Properties

Label 1925.1.dn.c.118.2
Level $1925$
Weight $1$
Character 1925.118
Analytic conductor $0.961$
Analytic rank $0$
Dimension $32$
Projective image $D_{30}$
CM discriminant -7
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1925,1,Mod(118,1925)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1925, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([15, 10, 6]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1925.118");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1925 = 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1925.dn (of order \(20\), degree \(8\), 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: \(0.960700149319\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(4\) over \(\Q(\zeta_{20})\)
Coefficient field: \(\Q(\zeta_{120})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{32} + x^{28} - x^{20} - x^{16} - x^{12} + x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{30}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{30} - \cdots)\)

Embedding invariants

Embedding label 118.2
Root \(-0.544639 - 0.838671i\) of defining polynomial
Character \(\chi\) \(=\) 1925.118
Dual form 1925.1.dn.c.832.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.32178 - 0.209350i) q^{2} +(0.752232 + 0.244415i) q^{4} +(-0.891007 + 0.453990i) q^{7} +(0.249279 + 0.127014i) q^{8} +(-0.587785 - 0.809017i) q^{9} +O(q^{10})\) \(q+(-1.32178 - 0.209350i) q^{2} +(0.752232 + 0.244415i) q^{4} +(-0.891007 + 0.453990i) q^{7} +(0.249279 + 0.127014i) q^{8} +(-0.587785 - 0.809017i) q^{9} +(0.913545 - 0.406737i) q^{11} +(1.27276 - 0.413545i) q^{14} +(-0.942790 - 0.684977i) q^{16} +(0.607558 + 1.19240i) q^{18} +(-1.29266 + 0.346367i) q^{22} +(-0.294032 - 0.294032i) q^{23} +(-0.781205 + 0.123731i) q^{28} +(-0.251377 + 0.773659i) q^{29} +(0.904936 + 0.904936i) q^{32} +(-0.244415 - 0.752232i) q^{36} +(-0.674761 - 1.32429i) q^{37} +(1.29195 - 1.29195i) q^{43} +(0.786610 - 0.0826761i) q^{44} +(0.327091 + 0.450202i) q^{46} +(0.587785 - 0.809017i) q^{49} +(-1.16110 - 0.183900i) q^{53} -0.279773 q^{56} +(0.494232 - 0.969985i) q^{58} +(0.891007 + 0.453990i) q^{63} +(-0.321706 - 0.442790i) q^{64} +(1.40647 - 1.40647i) q^{67} +(-1.47815 - 1.07394i) q^{71} +(-0.0437661 - 0.276328i) q^{72} +(0.614648 + 1.89169i) q^{74} +(-0.629320 + 0.777146i) q^{77} +(1.60917 - 1.16913i) q^{79} +(-0.309017 + 0.951057i) q^{81} +(-1.97815 + 1.43721i) q^{86} +(0.279389 + 0.0146422i) q^{88} +(-0.149314 - 0.293045i) q^{92} +(-0.946294 + 0.946294i) q^{98} +(-0.866025 - 0.500000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q+O(q^{10}) \) Copy content Toggle raw display \( 32 q + 4 q^{11} + 12 q^{16} - 40 q^{46} - 8 q^{56} - 12 q^{71} + 8 q^{81} - 28 q^{86}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1925\mathbb{Z}\right)^\times\).

\(n\) \(276\) \(1002\) \(1751\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{3}{10}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.32178 0.209350i −1.32178 0.209350i −0.544639 0.838671i \(-0.683333\pi\)
−0.777146 + 0.629320i \(0.783333\pi\)
\(3\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(4\) 0.752232 + 0.244415i 0.752232 + 0.244415i
\(5\) 0 0
\(6\) 0 0
\(7\) −0.891007 + 0.453990i −0.891007 + 0.453990i
\(8\) 0.249279 + 0.127014i 0.249279 + 0.127014i
\(9\) −0.587785 0.809017i −0.587785 0.809017i
\(10\) 0 0
\(11\) 0.913545 0.406737i 0.913545 0.406737i
\(12\) 0 0
\(13\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(14\) 1.27276 0.413545i 1.27276 0.413545i
\(15\) 0 0
\(16\) −0.942790 0.684977i −0.942790 0.684977i
\(17\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(18\) 0.607558 + 1.19240i 0.607558 + 1.19240i
\(19\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.29266 + 0.346367i −1.29266 + 0.346367i
\(23\) −0.294032 0.294032i −0.294032 0.294032i 0.544639 0.838671i \(-0.316667\pi\)
−0.838671 + 0.544639i \(0.816667\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) −0.781205 + 0.123731i −0.781205 + 0.123731i
\(29\) −0.251377 + 0.773659i −0.251377 + 0.773659i 0.743145 + 0.669131i \(0.233333\pi\)
−0.994522 + 0.104528i \(0.966667\pi\)
\(30\) 0 0
\(31\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(32\) 0.904936 + 0.904936i 0.904936 + 0.904936i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −0.244415 0.752232i −0.244415 0.752232i
\(37\) −0.674761 1.32429i −0.674761 1.32429i −0.933580 0.358368i \(-0.883333\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(42\) 0 0
\(43\) 1.29195 1.29195i 1.29195 1.29195i 0.358368 0.933580i \(-0.383333\pi\)
0.933580 0.358368i \(-0.116667\pi\)
\(44\) 0.786610 0.0826761i 0.786610 0.0826761i
\(45\) 0 0
\(46\) 0.327091 + 0.450202i 0.327091 + 0.450202i
\(47\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(48\) 0 0
\(49\) 0.587785 0.809017i 0.587785 0.809017i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.16110 0.183900i −1.16110 0.183900i −0.453990 0.891007i \(-0.650000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −0.279773 −0.279773
\(57\) 0 0
\(58\) 0.494232 0.969985i 0.494232 0.969985i
\(59\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(60\) 0 0
\(61\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(62\) 0 0
\(63\) 0.891007 + 0.453990i 0.891007 + 0.453990i
\(64\) −0.321706 0.442790i −0.321706 0.442790i
\(65\) 0 0
\(66\) 0 0
\(67\) 1.40647 1.40647i 1.40647 1.40647i 0.629320 0.777146i \(-0.283333\pi\)
0.777146 0.629320i \(-0.216667\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1.47815 1.07394i −1.47815 1.07394i −0.978148 0.207912i \(-0.933333\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(72\) −0.0437661 0.276328i −0.0437661 0.276328i
\(73\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(74\) 0.614648 + 1.89169i 0.614648 + 1.89169i
\(75\) 0 0
\(76\) 0 0
\(77\) −0.629320 + 0.777146i −0.629320 + 0.777146i
\(78\) 0 0
\(79\) 1.60917 1.16913i 1.60917 1.16913i 0.743145 0.669131i \(-0.233333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(80\) 0 0
\(81\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(82\) 0 0
\(83\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.97815 + 1.43721i −1.97815 + 1.43721i
\(87\) 0 0
\(88\) 0.279389 + 0.0146422i 0.279389 + 0.0146422i
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −0.149314 0.293045i −0.149314 0.293045i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(98\) −0.946294 + 0.946294i −0.946294 + 0.946294i
\(99\) −0.866025 0.500000i −0.866025 0.500000i
\(100\) 0 0
\(101\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(102\) 0 0
\(103\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 1.49622 + 0.486152i 1.49622 + 0.486152i
\(107\) −0.280582 + 0.550672i −0.280582 + 0.550672i −0.987688 0.156434i \(-0.950000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(108\) 0 0
\(109\) −1.48629 −1.48629 −0.743145 0.669131i \(-0.766667\pi\)
−0.743145 + 0.669131i \(0.766667\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.15100 + 0.182301i 1.15100 + 0.182301i
\(113\) 0.369309 0.724810i 0.369309 0.724810i −0.629320 0.777146i \(-0.716667\pi\)
0.998630 + 0.0523360i \(0.0166667\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −0.378188 + 0.520530i −0.378188 + 0.520530i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.669131 0.743145i 0.669131 0.743145i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) −1.08268 0.786610i −1.08268 0.786610i
\(127\) −0.306032 1.93221i −0.306032 1.93221i −0.358368 0.933580i \(-0.616667\pi\)
0.0523360 0.998630i \(-0.483333\pi\)
\(128\) −0.248477 0.487664i −0.248477 0.487664i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −2.15349 + 1.56460i −2.15349 + 1.56460i
\(135\) 0 0
\(136\) 0 0
\(137\) 1.87869 0.297556i 1.87869 0.297556i 0.891007 0.453990i \(-0.150000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(138\) 0 0
\(139\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 1.72896 + 1.72896i 1.72896 + 1.72896i
\(143\) 0 0
\(144\) 1.16535i 1.16535i
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −0.183900 1.16110i −0.183900 1.16110i
\(149\) −1.40126 1.01807i −1.40126 1.01807i −0.994522 0.104528i \(-0.966667\pi\)
−0.406737 0.913545i \(-0.633333\pi\)
\(150\) 0 0
\(151\) 0.395472 0.128496i 0.395472 0.128496i −0.104528 0.994522i \(-0.533333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0.994522 0.895472i 0.994522 0.895472i
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(158\) −2.37173 + 1.20846i −2.37173 + 1.20846i
\(159\) 0 0
\(160\) 0 0
\(161\) 0.395472 + 0.128496i 0.395472 + 0.128496i
\(162\) 0.607558 1.19240i 0.607558 1.19240i
\(163\) −1.87869 0.297556i −1.87869 0.297556i −0.891007 0.453990i \(-0.850000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(168\) 0 0
\(169\) −0.951057 0.309017i −0.951057 0.309017i
\(170\) 0 0
\(171\) 0 0
\(172\) 1.28762 0.656073i 1.28762 0.656073i
\(173\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.13989 0.242290i −1.13989 0.242290i
\(177\) 0 0
\(178\) 0 0
\(179\) 0.587785 0.190983i 0.587785 0.190983i 1.00000i \(-0.5\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(180\) 0 0
\(181\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −0.0359498 0.110642i −0.0359498 0.110642i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0.500000 1.53884i 0.500000 1.53884i −0.309017 0.951057i \(-0.600000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(192\) 0 0
\(193\) −0.206483 + 0.0327037i −0.206483 + 0.0327037i −0.258819 0.965926i \(-0.583333\pi\)
0.0523360 + 0.998630i \(0.483333\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0.639886 0.464905i 0.639886 0.464905i
\(197\) 1.38331 + 1.38331i 1.38331 + 1.38331i 0.838671 + 0.544639i \(0.183333\pi\)
0.544639 + 0.838671i \(0.316667\pi\)
\(198\) 1.04002 + 0.842195i 1.04002 + 0.842195i
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −0.127255 0.803458i −0.127255 0.803458i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −0.0650491 + 0.410704i −0.0650491 + 0.410704i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 1.11803 + 1.53884i 1.11803 + 1.53884i 0.809017 + 0.587785i \(0.200000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(212\) −0.828466 0.422125i −0.828466 0.422125i
\(213\) 0 0
\(214\) 0.486152 0.669131i 0.486152 0.669131i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 1.96456 + 0.311155i 1.96456 + 0.311155i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(224\) −1.21714 0.395472i −1.21714 0.395472i
\(225\) 0 0
\(226\) −0.639886 + 0.880728i −0.639886 + 0.880728i
\(227\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(228\) 0 0
\(229\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −0.160929 + 0.160929i −0.160929 + 0.160929i
\(233\) −0.156434 + 0.987688i −0.156434 + 0.987688i 0.777146 + 0.629320i \(0.216667\pi\)
−0.933580 + 0.358368i \(0.883333\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0.363271 + 1.11803i 0.363271 + 1.11803i 0.951057 + 0.309017i \(0.100000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) −1.04002 + 0.842195i −1.04002 + 0.842195i
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(252\) 0.559281 + 0.559281i 0.559281 + 0.559281i
\(253\) −0.388205 0.149018i −0.388205 0.149018i
\(254\) 2.61803i 2.61803i
\(255\) 0 0
\(256\) 0.395472 + 1.21714i 0.395472 + 1.21714i
\(257\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(258\) 0 0
\(259\) 1.20243 + 0.873619i 1.20243 + 0.873619i
\(260\) 0 0
\(261\) 0.773659 0.251377i 0.773659 0.251377i
\(262\) 0 0
\(263\) −0.147826 + 0.147826i −0.147826 + 0.147826i −0.777146 0.629320i \(-0.783333\pi\)
0.629320 + 0.777146i \(0.283333\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 1.40175 0.714227i 1.40175 0.714227i
\(269\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(270\) 0 0
\(271\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −2.54552 −2.54552
\(275\) 0 0
\(276\) 0 0
\(277\) 0.610425 + 0.0966818i 0.610425 + 0.0966818i 0.453990 0.891007i \(-0.350000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −1.16913 + 1.60917i −1.16913 + 1.60917i −0.500000 + 0.866025i \(0.666667\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(282\) 0 0
\(283\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(284\) −0.849423 1.16913i −0.849423 1.16913i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0.200201 1.26402i 0.200201 1.26402i
\(289\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0.415823i 0.415823i
\(297\) 0 0
\(298\) 1.63903 + 1.63903i 1.63903 + 1.63903i
\(299\) 0 0
\(300\) 0 0
\(301\) −0.564602 + 1.73767i −0.564602 + 1.73767i
\(302\) −0.549629 + 0.0870527i −0.549629 + 0.0870527i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(308\) −0.663341 + 0.430778i −0.663341 + 0.430778i
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(312\) 0 0
\(313\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 1.49622 0.486152i 1.49622 0.486152i
\(317\) −0.311155 + 1.96456i −0.311155 + 1.96456i −0.0523360 + 0.998630i \(0.516667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(318\) 0 0
\(319\) 0.0850311 + 0.809017i 0.0850311 + 0.809017i
\(320\) 0 0
\(321\) 0 0
\(322\) −0.495828 0.252637i −0.495828 0.252637i
\(323\) 0 0
\(324\) −0.464905 + 0.639886i −0.464905 + 0.639886i
\(325\) 0 0
\(326\) 2.42094 + 0.786610i 2.42094 + 0.786610i
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.95630 1.95630 0.978148 0.207912i \(-0.0666667\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(332\) 0 0
\(333\) −0.674761 + 1.32429i −0.674761 + 1.32429i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −0.550672 + 0.280582i −0.550672 + 0.280582i −0.707107 0.707107i \(-0.750000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(338\) 1.19240 + 0.607558i 1.19240 + 0.607558i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −0.156434 + 0.987688i −0.156434 + 0.987688i
\(344\) 0.486152 0.157960i 0.486152 0.157960i
\(345\) 0 0
\(346\) 0 0
\(347\) 0.0327037 + 0.206483i 0.0327037 + 0.206483i 0.998630 0.0523360i \(-0.0166667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(348\) 0 0
\(349\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.19477 + 0.458630i 1.19477 + 0.458630i
\(353\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −0.816908 + 0.129386i −0.816908 + 0.129386i
\(359\) −0.459289 + 1.41355i −0.459289 + 1.41355i 0.406737 + 0.913545i \(0.366667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(360\) 0 0
\(361\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(368\) 0.0758051 + 0.478615i 0.0758051 + 0.478615i
\(369\) 0 0
\(370\) 0 0
\(371\) 1.11803 0.363271i 1.11803 0.363271i
\(372\) 0 0
\(373\) 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0.786610 1.08268i 0.786610 1.08268i −0.207912 0.978148i \(-0.566667\pi\)
0.994522 0.104528i \(-0.0333333\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −0.983049 + 1.92934i −0.983049 + 1.92934i
\(383\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0.279773 0.279773
\(387\) −1.80460 0.285820i −1.80460 0.285820i
\(388\) 0 0
\(389\) 1.86055 + 0.604528i 1.86055 + 0.604528i 0.994522 + 0.104528i \(0.0333333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0.249279 0.127014i 0.249279 0.127014i
\(393\) 0 0
\(394\) −1.53884 2.11803i −1.53884 2.11803i
\(395\) 0 0
\(396\) −0.529244 0.587785i −0.529244 0.587785i
\(397\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.58268 + 1.14988i 1.58268 + 1.14988i 0.913545 + 0.406737i \(0.133333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 1.08864i 1.08864i
\(407\) −1.15506 0.935352i −1.15506 0.935352i
\(408\) 0 0
\(409\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0.171962 0.529244i 0.171962 0.529244i
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −0.413545 1.27276i −0.413545 1.27276i −0.913545 0.406737i \(-0.866667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(422\) −1.15564 2.26808i −1.15564 2.26808i
\(423\) 0 0
\(424\) −0.266080 0.193318i −0.266080 0.193318i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −0.345655 + 0.345655i −0.345655 + 0.345655i
\(429\) 0 0
\(430\) 0 0
\(431\) −1.11803 1.53884i −1.11803 1.53884i −0.809017 0.587785i \(-0.800000\pi\)
−0.309017 0.951057i \(-0.600000\pi\)
\(432\) 0 0
\(433\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −1.11803 0.363271i −1.11803 0.363271i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −1.00000 −1.00000
\(442\) 0 0
\(443\) −0.533698 + 1.04744i −0.533698 + 1.04744i 0.453990 + 0.891007i \(0.350000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0.487664 + 0.248477i 0.487664 + 0.248477i
\(449\) 0.786610 + 1.08268i 0.786610 + 1.08268i 0.994522 + 0.104528i \(0.0333333\pi\)
−0.207912 + 0.978148i \(0.566667\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0.454960 0.454960i 0.454960 0.454960i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0.306032 + 1.93221i 0.306032 + 1.93221i 0.358368 + 0.933580i \(0.383333\pi\)
−0.0523360 + 0.998630i \(0.516667\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 0.831254 + 0.831254i 0.831254 + 0.831254i 0.987688 0.156434i \(-0.0500000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(464\) 0.766934 0.557210i 0.766934 0.557210i
\(465\) 0 0
\(466\) 0.413545 1.27276i 0.413545 1.27276i
\(467\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(468\) 0 0
\(469\) −0.614648 + 1.89169i −0.614648 + 1.89169i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0.654771 1.70574i 0.654771 1.70574i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0.533698 + 1.04744i 0.533698 + 1.04744i
\(478\) −0.246106 1.55385i −0.246106 1.55385i
\(479\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0.684977 0.395472i 0.684977 0.395472i
\(485\) 0 0
\(486\) 0 0
\(487\) −0.724810 0.369309i −0.724810 0.369309i 0.0523360 0.998630i \(-0.483333\pi\)
−0.777146 + 0.629320i \(0.783333\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −1.64728 0.535233i −1.64728 0.535233i −0.669131 0.743145i \(-0.733333\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.80460 + 0.285820i 1.80460 + 0.285820i
\(498\) 0 0
\(499\) 1.53884 + 0.500000i 1.53884 + 0.500000i 0.951057 0.309017i \(-0.100000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(504\) 0.164446 + 0.226341i 0.164446 + 0.226341i
\(505\) 0 0
\(506\) 0.481926 + 0.278240i 0.481926 + 0.278240i
\(507\) 0 0
\(508\) 0.242054 1.52827i 0.242054 1.52827i
\(509\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −0.182301 1.15100i −0.182301 1.15100i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) −1.40647 1.40647i −1.40647 1.40647i
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(522\) −1.07524 + 0.170301i −1.07524 + 0.170301i
\(523\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0.226341 0.164446i 0.226341 0.164446i
\(527\) 0 0
\(528\) 0 0
\(529\) 0.827091i 0.827091i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0.529244 0.171962i 0.529244 0.171962i
\(537\) 0 0
\(538\) 0 0
\(539\) 0.207912 0.978148i 0.207912 0.978148i
\(540\) 0 0
\(541\) −0.244415 0.336408i −0.244415 0.336408i 0.669131 0.743145i \(-0.266667\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0.453990 0.891007i 0.453990 0.891007i −0.544639 0.838671i \(-0.683333\pi\)
0.998630 0.0523360i \(-0.0166667\pi\)
\(548\) 1.48594 + 0.235350i 1.48594 + 0.235350i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −0.903007 + 1.77225i −0.903007 + 1.77225i
\(554\) −0.786610 0.255585i −0.786610 0.255585i
\(555\) 0 0
\(556\) 0 0
\(557\) −0.186271 + 0.0949099i −0.186271 + 0.0949099i −0.544639 0.838671i \(-0.683333\pi\)
0.358368 + 0.933580i \(0.383333\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 1.88222 1.88222i 1.88222 1.88222i
\(563\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −0.156434 0.987688i −0.156434 0.987688i
\(568\) −0.232066 0.455456i −0.232066 0.455456i
\(569\) −0.535233 1.64728i −0.535233 1.64728i −0.743145 0.669131i \(-0.766667\pi\)
0.207912 0.978148i \(-0.433333\pi\)
\(570\) 0 0
\(571\) 0.415823i 0.415823i 0.978148 + 0.207912i \(0.0666667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −0.169131 + 0.520530i −0.169131 + 0.520530i
\(577\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(578\) 1.32178 0.209350i 1.32178 0.209350i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −1.13551 + 0.304260i −1.13551 + 0.304260i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −0.270952 + 1.71073i −0.270952 + 1.71073i
\(593\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −0.805239 1.10832i −0.805239 1.10832i
\(597\) 0 0
\(598\) 0 0
\(599\) 1.14988 1.58268i 1.14988 1.58268i 0.406737 0.913545i \(-0.366667\pi\)
0.743145 0.669131i \(-0.233333\pi\)
\(600\) 0 0
\(601\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(602\) 1.11006 2.17862i 1.11006 2.17862i
\(603\) −1.96456 0.311155i −1.96456 0.311155i
\(604\) 0.328893 0.328893
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.62795 0.829482i −1.62795 0.829482i −0.998630 0.0523360i \(-0.983333\pi\)
−0.629320 0.777146i \(-0.716667\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −0.255585 + 0.113794i −0.255585 + 0.113794i
\(617\) −1.40647 + 1.40647i −1.40647 + 1.40647i −0.629320 + 0.777146i \(0.716667\pi\)
−0.777146 + 0.629320i \(0.783333\pi\)
\(618\) 0 0
\(619\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −0.413545 + 1.27276i −0.413545 + 1.27276i 0.500000 + 0.866025i \(0.333333\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(632\) 0.549629 0.0870527i 0.549629 0.0870527i
\(633\) 0 0
\(634\) 0.822560 2.53158i 0.822560 2.53158i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0.0569750 1.08715i 0.0569750 1.08715i
\(639\) 1.82709i 1.82709i
\(640\) 0 0
\(641\) −0.564602 1.73767i −0.564602 1.73767i −0.669131 0.743145i \(-0.733333\pi\)
0.104528 0.994522i \(-0.466667\pi\)
\(642\) 0 0
\(643\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(644\) 0.266080 + 0.193318i 0.266080 + 0.193318i
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(648\) −0.197829 + 0.197829i −0.197829 + 0.197829i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −1.34049 0.683012i −1.34049 0.683012i
\(653\) 1.69480 0.863541i 1.69480 0.863541i 0.707107 0.707107i \(-0.250000\pi\)
0.987688 0.156434i \(-0.0500000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) −2.58580 0.409551i −2.58580 0.409551i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 1.16913 1.60917i 1.16913 1.60917i
\(667\) 0.301393 0.153567i 0.301393 0.153567i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −0.253116 + 1.59811i −0.253116 + 1.59811i 0.453990 + 0.891007i \(0.350000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(674\) 0.786610 0.255585i 0.786610 0.255585i
\(675\) 0 0
\(676\) −0.639886 0.464905i −0.639886 0.464905i
\(677\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1.05097 1.05097i −1.05097 1.05097i −0.998630 0.0523360i \(-0.983333\pi\)
−0.0523360 0.998630i \(-0.516667\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0.413545 1.27276i 0.413545 1.27276i
\(687\) 0 0
\(688\) −2.10299 + 0.333081i −2.10299 + 0.333081i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(692\) 0 0
\(693\) 0.998630 + 0.0523360i 0.998630 + 0.0523360i
\(694\) 0.279773i 0.279773i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −1.80902 + 0.587785i −1.80902 + 0.587785i −0.809017 + 0.587785i \(0.800000\pi\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −0.473991 0.273659i −0.473991 0.273659i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −0.951057 + 1.30902i −0.951057 + 1.30902i 1.00000i \(0.5\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(710\) 0 0
\(711\) −1.89169 0.614648i −1.89169 0.614648i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0.488830 0.488830
\(717\) 0 0
\(718\) 0.903007 1.77225i 0.903007 1.77225i
\(719\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.19240 0.607558i 1.19240 0.607558i
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(728\) 0 0
\(729\) 0.951057 0.309017i 0.951057 0.309017i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0.532159i 0.532159i
\(737\) 0.712810 1.85693i 0.712810 1.85693i
\(738\) 0 0
\(739\) 0.658114 0.478148i 0.658114 0.478148i −0.207912 0.978148i \(-0.566667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −1.55385 + 0.246106i −1.55385 + 0.246106i
\(743\) 1.59811 0.253116i 1.59811 0.253116i 0.707107 0.707107i \(-0.250000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −1.08268 + 0.786610i −1.08268 + 0.786610i
\(747\) 0 0
\(748\) 0 0
\(749\) 0.618034i 0.618034i
\(750\) 0 0
\(751\) −0.190983 0.587785i −0.190983 0.587785i 0.809017 0.587785i \(-0.200000\pi\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0.127255 0.803458i 0.127255 0.803458i −0.838671 0.544639i \(-0.816667\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(758\) −1.26639 + 1.26639i −1.26639 + 1.26639i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(762\) 0 0
\(763\) 1.32429 0.674761i 1.32429 0.674761i
\(764\) 0.752232 1.03536i 0.752232 1.03536i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −0.163316 0.0258668i −0.163316 0.0258668i
\(773\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(774\) 2.32545 + 0.755585i 2.32545 + 0.755585i
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −2.33269 1.18856i −2.33269 1.18856i
\(779\) 0 0
\(780\) 0 0
\(781\) −1.78716 0.379874i −1.78716 0.379874i
\(782\) 0 0
\(783\) 0 0
\(784\) −1.10832 + 0.360114i −1.10832 + 0.360114i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(788\) 0.702468 + 1.37867i 0.702468 + 1.37867i
\(789\) 0 0
\(790\) 0 0
\(791\) 0.813473i 0.813473i
\(792\) −0.152375 0.234637i −0.152375 0.234637i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) −1.85123 1.85123i −1.85123 1.85123i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1.20243 0.873619i −1.20243 0.873619i −0.207912 0.978148i \(-0.566667\pi\)
−0.994522 + 0.104528i \(0.966667\pi\)
\(810\) 0 0
\(811\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(812\) 0.100652 0.635490i 0.100652 0.635490i
\(813\) 0 0
\(814\) 1.33093 + 1.47815i 1.33093 + 1.47815i
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(822\) 0 0
\(823\) 0.803458 + 0.127255i 0.803458 + 0.127255i 0.544639 0.838671i \(-0.316667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0.987688 + 0.156434i 0.987688 + 0.156434i 0.629320 0.777146i \(-0.283333\pi\)
0.358368 + 0.933580i \(0.383333\pi\)
\(828\) −0.149314 + 0.293045i −0.149314 + 0.293045i
\(829\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(840\) 0 0
\(841\) 0.273659 + 0.198825i 0.273659 + 0.198825i
\(842\) 0.280165 + 1.76889i 0.280165 + 1.76889i
\(843\) 0 0
\(844\) 0.464905 + 1.43083i 0.464905 + 1.43083i
\(845\) 0 0
\(846\) 0 0
\(847\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(848\) 0.968703 + 0.968703i 0.968703 + 0.968703i
\(849\) 0 0
\(850\) 0 0
\(851\) −0.190983 + 0.587785i −0.190983 + 0.587785i
\(852\) 0 0
\(853\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −0.139886 + 0.101633i −0.139886 + 0.101633i
\(857\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 1.15564 + 2.26808i 1.15564 + 2.26808i
\(863\) 0.270952 + 1.71073i 0.270952 + 1.71073i 0.629320 + 0.777146i \(0.283333\pi\)
−0.358368 + 0.933580i \(0.616667\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0.994522 1.72256i 0.994522 1.72256i
\(870\) 0 0
\(871\) 0 0
\(872\) −0.370501 0.188780i −0.370501 0.188780i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0.734572 1.44168i 0.734572 1.44168i −0.156434 0.987688i \(-0.550000\pi\)
0.891007 0.453990i \(-0.150000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 1.32178 + 0.209350i 1.32178 + 0.209350i
\(883\) 0.786335 1.54327i 0.786335 1.54327i −0.0523360 0.998630i \(-0.516667\pi\)
0.838671 0.544639i \(-0.183333\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0.924716 1.27276i 0.924716 1.27276i
\(887\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(888\) 0 0
\(889\) 1.14988 + 1.58268i 1.14988 + 1.58268i
\(890\) 0 0
\(891\) 0.104528 + 0.994522i 0.104528 + 0.994522i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0.442790 + 0.321706i 0.442790 + 0.321706i
\(897\) 0 0
\(898\) −0.813071 1.59574i −0.813071 1.59574i
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0.184122 0.133773i 0.184122 0.133773i
\(905\) 0 0
\(906\) 0 0
\(907\) −1.16110 + 0.183900i −1.16110 + 0.183900i −0.707107 0.707107i \(-0.750000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0.809017 0.587785i 0.809017 0.587785i −0.104528 0.994522i \(-0.533333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 2.61803i 2.61803i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0.658114 + 0.478148i 0.658114 + 0.478148i 0.866025 0.500000i \(-0.166667\pi\)
−0.207912 + 0.978148i \(0.566667\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) −0.924716 1.27276i −0.924716 1.27276i
\(927\) 0 0
\(928\) −0.927592 + 0.472632i −0.927592 + 0.472632i
\(929\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −0.359081 + 0.704735i −0.359081 + 0.704735i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(938\) 1.20846 2.37173i 1.20846 2.37173i
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) −1.22256 + 2.11754i −1.22256 + 2.11754i
\(947\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0.888139 + 1.74307i 0.888139 + 1.74307i 0.629320 + 0.777146i \(0.283333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(954\) −0.486152 1.49622i −0.486152 1.49622i
\(955\) 0 0
\(956\) 0.929809i 0.929809i
\(957\) 0 0
\(958\) 0 0
\(959\) −1.53884 + 1.11803i −1.53884 + 1.11803i
\(960\) 0 0
\(961\) 0.309017 0.951057i 0.309017 0.951057i
\(962\) 0 0
\(963\) 0.610425 0.0966818i 0.610425 0.0966818i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −0.437016 0.437016i −0.437016 0.437016i 0.453990 0.891007i \(-0.350000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(968\) 0.261190 0.100262i 0.261190 0.100262i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0.880728 + 0.639886i 0.880728 + 0.639886i
\(975\) 0 0
\(976\) 0 0
\(977\) 0.0650491 0.410704i 0.0650491 0.410704i −0.933580 0.358368i \(-0.883333\pi\)
0.998630 0.0523360i \(-0.0166667\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0.873619 + 1.20243i 0.873619 + 1.20243i
\(982\) 2.06530 + 1.05232i 2.06530 + 1.05232i
\(983\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −0.759747 −0.759747
\(990\) 0 0
\(991\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −2.32545 0.755585i −2.32545 0.755585i
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(998\) −1.92934 0.983049i −1.92934 0.983049i
\(999\) 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 1925.1.dn.c.118.2 32
5.2 odd 4 inner 1925.1.dn.c.657.3 yes 32
5.3 odd 4 inner 1925.1.dn.c.657.2 yes 32
5.4 even 2 inner 1925.1.dn.c.118.3 yes 32
7.6 odd 2 CM 1925.1.dn.c.118.2 32
11.7 odd 10 inner 1925.1.dn.c.293.3 yes 32
35.13 even 4 inner 1925.1.dn.c.657.2 yes 32
35.27 even 4 inner 1925.1.dn.c.657.3 yes 32
35.34 odd 2 inner 1925.1.dn.c.118.3 yes 32
55.7 even 20 inner 1925.1.dn.c.832.2 yes 32
55.18 even 20 inner 1925.1.dn.c.832.3 yes 32
55.29 odd 10 inner 1925.1.dn.c.293.2 yes 32
77.62 even 10 inner 1925.1.dn.c.293.3 yes 32
385.62 odd 20 inner 1925.1.dn.c.832.2 yes 32
385.139 even 10 inner 1925.1.dn.c.293.2 yes 32
385.293 odd 20 inner 1925.1.dn.c.832.3 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1925.1.dn.c.118.2 32 1.1 even 1 trivial
1925.1.dn.c.118.2 32 7.6 odd 2 CM
1925.1.dn.c.118.3 yes 32 5.4 even 2 inner
1925.1.dn.c.118.3 yes 32 35.34 odd 2 inner
1925.1.dn.c.293.2 yes 32 55.29 odd 10 inner
1925.1.dn.c.293.2 yes 32 385.139 even 10 inner
1925.1.dn.c.293.3 yes 32 11.7 odd 10 inner
1925.1.dn.c.293.3 yes 32 77.62 even 10 inner
1925.1.dn.c.657.2 yes 32 5.3 odd 4 inner
1925.1.dn.c.657.2 yes 32 35.13 even 4 inner
1925.1.dn.c.657.3 yes 32 5.2 odd 4 inner
1925.1.dn.c.657.3 yes 32 35.27 even 4 inner
1925.1.dn.c.832.2 yes 32 55.7 even 20 inner
1925.1.dn.c.832.2 yes 32 385.62 odd 20 inner
1925.1.dn.c.832.3 yes 32 55.18 even 20 inner
1925.1.dn.c.832.3 yes 32 385.293 odd 20 inner