# Properties

 Label 950.2.a.h.1.2 Level $950$ Weight $2$ Character 950.1 Self dual yes Analytic conductor $7.586$ 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] = [950,2,Mod(1,950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(950, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("950.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.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: $$7.58578819202$$ 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 $$\chi$$ $$=$$ 950.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +2.56155 q^{3} +1.00000 q^{4} +2.56155 q^{6} +2.56155 q^{7} +1.00000 q^{8} +3.56155 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +2.56155 q^{3} +1.00000 q^{4} +2.56155 q^{6} +2.56155 q^{7} +1.00000 q^{8} +3.56155 q^{9} +4.00000 q^{11} +2.56155 q^{12} -5.68466 q^{13} +2.56155 q^{14} +1.00000 q^{16} -3.43845 q^{17} +3.56155 q^{18} -1.00000 q^{19} +6.56155 q^{21} +4.00000 q^{22} -7.68466 q^{23} +2.56155 q^{24} -5.68466 q^{26} +1.43845 q^{27} +2.56155 q^{28} -5.68466 q^{29} -5.12311 q^{31} +1.00000 q^{32} +10.2462 q^{33} -3.43845 q^{34} +3.56155 q^{36} +6.00000 q^{37} -1.00000 q^{38} -14.5616 q^{39} +12.2462 q^{41} +6.56155 q^{42} +2.87689 q^{43} +4.00000 q^{44} -7.68466 q^{46} -6.24621 q^{47} +2.56155 q^{48} -0.438447 q^{49} -8.80776 q^{51} -5.68466 q^{52} +4.56155 q^{53} +1.43845 q^{54} +2.56155 q^{56} -2.56155 q^{57} -5.68466 q^{58} +2.56155 q^{59} +11.1231 q^{61} -5.12311 q^{62} +9.12311 q^{63} +1.00000 q^{64} +10.2462 q^{66} +2.56155 q^{67} -3.43845 q^{68} -19.6847 q^{69} +10.2462 q^{71} +3.56155 q^{72} +1.68466 q^{73} +6.00000 q^{74} -1.00000 q^{76} +10.2462 q^{77} -14.5616 q^{78} -5.12311 q^{79} -7.00000 q^{81} +12.2462 q^{82} -2.87689 q^{83} +6.56155 q^{84} +2.87689 q^{86} -14.5616 q^{87} +4.00000 q^{88} +2.00000 q^{89} -14.5616 q^{91} -7.68466 q^{92} -13.1231 q^{93} -6.24621 q^{94} +2.56155 q^{96} -6.00000 q^{97} -0.438447 q^{98} +14.2462 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + q^{3} + 2 q^{4} + q^{6} + q^{7} + 2 q^{8} + 3 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + q^3 + 2 * q^4 + q^6 + q^7 + 2 * q^8 + 3 * q^9 $$2 q + 2 q^{2} + q^{3} + 2 q^{4} + q^{6} + q^{7} + 2 q^{8} + 3 q^{9} + 8 q^{11} + q^{12} + q^{13} + q^{14} + 2 q^{16} - 11 q^{17} + 3 q^{18} - 2 q^{19} + 9 q^{21} + 8 q^{22} - 3 q^{23} + q^{24} + q^{26} + 7 q^{27} + q^{28} + q^{29} - 2 q^{31} + 2 q^{32} + 4 q^{33} - 11 q^{34} + 3 q^{36} + 12 q^{37} - 2 q^{38} - 25 q^{39} + 8 q^{41} + 9 q^{42} + 14 q^{43} + 8 q^{44} - 3 q^{46} + 4 q^{47} + q^{48} - 5 q^{49} + 3 q^{51} + q^{52} + 5 q^{53} + 7 q^{54} + q^{56} - q^{57} + q^{58} + q^{59} + 14 q^{61} - 2 q^{62} + 10 q^{63} + 2 q^{64} + 4 q^{66} + q^{67} - 11 q^{68} - 27 q^{69} + 4 q^{71} + 3 q^{72} - 9 q^{73} + 12 q^{74} - 2 q^{76} + 4 q^{77} - 25 q^{78} - 2 q^{79} - 14 q^{81} + 8 q^{82} - 14 q^{83} + 9 q^{84} + 14 q^{86} - 25 q^{87} + 8 q^{88} + 4 q^{89} - 25 q^{91} - 3 q^{92} - 18 q^{93} + 4 q^{94} + q^{96} - 12 q^{97} - 5 q^{98} + 12 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + q^3 + 2 * q^4 + q^6 + q^7 + 2 * q^8 + 3 * q^9 + 8 * q^11 + q^12 + q^13 + q^14 + 2 * q^16 - 11 * q^17 + 3 * q^18 - 2 * q^19 + 9 * q^21 + 8 * q^22 - 3 * q^23 + q^24 + q^26 + 7 * q^27 + q^28 + q^29 - 2 * q^31 + 2 * q^32 + 4 * q^33 - 11 * q^34 + 3 * q^36 + 12 * q^37 - 2 * q^38 - 25 * q^39 + 8 * q^41 + 9 * q^42 + 14 * q^43 + 8 * q^44 - 3 * q^46 + 4 * q^47 + q^48 - 5 * q^49 + 3 * q^51 + q^52 + 5 * q^53 + 7 * q^54 + q^56 - q^57 + q^58 + q^59 + 14 * q^61 - 2 * q^62 + 10 * q^63 + 2 * q^64 + 4 * q^66 + q^67 - 11 * q^68 - 27 * q^69 + 4 * q^71 + 3 * q^72 - 9 * q^73 + 12 * q^74 - 2 * q^76 + 4 * q^77 - 25 * q^78 - 2 * q^79 - 14 * q^81 + 8 * q^82 - 14 * q^83 + 9 * q^84 + 14 * q^86 - 25 * q^87 + 8 * q^88 + 4 * q^89 - 25 * q^91 - 3 * q^92 - 18 * q^93 + 4 * q^94 + q^96 - 12 * q^97 - 5 * q^98 + 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 2.56155 1.47891 0.739457 0.673204i $$-0.235083\pi$$
0.739457 + 0.673204i $$0.235083\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 2.56155 1.04575
$$7$$ 2.56155 0.968176 0.484088 0.875019i $$-0.339151\pi$$
0.484088 + 0.875019i $$0.339151\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 3.56155 1.18718
$$10$$ 0 0
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 2.56155 0.739457
$$13$$ −5.68466 −1.57664 −0.788320 0.615265i $$-0.789049\pi$$
−0.788320 + 0.615265i $$0.789049\pi$$
$$14$$ 2.56155 0.684604
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −3.43845 −0.833946 −0.416973 0.908919i $$-0.636909\pi$$
−0.416973 + 0.908919i $$0.636909\pi$$
$$18$$ 3.56155 0.839466
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 6.56155 1.43185
$$22$$ 4.00000 0.852803
$$23$$ −7.68466 −1.60236 −0.801181 0.598422i $$-0.795795\pi$$
−0.801181 + 0.598422i $$0.795795\pi$$
$$24$$ 2.56155 0.522875
$$25$$ 0 0
$$26$$ −5.68466 −1.11485
$$27$$ 1.43845 0.276829
$$28$$ 2.56155 0.484088
$$29$$ −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 $$-0.652175\pi$$
−0.460068 + 0.887883i $$0.652175\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 10.2462 1.78364
$$34$$ −3.43845 −0.589689
$$35$$ 0 0
$$36$$ 3.56155 0.593592
$$37$$ 6.00000 0.986394 0.493197 0.869918i $$-0.335828\pi$$
0.493197 + 0.869918i $$0.335828\pi$$
$$38$$ −1.00000 −0.162221
$$39$$ −14.5616 −2.33171
$$40$$ 0 0
$$41$$ 12.2462 1.91254 0.956268 0.292490i $$-0.0944840\pi$$
0.956268 + 0.292490i $$0.0944840\pi$$
$$42$$ 6.56155 1.01247
$$43$$ 2.87689 0.438722 0.219361 0.975644i $$-0.429603\pi$$
0.219361 + 0.975644i $$0.429603\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ −7.68466 −1.13304
$$47$$ −6.24621 −0.911104 −0.455552 0.890209i $$-0.650558\pi$$
−0.455552 + 0.890209i $$0.650558\pi$$
$$48$$ 2.56155 0.369728
$$49$$ −0.438447 −0.0626353
$$50$$ 0 0
$$51$$ −8.80776 −1.23333
$$52$$ −5.68466 −0.788320
$$53$$ 4.56155 0.626577 0.313289 0.949658i $$-0.398569\pi$$
0.313289 + 0.949658i $$0.398569\pi$$
$$54$$ 1.43845 0.195748
$$55$$ 0 0
$$56$$ 2.56155 0.342302
$$57$$ −2.56155 −0.339286
$$58$$ −5.68466 −0.746432
$$59$$ 2.56155 0.333486 0.166743 0.986000i $$-0.446675\pi$$
0.166743 + 0.986000i $$0.446675\pi$$
$$60$$ 0 0
$$61$$ 11.1231 1.42417 0.712084 0.702094i $$-0.247752\pi$$
0.712084 + 0.702094i $$0.247752\pi$$
$$62$$ −5.12311 −0.650635
$$63$$ 9.12311 1.14940
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 10.2462 1.26122
$$67$$ 2.56155 0.312943 0.156472 0.987682i $$-0.449988\pi$$
0.156472 + 0.987682i $$0.449988\pi$$
$$68$$ −3.43845 −0.416973
$$69$$ −19.6847 −2.36975
$$70$$ 0 0
$$71$$ 10.2462 1.21600 0.608001 0.793936i $$-0.291972\pi$$
0.608001 + 0.793936i $$0.291972\pi$$
$$72$$ 3.56155 0.419733
$$73$$ 1.68466 0.197174 0.0985872 0.995128i $$-0.468568\pi$$
0.0985872 + 0.995128i $$0.468568\pi$$
$$74$$ 6.00000 0.697486
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ 10.2462 1.16766
$$78$$ −14.5616 −1.64877
$$79$$ −5.12311 −0.576394 −0.288197 0.957571i $$-0.593056\pi$$
−0.288197 + 0.957571i $$0.593056\pi$$
$$80$$ 0 0
$$81$$ −7.00000 −0.777778
$$82$$ 12.2462 1.35237
$$83$$ −2.87689 −0.315780 −0.157890 0.987457i $$-0.550469\pi$$
−0.157890 + 0.987457i $$0.550469\pi$$
$$84$$ 6.56155 0.715924
$$85$$ 0 0
$$86$$ 2.87689 0.310223
$$87$$ −14.5616 −1.56116
$$88$$ 4.00000 0.426401
$$89$$ 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
$$92$$ −7.68466 −0.801181
$$93$$ −13.1231 −1.36080
$$94$$ −6.24621 −0.644247
$$95$$ 0 0
$$96$$ 2.56155 0.261437
$$97$$ −6.00000 −0.609208 −0.304604 0.952479i $$-0.598524\pi$$
−0.304604 + 0.952479i $$0.598524\pi$$
$$98$$ −0.438447 −0.0442899
$$99$$ 14.2462 1.43180
$$100$$ 0 0
$$101$$ −17.3693 −1.72831 −0.864156 0.503224i $$-0.832147\pi$$
−0.864156 + 0.503224i $$0.832147\pi$$
$$102$$ −8.80776 −0.872099
$$103$$ −2.24621 −0.221326 −0.110663 0.993858i $$-0.535297\pi$$
−0.110663 + 0.993858i $$0.535297\pi$$
$$104$$ −5.68466 −0.557427
$$105$$ 0 0
$$106$$ 4.56155 0.443057
$$107$$ 5.43845 0.525755 0.262877 0.964829i $$-0.415329\pi$$
0.262877 + 0.964829i $$0.415329\pi$$
$$108$$ 1.43845 0.138415
$$109$$ −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
$$112$$ 2.56155 0.242044
$$113$$ −8.87689 −0.835068 −0.417534 0.908661i $$-0.637106\pi$$
−0.417534 + 0.908661i $$0.637106\pi$$
$$114$$ −2.56155 −0.239911
$$115$$ 0 0
$$116$$ −5.68466 −0.527807
$$117$$ −20.2462 −1.87176
$$118$$ 2.56155 0.235810
$$119$$ −8.80776 −0.807406
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 11.1231 1.00704
$$123$$ 31.3693 2.82848
$$124$$ −5.12311 −0.460068
$$125$$ 0 0
$$126$$ 9.12311 0.812751
$$127$$ −13.1231 −1.16449 −0.582244 0.813014i $$-0.697825\pi$$
−0.582244 + 0.813014i $$0.697825\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 7.36932 0.648832
$$130$$ 0 0
$$131$$ −16.4924 −1.44095 −0.720475 0.693481i $$-0.756076\pi$$
−0.720475 + 0.693481i $$0.756076\pi$$
$$132$$ 10.2462 0.891818
$$133$$ −2.56155 −0.222115
$$134$$ 2.56155 0.221284
$$135$$ 0 0
$$136$$ −3.43845 −0.294844
$$137$$ 14.8078 1.26511 0.632556 0.774514i $$-0.282006\pi$$
0.632556 + 0.774514i $$0.282006\pi$$
$$138$$ −19.6847 −1.67567
$$139$$ 16.4924 1.39887 0.699435 0.714697i $$-0.253435\pi$$
0.699435 + 0.714697i $$0.253435\pi$$
$$140$$ 0 0
$$141$$ −16.0000 −1.34744
$$142$$ 10.2462 0.859843
$$143$$ −22.7386 −1.90150
$$144$$ 3.56155 0.296796
$$145$$ 0 0
$$146$$ 1.68466 0.139423
$$147$$ −1.12311 −0.0926322
$$148$$ 6.00000 0.493197
$$149$$ 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 $$-0.433156\pi$$
0.208456 + 0.978032i $$0.433156\pi$$
$$152$$ −1.00000 −0.0811107
$$153$$ −12.2462 −0.990048
$$154$$ 10.2462 0.825663
$$155$$ 0 0
$$156$$ −14.5616 −1.16586
$$157$$ 20.2462 1.61582 0.807912 0.589303i $$-0.200598\pi$$
0.807912 + 0.589303i $$0.200598\pi$$
$$158$$ −5.12311 −0.407572
$$159$$ 11.6847 0.926654
$$160$$ 0 0
$$161$$ −19.6847 −1.55137
$$162$$ −7.00000 −0.549972
$$163$$ 15.3693 1.20382 0.601909 0.798565i $$-0.294407\pi$$
0.601909 + 0.798565i $$0.294407\pi$$
$$164$$ 12.2462 0.956268
$$165$$ 0 0
$$166$$ −2.87689 −0.223290
$$167$$ 7.36932 0.570255 0.285127 0.958490i $$-0.407964\pi$$
0.285127 + 0.958490i $$0.407964\pi$$
$$168$$ 6.56155 0.506235
$$169$$ 19.3153 1.48580
$$170$$ 0 0
$$171$$ −3.56155 −0.272359
$$172$$ 2.87689 0.219361
$$173$$ −20.2462 −1.53929 −0.769645 0.638471i $$-0.779567\pi$$
−0.769645 + 0.638471i $$0.779567\pi$$
$$174$$ −14.5616 −1.10391
$$175$$ 0 0
$$176$$ 4.00000 0.301511
$$177$$ 6.56155 0.493197
$$178$$ 2.00000 0.149906
$$179$$ −22.2462 −1.66276 −0.831380 0.555704i $$-0.812449\pi$$
−0.831380 + 0.555704i $$0.812449\pi$$
$$180$$ 0 0
$$181$$ −18.0000 −1.33793 −0.668965 0.743294i $$-0.733262\pi$$
−0.668965 + 0.743294i $$0.733262\pi$$
$$182$$ −14.5616 −1.07937
$$183$$ 28.4924 2.10622
$$184$$ −7.68466 −0.566521
$$185$$ 0 0
$$186$$ −13.1231 −0.962233
$$187$$ −13.7538 −1.00578
$$188$$ −6.24621 −0.455552
$$189$$ 3.68466 0.268019
$$190$$ 0 0
$$191$$ −3.68466 −0.266613 −0.133306 0.991075i $$-0.542559\pi$$
−0.133306 + 0.991075i $$0.542559\pi$$
$$192$$ 2.56155 0.184864
$$193$$ 14.4924 1.04319 0.521594 0.853194i $$-0.325338\pi$$
0.521594 + 0.853194i $$0.325338\pi$$
$$194$$ −6.00000 −0.430775
$$195$$ 0 0
$$196$$ −0.438447 −0.0313177
$$197$$ 20.2462 1.44248 0.721241 0.692684i $$-0.243572\pi$$
0.721241 + 0.692684i $$0.243572\pi$$
$$198$$ 14.2462 1.01243
$$199$$ 16.8078 1.19147 0.595735 0.803181i $$-0.296861\pi$$
0.595735 + 0.803181i $$0.296861\pi$$
$$200$$ 0 0
$$201$$ 6.56155 0.462816
$$202$$ −17.3693 −1.22210
$$203$$ −14.5616 −1.02202
$$204$$ −8.80776 −0.616667
$$205$$ 0 0
$$206$$ −2.24621 −0.156501
$$207$$ −27.3693 −1.90230
$$208$$ −5.68466 −0.394160
$$209$$ −4.00000 −0.276686
$$210$$ 0 0
$$211$$ 8.31534 0.572452 0.286226 0.958162i $$-0.407599\pi$$
0.286226 + 0.958162i $$0.407599\pi$$
$$212$$ 4.56155 0.313289
$$213$$ 26.2462 1.79836
$$214$$ 5.43845 0.371765
$$215$$ 0 0
$$216$$ 1.43845 0.0978739
$$217$$ −13.1231 −0.890854
$$218$$ −0.561553 −0.0380332
$$219$$ 4.31534 0.291604
$$220$$ 0 0
$$221$$ 19.5464 1.31483
$$222$$ 15.3693 1.03152
$$223$$ 23.3693 1.56493 0.782463 0.622698i $$-0.213963\pi$$
0.782463 + 0.622698i $$0.213963\pi$$
$$224$$ 2.56155 0.171151
$$225$$ 0 0
$$226$$ −8.87689 −0.590482
$$227$$ −25.9309 −1.72109 −0.860546 0.509373i $$-0.829878\pi$$
−0.860546 + 0.509373i $$0.829878\pi$$
$$228$$ −2.56155 −0.169643
$$229$$ −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
$$232$$ −5.68466 −0.373216
$$233$$ −10.0000 −0.655122 −0.327561 0.944830i $$-0.606227\pi$$
−0.327561 + 0.944830i $$0.606227\pi$$
$$234$$ −20.2462 −1.32354
$$235$$ 0 0
$$236$$ 2.56155 0.166743
$$237$$ −13.1231 −0.852437
$$238$$ −8.80776 −0.570923
$$239$$ −1.43845 −0.0930454 −0.0465227 0.998917i $$-0.514814\pi$$
−0.0465227 + 0.998917i $$0.514814\pi$$
$$240$$ 0 0
$$241$$ 23.1231 1.48949 0.744745 0.667349i $$-0.232571\pi$$
0.744745 + 0.667349i $$0.232571\pi$$
$$242$$ 5.00000 0.321412
$$243$$ −22.2462 −1.42710
$$244$$ 11.1231 0.712084
$$245$$ 0 0
$$246$$ 31.3693 2.00003
$$247$$ 5.68466 0.361706
$$248$$ −5.12311 −0.325318
$$249$$ −7.36932 −0.467011
$$250$$ 0 0
$$251$$ 6.24621 0.394257 0.197129 0.980378i $$-0.436838\pi$$
0.197129 + 0.980378i $$0.436838\pi$$
$$252$$ 9.12311 0.574702
$$253$$ −30.7386 −1.93252
$$254$$ −13.1231 −0.823417
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −14.0000 −0.873296 −0.436648 0.899632i $$-0.643834\pi$$
−0.436648 + 0.899632i $$0.643834\pi$$
$$258$$ 7.36932 0.458794
$$259$$ 15.3693 0.955003
$$260$$ 0 0
$$261$$ −20.2462 −1.25321
$$262$$ −16.4924 −1.01891
$$263$$ 22.2462 1.37176 0.685880 0.727715i $$-0.259417\pi$$
0.685880 + 0.727715i $$0.259417\pi$$
$$264$$ 10.2462 0.630611
$$265$$ 0 0
$$266$$ −2.56155 −0.157059
$$267$$ 5.12311 0.313529
$$268$$ 2.56155 0.156472
$$269$$ −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 $$-0.732039\pi$$
−0.666103 + 0.745860i $$0.732039\pi$$
$$272$$ −3.43845 −0.208486
$$273$$ −37.3002 −2.25751
$$274$$ 14.8078 0.894570
$$275$$ 0 0
$$276$$ −19.6847 −1.18488
$$277$$ −0.876894 −0.0526875 −0.0263437 0.999653i $$-0.508386\pi$$
−0.0263437 + 0.999653i $$0.508386\pi$$
$$278$$ 16.4924 0.989150
$$279$$ −18.2462 −1.09237
$$280$$ 0 0
$$281$$ 2.00000 0.119310 0.0596550 0.998219i $$-0.481000\pi$$
0.0596550 + 0.998219i $$0.481000\pi$$
$$282$$ −16.0000 −0.952786
$$283$$ 21.1231 1.25564 0.627819 0.778359i $$-0.283948\pi$$
0.627819 + 0.778359i $$0.283948\pi$$
$$284$$ 10.2462 0.608001
$$285$$ 0 0
$$286$$ −22.7386 −1.34456
$$287$$ 31.3693 1.85167
$$288$$ 3.56155 0.209867
$$289$$ −5.17708 −0.304534
$$290$$ 0 0
$$291$$ −15.3693 −0.900965
$$292$$ 1.68466 0.0985872
$$293$$ 22.1771 1.29560 0.647799 0.761811i $$-0.275689\pi$$
0.647799 + 0.761811i $$0.275689\pi$$
$$294$$ −1.12311 −0.0655009
$$295$$ 0 0
$$296$$ 6.00000 0.348743
$$297$$ 5.75379 0.333869
$$298$$ 13.3693 0.774464
$$299$$ 43.6847 2.52635
$$300$$ 0 0
$$301$$ 7.36932 0.424760
$$302$$ 5.12311 0.294802
$$303$$ −44.4924 −2.55602
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ −12.2462 −0.700069
$$307$$ −32.4924 −1.85444 −0.927220 0.374516i $$-0.877809\pi$$
−0.927220 + 0.374516i $$0.877809\pi$$
$$308$$ 10.2462 0.583832
$$309$$ −5.75379 −0.327322
$$310$$ 0 0
$$311$$ −3.68466 −0.208938 −0.104469 0.994528i $$-0.533314\pi$$
−0.104469 + 0.994528i $$0.533314\pi$$
$$312$$ −14.5616 −0.824386
$$313$$ −5.05398 −0.285668 −0.142834 0.989747i $$-0.545621\pi$$
−0.142834 + 0.989747i $$0.545621\pi$$
$$314$$ 20.2462 1.14256
$$315$$ 0 0
$$316$$ −5.12311 −0.288197
$$317$$ −13.0540 −0.733184 −0.366592 0.930382i $$-0.619476\pi$$
−0.366592 + 0.930382i $$0.619476\pi$$
$$318$$ 11.6847 0.655243
$$319$$ −22.7386 −1.27312
$$320$$ 0 0
$$321$$ 13.9309 0.777545
$$322$$ −19.6847 −1.09698
$$323$$ 3.43845 0.191320
$$324$$ −7.00000 −0.388889
$$325$$ 0 0
$$326$$ 15.3693 0.851228
$$327$$ −1.43845 −0.0795463
$$328$$ 12.2462 0.676184
$$329$$ −16.0000 −0.882109
$$330$$ 0 0
$$331$$ −2.56155 −0.140796 −0.0703978 0.997519i $$-0.522427\pi$$
−0.0703978 + 0.997519i $$0.522427\pi$$
$$332$$ −2.87689 −0.157890
$$333$$ 21.3693 1.17103
$$334$$ 7.36932 0.403231
$$335$$ 0 0
$$336$$ 6.56155 0.357962
$$337$$ 26.0000 1.41631 0.708155 0.706057i $$-0.249528\pi$$
0.708155 + 0.706057i $$0.249528\pi$$
$$338$$ 19.3153 1.05062
$$339$$ −22.7386 −1.23499
$$340$$ 0 0
$$341$$ −20.4924 −1.10973
$$342$$ −3.56155 −0.192587
$$343$$ −19.0540 −1.02882
$$344$$ 2.87689 0.155112
$$345$$ 0 0
$$346$$ −20.2462 −1.08844
$$347$$ 8.63068 0.463319 0.231660 0.972797i $$-0.425584\pi$$
0.231660 + 0.972797i $$0.425584\pi$$
$$348$$ −14.5616 −0.780581
$$349$$ 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
$$352$$ 4.00000 0.213201
$$353$$ 3.93087 0.209219 0.104610 0.994513i $$-0.466641\pi$$
0.104610 + 0.994513i $$0.466641\pi$$
$$354$$ 6.56155 0.348743
$$355$$ 0 0
$$356$$ 2.00000 0.106000
$$357$$ −22.5616 −1.19408
$$358$$ −22.2462 −1.17575
$$359$$ 1.43845 0.0759183 0.0379592 0.999279i $$-0.487914\pi$$
0.0379592 + 0.999279i $$0.487914\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −18.0000 −0.946059
$$363$$ 12.8078 0.672233
$$364$$ −14.5616 −0.763233
$$365$$ 0 0
$$366$$ 28.4924 1.48932
$$367$$ −6.24621 −0.326050 −0.163025 0.986622i $$-0.552125\pi$$
−0.163025 + 0.986622i $$0.552125\pi$$
$$368$$ −7.68466 −0.400591
$$369$$ 43.6155 2.27053
$$370$$ 0 0
$$371$$ 11.6847 0.606637
$$372$$ −13.1231 −0.680401
$$373$$ 23.4384 1.21360 0.606798 0.794856i $$-0.292454\pi$$
0.606798 + 0.794856i $$0.292454\pi$$
$$374$$ −13.7538 −0.711191
$$375$$ 0 0
$$376$$ −6.24621 −0.322124
$$377$$ 32.3153 1.66432
$$378$$ 3.68466 0.189518
$$379$$ 10.5616 0.542511 0.271255 0.962507i $$-0.412561\pi$$
0.271255 + 0.962507i $$0.412561\pi$$
$$380$$ 0 0
$$381$$ −33.6155 −1.72218
$$382$$ −3.68466 −0.188524
$$383$$ −13.7538 −0.702786 −0.351393 0.936228i $$-0.614292\pi$$
−0.351393 + 0.936228i $$0.614292\pi$$
$$384$$ 2.56155 0.130719
$$385$$ 0 0
$$386$$ 14.4924 0.737645
$$387$$ 10.2462 0.520844
$$388$$ −6.00000 −0.304604
$$389$$ −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
$$392$$ −0.438447 −0.0221449
$$393$$ −42.2462 −2.13104
$$394$$ 20.2462 1.01999
$$395$$ 0 0
$$396$$ 14.2462 0.715899
$$397$$ 7.12311 0.357498 0.178749 0.983895i $$-0.442795\pi$$
0.178749 + 0.983895i $$0.442795\pi$$
$$398$$ 16.8078 0.842497
$$399$$ −6.56155 −0.328489
$$400$$ 0 0
$$401$$ −3.75379 −0.187455 −0.0937276 0.995598i $$-0.529878\pi$$
−0.0937276 + 0.995598i $$0.529878\pi$$
$$402$$ 6.56155 0.327261
$$403$$ 29.1231 1.45073
$$404$$ −17.3693 −0.864156
$$405$$ 0 0
$$406$$ −14.5616 −0.722678
$$407$$ 24.0000 1.18964
$$408$$ −8.80776 −0.436049
$$409$$ 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
$$412$$ −2.24621 −0.110663
$$413$$ 6.56155 0.322873
$$414$$ −27.3693 −1.34513
$$415$$ 0 0
$$416$$ −5.68466 −0.278713
$$417$$ 42.2462 2.06881
$$418$$ −4.00000 −0.195646
$$419$$ 23.8617 1.16572 0.582861 0.812572i $$-0.301933\pi$$
0.582861 + 0.812572i $$0.301933\pi$$
$$420$$ 0 0
$$421$$ −23.9309 −1.16632 −0.583160 0.812358i $$-0.698184\pi$$
−0.583160 + 0.812358i $$0.698184\pi$$
$$422$$ 8.31534 0.404784
$$423$$ −22.2462 −1.08165
$$424$$ 4.56155 0.221529
$$425$$ 0 0
$$426$$ 26.2462 1.27163
$$427$$ 28.4924 1.37884
$$428$$ 5.43845 0.262877
$$429$$ −58.2462 −2.81215
$$430$$ 0 0
$$431$$ −16.0000 −0.770693 −0.385346 0.922772i $$-0.625918\pi$$
−0.385346 + 0.922772i $$0.625918\pi$$
$$432$$ 1.43845 0.0692073
$$433$$ −14.6307 −0.703106 −0.351553 0.936168i $$-0.614346\pi$$
−0.351553 + 0.936168i $$0.614346\pi$$
$$434$$ −13.1231 −0.629929
$$435$$ 0 0
$$436$$ −0.561553 −0.0268935
$$437$$ 7.68466 0.367607
$$438$$ 4.31534 0.206195
$$439$$ −13.1231 −0.626332 −0.313166 0.949698i $$-0.601390\pi$$
−0.313166 + 0.949698i $$0.601390\pi$$
$$440$$ 0 0
$$441$$ −1.56155 −0.0743597
$$442$$ 19.5464 0.929727
$$443$$ −2.24621 −0.106721 −0.0533604 0.998575i $$-0.516993\pi$$
−0.0533604 + 0.998575i $$0.516993\pi$$
$$444$$ 15.3693 0.729396
$$445$$ 0 0
$$446$$ 23.3693 1.10657
$$447$$ 34.2462 1.61979
$$448$$ 2.56155 0.121022
$$449$$ −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
$$452$$ −8.87689 −0.417534
$$453$$ 13.1231 0.616577
$$454$$ −25.9309 −1.21700
$$455$$ 0 0
$$456$$ −2.56155 −0.119956
$$457$$ −6.31534 −0.295419 −0.147710 0.989031i $$-0.547190\pi$$
−0.147710 + 0.989031i $$0.547190\pi$$
$$458$$ −14.4924 −0.677186
$$459$$ −4.94602 −0.230861
$$460$$ 0 0
$$461$$ 3.75379 0.174831 0.0874157 0.996172i $$-0.472139\pi$$
0.0874157 + 0.996172i $$0.472139\pi$$
$$462$$ 26.2462 1.22108
$$463$$ 30.2462 1.40566 0.702830 0.711358i $$-0.251919\pi$$
0.702830 + 0.711358i $$0.251919\pi$$
$$464$$ −5.68466 −0.263904
$$465$$ 0 0
$$466$$ −10.0000 −0.463241
$$467$$ −18.2462 −0.844334 −0.422167 0.906518i $$-0.638730\pi$$
−0.422167 + 0.906518i $$0.638730\pi$$
$$468$$ −20.2462 −0.935881
$$469$$ 6.56155 0.302984
$$470$$ 0 0
$$471$$ 51.8617 2.38966
$$472$$ 2.56155 0.117905
$$473$$ 11.5076 0.529119
$$474$$ −13.1231 −0.602764
$$475$$ 0 0
$$476$$ −8.80776 −0.403703
$$477$$ 16.2462 0.743863
$$478$$ −1.43845 −0.0657930
$$479$$ 32.0000 1.46212 0.731059 0.682315i $$-0.239027\pi$$
0.731059 + 0.682315i $$0.239027\pi$$
$$480$$ 0 0
$$481$$ −34.1080 −1.55519
$$482$$ 23.1231 1.05323
$$483$$ −50.4233 −2.29434
$$484$$ 5.00000 0.227273
$$485$$ 0 0
$$486$$ −22.2462 −1.00911
$$487$$ 17.6155 0.798236 0.399118 0.916900i $$-0.369316\pi$$
0.399118 + 0.916900i $$0.369316\pi$$
$$488$$ 11.1231 0.503519
$$489$$ 39.3693 1.78034
$$490$$ 0 0
$$491$$ −1.12311 −0.0506850 −0.0253425 0.999679i $$-0.508068\pi$$
−0.0253425 + 0.999679i $$0.508068\pi$$
$$492$$ 31.3693 1.41424
$$493$$ 19.5464 0.880325
$$494$$ 5.68466 0.255765
$$495$$ 0 0
$$496$$ −5.12311 −0.230034
$$497$$ 26.2462 1.17730
$$498$$ −7.36932 −0.330227
$$499$$ 42.1080 1.88501 0.942505 0.334191i $$-0.108463\pi$$
0.942505 + 0.334191i $$0.108463\pi$$
$$500$$ 0 0
$$501$$ 18.8769 0.843357
$$502$$ 6.24621 0.278782
$$503$$ 7.05398 0.314521 0.157261 0.987557i $$-0.449734\pi$$
0.157261 + 0.987557i $$0.449734\pi$$
$$504$$ 9.12311 0.406375
$$505$$ 0 0
$$506$$ −30.7386 −1.36650
$$507$$ 49.4773 2.19736
$$508$$ −13.1231 −0.582244
$$509$$ 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
$$512$$ 1.00000 0.0441942
$$513$$ −1.43845 −0.0635090
$$514$$ −14.0000 −0.617514
$$515$$ 0 0
$$516$$ 7.36932 0.324416
$$517$$ −24.9848 −1.09883
$$518$$ 15.3693 0.675289
$$519$$ −51.8617 −2.27648
$$520$$ 0 0
$$521$$ −3.12311 −0.136826 −0.0684129 0.997657i $$-0.521794\pi$$
−0.0684129 + 0.997657i $$0.521794\pi$$
$$522$$ −20.2462 −0.886153
$$523$$ 31.6847 1.38547 0.692737 0.721191i $$-0.256405\pi$$
0.692737 + 0.721191i $$0.256405\pi$$
$$524$$ −16.4924 −0.720475
$$525$$ 0 0
$$526$$ 22.2462 0.969981
$$527$$ 17.6155 0.767344
$$528$$ 10.2462 0.445909
$$529$$ 36.0540 1.56756
$$530$$ 0 0
$$531$$ 9.12311 0.395909
$$532$$ −2.56155 −0.111057
$$533$$ −69.6155 −3.01538
$$534$$ 5.12311 0.221698
$$535$$ 0 0
$$536$$ 2.56155 0.110642
$$537$$ −56.9848 −2.45908
$$538$$ −26.0000 −1.12094
$$539$$ −1.75379 −0.0755410
$$540$$ 0 0
$$541$$ −0.384472 −0.0165297 −0.00826487 0.999966i $$-0.502631\pi$$
−0.00826487 + 0.999966i $$0.502631\pi$$
$$542$$ −21.9309 −0.942012
$$543$$ −46.1080 −1.97868
$$544$$ −3.43845 −0.147422
$$545$$ 0 0
$$546$$ −37.3002 −1.59630
$$547$$ −16.4924 −0.705165 −0.352583 0.935781i $$-0.614696\pi$$
−0.352583 + 0.935781i $$0.614696\pi$$
$$548$$ 14.8078 0.632556
$$549$$ 39.6155 1.69075
$$550$$ 0 0
$$551$$ 5.68466 0.242175
$$552$$ −19.6847 −0.837835
$$553$$ −13.1231 −0.558051
$$554$$ −0.876894 −0.0372557
$$555$$ 0 0
$$556$$ 16.4924 0.699435
$$557$$ −39.6155 −1.67856 −0.839282 0.543697i $$-0.817024\pi$$
−0.839282 + 0.543697i $$0.817024\pi$$
$$558$$ −18.2462 −0.772424
$$559$$ −16.3542 −0.691707
$$560$$ 0 0
$$561$$ −35.2311 −1.48746
$$562$$ 2.00000 0.0843649
$$563$$ 8.49242 0.357913 0.178956 0.983857i $$-0.442728\pi$$
0.178956 + 0.983857i $$0.442728\pi$$
$$564$$ −16.0000 −0.673722
$$565$$ 0 0
$$566$$ 21.1231 0.887870
$$567$$ −17.9309 −0.753026
$$568$$ 10.2462 0.429921
$$569$$ −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 $$-0.350607\pi$$
0.452291 + 0.891870i $$0.350607\pi$$
$$572$$ −22.7386 −0.950750
$$573$$ −9.43845 −0.394297
$$574$$ 31.3693 1.30933
$$575$$ 0 0
$$576$$ 3.56155 0.148398
$$577$$ 10.3153 0.429433 0.214717 0.976676i $$-0.431117\pi$$
0.214717 + 0.976676i $$0.431117\pi$$
$$578$$ −5.17708 −0.215338
$$579$$ 37.1231 1.54278
$$580$$ 0 0
$$581$$ −7.36932 −0.305731
$$582$$ −15.3693 −0.637079
$$583$$ 18.2462 0.755681
$$584$$ 1.68466 0.0697117
$$585$$ 0 0
$$586$$ 22.1771 0.916127
$$587$$ 7.36932 0.304164 0.152082 0.988368i $$-0.451402\pi$$
0.152082 + 0.988368i $$0.451402\pi$$
$$588$$ −1.12311 −0.0463161
$$589$$ 5.12311 0.211094
$$590$$ 0 0
$$591$$ 51.8617 2.13331
$$592$$ 6.00000 0.246598
$$593$$ −7.75379 −0.318410 −0.159205 0.987246i $$-0.550893\pi$$
−0.159205 + 0.987246i $$0.550893\pi$$
$$594$$ 5.75379 0.236081
$$595$$ 0 0
$$596$$ 13.3693 0.547629
$$597$$ 43.0540 1.76208
$$598$$ 43.6847 1.78640
$$599$$ −11.8617 −0.484658 −0.242329 0.970194i $$-0.577911\pi$$
−0.242329 + 0.970194i $$0.577911\pi$$
$$600$$ 0 0
$$601$$ 18.0000 0.734235 0.367118 0.930175i $$-0.380345\pi$$
0.367118 + 0.930175i $$0.380345\pi$$
$$602$$ 7.36932 0.300351
$$603$$ 9.12311 0.371522
$$604$$ 5.12311 0.208456
$$605$$ 0 0
$$606$$ −44.4924 −1.80738
$$607$$ 21.1231 0.857360 0.428680 0.903456i $$-0.358979\pi$$
0.428680 + 0.903456i $$0.358979\pi$$
$$608$$ −1.00000 −0.0405554
$$609$$ −37.3002 −1.51148
$$610$$ 0 0
$$611$$ 35.5076 1.43648
$$612$$ −12.2462 −0.495024
$$613$$ −5.36932 −0.216865 −0.108432 0.994104i $$-0.534583\pi$$
−0.108432 + 0.994104i $$0.534583\pi$$
$$614$$ −32.4924 −1.31129
$$615$$ 0 0
$$616$$ 10.2462 0.412832
$$617$$ −12.2462 −0.493014 −0.246507 0.969141i $$-0.579283\pi$$
−0.246507 + 0.969141i $$0.579283\pi$$
$$618$$ −5.75379 −0.231451
$$619$$ −36.0000 −1.44696 −0.723481 0.690344i $$-0.757459\pi$$
−0.723481 + 0.690344i $$0.757459\pi$$
$$620$$ 0 0
$$621$$ −11.0540 −0.443581
$$622$$ −3.68466 −0.147741
$$623$$ 5.12311 0.205253
$$624$$ −14.5616 −0.582929
$$625$$ 0 0
$$626$$ −5.05398 −0.201997
$$627$$ −10.2462 −0.409194
$$628$$ 20.2462 0.807912
$$629$$ −20.6307 −0.822599
$$630$$ 0 0
$$631$$ −20.4924 −0.815790 −0.407895 0.913029i $$-0.633737\pi$$
−0.407895 + 0.913029i $$0.633737\pi$$
$$632$$ −5.12311 −0.203786
$$633$$ 21.3002 0.846606
$$634$$ −13.0540 −0.518440
$$635$$ 0 0
$$636$$ 11.6847 0.463327
$$637$$ 2.49242 0.0987534
$$638$$ −22.7386 −0.900231
$$639$$ 36.4924 1.44362
$$640$$ 0 0
$$641$$ 21.8617 0.863487 0.431743 0.901996i $$-0.357899\pi$$
0.431743 + 0.901996i $$0.357899\pi$$
$$642$$ 13.9309 0.549808
$$643$$ 33.6155 1.32567 0.662834 0.748767i $$-0.269354\pi$$
0.662834 + 0.748767i $$0.269354\pi$$
$$644$$ −19.6847 −0.775684
$$645$$ 0 0
$$646$$ 3.43845 0.135284
$$647$$ −5.43845 −0.213807 −0.106904 0.994269i $$-0.534094\pi$$
−0.106904 + 0.994269i $$0.534094\pi$$
$$648$$ −7.00000 −0.274986
$$649$$ 10.2462 0.402199
$$650$$ 0 0
$$651$$ −33.6155 −1.31750
$$652$$ 15.3693 0.601909
$$653$$ 7.12311 0.278749 0.139374 0.990240i $$-0.455491\pi$$
0.139374 + 0.990240i $$0.455491\pi$$
$$654$$ −1.43845 −0.0562477
$$655$$ 0 0
$$656$$ 12.2462 0.478134
$$657$$ 6.00000 0.234082
$$658$$ −16.0000 −0.623745
$$659$$ −14.0691 −0.548056 −0.274028 0.961722i $$-0.588356\pi$$
−0.274028 + 0.961722i $$0.588356\pi$$
$$660$$ 0 0
$$661$$ −10.8078 −0.420373 −0.210187 0.977661i $$-0.567407\pi$$
−0.210187 + 0.977661i $$0.567407\pi$$
$$662$$ −2.56155 −0.0995576
$$663$$ 50.0691 1.94452
$$664$$ −2.87689 −0.111645
$$665$$ 0 0
$$666$$ 21.3693 0.828044
$$667$$ 43.6847 1.69148
$$668$$ 7.36932 0.285127
$$669$$ 59.8617 2.31439
$$670$$ 0 0
$$671$$ 44.4924 1.71761
$$672$$ 6.56155 0.253117
$$673$$ −21.3693 −0.823727 −0.411863 0.911246i $$-0.635122\pi$$
−0.411863 + 0.911246i $$0.635122\pi$$
$$674$$ 26.0000 1.00148
$$675$$ 0 0
$$676$$ 19.3153 0.742898
$$677$$ −40.5616 −1.55891 −0.779454 0.626460i $$-0.784503\pi$$
−0.779454 + 0.626460i $$0.784503\pi$$
$$678$$ −22.7386 −0.873272
$$679$$ −15.3693 −0.589820
$$680$$ 0 0
$$681$$ −66.4233 −2.54535
$$682$$ −20.4924 −0.784695
$$683$$ −4.00000 −0.153056 −0.0765279 0.997067i $$-0.524383\pi$$
−0.0765279 + 0.997067i $$0.524383\pi$$
$$684$$ −3.56155 −0.136179
$$685$$ 0 0
$$686$$ −19.0540 −0.727484
$$687$$ −37.1231 −1.41633
$$688$$ 2.87689 0.109681
$$689$$ −25.9309 −0.987887
$$690$$ 0 0
$$691$$ −17.1231 −0.651394 −0.325697 0.945474i $$-0.605599\pi$$
−0.325697 + 0.945474i $$0.605599\pi$$
$$692$$ −20.2462 −0.769645
$$693$$ 36.4924 1.38623
$$694$$ 8.63068 0.327616
$$695$$ 0 0
$$696$$ −14.5616 −0.551954
$$697$$ −42.1080 −1.59495
$$698$$ 3.75379 0.142083
$$699$$ −25.6155 −0.968868
$$700$$ 0 0
$$701$$ −20.8769 −0.788509 −0.394255 0.919001i $$-0.628997\pi$$
−0.394255 + 0.919001i $$0.628997\pi$$
$$702$$ −8.17708 −0.308624
$$703$$ −6.00000 −0.226294
$$704$$ 4.00000 0.150756
$$705$$ 0 0
$$706$$ 3.93087 0.147940
$$707$$ −44.4924 −1.67331
$$708$$ 6.56155 0.246598
$$709$$ 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
$$712$$ 2.00000 0.0749532
$$713$$ 39.3693 1.47439
$$714$$ −22.5616 −0.844345
$$715$$ 0 0
$$716$$ −22.2462 −0.831380
$$717$$ −3.68466 −0.137606
$$718$$ 1.43845 0.0536824
$$719$$ −25.4384 −0.948694 −0.474347 0.880338i $$-0.657316\pi$$
−0.474347 + 0.880338i $$0.657316\pi$$
$$720$$ 0 0
$$721$$ −5.75379 −0.214282
$$722$$ 1.00000 0.0372161
$$723$$ 59.2311 2.20283
$$724$$ −18.0000 −0.668965
$$725$$ 0 0
$$726$$ 12.8078 0.475341
$$727$$ 24.3153 0.901806 0.450903 0.892573i $$-0.351102\pi$$
0.450903 + 0.892573i $$0.351102\pi$$
$$728$$ −14.5616 −0.539687
$$729$$ −35.9848 −1.33277
$$730$$ 0 0
$$731$$ −9.89205 −0.365871
$$732$$ 28.4924 1.05311
$$733$$ 36.8769 1.36208 0.681040 0.732247i $$-0.261528\pi$$
0.681040 + 0.732247i $$0.261528\pi$$
$$734$$ −6.24621 −0.230552
$$735$$ 0 0
$$736$$ −7.68466 −0.283260
$$737$$ 10.2462 0.377424
$$738$$ 43.6155 1.60551
$$739$$ 25.1231 0.924168 0.462084 0.886836i $$-0.347102\pi$$
0.462084 + 0.886836i $$0.347102\pi$$
$$740$$ 0 0
$$741$$ 14.5616 0.534932
$$742$$ 11.6847 0.428957
$$743$$ −18.8769 −0.692526 −0.346263 0.938137i $$-0.612550\pi$$
−0.346263 + 0.938137i $$0.612550\pi$$
$$744$$ −13.1231 −0.481116
$$745$$ 0 0
$$746$$ 23.4384 0.858143
$$747$$ −10.2462 −0.374889
$$748$$ −13.7538 −0.502888
$$749$$ 13.9309 0.509023
$$750$$ 0 0
$$751$$ −34.8769 −1.27268 −0.636338 0.771410i $$-0.719552\pi$$
−0.636338 + 0.771410i $$0.719552\pi$$
$$752$$ −6.24621 −0.227776
$$753$$ 16.0000 0.583072
$$754$$ 32.3153 1.17686
$$755$$ 0 0
$$756$$ 3.68466 0.134010
$$757$$ −42.4924 −1.54441 −0.772207 0.635371i $$-0.780847\pi$$
−0.772207 + 0.635371i $$0.780847\pi$$
$$758$$ 10.5616 0.383613
$$759$$ −78.7386 −2.85803
$$760$$ 0 0
$$761$$ 41.5464 1.50606 0.753028 0.657989i $$-0.228593\pi$$
0.753028 + 0.657989i $$0.228593\pi$$
$$762$$ −33.6155 −1.21776
$$763$$ −1.43845 −0.0520753
$$764$$ −3.68466 −0.133306
$$765$$ 0 0
$$766$$ −13.7538 −0.496945
$$767$$ −14.5616 −0.525787
$$768$$ 2.56155 0.0924321
$$769$$ 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
$$772$$ 14.4924 0.521594
$$773$$ −10.8078 −0.388728 −0.194364 0.980929i $$-0.562264\pi$$
−0.194364 + 0.980929i $$0.562264\pi$$
$$774$$ 10.2462 0.368292
$$775$$ 0 0
$$776$$ −6.00000 −0.215387
$$777$$ 39.3693 1.41237
$$778$$ −7.12311 −0.255376
$$779$$ −12.2462 −0.438766
$$780$$ 0 0
$$781$$ 40.9848 1.46655
$$782$$ 26.4233 0.944895
$$783$$ −8.17708 −0.292225
$$784$$ −0.438447 −0.0156588
$$785$$ 0 0
$$786$$ −42.2462 −1.50687
$$787$$ −11.1922 −0.398960 −0.199480 0.979902i $$-0.563925\pi$$
−0.199480 + 0.979902i $$0.563925\pi$$
$$788$$ 20.2462 0.721241
$$789$$ 56.9848 2.02871
$$790$$ 0 0
$$791$$ −22.7386 −0.808493
$$792$$ 14.2462 0.506217
$$793$$ −63.2311 −2.24540
$$794$$ 7.12311 0.252790
$$795$$ 0 0
$$796$$ 16.8078 0.595735
$$797$$ 11.3002 0.400273 0.200137 0.979768i $$-0.435861\pi$$
0.200137 + 0.979768i $$0.435861\pi$$
$$798$$ −6.56155 −0.232276
$$799$$ 21.4773 0.759811
$$800$$ 0 0
$$801$$ 7.12311 0.251683
$$802$$ −3.75379 −0.132551
$$803$$ 6.73863 0.237801
$$804$$ 6.56155 0.231408
$$805$$ 0 0
$$806$$ 29.1231 1.02582
$$807$$ −66.6004 −2.34444
$$808$$ −17.3693 −0.611050
$$809$$ −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 $$-0.619044\pi$$
−0.365330 + 0.930878i $$0.619044\pi$$
$$812$$ −14.5616 −0.511010
$$813$$ −56.1771 −1.97022
$$814$$ 24.0000 0.841200
$$815$$ 0 0
$$816$$ −8.80776 −0.308333
$$817$$ −2.87689 −0.100650
$$818$$ 24.7386 0.864966
$$819$$ −51.8617 −1.81220
$$820$$ 0 0
$$821$$ −17.3693 −0.606193 −0.303097 0.952960i $$-0.598021\pi$$
−0.303097 + 0.952960i $$0.598021\pi$$
$$822$$ 37.9309 1.32299
$$823$$ 29.4384 1.02616 0.513080 0.858341i $$-0.328504\pi$$
0.513080 + 0.858341i $$0.328504\pi$$
$$824$$ −2.24621 −0.0782505
$$825$$ 0 0
$$826$$ 6.56155 0.228306
$$827$$ 10.5616 0.367261 0.183631 0.982995i $$-0.441215\pi$$
0.183631 + 0.982995i $$0.441215\pi$$
$$828$$ −27.3693 −0.951150
$$829$$ −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
$$832$$ −5.68466 −0.197080
$$833$$ 1.50758 0.0522345
$$834$$ 42.2462 1.46287
$$835$$ 0 0
$$836$$ −4.00000 −0.138343
$$837$$ −7.36932 −0.254721
$$838$$ 23.8617 0.824290
$$839$$ 20.4924 0.707477 0.353738 0.935344i $$-0.384910\pi$$
0.353738 + 0.935344i $$0.384910\pi$$
$$840$$ 0 0
$$841$$ 3.31534 0.114322
$$842$$ −23.9309 −0.824712
$$843$$ 5.12311 0.176449
$$844$$ 8.31534 0.286226
$$845$$ 0 0
$$846$$ −22.2462 −0.764840
$$847$$ 12.8078 0.440080
$$848$$ 4.56155 0.156644
$$849$$ 54.1080 1.85698
$$850$$ 0 0
$$851$$ −46.1080 −1.58056
$$852$$ 26.2462 0.899180
$$853$$ 24.7386 0.847035 0.423517 0.905888i $$-0.360795\pi$$
0.423517 + 0.905888i $$0.360795\pi$$
$$854$$ 28.4924 0.974991
$$855$$ 0 0
$$856$$ 5.43845 0.185882
$$857$$ −14.6307 −0.499775 −0.249887 0.968275i $$-0.580394\pi$$
−0.249887 + 0.968275i $$0.580394\pi$$
$$858$$ −58.2462 −1.98849
$$859$$ −52.9848 −1.80782 −0.903910 0.427723i $$-0.859316\pi$$
−0.903910 + 0.427723i $$0.859316\pi$$
$$860$$ 0 0
$$861$$ 80.3542 2.73846
$$862$$ −16.0000 −0.544962
$$863$$ −2.24621 −0.0764619 −0.0382310 0.999269i $$-0.512172\pi$$
−0.0382310 + 0.999269i $$0.512172\pi$$
$$864$$ 1.43845 0.0489370
$$865$$ 0 0
$$866$$ −14.6307 −0.497171
$$867$$ −13.2614 −0.450380
$$868$$ −13.1231 −0.445427
$$869$$ −20.4924 −0.695158
$$870$$ 0 0
$$871$$ −14.5616 −0.493399
$$872$$ −0.561553 −0.0190166
$$873$$ −21.3693 −0.723242
$$874$$ 7.68466 0.259937
$$875$$ 0 0
$$876$$ 4.31534 0.145802
$$877$$ 3.93087 0.132736 0.0663680 0.997795i $$-0.478859\pi$$
0.0663680 + 0.997795i $$0.478859\pi$$
$$878$$ −13.1231 −0.442883
$$879$$ 56.8078 1.91608
$$880$$ 0 0
$$881$$ 42.9848 1.44820 0.724098 0.689697i $$-0.242256\pi$$
0.724098 + 0.689697i $$0.242256\pi$$
$$882$$ −1.56155 −0.0525802
$$883$$ 6.38447 0.214855 0.107427 0.994213i $$-0.465739\pi$$
0.107427 + 0.994213i $$0.465739\pi$$
$$884$$ 19.5464 0.657417
$$885$$ 0 0
$$886$$ −2.24621 −0.0754629
$$887$$ −28.4924 −0.956682 −0.478341 0.878174i $$-0.658762\pi$$
−0.478341 + 0.878174i $$0.658762\pi$$
$$888$$ 15.3693 0.515761
$$889$$ −33.6155 −1.12743
$$890$$ 0 0
$$891$$ −28.0000 −0.938035
$$892$$ 23.3693 0.782463
$$893$$ 6.24621 0.209021
$$894$$ 34.2462 1.14536
$$895$$ 0 0
$$896$$ 2.56155 0.0855755
$$897$$ 111.901 3.73625
$$898$$ −28.7386 −0.959021
$$899$$ 29.1231 0.971310
$$900$$ 0 0
$$901$$ −15.6847 −0.522532
$$902$$ 48.9848 1.63102
$$903$$ 18.8769 0.628184
$$904$$ −8.87689 −0.295241
$$905$$ 0 0
$$906$$ 13.1231 0.435986
$$907$$ 20.1771 0.669969 0.334984 0.942224i $$-0.391269\pi$$
0.334984 + 0.942224i $$0.391269\pi$$
$$908$$ −25.9309 −0.860546
$$909$$ −61.8617 −2.05182
$$910$$ 0 0
$$911$$ 4.49242 0.148841 0.0744203 0.997227i $$-0.476289\pi$$
0.0744203 + 0.997227i $$0.476289\pi$$
$$912$$ −2.56155 −0.0848215
$$913$$ −11.5076 −0.380845
$$914$$ −6.31534 −0.208893
$$915$$ 0 0
$$916$$ −14.4924 −0.478843
$$917$$ −42.2462 −1.39509
$$918$$ −4.94602 −0.163243
$$919$$ −2.06913 −0.0682543 −0.0341272 0.999417i $$-0.510865\pi$$
−0.0341272 + 0.999417i $$0.510865\pi$$
$$920$$ 0 0
$$921$$ −83.2311 −2.74256
$$922$$ 3.75379 0.123624
$$923$$ −58.2462 −1.91720
$$924$$ 26.2462 0.863437
$$925$$ 0 0
$$926$$ 30.2462 0.993952
$$927$$ −8.00000 −0.262754
$$928$$ −5.68466 −0.186608
$$929$$ −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
$$932$$ −10.0000 −0.327561
$$933$$ −9.43845 −0.309001
$$934$$ −18.2462 −0.597034
$$935$$ 0 0
$$936$$ −20.2462 −0.661768
$$937$$ −40.5616 −1.32509 −0.662544 0.749023i $$-0.730523\pi$$
−0.662544 + 0.749023i $$0.730523\pi$$
$$938$$ 6.56155 0.214242
$$939$$ −12.9460 −0.422478
$$940$$ 0 0
$$941$$ 54.8078 1.78668 0.893341 0.449379i $$-0.148355\pi$$
0.893341 + 0.449379i $$0.148355\pi$$
$$942$$ 51.8617 1.68975
$$943$$ −94.1080 −3.06458
$$944$$ 2.56155 0.0833714
$$945$$ 0 0
$$946$$ 11.5076 0.374144
$$947$$ 34.2462 1.11285 0.556426 0.830897i $$-0.312172\pi$$
0.556426 + 0.830897i $$0.312172\pi$$
$$948$$ −13.1231 −0.426219
$$949$$ −9.57671 −0.310873
$$950$$ 0 0
$$951$$ −33.4384 −1.08432
$$952$$ −8.80776 −0.285461
$$953$$ −44.1080 −1.42880 −0.714398 0.699739i $$-0.753300\pi$$
−0.714398 + 0.699739i $$0.753300\pi$$
$$954$$ 16.2462 0.525991
$$955$$ 0 0
$$956$$ −1.43845 −0.0465227
$$957$$ −58.2462 −1.88283
$$958$$ 32.0000 1.03387
$$959$$ 37.9309 1.22485
$$960$$ 0 0
$$961$$ −4.75379 −0.153348
$$962$$ −34.1080 −1.09968
$$963$$ 19.3693 0.624168
$$964$$ 23.1231 0.744745
$$965$$ 0 0
$$966$$ −50.4233 −1.62234
$$967$$ 15.5076 0.498690 0.249345 0.968415i $$-0.419785\pi$$
0.249345 + 0.968415i $$0.419785\pi$$
$$968$$ 5.00000 0.160706
$$969$$ 8.80776 0.282946
$$970$$ 0 0
$$971$$ −56.4924 −1.81293 −0.906464 0.422283i $$-0.861229\pi$$
−0.906464 + 0.422283i $$0.861229\pi$$
$$972$$ −22.2462 −0.713548
$$973$$ 42.2462 1.35435
$$974$$ 17.6155 0.564438
$$975$$ 0 0
$$976$$ 11.1231 0.356042
$$977$$ −28.7386 −0.919430 −0.459715 0.888066i $$-0.652048\pi$$
−0.459715 + 0.888066i $$0.652048\pi$$
$$978$$ 39.3693 1.25889
$$979$$ 8.00000 0.255681
$$980$$ 0 0
$$981$$ −2.00000 −0.0638551
$$982$$ −1.12311 −0.0358397
$$983$$ −18.8769 −0.602079 −0.301040 0.953612i $$-0.597334\pi$$
−0.301040 + 0.953612i $$0.597334\pi$$
$$984$$ 31.3693 1.00002
$$985$$ 0 0
$$986$$ 19.5464 0.622484
$$987$$ −40.9848 −1.30456
$$988$$ 5.68466 0.180853
$$989$$ −22.1080 −0.702992
$$990$$ 0 0
$$991$$ 2.87689 0.0913876 0.0456938 0.998955i $$-0.485450\pi$$
0.0456938 + 0.998955i $$0.485450\pi$$
$$992$$ −5.12311 −0.162659
$$993$$ −6.56155 −0.208225
$$994$$ 26.2462 0.832479
$$995$$ 0 0
$$996$$ −7.36932 −0.233506
$$997$$ 16.7386 0.530118 0.265059 0.964232i $$-0.414609\pi$$
0.265059 + 0.964232i $$0.414609\pi$$
$$998$$ 42.1080 1.33290
$$999$$ 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 950.2.a.h.1.2 2
3.2 odd 2 8550.2.a.br.1.2 2
4.3 odd 2 7600.2.a.y.1.1 2
5.2 odd 4 950.2.b.f.799.3 4
5.3 odd 4 950.2.b.f.799.2 4
5.4 even 2 190.2.a.d.1.1 2
15.14 odd 2 1710.2.a.w.1.1 2
20.19 odd 2 1520.2.a.n.1.2 2
35.34 odd 2 9310.2.a.bc.1.2 2
40.19 odd 2 6080.2.a.bb.1.1 2
40.29 even 2 6080.2.a.bh.1.2 2
95.94 odd 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 5.4 even 2
950.2.a.h.1.2 2 1.1 even 1 trivial
950.2.b.f.799.2 4 5.3 odd 4
950.2.b.f.799.3 4 5.2 odd 4
1520.2.a.n.1.2 2 20.19 odd 2
1710.2.a.w.1.1 2 15.14 odd 2
3610.2.a.t.1.2 2 95.94 odd 2
6080.2.a.bb.1.1 2 40.19 odd 2
6080.2.a.bh.1.2 2 40.29 even 2
7600.2.a.y.1.1 2 4.3 odd 2
8550.2.a.br.1.2 2 3.2 odd 2
9310.2.a.bc.1.2 2 35.34 odd 2