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Properties

Label 2.74.a.a.1.3
Level $2$
Weight $74$
Character 2.1
Self dual yes
Analytic conductor $67.497$
Analytic rank $1$
Dimension $3$
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] = [2,74,Mod(1,2)] 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("2.1"); S:= CuspForms(chi, 74); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2, base_ring=CyclotomicField(1)) chi = DirichletCharacter(H, H._module([])) N = Newforms(chi, 74, names="a")
 
Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 74 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3] 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: \(67.4967947474\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 413501459186944860372404680x - 2966140105783309949999568694815716833028 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{23}\cdot 3^{7}\cdot 5^{3} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.32599e13\) of defining polynomial
Character \(\chi\) \(=\) 2.1

$q$-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)
 
\(f(q)\) \(=\) \(q-6.87195e10 q^{2} +3.00585e17 q^{3} +4.72237e21 q^{4} -1.10375e25 q^{5} -2.06560e28 q^{6} +7.83423e29 q^{7} -3.24519e32 q^{8} +2.27660e34 q^{9} +7.58489e35 q^{10} +3.40887e36 q^{11} +1.41947e39 q^{12} +3.80927e40 q^{13} -5.38364e40 q^{14} -3.31769e42 q^{15} +2.23007e43 q^{16} -2.38211e44 q^{17} -1.56447e45 q^{18} -4.58511e46 q^{19} -5.21229e46 q^{20} +2.35485e47 q^{21} -2.34256e47 q^{22} -5.99384e49 q^{23} -9.75453e49 q^{24} -9.36966e50 q^{25} -2.61771e51 q^{26} -1.34720e52 q^{27} +3.69961e51 q^{28} +3.66647e53 q^{29} +2.27990e53 q^{30} +1.75180e54 q^{31} -1.53250e54 q^{32} +1.02465e54 q^{33} +1.63697e55 q^{34} -8.64700e54 q^{35} +1.07509e56 q^{36} +1.64193e57 q^{37} +3.15086e57 q^{38} +1.14501e58 q^{39} +3.58186e57 q^{40} +1.95654e58 q^{41} -1.61824e58 q^{42} -7.77238e59 q^{43} +1.60979e58 q^{44} -2.51279e59 q^{45} +4.11894e60 q^{46} -5.34256e60 q^{47} +6.70326e60 q^{48} -4.86080e61 q^{49} +6.43878e61 q^{50} -7.16026e61 q^{51} +1.79888e62 q^{52} +5.17851e62 q^{53} +9.25787e62 q^{54} -3.76253e61 q^{55} -2.54235e62 q^{56} -1.37821e64 q^{57} -2.51958e64 q^{58} -1.88861e63 q^{59} -1.56674e64 q^{60} +1.49443e65 q^{61} -1.20383e65 q^{62} +1.78354e64 q^{63} +1.05312e65 q^{64} -4.20447e65 q^{65} -7.04137e64 q^{66} -8.11124e66 q^{67} -1.12492e66 q^{68} -1.80166e67 q^{69} +5.94218e65 q^{70} -2.29663e67 q^{71} -7.38798e66 q^{72} -1.30543e68 q^{73} -1.12832e68 q^{74} -2.81638e68 q^{75} -2.16526e68 q^{76} +2.67059e66 q^{77} -7.86844e68 q^{78} -7.20690e67 q^{79} -2.46144e68 q^{80} -5.58811e69 q^{81} -1.34452e69 q^{82} -1.10136e70 q^{83} +1.11205e69 q^{84} +2.62925e69 q^{85} +5.34114e70 q^{86} +1.10208e71 q^{87} -1.10624e69 q^{88} +1.46084e71 q^{89} +1.72677e70 q^{90} +2.98427e70 q^{91} -2.83051e71 q^{92} +5.26565e71 q^{93} +3.67138e71 q^{94} +5.06080e71 q^{95} -4.60645e71 q^{96} +7.46916e71 q^{97} +3.34032e72 q^{98} +7.76062e70 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 206158430208 q^{2} - 30\!\cdots\!72 q^{3} + 14\!\cdots\!88 q^{4} - 32\!\cdots\!50 q^{5} + 20\!\cdots\!92 q^{6} - 23\!\cdots\!64 q^{7} - 97\!\cdots\!68 q^{8} + 74\!\cdots\!59 q^{9} + 22\!\cdots\!00 q^{10}+ \cdots + 32\!\cdots\!68 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −6.87195e10 −0.707107
\(3\) 3.00585e17 1.15622 0.578111 − 0.815958i \(-0.303790\pi\)
0.578111 + 0.815958i \(0.303790\pi\)
\(4\) 4.72237e21 0.500000
\(5\) −1.10375e25 −0.339206 −0.169603 − 0.985512i \(-0.554249\pi\)
−0.169603 + 0.985512i \(0.554249\pi\)
\(6\) −2.06560e28 −0.817572
\(7\) 7.83423e29 0.111665 0.0558326 − 0.998440i \(-0.482219\pi\)
0.0558326 + 0.998440i \(0.482219\pi\)
\(8\) −3.24519e32 −0.353553
\(9\) 2.27660e34 0.336848
\(10\) 7.58489e35 0.239855
\(11\) 3.40887e36 0.0332489 0.0166245 − 0.999862i \(-0.494708\pi\)
0.0166245 + 0.999862i \(0.494708\pi\)
\(12\) 1.41947e39 0.578111
\(13\) 3.80927e40 0.835429 0.417714 − 0.908578i \(-0.362831\pi\)
0.417714 + 0.908578i \(0.362831\pi\)
\(14\) −5.38364e40 −0.0789592
\(15\) −3.31769e42 −0.392198
\(16\) 2.23007e43 0.250000
\(17\) −2.38211e44 −0.292130 −0.146065 − 0.989275i \(-0.546661\pi\)
−0.146065 + 0.989275i \(0.546661\pi\)
\(18\) −1.56447e45 −0.238188
\(19\) −4.58511e46 −0.970160 −0.485080 − 0.874470i \(-0.661210\pi\)
−0.485080 + 0.874470i \(0.661210\pi\)
\(20\) −5.21229e46 −0.169603
\(21\) 2.35485e47 0.129110
\(22\) −2.34256e47 −0.0235105
\(23\) −5.99384e49 −1.18752 −0.593758 − 0.804644i \(-0.702356\pi\)
−0.593758 + 0.804644i \(0.702356\pi\)
\(24\) −9.75453e49 −0.408786
\(25\) −9.36966e50 −0.884939
\(26\) −2.61771e51 −0.590737
\(27\) −1.34720e52 −0.766750
\(28\) 3.69961e51 0.0558326
\(29\) 3.66647e53 1.53717 0.768583 − 0.639750i \(-0.220962\pi\)
0.768583 + 0.639750i \(0.220962\pi\)
\(30\) 2.27990e53 0.277326
\(31\) 1.75180e54 0.643847 0.321924 − 0.946766i \(-0.395671\pi\)
0.321924 + 0.946766i \(0.395671\pi\)
\(32\) −1.53250e54 −0.176777
\(33\) 1.02465e54 0.0384431
\(34\) 1.63697e55 0.206567
\(35\) −8.64700e54 −0.0378776
\(36\) 1.07509e56 0.168424
\(37\) 1.64193e57 0.946217 0.473109 − 0.881004i \(-0.343132\pi\)
0.473109 + 0.881004i \(0.343132\pi\)
\(38\) 3.15086e57 0.686007
\(39\) 1.14501e58 0.965941
\(40\) 3.58186e57 0.119928
\(41\) 1.95654e58 0.265998 0.132999 − 0.991116i \(-0.457539\pi\)
0.132999 + 0.991116i \(0.457539\pi\)
\(42\) −1.61824e58 −0.0912944
\(43\) −7.77238e59 −1.85761 −0.928806 − 0.370565i \(-0.879164\pi\)
−0.928806 + 0.370565i \(0.879164\pi\)
\(44\) 1.60979e58 0.0166245
\(45\) −2.51279e59 −0.114261
\(46\) 4.11894e60 0.839700
\(47\) −5.34256e60 −0.496795 −0.248398 − 0.968658i \(-0.579904\pi\)
−0.248398 + 0.968658i \(0.579904\pi\)
\(48\) 6.70326e60 0.289055
\(49\) −4.86080e61 −0.987531
\(50\) 6.43878e61 0.625746
\(51\) −7.16026e61 −0.337767
\(52\) 1.79888e62 0.417714
\(53\) 5.17851e62 0.599979 0.299989 − 0.953943i \(-0.403017\pi\)
0.299989 + 0.953943i \(0.403017\pi\)
\(54\) 9.25787e62 0.542174
\(55\) −3.76253e61 −0.0112782
\(56\) −2.54235e62 −0.0394796
\(57\) −1.37821e64 −1.12172
\(58\) −2.51958e64 −1.08694
\(59\) −1.88861e63 −0.0436560 −0.0218280 − 0.999762i \(-0.506949\pi\)
−0.0218280 + 0.999762i \(0.506949\pi\)
\(60\) −1.56674e64 −0.196099
\(61\) 1.49443e65 1.02315 0.511573 − 0.859240i \(-0.329063\pi\)
0.511573 + 0.859240i \(0.329063\pi\)
\(62\) −1.20383e65 −0.455269
\(63\) 1.78354e64 0.0376143
\(64\) 1.05312e65 0.125000
\(65\) −4.20447e65 −0.283383
\(66\) −7.04137e64 −0.0271834
\(67\) −8.11124e66 −1.80866 −0.904330 − 0.426834i \(-0.859629\pi\)
−0.904330 + 0.426834i \(0.859629\pi\)
\(68\) −1.12492e66 −0.146065
\(69\) −1.80166e67 −1.37303
\(70\) 5.94218e65 0.0267835
\(71\) −2.29663e67 −0.616823 −0.308412 − 0.951253i \(-0.599797\pi\)
−0.308412 + 0.951253i \(0.599797\pi\)
\(72\) −7.38798e66 −0.119094
\(73\) −1.30543e68 −1.27195 −0.635974 − 0.771711i \(-0.719401\pi\)
−0.635974 + 0.771711i \(0.719401\pi\)
\(74\) −1.12832e68 −0.669077
\(75\) −2.81638e68 −1.02319
\(76\) −2.16526e68 −0.485080
\(77\) 2.67059e66 0.00371275
\(78\) −7.86844e68 −0.683023
\(79\) −7.20690e67 −0.0392970 −0.0196485 − 0.999807i \(-0.506255\pi\)
−0.0196485 + 0.999807i \(0.506255\pi\)
\(80\) −2.46144e68 −0.0848016
\(81\) −5.58811e69 −1.22338
\(82\) −1.34452e69 −0.188089
\(83\) −1.10136e70 −0.989876 −0.494938 − 0.868928i \(-0.664809\pi\)
−0.494938 + 0.868928i \(0.664809\pi\)
\(84\) 1.11205e69 0.0645549
\(85\) 2.62925e69 0.0990924
\(86\) 5.34114e70 1.31353
\(87\) 1.10208e71 1.77731
\(88\) −1.10624e69 −0.0117553
\(89\) 1.46084e71 1.02770 0.513851 − 0.857879i \(-0.328218\pi\)
0.513851 + 0.857879i \(0.328218\pi\)
\(90\) 1.72677e70 0.0807949
\(91\) 2.98427e70 0.0932883
\(92\) −2.83051e71 −0.593758
\(93\) 5.26565e71 0.744430
\(94\) 3.67138e71 0.351287
\(95\) 5.06080e71 0.329085
\(96\) −4.60645e71 −0.204393
\(97\) 7.46916e71 0.227040 0.113520 − 0.993536i \(-0.463787\pi\)
0.113520 + 0.993536i \(0.463787\pi\)
\(98\) 3.34032e72 0.698290
\(99\) 7.76062e70 0.0111998
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 2.74.a.a.1.3 ✓ 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.74.a.a.1.3 ✓ 3 1.1 even 1 trivial