Properties

Label 1110.2.a.h.1.1
Level $1110$
Weight $2$
Character 1110.1
Self dual yes
Analytic conductor $8.863$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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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] = [1110,2,Mod(1,1110)] 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("1110.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1110, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1,1,1,1,-1,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(8.86339462436\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1110.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +6.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +4.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} +2.00000 q^{19} +1.00000 q^{20} -4.00000 q^{21} -6.00000 q^{22} -1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -4.00000 q^{28} +6.00000 q^{29} -1.00000 q^{30} +8.00000 q^{31} -1.00000 q^{32} +6.00000 q^{33} +6.00000 q^{34} -4.00000 q^{35} +1.00000 q^{36} +1.00000 q^{37} -2.00000 q^{38} +2.00000 q^{39} -1.00000 q^{40} -6.00000 q^{41} +4.00000 q^{42} +8.00000 q^{43} +6.00000 q^{44} +1.00000 q^{45} +6.00000 q^{47} +1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} -6.00000 q^{51} +2.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} +6.00000 q^{55} +4.00000 q^{56} +2.00000 q^{57} -6.00000 q^{58} -12.0000 q^{59} +1.00000 q^{60} +8.00000 q^{61} -8.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} -6.00000 q^{66} -4.00000 q^{67} -6.00000 q^{68} +4.00000 q^{70} -1.00000 q^{72} +14.0000 q^{73} -1.00000 q^{74} +1.00000 q^{75} +2.00000 q^{76} -24.0000 q^{77} -2.00000 q^{78} -16.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -12.0000 q^{83} -4.00000 q^{84} -6.00000 q^{85} -8.00000 q^{86} +6.00000 q^{87} -6.00000 q^{88} +6.00000 q^{89} -1.00000 q^{90} -8.00000 q^{91} +8.00000 q^{93} -6.00000 q^{94} +2.00000 q^{95} -1.00000 q^{96} +8.00000 q^{97} -9.00000 q^{98} +6.00000 q^{99} +O(q^{100})\)

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) −4.00000 −1.51186 −0.755929 − 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 6.00000 1.80907 0.904534 − 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 4.00000 1.06904
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) −1.00000 −0.235702
\(19\) 2.00000 0.458831 0.229416 − 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 1.00000 0.223607
\(21\) −4.00000 −0.872872
\(22\) −6.00000 −1.27920
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) 1.00000 0.192450
\(28\) −4.00000 −0.755929
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) −1.00000 −0.182574
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) −1.00000 −0.176777
\(33\) 6.00000 1.04447
\(34\) 6.00000 1.02899
\(35\) −4.00000 −0.676123
\(36\) 1.00000 0.166667
\(37\) 1.00000 0.164399
\(38\) −2.00000 −0.324443
\(39\) 2.00000 0.320256
\(40\) −1.00000 −0.158114
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 4.00000 0.617213
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 6.00000 0.904534
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 1.00000 0.144338
\(49\) 9.00000 1.28571
\(50\) −1.00000 −0.141421
\(51\) −6.00000 −0.840168
\(52\) 2.00000 0.277350
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −1.00000 −0.136083
\(55\) 6.00000 0.809040
\(56\) 4.00000 0.534522
\(57\) 2.00000 0.264906
\(58\) −6.00000 −0.787839
\(59\) −12.0000 −1.56227 −0.781133 − 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 1.00000 0.129099
\(61\) 8.00000 1.02430 0.512148 − 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) −8.00000 −1.01600
\(63\) −4.00000 −0.503953
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) −6.00000 −0.738549
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 4.00000 0.478091
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −1.00000 −0.117851
\(73\) 14.0000 1.63858 0.819288 − 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) −1.00000 −0.116248
\(75\) 1.00000 0.115470
\(76\) 2.00000 0.229416
\(77\) −24.0000 −2.73505
\(78\) −2.00000 −0.226455
\(79\) −16.0000 −1.80014 −0.900070 − 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) −4.00000 −0.436436
\(85\) −6.00000 −0.650791
\(86\) −8.00000 −0.862662
\(87\) 6.00000 0.643268
\(88\) −6.00000 −0.639602
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) −1.00000 −0.105409
\(91\) −8.00000 −0.838628
\(92\) 0 0
\(93\) 8.00000 0.829561
\(94\) −6.00000 −0.618853
\(95\) 2.00000 0.205196
\(96\) −1.00000 −0.102062
\(97\) 8.00000 0.812277 0.406138 − 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) −9.00000 −0.909137
\(99\) 6.00000 0.603023
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 1110.2.a.h.1.1 ✓ 1
3.2 odd 2 3330.2.a.m.1.1 1
4.3 odd 2 8880.2.a.n.1.1 1
5.4 even 2 5550.2.a.bd.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.h.1.1 ✓ 1 1.1 even 1 trivial
3330.2.a.m.1.1 1 3.2 odd 2
5550.2.a.bd.1.1 1 5.4 even 2
8880.2.a.n.1.1 1 4.3 odd 2