# Properties

 Label 950.2.a.f.1.1 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: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 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 1.1 Root $$-1.41421$$ 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)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -0.414214 q^{3} +1.00000 q^{4} +0.414214 q^{6} +4.41421 q^{7} -1.00000 q^{8} -2.82843 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -0.414214 q^{3} +1.00000 q^{4} +0.414214 q^{6} +4.41421 q^{7} -1.00000 q^{8} -2.82843 q^{9} -1.41421 q^{11} -0.414214 q^{12} +5.82843 q^{13} -4.41421 q^{14} +1.00000 q^{16} +1.00000 q^{17} +2.82843 q^{18} -1.00000 q^{19} -1.82843 q^{21} +1.41421 q^{22} +0.757359 q^{23} +0.414214 q^{24} -5.82843 q^{26} +2.41421 q^{27} +4.41421 q^{28} -0.171573 q^{29} +6.24264 q^{31} -1.00000 q^{32} +0.585786 q^{33} -1.00000 q^{34} -2.82843 q^{36} -8.48528 q^{37} +1.00000 q^{38} -2.41421 q^{39} -4.24264 q^{41} +1.82843 q^{42} +1.75736 q^{43} -1.41421 q^{44} -0.757359 q^{46} -0.414214 q^{48} +12.4853 q^{49} -0.414214 q^{51} +5.82843 q^{52} +5.48528 q^{53} -2.41421 q^{54} -4.41421 q^{56} +0.414214 q^{57} +0.171573 q^{58} +6.89949 q^{59} +14.2426 q^{61} -6.24264 q^{62} -12.4853 q^{63} +1.00000 q^{64} -0.585786 q^{66} +4.75736 q^{67} +1.00000 q^{68} -0.313708 q^{69} -13.4142 q^{71} +2.82843 q^{72} +11.4853 q^{73} +8.48528 q^{74} -1.00000 q^{76} -6.24264 q^{77} +2.41421 q^{78} -6.48528 q^{79} +7.48528 q^{81} +4.24264 q^{82} +14.4853 q^{83} -1.82843 q^{84} -1.75736 q^{86} +0.0710678 q^{87} +1.41421 q^{88} +7.07107 q^{89} +25.7279 q^{91} +0.757359 q^{92} -2.58579 q^{93} +0.414214 q^{96} -0.343146 q^{97} -12.4853 q^{98} +4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} + 6 q^{7} - 2 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 + 6 * q^7 - 2 * q^8 $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} + 6 q^{7} - 2 q^{8} + 2 q^{12} + 6 q^{13} - 6 q^{14} + 2 q^{16} + 2 q^{17} - 2 q^{19} + 2 q^{21} + 10 q^{23} - 2 q^{24} - 6 q^{26} + 2 q^{27} + 6 q^{28} - 6 q^{29} + 4 q^{31} - 2 q^{32} + 4 q^{33} - 2 q^{34} + 2 q^{38} - 2 q^{39} - 2 q^{42} + 12 q^{43} - 10 q^{46} + 2 q^{48} + 8 q^{49} + 2 q^{51} + 6 q^{52} - 6 q^{53} - 2 q^{54} - 6 q^{56} - 2 q^{57} + 6 q^{58} - 6 q^{59} + 20 q^{61} - 4 q^{62} - 8 q^{63} + 2 q^{64} - 4 q^{66} + 18 q^{67} + 2 q^{68} + 22 q^{69} - 24 q^{71} + 6 q^{73} - 2 q^{76} - 4 q^{77} + 2 q^{78} + 4 q^{79} - 2 q^{81} + 12 q^{83} + 2 q^{84} - 12 q^{86} - 14 q^{87} + 26 q^{91} + 10 q^{92} - 8 q^{93} - 2 q^{96} - 12 q^{97} - 8 q^{98} + 8 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 + 6 * q^7 - 2 * q^8 + 2 * q^12 + 6 * q^13 - 6 * q^14 + 2 * q^16 + 2 * q^17 - 2 * q^19 + 2 * q^21 + 10 * q^23 - 2 * q^24 - 6 * q^26 + 2 * q^27 + 6 * q^28 - 6 * q^29 + 4 * q^31 - 2 * q^32 + 4 * q^33 - 2 * q^34 + 2 * q^38 - 2 * q^39 - 2 * q^42 + 12 * q^43 - 10 * q^46 + 2 * q^48 + 8 * q^49 + 2 * q^51 + 6 * q^52 - 6 * q^53 - 2 * q^54 - 6 * q^56 - 2 * q^57 + 6 * q^58 - 6 * q^59 + 20 * q^61 - 4 * q^62 - 8 * q^63 + 2 * q^64 - 4 * q^66 + 18 * q^67 + 2 * q^68 + 22 * q^69 - 24 * q^71 + 6 * q^73 - 2 * q^76 - 4 * q^77 + 2 * q^78 + 4 * q^79 - 2 * q^81 + 12 * q^83 + 2 * q^84 - 12 * q^86 - 14 * q^87 + 26 * q^91 + 10 * q^92 - 8 * q^93 - 2 * q^96 - 12 * q^97 - 8 * q^98 + 8 * 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
$$3$$ −0.414214 −0.239146 −0.119573 0.992825i $$-0.538153\pi$$
−0.119573 + 0.992825i $$0.538153\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 0.414214 0.169102
$$7$$ 4.41421 1.66842 0.834208 0.551450i $$-0.185925\pi$$
0.834208 + 0.551450i $$0.185925\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ −2.82843 −0.942809
$$10$$ 0 0
$$11$$ −1.41421 −0.426401 −0.213201 0.977008i $$-0.568389\pi$$
−0.213201 + 0.977008i $$0.568389\pi$$
$$12$$ −0.414214 −0.119573
$$13$$ 5.82843 1.61651 0.808257 0.588829i $$-0.200411\pi$$
0.808257 + 0.588829i $$0.200411\pi$$
$$14$$ −4.41421 −1.17975
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 1.00000 0.242536 0.121268 0.992620i $$-0.461304\pi$$
0.121268 + 0.992620i $$0.461304\pi$$
$$18$$ 2.82843 0.666667
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −1.82843 −0.398996
$$22$$ 1.41421 0.301511
$$23$$ 0.757359 0.157920 0.0789602 0.996878i $$-0.474840\pi$$
0.0789602 + 0.996878i $$0.474840\pi$$
$$24$$ 0.414214 0.0845510
$$25$$ 0 0
$$26$$ −5.82843 −1.14305
$$27$$ 2.41421 0.464616
$$28$$ 4.41421 0.834208
$$29$$ −0.171573 −0.0318603 −0.0159301 0.999873i $$-0.505071\pi$$
−0.0159301 + 0.999873i $$0.505071\pi$$
$$30$$ 0 0
$$31$$ 6.24264 1.12121 0.560606 0.828083i $$-0.310568\pi$$
0.560606 + 0.828083i $$0.310568\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0.585786 0.101972
$$34$$ −1.00000 −0.171499
$$35$$ 0 0
$$36$$ −2.82843 −0.471405
$$37$$ −8.48528 −1.39497 −0.697486 0.716599i $$-0.745698\pi$$
−0.697486 + 0.716599i $$0.745698\pi$$
$$38$$ 1.00000 0.162221
$$39$$ −2.41421 −0.386584
$$40$$ 0 0
$$41$$ −4.24264 −0.662589 −0.331295 0.943527i $$-0.607485\pi$$
−0.331295 + 0.943527i $$0.607485\pi$$
$$42$$ 1.82843 0.282132
$$43$$ 1.75736 0.267995 0.133997 0.990982i $$-0.457219\pi$$
0.133997 + 0.990982i $$0.457219\pi$$
$$44$$ −1.41421 −0.213201
$$45$$ 0 0
$$46$$ −0.757359 −0.111667
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ −0.414214 −0.0597866
$$49$$ 12.4853 1.78361
$$50$$ 0 0
$$51$$ −0.414214 −0.0580015
$$52$$ 5.82843 0.808257
$$53$$ 5.48528 0.753461 0.376731 0.926323i $$-0.377048\pi$$
0.376731 + 0.926323i $$0.377048\pi$$
$$54$$ −2.41421 −0.328533
$$55$$ 0 0
$$56$$ −4.41421 −0.589874
$$57$$ 0.414214 0.0548639
$$58$$ 0.171573 0.0225286
$$59$$ 6.89949 0.898238 0.449119 0.893472i $$-0.351738\pi$$
0.449119 + 0.893472i $$0.351738\pi$$
$$60$$ 0 0
$$61$$ 14.2426 1.82358 0.911792 0.410653i $$-0.134699\pi$$
0.911792 + 0.410653i $$0.134699\pi$$
$$62$$ −6.24264 −0.792816
$$63$$ −12.4853 −1.57300
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −0.585786 −0.0721053
$$67$$ 4.75736 0.581204 0.290602 0.956844i $$-0.406144\pi$$
0.290602 + 0.956844i $$0.406144\pi$$
$$68$$ 1.00000 0.121268
$$69$$ −0.313708 −0.0377661
$$70$$ 0 0
$$71$$ −13.4142 −1.59197 −0.795987 0.605314i $$-0.793048\pi$$
−0.795987 + 0.605314i $$0.793048\pi$$
$$72$$ 2.82843 0.333333
$$73$$ 11.4853 1.34425 0.672125 0.740437i $$-0.265382\pi$$
0.672125 + 0.740437i $$0.265382\pi$$
$$74$$ 8.48528 0.986394
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ −6.24264 −0.711415
$$78$$ 2.41421 0.273356
$$79$$ −6.48528 −0.729651 −0.364826 0.931076i $$-0.618871\pi$$
−0.364826 + 0.931076i $$0.618871\pi$$
$$80$$ 0 0
$$81$$ 7.48528 0.831698
$$82$$ 4.24264 0.468521
$$83$$ 14.4853 1.58997 0.794983 0.606632i $$-0.207480\pi$$
0.794983 + 0.606632i $$0.207480\pi$$
$$84$$ −1.82843 −0.199498
$$85$$ 0 0
$$86$$ −1.75736 −0.189501
$$87$$ 0.0710678 0.00761927
$$88$$ 1.41421 0.150756
$$89$$ 7.07107 0.749532 0.374766 0.927119i $$-0.377723\pi$$
0.374766 + 0.927119i $$0.377723\pi$$
$$90$$ 0 0
$$91$$ 25.7279 2.69702
$$92$$ 0.757359 0.0789602
$$93$$ −2.58579 −0.268134
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0.414214 0.0422755
$$97$$ −0.343146 −0.0348412 −0.0174206 0.999848i $$-0.505545\pi$$
−0.0174206 + 0.999848i $$0.505545\pi$$
$$98$$ −12.4853 −1.26120
$$99$$ 4.00000 0.402015
$$100$$ 0 0
$$101$$ 13.0711 1.30062 0.650310 0.759669i $$-0.274639\pi$$
0.650310 + 0.759669i $$0.274639\pi$$
$$102$$ 0.414214 0.0410133
$$103$$ 4.24264 0.418040 0.209020 0.977911i $$-0.432973\pi$$
0.209020 + 0.977911i $$0.432973\pi$$
$$104$$ −5.82843 −0.571524
$$105$$ 0 0
$$106$$ −5.48528 −0.532778
$$107$$ 19.7279 1.90717 0.953585 0.301124i $$-0.0973617\pi$$
0.953585 + 0.301124i $$0.0973617\pi$$
$$108$$ 2.41421 0.232308
$$109$$ −17.9706 −1.72127 −0.860634 0.509224i $$-0.829932\pi$$
−0.860634 + 0.509224i $$0.829932\pi$$
$$110$$ 0 0
$$111$$ 3.51472 0.333602
$$112$$ 4.41421 0.417104
$$113$$ −10.2426 −0.963547 −0.481773 0.876296i $$-0.660007\pi$$
−0.481773 + 0.876296i $$0.660007\pi$$
$$114$$ −0.414214 −0.0387947
$$115$$ 0 0
$$116$$ −0.171573 −0.0159301
$$117$$ −16.4853 −1.52406
$$118$$ −6.89949 −0.635150
$$119$$ 4.41421 0.404650
$$120$$ 0 0
$$121$$ −9.00000 −0.818182
$$122$$ −14.2426 −1.28947
$$123$$ 1.75736 0.158456
$$124$$ 6.24264 0.560606
$$125$$ 0 0
$$126$$ 12.4853 1.11228
$$127$$ −2.48528 −0.220533 −0.110267 0.993902i $$-0.535170\pi$$
−0.110267 + 0.993902i $$0.535170\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −0.727922 −0.0640900
$$130$$ 0 0
$$131$$ −16.9706 −1.48272 −0.741362 0.671105i $$-0.765820\pi$$
−0.741362 + 0.671105i $$0.765820\pi$$
$$132$$ 0.585786 0.0509862
$$133$$ −4.41421 −0.382761
$$134$$ −4.75736 −0.410973
$$135$$ 0 0
$$136$$ −1.00000 −0.0857493
$$137$$ 13.0000 1.11066 0.555332 0.831628i $$-0.312591\pi$$
0.555332 + 0.831628i $$0.312591\pi$$
$$138$$ 0.313708 0.0267046
$$139$$ −12.0000 −1.01783 −0.508913 0.860818i $$-0.669953\pi$$
−0.508913 + 0.860818i $$0.669953\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 13.4142 1.12570
$$143$$ −8.24264 −0.689284
$$144$$ −2.82843 −0.235702
$$145$$ 0 0
$$146$$ −11.4853 −0.950529
$$147$$ −5.17157 −0.426544
$$148$$ −8.48528 −0.697486
$$149$$ 17.6569 1.44651 0.723253 0.690583i $$-0.242646\pi$$
0.723253 + 0.690583i $$0.242646\pi$$
$$150$$ 0 0
$$151$$ −10.4853 −0.853280 −0.426640 0.904422i $$-0.640303\pi$$
−0.426640 + 0.904422i $$0.640303\pi$$
$$152$$ 1.00000 0.0811107
$$153$$ −2.82843 −0.228665
$$154$$ 6.24264 0.503046
$$155$$ 0 0
$$156$$ −2.41421 −0.193292
$$157$$ 11.6569 0.930318 0.465159 0.885227i $$-0.345997\pi$$
0.465159 + 0.885227i $$0.345997\pi$$
$$158$$ 6.48528 0.515941
$$159$$ −2.27208 −0.180188
$$160$$ 0 0
$$161$$ 3.34315 0.263477
$$162$$ −7.48528 −0.588099
$$163$$ −10.2426 −0.802266 −0.401133 0.916020i $$-0.631383\pi$$
−0.401133 + 0.916020i $$0.631383\pi$$
$$164$$ −4.24264 −0.331295
$$165$$ 0 0
$$166$$ −14.4853 −1.12428
$$167$$ −18.2426 −1.41166 −0.705829 0.708382i $$-0.749425\pi$$
−0.705829 + 0.708382i $$0.749425\pi$$
$$168$$ 1.82843 0.141066
$$169$$ 20.9706 1.61312
$$170$$ 0 0
$$171$$ 2.82843 0.216295
$$172$$ 1.75736 0.133997
$$173$$ 0.485281 0.0368953 0.0184476 0.999830i $$-0.494128\pi$$
0.0184476 + 0.999830i $$0.494128\pi$$
$$174$$ −0.0710678 −0.00538764
$$175$$ 0 0
$$176$$ −1.41421 −0.106600
$$177$$ −2.85786 −0.214810
$$178$$ −7.07107 −0.529999
$$179$$ −11.6569 −0.871274 −0.435637 0.900122i $$-0.643477\pi$$
−0.435637 + 0.900122i $$0.643477\pi$$
$$180$$ 0 0
$$181$$ −8.48528 −0.630706 −0.315353 0.948974i $$-0.602123\pi$$
−0.315353 + 0.948974i $$0.602123\pi$$
$$182$$ −25.7279 −1.90708
$$183$$ −5.89949 −0.436103
$$184$$ −0.757359 −0.0558333
$$185$$ 0 0
$$186$$ 2.58579 0.189599
$$187$$ −1.41421 −0.103418
$$188$$ 0 0
$$189$$ 10.6569 0.775172
$$190$$ 0 0
$$191$$ 12.5563 0.908546 0.454273 0.890863i $$-0.349899\pi$$
0.454273 + 0.890863i $$0.349899\pi$$
$$192$$ −0.414214 −0.0298933
$$193$$ −0.343146 −0.0247002 −0.0123501 0.999924i $$-0.503931\pi$$
−0.0123501 + 0.999924i $$0.503931\pi$$
$$194$$ 0.343146 0.0246364
$$195$$ 0 0
$$196$$ 12.4853 0.891806
$$197$$ −11.7574 −0.837677 −0.418839 0.908061i $$-0.637563\pi$$
−0.418839 + 0.908061i $$0.637563\pi$$
$$198$$ −4.00000 −0.284268
$$199$$ 9.24264 0.655193 0.327597 0.944818i $$-0.393761\pi$$
0.327597 + 0.944818i $$0.393761\pi$$
$$200$$ 0 0
$$201$$ −1.97056 −0.138993
$$202$$ −13.0711 −0.919677
$$203$$ −0.757359 −0.0531562
$$204$$ −0.414214 −0.0290008
$$205$$ 0 0
$$206$$ −4.24264 −0.295599
$$207$$ −2.14214 −0.148889
$$208$$ 5.82843 0.404129
$$209$$ 1.41421 0.0978232
$$210$$ 0 0
$$211$$ −19.7279 −1.35813 −0.679063 0.734080i $$-0.737614\pi$$
−0.679063 + 0.734080i $$0.737614\pi$$
$$212$$ 5.48528 0.376731
$$213$$ 5.55635 0.380715
$$214$$ −19.7279 −1.34857
$$215$$ 0 0
$$216$$ −2.41421 −0.164266
$$217$$ 27.5563 1.87065
$$218$$ 17.9706 1.21712
$$219$$ −4.75736 −0.321473
$$220$$ 0 0
$$221$$ 5.82843 0.392062
$$222$$ −3.51472 −0.235892
$$223$$ −15.1716 −1.01596 −0.507982 0.861368i $$-0.669608\pi$$
−0.507982 + 0.861368i $$0.669608\pi$$
$$224$$ −4.41421 −0.294937
$$225$$ 0 0
$$226$$ 10.2426 0.681330
$$227$$ −16.7574 −1.11223 −0.556113 0.831107i $$-0.687708\pi$$
−0.556113 + 0.831107i $$0.687708\pi$$
$$228$$ 0.414214 0.0274320
$$229$$ 14.9706 0.989283 0.494641 0.869097i $$-0.335299\pi$$
0.494641 + 0.869097i $$0.335299\pi$$
$$230$$ 0 0
$$231$$ 2.58579 0.170132
$$232$$ 0.171573 0.0112643
$$233$$ −24.9706 −1.63588 −0.817938 0.575306i $$-0.804883\pi$$
−0.817938 + 0.575306i $$0.804883\pi$$
$$234$$ 16.4853 1.07768
$$235$$ 0 0
$$236$$ 6.89949 0.449119
$$237$$ 2.68629 0.174493
$$238$$ −4.41421 −0.286131
$$239$$ −6.89949 −0.446291 −0.223146 0.974785i $$-0.571633\pi$$
−0.223146 + 0.974785i $$0.571633\pi$$
$$240$$ 0 0
$$241$$ 8.97056 0.577845 0.288922 0.957353i $$-0.406703\pi$$
0.288922 + 0.957353i $$0.406703\pi$$
$$242$$ 9.00000 0.578542
$$243$$ −10.3431 −0.663513
$$244$$ 14.2426 0.911792
$$245$$ 0 0
$$246$$ −1.75736 −0.112045
$$247$$ −5.82843 −0.370854
$$248$$ −6.24264 −0.396408
$$249$$ −6.00000 −0.380235
$$250$$ 0 0
$$251$$ 3.55635 0.224475 0.112237 0.993681i $$-0.464198\pi$$
0.112237 + 0.993681i $$0.464198\pi$$
$$252$$ −12.4853 −0.786499
$$253$$ −1.07107 −0.0673375
$$254$$ 2.48528 0.155940
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 20.7279 1.29297 0.646486 0.762926i $$-0.276238\pi$$
0.646486 + 0.762926i $$0.276238\pi$$
$$258$$ 0.727922 0.0453184
$$259$$ −37.4558 −2.32739
$$260$$ 0 0
$$261$$ 0.485281 0.0300382
$$262$$ 16.9706 1.04844
$$263$$ −26.9706 −1.66308 −0.831538 0.555468i $$-0.812539\pi$$
−0.831538 + 0.555468i $$0.812539\pi$$
$$264$$ −0.585786 −0.0360527
$$265$$ 0 0
$$266$$ 4.41421 0.270653
$$267$$ −2.92893 −0.179248
$$268$$ 4.75736 0.290602
$$269$$ −16.6274 −1.01379 −0.506896 0.862007i $$-0.669207\pi$$
−0.506896 + 0.862007i $$0.669207\pi$$
$$270$$ 0 0
$$271$$ −27.2426 −1.65487 −0.827436 0.561560i $$-0.810202\pi$$
−0.827436 + 0.561560i $$0.810202\pi$$
$$272$$ 1.00000 0.0606339
$$273$$ −10.6569 −0.644982
$$274$$ −13.0000 −0.785359
$$275$$ 0 0
$$276$$ −0.313708 −0.0188830
$$277$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$278$$ 12.0000 0.719712
$$279$$ −17.6569 −1.05709
$$280$$ 0 0
$$281$$ 4.24264 0.253095 0.126547 0.991961i $$-0.459610\pi$$
0.126547 + 0.991961i $$0.459610\pi$$
$$282$$ 0 0
$$283$$ 32.1421 1.91065 0.955326 0.295555i $$-0.0955045\pi$$
0.955326 + 0.295555i $$0.0955045\pi$$
$$284$$ −13.4142 −0.795987
$$285$$ 0 0
$$286$$ 8.24264 0.487398
$$287$$ −18.7279 −1.10547
$$288$$ 2.82843 0.166667
$$289$$ −16.0000 −0.941176
$$290$$ 0 0
$$291$$ 0.142136 0.00833214
$$292$$ 11.4853 0.672125
$$293$$ −5.48528 −0.320454 −0.160227 0.987080i $$-0.551223\pi$$
−0.160227 + 0.987080i $$0.551223\pi$$
$$294$$ 5.17157 0.301612
$$295$$ 0 0
$$296$$ 8.48528 0.493197
$$297$$ −3.41421 −0.198113
$$298$$ −17.6569 −1.02283
$$299$$ 4.41421 0.255281
$$300$$ 0 0
$$301$$ 7.75736 0.447127
$$302$$ 10.4853 0.603360
$$303$$ −5.41421 −0.311038
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ 2.82843 0.161690
$$307$$ −17.6569 −1.00773 −0.503865 0.863782i $$-0.668089\pi$$
−0.503865 + 0.863782i $$0.668089\pi$$
$$308$$ −6.24264 −0.355707
$$309$$ −1.75736 −0.0999727
$$310$$ 0 0
$$311$$ −4.75736 −0.269765 −0.134883 0.990862i $$-0.543066\pi$$
−0.134883 + 0.990862i $$0.543066\pi$$
$$312$$ 2.41421 0.136678
$$313$$ −7.97056 −0.450523 −0.225261 0.974298i $$-0.572324\pi$$
−0.225261 + 0.974298i $$0.572324\pi$$
$$314$$ −11.6569 −0.657834
$$315$$ 0 0
$$316$$ −6.48528 −0.364826
$$317$$ −9.48528 −0.532746 −0.266373 0.963870i $$-0.585825\pi$$
−0.266373 + 0.963870i $$0.585825\pi$$
$$318$$ 2.27208 0.127412
$$319$$ 0.242641 0.0135853
$$320$$ 0 0
$$321$$ −8.17157 −0.456093
$$322$$ −3.34315 −0.186306
$$323$$ −1.00000 −0.0556415
$$324$$ 7.48528 0.415849
$$325$$ 0 0
$$326$$ 10.2426 0.567287
$$327$$ 7.44365 0.411635
$$328$$ 4.24264 0.234261
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −19.2426 −1.05767 −0.528836 0.848724i $$-0.677371\pi$$
−0.528836 + 0.848724i $$0.677371\pi$$
$$332$$ 14.4853 0.794983
$$333$$ 24.0000 1.31519
$$334$$ 18.2426 0.998193
$$335$$ 0 0
$$336$$ −1.82843 −0.0997489
$$337$$ 14.1005 0.768103 0.384052 0.923312i $$-0.374528\pi$$
0.384052 + 0.923312i $$0.374528\pi$$
$$338$$ −20.9706 −1.14065
$$339$$ 4.24264 0.230429
$$340$$ 0 0
$$341$$ −8.82843 −0.478086
$$342$$ −2.82843 −0.152944
$$343$$ 24.2132 1.30739
$$344$$ −1.75736 −0.0947505
$$345$$ 0 0
$$346$$ −0.485281 −0.0260889
$$347$$ −5.51472 −0.296046 −0.148023 0.988984i $$-0.547291\pi$$
−0.148023 + 0.988984i $$0.547291\pi$$
$$348$$ 0.0710678 0.00380963
$$349$$ 6.00000 0.321173 0.160586 0.987022i $$-0.448662\pi$$
0.160586 + 0.987022i $$0.448662\pi$$
$$350$$ 0 0
$$351$$ 14.0711 0.751058
$$352$$ 1.41421 0.0753778
$$353$$ −2.51472 −0.133845 −0.0669225 0.997758i $$-0.521318\pi$$
−0.0669225 + 0.997758i $$0.521318\pi$$
$$354$$ 2.85786 0.151894
$$355$$ 0 0
$$356$$ 7.07107 0.374766
$$357$$ −1.82843 −0.0967706
$$358$$ 11.6569 0.616084
$$359$$ −22.7574 −1.20109 −0.600544 0.799592i $$-0.705049\pi$$
−0.600544 + 0.799592i $$0.705049\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 8.48528 0.445976
$$363$$ 3.72792 0.195665
$$364$$ 25.7279 1.34851
$$365$$ 0 0
$$366$$ 5.89949 0.308372
$$367$$ −25.4558 −1.32878 −0.664392 0.747384i $$-0.731309\pi$$
−0.664392 + 0.747384i $$0.731309\pi$$
$$368$$ 0.757359 0.0394801
$$369$$ 12.0000 0.624695
$$370$$ 0 0
$$371$$ 24.2132 1.25709
$$372$$ −2.58579 −0.134067
$$373$$ 9.00000 0.466002 0.233001 0.972476i $$-0.425145\pi$$
0.233001 + 0.972476i $$0.425145\pi$$
$$374$$ 1.41421 0.0731272
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −1.00000 −0.0515026
$$378$$ −10.6569 −0.548129
$$379$$ 11.2426 0.577496 0.288748 0.957405i $$-0.406761\pi$$
0.288748 + 0.957405i $$0.406761\pi$$
$$380$$ 0 0
$$381$$ 1.02944 0.0527397
$$382$$ −12.5563 −0.642439
$$383$$ 12.2426 0.625570 0.312785 0.949824i $$-0.398738\pi$$
0.312785 + 0.949824i $$0.398738\pi$$
$$384$$ 0.414214 0.0211377
$$385$$ 0 0
$$386$$ 0.343146 0.0174657
$$387$$ −4.97056 −0.252668
$$388$$ −0.343146 −0.0174206
$$389$$ 22.9289 1.16254 0.581272 0.813710i $$-0.302555\pi$$
0.581272 + 0.813710i $$0.302555\pi$$
$$390$$ 0 0
$$391$$ 0.757359 0.0383013
$$392$$ −12.4853 −0.630602
$$393$$ 7.02944 0.354588
$$394$$ 11.7574 0.592327
$$395$$ 0 0
$$396$$ 4.00000 0.201008
$$397$$ 24.0000 1.20453 0.602263 0.798298i $$-0.294266\pi$$
0.602263 + 0.798298i $$0.294266\pi$$
$$398$$ −9.24264 −0.463292
$$399$$ 1.82843 0.0915358
$$400$$ 0 0
$$401$$ −25.4142 −1.26913 −0.634563 0.772871i $$-0.718820\pi$$
−0.634563 + 0.772871i $$0.718820\pi$$
$$402$$ 1.97056 0.0982827
$$403$$ 36.3848 1.81245
$$404$$ 13.0711 0.650310
$$405$$ 0 0
$$406$$ 0.757359 0.0375871
$$407$$ 12.0000 0.594818
$$408$$ 0.414214 0.0205066
$$409$$ 25.2132 1.24671 0.623356 0.781938i $$-0.285769\pi$$
0.623356 + 0.781938i $$0.285769\pi$$
$$410$$ 0 0
$$411$$ −5.38478 −0.265611
$$412$$ 4.24264 0.209020
$$413$$ 30.4558 1.49863
$$414$$ 2.14214 0.105280
$$415$$ 0 0
$$416$$ −5.82843 −0.285762
$$417$$ 4.97056 0.243410
$$418$$ −1.41421 −0.0691714
$$419$$ 19.4142 0.948446 0.474223 0.880405i $$-0.342729\pi$$
0.474223 + 0.880405i $$0.342729\pi$$
$$420$$ 0 0
$$421$$ 19.4853 0.949655 0.474827 0.880079i $$-0.342511\pi$$
0.474827 + 0.880079i $$0.342511\pi$$
$$422$$ 19.7279 0.960340
$$423$$ 0 0
$$424$$ −5.48528 −0.266389
$$425$$ 0 0
$$426$$ −5.55635 −0.269206
$$427$$ 62.8701 3.04250
$$428$$ 19.7279 0.953585
$$429$$ 3.41421 0.164840
$$430$$ 0 0
$$431$$ −6.38478 −0.307544 −0.153772 0.988106i $$-0.549142\pi$$
−0.153772 + 0.988106i $$0.549142\pi$$
$$432$$ 2.41421 0.116154
$$433$$ −9.55635 −0.459249 −0.229624 0.973279i $$-0.573750\pi$$
−0.229624 + 0.973279i $$0.573750\pi$$
$$434$$ −27.5563 −1.32275
$$435$$ 0 0
$$436$$ −17.9706 −0.860634
$$437$$ −0.757359 −0.0362294
$$438$$ 4.75736 0.227315
$$439$$ −5.75736 −0.274784 −0.137392 0.990517i $$-0.543872\pi$$
−0.137392 + 0.990517i $$0.543872\pi$$
$$440$$ 0 0
$$441$$ −35.3137 −1.68161
$$442$$ −5.82843 −0.277230
$$443$$ 4.24264 0.201574 0.100787 0.994908i $$-0.467864\pi$$
0.100787 + 0.994908i $$0.467864\pi$$
$$444$$ 3.51472 0.166801
$$445$$ 0 0
$$446$$ 15.1716 0.718395
$$447$$ −7.31371 −0.345927
$$448$$ 4.41421 0.208552
$$449$$ 17.3137 0.817084 0.408542 0.912739i $$-0.366037\pi$$
0.408542 + 0.912739i $$0.366037\pi$$
$$450$$ 0 0
$$451$$ 6.00000 0.282529
$$452$$ −10.2426 −0.481773
$$453$$ 4.34315 0.204059
$$454$$ 16.7574 0.786462
$$455$$ 0 0
$$456$$ −0.414214 −0.0193973
$$457$$ −3.00000 −0.140334 −0.0701670 0.997535i $$-0.522353\pi$$
−0.0701670 + 0.997535i $$0.522353\pi$$
$$458$$ −14.9706 −0.699528
$$459$$ 2.41421 0.112686
$$460$$ 0 0
$$461$$ −3.55635 −0.165636 −0.0828178 0.996565i $$-0.526392\pi$$
−0.0828178 + 0.996565i $$0.526392\pi$$
$$462$$ −2.58579 −0.120302
$$463$$ 14.1421 0.657241 0.328620 0.944462i $$-0.393416\pi$$
0.328620 + 0.944462i $$0.393416\pi$$
$$464$$ −0.171573 −0.00796507
$$465$$ 0 0
$$466$$ 24.9706 1.15674
$$467$$ 0.727922 0.0336842 0.0168421 0.999858i $$-0.494639\pi$$
0.0168421 + 0.999858i $$0.494639\pi$$
$$468$$ −16.4853 −0.762032
$$469$$ 21.0000 0.969690
$$470$$ 0 0
$$471$$ −4.82843 −0.222482
$$472$$ −6.89949 −0.317575
$$473$$ −2.48528 −0.114273
$$474$$ −2.68629 −0.123385
$$475$$ 0 0
$$476$$ 4.41421 0.202325
$$477$$ −15.5147 −0.710370
$$478$$ 6.89949 0.315576
$$479$$ −31.1127 −1.42158 −0.710788 0.703407i $$-0.751661\pi$$
−0.710788 + 0.703407i $$0.751661\pi$$
$$480$$ 0 0
$$481$$ −49.4558 −2.25499
$$482$$ −8.97056 −0.408598
$$483$$ −1.38478 −0.0630095
$$484$$ −9.00000 −0.409091
$$485$$ 0 0
$$486$$ 10.3431 0.469175
$$487$$ 37.7990 1.71284 0.856418 0.516283i $$-0.172685\pi$$
0.856418 + 0.516283i $$0.172685\pi$$
$$488$$ −14.2426 −0.644734
$$489$$ 4.24264 0.191859
$$490$$ 0 0
$$491$$ −33.5563 −1.51438 −0.757188 0.653197i $$-0.773428\pi$$
−0.757188 + 0.653197i $$0.773428\pi$$
$$492$$ 1.75736 0.0792279
$$493$$ −0.171573 −0.00772725
$$494$$ 5.82843 0.262233
$$495$$ 0 0
$$496$$ 6.24264 0.280303
$$497$$ −59.2132 −2.65608
$$498$$ 6.00000 0.268866
$$499$$ −25.7574 −1.15306 −0.576529 0.817077i $$-0.695593\pi$$
−0.576529 + 0.817077i $$0.695593\pi$$
$$500$$ 0 0
$$501$$ 7.55635 0.337593
$$502$$ −3.55635 −0.158728
$$503$$ 14.2721 0.636361 0.318180 0.948030i $$-0.396928\pi$$
0.318180 + 0.948030i $$0.396928\pi$$
$$504$$ 12.4853 0.556139
$$505$$ 0 0
$$506$$ 1.07107 0.0476148
$$507$$ −8.68629 −0.385772
$$508$$ −2.48528 −0.110267
$$509$$ 28.9706 1.28410 0.642049 0.766664i $$-0.278085\pi$$
0.642049 + 0.766664i $$0.278085\pi$$
$$510$$ 0 0
$$511$$ 50.6985 2.24277
$$512$$ −1.00000 −0.0441942
$$513$$ −2.41421 −0.106590
$$514$$ −20.7279 −0.914269
$$515$$ 0 0
$$516$$ −0.727922 −0.0320450
$$517$$ 0 0
$$518$$ 37.4558 1.64572
$$519$$ −0.201010 −0.00882337
$$520$$ 0 0
$$521$$ 23.3137 1.02139 0.510696 0.859761i $$-0.329388\pi$$
0.510696 + 0.859761i $$0.329388\pi$$
$$522$$ −0.485281 −0.0212402
$$523$$ −2.27208 −0.0993510 −0.0496755 0.998765i $$-0.515819\pi$$
−0.0496755 + 0.998765i $$0.515819\pi$$
$$524$$ −16.9706 −0.741362
$$525$$ 0 0
$$526$$ 26.9706 1.17597
$$527$$ 6.24264 0.271934
$$528$$ 0.585786 0.0254931
$$529$$ −22.4264 −0.975061
$$530$$ 0 0
$$531$$ −19.5147 −0.846867
$$532$$ −4.41421 −0.191380
$$533$$ −24.7279 −1.07109
$$534$$ 2.92893 0.126747
$$535$$ 0 0
$$536$$ −4.75736 −0.205487
$$537$$ 4.82843 0.208362
$$538$$ 16.6274 0.716859
$$539$$ −17.6569 −0.760535
$$540$$ 0 0
$$541$$ 9.75736 0.419502 0.209751 0.977755i $$-0.432735\pi$$
0.209751 + 0.977755i $$0.432735\pi$$
$$542$$ 27.2426 1.17017
$$543$$ 3.51472 0.150831
$$544$$ −1.00000 −0.0428746
$$545$$ 0 0
$$546$$ 10.6569 0.456071
$$547$$ −17.3137 −0.740281 −0.370140 0.928976i $$-0.620690\pi$$
−0.370140 + 0.928976i $$0.620690\pi$$
$$548$$ 13.0000 0.555332
$$549$$ −40.2843 −1.71929
$$550$$ 0 0
$$551$$ 0.171573 0.00730925
$$552$$ 0.313708 0.0133523
$$553$$ −28.6274 −1.21736
$$554$$ 0 0
$$555$$ 0 0
$$556$$ −12.0000 −0.508913
$$557$$ 16.0000 0.677942 0.338971 0.940797i $$-0.389921\pi$$
0.338971 + 0.940797i $$0.389921\pi$$
$$558$$ 17.6569 0.747474
$$559$$ 10.2426 0.433218
$$560$$ 0 0
$$561$$ 0.585786 0.0247319
$$562$$ −4.24264 −0.178965
$$563$$ 4.97056 0.209484 0.104742 0.994499i $$-0.466598\pi$$
0.104742 + 0.994499i $$0.466598\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −32.1421 −1.35103
$$567$$ 33.0416 1.38762
$$568$$ 13.4142 0.562848
$$569$$ −28.2843 −1.18574 −0.592869 0.805299i $$-0.702005\pi$$
−0.592869 + 0.805299i $$0.702005\pi$$
$$570$$ 0 0
$$571$$ −2.24264 −0.0938516 −0.0469258 0.998898i $$-0.514942\pi$$
−0.0469258 + 0.998898i $$0.514942\pi$$
$$572$$ −8.24264 −0.344642
$$573$$ −5.20101 −0.217275
$$574$$ 18.7279 0.781688
$$575$$ 0 0
$$576$$ −2.82843 −0.117851
$$577$$ −20.3137 −0.845671 −0.422835 0.906207i $$-0.638965\pi$$
−0.422835 + 0.906207i $$0.638965\pi$$
$$578$$ 16.0000 0.665512
$$579$$ 0.142136 0.00590695
$$580$$ 0 0
$$581$$ 63.9411 2.65272
$$582$$ −0.142136 −0.00589171
$$583$$ −7.75736 −0.321277
$$584$$ −11.4853 −0.475264
$$585$$ 0 0
$$586$$ 5.48528 0.226595
$$587$$ 30.2426 1.24825 0.624124 0.781326i $$-0.285456\pi$$
0.624124 + 0.781326i $$0.285456\pi$$
$$588$$ −5.17157 −0.213272
$$589$$ −6.24264 −0.257224
$$590$$ 0 0
$$591$$ 4.87006 0.200327
$$592$$ −8.48528 −0.348743
$$593$$ 22.0000 0.903432 0.451716 0.892162i $$-0.350812\pi$$
0.451716 + 0.892162i $$0.350812\pi$$
$$594$$ 3.41421 0.140087
$$595$$ 0 0
$$596$$ 17.6569 0.723253
$$597$$ −3.82843 −0.156687
$$598$$ −4.41421 −0.180511
$$599$$ −15.2132 −0.621595 −0.310797 0.950476i $$-0.600596\pi$$
−0.310797 + 0.950476i $$0.600596\pi$$
$$600$$ 0 0
$$601$$ −20.2426 −0.825715 −0.412857 0.910796i $$-0.635469\pi$$
−0.412857 + 0.910796i $$0.635469\pi$$
$$602$$ −7.75736 −0.316166
$$603$$ −13.4558 −0.547964
$$604$$ −10.4853 −0.426640
$$605$$ 0 0
$$606$$ 5.41421 0.219937
$$607$$ 3.17157 0.128730 0.0643651 0.997926i $$-0.479498\pi$$
0.0643651 + 0.997926i $$0.479498\pi$$
$$608$$ 1.00000 0.0405554
$$609$$ 0.313708 0.0127121
$$610$$ 0 0
$$611$$ 0 0
$$612$$ −2.82843 −0.114332
$$613$$ −11.6985 −0.472497 −0.236249 0.971693i $$-0.575918\pi$$
−0.236249 + 0.971693i $$0.575918\pi$$
$$614$$ 17.6569 0.712573
$$615$$ 0 0
$$616$$ 6.24264 0.251523
$$617$$ −4.48528 −0.180571 −0.0902853 0.995916i $$-0.528778\pi$$
−0.0902853 + 0.995916i $$0.528778\pi$$
$$618$$ 1.75736 0.0706914
$$619$$ 7.75736 0.311795 0.155897 0.987773i $$-0.450173\pi$$
0.155897 + 0.987773i $$0.450173\pi$$
$$620$$ 0 0
$$621$$ 1.82843 0.0733723
$$622$$ 4.75736 0.190753
$$623$$ 31.2132 1.25053
$$624$$ −2.41421 −0.0966459
$$625$$ 0 0
$$626$$ 7.97056 0.318568
$$627$$ −0.585786 −0.0233941
$$628$$ 11.6569 0.465159
$$629$$ −8.48528 −0.338330
$$630$$ 0 0
$$631$$ −38.9706 −1.55139 −0.775697 0.631106i $$-0.782601\pi$$
−0.775697 + 0.631106i $$0.782601\pi$$
$$632$$ 6.48528 0.257971
$$633$$ 8.17157 0.324791
$$634$$ 9.48528 0.376709
$$635$$ 0 0
$$636$$ −2.27208 −0.0900938
$$637$$ 72.7696 2.88323
$$638$$ −0.242641 −0.00960624
$$639$$ 37.9411 1.50093
$$640$$ 0 0
$$641$$ −18.0416 −0.712602 −0.356301 0.934371i $$-0.615962\pi$$
−0.356301 + 0.934371i $$0.615962\pi$$
$$642$$ 8.17157 0.322506
$$643$$ −14.4853 −0.571244 −0.285622 0.958342i $$-0.592200\pi$$
−0.285622 + 0.958342i $$0.592200\pi$$
$$644$$ 3.34315 0.131738
$$645$$ 0 0
$$646$$ 1.00000 0.0393445
$$647$$ −18.7574 −0.737428 −0.368714 0.929543i $$-0.620202\pi$$
−0.368714 + 0.929543i $$0.620202\pi$$
$$648$$ −7.48528 −0.294050
$$649$$ −9.75736 −0.383010
$$650$$ 0 0
$$651$$ −11.4142 −0.447358
$$652$$ −10.2426 −0.401133
$$653$$ −28.9706 −1.13371 −0.566853 0.823819i $$-0.691839\pi$$
−0.566853 + 0.823819i $$0.691839\pi$$
$$654$$ −7.44365 −0.291070
$$655$$ 0 0
$$656$$ −4.24264 −0.165647
$$657$$ −32.4853 −1.26737
$$658$$ 0 0
$$659$$ 18.8995 0.736220 0.368110 0.929782i $$-0.380005\pi$$
0.368110 + 0.929782i $$0.380005\pi$$
$$660$$ 0 0
$$661$$ −18.4558 −0.717849 −0.358925 0.933367i $$-0.616856\pi$$
−0.358925 + 0.933367i $$0.616856\pi$$
$$662$$ 19.2426 0.747886
$$663$$ −2.41421 −0.0937603
$$664$$ −14.4853 −0.562138
$$665$$ 0 0
$$666$$ −24.0000 −0.929981
$$667$$ −0.129942 −0.00503139
$$668$$ −18.2426 −0.705829
$$669$$ 6.28427 0.242964
$$670$$ 0 0
$$671$$ −20.1421 −0.777579
$$672$$ 1.82843 0.0705331
$$673$$ −12.0000 −0.462566 −0.231283 0.972887i $$-0.574292\pi$$
−0.231283 + 0.972887i $$0.574292\pi$$
$$674$$ −14.1005 −0.543131
$$675$$ 0 0
$$676$$ 20.9706 0.806560
$$677$$ −40.9411 −1.57350 −0.786748 0.617275i $$-0.788237\pi$$
−0.786748 + 0.617275i $$0.788237\pi$$
$$678$$ −4.24264 −0.162938
$$679$$ −1.51472 −0.0581296
$$680$$ 0 0
$$681$$ 6.94113 0.265985
$$682$$ 8.82843 0.338058
$$683$$ 12.0000 0.459167 0.229584 0.973289i $$-0.426264\pi$$
0.229584 + 0.973289i $$0.426264\pi$$
$$684$$ 2.82843 0.108148
$$685$$ 0 0
$$686$$ −24.2132 −0.924464
$$687$$ −6.20101 −0.236583
$$688$$ 1.75736 0.0669987
$$689$$ 31.9706 1.21798
$$690$$ 0 0
$$691$$ 22.4853 0.855380 0.427690 0.903925i $$-0.359327\pi$$
0.427690 + 0.903925i $$0.359327\pi$$
$$692$$ 0.485281 0.0184476
$$693$$ 17.6569 0.670728
$$694$$ 5.51472 0.209336
$$695$$ 0 0
$$696$$ −0.0710678 −0.00269382
$$697$$ −4.24264 −0.160701
$$698$$ −6.00000 −0.227103
$$699$$ 10.3431 0.391214
$$700$$ 0 0
$$701$$ 16.9706 0.640969 0.320485 0.947254i $$-0.396154\pi$$
0.320485 + 0.947254i $$0.396154\pi$$
$$702$$ −14.0711 −0.531078
$$703$$ 8.48528 0.320028
$$704$$ −1.41421 −0.0533002
$$705$$ 0 0
$$706$$ 2.51472 0.0946427
$$707$$ 57.6985 2.16997
$$708$$ −2.85786 −0.107405
$$709$$ −34.0000 −1.27690 −0.638448 0.769665i $$-0.720423\pi$$
−0.638448 + 0.769665i $$0.720423\pi$$
$$710$$ 0 0
$$711$$ 18.3431 0.687922
$$712$$ −7.07107 −0.264999
$$713$$ 4.72792 0.177062
$$714$$ 1.82843 0.0684272
$$715$$ 0 0
$$716$$ −11.6569 −0.435637
$$717$$ 2.85786 0.106729
$$718$$ 22.7574 0.849297
$$719$$ −5.10051 −0.190217 −0.0951084 0.995467i $$-0.530320\pi$$
−0.0951084 + 0.995467i $$0.530320\pi$$
$$720$$ 0 0
$$721$$ 18.7279 0.697464
$$722$$ −1.00000 −0.0372161
$$723$$ −3.71573 −0.138189
$$724$$ −8.48528 −0.315353
$$725$$ 0 0
$$726$$ −3.72792 −0.138356
$$727$$ 9.72792 0.360789 0.180394 0.983594i $$-0.442263\pi$$
0.180394 + 0.983594i $$0.442263\pi$$
$$728$$ −25.7279 −0.953540
$$729$$ −18.1716 −0.673021
$$730$$ 0 0
$$731$$ 1.75736 0.0649983
$$732$$ −5.89949 −0.218052
$$733$$ 12.0000 0.443230 0.221615 0.975134i $$-0.428867\pi$$
0.221615 + 0.975134i $$0.428867\pi$$
$$734$$ 25.4558 0.939592
$$735$$ 0 0
$$736$$ −0.757359 −0.0279166
$$737$$ −6.72792 −0.247826
$$738$$ −12.0000 −0.441726
$$739$$ 1.27208 0.0467941 0.0233971 0.999726i $$-0.492552\pi$$
0.0233971 + 0.999726i $$0.492552\pi$$
$$740$$ 0 0
$$741$$ 2.41421 0.0886884
$$742$$ −24.2132 −0.888895
$$743$$ 6.72792 0.246824 0.123412 0.992356i $$-0.460616\pi$$
0.123412 + 0.992356i $$0.460616\pi$$
$$744$$ 2.58579 0.0947995
$$745$$ 0 0
$$746$$ −9.00000 −0.329513
$$747$$ −40.9706 −1.49903
$$748$$ −1.41421 −0.0517088
$$749$$ 87.0833 3.18195
$$750$$ 0 0
$$751$$ −26.7279 −0.975316 −0.487658 0.873035i $$-0.662149\pi$$
−0.487658 + 0.873035i $$0.662149\pi$$
$$752$$ 0 0
$$753$$ −1.47309 −0.0536823
$$754$$ 1.00000 0.0364179
$$755$$ 0 0
$$756$$ 10.6569 0.387586
$$757$$ 12.3431 0.448619 0.224310 0.974518i $$-0.427987\pi$$
0.224310 + 0.974518i $$0.427987\pi$$
$$758$$ −11.2426 −0.408351
$$759$$ 0.443651 0.0161035
$$760$$ 0 0
$$761$$ 43.9706 1.59393 0.796966 0.604024i $$-0.206437\pi$$
0.796966 + 0.604024i $$0.206437\pi$$
$$762$$ −1.02944 −0.0372926
$$763$$ −79.3259 −2.87179
$$764$$ 12.5563 0.454273
$$765$$ 0 0
$$766$$ −12.2426 −0.442345
$$767$$ 40.2132 1.45201
$$768$$ −0.414214 −0.0149466
$$769$$ −36.4558 −1.31463 −0.657316 0.753615i $$-0.728308\pi$$
−0.657316 + 0.753615i $$0.728308\pi$$
$$770$$ 0 0
$$771$$ −8.58579 −0.309210
$$772$$ −0.343146 −0.0123501
$$773$$ 13.9706 0.502486 0.251243 0.967924i $$-0.419161\pi$$
0.251243 + 0.967924i $$0.419161\pi$$
$$774$$ 4.97056 0.178663
$$775$$ 0 0
$$776$$ 0.343146 0.0123182
$$777$$ 15.5147 0.556587
$$778$$ −22.9289 −0.822042
$$779$$ 4.24264 0.152008
$$780$$ 0 0
$$781$$ 18.9706 0.678820
$$782$$ −0.757359 −0.0270831
$$783$$ −0.414214 −0.0148028
$$784$$ 12.4853 0.445903
$$785$$ 0 0
$$786$$ −7.02944 −0.250732
$$787$$ −41.1838 −1.46804 −0.734021 0.679126i $$-0.762359\pi$$
−0.734021 + 0.679126i $$0.762359\pi$$
$$788$$ −11.7574 −0.418839
$$789$$ 11.1716 0.397719
$$790$$ 0 0
$$791$$ −45.2132 −1.60760
$$792$$ −4.00000 −0.142134
$$793$$ 83.0122 2.94785
$$794$$ −24.0000 −0.851728
$$795$$ 0 0
$$796$$ 9.24264 0.327597
$$797$$ −47.4853 −1.68201 −0.841007 0.541023i $$-0.818037\pi$$
−0.841007 + 0.541023i $$0.818037\pi$$
$$798$$ −1.82843 −0.0647256
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −20.0000 −0.706665
$$802$$ 25.4142 0.897407
$$803$$ −16.2426 −0.573190
$$804$$ −1.97056 −0.0694964
$$805$$ 0 0
$$806$$ −36.3848 −1.28160
$$807$$ 6.88730 0.242444
$$808$$ −13.0711 −0.459839
$$809$$ −28.7990 −1.01252 −0.506259 0.862381i $$-0.668972\pi$$
−0.506259 + 0.862381i $$0.668972\pi$$
$$810$$ 0 0
$$811$$ 52.6985 1.85049 0.925247 0.379365i $$-0.123858\pi$$
0.925247 + 0.379365i $$0.123858\pi$$
$$812$$ −0.757359 −0.0265781
$$813$$ 11.2843 0.395757
$$814$$ −12.0000 −0.420600
$$815$$ 0 0
$$816$$ −0.414214 −0.0145004
$$817$$ −1.75736 −0.0614822
$$818$$ −25.2132 −0.881559
$$819$$ −72.7696 −2.54277
$$820$$ 0 0
$$821$$ 6.34315 0.221377 0.110689 0.993855i $$-0.464694\pi$$
0.110689 + 0.993855i $$0.464694\pi$$
$$822$$ 5.38478 0.187816
$$823$$ −15.7279 −0.548241 −0.274120 0.961695i $$-0.588387\pi$$
−0.274120 + 0.961695i $$0.588387\pi$$
$$824$$ −4.24264 −0.147799
$$825$$ 0 0
$$826$$ −30.4558 −1.05969
$$827$$ −20.6985 −0.719757 −0.359878 0.932999i $$-0.617182\pi$$
−0.359878 + 0.932999i $$0.617182\pi$$
$$828$$ −2.14214 −0.0744444
$$829$$ 10.4558 0.363146 0.181573 0.983377i $$-0.441881\pi$$
0.181573 + 0.983377i $$0.441881\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 5.82843 0.202064
$$833$$ 12.4853 0.432589
$$834$$ −4.97056 −0.172117
$$835$$ 0 0
$$836$$ 1.41421 0.0489116
$$837$$ 15.0711 0.520932
$$838$$ −19.4142 −0.670653
$$839$$ 22.6274 0.781185 0.390593 0.920564i $$-0.372270\pi$$
0.390593 + 0.920564i $$0.372270\pi$$
$$840$$ 0 0
$$841$$ −28.9706 −0.998985
$$842$$ −19.4853 −0.671507
$$843$$ −1.75736 −0.0605267
$$844$$ −19.7279 −0.679063
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −39.7279 −1.36507
$$848$$ 5.48528 0.188365
$$849$$ −13.3137 −0.456925
$$850$$ 0 0
$$851$$ −6.42641 −0.220294
$$852$$ 5.55635 0.190357
$$853$$ −27.9411 −0.956686 −0.478343 0.878173i $$-0.658762\pi$$
−0.478343 + 0.878173i $$0.658762\pi$$
$$854$$ −62.8701 −2.15137
$$855$$ 0 0
$$856$$ −19.7279 −0.674286
$$857$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$858$$ −3.41421 −0.116559
$$859$$ 4.72792 0.161315 0.0806573 0.996742i $$-0.474298\pi$$
0.0806573 + 0.996742i $$0.474298\pi$$
$$860$$ 0 0
$$861$$ 7.75736 0.264370
$$862$$ 6.38478 0.217466
$$863$$ 0.727922 0.0247788 0.0123894 0.999923i $$-0.496056\pi$$
0.0123894 + 0.999923i $$0.496056\pi$$
$$864$$ −2.41421 −0.0821332
$$865$$ 0 0
$$866$$ 9.55635 0.324738
$$867$$ 6.62742 0.225079
$$868$$ 27.5563 0.935323
$$869$$ 9.17157 0.311124
$$870$$ 0 0
$$871$$ 27.7279 0.939525
$$872$$ 17.9706 0.608560
$$873$$ 0.970563 0.0328486
$$874$$ 0.757359 0.0256181
$$875$$ 0 0
$$876$$ −4.75736 −0.160736
$$877$$ −18.8579 −0.636785 −0.318392 0.947959i $$-0.603143\pi$$
−0.318392 + 0.947959i $$0.603143\pi$$
$$878$$ 5.75736 0.194301
$$879$$ 2.27208 0.0766353
$$880$$ 0 0
$$881$$ −39.5980 −1.33409 −0.667045 0.745018i $$-0.732441\pi$$
−0.667045 + 0.745018i $$0.732441\pi$$
$$882$$ 35.3137 1.18907
$$883$$ 14.4853 0.487469 0.243734 0.969842i $$-0.421628\pi$$
0.243734 + 0.969842i $$0.421628\pi$$
$$884$$ 5.82843 0.196031
$$885$$ 0 0
$$886$$ −4.24264 −0.142534
$$887$$ −45.2132 −1.51811 −0.759055 0.651026i $$-0.774339\pi$$
−0.759055 + 0.651026i $$0.774339\pi$$
$$888$$ −3.51472 −0.117946
$$889$$ −10.9706 −0.367941
$$890$$ 0 0
$$891$$ −10.5858 −0.354637
$$892$$ −15.1716 −0.507982
$$893$$ 0 0
$$894$$ 7.31371 0.244607
$$895$$ 0 0
$$896$$ −4.41421 −0.147469
$$897$$ −1.82843 −0.0610494
$$898$$ −17.3137 −0.577766
$$899$$ −1.07107 −0.0357221
$$900$$ 0 0
$$901$$ 5.48528 0.182741
$$902$$ −6.00000 −0.199778
$$903$$ −3.21320 −0.106929
$$904$$ 10.2426 0.340665
$$905$$ 0 0
$$906$$ −4.34315 −0.144291
$$907$$ 38.6985 1.28496 0.642481 0.766302i $$-0.277905\pi$$
0.642481 + 0.766302i $$0.277905\pi$$
$$908$$ −16.7574 −0.556113
$$909$$ −36.9706 −1.22624
$$910$$ 0 0
$$911$$ 40.2843 1.33468 0.667339 0.744754i $$-0.267433\pi$$
0.667339 + 0.744754i $$0.267433\pi$$
$$912$$ 0.414214 0.0137160
$$913$$ −20.4853 −0.677964
$$914$$ 3.00000 0.0992312
$$915$$ 0 0
$$916$$ 14.9706 0.494641
$$917$$ −74.9117 −2.47380
$$918$$ −2.41421 −0.0796809
$$919$$ −6.21320 −0.204955 −0.102477 0.994735i $$-0.532677\pi$$
−0.102477 + 0.994735i $$0.532677\pi$$
$$920$$ 0 0
$$921$$ 7.31371 0.240995
$$922$$ 3.55635 0.117122
$$923$$ −78.1838 −2.57345
$$924$$ 2.58579 0.0850661
$$925$$ 0 0
$$926$$ −14.1421 −0.464739
$$927$$ −12.0000 −0.394132
$$928$$ 0.171573 0.00563216
$$929$$ −12.1716 −0.399336 −0.199668 0.979864i $$-0.563986\pi$$
−0.199668 + 0.979864i $$0.563986\pi$$
$$930$$ 0 0
$$931$$ −12.4853 −0.409189
$$932$$ −24.9706 −0.817938
$$933$$ 1.97056 0.0645133
$$934$$ −0.727922 −0.0238183
$$935$$ 0 0
$$936$$ 16.4853 0.538838
$$937$$ 27.0000 0.882052 0.441026 0.897494i $$-0.354615\pi$$
0.441026 + 0.897494i $$0.354615\pi$$
$$938$$ −21.0000 −0.685674
$$939$$ 3.30152 0.107741
$$940$$ 0 0
$$941$$ 53.1421 1.73238 0.866192 0.499711i $$-0.166561\pi$$
0.866192 + 0.499711i $$0.166561\pi$$
$$942$$ 4.82843 0.157319
$$943$$ −3.21320 −0.104636
$$944$$ 6.89949 0.224559
$$945$$ 0 0
$$946$$ 2.48528 0.0808035
$$947$$ −12.0000 −0.389948 −0.194974 0.980808i $$-0.562462\pi$$
−0.194974 + 0.980808i $$0.562462\pi$$
$$948$$ 2.68629 0.0872467
$$949$$ 66.9411 2.17300
$$950$$ 0 0
$$951$$ 3.92893 0.127404
$$952$$ −4.41421 −0.143065
$$953$$ 18.7279 0.606657 0.303328 0.952886i $$-0.401902\pi$$
0.303328 + 0.952886i $$0.401902\pi$$
$$954$$ 15.5147 0.502308
$$955$$ 0 0
$$956$$ −6.89949 −0.223146
$$957$$ −0.100505 −0.00324887
$$958$$ 31.1127 1.00521
$$959$$ 57.3848 1.85305
$$960$$ 0 0
$$961$$ 7.97056 0.257115
$$962$$ 49.4558 1.59452
$$963$$ −55.7990 −1.79810
$$964$$ 8.97056 0.288922
$$965$$ 0 0
$$966$$ 1.38478 0.0445544
$$967$$ 43.1127 1.38641 0.693205 0.720740i $$-0.256198\pi$$
0.693205 + 0.720740i $$0.256198\pi$$
$$968$$ 9.00000 0.289271
$$969$$ 0.414214 0.0133065
$$970$$ 0 0
$$971$$ 41.6569 1.33683 0.668416 0.743788i $$-0.266973\pi$$
0.668416 + 0.743788i $$0.266973\pi$$
$$972$$ −10.3431 −0.331757
$$973$$ −52.9706 −1.69816
$$974$$ −37.7990 −1.21116
$$975$$ 0 0
$$976$$ 14.2426 0.455896
$$977$$ 0.242641 0.00776276 0.00388138 0.999992i $$-0.498765\pi$$
0.00388138 + 0.999992i $$0.498765\pi$$
$$978$$ −4.24264 −0.135665
$$979$$ −10.0000 −0.319601
$$980$$ 0 0
$$981$$ 50.8284 1.62283
$$982$$ 33.5563 1.07083
$$983$$ −51.4558 −1.64119 −0.820593 0.571513i $$-0.806357\pi$$
−0.820593 + 0.571513i $$0.806357\pi$$
$$984$$ −1.75736 −0.0560226
$$985$$ 0 0
$$986$$ 0.171573 0.00546399
$$987$$ 0 0
$$988$$ −5.82843 −0.185427
$$989$$ 1.33095 0.0423218
$$990$$ 0 0
$$991$$ 13.7574 0.437017 0.218508 0.975835i $$-0.429881\pi$$
0.218508 + 0.975835i $$0.429881\pi$$
$$992$$ −6.24264 −0.198204
$$993$$ 7.97056 0.252938
$$994$$ 59.2132 1.87813
$$995$$ 0 0
$$996$$ −6.00000 −0.190117
$$997$$ 48.7279 1.54323 0.771614 0.636091i $$-0.219450\pi$$
0.771614 + 0.636091i $$0.219450\pi$$
$$998$$ 25.7574 0.815335
$$999$$ −20.4853 −0.648126
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.f.1.1 2
3.2 odd 2 8550.2.a.cb.1.2 2
4.3 odd 2 7600.2.a.v.1.2 2
5.2 odd 4 190.2.b.a.39.2 4
5.3 odd 4 190.2.b.a.39.3 yes 4
5.4 even 2 950.2.a.g.1.2 2
15.2 even 4 1710.2.d.c.1369.3 4
15.8 even 4 1710.2.d.c.1369.1 4
15.14 odd 2 8550.2.a.bn.1.1 2
20.3 even 4 1520.2.d.e.609.3 4
20.7 even 4 1520.2.d.e.609.2 4
20.19 odd 2 7600.2.a.bg.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
190.2.b.a.39.2 4 5.2 odd 4
190.2.b.a.39.3 yes 4 5.3 odd 4
950.2.a.f.1.1 2 1.1 even 1 trivial
950.2.a.g.1.2 2 5.4 even 2
1520.2.d.e.609.2 4 20.7 even 4
1520.2.d.e.609.3 4 20.3 even 4
1710.2.d.c.1369.1 4 15.8 even 4
1710.2.d.c.1369.3 4 15.2 even 4
7600.2.a.v.1.2 2 4.3 odd 2
7600.2.a.bg.1.1 2 20.19 odd 2
8550.2.a.bn.1.1 2 15.14 odd 2
8550.2.a.cb.1.2 2 3.2 odd 2