Properties

Label 1875.2.b.e.1249.1
Level $1875$
Weight $2$
Character 1875.1249
Analytic conductor $14.972$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1875,2,Mod(1249,1875)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1875, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1875.1249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1875 = 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1875.b (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: \(14.9719503790\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 23x^{10} + 199x^{8} + 794x^{6} + 1399x^{4} + 783x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1249.1
Root \(-2.78712i\) of defining polynomial
Character \(\chi\) \(=\) 1875.1249
Dual form 1875.2.b.e.1249.12

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.78712i q^{2} -1.00000i q^{3} -5.76803 q^{4} -2.78712 q^{6} +3.15000i q^{7} +10.5020i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-2.78712i q^{2} -1.00000i q^{3} -5.76803 q^{4} -2.78712 q^{6} +3.15000i q^{7} +10.5020i q^{8} -1.00000 q^{9} +2.94681 q^{11} +5.76803i q^{12} +0.188171i q^{13} +8.77943 q^{14} +17.7341 q^{16} -2.62743i q^{17} +2.78712i q^{18} -1.94100 q^{19} +3.15000 q^{21} -8.21310i q^{22} -1.30447i q^{23} +10.5020 q^{24} +0.524454 q^{26} +1.00000i q^{27} -18.1693i q^{28} -1.23197 q^{29} +2.65174 q^{31} -28.4233i q^{32} -2.94681i q^{33} -7.32297 q^{34} +5.76803 q^{36} -10.1161i q^{37} +5.40980i q^{38} +0.188171 q^{39} +3.85940 q^{41} -8.77943i q^{42} -9.63734i q^{43} -16.9973 q^{44} -3.63570 q^{46} +11.9846i q^{47} -17.7341i q^{48} -2.92250 q^{49} -2.62743 q^{51} -1.08537i q^{52} +0.369521i q^{53} +2.78712 q^{54} -33.0812 q^{56} +1.94100i q^{57} +3.43364i q^{58} +6.90864 q^{59} +5.68643 q^{61} -7.39071i q^{62} -3.15000i q^{63} -43.7507 q^{64} -8.21310 q^{66} -3.24188i q^{67} +15.1551i q^{68} -1.30447 q^{69} +6.69982 q^{71} -10.5020i q^{72} -9.46938i q^{73} -28.1948 q^{74} +11.1957 q^{76} +9.28244i q^{77} -0.524454i q^{78} -0.420137 q^{79} +1.00000 q^{81} -10.7566i q^{82} -12.1635i q^{83} -18.1693 q^{84} -26.8604 q^{86} +1.23197i q^{87} +30.9473i q^{88} +15.1828 q^{89} -0.592737 q^{91} +7.52421i q^{92} -2.65174i q^{93} +33.4025 q^{94} -28.4233 q^{96} -13.9035i q^{97} +8.14536i q^{98} -2.94681 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 22 q^{4} - 2 q^{6} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 22 q^{4} - 2 q^{6} - 12 q^{9} + 8 q^{14} + 34 q^{16} + 4 q^{19} + 4 q^{21} + 12 q^{24} + 74 q^{26} - 62 q^{29} - 4 q^{31} - 74 q^{34} + 22 q^{36} + 66 q^{41} + 22 q^{44} - 24 q^{46} - 8 q^{49} - 4 q^{51} + 2 q^{54} - 60 q^{56} + 16 q^{59} + 68 q^{61} - 24 q^{64} - 18 q^{66} - 2 q^{69} - 6 q^{71} - 72 q^{74} + 54 q^{76} - 50 q^{79} + 12 q^{81} - 88 q^{84} - 60 q^{86} - 36 q^{89} + 56 q^{91} + 100 q^{94} - 66 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times\).

\(n\) \(626\) \(1252\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.78712i − 1.97079i −0.170281 0.985395i \(-0.554468\pi\)
0.170281 0.985395i \(-0.445532\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) −5.76803 −2.88402
\(5\) 0 0
\(6\) −2.78712 −1.13784
\(7\) 3.15000i 1.19059i 0.803508 + 0.595294i \(0.202964\pi\)
−0.803508 + 0.595294i \(0.797036\pi\)
\(8\) 10.5020i 3.71300i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 2.94681 0.888496 0.444248 0.895904i \(-0.353471\pi\)
0.444248 + 0.895904i \(0.353471\pi\)
\(12\) 5.76803i 1.66509i
\(13\) 0.188171i 0.0521891i 0.999659 + 0.0260946i \(0.00830710\pi\)
−0.999659 + 0.0260946i \(0.991693\pi\)
\(14\) 8.77943 2.34640
\(15\) 0 0
\(16\) 17.7341 4.43354
\(17\) − 2.62743i − 0.637246i −0.947882 0.318623i \(-0.896780\pi\)
0.947882 0.318623i \(-0.103220\pi\)
\(18\) 2.78712i 0.656930i
\(19\) −1.94100 −0.445296 −0.222648 0.974899i \(-0.571470\pi\)
−0.222648 + 0.974899i \(0.571470\pi\)
\(20\) 0 0
\(21\) 3.15000 0.687386
\(22\) − 8.21310i − 1.75104i
\(23\) − 1.30447i − 0.272000i −0.990709 0.136000i \(-0.956575\pi\)
0.990709 0.136000i \(-0.0434247\pi\)
\(24\) 10.5020 2.14370
\(25\) 0 0
\(26\) 0.524454 0.102854
\(27\) 1.00000i 0.192450i
\(28\) − 18.1693i − 3.43368i
\(29\) −1.23197 −0.228770 −0.114385 0.993436i \(-0.536490\pi\)
−0.114385 + 0.993436i \(0.536490\pi\)
\(30\) 0 0
\(31\) 2.65174 0.476266 0.238133 0.971233i \(-0.423465\pi\)
0.238133 + 0.971233i \(0.423465\pi\)
\(32\) − 28.4233i − 5.02457i
\(33\) − 2.94681i − 0.512973i
\(34\) −7.32297 −1.25588
\(35\) 0 0
\(36\) 5.76803 0.961339
\(37\) − 10.1161i − 1.66308i −0.555466 0.831540i \(-0.687460\pi\)
0.555466 0.831540i \(-0.312540\pi\)
\(38\) 5.40980i 0.877585i
\(39\) 0.188171 0.0301314
\(40\) 0 0
\(41\) 3.85940 0.602737 0.301368 0.953508i \(-0.402557\pi\)
0.301368 + 0.953508i \(0.402557\pi\)
\(42\) − 8.77943i − 1.35469i
\(43\) − 9.63734i − 1.46968i −0.678240 0.734840i \(-0.737257\pi\)
0.678240 0.734840i \(-0.262743\pi\)
\(44\) −16.9973 −2.56244
\(45\) 0 0
\(46\) −3.63570 −0.536055
\(47\) 11.9846i 1.74814i 0.485804 + 0.874068i \(0.338527\pi\)
−0.485804 + 0.874068i \(0.661473\pi\)
\(48\) − 17.7341i − 2.55970i
\(49\) −2.92250 −0.417500
\(50\) 0 0
\(51\) −2.62743 −0.367914
\(52\) − 1.08537i − 0.150514i
\(53\) 0.369521i 0.0507576i 0.999678 + 0.0253788i \(0.00807919\pi\)
−0.999678 + 0.0253788i \(0.991921\pi\)
\(54\) 2.78712 0.379279
\(55\) 0 0
\(56\) −33.0812 −4.42066
\(57\) 1.94100i 0.257092i
\(58\) 3.43364i 0.450859i
\(59\) 6.90864 0.899428 0.449714 0.893173i \(-0.351526\pi\)
0.449714 + 0.893173i \(0.351526\pi\)
\(60\) 0 0
\(61\) 5.68643 0.728073 0.364037 0.931385i \(-0.381398\pi\)
0.364037 + 0.931385i \(0.381398\pi\)
\(62\) − 7.39071i − 0.938621i
\(63\) − 3.15000i − 0.396863i
\(64\) −43.7507 −5.46884
\(65\) 0 0
\(66\) −8.21310 −1.01096
\(67\) − 3.24188i − 0.396058i −0.980196 0.198029i \(-0.936546\pi\)
0.980196 0.198029i \(-0.0634541\pi\)
\(68\) 15.1551i 1.83783i
\(69\) −1.30447 −0.157039
\(70\) 0 0
\(71\) 6.69982 0.795122 0.397561 0.917576i \(-0.369857\pi\)
0.397561 + 0.917576i \(0.369857\pi\)
\(72\) − 10.5020i − 1.23767i
\(73\) − 9.46938i − 1.10831i −0.832415 0.554153i \(-0.813042\pi\)
0.832415 0.554153i \(-0.186958\pi\)
\(74\) −28.1948 −3.27758
\(75\) 0 0
\(76\) 11.1957 1.28424
\(77\) 9.28244i 1.05783i
\(78\) − 0.524454i − 0.0593827i
\(79\) −0.420137 −0.0472691 −0.0236345 0.999721i \(-0.507524\pi\)
−0.0236345 + 0.999721i \(0.507524\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 10.7566i − 1.18787i
\(83\) − 12.1635i − 1.33512i −0.744557 0.667559i \(-0.767339\pi\)
0.744557 0.667559i \(-0.232661\pi\)
\(84\) −18.1693 −1.98243
\(85\) 0 0
\(86\) −26.8604 −2.89643
\(87\) 1.23197i 0.132081i
\(88\) 30.9473i 3.29899i
\(89\) 15.1828 1.60937 0.804687 0.593699i \(-0.202333\pi\)
0.804687 + 0.593699i \(0.202333\pi\)
\(90\) 0 0
\(91\) −0.592737 −0.0621358
\(92\) 7.52421i 0.784453i
\(93\) − 2.65174i − 0.274972i
\(94\) 33.4025 3.44521
\(95\) 0 0
\(96\) −28.4233 −2.90094
\(97\) − 13.9035i − 1.41169i −0.708367 0.705845i \(-0.750568\pi\)
0.708367 0.705845i \(-0.249432\pi\)
\(98\) 8.14536i 0.822805i
\(99\) −2.94681 −0.296165
\(100\) 0 0
\(101\) −9.67188 −0.962388 −0.481194 0.876614i \(-0.659797\pi\)
−0.481194 + 0.876614i \(0.659797\pi\)
\(102\) 7.32297i 0.725082i
\(103\) 8.69434i 0.856679i 0.903618 + 0.428339i \(0.140901\pi\)
−0.903618 + 0.428339i \(0.859099\pi\)
\(104\) −1.97616 −0.193778
\(105\) 0 0
\(106\) 1.02990 0.100033
\(107\) 2.56585i 0.248050i 0.992279 + 0.124025i \(0.0395804\pi\)
−0.992279 + 0.124025i \(0.960420\pi\)
\(108\) − 5.76803i − 0.555029i
\(109\) 15.2352 1.45926 0.729632 0.683840i \(-0.239691\pi\)
0.729632 + 0.683840i \(0.239691\pi\)
\(110\) 0 0
\(111\) −10.1161 −0.960179
\(112\) 55.8626i 5.27852i
\(113\) 2.20391i 0.207326i 0.994612 + 0.103663i \(0.0330564\pi\)
−0.994612 + 0.103663i \(0.966944\pi\)
\(114\) 5.40980 0.506674
\(115\) 0 0
\(116\) 7.10602 0.659778
\(117\) − 0.188171i − 0.0173964i
\(118\) − 19.2552i − 1.77258i
\(119\) 8.27641 0.758697
\(120\) 0 0
\(121\) −2.31633 −0.210575
\(122\) − 15.8488i − 1.43488i
\(123\) − 3.85940i − 0.347990i
\(124\) −15.2953 −1.37356
\(125\) 0 0
\(126\) −8.77943 −0.782133
\(127\) 14.9689i 1.32827i 0.747611 + 0.664136i \(0.231201\pi\)
−0.747611 + 0.664136i \(0.768799\pi\)
\(128\) 65.0920i 5.75337i
\(129\) −9.63734 −0.848521
\(130\) 0 0
\(131\) 11.2957 0.986911 0.493456 0.869771i \(-0.335734\pi\)
0.493456 + 0.869771i \(0.335734\pi\)
\(132\) 16.9973i 1.47942i
\(133\) − 6.11415i − 0.530164i
\(134\) −9.03550 −0.780548
\(135\) 0 0
\(136\) 27.5932 2.36610
\(137\) − 4.17877i − 0.357017i −0.983938 0.178508i \(-0.942873\pi\)
0.983938 0.178508i \(-0.0571271\pi\)
\(138\) 3.63570i 0.309492i
\(139\) 15.3552 1.30241 0.651207 0.758900i \(-0.274263\pi\)
0.651207 + 0.758900i \(0.274263\pi\)
\(140\) 0 0
\(141\) 11.9846 1.00929
\(142\) − 18.6732i − 1.56702i
\(143\) 0.554502i 0.0463698i
\(144\) −17.7341 −1.47785
\(145\) 0 0
\(146\) −26.3923 −2.18424
\(147\) 2.92250i 0.241044i
\(148\) 58.3501i 4.79635i
\(149\) −3.43364 −0.281294 −0.140647 0.990060i \(-0.544918\pi\)
−0.140647 + 0.990060i \(0.544918\pi\)
\(150\) 0 0
\(151\) 10.7716 0.876581 0.438290 0.898833i \(-0.355584\pi\)
0.438290 + 0.898833i \(0.355584\pi\)
\(152\) − 20.3843i − 1.65338i
\(153\) 2.62743i 0.212415i
\(154\) 25.8713 2.08477
\(155\) 0 0
\(156\) −1.08537 −0.0868995
\(157\) 13.4045i 1.06980i 0.844916 + 0.534900i \(0.179651\pi\)
−0.844916 + 0.534900i \(0.820349\pi\)
\(158\) 1.17097i 0.0931574i
\(159\) 0.369521 0.0293049
\(160\) 0 0
\(161\) 4.10907 0.323840
\(162\) − 2.78712i − 0.218977i
\(163\) − 15.4178i − 1.20762i −0.797129 0.603808i \(-0.793649\pi\)
0.797129 0.603808i \(-0.206351\pi\)
\(164\) −22.2611 −1.73830
\(165\) 0 0
\(166\) −33.9011 −2.63124
\(167\) − 17.4045i − 1.34680i −0.739276 0.673402i \(-0.764832\pi\)
0.739276 0.673402i \(-0.235168\pi\)
\(168\) 33.0812i 2.55227i
\(169\) 12.9646 0.997276
\(170\) 0 0
\(171\) 1.94100 0.148432
\(172\) 55.5885i 4.23858i
\(173\) − 19.2737i − 1.46535i −0.680578 0.732676i \(-0.738271\pi\)
0.680578 0.732676i \(-0.261729\pi\)
\(174\) 3.43364 0.260303
\(175\) 0 0
\(176\) 52.2591 3.93918
\(177\) − 6.90864i − 0.519285i
\(178\) − 42.3163i − 3.17174i
\(179\) −23.7940 −1.77845 −0.889223 0.457475i \(-0.848754\pi\)
−0.889223 + 0.457475i \(0.848754\pi\)
\(180\) 0 0
\(181\) −10.6891 −0.794516 −0.397258 0.917707i \(-0.630038\pi\)
−0.397258 + 0.917707i \(0.630038\pi\)
\(182\) 1.65203i 0.122457i
\(183\) − 5.68643i − 0.420353i
\(184\) 13.6995 1.00994
\(185\) 0 0
\(186\) −7.39071 −0.541913
\(187\) − 7.74253i − 0.566190i
\(188\) − 69.1277i − 5.04165i
\(189\) −3.15000 −0.229129
\(190\) 0 0
\(191\) −14.3581 −1.03892 −0.519458 0.854496i \(-0.673866\pi\)
−0.519458 + 0.854496i \(0.673866\pi\)
\(192\) 43.7507i 3.15744i
\(193\) 1.95508i 0.140730i 0.997521 + 0.0703648i \(0.0224163\pi\)
−0.997521 + 0.0703648i \(0.977584\pi\)
\(194\) −38.7508 −2.78214
\(195\) 0 0
\(196\) 16.8571 1.20408
\(197\) 1.17958i 0.0840417i 0.999117 + 0.0420209i \(0.0133796\pi\)
−0.999117 + 0.0420209i \(0.986620\pi\)
\(198\) 8.21310i 0.583680i
\(199\) 6.55820 0.464899 0.232449 0.972608i \(-0.425326\pi\)
0.232449 + 0.972608i \(0.425326\pi\)
\(200\) 0 0
\(201\) −3.24188 −0.228664
\(202\) 26.9567i 1.89666i
\(203\) − 3.88069i − 0.272371i
\(204\) 15.1551 1.06107
\(205\) 0 0
\(206\) 24.2322 1.68833
\(207\) 1.30447i 0.0906667i
\(208\) 3.33705i 0.231382i
\(209\) −5.71975 −0.395643
\(210\) 0 0
\(211\) 12.3573 0.850715 0.425357 0.905026i \(-0.360148\pi\)
0.425357 + 0.905026i \(0.360148\pi\)
\(212\) − 2.13141i − 0.146386i
\(213\) − 6.69982i − 0.459064i
\(214\) 7.15134 0.488856
\(215\) 0 0
\(216\) −10.5020 −0.714568
\(217\) 8.35298i 0.567037i
\(218\) − 42.4622i − 2.87591i
\(219\) −9.46938 −0.639881
\(220\) 0 0
\(221\) 0.494405 0.0332573
\(222\) 28.1948i 1.89231i
\(223\) − 7.30693i − 0.489308i −0.969610 0.244654i \(-0.921326\pi\)
0.969610 0.244654i \(-0.0786744\pi\)
\(224\) 89.5333 5.98219
\(225\) 0 0
\(226\) 6.14256 0.408597
\(227\) 21.0202i 1.39516i 0.716508 + 0.697579i \(0.245739\pi\)
−0.716508 + 0.697579i \(0.754261\pi\)
\(228\) − 11.1957i − 0.741457i
\(229\) −19.5544 −1.29219 −0.646094 0.763258i \(-0.723599\pi\)
−0.646094 + 0.763258i \(0.723599\pi\)
\(230\) 0 0
\(231\) 9.28244 0.610740
\(232\) − 12.9381i − 0.849425i
\(233\) 12.1472i 0.795792i 0.917430 + 0.397896i \(0.130260\pi\)
−0.917430 + 0.397896i \(0.869740\pi\)
\(234\) −0.524454 −0.0342846
\(235\) 0 0
\(236\) −39.8493 −2.59397
\(237\) 0.420137i 0.0272908i
\(238\) − 23.0673i − 1.49523i
\(239\) −13.4517 −0.870119 −0.435059 0.900402i \(-0.643273\pi\)
−0.435059 + 0.900402i \(0.643273\pi\)
\(240\) 0 0
\(241\) 19.9370 1.28426 0.642129 0.766597i \(-0.278051\pi\)
0.642129 + 0.766597i \(0.278051\pi\)
\(242\) 6.45588i 0.415000i
\(243\) − 1.00000i − 0.0641500i
\(244\) −32.7995 −2.09978
\(245\) 0 0
\(246\) −10.7566 −0.685816
\(247\) − 0.365239i − 0.0232396i
\(248\) 27.8485i 1.76838i
\(249\) −12.1635 −0.770830
\(250\) 0 0
\(251\) −2.76013 −0.174218 −0.0871088 0.996199i \(-0.527763\pi\)
−0.0871088 + 0.996199i \(0.527763\pi\)
\(252\) 18.1693i 1.14456i
\(253\) − 3.84401i − 0.241671i
\(254\) 41.7200 2.61775
\(255\) 0 0
\(256\) 93.9177 5.86986
\(257\) 17.2848i 1.07820i 0.842243 + 0.539098i \(0.181235\pi\)
−0.842243 + 0.539098i \(0.818765\pi\)
\(258\) 26.8604i 1.67226i
\(259\) 31.8658 1.98004
\(260\) 0 0
\(261\) 1.23197 0.0762568
\(262\) − 31.4825i − 1.94500i
\(263\) − 3.96578i − 0.244541i −0.992497 0.122270i \(-0.960983\pi\)
0.992497 0.122270i \(-0.0390175\pi\)
\(264\) 30.9473 1.90467
\(265\) 0 0
\(266\) −17.0409 −1.04484
\(267\) − 15.1828i − 0.929173i
\(268\) 18.6993i 1.14224i
\(269\) −1.19564 −0.0728993 −0.0364496 0.999335i \(-0.511605\pi\)
−0.0364496 + 0.999335i \(0.511605\pi\)
\(270\) 0 0
\(271\) −2.18295 −0.132605 −0.0663023 0.997800i \(-0.521120\pi\)
−0.0663023 + 0.997800i \(0.521120\pi\)
\(272\) − 46.5953i − 2.82525i
\(273\) 0.592737i 0.0358741i
\(274\) −11.6467 −0.703605
\(275\) 0 0
\(276\) 7.52421 0.452904
\(277\) − 1.95174i − 0.117268i −0.998280 0.0586342i \(-0.981325\pi\)
0.998280 0.0586342i \(-0.0186746\pi\)
\(278\) − 42.7969i − 2.56679i
\(279\) −2.65174 −0.158755
\(280\) 0 0
\(281\) 2.30131 0.137284 0.0686422 0.997641i \(-0.478133\pi\)
0.0686422 + 0.997641i \(0.478133\pi\)
\(282\) − 33.4025i − 1.98909i
\(283\) − 10.6459i − 0.632835i −0.948620 0.316417i \(-0.897520\pi\)
0.948620 0.316417i \(-0.102480\pi\)
\(284\) −38.6448 −2.29315
\(285\) 0 0
\(286\) 1.54546 0.0913852
\(287\) 12.1571i 0.717611i
\(288\) 28.4233i 1.67486i
\(289\) 10.0966 0.593918
\(290\) 0 0
\(291\) −13.9035 −0.815039
\(292\) 54.6197i 3.19638i
\(293\) 14.9591i 0.873921i 0.899481 + 0.436960i \(0.143945\pi\)
−0.899481 + 0.436960i \(0.856055\pi\)
\(294\) 8.14536 0.475047
\(295\) 0 0
\(296\) 106.239 6.17502
\(297\) 2.94681i 0.170991i
\(298\) 9.56995i 0.554373i
\(299\) 0.245462 0.0141954
\(300\) 0 0
\(301\) 30.3576 1.74978
\(302\) − 30.0217i − 1.72756i
\(303\) 9.67188i 0.555635i
\(304\) −34.4220 −1.97424
\(305\) 0 0
\(306\) 7.32297 0.418626
\(307\) 3.07333i 0.175404i 0.996147 + 0.0877021i \(0.0279524\pi\)
−0.996147 + 0.0877021i \(0.972048\pi\)
\(308\) − 53.5414i − 3.05081i
\(309\) 8.69434 0.494604
\(310\) 0 0
\(311\) 24.9095 1.41249 0.706244 0.707968i \(-0.250388\pi\)
0.706244 + 0.707968i \(0.250388\pi\)
\(312\) 1.97616i 0.111878i
\(313\) 9.28562i 0.524854i 0.964952 + 0.262427i \(0.0845230\pi\)
−0.964952 + 0.262427i \(0.915477\pi\)
\(314\) 37.3601 2.10835
\(315\) 0 0
\(316\) 2.42336 0.136325
\(317\) − 20.7685i − 1.16647i −0.812302 0.583237i \(-0.801786\pi\)
0.812302 0.583237i \(-0.198214\pi\)
\(318\) − 1.02990i − 0.0577538i
\(319\) −3.63037 −0.203261
\(320\) 0 0
\(321\) 2.56585 0.143212
\(322\) − 11.4525i − 0.638221i
\(323\) 5.09984i 0.283763i
\(324\) −5.76803 −0.320446
\(325\) 0 0
\(326\) −42.9713 −2.37996
\(327\) − 15.2352i − 0.842507i
\(328\) 40.5312i 2.23796i
\(329\) −37.7515 −2.08131
\(330\) 0 0
\(331\) −33.9172 −1.86426 −0.932128 0.362129i \(-0.882050\pi\)
−0.932128 + 0.362129i \(0.882050\pi\)
\(332\) 70.1595i 3.85050i
\(333\) 10.1161i 0.554360i
\(334\) −48.5085 −2.65427
\(335\) 0 0
\(336\) 55.8626 3.04755
\(337\) − 17.7792i − 0.968494i −0.874931 0.484247i \(-0.839094\pi\)
0.874931 0.484247i \(-0.160906\pi\)
\(338\) − 36.1339i − 1.96542i
\(339\) 2.20391 0.119700
\(340\) 0 0
\(341\) 7.81416 0.423161
\(342\) − 5.40980i − 0.292528i
\(343\) 12.8441i 0.693518i
\(344\) 101.211 5.45693
\(345\) 0 0
\(346\) −53.7181 −2.88790
\(347\) 12.2251i 0.656277i 0.944630 + 0.328138i \(0.106421\pi\)
−0.944630 + 0.328138i \(0.893579\pi\)
\(348\) − 7.10602i − 0.380923i
\(349\) −12.8425 −0.687441 −0.343720 0.939072i \(-0.611687\pi\)
−0.343720 + 0.939072i \(0.611687\pi\)
\(350\) 0 0
\(351\) −0.188171 −0.0100438
\(352\) − 83.7579i − 4.46431i
\(353\) 17.4163i 0.926976i 0.886103 + 0.463488i \(0.153402\pi\)
−0.886103 + 0.463488i \(0.846598\pi\)
\(354\) −19.2552 −1.02340
\(355\) 0 0
\(356\) −87.5749 −4.64146
\(357\) − 8.27641i − 0.438034i
\(358\) 66.3167i 3.50494i
\(359\) 3.82947 0.202111 0.101056 0.994881i \(-0.467778\pi\)
0.101056 + 0.994881i \(0.467778\pi\)
\(360\) 0 0
\(361\) −15.2325 −0.801712
\(362\) 29.7918i 1.56582i
\(363\) 2.31633i 0.121576i
\(364\) 3.41893 0.179201
\(365\) 0 0
\(366\) −15.8488 −0.828428
\(367\) 29.6282i 1.54658i 0.634054 + 0.773289i \(0.281390\pi\)
−0.634054 + 0.773289i \(0.718610\pi\)
\(368\) − 23.1336i − 1.20592i
\(369\) −3.85940 −0.200912
\(370\) 0 0
\(371\) −1.16399 −0.0604314
\(372\) 15.2953i 0.793025i
\(373\) 11.4595i 0.593350i 0.954978 + 0.296675i \(0.0958779\pi\)
−0.954978 + 0.296675i \(0.904122\pi\)
\(374\) −21.5794 −1.11584
\(375\) 0 0
\(376\) −125.862 −6.49083
\(377\) − 0.231820i − 0.0119393i
\(378\) 8.77943i 0.451565i
\(379\) 2.19307 0.112650 0.0563251 0.998412i \(-0.482062\pi\)
0.0563251 + 0.998412i \(0.482062\pi\)
\(380\) 0 0
\(381\) 14.9689 0.766879
\(382\) 40.0177i 2.04749i
\(383\) − 10.1997i − 0.521181i −0.965449 0.260591i \(-0.916083\pi\)
0.965449 0.260591i \(-0.0839173\pi\)
\(384\) 65.0920 3.32171
\(385\) 0 0
\(386\) 5.44904 0.277349
\(387\) 9.63734i 0.489894i
\(388\) 80.1960i 4.07134i
\(389\) −28.5482 −1.44745 −0.723726 0.690088i \(-0.757572\pi\)
−0.723726 + 0.690088i \(0.757572\pi\)
\(390\) 0 0
\(391\) −3.42740 −0.173331
\(392\) − 30.6920i − 1.55018i
\(393\) − 11.2957i − 0.569794i
\(394\) 3.28764 0.165629
\(395\) 0 0
\(396\) 16.9973 0.854146
\(397\) − 37.1827i − 1.86614i −0.359688 0.933072i \(-0.617117\pi\)
0.359688 0.933072i \(-0.382883\pi\)
\(398\) − 18.2785i − 0.916218i
\(399\) −6.11415 −0.306090
\(400\) 0 0
\(401\) 7.38648 0.368863 0.184432 0.982845i \(-0.440956\pi\)
0.184432 + 0.982845i \(0.440956\pi\)
\(402\) 9.03550i 0.450650i
\(403\) 0.498979i 0.0248559i
\(404\) 55.7877 2.77554
\(405\) 0 0
\(406\) −10.8160 −0.536787
\(407\) − 29.8102i − 1.47764i
\(408\) − 27.5932i − 1.36607i
\(409\) 5.19942 0.257095 0.128547 0.991703i \(-0.458969\pi\)
0.128547 + 0.991703i \(0.458969\pi\)
\(410\) 0 0
\(411\) −4.17877 −0.206124
\(412\) − 50.1492i − 2.47068i
\(413\) 21.7622i 1.07085i
\(414\) 3.63570 0.178685
\(415\) 0 0
\(416\) 5.34842 0.262228
\(417\) − 15.3552i − 0.751949i
\(418\) 15.9416i 0.779730i
\(419\) 14.9205 0.728914 0.364457 0.931220i \(-0.381255\pi\)
0.364457 + 0.931220i \(0.381255\pi\)
\(420\) 0 0
\(421\) −0.0666925 −0.00325039 −0.00162520 0.999999i \(-0.500517\pi\)
−0.00162520 + 0.999999i \(0.500517\pi\)
\(422\) − 34.4414i − 1.67658i
\(423\) − 11.9846i − 0.582712i
\(424\) −3.88069 −0.188463
\(425\) 0 0
\(426\) −18.6732 −0.904719
\(427\) 17.9123i 0.866835i
\(428\) − 14.7999i − 0.715382i
\(429\) 0.554502 0.0267716
\(430\) 0 0
\(431\) 27.6996 1.33424 0.667121 0.744950i \(-0.267527\pi\)
0.667121 + 0.744950i \(0.267527\pi\)
\(432\) 17.7341i 0.853235i
\(433\) − 30.5626i − 1.46874i −0.678748 0.734372i \(-0.737477\pi\)
0.678748 0.734372i \(-0.262523\pi\)
\(434\) 23.2807 1.11751
\(435\) 0 0
\(436\) −87.8770 −4.20854
\(437\) 2.53197i 0.121120i
\(438\) 26.3923i 1.26107i
\(439\) −11.0968 −0.529623 −0.264812 0.964300i \(-0.585310\pi\)
−0.264812 + 0.964300i \(0.585310\pi\)
\(440\) 0 0
\(441\) 2.92250 0.139167
\(442\) − 1.37797i − 0.0655432i
\(443\) − 9.82831i − 0.466957i −0.972362 0.233479i \(-0.924989\pi\)
0.972362 0.233479i \(-0.0750109\pi\)
\(444\) 58.3501 2.76917
\(445\) 0 0
\(446\) −20.3653 −0.964324
\(447\) 3.43364i 0.162405i
\(448\) − 137.815i − 6.51114i
\(449\) 19.1301 0.902805 0.451402 0.892320i \(-0.350924\pi\)
0.451402 + 0.892320i \(0.350924\pi\)
\(450\) 0 0
\(451\) 11.3729 0.535529
\(452\) − 12.7122i − 0.597933i
\(453\) − 10.7716i − 0.506094i
\(454\) 58.5858 2.74957
\(455\) 0 0
\(456\) −20.3843 −0.954582
\(457\) − 18.7578i − 0.877452i −0.898621 0.438726i \(-0.855430\pi\)
0.898621 0.438726i \(-0.144570\pi\)
\(458\) 54.5003i 2.54663i
\(459\) 2.62743 0.122638
\(460\) 0 0
\(461\) −37.6208 −1.75217 −0.876087 0.482153i \(-0.839855\pi\)
−0.876087 + 0.482153i \(0.839855\pi\)
\(462\) − 25.8713i − 1.20364i
\(463\) − 19.9275i − 0.926111i −0.886329 0.463056i \(-0.846753\pi\)
0.886329 0.463056i \(-0.153247\pi\)
\(464\) −21.8479 −1.01426
\(465\) 0 0
\(466\) 33.8558 1.56834
\(467\) − 27.3667i − 1.26638i −0.773997 0.633190i \(-0.781745\pi\)
0.773997 0.633190i \(-0.218255\pi\)
\(468\) 1.08537i 0.0501714i
\(469\) 10.2119 0.471542
\(470\) 0 0
\(471\) 13.4045 0.617649
\(472\) 72.5542i 3.33958i
\(473\) − 28.3994i − 1.30581i
\(474\) 1.17097 0.0537845
\(475\) 0 0
\(476\) −47.7386 −2.18810
\(477\) − 0.369521i − 0.0169192i
\(478\) 37.4915i 1.71482i
\(479\) 38.5742 1.76250 0.881250 0.472650i \(-0.156703\pi\)
0.881250 + 0.472650i \(0.156703\pi\)
\(480\) 0 0
\(481\) 1.90356 0.0867946
\(482\) − 55.5669i − 2.53100i
\(483\) − 4.10907i − 0.186969i
\(484\) 13.3607 0.607303
\(485\) 0 0
\(486\) −2.78712 −0.126426
\(487\) 18.4995i 0.838294i 0.907918 + 0.419147i \(0.137671\pi\)
−0.907918 + 0.419147i \(0.862329\pi\)
\(488\) 59.7187i 2.70334i
\(489\) −15.4178 −0.697218
\(490\) 0 0
\(491\) 19.7211 0.890000 0.445000 0.895531i \(-0.353204\pi\)
0.445000 + 0.895531i \(0.353204\pi\)
\(492\) 22.2611i 1.00361i
\(493\) 3.23691i 0.145783i
\(494\) −1.01796 −0.0458004
\(495\) 0 0
\(496\) 47.0263 2.11154
\(497\) 21.1044i 0.946663i
\(498\) 33.9011i 1.51915i
\(499\) 36.2906 1.62459 0.812296 0.583245i \(-0.198217\pi\)
0.812296 + 0.583245i \(0.198217\pi\)
\(500\) 0 0
\(501\) −17.4045 −0.777578
\(502\) 7.69280i 0.343347i
\(503\) 2.66163i 0.118676i 0.998238 + 0.0593382i \(0.0188990\pi\)
−0.998238 + 0.0593382i \(0.981101\pi\)
\(504\) 33.0812 1.47355
\(505\) 0 0
\(506\) −10.7137 −0.476283
\(507\) − 12.9646i − 0.575778i
\(508\) − 86.3410i − 3.83076i
\(509\) −22.8322 −1.01202 −0.506010 0.862528i \(-0.668880\pi\)
−0.506010 + 0.862528i \(0.668880\pi\)
\(510\) 0 0
\(511\) 29.8285 1.31954
\(512\) − 131.576i − 5.81488i
\(513\) − 1.94100i − 0.0856972i
\(514\) 48.1748 2.12490
\(515\) 0 0
\(516\) 55.5885 2.44715
\(517\) 35.3163i 1.55321i
\(518\) − 88.8137i − 3.90225i
\(519\) −19.2737 −0.846021
\(520\) 0 0
\(521\) −12.3029 −0.539001 −0.269501 0.963000i \(-0.586859\pi\)
−0.269501 + 0.963000i \(0.586859\pi\)
\(522\) − 3.43364i − 0.150286i
\(523\) − 33.3902i − 1.46005i −0.683420 0.730025i \(-0.739508\pi\)
0.683420 0.730025i \(-0.260492\pi\)
\(524\) −65.1541 −2.84627
\(525\) 0 0
\(526\) −11.0531 −0.481939
\(527\) − 6.96726i − 0.303499i
\(528\) − 52.2591i − 2.27429i
\(529\) 21.2984 0.926016
\(530\) 0 0
\(531\) −6.90864 −0.299809
\(532\) 35.2666i 1.52900i
\(533\) 0.726225i 0.0314563i
\(534\) −42.3163 −1.83121
\(535\) 0 0
\(536\) 34.0461 1.47057
\(537\) 23.7940i 1.02679i
\(538\) 3.33238i 0.143669i
\(539\) −8.61204 −0.370947
\(540\) 0 0
\(541\) 5.98673 0.257389 0.128695 0.991684i \(-0.458921\pi\)
0.128695 + 0.991684i \(0.458921\pi\)
\(542\) 6.08413i 0.261336i
\(543\) 10.6891i 0.458714i
\(544\) −74.6802 −3.20189
\(545\) 0 0
\(546\) 1.65203 0.0707003
\(547\) − 4.47889i − 0.191503i −0.995405 0.0957516i \(-0.969475\pi\)
0.995405 0.0957516i \(-0.0305255\pi\)
\(548\) 24.1033i 1.02964i
\(549\) −5.68643 −0.242691
\(550\) 0 0
\(551\) 2.39124 0.101870
\(552\) − 13.6995i − 0.583088i
\(553\) − 1.32343i − 0.0562780i
\(554\) −5.43972 −0.231112
\(555\) 0 0
\(556\) −88.5695 −3.75618
\(557\) 14.4278i 0.611326i 0.952140 + 0.305663i \(0.0988781\pi\)
−0.952140 + 0.305663i \(0.901122\pi\)
\(558\) 7.39071i 0.312874i
\(559\) 1.81346 0.0767014
\(560\) 0 0
\(561\) −7.74253 −0.326890
\(562\) − 6.41401i − 0.270559i
\(563\) 27.6973i 1.16730i 0.812004 + 0.583652i \(0.198377\pi\)
−0.812004 + 0.583652i \(0.801623\pi\)
\(564\) −69.1277 −2.91080
\(565\) 0 0
\(566\) −29.6715 −1.24719
\(567\) 3.15000i 0.132288i
\(568\) 70.3612i 2.95229i
\(569\) −35.6179 −1.49318 −0.746590 0.665285i \(-0.768310\pi\)
−0.746590 + 0.665285i \(0.768310\pi\)
\(570\) 0 0
\(571\) 5.85044 0.244833 0.122417 0.992479i \(-0.460936\pi\)
0.122417 + 0.992479i \(0.460936\pi\)
\(572\) − 3.19839i − 0.133731i
\(573\) 14.3581i 0.599818i
\(574\) 33.8833 1.41426
\(575\) 0 0
\(576\) 43.7507 1.82295
\(577\) − 12.0524i − 0.501746i −0.968020 0.250873i \(-0.919282\pi\)
0.968020 0.250873i \(-0.0807177\pi\)
\(578\) − 28.1404i − 1.17049i
\(579\) 1.95508 0.0812503
\(580\) 0 0
\(581\) 38.3150 1.58958
\(582\) 38.7508i 1.60627i
\(583\) 1.08891i 0.0450979i
\(584\) 99.4470 4.11515
\(585\) 0 0
\(586\) 41.6928 1.72232
\(587\) − 40.2350i − 1.66068i −0.557259 0.830339i \(-0.688147\pi\)
0.557259 0.830339i \(-0.311853\pi\)
\(588\) − 16.8571i − 0.695174i
\(589\) −5.14702 −0.212079
\(590\) 0 0
\(591\) 1.17958 0.0485215
\(592\) − 179.401i − 7.37332i
\(593\) 33.4086i 1.37193i 0.727636 + 0.685964i \(0.240619\pi\)
−0.727636 + 0.685964i \(0.759381\pi\)
\(594\) 8.21310 0.336988
\(595\) 0 0
\(596\) 19.8053 0.811258
\(597\) − 6.55820i − 0.268409i
\(598\) − 0.684132i − 0.0279763i
\(599\) 24.6421 1.00685 0.503424 0.864040i \(-0.332073\pi\)
0.503424 + 0.864040i \(0.332073\pi\)
\(600\) 0 0
\(601\) 16.6327 0.678461 0.339230 0.940703i \(-0.389833\pi\)
0.339230 + 0.940703i \(0.389833\pi\)
\(602\) − 84.6103i − 3.44846i
\(603\) 3.24188i 0.132019i
\(604\) −62.1310 −2.52807
\(605\) 0 0
\(606\) 26.9567 1.09504
\(607\) − 29.4057i − 1.19354i −0.802412 0.596770i \(-0.796450\pi\)
0.802412 0.596770i \(-0.203550\pi\)
\(608\) 55.1695i 2.23742i
\(609\) −3.88069 −0.157254
\(610\) 0 0
\(611\) −2.25515 −0.0912337
\(612\) − 15.1551i − 0.612609i
\(613\) 12.1702i 0.491549i 0.969327 + 0.245775i \(0.0790423\pi\)
−0.969327 + 0.245775i \(0.920958\pi\)
\(614\) 8.56574 0.345685
\(615\) 0 0
\(616\) −97.4838 −3.92774
\(617\) 39.1024i 1.57420i 0.616823 + 0.787102i \(0.288419\pi\)
−0.616823 + 0.787102i \(0.711581\pi\)
\(618\) − 24.2322i − 0.974761i
\(619\) 4.72717 0.190001 0.0950006 0.995477i \(-0.469715\pi\)
0.0950006 + 0.995477i \(0.469715\pi\)
\(620\) 0 0
\(621\) 1.30447 0.0523464
\(622\) − 69.4258i − 2.78372i
\(623\) 47.8258i 1.91610i
\(624\) 3.33705 0.133589
\(625\) 0 0
\(626\) 25.8801 1.03438
\(627\) 5.71975i 0.228425i
\(628\) − 77.3179i − 3.08532i
\(629\) −26.5794 −1.05979
\(630\) 0 0
\(631\) −33.8710 −1.34838 −0.674191 0.738557i \(-0.735508\pi\)
−0.674191 + 0.738557i \(0.735508\pi\)
\(632\) − 4.41226i − 0.175510i
\(633\) − 12.3573i − 0.491160i
\(634\) −57.8842 −2.29888
\(635\) 0 0
\(636\) −2.13141 −0.0845158
\(637\) − 0.549929i − 0.0217890i
\(638\) 10.1183i 0.400586i
\(639\) −6.69982 −0.265041
\(640\) 0 0
\(641\) 15.9571 0.630269 0.315135 0.949047i \(-0.397950\pi\)
0.315135 + 0.949047i \(0.397950\pi\)
\(642\) − 7.15134i − 0.282241i
\(643\) − 10.2436i − 0.403968i −0.979389 0.201984i \(-0.935261\pi\)
0.979389 0.201984i \(-0.0647389\pi\)
\(644\) −23.7013 −0.933960
\(645\) 0 0
\(646\) 14.2139 0.559237
\(647\) 6.00244i 0.235980i 0.993015 + 0.117990i \(0.0376451\pi\)
−0.993015 + 0.117990i \(0.962355\pi\)
\(648\) 10.5020i 0.412556i
\(649\) 20.3584 0.799138
\(650\) 0 0
\(651\) 8.35298 0.327379
\(652\) 88.9305i 3.48279i
\(653\) 33.1069i 1.29557i 0.761822 + 0.647787i \(0.224305\pi\)
−0.761822 + 0.647787i \(0.775695\pi\)
\(654\) −42.4622 −1.66040
\(655\) 0 0
\(656\) 68.4431 2.67226
\(657\) 9.46938i 0.369436i
\(658\) 105.218i 4.10183i
\(659\) −19.5593 −0.761922 −0.380961 0.924591i \(-0.624407\pi\)
−0.380961 + 0.924591i \(0.624407\pi\)
\(660\) 0 0
\(661\) 30.5290 1.18744 0.593721 0.804671i \(-0.297658\pi\)
0.593721 + 0.804671i \(0.297658\pi\)
\(662\) 94.5312i 3.67406i
\(663\) − 0.494405i − 0.0192011i
\(664\) 127.741 4.95730
\(665\) 0 0
\(666\) 28.1948 1.09253
\(667\) 1.60706i 0.0622255i
\(668\) 100.390i 3.88421i
\(669\) −7.30693 −0.282502
\(670\) 0 0
\(671\) 16.7568 0.646890
\(672\) − 89.5333i − 3.45382i
\(673\) 4.19687i 0.161778i 0.996723 + 0.0808888i \(0.0257759\pi\)
−0.996723 + 0.0808888i \(0.974224\pi\)
\(674\) −49.5527 −1.90870
\(675\) 0 0
\(676\) −74.7802 −2.87616
\(677\) − 9.66481i − 0.371449i −0.982602 0.185724i \(-0.940537\pi\)
0.982602 0.185724i \(-0.0594632\pi\)
\(678\) − 6.14256i − 0.235903i
\(679\) 43.7961 1.68074
\(680\) 0 0
\(681\) 21.0202 0.805495
\(682\) − 21.7790i − 0.833961i
\(683\) 9.03781i 0.345822i 0.984937 + 0.172911i \(0.0553173\pi\)
−0.984937 + 0.172911i \(0.944683\pi\)
\(684\) −11.1957 −0.428080
\(685\) 0 0
\(686\) 35.7981 1.36678
\(687\) 19.5544i 0.746045i
\(688\) − 170.910i − 6.51588i
\(689\) −0.0695329 −0.00264899
\(690\) 0 0
\(691\) −32.1359 −1.22251 −0.611254 0.791435i \(-0.709334\pi\)
−0.611254 + 0.791435i \(0.709334\pi\)
\(692\) 111.171i 4.22610i
\(693\) − 9.28244i − 0.352611i
\(694\) 34.0728 1.29338
\(695\) 0 0
\(696\) −12.9381 −0.490416
\(697\) − 10.1403i − 0.384091i
\(698\) 35.7935i 1.35480i
\(699\) 12.1472 0.459451
\(700\) 0 0
\(701\) −8.54116 −0.322595 −0.161298 0.986906i \(-0.551568\pi\)
−0.161298 + 0.986906i \(0.551568\pi\)
\(702\) 0.524454i 0.0197942i
\(703\) 19.6354i 0.740562i
\(704\) −128.925 −4.85904
\(705\) 0 0
\(706\) 48.5413 1.82688
\(707\) − 30.4664i − 1.14581i
\(708\) 39.8493i 1.49763i
\(709\) −5.04613 −0.189511 −0.0947556 0.995501i \(-0.530207\pi\)
−0.0947556 + 0.995501i \(0.530207\pi\)
\(710\) 0 0
\(711\) 0.420137 0.0157564
\(712\) 159.449i 5.97561i
\(713\) − 3.45910i − 0.129544i
\(714\) −23.0673 −0.863274
\(715\) 0 0
\(716\) 137.244 5.12907
\(717\) 13.4517i 0.502363i
\(718\) − 10.6732i − 0.398319i
\(719\) 19.5253 0.728170 0.364085 0.931366i \(-0.381382\pi\)
0.364085 + 0.931366i \(0.381382\pi\)
\(720\) 0 0
\(721\) −27.3872 −1.01995
\(722\) 42.4549i 1.58001i
\(723\) − 19.9370i − 0.741467i
\(724\) 61.6552 2.29140
\(725\) 0 0
\(726\) 6.45588 0.239600
\(727\) − 40.4464i − 1.50007i −0.661395 0.750037i \(-0.730035\pi\)
0.661395 0.750037i \(-0.269965\pi\)
\(728\) − 6.22490i − 0.230710i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −25.3215 −0.936548
\(732\) 32.7995i 1.21231i
\(733\) 29.3862i 1.08540i 0.839926 + 0.542702i \(0.182599\pi\)
−0.839926 + 0.542702i \(0.817401\pi\)
\(734\) 82.5772 3.04798
\(735\) 0 0
\(736\) −37.0772 −1.36668
\(737\) − 9.55318i − 0.351896i
\(738\) 10.7566i 0.395956i
\(739\) 30.7254 1.13025 0.565126 0.825005i \(-0.308828\pi\)
0.565126 + 0.825005i \(0.308828\pi\)
\(740\) 0 0
\(741\) −0.365239 −0.0134174
\(742\) 3.24418i 0.119098i
\(743\) 41.4841i 1.52190i 0.648808 + 0.760952i \(0.275268\pi\)
−0.648808 + 0.760952i \(0.724732\pi\)
\(744\) 27.8485 1.02097
\(745\) 0 0
\(746\) 31.9390 1.16937
\(747\) 12.1635i 0.445039i
\(748\) 44.6592i 1.63290i
\(749\) −8.08244 −0.295326
\(750\) 0 0
\(751\) 38.6123 1.40898 0.704491 0.709713i \(-0.251175\pi\)
0.704491 + 0.709713i \(0.251175\pi\)
\(752\) 212.537i 7.75042i
\(753\) 2.76013i 0.100585i
\(754\) −0.646109 −0.0235299
\(755\) 0 0
\(756\) 18.1693 0.660811
\(757\) 23.0860i 0.839074i 0.907738 + 0.419537i \(0.137807\pi\)
−0.907738 + 0.419537i \(0.862193\pi\)
\(758\) − 6.11234i − 0.222010i
\(759\) −3.84401 −0.139529
\(760\) 0 0
\(761\) −31.7092 −1.14946 −0.574728 0.818344i \(-0.694892\pi\)
−0.574728 + 0.818344i \(0.694892\pi\)
\(762\) − 41.7200i − 1.51136i
\(763\) 47.9908i 1.73738i
\(764\) 82.8180 2.99625
\(765\) 0 0
\(766\) −28.4278 −1.02714
\(767\) 1.30000i 0.0469404i
\(768\) − 93.9177i − 3.38896i
\(769\) −38.6475 −1.39366 −0.696832 0.717235i \(-0.745408\pi\)
−0.696832 + 0.717235i \(0.745408\pi\)
\(770\) 0 0
\(771\) 17.2848 0.622497
\(772\) − 11.2770i − 0.405867i
\(773\) 38.9958i 1.40258i 0.712875 + 0.701291i \(0.247393\pi\)
−0.712875 + 0.701291i \(0.752607\pi\)
\(774\) 26.8604 0.965478
\(775\) 0 0
\(776\) 146.014 5.24161
\(777\) − 31.8658i − 1.14318i
\(778\) 79.5672i 2.85262i
\(779\) −7.49109 −0.268396
\(780\) 0 0
\(781\) 19.7431 0.706463
\(782\) 9.55256i 0.341599i
\(783\) − 1.23197i − 0.0440269i
\(784\) −51.8281 −1.85100
\(785\) 0 0
\(786\) −31.4825 −1.12294
\(787\) 8.11107i 0.289128i 0.989495 + 0.144564i \(0.0461780\pi\)
−0.989495 + 0.144564i \(0.953822\pi\)
\(788\) − 6.80387i − 0.242378i
\(789\) −3.96578 −0.141186
\(790\) 0 0
\(791\) −6.94231 −0.246840
\(792\) − 30.9473i − 1.09966i
\(793\) 1.07002i 0.0379975i
\(794\) −103.633 −3.67778
\(795\) 0 0
\(796\) −37.8279 −1.34078
\(797\) 19.0152i 0.673554i 0.941584 + 0.336777i \(0.109337\pi\)
−0.941584 + 0.336777i \(0.890663\pi\)
\(798\) 17.0409i 0.603240i
\(799\) 31.4888 1.11399
\(800\) 0 0
\(801\) −15.1828 −0.536458
\(802\) − 20.5870i − 0.726952i
\(803\) − 27.9044i − 0.984726i
\(804\) 18.6993 0.659472
\(805\) 0 0
\(806\) 1.39071 0.0489858
\(807\) 1.19564i 0.0420884i
\(808\) − 101.574i − 3.57335i
\(809\) −49.9078 −1.75467 −0.877333 0.479882i \(-0.840679\pi\)
−0.877333 + 0.479882i \(0.840679\pi\)
\(810\) 0 0
\(811\) 30.9727 1.08760 0.543799 0.839215i \(-0.316985\pi\)
0.543799 + 0.839215i \(0.316985\pi\)
\(812\) 22.3840i 0.785523i
\(813\) 2.18295i 0.0765593i
\(814\) −83.0847 −2.91212
\(815\) 0 0
\(816\) −46.5953 −1.63116
\(817\) 18.7061i 0.654443i
\(818\) − 14.4914i − 0.506680i
\(819\) 0.592737 0.0207119
\(820\) 0 0
\(821\) 18.9246 0.660473 0.330236 0.943898i \(-0.392872\pi\)
0.330236 + 0.943898i \(0.392872\pi\)
\(822\) 11.6467i 0.406227i
\(823\) − 9.31122i − 0.324569i −0.986744 0.162284i \(-0.948114\pi\)
0.986744 0.162284i \(-0.0518862\pi\)
\(824\) −91.3076 −3.18085
\(825\) 0 0
\(826\) 60.6539 2.11042
\(827\) − 23.7383i − 0.825461i −0.910853 0.412731i \(-0.864575\pi\)
0.910853 0.412731i \(-0.135425\pi\)
\(828\) − 7.52421i − 0.261484i
\(829\) −1.91351 −0.0664590 −0.0332295 0.999448i \(-0.510579\pi\)
−0.0332295 + 0.999448i \(0.510579\pi\)
\(830\) 0 0
\(831\) −1.95174 −0.0677050
\(832\) − 8.23260i − 0.285414i
\(833\) 7.67867i 0.266050i
\(834\) −42.7969 −1.48193
\(835\) 0 0
\(836\) 32.9917 1.14104
\(837\) 2.65174i 0.0916575i
\(838\) − 41.5852i − 1.43654i
\(839\) −40.5105 −1.39858 −0.699289 0.714839i \(-0.746500\pi\)
−0.699289 + 0.714839i \(0.746500\pi\)
\(840\) 0 0
\(841\) −27.4823 −0.947664
\(842\) 0.185880i 0.00640584i
\(843\) − 2.30131i − 0.0792612i
\(844\) −71.2776 −2.45348
\(845\) 0 0
\(846\) −33.4025 −1.14840
\(847\) − 7.29643i − 0.250708i
\(848\) 6.55314i 0.225036i
\(849\) −10.6459 −0.365367
\(850\) 0 0
\(851\) −13.1961 −0.452358
\(852\) 38.6448i 1.32395i
\(853\) 18.7735i 0.642792i 0.946945 + 0.321396i \(0.104152\pi\)
−0.946945 + 0.321396i \(0.895848\pi\)
\(854\) 49.9236 1.70835
\(855\) 0 0
\(856\) −26.9465 −0.921012
\(857\) 10.0276i 0.342536i 0.985225 + 0.171268i \(0.0547863\pi\)
−0.985225 + 0.171268i \(0.945214\pi\)
\(858\) − 1.54546i − 0.0527613i
\(859\) 28.0443 0.956858 0.478429 0.878126i \(-0.341206\pi\)
0.478429 + 0.878126i \(0.341206\pi\)
\(860\) 0 0
\(861\) 12.1571 0.414313
\(862\) − 77.2020i − 2.62951i
\(863\) 9.84666i 0.335184i 0.985856 + 0.167592i \(0.0535992\pi\)
−0.985856 + 0.167592i \(0.946401\pi\)
\(864\) 28.4233 0.966979
\(865\) 0 0
\(866\) −85.1815 −2.89459
\(867\) − 10.0966i − 0.342899i
\(868\) − 48.1802i − 1.63534i
\(869\) −1.23806 −0.0419984
\(870\) 0 0
\(871\) 0.610026 0.0206699
\(872\) 159.999i 5.41826i
\(873\) 13.9035i 0.470563i
\(874\) 7.05690 0.238703
\(875\) 0 0
\(876\) 54.6197 1.84543
\(877\) − 36.9671i − 1.24829i −0.781308 0.624146i \(-0.785447\pi\)
0.781308 0.624146i \(-0.214553\pi\)
\(878\) 30.9282i 1.04378i
\(879\) 14.9591 0.504558
\(880\) 0 0
\(881\) −20.7137 −0.697863 −0.348932 0.937148i \(-0.613455\pi\)
−0.348932 + 0.937148i \(0.613455\pi\)
\(882\) − 8.14536i − 0.274268i
\(883\) 12.7254i 0.428243i 0.976807 + 0.214121i \(0.0686888\pi\)
−0.976807 + 0.214121i \(0.931311\pi\)
\(884\) −2.85175 −0.0959146
\(885\) 0 0
\(886\) −27.3927 −0.920275
\(887\) 21.6903i 0.728287i 0.931343 + 0.364144i \(0.118638\pi\)
−0.931343 + 0.364144i \(0.881362\pi\)
\(888\) − 106.239i − 3.56515i
\(889\) −47.1520 −1.58143
\(890\) 0 0
\(891\) 2.94681 0.0987218
\(892\) 42.1466i 1.41117i
\(893\) − 23.2621i − 0.778437i
\(894\) 9.56995 0.320067
\(895\) 0 0
\(896\) −205.040 −6.84990
\(897\) − 0.245462i − 0.00819574i
\(898\) − 53.3178i − 1.77924i
\(899\) −3.26685 −0.108956
\(900\) 0 0
\(901\) 0.970891 0.0323451
\(902\) − 31.6976i − 1.05542i
\(903\) − 30.3576i − 1.01024i
\(904\) −23.1454 −0.769803
\(905\) 0 0
\(906\) −30.0217 −0.997406
\(907\) 3.28462i 0.109064i 0.998512 + 0.0545320i \(0.0173667\pi\)
−0.998512 + 0.0545320i \(0.982633\pi\)
\(908\) − 121.245i − 4.02366i
\(909\) 9.67188 0.320796
\(910\) 0 0
\(911\) 28.8264 0.955060 0.477530 0.878615i \(-0.341532\pi\)
0.477530 + 0.878615i \(0.341532\pi\)
\(912\) 34.4220i 1.13983i
\(913\) − 35.8435i − 1.18625i
\(914\) −52.2802 −1.72927
\(915\) 0 0
\(916\) 112.790 3.72669
\(917\) 35.5815i 1.17500i
\(918\) − 7.32297i − 0.241694i
\(919\) 17.3985 0.573925 0.286962 0.957942i \(-0.407355\pi\)
0.286962 + 0.957942i \(0.407355\pi\)
\(920\) 0 0
\(921\) 3.07333 0.101270
\(922\) 104.854i 3.45317i
\(923\) 1.26071i 0.0414967i
\(924\) −53.5414 −1.76138
\(925\) 0 0
\(926\) −55.5404 −1.82517
\(927\) − 8.69434i − 0.285560i
\(928\) 35.0165i 1.14947i
\(929\) 24.3715 0.799603 0.399801 0.916602i \(-0.369079\pi\)
0.399801 + 0.916602i \(0.369079\pi\)
\(930\) 0 0
\(931\) 5.67257 0.185911
\(932\) − 70.0657i − 2.29508i
\(933\) − 24.9095i − 0.815501i
\(934\) −76.2742 −2.49577
\(935\) 0 0
\(936\) 1.97616 0.0645928
\(937\) − 14.5784i − 0.476257i −0.971234 0.238129i \(-0.923466\pi\)
0.971234 0.238129i \(-0.0765339\pi\)
\(938\) − 28.4618i − 0.929311i
\(939\) 9.28562 0.303025
\(940\) 0 0
\(941\) −4.19686 −0.136814 −0.0684068 0.997658i \(-0.521792\pi\)
−0.0684068 + 0.997658i \(0.521792\pi\)
\(942\) − 37.3601i − 1.21726i
\(943\) − 5.03445i − 0.163944i
\(944\) 122.519 3.98765
\(945\) 0 0
\(946\) −79.1525 −2.57347
\(947\) 5.53251i 0.179782i 0.995952 + 0.0898912i \(0.0286519\pi\)
−0.995952 + 0.0898912i \(0.971348\pi\)
\(948\) − 2.42336i − 0.0787071i
\(949\) 1.78186 0.0578416
\(950\) 0 0
\(951\) −20.7685 −0.673464
\(952\) 86.9185i 2.81705i
\(953\) 50.4779i 1.63514i 0.575830 + 0.817570i \(0.304679\pi\)
−0.575830 + 0.817570i \(0.695321\pi\)
\(954\) −1.02990 −0.0333442
\(955\) 0 0
\(956\) 77.5899 2.50944
\(957\) 3.63037i 0.117353i
\(958\) − 107.511i − 3.47352i
\(959\) 13.1631 0.425060
\(960\) 0 0
\(961\) −23.9683 −0.773170
\(962\) − 5.30544i − 0.171054i
\(963\) − 2.56585i − 0.0826835i
\(964\) −114.998 −3.70382
\(965\) 0 0
\(966\) −11.4525 −0.368477
\(967\) 17.0340i 0.547777i 0.961761 + 0.273889i \(0.0883100\pi\)
−0.961761 + 0.273889i \(0.911690\pi\)
\(968\) − 24.3260i − 0.781867i
\(969\) 5.09984 0.163831
\(970\) 0 0
\(971\) 23.4172 0.751494 0.375747 0.926722i \(-0.377386\pi\)
0.375747 + 0.926722i \(0.377386\pi\)
\(972\) 5.76803i 0.185010i
\(973\) 48.3690i 1.55064i
\(974\) 51.5604 1.65210
\(975\) 0 0
\(976\) 100.844 3.22794
\(977\) 20.7626i 0.664253i 0.943235 + 0.332127i \(0.107766\pi\)
−0.943235 + 0.332127i \(0.892234\pi\)
\(978\) 42.9713i 1.37407i
\(979\) 44.7408 1.42992
\(980\) 0 0
\(981\) −15.2352 −0.486422
\(982\) − 54.9650i − 1.75400i
\(983\) 42.1977i 1.34590i 0.739690 + 0.672948i \(0.234972\pi\)
−0.739690 + 0.672948i \(0.765028\pi\)
\(984\) 40.5312 1.29209
\(985\) 0 0
\(986\) 9.02164 0.287308
\(987\) 37.7515i 1.20164i
\(988\) 2.10671i 0.0670234i
\(989\) −12.5716 −0.399753
\(990\) 0 0
\(991\) −3.83926 −0.121958 −0.0609791 0.998139i \(-0.519422\pi\)
−0.0609791 + 0.998139i \(0.519422\pi\)
\(992\) − 75.3711i − 2.39303i
\(993\) 33.9172i 1.07633i
\(994\) 58.8206 1.86567
\(995\) 0 0
\(996\) 70.1595 2.22309
\(997\) − 42.2091i − 1.33678i −0.743812 0.668389i \(-0.766984\pi\)
0.743812 0.668389i \(-0.233016\pi\)
\(998\) − 101.146i − 3.20173i
\(999\) 10.1161 0.320060
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 1875.2.b.e.1249.1 12
5.2 odd 4 1875.2.a.l.1.6 yes 6
5.3 odd 4 1875.2.a.i.1.1 6
5.4 even 2 inner 1875.2.b.e.1249.12 12
15.2 even 4 5625.2.a.o.1.1 6
15.8 even 4 5625.2.a.r.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1875.2.a.i.1.1 6 5.3 odd 4
1875.2.a.l.1.6 yes 6 5.2 odd 4
1875.2.b.e.1249.1 12 1.1 even 1 trivial
1875.2.b.e.1249.12 12 5.4 even 2 inner
5625.2.a.o.1.1 6 15.2 even 4
5625.2.a.r.1.6 6 15.8 even 4