Properties

Label 448.3.r.b.191.1
Level $448$
Weight $3$
Character 448.191
Analytic conductor $12.207$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [448,3,Mod(191,448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(448, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("448.191");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 448.r (of order \(6\), degree \(2\), 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: \(12.2071158433\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) 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 224)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 191.1
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 448.191
Dual form 448.3.r.b.319.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.866025 + 0.500000i) q^{3} +(0.500000 - 0.866025i) q^{5} +7.00000i q^{7} +(-4.00000 + 6.92820i) q^{9} +O(q^{10})\) \(q+(-0.866025 + 0.500000i) q^{3} +(0.500000 - 0.866025i) q^{5} +7.00000i q^{7} +(-4.00000 + 6.92820i) q^{9} +(14.7224 - 8.50000i) q^{11} -24.0000 q^{13} +1.00000i q^{15} +(-0.500000 - 0.866025i) q^{17} +(-6.06218 - 3.50000i) q^{19} +(-3.50000 - 6.06218i) q^{21} +(-6.06218 - 3.50000i) q^{23} +(12.0000 + 20.7846i) q^{25} -17.0000i q^{27} -24.0000 q^{29} +(-35.5070 + 20.5000i) q^{31} +(-8.50000 + 14.7224i) q^{33} +(6.06218 + 3.50000i) q^{35} +(-24.5000 + 42.4352i) q^{37} +(20.7846 - 12.0000i) q^{39} -48.0000 q^{41} +24.0000i q^{43} +(4.00000 + 6.92820i) q^{45} +(47.6314 + 27.5000i) q^{47} -49.0000 q^{49} +(0.866025 + 0.500000i) q^{51} +(-12.5000 - 21.6506i) q^{53} -17.0000i q^{55} +7.00000 q^{57} +(-14.7224 + 8.50000i) q^{59} +(-0.500000 + 0.866025i) q^{61} +(-48.4974 - 28.0000i) q^{63} +(-12.0000 + 20.7846i) q^{65} +(-56.2917 + 32.5000i) q^{67} +7.00000 q^{69} -96.0000i q^{71} +(-47.5000 - 82.2724i) q^{73} +(-20.7846 - 12.0000i) q^{75} +(59.5000 + 103.057i) q^{77} +(35.5070 + 20.5000i) q^{79} +(-27.5000 - 47.6314i) q^{81} +72.0000i q^{83} -1.00000 q^{85} +(20.7846 - 12.0000i) q^{87} +(-47.5000 + 82.2724i) q^{89} -168.000i q^{91} +(20.5000 - 35.5070i) q^{93} +(-6.06218 + 3.50000i) q^{95} +144.000 q^{97} +136.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{5} - 16 q^{9} - 96 q^{13} - 2 q^{17} - 14 q^{21} + 48 q^{25} - 96 q^{29} - 34 q^{33} - 98 q^{37} - 192 q^{41} + 16 q^{45} - 196 q^{49} - 50 q^{53} + 28 q^{57} - 2 q^{61} - 48 q^{65} + 28 q^{69} - 190 q^{73} + 238 q^{77} - 110 q^{81} - 4 q^{85} - 190 q^{89} + 82 q^{93} + 576 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(1\)

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
\(3\) −0.866025 + 0.500000i −0.288675 + 0.166667i −0.637344 0.770579i \(-0.719967\pi\)
0.348669 + 0.937246i \(0.386634\pi\)
\(4\) 0 0
\(5\) 0.500000 0.866025i 0.100000 0.173205i −0.811684 0.584096i \(-0.801449\pi\)
0.911684 + 0.410891i \(0.134782\pi\)
\(6\) 0 0
\(7\) 7.00000i 1.00000i
\(8\) 0 0
\(9\) −4.00000 + 6.92820i −0.444444 + 0.769800i
\(10\) 0 0
\(11\) 14.7224 8.50000i 1.33840 0.772727i 0.351832 0.936063i \(-0.385559\pi\)
0.986571 + 0.163336i \(0.0522254\pi\)
\(12\) 0 0
\(13\) −24.0000 −1.84615 −0.923077 0.384615i \(-0.874334\pi\)
−0.923077 + 0.384615i \(0.874334\pi\)
\(14\) 0 0
\(15\) 1.00000i 0.0666667i
\(16\) 0 0
\(17\) −0.500000 0.866025i −0.0294118 0.0509427i 0.850945 0.525255i \(-0.176030\pi\)
−0.880357 + 0.474312i \(0.842697\pi\)
\(18\) 0 0
\(19\) −6.06218 3.50000i −0.319062 0.184211i 0.331912 0.943310i \(-0.392306\pi\)
−0.650974 + 0.759100i \(0.725639\pi\)
\(20\) 0 0
\(21\) −3.50000 6.06218i −0.166667 0.288675i
\(22\) 0 0
\(23\) −6.06218 3.50000i −0.263573 0.152174i 0.362390 0.932026i \(-0.381961\pi\)
−0.625963 + 0.779852i \(0.715294\pi\)
\(24\) 0 0
\(25\) 12.0000 + 20.7846i 0.480000 + 0.831384i
\(26\) 0 0
\(27\) 17.0000i 0.629630i
\(28\) 0 0
\(29\) −24.0000 −0.827586 −0.413793 0.910371i \(-0.635796\pi\)
−0.413793 + 0.910371i \(0.635796\pi\)
\(30\) 0 0
\(31\) −35.5070 + 20.5000i −1.14539 + 0.661290i −0.947759 0.318987i \(-0.896658\pi\)
−0.197629 + 0.980277i \(0.563324\pi\)
\(32\) 0 0
\(33\) −8.50000 + 14.7224i −0.257576 + 0.446134i
\(34\) 0 0
\(35\) 6.06218 + 3.50000i 0.173205 + 0.100000i
\(36\) 0 0
\(37\) −24.5000 + 42.4352i −0.662162 + 1.14690i 0.317884 + 0.948130i \(0.397028\pi\)
−0.980046 + 0.198769i \(0.936306\pi\)
\(38\) 0 0
\(39\) 20.7846 12.0000i 0.532939 0.307692i
\(40\) 0 0
\(41\) −48.0000 −1.17073 −0.585366 0.810769i \(-0.699049\pi\)
−0.585366 + 0.810769i \(0.699049\pi\)
\(42\) 0 0
\(43\) 24.0000i 0.558140i 0.960271 + 0.279070i \(0.0900261\pi\)
−0.960271 + 0.279070i \(0.909974\pi\)
\(44\) 0 0
\(45\) 4.00000 + 6.92820i 0.0888889 + 0.153960i
\(46\) 0 0
\(47\) 47.6314 + 27.5000i 1.01343 + 0.585106i 0.912195 0.409756i \(-0.134386\pi\)
0.101239 + 0.994862i \(0.467719\pi\)
\(48\) 0 0
\(49\) −49.0000 −1.00000
\(50\) 0 0
\(51\) 0.866025 + 0.500000i 0.0169809 + 0.00980392i
\(52\) 0 0
\(53\) −12.5000 21.6506i −0.235849 0.408503i 0.723670 0.690146i \(-0.242454\pi\)
−0.959519 + 0.281644i \(0.909120\pi\)
\(54\) 0 0
\(55\) 17.0000i 0.309091i
\(56\) 0 0
\(57\) 7.00000 0.122807
\(58\) 0 0
\(59\) −14.7224 + 8.50000i −0.249533 + 0.144068i −0.619550 0.784957i \(-0.712685\pi\)
0.370018 + 0.929025i \(0.379352\pi\)
\(60\) 0 0
\(61\) −0.500000 + 0.866025i −0.00819672 + 0.0141971i −0.870095 0.492885i \(-0.835942\pi\)
0.861898 + 0.507082i \(0.169276\pi\)
\(62\) 0 0
\(63\) −48.4974 28.0000i −0.769800 0.444444i
\(64\) 0 0
\(65\) −12.0000 + 20.7846i −0.184615 + 0.319763i
\(66\) 0 0
\(67\) −56.2917 + 32.5000i −0.840174 + 0.485075i −0.857323 0.514778i \(-0.827874\pi\)
0.0171494 + 0.999853i \(0.494541\pi\)
\(68\) 0 0
\(69\) 7.00000 0.101449
\(70\) 0 0
\(71\) 96.0000i 1.35211i −0.736850 0.676056i \(-0.763688\pi\)
0.736850 0.676056i \(-0.236312\pi\)
\(72\) 0 0
\(73\) −47.5000 82.2724i −0.650685 1.12702i −0.982957 0.183836i \(-0.941149\pi\)
0.332272 0.943184i \(-0.392185\pi\)
\(74\) 0 0
\(75\) −20.7846 12.0000i −0.277128 0.160000i
\(76\) 0 0
\(77\) 59.5000 + 103.057i 0.772727 + 1.33840i
\(78\) 0 0
\(79\) 35.5070 + 20.5000i 0.449456 + 0.259494i 0.707601 0.706613i \(-0.249778\pi\)
−0.258144 + 0.966106i \(0.583111\pi\)
\(80\) 0 0
\(81\) −27.5000 47.6314i −0.339506 0.588042i
\(82\) 0 0
\(83\) 72.0000i 0.867470i 0.901041 + 0.433735i \(0.142805\pi\)
−0.901041 + 0.433735i \(0.857195\pi\)
\(84\) 0 0
\(85\) −1.00000 −0.0117647
\(86\) 0 0
\(87\) 20.7846 12.0000i 0.238904 0.137931i
\(88\) 0 0
\(89\) −47.5000 + 82.2724i −0.533708 + 0.924409i 0.465517 + 0.885039i \(0.345868\pi\)
−0.999225 + 0.0393701i \(0.987465\pi\)
\(90\) 0 0
\(91\) 168.000i 1.84615i
\(92\) 0 0
\(93\) 20.5000 35.5070i 0.220430 0.381796i
\(94\) 0 0
\(95\) −6.06218 + 3.50000i −0.0638124 + 0.0368421i
\(96\) 0 0
\(97\) 144.000 1.48454 0.742268 0.670103i \(-0.233750\pi\)
0.742268 + 0.670103i \(0.233750\pi\)
\(98\) 0 0
\(99\) 136.000i 1.37374i
\(100\) 0 0
\(101\) 36.5000 + 63.2199i 0.361386 + 0.625939i 0.988189 0.153239i \(-0.0489703\pi\)
−0.626803 + 0.779178i \(0.715637\pi\)
\(102\) 0 0
\(103\) 77.0763 + 44.5000i 0.748313 + 0.432039i 0.825084 0.565010i \(-0.191128\pi\)
−0.0767709 + 0.997049i \(0.524461\pi\)
\(104\) 0 0
\(105\) −7.00000 −0.0666667
\(106\) 0 0
\(107\) 160.215 + 92.5000i 1.49733 + 0.864486i 0.999995 0.00307068i \(-0.000977429\pi\)
0.497338 + 0.867557i \(0.334311\pi\)
\(108\) 0 0
\(109\) −35.5000 61.4878i −0.325688 0.564108i 0.655963 0.754793i \(-0.272263\pi\)
−0.981651 + 0.190684i \(0.938929\pi\)
\(110\) 0 0
\(111\) 49.0000i 0.441441i
\(112\) 0 0
\(113\) 96.0000 0.849558 0.424779 0.905297i \(-0.360352\pi\)
0.424779 + 0.905297i \(0.360352\pi\)
\(114\) 0 0
\(115\) −6.06218 + 3.50000i −0.0527146 + 0.0304348i
\(116\) 0 0
\(117\) 96.0000 166.277i 0.820513 1.42117i
\(118\) 0 0
\(119\) 6.06218 3.50000i 0.0509427 0.0294118i
\(120\) 0 0
\(121\) 84.0000 145.492i 0.694215 1.20242i
\(122\) 0 0
\(123\) 41.5692 24.0000i 0.337961 0.195122i
\(124\) 0 0
\(125\) 49.0000 0.392000
\(126\) 0 0
\(127\) 144.000i 1.13386i 0.823767 + 0.566929i \(0.191869\pi\)
−0.823767 + 0.566929i \(0.808131\pi\)
\(128\) 0 0
\(129\) −12.0000 20.7846i −0.0930233 0.161121i
\(130\) 0 0
\(131\) −160.215 92.5000i −1.22301 0.706107i −0.257454 0.966291i \(-0.582884\pi\)
−0.965559 + 0.260184i \(0.916217\pi\)
\(132\) 0 0
\(133\) 24.5000 42.4352i 0.184211 0.319062i
\(134\) 0 0
\(135\) −14.7224 8.50000i −0.109055 0.0629630i
\(136\) 0 0
\(137\) 71.5000 + 123.842i 0.521898 + 0.903954i 0.999676 + 0.0254728i \(0.00810911\pi\)
−0.477778 + 0.878481i \(0.658558\pi\)
\(138\) 0 0
\(139\) 216.000i 1.55396i −0.629527 0.776978i \(-0.716751\pi\)
0.629527 0.776978i \(-0.283249\pi\)
\(140\) 0 0
\(141\) −55.0000 −0.390071
\(142\) 0 0
\(143\) −353.338 + 204.000i −2.47090 + 1.42657i
\(144\) 0 0
\(145\) −12.0000 + 20.7846i −0.0827586 + 0.143342i
\(146\) 0 0
\(147\) 42.4352 24.5000i 0.288675 0.166667i
\(148\) 0 0
\(149\) −23.5000 + 40.7032i −0.157718 + 0.273176i −0.934045 0.357154i \(-0.883747\pi\)
0.776327 + 0.630330i \(0.217080\pi\)
\(150\) 0 0
\(151\) 172.339 99.5000i 1.14132 0.658940i 0.194562 0.980890i \(-0.437672\pi\)
0.946757 + 0.321950i \(0.104338\pi\)
\(152\) 0 0
\(153\) 8.00000 0.0522876
\(154\) 0 0
\(155\) 41.0000i 0.264516i
\(156\) 0 0
\(157\) −36.5000 63.2199i −0.232484 0.402674i 0.726054 0.687637i \(-0.241352\pi\)
−0.958539 + 0.284963i \(0.908019\pi\)
\(158\) 0 0
\(159\) 21.6506 + 12.5000i 0.136168 + 0.0786164i
\(160\) 0 0
\(161\) 24.5000 42.4352i 0.152174 0.263573i
\(162\) 0 0
\(163\) 47.6314 + 27.5000i 0.292217 + 0.168712i 0.638941 0.769255i \(-0.279373\pi\)
−0.346724 + 0.937967i \(0.612706\pi\)
\(164\) 0 0
\(165\) 8.50000 + 14.7224i 0.0515152 + 0.0892269i
\(166\) 0 0
\(167\) 206.000i 1.23353i −0.787146 0.616766i \(-0.788442\pi\)
0.787146 0.616766i \(-0.211558\pi\)
\(168\) 0 0
\(169\) 407.000 2.40828
\(170\) 0 0
\(171\) 48.4974 28.0000i 0.283611 0.163743i
\(172\) 0 0
\(173\) 119.500 206.980i 0.690751 1.19642i −0.280841 0.959754i \(-0.590613\pi\)
0.971592 0.236662i \(-0.0760535\pi\)
\(174\) 0 0
\(175\) −145.492 + 84.0000i −0.831384 + 0.480000i
\(176\) 0 0
\(177\) 8.50000 14.7224i 0.0480226 0.0831776i
\(178\) 0 0
\(179\) 14.7224 8.50000i 0.0822482 0.0474860i −0.458312 0.888791i \(-0.651546\pi\)
0.540560 + 0.841305i \(0.318212\pi\)
\(180\) 0 0
\(181\) −70.0000 −0.386740 −0.193370 0.981126i \(-0.561942\pi\)
−0.193370 + 0.981126i \(0.561942\pi\)
\(182\) 0 0
\(183\) 1.00000i 0.00546448i
\(184\) 0 0
\(185\) 24.5000 + 42.4352i 0.132432 + 0.229380i
\(186\) 0 0
\(187\) −14.7224 8.50000i −0.0787296 0.0454545i
\(188\) 0 0
\(189\) 119.000 0.629630
\(190\) 0 0
\(191\) 172.339 + 99.5000i 0.902299 + 0.520942i 0.877945 0.478761i \(-0.158914\pi\)
0.0243534 + 0.999703i \(0.492247\pi\)
\(192\) 0 0
\(193\) 23.5000 + 40.7032i 0.121762 + 0.210897i 0.920462 0.390831i \(-0.127812\pi\)
−0.798701 + 0.601728i \(0.794479\pi\)
\(194\) 0 0
\(195\) 24.0000i 0.123077i
\(196\) 0 0
\(197\) −24.0000 −0.121827 −0.0609137 0.998143i \(-0.519401\pi\)
−0.0609137 + 0.998143i \(0.519401\pi\)
\(198\) 0 0
\(199\) −118.645 + 68.5000i −0.596208 + 0.344221i −0.767549 0.640991i \(-0.778524\pi\)
0.171340 + 0.985212i \(0.445190\pi\)
\(200\) 0 0
\(201\) 32.5000 56.2917i 0.161692 0.280058i
\(202\) 0 0
\(203\) 168.000i 0.827586i
\(204\) 0 0
\(205\) −24.0000 + 41.5692i −0.117073 + 0.202777i
\(206\) 0 0
\(207\) 48.4974 28.0000i 0.234287 0.135266i
\(208\) 0 0
\(209\) −119.000 −0.569378
\(210\) 0 0
\(211\) 264.000i 1.25118i 0.780150 + 0.625592i \(0.215143\pi\)
−0.780150 + 0.625592i \(0.784857\pi\)
\(212\) 0 0
\(213\) 48.0000 + 83.1384i 0.225352 + 0.390321i
\(214\) 0 0
\(215\) 20.7846 + 12.0000i 0.0966726 + 0.0558140i
\(216\) 0 0
\(217\) −143.500 248.549i −0.661290 1.14539i
\(218\) 0 0
\(219\) 82.2724 + 47.5000i 0.375673 + 0.216895i
\(220\) 0 0
\(221\) 12.0000 + 20.7846i 0.0542986 + 0.0940480i
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) −192.000 −0.853333
\(226\) 0 0
\(227\) −359.401 + 207.500i −1.58326 + 0.914097i −0.588883 + 0.808218i \(0.700432\pi\)
−0.994379 + 0.105879i \(0.966234\pi\)
\(228\) 0 0
\(229\) −71.5000 + 123.842i −0.312227 + 0.540793i −0.978844 0.204607i \(-0.934408\pi\)
0.666617 + 0.745400i \(0.267742\pi\)
\(230\) 0 0
\(231\) −103.057 59.5000i −0.446134 0.257576i
\(232\) 0 0
\(233\) −144.500 + 250.281i −0.620172 + 1.07417i 0.369282 + 0.929317i \(0.379604\pi\)
−0.989453 + 0.144851i \(0.953730\pi\)
\(234\) 0 0
\(235\) 47.6314 27.5000i 0.202687 0.117021i
\(236\) 0 0
\(237\) −41.0000 −0.172996
\(238\) 0 0
\(239\) 226.000i 0.945607i −0.881168 0.472803i \(-0.843242\pi\)
0.881168 0.472803i \(-0.156758\pi\)
\(240\) 0 0
\(241\) −47.5000 82.2724i −0.197095 0.341379i 0.750490 0.660882i \(-0.229818\pi\)
−0.947585 + 0.319502i \(0.896484\pi\)
\(242\) 0 0
\(243\) 180.133 + 104.000i 0.741289 + 0.427984i
\(244\) 0 0
\(245\) −24.5000 + 42.4352i −0.100000 + 0.173205i
\(246\) 0 0
\(247\) 145.492 + 84.0000i 0.589038 + 0.340081i
\(248\) 0 0
\(249\) −36.0000 62.3538i −0.144578 0.250417i
\(250\) 0 0
\(251\) 38.0000i 0.151394i −0.997131 0.0756972i \(-0.975882\pi\)
0.997131 0.0756972i \(-0.0241182\pi\)
\(252\) 0 0
\(253\) −119.000 −0.470356
\(254\) 0 0
\(255\) 0.866025 0.500000i 0.00339618 0.00196078i
\(256\) 0 0
\(257\) −96.5000 + 167.143i −0.375486 + 0.650361i −0.990400 0.138233i \(-0.955858\pi\)
0.614913 + 0.788595i \(0.289191\pi\)
\(258\) 0 0
\(259\) −297.047 171.500i −1.14690 0.662162i
\(260\) 0 0
\(261\) 96.0000 166.277i 0.367816 0.637076i
\(262\) 0 0
\(263\) 201.784 116.500i 0.767239 0.442966i −0.0646496 0.997908i \(-0.520593\pi\)
0.831889 + 0.554942i \(0.187260\pi\)
\(264\) 0 0
\(265\) −25.0000 −0.0943396
\(266\) 0 0
\(267\) 95.0000i 0.355805i
\(268\) 0 0
\(269\) 227.500 + 394.042i 0.845725 + 1.46484i 0.884990 + 0.465610i \(0.154165\pi\)
−0.0392653 + 0.999229i \(0.512502\pi\)
\(270\) 0 0
\(271\) 368.061 + 212.500i 1.35816 + 0.784133i 0.989375 0.145383i \(-0.0464414\pi\)
0.368782 + 0.929516i \(0.379775\pi\)
\(272\) 0 0
\(273\) 84.0000 + 145.492i 0.307692 + 0.532939i
\(274\) 0 0
\(275\) 353.338 + 204.000i 1.28487 + 0.741818i
\(276\) 0 0
\(277\) 83.5000 + 144.626i 0.301444 + 0.522116i 0.976463 0.215684i \(-0.0691980\pi\)
−0.675019 + 0.737800i \(0.735865\pi\)
\(278\) 0 0
\(279\) 328.000i 1.17563i
\(280\) 0 0
\(281\) −432.000 −1.53737 −0.768683 0.639630i \(-0.779088\pi\)
−0.768683 + 0.639630i \(0.779088\pi\)
\(282\) 0 0
\(283\) −193.124 + 111.500i −0.682416 + 0.393993i −0.800765 0.598979i \(-0.795573\pi\)
0.118349 + 0.992972i \(0.462240\pi\)
\(284\) 0 0
\(285\) 3.50000 6.06218i 0.0122807 0.0212708i
\(286\) 0 0
\(287\) 336.000i 1.17073i
\(288\) 0 0
\(289\) 144.000 249.415i 0.498270 0.863029i
\(290\) 0 0
\(291\) −124.708 + 72.0000i −0.428549 + 0.247423i
\(292\) 0 0
\(293\) −26.0000 −0.0887372 −0.0443686 0.999015i \(-0.514128\pi\)
−0.0443686 + 0.999015i \(0.514128\pi\)
\(294\) 0 0
\(295\) 17.0000i 0.0576271i
\(296\) 0 0
\(297\) −144.500 250.281i −0.486532 0.842698i
\(298\) 0 0
\(299\) 145.492 + 84.0000i 0.486596 + 0.280936i
\(300\) 0 0
\(301\) −168.000 −0.558140
\(302\) 0 0
\(303\) −63.2199 36.5000i −0.208646 0.120462i
\(304\) 0 0
\(305\) 0.500000 + 0.866025i 0.00163934 + 0.00283943i
\(306\) 0 0
\(307\) 264.000i 0.859935i −0.902844 0.429967i \(-0.858525\pi\)
0.902844 0.429967i \(-0.141475\pi\)
\(308\) 0 0
\(309\) −89.0000 −0.288026
\(310\) 0 0
\(311\) −297.047 + 171.500i −0.955134 + 0.551447i −0.894672 0.446724i \(-0.852591\pi\)
−0.0604621 + 0.998170i \(0.519257\pi\)
\(312\) 0 0
\(313\) −167.500 + 290.119i −0.535144 + 0.926896i 0.464013 + 0.885829i \(0.346409\pi\)
−0.999156 + 0.0410676i \(0.986924\pi\)
\(314\) 0 0
\(315\) −48.4974 + 28.0000i −0.153960 + 0.0888889i
\(316\) 0 0
\(317\) −264.500 + 458.127i −0.834385 + 1.44520i 0.0601454 + 0.998190i \(0.480844\pi\)
−0.894530 + 0.447007i \(0.852490\pi\)
\(318\) 0 0
\(319\) −353.338 + 204.000i −1.10764 + 0.639498i
\(320\) 0 0
\(321\) −185.000 −0.576324
\(322\) 0 0
\(323\) 7.00000i 0.0216718i
\(324\) 0 0
\(325\) −288.000 498.831i −0.886154 1.53486i
\(326\) 0 0
\(327\) 61.4878 + 35.5000i 0.188036 + 0.108563i
\(328\) 0 0
\(329\) −192.500 + 333.420i −0.585106 + 1.01343i
\(330\) 0 0
\(331\) 47.6314 + 27.5000i 0.143902 + 0.0830816i 0.570222 0.821491i \(-0.306857\pi\)
−0.426321 + 0.904572i \(0.640190\pi\)
\(332\) 0 0
\(333\) −196.000 339.482i −0.588589 1.01947i
\(334\) 0 0
\(335\) 65.0000i 0.194030i
\(336\) 0 0
\(337\) −240.000 −0.712166 −0.356083 0.934454i \(-0.615888\pi\)
−0.356083 + 0.934454i \(0.615888\pi\)
\(338\) 0 0
\(339\) −83.1384 + 48.0000i −0.245246 + 0.141593i
\(340\) 0 0
\(341\) −348.500 + 603.620i −1.02199 + 1.77015i
\(342\) 0 0
\(343\) 343.000i 1.00000i
\(344\) 0 0
\(345\) 3.50000 6.06218i 0.0101449 0.0175715i
\(346\) 0 0
\(347\) 26.8468 15.5000i 0.0773683 0.0446686i −0.460817 0.887495i \(-0.652443\pi\)
0.538185 + 0.842827i \(0.319110\pi\)
\(348\) 0 0
\(349\) 120.000 0.343840 0.171920 0.985111i \(-0.445003\pi\)
0.171920 + 0.985111i \(0.445003\pi\)
\(350\) 0 0
\(351\) 408.000i 1.16239i
\(352\) 0 0
\(353\) 263.500 + 456.395i 0.746459 + 1.29290i 0.949510 + 0.313737i \(0.101581\pi\)
−0.203051 + 0.979168i \(0.565086\pi\)
\(354\) 0 0
\(355\) −83.1384 48.0000i −0.234193 0.135211i
\(356\) 0 0
\(357\) −3.50000 + 6.06218i −0.00980392 + 0.0169809i
\(358\) 0 0
\(359\) 463.324 + 267.500i 1.29059 + 0.745125i 0.978760 0.205011i \(-0.0657230\pi\)
0.311835 + 0.950136i \(0.399056\pi\)
\(360\) 0 0
\(361\) −156.000 270.200i −0.432133 0.748476i
\(362\) 0 0
\(363\) 168.000i 0.462810i
\(364\) 0 0
\(365\) −95.0000 −0.260274
\(366\) 0 0
\(367\) 77.0763 44.5000i 0.210017 0.121253i −0.391302 0.920262i \(-0.627975\pi\)
0.601319 + 0.799009i \(0.294642\pi\)
\(368\) 0 0
\(369\) 192.000 332.554i 0.520325 0.901230i
\(370\) 0 0
\(371\) 151.554 87.5000i 0.408503 0.235849i
\(372\) 0 0
\(373\) 167.500 290.119i 0.449062 0.777798i −0.549264 0.835649i \(-0.685092\pi\)
0.998325 + 0.0578516i \(0.0184250\pi\)
\(374\) 0 0
\(375\) −42.4352 + 24.5000i −0.113161 + 0.0653333i
\(376\) 0 0
\(377\) 576.000 1.52785
\(378\) 0 0
\(379\) 38.0000i 0.100264i 0.998743 + 0.0501319i \(0.0159642\pi\)
−0.998743 + 0.0501319i \(0.984036\pi\)
\(380\) 0 0
\(381\) −72.0000 124.708i −0.188976 0.327317i
\(382\) 0 0
\(383\) −575.907 332.500i −1.50367 0.868146i −0.999991 0.00425732i \(-0.998645\pi\)
−0.503682 0.863889i \(-0.668022\pi\)
\(384\) 0 0
\(385\) 119.000 0.309091
\(386\) 0 0
\(387\) −166.277 96.0000i −0.429656 0.248062i
\(388\) 0 0
\(389\) 36.5000 + 63.2199i 0.0938303 + 0.162519i 0.909120 0.416535i \(-0.136756\pi\)
−0.815290 + 0.579054i \(0.803422\pi\)
\(390\) 0 0
\(391\) 7.00000i 0.0179028i
\(392\) 0 0
\(393\) 185.000 0.470738
\(394\) 0 0
\(395\) 35.5070 20.5000i 0.0898912 0.0518987i
\(396\) 0 0
\(397\) 0.500000 0.866025i 0.00125945 0.00218142i −0.865395 0.501090i \(-0.832932\pi\)
0.866654 + 0.498909i \(0.166266\pi\)
\(398\) 0 0
\(399\) 49.0000i 0.122807i
\(400\) 0 0
\(401\) −168.500 + 291.851i −0.420200 + 0.727807i −0.995959 0.0898124i \(-0.971373\pi\)
0.575759 + 0.817619i \(0.304707\pi\)
\(402\) 0 0
\(403\) 852.169 492.000i 2.11456 1.22084i
\(404\) 0 0
\(405\) −55.0000 −0.135802
\(406\) 0 0
\(407\) 833.000i 2.04668i
\(408\) 0 0
\(409\) 239.500 + 414.826i 0.585575 + 1.01424i 0.994804 + 0.101813i \(0.0324644\pi\)
−0.409229 + 0.912432i \(0.634202\pi\)
\(410\) 0 0
\(411\) −123.842 71.5000i −0.301318 0.173966i
\(412\) 0 0
\(413\) −59.5000 103.057i −0.144068 0.249533i
\(414\) 0 0
\(415\) 62.3538 + 36.0000i 0.150250 + 0.0867470i
\(416\) 0 0
\(417\) 108.000 + 187.061i 0.258993 + 0.448589i
\(418\) 0 0
\(419\) 552.000i 1.31742i −0.752396 0.658711i \(-0.771102\pi\)
0.752396 0.658711i \(-0.228898\pi\)
\(420\) 0 0
\(421\) 216.000 0.513064 0.256532 0.966536i \(-0.417420\pi\)
0.256532 + 0.966536i \(0.417420\pi\)
\(422\) 0 0
\(423\) −381.051 + 220.000i −0.900830 + 0.520095i
\(424\) 0 0
\(425\) 12.0000 20.7846i 0.0282353 0.0489050i
\(426\) 0 0
\(427\) −6.06218 3.50000i −0.0141971 0.00819672i
\(428\) 0 0
\(429\) 204.000 353.338i 0.475524 0.823633i
\(430\) 0 0
\(431\) 380.185 219.500i 0.882100 0.509281i 0.0107498 0.999942i \(-0.496578\pi\)
0.871350 + 0.490661i \(0.163245\pi\)
\(432\) 0 0
\(433\) −288.000 −0.665127 −0.332564 0.943081i \(-0.607914\pi\)
−0.332564 + 0.943081i \(0.607914\pi\)
\(434\) 0 0
\(435\) 24.0000i 0.0551724i
\(436\) 0 0
\(437\) 24.5000 + 42.4352i 0.0560641 + 0.0971058i
\(438\) 0 0
\(439\) −534.338 308.500i −1.21717 0.702733i −0.252858 0.967503i \(-0.581371\pi\)
−0.964312 + 0.264770i \(0.914704\pi\)
\(440\) 0 0
\(441\) 196.000 339.482i 0.444444 0.769800i
\(442\) 0 0
\(443\) −213.908 123.500i −0.482863 0.278781i 0.238746 0.971082i \(-0.423264\pi\)
−0.721609 + 0.692301i \(0.756597\pi\)
\(444\) 0 0
\(445\) 47.5000 + 82.2724i 0.106742 + 0.184882i
\(446\) 0 0
\(447\) 47.0000i 0.105145i
\(448\) 0 0
\(449\) −288.000 −0.641425 −0.320713 0.947177i \(-0.603922\pi\)
−0.320713 + 0.947177i \(0.603922\pi\)
\(450\) 0 0
\(451\) −706.677 + 408.000i −1.56691 + 0.904656i
\(452\) 0 0
\(453\) −99.5000 + 172.339i −0.219647 + 0.380439i
\(454\) 0 0
\(455\) −145.492 84.0000i −0.319763 0.184615i
\(456\) 0 0
\(457\) 48.5000 84.0045i 0.106127 0.183817i −0.808071 0.589085i \(-0.799488\pi\)
0.914198 + 0.405268i \(0.132822\pi\)
\(458\) 0 0
\(459\) −14.7224 + 8.50000i −0.0320750 + 0.0185185i
\(460\) 0 0
\(461\) −312.000 −0.676790 −0.338395 0.941004i \(-0.609884\pi\)
−0.338395 + 0.941004i \(0.609884\pi\)
\(462\) 0 0
\(463\) 192.000i 0.414687i −0.978268 0.207343i \(-0.933518\pi\)
0.978268 0.207343i \(-0.0664817\pi\)
\(464\) 0 0
\(465\) −20.5000 35.5070i −0.0440860 0.0763592i
\(466\) 0 0
\(467\) 255.477 + 147.500i 0.547061 + 0.315846i 0.747936 0.663771i \(-0.231045\pi\)
−0.200875 + 0.979617i \(0.564378\pi\)
\(468\) 0 0
\(469\) −227.500 394.042i −0.485075 0.840174i
\(470\) 0 0
\(471\) 63.2199 + 36.5000i 0.134225 + 0.0774947i
\(472\) 0 0
\(473\) 204.000 + 353.338i 0.431290 + 0.747016i
\(474\) 0 0
\(475\) 168.000i 0.353684i
\(476\) 0 0
\(477\) 200.000 0.419287
\(478\) 0 0
\(479\) 534.338 308.500i 1.11553 0.644050i 0.175272 0.984520i \(-0.443919\pi\)
0.940255 + 0.340470i \(0.110586\pi\)
\(480\) 0 0
\(481\) 588.000 1018.45i 1.22245 2.11735i
\(482\) 0 0
\(483\) 49.0000i 0.101449i
\(484\) 0 0
\(485\) 72.0000 124.708i 0.148454 0.257129i
\(486\) 0 0
\(487\) −213.908 + 123.500i −0.439237 + 0.253593i −0.703274 0.710919i \(-0.748279\pi\)
0.264037 + 0.964513i \(0.414946\pi\)
\(488\) 0 0
\(489\) −55.0000 −0.112474
\(490\) 0 0
\(491\) 134.000i 0.272912i 0.990646 + 0.136456i \(0.0435713\pi\)
−0.990646 + 0.136456i \(0.956429\pi\)
\(492\) 0 0
\(493\) 12.0000 + 20.7846i 0.0243408 + 0.0421595i
\(494\) 0 0
\(495\) 117.779 + 68.0000i 0.237938 + 0.137374i
\(496\) 0 0
\(497\) 672.000 1.35211
\(498\) 0 0
\(499\) 629.600 + 363.500i 1.26172 + 0.728457i 0.973408 0.229078i \(-0.0735712\pi\)
0.288316 + 0.957535i \(0.406905\pi\)
\(500\) 0 0
\(501\) 103.000 + 178.401i 0.205589 + 0.356090i
\(502\) 0 0
\(503\) 432.000i 0.858847i 0.903103 + 0.429423i \(0.141283\pi\)
−0.903103 + 0.429423i \(0.858717\pi\)
\(504\) 0 0
\(505\) 73.0000 0.144554
\(506\) 0 0
\(507\) −352.472 + 203.500i −0.695212 + 0.401381i
\(508\) 0 0
\(509\) 216.500 374.989i 0.425344 0.736717i −0.571109 0.820874i \(-0.693487\pi\)
0.996452 + 0.0841574i \(0.0268198\pi\)
\(510\) 0 0
\(511\) 575.907 332.500i 1.12702 0.650685i
\(512\) 0 0
\(513\) −59.5000 + 103.057i −0.115984 + 0.200891i
\(514\) 0 0
\(515\) 77.0763 44.5000i 0.149663 0.0864078i
\(516\) 0 0
\(517\) 935.000 1.80851
\(518\) 0 0
\(519\) 239.000i 0.460501i
\(520\) 0 0
\(521\) −239.500 414.826i −0.459693 0.796211i 0.539252 0.842145i \(-0.318707\pi\)
−0.998945 + 0.0459332i \(0.985374\pi\)
\(522\) 0 0
\(523\) 783.753 + 452.500i 1.49857 + 0.865201i 0.999999 0.00164693i \(-0.000524234\pi\)
0.498573 + 0.866848i \(0.333858\pi\)
\(524\) 0 0
\(525\) 84.0000 145.492i 0.160000 0.277128i
\(526\) 0 0
\(527\) 35.5070 + 20.5000i 0.0673758 + 0.0388994i
\(528\) 0 0
\(529\) −240.000 415.692i −0.453686 0.785808i
\(530\) 0 0
\(531\) 136.000i 0.256121i
\(532\) 0 0
\(533\) 1152.00 2.16135
\(534\) 0 0
\(535\) 160.215 92.5000i 0.299467 0.172897i
\(536\) 0 0
\(537\) −8.50000 + 14.7224i −0.0158287 + 0.0274161i
\(538\) 0 0
\(539\) −721.399 + 416.500i −1.33840 + 0.772727i
\(540\) 0 0
\(541\) −95.5000 + 165.411i −0.176525 + 0.305750i −0.940688 0.339273i \(-0.889819\pi\)
0.764163 + 0.645023i \(0.223152\pi\)
\(542\) 0 0
\(543\) 60.6218 35.0000i 0.111642 0.0644567i
\(544\) 0 0
\(545\) −71.0000 −0.130275
\(546\) 0 0
\(547\) 374.000i 0.683729i −0.939749 0.341865i \(-0.888942\pi\)
0.939749 0.341865i \(-0.111058\pi\)
\(548\) 0 0
\(549\) −4.00000 6.92820i −0.00728597 0.0126197i
\(550\) 0 0
\(551\) 145.492 + 84.0000i 0.264051 + 0.152450i
\(552\) 0 0
\(553\) −143.500 + 248.549i −0.259494 + 0.449456i
\(554\) 0 0
\(555\) −42.4352 24.5000i −0.0764599 0.0441441i
\(556\) 0 0
\(557\) 468.500 + 811.466i 0.841113 + 1.45685i 0.888954 + 0.457996i \(0.151432\pi\)
−0.0478413 + 0.998855i \(0.515234\pi\)
\(558\) 0 0
\(559\) 576.000i 1.03041i
\(560\) 0 0
\(561\) 17.0000 0.0303030
\(562\) 0 0
\(563\) 691.954 399.500i 1.22905 0.709591i 0.262218 0.965009i \(-0.415546\pi\)
0.966831 + 0.255417i \(0.0822129\pi\)
\(564\) 0 0
\(565\) 48.0000 83.1384i 0.0849558 0.147148i
\(566\) 0 0
\(567\) 333.420 192.500i 0.588042 0.339506i
\(568\) 0 0
\(569\) 264.500 458.127i 0.464851 0.805145i −0.534344 0.845267i \(-0.679441\pi\)
0.999195 + 0.0401221i \(0.0127747\pi\)
\(570\) 0 0
\(571\) −899.800 + 519.500i −1.57583 + 0.909807i −0.580401 + 0.814331i \(0.697104\pi\)
−0.995432 + 0.0954765i \(0.969563\pi\)
\(572\) 0 0
\(573\) −199.000 −0.347295
\(574\) 0 0
\(575\) 168.000i 0.292174i
\(576\) 0 0
\(577\) 264.500 + 458.127i 0.458406 + 0.793982i 0.998877 0.0473807i \(-0.0150874\pi\)
−0.540471 + 0.841362i \(0.681754\pi\)
\(578\) 0 0
\(579\) −40.7032 23.5000i −0.0702991 0.0405872i
\(580\) 0 0
\(581\) −504.000 −0.867470
\(582\) 0 0
\(583\) −368.061 212.500i −0.631322 0.364494i
\(584\) 0 0
\(585\) −96.0000 166.277i −0.164103 0.284234i
\(586\) 0 0
\(587\) 840.000i 1.43101i 0.698610 + 0.715503i \(0.253802\pi\)
−0.698610 + 0.715503i \(0.746198\pi\)
\(588\) 0 0
\(589\) 287.000 0.487267
\(590\) 0 0
\(591\) 20.7846 12.0000i 0.0351685 0.0203046i
\(592\) 0 0
\(593\) 191.500 331.688i 0.322934 0.559338i −0.658158 0.752880i \(-0.728664\pi\)
0.981092 + 0.193541i \(0.0619974\pi\)
\(594\) 0 0
\(595\) 7.00000i 0.0117647i
\(596\) 0 0
\(597\) 68.5000 118.645i 0.114740 0.198736i
\(598\) 0 0
\(599\) −297.047 + 171.500i −0.495904 + 0.286311i −0.727021 0.686616i \(-0.759096\pi\)
0.231116 + 0.972926i \(0.425762\pi\)
\(600\) 0 0
\(601\) −624.000 −1.03827 −0.519135 0.854692i \(-0.673746\pi\)
−0.519135 + 0.854692i \(0.673746\pi\)
\(602\) 0 0
\(603\) 520.000i 0.862355i
\(604\) 0 0
\(605\) −84.0000 145.492i −0.138843 0.240483i
\(606\) 0 0
\(607\) −118.645 68.5000i −0.195462 0.112850i 0.399075 0.916918i \(-0.369331\pi\)
−0.594537 + 0.804068i \(0.702665\pi\)
\(608\) 0 0
\(609\) 84.0000 + 145.492i 0.137931 + 0.238904i
\(610\) 0 0
\(611\) −1143.15 660.000i −1.87096 1.08020i
\(612\) 0 0
\(613\) −35.5000 61.4878i −0.0579119 0.100306i 0.835616 0.549314i \(-0.185111\pi\)
−0.893528 + 0.449008i \(0.851778\pi\)
\(614\) 0 0
\(615\) 48.0000i 0.0780488i
\(616\) 0 0
\(617\) 384.000 0.622366 0.311183 0.950350i \(-0.399275\pi\)
0.311183 + 0.950350i \(0.399275\pi\)
\(618\) 0 0
\(619\) −513.553 + 296.500i −0.829650 + 0.478998i −0.853733 0.520712i \(-0.825667\pi\)
0.0240831 + 0.999710i \(0.492333\pi\)
\(620\) 0 0
\(621\) −59.5000 + 103.057i −0.0958132 + 0.165953i
\(622\) 0 0
\(623\) −575.907 332.500i −0.924409 0.533708i
\(624\) 0 0
\(625\) −275.500 + 477.180i −0.440800 + 0.763488i
\(626\) 0 0
\(627\) 103.057 59.5000i 0.164365 0.0948963i
\(628\) 0 0
\(629\) 49.0000 0.0779014
\(630\) 0 0
\(631\) 384.000i 0.608558i −0.952583 0.304279i \(-0.901585\pi\)
0.952583 0.304279i \(-0.0984155\pi\)
\(632\) 0 0
\(633\) −132.000 228.631i −0.208531 0.361186i
\(634\) 0 0
\(635\) 124.708 + 72.0000i 0.196390 + 0.113386i
\(636\) 0 0
\(637\) 1176.00 1.84615
\(638\) 0 0
\(639\) 665.108 + 384.000i 1.04086 + 0.600939i
\(640\) 0 0
\(641\) −383.500 664.241i −0.598284 1.03626i −0.993074 0.117487i \(-0.962516\pi\)
0.394790 0.918771i \(-0.370817\pi\)
\(642\) 0 0
\(643\) 456.000i 0.709176i 0.935023 + 0.354588i \(0.115379\pi\)
−0.935023 + 0.354588i \(0.884621\pi\)
\(644\) 0 0
\(645\) −24.0000 −0.0372093
\(646\) 0 0
\(647\) 77.0763 44.5000i 0.119129 0.0687790i −0.439252 0.898364i \(-0.644756\pi\)
0.558380 + 0.829585i \(0.311423\pi\)
\(648\) 0 0
\(649\) −144.500 + 250.281i −0.222650 + 0.385642i
\(650\) 0 0
\(651\) 248.549 + 143.500i 0.381796 + 0.220430i
\(652\) 0 0
\(653\) 263.500 456.395i 0.403522 0.698921i −0.590626 0.806945i \(-0.701119\pi\)
0.994148 + 0.108024i \(0.0344525\pi\)
\(654\) 0 0
\(655\) −160.215 + 92.5000i −0.244603 + 0.141221i
\(656\) 0 0
\(657\) 760.000 1.15677
\(658\) 0 0
\(659\) 936.000i 1.42033i −0.704033 0.710167i \(-0.748619\pi\)
0.704033 0.710167i \(-0.251381\pi\)
\(660\) 0 0
\(661\) −372.500 645.189i −0.563540 0.976080i −0.997184 0.0749957i \(-0.976106\pi\)
0.433644 0.901084i \(-0.357228\pi\)
\(662\) 0 0
\(663\) −20.7846 12.0000i −0.0313493 0.0180995i
\(664\) 0 0
\(665\) −24.5000 42.4352i −0.0368421 0.0638124i
\(666\) 0 0
\(667\) 145.492 + 84.0000i 0.218129 + 0.125937i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 17.0000i 0.0253353i
\(672\) 0 0
\(673\) −720.000 −1.06984 −0.534918 0.844904i \(-0.679658\pi\)
−0.534918 + 0.844904i \(0.679658\pi\)
\(674\) 0 0
\(675\) 353.338 204.000i 0.523464 0.302222i
\(676\) 0 0
\(677\) −575.500 + 996.795i −0.850074 + 1.47237i 0.0310670 + 0.999517i \(0.490109\pi\)
−0.881141 + 0.472854i \(0.843224\pi\)
\(678\) 0 0
\(679\) 1008.00i 1.48454i
\(680\) 0 0
\(681\) 207.500 359.401i 0.304699 0.527754i
\(682\) 0 0
\(683\) 638.261 368.500i 0.934496 0.539531i 0.0462653 0.998929i \(-0.485268\pi\)
0.888231 + 0.459398i \(0.151935\pi\)
\(684\) 0 0
\(685\) 143.000 0.208759
\(686\) 0 0
\(687\) 143.000i 0.208151i
\(688\) 0 0
\(689\) 300.000 + 519.615i 0.435414 + 0.754159i
\(690\) 0 0
\(691\) 879.016 + 507.500i 1.27209 + 0.734443i 0.975381 0.220525i \(-0.0707769\pi\)
0.296711 + 0.954967i \(0.404110\pi\)
\(692\) 0 0
\(693\) −952.000 −1.37374
\(694\) 0 0
\(695\) −187.061 108.000i −0.269153 0.155396i
\(696\) 0 0
\(697\) 24.0000 + 41.5692i 0.0344333 + 0.0596402i
\(698\) 0 0
\(699\) 289.000i 0.413448i
\(700\) 0 0
\(701\) −1226.00 −1.74893 −0.874465 0.485089i \(-0.838787\pi\)
−0.874465 + 0.485089i \(0.838787\pi\)
\(702\) 0 0
\(703\) 297.047 171.500i 0.422542 0.243954i
\(704\) 0 0
\(705\) −27.5000 + 47.6314i −0.0390071 + 0.0675623i
\(706\) 0 0
\(707\) −442.539 + 255.500i −0.625939 + 0.361386i
\(708\) 0 0
\(709\) −312.500 + 541.266i −0.440762 + 0.763422i −0.997746 0.0671013i \(-0.978625\pi\)
0.556985 + 0.830523i \(0.311958\pi\)
\(710\) 0 0
\(711\) −284.056 + 164.000i −0.399517 + 0.230661i
\(712\) 0 0
\(713\) 287.000 0.402525
\(714\) 0 0
\(715\) 408.000i 0.570629i
\(716\) 0 0
\(717\) 113.000 + 195.722i 0.157601 + 0.272973i
\(718\) 0 0
\(719\) 118.645 + 68.5000i 0.165015 + 0.0952712i 0.580233 0.814451i \(-0.302962\pi\)
−0.415218 + 0.909722i \(0.636295\pi\)
\(720\) 0 0
\(721\) −311.500 + 539.534i −0.432039 + 0.748313i
\(722\) 0 0
\(723\) 82.2724 + 47.5000i 0.113793 + 0.0656985i
\(724\) 0 0
\(725\) −288.000 498.831i −0.397241 0.688042i
\(726\) 0 0
\(727\) 960.000i 1.32050i 0.751048 + 0.660248i \(0.229549\pi\)
−0.751048 + 0.660248i \(0.770451\pi\)
\(728\) 0 0
\(729\) 287.000 0.393690
\(730\) 0 0
\(731\) 20.7846 12.0000i 0.0284331 0.0164159i
\(732\) 0 0
\(733\) 239.500 414.826i 0.326739 0.565929i −0.655123 0.755522i \(-0.727383\pi\)
0.981863 + 0.189593i \(0.0607167\pi\)
\(734\) 0 0
\(735\) 49.0000i 0.0666667i
\(736\) 0 0
\(737\) −552.500 + 956.958i −0.749661 + 1.29845i
\(738\) 0 0
\(739\) −442.539 + 255.500i −0.598835 + 0.345737i −0.768583 0.639750i \(-0.779038\pi\)
0.169748 + 0.985487i \(0.445705\pi\)
\(740\) 0 0
\(741\) −168.000 −0.226721
\(742\) 0 0
\(743\) 528.000i 0.710633i 0.934746 + 0.355316i \(0.115627\pi\)
−0.934746 + 0.355316i \(0.884373\pi\)
\(744\) 0 0
\(745\) 23.5000 + 40.7032i 0.0315436 + 0.0546352i
\(746\) 0 0
\(747\) −498.831 288.000i −0.667779 0.385542i
\(748\) 0 0
\(749\) −647.500 + 1121.50i −0.864486 + 1.49733i
\(750\) 0 0
\(751\) −700.615 404.500i −0.932909 0.538615i −0.0451785 0.998979i \(-0.514386\pi\)
−0.887730 + 0.460364i \(0.847719\pi\)
\(752\) 0 0
\(753\) 19.0000 + 32.9090i 0.0252324 + 0.0437038i
\(754\) 0 0
\(755\) 199.000i 0.263576i
\(756\) 0 0
\(757\) 120.000 0.158520 0.0792602 0.996854i \(-0.474744\pi\)
0.0792602 + 0.996854i \(0.474744\pi\)
\(758\) 0 0
\(759\) 103.057 59.5000i 0.135780 0.0783926i
\(760\) 0 0
\(761\) −384.500 + 665.974i −0.505256 + 0.875129i 0.494725 + 0.869049i \(0.335269\pi\)
−0.999982 + 0.00608006i \(0.998065\pi\)
\(762\) 0 0
\(763\) 430.415 248.500i 0.564108 0.325688i
\(764\) 0 0
\(765\) 4.00000 6.92820i 0.00522876 0.00905647i
\(766\) 0 0
\(767\) 353.338 204.000i 0.460676 0.265971i
\(768\) 0 0
\(769\) −144.000 −0.187256 −0.0936281 0.995607i \(-0.529846\pi\)
−0.0936281 + 0.995607i \(0.529846\pi\)
\(770\) 0 0
\(771\) 193.000i 0.250324i
\(772\) 0 0
\(773\) −203.500 352.472i −0.263260 0.455980i 0.703846 0.710352i \(-0.251464\pi\)
−0.967106 + 0.254373i \(0.918131\pi\)
\(774\) 0 0
\(775\) −852.169 492.000i −1.09957 0.634839i
\(776\) 0 0
\(777\) 343.000 0.441441
\(778\) 0 0
\(779\) 290.985 + 168.000i 0.373536 + 0.215661i
\(780\) 0 0
\(781\) −816.000 1413.35i −1.04481 1.80967i
\(782\) 0 0
\(783\) 408.000i 0.521073i
\(784\) 0 0
\(785\) −73.0000 −0.0929936
\(786\) 0 0
\(787\) 982.939 567.500i 1.24897 0.721093i 0.278065 0.960562i \(-0.410307\pi\)
0.970904 + 0.239469i \(0.0769735\pi\)
\(788\) 0 0
\(789\) −116.500 + 201.784i −0.147655 + 0.255746i
\(790\) 0 0
\(791\) 672.000i 0.849558i
\(792\) 0 0
\(793\) 12.0000 20.7846i 0.0151324 0.0262101i
\(794\) 0 0
\(795\) 21.6506 12.5000i 0.0272335 0.0157233i
\(796\) 0 0
\(797\) −312.000 −0.391468 −0.195734 0.980657i \(-0.562709\pi\)
−0.195734 + 0.980657i \(0.562709\pi\)
\(798\) 0 0
\(799\) 55.0000i 0.0688360i
\(800\) 0 0
\(801\) −380.000 658.179i −0.474407 0.821697i
\(802\) 0 0
\(803\) −1398.63 807.500i −1.74176 1.00560i
\(804\) 0 0
\(805\) −24.5000 42.4352i −0.0304348 0.0527146i
\(806\) 0 0
\(807\) −394.042 227.500i −0.488280 0.281908i
\(808\) 0 0
\(809\) 648.500 + 1123.23i 0.801607 + 1.38842i 0.918558 + 0.395286i \(0.129355\pi\)
−0.116951 + 0.993138i \(0.537312\pi\)
\(810\) 0 0
\(811\) 1128.00i 1.39088i −0.718586 0.695438i \(-0.755211\pi\)
0.718586 0.695438i \(-0.244789\pi\)
\(812\) 0 0
\(813\) −425.000 −0.522755
\(814\) 0 0
\(815\) 47.6314 27.5000i 0.0584434 0.0337423i
\(816\) 0 0
\(817\) 84.0000 145.492i 0.102815 0.178081i
\(818\) 0 0
\(819\) 1163.94 + 672.000i 1.42117 + 0.820513i
\(820\) 0 0
\(821\) −191.500 + 331.688i −0.233252 + 0.404005i −0.958763 0.284206i \(-0.908270\pi\)
0.725511 + 0.688210i \(0.241603\pi\)
\(822\) 0 0
\(823\) −47.6314 + 27.5000i −0.0578753 + 0.0334143i −0.528658 0.848835i \(-0.677305\pi\)
0.470783 + 0.882249i \(0.343971\pi\)
\(824\) 0 0
\(825\) −408.000 −0.494545
\(826\) 0 0
\(827\) 696.000i 0.841596i 0.907154 + 0.420798i \(0.138250\pi\)
−0.907154 + 0.420798i \(0.861750\pi\)
\(828\) 0 0
\(829\) 252.500 + 437.343i 0.304584 + 0.527555i 0.977169 0.212465i \(-0.0681493\pi\)
−0.672585 + 0.740020i \(0.734816\pi\)
\(830\) 0 0
\(831\) −144.626 83.5000i −0.174039 0.100481i
\(832\) 0 0
\(833\) 24.5000 + 42.4352i 0.0294118 + 0.0509427i
\(834\) 0 0
\(835\) −178.401 103.000i −0.213654 0.123353i
\(836\) 0 0
\(837\) 348.500 + 603.620i 0.416368 + 0.721170i
\(838\) 0 0
\(839\) 48.0000i 0.0572110i 0.999591 + 0.0286055i \(0.00910665\pi\)
−0.999591 + 0.0286055i \(0.990893\pi\)
\(840\) 0 0
\(841\) −265.000 −0.315101
\(842\) 0 0
\(843\) 374.123 216.000i 0.443799 0.256228i
\(844\) 0 0
\(845\) 203.500 352.472i 0.240828 0.417127i
\(846\) 0 0
\(847\) 1018.45 + 588.000i 1.20242 + 0.694215i
\(848\) 0 0
\(849\) 111.500 193.124i 0.131331 0.227472i
\(850\) 0 0
\(851\) 297.047 171.500i 0.349056 0.201528i
\(852\) 0 0
\(853\) −696.000 −0.815944 −0.407972 0.912995i \(-0.633764\pi\)
−0.407972 + 0.912995i \(0.633764\pi\)
\(854\) 0 0
\(855\) 56.0000i 0.0654971i
\(856\) 0 0
\(857\) −599.500 1038.36i −0.699533 1.21163i −0.968628 0.248514i \(-0.920058\pi\)
0.269095 0.963114i \(-0.413275\pi\)
\(858\) 0 0
\(859\) −671.170 387.500i −0.781338 0.451106i 0.0555660 0.998455i \(-0.482304\pi\)
−0.836904 + 0.547349i \(0.815637\pi\)
\(860\) 0 0
\(861\) 168.000 + 290.985i 0.195122 + 0.337961i
\(862\) 0 0
\(863\) −297.047 171.500i −0.344202 0.198725i 0.317926 0.948115i \(-0.397014\pi\)
−0.662129 + 0.749390i \(0.730347\pi\)
\(864\) 0 0
\(865\) −119.500 206.980i −0.138150 0.239283i
\(866\) 0 0
\(867\) 288.000i 0.332180i
\(868\) 0 0
\(869\) 697.000 0.802071
\(870\) 0 0
\(871\) 1351.00 780.000i 1.55109 0.895522i
\(872\) 0 0
\(873\) −576.000 + 997.661i −0.659794 + 1.14280i
\(874\) 0 0
\(875\) 343.000i 0.392000i
\(876\) 0 0
\(877\) −575.500 + 996.795i −0.656214 + 1.13660i 0.325374 + 0.945586i \(0.394510\pi\)
−0.981588 + 0.191011i \(0.938823\pi\)
\(878\) 0 0
\(879\) 22.5167 13.0000i 0.0256162 0.0147895i
\(880\) 0 0
\(881\) −866.000 −0.982974 −0.491487 0.870885i \(-0.663546\pi\)
−0.491487 + 0.870885i \(0.663546\pi\)
\(882\) 0 0
\(883\) 648.000i 0.733862i 0.930248 + 0.366931i \(0.119591\pi\)
−0.930248 + 0.366931i \(0.880409\pi\)
\(884\) 0 0
\(885\) −8.50000 14.7224i −0.00960452 0.0166355i
\(886\) 0 0
\(887\) 588.031 + 339.500i 0.662944 + 0.382751i 0.793398 0.608704i \(-0.208310\pi\)
−0.130454 + 0.991454i \(0.541643\pi\)
\(888\) 0 0
\(889\) −1008.00 −1.13386
\(890\) 0 0
\(891\) −809.734 467.500i −0.908792 0.524691i
\(892\) 0 0
\(893\) −192.500 333.420i −0.215566 0.373370i
\(894\) 0 0
\(895\) 17.0000i 0.0189944i
\(896\) 0 0
\(897\) −168.000 −0.187291
\(898\) 0 0
\(899\) 852.169 492.000i 0.947908 0.547275i
\(900\) 0 0
\(901\) −12.5000 + 21.6506i −0.0138735 + 0.0240296i
\(902\) 0 0
\(903\) 145.492 84.0000i 0.161121 0.0930233i
\(904\) 0 0
\(905\) −35.0000 + 60.6218i −0.0386740 + 0.0669854i
\(906\) 0 0
\(907\) −305.707 + 176.500i −0.337053 + 0.194598i −0.658968 0.752171i \(-0.729007\pi\)
0.321915 + 0.946769i \(0.395673\pi\)
\(908\) 0 0
\(909\) −584.000 −0.642464
\(910\) 0 0
\(911\) 144.000i 0.158068i −0.996872 0.0790340i \(-0.974816\pi\)
0.996872 0.0790340i \(-0.0251836\pi\)
\(912\) 0 0
\(913\) 612.000 + 1060.02i 0.670318 + 1.16102i
\(914\) 0 0
\(915\) −0.866025 0.500000i −0.000946476 0.000546448i
\(916\) 0 0
\(917\) 647.500 1121.50i 0.706107 1.22301i
\(918\) 0 0
\(919\) −950.030 548.500i −1.03376 0.596844i −0.115704 0.993284i \(-0.536912\pi\)
−0.918061 + 0.396439i \(0.870246\pi\)
\(920\) 0 0
\(921\) 132.000 + 228.631i 0.143322 + 0.248242i
\(922\) 0 0
\(923\) 2304.00i 2.49621i
\(924\) 0 0
\(925\) −1176.00 −1.27135
\(926\) 0 0
\(927\) −616.610 + 356.000i −0.665167 + 0.384035i
\(928\) 0 0
\(929\) 264.500 458.127i 0.284715 0.493140i −0.687825 0.725876i \(-0.741434\pi\)
0.972540 + 0.232736i \(0.0747678\pi\)
\(930\) 0 0
\(931\) 297.047 + 171.500i 0.319062 + 0.184211i
\(932\) 0 0
\(933\) 171.500 297.047i 0.183816 0.318378i
\(934\) 0 0
\(935\) −14.7224 + 8.50000i −0.0157459 + 0.00909091i
\(936\) 0 0
\(937\) 146.000 0.155816 0.0779082 0.996961i \(-0.475176\pi\)
0.0779082 + 0.996961i \(0.475176\pi\)
\(938\) 0 0
\(939\) 335.000i 0.356763i
\(940\) 0 0
\(941\) 371.500 + 643.457i 0.394793 + 0.683801i 0.993075 0.117485i \(-0.0374831\pi\)
−0.598282 + 0.801286i \(0.704150\pi\)
\(942\) 0 0
\(943\) 290.985 + 168.000i 0.308573 + 0.178155i
\(944\) 0 0
\(945\) 59.5000 103.057i 0.0629630 0.109055i
\(946\) 0 0
\(947\) −451.199 260.500i −0.476451 0.275079i 0.242485 0.970155i \(-0.422037\pi\)
−0.718936 + 0.695076i \(0.755371\pi\)
\(948\) 0 0
\(949\) 1140.00 + 1974.54i 1.20126 + 2.08065i
\(950\) 0 0
\(951\) 529.000i 0.556257i
\(952\) 0 0
\(953\) 1440.00 1.51102 0.755509 0.655138i \(-0.227390\pi\)
0.755509 + 0.655138i \(0.227390\pi\)
\(954\) 0 0
\(955\) 172.339 99.5000i 0.180460 0.104188i
\(956\) 0 0
\(957\) 204.000 353.338i 0.213166 0.369215i
\(958\) 0 0
\(959\) −866.891 + 500.500i −0.903954 + 0.521898i
\(960\) 0 0
\(961\) 360.000 623.538i 0.374610 0.648843i
\(962\) 0 0
\(963\) −1281.72 + 740.000i −1.33096 + 0.768432i
\(964\) 0 0
\(965\) 47.0000 0.0487047
\(966\) 0 0
\(967\) 1920.00i 1.98552i −0.120106 0.992761i \(-0.538323\pi\)
0.120106 0.992761i \(-0.461677\pi\)
\(968\) 0 0
\(969\) −3.50000 6.06218i −0.00361197 0.00625612i
\(970\) 0 0
\(971\) −284.922 164.500i −0.293432 0.169413i 0.346057 0.938214i \(-0.387521\pi\)
−0.639489 + 0.768801i \(0.720854\pi\)
\(972\) 0 0
\(973\) 1512.00 1.55396
\(974\) 0 0
\(975\) 498.831 + 288.000i 0.511621 + 0.295385i
\(976\) 0 0
\(977\) −360.500 624.404i −0.368987 0.639104i 0.620421 0.784269i \(-0.286962\pi\)
−0.989407 + 0.145165i \(0.953629\pi\)
\(978\) 0 0
\(979\) 1615.00i 1.64964i
\(980\) 0 0
\(981\) 568.000 0.579001
\(982\) 0 0
\(983\) 546.462 315.500i 0.555913 0.320956i −0.195591 0.980686i \(-0.562662\pi\)
0.751503 + 0.659729i \(0.229329\pi\)
\(984\) 0 0
\(985\) −12.0000 + 20.7846i −0.0121827 + 0.0211011i
\(986\) 0 0
\(987\) 385.000i 0.390071i
\(988\) 0 0
\(989\) 84.0000 145.492i 0.0849343 0.147110i
\(990\) 0 0
\(991\) −629.600 + 363.500i −0.635318 + 0.366801i −0.782809 0.622262i \(-0.786214\pi\)
0.147491 + 0.989063i \(0.452880\pi\)
\(992\) 0 0
\(993\) −55.0000 −0.0553877
\(994\) 0 0
\(995\) 137.000i 0.137688i
\(996\) 0 0
\(997\) −12.5000 21.6506i −0.0125376 0.0217158i 0.859689 0.510819i \(-0.170658\pi\)
−0.872226 + 0.489103i \(0.837324\pi\)
\(998\) 0 0
\(999\) 721.399 + 416.500i 0.722121 + 0.416917i
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 448.3.r.b.191.1 4
4.3 odd 2 inner 448.3.r.b.191.2 4
7.4 even 3 inner 448.3.r.b.319.2 4
8.3 odd 2 224.3.r.b.191.1 yes 4
8.5 even 2 224.3.r.b.191.2 yes 4
28.11 odd 6 inner 448.3.r.b.319.1 4
56.5 odd 6 1568.3.d.c.1471.1 2
56.11 odd 6 224.3.r.b.95.2 yes 4
56.19 even 6 1568.3.d.c.1471.2 2
56.37 even 6 1568.3.d.d.1471.2 2
56.51 odd 6 1568.3.d.d.1471.1 2
56.53 even 6 224.3.r.b.95.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
224.3.r.b.95.1 4 56.53 even 6
224.3.r.b.95.2 yes 4 56.11 odd 6
224.3.r.b.191.1 yes 4 8.3 odd 2
224.3.r.b.191.2 yes 4 8.5 even 2
448.3.r.b.191.1 4 1.1 even 1 trivial
448.3.r.b.191.2 4 4.3 odd 2 inner
448.3.r.b.319.1 4 28.11 odd 6 inner
448.3.r.b.319.2 4 7.4 even 3 inner
1568.3.d.c.1471.1 2 56.5 odd 6
1568.3.d.c.1471.2 2 56.19 even 6
1568.3.d.d.1471.1 2 56.51 odd 6
1568.3.d.d.1471.2 2 56.37 even 6