# Properties

 Label 6960.2.a.co.1.4 Level $6960$ Weight $2$ Character 6960.1 Self dual yes Analytic conductor $55.576$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6960,2,Mod(1,6960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6960, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6960.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6960 = 2^{4} \cdot 3 \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6960.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: $$55.5758798068$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.2225.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 5x^{2} + 2x + 4$$ x^4 - x^3 - 5*x^2 + 2*x + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 435) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$2.43828$$ of defining polynomial Character $$\chi$$ $$=$$ 6960.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -1.00000 q^{5} +2.74301 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -1.00000 q^{5} +2.74301 q^{7} +1.00000 q^{9} -2.74301 q^{11} -5.14744 q^{13} -1.00000 q^{15} -3.72913 q^{17} +0.404431 q^{19} +2.74301 q^{21} +5.45825 q^{23} +1.00000 q^{25} +1.00000 q^{27} -1.00000 q^{29} -1.45825 q^{31} -2.74301 q^{33} -2.74301 q^{35} +6.76702 q^{37} -5.14744 q^{39} +9.78090 q^{41} -4.43220 q^{43} -1.00000 q^{45} -2.60569 q^{47} +0.524103 q^{49} -3.72913 q^{51} -6.43220 q^{53} +2.74301 q^{55} +0.404431 q^{57} +9.91822 q^{59} -13.0816 q^{61} +2.74301 q^{63} +5.14744 q^{65} -12.4961 q^{67} +5.45825 q^{69} +11.3487 q^{71} -10.7670 q^{73} +1.00000 q^{75} -7.52410 q^{77} -14.1576 q^{79} +1.00000 q^{81} -1.62334 q^{83} +3.72913 q^{85} -1.00000 q^{87} -8.87281 q^{89} -14.1195 q^{91} -1.45825 q^{93} -0.404431 q^{95} +7.82084 q^{97} -2.74301 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} - 4 q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^3 - 4 * q^5 - 2 * q^7 + 4 * q^9 $$4 q + 4 q^{3} - 4 q^{5} - 2 q^{7} + 4 q^{9} + 2 q^{11} - 8 q^{13} - 4 q^{15} - 10 q^{17} + 2 q^{19} - 2 q^{21} + 12 q^{23} + 4 q^{25} + 4 q^{27} - 4 q^{29} + 4 q^{31} + 2 q^{33} + 2 q^{35} - 16 q^{37} - 8 q^{39} - 12 q^{41} - 2 q^{43} - 4 q^{45} + 12 q^{47} + 6 q^{49} - 10 q^{51} - 10 q^{53} - 2 q^{55} + 2 q^{57} - 2 q^{59} - 26 q^{61} - 2 q^{63} + 8 q^{65} - 2 q^{67} + 12 q^{69} + 10 q^{71} + 4 q^{75} - 34 q^{77} - 22 q^{79} + 4 q^{81} + 10 q^{83} + 10 q^{85} - 4 q^{87} - 4 q^{89} + 8 q^{91} + 4 q^{93} - 2 q^{95} - 22 q^{97} + 2 q^{99}+O(q^{100})$$ 4 * q + 4 * q^3 - 4 * q^5 - 2 * q^7 + 4 * q^9 + 2 * q^11 - 8 * q^13 - 4 * q^15 - 10 * q^17 + 2 * q^19 - 2 * q^21 + 12 * q^23 + 4 * q^25 + 4 * q^27 - 4 * q^29 + 4 * q^31 + 2 * q^33 + 2 * q^35 - 16 * q^37 - 8 * q^39 - 12 * q^41 - 2 * q^43 - 4 * q^45 + 12 * q^47 + 6 * q^49 - 10 * q^51 - 10 * q^53 - 2 * q^55 + 2 * q^57 - 2 * q^59 - 26 * q^61 - 2 * q^63 + 8 * q^65 - 2 * q^67 + 12 * q^69 + 10 * q^71 + 4 * q^75 - 34 * q^77 - 22 * q^79 + 4 * q^81 + 10 * q^83 + 10 * q^85 - 4 * q^87 - 4 * q^89 + 8 * q^91 + 4 * q^93 - 2 * q^95 - 22 * q^97 + 2 * 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
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 2.74301 1.03676 0.518380 0.855150i $$-0.326535\pi$$
0.518380 + 0.855150i $$0.326535\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −2.74301 −0.827049 −0.413524 0.910493i $$-0.635702\pi$$
−0.413524 + 0.910493i $$0.635702\pi$$
$$12$$ 0 0
$$13$$ −5.14744 −1.42764 −0.713822 0.700328i $$-0.753037\pi$$
−0.713822 + 0.700328i $$0.753037\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 0 0
$$17$$ −3.72913 −0.904446 −0.452223 0.891905i $$-0.649369\pi$$
−0.452223 + 0.891905i $$0.649369\pi$$
$$18$$ 0 0
$$19$$ 0.404431 0.0927827 0.0463914 0.998923i $$-0.485228\pi$$
0.0463914 + 0.998923i $$0.485228\pi$$
$$20$$ 0 0
$$21$$ 2.74301 0.598574
$$22$$ 0 0
$$23$$ 5.45825 1.13812 0.569062 0.822295i $$-0.307306\pi$$
0.569062 + 0.822295i $$0.307306\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −1.00000 −0.185695
$$30$$ 0 0
$$31$$ −1.45825 −0.261910 −0.130955 0.991388i $$-0.541804\pi$$
−0.130955 + 0.991388i $$0.541804\pi$$
$$32$$ 0 0
$$33$$ −2.74301 −0.477497
$$34$$ 0 0
$$35$$ −2.74301 −0.463653
$$36$$ 0 0
$$37$$ 6.76702 1.11249 0.556245 0.831018i $$-0.312241\pi$$
0.556245 + 0.831018i $$0.312241\pi$$
$$38$$ 0 0
$$39$$ −5.14744 −0.824250
$$40$$ 0 0
$$41$$ 9.78090 1.52752 0.763760 0.645500i $$-0.223351\pi$$
0.763760 + 0.645500i $$0.223351\pi$$
$$42$$ 0 0
$$43$$ −4.43220 −0.675904 −0.337952 0.941163i $$-0.609734\pi$$
−0.337952 + 0.941163i $$0.609734\pi$$
$$44$$ 0 0
$$45$$ −1.00000 −0.149071
$$46$$ 0 0
$$47$$ −2.60569 −0.380079 −0.190040 0.981776i $$-0.560862\pi$$
−0.190040 + 0.981776i $$0.560862\pi$$
$$48$$ 0 0
$$49$$ 0.524103 0.0748719
$$50$$ 0 0
$$51$$ −3.72913 −0.522182
$$52$$ 0 0
$$53$$ −6.43220 −0.883530 −0.441765 0.897131i $$-0.645648\pi$$
−0.441765 + 0.897131i $$0.645648\pi$$
$$54$$ 0 0
$$55$$ 2.74301 0.369867
$$56$$ 0 0
$$57$$ 0.404431 0.0535681
$$58$$ 0 0
$$59$$ 9.91822 1.29124 0.645621 0.763658i $$-0.276599\pi$$
0.645621 + 0.763658i $$0.276599\pi$$
$$60$$ 0 0
$$61$$ −13.0816 −1.67493 −0.837463 0.546494i $$-0.815962\pi$$
−0.837463 + 0.546494i $$0.815962\pi$$
$$62$$ 0 0
$$63$$ 2.74301 0.345587
$$64$$ 0 0
$$65$$ 5.14744 0.638461
$$66$$ 0 0
$$67$$ −12.4961 −1.52665 −0.763323 0.646017i $$-0.776434\pi$$
−0.763323 + 0.646017i $$0.776434\pi$$
$$68$$ 0 0
$$69$$ 5.45825 0.657096
$$70$$ 0 0
$$71$$ 11.3487 1.34684 0.673422 0.739259i $$-0.264824\pi$$
0.673422 + 0.739259i $$0.264824\pi$$
$$72$$ 0 0
$$73$$ −10.7670 −1.26018 −0.630092 0.776520i $$-0.716983\pi$$
−0.630092 + 0.776520i $$0.716983\pi$$
$$74$$ 0 0
$$75$$ 1.00000 0.115470
$$76$$ 0 0
$$77$$ −7.52410 −0.857451
$$78$$ 0 0
$$79$$ −14.1576 −1.59285 −0.796425 0.604737i $$-0.793278\pi$$
−0.796425 + 0.604737i $$0.793278\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −1.62334 −0.178184 −0.0890922 0.996023i $$-0.528397\pi$$
−0.0890922 + 0.996023i $$0.528397\pi$$
$$84$$ 0 0
$$85$$ 3.72913 0.404481
$$86$$ 0 0
$$87$$ −1.00000 −0.107211
$$88$$ 0 0
$$89$$ −8.87281 −0.940516 −0.470258 0.882529i $$-0.655839\pi$$
−0.470258 + 0.882529i $$0.655839\pi$$
$$90$$ 0 0
$$91$$ −14.1195 −1.48012
$$92$$ 0 0
$$93$$ −1.45825 −0.151214
$$94$$ 0 0
$$95$$ −0.404431 −0.0414937
$$96$$ 0 0
$$97$$ 7.82084 0.794086 0.397043 0.917800i $$-0.370036\pi$$
0.397043 + 0.917800i $$0.370036\pi$$
$$98$$ 0 0
$$99$$ −2.74301 −0.275683
$$100$$ 0 0
$$101$$ −4.88033 −0.485611 −0.242805 0.970075i $$-0.578068\pi$$
−0.242805 + 0.970075i $$0.578068\pi$$
$$102$$ 0 0
$$103$$ 0.294881 0.0290555 0.0145277 0.999894i $$-0.495376\pi$$
0.0145277 + 0.999894i $$0.495376\pi$$
$$104$$ 0 0
$$105$$ −2.74301 −0.267690
$$106$$ 0 0
$$107$$ 13.7809 1.33225 0.666125 0.745840i $$-0.267952\pi$$
0.666125 + 0.745840i $$0.267952\pi$$
$$108$$ 0 0
$$109$$ −6.20126 −0.593973 −0.296987 0.954882i $$-0.595982\pi$$
−0.296987 + 0.954882i $$0.595982\pi$$
$$110$$ 0 0
$$111$$ 6.76702 0.642297
$$112$$ 0 0
$$113$$ −10.5658 −0.993943 −0.496971 0.867767i $$-0.665555\pi$$
−0.496971 + 0.867767i $$0.665555\pi$$
$$114$$ 0 0
$$115$$ −5.45825 −0.508985
$$116$$ 0 0
$$117$$ −5.14744 −0.475881
$$118$$ 0 0
$$119$$ −10.2290 −0.937694
$$120$$ 0 0
$$121$$ −3.47590 −0.315991
$$122$$ 0 0
$$123$$ 9.78090 0.881914
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −8.54175 −0.757958 −0.378979 0.925405i $$-0.623725\pi$$
−0.378979 + 0.925405i $$0.623725\pi$$
$$128$$ 0 0
$$129$$ −4.43220 −0.390233
$$130$$ 0 0
$$131$$ −13.3050 −1.16246 −0.581232 0.813738i $$-0.697429\pi$$
−0.581232 + 0.813738i $$0.697429\pi$$
$$132$$ 0 0
$$133$$ 1.10936 0.0961934
$$134$$ 0 0
$$135$$ −1.00000 −0.0860663
$$136$$ 0 0
$$137$$ −18.2253 −1.55709 −0.778545 0.627589i $$-0.784042\pi$$
−0.778545 + 0.627589i $$0.784042\pi$$
$$138$$ 0 0
$$139$$ 5.66142 0.480195 0.240098 0.970749i $$-0.422821\pi$$
0.240098 + 0.970749i $$0.422821\pi$$
$$140$$ 0 0
$$141$$ −2.60569 −0.219439
$$142$$ 0 0
$$143$$ 14.1195 1.18073
$$144$$ 0 0
$$145$$ 1.00000 0.0830455
$$146$$ 0 0
$$147$$ 0.524103 0.0432273
$$148$$ 0 0
$$149$$ −11.9182 −0.976378 −0.488189 0.872738i $$-0.662342\pi$$
−0.488189 + 0.872738i $$0.662342\pi$$
$$150$$ 0 0
$$151$$ −7.19114 −0.585207 −0.292603 0.956234i $$-0.594522\pi$$
−0.292603 + 0.956234i $$0.594522\pi$$
$$152$$ 0 0
$$153$$ −3.72913 −0.301482
$$154$$ 0 0
$$155$$ 1.45825 0.117130
$$156$$ 0 0
$$157$$ −4.31457 −0.344340 −0.172170 0.985067i $$-0.555078\pi$$
−0.172170 + 0.985067i $$0.555078\pi$$
$$158$$ 0 0
$$159$$ −6.43220 −0.510106
$$160$$ 0 0
$$161$$ 14.9720 1.17996
$$162$$ 0 0
$$163$$ −21.6436 −1.69526 −0.847628 0.530591i $$-0.821970\pi$$
−0.847628 + 0.530591i $$0.821970\pi$$
$$164$$ 0 0
$$165$$ 2.74301 0.213543
$$166$$ 0 0
$$167$$ 16.4303 1.27141 0.635707 0.771930i $$-0.280709\pi$$
0.635707 + 0.771930i $$0.280709\pi$$
$$168$$ 0 0
$$169$$ 13.4961 1.03816
$$170$$ 0 0
$$171$$ 0.404431 0.0309276
$$172$$ 0 0
$$173$$ −4.64939 −0.353487 −0.176743 0.984257i $$-0.556556\pi$$
−0.176743 + 0.984257i $$0.556556\pi$$
$$174$$ 0 0
$$175$$ 2.74301 0.207352
$$176$$ 0 0
$$177$$ 9.91822 0.745499
$$178$$ 0 0
$$179$$ −7.62334 −0.569795 −0.284897 0.958558i $$-0.591960\pi$$
−0.284897 + 0.958558i $$0.591960\pi$$
$$180$$ 0 0
$$181$$ −21.3050 −1.58359 −0.791794 0.610788i $$-0.790853\pi$$
−0.791794 + 0.610788i $$0.790853\pi$$
$$182$$ 0 0
$$183$$ −13.0816 −0.967019
$$184$$ 0 0
$$185$$ −6.76702 −0.497521
$$186$$ 0 0
$$187$$ 10.2290 0.748021
$$188$$ 0 0
$$189$$ 2.74301 0.199525
$$190$$ 0 0
$$191$$ −3.07597 −0.222570 −0.111285 0.993789i $$-0.535497\pi$$
−0.111285 + 0.993789i $$0.535497\pi$$
$$192$$ 0 0
$$193$$ 6.77454 0.487642 0.243821 0.969820i $$-0.421599\pi$$
0.243821 + 0.969820i $$0.421599\pi$$
$$194$$ 0 0
$$195$$ 5.14744 0.368616
$$196$$ 0 0
$$197$$ −13.6455 −0.972201 −0.486100 0.873903i $$-0.661581\pi$$
−0.486100 + 0.873903i $$0.661581\pi$$
$$198$$ 0 0
$$199$$ 6.33858 0.449330 0.224665 0.974436i $$-0.427871\pi$$
0.224665 + 0.974436i $$0.427871\pi$$
$$200$$ 0 0
$$201$$ −12.4961 −0.881410
$$202$$ 0 0
$$203$$ −2.74301 −0.192522
$$204$$ 0 0
$$205$$ −9.78090 −0.683128
$$206$$ 0 0
$$207$$ 5.45825 0.379375
$$208$$ 0 0
$$209$$ −1.10936 −0.0767358
$$210$$ 0 0
$$211$$ −2.00000 −0.137686 −0.0688428 0.997628i $$-0.521931\pi$$
−0.0688428 + 0.997628i $$0.521931\pi$$
$$212$$ 0 0
$$213$$ 11.3487 0.777600
$$214$$ 0 0
$$215$$ 4.43220 0.302273
$$216$$ 0 0
$$217$$ −4.00000 −0.271538
$$218$$ 0 0
$$219$$ −10.7670 −0.727568
$$220$$ 0 0
$$221$$ 19.1955 1.29123
$$222$$ 0 0
$$223$$ 2.20126 0.147407 0.0737037 0.997280i $$-0.476518\pi$$
0.0737037 + 0.997280i $$0.476518\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ −12.1853 −0.808769 −0.404384 0.914589i $$-0.632514\pi$$
−0.404384 + 0.914589i $$0.632514\pi$$
$$228$$ 0 0
$$229$$ −3.16337 −0.209041 −0.104521 0.994523i $$-0.533331\pi$$
−0.104521 + 0.994523i $$0.533331\pi$$
$$230$$ 0 0
$$231$$ −7.52410 −0.495050
$$232$$ 0 0
$$233$$ 23.8087 1.55976 0.779879 0.625930i $$-0.215281\pi$$
0.779879 + 0.625930i $$0.215281\pi$$
$$234$$ 0 0
$$235$$ 2.60569 0.169977
$$236$$ 0 0
$$237$$ −14.1576 −0.919633
$$238$$ 0 0
$$239$$ 6.02025 0.389417 0.194709 0.980861i $$-0.437624\pi$$
0.194709 + 0.980861i $$0.437624\pi$$
$$240$$ 0 0
$$241$$ 24.5517 1.58151 0.790756 0.612131i $$-0.209688\pi$$
0.790756 + 0.612131i $$0.209688\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ −0.524103 −0.0334837
$$246$$ 0 0
$$247$$ −2.08178 −0.132461
$$248$$ 0 0
$$249$$ −1.62334 −0.102875
$$250$$ 0 0
$$251$$ 27.5162 1.73681 0.868403 0.495858i $$-0.165146\pi$$
0.868403 + 0.495858i $$0.165146\pi$$
$$252$$ 0 0
$$253$$ −14.9720 −0.941284
$$254$$ 0 0
$$255$$ 3.72913 0.233527
$$256$$ 0 0
$$257$$ 15.6436 0.975820 0.487910 0.872894i $$-0.337759\pi$$
0.487910 + 0.872894i $$0.337759\pi$$
$$258$$ 0 0
$$259$$ 18.5620 1.15339
$$260$$ 0 0
$$261$$ −1.00000 −0.0618984
$$262$$ 0 0
$$263$$ 18.6175 1.14801 0.574003 0.818853i $$-0.305390\pi$$
0.574003 + 0.818853i $$0.305390\pi$$
$$264$$ 0 0
$$265$$ 6.43220 0.395127
$$266$$ 0 0
$$267$$ −8.87281 −0.543007
$$268$$ 0 0
$$269$$ 10.9006 0.664620 0.332310 0.943170i $$-0.392172\pi$$
0.332310 + 0.943170i $$0.392172\pi$$
$$270$$ 0 0
$$271$$ −17.7809 −1.08011 −0.540056 0.841629i $$-0.681597\pi$$
−0.540056 + 0.841629i $$0.681597\pi$$
$$272$$ 0 0
$$273$$ −14.1195 −0.854550
$$274$$ 0 0
$$275$$ −2.74301 −0.165410
$$276$$ 0 0
$$277$$ −20.6612 −1.24141 −0.620706 0.784043i $$-0.713154\pi$$
−0.620706 + 0.784043i $$0.713154\pi$$
$$278$$ 0 0
$$279$$ −1.45825 −0.0873033
$$280$$ 0 0
$$281$$ 28.2652 1.68616 0.843080 0.537787i $$-0.180740\pi$$
0.843080 + 0.537787i $$0.180740\pi$$
$$282$$ 0 0
$$283$$ 3.21139 0.190897 0.0954485 0.995434i $$-0.469571\pi$$
0.0954485 + 0.995434i $$0.469571\pi$$
$$284$$ 0 0
$$285$$ −0.404431 −0.0239564
$$286$$ 0 0
$$287$$ 26.8291 1.58367
$$288$$ 0 0
$$289$$ −3.09362 −0.181978
$$290$$ 0 0
$$291$$ 7.82084 0.458466
$$292$$ 0 0
$$293$$ −3.99624 −0.233463 −0.116731 0.993164i $$-0.537242\pi$$
−0.116731 + 0.993164i $$0.537242\pi$$
$$294$$ 0 0
$$295$$ −9.91822 −0.577461
$$296$$ 0 0
$$297$$ −2.74301 −0.159166
$$298$$ 0 0
$$299$$ −28.0960 −1.62484
$$300$$ 0 0
$$301$$ −12.1576 −0.700750
$$302$$ 0 0
$$303$$ −4.88033 −0.280367
$$304$$ 0 0
$$305$$ 13.0816 0.749050
$$306$$ 0 0
$$307$$ 12.0555 0.688046 0.344023 0.938961i $$-0.388210\pi$$
0.344023 + 0.938961i $$0.388210\pi$$
$$308$$ 0 0
$$309$$ 0.294881 0.0167752
$$310$$ 0 0
$$311$$ −15.6873 −0.889544 −0.444772 0.895644i $$-0.646715\pi$$
−0.444772 + 0.895644i $$0.646715\pi$$
$$312$$ 0 0
$$313$$ −33.8726 −1.91459 −0.957297 0.289107i $$-0.906641\pi$$
−0.957297 + 0.289107i $$0.906641\pi$$
$$314$$ 0 0
$$315$$ −2.74301 −0.154551
$$316$$ 0 0
$$317$$ 1.75689 0.0986770 0.0493385 0.998782i $$-0.484289\pi$$
0.0493385 + 0.998782i $$0.484289\pi$$
$$318$$ 0 0
$$319$$ 2.74301 0.153579
$$320$$ 0 0
$$321$$ 13.7809 0.769175
$$322$$ 0 0
$$323$$ −1.50817 −0.0839170
$$324$$ 0 0
$$325$$ −5.14744 −0.285529
$$326$$ 0 0
$$327$$ −6.20126 −0.342931
$$328$$ 0 0
$$329$$ −7.14744 −0.394051
$$330$$ 0 0
$$331$$ 22.4505 1.23399 0.616997 0.786966i $$-0.288349\pi$$
0.616997 + 0.786966i $$0.288349\pi$$
$$332$$ 0 0
$$333$$ 6.76702 0.370830
$$334$$ 0 0
$$335$$ 12.4961 0.682737
$$336$$ 0 0
$$337$$ 15.3886 0.838273 0.419136 0.907923i $$-0.362333\pi$$
0.419136 + 0.907923i $$0.362333\pi$$
$$338$$ 0 0
$$339$$ −10.5658 −0.573853
$$340$$ 0 0
$$341$$ 4.00000 0.216612
$$342$$ 0 0
$$343$$ −17.7634 −0.959136
$$344$$ 0 0
$$345$$ −5.45825 −0.293862
$$346$$ 0 0
$$347$$ 12.3504 0.663005 0.331503 0.943454i $$-0.392444\pi$$
0.331503 + 0.943454i $$0.392444\pi$$
$$348$$ 0 0
$$349$$ −1.94446 −0.104085 −0.0520424 0.998645i $$-0.516573\pi$$
−0.0520424 + 0.998645i $$0.516573\pi$$
$$350$$ 0 0
$$351$$ −5.14744 −0.274750
$$352$$ 0 0
$$353$$ 6.72517 0.357945 0.178972 0.983854i $$-0.442723\pi$$
0.178972 + 0.983854i $$0.442723\pi$$
$$354$$ 0 0
$$355$$ −11.3487 −0.602327
$$356$$ 0 0
$$357$$ −10.2290 −0.541378
$$358$$ 0 0
$$359$$ −15.2114 −0.802826 −0.401413 0.915897i $$-0.631481\pi$$
−0.401413 + 0.915897i $$0.631481\pi$$
$$360$$ 0 0
$$361$$ −18.8364 −0.991391
$$362$$ 0 0
$$363$$ −3.47590 −0.182437
$$364$$ 0 0
$$365$$ 10.7670 0.563571
$$366$$ 0 0
$$367$$ −13.2189 −0.690021 −0.345011 0.938599i $$-0.612125\pi$$
−0.345011 + 0.938599i $$0.612125\pi$$
$$368$$ 0 0
$$369$$ 9.78090 0.509173
$$370$$ 0 0
$$371$$ −17.6436 −0.916009
$$372$$ 0 0
$$373$$ 26.8050 1.38791 0.693956 0.720017i $$-0.255866\pi$$
0.693956 + 0.720017i $$0.255866\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ 0 0
$$377$$ 5.14744 0.265107
$$378$$ 0 0
$$379$$ 24.4228 1.25451 0.627257 0.778813i $$-0.284178\pi$$
0.627257 + 0.778813i $$0.284178\pi$$
$$380$$ 0 0
$$381$$ −8.54175 −0.437607
$$382$$ 0 0
$$383$$ −27.2372 −1.39176 −0.695879 0.718159i $$-0.744985\pi$$
−0.695879 + 0.718159i $$0.744985\pi$$
$$384$$ 0 0
$$385$$ 7.52410 0.383464
$$386$$ 0 0
$$387$$ −4.43220 −0.225301
$$388$$ 0 0
$$389$$ −12.1994 −0.618532 −0.309266 0.950976i $$-0.600083\pi$$
−0.309266 + 0.950976i $$0.600083\pi$$
$$390$$ 0 0
$$391$$ −20.3545 −1.02937
$$392$$ 0 0
$$393$$ −13.3050 −0.671149
$$394$$ 0 0
$$395$$ 14.1576 0.712344
$$396$$ 0 0
$$397$$ −17.7254 −0.889611 −0.444805 0.895627i $$-0.646727\pi$$
−0.444805 + 0.895627i $$0.646727\pi$$
$$398$$ 0 0
$$399$$ 1.10936 0.0555373
$$400$$ 0 0
$$401$$ −2.48773 −0.124231 −0.0621157 0.998069i $$-0.519785\pi$$
−0.0621157 + 0.998069i $$0.519785\pi$$
$$402$$ 0 0
$$403$$ 7.50627 0.373914
$$404$$ 0 0
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ −18.5620 −0.920084
$$408$$ 0 0
$$409$$ 37.7194 1.86510 0.932551 0.361038i $$-0.117577\pi$$
0.932551 + 0.361038i $$0.117577\pi$$
$$410$$ 0 0
$$411$$ −18.2253 −0.898986
$$412$$ 0 0
$$413$$ 27.2058 1.33871
$$414$$ 0 0
$$415$$ 1.62334 0.0796865
$$416$$ 0 0
$$417$$ 5.66142 0.277241
$$418$$ 0 0
$$419$$ 2.29488 0.112112 0.0560561 0.998428i $$-0.482147\pi$$
0.0560561 + 0.998428i $$0.482147\pi$$
$$420$$ 0 0
$$421$$ 35.9662 1.75289 0.876443 0.481505i $$-0.159910\pi$$
0.876443 + 0.481505i $$0.159910\pi$$
$$422$$ 0 0
$$423$$ −2.60569 −0.126693
$$424$$ 0 0
$$425$$ −3.72913 −0.180889
$$426$$ 0 0
$$427$$ −35.8829 −1.73650
$$428$$ 0 0
$$429$$ 14.1195 0.681695
$$430$$ 0 0
$$431$$ −22.6974 −1.09330 −0.546648 0.837363i $$-0.684096\pi$$
−0.546648 + 0.837363i $$0.684096\pi$$
$$432$$ 0 0
$$433$$ −11.4108 −0.548368 −0.274184 0.961677i $$-0.588408\pi$$
−0.274184 + 0.961677i $$0.588408\pi$$
$$434$$ 0 0
$$435$$ 1.00000 0.0479463
$$436$$ 0 0
$$437$$ 2.20748 0.105598
$$438$$ 0 0
$$439$$ 7.36654 0.351586 0.175793 0.984427i $$-0.443751\pi$$
0.175793 + 0.984427i $$0.443751\pi$$
$$440$$ 0 0
$$441$$ 0.524103 0.0249573
$$442$$ 0 0
$$443$$ −36.5423 −1.73617 −0.868087 0.496411i $$-0.834651\pi$$
−0.868087 + 0.496411i $$0.834651\pi$$
$$444$$ 0 0
$$445$$ 8.87281 0.420611
$$446$$ 0 0
$$447$$ −11.9182 −0.563712
$$448$$ 0 0
$$449$$ −39.6182 −1.86970 −0.934850 0.355043i $$-0.884466\pi$$
−0.934850 + 0.355043i $$0.884466\pi$$
$$450$$ 0 0
$$451$$ −26.8291 −1.26333
$$452$$ 0 0
$$453$$ −7.19114 −0.337869
$$454$$ 0 0
$$455$$ 14.1195 0.661931
$$456$$ 0 0
$$457$$ 28.5220 1.33420 0.667102 0.744967i $$-0.267535\pi$$
0.667102 + 0.744967i $$0.267535\pi$$
$$458$$ 0 0
$$459$$ −3.72913 −0.174061
$$460$$ 0 0
$$461$$ −22.6696 −1.05583 −0.527915 0.849297i $$-0.677026\pi$$
−0.527915 + 0.849297i $$0.677026\pi$$
$$462$$ 0 0
$$463$$ −28.5517 −1.32691 −0.663455 0.748217i $$-0.730910\pi$$
−0.663455 + 0.748217i $$0.730910\pi$$
$$464$$ 0 0
$$465$$ 1.45825 0.0676248
$$466$$ 0 0
$$467$$ 8.00000 0.370196 0.185098 0.982720i $$-0.440740\pi$$
0.185098 + 0.982720i $$0.440740\pi$$
$$468$$ 0 0
$$469$$ −34.2770 −1.58277
$$470$$ 0 0
$$471$$ −4.31457 −0.198805
$$472$$ 0 0
$$473$$ 12.1576 0.559005
$$474$$ 0 0
$$475$$ 0.404431 0.0185565
$$476$$ 0 0
$$477$$ −6.43220 −0.294510
$$478$$ 0 0
$$479$$ 4.43410 0.202599 0.101300 0.994856i $$-0.467700\pi$$
0.101300 + 0.994856i $$0.467700\pi$$
$$480$$ 0 0
$$481$$ −34.8328 −1.58824
$$482$$ 0 0
$$483$$ 14.9720 0.681251
$$484$$ 0 0
$$485$$ −7.82084 −0.355126
$$486$$ 0 0
$$487$$ −14.1035 −0.639093 −0.319546 0.947571i $$-0.603531\pi$$
−0.319546 + 0.947571i $$0.603531\pi$$
$$488$$ 0 0
$$489$$ −21.6436 −0.978757
$$490$$ 0 0
$$491$$ −39.6376 −1.78882 −0.894410 0.447249i $$-0.852404\pi$$
−0.894410 + 0.447249i $$0.852404\pi$$
$$492$$ 0 0
$$493$$ 3.72913 0.167951
$$494$$ 0 0
$$495$$ 2.74301 0.123289
$$496$$ 0 0
$$497$$ 31.1296 1.39635
$$498$$ 0 0
$$499$$ −6.33858 −0.283754 −0.141877 0.989884i $$-0.545314\pi$$
−0.141877 + 0.989884i $$0.545314\pi$$
$$500$$ 0 0
$$501$$ 16.4303 0.734051
$$502$$ 0 0
$$503$$ 19.0638 0.850011 0.425005 0.905191i $$-0.360272\pi$$
0.425005 + 0.905191i $$0.360272\pi$$
$$504$$ 0 0
$$505$$ 4.88033 0.217172
$$506$$ 0 0
$$507$$ 13.4961 0.599385
$$508$$ 0 0
$$509$$ −12.8145 −0.567992 −0.283996 0.958826i $$-0.591660\pi$$
−0.283996 + 0.958826i $$0.591660\pi$$
$$510$$ 0 0
$$511$$ −29.5340 −1.30651
$$512$$ 0 0
$$513$$ 0.404431 0.0178560
$$514$$ 0 0
$$515$$ −0.294881 −0.0129940
$$516$$ 0 0
$$517$$ 7.14744 0.314344
$$518$$ 0 0
$$519$$ −4.64939 −0.204086
$$520$$ 0 0
$$521$$ −35.4800 −1.55441 −0.777204 0.629249i $$-0.783363\pi$$
−0.777204 + 0.629249i $$0.783363\pi$$
$$522$$ 0 0
$$523$$ 14.9227 0.652525 0.326263 0.945279i $$-0.394211\pi$$
0.326263 + 0.945279i $$0.394211\pi$$
$$524$$ 0 0
$$525$$ 2.74301 0.119715
$$526$$ 0 0
$$527$$ 5.43801 0.236883
$$528$$ 0 0
$$529$$ 6.79252 0.295327
$$530$$ 0 0
$$531$$ 9.91822 0.430414
$$532$$ 0 0
$$533$$ −50.3466 −2.18075
$$534$$ 0 0
$$535$$ −13.7809 −0.595800
$$536$$ 0 0
$$537$$ −7.62334 −0.328971
$$538$$ 0 0
$$539$$ −1.43762 −0.0619227
$$540$$ 0 0
$$541$$ −22.8644 −0.983017 −0.491509 0.870873i $$-0.663554\pi$$
−0.491509 + 0.870873i $$0.663554\pi$$
$$542$$ 0 0
$$543$$ −21.3050 −0.914285
$$544$$ 0 0
$$545$$ 6.20126 0.265633
$$546$$ 0 0
$$547$$ 35.3290 1.51056 0.755279 0.655404i $$-0.227502\pi$$
0.755279 + 0.655404i $$0.227502\pi$$
$$548$$ 0 0
$$549$$ −13.0816 −0.558309
$$550$$ 0 0
$$551$$ −0.404431 −0.0172293
$$552$$ 0 0
$$553$$ −38.8343 −1.65140
$$554$$ 0 0
$$555$$ −6.76702 −0.287244
$$556$$ 0 0
$$557$$ 32.7994 1.38976 0.694878 0.719127i $$-0.255458\pi$$
0.694878 + 0.719127i $$0.255458\pi$$
$$558$$ 0 0
$$559$$ 22.8145 0.964950
$$560$$ 0 0
$$561$$ 10.2290 0.431870
$$562$$ 0 0
$$563$$ −16.8169 −0.708747 −0.354374 0.935104i $$-0.615306\pi$$
−0.354374 + 0.935104i $$0.615306\pi$$
$$564$$ 0 0
$$565$$ 10.5658 0.444505
$$566$$ 0 0
$$567$$ 2.74301 0.115196
$$568$$ 0 0
$$569$$ 8.88033 0.372283 0.186141 0.982523i $$-0.440402\pi$$
0.186141 + 0.982523i $$0.440402\pi$$
$$570$$ 0 0
$$571$$ −25.1240 −1.05141 −0.525703 0.850668i $$-0.676198\pi$$
−0.525703 + 0.850668i $$0.676198\pi$$
$$572$$ 0 0
$$573$$ −3.07597 −0.128501
$$574$$ 0 0
$$575$$ 5.45825 0.227625
$$576$$ 0 0
$$577$$ −13.9026 −0.578774 −0.289387 0.957212i $$-0.593451\pi$$
−0.289387 + 0.957212i $$0.593451\pi$$
$$578$$ 0 0
$$579$$ 6.77454 0.281540
$$580$$ 0 0
$$581$$ −4.45283 −0.184735
$$582$$ 0 0
$$583$$ 17.6436 0.730723
$$584$$ 0 0
$$585$$ 5.14744 0.212820
$$586$$ 0 0
$$587$$ 20.9942 0.866523 0.433262 0.901268i $$-0.357363\pi$$
0.433262 + 0.901268i $$0.357363\pi$$
$$588$$ 0 0
$$589$$ −0.589762 −0.0243007
$$590$$ 0 0
$$591$$ −13.6455 −0.561300
$$592$$ 0 0
$$593$$ 3.54003 0.145372 0.0726859 0.997355i $$-0.476843\pi$$
0.0726859 + 0.997355i $$0.476843\pi$$
$$594$$ 0 0
$$595$$ 10.2290 0.419349
$$596$$ 0 0
$$597$$ 6.33858 0.259421
$$598$$ 0 0
$$599$$ 32.4886 1.32745 0.663725 0.747977i $$-0.268975\pi$$
0.663725 + 0.747977i $$0.268975\pi$$
$$600$$ 0 0
$$601$$ −12.5620 −0.512414 −0.256207 0.966622i $$-0.582473\pi$$
−0.256207 + 0.966622i $$0.582473\pi$$
$$602$$ 0 0
$$603$$ −12.4961 −0.508882
$$604$$ 0 0
$$605$$ 3.47590 0.141315
$$606$$ 0 0
$$607$$ 0.369141 0.0149830 0.00749149 0.999972i $$-0.497615\pi$$
0.00749149 + 0.999972i $$0.497615\pi$$
$$608$$ 0 0
$$609$$ −2.74301 −0.111152
$$610$$ 0 0
$$611$$ 13.4126 0.542618
$$612$$ 0 0
$$613$$ 44.7017 1.80549 0.902743 0.430181i $$-0.141550\pi$$
0.902743 + 0.430181i $$0.141550\pi$$
$$614$$ 0 0
$$615$$ −9.78090 −0.394404
$$616$$ 0 0
$$617$$ −5.91014 −0.237933 −0.118967 0.992898i $$-0.537958\pi$$
−0.118967 + 0.992898i $$0.537958\pi$$
$$618$$ 0 0
$$619$$ 44.2187 1.77730 0.888650 0.458586i $$-0.151644\pi$$
0.888650 + 0.458586i $$0.151644\pi$$
$$620$$ 0 0
$$621$$ 5.45825 0.219032
$$622$$ 0 0
$$623$$ −24.3382 −0.975089
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −1.10936 −0.0443034
$$628$$ 0 0
$$629$$ −25.2351 −1.00619
$$630$$ 0 0
$$631$$ 29.5702 1.17717 0.588586 0.808435i $$-0.299685\pi$$
0.588586 + 0.808435i $$0.299685\pi$$
$$632$$ 0 0
$$633$$ −2.00000 −0.0794929
$$634$$ 0 0
$$635$$ 8.54175 0.338969
$$636$$ 0 0
$$637$$ −2.69779 −0.106890
$$638$$ 0 0
$$639$$ 11.3487 0.448948
$$640$$ 0 0
$$641$$ 26.6335 1.05196 0.525979 0.850497i $$-0.323699\pi$$
0.525979 + 0.850497i $$0.323699\pi$$
$$642$$ 0 0
$$643$$ −8.16597 −0.322035 −0.161017 0.986952i $$-0.551477\pi$$
−0.161017 + 0.986952i $$0.551477\pi$$
$$644$$ 0 0
$$645$$ 4.43220 0.174518
$$646$$ 0 0
$$647$$ −20.8381 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$648$$ 0 0
$$649$$ −27.2058 −1.06792
$$650$$ 0 0
$$651$$ −4.00000 −0.156772
$$652$$ 0 0
$$653$$ 13.4267 0.525428 0.262714 0.964874i $$-0.415382\pi$$
0.262714 + 0.964874i $$0.415382\pi$$
$$654$$ 0 0
$$655$$ 13.3050 0.519870
$$656$$ 0 0
$$657$$ −10.7670 −0.420061
$$658$$ 0 0
$$659$$ 42.9744 1.67405 0.837023 0.547167i $$-0.184294\pi$$
0.837023 + 0.547167i $$0.184294\pi$$
$$660$$ 0 0
$$661$$ −10.0178 −0.389649 −0.194824 0.980838i $$-0.562414\pi$$
−0.194824 + 0.980838i $$0.562414\pi$$
$$662$$ 0 0
$$663$$ 19.1955 0.745490
$$664$$ 0 0
$$665$$ −1.10936 −0.0430190
$$666$$ 0 0
$$667$$ −5.45825 −0.211344
$$668$$ 0 0
$$669$$ 2.20126 0.0851057
$$670$$ 0 0
$$671$$ 35.8829 1.38525
$$672$$ 0 0
$$673$$ 23.0878 0.889970 0.444985 0.895538i $$-0.353209\pi$$
0.444985 + 0.895538i $$0.353209\pi$$
$$674$$ 0 0
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ 4.95555 0.190457 0.0952287 0.995455i $$-0.469642\pi$$
0.0952287 + 0.995455i $$0.469642\pi$$
$$678$$ 0 0
$$679$$ 21.4526 0.823277
$$680$$ 0 0
$$681$$ −12.1853 −0.466943
$$682$$ 0 0
$$683$$ 36.8010 1.40815 0.704075 0.710126i $$-0.251362\pi$$
0.704075 + 0.710126i $$0.251362\pi$$
$$684$$ 0 0
$$685$$ 18.2253 0.696352
$$686$$ 0 0
$$687$$ −3.16337 −0.120690
$$688$$ 0 0
$$689$$ 33.1094 1.26137
$$690$$ 0 0
$$691$$ −15.8246 −0.601996 −0.300998 0.953625i $$-0.597320\pi$$
−0.300998 + 0.953625i $$0.597320\pi$$
$$692$$ 0 0
$$693$$ −7.52410 −0.285817
$$694$$ 0 0
$$695$$ −5.66142 −0.214750
$$696$$ 0 0
$$697$$ −36.4742 −1.38156
$$698$$ 0 0
$$699$$ 23.8087 0.900527
$$700$$ 0 0
$$701$$ −42.6156 −1.60957 −0.804785 0.593567i $$-0.797719\pi$$
−0.804785 + 0.593567i $$0.797719\pi$$
$$702$$ 0 0
$$703$$ 2.73679 0.103220
$$704$$ 0 0
$$705$$ 2.60569 0.0981361
$$706$$ 0 0
$$707$$ −13.3868 −0.503462
$$708$$ 0 0
$$709$$ 35.1795 1.32119 0.660597 0.750740i $$-0.270303\pi$$
0.660597 + 0.750740i $$0.270303\pi$$
$$710$$ 0 0
$$711$$ −14.1576 −0.530950
$$712$$ 0 0
$$713$$ −7.95951 −0.298086
$$714$$ 0 0
$$715$$ −14.1195 −0.528039
$$716$$ 0 0
$$717$$ 6.02025 0.224830
$$718$$ 0 0
$$719$$ 29.4563 1.09854 0.549268 0.835646i $$-0.314907\pi$$
0.549268 + 0.835646i $$0.314907\pi$$
$$720$$ 0 0
$$721$$ 0.808861 0.0301236
$$722$$ 0 0
$$723$$ 24.5517 0.913087
$$724$$ 0 0
$$725$$ −1.00000 −0.0371391
$$726$$ 0 0
$$727$$ −46.3391 −1.71862 −0.859311 0.511454i $$-0.829107\pi$$
−0.859311 + 0.511454i $$0.829107\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 16.5282 0.611319
$$732$$ 0 0
$$733$$ −3.73344 −0.137898 −0.0689489 0.997620i $$-0.521965\pi$$
−0.0689489 + 0.997620i $$0.521965\pi$$
$$734$$ 0 0
$$735$$ −0.524103 −0.0193318
$$736$$ 0 0
$$737$$ 34.2770 1.26261
$$738$$ 0 0
$$739$$ 6.48021 0.238378 0.119189 0.992872i $$-0.461970\pi$$
0.119189 + 0.992872i $$0.461970\pi$$
$$740$$ 0 0
$$741$$ −2.08178 −0.0764762
$$742$$ 0 0
$$743$$ 51.0638 1.87335 0.936674 0.350203i $$-0.113888\pi$$
0.936674 + 0.350203i $$0.113888\pi$$
$$744$$ 0 0
$$745$$ 11.9182 0.436650
$$746$$ 0 0
$$747$$ −1.62334 −0.0593948
$$748$$ 0 0
$$749$$ 37.8011 1.38122
$$750$$ 0 0
$$751$$ −24.6494 −0.899469 −0.449735 0.893162i $$-0.648481\pi$$
−0.449735 + 0.893162i $$0.648481\pi$$
$$752$$ 0 0
$$753$$ 27.5162 1.00275
$$754$$ 0 0
$$755$$ 7.19114 0.261712
$$756$$ 0 0
$$757$$ −38.6833 −1.40597 −0.702985 0.711205i $$-0.748150\pi$$
−0.702985 + 0.711205i $$0.748150\pi$$
$$758$$ 0 0
$$759$$ −14.9720 −0.543451
$$760$$ 0 0
$$761$$ −3.23725 −0.117350 −0.0586750 0.998277i $$-0.518688\pi$$
−0.0586750 + 0.998277i $$0.518688\pi$$
$$762$$ 0 0
$$763$$ −17.0101 −0.615808
$$764$$ 0 0
$$765$$ 3.72913 0.134827
$$766$$ 0 0
$$767$$ −51.0534 −1.84343
$$768$$ 0 0
$$769$$ 46.3166 1.67022 0.835111 0.550082i $$-0.185404\pi$$
0.835111 + 0.550082i $$0.185404\pi$$
$$770$$ 0 0
$$771$$ 15.6436 0.563390
$$772$$ 0 0
$$773$$ −27.0341 −0.972350 −0.486175 0.873861i $$-0.661608\pi$$
−0.486175 + 0.873861i $$0.661608\pi$$
$$774$$ 0 0
$$775$$ −1.45825 −0.0523820
$$776$$ 0 0
$$777$$ 18.5620 0.665908
$$778$$ 0 0
$$779$$ 3.95569 0.141727
$$780$$ 0 0
$$781$$ −31.1296 −1.11390
$$782$$ 0 0
$$783$$ −1.00000 −0.0357371
$$784$$ 0 0
$$785$$ 4.31457 0.153994
$$786$$ 0 0
$$787$$ 39.1870 1.39687 0.698434 0.715675i $$-0.253881\pi$$
0.698434 + 0.715675i $$0.253881\pi$$
$$788$$ 0 0
$$789$$ 18.6175 0.662802
$$790$$ 0 0
$$791$$ −28.9820 −1.03048
$$792$$ 0 0
$$793$$ 67.3367 2.39120
$$794$$ 0 0
$$795$$ 6.43220 0.228127
$$796$$ 0 0
$$797$$ −53.2933 −1.88775 −0.943873 0.330307i $$-0.892848\pi$$
−0.943873 + 0.330307i $$0.892848\pi$$
$$798$$ 0 0
$$799$$ 9.71696 0.343761
$$800$$ 0 0
$$801$$ −8.87281 −0.313505
$$802$$ 0 0
$$803$$ 29.5340 1.04223
$$804$$ 0 0
$$805$$ −14.9720 −0.527695
$$806$$ 0 0
$$807$$ 10.9006 0.383718
$$808$$ 0 0
$$809$$ 0.613214 0.0215595 0.0107797 0.999942i $$-0.496569\pi$$
0.0107797 + 0.999942i $$0.496569\pi$$
$$810$$ 0 0
$$811$$ 0.230936 0.00810927 0.00405463 0.999992i $$-0.498709\pi$$
0.00405463 + 0.999992i $$0.498709\pi$$
$$812$$ 0 0
$$813$$ −17.7809 −0.623603
$$814$$ 0 0
$$815$$ 21.6436 0.758142
$$816$$ 0 0
$$817$$ −1.79252 −0.0627122
$$818$$ 0 0
$$819$$ −14.1195 −0.493375
$$820$$ 0 0
$$821$$ 25.7254 0.897821 0.448911 0.893577i $$-0.351812\pi$$
0.448911 + 0.893577i $$0.351812\pi$$
$$822$$ 0 0
$$823$$ 38.2258 1.33247 0.666234 0.745743i $$-0.267905\pi$$
0.666234 + 0.745743i $$0.267905\pi$$
$$824$$ 0 0
$$825$$ −2.74301 −0.0954993
$$826$$ 0 0
$$827$$ −24.9199 −0.866551 −0.433275 0.901262i $$-0.642642\pi$$
−0.433275 + 0.901262i $$0.642642\pi$$
$$828$$ 0 0
$$829$$ −1.65301 −0.0574115 −0.0287057 0.999588i $$-0.509139\pi$$
−0.0287057 + 0.999588i $$0.509139\pi$$
$$830$$ 0 0
$$831$$ −20.6612 −0.716730
$$832$$ 0 0
$$833$$ −1.95445 −0.0677176
$$834$$ 0 0
$$835$$ −16.4303 −0.568594
$$836$$ 0 0
$$837$$ −1.45825 −0.0504046
$$838$$ 0 0
$$839$$ 31.4757 1.08666 0.543331 0.839519i $$-0.317163\pi$$
0.543331 + 0.839519i $$0.317163\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 0 0
$$843$$ 28.2652 0.973505
$$844$$ 0 0
$$845$$ −13.4961 −0.464281
$$846$$ 0 0
$$847$$ −9.53442 −0.327607
$$848$$ 0 0
$$849$$ 3.21139 0.110214
$$850$$ 0 0
$$851$$ 36.9361 1.26615
$$852$$ 0 0
$$853$$ −30.3753 −1.04003 −0.520015 0.854157i $$-0.674074\pi$$
−0.520015 + 0.854157i $$0.674074\pi$$
$$854$$ 0 0
$$855$$ −0.404431 −0.0138312
$$856$$ 0 0
$$857$$ −6.53423 −0.223205 −0.111602 0.993753i $$-0.535598\pi$$
−0.111602 + 0.993753i $$0.535598\pi$$
$$858$$ 0 0
$$859$$ −27.0220 −0.921977 −0.460989 0.887406i $$-0.652505\pi$$
−0.460989 + 0.887406i $$0.652505\pi$$
$$860$$ 0 0
$$861$$ 26.8291 0.914334
$$862$$ 0 0
$$863$$ −31.9587 −1.08789 −0.543944 0.839122i $$-0.683069\pi$$
−0.543944 + 0.839122i $$0.683069\pi$$
$$864$$ 0 0
$$865$$ 4.64939 0.158084
$$866$$ 0 0
$$867$$ −3.09362 −0.105065
$$868$$ 0 0
$$869$$ 38.8343 1.31736
$$870$$ 0 0
$$871$$ 64.3232 2.17951
$$872$$ 0 0
$$873$$ 7.82084 0.264695
$$874$$ 0 0
$$875$$ −2.74301 −0.0927307
$$876$$ 0 0
$$877$$ −50.0679 −1.69067 −0.845336 0.534235i $$-0.820600\pi$$
−0.845336 + 0.534235i $$0.820600\pi$$
$$878$$ 0 0
$$879$$ −3.99624 −0.134790
$$880$$ 0 0
$$881$$ −13.6817 −0.460947 −0.230474 0.973079i $$-0.574028\pi$$
−0.230474 + 0.973079i $$0.574028\pi$$
$$882$$ 0 0
$$883$$ −29.5693 −0.995087 −0.497543 0.867439i $$-0.665764\pi$$
−0.497543 + 0.867439i $$0.665764\pi$$
$$884$$ 0 0
$$885$$ −9.91822 −0.333397
$$886$$ 0 0
$$887$$ −4.35130 −0.146102 −0.0730512 0.997328i $$-0.523274\pi$$
−0.0730512 + 0.997328i $$0.523274\pi$$
$$888$$ 0 0
$$889$$ −23.4301 −0.785820
$$890$$ 0 0
$$891$$ −2.74301 −0.0918943
$$892$$ 0 0
$$893$$ −1.05382 −0.0352648
$$894$$ 0 0
$$895$$ 7.62334 0.254820
$$896$$ 0 0
$$897$$ −28.0960 −0.938099
$$898$$ 0 0
$$899$$ 1.45825 0.0486354
$$900$$ 0 0
$$901$$ 23.9865 0.799105
$$902$$ 0 0
$$903$$ −12.1576 −0.404578
$$904$$ 0 0
$$905$$ 21.3050 0.708202
$$906$$ 0 0
$$907$$ 52.3965 1.73980 0.869899 0.493230i $$-0.164184\pi$$
0.869899 + 0.493230i $$0.164184\pi$$
$$908$$ 0 0
$$909$$ −4.88033 −0.161870
$$910$$ 0 0
$$911$$ −7.85085 −0.260110 −0.130055 0.991507i $$-0.541515\pi$$
−0.130055 + 0.991507i $$0.541515\pi$$
$$912$$ 0 0
$$913$$ 4.45283 0.147367
$$914$$ 0 0
$$915$$ 13.0816 0.432464
$$916$$ 0 0
$$917$$ −36.4958 −1.20520
$$918$$ 0 0
$$919$$ −13.4979 −0.445253 −0.222627 0.974904i $$-0.571463\pi$$
−0.222627 + 0.974904i $$0.571463\pi$$
$$920$$ 0 0
$$921$$ 12.0555 0.397243
$$922$$ 0 0
$$923$$ −58.4168 −1.92281
$$924$$ 0 0
$$925$$ 6.76702 0.222498
$$926$$ 0 0
$$927$$ 0.294881 0.00968516
$$928$$ 0 0
$$929$$ 11.6177 0.381165 0.190583 0.981671i $$-0.438962\pi$$
0.190583 + 0.981671i $$0.438962\pi$$
$$930$$ 0 0
$$931$$ 0.211963 0.00694682
$$932$$ 0 0
$$933$$ −15.6873 −0.513579
$$934$$ 0 0
$$935$$ −10.2290 −0.334525
$$936$$ 0 0
$$937$$ −42.6740 −1.39410 −0.697049 0.717024i $$-0.745504\pi$$
−0.697049 + 0.717024i $$0.745504\pi$$
$$938$$ 0 0
$$939$$ −33.8726 −1.10539
$$940$$ 0 0
$$941$$ −5.91479 −0.192817 −0.0964083 0.995342i $$-0.530735\pi$$
−0.0964083 + 0.995342i $$0.530735\pi$$
$$942$$ 0 0
$$943$$ 53.3866 1.73851
$$944$$ 0 0
$$945$$ −2.74301 −0.0892301
$$946$$ 0 0
$$947$$ 45.0915 1.46528 0.732639 0.680618i $$-0.238289\pi$$
0.732639 + 0.680618i $$0.238289\pi$$
$$948$$ 0 0
$$949$$ 55.4226 1.79909
$$950$$ 0 0
$$951$$ 1.75689 0.0569712
$$952$$ 0 0
$$953$$ 30.3166 0.982053 0.491026 0.871145i $$-0.336622\pi$$
0.491026 + 0.871145i $$0.336622\pi$$
$$954$$ 0 0
$$955$$ 3.07597 0.0995362
$$956$$ 0 0
$$957$$ 2.74301 0.0886689
$$958$$ 0 0
$$959$$ −49.9921 −1.61433
$$960$$ 0 0
$$961$$ −28.8735 −0.931403
$$962$$ 0 0
$$963$$ 13.7809 0.444083
$$964$$ 0 0
$$965$$ −6.77454 −0.218080
$$966$$ 0 0
$$967$$ −5.99809 −0.192886 −0.0964428 0.995339i $$-0.530746\pi$$
−0.0964428 + 0.995339i $$0.530746\pi$$
$$968$$ 0 0
$$969$$ −1.50817 −0.0484495
$$970$$ 0 0
$$971$$ −28.5140 −0.915057 −0.457529 0.889195i $$-0.651265\pi$$
−0.457529 + 0.889195i $$0.651265\pi$$
$$972$$ 0 0
$$973$$ 15.5293 0.497848
$$974$$ 0 0
$$975$$ −5.14744 −0.164850
$$976$$ 0 0
$$977$$ −5.53813 −0.177180 −0.0885902 0.996068i $$-0.528236\pi$$
−0.0885902 + 0.996068i $$0.528236\pi$$
$$978$$ 0 0
$$979$$ 24.3382 0.777852
$$980$$ 0 0
$$981$$ −6.20126 −0.197991
$$982$$ 0 0
$$983$$ −7.37086 −0.235094 −0.117547 0.993067i $$-0.537503\pi$$
−0.117547 + 0.993067i $$0.537503\pi$$
$$984$$ 0 0
$$985$$ 13.6455 0.434781
$$986$$ 0 0
$$987$$ −7.14744 −0.227506
$$988$$ 0 0
$$989$$ −24.1921 −0.769263
$$990$$ 0 0
$$991$$ −3.68167 −0.116952 −0.0584760 0.998289i $$-0.518624\pi$$
−0.0584760 + 0.998289i $$0.518624\pi$$
$$992$$ 0 0
$$993$$ 22.4505 0.712446
$$994$$ 0 0
$$995$$ −6.33858 −0.200946
$$996$$ 0 0
$$997$$ 36.8747 1.16783 0.583916 0.811814i $$-0.301520\pi$$
0.583916 + 0.811814i $$0.301520\pi$$
$$998$$ 0 0
$$999$$ 6.76702 0.214099
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 6960.2.a.co.1.4 4
4.3 odd 2 435.2.a.j.1.4 4
12.11 even 2 1305.2.a.r.1.1 4
20.3 even 4 2175.2.c.n.349.3 8
20.7 even 4 2175.2.c.n.349.6 8
20.19 odd 2 2175.2.a.v.1.1 4
60.59 even 2 6525.2.a.bi.1.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.j.1.4 4 4.3 odd 2
1305.2.a.r.1.1 4 12.11 even 2
2175.2.a.v.1.1 4 20.19 odd 2
2175.2.c.n.349.3 8 20.3 even 4
2175.2.c.n.349.6 8 20.7 even 4
6525.2.a.bi.1.4 4 60.59 even 2
6960.2.a.co.1.4 4 1.1 even 1 trivial