# Properties

 Label 3332.2.a.o.1.2 Level $3332$ Weight $2$ Character 3332.1 Self dual yes Analytic conductor $26.606$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3332,2,Mod(1,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3332.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3332 = 2^{2} \cdot 7^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3332.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: $$26.6061539535$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 3332.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.61803 q^{3} +3.61803 q^{5} +3.85410 q^{9} +O(q^{10})$$ $$q+2.61803 q^{3} +3.61803 q^{5} +3.85410 q^{9} +1.23607 q^{13} +9.47214 q^{15} +1.00000 q^{17} +6.47214 q^{19} +0.763932 q^{23} +8.09017 q^{25} +2.23607 q^{27} +7.23607 q^{29} -10.5623 q^{31} -9.70820 q^{37} +3.23607 q^{39} -7.85410 q^{41} -1.85410 q^{43} +13.9443 q^{45} -6.94427 q^{47} +2.61803 q^{51} -3.32624 q^{53} +16.9443 q^{57} +0.763932 q^{59} -7.56231 q^{61} +4.47214 q^{65} +6.09017 q^{67} +2.00000 q^{69} -9.23607 q^{71} +3.85410 q^{73} +21.1803 q^{75} +11.7082 q^{79} -5.70820 q^{81} -10.4721 q^{83} +3.61803 q^{85} +18.9443 q^{87} +8.76393 q^{89} -27.6525 q^{93} +23.4164 q^{95} -2.85410 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{3} + 5 q^{5} + q^{9}+O(q^{10})$$ 2 * q + 3 * q^3 + 5 * q^5 + q^9 $$2 q + 3 q^{3} + 5 q^{5} + q^{9} - 2 q^{13} + 10 q^{15} + 2 q^{17} + 4 q^{19} + 6 q^{23} + 5 q^{25} + 10 q^{29} - q^{31} - 6 q^{37} + 2 q^{39} - 9 q^{41} + 3 q^{43} + 10 q^{45} + 4 q^{47} + 3 q^{51} + 9 q^{53} + 16 q^{57} + 6 q^{59} + 5 q^{61} + q^{67} + 4 q^{69} - 14 q^{71} + q^{73} + 20 q^{75} + 10 q^{79} + 2 q^{81} - 12 q^{83} + 5 q^{85} + 20 q^{87} + 22 q^{89} - 24 q^{93} + 20 q^{95} + q^{97}+O(q^{100})$$ 2 * q + 3 * q^3 + 5 * q^5 + q^9 - 2 * q^13 + 10 * q^15 + 2 * q^17 + 4 * q^19 + 6 * q^23 + 5 * q^25 + 10 * q^29 - q^31 - 6 * q^37 + 2 * q^39 - 9 * q^41 + 3 * q^43 + 10 * q^45 + 4 * q^47 + 3 * q^51 + 9 * q^53 + 16 * q^57 + 6 * q^59 + 5 * q^61 + q^67 + 4 * q^69 - 14 * q^71 + q^73 + 20 * q^75 + 10 * q^79 + 2 * q^81 - 12 * q^83 + 5 * q^85 + 20 * q^87 + 22 * q^89 - 24 * q^93 + 20 * q^95 + q^97

## 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
$$3$$ 2.61803 1.51152 0.755761 0.654847i $$-0.227267\pi$$
0.755761 + 0.654847i $$0.227267\pi$$
$$4$$ 0 0
$$5$$ 3.61803 1.61803 0.809017 0.587785i $$-0.200000\pi$$
0.809017 + 0.587785i $$0.200000\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 3.85410 1.28470
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ 1.23607 0.342824 0.171412 0.985199i $$-0.445167\pi$$
0.171412 + 0.985199i $$0.445167\pi$$
$$14$$ 0 0
$$15$$ 9.47214 2.44569
$$16$$ 0 0
$$17$$ 1.00000 0.242536
$$18$$ 0 0
$$19$$ 6.47214 1.48481 0.742405 0.669951i $$-0.233685\pi$$
0.742405 + 0.669951i $$0.233685\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0.763932 0.159291 0.0796454 0.996823i $$-0.474621\pi$$
0.0796454 + 0.996823i $$0.474621\pi$$
$$24$$ 0 0
$$25$$ 8.09017 1.61803
$$26$$ 0 0
$$27$$ 2.23607 0.430331
$$28$$ 0 0
$$29$$ 7.23607 1.34370 0.671852 0.740685i $$-0.265499\pi$$
0.671852 + 0.740685i $$0.265499\pi$$
$$30$$ 0 0
$$31$$ −10.5623 −1.89705 −0.948523 0.316708i $$-0.897422\pi$$
−0.948523 + 0.316708i $$0.897422\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −9.70820 −1.59602 −0.798009 0.602645i $$-0.794114\pi$$
−0.798009 + 0.602645i $$0.794114\pi$$
$$38$$ 0 0
$$39$$ 3.23607 0.518186
$$40$$ 0 0
$$41$$ −7.85410 −1.22660 −0.613302 0.789848i $$-0.710159\pi$$
−0.613302 + 0.789848i $$0.710159\pi$$
$$42$$ 0 0
$$43$$ −1.85410 −0.282748 −0.141374 0.989956i $$-0.545152\pi$$
−0.141374 + 0.989956i $$0.545152\pi$$
$$44$$ 0 0
$$45$$ 13.9443 2.07869
$$46$$ 0 0
$$47$$ −6.94427 −1.01293 −0.506463 0.862262i $$-0.669047\pi$$
−0.506463 + 0.862262i $$0.669047\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 2.61803 0.366598
$$52$$ 0 0
$$53$$ −3.32624 −0.456894 −0.228447 0.973556i $$-0.573365\pi$$
−0.228447 + 0.973556i $$0.573365\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 16.9443 2.24432
$$58$$ 0 0
$$59$$ 0.763932 0.0994555 0.0497277 0.998763i $$-0.484165\pi$$
0.0497277 + 0.998763i $$0.484165\pi$$
$$60$$ 0 0
$$61$$ −7.56231 −0.968254 −0.484127 0.874998i $$-0.660863\pi$$
−0.484127 + 0.874998i $$0.660863\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 4.47214 0.554700
$$66$$ 0 0
$$67$$ 6.09017 0.744033 0.372016 0.928226i $$-0.378667\pi$$
0.372016 + 0.928226i $$0.378667\pi$$
$$68$$ 0 0
$$69$$ 2.00000 0.240772
$$70$$ 0 0
$$71$$ −9.23607 −1.09612 −0.548060 0.836439i $$-0.684633\pi$$
−0.548060 + 0.836439i $$0.684633\pi$$
$$72$$ 0 0
$$73$$ 3.85410 0.451089 0.225544 0.974233i $$-0.427584\pi$$
0.225544 + 0.974233i $$0.427584\pi$$
$$74$$ 0 0
$$75$$ 21.1803 2.44569
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 11.7082 1.31728 0.658638 0.752460i $$-0.271133\pi$$
0.658638 + 0.752460i $$0.271133\pi$$
$$80$$ 0 0
$$81$$ −5.70820 −0.634245
$$82$$ 0 0
$$83$$ −10.4721 −1.14947 −0.574733 0.818341i $$-0.694894\pi$$
−0.574733 + 0.818341i $$0.694894\pi$$
$$84$$ 0 0
$$85$$ 3.61803 0.392431
$$86$$ 0 0
$$87$$ 18.9443 2.03104
$$88$$ 0 0
$$89$$ 8.76393 0.928975 0.464487 0.885580i $$-0.346239\pi$$
0.464487 + 0.885580i $$0.346239\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −27.6525 −2.86743
$$94$$ 0 0
$$95$$ 23.4164 2.40247
$$96$$ 0 0
$$97$$ −2.85410 −0.289790 −0.144895 0.989447i $$-0.546284\pi$$
−0.144895 + 0.989447i $$0.546284\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 7.52786 0.749050 0.374525 0.927217i $$-0.377806\pi$$
0.374525 + 0.927217i $$0.377806\pi$$
$$102$$ 0 0
$$103$$ 6.00000 0.591198 0.295599 0.955312i $$-0.404481\pi$$
0.295599 + 0.955312i $$0.404481\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ 12.1803 1.16666 0.583332 0.812233i $$-0.301748\pi$$
0.583332 + 0.812233i $$0.301748\pi$$
$$110$$ 0 0
$$111$$ −25.4164 −2.41242
$$112$$ 0 0
$$113$$ 12.9443 1.21769 0.608847 0.793287i $$-0.291632\pi$$
0.608847 + 0.793287i $$0.291632\pi$$
$$114$$ 0 0
$$115$$ 2.76393 0.257738
$$116$$ 0 0
$$117$$ 4.76393 0.440426
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ −20.5623 −1.85404
$$124$$ 0 0
$$125$$ 11.1803 1.00000
$$126$$ 0 0
$$127$$ −15.6180 −1.38588 −0.692938 0.720997i $$-0.743684\pi$$
−0.692938 + 0.720997i $$0.743684\pi$$
$$128$$ 0 0
$$129$$ −4.85410 −0.427380
$$130$$ 0 0
$$131$$ 6.47214 0.565473 0.282737 0.959198i $$-0.408758\pi$$
0.282737 + 0.959198i $$0.408758\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 8.09017 0.696291
$$136$$ 0 0
$$137$$ 11.0902 0.947497 0.473749 0.880660i $$-0.342901\pi$$
0.473749 + 0.880660i $$0.342901\pi$$
$$138$$ 0 0
$$139$$ −13.6180 −1.15507 −0.577533 0.816367i $$-0.695985\pi$$
−0.577533 + 0.816367i $$0.695985\pi$$
$$140$$ 0 0
$$141$$ −18.1803 −1.53106
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 26.1803 2.17416
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −3.38197 −0.277061 −0.138531 0.990358i $$-0.544238\pi$$
−0.138531 + 0.990358i $$0.544238\pi$$
$$150$$ 0 0
$$151$$ −21.2705 −1.73097 −0.865485 0.500935i $$-0.832989\pi$$
−0.865485 + 0.500935i $$0.832989\pi$$
$$152$$ 0 0
$$153$$ 3.85410 0.311586
$$154$$ 0 0
$$155$$ −38.2148 −3.06949
$$156$$ 0 0
$$157$$ 23.7082 1.89212 0.946060 0.323991i $$-0.105025\pi$$
0.946060 + 0.323991i $$0.105025\pi$$
$$158$$ 0 0
$$159$$ −8.70820 −0.690605
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −6.00000 −0.469956 −0.234978 0.972001i $$-0.575502\pi$$
−0.234978 + 0.972001i $$0.575502\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −0.437694 −0.0338698 −0.0169349 0.999857i $$-0.505391\pi$$
−0.0169349 + 0.999857i $$0.505391\pi$$
$$168$$ 0 0
$$169$$ −11.4721 −0.882472
$$170$$ 0 0
$$171$$ 24.9443 1.90754
$$172$$ 0 0
$$173$$ 22.3262 1.69743 0.848716 0.528849i $$-0.177376\pi$$
0.848716 + 0.528849i $$0.177376\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 2.00000 0.150329
$$178$$ 0 0
$$179$$ −3.09017 −0.230970 −0.115485 0.993309i $$-0.536842\pi$$
−0.115485 + 0.993309i $$0.536842\pi$$
$$180$$ 0 0
$$181$$ 14.0000 1.04061 0.520306 0.853980i $$-0.325818\pi$$
0.520306 + 0.853980i $$0.325818\pi$$
$$182$$ 0 0
$$183$$ −19.7984 −1.46354
$$184$$ 0 0
$$185$$ −35.1246 −2.58241
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 22.0902 1.59839 0.799194 0.601073i $$-0.205260\pi$$
0.799194 + 0.601073i $$0.205260\pi$$
$$192$$ 0 0
$$193$$ −5.52786 −0.397904 −0.198952 0.980009i $$-0.563754\pi$$
−0.198952 + 0.980009i $$0.563754\pi$$
$$194$$ 0 0
$$195$$ 11.7082 0.838442
$$196$$ 0 0
$$197$$ −15.7082 −1.11916 −0.559582 0.828775i $$-0.689038\pi$$
−0.559582 + 0.828775i $$0.689038\pi$$
$$198$$ 0 0
$$199$$ 12.1459 0.861000 0.430500 0.902591i $$-0.358337\pi$$
0.430500 + 0.902591i $$0.358337\pi$$
$$200$$ 0 0
$$201$$ 15.9443 1.12462
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −28.4164 −1.98469
$$206$$ 0 0
$$207$$ 2.94427 0.204641
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 23.5967 1.62447 0.812234 0.583332i $$-0.198251\pi$$
0.812234 + 0.583332i $$0.198251\pi$$
$$212$$ 0 0
$$213$$ −24.1803 −1.65681
$$214$$ 0 0
$$215$$ −6.70820 −0.457496
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 10.0902 0.681830
$$220$$ 0 0
$$221$$ 1.23607 0.0831469
$$222$$ 0 0
$$223$$ 15.8885 1.06398 0.531988 0.846752i $$-0.321445\pi$$
0.531988 + 0.846752i $$0.321445\pi$$
$$224$$ 0 0
$$225$$ 31.1803 2.07869
$$226$$ 0 0
$$227$$ −7.14590 −0.474290 −0.237145 0.971474i $$-0.576212\pi$$
−0.237145 + 0.971474i $$0.576212\pi$$
$$228$$ 0 0
$$229$$ −17.4164 −1.15091 −0.575454 0.817834i $$-0.695175\pi$$
−0.575454 + 0.817834i $$0.695175\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −2.18034 −0.142839 −0.0714194 0.997446i $$-0.522753\pi$$
−0.0714194 + 0.997446i $$0.522753\pi$$
$$234$$ 0 0
$$235$$ −25.1246 −1.63895
$$236$$ 0 0
$$237$$ 30.6525 1.99109
$$238$$ 0 0
$$239$$ −19.3820 −1.25372 −0.626858 0.779134i $$-0.715659\pi$$
−0.626858 + 0.779134i $$0.715659\pi$$
$$240$$ 0 0
$$241$$ 3.61803 0.233058 0.116529 0.993187i $$-0.462823\pi$$
0.116529 + 0.993187i $$0.462823\pi$$
$$242$$ 0 0
$$243$$ −21.6525 −1.38901
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 8.00000 0.509028
$$248$$ 0 0
$$249$$ −27.4164 −1.73744
$$250$$ 0 0
$$251$$ −20.1803 −1.27377 −0.636886 0.770958i $$-0.719778\pi$$
−0.636886 + 0.770958i $$0.719778\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 9.47214 0.593168
$$256$$ 0 0
$$257$$ 1.52786 0.0953055 0.0476528 0.998864i $$-0.484826\pi$$
0.0476528 + 0.998864i $$0.484826\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 27.8885 1.72626
$$262$$ 0 0
$$263$$ 19.4164 1.19727 0.598633 0.801023i $$-0.295711\pi$$
0.598633 + 0.801023i $$0.295711\pi$$
$$264$$ 0 0
$$265$$ −12.0344 −0.739270
$$266$$ 0 0
$$267$$ 22.9443 1.40417
$$268$$ 0 0
$$269$$ 13.4164 0.818013 0.409006 0.912532i $$-0.365875\pi$$
0.409006 + 0.912532i $$0.365875\pi$$
$$270$$ 0 0
$$271$$ −20.4721 −1.24359 −0.621797 0.783179i $$-0.713597\pi$$
−0.621797 + 0.783179i $$0.713597\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −31.1246 −1.87010 −0.935048 0.354520i $$-0.884644\pi$$
−0.935048 + 0.354520i $$0.884644\pi$$
$$278$$ 0 0
$$279$$ −40.7082 −2.43714
$$280$$ 0 0
$$281$$ −23.5066 −1.40228 −0.701142 0.713021i $$-0.747326\pi$$
−0.701142 + 0.713021i $$0.747326\pi$$
$$282$$ 0 0
$$283$$ 19.1459 1.13811 0.569053 0.822301i $$-0.307310\pi$$
0.569053 + 0.822301i $$0.307310\pi$$
$$284$$ 0 0
$$285$$ 61.3050 3.63139
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ −7.47214 −0.438024
$$292$$ 0 0
$$293$$ −27.5967 −1.61222 −0.806110 0.591766i $$-0.798431\pi$$
−0.806110 + 0.591766i $$0.798431\pi$$
$$294$$ 0 0
$$295$$ 2.76393 0.160922
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0.944272 0.0546087
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 19.7082 1.13221
$$304$$ 0 0
$$305$$ −27.3607 −1.56667
$$306$$ 0 0
$$307$$ −9.41641 −0.537423 −0.268711 0.963221i $$-0.586598\pi$$
−0.268711 + 0.963221i $$0.586598\pi$$
$$308$$ 0 0
$$309$$ 15.7082 0.893609
$$310$$ 0 0
$$311$$ −30.7426 −1.74326 −0.871628 0.490168i $$-0.836935\pi$$
−0.871628 + 0.490168i $$0.836935\pi$$
$$312$$ 0 0
$$313$$ 17.1459 0.969143 0.484572 0.874752i $$-0.338975\pi$$
0.484572 + 0.874752i $$0.338975\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 23.8885 1.34171 0.670857 0.741587i $$-0.265926\pi$$
0.670857 + 0.741587i $$0.265926\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 6.47214 0.360119
$$324$$ 0 0
$$325$$ 10.0000 0.554700
$$326$$ 0 0
$$327$$ 31.8885 1.76344
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 13.3262 0.732476 0.366238 0.930521i $$-0.380646\pi$$
0.366238 + 0.930521i $$0.380646\pi$$
$$332$$ 0 0
$$333$$ −37.4164 −2.05041
$$334$$ 0 0
$$335$$ 22.0344 1.20387
$$336$$ 0 0
$$337$$ −13.4164 −0.730838 −0.365419 0.930843i $$-0.619074\pi$$
−0.365419 + 0.930843i $$0.619074\pi$$
$$338$$ 0 0
$$339$$ 33.8885 1.84057
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 7.23607 0.389577
$$346$$ 0 0
$$347$$ 25.8885 1.38977 0.694885 0.719121i $$-0.255455\pi$$
0.694885 + 0.719121i $$0.255455\pi$$
$$348$$ 0 0
$$349$$ 8.47214 0.453503 0.226752 0.973953i $$-0.427189\pi$$
0.226752 + 0.973953i $$0.427189\pi$$
$$350$$ 0 0
$$351$$ 2.76393 0.147528
$$352$$ 0 0
$$353$$ 34.0689 1.81330 0.906652 0.421880i $$-0.138630\pi$$
0.906652 + 0.421880i $$0.138630\pi$$
$$354$$ 0 0
$$355$$ −33.4164 −1.77356
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 10.8541 0.572858 0.286429 0.958102i $$-0.407532\pi$$
0.286429 + 0.958102i $$0.407532\pi$$
$$360$$ 0 0
$$361$$ 22.8885 1.20466
$$362$$ 0 0
$$363$$ −28.7984 −1.51152
$$364$$ 0 0
$$365$$ 13.9443 0.729877
$$366$$ 0 0
$$367$$ −13.3820 −0.698533 −0.349266 0.937023i $$-0.613569\pi$$
−0.349266 + 0.937023i $$0.613569\pi$$
$$368$$ 0 0
$$369$$ −30.2705 −1.57582
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −30.5066 −1.57957 −0.789785 0.613383i $$-0.789808\pi$$
−0.789785 + 0.613383i $$0.789808\pi$$
$$374$$ 0 0
$$375$$ 29.2705 1.51152
$$376$$ 0 0
$$377$$ 8.94427 0.460653
$$378$$ 0 0
$$379$$ −9.12461 −0.468700 −0.234350 0.972152i $$-0.575296\pi$$
−0.234350 + 0.972152i $$0.575296\pi$$
$$380$$ 0 0
$$381$$ −40.8885 −2.09478
$$382$$ 0 0
$$383$$ −9.88854 −0.505281 −0.252640 0.967560i $$-0.581299\pi$$
−0.252640 + 0.967560i $$0.581299\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −7.14590 −0.363246
$$388$$ 0 0
$$389$$ 27.5066 1.39464 0.697319 0.716761i $$-0.254376\pi$$
0.697319 + 0.716761i $$0.254376\pi$$
$$390$$ 0 0
$$391$$ 0.763932 0.0386337
$$392$$ 0 0
$$393$$ 16.9443 0.854725
$$394$$ 0 0
$$395$$ 42.3607 2.13140
$$396$$ 0 0
$$397$$ −17.3262 −0.869579 −0.434789 0.900532i $$-0.643177\pi$$
−0.434789 + 0.900532i $$0.643177\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 11.0557 0.552097 0.276048 0.961144i $$-0.410975\pi$$
0.276048 + 0.961144i $$0.410975\pi$$
$$402$$ 0 0
$$403$$ −13.0557 −0.650352
$$404$$ 0 0
$$405$$ −20.6525 −1.02623
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 16.0000 0.791149 0.395575 0.918434i $$-0.370545\pi$$
0.395575 + 0.918434i $$0.370545\pi$$
$$410$$ 0 0
$$411$$ 29.0344 1.43216
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −37.8885 −1.85988
$$416$$ 0 0
$$417$$ −35.6525 −1.74591
$$418$$ 0 0
$$419$$ 35.5066 1.73461 0.867305 0.497777i $$-0.165850\pi$$
0.867305 + 0.497777i $$0.165850\pi$$
$$420$$ 0 0
$$421$$ −10.1459 −0.494481 −0.247240 0.968954i $$-0.579524\pi$$
−0.247240 + 0.968954i $$0.579524\pi$$
$$422$$ 0 0
$$423$$ −26.7639 −1.30131
$$424$$ 0 0
$$425$$ 8.09017 0.392431
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −2.94427 −0.141821 −0.0709103 0.997483i $$-0.522590\pi$$
−0.0709103 + 0.997483i $$0.522590\pi$$
$$432$$ 0 0
$$433$$ 30.0000 1.44171 0.720854 0.693087i $$-0.243750\pi$$
0.720854 + 0.693087i $$0.243750\pi$$
$$434$$ 0 0
$$435$$ 68.5410 3.28629
$$436$$ 0 0
$$437$$ 4.94427 0.236517
$$438$$ 0 0
$$439$$ −16.1459 −0.770602 −0.385301 0.922791i $$-0.625902\pi$$
−0.385301 + 0.922791i $$0.625902\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −7.05573 −0.335228 −0.167614 0.985853i $$-0.553606\pi$$
−0.167614 + 0.985853i $$0.553606\pi$$
$$444$$ 0 0
$$445$$ 31.7082 1.50311
$$446$$ 0 0
$$447$$ −8.85410 −0.418785
$$448$$ 0 0
$$449$$ 0.944272 0.0445629 0.0222815 0.999752i $$-0.492907\pi$$
0.0222815 + 0.999752i $$0.492907\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ −55.6869 −2.61640
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 12.2705 0.573990 0.286995 0.957932i $$-0.407344\pi$$
0.286995 + 0.957932i $$0.407344\pi$$
$$458$$ 0 0
$$459$$ 2.23607 0.104371
$$460$$ 0 0
$$461$$ −26.8328 −1.24973 −0.624864 0.780733i $$-0.714846\pi$$
−0.624864 + 0.780733i $$0.714846\pi$$
$$462$$ 0 0
$$463$$ −2.79837 −0.130051 −0.0650257 0.997884i $$-0.520713\pi$$
−0.0650257 + 0.997884i $$0.520713\pi$$
$$464$$ 0 0
$$465$$ −100.048 −4.63960
$$466$$ 0 0
$$467$$ −28.7639 −1.33104 −0.665518 0.746382i $$-0.731789\pi$$
−0.665518 + 0.746382i $$0.731789\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 62.0689 2.85998
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 52.3607 2.40247
$$476$$ 0 0
$$477$$ −12.8197 −0.586972
$$478$$ 0 0
$$479$$ −10.6180 −0.485150 −0.242575 0.970133i $$-0.577992\pi$$
−0.242575 + 0.970133i $$0.577992\pi$$
$$480$$ 0 0
$$481$$ −12.0000 −0.547153
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −10.3262 −0.468890
$$486$$ 0 0
$$487$$ 19.0557 0.863497 0.431749 0.901994i $$-0.357897\pi$$
0.431749 + 0.901994i $$0.357897\pi$$
$$488$$ 0 0
$$489$$ −15.7082 −0.710350
$$490$$ 0 0
$$491$$ −39.7984 −1.79608 −0.898038 0.439918i $$-0.855007\pi$$
−0.898038 + 0.439918i $$0.855007\pi$$
$$492$$ 0 0
$$493$$ 7.23607 0.325896
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −35.3050 −1.58047 −0.790233 0.612806i $$-0.790041\pi$$
−0.790233 + 0.612806i $$0.790041\pi$$
$$500$$ 0 0
$$501$$ −1.14590 −0.0511949
$$502$$ 0 0
$$503$$ 12.2705 0.547115 0.273557 0.961856i $$-0.411800\pi$$
0.273557 + 0.961856i $$0.411800\pi$$
$$504$$ 0 0
$$505$$ 27.2361 1.21199
$$506$$ 0 0
$$507$$ −30.0344 −1.33388
$$508$$ 0 0
$$509$$ −23.2361 −1.02992 −0.514960 0.857214i $$-0.672193\pi$$
−0.514960 + 0.857214i $$0.672193\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 14.4721 0.638960
$$514$$ 0 0
$$515$$ 21.7082 0.956578
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 58.4508 2.56571
$$520$$ 0 0
$$521$$ −13.0902 −0.573491 −0.286745 0.958007i $$-0.592573\pi$$
−0.286745 + 0.958007i $$0.592573\pi$$
$$522$$ 0 0
$$523$$ −7.34752 −0.321285 −0.160642 0.987013i $$-0.551357\pi$$
−0.160642 + 0.987013i $$0.551357\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −10.5623 −0.460101
$$528$$ 0 0
$$529$$ −22.4164 −0.974626
$$530$$ 0 0
$$531$$ 2.94427 0.127771
$$532$$ 0 0
$$533$$ −9.70820 −0.420509
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −8.09017 −0.349117
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −30.0000 −1.28980 −0.644900 0.764267i $$-0.723101\pi$$
−0.644900 + 0.764267i $$0.723101\pi$$
$$542$$ 0 0
$$543$$ 36.6525 1.57291
$$544$$ 0 0
$$545$$ 44.0689 1.88770
$$546$$ 0 0
$$547$$ −13.1246 −0.561168 −0.280584 0.959829i $$-0.590528\pi$$
−0.280584 + 0.959829i $$0.590528\pi$$
$$548$$ 0 0
$$549$$ −29.1459 −1.24392
$$550$$ 0 0
$$551$$ 46.8328 1.99515
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −91.9574 −3.90338
$$556$$ 0 0
$$557$$ 24.4721 1.03692 0.518459 0.855103i $$-0.326506\pi$$
0.518459 + 0.855103i $$0.326506\pi$$
$$558$$ 0 0
$$559$$ −2.29180 −0.0969326
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −17.1246 −0.721716 −0.360858 0.932621i $$-0.617516\pi$$
−0.360858 + 0.932621i $$0.617516\pi$$
$$564$$ 0 0
$$565$$ 46.8328 1.97027
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 7.09017 0.297235 0.148618 0.988895i $$-0.452518\pi$$
0.148618 + 0.988895i $$0.452518\pi$$
$$570$$ 0 0
$$571$$ 28.6525 1.19907 0.599534 0.800349i $$-0.295352\pi$$
0.599534 + 0.800349i $$0.295352\pi$$
$$572$$ 0 0
$$573$$ 57.8328 2.41600
$$574$$ 0 0
$$575$$ 6.18034 0.257738
$$576$$ 0 0
$$577$$ 45.4164 1.89071 0.945355 0.326043i $$-0.105715\pi$$
0.945355 + 0.326043i $$0.105715\pi$$
$$578$$ 0 0
$$579$$ −14.4721 −0.601441
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 17.2361 0.712624
$$586$$ 0 0
$$587$$ −30.0000 −1.23823 −0.619116 0.785299i $$-0.712509\pi$$
−0.619116 + 0.785299i $$0.712509\pi$$
$$588$$ 0 0
$$589$$ −68.3607 −2.81675
$$590$$ 0 0
$$591$$ −41.1246 −1.69164
$$592$$ 0 0
$$593$$ 29.5967 1.21539 0.607696 0.794169i $$-0.292094\pi$$
0.607696 + 0.794169i $$0.292094\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 31.7984 1.30142
$$598$$ 0 0
$$599$$ −12.0344 −0.491714 −0.245857 0.969306i $$-0.579069\pi$$
−0.245857 + 0.969306i $$0.579069\pi$$
$$600$$ 0 0
$$601$$ 23.5279 0.959722 0.479861 0.877345i $$-0.340687\pi$$
0.479861 + 0.877345i $$0.340687\pi$$
$$602$$ 0 0
$$603$$ 23.4721 0.955859
$$604$$ 0 0
$$605$$ −39.7984 −1.61803
$$606$$ 0 0
$$607$$ −24.8541 −1.00880 −0.504398 0.863471i $$-0.668286\pi$$
−0.504398 + 0.863471i $$0.668286\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −8.58359 −0.347255
$$612$$ 0 0
$$613$$ −0.854102 −0.0344969 −0.0172484 0.999851i $$-0.505491\pi$$
−0.0172484 + 0.999851i $$0.505491\pi$$
$$614$$ 0 0
$$615$$ −74.3951 −2.99990
$$616$$ 0 0
$$617$$ 45.5967 1.83566 0.917828 0.396978i $$-0.129941\pi$$
0.917828 + 0.396978i $$0.129941\pi$$
$$618$$ 0 0
$$619$$ 46.8328 1.88237 0.941185 0.337892i $$-0.109714\pi$$
0.941185 + 0.337892i $$0.109714\pi$$
$$620$$ 0 0
$$621$$ 1.70820 0.0685479
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −9.70820 −0.387091
$$630$$ 0 0
$$631$$ 15.9787 0.636103 0.318051 0.948074i $$-0.396972\pi$$
0.318051 + 0.948074i $$0.396972\pi$$
$$632$$ 0 0
$$633$$ 61.7771 2.45542
$$634$$ 0 0
$$635$$ −56.5066 −2.24240
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −35.5967 −1.40819
$$640$$ 0 0
$$641$$ 41.7771 1.65010 0.825048 0.565063i $$-0.191148\pi$$
0.825048 + 0.565063i $$0.191148\pi$$
$$642$$ 0 0
$$643$$ 29.3820 1.15871 0.579356 0.815075i $$-0.303304\pi$$
0.579356 + 0.815075i $$0.303304\pi$$
$$644$$ 0 0
$$645$$ −17.5623 −0.691515
$$646$$ 0 0
$$647$$ 27.5967 1.08494 0.542470 0.840075i $$-0.317489\pi$$
0.542470 + 0.840075i $$0.317489\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −25.2361 −0.987564 −0.493782 0.869586i $$-0.664386\pi$$
−0.493782 + 0.869586i $$0.664386\pi$$
$$654$$ 0 0
$$655$$ 23.4164 0.914955
$$656$$ 0 0
$$657$$ 14.8541 0.579514
$$658$$ 0 0
$$659$$ −9.90983 −0.386032 −0.193016 0.981196i $$-0.561827\pi$$
−0.193016 + 0.981196i $$0.561827\pi$$
$$660$$ 0 0
$$661$$ −4.58359 −0.178281 −0.0891405 0.996019i $$-0.528412\pi$$
−0.0891405 + 0.996019i $$0.528412\pi$$
$$662$$ 0 0
$$663$$ 3.23607 0.125678
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 5.52786 0.214040
$$668$$ 0 0
$$669$$ 41.5967 1.60822
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 0.944272 0.0363990 0.0181995 0.999834i $$-0.494207\pi$$
0.0181995 + 0.999834i $$0.494207\pi$$
$$674$$ 0 0
$$675$$ 18.0902 0.696291
$$676$$ 0 0
$$677$$ −20.8328 −0.800670 −0.400335 0.916369i $$-0.631106\pi$$
−0.400335 + 0.916369i $$0.631106\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −18.7082 −0.716900
$$682$$ 0 0
$$683$$ −45.8885 −1.75588 −0.877938 0.478774i $$-0.841081\pi$$
−0.877938 + 0.478774i $$0.841081\pi$$
$$684$$ 0 0
$$685$$ 40.1246 1.53308
$$686$$ 0 0
$$687$$ −45.5967 −1.73962
$$688$$ 0 0
$$689$$ −4.11146 −0.156634
$$690$$ 0 0
$$691$$ 10.9787 0.417650 0.208825 0.977953i $$-0.433036\pi$$
0.208825 + 0.977953i $$0.433036\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −49.2705 −1.86894
$$696$$ 0 0
$$697$$ −7.85410 −0.297495
$$698$$ 0 0
$$699$$ −5.70820 −0.215904
$$700$$ 0 0
$$701$$ −23.3050 −0.880216 −0.440108 0.897945i $$-0.645060\pi$$
−0.440108 + 0.897945i $$0.645060\pi$$
$$702$$ 0 0
$$703$$ −62.8328 −2.36978
$$704$$ 0 0
$$705$$ −65.7771 −2.47731
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −2.65248 −0.0996158 −0.0498079 0.998759i $$-0.515861\pi$$
−0.0498079 + 0.998759i $$0.515861\pi$$
$$710$$ 0 0
$$711$$ 45.1246 1.69231
$$712$$ 0 0
$$713$$ −8.06888 −0.302182
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −50.7426 −1.89502
$$718$$ 0 0
$$719$$ −10.9098 −0.406868 −0.203434 0.979089i $$-0.565210\pi$$
−0.203434 + 0.979089i $$0.565210\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 9.47214 0.352273
$$724$$ 0 0
$$725$$ 58.5410 2.17416
$$726$$ 0 0
$$727$$ −13.7082 −0.508409 −0.254205 0.967150i $$-0.581814\pi$$
−0.254205 + 0.967150i $$0.581814\pi$$
$$728$$ 0 0
$$729$$ −39.5623 −1.46527
$$730$$ 0 0
$$731$$ −1.85410 −0.0685764
$$732$$ 0 0
$$733$$ 46.2492 1.70825 0.854127 0.520064i $$-0.174092\pi$$
0.854127 + 0.520064i $$0.174092\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 7.38197 0.271550 0.135775 0.990740i $$-0.456648\pi$$
0.135775 + 0.990740i $$0.456648\pi$$
$$740$$ 0 0
$$741$$ 20.9443 0.769407
$$742$$ 0 0
$$743$$ 0.111456 0.00408893 0.00204447 0.999998i $$-0.499349\pi$$
0.00204447 + 0.999998i $$0.499349\pi$$
$$744$$ 0 0
$$745$$ −12.2361 −0.448295
$$746$$ 0 0
$$747$$ −40.3607 −1.47672
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 50.8328 1.85492 0.927458 0.373928i $$-0.121989\pi$$
0.927458 + 0.373928i $$0.121989\pi$$
$$752$$ 0 0
$$753$$ −52.8328 −1.92533
$$754$$ 0 0
$$755$$ −76.9574 −2.80077
$$756$$ 0 0
$$757$$ −5.56231 −0.202165 −0.101083 0.994878i $$-0.532231\pi$$
−0.101083 + 0.994878i $$0.532231\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 18.1115 0.656540 0.328270 0.944584i $$-0.393534\pi$$
0.328270 + 0.944584i $$0.393534\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 13.9443 0.504156
$$766$$ 0 0
$$767$$ 0.944272 0.0340957
$$768$$ 0 0
$$769$$ 35.5967 1.28365 0.641826 0.766850i $$-0.278177\pi$$
0.641826 + 0.766850i $$0.278177\pi$$
$$770$$ 0 0
$$771$$ 4.00000 0.144056
$$772$$ 0 0
$$773$$ −11.5279 −0.414628 −0.207314 0.978274i $$-0.566472\pi$$
−0.207314 + 0.978274i $$0.566472\pi$$
$$774$$ 0 0
$$775$$ −85.4508 −3.06949
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −50.8328 −1.82127
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 16.1803 0.578238
$$784$$ 0 0
$$785$$ 85.7771 3.06152
$$786$$ 0 0
$$787$$ 8.00000 0.285169 0.142585 0.989783i $$-0.454459\pi$$
0.142585 + 0.989783i $$0.454459\pi$$
$$788$$ 0 0
$$789$$ 50.8328 1.80970
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −9.34752 −0.331940
$$794$$ 0 0
$$795$$ −31.5066 −1.11742
$$796$$ 0 0
$$797$$ 12.0000 0.425062 0.212531 0.977154i $$-0.431829\pi$$
0.212531 + 0.977154i $$0.431829\pi$$
$$798$$ 0 0
$$799$$ −6.94427 −0.245671
$$800$$ 0 0
$$801$$ 33.7771 1.19345
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 35.1246 1.23644
$$808$$ 0 0
$$809$$ −41.1246 −1.44586 −0.722932 0.690919i $$-0.757206\pi$$
−0.722932 + 0.690919i $$0.757206\pi$$
$$810$$ 0 0
$$811$$ −14.9787 −0.525974 −0.262987 0.964799i $$-0.584708\pi$$
−0.262987 + 0.964799i $$0.584708\pi$$
$$812$$ 0 0
$$813$$ −53.5967 −1.87972
$$814$$ 0 0
$$815$$ −21.7082 −0.760405
$$816$$ 0 0
$$817$$ −12.0000 −0.419827
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 37.5279 1.30973 0.654866 0.755745i $$-0.272725\pi$$
0.654866 + 0.755745i $$0.272725\pi$$
$$822$$ 0 0
$$823$$ −42.5410 −1.48289 −0.741443 0.671015i $$-0.765858\pi$$
−0.741443 + 0.671015i $$0.765858\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −36.7639 −1.27841 −0.639204 0.769038i $$-0.720736\pi$$
−0.639204 + 0.769038i $$0.720736\pi$$
$$828$$ 0 0
$$829$$ −11.5967 −0.402772 −0.201386 0.979512i $$-0.564545\pi$$
−0.201386 + 0.979512i $$0.564545\pi$$
$$830$$ 0 0
$$831$$ −81.4853 −2.82669
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −1.58359 −0.0548025
$$836$$ 0 0
$$837$$ −23.6180 −0.816359
$$838$$ 0 0
$$839$$ −32.9443 −1.13736 −0.568681 0.822558i $$-0.692546\pi$$
−0.568681 + 0.822558i $$0.692546\pi$$
$$840$$ 0 0
$$841$$ 23.3607 0.805541
$$842$$ 0 0
$$843$$ −61.5410 −2.11959
$$844$$ 0 0
$$845$$ −41.5066 −1.42787
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 50.1246 1.72027
$$850$$ 0 0
$$851$$ −7.41641 −0.254231
$$852$$ 0 0
$$853$$ −25.7771 −0.882591 −0.441295 0.897362i $$-0.645481\pi$$
−0.441295 + 0.897362i $$0.645481\pi$$
$$854$$ 0 0
$$855$$ 90.2492 3.08646
$$856$$ 0 0
$$857$$ −20.2016 −0.690074 −0.345037 0.938589i $$-0.612134\pi$$
−0.345037 + 0.938589i $$0.612134\pi$$
$$858$$ 0 0
$$859$$ 16.8328 0.574328 0.287164 0.957881i $$-0.407287\pi$$
0.287164 + 0.957881i $$0.407287\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −10.6738 −0.363339 −0.181670 0.983360i $$-0.558150\pi$$
−0.181670 + 0.983360i $$0.558150\pi$$
$$864$$ 0 0
$$865$$ 80.7771 2.74650
$$866$$ 0 0
$$867$$ 2.61803 0.0889131
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 7.52786 0.255072
$$872$$ 0 0
$$873$$ −11.0000 −0.372294
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 45.3050 1.52984 0.764920 0.644126i $$-0.222779\pi$$
0.764920 + 0.644126i $$0.222779\pi$$
$$878$$ 0 0
$$879$$ −72.2492 −2.43691
$$880$$ 0 0
$$881$$ 48.1591 1.62252 0.811260 0.584686i $$-0.198782\pi$$
0.811260 + 0.584686i $$0.198782\pi$$
$$882$$ 0 0
$$883$$ 45.6869 1.53749 0.768744 0.639557i $$-0.220882\pi$$
0.768744 + 0.639557i $$0.220882\pi$$
$$884$$ 0 0
$$885$$ 7.23607 0.243238
$$886$$ 0 0
$$887$$ 30.7426 1.03224 0.516119 0.856517i $$-0.327376\pi$$
0.516119 + 0.856517i $$0.327376\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −44.9443 −1.50400
$$894$$ 0 0
$$895$$ −11.1803 −0.373718
$$896$$ 0 0
$$897$$ 2.47214 0.0825422
$$898$$ 0 0
$$899$$ −76.4296 −2.54907
$$900$$ 0 0
$$901$$ −3.32624 −0.110813
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 50.6525 1.68375
$$906$$ 0 0
$$907$$ −13.8197 −0.458874 −0.229437 0.973323i $$-0.573689\pi$$
−0.229437 + 0.973323i $$0.573689\pi$$
$$908$$ 0 0
$$909$$ 29.0132 0.962306
$$910$$ 0 0
$$911$$ −38.7214 −1.28290 −0.641448 0.767167i $$-0.721666\pi$$
−0.641448 + 0.767167i $$0.721666\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ −71.6312 −2.36805
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 36.6869 1.21019 0.605095 0.796153i $$-0.293135\pi$$
0.605095 + 0.796153i $$0.293135\pi$$
$$920$$ 0 0
$$921$$ −24.6525 −0.812327
$$922$$ 0 0
$$923$$ −11.4164 −0.375776
$$924$$ 0 0
$$925$$ −78.5410 −2.58241
$$926$$ 0 0
$$927$$ 23.1246 0.759512
$$928$$ 0 0
$$929$$ −20.2016 −0.662794 −0.331397 0.943491i $$-0.607520\pi$$
−0.331397 + 0.943491i $$0.607520\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −80.4853 −2.63497
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 4.58359 0.149739 0.0748697 0.997193i $$-0.476146\pi$$
0.0748697 + 0.997193i $$0.476146\pi$$
$$938$$ 0 0
$$939$$ 44.8885 1.46488
$$940$$ 0 0
$$941$$ −19.2016 −0.625955 −0.312978 0.949761i $$-0.601326\pi$$
−0.312978 + 0.949761i $$0.601326\pi$$
$$942$$ 0 0
$$943$$ −6.00000 −0.195387
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −41.1935 −1.33861 −0.669304 0.742988i $$-0.733408\pi$$
−0.669304 + 0.742988i $$0.733408\pi$$
$$948$$ 0 0
$$949$$ 4.76393 0.154644
$$950$$ 0 0
$$951$$ 62.5410 2.02803
$$952$$ 0 0
$$953$$ −19.9098 −0.644943 −0.322471 0.946579i $$-0.604514\pi$$
−0.322471 + 0.946579i $$0.604514\pi$$
$$954$$ 0 0
$$955$$ 79.9230 2.58625
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 80.5623 2.59878
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −20.0000 −0.643823
$$966$$ 0 0
$$967$$ −24.9098 −0.801046 −0.400523 0.916287i $$-0.631172\pi$$
−0.400523 + 0.916287i $$0.631172\pi$$
$$968$$ 0 0
$$969$$ 16.9443 0.544328
$$970$$ 0 0
$$971$$ 22.7639 0.730529 0.365265 0.930904i $$-0.380978\pi$$
0.365265 + 0.930904i $$0.380978\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 26.1803 0.838442
$$976$$ 0 0
$$977$$ −36.4508 −1.16617 −0.583083 0.812413i $$-0.698154\pi$$
−0.583083 + 0.812413i $$0.698154\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 46.9443 1.49882
$$982$$ 0 0
$$983$$ 0.270510 0.00862792 0.00431396 0.999991i $$-0.498627\pi$$
0.00431396 + 0.999991i $$0.498627\pi$$
$$984$$ 0 0
$$985$$ −56.8328 −1.81084
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −1.41641 −0.0450391
$$990$$ 0 0
$$991$$ 8.54102 0.271314 0.135657 0.990756i $$-0.456685\pi$$
0.135657 + 0.990756i $$0.456685\pi$$
$$992$$ 0 0
$$993$$ 34.8885 1.10715
$$994$$ 0 0
$$995$$ 43.9443 1.39313
$$996$$ 0 0
$$997$$ 29.7426 0.941959 0.470980 0.882144i $$-0.343901\pi$$
0.470980 + 0.882144i $$0.343901\pi$$
$$998$$ 0 0
$$999$$ −21.7082 −0.686817
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 3332.2.a.o.1.2 yes 2
7.6 odd 2 3332.2.a.g.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
3332.2.a.g.1.1 2 7.6 odd 2
3332.2.a.o.1.2 yes 2 1.1 even 1 trivial