# Properties

 Label 230.2.a.b.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: $$\Q(\sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 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 $$2.30278$$ 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)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +3.30278 q^{3} +1.00000 q^{4} +1.00000 q^{5} -3.30278 q^{6} -0.302776 q^{7} -1.00000 q^{8} +7.90833 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +3.30278 q^{3} +1.00000 q^{4} +1.00000 q^{5} -3.30278 q^{6} -0.302776 q^{7} -1.00000 q^{8} +7.90833 q^{9} -1.00000 q^{10} -5.30278 q^{11} +3.30278 q^{12} -0.302776 q^{13} +0.302776 q^{14} +3.30278 q^{15} +1.00000 q^{16} -3.90833 q^{17} -7.90833 q^{18} -4.90833 q^{19} +1.00000 q^{20} -1.00000 q^{21} +5.30278 q^{22} -1.00000 q^{23} -3.30278 q^{24} +1.00000 q^{25} +0.302776 q^{26} +16.2111 q^{27} -0.302776 q^{28} +4.60555 q^{29} -3.30278 q^{30} +2.90833 q^{31} -1.00000 q^{32} -17.5139 q^{33} +3.90833 q^{34} -0.302776 q^{35} +7.90833 q^{36} +8.00000 q^{37} +4.90833 q^{38} -1.00000 q^{39} -1.00000 q^{40} -9.90833 q^{41} +1.00000 q^{42} +5.21110 q^{43} -5.30278 q^{44} +7.90833 q^{45} +1.00000 q^{46} +4.60555 q^{47} +3.30278 q^{48} -6.90833 q^{49} -1.00000 q^{50} -12.9083 q^{51} -0.302776 q^{52} +3.21110 q^{53} -16.2111 q^{54} -5.30278 q^{55} +0.302776 q^{56} -16.2111 q^{57} -4.60555 q^{58} -10.6056 q^{59} +3.30278 q^{60} -6.51388 q^{61} -2.90833 q^{62} -2.39445 q^{63} +1.00000 q^{64} -0.302776 q^{65} +17.5139 q^{66} -4.00000 q^{67} -3.90833 q^{68} -3.30278 q^{69} +0.302776 q^{70} -12.6972 q^{71} -7.90833 q^{72} +15.8167 q^{73} -8.00000 q^{74} +3.30278 q^{75} -4.90833 q^{76} +1.60555 q^{77} +1.00000 q^{78} +14.4222 q^{79} +1.00000 q^{80} +29.8167 q^{81} +9.90833 q^{82} -3.21110 q^{83} -1.00000 q^{84} -3.90833 q^{85} -5.21110 q^{86} +15.2111 q^{87} +5.30278 q^{88} -7.90833 q^{90} +0.0916731 q^{91} -1.00000 q^{92} +9.60555 q^{93} -4.60555 q^{94} -4.90833 q^{95} -3.30278 q^{96} +2.69722 q^{97} +6.90833 q^{98} -41.9361 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 3 q^{3} + 2 q^{4} + 2 q^{5} - 3 q^{6} + 3 q^{7} - 2 q^{8} + 5 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 3 * q^3 + 2 * q^4 + 2 * q^5 - 3 * q^6 + 3 * q^7 - 2 * q^8 + 5 * q^9 $$2 q - 2 q^{2} + 3 q^{3} + 2 q^{4} + 2 q^{5} - 3 q^{6} + 3 q^{7} - 2 q^{8} + 5 q^{9} - 2 q^{10} - 7 q^{11} + 3 q^{12} + 3 q^{13} - 3 q^{14} + 3 q^{15} + 2 q^{16} + 3 q^{17} - 5 q^{18} + q^{19} + 2 q^{20} - 2 q^{21} + 7 q^{22} - 2 q^{23} - 3 q^{24} + 2 q^{25} - 3 q^{26} + 18 q^{27} + 3 q^{28} + 2 q^{29} - 3 q^{30} - 5 q^{31} - 2 q^{32} - 17 q^{33} - 3 q^{34} + 3 q^{35} + 5 q^{36} + 16 q^{37} - q^{38} - 2 q^{39} - 2 q^{40} - 9 q^{41} + 2 q^{42} - 4 q^{43} - 7 q^{44} + 5 q^{45} + 2 q^{46} + 2 q^{47} + 3 q^{48} - 3 q^{49} - 2 q^{50} - 15 q^{51} + 3 q^{52} - 8 q^{53} - 18 q^{54} - 7 q^{55} - 3 q^{56} - 18 q^{57} - 2 q^{58} - 14 q^{59} + 3 q^{60} + 5 q^{61} + 5 q^{62} - 12 q^{63} + 2 q^{64} + 3 q^{65} + 17 q^{66} - 8 q^{67} + 3 q^{68} - 3 q^{69} - 3 q^{70} - 29 q^{71} - 5 q^{72} + 10 q^{73} - 16 q^{74} + 3 q^{75} + q^{76} - 4 q^{77} + 2 q^{78} + 2 q^{80} + 38 q^{81} + 9 q^{82} + 8 q^{83} - 2 q^{84} + 3 q^{85} + 4 q^{86} + 16 q^{87} + 7 q^{88} - 5 q^{90} + 11 q^{91} - 2 q^{92} + 12 q^{93} - 2 q^{94} + q^{95} - 3 q^{96} + 9 q^{97} + 3 q^{98} - 37 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + 3 * q^3 + 2 * q^4 + 2 * q^5 - 3 * q^6 + 3 * q^7 - 2 * q^8 + 5 * q^9 - 2 * q^10 - 7 * q^11 + 3 * q^12 + 3 * q^13 - 3 * q^14 + 3 * q^15 + 2 * q^16 + 3 * q^17 - 5 * q^18 + q^19 + 2 * q^20 - 2 * q^21 + 7 * q^22 - 2 * q^23 - 3 * q^24 + 2 * q^25 - 3 * q^26 + 18 * q^27 + 3 * q^28 + 2 * q^29 - 3 * q^30 - 5 * q^31 - 2 * q^32 - 17 * q^33 - 3 * q^34 + 3 * q^35 + 5 * q^36 + 16 * q^37 - q^38 - 2 * q^39 - 2 * q^40 - 9 * q^41 + 2 * q^42 - 4 * q^43 - 7 * q^44 + 5 * q^45 + 2 * q^46 + 2 * q^47 + 3 * q^48 - 3 * q^49 - 2 * q^50 - 15 * q^51 + 3 * q^52 - 8 * q^53 - 18 * q^54 - 7 * q^55 - 3 * q^56 - 18 * q^57 - 2 * q^58 - 14 * q^59 + 3 * q^60 + 5 * q^61 + 5 * q^62 - 12 * q^63 + 2 * q^64 + 3 * q^65 + 17 * q^66 - 8 * q^67 + 3 * q^68 - 3 * q^69 - 3 * q^70 - 29 * q^71 - 5 * q^72 + 10 * q^73 - 16 * q^74 + 3 * q^75 + q^76 - 4 * q^77 + 2 * q^78 + 2 * q^80 + 38 * q^81 + 9 * q^82 + 8 * q^83 - 2 * q^84 + 3 * q^85 + 4 * q^86 + 16 * q^87 + 7 * q^88 - 5 * q^90 + 11 * q^91 - 2 * q^92 + 12 * q^93 - 2 * q^94 + q^95 - 3 * q^96 + 9 * q^97 + 3 * q^98 - 37 * 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
$$3$$ 3.30278 1.90686 0.953429 0.301617i $$-0.0975264\pi$$
0.953429 + 0.301617i $$0.0975264\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ −3.30278 −1.34835
$$7$$ −0.302776 −0.114438 −0.0572192 0.998362i $$-0.518223\pi$$
−0.0572192 + 0.998362i $$0.518223\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 7.90833 2.63611
$$10$$ −1.00000 −0.316228
$$11$$ −5.30278 −1.59885 −0.799424 0.600768i $$-0.794862\pi$$
−0.799424 + 0.600768i $$0.794862\pi$$
$$12$$ 3.30278 0.953429
$$13$$ −0.302776 −0.0839749 −0.0419874 0.999118i $$-0.513369\pi$$
−0.0419874 + 0.999118i $$0.513369\pi$$
$$14$$ 0.302776 0.0809202
$$15$$ 3.30278 0.852773
$$16$$ 1.00000 0.250000
$$17$$ −3.90833 −0.947909 −0.473954 0.880549i $$-0.657174\pi$$
−0.473954 + 0.880549i $$0.657174\pi$$
$$18$$ −7.90833 −1.86401
$$19$$ −4.90833 −1.12605 −0.563024 0.826441i $$-0.690362\pi$$
−0.563024 + 0.826441i $$0.690362\pi$$
$$20$$ 1.00000 0.223607
$$21$$ −1.00000 −0.218218
$$22$$ 5.30278 1.13056
$$23$$ −1.00000 −0.208514
$$24$$ −3.30278 −0.674176
$$25$$ 1.00000 0.200000
$$26$$ 0.302776 0.0593792
$$27$$ 16.2111 3.11983
$$28$$ −0.302776 −0.0572192
$$29$$ 4.60555 0.855229 0.427615 0.903961i $$-0.359354\pi$$
0.427615 + 0.903961i $$0.359354\pi$$
$$30$$ −3.30278 −0.603002
$$31$$ 2.90833 0.522351 0.261175 0.965291i $$-0.415890\pi$$
0.261175 + 0.965291i $$0.415890\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −17.5139 −3.04877
$$34$$ 3.90833 0.670273
$$35$$ −0.302776 −0.0511784
$$36$$ 7.90833 1.31805
$$37$$ 8.00000 1.31519 0.657596 0.753371i $$-0.271573\pi$$
0.657596 + 0.753371i $$0.271573\pi$$
$$38$$ 4.90833 0.796236
$$39$$ −1.00000 −0.160128
$$40$$ −1.00000 −0.158114
$$41$$ −9.90833 −1.54742 −0.773710 0.633540i $$-0.781601\pi$$
−0.773710 + 0.633540i $$0.781601\pi$$
$$42$$ 1.00000 0.154303
$$43$$ 5.21110 0.794686 0.397343 0.917670i $$-0.369932\pi$$
0.397343 + 0.917670i $$0.369932\pi$$
$$44$$ −5.30278 −0.799424
$$45$$ 7.90833 1.17890
$$46$$ 1.00000 0.147442
$$47$$ 4.60555 0.671789 0.335894 0.941900i $$-0.390961\pi$$
0.335894 + 0.941900i $$0.390961\pi$$
$$48$$ 3.30278 0.476715
$$49$$ −6.90833 −0.986904
$$50$$ −1.00000 −0.141421
$$51$$ −12.9083 −1.80753
$$52$$ −0.302776 −0.0419874
$$53$$ 3.21110 0.441079 0.220539 0.975378i $$-0.429218\pi$$
0.220539 + 0.975378i $$0.429218\pi$$
$$54$$ −16.2111 −2.20605
$$55$$ −5.30278 −0.715026
$$56$$ 0.302776 0.0404601
$$57$$ −16.2111 −2.14721
$$58$$ −4.60555 −0.604739
$$59$$ −10.6056 −1.38073 −0.690363 0.723464i $$-0.742549\pi$$
−0.690363 + 0.723464i $$0.742549\pi$$
$$60$$ 3.30278 0.426387
$$61$$ −6.51388 −0.834017 −0.417008 0.908903i $$-0.636921\pi$$
−0.417008 + 0.908903i $$0.636921\pi$$
$$62$$ −2.90833 −0.369358
$$63$$ −2.39445 −0.301672
$$64$$ 1.00000 0.125000
$$65$$ −0.302776 −0.0375547
$$66$$ 17.5139 2.15581
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ −3.90833 −0.473954
$$69$$ −3.30278 −0.397607
$$70$$ 0.302776 0.0361886
$$71$$ −12.6972 −1.50688 −0.753442 0.657515i $$-0.771608\pi$$
−0.753442 + 0.657515i $$0.771608\pi$$
$$72$$ −7.90833 −0.932005
$$73$$ 15.8167 1.85120 0.925600 0.378504i $$-0.123561\pi$$
0.925600 + 0.378504i $$0.123561\pi$$
$$74$$ −8.00000 −0.929981
$$75$$ 3.30278 0.381372
$$76$$ −4.90833 −0.563024
$$77$$ 1.60555 0.182970
$$78$$ 1.00000 0.113228
$$79$$ 14.4222 1.62262 0.811312 0.584613i $$-0.198754\pi$$
0.811312 + 0.584613i $$0.198754\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 29.8167 3.31296
$$82$$ 9.90833 1.09419
$$83$$ −3.21110 −0.352464 −0.176232 0.984349i $$-0.556391\pi$$
−0.176232 + 0.984349i $$0.556391\pi$$
$$84$$ −1.00000 −0.109109
$$85$$ −3.90833 −0.423918
$$86$$ −5.21110 −0.561928
$$87$$ 15.2111 1.63080
$$88$$ 5.30278 0.565278
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ −7.90833 −0.833611
$$91$$ 0.0916731 0.00960995
$$92$$ −1.00000 −0.104257
$$93$$ 9.60555 0.996049
$$94$$ −4.60555 −0.475026
$$95$$ −4.90833 −0.503584
$$96$$ −3.30278 −0.337088
$$97$$ 2.69722 0.273862 0.136931 0.990581i $$-0.456276\pi$$
0.136931 + 0.990581i $$0.456276\pi$$
$$98$$ 6.90833 0.697846
$$99$$ −41.9361 −4.21473
$$100$$ 1.00000 0.100000
$$101$$ −4.60555 −0.458269 −0.229135 0.973395i $$-0.573590\pi$$
−0.229135 + 0.973395i $$0.573590\pi$$
$$102$$ 12.9083 1.27811
$$103$$ −17.1194 −1.68683 −0.843414 0.537265i $$-0.819458\pi$$
−0.843414 + 0.537265i $$0.819458\pi$$
$$104$$ 0.302776 0.0296896
$$105$$ −1.00000 −0.0975900
$$106$$ −3.21110 −0.311890
$$107$$ 4.60555 0.445235 0.222618 0.974906i $$-0.428540\pi$$
0.222618 + 0.974906i $$0.428540\pi$$
$$108$$ 16.2111 1.55991
$$109$$ 19.5139 1.86909 0.934545 0.355844i $$-0.115807\pi$$
0.934545 + 0.355844i $$0.115807\pi$$
$$110$$ 5.30278 0.505600
$$111$$ 26.4222 2.50788
$$112$$ −0.302776 −0.0286096
$$113$$ 12.4222 1.16858 0.584291 0.811544i $$-0.301373\pi$$
0.584291 + 0.811544i $$0.301373\pi$$
$$114$$ 16.2111 1.51831
$$115$$ −1.00000 −0.0932505
$$116$$ 4.60555 0.427615
$$117$$ −2.39445 −0.221367
$$118$$ 10.6056 0.976320
$$119$$ 1.18335 0.108477
$$120$$ −3.30278 −0.301501
$$121$$ 17.1194 1.55631
$$122$$ 6.51388 0.589739
$$123$$ −32.7250 −2.95071
$$124$$ 2.90833 0.261175
$$125$$ 1.00000 0.0894427
$$126$$ 2.39445 0.213314
$$127$$ −11.8167 −1.04856 −0.524279 0.851546i $$-0.675665\pi$$
−0.524279 + 0.851546i $$0.675665\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 17.2111 1.51535
$$130$$ 0.302776 0.0265552
$$131$$ 3.21110 0.280555 0.140278 0.990112i $$-0.455200\pi$$
0.140278 + 0.990112i $$0.455200\pi$$
$$132$$ −17.5139 −1.52439
$$133$$ 1.48612 0.128863
$$134$$ 4.00000 0.345547
$$135$$ 16.2111 1.39523
$$136$$ 3.90833 0.335136
$$137$$ −6.90833 −0.590218 −0.295109 0.955464i $$-0.595356\pi$$
−0.295109 + 0.955464i $$0.595356\pi$$
$$138$$ 3.30278 0.281151
$$139$$ −5.39445 −0.457551 −0.228776 0.973479i $$-0.573472\pi$$
−0.228776 + 0.973479i $$0.573472\pi$$
$$140$$ −0.302776 −0.0255892
$$141$$ 15.2111 1.28101
$$142$$ 12.6972 1.06553
$$143$$ 1.60555 0.134263
$$144$$ 7.90833 0.659027
$$145$$ 4.60555 0.382470
$$146$$ −15.8167 −1.30900
$$147$$ −22.8167 −1.88189
$$148$$ 8.00000 0.657596
$$149$$ 9.69722 0.794428 0.397214 0.917726i $$-0.369977\pi$$
0.397214 + 0.917726i $$0.369977\pi$$
$$150$$ −3.30278 −0.269671
$$151$$ −1.90833 −0.155297 −0.0776487 0.996981i $$-0.524741\pi$$
−0.0776487 + 0.996981i $$0.524741\pi$$
$$152$$ 4.90833 0.398118
$$153$$ −30.9083 −2.49879
$$154$$ −1.60555 −0.129379
$$155$$ 2.90833 0.233602
$$156$$ −1.00000 −0.0800641
$$157$$ −11.3944 −0.909376 −0.454688 0.890651i $$-0.650249\pi$$
−0.454688 + 0.890651i $$0.650249\pi$$
$$158$$ −14.4222 −1.14737
$$159$$ 10.6056 0.841075
$$160$$ −1.00000 −0.0790569
$$161$$ 0.302776 0.0238621
$$162$$ −29.8167 −2.34262
$$163$$ 5.69722 0.446241 0.223121 0.974791i $$-0.428376\pi$$
0.223121 + 0.974791i $$0.428376\pi$$
$$164$$ −9.90833 −0.773710
$$165$$ −17.5139 −1.36345
$$166$$ 3.21110 0.249230
$$167$$ 21.2111 1.64136 0.820682 0.571385i $$-0.193594\pi$$
0.820682 + 0.571385i $$0.193594\pi$$
$$168$$ 1.00000 0.0771517
$$169$$ −12.9083 −0.992948
$$170$$ 3.90833 0.299755
$$171$$ −38.8167 −2.96838
$$172$$ 5.21110 0.397343
$$173$$ 23.3028 1.77168 0.885839 0.463993i $$-0.153584\pi$$
0.885839 + 0.463993i $$0.153584\pi$$
$$174$$ −15.2111 −1.15315
$$175$$ −0.302776 −0.0228877
$$176$$ −5.30278 −0.399712
$$177$$ −35.0278 −2.63285
$$178$$ 0 0
$$179$$ 16.6056 1.24116 0.620579 0.784144i $$-0.286898\pi$$
0.620579 + 0.784144i $$0.286898\pi$$
$$180$$ 7.90833 0.589452
$$181$$ −8.11943 −0.603512 −0.301756 0.953385i $$-0.597573\pi$$
−0.301756 + 0.953385i $$0.597573\pi$$
$$182$$ −0.0916731 −0.00679526
$$183$$ −21.5139 −1.59035
$$184$$ 1.00000 0.0737210
$$185$$ 8.00000 0.588172
$$186$$ −9.60555 −0.704313
$$187$$ 20.7250 1.51556
$$188$$ 4.60555 0.335894
$$189$$ −4.90833 −0.357028
$$190$$ 4.90833 0.356087
$$191$$ −1.39445 −0.100899 −0.0504494 0.998727i $$-0.516065\pi$$
−0.0504494 + 0.998727i $$0.516065\pi$$
$$192$$ 3.30278 0.238357
$$193$$ 3.81665 0.274729 0.137364 0.990521i $$-0.456137\pi$$
0.137364 + 0.990521i $$0.456137\pi$$
$$194$$ −2.69722 −0.193649
$$195$$ −1.00000 −0.0716115
$$196$$ −6.90833 −0.493452
$$197$$ 0.697224 0.0496752 0.0248376 0.999691i $$-0.492093\pi$$
0.0248376 + 0.999691i $$0.492093\pi$$
$$198$$ 41.9361 2.98027
$$199$$ 8.42221 0.597034 0.298517 0.954404i $$-0.403508\pi$$
0.298517 + 0.954404i $$0.403508\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ −13.2111 −0.931839
$$202$$ 4.60555 0.324045
$$203$$ −1.39445 −0.0978711
$$204$$ −12.9083 −0.903764
$$205$$ −9.90833 −0.692028
$$206$$ 17.1194 1.19277
$$207$$ −7.90833 −0.549667
$$208$$ −0.302776 −0.0209937
$$209$$ 26.0278 1.80038
$$210$$ 1.00000 0.0690066
$$211$$ −7.21110 −0.496433 −0.248216 0.968705i $$-0.579844\pi$$
−0.248216 + 0.968705i $$0.579844\pi$$
$$212$$ 3.21110 0.220539
$$213$$ −41.9361 −2.87341
$$214$$ −4.60555 −0.314829
$$215$$ 5.21110 0.355394
$$216$$ −16.2111 −1.10303
$$217$$ −0.880571 −0.0597770
$$218$$ −19.5139 −1.32165
$$219$$ 52.2389 3.52997
$$220$$ −5.30278 −0.357513
$$221$$ 1.18335 0.0796005
$$222$$ −26.4222 −1.77334
$$223$$ −4.00000 −0.267860 −0.133930 0.990991i $$-0.542760\pi$$
−0.133930 + 0.990991i $$0.542760\pi$$
$$224$$ 0.302776 0.0202300
$$225$$ 7.90833 0.527222
$$226$$ −12.4222 −0.826313
$$227$$ −7.39445 −0.490787 −0.245393 0.969424i $$-0.578917\pi$$
−0.245393 + 0.969424i $$0.578917\pi$$
$$228$$ −16.2111 −1.07361
$$229$$ 2.00000 0.132164 0.0660819 0.997814i $$-0.478950\pi$$
0.0660819 + 0.997814i $$0.478950\pi$$
$$230$$ 1.00000 0.0659380
$$231$$ 5.30278 0.348897
$$232$$ −4.60555 −0.302369
$$233$$ 4.18335 0.274060 0.137030 0.990567i $$-0.456244\pi$$
0.137030 + 0.990567i $$0.456244\pi$$
$$234$$ 2.39445 0.156530
$$235$$ 4.60555 0.300433
$$236$$ −10.6056 −0.690363
$$237$$ 47.6333 3.09412
$$238$$ −1.18335 −0.0767049
$$239$$ −9.21110 −0.595817 −0.297908 0.954594i $$-0.596289\pi$$
−0.297908 + 0.954594i $$0.596289\pi$$
$$240$$ 3.30278 0.213193
$$241$$ 14.4222 0.929016 0.464508 0.885569i $$-0.346231\pi$$
0.464508 + 0.885569i $$0.346231\pi$$
$$242$$ −17.1194 −1.10048
$$243$$ 49.8444 3.19752
$$244$$ −6.51388 −0.417008
$$245$$ −6.90833 −0.441357
$$246$$ 32.7250 2.08647
$$247$$ 1.48612 0.0945597
$$248$$ −2.90833 −0.184679
$$249$$ −10.6056 −0.672100
$$250$$ −1.00000 −0.0632456
$$251$$ −5.51388 −0.348033 −0.174016 0.984743i $$-0.555675\pi$$
−0.174016 + 0.984743i $$0.555675\pi$$
$$252$$ −2.39445 −0.150836
$$253$$ 5.30278 0.333383
$$254$$ 11.8167 0.741443
$$255$$ −12.9083 −0.808351
$$256$$ 1.00000 0.0625000
$$257$$ 19.8167 1.23613 0.618064 0.786127i $$-0.287917\pi$$
0.618064 + 0.786127i $$0.287917\pi$$
$$258$$ −17.2111 −1.07152
$$259$$ −2.42221 −0.150509
$$260$$ −0.302776 −0.0187773
$$261$$ 36.4222 2.25448
$$262$$ −3.21110 −0.198383
$$263$$ −14.5139 −0.894964 −0.447482 0.894293i $$-0.647679\pi$$
−0.447482 + 0.894293i $$0.647679\pi$$
$$264$$ 17.5139 1.07790
$$265$$ 3.21110 0.197256
$$266$$ −1.48612 −0.0911200
$$267$$ 0 0
$$268$$ −4.00000 −0.244339
$$269$$ 25.8167 1.57407 0.787035 0.616909i $$-0.211615\pi$$
0.787035 + 0.616909i $$0.211615\pi$$
$$270$$ −16.2111 −0.986576
$$271$$ −6.30278 −0.382866 −0.191433 0.981506i $$-0.561314\pi$$
−0.191433 + 0.981506i $$0.561314\pi$$
$$272$$ −3.90833 −0.236977
$$273$$ 0.302776 0.0183248
$$274$$ 6.90833 0.417347
$$275$$ −5.30278 −0.319769
$$276$$ −3.30278 −0.198804
$$277$$ −12.7889 −0.768410 −0.384205 0.923248i $$-0.625524\pi$$
−0.384205 + 0.923248i $$0.625524\pi$$
$$278$$ 5.39445 0.323538
$$279$$ 23.0000 1.37697
$$280$$ 0.302776 0.0180943
$$281$$ −19.3944 −1.15698 −0.578488 0.815691i $$-0.696357\pi$$
−0.578488 + 0.815691i $$0.696357\pi$$
$$282$$ −15.2111 −0.905808
$$283$$ 2.00000 0.118888 0.0594438 0.998232i $$-0.481067\pi$$
0.0594438 + 0.998232i $$0.481067\pi$$
$$284$$ −12.6972 −0.753442
$$285$$ −16.2111 −0.960263
$$286$$ −1.60555 −0.0949382
$$287$$ 3.00000 0.177084
$$288$$ −7.90833 −0.466003
$$289$$ −1.72498 −0.101469
$$290$$ −4.60555 −0.270447
$$291$$ 8.90833 0.522215
$$292$$ 15.8167 0.925600
$$293$$ 8.78890 0.513453 0.256726 0.966484i $$-0.417356\pi$$
0.256726 + 0.966484i $$0.417356\pi$$
$$294$$ 22.8167 1.33069
$$295$$ −10.6056 −0.617479
$$296$$ −8.00000 −0.464991
$$297$$ −85.9638 −4.98813
$$298$$ −9.69722 −0.561745
$$299$$ 0.302776 0.0175100
$$300$$ 3.30278 0.190686
$$301$$ −1.57779 −0.0909426
$$302$$ 1.90833 0.109812
$$303$$ −15.2111 −0.873855
$$304$$ −4.90833 −0.281512
$$305$$ −6.51388 −0.372984
$$306$$ 30.9083 1.76691
$$307$$ −15.3028 −0.873376 −0.436688 0.899613i $$-0.643849\pi$$
−0.436688 + 0.899613i $$0.643849\pi$$
$$308$$ 1.60555 0.0914848
$$309$$ −56.5416 −3.21654
$$310$$ −2.90833 −0.165182
$$311$$ −6.42221 −0.364170 −0.182085 0.983283i $$-0.558285\pi$$
−0.182085 + 0.983283i $$0.558285\pi$$
$$312$$ 1.00000 0.0566139
$$313$$ −12.7250 −0.719258 −0.359629 0.933095i $$-0.617097\pi$$
−0.359629 + 0.933095i $$0.617097\pi$$
$$314$$ 11.3944 0.643026
$$315$$ −2.39445 −0.134912
$$316$$ 14.4222 0.811312
$$317$$ −14.7250 −0.827037 −0.413519 0.910496i $$-0.635700\pi$$
−0.413519 + 0.910496i $$0.635700\pi$$
$$318$$ −10.6056 −0.594730
$$319$$ −24.4222 −1.36738
$$320$$ 1.00000 0.0559017
$$321$$ 15.2111 0.849001
$$322$$ −0.302776 −0.0168730
$$323$$ 19.1833 1.06739
$$324$$ 29.8167 1.65648
$$325$$ −0.302776 −0.0167950
$$326$$ −5.69722 −0.315540
$$327$$ 64.4500 3.56409
$$328$$ 9.90833 0.547096
$$329$$ −1.39445 −0.0768784
$$330$$ 17.5139 0.964107
$$331$$ 9.39445 0.516366 0.258183 0.966096i $$-0.416876\pi$$
0.258183 + 0.966096i $$0.416876\pi$$
$$332$$ −3.21110 −0.176232
$$333$$ 63.2666 3.46699
$$334$$ −21.2111 −1.16062
$$335$$ −4.00000 −0.218543
$$336$$ −1.00000 −0.0545545
$$337$$ −4.48612 −0.244375 −0.122187 0.992507i $$-0.538991\pi$$
−0.122187 + 0.992507i $$0.538991\pi$$
$$338$$ 12.9083 0.702120
$$339$$ 41.0278 2.22832
$$340$$ −3.90833 −0.211959
$$341$$ −15.4222 −0.835159
$$342$$ 38.8167 2.09896
$$343$$ 4.21110 0.227378
$$344$$ −5.21110 −0.280964
$$345$$ −3.30278 −0.177815
$$346$$ −23.3028 −1.25276
$$347$$ −25.5416 −1.37115 −0.685573 0.728004i $$-0.740448\pi$$
−0.685573 + 0.728004i $$0.740448\pi$$
$$348$$ 15.2111 0.815401
$$349$$ −12.7889 −0.684574 −0.342287 0.939595i $$-0.611202\pi$$
−0.342287 + 0.939595i $$0.611202\pi$$
$$350$$ 0.302776 0.0161840
$$351$$ −4.90833 −0.261987
$$352$$ 5.30278 0.282639
$$353$$ 18.4222 0.980515 0.490258 0.871578i $$-0.336903\pi$$
0.490258 + 0.871578i $$0.336903\pi$$
$$354$$ 35.0278 1.86170
$$355$$ −12.6972 −0.673899
$$356$$ 0 0
$$357$$ 3.90833 0.206851
$$358$$ −16.6056 −0.877631
$$359$$ −3.21110 −0.169476 −0.0847378 0.996403i $$-0.527005\pi$$
−0.0847378 + 0.996403i $$0.527005\pi$$
$$360$$ −7.90833 −0.416805
$$361$$ 5.09167 0.267983
$$362$$ 8.11943 0.426748
$$363$$ 56.5416 2.96767
$$364$$ 0.0916731 0.00480498
$$365$$ 15.8167 0.827881
$$366$$ 21.5139 1.12455
$$367$$ 29.2111 1.52481 0.762404 0.647102i $$-0.224019\pi$$
0.762404 + 0.647102i $$0.224019\pi$$
$$368$$ −1.00000 −0.0521286
$$369$$ −78.3583 −4.07917
$$370$$ −8.00000 −0.415900
$$371$$ −0.972244 −0.0504764
$$372$$ 9.60555 0.498025
$$373$$ −2.60555 −0.134910 −0.0674552 0.997722i $$-0.521488\pi$$
−0.0674552 + 0.997722i $$0.521488\pi$$
$$374$$ −20.7250 −1.07166
$$375$$ 3.30278 0.170555
$$376$$ −4.60555 −0.237513
$$377$$ −1.39445 −0.0718178
$$378$$ 4.90833 0.252457
$$379$$ 4.09167 0.210175 0.105088 0.994463i $$-0.466488\pi$$
0.105088 + 0.994463i $$0.466488\pi$$
$$380$$ −4.90833 −0.251792
$$381$$ −39.0278 −1.99945
$$382$$ 1.39445 0.0713462
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ −3.30278 −0.168544
$$385$$ 1.60555 0.0818265
$$386$$ −3.81665 −0.194263
$$387$$ 41.2111 2.09488
$$388$$ 2.69722 0.136931
$$389$$ 20.9361 1.06150 0.530751 0.847528i $$-0.321910\pi$$
0.530751 + 0.847528i $$0.321910\pi$$
$$390$$ 1.00000 0.0506370
$$391$$ 3.90833 0.197653
$$392$$ 6.90833 0.348923
$$393$$ 10.6056 0.534979
$$394$$ −0.697224 −0.0351257
$$395$$ 14.4222 0.725660
$$396$$ −41.9361 −2.10737
$$397$$ −21.7250 −1.09035 −0.545173 0.838324i $$-0.683536\pi$$
−0.545173 + 0.838324i $$0.683536\pi$$
$$398$$ −8.42221 −0.422167
$$399$$ 4.90833 0.245724
$$400$$ 1.00000 0.0500000
$$401$$ −1.39445 −0.0696354 −0.0348177 0.999394i $$-0.511085\pi$$
−0.0348177 + 0.999394i $$0.511085\pi$$
$$402$$ 13.2111 0.658910
$$403$$ −0.880571 −0.0438643
$$404$$ −4.60555 −0.229135
$$405$$ 29.8167 1.48160
$$406$$ 1.39445 0.0692053
$$407$$ −42.4222 −2.10279
$$408$$ 12.9083 0.639057
$$409$$ −15.0917 −0.746235 −0.373118 0.927784i $$-0.621711\pi$$
−0.373118 + 0.927784i $$0.621711\pi$$
$$410$$ 9.90833 0.489337
$$411$$ −22.8167 −1.12546
$$412$$ −17.1194 −0.843414
$$413$$ 3.21110 0.158008
$$414$$ 7.90833 0.388673
$$415$$ −3.21110 −0.157627
$$416$$ 0.302776 0.0148448
$$417$$ −17.8167 −0.872485
$$418$$ −26.0278 −1.27306
$$419$$ −39.6333 −1.93621 −0.968107 0.250538i $$-0.919393\pi$$
−0.968107 + 0.250538i $$0.919393\pi$$
$$420$$ −1.00000 −0.0487950
$$421$$ 34.3028 1.67181 0.835907 0.548870i $$-0.184942\pi$$
0.835907 + 0.548870i $$0.184942\pi$$
$$422$$ 7.21110 0.351031
$$423$$ 36.4222 1.77091
$$424$$ −3.21110 −0.155945
$$425$$ −3.90833 −0.189582
$$426$$ 41.9361 2.03181
$$427$$ 1.97224 0.0954436
$$428$$ 4.60555 0.222618
$$429$$ 5.30278 0.256020
$$430$$ −5.21110 −0.251302
$$431$$ 20.2389 0.974872 0.487436 0.873159i $$-0.337932\pi$$
0.487436 + 0.873159i $$0.337932\pi$$
$$432$$ 16.2111 0.779957
$$433$$ −34.9083 −1.67759 −0.838794 0.544450i $$-0.816739\pi$$
−0.838794 + 0.544450i $$0.816739\pi$$
$$434$$ 0.880571 0.0422687
$$435$$ 15.2111 0.729317
$$436$$ 19.5139 0.934545
$$437$$ 4.90833 0.234797
$$438$$ −52.2389 −2.49607
$$439$$ −18.3028 −0.873544 −0.436772 0.899572i $$-0.643878\pi$$
−0.436772 + 0.899572i $$0.643878\pi$$
$$440$$ 5.30278 0.252800
$$441$$ −54.6333 −2.60159
$$442$$ −1.18335 −0.0562860
$$443$$ 35.5139 1.68732 0.843658 0.536882i $$-0.180398\pi$$
0.843658 + 0.536882i $$0.180398\pi$$
$$444$$ 26.4222 1.25394
$$445$$ 0 0
$$446$$ 4.00000 0.189405
$$447$$ 32.0278 1.51486
$$448$$ −0.302776 −0.0143048
$$449$$ −12.9083 −0.609182 −0.304591 0.952483i $$-0.598520\pi$$
−0.304591 + 0.952483i $$0.598520\pi$$
$$450$$ −7.90833 −0.372802
$$451$$ 52.5416 2.47409
$$452$$ 12.4222 0.584291
$$453$$ −6.30278 −0.296130
$$454$$ 7.39445 0.347039
$$455$$ 0.0916731 0.00429770
$$456$$ 16.2111 0.759154
$$457$$ −3.57779 −0.167362 −0.0836811 0.996493i $$-0.526668\pi$$
−0.0836811 + 0.996493i $$0.526668\pi$$
$$458$$ −2.00000 −0.0934539
$$459$$ −63.3583 −2.95731
$$460$$ −1.00000 −0.0466252
$$461$$ 31.8167 1.48185 0.740925 0.671588i $$-0.234388\pi$$
0.740925 + 0.671588i $$0.234388\pi$$
$$462$$ −5.30278 −0.246707
$$463$$ −25.6333 −1.19128 −0.595640 0.803251i $$-0.703102\pi$$
−0.595640 + 0.803251i $$0.703102\pi$$
$$464$$ 4.60555 0.213807
$$465$$ 9.60555 0.445447
$$466$$ −4.18335 −0.193790
$$467$$ 19.8167 0.917005 0.458503 0.888693i $$-0.348386\pi$$
0.458503 + 0.888693i $$0.348386\pi$$
$$468$$ −2.39445 −0.110683
$$469$$ 1.21110 0.0559235
$$470$$ −4.60555 −0.212438
$$471$$ −37.6333 −1.73405
$$472$$ 10.6056 0.488160
$$473$$ −27.6333 −1.27058
$$474$$ −47.6333 −2.18787
$$475$$ −4.90833 −0.225209
$$476$$ 1.18335 0.0542386
$$477$$ 25.3944 1.16273
$$478$$ 9.21110 0.421306
$$479$$ −30.0000 −1.37073 −0.685367 0.728197i $$-0.740358\pi$$
−0.685367 + 0.728197i $$0.740358\pi$$
$$480$$ −3.30278 −0.150750
$$481$$ −2.42221 −0.110443
$$482$$ −14.4222 −0.656913
$$483$$ 1.00000 0.0455016
$$484$$ 17.1194 0.778156
$$485$$ 2.69722 0.122475
$$486$$ −49.8444 −2.26099
$$487$$ −11.8167 −0.535464 −0.267732 0.963493i $$-0.586274\pi$$
−0.267732 + 0.963493i $$0.586274\pi$$
$$488$$ 6.51388 0.294869
$$489$$ 18.8167 0.850918
$$490$$ 6.90833 0.312086
$$491$$ −25.8167 −1.16509 −0.582545 0.812799i $$-0.697943\pi$$
−0.582545 + 0.812799i $$0.697943\pi$$
$$492$$ −32.7250 −1.47536
$$493$$ −18.0000 −0.810679
$$494$$ −1.48612 −0.0668638
$$495$$ −41.9361 −1.88489
$$496$$ 2.90833 0.130588
$$497$$ 3.84441 0.172445
$$498$$ 10.6056 0.475246
$$499$$ 11.6333 0.520778 0.260389 0.965504i $$-0.416149\pi$$
0.260389 + 0.965504i $$0.416149\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 70.0555 3.12985
$$502$$ 5.51388 0.246096
$$503$$ −2.72498 −0.121501 −0.0607504 0.998153i $$-0.519349\pi$$
−0.0607504 + 0.998153i $$0.519349\pi$$
$$504$$ 2.39445 0.106657
$$505$$ −4.60555 −0.204944
$$506$$ −5.30278 −0.235737
$$507$$ −42.6333 −1.89341
$$508$$ −11.8167 −0.524279
$$509$$ 29.4500 1.30535 0.652673 0.757639i $$-0.273647\pi$$
0.652673 + 0.757639i $$0.273647\pi$$
$$510$$ 12.9083 0.571590
$$511$$ −4.78890 −0.211848
$$512$$ −1.00000 −0.0441942
$$513$$ −79.5694 −3.51307
$$514$$ −19.8167 −0.874075
$$515$$ −17.1194 −0.754372
$$516$$ 17.2111 0.757677
$$517$$ −24.4222 −1.07409
$$518$$ 2.42221 0.106426
$$519$$ 76.9638 3.37834
$$520$$ 0.302776 0.0132776
$$521$$ 6.00000 0.262865 0.131432 0.991325i $$-0.458042\pi$$
0.131432 + 0.991325i $$0.458042\pi$$
$$522$$ −36.4222 −1.59416
$$523$$ 8.42221 0.368277 0.184139 0.982900i $$-0.441050\pi$$
0.184139 + 0.982900i $$0.441050\pi$$
$$524$$ 3.21110 0.140278
$$525$$ −1.00000 −0.0436436
$$526$$ 14.5139 0.632835
$$527$$ −11.3667 −0.495141
$$528$$ −17.5139 −0.762194
$$529$$ 1.00000 0.0434783
$$530$$ −3.21110 −0.139481
$$531$$ −83.8722 −3.63974
$$532$$ 1.48612 0.0644316
$$533$$ 3.00000 0.129944
$$534$$ 0 0
$$535$$ 4.60555 0.199115
$$536$$ 4.00000 0.172774
$$537$$ 54.8444 2.36671
$$538$$ −25.8167 −1.11303
$$539$$ 36.6333 1.57791
$$540$$ 16.2111 0.697615
$$541$$ −28.8444 −1.24012 −0.620059 0.784555i $$-0.712891\pi$$
−0.620059 + 0.784555i $$0.712891\pi$$
$$542$$ 6.30278 0.270727
$$543$$ −26.8167 −1.15081
$$544$$ 3.90833 0.167568
$$545$$ 19.5139 0.835883
$$546$$ −0.302776 −0.0129576
$$547$$ 7.51388 0.321270 0.160635 0.987014i $$-0.448646\pi$$
0.160635 + 0.987014i $$0.448646\pi$$
$$548$$ −6.90833 −0.295109
$$549$$ −51.5139 −2.19856
$$550$$ 5.30278 0.226111
$$551$$ −22.6056 −0.963029
$$552$$ 3.30278 0.140575
$$553$$ −4.36669 −0.185691
$$554$$ 12.7889 0.543348
$$555$$ 26.4222 1.12156
$$556$$ −5.39445 −0.228776
$$557$$ −6.42221 −0.272118 −0.136059 0.990701i $$-0.543444\pi$$
−0.136059 + 0.990701i $$0.543444\pi$$
$$558$$ −23.0000 −0.973668
$$559$$ −1.57779 −0.0667336
$$560$$ −0.302776 −0.0127946
$$561$$ 68.4500 2.88996
$$562$$ 19.3944 0.818105
$$563$$ 39.6333 1.67034 0.835172 0.549988i $$-0.185368\pi$$
0.835172 + 0.549988i $$0.185368\pi$$
$$564$$ 15.2111 0.640503
$$565$$ 12.4222 0.522606
$$566$$ −2.00000 −0.0840663
$$567$$ −9.02776 −0.379130
$$568$$ 12.6972 0.532764
$$569$$ −0.422205 −0.0176998 −0.00884988 0.999961i $$-0.502817\pi$$
−0.00884988 + 0.999961i $$0.502817\pi$$
$$570$$ 16.2111 0.679008
$$571$$ 9.11943 0.381636 0.190818 0.981625i $$-0.438886\pi$$
0.190818 + 0.981625i $$0.438886\pi$$
$$572$$ 1.60555 0.0671315
$$573$$ −4.60555 −0.192400
$$574$$ −3.00000 −0.125218
$$575$$ −1.00000 −0.0417029
$$576$$ 7.90833 0.329514
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ 1.72498 0.0717497
$$579$$ 12.6056 0.523869
$$580$$ 4.60555 0.191235
$$581$$ 0.972244 0.0403355
$$582$$ −8.90833 −0.369262
$$583$$ −17.0278 −0.705218
$$584$$ −15.8167 −0.654498
$$585$$ −2.39445 −0.0989983
$$586$$ −8.78890 −0.363066
$$587$$ −37.5416 −1.54951 −0.774755 0.632262i $$-0.782127\pi$$
−0.774755 + 0.632262i $$0.782127\pi$$
$$588$$ −22.8167 −0.940943
$$589$$ −14.2750 −0.588192
$$590$$ 10.6056 0.436624
$$591$$ 2.30278 0.0947235
$$592$$ 8.00000 0.328798
$$593$$ 19.8167 0.813772 0.406886 0.913479i $$-0.366615\pi$$
0.406886 + 0.913479i $$0.366615\pi$$
$$594$$ 85.9638 3.52714
$$595$$ 1.18335 0.0485125
$$596$$ 9.69722 0.397214
$$597$$ 27.8167 1.13846
$$598$$ −0.302776 −0.0123814
$$599$$ 4.33053 0.176941 0.0884704 0.996079i $$-0.471802\pi$$
0.0884704 + 0.996079i $$0.471802\pi$$
$$600$$ −3.30278 −0.134835
$$601$$ −3.93608 −0.160556 −0.0802781 0.996773i $$-0.525581\pi$$
−0.0802781 + 0.996773i $$0.525581\pi$$
$$602$$ 1.57779 0.0643061
$$603$$ −31.6333 −1.28821
$$604$$ −1.90833 −0.0776487
$$605$$ 17.1194 0.696004
$$606$$ 15.2111 0.617909
$$607$$ −26.0555 −1.05756 −0.528780 0.848759i $$-0.677350\pi$$
−0.528780 + 0.848759i $$0.677350\pi$$
$$608$$ 4.90833 0.199059
$$609$$ −4.60555 −0.186626
$$610$$ 6.51388 0.263739
$$611$$ −1.39445 −0.0564134
$$612$$ −30.9083 −1.24940
$$613$$ 32.4222 1.30952 0.654760 0.755837i $$-0.272770\pi$$
0.654760 + 0.755837i $$0.272770\pi$$
$$614$$ 15.3028 0.617570
$$615$$ −32.7250 −1.31960
$$616$$ −1.60555 −0.0646895
$$617$$ 8.09167 0.325758 0.162879 0.986646i $$-0.447922\pi$$
0.162879 + 0.986646i $$0.447922\pi$$
$$618$$ 56.5416 2.27444
$$619$$ 27.3305 1.09851 0.549253 0.835656i $$-0.314912\pi$$
0.549253 + 0.835656i $$0.314912\pi$$
$$620$$ 2.90833 0.116801
$$621$$ −16.2111 −0.650529
$$622$$ 6.42221 0.257507
$$623$$ 0 0
$$624$$ −1.00000 −0.0400320
$$625$$ 1.00000 0.0400000
$$626$$ 12.7250 0.508593
$$627$$ 85.9638 3.43307
$$628$$ −11.3944 −0.454688
$$629$$ −31.2666 −1.24668
$$630$$ 2.39445 0.0953971
$$631$$ 30.6056 1.21839 0.609194 0.793021i $$-0.291493\pi$$
0.609194 + 0.793021i $$0.291493\pi$$
$$632$$ −14.4222 −0.573685
$$633$$ −23.8167 −0.946627
$$634$$ 14.7250 0.584804
$$635$$ −11.8167 −0.468930
$$636$$ 10.6056 0.420537
$$637$$ 2.09167 0.0828751
$$638$$ 24.4222 0.966884
$$639$$ −100.414 −3.97231
$$640$$ −1.00000 −0.0395285
$$641$$ −36.0000 −1.42191 −0.710957 0.703235i $$-0.751738\pi$$
−0.710957 + 0.703235i $$0.751738\pi$$
$$642$$ −15.2111 −0.600334
$$643$$ 16.2389 0.640398 0.320199 0.947350i $$-0.396250\pi$$
0.320199 + 0.947350i $$0.396250\pi$$
$$644$$ 0.302776 0.0119310
$$645$$ 17.2111 0.677687
$$646$$ −19.1833 −0.754759
$$647$$ −30.8444 −1.21262 −0.606309 0.795229i $$-0.707351\pi$$
−0.606309 + 0.795229i $$0.707351\pi$$
$$648$$ −29.8167 −1.17131
$$649$$ 56.2389 2.20757
$$650$$ 0.302776 0.0118758
$$651$$ −2.90833 −0.113986
$$652$$ 5.69722 0.223121
$$653$$ 9.27502 0.362960 0.181480 0.983395i $$-0.441911\pi$$
0.181480 + 0.983395i $$0.441911\pi$$
$$654$$ −64.4500 −2.52019
$$655$$ 3.21110 0.125468
$$656$$ −9.90833 −0.386855
$$657$$ 125.083 4.87996
$$658$$ 1.39445 0.0543613
$$659$$ −27.6333 −1.07644 −0.538220 0.842804i $$-0.680903\pi$$
−0.538220 + 0.842804i $$0.680903\pi$$
$$660$$ −17.5139 −0.681727
$$661$$ −24.0917 −0.937057 −0.468529 0.883448i $$-0.655216\pi$$
−0.468529 + 0.883448i $$0.655216\pi$$
$$662$$ −9.39445 −0.365126
$$663$$ 3.90833 0.151787
$$664$$ 3.21110 0.124615
$$665$$ 1.48612 0.0576293
$$666$$ −63.2666 −2.45153
$$667$$ −4.60555 −0.178328
$$668$$ 21.2111 0.820682
$$669$$ −13.2111 −0.510771
$$670$$ 4.00000 0.154533
$$671$$ 34.5416 1.33347
$$672$$ 1.00000 0.0385758
$$673$$ 5.63331 0.217148 0.108574 0.994088i $$-0.465372\pi$$
0.108574 + 0.994088i $$0.465372\pi$$
$$674$$ 4.48612 0.172799
$$675$$ 16.2111 0.623966
$$676$$ −12.9083 −0.496474
$$677$$ −12.4222 −0.477424 −0.238712 0.971090i $$-0.576725\pi$$
−0.238712 + 0.971090i $$0.576725\pi$$
$$678$$ −41.0278 −1.57566
$$679$$ −0.816654 −0.0313403
$$680$$ 3.90833 0.149877
$$681$$ −24.4222 −0.935861
$$682$$ 15.4222 0.590547
$$683$$ −32.7250 −1.25219 −0.626093 0.779748i $$-0.715347\pi$$
−0.626093 + 0.779748i $$0.715347\pi$$
$$684$$ −38.8167 −1.48419
$$685$$ −6.90833 −0.263954
$$686$$ −4.21110 −0.160781
$$687$$ 6.60555 0.252018
$$688$$ 5.21110 0.198671
$$689$$ −0.972244 −0.0370395
$$690$$ 3.30278 0.125735
$$691$$ 30.1833 1.14823 0.574114 0.818775i $$-0.305347\pi$$
0.574114 + 0.818775i $$0.305347\pi$$
$$692$$ 23.3028 0.885839
$$693$$ 12.6972 0.482328
$$694$$ 25.5416 0.969547
$$695$$ −5.39445 −0.204623
$$696$$ −15.2111 −0.576575
$$697$$ 38.7250 1.46681
$$698$$ 12.7889 0.484067
$$699$$ 13.8167 0.522594
$$700$$ −0.302776 −0.0114438
$$701$$ 42.9083 1.62063 0.810313 0.585998i $$-0.199297\pi$$
0.810313 + 0.585998i $$0.199297\pi$$
$$702$$ 4.90833 0.185253
$$703$$ −39.2666 −1.48097
$$704$$ −5.30278 −0.199856
$$705$$ 15.2111 0.572883
$$706$$ −18.4222 −0.693329
$$707$$ 1.39445 0.0524436
$$708$$ −35.0278 −1.31642
$$709$$ −41.1194 −1.54427 −0.772136 0.635457i $$-0.780812\pi$$
−0.772136 + 0.635457i $$0.780812\pi$$
$$710$$ 12.6972 0.476518
$$711$$ 114.056 4.27742
$$712$$ 0 0
$$713$$ −2.90833 −0.108918
$$714$$ −3.90833 −0.146265
$$715$$ 1.60555 0.0600442
$$716$$ 16.6056 0.620579
$$717$$ −30.4222 −1.13614
$$718$$ 3.21110 0.119837
$$719$$ 14.3028 0.533404 0.266702 0.963779i $$-0.414066\pi$$
0.266702 + 0.963779i $$0.414066\pi$$
$$720$$ 7.90833 0.294726
$$721$$ 5.18335 0.193038
$$722$$ −5.09167 −0.189492
$$723$$ 47.6333 1.77150
$$724$$ −8.11943 −0.301756
$$725$$ 4.60555 0.171046
$$726$$ −56.5416 −2.09846
$$727$$ −7.90833 −0.293304 −0.146652 0.989188i $$-0.546850\pi$$
−0.146652 + 0.989188i $$0.546850\pi$$
$$728$$ −0.0916731 −0.00339763
$$729$$ 75.1749 2.78426
$$730$$ −15.8167 −0.585401
$$731$$ −20.3667 −0.753289
$$732$$ −21.5139 −0.795176
$$733$$ −13.6333 −0.503558 −0.251779 0.967785i $$-0.581016\pi$$
−0.251779 + 0.967785i $$0.581016\pi$$
$$734$$ −29.2111 −1.07820
$$735$$ −22.8167 −0.841605
$$736$$ 1.00000 0.0368605
$$737$$ 21.2111 0.781321
$$738$$ 78.3583 2.88441
$$739$$ −7.63331 −0.280796 −0.140398 0.990095i $$-0.544838\pi$$
−0.140398 + 0.990095i $$0.544838\pi$$
$$740$$ 8.00000 0.294086
$$741$$ 4.90833 0.180312
$$742$$ 0.972244 0.0356922
$$743$$ 7.33053 0.268931 0.134466 0.990918i $$-0.457068\pi$$
0.134466 + 0.990918i $$0.457068\pi$$
$$744$$ −9.60555 −0.352157
$$745$$ 9.69722 0.355279
$$746$$ 2.60555 0.0953960
$$747$$ −25.3944 −0.929134
$$748$$ 20.7250 0.757780
$$749$$ −1.39445 −0.0509520
$$750$$ −3.30278 −0.120600
$$751$$ 0.183346 0.00669040 0.00334520 0.999994i $$-0.498935\pi$$
0.00334520 + 0.999994i $$0.498935\pi$$
$$752$$ 4.60555 0.167947
$$753$$ −18.2111 −0.663649
$$754$$ 1.39445 0.0507828
$$755$$ −1.90833 −0.0694511
$$756$$ −4.90833 −0.178514
$$757$$ −1.21110 −0.0440183 −0.0220091 0.999758i $$-0.507006\pi$$
−0.0220091 + 0.999758i $$0.507006\pi$$
$$758$$ −4.09167 −0.148616
$$759$$ 17.5139 0.635714
$$760$$ 4.90833 0.178044
$$761$$ 4.54163 0.164634 0.0823171 0.996606i $$-0.473768\pi$$
0.0823171 + 0.996606i $$0.473768\pi$$
$$762$$ 39.0278 1.41383
$$763$$ −5.90833 −0.213896
$$764$$ −1.39445 −0.0504494
$$765$$ −30.9083 −1.11749
$$766$$ 0 0
$$767$$ 3.21110 0.115946
$$768$$ 3.30278 0.119179
$$769$$ −41.2666 −1.48811 −0.744056 0.668117i $$-0.767100\pi$$
−0.744056 + 0.668117i $$0.767100\pi$$
$$770$$ −1.60555 −0.0578601
$$771$$ 65.4500 2.35712
$$772$$ 3.81665 0.137364
$$773$$ 12.0000 0.431610 0.215805 0.976436i $$-0.430762\pi$$
0.215805 + 0.976436i $$0.430762\pi$$
$$774$$ −41.2111 −1.48130
$$775$$ 2.90833 0.104470
$$776$$ −2.69722 −0.0968247
$$777$$ −8.00000 −0.286998
$$778$$ −20.9361 −0.750595
$$779$$ 48.6333 1.74247
$$780$$ −1.00000 −0.0358057
$$781$$ 67.3305 2.40928
$$782$$ −3.90833 −0.139761
$$783$$ 74.6611 2.66817
$$784$$ −6.90833 −0.246726
$$785$$ −11.3944 −0.406685
$$786$$ −10.6056 −0.378287
$$787$$ −27.4500 −0.978485 −0.489243 0.872148i $$-0.662727\pi$$
−0.489243 + 0.872148i $$0.662727\pi$$
$$788$$ 0.697224 0.0248376
$$789$$ −47.9361 −1.70657
$$790$$ −14.4222 −0.513119
$$791$$ −3.76114 −0.133731
$$792$$ 41.9361 1.49013
$$793$$ 1.97224 0.0700364
$$794$$ 21.7250 0.770991
$$795$$ 10.6056 0.376140
$$796$$ 8.42221 0.298517
$$797$$ −19.8167 −0.701942 −0.350971 0.936386i $$-0.614148\pi$$
−0.350971 + 0.936386i $$0.614148\pi$$
$$798$$ −4.90833 −0.173753
$$799$$ −18.0000 −0.636794
$$800$$ −1.00000 −0.0353553
$$801$$ 0 0
$$802$$ 1.39445 0.0492397
$$803$$ −83.8722 −2.95978
$$804$$ −13.2111 −0.465920
$$805$$ 0.302776 0.0106714
$$806$$ 0.880571 0.0310168
$$807$$ 85.2666 3.00153
$$808$$ 4.60555 0.162023
$$809$$ 18.2750 0.642515 0.321258 0.946992i $$-0.395894\pi$$
0.321258 + 0.946992i $$0.395894\pi$$
$$810$$ −29.8167 −1.04765
$$811$$ −4.97224 −0.174599 −0.0872995 0.996182i $$-0.527824\pi$$
−0.0872995 + 0.996182i $$0.527824\pi$$
$$812$$ −1.39445 −0.0489356
$$813$$ −20.8167 −0.730072
$$814$$ 42.4222 1.48690
$$815$$ 5.69722 0.199565
$$816$$ −12.9083 −0.451882
$$817$$ −25.5778 −0.894854
$$818$$ 15.0917 0.527668
$$819$$ 0.724981 0.0253329
$$820$$ −9.90833 −0.346014
$$821$$ −30.0000 −1.04701 −0.523504 0.852023i $$-0.675375\pi$$
−0.523504 + 0.852023i $$0.675375\pi$$
$$822$$ 22.8167 0.795822
$$823$$ −0.788897 −0.0274992 −0.0137496 0.999905i $$-0.504377\pi$$
−0.0137496 + 0.999905i $$0.504377\pi$$
$$824$$ 17.1194 0.596384
$$825$$ −17.5139 −0.609755
$$826$$ −3.21110 −0.111729
$$827$$ 35.4500 1.23272 0.616358 0.787466i $$-0.288607\pi$$
0.616358 + 0.787466i $$0.288607\pi$$
$$828$$ −7.90833 −0.274833
$$829$$ 16.7889 0.583103 0.291551 0.956555i $$-0.405829\pi$$
0.291551 + 0.956555i $$0.405829\pi$$
$$830$$ 3.21110 0.111459
$$831$$ −42.2389 −1.46525
$$832$$ −0.302776 −0.0104969
$$833$$ 27.0000 0.935495
$$834$$ 17.8167 0.616940
$$835$$ 21.2111 0.734040
$$836$$ 26.0278 0.900189
$$837$$ 47.1472 1.62965
$$838$$ 39.6333 1.36911
$$839$$ −22.1833 −0.765854 −0.382927 0.923779i $$-0.625084\pi$$
−0.382927 + 0.923779i $$0.625084\pi$$
$$840$$ 1.00000 0.0345033
$$841$$ −7.78890 −0.268583
$$842$$ −34.3028 −1.18215
$$843$$ −64.0555 −2.20619
$$844$$ −7.21110 −0.248216
$$845$$ −12.9083 −0.444060
$$846$$ −36.4222 −1.25222
$$847$$ −5.18335 −0.178102
$$848$$ 3.21110 0.110270
$$849$$ 6.60555 0.226702
$$850$$ 3.90833 0.134055
$$851$$ −8.00000 −0.274236
$$852$$ −41.9361 −1.43671
$$853$$ 10.7250 0.367216 0.183608 0.983000i $$-0.441222\pi$$
0.183608 + 0.983000i $$0.441222\pi$$
$$854$$ −1.97224 −0.0674888
$$855$$ −38.8167 −1.32750
$$856$$ −4.60555 −0.157415
$$857$$ −33.6333 −1.14889 −0.574446 0.818543i $$-0.694782\pi$$
−0.574446 + 0.818543i $$0.694782\pi$$
$$858$$ −5.30278 −0.181034
$$859$$ −14.1833 −0.483930 −0.241965 0.970285i $$-0.577792\pi$$
−0.241965 + 0.970285i $$0.577792\pi$$
$$860$$ 5.21110 0.177697
$$861$$ 9.90833 0.337675
$$862$$ −20.2389 −0.689338
$$863$$ 23.4500 0.798246 0.399123 0.916897i $$-0.369315\pi$$
0.399123 + 0.916897i $$0.369315\pi$$
$$864$$ −16.2111 −0.551513
$$865$$ 23.3028 0.792318
$$866$$ 34.9083 1.18623
$$867$$ −5.69722 −0.193488
$$868$$ −0.880571 −0.0298885
$$869$$ −76.4777 −2.59433
$$870$$ −15.2111 −0.515705
$$871$$ 1.21110 0.0410366
$$872$$ −19.5139 −0.660823
$$873$$ 21.3305 0.721929
$$874$$ −4.90833 −0.166027
$$875$$ −0.302776 −0.0102357
$$876$$ 52.2389 1.76499
$$877$$ 49.1749 1.66052 0.830260 0.557376i $$-0.188192\pi$$
0.830260 + 0.557376i $$0.188192\pi$$
$$878$$ 18.3028 0.617689
$$879$$ 29.0278 0.979082
$$880$$ −5.30278 −0.178757
$$881$$ −31.2666 −1.05340 −0.526700 0.850052i $$-0.676571\pi$$
−0.526700 + 0.850052i $$0.676571\pi$$
$$882$$ 54.6333 1.83960
$$883$$ 40.7250 1.37050 0.685252 0.728306i $$-0.259692\pi$$
0.685252 + 0.728306i $$0.259692\pi$$
$$884$$ 1.18335 0.0398002
$$885$$ −35.0278 −1.17745
$$886$$ −35.5139 −1.19311
$$887$$ −15.6333 −0.524915 −0.262458 0.964944i $$-0.584533\pi$$
−0.262458 + 0.964944i $$0.584533\pi$$
$$888$$ −26.4222 −0.886671
$$889$$ 3.57779 0.119995
$$890$$ 0 0
$$891$$ −158.111 −5.29692
$$892$$ −4.00000 −0.133930
$$893$$ −22.6056 −0.756466
$$894$$ −32.0278 −1.07117
$$895$$ 16.6056 0.555062
$$896$$ 0.302776 0.0101150
$$897$$ 1.00000 0.0333890
$$898$$ 12.9083 0.430756
$$899$$ 13.3944 0.446730
$$900$$ 7.90833 0.263611
$$901$$ −12.5500 −0.418102
$$902$$ −52.5416 −1.74945
$$903$$ −5.21110 −0.173415
$$904$$ −12.4222 −0.413156
$$905$$ −8.11943 −0.269899
$$906$$ 6.30278 0.209396
$$907$$ −30.6611 −1.01808 −0.509042 0.860742i $$-0.670000\pi$$
−0.509042 + 0.860742i $$0.670000\pi$$
$$908$$ −7.39445 −0.245393
$$909$$ −36.4222 −1.20805
$$910$$ −0.0916731 −0.00303893
$$911$$ −25.8167 −0.855344 −0.427672 0.903934i $$-0.640666\pi$$
−0.427672 + 0.903934i $$0.640666\pi$$
$$912$$ −16.2111 −0.536803
$$913$$ 17.0278 0.563536
$$914$$ 3.57779 0.118343
$$915$$ −21.5139 −0.711227
$$916$$ 2.00000 0.0660819
$$917$$ −0.972244 −0.0321063
$$918$$ 63.3583 2.09114
$$919$$ 44.0000 1.45143 0.725713 0.687998i $$-0.241510\pi$$
0.725713 + 0.687998i $$0.241510\pi$$
$$920$$ 1.00000 0.0329690
$$921$$ −50.5416 −1.66540
$$922$$ −31.8167 −1.04783
$$923$$ 3.84441 0.126540
$$924$$ 5.30278 0.174449
$$925$$ 8.00000 0.263038
$$926$$ 25.6333 0.842363
$$927$$ −135.386 −4.44666
$$928$$ −4.60555 −0.151185
$$929$$ −57.6333 −1.89089 −0.945444 0.325785i $$-0.894371\pi$$
−0.945444 + 0.325785i $$0.894371\pi$$
$$930$$ −9.60555 −0.314978
$$931$$ 33.9083 1.11130
$$932$$ 4.18335 0.137030
$$933$$ −21.2111 −0.694420
$$934$$ −19.8167 −0.648421
$$935$$ 20.7250 0.677779
$$936$$ 2.39445 0.0782650
$$937$$ −44.9638 −1.46890 −0.734452 0.678660i $$-0.762561\pi$$
−0.734452 + 0.678660i $$0.762561\pi$$
$$938$$ −1.21110 −0.0395439
$$939$$ −42.0278 −1.37152
$$940$$ 4.60555 0.150217
$$941$$ −20.9361 −0.682497 −0.341248 0.939973i $$-0.610850\pi$$
−0.341248 + 0.939973i $$0.610850\pi$$
$$942$$ 37.6333 1.22616
$$943$$ 9.90833 0.322660
$$944$$ −10.6056 −0.345181
$$945$$ −4.90833 −0.159668
$$946$$ 27.6333 0.898436
$$947$$ 41.9361 1.36274 0.681370 0.731939i $$-0.261385\pi$$
0.681370 + 0.731939i $$0.261385\pi$$
$$948$$ 47.6333 1.54706
$$949$$ −4.78890 −0.155454
$$950$$ 4.90833 0.159247
$$951$$ −48.6333 −1.57704
$$952$$ −1.18335 −0.0383525
$$953$$ −1.66947 −0.0540794 −0.0270397 0.999634i $$-0.508608\pi$$
−0.0270397 + 0.999634i $$0.508608\pi$$
$$954$$ −25.3944 −0.822176
$$955$$ −1.39445 −0.0451233
$$956$$ −9.21110 −0.297908
$$957$$ −80.6611 −2.60740
$$958$$ 30.0000 0.969256
$$959$$ 2.09167 0.0675436
$$960$$ 3.30278 0.106597
$$961$$ −22.5416 −0.727150
$$962$$ 2.42221 0.0780950
$$963$$ 36.4222 1.17369
$$964$$ 14.4222 0.464508
$$965$$ 3.81665 0.122862
$$966$$ −1.00000 −0.0321745
$$967$$ −5.39445 −0.173474 −0.0867369 0.996231i $$-0.527644\pi$$
−0.0867369 + 0.996231i $$0.527644\pi$$
$$968$$ −17.1194 −0.550239
$$969$$ 63.3583 2.03536
$$970$$ −2.69722 −0.0866027
$$971$$ −27.9083 −0.895621 −0.447810 0.894129i $$-0.647796\pi$$
−0.447810 + 0.894129i $$0.647796\pi$$
$$972$$ 49.8444 1.59876
$$973$$ 1.63331 0.0523614
$$974$$ 11.8167 0.378630
$$975$$ −1.00000 −0.0320256
$$976$$ −6.51388 −0.208504
$$977$$ −11.5139 −0.368362 −0.184181 0.982892i $$-0.558963\pi$$
−0.184181 + 0.982892i $$0.558963\pi$$
$$978$$ −18.8167 −0.601690
$$979$$ 0 0
$$980$$ −6.90833 −0.220678
$$981$$ 154.322 4.92713
$$982$$ 25.8167 0.823843
$$983$$ −19.5416 −0.623281 −0.311641 0.950200i $$-0.600879\pi$$
−0.311641 + 0.950200i $$0.600879\pi$$
$$984$$ 32.7250 1.04323
$$985$$ 0.697224 0.0222154
$$986$$ 18.0000 0.573237
$$987$$ −4.60555 −0.146596
$$988$$ 1.48612 0.0472798
$$989$$ −5.21110 −0.165703
$$990$$ 41.9361 1.33282
$$991$$ 24.3305 0.772885 0.386442 0.922314i $$-0.373704\pi$$
0.386442 + 0.922314i $$0.373704\pi$$
$$992$$ −2.90833 −0.0923395
$$993$$ 31.0278 0.984636
$$994$$ −3.84441 −0.121937
$$995$$ 8.42221 0.267002
$$996$$ −10.6056 −0.336050
$$997$$ −31.2111 −0.988466 −0.494233 0.869330i $$-0.664551\pi$$
−0.494233 + 0.869330i $$0.664551\pi$$
$$998$$ −11.6333 −0.368246
$$999$$ 129.689 4.10317
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.b.1.2 2
3.2 odd 2 2070.2.a.w.1.1 2
4.3 odd 2 1840.2.a.j.1.1 2
5.2 odd 4 1150.2.b.f.599.1 4
5.3 odd 4 1150.2.b.f.599.4 4
5.4 even 2 1150.2.a.m.1.1 2
8.3 odd 2 7360.2.a.bu.1.2 2
8.5 even 2 7360.2.a.bc.1.1 2
20.19 odd 2 9200.2.a.ca.1.2 2
23.22 odd 2 5290.2.a.j.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.b.1.2 2 1.1 even 1 trivial
1150.2.a.m.1.1 2 5.4 even 2
1150.2.b.f.599.1 4 5.2 odd 4
1150.2.b.f.599.4 4 5.3 odd 4
1840.2.a.j.1.1 2 4.3 odd 2
2070.2.a.w.1.1 2 3.2 odd 2
5290.2.a.j.1.2 2 23.22 odd 2
7360.2.a.bc.1.1 2 8.5 even 2
7360.2.a.bu.1.2 2 8.3 odd 2
9200.2.a.ca.1.2 2 20.19 odd 2