Properties

Label 175.6.b.c.99.2
Level $175$
Weight $6$
Character 175.99
Analytic conductor $28.067$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [175,6,Mod(99,175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(175, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.99");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 175.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: \(28.0671684673\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{57})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 29x^{2} + 196 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.2
Root \(4.27492i\) of defining polynomial
Character \(\chi\) \(=\) 175.99
Dual form 175.6.b.c.99.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.725083i q^{2} +19.6495i q^{3} +31.4743 q^{4} +14.2475 q^{6} -49.0000i q^{7} -46.0241i q^{8} -143.103 q^{9} +O(q^{10})\) \(q-0.725083i q^{2} +19.6495i q^{3} +31.4743 q^{4} +14.2475 q^{6} -49.0000i q^{7} -46.0241i q^{8} -143.103 q^{9} +666.090 q^{11} +618.453i q^{12} -650.640i q^{13} -35.5291 q^{14} +973.805 q^{16} -1186.89i q^{17} +103.762i q^{18} +1565.05 q^{19} +962.826 q^{21} -482.970i q^{22} -1100.15i q^{23} +904.350 q^{24} -471.768 q^{26} +1962.93i q^{27} -1542.24i q^{28} -2396.72 q^{29} -2048.46 q^{31} -2178.86i q^{32} +13088.3i q^{33} -860.596 q^{34} -4504.06 q^{36} -1077.54i q^{37} -1134.79i q^{38} +12784.7 q^{39} +1098.21 q^{41} -698.128i q^{42} +16564.3i q^{43} +20964.7 q^{44} -797.702 q^{46} +8298.39i q^{47} +19134.8i q^{48} -2401.00 q^{49} +23321.9 q^{51} -20478.4i q^{52} +5519.18i q^{53} +1423.28 q^{54} -2255.18 q^{56} +30752.5i q^{57} +1737.82i q^{58} +14230.4 q^{59} -14234.7 q^{61} +1485.30i q^{62} +7012.05i q^{63} +29581.9 q^{64} +9490.12 q^{66} -19730.4i q^{67} -37356.6i q^{68} +21617.5 q^{69} +64562.7 q^{71} +6586.18i q^{72} +28567.0i q^{73} -781.309 q^{74} +49258.8 q^{76} -32638.4i q^{77} -9270.00i q^{78} +30633.4 q^{79} -73344.6 q^{81} -796.293i q^{82} -675.946i q^{83} +30304.2 q^{84} +12010.5 q^{86} -47094.4i q^{87} -30656.2i q^{88} -125971. q^{89} -31881.3 q^{91} -34626.5i q^{92} -40251.1i q^{93} +6017.02 q^{94} +42813.5 q^{96} +22906.8i q^{97} +1740.92i q^{98} -95319.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{4} - 396 q^{6} - 1116 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 10 q^{4} - 396 q^{6} - 1116 q^{9} + 792 q^{11} - 882 q^{14} + 226 q^{16} + 6532 q^{19} - 588 q^{21} + 3708 q^{24} + 4032 q^{26} - 13392 q^{29} - 40 q^{31} - 11868 q^{34} + 21258 q^{36} + 40992 q^{39} - 12096 q^{41} + 61632 q^{44} + 51168 q^{46} - 9604 q^{49} + 15192 q^{51} + 75816 q^{54} - 882 q^{56} + 87876 q^{59} - 129508 q^{61} + 141566 q^{64} + 133632 q^{66} + 206784 q^{69} + 194832 q^{71} - 86868 q^{74} - 25564 q^{76} - 102512 q^{79} - 122148 q^{81} + 152292 q^{84} - 300096 q^{86} - 168552 q^{89} - 34300 q^{91} - 318936 q^{94} + 511812 q^{96} + 33480 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 0.725083i − 0.128178i −0.997944 0.0640889i \(-0.979586\pi\)
0.997944 0.0640889i \(-0.0204141\pi\)
\(3\) 19.6495i 1.26052i 0.776386 + 0.630258i \(0.217051\pi\)
−0.776386 + 0.630258i \(0.782949\pi\)
\(4\) 31.4743 0.983570
\(5\) 0 0
\(6\) 14.2475 0.161570
\(7\) − 49.0000i − 0.377964i
\(8\) − 46.0241i − 0.254250i
\(9\) −143.103 −0.588901
\(10\) 0 0
\(11\) 666.090 1.65978 0.829891 0.557926i \(-0.188403\pi\)
0.829891 + 0.557926i \(0.188403\pi\)
\(12\) 618.453i 1.23981i
\(13\) − 650.640i − 1.06778i −0.845554 0.533890i \(-0.820729\pi\)
0.845554 0.533890i \(-0.179271\pi\)
\(14\) −35.5291 −0.0484466
\(15\) 0 0
\(16\) 973.805 0.950981
\(17\) − 1186.89i − 0.996069i −0.867157 0.498035i \(-0.834055\pi\)
0.867157 0.498035i \(-0.165945\pi\)
\(18\) 103.762i 0.0754840i
\(19\) 1565.05 0.994591 0.497296 0.867581i \(-0.334326\pi\)
0.497296 + 0.867581i \(0.334326\pi\)
\(20\) 0 0
\(21\) 962.826 0.476430
\(22\) − 482.970i − 0.212747i
\(23\) − 1100.15i − 0.433644i −0.976211 0.216822i \(-0.930431\pi\)
0.976211 0.216822i \(-0.0695691\pi\)
\(24\) 904.350 0.320486
\(25\) 0 0
\(26\) −471.768 −0.136866
\(27\) 1962.93i 0.518197i
\(28\) − 1542.24i − 0.371755i
\(29\) −2396.72 −0.529203 −0.264602 0.964358i \(-0.585240\pi\)
−0.264602 + 0.964358i \(0.585240\pi\)
\(30\) 0 0
\(31\) −2048.46 −0.382844 −0.191422 0.981508i \(-0.561310\pi\)
−0.191422 + 0.981508i \(0.561310\pi\)
\(32\) − 2178.86i − 0.376144i
\(33\) 13088.3i 2.09218i
\(34\) −860.596 −0.127674
\(35\) 0 0
\(36\) −4504.06 −0.579226
\(37\) − 1077.54i − 0.129399i −0.997905 0.0646995i \(-0.979391\pi\)
0.997905 0.0646995i \(-0.0206089\pi\)
\(38\) − 1134.79i − 0.127484i
\(39\) 12784.7 1.34595
\(40\) 0 0
\(41\) 1098.21 0.102029 0.0510147 0.998698i \(-0.483754\pi\)
0.0510147 + 0.998698i \(0.483754\pi\)
\(42\) − 698.128i − 0.0610678i
\(43\) 16564.3i 1.36616i 0.730343 + 0.683081i \(0.239360\pi\)
−0.730343 + 0.683081i \(0.760640\pi\)
\(44\) 20964.7 1.63251
\(45\) 0 0
\(46\) −797.702 −0.0555835
\(47\) 8298.39i 0.547960i 0.961735 + 0.273980i \(0.0883403\pi\)
−0.961735 + 0.273980i \(0.911660\pi\)
\(48\) 19134.8i 1.19873i
\(49\) −2401.00 −0.142857
\(50\) 0 0
\(51\) 23321.9 1.25556
\(52\) − 20478.4i − 1.05024i
\(53\) 5519.18i 0.269889i 0.990853 + 0.134944i \(0.0430856\pi\)
−0.990853 + 0.134944i \(0.956914\pi\)
\(54\) 1423.28 0.0664213
\(55\) 0 0
\(56\) −2255.18 −0.0960973
\(57\) 30752.5i 1.25370i
\(58\) 1737.82i 0.0678321i
\(59\) 14230.4 0.532216 0.266108 0.963943i \(-0.414262\pi\)
0.266108 + 0.963943i \(0.414262\pi\)
\(60\) 0 0
\(61\) −14234.7 −0.489807 −0.244904 0.969547i \(-0.578756\pi\)
−0.244904 + 0.969547i \(0.578756\pi\)
\(62\) 1485.30i 0.0490721i
\(63\) 7012.05i 0.222584i
\(64\) 29581.9 0.902768
\(65\) 0 0
\(66\) 9490.12 0.268171
\(67\) − 19730.4i − 0.536970i −0.963284 0.268485i \(-0.913477\pi\)
0.963284 0.268485i \(-0.0865229\pi\)
\(68\) − 37356.6i − 0.979704i
\(69\) 21617.5 0.546615
\(70\) 0 0
\(71\) 64562.7 1.51997 0.759986 0.649940i \(-0.225206\pi\)
0.759986 + 0.649940i \(0.225206\pi\)
\(72\) 6586.18i 0.149728i
\(73\) 28567.0i 0.627418i 0.949519 + 0.313709i \(0.101572\pi\)
−0.949519 + 0.313709i \(0.898428\pi\)
\(74\) −781.309 −0.0165861
\(75\) 0 0
\(76\) 49258.8 0.978251
\(77\) − 32638.4i − 0.627339i
\(78\) − 9270.00i − 0.172521i
\(79\) 30633.4 0.552239 0.276119 0.961123i \(-0.410951\pi\)
0.276119 + 0.961123i \(0.410951\pi\)
\(80\) 0 0
\(81\) −73344.6 −1.24210
\(82\) − 796.293i − 0.0130779i
\(83\) − 675.946i − 0.0107700i −0.999986 0.00538501i \(-0.998286\pi\)
0.999986 0.00538501i \(-0.00171411\pi\)
\(84\) 30304.2 0.468603
\(85\) 0 0
\(86\) 12010.5 0.175111
\(87\) − 47094.4i − 0.667069i
\(88\) − 30656.2i − 0.421999i
\(89\) −125971. −1.68576 −0.842882 0.538098i \(-0.819143\pi\)
−0.842882 + 0.538098i \(0.819143\pi\)
\(90\) 0 0
\(91\) −31881.3 −0.403583
\(92\) − 34626.5i − 0.426520i
\(93\) − 40251.1i − 0.482582i
\(94\) 6017.02 0.0702363
\(95\) 0 0
\(96\) 42813.5 0.474136
\(97\) 22906.8i 0.247192i 0.992333 + 0.123596i \(0.0394427\pi\)
−0.992333 + 0.123596i \(0.960557\pi\)
\(98\) 1740.92i 0.0183111i
\(99\) −95319.4 −0.977447
\(100\) 0 0
\(101\) −181474. −1.77015 −0.885077 0.465444i \(-0.845895\pi\)
−0.885077 + 0.465444i \(0.845895\pi\)
\(102\) − 16910.3i − 0.160935i
\(103\) 64772.0i 0.601581i 0.953690 + 0.300791i \(0.0972506\pi\)
−0.953690 + 0.300791i \(0.902749\pi\)
\(104\) −29945.1 −0.271483
\(105\) 0 0
\(106\) 4001.86 0.0345938
\(107\) 148170.i 1.25112i 0.780175 + 0.625562i \(0.215130\pi\)
−0.780175 + 0.625562i \(0.784870\pi\)
\(108\) 61781.7i 0.509683i
\(109\) 111294. 0.897237 0.448618 0.893723i \(-0.351916\pi\)
0.448618 + 0.893723i \(0.351916\pi\)
\(110\) 0 0
\(111\) 21173.2 0.163110
\(112\) − 47716.4i − 0.359437i
\(113\) − 43175.5i − 0.318084i −0.987272 0.159042i \(-0.949160\pi\)
0.987272 0.159042i \(-0.0508405\pi\)
\(114\) 22298.1 0.160696
\(115\) 0 0
\(116\) −75435.0 −0.520509
\(117\) 93108.5i 0.628817i
\(118\) − 10318.2i − 0.0682182i
\(119\) −58157.8 −0.376479
\(120\) 0 0
\(121\) 282625. 1.75488
\(122\) 10321.4i 0.0627824i
\(123\) 21579.3i 0.128610i
\(124\) −64473.6 −0.376554
\(125\) 0 0
\(126\) 5084.31 0.0285303
\(127\) 131449.i 0.723182i 0.932337 + 0.361591i \(0.117766\pi\)
−0.932337 + 0.361591i \(0.882234\pi\)
\(128\) − 91172.8i − 0.491859i
\(129\) −325480. −1.72207
\(130\) 0 0
\(131\) −349458. −1.77916 −0.889582 0.456775i \(-0.849005\pi\)
−0.889582 + 0.456775i \(0.849005\pi\)
\(132\) 411946.i 2.05781i
\(133\) − 76687.5i − 0.375920i
\(134\) −14306.2 −0.0688276
\(135\) 0 0
\(136\) −54625.7 −0.253250
\(137\) − 386434.i − 1.75903i −0.475869 0.879516i \(-0.657866\pi\)
0.475869 0.879516i \(-0.342134\pi\)
\(138\) − 15674.4i − 0.0700639i
\(139\) −17289.3 −0.0758997 −0.0379498 0.999280i \(-0.512083\pi\)
−0.0379498 + 0.999280i \(0.512083\pi\)
\(140\) 0 0
\(141\) −163059. −0.690713
\(142\) − 46813.3i − 0.194827i
\(143\) − 433384.i − 1.77228i
\(144\) −139354. −0.560034
\(145\) 0 0
\(146\) 20713.4 0.0804210
\(147\) − 47178.5i − 0.180074i
\(148\) − 33914.9i − 0.127273i
\(149\) 112171. 0.413917 0.206959 0.978350i \(-0.433643\pi\)
0.206959 + 0.978350i \(0.433643\pi\)
\(150\) 0 0
\(151\) 30495.4 0.108841 0.0544205 0.998518i \(-0.482669\pi\)
0.0544205 + 0.998518i \(0.482669\pi\)
\(152\) − 72030.1i − 0.252874i
\(153\) 169848.i 0.586586i
\(154\) −23665.5 −0.0804108
\(155\) 0 0
\(156\) 402390. 1.32384
\(157\) − 523509.i − 1.69502i −0.530780 0.847510i \(-0.678101\pi\)
0.530780 0.847510i \(-0.321899\pi\)
\(158\) − 22211.7i − 0.0707847i
\(159\) −108449. −0.340199
\(160\) 0 0
\(161\) −53907.5 −0.163902
\(162\) 53180.9i 0.159209i
\(163\) − 439646.i − 1.29609i −0.761604 0.648043i \(-0.775588\pi\)
0.761604 0.648043i \(-0.224412\pi\)
\(164\) 34565.3 0.100353
\(165\) 0 0
\(166\) −490.117 −0.00138048
\(167\) − 279353.i − 0.775107i −0.921847 0.387554i \(-0.873320\pi\)
0.921847 0.387554i \(-0.126680\pi\)
\(168\) − 44313.2i − 0.121132i
\(169\) −52038.8 −0.140156
\(170\) 0 0
\(171\) −223964. −0.585716
\(172\) 521349.i 1.34372i
\(173\) 99699.4i 0.253266i 0.991950 + 0.126633i \(0.0404171\pi\)
−0.991950 + 0.126633i \(0.959583\pi\)
\(174\) −34147.3 −0.0855034
\(175\) 0 0
\(176\) 648641. 1.57842
\(177\) 279621.i 0.670867i
\(178\) 91339.7i 0.216077i
\(179\) −329980. −0.769760 −0.384880 0.922967i \(-0.625757\pi\)
−0.384880 + 0.922967i \(0.625757\pi\)
\(180\) 0 0
\(181\) −505810. −1.14760 −0.573800 0.818995i \(-0.694531\pi\)
−0.573800 + 0.818995i \(0.694531\pi\)
\(182\) 23116.6i 0.0517304i
\(183\) − 279706.i − 0.617410i
\(184\) −50633.5 −0.110254
\(185\) 0 0
\(186\) −29185.4 −0.0618562
\(187\) − 790578.i − 1.65326i
\(188\) 261186.i 0.538958i
\(189\) 96183.4 0.195860
\(190\) 0 0
\(191\) −63835.6 −0.126613 −0.0633067 0.997994i \(-0.520165\pi\)
−0.0633067 + 0.997994i \(0.520165\pi\)
\(192\) 581270.i 1.13795i
\(193\) 469355.i 0.907001i 0.891256 + 0.453501i \(0.149825\pi\)
−0.891256 + 0.453501i \(0.850175\pi\)
\(194\) 16609.3 0.0316845
\(195\) 0 0
\(196\) −75569.7 −0.140510
\(197\) − 268021.i − 0.492043i −0.969264 0.246021i \(-0.920877\pi\)
0.969264 0.246021i \(-0.0791234\pi\)
\(198\) 69114.5i 0.125287i
\(199\) 605167. 1.08328 0.541642 0.840609i \(-0.317803\pi\)
0.541642 + 0.840609i \(0.317803\pi\)
\(200\) 0 0
\(201\) 387693. 0.676859
\(202\) 131584.i 0.226894i
\(203\) 117439.i 0.200020i
\(204\) 734039. 1.23493
\(205\) 0 0
\(206\) 46965.1 0.0771094
\(207\) 157435.i 0.255374i
\(208\) − 633596.i − 1.01544i
\(209\) 1.04246e6 1.65080
\(210\) 0 0
\(211\) 335389. 0.518612 0.259306 0.965795i \(-0.416506\pi\)
0.259306 + 0.965795i \(0.416506\pi\)
\(212\) 173712.i 0.265455i
\(213\) 1.26862e6i 1.91595i
\(214\) 107435. 0.160366
\(215\) 0 0
\(216\) 90341.9 0.131751
\(217\) 100374.i 0.144702i
\(218\) − 80697.7i − 0.115006i
\(219\) −561327. −0.790871
\(220\) 0 0
\(221\) −772240. −1.06358
\(222\) − 15352.3i − 0.0209070i
\(223\) 1.02526e6i 1.38061i 0.723518 + 0.690305i \(0.242524\pi\)
−0.723518 + 0.690305i \(0.757476\pi\)
\(224\) −106764. −0.142169
\(225\) 0 0
\(226\) −31305.8 −0.0407713
\(227\) 504226.i 0.649473i 0.945805 + 0.324736i \(0.105276\pi\)
−0.945805 + 0.324736i \(0.894724\pi\)
\(228\) 967912.i 1.23310i
\(229\) 1.11939e6 1.41057 0.705283 0.708925i \(-0.250820\pi\)
0.705283 + 0.708925i \(0.250820\pi\)
\(230\) 0 0
\(231\) 641328. 0.790770
\(232\) 110307.i 0.134550i
\(233\) 770703.i 0.930031i 0.885303 + 0.465015i \(0.153951\pi\)
−0.885303 + 0.465015i \(0.846049\pi\)
\(234\) 67511.3 0.0806004
\(235\) 0 0
\(236\) 447892. 0.523472
\(237\) 601930.i 0.696106i
\(238\) 42169.2i 0.0482562i
\(239\) 171646. 0.194374 0.0971871 0.995266i \(-0.469015\pi\)
0.0971871 + 0.995266i \(0.469015\pi\)
\(240\) 0 0
\(241\) −383779. −0.425637 −0.212818 0.977092i \(-0.568264\pi\)
−0.212818 + 0.977092i \(0.568264\pi\)
\(242\) − 204926.i − 0.224936i
\(243\) − 964193.i − 1.04749i
\(244\) −448028. −0.481760
\(245\) 0 0
\(246\) 15646.8 0.0164849
\(247\) − 1.01828e6i − 1.06201i
\(248\) 94278.3i 0.0973380i
\(249\) 13282.0 0.0135758
\(250\) 0 0
\(251\) −1.57046e6 −1.57342 −0.786708 0.617325i \(-0.788216\pi\)
−0.786708 + 0.617325i \(0.788216\pi\)
\(252\) 220699.i 0.218927i
\(253\) − 732801.i − 0.719755i
\(254\) 95311.4 0.0926959
\(255\) 0 0
\(256\) 880513. 0.839723
\(257\) − 790656.i − 0.746715i −0.927688 0.373357i \(-0.878207\pi\)
0.927688 0.373357i \(-0.121793\pi\)
\(258\) 236000.i 0.220731i
\(259\) −52799.7 −0.0489082
\(260\) 0 0
\(261\) 342978. 0.311648
\(262\) 253386.i 0.228049i
\(263\) 464416.i 0.414017i 0.978339 + 0.207008i \(0.0663728\pi\)
−0.978339 + 0.207008i \(0.933627\pi\)
\(264\) 602379. 0.531936
\(265\) 0 0
\(266\) −55604.8 −0.0481846
\(267\) − 2.47527e6i − 2.12493i
\(268\) − 621001.i − 0.528147i
\(269\) −1.99959e6 −1.68484 −0.842422 0.538818i \(-0.818871\pi\)
−0.842422 + 0.538818i \(0.818871\pi\)
\(270\) 0 0
\(271\) 1.61296e6 1.33414 0.667070 0.744995i \(-0.267548\pi\)
0.667070 + 0.744995i \(0.267548\pi\)
\(272\) − 1.15580e6i − 0.947243i
\(273\) − 626452.i − 0.508723i
\(274\) −280197. −0.225469
\(275\) 0 0
\(276\) 680393. 0.537635
\(277\) 2.08119e6i 1.62972i 0.579658 + 0.814860i \(0.303186\pi\)
−0.579658 + 0.814860i \(0.696814\pi\)
\(278\) 12536.2i 0.00972865i
\(279\) 293140. 0.225457
\(280\) 0 0
\(281\) −982035. −0.741927 −0.370964 0.928647i \(-0.620973\pi\)
−0.370964 + 0.928647i \(0.620973\pi\)
\(282\) 118231.i 0.0885340i
\(283\) − 1.39622e6i − 1.03630i −0.855289 0.518152i \(-0.826620\pi\)
0.855289 0.518152i \(-0.173380\pi\)
\(284\) 2.03206e6 1.49500
\(285\) 0 0
\(286\) −314240. −0.227167
\(287\) − 53812.3i − 0.0385635i
\(288\) 311801.i 0.221512i
\(289\) 11140.3 0.00784609
\(290\) 0 0
\(291\) −450107. −0.311590
\(292\) 899124.i 0.617110i
\(293\) 2.56205e6i 1.74348i 0.489965 + 0.871742i \(0.337010\pi\)
−0.489965 + 0.871742i \(0.662990\pi\)
\(294\) −34208.3 −0.0230814
\(295\) 0 0
\(296\) −49593.0 −0.0328996
\(297\) 1.30749e6i 0.860094i
\(298\) − 81333.0i − 0.0530550i
\(299\) −715803. −0.463037
\(300\) 0 0
\(301\) 811651. 0.516361
\(302\) − 22111.7i − 0.0139510i
\(303\) − 3.56588e6i − 2.23131i
\(304\) 1.52405e6 0.945838
\(305\) 0 0
\(306\) 123154. 0.0751873
\(307\) 884855.i 0.535829i 0.963443 + 0.267915i \(0.0863345\pi\)
−0.963443 + 0.267915i \(0.913666\pi\)
\(308\) − 1.02727e6i − 0.617032i
\(309\) −1.27274e6 −0.758303
\(310\) 0 0
\(311\) 1.80120e6 1.05599 0.527997 0.849246i \(-0.322943\pi\)
0.527997 + 0.849246i \(0.322943\pi\)
\(312\) − 588406.i − 0.342208i
\(313\) 950366.i 0.548315i 0.961685 + 0.274158i \(0.0883990\pi\)
−0.961685 + 0.274158i \(0.911601\pi\)
\(314\) −379587. −0.217264
\(315\) 0 0
\(316\) 964162. 0.543166
\(317\) − 3.04277e6i − 1.70068i −0.526237 0.850338i \(-0.676398\pi\)
0.526237 0.850338i \(-0.323602\pi\)
\(318\) 78634.6i 0.0436060i
\(319\) −1.59643e6 −0.878362
\(320\) 0 0
\(321\) −2.91146e6 −1.57706
\(322\) 39087.4i 0.0210086i
\(323\) − 1.85755e6i − 0.990682i
\(324\) −2.30847e6 −1.22169
\(325\) 0 0
\(326\) −318780. −0.166129
\(327\) 2.18688e6i 1.13098i
\(328\) − 50544.1i − 0.0259409i
\(329\) 406621. 0.207110
\(330\) 0 0
\(331\) −2.19616e6 −1.10178 −0.550889 0.834579i \(-0.685711\pi\)
−0.550889 + 0.834579i \(0.685711\pi\)
\(332\) − 21274.9i − 0.0105931i
\(333\) 154200.i 0.0762032i
\(334\) −202554. −0.0993515
\(335\) 0 0
\(336\) 937604. 0.453076
\(337\) 2.41491e6i 1.15832i 0.815216 + 0.579158i \(0.196618\pi\)
−0.815216 + 0.579158i \(0.803382\pi\)
\(338\) 37732.5i 0.0179648i
\(339\) 848377. 0.400950
\(340\) 0 0
\(341\) −1.36446e6 −0.635438
\(342\) 162392.i 0.0750757i
\(343\) 117649.i 0.0539949i
\(344\) 762357. 0.347346
\(345\) 0 0
\(346\) 72290.3 0.0324631
\(347\) 1.08833e6i 0.485219i 0.970124 + 0.242609i \(0.0780033\pi\)
−0.970124 + 0.242609i \(0.921997\pi\)
\(348\) − 1.48226e6i − 0.656110i
\(349\) −2.79267e6 −1.22731 −0.613657 0.789573i \(-0.710302\pi\)
−0.613657 + 0.789573i \(0.710302\pi\)
\(350\) 0 0
\(351\) 1.27716e6 0.553321
\(352\) − 1.45132e6i − 0.624317i
\(353\) 2.53134e6i 1.08122i 0.841273 + 0.540610i \(0.181807\pi\)
−0.841273 + 0.540610i \(0.818193\pi\)
\(354\) 202748. 0.0859902
\(355\) 0 0
\(356\) −3.96485e6 −1.65807
\(357\) − 1.14277e6i − 0.474558i
\(358\) 239263.i 0.0986661i
\(359\) −1.09028e6 −0.446480 −0.223240 0.974763i \(-0.571663\pi\)
−0.223240 + 0.974763i \(0.571663\pi\)
\(360\) 0 0
\(361\) −26712.8 −0.0107883
\(362\) 366754.i 0.147097i
\(363\) 5.55343e6i 2.21205i
\(364\) −1.00344e6 −0.396953
\(365\) 0 0
\(366\) −202810. −0.0791382
\(367\) − 188070.i − 0.0728879i −0.999336 0.0364439i \(-0.988397\pi\)
0.999336 0.0364439i \(-0.0116030\pi\)
\(368\) − 1.07133e6i − 0.412387i
\(369\) −157157. −0.0600853
\(370\) 0 0
\(371\) 270440. 0.102008
\(372\) − 1.26687e6i − 0.474653i
\(373\) − 1.79371e6i − 0.667545i −0.942654 0.333772i \(-0.891678\pi\)
0.942654 0.333772i \(-0.108322\pi\)
\(374\) −573234. −0.211911
\(375\) 0 0
\(376\) 381926. 0.139319
\(377\) 1.55940e6i 0.565073i
\(378\) − 69740.9i − 0.0251049i
\(379\) −3.58806e6 −1.28310 −0.641551 0.767080i \(-0.721709\pi\)
−0.641551 + 0.767080i \(0.721709\pi\)
\(380\) 0 0
\(381\) −2.58291e6 −0.911583
\(382\) 46286.1i 0.0162290i
\(383\) − 3.42457e6i − 1.19291i −0.802645 0.596457i \(-0.796575\pi\)
0.802645 0.596457i \(-0.203425\pi\)
\(384\) 1.79150e6 0.619996
\(385\) 0 0
\(386\) 340321. 0.116257
\(387\) − 2.37040e6i − 0.804534i
\(388\) 720974.i 0.243131i
\(389\) −8625.27 −0.00289001 −0.00144500 0.999999i \(-0.500460\pi\)
−0.00144500 + 0.999999i \(0.500460\pi\)
\(390\) 0 0
\(391\) −1.30576e6 −0.431940
\(392\) 110504.i 0.0363214i
\(393\) − 6.86667e6i − 2.24267i
\(394\) −194337. −0.0630690
\(395\) 0 0
\(396\) −3.00011e6 −0.961388
\(397\) − 1.25709e6i − 0.400306i −0.979765 0.200153i \(-0.935856\pi\)
0.979765 0.200153i \(-0.0641439\pi\)
\(398\) − 438796.i − 0.138853i
\(399\) 1.50687e6 0.473853
\(400\) 0 0
\(401\) 1.42670e6 0.443070 0.221535 0.975152i \(-0.428893\pi\)
0.221535 + 0.975152i \(0.428893\pi\)
\(402\) − 281110.i − 0.0867583i
\(403\) 1.33281e6i 0.408794i
\(404\) −5.71176e6 −1.74107
\(405\) 0 0
\(406\) 85153.2 0.0256381
\(407\) − 717741.i − 0.214774i
\(408\) − 1.07337e6i − 0.319226i
\(409\) 3.06529e6 0.906073 0.453036 0.891492i \(-0.350341\pi\)
0.453036 + 0.891492i \(0.350341\pi\)
\(410\) 0 0
\(411\) 7.59324e6 2.21729
\(412\) 2.03865e6i 0.591698i
\(413\) − 697291.i − 0.201159i
\(414\) 114154. 0.0327332
\(415\) 0 0
\(416\) −1.41765e6 −0.401640
\(417\) − 339726.i − 0.0956728i
\(418\) − 755873.i − 0.211596i
\(419\) 248240. 0.0690776 0.0345388 0.999403i \(-0.489004\pi\)
0.0345388 + 0.999403i \(0.489004\pi\)
\(420\) 0 0
\(421\) 5.96280e6 1.63963 0.819814 0.572630i \(-0.194077\pi\)
0.819814 + 0.572630i \(0.194077\pi\)
\(422\) − 243185.i − 0.0664746i
\(423\) − 1.18752e6i − 0.322695i
\(424\) 254015. 0.0686192
\(425\) 0 0
\(426\) 919857. 0.245582
\(427\) 697503.i 0.185130i
\(428\) 4.66353e6i 1.23057i
\(429\) 8.51579e6 2.23399
\(430\) 0 0
\(431\) −4.93538e6 −1.27976 −0.639879 0.768476i \(-0.721015\pi\)
−0.639879 + 0.768476i \(0.721015\pi\)
\(432\) 1.91151e6i 0.492795i
\(433\) 4.15513e6i 1.06504i 0.846418 + 0.532519i \(0.178755\pi\)
−0.846418 + 0.532519i \(0.821245\pi\)
\(434\) 72779.7 0.0185475
\(435\) 0 0
\(436\) 3.50291e6 0.882496
\(437\) − 1.72180e6i − 0.431299i
\(438\) 407008.i 0.101372i
\(439\) 227955. 0.0564531 0.0282265 0.999602i \(-0.491014\pi\)
0.0282265 + 0.999602i \(0.491014\pi\)
\(440\) 0 0
\(441\) 343590. 0.0841287
\(442\) 559938.i 0.136328i
\(443\) − 1.98462e6i − 0.480472i −0.970715 0.240236i \(-0.922775\pi\)
0.970715 0.240236i \(-0.0772248\pi\)
\(444\) 666411. 0.160430
\(445\) 0 0
\(446\) 743397. 0.176963
\(447\) 2.20410e6i 0.521749i
\(448\) − 1.44951e6i − 0.341214i
\(449\) −2.61077e6 −0.611157 −0.305579 0.952167i \(-0.598850\pi\)
−0.305579 + 0.952167i \(0.598850\pi\)
\(450\) 0 0
\(451\) 731506. 0.169347
\(452\) − 1.35892e6i − 0.312858i
\(453\) 599220.i 0.137196i
\(454\) 365606. 0.0832480
\(455\) 0 0
\(456\) 1.41536e6 0.318752
\(457\) 4.09917e6i 0.918132i 0.888402 + 0.459066i \(0.151816\pi\)
−0.888402 + 0.459066i \(0.848184\pi\)
\(458\) − 811652.i − 0.180803i
\(459\) 2.32979e6 0.516160
\(460\) 0 0
\(461\) −2.62378e6 −0.575009 −0.287505 0.957779i \(-0.592826\pi\)
−0.287505 + 0.957779i \(0.592826\pi\)
\(462\) − 465016.i − 0.101359i
\(463\) − 4.28563e6i − 0.929100i −0.885547 0.464550i \(-0.846216\pi\)
0.885547 0.464550i \(-0.153784\pi\)
\(464\) −2.33394e6 −0.503262
\(465\) 0 0
\(466\) 558824. 0.119209
\(467\) − 990118.i − 0.210085i −0.994468 0.105042i \(-0.966502\pi\)
0.994468 0.105042i \(-0.0334978\pi\)
\(468\) 2.93052e6i 0.618486i
\(469\) −966792. −0.202955
\(470\) 0 0
\(471\) 1.02867e7 2.13660
\(472\) − 654942.i − 0.135316i
\(473\) 1.10333e7i 2.26753i
\(474\) 436449. 0.0892253
\(475\) 0 0
\(476\) −1.83047e6 −0.370293
\(477\) − 789812.i − 0.158938i
\(478\) − 124457.i − 0.0249144i
\(479\) −2.57976e6 −0.513737 −0.256868 0.966446i \(-0.582691\pi\)
−0.256868 + 0.966446i \(0.582691\pi\)
\(480\) 0 0
\(481\) −701093. −0.138170
\(482\) 278272.i 0.0545571i
\(483\) − 1.05926e6i − 0.206601i
\(484\) 8.89540e6 1.72604
\(485\) 0 0
\(486\) −699120. −0.134264
\(487\) − 3.21474e6i − 0.614219i −0.951674 0.307109i \(-0.900638\pi\)
0.951674 0.307109i \(-0.0993618\pi\)
\(488\) 655141.i 0.124533i
\(489\) 8.63882e6 1.63374
\(490\) 0 0
\(491\) −7.86108e6 −1.47156 −0.735781 0.677220i \(-0.763185\pi\)
−0.735781 + 0.677220i \(0.763185\pi\)
\(492\) 679192.i 0.126497i
\(493\) 2.84465e6i 0.527123i
\(494\) −738341. −0.136125
\(495\) 0 0
\(496\) −1.99480e6 −0.364078
\(497\) − 3.16357e6i − 0.574495i
\(498\) − 9630.55i − 0.00174011i
\(499\) 1.35382e6 0.243395 0.121697 0.992567i \(-0.461166\pi\)
0.121697 + 0.992567i \(0.461166\pi\)
\(500\) 0 0
\(501\) 5.48914e6 0.977035
\(502\) 1.13872e6i 0.201677i
\(503\) 3.85775e6i 0.679851i 0.940452 + 0.339926i \(0.110402\pi\)
−0.940452 + 0.339926i \(0.889598\pi\)
\(504\) 322723. 0.0565918
\(505\) 0 0
\(506\) −531341. −0.0922565
\(507\) − 1.02254e6i − 0.176669i
\(508\) 4.13726e6i 0.711301i
\(509\) 1.06060e7 1.81451 0.907253 0.420585i \(-0.138175\pi\)
0.907253 + 0.420585i \(0.138175\pi\)
\(510\) 0 0
\(511\) 1.39978e6 0.237142
\(512\) − 3.55598e6i − 0.599493i
\(513\) 3.07208e6i 0.515394i
\(514\) −573291. −0.0957122
\(515\) 0 0
\(516\) −1.02443e7 −1.69378
\(517\) 5.52747e6i 0.909495i
\(518\) 38284.1i 0.00626895i
\(519\) −1.95904e6 −0.319246
\(520\) 0 0
\(521\) −9.17989e6 −1.48164 −0.740821 0.671703i \(-0.765563\pi\)
−0.740821 + 0.671703i \(0.765563\pi\)
\(522\) − 248687.i − 0.0399464i
\(523\) 9.05585e6i 1.44769i 0.689964 + 0.723844i \(0.257627\pi\)
−0.689964 + 0.723844i \(0.742373\pi\)
\(524\) −1.09989e7 −1.74993
\(525\) 0 0
\(526\) 336740. 0.0530677
\(527\) 2.43130e6i 0.381339i
\(528\) 1.27455e7i 1.98963i
\(529\) 5.22601e6 0.811953
\(530\) 0 0
\(531\) −2.03642e6 −0.313422
\(532\) − 2.41368e6i − 0.369744i
\(533\) − 714539.i − 0.108945i
\(534\) −1.79478e6 −0.272369
\(535\) 0 0
\(536\) −908075. −0.136524
\(537\) − 6.48394e6i − 0.970295i
\(538\) 1.44987e6i 0.215960i
\(539\) −1.59928e6 −0.237112
\(540\) 0 0
\(541\) 1.21783e7 1.78894 0.894468 0.447132i \(-0.147555\pi\)
0.894468 + 0.447132i \(0.147555\pi\)
\(542\) − 1.16953e6i − 0.171007i
\(543\) − 9.93891e6i − 1.44657i
\(544\) −2.58608e6 −0.374666
\(545\) 0 0
\(546\) −454230. −0.0652070
\(547\) 9.00451e6i 1.28674i 0.765554 + 0.643372i \(0.222465\pi\)
−0.765554 + 0.643372i \(0.777535\pi\)
\(548\) − 1.21627e7i − 1.73013i
\(549\) 2.03703e6 0.288448
\(550\) 0 0
\(551\) −3.75099e6 −0.526341
\(552\) − 994924.i − 0.138977i
\(553\) − 1.50103e6i − 0.208727i
\(554\) 1.50904e6 0.208894
\(555\) 0 0
\(556\) −544167. −0.0746527
\(557\) − 1.54461e6i − 0.210950i −0.994422 0.105475i \(-0.966364\pi\)
0.994422 0.105475i \(-0.0336363\pi\)
\(558\) − 212551.i − 0.0288986i
\(559\) 1.07774e7 1.45876
\(560\) 0 0
\(561\) 1.55345e7 2.08396
\(562\) 712057.i 0.0950986i
\(563\) − 1.18748e7i − 1.57890i −0.613816 0.789449i \(-0.710366\pi\)
0.613816 0.789449i \(-0.289634\pi\)
\(564\) −5.13217e6 −0.679365
\(565\) 0 0
\(566\) −1.01237e6 −0.132831
\(567\) 3.59388e6i 0.469468i
\(568\) − 2.97144e6i − 0.386452i
\(569\) 1.54308e6 0.199806 0.0999031 0.994997i \(-0.468147\pi\)
0.0999031 + 0.994997i \(0.468147\pi\)
\(570\) 0 0
\(571\) −7.01812e6 −0.900804 −0.450402 0.892826i \(-0.648719\pi\)
−0.450402 + 0.892826i \(0.648719\pi\)
\(572\) − 1.36404e7i − 1.74317i
\(573\) − 1.25434e6i − 0.159598i
\(574\) −39018.4 −0.00494298
\(575\) 0 0
\(576\) −4.23326e6 −0.531641
\(577\) 5.37543e6i 0.672162i 0.941833 + 0.336081i \(0.109102\pi\)
−0.941833 + 0.336081i \(0.890898\pi\)
\(578\) − 8077.66i − 0.00100569i
\(579\) −9.22259e6 −1.14329
\(580\) 0 0
\(581\) −33121.4 −0.00407069
\(582\) 326365.i 0.0399389i
\(583\) 3.67627e6i 0.447957i
\(584\) 1.31477e6 0.159521
\(585\) 0 0
\(586\) 1.85770e6 0.223476
\(587\) 2.06682e6i 0.247575i 0.992309 + 0.123788i \(0.0395041\pi\)
−0.992309 + 0.123788i \(0.960496\pi\)
\(588\) − 1.48491e6i − 0.177115i
\(589\) −3.20594e6 −0.380774
\(590\) 0 0
\(591\) 5.26648e6 0.620228
\(592\) − 1.04932e6i − 0.123056i
\(593\) − 5.46947e6i − 0.638717i −0.947634 0.319358i \(-0.896533\pi\)
0.947634 0.319358i \(-0.103467\pi\)
\(594\) 948035. 0.110245
\(595\) 0 0
\(596\) 3.53049e6 0.407117
\(597\) 1.18912e7i 1.36550i
\(598\) 519016.i 0.0593510i
\(599\) −1.09943e7 −1.25199 −0.625996 0.779826i \(-0.715307\pi\)
−0.625996 + 0.779826i \(0.715307\pi\)
\(600\) 0 0
\(601\) 1.58788e7 1.79322 0.896608 0.442826i \(-0.146024\pi\)
0.896608 + 0.442826i \(0.146024\pi\)
\(602\) − 588514.i − 0.0661859i
\(603\) 2.82348e6i 0.316222i
\(604\) 959820. 0.107053
\(605\) 0 0
\(606\) −2.58555e6 −0.286004
\(607\) − 5.33262e6i − 0.587447i −0.955890 0.293724i \(-0.905105\pi\)
0.955890 0.293724i \(-0.0948945\pi\)
\(608\) − 3.41003e6i − 0.374110i
\(609\) −2.30762e6 −0.252128
\(610\) 0 0
\(611\) 5.39926e6 0.585102
\(612\) 5.34584e6i 0.576949i
\(613\) − 8.91838e6i − 0.958594i −0.877653 0.479297i \(-0.840892\pi\)
0.877653 0.479297i \(-0.159108\pi\)
\(614\) 641593. 0.0686814
\(615\) 0 0
\(616\) −1.50215e6 −0.159501
\(617\) − 9.63586e6i − 1.01901i −0.860468 0.509504i \(-0.829829\pi\)
0.860468 0.509504i \(-0.170171\pi\)
\(618\) 922841.i 0.0971976i
\(619\) −1.21747e7 −1.27712 −0.638560 0.769572i \(-0.720470\pi\)
−0.638560 + 0.769572i \(0.720470\pi\)
\(620\) 0 0
\(621\) 2.15952e6 0.224713
\(622\) − 1.30602e6i − 0.135355i
\(623\) 6.17260e6i 0.637159i
\(624\) 1.24498e7 1.27998
\(625\) 0 0
\(626\) 689094. 0.0702818
\(627\) 2.04839e7i 2.08087i
\(628\) − 1.64770e7i − 1.66717i
\(629\) −1.27893e6 −0.128890
\(630\) 0 0
\(631\) −1.30854e7 −1.30832 −0.654161 0.756356i \(-0.726978\pi\)
−0.654161 + 0.756356i \(0.726978\pi\)
\(632\) − 1.40987e6i − 0.140407i
\(633\) 6.59023e6i 0.653719i
\(634\) −2.20626e6 −0.217989
\(635\) 0 0
\(636\) −3.41336e6 −0.334610
\(637\) 1.56219e6i 0.152540i
\(638\) 1.15754e6i 0.112586i
\(639\) −9.23911e6 −0.895113
\(640\) 0 0
\(641\) −441107. −0.0424032 −0.0212016 0.999775i \(-0.506749\pi\)
−0.0212016 + 0.999775i \(0.506749\pi\)
\(642\) 2.11105e6i 0.202144i
\(643\) − 4.18888e6i − 0.399550i −0.979842 0.199775i \(-0.935979\pi\)
0.979842 0.199775i \(-0.0640211\pi\)
\(644\) −1.69670e6 −0.161209
\(645\) 0 0
\(646\) −1.34688e6 −0.126983
\(647\) − 1.87822e7i − 1.76395i −0.471300 0.881973i \(-0.656215\pi\)
0.471300 0.881973i \(-0.343785\pi\)
\(648\) 3.37562e6i 0.315803i
\(649\) 9.47874e6 0.883362
\(650\) 0 0
\(651\) −1.97231e6 −0.182399
\(652\) − 1.38375e7i − 1.27479i
\(653\) − 1.51733e6i − 0.139251i −0.997573 0.0696254i \(-0.977820\pi\)
0.997573 0.0696254i \(-0.0221804\pi\)
\(654\) 1.58567e6 0.144967
\(655\) 0 0
\(656\) 1.06944e6 0.0970281
\(657\) − 4.08802e6i − 0.369487i
\(658\) − 294834.i − 0.0265468i
\(659\) −1.84809e7 −1.65772 −0.828859 0.559458i \(-0.811009\pi\)
−0.828859 + 0.559458i \(0.811009\pi\)
\(660\) 0 0
\(661\) −1.03952e7 −0.925403 −0.462702 0.886514i \(-0.653120\pi\)
−0.462702 + 0.886514i \(0.653120\pi\)
\(662\) 1.59240e6i 0.141223i
\(663\) − 1.51741e7i − 1.34066i
\(664\) −31109.8 −0.00273828
\(665\) 0 0
\(666\) 111808. 0.00976756
\(667\) 2.63676e6i 0.229486i
\(668\) − 8.79242e6i − 0.762372i
\(669\) −2.01458e7 −1.74028
\(670\) 0 0
\(671\) −9.48162e6 −0.812973
\(672\) − 2.09786e6i − 0.179207i
\(673\) − 1.10398e7i − 0.939556i −0.882785 0.469778i \(-0.844334\pi\)
0.882785 0.469778i \(-0.155666\pi\)
\(674\) 1.75101e6 0.148470
\(675\) 0 0
\(676\) −1.63788e6 −0.137853
\(677\) 8.23485e6i 0.690532i 0.938505 + 0.345266i \(0.112211\pi\)
−0.938505 + 0.345266i \(0.887789\pi\)
\(678\) − 615144.i − 0.0513928i
\(679\) 1.12243e6 0.0934298
\(680\) 0 0
\(681\) −9.90780e6 −0.818671
\(682\) 989343.i 0.0814490i
\(683\) 1.98437e7i 1.62768i 0.581086 + 0.813842i \(0.302628\pi\)
−0.581086 + 0.813842i \(0.697372\pi\)
\(684\) −7.04909e6 −0.576093
\(685\) 0 0
\(686\) 85305.3 0.00692095
\(687\) 2.19955e7i 1.77804i
\(688\) 1.61304e7i 1.29919i
\(689\) 3.59100e6 0.288182
\(690\) 0 0
\(691\) −2.37019e6 −0.188837 −0.0944185 0.995533i \(-0.530099\pi\)
−0.0944185 + 0.995533i \(0.530099\pi\)
\(692\) 3.13796e6i 0.249105i
\(693\) 4.67065e6i 0.369440i
\(694\) 789131. 0.0621943
\(695\) 0 0
\(696\) −2.16748e6 −0.169602
\(697\) − 1.30346e6i − 0.101628i
\(698\) 2.02491e6i 0.157314i
\(699\) −1.51439e7 −1.17232
\(700\) 0 0
\(701\) −1.56833e7 −1.20543 −0.602714 0.797957i \(-0.705914\pi\)
−0.602714 + 0.797957i \(0.705914\pi\)
\(702\) − 926045.i − 0.0709234i
\(703\) − 1.68641e6i − 0.128699i
\(704\) 1.97042e7 1.49840
\(705\) 0 0
\(706\) 1.83543e6 0.138588
\(707\) 8.89223e6i 0.669056i
\(708\) 8.80085e6i 0.659844i
\(709\) −3.68544e6 −0.275343 −0.137671 0.990478i \(-0.543962\pi\)
−0.137671 + 0.990478i \(0.543962\pi\)
\(710\) 0 0
\(711\) −4.38373e6 −0.325214
\(712\) 5.79772e6i 0.428605i
\(713\) 2.25361e6i 0.166018i
\(714\) −828604. −0.0608277
\(715\) 0 0
\(716\) −1.03859e7 −0.757113
\(717\) 3.37276e6i 0.245012i
\(718\) 790544.i 0.0572289i
\(719\) 1.56427e7 1.12847 0.564233 0.825615i \(-0.309172\pi\)
0.564233 + 0.825615i \(0.309172\pi\)
\(720\) 0 0
\(721\) 3.17383e6 0.227376
\(722\) 19369.0i 0.00138282i
\(723\) − 7.54107e6i − 0.536522i
\(724\) −1.59200e7 −1.12875
\(725\) 0 0
\(726\) 4.02670e6 0.283536
\(727\) − 1.85908e7i − 1.30456i −0.757979 0.652279i \(-0.773813\pi\)
0.757979 0.652279i \(-0.226187\pi\)
\(728\) 1.46731e6i 0.102611i
\(729\) 1.12318e6 0.0782767
\(730\) 0 0
\(731\) 1.96601e7 1.36079
\(732\) − 8.80353e6i − 0.607266i
\(733\) 2.49466e7i 1.71495i 0.514524 + 0.857476i \(0.327969\pi\)
−0.514524 + 0.857476i \(0.672031\pi\)
\(734\) −136367. −0.00934260
\(735\) 0 0
\(736\) −2.39708e6 −0.163113
\(737\) − 1.31422e7i − 0.891253i
\(738\) 113952.i 0.00770159i
\(739\) 2.42944e7 1.63642 0.818211 0.574918i \(-0.194966\pi\)
0.818211 + 0.574918i \(0.194966\pi\)
\(740\) 0 0
\(741\) 2.00088e7 1.33868
\(742\) − 196091.i − 0.0130752i
\(743\) 4.16541e6i 0.276812i 0.990376 + 0.138406i \(0.0441980\pi\)
−0.990376 + 0.138406i \(0.955802\pi\)
\(744\) −1.85252e6 −0.122696
\(745\) 0 0
\(746\) −1.30059e6 −0.0855644
\(747\) 96729.9i 0.00634248i
\(748\) − 2.48828e7i − 1.62610i
\(749\) 7.26032e6 0.472880
\(750\) 0 0
\(751\) −2.36434e7 −1.52972 −0.764858 0.644199i \(-0.777191\pi\)
−0.764858 + 0.644199i \(0.777191\pi\)
\(752\) 8.08101e6i 0.521100i
\(753\) − 3.08588e7i − 1.98332i
\(754\) 1.13070e6 0.0724298
\(755\) 0 0
\(756\) 3.02730e6 0.192642
\(757\) − 1.50108e7i − 0.952062i −0.879429 0.476031i \(-0.842075\pi\)
0.879429 0.476031i \(-0.157925\pi\)
\(758\) 2.60164e6i 0.164465i
\(759\) 1.43992e7 0.907262
\(760\) 0 0
\(761\) 2.92191e6 0.182897 0.0914483 0.995810i \(-0.470850\pi\)
0.0914483 + 0.995810i \(0.470850\pi\)
\(762\) 1.87282e6i 0.116845i
\(763\) − 5.45343e6i − 0.339124i
\(764\) −2.00918e6 −0.124533
\(765\) 0 0
\(766\) −2.48310e6 −0.152905
\(767\) − 9.25887e6i − 0.568290i
\(768\) 1.73016e7i 1.05848i
\(769\) −1.42847e7 −0.871073 −0.435536 0.900171i \(-0.643441\pi\)
−0.435536 + 0.900171i \(0.643441\pi\)
\(770\) 0 0
\(771\) 1.55360e7 0.941246
\(772\) 1.47726e7i 0.892100i
\(773\) − 1.09012e7i − 0.656186i −0.944645 0.328093i \(-0.893594\pi\)
0.944645 0.328093i \(-0.106406\pi\)
\(774\) −1.71874e6 −0.103123
\(775\) 0 0
\(776\) 1.05426e6 0.0628485
\(777\) − 1.03749e6i − 0.0616496i
\(778\) 6254.04i 0 0.000370434i
\(779\) 1.71875e6 0.101478
\(780\) 0 0
\(781\) 4.30045e7 2.52282
\(782\) 946787.i 0.0553650i
\(783\) − 4.70459e6i − 0.274231i
\(784\) −2.33811e6 −0.135854
\(785\) 0 0
\(786\) −4.97890e6 −0.287460
\(787\) 2.56449e7i 1.47593i 0.674841 + 0.737963i \(0.264212\pi\)
−0.674841 + 0.737963i \(0.735788\pi\)
\(788\) − 8.43576e6i − 0.483959i
\(789\) −9.12555e6 −0.521875
\(790\) 0 0
\(791\) −2.11560e6 −0.120224
\(792\) 4.38699e6i 0.248516i
\(793\) 9.26169e6i 0.523007i
\(794\) −911498. −0.0513103
\(795\) 0 0
\(796\) 1.90472e7 1.06549
\(797\) − 7.73086e6i − 0.431104i −0.976492 0.215552i \(-0.930845\pi\)
0.976492 0.215552i \(-0.0691551\pi\)
\(798\) − 1.09261e6i − 0.0607375i
\(799\) 9.84931e6 0.545807
\(800\) 0 0
\(801\) 1.80269e7 0.992748
\(802\) − 1.03448e6i − 0.0567917i
\(803\) 1.90282e7i 1.04138i
\(804\) 1.22024e7 0.665738
\(805\) 0 0
\(806\) 966395. 0.0523983
\(807\) − 3.92909e7i − 2.12377i
\(808\) 8.35218e6i 0.450061i
\(809\) 1.87811e7 1.00890 0.504452 0.863440i \(-0.331695\pi\)
0.504452 + 0.863440i \(0.331695\pi\)
\(810\) 0 0
\(811\) −9.00729e6 −0.480886 −0.240443 0.970663i \(-0.577293\pi\)
−0.240443 + 0.970663i \(0.577293\pi\)
\(812\) 3.69632e6i 0.196734i
\(813\) 3.16939e7i 1.68170i
\(814\) −520422. −0.0275293
\(815\) 0 0
\(816\) 2.27110e7 1.19402
\(817\) 2.59240e7i 1.35877i
\(818\) − 2.22259e6i − 0.116138i
\(819\) 4.56231e6 0.237671
\(820\) 0 0
\(821\) −9.27965e6 −0.480478 −0.240239 0.970714i \(-0.577226\pi\)
−0.240239 + 0.970714i \(0.577226\pi\)
\(822\) − 5.50572e6i − 0.284207i
\(823\) − 1.08308e7i − 0.557393i −0.960379 0.278697i \(-0.910098\pi\)
0.960379 0.278697i \(-0.0899024\pi\)
\(824\) 2.98107e6 0.152952
\(825\) 0 0
\(826\) −505593. −0.0257841
\(827\) 2.05230e7i 1.04346i 0.853110 + 0.521731i \(0.174714\pi\)
−0.853110 + 0.521731i \(0.825286\pi\)
\(828\) 4.95515e6i 0.251178i
\(829\) 1.42216e7 0.718724 0.359362 0.933198i \(-0.382994\pi\)
0.359362 + 0.933198i \(0.382994\pi\)
\(830\) 0 0
\(831\) −4.08944e7 −2.05429
\(832\) − 1.92472e7i − 0.963958i
\(833\) 2.84973e6i 0.142296i
\(834\) −246329. −0.0122631
\(835\) 0 0
\(836\) 3.28108e7 1.62368
\(837\) − 4.02097e6i − 0.198389i
\(838\) − 179995.i − 0.00885421i
\(839\) 9.29934e6 0.456087 0.228043 0.973651i \(-0.426767\pi\)
0.228043 + 0.973651i \(0.426767\pi\)
\(840\) 0 0
\(841\) −1.47669e7 −0.719944
\(842\) − 4.32353e6i − 0.210164i
\(843\) − 1.92965e7i − 0.935211i
\(844\) 1.05561e7 0.510092
\(845\) 0 0
\(846\) −861053. −0.0413623
\(847\) − 1.38486e7i − 0.663281i
\(848\) 5.37461e6i 0.256659i
\(849\) 2.74350e7 1.30628
\(850\) 0 0
\(851\) −1.18546e6 −0.0561131
\(852\) 3.99290e7i 1.88447i
\(853\) − 3.07436e6i − 0.144671i −0.997380 0.0723357i \(-0.976955\pi\)
0.997380 0.0723357i \(-0.0230453\pi\)
\(854\) 505747. 0.0237295
\(855\) 0 0
\(856\) 6.81938e6 0.318098
\(857\) − 3.45835e7i − 1.60848i −0.594302 0.804242i \(-0.702572\pi\)
0.594302 0.804242i \(-0.297428\pi\)
\(858\) − 6.17465e6i − 0.286348i
\(859\) −1.63022e7 −0.753814 −0.376907 0.926251i \(-0.623012\pi\)
−0.376907 + 0.926251i \(0.623012\pi\)
\(860\) 0 0
\(861\) 1.05738e6 0.0486099
\(862\) 3.57856e6i 0.164036i
\(863\) 2.56962e7i 1.17447i 0.809416 + 0.587235i \(0.199784\pi\)
−0.809416 + 0.587235i \(0.800216\pi\)
\(864\) 4.27694e6 0.194917
\(865\) 0 0
\(866\) 3.01281e6 0.136514
\(867\) 218902.i 0.00989012i
\(868\) 3.15921e6i 0.142324i
\(869\) 2.04046e7 0.916596
\(870\) 0 0
\(871\) −1.28374e7 −0.573366
\(872\) − 5.12222e6i − 0.228122i
\(873\) − 3.27803e6i − 0.145572i
\(874\) −1.24844e6 −0.0552829
\(875\) 0 0
\(876\) −1.76673e7 −0.777877
\(877\) − 3.30060e7i − 1.44908i −0.689230 0.724542i \(-0.742051\pi\)
0.689230 0.724542i \(-0.257949\pi\)
\(878\) − 165286.i − 0.00723603i
\(879\) −5.03429e7 −2.19769
\(880\) 0 0
\(881\) 2.26705e6 0.0984060 0.0492030 0.998789i \(-0.484332\pi\)
0.0492030 + 0.998789i \(0.484332\pi\)
\(882\) − 249131.i − 0.0107834i
\(883\) 1.97779e7i 0.853649i 0.904334 + 0.426825i \(0.140368\pi\)
−0.904334 + 0.426825i \(0.859632\pi\)
\(884\) −2.43057e7 −1.04611
\(885\) 0 0
\(886\) −1.43901e6 −0.0615858
\(887\) 36468.3i 0.00155635i 1.00000 0.000778173i \(0.000247700\pi\)
−1.00000 0.000778173i \(0.999752\pi\)
\(888\) − 974478.i − 0.0414705i
\(889\) 6.44100e6 0.273337
\(890\) 0 0
\(891\) −4.88541e7 −2.06161
\(892\) 3.22692e7i 1.35793i
\(893\) 1.29874e7i 0.544997i
\(894\) 1.59815e6 0.0668767
\(895\) 0 0
\(896\) −4.46747e6 −0.185905
\(897\) − 1.40652e7i − 0.583665i
\(898\) 1.89303e6i 0.0783367i
\(899\) 4.90958e6 0.202602
\(900\) 0 0
\(901\) 6.55068e6 0.268828
\(902\) − 530403.i − 0.0217065i
\(903\) 1.59485e7i 0.650881i
\(904\) −1.98711e6 −0.0808727
\(905\) 0 0
\(906\) 434484. 0.0175854
\(907\) − 1.62298e7i − 0.655079i −0.944837 0.327540i \(-0.893781\pi\)
0.944837 0.327540i \(-0.106219\pi\)
\(908\) 1.58702e7i 0.638802i
\(909\) 2.59695e7 1.04245
\(910\) 0 0
\(911\) −2.61699e7 −1.04474 −0.522368 0.852720i \(-0.674951\pi\)
−0.522368 + 0.852720i \(0.674951\pi\)
\(912\) 2.99469e7i 1.19224i
\(913\) − 450241.i − 0.0178759i
\(914\) 2.97224e6 0.117684
\(915\) 0 0
\(916\) 3.52320e7 1.38739
\(917\) 1.71234e7i 0.672461i
\(918\) − 1.68929e6i − 0.0661602i
\(919\) −4.05973e6 −0.158565 −0.0792826 0.996852i \(-0.525263\pi\)
−0.0792826 + 0.996852i \(0.525263\pi\)
\(920\) 0 0
\(921\) −1.73870e7 −0.675421
\(922\) 1.90246e6i 0.0737034i
\(923\) − 4.20070e7i − 1.62300i
\(924\) 2.01853e7 0.777778
\(925\) 0 0
\(926\) −3.10744e6 −0.119090
\(927\) − 9.26907e6i − 0.354272i
\(928\) 5.22212e6i 0.199057i
\(929\) −3.69518e6 −0.140474 −0.0702370 0.997530i \(-0.522376\pi\)
−0.0702370 + 0.997530i \(0.522376\pi\)
\(930\) 0 0
\(931\) −3.75769e6 −0.142084
\(932\) 2.42573e7i 0.914751i
\(933\) 3.53927e7i 1.33110i
\(934\) −717918. −0.0269282
\(935\) 0 0
\(936\) 4.28523e6 0.159877
\(937\) 1.91384e7i 0.712126i 0.934462 + 0.356063i \(0.115881\pi\)
−0.934462 + 0.356063i \(0.884119\pi\)
\(938\) 701004.i 0.0260144i
\(939\) −1.86742e7 −0.691160
\(940\) 0 0
\(941\) 1.20954e7 0.445294 0.222647 0.974899i \(-0.428530\pi\)
0.222647 + 0.974899i \(0.428530\pi\)
\(942\) − 7.45870e6i − 0.273865i
\(943\) − 1.20820e6i − 0.0442445i
\(944\) 1.38577e7 0.506127
\(945\) 0 0
\(946\) 8.00007e6 0.290647
\(947\) 1.95969e7i 0.710087i 0.934850 + 0.355044i \(0.115534\pi\)
−0.934850 + 0.355044i \(0.884466\pi\)
\(948\) 1.89453e7i 0.684669i
\(949\) 1.85868e7 0.669945
\(950\) 0 0
\(951\) 5.97890e7 2.14373
\(952\) 2.67666e6i 0.0957196i
\(953\) 4.23265e7i 1.50966i 0.655918 + 0.754832i \(0.272282\pi\)
−0.655918 + 0.754832i \(0.727718\pi\)
\(954\) −572679. −0.0203723
\(955\) 0 0
\(956\) 5.40243e6 0.191181
\(957\) − 3.13691e7i − 1.10719i
\(958\) 1.87054e6i 0.0658496i
\(959\) −1.89353e7 −0.664852
\(960\) 0 0
\(961\) −2.44330e7 −0.853430
\(962\) 508351.i 0.0177103i
\(963\) − 2.12035e7i − 0.736788i
\(964\) −1.20792e7 −0.418644
\(965\) 0 0
\(966\) −768048. −0.0264817
\(967\) − 1.39211e6i − 0.0478749i −0.999713 0.0239375i \(-0.992380\pi\)
0.999713 0.0239375i \(-0.00762026\pi\)
\(968\) − 1.30075e7i − 0.446176i
\(969\) 3.64999e7 1.24877
\(970\) 0 0
\(971\) 4.41877e7 1.50402 0.752009 0.659153i \(-0.229085\pi\)
0.752009 + 0.659153i \(0.229085\pi\)
\(972\) − 3.03473e7i − 1.03028i
\(973\) 847175.i 0.0286874i
\(974\) −2.33095e6 −0.0787292
\(975\) 0 0
\(976\) −1.38619e7 −0.465798
\(977\) 5.60457e7i 1.87848i 0.343264 + 0.939239i \(0.388467\pi\)
−0.343264 + 0.939239i \(0.611533\pi\)
\(978\) − 6.26386e6i − 0.209409i
\(979\) −8.39082e7 −2.79800
\(980\) 0 0
\(981\) −1.59266e7 −0.528384
\(982\) 5.69993e6i 0.188621i
\(983\) − 5.00560e7i − 1.65224i −0.563497 0.826118i \(-0.690544\pi\)
0.563497 0.826118i \(-0.309456\pi\)
\(984\) 993166. 0.0326990
\(985\) 0 0
\(986\) 2.06261e6 0.0675654
\(987\) 7.98990e6i 0.261065i
\(988\) − 3.20497e7i − 1.04456i
\(989\) 1.82233e7 0.592428
\(990\) 0 0
\(991\) 1.59116e7 0.514670 0.257335 0.966322i \(-0.417156\pi\)
0.257335 + 0.966322i \(0.417156\pi\)
\(992\) 4.46330e6i 0.144005i
\(993\) − 4.31534e7i − 1.38881i
\(994\) −2.29385e6 −0.0736375
\(995\) 0 0
\(996\) 418041. 0.0133528
\(997\) − 4.25995e7i − 1.35727i −0.734476 0.678635i \(-0.762572\pi\)
0.734476 0.678635i \(-0.237428\pi\)
\(998\) − 981634.i − 0.0311978i
\(999\) 2.11514e6 0.0670542
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 175.6.b.c.99.2 4
5.2 odd 4 7.6.a.b.1.1 2
5.3 odd 4 175.6.a.c.1.2 2
5.4 even 2 inner 175.6.b.c.99.3 4
15.2 even 4 63.6.a.f.1.2 2
20.7 even 4 112.6.a.h.1.1 2
35.2 odd 12 49.6.c.e.18.2 4
35.12 even 12 49.6.c.d.18.2 4
35.17 even 12 49.6.c.d.30.2 4
35.27 even 4 49.6.a.f.1.1 2
35.32 odd 12 49.6.c.e.30.2 4
40.27 even 4 448.6.a.u.1.2 2
40.37 odd 4 448.6.a.w.1.1 2
55.32 even 4 847.6.a.c.1.2 2
60.47 odd 4 1008.6.a.bq.1.2 2
105.62 odd 4 441.6.a.l.1.2 2
140.27 odd 4 784.6.a.v.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.6.a.b.1.1 2 5.2 odd 4
49.6.a.f.1.1 2 35.27 even 4
49.6.c.d.18.2 4 35.12 even 12
49.6.c.d.30.2 4 35.17 even 12
49.6.c.e.18.2 4 35.2 odd 12
49.6.c.e.30.2 4 35.32 odd 12
63.6.a.f.1.2 2 15.2 even 4
112.6.a.h.1.1 2 20.7 even 4
175.6.a.c.1.2 2 5.3 odd 4
175.6.b.c.99.2 4 1.1 even 1 trivial
175.6.b.c.99.3 4 5.4 even 2 inner
441.6.a.l.1.2 2 105.62 odd 4
448.6.a.u.1.2 2 40.27 even 4
448.6.a.w.1.1 2 40.37 odd 4
784.6.a.v.1.2 2 140.27 odd 4
847.6.a.c.1.2 2 55.32 even 4
1008.6.a.bq.1.2 2 60.47 odd 4