# Properties

 Label 2738.2.a.i.1.1 Level $2738$ Weight $2$ Character 2738.1 Self dual yes Analytic conductor $21.863$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2738,2,Mod(1,2738)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2738, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2738.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2738 = 2 \cdot 37^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2738.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: $$21.8630400734$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 74) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 2738.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -2.73205 q^{3} +1.00000 q^{4} +1.73205 q^{5} -2.73205 q^{6} -2.00000 q^{7} +1.00000 q^{8} +4.46410 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -2.73205 q^{3} +1.00000 q^{4} +1.73205 q^{5} -2.73205 q^{6} -2.00000 q^{7} +1.00000 q^{8} +4.46410 q^{9} +1.73205 q^{10} +4.73205 q^{11} -2.73205 q^{12} -3.46410 q^{13} -2.00000 q^{14} -4.73205 q^{15} +1.00000 q^{16} -7.73205 q^{17} +4.46410 q^{18} +1.26795 q^{19} +1.73205 q^{20} +5.46410 q^{21} +4.73205 q^{22} -4.73205 q^{23} -2.73205 q^{24} -2.00000 q^{25} -3.46410 q^{26} -4.00000 q^{27} -2.00000 q^{28} +8.66025 q^{29} -4.73205 q^{30} -1.26795 q^{31} +1.00000 q^{32} -12.9282 q^{33} -7.73205 q^{34} -3.46410 q^{35} +4.46410 q^{36} +1.26795 q^{38} +9.46410 q^{39} +1.73205 q^{40} -9.92820 q^{41} +5.46410 q^{42} +0.928203 q^{43} +4.73205 q^{44} +7.73205 q^{45} -4.73205 q^{46} +4.73205 q^{47} -2.73205 q^{48} -3.00000 q^{49} -2.00000 q^{50} +21.1244 q^{51} -3.46410 q^{52} +2.53590 q^{53} -4.00000 q^{54} +8.19615 q^{55} -2.00000 q^{56} -3.46410 q^{57} +8.66025 q^{58} -2.53590 q^{59} -4.73205 q^{60} -1.73205 q^{61} -1.26795 q^{62} -8.92820 q^{63} +1.00000 q^{64} -6.00000 q^{65} -12.9282 q^{66} +10.1962 q^{67} -7.73205 q^{68} +12.9282 q^{69} -3.46410 q^{70} +3.46410 q^{71} +4.46410 q^{72} -4.00000 q^{73} +5.46410 q^{75} +1.26795 q^{76} -9.46410 q^{77} +9.46410 q^{78} -13.2679 q^{79} +1.73205 q^{80} -2.46410 q^{81} -9.92820 q^{82} -5.66025 q^{83} +5.46410 q^{84} -13.3923 q^{85} +0.928203 q^{86} -23.6603 q^{87} +4.73205 q^{88} -6.80385 q^{89} +7.73205 q^{90} +6.92820 q^{91} -4.73205 q^{92} +3.46410 q^{93} +4.73205 q^{94} +2.19615 q^{95} -2.73205 q^{96} -7.73205 q^{97} -3.00000 q^{98} +21.1244 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} - 4 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^6 - 4 * q^7 + 2 * q^8 + 2 * q^9 $$2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} - 4 q^{7} + 2 q^{8} + 2 q^{9} + 6 q^{11} - 2 q^{12} - 4 q^{14} - 6 q^{15} + 2 q^{16} - 12 q^{17} + 2 q^{18} + 6 q^{19} + 4 q^{21} + 6 q^{22} - 6 q^{23} - 2 q^{24} - 4 q^{25} - 8 q^{27} - 4 q^{28} - 6 q^{30} - 6 q^{31} + 2 q^{32} - 12 q^{33} - 12 q^{34} + 2 q^{36} + 6 q^{38} + 12 q^{39} - 6 q^{41} + 4 q^{42} - 12 q^{43} + 6 q^{44} + 12 q^{45} - 6 q^{46} + 6 q^{47} - 2 q^{48} - 6 q^{49} - 4 q^{50} + 18 q^{51} + 12 q^{53} - 8 q^{54} + 6 q^{55} - 4 q^{56} - 12 q^{59} - 6 q^{60} - 6 q^{62} - 4 q^{63} + 2 q^{64} - 12 q^{65} - 12 q^{66} + 10 q^{67} - 12 q^{68} + 12 q^{69} + 2 q^{72} - 8 q^{73} + 4 q^{75} + 6 q^{76} - 12 q^{77} + 12 q^{78} - 30 q^{79} + 2 q^{81} - 6 q^{82} + 6 q^{83} + 4 q^{84} - 6 q^{85} - 12 q^{86} - 30 q^{87} + 6 q^{88} - 24 q^{89} + 12 q^{90} - 6 q^{92} + 6 q^{94} - 6 q^{95} - 2 q^{96} - 12 q^{97} - 6 q^{98} + 18 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^6 - 4 * q^7 + 2 * q^8 + 2 * q^9 + 6 * q^11 - 2 * q^12 - 4 * q^14 - 6 * q^15 + 2 * q^16 - 12 * q^17 + 2 * q^18 + 6 * q^19 + 4 * q^21 + 6 * q^22 - 6 * q^23 - 2 * q^24 - 4 * q^25 - 8 * q^27 - 4 * q^28 - 6 * q^30 - 6 * q^31 + 2 * q^32 - 12 * q^33 - 12 * q^34 + 2 * q^36 + 6 * q^38 + 12 * q^39 - 6 * q^41 + 4 * q^42 - 12 * q^43 + 6 * q^44 + 12 * q^45 - 6 * q^46 + 6 * q^47 - 2 * q^48 - 6 * q^49 - 4 * q^50 + 18 * q^51 + 12 * q^53 - 8 * q^54 + 6 * q^55 - 4 * q^56 - 12 * q^59 - 6 * q^60 - 6 * q^62 - 4 * q^63 + 2 * q^64 - 12 * q^65 - 12 * q^66 + 10 * q^67 - 12 * q^68 + 12 * q^69 + 2 * q^72 - 8 * q^73 + 4 * q^75 + 6 * q^76 - 12 * q^77 + 12 * q^78 - 30 * q^79 + 2 * q^81 - 6 * q^82 + 6 * q^83 + 4 * q^84 - 6 * q^85 - 12 * q^86 - 30 * q^87 + 6 * q^88 - 24 * q^89 + 12 * q^90 - 6 * q^92 + 6 * q^94 - 6 * q^95 - 2 * q^96 - 12 * q^97 - 6 * q^98 + 18 * 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
$$3$$ −2.73205 −1.57735 −0.788675 0.614810i $$-0.789233\pi$$
−0.788675 + 0.614810i $$0.789233\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 1.73205 0.774597 0.387298 0.921954i $$-0.373408\pi$$
0.387298 + 0.921954i $$0.373408\pi$$
$$6$$ −2.73205 −1.11536
$$7$$ −2.00000 −0.755929 −0.377964 0.925820i $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 4.46410 1.48803
$$10$$ 1.73205 0.547723
$$11$$ 4.73205 1.42677 0.713384 0.700774i $$-0.247162\pi$$
0.713384 + 0.700774i $$0.247162\pi$$
$$12$$ −2.73205 −0.788675
$$13$$ −3.46410 −0.960769 −0.480384 0.877058i $$-0.659503\pi$$
−0.480384 + 0.877058i $$0.659503\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ −4.73205 −1.22181
$$16$$ 1.00000 0.250000
$$17$$ −7.73205 −1.87530 −0.937649 0.347584i $$-0.887002\pi$$
−0.937649 + 0.347584i $$0.887002\pi$$
$$18$$ 4.46410 1.05220
$$19$$ 1.26795 0.290887 0.145444 0.989367i $$-0.453539\pi$$
0.145444 + 0.989367i $$0.453539\pi$$
$$20$$ 1.73205 0.387298
$$21$$ 5.46410 1.19236
$$22$$ 4.73205 1.00888
$$23$$ −4.73205 −0.986701 −0.493350 0.869831i $$-0.664228\pi$$
−0.493350 + 0.869831i $$0.664228\pi$$
$$24$$ −2.73205 −0.557678
$$25$$ −2.00000 −0.400000
$$26$$ −3.46410 −0.679366
$$27$$ −4.00000 −0.769800
$$28$$ −2.00000 −0.377964
$$29$$ 8.66025 1.60817 0.804084 0.594515i $$-0.202656\pi$$
0.804084 + 0.594515i $$0.202656\pi$$
$$30$$ −4.73205 −0.863950
$$31$$ −1.26795 −0.227730 −0.113865 0.993496i $$-0.536323\pi$$
−0.113865 + 0.993496i $$0.536323\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −12.9282 −2.25051
$$34$$ −7.73205 −1.32604
$$35$$ −3.46410 −0.585540
$$36$$ 4.46410 0.744017
$$37$$ 0 0
$$38$$ 1.26795 0.205689
$$39$$ 9.46410 1.51547
$$40$$ 1.73205 0.273861
$$41$$ −9.92820 −1.55052 −0.775262 0.631639i $$-0.782382\pi$$
−0.775262 + 0.631639i $$0.782382\pi$$
$$42$$ 5.46410 0.843129
$$43$$ 0.928203 0.141550 0.0707748 0.997492i $$-0.477453\pi$$
0.0707748 + 0.997492i $$0.477453\pi$$
$$44$$ 4.73205 0.713384
$$45$$ 7.73205 1.15263
$$46$$ −4.73205 −0.697703
$$47$$ 4.73205 0.690241 0.345120 0.938558i $$-0.387838\pi$$
0.345120 + 0.938558i $$0.387838\pi$$
$$48$$ −2.73205 −0.394338
$$49$$ −3.00000 −0.428571
$$50$$ −2.00000 −0.282843
$$51$$ 21.1244 2.95800
$$52$$ −3.46410 −0.480384
$$53$$ 2.53590 0.348332 0.174166 0.984716i $$-0.444277\pi$$
0.174166 + 0.984716i $$0.444277\pi$$
$$54$$ −4.00000 −0.544331
$$55$$ 8.19615 1.10517
$$56$$ −2.00000 −0.267261
$$57$$ −3.46410 −0.458831
$$58$$ 8.66025 1.13715
$$59$$ −2.53590 −0.330146 −0.165073 0.986281i $$-0.552786\pi$$
−0.165073 + 0.986281i $$0.552786\pi$$
$$60$$ −4.73205 −0.610905
$$61$$ −1.73205 −0.221766 −0.110883 0.993833i $$-0.535368\pi$$
−0.110883 + 0.993833i $$0.535368\pi$$
$$62$$ −1.26795 −0.161030
$$63$$ −8.92820 −1.12485
$$64$$ 1.00000 0.125000
$$65$$ −6.00000 −0.744208
$$66$$ −12.9282 −1.59135
$$67$$ 10.1962 1.24566 0.622829 0.782358i $$-0.285983\pi$$
0.622829 + 0.782358i $$0.285983\pi$$
$$68$$ −7.73205 −0.937649
$$69$$ 12.9282 1.55637
$$70$$ −3.46410 −0.414039
$$71$$ 3.46410 0.411113 0.205557 0.978645i $$-0.434100\pi$$
0.205557 + 0.978645i $$0.434100\pi$$
$$72$$ 4.46410 0.526099
$$73$$ −4.00000 −0.468165 −0.234082 0.972217i $$-0.575209\pi$$
−0.234082 + 0.972217i $$0.575209\pi$$
$$74$$ 0 0
$$75$$ 5.46410 0.630940
$$76$$ 1.26795 0.145444
$$77$$ −9.46410 −1.07853
$$78$$ 9.46410 1.07160
$$79$$ −13.2679 −1.49276 −0.746380 0.665520i $$-0.768210\pi$$
−0.746380 + 0.665520i $$0.768210\pi$$
$$80$$ 1.73205 0.193649
$$81$$ −2.46410 −0.273789
$$82$$ −9.92820 −1.09639
$$83$$ −5.66025 −0.621294 −0.310647 0.950525i $$-0.600546\pi$$
−0.310647 + 0.950525i $$0.600546\pi$$
$$84$$ 5.46410 0.596182
$$85$$ −13.3923 −1.45260
$$86$$ 0.928203 0.100091
$$87$$ −23.6603 −2.53665
$$88$$ 4.73205 0.504438
$$89$$ −6.80385 −0.721206 −0.360603 0.932719i $$-0.617429\pi$$
−0.360603 + 0.932719i $$0.617429\pi$$
$$90$$ 7.73205 0.815030
$$91$$ 6.92820 0.726273
$$92$$ −4.73205 −0.493350
$$93$$ 3.46410 0.359211
$$94$$ 4.73205 0.488074
$$95$$ 2.19615 0.225320
$$96$$ −2.73205 −0.278839
$$97$$ −7.73205 −0.785071 −0.392535 0.919737i $$-0.628402\pi$$
−0.392535 + 0.919737i $$0.628402\pi$$
$$98$$ −3.00000 −0.303046
$$99$$ 21.1244 2.12308
$$100$$ −2.00000 −0.200000
$$101$$ −12.4641 −1.24022 −0.620112 0.784513i $$-0.712913\pi$$
−0.620112 + 0.784513i $$0.712913\pi$$
$$102$$ 21.1244 2.09162
$$103$$ −8.53590 −0.841067 −0.420534 0.907277i $$-0.638157\pi$$
−0.420534 + 0.907277i $$0.638157\pi$$
$$104$$ −3.46410 −0.339683
$$105$$ 9.46410 0.923602
$$106$$ 2.53590 0.246308
$$107$$ −10.3923 −1.00466 −0.502331 0.864675i $$-0.667524\pi$$
−0.502331 + 0.864675i $$0.667524\pi$$
$$108$$ −4.00000 −0.384900
$$109$$ −2.66025 −0.254806 −0.127403 0.991851i $$-0.540664\pi$$
−0.127403 + 0.991851i $$0.540664\pi$$
$$110$$ 8.19615 0.781472
$$111$$ 0 0
$$112$$ −2.00000 −0.188982
$$113$$ −3.46410 −0.325875 −0.162938 0.986636i $$-0.552097\pi$$
−0.162938 + 0.986636i $$0.552097\pi$$
$$114$$ −3.46410 −0.324443
$$115$$ −8.19615 −0.764295
$$116$$ 8.66025 0.804084
$$117$$ −15.4641 −1.42966
$$118$$ −2.53590 −0.233448
$$119$$ 15.4641 1.41759
$$120$$ −4.73205 −0.431975
$$121$$ 11.3923 1.03566
$$122$$ −1.73205 −0.156813
$$123$$ 27.1244 2.44572
$$124$$ −1.26795 −0.113865
$$125$$ −12.1244 −1.08444
$$126$$ −8.92820 −0.795388
$$127$$ −7.80385 −0.692479 −0.346240 0.938146i $$-0.612542\pi$$
−0.346240 + 0.938146i $$0.612542\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −2.53590 −0.223273
$$130$$ −6.00000 −0.526235
$$131$$ 9.80385 0.856566 0.428283 0.903645i $$-0.359119\pi$$
0.428283 + 0.903645i $$0.359119\pi$$
$$132$$ −12.9282 −1.12526
$$133$$ −2.53590 −0.219890
$$134$$ 10.1962 0.880813
$$135$$ −6.92820 −0.596285
$$136$$ −7.73205 −0.663018
$$137$$ −1.39230 −0.118953 −0.0594763 0.998230i $$-0.518943\pi$$
−0.0594763 + 0.998230i $$0.518943\pi$$
$$138$$ 12.9282 1.10052
$$139$$ 20.5885 1.74629 0.873145 0.487460i $$-0.162077\pi$$
0.873145 + 0.487460i $$0.162077\pi$$
$$140$$ −3.46410 −0.292770
$$141$$ −12.9282 −1.08875
$$142$$ 3.46410 0.290701
$$143$$ −16.3923 −1.37079
$$144$$ 4.46410 0.372008
$$145$$ 15.0000 1.24568
$$146$$ −4.00000 −0.331042
$$147$$ 8.19615 0.676007
$$148$$ 0 0
$$149$$ −15.9282 −1.30489 −0.652445 0.757836i $$-0.726257\pi$$
−0.652445 + 0.757836i $$0.726257\pi$$
$$150$$ 5.46410 0.446142
$$151$$ −4.19615 −0.341478 −0.170739 0.985316i $$-0.554616\pi$$
−0.170739 + 0.985316i $$0.554616\pi$$
$$152$$ 1.26795 0.102844
$$153$$ −34.5167 −2.79051
$$154$$ −9.46410 −0.762639
$$155$$ −2.19615 −0.176399
$$156$$ 9.46410 0.757735
$$157$$ 1.00000 0.0798087 0.0399043 0.999204i $$-0.487295\pi$$
0.0399043 + 0.999204i $$0.487295\pi$$
$$158$$ −13.2679 −1.05554
$$159$$ −6.92820 −0.549442
$$160$$ 1.73205 0.136931
$$161$$ 9.46410 0.745876
$$162$$ −2.46410 −0.193598
$$163$$ 12.9282 1.01262 0.506308 0.862353i $$-0.331010\pi$$
0.506308 + 0.862353i $$0.331010\pi$$
$$164$$ −9.92820 −0.775262
$$165$$ −22.3923 −1.74324
$$166$$ −5.66025 −0.439321
$$167$$ −13.8564 −1.07224 −0.536120 0.844141i $$-0.680111\pi$$
−0.536120 + 0.844141i $$0.680111\pi$$
$$168$$ 5.46410 0.421565
$$169$$ −1.00000 −0.0769231
$$170$$ −13.3923 −1.02714
$$171$$ 5.66025 0.432850
$$172$$ 0.928203 0.0707748
$$173$$ −5.53590 −0.420887 −0.210443 0.977606i $$-0.567491\pi$$
−0.210443 + 0.977606i $$0.567491\pi$$
$$174$$ −23.6603 −1.79368
$$175$$ 4.00000 0.302372
$$176$$ 4.73205 0.356692
$$177$$ 6.92820 0.520756
$$178$$ −6.80385 −0.509970
$$179$$ −9.46410 −0.707380 −0.353690 0.935363i $$-0.615073\pi$$
−0.353690 + 0.935363i $$0.615073\pi$$
$$180$$ 7.73205 0.576313
$$181$$ −9.39230 −0.698125 −0.349062 0.937100i $$-0.613500\pi$$
−0.349062 + 0.937100i $$0.613500\pi$$
$$182$$ 6.92820 0.513553
$$183$$ 4.73205 0.349803
$$184$$ −4.73205 −0.348851
$$185$$ 0 0
$$186$$ 3.46410 0.254000
$$187$$ −36.5885 −2.67561
$$188$$ 4.73205 0.345120
$$189$$ 8.00000 0.581914
$$190$$ 2.19615 0.159326
$$191$$ −2.19615 −0.158908 −0.0794540 0.996839i $$-0.525318\pi$$
−0.0794540 + 0.996839i $$0.525318\pi$$
$$192$$ −2.73205 −0.197169
$$193$$ −11.1962 −0.805917 −0.402958 0.915218i $$-0.632018\pi$$
−0.402958 + 0.915218i $$0.632018\pi$$
$$194$$ −7.73205 −0.555129
$$195$$ 16.3923 1.17388
$$196$$ −3.00000 −0.214286
$$197$$ 6.46410 0.460548 0.230274 0.973126i $$-0.426038\pi$$
0.230274 + 0.973126i $$0.426038\pi$$
$$198$$ 21.1244 1.50124
$$199$$ −10.3923 −0.736691 −0.368345 0.929689i $$-0.620076\pi$$
−0.368345 + 0.929689i $$0.620076\pi$$
$$200$$ −2.00000 −0.141421
$$201$$ −27.8564 −1.96484
$$202$$ −12.4641 −0.876971
$$203$$ −17.3205 −1.21566
$$204$$ 21.1244 1.47900
$$205$$ −17.1962 −1.20103
$$206$$ −8.53590 −0.594724
$$207$$ −21.1244 −1.46824
$$208$$ −3.46410 −0.240192
$$209$$ 6.00000 0.415029
$$210$$ 9.46410 0.653085
$$211$$ −2.39230 −0.164693 −0.0823465 0.996604i $$-0.526241\pi$$
−0.0823465 + 0.996604i $$0.526241\pi$$
$$212$$ 2.53590 0.174166
$$213$$ −9.46410 −0.648470
$$214$$ −10.3923 −0.710403
$$215$$ 1.60770 0.109644
$$216$$ −4.00000 −0.272166
$$217$$ 2.53590 0.172148
$$218$$ −2.66025 −0.180175
$$219$$ 10.9282 0.738460
$$220$$ 8.19615 0.552584
$$221$$ 26.7846 1.80173
$$222$$ 0 0
$$223$$ 16.1962 1.08457 0.542287 0.840193i $$-0.317558\pi$$
0.542287 + 0.840193i $$0.317558\pi$$
$$224$$ −2.00000 −0.133631
$$225$$ −8.92820 −0.595214
$$226$$ −3.46410 −0.230429
$$227$$ 1.26795 0.0841567 0.0420784 0.999114i $$-0.486602\pi$$
0.0420784 + 0.999114i $$0.486602\pi$$
$$228$$ −3.46410 −0.229416
$$229$$ 27.3923 1.81013 0.905067 0.425269i $$-0.139820\pi$$
0.905067 + 0.425269i $$0.139820\pi$$
$$230$$ −8.19615 −0.540438
$$231$$ 25.8564 1.70123
$$232$$ 8.66025 0.568574
$$233$$ −15.0000 −0.982683 −0.491341 0.870967i $$-0.663493\pi$$
−0.491341 + 0.870967i $$0.663493\pi$$
$$234$$ −15.4641 −1.01092
$$235$$ 8.19615 0.534658
$$236$$ −2.53590 −0.165073
$$237$$ 36.2487 2.35461
$$238$$ 15.4641 1.00239
$$239$$ 17.3205 1.12037 0.560185 0.828367i $$-0.310730\pi$$
0.560185 + 0.828367i $$0.310730\pi$$
$$240$$ −4.73205 −0.305453
$$241$$ 15.4641 0.996130 0.498065 0.867140i $$-0.334044\pi$$
0.498065 + 0.867140i $$0.334044\pi$$
$$242$$ 11.3923 0.732325
$$243$$ 18.7321 1.20166
$$244$$ −1.73205 −0.110883
$$245$$ −5.19615 −0.331970
$$246$$ 27.1244 1.72939
$$247$$ −4.39230 −0.279476
$$248$$ −1.26795 −0.0805149
$$249$$ 15.4641 0.979998
$$250$$ −12.1244 −0.766812
$$251$$ −17.3205 −1.09326 −0.546630 0.837374i $$-0.684090\pi$$
−0.546630 + 0.837374i $$0.684090\pi$$
$$252$$ −8.92820 −0.562424
$$253$$ −22.3923 −1.40779
$$254$$ −7.80385 −0.489657
$$255$$ 36.5885 2.29126
$$256$$ 1.00000 0.0625000
$$257$$ −6.80385 −0.424412 −0.212206 0.977225i $$-0.568065\pi$$
−0.212206 + 0.977225i $$0.568065\pi$$
$$258$$ −2.53590 −0.157878
$$259$$ 0 0
$$260$$ −6.00000 −0.372104
$$261$$ 38.6603 2.39301
$$262$$ 9.80385 0.605684
$$263$$ −21.4641 −1.32353 −0.661767 0.749710i $$-0.730193\pi$$
−0.661767 + 0.749710i $$0.730193\pi$$
$$264$$ −12.9282 −0.795676
$$265$$ 4.39230 0.269817
$$266$$ −2.53590 −0.155486
$$267$$ 18.5885 1.13760
$$268$$ 10.1962 0.622829
$$269$$ −4.39230 −0.267804 −0.133902 0.990995i $$-0.542751\pi$$
−0.133902 + 0.990995i $$0.542751\pi$$
$$270$$ −6.92820 −0.421637
$$271$$ −2.58846 −0.157238 −0.0786188 0.996905i $$-0.525051\pi$$
−0.0786188 + 0.996905i $$0.525051\pi$$
$$272$$ −7.73205 −0.468824
$$273$$ −18.9282 −1.14559
$$274$$ −1.39230 −0.0841122
$$275$$ −9.46410 −0.570707
$$276$$ 12.9282 0.778186
$$277$$ −11.1962 −0.672712 −0.336356 0.941735i $$-0.609194\pi$$
−0.336356 + 0.941735i $$0.609194\pi$$
$$278$$ 20.5885 1.23481
$$279$$ −5.66025 −0.338871
$$280$$ −3.46410 −0.207020
$$281$$ 27.5885 1.64579 0.822895 0.568194i $$-0.192358\pi$$
0.822895 + 0.568194i $$0.192358\pi$$
$$282$$ −12.9282 −0.769863
$$283$$ 7.85641 0.467015 0.233507 0.972355i $$-0.424980\pi$$
0.233507 + 0.972355i $$0.424980\pi$$
$$284$$ 3.46410 0.205557
$$285$$ −6.00000 −0.355409
$$286$$ −16.3923 −0.969297
$$287$$ 19.8564 1.17209
$$288$$ 4.46410 0.263050
$$289$$ 42.7846 2.51674
$$290$$ 15.0000 0.880830
$$291$$ 21.1244 1.23833
$$292$$ −4.00000 −0.234082
$$293$$ −1.39230 −0.0813393 −0.0406697 0.999173i $$-0.512949\pi$$
−0.0406697 + 0.999173i $$0.512949\pi$$
$$294$$ 8.19615 0.478009
$$295$$ −4.39230 −0.255730
$$296$$ 0 0
$$297$$ −18.9282 −1.09833
$$298$$ −15.9282 −0.922696
$$299$$ 16.3923 0.947991
$$300$$ 5.46410 0.315470
$$301$$ −1.85641 −0.107001
$$302$$ −4.19615 −0.241461
$$303$$ 34.0526 1.95627
$$304$$ 1.26795 0.0727219
$$305$$ −3.00000 −0.171780
$$306$$ −34.5167 −1.97319
$$307$$ 22.0000 1.25561 0.627803 0.778372i $$-0.283954\pi$$
0.627803 + 0.778372i $$0.283954\pi$$
$$308$$ −9.46410 −0.539267
$$309$$ 23.3205 1.32666
$$310$$ −2.19615 −0.124733
$$311$$ −18.0000 −1.02069 −0.510343 0.859971i $$-0.670482\pi$$
−0.510343 + 0.859971i $$0.670482\pi$$
$$312$$ 9.46410 0.535799
$$313$$ −24.1244 −1.36359 −0.681795 0.731544i $$-0.738800\pi$$
−0.681795 + 0.731544i $$0.738800\pi$$
$$314$$ 1.00000 0.0564333
$$315$$ −15.4641 −0.871303
$$316$$ −13.2679 −0.746380
$$317$$ −7.39230 −0.415193 −0.207597 0.978215i $$-0.566564\pi$$
−0.207597 + 0.978215i $$0.566564\pi$$
$$318$$ −6.92820 −0.388514
$$319$$ 40.9808 2.29448
$$320$$ 1.73205 0.0968246
$$321$$ 28.3923 1.58470
$$322$$ 9.46410 0.527414
$$323$$ −9.80385 −0.545501
$$324$$ −2.46410 −0.136895
$$325$$ 6.92820 0.384308
$$326$$ 12.9282 0.716027
$$327$$ 7.26795 0.401919
$$328$$ −9.92820 −0.548193
$$329$$ −9.46410 −0.521773
$$330$$ −22.3923 −1.23266
$$331$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$332$$ −5.66025 −0.310647
$$333$$ 0 0
$$334$$ −13.8564 −0.758189
$$335$$ 17.6603 0.964883
$$336$$ 5.46410 0.298091
$$337$$ 32.1769 1.75279 0.876394 0.481595i $$-0.159942\pi$$
0.876394 + 0.481595i $$0.159942\pi$$
$$338$$ −1.00000 −0.0543928
$$339$$ 9.46410 0.514019
$$340$$ −13.3923 −0.726300
$$341$$ −6.00000 −0.324918
$$342$$ 5.66025 0.306071
$$343$$ 20.0000 1.07990
$$344$$ 0.928203 0.0500454
$$345$$ 22.3923 1.20556
$$346$$ −5.53590 −0.297612
$$347$$ 9.12436 0.489821 0.244911 0.969546i $$-0.421241\pi$$
0.244911 + 0.969546i $$0.421241\pi$$
$$348$$ −23.6603 −1.26832
$$349$$ 11.3923 0.609816 0.304908 0.952382i $$-0.401374\pi$$
0.304908 + 0.952382i $$0.401374\pi$$
$$350$$ 4.00000 0.213809
$$351$$ 13.8564 0.739600
$$352$$ 4.73205 0.252219
$$353$$ −19.7321 −1.05023 −0.525116 0.851031i $$-0.675978\pi$$
−0.525116 + 0.851031i $$0.675978\pi$$
$$354$$ 6.92820 0.368230
$$355$$ 6.00000 0.318447
$$356$$ −6.80385 −0.360603
$$357$$ −42.2487 −2.23604
$$358$$ −9.46410 −0.500193
$$359$$ 35.3205 1.86415 0.932073 0.362272i $$-0.117999\pi$$
0.932073 + 0.362272i $$0.117999\pi$$
$$360$$ 7.73205 0.407515
$$361$$ −17.3923 −0.915384
$$362$$ −9.39230 −0.493649
$$363$$ −31.1244 −1.63361
$$364$$ 6.92820 0.363137
$$365$$ −6.92820 −0.362639
$$366$$ 4.73205 0.247348
$$367$$ 12.3923 0.646873 0.323437 0.946250i $$-0.395162\pi$$
0.323437 + 0.946250i $$0.395162\pi$$
$$368$$ −4.73205 −0.246675
$$369$$ −44.3205 −2.30723
$$370$$ 0 0
$$371$$ −5.07180 −0.263315
$$372$$ 3.46410 0.179605
$$373$$ −7.00000 −0.362446 −0.181223 0.983442i $$-0.558006\pi$$
−0.181223 + 0.983442i $$0.558006\pi$$
$$374$$ −36.5885 −1.89194
$$375$$ 33.1244 1.71053
$$376$$ 4.73205 0.244037
$$377$$ −30.0000 −1.54508
$$378$$ 8.00000 0.411476
$$379$$ −34.7846 −1.78677 −0.893383 0.449297i $$-0.851675\pi$$
−0.893383 + 0.449297i $$0.851675\pi$$
$$380$$ 2.19615 0.112660
$$381$$ 21.3205 1.09228
$$382$$ −2.19615 −0.112365
$$383$$ −3.46410 −0.177007 −0.0885037 0.996076i $$-0.528208\pi$$
−0.0885037 + 0.996076i $$0.528208\pi$$
$$384$$ −2.73205 −0.139419
$$385$$ −16.3923 −0.835429
$$386$$ −11.1962 −0.569869
$$387$$ 4.14359 0.210631
$$388$$ −7.73205 −0.392535
$$389$$ −25.9808 −1.31728 −0.658638 0.752460i $$-0.728867\pi$$
−0.658638 + 0.752460i $$0.728867\pi$$
$$390$$ 16.3923 0.830057
$$391$$ 36.5885 1.85036
$$392$$ −3.00000 −0.151523
$$393$$ −26.7846 −1.35110
$$394$$ 6.46410 0.325657
$$395$$ −22.9808 −1.15629
$$396$$ 21.1244 1.06154
$$397$$ 23.0000 1.15434 0.577168 0.816625i $$-0.304158\pi$$
0.577168 + 0.816625i $$0.304158\pi$$
$$398$$ −10.3923 −0.520919
$$399$$ 6.92820 0.346844
$$400$$ −2.00000 −0.100000
$$401$$ −10.3923 −0.518967 −0.259483 0.965748i $$-0.583552\pi$$
−0.259483 + 0.965748i $$0.583552\pi$$
$$402$$ −27.8564 −1.38935
$$403$$ 4.39230 0.218796
$$404$$ −12.4641 −0.620112
$$405$$ −4.26795 −0.212076
$$406$$ −17.3205 −0.859602
$$407$$ 0 0
$$408$$ 21.1244 1.04581
$$409$$ 10.5167 0.520015 0.260008 0.965607i $$-0.416275\pi$$
0.260008 + 0.965607i $$0.416275\pi$$
$$410$$ −17.1962 −0.849257
$$411$$ 3.80385 0.187630
$$412$$ −8.53590 −0.420534
$$413$$ 5.07180 0.249567
$$414$$ −21.1244 −1.03821
$$415$$ −9.80385 −0.481252
$$416$$ −3.46410 −0.169842
$$417$$ −56.2487 −2.75451
$$418$$ 6.00000 0.293470
$$419$$ −4.14359 −0.202428 −0.101214 0.994865i $$-0.532273\pi$$
−0.101214 + 0.994865i $$0.532273\pi$$
$$420$$ 9.46410 0.461801
$$421$$ −12.1244 −0.590905 −0.295452 0.955357i $$-0.595470\pi$$
−0.295452 + 0.955357i $$0.595470\pi$$
$$422$$ −2.39230 −0.116456
$$423$$ 21.1244 1.02710
$$424$$ 2.53590 0.123154
$$425$$ 15.4641 0.750119
$$426$$ −9.46410 −0.458537
$$427$$ 3.46410 0.167640
$$428$$ −10.3923 −0.502331
$$429$$ 44.7846 2.16222
$$430$$ 1.60770 0.0775299
$$431$$ 17.6603 0.850665 0.425332 0.905037i $$-0.360157\pi$$
0.425332 + 0.905037i $$0.360157\pi$$
$$432$$ −4.00000 −0.192450
$$433$$ 15.7846 0.758560 0.379280 0.925282i $$-0.376172\pi$$
0.379280 + 0.925282i $$0.376172\pi$$
$$434$$ 2.53590 0.121727
$$435$$ −40.9808 −1.96488
$$436$$ −2.66025 −0.127403
$$437$$ −6.00000 −0.287019
$$438$$ 10.9282 0.522170
$$439$$ 18.2487 0.870963 0.435482 0.900198i $$-0.356578\pi$$
0.435482 + 0.900198i $$0.356578\pi$$
$$440$$ 8.19615 0.390736
$$441$$ −13.3923 −0.637729
$$442$$ 26.7846 1.27401
$$443$$ 14.5359 0.690621 0.345311 0.938488i $$-0.387774\pi$$
0.345311 + 0.938488i $$0.387774\pi$$
$$444$$ 0 0
$$445$$ −11.7846 −0.558644
$$446$$ 16.1962 0.766910
$$447$$ 43.5167 2.05827
$$448$$ −2.00000 −0.0944911
$$449$$ 20.5359 0.969149 0.484574 0.874750i $$-0.338974\pi$$
0.484574 + 0.874750i $$0.338974\pi$$
$$450$$ −8.92820 −0.420880
$$451$$ −46.9808 −2.21224
$$452$$ −3.46410 −0.162938
$$453$$ 11.4641 0.538630
$$454$$ 1.26795 0.0595078
$$455$$ 12.0000 0.562569
$$456$$ −3.46410 −0.162221
$$457$$ −37.9808 −1.77667 −0.888333 0.459201i $$-0.848136\pi$$
−0.888333 + 0.459201i $$0.848136\pi$$
$$458$$ 27.3923 1.27996
$$459$$ 30.9282 1.44360
$$460$$ −8.19615 −0.382148
$$461$$ 39.4641 1.83803 0.919013 0.394227i $$-0.128988\pi$$
0.919013 + 0.394227i $$0.128988\pi$$
$$462$$ 25.8564 1.20295
$$463$$ −30.0000 −1.39422 −0.697109 0.716965i $$-0.745531\pi$$
−0.697109 + 0.716965i $$0.745531\pi$$
$$464$$ 8.66025 0.402042
$$465$$ 6.00000 0.278243
$$466$$ −15.0000 −0.694862
$$467$$ 2.53590 0.117347 0.0586737 0.998277i $$-0.481313\pi$$
0.0586737 + 0.998277i $$0.481313\pi$$
$$468$$ −15.4641 −0.714828
$$469$$ −20.3923 −0.941629
$$470$$ 8.19615 0.378060
$$471$$ −2.73205 −0.125886
$$472$$ −2.53590 −0.116724
$$473$$ 4.39230 0.201958
$$474$$ 36.2487 1.66496
$$475$$ −2.53590 −0.116355
$$476$$ 15.4641 0.708796
$$477$$ 11.3205 0.518330
$$478$$ 17.3205 0.792222
$$479$$ −9.46410 −0.432426 −0.216213 0.976346i $$-0.569371\pi$$
−0.216213 + 0.976346i $$0.569371\pi$$
$$480$$ −4.73205 −0.215988
$$481$$ 0 0
$$482$$ 15.4641 0.704371
$$483$$ −25.8564 −1.17651
$$484$$ 11.3923 0.517832
$$485$$ −13.3923 −0.608113
$$486$$ 18.7321 0.849703
$$487$$ −16.0526 −0.727411 −0.363705 0.931514i $$-0.618489\pi$$
−0.363705 + 0.931514i $$0.618489\pi$$
$$488$$ −1.73205 −0.0784063
$$489$$ −35.3205 −1.59725
$$490$$ −5.19615 −0.234738
$$491$$ 26.1962 1.18222 0.591108 0.806592i $$-0.298691\pi$$
0.591108 + 0.806592i $$0.298691\pi$$
$$492$$ 27.1244 1.22286
$$493$$ −66.9615 −3.01580
$$494$$ −4.39230 −0.197619
$$495$$ 36.5885 1.64453
$$496$$ −1.26795 −0.0569326
$$497$$ −6.92820 −0.310772
$$498$$ 15.4641 0.692963
$$499$$ 20.1962 0.904104 0.452052 0.891992i $$-0.350692\pi$$
0.452052 + 0.891992i $$0.350692\pi$$
$$500$$ −12.1244 −0.542218
$$501$$ 37.8564 1.69130
$$502$$ −17.3205 −0.773052
$$503$$ 20.1962 0.900502 0.450251 0.892902i $$-0.351335\pi$$
0.450251 + 0.892902i $$0.351335\pi$$
$$504$$ −8.92820 −0.397694
$$505$$ −21.5885 −0.960674
$$506$$ −22.3923 −0.995459
$$507$$ 2.73205 0.121335
$$508$$ −7.80385 −0.346240
$$509$$ 6.46410 0.286516 0.143258 0.989685i $$-0.454242\pi$$
0.143258 + 0.989685i $$0.454242\pi$$
$$510$$ 36.5885 1.62016
$$511$$ 8.00000 0.353899
$$512$$ 1.00000 0.0441942
$$513$$ −5.07180 −0.223925
$$514$$ −6.80385 −0.300105
$$515$$ −14.7846 −0.651488
$$516$$ −2.53590 −0.111637
$$517$$ 22.3923 0.984812
$$518$$ 0 0
$$519$$ 15.1244 0.663886
$$520$$ −6.00000 −0.263117
$$521$$ −5.07180 −0.222199 −0.111100 0.993809i $$-0.535437\pi$$
−0.111100 + 0.993809i $$0.535437\pi$$
$$522$$ 38.6603 1.69211
$$523$$ 20.4449 0.893991 0.446996 0.894536i $$-0.352494\pi$$
0.446996 + 0.894536i $$0.352494\pi$$
$$524$$ 9.80385 0.428283
$$525$$ −10.9282 −0.476946
$$526$$ −21.4641 −0.935879
$$527$$ 9.80385 0.427062
$$528$$ −12.9282 −0.562628
$$529$$ −0.607695 −0.0264215
$$530$$ 4.39230 0.190790
$$531$$ −11.3205 −0.491268
$$532$$ −2.53590 −0.109945
$$533$$ 34.3923 1.48970
$$534$$ 18.5885 0.804401
$$535$$ −18.0000 −0.778208
$$536$$ 10.1962 0.440407
$$537$$ 25.8564 1.11579
$$538$$ −4.39230 −0.189366
$$539$$ −14.1962 −0.611472
$$540$$ −6.92820 −0.298142
$$541$$ −19.0526 −0.819133 −0.409567 0.912280i $$-0.634320\pi$$
−0.409567 + 0.912280i $$0.634320\pi$$
$$542$$ −2.58846 −0.111184
$$543$$ 25.6603 1.10119
$$544$$ −7.73205 −0.331509
$$545$$ −4.60770 −0.197372
$$546$$ −18.9282 −0.810052
$$547$$ 41.6603 1.78126 0.890632 0.454725i $$-0.150262\pi$$
0.890632 + 0.454725i $$0.150262\pi$$
$$548$$ −1.39230 −0.0594763
$$549$$ −7.73205 −0.329996
$$550$$ −9.46410 −0.403551
$$551$$ 10.9808 0.467796
$$552$$ 12.9282 0.550261
$$553$$ 26.5359 1.12842
$$554$$ −11.1962 −0.475679
$$555$$ 0 0
$$556$$ 20.5885 0.873145
$$557$$ 23.4449 0.993391 0.496695 0.867925i $$-0.334547\pi$$
0.496695 + 0.867925i $$0.334547\pi$$
$$558$$ −5.66025 −0.239618
$$559$$ −3.21539 −0.135997
$$560$$ −3.46410 −0.146385
$$561$$ 99.9615 4.22038
$$562$$ 27.5885 1.16375
$$563$$ 18.5885 0.783410 0.391705 0.920091i $$-0.371885\pi$$
0.391705 + 0.920091i $$0.371885\pi$$
$$564$$ −12.9282 −0.544376
$$565$$ −6.00000 −0.252422
$$566$$ 7.85641 0.330229
$$567$$ 4.92820 0.206965
$$568$$ 3.46410 0.145350
$$569$$ −5.87564 −0.246320 −0.123160 0.992387i $$-0.539303\pi$$
−0.123160 + 0.992387i $$0.539303\pi$$
$$570$$ −6.00000 −0.251312
$$571$$ 37.3731 1.56401 0.782007 0.623270i $$-0.214196\pi$$
0.782007 + 0.623270i $$0.214196\pi$$
$$572$$ −16.3923 −0.685397
$$573$$ 6.00000 0.250654
$$574$$ 19.8564 0.828790
$$575$$ 9.46410 0.394680
$$576$$ 4.46410 0.186004
$$577$$ 0.928203 0.0386416 0.0193208 0.999813i $$-0.493850\pi$$
0.0193208 + 0.999813i $$0.493850\pi$$
$$578$$ 42.7846 1.77961
$$579$$ 30.5885 1.27121
$$580$$ 15.0000 0.622841
$$581$$ 11.3205 0.469654
$$582$$ 21.1244 0.875633
$$583$$ 12.0000 0.496989
$$584$$ −4.00000 −0.165521
$$585$$ −26.7846 −1.10741
$$586$$ −1.39230 −0.0575156
$$587$$ −33.1244 −1.36719 −0.683594 0.729862i $$-0.739584\pi$$
−0.683594 + 0.729862i $$0.739584\pi$$
$$588$$ 8.19615 0.338004
$$589$$ −1.60770 −0.0662439
$$590$$ −4.39230 −0.180828
$$591$$ −17.6603 −0.726446
$$592$$ 0 0
$$593$$ 43.6410 1.79212 0.896061 0.443931i $$-0.146417\pi$$
0.896061 + 0.443931i $$0.146417\pi$$
$$594$$ −18.9282 −0.776634
$$595$$ 26.7846 1.09806
$$596$$ −15.9282 −0.652445
$$597$$ 28.3923 1.16202
$$598$$ 16.3923 0.670331
$$599$$ 47.6603 1.94735 0.973673 0.227951i $$-0.0732026\pi$$
0.973673 + 0.227951i $$0.0732026\pi$$
$$600$$ 5.46410 0.223071
$$601$$ −12.6077 −0.514279 −0.257139 0.966374i $$-0.582780\pi$$
−0.257139 + 0.966374i $$0.582780\pi$$
$$602$$ −1.85641 −0.0756615
$$603$$ 45.5167 1.85358
$$604$$ −4.19615 −0.170739
$$605$$ 19.7321 0.802222
$$606$$ 34.0526 1.38329
$$607$$ −23.9090 −0.970435 −0.485217 0.874394i $$-0.661260\pi$$
−0.485217 + 0.874394i $$0.661260\pi$$
$$608$$ 1.26795 0.0514221
$$609$$ 47.3205 1.91752
$$610$$ −3.00000 −0.121466
$$611$$ −16.3923 −0.663162
$$612$$ −34.5167 −1.39525
$$613$$ −7.78461 −0.314417 −0.157209 0.987565i $$-0.550250\pi$$
−0.157209 + 0.987565i $$0.550250\pi$$
$$614$$ 22.0000 0.887848
$$615$$ 46.9808 1.89445
$$616$$ −9.46410 −0.381320
$$617$$ −12.6795 −0.510457 −0.255229 0.966881i $$-0.582151\pi$$
−0.255229 + 0.966881i $$0.582151\pi$$
$$618$$ 23.3205 0.938088
$$619$$ −30.3923 −1.22157 −0.610785 0.791797i $$-0.709146\pi$$
−0.610785 + 0.791797i $$0.709146\pi$$
$$620$$ −2.19615 −0.0881996
$$621$$ 18.9282 0.759563
$$622$$ −18.0000 −0.721734
$$623$$ 13.6077 0.545181
$$624$$ 9.46410 0.378867
$$625$$ −11.0000 −0.440000
$$626$$ −24.1244 −0.964203
$$627$$ −16.3923 −0.654646
$$628$$ 1.00000 0.0399043
$$629$$ 0 0
$$630$$ −15.4641 −0.616105
$$631$$ −34.9808 −1.39256 −0.696281 0.717769i $$-0.745163\pi$$
−0.696281 + 0.717769i $$0.745163\pi$$
$$632$$ −13.2679 −0.527771
$$633$$ 6.53590 0.259779
$$634$$ −7.39230 −0.293586
$$635$$ −13.5167 −0.536392
$$636$$ −6.92820 −0.274721
$$637$$ 10.3923 0.411758
$$638$$ 40.9808 1.62244
$$639$$ 15.4641 0.611750
$$640$$ 1.73205 0.0684653
$$641$$ 1.14359 0.0451692 0.0225846 0.999745i $$-0.492810\pi$$
0.0225846 + 0.999745i $$0.492810\pi$$
$$642$$ 28.3923 1.12055
$$643$$ −13.2679 −0.523237 −0.261618 0.965171i $$-0.584256\pi$$
−0.261618 + 0.965171i $$0.584256\pi$$
$$644$$ 9.46410 0.372938
$$645$$ −4.39230 −0.172947
$$646$$ −9.80385 −0.385727
$$647$$ −23.0718 −0.907046 −0.453523 0.891245i $$-0.649833\pi$$
−0.453523 + 0.891245i $$0.649833\pi$$
$$648$$ −2.46410 −0.0967991
$$649$$ −12.0000 −0.471041
$$650$$ 6.92820 0.271746
$$651$$ −6.92820 −0.271538
$$652$$ 12.9282 0.506308
$$653$$ 43.9808 1.72110 0.860550 0.509366i $$-0.170120\pi$$
0.860550 + 0.509366i $$0.170120\pi$$
$$654$$ 7.26795 0.284199
$$655$$ 16.9808 0.663493
$$656$$ −9.92820 −0.387631
$$657$$ −17.8564 −0.696645
$$658$$ −9.46410 −0.368949
$$659$$ 9.46410 0.368669 0.184335 0.982864i $$-0.440987\pi$$
0.184335 + 0.982864i $$0.440987\pi$$
$$660$$ −22.3923 −0.871619
$$661$$ −1.73205 −0.0673690 −0.0336845 0.999433i $$-0.510724\pi$$
−0.0336845 + 0.999433i $$0.510724\pi$$
$$662$$ 0 0
$$663$$ −73.1769 −2.84196
$$664$$ −5.66025 −0.219660
$$665$$ −4.39230 −0.170326
$$666$$ 0 0
$$667$$ −40.9808 −1.58678
$$668$$ −13.8564 −0.536120
$$669$$ −44.2487 −1.71075
$$670$$ 17.6603 0.682275
$$671$$ −8.19615 −0.316409
$$672$$ 5.46410 0.210782
$$673$$ 16.0000 0.616755 0.308377 0.951264i $$-0.400214\pi$$
0.308377 + 0.951264i $$0.400214\pi$$
$$674$$ 32.1769 1.23941
$$675$$ 8.00000 0.307920
$$676$$ −1.00000 −0.0384615
$$677$$ −33.9282 −1.30397 −0.651983 0.758233i $$-0.726063\pi$$
−0.651983 + 0.758233i $$0.726063\pi$$
$$678$$ 9.46410 0.363467
$$679$$ 15.4641 0.593458
$$680$$ −13.3923 −0.513571
$$681$$ −3.46410 −0.132745
$$682$$ −6.00000 −0.229752
$$683$$ 20.5359 0.785784 0.392892 0.919585i $$-0.371475\pi$$
0.392892 + 0.919585i $$0.371475\pi$$
$$684$$ 5.66025 0.216425
$$685$$ −2.41154 −0.0921403
$$686$$ 20.0000 0.763604
$$687$$ −74.8372 −2.85522
$$688$$ 0.928203 0.0353874
$$689$$ −8.78461 −0.334667
$$690$$ 22.3923 0.852460
$$691$$ −26.9808 −1.02640 −0.513198 0.858270i $$-0.671539\pi$$
−0.513198 + 0.858270i $$0.671539\pi$$
$$692$$ −5.53590 −0.210443
$$693$$ −42.2487 −1.60490
$$694$$ 9.12436 0.346356
$$695$$ 35.6603 1.35267
$$696$$ −23.6603 −0.896840
$$697$$ 76.7654 2.90770
$$698$$ 11.3923 0.431205
$$699$$ 40.9808 1.55003
$$700$$ 4.00000 0.151186
$$701$$ −18.9282 −0.714908 −0.357454 0.933931i $$-0.616355\pi$$
−0.357454 + 0.933931i $$0.616355\pi$$
$$702$$ 13.8564 0.522976
$$703$$ 0 0
$$704$$ 4.73205 0.178346
$$705$$ −22.3923 −0.843343
$$706$$ −19.7321 −0.742626
$$707$$ 24.9282 0.937522
$$708$$ 6.92820 0.260378
$$709$$ 9.71281 0.364772 0.182386 0.983227i $$-0.441618\pi$$
0.182386 + 0.983227i $$0.441618\pi$$
$$710$$ 6.00000 0.225176
$$711$$ −59.2295 −2.22128
$$712$$ −6.80385 −0.254985
$$713$$ 6.00000 0.224702
$$714$$ −42.2487 −1.58112
$$715$$ −28.3923 −1.06181
$$716$$ −9.46410 −0.353690
$$717$$ −47.3205 −1.76722
$$718$$ 35.3205 1.31815
$$719$$ 37.2679 1.38986 0.694930 0.719077i $$-0.255435\pi$$
0.694930 + 0.719077i $$0.255435\pi$$
$$720$$ 7.73205 0.288157
$$721$$ 17.0718 0.635787
$$722$$ −17.3923 −0.647275
$$723$$ −42.2487 −1.57125
$$724$$ −9.39230 −0.349062
$$725$$ −17.3205 −0.643268
$$726$$ −31.1244 −1.15513
$$727$$ −39.7128 −1.47287 −0.736433 0.676510i $$-0.763492\pi$$
−0.736433 + 0.676510i $$0.763492\pi$$
$$728$$ 6.92820 0.256776
$$729$$ −43.7846 −1.62165
$$730$$ −6.92820 −0.256424
$$731$$ −7.17691 −0.265448
$$732$$ 4.73205 0.174902
$$733$$ 33.5692 1.23991 0.619954 0.784638i $$-0.287151\pi$$
0.619954 + 0.784638i $$0.287151\pi$$
$$734$$ 12.3923 0.457408
$$735$$ 14.1962 0.523633
$$736$$ −4.73205 −0.174426
$$737$$ 48.2487 1.77726
$$738$$ −44.3205 −1.63146
$$739$$ −32.5885 −1.19879 −0.599393 0.800455i $$-0.704591\pi$$
−0.599393 + 0.800455i $$0.704591\pi$$
$$740$$ 0 0
$$741$$ 12.0000 0.440831
$$742$$ −5.07180 −0.186192
$$743$$ −1.51666 −0.0556409 −0.0278204 0.999613i $$-0.508857\pi$$
−0.0278204 + 0.999613i $$0.508857\pi$$
$$744$$ 3.46410 0.127000
$$745$$ −27.5885 −1.01076
$$746$$ −7.00000 −0.256288
$$747$$ −25.2679 −0.924506
$$748$$ −36.5885 −1.33781
$$749$$ 20.7846 0.759453
$$750$$ 33.1244 1.20953
$$751$$ −29.6077 −1.08040 −0.540200 0.841537i $$-0.681651\pi$$
−0.540200 + 0.841537i $$0.681651\pi$$
$$752$$ 4.73205 0.172560
$$753$$ 47.3205 1.72446
$$754$$ −30.0000 −1.09254
$$755$$ −7.26795 −0.264508
$$756$$ 8.00000 0.290957
$$757$$ −17.1962 −0.625005 −0.312502 0.949917i $$-0.601167\pi$$
−0.312502 + 0.949917i $$0.601167\pi$$
$$758$$ −34.7846 −1.26343
$$759$$ 61.1769 2.22058
$$760$$ 2.19615 0.0796628
$$761$$ 2.32051 0.0841184 0.0420592 0.999115i $$-0.486608\pi$$
0.0420592 + 0.999115i $$0.486608\pi$$
$$762$$ 21.3205 0.772361
$$763$$ 5.32051 0.192615
$$764$$ −2.19615 −0.0794540
$$765$$ −59.7846 −2.16152
$$766$$ −3.46410 −0.125163
$$767$$ 8.78461 0.317194
$$768$$ −2.73205 −0.0985844
$$769$$ 13.6077 0.490706 0.245353 0.969434i $$-0.421096\pi$$
0.245353 + 0.969434i $$0.421096\pi$$
$$770$$ −16.3923 −0.590738
$$771$$ 18.5885 0.669447
$$772$$ −11.1962 −0.402958
$$773$$ 33.9282 1.22031 0.610156 0.792281i $$-0.291107\pi$$
0.610156 + 0.792281i $$0.291107\pi$$
$$774$$ 4.14359 0.148938
$$775$$ 2.53590 0.0910922
$$776$$ −7.73205 −0.277564
$$777$$ 0 0
$$778$$ −25.9808 −0.931455
$$779$$ −12.5885 −0.451028
$$780$$ 16.3923 0.586939
$$781$$ 16.3923 0.586563
$$782$$ 36.5885 1.30840
$$783$$ −34.6410 −1.23797
$$784$$ −3.00000 −0.107143
$$785$$ 1.73205 0.0618195
$$786$$ −26.7846 −0.955375
$$787$$ −41.1769 −1.46780 −0.733899 0.679258i $$-0.762302\pi$$
−0.733899 + 0.679258i $$0.762302\pi$$
$$788$$ 6.46410 0.230274
$$789$$ 58.6410 2.08768
$$790$$ −22.9808 −0.817619
$$791$$ 6.92820 0.246339
$$792$$ 21.1244 0.750621
$$793$$ 6.00000 0.213066
$$794$$ 23.0000 0.816239
$$795$$ −12.0000 −0.425596
$$796$$ −10.3923 −0.368345
$$797$$ 8.78461 0.311167 0.155583 0.987823i $$-0.450274\pi$$
0.155583 + 0.987823i $$0.450274\pi$$
$$798$$ 6.92820 0.245256
$$799$$ −36.5885 −1.29441
$$800$$ −2.00000 −0.0707107
$$801$$ −30.3731 −1.07318
$$802$$ −10.3923 −0.366965
$$803$$ −18.9282 −0.667962
$$804$$ −27.8564 −0.982420
$$805$$ 16.3923 0.577753
$$806$$ 4.39230 0.154712
$$807$$ 12.0000 0.422420
$$808$$ −12.4641 −0.438486
$$809$$ −15.7128 −0.552433 −0.276217 0.961095i $$-0.589081\pi$$
−0.276217 + 0.961095i $$0.589081\pi$$
$$810$$ −4.26795 −0.149960
$$811$$ 20.3923 0.716071 0.358035 0.933708i $$-0.383447\pi$$
0.358035 + 0.933708i $$0.383447\pi$$
$$812$$ −17.3205 −0.607831
$$813$$ 7.07180 0.248019
$$814$$ 0 0
$$815$$ 22.3923 0.784368
$$816$$ 21.1244 0.739500
$$817$$ 1.17691 0.0411750
$$818$$ 10.5167 0.367706
$$819$$ 30.9282 1.08072
$$820$$ −17.1962 −0.600516
$$821$$ 2.53590 0.0885035 0.0442517 0.999020i $$-0.485910\pi$$
0.0442517 + 0.999020i $$0.485910\pi$$
$$822$$ 3.80385 0.132674
$$823$$ 23.8038 0.829750 0.414875 0.909878i $$-0.363825\pi$$
0.414875 + 0.909878i $$0.363825\pi$$
$$824$$ −8.53590 −0.297362
$$825$$ 25.8564 0.900205
$$826$$ 5.07180 0.176470
$$827$$ −1.85641 −0.0645536 −0.0322768 0.999479i $$-0.510276\pi$$
−0.0322768 + 0.999479i $$0.510276\pi$$
$$828$$ −21.1244 −0.734122
$$829$$ 10.3923 0.360940 0.180470 0.983581i $$-0.442238\pi$$
0.180470 + 0.983581i $$0.442238\pi$$
$$830$$ −9.80385 −0.340297
$$831$$ 30.5885 1.06110
$$832$$ −3.46410 −0.120096
$$833$$ 23.1962 0.803699
$$834$$ −56.2487 −1.94773
$$835$$ −24.0000 −0.830554
$$836$$ 6.00000 0.207514
$$837$$ 5.07180 0.175307
$$838$$ −4.14359 −0.143138
$$839$$ 52.9808 1.82910 0.914550 0.404474i $$-0.132545\pi$$
0.914550 + 0.404474i $$0.132545\pi$$
$$840$$ 9.46410 0.326543
$$841$$ 46.0000 1.58621
$$842$$ −12.1244 −0.417833
$$843$$ −75.3731 −2.59599
$$844$$ −2.39230 −0.0823465
$$845$$ −1.73205 −0.0595844
$$846$$ 21.1244 0.726270
$$847$$ −22.7846 −0.782888
$$848$$ 2.53590 0.0870831
$$849$$ −21.4641 −0.736646
$$850$$ 15.4641 0.530414
$$851$$ 0 0
$$852$$ −9.46410 −0.324235
$$853$$ −10.2679 −0.351568 −0.175784 0.984429i $$-0.556246\pi$$
−0.175784 + 0.984429i $$0.556246\pi$$
$$854$$ 3.46410 0.118539
$$855$$ 9.80385 0.335285
$$856$$ −10.3923 −0.355202
$$857$$ −2.66025 −0.0908725 −0.0454363 0.998967i $$-0.514468\pi$$
−0.0454363 + 0.998967i $$0.514468\pi$$
$$858$$ 44.7846 1.52892
$$859$$ −30.0000 −1.02359 −0.511793 0.859109i $$-0.671019\pi$$
−0.511793 + 0.859109i $$0.671019\pi$$
$$860$$ 1.60770 0.0548219
$$861$$ −54.2487 −1.84879
$$862$$ 17.6603 0.601511
$$863$$ 4.39230 0.149516 0.0747579 0.997202i $$-0.476182\pi$$
0.0747579 + 0.997202i $$0.476182\pi$$
$$864$$ −4.00000 −0.136083
$$865$$ −9.58846 −0.326017
$$866$$ 15.7846 0.536383
$$867$$ −116.890 −3.96978
$$868$$ 2.53590 0.0860740
$$869$$ −62.7846 −2.12982
$$870$$ −40.9808 −1.38938
$$871$$ −35.3205 −1.19679
$$872$$ −2.66025 −0.0900876
$$873$$ −34.5167 −1.16821
$$874$$ −6.00000 −0.202953
$$875$$ 24.2487 0.819756
$$876$$ 10.9282 0.369230
$$877$$ 38.1769 1.28914 0.644571 0.764544i $$-0.277036\pi$$
0.644571 + 0.764544i $$0.277036\pi$$
$$878$$ 18.2487 0.615864
$$879$$ 3.80385 0.128301
$$880$$ 8.19615 0.276292
$$881$$ −28.6077 −0.963818 −0.481909 0.876221i $$-0.660056\pi$$
−0.481909 + 0.876221i $$0.660056\pi$$
$$882$$ −13.3923 −0.450942
$$883$$ 16.0526 0.540212 0.270106 0.962831i $$-0.412941\pi$$
0.270106 + 0.962831i $$0.412941\pi$$
$$884$$ 26.7846 0.900864
$$885$$ 12.0000 0.403376
$$886$$ 14.5359 0.488343
$$887$$ 46.0526 1.54629 0.773147 0.634227i $$-0.218682\pi$$
0.773147 + 0.634227i $$0.218682\pi$$
$$888$$ 0 0
$$889$$ 15.6077 0.523465
$$890$$ −11.7846 −0.395021
$$891$$ −11.6603 −0.390633
$$892$$ 16.1962 0.542287
$$893$$ 6.00000 0.200782
$$894$$ 43.5167 1.45541
$$895$$ −16.3923 −0.547934
$$896$$ −2.00000 −0.0668153
$$897$$ −44.7846 −1.49531
$$898$$ 20.5359 0.685292
$$899$$ −10.9808 −0.366229
$$900$$ −8.92820 −0.297607
$$901$$ −19.6077 −0.653227
$$902$$ −46.9808 −1.56429
$$903$$ 5.07180 0.168779
$$904$$ −3.46410 −0.115214
$$905$$ −16.2679 −0.540765
$$906$$ 11.4641 0.380869
$$907$$ 11.4115 0.378914 0.189457 0.981889i $$-0.439327\pi$$
0.189457 + 0.981889i $$0.439327\pi$$
$$908$$ 1.26795 0.0420784
$$909$$ −55.6410 −1.84550
$$910$$ 12.0000 0.397796
$$911$$ −35.6603 −1.18148 −0.590738 0.806863i $$-0.701163\pi$$
−0.590738 + 0.806863i $$0.701163\pi$$
$$912$$ −3.46410 −0.114708
$$913$$ −26.7846 −0.886441
$$914$$ −37.9808 −1.25629
$$915$$ 8.19615 0.270956
$$916$$ 27.3923 0.905067
$$917$$ −19.6077 −0.647503
$$918$$ 30.9282 1.02078
$$919$$ −21.1244 −0.696828 −0.348414 0.937341i $$-0.613280\pi$$
−0.348414 + 0.937341i $$0.613280\pi$$
$$920$$ −8.19615 −0.270219
$$921$$ −60.1051 −1.98053
$$922$$ 39.4641 1.29968
$$923$$ −12.0000 −0.394985
$$924$$ 25.8564 0.850613
$$925$$ 0 0
$$926$$ −30.0000 −0.985861
$$927$$ −38.1051 −1.25154
$$928$$ 8.66025 0.284287
$$929$$ 25.3923 0.833094 0.416547 0.909114i $$-0.363240\pi$$
0.416547 + 0.909114i $$0.363240\pi$$
$$930$$ 6.00000 0.196748
$$931$$ −3.80385 −0.124666
$$932$$ −15.0000 −0.491341
$$933$$ 49.1769 1.60998
$$934$$ 2.53590 0.0829771
$$935$$ −63.3731 −2.07252
$$936$$ −15.4641 −0.505460
$$937$$ 27.3923 0.894868 0.447434 0.894317i $$-0.352338\pi$$
0.447434 + 0.894317i $$0.352338\pi$$
$$938$$ −20.3923 −0.665832
$$939$$ 65.9090 2.15086
$$940$$ 8.19615 0.267329
$$941$$ 29.7846 0.970951 0.485475 0.874250i $$-0.338647\pi$$
0.485475 + 0.874250i $$0.338647\pi$$
$$942$$ −2.73205 −0.0890150
$$943$$ 46.9808 1.52990
$$944$$ −2.53590 −0.0825365
$$945$$ 13.8564 0.450749
$$946$$ 4.39230 0.142806
$$947$$ −37.1769 −1.20809 −0.604044 0.796951i $$-0.706445\pi$$
−0.604044 + 0.796951i $$0.706445\pi$$
$$948$$ 36.2487 1.17730
$$949$$ 13.8564 0.449798
$$950$$ −2.53590 −0.0822754
$$951$$ 20.1962 0.654905
$$952$$ 15.4641 0.501194
$$953$$ −19.8564 −0.643212 −0.321606 0.946874i $$-0.604223\pi$$
−0.321606 + 0.946874i $$0.604223\pi$$
$$954$$ 11.3205 0.366515
$$955$$ −3.80385 −0.123090
$$956$$ 17.3205 0.560185
$$957$$ −111.962 −3.61920
$$958$$ −9.46410 −0.305771
$$959$$ 2.78461 0.0899197
$$960$$ −4.73205 −0.152726
$$961$$ −29.3923 −0.948139
$$962$$ 0 0
$$963$$ −46.3923 −1.49497
$$964$$ 15.4641 0.498065
$$965$$ −19.3923 −0.624260
$$966$$ −25.8564 −0.831916
$$967$$ −48.0000 −1.54358 −0.771788 0.635880i $$-0.780637\pi$$
−0.771788 + 0.635880i $$0.780637\pi$$
$$968$$ 11.3923 0.366163
$$969$$ 26.7846 0.860446
$$970$$ −13.3923 −0.430001
$$971$$ −7.01924 −0.225258 −0.112629 0.993637i $$-0.535927\pi$$
−0.112629 + 0.993637i $$0.535927\pi$$
$$972$$ 18.7321 0.600831
$$973$$ −41.1769 −1.32007
$$974$$ −16.0526 −0.514357
$$975$$ −18.9282 −0.606188
$$976$$ −1.73205 −0.0554416
$$977$$ −3.46410 −0.110826 −0.0554132 0.998464i $$-0.517648\pi$$
−0.0554132 + 0.998464i $$0.517648\pi$$
$$978$$ −35.3205 −1.12943
$$979$$ −32.1962 −1.02899
$$980$$ −5.19615 −0.165985
$$981$$ −11.8756 −0.379160
$$982$$ 26.1962 0.835953
$$983$$ −45.1244 −1.43924 −0.719622 0.694366i $$-0.755685\pi$$
−0.719622 + 0.694366i $$0.755685\pi$$
$$984$$ 27.1244 0.864693
$$985$$ 11.1962 0.356739
$$986$$ −66.9615 −2.13249
$$987$$ 25.8564 0.823018
$$988$$ −4.39230 −0.139738
$$989$$ −4.39230 −0.139667
$$990$$ 36.5885 1.16286
$$991$$ −16.6410 −0.528619 −0.264310 0.964438i $$-0.585144\pi$$
−0.264310 + 0.964438i $$0.585144\pi$$
$$992$$ −1.26795 −0.0402574
$$993$$ 0 0
$$994$$ −6.92820 −0.219749
$$995$$ −18.0000 −0.570638
$$996$$ 15.4641 0.489999
$$997$$ 10.3923 0.329128 0.164564 0.986366i $$-0.447378\pi$$
0.164564 + 0.986366i $$0.447378\pi$$
$$998$$ 20.1962 0.639298
$$999$$ 0 0
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 2738.2.a.i.1.1 2
37.8 odd 12 74.2.e.b.27.2 yes 4
37.14 odd 12 74.2.e.b.11.2 4
37.36 even 2 2738.2.a.e.1.1 2
111.8 even 12 666.2.s.a.397.1 4
111.14 even 12 666.2.s.a.307.1 4
148.51 even 12 592.2.w.e.529.2 4
148.119 even 12 592.2.w.e.545.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.e.b.11.2 4 37.14 odd 12
74.2.e.b.27.2 yes 4 37.8 odd 12
592.2.w.e.529.2 4 148.51 even 12
592.2.w.e.545.2 4 148.119 even 12
666.2.s.a.307.1 4 111.14 even 12
666.2.s.a.397.1 4 111.8 even 12
2738.2.a.e.1.1 2 37.36 even 2
2738.2.a.i.1.1 2 1.1 even 1 trivial