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Properties

Label 5610.2.a.w.1.1
Level $5610$
Weight $2$
Character 5610.1
Self dual yes
Analytic conductor $44.796$
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] = [5610,2,Mod(1,5610)] 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("5610.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5610, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5610.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,1,-1,1,-1,-1,-4,1,1,-1,1,-1,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(44.7960755339\)
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 \(\chi\) \(=\) 5610.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} +4.00000 q^{13} -4.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{20} +4.00000 q^{21} +1.00000 q^{22} -8.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +4.00000 q^{26} -1.00000 q^{27} -4.00000 q^{28} -4.00000 q^{29} +1.00000 q^{30} +6.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} +4.00000 q^{35} +1.00000 q^{36} +4.00000 q^{37} +4.00000 q^{38} -4.00000 q^{39} -1.00000 q^{40} -10.0000 q^{41} +4.00000 q^{42} +4.00000 q^{43} +1.00000 q^{44} -1.00000 q^{45} -8.00000 q^{46} -2.00000 q^{47} -1.00000 q^{48} +9.00000 q^{49} +1.00000 q^{50} +1.00000 q^{51} +4.00000 q^{52} -4.00000 q^{53} -1.00000 q^{54} -1.00000 q^{55} -4.00000 q^{56} -4.00000 q^{57} -4.00000 q^{58} +4.00000 q^{59} +1.00000 q^{60} -14.0000 q^{61} +6.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} -1.00000 q^{66} -4.00000 q^{67} -1.00000 q^{68} +8.00000 q^{69} +4.00000 q^{70} +12.0000 q^{71} +1.00000 q^{72} +14.0000 q^{73} +4.00000 q^{74} -1.00000 q^{75} +4.00000 q^{76} -4.00000 q^{77} -4.00000 q^{78} +12.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} -4.00000 q^{83} +4.00000 q^{84} +1.00000 q^{85} +4.00000 q^{86} +4.00000 q^{87} +1.00000 q^{88} +2.00000 q^{89} -1.00000 q^{90} -16.0000 q^{91} -8.00000 q^{92} -6.00000 q^{93} -2.00000 q^{94} -4.00000 q^{95} -1.00000 q^{96} -10.0000 q^{97} +9.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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
\(11\) 1.00000 0.301511
\(12\) −1.00000 −0.288675
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) −4.00000 −1.06904
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536
\(18\) 1.00000 0.235702
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −1.00000 −0.223607
\(21\) 4.00000 0.872872
\(22\) 1.00000 0.213201
\(23\) −8.00000 −1.66812 −0.834058 − 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 4.00000 0.784465
\(27\) −1.00000 −0.192450
\(28\) −4.00000 −0.755929
\(29\) −4.00000 −0.742781 −0.371391 − 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 1.00000 0.182574
\(31\) 6.00000 1.07763 0.538816 − 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.00000 −0.174078
\(34\) −1.00000 −0.171499
\(35\) 4.00000 0.676123
\(36\) 1.00000 0.166667
\(37\) 4.00000 0.657596 0.328798 − 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 4.00000 0.648886
\(39\) −4.00000 −0.640513
\(40\) −1.00000 −0.158114
\(41\) −10.0000 −1.56174 −0.780869 − 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 4.00000 0.617213
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 1.00000 0.150756
\(45\) −1.00000 −0.149071
\(46\) −8.00000 −1.17954
\(47\) −2.00000 −0.291730 −0.145865 − 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) −1.00000 −0.144338
\(49\) 9.00000 1.28571
\(50\) 1.00000 0.141421
\(51\) 1.00000 0.140028
\(52\) 4.00000 0.554700
\(53\) −4.00000 −0.549442 −0.274721 − 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) −1.00000 −0.136083
\(55\) −1.00000 −0.134840
\(56\) −4.00000 −0.534522
\(57\) −4.00000 −0.529813
\(58\) −4.00000 −0.525226
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 1.00000 0.129099
\(61\) −14.0000 −1.79252 −0.896258 − 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 6.00000 0.762001
\(63\) −4.00000 −0.503953
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) −1.00000 −0.123091
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −1.00000 −0.121268
\(69\) 8.00000 0.963087
\(70\) 4.00000 0.478091
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 1.00000 0.117851
\(73\) 14.0000 1.63858 0.819288 − 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) 4.00000 0.464991
\(75\) −1.00000 −0.115470
\(76\) 4.00000 0.458831
\(77\) −4.00000 −0.455842
\(78\) −4.00000 −0.452911
\(79\) 12.0000 1.35011 0.675053 − 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −10.0000 −1.10432
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 4.00000 0.436436
\(85\) 1.00000 0.108465
\(86\) 4.00000 0.431331
\(87\) 4.00000 0.428845
\(88\) 1.00000 0.106600
\(89\) 2.00000 0.212000 0.106000 − 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) −1.00000 −0.105409
\(91\) −16.0000 −1.67726
\(92\) −8.00000 −0.834058
\(93\) −6.00000 −0.622171
\(94\) −2.00000 −0.206284
\(95\) −4.00000 −0.410391
\(96\) −1.00000 −0.102062
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 9.00000 0.909137
\(99\) 1.00000 0.100504
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 5610.2.a.w.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5610.2.a.w.1.1 ✓ 1 1.1 even 1 trivial