# Properties

 Label 5610.2.a.bo.1.1 Level $5610$ Weight $2$ Character 5610.1 Self dual yes Analytic conductor $44.796$ 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] = [5610,2,Mod(1,5610)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5610, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5610.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5610.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: $$44.7960755339$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{41})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 10$$ x^2 - x - 10 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$3.70156$$ of defining polynomial Character $$\chi$$ $$=$$ 5610.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -3.70156 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -3.70156 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} -1.70156 q^{13} +3.70156 q^{14} +1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +1.70156 q^{19} -1.00000 q^{20} +3.70156 q^{21} -1.00000 q^{22} -2.29844 q^{23} +1.00000 q^{24} +1.00000 q^{25} +1.70156 q^{26} -1.00000 q^{27} -3.70156 q^{28} +2.00000 q^{29} -1.00000 q^{30} -5.70156 q^{31} -1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} +3.70156 q^{35} +1.00000 q^{36} +4.29844 q^{37} -1.70156 q^{38} +1.70156 q^{39} +1.00000 q^{40} +3.40312 q^{41} -3.70156 q^{42} +5.40312 q^{43} +1.00000 q^{44} -1.00000 q^{45} +2.29844 q^{46} +3.40312 q^{47} -1.00000 q^{48} +6.70156 q^{49} -1.00000 q^{50} -1.00000 q^{51} -1.70156 q^{52} +2.00000 q^{53} +1.00000 q^{54} -1.00000 q^{55} +3.70156 q^{56} -1.70156 q^{57} -2.00000 q^{58} -13.4031 q^{59} +1.00000 q^{60} +7.70156 q^{61} +5.70156 q^{62} -3.70156 q^{63} +1.00000 q^{64} +1.70156 q^{65} +1.00000 q^{66} -8.29844 q^{67} +1.00000 q^{68} +2.29844 q^{69} -3.70156 q^{70} +12.8062 q^{71} -1.00000 q^{72} -8.00000 q^{73} -4.29844 q^{74} -1.00000 q^{75} +1.70156 q^{76} -3.70156 q^{77} -1.70156 q^{78} +7.40312 q^{79} -1.00000 q^{80} +1.00000 q^{81} -3.40312 q^{82} -9.70156 q^{83} +3.70156 q^{84} -1.00000 q^{85} -5.40312 q^{86} -2.00000 q^{87} -1.00000 q^{88} +14.0000 q^{89} +1.00000 q^{90} +6.29844 q^{91} -2.29844 q^{92} +5.70156 q^{93} -3.40312 q^{94} -1.70156 q^{95} +1.00000 q^{96} +9.70156 q^{97} -6.70156 q^{98} +1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 - q^7 - 2 * q^8 + 2 * q^9 $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{11} - 2 q^{12} + 3 q^{13} + q^{14} + 2 q^{15} + 2 q^{16} + 2 q^{17} - 2 q^{18} - 3 q^{19} - 2 q^{20} + q^{21} - 2 q^{22} - 11 q^{23} + 2 q^{24} + 2 q^{25} - 3 q^{26} - 2 q^{27} - q^{28} + 4 q^{29} - 2 q^{30} - 5 q^{31} - 2 q^{32} - 2 q^{33} - 2 q^{34} + q^{35} + 2 q^{36} + 15 q^{37} + 3 q^{38} - 3 q^{39} + 2 q^{40} - 6 q^{41} - q^{42} - 2 q^{43} + 2 q^{44} - 2 q^{45} + 11 q^{46} - 6 q^{47} - 2 q^{48} + 7 q^{49} - 2 q^{50} - 2 q^{51} + 3 q^{52} + 4 q^{53} + 2 q^{54} - 2 q^{55} + q^{56} + 3 q^{57} - 4 q^{58} - 14 q^{59} + 2 q^{60} + 9 q^{61} + 5 q^{62} - q^{63} + 2 q^{64} - 3 q^{65} + 2 q^{66} - 23 q^{67} + 2 q^{68} + 11 q^{69} - q^{70} - 2 q^{72} - 16 q^{73} - 15 q^{74} - 2 q^{75} - 3 q^{76} - q^{77} + 3 q^{78} + 2 q^{79} - 2 q^{80} + 2 q^{81} + 6 q^{82} - 13 q^{83} + q^{84} - 2 q^{85} + 2 q^{86} - 4 q^{87} - 2 q^{88} + 28 q^{89} + 2 q^{90} + 19 q^{91} - 11 q^{92} + 5 q^{93} + 6 q^{94} + 3 q^{95} + 2 q^{96} + 13 q^{97} - 7 q^{98} + 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 - q^7 - 2 * q^8 + 2 * q^9 + 2 * q^10 + 2 * q^11 - 2 * q^12 + 3 * q^13 + q^14 + 2 * q^15 + 2 * q^16 + 2 * q^17 - 2 * q^18 - 3 * q^19 - 2 * q^20 + q^21 - 2 * q^22 - 11 * q^23 + 2 * q^24 + 2 * q^25 - 3 * q^26 - 2 * q^27 - q^28 + 4 * q^29 - 2 * q^30 - 5 * q^31 - 2 * q^32 - 2 * q^33 - 2 * q^34 + q^35 + 2 * q^36 + 15 * q^37 + 3 * q^38 - 3 * q^39 + 2 * q^40 - 6 * q^41 - q^42 - 2 * q^43 + 2 * q^44 - 2 * q^45 + 11 * q^46 - 6 * q^47 - 2 * q^48 + 7 * q^49 - 2 * q^50 - 2 * q^51 + 3 * q^52 + 4 * q^53 + 2 * q^54 - 2 * q^55 + q^56 + 3 * q^57 - 4 * q^58 - 14 * q^59 + 2 * q^60 + 9 * q^61 + 5 * q^62 - q^63 + 2 * q^64 - 3 * q^65 + 2 * q^66 - 23 * q^67 + 2 * q^68 + 11 * q^69 - q^70 - 2 * q^72 - 16 * q^73 - 15 * q^74 - 2 * q^75 - 3 * q^76 - q^77 + 3 * q^78 + 2 * q^79 - 2 * q^80 + 2 * q^81 + 6 * q^82 - 13 * q^83 + q^84 - 2 * q^85 + 2 * q^86 - 4 * q^87 - 2 * q^88 + 28 * q^89 + 2 * q^90 + 19 * q^91 - 11 * q^92 + 5 * q^93 + 6 * q^94 + 3 * q^95 + 2 * q^96 + 13 * q^97 - 7 * q^98 + 2 * q^99

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ 1.00000 0.408248
$$7$$ −3.70156 −1.39906 −0.699529 0.714604i $$-0.746607\pi$$
−0.699529 + 0.714604i $$0.746607\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ 1.00000 0.301511
$$12$$ −1.00000 −0.288675
$$13$$ −1.70156 −0.471928 −0.235964 0.971762i $$-0.575825\pi$$
−0.235964 + 0.971762i $$0.575825\pi$$
$$14$$ 3.70156 0.989284
$$15$$ 1.00000 0.258199
$$16$$ 1.00000 0.250000
$$17$$ 1.00000 0.242536
$$18$$ −1.00000 −0.235702
$$19$$ 1.70156 0.390365 0.195183 0.980767i $$-0.437470\pi$$
0.195183 + 0.980767i $$0.437470\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 3.70156 0.807747
$$22$$ −1.00000 −0.213201
$$23$$ −2.29844 −0.479257 −0.239629 0.970865i $$-0.577026\pi$$
−0.239629 + 0.970865i $$0.577026\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 1.00000 0.200000
$$26$$ 1.70156 0.333704
$$27$$ −1.00000 −0.192450
$$28$$ −3.70156 −0.699529
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ −5.70156 −1.02403 −0.512015 0.858976i $$-0.671101\pi$$
−0.512015 + 0.858976i $$0.671101\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −1.00000 −0.174078
$$34$$ −1.00000 −0.171499
$$35$$ 3.70156 0.625678
$$36$$ 1.00000 0.166667
$$37$$ 4.29844 0.706659 0.353329 0.935499i $$-0.385049\pi$$
0.353329 + 0.935499i $$0.385049\pi$$
$$38$$ −1.70156 −0.276030
$$39$$ 1.70156 0.272468
$$40$$ 1.00000 0.158114
$$41$$ 3.40312 0.531479 0.265739 0.964045i $$-0.414384\pi$$
0.265739 + 0.964045i $$0.414384\pi$$
$$42$$ −3.70156 −0.571163
$$43$$ 5.40312 0.823969 0.411984 0.911191i $$-0.364836\pi$$
0.411984 + 0.911191i $$0.364836\pi$$
$$44$$ 1.00000 0.150756
$$45$$ −1.00000 −0.149071
$$46$$ 2.29844 0.338886
$$47$$ 3.40312 0.496397 0.248198 0.968709i $$-0.420162\pi$$
0.248198 + 0.968709i $$0.420162\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 6.70156 0.957366
$$50$$ −1.00000 −0.141421
$$51$$ −1.00000 −0.140028
$$52$$ −1.70156 −0.235964
$$53$$ 2.00000 0.274721 0.137361 0.990521i $$-0.456138\pi$$
0.137361 + 0.990521i $$0.456138\pi$$
$$54$$ 1.00000 0.136083
$$55$$ −1.00000 −0.134840
$$56$$ 3.70156 0.494642
$$57$$ −1.70156 −0.225377
$$58$$ −2.00000 −0.262613
$$59$$ −13.4031 −1.74494 −0.872469 0.488669i $$-0.837482\pi$$
−0.872469 + 0.488669i $$0.837482\pi$$
$$60$$ 1.00000 0.129099
$$61$$ 7.70156 0.986084 0.493042 0.870006i $$-0.335885\pi$$
0.493042 + 0.870006i $$0.335885\pi$$
$$62$$ 5.70156 0.724099
$$63$$ −3.70156 −0.466353
$$64$$ 1.00000 0.125000
$$65$$ 1.70156 0.211053
$$66$$ 1.00000 0.123091
$$67$$ −8.29844 −1.01382 −0.506908 0.862000i $$-0.669212\pi$$
−0.506908 + 0.862000i $$0.669212\pi$$
$$68$$ 1.00000 0.121268
$$69$$ 2.29844 0.276699
$$70$$ −3.70156 −0.442421
$$71$$ 12.8062 1.51982 0.759911 0.650027i $$-0.225242\pi$$
0.759911 + 0.650027i $$0.225242\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ −8.00000 −0.936329 −0.468165 0.883641i $$-0.655085\pi$$
−0.468165 + 0.883641i $$0.655085\pi$$
$$74$$ −4.29844 −0.499683
$$75$$ −1.00000 −0.115470
$$76$$ 1.70156 0.195183
$$77$$ −3.70156 −0.421832
$$78$$ −1.70156 −0.192664
$$79$$ 7.40312 0.832917 0.416458 0.909155i $$-0.363271\pi$$
0.416458 + 0.909155i $$0.363271\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 1.00000 0.111111
$$82$$ −3.40312 −0.375812
$$83$$ −9.70156 −1.06488 −0.532442 0.846466i $$-0.678726\pi$$
−0.532442 + 0.846466i $$0.678726\pi$$
$$84$$ 3.70156 0.403874
$$85$$ −1.00000 −0.108465
$$86$$ −5.40312 −0.582634
$$87$$ −2.00000 −0.214423
$$88$$ −1.00000 −0.106600
$$89$$ 14.0000 1.48400 0.741999 0.670402i $$-0.233878\pi$$
0.741999 + 0.670402i $$0.233878\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 6.29844 0.660256
$$92$$ −2.29844 −0.239629
$$93$$ 5.70156 0.591224
$$94$$ −3.40312 −0.351005
$$95$$ −1.70156 −0.174577
$$96$$ 1.00000 0.102062
$$97$$ 9.70156 0.985044 0.492522 0.870300i $$-0.336075\pi$$
0.492522 + 0.870300i $$0.336075\pi$$
$$98$$ −6.70156 −0.676960
$$99$$ 1.00000 0.100504
$$100$$ 1.00000 0.100000
$$101$$ −3.40312 −0.338624 −0.169312 0.985563i $$-0.554154\pi$$
−0.169312 + 0.985563i $$0.554154\pi$$
$$102$$ 1.00000 0.0990148
$$103$$ 17.1047 1.68537 0.842687 0.538403i $$-0.180972\pi$$
0.842687 + 0.538403i $$0.180972\pi$$
$$104$$ 1.70156 0.166852
$$105$$ −3.70156 −0.361235
$$106$$ −2.00000 −0.194257
$$107$$ −7.40312 −0.715687 −0.357844 0.933782i $$-0.616488\pi$$
−0.357844 + 0.933782i $$0.616488\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 15.1047 1.44677 0.723383 0.690447i $$-0.242586\pi$$
0.723383 + 0.690447i $$0.242586\pi$$
$$110$$ 1.00000 0.0953463
$$111$$ −4.29844 −0.407990
$$112$$ −3.70156 −0.349765
$$113$$ 2.00000 0.188144 0.0940721 0.995565i $$-0.470012\pi$$
0.0940721 + 0.995565i $$0.470012\pi$$
$$114$$ 1.70156 0.159366
$$115$$ 2.29844 0.214330
$$116$$ 2.00000 0.185695
$$117$$ −1.70156 −0.157309
$$118$$ 13.4031 1.23386
$$119$$ −3.70156 −0.339322
$$120$$ −1.00000 −0.0912871
$$121$$ 1.00000 0.0909091
$$122$$ −7.70156 −0.697267
$$123$$ −3.40312 −0.306849
$$124$$ −5.70156 −0.512015
$$125$$ −1.00000 −0.0894427
$$126$$ 3.70156 0.329761
$$127$$ 4.00000 0.354943 0.177471 0.984126i $$-0.443208\pi$$
0.177471 + 0.984126i $$0.443208\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −5.40312 −0.475719
$$130$$ −1.70156 −0.149237
$$131$$ 13.1047 1.14496 0.572481 0.819918i $$-0.305981\pi$$
0.572481 + 0.819918i $$0.305981\pi$$
$$132$$ −1.00000 −0.0870388
$$133$$ −6.29844 −0.546144
$$134$$ 8.29844 0.716876
$$135$$ 1.00000 0.0860663
$$136$$ −1.00000 −0.0857493
$$137$$ −19.1047 −1.63222 −0.816112 0.577894i $$-0.803875\pi$$
−0.816112 + 0.577894i $$0.803875\pi$$
$$138$$ −2.29844 −0.195656
$$139$$ 8.59688 0.729177 0.364589 0.931169i $$-0.381210\pi$$
0.364589 + 0.931169i $$0.381210\pi$$
$$140$$ 3.70156 0.312839
$$141$$ −3.40312 −0.286595
$$142$$ −12.8062 −1.07468
$$143$$ −1.70156 −0.142292
$$144$$ 1.00000 0.0833333
$$145$$ −2.00000 −0.166091
$$146$$ 8.00000 0.662085
$$147$$ −6.70156 −0.552736
$$148$$ 4.29844 0.353329
$$149$$ −6.29844 −0.515988 −0.257994 0.966147i $$-0.583062\pi$$
−0.257994 + 0.966147i $$0.583062\pi$$
$$150$$ 1.00000 0.0816497
$$151$$ −16.5078 −1.34339 −0.671693 0.740829i $$-0.734433\pi$$
−0.671693 + 0.740829i $$0.734433\pi$$
$$152$$ −1.70156 −0.138015
$$153$$ 1.00000 0.0808452
$$154$$ 3.70156 0.298280
$$155$$ 5.70156 0.457960
$$156$$ 1.70156 0.136234
$$157$$ −12.0000 −0.957704 −0.478852 0.877896i $$-0.658947\pi$$
−0.478852 + 0.877896i $$0.658947\pi$$
$$158$$ −7.40312 −0.588961
$$159$$ −2.00000 −0.158610
$$160$$ 1.00000 0.0790569
$$161$$ 8.50781 0.670509
$$162$$ −1.00000 −0.0785674
$$163$$ −14.8062 −1.15971 −0.579857 0.814718i $$-0.696892\pi$$
−0.579857 + 0.814718i $$0.696892\pi$$
$$164$$ 3.40312 0.265739
$$165$$ 1.00000 0.0778499
$$166$$ 9.70156 0.752987
$$167$$ −18.8062 −1.45527 −0.727636 0.685964i $$-0.759381\pi$$
−0.727636 + 0.685964i $$0.759381\pi$$
$$168$$ −3.70156 −0.285582
$$169$$ −10.1047 −0.777284
$$170$$ 1.00000 0.0766965
$$171$$ 1.70156 0.130122
$$172$$ 5.40312 0.411984
$$173$$ 6.50781 0.494780 0.247390 0.968916i $$-0.420427\pi$$
0.247390 + 0.968916i $$0.420427\pi$$
$$174$$ 2.00000 0.151620
$$175$$ −3.70156 −0.279812
$$176$$ 1.00000 0.0753778
$$177$$ 13.4031 1.00744
$$178$$ −14.0000 −1.04934
$$179$$ −3.10469 −0.232055 −0.116028 0.993246i $$-0.537016\pi$$
−0.116028 + 0.993246i $$0.537016\pi$$
$$180$$ −1.00000 −0.0745356
$$181$$ −9.40312 −0.698929 −0.349464 0.936950i $$-0.613636\pi$$
−0.349464 + 0.936950i $$0.613636\pi$$
$$182$$ −6.29844 −0.466871
$$183$$ −7.70156 −0.569316
$$184$$ 2.29844 0.169443
$$185$$ −4.29844 −0.316027
$$186$$ −5.70156 −0.418059
$$187$$ 1.00000 0.0731272
$$188$$ 3.40312 0.248198
$$189$$ 3.70156 0.269249
$$190$$ 1.70156 0.123444
$$191$$ −26.8062 −1.93963 −0.969816 0.243838i $$-0.921594\pi$$
−0.969816 + 0.243838i $$0.921594\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −12.0000 −0.863779 −0.431889 0.901927i $$-0.642153\pi$$
−0.431889 + 0.901927i $$0.642153\pi$$
$$194$$ −9.70156 −0.696532
$$195$$ −1.70156 −0.121851
$$196$$ 6.70156 0.478683
$$197$$ −16.2984 −1.16122 −0.580608 0.814183i $$-0.697185\pi$$
−0.580608 + 0.814183i $$0.697185\pi$$
$$198$$ −1.00000 −0.0710669
$$199$$ 2.29844 0.162932 0.0814660 0.996676i $$-0.474040\pi$$
0.0814660 + 0.996676i $$0.474040\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 8.29844 0.585327
$$202$$ 3.40312 0.239443
$$203$$ −7.40312 −0.519597
$$204$$ −1.00000 −0.0700140
$$205$$ −3.40312 −0.237685
$$206$$ −17.1047 −1.19174
$$207$$ −2.29844 −0.159752
$$208$$ −1.70156 −0.117982
$$209$$ 1.70156 0.117700
$$210$$ 3.70156 0.255432
$$211$$ 22.2094 1.52896 0.764478 0.644650i $$-0.222997\pi$$
0.764478 + 0.644650i $$0.222997\pi$$
$$212$$ 2.00000 0.137361
$$213$$ −12.8062 −0.877470
$$214$$ 7.40312 0.506067
$$215$$ −5.40312 −0.368490
$$216$$ 1.00000 0.0680414
$$217$$ 21.1047 1.43268
$$218$$ −15.1047 −1.02302
$$219$$ 8.00000 0.540590
$$220$$ −1.00000 −0.0674200
$$221$$ −1.70156 −0.114459
$$222$$ 4.29844 0.288492
$$223$$ 12.5078 0.837585 0.418792 0.908082i $$-0.362453\pi$$
0.418792 + 0.908082i $$0.362453\pi$$
$$224$$ 3.70156 0.247321
$$225$$ 1.00000 0.0666667
$$226$$ −2.00000 −0.133038
$$227$$ 14.2094 0.943109 0.471555 0.881837i $$-0.343693\pi$$
0.471555 + 0.881837i $$0.343693\pi$$
$$228$$ −1.70156 −0.112689
$$229$$ 7.70156 0.508934 0.254467 0.967082i $$-0.418100\pi$$
0.254467 + 0.967082i $$0.418100\pi$$
$$230$$ −2.29844 −0.151555
$$231$$ 3.70156 0.243545
$$232$$ −2.00000 −0.131306
$$233$$ −12.8062 −0.838965 −0.419483 0.907763i $$-0.637788\pi$$
−0.419483 + 0.907763i $$0.637788\pi$$
$$234$$ 1.70156 0.111235
$$235$$ −3.40312 −0.221995
$$236$$ −13.4031 −0.872469
$$237$$ −7.40312 −0.480885
$$238$$ 3.70156 0.239937
$$239$$ 3.40312 0.220130 0.110065 0.993924i $$-0.464894\pi$$
0.110065 + 0.993924i $$0.464894\pi$$
$$240$$ 1.00000 0.0645497
$$241$$ 4.29844 0.276887 0.138443 0.990370i $$-0.455790\pi$$
0.138443 + 0.990370i $$0.455790\pi$$
$$242$$ −1.00000 −0.0642824
$$243$$ −1.00000 −0.0641500
$$244$$ 7.70156 0.493042
$$245$$ −6.70156 −0.428147
$$246$$ 3.40312 0.216975
$$247$$ −2.89531 −0.184224
$$248$$ 5.70156 0.362050
$$249$$ 9.70156 0.614812
$$250$$ 1.00000 0.0632456
$$251$$ −31.1047 −1.96331 −0.981655 0.190665i $$-0.938936\pi$$
−0.981655 + 0.190665i $$0.938936\pi$$
$$252$$ −3.70156 −0.233176
$$253$$ −2.29844 −0.144502
$$254$$ −4.00000 −0.250982
$$255$$ 1.00000 0.0626224
$$256$$ 1.00000 0.0625000
$$257$$ −23.6125 −1.47291 −0.736454 0.676488i $$-0.763501\pi$$
−0.736454 + 0.676488i $$0.763501\pi$$
$$258$$ 5.40312 0.336384
$$259$$ −15.9109 −0.988657
$$260$$ 1.70156 0.105526
$$261$$ 2.00000 0.123797
$$262$$ −13.1047 −0.809610
$$263$$ −2.89531 −0.178533 −0.0892663 0.996008i $$-0.528452\pi$$
−0.0892663 + 0.996008i $$0.528452\pi$$
$$264$$ 1.00000 0.0615457
$$265$$ −2.00000 −0.122859
$$266$$ 6.29844 0.386182
$$267$$ −14.0000 −0.856786
$$268$$ −8.29844 −0.506908
$$269$$ 26.5078 1.61621 0.808105 0.589039i $$-0.200493\pi$$
0.808105 + 0.589039i $$0.200493\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ 14.8062 0.899416 0.449708 0.893176i $$-0.351528\pi$$
0.449708 + 0.893176i $$0.351528\pi$$
$$272$$ 1.00000 0.0606339
$$273$$ −6.29844 −0.381199
$$274$$ 19.1047 1.15416
$$275$$ 1.00000 0.0603023
$$276$$ 2.29844 0.138350
$$277$$ 13.4031 0.805316 0.402658 0.915351i $$-0.368086\pi$$
0.402658 + 0.915351i $$0.368086\pi$$
$$278$$ −8.59688 −0.515606
$$279$$ −5.70156 −0.341344
$$280$$ −3.70156 −0.221211
$$281$$ −10.0000 −0.596550 −0.298275 0.954480i $$-0.596411\pi$$
−0.298275 + 0.954480i $$0.596411\pi$$
$$282$$ 3.40312 0.202653
$$283$$ 12.0000 0.713326 0.356663 0.934233i $$-0.383914\pi$$
0.356663 + 0.934233i $$0.383914\pi$$
$$284$$ 12.8062 0.759911
$$285$$ 1.70156 0.100792
$$286$$ 1.70156 0.100615
$$287$$ −12.5969 −0.743570
$$288$$ −1.00000 −0.0589256
$$289$$ 1.00000 0.0588235
$$290$$ 2.00000 0.117444
$$291$$ −9.70156 −0.568716
$$292$$ −8.00000 −0.468165
$$293$$ 21.4031 1.25038 0.625192 0.780471i $$-0.285021\pi$$
0.625192 + 0.780471i $$0.285021\pi$$
$$294$$ 6.70156 0.390843
$$295$$ 13.4031 0.780360
$$296$$ −4.29844 −0.249842
$$297$$ −1.00000 −0.0580259
$$298$$ 6.29844 0.364859
$$299$$ 3.91093 0.226175
$$300$$ −1.00000 −0.0577350
$$301$$ −20.0000 −1.15278
$$302$$ 16.5078 0.949918
$$303$$ 3.40312 0.195504
$$304$$ 1.70156 0.0975913
$$305$$ −7.70156 −0.440990
$$306$$ −1.00000 −0.0571662
$$307$$ −2.59688 −0.148212 −0.0741058 0.997250i $$-0.523610\pi$$
−0.0741058 + 0.997250i $$0.523610\pi$$
$$308$$ −3.70156 −0.210916
$$309$$ −17.1047 −0.973052
$$310$$ −5.70156 −0.323827
$$311$$ −4.80625 −0.272537 −0.136269 0.990672i $$-0.543511\pi$$
−0.136269 + 0.990672i $$0.543511\pi$$
$$312$$ −1.70156 −0.0963320
$$313$$ 19.9109 1.12543 0.562716 0.826650i $$-0.309756\pi$$
0.562716 + 0.826650i $$0.309756\pi$$
$$314$$ 12.0000 0.677199
$$315$$ 3.70156 0.208559
$$316$$ 7.40312 0.416458
$$317$$ 14.0000 0.786318 0.393159 0.919470i $$-0.371382\pi$$
0.393159 + 0.919470i $$0.371382\pi$$
$$318$$ 2.00000 0.112154
$$319$$ 2.00000 0.111979
$$320$$ −1.00000 −0.0559017
$$321$$ 7.40312 0.413202
$$322$$ −8.50781 −0.474122
$$323$$ 1.70156 0.0946774
$$324$$ 1.00000 0.0555556
$$325$$ −1.70156 −0.0943857
$$326$$ 14.8062 0.820042
$$327$$ −15.1047 −0.835291
$$328$$ −3.40312 −0.187906
$$329$$ −12.5969 −0.694488
$$330$$ −1.00000 −0.0550482
$$331$$ −21.6125 −1.18793 −0.593965 0.804491i $$-0.702438\pi$$
−0.593965 + 0.804491i $$0.702438\pi$$
$$332$$ −9.70156 −0.532442
$$333$$ 4.29844 0.235553
$$334$$ 18.8062 1.02903
$$335$$ 8.29844 0.453392
$$336$$ 3.70156 0.201937
$$337$$ −14.8062 −0.806548 −0.403274 0.915079i $$-0.632128\pi$$
−0.403274 + 0.915079i $$0.632128\pi$$
$$338$$ 10.1047 0.549622
$$339$$ −2.00000 −0.108625
$$340$$ −1.00000 −0.0542326
$$341$$ −5.70156 −0.308757
$$342$$ −1.70156 −0.0920099
$$343$$ 1.10469 0.0596475
$$344$$ −5.40312 −0.291317
$$345$$ −2.29844 −0.123744
$$346$$ −6.50781 −0.349862
$$347$$ −7.40312 −0.397421 −0.198710 0.980058i $$-0.563675\pi$$
−0.198710 + 0.980058i $$0.563675\pi$$
$$348$$ −2.00000 −0.107211
$$349$$ 19.6125 1.04983 0.524916 0.851154i $$-0.324097\pi$$
0.524916 + 0.851154i $$0.324097\pi$$
$$350$$ 3.70156 0.197857
$$351$$ 1.70156 0.0908227
$$352$$ −1.00000 −0.0533002
$$353$$ −2.50781 −0.133477 −0.0667386 0.997770i $$-0.521259\pi$$
−0.0667386 + 0.997770i $$0.521259\pi$$
$$354$$ −13.4031 −0.712368
$$355$$ −12.8062 −0.679685
$$356$$ 14.0000 0.741999
$$357$$ 3.70156 0.195907
$$358$$ 3.10469 0.164088
$$359$$ −29.0156 −1.53139 −0.765693 0.643206i $$-0.777604\pi$$
−0.765693 + 0.643206i $$0.777604\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ −16.1047 −0.847615
$$362$$ 9.40312 0.494217
$$363$$ −1.00000 −0.0524864
$$364$$ 6.29844 0.330128
$$365$$ 8.00000 0.418739
$$366$$ 7.70156 0.402567
$$367$$ −6.59688 −0.344354 −0.172177 0.985066i $$-0.555080\pi$$
−0.172177 + 0.985066i $$0.555080\pi$$
$$368$$ −2.29844 −0.119814
$$369$$ 3.40312 0.177160
$$370$$ 4.29844 0.223465
$$371$$ −7.40312 −0.384351
$$372$$ 5.70156 0.295612
$$373$$ −15.4031 −0.797544 −0.398772 0.917050i $$-0.630563\pi$$
−0.398772 + 0.917050i $$0.630563\pi$$
$$374$$ −1.00000 −0.0517088
$$375$$ 1.00000 0.0516398
$$376$$ −3.40312 −0.175503
$$377$$ −3.40312 −0.175270
$$378$$ −3.70156 −0.190388
$$379$$ −27.3141 −1.40303 −0.701514 0.712655i $$-0.747492\pi$$
−0.701514 + 0.712655i $$0.747492\pi$$
$$380$$ −1.70156 −0.0872883
$$381$$ −4.00000 −0.204926
$$382$$ 26.8062 1.37153
$$383$$ −35.4031 −1.80902 −0.904508 0.426458i $$-0.859761\pi$$
−0.904508 + 0.426458i $$0.859761\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 3.70156 0.188649
$$386$$ 12.0000 0.610784
$$387$$ 5.40312 0.274656
$$388$$ 9.70156 0.492522
$$389$$ −12.0000 −0.608424 −0.304212 0.952604i $$-0.598393\pi$$
−0.304212 + 0.952604i $$0.598393\pi$$
$$390$$ 1.70156 0.0861619
$$391$$ −2.29844 −0.116237
$$392$$ −6.70156 −0.338480
$$393$$ −13.1047 −0.661044
$$394$$ 16.2984 0.821103
$$395$$ −7.40312 −0.372492
$$396$$ 1.00000 0.0502519
$$397$$ 12.8062 0.642727 0.321364 0.946956i $$-0.395859\pi$$
0.321364 + 0.946956i $$0.395859\pi$$
$$398$$ −2.29844 −0.115210
$$399$$ 6.29844 0.315316
$$400$$ 1.00000 0.0500000
$$401$$ −13.1047 −0.654417 −0.327208 0.944952i $$-0.606108\pi$$
−0.327208 + 0.944952i $$0.606108\pi$$
$$402$$ −8.29844 −0.413888
$$403$$ 9.70156 0.483269
$$404$$ −3.40312 −0.169312
$$405$$ −1.00000 −0.0496904
$$406$$ 7.40312 0.367411
$$407$$ 4.29844 0.213066
$$408$$ 1.00000 0.0495074
$$409$$ 6.00000 0.296681 0.148340 0.988936i $$-0.452607\pi$$
0.148340 + 0.988936i $$0.452607\pi$$
$$410$$ 3.40312 0.168068
$$411$$ 19.1047 0.942365
$$412$$ 17.1047 0.842687
$$413$$ 49.6125 2.44127
$$414$$ 2.29844 0.112962
$$415$$ 9.70156 0.476231
$$416$$ 1.70156 0.0834259
$$417$$ −8.59688 −0.420991
$$418$$ −1.70156 −0.0832261
$$419$$ −30.8062 −1.50498 −0.752492 0.658602i $$-0.771148\pi$$
−0.752492 + 0.658602i $$0.771148\pi$$
$$420$$ −3.70156 −0.180618
$$421$$ −8.29844 −0.404441 −0.202221 0.979340i $$-0.564816\pi$$
−0.202221 + 0.979340i $$0.564816\pi$$
$$422$$ −22.2094 −1.08114
$$423$$ 3.40312 0.165466
$$424$$ −2.00000 −0.0971286
$$425$$ 1.00000 0.0485071
$$426$$ 12.8062 0.620465
$$427$$ −28.5078 −1.37959
$$428$$ −7.40312 −0.357844
$$429$$ 1.70156 0.0821522
$$430$$ 5.40312 0.260562
$$431$$ −36.2094 −1.74414 −0.872072 0.489377i $$-0.837224\pi$$
−0.872072 + 0.489377i $$0.837224\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −18.0000 −0.865025 −0.432512 0.901628i $$-0.642373\pi$$
−0.432512 + 0.901628i $$0.642373\pi$$
$$434$$ −21.1047 −1.01306
$$435$$ 2.00000 0.0958927
$$436$$ 15.1047 0.723383
$$437$$ −3.91093 −0.187085
$$438$$ −8.00000 −0.382255
$$439$$ 1.79063 0.0854620 0.0427310 0.999087i $$-0.486394\pi$$
0.0427310 + 0.999087i $$0.486394\pi$$
$$440$$ 1.00000 0.0476731
$$441$$ 6.70156 0.319122
$$442$$ 1.70156 0.0809351
$$443$$ 14.2094 0.675108 0.337554 0.941306i $$-0.390400\pi$$
0.337554 + 0.941306i $$0.390400\pi$$
$$444$$ −4.29844 −0.203995
$$445$$ −14.0000 −0.663664
$$446$$ −12.5078 −0.592262
$$447$$ 6.29844 0.297906
$$448$$ −3.70156 −0.174882
$$449$$ −33.7016 −1.59048 −0.795238 0.606298i $$-0.792654\pi$$
−0.795238 + 0.606298i $$0.792654\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ 3.40312 0.160247
$$452$$ 2.00000 0.0940721
$$453$$ 16.5078 0.775605
$$454$$ −14.2094 −0.666879
$$455$$ −6.29844 −0.295275
$$456$$ 1.70156 0.0796829
$$457$$ 3.10469 0.145231 0.0726156 0.997360i $$-0.476865\pi$$
0.0726156 + 0.997360i $$0.476865\pi$$
$$458$$ −7.70156 −0.359870
$$459$$ −1.00000 −0.0466760
$$460$$ 2.29844 0.107165
$$461$$ −24.5969 −1.14559 −0.572795 0.819698i $$-0.694141\pi$$
−0.572795 + 0.819698i $$0.694141\pi$$
$$462$$ −3.70156 −0.172212
$$463$$ −18.2984 −0.850401 −0.425200 0.905099i $$-0.639796\pi$$
−0.425200 + 0.905099i $$0.639796\pi$$
$$464$$ 2.00000 0.0928477
$$465$$ −5.70156 −0.264404
$$466$$ 12.8062 0.593238
$$467$$ −9.19375 −0.425436 −0.212718 0.977114i $$-0.568232\pi$$
−0.212718 + 0.977114i $$0.568232\pi$$
$$468$$ −1.70156 −0.0786547
$$469$$ 30.7172 1.41839
$$470$$ 3.40312 0.156974
$$471$$ 12.0000 0.552931
$$472$$ 13.4031 0.616929
$$473$$ 5.40312 0.248436
$$474$$ 7.40312 0.340037
$$475$$ 1.70156 0.0780730
$$476$$ −3.70156 −0.169661
$$477$$ 2.00000 0.0915737
$$478$$ −3.40312 −0.155655
$$479$$ 29.9109 1.36667 0.683333 0.730107i $$-0.260530\pi$$
0.683333 + 0.730107i $$0.260530\pi$$
$$480$$ −1.00000 −0.0456435
$$481$$ −7.31406 −0.333492
$$482$$ −4.29844 −0.195788
$$483$$ −8.50781 −0.387119
$$484$$ 1.00000 0.0454545
$$485$$ −9.70156 −0.440525
$$486$$ 1.00000 0.0453609
$$487$$ 20.2094 0.915774 0.457887 0.889010i $$-0.348606\pi$$
0.457887 + 0.889010i $$0.348606\pi$$
$$488$$ −7.70156 −0.348633
$$489$$ 14.8062 0.669562
$$490$$ 6.70156 0.302746
$$491$$ 12.8062 0.577938 0.288969 0.957338i $$-0.406688\pi$$
0.288969 + 0.957338i $$0.406688\pi$$
$$492$$ −3.40312 −0.153425
$$493$$ 2.00000 0.0900755
$$494$$ 2.89531 0.130266
$$495$$ −1.00000 −0.0449467
$$496$$ −5.70156 −0.256008
$$497$$ −47.4031 −2.12632
$$498$$ −9.70156 −0.434737
$$499$$ −36.0000 −1.61158 −0.805791 0.592200i $$-0.798259\pi$$
−0.805791 + 0.592200i $$0.798259\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 18.8062 0.840201
$$502$$ 31.1047 1.38827
$$503$$ −14.8062 −0.660178 −0.330089 0.943950i $$-0.607079\pi$$
−0.330089 + 0.943950i $$0.607079\pi$$
$$504$$ 3.70156 0.164881
$$505$$ 3.40312 0.151437
$$506$$ 2.29844 0.102178
$$507$$ 10.1047 0.448765
$$508$$ 4.00000 0.177471
$$509$$ 30.2094 1.33901 0.669503 0.742809i $$-0.266507\pi$$
0.669503 + 0.742809i $$0.266507\pi$$
$$510$$ −1.00000 −0.0442807
$$511$$ 29.6125 1.30998
$$512$$ −1.00000 −0.0441942
$$513$$ −1.70156 −0.0751258
$$514$$ 23.6125 1.04150
$$515$$ −17.1047 −0.753723
$$516$$ −5.40312 −0.237859
$$517$$ 3.40312 0.149669
$$518$$ 15.9109 0.699086
$$519$$ −6.50781 −0.285661
$$520$$ −1.70156 −0.0746184
$$521$$ 10.8953 0.477332 0.238666 0.971102i $$-0.423290\pi$$
0.238666 + 0.971102i $$0.423290\pi$$
$$522$$ −2.00000 −0.0875376
$$523$$ 16.2094 0.708786 0.354393 0.935097i $$-0.384687\pi$$
0.354393 + 0.935097i $$0.384687\pi$$
$$524$$ 13.1047 0.572481
$$525$$ 3.70156 0.161549
$$526$$ 2.89531 0.126242
$$527$$ −5.70156 −0.248364
$$528$$ −1.00000 −0.0435194
$$529$$ −17.7172 −0.770312
$$530$$ 2.00000 0.0868744
$$531$$ −13.4031 −0.581646
$$532$$ −6.29844 −0.273072
$$533$$ −5.79063 −0.250820
$$534$$ 14.0000 0.605839
$$535$$ 7.40312 0.320065
$$536$$ 8.29844 0.358438
$$537$$ 3.10469 0.133977
$$538$$ −26.5078 −1.14283
$$539$$ 6.70156 0.288657
$$540$$ 1.00000 0.0430331
$$541$$ −8.80625 −0.378610 −0.189305 0.981918i $$-0.560624\pi$$
−0.189305 + 0.981918i $$0.560624\pi$$
$$542$$ −14.8062 −0.635983
$$543$$ 9.40312 0.403527
$$544$$ −1.00000 −0.0428746
$$545$$ −15.1047 −0.647014
$$546$$ 6.29844 0.269548
$$547$$ 9.10469 0.389288 0.194644 0.980874i $$-0.437645\pi$$
0.194644 + 0.980874i $$0.437645\pi$$
$$548$$ −19.1047 −0.816112
$$549$$ 7.70156 0.328695
$$550$$ −1.00000 −0.0426401
$$551$$ 3.40312 0.144978
$$552$$ −2.29844 −0.0978280
$$553$$ −27.4031 −1.16530
$$554$$ −13.4031 −0.569444
$$555$$ 4.29844 0.182459
$$556$$ 8.59688 0.364589
$$557$$ −28.2094 −1.19527 −0.597635 0.801768i $$-0.703893\pi$$
−0.597635 + 0.801768i $$0.703893\pi$$
$$558$$ 5.70156 0.241366
$$559$$ −9.19375 −0.388854
$$560$$ 3.70156 0.156420
$$561$$ −1.00000 −0.0422200
$$562$$ 10.0000 0.421825
$$563$$ −37.1047 −1.56378 −0.781888 0.623419i $$-0.785743\pi$$
−0.781888 + 0.623419i $$0.785743\pi$$
$$564$$ −3.40312 −0.143297
$$565$$ −2.00000 −0.0841406
$$566$$ −12.0000 −0.504398
$$567$$ −3.70156 −0.155451
$$568$$ −12.8062 −0.537338
$$569$$ 42.5078 1.78202 0.891010 0.453984i $$-0.149998\pi$$
0.891010 + 0.453984i $$0.149998\pi$$
$$570$$ −1.70156 −0.0712706
$$571$$ −32.5969 −1.36414 −0.682068 0.731288i $$-0.738919\pi$$
−0.682068 + 0.731288i $$0.738919\pi$$
$$572$$ −1.70156 −0.0711459
$$573$$ 26.8062 1.11985
$$574$$ 12.5969 0.525783
$$575$$ −2.29844 −0.0958515
$$576$$ 1.00000 0.0416667
$$577$$ 27.6125 1.14952 0.574762 0.818321i $$-0.305095\pi$$
0.574762 + 0.818321i $$0.305095\pi$$
$$578$$ −1.00000 −0.0415945
$$579$$ 12.0000 0.498703
$$580$$ −2.00000 −0.0830455
$$581$$ 35.9109 1.48984
$$582$$ 9.70156 0.402143
$$583$$ 2.00000 0.0828315
$$584$$ 8.00000 0.331042
$$585$$ 1.70156 0.0703509
$$586$$ −21.4031 −0.884155
$$587$$ −24.5969 −1.01522 −0.507611 0.861586i $$-0.669471\pi$$
−0.507611 + 0.861586i $$0.669471\pi$$
$$588$$ −6.70156 −0.276368
$$589$$ −9.70156 −0.399746
$$590$$ −13.4031 −0.551798
$$591$$ 16.2984 0.670428
$$592$$ 4.29844 0.176665
$$593$$ −40.8062 −1.67571 −0.837856 0.545891i $$-0.816191\pi$$
−0.837856 + 0.545891i $$0.816191\pi$$
$$594$$ 1.00000 0.0410305
$$595$$ 3.70156 0.151749
$$596$$ −6.29844 −0.257994
$$597$$ −2.29844 −0.0940688
$$598$$ −3.91093 −0.159930
$$599$$ −2.29844 −0.0939116 −0.0469558 0.998897i $$-0.514952\pi$$
−0.0469558 + 0.998897i $$0.514952\pi$$
$$600$$ 1.00000 0.0408248
$$601$$ −30.5078 −1.24444 −0.622220 0.782843i $$-0.713769\pi$$
−0.622220 + 0.782843i $$0.713769\pi$$
$$602$$ 20.0000 0.815139
$$603$$ −8.29844 −0.337939
$$604$$ −16.5078 −0.671693
$$605$$ −1.00000 −0.0406558
$$606$$ −3.40312 −0.138242
$$607$$ 12.2984 0.499178 0.249589 0.968352i $$-0.419704\pi$$
0.249589 + 0.968352i $$0.419704\pi$$
$$608$$ −1.70156 −0.0690075
$$609$$ 7.40312 0.299990
$$610$$ 7.70156 0.311827
$$611$$ −5.79063 −0.234264
$$612$$ 1.00000 0.0404226
$$613$$ 39.4031 1.59148 0.795739 0.605640i $$-0.207083\pi$$
0.795739 + 0.605640i $$0.207083\pi$$
$$614$$ 2.59688 0.104801
$$615$$ 3.40312 0.137227
$$616$$ 3.70156 0.149140
$$617$$ −47.6125 −1.91681 −0.958403 0.285417i $$-0.907868\pi$$
−0.958403 + 0.285417i $$0.907868\pi$$
$$618$$ 17.1047 0.688051
$$619$$ 43.3141 1.74094 0.870470 0.492222i $$-0.163815\pi$$
0.870470 + 0.492222i $$0.163815\pi$$
$$620$$ 5.70156 0.228980
$$621$$ 2.29844 0.0922331
$$622$$ 4.80625 0.192713
$$623$$ −51.8219 −2.07620
$$624$$ 1.70156 0.0681170
$$625$$ 1.00000 0.0400000
$$626$$ −19.9109 −0.795801
$$627$$ −1.70156 −0.0679538
$$628$$ −12.0000 −0.478852
$$629$$ 4.29844 0.171390
$$630$$ −3.70156 −0.147474
$$631$$ 2.20937 0.0879537 0.0439769 0.999033i $$-0.485997\pi$$
0.0439769 + 0.999033i $$0.485997\pi$$
$$632$$ −7.40312 −0.294480
$$633$$ −22.2094 −0.882743
$$634$$ −14.0000 −0.556011
$$635$$ −4.00000 −0.158735
$$636$$ −2.00000 −0.0793052
$$637$$ −11.4031 −0.451808
$$638$$ −2.00000 −0.0791808
$$639$$ 12.8062 0.506607
$$640$$ 1.00000 0.0395285
$$641$$ 34.2094 1.35119 0.675594 0.737273i $$-0.263887\pi$$
0.675594 + 0.737273i $$0.263887\pi$$
$$642$$ −7.40312 −0.292178
$$643$$ −29.6125 −1.16780 −0.583901 0.811825i $$-0.698475\pi$$
−0.583901 + 0.811825i $$0.698475\pi$$
$$644$$ 8.50781 0.335255
$$645$$ 5.40312 0.212748
$$646$$ −1.70156 −0.0669471
$$647$$ 28.5969 1.12426 0.562130 0.827049i $$-0.309982\pi$$
0.562130 + 0.827049i $$0.309982\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ −13.4031 −0.526119
$$650$$ 1.70156 0.0667408
$$651$$ −21.1047 −0.827158
$$652$$ −14.8062 −0.579857
$$653$$ 12.2094 0.477790 0.238895 0.971045i $$-0.423215\pi$$
0.238895 + 0.971045i $$0.423215\pi$$
$$654$$ 15.1047 0.590640
$$655$$ −13.1047 −0.512042
$$656$$ 3.40312 0.132870
$$657$$ −8.00000 −0.312110
$$658$$ 12.5969 0.491077
$$659$$ −0.806248 −0.0314070 −0.0157035 0.999877i $$-0.504999\pi$$
−0.0157035 + 0.999877i $$0.504999\pi$$
$$660$$ 1.00000 0.0389249
$$661$$ 0.298438 0.0116079 0.00580394 0.999983i $$-0.498153\pi$$
0.00580394 + 0.999983i $$0.498153\pi$$
$$662$$ 21.6125 0.839994
$$663$$ 1.70156 0.0660832
$$664$$ 9.70156 0.376494
$$665$$ 6.29844 0.244243
$$666$$ −4.29844 −0.166561
$$667$$ −4.59688 −0.177992
$$668$$ −18.8062 −0.727636
$$669$$ −12.5078 −0.483580
$$670$$ −8.29844 −0.320597
$$671$$ 7.70156 0.297316
$$672$$ −3.70156 −0.142791
$$673$$ 4.00000 0.154189 0.0770943 0.997024i $$-0.475436\pi$$
0.0770943 + 0.997024i $$0.475436\pi$$
$$674$$ 14.8062 0.570315
$$675$$ −1.00000 −0.0384900
$$676$$ −10.1047 −0.388642
$$677$$ 26.4187 1.01535 0.507677 0.861547i $$-0.330504\pi$$
0.507677 + 0.861547i $$0.330504\pi$$
$$678$$ 2.00000 0.0768095
$$679$$ −35.9109 −1.37814
$$680$$ 1.00000 0.0383482
$$681$$ −14.2094 −0.544504
$$682$$ 5.70156 0.218324
$$683$$ 4.50781 0.172487 0.0862433 0.996274i $$-0.472514\pi$$
0.0862433 + 0.996274i $$0.472514\pi$$
$$684$$ 1.70156 0.0650609
$$685$$ 19.1047 0.729953
$$686$$ −1.10469 −0.0421771
$$687$$ −7.70156 −0.293833
$$688$$ 5.40312 0.205992
$$689$$ −3.40312 −0.129649
$$690$$ 2.29844 0.0875000
$$691$$ −30.7172 −1.16854 −0.584268 0.811561i $$-0.698618\pi$$
−0.584268 + 0.811561i $$0.698618\pi$$
$$692$$ 6.50781 0.247390
$$693$$ −3.70156 −0.140611
$$694$$ 7.40312 0.281019
$$695$$ −8.59688 −0.326098
$$696$$ 2.00000 0.0758098
$$697$$ 3.40312 0.128903
$$698$$ −19.6125 −0.742344
$$699$$ 12.8062 0.484377
$$700$$ −3.70156 −0.139906
$$701$$ −38.2094 −1.44315 −0.721574 0.692337i $$-0.756581\pi$$
−0.721574 + 0.692337i $$0.756581\pi$$
$$702$$ −1.70156 −0.0642213
$$703$$ 7.31406 0.275855
$$704$$ 1.00000 0.0376889
$$705$$ 3.40312 0.128169
$$706$$ 2.50781 0.0943827
$$707$$ 12.5969 0.473754
$$708$$ 13.4031 0.503720
$$709$$ 6.00000 0.225335 0.112667 0.993633i $$-0.464061\pi$$
0.112667 + 0.993633i $$0.464061\pi$$
$$710$$ 12.8062 0.480610
$$711$$ 7.40312 0.277639
$$712$$ −14.0000 −0.524672
$$713$$ 13.1047 0.490774
$$714$$ −3.70156 −0.138527
$$715$$ 1.70156 0.0636348
$$716$$ −3.10469 −0.116028
$$717$$ −3.40312 −0.127092
$$718$$ 29.0156 1.08285
$$719$$ −30.4187 −1.13443 −0.567214 0.823571i $$-0.691979\pi$$
−0.567214 + 0.823571i $$0.691979\pi$$
$$720$$ −1.00000 −0.0372678
$$721$$ −63.3141 −2.35794
$$722$$ 16.1047 0.599354
$$723$$ −4.29844 −0.159861
$$724$$ −9.40312 −0.349464
$$725$$ 2.00000 0.0742781
$$726$$ 1.00000 0.0371135
$$727$$ 5.19375 0.192626 0.0963128 0.995351i $$-0.469295\pi$$
0.0963128 + 0.995351i $$0.469295\pi$$
$$728$$ −6.29844 −0.233436
$$729$$ 1.00000 0.0370370
$$730$$ −8.00000 −0.296093
$$731$$ 5.40312 0.199842
$$732$$ −7.70156 −0.284658
$$733$$ 0.0890652 0.00328970 0.00164485 0.999999i $$-0.499476\pi$$
0.00164485 + 0.999999i $$0.499476\pi$$
$$734$$ 6.59688 0.243495
$$735$$ 6.70156 0.247191
$$736$$ 2.29844 0.0847215
$$737$$ −8.29844 −0.305677
$$738$$ −3.40312 −0.125271
$$739$$ −4.50781 −0.165822 −0.0829112 0.996557i $$-0.526422\pi$$
−0.0829112 + 0.996557i $$0.526422\pi$$
$$740$$ −4.29844 −0.158014
$$741$$ 2.89531 0.106362
$$742$$ 7.40312 0.271777
$$743$$ −32.4187 −1.18933 −0.594664 0.803974i $$-0.702715\pi$$
−0.594664 + 0.803974i $$0.702715\pi$$
$$744$$ −5.70156 −0.209029
$$745$$ 6.29844 0.230757
$$746$$ 15.4031 0.563948
$$747$$ −9.70156 −0.354962
$$748$$ 1.00000 0.0365636
$$749$$ 27.4031 1.00129
$$750$$ −1.00000 −0.0365148
$$751$$ −29.6125 −1.08058 −0.540288 0.841480i $$-0.681685\pi$$
−0.540288 + 0.841480i $$0.681685\pi$$
$$752$$ 3.40312 0.124099
$$753$$ 31.1047 1.13352
$$754$$ 3.40312 0.123934
$$755$$ 16.5078 0.600781
$$756$$ 3.70156 0.134625
$$757$$ 23.4031 0.850601 0.425301 0.905052i $$-0.360168\pi$$
0.425301 + 0.905052i $$0.360168\pi$$
$$758$$ 27.3141 0.992091
$$759$$ 2.29844 0.0834280
$$760$$ 1.70156 0.0617221
$$761$$ −11.1047 −0.402545 −0.201272 0.979535i $$-0.564508\pi$$
−0.201272 + 0.979535i $$0.564508\pi$$
$$762$$ 4.00000 0.144905
$$763$$ −55.9109 −2.02411
$$764$$ −26.8062 −0.969816
$$765$$ −1.00000 −0.0361551
$$766$$ 35.4031 1.27917
$$767$$ 22.8062 0.823486
$$768$$ −1.00000 −0.0360844
$$769$$ 32.2094 1.16150 0.580750 0.814082i $$-0.302759\pi$$
0.580750 + 0.814082i $$0.302759\pi$$
$$770$$ −3.70156 −0.133395
$$771$$ 23.6125 0.850383
$$772$$ −12.0000 −0.431889
$$773$$ −14.5078 −0.521810 −0.260905 0.965365i $$-0.584021\pi$$
−0.260905 + 0.965365i $$0.584021\pi$$
$$774$$ −5.40312 −0.194211
$$775$$ −5.70156 −0.204806
$$776$$ −9.70156 −0.348266
$$777$$ 15.9109 0.570802
$$778$$ 12.0000 0.430221
$$779$$ 5.79063 0.207471
$$780$$ −1.70156 −0.0609257
$$781$$ 12.8062 0.458244
$$782$$ 2.29844 0.0821920
$$783$$ −2.00000 −0.0714742
$$784$$ 6.70156 0.239342
$$785$$ 12.0000 0.428298
$$786$$ 13.1047 0.467429
$$787$$ −42.7172 −1.52270 −0.761352 0.648339i $$-0.775464\pi$$
−0.761352 + 0.648339i $$0.775464\pi$$
$$788$$ −16.2984 −0.580608
$$789$$ 2.89531 0.103076
$$790$$ 7.40312 0.263391
$$791$$ −7.40312 −0.263225
$$792$$ −1.00000 −0.0355335
$$793$$ −13.1047 −0.465361
$$794$$ −12.8062 −0.454477
$$795$$ 2.00000 0.0709327
$$796$$ 2.29844 0.0814660
$$797$$ 13.3141 0.471608 0.235804 0.971801i $$-0.424228\pi$$
0.235804 + 0.971801i $$0.424228\pi$$
$$798$$ −6.29844 −0.222962
$$799$$ 3.40312 0.120394
$$800$$ −1.00000 −0.0353553
$$801$$ 14.0000 0.494666
$$802$$ 13.1047 0.462743
$$803$$ −8.00000 −0.282314
$$804$$ 8.29844 0.292663
$$805$$ −8.50781 −0.299861
$$806$$ −9.70156 −0.341723
$$807$$ −26.5078 −0.933119
$$808$$ 3.40312 0.119721
$$809$$ 12.5969 0.442883 0.221441 0.975174i $$-0.428924\pi$$
0.221441 + 0.975174i $$0.428924\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ 24.5969 0.863713 0.431857 0.901942i $$-0.357859\pi$$
0.431857 + 0.901942i $$0.357859\pi$$
$$812$$ −7.40312 −0.259799
$$813$$ −14.8062 −0.519278
$$814$$ −4.29844 −0.150660
$$815$$ 14.8062 0.518640
$$816$$ −1.00000 −0.0350070
$$817$$ 9.19375 0.321649
$$818$$ −6.00000 −0.209785
$$819$$ 6.29844 0.220085
$$820$$ −3.40312 −0.118842
$$821$$ 19.6125 0.684481 0.342240 0.939612i $$-0.388814\pi$$
0.342240 + 0.939612i $$0.388814\pi$$
$$822$$ −19.1047 −0.666352
$$823$$ 6.00000 0.209147 0.104573 0.994517i $$-0.466652\pi$$
0.104573 + 0.994517i $$0.466652\pi$$
$$824$$ −17.1047 −0.595870
$$825$$ −1.00000 −0.0348155
$$826$$ −49.6125 −1.72624
$$827$$ −30.2094 −1.05048 −0.525241 0.850953i $$-0.676025\pi$$
−0.525241 + 0.850953i $$0.676025\pi$$
$$828$$ −2.29844 −0.0798762
$$829$$ −43.1047 −1.49709 −0.748544 0.663085i $$-0.769247\pi$$
−0.748544 + 0.663085i $$0.769247\pi$$
$$830$$ −9.70156 −0.336746
$$831$$ −13.4031 −0.464949
$$832$$ −1.70156 −0.0589911
$$833$$ 6.70156 0.232195
$$834$$ 8.59688 0.297685
$$835$$ 18.8062 0.650817
$$836$$ 1.70156 0.0588498
$$837$$ 5.70156 0.197075
$$838$$ 30.8062 1.06418
$$839$$ 15.7906 0.545153 0.272576 0.962134i $$-0.412124\pi$$
0.272576 + 0.962134i $$0.412124\pi$$
$$840$$ 3.70156 0.127716
$$841$$ −25.0000 −0.862069
$$842$$ 8.29844 0.285983
$$843$$ 10.0000 0.344418
$$844$$ 22.2094 0.764478
$$845$$ 10.1047 0.347612
$$846$$ −3.40312 −0.117002
$$847$$ −3.70156 −0.127187
$$848$$ 2.00000 0.0686803
$$849$$ −12.0000 −0.411839
$$850$$ −1.00000 −0.0342997
$$851$$ −9.87969 −0.338671
$$852$$ −12.8062 −0.438735
$$853$$ −1.40312 −0.0480421 −0.0240210 0.999711i $$-0.507647\pi$$
−0.0240210 + 0.999711i $$0.507647\pi$$
$$854$$ 28.5078 0.975517
$$855$$ −1.70156 −0.0581922
$$856$$ 7.40312 0.253034
$$857$$ 51.7016 1.76609 0.883046 0.469287i $$-0.155489\pi$$
0.883046 + 0.469287i $$0.155489\pi$$
$$858$$ −1.70156 −0.0580904
$$859$$ −20.4187 −0.696679 −0.348339 0.937369i $$-0.613254\pi$$
−0.348339 + 0.937369i $$0.613254\pi$$
$$860$$ −5.40312 −0.184245
$$861$$ 12.5969 0.429300
$$862$$ 36.2094 1.23330
$$863$$ −16.0000 −0.544646 −0.272323 0.962206i $$-0.587792\pi$$
−0.272323 + 0.962206i $$0.587792\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ −6.50781 −0.221272
$$866$$ 18.0000 0.611665
$$867$$ −1.00000 −0.0339618
$$868$$ 21.1047 0.716340
$$869$$ 7.40312 0.251134
$$870$$ −2.00000 −0.0678064
$$871$$ 14.1203 0.478448
$$872$$ −15.1047 −0.511509
$$873$$ 9.70156 0.328348
$$874$$ 3.91093 0.132289
$$875$$ 3.70156 0.125136
$$876$$ 8.00000 0.270295
$$877$$ 5.40312 0.182451 0.0912253 0.995830i $$-0.470922\pi$$
0.0912253 + 0.995830i $$0.470922\pi$$
$$878$$ −1.79063 −0.0604307
$$879$$ −21.4031 −0.721909
$$880$$ −1.00000 −0.0337100
$$881$$ −4.59688 −0.154873 −0.0774363 0.996997i $$-0.524673\pi$$
−0.0774363 + 0.996997i $$0.524673\pi$$
$$882$$ −6.70156 −0.225653
$$883$$ 11.0156 0.370705 0.185353 0.982672i $$-0.440657\pi$$
0.185353 + 0.982672i $$0.440657\pi$$
$$884$$ −1.70156 −0.0572297
$$885$$ −13.4031 −0.450541
$$886$$ −14.2094 −0.477373
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 4.29844 0.144246
$$889$$ −14.8062 −0.496586
$$890$$ 14.0000 0.469281
$$891$$ 1.00000 0.0335013
$$892$$ 12.5078 0.418792
$$893$$ 5.79063 0.193776
$$894$$ −6.29844 −0.210651
$$895$$ 3.10469 0.103778
$$896$$ 3.70156 0.123661
$$897$$ −3.91093 −0.130582
$$898$$ 33.7016 1.12464
$$899$$ −11.4031 −0.380315
$$900$$ 1.00000 0.0333333
$$901$$ 2.00000 0.0666297
$$902$$ −3.40312 −0.113312
$$903$$ 20.0000 0.665558
$$904$$ −2.00000 −0.0665190
$$905$$ 9.40312 0.312570
$$906$$ −16.5078 −0.548435
$$907$$ −48.0000 −1.59381 −0.796907 0.604102i $$-0.793532\pi$$
−0.796907 + 0.604102i $$0.793532\pi$$
$$908$$ 14.2094 0.471555
$$909$$ −3.40312 −0.112875
$$910$$ 6.29844 0.208791
$$911$$ −22.5969 −0.748668 −0.374334 0.927294i $$-0.622129\pi$$
−0.374334 + 0.927294i $$0.622129\pi$$
$$912$$ −1.70156 −0.0563444
$$913$$ −9.70156 −0.321075
$$914$$ −3.10469 −0.102694
$$915$$ 7.70156 0.254606
$$916$$ 7.70156 0.254467
$$917$$ −48.5078 −1.60187
$$918$$ 1.00000 0.0330049
$$919$$ 22.2984 0.735558 0.367779 0.929913i $$-0.380118\pi$$
0.367779 + 0.929913i $$0.380118\pi$$
$$920$$ −2.29844 −0.0757773
$$921$$ 2.59688 0.0855700
$$922$$ 24.5969 0.810055
$$923$$ −21.7906 −0.717247
$$924$$ 3.70156 0.121772
$$925$$ 4.29844 0.141332
$$926$$ 18.2984 0.601324
$$927$$ 17.1047 0.561792
$$928$$ −2.00000 −0.0656532
$$929$$ 26.2984 0.862824 0.431412 0.902155i $$-0.358016\pi$$
0.431412 + 0.902155i $$0.358016\pi$$
$$930$$ 5.70156 0.186962
$$931$$ 11.4031 0.373722
$$932$$ −12.8062 −0.419483
$$933$$ 4.80625 0.157350
$$934$$ 9.19375 0.300829
$$935$$ −1.00000 −0.0327035
$$936$$ 1.70156 0.0556173
$$937$$ −2.50781 −0.0819266 −0.0409633 0.999161i $$-0.513043\pi$$
−0.0409633 + 0.999161i $$0.513043\pi$$
$$938$$ −30.7172 −1.00295
$$939$$ −19.9109 −0.649769
$$940$$ −3.40312 −0.110998
$$941$$ 2.59688 0.0846557 0.0423279 0.999104i $$-0.486523\pi$$
0.0423279 + 0.999104i $$0.486523\pi$$
$$942$$ −12.0000 −0.390981
$$943$$ −7.82187 −0.254715
$$944$$ −13.4031 −0.436235
$$945$$ −3.70156 −0.120412
$$946$$ −5.40312 −0.175671
$$947$$ 28.0000 0.909878 0.454939 0.890523i $$-0.349661\pi$$
0.454939 + 0.890523i $$0.349661\pi$$
$$948$$ −7.40312 −0.240442
$$949$$ 13.6125 0.441880
$$950$$ −1.70156 −0.0552060
$$951$$ −14.0000 −0.453981
$$952$$ 3.70156 0.119968
$$953$$ 49.8219 1.61389 0.806944 0.590628i $$-0.201120\pi$$
0.806944 + 0.590628i $$0.201120\pi$$
$$954$$ −2.00000 −0.0647524
$$955$$ 26.8062 0.867430
$$956$$ 3.40312 0.110065
$$957$$ −2.00000 −0.0646508
$$958$$ −29.9109 −0.966378
$$959$$ 70.7172 2.28358
$$960$$ 1.00000 0.0322749
$$961$$ 1.50781 0.0486391
$$962$$ 7.31406 0.235815
$$963$$ −7.40312 −0.238562
$$964$$ 4.29844 0.138443
$$965$$ 12.0000 0.386294
$$966$$ 8.50781 0.273734
$$967$$ 34.2094 1.10010 0.550050 0.835132i $$-0.314609\pi$$
0.550050 + 0.835132i $$0.314609\pi$$
$$968$$ −1.00000 −0.0321412
$$969$$ −1.70156 −0.0546621
$$970$$ 9.70156 0.311498
$$971$$ 36.7172 1.17831 0.589155 0.808020i $$-0.299461\pi$$
0.589155 + 0.808020i $$0.299461\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ −31.8219 −1.02016
$$974$$ −20.2094 −0.647550
$$975$$ 1.70156 0.0544936
$$976$$ 7.70156 0.246521
$$977$$ 47.5234 1.52041 0.760205 0.649684i $$-0.225099\pi$$
0.760205 + 0.649684i $$0.225099\pi$$
$$978$$ −14.8062 −0.473452
$$979$$ 14.0000 0.447442
$$980$$ −6.70156 −0.214074
$$981$$ 15.1047 0.482256
$$982$$ −12.8062 −0.408664
$$983$$ −1.19375 −0.0380748 −0.0190374 0.999819i $$-0.506060\pi$$
−0.0190374 + 0.999819i $$0.506060\pi$$
$$984$$ 3.40312 0.108488
$$985$$ 16.2984 0.519311
$$986$$ −2.00000 −0.0636930
$$987$$ 12.5969 0.400963
$$988$$ −2.89531 −0.0921122
$$989$$ −12.4187 −0.394893
$$990$$ 1.00000 0.0317821
$$991$$ 28.5078 0.905580 0.452790 0.891617i $$-0.350429\pi$$
0.452790 + 0.891617i $$0.350429\pi$$
$$992$$ 5.70156 0.181025
$$993$$ 21.6125 0.685852
$$994$$ 47.4031 1.50354
$$995$$ −2.29844 −0.0728654
$$996$$ 9.70156 0.307406
$$997$$ −18.0000 −0.570066 −0.285033 0.958518i $$-0.592005\pi$$
−0.285033 + 0.958518i $$0.592005\pi$$
$$998$$ 36.0000 1.13956
$$999$$ −4.29844 −0.135997
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 5610.2.a.bo.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
5610.2.a.bo.1.1 2 1.1 even 1 trivial