# Properties

 Label 4560.2.a.bh.1.2 Level $4560$ Weight $2$ Character 4560.1 Self dual yes Analytic conductor $36.412$ 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] = [4560,2,Mod(1,4560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4560, 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("4560.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4560.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: $$36.4117833217$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 285) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 4560.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} +1.00000 q^{5} +2.73205 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} +1.00000 q^{5} +2.73205 q^{7} +1.00000 q^{9} -4.73205 q^{11} +0.732051 q^{13} -1.00000 q^{15} -1.00000 q^{19} -2.73205 q^{21} -3.46410 q^{23} +1.00000 q^{25} -1.00000 q^{27} +8.19615 q^{29} -8.92820 q^{31} +4.73205 q^{33} +2.73205 q^{35} -6.19615 q^{37} -0.732051 q^{39} +1.26795 q^{41} -4.19615 q^{43} +1.00000 q^{45} +3.46410 q^{47} +0.464102 q^{49} -9.46410 q^{53} -4.73205 q^{55} +1.00000 q^{57} -2.53590 q^{59} -6.53590 q^{61} +2.73205 q^{63} +0.732051 q^{65} -8.00000 q^{67} +3.46410 q^{69} +4.39230 q^{71} -16.9282 q^{73} -1.00000 q^{75} -12.9282 q^{77} +10.9282 q^{79} +1.00000 q^{81} +12.9282 q^{83} -8.19615 q^{87} +10.7321 q^{89} +2.00000 q^{91} +8.92820 q^{93} -1.00000 q^{95} -6.19615 q^{97} -4.73205 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 $$2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} - 6 q^{11} - 2 q^{13} - 2 q^{15} - 2 q^{19} - 2 q^{21} + 2 q^{25} - 2 q^{27} + 6 q^{29} - 4 q^{31} + 6 q^{33} + 2 q^{35} - 2 q^{37} + 2 q^{39} + 6 q^{41} + 2 q^{43} + 2 q^{45} - 6 q^{49} - 12 q^{53} - 6 q^{55} + 2 q^{57} - 12 q^{59} - 20 q^{61} + 2 q^{63} - 2 q^{65} - 16 q^{67} - 12 q^{71} - 20 q^{73} - 2 q^{75} - 12 q^{77} + 8 q^{79} + 2 q^{81} + 12 q^{83} - 6 q^{87} + 18 q^{89} + 4 q^{91} + 4 q^{93} - 2 q^{95} - 2 q^{97} - 6 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 - 6 * q^11 - 2 * q^13 - 2 * q^15 - 2 * q^19 - 2 * q^21 + 2 * q^25 - 2 * q^27 + 6 * q^29 - 4 * q^31 + 6 * q^33 + 2 * q^35 - 2 * q^37 + 2 * q^39 + 6 * q^41 + 2 * q^43 + 2 * q^45 - 6 * q^49 - 12 * q^53 - 6 * q^55 + 2 * q^57 - 12 * q^59 - 20 * q^61 + 2 * q^63 - 2 * q^65 - 16 * q^67 - 12 * q^71 - 20 * q^73 - 2 * q^75 - 12 * q^77 + 8 * q^79 + 2 * q^81 + 12 * q^83 - 6 * q^87 + 18 * q^89 + 4 * q^91 + 4 * q^93 - 2 * q^95 - 2 * q^97 - 6 * 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
$$7$$ 2.73205 1.03262 0.516309 0.856402i $$-0.327306\pi$$
0.516309 + 0.856402i $$0.327306\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −4.73205 −1.42677 −0.713384 0.700774i $$-0.752838\pi$$
−0.713384 + 0.700774i $$0.752838\pi$$
$$12$$ 0 0
$$13$$ 0.732051 0.203034 0.101517 0.994834i $$-0.467630\pi$$
0.101517 + 0.994834i $$0.467630\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 0 0
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −2.73205 −0.596182
$$22$$ 0 0
$$23$$ −3.46410 −0.722315 −0.361158 0.932505i $$-0.617618\pi$$
−0.361158 + 0.932505i $$0.617618\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 8.19615 1.52199 0.760994 0.648759i $$-0.224712\pi$$
0.760994 + 0.648759i $$0.224712\pi$$
$$30$$ 0 0
$$31$$ −8.92820 −1.60355 −0.801776 0.597624i $$-0.796111\pi$$
−0.801776 + 0.597624i $$0.796111\pi$$
$$32$$ 0 0
$$33$$ 4.73205 0.823744
$$34$$ 0 0
$$35$$ 2.73205 0.461801
$$36$$ 0 0
$$37$$ −6.19615 −1.01864 −0.509321 0.860577i $$-0.670103\pi$$
−0.509321 + 0.860577i $$0.670103\pi$$
$$38$$ 0 0
$$39$$ −0.732051 −0.117222
$$40$$ 0 0
$$41$$ 1.26795 0.198020 0.0990102 0.995086i $$-0.468432\pi$$
0.0990102 + 0.995086i $$0.468432\pi$$
$$42$$ 0 0
$$43$$ −4.19615 −0.639907 −0.319954 0.947433i $$-0.603667\pi$$
−0.319954 + 0.947433i $$0.603667\pi$$
$$44$$ 0 0
$$45$$ 1.00000 0.149071
$$46$$ 0 0
$$47$$ 3.46410 0.505291 0.252646 0.967559i $$-0.418699\pi$$
0.252646 + 0.967559i $$0.418699\pi$$
$$48$$ 0 0
$$49$$ 0.464102 0.0663002
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −9.46410 −1.29999 −0.649997 0.759937i $$-0.725230\pi$$
−0.649997 + 0.759937i $$0.725230\pi$$
$$54$$ 0 0
$$55$$ −4.73205 −0.638070
$$56$$ 0 0
$$57$$ 1.00000 0.132453
$$58$$ 0 0
$$59$$ −2.53590 −0.330146 −0.165073 0.986281i $$-0.552786\pi$$
−0.165073 + 0.986281i $$0.552786\pi$$
$$60$$ 0 0
$$61$$ −6.53590 −0.836836 −0.418418 0.908255i $$-0.637415\pi$$
−0.418418 + 0.908255i $$0.637415\pi$$
$$62$$ 0 0
$$63$$ 2.73205 0.344206
$$64$$ 0 0
$$65$$ 0.732051 0.0907997
$$66$$ 0 0
$$67$$ −8.00000 −0.977356 −0.488678 0.872464i $$-0.662521\pi$$
−0.488678 + 0.872464i $$0.662521\pi$$
$$68$$ 0 0
$$69$$ 3.46410 0.417029
$$70$$ 0 0
$$71$$ 4.39230 0.521271 0.260635 0.965437i $$-0.416068\pi$$
0.260635 + 0.965437i $$0.416068\pi$$
$$72$$ 0 0
$$73$$ −16.9282 −1.98130 −0.990648 0.136441i $$-0.956434\pi$$
−0.990648 + 0.136441i $$0.956434\pi$$
$$74$$ 0 0
$$75$$ −1.00000 −0.115470
$$76$$ 0 0
$$77$$ −12.9282 −1.47331
$$78$$ 0 0
$$79$$ 10.9282 1.22952 0.614759 0.788715i $$-0.289253\pi$$
0.614759 + 0.788715i $$0.289253\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 12.9282 1.41905 0.709527 0.704678i $$-0.248908\pi$$
0.709527 + 0.704678i $$0.248908\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −8.19615 −0.878720
$$88$$ 0 0
$$89$$ 10.7321 1.13760 0.568798 0.822478i $$-0.307409\pi$$
0.568798 + 0.822478i $$0.307409\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ 0 0
$$93$$ 8.92820 0.925812
$$94$$ 0 0
$$95$$ −1.00000 −0.102598
$$96$$ 0 0
$$97$$ −6.19615 −0.629124 −0.314562 0.949237i $$-0.601858\pi$$
−0.314562 + 0.949237i $$0.601858\pi$$
$$98$$ 0 0
$$99$$ −4.73205 −0.475589
$$100$$ 0 0
$$101$$ −10.3923 −1.03407 −0.517036 0.855963i $$-0.672965\pi$$
−0.517036 + 0.855963i $$0.672965\pi$$
$$102$$ 0 0
$$103$$ −9.85641 −0.971181 −0.485590 0.874187i $$-0.661395\pi$$
−0.485590 + 0.874187i $$0.661395\pi$$
$$104$$ 0 0
$$105$$ −2.73205 −0.266621
$$106$$ 0 0
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ −14.3923 −1.37853 −0.689266 0.724508i $$-0.742067\pi$$
−0.689266 + 0.724508i $$0.742067\pi$$
$$110$$ 0 0
$$111$$ 6.19615 0.588113
$$112$$ 0 0
$$113$$ 18.9282 1.78062 0.890308 0.455359i $$-0.150489\pi$$
0.890308 + 0.455359i $$0.150489\pi$$
$$114$$ 0 0
$$115$$ −3.46410 −0.323029
$$116$$ 0 0
$$117$$ 0.732051 0.0676781
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 11.3923 1.03566
$$122$$ 0 0
$$123$$ −1.26795 −0.114327
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 4.00000 0.354943 0.177471 0.984126i $$-0.443208\pi$$
0.177471 + 0.984126i $$0.443208\pi$$
$$128$$ 0 0
$$129$$ 4.19615 0.369451
$$130$$ 0 0
$$131$$ 9.12436 0.797199 0.398599 0.917125i $$-0.369496\pi$$
0.398599 + 0.917125i $$0.369496\pi$$
$$132$$ 0 0
$$133$$ −2.73205 −0.236899
$$134$$ 0 0
$$135$$ −1.00000 −0.0860663
$$136$$ 0 0
$$137$$ 19.8564 1.69645 0.848224 0.529638i $$-0.177672\pi$$
0.848224 + 0.529638i $$0.177672\pi$$
$$138$$ 0 0
$$139$$ 8.39230 0.711826 0.355913 0.934519i $$-0.384170\pi$$
0.355913 + 0.934519i $$0.384170\pi$$
$$140$$ 0 0
$$141$$ −3.46410 −0.291730
$$142$$ 0 0
$$143$$ −3.46410 −0.289683
$$144$$ 0 0
$$145$$ 8.19615 0.680653
$$146$$ 0 0
$$147$$ −0.464102 −0.0382785
$$148$$ 0 0
$$149$$ −19.8564 −1.62670 −0.813350 0.581775i $$-0.802359\pi$$
−0.813350 + 0.581775i $$0.802359\pi$$
$$150$$ 0 0
$$151$$ −14.0000 −1.13930 −0.569652 0.821886i $$-0.692922\pi$$
−0.569652 + 0.821886i $$0.692922\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −8.92820 −0.717131
$$156$$ 0 0
$$157$$ 6.39230 0.510161 0.255081 0.966920i $$-0.417898\pi$$
0.255081 + 0.966920i $$0.417898\pi$$
$$158$$ 0 0
$$159$$ 9.46410 0.750552
$$160$$ 0 0
$$161$$ −9.46410 −0.745876
$$162$$ 0 0
$$163$$ −9.26795 −0.725922 −0.362961 0.931804i $$-0.618234\pi$$
−0.362961 + 0.931804i $$0.618234\pi$$
$$164$$ 0 0
$$165$$ 4.73205 0.368390
$$166$$ 0 0
$$167$$ −3.46410 −0.268060 −0.134030 0.990977i $$-0.542792\pi$$
−0.134030 + 0.990977i $$0.542792\pi$$
$$168$$ 0 0
$$169$$ −12.4641 −0.958777
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ 0 0
$$173$$ 6.92820 0.526742 0.263371 0.964695i $$-0.415166\pi$$
0.263371 + 0.964695i $$0.415166\pi$$
$$174$$ 0 0
$$175$$ 2.73205 0.206524
$$176$$ 0 0
$$177$$ 2.53590 0.190610
$$178$$ 0 0
$$179$$ −23.3205 −1.74306 −0.871528 0.490345i $$-0.836871\pi$$
−0.871528 + 0.490345i $$0.836871\pi$$
$$180$$ 0 0
$$181$$ −2.39230 −0.177819 −0.0889093 0.996040i $$-0.528338\pi$$
−0.0889093 + 0.996040i $$0.528338\pi$$
$$182$$ 0 0
$$183$$ 6.53590 0.483148
$$184$$ 0 0
$$185$$ −6.19615 −0.455550
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −2.73205 −0.198727
$$190$$ 0 0
$$191$$ −0.339746 −0.0245832 −0.0122916 0.999924i $$-0.503913\pi$$
−0.0122916 + 0.999924i $$0.503913\pi$$
$$192$$ 0 0
$$193$$ 17.1244 1.23264 0.616319 0.787497i $$-0.288623\pi$$
0.616319 + 0.787497i $$0.288623\pi$$
$$194$$ 0 0
$$195$$ −0.732051 −0.0524232
$$196$$ 0 0
$$197$$ −24.0000 −1.70993 −0.854965 0.518686i $$-0.826421\pi$$
−0.854965 + 0.518686i $$0.826421\pi$$
$$198$$ 0 0
$$199$$ 15.3205 1.08604 0.543021 0.839719i $$-0.317280\pi$$
0.543021 + 0.839719i $$0.317280\pi$$
$$200$$ 0 0
$$201$$ 8.00000 0.564276
$$202$$ 0 0
$$203$$ 22.3923 1.57163
$$204$$ 0 0
$$205$$ 1.26795 0.0885574
$$206$$ 0 0
$$207$$ −3.46410 −0.240772
$$208$$ 0 0
$$209$$ 4.73205 0.327323
$$210$$ 0 0
$$211$$ −1.07180 −0.0737855 −0.0368928 0.999319i $$-0.511746\pi$$
−0.0368928 + 0.999319i $$0.511746\pi$$
$$212$$ 0 0
$$213$$ −4.39230 −0.300956
$$214$$ 0 0
$$215$$ −4.19615 −0.286175
$$216$$ 0 0
$$217$$ −24.3923 −1.65586
$$218$$ 0 0
$$219$$ 16.9282 1.14390
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 17.8564 1.19575 0.597877 0.801588i $$-0.296011\pi$$
0.597877 + 0.801588i $$0.296011\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ −10.3923 −0.689761 −0.344881 0.938647i $$-0.612081\pi$$
−0.344881 + 0.938647i $$0.612081\pi$$
$$228$$ 0 0
$$229$$ −18.5359 −1.22489 −0.612443 0.790515i $$-0.709813\pi$$
−0.612443 + 0.790515i $$0.709813\pi$$
$$230$$ 0 0
$$231$$ 12.9282 0.850613
$$232$$ 0 0
$$233$$ −7.85641 −0.514690 −0.257345 0.966320i $$-0.582848\pi$$
−0.257345 + 0.966320i $$0.582848\pi$$
$$234$$ 0 0
$$235$$ 3.46410 0.225973
$$236$$ 0 0
$$237$$ −10.9282 −0.709863
$$238$$ 0 0
$$239$$ 9.80385 0.634158 0.317079 0.948399i $$-0.397298\pi$$
0.317079 + 0.948399i $$0.397298\pi$$
$$240$$ 0 0
$$241$$ −3.07180 −0.197872 −0.0989359 0.995094i $$-0.531544\pi$$
−0.0989359 + 0.995094i $$0.531544\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 0.464102 0.0296504
$$246$$ 0 0
$$247$$ −0.732051 −0.0465793
$$248$$ 0 0
$$249$$ −12.9282 −0.819292
$$250$$ 0 0
$$251$$ −28.0526 −1.77066 −0.885331 0.464961i $$-0.846068\pi$$
−0.885331 + 0.464961i $$0.846068\pi$$
$$252$$ 0 0
$$253$$ 16.3923 1.03058
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −24.0000 −1.49708 −0.748539 0.663090i $$-0.769245\pi$$
−0.748539 + 0.663090i $$0.769245\pi$$
$$258$$ 0 0
$$259$$ −16.9282 −1.05187
$$260$$ 0 0
$$261$$ 8.19615 0.507329
$$262$$ 0 0
$$263$$ −6.00000 −0.369976 −0.184988 0.982741i $$-0.559225\pi$$
−0.184988 + 0.982741i $$0.559225\pi$$
$$264$$ 0 0
$$265$$ −9.46410 −0.581375
$$266$$ 0 0
$$267$$ −10.7321 −0.656791
$$268$$ 0 0
$$269$$ 0.588457 0.0358789 0.0179394 0.999839i $$-0.494289\pi$$
0.0179394 + 0.999839i $$0.494289\pi$$
$$270$$ 0 0
$$271$$ −0.392305 −0.0238308 −0.0119154 0.999929i $$-0.503793\pi$$
−0.0119154 + 0.999929i $$0.503793\pi$$
$$272$$ 0 0
$$273$$ −2.00000 −0.121046
$$274$$ 0 0
$$275$$ −4.73205 −0.285353
$$276$$ 0 0
$$277$$ 2.00000 0.120168 0.0600842 0.998193i $$-0.480863\pi$$
0.0600842 + 0.998193i $$0.480863\pi$$
$$278$$ 0 0
$$279$$ −8.92820 −0.534518
$$280$$ 0 0
$$281$$ 1.26795 0.0756395 0.0378198 0.999285i $$-0.487959\pi$$
0.0378198 + 0.999285i $$0.487959\pi$$
$$282$$ 0 0
$$283$$ −24.9808 −1.48495 −0.742476 0.669873i $$-0.766349\pi$$
−0.742476 + 0.669873i $$0.766349\pi$$
$$284$$ 0 0
$$285$$ 1.00000 0.0592349
$$286$$ 0 0
$$287$$ 3.46410 0.204479
$$288$$ 0 0
$$289$$ −17.0000 −1.00000
$$290$$ 0 0
$$291$$ 6.19615 0.363225
$$292$$ 0 0
$$293$$ 27.7128 1.61900 0.809500 0.587120i $$-0.199738\pi$$
0.809500 + 0.587120i $$0.199738\pi$$
$$294$$ 0 0
$$295$$ −2.53590 −0.147646
$$296$$ 0 0
$$297$$ 4.73205 0.274581
$$298$$ 0 0
$$299$$ −2.53590 −0.146655
$$300$$ 0 0
$$301$$ −11.4641 −0.660780
$$302$$ 0 0
$$303$$ 10.3923 0.597022
$$304$$ 0 0
$$305$$ −6.53590 −0.374244
$$306$$ 0 0
$$307$$ 32.3923 1.84873 0.924363 0.381514i $$-0.124597\pi$$
0.924363 + 0.381514i $$0.124597\pi$$
$$308$$ 0 0
$$309$$ 9.85641 0.560711
$$310$$ 0 0
$$311$$ −32.4449 −1.83978 −0.919890 0.392177i $$-0.871722\pi$$
−0.919890 + 0.392177i $$0.871722\pi$$
$$312$$ 0 0
$$313$$ 6.39230 0.361314 0.180657 0.983546i $$-0.442178\pi$$
0.180657 + 0.983546i $$0.442178\pi$$
$$314$$ 0 0
$$315$$ 2.73205 0.153934
$$316$$ 0 0
$$317$$ −11.3205 −0.635823 −0.317912 0.948120i $$-0.602982\pi$$
−0.317912 + 0.948120i $$0.602982\pi$$
$$318$$ 0 0
$$319$$ −38.7846 −2.17152
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0.732051 0.0406069
$$326$$ 0 0
$$327$$ 14.3923 0.795896
$$328$$ 0 0
$$329$$ 9.46410 0.521773
$$330$$ 0 0
$$331$$ 25.7128 1.41330 0.706652 0.707561i $$-0.250205\pi$$
0.706652 + 0.707561i $$0.250205\pi$$
$$332$$ 0 0
$$333$$ −6.19615 −0.339547
$$334$$ 0 0
$$335$$ −8.00000 −0.437087
$$336$$ 0 0
$$337$$ 5.12436 0.279141 0.139571 0.990212i $$-0.455428\pi$$
0.139571 + 0.990212i $$0.455428\pi$$
$$338$$ 0 0
$$339$$ −18.9282 −1.02804
$$340$$ 0 0
$$341$$ 42.2487 2.28790
$$342$$ 0 0
$$343$$ −17.8564 −0.964155
$$344$$ 0 0
$$345$$ 3.46410 0.186501
$$346$$ 0 0
$$347$$ 0.928203 0.0498286 0.0249143 0.999690i $$-0.492069\pi$$
0.0249143 + 0.999690i $$0.492069\pi$$
$$348$$ 0 0
$$349$$ −22.0000 −1.17763 −0.588817 0.808267i $$-0.700406\pi$$
−0.588817 + 0.808267i $$0.700406\pi$$
$$350$$ 0 0
$$351$$ −0.732051 −0.0390740
$$352$$ 0 0
$$353$$ −14.7846 −0.786905 −0.393453 0.919345i $$-0.628719\pi$$
−0.393453 + 0.919345i $$0.628719\pi$$
$$354$$ 0 0
$$355$$ 4.39230 0.233119
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −0.339746 −0.0179311 −0.00896555 0.999960i $$-0.502854\pi$$
−0.00896555 + 0.999960i $$0.502854\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −11.3923 −0.597941
$$364$$ 0 0
$$365$$ −16.9282 −0.886063
$$366$$ 0 0
$$367$$ −16.1962 −0.845432 −0.422716 0.906262i $$-0.638923\pi$$
−0.422716 + 0.906262i $$0.638923\pi$$
$$368$$ 0 0
$$369$$ 1.26795 0.0660068
$$370$$ 0 0
$$371$$ −25.8564 −1.34240
$$372$$ 0 0
$$373$$ −6.19615 −0.320825 −0.160412 0.987050i $$-0.551282\pi$$
−0.160412 + 0.987050i $$0.551282\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ 0 0
$$377$$ 6.00000 0.309016
$$378$$ 0 0
$$379$$ −20.9282 −1.07501 −0.537505 0.843261i $$-0.680633\pi$$
−0.537505 + 0.843261i $$0.680633\pi$$
$$380$$ 0 0
$$381$$ −4.00000 −0.204926
$$382$$ 0 0
$$383$$ 17.0718 0.872328 0.436164 0.899867i $$-0.356337\pi$$
0.436164 + 0.899867i $$0.356337\pi$$
$$384$$ 0 0
$$385$$ −12.9282 −0.658882
$$386$$ 0 0
$$387$$ −4.19615 −0.213302
$$388$$ 0 0
$$389$$ 7.85641 0.398336 0.199168 0.979965i $$-0.436176\pi$$
0.199168 + 0.979965i $$0.436176\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ −9.12436 −0.460263
$$394$$ 0 0
$$395$$ 10.9282 0.549858
$$396$$ 0 0
$$397$$ 8.92820 0.448094 0.224047 0.974578i $$-0.428073\pi$$
0.224047 + 0.974578i $$0.428073\pi$$
$$398$$ 0 0
$$399$$ 2.73205 0.136774
$$400$$ 0 0
$$401$$ −34.0526 −1.70050 −0.850252 0.526376i $$-0.823550\pi$$
−0.850252 + 0.526376i $$0.823550\pi$$
$$402$$ 0 0
$$403$$ −6.53590 −0.325576
$$404$$ 0 0
$$405$$ 1.00000 0.0496904
$$406$$ 0 0
$$407$$ 29.3205 1.45336
$$408$$ 0 0
$$409$$ −26.3923 −1.30502 −0.652508 0.757782i $$-0.726283\pi$$
−0.652508 + 0.757782i $$0.726283\pi$$
$$410$$ 0 0
$$411$$ −19.8564 −0.979444
$$412$$ 0 0
$$413$$ −6.92820 −0.340915
$$414$$ 0 0
$$415$$ 12.9282 0.634621
$$416$$ 0 0
$$417$$ −8.39230 −0.410973
$$418$$ 0 0
$$419$$ 28.0526 1.37046 0.685229 0.728328i $$-0.259702\pi$$
0.685229 + 0.728328i $$0.259702\pi$$
$$420$$ 0 0
$$421$$ −18.7846 −0.915506 −0.457753 0.889079i $$-0.651346\pi$$
−0.457753 + 0.889079i $$0.651346\pi$$
$$422$$ 0 0
$$423$$ 3.46410 0.168430
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −17.8564 −0.864132
$$428$$ 0 0
$$429$$ 3.46410 0.167248
$$430$$ 0 0
$$431$$ 11.3205 0.545290 0.272645 0.962115i $$-0.412102\pi$$
0.272645 + 0.962115i $$0.412102\pi$$
$$432$$ 0 0
$$433$$ −10.5885 −0.508849 −0.254424 0.967093i $$-0.581886\pi$$
−0.254424 + 0.967093i $$0.581886\pi$$
$$434$$ 0 0
$$435$$ −8.19615 −0.392975
$$436$$ 0 0
$$437$$ 3.46410 0.165710
$$438$$ 0 0
$$439$$ −26.9282 −1.28521 −0.642607 0.766196i $$-0.722147\pi$$
−0.642607 + 0.766196i $$0.722147\pi$$
$$440$$ 0 0
$$441$$ 0.464102 0.0221001
$$442$$ 0 0
$$443$$ 5.32051 0.252785 0.126392 0.991980i $$-0.459660\pi$$
0.126392 + 0.991980i $$0.459660\pi$$
$$444$$ 0 0
$$445$$ 10.7321 0.508748
$$446$$ 0 0
$$447$$ 19.8564 0.939176
$$448$$ 0 0
$$449$$ −5.66025 −0.267124 −0.133562 0.991040i $$-0.542642\pi$$
−0.133562 + 0.991040i $$0.542642\pi$$
$$450$$ 0 0
$$451$$ −6.00000 −0.282529
$$452$$ 0 0
$$453$$ 14.0000 0.657777
$$454$$ 0 0
$$455$$ 2.00000 0.0937614
$$456$$ 0 0
$$457$$ 4.53590 0.212180 0.106090 0.994357i $$-0.466167\pi$$
0.106090 + 0.994357i $$0.466167\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 6.00000 0.279448 0.139724 0.990190i $$-0.455378\pi$$
0.139724 + 0.990190i $$0.455378\pi$$
$$462$$ 0 0
$$463$$ 35.5167 1.65060 0.825300 0.564695i $$-0.191006\pi$$
0.825300 + 0.564695i $$0.191006\pi$$
$$464$$ 0 0
$$465$$ 8.92820 0.414036
$$466$$ 0 0
$$467$$ −20.5359 −0.950288 −0.475144 0.879908i $$-0.657604\pi$$
−0.475144 + 0.879908i $$0.657604\pi$$
$$468$$ 0 0
$$469$$ −21.8564 −1.00924
$$470$$ 0 0
$$471$$ −6.39230 −0.294542
$$472$$ 0 0
$$473$$ 19.8564 0.912999
$$474$$ 0 0
$$475$$ −1.00000 −0.0458831
$$476$$ 0 0
$$477$$ −9.46410 −0.433331
$$478$$ 0 0
$$479$$ −25.5167 −1.16589 −0.582943 0.812513i $$-0.698099\pi$$
−0.582943 + 0.812513i $$0.698099\pi$$
$$480$$ 0 0
$$481$$ −4.53590 −0.206819
$$482$$ 0 0
$$483$$ 9.46410 0.430632
$$484$$ 0 0
$$485$$ −6.19615 −0.281353
$$486$$ 0 0
$$487$$ 32.3923 1.46784 0.733918 0.679238i $$-0.237690\pi$$
0.733918 + 0.679238i $$0.237690\pi$$
$$488$$ 0 0
$$489$$ 9.26795 0.419111
$$490$$ 0 0
$$491$$ −16.0526 −0.724442 −0.362221 0.932092i $$-0.617981\pi$$
−0.362221 + 0.932092i $$0.617981\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ −4.73205 −0.212690
$$496$$ 0 0
$$497$$ 12.0000 0.538274
$$498$$ 0 0
$$499$$ −10.5359 −0.471652 −0.235826 0.971795i $$-0.575779\pi$$
−0.235826 + 0.971795i $$0.575779\pi$$
$$500$$ 0 0
$$501$$ 3.46410 0.154765
$$502$$ 0 0
$$503$$ 23.0718 1.02872 0.514360 0.857574i $$-0.328029\pi$$
0.514360 + 0.857574i $$0.328029\pi$$
$$504$$ 0 0
$$505$$ −10.3923 −0.462451
$$506$$ 0 0
$$507$$ 12.4641 0.553550
$$508$$ 0 0
$$509$$ −10.0526 −0.445572 −0.222786 0.974867i $$-0.571515\pi$$
−0.222786 + 0.974867i $$0.571515\pi$$
$$510$$ 0 0
$$511$$ −46.2487 −2.04592
$$512$$ 0 0
$$513$$ 1.00000 0.0441511
$$514$$ 0 0
$$515$$ −9.85641 −0.434325
$$516$$ 0 0
$$517$$ −16.3923 −0.720933
$$518$$ 0 0
$$519$$ −6.92820 −0.304114
$$520$$ 0 0
$$521$$ 37.2679 1.63274 0.816369 0.577530i $$-0.195983\pi$$
0.816369 + 0.577530i $$0.195983\pi$$
$$522$$ 0 0
$$523$$ −8.67949 −0.379528 −0.189764 0.981830i $$-0.560772\pi$$
−0.189764 + 0.981830i $$0.560772\pi$$
$$524$$ 0 0
$$525$$ −2.73205 −0.119236
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −11.0000 −0.478261
$$530$$ 0 0
$$531$$ −2.53590 −0.110049
$$532$$ 0 0
$$533$$ 0.928203 0.0402049
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 23.3205 1.00635
$$538$$ 0 0
$$539$$ −2.19615 −0.0945950
$$540$$ 0 0
$$541$$ 41.7128 1.79337 0.896687 0.442665i $$-0.145967\pi$$
0.896687 + 0.442665i $$0.145967\pi$$
$$542$$ 0 0
$$543$$ 2.39230 0.102664
$$544$$ 0 0
$$545$$ −14.3923 −0.616499
$$546$$ 0 0
$$547$$ −43.3205 −1.85225 −0.926126 0.377215i $$-0.876882\pi$$
−0.926126 + 0.377215i $$0.876882\pi$$
$$548$$ 0 0
$$549$$ −6.53590 −0.278945
$$550$$ 0 0
$$551$$ −8.19615 −0.349168
$$552$$ 0 0
$$553$$ 29.8564 1.26962
$$554$$ 0 0
$$555$$ 6.19615 0.263012
$$556$$ 0 0
$$557$$ 0.928203 0.0393292 0.0196646 0.999807i $$-0.493740\pi$$
0.0196646 + 0.999807i $$0.493740\pi$$
$$558$$ 0 0
$$559$$ −3.07180 −0.129923
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 27.4641 1.15747 0.578737 0.815514i $$-0.303546\pi$$
0.578737 + 0.815514i $$0.303546\pi$$
$$564$$ 0 0
$$565$$ 18.9282 0.796315
$$566$$ 0 0
$$567$$ 2.73205 0.114735
$$568$$ 0 0
$$569$$ 22.0526 0.924491 0.462246 0.886752i $$-0.347044\pi$$
0.462246 + 0.886752i $$0.347044\pi$$
$$570$$ 0 0
$$571$$ 34.2487 1.43326 0.716632 0.697452i $$-0.245683\pi$$
0.716632 + 0.697452i $$0.245683\pi$$
$$572$$ 0 0
$$573$$ 0.339746 0.0141931
$$574$$ 0 0
$$575$$ −3.46410 −0.144463
$$576$$ 0 0
$$577$$ 15.1769 0.631823 0.315912 0.948789i $$-0.397690\pi$$
0.315912 + 0.948789i $$0.397690\pi$$
$$578$$ 0 0
$$579$$ −17.1244 −0.711664
$$580$$ 0 0
$$581$$ 35.3205 1.46534
$$582$$ 0 0
$$583$$ 44.7846 1.85479
$$584$$ 0 0
$$585$$ 0.732051 0.0302666
$$586$$ 0 0
$$587$$ 3.46410 0.142979 0.0714894 0.997441i $$-0.477225\pi$$
0.0714894 + 0.997441i $$0.477225\pi$$
$$588$$ 0 0
$$589$$ 8.92820 0.367880
$$590$$ 0 0
$$591$$ 24.0000 0.987228
$$592$$ 0 0
$$593$$ −38.7846 −1.59269 −0.796347 0.604841i $$-0.793237\pi$$
−0.796347 + 0.604841i $$0.793237\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −15.3205 −0.627027
$$598$$ 0 0
$$599$$ 13.8564 0.566157 0.283079 0.959097i $$-0.408644\pi$$
0.283079 + 0.959097i $$0.408644\pi$$
$$600$$ 0 0
$$601$$ −47.1769 −1.92439 −0.962193 0.272368i $$-0.912193\pi$$
−0.962193 + 0.272368i $$0.912193\pi$$
$$602$$ 0 0
$$603$$ −8.00000 −0.325785
$$604$$ 0 0
$$605$$ 11.3923 0.463163
$$606$$ 0 0
$$607$$ 11.6077 0.471142 0.235571 0.971857i $$-0.424304\pi$$
0.235571 + 0.971857i $$0.424304\pi$$
$$608$$ 0 0
$$609$$ −22.3923 −0.907382
$$610$$ 0 0
$$611$$ 2.53590 0.102591
$$612$$ 0 0
$$613$$ 42.3923 1.71221 0.856105 0.516803i $$-0.172878\pi$$
0.856105 + 0.516803i $$0.172878\pi$$
$$614$$ 0 0
$$615$$ −1.26795 −0.0511286
$$616$$ 0 0
$$617$$ 27.7128 1.11568 0.557838 0.829950i $$-0.311631\pi$$
0.557838 + 0.829950i $$0.311631\pi$$
$$618$$ 0 0
$$619$$ 15.3205 0.615783 0.307892 0.951421i $$-0.400377\pi$$
0.307892 + 0.951421i $$0.400377\pi$$
$$620$$ 0 0
$$621$$ 3.46410 0.139010
$$622$$ 0 0
$$623$$ 29.3205 1.17470
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −4.73205 −0.188980
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 34.9282 1.39047 0.695235 0.718783i $$-0.255300\pi$$
0.695235 + 0.718783i $$0.255300\pi$$
$$632$$ 0 0
$$633$$ 1.07180 0.0426001
$$634$$ 0 0
$$635$$ 4.00000 0.158735
$$636$$ 0 0
$$637$$ 0.339746 0.0134612
$$638$$ 0 0
$$639$$ 4.39230 0.173757
$$640$$ 0 0
$$641$$ 48.5885 1.91913 0.959564 0.281489i $$-0.0908284\pi$$
0.959564 + 0.281489i $$0.0908284\pi$$
$$642$$ 0 0
$$643$$ 12.1962 0.480969 0.240485 0.970653i $$-0.422694\pi$$
0.240485 + 0.970653i $$0.422694\pi$$
$$644$$ 0 0
$$645$$ 4.19615 0.165223
$$646$$ 0 0
$$647$$ 4.14359 0.162901 0.0814507 0.996677i $$-0.474045\pi$$
0.0814507 + 0.996677i $$0.474045\pi$$
$$648$$ 0 0
$$649$$ 12.0000 0.471041
$$650$$ 0 0
$$651$$ 24.3923 0.956010
$$652$$ 0 0
$$653$$ 17.0718 0.668071 0.334036 0.942560i $$-0.391589\pi$$
0.334036 + 0.942560i $$0.391589\pi$$
$$654$$ 0 0
$$655$$ 9.12436 0.356518
$$656$$ 0 0
$$657$$ −16.9282 −0.660432
$$658$$ 0 0
$$659$$ −5.07180 −0.197569 −0.0987846 0.995109i $$-0.531495\pi$$
−0.0987846 + 0.995109i $$0.531495\pi$$
$$660$$ 0 0
$$661$$ 39.1769 1.52381 0.761903 0.647692i $$-0.224265\pi$$
0.761903 + 0.647692i $$0.224265\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −2.73205 −0.105944
$$666$$ 0 0
$$667$$ −28.3923 −1.09935
$$668$$ 0 0
$$669$$ −17.8564 −0.690369
$$670$$ 0 0
$$671$$ 30.9282 1.19397
$$672$$ 0 0
$$673$$ 17.1244 0.660095 0.330048 0.943964i $$-0.392935\pi$$
0.330048 + 0.943964i $$0.392935\pi$$
$$674$$ 0 0
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ 0.679492 0.0261150 0.0130575 0.999915i $$-0.495844\pi$$
0.0130575 + 0.999915i $$0.495844\pi$$
$$678$$ 0 0
$$679$$ −16.9282 −0.649645
$$680$$ 0 0
$$681$$ 10.3923 0.398234
$$682$$ 0 0
$$683$$ 5.07180 0.194067 0.0970335 0.995281i $$-0.469065\pi$$
0.0970335 + 0.995281i $$0.469065\pi$$
$$684$$ 0 0
$$685$$ 19.8564 0.758674
$$686$$ 0 0
$$687$$ 18.5359 0.707189
$$688$$ 0 0
$$689$$ −6.92820 −0.263944
$$690$$ 0 0
$$691$$ −12.3923 −0.471425 −0.235713 0.971823i $$-0.575742\pi$$
−0.235713 + 0.971823i $$0.575742\pi$$
$$692$$ 0 0
$$693$$ −12.9282 −0.491102
$$694$$ 0 0
$$695$$ 8.39230 0.318338
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 7.85641 0.297157
$$700$$ 0 0
$$701$$ −33.7128 −1.27332 −0.636658 0.771147i $$-0.719684\pi$$
−0.636658 + 0.771147i $$0.719684\pi$$
$$702$$ 0 0
$$703$$ 6.19615 0.233692
$$704$$ 0 0
$$705$$ −3.46410 −0.130466
$$706$$ 0 0
$$707$$ −28.3923 −1.06780
$$708$$ 0 0
$$709$$ −29.1769 −1.09576 −0.547881 0.836556i $$-0.684565\pi$$
−0.547881 + 0.836556i $$0.684565\pi$$
$$710$$ 0 0
$$711$$ 10.9282 0.409840
$$712$$ 0 0
$$713$$ 30.9282 1.15827
$$714$$ 0 0
$$715$$ −3.46410 −0.129550
$$716$$ 0 0
$$717$$ −9.80385 −0.366131
$$718$$ 0 0
$$719$$ 11.6603 0.434854 0.217427 0.976077i $$-0.430234\pi$$
0.217427 + 0.976077i $$0.430234\pi$$
$$720$$ 0 0
$$721$$ −26.9282 −1.00286
$$722$$ 0 0
$$723$$ 3.07180 0.114241
$$724$$ 0 0
$$725$$ 8.19615 0.304397
$$726$$ 0 0
$$727$$ −25.6603 −0.951686 −0.475843 0.879530i $$-0.657857\pi$$
−0.475843 + 0.879530i $$0.657857\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −18.7846 −0.693825 −0.346913 0.937897i $$-0.612770\pi$$
−0.346913 + 0.937897i $$0.612770\pi$$
$$734$$ 0 0
$$735$$ −0.464102 −0.0171186
$$736$$ 0 0
$$737$$ 37.8564 1.39446
$$738$$ 0 0
$$739$$ −6.14359 −0.225996 −0.112998 0.993595i $$-0.536045\pi$$
−0.112998 + 0.993595i $$0.536045\pi$$
$$740$$ 0 0
$$741$$ 0.732051 0.0268926
$$742$$ 0 0
$$743$$ 3.21539 0.117961 0.0589806 0.998259i $$-0.481215\pi$$
0.0589806 + 0.998259i $$0.481215\pi$$
$$744$$ 0 0
$$745$$ −19.8564 −0.727482
$$746$$ 0 0
$$747$$ 12.9282 0.473018
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −26.0000 −0.948753 −0.474377 0.880322i $$-0.657327\pi$$
−0.474377 + 0.880322i $$0.657327\pi$$
$$752$$ 0 0
$$753$$ 28.0526 1.02229
$$754$$ 0 0
$$755$$ −14.0000 −0.509512
$$756$$ 0 0
$$757$$ 32.2487 1.17210 0.586050 0.810275i $$-0.300682\pi$$
0.586050 + 0.810275i $$0.300682\pi$$
$$758$$ 0 0
$$759$$ −16.3923 −0.595003
$$760$$ 0 0
$$761$$ 6.00000 0.217500 0.108750 0.994069i $$-0.465315\pi$$
0.108750 + 0.994069i $$0.465315\pi$$
$$762$$ 0 0
$$763$$ −39.3205 −1.42350
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −1.85641 −0.0670310
$$768$$ 0 0
$$769$$ −20.6410 −0.744334 −0.372167 0.928166i $$-0.621385\pi$$
−0.372167 + 0.928166i $$0.621385\pi$$
$$770$$ 0 0
$$771$$ 24.0000 0.864339
$$772$$ 0 0
$$773$$ −25.1769 −0.905551 −0.452775 0.891625i $$-0.649566\pi$$
−0.452775 + 0.891625i $$0.649566\pi$$
$$774$$ 0 0
$$775$$ −8.92820 −0.320711
$$776$$ 0 0
$$777$$ 16.9282 0.607296
$$778$$ 0 0
$$779$$ −1.26795 −0.0454290
$$780$$ 0 0
$$781$$ −20.7846 −0.743732
$$782$$ 0 0
$$783$$ −8.19615 −0.292907
$$784$$ 0 0
$$785$$ 6.39230 0.228151
$$786$$ 0 0
$$787$$ −8.67949 −0.309390 −0.154695 0.987962i $$-0.549440\pi$$
−0.154695 + 0.987962i $$0.549440\pi$$
$$788$$ 0 0
$$789$$ 6.00000 0.213606
$$790$$ 0 0
$$791$$ 51.7128 1.83870
$$792$$ 0 0
$$793$$ −4.78461 −0.169906
$$794$$ 0 0
$$795$$ 9.46410 0.335657
$$796$$ 0 0
$$797$$ 44.7846 1.58635 0.793176 0.608992i $$-0.208426\pi$$
0.793176 + 0.608992i $$0.208426\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 10.7321 0.379198
$$802$$ 0 0
$$803$$ 80.1051 2.82685
$$804$$ 0 0
$$805$$ −9.46410 −0.333566
$$806$$ 0 0
$$807$$ −0.588457 −0.0207147
$$808$$ 0 0
$$809$$ 14.7846 0.519799 0.259900 0.965636i $$-0.416311\pi$$
0.259900 + 0.965636i $$0.416311\pi$$
$$810$$ 0 0
$$811$$ −37.5692 −1.31923 −0.659617 0.751602i $$-0.729281\pi$$
−0.659617 + 0.751602i $$0.729281\pi$$
$$812$$ 0 0
$$813$$ 0.392305 0.0137587
$$814$$ 0 0
$$815$$ −9.26795 −0.324642
$$816$$ 0 0
$$817$$ 4.19615 0.146805
$$818$$ 0 0
$$819$$ 2.00000 0.0698857
$$820$$ 0 0
$$821$$ −32.5359 −1.13551 −0.567755 0.823197i $$-0.692188\pi$$
−0.567755 + 0.823197i $$0.692188\pi$$
$$822$$ 0 0
$$823$$ −12.9808 −0.452481 −0.226240 0.974071i $$-0.572644\pi$$
−0.226240 + 0.974071i $$0.572644\pi$$
$$824$$ 0 0
$$825$$ 4.73205 0.164749
$$826$$ 0 0
$$827$$ 5.32051 0.185012 0.0925061 0.995712i $$-0.470512\pi$$
0.0925061 + 0.995712i $$0.470512\pi$$
$$828$$ 0 0
$$829$$ 34.1051 1.18452 0.592260 0.805747i $$-0.298236\pi$$
0.592260 + 0.805747i $$0.298236\pi$$
$$830$$ 0 0
$$831$$ −2.00000 −0.0693792
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −3.46410 −0.119880
$$836$$ 0 0
$$837$$ 8.92820 0.308604
$$838$$ 0 0
$$839$$ 19.6077 0.676933 0.338466 0.940978i $$-0.390092\pi$$
0.338466 + 0.940978i $$0.390092\pi$$
$$840$$ 0 0
$$841$$ 38.1769 1.31645
$$842$$ 0 0
$$843$$ −1.26795 −0.0436705
$$844$$ 0 0
$$845$$ −12.4641 −0.428778
$$846$$ 0 0
$$847$$ 31.1244 1.06945
$$848$$ 0 0
$$849$$ 24.9808 0.857338
$$850$$ 0 0
$$851$$ 21.4641 0.735780
$$852$$ 0 0
$$853$$ 27.1769 0.930520 0.465260 0.885174i $$-0.345961\pi$$
0.465260 + 0.885174i $$0.345961\pi$$
$$854$$ 0 0
$$855$$ −1.00000 −0.0341993
$$856$$ 0 0
$$857$$ −42.2487 −1.44319 −0.721594 0.692316i $$-0.756590\pi$$
−0.721594 + 0.692316i $$0.756590\pi$$
$$858$$ 0 0
$$859$$ −32.0000 −1.09183 −0.545913 0.837842i $$-0.683817\pi$$
−0.545913 + 0.837842i $$0.683817\pi$$
$$860$$ 0 0
$$861$$ −3.46410 −0.118056
$$862$$ 0 0
$$863$$ 12.0000 0.408485 0.204242 0.978920i $$-0.434527\pi$$
0.204242 + 0.978920i $$0.434527\pi$$
$$864$$ 0 0
$$865$$ 6.92820 0.235566
$$866$$ 0 0
$$867$$ 17.0000 0.577350
$$868$$ 0 0
$$869$$ −51.7128 −1.75424
$$870$$ 0 0
$$871$$ −5.85641 −0.198437
$$872$$ 0 0
$$873$$ −6.19615 −0.209708
$$874$$ 0 0
$$875$$ 2.73205 0.0923602
$$876$$ 0 0
$$877$$ 53.1244 1.79388 0.896941 0.442150i $$-0.145784\pi$$
0.896941 + 0.442150i $$0.145784\pi$$
$$878$$ 0 0
$$879$$ −27.7128 −0.934730
$$880$$ 0 0
$$881$$ 8.53590 0.287582 0.143791 0.989608i $$-0.454071\pi$$
0.143791 + 0.989608i $$0.454071\pi$$
$$882$$ 0 0
$$883$$ −36.9808 −1.24450 −0.622251 0.782818i $$-0.713782\pi$$
−0.622251 + 0.782818i $$0.713782\pi$$
$$884$$ 0 0
$$885$$ 2.53590 0.0852433
$$886$$ 0 0
$$887$$ 24.0000 0.805841 0.402921 0.915235i $$-0.367995\pi$$
0.402921 + 0.915235i $$0.367995\pi$$
$$888$$ 0 0
$$889$$ 10.9282 0.366520
$$890$$ 0 0
$$891$$ −4.73205 −0.158530
$$892$$ 0 0
$$893$$ −3.46410 −0.115922
$$894$$ 0 0
$$895$$ −23.3205 −0.779519
$$896$$ 0 0
$$897$$ 2.53590 0.0846712
$$898$$ 0 0
$$899$$ −73.1769 −2.44059
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 11.4641 0.381501
$$904$$ 0 0
$$905$$ −2.39230 −0.0795229
$$906$$ 0 0
$$907$$ 32.3923 1.07557 0.537784 0.843082i $$-0.319261\pi$$
0.537784 + 0.843082i $$0.319261\pi$$
$$908$$ 0 0
$$909$$ −10.3923 −0.344691
$$910$$ 0 0
$$911$$ 54.9282 1.81985 0.909926 0.414770i $$-0.136138\pi$$
0.909926 + 0.414770i $$0.136138\pi$$
$$912$$ 0 0
$$913$$ −61.1769 −2.02466
$$914$$ 0 0
$$915$$ 6.53590 0.216070
$$916$$ 0 0
$$917$$ 24.9282 0.823202
$$918$$ 0 0
$$919$$ −51.4256 −1.69637 −0.848187 0.529696i $$-0.822306\pi$$
−0.848187 + 0.529696i $$0.822306\pi$$
$$920$$ 0 0
$$921$$ −32.3923 −1.06736
$$922$$ 0 0
$$923$$ 3.21539 0.105836
$$924$$ 0 0
$$925$$ −6.19615 −0.203728
$$926$$ 0 0
$$927$$ −9.85641 −0.323727
$$928$$ 0 0
$$929$$ −1.60770 −0.0527468 −0.0263734 0.999652i $$-0.508396\pi$$
−0.0263734 + 0.999652i $$0.508396\pi$$
$$930$$ 0 0
$$931$$ −0.464102 −0.0152103
$$932$$ 0 0
$$933$$ 32.4449 1.06220
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −16.2487 −0.530822 −0.265411 0.964135i $$-0.585508\pi$$
−0.265411 + 0.964135i $$0.585508\pi$$
$$938$$ 0 0
$$939$$ −6.39230 −0.208605
$$940$$ 0 0
$$941$$ 0.588457 0.0191832 0.00959158 0.999954i $$-0.496947\pi$$
0.00959158 + 0.999954i $$0.496947\pi$$
$$942$$ 0 0
$$943$$ −4.39230 −0.143033
$$944$$ 0 0
$$945$$ −2.73205 −0.0888736
$$946$$ 0 0
$$947$$ −28.1436 −0.914544 −0.457272 0.889327i $$-0.651173\pi$$
−0.457272 + 0.889327i $$0.651173\pi$$
$$948$$ 0 0
$$949$$ −12.3923 −0.402271
$$950$$ 0 0
$$951$$ 11.3205 0.367093
$$952$$ 0 0
$$953$$ 37.8564 1.22629 0.613145 0.789971i $$-0.289904\pi$$
0.613145 + 0.789971i $$0.289904\pi$$
$$954$$ 0 0
$$955$$ −0.339746 −0.0109939
$$956$$ 0 0
$$957$$ 38.7846 1.25373
$$958$$ 0 0
$$959$$ 54.2487 1.75178
$$960$$ 0 0
$$961$$ 48.7128 1.57138
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 17.1244 0.551253
$$966$$ 0 0
$$967$$ −4.87564 −0.156790 −0.0783951 0.996922i $$-0.524980\pi$$
−0.0783951 + 0.996922i $$0.524980\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 27.7128 0.889346 0.444673 0.895693i $$-0.353320\pi$$
0.444673 + 0.895693i $$0.353320\pi$$
$$972$$ 0 0
$$973$$ 22.9282 0.735044
$$974$$ 0 0
$$975$$ −0.732051 −0.0234444
$$976$$ 0 0
$$977$$ −39.0333 −1.24879 −0.624393 0.781110i $$-0.714654\pi$$
−0.624393 + 0.781110i $$0.714654\pi$$
$$978$$ 0 0
$$979$$ −50.7846 −1.62308
$$980$$ 0 0
$$981$$ −14.3923 −0.459511
$$982$$ 0 0
$$983$$ 41.3205 1.31792 0.658960 0.752178i $$-0.270997\pi$$
0.658960 + 0.752178i $$0.270997\pi$$
$$984$$ 0 0
$$985$$ −24.0000 −0.764704
$$986$$ 0 0
$$987$$ −9.46410 −0.301246
$$988$$ 0 0
$$989$$ 14.5359 0.462215
$$990$$ 0 0
$$991$$ −13.0718 −0.415239 −0.207620 0.978210i $$-0.566572\pi$$
−0.207620 + 0.978210i $$0.566572\pi$$
$$992$$ 0 0
$$993$$ −25.7128 −0.815971
$$994$$ 0 0
$$995$$ 15.3205 0.485693
$$996$$ 0 0
$$997$$ −17.6077 −0.557641 −0.278821 0.960343i $$-0.589944\pi$$
−0.278821 + 0.960343i $$0.589944\pi$$
$$998$$ 0 0
$$999$$ 6.19615 0.196038
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 4560.2.a.bh.1.2 2
4.3 odd 2 285.2.a.e.1.1 2
12.11 even 2 855.2.a.f.1.2 2
20.3 even 4 1425.2.c.k.799.4 4
20.7 even 4 1425.2.c.k.799.1 4
20.19 odd 2 1425.2.a.o.1.2 2
60.59 even 2 4275.2.a.t.1.1 2
76.75 even 2 5415.2.a.r.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.e.1.1 2 4.3 odd 2
855.2.a.f.1.2 2 12.11 even 2
1425.2.a.o.1.2 2 20.19 odd 2
1425.2.c.k.799.1 4 20.7 even 4
1425.2.c.k.799.4 4 20.3 even 4
4275.2.a.t.1.1 2 60.59 even 2
4560.2.a.bh.1.2 2 1.1 even 1 trivial
5415.2.a.r.1.2 2 76.75 even 2