# Properties

 Label 6080.2.a.bf.1.1 Level $6080$ Weight $2$ Character 6080.1 Self dual yes Analytic conductor $48.549$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6080,2,Mod(1,6080)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6080, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6080.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6080 = 2^{6} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6080.a (trivial)

## 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: yes Analytic conductor: $$48.5490444289$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 760) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 6080.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.41421 q^{3} +1.00000 q^{5} +2.82843 q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q-1.41421 q^{3} +1.00000 q^{5} +2.82843 q^{7} -1.00000 q^{9} -4.82843 q^{11} -0.585786 q^{13} -1.41421 q^{15} -0.828427 q^{17} +1.00000 q^{19} -4.00000 q^{21} +4.00000 q^{23} +1.00000 q^{25} +5.65685 q^{27} -4.82843 q^{29} +6.82843 q^{33} +2.82843 q^{35} -1.75736 q^{37} +0.828427 q^{39} +4.82843 q^{41} +2.82843 q^{43} -1.00000 q^{45} +8.48528 q^{47} +1.00000 q^{49} +1.17157 q^{51} +1.07107 q^{53} -4.82843 q^{55} -1.41421 q^{57} +2.82843 q^{59} -9.65685 q^{61} -2.82843 q^{63} -0.585786 q^{65} +6.58579 q^{67} -5.65685 q^{69} -7.31371 q^{71} -16.1421 q^{73} -1.41421 q^{75} -13.6569 q^{77} -14.8284 q^{79} -5.00000 q^{81} -8.00000 q^{83} -0.828427 q^{85} +6.82843 q^{87} -8.82843 q^{89} -1.65685 q^{91} +1.00000 q^{95} +6.24264 q^{97} +4.82843 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^5 - 2 * q^9 $$2 q + 2 q^{5} - 2 q^{9} - 4 q^{11} - 4 q^{13} + 4 q^{17} + 2 q^{19} - 8 q^{21} + 8 q^{23} + 2 q^{25} - 4 q^{29} + 8 q^{33} - 12 q^{37} - 4 q^{39} + 4 q^{41} - 2 q^{45} + 2 q^{49} + 8 q^{51} - 12 q^{53} - 4 q^{55} - 8 q^{61} - 4 q^{65} + 16 q^{67} + 8 q^{71} - 4 q^{73} - 16 q^{77} - 24 q^{79} - 10 q^{81} - 16 q^{83} + 4 q^{85} + 8 q^{87} - 12 q^{89} + 8 q^{91} + 2 q^{95} + 4 q^{97} + 4 q^{99}+O(q^{100})$$ 2 * q + 2 * q^5 - 2 * q^9 - 4 * q^11 - 4 * q^13 + 4 * q^17 + 2 * q^19 - 8 * q^21 + 8 * q^23 + 2 * q^25 - 4 * q^29 + 8 * q^33 - 12 * q^37 - 4 * q^39 + 4 * q^41 - 2 * q^45 + 2 * q^49 + 8 * q^51 - 12 * q^53 - 4 * q^55 - 8 * q^61 - 4 * q^65 + 16 * q^67 + 8 * q^71 - 4 * q^73 - 16 * q^77 - 24 * q^79 - 10 * q^81 - 16 * q^83 + 4 * q^85 + 8 * q^87 - 12 * q^89 + 8 * q^91 + 2 * q^95 + 4 * q^97 + 4 * q^99

## 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 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −1.41421 −0.816497 −0.408248 0.912871i $$-0.633860\pi$$
−0.408248 + 0.912871i $$0.633860\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 2.82843 1.06904 0.534522 0.845154i $$-0.320491\pi$$
0.534522 + 0.845154i $$0.320491\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −4.82843 −1.45583 −0.727913 0.685670i $$-0.759509\pi$$
−0.727913 + 0.685670i $$0.759509\pi$$
$$12$$ 0 0
$$13$$ −0.585786 −0.162468 −0.0812340 0.996695i $$-0.525886\pi$$
−0.0812340 + 0.996695i $$0.525886\pi$$
$$14$$ 0 0
$$15$$ −1.41421 −0.365148
$$16$$ 0 0
$$17$$ −0.828427 −0.200923 −0.100462 0.994941i $$-0.532032\pi$$
−0.100462 + 0.994941i $$0.532032\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −4.00000 −0.872872
$$22$$ 0 0
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 5.65685 1.08866
$$28$$ 0 0
$$29$$ −4.82843 −0.896616 −0.448308 0.893879i $$-0.647973\pi$$
−0.448308 + 0.893879i $$0.647973\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 0 0
$$33$$ 6.82843 1.18868
$$34$$ 0 0
$$35$$ 2.82843 0.478091
$$36$$ 0 0
$$37$$ −1.75736 −0.288908 −0.144454 0.989512i $$-0.546143\pi$$
−0.144454 + 0.989512i $$0.546143\pi$$
$$38$$ 0 0
$$39$$ 0.828427 0.132655
$$40$$ 0 0
$$41$$ 4.82843 0.754074 0.377037 0.926198i $$-0.376943\pi$$
0.377037 + 0.926198i $$0.376943\pi$$
$$42$$ 0 0
$$43$$ 2.82843 0.431331 0.215666 0.976467i $$-0.430808\pi$$
0.215666 + 0.976467i $$0.430808\pi$$
$$44$$ 0 0
$$45$$ −1.00000 −0.149071
$$46$$ 0 0
$$47$$ 8.48528 1.23771 0.618853 0.785507i $$-0.287598\pi$$
0.618853 + 0.785507i $$0.287598\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 1.17157 0.164053
$$52$$ 0 0
$$53$$ 1.07107 0.147122 0.0735612 0.997291i $$-0.476564\pi$$
0.0735612 + 0.997291i $$0.476564\pi$$
$$54$$ 0 0
$$55$$ −4.82843 −0.651065
$$56$$ 0 0
$$57$$ −1.41421 −0.187317
$$58$$ 0 0
$$59$$ 2.82843 0.368230 0.184115 0.982905i $$-0.441058\pi$$
0.184115 + 0.982905i $$0.441058\pi$$
$$60$$ 0 0
$$61$$ −9.65685 −1.23643 −0.618217 0.786008i $$-0.712145\pi$$
−0.618217 + 0.786008i $$0.712145\pi$$
$$62$$ 0 0
$$63$$ −2.82843 −0.356348
$$64$$ 0 0
$$65$$ −0.585786 −0.0726579
$$66$$ 0 0
$$67$$ 6.58579 0.804582 0.402291 0.915512i $$-0.368214\pi$$
0.402291 + 0.915512i $$0.368214\pi$$
$$68$$ 0 0
$$69$$ −5.65685 −0.681005
$$70$$ 0 0
$$71$$ −7.31371 −0.867978 −0.433989 0.900918i $$-0.642894\pi$$
−0.433989 + 0.900918i $$0.642894\pi$$
$$72$$ 0 0
$$73$$ −16.1421 −1.88929 −0.944647 0.328088i $$-0.893596\pi$$
−0.944647 + 0.328088i $$0.893596\pi$$
$$74$$ 0 0
$$75$$ −1.41421 −0.163299
$$76$$ 0 0
$$77$$ −13.6569 −1.55634
$$78$$ 0 0
$$79$$ −14.8284 −1.66833 −0.834164 0.551516i $$-0.814049\pi$$
−0.834164 + 0.551516i $$0.814049\pi$$
$$80$$ 0 0
$$81$$ −5.00000 −0.555556
$$82$$ 0 0
$$83$$ −8.00000 −0.878114 −0.439057 0.898459i $$-0.644687\pi$$
−0.439057 + 0.898459i $$0.644687\pi$$
$$84$$ 0 0
$$85$$ −0.828427 −0.0898555
$$86$$ 0 0
$$87$$ 6.82843 0.732084
$$88$$ 0 0
$$89$$ −8.82843 −0.935811 −0.467906 0.883778i $$-0.654991\pi$$
−0.467906 + 0.883778i $$0.654991\pi$$
$$90$$ 0 0
$$91$$ −1.65685 −0.173686
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 1.00000 0.102598
$$96$$ 0 0
$$97$$ 6.24264 0.633844 0.316922 0.948452i $$-0.397351\pi$$
0.316922 + 0.948452i $$0.397351\pi$$
$$98$$ 0 0
$$99$$ 4.82843 0.485275
$$100$$ 0 0
$$101$$ −6.34315 −0.631167 −0.315583 0.948898i $$-0.602200\pi$$
−0.315583 + 0.948898i $$0.602200\pi$$
$$102$$ 0 0
$$103$$ −0.242641 −0.0239081 −0.0119540 0.999929i $$-0.503805\pi$$
−0.0119540 + 0.999929i $$0.503805\pi$$
$$104$$ 0 0
$$105$$ −4.00000 −0.390360
$$106$$ 0 0
$$107$$ 1.41421 0.136717 0.0683586 0.997661i $$-0.478224\pi$$
0.0683586 + 0.997661i $$0.478224\pi$$
$$108$$ 0 0
$$109$$ 7.65685 0.733394 0.366697 0.930341i $$-0.380489\pi$$
0.366697 + 0.930341i $$0.380489\pi$$
$$110$$ 0 0
$$111$$ 2.48528 0.235892
$$112$$ 0 0
$$113$$ 10.7279 1.00920 0.504599 0.863354i $$-0.331640\pi$$
0.504599 + 0.863354i $$0.331640\pi$$
$$114$$ 0 0
$$115$$ 4.00000 0.373002
$$116$$ 0 0
$$117$$ 0.585786 0.0541560
$$118$$ 0 0
$$119$$ −2.34315 −0.214796
$$120$$ 0 0
$$121$$ 12.3137 1.11943
$$122$$ 0 0
$$123$$ −6.82843 −0.615699
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −10.5858 −0.939337 −0.469668 0.882843i $$-0.655626\pi$$
−0.469668 + 0.882843i $$0.655626\pi$$
$$128$$ 0 0
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ 15.3137 1.33796 0.668982 0.743278i $$-0.266730\pi$$
0.668982 + 0.743278i $$0.266730\pi$$
$$132$$ 0 0
$$133$$ 2.82843 0.245256
$$134$$ 0 0
$$135$$ 5.65685 0.486864
$$136$$ 0 0
$$137$$ −12.8284 −1.09601 −0.548003 0.836476i $$-0.684612\pi$$
−0.548003 + 0.836476i $$0.684612\pi$$
$$138$$ 0 0
$$139$$ 8.82843 0.748817 0.374409 0.927264i $$-0.377846\pi$$
0.374409 + 0.927264i $$0.377846\pi$$
$$140$$ 0 0
$$141$$ −12.0000 −1.01058
$$142$$ 0 0
$$143$$ 2.82843 0.236525
$$144$$ 0 0
$$145$$ −4.82843 −0.400979
$$146$$ 0 0
$$147$$ −1.41421 −0.116642
$$148$$ 0 0
$$149$$ 9.65685 0.791120 0.395560 0.918440i $$-0.370550\pi$$
0.395560 + 0.918440i $$0.370550\pi$$
$$150$$ 0 0
$$151$$ −22.1421 −1.80190 −0.900951 0.433921i $$-0.857130\pi$$
−0.900951 + 0.433921i $$0.857130\pi$$
$$152$$ 0 0
$$153$$ 0.828427 0.0669744
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −6.48528 −0.517582 −0.258791 0.965933i $$-0.583324\pi$$
−0.258791 + 0.965933i $$0.583324\pi$$
$$158$$ 0 0
$$159$$ −1.51472 −0.120125
$$160$$ 0 0
$$161$$ 11.3137 0.891645
$$162$$ 0 0
$$163$$ −9.17157 −0.718373 −0.359187 0.933266i $$-0.616946\pi$$
−0.359187 + 0.933266i $$0.616946\pi$$
$$164$$ 0 0
$$165$$ 6.82843 0.531592
$$166$$ 0 0
$$167$$ 21.8995 1.69463 0.847317 0.531088i $$-0.178217\pi$$
0.847317 + 0.531088i $$0.178217\pi$$
$$168$$ 0 0
$$169$$ −12.6569 −0.973604
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ 0 0
$$173$$ −14.7279 −1.11974 −0.559872 0.828579i $$-0.689150\pi$$
−0.559872 + 0.828579i $$0.689150\pi$$
$$174$$ 0 0
$$175$$ 2.82843 0.213809
$$176$$ 0 0
$$177$$ −4.00000 −0.300658
$$178$$ 0 0
$$179$$ 8.48528 0.634220 0.317110 0.948389i $$-0.397288\pi$$
0.317110 + 0.948389i $$0.397288\pi$$
$$180$$ 0 0
$$181$$ −17.3137 −1.28692 −0.643459 0.765481i $$-0.722501\pi$$
−0.643459 + 0.765481i $$0.722501\pi$$
$$182$$ 0 0
$$183$$ 13.6569 1.00954
$$184$$ 0 0
$$185$$ −1.75736 −0.129204
$$186$$ 0 0
$$187$$ 4.00000 0.292509
$$188$$ 0 0
$$189$$ 16.0000 1.16383
$$190$$ 0 0
$$191$$ −19.3137 −1.39749 −0.698745 0.715370i $$-0.746258\pi$$
−0.698745 + 0.715370i $$0.746258\pi$$
$$192$$ 0 0
$$193$$ −21.0711 −1.51673 −0.758364 0.651831i $$-0.774001\pi$$
−0.758364 + 0.651831i $$0.774001\pi$$
$$194$$ 0 0
$$195$$ 0.828427 0.0593249
$$196$$ 0 0
$$197$$ −2.68629 −0.191390 −0.0956952 0.995411i $$-0.530507\pi$$
−0.0956952 + 0.995411i $$0.530507\pi$$
$$198$$ 0 0
$$199$$ 5.65685 0.401004 0.200502 0.979693i $$-0.435743\pi$$
0.200502 + 0.979693i $$0.435743\pi$$
$$200$$ 0 0
$$201$$ −9.31371 −0.656938
$$202$$ 0 0
$$203$$ −13.6569 −0.958523
$$204$$ 0 0
$$205$$ 4.82843 0.337232
$$206$$ 0 0
$$207$$ −4.00000 −0.278019
$$208$$ 0 0
$$209$$ −4.82843 −0.333989
$$210$$ 0 0
$$211$$ −0.485281 −0.0334081 −0.0167041 0.999860i $$-0.505317\pi$$
−0.0167041 + 0.999860i $$0.505317\pi$$
$$212$$ 0 0
$$213$$ 10.3431 0.708701
$$214$$ 0 0
$$215$$ 2.82843 0.192897
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 22.8284 1.54260
$$220$$ 0 0
$$221$$ 0.485281 0.0326436
$$222$$ 0 0
$$223$$ 10.5858 0.708877 0.354438 0.935079i $$-0.384672\pi$$
0.354438 + 0.935079i $$0.384672\pi$$
$$224$$ 0 0
$$225$$ −1.00000 −0.0666667
$$226$$ 0 0
$$227$$ 24.0416 1.59570 0.797850 0.602857i $$-0.205971\pi$$
0.797850 + 0.602857i $$0.205971\pi$$
$$228$$ 0 0
$$229$$ 11.3137 0.747631 0.373815 0.927503i $$-0.378049\pi$$
0.373815 + 0.927503i $$0.378049\pi$$
$$230$$ 0 0
$$231$$ 19.3137 1.27075
$$232$$ 0 0
$$233$$ −10.0000 −0.655122 −0.327561 0.944830i $$-0.606227\pi$$
−0.327561 + 0.944830i $$0.606227\pi$$
$$234$$ 0 0
$$235$$ 8.48528 0.553519
$$236$$ 0 0
$$237$$ 20.9706 1.36218
$$238$$ 0 0
$$239$$ 8.00000 0.517477 0.258738 0.965947i $$-0.416693\pi$$
0.258738 + 0.965947i $$0.416693\pi$$
$$240$$ 0 0
$$241$$ −20.1421 −1.29747 −0.648735 0.761015i $$-0.724701\pi$$
−0.648735 + 0.761015i $$0.724701\pi$$
$$242$$ 0 0
$$243$$ −9.89949 −0.635053
$$244$$ 0 0
$$245$$ 1.00000 0.0638877
$$246$$ 0 0
$$247$$ −0.585786 −0.0372727
$$248$$ 0 0
$$249$$ 11.3137 0.716977
$$250$$ 0 0
$$251$$ 4.97056 0.313739 0.156870 0.987619i $$-0.449860\pi$$
0.156870 + 0.987619i $$0.449860\pi$$
$$252$$ 0 0
$$253$$ −19.3137 −1.21424
$$254$$ 0 0
$$255$$ 1.17157 0.0733667
$$256$$ 0 0
$$257$$ −23.8995 −1.49081 −0.745405 0.666612i $$-0.767744\pi$$
−0.745405 + 0.666612i $$0.767744\pi$$
$$258$$ 0 0
$$259$$ −4.97056 −0.308856
$$260$$ 0 0
$$261$$ 4.82843 0.298872
$$262$$ 0 0
$$263$$ 4.68629 0.288969 0.144485 0.989507i $$-0.453848\pi$$
0.144485 + 0.989507i $$0.453848\pi$$
$$264$$ 0 0
$$265$$ 1.07107 0.0657952
$$266$$ 0 0
$$267$$ 12.4853 0.764087
$$268$$ 0 0
$$269$$ 6.68629 0.407670 0.203835 0.979005i $$-0.434659\pi$$
0.203835 + 0.979005i $$0.434659\pi$$
$$270$$ 0 0
$$271$$ −26.4853 −1.60887 −0.804433 0.594043i $$-0.797531\pi$$
−0.804433 + 0.594043i $$0.797531\pi$$
$$272$$ 0 0
$$273$$ 2.34315 0.141814
$$274$$ 0 0
$$275$$ −4.82843 −0.291165
$$276$$ 0 0
$$277$$ −24.8284 −1.49180 −0.745898 0.666060i $$-0.767979\pi$$
−0.745898 + 0.666060i $$0.767979\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 16.8284 1.00390 0.501950 0.864897i $$-0.332616\pi$$
0.501950 + 0.864897i $$0.332616\pi$$
$$282$$ 0 0
$$283$$ −17.6569 −1.04959 −0.524796 0.851228i $$-0.675858\pi$$
−0.524796 + 0.851228i $$0.675858\pi$$
$$284$$ 0 0
$$285$$ −1.41421 −0.0837708
$$286$$ 0 0
$$287$$ 13.6569 0.806139
$$288$$ 0 0
$$289$$ −16.3137 −0.959630
$$290$$ 0 0
$$291$$ −8.82843 −0.517532
$$292$$ 0 0
$$293$$ 4.38478 0.256161 0.128081 0.991764i $$-0.459118\pi$$
0.128081 + 0.991764i $$0.459118\pi$$
$$294$$ 0 0
$$295$$ 2.82843 0.164677
$$296$$ 0 0
$$297$$ −27.3137 −1.58490
$$298$$ 0 0
$$299$$ −2.34315 −0.135508
$$300$$ 0 0
$$301$$ 8.00000 0.461112
$$302$$ 0 0
$$303$$ 8.97056 0.515345
$$304$$ 0 0
$$305$$ −9.65685 −0.552950
$$306$$ 0 0
$$307$$ 1.41421 0.0807134 0.0403567 0.999185i $$-0.487151\pi$$
0.0403567 + 0.999185i $$0.487151\pi$$
$$308$$ 0 0
$$309$$ 0.343146 0.0195209
$$310$$ 0 0
$$311$$ 24.8284 1.40789 0.703945 0.710254i $$-0.251420\pi$$
0.703945 + 0.710254i $$0.251420\pi$$
$$312$$ 0 0
$$313$$ 17.3137 0.978629 0.489314 0.872107i $$-0.337247\pi$$
0.489314 + 0.872107i $$0.337247\pi$$
$$314$$ 0 0
$$315$$ −2.82843 −0.159364
$$316$$ 0 0
$$317$$ −19.8995 −1.11767 −0.558833 0.829280i $$-0.688751\pi$$
−0.558833 + 0.829280i $$0.688751\pi$$
$$318$$ 0 0
$$319$$ 23.3137 1.30532
$$320$$ 0 0
$$321$$ −2.00000 −0.111629
$$322$$ 0 0
$$323$$ −0.828427 −0.0460949
$$324$$ 0 0
$$325$$ −0.585786 −0.0324936
$$326$$ 0 0
$$327$$ −10.8284 −0.598813
$$328$$ 0 0
$$329$$ 24.0000 1.32316
$$330$$ 0 0
$$331$$ −33.4558 −1.83890 −0.919450 0.393208i $$-0.871365\pi$$
−0.919450 + 0.393208i $$0.871365\pi$$
$$332$$ 0 0
$$333$$ 1.75736 0.0963027
$$334$$ 0 0
$$335$$ 6.58579 0.359820
$$336$$ 0 0
$$337$$ 16.3848 0.892536 0.446268 0.894899i $$-0.352753\pi$$
0.446268 + 0.894899i $$0.352753\pi$$
$$338$$ 0 0
$$339$$ −15.1716 −0.824007
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −16.9706 −0.916324
$$344$$ 0 0
$$345$$ −5.65685 −0.304555
$$346$$ 0 0
$$347$$ −36.2843 −1.94784 −0.973921 0.226888i $$-0.927145\pi$$
−0.973921 + 0.226888i $$0.927145\pi$$
$$348$$ 0 0
$$349$$ −23.6569 −1.26632 −0.633161 0.774020i $$-0.718243\pi$$
−0.633161 + 0.774020i $$0.718243\pi$$
$$350$$ 0 0
$$351$$ −3.31371 −0.176873
$$352$$ 0 0
$$353$$ 16.1421 0.859159 0.429580 0.903029i $$-0.358662\pi$$
0.429580 + 0.903029i $$0.358662\pi$$
$$354$$ 0 0
$$355$$ −7.31371 −0.388171
$$356$$ 0 0
$$357$$ 3.31371 0.175380
$$358$$ 0 0
$$359$$ 31.4558 1.66018 0.830088 0.557632i $$-0.188290\pi$$
0.830088 + 0.557632i $$0.188290\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −17.4142 −0.914009
$$364$$ 0 0
$$365$$ −16.1421 −0.844918
$$366$$ 0 0
$$367$$ −31.1127 −1.62407 −0.812035 0.583609i $$-0.801640\pi$$
−0.812035 + 0.583609i $$0.801640\pi$$
$$368$$ 0 0
$$369$$ −4.82843 −0.251358
$$370$$ 0 0
$$371$$ 3.02944 0.157281
$$372$$ 0 0
$$373$$ 2.44365 0.126527 0.0632637 0.997997i $$-0.479849\pi$$
0.0632637 + 0.997997i $$0.479849\pi$$
$$374$$ 0 0
$$375$$ −1.41421 −0.0730297
$$376$$ 0 0
$$377$$ 2.82843 0.145671
$$378$$ 0 0
$$379$$ 1.65685 0.0851069 0.0425534 0.999094i $$-0.486451\pi$$
0.0425534 + 0.999094i $$0.486451\pi$$
$$380$$ 0 0
$$381$$ 14.9706 0.766965
$$382$$ 0 0
$$383$$ 7.27208 0.371586 0.185793 0.982589i $$-0.440515\pi$$
0.185793 + 0.982589i $$0.440515\pi$$
$$384$$ 0 0
$$385$$ −13.6569 −0.696018
$$386$$ 0 0
$$387$$ −2.82843 −0.143777
$$388$$ 0 0
$$389$$ 13.3137 0.675032 0.337516 0.941320i $$-0.390413\pi$$
0.337516 + 0.941320i $$0.390413\pi$$
$$390$$ 0 0
$$391$$ −3.31371 −0.167581
$$392$$ 0 0
$$393$$ −21.6569 −1.09244
$$394$$ 0 0
$$395$$ −14.8284 −0.746099
$$396$$ 0 0
$$397$$ −31.1716 −1.56446 −0.782228 0.622992i $$-0.785917\pi$$
−0.782228 + 0.622992i $$0.785917\pi$$
$$398$$ 0 0
$$399$$ −4.00000 −0.200250
$$400$$ 0 0
$$401$$ −18.9706 −0.947345 −0.473672 0.880701i $$-0.657072\pi$$
−0.473672 + 0.880701i $$0.657072\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ −5.00000 −0.248452
$$406$$ 0 0
$$407$$ 8.48528 0.420600
$$408$$ 0 0
$$409$$ 26.2843 1.29967 0.649837 0.760074i $$-0.274837\pi$$
0.649837 + 0.760074i $$0.274837\pi$$
$$410$$ 0 0
$$411$$ 18.1421 0.894886
$$412$$ 0 0
$$413$$ 8.00000 0.393654
$$414$$ 0 0
$$415$$ −8.00000 −0.392705
$$416$$ 0 0
$$417$$ −12.4853 −0.611407
$$418$$ 0 0
$$419$$ 9.65685 0.471768 0.235884 0.971781i $$-0.424201\pi$$
0.235884 + 0.971781i $$0.424201\pi$$
$$420$$ 0 0
$$421$$ 15.1716 0.739417 0.369709 0.929148i $$-0.379458\pi$$
0.369709 + 0.929148i $$0.379458\pi$$
$$422$$ 0 0
$$423$$ −8.48528 −0.412568
$$424$$ 0 0
$$425$$ −0.828427 −0.0401846
$$426$$ 0 0
$$427$$ −27.3137 −1.32180
$$428$$ 0 0
$$429$$ −4.00000 −0.193122
$$430$$ 0 0
$$431$$ −9.17157 −0.441779 −0.220890 0.975299i $$-0.570896\pi$$
−0.220890 + 0.975299i $$0.570896\pi$$
$$432$$ 0 0
$$433$$ 30.0416 1.44371 0.721854 0.692045i $$-0.243290\pi$$
0.721854 + 0.692045i $$0.243290\pi$$
$$434$$ 0 0
$$435$$ 6.82843 0.327398
$$436$$ 0 0
$$437$$ 4.00000 0.191346
$$438$$ 0 0
$$439$$ −14.1421 −0.674967 −0.337484 0.941331i $$-0.609576\pi$$
−0.337484 + 0.941331i $$0.609576\pi$$
$$440$$ 0 0
$$441$$ −1.00000 −0.0476190
$$442$$ 0 0
$$443$$ −23.3137 −1.10767 −0.553834 0.832627i $$-0.686836\pi$$
−0.553834 + 0.832627i $$0.686836\pi$$
$$444$$ 0 0
$$445$$ −8.82843 −0.418508
$$446$$ 0 0
$$447$$ −13.6569 −0.645947
$$448$$ 0 0
$$449$$ −30.2843 −1.42920 −0.714602 0.699532i $$-0.753392\pi$$
−0.714602 + 0.699532i $$0.753392\pi$$
$$450$$ 0 0
$$451$$ −23.3137 −1.09780
$$452$$ 0 0
$$453$$ 31.3137 1.47125
$$454$$ 0 0
$$455$$ −1.65685 −0.0776745
$$456$$ 0 0
$$457$$ 22.2843 1.04241 0.521207 0.853430i $$-0.325482\pi$$
0.521207 + 0.853430i $$0.325482\pi$$
$$458$$ 0 0
$$459$$ −4.68629 −0.218737
$$460$$ 0 0
$$461$$ −36.6274 −1.70591 −0.852954 0.521985i $$-0.825192\pi$$
−0.852954 + 0.521985i $$0.825192\pi$$
$$462$$ 0 0
$$463$$ −18.6274 −0.865689 −0.432845 0.901468i $$-0.642490\pi$$
−0.432845 + 0.901468i $$0.642490\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −24.0000 −1.11059 −0.555294 0.831654i $$-0.687394\pi$$
−0.555294 + 0.831654i $$0.687394\pi$$
$$468$$ 0 0
$$469$$ 18.6274 0.860134
$$470$$ 0 0
$$471$$ 9.17157 0.422604
$$472$$ 0 0
$$473$$ −13.6569 −0.627943
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ −1.07107 −0.0490408
$$478$$ 0 0
$$479$$ −27.4558 −1.25449 −0.627245 0.778822i $$-0.715817\pi$$
−0.627245 + 0.778822i $$0.715817\pi$$
$$480$$ 0 0
$$481$$ 1.02944 0.0469383
$$482$$ 0 0
$$483$$ −16.0000 −0.728025
$$484$$ 0 0
$$485$$ 6.24264 0.283464
$$486$$ 0 0
$$487$$ −3.55635 −0.161154 −0.0805768 0.996748i $$-0.525676\pi$$
−0.0805768 + 0.996748i $$0.525676\pi$$
$$488$$ 0 0
$$489$$ 12.9706 0.586549
$$490$$ 0 0
$$491$$ −20.9706 −0.946388 −0.473194 0.880958i $$-0.656899\pi$$
−0.473194 + 0.880958i $$0.656899\pi$$
$$492$$ 0 0
$$493$$ 4.00000 0.180151
$$494$$ 0 0
$$495$$ 4.82843 0.217022
$$496$$ 0 0
$$497$$ −20.6863 −0.927907
$$498$$ 0 0
$$499$$ 24.1421 1.08075 0.540375 0.841424i $$-0.318282\pi$$
0.540375 + 0.841424i $$0.318282\pi$$
$$500$$ 0 0
$$501$$ −30.9706 −1.38366
$$502$$ 0 0
$$503$$ 9.65685 0.430578 0.215289 0.976550i $$-0.430931\pi$$
0.215289 + 0.976550i $$0.430931\pi$$
$$504$$ 0 0
$$505$$ −6.34315 −0.282266
$$506$$ 0 0
$$507$$ 17.8995 0.794944
$$508$$ 0 0
$$509$$ −35.4558 −1.57155 −0.785776 0.618511i $$-0.787736\pi$$
−0.785776 + 0.618511i $$0.787736\pi$$
$$510$$ 0 0
$$511$$ −45.6569 −2.01974
$$512$$ 0 0
$$513$$ 5.65685 0.249756
$$514$$ 0 0
$$515$$ −0.242641 −0.0106920
$$516$$ 0 0
$$517$$ −40.9706 −1.80188
$$518$$ 0 0
$$519$$ 20.8284 0.914266
$$520$$ 0 0
$$521$$ 37.3137 1.63474 0.817372 0.576111i $$-0.195430\pi$$
0.817372 + 0.576111i $$0.195430\pi$$
$$522$$ 0 0
$$523$$ 27.7574 1.21374 0.606872 0.794799i $$-0.292424\pi$$
0.606872 + 0.794799i $$0.292424\pi$$
$$524$$ 0 0
$$525$$ −4.00000 −0.174574
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ −2.82843 −0.122743
$$532$$ 0 0
$$533$$ −2.82843 −0.122513
$$534$$ 0 0
$$535$$ 1.41421 0.0611418
$$536$$ 0 0
$$537$$ −12.0000 −0.517838
$$538$$ 0 0
$$539$$ −4.82843 −0.207975
$$540$$ 0 0
$$541$$ 20.2843 0.872089 0.436044 0.899925i $$-0.356379\pi$$
0.436044 + 0.899925i $$0.356379\pi$$
$$542$$ 0 0
$$543$$ 24.4853 1.05076
$$544$$ 0 0
$$545$$ 7.65685 0.327984
$$546$$ 0 0
$$547$$ 24.9289 1.06588 0.532942 0.846152i $$-0.321086\pi$$
0.532942 + 0.846152i $$0.321086\pi$$
$$548$$ 0 0
$$549$$ 9.65685 0.412144
$$550$$ 0 0
$$551$$ −4.82843 −0.205698
$$552$$ 0 0
$$553$$ −41.9411 −1.78352
$$554$$ 0 0
$$555$$ 2.48528 0.105494
$$556$$ 0 0
$$557$$ −33.7990 −1.43211 −0.716055 0.698044i $$-0.754054\pi$$
−0.716055 + 0.698044i $$0.754054\pi$$
$$558$$ 0 0
$$559$$ −1.65685 −0.0700775
$$560$$ 0 0
$$561$$ −5.65685 −0.238833
$$562$$ 0 0
$$563$$ −4.24264 −0.178806 −0.0894030 0.995996i $$-0.528496\pi$$
−0.0894030 + 0.995996i $$0.528496\pi$$
$$564$$ 0 0
$$565$$ 10.7279 0.451327
$$566$$ 0 0
$$567$$ −14.1421 −0.593914
$$568$$ 0 0
$$569$$ 35.9411 1.50673 0.753365 0.657602i $$-0.228429\pi$$
0.753365 + 0.657602i $$0.228429\pi$$
$$570$$ 0 0
$$571$$ 13.5147 0.565573 0.282787 0.959183i $$-0.408741\pi$$
0.282787 + 0.959183i $$0.408741\pi$$
$$572$$ 0 0
$$573$$ 27.3137 1.14105
$$574$$ 0 0
$$575$$ 4.00000 0.166812
$$576$$ 0 0
$$577$$ −0.627417 −0.0261197 −0.0130599 0.999915i $$-0.504157\pi$$
−0.0130599 + 0.999915i $$0.504157\pi$$
$$578$$ 0 0
$$579$$ 29.7990 1.23840
$$580$$ 0 0
$$581$$ −22.6274 −0.938743
$$582$$ 0 0
$$583$$ −5.17157 −0.214185
$$584$$ 0 0
$$585$$ 0.585786 0.0242193
$$586$$ 0 0
$$587$$ −5.65685 −0.233483 −0.116742 0.993162i $$-0.537245\pi$$
−0.116742 + 0.993162i $$0.537245\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 3.79899 0.156270
$$592$$ 0 0
$$593$$ −28.6274 −1.17559 −0.587794 0.809011i $$-0.700003\pi$$
−0.587794 + 0.809011i $$0.700003\pi$$
$$594$$ 0 0
$$595$$ −2.34315 −0.0960596
$$596$$ 0 0
$$597$$ −8.00000 −0.327418
$$598$$ 0 0
$$599$$ −22.1421 −0.904703 −0.452352 0.891840i $$-0.649415\pi$$
−0.452352 + 0.891840i $$0.649415\pi$$
$$600$$ 0 0
$$601$$ 2.48528 0.101377 0.0506884 0.998715i $$-0.483858\pi$$
0.0506884 + 0.998715i $$0.483858\pi$$
$$602$$ 0 0
$$603$$ −6.58579 −0.268194
$$604$$ 0 0
$$605$$ 12.3137 0.500623
$$606$$ 0 0
$$607$$ −16.2426 −0.659268 −0.329634 0.944109i $$-0.606925\pi$$
−0.329634 + 0.944109i $$0.606925\pi$$
$$608$$ 0 0
$$609$$ 19.3137 0.782631
$$610$$ 0 0
$$611$$ −4.97056 −0.201087
$$612$$ 0 0
$$613$$ 21.7990 0.880453 0.440226 0.897887i $$-0.354898\pi$$
0.440226 + 0.897887i $$0.354898\pi$$
$$614$$ 0 0
$$615$$ −6.82843 −0.275349
$$616$$ 0 0
$$617$$ 1.79899 0.0724246 0.0362123 0.999344i $$-0.488471\pi$$
0.0362123 + 0.999344i $$0.488471\pi$$
$$618$$ 0 0
$$619$$ −11.8579 −0.476608 −0.238304 0.971191i $$-0.576591\pi$$
−0.238304 + 0.971191i $$0.576591\pi$$
$$620$$ 0 0
$$621$$ 22.6274 0.908007
$$622$$ 0 0
$$623$$ −24.9706 −1.00042
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 6.82843 0.272701
$$628$$ 0 0
$$629$$ 1.45584 0.0580483
$$630$$ 0 0
$$631$$ −17.5147 −0.697250 −0.348625 0.937262i $$-0.613351\pi$$
−0.348625 + 0.937262i $$0.613351\pi$$
$$632$$ 0 0
$$633$$ 0.686292 0.0272776
$$634$$ 0 0
$$635$$ −10.5858 −0.420084
$$636$$ 0 0
$$637$$ −0.585786 −0.0232097
$$638$$ 0 0
$$639$$ 7.31371 0.289326
$$640$$ 0 0
$$641$$ −23.4558 −0.926450 −0.463225 0.886241i $$-0.653308\pi$$
−0.463225 + 0.886241i $$0.653308\pi$$
$$642$$ 0 0
$$643$$ 12.6863 0.500298 0.250149 0.968207i $$-0.419520\pi$$
0.250149 + 0.968207i $$0.419520\pi$$
$$644$$ 0 0
$$645$$ −4.00000 −0.157500
$$646$$ 0 0
$$647$$ −39.3137 −1.54558 −0.772791 0.634661i $$-0.781140\pi$$
−0.772791 + 0.634661i $$0.781140\pi$$
$$648$$ 0 0
$$649$$ −13.6569 −0.536078
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 16.1421 0.631691 0.315845 0.948811i $$-0.397712\pi$$
0.315845 + 0.948811i $$0.397712\pi$$
$$654$$ 0 0
$$655$$ 15.3137 0.598356
$$656$$ 0 0
$$657$$ 16.1421 0.629765
$$658$$ 0 0
$$659$$ 11.5147 0.448550 0.224275 0.974526i $$-0.427999\pi$$
0.224275 + 0.974526i $$0.427999\pi$$
$$660$$ 0 0
$$661$$ −3.17157 −0.123360 −0.0616799 0.998096i $$-0.519646\pi$$
−0.0616799 + 0.998096i $$0.519646\pi$$
$$662$$ 0 0
$$663$$ −0.686292 −0.0266534
$$664$$ 0 0
$$665$$ 2.82843 0.109682
$$666$$ 0 0
$$667$$ −19.3137 −0.747830
$$668$$ 0 0
$$669$$ −14.9706 −0.578795
$$670$$ 0 0
$$671$$ 46.6274 1.80003
$$672$$ 0 0
$$673$$ −28.3848 −1.09415 −0.547076 0.837083i $$-0.684259\pi$$
−0.547076 + 0.837083i $$0.684259\pi$$
$$674$$ 0 0
$$675$$ 5.65685 0.217732
$$676$$ 0 0
$$677$$ 44.3848 1.70585 0.852923 0.522037i $$-0.174828\pi$$
0.852923 + 0.522037i $$0.174828\pi$$
$$678$$ 0 0
$$679$$ 17.6569 0.677608
$$680$$ 0 0
$$681$$ −34.0000 −1.30288
$$682$$ 0 0
$$683$$ −9.89949 −0.378794 −0.189397 0.981901i $$-0.560653\pi$$
−0.189397 + 0.981901i $$0.560653\pi$$
$$684$$ 0 0
$$685$$ −12.8284 −0.490149
$$686$$ 0 0
$$687$$ −16.0000 −0.610438
$$688$$ 0 0
$$689$$ −0.627417 −0.0239027
$$690$$ 0 0
$$691$$ 40.8284 1.55319 0.776593 0.630002i $$-0.216946\pi$$
0.776593 + 0.630002i $$0.216946\pi$$
$$692$$ 0 0
$$693$$ 13.6569 0.518781
$$694$$ 0 0
$$695$$ 8.82843 0.334881
$$696$$ 0 0
$$697$$ −4.00000 −0.151511
$$698$$ 0 0
$$699$$ 14.1421 0.534905
$$700$$ 0 0
$$701$$ 24.0000 0.906467 0.453234 0.891392i $$-0.350270\pi$$
0.453234 + 0.891392i $$0.350270\pi$$
$$702$$ 0 0
$$703$$ −1.75736 −0.0662801
$$704$$ 0 0
$$705$$ −12.0000 −0.451946
$$706$$ 0 0
$$707$$ −17.9411 −0.674745
$$708$$ 0 0
$$709$$ −38.2843 −1.43780 −0.718898 0.695116i $$-0.755353\pi$$
−0.718898 + 0.695116i $$0.755353\pi$$
$$710$$ 0 0
$$711$$ 14.8284 0.556109
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 2.82843 0.105777
$$716$$ 0 0
$$717$$ −11.3137 −0.422518
$$718$$ 0 0
$$719$$ 19.4558 0.725581 0.362790 0.931871i $$-0.381824\pi$$
0.362790 + 0.931871i $$0.381824\pi$$
$$720$$ 0 0
$$721$$ −0.686292 −0.0255588
$$722$$ 0 0
$$723$$ 28.4853 1.05938
$$724$$ 0 0
$$725$$ −4.82843 −0.179323
$$726$$ 0 0
$$727$$ 9.45584 0.350698 0.175349 0.984506i $$-0.443895\pi$$
0.175349 + 0.984506i $$0.443895\pi$$
$$728$$ 0 0
$$729$$ 29.0000 1.07407
$$730$$ 0 0
$$731$$ −2.34315 −0.0866644
$$732$$ 0 0
$$733$$ −15.6569 −0.578299 −0.289150 0.957284i $$-0.593372\pi$$
−0.289150 + 0.957284i $$0.593372\pi$$
$$734$$ 0 0
$$735$$ −1.41421 −0.0521641
$$736$$ 0 0
$$737$$ −31.7990 −1.17133
$$738$$ 0 0
$$739$$ −23.3137 −0.857609 −0.428804 0.903397i $$-0.641065\pi$$
−0.428804 + 0.903397i $$0.641065\pi$$
$$740$$ 0 0
$$741$$ 0.828427 0.0304330
$$742$$ 0 0
$$743$$ −24.2426 −0.889376 −0.444688 0.895685i $$-0.646685\pi$$
−0.444688 + 0.895685i $$0.646685\pi$$
$$744$$ 0 0
$$745$$ 9.65685 0.353800
$$746$$ 0 0
$$747$$ 8.00000 0.292705
$$748$$ 0 0
$$749$$ 4.00000 0.146157
$$750$$ 0 0
$$751$$ −27.5147 −1.00403 −0.502013 0.864860i $$-0.667407\pi$$
−0.502013 + 0.864860i $$0.667407\pi$$
$$752$$ 0 0
$$753$$ −7.02944 −0.256167
$$754$$ 0 0
$$755$$ −22.1421 −0.805835
$$756$$ 0 0
$$757$$ −34.2843 −1.24608 −0.623042 0.782189i $$-0.714103\pi$$
−0.623042 + 0.782189i $$0.714103\pi$$
$$758$$ 0 0
$$759$$ 27.3137 0.991425
$$760$$ 0 0
$$761$$ 46.6274 1.69024 0.845121 0.534575i $$-0.179528\pi$$
0.845121 + 0.534575i $$0.179528\pi$$
$$762$$ 0 0
$$763$$ 21.6569 0.784031
$$764$$ 0 0
$$765$$ 0.828427 0.0299518
$$766$$ 0 0
$$767$$ −1.65685 −0.0598255
$$768$$ 0 0
$$769$$ 40.0000 1.44244 0.721218 0.692708i $$-0.243582\pi$$
0.721218 + 0.692708i $$0.243582\pi$$
$$770$$ 0 0
$$771$$ 33.7990 1.21724
$$772$$ 0 0
$$773$$ −4.87006 −0.175164 −0.0875819 0.996157i $$-0.527914\pi$$
−0.0875819 + 0.996157i $$0.527914\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 7.02944 0.252180
$$778$$ 0 0
$$779$$ 4.82843 0.172996
$$780$$ 0 0
$$781$$ 35.3137 1.26362
$$782$$ 0 0
$$783$$ −27.3137 −0.976112
$$784$$ 0 0
$$785$$ −6.48528 −0.231470
$$786$$ 0 0
$$787$$ −1.41421 −0.0504113 −0.0252056 0.999682i $$-0.508024\pi$$
−0.0252056 + 0.999682i $$0.508024\pi$$
$$788$$ 0 0
$$789$$ −6.62742 −0.235942
$$790$$ 0 0
$$791$$ 30.3431 1.07888
$$792$$ 0 0
$$793$$ 5.65685 0.200881
$$794$$ 0 0
$$795$$ −1.51472 −0.0537215
$$796$$ 0 0
$$797$$ 31.6985 1.12282 0.561409 0.827538i $$-0.310259\pi$$
0.561409 + 0.827538i $$0.310259\pi$$
$$798$$ 0 0
$$799$$ −7.02944 −0.248684
$$800$$ 0 0
$$801$$ 8.82843 0.311937
$$802$$ 0 0
$$803$$ 77.9411 2.75048
$$804$$ 0 0
$$805$$ 11.3137 0.398756
$$806$$ 0 0
$$807$$ −9.45584 −0.332861
$$808$$ 0 0
$$809$$ 38.0000 1.33601 0.668004 0.744157i $$-0.267149\pi$$
0.668004 + 0.744157i $$0.267149\pi$$
$$810$$ 0 0
$$811$$ 23.3137 0.818655 0.409328 0.912388i $$-0.365763\pi$$
0.409328 + 0.912388i $$0.365763\pi$$
$$812$$ 0 0
$$813$$ 37.4558 1.31363
$$814$$ 0 0
$$815$$ −9.17157 −0.321266
$$816$$ 0 0
$$817$$ 2.82843 0.0989541
$$818$$ 0 0
$$819$$ 1.65685 0.0578952
$$820$$ 0 0
$$821$$ −29.5980 −1.03298 −0.516488 0.856294i $$-0.672761\pi$$
−0.516488 + 0.856294i $$0.672761\pi$$
$$822$$ 0 0
$$823$$ 0.485281 0.0169158 0.00845792 0.999964i $$-0.497308\pi$$
0.00845792 + 0.999964i $$0.497308\pi$$
$$824$$ 0 0
$$825$$ 6.82843 0.237735
$$826$$ 0 0
$$827$$ 45.6985 1.58909 0.794546 0.607204i $$-0.207709\pi$$
0.794546 + 0.607204i $$0.207709\pi$$
$$828$$ 0 0
$$829$$ −29.5980 −1.02798 −0.513990 0.857796i $$-0.671833\pi$$
−0.513990 + 0.857796i $$0.671833\pi$$
$$830$$ 0 0
$$831$$ 35.1127 1.21805
$$832$$ 0 0
$$833$$ −0.828427 −0.0287033
$$834$$ 0 0
$$835$$ 21.8995 0.757863
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 39.7990 1.37401 0.687007 0.726651i $$-0.258924\pi$$
0.687007 + 0.726651i $$0.258924\pi$$
$$840$$ 0 0
$$841$$ −5.68629 −0.196079
$$842$$ 0 0
$$843$$ −23.7990 −0.819681
$$844$$ 0 0
$$845$$ −12.6569 −0.435409
$$846$$ 0 0
$$847$$ 34.8284 1.19672
$$848$$ 0 0
$$849$$ 24.9706 0.856987
$$850$$ 0 0
$$851$$ −7.02944 −0.240966
$$852$$ 0 0
$$853$$ −39.9411 −1.36756 −0.683779 0.729689i $$-0.739665\pi$$
−0.683779 + 0.729689i $$0.739665\pi$$
$$854$$ 0 0
$$855$$ −1.00000 −0.0341993
$$856$$ 0 0
$$857$$ −11.2132 −0.383036 −0.191518 0.981489i $$-0.561341\pi$$
−0.191518 + 0.981489i $$0.561341\pi$$
$$858$$ 0 0
$$859$$ −17.6569 −0.602444 −0.301222 0.953554i $$-0.597395\pi$$
−0.301222 + 0.953554i $$0.597395\pi$$
$$860$$ 0 0
$$861$$ −19.3137 −0.658209
$$862$$ 0 0
$$863$$ 4.04163 0.137579 0.0687894 0.997631i $$-0.478086\pi$$
0.0687894 + 0.997631i $$0.478086\pi$$
$$864$$ 0 0
$$865$$ −14.7279 −0.500764
$$866$$ 0 0
$$867$$ 23.0711 0.783535
$$868$$ 0 0
$$869$$ 71.5980 2.42880
$$870$$ 0 0
$$871$$ −3.85786 −0.130719
$$872$$ 0 0
$$873$$ −6.24264 −0.211281
$$874$$ 0 0
$$875$$ 2.82843 0.0956183
$$876$$ 0 0
$$877$$ 6.72792 0.227186 0.113593 0.993527i $$-0.463764\pi$$
0.113593 + 0.993527i $$0.463764\pi$$
$$878$$ 0 0
$$879$$ −6.20101 −0.209155
$$880$$ 0 0
$$881$$ 9.65685 0.325348 0.162674 0.986680i $$-0.447988\pi$$
0.162674 + 0.986680i $$0.447988\pi$$
$$882$$ 0 0
$$883$$ 5.17157 0.174037 0.0870186 0.996207i $$-0.472266\pi$$
0.0870186 + 0.996207i $$0.472266\pi$$
$$884$$ 0 0
$$885$$ −4.00000 −0.134459
$$886$$ 0 0
$$887$$ −40.7279 −1.36751 −0.683755 0.729712i $$-0.739654\pi$$
−0.683755 + 0.729712i $$0.739654\pi$$
$$888$$ 0 0
$$889$$ −29.9411 −1.00419
$$890$$ 0 0
$$891$$ 24.1421 0.808792
$$892$$ 0 0
$$893$$ 8.48528 0.283949
$$894$$ 0 0
$$895$$ 8.48528 0.283632
$$896$$ 0 0
$$897$$ 3.31371 0.110642
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −0.887302 −0.0295603
$$902$$ 0 0
$$903$$ −11.3137 −0.376497
$$904$$ 0 0
$$905$$ −17.3137 −0.575527
$$906$$ 0 0
$$907$$ 0.0416306 0.00138232 0.000691160 1.00000i $$-0.499780\pi$$
0.000691160 1.00000i $$0.499780\pi$$
$$908$$ 0 0
$$909$$ 6.34315 0.210389
$$910$$ 0 0
$$911$$ −14.8284 −0.491288 −0.245644 0.969360i $$-0.578999\pi$$
−0.245644 + 0.969360i $$0.578999\pi$$
$$912$$ 0 0
$$913$$ 38.6274 1.27838
$$914$$ 0 0
$$915$$ 13.6569 0.451482
$$916$$ 0 0
$$917$$ 43.3137 1.43034
$$918$$ 0 0
$$919$$ 49.9411 1.64741 0.823703 0.567022i $$-0.191904\pi$$
0.823703 + 0.567022i $$0.191904\pi$$
$$920$$ 0 0
$$921$$ −2.00000 −0.0659022
$$922$$ 0 0
$$923$$ 4.28427 0.141019
$$924$$ 0 0
$$925$$ −1.75736 −0.0577816
$$926$$ 0 0
$$927$$ 0.242641 0.00796937
$$928$$ 0 0
$$929$$ 18.6863 0.613077 0.306539 0.951858i $$-0.400829\pi$$
0.306539 + 0.951858i $$0.400829\pi$$
$$930$$ 0 0
$$931$$ 1.00000 0.0327737
$$932$$ 0 0
$$933$$ −35.1127 −1.14954
$$934$$ 0 0
$$935$$ 4.00000 0.130814
$$936$$ 0 0
$$937$$ 4.54416 0.148451 0.0742256 0.997241i $$-0.476352\pi$$
0.0742256 + 0.997241i $$0.476352\pi$$
$$938$$ 0 0
$$939$$ −24.4853 −0.799047
$$940$$ 0 0
$$941$$ 57.5980 1.87764 0.938820 0.344408i $$-0.111920\pi$$
0.938820 + 0.344408i $$0.111920\pi$$
$$942$$ 0 0
$$943$$ 19.3137 0.628941
$$944$$ 0 0
$$945$$ 16.0000 0.520480
$$946$$ 0 0
$$947$$ −24.6863 −0.802197 −0.401098 0.916035i $$-0.631371\pi$$
−0.401098 + 0.916035i $$0.631371\pi$$
$$948$$ 0 0
$$949$$ 9.45584 0.306950
$$950$$ 0 0
$$951$$ 28.1421 0.912571
$$952$$ 0 0
$$953$$ −9.27208 −0.300352 −0.150176 0.988659i $$-0.547984\pi$$
−0.150176 + 0.988659i $$0.547984\pi$$
$$954$$ 0 0
$$955$$ −19.3137 −0.624977
$$956$$ 0 0
$$957$$ −32.9706 −1.06579
$$958$$ 0 0
$$959$$ −36.2843 −1.17168
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ −1.41421 −0.0455724
$$964$$ 0 0
$$965$$ −21.0711 −0.678302
$$966$$ 0 0
$$967$$ 26.3431 0.847138 0.423569 0.905864i $$-0.360777\pi$$
0.423569 + 0.905864i $$0.360777\pi$$
$$968$$ 0 0
$$969$$ 1.17157 0.0376363
$$970$$ 0 0
$$971$$ 41.9411 1.34595 0.672977 0.739663i $$-0.265015\pi$$
0.672977 + 0.739663i $$0.265015\pi$$
$$972$$ 0 0
$$973$$ 24.9706 0.800519
$$974$$ 0 0
$$975$$ 0.828427 0.0265309
$$976$$ 0 0
$$977$$ 33.8406 1.08266 0.541329 0.840811i $$-0.317921\pi$$
0.541329 + 0.840811i $$0.317921\pi$$
$$978$$ 0 0
$$979$$ 42.6274 1.36238
$$980$$ 0 0
$$981$$ −7.65685 −0.244465
$$982$$ 0 0
$$983$$ −44.0416 −1.40471 −0.702355 0.711827i $$-0.747868\pi$$
−0.702355 + 0.711827i $$0.747868\pi$$
$$984$$ 0 0
$$985$$ −2.68629 −0.0855924
$$986$$ 0 0
$$987$$ −33.9411 −1.08036
$$988$$ 0 0
$$989$$ 11.3137 0.359755
$$990$$ 0 0
$$991$$ −28.7696 −0.913895 −0.456947 0.889494i $$-0.651057\pi$$
−0.456947 + 0.889494i $$0.651057\pi$$
$$992$$ 0 0
$$993$$ 47.3137 1.50146
$$994$$ 0 0
$$995$$ 5.65685 0.179334
$$996$$ 0 0
$$997$$ −7.17157 −0.227126 −0.113563 0.993531i $$-0.536226\pi$$
−0.113563 + 0.993531i $$0.536226\pi$$
$$998$$ 0 0
$$999$$ −9.94113 −0.314523
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6080.2.a.bf.1.1 2
4.3 odd 2 6080.2.a.bg.1.2 2
8.3 odd 2 1520.2.a.m.1.1 2
8.5 even 2 760.2.a.f.1.2 2
24.5 odd 2 6840.2.a.z.1.2 2
40.13 odd 4 3800.2.d.i.3649.4 4
40.19 odd 2 7600.2.a.ba.1.2 2
40.29 even 2 3800.2.a.n.1.1 2
40.37 odd 4 3800.2.d.i.3649.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.f.1.2 2 8.5 even 2
1520.2.a.m.1.1 2 8.3 odd 2
3800.2.a.n.1.1 2 40.29 even 2
3800.2.d.i.3649.2 4 40.37 odd 4
3800.2.d.i.3649.4 4 40.13 odd 4
6080.2.a.bf.1.1 2 1.1 even 1 trivial
6080.2.a.bg.1.2 2 4.3 odd 2
6840.2.a.z.1.2 2 24.5 odd 2
7600.2.a.ba.1.2 2 40.19 odd 2