Properties

Label 400.4.c.m.49.2
Level $400$
Weight $4$
Character 400.49
Analytic conductor $23.601$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [400,4,Mod(49,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 400.c (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(23.6007640023\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 200)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 400.49
Dual form 400.4.c.m.49.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} +6.00000i q^{7} +26.0000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +6.00000i q^{7} +26.0000 q^{9} +19.0000 q^{11} -12.0000i q^{13} -75.0000i q^{17} -91.0000 q^{19} -6.00000 q^{21} +174.000i q^{23} +53.0000i q^{27} +272.000 q^{29} +230.000 q^{31} +19.0000i q^{33} -182.000i q^{37} +12.0000 q^{39} +117.000 q^{41} +372.000i q^{43} +52.0000i q^{47} +307.000 q^{49} +75.0000 q^{51} +402.000i q^{53} -91.0000i q^{57} +312.000 q^{59} +170.000 q^{61} +156.000i q^{63} -763.000i q^{67} -174.000 q^{69} +52.0000 q^{71} +981.000i q^{73} +114.000i q^{77} +1054.00 q^{79} +649.000 q^{81} +351.000i q^{83} +272.000i q^{87} -799.000 q^{89} +72.0000 q^{91} +230.000i q^{93} +962.000i q^{97} +494.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 52 q^{9} + 38 q^{11} - 182 q^{19} - 12 q^{21} + 544 q^{29} + 460 q^{31} + 24 q^{39} + 234 q^{41} + 614 q^{49} + 150 q^{51} + 624 q^{59} + 340 q^{61} - 348 q^{69} + 104 q^{71} + 2108 q^{79} + 1298 q^{81} - 1598 q^{89} + 144 q^{91} + 988 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000i 0.192450i 0.995360 + 0.0962250i \(0.0306768\pi\)
−0.995360 + 0.0962250i \(0.969323\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 6.00000i 0.323970i 0.986793 + 0.161985i \(0.0517895\pi\)
−0.986793 + 0.161985i \(0.948210\pi\)
\(8\) 0 0
\(9\) 26.0000 0.962963
\(10\) 0 0
\(11\) 19.0000 0.520792 0.260396 0.965502i \(-0.416147\pi\)
0.260396 + 0.965502i \(0.416147\pi\)
\(12\) 0 0
\(13\) − 12.0000i − 0.256015i −0.991773 0.128008i \(-0.959142\pi\)
0.991773 0.128008i \(-0.0408582\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 75.0000i − 1.07001i −0.844849 0.535005i \(-0.820310\pi\)
0.844849 0.535005i \(-0.179690\pi\)
\(18\) 0 0
\(19\) −91.0000 −1.09878 −0.549390 0.835566i \(-0.685140\pi\)
−0.549390 + 0.835566i \(0.685140\pi\)
\(20\) 0 0
\(21\) −6.00000 −0.0623480
\(22\) 0 0
\(23\) 174.000i 1.57746i 0.614742 + 0.788728i \(0.289260\pi\)
−0.614742 + 0.788728i \(0.710740\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 53.0000i 0.377772i
\(28\) 0 0
\(29\) 272.000 1.74169 0.870847 0.491554i \(-0.163571\pi\)
0.870847 + 0.491554i \(0.163571\pi\)
\(30\) 0 0
\(31\) 230.000 1.33256 0.666278 0.745704i \(-0.267887\pi\)
0.666278 + 0.745704i \(0.267887\pi\)
\(32\) 0 0
\(33\) 19.0000i 0.100227i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 182.000i − 0.808665i −0.914612 0.404333i \(-0.867504\pi\)
0.914612 0.404333i \(-0.132496\pi\)
\(38\) 0 0
\(39\) 12.0000 0.0492702
\(40\) 0 0
\(41\) 117.000 0.445667 0.222833 0.974857i \(-0.428469\pi\)
0.222833 + 0.974857i \(0.428469\pi\)
\(42\) 0 0
\(43\) 372.000i 1.31929i 0.751577 + 0.659645i \(0.229293\pi\)
−0.751577 + 0.659645i \(0.770707\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 52.0000i 0.161383i 0.996739 + 0.0806913i \(0.0257128\pi\)
−0.996739 + 0.0806913i \(0.974287\pi\)
\(48\) 0 0
\(49\) 307.000 0.895044
\(50\) 0 0
\(51\) 75.0000 0.205924
\(52\) 0 0
\(53\) 402.000i 1.04187i 0.853597 + 0.520933i \(0.174416\pi\)
−0.853597 + 0.520933i \(0.825584\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 91.0000i − 0.211460i
\(58\) 0 0
\(59\) 312.000 0.688457 0.344228 0.938886i \(-0.388141\pi\)
0.344228 + 0.938886i \(0.388141\pi\)
\(60\) 0 0
\(61\) 170.000 0.356824 0.178412 0.983956i \(-0.442904\pi\)
0.178412 + 0.983956i \(0.442904\pi\)
\(62\) 0 0
\(63\) 156.000i 0.311971i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 763.000i − 1.39127i −0.718394 0.695636i \(-0.755122\pi\)
0.718394 0.695636i \(-0.244878\pi\)
\(68\) 0 0
\(69\) −174.000 −0.303582
\(70\) 0 0
\(71\) 52.0000 0.0869192 0.0434596 0.999055i \(-0.486162\pi\)
0.0434596 + 0.999055i \(0.486162\pi\)
\(72\) 0 0
\(73\) 981.000i 1.57284i 0.617692 + 0.786420i \(0.288068\pi\)
−0.617692 + 0.786420i \(0.711932\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 114.000i 0.168721i
\(78\) 0 0
\(79\) 1054.00 1.50107 0.750533 0.660833i \(-0.229797\pi\)
0.750533 + 0.660833i \(0.229797\pi\)
\(80\) 0 0
\(81\) 649.000 0.890261
\(82\) 0 0
\(83\) 351.000i 0.464184i 0.972694 + 0.232092i \(0.0745570\pi\)
−0.972694 + 0.232092i \(0.925443\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 272.000i 0.335189i
\(88\) 0 0
\(89\) −799.000 −0.951616 −0.475808 0.879549i \(-0.657844\pi\)
−0.475808 + 0.879549i \(0.657844\pi\)
\(90\) 0 0
\(91\) 72.0000 0.0829412
\(92\) 0 0
\(93\) 230.000i 0.256450i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 962.000i 1.00697i 0.864003 + 0.503486i \(0.167949\pi\)
−0.864003 + 0.503486i \(0.832051\pi\)
\(98\) 0 0
\(99\) 494.000 0.501504
\(100\) 0 0
\(101\) 486.000 0.478800 0.239400 0.970921i \(-0.423049\pi\)
0.239400 + 0.970921i \(0.423049\pi\)
\(102\) 0 0
\(103\) − 1188.00i − 1.13648i −0.822864 0.568238i \(-0.807625\pi\)
0.822864 0.568238i \(-0.192375\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 1325.00i − 1.19713i −0.801075 0.598563i \(-0.795738\pi\)
0.801075 0.598563i \(-0.204262\pi\)
\(108\) 0 0
\(109\) −126.000 −0.110721 −0.0553606 0.998466i \(-0.517631\pi\)
−0.0553606 + 0.998466i \(0.517631\pi\)
\(110\) 0 0
\(111\) 182.000 0.155628
\(112\) 0 0
\(113\) − 183.000i − 0.152347i −0.997095 0.0761734i \(-0.975730\pi\)
0.997095 0.0761734i \(-0.0242703\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 312.000i − 0.246533i
\(118\) 0 0
\(119\) 450.000 0.346651
\(120\) 0 0
\(121\) −970.000 −0.728775
\(122\) 0 0
\(123\) 117.000i 0.0857686i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 902.000i 0.630233i 0.949053 + 0.315116i \(0.102044\pi\)
−0.949053 + 0.315116i \(0.897956\pi\)
\(128\) 0 0
\(129\) −372.000 −0.253897
\(130\) 0 0
\(131\) −2068.00 −1.37925 −0.689626 0.724166i \(-0.742225\pi\)
−0.689626 + 0.724166i \(0.742225\pi\)
\(132\) 0 0
\(133\) − 546.000i − 0.355971i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 1339.00i − 0.835025i −0.908671 0.417513i \(-0.862902\pi\)
0.908671 0.417513i \(-0.137098\pi\)
\(138\) 0 0
\(139\) −2939.00 −1.79340 −0.896700 0.442638i \(-0.854043\pi\)
−0.896700 + 0.442638i \(0.854043\pi\)
\(140\) 0 0
\(141\) −52.0000 −0.0310581
\(142\) 0 0
\(143\) − 228.000i − 0.133331i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 307.000i 0.172251i
\(148\) 0 0
\(149\) 208.000 0.114363 0.0571813 0.998364i \(-0.481789\pi\)
0.0571813 + 0.998364i \(0.481789\pi\)
\(150\) 0 0
\(151\) 2678.00 1.44326 0.721631 0.692278i \(-0.243393\pi\)
0.721631 + 0.692278i \(0.243393\pi\)
\(152\) 0 0
\(153\) − 1950.00i − 1.03038i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1482.00i 0.753353i 0.926345 + 0.376677i \(0.122933\pi\)
−0.926345 + 0.376677i \(0.877067\pi\)
\(158\) 0 0
\(159\) −402.000 −0.200507
\(160\) 0 0
\(161\) −1044.00 −0.511048
\(162\) 0 0
\(163\) − 1469.00i − 0.705895i −0.935643 0.352948i \(-0.885179\pi\)
0.935643 0.352948i \(-0.114821\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 4004.00i 1.85532i 0.373423 + 0.927661i \(0.378184\pi\)
−0.373423 + 0.927661i \(0.621816\pi\)
\(168\) 0 0
\(169\) 2053.00 0.934456
\(170\) 0 0
\(171\) −2366.00 −1.05809
\(172\) 0 0
\(173\) − 3224.00i − 1.41686i −0.705783 0.708428i \(-0.749405\pi\)
0.705783 0.708428i \(-0.250595\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 312.000i 0.132494i
\(178\) 0 0
\(179\) −4191.00 −1.75000 −0.875000 0.484123i \(-0.839139\pi\)
−0.875000 + 0.484123i \(0.839139\pi\)
\(180\) 0 0
\(181\) −3718.00 −1.52683 −0.763416 0.645907i \(-0.776480\pi\)
−0.763416 + 0.645907i \(0.776480\pi\)
\(182\) 0 0
\(183\) 170.000i 0.0686708i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 1425.00i − 0.557253i
\(188\) 0 0
\(189\) −318.000 −0.122387
\(190\) 0 0
\(191\) −870.000 −0.329586 −0.164793 0.986328i \(-0.552696\pi\)
−0.164793 + 0.986328i \(0.552696\pi\)
\(192\) 0 0
\(193\) − 2197.00i − 0.819396i −0.912221 0.409698i \(-0.865634\pi\)
0.912221 0.409698i \(-0.134366\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 2314.00i − 0.836882i −0.908244 0.418441i \(-0.862577\pi\)
0.908244 0.418441i \(-0.137423\pi\)
\(198\) 0 0
\(199\) −252.000 −0.0897679 −0.0448839 0.998992i \(-0.514292\pi\)
−0.0448839 + 0.998992i \(0.514292\pi\)
\(200\) 0 0
\(201\) 763.000 0.267751
\(202\) 0 0
\(203\) 1632.00i 0.564256i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 4524.00i 1.51903i
\(208\) 0 0
\(209\) −1729.00 −0.572237
\(210\) 0 0
\(211\) 741.000 0.241766 0.120883 0.992667i \(-0.461427\pi\)
0.120883 + 0.992667i \(0.461427\pi\)
\(212\) 0 0
\(213\) 52.0000i 0.0167276i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 1380.00i 0.431707i
\(218\) 0 0
\(219\) −981.000 −0.302693
\(220\) 0 0
\(221\) −900.000 −0.273939
\(222\) 0 0
\(223\) − 5092.00i − 1.52908i −0.644574 0.764542i \(-0.722965\pi\)
0.644574 0.764542i \(-0.277035\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 5876.00i − 1.71808i −0.511910 0.859039i \(-0.671062\pi\)
0.511910 0.859039i \(-0.328938\pi\)
\(228\) 0 0
\(229\) 604.000 0.174295 0.0871473 0.996195i \(-0.472225\pi\)
0.0871473 + 0.996195i \(0.472225\pi\)
\(230\) 0 0
\(231\) −114.000 −0.0324703
\(232\) 0 0
\(233\) − 278.000i − 0.0781647i −0.999236 0.0390824i \(-0.987557\pi\)
0.999236 0.0390824i \(-0.0124435\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1054.00i 0.288880i
\(238\) 0 0
\(239\) 2496.00 0.675535 0.337767 0.941230i \(-0.390328\pi\)
0.337767 + 0.941230i \(0.390328\pi\)
\(240\) 0 0
\(241\) −2567.00 −0.686120 −0.343060 0.939313i \(-0.611463\pi\)
−0.343060 + 0.939313i \(0.611463\pi\)
\(242\) 0 0
\(243\) 2080.00i 0.549103i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1092.00i 0.281305i
\(248\) 0 0
\(249\) −351.000 −0.0893322
\(250\) 0 0
\(251\) −5395.00 −1.35669 −0.678345 0.734743i \(-0.737303\pi\)
−0.678345 + 0.734743i \(0.737303\pi\)
\(252\) 0 0
\(253\) 3306.00i 0.821527i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 1490.00i − 0.361648i −0.983515 0.180824i \(-0.942123\pi\)
0.983515 0.180824i \(-0.0578765\pi\)
\(258\) 0 0
\(259\) 1092.00 0.261983
\(260\) 0 0
\(261\) 7072.00 1.67719
\(262\) 0 0
\(263\) − 3330.00i − 0.780748i −0.920656 0.390374i \(-0.872346\pi\)
0.920656 0.390374i \(-0.127654\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 799.000i − 0.183139i
\(268\) 0 0
\(269\) −6096.00 −1.38171 −0.690854 0.722994i \(-0.742765\pi\)
−0.690854 + 0.722994i \(0.742765\pi\)
\(270\) 0 0
\(271\) 6006.00 1.34627 0.673134 0.739521i \(-0.264948\pi\)
0.673134 + 0.739521i \(0.264948\pi\)
\(272\) 0 0
\(273\) 72.0000i 0.0159620i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 6976.00i 1.51317i 0.653897 + 0.756583i \(0.273133\pi\)
−0.653897 + 0.756583i \(0.726867\pi\)
\(278\) 0 0
\(279\) 5980.00 1.28320
\(280\) 0 0
\(281\) 3998.00 0.848757 0.424378 0.905485i \(-0.360493\pi\)
0.424378 + 0.905485i \(0.360493\pi\)
\(282\) 0 0
\(283\) − 13.0000i − 0.00273064i −0.999999 0.00136532i \(-0.999565\pi\)
0.999999 0.00136532i \(-0.000434594\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 702.000i 0.144382i
\(288\) 0 0
\(289\) −712.000 −0.144922
\(290\) 0 0
\(291\) −962.000 −0.193792
\(292\) 0 0
\(293\) 4466.00i 0.890466i 0.895415 + 0.445233i \(0.146879\pi\)
−0.895415 + 0.445233i \(0.853121\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 1007.00i 0.196741i
\(298\) 0 0
\(299\) 2088.00 0.403853
\(300\) 0 0
\(301\) −2232.00 −0.427410
\(302\) 0 0
\(303\) 486.000i 0.0921451i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 9041.00i − 1.68077i −0.541988 0.840386i \(-0.682328\pi\)
0.541988 0.840386i \(-0.317672\pi\)
\(308\) 0 0
\(309\) 1188.00 0.218715
\(310\) 0 0
\(311\) 346.000 0.0630864 0.0315432 0.999502i \(-0.489958\pi\)
0.0315432 + 0.999502i \(0.489958\pi\)
\(312\) 0 0
\(313\) 10646.0i 1.92252i 0.275650 + 0.961258i \(0.411107\pi\)
−0.275650 + 0.961258i \(0.588893\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 8116.00i − 1.43798i −0.695020 0.718990i \(-0.744604\pi\)
0.695020 0.718990i \(-0.255396\pi\)
\(318\) 0 0
\(319\) 5168.00 0.907061
\(320\) 0 0
\(321\) 1325.00 0.230387
\(322\) 0 0
\(323\) 6825.00i 1.17571i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 126.000i − 0.0213083i
\(328\) 0 0
\(329\) −312.000 −0.0522830
\(330\) 0 0
\(331\) 3007.00 0.499334 0.249667 0.968332i \(-0.419679\pi\)
0.249667 + 0.968332i \(0.419679\pi\)
\(332\) 0 0
\(333\) − 4732.00i − 0.778715i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 83.0000i 0.0134163i 0.999978 + 0.00670816i \(0.00213529\pi\)
−0.999978 + 0.00670816i \(0.997865\pi\)
\(338\) 0 0
\(339\) 183.000 0.0293192
\(340\) 0 0
\(341\) 4370.00 0.693985
\(342\) 0 0
\(343\) 3900.00i 0.613936i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 6189.00i − 0.957472i −0.877959 0.478736i \(-0.841095\pi\)
0.877959 0.478736i \(-0.158905\pi\)
\(348\) 0 0
\(349\) 5362.00 0.822411 0.411205 0.911543i \(-0.365108\pi\)
0.411205 + 0.911543i \(0.365108\pi\)
\(350\) 0 0
\(351\) 636.000 0.0967156
\(352\) 0 0
\(353\) 1690.00i 0.254815i 0.991850 + 0.127407i \(0.0406656\pi\)
−0.991850 + 0.127407i \(0.959334\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 450.000i 0.0667130i
\(358\) 0 0
\(359\) 1638.00 0.240809 0.120404 0.992725i \(-0.461581\pi\)
0.120404 + 0.992725i \(0.461581\pi\)
\(360\) 0 0
\(361\) 1422.00 0.207319
\(362\) 0 0
\(363\) − 970.000i − 0.140253i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 7580.00i 1.07813i 0.842265 + 0.539064i \(0.181222\pi\)
−0.842265 + 0.539064i \(0.818778\pi\)
\(368\) 0 0
\(369\) 3042.00 0.429160
\(370\) 0 0
\(371\) −2412.00 −0.337533
\(372\) 0 0
\(373\) 5630.00i 0.781529i 0.920491 + 0.390765i \(0.127789\pi\)
−0.920491 + 0.390765i \(0.872211\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 3264.00i − 0.445901i
\(378\) 0 0
\(379\) −4385.00 −0.594307 −0.297153 0.954830i \(-0.596037\pi\)
−0.297153 + 0.954830i \(0.596037\pi\)
\(380\) 0 0
\(381\) −902.000 −0.121288
\(382\) 0 0
\(383\) 12558.0i 1.67541i 0.546119 + 0.837707i \(0.316104\pi\)
−0.546119 + 0.837707i \(0.683896\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 9672.00i 1.27043i
\(388\) 0 0
\(389\) 6570.00 0.856330 0.428165 0.903701i \(-0.359160\pi\)
0.428165 + 0.903701i \(0.359160\pi\)
\(390\) 0 0
\(391\) 13050.0 1.68789
\(392\) 0 0
\(393\) − 2068.00i − 0.265437i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 1268.00i 0.160300i 0.996783 + 0.0801500i \(0.0255399\pi\)
−0.996783 + 0.0801500i \(0.974460\pi\)
\(398\) 0 0
\(399\) 546.000 0.0685067
\(400\) 0 0
\(401\) −6299.00 −0.784432 −0.392216 0.919873i \(-0.628291\pi\)
−0.392216 + 0.919873i \(0.628291\pi\)
\(402\) 0 0
\(403\) − 2760.00i − 0.341155i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 3458.00i − 0.421147i
\(408\) 0 0
\(409\) 13459.0 1.62715 0.813575 0.581459i \(-0.197518\pi\)
0.813575 + 0.581459i \(0.197518\pi\)
\(410\) 0 0
\(411\) 1339.00 0.160701
\(412\) 0 0
\(413\) 1872.00i 0.223039i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 2939.00i − 0.345140i
\(418\) 0 0
\(419\) −9567.00 −1.11546 −0.557731 0.830022i \(-0.688328\pi\)
−0.557731 + 0.830022i \(0.688328\pi\)
\(420\) 0 0
\(421\) 2708.00 0.313491 0.156746 0.987639i \(-0.449900\pi\)
0.156746 + 0.987639i \(0.449900\pi\)
\(422\) 0 0
\(423\) 1352.00i 0.155405i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1020.00i 0.115600i
\(428\) 0 0
\(429\) 228.000 0.0256595
\(430\) 0 0
\(431\) −5126.00 −0.572879 −0.286439 0.958098i \(-0.592472\pi\)
−0.286439 + 0.958098i \(0.592472\pi\)
\(432\) 0 0
\(433\) − 11445.0i − 1.27023i −0.772416 0.635117i \(-0.780952\pi\)
0.772416 0.635117i \(-0.219048\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 15834.0i − 1.73328i
\(438\) 0 0
\(439\) −5096.00 −0.554029 −0.277015 0.960866i \(-0.589345\pi\)
−0.277015 + 0.960866i \(0.589345\pi\)
\(440\) 0 0
\(441\) 7982.00 0.861894
\(442\) 0 0
\(443\) 13247.0i 1.42073i 0.703833 + 0.710366i \(0.251470\pi\)
−0.703833 + 0.710366i \(0.748530\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 208.000i 0.0220091i
\(448\) 0 0
\(449\) −7449.00 −0.782940 −0.391470 0.920191i \(-0.628033\pi\)
−0.391470 + 0.920191i \(0.628033\pi\)
\(450\) 0 0
\(451\) 2223.00 0.232100
\(452\) 0 0
\(453\) 2678.00i 0.277756i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 3497.00i 0.357949i 0.983854 + 0.178975i \(0.0572780\pi\)
−0.983854 + 0.178975i \(0.942722\pi\)
\(458\) 0 0
\(459\) 3975.00 0.404220
\(460\) 0 0
\(461\) −13108.0 −1.32430 −0.662148 0.749373i \(-0.730355\pi\)
−0.662148 + 0.749373i \(0.730355\pi\)
\(462\) 0 0
\(463\) − 5428.00i − 0.544839i −0.962179 0.272420i \(-0.912176\pi\)
0.962179 0.272420i \(-0.0878239\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 1516.00i 0.150219i 0.997175 + 0.0751093i \(0.0239306\pi\)
−0.997175 + 0.0751093i \(0.976069\pi\)
\(468\) 0 0
\(469\) 4578.00 0.450730
\(470\) 0 0
\(471\) −1482.00 −0.144983
\(472\) 0 0
\(473\) 7068.00i 0.687076i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 10452.0i 1.00328i
\(478\) 0 0
\(479\) −14762.0 −1.40813 −0.704064 0.710137i \(-0.748633\pi\)
−0.704064 + 0.710137i \(0.748633\pi\)
\(480\) 0 0
\(481\) −2184.00 −0.207031
\(482\) 0 0
\(483\) − 1044.00i − 0.0983512i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 3926.00i 0.365306i 0.983177 + 0.182653i \(0.0584685\pi\)
−0.983177 + 0.182653i \(0.941532\pi\)
\(488\) 0 0
\(489\) 1469.00 0.135850
\(490\) 0 0
\(491\) −996.000 −0.0915455 −0.0457728 0.998952i \(-0.514575\pi\)
−0.0457728 + 0.998952i \(0.514575\pi\)
\(492\) 0 0
\(493\) − 20400.0i − 1.86363i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 312.000i 0.0281592i
\(498\) 0 0
\(499\) 7804.00 0.700110 0.350055 0.936729i \(-0.386163\pi\)
0.350055 + 0.936729i \(0.386163\pi\)
\(500\) 0 0
\(501\) −4004.00 −0.357057
\(502\) 0 0
\(503\) − 16732.0i − 1.48319i −0.670850 0.741593i \(-0.734070\pi\)
0.670850 0.741593i \(-0.265930\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 2053.00i 0.179836i
\(508\) 0 0
\(509\) −10426.0 −0.907906 −0.453953 0.891026i \(-0.649987\pi\)
−0.453953 + 0.891026i \(0.649987\pi\)
\(510\) 0 0
\(511\) −5886.00 −0.509552
\(512\) 0 0
\(513\) − 4823.00i − 0.415089i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 988.000i 0.0840468i
\(518\) 0 0
\(519\) 3224.00 0.272674
\(520\) 0 0
\(521\) −2235.00 −0.187941 −0.0939704 0.995575i \(-0.529956\pi\)
−0.0939704 + 0.995575i \(0.529956\pi\)
\(522\) 0 0
\(523\) − 10855.0i − 0.907564i −0.891113 0.453782i \(-0.850074\pi\)
0.891113 0.453782i \(-0.149926\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 17250.0i − 1.42585i
\(528\) 0 0
\(529\) −18109.0 −1.48837
\(530\) 0 0
\(531\) 8112.00 0.662958
\(532\) 0 0
\(533\) − 1404.00i − 0.114098i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 4191.00i − 0.336788i
\(538\) 0 0
\(539\) 5833.00 0.466132
\(540\) 0 0
\(541\) 10608.0 0.843019 0.421510 0.906824i \(-0.361500\pi\)
0.421510 + 0.906824i \(0.361500\pi\)
\(542\) 0 0
\(543\) − 3718.00i − 0.293839i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 11583.0i 0.905399i 0.891663 + 0.452700i \(0.149539\pi\)
−0.891663 + 0.452700i \(0.850461\pi\)
\(548\) 0 0
\(549\) 4420.00 0.343608
\(550\) 0 0
\(551\) −24752.0 −1.91374
\(552\) 0 0
\(553\) 6324.00i 0.486300i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 17244.0i 1.31176i 0.754864 + 0.655881i \(0.227703\pi\)
−0.754864 + 0.655881i \(0.772297\pi\)
\(558\) 0 0
\(559\) 4464.00 0.337759
\(560\) 0 0
\(561\) 1425.00 0.107243
\(562\) 0 0
\(563\) − 18416.0i − 1.37858i −0.724484 0.689291i \(-0.757922\pi\)
0.724484 0.689291i \(-0.242078\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 3894.00i 0.288417i
\(568\) 0 0
\(569\) −11913.0 −0.877713 −0.438857 0.898557i \(-0.644616\pi\)
−0.438857 + 0.898557i \(0.644616\pi\)
\(570\) 0 0
\(571\) −3900.00 −0.285832 −0.142916 0.989735i \(-0.545648\pi\)
−0.142916 + 0.989735i \(0.545648\pi\)
\(572\) 0 0
\(573\) − 870.000i − 0.0634289i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 24899.0i 1.79646i 0.439523 + 0.898231i \(0.355147\pi\)
−0.439523 + 0.898231i \(0.644853\pi\)
\(578\) 0 0
\(579\) 2197.00 0.157693
\(580\) 0 0
\(581\) −2106.00 −0.150381
\(582\) 0 0
\(583\) 7638.00i 0.542596i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 1751.00i − 0.123120i −0.998103 0.0615601i \(-0.980392\pi\)
0.998103 0.0615601i \(-0.0196076\pi\)
\(588\) 0 0
\(589\) −20930.0 −1.46419
\(590\) 0 0
\(591\) 2314.00 0.161058
\(592\) 0 0
\(593\) − 10887.0i − 0.753922i −0.926229 0.376961i \(-0.876969\pi\)
0.926229 0.376961i \(-0.123031\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 252.000i − 0.0172758i
\(598\) 0 0
\(599\) −14650.0 −0.999303 −0.499652 0.866226i \(-0.666539\pi\)
−0.499652 + 0.866226i \(0.666539\pi\)
\(600\) 0 0
\(601\) −4237.00 −0.287572 −0.143786 0.989609i \(-0.545928\pi\)
−0.143786 + 0.989609i \(0.545928\pi\)
\(602\) 0 0
\(603\) − 19838.0i − 1.33974i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 11440.0i − 0.764968i −0.923962 0.382484i \(-0.875069\pi\)
0.923962 0.382484i \(-0.124931\pi\)
\(608\) 0 0
\(609\) −1632.00 −0.108591
\(610\) 0 0
\(611\) 624.000 0.0413164
\(612\) 0 0
\(613\) 19370.0i 1.27626i 0.769929 + 0.638130i \(0.220292\pi\)
−0.769929 + 0.638130i \(0.779708\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 21346.0i − 1.39280i −0.717654 0.696400i \(-0.754784\pi\)
0.717654 0.696400i \(-0.245216\pi\)
\(618\) 0 0
\(619\) 7436.00 0.482840 0.241420 0.970421i \(-0.422387\pi\)
0.241420 + 0.970421i \(0.422387\pi\)
\(620\) 0 0
\(621\) −9222.00 −0.595920
\(622\) 0 0
\(623\) − 4794.00i − 0.308295i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 1729.00i − 0.110127i
\(628\) 0 0
\(629\) −13650.0 −0.865280
\(630\) 0 0
\(631\) −22490.0 −1.41888 −0.709440 0.704766i \(-0.751052\pi\)
−0.709440 + 0.704766i \(0.751052\pi\)
\(632\) 0 0
\(633\) 741.000i 0.0465278i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 3684.00i − 0.229145i
\(638\) 0 0
\(639\) 1352.00 0.0837000
\(640\) 0 0
\(641\) −16086.0 −0.991199 −0.495600 0.868551i \(-0.665052\pi\)
−0.495600 + 0.868551i \(0.665052\pi\)
\(642\) 0 0
\(643\) − 2396.00i − 0.146950i −0.997297 0.0734751i \(-0.976591\pi\)
0.997297 0.0734751i \(-0.0234090\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 23244.0i − 1.41239i −0.708018 0.706195i \(-0.750410\pi\)
0.708018 0.706195i \(-0.249590\pi\)
\(648\) 0 0
\(649\) 5928.00 0.358543
\(650\) 0 0
\(651\) −1380.00 −0.0830821
\(652\) 0 0
\(653\) − 13598.0i − 0.814902i −0.913227 0.407451i \(-0.866418\pi\)
0.913227 0.407451i \(-0.133582\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 25506.0i 1.51459i
\(658\) 0 0
\(659\) 9751.00 0.576396 0.288198 0.957571i \(-0.406944\pi\)
0.288198 + 0.957571i \(0.406944\pi\)
\(660\) 0 0
\(661\) 19104.0 1.12414 0.562072 0.827088i \(-0.310004\pi\)
0.562072 + 0.827088i \(0.310004\pi\)
\(662\) 0 0
\(663\) − 900.000i − 0.0527196i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 47328.0i 2.74745i
\(668\) 0 0
\(669\) 5092.00 0.294272
\(670\) 0 0
\(671\) 3230.00 0.185831
\(672\) 0 0
\(673\) − 25402.0i − 1.45494i −0.686139 0.727470i \(-0.740696\pi\)
0.686139 0.727470i \(-0.259304\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 2224.00i − 0.126256i −0.998005 0.0631280i \(-0.979892\pi\)
0.998005 0.0631280i \(-0.0201076\pi\)
\(678\) 0 0
\(679\) −5772.00 −0.326228
\(680\) 0 0
\(681\) 5876.00 0.330644
\(682\) 0 0
\(683\) 8281.00i 0.463929i 0.972724 + 0.231965i \(0.0745154\pi\)
−0.972724 + 0.231965i \(0.925485\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 604.000i 0.0335430i
\(688\) 0 0
\(689\) 4824.00 0.266734
\(690\) 0 0
\(691\) −481.000 −0.0264806 −0.0132403 0.999912i \(-0.504215\pi\)
−0.0132403 + 0.999912i \(0.504215\pi\)
\(692\) 0 0
\(693\) 2964.00i 0.162472i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 8775.00i − 0.476868i
\(698\) 0 0
\(699\) 278.000 0.0150428
\(700\) 0 0
\(701\) −16788.0 −0.904528 −0.452264 0.891884i \(-0.649383\pi\)
−0.452264 + 0.891884i \(0.649383\pi\)
\(702\) 0 0
\(703\) 16562.0i 0.888546i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 2916.00i 0.155117i
\(708\) 0 0
\(709\) 23452.0 1.24225 0.621127 0.783710i \(-0.286675\pi\)
0.621127 + 0.783710i \(0.286675\pi\)
\(710\) 0 0
\(711\) 27404.0 1.44547
\(712\) 0 0
\(713\) 40020.0i 2.10205i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 2496.00i 0.130007i
\(718\) 0 0
\(719\) −20886.0 −1.08333 −0.541666 0.840594i \(-0.682206\pi\)
−0.541666 + 0.840594i \(0.682206\pi\)
\(720\) 0 0
\(721\) 7128.00 0.368184
\(722\) 0 0
\(723\) − 2567.00i − 0.132044i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 22576.0i − 1.15172i −0.817550 0.575858i \(-0.804668\pi\)
0.817550 0.575858i \(-0.195332\pi\)
\(728\) 0 0
\(729\) 15443.0 0.784586
\(730\) 0 0
\(731\) 27900.0 1.41165
\(732\) 0 0
\(733\) − 35308.0i − 1.77917i −0.456771 0.889584i \(-0.650994\pi\)
0.456771 0.889584i \(-0.349006\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 14497.0i − 0.724564i
\(738\) 0 0
\(739\) 188.000 0.00935818 0.00467909 0.999989i \(-0.498511\pi\)
0.00467909 + 0.999989i \(0.498511\pi\)
\(740\) 0 0
\(741\) −1092.00 −0.0541371
\(742\) 0 0
\(743\) 29870.0i 1.47486i 0.675421 + 0.737432i \(0.263962\pi\)
−0.675421 + 0.737432i \(0.736038\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 9126.00i 0.446992i
\(748\) 0 0
\(749\) 7950.00 0.387833
\(750\) 0 0
\(751\) −22784.0 −1.10706 −0.553529 0.832830i \(-0.686719\pi\)
−0.553529 + 0.832830i \(0.686719\pi\)
\(752\) 0 0
\(753\) − 5395.00i − 0.261095i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 28496.0i − 1.36817i −0.729402 0.684085i \(-0.760202\pi\)
0.729402 0.684085i \(-0.239798\pi\)
\(758\) 0 0
\(759\) −3306.00 −0.158103
\(760\) 0 0
\(761\) 7397.00 0.352354 0.176177 0.984359i \(-0.443627\pi\)
0.176177 + 0.984359i \(0.443627\pi\)
\(762\) 0 0
\(763\) − 756.000i − 0.0358703i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 3744.00i − 0.176256i
\(768\) 0 0
\(769\) 8883.00 0.416553 0.208276 0.978070i \(-0.433215\pi\)
0.208276 + 0.978070i \(0.433215\pi\)
\(770\) 0 0
\(771\) 1490.00 0.0695993
\(772\) 0 0
\(773\) 33960.0i 1.58015i 0.613010 + 0.790075i \(0.289959\pi\)
−0.613010 + 0.790075i \(0.710041\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 1092.00i 0.0504186i
\(778\) 0 0
\(779\) −10647.0 −0.489690
\(780\) 0 0
\(781\) 988.000 0.0452669
\(782\) 0 0
\(783\) 14416.0i 0.657964i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 32084.0i 1.45320i 0.687059 + 0.726602i \(0.258901\pi\)
−0.687059 + 0.726602i \(0.741099\pi\)
\(788\) 0 0
\(789\) 3330.00 0.150255
\(790\) 0 0
\(791\) 1098.00 0.0493557
\(792\) 0 0
\(793\) − 2040.00i − 0.0913525i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 4766.00i − 0.211820i −0.994376 0.105910i \(-0.966224\pi\)
0.994376 0.105910i \(-0.0337755\pi\)
\(798\) 0 0
\(799\) 3900.00 0.172681
\(800\) 0 0
\(801\) −20774.0 −0.916371
\(802\) 0 0
\(803\) 18639.0i 0.819123i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 6096.00i − 0.265910i
\(808\) 0 0
\(809\) 31278.0 1.35930 0.679651 0.733535i \(-0.262131\pi\)
0.679651 + 0.733535i \(0.262131\pi\)
\(810\) 0 0
\(811\) 29956.0 1.29704 0.648519 0.761199i \(-0.275389\pi\)
0.648519 + 0.761199i \(0.275389\pi\)
\(812\) 0 0
\(813\) 6006.00i 0.259089i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 33852.0i − 1.44961i
\(818\) 0 0
\(819\) 1872.00 0.0798693
\(820\) 0 0
\(821\) 14642.0 0.622423 0.311212 0.950341i \(-0.399265\pi\)
0.311212 + 0.950341i \(0.399265\pi\)
\(822\) 0 0
\(823\) 20844.0i 0.882839i 0.897301 + 0.441419i \(0.145525\pi\)
−0.897301 + 0.441419i \(0.854475\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 23751.0i − 0.998674i −0.866408 0.499337i \(-0.833577\pi\)
0.866408 0.499337i \(-0.166423\pi\)
\(828\) 0 0
\(829\) −11380.0 −0.476772 −0.238386 0.971171i \(-0.576618\pi\)
−0.238386 + 0.971171i \(0.576618\pi\)
\(830\) 0 0
\(831\) −6976.00 −0.291209
\(832\) 0 0
\(833\) − 23025.0i − 0.957706i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 12190.0i 0.503403i
\(838\) 0 0
\(839\) −29744.0 −1.22393 −0.611965 0.790885i \(-0.709621\pi\)
−0.611965 + 0.790885i \(0.709621\pi\)
\(840\) 0 0
\(841\) 49595.0 2.03350
\(842\) 0 0
\(843\) 3998.00i 0.163343i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 5820.00i − 0.236101i
\(848\) 0 0
\(849\) 13.0000 0.000525511 0
\(850\) 0 0
\(851\) 31668.0 1.27563
\(852\) 0 0
\(853\) 37726.0i 1.51432i 0.653230 + 0.757159i \(0.273413\pi\)
−0.653230 + 0.757159i \(0.726587\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 5429.00i 0.216396i 0.994129 + 0.108198i \(0.0345080\pi\)
−0.994129 + 0.108198i \(0.965492\pi\)
\(858\) 0 0
\(859\) 32149.0 1.27696 0.638481 0.769638i \(-0.279563\pi\)
0.638481 + 0.769638i \(0.279563\pi\)
\(860\) 0 0
\(861\) −702.000 −0.0277864
\(862\) 0 0
\(863\) − 29176.0i − 1.15083i −0.817863 0.575413i \(-0.804841\pi\)
0.817863 0.575413i \(-0.195159\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 712.000i − 0.0278902i
\(868\) 0 0
\(869\) 20026.0 0.781744
\(870\) 0 0
\(871\) −9156.00 −0.356187
\(872\) 0 0
\(873\) 25012.0i 0.969677i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 20068.0i 0.772689i 0.922355 + 0.386344i \(0.126262\pi\)
−0.922355 + 0.386344i \(0.873738\pi\)
\(878\) 0 0
\(879\) −4466.00 −0.171370
\(880\) 0 0
\(881\) −36850.0 −1.40920 −0.704602 0.709603i \(-0.748874\pi\)
−0.704602 + 0.709603i \(0.748874\pi\)
\(882\) 0 0
\(883\) 30025.0i 1.14431i 0.820147 + 0.572153i \(0.193892\pi\)
−0.820147 + 0.572153i \(0.806108\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 156.000i − 0.00590526i −0.999996 0.00295263i \(-0.999060\pi\)
0.999996 0.00295263i \(-0.000939853\pi\)
\(888\) 0 0
\(889\) −5412.00 −0.204176
\(890\) 0 0
\(891\) 12331.0 0.463641
\(892\) 0 0
\(893\) − 4732.00i − 0.177324i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 2088.00i 0.0777216i
\(898\) 0 0
\(899\) 62560.0 2.32090
\(900\) 0 0
\(901\) 30150.0 1.11481
\(902\) 0 0
\(903\) − 2232.00i − 0.0822550i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 356.000i 0.0130328i 0.999979 + 0.00651642i \(0.00207426\pi\)
−0.999979 + 0.00651642i \(0.997926\pi\)
\(908\) 0 0
\(909\) 12636.0 0.461067
\(910\) 0 0
\(911\) −8748.00 −0.318149 −0.159075 0.987267i \(-0.550851\pi\)
−0.159075 + 0.987267i \(0.550851\pi\)
\(912\) 0 0
\(913\) 6669.00i 0.241743i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 12408.0i − 0.446836i
\(918\) 0 0
\(919\) 36974.0 1.32716 0.663580 0.748105i \(-0.269036\pi\)
0.663580 + 0.748105i \(0.269036\pi\)
\(920\) 0 0
\(921\) 9041.00 0.323465
\(922\) 0 0
\(923\) − 624.000i − 0.0222527i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 30888.0i − 1.09439i
\(928\) 0 0
\(929\) −44382.0 −1.56741 −0.783706 0.621132i \(-0.786673\pi\)
−0.783706 + 0.621132i \(0.786673\pi\)
\(930\) 0 0
\(931\) −27937.0 −0.983457
\(932\) 0 0
\(933\) 346.000i 0.0121410i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 2445.00i − 0.0852451i −0.999091 0.0426226i \(-0.986429\pi\)
0.999091 0.0426226i \(-0.0135713\pi\)
\(938\) 0 0
\(939\) −10646.0 −0.369988
\(940\) 0 0
\(941\) −7076.00 −0.245134 −0.122567 0.992460i \(-0.539113\pi\)
−0.122567 + 0.992460i \(0.539113\pi\)
\(942\) 0 0
\(943\) 20358.0i 0.703020i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 1560.00i − 0.0535303i −0.999642 0.0267651i \(-0.991479\pi\)
0.999642 0.0267651i \(-0.00852063\pi\)
\(948\) 0 0
\(949\) 11772.0 0.402672
\(950\) 0 0
\(951\) 8116.00 0.276740
\(952\) 0 0
\(953\) 35087.0i 1.19263i 0.802749 + 0.596317i \(0.203370\pi\)
−0.802749 + 0.596317i \(0.796630\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 5168.00i 0.174564i
\(958\) 0 0
\(959\) 8034.00 0.270523
\(960\) 0 0
\(961\) 23109.0 0.775704
\(962\) 0 0
\(963\) − 34450.0i − 1.15279i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 9360.00i 0.311269i 0.987815 + 0.155635i \(0.0497422\pi\)
−0.987815 + 0.155635i \(0.950258\pi\)
\(968\) 0 0
\(969\) −6825.00 −0.226265
\(970\) 0 0
\(971\) −22269.0 −0.735990 −0.367995 0.929828i \(-0.619956\pi\)
−0.367995 + 0.929828i \(0.619956\pi\)
\(972\) 0 0
\(973\) − 17634.0i − 0.581007i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 37249.0i 1.21976i 0.792496 + 0.609878i \(0.208781\pi\)
−0.792496 + 0.609878i \(0.791219\pi\)
\(978\) 0 0
\(979\) −15181.0 −0.495594
\(980\) 0 0
\(981\) −3276.00 −0.106620
\(982\) 0 0
\(983\) − 17602.0i − 0.571126i −0.958360 0.285563i \(-0.907819\pi\)
0.958360 0.285563i \(-0.0921805\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 312.000i − 0.0100619i
\(988\) 0 0
\(989\) −64728.0 −2.08112
\(990\) 0 0
\(991\) 26402.0 0.846304 0.423152 0.906059i \(-0.360924\pi\)
0.423152 + 0.906059i \(0.360924\pi\)
\(992\) 0 0
\(993\) 3007.00i 0.0960969i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 16836.0i 0.534806i 0.963585 + 0.267403i \(0.0861655\pi\)
−0.963585 + 0.267403i \(0.913835\pi\)
\(998\) 0 0
\(999\) 9646.00 0.305491
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 400.4.c.m.49.2 2
4.3 odd 2 200.4.c.g.49.1 2
5.2 odd 4 400.4.a.k.1.1 1
5.3 odd 4 400.4.a.j.1.1 1
5.4 even 2 inner 400.4.c.m.49.1 2
12.11 even 2 1800.4.f.p.649.1 2
20.3 even 4 200.4.a.f.1.1 yes 1
20.7 even 4 200.4.a.e.1.1 1
20.19 odd 2 200.4.c.g.49.2 2
40.3 even 4 1600.4.a.w.1.1 1
40.13 odd 4 1600.4.a.be.1.1 1
40.27 even 4 1600.4.a.bf.1.1 1
40.37 odd 4 1600.4.a.v.1.1 1
60.23 odd 4 1800.4.a.l.1.1 1
60.47 odd 4 1800.4.a.w.1.1 1
60.59 even 2 1800.4.f.p.649.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
200.4.a.e.1.1 1 20.7 even 4
200.4.a.f.1.1 yes 1 20.3 even 4
200.4.c.g.49.1 2 4.3 odd 2
200.4.c.g.49.2 2 20.19 odd 2
400.4.a.j.1.1 1 5.3 odd 4
400.4.a.k.1.1 1 5.2 odd 4
400.4.c.m.49.1 2 5.4 even 2 inner
400.4.c.m.49.2 2 1.1 even 1 trivial
1600.4.a.v.1.1 1 40.37 odd 4
1600.4.a.w.1.1 1 40.3 even 4
1600.4.a.be.1.1 1 40.13 odd 4
1600.4.a.bf.1.1 1 40.27 even 4
1800.4.a.l.1.1 1 60.23 odd 4
1800.4.a.w.1.1 1 60.47 odd 4
1800.4.f.p.649.1 2 12.11 even 2
1800.4.f.p.649.2 2 60.59 even 2