# Properties

 Label 1520.2.a.n.1.2 Level $1520$ Weight $2$ Character 1520.1 Self dual yes Analytic conductor $12.137$ Analytic rank $0$ 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] = [1520,2,Mod(1,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, 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("1520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1520 = 2^{4} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1520.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: $$12.1372611072$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 190) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 1520.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.56155 q^{3} +1.00000 q^{5} +2.56155 q^{7} +3.56155 q^{9} +O(q^{10})$$ $$q+2.56155 q^{3} +1.00000 q^{5} +2.56155 q^{7} +3.56155 q^{9} -4.00000 q^{11} +5.68466 q^{13} +2.56155 q^{15} +3.43845 q^{17} +1.00000 q^{19} +6.56155 q^{21} -7.68466 q^{23} +1.00000 q^{25} +1.43845 q^{27} -5.68466 q^{29} +5.12311 q^{31} -10.2462 q^{33} +2.56155 q^{35} -6.00000 q^{37} +14.5616 q^{39} +12.2462 q^{41} +2.87689 q^{43} +3.56155 q^{45} -6.24621 q^{47} -0.438447 q^{49} +8.80776 q^{51} -4.56155 q^{53} -4.00000 q^{55} +2.56155 q^{57} -2.56155 q^{59} +11.1231 q^{61} +9.12311 q^{63} +5.68466 q^{65} +2.56155 q^{67} -19.6847 q^{69} -10.2462 q^{71} -1.68466 q^{73} +2.56155 q^{75} -10.2462 q^{77} +5.12311 q^{79} -7.00000 q^{81} -2.87689 q^{83} +3.43845 q^{85} -14.5616 q^{87} +2.00000 q^{89} +14.5616 q^{91} +13.1231 q^{93} +1.00000 q^{95} +6.00000 q^{97} -14.2462 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + 2 q^{5} + q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q + q^3 + 2 * q^5 + q^7 + 3 * q^9 $$2 q + q^{3} + 2 q^{5} + q^{7} + 3 q^{9} - 8 q^{11} - q^{13} + q^{15} + 11 q^{17} + 2 q^{19} + 9 q^{21} - 3 q^{23} + 2 q^{25} + 7 q^{27} + q^{29} + 2 q^{31} - 4 q^{33} + q^{35} - 12 q^{37} + 25 q^{39} + 8 q^{41} + 14 q^{43} + 3 q^{45} + 4 q^{47} - 5 q^{49} - 3 q^{51} - 5 q^{53} - 8 q^{55} + q^{57} - q^{59} + 14 q^{61} + 10 q^{63} - q^{65} + q^{67} - 27 q^{69} - 4 q^{71} + 9 q^{73} + q^{75} - 4 q^{77} + 2 q^{79} - 14 q^{81} - 14 q^{83} + 11 q^{85} - 25 q^{87} + 4 q^{89} + 25 q^{91} + 18 q^{93} + 2 q^{95} + 12 q^{97} - 12 q^{99}+O(q^{100})$$ 2 * q + q^3 + 2 * q^5 + q^7 + 3 * q^9 - 8 * q^11 - q^13 + q^15 + 11 * q^17 + 2 * q^19 + 9 * q^21 - 3 * q^23 + 2 * q^25 + 7 * q^27 + q^29 + 2 * q^31 - 4 * q^33 + q^35 - 12 * q^37 + 25 * q^39 + 8 * q^41 + 14 * q^43 + 3 * q^45 + 4 * q^47 - 5 * q^49 - 3 * q^51 - 5 * q^53 - 8 * q^55 + q^57 - q^59 + 14 * q^61 + 10 * q^63 - q^65 + q^67 - 27 * q^69 - 4 * q^71 + 9 * q^73 + q^75 - 4 * q^77 + 2 * q^79 - 14 * q^81 - 14 * q^83 + 11 * q^85 - 25 * q^87 + 4 * q^89 + 25 * q^91 + 18 * q^93 + 2 * q^95 + 12 * q^97 - 12 * 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
$$3$$ 2.56155 1.47891 0.739457 0.673204i $$-0.235083\pi$$
0.739457 + 0.673204i $$0.235083\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 2.56155 0.968176 0.484088 0.875019i $$-0.339151\pi$$
0.484088 + 0.875019i $$0.339151\pi$$
$$8$$ 0 0
$$9$$ 3.56155 1.18718
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 0 0
$$13$$ 5.68466 1.57664 0.788320 0.615265i $$-0.210951\pi$$
0.788320 + 0.615265i $$0.210951\pi$$
$$14$$ 0 0
$$15$$ 2.56155 0.661390
$$16$$ 0 0
$$17$$ 3.43845 0.833946 0.416973 0.908919i $$-0.363091\pi$$
0.416973 + 0.908919i $$0.363091\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 6.56155 1.43185
$$22$$ 0 0
$$23$$ −7.68466 −1.60236 −0.801181 0.598422i $$-0.795795\pi$$
−0.801181 + 0.598422i $$0.795795\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.43845 0.276829
$$28$$ 0 0
$$29$$ −5.68466 −1.05561 −0.527807 0.849364i $$-0.676986\pi$$
−0.527807 + 0.849364i $$0.676986\pi$$
$$30$$ 0 0
$$31$$ 5.12311 0.920137 0.460068 0.887883i $$-0.347825\pi$$
0.460068 + 0.887883i $$0.347825\pi$$
$$32$$ 0 0
$$33$$ −10.2462 −1.78364
$$34$$ 0 0
$$35$$ 2.56155 0.432981
$$36$$ 0 0
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ 0 0
$$39$$ 14.5616 2.33171
$$40$$ 0 0
$$41$$ 12.2462 1.91254 0.956268 0.292490i $$-0.0944840\pi$$
0.956268 + 0.292490i $$0.0944840\pi$$
$$42$$ 0 0
$$43$$ 2.87689 0.438722 0.219361 0.975644i $$-0.429603\pi$$
0.219361 + 0.975644i $$0.429603\pi$$
$$44$$ 0 0
$$45$$ 3.56155 0.530925
$$46$$ 0 0
$$47$$ −6.24621 −0.911104 −0.455552 0.890209i $$-0.650558\pi$$
−0.455552 + 0.890209i $$0.650558\pi$$
$$48$$ 0 0
$$49$$ −0.438447 −0.0626353
$$50$$ 0 0
$$51$$ 8.80776 1.23333
$$52$$ 0 0
$$53$$ −4.56155 −0.626577 −0.313289 0.949658i $$-0.601431\pi$$
−0.313289 + 0.949658i $$0.601431\pi$$
$$54$$ 0 0
$$55$$ −4.00000 −0.539360
$$56$$ 0 0
$$57$$ 2.56155 0.339286
$$58$$ 0 0
$$59$$ −2.56155 −0.333486 −0.166743 0.986000i $$-0.553325\pi$$
−0.166743 + 0.986000i $$0.553325\pi$$
$$60$$ 0 0
$$61$$ 11.1231 1.42417 0.712084 0.702094i $$-0.247752\pi$$
0.712084 + 0.702094i $$0.247752\pi$$
$$62$$ 0 0
$$63$$ 9.12311 1.14940
$$64$$ 0 0
$$65$$ 5.68466 0.705095
$$66$$ 0 0
$$67$$ 2.56155 0.312943 0.156472 0.987682i $$-0.449988\pi$$
0.156472 + 0.987682i $$0.449988\pi$$
$$68$$ 0 0
$$69$$ −19.6847 −2.36975
$$70$$ 0 0
$$71$$ −10.2462 −1.21600 −0.608001 0.793936i $$-0.708028\pi$$
−0.608001 + 0.793936i $$0.708028\pi$$
$$72$$ 0 0
$$73$$ −1.68466 −0.197174 −0.0985872 0.995128i $$-0.531432\pi$$
−0.0985872 + 0.995128i $$0.531432\pi$$
$$74$$ 0 0
$$75$$ 2.56155 0.295783
$$76$$ 0 0
$$77$$ −10.2462 −1.16766
$$78$$ 0 0
$$79$$ 5.12311 0.576394 0.288197 0.957571i $$-0.406944\pi$$
0.288197 + 0.957571i $$0.406944\pi$$
$$80$$ 0 0
$$81$$ −7.00000 −0.777778
$$82$$ 0 0
$$83$$ −2.87689 −0.315780 −0.157890 0.987457i $$-0.550469\pi$$
−0.157890 + 0.987457i $$0.550469\pi$$
$$84$$ 0 0
$$85$$ 3.43845 0.372952
$$86$$ 0 0
$$87$$ −14.5616 −1.56116
$$88$$ 0 0
$$89$$ 2.00000 0.212000 0.106000 0.994366i $$-0.466196\pi$$
0.106000 + 0.994366i $$0.466196\pi$$
$$90$$ 0 0
$$91$$ 14.5616 1.52647
$$92$$ 0 0
$$93$$ 13.1231 1.36080
$$94$$ 0 0
$$95$$ 1.00000 0.102598
$$96$$ 0 0
$$97$$ 6.00000 0.609208 0.304604 0.952479i $$-0.401476\pi$$
0.304604 + 0.952479i $$0.401476\pi$$
$$98$$ 0 0
$$99$$ −14.2462 −1.43180
$$100$$ 0 0
$$101$$ −17.3693 −1.72831 −0.864156 0.503224i $$-0.832147\pi$$
−0.864156 + 0.503224i $$0.832147\pi$$
$$102$$ 0 0
$$103$$ −2.24621 −0.221326 −0.110663 0.993858i $$-0.535297\pi$$
−0.110663 + 0.993858i $$0.535297\pi$$
$$104$$ 0 0
$$105$$ 6.56155 0.640342
$$106$$ 0 0
$$107$$ 5.43845 0.525755 0.262877 0.964829i $$-0.415329\pi$$
0.262877 + 0.964829i $$0.415329\pi$$
$$108$$ 0 0
$$109$$ −0.561553 −0.0537870 −0.0268935 0.999638i $$-0.508561\pi$$
−0.0268935 + 0.999638i $$0.508561\pi$$
$$110$$ 0 0
$$111$$ −15.3693 −1.45879
$$112$$ 0 0
$$113$$ 8.87689 0.835068 0.417534 0.908661i $$-0.362894\pi$$
0.417534 + 0.908661i $$0.362894\pi$$
$$114$$ 0 0
$$115$$ −7.68466 −0.716598
$$116$$ 0 0
$$117$$ 20.2462 1.87176
$$118$$ 0 0
$$119$$ 8.80776 0.807406
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 0 0
$$123$$ 31.3693 2.82848
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −13.1231 −1.16449 −0.582244 0.813014i $$-0.697825\pi$$
−0.582244 + 0.813014i $$0.697825\pi$$
$$128$$ 0 0
$$129$$ 7.36932 0.648832
$$130$$ 0 0
$$131$$ 16.4924 1.44095 0.720475 0.693481i $$-0.243924\pi$$
0.720475 + 0.693481i $$0.243924\pi$$
$$132$$ 0 0
$$133$$ 2.56155 0.222115
$$134$$ 0 0
$$135$$ 1.43845 0.123802
$$136$$ 0 0
$$137$$ −14.8078 −1.26511 −0.632556 0.774514i $$-0.717994\pi$$
−0.632556 + 0.774514i $$0.717994\pi$$
$$138$$ 0 0
$$139$$ −16.4924 −1.39887 −0.699435 0.714697i $$-0.746565\pi$$
−0.699435 + 0.714697i $$0.746565\pi$$
$$140$$ 0 0
$$141$$ −16.0000 −1.34744
$$142$$ 0 0
$$143$$ −22.7386 −1.90150
$$144$$ 0 0
$$145$$ −5.68466 −0.472085
$$146$$ 0 0
$$147$$ −1.12311 −0.0926322
$$148$$ 0 0
$$149$$ 13.3693 1.09526 0.547629 0.836722i $$-0.315531\pi$$
0.547629 + 0.836722i $$0.315531\pi$$
$$150$$ 0 0
$$151$$ −5.12311 −0.416912 −0.208456 0.978032i $$-0.566844\pi$$
−0.208456 + 0.978032i $$0.566844\pi$$
$$152$$ 0 0
$$153$$ 12.2462 0.990048
$$154$$ 0 0
$$155$$ 5.12311 0.411498
$$156$$ 0 0
$$157$$ −20.2462 −1.61582 −0.807912 0.589303i $$-0.799402\pi$$
−0.807912 + 0.589303i $$0.799402\pi$$
$$158$$ 0 0
$$159$$ −11.6847 −0.926654
$$160$$ 0 0
$$161$$ −19.6847 −1.55137
$$162$$ 0 0
$$163$$ 15.3693 1.20382 0.601909 0.798565i $$-0.294407\pi$$
0.601909 + 0.798565i $$0.294407\pi$$
$$164$$ 0 0
$$165$$ −10.2462 −0.797666
$$166$$ 0 0
$$167$$ 7.36932 0.570255 0.285127 0.958490i $$-0.407964\pi$$
0.285127 + 0.958490i $$0.407964\pi$$
$$168$$ 0 0
$$169$$ 19.3153 1.48580
$$170$$ 0 0
$$171$$ 3.56155 0.272359
$$172$$ 0 0
$$173$$ 20.2462 1.53929 0.769645 0.638471i $$-0.220433\pi$$
0.769645 + 0.638471i $$0.220433\pi$$
$$174$$ 0 0
$$175$$ 2.56155 0.193635
$$176$$ 0 0
$$177$$ −6.56155 −0.493197
$$178$$ 0 0
$$179$$ 22.2462 1.66276 0.831380 0.555704i $$-0.187551\pi$$
0.831380 + 0.555704i $$0.187551\pi$$
$$180$$ 0 0
$$181$$ −18.0000 −1.33793 −0.668965 0.743294i $$-0.733262\pi$$
−0.668965 + 0.743294i $$0.733262\pi$$
$$182$$ 0 0
$$183$$ 28.4924 2.10622
$$184$$ 0 0
$$185$$ −6.00000 −0.441129
$$186$$ 0 0
$$187$$ −13.7538 −1.00578
$$188$$ 0 0
$$189$$ 3.68466 0.268019
$$190$$ 0 0
$$191$$ 3.68466 0.266613 0.133306 0.991075i $$-0.457441\pi$$
0.133306 + 0.991075i $$0.457441\pi$$
$$192$$ 0 0
$$193$$ −14.4924 −1.04319 −0.521594 0.853194i $$-0.674662\pi$$
−0.521594 + 0.853194i $$0.674662\pi$$
$$194$$ 0 0
$$195$$ 14.5616 1.04277
$$196$$ 0 0
$$197$$ −20.2462 −1.44248 −0.721241 0.692684i $$-0.756428\pi$$
−0.721241 + 0.692684i $$0.756428\pi$$
$$198$$ 0 0
$$199$$ −16.8078 −1.19147 −0.595735 0.803181i $$-0.703139\pi$$
−0.595735 + 0.803181i $$0.703139\pi$$
$$200$$ 0 0
$$201$$ 6.56155 0.462816
$$202$$ 0 0
$$203$$ −14.5616 −1.02202
$$204$$ 0 0
$$205$$ 12.2462 0.855312
$$206$$ 0 0
$$207$$ −27.3693 −1.90230
$$208$$ 0 0
$$209$$ −4.00000 −0.276686
$$210$$ 0 0
$$211$$ −8.31534 −0.572452 −0.286226 0.958162i $$-0.592401\pi$$
−0.286226 + 0.958162i $$0.592401\pi$$
$$212$$ 0 0
$$213$$ −26.2462 −1.79836
$$214$$ 0 0
$$215$$ 2.87689 0.196203
$$216$$ 0 0
$$217$$ 13.1231 0.890854
$$218$$ 0 0
$$219$$ −4.31534 −0.291604
$$220$$ 0 0
$$221$$ 19.5464 1.31483
$$222$$ 0 0
$$223$$ 23.3693 1.56493 0.782463 0.622698i $$-0.213963\pi$$
0.782463 + 0.622698i $$0.213963\pi$$
$$224$$ 0 0
$$225$$ 3.56155 0.237437
$$226$$ 0 0
$$227$$ −25.9309 −1.72109 −0.860546 0.509373i $$-0.829878\pi$$
−0.860546 + 0.509373i $$0.829878\pi$$
$$228$$ 0 0
$$229$$ −14.4924 −0.957686 −0.478843 0.877900i $$-0.658944\pi$$
−0.478843 + 0.877900i $$0.658944\pi$$
$$230$$ 0 0
$$231$$ −26.2462 −1.72687
$$232$$ 0 0
$$233$$ 10.0000 0.655122 0.327561 0.944830i $$-0.393773\pi$$
0.327561 + 0.944830i $$0.393773\pi$$
$$234$$ 0 0
$$235$$ −6.24621 −0.407458
$$236$$ 0 0
$$237$$ 13.1231 0.852437
$$238$$ 0 0
$$239$$ 1.43845 0.0930454 0.0465227 0.998917i $$-0.485186\pi$$
0.0465227 + 0.998917i $$0.485186\pi$$
$$240$$ 0 0
$$241$$ 23.1231 1.48949 0.744745 0.667349i $$-0.232571\pi$$
0.744745 + 0.667349i $$0.232571\pi$$
$$242$$ 0 0
$$243$$ −22.2462 −1.42710
$$244$$ 0 0
$$245$$ −0.438447 −0.0280114
$$246$$ 0 0
$$247$$ 5.68466 0.361706
$$248$$ 0 0
$$249$$ −7.36932 −0.467011
$$250$$ 0 0
$$251$$ −6.24621 −0.394257 −0.197129 0.980378i $$-0.563162\pi$$
−0.197129 + 0.980378i $$0.563162\pi$$
$$252$$ 0 0
$$253$$ 30.7386 1.93252
$$254$$ 0 0
$$255$$ 8.80776 0.551564
$$256$$ 0 0
$$257$$ 14.0000 0.873296 0.436648 0.899632i $$-0.356166\pi$$
0.436648 + 0.899632i $$0.356166\pi$$
$$258$$ 0 0
$$259$$ −15.3693 −0.955003
$$260$$ 0 0
$$261$$ −20.2462 −1.25321
$$262$$ 0 0
$$263$$ 22.2462 1.37176 0.685880 0.727715i $$-0.259417\pi$$
0.685880 + 0.727715i $$0.259417\pi$$
$$264$$ 0 0
$$265$$ −4.56155 −0.280214
$$266$$ 0 0
$$267$$ 5.12311 0.313529
$$268$$ 0 0
$$269$$ −26.0000 −1.58525 −0.792624 0.609711i $$-0.791286\pi$$
−0.792624 + 0.609711i $$0.791286\pi$$
$$270$$ 0 0
$$271$$ 21.9309 1.33221 0.666103 0.745860i $$-0.267961\pi$$
0.666103 + 0.745860i $$0.267961\pi$$
$$272$$ 0 0
$$273$$ 37.3002 2.25751
$$274$$ 0 0
$$275$$ −4.00000 −0.241209
$$276$$ 0 0
$$277$$ 0.876894 0.0526875 0.0263437 0.999653i $$-0.491614\pi$$
0.0263437 + 0.999653i $$0.491614\pi$$
$$278$$ 0 0
$$279$$ 18.2462 1.09237
$$280$$ 0 0
$$281$$ 2.00000 0.119310 0.0596550 0.998219i $$-0.481000\pi$$
0.0596550 + 0.998219i $$0.481000\pi$$
$$282$$ 0 0
$$283$$ 21.1231 1.25564 0.627819 0.778359i $$-0.283948\pi$$
0.627819 + 0.778359i $$0.283948\pi$$
$$284$$ 0 0
$$285$$ 2.56155 0.151733
$$286$$ 0 0
$$287$$ 31.3693 1.85167
$$288$$ 0 0
$$289$$ −5.17708 −0.304534
$$290$$ 0 0
$$291$$ 15.3693 0.900965
$$292$$ 0 0
$$293$$ −22.1771 −1.29560 −0.647799 0.761811i $$-0.724311\pi$$
−0.647799 + 0.761811i $$0.724311\pi$$
$$294$$ 0 0
$$295$$ −2.56155 −0.149139
$$296$$ 0 0
$$297$$ −5.75379 −0.333869
$$298$$ 0 0
$$299$$ −43.6847 −2.52635
$$300$$ 0 0
$$301$$ 7.36932 0.424760
$$302$$ 0 0
$$303$$ −44.4924 −2.55602
$$304$$ 0 0
$$305$$ 11.1231 0.636907
$$306$$ 0 0
$$307$$ −32.4924 −1.85444 −0.927220 0.374516i $$-0.877809\pi$$
−0.927220 + 0.374516i $$0.877809\pi$$
$$308$$ 0 0
$$309$$ −5.75379 −0.327322
$$310$$ 0 0
$$311$$ 3.68466 0.208938 0.104469 0.994528i $$-0.466686\pi$$
0.104469 + 0.994528i $$0.466686\pi$$
$$312$$ 0 0
$$313$$ 5.05398 0.285668 0.142834 0.989747i $$-0.454379\pi$$
0.142834 + 0.989747i $$0.454379\pi$$
$$314$$ 0 0
$$315$$ 9.12311 0.514029
$$316$$ 0 0
$$317$$ 13.0540 0.733184 0.366592 0.930382i $$-0.380524\pi$$
0.366592 + 0.930382i $$0.380524\pi$$
$$318$$ 0 0
$$319$$ 22.7386 1.27312
$$320$$ 0 0
$$321$$ 13.9309 0.777545
$$322$$ 0 0
$$323$$ 3.43845 0.191320
$$324$$ 0 0
$$325$$ 5.68466 0.315328
$$326$$ 0 0
$$327$$ −1.43845 −0.0795463
$$328$$ 0 0
$$329$$ −16.0000 −0.882109
$$330$$ 0 0
$$331$$ 2.56155 0.140796 0.0703978 0.997519i $$-0.477573\pi$$
0.0703978 + 0.997519i $$0.477573\pi$$
$$332$$ 0 0
$$333$$ −21.3693 −1.17103
$$334$$ 0 0
$$335$$ 2.56155 0.139953
$$336$$ 0 0
$$337$$ −26.0000 −1.41631 −0.708155 0.706057i $$-0.750472\pi$$
−0.708155 + 0.706057i $$0.750472\pi$$
$$338$$ 0 0
$$339$$ 22.7386 1.23499
$$340$$ 0 0
$$341$$ −20.4924 −1.10973
$$342$$ 0 0
$$343$$ −19.0540 −1.02882
$$344$$ 0 0
$$345$$ −19.6847 −1.05979
$$346$$ 0 0
$$347$$ 8.63068 0.463319 0.231660 0.972797i $$-0.425584\pi$$
0.231660 + 0.972797i $$0.425584\pi$$
$$348$$ 0 0
$$349$$ 3.75379 0.200936 0.100468 0.994940i $$-0.467966\pi$$
0.100468 + 0.994940i $$0.467966\pi$$
$$350$$ 0 0
$$351$$ 8.17708 0.436460
$$352$$ 0 0
$$353$$ −3.93087 −0.209219 −0.104610 0.994513i $$-0.533359\pi$$
−0.104610 + 0.994513i $$0.533359\pi$$
$$354$$ 0 0
$$355$$ −10.2462 −0.543812
$$356$$ 0 0
$$357$$ 22.5616 1.19408
$$358$$ 0 0
$$359$$ −1.43845 −0.0759183 −0.0379592 0.999279i $$-0.512086\pi$$
−0.0379592 + 0.999279i $$0.512086\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 12.8078 0.672233
$$364$$ 0 0
$$365$$ −1.68466 −0.0881791
$$366$$ 0 0
$$367$$ −6.24621 −0.326050 −0.163025 0.986622i $$-0.552125\pi$$
−0.163025 + 0.986622i $$0.552125\pi$$
$$368$$ 0 0
$$369$$ 43.6155 2.27053
$$370$$ 0 0
$$371$$ −11.6847 −0.606637
$$372$$ 0 0
$$373$$ −23.4384 −1.21360 −0.606798 0.794856i $$-0.707546\pi$$
−0.606798 + 0.794856i $$0.707546\pi$$
$$374$$ 0 0
$$375$$ 2.56155 0.132278
$$376$$ 0 0
$$377$$ −32.3153 −1.66432
$$378$$ 0 0
$$379$$ −10.5616 −0.542511 −0.271255 0.962507i $$-0.587439\pi$$
−0.271255 + 0.962507i $$0.587439\pi$$
$$380$$ 0 0
$$381$$ −33.6155 −1.72218
$$382$$ 0 0
$$383$$ −13.7538 −0.702786 −0.351393 0.936228i $$-0.614292\pi$$
−0.351393 + 0.936228i $$0.614292\pi$$
$$384$$ 0 0
$$385$$ −10.2462 −0.522195
$$386$$ 0 0
$$387$$ 10.2462 0.520844
$$388$$ 0 0
$$389$$ −7.12311 −0.361156 −0.180578 0.983561i $$-0.557797\pi$$
−0.180578 + 0.983561i $$0.557797\pi$$
$$390$$ 0 0
$$391$$ −26.4233 −1.33628
$$392$$ 0 0
$$393$$ 42.2462 2.13104
$$394$$ 0 0
$$395$$ 5.12311 0.257771
$$396$$ 0 0
$$397$$ −7.12311 −0.357498 −0.178749 0.983895i $$-0.557205\pi$$
−0.178749 + 0.983895i $$0.557205\pi$$
$$398$$ 0 0
$$399$$ 6.56155 0.328489
$$400$$ 0 0
$$401$$ −3.75379 −0.187455 −0.0937276 0.995598i $$-0.529878\pi$$
−0.0937276 + 0.995598i $$0.529878\pi$$
$$402$$ 0 0
$$403$$ 29.1231 1.45073
$$404$$ 0 0
$$405$$ −7.00000 −0.347833
$$406$$ 0 0
$$407$$ 24.0000 1.18964
$$408$$ 0 0
$$409$$ 24.7386 1.22325 0.611623 0.791149i $$-0.290517\pi$$
0.611623 + 0.791149i $$0.290517\pi$$
$$410$$ 0 0
$$411$$ −37.9309 −1.87099
$$412$$ 0 0
$$413$$ −6.56155 −0.322873
$$414$$ 0 0
$$415$$ −2.87689 −0.141221
$$416$$ 0 0
$$417$$ −42.2462 −2.06881
$$418$$ 0 0
$$419$$ −23.8617 −1.16572 −0.582861 0.812572i $$-0.698067\pi$$
−0.582861 + 0.812572i $$0.698067\pi$$
$$420$$ 0 0
$$421$$ −23.9309 −1.16632 −0.583160 0.812358i $$-0.698184\pi$$
−0.583160 + 0.812358i $$0.698184\pi$$
$$422$$ 0 0
$$423$$ −22.2462 −1.08165
$$424$$ 0 0
$$425$$ 3.43845 0.166789
$$426$$ 0 0
$$427$$ 28.4924 1.37884
$$428$$ 0 0
$$429$$ −58.2462 −2.81215
$$430$$ 0 0
$$431$$ 16.0000 0.770693 0.385346 0.922772i $$-0.374082\pi$$
0.385346 + 0.922772i $$0.374082\pi$$
$$432$$ 0 0
$$433$$ 14.6307 0.703106 0.351553 0.936168i $$-0.385654\pi$$
0.351553 + 0.936168i $$0.385654\pi$$
$$434$$ 0 0
$$435$$ −14.5616 −0.698173
$$436$$ 0 0
$$437$$ −7.68466 −0.367607
$$438$$ 0 0
$$439$$ 13.1231 0.626332 0.313166 0.949698i $$-0.398610\pi$$
0.313166 + 0.949698i $$0.398610\pi$$
$$440$$ 0 0
$$441$$ −1.56155 −0.0743597
$$442$$ 0 0
$$443$$ −2.24621 −0.106721 −0.0533604 0.998575i $$-0.516993\pi$$
−0.0533604 + 0.998575i $$0.516993\pi$$
$$444$$ 0 0
$$445$$ 2.00000 0.0948091
$$446$$ 0 0
$$447$$ 34.2462 1.61979
$$448$$ 0 0
$$449$$ −28.7386 −1.35626 −0.678130 0.734942i $$-0.737209\pi$$
−0.678130 + 0.734942i $$0.737209\pi$$
$$450$$ 0 0
$$451$$ −48.9848 −2.30661
$$452$$ 0 0
$$453$$ −13.1231 −0.616577
$$454$$ 0 0
$$455$$ 14.5616 0.682656
$$456$$ 0 0
$$457$$ 6.31534 0.295419 0.147710 0.989031i $$-0.452810\pi$$
0.147710 + 0.989031i $$0.452810\pi$$
$$458$$ 0 0
$$459$$ 4.94602 0.230861
$$460$$ 0 0
$$461$$ 3.75379 0.174831 0.0874157 0.996172i $$-0.472139\pi$$
0.0874157 + 0.996172i $$0.472139\pi$$
$$462$$ 0 0
$$463$$ 30.2462 1.40566 0.702830 0.711358i $$-0.251919\pi$$
0.702830 + 0.711358i $$0.251919\pi$$
$$464$$ 0 0
$$465$$ 13.1231 0.608569
$$466$$ 0 0
$$467$$ −18.2462 −0.844334 −0.422167 0.906518i $$-0.638730\pi$$
−0.422167 + 0.906518i $$0.638730\pi$$
$$468$$ 0 0
$$469$$ 6.56155 0.302984
$$470$$ 0 0
$$471$$ −51.8617 −2.38966
$$472$$ 0 0
$$473$$ −11.5076 −0.529119
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ −16.2462 −0.743863
$$478$$ 0 0
$$479$$ −32.0000 −1.46212 −0.731059 0.682315i $$-0.760973\pi$$
−0.731059 + 0.682315i $$0.760973\pi$$
$$480$$ 0 0
$$481$$ −34.1080 −1.55519
$$482$$ 0 0
$$483$$ −50.4233 −2.29434
$$484$$ 0 0
$$485$$ 6.00000 0.272446
$$486$$ 0 0
$$487$$ 17.6155 0.798236 0.399118 0.916900i $$-0.369316\pi$$
0.399118 + 0.916900i $$0.369316\pi$$
$$488$$ 0 0
$$489$$ 39.3693 1.78034
$$490$$ 0 0
$$491$$ 1.12311 0.0506850 0.0253425 0.999679i $$-0.491932\pi$$
0.0253425 + 0.999679i $$0.491932\pi$$
$$492$$ 0 0
$$493$$ −19.5464 −0.880325
$$494$$ 0 0
$$495$$ −14.2462 −0.640320
$$496$$ 0 0
$$497$$ −26.2462 −1.17730
$$498$$ 0 0
$$499$$ −42.1080 −1.88501 −0.942505 0.334191i $$-0.891537\pi$$
−0.942505 + 0.334191i $$0.891537\pi$$
$$500$$ 0 0
$$501$$ 18.8769 0.843357
$$502$$ 0 0
$$503$$ 7.05398 0.314521 0.157261 0.987557i $$-0.449734\pi$$
0.157261 + 0.987557i $$0.449734\pi$$
$$504$$ 0 0
$$505$$ −17.3693 −0.772924
$$506$$ 0 0
$$507$$ 49.4773 2.19736
$$508$$ 0 0
$$509$$ 2.49242 0.110475 0.0552373 0.998473i $$-0.482408\pi$$
0.0552373 + 0.998473i $$0.482408\pi$$
$$510$$ 0 0
$$511$$ −4.31534 −0.190899
$$512$$ 0 0
$$513$$ 1.43845 0.0635090
$$514$$ 0 0
$$515$$ −2.24621 −0.0989799
$$516$$ 0 0
$$517$$ 24.9848 1.09883
$$518$$ 0 0
$$519$$ 51.8617 2.27648
$$520$$ 0 0
$$521$$ −3.12311 −0.136826 −0.0684129 0.997657i $$-0.521794\pi$$
−0.0684129 + 0.997657i $$0.521794\pi$$
$$522$$ 0 0
$$523$$ 31.6847 1.38547 0.692737 0.721191i $$-0.256405\pi$$
0.692737 + 0.721191i $$0.256405\pi$$
$$524$$ 0 0
$$525$$ 6.56155 0.286370
$$526$$ 0 0
$$527$$ 17.6155 0.767344
$$528$$ 0 0
$$529$$ 36.0540 1.56756
$$530$$ 0 0
$$531$$ −9.12311 −0.395909
$$532$$ 0 0
$$533$$ 69.6155 3.01538
$$534$$ 0 0
$$535$$ 5.43845 0.235125
$$536$$ 0 0
$$537$$ 56.9848 2.45908
$$538$$ 0 0
$$539$$ 1.75379 0.0755410
$$540$$ 0 0
$$541$$ −0.384472 −0.0165297 −0.00826487 0.999966i $$-0.502631\pi$$
−0.00826487 + 0.999966i $$0.502631\pi$$
$$542$$ 0 0
$$543$$ −46.1080 −1.97868
$$544$$ 0 0
$$545$$ −0.561553 −0.0240543
$$546$$ 0 0
$$547$$ −16.4924 −0.705165 −0.352583 0.935781i $$-0.614696\pi$$
−0.352583 + 0.935781i $$0.614696\pi$$
$$548$$ 0 0
$$549$$ 39.6155 1.69075
$$550$$ 0 0
$$551$$ −5.68466 −0.242175
$$552$$ 0 0
$$553$$ 13.1231 0.558051
$$554$$ 0 0
$$555$$ −15.3693 −0.652391
$$556$$ 0 0
$$557$$ 39.6155 1.67856 0.839282 0.543697i $$-0.182976\pi$$
0.839282 + 0.543697i $$0.182976\pi$$
$$558$$ 0 0
$$559$$ 16.3542 0.691707
$$560$$ 0 0
$$561$$ −35.2311 −1.48746
$$562$$ 0 0
$$563$$ 8.49242 0.357913 0.178956 0.983857i $$-0.442728\pi$$
0.178956 + 0.983857i $$0.442728\pi$$
$$564$$ 0 0
$$565$$ 8.87689 0.373454
$$566$$ 0 0
$$567$$ −17.9309 −0.753026
$$568$$ 0 0
$$569$$ −3.12311 −0.130927 −0.0654637 0.997855i $$-0.520853\pi$$
−0.0654637 + 0.997855i $$0.520853\pi$$
$$570$$ 0 0
$$571$$ −21.6155 −0.904582 −0.452291 0.891870i $$-0.649393\pi$$
−0.452291 + 0.891870i $$0.649393\pi$$
$$572$$ 0 0
$$573$$ 9.43845 0.394297
$$574$$ 0 0
$$575$$ −7.68466 −0.320472
$$576$$ 0 0
$$577$$ −10.3153 −0.429433 −0.214717 0.976676i $$-0.568883\pi$$
−0.214717 + 0.976676i $$0.568883\pi$$
$$578$$ 0 0
$$579$$ −37.1231 −1.54278
$$580$$ 0 0
$$581$$ −7.36932 −0.305731
$$582$$ 0 0
$$583$$ 18.2462 0.755681
$$584$$ 0 0
$$585$$ 20.2462 0.837078
$$586$$ 0 0
$$587$$ 7.36932 0.304164 0.152082 0.988368i $$-0.451402\pi$$
0.152082 + 0.988368i $$0.451402\pi$$
$$588$$ 0 0
$$589$$ 5.12311 0.211094
$$590$$ 0 0
$$591$$ −51.8617 −2.13331
$$592$$ 0 0
$$593$$ 7.75379 0.318410 0.159205 0.987246i $$-0.449107\pi$$
0.159205 + 0.987246i $$0.449107\pi$$
$$594$$ 0 0
$$595$$ 8.80776 0.361083
$$596$$ 0 0
$$597$$ −43.0540 −1.76208
$$598$$ 0 0
$$599$$ 11.8617 0.484658 0.242329 0.970194i $$-0.422089\pi$$
0.242329 + 0.970194i $$0.422089\pi$$
$$600$$ 0 0
$$601$$ 18.0000 0.734235 0.367118 0.930175i $$-0.380345\pi$$
0.367118 + 0.930175i $$0.380345\pi$$
$$602$$ 0 0
$$603$$ 9.12311 0.371522
$$604$$ 0 0
$$605$$ 5.00000 0.203279
$$606$$ 0 0
$$607$$ 21.1231 0.857360 0.428680 0.903456i $$-0.358979\pi$$
0.428680 + 0.903456i $$0.358979\pi$$
$$608$$ 0 0
$$609$$ −37.3002 −1.51148
$$610$$ 0 0
$$611$$ −35.5076 −1.43648
$$612$$ 0 0
$$613$$ 5.36932 0.216865 0.108432 0.994104i $$-0.465417\pi$$
0.108432 + 0.994104i $$0.465417\pi$$
$$614$$ 0 0
$$615$$ 31.3693 1.26493
$$616$$ 0 0
$$617$$ 12.2462 0.493014 0.246507 0.969141i $$-0.420717\pi$$
0.246507 + 0.969141i $$0.420717\pi$$
$$618$$ 0 0
$$619$$ 36.0000 1.44696 0.723481 0.690344i $$-0.242541\pi$$
0.723481 + 0.690344i $$0.242541\pi$$
$$620$$ 0 0
$$621$$ −11.0540 −0.443581
$$622$$ 0 0
$$623$$ 5.12311 0.205253
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −10.2462 −0.409194
$$628$$ 0 0
$$629$$ −20.6307 −0.822599
$$630$$ 0 0
$$631$$ 20.4924 0.815790 0.407895 0.913029i $$-0.366263\pi$$
0.407895 + 0.913029i $$0.366263\pi$$
$$632$$ 0 0
$$633$$ −21.3002 −0.846606
$$634$$ 0 0
$$635$$ −13.1231 −0.520775
$$636$$ 0 0
$$637$$ −2.49242 −0.0987534
$$638$$ 0 0
$$639$$ −36.4924 −1.44362
$$640$$ 0 0
$$641$$ 21.8617 0.863487 0.431743 0.901996i $$-0.357899\pi$$
0.431743 + 0.901996i $$0.357899\pi$$
$$642$$ 0 0
$$643$$ 33.6155 1.32567 0.662834 0.748767i $$-0.269354\pi$$
0.662834 + 0.748767i $$0.269354\pi$$
$$644$$ 0 0
$$645$$ 7.36932 0.290167
$$646$$ 0 0
$$647$$ −5.43845 −0.213807 −0.106904 0.994269i $$-0.534094\pi$$
−0.106904 + 0.994269i $$0.534094\pi$$
$$648$$ 0 0
$$649$$ 10.2462 0.402199
$$650$$ 0 0
$$651$$ 33.6155 1.31750
$$652$$ 0 0
$$653$$ −7.12311 −0.278749 −0.139374 0.990240i $$-0.544509\pi$$
−0.139374 + 0.990240i $$0.544509\pi$$
$$654$$ 0 0
$$655$$ 16.4924 0.644412
$$656$$ 0 0
$$657$$ −6.00000 −0.234082
$$658$$ 0 0
$$659$$ 14.0691 0.548056 0.274028 0.961722i $$-0.411644\pi$$
0.274028 + 0.961722i $$0.411644\pi$$
$$660$$ 0 0
$$661$$ −10.8078 −0.420373 −0.210187 0.977661i $$-0.567407\pi$$
−0.210187 + 0.977661i $$0.567407\pi$$
$$662$$ 0 0
$$663$$ 50.0691 1.94452
$$664$$ 0 0
$$665$$ 2.56155 0.0993328
$$666$$ 0 0
$$667$$ 43.6847 1.69148
$$668$$ 0 0
$$669$$ 59.8617 2.31439
$$670$$ 0 0
$$671$$ −44.4924 −1.71761
$$672$$ 0 0
$$673$$ 21.3693 0.823727 0.411863 0.911246i $$-0.364878\pi$$
0.411863 + 0.911246i $$0.364878\pi$$
$$674$$ 0 0
$$675$$ 1.43845 0.0553659
$$676$$ 0 0
$$677$$ 40.5616 1.55891 0.779454 0.626460i $$-0.215497\pi$$
0.779454 + 0.626460i $$0.215497\pi$$
$$678$$ 0 0
$$679$$ 15.3693 0.589820
$$680$$ 0 0
$$681$$ −66.4233 −2.54535
$$682$$ 0 0
$$683$$ −4.00000 −0.153056 −0.0765279 0.997067i $$-0.524383\pi$$
−0.0765279 + 0.997067i $$0.524383\pi$$
$$684$$ 0 0
$$685$$ −14.8078 −0.565776
$$686$$ 0 0
$$687$$ −37.1231 −1.41633
$$688$$ 0 0
$$689$$ −25.9309 −0.987887
$$690$$ 0 0
$$691$$ 17.1231 0.651394 0.325697 0.945474i $$-0.394401\pi$$
0.325697 + 0.945474i $$0.394401\pi$$
$$692$$ 0 0
$$693$$ −36.4924 −1.38623
$$694$$ 0 0
$$695$$ −16.4924 −0.625593
$$696$$ 0 0
$$697$$ 42.1080 1.59495
$$698$$ 0 0
$$699$$ 25.6155 0.968868
$$700$$ 0 0
$$701$$ −20.8769 −0.788509 −0.394255 0.919001i $$-0.628997\pi$$
−0.394255 + 0.919001i $$0.628997\pi$$
$$702$$ 0 0
$$703$$ −6.00000 −0.226294
$$704$$ 0 0
$$705$$ −16.0000 −0.602595
$$706$$ 0 0
$$707$$ −44.4924 −1.67331
$$708$$ 0 0
$$709$$ 38.0000 1.42712 0.713560 0.700594i $$-0.247082\pi$$
0.713560 + 0.700594i $$0.247082\pi$$
$$710$$ 0 0
$$711$$ 18.2462 0.684286
$$712$$ 0 0
$$713$$ −39.3693 −1.47439
$$714$$ 0 0
$$715$$ −22.7386 −0.850377
$$716$$ 0 0
$$717$$ 3.68466 0.137606
$$718$$ 0 0
$$719$$ 25.4384 0.948694 0.474347 0.880338i $$-0.342684\pi$$
0.474347 + 0.880338i $$0.342684\pi$$
$$720$$ 0 0
$$721$$ −5.75379 −0.214282
$$722$$ 0 0
$$723$$ 59.2311 2.20283
$$724$$ 0 0
$$725$$ −5.68466 −0.211123
$$726$$ 0 0
$$727$$ 24.3153 0.901806 0.450903 0.892573i $$-0.351102\pi$$
0.450903 + 0.892573i $$0.351102\pi$$
$$728$$ 0 0
$$729$$ −35.9848 −1.33277
$$730$$ 0 0
$$731$$ 9.89205 0.365871
$$732$$ 0 0
$$733$$ −36.8769 −1.36208 −0.681040 0.732247i $$-0.738472\pi$$
−0.681040 + 0.732247i $$0.738472\pi$$
$$734$$ 0 0
$$735$$ −1.12311 −0.0414264
$$736$$ 0 0
$$737$$ −10.2462 −0.377424
$$738$$ 0 0
$$739$$ −25.1231 −0.924168 −0.462084 0.886836i $$-0.652898\pi$$
−0.462084 + 0.886836i $$0.652898\pi$$
$$740$$ 0 0
$$741$$ 14.5616 0.534932
$$742$$ 0 0
$$743$$ −18.8769 −0.692526 −0.346263 0.938137i $$-0.612550\pi$$
−0.346263 + 0.938137i $$0.612550\pi$$
$$744$$ 0 0
$$745$$ 13.3693 0.489814
$$746$$ 0 0
$$747$$ −10.2462 −0.374889
$$748$$ 0 0
$$749$$ 13.9309 0.509023
$$750$$ 0 0
$$751$$ 34.8769 1.27268 0.636338 0.771410i $$-0.280448\pi$$
0.636338 + 0.771410i $$0.280448\pi$$
$$752$$ 0 0
$$753$$ −16.0000 −0.583072
$$754$$ 0 0
$$755$$ −5.12311 −0.186449
$$756$$ 0 0
$$757$$ 42.4924 1.54441 0.772207 0.635371i $$-0.219153\pi$$
0.772207 + 0.635371i $$0.219153\pi$$
$$758$$ 0 0
$$759$$ 78.7386 2.85803
$$760$$ 0 0
$$761$$ 41.5464 1.50606 0.753028 0.657989i $$-0.228593\pi$$
0.753028 + 0.657989i $$0.228593\pi$$
$$762$$ 0 0
$$763$$ −1.43845 −0.0520753
$$764$$ 0 0
$$765$$ 12.2462 0.442763
$$766$$ 0 0
$$767$$ −14.5616 −0.525787
$$768$$ 0 0
$$769$$ 27.4384 0.989456 0.494728 0.869048i $$-0.335268\pi$$
0.494728 + 0.869048i $$0.335268\pi$$
$$770$$ 0 0
$$771$$ 35.8617 1.29153
$$772$$ 0 0
$$773$$ 10.8078 0.388728 0.194364 0.980929i $$-0.437736\pi$$
0.194364 + 0.980929i $$0.437736\pi$$
$$774$$ 0 0
$$775$$ 5.12311 0.184027
$$776$$ 0 0
$$777$$ −39.3693 −1.41237
$$778$$ 0 0
$$779$$ 12.2462 0.438766
$$780$$ 0 0
$$781$$ 40.9848 1.46655
$$782$$ 0 0
$$783$$ −8.17708 −0.292225
$$784$$ 0 0
$$785$$ −20.2462 −0.722618
$$786$$ 0 0
$$787$$ −11.1922 −0.398960 −0.199480 0.979902i $$-0.563925\pi$$
−0.199480 + 0.979902i $$0.563925\pi$$
$$788$$ 0 0
$$789$$ 56.9848 2.02871
$$790$$ 0 0
$$791$$ 22.7386 0.808493
$$792$$ 0 0
$$793$$ 63.2311 2.24540
$$794$$ 0 0
$$795$$ −11.6847 −0.414412
$$796$$ 0 0
$$797$$ −11.3002 −0.400273 −0.200137 0.979768i $$-0.564139\pi$$
−0.200137 + 0.979768i $$0.564139\pi$$
$$798$$ 0 0
$$799$$ −21.4773 −0.759811
$$800$$ 0 0
$$801$$ 7.12311 0.251683
$$802$$ 0 0
$$803$$ 6.73863 0.237801
$$804$$ 0 0
$$805$$ −19.6847 −0.693793
$$806$$ 0 0
$$807$$ −66.6004 −2.34444
$$808$$ 0 0
$$809$$ −12.5616 −0.441641 −0.220820 0.975315i $$-0.570873\pi$$
−0.220820 + 0.975315i $$0.570873\pi$$
$$810$$ 0 0
$$811$$ 20.8078 0.730659 0.365330 0.930878i $$-0.380956\pi$$
0.365330 + 0.930878i $$0.380956\pi$$
$$812$$ 0 0
$$813$$ 56.1771 1.97022
$$814$$ 0 0
$$815$$ 15.3693 0.538364
$$816$$ 0 0
$$817$$ 2.87689 0.100650
$$818$$ 0 0
$$819$$ 51.8617 1.81220
$$820$$ 0 0
$$821$$ −17.3693 −0.606193 −0.303097 0.952960i $$-0.598021\pi$$
−0.303097 + 0.952960i $$0.598021\pi$$
$$822$$ 0 0
$$823$$ 29.4384 1.02616 0.513080 0.858341i $$-0.328504\pi$$
0.513080 + 0.858341i $$0.328504\pi$$
$$824$$ 0 0
$$825$$ −10.2462 −0.356727
$$826$$ 0 0
$$827$$ 10.5616 0.367261 0.183631 0.982995i $$-0.441215\pi$$
0.183631 + 0.982995i $$0.441215\pi$$
$$828$$ 0 0
$$829$$ −21.0540 −0.731235 −0.365617 0.930765i $$-0.619142\pi$$
−0.365617 + 0.930765i $$0.619142\pi$$
$$830$$ 0 0
$$831$$ 2.24621 0.0779202
$$832$$ 0 0
$$833$$ −1.50758 −0.0522345
$$834$$ 0 0
$$835$$ 7.36932 0.255026
$$836$$ 0 0
$$837$$ 7.36932 0.254721
$$838$$ 0 0
$$839$$ −20.4924 −0.707477 −0.353738 0.935344i $$-0.615090\pi$$
−0.353738 + 0.935344i $$0.615090\pi$$
$$840$$ 0 0
$$841$$ 3.31534 0.114322
$$842$$ 0 0
$$843$$ 5.12311 0.176449
$$844$$ 0 0
$$845$$ 19.3153 0.664468
$$846$$ 0 0
$$847$$ 12.8078 0.440080
$$848$$ 0 0
$$849$$ 54.1080 1.85698
$$850$$ 0 0
$$851$$ 46.1080 1.58056
$$852$$ 0 0
$$853$$ −24.7386 −0.847035 −0.423517 0.905888i $$-0.639205\pi$$
−0.423517 + 0.905888i $$0.639205\pi$$
$$854$$ 0 0
$$855$$ 3.56155 0.121803
$$856$$ 0 0
$$857$$ 14.6307 0.499775 0.249887 0.968275i $$-0.419606\pi$$
0.249887 + 0.968275i $$0.419606\pi$$
$$858$$ 0 0
$$859$$ 52.9848 1.80782 0.903910 0.427723i $$-0.140684\pi$$
0.903910 + 0.427723i $$0.140684\pi$$
$$860$$ 0 0
$$861$$ 80.3542 2.73846
$$862$$ 0 0
$$863$$ −2.24621 −0.0764619 −0.0382310 0.999269i $$-0.512172\pi$$
−0.0382310 + 0.999269i $$0.512172\pi$$
$$864$$ 0 0
$$865$$ 20.2462 0.688392
$$866$$ 0 0
$$867$$ −13.2614 −0.450380
$$868$$ 0 0
$$869$$ −20.4924 −0.695158
$$870$$ 0 0
$$871$$ 14.5616 0.493399
$$872$$ 0 0
$$873$$ 21.3693 0.723242
$$874$$ 0 0
$$875$$ 2.56155 0.0865963
$$876$$ 0 0
$$877$$ −3.93087 −0.132736 −0.0663680 0.997795i $$-0.521141\pi$$
−0.0663680 + 0.997795i $$0.521141\pi$$
$$878$$ 0 0
$$879$$ −56.8078 −1.91608
$$880$$ 0 0
$$881$$ 42.9848 1.44820 0.724098 0.689697i $$-0.242256\pi$$
0.724098 + 0.689697i $$0.242256\pi$$
$$882$$ 0 0
$$883$$ 6.38447 0.214855 0.107427 0.994213i $$-0.465739\pi$$
0.107427 + 0.994213i $$0.465739\pi$$
$$884$$ 0 0
$$885$$ −6.56155 −0.220564
$$886$$ 0 0
$$887$$ −28.4924 −0.956682 −0.478341 0.878174i $$-0.658762\pi$$
−0.478341 + 0.878174i $$0.658762\pi$$
$$888$$ 0 0
$$889$$ −33.6155 −1.12743
$$890$$ 0 0
$$891$$ 28.0000 0.938035
$$892$$ 0 0
$$893$$ −6.24621 −0.209021
$$894$$ 0 0
$$895$$ 22.2462 0.743609
$$896$$ 0 0
$$897$$ −111.901 −3.73625
$$898$$ 0 0
$$899$$ −29.1231 −0.971310
$$900$$ 0 0
$$901$$ −15.6847 −0.522532
$$902$$ 0 0
$$903$$ 18.8769 0.628184
$$904$$ 0 0
$$905$$ −18.0000 −0.598340
$$906$$ 0 0
$$907$$ 20.1771 0.669969 0.334984 0.942224i $$-0.391269\pi$$
0.334984 + 0.942224i $$0.391269\pi$$
$$908$$ 0 0
$$909$$ −61.8617 −2.05182
$$910$$ 0 0
$$911$$ −4.49242 −0.148841 −0.0744203 0.997227i $$-0.523711\pi$$
−0.0744203 + 0.997227i $$0.523711\pi$$
$$912$$ 0 0
$$913$$ 11.5076 0.380845
$$914$$ 0 0
$$915$$ 28.4924 0.941930
$$916$$ 0 0
$$917$$ 42.2462 1.39509
$$918$$ 0 0
$$919$$ 2.06913 0.0682543 0.0341272 0.999417i $$-0.489135\pi$$
0.0341272 + 0.999417i $$0.489135\pi$$
$$920$$ 0 0
$$921$$ −83.2311 −2.74256
$$922$$ 0 0
$$923$$ −58.2462 −1.91720
$$924$$ 0 0
$$925$$ −6.00000 −0.197279
$$926$$ 0 0
$$927$$ −8.00000 −0.262754
$$928$$ 0 0
$$929$$ −19.3002 −0.633219 −0.316609 0.948556i $$-0.602544\pi$$
−0.316609 + 0.948556i $$0.602544\pi$$
$$930$$ 0 0
$$931$$ −0.438447 −0.0143695
$$932$$ 0 0
$$933$$ 9.43845 0.309001
$$934$$ 0 0
$$935$$ −13.7538 −0.449797
$$936$$ 0 0
$$937$$ 40.5616 1.32509 0.662544 0.749023i $$-0.269477\pi$$
0.662544 + 0.749023i $$0.269477\pi$$
$$938$$ 0 0
$$939$$ 12.9460 0.422478
$$940$$ 0 0
$$941$$ 54.8078 1.78668 0.893341 0.449379i $$-0.148355\pi$$
0.893341 + 0.449379i $$0.148355\pi$$
$$942$$ 0 0
$$943$$ −94.1080 −3.06458
$$944$$ 0 0
$$945$$ 3.68466 0.119862
$$946$$ 0 0
$$947$$ 34.2462 1.11285 0.556426 0.830897i $$-0.312172\pi$$
0.556426 + 0.830897i $$0.312172\pi$$
$$948$$ 0 0
$$949$$ −9.57671 −0.310873
$$950$$ 0 0
$$951$$ 33.4384 1.08432
$$952$$ 0 0
$$953$$ 44.1080 1.42880 0.714398 0.699739i $$-0.246700\pi$$
0.714398 + 0.699739i $$0.246700\pi$$
$$954$$ 0 0
$$955$$ 3.68466 0.119233
$$956$$ 0 0
$$957$$ 58.2462 1.88283
$$958$$ 0 0
$$959$$ −37.9309 −1.22485
$$960$$ 0 0
$$961$$ −4.75379 −0.153348
$$962$$ 0 0
$$963$$ 19.3693 0.624168
$$964$$ 0 0
$$965$$ −14.4924 −0.466528
$$966$$ 0 0
$$967$$ 15.5076 0.498690 0.249345 0.968415i $$-0.419785\pi$$
0.249345 + 0.968415i $$0.419785\pi$$
$$968$$ 0 0
$$969$$ 8.80776 0.282946
$$970$$ 0 0
$$971$$ 56.4924 1.81293 0.906464 0.422283i $$-0.138771\pi$$
0.906464 + 0.422283i $$0.138771\pi$$
$$972$$ 0 0
$$973$$ −42.2462 −1.35435
$$974$$ 0 0
$$975$$ 14.5616 0.466343
$$976$$ 0 0
$$977$$ 28.7386 0.919430 0.459715 0.888066i $$-0.347952\pi$$
0.459715 + 0.888066i $$0.347952\pi$$
$$978$$ 0 0
$$979$$ −8.00000 −0.255681
$$980$$ 0 0
$$981$$ −2.00000 −0.0638551
$$982$$ 0 0
$$983$$ −18.8769 −0.602079 −0.301040 0.953612i $$-0.597334\pi$$
−0.301040 + 0.953612i $$0.597334\pi$$
$$984$$ 0 0
$$985$$ −20.2462 −0.645098
$$986$$ 0 0
$$987$$ −40.9848 −1.30456
$$988$$ 0 0
$$989$$ −22.1080 −0.702992
$$990$$ 0 0
$$991$$ −2.87689 −0.0913876 −0.0456938 0.998955i $$-0.514550\pi$$
−0.0456938 + 0.998955i $$0.514550\pi$$
$$992$$ 0 0
$$993$$ 6.56155 0.208225
$$994$$ 0 0
$$995$$ −16.8078 −0.532842
$$996$$ 0 0
$$997$$ −16.7386 −0.530118 −0.265059 0.964232i $$-0.585391\pi$$
−0.265059 + 0.964232i $$0.585391\pi$$
$$998$$ 0 0
$$999$$ −8.63068 −0.273063
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 1520.2.a.n.1.2 2
4.3 odd 2 190.2.a.d.1.1 2
5.4 even 2 7600.2.a.y.1.1 2
8.3 odd 2 6080.2.a.bh.1.2 2
8.5 even 2 6080.2.a.bb.1.1 2
12.11 even 2 1710.2.a.w.1.1 2
20.3 even 4 950.2.b.f.799.3 4
20.7 even 4 950.2.b.f.799.2 4
20.19 odd 2 950.2.a.h.1.2 2
28.27 even 2 9310.2.a.bc.1.2 2
60.59 even 2 8550.2.a.br.1.2 2
76.75 even 2 3610.2.a.t.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
190.2.a.d.1.1 2 4.3 odd 2
950.2.a.h.1.2 2 20.19 odd 2
950.2.b.f.799.2 4 20.7 even 4
950.2.b.f.799.3 4 20.3 even 4
1520.2.a.n.1.2 2 1.1 even 1 trivial
1710.2.a.w.1.1 2 12.11 even 2
3610.2.a.t.1.2 2 76.75 even 2
6080.2.a.bb.1.1 2 8.5 even 2
6080.2.a.bh.1.2 2 8.3 odd 2
7600.2.a.y.1.1 2 5.4 even 2
8550.2.a.br.1.2 2 60.59 even 2
9310.2.a.bc.1.2 2 28.27 even 2