# Properties

 Label 5625.2.a.g.1.1 Level $5625$ Weight $2$ Character 5625.1 Self dual yes Analytic conductor $44.916$ 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] = [5625,2,Mod(1,5625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5625, 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("5625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5625 = 3^{2} \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5625.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$44.9158511370$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1875) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-0.618034$$ of defining polynomial Character $$\chi$$ $$=$$ 5625.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.618034 q^{2} -1.61803 q^{4} +2.00000 q^{7} +2.23607 q^{8} +O(q^{10})$$ $$q-0.618034 q^{2} -1.61803 q^{4} +2.00000 q^{7} +2.23607 q^{8} +3.00000 q^{11} +1.00000 q^{13} -1.23607 q^{14} +1.85410 q^{16} +0.236068 q^{17} +6.70820 q^{19} -1.85410 q^{22} +7.61803 q^{23} -0.618034 q^{26} -3.23607 q^{28} +1.38197 q^{29} -4.70820 q^{31} -5.61803 q^{32} -0.145898 q^{34} +2.00000 q^{37} -4.14590 q^{38} +11.6180 q^{41} +9.61803 q^{43} -4.85410 q^{44} -4.70820 q^{46} -9.23607 q^{47} -3.00000 q^{49} -1.61803 q^{52} +6.76393 q^{53} +4.47214 q^{56} -0.854102 q^{58} +13.9443 q^{59} -4.70820 q^{61} +2.90983 q^{62} -0.236068 q^{64} -9.18034 q^{67} -0.381966 q^{68} +1.09017 q^{71} -2.29180 q^{73} -1.23607 q^{74} -10.8541 q^{76} +6.00000 q^{77} -15.8541 q^{79} -7.18034 q^{82} +9.00000 q^{83} -5.94427 q^{86} +6.70820 q^{88} -11.1803 q^{89} +2.00000 q^{91} -12.3262 q^{92} +5.70820 q^{94} +2.85410 q^{97} +1.85410 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - q^{4} + 4 q^{7}+O(q^{10})$$ 2 * q + q^2 - q^4 + 4 * q^7 $$2 q + q^{2} - q^{4} + 4 q^{7} + 6 q^{11} + 2 q^{13} + 2 q^{14} - 3 q^{16} - 4 q^{17} + 3 q^{22} + 13 q^{23} + q^{26} - 2 q^{28} + 5 q^{29} + 4 q^{31} - 9 q^{32} - 7 q^{34} + 4 q^{37} - 15 q^{38} + 21 q^{41} + 17 q^{43} - 3 q^{44} + 4 q^{46} - 14 q^{47} - 6 q^{49} - q^{52} + 18 q^{53} + 5 q^{58} + 10 q^{59} + 4 q^{61} + 17 q^{62} + 4 q^{64} + 4 q^{67} - 3 q^{68} - 9 q^{71} - 18 q^{73} + 2 q^{74} - 15 q^{76} + 12 q^{77} - 25 q^{79} + 8 q^{82} + 18 q^{83} + 6 q^{86} + 4 q^{91} - 9 q^{92} - 2 q^{94} - q^{97} - 3 q^{98}+O(q^{100})$$ 2 * q + q^2 - q^4 + 4 * q^7 + 6 * q^11 + 2 * q^13 + 2 * q^14 - 3 * q^16 - 4 * q^17 + 3 * q^22 + 13 * q^23 + q^26 - 2 * q^28 + 5 * q^29 + 4 * q^31 - 9 * q^32 - 7 * q^34 + 4 * q^37 - 15 * q^38 + 21 * q^41 + 17 * q^43 - 3 * q^44 + 4 * q^46 - 14 * q^47 - 6 * q^49 - q^52 + 18 * q^53 + 5 * q^58 + 10 * q^59 + 4 * q^61 + 17 * q^62 + 4 * q^64 + 4 * q^67 - 3 * q^68 - 9 * q^71 - 18 * q^73 + 2 * q^74 - 15 * q^76 + 12 * q^77 - 25 * q^79 + 8 * q^82 + 18 * q^83 + 6 * q^86 + 4 * q^91 - 9 * q^92 - 2 * q^94 - q^97 - 3 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.618034 −0.437016 −0.218508 0.975835i $$-0.570119\pi$$
−0.218508 + 0.975835i $$0.570119\pi$$
$$3$$ 0 0
$$4$$ −1.61803 −0.809017
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ 2.23607 0.790569
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 3.00000 0.904534 0.452267 0.891883i $$-0.350615\pi$$
0.452267 + 0.891883i $$0.350615\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350 0.138675 0.990338i $$-0.455716\pi$$
0.138675 + 0.990338i $$0.455716\pi$$
$$14$$ −1.23607 −0.330353
$$15$$ 0 0
$$16$$ 1.85410 0.463525
$$17$$ 0.236068 0.0572549 0.0286274 0.999590i $$-0.490886\pi$$
0.0286274 + 0.999590i $$0.490886\pi$$
$$18$$ 0 0
$$19$$ 6.70820 1.53897 0.769484 0.638666i $$-0.220514\pi$$
0.769484 + 0.638666i $$0.220514\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −1.85410 −0.395296
$$23$$ 7.61803 1.58847 0.794235 0.607611i $$-0.207872\pi$$
0.794235 + 0.607611i $$0.207872\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −0.618034 −0.121206
$$27$$ 0 0
$$28$$ −3.23607 −0.611559
$$29$$ 1.38197 0.256625 0.128312 0.991734i $$-0.459044\pi$$
0.128312 + 0.991734i $$0.459044\pi$$
$$30$$ 0 0
$$31$$ −4.70820 −0.845618 −0.422809 0.906219i $$-0.638956\pi$$
−0.422809 + 0.906219i $$0.638956\pi$$
$$32$$ −5.61803 −0.993137
$$33$$ 0 0
$$34$$ −0.145898 −0.0250213
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ −4.14590 −0.672553
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 11.6180 1.81443 0.907216 0.420665i $$-0.138203\pi$$
0.907216 + 0.420665i $$0.138203\pi$$
$$42$$ 0 0
$$43$$ 9.61803 1.46674 0.733368 0.679832i $$-0.237947\pi$$
0.733368 + 0.679832i $$0.237947\pi$$
$$44$$ −4.85410 −0.731783
$$45$$ 0 0
$$46$$ −4.70820 −0.694187
$$47$$ −9.23607 −1.34722 −0.673609 0.739087i $$-0.735257\pi$$
−0.673609 + 0.739087i $$0.735257\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −1.61803 −0.224381
$$53$$ 6.76393 0.929098 0.464549 0.885548i $$-0.346217\pi$$
0.464549 + 0.885548i $$0.346217\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 4.47214 0.597614
$$57$$ 0 0
$$58$$ −0.854102 −0.112149
$$59$$ 13.9443 1.81539 0.907695 0.419631i $$-0.137841\pi$$
0.907695 + 0.419631i $$0.137841\pi$$
$$60$$ 0 0
$$61$$ −4.70820 −0.602824 −0.301412 0.953494i $$-0.597458\pi$$
−0.301412 + 0.953494i $$0.597458\pi$$
$$62$$ 2.90983 0.369549
$$63$$ 0 0
$$64$$ −0.236068 −0.0295085
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −9.18034 −1.12156 −0.560779 0.827966i $$-0.689498\pi$$
−0.560779 + 0.827966i $$0.689498\pi$$
$$68$$ −0.381966 −0.0463202
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 1.09017 0.129379 0.0646897 0.997905i $$-0.479394\pi$$
0.0646897 + 0.997905i $$0.479394\pi$$
$$72$$ 0 0
$$73$$ −2.29180 −0.268234 −0.134117 0.990965i $$-0.542820\pi$$
−0.134117 + 0.990965i $$0.542820\pi$$
$$74$$ −1.23607 −0.143690
$$75$$ 0 0
$$76$$ −10.8541 −1.24505
$$77$$ 6.00000 0.683763
$$78$$ 0 0
$$79$$ −15.8541 −1.78373 −0.891863 0.452306i $$-0.850602\pi$$
−0.891863 + 0.452306i $$0.850602\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −7.18034 −0.792936
$$83$$ 9.00000 0.987878 0.493939 0.869496i $$-0.335557\pi$$
0.493939 + 0.869496i $$0.335557\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −5.94427 −0.640987
$$87$$ 0 0
$$88$$ 6.70820 0.715097
$$89$$ −11.1803 −1.18511 −0.592557 0.805529i $$-0.701881\pi$$
−0.592557 + 0.805529i $$0.701881\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ −12.3262 −1.28510
$$93$$ 0 0
$$94$$ 5.70820 0.588756
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 2.85410 0.289790 0.144895 0.989447i $$-0.453716\pi$$
0.144895 + 0.989447i $$0.453716\pi$$
$$98$$ 1.85410 0.187293
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 11.6180 1.15604 0.578019 0.816023i $$-0.303826\pi$$
0.578019 + 0.816023i $$0.303826\pi$$
$$102$$ 0 0
$$103$$ −12.4164 −1.22343 −0.611713 0.791080i $$-0.709519\pi$$
−0.611713 + 0.791080i $$0.709519\pi$$
$$104$$ 2.23607 0.219265
$$105$$ 0 0
$$106$$ −4.18034 −0.406031
$$107$$ −7.85410 −0.759285 −0.379642 0.925133i $$-0.623953\pi$$
−0.379642 + 0.925133i $$0.623953\pi$$
$$108$$ 0 0
$$109$$ −10.8541 −1.03963 −0.519817 0.854278i $$-0.674000\pi$$
−0.519817 + 0.854278i $$0.674000\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 3.70820 0.350392
$$113$$ −8.23607 −0.774784 −0.387392 0.921915i $$-0.626624\pi$$
−0.387392 + 0.921915i $$0.626624\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −2.23607 −0.207614
$$117$$ 0 0
$$118$$ −8.61803 −0.793354
$$119$$ 0.472136 0.0432806
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 2.90983 0.263444
$$123$$ 0 0
$$124$$ 7.61803 0.684120
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 17.6525 1.56640 0.783202 0.621767i $$-0.213585\pi$$
0.783202 + 0.621767i $$0.213585\pi$$
$$128$$ 11.3820 1.00603
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −8.18034 −0.714720 −0.357360 0.933967i $$-0.616323\pi$$
−0.357360 + 0.933967i $$0.616323\pi$$
$$132$$ 0 0
$$133$$ 13.4164 1.16335
$$134$$ 5.67376 0.490138
$$135$$ 0 0
$$136$$ 0.527864 0.0452640
$$137$$ 20.5623 1.75676 0.878378 0.477966i $$-0.158626\pi$$
0.878378 + 0.477966i $$0.158626\pi$$
$$138$$ 0 0
$$139$$ −13.4164 −1.13796 −0.568982 0.822350i $$-0.692663\pi$$
−0.568982 + 0.822350i $$0.692663\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −0.673762 −0.0565409
$$143$$ 3.00000 0.250873
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 1.41641 0.117223
$$147$$ 0 0
$$148$$ −3.23607 −0.266003
$$149$$ 1.90983 0.156459 0.0782297 0.996935i $$-0.475073\pi$$
0.0782297 + 0.996935i $$0.475073\pi$$
$$150$$ 0 0
$$151$$ −4.38197 −0.356599 −0.178300 0.983976i $$-0.557060\pi$$
−0.178300 + 0.983976i $$0.557060\pi$$
$$152$$ 15.0000 1.21666
$$153$$ 0 0
$$154$$ −3.70820 −0.298816
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −3.85410 −0.307591 −0.153795 0.988103i $$-0.549150\pi$$
−0.153795 + 0.988103i $$0.549150\pi$$
$$158$$ 9.79837 0.779517
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 15.2361 1.20077
$$162$$ 0 0
$$163$$ 15.2705 1.19608 0.598039 0.801467i $$-0.295947\pi$$
0.598039 + 0.801467i $$0.295947\pi$$
$$164$$ −18.7984 −1.46791
$$165$$ 0 0
$$166$$ −5.56231 −0.431719
$$167$$ −6.79837 −0.526074 −0.263037 0.964786i $$-0.584724\pi$$
−0.263037 + 0.964786i $$0.584724\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −15.5623 −1.18661
$$173$$ 12.0902 0.919199 0.459599 0.888126i $$-0.347993\pi$$
0.459599 + 0.888126i $$0.347993\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 5.56231 0.419275
$$177$$ 0 0
$$178$$ 6.90983 0.517914
$$179$$ −15.6525 −1.16992 −0.584960 0.811062i $$-0.698890\pi$$
−0.584960 + 0.811062i $$0.698890\pi$$
$$180$$ 0 0
$$181$$ −3.52786 −0.262224 −0.131112 0.991368i $$-0.541855\pi$$
−0.131112 + 0.991368i $$0.541855\pi$$
$$182$$ −1.23607 −0.0916235
$$183$$ 0 0
$$184$$ 17.0344 1.25580
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0.708204 0.0517890
$$188$$ 14.9443 1.08992
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −1.67376 −0.121109 −0.0605546 0.998165i $$-0.519287\pi$$
−0.0605546 + 0.998165i $$0.519287\pi$$
$$192$$ 0 0
$$193$$ 11.0000 0.791797 0.395899 0.918294i $$-0.370433\pi$$
0.395899 + 0.918294i $$0.370433\pi$$
$$194$$ −1.76393 −0.126643
$$195$$ 0 0
$$196$$ 4.85410 0.346722
$$197$$ 11.0902 0.790142 0.395071 0.918651i $$-0.370720\pi$$
0.395071 + 0.918651i $$0.370720\pi$$
$$198$$ 0 0
$$199$$ −1.70820 −0.121091 −0.0605457 0.998165i $$-0.519284\pi$$
−0.0605457 + 0.998165i $$0.519284\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −7.18034 −0.505207
$$203$$ 2.76393 0.193990
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 7.67376 0.534656
$$207$$ 0 0
$$208$$ 1.85410 0.128559
$$209$$ 20.1246 1.39205
$$210$$ 0 0
$$211$$ −3.00000 −0.206529 −0.103264 0.994654i $$-0.532929\pi$$
−0.103264 + 0.994654i $$0.532929\pi$$
$$212$$ −10.9443 −0.751656
$$213$$ 0 0
$$214$$ 4.85410 0.331820
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −9.41641 −0.639227
$$218$$ 6.70820 0.454337
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0.236068 0.0158797
$$222$$ 0 0
$$223$$ 16.8541 1.12863 0.564317 0.825558i $$-0.309140\pi$$
0.564317 + 0.825558i $$0.309140\pi$$
$$224$$ −11.2361 −0.750741
$$225$$ 0 0
$$226$$ 5.09017 0.338593
$$227$$ 10.2361 0.679392 0.339696 0.940535i $$-0.389676\pi$$
0.339696 + 0.940535i $$0.389676\pi$$
$$228$$ 0 0
$$229$$ 6.18034 0.408408 0.204204 0.978928i $$-0.434539\pi$$
0.204204 + 0.978928i $$0.434539\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 3.09017 0.202880
$$233$$ −12.1803 −0.797961 −0.398980 0.916959i $$-0.630636\pi$$
−0.398980 + 0.916959i $$0.630636\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −22.5623 −1.46868
$$237$$ 0 0
$$238$$ −0.291796 −0.0189143
$$239$$ −23.6180 −1.52772 −0.763862 0.645380i $$-0.776699\pi$$
−0.763862 + 0.645380i $$0.776699\pi$$
$$240$$ 0 0
$$241$$ −8.32624 −0.536340 −0.268170 0.963372i $$-0.586419\pi$$
−0.268170 + 0.963372i $$0.586419\pi$$
$$242$$ 1.23607 0.0794575
$$243$$ 0 0
$$244$$ 7.61803 0.487695
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 6.70820 0.426833
$$248$$ −10.5279 −0.668520
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −27.9787 −1.76600 −0.883000 0.469372i $$-0.844480\pi$$
−0.883000 + 0.469372i $$0.844480\pi$$
$$252$$ 0 0
$$253$$ 22.8541 1.43683
$$254$$ −10.9098 −0.684544
$$255$$ 0 0
$$256$$ −6.56231 −0.410144
$$257$$ −20.2148 −1.26096 −0.630482 0.776204i $$-0.717143\pi$$
−0.630482 + 0.776204i $$0.717143\pi$$
$$258$$ 0 0
$$259$$ 4.00000 0.248548
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 5.05573 0.312344
$$263$$ 25.5066 1.57280 0.786401 0.617716i $$-0.211942\pi$$
0.786401 + 0.617716i $$0.211942\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −8.29180 −0.508403
$$267$$ 0 0
$$268$$ 14.8541 0.907359
$$269$$ 29.4721 1.79695 0.898474 0.439027i $$-0.144677\pi$$
0.898474 + 0.439027i $$0.144677\pi$$
$$270$$ 0 0
$$271$$ 15.4164 0.936480 0.468240 0.883601i $$-0.344888\pi$$
0.468240 + 0.883601i $$0.344888\pi$$
$$272$$ 0.437694 0.0265391
$$273$$ 0 0
$$274$$ −12.7082 −0.767731
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 30.9443 1.85926 0.929631 0.368493i $$-0.120126\pi$$
0.929631 + 0.368493i $$0.120126\pi$$
$$278$$ 8.29180 0.497309
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −8.18034 −0.487998 −0.243999 0.969775i $$-0.578459\pi$$
−0.243999 + 0.969775i $$0.578459\pi$$
$$282$$ 0 0
$$283$$ −15.7082 −0.933756 −0.466878 0.884322i $$-0.654621\pi$$
−0.466878 + 0.884322i $$0.654621\pi$$
$$284$$ −1.76393 −0.104670
$$285$$ 0 0
$$286$$ −1.85410 −0.109635
$$287$$ 23.2361 1.37158
$$288$$ 0 0
$$289$$ −16.9443 −0.996722
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 3.70820 0.217006
$$293$$ 9.32624 0.544845 0.272422 0.962178i $$-0.412175\pi$$
0.272422 + 0.962178i $$0.412175\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 4.47214 0.259938
$$297$$ 0 0
$$298$$ −1.18034 −0.0683753
$$299$$ 7.61803 0.440562
$$300$$ 0 0
$$301$$ 19.2361 1.10875
$$302$$ 2.70820 0.155840
$$303$$ 0 0
$$304$$ 12.4377 0.713351
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −2.14590 −0.122473 −0.0612364 0.998123i $$-0.519504\pi$$
−0.0612364 + 0.998123i $$0.519504\pi$$
$$308$$ −9.70820 −0.553176
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 22.4721 1.27428 0.637139 0.770749i $$-0.280118\pi$$
0.637139 + 0.770749i $$0.280118\pi$$
$$312$$ 0 0
$$313$$ −15.7082 −0.887880 −0.443940 0.896056i $$-0.646420\pi$$
−0.443940 + 0.896056i $$0.646420\pi$$
$$314$$ 2.38197 0.134422
$$315$$ 0 0
$$316$$ 25.6525 1.44306
$$317$$ 0.437694 0.0245833 0.0122917 0.999924i $$-0.496087\pi$$
0.0122917 + 0.999924i $$0.496087\pi$$
$$318$$ 0 0
$$319$$ 4.14590 0.232126
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −9.41641 −0.524756
$$323$$ 1.58359 0.0881134
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −9.43769 −0.522706
$$327$$ 0 0
$$328$$ 25.9787 1.43443
$$329$$ −18.4721 −1.01840
$$330$$ 0 0
$$331$$ 29.6869 1.63174 0.815870 0.578235i $$-0.196258\pi$$
0.815870 + 0.578235i $$0.196258\pi$$
$$332$$ −14.5623 −0.799210
$$333$$ 0 0
$$334$$ 4.20163 0.229903
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 10.8197 0.589384 0.294692 0.955592i $$-0.404783\pi$$
0.294692 + 0.955592i $$0.404783\pi$$
$$338$$ 7.41641 0.403399
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −14.1246 −0.764891
$$342$$ 0 0
$$343$$ −20.0000 −1.07990
$$344$$ 21.5066 1.15956
$$345$$ 0 0
$$346$$ −7.47214 −0.401705
$$347$$ −21.2705 −1.14186 −0.570930 0.820998i $$-0.693417\pi$$
−0.570930 + 0.820998i $$0.693417\pi$$
$$348$$ 0 0
$$349$$ 2.76393 0.147950 0.0739749 0.997260i $$-0.476432\pi$$
0.0739749 + 0.997260i $$0.476432\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −16.8541 −0.898327
$$353$$ −14.6180 −0.778039 −0.389020 0.921229i $$-0.627186\pi$$
−0.389020 + 0.921229i $$0.627186\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 18.0902 0.958777
$$357$$ 0 0
$$358$$ 9.67376 0.511274
$$359$$ −6.05573 −0.319609 −0.159805 0.987149i $$-0.551086\pi$$
−0.159805 + 0.987149i $$0.551086\pi$$
$$360$$ 0 0
$$361$$ 26.0000 1.36842
$$362$$ 2.18034 0.114596
$$363$$ 0 0
$$364$$ −3.23607 −0.169616
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 21.4721 1.12084 0.560418 0.828210i $$-0.310640\pi$$
0.560418 + 0.828210i $$0.310640\pi$$
$$368$$ 14.1246 0.736296
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 13.5279 0.702332
$$372$$ 0 0
$$373$$ 9.41641 0.487563 0.243782 0.969830i $$-0.421612\pi$$
0.243782 + 0.969830i $$0.421612\pi$$
$$374$$ −0.437694 −0.0226326
$$375$$ 0 0
$$376$$ −20.6525 −1.06507
$$377$$ 1.38197 0.0711749
$$378$$ 0 0
$$379$$ −11.3820 −0.584652 −0.292326 0.956319i $$-0.594429\pi$$
−0.292326 + 0.956319i $$0.594429\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 1.03444 0.0529266
$$383$$ 22.9443 1.17240 0.586199 0.810167i $$-0.300624\pi$$
0.586199 + 0.810167i $$0.300624\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −6.79837 −0.346028
$$387$$ 0 0
$$388$$ −4.61803 −0.234445
$$389$$ 30.6525 1.55414 0.777071 0.629413i $$-0.216705\pi$$
0.777071 + 0.629413i $$0.216705\pi$$
$$390$$ 0 0
$$391$$ 1.79837 0.0909477
$$392$$ −6.70820 −0.338815
$$393$$ 0 0
$$394$$ −6.85410 −0.345305
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 11.4721 0.575770 0.287885 0.957665i $$-0.407048\pi$$
0.287885 + 0.957665i $$0.407048\pi$$
$$398$$ 1.05573 0.0529189
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −2.72949 −0.136304 −0.0681521 0.997675i $$-0.521710\pi$$
−0.0681521 + 0.997675i $$0.521710\pi$$
$$402$$ 0 0
$$403$$ −4.70820 −0.234532
$$404$$ −18.7984 −0.935254
$$405$$ 0 0
$$406$$ −1.70820 −0.0847767
$$407$$ 6.00000 0.297409
$$408$$ 0 0
$$409$$ 35.1246 1.73680 0.868400 0.495864i $$-0.165149\pi$$
0.868400 + 0.495864i $$0.165149\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 20.0902 0.989772
$$413$$ 27.8885 1.37231
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −5.61803 −0.275447
$$417$$ 0 0
$$418$$ −12.4377 −0.608348
$$419$$ 15.3262 0.748736 0.374368 0.927280i $$-0.377860\pi$$
0.374368 + 0.927280i $$0.377860\pi$$
$$420$$ 0 0
$$421$$ 14.3607 0.699897 0.349948 0.936769i $$-0.386199\pi$$
0.349948 + 0.936769i $$0.386199\pi$$
$$422$$ 1.85410 0.0902563
$$423$$ 0 0
$$424$$ 15.1246 0.734516
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −9.41641 −0.455692
$$428$$ 12.7082 0.614274
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −34.2361 −1.64909 −0.824547 0.565794i $$-0.808570\pi$$
−0.824547 + 0.565794i $$0.808570\pi$$
$$432$$ 0 0
$$433$$ 5.47214 0.262974 0.131487 0.991318i $$-0.458025\pi$$
0.131487 + 0.991318i $$0.458025\pi$$
$$434$$ 5.81966 0.279353
$$435$$ 0 0
$$436$$ 17.5623 0.841082
$$437$$ 51.1033 2.44460
$$438$$ 0 0
$$439$$ −2.96556 −0.141538 −0.0707692 0.997493i $$-0.522545\pi$$
−0.0707692 + 0.997493i $$0.522545\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −0.145898 −0.00693966
$$443$$ 7.41641 0.352364 0.176182 0.984358i $$-0.443625\pi$$
0.176182 + 0.984358i $$0.443625\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −10.4164 −0.493231
$$447$$ 0 0
$$448$$ −0.472136 −0.0223063
$$449$$ 21.5066 1.01496 0.507479 0.861664i $$-0.330577\pi$$
0.507479 + 0.861664i $$0.330577\pi$$
$$450$$ 0 0
$$451$$ 34.8541 1.64122
$$452$$ 13.3262 0.626814
$$453$$ 0 0
$$454$$ −6.32624 −0.296905
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −25.8885 −1.21102 −0.605508 0.795840i $$-0.707030\pi$$
−0.605508 + 0.795840i $$0.707030\pi$$
$$458$$ −3.81966 −0.178481
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −3.18034 −0.148123 −0.0740616 0.997254i $$-0.523596\pi$$
−0.0740616 + 0.997254i $$0.523596\pi$$
$$462$$ 0 0
$$463$$ −26.6869 −1.24025 −0.620123 0.784505i $$-0.712917\pi$$
−0.620123 + 0.784505i $$0.712917\pi$$
$$464$$ 2.56231 0.118952
$$465$$ 0 0
$$466$$ 7.52786 0.348722
$$467$$ 16.4164 0.759661 0.379830 0.925056i $$-0.375982\pi$$
0.379830 + 0.925056i $$0.375982\pi$$
$$468$$ 0 0
$$469$$ −18.3607 −0.847817
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 31.1803 1.43519
$$473$$ 28.8541 1.32671
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −0.763932 −0.0350148
$$477$$ 0 0
$$478$$ 14.5967 0.667640
$$479$$ 19.7984 0.904611 0.452305 0.891863i $$-0.350602\pi$$
0.452305 + 0.891863i $$0.350602\pi$$
$$480$$ 0 0
$$481$$ 2.00000 0.0911922
$$482$$ 5.14590 0.234389
$$483$$ 0 0
$$484$$ 3.23607 0.147094
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −14.3820 −0.651709 −0.325855 0.945420i $$-0.605652\pi$$
−0.325855 + 0.945420i $$0.605652\pi$$
$$488$$ −10.5279 −0.476574
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −6.67376 −0.301183 −0.150591 0.988596i $$-0.548118\pi$$
−0.150591 + 0.988596i $$0.548118\pi$$
$$492$$ 0 0
$$493$$ 0.326238 0.0146930
$$494$$ −4.14590 −0.186533
$$495$$ 0 0
$$496$$ −8.72949 −0.391966
$$497$$ 2.18034 0.0978016
$$498$$ 0 0
$$499$$ −15.0000 −0.671492 −0.335746 0.941953i $$-0.608988\pi$$
−0.335746 + 0.941953i $$0.608988\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 17.2918 0.771771
$$503$$ −33.0344 −1.47293 −0.736466 0.676474i $$-0.763507\pi$$
−0.736466 + 0.676474i $$0.763507\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −14.1246 −0.627916
$$507$$ 0 0
$$508$$ −28.5623 −1.26725
$$509$$ 2.88854 0.128032 0.0640162 0.997949i $$-0.479609\pi$$
0.0640162 + 0.997949i $$0.479609\pi$$
$$510$$ 0 0
$$511$$ −4.58359 −0.202766
$$512$$ −18.7082 −0.826794
$$513$$ 0 0
$$514$$ 12.4934 0.551061
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −27.7082 −1.21861
$$518$$ −2.47214 −0.108619
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −28.9098 −1.26656 −0.633281 0.773922i $$-0.718292\pi$$
−0.633281 + 0.773922i $$0.718292\pi$$
$$522$$ 0 0
$$523$$ 18.5623 0.811673 0.405836 0.913946i $$-0.366980\pi$$
0.405836 + 0.913946i $$0.366980\pi$$
$$524$$ 13.2361 0.578220
$$525$$ 0 0
$$526$$ −15.7639 −0.687340
$$527$$ −1.11146 −0.0484158
$$528$$ 0 0
$$529$$ 35.0344 1.52324
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −21.7082 −0.941170
$$533$$ 11.6180 0.503233
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −20.5279 −0.886669
$$537$$ 0 0
$$538$$ −18.2148 −0.785295
$$539$$ −9.00000 −0.387657
$$540$$ 0 0
$$541$$ −39.7082 −1.70719 −0.853595 0.520938i $$-0.825582\pi$$
−0.853595 + 0.520938i $$0.825582\pi$$
$$542$$ −9.52786 −0.409257
$$543$$ 0 0
$$544$$ −1.32624 −0.0568620
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −11.2918 −0.482802 −0.241401 0.970425i $$-0.577607\pi$$
−0.241401 + 0.970425i $$0.577607\pi$$
$$548$$ −33.2705 −1.42125
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 9.27051 0.394937
$$552$$ 0 0
$$553$$ −31.7082 −1.34837
$$554$$ −19.1246 −0.812527
$$555$$ 0 0
$$556$$ 21.7082 0.920633
$$557$$ −6.34752 −0.268953 −0.134477 0.990917i $$-0.542935\pi$$
−0.134477 + 0.990917i $$0.542935\pi$$
$$558$$ 0 0
$$559$$ 9.61803 0.406799
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 5.05573 0.213263
$$563$$ 9.00000 0.379305 0.189652 0.981851i $$-0.439264\pi$$
0.189652 + 0.981851i $$0.439264\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 9.70820 0.408066
$$567$$ 0 0
$$568$$ 2.43769 0.102283
$$569$$ 4.14590 0.173805 0.0869025 0.996217i $$-0.472303\pi$$
0.0869025 + 0.996217i $$0.472303\pi$$
$$570$$ 0 0
$$571$$ 2.12461 0.0889122 0.0444561 0.999011i $$-0.485845\pi$$
0.0444561 + 0.999011i $$0.485845\pi$$
$$572$$ −4.85410 −0.202960
$$573$$ 0 0
$$574$$ −14.3607 −0.599403
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −37.2705 −1.55159 −0.775796 0.630984i $$-0.782651\pi$$
−0.775796 + 0.630984i $$0.782651\pi$$
$$578$$ 10.4721 0.435583
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 18.0000 0.746766
$$582$$ 0 0
$$583$$ 20.2918 0.840400
$$584$$ −5.12461 −0.212058
$$585$$ 0 0
$$586$$ −5.76393 −0.238106
$$587$$ −23.3050 −0.961898 −0.480949 0.876748i $$-0.659708\pi$$
−0.480949 + 0.876748i $$0.659708\pi$$
$$588$$ 0 0
$$589$$ −31.5836 −1.30138
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 3.70820 0.152406
$$593$$ 15.3820 0.631662 0.315831 0.948816i $$-0.397717\pi$$
0.315831 + 0.948816i $$0.397717\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −3.09017 −0.126578
$$597$$ 0 0
$$598$$ −4.70820 −0.192533
$$599$$ −5.72949 −0.234101 −0.117050 0.993126i $$-0.537344\pi$$
−0.117050 + 0.993126i $$0.537344\pi$$
$$600$$ 0 0
$$601$$ −11.2918 −0.460602 −0.230301 0.973119i $$-0.573971\pi$$
−0.230301 + 0.973119i $$0.573971\pi$$
$$602$$ −11.8885 −0.484541
$$603$$ 0 0
$$604$$ 7.09017 0.288495
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 16.1459 0.655342 0.327671 0.944792i $$-0.393736\pi$$
0.327671 + 0.944792i $$0.393736\pi$$
$$608$$ −37.6869 −1.52841
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −9.23607 −0.373651
$$612$$ 0 0
$$613$$ 46.1246 1.86296 0.931478 0.363798i $$-0.118520\pi$$
0.931478 + 0.363798i $$0.118520\pi$$
$$614$$ 1.32624 0.0535226
$$615$$ 0 0
$$616$$ 13.4164 0.540562
$$617$$ 20.7639 0.835924 0.417962 0.908464i $$-0.362744\pi$$
0.417962 + 0.908464i $$0.362744\pi$$
$$618$$ 0 0
$$619$$ 0.729490 0.0293207 0.0146603 0.999893i $$-0.495333\pi$$
0.0146603 + 0.999893i $$0.495333\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −13.8885 −0.556880
$$623$$ −22.3607 −0.895862
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 9.70820 0.388018
$$627$$ 0 0
$$628$$ 6.23607 0.248846
$$629$$ 0.472136 0.0188253
$$630$$ 0 0
$$631$$ −15.2361 −0.606538 −0.303269 0.952905i $$-0.598078\pi$$
−0.303269 + 0.952905i $$0.598078\pi$$
$$632$$ −35.4508 −1.41016
$$633$$ 0 0
$$634$$ −0.270510 −0.0107433
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −3.00000 −0.118864
$$638$$ −2.56231 −0.101443
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 7.67376 0.303095 0.151548 0.988450i $$-0.451574\pi$$
0.151548 + 0.988450i $$0.451574\pi$$
$$642$$ 0 0
$$643$$ −17.0902 −0.673971 −0.336985 0.941510i $$-0.609407\pi$$
−0.336985 + 0.941510i $$0.609407\pi$$
$$644$$ −24.6525 −0.971444
$$645$$ 0 0
$$646$$ −0.978714 −0.0385070
$$647$$ 10.0344 0.394495 0.197247 0.980354i $$-0.436800\pi$$
0.197247 + 0.980354i $$0.436800\pi$$
$$648$$ 0 0
$$649$$ 41.8328 1.64208
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −24.7082 −0.967648
$$653$$ −1.65248 −0.0646664 −0.0323332 0.999477i $$-0.510294\pi$$
−0.0323332 + 0.999477i $$0.510294\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 21.5410 0.841036
$$657$$ 0 0
$$658$$ 11.4164 0.445058
$$659$$ −2.23607 −0.0871048 −0.0435524 0.999051i $$-0.513868\pi$$
−0.0435524 + 0.999051i $$0.513868\pi$$
$$660$$ 0 0
$$661$$ −30.8885 −1.20143 −0.600713 0.799465i $$-0.705116\pi$$
−0.600713 + 0.799465i $$0.705116\pi$$
$$662$$ −18.3475 −0.713097
$$663$$ 0 0
$$664$$ 20.1246 0.780986
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 10.5279 0.407641
$$668$$ 11.0000 0.425603
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −14.1246 −0.545275
$$672$$ 0 0
$$673$$ −24.7771 −0.955087 −0.477543 0.878608i $$-0.658473\pi$$
−0.477543 + 0.878608i $$0.658473\pi$$
$$674$$ −6.68692 −0.257570
$$675$$ 0 0
$$676$$ 19.4164 0.746785
$$677$$ −9.11146 −0.350182 −0.175091 0.984552i $$-0.556022\pi$$
−0.175091 + 0.984552i $$0.556022\pi$$
$$678$$ 0 0
$$679$$ 5.70820 0.219061
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 8.72949 0.334269
$$683$$ 48.5967 1.85950 0.929751 0.368188i $$-0.120022\pi$$
0.929751 + 0.368188i $$0.120022\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 12.3607 0.471933
$$687$$ 0 0
$$688$$ 17.8328 0.679870
$$689$$ 6.76393 0.257685
$$690$$ 0 0
$$691$$ 3.90983 0.148737 0.0743685 0.997231i $$-0.476306\pi$$
0.0743685 + 0.997231i $$0.476306\pi$$
$$692$$ −19.5623 −0.743647
$$693$$ 0 0
$$694$$ 13.1459 0.499011
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 2.74265 0.103885
$$698$$ −1.70820 −0.0646565
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 17.3475 0.655207 0.327603 0.944815i $$-0.393759\pi$$
0.327603 + 0.944815i $$0.393759\pi$$
$$702$$ 0 0
$$703$$ 13.4164 0.506009
$$704$$ −0.708204 −0.0266914
$$705$$ 0 0
$$706$$ 9.03444 0.340016
$$707$$ 23.2361 0.873882
$$708$$ 0 0
$$709$$ −29.7984 −1.11910 −0.559551 0.828796i $$-0.689026\pi$$
−0.559551 + 0.828796i $$0.689026\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −25.0000 −0.936915
$$713$$ −35.8673 −1.34324
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 25.3262 0.946486
$$717$$ 0 0
$$718$$ 3.74265 0.139674
$$719$$ −5.12461 −0.191116 −0.0955579 0.995424i $$-0.530464\pi$$
−0.0955579 + 0.995424i $$0.530464\pi$$
$$720$$ 0 0
$$721$$ −24.8328 −0.924822
$$722$$ −16.0689 −0.598022
$$723$$ 0 0
$$724$$ 5.70820 0.212144
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 34.5623 1.28184 0.640922 0.767606i $$-0.278552\pi$$
0.640922 + 0.767606i $$0.278552\pi$$
$$728$$ 4.47214 0.165748
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 2.27051 0.0839778
$$732$$ 0 0
$$733$$ 16.8541 0.622520 0.311260 0.950325i $$-0.399249\pi$$
0.311260 + 0.950325i $$0.399249\pi$$
$$734$$ −13.2705 −0.489823
$$735$$ 0 0
$$736$$ −42.7984 −1.57757
$$737$$ −27.5410 −1.01449
$$738$$ 0 0
$$739$$ 11.7082 0.430693 0.215347 0.976538i $$-0.430912\pi$$
0.215347 + 0.976538i $$0.430912\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −8.36068 −0.306930
$$743$$ −25.4721 −0.934482 −0.467241 0.884130i $$-0.654752\pi$$
−0.467241 + 0.884130i $$0.654752\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −5.81966 −0.213073
$$747$$ 0 0
$$748$$ −1.14590 −0.0418982
$$749$$ −15.7082 −0.573965
$$750$$ 0 0
$$751$$ 28.7082 1.04758 0.523789 0.851848i $$-0.324518\pi$$
0.523789 + 0.851848i $$0.324518\pi$$
$$752$$ −17.1246 −0.624470
$$753$$ 0 0
$$754$$ −0.854102 −0.0311046
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 1.27051 0.0461775 0.0230887 0.999733i $$-0.492650\pi$$
0.0230887 + 0.999733i $$0.492650\pi$$
$$758$$ 7.03444 0.255502
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 19.1803 0.695287 0.347643 0.937627i $$-0.386982\pi$$
0.347643 + 0.937627i $$0.386982\pi$$
$$762$$ 0 0
$$763$$ −21.7082 −0.785890
$$764$$ 2.70820 0.0979794
$$765$$ 0 0
$$766$$ −14.1803 −0.512357
$$767$$ 13.9443 0.503498
$$768$$ 0 0
$$769$$ 26.3050 0.948581 0.474290 0.880368i $$-0.342705\pi$$
0.474290 + 0.880368i $$0.342705\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −17.7984 −0.640577
$$773$$ 29.7771 1.07101 0.535504 0.844533i $$-0.320122\pi$$
0.535504 + 0.844533i $$0.320122\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 6.38197 0.229099
$$777$$ 0 0
$$778$$ −18.9443 −0.679185
$$779$$ 77.9361 2.79235
$$780$$ 0 0
$$781$$ 3.27051 0.117028
$$782$$ −1.11146 −0.0397456
$$783$$ 0 0
$$784$$ −5.56231 −0.198654
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −23.8541 −0.850307 −0.425153 0.905121i $$-0.639780\pi$$
−0.425153 + 0.905121i $$0.639780\pi$$
$$788$$ −17.9443 −0.639238
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −16.4721 −0.585682
$$792$$ 0 0
$$793$$ −4.70820 −0.167193
$$794$$ −7.09017 −0.251621
$$795$$ 0 0
$$796$$ 2.76393 0.0979650
$$797$$ 46.0132 1.62987 0.814935 0.579553i $$-0.196773\pi$$
0.814935 + 0.579553i $$0.196773\pi$$
$$798$$ 0 0
$$799$$ −2.18034 −0.0771349
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 1.68692 0.0595671
$$803$$ −6.87539 −0.242627
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 2.90983 0.102494
$$807$$ 0 0
$$808$$ 25.9787 0.913928
$$809$$ −24.9230 −0.876246 −0.438123 0.898915i $$-0.644356\pi$$
−0.438123 + 0.898915i $$0.644356\pi$$
$$810$$ 0 0
$$811$$ 37.7771 1.32653 0.663266 0.748383i $$-0.269170\pi$$
0.663266 + 0.748383i $$0.269170\pi$$
$$812$$ −4.47214 −0.156941
$$813$$ 0 0
$$814$$ −3.70820 −0.129972
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 64.5197 2.25726
$$818$$ −21.7082 −0.759010
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 11.9443 0.416858 0.208429 0.978038i $$-0.433165\pi$$
0.208429 + 0.978038i $$0.433165\pi$$
$$822$$ 0 0
$$823$$ 8.43769 0.294120 0.147060 0.989128i $$-0.453019\pi$$
0.147060 + 0.989128i $$0.453019\pi$$
$$824$$ −27.7639 −0.967202
$$825$$ 0 0
$$826$$ −17.2361 −0.599720
$$827$$ 2.02129 0.0702870 0.0351435 0.999382i $$-0.488811\pi$$
0.0351435 + 0.999382i $$0.488811\pi$$
$$828$$ 0 0
$$829$$ −9.87539 −0.342986 −0.171493 0.985185i $$-0.554859\pi$$
−0.171493 + 0.985185i $$0.554859\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −0.236068 −0.00818418
$$833$$ −0.708204 −0.0245378
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −32.5623 −1.12619
$$837$$ 0 0
$$838$$ −9.47214 −0.327210
$$839$$ −48.2148 −1.66456 −0.832280 0.554356i $$-0.812965\pi$$
−0.832280 + 0.554356i $$0.812965\pi$$
$$840$$ 0 0
$$841$$ −27.0902 −0.934144
$$842$$ −8.87539 −0.305866
$$843$$ 0 0
$$844$$ 4.85410 0.167085
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −4.00000 −0.137442
$$848$$ 12.5410 0.430660
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 15.2361 0.522286
$$852$$ 0 0
$$853$$ −53.3951 −1.82821 −0.914107 0.405473i $$-0.867107\pi$$
−0.914107 + 0.405473i $$0.867107\pi$$
$$854$$ 5.81966 0.199145
$$855$$ 0 0
$$856$$ −17.5623 −0.600267
$$857$$ 26.9443 0.920399 0.460199 0.887816i $$-0.347778\pi$$
0.460199 + 0.887816i $$0.347778\pi$$
$$858$$ 0 0
$$859$$ −25.1246 −0.857241 −0.428620 0.903485i $$-0.641000\pi$$
−0.428620 + 0.903485i $$0.641000\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 21.1591 0.720680
$$863$$ −45.0689 −1.53416 −0.767081 0.641550i $$-0.778292\pi$$
−0.767081 + 0.641550i $$0.778292\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −3.38197 −0.114924
$$867$$ 0 0
$$868$$ 15.2361 0.517146
$$869$$ −47.5623 −1.61344
$$870$$ 0 0
$$871$$ −9.18034 −0.311064
$$872$$ −24.2705 −0.821903
$$873$$ 0 0
$$874$$ −31.5836 −1.06833
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −2.87539 −0.0970950 −0.0485475 0.998821i $$-0.515459\pi$$
−0.0485475 + 0.998821i $$0.515459\pi$$
$$878$$ 1.83282 0.0618545
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −3.90983 −0.131726 −0.0658628 0.997829i $$-0.520980\pi$$
−0.0658628 + 0.997829i $$0.520980\pi$$
$$882$$ 0 0
$$883$$ −53.7984 −1.81046 −0.905230 0.424923i $$-0.860301\pi$$
−0.905230 + 0.424923i $$0.860301\pi$$
$$884$$ −0.381966 −0.0128469
$$885$$ 0 0
$$886$$ −4.58359 −0.153989
$$887$$ −21.9230 −0.736102 −0.368051 0.929806i $$-0.619975\pi$$
−0.368051 + 0.929806i $$0.619975\pi$$
$$888$$ 0 0
$$889$$ 35.3050 1.18409
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −27.2705 −0.913084
$$893$$ −61.9574 −2.07333
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 22.7639 0.760490
$$897$$ 0 0
$$898$$ −13.2918 −0.443553
$$899$$ −6.50658 −0.217007
$$900$$ 0 0
$$901$$ 1.59675 0.0531954
$$902$$ −21.5410 −0.717238
$$903$$ 0 0
$$904$$ −18.4164 −0.612521
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 17.0000 0.564476 0.282238 0.959344i $$-0.408923\pi$$
0.282238 + 0.959344i $$0.408923\pi$$
$$908$$ −16.5623 −0.549639
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 20.8885 0.692068 0.346034 0.938222i $$-0.387528\pi$$
0.346034 + 0.938222i $$0.387528\pi$$
$$912$$ 0 0
$$913$$ 27.0000 0.893570
$$914$$ 16.0000 0.529233
$$915$$ 0 0
$$916$$ −10.0000 −0.330409
$$917$$ −16.3607 −0.540277
$$918$$ 0 0
$$919$$ −5.00000 −0.164935 −0.0824674 0.996594i $$-0.526280\pi$$
−0.0824674 + 0.996594i $$0.526280\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 1.96556 0.0647322
$$923$$ 1.09017 0.0358834
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 16.4934 0.542007
$$927$$ 0 0
$$928$$ −7.76393 −0.254864
$$929$$ −19.5967 −0.642948 −0.321474 0.946918i $$-0.604178\pi$$
−0.321474 + 0.946918i $$0.604178\pi$$
$$930$$ 0 0
$$931$$ −20.1246 −0.659558
$$932$$ 19.7082 0.645564
$$933$$ 0 0
$$934$$ −10.1459 −0.331984
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −16.4164 −0.536301 −0.268150 0.963377i $$-0.586412\pi$$
−0.268150 + 0.963377i $$0.586412\pi$$
$$938$$ 11.3475 0.370510
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −11.0213 −0.359284 −0.179642 0.983732i $$-0.557494\pi$$
−0.179642 + 0.983732i $$0.557494\pi$$
$$942$$ 0 0
$$943$$ 88.5066 2.88217
$$944$$ 25.8541 0.841479
$$945$$ 0 0
$$946$$ −17.8328 −0.579795
$$947$$ 29.8328 0.969436 0.484718 0.874670i $$-0.338922\pi$$
0.484718 + 0.874670i $$0.338922\pi$$
$$948$$ 0 0
$$949$$ −2.29180 −0.0743948
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 1.05573 0.0342163
$$953$$ −59.9443 −1.94179 −0.970893 0.239515i $$-0.923012\pi$$
−0.970893 + 0.239515i $$0.923012\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 38.2148 1.23595
$$957$$ 0 0
$$958$$ −12.2361 −0.395329
$$959$$ 41.1246 1.32798
$$960$$ 0 0
$$961$$ −8.83282 −0.284930
$$962$$ −1.23607 −0.0398524
$$963$$ 0 0
$$964$$ 13.4721 0.433908
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 8.58359 0.276030 0.138015 0.990430i $$-0.455928\pi$$
0.138015 + 0.990430i $$0.455928\pi$$
$$968$$ −4.47214 −0.143740
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 5.88854 0.188972 0.0944862 0.995526i $$-0.469879\pi$$
0.0944862 + 0.995526i $$0.469879\pi$$
$$972$$ 0 0
$$973$$ −26.8328 −0.860221
$$974$$ 8.88854 0.284807
$$975$$ 0 0
$$976$$ −8.72949 −0.279424
$$977$$ −6.34752 −0.203075 −0.101538 0.994832i $$-0.532376\pi$$
−0.101538 + 0.994832i $$0.532376\pi$$
$$978$$ 0 0
$$979$$ −33.5410 −1.07198
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 4.12461 0.131622
$$983$$ −22.3820 −0.713874 −0.356937 0.934128i $$-0.616179\pi$$
−0.356937 + 0.934128i $$0.616179\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −0.201626 −0.00642108
$$987$$ 0 0
$$988$$ −10.8541 −0.345315
$$989$$ 73.2705 2.32987
$$990$$ 0 0
$$991$$ 32.0000 1.01651 0.508257 0.861206i $$-0.330290\pi$$
0.508257 + 0.861206i $$0.330290\pi$$
$$992$$ 26.4508 0.839815
$$993$$ 0 0
$$994$$ −1.34752 −0.0427409
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 61.0689 1.93407 0.967035 0.254642i $$-0.0819575\pi$$
0.967035 + 0.254642i $$0.0819575\pi$$
$$998$$ 9.27051 0.293453
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 5625.2.a.g.1.1 2
3.2 odd 2 1875.2.a.b.1.2 2
5.4 even 2 5625.2.a.b.1.2 2
15.2 even 4 1875.2.b.a.1249.3 4
15.8 even 4 1875.2.b.a.1249.2 4
15.14 odd 2 1875.2.a.c.1.1 yes 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1875.2.a.b.1.2 2 3.2 odd 2
1875.2.a.c.1.1 yes 2 15.14 odd 2
1875.2.b.a.1249.2 4 15.8 even 4
1875.2.b.a.1249.3 4 15.2 even 4
5625.2.a.b.1.2 2 5.4 even 2
5625.2.a.g.1.1 2 1.1 even 1 trivial