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Properties

Label 1.76.a.a.1.4
Level $1$
Weight $76$
Character 1.1
Self dual yes
Analytic conductor $35.623$
Analytic rank $0$
Dimension $6$
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] = [1,76,Mod(1,1)] 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("1.1"); S:= CuspForms(chi, 76); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1, base_ring=CyclotomicField(1)) chi = DirichletCharacter(H, H._module([])) N = Newforms(chi, 76, names="a")
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 76 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(35.6228392822\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3 x^{5} + \cdots - 67\!\cdots\!50 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{58}\cdot 3^{26}\cdot 5^{7}\cdot 7^{3}\cdot 11\cdot 13\cdot 19 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.46671e9\) of defining polynomial
Character \(\chi\) \(=\) 1.1

$q$-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)
 
\(f(q)\) \(=\) \(q+9.60898e10 q^{2} -6.47242e17 q^{3} -2.85457e22 q^{4} -1.49650e26 q^{5} -6.21934e28 q^{6} -3.16100e31 q^{7} -6.37312e33 q^{8} -1.89344e35 q^{9} -1.43799e37 q^{10} -9.04175e38 q^{11} +1.84760e40 q^{12} -1.02333e42 q^{13} -3.03740e42 q^{14} +9.68599e43 q^{15} +4.66033e44 q^{16} +2.82749e45 q^{17} -1.81941e46 q^{18} +5.13524e47 q^{19} +4.27187e48 q^{20} +2.04593e49 q^{21} -8.68820e49 q^{22} -1.06901e51 q^{23} +4.12495e51 q^{24} -4.07461e51 q^{25} -9.83316e52 q^{26} +5.16248e53 q^{27} +9.02328e53 q^{28} -5.54224e54 q^{29} +9.30725e54 q^{30} -6.85555e55 q^{31} +2.85551e56 q^{32} +5.85220e56 q^{33} +2.71693e56 q^{34} +4.73044e57 q^{35} +5.40496e57 q^{36} +1.00154e59 q^{37} +4.93445e58 q^{38} +6.62342e59 q^{39} +9.53739e59 q^{40} +4.31368e60 q^{41} +1.96593e60 q^{42} -1.74752e61 q^{43} +2.58103e61 q^{44} +2.83354e61 q^{45} -1.02721e62 q^{46} -9.73598e62 q^{47} -3.01636e62 q^{48} -1.41267e63 q^{49} -3.91528e62 q^{50} -1.83007e63 q^{51} +2.92116e64 q^{52} -2.55811e63 q^{53} +4.96061e64 q^{54} +1.35310e65 q^{55} +2.01454e65 q^{56} -3.32375e65 q^{57} -5.32553e65 q^{58} +5.98212e65 q^{59} -2.76493e66 q^{60} -1.23057e67 q^{61} -6.58749e66 q^{62} +5.98517e66 q^{63} +9.83229e66 q^{64} +1.53141e68 q^{65} +5.62337e67 q^{66} -9.36690e67 q^{67} -8.07127e67 q^{68} +6.91907e68 q^{69} +4.54547e68 q^{70} -3.34843e69 q^{71} +1.20671e69 q^{72} -1.44439e70 q^{73} +9.62381e69 q^{74} +2.63726e69 q^{75} -1.46589e70 q^{76} +2.85810e70 q^{77} +6.36444e70 q^{78} +1.89027e71 q^{79} -6.97419e70 q^{80} -2.18965e71 q^{81} +4.14501e71 q^{82} -1.25156e72 q^{83} -5.84025e71 q^{84} -4.23135e71 q^{85} -1.67919e72 q^{86} +3.58717e72 q^{87} +5.76242e72 q^{88} -1.25211e73 q^{89} +2.72275e72 q^{90} +3.23474e73 q^{91} +3.05156e73 q^{92} +4.43720e73 q^{93} -9.35529e73 q^{94} -7.68490e73 q^{95} -1.84820e74 q^{96} -8.39965e73 q^{97} -1.35744e74 q^{98} +1.71200e74 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 57080822040 q^{2} - 78\!\cdots\!40 q^{3} + 17\!\cdots\!28 q^{4} - 38\!\cdots\!40 q^{5} + 31\!\cdots\!92 q^{6} + 19\!\cdots\!00 q^{7} + 44\!\cdots\!20 q^{8} + 21\!\cdots\!82 q^{9} + 13\!\cdots\!60 q^{10}+ \cdots - 18\!\cdots\!36 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 9.60898e10 0.494371 0.247185 − 0.968968i \(-0.420494\pi\)
0.247185 + 0.968968i \(0.420494\pi\)
\(3\) −6.47242e17 −0.829888 −0.414944 − 0.909847i \(-0.636199\pi\)
−0.414944 + 0.909847i \(0.636199\pi\)
\(4\) −2.85457e22 −0.755598
\(5\) −1.49650e26 −0.919818 −0.459909 − 0.887966i \(-0.652118\pi\)
−0.459909 + 0.887966i \(0.652118\pi\)
\(6\) −6.21934e28 −0.410272
\(7\) −3.16100e31 −0.643647 −0.321824 − 0.946800i \(-0.604296\pi\)
−0.321824 + 0.946800i \(0.604296\pi\)
\(8\) −6.37312e33 −0.867916
\(9\) −1.89344e35 −0.311285
\(10\) −1.43799e37 −0.454731
\(11\) −9.04175e38 −0.801727 −0.400864 − 0.916138i \(-0.631290\pi\)
−0.400864 + 0.916138i \(0.631290\pi\)
\(12\) 1.84760e40 0.627062
\(13\) −1.02333e42 −1.72639 −0.863197 − 0.504867i \(-0.831541\pi\)
−0.863197 + 0.504867i \(0.831541\pi\)
\(14\) −3.03740e42 −0.318200
\(15\) 9.68599e43 0.763347
\(16\) 4.66033e44 0.326526
\(17\) 2.82749e45 0.203970 0.101985 − 0.994786i \(-0.467481\pi\)
0.101985 + 0.994786i \(0.467481\pi\)
\(18\) −1.81941e46 −0.153890
\(19\) 5.13524e47 0.571875 0.285937 − 0.958248i \(-0.407695\pi\)
0.285937 + 0.958248i \(0.407695\pi\)
\(20\) 4.27187e48 0.695013
\(21\) 2.04593e49 0.534155
\(22\) −8.68820e49 −0.396350
\(23\) −1.06901e51 −0.920846 −0.460423 − 0.887700i \(-0.652302\pi\)
−0.460423 + 0.887700i \(0.652302\pi\)
\(24\) 4.12495e51 0.720273
\(25\) −4.07461e51 −0.153934
\(26\) −9.83316e52 −0.853478
\(27\) 5.16248e53 1.08822
\(28\) 9.02328e53 0.486338
\(29\) −5.54224e54 −0.801236 −0.400618 − 0.916245i \(-0.631204\pi\)
−0.400618 + 0.916245i \(0.631204\pi\)
\(30\) 9.30725e54 0.377376
\(31\) −6.85555e55 −0.812791 −0.406396 − 0.913697i \(-0.633215\pi\)
−0.406396 + 0.913697i \(0.633215\pi\)
\(32\) 2.85551e56 1.02934
\(33\) 5.85220e56 0.665344
\(34\) 2.71693e56 0.100837
\(35\) 4.73044e57 0.592038
\(36\) 5.40496e57 0.235206
\(37\) 1.00154e59 1.55993 0.779965 − 0.625823i \(-0.215237\pi\)
0.779965 + 0.625823i \(0.215237\pi\)
\(38\) 4.93445e58 0.282718
\(39\) 6.62342e59 1.43271
\(40\) 9.53739e59 0.798325
\(41\) 4.31368e60 1.43039 0.715193 − 0.698927i \(-0.246339\pi\)
0.715193 + 0.698927i \(0.246339\pi\)
\(42\) 1.96593e60 0.264071
\(43\) −1.74752e61 −0.971303 −0.485651 − 0.874153i \(-0.661418\pi\)
−0.485651 + 0.874153i \(0.661418\pi\)
\(44\) 2.58103e61 0.605783
\(45\) 2.83354e61 0.286326
\(46\) −1.02721e62 −0.455239
\(47\) −9.73598e62 −1.92624 −0.963119 − 0.269076i \(-0.913282\pi\)
−0.963119 + 0.269076i \(0.913282\pi\)
\(48\) −3.01636e62 −0.270980
\(49\) −1.41267e63 −0.585718
\(50\) −3.91528e62 −0.0761006
\(51\) −1.83007e63 −0.169273
\(52\) 2.92116e64 1.30446
\(53\) −2.55811e63 −0.0559209 −0.0279604 − 0.999609i \(-0.508901\pi\)
−0.0279604 + 0.999609i \(0.508901\pi\)
\(54\) 4.96061e64 0.537984
\(55\) 1.35310e65 0.737444
\(56\) 2.01454e65 0.558631
\(57\) −3.32375e65 −0.474592
\(58\) −5.32553e65 −0.396107
\(59\) 5.98212e65 0.234372 0.117186 − 0.993110i \(-0.462613\pi\)
0.117186 + 0.993110i \(0.462613\pi\)
\(60\) −2.76493e66 −0.576783
\(61\) −1.23057e67 −1.38114 −0.690571 − 0.723265i \(-0.742641\pi\)
−0.690571 + 0.723265i \(0.742641\pi\)
\(62\) −6.58749e66 −0.401820
\(63\) 5.98517e66 0.200358
\(64\) 9.83229e66 0.182350
\(65\) 1.53141e68 1.58797
\(66\) 5.62337e67 0.328927
\(67\) −9.36690e67 −0.311739 −0.155869 − 0.987778i \(-0.549818\pi\)
−0.155869 + 0.987778i \(0.549818\pi\)
\(68\) −8.07127e67 −0.154120
\(69\) 6.91907e68 0.764200
\(70\) 4.54547e68 0.292686
\(71\) −3.34843e69 −1.26664 −0.633320 − 0.773890i \(-0.718308\pi\)
−0.633320 + 0.773890i \(0.718308\pi\)
\(72\) 1.20671e69 0.270169
\(73\) −1.44439e70 −1.92787 −0.963934 − 0.266141i \(-0.914251\pi\)
−0.963934 + 0.266141i \(0.914251\pi\)
\(74\) 9.62381e69 0.771183
\(75\) 2.63726e69 0.127748
\(76\) −1.46589e70 −0.432107
\(77\) 2.85810e70 0.516029
\(78\) 6.36444e70 0.708292
\(79\) 1.89027e71 1.30469 0.652347 − 0.757920i \(-0.273784\pi\)
0.652347 + 0.757920i \(0.273784\pi\)
\(80\) −6.97419e70 −0.300344
\(81\) −2.18965e71 −0.591816
\(82\) 4.14501e71 0.707140
\(83\) −1.25156e72 −1.35526 −0.677630 − 0.735403i \(-0.736993\pi\)
−0.677630 + 0.735403i \(0.736993\pi\)
\(84\) −5.84025e71 −0.403607
\(85\) −4.23135e71 −0.187616
\(86\) −1.67919e72 −0.480184
\(87\) 3.58717e72 0.664936
\(88\) 5.76242e72 0.695832
\(89\) −1.25211e73 −0.989728 −0.494864 − 0.868971i \(-0.664782\pi\)
−0.494864 + 0.868971i \(0.664782\pi\)
\(90\) 2.72275e72 0.141551
\(91\) 3.23474e73 1.11119
\(92\) 3.05156e73 0.695789
\(93\) 4.43720e73 0.674526
\(94\) −9.35529e73 −0.952275
\(95\) −7.68490e73 −0.526021
\(96\) −1.84820e74 −0.854238
\(97\) −8.39965e73 −0.263221 −0.131611 − 0.991301i \(-0.542015\pi\)
−0.131611 + 0.991301i \(0.542015\pi\)
\(98\) −1.35744e74 −0.289562
\(99\) 1.71200e74 0.249566
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 1.76.a.a.1.4 ✓ 6
3.2 odd 2 9.76.a.c.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.76.a.a.1.4 ✓ 6 1.1 even 1 trivial
9.76.a.c.1.3 6 3.2 odd 2