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Properties

Label 1150.2.a.e.1.1
Level $1150$
Weight $2$
Character 1150.1
Self dual yes
Analytic conductor $9.183$
Analytic rank $1$
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] = [1150,2,Mod(1,1150)] 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("1150.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1150.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,1,-3,1,0,-3,-4,1,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(9.18279623245\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 1150.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} -3.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +6.00000 q^{9} +3.00000 q^{11} -3.00000 q^{12} +6.00000 q^{13} -4.00000 q^{14} +1.00000 q^{16} -5.00000 q^{17} +6.00000 q^{18} -1.00000 q^{19} +12.0000 q^{21} +3.00000 q^{22} -1.00000 q^{23} -3.00000 q^{24} +6.00000 q^{26} -9.00000 q^{27} -4.00000 q^{28} -8.00000 q^{29} -8.00000 q^{31} +1.00000 q^{32} -9.00000 q^{33} -5.00000 q^{34} +6.00000 q^{36} -2.00000 q^{37} -1.00000 q^{38} -18.0000 q^{39} -7.00000 q^{41} +12.0000 q^{42} -4.00000 q^{43} +3.00000 q^{44} -1.00000 q^{46} -10.0000 q^{47} -3.00000 q^{48} +9.00000 q^{49} +15.0000 q^{51} +6.00000 q^{52} +12.0000 q^{53} -9.00000 q^{54} -4.00000 q^{56} +3.00000 q^{57} -8.00000 q^{58} +4.00000 q^{59} -8.00000 q^{61} -8.00000 q^{62} -24.0000 q^{63} +1.00000 q^{64} -9.00000 q^{66} -3.00000 q^{67} -5.00000 q^{68} +3.00000 q^{69} +4.00000 q^{71} +6.00000 q^{72} +7.00000 q^{73} -2.00000 q^{74} -1.00000 q^{76} -12.0000 q^{77} -18.0000 q^{78} -6.00000 q^{79} +9.00000 q^{81} -7.00000 q^{82} -11.0000 q^{83} +12.0000 q^{84} -4.00000 q^{86} +24.0000 q^{87} +3.00000 q^{88} -3.00000 q^{89} -24.0000 q^{91} -1.00000 q^{92} +24.0000 q^{93} -10.0000 q^{94} -3.00000 q^{96} +14.0000 q^{97} +9.00000 q^{98} +18.0000 q^{99} +O(q^{100})\)

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
\(3\) −3.00000 −1.73205 −0.866025 − 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −3.00000 −1.22474
\(7\) −4.00000 −1.51186 −0.755929 − 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 1.00000 0.353553
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 − 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) −3.00000 −0.866025
\(13\) 6.00000 1.66410 0.832050 − 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −5.00000 −1.21268 −0.606339 − 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(18\) 6.00000 1.41421
\(19\) −1.00000 −0.229416 −0.114708 − 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 12.0000 2.61861
\(22\) 3.00000 0.639602
\(23\) −1.00000 −0.208514
\(24\) −3.00000 −0.612372
\(25\) 0 0
\(26\) 6.00000 1.17670
\(27\) −9.00000 −1.73205
\(28\) −4.00000 −0.755929
\(29\) −8.00000 −1.48556 −0.742781 − 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 1.00000 0.176777
\(33\) −9.00000 −1.56670
\(34\) −5.00000 −0.857493
\(35\) 0 0
\(36\) 6.00000 1.00000
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −1.00000 −0.162221
\(39\) −18.0000 −2.88231
\(40\) 0 0
\(41\) −7.00000 −1.09322 −0.546608 − 0.837389i \(-0.684081\pi\)
−0.546608 + 0.837389i \(0.684081\pi\)
\(42\) 12.0000 1.85164
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) −10.0000 −1.45865 −0.729325 − 0.684167i \(-0.760166\pi\)
−0.729325 + 0.684167i \(0.760166\pi\)
\(48\) −3.00000 −0.433013
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 15.0000 2.10042
\(52\) 6.00000 0.832050
\(53\) 12.0000 1.64833 0.824163 − 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) −9.00000 −1.22474
\(55\) 0 0
\(56\) −4.00000 −0.534522
\(57\) 3.00000 0.397360
\(58\) −8.00000 −1.05045
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 − 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) −8.00000 −1.01600
\(63\) −24.0000 −3.02372
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −9.00000 −1.10782
\(67\) −3.00000 −0.366508 −0.183254 − 0.983066i \(-0.558663\pi\)
−0.183254 + 0.983066i \(0.558663\pi\)
\(68\) −5.00000 −0.606339
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 − 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 6.00000 0.707107
\(73\) 7.00000 0.819288 0.409644 − 0.912245i \(-0.365653\pi\)
0.409644 + 0.912245i \(0.365653\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) −12.0000 −1.36753
\(78\) −18.0000 −2.03810
\(79\) −6.00000 −0.675053 −0.337526 − 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) −7.00000 −0.773021
\(83\) −11.0000 −1.20741 −0.603703 − 0.797209i \(-0.706309\pi\)
−0.603703 + 0.797209i \(0.706309\pi\)
\(84\) 12.0000 1.30931
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 24.0000 2.57307
\(88\) 3.00000 0.319801
\(89\) −3.00000 −0.317999 −0.159000 − 0.987279i \(-0.550827\pi\)
−0.159000 + 0.987279i \(0.550827\pi\)
\(90\) 0 0
\(91\) −24.0000 −2.51588
\(92\) −1.00000 −0.104257
\(93\) 24.0000 2.48868
\(94\) −10.0000 −1.03142
\(95\) 0 0
\(96\) −3.00000 −0.306186
\(97\) 14.0000 1.42148 0.710742 − 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 9.00000 0.909137
\(99\) 18.0000 1.80907
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 1150.2.a.e.1.1 yes 1
4.3 odd 2 9200.2.a.bl.1.1 1
5.2 odd 4 1150.2.b.a.599.2 2
5.3 odd 4 1150.2.b.a.599.1 2
5.4 even 2 1150.2.a.d.1.1 ✓ 1
20.19 odd 2 9200.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1150.2.a.d.1.1 ✓ 1 5.4 even 2
1150.2.a.e.1.1 yes 1 1.1 even 1 trivial
1150.2.b.a.599.1 2 5.3 odd 4
1150.2.b.a.599.2 2 5.2 odd 4
9200.2.a.a.1.1 1 20.19 odd 2
9200.2.a.bl.1.1 1 4.3 odd 2