Properties

 Label 435.2.a.f.1.2 Level $435$ Weight $2$ Character 435.1 Self dual yes Analytic conductor $3.473$ 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] = [435,2,Mod(1,435)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(435, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("435.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$435 = 3 \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 435.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: $$3.47349248793$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{21})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 5$$ x^2 - x - 5 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 1.2 Root $$-1.79129$$ of defining polynomial Character $$\chi$$ $$=$$ 435.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.79129 q^{2} +1.00000 q^{3} +1.20871 q^{4} +1.00000 q^{5} +1.79129 q^{6} +1.00000 q^{7} -1.41742 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.79129 q^{2} +1.00000 q^{3} +1.20871 q^{4} +1.00000 q^{5} +1.79129 q^{6} +1.00000 q^{7} -1.41742 q^{8} +1.00000 q^{9} +1.79129 q^{10} +5.00000 q^{11} +1.20871 q^{12} -4.58258 q^{13} +1.79129 q^{14} +1.00000 q^{15} -4.95644 q^{16} -3.00000 q^{17} +1.79129 q^{18} +3.58258 q^{19} +1.20871 q^{20} +1.00000 q^{21} +8.95644 q^{22} -4.00000 q^{23} -1.41742 q^{24} +1.00000 q^{25} -8.20871 q^{26} +1.00000 q^{27} +1.20871 q^{28} +1.00000 q^{29} +1.79129 q^{30} +4.00000 q^{31} -6.04356 q^{32} +5.00000 q^{33} -5.37386 q^{34} +1.00000 q^{35} +1.20871 q^{36} -4.00000 q^{37} +6.41742 q^{38} -4.58258 q^{39} -1.41742 q^{40} -9.16515 q^{41} +1.79129 q^{42} -9.58258 q^{43} +6.04356 q^{44} +1.00000 q^{45} -7.16515 q^{46} +10.5826 q^{47} -4.95644 q^{48} -6.00000 q^{49} +1.79129 q^{50} -3.00000 q^{51} -5.53901 q^{52} +0.417424 q^{53} +1.79129 q^{54} +5.00000 q^{55} -1.41742 q^{56} +3.58258 q^{57} +1.79129 q^{58} -7.58258 q^{59} +1.20871 q^{60} +12.7477 q^{61} +7.16515 q^{62} +1.00000 q^{63} -0.912878 q^{64} -4.58258 q^{65} +8.95644 q^{66} -4.16515 q^{67} -3.62614 q^{68} -4.00000 q^{69} +1.79129 q^{70} -9.58258 q^{71} -1.41742 q^{72} +4.00000 q^{73} -7.16515 q^{74} +1.00000 q^{75} +4.33030 q^{76} +5.00000 q^{77} -8.20871 q^{78} +7.58258 q^{79} -4.95644 q^{80} +1.00000 q^{81} -16.4174 q^{82} -11.5826 q^{83} +1.20871 q^{84} -3.00000 q^{85} -17.1652 q^{86} +1.00000 q^{87} -7.08712 q^{88} +1.41742 q^{89} +1.79129 q^{90} -4.58258 q^{91} -4.83485 q^{92} +4.00000 q^{93} +18.9564 q^{94} +3.58258 q^{95} -6.04356 q^{96} +11.5826 q^{97} -10.7477 q^{98} +5.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 2 q^{3} + 7 q^{4} + 2 q^{5} - q^{6} + 2 q^{7} - 12 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - q^2 + 2 * q^3 + 7 * q^4 + 2 * q^5 - q^6 + 2 * q^7 - 12 * q^8 + 2 * q^9 $$2 q - q^{2} + 2 q^{3} + 7 q^{4} + 2 q^{5} - q^{6} + 2 q^{7} - 12 q^{8} + 2 q^{9} - q^{10} + 10 q^{11} + 7 q^{12} - q^{14} + 2 q^{15} + 13 q^{16} - 6 q^{17} - q^{18} - 2 q^{19} + 7 q^{20} + 2 q^{21} - 5 q^{22} - 8 q^{23} - 12 q^{24} + 2 q^{25} - 21 q^{26} + 2 q^{27} + 7 q^{28} + 2 q^{29} - q^{30} + 8 q^{31} - 35 q^{32} + 10 q^{33} + 3 q^{34} + 2 q^{35} + 7 q^{36} - 8 q^{37} + 22 q^{38} - 12 q^{40} - q^{42} - 10 q^{43} + 35 q^{44} + 2 q^{45} + 4 q^{46} + 12 q^{47} + 13 q^{48} - 12 q^{49} - q^{50} - 6 q^{51} + 21 q^{52} + 10 q^{53} - q^{54} + 10 q^{55} - 12 q^{56} - 2 q^{57} - q^{58} - 6 q^{59} + 7 q^{60} - 2 q^{61} - 4 q^{62} + 2 q^{63} + 44 q^{64} - 5 q^{66} + 10 q^{67} - 21 q^{68} - 8 q^{69} - q^{70} - 10 q^{71} - 12 q^{72} + 8 q^{73} + 4 q^{74} + 2 q^{75} - 28 q^{76} + 10 q^{77} - 21 q^{78} + 6 q^{79} + 13 q^{80} + 2 q^{81} - 42 q^{82} - 14 q^{83} + 7 q^{84} - 6 q^{85} - 16 q^{86} + 2 q^{87} - 60 q^{88} + 12 q^{89} - q^{90} - 28 q^{92} + 8 q^{93} + 15 q^{94} - 2 q^{95} - 35 q^{96} + 14 q^{97} + 6 q^{98} + 10 q^{99}+O(q^{100})$$ 2 * q - q^2 + 2 * q^3 + 7 * q^4 + 2 * q^5 - q^6 + 2 * q^7 - 12 * q^8 + 2 * q^9 - q^10 + 10 * q^11 + 7 * q^12 - q^14 + 2 * q^15 + 13 * q^16 - 6 * q^17 - q^18 - 2 * q^19 + 7 * q^20 + 2 * q^21 - 5 * q^22 - 8 * q^23 - 12 * q^24 + 2 * q^25 - 21 * q^26 + 2 * q^27 + 7 * q^28 + 2 * q^29 - q^30 + 8 * q^31 - 35 * q^32 + 10 * q^33 + 3 * q^34 + 2 * q^35 + 7 * q^36 - 8 * q^37 + 22 * q^38 - 12 * q^40 - q^42 - 10 * q^43 + 35 * q^44 + 2 * q^45 + 4 * q^46 + 12 * q^47 + 13 * q^48 - 12 * q^49 - q^50 - 6 * q^51 + 21 * q^52 + 10 * q^53 - q^54 + 10 * q^55 - 12 * q^56 - 2 * q^57 - q^58 - 6 * q^59 + 7 * q^60 - 2 * q^61 - 4 * q^62 + 2 * q^63 + 44 * q^64 - 5 * q^66 + 10 * q^67 - 21 * q^68 - 8 * q^69 - q^70 - 10 * q^71 - 12 * q^72 + 8 * q^73 + 4 * q^74 + 2 * q^75 - 28 * q^76 + 10 * q^77 - 21 * q^78 + 6 * q^79 + 13 * q^80 + 2 * q^81 - 42 * q^82 - 14 * q^83 + 7 * q^84 - 6 * q^85 - 16 * q^86 + 2 * q^87 - 60 * q^88 + 12 * q^89 - q^90 - 28 * q^92 + 8 * q^93 + 15 * q^94 - 2 * q^95 - 35 * q^96 + 14 * q^97 + 6 * q^98 + 10 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.79129 1.26663 0.633316 0.773893i $$-0.281693\pi$$
0.633316 + 0.773893i $$0.281693\pi$$
$$3$$ 1.00000 0.577350
$$4$$ 1.20871 0.604356
$$5$$ 1.00000 0.447214
$$6$$ 1.79129 0.731290
$$7$$ 1.00000 0.377964 0.188982 0.981981i $$-0.439481\pi$$
0.188982 + 0.981981i $$0.439481\pi$$
$$8$$ −1.41742 −0.501135
$$9$$ 1.00000 0.333333
$$10$$ 1.79129 0.566455
$$11$$ 5.00000 1.50756 0.753778 0.657129i $$-0.228229\pi$$
0.753778 + 0.657129i $$0.228229\pi$$
$$12$$ 1.20871 0.348925
$$13$$ −4.58258 −1.27098 −0.635489 0.772110i $$-0.719201\pi$$
−0.635489 + 0.772110i $$0.719201\pi$$
$$14$$ 1.79129 0.478742
$$15$$ 1.00000 0.258199
$$16$$ −4.95644 −1.23911
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ 1.79129 0.422211
$$19$$ 3.58258 0.821899 0.410950 0.911658i $$-0.365197\pi$$
0.410950 + 0.911658i $$0.365197\pi$$
$$20$$ 1.20871 0.270276
$$21$$ 1.00000 0.218218
$$22$$ 8.95644 1.90952
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ −1.41742 −0.289331
$$25$$ 1.00000 0.200000
$$26$$ −8.20871 −1.60986
$$27$$ 1.00000 0.192450
$$28$$ 1.20871 0.228425
$$29$$ 1.00000 0.185695
$$30$$ 1.79129 0.327043
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ −6.04356 −1.06836
$$33$$ 5.00000 0.870388
$$34$$ −5.37386 −0.921610
$$35$$ 1.00000 0.169031
$$36$$ 1.20871 0.201452
$$37$$ −4.00000 −0.657596 −0.328798 0.944400i $$-0.606644\pi$$
−0.328798 + 0.944400i $$0.606644\pi$$
$$38$$ 6.41742 1.04104
$$39$$ −4.58258 −0.733799
$$40$$ −1.41742 −0.224114
$$41$$ −9.16515 −1.43136 −0.715678 0.698430i $$-0.753882\pi$$
−0.715678 + 0.698430i $$0.753882\pi$$
$$42$$ 1.79129 0.276402
$$43$$ −9.58258 −1.46133 −0.730665 0.682737i $$-0.760790\pi$$
−0.730665 + 0.682737i $$0.760790\pi$$
$$44$$ 6.04356 0.911101
$$45$$ 1.00000 0.149071
$$46$$ −7.16515 −1.05644
$$47$$ 10.5826 1.54363 0.771814 0.635849i $$-0.219350\pi$$
0.771814 + 0.635849i $$0.219350\pi$$
$$48$$ −4.95644 −0.715400
$$49$$ −6.00000 −0.857143
$$50$$ 1.79129 0.253326
$$51$$ −3.00000 −0.420084
$$52$$ −5.53901 −0.768123
$$53$$ 0.417424 0.0573376 0.0286688 0.999589i $$-0.490873\pi$$
0.0286688 + 0.999589i $$0.490873\pi$$
$$54$$ 1.79129 0.243763
$$55$$ 5.00000 0.674200
$$56$$ −1.41742 −0.189411
$$57$$ 3.58258 0.474524
$$58$$ 1.79129 0.235208
$$59$$ −7.58258 −0.987167 −0.493584 0.869698i $$-0.664313\pi$$
−0.493584 + 0.869698i $$0.664313\pi$$
$$60$$ 1.20871 0.156044
$$61$$ 12.7477 1.63218 0.816090 0.577925i $$-0.196138\pi$$
0.816090 + 0.577925i $$0.196138\pi$$
$$62$$ 7.16515 0.909975
$$63$$ 1.00000 0.125988
$$64$$ −0.912878 −0.114110
$$65$$ −4.58258 −0.568399
$$66$$ 8.95644 1.10246
$$67$$ −4.16515 −0.508854 −0.254427 0.967092i $$-0.581887\pi$$
−0.254427 + 0.967092i $$0.581887\pi$$
$$68$$ −3.62614 −0.439734
$$69$$ −4.00000 −0.481543
$$70$$ 1.79129 0.214100
$$71$$ −9.58258 −1.13724 −0.568621 0.822599i $$-0.692523\pi$$
−0.568621 + 0.822599i $$0.692523\pi$$
$$72$$ −1.41742 −0.167045
$$73$$ 4.00000 0.468165 0.234082 0.972217i $$-0.424791\pi$$
0.234082 + 0.972217i $$0.424791\pi$$
$$74$$ −7.16515 −0.832932
$$75$$ 1.00000 0.115470
$$76$$ 4.33030 0.496720
$$77$$ 5.00000 0.569803
$$78$$ −8.20871 −0.929454
$$79$$ 7.58258 0.853106 0.426553 0.904462i $$-0.359728\pi$$
0.426553 + 0.904462i $$0.359728\pi$$
$$80$$ −4.95644 −0.554147
$$81$$ 1.00000 0.111111
$$82$$ −16.4174 −1.81300
$$83$$ −11.5826 −1.27135 −0.635676 0.771956i $$-0.719279\pi$$
−0.635676 + 0.771956i $$0.719279\pi$$
$$84$$ 1.20871 0.131881
$$85$$ −3.00000 −0.325396
$$86$$ −17.1652 −1.85097
$$87$$ 1.00000 0.107211
$$88$$ −7.08712 −0.755490
$$89$$ 1.41742 0.150247 0.0751233 0.997174i $$-0.476065\pi$$
0.0751233 + 0.997174i $$0.476065\pi$$
$$90$$ 1.79129 0.188818
$$91$$ −4.58258 −0.480384
$$92$$ −4.83485 −0.504068
$$93$$ 4.00000 0.414781
$$94$$ 18.9564 1.95521
$$95$$ 3.58258 0.367565
$$96$$ −6.04356 −0.616818
$$97$$ 11.5826 1.17603 0.588016 0.808849i $$-0.299909\pi$$
0.588016 + 0.808849i $$0.299909\pi$$
$$98$$ −10.7477 −1.08568
$$99$$ 5.00000 0.502519
$$100$$ 1.20871 0.120871
$$101$$ −0.582576 −0.0579684 −0.0289842 0.999580i $$-0.509227\pi$$
−0.0289842 + 0.999580i $$0.509227\pi$$
$$102$$ −5.37386 −0.532092
$$103$$ 15.1652 1.49427 0.747133 0.664674i $$-0.231430\pi$$
0.747133 + 0.664674i $$0.231430\pi$$
$$104$$ 6.49545 0.636932
$$105$$ 1.00000 0.0975900
$$106$$ 0.747727 0.0726257
$$107$$ 5.16515 0.499334 0.249667 0.968332i $$-0.419679\pi$$
0.249667 + 0.968332i $$0.419679\pi$$
$$108$$ 1.20871 0.116308
$$109$$ 14.1652 1.35678 0.678388 0.734704i $$-0.262679\pi$$
0.678388 + 0.734704i $$0.262679\pi$$
$$110$$ 8.95644 0.853963
$$111$$ −4.00000 −0.379663
$$112$$ −4.95644 −0.468339
$$113$$ −14.1652 −1.33255 −0.666273 0.745708i $$-0.732111\pi$$
−0.666273 + 0.745708i $$0.732111\pi$$
$$114$$ 6.41742 0.601047
$$115$$ −4.00000 −0.373002
$$116$$ 1.20871 0.112226
$$117$$ −4.58258 −0.423659
$$118$$ −13.5826 −1.25038
$$119$$ −3.00000 −0.275010
$$120$$ −1.41742 −0.129393
$$121$$ 14.0000 1.27273
$$122$$ 22.8348 2.06737
$$123$$ −9.16515 −0.826394
$$124$$ 4.83485 0.434182
$$125$$ 1.00000 0.0894427
$$126$$ 1.79129 0.159581
$$127$$ 2.00000 0.177471 0.0887357 0.996055i $$-0.471717\pi$$
0.0887357 + 0.996055i $$0.471717\pi$$
$$128$$ 10.4519 0.923826
$$129$$ −9.58258 −0.843699
$$130$$ −8.20871 −0.719952
$$131$$ −15.0000 −1.31056 −0.655278 0.755388i $$-0.727449\pi$$
−0.655278 + 0.755388i $$0.727449\pi$$
$$132$$ 6.04356 0.526024
$$133$$ 3.58258 0.310649
$$134$$ −7.46099 −0.644531
$$135$$ 1.00000 0.0860663
$$136$$ 4.25227 0.364629
$$137$$ 16.3303 1.39519 0.697596 0.716491i $$-0.254253\pi$$
0.697596 + 0.716491i $$0.254253\pi$$
$$138$$ −7.16515 −0.609938
$$139$$ −9.41742 −0.798776 −0.399388 0.916782i $$-0.630777\pi$$
−0.399388 + 0.916782i $$0.630777\pi$$
$$140$$ 1.20871 0.102155
$$141$$ 10.5826 0.891214
$$142$$ −17.1652 −1.44047
$$143$$ −22.9129 −1.91607
$$144$$ −4.95644 −0.413037
$$145$$ 1.00000 0.0830455
$$146$$ 7.16515 0.592992
$$147$$ −6.00000 −0.494872
$$148$$ −4.83485 −0.397422
$$149$$ 16.7477 1.37203 0.686014 0.727589i $$-0.259359\pi$$
0.686014 + 0.727589i $$0.259359\pi$$
$$150$$ 1.79129 0.146258
$$151$$ 7.16515 0.583092 0.291546 0.956557i $$-0.405830\pi$$
0.291546 + 0.956557i $$0.405830\pi$$
$$152$$ −5.07803 −0.411883
$$153$$ −3.00000 −0.242536
$$154$$ 8.95644 0.721730
$$155$$ 4.00000 0.321288
$$156$$ −5.53901 −0.443476
$$157$$ 10.7477 0.857762 0.428881 0.903361i $$-0.358908\pi$$
0.428881 + 0.903361i $$0.358908\pi$$
$$158$$ 13.5826 1.08057
$$159$$ 0.417424 0.0331039
$$160$$ −6.04356 −0.477785
$$161$$ −4.00000 −0.315244
$$162$$ 1.79129 0.140737
$$163$$ 7.58258 0.593913 0.296957 0.954891i $$-0.404028\pi$$
0.296957 + 0.954891i $$0.404028\pi$$
$$164$$ −11.0780 −0.865049
$$165$$ 5.00000 0.389249
$$166$$ −20.7477 −1.61034
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ −1.41742 −0.109357
$$169$$ 8.00000 0.615385
$$170$$ −5.37386 −0.412157
$$171$$ 3.58258 0.273966
$$172$$ −11.5826 −0.883163
$$173$$ −3.16515 −0.240642 −0.120321 0.992735i $$-0.538392\pi$$
−0.120321 + 0.992735i $$0.538392\pi$$
$$174$$ 1.79129 0.135797
$$175$$ 1.00000 0.0755929
$$176$$ −24.7822 −1.86803
$$177$$ −7.58258 −0.569941
$$178$$ 2.53901 0.190307
$$179$$ 4.74773 0.354862 0.177431 0.984133i $$-0.443221\pi$$
0.177431 + 0.984133i $$0.443221\pi$$
$$180$$ 1.20871 0.0900921
$$181$$ 16.1652 1.20155 0.600773 0.799420i $$-0.294860\pi$$
0.600773 + 0.799420i $$0.294860\pi$$
$$182$$ −8.20871 −0.608470
$$183$$ 12.7477 0.942339
$$184$$ 5.66970 0.417976
$$185$$ −4.00000 −0.294086
$$186$$ 7.16515 0.525374
$$187$$ −15.0000 −1.09691
$$188$$ 12.7913 0.932901
$$189$$ 1.00000 0.0727393
$$190$$ 6.41742 0.465569
$$191$$ −4.00000 −0.289430 −0.144715 0.989473i $$-0.546227\pi$$
−0.144715 + 0.989473i $$0.546227\pi$$
$$192$$ −0.912878 −0.0658813
$$193$$ −20.3303 −1.46341 −0.731704 0.681623i $$-0.761274\pi$$
−0.731704 + 0.681623i $$0.761274\pi$$
$$194$$ 20.7477 1.48960
$$195$$ −4.58258 −0.328165
$$196$$ −7.25227 −0.518019
$$197$$ −16.3303 −1.16349 −0.581743 0.813373i $$-0.697629\pi$$
−0.581743 + 0.813373i $$0.697629\pi$$
$$198$$ 8.95644 0.636506
$$199$$ 13.4174 0.951136 0.475568 0.879679i $$-0.342243\pi$$
0.475568 + 0.879679i $$0.342243\pi$$
$$200$$ −1.41742 −0.100227
$$201$$ −4.16515 −0.293787
$$202$$ −1.04356 −0.0734247
$$203$$ 1.00000 0.0701862
$$204$$ −3.62614 −0.253880
$$205$$ −9.16515 −0.640122
$$206$$ 27.1652 1.89269
$$207$$ −4.00000 −0.278019
$$208$$ 22.7133 1.57488
$$209$$ 17.9129 1.23906
$$210$$ 1.79129 0.123611
$$211$$ 20.3303 1.39960 0.699798 0.714341i $$-0.253273\pi$$
0.699798 + 0.714341i $$0.253273\pi$$
$$212$$ 0.504546 0.0346523
$$213$$ −9.58258 −0.656587
$$214$$ 9.25227 0.632472
$$215$$ −9.58258 −0.653526
$$216$$ −1.41742 −0.0964435
$$217$$ 4.00000 0.271538
$$218$$ 25.3739 1.71853
$$219$$ 4.00000 0.270295
$$220$$ 6.04356 0.407457
$$221$$ 13.7477 0.924772
$$222$$ −7.16515 −0.480893
$$223$$ 7.00000 0.468755 0.234377 0.972146i $$-0.424695\pi$$
0.234377 + 0.972146i $$0.424695\pi$$
$$224$$ −6.04356 −0.403802
$$225$$ 1.00000 0.0666667
$$226$$ −25.3739 −1.68784
$$227$$ −7.58258 −0.503273 −0.251637 0.967822i $$-0.580969\pi$$
−0.251637 + 0.967822i $$0.580969\pi$$
$$228$$ 4.33030 0.286781
$$229$$ −17.1652 −1.13431 −0.567153 0.823613i $$-0.691955\pi$$
−0.567153 + 0.823613i $$0.691955\pi$$
$$230$$ −7.16515 −0.472456
$$231$$ 5.00000 0.328976
$$232$$ −1.41742 −0.0930585
$$233$$ −13.1652 −0.862478 −0.431239 0.902238i $$-0.641923\pi$$
−0.431239 + 0.902238i $$0.641923\pi$$
$$234$$ −8.20871 −0.536620
$$235$$ 10.5826 0.690331
$$236$$ −9.16515 −0.596601
$$237$$ 7.58258 0.492541
$$238$$ −5.37386 −0.348336
$$239$$ −26.0000 −1.68180 −0.840900 0.541190i $$-0.817974\pi$$
−0.840900 + 0.541190i $$0.817974\pi$$
$$240$$ −4.95644 −0.319937
$$241$$ 29.3303 1.88933 0.944665 0.328035i $$-0.106387\pi$$
0.944665 + 0.328035i $$0.106387\pi$$
$$242$$ 25.0780 1.61208
$$243$$ 1.00000 0.0641500
$$244$$ 15.4083 0.986417
$$245$$ −6.00000 −0.383326
$$246$$ −16.4174 −1.04674
$$247$$ −16.4174 −1.04462
$$248$$ −5.66970 −0.360026
$$249$$ −11.5826 −0.734016
$$250$$ 1.79129 0.113291
$$251$$ 16.1652 1.02034 0.510168 0.860075i $$-0.329583\pi$$
0.510168 + 0.860075i $$0.329583\pi$$
$$252$$ 1.20871 0.0761417
$$253$$ −20.0000 −1.25739
$$254$$ 3.58258 0.224791
$$255$$ −3.00000 −0.187867
$$256$$ 20.5481 1.28426
$$257$$ 12.7477 0.795181 0.397591 0.917563i $$-0.369846\pi$$
0.397591 + 0.917563i $$0.369846\pi$$
$$258$$ −17.1652 −1.06866
$$259$$ −4.00000 −0.248548
$$260$$ −5.53901 −0.343515
$$261$$ 1.00000 0.0618984
$$262$$ −26.8693 −1.65999
$$263$$ 30.3303 1.87025 0.935123 0.354322i $$-0.115288\pi$$
0.935123 + 0.354322i $$0.115288\pi$$
$$264$$ −7.08712 −0.436182
$$265$$ 0.417424 0.0256422
$$266$$ 6.41742 0.393478
$$267$$ 1.41742 0.0867450
$$268$$ −5.03447 −0.307529
$$269$$ 22.5826 1.37688 0.688442 0.725291i $$-0.258295\pi$$
0.688442 + 0.725291i $$0.258295\pi$$
$$270$$ 1.79129 0.109014
$$271$$ 1.16515 0.0707779 0.0353890 0.999374i $$-0.488733\pi$$
0.0353890 + 0.999374i $$0.488733\pi$$
$$272$$ 14.8693 0.901585
$$273$$ −4.58258 −0.277350
$$274$$ 29.2523 1.76719
$$275$$ 5.00000 0.301511
$$276$$ −4.83485 −0.291024
$$277$$ −24.9129 −1.49687 −0.748435 0.663208i $$-0.769194\pi$$
−0.748435 + 0.663208i $$0.769194\pi$$
$$278$$ −16.8693 −1.01175
$$279$$ 4.00000 0.239474
$$280$$ −1.41742 −0.0847073
$$281$$ −0.417424 −0.0249014 −0.0124507 0.999922i $$-0.503963\pi$$
−0.0124507 + 0.999922i $$0.503963\pi$$
$$282$$ 18.9564 1.12884
$$283$$ −0.834849 −0.0496266 −0.0248133 0.999692i $$-0.507899\pi$$
−0.0248133 + 0.999692i $$0.507899\pi$$
$$284$$ −11.5826 −0.687299
$$285$$ 3.58258 0.212213
$$286$$ −41.0436 −2.42696
$$287$$ −9.16515 −0.541002
$$288$$ −6.04356 −0.356120
$$289$$ −8.00000 −0.470588
$$290$$ 1.79129 0.105188
$$291$$ 11.5826 0.678983
$$292$$ 4.83485 0.282938
$$293$$ 30.1652 1.76227 0.881133 0.472868i $$-0.156781\pi$$
0.881133 + 0.472868i $$0.156781\pi$$
$$294$$ −10.7477 −0.626820
$$295$$ −7.58258 −0.441475
$$296$$ 5.66970 0.329544
$$297$$ 5.00000 0.290129
$$298$$ 30.0000 1.73785
$$299$$ 18.3303 1.06007
$$300$$ 1.20871 0.0697850
$$301$$ −9.58258 −0.552330
$$302$$ 12.8348 0.738563
$$303$$ −0.582576 −0.0334681
$$304$$ −17.7568 −1.01842
$$305$$ 12.7477 0.729933
$$306$$ −5.37386 −0.307203
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ 6.04356 0.344364
$$309$$ 15.1652 0.862715
$$310$$ 7.16515 0.406953
$$311$$ −3.00000 −0.170114 −0.0850572 0.996376i $$-0.527107\pi$$
−0.0850572 + 0.996376i $$0.527107\pi$$
$$312$$ 6.49545 0.367733
$$313$$ 10.5826 0.598163 0.299081 0.954228i $$-0.403320\pi$$
0.299081 + 0.954228i $$0.403320\pi$$
$$314$$ 19.2523 1.08647
$$315$$ 1.00000 0.0563436
$$316$$ 9.16515 0.515580
$$317$$ −25.0000 −1.40414 −0.702070 0.712108i $$-0.747741\pi$$
−0.702070 + 0.712108i $$0.747741\pi$$
$$318$$ 0.747727 0.0419305
$$319$$ 5.00000 0.279946
$$320$$ −0.912878 −0.0510315
$$321$$ 5.16515 0.288291
$$322$$ −7.16515 −0.399298
$$323$$ −10.7477 −0.598020
$$324$$ 1.20871 0.0671507
$$325$$ −4.58258 −0.254196
$$326$$ 13.5826 0.752269
$$327$$ 14.1652 0.783335
$$328$$ 12.9909 0.717303
$$329$$ 10.5826 0.583436
$$330$$ 8.95644 0.493036
$$331$$ −8.33030 −0.457875 −0.228937 0.973441i $$-0.573525\pi$$
−0.228937 + 0.973441i $$0.573525\pi$$
$$332$$ −14.0000 −0.768350
$$333$$ −4.00000 −0.219199
$$334$$ 0 0
$$335$$ −4.16515 −0.227567
$$336$$ −4.95644 −0.270396
$$337$$ −21.1652 −1.15294 −0.576470 0.817119i $$-0.695570\pi$$
−0.576470 + 0.817119i $$0.695570\pi$$
$$338$$ 14.3303 0.779466
$$339$$ −14.1652 −0.769345
$$340$$ −3.62614 −0.196655
$$341$$ 20.0000 1.08306
$$342$$ 6.41742 0.347015
$$343$$ −13.0000 −0.701934
$$344$$ 13.5826 0.732323
$$345$$ −4.00000 −0.215353
$$346$$ −5.66970 −0.304805
$$347$$ 7.16515 0.384645 0.192323 0.981332i $$-0.438398\pi$$
0.192323 + 0.981332i $$0.438398\pi$$
$$348$$ 1.20871 0.0647938
$$349$$ −4.33030 −0.231796 −0.115898 0.993261i $$-0.536975\pi$$
−0.115898 + 0.993261i $$0.536975\pi$$
$$350$$ 1.79129 0.0957484
$$351$$ −4.58258 −0.244600
$$352$$ −30.2178 −1.61061
$$353$$ −14.8348 −0.789579 −0.394790 0.918772i $$-0.629183\pi$$
−0.394790 + 0.918772i $$0.629183\pi$$
$$354$$ −13.5826 −0.721906
$$355$$ −9.58258 −0.508590
$$356$$ 1.71326 0.0908025
$$357$$ −3.00000 −0.158777
$$358$$ 8.50455 0.449479
$$359$$ −27.1652 −1.43372 −0.716861 0.697216i $$-0.754422\pi$$
−0.716861 + 0.697216i $$0.754422\pi$$
$$360$$ −1.41742 −0.0747048
$$361$$ −6.16515 −0.324482
$$362$$ 28.9564 1.52192
$$363$$ 14.0000 0.734809
$$364$$ −5.53901 −0.290323
$$365$$ 4.00000 0.209370
$$366$$ 22.8348 1.19360
$$367$$ −37.4955 −1.95725 −0.978623 0.205661i $$-0.934066\pi$$
−0.978623 + 0.205661i $$0.934066\pi$$
$$368$$ 19.8258 1.03349
$$369$$ −9.16515 −0.477119
$$370$$ −7.16515 −0.372498
$$371$$ 0.417424 0.0216716
$$372$$ 4.83485 0.250675
$$373$$ 28.3303 1.46689 0.733444 0.679750i $$-0.237912\pi$$
0.733444 + 0.679750i $$0.237912\pi$$
$$374$$ −26.8693 −1.38938
$$375$$ 1.00000 0.0516398
$$376$$ −15.0000 −0.773566
$$377$$ −4.58258 −0.236015
$$378$$ 1.79129 0.0921339
$$379$$ −26.0000 −1.33553 −0.667765 0.744372i $$-0.732749\pi$$
−0.667765 + 0.744372i $$0.732749\pi$$
$$380$$ 4.33030 0.222140
$$381$$ 2.00000 0.102463
$$382$$ −7.16515 −0.366601
$$383$$ 3.58258 0.183061 0.0915305 0.995802i $$-0.470824\pi$$
0.0915305 + 0.995802i $$0.470824\pi$$
$$384$$ 10.4519 0.533371
$$385$$ 5.00000 0.254824
$$386$$ −36.4174 −1.85360
$$387$$ −9.58258 −0.487110
$$388$$ 14.0000 0.710742
$$389$$ −6.58258 −0.333750 −0.166875 0.985978i $$-0.553368\pi$$
−0.166875 + 0.985978i $$0.553368\pi$$
$$390$$ −8.20871 −0.415664
$$391$$ 12.0000 0.606866
$$392$$ 8.50455 0.429544
$$393$$ −15.0000 −0.756650
$$394$$ −29.2523 −1.47371
$$395$$ 7.58258 0.381521
$$396$$ 6.04356 0.303700
$$397$$ 10.8348 0.543785 0.271893 0.962328i $$-0.412350\pi$$
0.271893 + 0.962328i $$0.412350\pi$$
$$398$$ 24.0345 1.20474
$$399$$ 3.58258 0.179353
$$400$$ −4.95644 −0.247822
$$401$$ 12.4174 0.620097 0.310048 0.950721i $$-0.399655\pi$$
0.310048 + 0.950721i $$0.399655\pi$$
$$402$$ −7.46099 −0.372120
$$403$$ −18.3303 −0.913097
$$404$$ −0.704166 −0.0350336
$$405$$ 1.00000 0.0496904
$$406$$ 1.79129 0.0889001
$$407$$ −20.0000 −0.991363
$$408$$ 4.25227 0.210519
$$409$$ −2.74773 −0.135866 −0.0679332 0.997690i $$-0.521640\pi$$
−0.0679332 + 0.997690i $$0.521640\pi$$
$$410$$ −16.4174 −0.810799
$$411$$ 16.3303 0.805514
$$412$$ 18.3303 0.903069
$$413$$ −7.58258 −0.373114
$$414$$ −7.16515 −0.352148
$$415$$ −11.5826 −0.568566
$$416$$ 27.6951 1.35786
$$417$$ −9.41742 −0.461173
$$418$$ 32.0871 1.56943
$$419$$ −37.1652 −1.81564 −0.907818 0.419364i $$-0.862253\pi$$
−0.907818 + 0.419364i $$0.862253\pi$$
$$420$$ 1.20871 0.0589791
$$421$$ 4.41742 0.215292 0.107646 0.994189i $$-0.465669\pi$$
0.107646 + 0.994189i $$0.465669\pi$$
$$422$$ 36.4174 1.77277
$$423$$ 10.5826 0.514542
$$424$$ −0.591667 −0.0287339
$$425$$ −3.00000 −0.145521
$$426$$ −17.1652 −0.831654
$$427$$ 12.7477 0.616906
$$428$$ 6.24318 0.301776
$$429$$ −22.9129 −1.10624
$$430$$ −17.1652 −0.827777
$$431$$ −39.1652 −1.88652 −0.943259 0.332057i $$-0.892258\pi$$
−0.943259 + 0.332057i $$0.892258\pi$$
$$432$$ −4.95644 −0.238467
$$433$$ 25.0780 1.20517 0.602587 0.798053i $$-0.294137\pi$$
0.602587 + 0.798053i $$0.294137\pi$$
$$434$$ 7.16515 0.343938
$$435$$ 1.00000 0.0479463
$$436$$ 17.1216 0.819975
$$437$$ −14.3303 −0.685511
$$438$$ 7.16515 0.342364
$$439$$ −16.9129 −0.807208 −0.403604 0.914934i $$-0.632243\pi$$
−0.403604 + 0.914934i $$0.632243\pi$$
$$440$$ −7.08712 −0.337865
$$441$$ −6.00000 −0.285714
$$442$$ 24.6261 1.17135
$$443$$ 0.582576 0.0276790 0.0138395 0.999904i $$-0.495595\pi$$
0.0138395 + 0.999904i $$0.495595\pi$$
$$444$$ −4.83485 −0.229452
$$445$$ 1.41742 0.0671924
$$446$$ 12.5390 0.593740
$$447$$ 16.7477 0.792140
$$448$$ −0.912878 −0.0431295
$$449$$ −30.0780 −1.41947 −0.709735 0.704469i $$-0.751185\pi$$
−0.709735 + 0.704469i $$0.751185\pi$$
$$450$$ 1.79129 0.0844421
$$451$$ −45.8258 −2.15785
$$452$$ −17.1216 −0.805332
$$453$$ 7.16515 0.336648
$$454$$ −13.5826 −0.637462
$$455$$ −4.58258 −0.214834
$$456$$ −5.07803 −0.237801
$$457$$ −3.74773 −0.175311 −0.0876556 0.996151i $$-0.527938\pi$$
−0.0876556 + 0.996151i $$0.527938\pi$$
$$458$$ −30.7477 −1.43675
$$459$$ −3.00000 −0.140028
$$460$$ −4.83485 −0.225426
$$461$$ −9.16515 −0.426864 −0.213432 0.976958i $$-0.568464\pi$$
−0.213432 + 0.976958i $$0.568464\pi$$
$$462$$ 8.95644 0.416691
$$463$$ −0.165151 −0.00767524 −0.00383762 0.999993i $$-0.501222\pi$$
−0.00383762 + 0.999993i $$0.501222\pi$$
$$464$$ −4.95644 −0.230097
$$465$$ 4.00000 0.185496
$$466$$ −23.5826 −1.09244
$$467$$ −8.00000 −0.370196 −0.185098 0.982720i $$-0.559260\pi$$
−0.185098 + 0.982720i $$0.559260\pi$$
$$468$$ −5.53901 −0.256041
$$469$$ −4.16515 −0.192329
$$470$$ 18.9564 0.874395
$$471$$ 10.7477 0.495229
$$472$$ 10.7477 0.494704
$$473$$ −47.9129 −2.20304
$$474$$ 13.5826 0.623868
$$475$$ 3.58258 0.164380
$$476$$ −3.62614 −0.166204
$$477$$ 0.417424 0.0191125
$$478$$ −46.5735 −2.13022
$$479$$ −19.1652 −0.875678 −0.437839 0.899053i $$-0.644256\pi$$
−0.437839 + 0.899053i $$0.644256\pi$$
$$480$$ −6.04356 −0.275850
$$481$$ 18.3303 0.835790
$$482$$ 52.5390 2.39309
$$483$$ −4.00000 −0.182006
$$484$$ 16.9220 0.769180
$$485$$ 11.5826 0.525938
$$486$$ 1.79129 0.0812545
$$487$$ 2.33030 0.105596 0.0527980 0.998605i $$-0.483186\pi$$
0.0527980 + 0.998605i $$0.483186\pi$$
$$488$$ −18.0689 −0.817942
$$489$$ 7.58258 0.342896
$$490$$ −10.7477 −0.485533
$$491$$ 16.0000 0.722070 0.361035 0.932552i $$-0.382424\pi$$
0.361035 + 0.932552i $$0.382424\pi$$
$$492$$ −11.0780 −0.499436
$$493$$ −3.00000 −0.135113
$$494$$ −29.4083 −1.32314
$$495$$ 5.00000 0.224733
$$496$$ −19.8258 −0.890203
$$497$$ −9.58258 −0.429837
$$498$$ −20.7477 −0.929728
$$499$$ −13.4174 −0.600646 −0.300323 0.953837i $$-0.597095\pi$$
−0.300323 + 0.953837i $$0.597095\pi$$
$$500$$ 1.20871 0.0540553
$$501$$ 0 0
$$502$$ 28.9564 1.29239
$$503$$ −22.9129 −1.02163 −0.510817 0.859689i $$-0.670657\pi$$
−0.510817 + 0.859689i $$0.670657\pi$$
$$504$$ −1.41742 −0.0631371
$$505$$ −0.582576 −0.0259243
$$506$$ −35.8258 −1.59265
$$507$$ 8.00000 0.355292
$$508$$ 2.41742 0.107256
$$509$$ 26.7477 1.18557 0.592786 0.805360i $$-0.298028\pi$$
0.592786 + 0.805360i $$0.298028\pi$$
$$510$$ −5.37386 −0.237959
$$511$$ 4.00000 0.176950
$$512$$ 15.9038 0.702855
$$513$$ 3.58258 0.158175
$$514$$ 22.8348 1.00720
$$515$$ 15.1652 0.668256
$$516$$ −11.5826 −0.509894
$$517$$ 52.9129 2.32711
$$518$$ −7.16515 −0.314819
$$519$$ −3.16515 −0.138935
$$520$$ 6.49545 0.284845
$$521$$ 43.0780 1.88728 0.943641 0.330970i $$-0.107376\pi$$
0.943641 + 0.330970i $$0.107376\pi$$
$$522$$ 1.79129 0.0784025
$$523$$ −33.3303 −1.45743 −0.728716 0.684816i $$-0.759883\pi$$
−0.728716 + 0.684816i $$0.759883\pi$$
$$524$$ −18.1307 −0.792043
$$525$$ 1.00000 0.0436436
$$526$$ 54.3303 2.36891
$$527$$ −12.0000 −0.522728
$$528$$ −24.7822 −1.07851
$$529$$ −7.00000 −0.304348
$$530$$ 0.747727 0.0324792
$$531$$ −7.58258 −0.329056
$$532$$ 4.33030 0.187742
$$533$$ 42.0000 1.81922
$$534$$ 2.53901 0.109874
$$535$$ 5.16515 0.223309
$$536$$ 5.90379 0.255005
$$537$$ 4.74773 0.204880
$$538$$ 40.4519 1.74400
$$539$$ −30.0000 −1.29219
$$540$$ 1.20871 0.0520147
$$541$$ −31.4955 −1.35410 −0.677048 0.735939i $$-0.736741\pi$$
−0.677048 + 0.735939i $$0.736741\pi$$
$$542$$ 2.08712 0.0896495
$$543$$ 16.1652 0.693713
$$544$$ 18.1307 0.777347
$$545$$ 14.1652 0.606768
$$546$$ −8.20871 −0.351300
$$547$$ 4.16515 0.178089 0.0890445 0.996028i $$-0.471619\pi$$
0.0890445 + 0.996028i $$0.471619\pi$$
$$548$$ 19.7386 0.843193
$$549$$ 12.7477 0.544060
$$550$$ 8.95644 0.381904
$$551$$ 3.58258 0.152623
$$552$$ 5.66970 0.241318
$$553$$ 7.58258 0.322444
$$554$$ −44.6261 −1.89598
$$555$$ −4.00000 −0.169791
$$556$$ −11.3830 −0.482745
$$557$$ −22.7477 −0.963852 −0.481926 0.876212i $$-0.660063\pi$$
−0.481926 + 0.876212i $$0.660063\pi$$
$$558$$ 7.16515 0.303325
$$559$$ 43.9129 1.85732
$$560$$ −4.95644 −0.209448
$$561$$ −15.0000 −0.633300
$$562$$ −0.747727 −0.0315410
$$563$$ −14.5826 −0.614582 −0.307291 0.951616i $$-0.599423\pi$$
−0.307291 + 0.951616i $$0.599423\pi$$
$$564$$ 12.7913 0.538610
$$565$$ −14.1652 −0.595932
$$566$$ −1.49545 −0.0628586
$$567$$ 1.00000 0.0419961
$$568$$ 13.5826 0.569912
$$569$$ 28.5826 1.19824 0.599122 0.800658i $$-0.295516\pi$$
0.599122 + 0.800658i $$0.295516\pi$$
$$570$$ 6.41742 0.268796
$$571$$ 12.0000 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ −27.6951 −1.15799
$$573$$ −4.00000 −0.167102
$$574$$ −16.4174 −0.685250
$$575$$ −4.00000 −0.166812
$$576$$ −0.912878 −0.0380366
$$577$$ −19.1652 −0.797856 −0.398928 0.916982i $$-0.630618\pi$$
−0.398928 + 0.916982i $$0.630618\pi$$
$$578$$ −14.3303 −0.596062
$$579$$ −20.3303 −0.844899
$$580$$ 1.20871 0.0501890
$$581$$ −11.5826 −0.480526
$$582$$ 20.7477 0.860021
$$583$$ 2.08712 0.0864397
$$584$$ −5.66970 −0.234614
$$585$$ −4.58258 −0.189466
$$586$$ 54.0345 2.23214
$$587$$ −11.5826 −0.478064 −0.239032 0.971012i $$-0.576830\pi$$
−0.239032 + 0.971012i $$0.576830\pi$$
$$588$$ −7.25227 −0.299079
$$589$$ 14.3303 0.590470
$$590$$ −13.5826 −0.559186
$$591$$ −16.3303 −0.671739
$$592$$ 19.8258 0.814834
$$593$$ 8.41742 0.345662 0.172831 0.984951i $$-0.444709\pi$$
0.172831 + 0.984951i $$0.444709\pi$$
$$594$$ 8.95644 0.367487
$$595$$ −3.00000 −0.122988
$$596$$ 20.2432 0.829193
$$597$$ 13.4174 0.549139
$$598$$ 32.8348 1.34272
$$599$$ 0.165151 0.00674790 0.00337395 0.999994i $$-0.498926\pi$$
0.00337395 + 0.999994i $$0.498926\pi$$
$$600$$ −1.41742 −0.0578661
$$601$$ −32.3303 −1.31878 −0.659390 0.751801i $$-0.729185\pi$$
−0.659390 + 0.751801i $$0.729185\pi$$
$$602$$ −17.1652 −0.699599
$$603$$ −4.16515 −0.169618
$$604$$ 8.66061 0.352395
$$605$$ 14.0000 0.569181
$$606$$ −1.04356 −0.0423918
$$607$$ 3.58258 0.145412 0.0727061 0.997353i $$-0.476836\pi$$
0.0727061 + 0.997353i $$0.476836\pi$$
$$608$$ −21.6515 −0.878085
$$609$$ 1.00000 0.0405220
$$610$$ 22.8348 0.924556
$$611$$ −48.4955 −1.96192
$$612$$ −3.62614 −0.146578
$$613$$ −43.7477 −1.76695 −0.883477 0.468474i $$-0.844804\pi$$
−0.883477 + 0.468474i $$0.844804\pi$$
$$614$$ 0 0
$$615$$ −9.16515 −0.369575
$$616$$ −7.08712 −0.285548
$$617$$ 15.4955 0.623823 0.311912 0.950111i $$-0.399031\pi$$
0.311912 + 0.950111i $$0.399031\pi$$
$$618$$ 27.1652 1.09274
$$619$$ 13.1652 0.529152 0.264576 0.964365i $$-0.414768\pi$$
0.264576 + 0.964365i $$0.414768\pi$$
$$620$$ 4.83485 0.194172
$$621$$ −4.00000 −0.160514
$$622$$ −5.37386 −0.215472
$$623$$ 1.41742 0.0567879
$$624$$ 22.7133 0.909258
$$625$$ 1.00000 0.0400000
$$626$$ 18.9564 0.757652
$$627$$ 17.9129 0.715371
$$628$$ 12.9909 0.518394
$$629$$ 12.0000 0.478471
$$630$$ 1.79129 0.0713666
$$631$$ 10.9129 0.434435 0.217217 0.976123i $$-0.430302\pi$$
0.217217 + 0.976123i $$0.430302\pi$$
$$632$$ −10.7477 −0.427522
$$633$$ 20.3303 0.808057
$$634$$ −44.7822 −1.77853
$$635$$ 2.00000 0.0793676
$$636$$ 0.504546 0.0200065
$$637$$ 27.4955 1.08941
$$638$$ 8.95644 0.354589
$$639$$ −9.58258 −0.379081
$$640$$ 10.4519 0.413147
$$641$$ −6.58258 −0.259996 −0.129998 0.991514i $$-0.541497\pi$$
−0.129998 + 0.991514i $$0.541497\pi$$
$$642$$ 9.25227 0.365158
$$643$$ 43.6606 1.72181 0.860903 0.508769i $$-0.169899\pi$$
0.860903 + 0.508769i $$0.169899\pi$$
$$644$$ −4.83485 −0.190520
$$645$$ −9.58258 −0.377314
$$646$$ −19.2523 −0.757471
$$647$$ 24.7477 0.972934 0.486467 0.873699i $$-0.338285\pi$$
0.486467 + 0.873699i $$0.338285\pi$$
$$648$$ −1.41742 −0.0556817
$$649$$ −37.9129 −1.48821
$$650$$ −8.20871 −0.321972
$$651$$ 4.00000 0.156772
$$652$$ 9.16515 0.358935
$$653$$ 26.1652 1.02392 0.511961 0.859009i $$-0.328919\pi$$
0.511961 + 0.859009i $$0.328919\pi$$
$$654$$ 25.3739 0.992197
$$655$$ −15.0000 −0.586098
$$656$$ 45.4265 1.77361
$$657$$ 4.00000 0.156055
$$658$$ 18.9564 0.738999
$$659$$ 18.1652 0.707614 0.353807 0.935318i $$-0.384887\pi$$
0.353807 + 0.935318i $$0.384887\pi$$
$$660$$ 6.04356 0.235245
$$661$$ 25.0000 0.972387 0.486194 0.873851i $$-0.338385\pi$$
0.486194 + 0.873851i $$0.338385\pi$$
$$662$$ −14.9220 −0.579959
$$663$$ 13.7477 0.533917
$$664$$ 16.4174 0.637120
$$665$$ 3.58258 0.138926
$$666$$ −7.16515 −0.277644
$$667$$ −4.00000 −0.154881
$$668$$ 0 0
$$669$$ 7.00000 0.270636
$$670$$ −7.46099 −0.288243
$$671$$ 63.7386 2.46060
$$672$$ −6.04356 −0.233135
$$673$$ 1.74773 0.0673699 0.0336850 0.999433i $$-0.489276\pi$$
0.0336850 + 0.999433i $$0.489276\pi$$
$$674$$ −37.9129 −1.46035
$$675$$ 1.00000 0.0384900
$$676$$ 9.66970 0.371911
$$677$$ 44.8258 1.72279 0.861397 0.507932i $$-0.169590\pi$$
0.861397 + 0.507932i $$0.169590\pi$$
$$678$$ −25.3739 −0.974477
$$679$$ 11.5826 0.444498
$$680$$ 4.25227 0.163067
$$681$$ −7.58258 −0.290565
$$682$$ 35.8258 1.37184
$$683$$ 7.16515 0.274167 0.137083 0.990560i $$-0.456227\pi$$
0.137083 + 0.990560i $$0.456227\pi$$
$$684$$ 4.33030 0.165573
$$685$$ 16.3303 0.623949
$$686$$ −23.2867 −0.889092
$$687$$ −17.1652 −0.654891
$$688$$ 47.4955 1.81075
$$689$$ −1.91288 −0.0728749
$$690$$ −7.16515 −0.272773
$$691$$ 42.9129 1.63248 0.816241 0.577711i $$-0.196054\pi$$
0.816241 + 0.577711i $$0.196054\pi$$
$$692$$ −3.82576 −0.145433
$$693$$ 5.00000 0.189934
$$694$$ 12.8348 0.487204
$$695$$ −9.41742 −0.357223
$$696$$ −1.41742 −0.0537273
$$697$$ 27.4955 1.04146
$$698$$ −7.75682 −0.293600
$$699$$ −13.1652 −0.497952
$$700$$ 1.20871 0.0456850
$$701$$ −23.0780 −0.871645 −0.435823 0.900033i $$-0.643542\pi$$
−0.435823 + 0.900033i $$0.643542\pi$$
$$702$$ −8.20871 −0.309818
$$703$$ −14.3303 −0.540478
$$704$$ −4.56439 −0.172027
$$705$$ 10.5826 0.398563
$$706$$ −26.5735 −1.00011
$$707$$ −0.582576 −0.0219100
$$708$$ −9.16515 −0.344447
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ −17.1652 −0.644197
$$711$$ 7.58258 0.284369
$$712$$ −2.00909 −0.0752939
$$713$$ −16.0000 −0.599205
$$714$$ −5.37386 −0.201112
$$715$$ −22.9129 −0.856893
$$716$$ 5.73864 0.214463
$$717$$ −26.0000 −0.970988
$$718$$ −48.6606 −1.81600
$$719$$ 7.91288 0.295101 0.147550 0.989055i $$-0.452861\pi$$
0.147550 + 0.989055i $$0.452861\pi$$
$$720$$ −4.95644 −0.184716
$$721$$ 15.1652 0.564780
$$722$$ −11.0436 −0.410999
$$723$$ 29.3303 1.09081
$$724$$ 19.5390 0.726162
$$725$$ 1.00000 0.0371391
$$726$$ 25.0780 0.930733
$$727$$ 9.66970 0.358629 0.179315 0.983792i $$-0.442612\pi$$
0.179315 + 0.983792i $$0.442612\pi$$
$$728$$ 6.49545 0.240738
$$729$$ 1.00000 0.0370370
$$730$$ 7.16515 0.265194
$$731$$ 28.7477 1.06327
$$732$$ 15.4083 0.569508
$$733$$ −38.4174 −1.41898 −0.709490 0.704716i $$-0.751075\pi$$
−0.709490 + 0.704716i $$0.751075\pi$$
$$734$$ −67.1652 −2.47911
$$735$$ −6.00000 −0.221313
$$736$$ 24.1742 0.891074
$$737$$ −20.8258 −0.767127
$$738$$ −16.4174 −0.604334
$$739$$ 19.2523 0.708206 0.354103 0.935206i $$-0.384786\pi$$
0.354103 + 0.935206i $$0.384786\pi$$
$$740$$ −4.83485 −0.177733
$$741$$ −16.4174 −0.603109
$$742$$ 0.747727 0.0274499
$$743$$ 29.7477 1.09134 0.545669 0.838001i $$-0.316276\pi$$
0.545669 + 0.838001i $$0.316276\pi$$
$$744$$ −5.66970 −0.207861
$$745$$ 16.7477 0.613589
$$746$$ 50.7477 1.85801
$$747$$ −11.5826 −0.423784
$$748$$ −18.1307 −0.662923
$$749$$ 5.16515 0.188731
$$750$$ 1.79129 0.0654086
$$751$$ −17.4955 −0.638418 −0.319209 0.947684i $$-0.603417\pi$$
−0.319209 + 0.947684i $$0.603417\pi$$
$$752$$ −52.4519 −1.91272
$$753$$ 16.1652 0.589091
$$754$$ −8.20871 −0.298944
$$755$$ 7.16515 0.260767
$$756$$ 1.20871 0.0439604
$$757$$ −2.33030 −0.0846963 −0.0423481 0.999103i $$-0.513484\pi$$
−0.0423481 + 0.999103i $$0.513484\pi$$
$$758$$ −46.5735 −1.69163
$$759$$ −20.0000 −0.725954
$$760$$ −5.07803 −0.184200
$$761$$ 36.4174 1.32013 0.660065 0.751208i $$-0.270529\pi$$
0.660065 + 0.751208i $$0.270529\pi$$
$$762$$ 3.58258 0.129783
$$763$$ 14.1652 0.512813
$$764$$ −4.83485 −0.174919
$$765$$ −3.00000 −0.108465
$$766$$ 6.41742 0.231871
$$767$$ 34.7477 1.25467
$$768$$ 20.5481 0.741466
$$769$$ 25.5826 0.922531 0.461266 0.887262i $$-0.347396\pi$$
0.461266 + 0.887262i $$0.347396\pi$$
$$770$$ 8.95644 0.322768
$$771$$ 12.7477 0.459098
$$772$$ −24.5735 −0.884419
$$773$$ −5.16515 −0.185778 −0.0928888 0.995676i $$-0.529610\pi$$
−0.0928888 + 0.995676i $$0.529610\pi$$
$$774$$ −17.1652 −0.616989
$$775$$ 4.00000 0.143684
$$776$$ −16.4174 −0.589351
$$777$$ −4.00000 −0.143499
$$778$$ −11.7913 −0.422738
$$779$$ −32.8348 −1.17643
$$780$$ −5.53901 −0.198329
$$781$$ −47.9129 −1.71446
$$782$$ 21.4955 0.768676
$$783$$ 1.00000 0.0357371
$$784$$ 29.7386 1.06209
$$785$$ 10.7477 0.383603
$$786$$ −26.8693 −0.958397
$$787$$ −24.0000 −0.855508 −0.427754 0.903895i $$-0.640695\pi$$
−0.427754 + 0.903895i $$0.640695\pi$$
$$788$$ −19.7386 −0.703160
$$789$$ 30.3303 1.07979
$$790$$ 13.5826 0.483246
$$791$$ −14.1652 −0.503655
$$792$$ −7.08712 −0.251830
$$793$$ −58.4174 −2.07446
$$794$$ 19.4083 0.688776
$$795$$ 0.417424 0.0148045
$$796$$ 16.2178 0.574825
$$797$$ 32.3303 1.14520 0.572599 0.819836i $$-0.305935\pi$$
0.572599 + 0.819836i $$0.305935\pi$$
$$798$$ 6.41742 0.227174
$$799$$ −31.7477 −1.12315
$$800$$ −6.04356 −0.213672
$$801$$ 1.41742 0.0500822
$$802$$ 22.2432 0.785434
$$803$$ 20.0000 0.705785
$$804$$ −5.03447 −0.177552
$$805$$ −4.00000 −0.140981
$$806$$ −32.8348 −1.15656
$$807$$ 22.5826 0.794944
$$808$$ 0.825757 0.0290500
$$809$$ 44.0780 1.54970 0.774851 0.632145i $$-0.217825\pi$$
0.774851 + 0.632145i $$0.217825\pi$$
$$810$$ 1.79129 0.0629394
$$811$$ 32.5826 1.14413 0.572064 0.820209i $$-0.306143\pi$$
0.572064 + 0.820209i $$0.306143\pi$$
$$812$$ 1.20871 0.0424175
$$813$$ 1.16515 0.0408636
$$814$$ −35.8258 −1.25569
$$815$$ 7.58258 0.265606
$$816$$ 14.8693 0.520530
$$817$$ −34.3303 −1.20107
$$818$$ −4.92197 −0.172093
$$819$$ −4.58258 −0.160128
$$820$$ −11.0780 −0.386862
$$821$$ −31.4955 −1.09920 −0.549599 0.835428i $$-0.685220\pi$$
−0.549599 + 0.835428i $$0.685220\pi$$
$$822$$ 29.2523 1.02029
$$823$$ 51.0780 1.78047 0.890234 0.455503i $$-0.150541\pi$$
0.890234 + 0.455503i $$0.150541\pi$$
$$824$$ −21.4955 −0.748830
$$825$$ 5.00000 0.174078
$$826$$ −13.5826 −0.472598
$$827$$ −43.8258 −1.52397 −0.761985 0.647594i $$-0.775775\pi$$
−0.761985 + 0.647594i $$0.775775\pi$$
$$828$$ −4.83485 −0.168023
$$829$$ 4.00000 0.138926 0.0694629 0.997585i $$-0.477871\pi$$
0.0694629 + 0.997585i $$0.477871\pi$$
$$830$$ −20.7477 −0.720164
$$831$$ −24.9129 −0.864218
$$832$$ 4.18333 0.145031
$$833$$ 18.0000 0.623663
$$834$$ −16.8693 −0.584137
$$835$$ 0 0
$$836$$ 21.6515 0.748833
$$837$$ 4.00000 0.138260
$$838$$ −66.5735 −2.29974
$$839$$ −48.4955 −1.67425 −0.837125 0.547012i $$-0.815765\pi$$
−0.837125 + 0.547012i $$0.815765\pi$$
$$840$$ −1.41742 −0.0489058
$$841$$ 1.00000 0.0344828
$$842$$ 7.91288 0.272696
$$843$$ −0.417424 −0.0143769
$$844$$ 24.5735 0.845854
$$845$$ 8.00000 0.275208
$$846$$ 18.9564 0.651736
$$847$$ 14.0000 0.481046
$$848$$ −2.06894 −0.0710476
$$849$$ −0.834849 −0.0286519
$$850$$ −5.37386 −0.184322
$$851$$ 16.0000 0.548473
$$852$$ −11.5826 −0.396813
$$853$$ 20.7477 0.710389 0.355194 0.934792i $$-0.384415\pi$$
0.355194 + 0.934792i $$0.384415\pi$$
$$854$$ 22.8348 0.781392
$$855$$ 3.58258 0.122522
$$856$$ −7.32121 −0.250234
$$857$$ −46.6606 −1.59390 −0.796948 0.604048i $$-0.793554\pi$$
−0.796948 + 0.604048i $$0.793554\pi$$
$$858$$ −41.0436 −1.40120
$$859$$ −47.5826 −1.62350 −0.811748 0.584007i $$-0.801484\pi$$
−0.811748 + 0.584007i $$0.801484\pi$$
$$860$$ −11.5826 −0.394963
$$861$$ −9.16515 −0.312348
$$862$$ −70.1561 −2.38952
$$863$$ −6.41742 −0.218452 −0.109226 0.994017i $$-0.534837\pi$$
−0.109226 + 0.994017i $$0.534837\pi$$
$$864$$ −6.04356 −0.205606
$$865$$ −3.16515 −0.107618
$$866$$ 44.9220 1.52651
$$867$$ −8.00000 −0.271694
$$868$$ 4.83485 0.164105
$$869$$ 37.9129 1.28611
$$870$$ 1.79129 0.0607303
$$871$$ 19.0871 0.646742
$$872$$ −20.0780 −0.679928
$$873$$ 11.5826 0.392011
$$874$$ −25.6697 −0.868290
$$875$$ 1.00000 0.0338062
$$876$$ 4.83485 0.163354
$$877$$ 19.6697 0.664198 0.332099 0.943244i $$-0.392243\pi$$
0.332099 + 0.943244i $$0.392243\pi$$
$$878$$ −30.2958 −1.02243
$$879$$ 30.1652 1.01745
$$880$$ −24.7822 −0.835408
$$881$$ −32.0780 −1.08074 −0.540368 0.841429i $$-0.681715\pi$$
−0.540368 + 0.841429i $$0.681715\pi$$
$$882$$ −10.7477 −0.361895
$$883$$ 16.0000 0.538443 0.269221 0.963078i $$-0.413234\pi$$
0.269221 + 0.963078i $$0.413234\pi$$
$$884$$ 16.6170 0.558892
$$885$$ −7.58258 −0.254885
$$886$$ 1.04356 0.0350591
$$887$$ −44.9129 −1.50803 −0.754013 0.656859i $$-0.771885\pi$$
−0.754013 + 0.656859i $$0.771885\pi$$
$$888$$ 5.66970 0.190263
$$889$$ 2.00000 0.0670778
$$890$$ 2.53901 0.0851080
$$891$$ 5.00000 0.167506
$$892$$ 8.46099 0.283295
$$893$$ 37.9129 1.26871
$$894$$ 30.0000 1.00335
$$895$$ 4.74773 0.158699
$$896$$ 10.4519 0.349173
$$897$$ 18.3303 0.612031
$$898$$ −53.8784 −1.79795
$$899$$ 4.00000 0.133407
$$900$$ 1.20871 0.0402904
$$901$$ −1.25227 −0.0417193
$$902$$ −82.0871 −2.73320
$$903$$ −9.58258 −0.318888
$$904$$ 20.0780 0.667785
$$905$$ 16.1652 0.537348
$$906$$ 12.8348 0.426409
$$907$$ 23.5826 0.783047 0.391523 0.920168i $$-0.371948\pi$$
0.391523 + 0.920168i $$0.371948\pi$$
$$908$$ −9.16515 −0.304156
$$909$$ −0.582576 −0.0193228
$$910$$ −8.20871 −0.272116
$$911$$ −50.8258 −1.68393 −0.841966 0.539530i $$-0.818602\pi$$
−0.841966 + 0.539530i $$0.818602\pi$$
$$912$$ −17.7568 −0.587987
$$913$$ −57.9129 −1.91664
$$914$$ −6.71326 −0.222055
$$915$$ 12.7477 0.421427
$$916$$ −20.7477 −0.685524
$$917$$ −15.0000 −0.495344
$$918$$ −5.37386 −0.177364
$$919$$ 18.9129 0.623878 0.311939 0.950102i $$-0.399021\pi$$
0.311939 + 0.950102i $$0.399021\pi$$
$$920$$ 5.66970 0.186924
$$921$$ 0 0
$$922$$ −16.4174 −0.540679
$$923$$ 43.9129 1.44541
$$924$$ 6.04356 0.198819
$$925$$ −4.00000 −0.131519
$$926$$ −0.295834 −0.00972170
$$927$$ 15.1652 0.498089
$$928$$ −6.04356 −0.198390
$$929$$ 15.6697 0.514106 0.257053 0.966397i $$-0.417248\pi$$
0.257053 + 0.966397i $$0.417248\pi$$
$$930$$ 7.16515 0.234955
$$931$$ −21.4955 −0.704485
$$932$$ −15.9129 −0.521244
$$933$$ −3.00000 −0.0982156
$$934$$ −14.3303 −0.468902
$$935$$ −15.0000 −0.490552
$$936$$ 6.49545 0.212311
$$937$$ −18.5826 −0.607066 −0.303533 0.952821i $$-0.598166\pi$$
−0.303533 + 0.952821i $$0.598166\pi$$
$$938$$ −7.46099 −0.243610
$$939$$ 10.5826 0.345349
$$940$$ 12.7913 0.417206
$$941$$ 19.9129 0.649141 0.324571 0.945861i $$-0.394780\pi$$
0.324571 + 0.945861i $$0.394780\pi$$
$$942$$ 19.2523 0.627273
$$943$$ 36.6606 1.19383
$$944$$ 37.5826 1.22321
$$945$$ 1.00000 0.0325300
$$946$$ −85.8258 −2.79044
$$947$$ −54.9129 −1.78443 −0.892214 0.451612i $$-0.850849\pi$$
−0.892214 + 0.451612i $$0.850849\pi$$
$$948$$ 9.16515 0.297670
$$949$$ −18.3303 −0.595027
$$950$$ 6.41742 0.208209
$$951$$ −25.0000 −0.810681
$$952$$ 4.25227 0.137817
$$953$$ −37.5826 −1.21742 −0.608710 0.793393i $$-0.708313\pi$$
−0.608710 + 0.793393i $$0.708313\pi$$
$$954$$ 0.747727 0.0242086
$$955$$ −4.00000 −0.129437
$$956$$ −31.4265 −1.01641
$$957$$ 5.00000 0.161627
$$958$$ −34.3303 −1.10916
$$959$$ 16.3303 0.527333
$$960$$ −0.912878 −0.0294630
$$961$$ −15.0000 −0.483871
$$962$$ 32.8348 1.05864
$$963$$ 5.16515 0.166445
$$964$$ 35.4519 1.14183
$$965$$ −20.3303 −0.654456
$$966$$ −7.16515 −0.230535
$$967$$ 9.25227 0.297533 0.148767 0.988872i $$-0.452470\pi$$
0.148767 + 0.988872i $$0.452470\pi$$
$$968$$ −19.8439 −0.637808
$$969$$ −10.7477 −0.345267
$$970$$ 20.7477 0.666169
$$971$$ 20.0000 0.641831 0.320915 0.947108i $$-0.396010\pi$$
0.320915 + 0.947108i $$0.396010\pi$$
$$972$$ 1.20871 0.0387695
$$973$$ −9.41742 −0.301909
$$974$$ 4.17424 0.133751
$$975$$ −4.58258 −0.146760
$$976$$ −63.1833 −2.02245
$$977$$ −2.50455 −0.0801275 −0.0400638 0.999197i $$-0.512756\pi$$
−0.0400638 + 0.999197i $$0.512756\pi$$
$$978$$ 13.5826 0.434323
$$979$$ 7.08712 0.226505
$$980$$ −7.25227 −0.231665
$$981$$ 14.1652 0.452258
$$982$$ 28.6606 0.914597
$$983$$ 55.1652 1.75950 0.879748 0.475441i $$-0.157712\pi$$
0.879748 + 0.475441i $$0.157712\pi$$
$$984$$ 12.9909 0.414135
$$985$$ −16.3303 −0.520327
$$986$$ −5.37386 −0.171139
$$987$$ 10.5826 0.336847
$$988$$ −19.8439 −0.631320
$$989$$ 38.3303 1.21883
$$990$$ 8.95644 0.284654
$$991$$ 16.0780 0.510735 0.255368 0.966844i $$-0.417803\pi$$
0.255368 + 0.966844i $$0.417803\pi$$
$$992$$ −24.1742 −0.767533
$$993$$ −8.33030 −0.264354
$$994$$ −17.1652 −0.544446
$$995$$ 13.4174 0.425361
$$996$$ −14.0000 −0.443607
$$997$$ −0.834849 −0.0264399 −0.0132200 0.999913i $$-0.504208\pi$$
−0.0132200 + 0.999913i $$0.504208\pi$$
$$998$$ −24.0345 −0.760798
$$999$$ −4.00000 −0.126554
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 435.2.a.f.1.2 2
3.2 odd 2 1305.2.a.m.1.1 2
4.3 odd 2 6960.2.a.bw.1.1 2
5.2 odd 4 2175.2.c.f.349.3 4
5.3 odd 4 2175.2.c.f.349.2 4
5.4 even 2 2175.2.a.r.1.1 2
15.14 odd 2 6525.2.a.t.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.f.1.2 2 1.1 even 1 trivial
1305.2.a.m.1.1 2 3.2 odd 2
2175.2.a.r.1.1 2 5.4 even 2
2175.2.c.f.349.2 4 5.3 odd 4
2175.2.c.f.349.3 4 5.2 odd 4
6525.2.a.t.1.2 2 15.14 odd 2
6960.2.a.bw.1.1 2 4.3 odd 2