# Properties

 Label 230.2.a.c.1.2 Level $230$ Weight $2$ Character 230.1 Self dual yes Analytic conductor $1.837$ 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] = [230,2,Mod(1,230)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(230, 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("230.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$230 = 2 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 230.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: $$1.83655924649$$ Analytic rank: $$0$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 230.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.61803 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.61803 q^{6} -0.618034 q^{7} +1.00000 q^{8} -0.381966 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.61803 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.61803 q^{6} -0.618034 q^{7} +1.00000 q^{8} -0.381966 q^{9} +1.00000 q^{10} -2.85410 q^{11} +1.61803 q^{12} -7.09017 q^{13} -0.618034 q^{14} +1.61803 q^{15} +1.00000 q^{16} +6.09017 q^{17} -0.381966 q^{18} +1.85410 q^{19} +1.00000 q^{20} -1.00000 q^{21} -2.85410 q^{22} +1.00000 q^{23} +1.61803 q^{24} +1.00000 q^{25} -7.09017 q^{26} -5.47214 q^{27} -0.618034 q^{28} -9.23607 q^{29} +1.61803 q^{30} +9.09017 q^{31} +1.00000 q^{32} -4.61803 q^{33} +6.09017 q^{34} -0.618034 q^{35} -0.381966 q^{36} +6.47214 q^{37} +1.85410 q^{38} -11.4721 q^{39} +1.00000 q^{40} +3.32624 q^{41} -1.00000 q^{42} -2.85410 q^{44} -0.381966 q^{45} +1.00000 q^{46} -3.70820 q^{47} +1.61803 q^{48} -6.61803 q^{49} +1.00000 q^{50} +9.85410 q^{51} -7.09017 q^{52} +0.472136 q^{53} -5.47214 q^{54} -2.85410 q^{55} -0.618034 q^{56} +3.00000 q^{57} -9.23607 q^{58} +1.70820 q^{59} +1.61803 q^{60} -9.32624 q^{61} +9.09017 q^{62} +0.236068 q^{63} +1.00000 q^{64} -7.09017 q^{65} -4.61803 q^{66} +14.4721 q^{67} +6.09017 q^{68} +1.61803 q^{69} -0.618034 q^{70} -4.09017 q^{71} -0.381966 q^{72} +3.23607 q^{73} +6.47214 q^{74} +1.61803 q^{75} +1.85410 q^{76} +1.76393 q^{77} -11.4721 q^{78} +1.52786 q^{79} +1.00000 q^{80} -7.70820 q^{81} +3.32624 q^{82} -6.94427 q^{83} -1.00000 q^{84} +6.09017 q^{85} -14.9443 q^{87} -2.85410 q^{88} -10.4721 q^{89} -0.381966 q^{90} +4.38197 q^{91} +1.00000 q^{92} +14.7082 q^{93} -3.70820 q^{94} +1.85410 q^{95} +1.61803 q^{96} +12.3820 q^{97} -6.61803 q^{98} +1.09017 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + q^{3} + 2 q^{4} + 2 q^{5} + q^{6} + q^{7} + 2 q^{8} - 3 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + q^3 + 2 * q^4 + 2 * q^5 + q^6 + q^7 + 2 * q^8 - 3 * q^9 $$2 q + 2 q^{2} + q^{3} + 2 q^{4} + 2 q^{5} + q^{6} + q^{7} + 2 q^{8} - 3 q^{9} + 2 q^{10} + q^{11} + q^{12} - 3 q^{13} + q^{14} + q^{15} + 2 q^{16} + q^{17} - 3 q^{18} - 3 q^{19} + 2 q^{20} - 2 q^{21} + q^{22} + 2 q^{23} + q^{24} + 2 q^{25} - 3 q^{26} - 2 q^{27} + q^{28} - 14 q^{29} + q^{30} + 7 q^{31} + 2 q^{32} - 7 q^{33} + q^{34} + q^{35} - 3 q^{36} + 4 q^{37} - 3 q^{38} - 14 q^{39} + 2 q^{40} - 9 q^{41} - 2 q^{42} + q^{44} - 3 q^{45} + 2 q^{46} + 6 q^{47} + q^{48} - 11 q^{49} + 2 q^{50} + 13 q^{51} - 3 q^{52} - 8 q^{53} - 2 q^{54} + q^{55} + q^{56} + 6 q^{57} - 14 q^{58} - 10 q^{59} + q^{60} - 3 q^{61} + 7 q^{62} - 4 q^{63} + 2 q^{64} - 3 q^{65} - 7 q^{66} + 20 q^{67} + q^{68} + q^{69} + q^{70} + 3 q^{71} - 3 q^{72} + 2 q^{73} + 4 q^{74} + q^{75} - 3 q^{76} + 8 q^{77} - 14 q^{78} + 12 q^{79} + 2 q^{80} - 2 q^{81} - 9 q^{82} + 4 q^{83} - 2 q^{84} + q^{85} - 12 q^{87} + q^{88} - 12 q^{89} - 3 q^{90} + 11 q^{91} + 2 q^{92} + 16 q^{93} + 6 q^{94} - 3 q^{95} + q^{96} + 27 q^{97} - 11 q^{98} - 9 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + q^3 + 2 * q^4 + 2 * q^5 + q^6 + q^7 + 2 * q^8 - 3 * q^9 + 2 * q^10 + q^11 + q^12 - 3 * q^13 + q^14 + q^15 + 2 * q^16 + q^17 - 3 * q^18 - 3 * q^19 + 2 * q^20 - 2 * q^21 + q^22 + 2 * q^23 + q^24 + 2 * q^25 - 3 * q^26 - 2 * q^27 + q^28 - 14 * q^29 + q^30 + 7 * q^31 + 2 * q^32 - 7 * q^33 + q^34 + q^35 - 3 * q^36 + 4 * q^37 - 3 * q^38 - 14 * q^39 + 2 * q^40 - 9 * q^41 - 2 * q^42 + q^44 - 3 * q^45 + 2 * q^46 + 6 * q^47 + q^48 - 11 * q^49 + 2 * q^50 + 13 * q^51 - 3 * q^52 - 8 * q^53 - 2 * q^54 + q^55 + q^56 + 6 * q^57 - 14 * q^58 - 10 * q^59 + q^60 - 3 * q^61 + 7 * q^62 - 4 * q^63 + 2 * q^64 - 3 * q^65 - 7 * q^66 + 20 * q^67 + q^68 + q^69 + q^70 + 3 * q^71 - 3 * q^72 + 2 * q^73 + 4 * q^74 + q^75 - 3 * q^76 + 8 * q^77 - 14 * q^78 + 12 * q^79 + 2 * q^80 - 2 * q^81 - 9 * q^82 + 4 * q^83 - 2 * q^84 + q^85 - 12 * q^87 + q^88 - 12 * q^89 - 3 * q^90 + 11 * q^91 + 2 * q^92 + 16 * q^93 + 6 * q^94 - 3 * q^95 + q^96 + 27 * q^97 - 11 * q^98 - 9 * 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
$$3$$ 1.61803 0.934172 0.467086 0.884212i $$-0.345304\pi$$
0.467086 + 0.884212i $$0.345304\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ 1.61803 0.660560
$$7$$ −0.618034 −0.233595 −0.116797 0.993156i $$-0.537263\pi$$
−0.116797 + 0.993156i $$0.537263\pi$$
$$8$$ 1.00000 0.353553
$$9$$ −0.381966 −0.127322
$$10$$ 1.00000 0.316228
$$11$$ −2.85410 −0.860544 −0.430272 0.902699i $$-0.641582\pi$$
−0.430272 + 0.902699i $$0.641582\pi$$
$$12$$ 1.61803 0.467086
$$13$$ −7.09017 −1.96646 −0.983230 0.182372i $$-0.941623\pi$$
−0.983230 + 0.182372i $$0.941623\pi$$
$$14$$ −0.618034 −0.165177
$$15$$ 1.61803 0.417775
$$16$$ 1.00000 0.250000
$$17$$ 6.09017 1.47708 0.738542 0.674208i $$-0.235515\pi$$
0.738542 + 0.674208i $$0.235515\pi$$
$$18$$ −0.381966 −0.0900303
$$19$$ 1.85410 0.425360 0.212680 0.977122i $$-0.431781\pi$$
0.212680 + 0.977122i $$0.431781\pi$$
$$20$$ 1.00000 0.223607
$$21$$ −1.00000 −0.218218
$$22$$ −2.85410 −0.608497
$$23$$ 1.00000 0.208514
$$24$$ 1.61803 0.330280
$$25$$ 1.00000 0.200000
$$26$$ −7.09017 −1.39050
$$27$$ −5.47214 −1.05311
$$28$$ −0.618034 −0.116797
$$29$$ −9.23607 −1.71509 −0.857547 0.514405i $$-0.828013\pi$$
−0.857547 + 0.514405i $$0.828013\pi$$
$$30$$ 1.61803 0.295411
$$31$$ 9.09017 1.63264 0.816321 0.577598i $$-0.196010\pi$$
0.816321 + 0.577598i $$0.196010\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −4.61803 −0.803897
$$34$$ 6.09017 1.04446
$$35$$ −0.618034 −0.104467
$$36$$ −0.381966 −0.0636610
$$37$$ 6.47214 1.06401 0.532006 0.846740i $$-0.321438\pi$$
0.532006 + 0.846740i $$0.321438\pi$$
$$38$$ 1.85410 0.300775
$$39$$ −11.4721 −1.83701
$$40$$ 1.00000 0.158114
$$41$$ 3.32624 0.519471 0.259736 0.965680i $$-0.416365\pi$$
0.259736 + 0.965680i $$0.416365\pi$$
$$42$$ −1.00000 −0.154303
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ −2.85410 −0.430272
$$45$$ −0.381966 −0.0569401
$$46$$ 1.00000 0.147442
$$47$$ −3.70820 −0.540897 −0.270449 0.962734i $$-0.587172\pi$$
−0.270449 + 0.962734i $$0.587172\pi$$
$$48$$ 1.61803 0.233543
$$49$$ −6.61803 −0.945433
$$50$$ 1.00000 0.141421
$$51$$ 9.85410 1.37985
$$52$$ −7.09017 −0.983230
$$53$$ 0.472136 0.0648529 0.0324264 0.999474i $$-0.489677\pi$$
0.0324264 + 0.999474i $$0.489677\pi$$
$$54$$ −5.47214 −0.744663
$$55$$ −2.85410 −0.384847
$$56$$ −0.618034 −0.0825883
$$57$$ 3.00000 0.397360
$$58$$ −9.23607 −1.21276
$$59$$ 1.70820 0.222389 0.111195 0.993799i $$-0.464532\pi$$
0.111195 + 0.993799i $$0.464532\pi$$
$$60$$ 1.61803 0.208887
$$61$$ −9.32624 −1.19410 −0.597051 0.802203i $$-0.703661\pi$$
−0.597051 + 0.802203i $$0.703661\pi$$
$$62$$ 9.09017 1.15445
$$63$$ 0.236068 0.0297418
$$64$$ 1.00000 0.125000
$$65$$ −7.09017 −0.879427
$$66$$ −4.61803 −0.568441
$$67$$ 14.4721 1.76805 0.884026 0.467437i $$-0.154823\pi$$
0.884026 + 0.467437i $$0.154823\pi$$
$$68$$ 6.09017 0.738542
$$69$$ 1.61803 0.194788
$$70$$ −0.618034 −0.0738692
$$71$$ −4.09017 −0.485414 −0.242707 0.970100i $$-0.578035\pi$$
−0.242707 + 0.970100i $$0.578035\pi$$
$$72$$ −0.381966 −0.0450151
$$73$$ 3.23607 0.378753 0.189377 0.981905i $$-0.439353\pi$$
0.189377 + 0.981905i $$0.439353\pi$$
$$74$$ 6.47214 0.752371
$$75$$ 1.61803 0.186834
$$76$$ 1.85410 0.212680
$$77$$ 1.76393 0.201019
$$78$$ −11.4721 −1.29896
$$79$$ 1.52786 0.171898 0.0859491 0.996300i $$-0.472608\pi$$
0.0859491 + 0.996300i $$0.472608\pi$$
$$80$$ 1.00000 0.111803
$$81$$ −7.70820 −0.856467
$$82$$ 3.32624 0.367322
$$83$$ −6.94427 −0.762233 −0.381116 0.924527i $$-0.624460\pi$$
−0.381116 + 0.924527i $$0.624460\pi$$
$$84$$ −1.00000 −0.109109
$$85$$ 6.09017 0.660572
$$86$$ 0 0
$$87$$ −14.9443 −1.60219
$$88$$ −2.85410 −0.304248
$$89$$ −10.4721 −1.11004 −0.555022 0.831836i $$-0.687290\pi$$
−0.555022 + 0.831836i $$0.687290\pi$$
$$90$$ −0.381966 −0.0402628
$$91$$ 4.38197 0.459355
$$92$$ 1.00000 0.104257
$$93$$ 14.7082 1.52517
$$94$$ −3.70820 −0.382472
$$95$$ 1.85410 0.190227
$$96$$ 1.61803 0.165140
$$97$$ 12.3820 1.25720 0.628599 0.777730i $$-0.283629\pi$$
0.628599 + 0.777730i $$0.283629\pi$$
$$98$$ −6.61803 −0.668522
$$99$$ 1.09017 0.109566
$$100$$ 1.00000 0.100000
$$101$$ −0.291796 −0.0290348 −0.0145174 0.999895i $$-0.504621\pi$$
−0.0145174 + 0.999895i $$0.504621\pi$$
$$102$$ 9.85410 0.975701
$$103$$ 16.5623 1.63193 0.815966 0.578100i $$-0.196205\pi$$
0.815966 + 0.578100i $$0.196205\pi$$
$$104$$ −7.09017 −0.695248
$$105$$ −1.00000 −0.0975900
$$106$$ 0.472136 0.0458579
$$107$$ 18.1803 1.75756 0.878780 0.477227i $$-0.158358\pi$$
0.878780 + 0.477227i $$0.158358\pi$$
$$108$$ −5.47214 −0.526557
$$109$$ −11.5623 −1.10747 −0.553734 0.832694i $$-0.686798\pi$$
−0.553734 + 0.832694i $$0.686798\pi$$
$$110$$ −2.85410 −0.272128
$$111$$ 10.4721 0.993971
$$112$$ −0.618034 −0.0583987
$$113$$ 1.05573 0.0993145 0.0496573 0.998766i $$-0.484187\pi$$
0.0496573 + 0.998766i $$0.484187\pi$$
$$114$$ 3.00000 0.280976
$$115$$ 1.00000 0.0932505
$$116$$ −9.23607 −0.857547
$$117$$ 2.70820 0.250374
$$118$$ 1.70820 0.157253
$$119$$ −3.76393 −0.345039
$$120$$ 1.61803 0.147706
$$121$$ −2.85410 −0.259464
$$122$$ −9.32624 −0.844358
$$123$$ 5.38197 0.485276
$$124$$ 9.09017 0.816321
$$125$$ 1.00000 0.0894427
$$126$$ 0.236068 0.0210306
$$127$$ 16.1803 1.43577 0.717886 0.696160i $$-0.245110\pi$$
0.717886 + 0.696160i $$0.245110\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ −7.09017 −0.621849
$$131$$ 2.94427 0.257242 0.128621 0.991694i $$-0.458945\pi$$
0.128621 + 0.991694i $$0.458945\pi$$
$$132$$ −4.61803 −0.401948
$$133$$ −1.14590 −0.0993620
$$134$$ 14.4721 1.25020
$$135$$ −5.47214 −0.470966
$$136$$ 6.09017 0.522228
$$137$$ −10.3262 −0.882230 −0.441115 0.897451i $$-0.645417\pi$$
−0.441115 + 0.897451i $$0.645417\pi$$
$$138$$ 1.61803 0.137736
$$139$$ 12.7639 1.08262 0.541311 0.840822i $$-0.317928\pi$$
0.541311 + 0.840822i $$0.317928\pi$$
$$140$$ −0.618034 −0.0522334
$$141$$ −6.00000 −0.505291
$$142$$ −4.09017 −0.343239
$$143$$ 20.2361 1.69223
$$144$$ −0.381966 −0.0318305
$$145$$ −9.23607 −0.767014
$$146$$ 3.23607 0.267819
$$147$$ −10.7082 −0.883198
$$148$$ 6.47214 0.532006
$$149$$ −7.85410 −0.643433 −0.321717 0.946836i $$-0.604260\pi$$
−0.321717 + 0.946836i $$0.604260\pi$$
$$150$$ 1.61803 0.132112
$$151$$ −2.56231 −0.208517 −0.104259 0.994550i $$-0.533247\pi$$
−0.104259 + 0.994550i $$0.533247\pi$$
$$152$$ 1.85410 0.150388
$$153$$ −2.32624 −0.188065
$$154$$ 1.76393 0.142142
$$155$$ 9.09017 0.730140
$$156$$ −11.4721 −0.918506
$$157$$ −3.70820 −0.295947 −0.147973 0.988991i $$-0.547275\pi$$
−0.147973 + 0.988991i $$0.547275\pi$$
$$158$$ 1.52786 0.121550
$$159$$ 0.763932 0.0605838
$$160$$ 1.00000 0.0790569
$$161$$ −0.618034 −0.0487079
$$162$$ −7.70820 −0.605614
$$163$$ 1.38197 0.108244 0.0541220 0.998534i $$-0.482764\pi$$
0.0541220 + 0.998534i $$0.482764\pi$$
$$164$$ 3.32624 0.259736
$$165$$ −4.61803 −0.359513
$$166$$ −6.94427 −0.538980
$$167$$ −8.00000 −0.619059 −0.309529 0.950890i $$-0.600171\pi$$
−0.309529 + 0.950890i $$0.600171\pi$$
$$168$$ −1.00000 −0.0771517
$$169$$ 37.2705 2.86696
$$170$$ 6.09017 0.467095
$$171$$ −0.708204 −0.0541577
$$172$$ 0 0
$$173$$ −1.43769 −0.109306 −0.0546529 0.998505i $$-0.517405\pi$$
−0.0546529 + 0.998505i $$0.517405\pi$$
$$174$$ −14.9443 −1.13292
$$175$$ −0.618034 −0.0467190
$$176$$ −2.85410 −0.215136
$$177$$ 2.76393 0.207750
$$178$$ −10.4721 −0.784920
$$179$$ 2.18034 0.162966 0.0814831 0.996675i $$-0.474034\pi$$
0.0814831 + 0.996675i $$0.474034\pi$$
$$180$$ −0.381966 −0.0284701
$$181$$ −12.1459 −0.902797 −0.451399 0.892322i $$-0.649075\pi$$
−0.451399 + 0.892322i $$0.649075\pi$$
$$182$$ 4.38197 0.324813
$$183$$ −15.0902 −1.11550
$$184$$ 1.00000 0.0737210
$$185$$ 6.47214 0.475841
$$186$$ 14.7082 1.07846
$$187$$ −17.3820 −1.27110
$$188$$ −3.70820 −0.270449
$$189$$ 3.38197 0.246002
$$190$$ 1.85410 0.134511
$$191$$ −13.7082 −0.991891 −0.495945 0.868354i $$-0.665178\pi$$
−0.495945 + 0.868354i $$0.665178\pi$$
$$192$$ 1.61803 0.116772
$$193$$ 0.763932 0.0549890 0.0274945 0.999622i $$-0.491247\pi$$
0.0274945 + 0.999622i $$0.491247\pi$$
$$194$$ 12.3820 0.888973
$$195$$ −11.4721 −0.821537
$$196$$ −6.61803 −0.472717
$$197$$ −22.5623 −1.60750 −0.803749 0.594969i $$-0.797164\pi$$
−0.803749 + 0.594969i $$0.797164\pi$$
$$198$$ 1.09017 0.0774750
$$199$$ 2.00000 0.141776 0.0708881 0.997484i $$-0.477417\pi$$
0.0708881 + 0.997484i $$0.477417\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 23.4164 1.65167
$$202$$ −0.291796 −0.0205307
$$203$$ 5.70820 0.400637
$$204$$ 9.85410 0.689925
$$205$$ 3.32624 0.232315
$$206$$ 16.5623 1.15395
$$207$$ −0.381966 −0.0265485
$$208$$ −7.09017 −0.491615
$$209$$ −5.29180 −0.366041
$$210$$ −1.00000 −0.0690066
$$211$$ 14.0000 0.963800 0.481900 0.876226i $$-0.339947\pi$$
0.481900 + 0.876226i $$0.339947\pi$$
$$212$$ 0.472136 0.0324264
$$213$$ −6.61803 −0.453460
$$214$$ 18.1803 1.24278
$$215$$ 0 0
$$216$$ −5.47214 −0.372332
$$217$$ −5.61803 −0.381377
$$218$$ −11.5623 −0.783098
$$219$$ 5.23607 0.353821
$$220$$ −2.85410 −0.192424
$$221$$ −43.1803 −2.90462
$$222$$ 10.4721 0.702844
$$223$$ −20.9443 −1.40253 −0.701266 0.712900i $$-0.747382\pi$$
−0.701266 + 0.712900i $$0.747382\pi$$
$$224$$ −0.618034 −0.0412941
$$225$$ −0.381966 −0.0254644
$$226$$ 1.05573 0.0702260
$$227$$ −18.7639 −1.24541 −0.622703 0.782458i $$-0.713965\pi$$
−0.622703 + 0.782458i $$0.713965\pi$$
$$228$$ 3.00000 0.198680
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ 1.00000 0.0659380
$$231$$ 2.85410 0.187786
$$232$$ −9.23607 −0.606378
$$233$$ 6.29180 0.412189 0.206095 0.978532i $$-0.433925\pi$$
0.206095 + 0.978532i $$0.433925\pi$$
$$234$$ 2.70820 0.177041
$$235$$ −3.70820 −0.241897
$$236$$ 1.70820 0.111195
$$237$$ 2.47214 0.160582
$$238$$ −3.76393 −0.243979
$$239$$ −20.3607 −1.31702 −0.658511 0.752571i $$-0.728814\pi$$
−0.658511 + 0.752571i $$0.728814\pi$$
$$240$$ 1.61803 0.104444
$$241$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$242$$ −2.85410 −0.183469
$$243$$ 3.94427 0.253025
$$244$$ −9.32624 −0.597051
$$245$$ −6.61803 −0.422811
$$246$$ 5.38197 0.343142
$$247$$ −13.1459 −0.836453
$$248$$ 9.09017 0.577226
$$249$$ −11.2361 −0.712057
$$250$$ 1.00000 0.0632456
$$251$$ 6.14590 0.387926 0.193963 0.981009i $$-0.437866\pi$$
0.193963 + 0.981009i $$0.437866\pi$$
$$252$$ 0.236068 0.0148709
$$253$$ −2.85410 −0.179436
$$254$$ 16.1803 1.01524
$$255$$ 9.85410 0.617088
$$256$$ 1.00000 0.0625000
$$257$$ −7.81966 −0.487777 −0.243888 0.969803i $$-0.578423\pi$$
−0.243888 + 0.969803i $$0.578423\pi$$
$$258$$ 0 0
$$259$$ −4.00000 −0.248548
$$260$$ −7.09017 −0.439714
$$261$$ 3.52786 0.218369
$$262$$ 2.94427 0.181898
$$263$$ 20.7426 1.27905 0.639523 0.768772i $$-0.279132\pi$$
0.639523 + 0.768772i $$0.279132\pi$$
$$264$$ −4.61803 −0.284220
$$265$$ 0.472136 0.0290031
$$266$$ −1.14590 −0.0702595
$$267$$ −16.9443 −1.03697
$$268$$ 14.4721 0.884026
$$269$$ −14.1803 −0.864591 −0.432295 0.901732i $$-0.642296\pi$$
−0.432295 + 0.901732i $$0.642296\pi$$
$$270$$ −5.47214 −0.333024
$$271$$ −30.3262 −1.84219 −0.921094 0.389341i $$-0.872703\pi$$
−0.921094 + 0.389341i $$0.872703\pi$$
$$272$$ 6.09017 0.369271
$$273$$ 7.09017 0.429117
$$274$$ −10.3262 −0.623831
$$275$$ −2.85410 −0.172109
$$276$$ 1.61803 0.0973942
$$277$$ 29.4164 1.76746 0.883730 0.467996i $$-0.155024\pi$$
0.883730 + 0.467996i $$0.155024\pi$$
$$278$$ 12.7639 0.765530
$$279$$ −3.47214 −0.207871
$$280$$ −0.618034 −0.0369346
$$281$$ 22.7639 1.35798 0.678991 0.734146i $$-0.262417\pi$$
0.678991 + 0.734146i $$0.262417\pi$$
$$282$$ −6.00000 −0.357295
$$283$$ −26.9443 −1.60167 −0.800835 0.598885i $$-0.795611\pi$$
−0.800835 + 0.598885i $$0.795611\pi$$
$$284$$ −4.09017 −0.242707
$$285$$ 3.00000 0.177705
$$286$$ 20.2361 1.19658
$$287$$ −2.05573 −0.121346
$$288$$ −0.381966 −0.0225076
$$289$$ 20.0902 1.18177
$$290$$ −9.23607 −0.542361
$$291$$ 20.0344 1.17444
$$292$$ 3.23607 0.189377
$$293$$ −19.8885 −1.16190 −0.580951 0.813939i $$-0.697319\pi$$
−0.580951 + 0.813939i $$0.697319\pi$$
$$294$$ −10.7082 −0.624515
$$295$$ 1.70820 0.0994555
$$296$$ 6.47214 0.376185
$$297$$ 15.6180 0.906250
$$298$$ −7.85410 −0.454976
$$299$$ −7.09017 −0.410035
$$300$$ 1.61803 0.0934172
$$301$$ 0 0
$$302$$ −2.56231 −0.147444
$$303$$ −0.472136 −0.0271235
$$304$$ 1.85410 0.106340
$$305$$ −9.32624 −0.534019
$$306$$ −2.32624 −0.132982
$$307$$ −28.4508 −1.62378 −0.811888 0.583813i $$-0.801560\pi$$
−0.811888 + 0.583813i $$0.801560\pi$$
$$308$$ 1.76393 0.100509
$$309$$ 26.7984 1.52451
$$310$$ 9.09017 0.516287
$$311$$ 4.00000 0.226819 0.113410 0.993548i $$-0.463823\pi$$
0.113410 + 0.993548i $$0.463823\pi$$
$$312$$ −11.4721 −0.649482
$$313$$ 12.7984 0.723407 0.361703 0.932293i $$-0.382195\pi$$
0.361703 + 0.932293i $$0.382195\pi$$
$$314$$ −3.70820 −0.209266
$$315$$ 0.236068 0.0133009
$$316$$ 1.52786 0.0859491
$$317$$ −11.0902 −0.622886 −0.311443 0.950265i $$-0.600812\pi$$
−0.311443 + 0.950265i $$0.600812\pi$$
$$318$$ 0.763932 0.0428392
$$319$$ 26.3607 1.47591
$$320$$ 1.00000 0.0559017
$$321$$ 29.4164 1.64186
$$322$$ −0.618034 −0.0344417
$$323$$ 11.2918 0.628292
$$324$$ −7.70820 −0.428234
$$325$$ −7.09017 −0.393292
$$326$$ 1.38197 0.0765400
$$327$$ −18.7082 −1.03457
$$328$$ 3.32624 0.183661
$$329$$ 2.29180 0.126351
$$330$$ −4.61803 −0.254214
$$331$$ 19.2361 1.05731 0.528655 0.848837i $$-0.322697\pi$$
0.528655 + 0.848837i $$0.322697\pi$$
$$332$$ −6.94427 −0.381116
$$333$$ −2.47214 −0.135472
$$334$$ −8.00000 −0.437741
$$335$$ 14.4721 0.790697
$$336$$ −1.00000 −0.0545545
$$337$$ 13.6738 0.744857 0.372429 0.928061i $$-0.378525\pi$$
0.372429 + 0.928061i $$0.378525\pi$$
$$338$$ 37.2705 2.02725
$$339$$ 1.70820 0.0927769
$$340$$ 6.09017 0.330286
$$341$$ −25.9443 −1.40496
$$342$$ −0.708204 −0.0382953
$$343$$ 8.41641 0.454443
$$344$$ 0 0
$$345$$ 1.61803 0.0871120
$$346$$ −1.43769 −0.0772909
$$347$$ −6.38197 −0.342602 −0.171301 0.985219i $$-0.554797\pi$$
−0.171301 + 0.985219i $$0.554797\pi$$
$$348$$ −14.9443 −0.801097
$$349$$ −2.00000 −0.107058 −0.0535288 0.998566i $$-0.517047\pi$$
−0.0535288 + 0.998566i $$0.517047\pi$$
$$350$$ −0.618034 −0.0330353
$$351$$ 38.7984 2.07090
$$352$$ −2.85410 −0.152124
$$353$$ 24.0000 1.27739 0.638696 0.769460i $$-0.279474\pi$$
0.638696 + 0.769460i $$0.279474\pi$$
$$354$$ 2.76393 0.146901
$$355$$ −4.09017 −0.217084
$$356$$ −10.4721 −0.555022
$$357$$ −6.09017 −0.322326
$$358$$ 2.18034 0.115235
$$359$$ 26.3607 1.39126 0.695632 0.718399i $$-0.255125\pi$$
0.695632 + 0.718399i $$0.255125\pi$$
$$360$$ −0.381966 −0.0201314
$$361$$ −15.5623 −0.819069
$$362$$ −12.1459 −0.638374
$$363$$ −4.61803 −0.242384
$$364$$ 4.38197 0.229677
$$365$$ 3.23607 0.169384
$$366$$ −15.0902 −0.788776
$$367$$ 6.47214 0.337843 0.168921 0.985630i $$-0.445972\pi$$
0.168921 + 0.985630i $$0.445972\pi$$
$$368$$ 1.00000 0.0521286
$$369$$ −1.27051 −0.0661401
$$370$$ 6.47214 0.336470
$$371$$ −0.291796 −0.0151493
$$372$$ 14.7082 0.762585
$$373$$ 20.1803 1.04490 0.522449 0.852670i $$-0.325018\pi$$
0.522449 + 0.852670i $$0.325018\pi$$
$$374$$ −17.3820 −0.898800
$$375$$ 1.61803 0.0835549
$$376$$ −3.70820 −0.191236
$$377$$ 65.4853 3.37266
$$378$$ 3.38197 0.173950
$$379$$ −22.4508 −1.15322 −0.576611 0.817019i $$-0.695625\pi$$
−0.576611 + 0.817019i $$0.695625\pi$$
$$380$$ 1.85410 0.0951134
$$381$$ 26.1803 1.34126
$$382$$ −13.7082 −0.701373
$$383$$ −17.8885 −0.914062 −0.457031 0.889451i $$-0.651087\pi$$
−0.457031 + 0.889451i $$0.651087\pi$$
$$384$$ 1.61803 0.0825700
$$385$$ 1.76393 0.0898983
$$386$$ 0.763932 0.0388831
$$387$$ 0 0
$$388$$ 12.3820 0.628599
$$389$$ 21.3262 1.08128 0.540642 0.841253i $$-0.318182\pi$$
0.540642 + 0.841253i $$0.318182\pi$$
$$390$$ −11.4721 −0.580914
$$391$$ 6.09017 0.307993
$$392$$ −6.61803 −0.334261
$$393$$ 4.76393 0.240309
$$394$$ −22.5623 −1.13667
$$395$$ 1.52786 0.0768752
$$396$$ 1.09017 0.0547831
$$397$$ 7.32624 0.367693 0.183847 0.982955i $$-0.441145\pi$$
0.183847 + 0.982955i $$0.441145\pi$$
$$398$$ 2.00000 0.100251
$$399$$ −1.85410 −0.0928212
$$400$$ 1.00000 0.0500000
$$401$$ −1.70820 −0.0853036 −0.0426518 0.999090i $$-0.513581\pi$$
−0.0426518 + 0.999090i $$0.513581\pi$$
$$402$$ 23.4164 1.16790
$$403$$ −64.4508 −3.21053
$$404$$ −0.291796 −0.0145174
$$405$$ −7.70820 −0.383024
$$406$$ 5.70820 0.283293
$$407$$ −18.4721 −0.915630
$$408$$ 9.85410 0.487851
$$409$$ −30.2148 −1.49402 −0.747012 0.664810i $$-0.768512\pi$$
−0.747012 + 0.664810i $$0.768512\pi$$
$$410$$ 3.32624 0.164271
$$411$$ −16.7082 −0.824155
$$412$$ 16.5623 0.815966
$$413$$ −1.05573 −0.0519490
$$414$$ −0.381966 −0.0187726
$$415$$ −6.94427 −0.340881
$$416$$ −7.09017 −0.347624
$$417$$ 20.6525 1.01136
$$418$$ −5.29180 −0.258830
$$419$$ 14.4721 0.707010 0.353505 0.935433i $$-0.384990\pi$$
0.353505 + 0.935433i $$0.384990\pi$$
$$420$$ −1.00000 −0.0487950
$$421$$ 13.7426 0.669776 0.334888 0.942258i $$-0.391302\pi$$
0.334888 + 0.942258i $$0.391302\pi$$
$$422$$ 14.0000 0.681509
$$423$$ 1.41641 0.0688681
$$424$$ 0.472136 0.0229289
$$425$$ 6.09017 0.295417
$$426$$ −6.61803 −0.320645
$$427$$ 5.76393 0.278936
$$428$$ 18.1803 0.878780
$$429$$ 32.7426 1.58083
$$430$$ 0 0
$$431$$ 3.34752 0.161245 0.0806223 0.996745i $$-0.474309\pi$$
0.0806223 + 0.996745i $$0.474309\pi$$
$$432$$ −5.47214 −0.263278
$$433$$ 8.50658 0.408800 0.204400 0.978887i $$-0.434476\pi$$
0.204400 + 0.978887i $$0.434476\pi$$
$$434$$ −5.61803 −0.269674
$$435$$ −14.9443 −0.716523
$$436$$ −11.5623 −0.553734
$$437$$ 1.85410 0.0886937
$$438$$ 5.23607 0.250189
$$439$$ −13.3820 −0.638686 −0.319343 0.947639i $$-0.603462\pi$$
−0.319343 + 0.947639i $$0.603462\pi$$
$$440$$ −2.85410 −0.136064
$$441$$ 2.52786 0.120374
$$442$$ −43.1803 −2.05388
$$443$$ 25.0902 1.19207 0.596035 0.802958i $$-0.296742\pi$$
0.596035 + 0.802958i $$0.296742\pi$$
$$444$$ 10.4721 0.496986
$$445$$ −10.4721 −0.496427
$$446$$ −20.9443 −0.991740
$$447$$ −12.7082 −0.601077
$$448$$ −0.618034 −0.0291994
$$449$$ −1.56231 −0.0737298 −0.0368649 0.999320i $$-0.511737\pi$$
−0.0368649 + 0.999320i $$0.511737\pi$$
$$450$$ −0.381966 −0.0180061
$$451$$ −9.49342 −0.447028
$$452$$ 1.05573 0.0496573
$$453$$ −4.14590 −0.194791
$$454$$ −18.7639 −0.880635
$$455$$ 4.38197 0.205430
$$456$$ 3.00000 0.140488
$$457$$ −37.7771 −1.76714 −0.883569 0.468301i $$-0.844866\pi$$
−0.883569 + 0.468301i $$0.844866\pi$$
$$458$$ 10.0000 0.467269
$$459$$ −33.3262 −1.55554
$$460$$ 1.00000 0.0466252
$$461$$ −39.2361 −1.82741 −0.913703 0.406383i $$-0.866790\pi$$
−0.913703 + 0.406383i $$0.866790\pi$$
$$462$$ 2.85410 0.132785
$$463$$ 2.00000 0.0929479 0.0464739 0.998920i $$-0.485202\pi$$
0.0464739 + 0.998920i $$0.485202\pi$$
$$464$$ −9.23607 −0.428774
$$465$$ 14.7082 0.682077
$$466$$ 6.29180 0.291462
$$467$$ −17.1246 −0.792433 −0.396216 0.918157i $$-0.629677\pi$$
−0.396216 + 0.918157i $$0.629677\pi$$
$$468$$ 2.70820 0.125187
$$469$$ −8.94427 −0.413008
$$470$$ −3.70820 −0.171047
$$471$$ −6.00000 −0.276465
$$472$$ 1.70820 0.0786265
$$473$$ 0 0
$$474$$ 2.47214 0.113549
$$475$$ 1.85410 0.0850720
$$476$$ −3.76393 −0.172520
$$477$$ −0.180340 −0.00825720
$$478$$ −20.3607 −0.931276
$$479$$ −31.8885 −1.45702 −0.728512 0.685033i $$-0.759788\pi$$
−0.728512 + 0.685033i $$0.759788\pi$$
$$480$$ 1.61803 0.0738528
$$481$$ −45.8885 −2.09234
$$482$$ 0 0
$$483$$ −1.00000 −0.0455016
$$484$$ −2.85410 −0.129732
$$485$$ 12.3820 0.562236
$$486$$ 3.94427 0.178916
$$487$$ −19.8197 −0.898115 −0.449057 0.893503i $$-0.648240\pi$$
−0.449057 + 0.893503i $$0.648240\pi$$
$$488$$ −9.32624 −0.422179
$$489$$ 2.23607 0.101118
$$490$$ −6.61803 −0.298972
$$491$$ 6.18034 0.278915 0.139457 0.990228i $$-0.455464\pi$$
0.139457 + 0.990228i $$0.455464\pi$$
$$492$$ 5.38197 0.242638
$$493$$ −56.2492 −2.53334
$$494$$ −13.1459 −0.591462
$$495$$ 1.09017 0.0489995
$$496$$ 9.09017 0.408161
$$497$$ 2.52786 0.113390
$$498$$ −11.2361 −0.503500
$$499$$ 12.3607 0.553340 0.276670 0.960965i $$-0.410769\pi$$
0.276670 + 0.960965i $$0.410769\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ −12.9443 −0.578307
$$502$$ 6.14590 0.274305
$$503$$ 36.3262 1.61971 0.809853 0.586632i $$-0.199547\pi$$
0.809853 + 0.586632i $$0.199547\pi$$
$$504$$ 0.236068 0.0105153
$$505$$ −0.291796 −0.0129848
$$506$$ −2.85410 −0.126880
$$507$$ 60.3050 2.67824
$$508$$ 16.1803 0.717886
$$509$$ 36.6525 1.62459 0.812296 0.583245i $$-0.198217\pi$$
0.812296 + 0.583245i $$0.198217\pi$$
$$510$$ 9.85410 0.436347
$$511$$ −2.00000 −0.0884748
$$512$$ 1.00000 0.0441942
$$513$$ −10.1459 −0.447952
$$514$$ −7.81966 −0.344910
$$515$$ 16.5623 0.729822
$$516$$ 0 0
$$517$$ 10.5836 0.465466
$$518$$ −4.00000 −0.175750
$$519$$ −2.32624 −0.102111
$$520$$ −7.09017 −0.310925
$$521$$ −15.5279 −0.680288 −0.340144 0.940373i $$-0.610476\pi$$
−0.340144 + 0.940373i $$0.610476\pi$$
$$522$$ 3.52786 0.154410
$$523$$ −26.0000 −1.13690 −0.568450 0.822718i $$-0.692457\pi$$
−0.568450 + 0.822718i $$0.692457\pi$$
$$524$$ 2.94427 0.128621
$$525$$ −1.00000 −0.0436436
$$526$$ 20.7426 0.904422
$$527$$ 55.3607 2.41155
$$528$$ −4.61803 −0.200974
$$529$$ 1.00000 0.0434783
$$530$$ 0.472136 0.0205083
$$531$$ −0.652476 −0.0283150
$$532$$ −1.14590 −0.0496810
$$533$$ −23.5836 −1.02152
$$534$$ −16.9443 −0.733250
$$535$$ 18.1803 0.786005
$$536$$ 14.4721 0.625101
$$537$$ 3.52786 0.152239
$$538$$ −14.1803 −0.611358
$$539$$ 18.8885 0.813587
$$540$$ −5.47214 −0.235483
$$541$$ 22.8328 0.981659 0.490830 0.871256i $$-0.336694\pi$$
0.490830 + 0.871256i $$0.336694\pi$$
$$542$$ −30.3262 −1.30262
$$543$$ −19.6525 −0.843368
$$544$$ 6.09017 0.261114
$$545$$ −11.5623 −0.495275
$$546$$ 7.09017 0.303431
$$547$$ 27.9230 1.19390 0.596950 0.802278i $$-0.296379\pi$$
0.596950 + 0.802278i $$0.296379\pi$$
$$548$$ −10.3262 −0.441115
$$549$$ 3.56231 0.152036
$$550$$ −2.85410 −0.121699
$$551$$ −17.1246 −0.729533
$$552$$ 1.61803 0.0688681
$$553$$ −0.944272 −0.0401545
$$554$$ 29.4164 1.24978
$$555$$ 10.4721 0.444517
$$556$$ 12.7639 0.541311
$$557$$ −22.8328 −0.967457 −0.483729 0.875218i $$-0.660718\pi$$
−0.483729 + 0.875218i $$0.660718\pi$$
$$558$$ −3.47214 −0.146987
$$559$$ 0 0
$$560$$ −0.618034 −0.0261167
$$561$$ −28.1246 −1.18742
$$562$$ 22.7639 0.960239
$$563$$ −13.8885 −0.585332 −0.292666 0.956215i $$-0.594542\pi$$
−0.292666 + 0.956215i $$0.594542\pi$$
$$564$$ −6.00000 −0.252646
$$565$$ 1.05573 0.0444148
$$566$$ −26.9443 −1.13255
$$567$$ 4.76393 0.200066
$$568$$ −4.09017 −0.171620
$$569$$ −2.00000 −0.0838444 −0.0419222 0.999121i $$-0.513348\pi$$
−0.0419222 + 0.999121i $$0.513348\pi$$
$$570$$ 3.00000 0.125656
$$571$$ 15.9787 0.668688 0.334344 0.942451i $$-0.391485\pi$$
0.334344 + 0.942451i $$0.391485\pi$$
$$572$$ 20.2361 0.846113
$$573$$ −22.1803 −0.926597
$$574$$ −2.05573 −0.0858044
$$575$$ 1.00000 0.0417029
$$576$$ −0.381966 −0.0159153
$$577$$ −3.52786 −0.146867 −0.0734335 0.997300i $$-0.523396\pi$$
−0.0734335 + 0.997300i $$0.523396\pi$$
$$578$$ 20.0902 0.835641
$$579$$ 1.23607 0.0513692
$$580$$ −9.23607 −0.383507
$$581$$ 4.29180 0.178054
$$582$$ 20.0344 0.830454
$$583$$ −1.34752 −0.0558087
$$584$$ 3.23607 0.133909
$$585$$ 2.70820 0.111970
$$586$$ −19.8885 −0.821588
$$587$$ 13.6180 0.562076 0.281038 0.959697i $$-0.409321\pi$$
0.281038 + 0.959697i $$0.409321\pi$$
$$588$$ −10.7082 −0.441599
$$589$$ 16.8541 0.694461
$$590$$ 1.70820 0.0703256
$$591$$ −36.5066 −1.50168
$$592$$ 6.47214 0.266003
$$593$$ 39.2361 1.61123 0.805616 0.592438i $$-0.201834\pi$$
0.805616 + 0.592438i $$0.201834\pi$$
$$594$$ 15.6180 0.640816
$$595$$ −3.76393 −0.154306
$$596$$ −7.85410 −0.321717
$$597$$ 3.23607 0.132443
$$598$$ −7.09017 −0.289939
$$599$$ −18.3820 −0.751067 −0.375533 0.926809i $$-0.622540\pi$$
−0.375533 + 0.926809i $$0.622540\pi$$
$$600$$ 1.61803 0.0660560
$$601$$ −33.2705 −1.35713 −0.678566 0.734539i $$-0.737398\pi$$
−0.678566 + 0.734539i $$0.737398\pi$$
$$602$$ 0 0
$$603$$ −5.52786 −0.225112
$$604$$ −2.56231 −0.104259
$$605$$ −2.85410 −0.116036
$$606$$ −0.472136 −0.0191792
$$607$$ 26.4721 1.07447 0.537235 0.843432i $$-0.319469\pi$$
0.537235 + 0.843432i $$0.319469\pi$$
$$608$$ 1.85410 0.0751938
$$609$$ 9.23607 0.374264
$$610$$ −9.32624 −0.377608
$$611$$ 26.2918 1.06365
$$612$$ −2.32624 −0.0940326
$$613$$ 19.3050 0.779720 0.389860 0.920874i $$-0.372523\pi$$
0.389860 + 0.920874i $$0.372523\pi$$
$$614$$ −28.4508 −1.14818
$$615$$ 5.38197 0.217022
$$616$$ 1.76393 0.0710708
$$617$$ 34.0902 1.37242 0.686209 0.727404i $$-0.259273\pi$$
0.686209 + 0.727404i $$0.259273\pi$$
$$618$$ 26.7984 1.07799
$$619$$ −2.79837 −0.112476 −0.0562381 0.998417i $$-0.517911\pi$$
−0.0562381 + 0.998417i $$0.517911\pi$$
$$620$$ 9.09017 0.365070
$$621$$ −5.47214 −0.219589
$$622$$ 4.00000 0.160385
$$623$$ 6.47214 0.259301
$$624$$ −11.4721 −0.459253
$$625$$ 1.00000 0.0400000
$$626$$ 12.7984 0.511526
$$627$$ −8.56231 −0.341946
$$628$$ −3.70820 −0.147973
$$629$$ 39.4164 1.57164
$$630$$ 0.236068 0.00940517
$$631$$ 42.0689 1.67474 0.837368 0.546640i $$-0.184093\pi$$
0.837368 + 0.546640i $$0.184093\pi$$
$$632$$ 1.52786 0.0607752
$$633$$ 22.6525 0.900355
$$634$$ −11.0902 −0.440447
$$635$$ 16.1803 0.642097
$$636$$ 0.763932 0.0302919
$$637$$ 46.9230 1.85916
$$638$$ 26.3607 1.04363
$$639$$ 1.56231 0.0618039
$$640$$ 1.00000 0.0395285
$$641$$ 0.360680 0.0142460 0.00712300 0.999975i $$-0.497733\pi$$
0.00712300 + 0.999975i $$0.497733\pi$$
$$642$$ 29.4164 1.16097
$$643$$ −8.29180 −0.326997 −0.163498 0.986544i $$-0.552278\pi$$
−0.163498 + 0.986544i $$0.552278\pi$$
$$644$$ −0.618034 −0.0243540
$$645$$ 0 0
$$646$$ 11.2918 0.444270
$$647$$ −36.2492 −1.42510 −0.712552 0.701619i $$-0.752461\pi$$
−0.712552 + 0.701619i $$0.752461\pi$$
$$648$$ −7.70820 −0.302807
$$649$$ −4.87539 −0.191376
$$650$$ −7.09017 −0.278099
$$651$$ −9.09017 −0.356272
$$652$$ 1.38197 0.0541220
$$653$$ −8.03444 −0.314412 −0.157206 0.987566i $$-0.550249\pi$$
−0.157206 + 0.987566i $$0.550249\pi$$
$$654$$ −18.7082 −0.731549
$$655$$ 2.94427 0.115042
$$656$$ 3.32624 0.129868
$$657$$ −1.23607 −0.0482236
$$658$$ 2.29180 0.0893435
$$659$$ −46.2492 −1.80161 −0.900807 0.434220i $$-0.857024\pi$$
−0.900807 + 0.434220i $$0.857024\pi$$
$$660$$ −4.61803 −0.179757
$$661$$ 18.6738 0.726325 0.363163 0.931726i $$-0.381697\pi$$
0.363163 + 0.931726i $$0.381697\pi$$
$$662$$ 19.2361 0.747631
$$663$$ −69.8673 −2.71342
$$664$$ −6.94427 −0.269490
$$665$$ −1.14590 −0.0444360
$$666$$ −2.47214 −0.0957933
$$667$$ −9.23607 −0.357622
$$668$$ −8.00000 −0.309529
$$669$$ −33.8885 −1.31021
$$670$$ 14.4721 0.559107
$$671$$ 26.6180 1.02758
$$672$$ −1.00000 −0.0385758
$$673$$ −10.9443 −0.421871 −0.210935 0.977500i $$-0.567651\pi$$
−0.210935 + 0.977500i $$0.567651\pi$$
$$674$$ 13.6738 0.526694
$$675$$ −5.47214 −0.210623
$$676$$ 37.2705 1.43348
$$677$$ −50.9443 −1.95795 −0.978974 0.203986i $$-0.934610\pi$$
−0.978974 + 0.203986i $$0.934610\pi$$
$$678$$ 1.70820 0.0656032
$$679$$ −7.65248 −0.293675
$$680$$ 6.09017 0.233547
$$681$$ −30.3607 −1.16342
$$682$$ −25.9443 −0.993458
$$683$$ −31.5623 −1.20770 −0.603849 0.797099i $$-0.706367\pi$$
−0.603849 + 0.797099i $$0.706367\pi$$
$$684$$ −0.708204 −0.0270789
$$685$$ −10.3262 −0.394545
$$686$$ 8.41641 0.321340
$$687$$ 16.1803 0.617318
$$688$$ 0 0
$$689$$ −3.34752 −0.127531
$$690$$ 1.61803 0.0615975
$$691$$ −29.2361 −1.11219 −0.556096 0.831118i $$-0.687701\pi$$
−0.556096 + 0.831118i $$0.687701\pi$$
$$692$$ −1.43769 −0.0546529
$$693$$ −0.673762 −0.0255941
$$694$$ −6.38197 −0.242256
$$695$$ 12.7639 0.484164
$$696$$ −14.9443 −0.566461
$$697$$ 20.2574 0.767302
$$698$$ −2.00000 −0.0757011
$$699$$ 10.1803 0.385056
$$700$$ −0.618034 −0.0233595
$$701$$ 43.3394 1.63691 0.818453 0.574573i $$-0.194832\pi$$
0.818453 + 0.574573i $$0.194832\pi$$
$$702$$ 38.7984 1.46435
$$703$$ 12.0000 0.452589
$$704$$ −2.85410 −0.107568
$$705$$ −6.00000 −0.225973
$$706$$ 24.0000 0.903252
$$707$$ 0.180340 0.00678238
$$708$$ 2.76393 0.103875
$$709$$ −26.0902 −0.979837 −0.489918 0.871768i $$-0.662973\pi$$
−0.489918 + 0.871768i $$0.662973\pi$$
$$710$$ −4.09017 −0.153501
$$711$$ −0.583592 −0.0218864
$$712$$ −10.4721 −0.392460
$$713$$ 9.09017 0.340430
$$714$$ −6.09017 −0.227919
$$715$$ 20.2361 0.756786
$$716$$ 2.18034 0.0814831
$$717$$ −32.9443 −1.23033
$$718$$ 26.3607 0.983772
$$719$$ 35.2705 1.31537 0.657684 0.753294i $$-0.271536\pi$$
0.657684 + 0.753294i $$0.271536\pi$$
$$720$$ −0.381966 −0.0142350
$$721$$ −10.2361 −0.381211
$$722$$ −15.5623 −0.579169
$$723$$ 0 0
$$724$$ −12.1459 −0.451399
$$725$$ −9.23607 −0.343019
$$726$$ −4.61803 −0.171391
$$727$$ 28.2016 1.04594 0.522970 0.852351i $$-0.324824\pi$$
0.522970 + 0.852351i $$0.324824\pi$$
$$728$$ 4.38197 0.162406
$$729$$ 29.5066 1.09284
$$730$$ 3.23607 0.119772
$$731$$ 0 0
$$732$$ −15.0902 −0.557749
$$733$$ −29.4164 −1.08652 −0.543260 0.839565i $$-0.682810\pi$$
−0.543260 + 0.839565i $$0.682810\pi$$
$$734$$ 6.47214 0.238891
$$735$$ −10.7082 −0.394978
$$736$$ 1.00000 0.0368605
$$737$$ −41.3050 −1.52149
$$738$$ −1.27051 −0.0467681
$$739$$ −13.8885 −0.510898 −0.255449 0.966822i $$-0.582223\pi$$
−0.255449 + 0.966822i $$0.582223\pi$$
$$740$$ 6.47214 0.237920
$$741$$ −21.2705 −0.781392
$$742$$ −0.291796 −0.0107122
$$743$$ 33.6312 1.23381 0.616904 0.787038i $$-0.288387\pi$$
0.616904 + 0.787038i $$0.288387\pi$$
$$744$$ 14.7082 0.539229
$$745$$ −7.85410 −0.287752
$$746$$ 20.1803 0.738855
$$747$$ 2.65248 0.0970490
$$748$$ −17.3820 −0.635548
$$749$$ −11.2361 −0.410557
$$750$$ 1.61803 0.0590822
$$751$$ −47.0132 −1.71553 −0.857767 0.514038i $$-0.828149\pi$$
−0.857767 + 0.514038i $$0.828149\pi$$
$$752$$ −3.70820 −0.135224
$$753$$ 9.94427 0.362389
$$754$$ 65.4853 2.38483
$$755$$ −2.56231 −0.0932519
$$756$$ 3.38197 0.123001
$$757$$ −17.8885 −0.650170 −0.325085 0.945685i $$-0.605393\pi$$
−0.325085 + 0.945685i $$0.605393\pi$$
$$758$$ −22.4508 −0.815452
$$759$$ −4.61803 −0.167624
$$760$$ 1.85410 0.0672553
$$761$$ 46.8673 1.69894 0.849468 0.527640i $$-0.176923\pi$$
0.849468 + 0.527640i $$0.176923\pi$$
$$762$$ 26.1803 0.948414
$$763$$ 7.14590 0.258699
$$764$$ −13.7082 −0.495945
$$765$$ −2.32624 −0.0841053
$$766$$ −17.8885 −0.646339
$$767$$ −12.1115 −0.437319
$$768$$ 1.61803 0.0583858
$$769$$ −6.58359 −0.237410 −0.118705 0.992930i $$-0.537874\pi$$
−0.118705 + 0.992930i $$0.537874\pi$$
$$770$$ 1.76393 0.0635677
$$771$$ −12.6525 −0.455668
$$772$$ 0.763932 0.0274945
$$773$$ 28.9443 1.04105 0.520527 0.853845i $$-0.325736\pi$$
0.520527 + 0.853845i $$0.325736\pi$$
$$774$$ 0 0
$$775$$ 9.09017 0.326529
$$776$$ 12.3820 0.444487
$$777$$ −6.47214 −0.232187
$$778$$ 21.3262 0.764583
$$779$$ 6.16718 0.220962
$$780$$ −11.4721 −0.410768
$$781$$ 11.6738 0.417720
$$782$$ 6.09017 0.217784
$$783$$ 50.5410 1.80619
$$784$$ −6.61803 −0.236358
$$785$$ −3.70820 −0.132351
$$786$$ 4.76393 0.169924
$$787$$ −2.87539 −0.102497 −0.0512483 0.998686i $$-0.516320\pi$$
−0.0512483 + 0.998686i $$0.516320\pi$$
$$788$$ −22.5623 −0.803749
$$789$$ 33.5623 1.19485
$$790$$ 1.52786 0.0543590
$$791$$ −0.652476 −0.0231994
$$792$$ 1.09017 0.0387375
$$793$$ 66.1246 2.34815
$$794$$ 7.32624 0.259998
$$795$$ 0.763932 0.0270939
$$796$$ 2.00000 0.0708881
$$797$$ 13.7082 0.485569 0.242785 0.970080i $$-0.421939\pi$$
0.242785 + 0.970080i $$0.421939\pi$$
$$798$$ −1.85410 −0.0656345
$$799$$ −22.5836 −0.798950
$$800$$ 1.00000 0.0353553
$$801$$ 4.00000 0.141333
$$802$$ −1.70820 −0.0603188
$$803$$ −9.23607 −0.325934
$$804$$ 23.4164 0.825833
$$805$$ −0.618034 −0.0217828
$$806$$ −64.4508 −2.27018
$$807$$ −22.9443 −0.807677
$$808$$ −0.291796 −0.0102653
$$809$$ −46.7426 −1.64338 −0.821692 0.569932i $$-0.806970\pi$$
−0.821692 + 0.569932i $$0.806970\pi$$
$$810$$ −7.70820 −0.270839
$$811$$ 21.8197 0.766192 0.383096 0.923709i $$-0.374858\pi$$
0.383096 + 0.923709i $$0.374858\pi$$
$$812$$ 5.70820 0.200319
$$813$$ −49.0689 −1.72092
$$814$$ −18.4721 −0.647448
$$815$$ 1.38197 0.0484082
$$816$$ 9.85410 0.344963
$$817$$ 0 0
$$818$$ −30.2148 −1.05644
$$819$$ −1.67376 −0.0584860
$$820$$ 3.32624 0.116157
$$821$$ −33.0557 −1.15365 −0.576826 0.816867i $$-0.695709\pi$$
−0.576826 + 0.816867i $$0.695709\pi$$
$$822$$ −16.7082 −0.582766
$$823$$ 25.4164 0.885960 0.442980 0.896531i $$-0.353921\pi$$
0.442980 + 0.896531i $$0.353921\pi$$
$$824$$ 16.5623 0.576975
$$825$$ −4.61803 −0.160779
$$826$$ −1.05573 −0.0367335
$$827$$ 21.7082 0.754868 0.377434 0.926036i $$-0.376806\pi$$
0.377434 + 0.926036i $$0.376806\pi$$
$$828$$ −0.381966 −0.0132742
$$829$$ 18.9443 0.657962 0.328981 0.944337i $$-0.393295\pi$$
0.328981 + 0.944337i $$0.393295\pi$$
$$830$$ −6.94427 −0.241039
$$831$$ 47.5967 1.65111
$$832$$ −7.09017 −0.245807
$$833$$ −40.3050 −1.39648
$$834$$ 20.6525 0.715137
$$835$$ −8.00000 −0.276851
$$836$$ −5.29180 −0.183021
$$837$$ −49.7426 −1.71936
$$838$$ 14.4721 0.499932
$$839$$ 33.0132 1.13974 0.569870 0.821735i $$-0.306993\pi$$
0.569870 + 0.821735i $$0.306993\pi$$
$$840$$ −1.00000 −0.0345033
$$841$$ 56.3050 1.94155
$$842$$ 13.7426 0.473603
$$843$$ 36.8328 1.26859
$$844$$ 14.0000 0.481900
$$845$$ 37.2705 1.28214
$$846$$ 1.41641 0.0486971
$$847$$ 1.76393 0.0606094
$$848$$ 0.472136 0.0162132
$$849$$ −43.5967 −1.49624
$$850$$ 6.09017 0.208891
$$851$$ 6.47214 0.221862
$$852$$ −6.61803 −0.226730
$$853$$ −10.7984 −0.369729 −0.184865 0.982764i $$-0.559185\pi$$
−0.184865 + 0.982764i $$0.559185\pi$$
$$854$$ 5.76393 0.197238
$$855$$ −0.708204 −0.0242201
$$856$$ 18.1803 0.621391
$$857$$ −6.58359 −0.224891 −0.112446 0.993658i $$-0.535868\pi$$
−0.112446 + 0.993658i $$0.535868\pi$$
$$858$$ 32.7426 1.11782
$$859$$ −24.0689 −0.821220 −0.410610 0.911811i $$-0.634684\pi$$
−0.410610 + 0.911811i $$0.634684\pi$$
$$860$$ 0 0
$$861$$ −3.32624 −0.113358
$$862$$ 3.34752 0.114017
$$863$$ 32.7639 1.11530 0.557649 0.830077i $$-0.311704\pi$$
0.557649 + 0.830077i $$0.311704\pi$$
$$864$$ −5.47214 −0.186166
$$865$$ −1.43769 −0.0488831
$$866$$ 8.50658 0.289065
$$867$$ 32.5066 1.10398
$$868$$ −5.61803 −0.190688
$$869$$ −4.36068 −0.147926
$$870$$ −14.9443 −0.506658
$$871$$ −102.610 −3.47680
$$872$$ −11.5623 −0.391549
$$873$$ −4.72949 −0.160069
$$874$$ 1.85410 0.0627159
$$875$$ −0.618034 −0.0208934
$$876$$ 5.23607 0.176910
$$877$$ 18.7426 0.632894 0.316447 0.948610i $$-0.397510\pi$$
0.316447 + 0.948610i $$0.397510\pi$$
$$878$$ −13.3820 −0.451619
$$879$$ −32.1803 −1.08542
$$880$$ −2.85410 −0.0962118
$$881$$ 8.58359 0.289189 0.144594 0.989491i $$-0.453812\pi$$
0.144594 + 0.989491i $$0.453812\pi$$
$$882$$ 2.52786 0.0851176
$$883$$ 15.5623 0.523713 0.261857 0.965107i $$-0.415665\pi$$
0.261857 + 0.965107i $$0.415665\pi$$
$$884$$ −43.1803 −1.45231
$$885$$ 2.76393 0.0929086
$$886$$ 25.0902 0.842921
$$887$$ −5.16718 −0.173497 −0.0867485 0.996230i $$-0.527648\pi$$
−0.0867485 + 0.996230i $$0.527648\pi$$
$$888$$ 10.4721 0.351422
$$889$$ −10.0000 −0.335389
$$890$$ −10.4721 −0.351027
$$891$$ 22.0000 0.737028
$$892$$ −20.9443 −0.701266
$$893$$ −6.87539 −0.230076
$$894$$ −12.7082 −0.425026
$$895$$ 2.18034 0.0728807
$$896$$ −0.618034 −0.0206471
$$897$$ −11.4721 −0.383043
$$898$$ −1.56231 −0.0521348
$$899$$ −83.9574 −2.80014
$$900$$ −0.381966 −0.0127322
$$901$$ 2.87539 0.0957931
$$902$$ −9.49342 −0.316096
$$903$$ 0 0
$$904$$ 1.05573 0.0351130
$$905$$ −12.1459 −0.403743
$$906$$ −4.14590 −0.137738
$$907$$ 7.12461 0.236569 0.118284 0.992980i $$-0.462261\pi$$
0.118284 + 0.992980i $$0.462261\pi$$
$$908$$ −18.7639 −0.622703
$$909$$ 0.111456 0.00369677
$$910$$ 4.38197 0.145261
$$911$$ 36.0689 1.19502 0.597508 0.801863i $$-0.296158\pi$$
0.597508 + 0.801863i $$0.296158\pi$$
$$912$$ 3.00000 0.0993399
$$913$$ 19.8197 0.655935
$$914$$ −37.7771 −1.24955
$$915$$ −15.0902 −0.498866
$$916$$ 10.0000 0.330409
$$917$$ −1.81966 −0.0600905
$$918$$ −33.3262 −1.09993
$$919$$ 40.0000 1.31948 0.659739 0.751495i $$-0.270667\pi$$
0.659739 + 0.751495i $$0.270667\pi$$
$$920$$ 1.00000 0.0329690
$$921$$ −46.0344 −1.51689
$$922$$ −39.2361 −1.29217
$$923$$ 29.0000 0.954547
$$924$$ 2.85410 0.0938931
$$925$$ 6.47214 0.212803
$$926$$ 2.00000 0.0657241
$$927$$ −6.32624 −0.207781
$$928$$ −9.23607 −0.303189
$$929$$ −3.52786 −0.115745 −0.0578727 0.998324i $$-0.518432\pi$$
−0.0578727 + 0.998324i $$0.518432\pi$$
$$930$$ 14.7082 0.482301
$$931$$ −12.2705 −0.402150
$$932$$ 6.29180 0.206095
$$933$$ 6.47214 0.211888
$$934$$ −17.1246 −0.560334
$$935$$ −17.3820 −0.568451
$$936$$ 2.70820 0.0885204
$$937$$ −36.7984 −1.20215 −0.601075 0.799192i $$-0.705261\pi$$
−0.601075 + 0.799192i $$0.705261\pi$$
$$938$$ −8.94427 −0.292041
$$939$$ 20.7082 0.675787
$$940$$ −3.70820 −0.120948
$$941$$ −22.4934 −0.733265 −0.366632 0.930366i $$-0.619489\pi$$
−0.366632 + 0.930366i $$0.619489\pi$$
$$942$$ −6.00000 −0.195491
$$943$$ 3.32624 0.108317
$$944$$ 1.70820 0.0555973
$$945$$ 3.38197 0.110015
$$946$$ 0 0
$$947$$ −54.6869 −1.77709 −0.888543 0.458793i $$-0.848282\pi$$
−0.888543 + 0.458793i $$0.848282\pi$$
$$948$$ 2.47214 0.0802912
$$949$$ −22.9443 −0.744803
$$950$$ 1.85410 0.0601550
$$951$$ −17.9443 −0.581883
$$952$$ −3.76393 −0.121990
$$953$$ 3.79837 0.123041 0.0615207 0.998106i $$-0.480405\pi$$
0.0615207 + 0.998106i $$0.480405\pi$$
$$954$$ −0.180340 −0.00583872
$$955$$ −13.7082 −0.443587
$$956$$ −20.3607 −0.658511
$$957$$ 42.6525 1.37876
$$958$$ −31.8885 −1.03027
$$959$$ 6.38197 0.206084
$$960$$ 1.61803 0.0522218
$$961$$ 51.6312 1.66552
$$962$$ −45.8885 −1.47951
$$963$$ −6.94427 −0.223776
$$964$$ 0 0
$$965$$ 0.763932 0.0245918
$$966$$ −1.00000 −0.0321745
$$967$$ −16.5410 −0.531923 −0.265962 0.963984i $$-0.585689\pi$$
−0.265962 + 0.963984i $$0.585689\pi$$
$$968$$ −2.85410 −0.0917343
$$969$$ 18.2705 0.586933
$$970$$ 12.3820 0.397561
$$971$$ 34.2705 1.09979 0.549896 0.835233i $$-0.314667\pi$$
0.549896 + 0.835233i $$0.314667\pi$$
$$972$$ 3.94427 0.126513
$$973$$ −7.88854 −0.252895
$$974$$ −19.8197 −0.635063
$$975$$ −11.4721 −0.367402
$$976$$ −9.32624 −0.298526
$$977$$ −23.5623 −0.753825 −0.376912 0.926249i $$-0.623014\pi$$
−0.376912 + 0.926249i $$0.623014\pi$$
$$978$$ 2.23607 0.0715016
$$979$$ 29.8885 0.955242
$$980$$ −6.61803 −0.211405
$$981$$ 4.41641 0.141005
$$982$$ 6.18034 0.197223
$$983$$ −14.2705 −0.455159 −0.227579 0.973760i $$-0.573081\pi$$
−0.227579 + 0.973760i $$0.573081\pi$$
$$984$$ 5.38197 0.171571
$$985$$ −22.5623 −0.718895
$$986$$ −56.2492 −1.79134
$$987$$ 3.70820 0.118033
$$988$$ −13.1459 −0.418227
$$989$$ 0 0
$$990$$ 1.09017 0.0346479
$$991$$ 27.5066 0.873775 0.436888 0.899516i $$-0.356081\pi$$
0.436888 + 0.899516i $$0.356081\pi$$
$$992$$ 9.09017 0.288613
$$993$$ 31.1246 0.987710
$$994$$ 2.52786 0.0801790
$$995$$ 2.00000 0.0634043
$$996$$ −11.2361 −0.356028
$$997$$ 57.1935 1.81134 0.905668 0.423987i $$-0.139370\pi$$
0.905668 + 0.423987i $$0.139370\pi$$
$$998$$ 12.3607 0.391270
$$999$$ −35.4164 −1.12053
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 230.2.a.c.1.2 2
3.2 odd 2 2070.2.a.u.1.1 2
4.3 odd 2 1840.2.a.l.1.1 2
5.2 odd 4 1150.2.b.i.599.3 4
5.3 odd 4 1150.2.b.i.599.2 4
5.4 even 2 1150.2.a.j.1.1 2
8.3 odd 2 7360.2.a.bn.1.2 2
8.5 even 2 7360.2.a.bh.1.1 2
20.19 odd 2 9200.2.a.bu.1.2 2
23.22 odd 2 5290.2.a.o.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.c.1.2 2 1.1 even 1 trivial
1150.2.a.j.1.1 2 5.4 even 2
1150.2.b.i.599.2 4 5.3 odd 4
1150.2.b.i.599.3 4 5.2 odd 4
1840.2.a.l.1.1 2 4.3 odd 2
2070.2.a.u.1.1 2 3.2 odd 2
5290.2.a.o.1.2 2 23.22 odd 2
7360.2.a.bh.1.1 2 8.5 even 2
7360.2.a.bn.1.2 2 8.3 odd 2
9200.2.a.bu.1.2 2 20.19 odd 2