000 - 5.19615i) q^{90} +1.73205i q^{91} -5.19615i q^{92} +(9.00000 + 15.5885i) q^{93} -3.46410i q^{94} +(6.00000 - 13.8564i) q^{95} +(-1.50000 + 0.866025i) q^{96} +13.8564i q^{97} +6.00000 q^{98} +(-9.00000 - 5.19615i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 3 q^{3} + 2 q^{4} - 3 q^{6} + 2 q^{7} - 2 q^{8} + 3 q^{9} + 3 q^{12} - 2 q^{14} + 6 q^{15} + 2 q^{16} - 3 q^{18} - 8 q^{19} + 3 q^{21} - 3 q^{24} - 14 q^{25} + 2 q^{28} - 18 q^{29} - 6 q^{30}+ \cdots - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.50000 − 0.866025i 0.866025 − 0.500000i
\(4\) 1.00000 0.500000
\(5\) 3.46410i 1.54919i 0.632456 + 0.774597i \(0.282047\pi\)
−0.632456 + 0.774597i \(0.717953\pi\)
\(6\) −1.50000 + 0.866025i −0.612372 + 0.353553i
\(7\) 1.00000 0.377964 0.188982 − 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.50000 − 2.59808i 0.500000 − 0.866025i
\(10\) − 3.46410i − 1.09545i
\(11\) − 3.46410i − 1.04447i −0.852803 − 0.522233i \(-0.825099\pi\)
0.852803 − 0.522233i \(-0.174901\pi\)
\(12\) 1.50000 − 0.866025i 0.433013 − 0.250000i
\(13\) 1.73205i 0.480384i 0.970725 + 0.240192i \(0.0772105\pi\)
−0.970725 + 0.240192i \(0.922790\pi\)
\(14\) −1.00000 −0.267261
\(15\) 3.00000 + 5.19615i 0.774597 + 1.34164i
\(16\) 1.00000 0.250000
\(17\) 1.73205i 0.420084i 0.977692 + 0.210042i \(0.0673601\pi\)
−0.977692 + 0.210042i \(0.932640\pi\)
\(18\) −1.50000 + 2.59808i −0.353553 + 0.612372i
\(19\) −4.00000 − 1.73205i −0.917663 − 0.397360i
\(20\) 3.46410i 0.774597i
\(21\) 1.50000 − 0.866025i 0.327327 − 0.188982i
\(22\) 3.46410i 0.738549i
\(23\) − 5.19615i − 1.08347i −0.840548 − 0.541736i \(-0.817767\pi\)
0.840548 − 0.541736i \(-0.182233\pi\)
\(24\) −1.50000 + 0.866025i −0.306186 + 0.176777i
\(25\) −7.00000 −1.40000
\(26\) − 1.73205i − 0.339683i
\(27\) − 5.19615i − 1.00000i
\(28\) 1.00000 0.188982
\(29\) −9.00000 −1.67126 −0.835629 − 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) −3.00000 − 5.19615i −0.547723 − 0.948683i
\(31\) 10.3923i 1.86651i 0.359211 + 0.933257i \(0.383046\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) −3.00000 − 5.19615i −0.522233 − 0.904534i
\(34\) − 1.73205i − 0.297044i
\(35\) 3.46410i 0.585540i
\(36\) 1.50000 − 2.59808i 0.250000 − 0.433013i
\(37\) − 6.92820i − 1.13899i −0.821995 − 0.569495i \(-0.807139\pi\)
0.821995 − 0.569495i \(-0.192861\pi\)
\(38\) 4.00000 + 1.73205i 0.648886 + 0.280976i
\(39\) 1.50000 + 2.59808i 0.240192 + 0.416025i
\(40\) − 3.46410i − 0.547723i
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) −1.50000 + 0.866025i −0.231455 + 0.133631i
\(43\) 2.00000 0.304997 0.152499 − 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) − 3.46410i − 0.522233i
\(45\) 9.00000 + 5.19615i 1.34164 + 0.774597i
\(46\) 5.19615i 0.766131i
\(47\) 3.46410i 0.505291i 0.967559 + 0.252646i \(0.0813007\pi\)
−0.967559 + 0.252646i \(0.918699\pi\)
\(48\) 1.50000 − 0.866025i 0.216506 − 0.125000i
\(49\) −6.00000 −0.857143
\(50\) 7.00000 0.989949
\(51\) 1.50000 + 2.59808i 0.210042 + 0.363803i
\(52\) 1.73205i 0.240192i
\(53\) 9.00000 1.23625 0.618123 − 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) 5.19615i 0.707107i
\(55\) 12.0000 1.61808
\(56\) −1.00000 −0.133631
\(57\) −7.50000 + 0.866025i −0.993399 + 0.114708i
\(58\) 9.00000 1.18176
\(59\) 3.00000 0.390567 0.195283 − 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 3.00000 + 5.19615i 0.387298 + 0.670820i
\(61\) −8.00000 −1.02430 −0.512148 − 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) − 10.3923i − 1.31982i
\(63\) 1.50000 − 2.59808i 0.188982 − 0.327327i
\(64\) 1.00000 0.125000
\(65\) −6.00000 −0.744208
\(66\) 3.00000 + 5.19615i 0.369274 + 0.639602i
\(67\) − 8.66025i − 1.05802i −0.848616 − 0.529009i \(-0.822564\pi\)
0.848616 − 0.529009i \(-0.177436\pi\)
\(68\) 1.73205i 0.210042i
\(69\) −4.50000 − 7.79423i −0.541736 − 0.938315i
\(70\) − 3.46410i − 0.414039i
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) −1.50000 + 2.59808i −0.176777 + 0.306186i
\(73\) 11.0000 1.28745 0.643726 − 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) 6.92820i 0.805387i
\(75\) −10.5000 + 6.06218i −1.21244 + 0.700000i
\(76\) −4.00000 − 1.73205i −0.458831 − 0.198680i
\(77\) − 3.46410i − 0.394771i
\(78\) −1.50000 − 2.59808i −0.169842 − 0.294174i
\(79\) − 6.92820i − 0.779484i −0.920924 − 0.389742i \(-0.872564\pi\)
0.920924 − 0.389742i \(-0.127436\pi\)
\(80\) 3.46410i 0.387298i
\(81\) −4.50000 − 7.79423i −0.500000 − 0.866025i
\(82\) 0 0
\(83\) 10.3923i 1.14070i 0.821401 + 0.570352i \(0.193193\pi\)
−0.821401 + 0.570352i \(0.806807\pi\)
\(84\) 1.50000 − 0.866025i 0.163663 − 0.0944911i
\(85\) −6.00000 −0.650791
\(86\) −2.00000 −0.215666
\(87\) −13.5000 + 7.79423i −1.44735 + 0.835629i
\(88\) 3.46410i 0.369274i
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) −9.00000 − 5.19615i −0.948683 − 0.547723i
\(91\) 1.73205i 0.181568i
\(92\) − 5.19615i − 0.541736i
\(93\) 9.00000 + 15.5885i 0.933257 + 1.61645i
\(94\) − 3.46410i − 0.357295i
\(95\) 6.00000 − 13.8564i 0.615587 − 1.42164i
\(96\) −1.50000 + 0.866025i −0.153093 + 0.0883883i
\(97\) 13.8564i 1.40690i 0.710742 + 0.703452i \(0.248359\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 6.00000 0.606092
\(99\) −9.00000 − 5.19615i −0.904534 − 0.522233i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 114.2.b.b.113.1 ✓ 2
3.2 odd 2 114.2.b.c.113.1 yes 2
4.3 odd 2 912.2.f.a.113.2 2
12.11 even 2 912.2.f.e.113.2 2
19.18 odd 2 114.2.b.c.113.2 yes 2
57.56 even 2 inner 114.2.b.b.113.2 yes 2
76.75 even 2 912.2.f.e.113.1 2
228.227 odd 2 912.2.f.a.113.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.b.b.113.1 ✓ 2 1.1 even 1 trivial
114.2.b.b.113.2 yes 2 57.56 even 2 inner
114.2.b.c.113.1 yes 2 3.2 odd 2
114.2.b.c.113.2 yes 2 19.18 odd 2
912.2.f.a.113.1 2 228.227 odd 2
912.2.f.a.113.2 2 4.3 odd 2
912.2.f.e.113.1 2 76.75 even 2
912.2.f.e.113.2 2 12.11 even 2