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Properties

Label 1.76.a.a.1.3
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 1.3
Root \(-1.60397e9\) 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)
 
\(f(q)\) \(=\) \(q-1.25000e11 q^{2} +1.35148e17 q^{3} -2.21540e22 q^{4} +1.80927e26 q^{5} -1.68935e28 q^{6} +2.76229e31 q^{7} +7.49160e33 q^{8} -5.90002e35 q^{9} -2.26158e37 q^{10} +7.60456e38 q^{11} -2.99408e39 q^{12} +2.57727e41 q^{13} -3.45284e42 q^{14} +2.44520e43 q^{15} -9.94899e43 q^{16} -1.39330e46 q^{17} +7.37500e46 q^{18} -1.16447e48 q^{19} -4.00827e48 q^{20} +3.73318e48 q^{21} -9.50567e49 q^{22} +9.81970e50 q^{23} +1.01248e51 q^{24} +6.26484e51 q^{25} -3.22158e52 q^{26} -1.61944e53 q^{27} -6.11958e53 q^{28} +3.47815e54 q^{29} -3.05649e54 q^{30} +1.19579e56 q^{31} -2.70588e56 q^{32} +1.02774e56 q^{33} +1.74162e57 q^{34} +4.99772e57 q^{35} +1.30709e58 q^{36} +1.03750e59 q^{37} +1.45558e59 q^{38} +3.48314e58 q^{39} +1.35543e60 q^{40} +4.07748e60 q^{41} -4.66646e59 q^{42} +2.20097e61 q^{43} -1.68472e61 q^{44} -1.06747e62 q^{45} -1.22746e62 q^{46} -4.37785e62 q^{47} -1.34459e61 q^{48} -1.64884e63 q^{49} -7.83102e62 q^{50} -1.88302e63 q^{51} -5.70970e63 q^{52} +7.68796e64 q^{53} +2.02429e64 q^{54} +1.37587e65 q^{55} +2.06939e65 q^{56} -1.57376e65 q^{57} -4.34767e65 q^{58} -3.79588e66 q^{59} -5.41711e65 q^{60} +5.02447e66 q^{61} -1.49473e67 q^{62} -1.62975e67 q^{63} +3.75820e67 q^{64} +4.66298e67 q^{65} -1.28468e67 q^{66} +3.39062e68 q^{67} +3.08672e68 q^{68} +1.32712e68 q^{69} -6.24713e68 q^{70} -3.17915e69 q^{71} -4.42005e69 q^{72} +7.11742e69 q^{73} -1.29687e70 q^{74} +8.46682e68 q^{75} +2.57978e70 q^{76} +2.10060e70 q^{77} -4.35391e69 q^{78} +9.47303e70 q^{79} -1.80004e70 q^{80} +3.36992e71 q^{81} -5.09684e71 q^{82} -1.43323e71 q^{83} -8.27051e70 q^{84} -2.52086e72 q^{85} -2.75120e72 q^{86} +4.70066e71 q^{87} +5.69703e72 q^{88} +9.94475e72 q^{89} +1.33434e73 q^{90} +7.11916e72 q^{91} -2.17546e73 q^{92} +1.61609e73 q^{93} +5.47229e73 q^{94} -2.10685e74 q^{95} -3.65695e73 q^{96} -7.42743e73 q^{97} +2.06105e74 q^{98} -4.48671e74 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 57080822040 q^{2} - 78\!\cdots\!40 q^{3} + 17\!\cdots\!28 q^{4} - 38\!\cdots\!40 q^{5} + 31\!\cdots\!92 q^{6} + 19\!\cdots\!00 q^{7} + 44\!\cdots\!20 q^{8} + 21\!\cdots\!82 q^{9} + 13\!\cdots\!60 q^{10}+ \cdots - 18\!\cdots\!36 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.25000e11 −0.643108 −0.321554 − 0.946891i \(-0.604205\pi\)
−0.321554 + 0.946891i \(0.604205\pi\)
\(3\) 1.35148e17 0.173286 0.0866430 − 0.996239i \(-0.472386\pi\)
0.0866430 + 0.996239i \(0.472386\pi\)
\(4\) −2.21540e22 −0.586413
\(5\) 1.80927e26 1.11206 0.556030 − 0.831162i \(-0.312324\pi\)
0.556030 + 0.831162i \(0.312324\pi\)
\(6\) −1.68935e28 −0.111442
\(7\) 2.76229e31 0.562461 0.281230 − 0.959640i \(-0.409258\pi\)
0.281230 + 0.959640i \(0.409258\pi\)
\(8\) 7.49160e33 1.02023
\(9\) −5.90002e35 −0.969972
\(10\) −2.26158e37 −0.715175
\(11\) 7.60456e38 0.674293 0.337146 − 0.941452i \(-0.390538\pi\)
0.337146 + 0.941452i \(0.390538\pi\)
\(12\) −2.99408e39 −0.101617
\(13\) 2.57727e41 0.434795 0.217397 − 0.976083i \(-0.430243\pi\)
0.217397 + 0.976083i \(0.430243\pi\)
\(14\) −3.45284e42 −0.361723
\(15\) 2.44520e43 0.192705
\(16\) −9.94899e43 −0.0697076
\(17\) −1.39330e46 −1.00510 −0.502551 − 0.864548i \(-0.667605\pi\)
−0.502551 + 0.864548i \(0.667605\pi\)
\(18\) 7.37500e46 0.623796
\(19\) −1.16447e48 −1.29679 −0.648394 − 0.761305i \(-0.724559\pi\)
−0.648394 + 0.761305i \(0.724559\pi\)
\(20\) −4.00827e48 −0.652126
\(21\) 3.73318e48 0.0974666
\(22\) −9.50567e49 −0.433643
\(23\) 9.81970e50 0.845872 0.422936 − 0.906160i \(-0.361000\pi\)
0.422936 + 0.906160i \(0.361000\pi\)
\(24\) 1.01248e51 0.176792
\(25\) 6.26484e51 0.236679
\(26\) −3.22158e52 −0.279620
\(27\) −1.61944e53 −0.341369
\(28\) −6.11958e53 −0.329834
\(29\) 3.47815e54 0.502832 0.251416 − 0.967879i \(-0.419104\pi\)
0.251416 + 0.967879i \(0.419104\pi\)
\(30\) −3.05649e54 −0.123930
\(31\) 1.19579e56 1.41773 0.708863 − 0.705346i \(-0.249209\pi\)
0.708863 + 0.705346i \(0.249209\pi\)
\(32\) −2.70588e56 −0.975405
\(33\) 1.02774e56 0.116846
\(34\) 1.74162e57 0.646388
\(35\) 4.99772e57 0.625490
\(36\) 1.30709e58 0.568804
\(37\) 1.03750e59 1.61593 0.807963 − 0.589233i \(-0.200570\pi\)
0.807963 + 0.589233i \(0.200570\pi\)
\(38\) 1.45558e59 0.833974
\(39\) 3.48314e58 0.0753439
\(40\) 1.35543e60 1.13456
\(41\) 4.07748e60 1.35207 0.676033 − 0.736872i \(-0.263698\pi\)
0.676033 + 0.736872i \(0.263698\pi\)
\(42\) −4.66646e59 −0.0626815
\(43\) 2.20097e61 1.22334 0.611670 − 0.791113i \(-0.290498\pi\)
0.611670 + 0.791113i \(0.290498\pi\)
\(44\) −1.68472e61 −0.395414
\(45\) −1.06747e62 −1.07867
\(46\) −1.22746e62 −0.543987
\(47\) −4.37785e62 −0.866146 −0.433073 − 0.901359i \(-0.642571\pi\)
−0.433073 + 0.901359i \(0.642571\pi\)
\(48\) −1.34459e61 −0.0120793
\(49\) −1.64884e63 −0.683638
\(50\) −7.83102e62 −0.152210
\(51\) −1.88302e63 −0.174170
\(52\) −5.70970e63 −0.254969
\(53\) 7.68796e64 1.68061 0.840303 − 0.542117i \(-0.182377\pi\)
0.840303 + 0.542117i \(0.182377\pi\)
\(54\) 2.02429e64 0.219537
\(55\) 1.37587e65 0.749855
\(56\) 2.06939e65 0.573841
\(57\) −1.57376e65 −0.224715
\(58\) −4.34767e65 −0.323375
\(59\) −3.79588e66 −1.48717 −0.743587 − 0.668639i \(-0.766877\pi\)
−0.743587 + 0.668639i \(0.766877\pi\)
\(60\) −5.41711e65 −0.113004
\(61\) 5.02447e66 0.563925 0.281962 − 0.959425i \(-0.409015\pi\)
0.281962 + 0.959425i \(0.409015\pi\)
\(62\) −1.49473e67 −0.911750
\(63\) −1.62975e67 −0.545571
\(64\) 3.75820e67 0.696998
\(65\) 4.66298e67 0.483518
\(66\) −1.28468e67 −0.0751443
\(67\) 3.39062e68 1.12843 0.564214 − 0.825629i \(-0.309179\pi\)
0.564214 + 0.825629i \(0.309179\pi\)
\(68\) 3.08672e68 0.589404
\(69\) 1.32712e68 0.146578
\(70\) −6.24713e68 −0.402258
\(71\) −3.17915e69 −1.20261 −0.601303 − 0.799021i \(-0.705352\pi\)
−0.601303 + 0.799021i \(0.705352\pi\)
\(72\) −4.42005e69 −0.989598
\(73\) 7.11742e69 0.949982 0.474991 − 0.879991i \(-0.342451\pi\)
0.474991 + 0.879991i \(0.342451\pi\)
\(74\) −1.29687e70 −1.03921
\(75\) 8.46682e68 0.0410131
\(76\) 2.57978e70 0.760453
\(77\) 2.10060e70 0.379263
\(78\) −4.35391e69 −0.0484542
\(79\) 9.47303e70 0.653842 0.326921 − 0.945052i \(-0.393989\pi\)
0.326921 + 0.945052i \(0.393989\pi\)
\(80\) −1.80004e70 −0.0775190
\(81\) 3.36992e71 0.910818
\(82\) −5.09684e71 −0.869523
\(83\) −1.43323e71 −0.155198 −0.0775992 − 0.996985i \(-0.524725\pi\)
−0.0775992 + 0.996985i \(0.524725\pi\)
\(84\) −8.27051e70 −0.0571556
\(85\) −2.52086e72 −1.11773
\(86\) −2.75120e72 −0.786739
\(87\) 4.70066e71 0.0871338
\(88\) 5.69703e72 0.687937
\(89\) 9.94475e72 0.786083 0.393042 − 0.919521i \(-0.371423\pi\)
0.393042 + 0.919521i \(0.371423\pi\)
\(90\) 1.33434e73 0.693699
\(91\) 7.11916e72 0.244555
\(92\) −2.17546e73 −0.496030
\(93\) 1.61609e73 0.245672
\(94\) 5.47229e73 0.557025
\(95\) −2.10685e74 −1.44211
\(96\) −3.65695e73 −0.169024
\(97\) −7.42743e73 −0.232755 −0.116377 − 0.993205i \(-0.537128\pi\)
−0.116377 + 0.993205i \(0.537128\pi\)
\(98\) 2.06105e74 0.439653
\(99\) −4.48671e74 −0.654045
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.1.3 ✓ 6
3.2 odd 2 9.76.a.c.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.76.a.a.1.3 ✓ 6 1.1 even 1 trivial
9.76.a.c.1.4 6 3.2 odd 2