# Properties

 Label 1875.2.b.a.1249.3 Level $1875$ Weight $2$ Character 1875.1249 Analytic conductor $14.972$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1875,2,Mod(1249,1875)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1875, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("1875.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1875 = 3 \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1875.b (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$14.9719503790$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1249.3 Root $$0.618034i$$ of defining polynomial Character $$\chi$$ $$=$$ 1875.1249 Dual form 1875.2.b.a.1249.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.618034i q^{2} +1.00000i q^{3} +1.61803 q^{4} -0.618034 q^{6} +2.00000i q^{7} +2.23607i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+0.618034i q^{2} +1.00000i q^{3} +1.61803 q^{4} -0.618034 q^{6} +2.00000i q^{7} +2.23607i q^{8} -1.00000 q^{9} -3.00000 q^{11} +1.61803i q^{12} -1.00000i q^{13} -1.23607 q^{14} +1.85410 q^{16} -0.236068i q^{17} -0.618034i q^{18} -6.70820 q^{19} -2.00000 q^{21} -1.85410i q^{22} +7.61803i q^{23} -2.23607 q^{24} +0.618034 q^{26} -1.00000i q^{27} +3.23607i q^{28} +1.38197 q^{29} -4.70820 q^{31} +5.61803i q^{32} -3.00000i q^{33} +0.145898 q^{34} -1.61803 q^{36} +2.00000i q^{37} -4.14590i q^{38} +1.00000 q^{39} -11.6180 q^{41} -1.23607i q^{42} -9.61803i q^{43} -4.85410 q^{44} -4.70820 q^{46} +9.23607i q^{47} +1.85410i q^{48} +3.00000 q^{49} +0.236068 q^{51} -1.61803i q^{52} +6.76393i q^{53} +0.618034 q^{54} -4.47214 q^{56} -6.70820i q^{57} +0.854102i q^{58} +13.9443 q^{59} -4.70820 q^{61} -2.90983i q^{62} -2.00000i q^{63} +0.236068 q^{64} +1.85410 q^{66} -9.18034i q^{67} -0.381966i q^{68} -7.61803 q^{69} -1.09017 q^{71} -2.23607i q^{72} +2.29180i q^{73} -1.23607 q^{74} -10.8541 q^{76} -6.00000i q^{77} +0.618034i q^{78} +15.8541 q^{79} +1.00000 q^{81} -7.18034i q^{82} +9.00000i q^{83} -3.23607 q^{84} +5.94427 q^{86} +1.38197i q^{87} -6.70820i q^{88} -11.1803 q^{89} +2.00000 q^{91} +12.3262i q^{92} -4.70820i q^{93} -5.70820 q^{94} -5.61803 q^{96} +2.85410i q^{97} +1.85410i q^{98} +3.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4} + 2 q^{6} - 4 q^{9}+O(q^{10})$$ 4 * q + 2 * q^4 + 2 * q^6 - 4 * q^9 $$4 q + 2 q^{4} + 2 q^{6} - 4 q^{9} - 12 q^{11} + 4 q^{14} - 6 q^{16} - 8 q^{21} - 2 q^{26} + 10 q^{29} + 8 q^{31} + 14 q^{34} - 2 q^{36} + 4 q^{39} - 42 q^{41} - 6 q^{44} + 8 q^{46} + 12 q^{49} - 8 q^{51} - 2 q^{54} + 20 q^{59} + 8 q^{61} - 8 q^{64} - 6 q^{66} - 26 q^{69} + 18 q^{71} + 4 q^{74} - 30 q^{76} + 50 q^{79} + 4 q^{81} - 4 q^{84} - 12 q^{86} + 8 q^{91} + 4 q^{94} - 18 q^{96} + 12 q^{99}+O(q^{100})$$ 4 * q + 2 * q^4 + 2 * q^6 - 4 * q^9 - 12 * q^11 + 4 * q^14 - 6 * q^16 - 8 * q^21 - 2 * q^26 + 10 * q^29 + 8 * q^31 + 14 * q^34 - 2 * q^36 + 4 * q^39 - 42 * q^41 - 6 * q^44 + 8 * q^46 + 12 * q^49 - 8 * q^51 - 2 * q^54 + 20 * q^59 + 8 * q^61 - 8 * q^64 - 6 * q^66 - 26 * q^69 + 18 * q^71 + 4 * q^74 - 30 * q^76 + 50 * q^79 + 4 * q^81 - 4 * q^84 - 12 * q^86 + 8 * q^91 + 4 * q^94 - 18 * q^96 + 12 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times$$.

 $$n$$ $$626$$ $$1252$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.618034i 0.437016i 0.975835 + 0.218508i $$0.0701190\pi$$
−0.975835 + 0.218508i $$0.929881\pi$$
$$3$$ 1.00000i 0.577350i
$$4$$ 1.61803 0.809017
$$5$$ 0 0
$$6$$ −0.618034 −0.252311
$$7$$ 2.00000i 0.755929i 0.925820 + 0.377964i $$0.123376\pi$$
−0.925820 + 0.377964i $$0.876624\pi$$
$$8$$ 2.23607i 0.790569i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −3.00000 −0.904534 −0.452267 0.891883i $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ 1.61803i 0.467086i
$$13$$ − 1.00000i − 0.277350i −0.990338 0.138675i $$-0.955716\pi$$
0.990338 0.138675i $$-0.0442844\pi$$
$$14$$ −1.23607 −0.330353
$$15$$ 0 0
$$16$$ 1.85410 0.463525
$$17$$ − 0.236068i − 0.0572549i −0.999590 0.0286274i $$-0.990886\pi$$
0.999590 0.0286274i $$-0.00911364\pi$$
$$18$$ − 0.618034i − 0.145672i
$$19$$ −6.70820 −1.53897 −0.769484 0.638666i $$-0.779486\pi$$
−0.769484 + 0.638666i $$0.779486\pi$$
$$20$$ 0 0
$$21$$ −2.00000 −0.436436
$$22$$ − 1.85410i − 0.395296i
$$23$$ 7.61803i 1.58847i 0.607611 + 0.794235i $$0.292128\pi$$
−0.607611 + 0.794235i $$0.707872\pi$$
$$24$$ −2.23607 −0.456435
$$25$$ 0 0
$$26$$ 0.618034 0.121206
$$27$$ − 1.00000i − 0.192450i
$$28$$ 3.23607i 0.611559i
$$29$$ 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 $$0.638956\pi$$
$$32$$ 5.61803i 0.993137i
$$33$$ − 3.00000i − 0.522233i
$$34$$ 0.145898 0.0250213
$$35$$ 0 0
$$36$$ −1.61803 −0.269672
$$37$$ 2.00000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ − 4.14590i − 0.672553i
$$39$$ 1.00000 0.160128
$$40$$ 0 0
$$41$$ −11.6180 −1.81443 −0.907216 0.420665i $$-0.861797\pi$$
−0.907216 + 0.420665i $$0.861797\pi$$
$$42$$ − 1.23607i − 0.190729i
$$43$$ − 9.61803i − 1.46674i −0.679832 0.733368i $$-0.737947\pi$$
0.679832 0.733368i $$-0.262053\pi$$
$$44$$ −4.85410 −0.731783
$$45$$ 0 0
$$46$$ −4.70820 −0.694187
$$47$$ 9.23607i 1.34722i 0.739087 + 0.673609i $$0.235257\pi$$
−0.739087 + 0.673609i $$0.764743\pi$$
$$48$$ 1.85410i 0.267617i
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ 0.236068 0.0330561
$$52$$ − 1.61803i − 0.224381i
$$53$$ 6.76393i 0.929098i 0.885548 + 0.464549i $$0.153783\pi$$
−0.885548 + 0.464549i $$0.846217\pi$$
$$54$$ 0.618034 0.0841038
$$55$$ 0 0
$$56$$ −4.47214 −0.597614
$$57$$ − 6.70820i − 0.888523i
$$58$$ 0.854102i 0.112149i
$$59$$ 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 $$0.597458\pi$$
$$62$$ − 2.90983i − 0.369549i
$$63$$ − 2.00000i − 0.251976i
$$64$$ 0.236068 0.0295085
$$65$$ 0 0
$$66$$ 1.85410 0.228224
$$67$$ − 9.18034i − 1.12156i −0.827966 0.560779i $$-0.810502\pi$$
0.827966 0.560779i $$-0.189498\pi$$
$$68$$ − 0.381966i − 0.0463202i
$$69$$ −7.61803 −0.917104
$$70$$ 0 0
$$71$$ −1.09017 −0.129379 −0.0646897 0.997905i $$-0.520606\pi$$
−0.0646897 + 0.997905i $$0.520606\pi$$
$$72$$ − 2.23607i − 0.263523i
$$73$$ 2.29180i 0.268234i 0.990965 + 0.134117i $$0.0428199\pi$$
−0.990965 + 0.134117i $$0.957180\pi$$
$$74$$ −1.23607 −0.143690
$$75$$ 0 0
$$76$$ −10.8541 −1.24505
$$77$$ − 6.00000i − 0.683763i
$$78$$ 0.618034i 0.0699786i
$$79$$ 15.8541 1.78373 0.891863 0.452306i $$-0.149398\pi$$
0.891863 + 0.452306i $$0.149398\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 7.18034i − 0.792936i
$$83$$ 9.00000i 0.987878i 0.869496 + 0.493939i $$0.164443\pi$$
−0.869496 + 0.493939i $$0.835557\pi$$
$$84$$ −3.23607 −0.353084
$$85$$ 0 0
$$86$$ 5.94427 0.640987
$$87$$ 1.38197i 0.148162i
$$88$$ − 6.70820i − 0.715097i
$$89$$ −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
$$92$$ 12.3262i 1.28510i
$$93$$ − 4.70820i − 0.488218i
$$94$$ −5.70820 −0.588756
$$95$$ 0 0
$$96$$ −5.61803 −0.573388
$$97$$ 2.85410i 0.289790i 0.989447 + 0.144895i $$0.0462845\pi$$
−0.989447 + 0.144895i $$0.953716\pi$$
$$98$$ 1.85410i 0.187293i
$$99$$ 3.00000 0.301511
$$100$$ 0 0
$$101$$ −11.6180 −1.15604 −0.578019 0.816023i $$-0.696174\pi$$
−0.578019 + 0.816023i $$0.696174\pi$$
$$102$$ 0.145898i 0.0144461i
$$103$$ 12.4164i 1.22343i 0.791080 + 0.611713i $$0.209519\pi$$
−0.791080 + 0.611713i $$0.790481\pi$$
$$104$$ 2.23607 0.219265
$$105$$ 0 0
$$106$$ −4.18034 −0.406031
$$107$$ 7.85410i 0.759285i 0.925133 + 0.379642i $$0.123953\pi$$
−0.925133 + 0.379642i $$0.876047\pi$$
$$108$$ − 1.61803i − 0.155695i
$$109$$ 10.8541 1.03963 0.519817 0.854278i $$-0.326000\pi$$
0.519817 + 0.854278i $$0.326000\pi$$
$$110$$ 0 0
$$111$$ −2.00000 −0.189832
$$112$$ 3.70820i 0.350392i
$$113$$ − 8.23607i − 0.774784i −0.921915 0.387392i $$-0.873376\pi$$
0.921915 0.387392i $$-0.126624\pi$$
$$114$$ 4.14590 0.388299
$$115$$ 0 0
$$116$$ 2.23607 0.207614
$$117$$ 1.00000i 0.0924500i
$$118$$ 8.61803i 0.793354i
$$119$$ 0.472136 0.0432806
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ − 2.90983i − 0.263444i
$$123$$ − 11.6180i − 1.04756i
$$124$$ −7.61803 −0.684120
$$125$$ 0 0
$$126$$ 1.23607 0.110118
$$127$$ 17.6525i 1.56640i 0.621767 + 0.783202i $$0.286415\pi$$
−0.621767 + 0.783202i $$0.713585\pi$$
$$128$$ 11.3820i 1.00603i
$$129$$ 9.61803 0.846821
$$130$$ 0 0
$$131$$ 8.18034 0.714720 0.357360 0.933967i $$-0.383677\pi$$
0.357360 + 0.933967i $$0.383677\pi$$
$$132$$ − 4.85410i − 0.422495i
$$133$$ − 13.4164i − 1.16335i
$$134$$ 5.67376 0.490138
$$135$$ 0 0
$$136$$ 0.527864 0.0452640
$$137$$ − 20.5623i − 1.75676i −0.477966 0.878378i $$-0.658626\pi$$
0.477966 0.878378i $$-0.341374\pi$$
$$138$$ − 4.70820i − 0.400789i
$$139$$ 13.4164 1.13796 0.568982 0.822350i $$-0.307337\pi$$
0.568982 + 0.822350i $$0.307337\pi$$
$$140$$ 0 0
$$141$$ −9.23607 −0.777817
$$142$$ − 0.673762i − 0.0565409i
$$143$$ 3.00000i 0.250873i
$$144$$ −1.85410 −0.154508
$$145$$ 0 0
$$146$$ −1.41641 −0.117223
$$147$$ 3.00000i 0.247436i
$$148$$ 3.23607i 0.266003i
$$149$$ 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 $$0.557060\pi$$
$$152$$ − 15.0000i − 1.21666i
$$153$$ 0.236068i 0.0190850i
$$154$$ 3.70820 0.298816
$$155$$ 0 0
$$156$$ 1.61803 0.129546
$$157$$ − 3.85410i − 0.307591i −0.988103 0.153795i $$-0.950850\pi$$
0.988103 0.153795i $$-0.0491497\pi$$
$$158$$ 9.79837i 0.779517i
$$159$$ −6.76393 −0.536415
$$160$$ 0 0
$$161$$ −15.2361 −1.20077
$$162$$ 0.618034i 0.0485573i
$$163$$ − 15.2705i − 1.19608i −0.801467 0.598039i $$-0.795947\pi$$
0.801467 0.598039i $$-0.204053\pi$$
$$164$$ −18.7984 −1.46791
$$165$$ 0 0
$$166$$ −5.56231 −0.431719
$$167$$ 6.79837i 0.526074i 0.964786 + 0.263037i $$0.0847241\pi$$
−0.964786 + 0.263037i $$0.915276\pi$$
$$168$$ − 4.47214i − 0.345033i
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ 6.70820 0.512989
$$172$$ − 15.5623i − 1.18661i
$$173$$ 12.0902i 0.919199i 0.888126 + 0.459599i $$0.152007\pi$$
−0.888126 + 0.459599i $$0.847993\pi$$
$$174$$ −0.854102 −0.0647493
$$175$$ 0 0
$$176$$ −5.56231 −0.419275
$$177$$ 13.9443i 1.04812i
$$178$$ − 6.90983i − 0.517914i
$$179$$ −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 $$0.541855\pi$$
$$182$$ 1.23607i 0.0916235i
$$183$$ − 4.70820i − 0.348040i
$$184$$ −17.0344 −1.25580
$$185$$ 0 0
$$186$$ 2.90983 0.213359
$$187$$ 0.708204i 0.0517890i
$$188$$ 14.9443i 1.08992i
$$189$$ 2.00000 0.145479
$$190$$ 0 0
$$191$$ 1.67376 0.121109 0.0605546 0.998165i $$-0.480713\pi$$
0.0605546 + 0.998165i $$0.480713\pi$$
$$192$$ 0.236068i 0.0170367i
$$193$$ − 11.0000i − 0.791797i −0.918294 0.395899i $$-0.870433\pi$$
0.918294 0.395899i $$-0.129567\pi$$
$$194$$ −1.76393 −0.126643
$$195$$ 0 0
$$196$$ 4.85410 0.346722
$$197$$ − 11.0902i − 0.790142i −0.918651 0.395071i $$-0.870720\pi$$
0.918651 0.395071i $$-0.129280\pi$$
$$198$$ 1.85410i 0.131765i
$$199$$ 1.70820 0.121091 0.0605457 0.998165i $$-0.480716\pi$$
0.0605457 + 0.998165i $$0.480716\pi$$
$$200$$ 0 0
$$201$$ 9.18034 0.647531
$$202$$ − 7.18034i − 0.505207i
$$203$$ 2.76393i 0.193990i
$$204$$ 0.381966 0.0267430
$$205$$ 0 0
$$206$$ −7.67376 −0.534656
$$207$$ − 7.61803i − 0.529490i
$$208$$ − 1.85410i − 0.128559i
$$209$$ 20.1246 1.39205
$$210$$ 0 0
$$211$$ −3.00000 −0.206529 −0.103264 0.994654i $$-0.532929\pi$$
−0.103264 + 0.994654i $$0.532929\pi$$
$$212$$ 10.9443i 0.751656i
$$213$$ − 1.09017i − 0.0746972i
$$214$$ −4.85410 −0.331820
$$215$$ 0 0
$$216$$ 2.23607 0.152145
$$217$$ − 9.41641i − 0.639227i
$$218$$ 6.70820i 0.454337i
$$219$$ −2.29180 −0.154865
$$220$$ 0 0
$$221$$ −0.236068 −0.0158797
$$222$$ − 1.23607i − 0.0829595i
$$223$$ − 16.8541i − 1.12863i −0.825558 0.564317i $$-0.809140\pi$$
0.825558 0.564317i $$-0.190860\pi$$
$$224$$ −11.2361 −0.750741
$$225$$ 0 0
$$226$$ 5.09017 0.338593
$$227$$ − 10.2361i − 0.679392i −0.940535 0.339696i $$-0.889676\pi$$
0.940535 0.339696i $$-0.110324\pi$$
$$228$$ − 10.8541i − 0.718830i
$$229$$ −6.18034 −0.408408 −0.204204 0.978928i $$-0.565461\pi$$
−0.204204 + 0.978928i $$0.565461\pi$$
$$230$$ 0 0
$$231$$ 6.00000 0.394771
$$232$$ 3.09017i 0.202880i
$$233$$ − 12.1803i − 0.797961i −0.916959 0.398980i $$-0.869364\pi$$
0.916959 0.398980i $$-0.130636\pi$$
$$234$$ −0.618034 −0.0404021
$$235$$ 0 0
$$236$$ 22.5623 1.46868
$$237$$ 15.8541i 1.02983i
$$238$$ 0.291796i 0.0189143i
$$239$$ −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 $$0.586419\pi$$
$$242$$ − 1.23607i − 0.0794575i
$$243$$ 1.00000i 0.0641500i
$$244$$ −7.61803 −0.487695
$$245$$ 0 0
$$246$$ 7.18034 0.457802
$$247$$ 6.70820i 0.426833i
$$248$$ − 10.5279i − 0.668520i
$$249$$ −9.00000 −0.570352
$$250$$ 0 0
$$251$$ 27.9787 1.76600 0.883000 0.469372i $$-0.155520\pi$$
0.883000 + 0.469372i $$0.155520\pi$$
$$252$$ − 3.23607i − 0.203853i
$$253$$ − 22.8541i − 1.43683i
$$254$$ −10.9098 −0.684544
$$255$$ 0 0
$$256$$ −6.56231 −0.410144
$$257$$ 20.2148i 1.26096i 0.776204 + 0.630482i $$0.217143\pi$$
−0.776204 + 0.630482i $$0.782857\pi$$
$$258$$ 5.94427i 0.370074i
$$259$$ −4.00000 −0.248548
$$260$$ 0 0
$$261$$ −1.38197 −0.0855415
$$262$$ 5.05573i 0.312344i
$$263$$ 25.5066i 1.57280i 0.617716 + 0.786401i $$0.288058\pi$$
−0.617716 + 0.786401i $$0.711942\pi$$
$$264$$ 6.70820 0.412861
$$265$$ 0 0
$$266$$ 8.29180 0.508403
$$267$$ − 11.1803i − 0.684226i
$$268$$ − 14.8541i − 0.907359i
$$269$$ 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 $$0.344888\pi$$
$$272$$ − 0.437694i − 0.0265391i
$$273$$ 2.00000i 0.121046i
$$274$$ 12.7082 0.767731
$$275$$ 0 0
$$276$$ −12.3262 −0.741952
$$277$$ 30.9443i 1.85926i 0.368493 + 0.929631i $$0.379874\pi$$
−0.368493 + 0.929631i $$0.620126\pi$$
$$278$$ 8.29180i 0.497309i
$$279$$ 4.70820 0.281873
$$280$$ 0 0
$$281$$ 8.18034 0.487998 0.243999 0.969775i $$-0.421541\pi$$
0.243999 + 0.969775i $$0.421541\pi$$
$$282$$ − 5.70820i − 0.339919i
$$283$$ 15.7082i 0.933756i 0.884322 + 0.466878i $$0.154621\pi$$
−0.884322 + 0.466878i $$0.845379\pi$$
$$284$$ −1.76393 −0.104670
$$285$$ 0 0
$$286$$ −1.85410 −0.109635
$$287$$ − 23.2361i − 1.37158i
$$288$$ − 5.61803i − 0.331046i
$$289$$ 16.9443 0.996722
$$290$$ 0 0
$$291$$ −2.85410 −0.167310
$$292$$ 3.70820i 0.217006i
$$293$$ 9.32624i 0.544845i 0.962178 + 0.272422i $$0.0878248\pi$$
−0.962178 + 0.272422i $$0.912175\pi$$
$$294$$ −1.85410 −0.108133
$$295$$ 0 0
$$296$$ −4.47214 −0.259938
$$297$$ 3.00000i 0.174078i
$$298$$ 1.18034i 0.0683753i
$$299$$ 7.61803 0.440562
$$300$$ 0 0
$$301$$ 19.2361 1.10875
$$302$$ − 2.70820i − 0.155840i
$$303$$ − 11.6180i − 0.667439i
$$304$$ −12.4377 −0.713351
$$305$$ 0 0
$$306$$ −0.145898 −0.00834044
$$307$$ − 2.14590i − 0.122473i −0.998123 0.0612364i $$-0.980496\pi$$
0.998123 0.0612364i $$-0.0195044\pi$$
$$308$$ − 9.70820i − 0.553176i
$$309$$ −12.4164 −0.706345
$$310$$ 0 0
$$311$$ −22.4721 −1.27428 −0.637139 0.770749i $$-0.719882\pi$$
−0.637139 + 0.770749i $$0.719882\pi$$
$$312$$ 2.23607i 0.126592i
$$313$$ 15.7082i 0.887880i 0.896056 + 0.443940i $$0.146420\pi$$
−0.896056 + 0.443940i $$0.853580\pi$$
$$314$$ 2.38197 0.134422
$$315$$ 0 0
$$316$$ 25.6525 1.44306
$$317$$ − 0.437694i − 0.0245833i −0.999924 0.0122917i $$-0.996087\pi$$
0.999924 0.0122917i $$-0.00391266\pi$$
$$318$$ − 4.18034i − 0.234422i
$$319$$ −4.14590 −0.232126
$$320$$ 0 0
$$321$$ −7.85410 −0.438373
$$322$$ − 9.41641i − 0.524756i
$$323$$ 1.58359i 0.0881134i
$$324$$ 1.61803 0.0898908
$$325$$ 0 0
$$326$$ 9.43769 0.522706
$$327$$ 10.8541i 0.600233i
$$328$$ − 25.9787i − 1.43443i
$$329$$ −18.4721 −1.01840
$$330$$ 0 0
$$331$$ 29.6869 1.63174 0.815870 0.578235i $$-0.196258\pi$$
0.815870 + 0.578235i $$0.196258\pi$$
$$332$$ 14.5623i 0.799210i
$$333$$ − 2.00000i − 0.109599i
$$334$$ −4.20163 −0.229903
$$335$$ 0 0
$$336$$ −3.70820 −0.202299
$$337$$ 10.8197i 0.589384i 0.955592 + 0.294692i $$0.0952171\pi$$
−0.955592 + 0.294692i $$0.904783\pi$$
$$338$$ 7.41641i 0.403399i
$$339$$ 8.23607 0.447322
$$340$$ 0 0
$$341$$ 14.1246 0.764891
$$342$$ 4.14590i 0.224184i
$$343$$ 20.0000i 1.07990i
$$344$$ 21.5066 1.15956
$$345$$ 0 0
$$346$$ −7.47214 −0.401705
$$347$$ 21.2705i 1.14186i 0.820998 + 0.570930i $$0.193417\pi$$
−0.820998 + 0.570930i $$0.806583\pi$$
$$348$$ 2.23607i 0.119866i
$$349$$ −2.76393 −0.147950 −0.0739749 0.997260i $$-0.523568\pi$$
−0.0739749 + 0.997260i $$0.523568\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ − 16.8541i − 0.898327i
$$353$$ − 14.6180i − 0.778039i −0.921229 0.389020i $$-0.872814\pi$$
0.921229 0.389020i $$-0.127186\pi$$
$$354$$ −8.61803 −0.458043
$$355$$ 0 0
$$356$$ −18.0902 −0.958777
$$357$$ 0.472136i 0.0249881i
$$358$$ − 9.67376i − 0.511274i
$$359$$ −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
$$362$$ − 2.18034i − 0.114596i
$$363$$ − 2.00000i − 0.104973i
$$364$$ 3.23607 0.169616
$$365$$ 0 0
$$366$$ 2.90983 0.152099
$$367$$ 21.4721i 1.12084i 0.828210 + 0.560418i $$0.189360\pi$$
−0.828210 + 0.560418i $$0.810640\pi$$
$$368$$ 14.1246i 0.736296i
$$369$$ 11.6180 0.604811
$$370$$ 0 0
$$371$$ −13.5279 −0.702332
$$372$$ − 7.61803i − 0.394977i
$$373$$ − 9.41641i − 0.487563i −0.969830 0.243782i $$-0.921612\pi$$
0.969830 0.243782i $$-0.0783880\pi$$
$$374$$ −0.437694 −0.0226326
$$375$$ 0 0
$$376$$ −20.6525 −1.06507
$$377$$ − 1.38197i − 0.0711749i
$$378$$ 1.23607i 0.0635765i
$$379$$ 11.3820 0.584652 0.292326 0.956319i $$-0.405571\pi$$
0.292326 + 0.956319i $$0.405571\pi$$
$$380$$ 0 0
$$381$$ −17.6525 −0.904364
$$382$$ 1.03444i 0.0529266i
$$383$$ 22.9443i 1.17240i 0.810167 + 0.586199i $$0.199376\pi$$
−0.810167 + 0.586199i $$0.800624\pi$$
$$384$$ −11.3820 −0.580834
$$385$$ 0 0
$$386$$ 6.79837 0.346028
$$387$$ 9.61803i 0.488912i
$$388$$ 4.61803i 0.234445i
$$389$$ 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
$$392$$ 6.70820i 0.338815i
$$393$$ 8.18034i 0.412644i
$$394$$ 6.85410 0.345305
$$395$$ 0 0
$$396$$ 4.85410 0.243928
$$397$$ 11.4721i 0.575770i 0.957665 + 0.287885i $$0.0929521\pi$$
−0.957665 + 0.287885i $$0.907048\pi$$
$$398$$ 1.05573i 0.0529189i
$$399$$ 13.4164 0.671660
$$400$$ 0 0
$$401$$ 2.72949 0.136304 0.0681521 0.997675i $$-0.478290\pi$$
0.0681521 + 0.997675i $$0.478290\pi$$
$$402$$ 5.67376i 0.282982i
$$403$$ 4.70820i 0.234532i
$$404$$ −18.7984 −0.935254
$$405$$ 0 0
$$406$$ −1.70820 −0.0847767
$$407$$ − 6.00000i − 0.297409i
$$408$$ 0.527864i 0.0261332i
$$409$$ −35.1246 −1.73680 −0.868400 0.495864i $$-0.834851\pi$$
−0.868400 + 0.495864i $$0.834851\pi$$
$$410$$ 0 0
$$411$$ 20.5623 1.01426
$$412$$ 20.0902i 0.989772i
$$413$$ 27.8885i 1.37231i
$$414$$ 4.70820 0.231396
$$415$$ 0 0
$$416$$ 5.61803 0.275447
$$417$$ 13.4164i 0.657004i
$$418$$ 12.4377i 0.608348i
$$419$$ 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 $$0.386199\pi$$
$$422$$ − 1.85410i − 0.0902563i
$$423$$ − 9.23607i − 0.449073i
$$424$$ −15.1246 −0.734516
$$425$$ 0 0
$$426$$ 0.673762 0.0326439
$$427$$ − 9.41641i − 0.455692i
$$428$$ 12.7082i 0.614274i
$$429$$ −3.00000 −0.144841
$$430$$ 0 0
$$431$$ 34.2361 1.64909 0.824547 0.565794i $$-0.191430\pi$$
0.824547 + 0.565794i $$0.191430\pi$$
$$432$$ − 1.85410i − 0.0892055i
$$433$$ − 5.47214i − 0.262974i −0.991318 0.131487i $$-0.958025\pi$$
0.991318 0.131487i $$-0.0419752\pi$$
$$434$$ 5.81966 0.279353
$$435$$ 0 0
$$436$$ 17.5623 0.841082
$$437$$ − 51.1033i − 2.44460i
$$438$$ − 1.41641i − 0.0676786i
$$439$$ 2.96556 0.141538 0.0707692 0.997493i $$-0.477455\pi$$
0.0707692 + 0.997493i $$0.477455\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ − 0.145898i − 0.00693966i
$$443$$ 7.41641i 0.352364i 0.984358 + 0.176182i $$0.0563748\pi$$
−0.984358 + 0.176182i $$0.943625\pi$$
$$444$$ −3.23607 −0.153577
$$445$$ 0 0
$$446$$ 10.4164 0.493231
$$447$$ 1.90983i 0.0903319i
$$448$$ 0.472136i 0.0223063i
$$449$$ 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
$$452$$ − 13.3262i − 0.626814i
$$453$$ − 4.38197i − 0.205883i
$$454$$ 6.32624 0.296905
$$455$$ 0 0
$$456$$ 15.0000 0.702439
$$457$$ − 25.8885i − 1.21102i −0.795840 0.605508i $$-0.792970\pi$$
0.795840 0.605508i $$-0.207030\pi$$
$$458$$ − 3.81966i − 0.178481i
$$459$$ −0.236068 −0.0110187
$$460$$ 0 0
$$461$$ 3.18034 0.148123 0.0740616 0.997254i $$-0.476404\pi$$
0.0740616 + 0.997254i $$0.476404\pi$$
$$462$$ 3.70820i 0.172521i
$$463$$ 26.6869i 1.24025i 0.784505 + 0.620123i $$0.212917\pi$$
−0.784505 + 0.620123i $$0.787083\pi$$
$$464$$ 2.56231 0.118952
$$465$$ 0 0
$$466$$ 7.52786 0.348722
$$467$$ − 16.4164i − 0.759661i −0.925056 0.379830i $$-0.875982\pi$$
0.925056 0.379830i $$-0.124018\pi$$
$$468$$ 1.61803i 0.0747936i
$$469$$ 18.3607 0.847817
$$470$$ 0 0
$$471$$ 3.85410 0.177588
$$472$$ 31.1803i 1.43519i
$$473$$ 28.8541i 1.32671i
$$474$$ −9.79837 −0.450054
$$475$$ 0 0
$$476$$ 0.763932 0.0350148
$$477$$ − 6.76393i − 0.309699i
$$478$$ − 14.5967i − 0.667640i
$$479$$ 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
$$482$$ − 5.14590i − 0.234389i
$$483$$ − 15.2361i − 0.693265i
$$484$$ −3.23607 −0.147094
$$485$$ 0 0
$$486$$ −0.618034 −0.0280346
$$487$$ − 14.3820i − 0.651709i −0.945420 0.325855i $$-0.894348\pi$$
0.945420 0.325855i $$-0.105652\pi$$
$$488$$ − 10.5279i − 0.476574i
$$489$$ 15.2705 0.690556
$$490$$ 0 0
$$491$$ 6.67376 0.301183 0.150591 0.988596i $$-0.451882\pi$$
0.150591 + 0.988596i $$0.451882\pi$$
$$492$$ − 18.7984i − 0.847496i
$$493$$ − 0.326238i − 0.0146930i
$$494$$ −4.14590 −0.186533
$$495$$ 0 0
$$496$$ −8.72949 −0.391966
$$497$$ − 2.18034i − 0.0978016i
$$498$$ − 5.56231i − 0.249253i
$$499$$ 15.0000 0.671492 0.335746 0.941953i $$-0.391012\pi$$
0.335746 + 0.941953i $$0.391012\pi$$
$$500$$ 0 0
$$501$$ −6.79837 −0.303729
$$502$$ 17.2918i 0.771771i
$$503$$ − 33.0344i − 1.47293i −0.676474 0.736466i $$-0.736493\pi$$
0.676474 0.736466i $$-0.263507\pi$$
$$504$$ 4.47214 0.199205
$$505$$ 0 0
$$506$$ 14.1246 0.627916
$$507$$ 12.0000i 0.532939i
$$508$$ 28.5623i 1.26725i
$$509$$ 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
$$512$$ 18.7082i 0.826794i
$$513$$ 6.70820i 0.296174i
$$514$$ −12.4934 −0.551061
$$515$$ 0 0
$$516$$ 15.5623 0.685092
$$517$$ − 27.7082i − 1.21861i
$$518$$ − 2.47214i − 0.108619i
$$519$$ −12.0902 −0.530700
$$520$$ 0 0
$$521$$ 28.9098 1.26656 0.633281 0.773922i $$-0.281708\pi$$
0.633281 + 0.773922i $$0.281708\pi$$
$$522$$ − 0.854102i − 0.0373830i
$$523$$ − 18.5623i − 0.811673i −0.913946 0.405836i $$-0.866980\pi$$
0.913946 0.405836i $$-0.133020\pi$$
$$524$$ 13.2361 0.578220
$$525$$ 0 0
$$526$$ −15.7639 −0.687340
$$527$$ 1.11146i 0.0484158i
$$528$$ − 5.56231i − 0.242068i
$$529$$ −35.0344 −1.52324
$$530$$ 0 0
$$531$$ −13.9443 −0.605130
$$532$$ − 21.7082i − 0.941170i
$$533$$ 11.6180i 0.503233i
$$534$$ 6.90983 0.299018
$$535$$ 0 0
$$536$$ 20.5279 0.886669
$$537$$ − 15.6525i − 0.675454i
$$538$$ 18.2148i 0.785295i
$$539$$ −9.00000 −0.387657
$$540$$ 0 0
$$541$$ −39.7082 −1.70719 −0.853595 0.520938i $$-0.825582\pi$$
−0.853595 + 0.520938i $$0.825582\pi$$
$$542$$ 9.52786i 0.409257i
$$543$$ − 3.52786i − 0.151395i
$$544$$ 1.32624 0.0568620
$$545$$ 0 0
$$546$$ −1.23607 −0.0528988
$$547$$ − 11.2918i − 0.482802i −0.970425 0.241401i $$-0.922393\pi$$
0.970425 0.241401i $$-0.0776070\pi$$
$$548$$ − 33.2705i − 1.42125i
$$549$$ 4.70820 0.200941
$$550$$ 0 0
$$551$$ −9.27051 −0.394937
$$552$$ − 17.0344i − 0.725034i
$$553$$ 31.7082i 1.34837i
$$554$$ −19.1246 −0.812527
$$555$$ 0 0
$$556$$ 21.7082 0.920633
$$557$$ 6.34752i 0.268953i 0.990917 + 0.134477i $$0.0429353\pi$$
−0.990917 + 0.134477i $$0.957065\pi$$
$$558$$ 2.90983i 0.123183i
$$559$$ −9.61803 −0.406799
$$560$$ 0 0
$$561$$ −0.708204 −0.0299004
$$562$$ 5.05573i 0.213263i
$$563$$ 9.00000i 0.379305i 0.981851 + 0.189652i $$0.0607361\pi$$
−0.981851 + 0.189652i $$0.939264\pi$$
$$564$$ −14.9443 −0.629267
$$565$$ 0 0
$$566$$ −9.70820 −0.408066
$$567$$ 2.00000i 0.0839921i
$$568$$ − 2.43769i − 0.102283i
$$569$$ 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 $$0.485845\pi$$
$$572$$ 4.85410i 0.202960i
$$573$$ 1.67376i 0.0699224i
$$574$$ 14.3607 0.599403
$$575$$ 0 0
$$576$$ −0.236068 −0.00983617
$$577$$ − 37.2705i − 1.55159i −0.630984 0.775796i $$-0.717349\pi$$
0.630984 0.775796i $$-0.282651\pi$$
$$578$$ 10.4721i 0.435583i
$$579$$ 11.0000 0.457144
$$580$$ 0 0
$$581$$ −18.0000 −0.746766
$$582$$ − 1.76393i − 0.0731173i
$$583$$ − 20.2918i − 0.840400i
$$584$$ −5.12461 −0.212058
$$585$$ 0 0
$$586$$ −5.76393 −0.238106
$$587$$ 23.3050i 0.961898i 0.876748 + 0.480949i $$0.159708\pi$$
−0.876748 + 0.480949i $$0.840292\pi$$
$$588$$ 4.85410i 0.200180i
$$589$$ 31.5836 1.30138
$$590$$ 0 0
$$591$$ 11.0902 0.456189
$$592$$ 3.70820i 0.152406i
$$593$$ 15.3820i 0.631662i 0.948816 + 0.315831i $$0.102283\pi$$
−0.948816 + 0.315831i $$0.897717\pi$$
$$594$$ −1.85410 −0.0760747
$$595$$ 0 0
$$596$$ 3.09017 0.126578
$$597$$ 1.70820i 0.0699121i
$$598$$ 4.70820i 0.192533i
$$599$$ −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 $$0.573971\pi$$
$$602$$ 11.8885i 0.484541i
$$603$$ 9.18034i 0.373852i
$$604$$ −7.09017 −0.288495
$$605$$ 0 0
$$606$$ 7.18034 0.291681
$$607$$ 16.1459i 0.655342i 0.944792 + 0.327671i $$0.106264\pi$$
−0.944792 + 0.327671i $$0.893736\pi$$
$$608$$ − 37.6869i − 1.52841i
$$609$$ −2.76393 −0.112000
$$610$$ 0 0
$$611$$ 9.23607 0.373651
$$612$$ 0.381966i 0.0154401i
$$613$$ − 46.1246i − 1.86296i −0.363798 0.931478i $$-0.618520\pi$$
0.363798 0.931478i $$-0.381480\pi$$
$$614$$ 1.32624 0.0535226
$$615$$ 0 0
$$616$$ 13.4164 0.540562
$$617$$ − 20.7639i − 0.835924i −0.908464 0.417962i $$-0.862744\pi$$
0.908464 0.417962i $$-0.137256\pi$$
$$618$$ − 7.67376i − 0.308684i
$$619$$ −0.729490 −0.0293207 −0.0146603 0.999893i $$-0.504667\pi$$
−0.0146603 + 0.999893i $$0.504667\pi$$
$$620$$ 0 0
$$621$$ 7.61803 0.305701
$$622$$ − 13.8885i − 0.556880i
$$623$$ − 22.3607i − 0.895862i
$$624$$ 1.85410 0.0742235
$$625$$ 0 0
$$626$$ −9.70820 −0.388018
$$627$$ 20.1246i 0.803700i
$$628$$ − 6.23607i − 0.248846i
$$629$$ 0.472136 0.0188253
$$630$$ 0 0
$$631$$ −15.2361 −0.606538 −0.303269 0.952905i $$-0.598078\pi$$
−0.303269 + 0.952905i $$0.598078\pi$$
$$632$$ 35.4508i 1.41016i
$$633$$ − 3.00000i − 0.119239i
$$634$$ 0.270510 0.0107433
$$635$$ 0 0
$$636$$ −10.9443 −0.433969
$$637$$ − 3.00000i − 0.118864i
$$638$$ − 2.56231i − 0.101443i
$$639$$ 1.09017 0.0431265
$$640$$ 0 0
$$641$$ −7.67376 −0.303095 −0.151548 0.988450i $$-0.548426\pi$$
−0.151548 + 0.988450i $$0.548426\pi$$
$$642$$ − 4.85410i − 0.191576i
$$643$$ 17.0902i 0.673971i 0.941510 + 0.336985i $$0.109407\pi$$
−0.941510 + 0.336985i $$0.890593\pi$$
$$644$$ −24.6525 −0.971444
$$645$$ 0 0
$$646$$ −0.978714 −0.0385070
$$647$$ − 10.0344i − 0.394495i −0.980354 0.197247i $$-0.936800\pi$$
0.980354 0.197247i $$-0.0632002\pi$$
$$648$$ 2.23607i 0.0878410i
$$649$$ −41.8328 −1.64208
$$650$$ 0 0
$$651$$ 9.41641 0.369058
$$652$$ − 24.7082i − 0.967648i
$$653$$ − 1.65248i − 0.0646664i −0.999477 0.0323332i $$-0.989706\pi$$
0.999477 0.0323332i $$-0.0102938\pi$$
$$654$$ −6.70820 −0.262312
$$655$$ 0 0
$$656$$ −21.5410 −0.841036
$$657$$ − 2.29180i − 0.0894115i
$$658$$ − 11.4164i − 0.445058i
$$659$$ −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 $$0.705116\pi$$
$$662$$ 18.3475i 0.713097i
$$663$$ − 0.236068i − 0.00916812i
$$664$$ −20.1246 −0.780986
$$665$$ 0 0
$$666$$ 1.23607 0.0478967
$$667$$ 10.5279i 0.407641i
$$668$$ 11.0000i 0.425603i
$$669$$ 16.8541 0.651617
$$670$$ 0 0
$$671$$ 14.1246 0.545275
$$672$$ − 11.2361i − 0.433441i
$$673$$ 24.7771i 0.955087i 0.878608 + 0.477543i $$0.158473\pi$$
−0.878608 + 0.477543i $$0.841527\pi$$
$$674$$ −6.68692 −0.257570
$$675$$ 0 0
$$676$$ 19.4164 0.746785
$$677$$ 9.11146i 0.350182i 0.984552 + 0.175091i $$0.0560219\pi$$
−0.984552 + 0.175091i $$0.943978\pi$$
$$678$$ 5.09017i 0.195487i
$$679$$ −5.70820 −0.219061
$$680$$ 0 0
$$681$$ 10.2361 0.392247
$$682$$ 8.72949i 0.334269i
$$683$$ 48.5967i 1.85950i 0.368188 + 0.929751i $$0.379978\pi$$
−0.368188 + 0.929751i $$0.620022\pi$$
$$684$$ 10.8541 0.415017
$$685$$ 0 0
$$686$$ −12.3607 −0.471933
$$687$$ − 6.18034i − 0.235795i
$$688$$ − 17.8328i − 0.679870i
$$689$$ 6.76393 0.257685
$$690$$ 0 0
$$691$$ 3.90983 0.148737 0.0743685 0.997231i $$-0.476306\pi$$
0.0743685 + 0.997231i $$0.476306\pi$$
$$692$$ 19.5623i 0.743647i
$$693$$ 6.00000i 0.227921i
$$694$$ −13.1459 −0.499011
$$695$$ 0 0
$$696$$ −3.09017 −0.117133
$$697$$ 2.74265i 0.103885i
$$698$$ − 1.70820i − 0.0646565i
$$699$$ 12.1803 0.460703
$$700$$ 0 0
$$701$$ −17.3475 −0.655207 −0.327603 0.944815i $$-0.606241\pi$$
−0.327603 + 0.944815i $$0.606241\pi$$
$$702$$ − 0.618034i − 0.0233262i
$$703$$ − 13.4164i − 0.506009i
$$704$$ −0.708204 −0.0266914
$$705$$ 0 0
$$706$$ 9.03444 0.340016
$$707$$ − 23.2361i − 0.873882i
$$708$$ 22.5623i 0.847943i
$$709$$ 29.7984 1.11910 0.559551 0.828796i $$-0.310974\pi$$
0.559551 + 0.828796i $$0.310974\pi$$
$$710$$ 0 0
$$711$$ −15.8541 −0.594575
$$712$$ − 25.0000i − 0.936915i
$$713$$ − 35.8673i − 1.34324i
$$714$$ −0.291796 −0.0109202
$$715$$ 0 0
$$716$$ −25.3262 −0.946486
$$717$$ − 23.6180i − 0.882032i
$$718$$ − 3.74265i − 0.139674i
$$719$$ −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
$$722$$ 16.0689i 0.598022i
$$723$$ − 8.32624i − 0.309656i
$$724$$ −5.70820 −0.212144
$$725$$ 0 0
$$726$$ 1.23607 0.0458748
$$727$$ 34.5623i 1.28184i 0.767606 + 0.640922i $$0.221448\pi$$
−0.767606 + 0.640922i $$0.778552\pi$$
$$728$$ 4.47214i 0.165748i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −2.27051 −0.0839778
$$732$$ − 7.61803i − 0.281571i
$$733$$ − 16.8541i − 0.622520i −0.950325 0.311260i $$-0.899249\pi$$
0.950325 0.311260i $$-0.100751\pi$$
$$734$$ −13.2705 −0.489823
$$735$$ 0 0
$$736$$ −42.7984 −1.57757
$$737$$ 27.5410i 1.01449i
$$738$$ 7.18034i 0.264312i
$$739$$ −11.7082 −0.430693 −0.215347 0.976538i $$-0.569088\pi$$
−0.215347 + 0.976538i $$0.569088\pi$$
$$740$$ 0 0
$$741$$ −6.70820 −0.246432
$$742$$ − 8.36068i − 0.306930i
$$743$$ − 25.4721i − 0.934482i −0.884130 0.467241i $$-0.845248\pi$$
0.884130 0.467241i $$-0.154752\pi$$
$$744$$ 10.5279 0.385970
$$745$$ 0 0
$$746$$ 5.81966 0.213073
$$747$$ − 9.00000i − 0.329293i
$$748$$ 1.14590i 0.0418982i
$$749$$ −15.7082 −0.573965
$$750$$ 0 0
$$751$$ 28.7082 1.04758 0.523789 0.851848i $$-0.324518\pi$$
0.523789 + 0.851848i $$0.324518\pi$$
$$752$$ 17.1246i 0.624470i
$$753$$ 27.9787i 1.01960i
$$754$$ 0.854102 0.0311046
$$755$$ 0 0
$$756$$ 3.23607 0.117695
$$757$$ 1.27051i 0.0461775i 0.999733 + 0.0230887i $$0.00735002\pi$$
−0.999733 + 0.0230887i $$0.992650\pi$$
$$758$$ 7.03444i 0.255502i
$$759$$ 22.8541 0.829551
$$760$$ 0 0
$$761$$ −19.1803 −0.695287 −0.347643 0.937627i $$-0.613018\pi$$
−0.347643 + 0.937627i $$0.613018\pi$$
$$762$$ − 10.9098i − 0.395221i
$$763$$ 21.7082i 0.785890i
$$764$$ 2.70820 0.0979794
$$765$$ 0 0
$$766$$ −14.1803 −0.512357
$$767$$ − 13.9443i − 0.503498i
$$768$$ − 6.56231i − 0.236797i
$$769$$ −26.3050 −0.948581 −0.474290 0.880368i $$-0.657295\pi$$
−0.474290 + 0.880368i $$0.657295\pi$$
$$770$$ 0 0
$$771$$ −20.2148 −0.728018
$$772$$ − 17.7984i − 0.640577i
$$773$$ 29.7771i 1.07101i 0.844533 + 0.535504i $$0.179878\pi$$
−0.844533 + 0.535504i $$0.820122\pi$$
$$774$$ −5.94427 −0.213662
$$775$$ 0 0
$$776$$ −6.38197 −0.229099
$$777$$ − 4.00000i − 0.143499i
$$778$$ 18.9443i 0.679185i
$$779$$ 77.9361 2.79235
$$780$$ 0 0
$$781$$ 3.27051 0.117028
$$782$$ 1.11146i 0.0397456i
$$783$$ − 1.38197i − 0.0493874i
$$784$$ 5.56231 0.198654
$$785$$ 0 0
$$786$$ −5.05573 −0.180332
$$787$$ − 23.8541i − 0.850307i −0.905121 0.425153i $$-0.860220\pi$$
0.905121 0.425153i $$-0.139780\pi$$
$$788$$ − 17.9443i − 0.639238i
$$789$$ −25.5066 −0.908058
$$790$$ 0 0
$$791$$ 16.4721 0.585682
$$792$$ 6.70820i 0.238366i
$$793$$ 4.70820i 0.167193i
$$794$$ −7.09017 −0.251621
$$795$$ 0 0
$$796$$ 2.76393 0.0979650
$$797$$ − 46.0132i − 1.62987i −0.579553 0.814935i $$-0.696773\pi$$
0.579553 0.814935i $$-0.303227\pi$$
$$798$$ 8.29180i 0.293526i
$$799$$ 2.18034 0.0771349
$$800$$ 0 0
$$801$$ 11.1803 0.395038
$$802$$ 1.68692i 0.0595671i
$$803$$ − 6.87539i − 0.242627i
$$804$$ 14.8541 0.523864
$$805$$ 0 0
$$806$$ −2.90983 −0.102494
$$807$$ 29.4721i 1.03747i
$$808$$ − 25.9787i − 0.913928i
$$809$$ −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 $$0.269170\pi$$
$$812$$ 4.47214i 0.156941i
$$813$$ 15.4164i 0.540677i
$$814$$ 3.70820 0.129972
$$815$$ 0 0
$$816$$ 0.437694 0.0153224
$$817$$ 64.5197i 2.25726i
$$818$$ − 21.7082i − 0.759010i
$$819$$ −2.00000 −0.0698857
$$820$$ 0 0
$$821$$ −11.9443 −0.416858 −0.208429 0.978038i $$-0.566835\pi$$
−0.208429 + 0.978038i $$0.566835\pi$$
$$822$$ 12.7082i 0.443250i
$$823$$ − 8.43769i − 0.294120i −0.989128 0.147060i $$-0.953019\pi$$
0.989128 0.147060i $$-0.0469810\pi$$
$$824$$ −27.7639 −0.967202
$$825$$ 0 0
$$826$$ −17.2361 −0.599720
$$827$$ − 2.02129i − 0.0702870i −0.999382 0.0351435i $$-0.988811\pi$$
0.999382 0.0351435i $$-0.0111888\pi$$
$$828$$ − 12.3262i − 0.428366i
$$829$$ 9.87539 0.342986 0.171493 0.985185i $$-0.445141\pi$$
0.171493 + 0.985185i $$0.445141\pi$$
$$830$$ 0 0
$$831$$ −30.9443 −1.07344
$$832$$ − 0.236068i − 0.00818418i
$$833$$ − 0.708204i − 0.0245378i
$$834$$ −8.29180 −0.287121
$$835$$ 0 0
$$836$$ 32.5623 1.12619
$$837$$ 4.70820i 0.162739i
$$838$$ 9.47214i 0.327210i
$$839$$ −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
$$842$$ 8.87539i 0.305866i
$$843$$ 8.18034i 0.281746i
$$844$$ −4.85410 −0.167085
$$845$$ 0 0
$$846$$ 5.70820 0.196252
$$847$$ − 4.00000i − 0.137442i
$$848$$ 12.5410i 0.430660i
$$849$$ −15.7082 −0.539104
$$850$$ 0 0
$$851$$ −15.2361 −0.522286
$$852$$ − 1.76393i − 0.0604313i
$$853$$ 53.3951i 1.82821i 0.405473 + 0.914107i $$0.367107\pi$$
−0.405473 + 0.914107i $$0.632893\pi$$
$$854$$ 5.81966 0.199145
$$855$$ 0 0
$$856$$ −17.5623 −0.600267
$$857$$ − 26.9443i − 0.920399i −0.887816 0.460199i $$-0.847778\pi$$
0.887816 0.460199i $$-0.152222\pi$$
$$858$$ − 1.85410i − 0.0632980i
$$859$$ 25.1246 0.857241 0.428620 0.903485i $$-0.359000\pi$$
0.428620 + 0.903485i $$0.359000\pi$$
$$860$$ 0 0
$$861$$ 23.2361 0.791883
$$862$$ 21.1591i 0.720680i
$$863$$ − 45.0689i − 1.53416i −0.641550 0.767081i $$-0.721708\pi$$
0.641550 0.767081i $$-0.278292\pi$$
$$864$$ 5.61803 0.191129
$$865$$ 0 0
$$866$$ 3.38197 0.114924
$$867$$ 16.9443i 0.575458i
$$868$$ − 15.2361i − 0.517146i
$$869$$ −47.5623 −1.61344
$$870$$ 0 0
$$871$$ −9.18034 −0.311064
$$872$$ 24.2705i 0.821903i
$$873$$ − 2.85410i − 0.0965967i
$$874$$ 31.5836 1.06833
$$875$$ 0 0
$$876$$ −3.70820 −0.125289
$$877$$ − 2.87539i − 0.0970950i −0.998821 0.0485475i $$-0.984541\pi$$
0.998821 0.0485475i $$-0.0154592\pi$$
$$878$$ 1.83282i 0.0618545i
$$879$$ −9.32624 −0.314566
$$880$$ 0 0
$$881$$ 3.90983 0.131726 0.0658628 0.997829i $$-0.479020\pi$$
0.0658628 + 0.997829i $$0.479020\pi$$
$$882$$ − 1.85410i − 0.0624309i
$$883$$ 53.7984i 1.81046i 0.424923 + 0.905230i $$0.360301\pi$$
−0.424923 + 0.905230i $$0.639699\pi$$
$$884$$ −0.381966 −0.0128469
$$885$$ 0 0
$$886$$ −4.58359 −0.153989
$$887$$ 21.9230i 0.736102i 0.929806 + 0.368051i $$0.119975\pi$$
−0.929806 + 0.368051i $$0.880025\pi$$
$$888$$ − 4.47214i − 0.150075i
$$889$$ −35.3050 −1.18409
$$890$$ 0 0
$$891$$ −3.00000 −0.100504
$$892$$ − 27.2705i − 0.913084i
$$893$$ − 61.9574i − 2.07333i
$$894$$ −1.18034 −0.0394765
$$895$$ 0 0
$$896$$ −22.7639 −0.760490
$$897$$ 7.61803i 0.254359i
$$898$$ 13.2918i 0.443553i
$$899$$ −6.50658 −0.217007
$$900$$ 0 0
$$901$$ 1.59675 0.0531954
$$902$$ 21.5410i 0.717238i
$$903$$ 19.2361i 0.640136i
$$904$$ 18.4164 0.612521
$$905$$ 0 0
$$906$$ 2.70820 0.0899740
$$907$$ 17.0000i 0.564476i 0.959344 + 0.282238i $$0.0910767\pi$$
−0.959344 + 0.282238i $$0.908923\pi$$
$$908$$ − 16.5623i − 0.549639i
$$909$$ 11.6180 0.385346
$$910$$ 0 0
$$911$$ −20.8885 −0.692068 −0.346034 0.938222i $$-0.612472\pi$$
−0.346034 + 0.938222i $$0.612472\pi$$
$$912$$ − 12.4377i − 0.411853i
$$913$$ − 27.0000i − 0.893570i
$$914$$ 16.0000 0.529233
$$915$$ 0 0
$$916$$ −10.0000 −0.330409
$$917$$ 16.3607i 0.540277i
$$918$$ − 0.145898i − 0.00481535i
$$919$$ 5.00000 0.164935 0.0824674 0.996594i $$-0.473720\pi$$
0.0824674 + 0.996594i $$0.473720\pi$$
$$920$$ 0 0
$$921$$ 2.14590 0.0707097
$$922$$ 1.96556i 0.0647322i
$$923$$ 1.09017i 0.0358834i
$$924$$ 9.70820 0.319376
$$925$$ 0 0
$$926$$ −16.4934 −0.542007
$$927$$ − 12.4164i − 0.407808i
$$928$$ 7.76393i 0.254864i
$$929$$ −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
$$932$$ − 19.7082i − 0.645564i
$$933$$ − 22.4721i − 0.735705i
$$934$$ 10.1459 0.331984
$$935$$ 0 0
$$936$$ −2.23607 −0.0730882
$$937$$ − 16.4164i − 0.536301i −0.963377 0.268150i $$-0.913588\pi$$
0.963377 0.268150i $$-0.0864124\pi$$
$$938$$ 11.3475i 0.370510i
$$939$$ −15.7082 −0.512618
$$940$$ 0 0
$$941$$ 11.0213 0.359284 0.179642 0.983732i $$-0.442506\pi$$
0.179642 + 0.983732i $$0.442506\pi$$
$$942$$ 2.38197i 0.0776086i
$$943$$ − 88.5066i − 2.88217i
$$944$$ 25.8541 0.841479
$$945$$ 0 0
$$946$$ −17.8328 −0.579795
$$947$$ − 29.8328i − 0.969436i −0.874670 0.484718i $$-0.838922\pi$$
0.874670 0.484718i $$-0.161078\pi$$
$$948$$ 25.6525i 0.833154i
$$949$$ 2.29180 0.0743948
$$950$$ 0 0
$$951$$ 0.437694 0.0141932
$$952$$ 1.05573i 0.0342163i
$$953$$ − 59.9443i − 1.94179i −0.239515 0.970893i $$-0.576988\pi$$
0.239515 0.970893i $$-0.423012\pi$$
$$954$$ 4.18034 0.135344
$$955$$ 0 0
$$956$$ −38.2148 −1.23595
$$957$$ − 4.14590i − 0.134018i
$$958$$ 12.2361i 0.395329i
$$959$$ 41.1246 1.32798
$$960$$ 0 0
$$961$$ −8.83282 −0.284930
$$962$$ 1.23607i 0.0398524i
$$963$$ − 7.85410i − 0.253095i
$$964$$ −13.4721 −0.433908
$$965$$ 0 0
$$966$$ 9.41641 0.302968
$$967$$ 8.58359i 0.276030i 0.990430 + 0.138015i $$0.0440722\pi$$
−0.990430 + 0.138015i $$0.955928\pi$$
$$968$$ − 4.47214i − 0.143740i
$$969$$ −1.58359 −0.0508723
$$970$$ 0 0
$$971$$ −5.88854 −0.188972 −0.0944862 0.995526i $$-0.530121\pi$$
−0.0944862 + 0.995526i $$0.530121\pi$$
$$972$$ 1.61803i 0.0518985i
$$973$$ 26.8328i 0.860221i
$$974$$ 8.88854 0.284807
$$975$$ 0 0
$$976$$ −8.72949 −0.279424
$$977$$ 6.34752i 0.203075i 0.994832 + 0.101538i $$0.0323762\pi$$
−0.994832 + 0.101538i $$0.967624\pi$$
$$978$$ 9.43769i 0.301784i
$$979$$ 33.5410 1.07198
$$980$$ 0 0
$$981$$ −10.8541 −0.346545
$$982$$ 4.12461i 0.131622i
$$983$$ − 22.3820i − 0.713874i −0.934128 0.356937i $$-0.883821\pi$$
0.934128 0.356937i $$-0.116179\pi$$
$$984$$ 25.9787 0.828171
$$985$$ 0 0
$$986$$ 0.201626 0.00642108
$$987$$ − 18.4721i − 0.587975i
$$988$$ 10.8541i 0.345315i
$$989$$ 73.2705 2.32987
$$990$$ 0 0
$$991$$ 32.0000 1.01651 0.508257 0.861206i $$-0.330290\pi$$
0.508257 + 0.861206i $$0.330290\pi$$
$$992$$ − 26.4508i − 0.839815i
$$993$$ 29.6869i 0.942086i
$$994$$ 1.34752 0.0427409
$$995$$ 0 0
$$996$$ −14.5623 −0.461424
$$997$$ 61.0689i 1.93407i 0.254642 + 0.967035i $$0.418042\pi$$
−0.254642 + 0.967035i $$0.581958\pi$$
$$998$$ 9.27051i 0.293453i
$$999$$ 2.00000 0.0632772
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 1875.2.b.a.1249.3 4
5.2 odd 4 1875.2.a.c.1.1 yes 2
5.3 odd 4 1875.2.a.b.1.2 2
5.4 even 2 inner 1875.2.b.a.1249.2 4
15.2 even 4 5625.2.a.b.1.2 2
15.8 even 4 5625.2.a.g.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1875.2.a.b.1.2 2 5.3 odd 4
1875.2.a.c.1.1 yes 2 5.2 odd 4
1875.2.b.a.1249.2 4 5.4 even 2 inner
1875.2.b.a.1249.3 4 1.1 even 1 trivial
5625.2.a.b.1.2 2 15.2 even 4
5625.2.a.g.1.1 2 15.8 even 4