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Properties

Label 2.76.a.b.1.3
Level $2$
Weight $76$
Character 2.1
Self dual yes
Analytic conductor $71.246$
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] = [2,76,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, 76); 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, 76, names="a")
 
Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 76 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,412316860416] 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: \(71.2456785644\)
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: \( x^{3} - x^{2} - 69170275937846809816619130x + 194820240227429941309097792198860377672 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{8}\cdot 5^{3}\cdot 7\cdot 11 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(6.10323e12\) 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)
 
\(f(q)\) \(=\) \(q+1.37439e11 q^{2} +7.85531e17 q^{3} +1.88895e22 q^{4} +1.02529e26 q^{5} +1.07962e29 q^{6} -9.01778e31 q^{7} +2.59615e33 q^{8} +8.79144e33 q^{9} +1.40915e37 q^{10} +1.29798e39 q^{11} +1.48383e40 q^{12} -8.54396e41 q^{13} -1.23939e43 q^{14} +8.05398e43 q^{15} +3.56812e44 q^{16} -1.96919e46 q^{17} +1.20829e45 q^{18} +1.45015e47 q^{19} +1.93672e48 q^{20} -7.08374e49 q^{21} +1.78393e50 q^{22} -9.62630e50 q^{23} +2.03935e51 q^{24} -1.59575e52 q^{25} -1.17427e53 q^{26} -4.70906e53 q^{27} -1.70341e54 q^{28} +6.83539e54 q^{29} +1.10693e55 q^{30} -2.27798e55 q^{31} +4.90399e55 q^{32} +1.01960e57 q^{33} -2.70643e57 q^{34} -9.24586e57 q^{35} +1.66066e56 q^{36} -6.31677e58 q^{37} +1.99307e58 q^{38} -6.71154e59 q^{39} +2.66181e59 q^{40} +3.47026e60 q^{41} -9.73582e60 q^{42} -1.32858e61 q^{43} +2.45182e61 q^{44} +9.01379e59 q^{45} -1.32303e62 q^{46} +4.14273e62 q^{47} +2.80287e62 q^{48} +5.72018e63 q^{49} -2.19319e63 q^{50} -1.54686e64 q^{51} -1.61391e64 q^{52} +1.50768e64 q^{53} -6.47209e64 q^{54} +1.33081e65 q^{55} -2.34115e65 q^{56} +1.13913e65 q^{57} +9.39448e65 q^{58} -1.88385e66 q^{59} +1.52135e66 q^{60} +2.68011e66 q^{61} -3.13083e66 q^{62} -7.92793e65 q^{63} +6.73999e66 q^{64} -8.76006e67 q^{65} +1.40133e68 q^{66} -3.70871e68 q^{67} -3.71969e68 q^{68} -7.56175e68 q^{69} -1.27074e69 q^{70} +2.25253e69 q^{71} +2.28239e67 q^{72} -2.96144e69 q^{73} -8.68170e69 q^{74} -1.25351e70 q^{75} +2.73925e69 q^{76} -1.17049e71 q^{77} -9.22428e70 q^{78} -2.41694e71 q^{79} +3.65836e70 q^{80} -3.75259e71 q^{81} +4.76949e71 q^{82} +9.02951e70 q^{83} -1.33808e72 q^{84} -2.01899e72 q^{85} -1.82599e72 q^{86} +5.36941e72 q^{87} +3.36975e72 q^{88} -1.80794e73 q^{89} +1.23885e71 q^{90} +7.70476e73 q^{91} -1.81836e73 q^{92} -1.78942e73 q^{93} +5.69372e73 q^{94} +1.48682e73 q^{95} +3.85223e73 q^{96} +5.17925e74 q^{97} +7.86175e74 q^{98} +1.14111e73 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 412316860416 q^{2} + 45\!\cdots\!16 q^{3} + 56\!\cdots\!52 q^{4} - 19\!\cdots\!90 q^{5} + 62\!\cdots\!52 q^{6} - 37\!\cdots\!52 q^{7} + 77\!\cdots\!44 q^{8} - 26\!\cdots\!69 q^{9} - 26\!\cdots\!80 q^{10}+ \cdots + 90\!\cdots\!32 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.37439e11 0.707107
\(3\) 7.85531e17 1.00720 0.503600 − 0.863937i \(-0.332008\pi\)
0.503600 + 0.863937i \(0.332008\pi\)
\(4\) 1.88895e22 0.500000
\(5\) 1.02529e26 0.630191 0.315096 − 0.949060i \(-0.397963\pi\)
0.315096 + 0.949060i \(0.397963\pi\)
\(6\) 1.07962e29 0.712198
\(7\) −9.01778e31 −1.83621 −0.918107 − 0.396332i \(-0.870283\pi\)
−0.918107 + 0.396332i \(0.870283\pi\)
\(8\) 2.59615e33 0.353553
\(9\) 8.79144e33 0.0144533
\(10\) 1.40915e37 0.445613
\(11\) 1.29798e39 1.15091 0.575457 − 0.817832i \(-0.304824\pi\)
0.575457 + 0.817832i \(0.304824\pi\)
\(12\) 1.48383e40 0.503600
\(13\) −8.54396e41 −1.44140 −0.720699 − 0.693249i \(-0.756179\pi\)
−0.720699 + 0.693249i \(0.756179\pi\)
\(14\) −1.23939e43 −1.29840
\(15\) 8.05398e43 0.634729
\(16\) 3.56812e44 0.250000
\(17\) −1.96919e46 −1.42054 −0.710269 − 0.703931i \(-0.751427\pi\)
−0.710269 + 0.703931i \(0.751427\pi\)
\(18\) 1.20829e45 0.0102200
\(19\) 1.45015e47 0.161492 0.0807462 − 0.996735i \(-0.474270\pi\)
0.0807462 + 0.996735i \(0.474270\pi\)
\(20\) 1.93672e48 0.315096
\(21\) −7.08374e49 −1.84944
\(22\) 1.78393e50 0.813819
\(23\) −9.62630e50 −0.829212 −0.414606 − 0.910001i \(-0.636081\pi\)
−0.414606 + 0.910001i \(0.636081\pi\)
\(24\) 2.03935e51 0.356099
\(25\) −1.59575e52 −0.602859
\(26\) −1.17427e53 −1.01922
\(27\) −4.70906e53 −0.992643
\(28\) −1.70341e54 −0.918107
\(29\) 6.83539e54 0.988185 0.494092 − 0.869409i \(-0.335500\pi\)
0.494092 + 0.869409i \(0.335500\pi\)
\(30\) 1.10693e55 0.448821
\(31\) −2.27798e55 −0.270076 −0.135038 − 0.990840i \(-0.543116\pi\)
−0.135038 + 0.990840i \(0.543116\pi\)
\(32\) 4.90399e55 0.176777
\(33\) 1.01960e57 1.15920
\(34\) −2.70643e57 −1.00447
\(35\) −9.24586e57 −1.15717
\(36\) 1.66066e56 0.00722663
\(37\) −6.31677e58 −0.983853 −0.491927 − 0.870637i \(-0.663707\pi\)
−0.491927 + 0.870637i \(0.663707\pi\)
\(38\) 1.99307e58 0.114192
\(39\) −6.71154e59 −1.45178
\(40\) 2.66181e59 0.222806
\(41\) 3.47026e60 1.15071 0.575357 − 0.817903i \(-0.304863\pi\)
0.575357 + 0.817903i \(0.304863\pi\)
\(42\) −9.73582e60 −1.30775
\(43\) −1.32858e61 −0.738450 −0.369225 − 0.929340i \(-0.620377\pi\)
−0.369225 + 0.929340i \(0.620377\pi\)
\(44\) 2.45182e61 0.575457
\(45\) 9.01379e59 0.00910832
\(46\) −1.32303e62 −0.586341
\(47\) 4.14273e62 0.819628 0.409814 − 0.912169i \(-0.365594\pi\)
0.409814 + 0.912169i \(0.365594\pi\)
\(48\) 2.80287e62 0.251800
\(49\) 5.72018e63 2.37168
\(50\) −2.19319e63 −0.426286
\(51\) −1.54686e64 −1.43077
\(52\) −1.61391e64 −0.720699
\(53\) 1.50768e64 0.329582 0.164791 − 0.986328i \(-0.447305\pi\)
0.164791 + 0.986328i \(0.447305\pi\)
\(54\) −6.47209e64 −0.701905
\(55\) 1.33081e65 0.725296
\(56\) −2.34115e65 −0.649200
\(57\) 1.13913e65 0.162655
\(58\) 9.39448e65 0.698752
\(59\) −1.88385e66 −0.738066 −0.369033 − 0.929416i \(-0.620311\pi\)
−0.369033 + 0.929416i \(0.620311\pi\)
\(60\) 1.52135e66 0.317365
\(61\) 2.68011e66 0.300804 0.150402 − 0.988625i \(-0.451943\pi\)
0.150402 + 0.988625i \(0.451943\pi\)
\(62\) −3.13083e66 −0.190973
\(63\) −7.92793e65 −0.0265393
\(64\) 6.73999e66 0.125000
\(65\) −8.76006e67 −0.908356
\(66\) 1.40133e68 0.819679
\(67\) −3.70871e68 −1.23429 −0.617146 − 0.786849i \(-0.711711\pi\)
−0.617146 + 0.786849i \(0.711711\pi\)
\(68\) −3.71969e68 −0.710269
\(69\) −7.56175e68 −0.835182
\(70\) −1.27074e69 −0.818240
\(71\) 2.25253e69 0.852083 0.426041 − 0.904704i \(-0.359908\pi\)
0.426041 + 0.904704i \(0.359908\pi\)
\(72\) 2.28239e67 0.00511000
\(73\) −2.96144e69 −0.395271 −0.197636 − 0.980276i \(-0.563326\pi\)
−0.197636 + 0.980276i \(0.563326\pi\)
\(74\) −8.68170e69 −0.695689
\(75\) −1.25351e70 −0.607200
\(76\) 2.73925e69 0.0807462
\(77\) −1.17049e71 −2.11333
\(78\) −9.22428e70 −1.02656
\(79\) −2.41694e71 −1.66821 −0.834103 − 0.551608i \(-0.814014\pi\)
−0.834103 + 0.551608i \(0.814014\pi\)
\(80\) 3.65836e70 0.157548
\(81\) −3.75259e71 −1.01424
\(82\) 4.76949e71 0.813677
\(83\) 9.02951e70 0.0977768 0.0488884 − 0.998804i \(-0.484432\pi\)
0.0488884 + 0.998804i \(0.484432\pi\)
\(84\) −1.33808e72 −0.924718
\(85\) −2.01899e72 −0.895210
\(86\) −1.82599e72 −0.522163
\(87\) 5.36941e72 0.995301
\(88\) 3.36975e72 0.406910
\(89\) −1.80794e73 −1.42909 −0.714545 − 0.699590i \(-0.753366\pi\)
−0.714545 + 0.699590i \(0.753366\pi\)
\(90\) 1.23885e71 0.00644055
\(91\) 7.70476e73 2.64671
\(92\) −1.81836e73 −0.414606
\(93\) −1.78942e73 −0.272021
\(94\) 5.69372e73 0.579565
\(95\) 1.48682e73 0.101771
\(96\) 3.85223e73 0.178050
\(97\) 5.17925e74 1.62303 0.811516 − 0.584331i \(-0.198643\pi\)
0.811516 + 0.584331i \(0.198643\pi\)
\(98\) 7.86175e74 1.67703
\(99\) 1.14111e73 0.0166345
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 2.76.a.b.1.3 ✓ 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.76.a.b.1.3 ✓ 3 1.1 even 1 trivial