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Properties

Label 55.4.a.a.1.1
Level $55$
Weight $4$
Character 55.1
Self dual yes
Analytic conductor $3.245$
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] = [55,4,Mod(1,55)] 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("55.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(55, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 55 = 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 55.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(3.24510505032\)
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 \(\chi\) \(=\) 55.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -3.00000 q^{3} -7.00000 q^{4} -5.00000 q^{5} -3.00000 q^{6} -9.00000 q^{7} -15.0000 q^{8} -18.0000 q^{9} -5.00000 q^{10} +11.0000 q^{11} +21.0000 q^{12} +2.00000 q^{13} -9.00000 q^{14} +15.0000 q^{15} +41.0000 q^{16} +21.0000 q^{17} -18.0000 q^{18} -85.0000 q^{19} +35.0000 q^{20} +27.0000 q^{21} +11.0000 q^{22} +22.0000 q^{23} +45.0000 q^{24} +25.0000 q^{25} +2.00000 q^{26} +135.000 q^{27} +63.0000 q^{28} -165.000 q^{29} +15.0000 q^{30} -83.0000 q^{31} +161.000 q^{32} -33.0000 q^{33} +21.0000 q^{34} +45.0000 q^{35} +126.000 q^{36} +1.00000 q^{37} -85.0000 q^{38} -6.00000 q^{39} +75.0000 q^{40} -478.000 q^{41} +27.0000 q^{42} -8.00000 q^{43} -77.0000 q^{44} +90.0000 q^{45} +22.0000 q^{46} +126.000 q^{47} -123.000 q^{48} -262.000 q^{49} +25.0000 q^{50} -63.0000 q^{51} -14.0000 q^{52} -683.000 q^{53} +135.000 q^{54} -55.0000 q^{55} +135.000 q^{56} +255.000 q^{57} -165.000 q^{58} -290.000 q^{59} -105.000 q^{60} +257.000 q^{61} -83.0000 q^{62} +162.000 q^{63} -167.000 q^{64} -10.0000 q^{65} -33.0000 q^{66} +776.000 q^{67} -147.000 q^{68} -66.0000 q^{69} +45.0000 q^{70} -313.000 q^{71} +270.000 q^{72} +902.000 q^{73} +1.00000 q^{74} -75.0000 q^{75} +595.000 q^{76} -99.0000 q^{77} -6.00000 q^{78} +830.000 q^{79} -205.000 q^{80} +81.0000 q^{81} -478.000 q^{82} +842.000 q^{83} -189.000 q^{84} -105.000 q^{85} -8.00000 q^{86} +495.000 q^{87} -165.000 q^{88} +25.0000 q^{89} +90.0000 q^{90} -18.0000 q^{91} -154.000 q^{92} +249.000 q^{93} +126.000 q^{94} +425.000 q^{95} -483.000 q^{96} -1784.00 q^{97} -262.000 q^{98} -198.000 q^{99} +O(q^{100})\)

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 0.353553 0.176777 − 0.984251i \(-0.443433\pi\)
0.176777 + 0.984251i \(0.443433\pi\)
\(3\) −3.00000 −0.577350 −0.288675 − 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) −7.00000 −0.875000
\(5\) −5.00000 −0.447214
\(6\) −3.00000 −0.204124
\(7\) −9.00000 −0.485954 −0.242977 − 0.970032i \(-0.578124\pi\)
−0.242977 + 0.970032i \(0.578124\pi\)
\(8\) −15.0000 −0.662913
\(9\) −18.0000 −0.666667
\(10\) −5.00000 −0.158114
\(11\) 11.0000 0.301511
\(12\) 21.0000 0.505181
\(13\) 2.00000 0.0426692 0.0213346 − 0.999772i \(-0.493208\pi\)
0.0213346 + 0.999772i \(0.493208\pi\)
\(14\) −9.00000 −0.171811
\(15\) 15.0000 0.258199
\(16\) 41.0000 0.640625
\(17\) 21.0000 0.299603 0.149801 − 0.988716i \(-0.452137\pi\)
0.149801 + 0.988716i \(0.452137\pi\)
\(18\) −18.0000 −0.235702
\(19\) −85.0000 −1.02633 −0.513167 − 0.858289i \(-0.671528\pi\)
−0.513167 + 0.858289i \(0.671528\pi\)
\(20\) 35.0000 0.391312
\(21\) 27.0000 0.280566
\(22\) 11.0000 0.106600
\(23\) 22.0000 0.199449 0.0997243 − 0.995015i \(-0.468204\pi\)
0.0997243 + 0.995015i \(0.468204\pi\)
\(24\) 45.0000 0.382733
\(25\) 25.0000 0.200000
\(26\) 2.00000 0.0150859
\(27\) 135.000 0.962250
\(28\) 63.0000 0.425210
\(29\) −165.000 −1.05654 −0.528271 − 0.849076i \(-0.677160\pi\)
−0.528271 + 0.849076i \(0.677160\pi\)
\(30\) 15.0000 0.0912871
\(31\) −83.0000 −0.480879 −0.240439 − 0.970664i \(-0.577292\pi\)
−0.240439 + 0.970664i \(0.577292\pi\)
\(32\) 161.000 0.889408
\(33\) −33.0000 −0.174078
\(34\) 21.0000 0.105926
\(35\) 45.0000 0.217325
\(36\) 126.000 0.583333
\(37\) 1.00000 0.00444322 0.00222161 − 0.999998i \(-0.499293\pi\)
0.00222161 + 0.999998i \(0.499293\pi\)
\(38\) −85.0000 −0.362864
\(39\) −6.00000 −0.0246351
\(40\) 75.0000 0.296464
\(41\) −478.000 −1.82076 −0.910379 − 0.413776i \(-0.864210\pi\)
−0.910379 + 0.413776i \(0.864210\pi\)
\(42\) 27.0000 0.0991950
\(43\) −8.00000 −0.0283718 −0.0141859 − 0.999899i \(-0.504516\pi\)
−0.0141859 + 0.999899i \(0.504516\pi\)
\(44\) −77.0000 −0.263822
\(45\) 90.0000 0.298142
\(46\) 22.0000 0.0705157
\(47\) 126.000 0.391042 0.195521 − 0.980699i \(-0.437360\pi\)
0.195521 + 0.980699i \(0.437360\pi\)
\(48\) −123.000 −0.369865
\(49\) −262.000 −0.763848
\(50\) 25.0000 0.0707107
\(51\) −63.0000 −0.172976
\(52\) −14.0000 −0.0373356
\(53\) −683.000 −1.77014 −0.885069 − 0.465461i \(-0.845889\pi\)
−0.885069 + 0.465461i \(0.845889\pi\)
\(54\) 135.000 0.340207
\(55\) −55.0000 −0.134840
\(56\) 135.000 0.322145
\(57\) 255.000 0.592554
\(58\) −165.000 −0.373544
\(59\) −290.000 −0.639912 −0.319956 − 0.947432i \(-0.603668\pi\)
−0.319956 + 0.947432i \(0.603668\pi\)
\(60\) −105.000 −0.225924
\(61\) 257.000 0.539434 0.269717 − 0.962940i \(-0.413070\pi\)
0.269717 + 0.962940i \(0.413070\pi\)
\(62\) −83.0000 −0.170016
\(63\) 162.000 0.323970
\(64\) −167.000 −0.326172
\(65\) −10.0000 −0.0190823
\(66\) −33.0000 −0.0615457
\(67\) 776.000 1.41498 0.707489 − 0.706725i \(-0.249828\pi\)
0.707489 + 0.706725i \(0.249828\pi\)
\(68\) −147.000 −0.262152
\(69\) −66.0000 −0.115152
\(70\) 45.0000 0.0768361
\(71\) −313.000 −0.523187 −0.261593 − 0.965178i \(-0.584248\pi\)
−0.261593 + 0.965178i \(0.584248\pi\)
\(72\) 270.000 0.441942
\(73\) 902.000 1.44618 0.723090 − 0.690754i \(-0.242721\pi\)
0.723090 + 0.690754i \(0.242721\pi\)
\(74\) 1.00000 0.00157091
\(75\) −75.0000 −0.115470
\(76\) 595.000 0.898042
\(77\) −99.0000 −0.146521
\(78\) −6.00000 −0.00870982
\(79\) 830.000 1.18205 0.591027 − 0.806652i \(-0.298723\pi\)
0.591027 + 0.806652i \(0.298723\pi\)
\(80\) −205.000 −0.286496
\(81\) 81.0000 0.111111
\(82\) −478.000 −0.643735
\(83\) 842.000 1.11351 0.556756 − 0.830676i \(-0.312046\pi\)
0.556756 + 0.830676i \(0.312046\pi\)
\(84\) −189.000 −0.245495
\(85\) −105.000 −0.133986
\(86\) −8.00000 −0.0100310
\(87\) 495.000 0.609995
\(88\) −165.000 −0.199876
\(89\) 25.0000 0.0297752 0.0148876 − 0.999889i \(-0.495261\pi\)
0.0148876 + 0.999889i \(0.495261\pi\)
\(90\) 90.0000 0.105409
\(91\) −18.0000 −0.0207353
\(92\) −154.000 −0.174517
\(93\) 249.000 0.277635
\(94\) 126.000 0.138254
\(95\) 425.000 0.458990
\(96\) −483.000 −0.513500
\(97\) −1784.00 −1.86740 −0.933700 − 0.358057i \(-0.883439\pi\)
−0.933700 + 0.358057i \(0.883439\pi\)
\(98\) −262.000 −0.270061
\(99\) −198.000 −0.201008
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 55.4.a.a.1.1 ✓ 1
3.2 odd 2 495.4.a.a.1.1 1
4.3 odd 2 880.4.a.j.1.1 1
5.2 odd 4 275.4.b.a.199.2 2
5.3 odd 4 275.4.b.a.199.1 2
5.4 even 2 275.4.a.a.1.1 1
11.10 odd 2 605.4.a.b.1.1 1
15.14 odd 2 2475.4.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.4.a.a.1.1 ✓ 1 1.1 even 1 trivial
275.4.a.a.1.1 1 5.4 even 2
275.4.b.a.199.1 2 5.3 odd 4
275.4.b.a.199.2 2 5.2 odd 4
495.4.a.a.1.1 1 3.2 odd 2
605.4.a.b.1.1 1 11.10 odd 2
880.4.a.j.1.1 1 4.3 odd 2
2475.4.a.h.1.1 1 15.14 odd 2