# Properties

 Label 103.2.a.a.1.1 Level $103$ Weight $2$ Character 103.1 Self dual yes Analytic conductor $0.822$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [103,2,Mod(1,103)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(103, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("103.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$103$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 103.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: $$0.822459140819$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 103.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.61803 q^{2} -1.00000 q^{3} +4.85410 q^{4} -0.381966 q^{5} +2.61803 q^{6} -1.00000 q^{7} -7.47214 q^{8} -2.00000 q^{9} +O(q^{10})$$ $$q-2.61803 q^{2} -1.00000 q^{3} +4.85410 q^{4} -0.381966 q^{5} +2.61803 q^{6} -1.00000 q^{7} -7.47214 q^{8} -2.00000 q^{9} +1.00000 q^{10} -2.61803 q^{11} -4.85410 q^{12} -4.85410 q^{13} +2.61803 q^{14} +0.381966 q^{15} +9.85410 q^{16} -5.61803 q^{17} +5.23607 q^{18} +5.85410 q^{19} -1.85410 q^{20} +1.00000 q^{21} +6.85410 q^{22} +4.47214 q^{23} +7.47214 q^{24} -4.85410 q^{25} +12.7082 q^{26} +5.00000 q^{27} -4.85410 q^{28} -5.23607 q^{29} -1.00000 q^{30} -6.70820 q^{31} -10.8541 q^{32} +2.61803 q^{33} +14.7082 q^{34} +0.381966 q^{35} -9.70820 q^{36} +6.70820 q^{37} -15.3262 q^{38} +4.85410 q^{39} +2.85410 q^{40} +8.94427 q^{41} -2.61803 q^{42} -8.70820 q^{43} -12.7082 q^{44} +0.763932 q^{45} -11.7082 q^{46} +4.09017 q^{47} -9.85410 q^{48} -6.00000 q^{49} +12.7082 q^{50} +5.61803 q^{51} -23.5623 q^{52} +1.09017 q^{53} -13.0902 q^{54} +1.00000 q^{55} +7.47214 q^{56} -5.85410 q^{57} +13.7082 q^{58} +6.38197 q^{59} +1.85410 q^{60} +4.14590 q^{61} +17.5623 q^{62} +2.00000 q^{63} +8.70820 q^{64} +1.85410 q^{65} -6.85410 q^{66} +14.4164 q^{67} -27.2705 q^{68} -4.47214 q^{69} -1.00000 q^{70} -4.09017 q^{71} +14.9443 q^{72} -10.8541 q^{73} -17.5623 q^{74} +4.85410 q^{75} +28.4164 q^{76} +2.61803 q^{77} -12.7082 q^{78} -6.56231 q^{79} -3.76393 q^{80} +1.00000 q^{81} -23.4164 q^{82} -6.32624 q^{83} +4.85410 q^{84} +2.14590 q^{85} +22.7984 q^{86} +5.23607 q^{87} +19.5623 q^{88} -2.29180 q^{89} -2.00000 q^{90} +4.85410 q^{91} +21.7082 q^{92} +6.70820 q^{93} -10.7082 q^{94} -2.23607 q^{95} +10.8541 q^{96} -1.70820 q^{97} +15.7082 q^{98} +5.23607 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} - 2 q^{3} + 3 q^{4} - 3 q^{5} + 3 q^{6} - 2 q^{7} - 6 q^{8} - 4 q^{9}+O(q^{10})$$ 2 * q - 3 * q^2 - 2 * q^3 + 3 * q^4 - 3 * q^5 + 3 * q^6 - 2 * q^7 - 6 * q^8 - 4 * q^9 $$2 q - 3 q^{2} - 2 q^{3} + 3 q^{4} - 3 q^{5} + 3 q^{6} - 2 q^{7} - 6 q^{8} - 4 q^{9} + 2 q^{10} - 3 q^{11} - 3 q^{12} - 3 q^{13} + 3 q^{14} + 3 q^{15} + 13 q^{16} - 9 q^{17} + 6 q^{18} + 5 q^{19} + 3 q^{20} + 2 q^{21} + 7 q^{22} + 6 q^{24} - 3 q^{25} + 12 q^{26} + 10 q^{27} - 3 q^{28} - 6 q^{29} - 2 q^{30} - 15 q^{32} + 3 q^{33} + 16 q^{34} + 3 q^{35} - 6 q^{36} - 15 q^{38} + 3 q^{39} - q^{40} - 3 q^{42} - 4 q^{43} - 12 q^{44} + 6 q^{45} - 10 q^{46} - 3 q^{47} - 13 q^{48} - 12 q^{49} + 12 q^{50} + 9 q^{51} - 27 q^{52} - 9 q^{53} - 15 q^{54} + 2 q^{55} + 6 q^{56} - 5 q^{57} + 14 q^{58} + 15 q^{59} - 3 q^{60} + 15 q^{61} + 15 q^{62} + 4 q^{63} + 4 q^{64} - 3 q^{65} - 7 q^{66} + 2 q^{67} - 21 q^{68} - 2 q^{70} + 3 q^{71} + 12 q^{72} - 15 q^{73} - 15 q^{74} + 3 q^{75} + 30 q^{76} + 3 q^{77} - 12 q^{78} + 7 q^{79} - 12 q^{80} + 2 q^{81} - 20 q^{82} + 3 q^{83} + 3 q^{84} + 11 q^{85} + 21 q^{86} + 6 q^{87} + 19 q^{88} - 18 q^{89} - 4 q^{90} + 3 q^{91} + 30 q^{92} - 8 q^{94} + 15 q^{96} + 10 q^{97} + 18 q^{98} + 6 q^{99}+O(q^{100})$$ 2 * q - 3 * q^2 - 2 * q^3 + 3 * q^4 - 3 * q^5 + 3 * q^6 - 2 * q^7 - 6 * q^8 - 4 * q^9 + 2 * q^10 - 3 * q^11 - 3 * q^12 - 3 * q^13 + 3 * q^14 + 3 * q^15 + 13 * q^16 - 9 * q^17 + 6 * q^18 + 5 * q^19 + 3 * q^20 + 2 * q^21 + 7 * q^22 + 6 * q^24 - 3 * q^25 + 12 * q^26 + 10 * q^27 - 3 * q^28 - 6 * q^29 - 2 * q^30 - 15 * q^32 + 3 * q^33 + 16 * q^34 + 3 * q^35 - 6 * q^36 - 15 * q^38 + 3 * q^39 - q^40 - 3 * q^42 - 4 * q^43 - 12 * q^44 + 6 * q^45 - 10 * q^46 - 3 * q^47 - 13 * q^48 - 12 * q^49 + 12 * q^50 + 9 * q^51 - 27 * q^52 - 9 * q^53 - 15 * q^54 + 2 * q^55 + 6 * q^56 - 5 * q^57 + 14 * q^58 + 15 * q^59 - 3 * q^60 + 15 * q^61 + 15 * q^62 + 4 * q^63 + 4 * q^64 - 3 * q^65 - 7 * q^66 + 2 * q^67 - 21 * q^68 - 2 * q^70 + 3 * q^71 + 12 * q^72 - 15 * q^73 - 15 * q^74 + 3 * q^75 + 30 * q^76 + 3 * q^77 - 12 * q^78 + 7 * q^79 - 12 * q^80 + 2 * q^81 - 20 * q^82 + 3 * q^83 + 3 * q^84 + 11 * q^85 + 21 * q^86 + 6 * q^87 + 19 * q^88 - 18 * q^89 - 4 * q^90 + 3 * q^91 + 30 * q^92 - 8 * q^94 + 15 * q^96 + 10 * q^97 + 18 * q^98 + 6 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.61803 −1.85123 −0.925615 0.378467i $$-0.876451\pi$$
−0.925615 + 0.378467i $$0.876451\pi$$
$$3$$ −1.00000 −0.577350 −0.288675 0.957427i $$-0.593215\pi$$
−0.288675 + 0.957427i $$0.593215\pi$$
$$4$$ 4.85410 2.42705
$$5$$ −0.381966 −0.170820 −0.0854102 0.996346i $$-0.527220\pi$$
−0.0854102 + 0.996346i $$0.527220\pi$$
$$6$$ 2.61803 1.06881
$$7$$ −1.00000 −0.377964 −0.188982 0.981981i $$-0.560519\pi$$
−0.188982 + 0.981981i $$0.560519\pi$$
$$8$$ −7.47214 −2.64180
$$9$$ −2.00000 −0.666667
$$10$$ 1.00000 0.316228
$$11$$ −2.61803 −0.789367 −0.394683 0.918817i $$-0.629146\pi$$
−0.394683 + 0.918817i $$0.629146\pi$$
$$12$$ −4.85410 −1.40126
$$13$$ −4.85410 −1.34629 −0.673143 0.739512i $$-0.735056\pi$$
−0.673143 + 0.739512i $$0.735056\pi$$
$$14$$ 2.61803 0.699699
$$15$$ 0.381966 0.0986232
$$16$$ 9.85410 2.46353
$$17$$ −5.61803 −1.36257 −0.681287 0.732017i $$-0.738579\pi$$
−0.681287 + 0.732017i $$0.738579\pi$$
$$18$$ 5.23607 1.23415
$$19$$ 5.85410 1.34302 0.671512 0.740994i $$-0.265645\pi$$
0.671512 + 0.740994i $$0.265645\pi$$
$$20$$ −1.85410 −0.414590
$$21$$ 1.00000 0.218218
$$22$$ 6.85410 1.46130
$$23$$ 4.47214 0.932505 0.466252 0.884652i $$-0.345604\pi$$
0.466252 + 0.884652i $$0.345604\pi$$
$$24$$ 7.47214 1.52524
$$25$$ −4.85410 −0.970820
$$26$$ 12.7082 2.49228
$$27$$ 5.00000 0.962250
$$28$$ −4.85410 −0.917339
$$29$$ −5.23607 −0.972313 −0.486157 0.873872i $$-0.661602\pi$$
−0.486157 + 0.873872i $$0.661602\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ −6.70820 −1.20483 −0.602414 0.798183i $$-0.705795\pi$$
−0.602414 + 0.798183i $$0.705795\pi$$
$$32$$ −10.8541 −1.91875
$$33$$ 2.61803 0.455741
$$34$$ 14.7082 2.52244
$$35$$ 0.381966 0.0645640
$$36$$ −9.70820 −1.61803
$$37$$ 6.70820 1.10282 0.551411 0.834234i $$-0.314090\pi$$
0.551411 + 0.834234i $$0.314090\pi$$
$$38$$ −15.3262 −2.48624
$$39$$ 4.85410 0.777278
$$40$$ 2.85410 0.451273
$$41$$ 8.94427 1.39686 0.698430 0.715678i $$-0.253882\pi$$
0.698430 + 0.715678i $$0.253882\pi$$
$$42$$ −2.61803 −0.403971
$$43$$ −8.70820 −1.32799 −0.663994 0.747738i $$-0.731140\pi$$
−0.663994 + 0.747738i $$0.731140\pi$$
$$44$$ −12.7082 −1.91583
$$45$$ 0.763932 0.113880
$$46$$ −11.7082 −1.72628
$$47$$ 4.09017 0.596613 0.298306 0.954470i $$-0.403578\pi$$
0.298306 + 0.954470i $$0.403578\pi$$
$$48$$ −9.85410 −1.42232
$$49$$ −6.00000 −0.857143
$$50$$ 12.7082 1.79721
$$51$$ 5.61803 0.786682
$$52$$ −23.5623 −3.26750
$$53$$ 1.09017 0.149746 0.0748732 0.997193i $$-0.476145\pi$$
0.0748732 + 0.997193i $$0.476145\pi$$
$$54$$ −13.0902 −1.78135
$$55$$ 1.00000 0.134840
$$56$$ 7.47214 0.998506
$$57$$ −5.85410 −0.775395
$$58$$ 13.7082 1.79998
$$59$$ 6.38197 0.830861 0.415431 0.909625i $$-0.363631\pi$$
0.415431 + 0.909625i $$0.363631\pi$$
$$60$$ 1.85410 0.239364
$$61$$ 4.14590 0.530828 0.265414 0.964135i $$-0.414491\pi$$
0.265414 + 0.964135i $$0.414491\pi$$
$$62$$ 17.5623 2.23042
$$63$$ 2.00000 0.251976
$$64$$ 8.70820 1.08853
$$65$$ 1.85410 0.229973
$$66$$ −6.85410 −0.843682
$$67$$ 14.4164 1.76124 0.880622 0.473819i $$-0.157125\pi$$
0.880622 + 0.473819i $$0.157125\pi$$
$$68$$ −27.2705 −3.30704
$$69$$ −4.47214 −0.538382
$$70$$ −1.00000 −0.119523
$$71$$ −4.09017 −0.485414 −0.242707 0.970100i $$-0.578035\pi$$
−0.242707 + 0.970100i $$0.578035\pi$$
$$72$$ 14.9443 1.76120
$$73$$ −10.8541 −1.27038 −0.635188 0.772357i $$-0.719077\pi$$
−0.635188 + 0.772357i $$0.719077\pi$$
$$74$$ −17.5623 −2.04158
$$75$$ 4.85410 0.560503
$$76$$ 28.4164 3.25959
$$77$$ 2.61803 0.298353
$$78$$ −12.7082 −1.43892
$$79$$ −6.56231 −0.738317 −0.369159 0.929366i $$-0.620354\pi$$
−0.369159 + 0.929366i $$0.620354\pi$$
$$80$$ −3.76393 −0.420820
$$81$$ 1.00000 0.111111
$$82$$ −23.4164 −2.58591
$$83$$ −6.32624 −0.694395 −0.347197 0.937792i $$-0.612867\pi$$
−0.347197 + 0.937792i $$0.612867\pi$$
$$84$$ 4.85410 0.529626
$$85$$ 2.14590 0.232755
$$86$$ 22.7984 2.45841
$$87$$ 5.23607 0.561365
$$88$$ 19.5623 2.08535
$$89$$ −2.29180 −0.242930 −0.121465 0.992596i $$-0.538759\pi$$
−0.121465 + 0.992596i $$0.538759\pi$$
$$90$$ −2.00000 −0.210819
$$91$$ 4.85410 0.508848
$$92$$ 21.7082 2.26324
$$93$$ 6.70820 0.695608
$$94$$ −10.7082 −1.10447
$$95$$ −2.23607 −0.229416
$$96$$ 10.8541 1.10779
$$97$$ −1.70820 −0.173442 −0.0867209 0.996233i $$-0.527639\pi$$
−0.0867209 + 0.996233i $$0.527639\pi$$
$$98$$ 15.7082 1.58677
$$99$$ 5.23607 0.526245
$$100$$ −23.5623 −2.35623
$$101$$ 1.90983 0.190035 0.0950176 0.995476i $$-0.469709\pi$$
0.0950176 + 0.995476i $$0.469709\pi$$
$$102$$ −14.7082 −1.45633
$$103$$ −1.00000 −0.0985329
$$104$$ 36.2705 3.55662
$$105$$ −0.381966 −0.0372761
$$106$$ −2.85410 −0.277215
$$107$$ −7.09017 −0.685433 −0.342716 0.939439i $$-0.611347\pi$$
−0.342716 + 0.939439i $$0.611347\pi$$
$$108$$ 24.2705 2.33543
$$109$$ −9.56231 −0.915903 −0.457951 0.888977i $$-0.651417\pi$$
−0.457951 + 0.888977i $$0.651417\pi$$
$$110$$ −2.61803 −0.249620
$$111$$ −6.70820 −0.636715
$$112$$ −9.85410 −0.931125
$$113$$ −15.0000 −1.41108 −0.705541 0.708669i $$-0.749296\pi$$
−0.705541 + 0.708669i $$0.749296\pi$$
$$114$$ 15.3262 1.43543
$$115$$ −1.70820 −0.159291
$$116$$ −25.4164 −2.35985
$$117$$ 9.70820 0.897524
$$118$$ −16.7082 −1.53811
$$119$$ 5.61803 0.515004
$$120$$ −2.85410 −0.260543
$$121$$ −4.14590 −0.376900
$$122$$ −10.8541 −0.982684
$$123$$ −8.94427 −0.806478
$$124$$ −32.5623 −2.92418
$$125$$ 3.76393 0.336656
$$126$$ −5.23607 −0.466466
$$127$$ 15.2705 1.35504 0.677519 0.735505i $$-0.263055\pi$$
0.677519 + 0.735505i $$0.263055\pi$$
$$128$$ −1.09017 −0.0963583
$$129$$ 8.70820 0.766715
$$130$$ −4.85410 −0.425733
$$131$$ −2.23607 −0.195366 −0.0976831 0.995218i $$-0.531143\pi$$
−0.0976831 + 0.995218i $$0.531143\pi$$
$$132$$ 12.7082 1.10611
$$133$$ −5.85410 −0.507615
$$134$$ −37.7426 −3.26047
$$135$$ −1.90983 −0.164372
$$136$$ 41.9787 3.59965
$$137$$ −12.7082 −1.08574 −0.542868 0.839818i $$-0.682661\pi$$
−0.542868 + 0.839818i $$0.682661\pi$$
$$138$$ 11.7082 0.996669
$$139$$ −11.1459 −0.945383 −0.472691 0.881228i $$-0.656717\pi$$
−0.472691 + 0.881228i $$0.656717\pi$$
$$140$$ 1.85410 0.156700
$$141$$ −4.09017 −0.344454
$$142$$ 10.7082 0.898613
$$143$$ 12.7082 1.06271
$$144$$ −19.7082 −1.64235
$$145$$ 2.00000 0.166091
$$146$$ 28.4164 2.35176
$$147$$ 6.00000 0.494872
$$148$$ 32.5623 2.67661
$$149$$ −7.47214 −0.612141 −0.306071 0.952009i $$-0.599014\pi$$
−0.306071 + 0.952009i $$0.599014\pi$$
$$150$$ −12.7082 −1.03762
$$151$$ −19.0000 −1.54620 −0.773099 0.634285i $$-0.781294\pi$$
−0.773099 + 0.634285i $$0.781294\pi$$
$$152$$ −43.7426 −3.54800
$$153$$ 11.2361 0.908382
$$154$$ −6.85410 −0.552319
$$155$$ 2.56231 0.205809
$$156$$ 23.5623 1.88649
$$157$$ 16.7082 1.33346 0.666730 0.745299i $$-0.267693\pi$$
0.666730 + 0.745299i $$0.267693\pi$$
$$158$$ 17.1803 1.36679
$$159$$ −1.09017 −0.0864561
$$160$$ 4.14590 0.327762
$$161$$ −4.47214 −0.352454
$$162$$ −2.61803 −0.205692
$$163$$ 2.70820 0.212123 0.106061 0.994360i $$-0.466176\pi$$
0.106061 + 0.994360i $$0.466176\pi$$
$$164$$ 43.4164 3.39025
$$165$$ −1.00000 −0.0778499
$$166$$ 16.5623 1.28548
$$167$$ 9.00000 0.696441 0.348220 0.937413i $$-0.386786\pi$$
0.348220 + 0.937413i $$0.386786\pi$$
$$168$$ −7.47214 −0.576488
$$169$$ 10.5623 0.812485
$$170$$ −5.61803 −0.430884
$$171$$ −11.7082 −0.895349
$$172$$ −42.2705 −3.22310
$$173$$ 16.0344 1.21908 0.609538 0.792757i $$-0.291355\pi$$
0.609538 + 0.792757i $$0.291355\pi$$
$$174$$ −13.7082 −1.03922
$$175$$ 4.85410 0.366936
$$176$$ −25.7984 −1.94463
$$177$$ −6.38197 −0.479698
$$178$$ 6.00000 0.449719
$$179$$ −7.85410 −0.587043 −0.293522 0.955952i $$-0.594827\pi$$
−0.293522 + 0.955952i $$0.594827\pi$$
$$180$$ 3.70820 0.276393
$$181$$ 3.85410 0.286473 0.143237 0.989688i $$-0.454249\pi$$
0.143237 + 0.989688i $$0.454249\pi$$
$$182$$ −12.7082 −0.941995
$$183$$ −4.14590 −0.306474
$$184$$ −33.4164 −2.46349
$$185$$ −2.56231 −0.188384
$$186$$ −17.5623 −1.28773
$$187$$ 14.7082 1.07557
$$188$$ 19.8541 1.44801
$$189$$ −5.00000 −0.363696
$$190$$ 5.85410 0.424701
$$191$$ 5.61803 0.406507 0.203253 0.979126i $$-0.434848\pi$$
0.203253 + 0.979126i $$0.434848\pi$$
$$192$$ −8.70820 −0.628460
$$193$$ 20.1246 1.44860 0.724301 0.689484i $$-0.242163\pi$$
0.724301 + 0.689484i $$0.242163\pi$$
$$194$$ 4.47214 0.321081
$$195$$ −1.85410 −0.132775
$$196$$ −29.1246 −2.08033
$$197$$ −16.4164 −1.16962 −0.584810 0.811170i $$-0.698831\pi$$
−0.584810 + 0.811170i $$0.698831\pi$$
$$198$$ −13.7082 −0.974200
$$199$$ −23.4164 −1.65995 −0.829973 0.557804i $$-0.811644\pi$$
−0.829973 + 0.557804i $$0.811644\pi$$
$$200$$ 36.2705 2.56471
$$201$$ −14.4164 −1.01686
$$202$$ −5.00000 −0.351799
$$203$$ 5.23607 0.367500
$$204$$ 27.2705 1.90932
$$205$$ −3.41641 −0.238612
$$206$$ 2.61803 0.182407
$$207$$ −8.94427 −0.621670
$$208$$ −47.8328 −3.31661
$$209$$ −15.3262 −1.06014
$$210$$ 1.00000 0.0690066
$$211$$ 14.8541 1.02260 0.511299 0.859403i $$-0.329164\pi$$
0.511299 + 0.859403i $$0.329164\pi$$
$$212$$ 5.29180 0.363442
$$213$$ 4.09017 0.280254
$$214$$ 18.5623 1.26889
$$215$$ 3.32624 0.226848
$$216$$ −37.3607 −2.54207
$$217$$ 6.70820 0.455383
$$218$$ 25.0344 1.69555
$$219$$ 10.8541 0.733452
$$220$$ 4.85410 0.327263
$$221$$ 27.2705 1.83441
$$222$$ 17.5623 1.17870
$$223$$ 7.70820 0.516180 0.258090 0.966121i $$-0.416907\pi$$
0.258090 + 0.966121i $$0.416907\pi$$
$$224$$ 10.8541 0.725220
$$225$$ 9.70820 0.647214
$$226$$ 39.2705 2.61224
$$227$$ 2.94427 0.195418 0.0977091 0.995215i $$-0.468849\pi$$
0.0977091 + 0.995215i $$0.468849\pi$$
$$228$$ −28.4164 −1.88192
$$229$$ 6.70820 0.443291 0.221645 0.975127i $$-0.428857\pi$$
0.221645 + 0.975127i $$0.428857\pi$$
$$230$$ 4.47214 0.294884
$$231$$ −2.61803 −0.172254
$$232$$ 39.1246 2.56866
$$233$$ −26.8885 −1.76153 −0.880764 0.473556i $$-0.842970\pi$$
−0.880764 + 0.473556i $$0.842970\pi$$
$$234$$ −25.4164 −1.66152
$$235$$ −1.56231 −0.101914
$$236$$ 30.9787 2.01654
$$237$$ 6.56231 0.426268
$$238$$ −14.7082 −0.953391
$$239$$ 8.67376 0.561059 0.280530 0.959845i $$-0.409490\pi$$
0.280530 + 0.959845i $$0.409490\pi$$
$$240$$ 3.76393 0.242961
$$241$$ −8.27051 −0.532750 −0.266375 0.963869i $$-0.585826\pi$$
−0.266375 + 0.963869i $$0.585826\pi$$
$$242$$ 10.8541 0.697728
$$243$$ −16.0000 −1.02640
$$244$$ 20.1246 1.28835
$$245$$ 2.29180 0.146417
$$246$$ 23.4164 1.49298
$$247$$ −28.4164 −1.80809
$$248$$ 50.1246 3.18292
$$249$$ 6.32624 0.400909
$$250$$ −9.85410 −0.623228
$$251$$ 11.2361 0.709214 0.354607 0.935015i $$-0.384615\pi$$
0.354607 + 0.935015i $$0.384615\pi$$
$$252$$ 9.70820 0.611559
$$253$$ −11.7082 −0.736088
$$254$$ −39.9787 −2.50849
$$255$$ −2.14590 −0.134381
$$256$$ −14.5623 −0.910144
$$257$$ −13.4721 −0.840369 −0.420184 0.907439i $$-0.638035\pi$$
−0.420184 + 0.907439i $$0.638035\pi$$
$$258$$ −22.7984 −1.41936
$$259$$ −6.70820 −0.416828
$$260$$ 9.00000 0.558156
$$261$$ 10.4721 0.648209
$$262$$ 5.85410 0.361668
$$263$$ −20.6180 −1.27136 −0.635681 0.771952i $$-0.719281\pi$$
−0.635681 + 0.771952i $$0.719281\pi$$
$$264$$ −19.5623 −1.20398
$$265$$ −0.416408 −0.0255797
$$266$$ 15.3262 0.939712
$$267$$ 2.29180 0.140256
$$268$$ 69.9787 4.27463
$$269$$ 12.3262 0.751544 0.375772 0.926712i $$-0.377378\pi$$
0.375772 + 0.926712i $$0.377378\pi$$
$$270$$ 5.00000 0.304290
$$271$$ 1.00000 0.0607457 0.0303728 0.999539i $$-0.490331\pi$$
0.0303728 + 0.999539i $$0.490331\pi$$
$$272$$ −55.3607 −3.35673
$$273$$ −4.85410 −0.293784
$$274$$ 33.2705 2.00995
$$275$$ 12.7082 0.766334
$$276$$ −21.7082 −1.30668
$$277$$ 4.70820 0.282889 0.141444 0.989946i $$-0.454825\pi$$
0.141444 + 0.989946i $$0.454825\pi$$
$$278$$ 29.1803 1.75012
$$279$$ 13.4164 0.803219
$$280$$ −2.85410 −0.170565
$$281$$ −31.4721 −1.87747 −0.938735 0.344640i $$-0.888001\pi$$
−0.938735 + 0.344640i $$0.888001\pi$$
$$282$$ 10.7082 0.637664
$$283$$ −19.7082 −1.17153 −0.585766 0.810481i $$-0.699206\pi$$
−0.585766 + 0.810481i $$0.699206\pi$$
$$284$$ −19.8541 −1.17812
$$285$$ 2.23607 0.132453
$$286$$ −33.2705 −1.96733
$$287$$ −8.94427 −0.527964
$$288$$ 21.7082 1.27917
$$289$$ 14.5623 0.856606
$$290$$ −5.23607 −0.307472
$$291$$ 1.70820 0.100137
$$292$$ −52.6869 −3.08327
$$293$$ −21.6525 −1.26495 −0.632476 0.774580i $$-0.717961\pi$$
−0.632476 + 0.774580i $$0.717961\pi$$
$$294$$ −15.7082 −0.916121
$$295$$ −2.43769 −0.141928
$$296$$ −50.1246 −2.91343
$$297$$ −13.0902 −0.759569
$$298$$ 19.5623 1.13321
$$299$$ −21.7082 −1.25542
$$300$$ 23.5623 1.36037
$$301$$ 8.70820 0.501933
$$302$$ 49.7426 2.86237
$$303$$ −1.90983 −0.109717
$$304$$ 57.6869 3.30857
$$305$$ −1.58359 −0.0906762
$$306$$ −29.4164 −1.68162
$$307$$ −2.85410 −0.162892 −0.0814461 0.996678i $$-0.525954\pi$$
−0.0814461 + 0.996678i $$0.525954\pi$$
$$308$$ 12.7082 0.724117
$$309$$ 1.00000 0.0568880
$$310$$ −6.70820 −0.381000
$$311$$ 2.88854 0.163794 0.0818971 0.996641i $$-0.473902\pi$$
0.0818971 + 0.996641i $$0.473902\pi$$
$$312$$ −36.2705 −2.05341
$$313$$ 16.2918 0.920867 0.460433 0.887694i $$-0.347694\pi$$
0.460433 + 0.887694i $$0.347694\pi$$
$$314$$ −43.7426 −2.46854
$$315$$ −0.763932 −0.0430427
$$316$$ −31.8541 −1.79193
$$317$$ 28.4164 1.59602 0.798012 0.602641i $$-0.205885\pi$$
0.798012 + 0.602641i $$0.205885\pi$$
$$318$$ 2.85410 0.160050
$$319$$ 13.7082 0.767512
$$320$$ −3.32624 −0.185942
$$321$$ 7.09017 0.395735
$$322$$ 11.7082 0.652473
$$323$$ −32.8885 −1.82997
$$324$$ 4.85410 0.269672
$$325$$ 23.5623 1.30700
$$326$$ −7.09017 −0.392688
$$327$$ 9.56231 0.528797
$$328$$ −66.8328 −3.69022
$$329$$ −4.09017 −0.225498
$$330$$ 2.61803 0.144118
$$331$$ −16.1459 −0.887459 −0.443729 0.896161i $$-0.646345\pi$$
−0.443729 + 0.896161i $$0.646345\pi$$
$$332$$ −30.7082 −1.68533
$$333$$ −13.4164 −0.735215
$$334$$ −23.5623 −1.28927
$$335$$ −5.50658 −0.300856
$$336$$ 9.85410 0.537585
$$337$$ −2.43769 −0.132790 −0.0663948 0.997793i $$-0.521150\pi$$
−0.0663948 + 0.997793i $$0.521150\pi$$
$$338$$ −27.6525 −1.50410
$$339$$ 15.0000 0.814688
$$340$$ 10.4164 0.564909
$$341$$ 17.5623 0.951052
$$342$$ 30.6525 1.65750
$$343$$ 13.0000 0.701934
$$344$$ 65.0689 3.50828
$$345$$ 1.70820 0.0919666
$$346$$ −41.9787 −2.25679
$$347$$ −1.47214 −0.0790284 −0.0395142 0.999219i $$-0.512581\pi$$
−0.0395142 + 0.999219i $$0.512581\pi$$
$$348$$ 25.4164 1.36246
$$349$$ 15.4164 0.825221 0.412611 0.910907i $$-0.364617\pi$$
0.412611 + 0.910907i $$0.364617\pi$$
$$350$$ −12.7082 −0.679282
$$351$$ −24.2705 −1.29546
$$352$$ 28.4164 1.51460
$$353$$ −25.0344 −1.33245 −0.666224 0.745751i $$-0.732091\pi$$
−0.666224 + 0.745751i $$0.732091\pi$$
$$354$$ 16.7082 0.888031
$$355$$ 1.56231 0.0829186
$$356$$ −11.1246 −0.589603
$$357$$ −5.61803 −0.297338
$$358$$ 20.5623 1.08675
$$359$$ 14.6738 0.774452 0.387226 0.921985i $$-0.373433\pi$$
0.387226 + 0.921985i $$0.373433\pi$$
$$360$$ −5.70820 −0.300849
$$361$$ 15.2705 0.803711
$$362$$ −10.0902 −0.530328
$$363$$ 4.14590 0.217603
$$364$$ 23.5623 1.23500
$$365$$ 4.14590 0.217006
$$366$$ 10.8541 0.567353
$$367$$ 16.4377 0.858041 0.429020 0.903295i $$-0.358859\pi$$
0.429020 + 0.903295i $$0.358859\pi$$
$$368$$ 44.0689 2.29725
$$369$$ −17.8885 −0.931240
$$370$$ 6.70820 0.348743
$$371$$ −1.09017 −0.0565988
$$372$$ 32.5623 1.68828
$$373$$ −22.6869 −1.17468 −0.587342 0.809339i $$-0.699826\pi$$
−0.587342 + 0.809339i $$0.699826\pi$$
$$374$$ −38.5066 −1.99113
$$375$$ −3.76393 −0.194369
$$376$$ −30.5623 −1.57613
$$377$$ 25.4164 1.30901
$$378$$ 13.0902 0.673286
$$379$$ 5.00000 0.256833 0.128416 0.991720i $$-0.459011\pi$$
0.128416 + 0.991720i $$0.459011\pi$$
$$380$$ −10.8541 −0.556804
$$381$$ −15.2705 −0.782332
$$382$$ −14.7082 −0.752537
$$383$$ 0.819660 0.0418827 0.0209413 0.999781i $$-0.493334\pi$$
0.0209413 + 0.999781i $$0.493334\pi$$
$$384$$ 1.09017 0.0556325
$$385$$ −1.00000 −0.0509647
$$386$$ −52.6869 −2.68169
$$387$$ 17.4164 0.885326
$$388$$ −8.29180 −0.420952
$$389$$ 7.41641 0.376027 0.188013 0.982166i $$-0.439795\pi$$
0.188013 + 0.982166i $$0.439795\pi$$
$$390$$ 4.85410 0.245797
$$391$$ −25.1246 −1.27061
$$392$$ 44.8328 2.26440
$$393$$ 2.23607 0.112795
$$394$$ 42.9787 2.16524
$$395$$ 2.50658 0.126120
$$396$$ 25.4164 1.27722
$$397$$ 20.0000 1.00377 0.501886 0.864934i $$-0.332640\pi$$
0.501886 + 0.864934i $$0.332640\pi$$
$$398$$ 61.3050 3.07294
$$399$$ 5.85410 0.293072
$$400$$ −47.8328 −2.39164
$$401$$ −23.8885 −1.19294 −0.596468 0.802637i $$-0.703430\pi$$
−0.596468 + 0.802637i $$0.703430\pi$$
$$402$$ 37.7426 1.88243
$$403$$ 32.5623 1.62204
$$404$$ 9.27051 0.461225
$$405$$ −0.381966 −0.0189800
$$406$$ −13.7082 −0.680327
$$407$$ −17.5623 −0.870531
$$408$$ −41.9787 −2.07826
$$409$$ 36.7082 1.81510 0.907552 0.419940i $$-0.137949\pi$$
0.907552 + 0.419940i $$0.137949\pi$$
$$410$$ 8.94427 0.441726
$$411$$ 12.7082 0.626849
$$412$$ −4.85410 −0.239144
$$413$$ −6.38197 −0.314036
$$414$$ 23.4164 1.15085
$$415$$ 2.41641 0.118617
$$416$$ 52.6869 2.58319
$$417$$ 11.1459 0.545817
$$418$$ 40.1246 1.96256
$$419$$ −4.09017 −0.199818 −0.0999089 0.994997i $$-0.531855\pi$$
−0.0999089 + 0.994997i $$0.531855\pi$$
$$420$$ −1.85410 −0.0904709
$$421$$ 3.00000 0.146211 0.0731055 0.997324i $$-0.476709\pi$$
0.0731055 + 0.997324i $$0.476709\pi$$
$$422$$ −38.8885 −1.89306
$$423$$ −8.18034 −0.397742
$$424$$ −8.14590 −0.395600
$$425$$ 27.2705 1.32281
$$426$$ −10.7082 −0.518814
$$427$$ −4.14590 −0.200634
$$428$$ −34.4164 −1.66358
$$429$$ −12.7082 −0.613558
$$430$$ −8.70820 −0.419947
$$431$$ 34.3607 1.65510 0.827548 0.561395i $$-0.189735\pi$$
0.827548 + 0.561395i $$0.189735\pi$$
$$432$$ 49.2705 2.37053
$$433$$ 14.4164 0.692808 0.346404 0.938085i $$-0.387403\pi$$
0.346404 + 0.938085i $$0.387403\pi$$
$$434$$ −17.5623 −0.843018
$$435$$ −2.00000 −0.0958927
$$436$$ −46.4164 −2.22294
$$437$$ 26.1803 1.25238
$$438$$ −28.4164 −1.35779
$$439$$ 29.5623 1.41093 0.705466 0.708744i $$-0.250738\pi$$
0.705466 + 0.708744i $$0.250738\pi$$
$$440$$ −7.47214 −0.356220
$$441$$ 12.0000 0.571429
$$442$$ −71.3951 −3.39592
$$443$$ −15.4377 −0.733467 −0.366733 0.930326i $$-0.619524\pi$$
−0.366733 + 0.930326i $$0.619524\pi$$
$$444$$ −32.5623 −1.54534
$$445$$ 0.875388 0.0414974
$$446$$ −20.1803 −0.955567
$$447$$ 7.47214 0.353420
$$448$$ −8.70820 −0.411424
$$449$$ −13.3607 −0.630529 −0.315265 0.949004i $$-0.602093\pi$$
−0.315265 + 0.949004i $$0.602093\pi$$
$$450$$ −25.4164 −1.19814
$$451$$ −23.4164 −1.10264
$$452$$ −72.8115 −3.42477
$$453$$ 19.0000 0.892698
$$454$$ −7.70820 −0.361764
$$455$$ −1.85410 −0.0869216
$$456$$ 43.7426 2.04844
$$457$$ −9.85410 −0.460955 −0.230478 0.973078i $$-0.574029\pi$$
−0.230478 + 0.973078i $$0.574029\pi$$
$$458$$ −17.5623 −0.820633
$$459$$ −28.0902 −1.31114
$$460$$ −8.29180 −0.386607
$$461$$ 12.2148 0.568899 0.284450 0.958691i $$-0.408189\pi$$
0.284450 + 0.958691i $$0.408189\pi$$
$$462$$ 6.85410 0.318882
$$463$$ −21.4164 −0.995305 −0.497652 0.867377i $$-0.665804\pi$$
−0.497652 + 0.867377i $$0.665804\pi$$
$$464$$ −51.5967 −2.39532
$$465$$ −2.56231 −0.118824
$$466$$ 70.3951 3.26099
$$467$$ −3.65248 −0.169016 −0.0845082 0.996423i $$-0.526932\pi$$
−0.0845082 + 0.996423i $$0.526932\pi$$
$$468$$ 47.1246 2.17834
$$469$$ −14.4164 −0.665688
$$470$$ 4.09017 0.188665
$$471$$ −16.7082 −0.769873
$$472$$ −47.6869 −2.19497
$$473$$ 22.7984 1.04827
$$474$$ −17.1803 −0.789119
$$475$$ −28.4164 −1.30383
$$476$$ 27.2705 1.24994
$$477$$ −2.18034 −0.0998309
$$478$$ −22.7082 −1.03865
$$479$$ 8.18034 0.373769 0.186885 0.982382i $$-0.440161\pi$$
0.186885 + 0.982382i $$0.440161\pi$$
$$480$$ −4.14590 −0.189233
$$481$$ −32.5623 −1.48471
$$482$$ 21.6525 0.986243
$$483$$ 4.47214 0.203489
$$484$$ −20.1246 −0.914755
$$485$$ 0.652476 0.0296274
$$486$$ 41.8885 1.90010
$$487$$ −23.0000 −1.04223 −0.521115 0.853487i $$-0.674484\pi$$
−0.521115 + 0.853487i $$0.674484\pi$$
$$488$$ −30.9787 −1.40234
$$489$$ −2.70820 −0.122469
$$490$$ −6.00000 −0.271052
$$491$$ 35.7771 1.61460 0.807299 0.590143i $$-0.200929\pi$$
0.807299 + 0.590143i $$0.200929\pi$$
$$492$$ −43.4164 −1.95736
$$493$$ 29.4164 1.32485
$$494$$ 74.3951 3.34719
$$495$$ −2.00000 −0.0898933
$$496$$ −66.1033 −2.96813
$$497$$ 4.09017 0.183469
$$498$$ −16.5623 −0.742175
$$499$$ 13.2705 0.594070 0.297035 0.954867i $$-0.404002\pi$$
0.297035 + 0.954867i $$0.404002\pi$$
$$500$$ 18.2705 0.817082
$$501$$ −9.00000 −0.402090
$$502$$ −29.4164 −1.31292
$$503$$ 13.3607 0.595723 0.297862 0.954609i $$-0.403727\pi$$
0.297862 + 0.954609i $$0.403727\pi$$
$$504$$ −14.9443 −0.665671
$$505$$ −0.729490 −0.0324619
$$506$$ 30.6525 1.36267
$$507$$ −10.5623 −0.469088
$$508$$ 74.1246 3.28875
$$509$$ 26.6180 1.17982 0.589912 0.807468i $$-0.299163\pi$$
0.589912 + 0.807468i $$0.299163\pi$$
$$510$$ 5.61803 0.248771
$$511$$ 10.8541 0.480157
$$512$$ 40.3050 1.78124
$$513$$ 29.2705 1.29232
$$514$$ 35.2705 1.55572
$$515$$ 0.381966 0.0168314
$$516$$ 42.2705 1.86086
$$517$$ −10.7082 −0.470946
$$518$$ 17.5623 0.771643
$$519$$ −16.0344 −0.703834
$$520$$ −13.8541 −0.607543
$$521$$ 17.1803 0.752684 0.376342 0.926481i $$-0.377182\pi$$
0.376342 + 0.926481i $$0.377182\pi$$
$$522$$ −27.4164 −1.19998
$$523$$ −10.5836 −0.462788 −0.231394 0.972860i $$-0.574329\pi$$
−0.231394 + 0.972860i $$0.574329\pi$$
$$524$$ −10.8541 −0.474164
$$525$$ −4.85410 −0.211850
$$526$$ 53.9787 2.35358
$$527$$ 37.6869 1.64167
$$528$$ 25.7984 1.12273
$$529$$ −3.00000 −0.130435
$$530$$ 1.09017 0.0473540
$$531$$ −12.7639 −0.553907
$$532$$ −28.4164 −1.23201
$$533$$ −43.4164 −1.88057
$$534$$ −6.00000 −0.259645
$$535$$ 2.70820 0.117086
$$536$$ −107.721 −4.65285
$$537$$ 7.85410 0.338930
$$538$$ −32.2705 −1.39128
$$539$$ 15.7082 0.676600
$$540$$ −9.27051 −0.398939
$$541$$ 9.14590 0.393213 0.196606 0.980482i $$-0.437008\pi$$
0.196606 + 0.980482i $$0.437008\pi$$
$$542$$ −2.61803 −0.112454
$$543$$ −3.85410 −0.165395
$$544$$ 60.9787 2.61444
$$545$$ 3.65248 0.156455
$$546$$ 12.7082 0.543861
$$547$$ −2.27051 −0.0970800 −0.0485400 0.998821i $$-0.515457\pi$$
−0.0485400 + 0.998821i $$0.515457\pi$$
$$548$$ −61.6869 −2.63513
$$549$$ −8.29180 −0.353885
$$550$$ −33.2705 −1.41866
$$551$$ −30.6525 −1.30584
$$552$$ 33.4164 1.42230
$$553$$ 6.56231 0.279058
$$554$$ −12.3262 −0.523692
$$555$$ 2.56231 0.108764
$$556$$ −54.1033 −2.29449
$$557$$ −31.6869 −1.34262 −0.671309 0.741178i $$-0.734268\pi$$
−0.671309 + 0.741178i $$0.734268\pi$$
$$558$$ −35.1246 −1.48694
$$559$$ 42.2705 1.78785
$$560$$ 3.76393 0.159055
$$561$$ −14.7082 −0.620981
$$562$$ 82.3951 3.47563
$$563$$ −1.20163 −0.0506425 −0.0253213 0.999679i $$-0.508061\pi$$
−0.0253213 + 0.999679i $$0.508061\pi$$
$$564$$ −19.8541 −0.836009
$$565$$ 5.72949 0.241041
$$566$$ 51.5967 2.16877
$$567$$ −1.00000 −0.0419961
$$568$$ 30.5623 1.28237
$$569$$ −27.1591 −1.13857 −0.569283 0.822141i $$-0.692779\pi$$
−0.569283 + 0.822141i $$0.692779\pi$$
$$570$$ −5.85410 −0.245201
$$571$$ −22.5623 −0.944203 −0.472102 0.881544i $$-0.656504\pi$$
−0.472102 + 0.881544i $$0.656504\pi$$
$$572$$ 61.6869 2.57926
$$573$$ −5.61803 −0.234697
$$574$$ 23.4164 0.977382
$$575$$ −21.7082 −0.905295
$$576$$ −17.4164 −0.725684
$$577$$ −36.8328 −1.53337 −0.766685 0.642023i $$-0.778095\pi$$
−0.766685 + 0.642023i $$0.778095\pi$$
$$578$$ −38.1246 −1.58577
$$579$$ −20.1246 −0.836350
$$580$$ 9.70820 0.403111
$$581$$ 6.32624 0.262457
$$582$$ −4.47214 −0.185376
$$583$$ −2.85410 −0.118205
$$584$$ 81.1033 3.35608
$$585$$ −3.70820 −0.153315
$$586$$ 56.6869 2.34171
$$587$$ −47.0689 −1.94274 −0.971370 0.237570i $$-0.923649\pi$$
−0.971370 + 0.237570i $$0.923649\pi$$
$$588$$ 29.1246 1.20108
$$589$$ −39.2705 −1.61811
$$590$$ 6.38197 0.262741
$$591$$ 16.4164 0.675281
$$592$$ 66.1033 2.71683
$$593$$ 2.18034 0.0895358 0.0447679 0.998997i $$-0.485745\pi$$
0.0447679 + 0.998997i $$0.485745\pi$$
$$594$$ 34.2705 1.40614
$$595$$ −2.14590 −0.0879732
$$596$$ −36.2705 −1.48570
$$597$$ 23.4164 0.958370
$$598$$ 56.8328 2.32407
$$599$$ −35.4508 −1.44848 −0.724241 0.689547i $$-0.757810\pi$$
−0.724241 + 0.689547i $$0.757810\pi$$
$$600$$ −36.2705 −1.48074
$$601$$ 3.56231 0.145309 0.0726547 0.997357i $$-0.476853\pi$$
0.0726547 + 0.997357i $$0.476853\pi$$
$$602$$ −22.7984 −0.929192
$$603$$ −28.8328 −1.17416
$$604$$ −92.2279 −3.75270
$$605$$ 1.58359 0.0643822
$$606$$ 5.00000 0.203111
$$607$$ 5.70820 0.231689 0.115844 0.993267i $$-0.463043\pi$$
0.115844 + 0.993267i $$0.463043\pi$$
$$608$$ −63.5410 −2.57693
$$609$$ −5.23607 −0.212176
$$610$$ 4.14590 0.167863
$$611$$ −19.8541 −0.803211
$$612$$ 54.5410 2.20469
$$613$$ 24.4164 0.986169 0.493085 0.869981i $$-0.335869\pi$$
0.493085 + 0.869981i $$0.335869\pi$$
$$614$$ 7.47214 0.301551
$$615$$ 3.41641 0.137763
$$616$$ −19.5623 −0.788188
$$617$$ 3.27051 0.131666 0.0658329 0.997831i $$-0.479030\pi$$
0.0658329 + 0.997831i $$0.479030\pi$$
$$618$$ −2.61803 −0.105313
$$619$$ −28.6869 −1.15302 −0.576512 0.817088i $$-0.695587\pi$$
−0.576512 + 0.817088i $$0.695587\pi$$
$$620$$ 12.4377 0.499510
$$621$$ 22.3607 0.897303
$$622$$ −7.56231 −0.303221
$$623$$ 2.29180 0.0918189
$$624$$ 47.8328 1.91485
$$625$$ 22.8328 0.913313
$$626$$ −42.6525 −1.70474
$$627$$ 15.3262 0.612071
$$628$$ 81.1033 3.23638
$$629$$ −37.6869 −1.50268
$$630$$ 2.00000 0.0796819
$$631$$ 42.2705 1.68276 0.841381 0.540442i $$-0.181743\pi$$
0.841381 + 0.540442i $$0.181743\pi$$
$$632$$ 49.0344 1.95049
$$633$$ −14.8541 −0.590398
$$634$$ −74.3951 −2.95461
$$635$$ −5.83282 −0.231468
$$636$$ −5.29180 −0.209833
$$637$$ 29.1246 1.15396
$$638$$ −35.8885 −1.42084
$$639$$ 8.18034 0.323609
$$640$$ 0.416408 0.0164600
$$641$$ −15.0000 −0.592464 −0.296232 0.955116i $$-0.595730\pi$$
−0.296232 + 0.955116i $$0.595730\pi$$
$$642$$ −18.5623 −0.732596
$$643$$ 5.00000 0.197181 0.0985904 0.995128i $$-0.468567\pi$$
0.0985904 + 0.995128i $$0.468567\pi$$
$$644$$ −21.7082 −0.855423
$$645$$ −3.32624 −0.130970
$$646$$ 86.1033 3.38769
$$647$$ 1.25735 0.0494317 0.0247158 0.999695i $$-0.492132\pi$$
0.0247158 + 0.999695i $$0.492132\pi$$
$$648$$ −7.47214 −0.293533
$$649$$ −16.7082 −0.655854
$$650$$ −61.6869 −2.41956
$$651$$ −6.70820 −0.262915
$$652$$ 13.1459 0.514833
$$653$$ −5.23607 −0.204903 −0.102452 0.994738i $$-0.532669\pi$$
−0.102452 + 0.994738i $$0.532669\pi$$
$$654$$ −25.0344 −0.978924
$$655$$ 0.854102 0.0333725
$$656$$ 88.1378 3.44120
$$657$$ 21.7082 0.846918
$$658$$ 10.7082 0.417449
$$659$$ 36.5967 1.42561 0.712803 0.701364i $$-0.247425\pi$$
0.712803 + 0.701364i $$0.247425\pi$$
$$660$$ −4.85410 −0.188946
$$661$$ −31.5623 −1.22763 −0.613816 0.789449i $$-0.710366\pi$$
−0.613816 + 0.789449i $$0.710366\pi$$
$$662$$ 42.2705 1.64289
$$663$$ −27.2705 −1.05910
$$664$$ 47.2705 1.83445
$$665$$ 2.23607 0.0867110
$$666$$ 35.1246 1.36105
$$667$$ −23.4164 −0.906687
$$668$$ 43.6869 1.69030
$$669$$ −7.70820 −0.298016
$$670$$ 14.4164 0.556954
$$671$$ −10.8541 −0.419018
$$672$$ −10.8541 −0.418706
$$673$$ −24.2918 −0.936380 −0.468190 0.883628i $$-0.655094\pi$$
−0.468190 + 0.883628i $$0.655094\pi$$
$$674$$ 6.38197 0.245824
$$675$$ −24.2705 −0.934172
$$676$$ 51.2705 1.97194
$$677$$ 40.0344 1.53865 0.769324 0.638858i $$-0.220593\pi$$
0.769324 + 0.638858i $$0.220593\pi$$
$$678$$ −39.2705 −1.50817
$$679$$ 1.70820 0.0655549
$$680$$ −16.0344 −0.614893
$$681$$ −2.94427 −0.112825
$$682$$ −45.9787 −1.76062
$$683$$ 47.2361 1.80744 0.903719 0.428126i $$-0.140826\pi$$
0.903719 + 0.428126i $$0.140826\pi$$
$$684$$ −56.8328 −2.17306
$$685$$ 4.85410 0.185466
$$686$$ −34.0344 −1.29944
$$687$$ −6.70820 −0.255934
$$688$$ −85.8115 −3.27153
$$689$$ −5.29180 −0.201601
$$690$$ −4.47214 −0.170251
$$691$$ 7.85410 0.298784 0.149392 0.988778i $$-0.452268\pi$$
0.149392 + 0.988778i $$0.452268\pi$$
$$692$$ 77.8328 2.95876
$$693$$ −5.23607 −0.198902
$$694$$ 3.85410 0.146300
$$695$$ 4.25735 0.161491
$$696$$ −39.1246 −1.48301
$$697$$ −50.2492 −1.90333
$$698$$ −40.3607 −1.52767
$$699$$ 26.8885 1.01702
$$700$$ 23.5623 0.890571
$$701$$ −25.7984 −0.974391 −0.487196 0.873293i $$-0.661980\pi$$
−0.487196 + 0.873293i $$0.661980\pi$$
$$702$$ 63.5410 2.39820
$$703$$ 39.2705 1.48112
$$704$$ −22.7984 −0.859246
$$705$$ 1.56231 0.0588398
$$706$$ 65.5410 2.46667
$$707$$ −1.90983 −0.0718266
$$708$$ −30.9787 −1.16425
$$709$$ −1.02129 −0.0383552 −0.0191776 0.999816i $$-0.506105\pi$$
−0.0191776 + 0.999816i $$0.506105\pi$$
$$710$$ −4.09017 −0.153501
$$711$$ 13.1246 0.492211
$$712$$ 17.1246 0.641772
$$713$$ −30.0000 −1.12351
$$714$$ 14.7082 0.550441
$$715$$ −4.85410 −0.181533
$$716$$ −38.1246 −1.42478
$$717$$ −8.67376 −0.323928
$$718$$ −38.4164 −1.43369
$$719$$ 39.3262 1.46662 0.733311 0.679894i $$-0.237974\pi$$
0.733311 + 0.679894i $$0.237974\pi$$
$$720$$ 7.52786 0.280547
$$721$$ 1.00000 0.0372419
$$722$$ −39.9787 −1.48785
$$723$$ 8.27051 0.307584
$$724$$ 18.7082 0.695285
$$725$$ 25.4164 0.943942
$$726$$ −10.8541 −0.402834
$$727$$ −4.72949 −0.175407 −0.0877035 0.996147i $$-0.527953\pi$$
−0.0877035 + 0.996147i $$0.527953\pi$$
$$728$$ −36.2705 −1.34427
$$729$$ 13.0000 0.481481
$$730$$ −10.8541 −0.401728
$$731$$ 48.9230 1.80948
$$732$$ −20.1246 −0.743827
$$733$$ 14.7082 0.543260 0.271630 0.962402i $$-0.412437\pi$$
0.271630 + 0.962402i $$0.412437\pi$$
$$734$$ −43.0344 −1.58843
$$735$$ −2.29180 −0.0845342
$$736$$ −48.5410 −1.78925
$$737$$ −37.7426 −1.39027
$$738$$ 46.8328 1.72394
$$739$$ −16.8328 −0.619205 −0.309603 0.950866i $$-0.600196\pi$$
−0.309603 + 0.950866i $$0.600196\pi$$
$$740$$ −12.4377 −0.457219
$$741$$ 28.4164 1.04390
$$742$$ 2.85410 0.104777
$$743$$ 30.7082 1.12657 0.563287 0.826261i $$-0.309536\pi$$
0.563287 + 0.826261i $$0.309536\pi$$
$$744$$ −50.1246 −1.83766
$$745$$ 2.85410 0.104566
$$746$$ 59.3951 2.17461
$$747$$ 12.6525 0.462930
$$748$$ 71.3951 2.61046
$$749$$ 7.09017 0.259069
$$750$$ 9.85410 0.359821
$$751$$ −44.1246 −1.61013 −0.805065 0.593187i $$-0.797870\pi$$
−0.805065 + 0.593187i $$0.797870\pi$$
$$752$$ 40.3050 1.46977
$$753$$ −11.2361 −0.409465
$$754$$ −66.5410 −2.42328
$$755$$ 7.25735 0.264122
$$756$$ −24.2705 −0.882710
$$757$$ −21.2918 −0.773863 −0.386932 0.922108i $$-0.626465\pi$$
−0.386932 + 0.922108i $$0.626465\pi$$
$$758$$ −13.0902 −0.475456
$$759$$ 11.7082 0.424981
$$760$$ 16.7082 0.606070
$$761$$ −7.47214 −0.270865 −0.135432 0.990787i $$-0.543242\pi$$
−0.135432 + 0.990787i $$0.543242\pi$$
$$762$$ 39.9787 1.44828
$$763$$ 9.56231 0.346179
$$764$$ 27.2705 0.986612
$$765$$ −4.29180 −0.155170
$$766$$ −2.14590 −0.0775344
$$767$$ −30.9787 −1.11858
$$768$$ 14.5623 0.525472
$$769$$ 31.3951 1.13214 0.566069 0.824358i $$-0.308464\pi$$
0.566069 + 0.824358i $$0.308464\pi$$
$$770$$ 2.61803 0.0943474
$$771$$ 13.4721 0.485187
$$772$$ 97.6869 3.51583
$$773$$ −16.5279 −0.594466 −0.297233 0.954805i $$-0.596064\pi$$
−0.297233 + 0.954805i $$0.596064\pi$$
$$774$$ −45.5967 −1.63894
$$775$$ 32.5623 1.16967
$$776$$ 12.7639 0.458198
$$777$$ 6.70820 0.240655
$$778$$ −19.4164 −0.696112
$$779$$ 52.3607 1.87602
$$780$$ −9.00000 −0.322252
$$781$$ 10.7082 0.383170
$$782$$ 65.7771 2.35218
$$783$$ −26.1803 −0.935609
$$784$$ −59.1246 −2.11159
$$785$$ −6.38197 −0.227782
$$786$$ −5.85410 −0.208809
$$787$$ 43.4164 1.54763 0.773814 0.633413i $$-0.218347\pi$$
0.773814 + 0.633413i $$0.218347\pi$$
$$788$$ −79.6869 −2.83873
$$789$$ 20.6180 0.734021
$$790$$ −6.56231 −0.233476
$$791$$ 15.0000 0.533339
$$792$$ −39.1246 −1.39023
$$793$$ −20.1246 −0.714646
$$794$$ −52.3607 −1.85821
$$795$$ 0.416408 0.0147685
$$796$$ −113.666 −4.02877
$$797$$ −50.1246 −1.77550 −0.887752 0.460321i $$-0.847734\pi$$
−0.887752 + 0.460321i $$0.847734\pi$$
$$798$$ −15.3262 −0.542543
$$799$$ −22.9787 −0.812928
$$800$$ 52.6869 1.86276
$$801$$ 4.58359 0.161953
$$802$$ 62.5410 2.20840
$$803$$ 28.4164 1.00279
$$804$$ −69.9787 −2.46796
$$805$$ 1.70820 0.0602063
$$806$$ −85.2492 −3.00278
$$807$$ −12.3262 −0.433904
$$808$$ −14.2705 −0.502035
$$809$$ 2.94427 0.103515 0.0517575 0.998660i $$-0.483518\pi$$
0.0517575 + 0.998660i $$0.483518\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ 21.5410 0.756408 0.378204 0.925722i $$-0.376542\pi$$
0.378204 + 0.925722i $$0.376542\pi$$
$$812$$ 25.4164 0.891941
$$813$$ −1.00000 −0.0350715
$$814$$ 45.9787 1.61155
$$815$$ −1.03444 −0.0362349
$$816$$ 55.3607 1.93801
$$817$$ −50.9787 −1.78352
$$818$$ −96.1033 −3.36017
$$819$$ −9.70820 −0.339232
$$820$$ −16.5836 −0.579124
$$821$$ 33.0000 1.15171 0.575854 0.817553i $$-0.304670\pi$$
0.575854 + 0.817553i $$0.304670\pi$$
$$822$$ −33.2705 −1.16044
$$823$$ 3.43769 0.119830 0.0599152 0.998203i $$-0.480917\pi$$
0.0599152 + 0.998203i $$0.480917\pi$$
$$824$$ 7.47214 0.260304
$$825$$ −12.7082 −0.442443
$$826$$ 16.7082 0.581353
$$827$$ 6.70820 0.233267 0.116634 0.993175i $$-0.462790\pi$$
0.116634 + 0.993175i $$0.462790\pi$$
$$828$$ −43.4164 −1.50882
$$829$$ 14.2705 0.495635 0.247818 0.968807i $$-0.420287\pi$$
0.247818 + 0.968807i $$0.420287\pi$$
$$830$$ −6.32624 −0.219587
$$831$$ −4.70820 −0.163326
$$832$$ −42.2705 −1.46547
$$833$$ 33.7082 1.16792
$$834$$ −29.1803 −1.01043
$$835$$ −3.43769 −0.118966
$$836$$ −74.3951 −2.57301
$$837$$ −33.5410 −1.15935
$$838$$ 10.7082 0.369909
$$839$$ 23.6180 0.815385 0.407693 0.913119i $$-0.366334\pi$$
0.407693 + 0.913119i $$0.366334\pi$$
$$840$$ 2.85410 0.0984759
$$841$$ −1.58359 −0.0546066
$$842$$ −7.85410 −0.270670
$$843$$ 31.4721 1.08396
$$844$$ 72.1033 2.48190
$$845$$ −4.03444 −0.138789
$$846$$ 21.4164 0.736311
$$847$$ 4.14590 0.142455
$$848$$ 10.7426 0.368904
$$849$$ 19.7082 0.676384
$$850$$ −71.3951 −2.44883
$$851$$ 30.0000 1.02839
$$852$$ 19.8541 0.680190
$$853$$ −44.2705 −1.51579 −0.757897 0.652375i $$-0.773773\pi$$
−0.757897 + 0.652375i $$0.773773\pi$$
$$854$$ 10.8541 0.371420
$$855$$ 4.47214 0.152944
$$856$$ 52.9787 1.81078
$$857$$ 8.23607 0.281339 0.140669 0.990057i $$-0.455075\pi$$
0.140669 + 0.990057i $$0.455075\pi$$
$$858$$ 33.2705 1.13584
$$859$$ 10.5623 0.360381 0.180191 0.983632i $$-0.442329\pi$$
0.180191 + 0.983632i $$0.442329\pi$$
$$860$$ 16.1459 0.550571
$$861$$ 8.94427 0.304820
$$862$$ −89.9574 −3.06396
$$863$$ −21.7082 −0.738956 −0.369478 0.929240i $$-0.620463\pi$$
−0.369478 + 0.929240i $$0.620463\pi$$
$$864$$ −54.2705 −1.84632
$$865$$ −6.12461 −0.208243
$$866$$ −37.7426 −1.28255
$$867$$ −14.5623 −0.494562
$$868$$ 32.5623 1.10524
$$869$$ 17.1803 0.582803
$$870$$ 5.23607 0.177519
$$871$$ −69.9787 −2.37114
$$872$$ 71.4508 2.41963
$$873$$ 3.41641 0.115628
$$874$$ −68.5410 −2.31843
$$875$$ −3.76393 −0.127244
$$876$$ 52.6869 1.78013
$$877$$ −13.0000 −0.438979 −0.219489 0.975615i $$-0.570439\pi$$
−0.219489 + 0.975615i $$0.570439\pi$$
$$878$$ −77.3951 −2.61196
$$879$$ 21.6525 0.730320
$$880$$ 9.85410 0.332182
$$881$$ 29.8885 1.00697 0.503485 0.864004i $$-0.332051\pi$$
0.503485 + 0.864004i $$0.332051\pi$$
$$882$$ −31.4164 −1.05785
$$883$$ 5.87539 0.197723 0.0988613 0.995101i $$-0.468480\pi$$
0.0988613 + 0.995101i $$0.468480\pi$$
$$884$$ 132.374 4.45221
$$885$$ 2.43769 0.0819422
$$886$$ 40.4164 1.35782
$$887$$ 40.1935 1.34957 0.674783 0.738016i $$-0.264237\pi$$
0.674783 + 0.738016i $$0.264237\pi$$
$$888$$ 50.1246 1.68207
$$889$$ −15.2705 −0.512156
$$890$$ −2.29180 −0.0768212
$$891$$ −2.61803 −0.0877074
$$892$$ 37.4164 1.25279
$$893$$ 23.9443 0.801265
$$894$$ −19.5623 −0.654261
$$895$$ 3.00000 0.100279
$$896$$ 1.09017 0.0364200
$$897$$ 21.7082 0.724816
$$898$$ 34.9787 1.16725
$$899$$ 35.1246 1.17147
$$900$$ 47.1246 1.57082
$$901$$ −6.12461 −0.204040
$$902$$ 61.3050 2.04123
$$903$$ −8.70820 −0.289791
$$904$$ 112.082 3.72779
$$905$$ −1.47214 −0.0489355
$$906$$ −49.7426 −1.65259
$$907$$ −7.12461 −0.236569 −0.118284 0.992980i $$-0.537739\pi$$
−0.118284 + 0.992980i $$0.537739\pi$$
$$908$$ 14.2918 0.474290
$$909$$ −3.81966 −0.126690
$$910$$ 4.85410 0.160912
$$911$$ −10.0344 −0.332456 −0.166228 0.986087i $$-0.553159\pi$$
−0.166228 + 0.986087i $$0.553159\pi$$
$$912$$ −57.6869 −1.91020
$$913$$ 16.5623 0.548132
$$914$$ 25.7984 0.853334
$$915$$ 1.58359 0.0523519
$$916$$ 32.5623 1.07589
$$917$$ 2.23607 0.0738415
$$918$$ 73.5410 2.42722
$$919$$ −27.9787 −0.922933 −0.461466 0.887158i $$-0.652676\pi$$
−0.461466 + 0.887158i $$0.652676\pi$$
$$920$$ 12.7639 0.420814
$$921$$ 2.85410 0.0940459
$$922$$ −31.9787 −1.05316
$$923$$ 19.8541 0.653506
$$924$$ −12.7082 −0.418069
$$925$$ −32.5623 −1.07064
$$926$$ 56.0689 1.84254
$$927$$ 2.00000 0.0656886
$$928$$ 56.8328 1.86563
$$929$$ 14.9443 0.490306 0.245153 0.969484i $$-0.421162\pi$$
0.245153 + 0.969484i $$0.421162\pi$$
$$930$$ 6.70820 0.219971
$$931$$ −35.1246 −1.15116
$$932$$ −130.520 −4.27532
$$933$$ −2.88854 −0.0945667
$$934$$ 9.56231 0.312888
$$935$$ −5.61803 −0.183729
$$936$$ −72.5410 −2.37108
$$937$$ −11.0000 −0.359354 −0.179677 0.983726i $$-0.557505\pi$$
−0.179677 + 0.983726i $$0.557505\pi$$
$$938$$ 37.7426 1.23234
$$939$$ −16.2918 −0.531663
$$940$$ −7.58359 −0.247350
$$941$$ −23.3951 −0.762659 −0.381330 0.924439i $$-0.624534\pi$$
−0.381330 + 0.924439i $$0.624534\pi$$
$$942$$ 43.7426 1.42521
$$943$$ 40.0000 1.30258
$$944$$ 62.8885 2.04685
$$945$$ 1.90983 0.0621268
$$946$$ −59.6869 −1.94059
$$947$$ −41.0132 −1.33275 −0.666374 0.745617i $$-0.732155\pi$$
−0.666374 + 0.745617i $$0.732155\pi$$
$$948$$ 31.8541 1.03457
$$949$$ 52.6869 1.71029
$$950$$ 74.3951 2.41370
$$951$$ −28.4164 −0.921465
$$952$$ −41.9787 −1.36054
$$953$$ 13.3607 0.432795 0.216397 0.976305i $$-0.430569\pi$$
0.216397 + 0.976305i $$0.430569\pi$$
$$954$$ 5.70820 0.184810
$$955$$ −2.14590 −0.0694396
$$956$$ 42.1033 1.36172
$$957$$ −13.7082 −0.443123
$$958$$ −21.4164 −0.691933
$$959$$ 12.7082 0.410369
$$960$$ 3.32624 0.107354
$$961$$ 14.0000 0.451613
$$962$$ 85.2492 2.74855
$$963$$ 14.1803 0.456955
$$964$$ −40.1459 −1.29301
$$965$$ −7.68692 −0.247451
$$966$$ −11.7082 −0.376705
$$967$$ −14.4164 −0.463600 −0.231800 0.972763i $$-0.574462\pi$$
−0.231800 + 0.972763i $$0.574462\pi$$
$$968$$ 30.9787 0.995694
$$969$$ 32.8885 1.05653
$$970$$ −1.70820 −0.0548471
$$971$$ −53.0132 −1.70127 −0.850637 0.525754i $$-0.823783\pi$$
−0.850637 + 0.525754i $$0.823783\pi$$
$$972$$ −77.6656 −2.49113
$$973$$ 11.1459 0.357321
$$974$$ 60.2148 1.92941
$$975$$ −23.5623 −0.754598
$$976$$ 40.8541 1.30771
$$977$$ 46.7426 1.49543 0.747715 0.664020i $$-0.231151\pi$$
0.747715 + 0.664020i $$0.231151\pi$$
$$978$$ 7.09017 0.226719
$$979$$ 6.00000 0.191761
$$980$$ 11.1246 0.355363
$$981$$ 19.1246 0.610602
$$982$$ −93.6656 −2.98899
$$983$$ 18.6525 0.594922 0.297461 0.954734i $$-0.403860\pi$$
0.297461 + 0.954734i $$0.403860\pi$$
$$984$$ 66.8328 2.13055
$$985$$ 6.27051 0.199795
$$986$$ −77.0132 −2.45260
$$987$$ 4.09017 0.130192
$$988$$ −137.936 −4.38833
$$989$$ −38.9443 −1.23836
$$990$$ 5.23607 0.166413
$$991$$ 24.2705 0.770978 0.385489 0.922712i $$-0.374033\pi$$
0.385489 + 0.922712i $$0.374033\pi$$
$$992$$ 72.8115 2.31177
$$993$$ 16.1459 0.512375
$$994$$ −10.7082 −0.339644
$$995$$ 8.94427 0.283552
$$996$$ 30.7082 0.973027
$$997$$ −39.2918 −1.24438 −0.622192 0.782865i $$-0.713758\pi$$
−0.622192 + 0.782865i $$0.713758\pi$$
$$998$$ −34.7426 −1.09976
$$999$$ 33.5410 1.06119
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 103.2.a.a.1.1 2
3.2 odd 2 927.2.a.b.1.2 2
4.3 odd 2 1648.2.a.f.1.2 2
5.4 even 2 2575.2.a.g.1.2 2
7.6 odd 2 5047.2.a.a.1.1 2
8.3 odd 2 6592.2.a.h.1.1 2
8.5 even 2 6592.2.a.t.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
103.2.a.a.1.1 2 1.1 even 1 trivial
927.2.a.b.1.2 2 3.2 odd 2
1648.2.a.f.1.2 2 4.3 odd 2
2575.2.a.g.1.2 2 5.4 even 2
5047.2.a.a.1.1 2 7.6 odd 2
6592.2.a.h.1.1 2 8.3 odd 2
6592.2.a.t.1.1 2 8.5 even 2