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Properties

Label 605.4.a.b.1.1
Level $605$
Weight $4$
Character 605.1
Self dual yes
Analytic conductor $35.696$
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] = [605,4,Mod(1,605)] 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("605.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(605, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 605.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1,-3,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(35.6961555535\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 605.1

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

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 \(-0.556567\pi\)
−0.176777 + 0.984251i \(0.556567\pi\)
\(3\) −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 \(-0.421876\pi\)
0.242977 + 0.970032i \(0.421876\pi\)
\(8\) 15.0000 0.662913
\(9\) −18.0000 −0.666667
\(10\) 5.00000 0.158114
\(11\) 0 0
\(12\) 21.0000 0.505181
\(13\) −2.00000 −0.0426692 −0.0213346 − 0.999772i \(-0.506792\pi\)
−0.0213346 + 0.999772i \(0.506792\pi\)
\(14\) −9.00000 −0.171811
\(15\) 15.0000 0.258199
\(16\) 41.0000 0.640625
\(17\) −21.0000 −0.299603 −0.149801 − 0.988716i \(-0.547863\pi\)
−0.149801 + 0.988716i \(0.547863\pi\)
\(18\) 18.0000 0.235702
\(19\) 85.0000 1.02633 0.513167 − 0.858289i \(-0.328472\pi\)
0.513167 + 0.858289i \(0.328472\pi\)
\(20\) 35.0000 0.391312
\(21\) −27.0000 −0.280566
\(22\) 0 0
\(23\) 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 \(-0.322840\pi\)
0.528271 + 0.849076i \(0.322840\pi\)
\(30\) −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
\(33\) 0 0
\(34\) 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 \(-0.135790\pi\)
0.910379 + 0.413776i \(0.135790\pi\)
\(42\) 27.0000 0.0991950
\(43\) 8.00000 0.0283718 0.0141859 − 0.999899i \(-0.495484\pi\)
0.0141859 + 0.999899i \(0.495484\pi\)
\(44\) 0 0
\(45\) 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
\(55\) 0 0
\(56\) 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 \(-0.586930\pi\)
−0.269717 + 0.962940i \(0.586930\pi\)
\(62\) 83.0000 0.170016
\(63\) −162.000 −0.323970
\(64\) −167.000 −0.326172
\(65\) 10.0000 0.0190823
\(66\) 0 0
\(67\) 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 \(-0.757279\pi\)
−0.723090 + 0.690754i \(0.757279\pi\)
\(74\) −1.00000 −0.00157091
\(75\) −75.0000 −0.115470
\(76\) −595.000 −0.898042
\(77\) 0 0
\(78\) −6.00000 −0.00870982
\(79\) −830.000 −1.18205 −0.591027 − 0.806652i \(-0.701277\pi\)
−0.591027 + 0.806652i \(0.701277\pi\)
\(80\) −205.000 −0.286496
\(81\) 81.0000 0.111111
\(82\) −478.000 −0.643735
\(83\) −842.000 −1.11351 −0.556756 − 0.830676i \(-0.687954\pi\)
−0.556756 + 0.830676i \(0.687954\pi\)
\(84\) 189.000 0.245495
\(85\) 105.000 0.133986
\(86\) −8.00000 −0.0100310
\(87\) −495.000 −0.609995
\(88\) 0 0
\(89\) 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
\(99\) 0 0
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 605.4.a.b.1.1 1
11.10 odd 2 55.4.a.a.1.1 ✓ 1
33.32 even 2 495.4.a.a.1.1 1
44.43 even 2 880.4.a.j.1.1 1
55.32 even 4 275.4.b.a.199.2 2
55.43 even 4 275.4.b.a.199.1 2
55.54 odd 2 275.4.a.a.1.1 1
165.164 even 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 11.10 odd 2
275.4.a.a.1.1 1 55.54 odd 2
275.4.b.a.199.1 2 55.43 even 4
275.4.b.a.199.2 2 55.32 even 4
495.4.a.a.1.1 1 33.32 even 2
605.4.a.b.1.1 1 1.1 even 1 trivial
880.4.a.j.1.1 1 44.43 even 2
2475.4.a.h.1.1 1 165.164 even 2