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Properties

Label 1.76.a.a.1.2
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 1.2
Root \(-3.89081e9\) 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)
 
\(f(q)\) \(=\) \(q-2.89652e11 q^{2} +1.03577e18 q^{3} +4.61192e22 q^{4} -2.47463e26 q^{5} -3.00013e29 q^{6} -6.83079e31 q^{7} -2.41577e33 q^{8} +4.64556e35 q^{9} +7.16781e37 q^{10} -1.33402e39 q^{11} +4.77689e40 q^{12} +2.92609e41 q^{13} +1.97855e43 q^{14} -2.56315e44 q^{15} -1.04260e45 q^{16} +4.46232e44 q^{17} -1.34559e47 q^{18} -8.32041e47 q^{19} -1.14128e49 q^{20} -7.07514e49 q^{21} +3.86400e50 q^{22} +1.13411e51 q^{23} -2.50218e51 q^{24} +3.47681e52 q^{25} -8.47548e52 q^{26} -1.48852e53 q^{27} -3.15030e54 q^{28} +3.24261e54 q^{29} +7.42421e55 q^{30} -1.48783e56 q^{31} +3.93257e56 q^{32} -1.38174e57 q^{33} -1.29252e56 q^{34} +1.69037e58 q^{35} +2.14249e58 q^{36} +2.52569e58 q^{37} +2.41002e59 q^{38} +3.03076e59 q^{39} +5.97813e59 q^{40} -1.57021e60 q^{41} +2.04933e61 q^{42} +2.24744e61 q^{43} -6.15237e61 q^{44} -1.14960e62 q^{45} -3.28497e62 q^{46} +3.99901e62 q^{47} -1.07990e63 q^{48} +2.25410e63 q^{49} -1.00706e64 q^{50} +4.62195e62 q^{51} +1.34949e64 q^{52} +2.56197e64 q^{53} +4.31152e64 q^{54} +3.30120e65 q^{55} +1.65016e65 q^{56} -8.61804e65 q^{57} -9.39227e65 q^{58} +1.98202e66 q^{59} -1.18210e67 q^{60} -1.77334e66 q^{61} +4.30953e67 q^{62} -3.17328e67 q^{63} -7.45191e67 q^{64} -7.24099e67 q^{65} +4.00222e68 q^{66} +3.79112e68 q^{67} +2.05799e67 q^{68} +1.17468e69 q^{69} -4.89618e69 q^{70} +1.15990e69 q^{71} -1.12226e69 q^{72} -5.75070e69 q^{73} -7.31570e69 q^{74} +3.60118e70 q^{75} -3.83730e70 q^{76} +9.11239e70 q^{77} -8.77866e70 q^{78} +6.59713e70 q^{79} +2.58005e71 q^{80} -4.36750e71 q^{81} +4.54813e71 q^{82} +1.04387e72 q^{83} -3.26300e72 q^{84} -1.10426e71 q^{85} -6.50974e72 q^{86} +3.35860e72 q^{87} +3.22267e72 q^{88} +1.34676e73 q^{89} +3.32984e73 q^{90} -1.99875e73 q^{91} +5.23043e73 q^{92} -1.54105e74 q^{93} -1.15832e74 q^{94} +2.05899e74 q^{95} +4.07324e74 q^{96} -5.17927e74 q^{97} -6.52905e74 q^{98} -6.19725e74 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 57080822040 q^{2} - 78\!\cdots\!40 q^{3} + 17\!\cdots\!28 q^{4} - 38\!\cdots\!40 q^{5} + 31\!\cdots\!92 q^{6} + 19\!\cdots\!00 q^{7} + 44\!\cdots\!20 q^{8} + 21\!\cdots\!82 q^{9} + 13\!\cdots\!60 q^{10}+ \cdots - 18\!\cdots\!36 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.89652e11 −1.49022 −0.745112 − 0.666940i \(-0.767604\pi\)
−0.745112 + 0.666940i \(0.767604\pi\)
\(3\) 1.03577e18 1.32806 0.664029 − 0.747707i \(-0.268845\pi\)
0.664029 + 0.747707i \(0.268845\pi\)
\(4\) 4.61192e22 1.22076
\(5\) −2.47463e26 −1.52102 −0.760510 − 0.649326i \(-0.775051\pi\)
−0.760510 + 0.649326i \(0.775051\pi\)
\(6\) −3.00013e29 −1.97910
\(7\) −6.83079e31 −1.39090 −0.695448 − 0.718577i \(-0.744794\pi\)
−0.695448 + 0.718577i \(0.744794\pi\)
\(8\) −2.41577e33 −0.328989
\(9\) 4.64556e35 0.763737
\(10\) 7.16781e37 2.26666
\(11\) −1.33402e39 −1.18287 −0.591433 − 0.806354i \(-0.701438\pi\)
−0.591433 + 0.806354i \(0.701438\pi\)
\(12\) 4.77689e40 1.62125
\(13\) 2.92609e41 0.493642 0.246821 − 0.969061i \(-0.420614\pi\)
0.246821 + 0.969061i \(0.420614\pi\)
\(14\) 1.97855e43 2.07274
\(15\) −2.56315e44 −2.02000
\(16\) −1.04260e45 −0.730498
\(17\) 4.46232e44 0.0321904 0.0160952 − 0.999870i \(-0.494877\pi\)
0.0160952 + 0.999870i \(0.494877\pi\)
\(18\) −1.34559e47 −1.13814
\(19\) −8.32041e47 −0.926583 −0.463292 − 0.886206i \(-0.653332\pi\)
−0.463292 + 0.886206i \(0.653332\pi\)
\(20\) −1.14128e49 −1.85681
\(21\) −7.07514e49 −1.84719
\(22\) 3.86400e50 1.76273
\(23\) 1.13411e51 0.976927 0.488463 − 0.872584i \(-0.337558\pi\)
0.488463 + 0.872584i \(0.337558\pi\)
\(24\) −2.50218e51 −0.436916
\(25\) 3.47681e52 1.31350
\(26\) −8.47548e52 −0.735637
\(27\) −1.48852e53 −0.313771
\(28\) −3.15030e54 −1.69796
\(29\) 3.24261e54 0.468781 0.234390 − 0.972143i \(-0.424691\pi\)
0.234390 + 0.972143i \(0.424691\pi\)
\(30\) 7.42421e55 3.01025
\(31\) −1.48783e56 −1.76397 −0.881983 − 0.471282i \(-0.843792\pi\)
−0.881983 + 0.471282i \(0.843792\pi\)
\(32\) 3.93257e56 1.41759
\(33\) −1.38174e57 −1.57091
\(34\) −1.29252e56 −0.0479709
\(35\) 1.69037e58 2.11558
\(36\) 2.14249e58 0.932343
\(37\) 2.52569e58 0.393383 0.196691 − 0.980465i \(-0.436980\pi\)
0.196691 + 0.980465i \(0.436980\pi\)
\(38\) 2.41002e59 1.38082
\(39\) 3.03076e59 0.655585
\(40\) 5.97813e59 0.500398
\(41\) −1.57021e60 −0.520669 −0.260335 − 0.965518i \(-0.583833\pi\)
−0.260335 + 0.965518i \(0.583833\pi\)
\(42\) 2.04933e61 2.75272
\(43\) 2.24744e61 1.24917 0.624584 − 0.780958i \(-0.285269\pi\)
0.624584 + 0.780958i \(0.285269\pi\)
\(44\) −6.15237e61 −1.44400
\(45\) −1.14960e62 −1.16166
\(46\) −3.28497e62 −1.45584
\(47\) 3.99901e62 0.791194 0.395597 − 0.918424i \(-0.370538\pi\)
0.395597 + 0.918424i \(0.370538\pi\)
\(48\) −1.07990e63 −0.970144
\(49\) 2.25410e63 0.934590
\(50\) −1.00706e64 −1.95741
\(51\) 4.62195e62 0.0427507
\(52\) 1.34949e64 0.602621
\(53\) 2.56197e64 0.560053 0.280027 − 0.959992i \(-0.409657\pi\)
0.280027 + 0.959992i \(0.409657\pi\)
\(54\) 4.31152e64 0.467589
\(55\) 3.30120e65 1.79916
\(56\) 1.65016e65 0.457589
\(57\) −8.61804e65 −1.23056
\(58\) −9.39227e65 −0.698588
\(59\) 1.98202e66 0.776528 0.388264 − 0.921548i \(-0.373075\pi\)
0.388264 + 0.921548i \(0.373075\pi\)
\(60\) −1.18210e67 −2.46595
\(61\) −1.77334e66 −0.199032 −0.0995159 − 0.995036i \(-0.531729\pi\)
−0.0995159 + 0.995036i \(0.531729\pi\)
\(62\) 4.30953e67 2.62870
\(63\) −3.17328e67 −1.06228
\(64\) −7.45191e67 −1.38203
\(65\) −7.24099e67 −0.750840
\(66\) 4.00222e68 2.34101
\(67\) 3.79112e68 1.26172 0.630860 − 0.775897i \(-0.282702\pi\)
0.630860 + 0.775897i \(0.282702\pi\)
\(68\) 2.05799e67 0.0392969
\(69\) 1.17468e69 1.29741
\(70\) −4.89618e69 −3.15269
\(71\) 1.15990e69 0.438765 0.219383 − 0.975639i \(-0.429596\pi\)
0.219383 + 0.975639i \(0.429596\pi\)
\(72\) −1.12226e69 −0.251261
\(73\) −5.75070e69 −0.767562 −0.383781 − 0.923424i \(-0.625378\pi\)
−0.383781 + 0.923424i \(0.625378\pi\)
\(74\) −7.31570e69 −0.586228
\(75\) 3.60118e70 1.74441
\(76\) −3.83730e70 −1.13114
\(77\) 9.11239e70 1.64524
\(78\) −8.77866e70 −0.976968
\(79\) 6.59713e70 0.455344 0.227672 − 0.973738i \(-0.426889\pi\)
0.227672 + 0.973738i \(0.426889\pi\)
\(80\) 2.58005e71 1.11110
\(81\) −4.36750e71 −1.18044
\(82\) 4.54813e71 0.775913
\(83\) 1.04387e72 1.13036 0.565182 − 0.824966i \(-0.308806\pi\)
0.565182 + 0.824966i \(0.308806\pi\)
\(84\) −3.26300e72 −2.25498
\(85\) −1.10426e71 −0.0489622
\(86\) −6.50974e72 −1.86154
\(87\) 3.35860e72 0.622567
\(88\) 3.22267e72 0.389149
\(89\) 1.34676e73 1.06455 0.532274 − 0.846572i \(-0.321338\pi\)
0.532274 + 0.846572i \(0.321338\pi\)
\(90\) 3.32984e73 1.73113
\(91\) −1.99875e73 −0.686605
\(92\) 5.23043e73 1.19260
\(93\) −1.54105e74 −2.34265
\(94\) −1.15832e74 −1.17906
\(95\) 2.05899e74 1.40935
\(96\) 4.07324e74 1.88265
\(97\) −5.17927e74 −1.62304 −0.811519 − 0.584326i \(-0.801359\pi\)
−0.811519 + 0.584326i \(0.801359\pi\)
\(98\) −6.52905e74 −1.39275
\(99\) −6.19725e74 −0.903398
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.1.2 ✓ 6
3.2 odd 2 9.76.a.c.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.76.a.a.1.2 ✓ 6 1.1 even 1 trivial
9.76.a.c.1.5 6 3.2 odd 2