Properties

 Label 1850.2.a.h.1.1 Level $1850$ Weight $2$ Character 1850.1 Self dual yes Analytic conductor $14.772$ Analytic rank $1$ Dimension $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 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)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} -3.00000 q^{6} +1.00000 q^{8} +6.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} -3.00000 q^{6} +1.00000 q^{8} +6.00000 q^{9} -1.00000 q^{11} -3.00000 q^{12} -2.00000 q^{13} +1.00000 q^{16} -7.00000 q^{17} +6.00000 q^{18} +5.00000 q^{19} -1.00000 q^{22} +6.00000 q^{23} -3.00000 q^{24} -2.00000 q^{26} -9.00000 q^{27} -4.00000 q^{31} +1.00000 q^{32} +3.00000 q^{33} -7.00000 q^{34} +6.00000 q^{36} +1.00000 q^{37} +5.00000 q^{38} +6.00000 q^{39} -3.00000 q^{41} -4.00000 q^{43} -1.00000 q^{44} +6.00000 q^{46} -4.00000 q^{47} -3.00000 q^{48} -7.00000 q^{49} +21.0000 q^{51} -2.00000 q^{52} +2.00000 q^{53} -9.00000 q^{54} -15.0000 q^{57} +4.00000 q^{59} -8.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +3.00000 q^{66} -13.0000 q^{67} -7.00000 q^{68} -18.0000 q^{69} -6.00000 q^{71} +6.00000 q^{72} -7.00000 q^{73} +1.00000 q^{74} +5.00000 q^{76} +6.00000 q^{78} +14.0000 q^{79} +9.00000 q^{81} -3.00000 q^{82} +3.00000 q^{83} -4.00000 q^{86} -1.00000 q^{88} -7.00000 q^{89} +6.00000 q^{92} +12.0000 q^{93} -4.00000 q^{94} -3.00000 q^{96} -18.0000 q^{97} -7.00000 q^{98} -6.00000 q^{99} +O(q^{100})$$

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
$$3$$ −3.00000 −1.73205 −0.866025 0.500000i $$-0.833333\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ −3.00000 −1.22474
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 6.00000 2.00000
$$10$$ 0 0
$$11$$ −1.00000 −0.301511 −0.150756 0.988571i $$-0.548171\pi$$
−0.150756 + 0.988571i $$0.548171\pi$$
$$12$$ −3.00000 −0.866025
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −7.00000 −1.69775 −0.848875 0.528594i $$-0.822719\pi$$
−0.848875 + 0.528594i $$0.822719\pi$$
$$18$$ 6.00000 1.41421
$$19$$ 5.00000 1.14708 0.573539 0.819178i $$-0.305570\pi$$
0.573539 + 0.819178i $$0.305570\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −1.00000 −0.213201
$$23$$ 6.00000 1.25109 0.625543 0.780189i $$-0.284877\pi$$
0.625543 + 0.780189i $$0.284877\pi$$
$$24$$ −3.00000 −0.612372
$$25$$ 0 0
$$26$$ −2.00000 −0.392232
$$27$$ −9.00000 −1.73205
$$28$$ 0 0
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 3.00000 0.522233
$$34$$ −7.00000 −1.20049
$$35$$ 0 0
$$36$$ 6.00000 1.00000
$$37$$ 1.00000 0.164399
$$38$$ 5.00000 0.811107
$$39$$ 6.00000 0.960769
$$40$$ 0 0
$$41$$ −3.00000 −0.468521 −0.234261 0.972174i $$-0.575267\pi$$
−0.234261 + 0.972174i $$0.575267\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 0 0
$$46$$ 6.00000 0.884652
$$47$$ −4.00000 −0.583460 −0.291730 0.956501i $$-0.594231\pi$$
−0.291730 + 0.956501i $$0.594231\pi$$
$$48$$ −3.00000 −0.433013
$$49$$ −7.00000 −1.00000
$$50$$ 0 0
$$51$$ 21.0000 2.94059
$$52$$ −2.00000 −0.277350
$$53$$ 2.00000 0.274721 0.137361 0.990521i $$-0.456138\pi$$
0.137361 + 0.990521i $$0.456138\pi$$
$$54$$ −9.00000 −1.22474
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −15.0000 −1.98680
$$58$$ 0 0
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ −8.00000 −1.02430 −0.512148 0.858898i $$-0.671150\pi$$
−0.512148 + 0.858898i $$0.671150\pi$$
$$62$$ −4.00000 −0.508001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 3.00000 0.369274
$$67$$ −13.0000 −1.58820 −0.794101 0.607785i $$-0.792058\pi$$
−0.794101 + 0.607785i $$0.792058\pi$$
$$68$$ −7.00000 −0.848875
$$69$$ −18.0000 −2.16695
$$70$$ 0 0
$$71$$ −6.00000 −0.712069 −0.356034 0.934473i $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ 6.00000 0.707107
$$73$$ −7.00000 −0.819288 −0.409644 0.912245i $$-0.634347\pi$$
−0.409644 + 0.912245i $$0.634347\pi$$
$$74$$ 1.00000 0.116248
$$75$$ 0 0
$$76$$ 5.00000 0.573539
$$77$$ 0 0
$$78$$ 6.00000 0.679366
$$79$$ 14.0000 1.57512 0.787562 0.616236i $$-0.211343\pi$$
0.787562 + 0.616236i $$0.211343\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ −3.00000 −0.331295
$$83$$ 3.00000 0.329293 0.164646 0.986353i $$-0.447352\pi$$
0.164646 + 0.986353i $$0.447352\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ 0 0
$$88$$ −1.00000 −0.106600
$$89$$ −7.00000 −0.741999 −0.370999 0.928633i $$-0.620985\pi$$
−0.370999 + 0.928633i $$0.620985\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 6.00000 0.625543
$$93$$ 12.0000 1.24434
$$94$$ −4.00000 −0.412568
$$95$$ 0 0
$$96$$ −3.00000 −0.306186
$$97$$ −18.0000 −1.82762 −0.913812 0.406138i $$-0.866875\pi$$
−0.913812 + 0.406138i $$0.866875\pi$$
$$98$$ −7.00000 −0.707107
$$99$$ −6.00000 −0.603023
$$100$$ 0 0
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 21.0000 2.07931
$$103$$ −18.0000 −1.77359 −0.886796 0.462160i $$-0.847074\pi$$
−0.886796 + 0.462160i $$0.847074\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 0 0
$$106$$ 2.00000 0.194257
$$107$$ 5.00000 0.483368 0.241684 0.970355i $$-0.422300\pi$$
0.241684 + 0.970355i $$0.422300\pi$$
$$108$$ −9.00000 −0.866025
$$109$$ −6.00000 −0.574696 −0.287348 0.957826i $$-0.592774\pi$$
−0.287348 + 0.957826i $$0.592774\pi$$
$$110$$ 0 0
$$111$$ −3.00000 −0.284747
$$112$$ 0 0
$$113$$ −17.0000 −1.59923 −0.799613 0.600516i $$-0.794962\pi$$
−0.799613 + 0.600516i $$0.794962\pi$$
$$114$$ −15.0000 −1.40488
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −12.0000 −1.10940
$$118$$ 4.00000 0.368230
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −10.0000 −0.909091
$$122$$ −8.00000 −0.724286
$$123$$ 9.00000 0.811503
$$124$$ −4.00000 −0.359211
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 4.00000 0.354943 0.177471 0.984126i $$-0.443208\pi$$
0.177471 + 0.984126i $$0.443208\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 12.0000 1.05654
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 3.00000 0.261116
$$133$$ 0 0
$$134$$ −13.0000 −1.12303
$$135$$ 0 0
$$136$$ −7.00000 −0.600245
$$137$$ 15.0000 1.28154 0.640768 0.767734i $$-0.278616\pi$$
0.640768 + 0.767734i $$0.278616\pi$$
$$138$$ −18.0000 −1.53226
$$139$$ −1.00000 −0.0848189 −0.0424094 0.999100i $$-0.513503\pi$$
−0.0424094 + 0.999100i $$0.513503\pi$$
$$140$$ 0 0
$$141$$ 12.0000 1.01058
$$142$$ −6.00000 −0.503509
$$143$$ 2.00000 0.167248
$$144$$ 6.00000 0.500000
$$145$$ 0 0
$$146$$ −7.00000 −0.579324
$$147$$ 21.0000 1.73205
$$148$$ 1.00000 0.0821995
$$149$$ −22.0000 −1.80231 −0.901155 0.433497i $$-0.857280\pi$$
−0.901155 + 0.433497i $$0.857280\pi$$
$$150$$ 0 0
$$151$$ −18.0000 −1.46482 −0.732410 0.680864i $$-0.761604\pi$$
−0.732410 + 0.680864i $$0.761604\pi$$
$$152$$ 5.00000 0.405554
$$153$$ −42.0000 −3.39550
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 6.00000 0.480384
$$157$$ 12.0000 0.957704 0.478852 0.877896i $$-0.341053\pi$$
0.478852 + 0.877896i $$0.341053\pi$$
$$158$$ 14.0000 1.11378
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 9.00000 0.707107
$$163$$ 1.00000 0.0783260 0.0391630 0.999233i $$-0.487531\pi$$
0.0391630 + 0.999233i $$0.487531\pi$$
$$164$$ −3.00000 −0.234261
$$165$$ 0 0
$$166$$ 3.00000 0.232845
$$167$$ −2.00000 −0.154765 −0.0773823 0.997001i $$-0.524656\pi$$
−0.0773823 + 0.997001i $$0.524656\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 30.0000 2.29416
$$172$$ −4.00000 −0.304997
$$173$$ 8.00000 0.608229 0.304114 0.952636i $$-0.401639\pi$$
0.304114 + 0.952636i $$0.401639\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −1.00000 −0.0753778
$$177$$ −12.0000 −0.901975
$$178$$ −7.00000 −0.524672
$$179$$ 13.0000 0.971666 0.485833 0.874052i $$-0.338516\pi$$
0.485833 + 0.874052i $$0.338516\pi$$
$$180$$ 0 0
$$181$$ −2.00000 −0.148659 −0.0743294 0.997234i $$-0.523682\pi$$
−0.0743294 + 0.997234i $$0.523682\pi$$
$$182$$ 0 0
$$183$$ 24.0000 1.77413
$$184$$ 6.00000 0.442326
$$185$$ 0 0
$$186$$ 12.0000 0.879883
$$187$$ 7.00000 0.511891
$$188$$ −4.00000 −0.291730
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 6.00000 0.434145 0.217072 0.976156i $$-0.430349\pi$$
0.217072 + 0.976156i $$0.430349\pi$$
$$192$$ −3.00000 −0.216506
$$193$$ 25.0000 1.79954 0.899770 0.436365i $$-0.143734\pi$$
0.899770 + 0.436365i $$0.143734\pi$$
$$194$$ −18.0000 −1.29232
$$195$$ 0 0
$$196$$ −7.00000 −0.500000
$$197$$ −8.00000 −0.569976 −0.284988 0.958531i $$-0.591990\pi$$
−0.284988 + 0.958531i $$0.591990\pi$$
$$198$$ −6.00000 −0.426401
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 0 0
$$201$$ 39.0000 2.75085
$$202$$ 6.00000 0.422159
$$203$$ 0 0
$$204$$ 21.0000 1.47029
$$205$$ 0 0
$$206$$ −18.0000 −1.25412
$$207$$ 36.0000 2.50217
$$208$$ −2.00000 −0.138675
$$209$$ −5.00000 −0.345857
$$210$$ 0 0
$$211$$ 13.0000 0.894957 0.447478 0.894295i $$-0.352322\pi$$
0.447478 + 0.894295i $$0.352322\pi$$
$$212$$ 2.00000 0.137361
$$213$$ 18.0000 1.23334
$$214$$ 5.00000 0.341793
$$215$$ 0 0
$$216$$ −9.00000 −0.612372
$$217$$ 0 0
$$218$$ −6.00000 −0.406371
$$219$$ 21.0000 1.41905
$$220$$ 0 0
$$221$$ 14.0000 0.941742
$$222$$ −3.00000 −0.201347
$$223$$ 16.0000 1.07144 0.535720 0.844396i $$-0.320040\pi$$
0.535720 + 0.844396i $$0.320040\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −17.0000 −1.13082
$$227$$ 28.0000 1.85843 0.929213 0.369546i $$-0.120487\pi$$
0.929213 + 0.369546i $$0.120487\pi$$
$$228$$ −15.0000 −0.993399
$$229$$ 24.0000 1.58596 0.792982 0.609245i $$-0.208527\pi$$
0.792982 + 0.609245i $$0.208527\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −14.0000 −0.917170 −0.458585 0.888650i $$-0.651644\pi$$
−0.458585 + 0.888650i $$0.651644\pi$$
$$234$$ −12.0000 −0.784465
$$235$$ 0 0
$$236$$ 4.00000 0.260378
$$237$$ −42.0000 −2.72819
$$238$$ 0 0
$$239$$ 12.0000 0.776215 0.388108 0.921614i $$-0.373129\pi$$
0.388108 + 0.921614i $$0.373129\pi$$
$$240$$ 0 0
$$241$$ −25.0000 −1.61039 −0.805196 0.593009i $$-0.797940\pi$$
−0.805196 + 0.593009i $$0.797940\pi$$
$$242$$ −10.0000 −0.642824
$$243$$ 0 0
$$244$$ −8.00000 −0.512148
$$245$$ 0 0
$$246$$ 9.00000 0.573819
$$247$$ −10.0000 −0.636285
$$248$$ −4.00000 −0.254000
$$249$$ −9.00000 −0.570352
$$250$$ 0 0
$$251$$ −1.00000 −0.0631194 −0.0315597 0.999502i $$-0.510047\pi$$
−0.0315597 + 0.999502i $$0.510047\pi$$
$$252$$ 0 0
$$253$$ −6.00000 −0.377217
$$254$$ 4.00000 0.250982
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −2.00000 −0.124757 −0.0623783 0.998053i $$-0.519869\pi$$
−0.0623783 + 0.998053i $$0.519869\pi$$
$$258$$ 12.0000 0.747087
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −12.0000 −0.741362
$$263$$ 14.0000 0.863277 0.431638 0.902047i $$-0.357936\pi$$
0.431638 + 0.902047i $$0.357936\pi$$
$$264$$ 3.00000 0.184637
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 21.0000 1.28518
$$268$$ −13.0000 −0.794101
$$269$$ −4.00000 −0.243884 −0.121942 0.992537i $$-0.538912\pi$$
−0.121942 + 0.992537i $$0.538912\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ −7.00000 −0.424437
$$273$$ 0 0
$$274$$ 15.0000 0.906183
$$275$$ 0 0
$$276$$ −18.0000 −1.08347
$$277$$ −20.0000 −1.20168 −0.600842 0.799368i $$-0.705168\pi$$
−0.600842 + 0.799368i $$0.705168\pi$$
$$278$$ −1.00000 −0.0599760
$$279$$ −24.0000 −1.43684
$$280$$ 0 0
$$281$$ −22.0000 −1.31241 −0.656205 0.754583i $$-0.727839\pi$$
−0.656205 + 0.754583i $$0.727839\pi$$
$$282$$ 12.0000 0.714590
$$283$$ −7.00000 −0.416107 −0.208053 0.978117i $$-0.566713\pi$$
−0.208053 + 0.978117i $$0.566713\pi$$
$$284$$ −6.00000 −0.356034
$$285$$ 0 0
$$286$$ 2.00000 0.118262
$$287$$ 0 0
$$288$$ 6.00000 0.353553
$$289$$ 32.0000 1.88235
$$290$$ 0 0
$$291$$ 54.0000 3.16554
$$292$$ −7.00000 −0.409644
$$293$$ 24.0000 1.40209 0.701047 0.713115i $$-0.252716\pi$$
0.701047 + 0.713115i $$0.252716\pi$$
$$294$$ 21.0000 1.22474
$$295$$ 0 0
$$296$$ 1.00000 0.0581238
$$297$$ 9.00000 0.522233
$$298$$ −22.0000 −1.27443
$$299$$ −12.0000 −0.693978
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −18.0000 −1.03578
$$303$$ −18.0000 −1.03407
$$304$$ 5.00000 0.286770
$$305$$ 0 0
$$306$$ −42.0000 −2.40098
$$307$$ −7.00000 −0.399511 −0.199756 0.979846i $$-0.564015\pi$$
−0.199756 + 0.979846i $$0.564015\pi$$
$$308$$ 0 0
$$309$$ 54.0000 3.07195
$$310$$ 0 0
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ 6.00000 0.339683
$$313$$ 10.0000 0.565233 0.282617 0.959233i $$-0.408798\pi$$
0.282617 + 0.959233i $$0.408798\pi$$
$$314$$ 12.0000 0.677199
$$315$$ 0 0
$$316$$ 14.0000 0.787562
$$317$$ −2.00000 −0.112331 −0.0561656 0.998421i $$-0.517887\pi$$
−0.0561656 + 0.998421i $$0.517887\pi$$
$$318$$ −6.00000 −0.336463
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −15.0000 −0.837218
$$322$$ 0 0
$$323$$ −35.0000 −1.94745
$$324$$ 9.00000 0.500000
$$325$$ 0 0
$$326$$ 1.00000 0.0553849
$$327$$ 18.0000 0.995402
$$328$$ −3.00000 −0.165647
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −19.0000 −1.04433 −0.522167 0.852843i $$-0.674876\pi$$
−0.522167 + 0.852843i $$0.674876\pi$$
$$332$$ 3.00000 0.164646
$$333$$ 6.00000 0.328798
$$334$$ −2.00000 −0.109435
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −31.0000 −1.68868 −0.844339 0.535810i $$-0.820006\pi$$
−0.844339 + 0.535810i $$0.820006\pi$$
$$338$$ −9.00000 −0.489535
$$339$$ 51.0000 2.76994
$$340$$ 0 0
$$341$$ 4.00000 0.216612
$$342$$ 30.0000 1.62221
$$343$$ 0 0
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ 8.00000 0.430083
$$347$$ 23.0000 1.23470 0.617352 0.786687i $$-0.288205\pi$$
0.617352 + 0.786687i $$0.288205\pi$$
$$348$$ 0 0
$$349$$ −12.0000 −0.642345 −0.321173 0.947021i $$-0.604077\pi$$
−0.321173 + 0.947021i $$0.604077\pi$$
$$350$$ 0 0
$$351$$ 18.0000 0.960769
$$352$$ −1.00000 −0.0533002
$$353$$ 6.00000 0.319348 0.159674 0.987170i $$-0.448956\pi$$
0.159674 + 0.987170i $$0.448956\pi$$
$$354$$ −12.0000 −0.637793
$$355$$ 0 0
$$356$$ −7.00000 −0.370999
$$357$$ 0 0
$$358$$ 13.0000 0.687071
$$359$$ −6.00000 −0.316668 −0.158334 0.987386i $$-0.550612\pi$$
−0.158334 + 0.987386i $$0.550612\pi$$
$$360$$ 0 0
$$361$$ 6.00000 0.315789
$$362$$ −2.00000 −0.105118
$$363$$ 30.0000 1.57459
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 24.0000 1.25450
$$367$$ −26.0000 −1.35719 −0.678594 0.734513i $$-0.737411\pi$$
−0.678594 + 0.734513i $$0.737411\pi$$
$$368$$ 6.00000 0.312772
$$369$$ −18.0000 −0.937043
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 12.0000 0.622171
$$373$$ 32.0000 1.65690 0.828449 0.560065i $$-0.189224\pi$$
0.828449 + 0.560065i $$0.189224\pi$$
$$374$$ 7.00000 0.361961
$$375$$ 0 0
$$376$$ −4.00000 −0.206284
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −19.0000 −0.975964 −0.487982 0.872854i $$-0.662267\pi$$
−0.487982 + 0.872854i $$0.662267\pi$$
$$380$$ 0 0
$$381$$ −12.0000 −0.614779
$$382$$ 6.00000 0.306987
$$383$$ 36.0000 1.83951 0.919757 0.392488i $$-0.128386\pi$$
0.919757 + 0.392488i $$0.128386\pi$$
$$384$$ −3.00000 −0.153093
$$385$$ 0 0
$$386$$ 25.0000 1.27247
$$387$$ −24.0000 −1.21999
$$388$$ −18.0000 −0.913812
$$389$$ −8.00000 −0.405616 −0.202808 0.979219i $$-0.565007\pi$$
−0.202808 + 0.979219i $$0.565007\pi$$
$$390$$ 0 0
$$391$$ −42.0000 −2.12403
$$392$$ −7.00000 −0.353553
$$393$$ 36.0000 1.81596
$$394$$ −8.00000 −0.403034
$$395$$ 0 0
$$396$$ −6.00000 −0.301511
$$397$$ 2.00000 0.100377 0.0501886 0.998740i $$-0.484018\pi$$
0.0501886 + 0.998740i $$0.484018\pi$$
$$398$$ 16.0000 0.802008
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 23.0000 1.14857 0.574283 0.818657i $$-0.305281\pi$$
0.574283 + 0.818657i $$0.305281\pi$$
$$402$$ 39.0000 1.94514
$$403$$ 8.00000 0.398508
$$404$$ 6.00000 0.298511
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −1.00000 −0.0495682
$$408$$ 21.0000 1.03965
$$409$$ 11.0000 0.543915 0.271957 0.962309i $$-0.412329\pi$$
0.271957 + 0.962309i $$0.412329\pi$$
$$410$$ 0 0
$$411$$ −45.0000 −2.21969
$$412$$ −18.0000 −0.886796
$$413$$ 0 0
$$414$$ 36.0000 1.76930
$$415$$ 0 0
$$416$$ −2.00000 −0.0980581
$$417$$ 3.00000 0.146911
$$418$$ −5.00000 −0.244558
$$419$$ −17.0000 −0.830504 −0.415252 0.909706i $$-0.636307\pi$$
−0.415252 + 0.909706i $$0.636307\pi$$
$$420$$ 0 0
$$421$$ 24.0000 1.16969 0.584844 0.811146i $$-0.301156\pi$$
0.584844 + 0.811146i $$0.301156\pi$$
$$422$$ 13.0000 0.632830
$$423$$ −24.0000 −1.16692
$$424$$ 2.00000 0.0971286
$$425$$ 0 0
$$426$$ 18.0000 0.872103
$$427$$ 0 0
$$428$$ 5.00000 0.241684
$$429$$ −6.00000 −0.289683
$$430$$ 0 0
$$431$$ −38.0000 −1.83040 −0.915198 0.403005i $$-0.867966\pi$$
−0.915198 + 0.403005i $$0.867966\pi$$
$$432$$ −9.00000 −0.433013
$$433$$ 11.0000 0.528626 0.264313 0.964437i $$-0.414855\pi$$
0.264313 + 0.964437i $$0.414855\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −6.00000 −0.287348
$$437$$ 30.0000 1.43509
$$438$$ 21.0000 1.00342
$$439$$ −32.0000 −1.52728 −0.763638 0.645644i $$-0.776589\pi$$
−0.763638 + 0.645644i $$0.776589\pi$$
$$440$$ 0 0
$$441$$ −42.0000 −2.00000
$$442$$ 14.0000 0.665912
$$443$$ 15.0000 0.712672 0.356336 0.934358i $$-0.384026\pi$$
0.356336 + 0.934358i $$0.384026\pi$$
$$444$$ −3.00000 −0.142374
$$445$$ 0 0
$$446$$ 16.0000 0.757622
$$447$$ 66.0000 3.12169
$$448$$ 0 0
$$449$$ 15.0000 0.707894 0.353947 0.935266i $$-0.384839\pi$$
0.353947 + 0.935266i $$0.384839\pi$$
$$450$$ 0 0
$$451$$ 3.00000 0.141264
$$452$$ −17.0000 −0.799613
$$453$$ 54.0000 2.53714
$$454$$ 28.0000 1.31411
$$455$$ 0 0
$$456$$ −15.0000 −0.702439
$$457$$ 21.0000 0.982339 0.491169 0.871064i $$-0.336570\pi$$
0.491169 + 0.871064i $$0.336570\pi$$
$$458$$ 24.0000 1.12145
$$459$$ 63.0000 2.94059
$$460$$ 0 0
$$461$$ 4.00000 0.186299 0.0931493 0.995652i $$-0.470307\pi$$
0.0931493 + 0.995652i $$0.470307\pi$$
$$462$$ 0 0
$$463$$ −26.0000 −1.20832 −0.604161 0.796862i $$-0.706492\pi$$
−0.604161 + 0.796862i $$0.706492\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ −14.0000 −0.648537
$$467$$ −4.00000 −0.185098 −0.0925490 0.995708i $$-0.529501\pi$$
−0.0925490 + 0.995708i $$0.529501\pi$$
$$468$$ −12.0000 −0.554700
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −36.0000 −1.65879
$$472$$ 4.00000 0.184115
$$473$$ 4.00000 0.183920
$$474$$ −42.0000 −1.92912
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 12.0000 0.549442
$$478$$ 12.0000 0.548867
$$479$$ 4.00000 0.182765 0.0913823 0.995816i $$-0.470871\pi$$
0.0913823 + 0.995816i $$0.470871\pi$$
$$480$$ 0 0
$$481$$ −2.00000 −0.0911922
$$482$$ −25.0000 −1.13872
$$483$$ 0 0
$$484$$ −10.0000 −0.454545
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 16.0000 0.725029 0.362515 0.931978i $$-0.381918\pi$$
0.362515 + 0.931978i $$0.381918\pi$$
$$488$$ −8.00000 −0.362143
$$489$$ −3.00000 −0.135665
$$490$$ 0 0
$$491$$ 44.0000 1.98569 0.992846 0.119401i $$-0.0380974\pi$$
0.992846 + 0.119401i $$0.0380974\pi$$
$$492$$ 9.00000 0.405751
$$493$$ 0 0
$$494$$ −10.0000 −0.449921
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ 0 0
$$498$$ −9.00000 −0.403300
$$499$$ 16.0000 0.716258 0.358129 0.933672i $$-0.383415\pi$$
0.358129 + 0.933672i $$0.383415\pi$$
$$500$$ 0 0
$$501$$ 6.00000 0.268060
$$502$$ −1.00000 −0.0446322
$$503$$ −36.0000 −1.60516 −0.802580 0.596544i $$-0.796540\pi$$
−0.802580 + 0.596544i $$0.796540\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −6.00000 −0.266733
$$507$$ 27.0000 1.19911
$$508$$ 4.00000 0.177471
$$509$$ −36.0000 −1.59567 −0.797836 0.602875i $$-0.794022\pi$$
−0.797836 + 0.602875i $$0.794022\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ −45.0000 −1.98680
$$514$$ −2.00000 −0.0882162
$$515$$ 0 0
$$516$$ 12.0000 0.528271
$$517$$ 4.00000 0.175920
$$518$$ 0 0
$$519$$ −24.0000 −1.05348
$$520$$ 0 0
$$521$$ −35.0000 −1.53338 −0.766689 0.642019i $$-0.778097\pi$$
−0.766689 + 0.642019i $$0.778097\pi$$
$$522$$ 0 0
$$523$$ −45.0000 −1.96771 −0.983856 0.178960i $$-0.942727\pi$$
−0.983856 + 0.178960i $$0.942727\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ 14.0000 0.610429
$$527$$ 28.0000 1.21970
$$528$$ 3.00000 0.130558
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ 24.0000 1.04151
$$532$$ 0 0
$$533$$ 6.00000 0.259889
$$534$$ 21.0000 0.908759
$$535$$ 0 0
$$536$$ −13.0000 −0.561514
$$537$$ −39.0000 −1.68297
$$538$$ −4.00000 −0.172452
$$539$$ 7.00000 0.301511
$$540$$ 0 0
$$541$$ −30.0000 −1.28980 −0.644900 0.764267i $$-0.723101\pi$$
−0.644900 + 0.764267i $$0.723101\pi$$
$$542$$ 16.0000 0.687259
$$543$$ 6.00000 0.257485
$$544$$ −7.00000 −0.300123
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 11.0000 0.470326 0.235163 0.971956i $$-0.424438\pi$$
0.235163 + 0.971956i $$0.424438\pi$$
$$548$$ 15.0000 0.640768
$$549$$ −48.0000 −2.04859
$$550$$ 0 0
$$551$$ 0 0
$$552$$ −18.0000 −0.766131
$$553$$ 0 0
$$554$$ −20.0000 −0.849719
$$555$$ 0 0
$$556$$ −1.00000 −0.0424094
$$557$$ 40.0000 1.69485 0.847427 0.530912i $$-0.178150\pi$$
0.847427 + 0.530912i $$0.178150\pi$$
$$558$$ −24.0000 −1.01600
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ −21.0000 −0.886621
$$562$$ −22.0000 −0.928014
$$563$$ 12.0000 0.505740 0.252870 0.967500i $$-0.418626\pi$$
0.252870 + 0.967500i $$0.418626\pi$$
$$564$$ 12.0000 0.505291
$$565$$ 0 0
$$566$$ −7.00000 −0.294232
$$567$$ 0 0
$$568$$ −6.00000 −0.251754
$$569$$ 39.0000 1.63497 0.817483 0.575953i $$-0.195369\pi$$
0.817483 + 0.575953i $$0.195369\pi$$
$$570$$ 0 0
$$571$$ −36.0000 −1.50655 −0.753277 0.657704i $$-0.771528\pi$$
−0.753277 + 0.657704i $$0.771528\pi$$
$$572$$ 2.00000 0.0836242
$$573$$ −18.0000 −0.751961
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 6.00000 0.250000
$$577$$ 11.0000 0.457936 0.228968 0.973434i $$-0.426465\pi$$
0.228968 + 0.973434i $$0.426465\pi$$
$$578$$ 32.0000 1.33102
$$579$$ −75.0000 −3.11689
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 54.0000 2.23837
$$583$$ −2.00000 −0.0828315
$$584$$ −7.00000 −0.289662
$$585$$ 0 0
$$586$$ 24.0000 0.991431
$$587$$ −3.00000 −0.123823 −0.0619116 0.998082i $$-0.519720\pi$$
−0.0619116 + 0.998082i $$0.519720\pi$$
$$588$$ 21.0000 0.866025
$$589$$ −20.0000 −0.824086
$$590$$ 0 0
$$591$$ 24.0000 0.987228
$$592$$ 1.00000 0.0410997
$$593$$ 41.0000 1.68367 0.841834 0.539736i $$-0.181476\pi$$
0.841834 + 0.539736i $$0.181476\pi$$
$$594$$ 9.00000 0.369274
$$595$$ 0 0
$$596$$ −22.0000 −0.901155
$$597$$ −48.0000 −1.96451
$$598$$ −12.0000 −0.490716
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ −17.0000 −0.693444 −0.346722 0.937968i $$-0.612705\pi$$
−0.346722 + 0.937968i $$0.612705\pi$$
$$602$$ 0 0
$$603$$ −78.0000 −3.17641
$$604$$ −18.0000 −0.732410
$$605$$ 0 0
$$606$$ −18.0000 −0.731200
$$607$$ −18.0000 −0.730597 −0.365299 0.930890i $$-0.619033\pi$$
−0.365299 + 0.930890i $$0.619033\pi$$
$$608$$ 5.00000 0.202777
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 8.00000 0.323645
$$612$$ −42.0000 −1.69775
$$613$$ −42.0000 −1.69636 −0.848182 0.529705i $$-0.822303\pi$$
−0.848182 + 0.529705i $$0.822303\pi$$
$$614$$ −7.00000 −0.282497
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 18.0000 0.724653 0.362326 0.932051i $$-0.381983\pi$$
0.362326 + 0.932051i $$0.381983\pi$$
$$618$$ 54.0000 2.17220
$$619$$ 40.0000 1.60774 0.803868 0.594808i $$-0.202772\pi$$
0.803868 + 0.594808i $$0.202772\pi$$
$$620$$ 0 0
$$621$$ −54.0000 −2.16695
$$622$$ −24.0000 −0.962312
$$623$$ 0 0
$$624$$ 6.00000 0.240192
$$625$$ 0 0
$$626$$ 10.0000 0.399680
$$627$$ 15.0000 0.599042
$$628$$ 12.0000 0.478852
$$629$$ −7.00000 −0.279108
$$630$$ 0 0
$$631$$ 30.0000 1.19428 0.597141 0.802137i $$-0.296303\pi$$
0.597141 + 0.802137i $$0.296303\pi$$
$$632$$ 14.0000 0.556890
$$633$$ −39.0000 −1.55011
$$634$$ −2.00000 −0.0794301
$$635$$ 0 0
$$636$$ −6.00000 −0.237915
$$637$$ 14.0000 0.554700
$$638$$ 0 0
$$639$$ −36.0000 −1.42414
$$640$$ 0 0
$$641$$ 26.0000 1.02694 0.513469 0.858108i $$-0.328360\pi$$
0.513469 + 0.858108i $$0.328360\pi$$
$$642$$ −15.0000 −0.592003
$$643$$ −28.0000 −1.10421 −0.552106 0.833774i $$-0.686176\pi$$
−0.552106 + 0.833774i $$0.686176\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −35.0000 −1.37706
$$647$$ −38.0000 −1.49393 −0.746967 0.664861i $$-0.768491\pi$$
−0.746967 + 0.664861i $$0.768491\pi$$
$$648$$ 9.00000 0.353553
$$649$$ −4.00000 −0.157014
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 1.00000 0.0391630
$$653$$ −40.0000 −1.56532 −0.782660 0.622449i $$-0.786138\pi$$
−0.782660 + 0.622449i $$0.786138\pi$$
$$654$$ 18.0000 0.703856
$$655$$ 0 0
$$656$$ −3.00000 −0.117130
$$657$$ −42.0000 −1.63858
$$658$$ 0 0
$$659$$ 5.00000 0.194772 0.0973862 0.995247i $$-0.468952\pi$$
0.0973862 + 0.995247i $$0.468952\pi$$
$$660$$ 0 0
$$661$$ −16.0000 −0.622328 −0.311164 0.950356i $$-0.600719\pi$$
−0.311164 + 0.950356i $$0.600719\pi$$
$$662$$ −19.0000 −0.738456
$$663$$ −42.0000 −1.63114
$$664$$ 3.00000 0.116423
$$665$$ 0 0
$$666$$ 6.00000 0.232495
$$667$$ 0 0
$$668$$ −2.00000 −0.0773823
$$669$$ −48.0000 −1.85579
$$670$$ 0 0
$$671$$ 8.00000 0.308837
$$672$$ 0 0
$$673$$ −46.0000 −1.77317 −0.886585 0.462566i $$-0.846929\pi$$
−0.886585 + 0.462566i $$0.846929\pi$$
$$674$$ −31.0000 −1.19408
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ −8.00000 −0.307465 −0.153732 0.988113i $$-0.549129\pi$$
−0.153732 + 0.988113i $$0.549129\pi$$
$$678$$ 51.0000 1.95864
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −84.0000 −3.21889
$$682$$ 4.00000 0.153168
$$683$$ −9.00000 −0.344375 −0.172188 0.985064i $$-0.555084\pi$$
−0.172188 + 0.985064i $$0.555084\pi$$
$$684$$ 30.0000 1.14708
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −72.0000 −2.74697
$$688$$ −4.00000 −0.152499
$$689$$ −4.00000 −0.152388
$$690$$ 0 0
$$691$$ −5.00000 −0.190209 −0.0951045 0.995467i $$-0.530319\pi$$
−0.0951045 + 0.995467i $$0.530319\pi$$
$$692$$ 8.00000 0.304114
$$693$$ 0 0
$$694$$ 23.0000 0.873068
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 21.0000 0.795432
$$698$$ −12.0000 −0.454207
$$699$$ 42.0000 1.58859
$$700$$ 0 0
$$701$$ 50.0000 1.88847 0.944237 0.329267i $$-0.106802\pi$$
0.944237 + 0.329267i $$0.106802\pi$$
$$702$$ 18.0000 0.679366
$$703$$ 5.00000 0.188579
$$704$$ −1.00000 −0.0376889
$$705$$ 0 0
$$706$$ 6.00000 0.225813
$$707$$ 0 0
$$708$$ −12.0000 −0.450988
$$709$$ 24.0000 0.901339 0.450669 0.892691i $$-0.351185\pi$$
0.450669 + 0.892691i $$0.351185\pi$$
$$710$$ 0 0
$$711$$ 84.0000 3.15025
$$712$$ −7.00000 −0.262336
$$713$$ −24.0000 −0.898807
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 13.0000 0.485833
$$717$$ −36.0000 −1.34444
$$718$$ −6.00000 −0.223918
$$719$$ 8.00000 0.298350 0.149175 0.988811i $$-0.452338\pi$$
0.149175 + 0.988811i $$0.452338\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 6.00000 0.223297
$$723$$ 75.0000 2.78928
$$724$$ −2.00000 −0.0743294
$$725$$ 0 0
$$726$$ 30.0000 1.11340
$$727$$ 20.0000 0.741759 0.370879 0.928681i $$-0.379056\pi$$
0.370879 + 0.928681i $$0.379056\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −1.00000
$$730$$ 0 0
$$731$$ 28.0000 1.03562
$$732$$ 24.0000 0.887066
$$733$$ 32.0000 1.18195 0.590973 0.806691i $$-0.298744\pi$$
0.590973 + 0.806691i $$0.298744\pi$$
$$734$$ −26.0000 −0.959678
$$735$$ 0 0
$$736$$ 6.00000 0.221163
$$737$$ 13.0000 0.478861
$$738$$ −18.0000 −0.662589
$$739$$ −12.0000 −0.441427 −0.220714 0.975339i $$-0.570839\pi$$
−0.220714 + 0.975339i $$0.570839\pi$$
$$740$$ 0 0
$$741$$ 30.0000 1.10208
$$742$$ 0 0
$$743$$ −54.0000 −1.98107 −0.990534 0.137268i $$-0.956168\pi$$
−0.990534 + 0.137268i $$0.956168\pi$$
$$744$$ 12.0000 0.439941
$$745$$ 0 0
$$746$$ 32.0000 1.17160
$$747$$ 18.0000 0.658586
$$748$$ 7.00000 0.255945
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −4.00000 −0.145962 −0.0729810 0.997333i $$-0.523251\pi$$
−0.0729810 + 0.997333i $$0.523251\pi$$
$$752$$ −4.00000 −0.145865
$$753$$ 3.00000 0.109326
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 32.0000 1.16306 0.581530 0.813525i $$-0.302454\pi$$
0.581530 + 0.813525i $$0.302454\pi$$
$$758$$ −19.0000 −0.690111
$$759$$ 18.0000 0.653359
$$760$$ 0 0
$$761$$ −27.0000 −0.978749 −0.489375 0.872074i $$-0.662775\pi$$
−0.489375 + 0.872074i $$0.662775\pi$$
$$762$$ −12.0000 −0.434714
$$763$$ 0 0
$$764$$ 6.00000 0.217072
$$765$$ 0 0
$$766$$ 36.0000 1.30073
$$767$$ −8.00000 −0.288863
$$768$$ −3.00000 −0.108253
$$769$$ −21.0000 −0.757279 −0.378640 0.925544i $$-0.623608\pi$$
−0.378640 + 0.925544i $$0.623608\pi$$
$$770$$ 0 0
$$771$$ 6.00000 0.216085
$$772$$ 25.0000 0.899770
$$773$$ 22.0000 0.791285 0.395643 0.918405i $$-0.370522\pi$$
0.395643 + 0.918405i $$0.370522\pi$$
$$774$$ −24.0000 −0.862662
$$775$$ 0 0
$$776$$ −18.0000 −0.646162
$$777$$ 0 0
$$778$$ −8.00000 −0.286814
$$779$$ −15.0000 −0.537431
$$780$$ 0 0
$$781$$ 6.00000 0.214697
$$782$$ −42.0000 −1.50192
$$783$$ 0 0
$$784$$ −7.00000 −0.250000
$$785$$ 0 0
$$786$$ 36.0000 1.28408
$$787$$ −24.0000 −0.855508 −0.427754 0.903895i $$-0.640695\pi$$
−0.427754 + 0.903895i $$0.640695\pi$$
$$788$$ −8.00000 −0.284988
$$789$$ −42.0000 −1.49524
$$790$$ 0 0
$$791$$ 0 0
$$792$$ −6.00000 −0.213201
$$793$$ 16.0000 0.568177
$$794$$ 2.00000 0.0709773
$$795$$ 0 0
$$796$$ 16.0000 0.567105
$$797$$ 12.0000 0.425062 0.212531 0.977154i $$-0.431829\pi$$
0.212531 + 0.977154i $$0.431829\pi$$
$$798$$ 0 0
$$799$$ 28.0000 0.990569
$$800$$ 0 0
$$801$$ −42.0000 −1.48400
$$802$$ 23.0000 0.812158
$$803$$ 7.00000 0.247025
$$804$$ 39.0000 1.37542
$$805$$ 0 0
$$806$$ 8.00000 0.281788
$$807$$ 12.0000 0.422420
$$808$$ 6.00000 0.211079
$$809$$ −22.0000 −0.773479 −0.386739 0.922189i $$-0.626399\pi$$
−0.386739 + 0.922189i $$0.626399\pi$$
$$810$$ 0 0
$$811$$ 4.00000 0.140459 0.0702295 0.997531i $$-0.477627\pi$$
0.0702295 + 0.997531i $$0.477627\pi$$
$$812$$ 0 0
$$813$$ −48.0000 −1.68343
$$814$$ −1.00000 −0.0350500
$$815$$ 0 0
$$816$$ 21.0000 0.735147
$$817$$ −20.0000 −0.699711
$$818$$ 11.0000 0.384606
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$822$$ −45.0000 −1.56956
$$823$$ 34.0000 1.18517 0.592583 0.805510i $$-0.298108\pi$$
0.592583 + 0.805510i $$0.298108\pi$$
$$824$$ −18.0000 −0.627060
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 33.0000 1.14752 0.573761 0.819023i $$-0.305484\pi$$
0.573761 + 0.819023i $$0.305484\pi$$
$$828$$ 36.0000 1.25109
$$829$$ 34.0000 1.18087 0.590434 0.807086i $$-0.298956\pi$$
0.590434 + 0.807086i $$0.298956\pi$$
$$830$$ 0 0
$$831$$ 60.0000 2.08138
$$832$$ −2.00000 −0.0693375
$$833$$ 49.0000 1.69775
$$834$$ 3.00000 0.103882
$$835$$ 0 0
$$836$$ −5.00000 −0.172929
$$837$$ 36.0000 1.24434
$$838$$ −17.0000 −0.587255
$$839$$ −6.00000 −0.207143 −0.103572 0.994622i $$-0.533027\pi$$
−0.103572 + 0.994622i $$0.533027\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ 24.0000 0.827095
$$843$$ 66.0000 2.27316
$$844$$ 13.0000 0.447478
$$845$$ 0 0
$$846$$ −24.0000 −0.825137
$$847$$ 0 0
$$848$$ 2.00000 0.0686803
$$849$$ 21.0000 0.720718
$$850$$ 0 0
$$851$$ 6.00000 0.205677
$$852$$ 18.0000 0.616670
$$853$$ 20.0000 0.684787 0.342393 0.939557i $$-0.388762\pi$$
0.342393 + 0.939557i $$0.388762\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 5.00000 0.170896
$$857$$ 9.00000 0.307434 0.153717 0.988115i $$-0.450876\pi$$
0.153717 + 0.988115i $$0.450876\pi$$
$$858$$ −6.00000 −0.204837
$$859$$ 41.0000 1.39890 0.699451 0.714681i $$-0.253428\pi$$
0.699451 + 0.714681i $$0.253428\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −38.0000 −1.29429
$$863$$ 6.00000 0.204242 0.102121 0.994772i $$-0.467437\pi$$
0.102121 + 0.994772i $$0.467437\pi$$
$$864$$ −9.00000 −0.306186
$$865$$ 0 0
$$866$$ 11.0000 0.373795
$$867$$ −96.0000 −3.26033
$$868$$ 0 0
$$869$$ −14.0000 −0.474917
$$870$$ 0 0
$$871$$ 26.0000 0.880976
$$872$$ −6.00000 −0.203186
$$873$$ −108.000 −3.65525
$$874$$ 30.0000 1.01477
$$875$$ 0 0
$$876$$ 21.0000 0.709524
$$877$$ 42.0000 1.41824 0.709120 0.705088i $$-0.249093\pi$$
0.709120 + 0.705088i $$0.249093\pi$$
$$878$$ −32.0000 −1.07995
$$879$$ −72.0000 −2.42850
$$880$$ 0 0
$$881$$ 30.0000 1.01073 0.505363 0.862907i $$-0.331359\pi$$
0.505363 + 0.862907i $$0.331359\pi$$
$$882$$ −42.0000 −1.41421
$$883$$ 7.00000 0.235569 0.117784 0.993039i $$-0.462421\pi$$
0.117784 + 0.993039i $$0.462421\pi$$
$$884$$ 14.0000 0.470871
$$885$$ 0 0
$$886$$ 15.0000 0.503935
$$887$$ 52.0000 1.74599 0.872995 0.487730i $$-0.162175\pi$$
0.872995 + 0.487730i $$0.162175\pi$$
$$888$$ −3.00000 −0.100673
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −9.00000 −0.301511
$$892$$ 16.0000 0.535720
$$893$$ −20.0000 −0.669274
$$894$$ 66.0000 2.20737
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 36.0000 1.20201
$$898$$ 15.0000 0.500556
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −14.0000 −0.466408
$$902$$ 3.00000 0.0998891
$$903$$ 0 0
$$904$$ −17.0000 −0.565412
$$905$$ 0 0
$$906$$ 54.0000 1.79403
$$907$$ −28.0000 −0.929725 −0.464862 0.885383i $$-0.653896\pi$$
−0.464862 + 0.885383i $$0.653896\pi$$
$$908$$ 28.0000 0.929213
$$909$$ 36.0000 1.19404
$$910$$ 0 0
$$911$$ 30.0000 0.993944 0.496972 0.867766i $$-0.334445\pi$$
0.496972 + 0.867766i $$0.334445\pi$$
$$912$$ −15.0000 −0.496700
$$913$$ −3.00000 −0.0992855
$$914$$ 21.0000 0.694618
$$915$$ 0 0
$$916$$ 24.0000 0.792982
$$917$$ 0 0
$$918$$ 63.0000 2.07931
$$919$$ 14.0000 0.461817 0.230909 0.972975i $$-0.425830\pi$$
0.230909 + 0.972975i $$0.425830\pi$$
$$920$$ 0 0
$$921$$ 21.0000 0.691974
$$922$$ 4.00000 0.131733
$$923$$ 12.0000 0.394985
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −26.0000 −0.854413
$$927$$ −108.000 −3.54719
$$928$$ 0 0
$$929$$ −6.00000 −0.196854 −0.0984268 0.995144i $$-0.531381\pi$$
−0.0984268 + 0.995144i $$0.531381\pi$$
$$930$$ 0 0
$$931$$ −35.0000 −1.14708
$$932$$ −14.0000 −0.458585
$$933$$ 72.0000 2.35717
$$934$$ −4.00000 −0.130884
$$935$$ 0 0
$$936$$ −12.0000 −0.392232
$$937$$ −7.00000 −0.228680 −0.114340 0.993442i $$-0.536475\pi$$
−0.114340 + 0.993442i $$0.536475\pi$$
$$938$$ 0 0
$$939$$ −30.0000 −0.979013
$$940$$ 0 0
$$941$$ −24.0000 −0.782378 −0.391189 0.920310i $$-0.627936\pi$$
−0.391189 + 0.920310i $$0.627936\pi$$
$$942$$ −36.0000 −1.17294
$$943$$ −18.0000 −0.586161
$$944$$ 4.00000 0.130189
$$945$$ 0 0
$$946$$ 4.00000 0.130051
$$947$$ −56.0000 −1.81976 −0.909878 0.414876i $$-0.863825\pi$$
−0.909878 + 0.414876i $$0.863825\pi$$
$$948$$ −42.0000 −1.36410
$$949$$ 14.0000 0.454459
$$950$$ 0 0
$$951$$ 6.00000 0.194563
$$952$$ 0 0
$$953$$ −9.00000 −0.291539 −0.145769 0.989319i $$-0.546566\pi$$
−0.145769 + 0.989319i $$0.546566\pi$$
$$954$$ 12.0000 0.388514
$$955$$ 0 0
$$956$$ 12.0000 0.388108
$$957$$ 0 0
$$958$$ 4.00000 0.129234
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ −2.00000 −0.0644826
$$963$$ 30.0000 0.966736
$$964$$ −25.0000 −0.805196
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 32.0000 1.02905 0.514525 0.857475i $$-0.327968\pi$$
0.514525 + 0.857475i $$0.327968\pi$$
$$968$$ −10.0000 −0.321412
$$969$$ 105.000 3.37309
$$970$$ 0 0
$$971$$ 15.0000 0.481373 0.240686 0.970603i $$-0.422627\pi$$
0.240686 + 0.970603i $$0.422627\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 16.0000 0.512673
$$975$$ 0 0
$$976$$ −8.00000 −0.256074
$$977$$ −43.0000 −1.37569 −0.687846 0.725857i $$-0.741444\pi$$
−0.687846 + 0.725857i $$0.741444\pi$$
$$978$$ −3.00000 −0.0959294
$$979$$ 7.00000 0.223721
$$980$$ 0 0
$$981$$ −36.0000 −1.14939
$$982$$ 44.0000 1.40410
$$983$$ −42.0000 −1.33959 −0.669796 0.742545i $$-0.733618\pi$$
−0.669796 + 0.742545i $$0.733618\pi$$
$$984$$ 9.00000 0.286910
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ −10.0000 −0.318142
$$989$$ −24.0000 −0.763156
$$990$$ 0 0
$$991$$ 32.0000 1.01651 0.508257 0.861206i $$-0.330290\pi$$
0.508257 + 0.861206i $$0.330290\pi$$
$$992$$ −4.00000 −0.127000
$$993$$ 57.0000 1.80884
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −9.00000 −0.285176
$$997$$ −10.0000 −0.316703 −0.158352 0.987383i $$-0.550618\pi$$
−0.158352 + 0.987383i $$0.550618\pi$$
$$998$$ 16.0000 0.506471
$$999$$ −9.00000 −0.284747
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.h.1.1 yes 1
5.2 odd 4 1850.2.b.a.149.2 2
5.3 odd 4 1850.2.b.a.149.1 2
5.4 even 2 1850.2.a.g.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
1850.2.a.g.1.1 1 5.4 even 2
1850.2.a.h.1.1 yes 1 1.1 even 1 trivial
1850.2.b.a.149.1 2 5.3 odd 4
1850.2.b.a.149.2 2 5.2 odd 4