# Properties

 Label 1850.2.a.t.1.2 Level $1850$ Weight $2$ Character 1850.1 Self dual yes Analytic conductor $14.772$ 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] = [1850,2,Mod(1,1850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1850, 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("1850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1850 = 2 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1850.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: $$14.7723243739$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 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 1.2 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 1850.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.61803 q^{3} +1.00000 q^{4} -1.61803 q^{6} +3.23607 q^{7} -1.00000 q^{8} -0.381966 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.61803 q^{3} +1.00000 q^{4} -1.61803 q^{6} +3.23607 q^{7} -1.00000 q^{8} -0.381966 q^{9} -1.38197 q^{11} +1.61803 q^{12} +2.85410 q^{13} -3.23607 q^{14} +1.00000 q^{16} +4.47214 q^{17} +0.381966 q^{18} +4.47214 q^{19} +5.23607 q^{21} +1.38197 q^{22} -2.85410 q^{23} -1.61803 q^{24} -2.85410 q^{26} -5.47214 q^{27} +3.23607 q^{28} -9.32624 q^{29} +7.38197 q^{31} -1.00000 q^{32} -2.23607 q^{33} -4.47214 q^{34} -0.381966 q^{36} +1.00000 q^{37} -4.47214 q^{38} +4.61803 q^{39} +9.61803 q^{41} -5.23607 q^{42} +5.23607 q^{43} -1.38197 q^{44} +2.85410 q^{46} +1.23607 q^{47} +1.61803 q^{48} +3.47214 q^{49} +7.23607 q^{51} +2.85410 q^{52} -0.472136 q^{53} +5.47214 q^{54} -3.23607 q^{56} +7.23607 q^{57} +9.32624 q^{58} -4.76393 q^{59} +10.6180 q^{61} -7.38197 q^{62} -1.23607 q^{63} +1.00000 q^{64} +2.23607 q^{66} -1.09017 q^{67} +4.47214 q^{68} -4.61803 q^{69} +2.94427 q^{71} +0.381966 q^{72} -7.09017 q^{73} -1.00000 q^{74} +4.47214 q^{76} -4.47214 q^{77} -4.61803 q^{78} -8.56231 q^{79} -7.70820 q^{81} -9.61803 q^{82} +14.4721 q^{83} +5.23607 q^{84} -5.23607 q^{86} -15.0902 q^{87} +1.38197 q^{88} -1.52786 q^{89} +9.23607 q^{91} -2.85410 q^{92} +11.9443 q^{93} -1.23607 q^{94} -1.61803 q^{96} +0.472136 q^{97} -3.47214 q^{98} +0.527864 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + q^{3} + 2 q^{4} - q^{6} + 2 q^{7} - 2 q^{8} - 3 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + q^3 + 2 * q^4 - q^6 + 2 * q^7 - 2 * q^8 - 3 * q^9 $$2 q - 2 q^{2} + q^{3} + 2 q^{4} - q^{6} + 2 q^{7} - 2 q^{8} - 3 q^{9} - 5 q^{11} + q^{12} - q^{13} - 2 q^{14} + 2 q^{16} + 3 q^{18} + 6 q^{21} + 5 q^{22} + q^{23} - q^{24} + q^{26} - 2 q^{27} + 2 q^{28} - 3 q^{29} + 17 q^{31} - 2 q^{32} - 3 q^{36} + 2 q^{37} + 7 q^{39} + 17 q^{41} - 6 q^{42} + 6 q^{43} - 5 q^{44} - q^{46} - 2 q^{47} + q^{48} - 2 q^{49} + 10 q^{51} - q^{52} + 8 q^{53} + 2 q^{54} - 2 q^{56} + 10 q^{57} + 3 q^{58} - 14 q^{59} + 19 q^{61} - 17 q^{62} + 2 q^{63} + 2 q^{64} + 9 q^{67} - 7 q^{69} - 12 q^{71} + 3 q^{72} - 3 q^{73} - 2 q^{74} - 7 q^{78} + 3 q^{79} - 2 q^{81} - 17 q^{82} + 20 q^{83} + 6 q^{84} - 6 q^{86} - 19 q^{87} + 5 q^{88} - 12 q^{89} + 14 q^{91} + q^{92} + 6 q^{93} + 2 q^{94} - q^{96} - 8 q^{97} + 2 q^{98} + 10 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + q^3 + 2 * q^4 - q^6 + 2 * q^7 - 2 * q^8 - 3 * q^9 - 5 * q^11 + q^12 - q^13 - 2 * q^14 + 2 * q^16 + 3 * q^18 + 6 * q^21 + 5 * q^22 + q^23 - q^24 + q^26 - 2 * q^27 + 2 * q^28 - 3 * q^29 + 17 * q^31 - 2 * q^32 - 3 * q^36 + 2 * q^37 + 7 * q^39 + 17 * q^41 - 6 * q^42 + 6 * q^43 - 5 * q^44 - q^46 - 2 * q^47 + q^48 - 2 * q^49 + 10 * q^51 - q^52 + 8 * q^53 + 2 * q^54 - 2 * q^56 + 10 * q^57 + 3 * q^58 - 14 * q^59 + 19 * q^61 - 17 * q^62 + 2 * q^63 + 2 * q^64 + 9 * q^67 - 7 * q^69 - 12 * q^71 + 3 * q^72 - 3 * q^73 - 2 * q^74 - 7 * q^78 + 3 * q^79 - 2 * q^81 - 17 * q^82 + 20 * q^83 + 6 * q^84 - 6 * q^86 - 19 * q^87 + 5 * q^88 - 12 * q^89 + 14 * q^91 + q^92 + 6 * q^93 + 2 * q^94 - q^96 - 8 * q^97 + 2 * q^98 + 10 * q^99

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 1.61803 0.934172 0.467086 0.884212i $$-0.345304\pi$$
0.467086 + 0.884212i $$0.345304\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ −1.61803 −0.660560
$$7$$ 3.23607 1.22312 0.611559 0.791199i $$-0.290543\pi$$
0.611559 + 0.791199i $$0.290543\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ −0.381966 −0.127322
$$10$$ 0 0
$$11$$ −1.38197 −0.416678 −0.208339 0.978057i $$-0.566806\pi$$
−0.208339 + 0.978057i $$0.566806\pi$$
$$12$$ 1.61803 0.467086
$$13$$ 2.85410 0.791585 0.395793 0.918340i $$-0.370470\pi$$
0.395793 + 0.918340i $$0.370470\pi$$
$$14$$ −3.23607 −0.864876
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 4.47214 1.08465 0.542326 0.840168i $$-0.317544\pi$$
0.542326 + 0.840168i $$0.317544\pi$$
$$18$$ 0.381966 0.0900303
$$19$$ 4.47214 1.02598 0.512989 0.858395i $$-0.328538\pi$$
0.512989 + 0.858395i $$0.328538\pi$$
$$20$$ 0 0
$$21$$ 5.23607 1.14260
$$22$$ 1.38197 0.294636
$$23$$ −2.85410 −0.595121 −0.297561 0.954703i $$-0.596173\pi$$
−0.297561 + 0.954703i $$0.596173\pi$$
$$24$$ −1.61803 −0.330280
$$25$$ 0 0
$$26$$ −2.85410 −0.559735
$$27$$ −5.47214 −1.05311
$$28$$ 3.23607 0.611559
$$29$$ −9.32624 −1.73184 −0.865919 0.500183i $$-0.833266\pi$$
−0.865919 + 0.500183i $$0.833266\pi$$
$$30$$ 0 0
$$31$$ 7.38197 1.32584 0.662920 0.748690i $$-0.269317\pi$$
0.662920 + 0.748690i $$0.269317\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −2.23607 −0.389249
$$34$$ −4.47214 −0.766965
$$35$$ 0 0
$$36$$ −0.381966 −0.0636610
$$37$$ 1.00000 0.164399
$$38$$ −4.47214 −0.725476
$$39$$ 4.61803 0.739477
$$40$$ 0 0
$$41$$ 9.61803 1.50208 0.751042 0.660254i $$-0.229551\pi$$
0.751042 + 0.660254i $$0.229551\pi$$
$$42$$ −5.23607 −0.807943
$$43$$ 5.23607 0.798493 0.399246 0.916844i $$-0.369272\pi$$
0.399246 + 0.916844i $$0.369272\pi$$
$$44$$ −1.38197 −0.208339
$$45$$ 0 0
$$46$$ 2.85410 0.420814
$$47$$ 1.23607 0.180299 0.0901495 0.995928i $$-0.471266\pi$$
0.0901495 + 0.995928i $$0.471266\pi$$
$$48$$ 1.61803 0.233543
$$49$$ 3.47214 0.496019
$$50$$ 0 0
$$51$$ 7.23607 1.01325
$$52$$ 2.85410 0.395793
$$53$$ −0.472136 −0.0648529 −0.0324264 0.999474i $$-0.510323\pi$$
−0.0324264 + 0.999474i $$0.510323\pi$$
$$54$$ 5.47214 0.744663
$$55$$ 0 0
$$56$$ −3.23607 −0.432438
$$57$$ 7.23607 0.958441
$$58$$ 9.32624 1.22460
$$59$$ −4.76393 −0.620211 −0.310106 0.950702i $$-0.600364\pi$$
−0.310106 + 0.950702i $$0.600364\pi$$
$$60$$ 0 0
$$61$$ 10.6180 1.35950 0.679750 0.733444i $$-0.262088\pi$$
0.679750 + 0.733444i $$0.262088\pi$$
$$62$$ −7.38197 −0.937511
$$63$$ −1.23607 −0.155730
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 2.23607 0.275241
$$67$$ −1.09017 −0.133185 −0.0665927 0.997780i $$-0.521213\pi$$
−0.0665927 + 0.997780i $$0.521213\pi$$
$$68$$ 4.47214 0.542326
$$69$$ −4.61803 −0.555946
$$70$$ 0 0
$$71$$ 2.94427 0.349421 0.174710 0.984620i $$-0.444101\pi$$
0.174710 + 0.984620i $$0.444101\pi$$
$$72$$ 0.381966 0.0450151
$$73$$ −7.09017 −0.829842 −0.414921 0.909858i $$-0.636191\pi$$
−0.414921 + 0.909858i $$0.636191\pi$$
$$74$$ −1.00000 −0.116248
$$75$$ 0 0
$$76$$ 4.47214 0.512989
$$77$$ −4.47214 −0.509647
$$78$$ −4.61803 −0.522889
$$79$$ −8.56231 −0.963335 −0.481667 0.876354i $$-0.659969\pi$$
−0.481667 + 0.876354i $$0.659969\pi$$
$$80$$ 0 0
$$81$$ −7.70820 −0.856467
$$82$$ −9.61803 −1.06213
$$83$$ 14.4721 1.58852 0.794262 0.607576i $$-0.207858\pi$$
0.794262 + 0.607576i $$0.207858\pi$$
$$84$$ 5.23607 0.571302
$$85$$ 0 0
$$86$$ −5.23607 −0.564620
$$87$$ −15.0902 −1.61784
$$88$$ 1.38197 0.147318
$$89$$ −1.52786 −0.161953 −0.0809766 0.996716i $$-0.525804\pi$$
−0.0809766 + 0.996716i $$0.525804\pi$$
$$90$$ 0 0
$$91$$ 9.23607 0.968203
$$92$$ −2.85410 −0.297561
$$93$$ 11.9443 1.23856
$$94$$ −1.23607 −0.127491
$$95$$ 0 0
$$96$$ −1.61803 −0.165140
$$97$$ 0.472136 0.0479381 0.0239691 0.999713i $$-0.492370\pi$$
0.0239691 + 0.999713i $$0.492370\pi$$
$$98$$ −3.47214 −0.350739
$$99$$ 0.527864 0.0530523
$$100$$ 0 0
$$101$$ 3.52786 0.351036 0.175518 0.984476i $$-0.443840\pi$$
0.175518 + 0.984476i $$0.443840\pi$$
$$102$$ −7.23607 −0.716477
$$103$$ −17.2705 −1.70171 −0.850857 0.525397i $$-0.823917\pi$$
−0.850857 + 0.525397i $$0.823917\pi$$
$$104$$ −2.85410 −0.279868
$$105$$ 0 0
$$106$$ 0.472136 0.0458579
$$107$$ 7.32624 0.708254 0.354127 0.935197i $$-0.384778\pi$$
0.354127 + 0.935197i $$0.384778\pi$$
$$108$$ −5.47214 −0.526557
$$109$$ 2.94427 0.282010 0.141005 0.990009i $$-0.454967\pi$$
0.141005 + 0.990009i $$0.454967\pi$$
$$110$$ 0 0
$$111$$ 1.61803 0.153577
$$112$$ 3.23607 0.305780
$$113$$ 6.94427 0.653262 0.326631 0.945152i $$-0.394087\pi$$
0.326631 + 0.945152i $$0.394087\pi$$
$$114$$ −7.23607 −0.677720
$$115$$ 0 0
$$116$$ −9.32624 −0.865919
$$117$$ −1.09017 −0.100786
$$118$$ 4.76393 0.438555
$$119$$ 14.4721 1.32666
$$120$$ 0 0
$$121$$ −9.09017 −0.826379
$$122$$ −10.6180 −0.961312
$$123$$ 15.5623 1.40321
$$124$$ 7.38197 0.662920
$$125$$ 0 0
$$126$$ 1.23607 0.110118
$$127$$ 8.47214 0.751780 0.375890 0.926664i $$-0.377337\pi$$
0.375890 + 0.926664i $$0.377337\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 8.47214 0.745930
$$130$$ 0 0
$$131$$ −22.6525 −1.97916 −0.989578 0.143998i $$-0.954004\pi$$
−0.989578 + 0.143998i $$0.954004\pi$$
$$132$$ −2.23607 −0.194625
$$133$$ 14.4721 1.25489
$$134$$ 1.09017 0.0941763
$$135$$ 0 0
$$136$$ −4.47214 −0.383482
$$137$$ −3.67376 −0.313871 −0.156935 0.987609i $$-0.550161\pi$$
−0.156935 + 0.987609i $$0.550161\pi$$
$$138$$ 4.61803 0.393113
$$139$$ 4.85410 0.411720 0.205860 0.978581i $$-0.434001\pi$$
0.205860 + 0.978581i $$0.434001\pi$$
$$140$$ 0 0
$$141$$ 2.00000 0.168430
$$142$$ −2.94427 −0.247078
$$143$$ −3.94427 −0.329837
$$144$$ −0.381966 −0.0318305
$$145$$ 0 0
$$146$$ 7.09017 0.586787
$$147$$ 5.61803 0.463368
$$148$$ 1.00000 0.0821995
$$149$$ 16.1803 1.32555 0.662773 0.748821i $$-0.269380\pi$$
0.662773 + 0.748821i $$0.269380\pi$$
$$150$$ 0 0
$$151$$ 4.29180 0.349261 0.174631 0.984634i $$-0.444127\pi$$
0.174631 + 0.984634i $$0.444127\pi$$
$$152$$ −4.47214 −0.362738
$$153$$ −1.70820 −0.138100
$$154$$ 4.47214 0.360375
$$155$$ 0 0
$$156$$ 4.61803 0.369739
$$157$$ 16.4721 1.31462 0.657310 0.753620i $$-0.271694\pi$$
0.657310 + 0.753620i $$0.271694\pi$$
$$158$$ 8.56231 0.681180
$$159$$ −0.763932 −0.0605838
$$160$$ 0 0
$$161$$ −9.23607 −0.727904
$$162$$ 7.70820 0.605614
$$163$$ 3.52786 0.276324 0.138162 0.990410i $$-0.455881\pi$$
0.138162 + 0.990410i $$0.455881\pi$$
$$164$$ 9.61803 0.751042
$$165$$ 0 0
$$166$$ −14.4721 −1.12326
$$167$$ −13.8541 −1.07206 −0.536031 0.844198i $$-0.680077\pi$$
−0.536031 + 0.844198i $$0.680077\pi$$
$$168$$ −5.23607 −0.403971
$$169$$ −4.85410 −0.373392
$$170$$ 0 0
$$171$$ −1.70820 −0.130630
$$172$$ 5.23607 0.399246
$$173$$ 0.472136 0.0358958 0.0179479 0.999839i $$-0.494287\pi$$
0.0179479 + 0.999839i $$0.494287\pi$$
$$174$$ 15.0902 1.14398
$$175$$ 0 0
$$176$$ −1.38197 −0.104170
$$177$$ −7.70820 −0.579384
$$178$$ 1.52786 0.114518
$$179$$ −12.6525 −0.945690 −0.472845 0.881146i $$-0.656773\pi$$
−0.472845 + 0.881146i $$0.656773\pi$$
$$180$$ 0 0
$$181$$ 14.4721 1.07571 0.537853 0.843039i $$-0.319236\pi$$
0.537853 + 0.843039i $$0.319236\pi$$
$$182$$ −9.23607 −0.684623
$$183$$ 17.1803 1.27001
$$184$$ 2.85410 0.210407
$$185$$ 0 0
$$186$$ −11.9443 −0.875797
$$187$$ −6.18034 −0.451951
$$188$$ 1.23607 0.0901495
$$189$$ −17.7082 −1.28808
$$190$$ 0 0
$$191$$ −7.09017 −0.513027 −0.256513 0.966541i $$-0.582574\pi$$
−0.256513 + 0.966541i $$0.582574\pi$$
$$192$$ 1.61803 0.116772
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ −0.472136 −0.0338974
$$195$$ 0 0
$$196$$ 3.47214 0.248010
$$197$$ 7.52786 0.536338 0.268169 0.963372i $$-0.413581\pi$$
0.268169 + 0.963372i $$0.413581\pi$$
$$198$$ −0.527864 −0.0375137
$$199$$ 3.05573 0.216615 0.108307 0.994117i $$-0.465457\pi$$
0.108307 + 0.994117i $$0.465457\pi$$
$$200$$ 0 0
$$201$$ −1.76393 −0.124418
$$202$$ −3.52786 −0.248220
$$203$$ −30.1803 −2.11824
$$204$$ 7.23607 0.506626
$$205$$ 0 0
$$206$$ 17.2705 1.20329
$$207$$ 1.09017 0.0757720
$$208$$ 2.85410 0.197896
$$209$$ −6.18034 −0.427503
$$210$$ 0 0
$$211$$ 11.2705 0.775894 0.387947 0.921682i $$-0.373184\pi$$
0.387947 + 0.921682i $$0.373184\pi$$
$$212$$ −0.472136 −0.0324264
$$213$$ 4.76393 0.326419
$$214$$ −7.32624 −0.500811
$$215$$ 0 0
$$216$$ 5.47214 0.372332
$$217$$ 23.8885 1.62166
$$218$$ −2.94427 −0.199411
$$219$$ −11.4721 −0.775215
$$220$$ 0 0
$$221$$ 12.7639 0.858595
$$222$$ −1.61803 −0.108595
$$223$$ −14.1803 −0.949586 −0.474793 0.880098i $$-0.657477\pi$$
−0.474793 + 0.880098i $$0.657477\pi$$
$$224$$ −3.23607 −0.216219
$$225$$ 0 0
$$226$$ −6.94427 −0.461926
$$227$$ 4.29180 0.284857 0.142428 0.989805i $$-0.454509\pi$$
0.142428 + 0.989805i $$0.454509\pi$$
$$228$$ 7.23607 0.479220
$$229$$ −23.1246 −1.52812 −0.764059 0.645147i $$-0.776796\pi$$
−0.764059 + 0.645147i $$0.776796\pi$$
$$230$$ 0 0
$$231$$ −7.23607 −0.476098
$$232$$ 9.32624 0.612298
$$233$$ 6.56231 0.429911 0.214955 0.976624i $$-0.431039\pi$$
0.214955 + 0.976624i $$0.431039\pi$$
$$234$$ 1.09017 0.0712666
$$235$$ 0 0
$$236$$ −4.76393 −0.310106
$$237$$ −13.8541 −0.899921
$$238$$ −14.4721 −0.938089
$$239$$ 9.85410 0.637409 0.318704 0.947854i $$-0.396752\pi$$
0.318704 + 0.947854i $$0.396752\pi$$
$$240$$ 0 0
$$241$$ −1.52786 −0.0984184 −0.0492092 0.998788i $$-0.515670\pi$$
−0.0492092 + 0.998788i $$0.515670\pi$$
$$242$$ 9.09017 0.584338
$$243$$ 3.94427 0.253025
$$244$$ 10.6180 0.679750
$$245$$ 0 0
$$246$$ −15.5623 −0.992216
$$247$$ 12.7639 0.812150
$$248$$ −7.38197 −0.468755
$$249$$ 23.4164 1.48395
$$250$$ 0 0
$$251$$ −20.9443 −1.32199 −0.660995 0.750390i $$-0.729866\pi$$
−0.660995 + 0.750390i $$0.729866\pi$$
$$252$$ −1.23607 −0.0778650
$$253$$ 3.94427 0.247974
$$254$$ −8.47214 −0.531589
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 1.05573 0.0658545 0.0329273 0.999458i $$-0.489517\pi$$
0.0329273 + 0.999458i $$0.489517\pi$$
$$258$$ −8.47214 −0.527452
$$259$$ 3.23607 0.201079
$$260$$ 0 0
$$261$$ 3.56231 0.220501
$$262$$ 22.6525 1.39947
$$263$$ 13.2361 0.816171 0.408085 0.912944i $$-0.366197\pi$$
0.408085 + 0.912944i $$0.366197\pi$$
$$264$$ 2.23607 0.137620
$$265$$ 0 0
$$266$$ −14.4721 −0.887344
$$267$$ −2.47214 −0.151292
$$268$$ −1.09017 −0.0665927
$$269$$ 4.00000 0.243884 0.121942 0.992537i $$-0.461088\pi$$
0.121942 + 0.992537i $$0.461088\pi$$
$$270$$ 0 0
$$271$$ 12.9443 0.786309 0.393154 0.919473i $$-0.371384\pi$$
0.393154 + 0.919473i $$0.371384\pi$$
$$272$$ 4.47214 0.271163
$$273$$ 14.9443 0.904468
$$274$$ 3.67376 0.221940
$$275$$ 0 0
$$276$$ −4.61803 −0.277973
$$277$$ −16.7984 −1.00932 −0.504658 0.863319i $$-0.668381\pi$$
−0.504658 + 0.863319i $$0.668381\pi$$
$$278$$ −4.85410 −0.291130
$$279$$ −2.81966 −0.168809
$$280$$ 0 0
$$281$$ 29.8885 1.78300 0.891501 0.453020i $$-0.149653\pi$$
0.891501 + 0.453020i $$0.149653\pi$$
$$282$$ −2.00000 −0.119098
$$283$$ −6.76393 −0.402074 −0.201037 0.979584i $$-0.564431\pi$$
−0.201037 + 0.979584i $$0.564431\pi$$
$$284$$ 2.94427 0.174710
$$285$$ 0 0
$$286$$ 3.94427 0.233230
$$287$$ 31.1246 1.83723
$$288$$ 0.381966 0.0225076
$$289$$ 3.00000 0.176471
$$290$$ 0 0
$$291$$ 0.763932 0.0447825
$$292$$ −7.09017 −0.414921
$$293$$ 12.6525 0.739166 0.369583 0.929198i $$-0.379501\pi$$
0.369583 + 0.929198i $$0.379501\pi$$
$$294$$ −5.61803 −0.327650
$$295$$ 0 0
$$296$$ −1.00000 −0.0581238
$$297$$ 7.56231 0.438809
$$298$$ −16.1803 −0.937302
$$299$$ −8.14590 −0.471089
$$300$$ 0 0
$$301$$ 16.9443 0.976652
$$302$$ −4.29180 −0.246965
$$303$$ 5.70820 0.327928
$$304$$ 4.47214 0.256495
$$305$$ 0 0
$$306$$ 1.70820 0.0976515
$$307$$ 12.8541 0.733622 0.366811 0.930295i $$-0.380450\pi$$
0.366811 + 0.930295i $$0.380450\pi$$
$$308$$ −4.47214 −0.254824
$$309$$ −27.9443 −1.58969
$$310$$ 0 0
$$311$$ −27.0344 −1.53298 −0.766491 0.642255i $$-0.777999\pi$$
−0.766491 + 0.642255i $$0.777999\pi$$
$$312$$ −4.61803 −0.261445
$$313$$ −28.1803 −1.59285 −0.796423 0.604739i $$-0.793277\pi$$
−0.796423 + 0.604739i $$0.793277\pi$$
$$314$$ −16.4721 −0.929576
$$315$$ 0 0
$$316$$ −8.56231 −0.481667
$$317$$ −20.9443 −1.17635 −0.588174 0.808735i $$-0.700153\pi$$
−0.588174 + 0.808735i $$0.700153\pi$$
$$318$$ 0.763932 0.0428392
$$319$$ 12.8885 0.721620
$$320$$ 0 0
$$321$$ 11.8541 0.661631
$$322$$ 9.23607 0.514706
$$323$$ 20.0000 1.11283
$$324$$ −7.70820 −0.428234
$$325$$ 0 0
$$326$$ −3.52786 −0.195390
$$327$$ 4.76393 0.263446
$$328$$ −9.61803 −0.531067
$$329$$ 4.00000 0.220527
$$330$$ 0 0
$$331$$ 28.0000 1.53902 0.769510 0.638635i $$-0.220501\pi$$
0.769510 + 0.638635i $$0.220501\pi$$
$$332$$ 14.4721 0.794262
$$333$$ −0.381966 −0.0209316
$$334$$ 13.8541 0.758063
$$335$$ 0 0
$$336$$ 5.23607 0.285651
$$337$$ −12.0344 −0.655558 −0.327779 0.944754i $$-0.606300\pi$$
−0.327779 + 0.944754i $$0.606300\pi$$
$$338$$ 4.85410 0.264028
$$339$$ 11.2361 0.610259
$$340$$ 0 0
$$341$$ −10.2016 −0.552449
$$342$$ 1.70820 0.0923691
$$343$$ −11.4164 −0.616428
$$344$$ −5.23607 −0.282310
$$345$$ 0 0
$$346$$ −0.472136 −0.0253822
$$347$$ −17.2361 −0.925281 −0.462640 0.886546i $$-0.653098\pi$$
−0.462640 + 0.886546i $$0.653098\pi$$
$$348$$ −15.0902 −0.808918
$$349$$ −10.1803 −0.544941 −0.272471 0.962164i $$-0.587841\pi$$
−0.272471 + 0.962164i $$0.587841\pi$$
$$350$$ 0 0
$$351$$ −15.6180 −0.833629
$$352$$ 1.38197 0.0736590
$$353$$ −16.2918 −0.867125 −0.433562 0.901124i $$-0.642744\pi$$
−0.433562 + 0.901124i $$0.642744\pi$$
$$354$$ 7.70820 0.409686
$$355$$ 0 0
$$356$$ −1.52786 −0.0809766
$$357$$ 23.4164 1.23933
$$358$$ 12.6525 0.668704
$$359$$ −4.47214 −0.236030 −0.118015 0.993012i $$-0.537653\pi$$
−0.118015 + 0.993012i $$0.537653\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −14.4721 −0.760639
$$363$$ −14.7082 −0.771980
$$364$$ 9.23607 0.484102
$$365$$ 0 0
$$366$$ −17.1803 −0.898031
$$367$$ −13.1246 −0.685099 −0.342550 0.939500i $$-0.611290\pi$$
−0.342550 + 0.939500i $$0.611290\pi$$
$$368$$ −2.85410 −0.148780
$$369$$ −3.67376 −0.191248
$$370$$ 0 0
$$371$$ −1.52786 −0.0793227
$$372$$ 11.9443 0.619282
$$373$$ −27.7082 −1.43468 −0.717338 0.696725i $$-0.754640\pi$$
−0.717338 + 0.696725i $$0.754640\pi$$
$$374$$ 6.18034 0.319578
$$375$$ 0 0
$$376$$ −1.23607 −0.0637453
$$377$$ −26.6180 −1.37090
$$378$$ 17.7082 0.910812
$$379$$ 28.0902 1.44290 0.721448 0.692469i $$-0.243477\pi$$
0.721448 + 0.692469i $$0.243477\pi$$
$$380$$ 0 0
$$381$$ 13.7082 0.702293
$$382$$ 7.09017 0.362765
$$383$$ −17.8885 −0.914062 −0.457031 0.889451i $$-0.651087\pi$$
−0.457031 + 0.889451i $$0.651087\pi$$
$$384$$ −1.61803 −0.0825700
$$385$$ 0 0
$$386$$ 4.00000 0.203595
$$387$$ −2.00000 −0.101666
$$388$$ 0.472136 0.0239691
$$389$$ −6.85410 −0.347517 −0.173758 0.984788i $$-0.555591\pi$$
−0.173758 + 0.984788i $$0.555591\pi$$
$$390$$ 0 0
$$391$$ −12.7639 −0.645500
$$392$$ −3.47214 −0.175369
$$393$$ −36.6525 −1.84887
$$394$$ −7.52786 −0.379248
$$395$$ 0 0
$$396$$ 0.527864 0.0265262
$$397$$ −20.6525 −1.03652 −0.518259 0.855224i $$-0.673420\pi$$
−0.518259 + 0.855224i $$0.673420\pi$$
$$398$$ −3.05573 −0.153170
$$399$$ 23.4164 1.17229
$$400$$ 0 0
$$401$$ −4.76393 −0.237899 −0.118950 0.992900i $$-0.537953\pi$$
−0.118950 + 0.992900i $$0.537953\pi$$
$$402$$ 1.76393 0.0879769
$$403$$ 21.0689 1.04952
$$404$$ 3.52786 0.175518
$$405$$ 0 0
$$406$$ 30.1803 1.49783
$$407$$ −1.38197 −0.0685015
$$408$$ −7.23607 −0.358239
$$409$$ −26.1803 −1.29453 −0.647267 0.762263i $$-0.724088\pi$$
−0.647267 + 0.762263i $$0.724088\pi$$
$$410$$ 0 0
$$411$$ −5.94427 −0.293209
$$412$$ −17.2705 −0.850857
$$413$$ −15.4164 −0.758592
$$414$$ −1.09017 −0.0535789
$$415$$ 0 0
$$416$$ −2.85410 −0.139934
$$417$$ 7.85410 0.384617
$$418$$ 6.18034 0.302290
$$419$$ −10.5623 −0.516002 −0.258001 0.966145i $$-0.583064\pi$$
−0.258001 + 0.966145i $$0.583064\pi$$
$$420$$ 0 0
$$421$$ −31.0344 −1.51253 −0.756263 0.654268i $$-0.772977\pi$$
−0.756263 + 0.654268i $$0.772977\pi$$
$$422$$ −11.2705 −0.548640
$$423$$ −0.472136 −0.0229560
$$424$$ 0.472136 0.0229289
$$425$$ 0 0
$$426$$ −4.76393 −0.230813
$$427$$ 34.3607 1.66283
$$428$$ 7.32624 0.354127
$$429$$ −6.38197 −0.308124
$$430$$ 0 0
$$431$$ 12.3607 0.595393 0.297696 0.954661i $$-0.403782\pi$$
0.297696 + 0.954661i $$0.403782\pi$$
$$432$$ −5.47214 −0.263278
$$433$$ 20.6738 0.993518 0.496759 0.867889i $$-0.334523\pi$$
0.496759 + 0.867889i $$0.334523\pi$$
$$434$$ −23.8885 −1.14669
$$435$$ 0 0
$$436$$ 2.94427 0.141005
$$437$$ −12.7639 −0.610582
$$438$$ 11.4721 0.548160
$$439$$ 7.79837 0.372196 0.186098 0.982531i $$-0.440416\pi$$
0.186098 + 0.982531i $$0.440416\pi$$
$$440$$ 0 0
$$441$$ −1.32624 −0.0631542
$$442$$ −12.7639 −0.607118
$$443$$ −15.2705 −0.725524 −0.362762 0.931882i $$-0.618166\pi$$
−0.362762 + 0.931882i $$0.618166\pi$$
$$444$$ 1.61803 0.0767885
$$445$$ 0 0
$$446$$ 14.1803 0.671459
$$447$$ 26.1803 1.23829
$$448$$ 3.23607 0.152890
$$449$$ −28.4721 −1.34368 −0.671842 0.740695i $$-0.734496\pi$$
−0.671842 + 0.740695i $$0.734496\pi$$
$$450$$ 0 0
$$451$$ −13.2918 −0.625886
$$452$$ 6.94427 0.326631
$$453$$ 6.94427 0.326270
$$454$$ −4.29180 −0.201424
$$455$$ 0 0
$$456$$ −7.23607 −0.338860
$$457$$ 24.7639 1.15841 0.579204 0.815183i $$-0.303363\pi$$
0.579204 + 0.815183i $$0.303363\pi$$
$$458$$ 23.1246 1.08054
$$459$$ −24.4721 −1.14226
$$460$$ 0 0
$$461$$ −38.9443 −1.81382 −0.906908 0.421329i $$-0.861564\pi$$
−0.906908 + 0.421329i $$0.861564\pi$$
$$462$$ 7.23607 0.336652
$$463$$ 4.56231 0.212028 0.106014 0.994365i $$-0.466191\pi$$
0.106014 + 0.994365i $$0.466191\pi$$
$$464$$ −9.32624 −0.432960
$$465$$ 0 0
$$466$$ −6.56231 −0.303993
$$467$$ 24.3607 1.12728 0.563639 0.826021i $$-0.309401\pi$$
0.563639 + 0.826021i $$0.309401\pi$$
$$468$$ −1.09017 −0.0503931
$$469$$ −3.52786 −0.162902
$$470$$ 0 0
$$471$$ 26.6525 1.22808
$$472$$ 4.76393 0.219278
$$473$$ −7.23607 −0.332715
$$474$$ 13.8541 0.636340
$$475$$ 0 0
$$476$$ 14.4721 0.663329
$$477$$ 0.180340 0.00825720
$$478$$ −9.85410 −0.450716
$$479$$ 36.5623 1.67057 0.835287 0.549814i $$-0.185301\pi$$
0.835287 + 0.549814i $$0.185301\pi$$
$$480$$ 0 0
$$481$$ 2.85410 0.130136
$$482$$ 1.52786 0.0695923
$$483$$ −14.9443 −0.679988
$$484$$ −9.09017 −0.413190
$$485$$ 0 0
$$486$$ −3.94427 −0.178916
$$487$$ −37.3050 −1.69045 −0.845224 0.534412i $$-0.820533\pi$$
−0.845224 + 0.534412i $$0.820533\pi$$
$$488$$ −10.6180 −0.480656
$$489$$ 5.70820 0.258134
$$490$$ 0 0
$$491$$ −28.4508 −1.28397 −0.641984 0.766718i $$-0.721889\pi$$
−0.641984 + 0.766718i $$0.721889\pi$$
$$492$$ 15.5623 0.701603
$$493$$ −41.7082 −1.87844
$$494$$ −12.7639 −0.574276
$$495$$ 0 0
$$496$$ 7.38197 0.331460
$$497$$ 9.52786 0.427383
$$498$$ −23.4164 −1.04931
$$499$$ 10.2918 0.460724 0.230362 0.973105i $$-0.426009\pi$$
0.230362 + 0.973105i $$0.426009\pi$$
$$500$$ 0 0
$$501$$ −22.4164 −1.00149
$$502$$ 20.9443 0.934789
$$503$$ −19.0902 −0.851189 −0.425594 0.904914i $$-0.639935\pi$$
−0.425594 + 0.904914i $$0.639935\pi$$
$$504$$ 1.23607 0.0550588
$$505$$ 0 0
$$506$$ −3.94427 −0.175344
$$507$$ −7.85410 −0.348813
$$508$$ 8.47214 0.375890
$$509$$ −17.7082 −0.784902 −0.392451 0.919773i $$-0.628373\pi$$
−0.392451 + 0.919773i $$0.628373\pi$$
$$510$$ 0 0
$$511$$ −22.9443 −1.01499
$$512$$ −1.00000 −0.0441942
$$513$$ −24.4721 −1.08047
$$514$$ −1.05573 −0.0465662
$$515$$ 0 0
$$516$$ 8.47214 0.372965
$$517$$ −1.70820 −0.0751267
$$518$$ −3.23607 −0.142185
$$519$$ 0.763932 0.0335329
$$520$$ 0 0
$$521$$ −1.41641 −0.0620540 −0.0310270 0.999519i $$-0.509878\pi$$
−0.0310270 + 0.999519i $$0.509878\pi$$
$$522$$ −3.56231 −0.155918
$$523$$ −11.8197 −0.516838 −0.258419 0.966033i $$-0.583201\pi$$
−0.258419 + 0.966033i $$0.583201\pi$$
$$524$$ −22.6525 −0.989578
$$525$$ 0 0
$$526$$ −13.2361 −0.577120
$$527$$ 33.0132 1.43808
$$528$$ −2.23607 −0.0973124
$$529$$ −14.8541 −0.645831
$$530$$ 0 0
$$531$$ 1.81966 0.0789665
$$532$$ 14.4721 0.627447
$$533$$ 27.4508 1.18903
$$534$$ 2.47214 0.106980
$$535$$ 0 0
$$536$$ 1.09017 0.0470882
$$537$$ −20.4721 −0.883438
$$538$$ −4.00000 −0.172452
$$539$$ −4.79837 −0.206681
$$540$$ 0 0
$$541$$ −10.3262 −0.443960 −0.221980 0.975051i $$-0.571252\pi$$
−0.221980 + 0.975051i $$0.571252\pi$$
$$542$$ −12.9443 −0.556004
$$543$$ 23.4164 1.00489
$$544$$ −4.47214 −0.191741
$$545$$ 0 0
$$546$$ −14.9443 −0.639556
$$547$$ 16.0689 0.687056 0.343528 0.939142i $$-0.388378\pi$$
0.343528 + 0.939142i $$0.388378\pi$$
$$548$$ −3.67376 −0.156935
$$549$$ −4.05573 −0.173094
$$550$$ 0 0
$$551$$ −41.7082 −1.77683
$$552$$ 4.61803 0.196557
$$553$$ −27.7082 −1.17827
$$554$$ 16.7984 0.713695
$$555$$ 0 0
$$556$$ 4.85410 0.205860
$$557$$ −19.5623 −0.828882 −0.414441 0.910076i $$-0.636023\pi$$
−0.414441 + 0.910076i $$0.636023\pi$$
$$558$$ 2.81966 0.119366
$$559$$ 14.9443 0.632075
$$560$$ 0 0
$$561$$ −10.0000 −0.422200
$$562$$ −29.8885 −1.26077
$$563$$ −7.88854 −0.332462 −0.166231 0.986087i $$-0.553160\pi$$
−0.166231 + 0.986087i $$0.553160\pi$$
$$564$$ 2.00000 0.0842152
$$565$$ 0 0
$$566$$ 6.76393 0.284309
$$567$$ −24.9443 −1.04756
$$568$$ −2.94427 −0.123539
$$569$$ 13.8885 0.582238 0.291119 0.956687i $$-0.405972\pi$$
0.291119 + 0.956687i $$0.405972\pi$$
$$570$$ 0 0
$$571$$ −10.5623 −0.442019 −0.221009 0.975272i $$-0.570935\pi$$
−0.221009 + 0.975272i $$0.570935\pi$$
$$572$$ −3.94427 −0.164918
$$573$$ −11.4721 −0.479255
$$574$$ −31.1246 −1.29912
$$575$$ 0 0
$$576$$ −0.381966 −0.0159153
$$577$$ −10.6525 −0.443468 −0.221734 0.975107i $$-0.571172\pi$$
−0.221734 + 0.975107i $$0.571172\pi$$
$$578$$ −3.00000 −0.124784
$$579$$ −6.47214 −0.268973
$$580$$ 0 0
$$581$$ 46.8328 1.94295
$$582$$ −0.763932 −0.0316660
$$583$$ 0.652476 0.0270228
$$584$$ 7.09017 0.293393
$$585$$ 0 0
$$586$$ −12.6525 −0.522669
$$587$$ 13.0557 0.538868 0.269434 0.963019i $$-0.413163\pi$$
0.269434 + 0.963019i $$0.413163\pi$$
$$588$$ 5.61803 0.231684
$$589$$ 33.0132 1.36028
$$590$$ 0 0
$$591$$ 12.1803 0.501032
$$592$$ 1.00000 0.0410997
$$593$$ −17.5623 −0.721197 −0.360599 0.932721i $$-0.617428\pi$$
−0.360599 + 0.932721i $$0.617428\pi$$
$$594$$ −7.56231 −0.310285
$$595$$ 0 0
$$596$$ 16.1803 0.662773
$$597$$ 4.94427 0.202356
$$598$$ 8.14590 0.333111
$$599$$ −6.36068 −0.259890 −0.129945 0.991521i $$-0.541480\pi$$
−0.129945 + 0.991521i $$0.541480\pi$$
$$600$$ 0 0
$$601$$ −24.6869 −1.00700 −0.503500 0.863995i $$-0.667955\pi$$
−0.503500 + 0.863995i $$0.667955\pi$$
$$602$$ −16.9443 −0.690597
$$603$$ 0.416408 0.0169574
$$604$$ 4.29180 0.174631
$$605$$ 0 0
$$606$$ −5.70820 −0.231880
$$607$$ −35.0344 −1.42200 −0.711002 0.703190i $$-0.751758\pi$$
−0.711002 + 0.703190i $$0.751758\pi$$
$$608$$ −4.47214 −0.181369
$$609$$ −48.8328 −1.97881
$$610$$ 0 0
$$611$$ 3.52786 0.142722
$$612$$ −1.70820 −0.0690501
$$613$$ 13.8197 0.558171 0.279085 0.960266i $$-0.409969\pi$$
0.279085 + 0.960266i $$0.409969\pi$$
$$614$$ −12.8541 −0.518749
$$615$$ 0 0
$$616$$ 4.47214 0.180187
$$617$$ −0.0901699 −0.00363011 −0.00181505 0.999998i $$-0.500578\pi$$
−0.00181505 + 0.999998i $$0.500578\pi$$
$$618$$ 27.9443 1.12408
$$619$$ −15.2705 −0.613774 −0.306887 0.951746i $$-0.599287\pi$$
−0.306887 + 0.951746i $$0.599287\pi$$
$$620$$ 0 0
$$621$$ 15.6180 0.626730
$$622$$ 27.0344 1.08398
$$623$$ −4.94427 −0.198088
$$624$$ 4.61803 0.184869
$$625$$ 0 0
$$626$$ 28.1803 1.12631
$$627$$ −10.0000 −0.399362
$$628$$ 16.4721 0.657310
$$629$$ 4.47214 0.178316
$$630$$ 0 0
$$631$$ −47.3951 −1.88677 −0.943385 0.331700i $$-0.892378\pi$$
−0.943385 + 0.331700i $$0.892378\pi$$
$$632$$ 8.56231 0.340590
$$633$$ 18.2361 0.724819
$$634$$ 20.9443 0.831803
$$635$$ 0 0
$$636$$ −0.763932 −0.0302919
$$637$$ 9.90983 0.392642
$$638$$ −12.8885 −0.510262
$$639$$ −1.12461 −0.0444890
$$640$$ 0 0
$$641$$ 15.5066 0.612473 0.306236 0.951955i $$-0.400930\pi$$
0.306236 + 0.951955i $$0.400930\pi$$
$$642$$ −11.8541 −0.467844
$$643$$ 28.7639 1.13434 0.567169 0.823601i $$-0.308038\pi$$
0.567169 + 0.823601i $$0.308038\pi$$
$$644$$ −9.23607 −0.363952
$$645$$ 0 0
$$646$$ −20.0000 −0.786889
$$647$$ −30.0902 −1.18297 −0.591483 0.806317i $$-0.701457\pi$$
−0.591483 + 0.806317i $$0.701457\pi$$
$$648$$ 7.70820 0.302807
$$649$$ 6.58359 0.258429
$$650$$ 0 0
$$651$$ 38.6525 1.51491
$$652$$ 3.52786 0.138162
$$653$$ 38.2705 1.49764 0.748820 0.662773i $$-0.230621\pi$$
0.748820 + 0.662773i $$0.230621\pi$$
$$654$$ −4.76393 −0.186284
$$655$$ 0 0
$$656$$ 9.61803 0.375521
$$657$$ 2.70820 0.105657
$$658$$ −4.00000 −0.155936
$$659$$ 40.4508 1.57574 0.787871 0.615841i $$-0.211184\pi$$
0.787871 + 0.615841i $$0.211184\pi$$
$$660$$ 0 0
$$661$$ 17.3262 0.673913 0.336956 0.941520i $$-0.390603\pi$$
0.336956 + 0.941520i $$0.390603\pi$$
$$662$$ −28.0000 −1.08825
$$663$$ 20.6525 0.802076
$$664$$ −14.4721 −0.561628
$$665$$ 0 0
$$666$$ 0.381966 0.0148009
$$667$$ 26.6180 1.03065
$$668$$ −13.8541 −0.536031
$$669$$ −22.9443 −0.887077
$$670$$ 0 0
$$671$$ −14.6738 −0.566474
$$672$$ −5.23607 −0.201986
$$673$$ −11.1459 −0.429643 −0.214821 0.976653i $$-0.568917\pi$$
−0.214821 + 0.976653i $$0.568917\pi$$
$$674$$ 12.0344 0.463549
$$675$$ 0 0
$$676$$ −4.85410 −0.186696
$$677$$ −34.6525 −1.33180 −0.665901 0.746040i $$-0.731953\pi$$
−0.665901 + 0.746040i $$0.731953\pi$$
$$678$$ −11.2361 −0.431519
$$679$$ 1.52786 0.0586340
$$680$$ 0 0
$$681$$ 6.94427 0.266105
$$682$$ 10.2016 0.390640
$$683$$ 8.58359 0.328442 0.164221 0.986424i $$-0.447489\pi$$
0.164221 + 0.986424i $$0.447489\pi$$
$$684$$ −1.70820 −0.0653148
$$685$$ 0 0
$$686$$ 11.4164 0.435880
$$687$$ −37.4164 −1.42752
$$688$$ 5.23607 0.199623
$$689$$ −1.34752 −0.0513366
$$690$$ 0 0
$$691$$ 35.7771 1.36102 0.680512 0.732737i $$-0.261757\pi$$
0.680512 + 0.732737i $$0.261757\pi$$
$$692$$ 0.472136 0.0179479
$$693$$ 1.70820 0.0648893
$$694$$ 17.2361 0.654272
$$695$$ 0 0
$$696$$ 15.0902 0.571991
$$697$$ 43.0132 1.62924
$$698$$ 10.1803 0.385332
$$699$$ 10.6180 0.401611
$$700$$ 0 0
$$701$$ −42.9787 −1.62328 −0.811642 0.584155i $$-0.801426\pi$$
−0.811642 + 0.584155i $$0.801426\pi$$
$$702$$ 15.6180 0.589465
$$703$$ 4.47214 0.168670
$$704$$ −1.38197 −0.0520848
$$705$$ 0 0
$$706$$ 16.2918 0.613150
$$707$$ 11.4164 0.429358
$$708$$ −7.70820 −0.289692
$$709$$ −8.21478 −0.308513 −0.154256 0.988031i $$-0.549298\pi$$
−0.154256 + 0.988031i $$0.549298\pi$$
$$710$$ 0 0
$$711$$ 3.27051 0.122654
$$712$$ 1.52786 0.0572591
$$713$$ −21.0689 −0.789036
$$714$$ −23.4164 −0.876337
$$715$$ 0 0
$$716$$ −12.6525 −0.472845
$$717$$ 15.9443 0.595450
$$718$$ 4.47214 0.166899
$$719$$ 27.4164 1.02246 0.511230 0.859444i $$-0.329190\pi$$
0.511230 + 0.859444i $$0.329190\pi$$
$$720$$ 0 0
$$721$$ −55.8885 −2.08140
$$722$$ −1.00000 −0.0372161
$$723$$ −2.47214 −0.0919397
$$724$$ 14.4721 0.537853
$$725$$ 0 0
$$726$$ 14.7082 0.545873
$$727$$ 29.8541 1.10723 0.553614 0.832774i $$-0.313248\pi$$
0.553614 + 0.832774i $$0.313248\pi$$
$$728$$ −9.23607 −0.342311
$$729$$ 29.5066 1.09284
$$730$$ 0 0
$$731$$ 23.4164 0.866087
$$732$$ 17.1803 0.635004
$$733$$ −27.5279 −1.01676 −0.508382 0.861131i $$-0.669756\pi$$
−0.508382 + 0.861131i $$0.669756\pi$$
$$734$$ 13.1246 0.484438
$$735$$ 0 0
$$736$$ 2.85410 0.105204
$$737$$ 1.50658 0.0554955
$$738$$ 3.67376 0.135233
$$739$$ −10.9098 −0.401325 −0.200662 0.979660i $$-0.564309\pi$$
−0.200662 + 0.979660i $$0.564309\pi$$
$$740$$ 0 0
$$741$$ 20.6525 0.758688
$$742$$ 1.52786 0.0560897
$$743$$ 48.0689 1.76348 0.881738 0.471739i $$-0.156374\pi$$
0.881738 + 0.471739i $$0.156374\pi$$
$$744$$ −11.9443 −0.437898
$$745$$ 0 0
$$746$$ 27.7082 1.01447
$$747$$ −5.52786 −0.202254
$$748$$ −6.18034 −0.225976
$$749$$ 23.7082 0.866279
$$750$$ 0 0
$$751$$ −1.05573 −0.0385241 −0.0192620 0.999814i $$-0.506132\pi$$
−0.0192620 + 0.999814i $$0.506132\pi$$
$$752$$ 1.23607 0.0450748
$$753$$ −33.8885 −1.23497
$$754$$ 26.6180 0.969372
$$755$$ 0 0
$$756$$ −17.7082 −0.644041
$$757$$ 4.14590 0.150685 0.0753426 0.997158i $$-0.475995\pi$$
0.0753426 + 0.997158i $$0.475995\pi$$
$$758$$ −28.0902 −1.02028
$$759$$ 6.38197 0.231651
$$760$$ 0 0
$$761$$ −19.1459 −0.694038 −0.347019 0.937858i $$-0.612806\pi$$
−0.347019 + 0.937858i $$0.612806\pi$$
$$762$$ −13.7082 −0.496596
$$763$$ 9.52786 0.344932
$$764$$ −7.09017 −0.256513
$$765$$ 0 0
$$766$$ 17.8885 0.646339
$$767$$ −13.5967 −0.490950
$$768$$ 1.61803 0.0583858
$$769$$ 23.8885 0.861443 0.430721 0.902485i $$-0.358259\pi$$
0.430721 + 0.902485i $$0.358259\pi$$
$$770$$ 0 0
$$771$$ 1.70820 0.0615195
$$772$$ −4.00000 −0.143963
$$773$$ 24.0000 0.863220 0.431610 0.902060i $$-0.357946\pi$$
0.431610 + 0.902060i $$0.357946\pi$$
$$774$$ 2.00000 0.0718885
$$775$$ 0 0
$$776$$ −0.472136 −0.0169487
$$777$$ 5.23607 0.187843
$$778$$ 6.85410 0.245731
$$779$$ 43.0132 1.54111
$$780$$ 0 0
$$781$$ −4.06888 −0.145596
$$782$$ 12.7639 0.456437
$$783$$ 51.0344 1.82382
$$784$$ 3.47214 0.124005
$$785$$ 0 0
$$786$$ 36.6525 1.30735
$$787$$ 25.5279 0.909970 0.454985 0.890499i $$-0.349645\pi$$
0.454985 + 0.890499i $$0.349645\pi$$
$$788$$ 7.52786 0.268169
$$789$$ 21.4164 0.762444
$$790$$ 0 0
$$791$$ 22.4721 0.799017
$$792$$ −0.527864 −0.0187568
$$793$$ 30.3050 1.07616
$$794$$ 20.6525 0.732929
$$795$$ 0 0
$$796$$ 3.05573 0.108307
$$797$$ −46.2705 −1.63899 −0.819493 0.573090i $$-0.805745\pi$$
−0.819493 + 0.573090i $$0.805745\pi$$
$$798$$ −23.4164 −0.828932
$$799$$ 5.52786 0.195562
$$800$$ 0 0
$$801$$ 0.583592 0.0206202
$$802$$ 4.76393 0.168220
$$803$$ 9.79837 0.345777
$$804$$ −1.76393 −0.0622091
$$805$$ 0 0
$$806$$ −21.0689 −0.742120
$$807$$ 6.47214 0.227830
$$808$$ −3.52786 −0.124110
$$809$$ −13.1246 −0.461437 −0.230718 0.973021i $$-0.574108\pi$$
−0.230718 + 0.973021i $$0.574108\pi$$
$$810$$ 0 0
$$811$$ −34.1033 −1.19753 −0.598765 0.800925i $$-0.704342\pi$$
−0.598765 + 0.800925i $$0.704342\pi$$
$$812$$ −30.1803 −1.05912
$$813$$ 20.9443 0.734548
$$814$$ 1.38197 0.0484379
$$815$$ 0 0
$$816$$ 7.23607 0.253313
$$817$$ 23.4164 0.819236
$$818$$ 26.1803 0.915374
$$819$$ −3.52786 −0.123274
$$820$$ 0 0
$$821$$ −5.41641 −0.189034 −0.0945170 0.995523i $$-0.530131\pi$$
−0.0945170 + 0.995523i $$0.530131\pi$$
$$822$$ 5.94427 0.207330
$$823$$ −1.88854 −0.0658305 −0.0329152 0.999458i $$-0.510479\pi$$
−0.0329152 + 0.999458i $$0.510479\pi$$
$$824$$ 17.2705 0.601647
$$825$$ 0 0
$$826$$ 15.4164 0.536405
$$827$$ 2.06888 0.0719421 0.0359711 0.999353i $$-0.488548\pi$$
0.0359711 + 0.999353i $$0.488548\pi$$
$$828$$ 1.09017 0.0378860
$$829$$ 55.7984 1.93796 0.968979 0.247144i $$-0.0794920\pi$$
0.968979 + 0.247144i $$0.0794920\pi$$
$$830$$ 0 0
$$831$$ −27.1803 −0.942876
$$832$$ 2.85410 0.0989482
$$833$$ 15.5279 0.538009
$$834$$ −7.85410 −0.271965
$$835$$ 0 0
$$836$$ −6.18034 −0.213752
$$837$$ −40.3951 −1.39626
$$838$$ 10.5623 0.364869
$$839$$ −36.6525 −1.26538 −0.632692 0.774404i $$-0.718050\pi$$
−0.632692 + 0.774404i $$0.718050\pi$$
$$840$$ 0 0
$$841$$ 57.9787 1.99927
$$842$$ 31.0344 1.06952
$$843$$ 48.3607 1.66563
$$844$$ 11.2705 0.387947
$$845$$ 0 0
$$846$$ 0.472136 0.0162324
$$847$$ −29.4164 −1.01076
$$848$$ −0.472136 −0.0162132
$$849$$ −10.9443 −0.375606
$$850$$ 0 0
$$851$$ −2.85410 −0.0978374
$$852$$ 4.76393 0.163210
$$853$$ −29.7426 −1.01837 −0.509184 0.860657i $$-0.670053\pi$$
−0.509184 + 0.860657i $$0.670053\pi$$
$$854$$ −34.3607 −1.17580
$$855$$ 0 0
$$856$$ −7.32624 −0.250406
$$857$$ 9.05573 0.309338 0.154669 0.987966i $$-0.450569\pi$$
0.154669 + 0.987966i $$0.450569\pi$$
$$858$$ 6.38197 0.217877
$$859$$ 53.4164 1.82254 0.911272 0.411805i $$-0.135101\pi$$
0.911272 + 0.411805i $$0.135101\pi$$
$$860$$ 0 0
$$861$$ 50.3607 1.71629
$$862$$ −12.3607 −0.421006
$$863$$ −19.4164 −0.660942 −0.330471 0.943816i $$-0.607208\pi$$
−0.330471 + 0.943816i $$0.607208\pi$$
$$864$$ 5.47214 0.186166
$$865$$ 0 0
$$866$$ −20.6738 −0.702523
$$867$$ 4.85410 0.164854
$$868$$ 23.8885 0.810830
$$869$$ 11.8328 0.401401
$$870$$ 0 0
$$871$$ −3.11146 −0.105428
$$872$$ −2.94427 −0.0997056
$$873$$ −0.180340 −0.00610358
$$874$$ 12.7639 0.431746
$$875$$ 0 0
$$876$$ −11.4721 −0.387608
$$877$$ 16.8328 0.568404 0.284202 0.958764i $$-0.408271\pi$$
0.284202 + 0.958764i $$0.408271\pi$$
$$878$$ −7.79837 −0.263182
$$879$$ 20.4721 0.690508
$$880$$ 0 0
$$881$$ 19.7426 0.665147 0.332573 0.943077i $$-0.392083\pi$$
0.332573 + 0.943077i $$0.392083\pi$$
$$882$$ 1.32624 0.0446568
$$883$$ 33.3050 1.12080 0.560400 0.828222i $$-0.310647\pi$$
0.560400 + 0.828222i $$0.310647\pi$$
$$884$$ 12.7639 0.429297
$$885$$ 0 0
$$886$$ 15.2705 0.513023
$$887$$ −27.8885 −0.936406 −0.468203 0.883621i $$-0.655098\pi$$
−0.468203 + 0.883621i $$0.655098\pi$$
$$888$$ −1.61803 −0.0542977
$$889$$ 27.4164 0.919517
$$890$$ 0 0
$$891$$ 10.6525 0.356871
$$892$$ −14.1803 −0.474793
$$893$$ 5.52786 0.184983
$$894$$ −26.1803 −0.875602
$$895$$ 0 0
$$896$$ −3.23607 −0.108109
$$897$$ −13.1803 −0.440079
$$898$$ 28.4721 0.950127
$$899$$ −68.8460 −2.29614
$$900$$ 0 0
$$901$$ −2.11146 −0.0703428
$$902$$ 13.2918 0.442568
$$903$$ 27.4164 0.912361
$$904$$ −6.94427 −0.230963
$$905$$ 0 0
$$906$$ −6.94427 −0.230708
$$907$$ 55.8885 1.85575 0.927874 0.372893i $$-0.121634\pi$$
0.927874 + 0.372893i $$0.121634\pi$$
$$908$$ 4.29180 0.142428
$$909$$ −1.34752 −0.0446946
$$910$$ 0 0
$$911$$ 13.8885 0.460148 0.230074 0.973173i $$-0.426103\pi$$
0.230074 + 0.973173i $$0.426103\pi$$
$$912$$ 7.23607 0.239610
$$913$$ −20.0000 −0.661903
$$914$$ −24.7639 −0.819118
$$915$$ 0 0
$$916$$ −23.1246 −0.764059
$$917$$ −73.3050 −2.42074
$$918$$ 24.4721 0.807701
$$919$$ −5.88854 −0.194245 −0.0971226 0.995272i $$-0.530964\pi$$
−0.0971226 + 0.995272i $$0.530964\pi$$
$$920$$ 0 0
$$921$$ 20.7984 0.685330
$$922$$ 38.9443 1.28256
$$923$$ 8.40325 0.276596
$$924$$ −7.23607 −0.238049
$$925$$ 0 0
$$926$$ −4.56231 −0.149927
$$927$$ 6.59675 0.216666
$$928$$ 9.32624 0.306149
$$929$$ −27.4508 −0.900633 −0.450317 0.892869i $$-0.648689\pi$$
−0.450317 + 0.892869i $$0.648689\pi$$
$$930$$ 0 0
$$931$$ 15.5279 0.508905
$$932$$ 6.56231 0.214955
$$933$$ −43.7426 −1.43207
$$934$$ −24.3607 −0.797106
$$935$$ 0 0
$$936$$ 1.09017 0.0356333
$$937$$ 48.0476 1.56965 0.784823 0.619720i $$-0.212754\pi$$
0.784823 + 0.619720i $$0.212754\pi$$
$$938$$ 3.52786 0.115189
$$939$$ −45.5967 −1.48799
$$940$$ 0 0
$$941$$ −26.1803 −0.853455 −0.426727 0.904380i $$-0.640334\pi$$
−0.426727 + 0.904380i $$0.640334\pi$$
$$942$$ −26.6525 −0.868385
$$943$$ −27.4508 −0.893923
$$944$$ −4.76393 −0.155053
$$945$$ 0 0
$$946$$ 7.23607 0.235265
$$947$$ −18.8328 −0.611984 −0.305992 0.952034i $$-0.598988\pi$$
−0.305992 + 0.952034i $$0.598988\pi$$
$$948$$ −13.8541 −0.449960
$$949$$ −20.2361 −0.656891
$$950$$ 0 0
$$951$$ −33.8885 −1.09891
$$952$$ −14.4721 −0.469045
$$953$$ −11.4508 −0.370929 −0.185465 0.982651i $$-0.559379\pi$$
−0.185465 + 0.982651i $$0.559379\pi$$
$$954$$ −0.180340 −0.00583872
$$955$$ 0 0
$$956$$ 9.85410 0.318704
$$957$$ 20.8541 0.674117
$$958$$ −36.5623 −1.18127
$$959$$ −11.8885 −0.383901
$$960$$ 0 0
$$961$$ 23.4934 0.757852
$$962$$ −2.85410 −0.0920199
$$963$$ −2.79837 −0.0901763
$$964$$ −1.52786 −0.0492092
$$965$$ 0 0
$$966$$ 14.9443 0.480824
$$967$$ −45.2705 −1.45580 −0.727901 0.685683i $$-0.759504\pi$$
−0.727901 + 0.685683i $$0.759504\pi$$
$$968$$ 9.09017 0.292169
$$969$$ 32.3607 1.03957
$$970$$ 0 0
$$971$$ 42.3262 1.35831 0.679157 0.733993i $$-0.262346\pi$$
0.679157 + 0.733993i $$0.262346\pi$$
$$972$$ 3.94427 0.126513
$$973$$ 15.7082 0.503582
$$974$$ 37.3050 1.19533
$$975$$ 0 0
$$976$$ 10.6180 0.339875
$$977$$ −43.5279 −1.39258 −0.696290 0.717761i $$-0.745167\pi$$
−0.696290 + 0.717761i $$0.745167\pi$$
$$978$$ −5.70820 −0.182528
$$979$$ 2.11146 0.0674824
$$980$$ 0 0
$$981$$ −1.12461 −0.0359061
$$982$$ 28.4508 0.907903
$$983$$ 31.7771 1.01353 0.506766 0.862084i $$-0.330841\pi$$
0.506766 + 0.862084i $$0.330841\pi$$
$$984$$ −15.5623 −0.496108
$$985$$ 0 0
$$986$$ 41.7082 1.32826
$$987$$ 6.47214 0.206010
$$988$$ 12.7639 0.406075
$$989$$ −14.9443 −0.475200
$$990$$ 0 0
$$991$$ 33.1033 1.05156 0.525781 0.850620i $$-0.323773\pi$$
0.525781 + 0.850620i $$0.323773\pi$$
$$992$$ −7.38197 −0.234378
$$993$$ 45.3050 1.43771
$$994$$ −9.52786 −0.302205
$$995$$ 0 0
$$996$$ 23.4164 0.741977
$$997$$ 17.7771 0.563006 0.281503 0.959560i $$-0.409167\pi$$
0.281503 + 0.959560i $$0.409167\pi$$
$$998$$ −10.2918 −0.325781
$$999$$ −5.47214 −0.173131
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 1850.2.a.t.1.2 2
5.2 odd 4 1850.2.b.j.149.1 4
5.3 odd 4 1850.2.b.j.149.4 4
5.4 even 2 74.2.a.b.1.1 2
15.14 odd 2 666.2.a.i.1.1 2
20.19 odd 2 592.2.a.g.1.2 2
35.34 odd 2 3626.2.a.s.1.2 2
40.19 odd 2 2368.2.a.u.1.1 2
40.29 even 2 2368.2.a.y.1.2 2
55.54 odd 2 8954.2.a.j.1.1 2
60.59 even 2 5328.2.a.bc.1.1 2
185.184 even 2 2738.2.a.g.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.b.1.1 2 5.4 even 2
592.2.a.g.1.2 2 20.19 odd 2
666.2.a.i.1.1 2 15.14 odd 2
1850.2.a.t.1.2 2 1.1 even 1 trivial
1850.2.b.j.149.1 4 5.2 odd 4
1850.2.b.j.149.4 4 5.3 odd 4
2368.2.a.u.1.1 2 40.19 odd 2
2368.2.a.y.1.2 2 40.29 even 2
2738.2.a.g.1.1 2 185.184 even 2
3626.2.a.s.1.2 2 35.34 odd 2
5328.2.a.bc.1.1 2 60.59 even 2
8954.2.a.j.1.1 2 55.54 odd 2