# Properties

 Label 920.2.a.i.1.2 Level $920$ Weight $2$ Character 920.1 Self dual yes Analytic conductor $7.346$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [920,2,Mod(1,920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(920, 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("920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$920 = 2^{3} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 920.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: $$7.34623698596$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.229.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 4x - 1$$ x^3 - 4*x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.86081$$ of defining polynomial Character $$\chi$$ $$=$$ 920.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.46260 q^{3} +1.00000 q^{5} +1.53740 q^{7} -0.860806 q^{9} +O(q^{10})$$ $$q+1.46260 q^{3} +1.00000 q^{5} +1.53740 q^{7} -0.860806 q^{9} -0.860806 q^{11} +0.139194 q^{13} +1.46260 q^{15} +5.50761 q^{17} +5.25901 q^{19} +2.24860 q^{21} -1.00000 q^{23} +1.00000 q^{25} -5.64681 q^{27} +9.76663 q^{29} +6.78600 q^{31} -1.25901 q^{33} +1.53740 q^{35} -12.0900 q^{37} +0.203585 q^{39} -9.98062 q^{41} +11.4432 q^{43} -0.860806 q^{45} -2.32340 q^{47} -4.63640 q^{49} +8.05543 q^{51} +0.149606 q^{53} -0.860806 q^{55} +7.69182 q^{57} -11.0152 q^{59} +4.43281 q^{61} -1.32340 q^{63} +0.139194 q^{65} -10.4972 q^{67} -1.46260 q^{69} +7.31299 q^{71} +7.11982 q^{73} +1.46260 q^{75} -1.32340 q^{77} +6.79641 q^{79} -5.67660 q^{81} +10.6468 q^{83} +5.50761 q^{85} +14.2847 q^{87} -17.2936 q^{89} +0.213997 q^{91} +9.92520 q^{93} +5.25901 q^{95} +1.93561 q^{97} +0.740987 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{3} + 3 q^{5} + 7 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q + 2 * q^3 + 3 * q^5 + 7 * q^7 + 3 * q^9 $$3 q + 2 q^{3} + 3 q^{5} + 7 q^{7} + 3 q^{9} + 3 q^{11} + 6 q^{13} + 2 q^{15} - 5 q^{17} + 7 q^{19} - 6 q^{21} - 3 q^{23} + 3 q^{25} - q^{27} - q^{29} + 10 q^{31} + 5 q^{33} + 7 q^{35} + 2 q^{37} + 7 q^{39} - 10 q^{41} + 12 q^{43} + 3 q^{45} + q^{47} + 6 q^{49} + 9 q^{51} + 10 q^{53} + 3 q^{55} - 12 q^{57} + 10 q^{59} - 13 q^{61} + 4 q^{63} + 6 q^{65} - 6 q^{67} - 2 q^{69} + 10 q^{71} + 7 q^{73} + 2 q^{75} + 4 q^{77} + 14 q^{79} - 25 q^{81} + 16 q^{83} - 5 q^{85} - 5 q^{87} - 20 q^{89} + 11 q^{91} + 25 q^{93} + 7 q^{95} + 5 q^{97} + 11 q^{99}+O(q^{100})$$ 3 * q + 2 * q^3 + 3 * q^5 + 7 * q^7 + 3 * q^9 + 3 * q^11 + 6 * q^13 + 2 * q^15 - 5 * q^17 + 7 * q^19 - 6 * q^21 - 3 * q^23 + 3 * q^25 - q^27 - q^29 + 10 * q^31 + 5 * q^33 + 7 * q^35 + 2 * q^37 + 7 * q^39 - 10 * q^41 + 12 * q^43 + 3 * q^45 + q^47 + 6 * q^49 + 9 * q^51 + 10 * q^53 + 3 * q^55 - 12 * q^57 + 10 * q^59 - 13 * q^61 + 4 * q^63 + 6 * q^65 - 6 * q^67 - 2 * q^69 + 10 * q^71 + 7 * q^73 + 2 * q^75 + 4 * q^77 + 14 * q^79 - 25 * q^81 + 16 * q^83 - 5 * q^85 - 5 * q^87 - 20 * q^89 + 11 * q^91 + 25 * q^93 + 7 * q^95 + 5 * q^97 + 11 * 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
$$3$$ 1.46260 0.844432 0.422216 0.906495i $$-0.361252\pi$$
0.422216 + 0.906495i $$0.361252\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 1.53740 0.581083 0.290542 0.956862i $$-0.406165\pi$$
0.290542 + 0.956862i $$0.406165\pi$$
$$8$$ 0 0
$$9$$ −0.860806 −0.286935
$$10$$ 0 0
$$11$$ −0.860806 −0.259543 −0.129771 0.991544i $$-0.541424\pi$$
−0.129771 + 0.991544i $$0.541424\pi$$
$$12$$ 0 0
$$13$$ 0.139194 0.0386055 0.0193028 0.999814i $$-0.493855\pi$$
0.0193028 + 0.999814i $$0.493855\pi$$
$$14$$ 0 0
$$15$$ 1.46260 0.377641
$$16$$ 0 0
$$17$$ 5.50761 1.33579 0.667896 0.744254i $$-0.267195\pi$$
0.667896 + 0.744254i $$0.267195\pi$$
$$18$$ 0 0
$$19$$ 5.25901 1.20650 0.603250 0.797552i $$-0.293872\pi$$
0.603250 + 0.797552i $$0.293872\pi$$
$$20$$ 0 0
$$21$$ 2.24860 0.490685
$$22$$ 0 0
$$23$$ −1.00000 −0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −5.64681 −1.08673
$$28$$ 0 0
$$29$$ 9.76663 1.81362 0.906809 0.421543i $$-0.138511\pi$$
0.906809 + 0.421543i $$0.138511\pi$$
$$30$$ 0 0
$$31$$ 6.78600 1.21880 0.609401 0.792862i $$-0.291410\pi$$
0.609401 + 0.792862i $$0.291410\pi$$
$$32$$ 0 0
$$33$$ −1.25901 −0.219166
$$34$$ 0 0
$$35$$ 1.53740 0.259868
$$36$$ 0 0
$$37$$ −12.0900 −1.98759 −0.993795 0.111232i $$-0.964520\pi$$
−0.993795 + 0.111232i $$0.964520\pi$$
$$38$$ 0 0
$$39$$ 0.203585 0.0325997
$$40$$ 0 0
$$41$$ −9.98062 −1.55871 −0.779356 0.626582i $$-0.784454\pi$$
−0.779356 + 0.626582i $$0.784454\pi$$
$$42$$ 0 0
$$43$$ 11.4432 1.74508 0.872538 0.488547i $$-0.162473\pi$$
0.872538 + 0.488547i $$0.162473\pi$$
$$44$$ 0 0
$$45$$ −0.860806 −0.128321
$$46$$ 0 0
$$47$$ −2.32340 −0.338903 −0.169452 0.985538i $$-0.554200\pi$$
−0.169452 + 0.985538i $$0.554200\pi$$
$$48$$ 0 0
$$49$$ −4.63640 −0.662342
$$50$$ 0 0
$$51$$ 8.05543 1.12799
$$52$$ 0 0
$$53$$ 0.149606 0.0205500 0.0102750 0.999947i $$-0.496729\pi$$
0.0102750 + 0.999947i $$0.496729\pi$$
$$54$$ 0 0
$$55$$ −0.860806 −0.116071
$$56$$ 0 0
$$57$$ 7.69182 1.01881
$$58$$ 0 0
$$59$$ −11.0152 −1.43406 −0.717030 0.697042i $$-0.754499\pi$$
−0.717030 + 0.697042i $$0.754499\pi$$
$$60$$ 0 0
$$61$$ 4.43281 0.567563 0.283782 0.958889i $$-0.408411\pi$$
0.283782 + 0.958889i $$0.408411\pi$$
$$62$$ 0 0
$$63$$ −1.32340 −0.166733
$$64$$ 0 0
$$65$$ 0.139194 0.0172649
$$66$$ 0 0
$$67$$ −10.4972 −1.28244 −0.641219 0.767358i $$-0.721571\pi$$
−0.641219 + 0.767358i $$0.721571\pi$$
$$68$$ 0 0
$$69$$ −1.46260 −0.176076
$$70$$ 0 0
$$71$$ 7.31299 0.867892 0.433946 0.900939i $$-0.357121\pi$$
0.433946 + 0.900939i $$0.357121\pi$$
$$72$$ 0 0
$$73$$ 7.11982 0.833312 0.416656 0.909064i $$-0.363202\pi$$
0.416656 + 0.909064i $$0.363202\pi$$
$$74$$ 0 0
$$75$$ 1.46260 0.168886
$$76$$ 0 0
$$77$$ −1.32340 −0.150816
$$78$$ 0 0
$$79$$ 6.79641 0.764656 0.382328 0.924027i $$-0.375122\pi$$
0.382328 + 0.924027i $$0.375122\pi$$
$$80$$ 0 0
$$81$$ −5.67660 −0.630733
$$82$$ 0 0
$$83$$ 10.6468 1.16864 0.584320 0.811524i $$-0.301361\pi$$
0.584320 + 0.811524i $$0.301361\pi$$
$$84$$ 0 0
$$85$$ 5.50761 0.597385
$$86$$ 0 0
$$87$$ 14.2847 1.53148
$$88$$ 0 0
$$89$$ −17.2936 −1.83312 −0.916560 0.399897i $$-0.869046\pi$$
−0.916560 + 0.399897i $$0.869046\pi$$
$$90$$ 0 0
$$91$$ 0.213997 0.0224330
$$92$$ 0 0
$$93$$ 9.92520 1.02919
$$94$$ 0 0
$$95$$ 5.25901 0.539563
$$96$$ 0 0
$$97$$ 1.93561 0.196531 0.0982657 0.995160i $$-0.468671\pi$$
0.0982657 + 0.995160i $$0.468671\pi$$
$$98$$ 0 0
$$99$$ 0.740987 0.0744720
$$100$$ 0 0
$$101$$ −15.4224 −1.53459 −0.767293 0.641297i $$-0.778397\pi$$
−0.767293 + 0.641297i $$0.778397\pi$$
$$102$$ 0 0
$$103$$ 7.91478 0.779867 0.389933 0.920843i $$-0.372498\pi$$
0.389933 + 0.920843i $$0.372498\pi$$
$$104$$ 0 0
$$105$$ 2.24860 0.219441
$$106$$ 0 0
$$107$$ −4.27839 −0.413607 −0.206804 0.978382i $$-0.566306\pi$$
−0.206804 + 0.978382i $$0.566306\pi$$
$$108$$ 0 0
$$109$$ −17.7770 −1.70273 −0.851366 0.524572i $$-0.824225\pi$$
−0.851366 + 0.524572i $$0.824225\pi$$
$$110$$ 0 0
$$111$$ −17.6829 −1.67838
$$112$$ 0 0
$$113$$ −3.20359 −0.301368 −0.150684 0.988582i $$-0.548148\pi$$
−0.150684 + 0.988582i $$0.548148\pi$$
$$114$$ 0 0
$$115$$ −1.00000 −0.0932505
$$116$$ 0 0
$$117$$ −0.119819 −0.0110773
$$118$$ 0 0
$$119$$ 8.46742 0.776207
$$120$$ 0 0
$$121$$ −10.2590 −0.932638
$$122$$ 0 0
$$123$$ −14.5976 −1.31623
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −15.2099 −1.34966 −0.674828 0.737975i $$-0.735782\pi$$
−0.674828 + 0.737975i $$0.735782\pi$$
$$128$$ 0 0
$$129$$ 16.7368 1.47360
$$130$$ 0 0
$$131$$ 12.6918 1.10889 0.554445 0.832220i $$-0.312931\pi$$
0.554445 + 0.832220i $$0.312931\pi$$
$$132$$ 0 0
$$133$$ 8.08522 0.701077
$$134$$ 0 0
$$135$$ −5.64681 −0.486000
$$136$$ 0 0
$$137$$ −14.0346 −1.19906 −0.599529 0.800353i $$-0.704645\pi$$
−0.599529 + 0.800353i $$0.704645\pi$$
$$138$$ 0 0
$$139$$ 14.2638 1.20984 0.604921 0.796285i $$-0.293205\pi$$
0.604921 + 0.796285i $$0.293205\pi$$
$$140$$ 0 0
$$141$$ −3.39821 −0.286181
$$142$$ 0 0
$$143$$ −0.119819 −0.0100198
$$144$$ 0 0
$$145$$ 9.76663 0.811074
$$146$$ 0 0
$$147$$ −6.78119 −0.559303
$$148$$ 0 0
$$149$$ −4.46260 −0.365590 −0.182795 0.983151i $$-0.558515\pi$$
−0.182795 + 0.983151i $$0.558515\pi$$
$$150$$ 0 0
$$151$$ 10.7562 0.875328 0.437664 0.899139i $$-0.355806\pi$$
0.437664 + 0.899139i $$0.355806\pi$$
$$152$$ 0 0
$$153$$ −4.74099 −0.383286
$$154$$ 0 0
$$155$$ 6.78600 0.545065
$$156$$ 0 0
$$157$$ 14.5180 1.15866 0.579332 0.815091i $$-0.303313\pi$$
0.579332 + 0.815091i $$0.303313\pi$$
$$158$$ 0 0
$$159$$ 0.218814 0.0173531
$$160$$ 0 0
$$161$$ −1.53740 −0.121164
$$162$$ 0 0
$$163$$ −1.30403 −0.102139 −0.0510697 0.998695i $$-0.516263\pi$$
−0.0510697 + 0.998695i $$0.516263\pi$$
$$164$$ 0 0
$$165$$ −1.25901 −0.0980141
$$166$$ 0 0
$$167$$ 3.44322 0.266445 0.133222 0.991086i $$-0.457468\pi$$
0.133222 + 0.991086i $$0.457468\pi$$
$$168$$ 0 0
$$169$$ −12.9806 −0.998510
$$170$$ 0 0
$$171$$ −4.52699 −0.346188
$$172$$ 0 0
$$173$$ 9.35801 0.711476 0.355738 0.934586i $$-0.384230\pi$$
0.355738 + 0.934586i $$0.384230\pi$$
$$174$$ 0 0
$$175$$ 1.53740 0.116217
$$176$$ 0 0
$$177$$ −16.1109 −1.21097
$$178$$ 0 0
$$179$$ −18.4134 −1.37628 −0.688142 0.725576i $$-0.741574\pi$$
−0.688142 + 0.725576i $$0.741574\pi$$
$$180$$ 0 0
$$181$$ −1.94457 −0.144539 −0.0722694 0.997385i $$-0.523024\pi$$
−0.0722694 + 0.997385i $$0.523024\pi$$
$$182$$ 0 0
$$183$$ 6.48342 0.479268
$$184$$ 0 0
$$185$$ −12.0900 −0.888877
$$186$$ 0 0
$$187$$ −4.74099 −0.346695
$$188$$ 0 0
$$189$$ −8.68141 −0.631480
$$190$$ 0 0
$$191$$ −0.775591 −0.0561198 −0.0280599 0.999606i $$-0.508933\pi$$
−0.0280599 + 0.999606i $$0.508933\pi$$
$$192$$ 0 0
$$193$$ −11.7666 −0.846980 −0.423490 0.905901i $$-0.639195\pi$$
−0.423490 + 0.905901i $$0.639195\pi$$
$$194$$ 0 0
$$195$$ 0.203585 0.0145790
$$196$$ 0 0
$$197$$ 9.68141 0.689772 0.344886 0.938645i $$-0.387918\pi$$
0.344886 + 0.938645i $$0.387918\pi$$
$$198$$ 0 0
$$199$$ −3.29362 −0.233478 −0.116739 0.993163i $$-0.537244\pi$$
−0.116739 + 0.993163i $$0.537244\pi$$
$$200$$ 0 0
$$201$$ −15.3532 −1.08293
$$202$$ 0 0
$$203$$ 15.0152 1.05386
$$204$$ 0 0
$$205$$ −9.98062 −0.697077
$$206$$ 0 0
$$207$$ 0.860806 0.0598301
$$208$$ 0 0
$$209$$ −4.52699 −0.313138
$$210$$ 0 0
$$211$$ −16.2396 −1.11798 −0.558991 0.829173i $$-0.688812\pi$$
−0.558991 + 0.829173i $$0.688812\pi$$
$$212$$ 0 0
$$213$$ 10.6960 0.732876
$$214$$ 0 0
$$215$$ 11.4432 0.780421
$$216$$ 0 0
$$217$$ 10.4328 0.708225
$$218$$ 0 0
$$219$$ 10.4134 0.703675
$$220$$ 0 0
$$221$$ 0.766628 0.0515690
$$222$$ 0 0
$$223$$ 21.2936 1.42593 0.712963 0.701202i $$-0.247353\pi$$
0.712963 + 0.701202i $$0.247353\pi$$
$$224$$ 0 0
$$225$$ −0.860806 −0.0573871
$$226$$ 0 0
$$227$$ −29.6620 −1.96874 −0.984369 0.176117i $$-0.943646\pi$$
−0.984369 + 0.176117i $$0.943646\pi$$
$$228$$ 0 0
$$229$$ 19.6829 1.30068 0.650340 0.759643i $$-0.274626\pi$$
0.650340 + 0.759643i $$0.274626\pi$$
$$230$$ 0 0
$$231$$ −1.93561 −0.127354
$$232$$ 0 0
$$233$$ −5.52699 −0.362085 −0.181043 0.983475i $$-0.557947\pi$$
−0.181043 + 0.983475i $$0.557947\pi$$
$$234$$ 0 0
$$235$$ −2.32340 −0.151562
$$236$$ 0 0
$$237$$ 9.94043 0.645700
$$238$$ 0 0
$$239$$ −17.3982 −1.12540 −0.562698 0.826662i $$-0.690237\pi$$
−0.562698 + 0.826662i $$0.690237\pi$$
$$240$$ 0 0
$$241$$ −16.0900 −1.03645 −0.518225 0.855244i $$-0.673407\pi$$
−0.518225 + 0.855244i $$0.673407\pi$$
$$242$$ 0 0
$$243$$ 8.63785 0.554118
$$244$$ 0 0
$$245$$ −4.63640 −0.296209
$$246$$ 0 0
$$247$$ 0.732024 0.0465776
$$248$$ 0 0
$$249$$ 15.5720 0.986836
$$250$$ 0 0
$$251$$ −29.0498 −1.83361 −0.916805 0.399336i $$-0.869241\pi$$
−0.916805 + 0.399336i $$0.869241\pi$$
$$252$$ 0 0
$$253$$ 0.860806 0.0541184
$$254$$ 0 0
$$255$$ 8.05543 0.504450
$$256$$ 0 0
$$257$$ −6.56304 −0.409391 −0.204696 0.978826i $$-0.565620\pi$$
−0.204696 + 0.978826i $$0.565620\pi$$
$$258$$ 0 0
$$259$$ −18.5872 −1.15495
$$260$$ 0 0
$$261$$ −8.40717 −0.520391
$$262$$ 0 0
$$263$$ −0.561593 −0.0346293 −0.0173147 0.999850i $$-0.505512\pi$$
−0.0173147 + 0.999850i $$0.505512\pi$$
$$264$$ 0 0
$$265$$ 0.149606 0.00919024
$$266$$ 0 0
$$267$$ −25.2936 −1.54794
$$268$$ 0 0
$$269$$ −7.26943 −0.443225 −0.221612 0.975135i $$-0.571132\pi$$
−0.221612 + 0.975135i $$0.571132\pi$$
$$270$$ 0 0
$$271$$ 0.184210 0.0111900 0.00559498 0.999984i $$-0.498219\pi$$
0.00559498 + 0.999984i $$0.498219\pi$$
$$272$$ 0 0
$$273$$ 0.312992 0.0189431
$$274$$ 0 0
$$275$$ −0.860806 −0.0519085
$$276$$ 0 0
$$277$$ 25.8954 1.55590 0.777952 0.628323i $$-0.216259\pi$$
0.777952 + 0.628323i $$0.216259\pi$$
$$278$$ 0 0
$$279$$ −5.84143 −0.349717
$$280$$ 0 0
$$281$$ −26.7756 −1.59730 −0.798649 0.601797i $$-0.794452\pi$$
−0.798649 + 0.601797i $$0.794452\pi$$
$$282$$ 0 0
$$283$$ 5.70079 0.338877 0.169438 0.985541i $$-0.445805\pi$$
0.169438 + 0.985541i $$0.445805\pi$$
$$284$$ 0 0
$$285$$ 7.69182 0.455624
$$286$$ 0 0
$$287$$ −15.3442 −0.905741
$$288$$ 0 0
$$289$$ 13.3338 0.784342
$$290$$ 0 0
$$291$$ 2.83102 0.165957
$$292$$ 0 0
$$293$$ 1.05398 0.0615741 0.0307871 0.999526i $$-0.490199\pi$$
0.0307871 + 0.999526i $$0.490199\pi$$
$$294$$ 0 0
$$295$$ −11.0152 −0.641331
$$296$$ 0 0
$$297$$ 4.86081 0.282053
$$298$$ 0 0
$$299$$ −0.139194 −0.00804981
$$300$$ 0 0
$$301$$ 17.5928 1.01403
$$302$$ 0 0
$$303$$ −22.5568 −1.29585
$$304$$ 0 0
$$305$$ 4.43281 0.253822
$$306$$ 0 0
$$307$$ 17.8760 1.02024 0.510120 0.860103i $$-0.329601\pi$$
0.510120 + 0.860103i $$0.329601\pi$$
$$308$$ 0 0
$$309$$ 11.5762 0.658544
$$310$$ 0 0
$$311$$ 16.9315 0.960095 0.480048 0.877242i $$-0.340619\pi$$
0.480048 + 0.877242i $$0.340619\pi$$
$$312$$ 0 0
$$313$$ −19.9959 −1.13023 −0.565116 0.825011i $$-0.691169\pi$$
−0.565116 + 0.825011i $$0.691169\pi$$
$$314$$ 0 0
$$315$$ −1.32340 −0.0745654
$$316$$ 0 0
$$317$$ 3.29776 0.185221 0.0926104 0.995702i $$-0.470479\pi$$
0.0926104 + 0.995702i $$0.470479\pi$$
$$318$$ 0 0
$$319$$ −8.40717 −0.470711
$$320$$ 0 0
$$321$$ −6.25756 −0.349263
$$322$$ 0 0
$$323$$ 28.9646 1.61163
$$324$$ 0 0
$$325$$ 0.139194 0.00772110
$$326$$ 0 0
$$327$$ −26.0007 −1.43784
$$328$$ 0 0
$$329$$ −3.57201 −0.196931
$$330$$ 0 0
$$331$$ −12.8802 −0.707959 −0.353979 0.935253i $$-0.615172\pi$$
−0.353979 + 0.935253i $$0.615172\pi$$
$$332$$ 0 0
$$333$$ 10.4072 0.570309
$$334$$ 0 0
$$335$$ −10.4972 −0.573523
$$336$$ 0 0
$$337$$ −9.22026 −0.502260 −0.251130 0.967953i $$-0.580802\pi$$
−0.251130 + 0.967953i $$0.580802\pi$$
$$338$$ 0 0
$$339$$ −4.68556 −0.254485
$$340$$ 0 0
$$341$$ −5.84143 −0.316331
$$342$$ 0 0
$$343$$ −17.8898 −0.965959
$$344$$ 0 0
$$345$$ −1.46260 −0.0787437
$$346$$ 0 0
$$347$$ 31.2984 1.68019 0.840094 0.542441i $$-0.182500\pi$$
0.840094 + 0.542441i $$0.182500\pi$$
$$348$$ 0 0
$$349$$ 9.89541 0.529689 0.264845 0.964291i $$-0.414679\pi$$
0.264845 + 0.964291i $$0.414679\pi$$
$$350$$ 0 0
$$351$$ −0.786003 −0.0419537
$$352$$ 0 0
$$353$$ 19.4287 1.03408 0.517042 0.855960i $$-0.327033\pi$$
0.517042 + 0.855960i $$0.327033\pi$$
$$354$$ 0 0
$$355$$ 7.31299 0.388133
$$356$$ 0 0
$$357$$ 12.3844 0.655453
$$358$$ 0 0
$$359$$ −13.1440 −0.693714 −0.346857 0.937918i $$-0.612751\pi$$
−0.346857 + 0.937918i $$0.612751\pi$$
$$360$$ 0 0
$$361$$ 8.65722 0.455643
$$362$$ 0 0
$$363$$ −15.0048 −0.787549
$$364$$ 0 0
$$365$$ 7.11982 0.372668
$$366$$ 0 0
$$367$$ −15.7424 −0.821748 −0.410874 0.911692i $$-0.634776\pi$$
−0.410874 + 0.911692i $$0.634776\pi$$
$$368$$ 0 0
$$369$$ 8.59138 0.447249
$$370$$ 0 0
$$371$$ 0.230005 0.0119413
$$372$$ 0 0
$$373$$ 27.6620 1.43229 0.716143 0.697954i $$-0.245906\pi$$
0.716143 + 0.697954i $$0.245906\pi$$
$$374$$ 0 0
$$375$$ 1.46260 0.0755283
$$376$$ 0 0
$$377$$ 1.35946 0.0700156
$$378$$ 0 0
$$379$$ 28.2140 1.44926 0.724628 0.689140i $$-0.242012\pi$$
0.724628 + 0.689140i $$0.242012\pi$$
$$380$$ 0 0
$$381$$ −22.2459 −1.13969
$$382$$ 0 0
$$383$$ −21.5928 −1.10334 −0.551671 0.834062i $$-0.686010\pi$$
−0.551671 + 0.834062i $$0.686010\pi$$
$$384$$ 0 0
$$385$$ −1.32340 −0.0674469
$$386$$ 0 0
$$387$$ −9.85039 −0.500724
$$388$$ 0 0
$$389$$ 17.2501 0.874612 0.437306 0.899313i $$-0.355933\pi$$
0.437306 + 0.899313i $$0.355933\pi$$
$$390$$ 0 0
$$391$$ −5.50761 −0.278532
$$392$$ 0 0
$$393$$ 18.5630 0.936382
$$394$$ 0 0
$$395$$ 6.79641 0.341965
$$396$$ 0 0
$$397$$ 34.8850 1.75083 0.875414 0.483374i $$-0.160589\pi$$
0.875414 + 0.483374i $$0.160589\pi$$
$$398$$ 0 0
$$399$$ 11.8254 0.592012
$$400$$ 0 0
$$401$$ 4.21881 0.210678 0.105339 0.994436i $$-0.466407\pi$$
0.105339 + 0.994436i $$0.466407\pi$$
$$402$$ 0 0
$$403$$ 0.944572 0.0470525
$$404$$ 0 0
$$405$$ −5.67660 −0.282072
$$406$$ 0 0
$$407$$ 10.4072 0.515864
$$408$$ 0 0
$$409$$ 38.8165 1.91935 0.959675 0.281111i $$-0.0907030\pi$$
0.959675 + 0.281111i $$0.0907030\pi$$
$$410$$ 0 0
$$411$$ −20.5270 −1.01252
$$412$$ 0 0
$$413$$ −16.9348 −0.833309
$$414$$ 0 0
$$415$$ 10.6468 0.522631
$$416$$ 0 0
$$417$$ 20.8623 1.02163
$$418$$ 0 0
$$419$$ 22.0305 1.07626 0.538129 0.842862i $$-0.319131\pi$$
0.538129 + 0.842862i $$0.319131\pi$$
$$420$$ 0 0
$$421$$ −17.2203 −0.839264 −0.419632 0.907694i $$-0.637841\pi$$
−0.419632 + 0.907694i $$0.637841\pi$$
$$422$$ 0 0
$$423$$ 2.00000 0.0972433
$$424$$ 0 0
$$425$$ 5.50761 0.267159
$$426$$ 0 0
$$427$$ 6.81501 0.329801
$$428$$ 0 0
$$429$$ −0.175247 −0.00846102
$$430$$ 0 0
$$431$$ −11.8116 −0.568947 −0.284473 0.958684i $$-0.591819\pi$$
−0.284473 + 0.958684i $$0.591819\pi$$
$$432$$ 0 0
$$433$$ 19.8642 0.954611 0.477306 0.878737i $$-0.341613\pi$$
0.477306 + 0.878737i $$0.341613\pi$$
$$434$$ 0 0
$$435$$ 14.2847 0.684897
$$436$$ 0 0
$$437$$ −5.25901 −0.251573
$$438$$ 0 0
$$439$$ −5.04647 −0.240855 −0.120427 0.992722i $$-0.538426\pi$$
−0.120427 + 0.992722i $$0.538426\pi$$
$$440$$ 0 0
$$441$$ 3.99104 0.190049
$$442$$ 0 0
$$443$$ −14.3193 −0.680328 −0.340164 0.940366i $$-0.610483\pi$$
−0.340164 + 0.940366i $$0.610483\pi$$
$$444$$ 0 0
$$445$$ −17.2936 −0.819796
$$446$$ 0 0
$$447$$ −6.52699 −0.308716
$$448$$ 0 0
$$449$$ −13.3699 −0.630963 −0.315482 0.948932i $$-0.602166\pi$$
−0.315482 + 0.948932i $$0.602166\pi$$
$$450$$ 0 0
$$451$$ 8.59138 0.404552
$$452$$ 0 0
$$453$$ 15.7320 0.739155
$$454$$ 0 0
$$455$$ 0.213997 0.0100323
$$456$$ 0 0
$$457$$ 2.38924 0.111764 0.0558821 0.998437i $$-0.482203\pi$$
0.0558821 + 0.998437i $$0.482203\pi$$
$$458$$ 0 0
$$459$$ −31.1004 −1.45164
$$460$$ 0 0
$$461$$ 21.2278 0.988676 0.494338 0.869270i $$-0.335410\pi$$
0.494338 + 0.869270i $$0.335410\pi$$
$$462$$ 0 0
$$463$$ 35.1261 1.63245 0.816224 0.577736i $$-0.196064\pi$$
0.816224 + 0.577736i $$0.196064\pi$$
$$464$$ 0 0
$$465$$ 9.92520 0.460270
$$466$$ 0 0
$$467$$ 16.7756 0.776282 0.388141 0.921600i $$-0.373117\pi$$
0.388141 + 0.921600i $$0.373117\pi$$
$$468$$ 0 0
$$469$$ −16.1384 −0.745203
$$470$$ 0 0
$$471$$ 21.2340 0.978413
$$472$$ 0 0
$$473$$ −9.85039 −0.452922
$$474$$ 0 0
$$475$$ 5.25901 0.241300
$$476$$ 0 0
$$477$$ −0.128782 −0.00589652
$$478$$ 0 0
$$479$$ −5.61076 −0.256362 −0.128181 0.991751i $$-0.540914\pi$$
−0.128181 + 0.991751i $$0.540914\pi$$
$$480$$ 0 0
$$481$$ −1.68286 −0.0767319
$$482$$ 0 0
$$483$$ −2.24860 −0.102315
$$484$$ 0 0
$$485$$ 1.93561 0.0878915
$$486$$ 0 0
$$487$$ 10.2638 0.465099 0.232549 0.972585i $$-0.425293\pi$$
0.232549 + 0.972585i $$0.425293\pi$$
$$488$$ 0 0
$$489$$ −1.90727 −0.0862498
$$490$$ 0 0
$$491$$ 9.28735 0.419132 0.209566 0.977794i $$-0.432795\pi$$
0.209566 + 0.977794i $$0.432795\pi$$
$$492$$ 0 0
$$493$$ 53.7908 2.42262
$$494$$ 0 0
$$495$$ 0.740987 0.0333049
$$496$$ 0 0
$$497$$ 11.2430 0.504318
$$498$$ 0 0
$$499$$ 13.3082 0.595756 0.297878 0.954604i $$-0.403721\pi$$
0.297878 + 0.954604i $$0.403721\pi$$
$$500$$ 0 0
$$501$$ 5.03605 0.224994
$$502$$ 0 0
$$503$$ −37.7383 −1.68267 −0.841334 0.540516i $$-0.818229\pi$$
−0.841334 + 0.540516i $$0.818229\pi$$
$$504$$ 0 0
$$505$$ −15.4224 −0.686288
$$506$$ 0 0
$$507$$ −18.9854 −0.843173
$$508$$ 0 0
$$509$$ −1.11982 −0.0496351 −0.0248176 0.999692i $$-0.507900\pi$$
−0.0248176 + 0.999692i $$0.507900\pi$$
$$510$$ 0 0
$$511$$ 10.9460 0.484223
$$512$$ 0 0
$$513$$ −29.6966 −1.31114
$$514$$ 0 0
$$515$$ 7.91478 0.348767
$$516$$ 0 0
$$517$$ 2.00000 0.0879599
$$518$$ 0 0
$$519$$ 13.6870 0.600793
$$520$$ 0 0
$$521$$ −13.9100 −0.609407 −0.304703 0.952447i $$-0.598557\pi$$
−0.304703 + 0.952447i $$0.598557\pi$$
$$522$$ 0 0
$$523$$ 8.19799 0.358473 0.179237 0.983806i $$-0.442637\pi$$
0.179237 + 0.983806i $$0.442637\pi$$
$$524$$ 0 0
$$525$$ 2.24860 0.0981370
$$526$$ 0 0
$$527$$ 37.3747 1.62807
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 9.48197 0.411483
$$532$$ 0 0
$$533$$ −1.38924 −0.0601749
$$534$$ 0 0
$$535$$ −4.27839 −0.184971
$$536$$ 0 0
$$537$$ −26.9315 −1.16218
$$538$$ 0 0
$$539$$ 3.99104 0.171906
$$540$$ 0 0
$$541$$ 23.5478 1.01240 0.506200 0.862416i $$-0.331050\pi$$
0.506200 + 0.862416i $$0.331050\pi$$
$$542$$ 0 0
$$543$$ −2.84413 −0.122053
$$544$$ 0 0
$$545$$ −17.7770 −0.761485
$$546$$ 0 0
$$547$$ 1.77077 0.0757128 0.0378564 0.999283i $$-0.487947\pi$$
0.0378564 + 0.999283i $$0.487947\pi$$
$$548$$ 0 0
$$549$$ −3.81579 −0.162854
$$550$$ 0 0
$$551$$ 51.3628 2.18813
$$552$$ 0 0
$$553$$ 10.4488 0.444329
$$554$$ 0 0
$$555$$ −17.6829 −0.750596
$$556$$ 0 0
$$557$$ −22.5872 −0.957052 −0.478526 0.878073i $$-0.658829\pi$$
−0.478526 + 0.878073i $$0.658829\pi$$
$$558$$ 0 0
$$559$$ 1.59283 0.0673695
$$560$$ 0 0
$$561$$ −6.93416 −0.292760
$$562$$ 0 0
$$563$$ 21.8504 0.920884 0.460442 0.887690i $$-0.347691\pi$$
0.460442 + 0.887690i $$0.347691\pi$$
$$564$$ 0 0
$$565$$ −3.20359 −0.134776
$$566$$ 0 0
$$567$$ −8.72721 −0.366508
$$568$$ 0 0
$$569$$ 11.4737 0.481002 0.240501 0.970649i $$-0.422688\pi$$
0.240501 + 0.970649i $$0.422688\pi$$
$$570$$ 0 0
$$571$$ 27.7085 1.15956 0.579782 0.814771i $$-0.303138\pi$$
0.579782 + 0.814771i $$0.303138\pi$$
$$572$$ 0 0
$$573$$ −1.13438 −0.0473893
$$574$$ 0 0
$$575$$ −1.00000 −0.0417029
$$576$$ 0 0
$$577$$ 20.3442 0.846941 0.423471 0.905910i $$-0.360812\pi$$
0.423471 + 0.905910i $$0.360812\pi$$
$$578$$ 0 0
$$579$$ −17.2099 −0.715217
$$580$$ 0 0
$$581$$ 16.3684 0.679076
$$582$$ 0 0
$$583$$ −0.128782 −0.00533360
$$584$$ 0 0
$$585$$ −0.119819 −0.00495391
$$586$$ 0 0
$$587$$ 12.1690 0.502268 0.251134 0.967952i $$-0.419197\pi$$
0.251134 + 0.967952i $$0.419197\pi$$
$$588$$ 0 0
$$589$$ 35.6877 1.47049
$$590$$ 0 0
$$591$$ 14.1600 0.582465
$$592$$ 0 0
$$593$$ −40.5180 −1.66388 −0.831938 0.554869i $$-0.812769\pi$$
−0.831938 + 0.554869i $$0.812769\pi$$
$$594$$ 0 0
$$595$$ 8.46742 0.347130
$$596$$ 0 0
$$597$$ −4.81724 −0.197156
$$598$$ 0 0
$$599$$ −28.8913 −1.18047 −0.590233 0.807233i $$-0.700964\pi$$
−0.590233 + 0.807233i $$0.700964\pi$$
$$600$$ 0 0
$$601$$ 5.34278 0.217937 0.108968 0.994045i $$-0.465245\pi$$
0.108968 + 0.994045i $$0.465245\pi$$
$$602$$ 0 0
$$603$$ 9.03605 0.367977
$$604$$ 0 0
$$605$$ −10.2590 −0.417088
$$606$$ 0 0
$$607$$ −40.4376 −1.64131 −0.820656 0.571422i $$-0.806392\pi$$
−0.820656 + 0.571422i $$0.806392\pi$$
$$608$$ 0 0
$$609$$ 21.9612 0.889915
$$610$$ 0 0
$$611$$ −0.323404 −0.0130835
$$612$$ 0 0
$$613$$ −5.31154 −0.214531 −0.107266 0.994230i $$-0.534210\pi$$
−0.107266 + 0.994230i $$0.534210\pi$$
$$614$$ 0 0
$$615$$ −14.5976 −0.588634
$$616$$ 0 0
$$617$$ 9.71680 0.391183 0.195592 0.980685i $$-0.437337\pi$$
0.195592 + 0.980685i $$0.437337\pi$$
$$618$$ 0 0
$$619$$ −23.7562 −0.954843 −0.477421 0.878674i $$-0.658428\pi$$
−0.477421 + 0.878674i $$0.658428\pi$$
$$620$$ 0 0
$$621$$ 5.64681 0.226599
$$622$$ 0 0
$$623$$ −26.5872 −1.06520
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −6.62117 −0.264424
$$628$$ 0 0
$$629$$ −66.5872 −2.65501
$$630$$ 0 0
$$631$$ 1.24234 0.0494566 0.0247283 0.999694i $$-0.492128\pi$$
0.0247283 + 0.999694i $$0.492128\pi$$
$$632$$ 0 0
$$633$$ −23.7521 −0.944060
$$634$$ 0 0
$$635$$ −15.2099 −0.603585
$$636$$ 0 0
$$637$$ −0.645359 −0.0255701
$$638$$ 0 0
$$639$$ −6.29507 −0.249029
$$640$$ 0 0
$$641$$ −34.8864 −1.37793 −0.688966 0.724794i $$-0.741935\pi$$
−0.688966 + 0.724794i $$0.741935\pi$$
$$642$$ 0 0
$$643$$ 32.1592 1.26824 0.634118 0.773236i $$-0.281363\pi$$
0.634118 + 0.773236i $$0.281363\pi$$
$$644$$ 0 0
$$645$$ 16.7368 0.659012
$$646$$ 0 0
$$647$$ 23.9550 0.941768 0.470884 0.882195i $$-0.343935\pi$$
0.470884 + 0.882195i $$0.343935\pi$$
$$648$$ 0 0
$$649$$ 9.48197 0.372200
$$650$$ 0 0
$$651$$ 15.2590 0.598048
$$652$$ 0 0
$$653$$ 4.31926 0.169026 0.0845128 0.996422i $$-0.473067\pi$$
0.0845128 + 0.996422i $$0.473067\pi$$
$$654$$ 0 0
$$655$$ 12.6918 0.495911
$$656$$ 0 0
$$657$$ −6.12878 −0.239107
$$658$$ 0 0
$$659$$ −6.14961 −0.239555 −0.119777 0.992801i $$-0.538218\pi$$
−0.119777 + 0.992801i $$0.538218\pi$$
$$660$$ 0 0
$$661$$ −15.7473 −0.612497 −0.306249 0.951952i $$-0.599074\pi$$
−0.306249 + 0.951952i $$0.599074\pi$$
$$662$$ 0 0
$$663$$ 1.12127 0.0435465
$$664$$ 0 0
$$665$$ 8.08522 0.313531
$$666$$ 0 0
$$667$$ −9.76663 −0.378165
$$668$$ 0 0
$$669$$ 31.1440 1.20410
$$670$$ 0 0
$$671$$ −3.81579 −0.147307
$$672$$ 0 0
$$673$$ 43.5783 1.67982 0.839909 0.542727i $$-0.182608\pi$$
0.839909 + 0.542727i $$0.182608\pi$$
$$674$$ 0 0
$$675$$ −5.64681 −0.217346
$$676$$ 0 0
$$677$$ 30.1801 1.15991 0.579957 0.814647i $$-0.303069\pi$$
0.579957 + 0.814647i $$0.303069\pi$$
$$678$$ 0 0
$$679$$ 2.97581 0.114201
$$680$$ 0 0
$$681$$ −43.3836 −1.66247
$$682$$ 0 0
$$683$$ 29.2832 1.12049 0.560245 0.828327i $$-0.310707\pi$$
0.560245 + 0.828327i $$0.310707\pi$$
$$684$$ 0 0
$$685$$ −14.0346 −0.536235
$$686$$ 0 0
$$687$$ 28.7881 1.09834
$$688$$ 0 0
$$689$$ 0.0208243 0.000793344 0
$$690$$ 0 0
$$691$$ −14.9557 −0.568940 −0.284470 0.958685i $$-0.591818\pi$$
−0.284470 + 0.958685i $$0.591818\pi$$
$$692$$ 0 0
$$693$$ 1.13919 0.0432744
$$694$$ 0 0
$$695$$ 14.2638 0.541058
$$696$$ 0 0
$$697$$ −54.9694 −2.08212
$$698$$ 0 0
$$699$$ −8.08377 −0.305756
$$700$$ 0 0
$$701$$ 24.9627 0.942828 0.471414 0.881912i $$-0.343744\pi$$
0.471414 + 0.881912i $$0.343744\pi$$
$$702$$ 0 0
$$703$$ −63.5816 −2.39803
$$704$$ 0 0
$$705$$ −3.39821 −0.127984
$$706$$ 0 0
$$707$$ −23.7104 −0.891722
$$708$$ 0 0
$$709$$ −38.9717 −1.46361 −0.731806 0.681513i $$-0.761322\pi$$
−0.731806 + 0.681513i $$0.761322\pi$$
$$710$$ 0 0
$$711$$ −5.85039 −0.219407
$$712$$ 0 0
$$713$$ −6.78600 −0.254138
$$714$$ 0 0
$$715$$ −0.119819 −0.00448098
$$716$$ 0 0
$$717$$ −25.4466 −0.950320
$$718$$ 0 0
$$719$$ −25.2978 −0.943447 −0.471724 0.881746i $$-0.656368\pi$$
−0.471724 + 0.881746i $$0.656368\pi$$
$$720$$ 0 0
$$721$$ 12.1682 0.453168
$$722$$ 0 0
$$723$$ −23.5333 −0.875211
$$724$$ 0 0
$$725$$ 9.76663 0.362723
$$726$$ 0 0
$$727$$ 12.7417 0.472562 0.236281 0.971685i $$-0.424071\pi$$
0.236281 + 0.971685i $$0.424071\pi$$
$$728$$ 0 0
$$729$$ 29.6635 1.09865
$$730$$ 0 0
$$731$$ 63.0249 2.33106
$$732$$ 0 0
$$733$$ −9.83247 −0.363170 −0.181585 0.983375i $$-0.558123\pi$$
−0.181585 + 0.983375i $$0.558123\pi$$
$$734$$ 0 0
$$735$$ −6.78119 −0.250128
$$736$$ 0 0
$$737$$ 9.03605 0.332847
$$738$$ 0 0
$$739$$ 22.6918 0.834732 0.417366 0.908738i $$-0.362953\pi$$
0.417366 + 0.908738i $$0.362953\pi$$
$$740$$ 0 0
$$741$$ 1.07066 0.0393316
$$742$$ 0 0
$$743$$ 3.08858 0.113309 0.0566546 0.998394i $$-0.481957\pi$$
0.0566546 + 0.998394i $$0.481957\pi$$
$$744$$ 0 0
$$745$$ −4.46260 −0.163497
$$746$$ 0 0
$$747$$ −9.16484 −0.335324
$$748$$ 0 0
$$749$$ −6.57760 −0.240340
$$750$$ 0 0
$$751$$ −10.7577 −0.392553 −0.196276 0.980549i $$-0.562885\pi$$
−0.196276 + 0.980549i $$0.562885\pi$$
$$752$$ 0 0
$$753$$ −42.4882 −1.54836
$$754$$ 0 0
$$755$$ 10.7562 0.391459
$$756$$ 0 0
$$757$$ −19.5333 −0.709948 −0.354974 0.934876i $$-0.615510\pi$$
−0.354974 + 0.934876i $$0.615510\pi$$
$$758$$ 0 0
$$759$$ 1.25901 0.0456993
$$760$$ 0 0
$$761$$ −15.6933 −0.568881 −0.284440 0.958694i $$-0.591808\pi$$
−0.284440 + 0.958694i $$0.591808\pi$$
$$762$$ 0 0
$$763$$ −27.3304 −0.989429
$$764$$ 0 0
$$765$$ −4.74099 −0.171411
$$766$$ 0 0
$$767$$ −1.53326 −0.0553626
$$768$$ 0 0
$$769$$ 11.4128 0.411555 0.205777 0.978599i $$-0.434028\pi$$
0.205777 + 0.978599i $$0.434028\pi$$
$$770$$ 0 0
$$771$$ −9.59910 −0.345703
$$772$$ 0 0
$$773$$ −22.5872 −0.812406 −0.406203 0.913783i $$-0.633147\pi$$
−0.406203 + 0.913783i $$0.633147\pi$$
$$774$$ 0 0
$$775$$ 6.78600 0.243760
$$776$$ 0 0
$$777$$ −27.1857 −0.975280
$$778$$ 0 0
$$779$$ −52.4882 −1.88059
$$780$$ 0 0
$$781$$ −6.29507 −0.225255
$$782$$ 0 0
$$783$$ −55.1503 −1.97091
$$784$$ 0 0
$$785$$ 14.5180 0.518171
$$786$$ 0 0
$$787$$ −4.42799 −0.157841 −0.0789205 0.996881i $$-0.525147\pi$$
−0.0789205 + 0.996881i $$0.525147\pi$$
$$788$$ 0 0
$$789$$ −0.821385 −0.0292421
$$790$$ 0 0
$$791$$ −4.92520 −0.175120
$$792$$ 0 0
$$793$$ 0.617021 0.0219111
$$794$$ 0 0
$$795$$ 0.218814 0.00776053
$$796$$ 0 0
$$797$$ 21.5124 0.762009 0.381005 0.924573i $$-0.375578\pi$$
0.381005 + 0.924573i $$0.375578\pi$$
$$798$$ 0 0
$$799$$ −12.7964 −0.452705
$$800$$ 0 0
$$801$$ 14.8864 0.525987
$$802$$ 0 0
$$803$$ −6.12878 −0.216280
$$804$$ 0 0
$$805$$ −1.53740 −0.0541863
$$806$$ 0 0
$$807$$ −10.6323 −0.374273
$$808$$ 0 0
$$809$$ 7.37883 0.259426 0.129713 0.991552i $$-0.458594\pi$$
0.129713 + 0.991552i $$0.458594\pi$$
$$810$$ 0 0
$$811$$ 29.0423 1.01981 0.509907 0.860230i $$-0.329680\pi$$
0.509907 + 0.860230i $$0.329680\pi$$
$$812$$ 0 0
$$813$$ 0.269425 0.00944916
$$814$$ 0 0
$$815$$ −1.30403 −0.0456782
$$816$$ 0 0
$$817$$ 60.1801 2.10543
$$818$$ 0 0
$$819$$ −0.184210 −0.00643682
$$820$$ 0 0
$$821$$ 5.70079 0.198959 0.0994794 0.995040i $$-0.468282\pi$$
0.0994794 + 0.995040i $$0.468282\pi$$
$$822$$ 0 0
$$823$$ −16.0034 −0.557842 −0.278921 0.960314i $$-0.589977\pi$$
−0.278921 + 0.960314i $$0.589977\pi$$
$$824$$ 0 0
$$825$$ −1.25901 −0.0438332
$$826$$ 0 0
$$827$$ 39.7937 1.38376 0.691882 0.722011i $$-0.256782\pi$$
0.691882 + 0.722011i $$0.256782\pi$$
$$828$$ 0 0
$$829$$ −23.3353 −0.810467 −0.405234 0.914213i $$-0.632810\pi$$
−0.405234 + 0.914213i $$0.632810\pi$$
$$830$$ 0 0
$$831$$ 37.8746 1.31385
$$832$$ 0 0
$$833$$ −25.5355 −0.884752
$$834$$ 0 0
$$835$$ 3.44322 0.119158
$$836$$ 0 0
$$837$$ −38.3193 −1.32451
$$838$$ 0 0
$$839$$ 37.1469 1.28245 0.641227 0.767351i $$-0.278426\pi$$
0.641227 + 0.767351i $$0.278426\pi$$
$$840$$ 0 0
$$841$$ 66.3870 2.28921
$$842$$ 0 0
$$843$$ −39.1619 −1.34881
$$844$$ 0 0
$$845$$ −12.9806 −0.446547
$$846$$ 0 0
$$847$$ −15.7722 −0.541940
$$848$$ 0 0
$$849$$ 8.33796 0.286158
$$850$$ 0 0
$$851$$ 12.0900 0.414441
$$852$$ 0 0
$$853$$ −53.7658 −1.84091 −0.920454 0.390851i $$-0.872181\pi$$
−0.920454 + 0.390851i $$0.872181\pi$$
$$854$$ 0 0
$$855$$ −4.52699 −0.154820
$$856$$ 0 0
$$857$$ −23.9675 −0.818715 −0.409357 0.912374i $$-0.634247\pi$$
−0.409357 + 0.912374i $$0.634247\pi$$
$$858$$ 0 0
$$859$$ 3.70705 0.126483 0.0632415 0.997998i $$-0.479856\pi$$
0.0632415 + 0.997998i $$0.479856\pi$$
$$860$$ 0 0
$$861$$ −22.4424 −0.764836
$$862$$ 0 0
$$863$$ 49.6296 1.68941 0.844705 0.535232i $$-0.179776\pi$$
0.844705 + 0.535232i $$0.179776\pi$$
$$864$$ 0 0
$$865$$ 9.35801 0.318182
$$866$$ 0 0
$$867$$ 19.5020 0.662323
$$868$$ 0 0
$$869$$ −5.85039 −0.198461
$$870$$ 0 0
$$871$$ −1.46115 −0.0495091
$$872$$ 0 0
$$873$$ −1.66618 −0.0563918
$$874$$ 0 0
$$875$$ 1.53740 0.0519737
$$876$$ 0 0
$$877$$ −13.1004 −0.442371 −0.221185 0.975232i $$-0.570993\pi$$
−0.221185 + 0.975232i $$0.570993\pi$$
$$878$$ 0 0
$$879$$ 1.54155 0.0519951
$$880$$ 0 0
$$881$$ 3.88085 0.130749 0.0653746 0.997861i $$-0.479176\pi$$
0.0653746 + 0.997861i $$0.479176\pi$$
$$882$$ 0 0
$$883$$ −10.8131 −0.363890 −0.181945 0.983309i $$-0.558239\pi$$
−0.181945 + 0.983309i $$0.558239\pi$$
$$884$$ 0 0
$$885$$ −16.1109 −0.541561
$$886$$ 0 0
$$887$$ −8.58387 −0.288218 −0.144109 0.989562i $$-0.546032\pi$$
−0.144109 + 0.989562i $$0.546032\pi$$
$$888$$ 0 0
$$889$$ −23.3836 −0.784262
$$890$$ 0 0
$$891$$ 4.88645 0.163702
$$892$$ 0 0
$$893$$ −12.2188 −0.408887
$$894$$ 0 0
$$895$$ −18.4134 −0.615493
$$896$$ 0 0
$$897$$ −0.203585 −0.00679751
$$898$$ 0 0
$$899$$ 66.2764 2.21044
$$900$$ 0 0
$$901$$ 0.823974 0.0274505
$$902$$ 0 0
$$903$$ 25.7312 0.856282
$$904$$ 0 0
$$905$$ −1.94457 −0.0646398
$$906$$ 0 0
$$907$$ −21.2728 −0.706351 −0.353176 0.935557i $$-0.614898\pi$$
−0.353176 + 0.935557i $$0.614898\pi$$
$$908$$ 0 0
$$909$$ 13.2757 0.440327
$$910$$ 0 0
$$911$$ −27.6441 −0.915890 −0.457945 0.888980i $$-0.651414\pi$$
−0.457945 + 0.888980i $$0.651414\pi$$
$$912$$ 0 0
$$913$$ −9.16484 −0.303312
$$914$$ 0 0
$$915$$ 6.48342 0.214335
$$916$$ 0 0
$$917$$ 19.5124 0.644357
$$918$$ 0 0
$$919$$ −0.994404 −0.0328024 −0.0164012 0.999865i $$-0.505221\pi$$
−0.0164012 + 0.999865i $$0.505221\pi$$
$$920$$ 0 0
$$921$$ 26.1455 0.861522
$$922$$ 0 0
$$923$$ 1.01793 0.0335054
$$924$$ 0 0
$$925$$ −12.0900 −0.397518
$$926$$ 0 0
$$927$$ −6.81309 −0.223771
$$928$$ 0 0
$$929$$ −18.3747 −0.602854 −0.301427 0.953489i $$-0.597463\pi$$
−0.301427 + 0.953489i $$0.597463\pi$$
$$930$$ 0 0
$$931$$ −24.3829 −0.799116
$$932$$ 0 0
$$933$$ 24.7639 0.810735
$$934$$ 0 0
$$935$$ −4.74099 −0.155047
$$936$$ 0 0
$$937$$ −24.0942 −0.787122 −0.393561 0.919298i $$-0.628757\pi$$
−0.393561 + 0.919298i $$0.628757\pi$$
$$938$$ 0 0
$$939$$ −29.2459 −0.954404
$$940$$ 0 0
$$941$$ 51.9244 1.69269 0.846344 0.532637i $$-0.178799\pi$$
0.846344 + 0.532637i $$0.178799\pi$$
$$942$$ 0 0
$$943$$ 9.98062 0.325014
$$944$$ 0 0
$$945$$ −8.68141 −0.282406
$$946$$ 0 0
$$947$$ −4.72643 −0.153588 −0.0767941 0.997047i $$-0.524468\pi$$
−0.0767941 + 0.997047i $$0.524468\pi$$
$$948$$ 0 0
$$949$$ 0.991037 0.0321704
$$950$$ 0 0
$$951$$ 4.82330 0.156406
$$952$$ 0 0
$$953$$ 3.57682 0.115865 0.0579323 0.998321i $$-0.481549\pi$$
0.0579323 + 0.998321i $$0.481549\pi$$
$$954$$ 0 0
$$955$$ −0.775591 −0.0250975
$$956$$ 0 0
$$957$$ −12.2963 −0.397483
$$958$$ 0 0
$$959$$ −21.5768 −0.696752
$$960$$ 0 0
$$961$$ 15.0498 0.485478
$$962$$ 0 0
$$963$$ 3.68286 0.118679
$$964$$ 0 0
$$965$$ −11.7666 −0.378781
$$966$$ 0 0
$$967$$ 33.9646 1.09223 0.546114 0.837711i $$-0.316106\pi$$
0.546114 + 0.837711i $$0.316106\pi$$
$$968$$ 0 0
$$969$$ 42.3636 1.36092
$$970$$ 0 0
$$971$$ −56.0561 −1.79893 −0.899463 0.436997i $$-0.856042\pi$$
−0.899463 + 0.436997i $$0.856042\pi$$
$$972$$ 0 0
$$973$$ 21.9292 0.703019
$$974$$ 0 0
$$975$$ 0.203585 0.00651994
$$976$$ 0 0
$$977$$ −53.1219 −1.69952 −0.849761 0.527169i $$-0.823254\pi$$
−0.849761 + 0.527169i $$0.823254\pi$$
$$978$$ 0 0
$$979$$ 14.8864 0.475773
$$980$$ 0 0
$$981$$ 15.3026 0.488574
$$982$$ 0 0
$$983$$ 28.7833 0.918045 0.459022 0.888425i $$-0.348200\pi$$
0.459022 + 0.888425i $$0.348200\pi$$
$$984$$ 0 0
$$985$$ 9.68141 0.308475
$$986$$ 0 0
$$987$$ −5.22441 −0.166295
$$988$$ 0 0
$$989$$ −11.4432 −0.363873
$$990$$ 0 0
$$991$$ 48.1641 1.52998 0.764991 0.644041i $$-0.222743\pi$$
0.764991 + 0.644041i $$0.222743\pi$$
$$992$$ 0 0
$$993$$ −18.8385 −0.597823
$$994$$ 0 0
$$995$$ −3.29362 −0.104415
$$996$$ 0 0
$$997$$ 43.5153 1.37814 0.689072 0.724693i $$-0.258018\pi$$
0.689072 + 0.724693i $$0.258018\pi$$
$$998$$ 0 0
$$999$$ 68.2701 2.15997
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 920.2.a.i.1.2 3
3.2 odd 2 8280.2.a.bl.1.2 3
4.3 odd 2 1840.2.a.q.1.2 3
5.2 odd 4 4600.2.e.q.4049.3 6
5.3 odd 4 4600.2.e.q.4049.4 6
5.4 even 2 4600.2.a.v.1.2 3
8.3 odd 2 7360.2.a.cf.1.2 3
8.5 even 2 7360.2.a.bw.1.2 3
20.19 odd 2 9200.2.a.ci.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.i.1.2 3 1.1 even 1 trivial
1840.2.a.q.1.2 3 4.3 odd 2
4600.2.a.v.1.2 3 5.4 even 2
4600.2.e.q.4049.3 6 5.2 odd 4
4600.2.e.q.4049.4 6 5.3 odd 4
7360.2.a.bw.1.2 3 8.5 even 2
7360.2.a.cf.1.2 3 8.3 odd 2
8280.2.a.bl.1.2 3 3.2 odd 2
9200.2.a.ci.1.2 3 20.19 odd 2