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Properties

Label 1110.2.a.f.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,-1,1] 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -5.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +6.00000 q^{19} -1.00000 q^{20} +1.00000 q^{21} +5.00000 q^{22} -1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} +9.00000 q^{29} +1.00000 q^{30} +3.00000 q^{31} -1.00000 q^{32} -5.00000 q^{33} -1.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} -1.00000 q^{37} -6.00000 q^{38} +2.00000 q^{39} +1.00000 q^{40} +9.00000 q^{41} -1.00000 q^{42} +1.00000 q^{43} -5.00000 q^{44} -1.00000 q^{45} +1.00000 q^{48} -6.00000 q^{49} -1.00000 q^{50} +1.00000 q^{51} +2.00000 q^{52} +9.00000 q^{53} -1.00000 q^{54} +5.00000 q^{55} -1.00000 q^{56} +6.00000 q^{57} -9.00000 q^{58} -10.0000 q^{59} -1.00000 q^{60} -5.00000 q^{61} -3.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +5.00000 q^{66} +16.0000 q^{67} +1.00000 q^{68} +1.00000 q^{70} -1.00000 q^{72} +12.0000 q^{73} +1.00000 q^{74} +1.00000 q^{75} +6.00000 q^{76} -5.00000 q^{77} -2.00000 q^{78} -12.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -9.00000 q^{82} +2.00000 q^{83} +1.00000 q^{84} -1.00000 q^{85} -1.00000 q^{86} +9.00000 q^{87} +5.00000 q^{88} -2.00000 q^{89} +1.00000 q^{90} +2.00000 q^{91} +3.00000 q^{93} -6.00000 q^{95} -1.00000 q^{96} +17.0000 q^{97} +6.00000 q^{98} -5.00000 q^{99} +O(q^{100})\)

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
\(7\) 1.00000 0.377964 0.188982 − 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −5.00000 −1.50756 −0.753778 − 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −1.00000 −0.267261
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536 0.121268 − 0.992620i \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) −1.00000 −0.235702
\(19\) 6.00000 1.37649 0.688247 − 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) −1.00000 −0.223607
\(21\) 1.00000 0.218218
\(22\) 5.00000 1.06600
\(23\) 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
\(28\) 1.00000 0.188982
\(29\) 9.00000 1.67126 0.835629 − 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 1.00000 0.182574
\(31\) 3.00000 0.538816 0.269408 − 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) −1.00000 −0.176777
\(33\) −5.00000 −0.870388
\(34\) −1.00000 −0.171499
\(35\) −1.00000 −0.169031
\(36\) 1.00000 0.166667
\(37\) −1.00000 −0.164399
\(38\) −6.00000 −0.973329
\(39\) 2.00000 0.320256
\(40\) 1.00000 0.158114
\(41\) 9.00000 1.40556 0.702782 − 0.711405i \(-0.251941\pi\)
0.702782 + 0.711405i \(0.251941\pi\)
\(42\) −1.00000 −0.154303
\(43\) 1.00000 0.152499 0.0762493 − 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) −5.00000 −0.753778
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.00000 −0.857143
\(50\) −1.00000 −0.141421
\(51\) 1.00000 0.140028
\(52\) 2.00000 0.277350
\(53\) 9.00000 1.23625 0.618123 − 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) −1.00000 −0.136083
\(55\) 5.00000 0.674200
\(56\) −1.00000 −0.133631
\(57\) 6.00000 0.794719
\(58\) −9.00000 −1.18176
\(59\) −10.0000 −1.30189 −0.650945 − 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) −1.00000 −0.129099
\(61\) −5.00000 −0.640184 −0.320092 − 0.947386i \(-0.603714\pi\)
−0.320092 + 0.947386i \(0.603714\pi\)
\(62\) −3.00000 −0.381000
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 5.00000 0.615457
\(67\) 16.0000 1.95471 0.977356 − 0.211604i \(-0.0678686\pi\)
0.977356 + 0.211604i \(0.0678686\pi\)
\(68\) 1.00000 0.121268
\(69\) 0 0
\(70\) 1.00000 0.119523
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −1.00000 −0.117851
\(73\) 12.0000 1.40449 0.702247 − 0.711934i \(-0.252180\pi\)
0.702247 + 0.711934i \(0.252180\pi\)
\(74\) 1.00000 0.116248
\(75\) 1.00000 0.115470
\(76\) 6.00000 0.688247
\(77\) −5.00000 −0.569803
\(78\) −2.00000 −0.226455
\(79\) −12.0000 −1.35011 −0.675053 − 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −9.00000 −0.993884
\(83\) 2.00000 0.219529 0.109764 − 0.993958i \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) 1.00000 0.109109
\(85\) −1.00000 −0.108465
\(86\) −1.00000 −0.107833
\(87\) 9.00000 0.964901
\(88\) 5.00000 0.533002
\(89\) −2.00000 −0.212000 −0.106000 − 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 1.00000 0.105409
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) 3.00000 0.311086
\(94\) 0 0
\(95\) −6.00000 −0.615587
\(96\) −1.00000 −0.102062
\(97\) 17.0000 1.72609 0.863044 − 0.505128i \(-0.168555\pi\)
0.863044 + 0.505128i \(0.168555\pi\)
\(98\) 6.00000 0.606092
\(99\) −5.00000 −0.502519
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.f.1.1 ✓ 1
3.2 odd 2 3330.2.a.y.1.1 1
4.3 odd 2 8880.2.a.d.1.1 1
5.4 even 2 5550.2.a.y.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.f.1.1 ✓ 1 1.1 even 1 trivial
3330.2.a.y.1.1 1 3.2 odd 2
5550.2.a.y.1.1 1 5.4 even 2
8880.2.a.d.1.1 1 4.3 odd 2