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Properties

Label 1.76.a.a.1.1
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 1.1
Root \(-4.40594e9\) 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)
 
\(f(q)\) \(=\) \(q-3.26741e11 q^{2} -1.51453e18 q^{3} +6.89808e22 q^{4} -4.33451e25 q^{5} +4.94860e29 q^{6} +4.24638e31 q^{7} -1.01949e34 q^{8} +1.68554e36 q^{9} +1.41626e37 q^{10} +1.56603e38 q^{11} -1.04474e41 q^{12} -1.88327e41 q^{13} -1.38747e43 q^{14} +6.56475e43 q^{15} +7.25086e44 q^{16} +9.86838e45 q^{17} -5.50735e47 q^{18} -1.40944e47 q^{19} -2.98998e48 q^{20} -6.43128e49 q^{21} -5.11687e49 q^{22} +8.97012e50 q^{23} +1.54406e52 q^{24} -2.45910e52 q^{25} +6.15343e52 q^{26} -1.63156e54 q^{27} +2.92919e54 q^{28} +7.48449e54 q^{29} -2.14497e55 q^{30} +9.83848e55 q^{31} +1.48239e56 q^{32} -2.37180e56 q^{33} -3.22441e57 q^{34} -1.84060e57 q^{35} +1.16270e59 q^{36} -4.42368e58 q^{37} +4.60523e58 q^{38} +2.85228e59 q^{39} +4.41901e59 q^{40} +3.18294e60 q^{41} +2.10136e61 q^{42} -2.58944e61 q^{43} +1.08026e61 q^{44} -7.30598e61 q^{45} -2.93091e62 q^{46} -4.72886e61 q^{47} -1.09817e63 q^{48} -6.08688e62 q^{49} +8.03489e63 q^{50} -1.49460e64 q^{51} -1.29910e64 q^{52} -7.75824e64 q^{53} +5.33099e65 q^{54} -6.78797e63 q^{55} -4.32917e65 q^{56} +2.13465e65 q^{57} -2.44549e66 q^{58} -8.17255e64 q^{59} +4.52842e66 q^{60} -6.09863e66 q^{61} -3.21464e67 q^{62} +7.15745e67 q^{63} -7.58287e67 q^{64} +8.16306e66 q^{65} +7.74966e67 q^{66} +4.31126e68 q^{67} +6.80729e68 q^{68} -1.35855e69 q^{69} +6.01399e68 q^{70} +2.46110e69 q^{71} -1.71840e70 q^{72} -4.09634e69 q^{73} +1.44540e70 q^{74} +3.72438e70 q^{75} -9.72245e69 q^{76} +6.64997e69 q^{77} -9.31956e70 q^{78} +9.26377e69 q^{79} -3.14289e70 q^{80} +1.44580e72 q^{81} -1.04000e72 q^{82} -2.78206e69 q^{83} -4.43635e72 q^{84} -4.27746e71 q^{85} +8.46075e72 q^{86} -1.13355e73 q^{87} -1.59656e72 q^{88} +1.96684e73 q^{89} +2.38716e73 q^{90} -7.99710e72 q^{91} +6.18767e73 q^{92} -1.49007e74 q^{93} +1.54511e73 q^{94} +6.10924e72 q^{95} -2.24512e74 q^{96} +2.01886e74 q^{97} +1.98883e74 q^{98} +2.63961e74 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 57080822040 q^{2} - 78\!\cdots\!40 q^{3} + 17\!\cdots\!28 q^{4} - 38\!\cdots\!40 q^{5} + 31\!\cdots\!92 q^{6} + 19\!\cdots\!00 q^{7} + 44\!\cdots\!20 q^{8} + 21\!\cdots\!82 q^{9} + 13\!\cdots\!60 q^{10}+ \cdots - 18\!\cdots\!36 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.26741e11 −1.68104 −0.840522 − 0.541778i \(-0.817751\pi\)
−0.840522 + 0.541778i \(0.817751\pi\)
\(3\) −1.51453e18 −1.94192 −0.970960 − 0.239242i \(-0.923101\pi\)
−0.970960 + 0.239242i \(0.923101\pi\)
\(4\) 6.89808e22 1.82591
\(5\) −4.33451e25 −0.266419 −0.133209 − 0.991088i \(-0.542528\pi\)
−0.133209 + 0.991088i \(0.542528\pi\)
\(6\) 4.94860e29 3.26445
\(7\) 4.24638e31 0.864655 0.432327 − 0.901717i \(-0.357692\pi\)
0.432327 + 0.901717i \(0.357692\pi\)
\(8\) −1.01949e34 −1.38839
\(9\) 1.68554e36 2.77105
\(10\) 1.41626e37 0.447861
\(11\) 1.56603e38 0.138859 0.0694296 − 0.997587i \(-0.477882\pi\)
0.0694296 + 0.997587i \(0.477882\pi\)
\(12\) −1.04474e41 −3.54577
\(13\) −1.88327e41 −0.317715 −0.158857 − 0.987302i \(-0.550781\pi\)
−0.158857 + 0.987302i \(0.550781\pi\)
\(14\) −1.38747e43 −1.45352
\(15\) 6.56475e43 0.517364
\(16\) 7.25086e44 0.508031
\(17\) 9.86838e45 0.711888 0.355944 − 0.934507i \(-0.384159\pi\)
0.355944 + 0.934507i \(0.384159\pi\)
\(18\) −5.50735e47 −4.65826
\(19\) −1.40944e47 −0.156959 −0.0784797 − 0.996916i \(-0.525007\pi\)
−0.0784797 + 0.996916i \(0.525007\pi\)
\(20\) −2.98998e48 −0.486456
\(21\) −6.43128e49 −1.67909
\(22\) −5.11687e49 −0.233428
\(23\) 8.97012e50 0.772689 0.386344 − 0.922355i \(-0.373738\pi\)
0.386344 + 0.922355i \(0.373738\pi\)
\(24\) 1.54406e52 2.69614
\(25\) −2.45910e52 −0.929021
\(26\) 6.15343e52 0.534093
\(27\) −1.63156e54 −3.43924
\(28\) 2.92919e54 1.57878
\(29\) 7.48449e54 1.08202 0.541012 − 0.841015i \(-0.318041\pi\)
0.541012 + 0.841015i \(0.318041\pi\)
\(30\) −2.14497e55 −0.869711
\(31\) 9.83848e55 1.16645 0.583223 − 0.812312i \(-0.301791\pi\)
0.583223 + 0.812312i \(0.301791\pi\)
\(32\) 1.48239e56 0.534365
\(33\) −2.37180e56 −0.269653
\(34\) −3.22441e57 −1.19671
\(35\) −1.84060e57 −0.230360
\(36\) 1.16270e59 5.05969
\(37\) −4.42368e58 −0.689000 −0.344500 − 0.938786i \(-0.611952\pi\)
−0.344500 + 0.938786i \(0.611952\pi\)
\(38\) 4.60523e58 0.263856
\(39\) 2.85228e59 0.616977
\(40\) 4.41901e59 0.369892
\(41\) 3.18294e60 1.05544 0.527721 − 0.849418i \(-0.323047\pi\)
0.527721 + 0.849418i \(0.323047\pi\)
\(42\) 2.10136e61 2.82262
\(43\) −2.58944e61 −1.43926 −0.719628 − 0.694360i \(-0.755688\pi\)
−0.719628 + 0.694360i \(0.755688\pi\)
\(44\) 1.08026e61 0.253544
\(45\) −7.30598e61 −0.738260
\(46\) −2.93091e62 −1.29892
\(47\) −4.72886e61 −0.0935593 −0.0467797 − 0.998905i \(-0.514896\pi\)
−0.0467797 + 0.998905i \(0.514896\pi\)
\(48\) −1.09817e63 −0.986555
\(49\) −6.08688e62 −0.252372
\(50\) 8.03489e63 1.56173
\(51\) −1.49460e64 −1.38243
\(52\) −1.29910e64 −0.580118
\(53\) −7.75824e64 −1.69597 −0.847985 − 0.530020i \(-0.822184\pi\)
−0.847985 + 0.530020i \(0.822184\pi\)
\(54\) 5.33099e65 5.78152
\(55\) −6.78797e63 −0.0369947
\(56\) −4.32917e65 −1.20048
\(57\) 2.13465e65 0.304803
\(58\) −2.44549e66 −1.81893
\(59\) −8.17255e64 −0.0320190 −0.0160095 − 0.999872i \(-0.505096\pi\)
−0.0160095 + 0.999872i \(0.505096\pi\)
\(60\) 4.52842e66 0.944658
\(61\) −6.09863e66 −0.684484 −0.342242 − 0.939612i \(-0.611186\pi\)
−0.342242 + 0.939612i \(0.611186\pi\)
\(62\) −3.21464e67 −1.96085
\(63\) 7.15745e67 2.39600
\(64\) −7.58287e67 −1.40632
\(65\) 8.16306e66 0.0846451
\(66\) 7.74966e67 0.453299
\(67\) 4.31126e68 1.43483 0.717413 − 0.696648i \(-0.245326\pi\)
0.717413 + 0.696648i \(0.245326\pi\)
\(68\) 6.80729e68 1.29984
\(69\) −1.35855e69 −1.50050
\(70\) 6.01399e68 0.387245
\(71\) 2.46110e69 0.930981 0.465491 − 0.885053i \(-0.345878\pi\)
0.465491 + 0.885053i \(0.345878\pi\)
\(72\) −1.71840e70 −3.84729
\(73\) −4.09634e69 −0.546749 −0.273375 − 0.961908i \(-0.588140\pi\)
−0.273375 + 0.961908i \(0.588140\pi\)
\(74\) 1.44540e70 1.15824
\(75\) 3.72438e70 1.80408
\(76\) −9.72245e69 −0.286593
\(77\) 6.64997e69 0.120065
\(78\) −9.31956e70 −1.03716
\(79\) 9.26377e69 0.0639399 0.0319699 − 0.999489i \(-0.489822\pi\)
0.0319699 + 0.999489i \(0.489822\pi\)
\(80\) −3.14289e70 −0.135349
\(81\) 1.44580e72 3.90768
\(82\) −1.04000e72 −1.77424
\(83\) −2.78206e69 −0.00301257 −0.00150629 − 0.999999i \(-0.500479\pi\)
−0.00150629 + 0.999999i \(0.500479\pi\)
\(84\) −4.43635e72 −3.06586
\(85\) −4.27746e71 −0.189660
\(86\) 8.46075e72 2.41945
\(87\) −1.13355e73 −2.10121
\(88\) −1.59656e72 −0.192790
\(89\) 1.96684e73 1.55469 0.777346 − 0.629073i \(-0.216565\pi\)
0.777346 + 0.629073i \(0.216565\pi\)
\(90\) 2.38716e73 1.24105
\(91\) −7.99710e72 −0.274714
\(92\) 6.18767e73 1.41086
\(93\) −1.49007e74 −2.26514
\(94\) 1.54511e73 0.157277
\(95\) 6.10924e72 0.0418169
\(96\) −2.24512e74 −1.03769
\(97\) 2.01886e74 0.632653 0.316326 − 0.948650i \(-0.397551\pi\)
0.316326 + 0.948650i \(0.397551\pi\)
\(98\) 1.98883e74 0.424249
\(99\) 2.63961e74 0.384786
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.1.1 ✓ 6
3.2 odd 2 9.76.a.c.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.76.a.a.1.1 ✓ 6 1.1 even 1 trivial
9.76.a.c.1.6 6 3.2 odd 2