Properties

Label 675.2.f.b.593.1
Level $675$
Weight $2$
Character 675.593
Analytic conductor $5.390$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [675,2,Mod(107,675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(675, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("675.107");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.f (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: \(5.38990213644\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) 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 593.1
Root \(-1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 675.593
Dual form 675.2.f.b.107.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.22474 - 1.22474i) q^{2} +1.00000i q^{4} +(1.22474 - 1.22474i) q^{7} +(-1.22474 + 1.22474i) q^{8} +O(q^{10})\) \(q+(-1.22474 - 1.22474i) q^{2} +1.00000i q^{4} +(1.22474 - 1.22474i) q^{7} +(-1.22474 + 1.22474i) q^{8} +6.00000i q^{11} -3.00000 q^{14} +5.00000 q^{16} +(-2.44949 - 2.44949i) q^{17} +7.00000i q^{19} +(7.34847 - 7.34847i) q^{22} +(-4.89898 + 4.89898i) q^{23} +(1.22474 + 1.22474i) q^{28} +6.00000 q^{29} -5.00000 q^{31} +(-3.67423 - 3.67423i) q^{32} +6.00000i q^{34} +(-1.22474 + 1.22474i) q^{37} +(8.57321 - 8.57321i) q^{38} -6.00000i q^{41} +(6.12372 + 6.12372i) q^{43} -6.00000 q^{44} +12.0000 q^{46} +(7.34847 + 7.34847i) q^{47} +4.00000i q^{49} +(-2.44949 + 2.44949i) q^{53} +3.00000i q^{56} +(-7.34847 - 7.34847i) q^{58} -1.00000 q^{61} +(6.12372 + 6.12372i) q^{62} -1.00000i q^{64} +(7.34847 - 7.34847i) q^{67} +(2.44949 - 2.44949i) q^{68} -12.0000i q^{71} +(6.12372 + 6.12372i) q^{73} +3.00000 q^{74} -7.00000 q^{76} +(7.34847 + 7.34847i) q^{77} -1.00000i q^{79} +(-7.34847 + 7.34847i) q^{82} +(2.44949 - 2.44949i) q^{83} -15.0000i q^{86} +(-7.34847 - 7.34847i) q^{88} +12.0000 q^{89} +(-4.89898 - 4.89898i) q^{92} -18.0000i q^{94} +(-8.57321 + 8.57321i) q^{97} +(4.89898 - 4.89898i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{14} + 20 q^{16} + 24 q^{29} - 20 q^{31} - 24 q^{44} + 48 q^{46} - 4 q^{61} + 12 q^{74} - 28 q^{76} + 48 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.22474 1.22474i −0.866025 0.866025i 0.126004 0.992030i \(-0.459785\pi\)
−0.992030 + 0.126004i \(0.959785\pi\)
\(3\) 0 0
\(4\) 1.00000i 0.500000i
\(5\) 0 0
\(6\) 0 0
\(7\) 1.22474 1.22474i 0.462910 0.462910i −0.436698 0.899608i \(-0.643852\pi\)
0.899608 + 0.436698i \(0.143852\pi\)
\(8\) −1.22474 + 1.22474i −0.433013 + 0.433013i
\(9\) 0 0
\(10\) 0 0
\(11\) 6.00000i 1.80907i 0.426401 + 0.904534i \(0.359781\pi\)
−0.426401 + 0.904534i \(0.640219\pi\)
\(12\) 0 0
\(13\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(14\) −3.00000 −0.801784
\(15\) 0 0
\(16\) 5.00000 1.25000
\(17\) −2.44949 2.44949i −0.594089 0.594089i 0.344645 0.938733i \(-0.387999\pi\)
−0.938733 + 0.344645i \(0.887999\pi\)
\(18\) 0 0
\(19\) 7.00000i 1.60591i 0.596040 + 0.802955i \(0.296740\pi\)
−0.596040 + 0.802955i \(0.703260\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 7.34847 7.34847i 1.56670 1.56670i
\(23\) −4.89898 + 4.89898i −1.02151 + 1.02151i −0.0217443 + 0.999764i \(0.506922\pi\)
−0.999764 + 0.0217443i \(0.993078\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 1.22474 + 1.22474i 0.231455 + 0.231455i
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −5.00000 −0.898027 −0.449013 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) −3.67423 3.67423i −0.649519 0.649519i
\(33\) 0 0
\(34\) 6.00000i 1.02899i
\(35\) 0 0
\(36\) 0 0
\(37\) −1.22474 + 1.22474i −0.201347 + 0.201347i −0.800577 0.599230i \(-0.795473\pi\)
0.599230 + 0.800577i \(0.295473\pi\)
\(38\) 8.57321 8.57321i 1.39076 1.39076i
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000i 0.937043i −0.883452 0.468521i \(-0.844787\pi\)
0.883452 0.468521i \(-0.155213\pi\)
\(42\) 0 0
\(43\) 6.12372 + 6.12372i 0.933859 + 0.933859i 0.997944 0.0640852i \(-0.0204129\pi\)
−0.0640852 + 0.997944i \(0.520413\pi\)
\(44\) −6.00000 −0.904534
\(45\) 0 0
\(46\) 12.0000 1.76930
\(47\) 7.34847 + 7.34847i 1.07188 + 1.07188i 0.997208 + 0.0746766i \(0.0237924\pi\)
0.0746766 + 0.997208i \(0.476208\pi\)
\(48\) 0 0
\(49\) 4.00000i 0.571429i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.44949 + 2.44949i −0.336463 + 0.336463i −0.855034 0.518571i \(-0.826464\pi\)
0.518571 + 0.855034i \(0.326464\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 3.00000i 0.400892i
\(57\) 0 0
\(58\) −7.34847 7.34847i −0.964901 0.964901i
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 6.12372 + 6.12372i 0.777714 + 0.777714i
\(63\) 0 0
\(64\) 1.00000i 0.125000i
\(65\) 0 0
\(66\) 0 0
\(67\) 7.34847 7.34847i 0.897758 0.897758i −0.0974792 0.995238i \(-0.531078\pi\)
0.995238 + 0.0974792i \(0.0310779\pi\)
\(68\) 2.44949 2.44949i 0.297044 0.297044i
\(69\) 0 0
\(70\) 0 0
\(71\) 12.0000i 1.42414i −0.702109 0.712069i \(-0.747758\pi\)
0.702109 0.712069i \(-0.252242\pi\)
\(72\) 0 0
\(73\) 6.12372 + 6.12372i 0.716728 + 0.716728i 0.967934 0.251206i \(-0.0808271\pi\)
−0.251206 + 0.967934i \(0.580827\pi\)
\(74\) 3.00000 0.348743
\(75\) 0 0
\(76\) −7.00000 −0.802955
\(77\) 7.34847 + 7.34847i 0.837436 + 0.837436i
\(78\) 0 0
\(79\) 1.00000i 0.112509i −0.998416 0.0562544i \(-0.982084\pi\)
0.998416 0.0562544i \(-0.0179158\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −7.34847 + 7.34847i −0.811503 + 0.811503i
\(83\) 2.44949 2.44949i 0.268866 0.268866i −0.559777 0.828643i \(-0.689113\pi\)
0.828643 + 0.559777i \(0.189113\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 15.0000i 1.61749i
\(87\) 0 0
\(88\) −7.34847 7.34847i −0.783349 0.783349i
\(89\) 12.0000 1.27200 0.635999 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −4.89898 4.89898i −0.510754 0.510754i
\(93\) 0 0
\(94\) 18.0000i 1.85656i
\(95\) 0 0
\(96\) 0 0
\(97\) −8.57321 + 8.57321i −0.870478 + 0.870478i −0.992524 0.122046i \(-0.961054\pi\)
0.122046 + 0.992524i \(0.461054\pi\)
\(98\) 4.89898 4.89898i 0.494872 0.494872i
\(99\) 0 0
\(100\) 0 0
\(101\) 6.00000i 0.597022i 0.954406 + 0.298511i \(0.0964900\pi\)
−0.954406 + 0.298511i \(0.903510\pi\)
\(102\) 0 0
\(103\) −6.12372 6.12372i −0.603388 0.603388i 0.337822 0.941210i \(-0.390310\pi\)
−0.941210 + 0.337822i \(0.890310\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(108\) 0 0
\(109\) 7.00000i 0.670478i −0.942133 0.335239i \(-0.891183\pi\)
0.942133 0.335239i \(-0.108817\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 6.12372 6.12372i 0.578638 0.578638i
\(113\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 6.00000i 0.557086i
\(117\) 0 0
\(118\) 0 0
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) −25.0000 −2.27273
\(122\) 1.22474 + 1.22474i 0.110883 + 0.110883i
\(123\) 0 0
\(124\) 5.00000i 0.449013i
\(125\) 0 0
\(126\) 0 0
\(127\) −7.34847 + 7.34847i −0.652071 + 0.652071i −0.953491 0.301420i \(-0.902539\pi\)
0.301420 + 0.953491i \(0.402539\pi\)
\(128\) −8.57321 + 8.57321i −0.757772 + 0.757772i
\(129\) 0 0
\(130\) 0 0
\(131\) 18.0000i 1.57267i 0.617802 + 0.786334i \(0.288023\pi\)
−0.617802 + 0.786334i \(0.711977\pi\)
\(132\) 0 0
\(133\) 8.57321 + 8.57321i 0.743392 + 0.743392i
\(134\) −18.0000 −1.55496
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) −2.44949 2.44949i −0.209274 0.209274i 0.594685 0.803959i \(-0.297277\pi\)
−0.803959 + 0.594685i \(0.797277\pi\)
\(138\) 0 0
\(139\) 7.00000i 0.593732i 0.954919 + 0.296866i \(0.0959415\pi\)
−0.954919 + 0.296866i \(0.904058\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −14.6969 + 14.6969i −1.23334 + 1.23334i
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 15.0000i 1.24141i
\(147\) 0 0
\(148\) −1.22474 1.22474i −0.100673 0.100673i
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −5.00000 −0.406894 −0.203447 0.979086i \(-0.565214\pi\)
−0.203447 + 0.979086i \(0.565214\pi\)
\(152\) −8.57321 8.57321i −0.695379 0.695379i
\(153\) 0 0
\(154\) 18.0000i 1.45048i
\(155\) 0 0
\(156\) 0 0
\(157\) −13.4722 + 13.4722i −1.07520 + 1.07520i −0.0782656 + 0.996933i \(0.524938\pi\)
−0.996933 + 0.0782656i \(0.975062\pi\)
\(158\) −1.22474 + 1.22474i −0.0974355 + 0.0974355i
\(159\) 0 0
\(160\) 0 0
\(161\) 12.0000i 0.945732i
\(162\) 0 0
\(163\) 7.34847 + 7.34847i 0.575577 + 0.575577i 0.933681 0.358105i \(-0.116577\pi\)
−0.358105 + 0.933681i \(0.616577\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) −6.00000 −0.465690
\(167\) −12.2474 12.2474i −0.947736 0.947736i 0.0509644 0.998700i \(-0.483770\pi\)
−0.998700 + 0.0509644i \(0.983770\pi\)
\(168\) 0 0
\(169\) 13.0000i 1.00000i
\(170\) 0 0
\(171\) 0 0
\(172\) −6.12372 + 6.12372i −0.466930 + 0.466930i
\(173\) −9.79796 + 9.79796i −0.744925 + 0.744925i −0.973521 0.228596i \(-0.926586\pi\)
0.228596 + 0.973521i \(0.426586\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 30.0000i 2.26134i
\(177\) 0 0
\(178\) −14.6969 14.6969i −1.10158 1.10158i
\(179\) 24.0000 1.79384 0.896922 0.442189i \(-0.145798\pi\)
0.896922 + 0.442189i \(0.145798\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 12.0000i 0.884652i
\(185\) 0 0
\(186\) 0 0
\(187\) 14.6969 14.6969i 1.07475 1.07475i
\(188\) −7.34847 + 7.34847i −0.535942 + 0.535942i
\(189\) 0 0
\(190\) 0 0
\(191\) 18.0000i 1.30243i −0.758891 0.651217i \(-0.774259\pi\)
0.758891 0.651217i \(-0.225741\pi\)
\(192\) 0 0
\(193\) −1.22474 1.22474i −0.0881591 0.0881591i 0.661652 0.749811i \(-0.269856\pi\)
−0.749811 + 0.661652i \(0.769856\pi\)
\(194\) 21.0000 1.50771
\(195\) 0 0
\(196\) −4.00000 −0.285714
\(197\) −12.2474 12.2474i −0.872595 0.872595i 0.120160 0.992755i \(-0.461659\pi\)
−0.992755 + 0.120160i \(0.961659\pi\)
\(198\) 0 0
\(199\) 20.0000i 1.41776i 0.705328 + 0.708881i \(0.250800\pi\)
−0.705328 + 0.708881i \(0.749200\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 7.34847 7.34847i 0.517036 0.517036i
\(203\) 7.34847 7.34847i 0.515761 0.515761i
\(204\) 0 0
\(205\) 0 0
\(206\) 15.0000i 1.04510i
\(207\) 0 0
\(208\) 0 0
\(209\) −42.0000 −2.90520
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) −2.44949 2.44949i −0.168232 0.168232i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −6.12372 + 6.12372i −0.415705 + 0.415705i
\(218\) −8.57321 + 8.57321i −0.580651 + 0.580651i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 8.57321 + 8.57321i 0.574105 + 0.574105i 0.933273 0.359168i \(-0.116940\pi\)
−0.359168 + 0.933273i \(0.616940\pi\)
\(224\) −9.00000 −0.601338
\(225\) 0 0
\(226\) 0 0
\(227\) −4.89898 4.89898i −0.325157 0.325157i 0.525585 0.850741i \(-0.323847\pi\)
−0.850741 + 0.525585i \(0.823847\pi\)
\(228\) 0 0
\(229\) 5.00000i 0.330409i −0.986259 0.165205i \(-0.947172\pi\)
0.986259 0.165205i \(-0.0528285\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −7.34847 + 7.34847i −0.482451 + 0.482451i
\(233\) 7.34847 7.34847i 0.481414 0.481414i −0.424169 0.905583i \(-0.639434\pi\)
0.905583 + 0.424169i \(0.139434\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 7.34847 + 7.34847i 0.476331 + 0.476331i
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 30.6186 + 30.6186i 1.96824 + 1.96824i
\(243\) 0 0
\(244\) 1.00000i 0.0640184i
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 6.12372 6.12372i 0.388857 0.388857i
\(249\) 0 0
\(250\) 0 0
\(251\) 6.00000i 0.378717i −0.981908 0.189358i \(-0.939359\pi\)
0.981908 0.189358i \(-0.0606408\pi\)
\(252\) 0 0
\(253\) −29.3939 29.3939i −1.84798 1.84798i
\(254\) 18.0000 1.12942
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) 12.2474 + 12.2474i 0.763975 + 0.763975i 0.977038 0.213063i \(-0.0683441\pi\)
−0.213063 + 0.977038i \(0.568344\pi\)
\(258\) 0 0
\(259\) 3.00000i 0.186411i
\(260\) 0 0
\(261\) 0 0
\(262\) 22.0454 22.0454i 1.36197 1.36197i
\(263\) −7.34847 + 7.34847i −0.453126 + 0.453126i −0.896391 0.443265i \(-0.853820\pi\)
0.443265 + 0.896391i \(0.353820\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 21.0000i 1.28759i
\(267\) 0 0
\(268\) 7.34847 + 7.34847i 0.448879 + 0.448879i
\(269\) −12.0000 −0.731653 −0.365826 0.930683i \(-0.619214\pi\)
−0.365826 + 0.930683i \(0.619214\pi\)
\(270\) 0 0
\(271\) 11.0000 0.668202 0.334101 0.942537i \(-0.391567\pi\)
0.334101 + 0.942537i \(0.391567\pi\)
\(272\) −12.2474 12.2474i −0.742611 0.742611i
\(273\) 0 0
\(274\) 6.00000i 0.362473i
\(275\) 0 0
\(276\) 0 0
\(277\) 6.12372 6.12372i 0.367939 0.367939i −0.498786 0.866725i \(-0.666221\pi\)
0.866725 + 0.498786i \(0.166221\pi\)
\(278\) 8.57321 8.57321i 0.514187 0.514187i
\(279\) 0 0
\(280\) 0 0
\(281\) 6.00000i 0.357930i −0.983855 0.178965i \(-0.942725\pi\)
0.983855 0.178965i \(-0.0572749\pi\)
\(282\) 0 0
\(283\) 1.22474 + 1.22474i 0.0728035 + 0.0728035i 0.742571 0.669767i \(-0.233606\pi\)
−0.669767 + 0.742571i \(0.733606\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) 0 0
\(287\) −7.34847 7.34847i −0.433766 0.433766i
\(288\) 0 0
\(289\) 5.00000i 0.294118i
\(290\) 0 0
\(291\) 0 0
\(292\) −6.12372 + 6.12372i −0.358364 + 0.358364i
\(293\) 19.5959 19.5959i 1.14481 1.14481i 0.157246 0.987559i \(-0.449738\pi\)
0.987559 0.157246i \(-0.0502617\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 3.00000i 0.174371i
\(297\) 0 0
\(298\) 7.34847 + 7.34847i 0.425685 + 0.425685i
\(299\) 0 0
\(300\) 0 0
\(301\) 15.0000 0.864586
\(302\) 6.12372 + 6.12372i 0.352381 + 0.352381i
\(303\) 0 0
\(304\) 35.0000i 2.00739i
\(305\) 0 0
\(306\) 0 0
\(307\) 1.22474 1.22474i 0.0698999 0.0698999i −0.671293 0.741192i \(-0.734261\pi\)
0.741192 + 0.671293i \(0.234261\pi\)
\(308\) −7.34847 + 7.34847i −0.418718 + 0.418718i
\(309\) 0 0
\(310\) 0 0
\(311\) 6.00000i 0.340229i −0.985424 0.170114i \(-0.945586\pi\)
0.985424 0.170114i \(-0.0544137\pi\)
\(312\) 0 0
\(313\) 14.6969 + 14.6969i 0.830720 + 0.830720i 0.987615 0.156895i \(-0.0501485\pi\)
−0.156895 + 0.987615i \(0.550148\pi\)
\(314\) 33.0000 1.86230
\(315\) 0 0
\(316\) 1.00000 0.0562544
\(317\) 17.1464 + 17.1464i 0.963039 + 0.963039i 0.999341 0.0363015i \(-0.0115577\pi\)
−0.0363015 + 0.999341i \(0.511558\pi\)
\(318\) 0 0
\(319\) 36.0000i 2.01561i
\(320\) 0 0
\(321\) 0 0
\(322\) 14.6969 14.6969i 0.819028 0.819028i
\(323\) 17.1464 17.1464i 0.954053 0.954053i
\(324\) 0 0
\(325\) 0 0
\(326\) 18.0000i 0.996928i
\(327\) 0 0
\(328\) 7.34847 + 7.34847i 0.405751 + 0.405751i
\(329\) 18.0000 0.992372
\(330\) 0 0
\(331\) 5.00000 0.274825 0.137412 0.990514i \(-0.456121\pi\)
0.137412 + 0.990514i \(0.456121\pi\)
\(332\) 2.44949 + 2.44949i 0.134433 + 0.134433i
\(333\) 0 0
\(334\) 30.0000i 1.64153i
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(338\) −15.9217 + 15.9217i −0.866025 + 0.866025i
\(339\) 0 0
\(340\) 0 0
\(341\) 30.0000i 1.62459i
\(342\) 0 0
\(343\) 13.4722 + 13.4722i 0.727430 + 0.727430i
\(344\) −15.0000 −0.808746
\(345\) 0 0
\(346\) 24.0000 1.29025
\(347\) 14.6969 + 14.6969i 0.788973 + 0.788973i 0.981326 0.192353i \(-0.0616118\pi\)
−0.192353 + 0.981326i \(0.561612\pi\)
\(348\) 0 0
\(349\) 35.0000i 1.87351i −0.349990 0.936754i \(-0.613815\pi\)
0.349990 0.936754i \(-0.386185\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 22.0454 22.0454i 1.17502 1.17502i
\(353\) 14.6969 14.6969i 0.782239 0.782239i −0.197969 0.980208i \(-0.563435\pi\)
0.980208 + 0.197969i \(0.0634346\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 12.0000i 0.635999i
\(357\) 0 0
\(358\) −29.3939 29.3939i −1.55351 1.55351i
\(359\) −18.0000 −0.950004 −0.475002 0.879985i \(-0.657553\pi\)
−0.475002 + 0.879985i \(0.657553\pi\)
\(360\) 0 0
\(361\) −30.0000 −1.57895
\(362\) 17.1464 + 17.1464i 0.901196 + 0.901196i
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 22.0454 22.0454i 1.15076 1.15076i 0.164361 0.986400i \(-0.447444\pi\)
0.986400 0.164361i \(-0.0525561\pi\)
\(368\) −24.4949 + 24.4949i −1.27688 + 1.27688i
\(369\) 0 0
\(370\) 0 0
\(371\) 6.00000i 0.311504i
\(372\) 0 0
\(373\) 1.22474 + 1.22474i 0.0634149 + 0.0634149i 0.738103 0.674688i \(-0.235722\pi\)
−0.674688 + 0.738103i \(0.735722\pi\)
\(374\) −36.0000 −1.86152
\(375\) 0 0
\(376\) −18.0000 −0.928279
\(377\) 0 0
\(378\) 0 0
\(379\) 4.00000i 0.205466i 0.994709 + 0.102733i \(0.0327588\pi\)
−0.994709 + 0.102733i \(0.967241\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −22.0454 + 22.0454i −1.12794 + 1.12794i
\(383\) −2.44949 + 2.44949i −0.125163 + 0.125163i −0.766914 0.641750i \(-0.778208\pi\)
0.641750 + 0.766914i \(0.278208\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 3.00000i 0.152696i
\(387\) 0 0
\(388\) −8.57321 8.57321i −0.435239 0.435239i
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) 24.0000 1.21373
\(392\) −4.89898 4.89898i −0.247436 0.247436i
\(393\) 0 0
\(394\) 30.0000i 1.51138i
\(395\) 0 0
\(396\) 0 0
\(397\) 13.4722 13.4722i 0.676150 0.676150i −0.282977 0.959127i \(-0.591322\pi\)
0.959127 + 0.282977i \(0.0913219\pi\)
\(398\) 24.4949 24.4949i 1.22782 1.22782i
\(399\) 0 0
\(400\) 0 0
\(401\) 12.0000i 0.599251i 0.954057 + 0.299626i \(0.0968618\pi\)
−0.954057 + 0.299626i \(0.903138\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) −18.0000 −0.893325
\(407\) −7.34847 7.34847i −0.364250 0.364250i
\(408\) 0 0
\(409\) 10.0000i 0.494468i 0.968956 + 0.247234i \(0.0795217\pi\)
−0.968956 + 0.247234i \(0.920478\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 6.12372 6.12372i 0.301694 0.301694i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 51.4393 + 51.4393i 2.51598 + 2.51598i
\(419\) −24.0000 −1.17248 −0.586238 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 0 0
\(421\) 19.0000 0.926003 0.463002 0.886357i \(-0.346772\pi\)
0.463002 + 0.886357i \(0.346772\pi\)
\(422\) −9.79796 9.79796i −0.476957 0.476957i
\(423\) 0 0
\(424\) 6.00000i 0.291386i
\(425\) 0 0
\(426\) 0 0
\(427\) −1.22474 + 1.22474i −0.0592696 + 0.0592696i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 30.0000i 1.44505i 0.691345 + 0.722525i \(0.257018\pi\)
−0.691345 + 0.722525i \(0.742982\pi\)
\(432\) 0 0
\(433\) −13.4722 13.4722i −0.647432 0.647432i 0.304939 0.952372i \(-0.401364\pi\)
−0.952372 + 0.304939i \(0.901364\pi\)
\(434\) 15.0000 0.720023
\(435\) 0 0
\(436\) 7.00000 0.335239
\(437\) −34.2929 34.2929i −1.64045 1.64045i
\(438\) 0 0
\(439\) 17.0000i 0.811366i 0.914014 + 0.405683i \(0.132966\pi\)
−0.914014 + 0.405683i \(0.867034\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4.89898 4.89898i 0.232758 0.232758i −0.581085 0.813843i \(-0.697372\pi\)
0.813843 + 0.581085i \(0.197372\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 21.0000i 0.994379i
\(447\) 0 0
\(448\) −1.22474 1.22474i −0.0578638 0.0578638i
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) 36.0000 1.69517
\(452\) 0 0
\(453\) 0 0
\(454\) 12.0000i 0.563188i
\(455\) 0 0
\(456\) 0 0
\(457\) −14.6969 + 14.6969i −0.687494 + 0.687494i −0.961677 0.274184i \(-0.911592\pi\)
0.274184 + 0.961677i \(0.411592\pi\)
\(458\) −6.12372 + 6.12372i −0.286143 + 0.286143i
\(459\) 0 0
\(460\) 0 0
\(461\) 24.0000i 1.11779i 0.829238 + 0.558896i \(0.188775\pi\)
−0.829238 + 0.558896i \(0.811225\pi\)
\(462\) 0 0
\(463\) −23.2702 23.2702i −1.08146 1.08146i −0.996374 0.0850817i \(-0.972885\pi\)
−0.0850817 0.996374i \(-0.527115\pi\)
\(464\) 30.0000 1.39272
\(465\) 0 0
\(466\) −18.0000 −0.833834
\(467\) −4.89898 4.89898i −0.226698 0.226698i 0.584614 0.811312i \(-0.301246\pi\)
−0.811312 + 0.584614i \(0.801246\pi\)
\(468\) 0 0
\(469\) 18.0000i 0.831163i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −36.7423 + 36.7423i −1.68941 + 1.68941i
\(474\) 0 0
\(475\) 0 0
\(476\) 6.00000i 0.275010i
\(477\) 0 0
\(478\) 7.34847 + 7.34847i 0.336111 + 0.336111i
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −2.44949 2.44949i −0.111571 0.111571i
\(483\) 0 0
\(484\) 25.0000i 1.13636i
\(485\) 0 0
\(486\) 0 0
\(487\) 22.0454 22.0454i 0.998973 0.998973i −0.00102669 0.999999i \(-0.500327\pi\)
0.999999 + 0.00102669i \(0.000326807\pi\)
\(488\) 1.22474 1.22474i 0.0554416 0.0554416i
\(489\) 0 0
\(490\) 0 0
\(491\) 30.0000i 1.35388i −0.736038 0.676941i \(-0.763305\pi\)
0.736038 0.676941i \(-0.236695\pi\)
\(492\) 0 0
\(493\) −14.6969 14.6969i −0.661917 0.661917i
\(494\) 0 0
\(495\) 0 0
\(496\) −25.0000 −1.12253
\(497\) −14.6969 14.6969i −0.659248 0.659248i
\(498\) 0 0
\(499\) 13.0000i 0.581960i −0.956729 0.290980i \(-0.906019\pi\)
0.956729 0.290980i \(-0.0939813\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −7.34847 + 7.34847i −0.327978 + 0.327978i
\(503\) −12.2474 + 12.2474i −0.546087 + 0.546087i −0.925307 0.379220i \(-0.876192\pi\)
0.379220 + 0.925307i \(0.376192\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 72.0000i 3.20079i
\(507\) 0 0
\(508\) −7.34847 7.34847i −0.326036 0.326036i
\(509\) −24.0000 −1.06378 −0.531891 0.846813i \(-0.678518\pi\)
−0.531891 + 0.846813i \(0.678518\pi\)
\(510\) 0 0
\(511\) 15.0000 0.663561
\(512\) −6.12372 6.12372i −0.270633 0.270633i
\(513\) 0 0
\(514\) 30.0000i 1.32324i
\(515\) 0 0
\(516\) 0 0
\(517\) −44.0908 + 44.0908i −1.93911 + 1.93911i
\(518\) 3.67423 3.67423i 0.161437 0.161437i
\(519\) 0 0
\(520\) 0 0
\(521\) 12.0000i 0.525730i −0.964833 0.262865i \(-0.915333\pi\)
0.964833 0.262865i \(-0.0846673\pi\)
\(522\) 0 0
\(523\) −6.12372 6.12372i −0.267772 0.267772i 0.560430 0.828202i \(-0.310636\pi\)
−0.828202 + 0.560430i \(0.810636\pi\)
\(524\) −18.0000 −0.786334
\(525\) 0 0
\(526\) 18.0000 0.784837
\(527\) 12.2474 + 12.2474i 0.533507 + 0.533507i
\(528\) 0 0
\(529\) 25.0000i 1.08696i
\(530\) 0 0
\(531\) 0 0
\(532\) −8.57321 + 8.57321i −0.371696 + 0.371696i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 18.0000i 0.777482i
\(537\) 0 0
\(538\) 14.6969 + 14.6969i 0.633630 + 0.633630i
\(539\) −24.0000 −1.03375
\(540\) 0 0
\(541\) −19.0000 −0.816874 −0.408437 0.912787i \(-0.633926\pi\)
−0.408437 + 0.912787i \(0.633926\pi\)
\(542\) −13.4722 13.4722i −0.578680 0.578680i
\(543\) 0 0
\(544\) 18.0000i 0.771744i
\(545\) 0 0
\(546\) 0 0
\(547\) 8.57321 8.57321i 0.366564 0.366564i −0.499658 0.866223i \(-0.666541\pi\)
0.866223 + 0.499658i \(0.166541\pi\)
\(548\) 2.44949 2.44949i 0.104637 0.104637i
\(549\) 0 0
\(550\) 0 0
\(551\) 42.0000i 1.78926i
\(552\) 0 0
\(553\) −1.22474 1.22474i −0.0520814 0.0520814i
\(554\) −15.0000 −0.637289
\(555\) 0 0
\(556\) −7.00000 −0.296866
\(557\) 7.34847 + 7.34847i 0.311365 + 0.311365i 0.845438 0.534073i \(-0.179339\pi\)
−0.534073 + 0.845438i \(0.679339\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −7.34847 + 7.34847i −0.309976 + 0.309976i
\(563\) −14.6969 + 14.6969i −0.619402 + 0.619402i −0.945378 0.325976i \(-0.894307\pi\)
0.325976 + 0.945378i \(0.394307\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 3.00000i 0.126099i
\(567\) 0 0
\(568\) 14.6969 + 14.6969i 0.616670 + 0.616670i
\(569\) 36.0000 1.50920 0.754599 0.656186i \(-0.227831\pi\)
0.754599 + 0.656186i \(0.227831\pi\)
\(570\) 0 0
\(571\) 5.00000 0.209243 0.104622 0.994512i \(-0.466637\pi\)
0.104622 + 0.994512i \(0.466637\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 18.0000i 0.751305i
\(575\) 0 0
\(576\) 0 0
\(577\) −1.22474 + 1.22474i −0.0509868 + 0.0509868i −0.732140 0.681154i \(-0.761479\pi\)
0.681154 + 0.732140i \(0.261479\pi\)
\(578\) −6.12372 + 6.12372i −0.254713 + 0.254713i
\(579\) 0 0
\(580\) 0 0
\(581\) 6.00000i 0.248922i
\(582\) 0 0
\(583\) −14.6969 14.6969i −0.608685 0.608685i
\(584\) −15.0000 −0.620704
\(585\) 0 0
\(586\) −48.0000 −1.98286
\(587\) 12.2474 + 12.2474i 0.505506 + 0.505506i 0.913144 0.407638i \(-0.133647\pi\)
−0.407638 + 0.913144i \(0.633647\pi\)
\(588\) 0 0
\(589\) 35.0000i 1.44215i
\(590\) 0 0
\(591\) 0 0
\(592\) −6.12372 + 6.12372i −0.251684 + 0.251684i
\(593\) −19.5959 + 19.5959i −0.804708 + 0.804708i −0.983827 0.179119i \(-0.942675\pi\)
0.179119 + 0.983827i \(0.442675\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6.00000i 0.245770i
\(597\) 0 0
\(598\) 0 0
\(599\) 36.0000 1.47092 0.735460 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) 0 0
\(601\) −35.0000 −1.42768 −0.713840 0.700309i \(-0.753046\pi\)
−0.713840 + 0.700309i \(0.753046\pi\)
\(602\) −18.3712 18.3712i −0.748753 0.748753i
\(603\) 0 0
\(604\) 5.00000i 0.203447i
\(605\) 0 0
\(606\) 0 0
\(607\) 6.12372 6.12372i 0.248554 0.248554i −0.571823 0.820377i \(-0.693764\pi\)
0.820377 + 0.571823i \(0.193764\pi\)
\(608\) 25.7196 25.7196i 1.04307 1.04307i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −13.4722 13.4722i −0.544137 0.544137i 0.380602 0.924739i \(-0.375717\pi\)
−0.924739 + 0.380602i \(0.875717\pi\)
\(614\) −3.00000 −0.121070
\(615\) 0 0
\(616\) −18.0000 −0.725241
\(617\) −9.79796 9.79796i −0.394451 0.394451i 0.481820 0.876270i \(-0.339976\pi\)
−0.876270 + 0.481820i \(0.839976\pi\)
\(618\) 0 0
\(619\) 31.0000i 1.24600i −0.782224 0.622998i \(-0.785915\pi\)
0.782224 0.622998i \(-0.214085\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −7.34847 + 7.34847i −0.294647 + 0.294647i
\(623\) 14.6969 14.6969i 0.588820 0.588820i
\(624\) 0 0
\(625\) 0 0
\(626\) 36.0000i 1.43885i
\(627\) 0 0
\(628\) −13.4722 13.4722i −0.537599 0.537599i
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 1.22474 + 1.22474i 0.0487177 + 0.0487177i
\(633\) 0 0
\(634\) 42.0000i 1.66803i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 44.0908 44.0908i 1.74557 1.74557i
\(639\) 0 0
\(640\) 0 0
\(641\) 24.0000i 0.947943i −0.880540 0.473972i \(-0.842820\pi\)
0.880540 0.473972i \(-0.157180\pi\)
\(642\) 0 0
\(643\) 7.34847 + 7.34847i 0.289795 + 0.289795i 0.836999 0.547204i \(-0.184308\pi\)
−0.547204 + 0.836999i \(0.684308\pi\)
\(644\) −12.0000 −0.472866
\(645\) 0 0
\(646\) −42.0000 −1.65247
\(647\) 24.4949 + 24.4949i 0.962994 + 0.962994i 0.999339 0.0363455i \(-0.0115717\pi\)
−0.0363455 + 0.999339i \(0.511572\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −7.34847 + 7.34847i −0.287788 + 0.287788i
\(653\) 4.89898 4.89898i 0.191712 0.191712i −0.604724 0.796435i \(-0.706716\pi\)
0.796435 + 0.604724i \(0.206716\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 30.0000i 1.17130i
\(657\) 0 0
\(658\) −22.0454 22.0454i −0.859419 0.859419i
\(659\) 6.00000 0.233727 0.116863 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) 0 0
\(661\) 25.0000 0.972387 0.486194 0.873851i \(-0.338385\pi\)
0.486194 + 0.873851i \(0.338385\pi\)
\(662\) −6.12372 6.12372i −0.238005 0.238005i
\(663\) 0 0
\(664\) 6.00000i 0.232845i
\(665\) 0 0
\(666\) 0 0
\(667\) −29.3939 + 29.3939i −1.13814 + 1.13814i
\(668\) 12.2474 12.2474i 0.473868 0.473868i
\(669\) 0 0
\(670\) 0 0
\(671\) 6.00000i 0.231627i
\(672\) 0 0
\(673\) 28.1691 + 28.1691i 1.08584 + 1.08584i 0.995952 + 0.0898884i \(0.0286510\pi\)
0.0898884 + 0.995952i \(0.471349\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 13.0000 0.500000
\(677\) 7.34847 + 7.34847i 0.282425 + 0.282425i 0.834075 0.551651i \(-0.186002\pi\)
−0.551651 + 0.834075i \(0.686002\pi\)
\(678\) 0 0
\(679\) 21.0000i 0.805906i
\(680\) 0 0
\(681\) 0 0
\(682\) −36.7423 + 36.7423i −1.40694 + 1.40694i
\(683\) −31.8434 + 31.8434i −1.21845 + 1.21845i −0.250279 + 0.968174i \(0.580522\pi\)
−0.968174 + 0.250279i \(0.919478\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 33.0000i 1.25995i
\(687\) 0 0
\(688\) 30.6186 + 30.6186i 1.16732 + 1.16732i
\(689\) 0 0
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) −9.79796 9.79796i −0.372463 0.372463i
\(693\) 0 0
\(694\) 36.0000i 1.36654i
\(695\) 0 0
\(696\) 0 0
\(697\) −14.6969 + 14.6969i −0.556686 + 0.556686i
\(698\) −42.8661 + 42.8661i −1.62250 + 1.62250i
\(699\) 0 0
\(700\) 0 0
\(701\) 24.0000i 0.906467i 0.891392 + 0.453234i \(0.149730\pi\)
−0.891392 + 0.453234i \(0.850270\pi\)
\(702\) 0 0
\(703\) −8.57321 8.57321i −0.323345 0.323345i
\(704\) 6.00000 0.226134
\(705\) 0 0
\(706\) −36.0000 −1.35488
\(707\) 7.34847 + 7.34847i 0.276368 + 0.276368i
\(708\) 0 0
\(709\) 5.00000i 0.187779i −0.995583 0.0938895i \(-0.970070\pi\)
0.995583 0.0938895i \(-0.0299300\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −14.6969 + 14.6969i −0.550791 + 0.550791i
\(713\) 24.4949 24.4949i 0.917341 0.917341i
\(714\) 0 0
\(715\) 0 0
\(716\) 24.0000i 0.896922i
\(717\) 0 0
\(718\) 22.0454 + 22.0454i 0.822727 + 0.822727i
\(719\) 30.0000 1.11881 0.559406 0.828894i \(-0.311029\pi\)
0.559406 + 0.828894i \(0.311029\pi\)
\(720\) 0 0
\(721\) −15.0000 −0.558629
\(722\) 36.7423 + 36.7423i 1.36741 + 1.36741i
\(723\) 0 0
\(724\) 14.0000i 0.520306i
\(725\) 0 0
\(726\) 0 0
\(727\) 23.2702 23.2702i 0.863042 0.863042i −0.128648 0.991690i \(-0.541064\pi\)
0.991690 + 0.128648i \(0.0410638\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 30.0000i 1.10959i
\(732\) 0 0
\(733\) −29.3939 29.3939i −1.08569 1.08569i −0.995967 0.0897206i \(-0.971403\pi\)
−0.0897206 0.995967i \(-0.528597\pi\)
\(734\) −54.0000 −1.99318
\(735\) 0 0
\(736\) 36.0000 1.32698
\(737\) 44.0908 + 44.0908i 1.62411 + 1.62411i
\(738\) 0 0
\(739\) 40.0000i 1.47142i 0.677295 + 0.735712i \(0.263152\pi\)
−0.677295 + 0.735712i \(0.736848\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 7.34847 7.34847i 0.269771 0.269771i
\(743\) 29.3939 29.3939i 1.07836 1.07836i 0.0816998 0.996657i \(-0.473965\pi\)
0.996657 0.0816998i \(-0.0260349\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 3.00000i 0.109838i
\(747\) 0 0
\(748\) 14.6969 + 14.6969i 0.537373 + 0.537373i
\(749\) 0 0
\(750\) 0 0
\(751\) −23.0000 −0.839282 −0.419641 0.907690i \(-0.637844\pi\)
−0.419641 + 0.907690i \(0.637844\pi\)
\(752\) 36.7423 + 36.7423i 1.33986 + 1.33986i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 8.57321 8.57321i 0.311599 0.311599i −0.533930 0.845529i \(-0.679285\pi\)
0.845529 + 0.533930i \(0.179285\pi\)
\(758\) 4.89898 4.89898i 0.177939 0.177939i
\(759\) 0 0
\(760\) 0 0
\(761\) 18.0000i 0.652499i 0.945284 + 0.326250i \(0.105785\pi\)
−0.945284 + 0.326250i \(0.894215\pi\)
\(762\) 0 0
\(763\) −8.57321 8.57321i −0.310371 0.310371i
\(764\) 18.0000 0.651217
\(765\) 0 0
\(766\) 6.00000 0.216789
\(767\) 0 0
\(768\) 0 0
\(769\) 10.0000i 0.360609i −0.983611 0.180305i \(-0.942292\pi\)
0.983611 0.180305i \(-0.0577084\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.22474 1.22474i 0.0440795 0.0440795i
\(773\) 17.1464 17.1464i 0.616714 0.616714i −0.327973 0.944687i \(-0.606365\pi\)
0.944687 + 0.327973i \(0.106365\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 21.0000i 0.753856i
\(777\) 0 0
\(778\) −36.7423 36.7423i −1.31728 1.31728i
\(779\) 42.0000 1.50481
\(780\) 0 0
\(781\) 72.0000 2.57636
\(782\) −29.3939 29.3939i −1.05112 1.05112i
\(783\) 0 0
\(784\) 20.0000i 0.714286i
\(785\) 0 0
\(786\) 0 0
\(787\) 37.9671 37.9671i 1.35338 1.35338i 0.471531 0.881849i \(-0.343701\pi\)
0.881849 0.471531i \(-0.156299\pi\)
\(788\) 12.2474 12.2474i 0.436297 0.436297i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) −33.0000 −1.17113
\(795\) 0 0
\(796\) −20.0000 −0.708881
\(797\) 9.79796 + 9.79796i 0.347062 + 0.347062i 0.859014 0.511952i \(-0.171078\pi\)
−0.511952 + 0.859014i \(0.671078\pi\)
\(798\) 0 0
\(799\) 36.0000i 1.27359i
\(800\) 0 0
\(801\) 0 0
\(802\) 14.6969 14.6969i 0.518967 0.518967i
\(803\) −36.7423 + 36.7423i −1.29661 + 1.29661i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) −7.34847 7.34847i −0.258518 0.258518i
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 1.00000 0.0351147 0.0175574 0.999846i \(-0.494411\pi\)
0.0175574 + 0.999846i \(0.494411\pi\)
\(812\) 7.34847 + 7.34847i 0.257881 + 0.257881i
\(813\) 0 0
\(814\) 18.0000i 0.630900i
\(815\) 0 0
\(816\) 0 0
\(817\) −42.8661 + 42.8661i −1.49969 + 1.49969i
\(818\) 12.2474 12.2474i 0.428222 0.428222i
\(819\) 0 0
\(820\) 0 0
\(821\) 30.0000i 1.04701i −0.852023 0.523504i \(-0.824625\pi\)
0.852023 0.523504i \(-0.175375\pi\)
\(822\) 0 0
\(823\) 7.34847 + 7.34847i 0.256152 + 0.256152i 0.823487 0.567335i \(-0.192026\pi\)
−0.567335 + 0.823487i \(0.692026\pi\)
\(824\) 15.0000 0.522550
\(825\) 0 0
\(826\) 0 0
\(827\) −17.1464 17.1464i −0.596240 0.596240i 0.343070 0.939310i \(-0.388533\pi\)
−0.939310 + 0.343070i \(0.888533\pi\)
\(828\) 0 0
\(829\) 53.0000i 1.84077i −0.391018 0.920383i \(-0.627877\pi\)
0.391018 0.920383i \(-0.372123\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 9.79796 9.79796i 0.339479 0.339479i
\(834\) 0 0
\(835\) 0 0
\(836\) 42.0000i 1.45260i
\(837\) 0 0
\(838\) 29.3939 + 29.3939i 1.01539 + 1.01539i
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −23.2702 23.2702i −0.801942 0.801942i
\(843\) 0 0
\(844\) 8.00000i 0.275371i
\(845\) 0 0
\(846\) 0 0
\(847\) −30.6186 + 30.6186i −1.05207 + 1.05207i
\(848\) −12.2474 + 12.2474i −0.420579 + 0.420579i
\(849\) 0 0
\(850\) 0 0
\(851\) 12.0000i 0.411355i
\(852\) 0 0
\(853\) −14.6969 14.6969i −0.503214 0.503214i 0.409221 0.912435i \(-0.365800\pi\)
−0.912435 + 0.409221i \(0.865800\pi\)
\(854\) 3.00000 0.102658
\(855\) 0 0
\(856\) 0 0
\(857\) −24.4949 24.4949i −0.836730 0.836730i 0.151697 0.988427i \(-0.451526\pi\)
−0.988427 + 0.151697i \(0.951526\pi\)
\(858\) 0 0
\(859\) 5.00000i 0.170598i 0.996355 + 0.0852989i \(0.0271845\pi\)
−0.996355 + 0.0852989i \(0.972815\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 36.7423 36.7423i 1.25145 1.25145i
\(863\) 12.2474 12.2474i 0.416908 0.416908i −0.467229 0.884137i \(-0.654747\pi\)
0.884137 + 0.467229i \(0.154747\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 33.0000i 1.12139i
\(867\) 0 0
\(868\) −6.12372 6.12372i −0.207853 0.207853i
\(869\) 6.00000 0.203536
\(870\) 0 0
\(871\) 0 0
\(872\) 8.57321 + 8.57321i 0.290326 + 0.290326i
\(873\) 0 0
\(874\) 84.0000i 2.84134i
\(875\) 0 0
\(876\) 0 0
\(877\) 30.6186 30.6186i 1.03392 1.03392i 0.0345131 0.999404i \(-0.489012\pi\)
0.999404 0.0345131i \(-0.0109881\pi\)
\(878\) 20.8207 20.8207i 0.702663 0.702663i
\(879\) 0 0
\(880\) 0 0
\(881\) 18.0000i 0.606435i 0.952921 + 0.303218i \(0.0980609\pi\)
−0.952921 + 0.303218i \(0.901939\pi\)
\(882\) 0 0
\(883\) 1.22474 + 1.22474i 0.0412159 + 0.0412159i 0.727414 0.686198i \(-0.240722\pi\)
−0.686198 + 0.727414i \(0.740722\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) −2.44949 2.44949i −0.0822458 0.0822458i 0.664787 0.747033i \(-0.268522\pi\)
−0.747033 + 0.664787i \(0.768522\pi\)
\(888\) 0 0
\(889\) 18.0000i 0.603701i
\(890\) 0 0
\(891\) 0 0
\(892\) −8.57321 + 8.57321i −0.287052 + 0.287052i
\(893\) −51.4393 + 51.4393i −1.72135 + 1.72135i
\(894\) 0 0
\(895\) 0 0
\(896\) 21.0000i 0.701561i
\(897\) 0 0
\(898\) −7.34847 7.34847i −0.245222 0.245222i
\(899\) −30.0000 −1.00056
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) −44.0908 44.0908i −1.46806 1.46806i
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 28.1691 28.1691i 0.935341 0.935341i −0.0626922 0.998033i \(-0.519969\pi\)
0.998033 + 0.0626922i \(0.0199687\pi\)
\(908\) 4.89898 4.89898i 0.162578 0.162578i
\(909\) 0 0
\(910\) 0 0
\(911\) 18.0000i 0.596367i −0.954509 0.298183i \(-0.903619\pi\)
0.954509 0.298183i \(-0.0963807\pi\)
\(912\) 0 0
\(913\) 14.6969 + 14.6969i 0.486398 + 0.486398i
\(914\) 36.0000 1.19077
\(915\) 0 0
\(916\) 5.00000 0.165205
\(917\) 22.0454 + 22.0454i 0.728003 + 0.728003i
\(918\) 0 0
\(919\) 1.00000i 0.0329870i 0.999864 + 0.0164935i \(0.00525028\pi\)
−0.999864 + 0.0164935i \(0.994750\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 29.3939 29.3939i 0.968036 0.968036i
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 57.0000i 1.87314i
\(927\) 0 0
\(928\) −22.0454 22.0454i −0.723676 0.723676i
\(929\) −24.0000 −0.787414 −0.393707 0.919236i \(-0.628808\pi\)
−0.393707 + 0.919236i \(0.628808\pi\)
\(930\) 0 0
\(931\) −28.0000 −0.917663
\(932\) 7.34847 + 7.34847i 0.240707 + 0.240707i
\(933\) 0 0
\(934\) 12.0000i 0.392652i
\(935\) 0 0
\(936\) 0 0
\(937\) 30.6186 30.6186i 1.00027 1.00027i 0.000266809 1.00000i \(-0.499915\pi\)
1.00000 0.000266809i \(-8.49279e-5\pi\)
\(938\) −22.0454 + 22.0454i −0.719808 + 0.719808i
\(939\) 0 0
\(940\) 0 0
\(941\) 30.0000i 0.977972i 0.872292 + 0.488986i \(0.162633\pi\)
−0.872292 + 0.488986i \(0.837367\pi\)
\(942\) 0 0
\(943\) 29.3939 + 29.3939i 0.957196 + 0.957196i
\(944\) 0 0
\(945\) 0 0
\(946\) 90.0000 2.92615
\(947\) −31.8434 31.8434i −1.03477 1.03477i −0.999373 0.0353971i \(-0.988730\pi\)
−0.0353971 0.999373i \(-0.511270\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 7.34847 7.34847i 0.238165 0.238165i
\(953\) −7.34847 + 7.34847i −0.238040 + 0.238040i −0.816038 0.577998i \(-0.803834\pi\)
0.577998 + 0.816038i \(0.303834\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 6.00000i 0.194054i
\(957\) 0 0
\(958\) 0 0
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) 0 0
\(963\) 0 0
\(964\) 2.00000i 0.0644157i
\(965\) 0 0
\(966\) 0 0
\(967\) 13.4722 13.4722i 0.433237 0.433237i −0.456491 0.889728i \(-0.650894\pi\)
0.889728 + 0.456491i \(0.150894\pi\)
\(968\) 30.6186 30.6186i 0.984120 0.984120i
\(969\) 0 0
\(970\) 0 0
\(971\) 36.0000i 1.15529i 0.816286 + 0.577647i \(0.196029\pi\)
−0.816286 + 0.577647i \(0.803971\pi\)
\(972\) 0 0
\(973\) 8.57321 + 8.57321i 0.274845 + 0.274845i
\(974\) −54.0000 −1.73027
\(975\) 0 0
\(976\) −5.00000 −0.160046
\(977\) 14.6969 + 14.6969i 0.470197 + 0.470197i 0.901978 0.431782i \(-0.142115\pi\)
−0.431782 + 0.901978i \(0.642115\pi\)
\(978\) 0 0
\(979\) 72.0000i 2.30113i
\(980\) 0 0
\(981\) 0 0
\(982\) −36.7423 + 36.7423i −1.17250 + 1.17250i
\(983\) 2.44949 2.44949i 0.0781266 0.0781266i −0.666964 0.745090i \(-0.732406\pi\)
0.745090 + 0.666964i \(0.232406\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 36.0000i 1.14647i
\(987\) 0 0
\(988\) 0 0
\(989\) −60.0000 −1.90789
\(990\) 0 0
\(991\) 47.0000 1.49300 0.746502 0.665383i \(-0.231732\pi\)
0.746502 + 0.665383i \(0.231732\pi\)
\(992\) 18.3712 + 18.3712i 0.583285 + 0.583285i
\(993\) 0 0
\(994\) 36.0000i 1.14185i
\(995\) 0 0
\(996\) 0 0
\(997\) −29.3939 + 29.3939i −0.930913 + 0.930913i −0.997763 0.0668496i \(-0.978705\pi\)
0.0668496 + 0.997763i \(0.478705\pi\)
\(998\) −15.9217 + 15.9217i −0.503992 + 0.503992i
\(999\) 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 675.2.f.b.593.1 yes 4
3.2 odd 2 675.2.f.e.593.2 yes 4
5.2 odd 4 675.2.f.e.107.2 yes 4
5.3 odd 4 675.2.f.e.107.1 yes 4
5.4 even 2 inner 675.2.f.b.593.2 yes 4
15.2 even 4 inner 675.2.f.b.107.1 4
15.8 even 4 inner 675.2.f.b.107.2 yes 4
15.14 odd 2 675.2.f.e.593.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
675.2.f.b.107.1 4 15.2 even 4 inner
675.2.f.b.107.2 yes 4 15.8 even 4 inner
675.2.f.b.593.1 yes 4 1.1 even 1 trivial
675.2.f.b.593.2 yes 4 5.4 even 2 inner
675.2.f.e.107.1 yes 4 5.3 odd 4
675.2.f.e.107.2 yes 4 5.2 odd 4
675.2.f.e.593.1 yes 4 15.14 odd 2
675.2.f.e.593.2 yes 4 3.2 odd 2