# Properties

 Label 5625.2.a.b.1.2 Level $5625$ Weight $2$ Character 5625.1 Self dual yes Analytic conductor $44.916$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [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: $$1$$ 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 1.2 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 $$-0.429881\pi$$
0.218508 + 0.975835i $$0.429881\pi$$
$$3$$ 0 0
$$4$$ −1.61803 −0.809017
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −2.00000 −0.755929 −0.377964 0.925820i $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ −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 $$-0.544284\pi$$
−0.138675 + 0.990338i $$0.544284\pi$$
$$14$$ −1.23607 −0.330353
$$15$$ 0 0
$$16$$ 1.85410 0.463525
$$17$$ −0.236068 −0.0572549 −0.0286274 0.999590i $$-0.509114\pi$$
−0.0286274 + 0.999590i $$0.509114\pi$$
$$18$$ 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 $$-0.792128\pi$$
−0.794235 + 0.607611i $$0.792128\pi$$
$$24$$ 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 $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 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 $$-0.762053\pi$$
−0.733368 + 0.679832i $$0.762053\pi$$
$$44$$ −4.85410 −0.731783
$$45$$ 0 0
$$46$$ −4.70820 −0.694187
$$47$$ 9.23607 1.34722 0.673609 0.739087i $$-0.264743\pi$$
0.673609 + 0.739087i $$0.264743\pi$$
$$48$$ 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 $$-0.653783\pi$$
−0.464549 + 0.885548i $$0.653783\pi$$
$$54$$ 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 $$-0.310502\pi$$
0.560779 + 0.827966i $$0.310502\pi$$
$$68$$ 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 $$-0.457180\pi$$
0.134117 + 0.990965i $$0.457180\pi$$
$$74$$ −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 $$-0.664443\pi$$
−0.493939 + 0.869496i $$0.664443\pi$$
$$84$$ 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 $$-0.546284\pi$$
−0.144895 + 0.989447i $$0.546284\pi$$
$$98$$ −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 $$-0.290481\pi$$
0.611713 + 0.791080i $$0.290481\pi$$
$$104$$ 2.23607 0.219265
$$105$$ 0 0
$$106$$ −4.18034 −0.406031
$$107$$ 7.85410 0.759285 0.379642 0.925133i $$-0.376047\pi$$
0.379642 + 0.925133i $$0.376047\pi$$
$$108$$ 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 $$-0.373376\pi$$
0.387392 + 0.921915i $$0.373376\pi$$
$$114$$ 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 $$-0.786415\pi$$
−0.783202 + 0.621767i $$0.786415\pi$$
$$128$$ −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 $$-0.841374\pi$$
−0.878378 + 0.477966i $$0.841374\pi$$
$$138$$ 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 $$-0.450850\pi$$
0.153795 + 0.988103i $$0.450850\pi$$
$$158$$ −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 $$-0.704053\pi$$
−0.598039 + 0.801467i $$0.704053\pi$$
$$164$$ −18.7984 −1.46791
$$165$$ 0 0
$$166$$ −5.56231 −0.431719
$$167$$ 6.79837 0.526074 0.263037 0.964786i $$-0.415276\pi$$
0.263037 + 0.964786i $$0.415276\pi$$
$$168$$ 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 $$-0.652007\pi$$
−0.459599 + 0.888126i $$0.652007\pi$$
$$174$$ 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 $$-0.629567\pi$$
−0.395899 + 0.918294i $$0.629567\pi$$
$$194$$ −1.76393 −0.126643
$$195$$ 0 0
$$196$$ 4.85410 0.346722
$$197$$ −11.0902 −0.790142 −0.395071 0.918651i $$-0.629280\pi$$
−0.395071 + 0.918651i $$0.629280\pi$$
$$198$$ 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 $$-0.690860\pi$$
−0.564317 + 0.825558i $$0.690860\pi$$
$$224$$ −11.2361 −0.750741
$$225$$ 0 0
$$226$$ 5.09017 0.338593
$$227$$ −10.2361 −0.679392 −0.339696 0.940535i $$-0.610324\pi$$
−0.339696 + 0.940535i $$0.610324\pi$$
$$228$$ 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 $$-0.369364\pi$$
0.398980 + 0.916959i $$0.369364\pi$$
$$234$$ 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 $$-0.282857\pi$$
0.630482 + 0.776204i $$0.282857\pi$$
$$258$$ 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 $$-0.788058\pi$$
−0.786401 + 0.617716i $$0.788058\pi$$
$$264$$ 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 $$-0.879874\pi$$
−0.929631 + 0.368493i $$0.879874\pi$$
$$278$$ −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 $$-0.345379\pi$$
0.466878 + 0.884322i $$0.345379\pi$$
$$284$$ −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 $$-0.587825\pi$$
−0.272422 + 0.962178i $$0.587825\pi$$
$$294$$ 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 $$-0.480496\pi$$
0.0612364 + 0.998123i $$0.480496\pi$$
$$308$$ 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 $$-0.353580\pi$$
0.443940 + 0.896056i $$0.353580\pi$$
$$314$$ 2.38197 0.134422
$$315$$ 0 0
$$316$$ 25.6525 1.44306
$$317$$ −0.437694 −0.0245833 −0.0122917 0.999924i $$-0.503913\pi$$
−0.0122917 + 0.999924i $$0.503913\pi$$
$$318$$ 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 $$-0.595217\pi$$
−0.294692 + 0.955592i $$0.595217\pi$$
$$338$$ −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 $$-0.306583\pi$$
0.570930 + 0.820998i $$0.306583\pi$$
$$348$$ 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 $$-0.372814\pi$$
0.389020 + 0.921229i $$0.372814\pi$$
$$354$$ 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 $$-0.689360\pi$$
−0.560418 + 0.828210i $$0.689360\pi$$
$$368$$ −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 $$-0.578388\pi$$
−0.243782 + 0.969830i $$0.578388\pi$$
$$374$$ −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 $$-0.699376\pi$$
−0.586199 + 0.810167i $$0.699376\pi$$
$$384$$ 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 $$-0.592952\pi$$
−0.287885 + 0.957665i $$0.592952\pi$$
$$398$$ −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 $$-0.541975\pi$$
−0.131487 + 0.991318i $$0.541975\pi$$
$$434$$ 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 $$-0.556375\pi$$
−0.176182 + 0.984358i $$0.556375\pi$$
$$444$$ 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 $$-0.292970\pi$$
0.605508 + 0.795840i $$0.292970\pi$$
$$458$$ 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 $$-0.287083\pi$$
0.620123 + 0.784505i $$0.287083\pi$$
$$464$$ 2.56231 0.118952
$$465$$ 0 0
$$466$$ 7.52786 0.348722
$$467$$ −16.4164 −0.759661 −0.379830 0.925056i $$-0.624018\pi$$
−0.379830 + 0.925056i $$0.624018\pi$$
$$468$$ 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 $$-0.394348\pi$$
0.325855 + 0.945420i $$0.394348\pi$$
$$488$$ 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 $$-0.236493\pi$$
0.736466 + 0.676474i $$0.236493\pi$$
$$504$$ 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 $$-0.633020\pi$$
−0.405836 + 0.913946i $$0.633020\pi$$
$$524$$ 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 $$-0.422393\pi$$
0.241401 + 0.970425i $$0.422393\pi$$
$$548$$ 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 $$-0.457065\pi$$
0.134477 + 0.990917i $$0.457065\pi$$
$$558$$ 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 $$-0.560736\pi$$
−0.189652 + 0.981851i $$0.560736\pi$$
$$564$$ 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 $$-0.217349\pi$$
0.775796 + 0.630984i $$0.217349\pi$$
$$578$$ −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 $$-0.340292\pi$$
0.480949 + 0.876748i $$0.340292\pi$$
$$588$$ 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 $$-0.602283\pi$$
−0.315831 + 0.948816i $$0.602283\pi$$
$$594$$ 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 $$-0.606264\pi$$
−0.327671 + 0.944792i $$0.606264\pi$$
$$608$$ 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 $$-0.881480\pi$$
−0.931478 + 0.363798i $$0.881480\pi$$
$$614$$ 1.32624 0.0535226
$$615$$ 0 0
$$616$$ 13.4164 0.540562
$$617$$ −20.7639 −0.835924 −0.417962 0.908464i $$-0.637256\pi$$
−0.417962 + 0.908464i $$0.637256\pi$$
$$618$$ 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 $$-0.390593\pi$$
0.336985 + 0.941510i $$0.390593\pi$$
$$644$$ −24.6525 −0.971444
$$645$$ 0 0
$$646$$ −0.978714 −0.0385070
$$647$$ −10.0344 −0.394495 −0.197247 0.980354i $$-0.563200\pi$$
−0.197247 + 0.980354i $$0.563200\pi$$
$$648$$ 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 $$-0.489706\pi$$
0.0323332 + 0.999477i $$0.489706\pi$$
$$654$$ 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 $$-0.341527\pi$$
0.477543 + 0.878608i $$0.341527\pi$$
$$674$$ −6.68692 −0.257570
$$675$$ 0 0
$$676$$ 19.4164 0.746785
$$677$$ 9.11146 0.350182 0.175091 0.984552i $$-0.443978\pi$$
0.175091 + 0.984552i $$0.443978\pi$$
$$678$$ 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 $$-0.879978\pi$$
−0.929751 + 0.368188i $$0.879978\pi$$
$$684$$ 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 $$-0.721448\pi$$
−0.640922 + 0.767606i $$0.721448\pi$$
$$728$$ −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 $$-0.600751\pi$$
−0.311260 + 0.950325i $$0.600751\pi$$
$$734$$ −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 $$-0.345248\pi$$
0.467241 + 0.884130i $$0.345248\pi$$
$$744$$ 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 $$-0.507350\pi$$
−0.0230887 + 0.999733i $$0.507350\pi$$
$$758$$ −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 $$-0.679878\pi$$
−0.535504 + 0.844533i $$0.679878\pi$$
$$774$$ 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 $$-0.360220\pi$$
0.425153 + 0.905121i $$0.360220\pi$$
$$788$$ 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 $$-0.803227\pi$$
−0.814935 + 0.579553i $$0.803227\pi$$
$$798$$ 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 $$-0.546981\pi$$
−0.147060 + 0.989128i $$0.546981\pi$$
$$824$$ −27.7639 −0.967202
$$825$$ 0 0
$$826$$ −17.2361 −0.599720
$$827$$ −2.02129 −0.0702870 −0.0351435 0.999382i $$-0.511189\pi$$
−0.0351435 + 0.999382i $$0.511189\pi$$
$$828$$ 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 $$-0.132893\pi$$
0.914107 + 0.405473i $$0.132893\pi$$
$$854$$ 5.81966 0.199145
$$855$$ 0 0
$$856$$ −17.5623 −0.600267
$$857$$ −26.9443 −0.920399 −0.460199 0.887816i $$-0.652222\pi$$
−0.460199 + 0.887816i $$0.652222\pi$$
$$858$$ 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 $$-0.221708\pi$$
0.767081 + 0.641550i $$0.221708\pi$$
$$864$$ 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 $$-0.484541\pi$$
0.0485475 + 0.998821i $$0.484541\pi$$
$$878$$ −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 $$-0.139699\pi$$
0.905230 + 0.424923i $$0.139699\pi$$
$$884$$ −0.381966 −0.0128469
$$885$$ 0 0
$$886$$ −4.58359 −0.153989
$$887$$ 21.9230 0.736102 0.368051 0.929806i $$-0.380025\pi$$
0.368051 + 0.929806i $$0.380025\pi$$
$$888$$ 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 $$-0.591077\pi$$
−0.282238 + 0.959344i $$0.591077\pi$$
$$908$$ 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 $$-0.413588\pi$$
0.268150 + 0.963377i $$0.413588\pi$$
$$938$$ −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 $$-0.661078\pi$$
−0.484718 + 0.874670i $$0.661078\pi$$
$$948$$ 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 $$-0.0769884\pi$$
0.970893 + 0.239515i $$0.0769884\pi$$
$$954$$ 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 $$-0.544072\pi$$
−0.138015 + 0.990430i $$0.544072\pi$$
$$968$$ 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 $$-0.467624\pi$$
0.101538 + 0.994832i $$0.467624\pi$$
$$978$$ 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 $$-0.383821\pi$$
0.356937 + 0.934128i $$0.383821\pi$$
$$984$$ 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 $$-0.918042\pi$$
−0.967035 + 0.254642i $$0.918042\pi$$
$$998$$ −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.b.1.2 2
3.2 odd 2 1875.2.a.c.1.1 yes 2
5.4 even 2 5625.2.a.g.1.1 2
15.2 even 4 1875.2.b.a.1249.2 4
15.8 even 4 1875.2.b.a.1249.3 4
15.14 odd 2 1875.2.a.b.1.2 2

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