# Properties

 Label 1800.4.a.m.1.1 Level $1800$ Weight $4$ Character 1800.1 Self dual yes Analytic conductor $106.203$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1800,4,Mod(1,1800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1800, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1800.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1800 = 2^{3} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1800.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: $$106.203438010$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 600) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1800.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-5.00000 q^{7} +O(q^{10})$$ $$q-5.00000 q^{7} -14.0000 q^{11} -1.00000 q^{13} +46.0000 q^{17} +19.0000 q^{19} -46.0000 q^{23} -14.0000 q^{29} +133.000 q^{31} -258.000 q^{37} -84.0000 q^{41} +167.000 q^{43} +410.000 q^{47} -318.000 q^{49} +456.000 q^{53} +194.000 q^{59} -17.0000 q^{61} -653.000 q^{67} -828.000 q^{71} -570.000 q^{73} +70.0000 q^{77} -552.000 q^{79} +142.000 q^{83} +1104.00 q^{89} +5.00000 q^{91} -841.000 q^{97} +O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −5.00000 −0.269975 −0.134987 0.990847i $$-0.543099\pi$$
−0.134987 + 0.990847i $$0.543099\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −14.0000 −0.383742 −0.191871 0.981420i $$-0.561455\pi$$
−0.191871 + 0.981420i $$0.561455\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.0213346 −0.0106673 0.999943i $$-0.503396\pi$$
−0.0106673 + 0.999943i $$0.503396\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 46.0000 0.656273 0.328136 0.944630i $$-0.393579\pi$$
0.328136 + 0.944630i $$0.393579\pi$$
$$18$$ 0 0
$$19$$ 19.0000 0.229416 0.114708 0.993399i $$-0.463407\pi$$
0.114708 + 0.993399i $$0.463407\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −46.0000 −0.417029 −0.208514 0.978019i $$-0.566863\pi$$
−0.208514 + 0.978019i $$0.566863\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −14.0000 −0.0896460 −0.0448230 0.998995i $$-0.514272\pi$$
−0.0448230 + 0.998995i $$0.514272\pi$$
$$30$$ 0 0
$$31$$ 133.000 0.770565 0.385282 0.922799i $$-0.374104\pi$$
0.385282 + 0.922799i $$0.374104\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −258.000 −1.14635 −0.573175 0.819433i $$-0.694288\pi$$
−0.573175 + 0.819433i $$0.694288\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −84.0000 −0.319966 −0.159983 0.987120i $$-0.551144\pi$$
−0.159983 + 0.987120i $$0.551144\pi$$
$$42$$ 0 0
$$43$$ 167.000 0.592262 0.296131 0.955147i $$-0.404304\pi$$
0.296131 + 0.955147i $$0.404304\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 410.000 1.27244 0.636220 0.771508i $$-0.280497\pi$$
0.636220 + 0.771508i $$0.280497\pi$$
$$48$$ 0 0
$$49$$ −318.000 −0.927114
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 456.000 1.18182 0.590910 0.806738i $$-0.298769\pi$$
0.590910 + 0.806738i $$0.298769\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 194.000 0.428079 0.214039 0.976825i $$-0.431338\pi$$
0.214039 + 0.976825i $$0.431338\pi$$
$$60$$ 0 0
$$61$$ −17.0000 −0.0356824 −0.0178412 0.999841i $$-0.505679\pi$$
−0.0178412 + 0.999841i $$0.505679\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −653.000 −1.19070 −0.595348 0.803468i $$-0.702986\pi$$
−0.595348 + 0.803468i $$0.702986\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −828.000 −1.38402 −0.692011 0.721887i $$-0.743275\pi$$
−0.692011 + 0.721887i $$0.743275\pi$$
$$72$$ 0 0
$$73$$ −570.000 −0.913883 −0.456941 0.889497i $$-0.651055\pi$$
−0.456941 + 0.889497i $$0.651055\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 70.0000 0.103601
$$78$$ 0 0
$$79$$ −552.000 −0.786137 −0.393069 0.919509i $$-0.628587\pi$$
−0.393069 + 0.919509i $$0.628587\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 142.000 0.187789 0.0938947 0.995582i $$-0.470068\pi$$
0.0938947 + 0.995582i $$0.470068\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 1104.00 1.31487 0.657437 0.753510i $$-0.271641\pi$$
0.657437 + 0.753510i $$0.271641\pi$$
$$90$$ 0 0
$$91$$ 5.00000 0.00575981
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −841.000 −0.880316 −0.440158 0.897920i $$-0.645077\pi$$
−0.440158 + 0.897920i $$0.645077\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −552.000 −0.543822 −0.271911 0.962322i $$-0.587656\pi$$
−0.271911 + 0.962322i $$0.587656\pi$$
$$102$$ 0 0
$$103$$ 308.000 0.294642 0.147321 0.989089i $$-0.452935\pi$$
0.147321 + 0.989089i $$0.452935\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 984.000 0.889036 0.444518 0.895770i $$-0.353375\pi$$
0.444518 + 0.895770i $$0.353375\pi$$
$$108$$ 0 0
$$109$$ −1843.00 −1.61952 −0.809759 0.586763i $$-0.800402\pi$$
−0.809759 + 0.586763i $$0.800402\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 876.000 0.729267 0.364633 0.931151i $$-0.381194\pi$$
0.364633 + 0.931151i $$0.381194\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −230.000 −0.177177
$$120$$ 0 0
$$121$$ −1135.00 −0.852742
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −2376.00 −1.66013 −0.830063 0.557670i $$-0.811695\pi$$
−0.830063 + 0.557670i $$0.811695\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −1056.00 −0.704299 −0.352149 0.935944i $$-0.614549\pi$$
−0.352149 + 0.935944i $$0.614549\pi$$
$$132$$ 0 0
$$133$$ −95.0000 −0.0619364
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −778.000 −0.485175 −0.242588 0.970129i $$-0.577996\pi$$
−0.242588 + 0.970129i $$0.577996\pi$$
$$138$$ 0 0
$$139$$ 1692.00 1.03247 0.516236 0.856446i $$-0.327333\pi$$
0.516236 + 0.856446i $$0.327333\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 14.0000 0.00818698
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 494.000 0.271611 0.135806 0.990736i $$-0.456638\pi$$
0.135806 + 0.990736i $$0.456638\pi$$
$$150$$ 0 0
$$151$$ −841.000 −0.453242 −0.226621 0.973983i $$-0.572768\pi$$
−0.226621 + 0.973983i $$0.572768\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −19.0000 −0.00965838 −0.00482919 0.999988i $$-0.501537\pi$$
−0.00482919 + 0.999988i $$0.501537\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 230.000 0.112587
$$162$$ 0 0
$$163$$ 2261.00 1.08647 0.543237 0.839580i $$-0.317199\pi$$
0.543237 + 0.839580i $$0.317199\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 2112.00 0.978632 0.489316 0.872107i $$-0.337247\pi$$
0.489316 + 0.872107i $$0.337247\pi$$
$$168$$ 0 0
$$169$$ −2196.00 −0.999545
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 562.000 0.246983 0.123492 0.992346i $$-0.460591\pi$$
0.123492 + 0.992346i $$0.460591\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −3718.00 −1.55249 −0.776247 0.630429i $$-0.782879\pi$$
−0.776247 + 0.630429i $$0.782879\pi$$
$$180$$ 0 0
$$181$$ −1639.00 −0.673071 −0.336536 0.941671i $$-0.609255\pi$$
−0.336536 + 0.941671i $$0.609255\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −644.000 −0.251839
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −2410.00 −0.912992 −0.456496 0.889725i $$-0.650896\pi$$
−0.456496 + 0.889725i $$0.650896\pi$$
$$192$$ 0 0
$$193$$ 2621.00 0.977532 0.488766 0.872415i $$-0.337447\pi$$
0.488766 + 0.872415i $$0.337447\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −4954.00 −1.79166 −0.895832 0.444392i $$-0.853420\pi$$
−0.895832 + 0.444392i $$0.853420\pi$$
$$198$$ 0 0
$$199$$ 1739.00 0.619470 0.309735 0.950823i $$-0.399760\pi$$
0.309735 + 0.950823i $$0.399760\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 70.0000 0.0242022
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −266.000 −0.0880364
$$210$$ 0 0
$$211$$ −4525.00 −1.47637 −0.738184 0.674599i $$-0.764317\pi$$
−0.738184 + 0.674599i $$0.764317\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −665.000 −0.208033
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −46.0000 −0.0140013
$$222$$ 0 0
$$223$$ 3211.00 0.964235 0.482118 0.876106i $$-0.339868\pi$$
0.482118 + 0.876106i $$0.339868\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −2484.00 −0.726295 −0.363147 0.931732i $$-0.618298\pi$$
−0.363147 + 0.931732i $$0.618298\pi$$
$$228$$ 0 0
$$229$$ −1847.00 −0.532983 −0.266492 0.963837i $$-0.585864\pi$$
−0.266492 + 0.963837i $$0.585864\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −1020.00 −0.286792 −0.143396 0.989665i $$-0.545802\pi$$
−0.143396 + 0.989665i $$0.545802\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −1176.00 −0.318281 −0.159140 0.987256i $$-0.550872\pi$$
−0.159140 + 0.987256i $$0.550872\pi$$
$$240$$ 0 0
$$241$$ −6967.00 −1.86217 −0.931087 0.364797i $$-0.881138\pi$$
−0.931087 + 0.364797i $$0.881138\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −19.0000 −0.00489450
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −1380.00 −0.347031 −0.173516 0.984831i $$-0.555513\pi$$
−0.173516 + 0.984831i $$0.555513\pi$$
$$252$$ 0 0
$$253$$ 644.000 0.160031
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −6924.00 −1.68057 −0.840286 0.542143i $$-0.817613\pi$$
−0.840286 + 0.542143i $$0.817613\pi$$
$$258$$ 0 0
$$259$$ 1290.00 0.309485
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 1884.00 0.441720 0.220860 0.975305i $$-0.429114\pi$$
0.220860 + 0.975305i $$0.429114\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −3610.00 −0.818236 −0.409118 0.912481i $$-0.634164\pi$$
−0.409118 + 0.912481i $$0.634164\pi$$
$$270$$ 0 0
$$271$$ −6072.00 −1.36106 −0.680531 0.732719i $$-0.738251\pi$$
−0.680531 + 0.732719i $$0.738251\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 2803.00 0.608000 0.304000 0.952672i $$-0.401678\pi$$
0.304000 + 0.952672i $$0.401678\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 6694.00 1.42111 0.710553 0.703644i $$-0.248445\pi$$
0.710553 + 0.703644i $$0.248445\pi$$
$$282$$ 0 0
$$283$$ 6481.00 1.36133 0.680663 0.732596i $$-0.261692\pi$$
0.680663 + 0.732596i $$0.261692\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 420.000 0.0863826
$$288$$ 0 0
$$289$$ −2797.00 −0.569306
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 3014.00 0.600955 0.300477 0.953789i $$-0.402854\pi$$
0.300477 + 0.953789i $$0.402854\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 46.0000 0.00889715
$$300$$ 0 0
$$301$$ −835.000 −0.159896
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 5369.00 0.998127 0.499064 0.866565i $$-0.333677\pi$$
0.499064 + 0.866565i $$0.333677\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −4846.00 −0.883574 −0.441787 0.897120i $$-0.645655\pi$$
−0.441787 + 0.897120i $$0.645655\pi$$
$$312$$ 0 0
$$313$$ 757.000 0.136703 0.0683517 0.997661i $$-0.478226\pi$$
0.0683517 + 0.997661i $$0.478226\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −7632.00 −1.35223 −0.676113 0.736798i $$-0.736337\pi$$
−0.676113 + 0.736798i $$0.736337\pi$$
$$318$$ 0 0
$$319$$ 196.000 0.0344009
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 874.000 0.150559
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −2050.00 −0.343526
$$330$$ 0 0
$$331$$ −6780.00 −1.12587 −0.562934 0.826502i $$-0.690328\pi$$
−0.562934 + 0.826502i $$0.690328\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −7849.00 −1.26873 −0.634365 0.773033i $$-0.718739\pi$$
−0.634365 + 0.773033i $$0.718739\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −1862.00 −0.295698
$$342$$ 0 0
$$343$$ 3305.00 0.520272
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 634.000 0.0980833 0.0490416 0.998797i $$-0.484383\pi$$
0.0490416 + 0.998797i $$0.484383\pi$$
$$348$$ 0 0
$$349$$ 930.000 0.142641 0.0713206 0.997453i $$-0.477279\pi$$
0.0713206 + 0.997453i $$0.477279\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −4286.00 −0.646234 −0.323117 0.946359i $$-0.604731\pi$$
−0.323117 + 0.946359i $$0.604731\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 4236.00 0.622751 0.311375 0.950287i $$-0.399210\pi$$
0.311375 + 0.950287i $$0.399210\pi$$
$$360$$ 0 0
$$361$$ −6498.00 −0.947368
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −1451.00 −0.206380 −0.103190 0.994662i $$-0.532905\pi$$
−0.103190 + 0.994662i $$0.532905\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −2280.00 −0.319061
$$372$$ 0 0
$$373$$ −3115.00 −0.432409 −0.216205 0.976348i $$-0.569368\pi$$
−0.216205 + 0.976348i $$0.569368\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 14.0000 0.00191256
$$378$$ 0 0
$$379$$ −1415.00 −0.191777 −0.0958887 0.995392i $$-0.530569\pi$$
−0.0958887 + 0.995392i $$0.530569\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −180.000 −0.0240145 −0.0120073 0.999928i $$-0.503822\pi$$
−0.0120073 + 0.999928i $$0.503822\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −12372.0 −1.61256 −0.806279 0.591535i $$-0.798522\pi$$
−0.806279 + 0.591535i $$0.798522\pi$$
$$390$$ 0 0
$$391$$ −2116.00 −0.273685
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −5767.00 −0.729062 −0.364531 0.931191i $$-0.618771\pi$$
−0.364531 + 0.931191i $$0.618771\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 3120.00 0.388542 0.194271 0.980948i $$-0.437766\pi$$
0.194271 + 0.980948i $$0.437766\pi$$
$$402$$ 0 0
$$403$$ −133.000 −0.0164397
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 3612.00 0.439902
$$408$$ 0 0
$$409$$ 1501.00 0.181466 0.0907331 0.995875i $$-0.471079\pi$$
0.0907331 + 0.995875i $$0.471079\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −970.000 −0.115570
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 9072.00 1.05775 0.528874 0.848701i $$-0.322614\pi$$
0.528874 + 0.848701i $$0.322614\pi$$
$$420$$ 0 0
$$421$$ 7350.00 0.850872 0.425436 0.904989i $$-0.360121\pi$$
0.425436 + 0.904989i $$0.360121\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 85.0000 0.00963334
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −5962.00 −0.666310 −0.333155 0.942872i $$-0.608113\pi$$
−0.333155 + 0.942872i $$0.608113\pi$$
$$432$$ 0 0
$$433$$ 10093.0 1.12018 0.560091 0.828431i $$-0.310766\pi$$
0.560091 + 0.828431i $$0.310766\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −874.000 −0.0956730
$$438$$ 0 0
$$439$$ −2555.00 −0.277776 −0.138888 0.990308i $$-0.544353\pi$$
−0.138888 + 0.990308i $$0.544353\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 6240.00 0.669236 0.334618 0.942354i $$-0.391393\pi$$
0.334618 + 0.942354i $$0.391393\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −3324.00 −0.349375 −0.174687 0.984624i $$-0.555891\pi$$
−0.174687 + 0.984624i $$0.555891\pi$$
$$450$$ 0 0
$$451$$ 1176.00 0.122784
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −16774.0 −1.71697 −0.858484 0.512840i $$-0.828593\pi$$
−0.858484 + 0.512840i $$0.828593\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −14304.0 −1.44513 −0.722564 0.691304i $$-0.757036\pi$$
−0.722564 + 0.691304i $$0.757036\pi$$
$$462$$ 0 0
$$463$$ −6936.00 −0.696206 −0.348103 0.937456i $$-0.613174\pi$$
−0.348103 + 0.937456i $$0.613174\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 15622.0 1.54797 0.773983 0.633207i $$-0.218262\pi$$
0.773983 + 0.633207i $$0.218262\pi$$
$$468$$ 0 0
$$469$$ 3265.00 0.321458
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −2338.00 −0.227276
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 13354.0 1.27382 0.636910 0.770938i $$-0.280212\pi$$
0.636910 + 0.770938i $$0.280212\pi$$
$$480$$ 0 0
$$481$$ 258.000 0.0244569
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 461.000 0.0428951 0.0214475 0.999770i $$-0.493173\pi$$
0.0214475 + 0.999770i $$0.493173\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −3768.00 −0.346329 −0.173164 0.984893i $$-0.555399\pi$$
−0.173164 + 0.984893i $$0.555399\pi$$
$$492$$ 0 0
$$493$$ −644.000 −0.0588323
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 4140.00 0.373651
$$498$$ 0 0
$$499$$ −14317.0 −1.28440 −0.642201 0.766536i $$-0.721979\pi$$
−0.642201 + 0.766536i $$0.721979\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 9228.00 0.818004 0.409002 0.912533i $$-0.365877\pi$$
0.409002 + 0.912533i $$0.365877\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −4574.00 −0.398308 −0.199154 0.979968i $$-0.563819\pi$$
−0.199154 + 0.979968i $$0.563819\pi$$
$$510$$ 0 0
$$511$$ 2850.00 0.246725
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −5740.00 −0.488288
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 8494.00 0.714259 0.357129 0.934055i $$-0.383755\pi$$
0.357129 + 0.934055i $$0.383755\pi$$
$$522$$ 0 0
$$523$$ −8263.00 −0.690852 −0.345426 0.938446i $$-0.612266\pi$$
−0.345426 + 0.938446i $$0.612266\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 6118.00 0.505701
$$528$$ 0 0
$$529$$ −10051.0 −0.826087
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 84.0000 0.00682635
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 4452.00 0.355772
$$540$$ 0 0
$$541$$ 21157.0 1.68135 0.840675 0.541540i $$-0.182158\pi$$
0.840675 + 0.541540i $$0.182158\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 4048.00 0.316417 0.158208 0.987406i $$-0.449428\pi$$
0.158208 + 0.987406i $$0.449428\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −266.000 −0.0205662
$$552$$ 0 0
$$553$$ 2760.00 0.212237
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −6758.00 −0.514086 −0.257043 0.966400i $$-0.582748\pi$$
−0.257043 + 0.966400i $$0.582748\pi$$
$$558$$ 0 0
$$559$$ −167.000 −0.0126357
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 24506.0 1.83447 0.917233 0.398350i $$-0.130417\pi$$
0.917233 + 0.398350i $$0.130417\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 1430.00 0.105358 0.0526790 0.998611i $$-0.483224\pi$$
0.0526790 + 0.998611i $$0.483224\pi$$
$$570$$ 0 0
$$571$$ 3691.00 0.270514 0.135257 0.990811i $$-0.456814\pi$$
0.135257 + 0.990811i $$0.456814\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −4571.00 −0.329798 −0.164899 0.986310i $$-0.552730\pi$$
−0.164899 + 0.986310i $$0.552730\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −710.000 −0.0506984
$$582$$ 0 0
$$583$$ −6384.00 −0.453513
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 14808.0 1.04121 0.520606 0.853797i $$-0.325706\pi$$
0.520606 + 0.853797i $$0.325706\pi$$
$$588$$ 0 0
$$589$$ 2527.00 0.176780
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −24588.0 −1.70271 −0.851356 0.524588i $$-0.824219\pi$$
−0.851356 + 0.524588i $$0.824219\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 27564.0 1.88019 0.940096 0.340911i $$-0.110735\pi$$
0.940096 + 0.340911i $$0.110735\pi$$
$$600$$ 0 0
$$601$$ 10987.0 0.745706 0.372853 0.927891i $$-0.378380\pi$$
0.372853 + 0.927891i $$0.378380\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 13200.0 0.882655 0.441327 0.897346i $$-0.354508\pi$$
0.441327 + 0.897346i $$0.354508\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −410.000 −0.0271470
$$612$$ 0 0
$$613$$ −21066.0 −1.38801 −0.694003 0.719972i $$-0.744155\pi$$
−0.694003 + 0.719972i $$0.744155\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 12336.0 0.804909 0.402454 0.915440i $$-0.368157\pi$$
0.402454 + 0.915440i $$0.368157\pi$$
$$618$$ 0 0
$$619$$ −1441.00 −0.0935681 −0.0467841 0.998905i $$-0.514897\pi$$
−0.0467841 + 0.998905i $$0.514897\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −5520.00 −0.354983
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −11868.0 −0.752318
$$630$$ 0 0
$$631$$ −9839.00 −0.620736 −0.310368 0.950616i $$-0.600452\pi$$
−0.310368 + 0.950616i $$0.600452\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 318.000 0.0197796
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 21564.0 1.32875 0.664373 0.747401i $$-0.268698\pi$$
0.664373 + 0.747401i $$0.268698\pi$$
$$642$$ 0 0
$$643$$ −8604.00 −0.527696 −0.263848 0.964564i $$-0.584992\pi$$
−0.263848 + 0.964564i $$0.584992\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 3444.00 0.209270 0.104635 0.994511i $$-0.466633\pi$$
0.104635 + 0.994511i $$0.466633\pi$$
$$648$$ 0 0
$$649$$ −2716.00 −0.164272
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −3518.00 −0.210827 −0.105413 0.994428i $$-0.533617\pi$$
−0.105413 + 0.994428i $$0.533617\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 12612.0 0.745514 0.372757 0.927929i $$-0.378412\pi$$
0.372757 + 0.927929i $$0.378412\pi$$
$$660$$ 0 0
$$661$$ 27090.0 1.59407 0.797034 0.603935i $$-0.206401\pi$$
0.797034 + 0.603935i $$0.206401\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 644.000 0.0373850
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 238.000 0.0136928
$$672$$ 0 0
$$673$$ 5442.00 0.311699 0.155850 0.987781i $$-0.450188\pi$$
0.155850 + 0.987781i $$0.450188\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −15226.0 −0.864376 −0.432188 0.901783i $$-0.642258\pi$$
−0.432188 + 0.901783i $$0.642258\pi$$
$$678$$ 0 0
$$679$$ 4205.00 0.237663
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −552.000 −0.0309249 −0.0154624 0.999880i $$-0.504922\pi$$
−0.0154624 + 0.999880i $$0.504922\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −456.000 −0.0252137
$$690$$ 0 0
$$691$$ 9776.00 0.538201 0.269100 0.963112i $$-0.413274\pi$$
0.269100 + 0.963112i $$0.413274\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −3864.00 −0.209985
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −13066.0 −0.703989 −0.351994 0.936002i $$-0.614496\pi$$
−0.351994 + 0.936002i $$0.614496\pi$$
$$702$$ 0 0
$$703$$ −4902.00 −0.262991
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 2760.00 0.146818
$$708$$ 0 0
$$709$$ 28985.0 1.53534 0.767669 0.640847i $$-0.221417\pi$$
0.767669 + 0.640847i $$0.221417\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −6118.00 −0.321348
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −15722.0 −0.815482 −0.407741 0.913098i $$-0.633683\pi$$
−0.407741 + 0.913098i $$0.633683\pi$$
$$720$$ 0 0
$$721$$ −1540.00 −0.0795459
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 32611.0 1.66365 0.831826 0.555036i $$-0.187296\pi$$
0.831826 + 0.555036i $$0.187296\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 7682.00 0.388685
$$732$$ 0 0
$$733$$ −8358.00 −0.421159 −0.210580 0.977577i $$-0.567535\pi$$
−0.210580 + 0.977577i $$0.567535\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 9142.00 0.456920
$$738$$ 0 0
$$739$$ 20604.0 1.02562 0.512808 0.858503i $$-0.328605\pi$$
0.512808 + 0.858503i $$0.328605\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −19476.0 −0.961649 −0.480824 0.876817i $$-0.659663\pi$$
−0.480824 + 0.876817i $$0.659663\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −4920.00 −0.240017
$$750$$ 0 0
$$751$$ 3864.00 0.187749 0.0938744 0.995584i $$-0.470075\pi$$
0.0938744 + 0.995584i $$0.470075\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 18871.0 0.906048 0.453024 0.891498i $$-0.350345\pi$$
0.453024 + 0.891498i $$0.350345\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 36372.0 1.73257 0.866284 0.499552i $$-0.166502\pi$$
0.866284 + 0.499552i $$0.166502\pi$$
$$762$$ 0 0
$$763$$ 9215.00 0.437229
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −194.000 −0.00913290
$$768$$ 0 0
$$769$$ 4603.00 0.215850 0.107925 0.994159i $$-0.465579\pi$$
0.107925 + 0.994159i $$0.465579\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −36.0000 −0.00167507 −0.000837536 1.00000i $$-0.500267\pi$$
−0.000837536 1.00000i $$0.500267\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −1596.00 −0.0734052
$$780$$ 0 0
$$781$$ 11592.0 0.531107
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 20281.0 0.918602 0.459301 0.888281i $$-0.348100\pi$$
0.459301 + 0.888281i $$0.348100\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −4380.00 −0.196884
$$792$$ 0 0
$$793$$ 17.0000 0.000761271 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −37524.0 −1.66771 −0.833857 0.551980i $$-0.813872\pi$$
−0.833857 + 0.551980i $$0.813872\pi$$
$$798$$ 0 0
$$799$$ 18860.0 0.835067
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 7980.00 0.350695
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −31224.0 −1.35696 −0.678478 0.734621i $$-0.737360\pi$$
−0.678478 + 0.734621i $$0.737360\pi$$
$$810$$ 0 0
$$811$$ 32579.0 1.41061 0.705304 0.708905i $$-0.250810\pi$$
0.705304 + 0.708905i $$0.250810\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 3173.00 0.135874
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 19810.0 0.842112 0.421056 0.907035i $$-0.361660\pi$$
0.421056 + 0.907035i $$0.361660\pi$$
$$822$$ 0 0
$$823$$ −10273.0 −0.435108 −0.217554 0.976048i $$-0.569808\pi$$
−0.217554 + 0.976048i $$0.569808\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 16656.0 0.700346 0.350173 0.936685i $$-0.386123\pi$$
0.350173 + 0.936685i $$0.386123\pi$$
$$828$$ 0 0
$$829$$ −4790.00 −0.200680 −0.100340 0.994953i $$-0.531993\pi$$
−0.100340 + 0.994953i $$0.531993\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −14628.0 −0.608440
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −7414.00 −0.305077 −0.152539 0.988298i $$-0.548745\pi$$
−0.152539 + 0.988298i $$0.548745\pi$$
$$840$$ 0 0
$$841$$ −24193.0 −0.991964
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 5675.00 0.230219
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 11868.0 0.478061
$$852$$ 0 0
$$853$$ −30155.0 −1.21042 −0.605210 0.796066i $$-0.706911\pi$$
−0.605210 + 0.796066i $$0.706911\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −8244.00 −0.328599 −0.164300 0.986410i $$-0.552536\pi$$
−0.164300 + 0.986410i $$0.552536\pi$$
$$858$$ 0 0
$$859$$ 17552.0 0.697167 0.348584 0.937278i $$-0.386663\pi$$
0.348584 + 0.937278i $$0.386663\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −34104.0 −1.34521 −0.672604 0.740003i $$-0.734824\pi$$
−0.672604 + 0.740003i $$0.734824\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 7728.00 0.301674
$$870$$ 0 0
$$871$$ 653.000 0.0254031
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 46229.0 1.77998 0.889990 0.455980i $$-0.150711\pi$$
0.889990 + 0.455980i $$0.150711\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 22440.0 0.858142 0.429071 0.903271i $$-0.358841\pi$$
0.429071 + 0.903271i $$0.358841\pi$$
$$882$$ 0 0
$$883$$ 17143.0 0.653350 0.326675 0.945137i $$-0.394072\pi$$
0.326675 + 0.945137i $$0.394072\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −23626.0 −0.894344 −0.447172 0.894448i $$-0.647569\pi$$
−0.447172 + 0.894448i $$0.647569\pi$$
$$888$$ 0 0
$$889$$ 11880.0 0.448192
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 7790.00 0.291918
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −1862.00 −0.0690781
$$900$$ 0 0
$$901$$ 20976.0 0.775596
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 11268.0 0.412511 0.206256 0.978498i $$-0.433872\pi$$
0.206256 + 0.978498i $$0.433872\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −10046.0 −0.365355 −0.182678 0.983173i $$-0.558476\pi$$
−0.182678 + 0.983173i $$0.558476\pi$$
$$912$$ 0 0
$$913$$ −1988.00 −0.0720626
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 5280.00 0.190143
$$918$$ 0 0
$$919$$ 15359.0 0.551302 0.275651 0.961258i $$-0.411107\pi$$
0.275651 + 0.961258i $$0.411107\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 828.000 0.0295276
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 39790.0 1.40524 0.702620 0.711565i $$-0.252014\pi$$
0.702620 + 0.711565i $$0.252014\pi$$
$$930$$ 0 0
$$931$$ −6042.00 −0.212694
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 17009.0 0.593020 0.296510 0.955030i $$-0.404177\pi$$
0.296510 + 0.955030i $$0.404177\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 11674.0 0.404422 0.202211 0.979342i $$-0.435187\pi$$
0.202211 + 0.979342i $$0.435187\pi$$
$$942$$ 0 0
$$943$$ 3864.00 0.133435
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 6026.00 0.206778 0.103389 0.994641i $$-0.467031\pi$$
0.103389 + 0.994641i $$0.467031\pi$$
$$948$$ 0 0
$$949$$ 570.000 0.0194973
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −14088.0 −0.478862 −0.239431 0.970913i $$-0.576961\pi$$
−0.239431 + 0.970913i $$0.576961\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 3890.00 0.130985
$$960$$ 0 0
$$961$$ −12102.0 −0.406230
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 11208.0 0.372725 0.186362 0.982481i $$-0.440330\pi$$
0.186362 + 0.982481i $$0.440330\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 26054.0 0.861084 0.430542 0.902571i $$-0.358322\pi$$
0.430542 + 0.902571i $$0.358322\pi$$
$$972$$ 0 0
$$973$$ −8460.00 −0.278741
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −26870.0 −0.879885 −0.439942 0.898026i $$-0.645001\pi$$
−0.439942 + 0.898026i $$0.645001\pi$$
$$978$$ 0 0
$$979$$ −15456.0 −0.504572
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −23388.0 −0.758862 −0.379431 0.925220i $$-0.623880\pi$$
−0.379431 + 0.925220i $$0.623880\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −7682.00 −0.246990
$$990$$ 0 0
$$991$$ 17345.0 0.555986 0.277993 0.960583i $$-0.410331\pi$$
0.277993 + 0.960583i $$0.410331\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 25998.0 0.825842 0.412921 0.910767i $$-0.364508\pi$$
0.412921 + 0.910767i $$0.364508\pi$$
$$998$$ 0 0
$$999$$ 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 1800.4.a.m.1.1 1
3.2 odd 2 600.4.a.e.1.1 1
5.2 odd 4 1800.4.f.l.649.1 2
5.3 odd 4 1800.4.f.l.649.2 2
5.4 even 2 1800.4.a.v.1.1 1
12.11 even 2 1200.4.a.bd.1.1 1
15.2 even 4 600.4.f.e.49.2 2
15.8 even 4 600.4.f.e.49.1 2
15.14 odd 2 600.4.a.n.1.1 yes 1
60.23 odd 4 1200.4.f.i.49.2 2
60.47 odd 4 1200.4.f.i.49.1 2
60.59 even 2 1200.4.a.g.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
600.4.a.e.1.1 1 3.2 odd 2
600.4.a.n.1.1 yes 1 15.14 odd 2
600.4.f.e.49.1 2 15.8 even 4
600.4.f.e.49.2 2 15.2 even 4
1200.4.a.g.1.1 1 60.59 even 2
1200.4.a.bd.1.1 1 12.11 even 2
1200.4.f.i.49.1 2 60.47 odd 4
1200.4.f.i.49.2 2 60.23 odd 4
1800.4.a.m.1.1 1 1.1 even 1 trivial
1800.4.a.v.1.1 1 5.4 even 2
1800.4.f.l.649.1 2 5.2 odd 4
1800.4.f.l.649.2 2 5.3 odd 4