Properties

Label 315.2.p.d.307.1
Level $315$
Weight $2$
Character 315.307
Analytic conductor $2.515$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [315,2,Mod(118,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.118");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 315.p (of order \(4\), 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: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.40960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 307.1
Root \(1.14412 + 1.14412i\) of defining polynomial
Character \(\chi\) \(=\) 315.307
Dual form 315.2.p.d.118.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.707107 - 0.707107i) q^{2} -1.00000i q^{4} -2.23607i q^{5} +(0.581139 - 2.58114i) q^{7} +(-2.12132 + 2.12132i) q^{8} +O(q^{10})\) \(q+(-0.707107 - 0.707107i) q^{2} -1.00000i q^{4} -2.23607i q^{5} +(0.581139 - 2.58114i) q^{7} +(-2.12132 + 2.12132i) q^{8} +(-1.58114 + 1.58114i) q^{10} -4.24264 q^{11} +(3.16228 + 3.16228i) q^{13} +(-2.23607 + 1.41421i) q^{14} +1.00000 q^{16} +(2.23607 - 2.23607i) q^{17} -3.16228 q^{19} -2.23607 q^{20} +(3.00000 + 3.00000i) q^{22} +(-1.41421 + 1.41421i) q^{23} -5.00000 q^{25} -4.47214i q^{26} +(-2.58114 - 0.581139i) q^{28} -5.65685i q^{29} -9.48683i q^{31} +(3.53553 + 3.53553i) q^{32} -3.16228 q^{34} +(-5.77160 - 1.29947i) q^{35} +(1.00000 + 1.00000i) q^{37} +(2.23607 + 2.23607i) q^{38} +(4.74342 + 4.74342i) q^{40} +8.94427i q^{41} +(-2.00000 + 2.00000i) q^{43} +4.24264i q^{44} +2.00000 q^{46} +(8.94427 - 8.94427i) q^{47} +(-6.32456 - 3.00000i) q^{49} +(3.53553 + 3.53553i) q^{50} +(3.16228 - 3.16228i) q^{52} +(7.07107 - 7.07107i) q^{53} +9.48683i q^{55} +(4.24264 + 6.70820i) q^{56} +(-4.00000 + 4.00000i) q^{58} +(-6.70820 + 6.70820i) q^{62} -7.00000i q^{64} +(7.07107 - 7.07107i) q^{65} +(4.00000 + 4.00000i) q^{67} +(-2.23607 - 2.23607i) q^{68} +(3.16228 + 5.00000i) q^{70} +4.24264 q^{71} +(3.16228 + 3.16228i) q^{73} -1.41421i q^{74} +3.16228i q^{76} +(-2.46556 + 10.9508i) q^{77} -12.0000i q^{79} -2.23607i q^{80} +(6.32456 - 6.32456i) q^{82} +(4.47214 + 4.47214i) q^{83} +(-5.00000 - 5.00000i) q^{85} +2.82843 q^{86} +(9.00000 - 9.00000i) q^{88} +13.4164 q^{89} +(10.0000 - 6.32456i) q^{91} +(1.41421 + 1.41421i) q^{92} -12.6491 q^{94} +7.07107i q^{95} +(3.16228 - 3.16228i) q^{97} +(2.35082 + 6.59346i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{7} + 8 q^{16} + 24 q^{22} - 40 q^{25} - 8 q^{28} + 8 q^{37} - 16 q^{43} + 16 q^{46} - 32 q^{58} + 32 q^{67} - 40 q^{85} + 72 q^{88} + 80 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.707107 0.707107i −0.500000 0.500000i 0.411438 0.911438i \(-0.365027\pi\)
−0.911438 + 0.411438i \(0.865027\pi\)
\(3\) 0 0
\(4\) 1.00000i 0.500000i
\(5\) 2.23607i 1.00000i
\(6\) 0 0
\(7\) 0.581139 2.58114i 0.219650 0.975579i
\(8\) −2.12132 + 2.12132i −0.750000 + 0.750000i
\(9\) 0 0
\(10\) −1.58114 + 1.58114i −0.500000 + 0.500000i
\(11\) −4.24264 −1.27920 −0.639602 0.768706i \(-0.720901\pi\)
−0.639602 + 0.768706i \(0.720901\pi\)
\(12\) 0 0
\(13\) 3.16228 + 3.16228i 0.877058 + 0.877058i 0.993229 0.116171i \(-0.0370621\pi\)
−0.116171 + 0.993229i \(0.537062\pi\)
\(14\) −2.23607 + 1.41421i −0.597614 + 0.377964i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.23607 2.23607i 0.542326 0.542326i −0.381884 0.924210i \(-0.624725\pi\)
0.924210 + 0.381884i \(0.124725\pi\)
\(18\) 0 0
\(19\) −3.16228 −0.725476 −0.362738 0.931891i \(-0.618158\pi\)
−0.362738 + 0.931891i \(0.618158\pi\)
\(20\) −2.23607 −0.500000
\(21\) 0 0
\(22\) 3.00000 + 3.00000i 0.639602 + 0.639602i
\(23\) −1.41421 + 1.41421i −0.294884 + 0.294884i −0.839006 0.544122i \(-0.816863\pi\)
0.544122 + 0.839006i \(0.316863\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 4.47214i 0.877058i
\(27\) 0 0
\(28\) −2.58114 0.581139i −0.487789 0.109825i
\(29\) 5.65685i 1.05045i −0.850963 0.525226i \(-0.823981\pi\)
0.850963 0.525226i \(-0.176019\pi\)
\(30\) 0 0
\(31\) 9.48683i 1.70389i −0.523635 0.851943i \(-0.675424\pi\)
0.523635 0.851943i \(-0.324576\pi\)
\(32\) 3.53553 + 3.53553i 0.625000 + 0.625000i
\(33\) 0 0
\(34\) −3.16228 −0.542326
\(35\) −5.77160 1.29947i −0.975579 0.219650i
\(36\) 0 0
\(37\) 1.00000 + 1.00000i 0.164399 + 0.164399i 0.784512 0.620113i \(-0.212913\pi\)
−0.620113 + 0.784512i \(0.712913\pi\)
\(38\) 2.23607 + 2.23607i 0.362738 + 0.362738i
\(39\) 0 0
\(40\) 4.74342 + 4.74342i 0.750000 + 0.750000i
\(41\) 8.94427i 1.39686i 0.715678 + 0.698430i \(0.246118\pi\)
−0.715678 + 0.698430i \(0.753882\pi\)
\(42\) 0 0
\(43\) −2.00000 + 2.00000i −0.304997 + 0.304997i −0.842965 0.537968i \(-0.819192\pi\)
0.537968 + 0.842965i \(0.319192\pi\)
\(44\) 4.24264i 0.639602i
\(45\) 0 0
\(46\) 2.00000 0.294884
\(47\) 8.94427 8.94427i 1.30466 1.30466i 0.379440 0.925216i \(-0.376117\pi\)
0.925216 0.379440i \(-0.123883\pi\)
\(48\) 0 0
\(49\) −6.32456 3.00000i −0.903508 0.428571i
\(50\) 3.53553 + 3.53553i 0.500000 + 0.500000i
\(51\) 0 0
\(52\) 3.16228 3.16228i 0.438529 0.438529i
\(53\) 7.07107 7.07107i 0.971286 0.971286i −0.0283132 0.999599i \(-0.509014\pi\)
0.999599 + 0.0283132i \(0.00901359\pi\)
\(54\) 0 0
\(55\) 9.48683i 1.27920i
\(56\) 4.24264 + 6.70820i 0.566947 + 0.896421i
\(57\) 0 0
\(58\) −4.00000 + 4.00000i −0.525226 + 0.525226i
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) −6.70820 + 6.70820i −0.851943 + 0.851943i
\(63\) 0 0
\(64\) 7.00000i 0.875000i
\(65\) 7.07107 7.07107i 0.877058 0.877058i
\(66\) 0 0
\(67\) 4.00000 + 4.00000i 0.488678 + 0.488678i 0.907889 0.419211i \(-0.137693\pi\)
−0.419211 + 0.907889i \(0.637693\pi\)
\(68\) −2.23607 2.23607i −0.271163 0.271163i
\(69\) 0 0
\(70\) 3.16228 + 5.00000i 0.377964 + 0.597614i
\(71\) 4.24264 0.503509 0.251754 0.967791i \(-0.418992\pi\)
0.251754 + 0.967791i \(0.418992\pi\)
\(72\) 0 0
\(73\) 3.16228 + 3.16228i 0.370117 + 0.370117i 0.867520 0.497403i \(-0.165713\pi\)
−0.497403 + 0.867520i \(0.665713\pi\)
\(74\) 1.41421i 0.164399i
\(75\) 0 0
\(76\) 3.16228i 0.362738i
\(77\) −2.46556 + 10.9508i −0.280977 + 1.24796i
\(78\) 0 0
\(79\) 12.0000i 1.35011i −0.737769 0.675053i \(-0.764121\pi\)
0.737769 0.675053i \(-0.235879\pi\)
\(80\) 2.23607i 0.250000i
\(81\) 0 0
\(82\) 6.32456 6.32456i 0.698430 0.698430i
\(83\) 4.47214 + 4.47214i 0.490881 + 0.490881i 0.908584 0.417703i \(-0.137165\pi\)
−0.417703 + 0.908584i \(0.637165\pi\)
\(84\) 0 0
\(85\) −5.00000 5.00000i −0.542326 0.542326i
\(86\) 2.82843 0.304997
\(87\) 0 0
\(88\) 9.00000 9.00000i 0.959403 0.959403i
\(89\) 13.4164 1.42214 0.711068 0.703123i \(-0.248212\pi\)
0.711068 + 0.703123i \(0.248212\pi\)
\(90\) 0 0
\(91\) 10.0000 6.32456i 1.04828 0.662994i
\(92\) 1.41421 + 1.41421i 0.147442 + 0.147442i
\(93\) 0 0
\(94\) −12.6491 −1.30466
\(95\) 7.07107i 0.725476i
\(96\) 0 0
\(97\) 3.16228 3.16228i 0.321081 0.321081i −0.528101 0.849182i \(-0.677096\pi\)
0.849182 + 0.528101i \(0.177096\pi\)
\(98\) 2.35082 + 6.59346i 0.237468 + 0.666040i
\(99\) 0 0
\(100\) 5.00000i 0.500000i
\(101\) 4.47214i 0.444994i −0.974933 0.222497i \(-0.928579\pi\)
0.974933 0.222497i \(-0.0714208\pi\)
\(102\) 0 0
\(103\) 3.16228 + 3.16228i 0.311588 + 0.311588i 0.845525 0.533936i \(-0.179288\pi\)
−0.533936 + 0.845525i \(0.679288\pi\)
\(104\) −13.4164 −1.31559
\(105\) 0 0
\(106\) −10.0000 −0.971286
\(107\) −7.07107 7.07107i −0.683586 0.683586i 0.277220 0.960806i \(-0.410587\pi\)
−0.960806 + 0.277220i \(0.910587\pi\)
\(108\) 0 0
\(109\) 12.0000i 1.14939i 0.818367 + 0.574696i \(0.194880\pi\)
−0.818367 + 0.574696i \(0.805120\pi\)
\(110\) 6.70820 6.70820i 0.639602 0.639602i
\(111\) 0 0
\(112\) 0.581139 2.58114i 0.0549125 0.243895i
\(113\) −1.41421 + 1.41421i −0.133038 + 0.133038i −0.770490 0.637452i \(-0.779988\pi\)
0.637452 + 0.770490i \(0.279988\pi\)
\(114\) 0 0
\(115\) 3.16228 + 3.16228i 0.294884 + 0.294884i
\(116\) −5.65685 −0.525226
\(117\) 0 0
\(118\) 0 0
\(119\) −4.47214 7.07107i −0.409960 0.648204i
\(120\) 0 0
\(121\) 7.00000 0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) −9.48683 −0.851943
\(125\) 11.1803i 1.00000i
\(126\) 0 0
\(127\) 4.00000 + 4.00000i 0.354943 + 0.354943i 0.861945 0.507002i \(-0.169246\pi\)
−0.507002 + 0.861945i \(0.669246\pi\)
\(128\) 2.12132 2.12132i 0.187500 0.187500i
\(129\) 0 0
\(130\) −10.0000 −0.877058
\(131\) 17.8885i 1.56293i −0.623949 0.781465i \(-0.714473\pi\)
0.623949 0.781465i \(-0.285527\pi\)
\(132\) 0 0
\(133\) −1.83772 + 8.16228i −0.159351 + 0.707759i
\(134\) 5.65685i 0.488678i
\(135\) 0 0
\(136\) 9.48683i 0.813489i
\(137\) −7.07107 7.07107i −0.604122 0.604122i 0.337282 0.941404i \(-0.390493\pi\)
−0.941404 + 0.337282i \(0.890493\pi\)
\(138\) 0 0
\(139\) −3.16228 −0.268221 −0.134110 0.990966i \(-0.542818\pi\)
−0.134110 + 0.990966i \(0.542818\pi\)
\(140\) −1.29947 + 5.77160i −0.109825 + 0.487789i
\(141\) 0 0
\(142\) −3.00000 3.00000i −0.251754 0.251754i
\(143\) −13.4164 13.4164i −1.12194 1.12194i
\(144\) 0 0
\(145\) −12.6491 −1.05045
\(146\) 4.47214i 0.370117i
\(147\) 0 0
\(148\) 1.00000 1.00000i 0.0821995 0.0821995i
\(149\) 19.7990i 1.62200i 0.585049 + 0.810998i \(0.301075\pi\)
−0.585049 + 0.810998i \(0.698925\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 6.70820 6.70820i 0.544107 0.544107i
\(153\) 0 0
\(154\) 9.48683 6.00000i 0.764471 0.483494i
\(155\) −21.2132 −1.70389
\(156\) 0 0
\(157\) 3.16228 3.16228i 0.252377 0.252377i −0.569567 0.821945i \(-0.692889\pi\)
0.821945 + 0.569567i \(0.192889\pi\)
\(158\) −8.48528 + 8.48528i −0.675053 + 0.675053i
\(159\) 0 0
\(160\) 7.90569 7.90569i 0.625000 0.625000i
\(161\) 2.82843 + 4.47214i 0.222911 + 0.352454i
\(162\) 0 0
\(163\) −14.0000 + 14.0000i −1.09656 + 1.09656i −0.101755 + 0.994809i \(0.532446\pi\)
−0.994809 + 0.101755i \(0.967554\pi\)
\(164\) 8.94427 0.698430
\(165\) 0 0
\(166\) 6.32456i 0.490881i
\(167\) −4.47214 + 4.47214i −0.346064 + 0.346064i −0.858641 0.512577i \(-0.828691\pi\)
0.512577 + 0.858641i \(0.328691\pi\)
\(168\) 0 0
\(169\) 7.00000i 0.538462i
\(170\) 7.07107i 0.542326i
\(171\) 0 0
\(172\) 2.00000 + 2.00000i 0.152499 + 0.152499i
\(173\) −2.23607 2.23607i −0.170005 0.170005i 0.616976 0.786982i \(-0.288357\pi\)
−0.786982 + 0.616976i \(0.788357\pi\)
\(174\) 0 0
\(175\) −2.90569 + 12.9057i −0.219650 + 0.975579i
\(176\) −4.24264 −0.319801
\(177\) 0 0
\(178\) −9.48683 9.48683i −0.711068 0.711068i
\(179\) 7.07107i 0.528516i 0.964452 + 0.264258i \(0.0851271\pi\)
−0.964452 + 0.264258i \(0.914873\pi\)
\(180\) 0 0
\(181\) 18.9737i 1.41030i −0.709057 0.705151i \(-0.750879\pi\)
0.709057 0.705151i \(-0.249121\pi\)
\(182\) −11.5432 2.59893i −0.855639 0.192646i
\(183\) 0 0
\(184\) 6.00000i 0.442326i
\(185\) 2.23607 2.23607i 0.164399 0.164399i
\(186\) 0 0
\(187\) −9.48683 + 9.48683i −0.693746 + 0.693746i
\(188\) −8.94427 8.94427i −0.652328 0.652328i
\(189\) 0 0
\(190\) 5.00000 5.00000i 0.362738 0.362738i
\(191\) −21.2132 −1.53493 −0.767467 0.641089i \(-0.778483\pi\)
−0.767467 + 0.641089i \(0.778483\pi\)
\(192\) 0 0
\(193\) 13.0000 13.0000i 0.935760 0.935760i −0.0622972 0.998058i \(-0.519843\pi\)
0.998058 + 0.0622972i \(0.0198427\pi\)
\(194\) −4.47214 −0.321081
\(195\) 0 0
\(196\) −3.00000 + 6.32456i −0.214286 + 0.451754i
\(197\) 9.89949 + 9.89949i 0.705310 + 0.705310i 0.965545 0.260235i \(-0.0838002\pi\)
−0.260235 + 0.965545i \(0.583800\pi\)
\(198\) 0 0
\(199\) 15.8114 1.12084 0.560420 0.828209i \(-0.310640\pi\)
0.560420 + 0.828209i \(0.310640\pi\)
\(200\) 10.6066 10.6066i 0.750000 0.750000i
\(201\) 0 0
\(202\) −3.16228 + 3.16228i −0.222497 + 0.222497i
\(203\) −14.6011 3.28742i −1.02480 0.230731i
\(204\) 0 0
\(205\) 20.0000 1.39686
\(206\) 4.47214i 0.311588i
\(207\) 0 0
\(208\) 3.16228 + 3.16228i 0.219265 + 0.219265i
\(209\) 13.4164 0.928032
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) −7.07107 7.07107i −0.485643 0.485643i
\(213\) 0 0
\(214\) 10.0000i 0.683586i
\(215\) 4.47214 + 4.47214i 0.304997 + 0.304997i
\(216\) 0 0
\(217\) −24.4868 5.51317i −1.66227 0.374258i
\(218\) 8.48528 8.48528i 0.574696 0.574696i
\(219\) 0 0
\(220\) 9.48683 0.639602
\(221\) 14.1421 0.951303
\(222\) 0 0
\(223\) −6.32456 6.32456i −0.423524 0.423524i 0.462891 0.886415i \(-0.346812\pi\)
−0.886415 + 0.462891i \(0.846812\pi\)
\(224\) 11.1803 7.07107i 0.747018 0.472456i
\(225\) 0 0
\(226\) 2.00000 0.133038
\(227\) −4.47214 + 4.47214i −0.296826 + 0.296826i −0.839769 0.542943i \(-0.817310\pi\)
0.542943 + 0.839769i \(0.317310\pi\)
\(228\) 0 0
\(229\) 25.2982 1.67175 0.835877 0.548917i \(-0.184960\pi\)
0.835877 + 0.548917i \(0.184960\pi\)
\(230\) 4.47214i 0.294884i
\(231\) 0 0
\(232\) 12.0000 + 12.0000i 0.787839 + 0.787839i
\(233\) 15.5563 15.5563i 1.01913 1.01913i 0.0193169 0.999813i \(-0.493851\pi\)
0.999813 0.0193169i \(-0.00614915\pi\)
\(234\) 0 0
\(235\) −20.0000 20.0000i −1.30466 1.30466i
\(236\) 0 0
\(237\) 0 0
\(238\) −1.83772 + 8.16228i −0.119122 + 0.529082i
\(239\) 7.07107i 0.457389i 0.973498 + 0.228695i \(0.0734457\pi\)
−0.973498 + 0.228695i \(0.926554\pi\)
\(240\) 0 0
\(241\) 18.9737i 1.22220i 0.791553 + 0.611101i \(0.209273\pi\)
−0.791553 + 0.611101i \(0.790727\pi\)
\(242\) −4.94975 4.94975i −0.318182 0.318182i
\(243\) 0 0
\(244\) 0 0
\(245\) −6.70820 + 14.1421i −0.428571 + 0.903508i
\(246\) 0 0
\(247\) −10.0000 10.0000i −0.636285 0.636285i
\(248\) 20.1246 + 20.1246i 1.27791 + 1.27791i
\(249\) 0 0
\(250\) 7.90569 7.90569i 0.500000 0.500000i
\(251\) 8.94427i 0.564557i 0.959332 + 0.282279i \(0.0910903\pi\)
−0.959332 + 0.282279i \(0.908910\pi\)
\(252\) 0 0
\(253\) 6.00000 6.00000i 0.377217 0.377217i
\(254\) 5.65685i 0.354943i
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) −11.1803 + 11.1803i −0.697410 + 0.697410i −0.963851 0.266441i \(-0.914152\pi\)
0.266441 + 0.963851i \(0.414152\pi\)
\(258\) 0 0
\(259\) 3.16228 2.00000i 0.196494 0.124274i
\(260\) −7.07107 7.07107i −0.438529 0.438529i
\(261\) 0 0
\(262\) −12.6491 + 12.6491i −0.781465 + 0.781465i
\(263\) −5.65685 + 5.65685i −0.348817 + 0.348817i −0.859669 0.510852i \(-0.829330\pi\)
0.510852 + 0.859669i \(0.329330\pi\)
\(264\) 0 0
\(265\) −15.8114 15.8114i −0.971286 0.971286i
\(266\) 7.07107 4.47214i 0.433555 0.274204i
\(267\) 0 0
\(268\) 4.00000 4.00000i 0.244339 0.244339i
\(269\) −13.4164 −0.818013 −0.409006 0.912532i \(-0.634125\pi\)
−0.409006 + 0.912532i \(0.634125\pi\)
\(270\) 0 0
\(271\) 9.48683i 0.576284i −0.957588 0.288142i \(-0.906962\pi\)
0.957588 0.288142i \(-0.0930375\pi\)
\(272\) 2.23607 2.23607i 0.135582 0.135582i
\(273\) 0 0
\(274\) 10.0000i 0.604122i
\(275\) 21.2132 1.27920
\(276\) 0 0
\(277\) 7.00000 + 7.00000i 0.420589 + 0.420589i 0.885407 0.464817i \(-0.153880\pi\)
−0.464817 + 0.885407i \(0.653880\pi\)
\(278\) 2.23607 + 2.23607i 0.134110 + 0.134110i
\(279\) 0 0
\(280\) 15.0000 9.48683i 0.896421 0.566947i
\(281\) 25.4558 1.51857 0.759284 0.650759i \(-0.225549\pi\)
0.759284 + 0.650759i \(0.225549\pi\)
\(282\) 0 0
\(283\) 12.6491 + 12.6491i 0.751912 + 0.751912i 0.974836 0.222924i \(-0.0715602\pi\)
−0.222924 + 0.974836i \(0.571560\pi\)
\(284\) 4.24264i 0.251754i
\(285\) 0 0
\(286\) 18.9737i 1.12194i
\(287\) 23.0864 + 5.19786i 1.36275 + 0.306820i
\(288\) 0 0
\(289\) 7.00000i 0.411765i
\(290\) 8.94427 + 8.94427i 0.525226 + 0.525226i
\(291\) 0 0
\(292\) 3.16228 3.16228i 0.185058 0.185058i
\(293\) 11.1803 + 11.1803i 0.653162 + 0.653162i 0.953753 0.300591i \(-0.0971838\pi\)
−0.300591 + 0.953753i \(0.597184\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −4.24264 −0.246598
\(297\) 0 0
\(298\) 14.0000 14.0000i 0.810998 0.810998i
\(299\) −8.94427 −0.517261
\(300\) 0 0
\(301\) 4.00000 + 6.32456i 0.230556 + 0.364541i
\(302\) 2.82843 + 2.82843i 0.162758 + 0.162758i
\(303\) 0 0
\(304\) −3.16228 −0.181369
\(305\) 0 0
\(306\) 0 0
\(307\) 3.16228 3.16228i 0.180481 0.180481i −0.611085 0.791565i \(-0.709266\pi\)
0.791565 + 0.611085i \(0.209266\pi\)
\(308\) 10.9508 + 2.46556i 0.623982 + 0.140489i
\(309\) 0 0
\(310\) 15.0000 + 15.0000i 0.851943 + 0.851943i
\(311\) 8.94427i 0.507183i 0.967311 + 0.253592i \(0.0816119\pi\)
−0.967311 + 0.253592i \(0.918388\pi\)
\(312\) 0 0
\(313\) −15.8114 15.8114i −0.893713 0.893713i 0.101158 0.994870i \(-0.467745\pi\)
−0.994870 + 0.101158i \(0.967745\pi\)
\(314\) −4.47214 −0.252377
\(315\) 0 0
\(316\) −12.0000 −0.675053
\(317\) 9.89949 + 9.89949i 0.556011 + 0.556011i 0.928169 0.372158i \(-0.121382\pi\)
−0.372158 + 0.928169i \(0.621382\pi\)
\(318\) 0 0
\(319\) 24.0000i 1.34374i
\(320\) −15.6525 −0.875000
\(321\) 0 0
\(322\) 1.16228 5.16228i 0.0647712 0.287682i
\(323\) −7.07107 + 7.07107i −0.393445 + 0.393445i
\(324\) 0 0
\(325\) −15.8114 15.8114i −0.877058 0.877058i
\(326\) 19.7990 1.09656
\(327\) 0 0
\(328\) −18.9737 18.9737i −1.04765 1.04765i
\(329\) −17.8885 28.2843i −0.986227 1.55936i
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 4.47214 4.47214i 0.245440 0.245440i
\(333\) 0 0
\(334\) 6.32456 0.346064
\(335\) 8.94427 8.94427i 0.488678 0.488678i
\(336\) 0 0
\(337\) −17.0000 17.0000i −0.926049 0.926049i 0.0713988 0.997448i \(-0.477254\pi\)
−0.997448 + 0.0713988i \(0.977254\pi\)
\(338\) 4.94975 4.94975i 0.269231 0.269231i
\(339\) 0 0
\(340\) −5.00000 + 5.00000i −0.271163 + 0.271163i
\(341\) 40.2492i 2.17962i
\(342\) 0 0
\(343\) −11.4189 + 14.5811i −0.616561 + 0.787307i
\(344\) 8.48528i 0.457496i
\(345\) 0 0
\(346\) 3.16228i 0.170005i
\(347\) −2.82843 2.82843i −0.151838 0.151838i 0.627100 0.778938i \(-0.284242\pi\)
−0.778938 + 0.627100i \(0.784242\pi\)
\(348\) 0 0
\(349\) 25.2982 1.35418 0.677091 0.735899i \(-0.263240\pi\)
0.677091 + 0.735899i \(0.263240\pi\)
\(350\) 11.1803 7.07107i 0.597614 0.377964i
\(351\) 0 0
\(352\) −15.0000 15.0000i −0.799503 0.799503i
\(353\) −2.23607 2.23607i −0.119014 0.119014i 0.645091 0.764105i \(-0.276819\pi\)
−0.764105 + 0.645091i \(0.776819\pi\)
\(354\) 0 0
\(355\) 9.48683i 0.503509i
\(356\) 13.4164i 0.711068i
\(357\) 0 0
\(358\) 5.00000 5.00000i 0.264258 0.264258i
\(359\) 15.5563i 0.821033i 0.911853 + 0.410516i \(0.134652\pi\)
−0.911853 + 0.410516i \(0.865348\pi\)
\(360\) 0 0
\(361\) −9.00000 −0.473684
\(362\) −13.4164 + 13.4164i −0.705151 + 0.705151i
\(363\) 0 0
\(364\) −6.32456 10.0000i −0.331497 0.524142i
\(365\) 7.07107 7.07107i 0.370117 0.370117i
\(366\) 0 0
\(367\) −6.32456 + 6.32456i −0.330139 + 0.330139i −0.852639 0.522500i \(-0.824999\pi\)
0.522500 + 0.852639i \(0.324999\pi\)
\(368\) −1.41421 + 1.41421i −0.0737210 + 0.0737210i
\(369\) 0 0
\(370\) −3.16228 −0.164399
\(371\) −14.1421 22.3607i −0.734223 1.16091i
\(372\) 0 0
\(373\) −23.0000 + 23.0000i −1.19089 + 1.19089i −0.214078 + 0.976816i \(0.568675\pi\)
−0.976816 + 0.214078i \(0.931325\pi\)
\(374\) 13.4164 0.693746
\(375\) 0 0
\(376\) 37.9473i 1.95698i
\(377\) 17.8885 17.8885i 0.921307 0.921307i
\(378\) 0 0
\(379\) 12.0000i 0.616399i 0.951322 + 0.308199i \(0.0997264\pi\)
−0.951322 + 0.308199i \(0.900274\pi\)
\(380\) 7.07107 0.362738
\(381\) 0 0
\(382\) 15.0000 + 15.0000i 0.767467 + 0.767467i
\(383\) 4.47214 + 4.47214i 0.228515 + 0.228515i 0.812072 0.583557i \(-0.198339\pi\)
−0.583557 + 0.812072i \(0.698339\pi\)
\(384\) 0 0
\(385\) 24.4868 + 5.51317i 1.24796 + 0.280977i
\(386\) −18.3848 −0.935760
\(387\) 0 0
\(388\) −3.16228 3.16228i −0.160540 0.160540i
\(389\) 19.7990i 1.00385i 0.864912 + 0.501924i \(0.167374\pi\)
−0.864912 + 0.501924i \(0.832626\pi\)
\(390\) 0 0
\(391\) 6.32456i 0.319847i
\(392\) 19.7804 7.05245i 0.999059 0.356202i
\(393\) 0 0
\(394\) 14.0000i 0.705310i
\(395\) −26.8328 −1.35011
\(396\) 0 0
\(397\) −15.8114 + 15.8114i −0.793551 + 0.793551i −0.982070 0.188519i \(-0.939631\pi\)
0.188519 + 0.982070i \(0.439631\pi\)
\(398\) −11.1803 11.1803i −0.560420 0.560420i
\(399\) 0 0
\(400\) −5.00000 −0.250000
\(401\) −16.9706 −0.847469 −0.423735 0.905786i \(-0.639281\pi\)
−0.423735 + 0.905786i \(0.639281\pi\)
\(402\) 0 0
\(403\) 30.0000 30.0000i 1.49441 1.49441i
\(404\) −4.47214 −0.222497
\(405\) 0 0
\(406\) 8.00000 + 12.6491i 0.397033 + 0.627765i
\(407\) −4.24264 4.24264i −0.210300 0.210300i
\(408\) 0 0
\(409\) 25.2982 1.25092 0.625458 0.780258i \(-0.284912\pi\)
0.625458 + 0.780258i \(0.284912\pi\)
\(410\) −14.1421 14.1421i −0.698430 0.698430i
\(411\) 0 0
\(412\) 3.16228 3.16228i 0.155794 0.155794i
\(413\) 0 0
\(414\) 0 0
\(415\) 10.0000 10.0000i 0.490881 0.490881i
\(416\) 22.3607i 1.09632i
\(417\) 0 0
\(418\) −9.48683 9.48683i −0.464016 0.464016i
\(419\) 26.8328 1.31087 0.655434 0.755252i \(-0.272486\pi\)
0.655434 + 0.755252i \(0.272486\pi\)
\(420\) 0 0
\(421\) −16.0000 −0.779792 −0.389896 0.920859i \(-0.627489\pi\)
−0.389896 + 0.920859i \(0.627489\pi\)
\(422\) −5.65685 5.65685i −0.275371 0.275371i
\(423\) 0 0
\(424\) 30.0000i 1.45693i
\(425\) −11.1803 + 11.1803i −0.542326 + 0.542326i
\(426\) 0 0
\(427\) 0 0
\(428\) −7.07107 + 7.07107i −0.341793 + 0.341793i
\(429\) 0 0
\(430\) 6.32456i 0.304997i
\(431\) −21.2132 −1.02180 −0.510902 0.859639i \(-0.670689\pi\)
−0.510902 + 0.859639i \(0.670689\pi\)
\(432\) 0 0
\(433\) 22.1359 + 22.1359i 1.06379 + 1.06379i 0.997822 + 0.0659635i \(0.0210121\pi\)
0.0659635 + 0.997822i \(0.478988\pi\)
\(434\) 13.4164 + 21.2132i 0.644008 + 1.01827i
\(435\) 0 0
\(436\) 12.0000 0.574696
\(437\) 4.47214 4.47214i 0.213931 0.213931i
\(438\) 0 0
\(439\) −22.1359 −1.05649 −0.528245 0.849092i \(-0.677150\pi\)
−0.528245 + 0.849092i \(0.677150\pi\)
\(440\) −20.1246 20.1246i −0.959403 0.959403i
\(441\) 0 0
\(442\) −10.0000 10.0000i −0.475651 0.475651i
\(443\) −14.1421 + 14.1421i −0.671913 + 0.671913i −0.958157 0.286244i \(-0.907593\pi\)
0.286244 + 0.958157i \(0.407593\pi\)
\(444\) 0 0
\(445\) 30.0000i 1.42214i
\(446\) 8.94427i 0.423524i
\(447\) 0 0
\(448\) −18.0680 4.06797i −0.853631 0.192194i
\(449\) 22.6274i 1.06785i −0.845531 0.533927i \(-0.820716\pi\)
0.845531 0.533927i \(-0.179284\pi\)
\(450\) 0 0
\(451\) 37.9473i 1.78687i
\(452\) 1.41421 + 1.41421i 0.0665190 + 0.0665190i
\(453\) 0 0
\(454\) 6.32456 0.296826
\(455\) −14.1421 22.3607i −0.662994 1.04828i
\(456\) 0 0
\(457\) 1.00000 + 1.00000i 0.0467780 + 0.0467780i 0.730109 0.683331i \(-0.239469\pi\)
−0.683331 + 0.730109i \(0.739469\pi\)
\(458\) −17.8885 17.8885i −0.835877 0.835877i
\(459\) 0 0
\(460\) 3.16228 3.16228i 0.147442 0.147442i
\(461\) 17.8885i 0.833153i −0.909101 0.416576i \(-0.863230\pi\)
0.909101 0.416576i \(-0.136770\pi\)
\(462\) 0 0
\(463\) 16.0000 16.0000i 0.743583 0.743583i −0.229683 0.973266i \(-0.573769\pi\)
0.973266 + 0.229683i \(0.0737688\pi\)
\(464\) 5.65685i 0.262613i
\(465\) 0 0
\(466\) −22.0000 −1.01913
\(467\) −4.47214 + 4.47214i −0.206946 + 0.206946i −0.802968 0.596022i \(-0.796747\pi\)
0.596022 + 0.802968i \(0.296747\pi\)
\(468\) 0 0
\(469\) 12.6491 8.00000i 0.584082 0.369406i
\(470\) 28.2843i 1.30466i
\(471\) 0 0
\(472\) 0 0
\(473\) 8.48528 8.48528i 0.390154 0.390154i
\(474\) 0 0
\(475\) 15.8114 0.725476
\(476\) −7.07107 + 4.47214i −0.324102 + 0.204980i
\(477\) 0 0
\(478\) 5.00000 5.00000i 0.228695 0.228695i
\(479\) −26.8328 −1.22602 −0.613011 0.790074i \(-0.710042\pi\)
−0.613011 + 0.790074i \(0.710042\pi\)
\(480\) 0 0
\(481\) 6.32456i 0.288375i
\(482\) 13.4164 13.4164i 0.611101 0.611101i
\(483\) 0 0
\(484\) 7.00000i 0.318182i
\(485\) −7.07107 7.07107i −0.321081 0.321081i
\(486\) 0 0
\(487\) 16.0000 + 16.0000i 0.725029 + 0.725029i 0.969625 0.244596i \(-0.0786553\pi\)
−0.244596 + 0.969625i \(0.578655\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 14.7434 5.25658i 0.666040 0.237468i
\(491\) 4.24264 0.191468 0.0957338 0.995407i \(-0.469480\pi\)
0.0957338 + 0.995407i \(0.469480\pi\)
\(492\) 0 0
\(493\) −12.6491 12.6491i −0.569687 0.569687i
\(494\) 14.1421i 0.636285i
\(495\) 0 0
\(496\) 9.48683i 0.425971i
\(497\) 2.46556 10.9508i 0.110596 0.491213i
\(498\) 0 0
\(499\) 24.0000i 1.07439i −0.843459 0.537194i \(-0.819484\pi\)
0.843459 0.537194i \(-0.180516\pi\)
\(500\) 11.1803 0.500000
\(501\) 0 0
\(502\) 6.32456 6.32456i 0.282279 0.282279i
\(503\) 4.47214 + 4.47214i 0.199403 + 0.199403i 0.799744 0.600341i \(-0.204969\pi\)
−0.600341 + 0.799744i \(0.704969\pi\)
\(504\) 0 0
\(505\) −10.0000 −0.444994
\(506\) −8.48528 −0.377217
\(507\) 0 0
\(508\) 4.00000 4.00000i 0.177471 0.177471i
\(509\) −26.8328 −1.18934 −0.594672 0.803969i \(-0.702718\pi\)
−0.594672 + 0.803969i \(0.702718\pi\)
\(510\) 0 0
\(511\) 10.0000 6.32456i 0.442374 0.279782i
\(512\) 7.77817 + 7.77817i 0.343750 + 0.343750i
\(513\) 0 0
\(514\) 15.8114 0.697410
\(515\) 7.07107 7.07107i 0.311588 0.311588i
\(516\) 0 0
\(517\) −37.9473 + 37.9473i −1.66892 + 1.66892i
\(518\) −3.65028 0.821854i −0.160384 0.0361102i
\(519\) 0 0
\(520\) 30.0000i 1.31559i
\(521\) 31.3050i 1.37149i −0.727840 0.685747i \(-0.759475\pi\)
0.727840 0.685747i \(-0.240525\pi\)
\(522\) 0 0
\(523\) 22.1359 + 22.1359i 0.967937 + 0.967937i 0.999502 0.0315645i \(-0.0100490\pi\)
−0.0315645 + 0.999502i \(0.510049\pi\)
\(524\) −17.8885 −0.781465
\(525\) 0 0
\(526\) 8.00000 0.348817
\(527\) −21.2132 21.2132i −0.924062 0.924062i
\(528\) 0 0
\(529\) 19.0000i 0.826087i
\(530\) 22.3607i 0.971286i
\(531\) 0 0
\(532\) 8.16228 + 1.83772i 0.353880 + 0.0796754i
\(533\) −28.2843 + 28.2843i −1.22513 + 1.22513i
\(534\) 0 0
\(535\) −15.8114 + 15.8114i −0.683586 + 0.683586i
\(536\) −16.9706 −0.733017
\(537\) 0 0
\(538\) 9.48683 + 9.48683i 0.409006 + 0.409006i
\(539\) 26.8328 + 12.7279i 1.15577 + 0.548230i
\(540\) 0 0
\(541\) 14.0000 0.601907 0.300954 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) −6.70820 + 6.70820i −0.288142 + 0.288142i
\(543\) 0 0
\(544\) 15.8114 0.677908
\(545\) 26.8328 1.14939
\(546\) 0 0
\(547\) 4.00000 + 4.00000i 0.171028 + 0.171028i 0.787431 0.616403i \(-0.211411\pi\)
−0.616403 + 0.787431i \(0.711411\pi\)
\(548\) −7.07107 + 7.07107i −0.302061 + 0.302061i
\(549\) 0 0
\(550\) −15.0000 15.0000i −0.639602 0.639602i
\(551\) 17.8885i 0.762078i
\(552\) 0 0
\(553\) −30.9737 6.97367i −1.31713 0.296550i
\(554\) 9.89949i 0.420589i
\(555\) 0 0
\(556\) 3.16228i 0.134110i
\(557\) 18.3848 + 18.3848i 0.778988 + 0.778988i 0.979659 0.200671i \(-0.0643121\pi\)
−0.200671 + 0.979659i \(0.564312\pi\)
\(558\) 0 0
\(559\) −12.6491 −0.535000
\(560\) −5.77160 1.29947i −0.243895 0.0549125i
\(561\) 0 0
\(562\) −18.0000 18.0000i −0.759284 0.759284i
\(563\) 17.8885 + 17.8885i 0.753912 + 0.753912i 0.975207 0.221295i \(-0.0710283\pi\)
−0.221295 + 0.975207i \(0.571028\pi\)
\(564\) 0 0
\(565\) 3.16228 + 3.16228i 0.133038 + 0.133038i
\(566\) 17.8885i 0.751912i
\(567\) 0 0
\(568\) −9.00000 + 9.00000i −0.377632 + 0.377632i
\(569\) 5.65685i 0.237148i −0.992945 0.118574i \(-0.962168\pi\)
0.992945 0.118574i \(-0.0378322\pi\)
\(570\) 0 0
\(571\) −16.0000 −0.669579 −0.334790 0.942293i \(-0.608665\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) −13.4164 + 13.4164i −0.560968 + 0.560968i
\(573\) 0 0
\(574\) −12.6491 20.0000i −0.527964 0.834784i
\(575\) 7.07107 7.07107i 0.294884 0.294884i
\(576\) 0 0
\(577\) 3.16228 3.16228i 0.131647 0.131647i −0.638213 0.769860i \(-0.720326\pi\)
0.769860 + 0.638213i \(0.220326\pi\)
\(578\) 4.94975 4.94975i 0.205882 0.205882i
\(579\) 0 0
\(580\) 12.6491i 0.525226i
\(581\) 14.1421 8.94427i 0.586715 0.371071i
\(582\) 0 0
\(583\) −30.0000 + 30.0000i −1.24247 + 1.24247i
\(584\) −13.4164 −0.555175
\(585\) 0 0
\(586\) 15.8114i 0.653162i
\(587\) 8.94427 8.94427i 0.369170 0.369170i −0.498005 0.867174i \(-0.665934\pi\)
0.867174 + 0.498005i \(0.165934\pi\)
\(588\) 0 0
\(589\) 30.0000i 1.23613i
\(590\) 0 0
\(591\) 0 0
\(592\) 1.00000 + 1.00000i 0.0410997 + 0.0410997i
\(593\) 24.5967 + 24.5967i 1.01007 + 1.01007i 0.999949 + 0.0101186i \(0.00322089\pi\)
0.0101186 + 0.999949i \(0.496779\pi\)
\(594\) 0 0
\(595\) −15.8114 + 10.0000i −0.648204 + 0.409960i
\(596\) 19.7990 0.810998
\(597\) 0 0
\(598\) 6.32456 + 6.32456i 0.258630 + 0.258630i
\(599\) 43.8406i 1.79128i −0.444781 0.895640i \(-0.646718\pi\)
0.444781 0.895640i \(-0.353282\pi\)
\(600\) 0 0
\(601\) 18.9737i 0.773952i 0.922090 + 0.386976i \(0.126480\pi\)
−0.922090 + 0.386976i \(0.873520\pi\)
\(602\) 1.64371 7.30056i 0.0669926 0.297549i
\(603\) 0 0
\(604\) 4.00000i 0.162758i
\(605\) 15.6525i 0.636364i
\(606\) 0 0
\(607\) 31.6228 31.6228i 1.28353 1.28353i 0.344883 0.938646i \(-0.387918\pi\)
0.938646 0.344883i \(-0.112082\pi\)
\(608\) −11.1803 11.1803i −0.453423 0.453423i
\(609\) 0 0
\(610\) 0 0
\(611\) 56.5685 2.28852
\(612\) 0 0
\(613\) 7.00000 7.00000i 0.282727 0.282727i −0.551468 0.834196i \(-0.685932\pi\)
0.834196 + 0.551468i \(0.185932\pi\)
\(614\) −4.47214 −0.180481
\(615\) 0 0
\(616\) −18.0000 28.4605i −0.725241 1.14671i
\(617\) −24.0416 24.0416i −0.967880 0.967880i 0.0316203 0.999500i \(-0.489933\pi\)
−0.999500 + 0.0316203i \(0.989933\pi\)
\(618\) 0 0
\(619\) −41.1096 −1.65233 −0.826167 0.563425i \(-0.809483\pi\)
−0.826167 + 0.563425i \(0.809483\pi\)
\(620\) 21.2132i 0.851943i
\(621\) 0 0
\(622\) 6.32456 6.32456i 0.253592 0.253592i
\(623\) 7.79680 34.6296i 0.312372 1.38741i
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 22.3607i 0.893713i
\(627\) 0 0
\(628\) −3.16228 3.16228i −0.126189 0.126189i
\(629\) 4.47214 0.178316
\(630\) 0 0
\(631\) 44.0000 1.75161 0.875806 0.482663i \(-0.160330\pi\)
0.875806 + 0.482663i \(0.160330\pi\)
\(632\) 25.4558 + 25.4558i 1.01258 + 1.01258i
\(633\) 0 0
\(634\) 14.0000i 0.556011i
\(635\) 8.94427 8.94427i 0.354943 0.354943i
\(636\) 0 0
\(637\) −10.5132 29.4868i −0.416547 1.16831i
\(638\) 16.9706 16.9706i 0.671871 0.671871i
\(639\) 0 0
\(640\) −4.74342 4.74342i −0.187500 0.187500i
\(641\) −25.4558 −1.00545 −0.502723 0.864448i \(-0.667668\pi\)
−0.502723 + 0.864448i \(0.667668\pi\)
\(642\) 0 0
\(643\) 3.16228 + 3.16228i 0.124708 + 0.124708i 0.766706 0.641998i \(-0.221894\pi\)
−0.641998 + 0.766706i \(0.721894\pi\)
\(644\) 4.47214 2.82843i 0.176227 0.111456i
\(645\) 0 0
\(646\) 10.0000 0.393445
\(647\) −31.3050 + 31.3050i −1.23072 + 1.23072i −0.267039 + 0.963686i \(0.586045\pi\)
−0.963686 + 0.267039i \(0.913955\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 22.3607i 0.877058i
\(651\) 0 0
\(652\) 14.0000 + 14.0000i 0.548282 + 0.548282i
\(653\) 7.07107 7.07107i 0.276712 0.276712i −0.555083 0.831795i \(-0.687313\pi\)
0.831795 + 0.555083i \(0.187313\pi\)
\(654\) 0 0
\(655\) −40.0000 −1.56293
\(656\) 8.94427i 0.349215i
\(657\) 0 0
\(658\) −7.35089 + 32.6491i −0.286568 + 1.27279i
\(659\) 7.07107i 0.275450i 0.990471 + 0.137725i \(0.0439790\pi\)
−0.990471 + 0.137725i \(0.956021\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 2.82843 + 2.82843i 0.109930 + 0.109930i
\(663\) 0 0
\(664\) −18.9737 −0.736321
\(665\) 18.2514 + 4.10927i 0.707759 + 0.159351i
\(666\) 0 0
\(667\) 8.00000 + 8.00000i 0.309761 + 0.309761i
\(668\) 4.47214 + 4.47214i 0.173032 + 0.173032i
\(669\) 0 0
\(670\) −12.6491 −0.488678
\(671\) 0 0
\(672\) 0 0
\(673\) 19.0000 19.0000i 0.732396 0.732396i −0.238698 0.971094i \(-0.576721\pi\)
0.971094 + 0.238698i \(0.0767205\pi\)
\(674\) 24.0416i 0.926049i
\(675\) 0 0
\(676\) 7.00000 0.269231
\(677\) −24.5967 + 24.5967i −0.945330 + 0.945330i −0.998581 0.0532513i \(-0.983042\pi\)
0.0532513 + 0.998581i \(0.483042\pi\)
\(678\) 0 0
\(679\) −6.32456 10.0000i −0.242714 0.383765i
\(680\) 21.2132 0.813489
\(681\) 0 0
\(682\) 28.4605 28.4605i 1.08981 1.08981i
\(683\) 19.7990 19.7990i 0.757587 0.757587i −0.218295 0.975883i \(-0.570050\pi\)
0.975883 + 0.218295i \(0.0700496\pi\)
\(684\) 0 0
\(685\) −15.8114 + 15.8114i −0.604122 + 0.604122i
\(686\) 18.3848 2.23607i 0.701934 0.0853735i
\(687\) 0 0
\(688\) −2.00000 + 2.00000i −0.0762493 + 0.0762493i
\(689\) 44.7214 1.70375
\(690\) 0 0
\(691\) 9.48683i 0.360896i −0.983585 0.180448i \(-0.942245\pi\)
0.983585 0.180448i \(-0.0577548\pi\)
\(692\) −2.23607 + 2.23607i −0.0850026 + 0.0850026i
\(693\) 0 0
\(694\) 4.00000i 0.151838i
\(695\) 7.07107i 0.268221i
\(696\) 0 0
\(697\) 20.0000 + 20.0000i 0.757554 + 0.757554i
\(698\) −17.8885 17.8885i −0.677091 0.677091i
\(699\) 0 0
\(700\) 12.9057 + 2.90569i 0.487789 + 0.109825i
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) −3.16228 3.16228i −0.119268 0.119268i
\(704\) 29.6985i 1.11930i
\(705\) 0 0
\(706\) 3.16228i 0.119014i
\(707\) −11.5432 2.59893i −0.434127 0.0977429i
\(708\) 0 0
\(709\) 24.0000i 0.901339i −0.892691 0.450669i \(-0.851185\pi\)
0.892691 0.450669i \(-0.148815\pi\)
\(710\) −6.70820 + 6.70820i −0.251754 + 0.251754i
\(711\) 0 0
\(712\) −28.4605 + 28.4605i −1.06660 + 1.06660i
\(713\) 13.4164 + 13.4164i 0.502448 + 0.502448i
\(714\) 0 0
\(715\) −30.0000 + 30.0000i −1.12194 + 1.12194i
\(716\) 7.07107 0.264258
\(717\) 0 0
\(718\) 11.0000 11.0000i 0.410516 0.410516i
\(719\) 26.8328 1.00070 0.500348 0.865825i \(-0.333206\pi\)
0.500348 + 0.865825i \(0.333206\pi\)
\(720\) 0 0
\(721\) 10.0000 6.32456i 0.372419 0.235539i
\(722\) 6.36396 + 6.36396i 0.236842 + 0.236842i
\(723\) 0 0
\(724\) −18.9737 −0.705151
\(725\) 28.2843i 1.05045i
\(726\) 0 0
\(727\) 31.6228 31.6228i 1.17282 1.17282i 0.191290 0.981533i \(-0.438733\pi\)
0.981533 0.191290i \(-0.0612672\pi\)
\(728\) −7.79680 + 34.6296i −0.288968 + 1.28346i
\(729\) 0 0
\(730\) −10.0000 −0.370117
\(731\) 8.94427i 0.330816i
\(732\) 0 0
\(733\) 22.1359 + 22.1359i 0.817610 + 0.817610i 0.985761 0.168151i \(-0.0537798\pi\)
−0.168151 + 0.985761i \(0.553780\pi\)
\(734\) 8.94427 0.330139
\(735\) 0 0
\(736\) −10.0000 −0.368605
\(737\) −16.9706 16.9706i −0.625119 0.625119i
\(738\) 0 0
\(739\) 12.0000i 0.441427i −0.975339 0.220714i \(-0.929161\pi\)
0.975339 0.220714i \(-0.0708386\pi\)
\(740\) −2.23607 2.23607i −0.0821995 0.0821995i
\(741\) 0 0
\(742\) −5.81139 + 25.8114i −0.213343 + 0.947566i
\(743\) 28.2843 28.2843i 1.03765 1.03765i 0.0383863 0.999263i \(-0.487778\pi\)
0.999263 0.0383863i \(-0.0122217\pi\)
\(744\) 0 0
\(745\) 44.2719 1.62200
\(746\) 32.5269 1.19089
\(747\) 0 0
\(748\) 9.48683 + 9.48683i 0.346873 + 0.346873i
\(749\) −22.3607 + 14.1421i −0.817041 + 0.516742i
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 8.94427 8.94427i 0.326164 0.326164i
\(753\) 0 0
\(754\) −25.2982 −0.921307
\(755\) 8.94427i 0.325515i
\(756\) 0 0
\(757\) 1.00000 + 1.00000i 0.0363456 + 0.0363456i 0.725046 0.688700i \(-0.241818\pi\)
−0.688700 + 0.725046i \(0.741818\pi\)
\(758\) 8.48528 8.48528i 0.308199 0.308199i
\(759\) 0 0
\(760\) −15.0000 15.0000i −0.544107 0.544107i
\(761\) 17.8885i 0.648459i −0.945978 0.324230i \(-0.894895\pi\)
0.945978 0.324230i \(-0.105105\pi\)
\(762\) 0 0
\(763\) 30.9737 + 6.97367i 1.12132 + 0.252464i
\(764\) 21.2132i 0.767467i
\(765\) 0 0
\(766\) 6.32456i 0.228515i
\(767\) 0 0
\(768\) 0 0
\(769\) 6.32456 0.228069 0.114035 0.993477i \(-0.463623\pi\)
0.114035 + 0.993477i \(0.463623\pi\)
\(770\) −13.4164 21.2132i −0.483494 0.764471i
\(771\) 0 0
\(772\) −13.0000 13.0000i −0.467880 0.467880i
\(773\) −29.0689 29.0689i −1.04554 1.04554i −0.998913 0.0466225i \(-0.985154\pi\)
−0.0466225 0.998913i \(-0.514846\pi\)
\(774\) 0 0
\(775\) 47.4342i 1.70389i
\(776\) 13.4164i 0.481621i
\(777\) 0 0
\(778\) 14.0000 14.0000i 0.501924 0.501924i
\(779\) 28.2843i 1.01339i
\(780\) 0 0
\(781\) −18.0000 −0.644091
\(782\) 4.47214 4.47214i 0.159923 0.159923i
\(783\) 0 0
\(784\) −6.32456 3.00000i −0.225877 0.107143i
\(785\) −7.07107 7.07107i −0.252377 0.252377i
\(786\) 0 0
\(787\) 3.16228 3.16228i 0.112723 0.112723i −0.648495 0.761219i \(-0.724601\pi\)
0.761219 + 0.648495i \(0.224601\pi\)
\(788\) 9.89949 9.89949i 0.352655 0.352655i
\(789\) 0 0
\(790\) 18.9737 + 18.9737i 0.675053 + 0.675053i
\(791\) 2.82843 + 4.47214i 0.100567 + 0.159011i
\(792\) 0 0
\(793\) 0 0
\(794\) 22.3607 0.793551
\(795\) 0 0
\(796\) 15.8114i 0.560420i
\(797\) −11.1803 + 11.1803i −0.396028 + 0.396028i −0.876829 0.480802i \(-0.840346\pi\)
0.480802 + 0.876829i \(0.340346\pi\)
\(798\) 0 0
\(799\) 40.0000i 1.41510i
\(800\) −17.6777 17.6777i −0.625000 0.625000i
\(801\) 0 0
\(802\) 12.0000 + 12.0000i 0.423735 + 0.423735i
\(803\) −13.4164 13.4164i −0.473455 0.473455i
\(804\) 0 0
\(805\) 10.0000 6.32456i 0.352454 0.222911i
\(806\) −42.4264 −1.49441
\(807\) 0 0
\(808\) 9.48683 + 9.48683i 0.333746 + 0.333746i
\(809\) 48.0833i 1.69052i −0.534357 0.845259i \(-0.679446\pi\)
0.534357 0.845259i \(-0.320554\pi\)
\(810\) 0 0
\(811\) 9.48683i 0.333128i −0.986031 0.166564i \(-0.946733\pi\)
0.986031 0.166564i \(-0.0532672\pi\)
\(812\) −3.28742 + 14.6011i −0.115366 + 0.512399i
\(813\) 0 0
\(814\) 6.00000i 0.210300i
\(815\) 31.3050 + 31.3050i 1.09656 + 1.09656i
\(816\) 0 0
\(817\) 6.32456 6.32456i 0.221268 0.221268i
\(818\) −17.8885 17.8885i −0.625458 0.625458i
\(819\) 0 0
\(820\) 20.0000i 0.698430i
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) −8.00000 + 8.00000i −0.278862 + 0.278862i −0.832655 0.553792i \(-0.813180\pi\)
0.553792 + 0.832655i \(0.313180\pi\)
\(824\) −13.4164 −0.467383
\(825\) 0 0
\(826\) 0 0
\(827\) 9.89949 + 9.89949i 0.344239 + 0.344239i 0.857958 0.513719i \(-0.171733\pi\)
−0.513719 + 0.857958i \(0.671733\pi\)
\(828\) 0 0
\(829\) −31.6228 −1.09830 −0.549152 0.835722i \(-0.685049\pi\)
−0.549152 + 0.835722i \(0.685049\pi\)
\(830\) −14.1421 −0.490881
\(831\) 0 0
\(832\) 22.1359 22.1359i 0.767426 0.767426i
\(833\) −20.8503 + 7.43393i −0.722421 + 0.257570i
\(834\) 0 0
\(835\) 10.0000 + 10.0000i 0.346064 + 0.346064i
\(836\) 13.4164i 0.464016i
\(837\) 0 0
\(838\) −18.9737 18.9737i −0.655434 0.655434i
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −3.00000 −0.103448
\(842\) 11.3137 + 11.3137i 0.389896 + 0.389896i
\(843\) 0 0
\(844\) 8.00000i 0.275371i
\(845\) 15.6525 0.538462
\(846\) 0 0
\(847\) 4.06797 18.0680i 0.139777 0.620823i
\(848\) 7.07107 7.07107i 0.242821 0.242821i
\(849\) 0 0
\(850\) 15.8114 0.542326
\(851\) −2.82843 −0.0969572
\(852\) 0 0
\(853\) −34.7851 34.7851i −1.19102 1.19102i −0.976782 0.214236i \(-0.931274\pi\)
−0.214236 0.976782i \(-0.568726\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 30.0000 1.02538
\(857\) 29.0689 29.0689i 0.992974 0.992974i −0.00700134 0.999975i \(-0.502229\pi\)
0.999975 + 0.00700134i \(0.00222861\pi\)
\(858\) 0 0
\(859\) −22.1359 −0.755269 −0.377634 0.925955i \(-0.623262\pi\)
−0.377634 + 0.925955i \(0.623262\pi\)
\(860\) 4.47214 4.47214i 0.152499 0.152499i
\(861\) 0 0
\(862\) 15.0000 + 15.0000i 0.510902 + 0.510902i
\(863\) −26.8701 + 26.8701i −0.914667 + 0.914667i −0.996635 0.0819676i \(-0.973880\pi\)
0.0819676 + 0.996635i \(0.473880\pi\)
\(864\) 0 0
\(865\) −5.00000 + 5.00000i −0.170005 + 0.170005i
\(866\) 31.3050i 1.06379i
\(867\) 0 0
\(868\) −5.51317 + 24.4868i −0.187129 + 0.831137i
\(869\) 50.9117i 1.72706i
\(870\) 0 0
\(871\) 25.2982i 0.857198i
\(872\) −25.4558 25.4558i −0.862044 0.862044i
\(873\) 0 0
\(874\) −6.32456 −0.213931
\(875\) 28.8580 + 6.49733i 0.975579 + 0.219650i
\(876\) 0 0
\(877\) 7.00000 + 7.00000i 0.236373 + 0.236373i 0.815347 0.578973i \(-0.196546\pi\)
−0.578973 + 0.815347i \(0.696546\pi\)
\(878\) 15.6525 + 15.6525i 0.528245 + 0.528245i
\(879\) 0 0
\(880\) 9.48683i 0.319801i
\(881\) 4.47214i 0.150670i −0.997158 0.0753350i \(-0.975997\pi\)
0.997158 0.0753350i \(-0.0240026\pi\)
\(882\) 0 0
\(883\) −32.0000 + 32.0000i −1.07689 + 1.07689i −0.0800988 + 0.996787i \(0.525524\pi\)
−0.996787 + 0.0800988i \(0.974476\pi\)
\(884\) 14.1421i 0.475651i
\(885\) 0 0
\(886\) 20.0000 0.671913
\(887\) 35.7771 35.7771i 1.20128 1.20128i 0.227499 0.973778i \(-0.426945\pi\)
0.973778 0.227499i \(-0.0730547\pi\)
\(888\) 0 0
\(889\) 12.6491 8.00000i 0.424238 0.268311i
\(890\) −21.2132 + 21.2132i −0.711068 + 0.711068i
\(891\) 0 0
\(892\) −6.32456 + 6.32456i −0.211762 + 0.211762i
\(893\) −28.2843 + 28.2843i −0.946497 + 0.946497i
\(894\) 0 0
\(895\) 15.8114 0.528516
\(896\) −4.24264 6.70820i −0.141737 0.224105i
\(897\) 0 0
\(898\) −16.0000 + 16.0000i −0.533927 + 0.533927i
\(899\) −53.6656 −1.78985
\(900\) 0 0
\(901\) 31.6228i 1.05351i
\(902\) −26.8328 + 26.8328i −0.893435 + 0.893435i
\(903\) 0 0
\(904\) 6.00000i 0.199557i
\(905\) −42.4264 −1.41030
\(906\) 0 0
\(907\) 28.0000 + 28.0000i 0.929725 + 0.929725i 0.997688 0.0679631i \(-0.0216500\pi\)
−0.0679631 + 0.997688i \(0.521650\pi\)
\(908\) 4.47214 + 4.47214i 0.148413 + 0.148413i
\(909\) 0 0
\(910\) −5.81139 + 25.8114i −0.192646 + 0.855639i
\(911\) −4.24264 −0.140565 −0.0702825 0.997527i \(-0.522390\pi\)
−0.0702825 + 0.997527i \(0.522390\pi\)
\(912\) 0 0
\(913\) −18.9737 18.9737i −0.627937 0.627937i
\(914\) 1.41421i 0.0467780i
\(915\) 0 0
\(916\) 25.2982i 0.835877i
\(917\) −46.1728 10.3957i −1.52476 0.343297i
\(918\) 0 0
\(919\) 48.0000i 1.58337i −0.610927 0.791687i \(-0.709203\pi\)
0.610927 0.791687i \(-0.290797\pi\)
\(920\) −13.4164 −0.442326
\(921\) 0 0
\(922\) −12.6491 + 12.6491i −0.416576 + 0.416576i
\(923\) 13.4164 + 13.4164i 0.441606 + 0.441606i
\(924\) 0 0
\(925\) −5.00000 5.00000i −0.164399 0.164399i
\(926\) −22.6274 −0.743583
\(927\) 0 0
\(928\) 20.0000 20.0000i 0.656532 0.656532i
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 20.0000 + 9.48683i 0.655474 + 0.310918i
\(932\) −15.5563 15.5563i −0.509565 0.509565i
\(933\) 0 0
\(934\) 6.32456 0.206946
\(935\) 21.2132 + 21.2132i 0.693746 + 0.693746i
\(936\) 0 0
\(937\) 3.16228 3.16228i 0.103307 0.103307i −0.653564 0.756871i \(-0.726727\pi\)
0.756871 + 0.653564i \(0.226727\pi\)
\(938\) −14.6011 3.28742i −0.476744 0.107338i
\(939\) 0 0
\(940\) −20.0000 + 20.0000i −0.652328 + 0.652328i
\(941\) 17.8885i 0.583150i −0.956548 0.291575i \(-0.905821\pi\)
0.956548 0.291575i \(-0.0941793\pi\)
\(942\) 0 0
\(943\) −12.6491 12.6491i −0.411912 0.411912i
\(944\) 0 0
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) 14.1421 + 14.1421i 0.459558 + 0.459558i 0.898510 0.438953i \(-0.144650\pi\)
−0.438953 + 0.898510i \(0.644650\pi\)
\(948\) 0 0
\(949\) 20.0000i 0.649227i
\(950\) −11.1803 11.1803i −0.362738 0.362738i
\(951\) 0 0
\(952\) 24.4868 + 5.51317i 0.793623 + 0.178683i
\(953\) −1.41421 + 1.41421i −0.0458109 + 0.0458109i −0.729641 0.683830i \(-0.760313\pi\)
0.683830 + 0.729641i \(0.260313\pi\)
\(954\) 0 0
\(955\) 47.4342i 1.53493i
\(956\) 7.07107 0.228695
\(957\) 0 0
\(958\) 18.9737 + 18.9737i 0.613011 + 0.613011i
\(959\) −22.3607 + 14.1421i −0.722064 + 0.456673i
\(960\) 0 0
\(961\) −59.0000 −1.90323
\(962\) 4.47214 4.47214i 0.144187 0.144187i
\(963\) 0 0
\(964\) 18.9737 0.611101
\(965\) −29.0689 29.0689i −0.935760 0.935760i
\(966\) 0 0
\(967\) −38.0000 38.0000i −1.22200 1.22200i −0.966921 0.255077i \(-0.917899\pi\)
−0.255077 0.966921i \(-0.582101\pi\)
\(968\) −14.8492 + 14.8492i −0.477273 + 0.477273i
\(969\) 0 0
\(970\) 10.0000i 0.321081i
\(971\) 44.7214i 1.43518i −0.696467 0.717588i \(-0.745246\pi\)
0.696467 0.717588i \(-0.254754\pi\)
\(972\) 0 0
\(973\) −1.83772 + 8.16228i −0.0589147 + 0.261671i
\(974\) 22.6274i 0.725029i
\(975\) 0 0
\(976\) 0 0
\(977\) 9.89949 + 9.89949i 0.316713 + 0.316713i 0.847503 0.530790i \(-0.178105\pi\)
−0.530790 + 0.847503i \(0.678105\pi\)
\(978\) 0 0
\(979\) −56.9210 −1.81920
\(980\) 14.1421 + 6.70820i 0.451754 + 0.214286i
\(981\) 0 0
\(982\) −3.00000 3.00000i −0.0957338 0.0957338i
\(983\) 17.8885 + 17.8885i 0.570556 + 0.570556i 0.932284 0.361728i \(-0.117813\pi\)
−0.361728 + 0.932284i \(0.617813\pi\)
\(984\) 0 0
\(985\) 22.1359 22.1359i 0.705310 0.705310i
\(986\) 17.8885i 0.569687i
\(987\) 0 0
\(988\) −10.0000 + 10.0000i −0.318142 + 0.318142i
\(989\) 5.65685i 0.179878i
\(990\) 0 0
\(991\) −4.00000 −0.127064 −0.0635321 0.997980i \(-0.520237\pi\)
−0.0635321 + 0.997980i \(0.520237\pi\)
\(992\) 33.5410 33.5410i 1.06493 1.06493i
\(993\) 0 0
\(994\) −9.48683 + 6.00000i −0.300904 + 0.190308i
\(995\) 35.3553i 1.12084i
\(996\) 0 0
\(997\) −15.8114 + 15.8114i −0.500752 + 0.500752i −0.911671 0.410920i \(-0.865208\pi\)
0.410920 + 0.911671i \(0.365208\pi\)
\(998\) −16.9706 + 16.9706i −0.537194 + 0.537194i
\(999\) 0 0
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 315.2.p.d.307.1 yes 8
3.2 odd 2 inner 315.2.p.d.307.4 yes 8
5.3 odd 4 inner 315.2.p.d.118.1 8
7.6 odd 2 inner 315.2.p.d.307.2 yes 8
15.8 even 4 inner 315.2.p.d.118.4 yes 8
21.20 even 2 inner 315.2.p.d.307.3 yes 8
35.13 even 4 inner 315.2.p.d.118.2 yes 8
105.83 odd 4 inner 315.2.p.d.118.3 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
315.2.p.d.118.1 8 5.3 odd 4 inner
315.2.p.d.118.2 yes 8 35.13 even 4 inner
315.2.p.d.118.3 yes 8 105.83 odd 4 inner
315.2.p.d.118.4 yes 8 15.8 even 4 inner
315.2.p.d.307.1 yes 8 1.1 even 1 trivial
315.2.p.d.307.2 yes 8 7.6 odd 2 inner
315.2.p.d.307.3 yes 8 21.20 even 2 inner
315.2.p.d.307.4 yes 8 3.2 odd 2 inner