Properties

Label 315.2.p.a.118.1
Level $315$
Weight $2$
Character 315.118
Analytic conductor $2.515$
Analytic rank $0$
Dimension $4$
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] = [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: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 118.1
Root \(-1.58114 + 1.58114i\) of defining polynomial
Character \(\chi\) \(=\) 315.118
Dual form 315.2.p.a.307.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.58114 + 1.58114i) q^{2} -3.00000i q^{4} +(-2.00000 - 1.00000i) q^{5} +(2.58114 + 0.581139i) q^{7} +(1.58114 + 1.58114i) q^{8} +O(q^{10})\) \(q+(-1.58114 + 1.58114i) q^{2} -3.00000i q^{4} +(-2.00000 - 1.00000i) q^{5} +(2.58114 + 0.581139i) q^{7} +(1.58114 + 1.58114i) q^{8} +(4.74342 - 1.58114i) q^{10} +3.16228 q^{11} +(-3.16228 + 3.16228i) q^{13} +(-5.00000 + 3.16228i) q^{14} +1.00000 q^{16} +(5.00000 + 5.00000i) q^{17} -3.16228 q^{19} +(-3.00000 + 6.00000i) q^{20} +(-5.00000 + 5.00000i) q^{22} +(3.16228 + 3.16228i) q^{23} +(3.00000 + 4.00000i) q^{25} -10.0000i q^{26} +(1.74342 - 7.74342i) q^{28} -3.16228i q^{31} +(-4.74342 + 4.74342i) q^{32} -15.8114 q^{34} +(-4.58114 - 3.74342i) q^{35} +(-3.00000 + 3.00000i) q^{37} +(5.00000 - 5.00000i) q^{38} +(-1.58114 - 4.74342i) q^{40} +(6.00000 + 6.00000i) q^{43} -9.48683i q^{44} -10.0000 q^{46} +(6.32456 + 3.00000i) q^{49} +(-11.0680 - 1.58114i) q^{50} +(9.48683 + 9.48683i) q^{52} +(-3.16228 - 3.16228i) q^{53} +(-6.32456 - 3.16228i) q^{55} +(3.16228 + 5.00000i) q^{56} +12.6491i q^{61} +(5.00000 + 5.00000i) q^{62} -13.0000i q^{64} +(9.48683 - 3.16228i) q^{65} +(8.00000 - 8.00000i) q^{67} +(15.0000 - 15.0000i) q^{68} +(13.1623 - 1.32456i) q^{70} +9.48683 q^{71} +(9.48683 - 9.48683i) q^{73} -9.48683i q^{74} +9.48683i q^{76} +(8.16228 + 1.83772i) q^{77} -4.00000i q^{79} +(-2.00000 - 1.00000i) q^{80} +(-10.0000 + 10.0000i) q^{83} +(-5.00000 - 15.0000i) q^{85} -18.9737 q^{86} +(5.00000 + 5.00000i) q^{88} +10.0000 q^{89} +(-10.0000 + 6.32456i) q^{91} +(9.48683 - 9.48683i) q^{92} +(6.32456 + 3.16228i) q^{95} +(-3.16228 - 3.16228i) q^{97} +(-14.7434 + 5.25658i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{5} + 4 q^{7} - 20 q^{14} + 4 q^{16} + 20 q^{17} - 12 q^{20} - 20 q^{22} + 12 q^{25} - 12 q^{28} - 12 q^{35} - 12 q^{37} + 20 q^{38} + 24 q^{43} - 40 q^{46} + 20 q^{62} + 32 q^{67} + 60 q^{68} + 40 q^{70} + 20 q^{77} - 8 q^{80} - 40 q^{83} - 20 q^{85} + 20 q^{88} + 40 q^{89} - 40 q^{91} - 40 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.58114 + 1.58114i −1.11803 + 1.11803i −0.126004 + 0.992030i \(0.540215\pi\)
−0.992030 + 0.126004i \(0.959785\pi\)
\(3\) 0 0
\(4\) 3.00000i 1.50000i
\(5\) −2.00000 1.00000i −0.894427 0.447214i
\(6\) 0 0
\(7\) 2.58114 + 0.581139i 0.975579 + 0.219650i
\(8\) 1.58114 + 1.58114i 0.559017 + 0.559017i
\(9\) 0 0
\(10\) 4.74342 1.58114i 1.50000 0.500000i
\(11\) 3.16228 0.953463 0.476731 0.879049i \(-0.341821\pi\)
0.476731 + 0.879049i \(0.341821\pi\)
\(12\) 0 0
\(13\) −3.16228 + 3.16228i −0.877058 + 0.877058i −0.993229 0.116171i \(-0.962938\pi\)
0.116171 + 0.993229i \(0.462938\pi\)
\(14\) −5.00000 + 3.16228i −1.33631 + 0.845154i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 5.00000 + 5.00000i 1.21268 + 1.21268i 0.970143 + 0.242536i \(0.0779791\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −3.16228 −0.725476 −0.362738 0.931891i \(-0.618158\pi\)
−0.362738 + 0.931891i \(0.618158\pi\)
\(20\) −3.00000 + 6.00000i −0.670820 + 1.34164i
\(21\) 0 0
\(22\) −5.00000 + 5.00000i −1.06600 + 1.06600i
\(23\) 3.16228 + 3.16228i 0.659380 + 0.659380i 0.955233 0.295853i \(-0.0956039\pi\)
−0.295853 + 0.955233i \(0.595604\pi\)
\(24\) 0 0
\(25\) 3.00000 + 4.00000i 0.600000 + 0.800000i
\(26\) 10.0000i 1.96116i
\(27\) 0 0
\(28\) 1.74342 7.74342i 0.329475 1.46337i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 3.16228i 0.567962i −0.958830 0.283981i \(-0.908345\pi\)
0.958830 0.283981i \(-0.0916552\pi\)
\(32\) −4.74342 + 4.74342i −0.838525 + 0.838525i
\(33\) 0 0
\(34\) −15.8114 −2.71163
\(35\) −4.58114 3.74342i −0.774354 0.632753i
\(36\) 0 0
\(37\) −3.00000 + 3.00000i −0.493197 + 0.493197i −0.909312 0.416115i \(-0.863391\pi\)
0.416115 + 0.909312i \(0.363391\pi\)
\(38\) 5.00000 5.00000i 0.811107 0.811107i
\(39\) 0 0
\(40\) −1.58114 4.74342i −0.250000 0.750000i
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 6.00000 + 6.00000i 0.914991 + 0.914991i 0.996660 0.0816682i \(-0.0260248\pi\)
−0.0816682 + 0.996660i \(0.526025\pi\)
\(44\) 9.48683i 1.43019i
\(45\) 0 0
\(46\) −10.0000 −1.47442
\(47\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(48\) 0 0
\(49\) 6.32456 + 3.00000i 0.903508 + 0.428571i
\(50\) −11.0680 1.58114i −1.56525 0.223607i
\(51\) 0 0
\(52\) 9.48683 + 9.48683i 1.31559 + 1.31559i
\(53\) −3.16228 3.16228i −0.434372 0.434372i 0.455740 0.890113i \(-0.349375\pi\)
−0.890113 + 0.455740i \(0.849375\pi\)
\(54\) 0 0
\(55\) −6.32456 3.16228i −0.852803 0.426401i
\(56\) 3.16228 + 5.00000i 0.422577 + 0.668153i
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 12.6491i 1.61955i 0.586739 + 0.809776i \(0.300412\pi\)
−0.586739 + 0.809776i \(0.699588\pi\)
\(62\) 5.00000 + 5.00000i 0.635001 + 0.635001i
\(63\) 0 0
\(64\) 13.0000i 1.62500i
\(65\) 9.48683 3.16228i 1.17670 0.392232i
\(66\) 0 0
\(67\) 8.00000 8.00000i 0.977356 0.977356i −0.0223937 0.999749i \(-0.507129\pi\)
0.999749 + 0.0223937i \(0.00712872\pi\)
\(68\) 15.0000 15.0000i 1.81902 1.81902i
\(69\) 0 0
\(70\) 13.1623 1.32456i 1.57319 0.158315i
\(71\) 9.48683 1.12588 0.562940 0.826498i \(-0.309670\pi\)
0.562940 + 0.826498i \(0.309670\pi\)
\(72\) 0 0
\(73\) 9.48683 9.48683i 1.11035 1.11035i 0.117247 0.993103i \(-0.462593\pi\)
0.993103 0.117247i \(-0.0374069\pi\)
\(74\) 9.48683i 1.10282i
\(75\) 0 0
\(76\) 9.48683i 1.08821i
\(77\) 8.16228 + 1.83772i 0.930178 + 0.209428i
\(78\) 0 0
\(79\) 4.00000i 0.450035i −0.974355 0.225018i \(-0.927756\pi\)
0.974355 0.225018i \(-0.0722440\pi\)
\(80\) −2.00000 1.00000i −0.223607 0.111803i
\(81\) 0 0
\(82\) 0 0
\(83\) −10.0000 + 10.0000i −1.09764 + 1.09764i −0.102957 + 0.994686i \(0.532830\pi\)
−0.994686 + 0.102957i \(0.967170\pi\)
\(84\) 0 0
\(85\) −5.00000 15.0000i −0.542326 1.62698i
\(86\) −18.9737 −2.04598
\(87\) 0 0
\(88\) 5.00000 + 5.00000i 0.533002 + 0.533002i
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) −10.0000 + 6.32456i −1.04828 + 0.662994i
\(92\) 9.48683 9.48683i 0.989071 0.989071i
\(93\) 0 0
\(94\) 0 0
\(95\) 6.32456 + 3.16228i 0.648886 + 0.324443i
\(96\) 0 0
\(97\) −3.16228 3.16228i −0.321081 0.321081i 0.528101 0.849182i \(-0.322904\pi\)
−0.849182 + 0.528101i \(0.822904\pi\)
\(98\) −14.7434 + 5.25658i −1.48931 + 0.530995i
\(99\) 0 0
\(100\) 12.0000 9.00000i 1.20000 0.900000i
\(101\) 10.0000i 0.995037i −0.867453 0.497519i \(-0.834245\pi\)
0.867453 0.497519i \(-0.165755\pi\)
\(102\) 0 0
\(103\) 3.16228 3.16228i 0.311588 0.311588i −0.533936 0.845525i \(-0.679288\pi\)
0.845525 + 0.533936i \(0.179288\pi\)
\(104\) −10.0000 −0.980581
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) −9.48683 + 9.48683i −0.917127 + 0.917127i −0.996819 0.0796927i \(-0.974606\pi\)
0.0796927 + 0.996819i \(0.474606\pi\)
\(108\) 0 0
\(109\) 4.00000i 0.383131i 0.981480 + 0.191565i \(0.0613564\pi\)
−0.981480 + 0.191565i \(0.938644\pi\)
\(110\) 15.0000 5.00000i 1.43019 0.476731i
\(111\) 0 0
\(112\) 2.58114 + 0.581139i 0.243895 + 0.0549125i
\(113\) −9.48683 9.48683i −0.892446 0.892446i 0.102307 0.994753i \(-0.467378\pi\)
−0.994753 + 0.102307i \(0.967378\pi\)
\(114\) 0 0
\(115\) −3.16228 9.48683i −0.294884 0.884652i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 10.0000 + 15.8114i 0.916698 + 1.44943i
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) −20.0000 20.0000i −1.81071 1.81071i
\(123\) 0 0
\(124\) −9.48683 −0.851943
\(125\) −2.00000 11.0000i −0.178885 0.983870i
\(126\) 0 0
\(127\) −12.0000 + 12.0000i −1.06483 + 1.06483i −0.0670802 + 0.997748i \(0.521368\pi\)
−0.997748 + 0.0670802i \(0.978632\pi\)
\(128\) 11.0680 + 11.0680i 0.978280 + 0.978280i
\(129\) 0 0
\(130\) −10.0000 + 20.0000i −0.877058 + 1.75412i
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) −8.16228 1.83772i −0.707759 0.159351i
\(134\) 25.2982i 2.18543i
\(135\) 0 0
\(136\) 15.8114i 1.35582i
\(137\) 3.16228 3.16228i 0.270172 0.270172i −0.558998 0.829169i \(-0.688814\pi\)
0.829169 + 0.558998i \(0.188814\pi\)
\(138\) 0 0
\(139\) 9.48683 0.804663 0.402331 0.915494i \(-0.368200\pi\)
0.402331 + 0.915494i \(0.368200\pi\)
\(140\) −11.2302 + 13.7434i −0.949129 + 1.16153i
\(141\) 0 0
\(142\) −15.0000 + 15.0000i −1.25877 + 1.25877i
\(143\) −10.0000 + 10.0000i −0.836242 + 0.836242i
\(144\) 0 0
\(145\) 0 0
\(146\) 30.0000i 2.48282i
\(147\) 0 0
\(148\) 9.00000 + 9.00000i 0.739795 + 0.739795i
\(149\) 18.9737i 1.55438i −0.629264 0.777192i \(-0.716644\pi\)
0.629264 0.777192i \(-0.283356\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) −5.00000 5.00000i −0.405554 0.405554i
\(153\) 0 0
\(154\) −15.8114 + 10.0000i −1.27412 + 0.805823i
\(155\) −3.16228 + 6.32456i −0.254000 + 0.508001i
\(156\) 0 0
\(157\) −15.8114 15.8114i −1.26189 1.26189i −0.950178 0.311708i \(-0.899099\pi\)
−0.311708 0.950178i \(-0.600901\pi\)
\(158\) 6.32456 + 6.32456i 0.503155 + 0.503155i
\(159\) 0 0
\(160\) 14.2302 4.74342i 1.12500 0.375000i
\(161\) 6.32456 + 10.0000i 0.498445 + 0.788110i
\(162\) 0 0
\(163\) −6.00000 6.00000i −0.469956 0.469956i 0.431944 0.901900i \(-0.357828\pi\)
−0.901900 + 0.431944i \(0.857828\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 31.6228i 2.45440i
\(167\) −10.0000 10.0000i −0.773823 0.773823i 0.204949 0.978773i \(-0.434297\pi\)
−0.978773 + 0.204949i \(0.934297\pi\)
\(168\) 0 0
\(169\) 7.00000i 0.538462i
\(170\) 31.6228 + 15.8114i 2.42536 + 1.21268i
\(171\) 0 0
\(172\) 18.0000 18.0000i 1.37249 1.37249i
\(173\) −5.00000 + 5.00000i −0.380143 + 0.380143i −0.871154 0.491011i \(-0.836628\pi\)
0.491011 + 0.871154i \(0.336628\pi\)
\(174\) 0 0
\(175\) 5.41886 + 12.0680i 0.409627 + 0.912253i
\(176\) 3.16228 0.238366
\(177\) 0 0
\(178\) −15.8114 + 15.8114i −1.18511 + 1.18511i
\(179\) 9.48683i 0.709079i 0.935041 + 0.354540i \(0.115362\pi\)
−0.935041 + 0.354540i \(0.884638\pi\)
\(180\) 0 0
\(181\) 6.32456i 0.470100i −0.971983 0.235050i \(-0.924475\pi\)
0.971983 0.235050i \(-0.0755255\pi\)
\(182\) 5.81139 25.8114i 0.430769 1.91327i
\(183\) 0 0
\(184\) 10.0000i 0.737210i
\(185\) 9.00000 3.00000i 0.661693 0.220564i
\(186\) 0 0
\(187\) 15.8114 + 15.8114i 1.15624 + 1.15624i
\(188\) 0 0
\(189\) 0 0
\(190\) −15.0000 + 5.00000i −1.08821 + 0.362738i
\(191\) −9.48683 −0.686443 −0.343222 0.939254i \(-0.611518\pi\)
−0.343222 + 0.939254i \(0.611518\pi\)
\(192\) 0 0
\(193\) 1.00000 + 1.00000i 0.0719816 + 0.0719816i 0.742181 0.670199i \(-0.233791\pi\)
−0.670199 + 0.742181i \(0.733791\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) 9.00000 18.9737i 0.642857 1.35526i
\(197\) 3.16228 3.16228i 0.225303 0.225303i −0.585424 0.810727i \(-0.699072\pi\)
0.810727 + 0.585424i \(0.199072\pi\)
\(198\) 0 0
\(199\) −9.48683 −0.672504 −0.336252 0.941772i \(-0.609159\pi\)
−0.336252 + 0.941772i \(0.609159\pi\)
\(200\) −1.58114 + 11.0680i −0.111803 + 0.782624i
\(201\) 0 0
\(202\) 15.8114 + 15.8114i 1.11249 + 1.11249i
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 10.0000i 0.696733i
\(207\) 0 0
\(208\) −3.16228 + 3.16228i −0.219265 + 0.219265i
\(209\) −10.0000 −0.691714
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) −9.48683 + 9.48683i −0.651558 + 0.651558i
\(213\) 0 0
\(214\) 30.0000i 2.05076i
\(215\) −6.00000 18.0000i −0.409197 1.22759i
\(216\) 0 0
\(217\) 1.83772 8.16228i 0.124753 0.554092i
\(218\) −6.32456 6.32456i −0.428353 0.428353i
\(219\) 0 0
\(220\) −9.48683 + 18.9737i −0.639602 + 1.27920i
\(221\) −31.6228 −2.12718
\(222\) 0 0
\(223\) 18.9737 18.9737i 1.27057 1.27057i 0.324782 0.945789i \(-0.394709\pi\)
0.945789 0.324782i \(-0.105291\pi\)
\(224\) −15.0000 + 9.48683i −1.00223 + 0.633866i
\(225\) 0 0
\(226\) 30.0000 1.99557
\(227\) 10.0000 + 10.0000i 0.663723 + 0.663723i 0.956256 0.292532i \(-0.0944979\pi\)
−0.292532 + 0.956256i \(0.594498\pi\)
\(228\) 0 0
\(229\) 12.6491 0.835877 0.417938 0.908475i \(-0.362753\pi\)
0.417938 + 0.908475i \(0.362753\pi\)
\(230\) 20.0000 + 10.0000i 1.31876 + 0.659380i
\(231\) 0 0
\(232\) 0 0
\(233\) −9.48683 9.48683i −0.621503 0.621503i 0.324413 0.945916i \(-0.394833\pi\)
−0.945916 + 0.324413i \(0.894833\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) −40.8114 9.18861i −2.64541 0.595609i
\(239\) 22.1359i 1.43186i 0.698175 + 0.715928i \(0.253996\pi\)
−0.698175 + 0.715928i \(0.746004\pi\)
\(240\) 0 0
\(241\) 6.32456i 0.407400i −0.979033 0.203700i \(-0.934703\pi\)
0.979033 0.203700i \(-0.0652968\pi\)
\(242\) 1.58114 1.58114i 0.101639 0.101639i
\(243\) 0 0
\(244\) 37.9473 2.42933
\(245\) −9.64911 12.3246i −0.616459 0.787387i
\(246\) 0 0
\(247\) 10.0000 10.0000i 0.636285 0.636285i
\(248\) 5.00000 5.00000i 0.317500 0.317500i
\(249\) 0 0
\(250\) 20.5548 + 14.2302i 1.30000 + 0.900000i
\(251\) 20.0000i 1.26239i −0.775625 0.631194i \(-0.782565\pi\)
0.775625 0.631194i \(-0.217435\pi\)
\(252\) 0 0
\(253\) 10.0000 + 10.0000i 0.628695 + 0.628695i
\(254\) 37.9473i 2.38103i
\(255\) 0 0
\(256\) −9.00000 −0.562500
\(257\) −5.00000 5.00000i −0.311891 0.311891i 0.533751 0.845642i \(-0.320782\pi\)
−0.845642 + 0.533751i \(0.820782\pi\)
\(258\) 0 0
\(259\) −9.48683 + 6.00000i −0.589483 + 0.372822i
\(260\) −9.48683 28.4605i −0.588348 1.76505i
\(261\) 0 0
\(262\) 0 0
\(263\) 6.32456 + 6.32456i 0.389989 + 0.389989i 0.874683 0.484695i \(-0.161069\pi\)
−0.484695 + 0.874683i \(0.661069\pi\)
\(264\) 0 0
\(265\) 3.16228 + 9.48683i 0.194257 + 0.582772i
\(266\) 15.8114 10.0000i 0.969458 0.613139i
\(267\) 0 0
\(268\) −24.0000 24.0000i −1.46603 1.46603i
\(269\) 30.0000 1.82913 0.914566 0.404436i \(-0.132532\pi\)
0.914566 + 0.404436i \(0.132532\pi\)
\(270\) 0 0
\(271\) 22.1359i 1.34466i 0.740250 + 0.672331i \(0.234707\pi\)
−0.740250 + 0.672331i \(0.765293\pi\)
\(272\) 5.00000 + 5.00000i 0.303170 + 0.303170i
\(273\) 0 0
\(274\) 10.0000i 0.604122i
\(275\) 9.48683 + 12.6491i 0.572078 + 0.762770i
\(276\) 0 0
\(277\) 3.00000 3.00000i 0.180253 0.180253i −0.611213 0.791466i \(-0.709318\pi\)
0.791466 + 0.611213i \(0.209318\pi\)
\(278\) −15.0000 + 15.0000i −0.899640 + 0.899640i
\(279\) 0 0
\(280\) −1.32456 13.1623i −0.0791573 0.786597i
\(281\) 18.9737 1.13187 0.565937 0.824448i \(-0.308515\pi\)
0.565937 + 0.824448i \(0.308515\pi\)
\(282\) 0 0
\(283\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(284\) 28.4605i 1.68882i
\(285\) 0 0
\(286\) 31.6228i 1.86989i
\(287\) 0 0
\(288\) 0 0
\(289\) 33.0000i 1.94118i
\(290\) 0 0
\(291\) 0 0
\(292\) −28.4605 28.4605i −1.66552 1.66552i
\(293\) 5.00000 5.00000i 0.292103 0.292103i −0.545807 0.837911i \(-0.683777\pi\)
0.837911 + 0.545807i \(0.183777\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −9.48683 −0.551411
\(297\) 0 0
\(298\) 30.0000 + 30.0000i 1.73785 + 1.73785i
\(299\) −20.0000 −1.15663
\(300\) 0 0
\(301\) 12.0000 + 18.9737i 0.691669 + 1.09362i
\(302\) 18.9737 18.9737i 1.09181 1.09181i
\(303\) 0 0
\(304\) −3.16228 −0.181369
\(305\) 12.6491 25.2982i 0.724286 1.44857i
\(306\) 0 0
\(307\) −9.48683 9.48683i −0.541442 0.541442i 0.382509 0.923952i \(-0.375060\pi\)
−0.923952 + 0.382509i \(0.875060\pi\)
\(308\) 5.51317 24.4868i 0.314142 1.39527i
\(309\) 0 0
\(310\) −5.00000 15.0000i −0.283981 0.851943i
\(311\) 20.0000i 1.13410i 0.823685 + 0.567048i \(0.191915\pi\)
−0.823685 + 0.567048i \(0.808085\pi\)
\(312\) 0 0
\(313\) −9.48683 + 9.48683i −0.536228 + 0.536228i −0.922419 0.386191i \(-0.873790\pi\)
0.386191 + 0.922419i \(0.373790\pi\)
\(314\) 50.0000 2.82166
\(315\) 0 0
\(316\) −12.0000 −0.675053
\(317\) −22.1359 + 22.1359i −1.24328 + 1.24328i −0.284646 + 0.958633i \(0.591876\pi\)
−0.958633 + 0.284646i \(0.908124\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −13.0000 + 26.0000i −0.726722 + 1.45344i
\(321\) 0 0
\(322\) −25.8114 5.81139i −1.43841 0.323856i
\(323\) −15.8114 15.8114i −0.879769 0.879769i
\(324\) 0 0
\(325\) −22.1359 3.16228i −1.22788 0.175412i
\(326\) 18.9737 1.05085
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 30.0000 + 30.0000i 1.64646 + 1.64646i
\(333\) 0 0
\(334\) 31.6228 1.73032
\(335\) −24.0000 + 8.00000i −1.31126 + 0.437087i
\(336\) 0 0
\(337\) 7.00000 7.00000i 0.381314 0.381314i −0.490261 0.871576i \(-0.663099\pi\)
0.871576 + 0.490261i \(0.163099\pi\)
\(338\) 11.0680 + 11.0680i 0.602018 + 0.602018i
\(339\) 0 0
\(340\) −45.0000 + 15.0000i −2.44047 + 0.813489i
\(341\) 10.0000i 0.541530i
\(342\) 0 0
\(343\) 14.5811 + 11.4189i 0.787307 + 0.616561i
\(344\) 18.9737i 1.02299i
\(345\) 0 0
\(346\) 15.8114i 0.850026i
\(347\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(348\) 0 0
\(349\) 12.6491 0.677091 0.338546 0.940950i \(-0.390065\pi\)
0.338546 + 0.940950i \(0.390065\pi\)
\(350\) −27.6491 10.5132i −1.47791 0.561952i
\(351\) 0 0
\(352\) −15.0000 + 15.0000i −0.799503 + 0.799503i
\(353\) 15.0000 15.0000i 0.798369 0.798369i −0.184469 0.982838i \(-0.559057\pi\)
0.982838 + 0.184469i \(0.0590565\pi\)
\(354\) 0 0
\(355\) −18.9737 9.48683i −1.00702 0.503509i
\(356\) 30.0000i 1.59000i
\(357\) 0 0
\(358\) −15.0000 15.0000i −0.792775 0.792775i
\(359\) 22.1359i 1.16829i −0.811649 0.584145i \(-0.801430\pi\)
0.811649 0.584145i \(-0.198570\pi\)
\(360\) 0 0
\(361\) −9.00000 −0.473684
\(362\) 10.0000 + 10.0000i 0.525588 + 0.525588i
\(363\) 0 0
\(364\) 18.9737 + 30.0000i 0.994490 + 1.57243i
\(365\) −28.4605 + 9.48683i −1.48969 + 0.496564i
\(366\) 0 0
\(367\) −6.32456 6.32456i −0.330139 0.330139i 0.522500 0.852639i \(-0.324999\pi\)
−0.852639 + 0.522500i \(0.824999\pi\)
\(368\) 3.16228 + 3.16228i 0.164845 + 0.164845i
\(369\) 0 0
\(370\) −9.48683 + 18.9737i −0.493197 + 0.986394i
\(371\) −6.32456 10.0000i −0.328355 0.519174i
\(372\) 0 0
\(373\) 9.00000 + 9.00000i 0.466002 + 0.466002i 0.900617 0.434614i \(-0.143115\pi\)
−0.434614 + 0.900617i \(0.643115\pi\)
\(374\) −50.0000 −2.58544
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 4.00000i 0.205466i 0.994709 + 0.102733i \(0.0327588\pi\)
−0.994709 + 0.102733i \(0.967241\pi\)
\(380\) 9.48683 18.9737i 0.486664 0.973329i
\(381\) 0 0
\(382\) 15.0000 15.0000i 0.767467 0.767467i
\(383\) 10.0000 10.0000i 0.510976 0.510976i −0.403849 0.914825i \(-0.632328\pi\)
0.914825 + 0.403849i \(0.132328\pi\)
\(384\) 0 0
\(385\) −14.4868 11.8377i −0.738317 0.603306i
\(386\) −3.16228 −0.160956
\(387\) 0 0
\(388\) −9.48683 + 9.48683i −0.481621 + 0.481621i
\(389\) 6.32456i 0.320668i 0.987063 + 0.160334i \(0.0512571\pi\)
−0.987063 + 0.160334i \(0.948743\pi\)
\(390\) 0 0
\(391\) 31.6228i 1.59923i
\(392\) 5.25658 + 14.7434i 0.265498 + 0.744655i
\(393\) 0 0
\(394\) 10.0000i 0.503793i
\(395\) −4.00000 + 8.00000i −0.201262 + 0.402524i
\(396\) 0 0
\(397\) −9.48683 9.48683i −0.476130 0.476130i 0.427761 0.903892i \(-0.359302\pi\)
−0.903892 + 0.427761i \(0.859302\pi\)
\(398\) 15.0000 15.0000i 0.751882 0.751882i
\(399\) 0 0
\(400\) 3.00000 + 4.00000i 0.150000 + 0.200000i
\(401\) 12.6491 0.631666 0.315833 0.948815i \(-0.397716\pi\)
0.315833 + 0.948815i \(0.397716\pi\)
\(402\) 0 0
\(403\) 10.0000 + 10.0000i 0.498135 + 0.498135i
\(404\) −30.0000 −1.49256
\(405\) 0 0
\(406\) 0 0
\(407\) −9.48683 + 9.48683i −0.470245 + 0.470245i
\(408\) 0 0
\(409\) 25.2982 1.25092 0.625458 0.780258i \(-0.284912\pi\)
0.625458 + 0.780258i \(0.284912\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −9.48683 9.48683i −0.467383 0.467383i
\(413\) 0 0
\(414\) 0 0
\(415\) 30.0000 10.0000i 1.47264 0.490881i
\(416\) 30.0000i 1.47087i
\(417\) 0 0
\(418\) 15.8114 15.8114i 0.773360 0.773360i
\(419\) −20.0000 −0.977064 −0.488532 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 0 0
\(421\) −8.00000 −0.389896 −0.194948 0.980814i \(-0.562454\pi\)
−0.194948 + 0.980814i \(0.562454\pi\)
\(422\) −12.6491 + 12.6491i −0.615749 + 0.615749i
\(423\) 0 0
\(424\) 10.0000i 0.485643i
\(425\) −5.00000 + 35.0000i −0.242536 + 1.69775i
\(426\) 0 0
\(427\) −7.35089 + 32.6491i −0.355734 + 1.58000i
\(428\) 28.4605 + 28.4605i 1.37569 + 1.37569i
\(429\) 0 0
\(430\) 37.9473 + 18.9737i 1.82998 + 0.914991i
\(431\) −22.1359 −1.06625 −0.533125 0.846036i \(-0.678983\pi\)
−0.533125 + 0.846036i \(0.678983\pi\)
\(432\) 0 0
\(433\) −9.48683 + 9.48683i −0.455908 + 0.455908i −0.897310 0.441402i \(-0.854481\pi\)
0.441402 + 0.897310i \(0.354481\pi\)
\(434\) 10.0000 + 15.8114i 0.480015 + 0.758971i
\(435\) 0 0
\(436\) 12.0000 0.574696
\(437\) −10.0000 10.0000i −0.478365 0.478365i
\(438\) 0 0
\(439\) 3.16228 0.150927 0.0754636 0.997149i \(-0.475956\pi\)
0.0754636 + 0.997149i \(0.475956\pi\)
\(440\) −5.00000 15.0000i −0.238366 0.715097i
\(441\) 0 0
\(442\) 50.0000 50.0000i 2.37826 2.37826i
\(443\) 25.2982 + 25.2982i 1.20195 + 1.20195i 0.973571 + 0.228384i \(0.0733440\pi\)
0.228384 + 0.973571i \(0.426656\pi\)
\(444\) 0 0
\(445\) −20.0000 10.0000i −0.948091 0.474045i
\(446\) 60.0000i 2.84108i
\(447\) 0 0
\(448\) 7.55480 33.5548i 0.356931 1.58532i
\(449\) 25.2982i 1.19390i −0.802280 0.596948i \(-0.796380\pi\)
0.802280 0.596948i \(-0.203620\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −28.4605 + 28.4605i −1.33867 + 1.33867i
\(453\) 0 0
\(454\) −31.6228 −1.48413
\(455\) 26.3246 2.64911i 1.23411 0.124192i
\(456\) 0 0
\(457\) 17.0000 17.0000i 0.795226 0.795226i −0.187112 0.982339i \(-0.559913\pi\)
0.982339 + 0.187112i \(0.0599128\pi\)
\(458\) −20.0000 + 20.0000i −0.934539 + 0.934539i
\(459\) 0 0
\(460\) −28.4605 + 9.48683i −1.32698 + 0.442326i
\(461\) 20.0000i 0.931493i −0.884918 0.465746i \(-0.845786\pi\)
0.884918 0.465746i \(-0.154214\pi\)
\(462\) 0 0
\(463\) −16.0000 16.0000i −0.743583 0.743583i 0.229683 0.973266i \(-0.426231\pi\)
−0.973266 + 0.229683i \(0.926231\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 30.0000 1.38972
\(467\) 10.0000 + 10.0000i 0.462745 + 0.462745i 0.899554 0.436809i \(-0.143892\pi\)
−0.436809 + 0.899554i \(0.643892\pi\)
\(468\) 0 0
\(469\) 25.2982 16.0000i 1.16816 0.738811i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 18.9737 + 18.9737i 0.872410 + 0.872410i
\(474\) 0 0
\(475\) −9.48683 12.6491i −0.435286 0.580381i
\(476\) 47.4342 30.0000i 2.17414 1.37505i
\(477\) 0 0
\(478\) −35.0000 35.0000i −1.60086 1.60086i
\(479\) −20.0000 −0.913823 −0.456912 0.889512i \(-0.651044\pi\)
−0.456912 + 0.889512i \(0.651044\pi\)
\(480\) 0 0
\(481\) 18.9737i 0.865125i
\(482\) 10.0000 + 10.0000i 0.455488 + 0.455488i
\(483\) 0 0
\(484\) 3.00000i 0.136364i
\(485\) 3.16228 + 9.48683i 0.143592 + 0.430775i
\(486\) 0 0
\(487\) −8.00000 + 8.00000i −0.362515 + 0.362515i −0.864738 0.502223i \(-0.832516\pi\)
0.502223 + 0.864738i \(0.332516\pi\)
\(488\) −20.0000 + 20.0000i −0.905357 + 0.905357i
\(489\) 0 0
\(490\) 34.7434 + 4.23025i 1.56955 + 0.191103i
\(491\) 34.7851 1.56983 0.784914 0.619605i \(-0.212707\pi\)
0.784914 + 0.619605i \(0.212707\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 31.6228i 1.42278i
\(495\) 0 0
\(496\) 3.16228i 0.141990i
\(497\) 24.4868 + 5.51317i 1.09838 + 0.247299i
\(498\) 0 0
\(499\) 16.0000i 0.716258i −0.933672 0.358129i \(-0.883415\pi\)
0.933672 0.358129i \(-0.116585\pi\)
\(500\) −33.0000 + 6.00000i −1.47580 + 0.268328i
\(501\) 0 0
\(502\) 31.6228 + 31.6228i 1.41139 + 1.41139i
\(503\) 10.0000 10.0000i 0.445878 0.445878i −0.448104 0.893982i \(-0.647900\pi\)
0.893982 + 0.448104i \(0.147900\pi\)
\(504\) 0 0
\(505\) −10.0000 + 20.0000i −0.444994 + 0.889988i
\(506\) −31.6228 −1.40580
\(507\) 0 0
\(508\) 36.0000 + 36.0000i 1.59724 + 1.59724i
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 30.0000 18.9737i 1.32712 0.839346i
\(512\) −7.90569 + 7.90569i −0.349386 + 0.349386i
\(513\) 0 0
\(514\) 15.8114 0.697410
\(515\) −9.48683 + 3.16228i −0.418040 + 0.139347i
\(516\) 0 0
\(517\) 0 0
\(518\) 5.51317 24.4868i 0.242235 1.07589i
\(519\) 0 0
\(520\) 20.0000 + 10.0000i 0.877058 + 0.438529i
\(521\) 10.0000i 0.438108i 0.975713 + 0.219054i \(0.0702971\pi\)
−0.975713 + 0.219054i \(0.929703\pi\)
\(522\) 0 0
\(523\) 9.48683 9.48683i 0.414830 0.414830i −0.468587 0.883417i \(-0.655237\pi\)
0.883417 + 0.468587i \(0.155237\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −20.0000 −0.872041
\(527\) 15.8114 15.8114i 0.688755 0.688755i
\(528\) 0 0
\(529\) 3.00000i 0.130435i
\(530\) −20.0000 10.0000i −0.868744 0.434372i
\(531\) 0 0
\(532\) −5.51317 + 24.4868i −0.239026 + 1.06164i
\(533\) 0 0
\(534\) 0 0
\(535\) 28.4605 9.48683i 1.23045 0.410152i
\(536\) 25.2982 1.09272
\(537\) 0 0
\(538\) −47.4342 + 47.4342i −2.04503 + 2.04503i
\(539\) 20.0000 + 9.48683i 0.861461 + 0.408627i
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) −35.0000 35.0000i −1.50338 1.50338i
\(543\) 0 0
\(544\) −47.4342 −2.03372
\(545\) 4.00000 8.00000i 0.171341 0.342682i
\(546\) 0 0
\(547\) 8.00000 8.00000i 0.342055 0.342055i −0.515084 0.857140i \(-0.672239\pi\)
0.857140 + 0.515084i \(0.172239\pi\)
\(548\) −9.48683 9.48683i −0.405257 0.405257i
\(549\) 0 0
\(550\) −35.0000 5.00000i −1.49241 0.213201i
\(551\) 0 0
\(552\) 0 0
\(553\) 2.32456 10.3246i 0.0988501 0.439045i
\(554\) 9.48683i 0.403057i
\(555\) 0 0
\(556\) 28.4605i 1.20699i
\(557\) 9.48683 9.48683i 0.401970 0.401970i −0.476957 0.878927i \(-0.658260\pi\)
0.878927 + 0.476957i \(0.158260\pi\)
\(558\) 0 0
\(559\) −37.9473 −1.60500
\(560\) −4.58114 3.74342i −0.193588 0.158188i
\(561\) 0 0
\(562\) −30.0000 + 30.0000i −1.26547 + 1.26547i
\(563\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(564\) 0 0
\(565\) 9.48683 + 28.4605i 0.399114 + 1.19734i
\(566\) 0 0
\(567\) 0 0
\(568\) 15.0000 + 15.0000i 0.629386 + 0.629386i
\(569\) 12.6491i 0.530278i 0.964210 + 0.265139i \(0.0854179\pi\)
−0.964210 + 0.265139i \(0.914582\pi\)
\(570\) 0 0
\(571\) −8.00000 −0.334790 −0.167395 0.985890i \(-0.553535\pi\)
−0.167395 + 0.985890i \(0.553535\pi\)
\(572\) 30.0000 + 30.0000i 1.25436 + 1.25436i
\(573\) 0 0
\(574\) 0 0
\(575\) −3.16228 + 22.1359i −0.131876 + 0.923133i
\(576\) 0 0
\(577\) 22.1359 + 22.1359i 0.921531 + 0.921531i 0.997138 0.0756064i \(-0.0240892\pi\)
−0.0756064 + 0.997138i \(0.524089\pi\)
\(578\) −52.1776 52.1776i −2.17030 2.17030i
\(579\) 0 0
\(580\) 0 0
\(581\) −31.6228 + 20.0000i −1.31193 + 0.829740i
\(582\) 0 0
\(583\) −10.0000 10.0000i −0.414158 0.414158i
\(584\) 30.0000 1.24141
\(585\) 0 0
\(586\) 15.8114i 0.653162i
\(587\) −20.0000 20.0000i −0.825488 0.825488i 0.161401 0.986889i \(-0.448399\pi\)
−0.986889 + 0.161401i \(0.948399\pi\)
\(588\) 0 0
\(589\) 10.0000i 0.412043i
\(590\) 0 0
\(591\) 0 0
\(592\) −3.00000 + 3.00000i −0.123299 + 0.123299i
\(593\) −25.0000 + 25.0000i −1.02663 + 1.02663i −0.0269913 + 0.999636i \(0.508593\pi\)
−0.999636 + 0.0269913i \(0.991407\pi\)
\(594\) 0 0
\(595\) −4.18861 41.6228i −0.171716 1.70637i
\(596\) −56.9210 −2.33157
\(597\) 0 0
\(598\) 31.6228 31.6228i 1.29315 1.29315i
\(599\) 15.8114i 0.646036i −0.946393 0.323018i \(-0.895303\pi\)
0.946393 0.323018i \(-0.104697\pi\)
\(600\) 0 0
\(601\) 31.6228i 1.28992i −0.764216 0.644960i \(-0.776874\pi\)
0.764216 0.644960i \(-0.223126\pi\)
\(602\) −48.9737 11.0263i −1.99602 0.449400i
\(603\) 0 0
\(604\) 36.0000i 1.46482i
\(605\) 2.00000 + 1.00000i 0.0813116 + 0.0406558i
\(606\) 0 0
\(607\) −18.9737 18.9737i −0.770117 0.770117i 0.208009 0.978127i \(-0.433302\pi\)
−0.978127 + 0.208009i \(0.933302\pi\)
\(608\) 15.0000 15.0000i 0.608330 0.608330i
\(609\) 0 0
\(610\) 20.0000 + 60.0000i 0.809776 + 2.42933i
\(611\) 0 0
\(612\) 0 0
\(613\) −1.00000 1.00000i −0.0403896 0.0403896i 0.686624 0.727013i \(-0.259092\pi\)
−0.727013 + 0.686624i \(0.759092\pi\)
\(614\) 30.0000 1.21070
\(615\) 0 0
\(616\) 10.0000 + 15.8114i 0.402911 + 0.637059i
\(617\) 15.8114 15.8114i 0.636543 0.636543i −0.313158 0.949701i \(-0.601387\pi\)
0.949701 + 0.313158i \(0.101387\pi\)
\(618\) 0 0
\(619\) −3.16228 −0.127103 −0.0635513 0.997979i \(-0.520243\pi\)
−0.0635513 + 0.997979i \(0.520243\pi\)
\(620\) 18.9737 + 9.48683i 0.762001 + 0.381000i
\(621\) 0 0
\(622\) −31.6228 31.6228i −1.26796 1.26796i
\(623\) 25.8114 + 5.81139i 1.03411 + 0.232828i
\(624\) 0 0
\(625\) −7.00000 + 24.0000i −0.280000 + 0.960000i
\(626\) 30.0000i 1.19904i
\(627\) 0 0
\(628\) −47.4342 + 47.4342i −1.89283 + 1.89283i
\(629\) −30.0000 −1.19618
\(630\) 0 0
\(631\) −28.0000 −1.11466 −0.557331 0.830290i \(-0.688175\pi\)
−0.557331 + 0.830290i \(0.688175\pi\)
\(632\) 6.32456 6.32456i 0.251577 0.251577i
\(633\) 0 0
\(634\) 70.0000i 2.78006i
\(635\) 36.0000 12.0000i 1.42862 0.476205i
\(636\) 0 0
\(637\) −29.4868 + 10.5132i −1.16831 + 0.416547i
\(638\) 0 0
\(639\) 0 0
\(640\) −11.0680 33.2039i −0.437500 1.31250i
\(641\) 31.6228 1.24902 0.624512 0.781015i \(-0.285298\pi\)
0.624512 + 0.781015i \(0.285298\pi\)
\(642\) 0 0
\(643\) −9.48683 + 9.48683i −0.374124 + 0.374124i −0.868977 0.494853i \(-0.835222\pi\)
0.494853 + 0.868977i \(0.335222\pi\)
\(644\) 30.0000 18.9737i 1.18217 0.747667i
\(645\) 0 0
\(646\) 50.0000 1.96722
\(647\) 10.0000 + 10.0000i 0.393141 + 0.393141i 0.875805 0.482665i \(-0.160331\pi\)
−0.482665 + 0.875805i \(0.660331\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 40.0000 30.0000i 1.56893 1.17670i
\(651\) 0 0
\(652\) −18.0000 + 18.0000i −0.704934 + 0.704934i
\(653\) −15.8114 15.8114i −0.618747 0.618747i 0.326463 0.945210i \(-0.394143\pi\)
−0.945210 + 0.326463i \(0.894143\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 41.1096i 1.60140i −0.599064 0.800702i \(-0.704460\pi\)
0.599064 0.800702i \(-0.295540\pi\)
\(660\) 0 0
\(661\) 12.6491i 0.491993i 0.969271 + 0.245997i \(0.0791152\pi\)
−0.969271 + 0.245997i \(0.920885\pi\)
\(662\) 18.9737 18.9737i 0.737432 0.737432i
\(663\) 0 0
\(664\) −31.6228 −1.22720
\(665\) 14.4868 + 11.8377i 0.561775 + 0.459047i
\(666\) 0 0
\(667\) 0 0
\(668\) −30.0000 + 30.0000i −1.16073 + 1.16073i
\(669\) 0 0
\(670\) 25.2982 50.5964i 0.977356 1.95471i
\(671\) 40.0000i 1.54418i
\(672\) 0 0
\(673\) −9.00000 9.00000i −0.346925 0.346925i 0.512038 0.858963i \(-0.328891\pi\)
−0.858963 + 0.512038i \(0.828891\pi\)
\(674\) 22.1359i 0.852645i
\(675\) 0 0
\(676\) −21.0000 −0.807692
\(677\) −15.0000 15.0000i −0.576497 0.576497i 0.357439 0.933936i \(-0.383650\pi\)
−0.933936 + 0.357439i \(0.883650\pi\)
\(678\) 0 0
\(679\) −6.32456 10.0000i −0.242714 0.383765i
\(680\) 15.8114 31.6228i 0.606339 1.21268i
\(681\) 0 0
\(682\) 15.8114 + 15.8114i 0.605449 + 0.605449i
\(683\) 12.6491 + 12.6491i 0.484005 + 0.484005i 0.906408 0.422403i \(-0.138813\pi\)
−0.422403 + 0.906408i \(0.638813\pi\)
\(684\) 0 0
\(685\) −9.48683 + 3.16228i −0.362473 + 0.120824i
\(686\) −41.1096 + 5.00000i −1.56957 + 0.190901i
\(687\) 0 0
\(688\) 6.00000 + 6.00000i 0.228748 + 0.228748i
\(689\) 20.0000 0.761939
\(690\) 0 0
\(691\) 3.16228i 0.120299i −0.998189 0.0601494i \(-0.980842\pi\)
0.998189 0.0601494i \(-0.0191577\pi\)
\(692\) 15.0000 + 15.0000i 0.570214 + 0.570214i
\(693\) 0 0
\(694\) 0 0
\(695\) −18.9737 9.48683i −0.719712 0.359856i
\(696\) 0 0
\(697\) 0 0
\(698\) −20.0000 + 20.0000i −0.757011 + 0.757011i
\(699\) 0 0
\(700\) 36.2039 16.2566i 1.36838 0.614441i
\(701\) 12.6491 0.477750 0.238875 0.971050i \(-0.423221\pi\)
0.238875 + 0.971050i \(0.423221\pi\)
\(702\) 0 0
\(703\) 9.48683 9.48683i 0.357803 0.357803i
\(704\) 41.1096i 1.54938i
\(705\) 0 0
\(706\) 47.4342i 1.78521i
\(707\) 5.81139 25.8114i 0.218560 0.970737i
\(708\) 0 0
\(709\) 24.0000i 0.901339i 0.892691 + 0.450669i \(0.148815\pi\)
−0.892691 + 0.450669i \(0.851185\pi\)
\(710\) 45.0000 15.0000i 1.68882 0.562940i
\(711\) 0 0
\(712\) 15.8114 + 15.8114i 0.592557 + 0.592557i
\(713\) 10.0000 10.0000i 0.374503 0.374503i
\(714\) 0 0
\(715\) 30.0000 10.0000i 1.12194 0.373979i
\(716\) 28.4605 1.06362
\(717\) 0 0
\(718\) 35.0000 + 35.0000i 1.30619 + 1.30619i
\(719\) −20.0000 −0.745874 −0.372937 0.927857i \(-0.621649\pi\)
−0.372937 + 0.927857i \(0.621649\pi\)
\(720\) 0 0
\(721\) 10.0000 6.32456i 0.372419 0.235539i
\(722\) 14.2302 14.2302i 0.529595 0.529595i
\(723\) 0 0
\(724\) −18.9737 −0.705151
\(725\) 0 0
\(726\) 0 0
\(727\) 18.9737 + 18.9737i 0.703694 + 0.703694i 0.965202 0.261507i \(-0.0842195\pi\)
−0.261507 + 0.965202i \(0.584220\pi\)
\(728\) −25.8114 5.81139i −0.956634 0.215384i
\(729\) 0 0
\(730\) 30.0000 60.0000i 1.11035 2.22070i
\(731\) 60.0000i 2.21918i
\(732\) 0 0
\(733\) 15.8114 15.8114i 0.584007 0.584007i −0.351995 0.936002i \(-0.614497\pi\)
0.936002 + 0.351995i \(0.114497\pi\)
\(734\) 20.0000 0.738213
\(735\) 0 0
\(736\) −30.0000 −1.10581
\(737\) 25.2982 25.2982i 0.931872 0.931872i
\(738\) 0 0
\(739\) 44.0000i 1.61857i −0.587419 0.809283i \(-0.699856\pi\)
0.587419 0.809283i \(-0.300144\pi\)
\(740\) −9.00000 27.0000i −0.330847 0.992540i
\(741\) 0 0
\(742\) 25.8114 + 5.81139i 0.947566 + 0.213343i
\(743\) −18.9737 18.9737i −0.696076 0.696076i 0.267486 0.963562i \(-0.413807\pi\)
−0.963562 + 0.267486i \(0.913807\pi\)
\(744\) 0 0
\(745\) −18.9737 + 37.9473i −0.695141 + 1.39028i
\(746\) −28.4605 −1.04201
\(747\) 0 0
\(748\) 47.4342 47.4342i 1.73436 1.73436i
\(749\) −30.0000 + 18.9737i −1.09618 + 0.693283i
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 24.0000 + 12.0000i 0.873449 + 0.436725i
\(756\) 0 0
\(757\) 17.0000 17.0000i 0.617876 0.617876i −0.327111 0.944986i \(-0.606075\pi\)
0.944986 + 0.327111i \(0.106075\pi\)
\(758\) −6.32456 6.32456i −0.229718 0.229718i
\(759\) 0 0
\(760\) 5.00000 + 15.0000i 0.181369 + 0.544107i
\(761\) 20.0000i 0.724999i 0.931984 + 0.362500i \(0.118077\pi\)
−0.931984 + 0.362500i \(0.881923\pi\)
\(762\) 0 0
\(763\) −2.32456 + 10.3246i −0.0841546 + 0.373774i
\(764\) 28.4605i 1.02966i
\(765\) 0 0
\(766\) 31.6228i 1.14258i
\(767\) 0 0
\(768\) 0 0
\(769\) −6.32456 −0.228069 −0.114035 0.993477i \(-0.536377\pi\)
−0.114035 + 0.993477i \(0.536377\pi\)
\(770\) 41.6228 4.18861i 1.49998 0.150947i
\(771\) 0 0
\(772\) 3.00000 3.00000i 0.107972 0.107972i
\(773\) 35.0000 35.0000i 1.25886 1.25886i 0.307226 0.951637i \(-0.400599\pi\)
0.951637 0.307226i \(-0.0994007\pi\)
\(774\) 0 0
\(775\) 12.6491 9.48683i 0.454369 0.340777i
\(776\) 10.0000i 0.358979i
\(777\) 0 0
\(778\) −10.0000 10.0000i −0.358517 0.358517i
\(779\) 0 0
\(780\) 0 0
\(781\) 30.0000 1.07348
\(782\) −50.0000 50.0000i −1.78800 1.78800i
\(783\) 0 0
\(784\) 6.32456 + 3.00000i 0.225877 + 0.107143i
\(785\) 15.8114 + 47.4342i 0.564333 + 1.69300i
\(786\) 0 0
\(787\) 15.8114 + 15.8114i 0.563615 + 0.563615i 0.930332 0.366717i \(-0.119518\pi\)
−0.366717 + 0.930332i \(0.619518\pi\)
\(788\) −9.48683 9.48683i −0.337954 0.337954i
\(789\) 0 0
\(790\) −6.32456 18.9737i −0.225018 0.675053i
\(791\) −18.9737 30.0000i −0.674626 1.06668i
\(792\) 0 0
\(793\) −40.0000 40.0000i −1.42044 1.42044i
\(794\) 30.0000 1.06466
\(795\) 0 0
\(796\) 28.4605i 1.00876i
\(797\) −25.0000 25.0000i −0.885545 0.885545i 0.108546 0.994091i \(-0.465381\pi\)
−0.994091 + 0.108546i \(0.965381\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −33.2039 4.74342i −1.17394 0.167705i
\(801\) 0 0
\(802\) −20.0000 + 20.0000i −0.706225 + 0.706225i
\(803\) 30.0000 30.0000i 1.05868 1.05868i
\(804\) 0 0
\(805\) −2.64911 26.3246i −0.0933689 0.927819i
\(806\) −31.6228 −1.11386
\(807\) 0 0
\(808\) 15.8114 15.8114i 0.556243 0.556243i
\(809\) 18.9737i 0.667079i 0.942736 + 0.333539i \(0.108243\pi\)
−0.942736 + 0.333539i \(0.891757\pi\)
\(810\) 0 0
\(811\) 47.4342i 1.66564i 0.553545 + 0.832819i \(0.313275\pi\)
−0.553545 + 0.832819i \(0.686725\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 30.0000i 1.05150i
\(815\) 6.00000 + 18.0000i 0.210171 + 0.630512i
\(816\) 0 0
\(817\) −18.9737 18.9737i −0.663805 0.663805i
\(818\) −40.0000 + 40.0000i −1.39857 + 1.39857i
\(819\) 0 0
\(820\) 0 0
\(821\) −12.6491 −0.441457 −0.220729 0.975335i \(-0.570843\pi\)
−0.220729 + 0.975335i \(0.570843\pi\)
\(822\) 0 0
\(823\) 24.0000 + 24.0000i 0.836587 + 0.836587i 0.988408 0.151821i \(-0.0485136\pi\)
−0.151821 + 0.988408i \(0.548514\pi\)
\(824\) 10.0000 0.348367
\(825\) 0 0
\(826\) 0 0
\(827\) −22.1359 + 22.1359i −0.769742 + 0.769742i −0.978061 0.208319i \(-0.933201\pi\)
0.208319 + 0.978061i \(0.433201\pi\)
\(828\) 0 0
\(829\) 18.9737 0.658983 0.329491 0.944159i \(-0.393123\pi\)
0.329491 + 0.944159i \(0.393123\pi\)
\(830\) −31.6228 + 63.2456i −1.09764 + 2.19529i
\(831\) 0 0
\(832\) 41.1096 + 41.1096i 1.42522 + 1.42522i
\(833\) 16.6228 + 46.6228i 0.575945 + 1.61538i
\(834\) 0 0
\(835\) 10.0000 + 30.0000i 0.346064 + 1.03819i
\(836\) 30.0000i 1.03757i
\(837\) 0 0
\(838\) 31.6228 31.6228i 1.09239 1.09239i
\(839\) 40.0000 1.38095 0.690477 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) 0 0
\(841\) 29.0000 1.00000
\(842\) 12.6491 12.6491i 0.435917 0.435917i
\(843\) 0 0
\(844\) 24.0000i 0.826114i
\(845\) −7.00000 + 14.0000i −0.240807 + 0.481615i
\(846\) 0 0
\(847\) −2.58114 0.581139i −0.0886890 0.0199682i
\(848\) −3.16228 3.16228i −0.108593 0.108593i
\(849\) 0 0
\(850\) −47.4342 63.2456i −1.62698 2.16930i
\(851\) −18.9737 −0.650409
\(852\) 0 0
\(853\) 9.48683 9.48683i 0.324823 0.324823i −0.525791 0.850614i \(-0.676231\pi\)
0.850614 + 0.525791i \(0.176231\pi\)
\(854\) −40.0000 63.2456i −1.36877 2.16422i
\(855\) 0 0
\(856\) −30.0000 −1.02538
\(857\) 5.00000 + 5.00000i 0.170797 + 0.170797i 0.787329 0.616533i \(-0.211463\pi\)
−0.616533 + 0.787329i \(0.711463\pi\)
\(858\) 0 0
\(859\) −22.1359 −0.755269 −0.377634 0.925955i \(-0.623262\pi\)
−0.377634 + 0.925955i \(0.623262\pi\)
\(860\) −54.0000 + 18.0000i −1.84138 + 0.613795i
\(861\) 0 0
\(862\) 35.0000 35.0000i 1.19210 1.19210i
\(863\) −15.8114 15.8114i −0.538226 0.538226i 0.384782 0.923008i \(-0.374277\pi\)
−0.923008 + 0.384782i \(0.874277\pi\)
\(864\) 0 0
\(865\) 15.0000 5.00000i 0.510015 0.170005i
\(866\) 30.0000i 1.01944i
\(867\) 0 0
\(868\) −24.4868 5.51317i −0.831137 0.187129i
\(869\) 12.6491i 0.429092i
\(870\) 0 0
\(871\) 50.5964i 1.71440i
\(872\) −6.32456 + 6.32456i −0.214176 + 0.214176i
\(873\) 0 0
\(874\) 31.6228 1.06966
\(875\) 1.23025 29.5548i 0.0415900 0.999135i
\(876\) 0 0
\(877\) 23.0000 23.0000i 0.776655 0.776655i −0.202606 0.979260i \(-0.564941\pi\)
0.979260 + 0.202606i \(0.0649409\pi\)
\(878\) −5.00000 + 5.00000i −0.168742 + 0.168742i
\(879\) 0 0
\(880\) −6.32456 3.16228i −0.213201 0.106600i
\(881\) 30.0000i 1.01073i 0.862907 + 0.505363i \(0.168641\pi\)
−0.862907 + 0.505363i \(0.831359\pi\)
\(882\) 0 0
\(883\) 36.0000 + 36.0000i 1.21150 + 1.21150i 0.970535 + 0.240962i \(0.0774629\pi\)
0.240962 + 0.970535i \(0.422537\pi\)
\(884\) 94.8683i 3.19077i
\(885\) 0 0
\(886\) −80.0000 −2.68765
\(887\) −20.0000 20.0000i −0.671534 0.671534i 0.286535 0.958070i \(-0.407496\pi\)
−0.958070 + 0.286535i \(0.907496\pi\)
\(888\) 0 0
\(889\) −37.9473 + 24.0000i −1.27271 + 0.804934i
\(890\) 47.4342 15.8114i 1.59000 0.529999i
\(891\) 0 0
\(892\) −56.9210 56.9210i −1.90586 1.90586i
\(893\) 0 0
\(894\) 0 0
\(895\) 9.48683 18.9737i 0.317110 0.634220i
\(896\) 22.1359 + 35.0000i 0.739510 + 1.16927i
\(897\) 0 0
\(898\) 40.0000 + 40.0000i 1.33482 + 1.33482i
\(899\) 0 0
\(900\) 0 0
\(901\) 31.6228i 1.05351i
\(902\) 0 0
\(903\) 0 0
\(904\) 30.0000i 0.997785i
\(905\) −6.32456 + 12.6491i −0.210235 + 0.420471i
\(906\) 0 0
\(907\) −8.00000 + 8.00000i −0.265636 + 0.265636i −0.827339 0.561703i \(-0.810146\pi\)
0.561703 + 0.827339i \(0.310146\pi\)
\(908\) 30.0000 30.0000i 0.995585 0.995585i
\(909\) 0 0
\(910\) −37.4342 + 45.8114i −1.24093 + 1.51863i
\(911\) 28.4605 0.942938 0.471469 0.881883i \(-0.343724\pi\)
0.471469 + 0.881883i \(0.343724\pi\)
\(912\) 0 0
\(913\) −31.6228 + 31.6228i −1.04656 + 1.04656i
\(914\) 53.7587i 1.77818i
\(915\) 0 0
\(916\) 37.9473i 1.25382i
\(917\) 0 0
\(918\) 0 0
\(919\) 24.0000i 0.791687i 0.918318 + 0.395843i \(0.129548\pi\)
−0.918318 + 0.395843i \(0.870452\pi\)
\(920\) 10.0000 20.0000i 0.329690 0.659380i
\(921\) 0 0
\(922\) 31.6228 + 31.6228i 1.04144 + 1.04144i
\(923\) −30.0000 + 30.0000i −0.987462 + 0.987462i
\(924\) 0 0
\(925\) −21.0000 3.00000i −0.690476 0.0986394i
\(926\) 50.5964 1.66270
\(927\) 0 0
\(928\) 0 0
\(929\) −20.0000 −0.656179 −0.328089 0.944647i \(-0.606405\pi\)
−0.328089 + 0.944647i \(0.606405\pi\)
\(930\) 0 0
\(931\) −20.0000 9.48683i −0.655474 0.310918i
\(932\) −28.4605 + 28.4605i −0.932255 + 0.932255i
\(933\) 0 0
\(934\) −31.6228 −1.03473
\(935\) −15.8114 47.4342i −0.517088 1.55126i
\(936\) 0 0
\(937\) 9.48683 + 9.48683i 0.309921 + 0.309921i 0.844879 0.534958i \(-0.179672\pi\)
−0.534958 + 0.844879i \(0.679672\pi\)
\(938\) −14.7018 + 65.2982i −0.480030 + 2.13206i
\(939\) 0 0
\(940\) 0 0
\(941\) 60.0000i 1.95594i −0.208736 0.977972i \(-0.566935\pi\)
0.208736 0.977972i \(-0.433065\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) −60.0000 −1.95077
\(947\) 25.2982 25.2982i 0.822082 0.822082i −0.164325 0.986406i \(-0.552544\pi\)
0.986406 + 0.164325i \(0.0525444\pi\)
\(948\) 0 0
\(949\) 60.0000i 1.94768i
\(950\) 35.0000 + 5.00000i 1.13555 + 0.162221i
\(951\) 0 0
\(952\) −9.18861 + 40.8114i −0.297805 + 1.32270i
\(953\) 28.4605 + 28.4605i 0.921926 + 0.921926i 0.997165 0.0752395i \(-0.0239721\pi\)
−0.0752395 + 0.997165i \(0.523972\pi\)
\(954\) 0 0
\(955\) 18.9737 + 9.48683i 0.613973 + 0.306987i
\(956\) 66.4078 2.14778
\(957\) 0 0
\(958\) 31.6228 31.6228i 1.02169 1.02169i
\(959\) 10.0000 6.32456i 0.322917 0.204231i
\(960\) 0 0
\(961\) 21.0000 0.677419
\(962\) 30.0000 + 30.0000i 0.967239 + 0.967239i
\(963\) 0 0
\(964\) −18.9737 −0.611101
\(965\) −1.00000 3.00000i −0.0321911 0.0965734i
\(966\) 0 0
\(967\) −42.0000 + 42.0000i −1.35063 + 1.35063i −0.465671 + 0.884958i \(0.654187\pi\)
−0.884958 + 0.465671i \(0.845813\pi\)
\(968\) −1.58114 1.58114i −0.0508197 0.0508197i
\(969\) 0 0
\(970\) −20.0000 10.0000i −0.642161 0.321081i
\(971\) 20.0000i 0.641831i −0.947108 0.320915i \(-0.896010\pi\)
0.947108 0.320915i \(-0.103990\pi\)
\(972\) 0 0
\(973\) 24.4868 + 5.51317i 0.785012 + 0.176744i
\(974\) 25.2982i 0.810607i
\(975\) 0 0
\(976\) 12.6491i 0.404888i
\(977\) 3.16228 3.16228i 0.101170 0.101170i −0.654710 0.755880i \(-0.727209\pi\)
0.755880 + 0.654710i \(0.227209\pi\)
\(978\) 0 0
\(979\) 31.6228 1.01067
\(980\) −36.9737 + 28.9473i −1.18108 + 0.924689i
\(981\) 0 0
\(982\) −55.0000 + 55.0000i −1.75512 + 1.75512i
\(983\) −20.0000 + 20.0000i −0.637901 + 0.637901i −0.950037 0.312136i \(-0.898955\pi\)
0.312136 + 0.950037i \(0.398955\pi\)
\(984\) 0 0
\(985\) −9.48683 + 3.16228i −0.302276 + 0.100759i
\(986\) 0 0
\(987\) 0 0
\(988\) −30.0000 30.0000i −0.954427 0.954427i
\(989\) 37.9473i 1.20665i
\(990\) 0 0
\(991\) 28.0000 0.889449 0.444725 0.895667i \(-0.353302\pi\)
0.444725 + 0.895667i \(0.353302\pi\)
\(992\) 15.0000 + 15.0000i 0.476250 + 0.476250i
\(993\) 0 0
\(994\) −47.4342 + 30.0000i −1.50452 + 0.951542i
\(995\) 18.9737 + 9.48683i 0.601506 + 0.300753i
\(996\) 0 0
\(997\) −9.48683 9.48683i −0.300451 0.300451i 0.540739 0.841190i \(-0.318145\pi\)
−0.841190 + 0.540739i \(0.818145\pi\)
\(998\) 25.2982 + 25.2982i 0.800801 + 0.800801i
\(999\) 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.a.118.1 4
3.2 odd 2 315.2.p.b.118.2 yes 4
5.2 odd 4 315.2.p.b.307.1 yes 4
7.6 odd 2 315.2.p.b.118.1 yes 4
15.2 even 4 inner 315.2.p.a.307.2 yes 4
21.20 even 2 inner 315.2.p.a.118.2 yes 4
35.27 even 4 inner 315.2.p.a.307.1 yes 4
105.62 odd 4 315.2.p.b.307.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
315.2.p.a.118.1 4 1.1 even 1 trivial
315.2.p.a.118.2 yes 4 21.20 even 2 inner
315.2.p.a.307.1 yes 4 35.27 even 4 inner
315.2.p.a.307.2 yes 4 15.2 even 4 inner
315.2.p.b.118.1 yes 4 7.6 odd 2
315.2.p.b.118.2 yes 4 3.2 odd 2
315.2.p.b.307.1 yes 4 5.2 odd 4
315.2.p.b.307.2 yes 4 105.62 odd 4