Properties

Label 900.3.p.c.401.5
Level $900$
Weight $3$
Character 900.401
Analytic conductor $24.523$
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] = [900,3,Mod(101,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.101");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.p (of order \(6\), degree \(2\), 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: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} - 11 x^{10} + 60 x^{9} - 120 x^{8} - 342 x^{7} + 2709 x^{6} - 3078 x^{5} - 9720 x^{4} + 43740 x^{3} - 72171 x^{2} - 118098 x + 531441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 401.5
Root \(-2.85525 + 0.920635i\) of defining polynomial
Character \(\chi\) \(=\) 900.401
Dual form 900.3.p.c.101.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.85525 - 0.920635i) q^{3} +(-0.594587 - 1.02985i) q^{7} +(7.30486 - 5.25728i) q^{9} +O(q^{10})\) \(q+(2.85525 - 0.920635i) q^{3} +(-0.594587 - 1.02985i) q^{7} +(7.30486 - 5.25728i) q^{9} +(3.63473 - 2.09851i) q^{11} +(-7.15614 + 12.3948i) q^{13} +18.7074i q^{17} +33.8986 q^{19} +(-2.64581 - 2.39309i) q^{21} +(11.6912 + 6.74989i) q^{23} +(16.0171 - 21.7359i) q^{27} +(30.6315 - 17.6851i) q^{29} +(4.97816 - 8.62242i) q^{31} +(8.44607 - 9.33802i) q^{33} +19.3047 q^{37} +(-9.02146 + 41.9784i) q^{39} +(-55.9648 - 32.3113i) q^{41} +(-20.7772 - 35.9872i) q^{43} +(58.2068 - 33.6057i) q^{47} +(23.7929 - 41.2106i) q^{49} +(17.2227 + 53.4143i) q^{51} +30.0712i q^{53} +(96.7889 - 31.2083i) q^{57} +(3.66554 + 2.11630i) q^{59} +(43.8210 + 75.9002i) q^{61} +(-9.75761 - 4.39704i) q^{63} +(-17.9040 + 31.0106i) q^{67} +(39.5953 + 8.50932i) q^{69} -24.5173i q^{71} -11.9305 q^{73} +(-4.32232 - 2.49549i) q^{77} +(19.0858 + 33.0575i) q^{79} +(25.7220 - 76.8074i) q^{81} +(-20.8692 + 12.0489i) q^{83} +(71.1789 - 78.6957i) q^{87} -44.4494i q^{89} +17.0198 q^{91} +(6.27576 - 29.2022i) q^{93} +(31.9672 + 55.3688i) q^{97} +(15.5187 - 34.4381i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{3} + 6 q^{7} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{3} + 6 q^{7} + 26 q^{9} + 48 q^{11} + 30 q^{13} + 72 q^{19} - 128 q^{21} + 78 q^{23} + 106 q^{27} + 150 q^{29} - 12 q^{31} - 96 q^{33} + 12 q^{37} + 40 q^{39} + 90 q^{41} - 114 q^{43} - 12 q^{47} + 48 q^{49} - 144 q^{51} + 158 q^{57} + 48 q^{59} - 78 q^{61} + 212 q^{63} + 168 q^{67} - 150 q^{69} + 24 q^{73} + 258 q^{77} + 120 q^{79} + 434 q^{81} - 114 q^{83} + 330 q^{87} + 120 q^{91} - 82 q^{93} - 96 q^{97} - 120 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\)

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
\(3\) 2.85525 0.920635i 0.951749 0.306878i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −0.594587 1.02985i −0.0849410 0.147122i 0.820425 0.571754i \(-0.193737\pi\)
−0.905366 + 0.424632i \(0.860404\pi\)
\(8\) 0 0
\(9\) 7.30486 5.25728i 0.811651 0.584142i
\(10\) 0 0
\(11\) 3.63473 2.09851i 0.330430 0.190774i −0.325602 0.945507i \(-0.605567\pi\)
0.656032 + 0.754733i \(0.272234\pi\)
\(12\) 0 0
\(13\) −7.15614 + 12.3948i −0.550473 + 0.953447i 0.447768 + 0.894150i \(0.352219\pi\)
−0.998240 + 0.0592967i \(0.981114\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 18.7074i 1.10044i 0.835021 + 0.550218i \(0.185455\pi\)
−0.835021 + 0.550218i \(0.814545\pi\)
\(18\) 0 0
\(19\) 33.8986 1.78414 0.892069 0.451899i \(-0.149253\pi\)
0.892069 + 0.451899i \(0.149253\pi\)
\(20\) 0 0
\(21\) −2.64581 2.39309i −0.125991 0.113957i
\(22\) 0 0
\(23\) 11.6912 + 6.74989i 0.508311 + 0.293474i 0.732139 0.681155i \(-0.238522\pi\)
−0.223828 + 0.974629i \(0.571855\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 16.0171 21.7359i 0.593227 0.805035i
\(28\) 0 0
\(29\) 30.6315 17.6851i 1.05626 0.609831i 0.131863 0.991268i \(-0.457904\pi\)
0.924395 + 0.381437i \(0.124571\pi\)
\(30\) 0 0
\(31\) 4.97816 8.62242i 0.160586 0.278143i −0.774493 0.632582i \(-0.781995\pi\)
0.935079 + 0.354440i \(0.115328\pi\)
\(32\) 0 0
\(33\) 8.44607 9.33802i 0.255942 0.282970i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 19.3047 0.521750 0.260875 0.965373i \(-0.415989\pi\)
0.260875 + 0.965373i \(0.415989\pi\)
\(38\) 0 0
\(39\) −9.02146 + 41.9784i −0.231319 + 1.07637i
\(40\) 0 0
\(41\) −55.9648 32.3113i −1.36499 0.788080i −0.374711 0.927142i \(-0.622258\pi\)
−0.990284 + 0.139062i \(0.955591\pi\)
\(42\) 0 0
\(43\) −20.7772 35.9872i −0.483191 0.836911i 0.516623 0.856213i \(-0.327189\pi\)
−0.999814 + 0.0193018i \(0.993856\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 58.2068 33.6057i 1.23844 0.715015i 0.269667 0.962954i \(-0.413087\pi\)
0.968776 + 0.247939i \(0.0797532\pi\)
\(48\) 0 0
\(49\) 23.7929 41.2106i 0.485570 0.841032i
\(50\) 0 0
\(51\) 17.2227 + 53.4143i 0.337700 + 1.04734i
\(52\) 0 0
\(53\) 30.0712i 0.567382i 0.958916 + 0.283691i \(0.0915589\pi\)
−0.958916 + 0.283691i \(0.908441\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 96.7889 31.2083i 1.69805 0.547513i
\(58\) 0 0
\(59\) 3.66554 + 2.11630i 0.0621278 + 0.0358695i 0.530742 0.847533i \(-0.321913\pi\)
−0.468614 + 0.883403i \(0.655247\pi\)
\(60\) 0 0
\(61\) 43.8210 + 75.9002i 0.718377 + 1.24427i 0.961643 + 0.274305i \(0.0884479\pi\)
−0.243266 + 0.969960i \(0.578219\pi\)
\(62\) 0 0
\(63\) −9.75761 4.39704i −0.154883 0.0697942i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −17.9040 + 31.0106i −0.267224 + 0.462845i −0.968144 0.250395i \(-0.919440\pi\)
0.700920 + 0.713240i \(0.252773\pi\)
\(68\) 0 0
\(69\) 39.5953 + 8.50932i 0.573845 + 0.123323i
\(70\) 0 0
\(71\) 24.5173i 0.345314i −0.984982 0.172657i \(-0.944765\pi\)
0.984982 0.172657i \(-0.0552352\pi\)
\(72\) 0 0
\(73\) −11.9305 −0.163432 −0.0817159 0.996656i \(-0.526040\pi\)
−0.0817159 + 0.996656i \(0.526040\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.32232 2.49549i −0.0561340 0.0324090i
\(78\) 0 0
\(79\) 19.0858 + 33.0575i 0.241592 + 0.418449i 0.961168 0.275964i \(-0.0889971\pi\)
−0.719576 + 0.694414i \(0.755664\pi\)
\(80\) 0 0
\(81\) 25.7220 76.8074i 0.317556 0.948240i
\(82\) 0 0
\(83\) −20.8692 + 12.0489i −0.251436 + 0.145167i −0.620422 0.784268i \(-0.713039\pi\)
0.368985 + 0.929435i \(0.379705\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 71.1789 78.6957i 0.818148 0.904548i
\(88\) 0 0
\(89\) 44.4494i 0.499431i −0.968319 0.249716i \(-0.919663\pi\)
0.968319 0.249716i \(-0.0803371\pi\)
\(90\) 0 0
\(91\) 17.0198 0.187031
\(92\) 0 0
\(93\) 6.27576 29.2022i 0.0674813 0.314002i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 31.9672 + 55.3688i 0.329558 + 0.570812i 0.982424 0.186662i \(-0.0597668\pi\)
−0.652866 + 0.757474i \(0.726433\pi\)
\(98\) 0 0
\(99\) 15.5187 34.4381i 0.156755 0.347860i
\(100\) 0 0
\(101\) 10.3591 5.98084i 0.102566 0.0592163i −0.447840 0.894114i \(-0.647807\pi\)
0.550405 + 0.834898i \(0.314473\pi\)
\(102\) 0 0
\(103\) −81.9841 + 142.001i −0.795963 + 1.37865i 0.126264 + 0.991997i \(0.459701\pi\)
−0.922226 + 0.386651i \(0.873632\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 102.401i 0.957019i −0.878082 0.478509i \(-0.841177\pi\)
0.878082 0.478509i \(-0.158823\pi\)
\(108\) 0 0
\(109\) −56.4485 −0.517876 −0.258938 0.965894i \(-0.583373\pi\)
−0.258938 + 0.965894i \(0.583373\pi\)
\(110\) 0 0
\(111\) 55.1198 17.7726i 0.496575 0.160114i
\(112\) 0 0
\(113\) −149.307 86.2022i −1.32130 0.762852i −0.337362 0.941375i \(-0.609535\pi\)
−0.983936 + 0.178523i \(0.942868\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 12.8883 + 128.164i 0.110157 + 1.09542i
\(118\) 0 0
\(119\) 19.2659 11.1232i 0.161899 0.0934722i
\(120\) 0 0
\(121\) −51.6925 + 89.5341i −0.427211 + 0.739951i
\(122\) 0 0
\(123\) −189.540 40.7335i −1.54098 0.331167i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 195.853 1.54215 0.771073 0.636746i \(-0.219720\pi\)
0.771073 + 0.636746i \(0.219720\pi\)
\(128\) 0 0
\(129\) −92.4551 83.6240i −0.716706 0.648248i
\(130\) 0 0
\(131\) −98.5249 56.8834i −0.752099 0.434224i 0.0743530 0.997232i \(-0.476311\pi\)
−0.826452 + 0.563008i \(0.809644\pi\)
\(132\) 0 0
\(133\) −20.1557 34.9107i −0.151546 0.262486i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −156.773 + 90.5130i −1.14433 + 0.660679i −0.947499 0.319759i \(-0.896398\pi\)
−0.196830 + 0.980438i \(0.563065\pi\)
\(138\) 0 0
\(139\) 114.555 198.414i 0.824134 1.42744i −0.0784454 0.996918i \(-0.524996\pi\)
0.902579 0.430524i \(-0.141671\pi\)
\(140\) 0 0
\(141\) 135.256 149.540i 0.959263 1.06057i
\(142\) 0 0
\(143\) 60.0689i 0.420063i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 29.9948 139.571i 0.204046 0.949462i
\(148\) 0 0
\(149\) 233.844 + 135.010i 1.56942 + 0.906107i 0.996236 + 0.0866818i \(0.0276263\pi\)
0.573187 + 0.819425i \(0.305707\pi\)
\(150\) 0 0
\(151\) 81.5705 + 141.284i 0.540202 + 0.935658i 0.998892 + 0.0470612i \(0.0149856\pi\)
−0.458690 + 0.888596i \(0.651681\pi\)
\(152\) 0 0
\(153\) 98.3501 + 136.655i 0.642811 + 0.893171i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −67.2998 + 116.567i −0.428661 + 0.742463i −0.996754 0.0805014i \(-0.974348\pi\)
0.568094 + 0.822964i \(0.307681\pi\)
\(158\) 0 0
\(159\) 27.6846 + 85.8607i 0.174117 + 0.540005i
\(160\) 0 0
\(161\) 16.0536i 0.0997118i
\(162\) 0 0
\(163\) 159.658 0.979500 0.489750 0.871863i \(-0.337088\pi\)
0.489750 + 0.871863i \(0.337088\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −96.2622 55.5770i −0.576420 0.332796i 0.183289 0.983059i \(-0.441326\pi\)
−0.759709 + 0.650263i \(0.774659\pi\)
\(168\) 0 0
\(169\) −17.9208 31.0397i −0.106040 0.183667i
\(170\) 0 0
\(171\) 247.625 178.215i 1.44810 1.04219i
\(172\) 0 0
\(173\) 127.610 73.6755i 0.737629 0.425870i −0.0835778 0.996501i \(-0.526635\pi\)
0.821207 + 0.570631i \(0.193301\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 12.4144 + 2.66793i 0.0701376 + 0.0150731i
\(178\) 0 0
\(179\) 253.022i 1.41353i 0.707449 + 0.706765i \(0.249846\pi\)
−0.707449 + 0.706765i \(0.750154\pi\)
\(180\) 0 0
\(181\) −212.017 −1.17136 −0.585682 0.810541i \(-0.699173\pi\)
−0.585682 + 0.810541i \(0.699173\pi\)
\(182\) 0 0
\(183\) 194.996 + 176.371i 1.06555 + 0.963773i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 39.2577 + 67.9963i 0.209934 + 0.363617i
\(188\) 0 0
\(189\) −31.9084 3.57142i −0.168828 0.0188964i
\(190\) 0 0
\(191\) −287.027 + 165.715i −1.50276 + 0.867619i −0.502765 + 0.864423i \(0.667684\pi\)
−0.999995 + 0.00319548i \(0.998983\pi\)
\(192\) 0 0
\(193\) 166.279 288.004i 0.861549 1.49225i −0.00888358 0.999961i \(-0.502828\pi\)
0.870433 0.492287i \(-0.163839\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 321.219i 1.63056i −0.579070 0.815278i \(-0.696584\pi\)
0.579070 0.815278i \(-0.303416\pi\)
\(198\) 0 0
\(199\) −208.913 −1.04981 −0.524907 0.851160i \(-0.675900\pi\)
−0.524907 + 0.851160i \(0.675900\pi\)
\(200\) 0 0
\(201\) −22.5708 + 105.026i −0.112293 + 0.522518i
\(202\) 0 0
\(203\) −36.4262 21.0307i −0.179439 0.103599i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 120.888 12.1567i 0.584002 0.0587278i
\(208\) 0 0
\(209\) 123.212 71.1366i 0.589532 0.340366i
\(210\) 0 0
\(211\) −110.304 + 191.053i −0.522769 + 0.905463i 0.476880 + 0.878969i \(0.341768\pi\)
−0.999649 + 0.0264945i \(0.991566\pi\)
\(212\) 0 0
\(213\) −22.5715 70.0028i −0.105969 0.328652i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −11.8398 −0.0545613
\(218\) 0 0
\(219\) −34.0646 + 10.9837i −0.155546 + 0.0501537i
\(220\) 0 0
\(221\) −231.875 133.873i −1.04921 0.605760i
\(222\) 0 0
\(223\) 143.626 + 248.767i 0.644062 + 1.11555i 0.984517 + 0.175287i \(0.0560855\pi\)
−0.340455 + 0.940261i \(0.610581\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −226.479 + 130.758i −0.997704 + 0.576024i −0.907568 0.419905i \(-0.862063\pi\)
−0.0901356 + 0.995930i \(0.528730\pi\)
\(228\) 0 0
\(229\) −99.2244 + 171.862i −0.433294 + 0.750488i −0.997155 0.0753822i \(-0.975982\pi\)
0.563860 + 0.825870i \(0.309316\pi\)
\(230\) 0 0
\(231\) −14.6387 3.14597i −0.0633711 0.0136189i
\(232\) 0 0
\(233\) 182.063i 0.781387i −0.920521 0.390693i \(-0.872235\pi\)
0.920521 0.390693i \(-0.127765\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 84.9284 + 76.8163i 0.358348 + 0.324119i
\(238\) 0 0
\(239\) −210.133 121.320i −0.879218 0.507617i −0.00881748 0.999961i \(-0.502807\pi\)
−0.870400 + 0.492344i \(0.836140\pi\)
\(240\) 0 0
\(241\) 158.882 + 275.192i 0.659263 + 1.14188i 0.980807 + 0.194983i \(0.0624650\pi\)
−0.321544 + 0.946895i \(0.604202\pi\)
\(242\) 0 0
\(243\) 2.73105 242.985i 0.0112389 0.999937i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −242.583 + 420.167i −0.982119 + 1.70108i
\(248\) 0 0
\(249\) −48.4942 + 53.6154i −0.194756 + 0.215323i
\(250\) 0 0
\(251\) 64.0280i 0.255092i −0.991833 0.127546i \(-0.959290\pi\)
0.991833 0.127546i \(-0.0407100\pi\)
\(252\) 0 0
\(253\) 56.6589 0.223948
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −402.010 232.101i −1.56424 0.903116i −0.996820 0.0796896i \(-0.974607\pi\)
−0.567423 0.823426i \(-0.692060\pi\)
\(258\) 0 0
\(259\) −11.4784 19.8811i −0.0443180 0.0767610i
\(260\) 0 0
\(261\) 130.783 290.225i 0.501085 1.11197i
\(262\) 0 0
\(263\) 48.5809 28.0482i 0.184718 0.106647i −0.404789 0.914410i \(-0.632655\pi\)
0.589508 + 0.807763i \(0.299322\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −40.9217 126.914i −0.153265 0.475333i
\(268\) 0 0
\(269\) 29.8910i 0.111119i −0.998455 0.0555594i \(-0.982306\pi\)
0.998455 0.0555594i \(-0.0176942\pi\)
\(270\) 0 0
\(271\) −166.826 −0.615594 −0.307797 0.951452i \(-0.599592\pi\)
−0.307797 + 0.951452i \(0.599592\pi\)
\(272\) 0 0
\(273\) 48.5957 15.6690i 0.178006 0.0573957i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 98.4054 + 170.443i 0.355254 + 0.615318i 0.987161 0.159726i \(-0.0510609\pi\)
−0.631907 + 0.775044i \(0.717728\pi\)
\(278\) 0 0
\(279\) −8.96574 89.1572i −0.0321353 0.319560i
\(280\) 0 0
\(281\) 288.196 166.390i 1.02561 0.592136i 0.109886 0.993944i \(-0.464951\pi\)
0.915724 + 0.401808i \(0.131618\pi\)
\(282\) 0 0
\(283\) −233.467 + 404.376i −0.824971 + 1.42889i 0.0769712 + 0.997033i \(0.475475\pi\)
−0.901942 + 0.431858i \(0.857858\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 76.8474i 0.267761i
\(288\) 0 0
\(289\) −60.9675 −0.210960
\(290\) 0 0
\(291\) 142.249 + 128.661i 0.488827 + 0.442135i
\(292\) 0 0
\(293\) 18.6397 + 10.7616i 0.0636167 + 0.0367291i 0.531471 0.847077i \(-0.321639\pi\)
−0.467854 + 0.883806i \(0.654973\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 12.6048 112.616i 0.0424405 0.379179i
\(298\) 0 0
\(299\) −167.327 + 96.6064i −0.559623 + 0.323098i
\(300\) 0 0
\(301\) −24.7077 + 42.7950i −0.0820854 + 0.142176i
\(302\) 0 0
\(303\) 24.0717 26.6137i 0.0794445 0.0878342i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −452.833 −1.47503 −0.737514 0.675332i \(-0.764000\pi\)
−0.737514 + 0.675332i \(0.764000\pi\)
\(308\) 0 0
\(309\) −103.354 + 480.924i −0.334479 + 1.55639i
\(310\) 0 0
\(311\) −488.695 282.148i −1.57137 0.907229i −0.996002 0.0893313i \(-0.971527\pi\)
−0.575364 0.817897i \(-0.695140\pi\)
\(312\) 0 0
\(313\) −230.660 399.515i −0.736933 1.27641i −0.953870 0.300220i \(-0.902940\pi\)
0.216937 0.976186i \(-0.430393\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −291.643 + 168.380i −0.920008 + 0.531167i −0.883638 0.468171i \(-0.844913\pi\)
−0.0363706 + 0.999338i \(0.511580\pi\)
\(318\) 0 0
\(319\) 74.2247 128.561i 0.232679 0.403012i
\(320\) 0 0
\(321\) −94.2740 292.380i −0.293688 0.910841i
\(322\) 0 0
\(323\) 634.156i 1.96333i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −161.174 + 51.9684i −0.492888 + 0.158925i
\(328\) 0 0
\(329\) −69.2180 39.9630i −0.210389 0.121468i
\(330\) 0 0
\(331\) 210.896 + 365.283i 0.637149 + 1.10357i 0.986055 + 0.166417i \(0.0532197\pi\)
−0.348907 + 0.937158i \(0.613447\pi\)
\(332\) 0 0
\(333\) 141.019 101.490i 0.423479 0.304776i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −59.2115 + 102.557i −0.175702 + 0.304325i −0.940404 0.340059i \(-0.889553\pi\)
0.764702 + 0.644384i \(0.222886\pi\)
\(338\) 0 0
\(339\) −505.668 108.672i −1.49165 0.320565i
\(340\) 0 0
\(341\) 41.7869i 0.122542i
\(342\) 0 0
\(343\) −114.857 −0.334861
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −147.224 85.0001i −0.424278 0.244957i 0.272628 0.962120i \(-0.412107\pi\)
−0.696906 + 0.717163i \(0.745441\pi\)
\(348\) 0 0
\(349\) −310.127 537.156i −0.888617 1.53913i −0.841511 0.540240i \(-0.818333\pi\)
−0.0471063 0.998890i \(-0.515000\pi\)
\(350\) 0 0
\(351\) 154.792 + 354.075i 0.441002 + 1.00876i
\(352\) 0 0
\(353\) 354.557 204.704i 1.00441 0.579897i 0.0948611 0.995491i \(-0.469759\pi\)
0.909551 + 0.415593i \(0.136426\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 44.7686 49.4963i 0.125402 0.138645i
\(358\) 0 0
\(359\) 138.441i 0.385630i 0.981235 + 0.192815i \(0.0617618\pi\)
−0.981235 + 0.192815i \(0.938238\pi\)
\(360\) 0 0
\(361\) 788.116 2.18315
\(362\) 0 0
\(363\) −65.1666 + 303.232i −0.179522 + 0.835349i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 39.5358 + 68.4781i 0.107727 + 0.186589i 0.914849 0.403796i \(-0.132309\pi\)
−0.807122 + 0.590385i \(0.798976\pi\)
\(368\) 0 0
\(369\) −578.684 + 58.1931i −1.56825 + 0.157705i
\(370\) 0 0
\(371\) 30.9690 17.8800i 0.0834744 0.0481940i
\(372\) 0 0
\(373\) 51.8849 89.8672i 0.139102 0.240931i −0.788055 0.615605i \(-0.788912\pi\)
0.927157 + 0.374674i \(0.122245\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 506.228i 1.34278i
\(378\) 0 0
\(379\) 315.487 0.832419 0.416210 0.909269i \(-0.363358\pi\)
0.416210 + 0.909269i \(0.363358\pi\)
\(380\) 0 0
\(381\) 559.208 180.309i 1.46774 0.473252i
\(382\) 0 0
\(383\) −358.695 207.092i −0.936540 0.540711i −0.0476657 0.998863i \(-0.515178\pi\)
−0.888874 + 0.458152i \(0.848512\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −340.969 153.650i −0.881058 0.397028i
\(388\) 0 0
\(389\) −59.9057 + 34.5866i −0.153999 + 0.0889115i −0.575019 0.818140i \(-0.695006\pi\)
0.421020 + 0.907051i \(0.361672\pi\)
\(390\) 0 0
\(391\) −126.273 + 218.711i −0.322949 + 0.559364i
\(392\) 0 0
\(393\) −333.682 71.7106i −0.849063 0.182470i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −347.995 −0.876562 −0.438281 0.898838i \(-0.644413\pi\)
−0.438281 + 0.898838i \(0.644413\pi\)
\(398\) 0 0
\(399\) −89.6894 81.1225i −0.224785 0.203315i
\(400\) 0 0
\(401\) −635.313 366.798i −1.58432 0.914709i −0.994218 0.107382i \(-0.965753\pi\)
−0.590104 0.807327i \(-0.700913\pi\)
\(402\) 0 0
\(403\) 71.2488 + 123.407i 0.176796 + 0.306220i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 70.1675 40.5112i 0.172402 0.0995361i
\(408\) 0 0
\(409\) −47.0727 + 81.5324i −0.115092 + 0.199346i −0.917817 0.397005i \(-0.870050\pi\)
0.802724 + 0.596350i \(0.203383\pi\)
\(410\) 0 0
\(411\) −364.296 + 402.768i −0.886366 + 0.979970i
\(412\) 0 0
\(413\) 5.03330i 0.0121872i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 144.414 671.985i 0.346317 1.61147i
\(418\) 0 0
\(419\) −90.6043 52.3104i −0.216239 0.124846i 0.387968 0.921673i \(-0.373177\pi\)
−0.604208 + 0.796827i \(0.706510\pi\)
\(420\) 0 0
\(421\) −276.542 478.984i −0.656868 1.13773i −0.981422 0.191862i \(-0.938547\pi\)
0.324553 0.945867i \(-0.394786\pi\)
\(422\) 0 0
\(423\) 248.518 551.494i 0.587513 1.30377i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 52.1108 90.2585i 0.122039 0.211378i
\(428\) 0 0
\(429\) 55.3016 + 171.512i 0.128908 + 0.399794i
\(430\) 0 0
\(431\) 469.242i 1.08873i 0.838849 + 0.544364i \(0.183229\pi\)
−0.838849 + 0.544364i \(0.816771\pi\)
\(432\) 0 0
\(433\) −426.644 −0.985321 −0.492660 0.870222i \(-0.663976\pi\)
−0.492660 + 0.870222i \(0.663976\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 396.314 + 228.812i 0.906898 + 0.523598i
\(438\) 0 0
\(439\) 286.573 + 496.359i 0.652786 + 1.13066i 0.982444 + 0.186558i \(0.0597332\pi\)
−0.329658 + 0.944100i \(0.606933\pi\)
\(440\) 0 0
\(441\) −42.8514 426.124i −0.0971688 0.966267i
\(442\) 0 0
\(443\) −90.5838 + 52.2986i −0.204478 + 0.118055i −0.598743 0.800942i \(-0.704333\pi\)
0.394264 + 0.918997i \(0.370999\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 791.977 + 170.201i 1.77176 + 0.380764i
\(448\) 0 0
\(449\) 401.878i 0.895051i −0.894271 0.447526i \(-0.852305\pi\)
0.894271 0.447526i \(-0.147695\pi\)
\(450\) 0 0
\(451\) −271.222 −0.601379
\(452\) 0 0
\(453\) 362.975 + 328.305i 0.801270 + 0.724735i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 48.1592 + 83.4142i 0.105381 + 0.182526i 0.913894 0.405953i \(-0.133060\pi\)
−0.808513 + 0.588479i \(0.799727\pi\)
\(458\) 0 0
\(459\) 406.623 + 299.639i 0.885890 + 0.652809i
\(460\) 0 0
\(461\) −64.4588 + 37.2153i −0.139824 + 0.0807274i −0.568280 0.822835i \(-0.692391\pi\)
0.428456 + 0.903563i \(0.359058\pi\)
\(462\) 0 0
\(463\) 198.930 344.558i 0.429655 0.744185i −0.567187 0.823589i \(-0.691968\pi\)
0.996843 + 0.0794042i \(0.0253018\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 643.763i 1.37851i −0.724520 0.689254i \(-0.757938\pi\)
0.724520 0.689254i \(-0.242062\pi\)
\(468\) 0 0
\(469\) 42.5819 0.0907931
\(470\) 0 0
\(471\) −84.8421 + 394.785i −0.180132 + 0.838185i
\(472\) 0 0
\(473\) −151.039 87.2024i −0.319321 0.184360i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 158.093 + 219.666i 0.331432 + 0.460516i
\(478\) 0 0
\(479\) −152.770 + 88.2019i −0.318936 + 0.184138i −0.650918 0.759148i \(-0.725616\pi\)
0.331982 + 0.943286i \(0.392283\pi\)
\(480\) 0 0
\(481\) −138.148 + 239.279i −0.287209 + 0.497461i
\(482\) 0 0
\(483\) −14.7795 45.8370i −0.0305994 0.0949006i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 494.859 1.01614 0.508069 0.861317i \(-0.330360\pi\)
0.508069 + 0.861317i \(0.330360\pi\)
\(488\) 0 0
\(489\) 455.864 146.987i 0.932238 0.300587i
\(490\) 0 0
\(491\) 159.777 + 92.2472i 0.325411 + 0.187876i 0.653802 0.756666i \(-0.273173\pi\)
−0.328391 + 0.944542i \(0.606506\pi\)
\(492\) 0 0
\(493\) 330.842 + 573.036i 0.671080 + 1.16234i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −25.2492 + 14.5777i −0.0508033 + 0.0293313i
\(498\) 0 0
\(499\) −162.276 + 281.071i −0.325203 + 0.563269i −0.981553 0.191188i \(-0.938766\pi\)
0.656350 + 0.754456i \(0.272099\pi\)
\(500\) 0 0
\(501\) −326.018 70.0636i −0.650735 0.139848i
\(502\) 0 0
\(503\) 626.131i 1.24479i 0.782702 + 0.622397i \(0.213841\pi\)
−0.782702 + 0.622397i \(0.786159\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −79.7446 72.1276i −0.157287 0.142263i
\(508\) 0 0
\(509\) 387.634 + 223.800i 0.761559 + 0.439686i 0.829855 0.557979i \(-0.188423\pi\)
−0.0682960 + 0.997665i \(0.521756\pi\)
\(510\) 0 0
\(511\) 7.09374 + 12.2867i 0.0138821 + 0.0240444i
\(512\) 0 0
\(513\) 542.959 736.818i 1.05840 1.43629i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 141.044 244.295i 0.272812 0.472524i
\(518\) 0 0
\(519\) 296.529 327.844i 0.571347 0.631684i
\(520\) 0 0
\(521\) 181.821i 0.348984i 0.984659 + 0.174492i \(0.0558284\pi\)
−0.984659 + 0.174492i \(0.944172\pi\)
\(522\) 0 0
\(523\) 293.719 0.561604 0.280802 0.959766i \(-0.409400\pi\)
0.280802 + 0.959766i \(0.409400\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 161.303 + 93.1285i 0.306078 + 0.176714i
\(528\) 0 0
\(529\) −173.378 300.299i −0.327746 0.567673i
\(530\) 0 0
\(531\) 37.9022 3.81149i 0.0713790 0.00717795i
\(532\) 0 0
\(533\) 800.984 462.448i 1.50278 0.867633i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 232.941 + 722.439i 0.433782 + 1.34532i
\(538\) 0 0
\(539\) 199.719i 0.370536i
\(540\) 0 0
\(541\) 343.593 0.635107 0.317553 0.948240i \(-0.397139\pi\)
0.317553 + 0.948240i \(0.397139\pi\)
\(542\) 0 0
\(543\) −605.360 + 195.190i −1.11484 + 0.359466i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −20.5788 35.6434i −0.0376211 0.0651617i 0.846602 0.532227i \(-0.178645\pi\)
−0.884223 + 0.467065i \(0.845311\pi\)
\(548\) 0 0
\(549\) 719.135 + 324.061i 1.30990 + 0.590275i
\(550\) 0 0
\(551\) 1038.36 599.500i 1.88451 1.08802i
\(552\) 0 0
\(553\) 22.6963 39.3111i 0.0410421 0.0710870i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 976.675i 1.75346i −0.480986 0.876728i \(-0.659721\pi\)
0.480986 0.876728i \(-0.340279\pi\)
\(558\) 0 0
\(559\) 594.739 1.06393
\(560\) 0 0
\(561\) 174.690 + 158.004i 0.311391 + 0.281647i
\(562\) 0 0
\(563\) −383.325 221.313i −0.680862 0.393096i 0.119318 0.992856i \(-0.461929\pi\)
−0.800180 + 0.599760i \(0.795263\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −94.3945 + 19.1788i −0.166481 + 0.0338250i
\(568\) 0 0
\(569\) 681.591 393.517i 1.19788 0.691594i 0.237795 0.971315i \(-0.423575\pi\)
0.960081 + 0.279721i \(0.0902421\pi\)
\(570\) 0 0
\(571\) 3.41373 5.91275i 0.00597851 0.0103551i −0.863021 0.505169i \(-0.831430\pi\)
0.868999 + 0.494814i \(0.164764\pi\)
\(572\) 0 0
\(573\) −666.970 + 737.405i −1.16400 + 1.28692i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 45.4328 0.0787396 0.0393698 0.999225i \(-0.487465\pi\)
0.0393698 + 0.999225i \(0.487465\pi\)
\(578\) 0 0
\(579\) 209.621 975.404i 0.362040 1.68464i
\(580\) 0 0
\(581\) 24.8171 + 14.3282i 0.0427145 + 0.0246612i
\(582\) 0 0
\(583\) 63.1048 + 109.301i 0.108241 + 0.187480i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −85.9520 + 49.6244i −0.146426 + 0.0845391i −0.571423 0.820656i \(-0.693608\pi\)
0.424997 + 0.905195i \(0.360275\pi\)
\(588\) 0 0
\(589\) 168.753 292.288i 0.286507 0.496245i
\(590\) 0 0
\(591\) −295.726 917.161i −0.500382 1.55188i
\(592\) 0 0
\(593\) 226.897i 0.382626i 0.981529 + 0.191313i \(0.0612745\pi\)
−0.981529 + 0.191313i \(0.938726\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −596.498 + 192.333i −0.999159 + 0.322165i
\(598\) 0 0
\(599\) 652.129 + 376.507i 1.08870 + 0.628559i 0.933229 0.359282i \(-0.116978\pi\)
0.155467 + 0.987841i \(0.450312\pi\)
\(600\) 0 0
\(601\) −342.525 593.270i −0.569924 0.987138i −0.996573 0.0827206i \(-0.973639\pi\)
0.426648 0.904418i \(-0.359694\pi\)
\(602\) 0 0
\(603\) 32.2454 + 320.655i 0.0534749 + 0.531766i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 454.475 787.174i 0.748724 1.29683i −0.199711 0.979855i \(-0.564000\pi\)
0.948435 0.316973i \(-0.102666\pi\)
\(608\) 0 0
\(609\) −123.367 26.5125i −0.202573 0.0435345i
\(610\) 0 0
\(611\) 961.949i 1.57438i
\(612\) 0 0
\(613\) −514.234 −0.838881 −0.419441 0.907783i \(-0.637774\pi\)
−0.419441 + 0.907783i \(0.637774\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 996.159 + 575.133i 1.61452 + 0.932144i 0.988304 + 0.152494i \(0.0487304\pi\)
0.626216 + 0.779650i \(0.284603\pi\)
\(618\) 0 0
\(619\) 69.9937 + 121.233i 0.113076 + 0.195853i 0.917009 0.398867i \(-0.130596\pi\)
−0.803933 + 0.594719i \(0.797263\pi\)
\(620\) 0 0
\(621\) 333.974 146.004i 0.537801 0.235112i
\(622\) 0 0
\(623\) −45.7764 + 26.4290i −0.0734774 + 0.0424222i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 286.310 316.546i 0.456635 0.504858i
\(628\) 0 0
\(629\) 361.142i 0.574153i
\(630\) 0 0
\(631\) −369.971 −0.586325 −0.293163 0.956063i \(-0.594708\pi\)
−0.293163 + 0.956063i \(0.594708\pi\)
\(632\) 0 0
\(633\) −139.056 + 647.053i −0.219678 + 1.02220i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 340.531 + 589.818i 0.534586 + 0.925930i
\(638\) 0 0
\(639\) −128.894 179.095i −0.201712 0.280274i
\(640\) 0 0
\(641\) 67.0156 38.6914i 0.104548 0.0603611i −0.446814 0.894627i \(-0.647441\pi\)
0.551363 + 0.834266i \(0.314108\pi\)
\(642\) 0 0
\(643\) 130.904 226.733i 0.203583 0.352617i −0.746097 0.665837i \(-0.768075\pi\)
0.949680 + 0.313220i \(0.101408\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 617.428i 0.954293i 0.878824 + 0.477147i \(0.158329\pi\)
−0.878824 + 0.477147i \(0.841671\pi\)
\(648\) 0 0
\(649\) 17.7643 0.0273718
\(650\) 0 0
\(651\) −33.8055 + 10.9001i −0.0519286 + 0.0167437i
\(652\) 0 0
\(653\) −584.115 337.239i −0.894510 0.516446i −0.0190951 0.999818i \(-0.506079\pi\)
−0.875415 + 0.483372i \(0.839412\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −87.1508 + 62.7221i −0.132650 + 0.0954675i
\(658\) 0 0
\(659\) 332.973 192.242i 0.505271 0.291718i −0.225617 0.974216i \(-0.572440\pi\)
0.730888 + 0.682498i \(0.239106\pi\)
\(660\) 0 0
\(661\) 151.713 262.775i 0.229520 0.397541i −0.728146 0.685422i \(-0.759618\pi\)
0.957666 + 0.287882i \(0.0929509\pi\)
\(662\) 0 0
\(663\) −785.308 168.768i −1.18448 0.254552i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 477.490 0.715877
\(668\) 0 0
\(669\) 639.111 + 578.065i 0.955323 + 0.864073i
\(670\) 0 0
\(671\) 318.554 + 183.918i 0.474746 + 0.274095i
\(672\) 0 0
\(673\) −280.062 485.081i −0.416139 0.720775i 0.579408 0.815038i \(-0.303284\pi\)
−0.995547 + 0.0942631i \(0.969951\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 880.370 508.282i 1.30040 0.750786i 0.319927 0.947442i \(-0.396342\pi\)
0.980472 + 0.196656i \(0.0630083\pi\)
\(678\) 0 0
\(679\) 38.0145 65.8431i 0.0559860 0.0969707i
\(680\) 0 0
\(681\) −526.273 + 581.849i −0.772794 + 0.854404i
\(682\) 0 0
\(683\) 88.7713i 0.129973i 0.997886 + 0.0649863i \(0.0207004\pi\)
−0.997886 + 0.0649863i \(0.979300\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −125.088 + 582.057i −0.182079 + 0.847245i
\(688\) 0 0
\(689\) −372.727 215.194i −0.540968 0.312328i
\(690\) 0 0
\(691\) −403.626 699.101i −0.584119 1.01172i −0.994985 0.100027i \(-0.968107\pi\)
0.410866 0.911696i \(-0.365226\pi\)
\(692\) 0 0
\(693\) −44.6935 + 4.49442i −0.0644927 + 0.00648546i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 604.460 1046.96i 0.867232 1.50209i
\(698\) 0 0
\(699\) −167.614 519.835i −0.239791 0.743684i
\(700\) 0 0
\(701\) 201.018i 0.286758i 0.989668 + 0.143379i \(0.0457968\pi\)
−0.989668 + 0.143379i \(0.954203\pi\)
\(702\) 0 0
\(703\) 654.404 0.930874
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −12.3188 7.11226i −0.0174240 0.0100598i
\(708\) 0 0
\(709\) −195.660 338.894i −0.275967 0.477989i 0.694412 0.719578i \(-0.255665\pi\)
−0.970379 + 0.241589i \(0.922331\pi\)
\(710\) 0 0
\(711\) 313.211 + 141.141i 0.440522 + 0.198511i
\(712\) 0 0
\(713\) 116.401 67.2041i 0.163255 0.0942554i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −711.674 152.944i −0.992571 0.213311i
\(718\) 0 0
\(719\) 1241.75i 1.72705i 0.504308 + 0.863524i \(0.331747\pi\)
−0.504308 + 0.863524i \(0.668253\pi\)
\(720\) 0 0
\(721\) 194.987 0.270439
\(722\) 0 0
\(723\) 707.000 + 639.469i 0.977870 + 0.884467i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −100.212 173.571i −0.137843 0.238750i 0.788837 0.614602i \(-0.210683\pi\)
−0.926680 + 0.375852i \(0.877350\pi\)
\(728\) 0 0
\(729\) −215.902 696.295i −0.296162 0.955138i
\(730\) 0 0
\(731\) 673.227 388.688i 0.920968 0.531721i
\(732\) 0 0
\(733\) −60.3607 + 104.548i −0.0823475 + 0.142630i −0.904258 0.426987i \(-0.859575\pi\)
0.821910 + 0.569617i \(0.192908\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 150.287i 0.203917i
\(738\) 0 0
\(739\) −471.461 −0.637972 −0.318986 0.947759i \(-0.603342\pi\)
−0.318986 + 0.947759i \(0.603342\pi\)
\(740\) 0 0
\(741\) −305.815 + 1423.01i −0.412706 + 1.92039i
\(742\) 0 0
\(743\) −1128.15 651.339i −1.51837 0.876634i −0.999766 0.0216220i \(-0.993117\pi\)
−0.518608 0.855012i \(-0.673550\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −89.1026 + 197.731i −0.119281 + 0.264700i
\(748\) 0 0
\(749\) −105.458 + 60.8863i −0.140799 + 0.0812901i
\(750\) 0 0
\(751\) 439.726 761.629i 0.585521 1.01415i −0.409289 0.912405i \(-0.634223\pi\)
0.994810 0.101748i \(-0.0324435\pi\)
\(752\) 0 0
\(753\) −58.9464 182.816i −0.0782821 0.242783i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 35.1814 0.0464748 0.0232374 0.999730i \(-0.492603\pi\)
0.0232374 + 0.999730i \(0.492603\pi\)
\(758\) 0 0
\(759\) 161.775 52.1622i 0.213142 0.0687248i
\(760\) 0 0
\(761\) 1102.94 + 636.785i 1.44933 + 0.836774i 0.998442 0.0558046i \(-0.0177724\pi\)
0.450893 + 0.892578i \(0.351106\pi\)
\(762\) 0 0
\(763\) 33.5635 + 58.1337i 0.0439889 + 0.0761910i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −52.4623 + 30.2891i −0.0683993 + 0.0394903i
\(768\) 0 0
\(769\) −379.742 + 657.733i −0.493813 + 0.855310i −0.999975 0.00712921i \(-0.997731\pi\)
0.506161 + 0.862439i \(0.331064\pi\)
\(770\) 0 0
\(771\) −1361.52 292.600i −1.76591 0.379507i
\(772\) 0 0
\(773\) 165.240i 0.213765i 0.994272 + 0.106883i \(0.0340869\pi\)
−0.994272 + 0.106883i \(0.965913\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −51.0767 46.1980i −0.0657358 0.0594569i
\(778\) 0 0
\(779\) −1897.13 1095.31i −2.43534 1.40604i
\(780\) 0 0
\(781\) −51.4497 89.1135i −0.0658767 0.114102i
\(782\) 0 0
\(783\) 106.227 949.069i 0.135666 1.21209i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 287.752 498.401i 0.365632 0.633293i −0.623246 0.782026i \(-0.714186\pi\)
0.988877 + 0.148733i \(0.0475197\pi\)
\(788\) 0 0
\(789\) 112.888 124.810i 0.143078 0.158187i
\(790\) 0 0
\(791\) 205.019i 0.259190i
\(792\) 0 0
\(793\) −1254.36 −1.58179
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 526.210 + 303.808i 0.660238 + 0.381189i 0.792368 0.610044i \(-0.208848\pi\)
−0.132129 + 0.991232i \(0.542181\pi\)
\(798\) 0 0
\(799\) 628.676 + 1088.90i 0.786828 + 1.36283i
\(800\) 0 0
\(801\) −233.683 324.696i −0.291739 0.405364i
\(802\) 0 0
\(803\) −43.3642 + 25.0363i −0.0540027 + 0.0311785i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −27.5187 85.3460i −0.0341000 0.105757i
\(808\) 0 0
\(809\) 962.401i 1.18962i −0.803867 0.594809i \(-0.797228\pi\)
0.803867 0.594809i \(-0.202772\pi\)
\(810\) 0 0
\(811\) −140.096 −0.172745 −0.0863723 0.996263i \(-0.527527\pi\)
−0.0863723 + 0.996263i \(0.527527\pi\)
\(812\) 0 0
\(813\) −476.329 + 153.586i −0.585891 + 0.188913i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −704.319 1219.92i −0.862079 1.49317i
\(818\) 0 0
\(819\) 124.327 89.4779i 0.151804 0.109253i
\(820\) 0 0
\(821\) −280.294 + 161.828i −0.341406 + 0.197111i −0.660894 0.750480i \(-0.729823\pi\)
0.319488 + 0.947590i \(0.396489\pi\)
\(822\) 0 0
\(823\) −96.6061 + 167.327i −0.117383 + 0.203313i −0.918730 0.394887i \(-0.870784\pi\)
0.801347 + 0.598200i \(0.204117\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 15.7916i 0.0190951i 0.999954 + 0.00954754i \(0.00303912\pi\)
−0.999954 + 0.00954754i \(0.996961\pi\)
\(828\) 0 0
\(829\) −265.576 −0.320357 −0.160178 0.987088i \(-0.551207\pi\)
−0.160178 + 0.987088i \(0.551207\pi\)
\(830\) 0 0
\(831\) 437.888 + 396.062i 0.526941 + 0.476609i
\(832\) 0 0
\(833\) 770.943 + 445.104i 0.925502 + 0.534339i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −107.681 246.312i −0.128651 0.294279i
\(838\) 0 0
\(839\) −761.943 + 439.908i −0.908156 + 0.524324i −0.879838 0.475275i \(-0.842349\pi\)
−0.0283189 + 0.999599i \(0.509015\pi\)
\(840\) 0 0
\(841\) 205.025 355.114i 0.243787 0.422252i
\(842\) 0 0
\(843\) 669.687 740.409i 0.794409 0.878303i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 122.943 0.145151
\(848\) 0 0
\(849\) −294.322 + 1369.53i −0.346669 + 1.61311i
\(850\) 0 0
\(851\) 225.695 + 130.305i 0.265211 + 0.153120i
\(852\) 0 0
\(853\) 102.073 + 176.795i 0.119663 + 0.207263i 0.919634 0.392776i \(-0.128485\pi\)
−0.799971 + 0.600039i \(0.795152\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −184.859 + 106.729i −0.215705 + 0.124537i −0.603960 0.797015i \(-0.706411\pi\)
0.388255 + 0.921552i \(0.373078\pi\)
\(858\) 0 0
\(859\) −323.677 + 560.625i −0.376806 + 0.652648i −0.990596 0.136822i \(-0.956311\pi\)
0.613789 + 0.789470i \(0.289644\pi\)
\(860\) 0 0
\(861\) 70.7485 + 219.418i 0.0821701 + 0.254841i
\(862\) 0 0
\(863\) 699.772i 0.810859i 0.914126 + 0.405430i \(0.132878\pi\)
−0.914126 + 0.405430i \(0.867122\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −174.077 + 56.1288i −0.200781 + 0.0647391i
\(868\) 0 0
\(869\) 138.743 + 80.1033i 0.159658 + 0.0921787i
\(870\) 0 0
\(871\) −256.247 443.833i −0.294199 0.509567i
\(872\) 0 0
\(873\) 524.605 + 236.401i 0.600922 + 0.270791i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −712.480 + 1234.05i −0.812406 + 1.40713i 0.0987694 + 0.995110i \(0.468509\pi\)
−0.911176 + 0.412018i \(0.864824\pi\)
\(878\) 0 0
\(879\) 63.1285 + 13.5668i 0.0718185 + 0.0154343i
\(880\) 0 0
\(881\) 1598.97i 1.81495i 0.420102 + 0.907477i \(0.361994\pi\)
−0.420102 + 0.907477i \(0.638006\pi\)
\(882\) 0 0
\(883\) 1117.03 1.26504 0.632522 0.774542i \(-0.282020\pi\)
0.632522 + 0.774542i \(0.282020\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1009.79 + 583.003i 1.13843 + 0.657275i 0.946042 0.324044i \(-0.105042\pi\)
0.192391 + 0.981318i \(0.438376\pi\)
\(888\) 0 0
\(889\) −116.451 201.700i −0.130992 0.226884i
\(890\) 0 0
\(891\) −67.6887 333.152i −0.0759693 0.373908i
\(892\) 0 0
\(893\) 1973.13 1139.19i 2.20955 1.27569i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −388.821 + 429.883i −0.433468 + 0.479245i
\(898\) 0 0
\(899\) 352.157i 0.391721i
\(900\) 0 0
\(901\) −562.555 −0.624367
\(902\) 0 0
\(903\) −31.1480 + 144.937i −0.0344939 + 0.160506i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −881.243 1526.36i −0.971602 1.68286i −0.690720 0.723122i \(-0.742706\pi\)
−0.280882 0.959742i \(-0.590627\pi\)
\(908\) 0 0
\(909\) 44.2290 98.1500i 0.0486568 0.107976i
\(910\) 0 0
\(911\) 1449.90 837.098i 1.59154 0.918878i 0.598501 0.801122i \(-0.295763\pi\)
0.993043 0.117755i \(-0.0375699\pi\)
\(912\) 0 0
\(913\) −50.5693 + 87.5885i −0.0553880 + 0.0959349i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 135.289i 0.147534i
\(918\) 0 0
\(919\) 1136.29 1.23644 0.618221 0.786004i \(-0.287854\pi\)
0.618221 + 0.786004i \(0.287854\pi\)
\(920\) 0 0
\(921\) −1292.95 + 416.894i −1.40386 + 0.452654i
\(922\) 0 0
\(923\) 303.887 + 175.449i 0.329238 + 0.190086i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 147.655 + 1468.31i 0.159282 + 1.58394i
\(928\) 0 0
\(929\) 932.936 538.631i 1.00424 0.579796i 0.0947376 0.995502i \(-0.469799\pi\)
0.909499 + 0.415706i \(0.136465\pi\)
\(930\) 0 0
\(931\) 806.548 1396.98i 0.866324 1.50052i
\(932\) 0 0
\(933\) −1655.10 355.693i −1.77395 0.381235i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −1063.87 −1.13540 −0.567702 0.823234i \(-0.692167\pi\)
−0.567702 + 0.823234i \(0.692167\pi\)
\(938\) 0 0
\(939\) −1026.40 928.360i −1.09308 0.988668i
\(940\) 0 0
\(941\) −994.783 574.338i −1.05716 0.610349i −0.132511 0.991182i \(-0.542304\pi\)
−0.924644 + 0.380833i \(0.875637\pi\)
\(942\) 0 0
\(943\) −436.195 755.513i −0.462561 0.801180i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 100.366 57.9461i 0.105983 0.0611891i −0.446072 0.894997i \(-0.647177\pi\)
0.552054 + 0.833808i \(0.313844\pi\)
\(948\) 0 0
\(949\) 85.3766 147.877i 0.0899648 0.155824i
\(950\) 0 0
\(951\) −677.695 + 749.263i −0.712613 + 0.787868i
\(952\) 0 0
\(953\) 1854.05i 1.94549i 0.231881 + 0.972744i \(0.425512\pi\)
−0.231881 + 0.972744i \(0.574488\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 93.5720 435.407i 0.0977764 0.454971i
\(958\) 0 0
\(959\) 186.431 + 107.636i 0.194401 + 0.112237i
\(960\) 0 0
\(961\) 430.936 + 746.403i 0.448424 + 0.776694i
\(962\) 0 0
\(963\) −538.351 748.025i −0.559035 0.776766i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −456.447 + 790.590i −0.472024 + 0.817569i −0.999488 0.0320081i \(-0.989810\pi\)
0.527464 + 0.849578i \(0.323143\pi\)
\(968\) 0 0
\(969\) 583.826 + 1810.67i 0.602504 + 1.86860i
\(970\) 0 0
\(971\) 1732.32i 1.78405i −0.451982 0.892027i \(-0.649283\pi\)
0.451982 0.892027i \(-0.350717\pi\)
\(972\) 0 0
\(973\) −272.451 −0.280011
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 200.352 + 115.673i 0.205069 + 0.118396i 0.599018 0.800736i \(-0.295558\pi\)
−0.393949 + 0.919132i \(0.628891\pi\)
\(978\) 0 0
\(979\) −93.2774 161.561i −0.0952783 0.165027i
\(980\) 0 0
\(981\) −412.348 + 296.765i −0.420335 + 0.302513i
\(982\) 0 0
\(983\) −674.808 + 389.601i −0.686478 + 0.396338i −0.802291 0.596933i \(-0.796386\pi\)
0.115813 + 0.993271i \(0.463053\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −234.426 50.3798i −0.237513 0.0510433i
\(988\) 0 0
\(989\) 560.976i 0.567215i
\(990\) 0 0
\(991\) 222.470 0.224490 0.112245 0.993681i \(-0.464196\pi\)
0.112245 + 0.993681i \(0.464196\pi\)
\(992\) 0 0
\(993\) 938.453 + 848.815i 0.945069 + 0.854798i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −226.712 392.676i −0.227394 0.393858i 0.729641 0.683830i \(-0.239687\pi\)
−0.957035 + 0.289972i \(0.906354\pi\)
\(998\) 0 0
\(999\) 309.207 419.607i 0.309516 0.420027i
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 900.3.p.c.401.5 12
3.2 odd 2 2700.3.p.c.2501.3 12
5.2 odd 4 900.3.u.c.149.4 24
5.3 odd 4 900.3.u.c.149.9 24
5.4 even 2 180.3.o.b.41.2 12
9.2 odd 6 inner 900.3.p.c.101.5 12
9.7 even 3 2700.3.p.c.1601.3 12
15.2 even 4 2700.3.u.c.449.7 24
15.8 even 4 2700.3.u.c.449.6 24
15.14 odd 2 540.3.o.b.341.5 12
20.19 odd 2 720.3.bs.b.401.5 12
45.2 even 12 900.3.u.c.749.9 24
45.4 even 6 1620.3.g.b.161.9 12
45.7 odd 12 2700.3.u.c.2249.6 24
45.14 odd 6 1620.3.g.b.161.3 12
45.29 odd 6 180.3.o.b.101.2 yes 12
45.34 even 6 540.3.o.b.521.5 12
45.38 even 12 900.3.u.c.749.4 24
45.43 odd 12 2700.3.u.c.2249.7 24
60.59 even 2 2160.3.bs.b.881.5 12
180.79 odd 6 2160.3.bs.b.1601.5 12
180.119 even 6 720.3.bs.b.641.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
180.3.o.b.41.2 12 5.4 even 2
180.3.o.b.101.2 yes 12 45.29 odd 6
540.3.o.b.341.5 12 15.14 odd 2
540.3.o.b.521.5 12 45.34 even 6
720.3.bs.b.401.5 12 20.19 odd 2
720.3.bs.b.641.5 12 180.119 even 6
900.3.p.c.101.5 12 9.2 odd 6 inner
900.3.p.c.401.5 12 1.1 even 1 trivial
900.3.u.c.149.4 24 5.2 odd 4
900.3.u.c.149.9 24 5.3 odd 4
900.3.u.c.749.4 24 45.38 even 12
900.3.u.c.749.9 24 45.2 even 12
1620.3.g.b.161.3 12 45.14 odd 6
1620.3.g.b.161.9 12 45.4 even 6
2160.3.bs.b.881.5 12 60.59 even 2
2160.3.bs.b.1601.5 12 180.79 odd 6
2700.3.p.c.1601.3 12 9.7 even 3
2700.3.p.c.2501.3 12 3.2 odd 2
2700.3.u.c.449.6 24 15.8 even 4
2700.3.u.c.449.7 24 15.2 even 4
2700.3.u.c.2249.6 24 45.7 odd 12
2700.3.u.c.2249.7 24 45.43 odd 12