Properties

Label 1027.4.a.c.1.9
Level $1027$
Weight $4$
Character 1027.1
Self dual yes
Analytic conductor $60.595$
Analytic rank $1$
Dimension $56$
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] = [1027,4,Mod(1,1027)] 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("1027.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1027, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1027 = 13 \cdot 79 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1027.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [56,-16] 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: \(60.5949615759\)
Analytic rank: \(1\)
Dimension: \(56\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Character \(\chi\) \(=\) 1027.1

$q$-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)
 
\(f(q)\) \(=\) \(q-4.26566 q^{2} +0.165031 q^{3} +10.1958 q^{4} +4.40634 q^{5} -0.703966 q^{6} -4.29535 q^{7} -9.36672 q^{8} -26.9728 q^{9} -18.7959 q^{10} +38.5709 q^{11} +1.68263 q^{12} -13.0000 q^{13} +18.3225 q^{14} +0.727183 q^{15} -41.6115 q^{16} +83.3169 q^{17} +115.057 q^{18} -120.206 q^{19} +44.9263 q^{20} -0.708867 q^{21} -164.530 q^{22} +129.411 q^{23} -1.54580 q^{24} -105.584 q^{25} +55.4536 q^{26} -8.90719 q^{27} -43.7947 q^{28} -4.58956 q^{29} -3.10191 q^{30} +228.781 q^{31} +252.434 q^{32} +6.36540 q^{33} -355.402 q^{34} -18.9268 q^{35} -275.010 q^{36} -295.564 q^{37} +512.758 q^{38} -2.14540 q^{39} -41.2729 q^{40} +32.5763 q^{41} +3.02378 q^{42} -85.5639 q^{43} +393.263 q^{44} -118.851 q^{45} -552.022 q^{46} -296.605 q^{47} -6.86720 q^{48} -324.550 q^{49} +450.386 q^{50} +13.7499 q^{51} -132.546 q^{52} +143.171 q^{53} +37.9950 q^{54} +169.956 q^{55} +40.2334 q^{56} -19.8377 q^{57} +19.5775 q^{58} +208.102 q^{59} +7.41424 q^{60} +542.996 q^{61} -975.903 q^{62} +115.858 q^{63} -743.906 q^{64} -57.2824 q^{65} -27.1526 q^{66} -779.701 q^{67} +849.487 q^{68} +21.3568 q^{69} +80.7351 q^{70} -1022.63 q^{71} +252.646 q^{72} +850.537 q^{73} +1260.78 q^{74} -17.4247 q^{75} -1225.60 q^{76} -165.676 q^{77} +9.15156 q^{78} +79.0000 q^{79} -183.354 q^{80} +726.795 q^{81} -138.960 q^{82} +218.038 q^{83} -7.22750 q^{84} +367.122 q^{85} +364.986 q^{86} -0.757420 q^{87} -361.283 q^{88} +698.224 q^{89} +506.978 q^{90} +55.8396 q^{91} +1319.45 q^{92} +37.7560 q^{93} +1265.22 q^{94} -529.668 q^{95} +41.6595 q^{96} +1499.75 q^{97} +1384.42 q^{98} -1040.36 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q - 16 q^{2} - 17 q^{3} + 208 q^{4} - 35 q^{5} + 29 q^{6} - 37 q^{7} - 156 q^{8} + 415 q^{9} - 67 q^{10} - 114 q^{11} - 196 q^{12} - 728 q^{13} - 183 q^{14} - 17 q^{15} + 680 q^{16} - 383 q^{17} - 342 q^{18}+ \cdots - 11354 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.26566 −1.50814 −0.754069 − 0.656795i \(-0.771912\pi\)
−0.754069 + 0.656795i \(0.771912\pi\)
\(3\) 0.165031 0.0317603 0.0158801 − 0.999874i \(-0.494945\pi\)
0.0158801 + 0.999874i \(0.494945\pi\)
\(4\) 10.1958 1.27448
\(5\) 4.40634 0.394115 0.197057 − 0.980392i \(-0.436861\pi\)
0.197057 + 0.980392i \(0.436861\pi\)
\(6\) −0.703966 −0.0478989
\(7\) −4.29535 −0.231927 −0.115964 − 0.993253i \(-0.536996\pi\)
−0.115964 + 0.993253i \(0.536996\pi\)
\(8\) −9.36672 −0.413954
\(9\) −26.9728 −0.998991
\(10\) −18.7959 −0.594379
\(11\) 38.5709 1.05723 0.528616 − 0.848861i \(-0.322711\pi\)
0.528616 + 0.848861i \(0.322711\pi\)
\(12\) 1.68263 0.0404778
\(13\) −13.0000 −0.277350
\(14\) 18.3225 0.349778
\(15\) 0.727183 0.0125172
\(16\) −41.6115 −0.650180
\(17\) 83.3169 1.18867 0.594333 − 0.804219i \(-0.297416\pi\)
0.594333 + 0.804219i \(0.297416\pi\)
\(18\) 115.057 1.50662
\(19\) −120.206 −1.45143 −0.725714 − 0.687996i \(-0.758490\pi\)
−0.725714 + 0.687996i \(0.758490\pi\)
\(20\) 44.9263 0.502291
\(21\) −0.708867 −0.00736607
\(22\) −164.530 −1.59445
\(23\) 129.411 1.17322 0.586609 − 0.809870i \(-0.300462\pi\)
0.586609 + 0.809870i \(0.300462\pi\)
\(24\) −1.54580 −0.0131473
\(25\) −105.584 −0.844674
\(26\) 55.4536 0.418282
\(27\) −8.90719 −0.0634885
\(28\) −43.7947 −0.295587
\(29\) −4.58956 −0.0293882 −0.0146941 − 0.999892i \(-0.504677\pi\)
−0.0146941 + 0.999892i \(0.504677\pi\)
\(30\) −3.10191 −0.0188776
\(31\) 228.781 1.32549 0.662747 − 0.748843i \(-0.269390\pi\)
0.662747 + 0.748843i \(0.269390\pi\)
\(32\) 252.434 1.39452
\(33\) 6.36540 0.0335780
\(34\) −355.402 −1.79267
\(35\) −18.9268 −0.0914059
\(36\) −275.010 −1.27319
\(37\) −295.564 −1.31326 −0.656628 − 0.754215i \(-0.728018\pi\)
−0.656628 + 0.754215i \(0.728018\pi\)
\(38\) 512.758 2.18895
\(39\) −2.14540 −0.00880871
\(40\) −41.2729 −0.163146
\(41\) 32.5763 0.124087 0.0620435 − 0.998073i \(-0.480238\pi\)
0.0620435 + 0.998073i \(0.480238\pi\)
\(42\) 3.02378 0.0111090
\(43\) −85.5639 −0.303450 −0.151725 − 0.988423i \(-0.548483\pi\)
−0.151725 + 0.988423i \(0.548483\pi\)
\(44\) 393.263 1.34742
\(45\) −118.851 −0.393717
\(46\) −552.022 −1.76938
\(47\) −296.605 −0.920517 −0.460259 − 0.887785i \(-0.652243\pi\)
−0.460259 + 0.887785i \(0.652243\pi\)
\(48\) −6.86720 −0.0206499
\(49\) −324.550 −0.946210
\(50\) 450.386 1.27388
\(51\) 13.7499 0.0377523
\(52\) −132.546 −0.353477
\(53\) 143.171 0.371059 0.185529 − 0.982639i \(-0.440600\pi\)
0.185529 + 0.982639i \(0.440600\pi\)
\(54\) 37.9950 0.0957494
\(55\) 169.956 0.416671
\(56\) 40.2334 0.0960073
\(57\) −19.8377 −0.0460977
\(58\) 19.5775 0.0443215
\(59\) 208.102 0.459195 0.229598 − 0.973286i \(-0.426259\pi\)
0.229598 + 0.973286i \(0.426259\pi\)
\(60\) 7.41424 0.0159529
\(61\) 542.996 1.13973 0.569865 − 0.821739i \(-0.306996\pi\)
0.569865 + 0.821739i \(0.306996\pi\)
\(62\) −975.903 −1.99903
\(63\) 115.858 0.231693
\(64\) −743.906 −1.45294
\(65\) −57.2824 −0.109308
\(66\) −27.1526 −0.0506402
\(67\) −779.701 −1.42173 −0.710863 − 0.703331i \(-0.751695\pi\)
−0.710863 + 0.703331i \(0.751695\pi\)
\(68\) 849.487 1.51493
\(69\) 21.3568 0.0372617
\(70\) 80.7351 0.137853
\(71\) −1022.63 −1.70935 −0.854673 − 0.519166i \(-0.826242\pi\)
−0.854673 + 0.519166i \(0.826242\pi\)
\(72\) 252.646 0.413537
\(73\) 850.537 1.36367 0.681834 − 0.731507i \(-0.261183\pi\)
0.681834 + 0.731507i \(0.261183\pi\)
\(74\) 1260.78 1.98057
\(75\) −17.4247 −0.0268271
\(76\) −1225.60 −1.84982
\(77\) −165.676 −0.245201
\(78\) 9.15156 0.0132848
\(79\) 79.0000 0.112509
\(80\) −183.354 −0.256245
\(81\) 726.795 0.996975
\(82\) −138.960 −0.187140
\(83\) 218.038 0.288347 0.144173 − 0.989552i \(-0.453948\pi\)
0.144173 + 0.989552i \(0.453948\pi\)
\(84\) −7.22750 −0.00938791
\(85\) 367.122 0.468471
\(86\) 364.986 0.457645
\(87\) −0.757420 −0.000933378 0
\(88\) −361.283 −0.437646
\(89\) 698.224 0.831591 0.415796 − 0.909458i \(-0.363503\pi\)
0.415796 + 0.909458i \(0.363503\pi\)
\(90\) 506.978 0.593780
\(91\) 55.8396 0.0643250
\(92\) 1319.45 1.49524
\(93\) 37.7560 0.0420980
\(94\) 1265.22 1.38827
\(95\) −529.668 −0.572029
\(96\) 41.6595 0.0442902
\(97\) 1499.75 1.56986 0.784930 − 0.619585i \(-0.212699\pi\)
0.784930 + 0.619585i \(0.212699\pi\)
\(98\) 1384.42 1.42701
\(99\) −1040.36 −1.05617
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 1027.4.a.c.1.9 ✓ 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1027.4.a.c.1.9 ✓ 56 1.1 even 1 trivial