Properties

Label 1350.2.e.a.901.1
Level $1350$
Weight $2$
Character 1350.901
Analytic conductor $10.780$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1350,2,Mod(451,1350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1350, 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("1350.451");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1350.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: \(10.7798042729\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 90)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 901.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1350.901
Dual form 1350.2.e.a.451.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{7} +1.00000 q^{8} +(-1.00000 - 1.73205i) q^{11} +(-3.00000 + 5.19615i) q^{13} +(-0.500000 + 0.866025i) q^{14} +(-0.500000 - 0.866025i) q^{16} -2.00000 q^{17} +6.00000 q^{19} +(-1.00000 + 1.73205i) q^{22} +(-0.500000 + 0.866025i) q^{23} +6.00000 q^{26} +1.00000 q^{28} +(4.50000 + 7.79423i) q^{29} +(1.00000 - 1.73205i) q^{31} +(-0.500000 + 0.866025i) q^{32} +(1.00000 + 1.73205i) q^{34} -2.00000 q^{37} +(-3.00000 - 5.19615i) q^{38} +(-5.50000 + 9.52628i) q^{41} +(-2.00000 - 3.46410i) q^{43} +2.00000 q^{44} +1.00000 q^{46} +(3.50000 + 6.06218i) q^{47} +(3.00000 - 5.19615i) q^{49} +(-3.00000 - 5.19615i) q^{52} +(-0.500000 - 0.866025i) q^{56} +(4.50000 - 7.79423i) q^{58} +(-2.00000 + 3.46410i) q^{59} +(3.50000 + 6.06218i) q^{61} -2.00000 q^{62} +1.00000 q^{64} +(-5.50000 + 9.52628i) q^{67} +(1.00000 - 1.73205i) q^{68} +6.00000 q^{71} +4.00000 q^{73} +(1.00000 + 1.73205i) q^{74} +(-3.00000 + 5.19615i) q^{76} +(-1.00000 + 1.73205i) q^{77} +(6.00000 + 10.3923i) q^{79} +11.0000 q^{82} +(5.50000 + 9.52628i) q^{83} +(-2.00000 + 3.46410i) q^{86} +(-1.00000 - 1.73205i) q^{88} -1.00000 q^{89} +6.00000 q^{91} +(-0.500000 - 0.866025i) q^{92} +(3.50000 - 6.06218i) q^{94} +(-4.00000 - 6.92820i) q^{97} -6.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} - q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{4} - q^{7} + 2 q^{8} - 2 q^{11} - 6 q^{13} - q^{14} - q^{16} - 4 q^{17} + 12 q^{19} - 2 q^{22} - q^{23} + 12 q^{26} + 2 q^{28} + 9 q^{29} + 2 q^{31} - q^{32} + 2 q^{34} - 4 q^{37} - 6 q^{38} - 11 q^{41} - 4 q^{43} + 4 q^{44} + 2 q^{46} + 7 q^{47} + 6 q^{49} - 6 q^{52} - q^{56} + 9 q^{58} - 4 q^{59} + 7 q^{61} - 4 q^{62} + 2 q^{64} - 11 q^{67} + 2 q^{68} + 12 q^{71} + 8 q^{73} + 2 q^{74} - 6 q^{76} - 2 q^{77} + 12 q^{79} + 22 q^{82} + 11 q^{83} - 4 q^{86} - 2 q^{88} - 2 q^{89} + 12 q^{91} - q^{92} + 7 q^{94} - 8 q^{97} - 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1027\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 0.866025i −0.353553 0.612372i
\(3\) 0 0
\(4\) −0.500000 + 0.866025i −0.250000 + 0.433013i
\(5\) 0 0
\(6\) 0 0
\(7\) −0.500000 0.866025i −0.188982 0.327327i 0.755929 0.654654i \(-0.227186\pi\)
−0.944911 + 0.327327i \(0.893852\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 1.73205i −0.301511 0.522233i 0.674967 0.737848i \(-0.264158\pi\)
−0.976478 + 0.215615i \(0.930824\pi\)
\(12\) 0 0
\(13\) −3.00000 + 5.19615i −0.832050 + 1.44115i 0.0643593 + 0.997927i \(0.479500\pi\)
−0.896410 + 0.443227i \(0.853834\pi\)
\(14\) −0.500000 + 0.866025i −0.133631 + 0.231455i
\(15\) 0 0
\(16\) −0.500000 0.866025i −0.125000 0.216506i
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.00000 + 1.73205i −0.213201 + 0.369274i
\(23\) −0.500000 + 0.866025i −0.104257 + 0.180579i −0.913434 0.406986i \(-0.866580\pi\)
0.809177 + 0.587565i \(0.199913\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 6.00000 1.17670
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 4.50000 + 7.79423i 0.835629 + 1.44735i 0.893517 + 0.449029i \(0.148230\pi\)
−0.0578882 + 0.998323i \(0.518437\pi\)
\(30\) 0 0
\(31\) 1.00000 1.73205i 0.179605 0.311086i −0.762140 0.647412i \(-0.775851\pi\)
0.941745 + 0.336327i \(0.109185\pi\)
\(32\) −0.500000 + 0.866025i −0.0883883 + 0.153093i
\(33\) 0 0
\(34\) 1.00000 + 1.73205i 0.171499 + 0.297044i
\(35\) 0 0
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −3.00000 5.19615i −0.486664 0.842927i
\(39\) 0 0
\(40\) 0 0
\(41\) −5.50000 + 9.52628i −0.858956 + 1.48775i 0.0139704 + 0.999902i \(0.495553\pi\)
−0.872926 + 0.487852i \(0.837780\pi\)
\(42\) 0 0
\(43\) −2.00000 3.46410i −0.304997 0.528271i 0.672264 0.740312i \(-0.265322\pi\)
−0.977261 + 0.212041i \(0.931989\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 3.50000 + 6.06218i 0.510527 + 0.884260i 0.999926 + 0.0121990i \(0.00388317\pi\)
−0.489398 + 0.872060i \(0.662783\pi\)
\(48\) 0 0
\(49\) 3.00000 5.19615i 0.428571 0.742307i
\(50\) 0 0
\(51\) 0 0
\(52\) −3.00000 5.19615i −0.416025 0.720577i
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −0.500000 0.866025i −0.0668153 0.115728i
\(57\) 0 0
\(58\) 4.50000 7.79423i 0.590879 1.02343i
\(59\) −2.00000 + 3.46410i −0.260378 + 0.450988i −0.966342 0.257260i \(-0.917180\pi\)
0.705965 + 0.708247i \(0.250514\pi\)
\(60\) 0 0
\(61\) 3.50000 + 6.06218i 0.448129 + 0.776182i 0.998264 0.0588933i \(-0.0187572\pi\)
−0.550135 + 0.835076i \(0.685424\pi\)
\(62\) −2.00000 −0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −5.50000 + 9.52628i −0.671932 + 1.16382i 0.305424 + 0.952217i \(0.401202\pi\)
−0.977356 + 0.211604i \(0.932131\pi\)
\(68\) 1.00000 1.73205i 0.121268 0.210042i
\(69\) 0 0
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 1.00000 + 1.73205i 0.116248 + 0.201347i
\(75\) 0 0
\(76\) −3.00000 + 5.19615i −0.344124 + 0.596040i
\(77\) −1.00000 + 1.73205i −0.113961 + 0.197386i
\(78\) 0 0
\(79\) 6.00000 + 10.3923i 0.675053 + 1.16923i 0.976453 + 0.215728i \(0.0692125\pi\)
−0.301401 + 0.953498i \(0.597454\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 11.0000 1.21475
\(83\) 5.50000 + 9.52628i 0.603703 + 1.04565i 0.992255 + 0.124218i \(0.0396422\pi\)
−0.388552 + 0.921427i \(0.627024\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2.00000 + 3.46410i −0.215666 + 0.373544i
\(87\) 0 0
\(88\) −1.00000 1.73205i −0.106600 0.184637i
\(89\) −1.00000 −0.106000 −0.0529999 0.998595i \(-0.516878\pi\)
−0.0529999 + 0.998595i \(0.516878\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) −0.500000 0.866025i −0.0521286 0.0902894i
\(93\) 0 0
\(94\) 3.50000 6.06218i 0.360997 0.625266i
\(95\) 0 0
\(96\) 0 0
\(97\) −4.00000 6.92820i −0.406138 0.703452i 0.588315 0.808632i \(-0.299792\pi\)
−0.994453 + 0.105180i \(0.966458\pi\)
\(98\) −6.00000 −0.606092
\(99\) 0 0
\(100\) 0 0
\(101\) 1.00000 + 1.73205i 0.0995037 + 0.172345i 0.911479 0.411346i \(-0.134941\pi\)
−0.811976 + 0.583691i \(0.801608\pi\)
\(102\) 0 0
\(103\) −4.00000 + 6.92820i −0.394132 + 0.682656i −0.992990 0.118199i \(-0.962288\pi\)
0.598858 + 0.800855i \(0.295621\pi\)
\(104\) −3.00000 + 5.19615i −0.294174 + 0.509525i
\(105\) 0 0
\(106\) 0 0
\(107\) 3.00000 0.290021 0.145010 0.989430i \(-0.453678\pi\)
0.145010 + 0.989430i \(0.453678\pi\)
\(108\) 0 0
\(109\) 7.00000 0.670478 0.335239 0.942133i \(-0.391183\pi\)
0.335239 + 0.942133i \(0.391183\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −0.500000 + 0.866025i −0.0472456 + 0.0818317i
\(113\) −6.00000 + 10.3923i −0.564433 + 0.977626i 0.432670 + 0.901553i \(0.357572\pi\)
−0.997102 + 0.0760733i \(0.975762\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −9.00000 −0.835629
\(117\) 0 0
\(118\) 4.00000 0.368230
\(119\) 1.00000 + 1.73205i 0.0916698 + 0.158777i
\(120\) 0 0
\(121\) 3.50000 6.06218i 0.318182 0.551107i
\(122\) 3.50000 6.06218i 0.316875 0.548844i
\(123\) 0 0
\(124\) 1.00000 + 1.73205i 0.0898027 + 0.155543i
\(125\) 0 0
\(126\) 0 0
\(127\) 19.0000 1.68598 0.842989 0.537931i \(-0.180794\pi\)
0.842989 + 0.537931i \(0.180794\pi\)
\(128\) −0.500000 0.866025i −0.0441942 0.0765466i
\(129\) 0 0
\(130\) 0 0
\(131\) 6.00000 10.3923i 0.524222 0.907980i −0.475380 0.879781i \(-0.657689\pi\)
0.999602 0.0281993i \(-0.00897729\pi\)
\(132\) 0 0
\(133\) −3.00000 5.19615i −0.260133 0.450564i
\(134\) 11.0000 0.950255
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) −6.00000 10.3923i −0.512615 0.887875i −0.999893 0.0146279i \(-0.995344\pi\)
0.487278 0.873247i \(-0.337990\pi\)
\(138\) 0 0
\(139\) −8.00000 + 13.8564i −0.678551 + 1.17529i 0.296866 + 0.954919i \(0.404058\pi\)
−0.975417 + 0.220366i \(0.929275\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −3.00000 5.19615i −0.251754 0.436051i
\(143\) 12.0000 1.00349
\(144\) 0 0
\(145\) 0 0
\(146\) −2.00000 3.46410i −0.165521 0.286691i
\(147\) 0 0
\(148\) 1.00000 1.73205i 0.0821995 0.142374i
\(149\) −0.500000 + 0.866025i −0.0409616 + 0.0709476i −0.885779 0.464107i \(-0.846375\pi\)
0.844818 + 0.535054i \(0.179709\pi\)
\(150\) 0 0
\(151\) −5.00000 8.66025i −0.406894 0.704761i 0.587646 0.809118i \(-0.300055\pi\)
−0.994540 + 0.104357i \(0.966722\pi\)
\(152\) 6.00000 0.486664
\(153\) 0 0
\(154\) 2.00000 0.161165
\(155\) 0 0
\(156\) 0 0
\(157\) 2.00000 3.46410i 0.159617 0.276465i −0.775113 0.631822i \(-0.782307\pi\)
0.934731 + 0.355357i \(0.115641\pi\)
\(158\) 6.00000 10.3923i 0.477334 0.826767i
\(159\) 0 0
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) −5.50000 9.52628i −0.429478 0.743877i
\(165\) 0 0
\(166\) 5.50000 9.52628i 0.426883 0.739383i
\(167\) 1.50000 2.59808i 0.116073 0.201045i −0.802135 0.597143i \(-0.796303\pi\)
0.918208 + 0.396098i \(0.129636\pi\)
\(168\) 0 0
\(169\) −11.5000 19.9186i −0.884615 1.53220i
\(170\) 0 0
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) 2.00000 + 3.46410i 0.152057 + 0.263371i 0.931984 0.362500i \(-0.118077\pi\)
−0.779926 + 0.625871i \(0.784744\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.00000 + 1.73205i −0.0753778 + 0.130558i
\(177\) 0 0
\(178\) 0.500000 + 0.866025i 0.0374766 + 0.0649113i
\(179\) −2.00000 −0.149487 −0.0747435 0.997203i \(-0.523814\pi\)
−0.0747435 + 0.997203i \(0.523814\pi\)
\(180\) 0 0
\(181\) −13.0000 −0.966282 −0.483141 0.875542i \(-0.660504\pi\)
−0.483141 + 0.875542i \(0.660504\pi\)
\(182\) −3.00000 5.19615i −0.222375 0.385164i
\(183\) 0 0
\(184\) −0.500000 + 0.866025i −0.0368605 + 0.0638442i
\(185\) 0 0
\(186\) 0 0
\(187\) 2.00000 + 3.46410i 0.146254 + 0.253320i
\(188\) −7.00000 −0.510527
\(189\) 0 0
\(190\) 0 0
\(191\) 3.00000 + 5.19615i 0.217072 + 0.375980i 0.953912 0.300088i \(-0.0970159\pi\)
−0.736839 + 0.676068i \(0.763683\pi\)
\(192\) 0 0
\(193\) −5.00000 + 8.66025i −0.359908 + 0.623379i −0.987945 0.154805i \(-0.950525\pi\)
0.628037 + 0.778183i \(0.283859\pi\)
\(194\) −4.00000 + 6.92820i −0.287183 + 0.497416i
\(195\) 0 0
\(196\) 3.00000 + 5.19615i 0.214286 + 0.371154i
\(197\) 8.00000 0.569976 0.284988 0.958531i \(-0.408010\pi\)
0.284988 + 0.958531i \(0.408010\pi\)
\(198\) 0 0
\(199\) −18.0000 −1.27599 −0.637993 0.770042i \(-0.720235\pi\)
−0.637993 + 0.770042i \(0.720235\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 1.00000 1.73205i 0.0703598 0.121867i
\(203\) 4.50000 7.79423i 0.315838 0.547048i
\(204\) 0 0
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) 0 0
\(208\) 6.00000 0.416025
\(209\) −6.00000 10.3923i −0.415029 0.718851i
\(210\) 0 0
\(211\) −9.00000 + 15.5885i −0.619586 + 1.07315i 0.369976 + 0.929041i \(0.379366\pi\)
−0.989561 + 0.144112i \(0.953967\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −1.50000 2.59808i −0.102538 0.177601i
\(215\) 0 0
\(216\) 0 0
\(217\) −2.00000 −0.135769
\(218\) −3.50000 6.06218i −0.237050 0.410582i
\(219\) 0 0
\(220\) 0 0
\(221\) 6.00000 10.3923i 0.403604 0.699062i
\(222\) 0 0
\(223\) 11.5000 + 19.9186i 0.770097 + 1.33385i 0.937509 + 0.347960i \(0.113126\pi\)
−0.167412 + 0.985887i \(0.553541\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 12.0000 0.798228
\(227\) 4.00000 + 6.92820i 0.265489 + 0.459841i 0.967692 0.252136i \(-0.0811332\pi\)
−0.702202 + 0.711977i \(0.747800\pi\)
\(228\) 0 0
\(229\) 3.50000 6.06218i 0.231287 0.400600i −0.726900 0.686743i \(-0.759040\pi\)
0.958187 + 0.286143i \(0.0923732\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 4.50000 + 7.79423i 0.295439 + 0.511716i
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −2.00000 3.46410i −0.130189 0.225494i
\(237\) 0 0
\(238\) 1.00000 1.73205i 0.0648204 0.112272i
\(239\) 14.0000 24.2487i 0.905585 1.56852i 0.0854543 0.996342i \(-0.472766\pi\)
0.820130 0.572177i \(-0.193901\pi\)
\(240\) 0 0
\(241\) −0.500000 0.866025i −0.0322078 0.0557856i 0.849472 0.527633i \(-0.176921\pi\)
−0.881680 + 0.471848i \(0.843587\pi\)
\(242\) −7.00000 −0.449977
\(243\) 0 0
\(244\) −7.00000 −0.448129
\(245\) 0 0
\(246\) 0 0
\(247\) −18.0000 + 31.1769i −1.14531 + 1.98374i
\(248\) 1.00000 1.73205i 0.0635001 0.109985i
\(249\) 0 0
\(250\) 0 0
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) 0 0
\(253\) 2.00000 0.125739
\(254\) −9.50000 16.4545i −0.596083 1.03245i
\(255\) 0 0
\(256\) −0.500000 + 0.866025i −0.0312500 + 0.0541266i
\(257\) −6.00000 + 10.3923i −0.374270 + 0.648254i −0.990217 0.139533i \(-0.955440\pi\)
0.615948 + 0.787787i \(0.288773\pi\)
\(258\) 0 0
\(259\) 1.00000 + 1.73205i 0.0621370 + 0.107624i
\(260\) 0 0
\(261\) 0 0
\(262\) −12.0000 −0.741362
\(263\) −8.00000 13.8564i −0.493301 0.854423i 0.506669 0.862141i \(-0.330877\pi\)
−0.999970 + 0.00771799i \(0.997543\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −3.00000 + 5.19615i −0.183942 + 0.318597i
\(267\) 0 0
\(268\) −5.50000 9.52628i −0.335966 0.581910i
\(269\) −3.00000 −0.182913 −0.0914566 0.995809i \(-0.529152\pi\)
−0.0914566 + 0.995809i \(0.529152\pi\)
\(270\) 0 0
\(271\) −14.0000 −0.850439 −0.425220 0.905090i \(-0.639803\pi\)
−0.425220 + 0.905090i \(0.639803\pi\)
\(272\) 1.00000 + 1.73205i 0.0606339 + 0.105021i
\(273\) 0 0
\(274\) −6.00000 + 10.3923i −0.362473 + 0.627822i
\(275\) 0 0
\(276\) 0 0
\(277\) −11.0000 19.0526i −0.660926 1.14476i −0.980373 0.197153i \(-0.936830\pi\)
0.319447 0.947604i \(-0.396503\pi\)
\(278\) 16.0000 0.959616
\(279\) 0 0
\(280\) 0 0
\(281\) −1.50000 2.59808i −0.0894825 0.154988i 0.817810 0.575488i \(-0.195188\pi\)
−0.907293 + 0.420500i \(0.861855\pi\)
\(282\) 0 0
\(283\) 0.500000 0.866025i 0.0297219 0.0514799i −0.850782 0.525519i \(-0.823871\pi\)
0.880504 + 0.474039i \(0.157204\pi\)
\(284\) −3.00000 + 5.19615i −0.178017 + 0.308335i
\(285\) 0 0
\(286\) −6.00000 10.3923i −0.354787 0.614510i
\(287\) 11.0000 0.649309
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) −2.00000 + 3.46410i −0.117041 + 0.202721i
\(293\) 9.00000 15.5885i 0.525786 0.910687i −0.473763 0.880652i \(-0.657105\pi\)
0.999549 0.0300351i \(-0.00956192\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) 0 0
\(298\) 1.00000 0.0579284
\(299\) −3.00000 5.19615i −0.173494 0.300501i
\(300\) 0 0
\(301\) −2.00000 + 3.46410i −0.115278 + 0.199667i
\(302\) −5.00000 + 8.66025i −0.287718 + 0.498342i
\(303\) 0 0
\(304\) −3.00000 5.19615i −0.172062 0.298020i
\(305\) 0 0
\(306\) 0 0
\(307\) 9.00000 0.513657 0.256829 0.966457i \(-0.417322\pi\)
0.256829 + 0.966457i \(0.417322\pi\)
\(308\) −1.00000 1.73205i −0.0569803 0.0986928i
\(309\) 0 0
\(310\) 0 0
\(311\) 3.00000 5.19615i 0.170114 0.294647i −0.768345 0.640036i \(-0.778920\pi\)
0.938460 + 0.345389i \(0.112253\pi\)
\(312\) 0 0
\(313\) −11.0000 19.0526i −0.621757 1.07691i −0.989158 0.146852i \(-0.953086\pi\)
0.367402 0.930062i \(-0.380247\pi\)
\(314\) −4.00000 −0.225733
\(315\) 0 0
\(316\) −12.0000 −0.675053
\(317\) 1.00000 + 1.73205i 0.0561656 + 0.0972817i 0.892741 0.450570i \(-0.148779\pi\)
−0.836576 + 0.547852i \(0.815446\pi\)
\(318\) 0 0
\(319\) 9.00000 15.5885i 0.503903 0.872786i
\(320\) 0 0
\(321\) 0 0
\(322\) −0.500000 0.866025i −0.0278639 0.0482617i
\(323\) −12.0000 −0.667698
\(324\) 0 0
\(325\) 0 0
\(326\) 2.00000 + 3.46410i 0.110770 + 0.191859i
\(327\) 0 0
\(328\) −5.50000 + 9.52628i −0.303687 + 0.526001i
\(329\) 3.50000 6.06218i 0.192961 0.334219i
\(330\) 0 0
\(331\) 4.00000 + 6.92820i 0.219860 + 0.380808i 0.954765 0.297361i \(-0.0961066\pi\)
−0.734905 + 0.678170i \(0.762773\pi\)
\(332\) −11.0000 −0.603703
\(333\) 0 0
\(334\) −3.00000 −0.164153
\(335\) 0 0
\(336\) 0 0
\(337\) 4.00000 6.92820i 0.217894 0.377403i −0.736270 0.676688i \(-0.763415\pi\)
0.954164 + 0.299285i \(0.0967480\pi\)
\(338\) −11.5000 + 19.9186i −0.625518 + 1.08343i
\(339\) 0 0
\(340\) 0 0
\(341\) −4.00000 −0.216612
\(342\) 0 0
\(343\) −13.0000 −0.701934
\(344\) −2.00000 3.46410i −0.107833 0.186772i
\(345\) 0 0
\(346\) 2.00000 3.46410i 0.107521 0.186231i
\(347\) 6.00000 10.3923i 0.322097 0.557888i −0.658824 0.752297i \(-0.728946\pi\)
0.980921 + 0.194409i \(0.0622790\pi\)
\(348\) 0 0
\(349\) 5.50000 + 9.52628i 0.294408 + 0.509930i 0.974847 0.222875i \(-0.0715441\pi\)
−0.680439 + 0.732805i \(0.738211\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2.00000 0.106600
\(353\) 8.00000 + 13.8564i 0.425797 + 0.737502i 0.996495 0.0836583i \(-0.0266604\pi\)
−0.570697 + 0.821160i \(0.693327\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0.500000 0.866025i 0.0264999 0.0458993i
\(357\) 0 0
\(358\) 1.00000 + 1.73205i 0.0528516 + 0.0915417i
\(359\) −30.0000 −1.58334 −0.791670 0.610949i \(-0.790788\pi\)
−0.791670 + 0.610949i \(0.790788\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 6.50000 + 11.2583i 0.341632 + 0.591725i
\(363\) 0 0
\(364\) −3.00000 + 5.19615i −0.157243 + 0.272352i
\(365\) 0 0
\(366\) 0 0
\(367\) −8.00000 13.8564i −0.417597 0.723299i 0.578101 0.815966i \(-0.303794\pi\)
−0.995697 + 0.0926670i \(0.970461\pi\)
\(368\) 1.00000 0.0521286
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 6.00000 10.3923i 0.310668 0.538093i −0.667839 0.744306i \(-0.732781\pi\)
0.978507 + 0.206213i \(0.0661139\pi\)
\(374\) 2.00000 3.46410i 0.103418 0.179124i
\(375\) 0 0
\(376\) 3.50000 + 6.06218i 0.180499 + 0.312633i
\(377\) −54.0000 −2.78114
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 3.00000 5.19615i 0.153493 0.265858i
\(383\) −16.0000 + 27.7128i −0.817562 + 1.41606i 0.0899119 + 0.995950i \(0.471341\pi\)
−0.907474 + 0.420109i \(0.861992\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 10.0000 0.508987
\(387\) 0 0
\(388\) 8.00000 0.406138
\(389\) 9.50000 + 16.4545i 0.481669 + 0.834275i 0.999779 0.0210389i \(-0.00669738\pi\)
−0.518110 + 0.855314i \(0.673364\pi\)
\(390\) 0 0
\(391\) 1.00000 1.73205i 0.0505722 0.0875936i
\(392\) 3.00000 5.19615i 0.151523 0.262445i
\(393\) 0 0
\(394\) −4.00000 6.92820i −0.201517 0.349038i
\(395\) 0 0
\(396\) 0 0
\(397\) 4.00000 0.200754 0.100377 0.994949i \(-0.467995\pi\)
0.100377 + 0.994949i \(0.467995\pi\)
\(398\) 9.00000 + 15.5885i 0.451129 + 0.781379i
\(399\) 0 0
\(400\) 0 0
\(401\) −5.00000 + 8.66025i −0.249688 + 0.432472i −0.963439 0.267927i \(-0.913661\pi\)
0.713751 + 0.700399i \(0.246995\pi\)
\(402\) 0 0
\(403\) 6.00000 + 10.3923i 0.298881 + 0.517678i
\(404\) −2.00000 −0.0995037
\(405\) 0 0
\(406\) −9.00000 −0.446663
\(407\) 2.00000 + 3.46410i 0.0991363 + 0.171709i
\(408\) 0 0
\(409\) −19.0000 + 32.9090i −0.939490 + 1.62724i −0.173064 + 0.984911i \(0.555367\pi\)
−0.766426 + 0.642333i \(0.777967\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −4.00000 6.92820i −0.197066 0.341328i
\(413\) 4.00000 0.196827
\(414\) 0 0
\(415\) 0 0
\(416\) −3.00000 5.19615i −0.147087 0.254762i
\(417\) 0 0
\(418\) −6.00000 + 10.3923i −0.293470 + 0.508304i
\(419\) 17.0000 29.4449i 0.830504 1.43848i −0.0671345 0.997744i \(-0.521386\pi\)
0.897639 0.440732i \(-0.145281\pi\)
\(420\) 0 0
\(421\) −11.0000 19.0526i −0.536107 0.928565i −0.999109 0.0422075i \(-0.986561\pi\)
0.463002 0.886357i \(-0.346772\pi\)
\(422\) 18.0000 0.876226
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 3.50000 6.06218i 0.169377 0.293369i
\(428\) −1.50000 + 2.59808i −0.0725052 + 0.125583i
\(429\) 0 0
\(430\) 0 0
\(431\) −16.0000 −0.770693 −0.385346 0.922772i \(-0.625918\pi\)
−0.385346 + 0.922772i \(0.625918\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 1.00000 + 1.73205i 0.0480015 + 0.0831411i
\(435\) 0 0
\(436\) −3.50000 + 6.06218i −0.167620 + 0.290326i
\(437\) −3.00000 + 5.19615i −0.143509 + 0.248566i
\(438\) 0 0
\(439\) −12.0000 20.7846i −0.572729 0.991995i −0.996284 0.0861252i \(-0.972552\pi\)
0.423556 0.905870i \(-0.360782\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −12.0000 −0.570782
\(443\) 4.50000 + 7.79423i 0.213801 + 0.370315i 0.952901 0.303281i \(-0.0980821\pi\)
−0.739100 + 0.673596i \(0.764749\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 11.5000 19.9186i 0.544541 0.943172i
\(447\) 0 0
\(448\) −0.500000 0.866025i −0.0236228 0.0409159i
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) 22.0000 1.03594
\(452\) −6.00000 10.3923i −0.282216 0.488813i
\(453\) 0 0
\(454\) 4.00000 6.92820i 0.187729 0.325157i
\(455\) 0 0
\(456\) 0 0
\(457\) −5.00000 8.66025i −0.233890 0.405110i 0.725059 0.688686i \(-0.241812\pi\)
−0.958950 + 0.283577i \(0.908479\pi\)
\(458\) −7.00000 −0.327089
\(459\) 0 0
\(460\) 0 0
\(461\) −10.5000 18.1865i −0.489034 0.847031i 0.510887 0.859648i \(-0.329317\pi\)
−0.999920 + 0.0126168i \(0.995984\pi\)
\(462\) 0 0
\(463\) 18.0000 31.1769i 0.836531 1.44891i −0.0562469 0.998417i \(-0.517913\pi\)
0.892778 0.450497i \(-0.148753\pi\)
\(464\) 4.50000 7.79423i 0.208907 0.361838i
\(465\) 0 0
\(466\) 5.00000 + 8.66025i 0.231621 + 0.401179i
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) 0 0
\(469\) 11.0000 0.507933
\(470\) 0 0
\(471\) 0 0
\(472\) −2.00000 + 3.46410i −0.0920575 + 0.159448i
\(473\) −4.00000 + 6.92820i −0.183920 + 0.318559i
\(474\) 0 0
\(475\) 0 0
\(476\) −2.00000 −0.0916698
\(477\) 0 0
\(478\) −28.0000 −1.28069
\(479\) 14.0000 + 24.2487i 0.639676 + 1.10795i 0.985504 + 0.169654i \(0.0542649\pi\)
−0.345827 + 0.938298i \(0.612402\pi\)
\(480\) 0 0
\(481\) 6.00000 10.3923i 0.273576 0.473848i
\(482\) −0.500000 + 0.866025i −0.0227744 + 0.0394464i
\(483\) 0 0
\(484\) 3.50000 + 6.06218i 0.159091 + 0.275554i
\(485\) 0 0
\(486\) 0 0
\(487\) 12.0000 0.543772 0.271886 0.962329i \(-0.412353\pi\)
0.271886 + 0.962329i \(0.412353\pi\)
\(488\) 3.50000 + 6.06218i 0.158438 + 0.274422i
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(492\) 0 0
\(493\) −9.00000 15.5885i −0.405340 0.702069i
\(494\) 36.0000 1.61972
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) −3.00000 5.19615i −0.134568 0.233079i
\(498\) 0 0
\(499\) 12.0000 20.7846i 0.537194 0.930447i −0.461860 0.886953i \(-0.652818\pi\)
0.999054 0.0434940i \(-0.0138489\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 9.00000 + 15.5885i 0.401690 + 0.695747i
\(503\) 27.0000 1.20387 0.601935 0.798545i \(-0.294397\pi\)
0.601935 + 0.798545i \(0.294397\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −1.00000 1.73205i −0.0444554 0.0769991i
\(507\) 0 0
\(508\) −9.50000 + 16.4545i −0.421494 + 0.730050i
\(509\) −7.50000 + 12.9904i −0.332432 + 0.575789i −0.982988 0.183669i \(-0.941202\pi\)
0.650556 + 0.759458i \(0.274536\pi\)
\(510\) 0 0
\(511\) −2.00000 3.46410i −0.0884748 0.153243i
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 12.0000 0.529297
\(515\) 0 0
\(516\) 0 0
\(517\) 7.00000 12.1244i 0.307860 0.533229i
\(518\) 1.00000 1.73205i 0.0439375 0.0761019i
\(519\) 0 0
\(520\) 0 0
\(521\) 37.0000 1.62100 0.810500 0.585739i \(-0.199196\pi\)
0.810500 + 0.585739i \(0.199196\pi\)
\(522\) 0 0
\(523\) 29.0000 1.26808 0.634041 0.773300i \(-0.281395\pi\)
0.634041 + 0.773300i \(0.281395\pi\)
\(524\) 6.00000 + 10.3923i 0.262111 + 0.453990i
\(525\) 0 0
\(526\) −8.00000 + 13.8564i −0.348817 + 0.604168i
\(527\) −2.00000 + 3.46410i −0.0871214 + 0.150899i
\(528\) 0 0
\(529\) 11.0000 + 19.0526i 0.478261 + 0.828372i
\(530\) 0 0
\(531\) 0 0
\(532\) 6.00000 0.260133
\(533\) −33.0000 57.1577i −1.42939 2.47577i
\(534\) 0 0
\(535\) 0 0
\(536\) −5.50000 + 9.52628i −0.237564 + 0.411473i
\(537\) 0 0
\(538\) 1.50000 + 2.59808i 0.0646696 + 0.112011i
\(539\) −12.0000 −0.516877
\(540\) 0 0
\(541\) −17.0000 −0.730887 −0.365444 0.930834i \(-0.619083\pi\)
−0.365444 + 0.930834i \(0.619083\pi\)
\(542\) 7.00000 + 12.1244i 0.300676 + 0.520786i
\(543\) 0 0
\(544\) 1.00000 1.73205i 0.0428746 0.0742611i
\(545\) 0 0
\(546\) 0 0
\(547\) 17.5000 + 30.3109i 0.748246 + 1.29600i 0.948663 + 0.316289i \(0.102437\pi\)
−0.200417 + 0.979711i \(0.564230\pi\)
\(548\) 12.0000 0.512615
\(549\) 0 0
\(550\) 0 0
\(551\) 27.0000 + 46.7654i 1.15024 + 1.99227i
\(552\) 0 0
\(553\) 6.00000 10.3923i 0.255146 0.441926i
\(554\) −11.0000 + 19.0526i −0.467345 + 0.809466i
\(555\) 0 0
\(556\) −8.00000 13.8564i −0.339276 0.587643i
\(557\) −24.0000 −1.01691 −0.508456 0.861088i \(-0.669784\pi\)
−0.508456 + 0.861088i \(0.669784\pi\)
\(558\) 0 0
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) −1.50000 + 2.59808i −0.0632737 + 0.109593i
\(563\) 18.5000 32.0429i 0.779682 1.35045i −0.152443 0.988312i \(-0.548714\pi\)
0.932125 0.362137i \(-0.117953\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −1.00000 −0.0420331
\(567\) 0 0
\(568\) 6.00000 0.251754
\(569\) −1.00000 1.73205i −0.0419222 0.0726113i 0.844303 0.535866i \(-0.180015\pi\)
−0.886225 + 0.463255i \(0.846681\pi\)
\(570\) 0 0
\(571\) 10.0000 17.3205i 0.418487 0.724841i −0.577301 0.816532i \(-0.695894\pi\)
0.995788 + 0.0916910i \(0.0292272\pi\)
\(572\) −6.00000 + 10.3923i −0.250873 + 0.434524i
\(573\) 0 0
\(574\) −5.50000 9.52628i −0.229566 0.397619i
\(575\) 0 0
\(576\) 0 0
\(577\) −32.0000 −1.33218 −0.666089 0.745873i \(-0.732033\pi\)
−0.666089 + 0.745873i \(0.732033\pi\)
\(578\) 6.50000 + 11.2583i 0.270364 + 0.468285i
\(579\) 0 0
\(580\) 0 0
\(581\) 5.50000 9.52628i 0.228178 0.395217i
\(582\) 0 0
\(583\) 0 0
\(584\) 4.00000 0.165521
\(585\) 0 0
\(586\) −18.0000 −0.743573
\(587\) −1.50000 2.59808i −0.0619116 0.107234i 0.833408 0.552658i \(-0.186386\pi\)
−0.895320 + 0.445424i \(0.853053\pi\)
\(588\) 0 0
\(589\) 6.00000 10.3923i 0.247226 0.428207i
\(590\) 0 0
\(591\) 0 0
\(592\) 1.00000 + 1.73205i 0.0410997 + 0.0711868i
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −0.500000 0.866025i −0.0204808 0.0354738i
\(597\) 0 0
\(598\) −3.00000 + 5.19615i −0.122679 + 0.212486i
\(599\) −6.00000 + 10.3923i −0.245153 + 0.424618i −0.962175 0.272433i \(-0.912172\pi\)
0.717021 + 0.697051i \(0.245505\pi\)
\(600\) 0 0
\(601\) 11.0000 + 19.0526i 0.448699 + 0.777170i 0.998302 0.0582563i \(-0.0185541\pi\)
−0.549602 + 0.835426i \(0.685221\pi\)
\(602\) 4.00000 0.163028
\(603\) 0 0
\(604\) 10.0000 0.406894
\(605\) 0 0
\(606\) 0 0
\(607\) 0.500000 0.866025i 0.0202944 0.0351509i −0.855700 0.517472i \(-0.826873\pi\)
0.875994 + 0.482322i \(0.160206\pi\)
\(608\) −3.00000 + 5.19615i −0.121666 + 0.210732i
\(609\) 0 0
\(610\) 0 0
\(611\) −42.0000 −1.69914
\(612\) 0 0
\(613\) 34.0000 1.37325 0.686624 0.727013i \(-0.259092\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(614\) −4.50000 7.79423i −0.181605 0.314549i
\(615\) 0 0
\(616\) −1.00000 + 1.73205i −0.0402911 + 0.0697863i
\(617\) 16.0000 27.7128i 0.644136 1.11568i −0.340365 0.940294i \(-0.610551\pi\)
0.984500 0.175382i \(-0.0561162\pi\)
\(618\) 0 0
\(619\) 5.00000 + 8.66025i 0.200967 + 0.348085i 0.948840 0.315757i \(-0.102258\pi\)
−0.747873 + 0.663842i \(0.768925\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −6.00000 −0.240578
\(623\) 0.500000 + 0.866025i 0.0200321 + 0.0346966i
\(624\) 0 0
\(625\) 0 0
\(626\) −11.0000 + 19.0526i −0.439648 + 0.761493i
\(627\) 0 0
\(628\) 2.00000 + 3.46410i 0.0798087 + 0.138233i
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 6.00000 + 10.3923i 0.238667 + 0.413384i
\(633\) 0 0
\(634\) 1.00000 1.73205i 0.0397151 0.0687885i
\(635\) 0 0
\(636\) 0 0
\(637\) 18.0000 + 31.1769i 0.713186 + 1.23527i
\(638\) −18.0000 −0.712627
\(639\) 0 0
\(640\) 0 0
\(641\) 6.50000 + 11.2583i 0.256735 + 0.444677i 0.965365 0.260902i \(-0.0840201\pi\)
−0.708631 + 0.705580i \(0.750687\pi\)
\(642\) 0 0
\(643\) −16.5000 + 28.5788i −0.650696 + 1.12704i 0.332258 + 0.943189i \(0.392190\pi\)
−0.982954 + 0.183851i \(0.941144\pi\)
\(644\) −0.500000 + 0.866025i −0.0197028 + 0.0341262i
\(645\) 0 0
\(646\) 6.00000 + 10.3923i 0.236067 + 0.408880i
\(647\) 33.0000 1.29736 0.648682 0.761060i \(-0.275321\pi\)
0.648682 + 0.761060i \(0.275321\pi\)
\(648\) 0 0
\(649\) 8.00000 0.314027
\(650\) 0 0
\(651\) 0 0
\(652\) 2.00000 3.46410i 0.0783260 0.135665i
\(653\) 13.0000 22.5167i 0.508729 0.881145i −0.491220 0.871036i \(-0.663449\pi\)
0.999949 0.0101092i \(-0.00321793\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 11.0000 0.429478
\(657\) 0 0
\(658\) −7.00000 −0.272888
\(659\) 10.0000 + 17.3205i 0.389545 + 0.674711i 0.992388 0.123148i \(-0.0392990\pi\)
−0.602844 + 0.797859i \(0.705966\pi\)
\(660\) 0 0
\(661\) 5.00000 8.66025i 0.194477 0.336845i −0.752252 0.658876i \(-0.771032\pi\)
0.946729 + 0.322031i \(0.104366\pi\)
\(662\) 4.00000 6.92820i 0.155464 0.269272i
\(663\) 0 0
\(664\) 5.50000 + 9.52628i 0.213441 + 0.369691i
\(665\) 0 0
\(666\) 0 0
\(667\) −9.00000 −0.348481
\(668\) 1.50000 + 2.59808i 0.0580367 + 0.100523i
\(669\) 0 0
\(670\) 0 0
\(671\) 7.00000 12.1244i 0.270232 0.468056i
\(672\) 0 0
\(673\) 3.00000 + 5.19615i 0.115642 + 0.200297i 0.918036 0.396497i \(-0.129774\pi\)
−0.802395 + 0.596794i \(0.796441\pi\)
\(674\) −8.00000 −0.308148
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) −11.0000 19.0526i −0.422764 0.732249i 0.573444 0.819244i \(-0.305607\pi\)
−0.996209 + 0.0869952i \(0.972274\pi\)
\(678\) 0 0
\(679\) −4.00000 + 6.92820i −0.153506 + 0.265880i
\(680\) 0 0
\(681\) 0 0
\(682\) 2.00000 + 3.46410i 0.0765840 + 0.132647i
\(683\) 4.00000 0.153056 0.0765279 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 6.50000 + 11.2583i 0.248171 + 0.429845i
\(687\) 0 0
\(688\) −2.00000 + 3.46410i −0.0762493 + 0.132068i
\(689\) 0 0
\(690\) 0 0
\(691\) 22.0000 + 38.1051i 0.836919 + 1.44959i 0.892458 + 0.451130i \(0.148979\pi\)
−0.0555386 + 0.998457i \(0.517688\pi\)
\(692\) −4.00000 −0.152057
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) 0 0
\(696\) 0 0
\(697\) 11.0000 19.0526i 0.416655 0.721667i
\(698\) 5.50000 9.52628i 0.208178 0.360575i
\(699\) 0 0
\(700\) 0 0
\(701\) −13.0000 −0.491003 −0.245502 0.969396i \(-0.578953\pi\)
−0.245502 + 0.969396i \(0.578953\pi\)
\(702\) 0 0
\(703\) −12.0000 −0.452589
\(704\) −1.00000 1.73205i −0.0376889 0.0652791i
\(705\) 0 0
\(706\) 8.00000 13.8564i 0.301084 0.521493i
\(707\) 1.00000 1.73205i 0.0376089 0.0651405i
\(708\) 0 0
\(709\) 13.5000 + 23.3827i 0.507003 + 0.878155i 0.999967 + 0.00810550i \(0.00258009\pi\)
−0.492964 + 0.870050i \(0.664087\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −1.00000 −0.0374766
\(713\) 1.00000 + 1.73205i 0.0374503 + 0.0648658i
\(714\) 0 0
\(715\) 0 0
\(716\) 1.00000 1.73205i 0.0373718 0.0647298i
\(717\) 0 0
\(718\) 15.0000 + 25.9808i 0.559795 + 0.969593i
\(719\) 44.0000 1.64092 0.820462 0.571702i \(-0.193717\pi\)
0.820462 + 0.571702i \(0.193717\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) −8.50000 14.7224i −0.316337 0.547912i
\(723\) 0 0
\(724\) 6.50000 11.2583i 0.241571 0.418413i
\(725\) 0 0
\(726\) 0 0
\(727\) 10.5000 + 18.1865i 0.389423 + 0.674501i 0.992372 0.123279i \(-0.0393409\pi\)
−0.602949 + 0.797780i \(0.706008\pi\)
\(728\) 6.00000 0.222375
\(729\) 0 0
\(730\) 0 0
\(731\) 4.00000 + 6.92820i 0.147945 + 0.256249i
\(732\) 0 0
\(733\) 2.00000 3.46410i 0.0738717 0.127950i −0.826723 0.562609i \(-0.809798\pi\)
0.900595 + 0.434659i \(0.143131\pi\)
\(734\) −8.00000 + 13.8564i −0.295285 + 0.511449i
\(735\) 0 0
\(736\) −0.500000 0.866025i −0.0184302 0.0319221i
\(737\) 22.0000 0.810380
\(738\) 0 0
\(739\) 40.0000 1.47142 0.735712 0.677295i \(-0.236848\pi\)
0.735712 + 0.677295i \(0.236848\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 7.50000 12.9904i 0.275148 0.476571i −0.695024 0.718986i \(-0.744606\pi\)
0.970173 + 0.242415i \(0.0779397\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −12.0000 −0.439351
\(747\) 0 0
\(748\) −4.00000 −0.146254
\(749\) −1.50000 2.59808i −0.0548088 0.0949316i
\(750\) 0 0
\(751\) −13.0000 + 22.5167i −0.474377 + 0.821645i −0.999570 0.0293387i \(-0.990660\pi\)
0.525193 + 0.850983i \(0.323993\pi\)
\(752\) 3.50000 6.06218i 0.127632 0.221065i
\(753\) 0 0
\(754\) 27.0000 + 46.7654i 0.983282 + 1.70309i
\(755\) 0 0
\(756\) 0 0
\(757\) 10.0000 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) 8.00000 + 13.8564i 0.290573 + 0.503287i
\(759\) 0 0
\(760\) 0 0
\(761\) −4.50000 + 7.79423i −0.163125 + 0.282541i −0.935988 0.352032i \(-0.885491\pi\)
0.772863 + 0.634573i \(0.218824\pi\)
\(762\) 0 0
\(763\) −3.50000 6.06218i −0.126709 0.219466i
\(764\) −6.00000 −0.217072
\(765\) 0 0
\(766\) 32.0000 1.15621
\(767\) −12.0000 20.7846i −0.433295 0.750489i
\(768\) 0 0
\(769\) 7.50000 12.9904i 0.270457 0.468445i −0.698522 0.715589i \(-0.746159\pi\)
0.968979 + 0.247143i \(0.0794919\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −5.00000 8.66025i −0.179954 0.311689i
\(773\) 12.0000 0.431610 0.215805 0.976436i \(-0.430762\pi\)
0.215805 + 0.976436i \(0.430762\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −4.00000 6.92820i −0.143592 0.248708i
\(777\) 0 0
\(778\) 9.50000 16.4545i 0.340592 0.589922i
\(779\) −33.0000 + 57.1577i −1.18235 + 2.04789i
\(780\) 0 0
\(781\) −6.00000 10.3923i −0.214697 0.371866i
\(782\) −2.00000 −0.0715199
\(783\) 0 0
\(784\) −6.00000 −0.214286
\(785\) 0 0
\(786\) 0 0
\(787\) 22.0000 38.1051i 0.784215 1.35830i −0.145251 0.989395i \(-0.546399\pi\)
0.929467 0.368906i \(-0.120268\pi\)
\(788\) −4.00000 + 6.92820i −0.142494 + 0.246807i
\(789\) 0 0
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 0 0
\(793\) −42.0000 −1.49146
\(794\) −2.00000 3.46410i −0.0709773 0.122936i
\(795\) 0 0
\(796\) 9.00000 15.5885i 0.318997 0.552518i
\(797\) 1.00000 1.73205i 0.0354218 0.0613524i −0.847771 0.530362i \(-0.822056\pi\)
0.883193 + 0.469010i \(0.155389\pi\)
\(798\) 0 0
\(799\) −7.00000 12.1244i −0.247642 0.428929i
\(800\) 0 0
\(801\) 0 0
\(802\) 10.0000 0.353112
\(803\) −4.00000 6.92820i −0.141157 0.244491i
\(804\) 0 0
\(805\) 0 0
\(806\) 6.00000 10.3923i 0.211341 0.366053i
\(807\) 0 0
\(808\) 1.00000 + 1.73205i 0.0351799 + 0.0609333i
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) −20.0000 −0.702295 −0.351147 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) 4.50000 + 7.79423i 0.157919 + 0.273524i
\(813\) 0 0
\(814\) 2.00000 3.46410i 0.0701000 0.121417i
\(815\) 0 0
\(816\) 0 0
\(817\) −12.0000 20.7846i −0.419827 0.727161i
\(818\) 38.0000 1.32864
\(819\) 0 0
\(820\) 0 0
\(821\) −15.5000 26.8468i −0.540954 0.936959i −0.998850 0.0479535i \(-0.984730\pi\)
0.457896 0.889006i \(-0.348603\pi\)
\(822\) 0 0
\(823\) 7.50000 12.9904i 0.261434 0.452816i −0.705190 0.709019i \(-0.749138\pi\)
0.966623 + 0.256203i \(0.0824714\pi\)
\(824\) −4.00000 + 6.92820i −0.139347 + 0.241355i
\(825\) 0 0
\(826\) −2.00000 3.46410i −0.0695889 0.120532i
\(827\) −35.0000 −1.21707 −0.608535 0.793527i \(-0.708242\pi\)
−0.608535 + 0.793527i \(0.708242\pi\)
\(828\) 0 0
\(829\) 15.0000 0.520972 0.260486 0.965478i \(-0.416117\pi\)
0.260486 + 0.965478i \(0.416117\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −3.00000 + 5.19615i −0.104006 + 0.180144i
\(833\) −6.00000 + 10.3923i −0.207888 + 0.360072i
\(834\) 0 0
\(835\) 0 0
\(836\) 12.0000 0.415029
\(837\) 0 0
\(838\) −34.0000 −1.17451
\(839\) 19.0000 + 32.9090i 0.655953 + 1.13614i 0.981654 + 0.190671i \(0.0610663\pi\)
−0.325701 + 0.945473i \(0.605600\pi\)
\(840\) 0 0
\(841\) −26.0000 + 45.0333i −0.896552 + 1.55287i
\(842\) −11.0000 + 19.0526i −0.379085 + 0.656595i
\(843\) 0 0
\(844\) −9.00000 15.5885i −0.309793 0.536577i
\(845\) 0 0
\(846\) 0 0
\(847\) −7.00000 −0.240523
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1.00000 1.73205i 0.0342796 0.0593739i
\(852\) 0 0
\(853\) 23.0000 + 39.8372i 0.787505 + 1.36400i 0.927491 + 0.373845i \(0.121961\pi\)
−0.139986 + 0.990153i \(0.544706\pi\)
\(854\) −7.00000 −0.239535
\(855\) 0 0
\(856\) 3.00000 0.102538
\(857\) 9.00000 + 15.5885i 0.307434 + 0.532492i 0.977800 0.209539i \(-0.0671963\pi\)
−0.670366 + 0.742030i \(0.733863\pi\)
\(858\) 0 0
\(859\) −7.00000 + 12.1244i −0.238837 + 0.413678i −0.960381 0.278691i \(-0.910099\pi\)
0.721544 + 0.692369i \(0.243433\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 8.00000 + 13.8564i 0.272481 + 0.471951i
\(863\) −45.0000 −1.53182 −0.765909 0.642949i \(-0.777711\pi\)
−0.765909 + 0.642949i \(0.777711\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 1.00000 + 1.73205i 0.0339814 + 0.0588575i
\(867\) 0 0
\(868\) 1.00000 1.73205i 0.0339422 0.0587896i
\(869\) 12.0000 20.7846i 0.407072 0.705070i
\(870\) 0 0
\(871\) −33.0000 57.1577i −1.11816 1.93671i
\(872\) 7.00000 0.237050
\(873\) 0 0
\(874\) 6.00000 0.202953
\(875\) 0 0
\(876\) 0 0
\(877\) 23.0000 39.8372i 0.776655 1.34521i −0.157205 0.987566i \(-0.550248\pi\)
0.933860 0.357640i \(-0.116418\pi\)
\(878\) −12.0000 + 20.7846i −0.404980 + 0.701447i
\(879\) 0 0
\(880\) 0 0
\(881\) 15.0000 0.505363 0.252681 0.967550i \(-0.418688\pi\)
0.252681 + 0.967550i \(0.418688\pi\)
\(882\) 0 0
\(883\) −3.00000 −0.100958 −0.0504790 0.998725i \(-0.516075\pi\)
−0.0504790 + 0.998725i \(0.516075\pi\)
\(884\) 6.00000 + 10.3923i 0.201802 + 0.349531i
\(885\) 0 0
\(886\) 4.50000 7.79423i 0.151180 0.261852i
\(887\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(888\) 0 0
\(889\) −9.50000 16.4545i −0.318620 0.551866i
\(890\) 0 0
\(891\) 0 0
\(892\) −23.0000 −0.770097
\(893\) 21.0000 + 36.3731i 0.702738 + 1.21718i
\(894\) 0 0
\(895\) 0 0
\(896\) −0.500000 + 0.866025i −0.0167038 + 0.0289319i
\(897\) 0 0
\(898\) 9.00000 + 15.5885i 0.300334 + 0.520194i
\(899\) 18.0000 0.600334
\(900\) 0 0
\(901\) 0 0
\(902\) −11.0000 19.0526i −0.366260 0.634381i
\(903\) 0 0
\(904\) −6.00000 + 10.3923i −0.199557 + 0.345643i
\(905\) 0 0
\(906\) 0 0
\(907\) 16.5000 + 28.5788i 0.547874 + 0.948945i 0.998420 + 0.0561918i \(0.0178958\pi\)
−0.450546 + 0.892753i \(0.648771\pi\)
\(908\) −8.00000 −0.265489
\(909\) 0 0
\(910\) 0 0
\(911\) 6.00000 + 10.3923i 0.198789 + 0.344312i 0.948136 0.317865i \(-0.102966\pi\)
−0.749347 + 0.662177i \(0.769633\pi\)
\(912\) 0 0
\(913\) 11.0000 19.0526i 0.364047 0.630548i
\(914\) −5.00000 + 8.66025i −0.165385 + 0.286456i
\(915\) 0 0
\(916\) 3.50000 + 6.06218i 0.115643 + 0.200300i
\(917\) −12.0000 −0.396275
\(918\) 0 0
\(919\) −36.0000 −1.18753 −0.593765 0.804638i \(-0.702359\pi\)
−0.593765 + 0.804638i \(0.702359\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −10.5000 + 18.1865i −0.345799 + 0.598942i
\(923\) −18.0000 + 31.1769i −0.592477 + 1.02620i
\(924\) 0 0
\(925\) 0 0
\(926\) −36.0000 −1.18303
\(927\) 0 0
\(928\) −9.00000 −0.295439
\(929\) −9.00000 15.5885i −0.295280 0.511441i 0.679770 0.733426i \(-0.262080\pi\)
−0.975050 + 0.221985i \(0.928746\pi\)
\(930\) 0 0
\(931\) 18.0000 31.1769i 0.589926 1.02178i
\(932\) 5.00000 8.66025i 0.163780 0.283676i
\(933\) 0 0
\(934\) −18.0000 31.1769i −0.588978 1.02014i
\(935\) 0 0
\(936\) 0 0
\(937\) −52.0000 −1.69877 −0.849383 0.527777i \(-0.823026\pi\)
−0.849383 + 0.527777i \(0.823026\pi\)
\(938\) −5.50000 9.52628i −0.179581 0.311044i
\(939\) 0 0
\(940\) 0 0
\(941\) −20.5000 + 35.5070i −0.668281 + 1.15750i 0.310104 + 0.950703i \(0.399636\pi\)
−0.978385 + 0.206794i \(0.933697\pi\)
\(942\) 0 0
\(943\) −5.50000 9.52628i −0.179105 0.310218i
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) 8.00000 0.260102
\(947\) 25.5000 + 44.1673i 0.828639 + 1.43524i 0.899106 + 0.437730i \(0.144217\pi\)
−0.0704677 + 0.997514i \(0.522449\pi\)
\(948\) 0 0
\(949\) −12.0000 + 20.7846i −0.389536 + 0.674697i
\(950\) 0 0
\(951\) 0 0
\(952\) 1.00000 + 1.73205i 0.0324102 + 0.0561361i
\(953\) 2.00000 0.0647864 0.0323932 0.999475i \(-0.489687\pi\)
0.0323932 + 0.999475i \(0.489687\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 14.0000 + 24.2487i 0.452792 + 0.784259i
\(957\) 0 0
\(958\) 14.0000 24.2487i 0.452319 0.783440i
\(959\) −6.00000 + 10.3923i −0.193750 + 0.335585i
\(960\) 0 0
\(961\) 13.5000 + 23.3827i 0.435484 + 0.754280i
\(962\) −12.0000 −0.386896
\(963\) 0 0
\(964\) 1.00000 0.0322078
\(965\) 0 0
\(966\) 0 0
\(967\) −8.50000 + 14.7224i −0.273342 + 0.473441i −0.969715 0.244238i \(-0.921462\pi\)
0.696374 + 0.717679i \(0.254796\pi\)
\(968\) 3.50000 6.06218i 0.112494 0.194846i
\(969\) 0 0
\(970\) 0 0
\(971\) 42.0000 1.34784 0.673922 0.738802i \(-0.264608\pi\)
0.673922 + 0.738802i \(0.264608\pi\)
\(972\) 0 0
\(973\) 16.0000 0.512936
\(974\) −6.00000 10.3923i −0.192252 0.332991i
\(975\) 0 0
\(976\) 3.50000 6.06218i 0.112032 0.194046i
\(977\) −27.0000 + 46.7654i −0.863807 + 1.49616i 0.00442082 + 0.999990i \(0.498593\pi\)
−0.868227 + 0.496167i \(0.834741\pi\)
\(978\) 0 0
\(979\) 1.00000 + 1.73205i 0.0319601 + 0.0553566i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 9.50000 + 16.4545i 0.303003 + 0.524816i 0.976815 0.214087i \(-0.0686775\pi\)
−0.673812 + 0.738903i \(0.735344\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −9.00000 + 15.5885i −0.286618 + 0.496438i
\(987\) 0 0
\(988\) −18.0000 31.1769i −0.572656 0.991870i
\(989\) 4.00000 0.127193
\(990\) 0 0
\(991\) 4.00000 0.127064 0.0635321 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) 1.00000 + 1.73205i 0.0317500 + 0.0549927i
\(993\) 0 0
\(994\) −3.00000 + 5.19615i −0.0951542 + 0.164812i
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(998\) −24.0000 −0.759707
\(999\) 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 1350.2.e.a.901.1 2
3.2 odd 2 450.2.e.g.301.1 2
5.2 odd 4 270.2.i.a.199.2 4
5.3 odd 4 270.2.i.a.199.1 4
5.4 even 2 1350.2.e.i.901.1 2
9.2 odd 6 450.2.e.g.151.1 2
9.4 even 3 4050.2.a.be.1.1 1
9.5 odd 6 4050.2.a.j.1.1 1
9.7 even 3 inner 1350.2.e.a.451.1 2
15.2 even 4 90.2.i.a.49.1 4
15.8 even 4 90.2.i.a.49.2 yes 4
15.14 odd 2 450.2.e.b.301.1 2
20.3 even 4 2160.2.by.b.1009.1 4
20.7 even 4 2160.2.by.b.1009.2 4
45.2 even 12 90.2.i.a.79.2 yes 4
45.4 even 6 4050.2.a.g.1.1 1
45.7 odd 12 270.2.i.a.19.1 4
45.13 odd 12 810.2.c.c.649.1 2
45.14 odd 6 4050.2.a.x.1.1 1
45.22 odd 12 810.2.c.c.649.2 2
45.23 even 12 810.2.c.b.649.2 2
45.29 odd 6 450.2.e.b.151.1 2
45.32 even 12 810.2.c.b.649.1 2
45.34 even 6 1350.2.e.i.451.1 2
45.38 even 12 90.2.i.a.79.1 yes 4
45.43 odd 12 270.2.i.a.19.2 4
60.23 odd 4 720.2.by.a.49.2 4
60.47 odd 4 720.2.by.a.49.1 4
180.7 even 12 2160.2.by.b.289.1 4
180.43 even 12 2160.2.by.b.289.2 4
180.47 odd 12 720.2.by.a.529.2 4
180.83 odd 12 720.2.by.a.529.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.2.i.a.49.1 4 15.2 even 4
90.2.i.a.49.2 yes 4 15.8 even 4
90.2.i.a.79.1 yes 4 45.38 even 12
90.2.i.a.79.2 yes 4 45.2 even 12
270.2.i.a.19.1 4 45.7 odd 12
270.2.i.a.19.2 4 45.43 odd 12
270.2.i.a.199.1 4 5.3 odd 4
270.2.i.a.199.2 4 5.2 odd 4
450.2.e.b.151.1 2 45.29 odd 6
450.2.e.b.301.1 2 15.14 odd 2
450.2.e.g.151.1 2 9.2 odd 6
450.2.e.g.301.1 2 3.2 odd 2
720.2.by.a.49.1 4 60.47 odd 4
720.2.by.a.49.2 4 60.23 odd 4
720.2.by.a.529.1 4 180.83 odd 12
720.2.by.a.529.2 4 180.47 odd 12
810.2.c.b.649.1 2 45.32 even 12
810.2.c.b.649.2 2 45.23 even 12
810.2.c.c.649.1 2 45.13 odd 12
810.2.c.c.649.2 2 45.22 odd 12
1350.2.e.a.451.1 2 9.7 even 3 inner
1350.2.e.a.901.1 2 1.1 even 1 trivial
1350.2.e.i.451.1 2 45.34 even 6
1350.2.e.i.901.1 2 5.4 even 2
2160.2.by.b.289.1 4 180.7 even 12
2160.2.by.b.289.2 4 180.43 even 12
2160.2.by.b.1009.1 4 20.3 even 4
2160.2.by.b.1009.2 4 20.7 even 4
4050.2.a.g.1.1 1 45.4 even 6
4050.2.a.j.1.1 1 9.5 odd 6
4050.2.a.x.1.1 1 45.14 odd 6
4050.2.a.be.1.1 1 9.4 even 3