Properties

Label 3750.2.a.g.1.1
Level $3750$
Weight $2$
Character 3750.1
Self dual yes
Analytic conductor $29.944$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3750,2,Mod(1,3750)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3750, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3750.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3750 = 2 \cdot 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3750.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(29.9439007580\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 150)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 3750.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -5.23607 q^{11} +1.00000 q^{12} +4.85410 q^{13} -2.00000 q^{14} +1.00000 q^{16} -7.85410 q^{17} +1.00000 q^{18} -2.76393 q^{19} -2.00000 q^{21} -5.23607 q^{22} -6.00000 q^{23} +1.00000 q^{24} +4.85410 q^{26} +1.00000 q^{27} -2.00000 q^{28} -1.38197 q^{29} +3.70820 q^{31} +1.00000 q^{32} -5.23607 q^{33} -7.85410 q^{34} +1.00000 q^{36} +2.14590 q^{37} -2.76393 q^{38} +4.85410 q^{39} -6.09017 q^{41} -2.00000 q^{42} +1.23607 q^{43} -5.23607 q^{44} -6.00000 q^{46} -4.76393 q^{47} +1.00000 q^{48} -3.00000 q^{49} -7.85410 q^{51} +4.85410 q^{52} -8.56231 q^{53} +1.00000 q^{54} -2.00000 q^{56} -2.76393 q^{57} -1.38197 q^{58} +8.94427 q^{59} -8.85410 q^{61} +3.70820 q^{62} -2.00000 q^{63} +1.00000 q^{64} -5.23607 q^{66} +9.70820 q^{67} -7.85410 q^{68} -6.00000 q^{69} -14.1803 q^{71} +1.00000 q^{72} +3.14590 q^{73} +2.14590 q^{74} -2.76393 q^{76} +10.4721 q^{77} +4.85410 q^{78} +1.00000 q^{81} -6.09017 q^{82} -6.00000 q^{83} -2.00000 q^{84} +1.23607 q^{86} -1.38197 q^{87} -5.23607 q^{88} +1.38197 q^{89} -9.70820 q^{91} -6.00000 q^{92} +3.70820 q^{93} -4.76393 q^{94} +1.00000 q^{96} +13.8541 q^{97} -3.00000 q^{98} -5.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} - 4 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} - 4 q^{7} + 2 q^{8} + 2 q^{9} - 6 q^{11} + 2 q^{12} + 3 q^{13} - 4 q^{14} + 2 q^{16} - 9 q^{17} + 2 q^{18} - 10 q^{19} - 4 q^{21} - 6 q^{22} - 12 q^{23} + 2 q^{24} + 3 q^{26} + 2 q^{27} - 4 q^{28} - 5 q^{29} - 6 q^{31} + 2 q^{32} - 6 q^{33} - 9 q^{34} + 2 q^{36} + 11 q^{37} - 10 q^{38} + 3 q^{39} - q^{41} - 4 q^{42} - 2 q^{43} - 6 q^{44} - 12 q^{46} - 14 q^{47} + 2 q^{48} - 6 q^{49} - 9 q^{51} + 3 q^{52} + 3 q^{53} + 2 q^{54} - 4 q^{56} - 10 q^{57} - 5 q^{58} - 11 q^{61} - 6 q^{62} - 4 q^{63} + 2 q^{64} - 6 q^{66} + 6 q^{67} - 9 q^{68} - 12 q^{69} - 6 q^{71} + 2 q^{72} + 13 q^{73} + 11 q^{74} - 10 q^{76} + 12 q^{77} + 3 q^{78} + 2 q^{81} - q^{82} - 12 q^{83} - 4 q^{84} - 2 q^{86} - 5 q^{87} - 6 q^{88} + 5 q^{89} - 6 q^{91} - 12 q^{92} - 6 q^{93} - 14 q^{94} + 2 q^{96} + 21 q^{97} - 6 q^{98} - 6 q^{99}+O(q^{100}) \) 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 \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.23607 −1.57873 −0.789367 0.613922i \(-0.789591\pi\)
−0.789367 + 0.613922i \(0.789591\pi\)
\(12\) 1.00000 0.288675
\(13\) 4.85410 1.34629 0.673143 0.739512i \(-0.264944\pi\)
0.673143 + 0.739512i \(0.264944\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −7.85410 −1.90490 −0.952450 0.304696i \(-0.901445\pi\)
−0.952450 + 0.304696i \(0.901445\pi\)
\(18\) 1.00000 0.235702
\(19\) −2.76393 −0.634089 −0.317045 0.948411i \(-0.602691\pi\)
−0.317045 + 0.948411i \(0.602691\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) −5.23607 −1.11633
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 4.85410 0.951968
\(27\) 1.00000 0.192450
\(28\) −2.00000 −0.377964
\(29\) −1.38197 −0.256625 −0.128312 0.991734i \(-0.540956\pi\)
−0.128312 + 0.991734i \(0.540956\pi\)
\(30\) 0 0
\(31\) 3.70820 0.666013 0.333007 0.942925i \(-0.391937\pi\)
0.333007 + 0.942925i \(0.391937\pi\)
\(32\) 1.00000 0.176777
\(33\) −5.23607 −0.911482
\(34\) −7.85410 −1.34697
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.14590 0.352783 0.176392 0.984320i \(-0.443557\pi\)
0.176392 + 0.984320i \(0.443557\pi\)
\(38\) −2.76393 −0.448369
\(39\) 4.85410 0.777278
\(40\) 0 0
\(41\) −6.09017 −0.951125 −0.475562 0.879682i \(-0.657755\pi\)
−0.475562 + 0.879682i \(0.657755\pi\)
\(42\) −2.00000 −0.308607
\(43\) 1.23607 0.188499 0.0942493 0.995549i \(-0.469955\pi\)
0.0942493 + 0.995549i \(0.469955\pi\)
\(44\) −5.23607 −0.789367
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) −4.76393 −0.694891 −0.347445 0.937700i \(-0.612951\pi\)
−0.347445 + 0.937700i \(0.612951\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −7.85410 −1.09979
\(52\) 4.85410 0.673143
\(53\) −8.56231 −1.17612 −0.588062 0.808816i \(-0.700109\pi\)
−0.588062 + 0.808816i \(0.700109\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) −2.76393 −0.366092
\(58\) −1.38197 −0.181461
\(59\) 8.94427 1.16445 0.582223 0.813029i \(-0.302183\pi\)
0.582223 + 0.813029i \(0.302183\pi\)
\(60\) 0 0
\(61\) −8.85410 −1.13365 −0.566826 0.823838i \(-0.691829\pi\)
−0.566826 + 0.823838i \(0.691829\pi\)
\(62\) 3.70820 0.470942
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −5.23607 −0.644515
\(67\) 9.70820 1.18605 0.593023 0.805186i \(-0.297934\pi\)
0.593023 + 0.805186i \(0.297934\pi\)
\(68\) −7.85410 −0.952450
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) −14.1803 −1.68290 −0.841448 0.540338i \(-0.818297\pi\)
−0.841448 + 0.540338i \(0.818297\pi\)
\(72\) 1.00000 0.117851
\(73\) 3.14590 0.368200 0.184100 0.982908i \(-0.441063\pi\)
0.184100 + 0.982908i \(0.441063\pi\)
\(74\) 2.14590 0.249456
\(75\) 0 0
\(76\) −2.76393 −0.317045
\(77\) 10.4721 1.19341
\(78\) 4.85410 0.549619
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −6.09017 −0.672547
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) 1.23607 0.133289
\(87\) −1.38197 −0.148162
\(88\) −5.23607 −0.558167
\(89\) 1.38197 0.146488 0.0732441 0.997314i \(-0.476665\pi\)
0.0732441 + 0.997314i \(0.476665\pi\)
\(90\) 0 0
\(91\) −9.70820 −1.01770
\(92\) −6.00000 −0.625543
\(93\) 3.70820 0.384523
\(94\) −4.76393 −0.491362
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 13.8541 1.40667 0.703335 0.710858i \(-0.251693\pi\)
0.703335 + 0.710858i \(0.251693\pi\)
\(98\) −3.00000 −0.303046
\(99\) −5.23607 −0.526245
\(100\) 0 0
\(101\) 1.67376 0.166546 0.0832728 0.996527i \(-0.473463\pi\)
0.0832728 + 0.996527i \(0.473463\pi\)
\(102\) −7.85410 −0.777672
\(103\) 2.29180 0.225817 0.112909 0.993605i \(-0.463983\pi\)
0.112909 + 0.993605i \(0.463983\pi\)
\(104\) 4.85410 0.475984
\(105\) 0 0
\(106\) −8.56231 −0.831645
\(107\) −10.9443 −1.05802 −0.529011 0.848615i \(-0.677437\pi\)
−0.529011 + 0.848615i \(0.677437\pi\)
\(108\) 1.00000 0.0962250
\(109\) 2.56231 0.245424 0.122712 0.992442i \(-0.460841\pi\)
0.122712 + 0.992442i \(0.460841\pi\)
\(110\) 0 0
\(111\) 2.14590 0.203680
\(112\) −2.00000 −0.188982
\(113\) 11.5623 1.08769 0.543845 0.839186i \(-0.316968\pi\)
0.543845 + 0.839186i \(0.316968\pi\)
\(114\) −2.76393 −0.258866
\(115\) 0 0
\(116\) −1.38197 −0.128312
\(117\) 4.85410 0.448762
\(118\) 8.94427 0.823387
\(119\) 15.7082 1.43997
\(120\) 0 0
\(121\) 16.4164 1.49240
\(122\) −8.85410 −0.801613
\(123\) −6.09017 −0.549132
\(124\) 3.70820 0.333007
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) −0.291796 −0.0258927 −0.0129464 0.999916i \(-0.504121\pi\)
−0.0129464 + 0.999916i \(0.504121\pi\)
\(128\) 1.00000 0.0883883
\(129\) 1.23607 0.108830
\(130\) 0 0
\(131\) −0.763932 −0.0667451 −0.0333725 0.999443i \(-0.510625\pi\)
−0.0333725 + 0.999443i \(0.510625\pi\)
\(132\) −5.23607 −0.455741
\(133\) 5.52786 0.479327
\(134\) 9.70820 0.838661
\(135\) 0 0
\(136\) −7.85410 −0.673484
\(137\) 15.0344 1.28448 0.642240 0.766504i \(-0.278005\pi\)
0.642240 + 0.766504i \(0.278005\pi\)
\(138\) −6.00000 −0.510754
\(139\) −13.4164 −1.13796 −0.568982 0.822350i \(-0.692663\pi\)
−0.568982 + 0.822350i \(0.692663\pi\)
\(140\) 0 0
\(141\) −4.76393 −0.401195
\(142\) −14.1803 −1.18999
\(143\) −25.4164 −2.12543
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 3.14590 0.260356
\(147\) −3.00000 −0.247436
\(148\) 2.14590 0.176392
\(149\) 7.03444 0.576284 0.288142 0.957588i \(-0.406962\pi\)
0.288142 + 0.957588i \(0.406962\pi\)
\(150\) 0 0
\(151\) −6.94427 −0.565117 −0.282558 0.959250i \(-0.591183\pi\)
−0.282558 + 0.959250i \(0.591183\pi\)
\(152\) −2.76393 −0.224184
\(153\) −7.85410 −0.634967
\(154\) 10.4721 0.843869
\(155\) 0 0
\(156\) 4.85410 0.388639
\(157\) −17.8541 −1.42491 −0.712456 0.701717i \(-0.752417\pi\)
−0.712456 + 0.701717i \(0.752417\pi\)
\(158\) 0 0
\(159\) −8.56231 −0.679035
\(160\) 0 0
\(161\) 12.0000 0.945732
\(162\) 1.00000 0.0785674
\(163\) −20.4721 −1.60350 −0.801751 0.597659i \(-0.796098\pi\)
−0.801751 + 0.597659i \(0.796098\pi\)
\(164\) −6.09017 −0.475562
\(165\) 0 0
\(166\) −6.00000 −0.465690
\(167\) −6.47214 −0.500829 −0.250414 0.968139i \(-0.580567\pi\)
−0.250414 + 0.968139i \(0.580567\pi\)
\(168\) −2.00000 −0.154303
\(169\) 10.5623 0.812485
\(170\) 0 0
\(171\) −2.76393 −0.211363
\(172\) 1.23607 0.0942493
\(173\) 10.9098 0.829459 0.414730 0.909945i \(-0.363876\pi\)
0.414730 + 0.909945i \(0.363876\pi\)
\(174\) −1.38197 −0.104767
\(175\) 0 0
\(176\) −5.23607 −0.394683
\(177\) 8.94427 0.672293
\(178\) 1.38197 0.103583
\(179\) −3.81966 −0.285495 −0.142747 0.989759i \(-0.545594\pi\)
−0.142747 + 0.989759i \(0.545594\pi\)
\(180\) 0 0
\(181\) −20.0344 −1.48915 −0.744574 0.667540i \(-0.767347\pi\)
−0.744574 + 0.667540i \(0.767347\pi\)
\(182\) −9.70820 −0.719620
\(183\) −8.85410 −0.654514
\(184\) −6.00000 −0.442326
\(185\) 0 0
\(186\) 3.70820 0.271899
\(187\) 41.1246 3.00733
\(188\) −4.76393 −0.347445
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) −0.763932 −0.0552762 −0.0276381 0.999618i \(-0.508799\pi\)
−0.0276381 + 0.999618i \(0.508799\pi\)
\(192\) 1.00000 0.0721688
\(193\) −2.38197 −0.171458 −0.0857288 0.996319i \(-0.527322\pi\)
−0.0857288 + 0.996319i \(0.527322\pi\)
\(194\) 13.8541 0.994667
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 13.3262 0.949455 0.474728 0.880133i \(-0.342547\pi\)
0.474728 + 0.880133i \(0.342547\pi\)
\(198\) −5.23607 −0.372111
\(199\) −6.18034 −0.438113 −0.219056 0.975712i \(-0.570298\pi\)
−0.219056 + 0.975712i \(0.570298\pi\)
\(200\) 0 0
\(201\) 9.70820 0.684764
\(202\) 1.67376 0.117765
\(203\) 2.76393 0.193990
\(204\) −7.85410 −0.549897
\(205\) 0 0
\(206\) 2.29180 0.159677
\(207\) −6.00000 −0.417029
\(208\) 4.85410 0.336571
\(209\) 14.4721 1.00106
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) −8.56231 −0.588062
\(213\) −14.1803 −0.971621
\(214\) −10.9443 −0.748135
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −7.41641 −0.503459
\(218\) 2.56231 0.173541
\(219\) 3.14590 0.212580
\(220\) 0 0
\(221\) −38.1246 −2.56454
\(222\) 2.14590 0.144023
\(223\) 2.29180 0.153470 0.0767350 0.997052i \(-0.475550\pi\)
0.0767350 + 0.997052i \(0.475550\pi\)
\(224\) −2.00000 −0.133631
\(225\) 0 0
\(226\) 11.5623 0.769113
\(227\) 16.9443 1.12463 0.562315 0.826923i \(-0.309911\pi\)
0.562315 + 0.826923i \(0.309911\pi\)
\(228\) −2.76393 −0.183046
\(229\) −24.2705 −1.60384 −0.801920 0.597431i \(-0.796188\pi\)
−0.801920 + 0.597431i \(0.796188\pi\)
\(230\) 0 0
\(231\) 10.4721 0.689016
\(232\) −1.38197 −0.0907305
\(233\) 22.0902 1.44718 0.723588 0.690233i \(-0.242492\pi\)
0.723588 + 0.690233i \(0.242492\pi\)
\(234\) 4.85410 0.317323
\(235\) 0 0
\(236\) 8.94427 0.582223
\(237\) 0 0
\(238\) 15.7082 1.01821
\(239\) 26.8328 1.73567 0.867835 0.496852i \(-0.165511\pi\)
0.867835 + 0.496852i \(0.165511\pi\)
\(240\) 0 0
\(241\) −13.8541 −0.892421 −0.446211 0.894928i \(-0.647227\pi\)
−0.446211 + 0.894928i \(0.647227\pi\)
\(242\) 16.4164 1.05529
\(243\) 1.00000 0.0641500
\(244\) −8.85410 −0.566826
\(245\) 0 0
\(246\) −6.09017 −0.388295
\(247\) −13.4164 −0.853666
\(248\) 3.70820 0.235471
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) −12.4721 −0.787234 −0.393617 0.919274i \(-0.628776\pi\)
−0.393617 + 0.919274i \(0.628776\pi\)
\(252\) −2.00000 −0.125988
\(253\) 31.4164 1.97513
\(254\) −0.291796 −0.0183089
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 11.6180 0.724713 0.362357 0.932040i \(-0.381972\pi\)
0.362357 + 0.932040i \(0.381972\pi\)
\(258\) 1.23607 0.0769542
\(259\) −4.29180 −0.266679
\(260\) 0 0
\(261\) −1.38197 −0.0855415
\(262\) −0.763932 −0.0471959
\(263\) −0.472136 −0.0291132 −0.0145566 0.999894i \(-0.504634\pi\)
−0.0145566 + 0.999894i \(0.504634\pi\)
\(264\) −5.23607 −0.322258
\(265\) 0 0
\(266\) 5.52786 0.338935
\(267\) 1.38197 0.0845749
\(268\) 9.70820 0.593023
\(269\) −1.90983 −0.116444 −0.0582222 0.998304i \(-0.518543\pi\)
−0.0582222 + 0.998304i \(0.518543\pi\)
\(270\) 0 0
\(271\) 28.1803 1.71183 0.855917 0.517113i \(-0.172993\pi\)
0.855917 + 0.517113i \(0.172993\pi\)
\(272\) −7.85410 −0.476225
\(273\) −9.70820 −0.587567
\(274\) 15.0344 0.908264
\(275\) 0 0
\(276\) −6.00000 −0.361158
\(277\) 22.1459 1.33062 0.665309 0.746568i \(-0.268300\pi\)
0.665309 + 0.746568i \(0.268300\pi\)
\(278\) −13.4164 −0.804663
\(279\) 3.70820 0.222004
\(280\) 0 0
\(281\) −6.09017 −0.363309 −0.181655 0.983362i \(-0.558145\pi\)
−0.181655 + 0.983362i \(0.558145\pi\)
\(282\) −4.76393 −0.283688
\(283\) −3.23607 −0.192364 −0.0961821 0.995364i \(-0.530663\pi\)
−0.0961821 + 0.995364i \(0.530663\pi\)
\(284\) −14.1803 −0.841448
\(285\) 0 0
\(286\) −25.4164 −1.50290
\(287\) 12.1803 0.718983
\(288\) 1.00000 0.0589256
\(289\) 44.6869 2.62864
\(290\) 0 0
\(291\) 13.8541 0.812142
\(292\) 3.14590 0.184100
\(293\) 28.7984 1.68242 0.841209 0.540709i \(-0.181844\pi\)
0.841209 + 0.540709i \(0.181844\pi\)
\(294\) −3.00000 −0.174964
\(295\) 0 0
\(296\) 2.14590 0.124728
\(297\) −5.23607 −0.303827
\(298\) 7.03444 0.407494
\(299\) −29.1246 −1.68432
\(300\) 0 0
\(301\) −2.47214 −0.142492
\(302\) −6.94427 −0.399598
\(303\) 1.67376 0.0961551
\(304\) −2.76393 −0.158522
\(305\) 0 0
\(306\) −7.85410 −0.448989
\(307\) 16.2918 0.929822 0.464911 0.885357i \(-0.346086\pi\)
0.464911 + 0.885357i \(0.346086\pi\)
\(308\) 10.4721 0.596705
\(309\) 2.29180 0.130376
\(310\) 0 0
\(311\) −18.6525 −1.05768 −0.528842 0.848720i \(-0.677374\pi\)
−0.528842 + 0.848720i \(0.677374\pi\)
\(312\) 4.85410 0.274809
\(313\) 26.3607 1.48999 0.744997 0.667068i \(-0.232451\pi\)
0.744997 + 0.667068i \(0.232451\pi\)
\(314\) −17.8541 −1.00757
\(315\) 0 0
\(316\) 0 0
\(317\) 6.94427 0.390029 0.195015 0.980800i \(-0.437525\pi\)
0.195015 + 0.980800i \(0.437525\pi\)
\(318\) −8.56231 −0.480150
\(319\) 7.23607 0.405142
\(320\) 0 0
\(321\) −10.9443 −0.610850
\(322\) 12.0000 0.668734
\(323\) 21.7082 1.20788
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −20.4721 −1.13385
\(327\) 2.56231 0.141696
\(328\) −6.09017 −0.336273
\(329\) 9.52786 0.525288
\(330\) 0 0
\(331\) −32.4721 −1.78483 −0.892415 0.451216i \(-0.850991\pi\)
−0.892415 + 0.451216i \(0.850991\pi\)
\(332\) −6.00000 −0.329293
\(333\) 2.14590 0.117594
\(334\) −6.47214 −0.354140
\(335\) 0 0
\(336\) −2.00000 −0.109109
\(337\) −28.8328 −1.57062 −0.785312 0.619100i \(-0.787497\pi\)
−0.785312 + 0.619100i \(0.787497\pi\)
\(338\) 10.5623 0.574514
\(339\) 11.5623 0.627978
\(340\) 0 0
\(341\) −19.4164 −1.05146
\(342\) −2.76393 −0.149456
\(343\) 20.0000 1.07990
\(344\) 1.23607 0.0666443
\(345\) 0 0
\(346\) 10.9098 0.586516
\(347\) 5.23607 0.281087 0.140543 0.990075i \(-0.455115\pi\)
0.140543 + 0.990075i \(0.455115\pi\)
\(348\) −1.38197 −0.0740812
\(349\) 14.7984 0.792139 0.396069 0.918221i \(-0.370374\pi\)
0.396069 + 0.918221i \(0.370374\pi\)
\(350\) 0 0
\(351\) 4.85410 0.259093
\(352\) −5.23607 −0.279083
\(353\) −20.4721 −1.08962 −0.544811 0.838559i \(-0.683399\pi\)
−0.544811 + 0.838559i \(0.683399\pi\)
\(354\) 8.94427 0.475383
\(355\) 0 0
\(356\) 1.38197 0.0732441
\(357\) 15.7082 0.831366
\(358\) −3.81966 −0.201875
\(359\) −27.8885 −1.47190 −0.735951 0.677035i \(-0.763264\pi\)
−0.735951 + 0.677035i \(0.763264\pi\)
\(360\) 0 0
\(361\) −11.3607 −0.597931
\(362\) −20.0344 −1.05299
\(363\) 16.4164 0.861638
\(364\) −9.70820 −0.508848
\(365\) 0 0
\(366\) −8.85410 −0.462811
\(367\) 12.4721 0.651040 0.325520 0.945535i \(-0.394461\pi\)
0.325520 + 0.945535i \(0.394461\pi\)
\(368\) −6.00000 −0.312772
\(369\) −6.09017 −0.317042
\(370\) 0 0
\(371\) 17.1246 0.889066
\(372\) 3.70820 0.192261
\(373\) −2.58359 −0.133773 −0.0668867 0.997761i \(-0.521307\pi\)
−0.0668867 + 0.997761i \(0.521307\pi\)
\(374\) 41.1246 2.12650
\(375\) 0 0
\(376\) −4.76393 −0.245681
\(377\) −6.70820 −0.345490
\(378\) −2.00000 −0.102869
\(379\) −3.41641 −0.175489 −0.0877445 0.996143i \(-0.527966\pi\)
−0.0877445 + 0.996143i \(0.527966\pi\)
\(380\) 0 0
\(381\) −0.291796 −0.0149492
\(382\) −0.763932 −0.0390862
\(383\) −20.4721 −1.04608 −0.523039 0.852309i \(-0.675202\pi\)
−0.523039 + 0.852309i \(0.675202\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −2.38197 −0.121239
\(387\) 1.23607 0.0628329
\(388\) 13.8541 0.703335
\(389\) 15.9787 0.810153 0.405076 0.914283i \(-0.367245\pi\)
0.405076 + 0.914283i \(0.367245\pi\)
\(390\) 0 0
\(391\) 47.1246 2.38319
\(392\) −3.00000 −0.151523
\(393\) −0.763932 −0.0385353
\(394\) 13.3262 0.671366
\(395\) 0 0
\(396\) −5.23607 −0.263122
\(397\) −28.8328 −1.44708 −0.723539 0.690284i \(-0.757486\pi\)
−0.723539 + 0.690284i \(0.757486\pi\)
\(398\) −6.18034 −0.309792
\(399\) 5.52786 0.276739
\(400\) 0 0
\(401\) −26.0902 −1.30288 −0.651440 0.758700i \(-0.725835\pi\)
−0.651440 + 0.758700i \(0.725835\pi\)
\(402\) 9.70820 0.484201
\(403\) 18.0000 0.896644
\(404\) 1.67376 0.0832728
\(405\) 0 0
\(406\) 2.76393 0.137172
\(407\) −11.2361 −0.556951
\(408\) −7.85410 −0.388836
\(409\) 20.9787 1.03733 0.518665 0.854977i \(-0.326429\pi\)
0.518665 + 0.854977i \(0.326429\pi\)
\(410\) 0 0
\(411\) 15.0344 0.741594
\(412\) 2.29180 0.112909
\(413\) −17.8885 −0.880238
\(414\) −6.00000 −0.294884
\(415\) 0 0
\(416\) 4.85410 0.237992
\(417\) −13.4164 −0.657004
\(418\) 14.4721 0.707855
\(419\) −37.8885 −1.85098 −0.925488 0.378776i \(-0.876345\pi\)
−0.925488 + 0.378776i \(0.876345\pi\)
\(420\) 0 0
\(421\) −11.0902 −0.540502 −0.270251 0.962790i \(-0.587107\pi\)
−0.270251 + 0.962790i \(0.587107\pi\)
\(422\) −8.00000 −0.389434
\(423\) −4.76393 −0.231630
\(424\) −8.56231 −0.415822
\(425\) 0 0
\(426\) −14.1803 −0.687040
\(427\) 17.7082 0.856960
\(428\) −10.9443 −0.529011
\(429\) −25.4164 −1.22712
\(430\) 0 0
\(431\) 3.70820 0.178618 0.0893089 0.996004i \(-0.471534\pi\)
0.0893089 + 0.996004i \(0.471534\pi\)
\(432\) 1.00000 0.0481125
\(433\) 23.1459 1.11232 0.556160 0.831075i \(-0.312274\pi\)
0.556160 + 0.831075i \(0.312274\pi\)
\(434\) −7.41641 −0.355999
\(435\) 0 0
\(436\) 2.56231 0.122712
\(437\) 16.5836 0.793301
\(438\) 3.14590 0.150317
\(439\) −25.1246 −1.19913 −0.599566 0.800325i \(-0.704660\pi\)
−0.599566 + 0.800325i \(0.704660\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) −38.1246 −1.81340
\(443\) −30.0689 −1.42862 −0.714308 0.699832i \(-0.753258\pi\)
−0.714308 + 0.699832i \(0.753258\pi\)
\(444\) 2.14590 0.101840
\(445\) 0 0
\(446\) 2.29180 0.108520
\(447\) 7.03444 0.332718
\(448\) −2.00000 −0.0944911
\(449\) 4.79837 0.226449 0.113225 0.993569i \(-0.463882\pi\)
0.113225 + 0.993569i \(0.463882\pi\)
\(450\) 0 0
\(451\) 31.8885 1.50157
\(452\) 11.5623 0.543845
\(453\) −6.94427 −0.326270
\(454\) 16.9443 0.795234
\(455\) 0 0
\(456\) −2.76393 −0.129433
\(457\) 12.4721 0.583422 0.291711 0.956507i \(-0.405775\pi\)
0.291711 + 0.956507i \(0.405775\pi\)
\(458\) −24.2705 −1.13409
\(459\) −7.85410 −0.366598
\(460\) 0 0
\(461\) 25.0902 1.16857 0.584283 0.811550i \(-0.301376\pi\)
0.584283 + 0.811550i \(0.301376\pi\)
\(462\) 10.4721 0.487208
\(463\) 12.2918 0.571248 0.285624 0.958342i \(-0.407799\pi\)
0.285624 + 0.958342i \(0.407799\pi\)
\(464\) −1.38197 −0.0641562
\(465\) 0 0
\(466\) 22.0902 1.02331
\(467\) −15.8197 −0.732047 −0.366023 0.930606i \(-0.619281\pi\)
−0.366023 + 0.930606i \(0.619281\pi\)
\(468\) 4.85410 0.224381
\(469\) −19.4164 −0.896566
\(470\) 0 0
\(471\) −17.8541 −0.822674
\(472\) 8.94427 0.411693
\(473\) −6.47214 −0.297589
\(474\) 0 0
\(475\) 0 0
\(476\) 15.7082 0.719984
\(477\) −8.56231 −0.392041
\(478\) 26.8328 1.22730
\(479\) 4.47214 0.204337 0.102169 0.994767i \(-0.467422\pi\)
0.102169 + 0.994767i \(0.467422\pi\)
\(480\) 0 0
\(481\) 10.4164 0.474947
\(482\) −13.8541 −0.631037
\(483\) 12.0000 0.546019
\(484\) 16.4164 0.746200
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −10.2918 −0.466366 −0.233183 0.972433i \(-0.574914\pi\)
−0.233183 + 0.972433i \(0.574914\pi\)
\(488\) −8.85410 −0.400806
\(489\) −20.4721 −0.925782
\(490\) 0 0
\(491\) 29.8885 1.34885 0.674426 0.738343i \(-0.264391\pi\)
0.674426 + 0.738343i \(0.264391\pi\)
\(492\) −6.09017 −0.274566
\(493\) 10.8541 0.488844
\(494\) −13.4164 −0.603633
\(495\) 0 0
\(496\) 3.70820 0.166503
\(497\) 28.3607 1.27215
\(498\) −6.00000 −0.268866
\(499\) −33.4164 −1.49592 −0.747962 0.663742i \(-0.768967\pi\)
−0.747962 + 0.663742i \(0.768967\pi\)
\(500\) 0 0
\(501\) −6.47214 −0.289154
\(502\) −12.4721 −0.556659
\(503\) −21.5279 −0.959880 −0.479940 0.877301i \(-0.659342\pi\)
−0.479940 + 0.877301i \(0.659342\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 0 0
\(506\) 31.4164 1.39663
\(507\) 10.5623 0.469088
\(508\) −0.291796 −0.0129464
\(509\) −13.7426 −0.609132 −0.304566 0.952491i \(-0.598511\pi\)
−0.304566 + 0.952491i \(0.598511\pi\)
\(510\) 0 0
\(511\) −6.29180 −0.278333
\(512\) 1.00000 0.0441942
\(513\) −2.76393 −0.122031
\(514\) 11.6180 0.512450
\(515\) 0 0
\(516\) 1.23607 0.0544149
\(517\) 24.9443 1.09705
\(518\) −4.29180 −0.188571
\(519\) 10.9098 0.478888
\(520\) 0 0
\(521\) 6.27051 0.274716 0.137358 0.990521i \(-0.456139\pi\)
0.137358 + 0.990521i \(0.456139\pi\)
\(522\) −1.38197 −0.0604870
\(523\) 2.29180 0.100213 0.0501066 0.998744i \(-0.484044\pi\)
0.0501066 + 0.998744i \(0.484044\pi\)
\(524\) −0.763932 −0.0333725
\(525\) 0 0
\(526\) −0.472136 −0.0205861
\(527\) −29.1246 −1.26869
\(528\) −5.23607 −0.227871
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 8.94427 0.388148
\(532\) 5.52786 0.239663
\(533\) −29.5623 −1.28049
\(534\) 1.38197 0.0598035
\(535\) 0 0
\(536\) 9.70820 0.419331
\(537\) −3.81966 −0.164831
\(538\) −1.90983 −0.0823386
\(539\) 15.7082 0.676600
\(540\) 0 0
\(541\) 21.2705 0.914491 0.457245 0.889341i \(-0.348836\pi\)
0.457245 + 0.889341i \(0.348836\pi\)
\(542\) 28.1803 1.21045
\(543\) −20.0344 −0.859760
\(544\) −7.85410 −0.336742
\(545\) 0 0
\(546\) −9.70820 −0.415473
\(547\) −2.00000 −0.0855138 −0.0427569 0.999086i \(-0.513614\pi\)
−0.0427569 + 0.999086i \(0.513614\pi\)
\(548\) 15.0344 0.642240
\(549\) −8.85410 −0.377884
\(550\) 0 0
\(551\) 3.81966 0.162723
\(552\) −6.00000 −0.255377
\(553\) 0 0
\(554\) 22.1459 0.940889
\(555\) 0 0
\(556\) −13.4164 −0.568982
\(557\) 27.2705 1.15549 0.577744 0.816218i \(-0.303933\pi\)
0.577744 + 0.816218i \(0.303933\pi\)
\(558\) 3.70820 0.156981
\(559\) 6.00000 0.253773
\(560\) 0 0
\(561\) 41.1246 1.73628
\(562\) −6.09017 −0.256898
\(563\) −10.4721 −0.441348 −0.220674 0.975348i \(-0.570826\pi\)
−0.220674 + 0.975348i \(0.570826\pi\)
\(564\) −4.76393 −0.200598
\(565\) 0 0
\(566\) −3.23607 −0.136022
\(567\) −2.00000 −0.0839921
\(568\) −14.1803 −0.594994
\(569\) 21.3820 0.896379 0.448189 0.893939i \(-0.352069\pi\)
0.448189 + 0.893939i \(0.352069\pi\)
\(570\) 0 0
\(571\) 33.7082 1.41064 0.705322 0.708887i \(-0.250802\pi\)
0.705322 + 0.708887i \(0.250802\pi\)
\(572\) −25.4164 −1.06271
\(573\) −0.763932 −0.0319137
\(574\) 12.1803 0.508398
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −28.8328 −1.20033 −0.600163 0.799878i \(-0.704898\pi\)
−0.600163 + 0.799878i \(0.704898\pi\)
\(578\) 44.6869 1.85873
\(579\) −2.38197 −0.0989911
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 13.8541 0.574271
\(583\) 44.8328 1.85679
\(584\) 3.14590 0.130178
\(585\) 0 0
\(586\) 28.7984 1.18965
\(587\) 15.8885 0.655790 0.327895 0.944714i \(-0.393661\pi\)
0.327895 + 0.944714i \(0.393661\pi\)
\(588\) −3.00000 −0.123718
\(589\) −10.2492 −0.422312
\(590\) 0 0
\(591\) 13.3262 0.548168
\(592\) 2.14590 0.0881959
\(593\) 6.03444 0.247805 0.123902 0.992294i \(-0.460459\pi\)
0.123902 + 0.992294i \(0.460459\pi\)
\(594\) −5.23607 −0.214838
\(595\) 0 0
\(596\) 7.03444 0.288142
\(597\) −6.18034 −0.252944
\(598\) −29.1246 −1.19099
\(599\) 1.05573 0.0431359 0.0215679 0.999767i \(-0.493134\pi\)
0.0215679 + 0.999767i \(0.493134\pi\)
\(600\) 0 0
\(601\) 7.32624 0.298843 0.149422 0.988774i \(-0.452259\pi\)
0.149422 + 0.988774i \(0.452259\pi\)
\(602\) −2.47214 −0.100757
\(603\) 9.70820 0.395349
\(604\) −6.94427 −0.282558
\(605\) 0 0
\(606\) 1.67376 0.0679919
\(607\) 1.81966 0.0738577 0.0369289 0.999318i \(-0.488243\pi\)
0.0369289 + 0.999318i \(0.488243\pi\)
\(608\) −2.76393 −0.112092
\(609\) 2.76393 0.112000
\(610\) 0 0
\(611\) −23.1246 −0.935522
\(612\) −7.85410 −0.317483
\(613\) 21.5623 0.870893 0.435447 0.900215i \(-0.356590\pi\)
0.435447 + 0.900215i \(0.356590\pi\)
\(614\) 16.2918 0.657483
\(615\) 0 0
\(616\) 10.4721 0.421934
\(617\) −8.38197 −0.337445 −0.168723 0.985664i \(-0.553964\pi\)
−0.168723 + 0.985664i \(0.553964\pi\)
\(618\) 2.29180 0.0921896
\(619\) 3.41641 0.137317 0.0686585 0.997640i \(-0.478128\pi\)
0.0686585 + 0.997640i \(0.478128\pi\)
\(620\) 0 0
\(621\) −6.00000 −0.240772
\(622\) −18.6525 −0.747896
\(623\) −2.76393 −0.110735
\(624\) 4.85410 0.194320
\(625\) 0 0
\(626\) 26.3607 1.05358
\(627\) 14.4721 0.577961
\(628\) −17.8541 −0.712456
\(629\) −16.8541 −0.672017
\(630\) 0 0
\(631\) −43.1246 −1.71676 −0.858382 0.513011i \(-0.828530\pi\)
−0.858382 + 0.513011i \(0.828530\pi\)
\(632\) 0 0
\(633\) −8.00000 −0.317971
\(634\) 6.94427 0.275792
\(635\) 0 0
\(636\) −8.56231 −0.339518
\(637\) −14.5623 −0.576980
\(638\) 7.23607 0.286479
\(639\) −14.1803 −0.560966
\(640\) 0 0
\(641\) −26.9443 −1.06423 −0.532117 0.846671i \(-0.678603\pi\)
−0.532117 + 0.846671i \(0.678603\pi\)
\(642\) −10.9443 −0.431936
\(643\) 39.7771 1.56866 0.784328 0.620347i \(-0.213008\pi\)
0.784328 + 0.620347i \(0.213008\pi\)
\(644\) 12.0000 0.472866
\(645\) 0 0
\(646\) 21.7082 0.854098
\(647\) 35.8885 1.41092 0.705462 0.708748i \(-0.250739\pi\)
0.705462 + 0.708748i \(0.250739\pi\)
\(648\) 1.00000 0.0392837
\(649\) −46.8328 −1.83835
\(650\) 0 0
\(651\) −7.41641 −0.290672
\(652\) −20.4721 −0.801751
\(653\) 32.2148 1.26066 0.630331 0.776327i \(-0.282919\pi\)
0.630331 + 0.776327i \(0.282919\pi\)
\(654\) 2.56231 0.100194
\(655\) 0 0
\(656\) −6.09017 −0.237781
\(657\) 3.14590 0.122733
\(658\) 9.52786 0.371435
\(659\) 38.9443 1.51705 0.758527 0.651642i \(-0.225920\pi\)
0.758527 + 0.651642i \(0.225920\pi\)
\(660\) 0 0
\(661\) −23.5279 −0.915128 −0.457564 0.889177i \(-0.651278\pi\)
−0.457564 + 0.889177i \(0.651278\pi\)
\(662\) −32.4721 −1.26207
\(663\) −38.1246 −1.48064
\(664\) −6.00000 −0.232845
\(665\) 0 0
\(666\) 2.14590 0.0831519
\(667\) 8.29180 0.321060
\(668\) −6.47214 −0.250414
\(669\) 2.29180 0.0886060
\(670\) 0 0
\(671\) 46.3607 1.78973
\(672\) −2.00000 −0.0771517
\(673\) 18.7984 0.724624 0.362312 0.932057i \(-0.381987\pi\)
0.362312 + 0.932057i \(0.381987\pi\)
\(674\) −28.8328 −1.11060
\(675\) 0 0
\(676\) 10.5623 0.406243
\(677\) −36.4721 −1.40174 −0.700869 0.713290i \(-0.747204\pi\)
−0.700869 + 0.713290i \(0.747204\pi\)
\(678\) 11.5623 0.444048
\(679\) −27.7082 −1.06334
\(680\) 0 0
\(681\) 16.9443 0.649306
\(682\) −19.4164 −0.743493
\(683\) −19.4164 −0.742948 −0.371474 0.928443i \(-0.621148\pi\)
−0.371474 + 0.928443i \(0.621148\pi\)
\(684\) −2.76393 −0.105682
\(685\) 0 0
\(686\) 20.0000 0.763604
\(687\) −24.2705 −0.925978
\(688\) 1.23607 0.0471246
\(689\) −41.5623 −1.58340
\(690\) 0 0
\(691\) −24.1803 −0.919863 −0.459932 0.887954i \(-0.652126\pi\)
−0.459932 + 0.887954i \(0.652126\pi\)
\(692\) 10.9098 0.414730
\(693\) 10.4721 0.397804
\(694\) 5.23607 0.198758
\(695\) 0 0
\(696\) −1.38197 −0.0523833
\(697\) 47.8328 1.81180
\(698\) 14.7984 0.560127
\(699\) 22.0902 0.835527
\(700\) 0 0
\(701\) −20.1591 −0.761397 −0.380698 0.924699i \(-0.624316\pi\)
−0.380698 + 0.924699i \(0.624316\pi\)
\(702\) 4.85410 0.183206
\(703\) −5.93112 −0.223696
\(704\) −5.23607 −0.197342
\(705\) 0 0
\(706\) −20.4721 −0.770479
\(707\) −3.34752 −0.125897
\(708\) 8.94427 0.336146
\(709\) −29.7984 −1.11910 −0.559551 0.828796i \(-0.689026\pi\)
−0.559551 + 0.828796i \(0.689026\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 1.38197 0.0517914
\(713\) −22.2492 −0.833240
\(714\) 15.7082 0.587865
\(715\) 0 0
\(716\) −3.81966 −0.142747
\(717\) 26.8328 1.00209
\(718\) −27.8885 −1.04079
\(719\) 22.3607 0.833913 0.416956 0.908927i \(-0.363097\pi\)
0.416956 + 0.908927i \(0.363097\pi\)
\(720\) 0 0
\(721\) −4.58359 −0.170702
\(722\) −11.3607 −0.422801
\(723\) −13.8541 −0.515240
\(724\) −20.0344 −0.744574
\(725\) 0 0
\(726\) 16.4164 0.609270
\(727\) −10.2918 −0.381702 −0.190851 0.981619i \(-0.561125\pi\)
−0.190851 + 0.981619i \(0.561125\pi\)
\(728\) −9.70820 −0.359810
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −9.70820 −0.359071
\(732\) −8.85410 −0.327257
\(733\) 6.36068 0.234937 0.117469 0.993077i \(-0.462522\pi\)
0.117469 + 0.993077i \(0.462522\pi\)
\(734\) 12.4721 0.460355
\(735\) 0 0
\(736\) −6.00000 −0.221163
\(737\) −50.8328 −1.87245
\(738\) −6.09017 −0.224182
\(739\) 8.29180 0.305019 0.152509 0.988302i \(-0.451265\pi\)
0.152509 + 0.988302i \(0.451265\pi\)
\(740\) 0 0
\(741\) −13.4164 −0.492864
\(742\) 17.1246 0.628664
\(743\) −6.65248 −0.244056 −0.122028 0.992527i \(-0.538940\pi\)
−0.122028 + 0.992527i \(0.538940\pi\)
\(744\) 3.70820 0.135949
\(745\) 0 0
\(746\) −2.58359 −0.0945920
\(747\) −6.00000 −0.219529
\(748\) 41.1246 1.50366
\(749\) 21.8885 0.799790
\(750\) 0 0
\(751\) 14.7639 0.538744 0.269372 0.963036i \(-0.413184\pi\)
0.269372 + 0.963036i \(0.413184\pi\)
\(752\) −4.76393 −0.173723
\(753\) −12.4721 −0.454510
\(754\) −6.70820 −0.244298
\(755\) 0 0
\(756\) −2.00000 −0.0727393
\(757\) 28.8541 1.04872 0.524360 0.851497i \(-0.324305\pi\)
0.524360 + 0.851497i \(0.324305\pi\)
\(758\) −3.41641 −0.124090
\(759\) 31.4164 1.14034
\(760\) 0 0
\(761\) 24.1591 0.875765 0.437883 0.899032i \(-0.355729\pi\)
0.437883 + 0.899032i \(0.355729\pi\)
\(762\) −0.291796 −0.0105707
\(763\) −5.12461 −0.185523
\(764\) −0.763932 −0.0276381
\(765\) 0 0
\(766\) −20.4721 −0.739688
\(767\) 43.4164 1.56768
\(768\) 1.00000 0.0360844
\(769\) −2.36068 −0.0851283 −0.0425641 0.999094i \(-0.513553\pi\)
−0.0425641 + 0.999094i \(0.513553\pi\)
\(770\) 0 0
\(771\) 11.6180 0.418413
\(772\) −2.38197 −0.0857288
\(773\) −14.0902 −0.506788 −0.253394 0.967363i \(-0.581547\pi\)
−0.253394 + 0.967363i \(0.581547\pi\)
\(774\) 1.23607 0.0444295
\(775\) 0 0
\(776\) 13.8541 0.497333
\(777\) −4.29180 −0.153967
\(778\) 15.9787 0.572865
\(779\) 16.8328 0.603098
\(780\) 0 0
\(781\) 74.2492 2.65685
\(782\) 47.1246 1.68517
\(783\) −1.38197 −0.0493874
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) −0.763932 −0.0272486
\(787\) −30.5410 −1.08867 −0.544335 0.838868i \(-0.683218\pi\)
−0.544335 + 0.838868i \(0.683218\pi\)
\(788\) 13.3262 0.474728
\(789\) −0.472136 −0.0168085
\(790\) 0 0
\(791\) −23.1246 −0.822217
\(792\) −5.23607 −0.186056
\(793\) −42.9787 −1.52622
\(794\) −28.8328 −1.02324
\(795\) 0 0
\(796\) −6.18034 −0.219056
\(797\) 13.3262 0.472040 0.236020 0.971748i \(-0.424157\pi\)
0.236020 + 0.971748i \(0.424157\pi\)
\(798\) 5.52786 0.195684
\(799\) 37.4164 1.32370
\(800\) 0 0
\(801\) 1.38197 0.0488294
\(802\) −26.0902 −0.921276
\(803\) −16.4721 −0.581289
\(804\) 9.70820 0.342382
\(805\) 0 0
\(806\) 18.0000 0.634023
\(807\) −1.90983 −0.0672292
\(808\) 1.67376 0.0588827
\(809\) 8.21478 0.288816 0.144408 0.989518i \(-0.453872\pi\)
0.144408 + 0.989518i \(0.453872\pi\)
\(810\) 0 0
\(811\) 10.2918 0.361394 0.180697 0.983539i \(-0.442165\pi\)
0.180697 + 0.983539i \(0.442165\pi\)
\(812\) 2.76393 0.0969950
\(813\) 28.1803 0.988328
\(814\) −11.2361 −0.393824
\(815\) 0 0
\(816\) −7.85410 −0.274949
\(817\) −3.41641 −0.119525
\(818\) 20.9787 0.733504
\(819\) −9.70820 −0.339232
\(820\) 0 0
\(821\) −44.8328 −1.56468 −0.782338 0.622854i \(-0.785973\pi\)
−0.782338 + 0.622854i \(0.785973\pi\)
\(822\) 15.0344 0.524386
\(823\) 23.5967 0.822531 0.411265 0.911516i \(-0.365087\pi\)
0.411265 + 0.911516i \(0.365087\pi\)
\(824\) 2.29180 0.0798385
\(825\) 0 0
\(826\) −17.8885 −0.622422
\(827\) −14.7639 −0.513392 −0.256696 0.966492i \(-0.582634\pi\)
−0.256696 + 0.966492i \(0.582634\pi\)
\(828\) −6.00000 −0.208514
\(829\) 39.1459 1.35959 0.679797 0.733401i \(-0.262068\pi\)
0.679797 + 0.733401i \(0.262068\pi\)
\(830\) 0 0
\(831\) 22.1459 0.768233
\(832\) 4.85410 0.168286
\(833\) 23.5623 0.816386
\(834\) −13.4164 −0.464572
\(835\) 0 0
\(836\) 14.4721 0.500529
\(837\) 3.70820 0.128174
\(838\) −37.8885 −1.30884
\(839\) −5.12461 −0.176921 −0.0884606 0.996080i \(-0.528195\pi\)
−0.0884606 + 0.996080i \(0.528195\pi\)
\(840\) 0 0
\(841\) −27.0902 −0.934144
\(842\) −11.0902 −0.382192
\(843\) −6.09017 −0.209757
\(844\) −8.00000 −0.275371
\(845\) 0 0
\(846\) −4.76393 −0.163787
\(847\) −32.8328 −1.12815
\(848\) −8.56231 −0.294031
\(849\) −3.23607 −0.111062
\(850\) 0 0
\(851\) −12.8754 −0.441363
\(852\) −14.1803 −0.485810
\(853\) −16.4508 −0.563266 −0.281633 0.959522i \(-0.590876\pi\)
−0.281633 + 0.959522i \(0.590876\pi\)
\(854\) 17.7082 0.605962
\(855\) 0 0
\(856\) −10.9443 −0.374068
\(857\) −23.3050 −0.796082 −0.398041 0.917368i \(-0.630310\pi\)
−0.398041 + 0.917368i \(0.630310\pi\)
\(858\) −25.4164 −0.867702
\(859\) 45.5279 1.55339 0.776695 0.629876i \(-0.216894\pi\)
0.776695 + 0.629876i \(0.216894\pi\)
\(860\) 0 0
\(861\) 12.1803 0.415105
\(862\) 3.70820 0.126302
\(863\) −12.1803 −0.414624 −0.207312 0.978275i \(-0.566471\pi\)
−0.207312 + 0.978275i \(0.566471\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 23.1459 0.786530
\(867\) 44.6869 1.51765
\(868\) −7.41641 −0.251729
\(869\) 0 0
\(870\) 0 0
\(871\) 47.1246 1.59676
\(872\) 2.56231 0.0867706
\(873\) 13.8541 0.468890
\(874\) 16.5836 0.560948
\(875\) 0 0
\(876\) 3.14590 0.106290
\(877\) −17.9787 −0.607098 −0.303549 0.952816i \(-0.598172\pi\)
−0.303549 + 0.952816i \(0.598172\pi\)
\(878\) −25.1246 −0.847915
\(879\) 28.7984 0.971345
\(880\) 0 0
\(881\) 46.7214 1.57408 0.787041 0.616900i \(-0.211612\pi\)
0.787041 + 0.616900i \(0.211612\pi\)
\(882\) −3.00000 −0.101015
\(883\) −42.1803 −1.41948 −0.709741 0.704463i \(-0.751188\pi\)
−0.709741 + 0.704463i \(0.751188\pi\)
\(884\) −38.1246 −1.28227
\(885\) 0 0
\(886\) −30.0689 −1.01018
\(887\) −25.8197 −0.866939 −0.433470 0.901168i \(-0.642711\pi\)
−0.433470 + 0.901168i \(0.642711\pi\)
\(888\) 2.14590 0.0720116
\(889\) 0.583592 0.0195731
\(890\) 0 0
\(891\) −5.23607 −0.175415
\(892\) 2.29180 0.0767350
\(893\) 13.1672 0.440623
\(894\) 7.03444 0.235267
\(895\) 0 0
\(896\) −2.00000 −0.0668153
\(897\) −29.1246 −0.972442
\(898\) 4.79837 0.160124
\(899\) −5.12461 −0.170915
\(900\) 0 0
\(901\) 67.2492 2.24040
\(902\) 31.8885 1.06177
\(903\) −2.47214 −0.0822675
\(904\) 11.5623 0.384557
\(905\) 0 0
\(906\) −6.94427 −0.230708
\(907\) −20.2918 −0.673778 −0.336889 0.941544i \(-0.609375\pi\)
−0.336889 + 0.941544i \(0.609375\pi\)
\(908\) 16.9443 0.562315
\(909\) 1.67376 0.0555152
\(910\) 0 0
\(911\) 56.0689 1.85764 0.928822 0.370525i \(-0.120822\pi\)
0.928822 + 0.370525i \(0.120822\pi\)
\(912\) −2.76393 −0.0915229
\(913\) 31.4164 1.03973
\(914\) 12.4721 0.412542
\(915\) 0 0
\(916\) −24.2705 −0.801920
\(917\) 1.52786 0.0504545
\(918\) −7.85410 −0.259224
\(919\) −1.05573 −0.0348253 −0.0174126 0.999848i \(-0.505543\pi\)
−0.0174126 + 0.999848i \(0.505543\pi\)
\(920\) 0 0
\(921\) 16.2918 0.536833
\(922\) 25.0902 0.826301
\(923\) −68.8328 −2.26566
\(924\) 10.4721 0.344508
\(925\) 0 0
\(926\) 12.2918 0.403933
\(927\) 2.29180 0.0752725
\(928\) −1.38197 −0.0453653
\(929\) −30.9787 −1.01638 −0.508189 0.861245i \(-0.669685\pi\)
−0.508189 + 0.861245i \(0.669685\pi\)
\(930\) 0 0
\(931\) 8.29180 0.271753
\(932\) 22.0902 0.723588
\(933\) −18.6525 −0.610655
\(934\) −15.8197 −0.517635
\(935\) 0 0
\(936\) 4.85410 0.158661
\(937\) −35.0902 −1.14635 −0.573173 0.819434i \(-0.694288\pi\)
−0.573173 + 0.819434i \(0.694288\pi\)
\(938\) −19.4164 −0.633968
\(939\) 26.3607 0.860248
\(940\) 0 0
\(941\) 21.6738 0.706544 0.353272 0.935521i \(-0.385069\pi\)
0.353272 + 0.935521i \(0.385069\pi\)
\(942\) −17.8541 −0.581718
\(943\) 36.5410 1.18994
\(944\) 8.94427 0.291111
\(945\) 0 0
\(946\) −6.47214 −0.210427
\(947\) −10.9443 −0.355641 −0.177821 0.984063i \(-0.556905\pi\)
−0.177821 + 0.984063i \(0.556905\pi\)
\(948\) 0 0
\(949\) 15.2705 0.495702
\(950\) 0 0
\(951\) 6.94427 0.225183
\(952\) 15.7082 0.509106
\(953\) 32.8673 1.06467 0.532337 0.846532i \(-0.321314\pi\)
0.532337 + 0.846532i \(0.321314\pi\)
\(954\) −8.56231 −0.277215
\(955\) 0 0
\(956\) 26.8328 0.867835
\(957\) 7.23607 0.233909
\(958\) 4.47214 0.144488
\(959\) −30.0689 −0.970975
\(960\) 0 0
\(961\) −17.2492 −0.556427
\(962\) 10.4164 0.335838
\(963\) −10.9443 −0.352674
\(964\) −13.8541 −0.446211
\(965\) 0 0
\(966\) 12.0000 0.386094
\(967\) −40.9443 −1.31668 −0.658340 0.752721i \(-0.728741\pi\)
−0.658340 + 0.752721i \(0.728741\pi\)
\(968\) 16.4164 0.527643
\(969\) 21.7082 0.697368
\(970\) 0 0
\(971\) −33.7771 −1.08396 −0.541979 0.840392i \(-0.682325\pi\)
−0.541979 + 0.840392i \(0.682325\pi\)
\(972\) 1.00000 0.0320750
\(973\) 26.8328 0.860221
\(974\) −10.2918 −0.329770
\(975\) 0 0
\(976\) −8.85410 −0.283413
\(977\) 20.5623 0.657846 0.328923 0.944357i \(-0.393314\pi\)
0.328923 + 0.944357i \(0.393314\pi\)
\(978\) −20.4721 −0.654627
\(979\) −7.23607 −0.231266
\(980\) 0 0
\(981\) 2.56231 0.0818081
\(982\) 29.8885 0.953782
\(983\) 42.5410 1.35685 0.678424 0.734671i \(-0.262663\pi\)
0.678424 + 0.734671i \(0.262663\pi\)
\(984\) −6.09017 −0.194148
\(985\) 0 0
\(986\) 10.8541 0.345665
\(987\) 9.52786 0.303275
\(988\) −13.4164 −0.426833
\(989\) −7.41641 −0.235828
\(990\) 0 0
\(991\) 60.5410 1.92315 0.961574 0.274544i \(-0.0885270\pi\)
0.961574 + 0.274544i \(0.0885270\pi\)
\(992\) 3.70820 0.117736
\(993\) −32.4721 −1.03047
\(994\) 28.3607 0.899546
\(995\) 0 0
\(996\) −6.00000 −0.190117
\(997\) 1.41641 0.0448581 0.0224290 0.999748i \(-0.492860\pi\)
0.0224290 + 0.999748i \(0.492860\pi\)
\(998\) −33.4164 −1.05778
\(999\) 2.14590 0.0678932
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3750.2.a.g.1.1 2
5.2 odd 4 3750.2.c.c.1249.3 4
5.3 odd 4 3750.2.c.c.1249.1 4
5.4 even 2 3750.2.a.b.1.1 2
25.2 odd 20 750.2.h.a.649.2 8
25.9 even 10 150.2.g.b.31.1 4
25.11 even 5 750.2.g.a.601.1 4
25.12 odd 20 750.2.h.a.349.1 8
25.13 odd 20 750.2.h.a.349.2 8
25.14 even 10 150.2.g.b.121.1 yes 4
25.16 even 5 750.2.g.a.151.1 4
25.23 odd 20 750.2.h.a.649.1 8
75.14 odd 10 450.2.h.b.271.1 4
75.59 odd 10 450.2.h.b.181.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
150.2.g.b.31.1 4 25.9 even 10
150.2.g.b.121.1 yes 4 25.14 even 10
450.2.h.b.181.1 4 75.59 odd 10
450.2.h.b.271.1 4 75.14 odd 10
750.2.g.a.151.1 4 25.16 even 5
750.2.g.a.601.1 4 25.11 even 5
750.2.h.a.349.1 8 25.12 odd 20
750.2.h.a.349.2 8 25.13 odd 20
750.2.h.a.649.1 8 25.23 odd 20
750.2.h.a.649.2 8 25.2 odd 20
3750.2.a.b.1.1 2 5.4 even 2
3750.2.a.g.1.1 2 1.1 even 1 trivial
3750.2.c.c.1249.1 4 5.3 odd 4
3750.2.c.c.1249.3 4 5.2 odd 4