# Properties

 Label 370.2.a.f.1.2 Level $370$ Weight $2$ Character 370.1 Self dual yes Analytic conductor $2.954$ 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] = [370,2,Mod(1,370)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(370, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("370.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$370 = 2 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 370.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: $$2.95446487479$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 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 1.2 Root $$-2.37228$$ of defining polynomial Character $$\chi$$ $$=$$ 370.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} +1.37228 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} +1.37228 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -3.37228 q^{11} +2.00000 q^{12} -4.74456 q^{13} +1.37228 q^{14} +2.00000 q^{15} +1.00000 q^{16} -5.37228 q^{17} +1.00000 q^{18} -2.00000 q^{19} +1.00000 q^{20} +2.74456 q^{21} -3.37228 q^{22} +6.74456 q^{23} +2.00000 q^{24} +1.00000 q^{25} -4.74456 q^{26} -4.00000 q^{27} +1.37228 q^{28} +8.11684 q^{29} +2.00000 q^{30} -2.62772 q^{31} +1.00000 q^{32} -6.74456 q^{33} -5.37228 q^{34} +1.37228 q^{35} +1.00000 q^{36} +1.00000 q^{37} -2.00000 q^{38} -9.48913 q^{39} +1.00000 q^{40} +5.37228 q^{41} +2.74456 q^{42} +7.37228 q^{43} -3.37228 q^{44} +1.00000 q^{45} +6.74456 q^{46} -8.74456 q^{47} +2.00000 q^{48} -5.11684 q^{49} +1.00000 q^{50} -10.7446 q^{51} -4.74456 q^{52} +1.37228 q^{53} -4.00000 q^{54} -3.37228 q^{55} +1.37228 q^{56} -4.00000 q^{57} +8.11684 q^{58} +12.7446 q^{59} +2.00000 q^{60} -5.37228 q^{61} -2.62772 q^{62} +1.37228 q^{63} +1.00000 q^{64} -4.74456 q^{65} -6.74456 q^{66} +4.74456 q^{67} -5.37228 q^{68} +13.4891 q^{69} +1.37228 q^{70} -6.74456 q^{71} +1.00000 q^{72} +8.74456 q^{73} +1.00000 q^{74} +2.00000 q^{75} -2.00000 q^{76} -4.62772 q^{77} -9.48913 q^{78} -4.74456 q^{79} +1.00000 q^{80} -11.0000 q^{81} +5.37228 q^{82} -0.744563 q^{83} +2.74456 q^{84} -5.37228 q^{85} +7.37228 q^{86} +16.2337 q^{87} -3.37228 q^{88} +10.0000 q^{89} +1.00000 q^{90} -6.51087 q^{91} +6.74456 q^{92} -5.25544 q^{93} -8.74456 q^{94} -2.00000 q^{95} +2.00000 q^{96} -0.116844 q^{97} -5.11684 q^{98} -3.37228 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 2 q^{5} + 4 q^{6} - 3 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 4 * q^3 + 2 * q^4 + 2 * q^5 + 4 * q^6 - 3 * q^7 + 2 * q^8 + 2 * q^9 $$2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 2 q^{5} + 4 q^{6} - 3 q^{7} + 2 q^{8} + 2 q^{9} + 2 q^{10} - q^{11} + 4 q^{12} + 2 q^{13} - 3 q^{14} + 4 q^{15} + 2 q^{16} - 5 q^{17} + 2 q^{18} - 4 q^{19} + 2 q^{20} - 6 q^{21} - q^{22} + 2 q^{23} + 4 q^{24} + 2 q^{25} + 2 q^{26} - 8 q^{27} - 3 q^{28} - q^{29} + 4 q^{30} - 11 q^{31} + 2 q^{32} - 2 q^{33} - 5 q^{34} - 3 q^{35} + 2 q^{36} + 2 q^{37} - 4 q^{38} + 4 q^{39} + 2 q^{40} + 5 q^{41} - 6 q^{42} + 9 q^{43} - q^{44} + 2 q^{45} + 2 q^{46} - 6 q^{47} + 4 q^{48} + 7 q^{49} + 2 q^{50} - 10 q^{51} + 2 q^{52} - 3 q^{53} - 8 q^{54} - q^{55} - 3 q^{56} - 8 q^{57} - q^{58} + 14 q^{59} + 4 q^{60} - 5 q^{61} - 11 q^{62} - 3 q^{63} + 2 q^{64} + 2 q^{65} - 2 q^{66} - 2 q^{67} - 5 q^{68} + 4 q^{69} - 3 q^{70} - 2 q^{71} + 2 q^{72} + 6 q^{73} + 2 q^{74} + 4 q^{75} - 4 q^{76} - 15 q^{77} + 4 q^{78} + 2 q^{79} + 2 q^{80} - 22 q^{81} + 5 q^{82} + 10 q^{83} - 6 q^{84} - 5 q^{85} + 9 q^{86} - 2 q^{87} - q^{88} + 20 q^{89} + 2 q^{90} - 36 q^{91} + 2 q^{92} - 22 q^{93} - 6 q^{94} - 4 q^{95} + 4 q^{96} + 17 q^{97} + 7 q^{98} - q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 4 * q^3 + 2 * q^4 + 2 * q^5 + 4 * q^6 - 3 * q^7 + 2 * q^8 + 2 * q^9 + 2 * q^10 - q^11 + 4 * q^12 + 2 * q^13 - 3 * q^14 + 4 * q^15 + 2 * q^16 - 5 * q^17 + 2 * q^18 - 4 * q^19 + 2 * q^20 - 6 * q^21 - q^22 + 2 * q^23 + 4 * q^24 + 2 * q^25 + 2 * q^26 - 8 * q^27 - 3 * q^28 - q^29 + 4 * q^30 - 11 * q^31 + 2 * q^32 - 2 * q^33 - 5 * q^34 - 3 * q^35 + 2 * q^36 + 2 * q^37 - 4 * q^38 + 4 * q^39 + 2 * q^40 + 5 * q^41 - 6 * q^42 + 9 * q^43 - q^44 + 2 * q^45 + 2 * q^46 - 6 * q^47 + 4 * q^48 + 7 * q^49 + 2 * q^50 - 10 * q^51 + 2 * q^52 - 3 * q^53 - 8 * q^54 - q^55 - 3 * q^56 - 8 * q^57 - q^58 + 14 * q^59 + 4 * q^60 - 5 * q^61 - 11 * q^62 - 3 * q^63 + 2 * q^64 + 2 * q^65 - 2 * q^66 - 2 * q^67 - 5 * q^68 + 4 * q^69 - 3 * q^70 - 2 * q^71 + 2 * q^72 + 6 * q^73 + 2 * q^74 + 4 * q^75 - 4 * q^76 - 15 * q^77 + 4 * q^78 + 2 * q^79 + 2 * q^80 - 22 * q^81 + 5 * q^82 + 10 * q^83 - 6 * q^84 - 5 * q^85 + 9 * q^86 - 2 * q^87 - q^88 + 20 * q^89 + 2 * q^90 - 36 * q^91 + 2 * q^92 - 22 * q^93 - 6 * q^94 - 4 * q^95 + 4 * q^96 + 17 * q^97 + 7 * q^98 - 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
$$3$$ 2.00000 1.15470 0.577350 0.816497i $$-0.304087\pi$$
0.577350 + 0.816497i $$0.304087\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ 2.00000 0.816497
$$7$$ 1.37228 0.518674 0.259337 0.965787i $$-0.416496\pi$$
0.259337 + 0.965787i $$0.416496\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ −3.37228 −1.01678 −0.508391 0.861127i $$-0.669759\pi$$
−0.508391 + 0.861127i $$0.669759\pi$$
$$12$$ 2.00000 0.577350
$$13$$ −4.74456 −1.31590 −0.657952 0.753059i $$-0.728577\pi$$
−0.657952 + 0.753059i $$0.728577\pi$$
$$14$$ 1.37228 0.366758
$$15$$ 2.00000 0.516398
$$16$$ 1.00000 0.250000
$$17$$ −5.37228 −1.30297 −0.651485 0.758662i $$-0.725854\pi$$
−0.651485 + 0.758662i $$0.725854\pi$$
$$18$$ 1.00000 0.235702
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 2.74456 0.598913
$$22$$ −3.37228 −0.718973
$$23$$ 6.74456 1.40634 0.703169 0.711022i $$-0.251768\pi$$
0.703169 + 0.711022i $$0.251768\pi$$
$$24$$ 2.00000 0.408248
$$25$$ 1.00000 0.200000
$$26$$ −4.74456 −0.930485
$$27$$ −4.00000 −0.769800
$$28$$ 1.37228 0.259337
$$29$$ 8.11684 1.50726 0.753630 0.657299i $$-0.228301\pi$$
0.753630 + 0.657299i $$0.228301\pi$$
$$30$$ 2.00000 0.365148
$$31$$ −2.62772 −0.471952 −0.235976 0.971759i $$-0.575829\pi$$
−0.235976 + 0.971759i $$0.575829\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −6.74456 −1.17408
$$34$$ −5.37228 −0.921339
$$35$$ 1.37228 0.231958
$$36$$ 1.00000 0.166667
$$37$$ 1.00000 0.164399
$$38$$ −2.00000 −0.324443
$$39$$ −9.48913 −1.51948
$$40$$ 1.00000 0.158114
$$41$$ 5.37228 0.839009 0.419505 0.907753i $$-0.362204\pi$$
0.419505 + 0.907753i $$0.362204\pi$$
$$42$$ 2.74456 0.423495
$$43$$ 7.37228 1.12426 0.562131 0.827048i $$-0.309982\pi$$
0.562131 + 0.827048i $$0.309982\pi$$
$$44$$ −3.37228 −0.508391
$$45$$ 1.00000 0.149071
$$46$$ 6.74456 0.994432
$$47$$ −8.74456 −1.27553 −0.637763 0.770233i $$-0.720140\pi$$
−0.637763 + 0.770233i $$0.720140\pi$$
$$48$$ 2.00000 0.288675
$$49$$ −5.11684 −0.730978
$$50$$ 1.00000 0.141421
$$51$$ −10.7446 −1.50454
$$52$$ −4.74456 −0.657952
$$53$$ 1.37228 0.188497 0.0942487 0.995549i $$-0.469955\pi$$
0.0942487 + 0.995549i $$0.469955\pi$$
$$54$$ −4.00000 −0.544331
$$55$$ −3.37228 −0.454718
$$56$$ 1.37228 0.183379
$$57$$ −4.00000 −0.529813
$$58$$ 8.11684 1.06579
$$59$$ 12.7446 1.65920 0.829600 0.558358i $$-0.188568\pi$$
0.829600 + 0.558358i $$0.188568\pi$$
$$60$$ 2.00000 0.258199
$$61$$ −5.37228 −0.687850 −0.343925 0.938997i $$-0.611757\pi$$
−0.343925 + 0.938997i $$0.611757\pi$$
$$62$$ −2.62772 −0.333721
$$63$$ 1.37228 0.172891
$$64$$ 1.00000 0.125000
$$65$$ −4.74456 −0.588491
$$66$$ −6.74456 −0.830198
$$67$$ 4.74456 0.579641 0.289820 0.957081i $$-0.406404\pi$$
0.289820 + 0.957081i $$0.406404\pi$$
$$68$$ −5.37228 −0.651485
$$69$$ 13.4891 1.62390
$$70$$ 1.37228 0.164019
$$71$$ −6.74456 −0.800432 −0.400216 0.916421i $$-0.631065\pi$$
−0.400216 + 0.916421i $$0.631065\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 8.74456 1.02347 0.511737 0.859142i $$-0.329002\pi$$
0.511737 + 0.859142i $$0.329002\pi$$
$$74$$ 1.00000 0.116248
$$75$$ 2.00000 0.230940
$$76$$ −2.00000 −0.229416
$$77$$ −4.62772 −0.527377
$$78$$ −9.48913 −1.07443
$$79$$ −4.74456 −0.533805 −0.266903 0.963724i $$-0.586000\pi$$
−0.266903 + 0.963724i $$0.586000\pi$$
$$80$$ 1.00000 0.111803
$$81$$ −11.0000 −1.22222
$$82$$ 5.37228 0.593269
$$83$$ −0.744563 −0.0817264 −0.0408632 0.999165i $$-0.513011\pi$$
−0.0408632 + 0.999165i $$0.513011\pi$$
$$84$$ 2.74456 0.299456
$$85$$ −5.37228 −0.582706
$$86$$ 7.37228 0.794974
$$87$$ 16.2337 1.74043
$$88$$ −3.37228 −0.359486
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 1.00000 0.105409
$$91$$ −6.51087 −0.682525
$$92$$ 6.74456 0.703169
$$93$$ −5.25544 −0.544963
$$94$$ −8.74456 −0.901933
$$95$$ −2.00000 −0.205196
$$96$$ 2.00000 0.204124
$$97$$ −0.116844 −0.0118637 −0.00593185 0.999982i $$-0.501888\pi$$
−0.00593185 + 0.999982i $$0.501888\pi$$
$$98$$ −5.11684 −0.516879
$$99$$ −3.37228 −0.338927
$$100$$ 1.00000 0.100000
$$101$$ −11.4891 −1.14321 −0.571605 0.820529i $$-0.693679\pi$$
−0.571605 + 0.820529i $$0.693679\pi$$
$$102$$ −10.7446 −1.06387
$$103$$ −9.48913 −0.934991 −0.467496 0.883995i $$-0.654844\pi$$
−0.467496 + 0.883995i $$0.654844\pi$$
$$104$$ −4.74456 −0.465243
$$105$$ 2.74456 0.267842
$$106$$ 1.37228 0.133288
$$107$$ −3.48913 −0.337306 −0.168653 0.985675i $$-0.553942\pi$$
−0.168653 + 0.985675i $$0.553942\pi$$
$$108$$ −4.00000 −0.384900
$$109$$ 0.116844 0.0111916 0.00559581 0.999984i $$-0.498219\pi$$
0.00559581 + 0.999984i $$0.498219\pi$$
$$110$$ −3.37228 −0.321534
$$111$$ 2.00000 0.189832
$$112$$ 1.37228 0.129668
$$113$$ 17.3723 1.63425 0.817123 0.576463i $$-0.195567\pi$$
0.817123 + 0.576463i $$0.195567\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ 6.74456 0.628934
$$116$$ 8.11684 0.753630
$$117$$ −4.74456 −0.438635
$$118$$ 12.7446 1.17323
$$119$$ −7.37228 −0.675816
$$120$$ 2.00000 0.182574
$$121$$ 0.372281 0.0338438
$$122$$ −5.37228 −0.486383
$$123$$ 10.7446 0.968805
$$124$$ −2.62772 −0.235976
$$125$$ 1.00000 0.0894427
$$126$$ 1.37228 0.122253
$$127$$ −16.7446 −1.48584 −0.742920 0.669380i $$-0.766560\pi$$
−0.742920 + 0.669380i $$0.766560\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 14.7446 1.29819
$$130$$ −4.74456 −0.416126
$$131$$ −3.25544 −0.284429 −0.142214 0.989836i $$-0.545422\pi$$
−0.142214 + 0.989836i $$0.545422\pi$$
$$132$$ −6.74456 −0.587039
$$133$$ −2.74456 −0.237984
$$134$$ 4.74456 0.409868
$$135$$ −4.00000 −0.344265
$$136$$ −5.37228 −0.460669
$$137$$ 16.7446 1.43058 0.715292 0.698825i $$-0.246294\pi$$
0.715292 + 0.698825i $$0.246294\pi$$
$$138$$ 13.4891 1.14827
$$139$$ −1.88316 −0.159727 −0.0798636 0.996806i $$-0.525448\pi$$
−0.0798636 + 0.996806i $$0.525448\pi$$
$$140$$ 1.37228 0.115979
$$141$$ −17.4891 −1.47285
$$142$$ −6.74456 −0.565991
$$143$$ 16.0000 1.33799
$$144$$ 1.00000 0.0833333
$$145$$ 8.11684 0.674067
$$146$$ 8.74456 0.723705
$$147$$ −10.2337 −0.844060
$$148$$ 1.00000 0.0821995
$$149$$ −11.4891 −0.941226 −0.470613 0.882340i $$-0.655967\pi$$
−0.470613 + 0.882340i $$0.655967\pi$$
$$150$$ 2.00000 0.163299
$$151$$ −20.0000 −1.62758 −0.813788 0.581161i $$-0.802599\pi$$
−0.813788 + 0.581161i $$0.802599\pi$$
$$152$$ −2.00000 −0.162221
$$153$$ −5.37228 −0.434323
$$154$$ −4.62772 −0.372912
$$155$$ −2.62772 −0.211063
$$156$$ −9.48913 −0.759738
$$157$$ 17.3723 1.38646 0.693229 0.720717i $$-0.256187\pi$$
0.693229 + 0.720717i $$0.256187\pi$$
$$158$$ −4.74456 −0.377457
$$159$$ 2.74456 0.217658
$$160$$ 1.00000 0.0790569
$$161$$ 9.25544 0.729431
$$162$$ −11.0000 −0.864242
$$163$$ 19.3723 1.51735 0.758677 0.651467i $$-0.225846\pi$$
0.758677 + 0.651467i $$0.225846\pi$$
$$164$$ 5.37228 0.419505
$$165$$ −6.74456 −0.525063
$$166$$ −0.744563 −0.0577893
$$167$$ 1.48913 0.115232 0.0576160 0.998339i $$-0.481650\pi$$
0.0576160 + 0.998339i $$0.481650\pi$$
$$168$$ 2.74456 0.211748
$$169$$ 9.51087 0.731606
$$170$$ −5.37228 −0.412035
$$171$$ −2.00000 −0.152944
$$172$$ 7.37228 0.562131
$$173$$ 22.8614 1.73812 0.869060 0.494706i $$-0.164724\pi$$
0.869060 + 0.494706i $$0.164724\pi$$
$$174$$ 16.2337 1.23067
$$175$$ 1.37228 0.103735
$$176$$ −3.37228 −0.254195
$$177$$ 25.4891 1.91588
$$178$$ 10.0000 0.749532
$$179$$ −26.2337 −1.96080 −0.980399 0.197023i $$-0.936873\pi$$
−0.980399 + 0.197023i $$0.936873\pi$$
$$180$$ 1.00000 0.0745356
$$181$$ 7.48913 0.556662 0.278331 0.960485i $$-0.410219\pi$$
0.278331 + 0.960485i $$0.410219\pi$$
$$182$$ −6.51087 −0.482618
$$183$$ −10.7446 −0.794261
$$184$$ 6.74456 0.497216
$$185$$ 1.00000 0.0735215
$$186$$ −5.25544 −0.385347
$$187$$ 18.1168 1.32483
$$188$$ −8.74456 −0.637763
$$189$$ −5.48913 −0.399275
$$190$$ −2.00000 −0.145095
$$191$$ 18.6277 1.34785 0.673927 0.738798i $$-0.264606\pi$$
0.673927 + 0.738798i $$0.264606\pi$$
$$192$$ 2.00000 0.144338
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ −0.116844 −0.00838891
$$195$$ −9.48913 −0.679530
$$196$$ −5.11684 −0.365489
$$197$$ 27.4891 1.95852 0.979260 0.202610i $$-0.0649423\pi$$
0.979260 + 0.202610i $$0.0649423\pi$$
$$198$$ −3.37228 −0.239658
$$199$$ −11.2554 −0.797877 −0.398938 0.916978i $$-0.630621\pi$$
−0.398938 + 0.916978i $$0.630621\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 9.48913 0.669311
$$202$$ −11.4891 −0.808372
$$203$$ 11.1386 0.781776
$$204$$ −10.7446 −0.752270
$$205$$ 5.37228 0.375216
$$206$$ −9.48913 −0.661139
$$207$$ 6.74456 0.468780
$$208$$ −4.74456 −0.328976
$$209$$ 6.74456 0.466531
$$210$$ 2.74456 0.189393
$$211$$ −14.1168 −0.971844 −0.485922 0.874002i $$-0.661516\pi$$
−0.485922 + 0.874002i $$0.661516\pi$$
$$212$$ 1.37228 0.0942487
$$213$$ −13.4891 −0.924260
$$214$$ −3.48913 −0.238512
$$215$$ 7.37228 0.502785
$$216$$ −4.00000 −0.272166
$$217$$ −3.60597 −0.244789
$$218$$ 0.116844 0.00791367
$$219$$ 17.4891 1.18181
$$220$$ −3.37228 −0.227359
$$221$$ 25.4891 1.71458
$$222$$ 2.00000 0.134231
$$223$$ −1.37228 −0.0918948 −0.0459474 0.998944i $$-0.514631\pi$$
−0.0459474 + 0.998944i $$0.514631\pi$$
$$224$$ 1.37228 0.0916894
$$225$$ 1.00000 0.0666667
$$226$$ 17.3723 1.15559
$$227$$ −11.3723 −0.754805 −0.377402 0.926049i $$-0.623183\pi$$
−0.377402 + 0.926049i $$0.623183\pi$$
$$228$$ −4.00000 −0.264906
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ 6.74456 0.444723
$$231$$ −9.25544 −0.608963
$$232$$ 8.11684 0.532897
$$233$$ 6.23369 0.408382 0.204191 0.978931i $$-0.434544\pi$$
0.204191 + 0.978931i $$0.434544\pi$$
$$234$$ −4.74456 −0.310162
$$235$$ −8.74456 −0.570432
$$236$$ 12.7446 0.829600
$$237$$ −9.48913 −0.616385
$$238$$ −7.37228 −0.477874
$$239$$ 17.6060 1.13884 0.569418 0.822048i $$-0.307169\pi$$
0.569418 + 0.822048i $$0.307169\pi$$
$$240$$ 2.00000 0.129099
$$241$$ −10.2337 −0.659210 −0.329605 0.944119i $$-0.606916\pi$$
−0.329605 + 0.944119i $$0.606916\pi$$
$$242$$ 0.372281 0.0239311
$$243$$ −10.0000 −0.641500
$$244$$ −5.37228 −0.343925
$$245$$ −5.11684 −0.326903
$$246$$ 10.7446 0.685048
$$247$$ 9.48913 0.603779
$$248$$ −2.62772 −0.166860
$$249$$ −1.48913 −0.0943695
$$250$$ 1.00000 0.0632456
$$251$$ 11.4891 0.725187 0.362594 0.931947i $$-0.381891\pi$$
0.362594 + 0.931947i $$0.381891\pi$$
$$252$$ 1.37228 0.0864456
$$253$$ −22.7446 −1.42994
$$254$$ −16.7446 −1.05065
$$255$$ −10.7446 −0.672851
$$256$$ 1.00000 0.0625000
$$257$$ −20.9783 −1.30859 −0.654294 0.756241i $$-0.727034\pi$$
−0.654294 + 0.756241i $$0.727034\pi$$
$$258$$ 14.7446 0.917956
$$259$$ 1.37228 0.0852694
$$260$$ −4.74456 −0.294245
$$261$$ 8.11684 0.502420
$$262$$ −3.25544 −0.201122
$$263$$ 0.116844 0.00720491 0.00360245 0.999994i $$-0.498853\pi$$
0.00360245 + 0.999994i $$0.498853\pi$$
$$264$$ −6.74456 −0.415099
$$265$$ 1.37228 0.0842986
$$266$$ −2.74456 −0.168280
$$267$$ 20.0000 1.22398
$$268$$ 4.74456 0.289820
$$269$$ 10.0000 0.609711 0.304855 0.952399i $$-0.401392\pi$$
0.304855 + 0.952399i $$0.401392\pi$$
$$270$$ −4.00000 −0.243432
$$271$$ −6.74456 −0.409703 −0.204852 0.978793i $$-0.565671\pi$$
−0.204852 + 0.978793i $$0.565671\pi$$
$$272$$ −5.37228 −0.325742
$$273$$ −13.0217 −0.788112
$$274$$ 16.7446 1.01158
$$275$$ −3.37228 −0.203356
$$276$$ 13.4891 0.811950
$$277$$ 0.744563 0.0447364 0.0223682 0.999750i $$-0.492879\pi$$
0.0223682 + 0.999750i $$0.492879\pi$$
$$278$$ −1.88316 −0.112944
$$279$$ −2.62772 −0.157317
$$280$$ 1.37228 0.0820095
$$281$$ 24.7446 1.47614 0.738068 0.674726i $$-0.235738\pi$$
0.738068 + 0.674726i $$0.235738\pi$$
$$282$$ −17.4891 −1.04146
$$283$$ 5.48913 0.326295 0.163147 0.986602i $$-0.447835\pi$$
0.163147 + 0.986602i $$0.447835\pi$$
$$284$$ −6.74456 −0.400216
$$285$$ −4.00000 −0.236940
$$286$$ 16.0000 0.946100
$$287$$ 7.37228 0.435172
$$288$$ 1.00000 0.0589256
$$289$$ 11.8614 0.697730
$$290$$ 8.11684 0.476637
$$291$$ −0.233688 −0.0136990
$$292$$ 8.74456 0.511737
$$293$$ −18.8614 −1.10190 −0.550948 0.834540i $$-0.685734\pi$$
−0.550948 + 0.834540i $$0.685734\pi$$
$$294$$ −10.2337 −0.596841
$$295$$ 12.7446 0.742017
$$296$$ 1.00000 0.0581238
$$297$$ 13.4891 0.782718
$$298$$ −11.4891 −0.665547
$$299$$ −32.0000 −1.85061
$$300$$ 2.00000 0.115470
$$301$$ 10.1168 0.583125
$$302$$ −20.0000 −1.15087
$$303$$ −22.9783 −1.32007
$$304$$ −2.00000 −0.114708
$$305$$ −5.37228 −0.307616
$$306$$ −5.37228 −0.307113
$$307$$ −23.4891 −1.34060 −0.670298 0.742092i $$-0.733834\pi$$
−0.670298 + 0.742092i $$0.733834\pi$$
$$308$$ −4.62772 −0.263689
$$309$$ −18.9783 −1.07963
$$310$$ −2.62772 −0.149244
$$311$$ −1.37228 −0.0778149 −0.0389075 0.999243i $$-0.512388\pi$$
−0.0389075 + 0.999243i $$0.512388\pi$$
$$312$$ −9.48913 −0.537216
$$313$$ −3.48913 −0.197217 −0.0986085 0.995126i $$-0.531439\pi$$
−0.0986085 + 0.995126i $$0.531439\pi$$
$$314$$ 17.3723 0.980375
$$315$$ 1.37228 0.0773193
$$316$$ −4.74456 −0.266903
$$317$$ 25.3723 1.42505 0.712525 0.701647i $$-0.247552\pi$$
0.712525 + 0.701647i $$0.247552\pi$$
$$318$$ 2.74456 0.153907
$$319$$ −27.3723 −1.53255
$$320$$ 1.00000 0.0559017
$$321$$ −6.97825 −0.389488
$$322$$ 9.25544 0.515785
$$323$$ 10.7446 0.597843
$$324$$ −11.0000 −0.611111
$$325$$ −4.74456 −0.263181
$$326$$ 19.3723 1.07293
$$327$$ 0.233688 0.0129230
$$328$$ 5.37228 0.296635
$$329$$ −12.0000 −0.661581
$$330$$ −6.74456 −0.371276
$$331$$ 23.4891 1.29108 0.645540 0.763727i $$-0.276633\pi$$
0.645540 + 0.763727i $$0.276633\pi$$
$$332$$ −0.744563 −0.0408632
$$333$$ 1.00000 0.0547997
$$334$$ 1.48913 0.0814813
$$335$$ 4.74456 0.259223
$$336$$ 2.74456 0.149728
$$337$$ −7.25544 −0.395229 −0.197614 0.980280i $$-0.563319\pi$$
−0.197614 + 0.980280i $$0.563319\pi$$
$$338$$ 9.51087 0.517323
$$339$$ 34.7446 1.88707
$$340$$ −5.37228 −0.291353
$$341$$ 8.86141 0.479872
$$342$$ −2.00000 −0.108148
$$343$$ −16.6277 −0.897812
$$344$$ 7.37228 0.397487
$$345$$ 13.4891 0.726230
$$346$$ 22.8614 1.22904
$$347$$ −22.9783 −1.23354 −0.616769 0.787145i $$-0.711559\pi$$
−0.616769 + 0.787145i $$0.711559\pi$$
$$348$$ 16.2337 0.870217
$$349$$ −22.0000 −1.17763 −0.588817 0.808267i $$-0.700406\pi$$
−0.588817 + 0.808267i $$0.700406\pi$$
$$350$$ 1.37228 0.0733515
$$351$$ 18.9783 1.01298
$$352$$ −3.37228 −0.179743
$$353$$ −22.8614 −1.21679 −0.608395 0.793634i $$-0.708186\pi$$
−0.608395 + 0.793634i $$0.708186\pi$$
$$354$$ 25.4891 1.35473
$$355$$ −6.74456 −0.357964
$$356$$ 10.0000 0.529999
$$357$$ −14.7446 −0.780365
$$358$$ −26.2337 −1.38649
$$359$$ −30.9783 −1.63497 −0.817485 0.575950i $$-0.804632\pi$$
−0.817485 + 0.575950i $$0.804632\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ −15.0000 −0.789474
$$362$$ 7.48913 0.393620
$$363$$ 0.744563 0.0390794
$$364$$ −6.51087 −0.341263
$$365$$ 8.74456 0.457711
$$366$$ −10.7446 −0.561627
$$367$$ 3.88316 0.202699 0.101350 0.994851i $$-0.467684\pi$$
0.101350 + 0.994851i $$0.467684\pi$$
$$368$$ 6.74456 0.351585
$$369$$ 5.37228 0.279670
$$370$$ 1.00000 0.0519875
$$371$$ 1.88316 0.0977686
$$372$$ −5.25544 −0.272482
$$373$$ −31.4891 −1.63045 −0.815223 0.579148i $$-0.803385\pi$$
−0.815223 + 0.579148i $$0.803385\pi$$
$$374$$ 18.1168 0.936800
$$375$$ 2.00000 0.103280
$$376$$ −8.74456 −0.450966
$$377$$ −38.5109 −1.98341
$$378$$ −5.48913 −0.282330
$$379$$ −8.00000 −0.410932 −0.205466 0.978664i $$-0.565871\pi$$
−0.205466 + 0.978664i $$0.565871\pi$$
$$380$$ −2.00000 −0.102598
$$381$$ −33.4891 −1.71570
$$382$$ 18.6277 0.953077
$$383$$ −13.4891 −0.689262 −0.344631 0.938738i $$-0.611996\pi$$
−0.344631 + 0.938738i $$0.611996\pi$$
$$384$$ 2.00000 0.102062
$$385$$ −4.62772 −0.235850
$$386$$ 2.00000 0.101797
$$387$$ 7.37228 0.374754
$$388$$ −0.116844 −0.00593185
$$389$$ 14.8614 0.753503 0.376752 0.926314i $$-0.377041\pi$$
0.376752 + 0.926314i $$0.377041\pi$$
$$390$$ −9.48913 −0.480501
$$391$$ −36.2337 −1.83242
$$392$$ −5.11684 −0.258440
$$393$$ −6.51087 −0.328430
$$394$$ 27.4891 1.38488
$$395$$ −4.74456 −0.238725
$$396$$ −3.37228 −0.169464
$$397$$ 24.9783 1.25362 0.626811 0.779171i $$-0.284360\pi$$
0.626811 + 0.779171i $$0.284360\pi$$
$$398$$ −11.2554 −0.564184
$$399$$ −5.48913 −0.274800
$$400$$ 1.00000 0.0500000
$$401$$ 10.0000 0.499376 0.249688 0.968326i $$-0.419672\pi$$
0.249688 + 0.968326i $$0.419672\pi$$
$$402$$ 9.48913 0.473275
$$403$$ 12.4674 0.621044
$$404$$ −11.4891 −0.571605
$$405$$ −11.0000 −0.546594
$$406$$ 11.1386 0.552799
$$407$$ −3.37228 −0.167158
$$408$$ −10.7446 −0.531935
$$409$$ 6.23369 0.308236 0.154118 0.988052i $$-0.450746\pi$$
0.154118 + 0.988052i $$0.450746\pi$$
$$410$$ 5.37228 0.265318
$$411$$ 33.4891 1.65190
$$412$$ −9.48913 −0.467496
$$413$$ 17.4891 0.860584
$$414$$ 6.74456 0.331477
$$415$$ −0.744563 −0.0365491
$$416$$ −4.74456 −0.232621
$$417$$ −3.76631 −0.184437
$$418$$ 6.74456 0.329887
$$419$$ 13.4891 0.658987 0.329493 0.944158i $$-0.393122\pi$$
0.329493 + 0.944158i $$0.393122\pi$$
$$420$$ 2.74456 0.133921
$$421$$ −7.48913 −0.364998 −0.182499 0.983206i $$-0.558419\pi$$
−0.182499 + 0.983206i $$0.558419\pi$$
$$422$$ −14.1168 −0.687197
$$423$$ −8.74456 −0.425175
$$424$$ 1.37228 0.0666439
$$425$$ −5.37228 −0.260594
$$426$$ −13.4891 −0.653550
$$427$$ −7.37228 −0.356770
$$428$$ −3.48913 −0.168653
$$429$$ 32.0000 1.54497
$$430$$ 7.37228 0.355523
$$431$$ −1.37228 −0.0661005 −0.0330502 0.999454i $$-0.510522\pi$$
−0.0330502 + 0.999454i $$0.510522\pi$$
$$432$$ −4.00000 −0.192450
$$433$$ 12.9783 0.623695 0.311847 0.950132i $$-0.399052\pi$$
0.311847 + 0.950132i $$0.399052\pi$$
$$434$$ −3.60597 −0.173092
$$435$$ 16.2337 0.778346
$$436$$ 0.116844 0.00559581
$$437$$ −13.4891 −0.645272
$$438$$ 17.4891 0.835663
$$439$$ −13.3723 −0.638224 −0.319112 0.947717i $$-0.603385\pi$$
−0.319112 + 0.947717i $$0.603385\pi$$
$$440$$ −3.37228 −0.160767
$$441$$ −5.11684 −0.243659
$$442$$ 25.4891 1.21239
$$443$$ −16.9783 −0.806661 −0.403331 0.915054i $$-0.632148\pi$$
−0.403331 + 0.915054i $$0.632148\pi$$
$$444$$ 2.00000 0.0949158
$$445$$ 10.0000 0.474045
$$446$$ −1.37228 −0.0649794
$$447$$ −22.9783 −1.08683
$$448$$ 1.37228 0.0648342
$$449$$ 18.0000 0.849473 0.424736 0.905317i $$-0.360367\pi$$
0.424736 + 0.905317i $$0.360367\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ −18.1168 −0.853089
$$452$$ 17.3723 0.817123
$$453$$ −40.0000 −1.87936
$$454$$ −11.3723 −0.533728
$$455$$ −6.51087 −0.305235
$$456$$ −4.00000 −0.187317
$$457$$ 18.8614 0.882299 0.441150 0.897434i $$-0.354571\pi$$
0.441150 + 0.897434i $$0.354571\pi$$
$$458$$ −10.0000 −0.467269
$$459$$ 21.4891 1.00303
$$460$$ 6.74456 0.314467
$$461$$ −39.0951 −1.82084 −0.910420 0.413685i $$-0.864241\pi$$
−0.910420 + 0.413685i $$0.864241\pi$$
$$462$$ −9.25544 −0.430602
$$463$$ −5.48913 −0.255101 −0.127551 0.991832i $$-0.540712\pi$$
−0.127551 + 0.991832i $$0.540712\pi$$
$$464$$ 8.11684 0.376815
$$465$$ −5.25544 −0.243715
$$466$$ 6.23369 0.288770
$$467$$ 30.3505 1.40446 0.702228 0.711953i $$-0.252189\pi$$
0.702228 + 0.711953i $$0.252189\pi$$
$$468$$ −4.74456 −0.219317
$$469$$ 6.51087 0.300644
$$470$$ −8.74456 −0.403357
$$471$$ 34.7446 1.60094
$$472$$ 12.7446 0.586616
$$473$$ −24.8614 −1.14313
$$474$$ −9.48913 −0.435850
$$475$$ −2.00000 −0.0917663
$$476$$ −7.37228 −0.337908
$$477$$ 1.37228 0.0628324
$$478$$ 17.6060 0.805278
$$479$$ 3.25544 0.148745 0.0743724 0.997231i $$-0.476305\pi$$
0.0743724 + 0.997231i $$0.476305\pi$$
$$480$$ 2.00000 0.0912871
$$481$$ −4.74456 −0.216333
$$482$$ −10.2337 −0.466132
$$483$$ 18.5109 0.842274
$$484$$ 0.372281 0.0169219
$$485$$ −0.116844 −0.00530561
$$486$$ −10.0000 −0.453609
$$487$$ −37.7228 −1.70938 −0.854692 0.519136i $$-0.826254\pi$$
−0.854692 + 0.519136i $$0.826254\pi$$
$$488$$ −5.37228 −0.243192
$$489$$ 38.7446 1.75209
$$490$$ −5.11684 −0.231155
$$491$$ 14.9783 0.675959 0.337979 0.941153i $$-0.390257\pi$$
0.337979 + 0.941153i $$0.390257\pi$$
$$492$$ 10.7446 0.484402
$$493$$ −43.6060 −1.96391
$$494$$ 9.48913 0.426936
$$495$$ −3.37228 −0.151573
$$496$$ −2.62772 −0.117988
$$497$$ −9.25544 −0.415163
$$498$$ −1.48913 −0.0667293
$$499$$ −24.9783 −1.11818 −0.559090 0.829107i $$-0.688849\pi$$
−0.559090 + 0.829107i $$0.688849\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 2.97825 0.133058
$$502$$ 11.4891 0.512785
$$503$$ 29.4891 1.31486 0.657428 0.753518i $$-0.271645\pi$$
0.657428 + 0.753518i $$0.271645\pi$$
$$504$$ 1.37228 0.0611263
$$505$$ −11.4891 −0.511259
$$506$$ −22.7446 −1.01112
$$507$$ 19.0217 0.844786
$$508$$ −16.7446 −0.742920
$$509$$ −11.2554 −0.498888 −0.249444 0.968389i $$-0.580248\pi$$
−0.249444 + 0.968389i $$0.580248\pi$$
$$510$$ −10.7446 −0.475777
$$511$$ 12.0000 0.530849
$$512$$ 1.00000 0.0441942
$$513$$ 8.00000 0.353209
$$514$$ −20.9783 −0.925311
$$515$$ −9.48913 −0.418141
$$516$$ 14.7446 0.649093
$$517$$ 29.4891 1.29693
$$518$$ 1.37228 0.0602946
$$519$$ 45.7228 2.00701
$$520$$ −4.74456 −0.208063
$$521$$ 27.0951 1.18706 0.593529 0.804813i $$-0.297734\pi$$
0.593529 + 0.804813i $$0.297734\pi$$
$$522$$ 8.11684 0.355265
$$523$$ 28.0000 1.22435 0.612177 0.790721i $$-0.290294\pi$$
0.612177 + 0.790721i $$0.290294\pi$$
$$524$$ −3.25544 −0.142214
$$525$$ 2.74456 0.119783
$$526$$ 0.116844 0.00509464
$$527$$ 14.1168 0.614939
$$528$$ −6.74456 −0.293519
$$529$$ 22.4891 0.977788
$$530$$ 1.37228 0.0596081
$$531$$ 12.7446 0.553067
$$532$$ −2.74456 −0.118992
$$533$$ −25.4891 −1.10406
$$534$$ 20.0000 0.865485
$$535$$ −3.48913 −0.150848
$$536$$ 4.74456 0.204934
$$537$$ −52.4674 −2.26413
$$538$$ 10.0000 0.431131
$$539$$ 17.2554 0.743244
$$540$$ −4.00000 −0.172133
$$541$$ 30.0000 1.28980 0.644900 0.764267i $$-0.276899\pi$$
0.644900 + 0.764267i $$0.276899\pi$$
$$542$$ −6.74456 −0.289704
$$543$$ 14.9783 0.642778
$$544$$ −5.37228 −0.230335
$$545$$ 0.116844 0.00500505
$$546$$ −13.0217 −0.557279
$$547$$ 39.3723 1.68344 0.841719 0.539916i $$-0.181544\pi$$
0.841719 + 0.539916i $$0.181544\pi$$
$$548$$ 16.7446 0.715292
$$549$$ −5.37228 −0.229283
$$550$$ −3.37228 −0.143795
$$551$$ −16.2337 −0.691578
$$552$$ 13.4891 0.574135
$$553$$ −6.51087 −0.276871
$$554$$ 0.744563 0.0316334
$$555$$ 2.00000 0.0848953
$$556$$ −1.88316 −0.0798636
$$557$$ 8.74456 0.370519 0.185260 0.982690i $$-0.440687\pi$$
0.185260 + 0.982690i $$0.440687\pi$$
$$558$$ −2.62772 −0.111240
$$559$$ −34.9783 −1.47942
$$560$$ 1.37228 0.0579895
$$561$$ 36.2337 1.52979
$$562$$ 24.7446 1.04379
$$563$$ 33.0951 1.39479 0.697396 0.716686i $$-0.254342\pi$$
0.697396 + 0.716686i $$0.254342\pi$$
$$564$$ −17.4891 −0.736425
$$565$$ 17.3723 0.730857
$$566$$ 5.48913 0.230725
$$567$$ −15.0951 −0.633934
$$568$$ −6.74456 −0.282996
$$569$$ −32.9783 −1.38252 −0.691260 0.722606i $$-0.742944\pi$$
−0.691260 + 0.722606i $$0.742944\pi$$
$$570$$ −4.00000 −0.167542
$$571$$ −27.6060 −1.15527 −0.577637 0.816294i $$-0.696025\pi$$
−0.577637 + 0.816294i $$0.696025\pi$$
$$572$$ 16.0000 0.668994
$$573$$ 37.2554 1.55637
$$574$$ 7.37228 0.307713
$$575$$ 6.74456 0.281268
$$576$$ 1.00000 0.0416667
$$577$$ −7.48913 −0.311776 −0.155888 0.987775i $$-0.549824\pi$$
−0.155888 + 0.987775i $$0.549824\pi$$
$$578$$ 11.8614 0.493369
$$579$$ 4.00000 0.166234
$$580$$ 8.11684 0.337034
$$581$$ −1.02175 −0.0423893
$$582$$ −0.233688 −0.00968668
$$583$$ −4.62772 −0.191661
$$584$$ 8.74456 0.361853
$$585$$ −4.74456 −0.196164
$$586$$ −18.8614 −0.779158
$$587$$ 47.8397 1.97455 0.987277 0.159010i $$-0.0508302\pi$$
0.987277 + 0.159010i $$0.0508302\pi$$
$$588$$ −10.2337 −0.422030
$$589$$ 5.25544 0.216547
$$590$$ 12.7446 0.524685
$$591$$ 54.9783 2.26150
$$592$$ 1.00000 0.0410997
$$593$$ −10.2337 −0.420247 −0.210124 0.977675i $$-0.567387\pi$$
−0.210124 + 0.977675i $$0.567387\pi$$
$$594$$ 13.4891 0.553466
$$595$$ −7.37228 −0.302234
$$596$$ −11.4891 −0.470613
$$597$$ −22.5109 −0.921309
$$598$$ −32.0000 −1.30858
$$599$$ 17.4891 0.714586 0.357293 0.933992i $$-0.383700\pi$$
0.357293 + 0.933992i $$0.383700\pi$$
$$600$$ 2.00000 0.0816497
$$601$$ −5.37228 −0.219140 −0.109570 0.993979i $$-0.534947\pi$$
−0.109570 + 0.993979i $$0.534947\pi$$
$$602$$ 10.1168 0.412332
$$603$$ 4.74456 0.193214
$$604$$ −20.0000 −0.813788
$$605$$ 0.372281 0.0151354
$$606$$ −22.9783 −0.933428
$$607$$ 17.7228 0.719347 0.359673 0.933078i $$-0.382888\pi$$
0.359673 + 0.933078i $$0.382888\pi$$
$$608$$ −2.00000 −0.0811107
$$609$$ 22.2772 0.902717
$$610$$ −5.37228 −0.217517
$$611$$ 41.4891 1.67847
$$612$$ −5.37228 −0.217162
$$613$$ 43.0951 1.74059 0.870297 0.492527i $$-0.163927\pi$$
0.870297 + 0.492527i $$0.163927\pi$$
$$614$$ −23.4891 −0.947944
$$615$$ 10.7446 0.433263
$$616$$ −4.62772 −0.186456
$$617$$ −31.7228 −1.27711 −0.638556 0.769575i $$-0.720468\pi$$
−0.638556 + 0.769575i $$0.720468\pi$$
$$618$$ −18.9783 −0.763417
$$619$$ 30.1168 1.21050 0.605249 0.796036i $$-0.293074\pi$$
0.605249 + 0.796036i $$0.293074\pi$$
$$620$$ −2.62772 −0.105532
$$621$$ −26.9783 −1.08260
$$622$$ −1.37228 −0.0550235
$$623$$ 13.7228 0.549793
$$624$$ −9.48913 −0.379869
$$625$$ 1.00000 0.0400000
$$626$$ −3.48913 −0.139453
$$627$$ 13.4891 0.538704
$$628$$ 17.3723 0.693229
$$629$$ −5.37228 −0.214207
$$630$$ 1.37228 0.0546730
$$631$$ −19.0951 −0.760164 −0.380082 0.924953i $$-0.624104\pi$$
−0.380082 + 0.924953i $$0.624104\pi$$
$$632$$ −4.74456 −0.188729
$$633$$ −28.2337 −1.12219
$$634$$ 25.3723 1.00766
$$635$$ −16.7446 −0.664488
$$636$$ 2.74456 0.108829
$$637$$ 24.2772 0.961897
$$638$$ −27.3723 −1.08368
$$639$$ −6.74456 −0.266811
$$640$$ 1.00000 0.0395285
$$641$$ −49.6060 −1.95932 −0.979659 0.200670i $$-0.935688\pi$$
−0.979659 + 0.200670i $$0.935688\pi$$
$$642$$ −6.97825 −0.275410
$$643$$ 3.37228 0.132990 0.0664949 0.997787i $$-0.478818\pi$$
0.0664949 + 0.997787i $$0.478818\pi$$
$$644$$ 9.25544 0.364715
$$645$$ 14.7446 0.580567
$$646$$ 10.7446 0.422739
$$647$$ −13.4891 −0.530312 −0.265156 0.964205i $$-0.585423\pi$$
−0.265156 + 0.964205i $$0.585423\pi$$
$$648$$ −11.0000 −0.432121
$$649$$ −42.9783 −1.68704
$$650$$ −4.74456 −0.186097
$$651$$ −7.21194 −0.282658
$$652$$ 19.3723 0.758677
$$653$$ 0.510875 0.0199921 0.00999604 0.999950i $$-0.496818\pi$$
0.00999604 + 0.999950i $$0.496818\pi$$
$$654$$ 0.233688 0.00913792
$$655$$ −3.25544 −0.127200
$$656$$ 5.37228 0.209752
$$657$$ 8.74456 0.341158
$$658$$ −12.0000 −0.467809
$$659$$ 12.0000 0.467454 0.233727 0.972302i $$-0.424908\pi$$
0.233727 + 0.972302i $$0.424908\pi$$
$$660$$ −6.74456 −0.262532
$$661$$ 4.35053 0.169216 0.0846080 0.996414i $$-0.473036\pi$$
0.0846080 + 0.996414i $$0.473036\pi$$
$$662$$ 23.4891 0.912931
$$663$$ 50.9783 1.97983
$$664$$ −0.744563 −0.0288946
$$665$$ −2.74456 −0.106430
$$666$$ 1.00000 0.0387492
$$667$$ 54.7446 2.11972
$$668$$ 1.48913 0.0576160
$$669$$ −2.74456 −0.106111
$$670$$ 4.74456 0.183298
$$671$$ 18.1168 0.699393
$$672$$ 2.74456 0.105874
$$673$$ −8.51087 −0.328070 −0.164035 0.986455i $$-0.552451\pi$$
−0.164035 + 0.986455i $$0.552451\pi$$
$$674$$ −7.25544 −0.279469
$$675$$ −4.00000 −0.153960
$$676$$ 9.51087 0.365803
$$677$$ −42.0000 −1.61419 −0.807096 0.590421i $$-0.798962\pi$$
−0.807096 + 0.590421i $$0.798962\pi$$
$$678$$ 34.7446 1.33436
$$679$$ −0.160343 −0.00615339
$$680$$ −5.37228 −0.206018
$$681$$ −22.7446 −0.871574
$$682$$ 8.86141 0.339321
$$683$$ 8.62772 0.330130 0.165065 0.986283i $$-0.447217\pi$$
0.165065 + 0.986283i $$0.447217\pi$$
$$684$$ −2.00000 −0.0764719
$$685$$ 16.7446 0.639777
$$686$$ −16.6277 −0.634849
$$687$$ −20.0000 −0.763048
$$688$$ 7.37228 0.281066
$$689$$ −6.51087 −0.248045
$$690$$ 13.4891 0.513522
$$691$$ 22.3505 0.850254 0.425127 0.905134i $$-0.360229\pi$$
0.425127 + 0.905134i $$0.360229\pi$$
$$692$$ 22.8614 0.869060
$$693$$ −4.62772 −0.175792
$$694$$ −22.9783 −0.872242
$$695$$ −1.88316 −0.0714322
$$696$$ 16.2337 0.615336
$$697$$ −28.8614 −1.09320
$$698$$ −22.0000 −0.832712
$$699$$ 12.4674 0.471559
$$700$$ 1.37228 0.0518674
$$701$$ −26.4674 −0.999659 −0.499829 0.866124i $$-0.666604\pi$$
−0.499829 + 0.866124i $$0.666604\pi$$
$$702$$ 18.9783 0.716288
$$703$$ −2.00000 −0.0754314
$$704$$ −3.37228 −0.127098
$$705$$ −17.4891 −0.658679
$$706$$ −22.8614 −0.860400
$$707$$ −15.7663 −0.592953
$$708$$ 25.4891 0.957940
$$709$$ 40.1168 1.50662 0.753310 0.657666i $$-0.228456\pi$$
0.753310 + 0.657666i $$0.228456\pi$$
$$710$$ −6.74456 −0.253119
$$711$$ −4.74456 −0.177935
$$712$$ 10.0000 0.374766
$$713$$ −17.7228 −0.663725
$$714$$ −14.7446 −0.551801
$$715$$ 16.0000 0.598366
$$716$$ −26.2337 −0.980399
$$717$$ 35.2119 1.31501
$$718$$ −30.9783 −1.15610
$$719$$ −34.7446 −1.29575 −0.647877 0.761745i $$-0.724343\pi$$
−0.647877 + 0.761745i $$0.724343\pi$$
$$720$$ 1.00000 0.0372678
$$721$$ −13.0217 −0.484955
$$722$$ −15.0000 −0.558242
$$723$$ −20.4674 −0.761190
$$724$$ 7.48913 0.278331
$$725$$ 8.11684 0.301452
$$726$$ 0.744563 0.0276333
$$727$$ −48.0000 −1.78022 −0.890111 0.455744i $$-0.849373\pi$$
−0.890111 + 0.455744i $$0.849373\pi$$
$$728$$ −6.51087 −0.241309
$$729$$ 13.0000 0.481481
$$730$$ 8.74456 0.323651
$$731$$ −39.6060 −1.46488
$$732$$ −10.7446 −0.397130
$$733$$ −29.3723 −1.08489 −0.542445 0.840091i $$-0.682501\pi$$
−0.542445 + 0.840091i $$0.682501\pi$$
$$734$$ 3.88316 0.143330
$$735$$ −10.2337 −0.377475
$$736$$ 6.74456 0.248608
$$737$$ −16.0000 −0.589368
$$738$$ 5.37228 0.197756
$$739$$ −36.8614 −1.35597 −0.677984 0.735076i $$-0.737146\pi$$
−0.677984 + 0.735076i $$0.737146\pi$$
$$740$$ 1.00000 0.0367607
$$741$$ 18.9783 0.697183
$$742$$ 1.88316 0.0691328
$$743$$ −5.37228 −0.197090 −0.0985449 0.995133i $$-0.531419\pi$$
−0.0985449 + 0.995133i $$0.531419\pi$$
$$744$$ −5.25544 −0.192674
$$745$$ −11.4891 −0.420929
$$746$$ −31.4891 −1.15290
$$747$$ −0.744563 −0.0272421
$$748$$ 18.1168 0.662417
$$749$$ −4.78806 −0.174952
$$750$$ 2.00000 0.0730297
$$751$$ 10.7446 0.392075 0.196037 0.980596i $$-0.437193\pi$$
0.196037 + 0.980596i $$0.437193\pi$$
$$752$$ −8.74456 −0.318881
$$753$$ 22.9783 0.837374
$$754$$ −38.5109 −1.40248
$$755$$ −20.0000 −0.727875
$$756$$ −5.48913 −0.199638
$$757$$ 43.9565 1.59763 0.798813 0.601579i $$-0.205462\pi$$
0.798813 + 0.601579i $$0.205462\pi$$
$$758$$ −8.00000 −0.290573
$$759$$ −45.4891 −1.65115
$$760$$ −2.00000 −0.0725476
$$761$$ 5.37228 0.194745 0.0973725 0.995248i $$-0.468956\pi$$
0.0973725 + 0.995248i $$0.468956\pi$$
$$762$$ −33.4891 −1.21318
$$763$$ 0.160343 0.00580480
$$764$$ 18.6277 0.673927
$$765$$ −5.37228 −0.194235
$$766$$ −13.4891 −0.487382
$$767$$ −60.4674 −2.18335
$$768$$ 2.00000 0.0721688
$$769$$ 46.2337 1.66723 0.833615 0.552346i $$-0.186267\pi$$
0.833615 + 0.552346i $$0.186267\pi$$
$$770$$ −4.62772 −0.166771
$$771$$ −41.9565 −1.51103
$$772$$ 2.00000 0.0719816
$$773$$ −24.3505 −0.875828 −0.437914 0.899017i $$-0.644283\pi$$
−0.437914 + 0.899017i $$0.644283\pi$$
$$774$$ 7.37228 0.264991
$$775$$ −2.62772 −0.0943904
$$776$$ −0.116844 −0.00419445
$$777$$ 2.74456 0.0984606
$$778$$ 14.8614 0.532807
$$779$$ −10.7446 −0.384964
$$780$$ −9.48913 −0.339765
$$781$$ 22.7446 0.813864
$$782$$ −36.2337 −1.29571
$$783$$ −32.4674 −1.16029
$$784$$ −5.11684 −0.182744
$$785$$ 17.3723 0.620043
$$786$$ −6.51087 −0.232235
$$787$$ −22.0000 −0.784215 −0.392108 0.919919i $$-0.628254\pi$$
−0.392108 + 0.919919i $$0.628254\pi$$
$$788$$ 27.4891 0.979260
$$789$$ 0.233688 0.00831951
$$790$$ −4.74456 −0.168804
$$791$$ 23.8397 0.847641
$$792$$ −3.37228 −0.119829
$$793$$ 25.4891 0.905145
$$794$$ 24.9783 0.886445
$$795$$ 2.74456 0.0973396
$$796$$ −11.2554 −0.398938
$$797$$ −27.4891 −0.973715 −0.486857 0.873481i $$-0.661857\pi$$
−0.486857 + 0.873481i $$0.661857\pi$$
$$798$$ −5.48913 −0.194313
$$799$$ 46.9783 1.66197
$$800$$ 1.00000 0.0353553
$$801$$ 10.0000 0.353333
$$802$$ 10.0000 0.353112
$$803$$ −29.4891 −1.04065
$$804$$ 9.48913 0.334656
$$805$$ 9.25544 0.326211
$$806$$ 12.4674 0.439145
$$807$$ 20.0000 0.704033
$$808$$ −11.4891 −0.404186
$$809$$ −51.4891 −1.81026 −0.905131 0.425134i $$-0.860227\pi$$
−0.905131 + 0.425134i $$0.860227\pi$$
$$810$$ −11.0000 −0.386501
$$811$$ 49.4891 1.73780 0.868899 0.494989i $$-0.164828\pi$$
0.868899 + 0.494989i $$0.164828\pi$$
$$812$$ 11.1386 0.390888
$$813$$ −13.4891 −0.473084
$$814$$ −3.37228 −0.118198
$$815$$ 19.3723 0.678581
$$816$$ −10.7446 −0.376135
$$817$$ −14.7446 −0.515847
$$818$$ 6.23369 0.217956
$$819$$ −6.51087 −0.227508
$$820$$ 5.37228 0.187608
$$821$$ −32.7446 −1.14279 −0.571397 0.820674i $$-0.693598\pi$$
−0.571397 + 0.820674i $$0.693598\pi$$
$$822$$ 33.4891 1.16807
$$823$$ −39.7228 −1.38465 −0.692325 0.721586i $$-0.743414\pi$$
−0.692325 + 0.721586i $$0.743414\pi$$
$$824$$ −9.48913 −0.330569
$$825$$ −6.74456 −0.234816
$$826$$ 17.4891 0.608524
$$827$$ −54.3505 −1.88995 −0.944977 0.327138i $$-0.893916\pi$$
−0.944977 + 0.327138i $$0.893916\pi$$
$$828$$ 6.74456 0.234390
$$829$$ −40.3505 −1.40143 −0.700716 0.713440i $$-0.747136\pi$$
−0.700716 + 0.713440i $$0.747136\pi$$
$$830$$ −0.744563 −0.0258441
$$831$$ 1.48913 0.0516572
$$832$$ −4.74456 −0.164488
$$833$$ 27.4891 0.952442
$$834$$ −3.76631 −0.130417
$$835$$ 1.48913 0.0515333
$$836$$ 6.74456 0.233266
$$837$$ 10.5109 0.363309
$$838$$ 13.4891 0.465974
$$839$$ −29.4891 −1.01808 −0.509039 0.860744i $$-0.669999\pi$$
−0.509039 + 0.860744i $$0.669999\pi$$
$$840$$ 2.74456 0.0946964
$$841$$ 36.8832 1.27183
$$842$$ −7.48913 −0.258092
$$843$$ 49.4891 1.70450
$$844$$ −14.1168 −0.485922
$$845$$ 9.51087 0.327184
$$846$$ −8.74456 −0.300644
$$847$$ 0.510875 0.0175539
$$848$$ 1.37228 0.0471243
$$849$$ 10.9783 0.376773
$$850$$ −5.37228 −0.184268
$$851$$ 6.74456 0.231201
$$852$$ −13.4891 −0.462130
$$853$$ 35.4891 1.21512 0.607562 0.794272i $$-0.292148\pi$$
0.607562 + 0.794272i $$0.292148\pi$$
$$854$$ −7.37228 −0.252274
$$855$$ −2.00000 −0.0683986
$$856$$ −3.48913 −0.119256
$$857$$ −29.1386 −0.995355 −0.497678 0.867362i $$-0.665814\pi$$
−0.497678 + 0.867362i $$0.665814\pi$$
$$858$$ 32.0000 1.09246
$$859$$ −19.7228 −0.672934 −0.336467 0.941695i $$-0.609232\pi$$
−0.336467 + 0.941695i $$0.609232\pi$$
$$860$$ 7.37228 0.251393
$$861$$ 14.7446 0.502493
$$862$$ −1.37228 −0.0467401
$$863$$ 53.8397 1.83272 0.916362 0.400352i $$-0.131112\pi$$
0.916362 + 0.400352i $$0.131112\pi$$
$$864$$ −4.00000 −0.136083
$$865$$ 22.8614 0.777311
$$866$$ 12.9783 0.441019
$$867$$ 23.7228 0.805669
$$868$$ −3.60597 −0.122395
$$869$$ 16.0000 0.542763
$$870$$ 16.2337 0.550374
$$871$$ −22.5109 −0.762752
$$872$$ 0.116844 0.00395684
$$873$$ −0.116844 −0.00395457
$$874$$ −13.4891 −0.456276
$$875$$ 1.37228 0.0463916
$$876$$ 17.4891 0.590903
$$877$$ 49.3723 1.66718 0.833592 0.552381i $$-0.186281\pi$$
0.833592 + 0.552381i $$0.186281\pi$$
$$878$$ −13.3723 −0.451293
$$879$$ −37.7228 −1.27236
$$880$$ −3.37228 −0.113680
$$881$$ 1.37228 0.0462333 0.0231167 0.999733i $$-0.492641\pi$$
0.0231167 + 0.999733i $$0.492641\pi$$
$$882$$ −5.11684 −0.172293
$$883$$ −11.3723 −0.382708 −0.191354 0.981521i $$-0.561288\pi$$
−0.191354 + 0.981521i $$0.561288\pi$$
$$884$$ 25.4891 0.857292
$$885$$ 25.4891 0.856808
$$886$$ −16.9783 −0.570395
$$887$$ −25.3723 −0.851918 −0.425959 0.904743i $$-0.640063\pi$$
−0.425959 + 0.904743i $$0.640063\pi$$
$$888$$ 2.00000 0.0671156
$$889$$ −22.9783 −0.770666
$$890$$ 10.0000 0.335201
$$891$$ 37.0951 1.24273
$$892$$ −1.37228 −0.0459474
$$893$$ 17.4891 0.585251
$$894$$ −22.9783 −0.768508
$$895$$ −26.2337 −0.876895
$$896$$ 1.37228 0.0458447
$$897$$ −64.0000 −2.13690
$$898$$ 18.0000 0.600668
$$899$$ −21.3288 −0.711355
$$900$$ 1.00000 0.0333333
$$901$$ −7.37228 −0.245606
$$902$$ −18.1168 −0.603225
$$903$$ 20.2337 0.673335
$$904$$ 17.3723 0.577793
$$905$$ 7.48913 0.248947
$$906$$ −40.0000 −1.32891
$$907$$ 40.4674 1.34370 0.671849 0.740689i $$-0.265501\pi$$
0.671849 + 0.740689i $$0.265501\pi$$
$$908$$ −11.3723 −0.377402
$$909$$ −11.4891 −0.381070
$$910$$ −6.51087 −0.215833
$$911$$ 17.7663 0.588624 0.294312 0.955709i $$-0.404909\pi$$
0.294312 + 0.955709i $$0.404909\pi$$
$$912$$ −4.00000 −0.132453
$$913$$ 2.51087 0.0830978
$$914$$ 18.8614 0.623880
$$915$$ −10.7446 −0.355204
$$916$$ −10.0000 −0.330409
$$917$$ −4.46738 −0.147526
$$918$$ 21.4891 0.709247
$$919$$ 7.72281 0.254752 0.127376 0.991854i $$-0.459344\pi$$
0.127376 + 0.991854i $$0.459344\pi$$
$$920$$ 6.74456 0.222362
$$921$$ −46.9783 −1.54799
$$922$$ −39.0951 −1.28753
$$923$$ 32.0000 1.05329
$$924$$ −9.25544 −0.304482
$$925$$ 1.00000 0.0328798
$$926$$ −5.48913 −0.180384
$$927$$ −9.48913 −0.311664
$$928$$ 8.11684 0.266448
$$929$$ 31.0951 1.02020 0.510098 0.860116i $$-0.329609\pi$$
0.510098 + 0.860116i $$0.329609\pi$$
$$930$$ −5.25544 −0.172333
$$931$$ 10.2337 0.335396
$$932$$ 6.23369 0.204191
$$933$$ −2.74456 −0.0898529
$$934$$ 30.3505 0.993100
$$935$$ 18.1168 0.592484
$$936$$ −4.74456 −0.155081
$$937$$ 36.9783 1.20803 0.604013 0.796974i $$-0.293567\pi$$
0.604013 + 0.796974i $$0.293567\pi$$
$$938$$ 6.51087 0.212588
$$939$$ −6.97825 −0.227727
$$940$$ −8.74456 −0.285216
$$941$$ 18.0000 0.586783 0.293392 0.955992i $$-0.405216\pi$$
0.293392 + 0.955992i $$0.405216\pi$$
$$942$$ 34.7446 1.13204
$$943$$ 36.2337 1.17993
$$944$$ 12.7446 0.414800
$$945$$ −5.48913 −0.178561
$$946$$ −24.8614 −0.808314
$$947$$ 37.0951 1.20543 0.602714 0.797957i $$-0.294086\pi$$
0.602714 + 0.797957i $$0.294086\pi$$
$$948$$ −9.48913 −0.308192
$$949$$ −41.4891 −1.34679
$$950$$ −2.00000 −0.0648886
$$951$$ 50.7446 1.64551
$$952$$ −7.37228 −0.238937
$$953$$ 46.2337 1.49766 0.748828 0.662764i $$-0.230617\pi$$
0.748828 + 0.662764i $$0.230617\pi$$
$$954$$ 1.37228 0.0444292
$$955$$ 18.6277 0.602779
$$956$$ 17.6060 0.569418
$$957$$ −54.7446 −1.76964
$$958$$ 3.25544 0.105178
$$959$$ 22.9783 0.742006
$$960$$ 2.00000 0.0645497
$$961$$ −24.0951 −0.777261
$$962$$ −4.74456 −0.152971
$$963$$ −3.48913 −0.112435
$$964$$ −10.2337 −0.329605
$$965$$ 2.00000 0.0643823
$$966$$ 18.5109 0.595578
$$967$$ −28.0000 −0.900419 −0.450210 0.892923i $$-0.648651\pi$$
−0.450210 + 0.892923i $$0.648651\pi$$
$$968$$ 0.372281 0.0119656
$$969$$ 21.4891 0.690330
$$970$$ −0.116844 −0.00375163
$$971$$ −19.6060 −0.629185 −0.314593 0.949227i $$-0.601868\pi$$
−0.314593 + 0.949227i $$0.601868\pi$$
$$972$$ −10.0000 −0.320750
$$973$$ −2.58422 −0.0828463
$$974$$ −37.7228 −1.20872
$$975$$ −9.48913 −0.303895
$$976$$ −5.37228 −0.171963
$$977$$ −11.8832 −0.380176 −0.190088 0.981767i $$-0.560877\pi$$
−0.190088 + 0.981767i $$0.560877\pi$$
$$978$$ 38.7446 1.23891
$$979$$ −33.7228 −1.07779
$$980$$ −5.11684 −0.163452
$$981$$ 0.116844 0.00373054
$$982$$ 14.9783 0.477975
$$983$$ 30.8614 0.984326 0.492163 0.870503i $$-0.336206\pi$$
0.492163 + 0.870503i $$0.336206\pi$$
$$984$$ 10.7446 0.342524
$$985$$ 27.4891 0.875876
$$986$$ −43.6060 −1.38870
$$987$$ −24.0000 −0.763928
$$988$$ 9.48913 0.301889
$$989$$ 49.7228 1.58109
$$990$$ −3.37228 −0.107178
$$991$$ 25.6060 0.813400 0.406700 0.913562i $$-0.366679\pi$$
0.406700 + 0.913562i $$0.366679\pi$$
$$992$$ −2.62772 −0.0834302
$$993$$ 46.9783 1.49081
$$994$$ −9.25544 −0.293565
$$995$$ −11.2554 −0.356821
$$996$$ −1.48913 −0.0471847
$$997$$ 26.2337 0.830829 0.415415 0.909632i $$-0.363637\pi$$
0.415415 + 0.909632i $$0.363637\pi$$
$$998$$ −24.9783 −0.790673
$$999$$ −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 370.2.a.f.1.2 2
3.2 odd 2 3330.2.a.bb.1.2 2
4.3 odd 2 2960.2.a.o.1.1 2
5.2 odd 4 1850.2.b.m.149.4 4
5.3 odd 4 1850.2.b.m.149.1 4
5.4 even 2 1850.2.a.q.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.f.1.2 2 1.1 even 1 trivial
1850.2.a.q.1.1 2 5.4 even 2
1850.2.b.m.149.1 4 5.3 odd 4
1850.2.b.m.149.4 4 5.2 odd 4
2960.2.a.o.1.1 2 4.3 odd 2
3330.2.a.bb.1.2 2 3.2 odd 2