Properties

Label 2100.2.bc.e.949.2
Level $2100$
Weight $2$
Character 2100.949
Analytic conductor $16.769$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2100,2,Mod(949,2100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2100, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2100.949");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2100.bc (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.49787136.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 949.2
Root \(-0.228425 - 1.39564i\) of defining polynomial
Character \(\chi\) \(=\) 2100.949
Dual form 2100.2.bc.e.1549.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.866025 - 0.500000i) q^{3} +(2.29129 - 1.32288i) q^{7} +(0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(-0.866025 - 0.500000i) q^{3} +(2.29129 - 1.32288i) q^{7} +(0.500000 + 0.866025i) q^{9} +(-1.82288 + 3.15731i) q^{11} +2.64575i q^{13} +(3.15731 + 1.82288i) q^{17} +(-1.14575 - 1.98450i) q^{19} -2.64575 q^{21} +(-3.15731 + 1.82288i) q^{23} -1.00000i q^{27} -2.35425 q^{29} +(-3.14575 + 5.44860i) q^{31} +(3.15731 - 1.82288i) q^{33} +(4.02334 - 2.32288i) q^{37} +(1.32288 - 2.29129i) q^{39} -10.9373 q^{41} +9.93725i q^{43} +(-5.19615 + 3.00000i) q^{47} +(3.50000 - 6.06218i) q^{49} +(-1.82288 - 3.15731i) q^{51} +(6.31463 + 3.64575i) q^{53} +2.29150i q^{57} +(-2.46863 + 4.27579i) q^{59} +(-4.00000 - 6.92820i) q^{61} +(2.29129 + 1.32288i) q^{63} +(-4.02334 - 2.32288i) q^{67} +3.64575 q^{69} +8.35425 q^{71} +(11.4564 + 6.61438i) q^{73} +9.64575i q^{77} +(-4.14575 - 7.18065i) q^{79} +(-0.500000 + 0.866025i) q^{81} +4.93725i q^{83} +(2.03884 + 1.17712i) q^{87} +(6.11438 + 10.5904i) q^{89} +(3.50000 + 6.06218i) q^{91} +(5.44860 - 3.14575i) q^{93} -8.00000i q^{97} -3.64575 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{9} - 4 q^{11} + 12 q^{19} - 40 q^{29} - 4 q^{31} - 24 q^{41} + 28 q^{49} - 4 q^{51} + 12 q^{59} - 32 q^{61} + 8 q^{69} + 88 q^{71} - 12 q^{79} - 4 q^{81} - 4 q^{89} + 28 q^{91} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times\).

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(e\left(\frac{2}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.866025 0.500000i −0.500000 0.288675i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.29129 1.32288i 0.866025 0.500000i
\(8\) 0 0
\(9\) 0.500000 + 0.866025i 0.166667 + 0.288675i
\(10\) 0 0
\(11\) −1.82288 + 3.15731i −0.549618 + 0.951966i 0.448683 + 0.893691i \(0.351893\pi\)
−0.998301 + 0.0582747i \(0.981440\pi\)
\(12\) 0 0
\(13\) 2.64575i 0.733799i 0.930261 + 0.366900i \(0.119581\pi\)
−0.930261 + 0.366900i \(0.880419\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.15731 + 1.82288i 0.765761 + 0.442112i 0.831360 0.555734i \(-0.187563\pi\)
−0.0655994 + 0.997846i \(0.520896\pi\)
\(18\) 0 0
\(19\) −1.14575 1.98450i −0.262853 0.455275i 0.704146 0.710056i \(-0.251330\pi\)
−0.966999 + 0.254780i \(0.917997\pi\)
\(20\) 0 0
\(21\) −2.64575 −0.577350
\(22\) 0 0
\(23\) −3.15731 + 1.82288i −0.658345 + 0.380096i −0.791646 0.610980i \(-0.790776\pi\)
0.133301 + 0.991076i \(0.457442\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −2.35425 −0.437173 −0.218587 0.975818i \(-0.570145\pi\)
−0.218587 + 0.975818i \(0.570145\pi\)
\(30\) 0 0
\(31\) −3.14575 + 5.44860i −0.564994 + 0.978598i 0.432057 + 0.901846i \(0.357788\pi\)
−0.997050 + 0.0767512i \(0.975545\pi\)
\(32\) 0 0
\(33\) 3.15731 1.82288i 0.549618 0.317322i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.02334 2.32288i 0.661433 0.381878i −0.131390 0.991331i \(-0.541944\pi\)
0.792823 + 0.609452i \(0.208611\pi\)
\(38\) 0 0
\(39\) 1.32288 2.29129i 0.211830 0.366900i
\(40\) 0 0
\(41\) −10.9373 −1.70811 −0.854056 0.520181i \(-0.825864\pi\)
−0.854056 + 0.520181i \(0.825864\pi\)
\(42\) 0 0
\(43\) 9.93725i 1.51542i 0.652593 + 0.757709i \(0.273681\pi\)
−0.652593 + 0.757709i \(0.726319\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.19615 + 3.00000i −0.757937 + 0.437595i −0.828554 0.559908i \(-0.810836\pi\)
0.0706177 + 0.997503i \(0.477503\pi\)
\(48\) 0 0
\(49\) 3.50000 6.06218i 0.500000 0.866025i
\(50\) 0 0
\(51\) −1.82288 3.15731i −0.255254 0.442112i
\(52\) 0 0
\(53\) 6.31463 + 3.64575i 0.867381 + 0.500782i 0.866477 0.499217i \(-0.166379\pi\)
0.000903738 1.00000i \(0.499712\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.29150i 0.303517i
\(58\) 0 0
\(59\) −2.46863 + 4.27579i −0.321388 + 0.556660i −0.980775 0.195144i \(-0.937483\pi\)
0.659387 + 0.751804i \(0.270816\pi\)
\(60\) 0 0
\(61\) −4.00000 6.92820i −0.512148 0.887066i −0.999901 0.0140840i \(-0.995517\pi\)
0.487753 0.872982i \(-0.337817\pi\)
\(62\) 0 0
\(63\) 2.29129 + 1.32288i 0.288675 + 0.166667i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −4.02334 2.32288i −0.491529 0.283784i 0.233680 0.972314i \(-0.424923\pi\)
−0.725209 + 0.688529i \(0.758257\pi\)
\(68\) 0 0
\(69\) 3.64575 0.438897
\(70\) 0 0
\(71\) 8.35425 0.991467 0.495733 0.868475i \(-0.334899\pi\)
0.495733 + 0.868475i \(0.334899\pi\)
\(72\) 0 0
\(73\) 11.4564 + 6.61438i 1.34087 + 0.774154i 0.986936 0.161114i \(-0.0515087\pi\)
0.353939 + 0.935269i \(0.384842\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 9.64575i 1.09924i
\(78\) 0 0
\(79\) −4.14575 7.18065i −0.466433 0.807886i 0.532831 0.846221i \(-0.321128\pi\)
−0.999265 + 0.0383349i \(0.987795\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 4.93725i 0.541934i 0.962589 + 0.270967i \(0.0873434\pi\)
−0.962589 + 0.270967i \(0.912657\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.03884 + 1.17712i 0.218587 + 0.126201i
\(88\) 0 0
\(89\) 6.11438 + 10.5904i 0.648123 + 1.12258i 0.983571 + 0.180523i \(0.0577790\pi\)
−0.335448 + 0.942059i \(0.608888\pi\)
\(90\) 0 0
\(91\) 3.50000 + 6.06218i 0.366900 + 0.635489i
\(92\) 0 0
\(93\) 5.44860 3.14575i 0.564994 0.326199i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.00000i 0.812277i −0.913812 0.406138i \(-0.866875\pi\)
0.913812 0.406138i \(-0.133125\pi\)
\(98\) 0 0
\(99\) −3.64575 −0.366412
\(100\) 0 0
\(101\) −5.46863 + 9.47194i −0.544149 + 0.942493i 0.454511 + 0.890741i \(0.349814\pi\)
−0.998660 + 0.0517522i \(0.983519\pi\)
\(102\) 0 0
\(103\) 12.6836 7.32288i 1.24975 0.721544i 0.278692 0.960381i \(-0.410099\pi\)
0.971060 + 0.238836i \(0.0767659\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0.920365 0.531373i 0.0889751 0.0513698i −0.454852 0.890567i \(-0.650308\pi\)
0.543827 + 0.839197i \(0.316975\pi\)
\(108\) 0 0
\(109\) −6.50000 + 11.2583i −0.622587 + 1.07835i 0.366415 + 0.930451i \(0.380585\pi\)
−0.989002 + 0.147901i \(0.952748\pi\)
\(110\) 0 0
\(111\) −4.64575 −0.440955
\(112\) 0 0
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −2.29129 + 1.32288i −0.211830 + 0.122300i
\(118\) 0 0
\(119\) 9.64575 0.884225
\(120\) 0 0
\(121\) −1.14575 1.98450i −0.104159 0.180409i
\(122\) 0 0
\(123\) 9.47194 + 5.46863i 0.854056 + 0.493089i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 17.9373i 1.59167i 0.605511 + 0.795837i \(0.292969\pi\)
−0.605511 + 0.795837i \(0.707031\pi\)
\(128\) 0 0
\(129\) 4.96863 8.60591i 0.437463 0.757709i
\(130\) 0 0
\(131\) −5.35425 9.27383i −0.467803 0.810258i 0.531520 0.847046i \(-0.321621\pi\)
−0.999323 + 0.0367872i \(0.988288\pi\)
\(132\) 0 0
\(133\) −5.25049 3.03137i −0.455275 0.262853i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 5.39426 + 3.11438i 0.460863 + 0.266079i 0.712407 0.701767i \(-0.247605\pi\)
−0.251544 + 0.967846i \(0.580938\pi\)
\(138\) 0 0
\(139\) −5.00000 −0.424094 −0.212047 0.977259i \(-0.568013\pi\)
−0.212047 + 0.977259i \(0.568013\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 0 0
\(143\) −8.35347 4.82288i −0.698552 0.403309i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −6.06218 + 3.50000i −0.500000 + 0.288675i
\(148\) 0 0
\(149\) 7.29150 + 12.6293i 0.597343 + 1.03463i 0.993212 + 0.116321i \(0.0371103\pi\)
−0.395868 + 0.918307i \(0.629556\pi\)
\(150\) 0 0
\(151\) −8.29150 + 14.3613i −0.674753 + 1.16871i 0.301788 + 0.953375i \(0.402416\pi\)
−0.976541 + 0.215331i \(0.930917\pi\)
\(152\) 0 0
\(153\) 3.64575i 0.294742i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 8.04668 + 4.64575i 0.642195 + 0.370771i 0.785459 0.618913i \(-0.212427\pi\)
−0.143265 + 0.989684i \(0.545760\pi\)
\(158\) 0 0
\(159\) −3.64575 6.31463i −0.289127 0.500782i
\(160\) 0 0
\(161\) −4.82288 + 8.35347i −0.380096 + 0.658345i
\(162\) 0 0
\(163\) −13.8564 + 8.00000i −1.08532 + 0.626608i −0.932326 0.361619i \(-0.882224\pi\)
−0.152992 + 0.988227i \(0.548891\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 14.3542i 1.11077i −0.831595 0.555383i \(-0.812572\pi\)
0.831595 0.555383i \(-0.187428\pi\)
\(168\) 0 0
\(169\) 6.00000 0.461538
\(170\) 0 0
\(171\) 1.14575 1.98450i 0.0876178 0.151758i
\(172\) 0 0
\(173\) −18.9439 + 10.9373i −1.44028 + 0.831544i −0.997868 0.0652695i \(-0.979209\pi\)
−0.442409 + 0.896813i \(0.645876\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 4.27579 2.46863i 0.321388 0.185553i
\(178\) 0 0
\(179\) −10.2915 + 17.8254i −0.769223 + 1.33233i 0.168762 + 0.985657i \(0.446023\pi\)
−0.937985 + 0.346676i \(0.887310\pi\)
\(180\) 0 0
\(181\) 6.29150 0.467644 0.233822 0.972279i \(-0.424877\pi\)
0.233822 + 0.972279i \(0.424877\pi\)
\(182\) 0 0
\(183\) 8.00000i 0.591377i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −11.5108 + 6.64575i −0.841752 + 0.485985i
\(188\) 0 0
\(189\) −1.32288 2.29129i −0.0962250 0.166667i
\(190\) 0 0
\(191\) 11.5830 + 20.0624i 0.838117 + 1.45166i 0.891467 + 0.453085i \(0.149677\pi\)
−0.0533504 + 0.998576i \(0.516990\pi\)
\(192\) 0 0
\(193\) 20.7303 + 11.9686i 1.49220 + 0.861521i 0.999960 0.00894034i \(-0.00284584\pi\)
0.492237 + 0.870461i \(0.336179\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 10.9373i 0.779247i −0.920974 0.389624i \(-0.872605\pi\)
0.920974 0.389624i \(-0.127395\pi\)
\(198\) 0 0
\(199\) 9.58301 16.5983i 0.679321 1.17662i −0.295864 0.955230i \(-0.595608\pi\)
0.975186 0.221389i \(-0.0710590\pi\)
\(200\) 0 0
\(201\) 2.32288 + 4.02334i 0.163843 + 0.283784i
\(202\) 0 0
\(203\) −5.39426 + 3.11438i −0.378603 + 0.218587i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −3.15731 1.82288i −0.219448 0.126699i
\(208\) 0 0
\(209\) 8.35425 0.577875
\(210\) 0 0
\(211\) −12.5830 −0.866250 −0.433125 0.901334i \(-0.642589\pi\)
−0.433125 + 0.901334i \(0.642589\pi\)
\(212\) 0 0
\(213\) −7.23499 4.17712i −0.495733 0.286212i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 16.6458i 1.12999i
\(218\) 0 0
\(219\) −6.61438 11.4564i −0.446958 0.774154i
\(220\) 0 0
\(221\) −4.82288 + 8.35347i −0.324422 + 0.561915i
\(222\) 0 0
\(223\) 13.8745i 0.929106i −0.885545 0.464553i \(-0.846215\pi\)
0.885545 0.464553i \(-0.153785\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.4311 + 7.17712i 0.825084 + 0.476362i 0.852167 0.523271i \(-0.175288\pi\)
−0.0270825 + 0.999633i \(0.508622\pi\)
\(228\) 0 0
\(229\) −3.50000 6.06218i −0.231287 0.400600i 0.726900 0.686743i \(-0.240960\pi\)
−0.958187 + 0.286143i \(0.907627\pi\)
\(230\) 0 0
\(231\) 4.82288 8.35347i 0.317322 0.549618i
\(232\) 0 0
\(233\) 9.27383 5.35425i 0.607549 0.350768i −0.164457 0.986384i \(-0.552587\pi\)
0.772006 + 0.635616i \(0.219254\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 8.29150i 0.538591i
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) 11.6458 20.1710i 0.750169 1.29933i −0.197572 0.980288i \(-0.563306\pi\)
0.947741 0.319042i \(-0.103361\pi\)
\(242\) 0 0
\(243\) 0.866025 0.500000i 0.0555556 0.0320750i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 5.25049 3.03137i 0.334081 0.192882i
\(248\) 0 0
\(249\) 2.46863 4.27579i 0.156443 0.270967i
\(250\) 0 0
\(251\) 4.93725 0.311637 0.155818 0.987786i \(-0.450199\pi\)
0.155818 + 0.987786i \(0.450199\pi\)
\(252\) 0 0
\(253\) 13.2915i 0.835630i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 27.2973 15.7601i 1.70276 0.983090i 0.759819 0.650135i \(-0.225288\pi\)
0.942943 0.332955i \(-0.108046\pi\)
\(258\) 0 0
\(259\) 6.14575 10.6448i 0.381878 0.661433i
\(260\) 0 0
\(261\) −1.17712 2.03884i −0.0728622 0.126201i
\(262\) 0 0
\(263\) −21.1808 12.2288i −1.30607 0.754057i −0.324629 0.945842i \(-0.605239\pi\)
−0.981437 + 0.191784i \(0.938573\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 12.2288i 0.748388i
\(268\) 0 0
\(269\) −1.93725 + 3.35542i −0.118116 + 0.204584i −0.919021 0.394208i \(-0.871019\pi\)
0.800905 + 0.598792i \(0.204352\pi\)
\(270\) 0 0
\(271\) −14.2915 24.7536i −0.868147 1.50367i −0.863888 0.503684i \(-0.831978\pi\)
−0.00425882 0.999991i \(-0.501356\pi\)
\(272\) 0 0
\(273\) 7.00000i 0.423659i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −13.2972 7.67712i −0.798949 0.461274i 0.0441542 0.999025i \(-0.485941\pi\)
−0.843104 + 0.537751i \(0.819274\pi\)
\(278\) 0 0
\(279\) −6.29150 −0.376662
\(280\) 0 0
\(281\) 24.0000 1.43172 0.715860 0.698244i \(-0.246035\pi\)
0.715860 + 0.698244i \(0.246035\pi\)
\(282\) 0 0
\(283\) 8.10102 + 4.67712i 0.481555 + 0.278026i 0.721064 0.692868i \(-0.243653\pi\)
−0.239509 + 0.970894i \(0.576986\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −25.0604 + 14.4686i −1.47927 + 0.854056i
\(288\) 0 0
\(289\) −1.85425 3.21165i −0.109073 0.188921i
\(290\) 0 0
\(291\) −4.00000 + 6.92820i −0.234484 + 0.406138i
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 3.15731 + 1.82288i 0.183206 + 0.105774i
\(298\) 0 0
\(299\) −4.82288 8.35347i −0.278914 0.483093i
\(300\) 0 0
\(301\) 13.1458 + 22.7691i 0.757709 + 1.31239i
\(302\) 0 0
\(303\) 9.47194 5.46863i 0.544149 0.314164i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 12.0627i 0.688457i −0.938886 0.344229i \(-0.888140\pi\)
0.938886 0.344229i \(-0.111860\pi\)
\(308\) 0 0
\(309\) −14.6458 −0.833168
\(310\) 0 0
\(311\) −4.17712 + 7.23499i −0.236863 + 0.410259i −0.959812 0.280642i \(-0.909453\pi\)
0.722949 + 0.690901i \(0.242786\pi\)
\(312\) 0 0
\(313\) 4.52824 2.61438i 0.255951 0.147773i −0.366535 0.930404i \(-0.619456\pi\)
0.622486 + 0.782631i \(0.286123\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 11.7089 6.76013i 0.657637 0.379687i −0.133739 0.991017i \(-0.542698\pi\)
0.791376 + 0.611330i \(0.209365\pi\)
\(318\) 0 0
\(319\) 4.29150 7.43310i 0.240278 0.416174i
\(320\) 0 0
\(321\) −1.06275 −0.0593167
\(322\) 0 0
\(323\) 8.35425i 0.464843i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 11.2583 6.50000i 0.622587 0.359451i
\(328\) 0 0
\(329\) −7.93725 + 13.7477i −0.437595 + 0.757937i
\(330\) 0 0
\(331\) −11.0830 19.1963i −0.609177 1.05513i −0.991376 0.131046i \(-0.958167\pi\)
0.382199 0.924080i \(-0.375167\pi\)
\(332\) 0 0
\(333\) 4.02334 + 2.32288i 0.220478 + 0.127293i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 6.77124i 0.368853i 0.982846 + 0.184427i \(0.0590428\pi\)
−0.982846 + 0.184427i \(0.940957\pi\)
\(338\) 0 0
\(339\) 3.00000 5.19615i 0.162938 0.282216i
\(340\) 0 0
\(341\) −11.4686 19.8642i −0.621061 1.07571i
\(342\) 0 0
\(343\) 18.5203i 1.00000i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −25.2585 14.5830i −1.35595 0.782857i −0.366873 0.930271i \(-0.619572\pi\)
−0.989075 + 0.147414i \(0.952905\pi\)
\(348\) 0 0
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) 2.64575 0.141220
\(352\) 0 0
\(353\) −19.1420 11.0516i −1.01883 0.588219i −0.105062 0.994466i \(-0.533504\pi\)
−0.913763 + 0.406247i \(0.866837\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −8.35347 4.82288i −0.442112 0.255254i
\(358\) 0 0
\(359\) 13.4059 + 23.2197i 0.707535 + 1.22549i 0.965769 + 0.259405i \(0.0835263\pi\)
−0.258233 + 0.966083i \(0.583140\pi\)
\(360\) 0 0
\(361\) 6.87451 11.9070i 0.361816 0.626684i
\(362\) 0 0
\(363\) 2.29150i 0.120273i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −16.6526 9.61438i −0.869258 0.501866i −0.00215655 0.999998i \(-0.500686\pi\)
−0.867102 + 0.498131i \(0.834020\pi\)
\(368\) 0 0
\(369\) −5.46863 9.47194i −0.284685 0.493089i
\(370\) 0 0
\(371\) 19.2915 1.00156
\(372\) 0 0
\(373\) 13.8021 7.96863i 0.714644 0.412600i −0.0981342 0.995173i \(-0.531287\pi\)
0.812778 + 0.582573i \(0.197954\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.22876i 0.320797i
\(378\) 0 0
\(379\) 26.2915 1.35050 0.675252 0.737587i \(-0.264035\pi\)
0.675252 + 0.737587i \(0.264035\pi\)
\(380\) 0 0
\(381\) 8.96863 15.5341i 0.459477 0.795837i
\(382\) 0 0
\(383\) −14.4700 + 8.35425i −0.739382 + 0.426882i −0.821844 0.569712i \(-0.807055\pi\)
0.0824628 + 0.996594i \(0.473721\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −8.60591 + 4.96863i −0.437463 + 0.252570i
\(388\) 0 0
\(389\) −18.7601 + 32.4935i −0.951176 + 1.64749i −0.208291 + 0.978067i \(0.566790\pi\)
−0.742885 + 0.669419i \(0.766543\pi\)
\(390\) 0 0
\(391\) −13.2915 −0.672180
\(392\) 0 0
\(393\) 10.7085i 0.540172i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 14.4156 8.32288i 0.723500 0.417713i −0.0925393 0.995709i \(-0.529498\pi\)
0.816040 + 0.577996i \(0.196165\pi\)
\(398\) 0 0
\(399\) 3.03137 + 5.25049i 0.151758 + 0.262853i
\(400\) 0 0
\(401\) −9.64575 16.7069i −0.481686 0.834304i 0.518093 0.855324i \(-0.326642\pi\)
−0.999779 + 0.0210198i \(0.993309\pi\)
\(402\) 0 0
\(403\) −14.4156 8.32288i −0.718094 0.414592i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 16.9373i 0.839549i
\(408\) 0 0
\(409\) −18.0830 + 31.3207i −0.894147 + 1.54871i −0.0592904 + 0.998241i \(0.518884\pi\)
−0.834857 + 0.550467i \(0.814450\pi\)
\(410\) 0 0
\(411\) −3.11438 5.39426i −0.153621 0.266079i
\(412\) 0 0
\(413\) 13.0627i 0.642776i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 4.33013 + 2.50000i 0.212047 + 0.122426i
\(418\) 0 0
\(419\) −10.7085 −0.523144 −0.261572 0.965184i \(-0.584241\pi\)
−0.261572 + 0.965184i \(0.584241\pi\)
\(420\) 0 0
\(421\) 1.58301 0.0771510 0.0385755 0.999256i \(-0.487718\pi\)
0.0385755 + 0.999256i \(0.487718\pi\)
\(422\) 0 0
\(423\) −5.19615 3.00000i −0.252646 0.145865i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −18.3303 10.5830i −0.887066 0.512148i
\(428\) 0 0
\(429\) 4.82288 + 8.35347i 0.232851 + 0.403309i
\(430\) 0 0
\(431\) −4.06275 + 7.03688i −0.195696 + 0.338955i −0.947128 0.320855i \(-0.896030\pi\)
0.751433 + 0.659810i \(0.229363\pi\)
\(432\) 0 0
\(433\) 8.64575i 0.415488i 0.978183 + 0.207744i \(0.0666121\pi\)
−0.978183 + 0.207744i \(0.933388\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 7.23499 + 4.17712i 0.346097 + 0.199819i
\(438\) 0 0
\(439\) −2.64575 4.58258i −0.126275 0.218714i 0.795956 0.605355i \(-0.206969\pi\)
−0.922231 + 0.386640i \(0.873635\pi\)
\(440\) 0 0
\(441\) 7.00000 0.333333
\(442\) 0 0
\(443\) −18.9439 + 10.9373i −0.900051 + 0.519645i −0.877217 0.480095i \(-0.840602\pi\)
−0.0228342 + 0.999739i \(0.507269\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 14.5830i 0.689752i
\(448\) 0 0
\(449\) −8.58301 −0.405057 −0.202529 0.979276i \(-0.564916\pi\)
−0.202529 + 0.979276i \(0.564916\pi\)
\(450\) 0 0
\(451\) 19.9373 34.5323i 0.938809 1.62606i
\(452\) 0 0
\(453\) 14.3613 8.29150i 0.674753 0.389569i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 8.10102 4.67712i 0.378950 0.218787i −0.298412 0.954437i \(-0.596457\pi\)
0.677361 + 0.735651i \(0.263123\pi\)
\(458\) 0 0
\(459\) 1.82288 3.15731i 0.0850845 0.147371i
\(460\) 0 0
\(461\) −14.3542 −0.668544 −0.334272 0.942477i \(-0.608490\pi\)
−0.334272 + 0.942477i \(0.608490\pi\)
\(462\) 0 0
\(463\) 26.5203i 1.23250i −0.787550 0.616250i \(-0.788651\pi\)
0.787550 0.616250i \(-0.211349\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −22.2993 + 12.8745i −1.03189 + 0.595761i −0.917525 0.397678i \(-0.869816\pi\)
−0.114363 + 0.993439i \(0.536483\pi\)
\(468\) 0 0
\(469\) −12.2915 −0.567569
\(470\) 0 0
\(471\) −4.64575 8.04668i −0.214065 0.370771i
\(472\) 0 0
\(473\) −31.3750 18.1144i −1.44263 0.832900i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 7.29150i 0.333855i
\(478\) 0 0
\(479\) 7.29150 12.6293i 0.333157 0.577045i −0.649972 0.759958i \(-0.725219\pi\)
0.983129 + 0.182913i \(0.0585527\pi\)
\(480\) 0 0
\(481\) 6.14575 + 10.6448i 0.280222 + 0.485359i
\(482\) 0 0
\(483\) 8.35347 4.82288i 0.380096 0.219448i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 2.29129 + 1.32288i 0.103828 + 0.0599452i 0.551015 0.834495i \(-0.314241\pi\)
−0.447187 + 0.894441i \(0.647574\pi\)
\(488\) 0 0
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) 33.8745 1.52874 0.764368 0.644781i \(-0.223051\pi\)
0.764368 + 0.644781i \(0.223051\pi\)
\(492\) 0 0
\(493\) −7.43310 4.29150i −0.334770 0.193280i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 19.1420 11.0516i 0.858636 0.495733i
\(498\) 0 0
\(499\) −12.7288 22.0469i −0.569817 0.986953i −0.996584 0.0825907i \(-0.973681\pi\)
0.426766 0.904362i \(-0.359653\pi\)
\(500\) 0 0
\(501\) −7.17712 + 12.4311i −0.320650 + 0.555383i
\(502\) 0 0
\(503\) 30.4575i 1.35803i −0.734123 0.679017i \(-0.762406\pi\)
0.734123 0.679017i \(-0.237594\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −5.19615 3.00000i −0.230769 0.133235i
\(508\) 0 0
\(509\) 1.93725 + 3.35542i 0.0858673 + 0.148726i 0.905760 0.423790i \(-0.139301\pi\)
−0.819893 + 0.572517i \(0.805967\pi\)
\(510\) 0 0
\(511\) 35.0000 1.54831
\(512\) 0 0
\(513\) −1.98450 + 1.14575i −0.0876178 + 0.0505862i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 21.8745i 0.962040i
\(518\) 0 0
\(519\) 21.8745 0.960184
\(520\) 0 0
\(521\) 16.9373 29.3362i 0.742035 1.28524i −0.209533 0.977802i \(-0.567194\pi\)
0.951567 0.307440i \(-0.0994723\pi\)
\(522\) 0 0
\(523\) 4.13202 2.38562i 0.180681 0.104316i −0.406932 0.913459i \(-0.633401\pi\)
0.587612 + 0.809143i \(0.300068\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −19.8642 + 11.4686i −0.865300 + 0.499581i
\(528\) 0 0
\(529\) −4.85425 + 8.40781i −0.211054 + 0.365557i
\(530\) 0 0
\(531\) −4.93725 −0.214259
\(532\) 0 0
\(533\) 28.9373i 1.25341i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 17.8254 10.2915i 0.769223 0.444111i
\(538\) 0 0
\(539\) 12.7601 + 22.1012i 0.549618 + 0.951966i
\(540\) 0 0
\(541\) 13.3745 + 23.1653i 0.575015 + 0.995955i 0.996040 + 0.0889062i \(0.0283371\pi\)
−0.421025 + 0.907049i \(0.638330\pi\)
\(542\) 0 0
\(543\) −5.44860 3.14575i −0.233822 0.134997i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 32.7085i 1.39851i 0.714870 + 0.699257i \(0.246486\pi\)
−0.714870 + 0.699257i \(0.753514\pi\)
\(548\) 0 0
\(549\) 4.00000 6.92820i 0.170716 0.295689i
\(550\) 0 0
\(551\) 2.69738 + 4.67201i 0.114912 + 0.199034i
\(552\) 0 0
\(553\) −18.9982 10.9686i −0.807886 0.466433i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −15.5885 9.00000i −0.660504 0.381342i 0.131965 0.991254i \(-0.457871\pi\)
−0.792469 + 0.609912i \(0.791205\pi\)
\(558\) 0 0
\(559\) −26.2915 −1.11201
\(560\) 0 0
\(561\) 13.2915 0.561168
\(562\) 0 0
\(563\) 13.3515 + 7.70850i 0.562699 + 0.324874i 0.754228 0.656613i \(-0.228011\pi\)
−0.191529 + 0.981487i \(0.561345\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 2.64575i 0.111111i
\(568\) 0 0
\(569\) −15.1144 26.1789i −0.633628 1.09748i −0.986804 0.161919i \(-0.948232\pi\)
0.353176 0.935557i \(-0.385102\pi\)
\(570\) 0 0
\(571\) −16.8542 + 29.1924i −0.705328 + 1.22166i 0.261245 + 0.965273i \(0.415867\pi\)
−0.966573 + 0.256392i \(0.917466\pi\)
\(572\) 0 0
\(573\) 23.1660i 0.967774i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −4.74559 2.73987i −0.197562 0.114062i 0.397956 0.917405i \(-0.369720\pi\)
−0.595518 + 0.803342i \(0.703053\pi\)
\(578\) 0 0
\(579\) −11.9686 20.7303i −0.497399 0.861521i
\(580\) 0 0
\(581\) 6.53137 + 11.3127i 0.270967 + 0.469329i
\(582\) 0 0
\(583\) −23.0216 + 13.2915i −0.953456 + 0.550478i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.06275i 0.0438642i −0.999759 0.0219321i \(-0.993018\pi\)
0.999759 0.0219321i \(-0.00698177\pi\)
\(588\) 0 0
\(589\) 14.4170 0.594042
\(590\) 0 0
\(591\) −5.46863 + 9.47194i −0.224949 + 0.389624i
\(592\) 0 0
\(593\) −35.4527 + 20.4686i −1.45587 + 0.840546i −0.998804 0.0488882i \(-0.984432\pi\)
−0.457064 + 0.889434i \(0.651099\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −16.5983 + 9.58301i −0.679321 + 0.392206i
\(598\) 0 0
\(599\) 16.9373 29.3362i 0.692037 1.19864i −0.279132 0.960253i \(-0.590047\pi\)
0.971169 0.238391i \(-0.0766200\pi\)
\(600\) 0 0
\(601\) −16.4170 −0.669663 −0.334832 0.942278i \(-0.608679\pi\)
−0.334832 + 0.942278i \(0.608679\pi\)
\(602\) 0 0
\(603\) 4.64575i 0.189190i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 37.4372 21.6144i 1.51953 0.877301i 0.519794 0.854292i \(-0.326009\pi\)
0.999735 0.0230088i \(-0.00732459\pi\)
\(608\) 0 0
\(609\) 6.22876 0.252402
\(610\) 0 0
\(611\) −7.93725 13.7477i −0.321107 0.556174i
\(612\) 0 0
\(613\) −1.00979 0.583005i −0.0407852 0.0235474i 0.479469 0.877559i \(-0.340829\pi\)
−0.520254 + 0.854012i \(0.674163\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.29150i 0.0519939i 0.999662 + 0.0259970i \(0.00827602\pi\)
−0.999662 + 0.0259970i \(0.991724\pi\)
\(618\) 0 0
\(619\) −9.50000 + 16.4545i −0.381837 + 0.661361i −0.991325 0.131434i \(-0.958042\pi\)
0.609488 + 0.792796i \(0.291375\pi\)
\(620\) 0 0
\(621\) 1.82288 + 3.15731i 0.0731495 + 0.126699i
\(622\) 0 0
\(623\) 28.0196 + 16.1771i 1.12258 + 0.648123i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −7.23499 4.17712i −0.288938 0.166818i
\(628\) 0 0
\(629\) 16.9373 0.675333
\(630\) 0 0
\(631\) 27.7490 1.10467 0.552335 0.833622i \(-0.313737\pi\)
0.552335 + 0.833622i \(0.313737\pi\)
\(632\) 0 0
\(633\) 10.8972 + 6.29150i 0.433125 + 0.250065i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 16.0390 + 9.26013i 0.635489 + 0.366900i
\(638\) 0 0
\(639\) 4.17712 + 7.23499i 0.165244 + 0.286212i
\(640\) 0 0
\(641\) −23.0516 + 39.9266i −0.910485 + 1.57701i −0.0971039 + 0.995274i \(0.530958\pi\)
−0.813381 + 0.581732i \(0.802375\pi\)
\(642\) 0 0
\(643\) 25.8118i 1.01792i 0.860791 + 0.508958i \(0.169969\pi\)
−0.860791 + 0.508958i \(0.830031\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 6.11652 + 3.53137i 0.240465 + 0.138833i 0.615390 0.788222i \(-0.288998\pi\)
−0.374925 + 0.927055i \(0.622332\pi\)
\(648\) 0 0
\(649\) −9.00000 15.5885i −0.353281 0.611900i
\(650\) 0 0
\(651\) 8.32288 14.4156i 0.326199 0.564994i
\(652\) 0 0
\(653\) 29.1381 16.8229i 1.14026 0.658330i 0.193766 0.981048i \(-0.437930\pi\)
0.946495 + 0.322718i \(0.104596\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 13.2288i 0.516103i
\(658\) 0 0
\(659\) 24.0000 0.934907 0.467454 0.884018i \(-0.345171\pi\)
0.467454 + 0.884018i \(0.345171\pi\)
\(660\) 0 0
\(661\) 4.56275 7.90291i 0.177470 0.307387i −0.763543 0.645757i \(-0.776542\pi\)
0.941013 + 0.338369i \(0.109875\pi\)
\(662\) 0 0
\(663\) 8.35347 4.82288i 0.324422 0.187305i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 7.43310 4.29150i 0.287811 0.166168i
\(668\) 0 0
\(669\) −6.93725 + 12.0157i −0.268210 + 0.464553i
\(670\) 0 0
\(671\) 29.1660 1.12594
\(672\) 0 0
\(673\) 10.6458i 0.410364i −0.978724 0.205182i \(-0.934221\pi\)
0.978724 0.205182i \(-0.0657786\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 23.9419 13.8229i 0.920163 0.531256i 0.0364758 0.999335i \(-0.488387\pi\)
0.883687 + 0.468078i \(0.155053\pi\)
\(678\) 0 0
\(679\) −10.5830 18.3303i −0.406138 0.703452i
\(680\) 0 0
\(681\) −7.17712 12.4311i −0.275028 0.476362i
\(682\) 0 0
\(683\) 9.47194 + 5.46863i 0.362434 + 0.209251i 0.670148 0.742228i \(-0.266231\pi\)
−0.307714 + 0.951479i \(0.599564\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 7.00000i 0.267067i
\(688\) 0 0
\(689\) −9.64575 + 16.7069i −0.367474 + 0.636483i
\(690\) 0 0
\(691\) −1.20850 2.09318i −0.0459734 0.0796283i 0.842123 0.539285i \(-0.181306\pi\)
−0.888096 + 0.459657i \(0.847972\pi\)
\(692\) 0 0
\(693\) −8.35347 + 4.82288i −0.317322 + 0.183206i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −34.5323 19.9373i −1.30801 0.755177i
\(698\) 0 0
\(699\) −10.7085 −0.405033
\(700\) 0 0
\(701\) 19.0627 0.719990 0.359995 0.932954i \(-0.382778\pi\)
0.359995 + 0.932954i \(0.382778\pi\)
\(702\) 0 0
\(703\) −9.21949 5.32288i −0.347720 0.200756i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 28.9373i 1.08830i
\(708\) 0 0
\(709\) −0.708497 1.22715i −0.0266082 0.0460867i 0.852415 0.522866i \(-0.175137\pi\)
−0.879023 + 0.476780i \(0.841804\pi\)
\(710\) 0 0
\(711\) 4.14575 7.18065i 0.155478 0.269295i
\(712\) 0 0
\(713\) 22.9373i 0.859007i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 5.19615 + 3.00000i 0.194054 + 0.112037i
\(718\) 0 0
\(719\) −12.6458 21.9031i −0.471607 0.816847i 0.527865 0.849328i \(-0.322993\pi\)
−0.999472 + 0.0324808i \(0.989659\pi\)
\(720\) 0 0
\(721\) 19.3745 33.5576i 0.721544 1.24975i
\(722\) 0 0
\(723\) −20.1710 + 11.6458i −0.750169 + 0.433110i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 13.8118i 0.512250i −0.966644 0.256125i \(-0.917554\pi\)
0.966644 0.256125i \(-0.0824459\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −18.1144 + 31.3750i −0.669984 + 1.16045i
\(732\) 0 0
\(733\) −16.6526 + 9.61438i −0.615078 + 0.355115i −0.774950 0.632023i \(-0.782225\pi\)
0.159873 + 0.987138i \(0.448892\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 14.6681 8.46863i 0.540306 0.311946i
\(738\) 0 0
\(739\) 14.5000 25.1147i 0.533391 0.923861i −0.465848 0.884865i \(-0.654251\pi\)
0.999239 0.0389959i \(-0.0124159\pi\)
\(740\) 0 0
\(741\) −6.06275 −0.222721
\(742\) 0 0
\(743\) 28.9373i 1.06160i −0.847496 0.530802i \(-0.821891\pi\)
0.847496 0.530802i \(-0.178109\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −4.27579 + 2.46863i −0.156443 + 0.0903223i
\(748\) 0 0
\(749\) 1.40588 2.43506i 0.0513698 0.0889751i
\(750\) 0 0
\(751\) 9.08301 + 15.7322i 0.331444 + 0.574077i 0.982795 0.184699i \(-0.0591310\pi\)
−0.651352 + 0.758776i \(0.725798\pi\)
\(752\) 0 0
\(753\) −4.27579 2.46863i −0.155818 0.0899618i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 51.1660i 1.85966i 0.367989 + 0.929830i \(0.380046\pi\)
−0.367989 + 0.929830i \(0.619954\pi\)
\(758\) 0 0
\(759\) −6.64575 + 11.5108i −0.241225 + 0.417815i
\(760\) 0 0
\(761\) −23.0516 39.9266i −0.835621 1.44734i −0.893524 0.449016i \(-0.851775\pi\)
0.0579028 0.998322i \(-0.481559\pi\)
\(762\) 0 0
\(763\) 34.3948i 1.24517i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −11.3127 6.53137i −0.408477 0.235834i
\(768\) 0 0
\(769\) 14.2915 0.515365 0.257682 0.966230i \(-0.417041\pi\)
0.257682 + 0.966230i \(0.417041\pi\)
\(770\) 0 0
\(771\) −31.5203 −1.13517
\(772\) 0 0
\(773\) 35.8489 + 20.6974i 1.28940 + 0.744433i 0.978547 0.206026i \(-0.0660531\pi\)
0.310850 + 0.950459i \(0.399386\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −10.6448 + 6.14575i −0.381878 + 0.220478i
\(778\) 0 0
\(779\) 12.5314 + 21.7050i 0.448983 + 0.777661i
\(780\) 0 0
\(781\) −15.2288 + 26.3770i −0.544928 + 0.943843i
\(782\) 0 0
\(783\) 2.35425i 0.0841340i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −22.4080 12.9373i −0.798758 0.461163i 0.0442785 0.999019i \(-0.485901\pi\)
−0.843037 + 0.537856i \(0.819234\pi\)
\(788\) 0 0
\(789\) 12.2288 + 21.1808i 0.435355 + 0.754057i
\(790\) 0 0
\(791\) 7.93725 + 13.7477i 0.282216 + 0.488813i
\(792\) 0 0
\(793\) 18.3303 10.5830i 0.650928 0.375814i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 10.9373i 0.387417i 0.981059 + 0.193709i \(0.0620517\pi\)
−0.981059 + 0.193709i \(0.937948\pi\)
\(798\) 0 0
\(799\) −21.8745 −0.773864
\(800\) 0 0
\(801\) −6.11438 + 10.5904i −0.216041 + 0.374194i
\(802\) 0 0
\(803\) −41.7673 + 24.1144i −1.47394 + 0.850978i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 3.35542 1.93725i 0.118116 0.0681946i
\(808\) 0 0
\(809\) −11.5830 + 20.0624i −0.407237 + 0.705355i −0.994579 0.103984i \(-0.966841\pi\)
0.587342 + 0.809339i \(0.300174\pi\)
\(810\) 0 0
\(811\) 6.70850 0.235567 0.117784 0.993039i \(-0.462421\pi\)
0.117784 + 0.993039i \(0.462421\pi\)
\(812\) 0 0
\(813\) 28.5830i 1.00245i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 19.7205 11.3856i 0.689932 0.398332i
\(818\) 0 0
\(819\) −3.50000 + 6.06218i −0.122300 + 0.211830i
\(820\) 0 0
\(821\) −18.9889 32.8897i −0.662717 1.14786i −0.979899 0.199494i \(-0.936070\pi\)
0.317182 0.948365i \(-0.397263\pi\)
\(822\) 0 0
\(823\) 29.8411 + 17.2288i 1.04019 + 0.600557i 0.919888 0.392180i \(-0.128279\pi\)
0.120306 + 0.992737i \(0.461612\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 30.4575i 1.05911i 0.848275 + 0.529556i \(0.177641\pi\)
−0.848275 + 0.529556i \(0.822359\pi\)
\(828\) 0 0
\(829\) 15.1458 26.2332i 0.526034 0.911117i −0.473506 0.880790i \(-0.657012\pi\)
0.999540 0.0303267i \(-0.00965476\pi\)
\(830\) 0 0
\(831\) 7.67712 + 13.2972i 0.266316 + 0.461274i
\(832\) 0 0
\(833\) 22.1012 12.7601i 0.765761 0.442112i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 5.44860 + 3.14575i 0.188331 + 0.108733i
\(838\) 0 0
\(839\) −21.6458 −0.747294 −0.373647 0.927571i \(-0.621893\pi\)
−0.373647 + 0.927571i \(0.621893\pi\)
\(840\) 0 0
\(841\) −23.4575 −0.808880
\(842\) 0 0
\(843\) −20.7846 12.0000i −0.715860 0.413302i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −5.25049 3.03137i −0.180409 0.104159i
\(848\) 0 0
\(849\) −4.67712 8.10102i −0.160518 0.278026i
\(850\) 0 0
\(851\) −8.46863 + 14.6681i −0.290301 + 0.502816i
\(852\) 0 0
\(853\) 16.7712i 0.574236i 0.957895 + 0.287118i \(0.0926973\pi\)
−0.957895 + 0.287118i \(0.907303\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −7.23499 4.17712i −0.247143 0.142688i 0.371313 0.928508i \(-0.378908\pi\)
−0.618455 + 0.785820i \(0.712241\pi\)
\(858\) 0 0
\(859\) 13.2288 + 22.9129i 0.451359 + 0.781777i 0.998471 0.0552825i \(-0.0176059\pi\)
−0.547111 + 0.837060i \(0.684273\pi\)
\(860\) 0 0
\(861\) 28.9373 0.986179
\(862\) 0 0
\(863\) 15.9847 9.22876i 0.544125 0.314151i −0.202624 0.979257i \(-0.564947\pi\)
0.746749 + 0.665106i \(0.231614\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 3.70850i 0.125947i
\(868\) 0 0
\(869\) 30.2288 1.02544
\(870\) 0 0
\(871\) 6.14575 10.6448i 0.208241 0.360684i
\(872\) 0 0
\(873\) 6.92820 4.00000i 0.234484 0.135379i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −12.1244 + 7.00000i −0.409410 + 0.236373i −0.690536 0.723298i \(-0.742625\pi\)
0.281126 + 0.959671i \(0.409292\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −41.1660 −1.38692 −0.693459 0.720496i \(-0.743914\pi\)
−0.693459 + 0.720496i \(0.743914\pi\)
\(882\) 0 0
\(883\) 25.8118i 0.868635i 0.900760 + 0.434317i \(0.143010\pi\)
−0.900760 + 0.434317i \(0.856990\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 12.4311 7.17712i 0.417397 0.240984i −0.276566 0.960995i \(-0.589196\pi\)
0.693963 + 0.720011i \(0.255863\pi\)
\(888\) 0 0
\(889\) 23.7288 + 41.0994i 0.795837 + 1.37843i
\(890\) 0 0
\(891\) −1.82288 3.15731i −0.0610686 0.105774i
\(892\) 0 0
\(893\) 11.9070 + 6.87451i 0.398452 + 0.230047i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 9.64575i 0.322062i
\(898\) 0 0
\(899\) 7.40588 12.8274i 0.247000 0.427816i
\(900\) 0 0
\(901\) 13.2915 + 23.0216i 0.442804 + 0.766959i
\(902\) 0 0
\(903\) 26.2915i 0.874926i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −13.2972 7.67712i −0.441525 0.254915i 0.262719 0.964872i \(-0.415381\pi\)
−0.704244 + 0.709958i \(0.748714\pi\)
\(908\) 0 0
\(909\) −10.9373 −0.362766
\(910\) 0 0
\(911\) 31.5203 1.04431 0.522156 0.852850i \(-0.325128\pi\)
0.522156 + 0.852850i \(0.325128\pi\)
\(912\) 0 0
\(913\) −15.5885 9.00000i −0.515903 0.297857i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −24.5362 14.1660i −0.810258 0.467803i
\(918\) 0 0
\(919\) 15.7915 + 27.3517i 0.520914 + 0.902249i 0.999704 + 0.0243197i \(0.00774197\pi\)
−0.478791 + 0.877929i \(0.658925\pi\)
\(920\) 0 0
\(921\) −6.03137 + 10.4466i −0.198740 + 0.344229i
\(922\) 0 0
\(923\) 22.1033i 0.727538i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 12.6836 + 7.32288i 0.416584 + 0.240515i
\(928\) 0 0
\(929\) −0.760130 1.31658i −0.0249390 0.0431957i 0.853287 0.521442i \(-0.174606\pi\)
−0.878226 + 0.478247i \(0.841273\pi\)
\(930\) 0 0
\(931\) −16.0405 −0.525707
\(932\) 0 0
\(933\) 7.23499 4.17712i 0.236863 0.136753i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 50.9778i 1.66537i 0.553746 + 0.832686i \(0.313198\pi\)
−0.553746 + 0.832686i \(0.686802\pi\)
\(938\) 0 0
\(939\) −5.22876 −0.170634
\(940\) 0 0
\(941\) −2.46863 + 4.27579i −0.0804749 + 0.139387i −0.903454 0.428685i \(-0.858977\pi\)
0.822979 + 0.568072i \(0.192310\pi\)
\(942\) 0 0
\(943\) 34.5323 19.9373i 1.12453 0.649246i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 39.5304 22.8229i 1.28456 0.741644i 0.306885 0.951746i \(-0.400713\pi\)
0.977679 + 0.210103i \(0.0673798\pi\)
\(948\) 0 0
\(949\) −17.5000 + 30.3109i −0.568074 + 0.983933i
\(950\) 0 0
\(951\) −13.5203 −0.438424
\(952\) 0 0
\(953\) 0.457513i 0.0148203i 0.999973 + 0.00741015i \(0.00235875\pi\)
−0.999973 + 0.00741015i \(0.997641\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −7.43310 + 4.29150i −0.240278 + 0.138725i
\(958\) 0 0
\(959\) 16.4797 0.532159
\(960\) 0 0
\(961\) −4.29150 7.43310i −0.138436 0.239777i
\(962\) 0 0
\(963\) 0.920365 + 0.531373i 0.0296584 + 0.0171233i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 40.3948i 1.29901i −0.760358 0.649504i \(-0.774977\pi\)
0.760358 0.649504i \(-0.225023\pi\)
\(968\) 0 0
\(969\) −4.17712 + 7.23499i −0.134189 + 0.232421i
\(970\) 0 0
\(971\) −26.5830 46.0431i −0.853089 1.47759i −0.878406 0.477914i \(-0.841393\pi\)
0.0253172 0.999679i \(-0.491940\pi\)
\(972\) 0 0
\(973\) −11.4564 + 6.61438i −0.367277 + 0.212047i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −35.4527 20.4686i −1.13423 0.654849i −0.189237 0.981932i \(-0.560601\pi\)
−0.944996 + 0.327082i \(0.893935\pi\)
\(978\) 0 0
\(979\) −44.5830 −1.42488
\(980\) 0 0
\(981\) −13.0000 −0.415058
\(982\) 0 0
\(983\) 47.3597 + 27.3431i 1.51054 + 0.872111i 0.999924 + 0.0123014i \(0.00391575\pi\)
0.510615 + 0.859809i \(0.329418\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 13.7477 7.93725i 0.437595 0.252646i
\(988\) 0 0
\(989\) −18.1144 31.3750i −0.576004 0.997668i
\(990\) 0 0
\(991\) −16.8542 + 29.1924i −0.535393 + 0.927328i 0.463751 + 0.885965i \(0.346503\pi\)
−0.999144 + 0.0413622i \(0.986830\pi\)
\(992\) 0 0
\(993\) 22.1660i 0.703417i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 14.5243 + 8.38562i 0.459990 + 0.265575i 0.712040 0.702139i \(-0.247771\pi\)
−0.252050 + 0.967714i \(0.581105\pi\)
\(998\) 0 0
\(999\) −2.32288 4.02334i −0.0734925 0.127293i
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 2100.2.bc.e.949.2 8
5.2 odd 4 420.2.q.c.361.2 yes 4
5.3 odd 4 2100.2.q.h.1201.1 4
5.4 even 2 inner 2100.2.bc.e.949.3 8
7.2 even 3 inner 2100.2.bc.e.1549.3 8
15.2 even 4 1260.2.s.f.361.2 4
20.7 even 4 1680.2.bg.q.1201.1 4
35.2 odd 12 420.2.q.c.121.2 4
35.9 even 6 inner 2100.2.bc.e.1549.2 8
35.12 even 12 2940.2.q.t.961.1 4
35.17 even 12 2940.2.a.m.1.2 2
35.23 odd 12 2100.2.q.h.1801.1 4
35.27 even 4 2940.2.q.t.361.1 4
35.32 odd 12 2940.2.a.s.1.2 2
105.2 even 12 1260.2.s.f.541.2 4
105.17 odd 12 8820.2.a.bj.1.1 2
105.32 even 12 8820.2.a.be.1.1 2
140.107 even 12 1680.2.bg.q.961.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.q.c.121.2 4 35.2 odd 12
420.2.q.c.361.2 yes 4 5.2 odd 4
1260.2.s.f.361.2 4 15.2 even 4
1260.2.s.f.541.2 4 105.2 even 12
1680.2.bg.q.961.1 4 140.107 even 12
1680.2.bg.q.1201.1 4 20.7 even 4
2100.2.q.h.1201.1 4 5.3 odd 4
2100.2.q.h.1801.1 4 35.23 odd 12
2100.2.bc.e.949.2 8 1.1 even 1 trivial
2100.2.bc.e.949.3 8 5.4 even 2 inner
2100.2.bc.e.1549.2 8 35.9 even 6 inner
2100.2.bc.e.1549.3 8 7.2 even 3 inner
2940.2.a.m.1.2 2 35.17 even 12
2940.2.a.s.1.2 2 35.32 odd 12
2940.2.q.t.361.1 4 35.27 even 4
2940.2.q.t.961.1 4 35.12 even 12
8820.2.a.be.1.1 2 105.32 even 12
8820.2.a.bj.1.1 2 105.17 odd 12