# Properties

 Label 1800.4.a.s.1.1 Level $1800$ Weight $4$ Character 1800.1 Self dual yes Analytic conductor $106.203$ Analytic rank $0$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 120) 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)

 $$f(q)$$ $$=$$ $$q+O(q^{10})$$ $$q-4.00000 q^{11} -54.0000 q^{13} +114.000 q^{17} +44.0000 q^{19} +96.0000 q^{23} -134.000 q^{29} -272.000 q^{31} +98.0000 q^{37} +6.00000 q^{41} -12.0000 q^{43} -200.000 q^{47} -343.000 q^{49} +654.000 q^{53} -36.0000 q^{59} -442.000 q^{61} +188.000 q^{67} +632.000 q^{71} +390.000 q^{73} +688.000 q^{79} +1188.00 q^{83} +694.000 q^{89} +1726.00 q^{97} +O(q^{100})$$

## 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
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −4.00000 −0.109640 −0.0548202 0.998496i $$-0.517459\pi$$
−0.0548202 + 0.998496i $$0.517459\pi$$
$$12$$ 0 0
$$13$$ −54.0000 −1.15207 −0.576035 0.817425i $$-0.695401\pi$$
−0.576035 + 0.817425i $$0.695401\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 114.000 1.62642 0.813208 0.581974i $$-0.197719\pi$$
0.813208 + 0.581974i $$0.197719\pi$$
$$18$$ 0 0
$$19$$ 44.0000 0.531279 0.265639 0.964072i $$-0.414417\pi$$
0.265639 + 0.964072i $$0.414417\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 96.0000 0.870321 0.435161 0.900353i $$-0.356692\pi$$
0.435161 + 0.900353i $$0.356692\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −134.000 −0.858041 −0.429020 0.903295i $$-0.641141\pi$$
−0.429020 + 0.903295i $$0.641141\pi$$
$$30$$ 0 0
$$31$$ −272.000 −1.57589 −0.787946 0.615745i $$-0.788855\pi$$
−0.787946 + 0.615745i $$0.788855\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 98.0000 0.435435 0.217718 0.976012i $$-0.430139\pi$$
0.217718 + 0.976012i $$0.430139\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 6.00000 0.0228547 0.0114273 0.999935i $$-0.496362\pi$$
0.0114273 + 0.999935i $$0.496362\pi$$
$$42$$ 0 0
$$43$$ −12.0000 −0.0425577 −0.0212789 0.999774i $$-0.506774\pi$$
−0.0212789 + 0.999774i $$0.506774\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −200.000 −0.620702 −0.310351 0.950622i $$-0.600447\pi$$
−0.310351 + 0.950622i $$0.600447\pi$$
$$48$$ 0 0
$$49$$ −343.000 −1.00000
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 654.000 1.69498 0.847489 0.530813i $$-0.178113\pi$$
0.847489 + 0.530813i $$0.178113\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −36.0000 −0.0794373 −0.0397187 0.999211i $$-0.512646\pi$$
−0.0397187 + 0.999211i $$0.512646\pi$$
$$60$$ 0 0
$$61$$ −442.000 −0.927743 −0.463871 0.885903i $$-0.653540\pi$$
−0.463871 + 0.885903i $$0.653540\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 188.000 0.342804 0.171402 0.985201i $$-0.445170\pi$$
0.171402 + 0.985201i $$0.445170\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 632.000 1.05640 0.528201 0.849119i $$-0.322867\pi$$
0.528201 + 0.849119i $$0.322867\pi$$
$$72$$ 0 0
$$73$$ 390.000 0.625288 0.312644 0.949870i $$-0.398785\pi$$
0.312644 + 0.949870i $$0.398785\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 688.000 0.979823 0.489912 0.871772i $$-0.337029\pi$$
0.489912 + 0.871772i $$0.337029\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 1188.00 1.57108 0.785542 0.618809i $$-0.212384\pi$$
0.785542 + 0.618809i $$0.212384\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 694.000 0.826560 0.413280 0.910604i $$-0.364383\pi$$
0.413280 + 0.910604i $$0.364383\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 1726.00 1.80669 0.903344 0.428917i $$-0.141105\pi$$
0.903344 + 0.428917i $$0.141105\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −1182.00 −1.16449 −0.582245 0.813014i $$-0.697825\pi$$
−0.582245 + 0.813014i $$0.697825\pi$$
$$102$$ 0 0
$$103$$ −1968.00 −1.88265 −0.941324 0.337503i $$-0.890418\pi$$
−0.941324 + 0.337503i $$0.890418\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 796.000 0.719180 0.359590 0.933110i $$-0.382917\pi$$
0.359590 + 0.933110i $$0.382917\pi$$
$$108$$ 0 0
$$109$$ 342.000 0.300529 0.150264 0.988646i $$-0.451987\pi$$
0.150264 + 0.988646i $$0.451987\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 114.000 0.0949046 0.0474523 0.998874i $$-0.484890\pi$$
0.0474523 + 0.998874i $$0.484890\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −1315.00 −0.987979
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −2344.00 −1.63777 −0.818883 0.573960i $$-0.805406\pi$$
−0.818883 + 0.573960i $$0.805406\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 2164.00 1.44328 0.721640 0.692269i $$-0.243389\pi$$
0.721640 + 0.692269i $$0.243389\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −2822.00 −1.75985 −0.879926 0.475111i $$-0.842408\pi$$
−0.879926 + 0.475111i $$0.842408\pi$$
$$138$$ 0 0
$$139$$ 1972.00 1.20333 0.601665 0.798749i $$-0.294504\pi$$
0.601665 + 0.798749i $$0.294504\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 216.000 0.126313
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 1394.00 0.766449 0.383225 0.923655i $$-0.374814\pi$$
0.383225 + 0.923655i $$0.374814\pi$$
$$150$$ 0 0
$$151$$ −2216.00 −1.19427 −0.597137 0.802139i $$-0.703695\pi$$
−0.597137 + 0.802139i $$0.703695\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 954.000 0.484952 0.242476 0.970157i $$-0.422040\pi$$
0.242476 + 0.970157i $$0.422040\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 3404.00 1.63572 0.817858 0.575419i $$-0.195161\pi$$
0.817858 + 0.575419i $$0.195161\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −832.000 −0.385522 −0.192761 0.981246i $$-0.561744\pi$$
−0.192761 + 0.981246i $$0.561744\pi$$
$$168$$ 0 0
$$169$$ 719.000 0.327264
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −362.000 −0.159089 −0.0795444 0.996831i $$-0.525347\pi$$
−0.0795444 + 0.996831i $$0.525347\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 3252.00 1.35791 0.678955 0.734180i $$-0.262433\pi$$
0.678955 + 0.734180i $$0.262433\pi$$
$$180$$ 0 0
$$181$$ 3086.00 1.26730 0.633648 0.773621i $$-0.281557\pi$$
0.633648 + 0.773621i $$0.281557\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −456.000 −0.178321
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 4080.00 1.54565 0.772823 0.634621i $$-0.218844\pi$$
0.772823 + 0.634621i $$0.218844\pi$$
$$192$$ 0 0
$$193$$ 2654.00 0.989840 0.494920 0.868939i $$-0.335197\pi$$
0.494920 + 0.868939i $$0.335197\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 1534.00 0.554787 0.277393 0.960756i $$-0.410529\pi$$
0.277393 + 0.960756i $$0.410529\pi$$
$$198$$ 0 0
$$199$$ 4344.00 1.54743 0.773714 0.633536i $$-0.218397\pi$$
0.773714 + 0.633536i $$0.218397\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −176.000 −0.0582496
$$210$$ 0 0
$$211$$ −1380.00 −0.450252 −0.225126 0.974330i $$-0.572279\pi$$
−0.225126 + 0.974330i $$0.572279\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −6156.00 −1.87374
$$222$$ 0 0
$$223$$ 5224.00 1.56872 0.784361 0.620305i $$-0.212991\pi$$
0.784361 + 0.620305i $$0.212991\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 3364.00 0.983597 0.491799 0.870709i $$-0.336340\pi$$
0.491799 + 0.870709i $$0.336340\pi$$
$$228$$ 0 0
$$229$$ 3998.00 1.15369 0.576846 0.816853i $$-0.304283\pi$$
0.576846 + 0.816853i $$0.304283\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −3590.00 −1.00939 −0.504697 0.863297i $$-0.668396\pi$$
−0.504697 + 0.863297i $$0.668396\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 1104.00 0.298794 0.149397 0.988777i $$-0.452267\pi$$
0.149397 + 0.988777i $$0.452267\pi$$
$$240$$ 0 0
$$241$$ 1618.00 0.432467 0.216233 0.976342i $$-0.430623\pi$$
0.216233 + 0.976342i $$0.430623\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −2376.00 −0.612070
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −5780.00 −1.45351 −0.726754 0.686898i $$-0.758972\pi$$
−0.726754 + 0.686898i $$0.758972\pi$$
$$252$$ 0 0
$$253$$ −384.000 −0.0954224
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 2594.00 0.629608 0.314804 0.949157i $$-0.398061\pi$$
0.314804 + 0.949157i $$0.398061\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 3696.00 0.866559 0.433280 0.901260i $$-0.357356\pi$$
0.433280 + 0.901260i $$0.357356\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 2250.00 0.509981 0.254991 0.966944i $$-0.417928\pi$$
0.254991 + 0.966944i $$0.417928\pi$$
$$270$$ 0 0
$$271$$ 2208.00 0.494932 0.247466 0.968897i $$-0.420402\pi$$
0.247466 + 0.968897i $$0.420402\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 1682.00 0.364843 0.182422 0.983220i $$-0.441606\pi$$
0.182422 + 0.983220i $$0.441606\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −7306.00 −1.55103 −0.775515 0.631329i $$-0.782510\pi$$
−0.775515 + 0.631329i $$0.782510\pi$$
$$282$$ 0 0
$$283$$ 8164.00 1.71484 0.857419 0.514618i $$-0.172066\pi$$
0.857419 + 0.514618i $$0.172066\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 8083.00 1.64523
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −514.000 −0.102485 −0.0512427 0.998686i $$-0.516318\pi$$
−0.0512427 + 0.998686i $$0.516318\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −5184.00 −1.00267
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 2476.00 0.460302 0.230151 0.973155i $$-0.426078\pi$$
0.230151 + 0.973155i $$0.426078\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −2296.00 −0.418631 −0.209315 0.977848i $$-0.567124\pi$$
−0.209315 + 0.977848i $$0.567124\pi$$
$$312$$ 0 0
$$313$$ 9878.00 1.78383 0.891913 0.452207i $$-0.149363\pi$$
0.891913 + 0.452207i $$0.149363\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −2138.00 −0.378808 −0.189404 0.981899i $$-0.560656\pi$$
−0.189404 + 0.981899i $$0.560656\pi$$
$$318$$ 0 0
$$319$$ 536.000 0.0940760
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 5016.00 0.864080
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −6460.00 −1.07273 −0.536365 0.843986i $$-0.680203\pi$$
−0.536365 + 0.843986i $$0.680203\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −626.000 −0.101188 −0.0505941 0.998719i $$-0.516111\pi$$
−0.0505941 + 0.998719i $$0.516111\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 1088.00 0.172782
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 876.000 0.135522 0.0677610 0.997702i $$-0.478414\pi$$
0.0677610 + 0.997702i $$0.478414\pi$$
$$348$$ 0 0
$$349$$ −9850.00 −1.51077 −0.755385 0.655282i $$-0.772550\pi$$
−0.755385 + 0.655282i $$0.772550\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −8894.00 −1.34102 −0.670510 0.741901i $$-0.733925\pi$$
−0.670510 + 0.741901i $$0.733925\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −1464.00 −0.215228 −0.107614 0.994193i $$-0.534321\pi$$
−0.107614 + 0.994193i $$0.534321\pi$$
$$360$$ 0 0
$$361$$ −4923.00 −0.717743
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 7016.00 0.997908 0.498954 0.866628i $$-0.333718\pi$$
0.498954 + 0.866628i $$0.333718\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 1010.00 0.140203 0.0701016 0.997540i $$-0.477668\pi$$
0.0701016 + 0.997540i $$0.477668\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 7236.00 0.988522
$$378$$ 0 0
$$379$$ 4900.00 0.664106 0.332053 0.943261i $$-0.392259\pi$$
0.332053 + 0.943261i $$0.392259\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 7800.00 1.04063 0.520315 0.853974i $$-0.325814\pi$$
0.520315 + 0.853974i $$0.325814\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 12258.0 1.59770 0.798850 0.601530i $$-0.205442\pi$$
0.798850 + 0.601530i $$0.205442\pi$$
$$390$$ 0 0
$$391$$ 10944.0 1.41550
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −5558.00 −0.702640 −0.351320 0.936255i $$-0.614267\pi$$
−0.351320 + 0.936255i $$0.614267\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −1970.00 −0.245329 −0.122665 0.992448i $$-0.539144\pi$$
−0.122665 + 0.992448i $$0.539144\pi$$
$$402$$ 0 0
$$403$$ 14688.0 1.81554
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −392.000 −0.0477413
$$408$$ 0 0
$$409$$ 15626.0 1.88913 0.944567 0.328318i $$-0.106482\pi$$
0.944567 + 0.328318i $$0.106482\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 5412.00 0.631011 0.315505 0.948924i $$-0.397826\pi$$
0.315505 + 0.948924i $$0.397826\pi$$
$$420$$ 0 0
$$421$$ −10690.0 −1.23753 −0.618763 0.785577i $$-0.712366\pi$$
−0.618763 + 0.785577i $$0.712366\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 14048.0 1.57000 0.784998 0.619498i $$-0.212664\pi$$
0.784998 + 0.619498i $$0.212664\pi$$
$$432$$ 0 0
$$433$$ −17778.0 −1.97311 −0.986554 0.163433i $$-0.947743\pi$$
−0.986554 + 0.163433i $$0.947743\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 4224.00 0.462383
$$438$$ 0 0
$$439$$ 7240.00 0.787122 0.393561 0.919299i $$-0.371243\pi$$
0.393561 + 0.919299i $$0.371243\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 11740.0 1.25911 0.629553 0.776957i $$-0.283238\pi$$
0.629553 + 0.776957i $$0.283238\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −15234.0 −1.60120 −0.800598 0.599202i $$-0.795485\pi$$
−0.800598 + 0.599202i $$0.795485\pi$$
$$450$$ 0 0
$$451$$ −24.0000 −0.00250580
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −3866.00 −0.395720 −0.197860 0.980230i $$-0.563399\pi$$
−0.197860 + 0.980230i $$0.563399\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 1706.00 0.172356 0.0861782 0.996280i $$-0.472535\pi$$
0.0861782 + 0.996280i $$0.472535\pi$$
$$462$$ 0 0
$$463$$ −3944.00 −0.395882 −0.197941 0.980214i $$-0.563425\pi$$
−0.197941 + 0.980214i $$0.563425\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −9452.00 −0.936588 −0.468294 0.883573i $$-0.655131\pi$$
−0.468294 + 0.883573i $$0.655131\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 48.0000 0.00466605
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 12544.0 1.19656 0.598278 0.801289i $$-0.295852\pi$$
0.598278 + 0.801289i $$0.295852\pi$$
$$480$$ 0 0
$$481$$ −5292.00 −0.501652
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −7936.00 −0.738428 −0.369214 0.929344i $$-0.620373\pi$$
−0.369214 + 0.929344i $$0.620373\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 8412.00 0.773174 0.386587 0.922253i $$-0.373654\pi$$
0.386587 + 0.922253i $$0.373654\pi$$
$$492$$ 0 0
$$493$$ −15276.0 −1.39553
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −15092.0 −1.35393 −0.676965 0.736016i $$-0.736705\pi$$
−0.676965 + 0.736016i $$0.736705\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 6112.00 0.541790 0.270895 0.962609i $$-0.412680\pi$$
0.270895 + 0.962609i $$0.412680\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −2534.00 −0.220663 −0.110332 0.993895i $$-0.535191\pi$$
−0.110332 + 0.993895i $$0.535191\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 800.000 0.0680541
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 9894.00 0.831985 0.415992 0.909368i $$-0.363434\pi$$
0.415992 + 0.909368i $$0.363434\pi$$
$$522$$ 0 0
$$523$$ −16172.0 −1.35211 −0.676054 0.736852i $$-0.736311\pi$$
−0.676054 + 0.736852i $$0.736311\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −31008.0 −2.56305
$$528$$ 0 0
$$529$$ −2951.00 −0.242541
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −324.000 −0.0263302
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 1372.00 0.109640
$$540$$ 0 0
$$541$$ −6138.00 −0.487788 −0.243894 0.969802i $$-0.578425\pi$$
−0.243894 + 0.969802i $$0.578425\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 21852.0 1.70809 0.854044 0.520201i $$-0.174143\pi$$
0.854044 + 0.520201i $$0.174143\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −5896.00 −0.455859
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −1962.00 −0.149251 −0.0746253 0.997212i $$-0.523776\pi$$
−0.0746253 + 0.997212i $$0.523776\pi$$
$$558$$ 0 0
$$559$$ 648.000 0.0490295
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −10876.0 −0.814154 −0.407077 0.913394i $$-0.633452\pi$$
−0.407077 + 0.913394i $$0.633452\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −5610.00 −0.413328 −0.206664 0.978412i $$-0.566261\pi$$
−0.206664 + 0.978412i $$0.566261\pi$$
$$570$$ 0 0
$$571$$ 5076.00 0.372021 0.186010 0.982548i $$-0.440444\pi$$
0.186010 + 0.982548i $$0.440444\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 6526.00 0.470851 0.235425 0.971892i $$-0.424352\pi$$
0.235425 + 0.971892i $$0.424352\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −2616.00 −0.185838
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 2332.00 0.163973 0.0819863 0.996633i $$-0.473874\pi$$
0.0819863 + 0.996633i $$0.473874\pi$$
$$588$$ 0 0
$$589$$ −11968.0 −0.837237
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −9582.00 −0.663551 −0.331775 0.943358i $$-0.607648\pi$$
−0.331775 + 0.943358i $$0.607648\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 17624.0 1.20217 0.601083 0.799187i $$-0.294736\pi$$
0.601083 + 0.799187i $$0.294736\pi$$
$$600$$ 0 0
$$601$$ −21238.0 −1.44146 −0.720729 0.693217i $$-0.756193\pi$$
−0.720729 + 0.693217i $$0.756193\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −13000.0 −0.869281 −0.434641 0.900604i $$-0.643125\pi$$
−0.434641 + 0.900604i $$0.643125\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 10800.0 0.715092
$$612$$ 0 0
$$613$$ −9214.00 −0.607096 −0.303548 0.952816i $$-0.598171\pi$$
−0.303548 + 0.952816i $$0.598171\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 4474.00 0.291923 0.145961 0.989290i $$-0.453372\pi$$
0.145961 + 0.989290i $$0.453372\pi$$
$$618$$ 0 0
$$619$$ −12556.0 −0.815296 −0.407648 0.913139i $$-0.633651\pi$$
−0.407648 + 0.913139i $$0.633651\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 11172.0 0.708198
$$630$$ 0 0
$$631$$ 26936.0 1.69937 0.849687 0.527287i $$-0.176791\pi$$
0.849687 + 0.527287i $$0.176791\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 18522.0 1.15207
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 19134.0 1.17901 0.589507 0.807764i $$-0.299322\pi$$
0.589507 + 0.807764i $$0.299322\pi$$
$$642$$ 0 0
$$643$$ −12436.0 −0.762718 −0.381359 0.924427i $$-0.624544\pi$$
−0.381359 + 0.924427i $$0.624544\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −2784.00 −0.169166 −0.0845829 0.996416i $$-0.526956\pi$$
−0.0845829 + 0.996416i $$0.526956\pi$$
$$648$$ 0 0
$$649$$ 144.000 0.00870954
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 7318.00 0.438554 0.219277 0.975663i $$-0.429630\pi$$
0.219277 + 0.975663i $$0.429630\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −8108.00 −0.479276 −0.239638 0.970862i $$-0.577029\pi$$
−0.239638 + 0.970862i $$0.577029\pi$$
$$660$$ 0 0
$$661$$ 1230.00 0.0723774 0.0361887 0.999345i $$-0.488478\pi$$
0.0361887 + 0.999345i $$0.488478\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −12864.0 −0.746771
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 1768.00 0.101718
$$672$$ 0 0
$$673$$ 14078.0 0.806340 0.403170 0.915125i $$-0.367908\pi$$
0.403170 + 0.915125i $$0.367908\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 25246.0 1.43321 0.716605 0.697480i $$-0.245695\pi$$
0.716605 + 0.697480i $$0.245695\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 24332.0 1.36316 0.681580 0.731744i $$-0.261293\pi$$
0.681580 + 0.731744i $$0.261293\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −35316.0 −1.95273
$$690$$ 0 0
$$691$$ 19036.0 1.04799 0.523997 0.851720i $$-0.324440\pi$$
0.523997 + 0.851720i $$0.324440\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 684.000 0.0371712
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −28806.0 −1.55205 −0.776025 0.630702i $$-0.782767\pi$$
−0.776025 + 0.630702i $$0.782767\pi$$
$$702$$ 0 0
$$703$$ 4312.00 0.231337
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −25090.0 −1.32902 −0.664510 0.747280i $$-0.731360\pi$$
−0.664510 + 0.747280i $$0.731360\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −26112.0 −1.37153
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −36432.0 −1.88969 −0.944843 0.327523i $$-0.893786\pi$$
−0.944843 + 0.327523i $$0.893786\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −21616.0 −1.10274 −0.551371 0.834260i $$-0.685895\pi$$
−0.551371 + 0.834260i $$0.685895\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −1368.00 −0.0692166
$$732$$ 0 0
$$733$$ −28102.0 −1.41606 −0.708029 0.706183i $$-0.750416\pi$$
−0.708029 + 0.706183i $$0.750416\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −752.000 −0.0375852
$$738$$ 0 0
$$739$$ 764.000 0.0380300 0.0190150 0.999819i $$-0.493947\pi$$
0.0190150 + 0.999819i $$0.493947\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 6256.00 0.308897 0.154448 0.988001i $$-0.450640\pi$$
0.154448 + 0.988001i $$0.450640\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 1184.00 0.0575297 0.0287648 0.999586i $$-0.490843\pi$$
0.0287648 + 0.999586i $$0.490843\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −26446.0 −1.26974 −0.634872 0.772617i $$-0.718947\pi$$
−0.634872 + 0.772617i $$0.718947\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −36778.0 −1.75191 −0.875954 0.482395i $$-0.839767\pi$$
−0.875954 + 0.482395i $$0.839767\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 1944.00 0.0915173
$$768$$ 0 0
$$769$$ −10302.0 −0.483094 −0.241547 0.970389i $$-0.577655\pi$$
−0.241547 + 0.970389i $$0.577655\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −4674.00 −0.217480 −0.108740 0.994070i $$-0.534682\pi$$
−0.108740 + 0.994070i $$0.534682\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 264.000 0.0121422
$$780$$ 0 0
$$781$$ −2528.00 −0.115825
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 23084.0 1.04556 0.522780 0.852468i $$-0.324895\pi$$
0.522780 + 0.852468i $$0.324895\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 23868.0 1.06882
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 10694.0 0.475283 0.237642 0.971353i $$-0.423626\pi$$
0.237642 + 0.971353i $$0.423626\pi$$
$$798$$ 0 0
$$799$$ −22800.0 −1.00952
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −1560.00 −0.0685569
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −9594.00 −0.416943 −0.208472 0.978028i $$-0.566849\pi$$
−0.208472 + 0.978028i $$0.566849\pi$$
$$810$$ 0 0
$$811$$ 10244.0 0.443546 0.221773 0.975098i $$-0.428816\pi$$
0.221773 + 0.975098i $$0.428816\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −528.000 −0.0226100
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −1390.00 −0.0590881 −0.0295441 0.999563i $$-0.509406\pi$$
−0.0295441 + 0.999563i $$0.509406\pi$$
$$822$$ 0 0
$$823$$ 8448.00 0.357811 0.178906 0.983866i $$-0.442744\pi$$
0.178906 + 0.983866i $$0.442744\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 41484.0 1.74430 0.872152 0.489234i $$-0.162724\pi$$
0.872152 + 0.489234i $$0.162724\pi$$
$$828$$ 0 0
$$829$$ −31610.0 −1.32432 −0.662160 0.749363i $$-0.730360\pi$$
−0.662160 + 0.749363i $$0.730360\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −39102.0 −1.62642
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −38264.0 −1.57452 −0.787259 0.616623i $$-0.788500\pi$$
−0.787259 + 0.616623i $$0.788500\pi$$
$$840$$ 0 0
$$841$$ −6433.00 −0.263766
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 9408.00 0.378968
$$852$$ 0 0
$$853$$ −30350.0 −1.21825 −0.609123 0.793076i $$-0.708479\pi$$
−0.609123 + 0.793076i $$0.708479\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −12566.0 −0.500871 −0.250435 0.968133i $$-0.580574\pi$$
−0.250435 + 0.968133i $$0.580574\pi$$
$$858$$ 0 0
$$859$$ 11812.0 0.469174 0.234587 0.972095i $$-0.424626\pi$$
0.234587 + 0.972095i $$0.424626\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −31496.0 −1.24234 −0.621168 0.783677i $$-0.713342\pi$$
−0.621168 + 0.783677i $$0.713342\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −2752.00 −0.107428
$$870$$ 0 0
$$871$$ −10152.0 −0.394934
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −7414.00 −0.285465 −0.142733 0.989761i $$-0.545589\pi$$
−0.142733 + 0.989761i $$0.545589\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 22190.0 0.848581 0.424291 0.905526i $$-0.360523\pi$$
0.424291 + 0.905526i $$0.360523\pi$$
$$882$$ 0 0
$$883$$ 10172.0 0.387673 0.193836 0.981034i $$-0.437907\pi$$
0.193836 + 0.981034i $$0.437907\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −20784.0 −0.786763 −0.393381 0.919375i $$-0.628695\pi$$
−0.393381 + 0.919375i $$0.628695\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −8800.00 −0.329766
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 36448.0 1.35218
$$900$$ 0 0
$$901$$ 74556.0 2.75674
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 7652.00 0.280133 0.140066 0.990142i $$-0.455268\pi$$
0.140066 + 0.990142i $$0.455268\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −19296.0 −0.701762 −0.350881 0.936420i $$-0.614118\pi$$
−0.350881 + 0.936420i $$0.614118\pi$$
$$912$$ 0 0
$$913$$ −4752.00 −0.172254
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −35896.0 −1.28847 −0.644233 0.764830i $$-0.722823\pi$$
−0.644233 + 0.764830i $$0.722823\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −34128.0 −1.21705
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 16350.0 0.577423 0.288712 0.957416i $$-0.406773\pi$$
0.288712 + 0.957416i $$0.406773\pi$$
$$930$$ 0 0
$$931$$ −15092.0 −0.531279
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 19686.0 0.686354 0.343177 0.939271i $$-0.388497\pi$$
0.343177 + 0.939271i $$0.388497\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −56246.0 −1.94853 −0.974265 0.225405i $$-0.927630\pi$$
−0.974265 + 0.225405i $$0.927630\pi$$
$$942$$ 0 0
$$943$$ 576.000 0.0198909
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −11436.0 −0.392418 −0.196209 0.980562i $$-0.562863\pi$$
−0.196209 + 0.980562i $$0.562863\pi$$
$$948$$ 0 0
$$949$$ −21060.0 −0.720376
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −22582.0 −0.767579 −0.383789 0.923421i $$-0.625381\pi$$
−0.383789 + 0.923421i $$0.625381\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 44193.0 1.48343
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 2112.00 0.0702351 0.0351175 0.999383i $$-0.488819\pi$$
0.0351175 + 0.999383i $$0.488819\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 47964.0 1.58521 0.792605 0.609736i $$-0.208725\pi$$
0.792605 + 0.609736i $$0.208725\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −10510.0 −0.344160 −0.172080 0.985083i $$-0.555049\pi$$
−0.172080 + 0.985083i $$0.555049\pi$$
$$978$$ 0 0
$$979$$ −2776.00 −0.0906245
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 11488.0 0.372747 0.186373 0.982479i $$-0.440327\pi$$
0.186373 + 0.982479i $$0.440327\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −1152.00 −0.0370389
$$990$$ 0 0
$$991$$ −23120.0 −0.741101 −0.370550 0.928812i $$-0.620831\pi$$
−0.370550 + 0.928812i $$0.620831\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −30078.0 −0.955446 −0.477723 0.878510i $$-0.658538\pi$$
−0.477723 + 0.878510i $$0.658538\pi$$
$$998$$ 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.s.1.1 1
3.2 odd 2 600.4.a.m.1.1 1
5.2 odd 4 1800.4.f.m.649.2 2
5.3 odd 4 1800.4.f.m.649.1 2
5.4 even 2 360.4.a.c.1.1 1
12.11 even 2 1200.4.a.j.1.1 1
15.2 even 4 600.4.f.d.49.1 2
15.8 even 4 600.4.f.d.49.2 2
15.14 odd 2 120.4.a.d.1.1 1
20.19 odd 2 720.4.a.i.1.1 1
60.23 odd 4 1200.4.f.l.49.1 2
60.47 odd 4 1200.4.f.l.49.2 2
60.59 even 2 240.4.a.k.1.1 1
120.29 odd 2 960.4.a.x.1.1 1
120.59 even 2 960.4.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.a.d.1.1 1 15.14 odd 2
240.4.a.k.1.1 1 60.59 even 2
360.4.a.c.1.1 1 5.4 even 2
600.4.a.m.1.1 1 3.2 odd 2
600.4.f.d.49.1 2 15.2 even 4
600.4.f.d.49.2 2 15.8 even 4
720.4.a.i.1.1 1 20.19 odd 2
960.4.a.e.1.1 1 120.59 even 2
960.4.a.x.1.1 1 120.29 odd 2
1200.4.a.j.1.1 1 12.11 even 2
1200.4.f.l.49.1 2 60.23 odd 4
1200.4.f.l.49.2 2 60.47 odd 4
1800.4.a.s.1.1 1 1.1 even 1 trivial
1800.4.f.m.649.1 2 5.3 odd 4
1800.4.f.m.649.2 2 5.2 odd 4