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Properties

Label 77.4.a.e.1.3
Level $77$
Weight $4$
Character 77.1
Self dual yes
Analytic conductor $4.543$
Analytic rank $0$
Dimension $5$
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] = [77,4,Mod(1,77)] 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("77.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(77, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 77 = 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 77.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(4.54314707044\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 42x^{3} + 18x^{2} + 368x + 352 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.22767\) of defining polynomial
Character \(\chi\) \(=\) 77.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.22767 q^{2} -7.89221 q^{3} -6.49284 q^{4} -2.21191 q^{5} +9.68899 q^{6} +7.00000 q^{7} +17.7924 q^{8} +35.2869 q^{9} +2.71549 q^{10} +11.0000 q^{11} +51.2428 q^{12} +12.8361 q^{13} -8.59366 q^{14} +17.4569 q^{15} +30.0996 q^{16} -45.5444 q^{17} -43.3205 q^{18} +11.0493 q^{19} +14.3616 q^{20} -55.2454 q^{21} -13.5043 q^{22} +177.525 q^{23} -140.421 q^{24} -120.107 q^{25} -15.7584 q^{26} -65.4021 q^{27} -45.4499 q^{28} +58.5230 q^{29} -21.4312 q^{30} +175.188 q^{31} -179.291 q^{32} -86.8143 q^{33} +55.9132 q^{34} -15.4834 q^{35} -229.112 q^{36} +221.135 q^{37} -13.5648 q^{38} -101.305 q^{39} -39.3551 q^{40} -307.706 q^{41} +67.8229 q^{42} +462.781 q^{43} -71.4212 q^{44} -78.0515 q^{45} -217.942 q^{46} -293.151 q^{47} -237.552 q^{48} +49.0000 q^{49} +147.452 q^{50} +359.445 q^{51} -83.3427 q^{52} +400.608 q^{53} +80.2919 q^{54} -24.3310 q^{55} +124.547 q^{56} -87.2032 q^{57} -71.8467 q^{58} +16.3417 q^{59} -113.344 q^{60} +509.546 q^{61} -215.072 q^{62} +247.008 q^{63} -20.6874 q^{64} -28.3923 q^{65} +106.579 q^{66} +483.585 q^{67} +295.712 q^{68} -1401.07 q^{69} +19.0084 q^{70} +202.883 q^{71} +627.838 q^{72} +885.910 q^{73} -271.479 q^{74} +947.913 q^{75} -71.7412 q^{76} +77.0000 q^{77} +124.369 q^{78} +289.526 q^{79} -66.5777 q^{80} -436.580 q^{81} +377.760 q^{82} +106.577 q^{83} +358.700 q^{84} +100.740 q^{85} -568.140 q^{86} -461.876 q^{87} +195.716 q^{88} -1586.10 q^{89} +95.8211 q^{90} +89.8527 q^{91} -1152.64 q^{92} -1382.62 q^{93} +359.891 q^{94} -24.4400 q^{95} +1415.00 q^{96} -990.599 q^{97} -60.1556 q^{98} +388.156 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + q^{2} + 2 q^{3} + 45 q^{4} - 24 q^{5} + 4 q^{6} + 35 q^{7} + 57 q^{8} + 63 q^{9} - 10 q^{10} + 55 q^{11} + 24 q^{12} - 50 q^{13} + 7 q^{14} - 146 q^{15} + 433 q^{16} + 222 q^{17} + 245 q^{18} + 160 q^{19}+ \cdots + 693 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.22767 −0.434045 −0.217023 − 0.976167i \(-0.569635\pi\)
−0.217023 + 0.976167i \(0.569635\pi\)
\(3\) −7.89221 −1.51886 −0.759428 − 0.650591i \(-0.774521\pi\)
−0.759428 + 0.650591i \(0.774521\pi\)
\(4\) −6.49284 −0.811605
\(5\) −2.21191 −0.197839 −0.0989196 − 0.995095i \(-0.531539\pi\)
−0.0989196 + 0.995095i \(0.531539\pi\)
\(6\) 9.68899 0.659252
\(7\) 7.00000 0.377964
\(8\) 17.7924 0.786319
\(9\) 35.2869 1.30692
\(10\) 2.71549 0.0858712
\(11\) 11.0000 0.301511
\(12\) 51.2428 1.23271
\(13\) 12.8361 0.273853 0.136927 − 0.990581i \(-0.456278\pi\)
0.136927 + 0.990581i \(0.456278\pi\)
\(14\) −8.59366 −0.164054
\(15\) 17.4569 0.300489
\(16\) 30.0996 0.470307
\(17\) −45.5444 −0.649772 −0.324886 − 0.945753i \(-0.605326\pi\)
−0.324886 + 0.945753i \(0.605326\pi\)
\(18\) −43.3205 −0.567264
\(19\) 11.0493 0.133415 0.0667074 − 0.997773i \(-0.478751\pi\)
0.0667074 + 0.997773i \(0.478751\pi\)
\(20\) 14.3616 0.160567
\(21\) −55.2454 −0.574074
\(22\) −13.5043 −0.130870
\(23\) 177.525 1.60942 0.804708 − 0.593671i \(-0.202322\pi\)
0.804708 + 0.593671i \(0.202322\pi\)
\(24\) −140.421 −1.19430
\(25\) −120.107 −0.960860
\(26\) −15.7584 −0.118865
\(27\) −65.4021 −0.466172
\(28\) −45.4499 −0.306758
\(29\) 58.5230 0.374740 0.187370 − 0.982289i \(-0.440004\pi\)
0.187370 + 0.982289i \(0.440004\pi\)
\(30\) −21.4312 −0.130426
\(31\) 175.188 1.01499 0.507495 − 0.861654i \(-0.330571\pi\)
0.507495 + 0.861654i \(0.330571\pi\)
\(32\) −179.291 −0.990453
\(33\) −86.8143 −0.457952
\(34\) 55.9132 0.282031
\(35\) −15.4834 −0.0747762
\(36\) −229.112 −1.06070
\(37\) 221.135 0.982549 0.491274 − 0.871005i \(-0.336531\pi\)
0.491274 + 0.871005i \(0.336531\pi\)
\(38\) −13.5648 −0.0579080
\(39\) −101.305 −0.415944
\(40\) −39.3551 −0.155565
\(41\) −307.706 −1.17209 −0.586043 − 0.810280i \(-0.699315\pi\)
−0.586043 + 0.810280i \(0.699315\pi\)
\(42\) 67.8229 0.249174
\(43\) 462.781 1.64124 0.820621 − 0.571473i \(-0.193628\pi\)
0.820621 + 0.571473i \(0.193628\pi\)
\(44\) −71.4212 −0.244708
\(45\) −78.0515 −0.258561
\(46\) −217.942 −0.698560
\(47\) −293.151 −0.909796 −0.454898 − 0.890544i \(-0.650324\pi\)
−0.454898 + 0.890544i \(0.650324\pi\)
\(48\) −237.552 −0.714328
\(49\) 49.0000 0.142857
\(50\) 147.452 0.417057
\(51\) 359.445 0.986910
\(52\) −83.3427 −0.222261
\(53\) 400.608 1.03826 0.519130 − 0.854696i \(-0.326256\pi\)
0.519130 + 0.854696i \(0.326256\pi\)
\(54\) 80.2919 0.202340
\(55\) −24.3310 −0.0596508
\(56\) 124.547 0.297201
\(57\) −87.2032 −0.202638
\(58\) −71.8467 −0.162654
\(59\) 16.3417 0.0360595 0.0180298 − 0.999837i \(-0.494261\pi\)
0.0180298 + 0.999837i \(0.494261\pi\)
\(60\) −113.344 −0.243879
\(61\) 509.546 1.06952 0.534760 − 0.845004i \(-0.320402\pi\)
0.534760 + 0.845004i \(0.320402\pi\)
\(62\) −215.072 −0.440552
\(63\) 247.008 0.493970
\(64\) −20.6874 −0.0404051
\(65\) −28.3923 −0.0541790
\(66\) 106.579 0.198772
\(67\) 483.585 0.881781 0.440891 − 0.897561i \(-0.354663\pi\)
0.440891 + 0.897561i \(0.354663\pi\)
\(68\) 295.712 0.527358
\(69\) −1401.07 −2.44447
\(70\) 19.0084 0.0324563
\(71\) 202.883 0.339124 0.169562 − 0.985519i \(-0.445765\pi\)
0.169562 + 0.985519i \(0.445765\pi\)
\(72\) 627.838 1.02766
\(73\) 885.910 1.42038 0.710191 − 0.704009i \(-0.248608\pi\)
0.710191 + 0.704009i \(0.248608\pi\)
\(74\) −271.479 −0.426471
\(75\) 947.913 1.45941
\(76\) −71.7412 −0.108280
\(77\) 77.0000 0.113961
\(78\) 124.369 0.180539
\(79\) 289.526 0.412331 0.206166 − 0.978517i \(-0.433901\pi\)
0.206166 + 0.978517i \(0.433901\pi\)
\(80\) −66.5777 −0.0930451
\(81\) −436.580 −0.598875
\(82\) 377.760 0.508739
\(83\) 106.577 0.140944 0.0704722 − 0.997514i \(-0.477549\pi\)
0.0704722 + 0.997514i \(0.477549\pi\)
\(84\) 358.700 0.465921
\(85\) 100.740 0.128550
\(86\) −568.140 −0.712373
\(87\) −461.876 −0.569175
\(88\) 195.716 0.237084
\(89\) −1586.10 −1.88906 −0.944528 − 0.328432i \(-0.893480\pi\)
−0.944528 + 0.328432i \(0.893480\pi\)
\(90\) 95.8211 0.112227
\(91\) 89.8527 0.103507
\(92\) −1152.64 −1.30621
\(93\) −1382.62 −1.54162
\(94\) 359.891 0.394893
\(95\) −24.4400 −0.0263947
\(96\) 1415.00 1.50436
\(97\) −990.599 −1.03691 −0.518454 − 0.855105i \(-0.673492\pi\)
−0.518454 + 0.855105i \(0.673492\pi\)
\(98\) −60.1556 −0.0620065
\(99\) 388.156 0.394052
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 77.4.a.e.1.3 ✓ 5
3.2 odd 2 693.4.a.o.1.3 5
4.3 odd 2 1232.4.a.y.1.5 5
5.4 even 2 1925.4.a.r.1.3 5
7.6 odd 2 539.4.a.h.1.3 5
11.10 odd 2 847.4.a.f.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.4.a.e.1.3 ✓ 5 1.1 even 1 trivial
539.4.a.h.1.3 5 7.6 odd 2
693.4.a.o.1.3 5 3.2 odd 2
847.4.a.f.1.3 5 11.10 odd 2
1232.4.a.y.1.5 5 4.3 odd 2
1925.4.a.r.1.3 5 5.4 even 2