Properties

Label 405.2.e.d.271.1
Level $405$
Weight $2$
Character 405.271
Analytic conductor $3.234$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [405,2,Mod(136,405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(405, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("405.136");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 405.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.23394128186\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 271.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 405.271
Dual form 405.2.e.d.136.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.00000 - 1.73205i) q^{4} +(-0.500000 + 0.866025i) q^{5} +(-1.00000 - 1.73205i) q^{7} +O(q^{10})\) \(q+(1.00000 - 1.73205i) q^{4} +(-0.500000 + 0.866025i) q^{5} +(-1.00000 - 1.73205i) q^{7} +(-1.50000 - 2.59808i) q^{11} +(2.00000 - 3.46410i) q^{13} +(-2.00000 - 3.46410i) q^{16} +6.00000 q^{17} -1.00000 q^{19} +(1.00000 + 1.73205i) q^{20} +(-3.00000 + 5.19615i) q^{23} +(-0.500000 - 0.866025i) q^{25} -4.00000 q^{28} +(-4.50000 - 7.79423i) q^{29} +(0.500000 - 0.866025i) q^{31} +2.00000 q^{35} +8.00000 q^{37} +(1.50000 - 2.59808i) q^{41} +(2.00000 + 3.46410i) q^{43} -6.00000 q^{44} +(6.00000 + 10.3923i) q^{47} +(1.50000 - 2.59808i) q^{49} +(-4.00000 - 6.92820i) q^{52} -6.00000 q^{53} +3.00000 q^{55} +(1.50000 - 2.59808i) q^{59} +(5.00000 + 8.66025i) q^{61} -8.00000 q^{64} +(2.00000 + 3.46410i) q^{65} +(-7.00000 + 12.1244i) q^{67} +(6.00000 - 10.3923i) q^{68} +3.00000 q^{71} +2.00000 q^{73} +(-1.00000 + 1.73205i) q^{76} +(-3.00000 + 5.19615i) q^{77} +(8.00000 + 13.8564i) q^{79} +4.00000 q^{80} +(-6.00000 - 10.3923i) q^{83} +(-3.00000 + 5.19615i) q^{85} -15.0000 q^{89} -8.00000 q^{91} +(6.00000 + 10.3923i) q^{92} +(0.500000 - 0.866025i) q^{95} +(2.00000 + 3.46410i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} - q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} - q^{5} - 2 q^{7} - 3 q^{11} + 4 q^{13} - 4 q^{16} + 12 q^{17} - 2 q^{19} + 2 q^{20} - 6 q^{23} - q^{25} - 8 q^{28} - 9 q^{29} + q^{31} + 4 q^{35} + 16 q^{37} + 3 q^{41} + 4 q^{43} - 12 q^{44} + 12 q^{47} + 3 q^{49} - 8 q^{52} - 12 q^{53} + 6 q^{55} + 3 q^{59} + 10 q^{61} - 16 q^{64} + 4 q^{65} - 14 q^{67} + 12 q^{68} + 6 q^{71} + 4 q^{73} - 2 q^{76} - 6 q^{77} + 16 q^{79} + 8 q^{80} - 12 q^{83} - 6 q^{85} - 30 q^{89} - 16 q^{91} + 12 q^{92} + q^{95} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).

\(n\) \(82\) \(326\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(3\) 0 0
\(4\) 1.00000 1.73205i 0.500000 0.866025i
\(5\) −0.500000 + 0.866025i −0.223607 + 0.387298i
\(6\) 0 0
\(7\) −1.00000 1.73205i −0.377964 0.654654i 0.612801 0.790237i \(-0.290043\pi\)
−0.990766 + 0.135583i \(0.956709\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.50000 2.59808i −0.452267 0.783349i 0.546259 0.837616i \(-0.316051\pi\)
−0.998526 + 0.0542666i \(0.982718\pi\)
\(12\) 0 0
\(13\) 2.00000 3.46410i 0.554700 0.960769i −0.443227 0.896410i \(-0.646166\pi\)
0.997927 0.0643593i \(-0.0205004\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −2.00000 3.46410i −0.500000 0.866025i
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 1.00000 + 1.73205i 0.223607 + 0.387298i
\(21\) 0 0
\(22\) 0 0
\(23\) −3.00000 + 5.19615i −0.625543 + 1.08347i 0.362892 + 0.931831i \(0.381789\pi\)
−0.988436 + 0.151642i \(0.951544\pi\)
\(24\) 0 0
\(25\) −0.500000 0.866025i −0.100000 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) −4.00000 −0.755929
\(29\) −4.50000 7.79423i −0.835629 1.44735i −0.893517 0.449029i \(-0.851770\pi\)
0.0578882 0.998323i \(-0.481563\pi\)
\(30\) 0 0
\(31\) 0.500000 0.866025i 0.0898027 0.155543i −0.817625 0.575751i \(-0.804710\pi\)
0.907428 + 0.420208i \(0.138043\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.50000 2.59808i 0.234261 0.405751i −0.724797 0.688963i \(-0.758066\pi\)
0.959058 + 0.283211i \(0.0913998\pi\)
\(42\) 0 0
\(43\) 2.00000 + 3.46410i 0.304997 + 0.528271i 0.977261 0.212041i \(-0.0680112\pi\)
−0.672264 + 0.740312i \(0.734678\pi\)
\(44\) −6.00000 −0.904534
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00000 + 10.3923i 0.875190 + 1.51587i 0.856560 + 0.516047i \(0.172597\pi\)
0.0186297 + 0.999826i \(0.494070\pi\)
\(48\) 0 0
\(49\) 1.50000 2.59808i 0.214286 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) −4.00000 6.92820i −0.554700 0.960769i
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.50000 2.59808i 0.195283 0.338241i −0.751710 0.659494i \(-0.770771\pi\)
0.946993 + 0.321253i \(0.104104\pi\)
\(60\) 0 0
\(61\) 5.00000 + 8.66025i 0.640184 + 1.10883i 0.985391 + 0.170305i \(0.0544754\pi\)
−0.345207 + 0.938527i \(0.612191\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 2.00000 + 3.46410i 0.248069 + 0.429669i
\(66\) 0 0
\(67\) −7.00000 + 12.1244i −0.855186 + 1.48123i 0.0212861 + 0.999773i \(0.493224\pi\)
−0.876472 + 0.481452i \(0.840109\pi\)
\(68\) 6.00000 10.3923i 0.727607 1.26025i
\(69\) 0 0
\(70\) 0 0
\(71\) 3.00000 0.356034 0.178017 0.984027i \(-0.443032\pi\)
0.178017 + 0.984027i \(0.443032\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −1.00000 + 1.73205i −0.114708 + 0.198680i
\(77\) −3.00000 + 5.19615i −0.341882 + 0.592157i
\(78\) 0 0
\(79\) 8.00000 + 13.8564i 0.900070 + 1.55897i 0.827401 + 0.561611i \(0.189818\pi\)
0.0726692 + 0.997356i \(0.476848\pi\)
\(80\) 4.00000 0.447214
\(81\) 0 0
\(82\) 0 0
\(83\) −6.00000 10.3923i −0.658586 1.14070i −0.980982 0.194099i \(-0.937822\pi\)
0.322396 0.946605i \(-0.395512\pi\)
\(84\) 0 0
\(85\) −3.00000 + 5.19615i −0.325396 + 0.563602i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −15.0000 −1.59000 −0.794998 0.606612i \(-0.792528\pi\)
−0.794998 + 0.606612i \(0.792528\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 6.00000 + 10.3923i 0.625543 + 1.08347i
\(93\) 0 0
\(94\) 0 0
\(95\) 0.500000 0.866025i 0.0512989 0.0888523i
\(96\) 0 0
\(97\) 2.00000 + 3.46410i 0.203069 + 0.351726i 0.949516 0.313719i \(-0.101575\pi\)
−0.746447 + 0.665445i \(0.768242\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −2.00000 −0.200000
\(101\) 4.50000 + 7.79423i 0.447767 + 0.775555i 0.998240 0.0592978i \(-0.0188862\pi\)
−0.550474 + 0.834853i \(0.685553\pi\)
\(102\) 0 0
\(103\) −7.00000 + 12.1244i −0.689730 + 1.19465i 0.282194 + 0.959357i \(0.408938\pi\)
−0.971925 + 0.235291i \(0.924396\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 18.0000 1.74013 0.870063 0.492941i \(-0.164078\pi\)
0.870063 + 0.492941i \(0.164078\pi\)
\(108\) 0 0
\(109\) −1.00000 −0.0957826 −0.0478913 0.998853i \(-0.515250\pi\)
−0.0478913 + 0.998853i \(0.515250\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −4.00000 + 6.92820i −0.377964 + 0.654654i
\(113\) 3.00000 5.19615i 0.282216 0.488813i −0.689714 0.724082i \(-0.742264\pi\)
0.971930 + 0.235269i \(0.0755971\pi\)
\(114\) 0 0
\(115\) −3.00000 5.19615i −0.279751 0.484544i
\(116\) −18.0000 −1.67126
\(117\) 0 0
\(118\) 0 0
\(119\) −6.00000 10.3923i −0.550019 0.952661i
\(120\) 0 0
\(121\) 1.00000 1.73205i 0.0909091 0.157459i
\(122\) 0 0
\(123\) 0 0
\(124\) −1.00000 1.73205i −0.0898027 0.155543i
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −4.50000 + 7.79423i −0.393167 + 0.680985i −0.992865 0.119241i \(-0.961954\pi\)
0.599699 + 0.800226i \(0.295287\pi\)
\(132\) 0 0
\(133\) 1.00000 + 1.73205i 0.0867110 + 0.150188i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.00000 + 10.3923i 0.512615 + 0.887875i 0.999893 + 0.0146279i \(0.00465636\pi\)
−0.487278 + 0.873247i \(0.662010\pi\)
\(138\) 0 0
\(139\) 3.50000 6.06218i 0.296866 0.514187i −0.678551 0.734553i \(-0.737392\pi\)
0.975417 + 0.220366i \(0.0707252\pi\)
\(140\) 2.00000 3.46410i 0.169031 0.292770i
\(141\) 0 0
\(142\) 0 0
\(143\) −12.0000 −1.00349
\(144\) 0 0
\(145\) 9.00000 0.747409
\(146\) 0 0
\(147\) 0 0
\(148\) 8.00000 13.8564i 0.657596 1.13899i
\(149\) 3.00000 5.19615i 0.245770 0.425685i −0.716578 0.697507i \(-0.754293\pi\)
0.962348 + 0.271821i \(0.0876260\pi\)
\(150\) 0 0
\(151\) 0.500000 + 0.866025i 0.0406894 + 0.0704761i 0.885653 0.464348i \(-0.153711\pi\)
−0.844963 + 0.534824i \(0.820378\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0.500000 + 0.866025i 0.0401610 + 0.0695608i
\(156\) 0 0
\(157\) 8.00000 13.8564i 0.638470 1.10586i −0.347299 0.937754i \(-0.612901\pi\)
0.985769 0.168107i \(-0.0537655\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 12.0000 0.945732
\(162\) 0 0
\(163\) 2.00000 0.156652 0.0783260 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) −3.00000 5.19615i −0.234261 0.405751i
\(165\) 0 0
\(166\) 0 0
\(167\) 3.00000 5.19615i 0.232147 0.402090i −0.726293 0.687386i \(-0.758758\pi\)
0.958440 + 0.285295i \(0.0920916\pi\)
\(168\) 0 0
\(169\) −1.50000 2.59808i −0.115385 0.199852i
\(170\) 0 0
\(171\) 0 0
\(172\) 8.00000 0.609994
\(173\) −3.00000 5.19615i −0.228086 0.395056i 0.729155 0.684349i \(-0.239913\pi\)
−0.957241 + 0.289292i \(0.906580\pi\)
\(174\) 0 0
\(175\) −1.00000 + 1.73205i −0.0755929 + 0.130931i
\(176\) −6.00000 + 10.3923i −0.452267 + 0.783349i
\(177\) 0 0
\(178\) 0 0
\(179\) 9.00000 0.672692 0.336346 0.941739i \(-0.390809\pi\)
0.336346 + 0.941739i \(0.390809\pi\)
\(180\) 0 0
\(181\) 11.0000 0.817624 0.408812 0.912619i \(-0.365943\pi\)
0.408812 + 0.912619i \(0.365943\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −4.00000 + 6.92820i −0.294086 + 0.509372i
\(186\) 0 0
\(187\) −9.00000 15.5885i −0.658145 1.13994i
\(188\) 24.0000 1.75038
\(189\) 0 0
\(190\) 0 0
\(191\) −7.50000 12.9904i −0.542681 0.939951i −0.998749 0.0500060i \(-0.984076\pi\)
0.456068 0.889945i \(-0.349257\pi\)
\(192\) 0 0
\(193\) 2.00000 3.46410i 0.143963 0.249351i −0.785022 0.619467i \(-0.787349\pi\)
0.928986 + 0.370116i \(0.120682\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −3.00000 5.19615i −0.214286 0.371154i
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −9.00000 + 15.5885i −0.631676 + 1.09410i
\(204\) 0 0
\(205\) 1.50000 + 2.59808i 0.104765 + 0.181458i
\(206\) 0 0
\(207\) 0 0
\(208\) −16.0000 −1.10940
\(209\) 1.50000 + 2.59808i 0.103757 + 0.179713i
\(210\) 0 0
\(211\) 6.50000 11.2583i 0.447478 0.775055i −0.550743 0.834675i \(-0.685655\pi\)
0.998221 + 0.0596196i \(0.0189888\pi\)
\(212\) −6.00000 + 10.3923i −0.412082 + 0.713746i
\(213\) 0 0
\(214\) 0 0
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) −2.00000 −0.135769
\(218\) 0 0
\(219\) 0 0
\(220\) 3.00000 5.19615i 0.202260 0.350325i
\(221\) 12.0000 20.7846i 0.807207 1.39812i
\(222\) 0 0
\(223\) −10.0000 17.3205i −0.669650 1.15987i −0.978002 0.208595i \(-0.933111\pi\)
0.308353 0.951272i \(-0.400222\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −9.00000 15.5885i −0.597351 1.03464i −0.993210 0.116331i \(-0.962887\pi\)
0.395860 0.918311i \(-0.370447\pi\)
\(228\) 0 0
\(229\) 5.00000 8.66025i 0.330409 0.572286i −0.652183 0.758062i \(-0.726147\pi\)
0.982592 + 0.185776i \(0.0594799\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −12.0000 −0.786146 −0.393073 0.919507i \(-0.628588\pi\)
−0.393073 + 0.919507i \(0.628588\pi\)
\(234\) 0 0
\(235\) −12.0000 −0.782794
\(236\) −3.00000 5.19615i −0.195283 0.338241i
\(237\) 0 0
\(238\) 0 0
\(239\) −12.0000 + 20.7846i −0.776215 + 1.34444i 0.157893 + 0.987456i \(0.449530\pi\)
−0.934109 + 0.356988i \(0.883804\pi\)
\(240\) 0 0
\(241\) −8.50000 14.7224i −0.547533 0.948355i −0.998443 0.0557856i \(-0.982234\pi\)
0.450910 0.892570i \(-0.351100\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 20.0000 1.28037
\(245\) 1.50000 + 2.59808i 0.0958315 + 0.165985i
\(246\) 0 0
\(247\) −2.00000 + 3.46410i −0.127257 + 0.220416i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 18.0000 1.13165
\(254\) 0 0
\(255\) 0 0
\(256\) −8.00000 + 13.8564i −0.500000 + 0.866025i
\(257\) 3.00000 5.19615i 0.187135 0.324127i −0.757159 0.653231i \(-0.773413\pi\)
0.944294 + 0.329104i \(0.106747\pi\)
\(258\) 0 0
\(259\) −8.00000 13.8564i −0.497096 0.860995i
\(260\) 8.00000 0.496139
\(261\) 0 0
\(262\) 0 0
\(263\) 9.00000 + 15.5885i 0.554964 + 0.961225i 0.997906 + 0.0646755i \(0.0206012\pi\)
−0.442943 + 0.896550i \(0.646065\pi\)
\(264\) 0 0
\(265\) 3.00000 5.19615i 0.184289 0.319197i
\(266\) 0 0
\(267\) 0 0
\(268\) 14.0000 + 24.2487i 0.855186 + 1.48123i
\(269\) 3.00000 0.182913 0.0914566 0.995809i \(-0.470848\pi\)
0.0914566 + 0.995809i \(0.470848\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) −12.0000 20.7846i −0.727607 1.26025i
\(273\) 0 0
\(274\) 0 0
\(275\) −1.50000 + 2.59808i −0.0904534 + 0.156670i
\(276\) 0 0
\(277\) 8.00000 + 13.8564i 0.480673 + 0.832551i 0.999754 0.0221745i \(-0.00705893\pi\)
−0.519081 + 0.854725i \(0.673726\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 3.00000 + 5.19615i 0.178965 + 0.309976i 0.941526 0.336939i \(-0.109392\pi\)
−0.762561 + 0.646916i \(0.776058\pi\)
\(282\) 0 0
\(283\) 5.00000 8.66025i 0.297219 0.514799i −0.678280 0.734804i \(-0.737274\pi\)
0.975499 + 0.220005i \(0.0706075\pi\)
\(284\) 3.00000 5.19615i 0.178017 0.308335i
\(285\) 0 0
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) 2.00000 3.46410i 0.117041 0.202721i
\(293\) −6.00000 + 10.3923i −0.350524 + 0.607125i −0.986341 0.164714i \(-0.947330\pi\)
0.635818 + 0.771839i \(0.280663\pi\)
\(294\) 0 0
\(295\) 1.50000 + 2.59808i 0.0873334 + 0.151266i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 12.0000 + 20.7846i 0.693978 + 1.20201i
\(300\) 0 0
\(301\) 4.00000 6.92820i 0.230556 0.399335i
\(302\) 0 0
\(303\) 0 0
\(304\) 2.00000 + 3.46410i 0.114708 + 0.198680i
\(305\) −10.0000 −0.572598
\(306\) 0 0
\(307\) −16.0000 −0.913168 −0.456584 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 6.00000 + 10.3923i 0.341882 + 0.592157i
\(309\) 0 0
\(310\) 0 0
\(311\) 13.5000 23.3827i 0.765515 1.32591i −0.174459 0.984664i \(-0.555818\pi\)
0.939974 0.341246i \(-0.110849\pi\)
\(312\) 0 0
\(313\) −4.00000 6.92820i −0.226093 0.391605i 0.730554 0.682855i \(-0.239262\pi\)
−0.956647 + 0.291250i \(0.905929\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 32.0000 1.80014
\(317\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(318\) 0 0
\(319\) −13.5000 + 23.3827i −0.755855 + 1.30918i
\(320\) 4.00000 6.92820i 0.223607 0.387298i
\(321\) 0 0
\(322\) 0 0
\(323\) −6.00000 −0.333849
\(324\) 0 0
\(325\) −4.00000 −0.221880
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 12.0000 20.7846i 0.661581 1.14589i
\(330\) 0 0
\(331\) 12.5000 + 21.6506i 0.687062 + 1.19003i 0.972784 + 0.231714i \(0.0744333\pi\)
−0.285722 + 0.958313i \(0.592233\pi\)
\(332\) −24.0000 −1.31717
\(333\) 0 0
\(334\) 0 0
\(335\) −7.00000 12.1244i −0.382451 0.662424i
\(336\) 0 0
\(337\) 2.00000 3.46410i 0.108947 0.188702i −0.806397 0.591375i \(-0.798585\pi\)
0.915344 + 0.402673i \(0.131919\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 6.00000 + 10.3923i 0.325396 + 0.563602i
\(341\) −3.00000 −0.162459
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 9.00000 15.5885i 0.483145 0.836832i −0.516667 0.856186i \(-0.672828\pi\)
0.999813 + 0.0193540i \(0.00616095\pi\)
\(348\) 0 0
\(349\) −2.50000 4.33013i −0.133822 0.231786i 0.791325 0.611396i \(-0.209392\pi\)
−0.925147 + 0.379610i \(0.876058\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 6.00000 + 10.3923i 0.319348 + 0.553127i 0.980352 0.197256i \(-0.0632029\pi\)
−0.661004 + 0.750382i \(0.729870\pi\)
\(354\) 0 0
\(355\) −1.50000 + 2.59808i −0.0796117 + 0.137892i
\(356\) −15.0000 + 25.9808i −0.794998 + 1.37698i
\(357\) 0 0
\(358\) 0 0
\(359\) 15.0000 0.791670 0.395835 0.918322i \(-0.370455\pi\)
0.395835 + 0.918322i \(0.370455\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) 0 0
\(364\) −8.00000 + 13.8564i −0.419314 + 0.726273i
\(365\) −1.00000 + 1.73205i −0.0523424 + 0.0906597i
\(366\) 0 0
\(367\) 5.00000 + 8.66025i 0.260998 + 0.452062i 0.966507 0.256639i \(-0.0826151\pi\)
−0.705509 + 0.708700i \(0.749282\pi\)
\(368\) 24.0000 1.25109
\(369\) 0 0
\(370\) 0 0
\(371\) 6.00000 + 10.3923i 0.311504 + 0.539542i
\(372\) 0 0
\(373\) −19.0000 + 32.9090i −0.983783 + 1.70396i −0.336557 + 0.941663i \(0.609263\pi\)
−0.647225 + 0.762299i \(0.724071\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −36.0000 −1.85409
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) −1.00000 1.73205i −0.0512989 0.0888523i
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(384\) 0 0
\(385\) −3.00000 5.19615i −0.152894 0.264820i
\(386\) 0 0
\(387\) 0 0
\(388\) 8.00000 0.406138
\(389\) 15.0000 + 25.9808i 0.760530 + 1.31728i 0.942578 + 0.333987i \(0.108394\pi\)
−0.182047 + 0.983290i \(0.558272\pi\)
\(390\) 0 0
\(391\) −18.0000 + 31.1769i −0.910299 + 1.57668i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −16.0000 −0.805047
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −2.00000 + 3.46410i −0.100000 + 0.173205i
\(401\) −3.00000 + 5.19615i −0.149813 + 0.259483i −0.931158 0.364615i \(-0.881200\pi\)
0.781345 + 0.624099i \(0.214534\pi\)
\(402\) 0 0
\(403\) −2.00000 3.46410i −0.0996271 0.172559i
\(404\) 18.0000 0.895533
\(405\) 0 0
\(406\) 0 0
\(407\) −12.0000 20.7846i −0.594818 1.03025i
\(408\) 0 0
\(409\) −7.00000 + 12.1244i −0.346128 + 0.599511i −0.985558 0.169338i \(-0.945837\pi\)
0.639430 + 0.768849i \(0.279170\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 14.0000 + 24.2487i 0.689730 + 1.19465i
\(413\) −6.00000 −0.295241
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −6.00000 + 10.3923i −0.293119 + 0.507697i −0.974546 0.224189i \(-0.928027\pi\)
0.681426 + 0.731887i \(0.261360\pi\)
\(420\) 0 0
\(421\) −2.50000 4.33013i −0.121843 0.211037i 0.798652 0.601793i \(-0.205547\pi\)
−0.920494 + 0.390756i \(0.872214\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.00000 5.19615i −0.145521 0.252050i
\(426\) 0 0
\(427\) 10.0000 17.3205i 0.483934 0.838198i
\(428\) 18.0000 31.1769i 0.870063 1.50699i
\(429\) 0 0
\(430\) 0 0
\(431\) −15.0000 −0.722525 −0.361262 0.932464i \(-0.617654\pi\)
−0.361262 + 0.932464i \(0.617654\pi\)
\(432\) 0 0
\(433\) 26.0000 1.24948 0.624740 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −1.00000 + 1.73205i −0.0478913 + 0.0829502i
\(437\) 3.00000 5.19615i 0.143509 0.248566i
\(438\) 0 0
\(439\) 0.500000 + 0.866025i 0.0238637 + 0.0413331i 0.877711 0.479191i \(-0.159070\pi\)
−0.853847 + 0.520524i \(0.825737\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −6.00000 10.3923i −0.285069 0.493753i 0.687557 0.726130i \(-0.258683\pi\)
−0.972626 + 0.232377i \(0.925350\pi\)
\(444\) 0 0
\(445\) 7.50000 12.9904i 0.355534 0.615803i
\(446\) 0 0
\(447\) 0 0
\(448\) 8.00000 + 13.8564i 0.377964 + 0.654654i
\(449\) 15.0000 0.707894 0.353947 0.935266i \(-0.384839\pi\)
0.353947 + 0.935266i \(0.384839\pi\)
\(450\) 0 0
\(451\) −9.00000 −0.423793
\(452\) −6.00000 10.3923i −0.282216 0.488813i
\(453\) 0 0
\(454\) 0 0
\(455\) 4.00000 6.92820i 0.187523 0.324799i
\(456\) 0 0
\(457\) 14.0000 + 24.2487i 0.654892 + 1.13431i 0.981921 + 0.189292i \(0.0606194\pi\)
−0.327028 + 0.945015i \(0.606047\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −12.0000 −0.559503
\(461\) 10.5000 + 18.1865i 0.489034 + 0.847031i 0.999920 0.0126168i \(-0.00401615\pi\)
−0.510887 + 0.859648i \(0.670683\pi\)
\(462\) 0 0
\(463\) −13.0000 + 22.5167i −0.604161 + 1.04644i 0.388022 + 0.921650i \(0.373158\pi\)
−0.992183 + 0.124788i \(0.960175\pi\)
\(464\) −18.0000 + 31.1769i −0.835629 + 1.44735i
\(465\) 0 0
\(466\) 0 0
\(467\) 6.00000 0.277647 0.138823 0.990317i \(-0.455668\pi\)
0.138823 + 0.990317i \(0.455668\pi\)
\(468\) 0 0
\(469\) 28.0000 1.29292
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 6.00000 10.3923i 0.275880 0.477839i
\(474\) 0 0
\(475\) 0.500000 + 0.866025i 0.0229416 + 0.0397360i
\(476\) −24.0000 −1.10004
\(477\) 0 0
\(478\) 0 0
\(479\) 16.5000 + 28.5788i 0.753904 + 1.30580i 0.945917 + 0.324408i \(0.105165\pi\)
−0.192013 + 0.981392i \(0.561502\pi\)
\(480\) 0 0
\(481\) 16.0000 27.7128i 0.729537 1.26360i
\(482\) 0 0
\(483\) 0 0
\(484\) −2.00000 3.46410i −0.0909091 0.157459i
\(485\) −4.00000 −0.181631
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −4.50000 + 7.79423i −0.203082 + 0.351749i −0.949520 0.313707i \(-0.898429\pi\)
0.746438 + 0.665455i \(0.231763\pi\)
\(492\) 0 0
\(493\) −27.0000 46.7654i −1.21602 2.10621i
\(494\) 0 0
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) −3.00000 5.19615i −0.134568 0.233079i
\(498\) 0 0
\(499\) 6.50000 11.2583i 0.290980 0.503992i −0.683062 0.730361i \(-0.739352\pi\)
0.974042 + 0.226369i \(0.0726854\pi\)
\(500\) 1.00000 1.73205i 0.0447214 0.0774597i
\(501\) 0 0
\(502\) 0 0
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 0 0
\(505\) −9.00000 −0.400495
\(506\) 0 0
\(507\) 0 0
\(508\) 2.00000 3.46410i 0.0887357 0.153695i
\(509\) 15.0000 25.9808i 0.664863 1.15158i −0.314459 0.949271i \(-0.601823\pi\)
0.979322 0.202306i \(-0.0648436\pi\)
\(510\) 0 0
\(511\) −2.00000 3.46410i −0.0884748 0.153243i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −7.00000 12.1244i −0.308457 0.534263i
\(516\) 0 0
\(517\) 18.0000 31.1769i 0.791639 1.37116i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 9.00000 + 15.5885i 0.393167 + 0.680985i
\(525\) 0 0
\(526\) 0 0
\(527\) 3.00000 5.19615i 0.130682 0.226348i
\(528\) 0 0
\(529\) −6.50000 11.2583i −0.282609 0.489493i
\(530\) 0 0
\(531\) 0 0
\(532\) 4.00000 0.173422
\(533\) −6.00000 10.3923i −0.259889 0.450141i
\(534\) 0 0
\(535\) −9.00000 + 15.5885i −0.389104 + 0.673948i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −9.00000 −0.387657
\(540\) 0 0
\(541\) 23.0000 0.988847 0.494424 0.869221i \(-0.335379\pi\)
0.494424 + 0.869221i \(0.335379\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0.500000 0.866025i 0.0214176 0.0370965i
\(546\) 0 0
\(547\) −4.00000 6.92820i −0.171028 0.296229i 0.767752 0.640747i \(-0.221375\pi\)
−0.938779 + 0.344519i \(0.888042\pi\)
\(548\) 24.0000 1.02523
\(549\) 0 0
\(550\) 0 0
\(551\) 4.50000 + 7.79423i 0.191706 + 0.332045i
\(552\) 0 0
\(553\) 16.0000 27.7128i 0.680389 1.17847i
\(554\) 0 0
\(555\) 0 0
\(556\) −7.00000 12.1244i −0.296866 0.514187i
\(557\) −24.0000 −1.01691 −0.508456 0.861088i \(-0.669784\pi\)
−0.508456 + 0.861088i \(0.669784\pi\)
\(558\) 0 0
\(559\) 16.0000 0.676728
\(560\) −4.00000 6.92820i −0.169031 0.292770i
\(561\) 0 0
\(562\) 0 0
\(563\) −12.0000 + 20.7846i −0.505740 + 0.875967i 0.494238 + 0.869326i \(0.335447\pi\)
−0.999978 + 0.00664037i \(0.997886\pi\)
\(564\) 0 0
\(565\) 3.00000 + 5.19615i 0.126211 + 0.218604i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −13.5000 23.3827i −0.565949 0.980253i −0.996961 0.0779066i \(-0.975176\pi\)
0.431011 0.902347i \(-0.358157\pi\)
\(570\) 0 0
\(571\) 9.50000 16.4545i 0.397563 0.688599i −0.595862 0.803087i \(-0.703189\pi\)
0.993425 + 0.114488i \(0.0365228\pi\)
\(572\) −12.0000 + 20.7846i −0.501745 + 0.869048i
\(573\) 0 0
\(574\) 0 0
\(575\) 6.00000 0.250217
\(576\) 0 0
\(577\) −34.0000 −1.41544 −0.707719 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 9.00000 15.5885i 0.373705 0.647275i
\(581\) −12.0000 + 20.7846i −0.497844 + 0.862291i
\(582\) 0 0
\(583\) 9.00000 + 15.5885i 0.372742 + 0.645608i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −24.0000 41.5692i −0.990586 1.71575i −0.613845 0.789427i \(-0.710378\pi\)
−0.376741 0.926319i \(-0.622955\pi\)
\(588\) 0 0
\(589\) −0.500000 + 0.866025i −0.0206021 + 0.0356840i
\(590\) 0 0
\(591\) 0 0
\(592\) −16.0000 27.7128i −0.657596 1.13899i
\(593\) −18.0000 −0.739171 −0.369586 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 0 0
\(595\) 12.0000 0.491952
\(596\) −6.00000 10.3923i −0.245770 0.425685i
\(597\) 0 0
\(598\) 0 0
\(599\) −10.5000 + 18.1865i −0.429018 + 0.743082i −0.996786 0.0801071i \(-0.974474\pi\)
0.567768 + 0.823189i \(0.307807\pi\)
\(600\) 0 0
\(601\) 15.5000 + 26.8468i 0.632258 + 1.09510i 0.987089 + 0.160173i \(0.0512051\pi\)
−0.354831 + 0.934931i \(0.615462\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 2.00000 0.0813788
\(605\) 1.00000 + 1.73205i 0.0406558 + 0.0704179i
\(606\) 0 0
\(607\) −10.0000 + 17.3205i −0.405887 + 0.703018i −0.994424 0.105453i \(-0.966371\pi\)
0.588537 + 0.808470i \(0.299704\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 48.0000 1.94187
\(612\) 0 0
\(613\) −16.0000 −0.646234 −0.323117 0.946359i \(-0.604731\pi\)
−0.323117 + 0.946359i \(0.604731\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −12.0000 + 20.7846i −0.483102 + 0.836757i −0.999812 0.0194037i \(-0.993823\pi\)
0.516710 + 0.856161i \(0.327157\pi\)
\(618\) 0 0
\(619\) −22.0000 38.1051i −0.884255 1.53157i −0.846566 0.532284i \(-0.821334\pi\)
−0.0376891 0.999290i \(-0.512000\pi\)
\(620\) 2.00000 0.0803219
\(621\) 0 0
\(622\) 0 0
\(623\) 15.0000 + 25.9808i 0.600962 + 1.04090i
\(624\) 0 0
\(625\) −0.500000 + 0.866025i −0.0200000 + 0.0346410i
\(626\) 0 0
\(627\) 0 0
\(628\) −16.0000 27.7128i −0.638470 1.10586i
\(629\) 48.0000 1.91389
\(630\) 0 0
\(631\) −25.0000 −0.995234 −0.497617 0.867397i \(-0.665792\pi\)
−0.497617 + 0.867397i \(0.665792\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1.00000 + 1.73205i −0.0396838 + 0.0687343i
\(636\) 0 0
\(637\) −6.00000 10.3923i −0.237729 0.411758i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −16.5000 28.5788i −0.651711 1.12880i −0.982708 0.185164i \(-0.940718\pi\)
0.330997 0.943632i \(-0.392615\pi\)
\(642\) 0 0
\(643\) 2.00000 3.46410i 0.0788723 0.136611i −0.823891 0.566748i \(-0.808201\pi\)
0.902764 + 0.430137i \(0.141535\pi\)
\(644\) 12.0000 20.7846i 0.472866 0.819028i
\(645\) 0 0
\(646\) 0 0
\(647\) −6.00000 −0.235884 −0.117942 0.993020i \(-0.537630\pi\)
−0.117942 + 0.993020i \(0.537630\pi\)
\(648\) 0 0
\(649\) −9.00000 −0.353281
\(650\) 0 0
\(651\) 0 0
\(652\) 2.00000 3.46410i 0.0783260 0.135665i
\(653\) −12.0000 + 20.7846i −0.469596 + 0.813365i −0.999396 0.0347583i \(-0.988934\pi\)
0.529799 + 0.848123i \(0.322267\pi\)
\(654\) 0 0
\(655\) −4.50000 7.79423i −0.175830 0.304546i
\(656\) −12.0000 −0.468521
\(657\) 0 0
\(658\) 0 0
\(659\) −12.0000 20.7846i −0.467454 0.809653i 0.531855 0.846836i \(-0.321495\pi\)
−0.999309 + 0.0371821i \(0.988162\pi\)
\(660\) 0 0
\(661\) −2.50000 + 4.33013i −0.0972387 + 0.168422i −0.910541 0.413419i \(-0.864334\pi\)
0.813302 + 0.581842i \(0.197668\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −2.00000 −0.0775567
\(666\) 0 0
\(667\) 54.0000 2.09089
\(668\) −6.00000 10.3923i −0.232147 0.402090i
\(669\) 0 0
\(670\) 0 0
\(671\) 15.0000 25.9808i 0.579069 1.00298i
\(672\) 0 0
\(673\) 17.0000 + 29.4449i 0.655302 + 1.13502i 0.981818 + 0.189824i \(0.0607919\pi\)
−0.326516 + 0.945192i \(0.605875\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −6.00000 −0.230769
\(677\) 15.0000 + 25.9808i 0.576497 + 0.998522i 0.995877 + 0.0907112i \(0.0289140\pi\)
−0.419380 + 0.907811i \(0.637753\pi\)
\(678\) 0 0
\(679\) 4.00000 6.92820i 0.153506 0.265880i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 0 0
\(685\) −12.0000 −0.458496
\(686\) 0 0
\(687\) 0 0
\(688\) 8.00000 13.8564i 0.304997 0.528271i
\(689\) −12.0000 + 20.7846i −0.457164 + 0.791831i
\(690\) 0 0
\(691\) −22.0000 38.1051i −0.836919 1.44959i −0.892458 0.451130i \(-0.851021\pi\)
0.0555386 0.998457i \(-0.482312\pi\)
\(692\) −12.0000 −0.456172
\(693\) 0 0
\(694\) 0 0
\(695\) 3.50000 + 6.06218i 0.132763 + 0.229952i
\(696\) 0 0
\(697\) 9.00000 15.5885i 0.340899 0.590455i
\(698\) 0 0
\(699\) 0 0
\(700\) 2.00000 + 3.46410i 0.0755929 + 0.130931i
\(701\) −15.0000 −0.566542 −0.283271 0.959040i \(-0.591420\pi\)
−0.283271 + 0.959040i \(0.591420\pi\)
\(702\) 0 0
\(703\) −8.00000 −0.301726
\(704\) 12.0000 + 20.7846i 0.452267 + 0.783349i
\(705\) 0 0
\(706\) 0 0
\(707\) 9.00000 15.5885i 0.338480 0.586264i
\(708\) 0 0
\(709\) 11.0000 + 19.0526i 0.413114 + 0.715534i 0.995228 0.0975728i \(-0.0311079\pi\)
−0.582115 + 0.813107i \(0.697775\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 3.00000 + 5.19615i 0.112351 + 0.194597i
\(714\) 0 0
\(715\) 6.00000 10.3923i 0.224387 0.388650i
\(716\) 9.00000 15.5885i 0.336346 0.582568i
\(717\) 0 0
\(718\) 0 0
\(719\) −27.0000 −1.00693 −0.503465 0.864016i \(-0.667942\pi\)
−0.503465 + 0.864016i \(0.667942\pi\)
\(720\) 0 0
\(721\) 28.0000 1.04277
\(722\) 0 0
\(723\) 0 0
\(724\) 11.0000 19.0526i 0.408812 0.708083i
\(725\) −4.50000 + 7.79423i −0.167126 + 0.289470i
\(726\) 0 0
\(727\) 14.0000 + 24.2487i 0.519231 + 0.899335i 0.999750 + 0.0223506i \(0.00711500\pi\)
−0.480519 + 0.876984i \(0.659552\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 12.0000 + 20.7846i 0.443836 + 0.768747i
\(732\) 0 0
\(733\) −7.00000 + 12.1244i −0.258551 + 0.447823i −0.965854 0.259087i \(-0.916578\pi\)
0.707303 + 0.706910i \(0.249912\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 42.0000 1.54709
\(738\) 0 0
\(739\) −7.00000 −0.257499 −0.128750 0.991677i \(-0.541096\pi\)
−0.128750 + 0.991677i \(0.541096\pi\)
\(740\) 8.00000 + 13.8564i 0.294086 + 0.509372i
\(741\) 0 0
\(742\) 0 0
\(743\) −9.00000 + 15.5885i −0.330178 + 0.571885i −0.982547 0.186017i \(-0.940442\pi\)
0.652369 + 0.757902i \(0.273775\pi\)
\(744\) 0 0
\(745\) 3.00000 + 5.19615i 0.109911 + 0.190372i
\(746\) 0 0
\(747\) 0 0
\(748\) −36.0000 −1.31629
\(749\) −18.0000 31.1769i −0.657706 1.13918i
\(750\) 0 0
\(751\) 8.00000 13.8564i 0.291924 0.505627i −0.682341 0.731034i \(-0.739038\pi\)
0.974265 + 0.225407i \(0.0723712\pi\)
\(752\) 24.0000 41.5692i 0.875190 1.51587i
\(753\) 0 0
\(754\) 0 0
\(755\) −1.00000 −0.0363937
\(756\) 0 0
\(757\) 20.0000 0.726912 0.363456 0.931611i \(-0.381597\pi\)
0.363456 + 0.931611i \(0.381597\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 16.5000 28.5788i 0.598125 1.03598i −0.394973 0.918693i \(-0.629246\pi\)
0.993098 0.117289i \(-0.0374205\pi\)
\(762\) 0 0
\(763\) 1.00000 + 1.73205i 0.0362024 + 0.0627044i
\(764\) −30.0000 −1.08536
\(765\) 0 0
\(766\) 0 0
\(767\) −6.00000 10.3923i −0.216647 0.375244i
\(768\) 0 0
\(769\) −20.5000 + 35.5070i −0.739249 + 1.28042i 0.213585 + 0.976924i \(0.431486\pi\)
−0.952834 + 0.303492i \(0.901847\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −4.00000 6.92820i −0.143963 0.249351i
\(773\) −24.0000 −0.863220 −0.431610 0.902060i \(-0.642054\pi\)
−0.431610 + 0.902060i \(0.642054\pi\)
\(774\) 0 0
\(775\) −1.00000 −0.0359211
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1.50000 + 2.59808i −0.0537431 + 0.0930857i
\(780\) 0 0
\(781\) −4.50000 7.79423i −0.161023 0.278899i
\(782\) 0 0
\(783\) 0 0
\(784\) −12.0000 −0.428571
\(785\) 8.00000 + 13.8564i 0.285532 + 0.494556i
\(786\) 0 0
\(787\) −7.00000 + 12.1244i −0.249523 + 0.432187i −0.963394 0.268091i \(-0.913607\pi\)
0.713871 + 0.700278i \(0.246941\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −12.0000 −0.426671
\(792\) 0 0
\(793\) 40.0000 1.42044
\(794\) 0 0
\(795\) 0 0
\(796\) −16.0000 + 27.7128i −0.567105 + 0.982255i
\(797\) −12.0000 + 20.7846i −0.425062 + 0.736229i −0.996426 0.0844678i \(-0.973081\pi\)
0.571364 + 0.820696i \(0.306414\pi\)
\(798\) 0 0
\(799\) 36.0000 + 62.3538i 1.27359 + 2.20592i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −3.00000 5.19615i −0.105868 0.183368i
\(804\) 0 0
\(805\) −6.00000 + 10.3923i −0.211472 + 0.366281i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −15.0000 −0.527372 −0.263686 0.964609i \(-0.584938\pi\)
−0.263686 + 0.964609i \(0.584938\pi\)
\(810\) 0 0
\(811\) 23.0000 0.807639 0.403820 0.914839i \(-0.367682\pi\)
0.403820 + 0.914839i \(0.367682\pi\)
\(812\) 18.0000 + 31.1769i 0.631676 + 1.09410i
\(813\) 0 0
\(814\) 0 0
\(815\) −1.00000 + 1.73205i −0.0350285 + 0.0606711i
\(816\) 0 0
\(817\) −2.00000 3.46410i −0.0699711 0.121194i
\(818\) 0 0
\(819\) 0 0
\(820\) 6.00000 0.209529
\(821\) −22.5000 38.9711i −0.785255 1.36010i −0.928846 0.370465i \(-0.879198\pi\)
0.143591 0.989637i \(-0.454135\pi\)
\(822\) 0 0
\(823\) 11.0000 19.0526i 0.383436 0.664130i −0.608115 0.793849i \(-0.708074\pi\)
0.991551 + 0.129719i \(0.0414074\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 0 0
\(829\) −13.0000 −0.451509 −0.225754 0.974184i \(-0.572485\pi\)
−0.225754 + 0.974184i \(0.572485\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −16.0000 + 27.7128i −0.554700 + 0.960769i
\(833\) 9.00000 15.5885i 0.311832 0.540108i
\(834\) 0 0
\(835\) 3.00000 + 5.19615i 0.103819 + 0.179820i
\(836\) 6.00000 0.207514
\(837\) 0 0
\(838\) 0 0
\(839\) 22.5000 + 38.9711i 0.776786 + 1.34543i 0.933785 + 0.357834i \(0.116485\pi\)
−0.156999 + 0.987599i \(0.550182\pi\)
\(840\) 0 0
\(841\) −26.0000 + 45.0333i −0.896552 + 1.55287i
\(842\) 0 0
\(843\) 0 0
\(844\) −13.0000 22.5167i −0.447478 0.775055i
\(845\) 3.00000 0.103203
\(846\) 0 0
\(847\) −4.00000 −0.137442
\(848\) 12.0000 + 20.7846i 0.412082 + 0.713746i
\(849\) 0 0
\(850\) 0 0
\(851\) −24.0000 + 41.5692i −0.822709 + 1.42497i
\(852\) 0 0
\(853\) 23.0000 + 39.8372i 0.787505 + 1.36400i 0.927491 + 0.373845i \(0.121961\pi\)
−0.139986 + 0.990153i \(0.544706\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 24.0000 + 41.5692i 0.819824 + 1.41998i 0.905811 + 0.423681i \(0.139262\pi\)
−0.0859870 + 0.996296i \(0.527404\pi\)
\(858\) 0 0
\(859\) 12.5000 21.6506i 0.426494 0.738710i −0.570064 0.821600i \(-0.693082\pi\)
0.996559 + 0.0828900i \(0.0264150\pi\)
\(860\) −4.00000 + 6.92820i −0.136399 + 0.236250i
\(861\) 0 0
\(862\) 0 0
\(863\) −30.0000 −1.02121 −0.510606 0.859815i \(-0.670579\pi\)
−0.510606 + 0.859815i \(0.670579\pi\)
\(864\) 0 0
\(865\) 6.00000 0.204006
\(866\) 0 0
\(867\) 0 0
\(868\) −2.00000 + 3.46410i −0.0678844 + 0.117579i
\(869\) 24.0000 41.5692i 0.814144 1.41014i
\(870\) 0 0
\(871\) 28.0000 + 48.4974i 0.948744 + 1.64327i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1.00000 1.73205i −0.0338062 0.0585540i
\(876\) 0 0
\(877\) −22.0000 + 38.1051i −0.742887 + 1.28672i 0.208288 + 0.978068i \(0.433211\pi\)
−0.951175 + 0.308651i \(0.900123\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) −6.00000 10.3923i −0.202260 0.350325i
\(881\) −39.0000 −1.31394 −0.656972 0.753915i \(-0.728163\pi\)
−0.656972 + 0.753915i \(0.728163\pi\)
\(882\) 0 0
\(883\) −46.0000 −1.54802 −0.774012 0.633171i \(-0.781753\pi\)
−0.774012 + 0.633171i \(0.781753\pi\)
\(884\) −24.0000 41.5692i −0.807207 1.39812i
\(885\) 0 0
\(886\) 0 0
\(887\) 18.0000 31.1769i 0.604381 1.04682i −0.387768 0.921757i \(-0.626754\pi\)
0.992149 0.125061i \(-0.0399128\pi\)
\(888\) 0 0
\(889\) −2.00000 3.46410i −0.0670778 0.116182i
\(890\) 0 0
\(891\) 0 0
\(892\) −40.0000 −1.33930
\(893\) −6.00000 10.3923i −0.200782 0.347765i
\(894\) 0 0
\(895\) −4.50000 + 7.79423i −0.150418 + 0.260532i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −9.00000 −0.300167
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −5.50000 + 9.52628i −0.182826 + 0.316664i
\(906\) 0 0
\(907\) −19.0000 32.9090i −0.630885 1.09272i −0.987371 0.158424i \(-0.949359\pi\)
0.356487 0.934300i \(-0.383975\pi\)
\(908\) −36.0000 −1.19470
\(909\) 0 0
\(910\) 0 0
\(911\) −4.50000 7.79423i −0.149092 0.258234i 0.781800 0.623529i \(-0.214302\pi\)
−0.930892 + 0.365295i \(0.880968\pi\)
\(912\) 0 0
\(913\) −18.0000 + 31.1769i −0.595713 + 1.03181i
\(914\) 0 0
\(915\) 0 0
\(916\) −10.0000 17.3205i −0.330409 0.572286i
\(917\) 18.0000 0.594412
\(918\) 0 0
\(919\) −7.00000 −0.230909 −0.115454 0.993313i \(-0.536832\pi\)
−0.115454 + 0.993313i \(0.536832\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 6.00000 10.3923i 0.197492 0.342067i
\(924\) 0 0
\(925\) −4.00000 6.92820i −0.131519 0.227798i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −19.5000 33.7750i −0.639774 1.10812i −0.985482 0.169779i \(-0.945695\pi\)
0.345708 0.938342i \(-0.387639\pi\)
\(930\) 0 0
\(931\) −1.50000 + 2.59808i −0.0491605 + 0.0851485i
\(932\) −12.0000 + 20.7846i −0.393073 + 0.680823i
\(933\) 0 0
\(934\) 0 0
\(935\) 18.0000 0.588663
\(936\) 0 0
\(937\) 14.0000 0.457360 0.228680 0.973502i \(-0.426559\pi\)
0.228680 + 0.973502i \(0.426559\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −12.0000 + 20.7846i −0.391397 + 0.677919i
\(941\) 15.0000 25.9808i 0.488986 0.846949i −0.510934 0.859620i \(-0.670700\pi\)
0.999920 + 0.0126715i \(0.00403357\pi\)
\(942\) 0 0
\(943\) 9.00000 + 15.5885i 0.293080 + 0.507630i
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) 0 0
\(947\) 15.0000 + 25.9808i 0.487435 + 0.844261i 0.999896 0.0144491i \(-0.00459946\pi\)
−0.512461 + 0.858710i \(0.671266\pi\)
\(948\) 0 0
\(949\) 4.00000 6.92820i 0.129845 0.224899i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 48.0000 1.55487 0.777436 0.628962i \(-0.216520\pi\)
0.777436 + 0.628962i \(0.216520\pi\)
\(954\) 0 0
\(955\) 15.0000 0.485389
\(956\) 24.0000 + 41.5692i 0.776215 + 1.34444i
\(957\) 0 0
\(958\) 0 0
\(959\) 12.0000 20.7846i 0.387500 0.671170i
\(960\) 0 0
\(961\) 15.0000 + 25.9808i 0.483871 + 0.838089i
\(962\) 0 0
\(963\) 0 0
\(964\) −34.0000 −1.09507
\(965\) 2.00000 + 3.46410i 0.0643823 + 0.111513i
\(966\) 0 0
\(967\) −25.0000 + 43.3013i −0.803946 + 1.39247i 0.113055 + 0.993589i \(0.463936\pi\)
−0.917000 + 0.398886i \(0.869397\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 3.00000 0.0962746 0.0481373 0.998841i \(-0.484672\pi\)
0.0481373 + 0.998841i \(0.484672\pi\)
\(972\) 0 0
\(973\) −14.0000 −0.448819
\(974\) 0 0
\(975\) 0 0
\(976\) 20.0000 34.6410i 0.640184 1.10883i
\(977\) −24.0000 + 41.5692i −0.767828 + 1.32992i 0.170910 + 0.985287i \(0.445329\pi\)
−0.938738 + 0.344631i \(0.888004\pi\)
\(978\) 0 0
\(979\) 22.5000 + 38.9711i 0.719103 + 1.24552i
\(980\) 6.00000 0.191663
\(981\) 0 0
\(982\) 0 0
\(983\) −3.00000 5.19615i −0.0956851 0.165732i 0.814209 0.580572i \(-0.197171\pi\)
−0.909894 + 0.414840i \(0.863838\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 4.00000 + 6.92820i 0.127257 + 0.220416i
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) 47.0000 1.49300 0.746502 0.665383i \(-0.231732\pi\)
0.746502 + 0.665383i \(0.231732\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 8.00000 13.8564i 0.253617 0.439278i
\(996\) 0 0
\(997\) 5.00000 + 8.66025i 0.158352 + 0.274273i 0.934274 0.356555i \(-0.116049\pi\)
−0.775923 + 0.630828i \(0.782715\pi\)
\(998\) 0 0
\(999\) 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 405.2.e.d.271.1 2
3.2 odd 2 405.2.e.e.271.1 2
9.2 odd 6 405.2.e.e.136.1 2
9.4 even 3 405.2.a.d.1.1 yes 1
9.5 odd 6 405.2.a.c.1.1 1
9.7 even 3 inner 405.2.e.d.136.1 2
36.23 even 6 6480.2.a.c.1.1 1
36.31 odd 6 6480.2.a.o.1.1 1
45.4 even 6 2025.2.a.d.1.1 1
45.13 odd 12 2025.2.b.f.649.1 2
45.14 odd 6 2025.2.a.c.1.1 1
45.22 odd 12 2025.2.b.f.649.2 2
45.23 even 12 2025.2.b.e.649.1 2
45.32 even 12 2025.2.b.e.649.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
405.2.a.c.1.1 1 9.5 odd 6
405.2.a.d.1.1 yes 1 9.4 even 3
405.2.e.d.136.1 2 9.7 even 3 inner
405.2.e.d.271.1 2 1.1 even 1 trivial
405.2.e.e.136.1 2 9.2 odd 6
405.2.e.e.271.1 2 3.2 odd 2
2025.2.a.c.1.1 1 45.14 odd 6
2025.2.a.d.1.1 1 45.4 even 6
2025.2.b.e.649.1 2 45.23 even 12
2025.2.b.e.649.2 2 45.32 even 12
2025.2.b.f.649.1 2 45.13 odd 12
2025.2.b.f.649.2 2 45.22 odd 12
6480.2.a.c.1.1 1 36.23 even 6
6480.2.a.o.1.1 1 36.31 odd 6