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Properties

Label 2.76.a.b.1.1
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 1.1
Root \(-9.47292e12\) 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)
 
\(f(q)\) \(=\) \(q+1.37439e11 q^{2} -8.29405e17 q^{3} +1.88895e22 q^{4} -2.22623e26 q^{5} -1.13993e29 q^{6} -1.55415e31 q^{7} +2.59615e33 q^{8} +7.96464e34 q^{9} -3.05970e37 q^{10} +1.87805e39 q^{11} -1.56670e40 q^{12} +9.98468e41 q^{13} -2.13601e42 q^{14} +1.84644e44 q^{15} +3.56812e44 q^{16} +4.07379e44 q^{17} +1.09465e46 q^{18} -1.17368e48 q^{19} -4.20522e48 q^{20} +1.28902e49 q^{21} +2.58118e50 q^{22} +6.36387e50 q^{23} -2.15326e51 q^{24} +2.30910e52 q^{25} +1.37228e53 q^{26} +4.38441e53 q^{27} -2.93571e53 q^{28} -6.06178e54 q^{29} +2.53773e55 q^{30} -8.48184e55 q^{31} +4.90399e55 q^{32} -1.55767e57 q^{33} +5.59898e55 q^{34} +3.45990e57 q^{35} +1.50448e57 q^{36} +4.74204e58 q^{37} -1.61309e59 q^{38} -8.28135e59 q^{39} -5.77961e59 q^{40} -1.53555e60 q^{41} +1.77162e60 q^{42} -7.65577e60 q^{43} +3.54754e61 q^{44} -1.77311e61 q^{45} +8.74644e61 q^{46} +5.36288e62 q^{47} -2.95942e62 q^{48} -2.17033e63 q^{49} +3.17361e63 q^{50} -3.37882e62 q^{51} +1.88605e64 q^{52} -3.23526e64 q^{53} +6.02588e64 q^{54} -4.18097e65 q^{55} -4.03481e64 q^{56} +9.73456e65 q^{57} -8.33125e65 q^{58} -1.88381e66 q^{59} +3.48783e66 q^{60} +1.66500e67 q^{61} -1.16574e67 q^{62} -1.23783e66 q^{63} +6.73999e66 q^{64} -2.22282e68 q^{65} -2.14084e68 q^{66} +4.39869e68 q^{67} +7.69517e66 q^{68} -5.27823e68 q^{69} +4.75524e68 q^{70} -4.24492e69 q^{71} +2.06774e68 q^{72} -8.11635e69 q^{73} +6.51741e69 q^{74} -1.91518e70 q^{75} -2.21702e70 q^{76} -2.91878e70 q^{77} -1.13818e71 q^{78} -1.61362e71 q^{79} -7.94344e70 q^{80} -4.12091e71 q^{81} -2.11045e71 q^{82} -6.54865e71 q^{83} +2.43490e71 q^{84} -9.06918e70 q^{85} -1.05220e72 q^{86} +5.02767e72 q^{87} +4.87571e72 q^{88} -1.38399e73 q^{89} -2.43694e72 q^{90} -1.55177e73 q^{91} +1.20210e73 q^{92} +7.03488e73 q^{93} +7.37069e73 q^{94} +2.61288e74 q^{95} -4.06739e73 q^{96} +5.00488e74 q^{97} -2.98287e74 q^{98} +1.49580e74 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 412316860416 q^{2} + 45\!\cdots\!16 q^{3} + 56\!\cdots\!52 q^{4} - 19\!\cdots\!90 q^{5} + 62\!\cdots\!52 q^{6} - 37\!\cdots\!52 q^{7} + 77\!\cdots\!44 q^{8} - 26\!\cdots\!69 q^{9} - 26\!\cdots\!80 q^{10}+ \cdots + 90\!\cdots\!32 q^{99}+O(q^{100}) \) 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
\(3\) −8.29405e17 −1.06346 −0.531728 − 0.846915i \(-0.678457\pi\)
−0.531728 + 0.846915i \(0.678457\pi\)
\(4\) 1.88895e22 0.500000
\(5\) −2.22623e26 −1.36834 −0.684170 − 0.729322i \(-0.739835\pi\)
−0.684170 + 0.729322i \(0.739835\pi\)
\(6\) −1.13993e29 −0.751977
\(7\) −1.55415e31 −0.316459 −0.158229 − 0.987402i \(-0.550579\pi\)
−0.158229 + 0.987402i \(0.550579\pi\)
\(8\) 2.59615e33 0.353553
\(9\) 7.96464e34 0.130940
\(10\) −3.05970e37 −0.967563
\(11\) 1.87805e39 1.66526 0.832631 − 0.553829i \(-0.186834\pi\)
0.832631 + 0.553829i \(0.186834\pi\)
\(12\) −1.56670e40 −0.531728
\(13\) 9.98468e41 1.68445 0.842226 − 0.539125i \(-0.181245\pi\)
0.842226 + 0.539125i \(0.181245\pi\)
\(14\) −2.13601e42 −0.223770
\(15\) 1.84644e44 1.45517
\(16\) 3.56812e44 0.250000
\(17\) 4.07379e44 0.0293876 0.0146938 − 0.999892i \(-0.495323\pi\)
0.0146938 + 0.999892i \(0.495323\pi\)
\(18\) 1.09465e46 0.0925886
\(19\) −1.17368e48 −1.30704 −0.653521 − 0.756908i \(-0.726709\pi\)
−0.653521 + 0.756908i \(0.726709\pi\)
\(20\) −4.20522e48 −0.684170
\(21\) 1.28902e49 0.336540
\(22\) 2.58118e50 1.17752
\(23\) 6.36387e50 0.548185 0.274093 − 0.961703i \(-0.411622\pi\)
0.274093 + 0.961703i \(0.411622\pi\)
\(24\) −2.15326e51 −0.375989
\(25\) 2.30910e52 0.872355
\(26\) 1.37228e53 1.19109
\(27\) 4.38441e53 0.924208
\(28\) −2.93571e53 −0.158229
\(29\) −6.06178e54 −0.876345 −0.438173 − 0.898891i \(-0.644374\pi\)
−0.438173 + 0.898891i \(0.644374\pi\)
\(30\) 2.53773e55 1.02896
\(31\) −8.48184e55 −1.00560 −0.502802 − 0.864402i \(-0.667697\pi\)
−0.502802 + 0.864402i \(0.667697\pi\)
\(32\) 4.90399e55 0.176777
\(33\) −1.55767e57 −1.77093
\(34\) 5.59898e55 0.0207802
\(35\) 3.45990e57 0.433023
\(36\) 1.50448e57 0.0654700
\(37\) 4.74204e58 0.738585 0.369293 − 0.929313i \(-0.379600\pi\)
0.369293 + 0.929313i \(0.379600\pi\)
\(38\) −1.61309e59 −0.924218
\(39\) −8.28135e59 −1.79134
\(40\) −5.77961e59 −0.483781
\(41\) −1.53555e60 −0.509179 −0.254590 − 0.967049i \(-0.581940\pi\)
−0.254590 + 0.967049i \(0.581940\pi\)
\(42\) 1.77162e60 0.237970
\(43\) −7.65577e60 −0.425522 −0.212761 − 0.977104i \(-0.568246\pi\)
−0.212761 + 0.977104i \(0.568246\pi\)
\(44\) 3.54754e61 0.832631
\(45\) −1.77311e61 −0.179170
\(46\) 8.74644e61 0.387626
\(47\) 5.36288e62 1.06103 0.530516 − 0.847675i \(-0.321998\pi\)
0.530516 + 0.847675i \(0.321998\pi\)
\(48\) −2.95942e62 −0.265864
\(49\) −2.17033e63 −0.899854
\(50\) 3.17361e63 0.616848
\(51\) −3.37882e62 −0.0312524
\(52\) 1.88605e64 0.842226
\(53\) −3.23526e64 −0.707236 −0.353618 − 0.935390i \(-0.615049\pi\)
−0.353618 + 0.935390i \(0.615049\pi\)
\(54\) 6.02588e64 0.653513
\(55\) −4.18097e65 −2.27864
\(56\) −4.03481e64 −0.111885
\(57\) 9.73456e65 1.38998
\(58\) −8.33125e65 −0.619670
\(59\) −1.88381e66 −0.738052 −0.369026 − 0.929419i \(-0.620309\pi\)
−0.369026 + 0.929419i \(0.620309\pi\)
\(60\) 3.48783e66 0.727585
\(61\) 1.66500e67 1.86873 0.934363 − 0.356322i \(-0.115969\pi\)
0.934363 + 0.356322i \(0.115969\pi\)
\(62\) −1.16574e67 −0.711069
\(63\) −1.23783e66 −0.0414371
\(64\) 6.73999e66 0.125000
\(65\) −2.22282e68 −2.30490
\(66\) −2.14084e68 −1.25224
\(67\) 4.39869e68 1.46392 0.731962 − 0.681345i \(-0.238605\pi\)
0.731962 + 0.681345i \(0.238605\pi\)
\(68\) 7.69517e66 0.0146938
\(69\) −5.27823e68 −0.582971
\(70\) 4.75524e68 0.306194
\(71\) −4.24492e69 −1.60576 −0.802881 − 0.596139i \(-0.796701\pi\)
−0.802881 + 0.596139i \(0.796701\pi\)
\(72\) 2.06774e68 0.0462943
\(73\) −8.11635e69 −1.08331 −0.541656 − 0.840600i \(-0.682203\pi\)
−0.541656 + 0.840600i \(0.682203\pi\)
\(74\) 6.51741e69 0.522259
\(75\) −1.91518e70 −0.927712
\(76\) −2.21702e70 −0.653521
\(77\) −2.91878e70 −0.526987
\(78\) −1.13818e71 −1.26667
\(79\) −1.61362e71 −1.11375 −0.556873 − 0.830598i \(-0.687999\pi\)
−0.556873 + 0.830598i \(0.687999\pi\)
\(80\) −7.94344e70 −0.342085
\(81\) −4.12091e71 −1.11379
\(82\) −2.11045e71 −0.360044
\(83\) −6.54865e71 −0.709126 −0.354563 − 0.935032i \(-0.615370\pi\)
−0.354563 + 0.935032i \(0.615370\pi\)
\(84\) 2.43490e71 0.168270
\(85\) −9.06918e70 −0.0402122
\(86\) −1.05220e72 −0.300890
\(87\) 5.02767e72 0.931955
\(88\) 4.87571e72 0.588759
\(89\) −1.38399e73 −1.09398 −0.546988 − 0.837140i \(-0.684226\pi\)
−0.546988 + 0.837140i \(0.684226\pi\)
\(90\) −2.43694e72 −0.126693
\(91\) −1.55177e73 −0.533060
\(92\) 1.20210e73 0.274093
\(93\) 7.03488e73 1.06942
\(94\) 7.37069e73 0.750263
\(95\) 2.61288e74 1.78848
\(96\) −4.06739e73 −0.187994
\(97\) 5.00488e74 1.56839 0.784194 − 0.620515i \(-0.213076\pi\)
0.784194 + 0.620515i \(0.213076\pi\)
\(98\) −2.98287e74 −0.636293
\(99\) 1.49580e74 0.218049
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.1.1 ✓ 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.76.a.b.1.1 ✓ 3 1.1 even 1 trivial