# Properties

 Label 1900.2.c.d.1749.2 Level $1900$ Weight $2$ Character 1900.1749 Analytic conductor $15.172$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,2,Mod(1749,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1900.1749");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1900 = 2^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1900.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$15.1715763840$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 380) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1749.2 Root $$0.707107 - 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 1900.1749 Dual form 1900.2.c.d.1749.3

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.585786i q^{3} +4.82843i q^{7} +2.65685 q^{9} +O(q^{10})$$ $$q-0.585786i q^{3} +4.82843i q^{7} +2.65685 q^{9} -2.00000 q^{11} +2.24264i q^{13} +4.82843i q^{17} +1.00000 q^{19} +2.82843 q^{21} -6.00000i q^{23} -3.31371i q^{27} -10.4853 q^{29} -1.17157 q^{31} +1.17157i q^{33} +10.2426i q^{37} +1.31371 q^{39} -7.65685 q^{41} -0.828427i q^{43} -0.828427i q^{47} -16.3137 q^{49} +2.82843 q^{51} -5.07107i q^{53} -0.585786i q^{57} -2.34315 q^{59} -2.34315 q^{61} +12.8284i q^{63} +0.585786i q^{67} -3.51472 q^{69} +10.8284 q^{71} +14.4853i q^{73} -9.65685i q^{77} +9.65685 q^{79} +6.02944 q^{81} +9.31371i q^{83} +6.14214i q^{87} -10.4853 q^{89} -10.8284 q^{91} +0.686292i q^{93} +1.75736i q^{97} -5.31371 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 12 q^{9}+O(q^{10})$$ 4 * q - 12 * q^9 $$4 q - 12 q^{9} - 8 q^{11} + 4 q^{19} - 8 q^{29} - 16 q^{31} - 40 q^{39} - 8 q^{41} - 20 q^{49} - 32 q^{59} - 32 q^{61} - 48 q^{69} + 32 q^{71} + 16 q^{79} + 92 q^{81} - 8 q^{89} - 32 q^{91} + 24 q^{99}+O(q^{100})$$ 4 * q - 12 * q^9 - 8 * q^11 + 4 * q^19 - 8 * q^29 - 16 * q^31 - 40 * q^39 - 8 * q^41 - 20 * q^49 - 32 * q^59 - 32 * q^61 - 48 * q^69 + 32 * q^71 + 16 * q^79 + 92 * q^81 - 8 * q^89 - 32 * q^91 + 24 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$951$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ − 0.585786i − 0.338204i −0.985599 0.169102i $$-0.945913\pi$$
0.985599 0.169102i $$-0.0540867\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 4.82843i 1.82497i 0.409106 + 0.912487i $$0.365841\pi$$
−0.409106 + 0.912487i $$0.634159\pi$$
$$8$$ 0 0
$$9$$ 2.65685 0.885618
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 0 0
$$13$$ 2.24264i 0.621997i 0.950410 + 0.310998i $$0.100663\pi$$
−0.950410 + 0.310998i $$0.899337\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 4.82843i 1.17107i 0.810649 + 0.585533i $$0.199115\pi$$
−0.810649 + 0.585533i $$0.800885\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 2.82843 0.617213
$$22$$ 0 0
$$23$$ − 6.00000i − 1.25109i −0.780189 0.625543i $$-0.784877\pi$$
0.780189 0.625543i $$-0.215123\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 3.31371i − 0.637723i
$$28$$ 0 0
$$29$$ −10.4853 −1.94707 −0.973534 0.228543i $$-0.926604\pi$$
−0.973534 + 0.228543i $$0.926604\pi$$
$$30$$ 0 0
$$31$$ −1.17157 −0.210421 −0.105210 0.994450i $$-0.533552\pi$$
−0.105210 + 0.994450i $$0.533552\pi$$
$$32$$ 0 0
$$33$$ 1.17157i 0.203945i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 10.2426i 1.68388i 0.539571 + 0.841940i $$0.318586\pi$$
−0.539571 + 0.841940i $$0.681414\pi$$
$$38$$ 0 0
$$39$$ 1.31371 0.210362
$$40$$ 0 0
$$41$$ −7.65685 −1.19580 −0.597900 0.801571i $$-0.703998\pi$$
−0.597900 + 0.801571i $$0.703998\pi$$
$$42$$ 0 0
$$43$$ − 0.828427i − 0.126334i −0.998003 0.0631670i $$-0.979880\pi$$
0.998003 0.0631670i $$-0.0201201\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 0.828427i − 0.120839i −0.998173 0.0604193i $$-0.980756\pi$$
0.998173 0.0604193i $$-0.0192438\pi$$
$$48$$ 0 0
$$49$$ −16.3137 −2.33053
$$50$$ 0 0
$$51$$ 2.82843 0.396059
$$52$$ 0 0
$$53$$ − 5.07107i − 0.696565i −0.937390 0.348282i $$-0.886765\pi$$
0.937390 0.348282i $$-0.113235\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 0.585786i − 0.0775893i
$$58$$ 0 0
$$59$$ −2.34315 −0.305052 −0.152526 0.988299i $$-0.548741\pi$$
−0.152526 + 0.988299i $$0.548741\pi$$
$$60$$ 0 0
$$61$$ −2.34315 −0.300009 −0.150005 0.988685i $$-0.547929\pi$$
−0.150005 + 0.988685i $$0.547929\pi$$
$$62$$ 0 0
$$63$$ 12.8284i 1.61623i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0.585786i 0.0715652i 0.999360 + 0.0357826i $$0.0113924\pi$$
−0.999360 + 0.0357826i $$0.988608\pi$$
$$68$$ 0 0
$$69$$ −3.51472 −0.423122
$$70$$ 0 0
$$71$$ 10.8284 1.28510 0.642549 0.766245i $$-0.277877\pi$$
0.642549 + 0.766245i $$0.277877\pi$$
$$72$$ 0 0
$$73$$ 14.4853i 1.69537i 0.530497 + 0.847687i $$0.322005\pi$$
−0.530497 + 0.847687i $$0.677995\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 9.65685i − 1.10050i
$$78$$ 0 0
$$79$$ 9.65685 1.08648 0.543240 0.839577i $$-0.317197\pi$$
0.543240 + 0.839577i $$0.317197\pi$$
$$80$$ 0 0
$$81$$ 6.02944 0.669937
$$82$$ 0 0
$$83$$ 9.31371i 1.02231i 0.859488 + 0.511156i $$0.170783\pi$$
−0.859488 + 0.511156i $$0.829217\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 6.14214i 0.658506i
$$88$$ 0 0
$$89$$ −10.4853 −1.11144 −0.555719 0.831370i $$-0.687557\pi$$
−0.555719 + 0.831370i $$0.687557\pi$$
$$90$$ 0 0
$$91$$ −10.8284 −1.13513
$$92$$ 0 0
$$93$$ 0.686292i 0.0711651i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 1.75736i 0.178433i 0.996012 + 0.0892164i $$0.0284363\pi$$
−0.996012 + 0.0892164i $$0.971564\pi$$
$$98$$ 0 0
$$99$$ −5.31371 −0.534048
$$100$$ 0 0
$$101$$ 4.00000 0.398015 0.199007 0.979998i $$-0.436228\pi$$
0.199007 + 0.979998i $$0.436228\pi$$
$$102$$ 0 0
$$103$$ − 15.8995i − 1.56662i −0.621629 0.783312i $$-0.713529\pi$$
0.621629 0.783312i $$-0.286471\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 4.58579i − 0.443325i −0.975123 0.221662i $$-0.928852\pi$$
0.975123 0.221662i $$-0.0711483\pi$$
$$108$$ 0 0
$$109$$ 8.82843 0.845610 0.422805 0.906221i $$-0.361046\pi$$
0.422805 + 0.906221i $$0.361046\pi$$
$$110$$ 0 0
$$111$$ 6.00000 0.569495
$$112$$ 0 0
$$113$$ 5.07107i 0.477046i 0.971137 + 0.238523i $$0.0766632\pi$$
−0.971137 + 0.238523i $$0.923337\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 5.95837i 0.550851i
$$118$$ 0 0
$$119$$ −23.3137 −2.13716
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ 4.48528i 0.404424i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 21.5563i 1.91282i 0.292033 + 0.956408i $$0.405668\pi$$
−0.292033 + 0.956408i $$0.594332\pi$$
$$128$$ 0 0
$$129$$ −0.485281 −0.0427266
$$130$$ 0 0
$$131$$ −5.65685 −0.494242 −0.247121 0.968985i $$-0.579484\pi$$
−0.247121 + 0.968985i $$0.579484\pi$$
$$132$$ 0 0
$$133$$ 4.82843i 0.418678i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 20.1421i 1.72086i 0.509570 + 0.860429i $$0.329805\pi$$
−0.509570 + 0.860429i $$0.670195\pi$$
$$138$$ 0 0
$$139$$ 17.3137 1.46853 0.734265 0.678863i $$-0.237527\pi$$
0.734265 + 0.678863i $$0.237527\pi$$
$$140$$ 0 0
$$141$$ −0.485281 −0.0408681
$$142$$ 0 0
$$143$$ − 4.48528i − 0.375078i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 9.55635i 0.788194i
$$148$$ 0 0
$$149$$ 8.00000 0.655386 0.327693 0.944784i $$-0.393729\pi$$
0.327693 + 0.944784i $$0.393729\pi$$
$$150$$ 0 0
$$151$$ 18.1421 1.47639 0.738193 0.674590i $$-0.235679\pi$$
0.738193 + 0.674590i $$0.235679\pi$$
$$152$$ 0 0
$$153$$ 12.8284i 1.03712i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 3.17157i − 0.253119i −0.991959 0.126560i $$-0.959607\pi$$
0.991959 0.126560i $$-0.0403935\pi$$
$$158$$ 0 0
$$159$$ −2.97056 −0.235581
$$160$$ 0 0
$$161$$ 28.9706 2.28320
$$162$$ 0 0
$$163$$ − 2.48528i − 0.194662i −0.995252 0.0973311i $$-0.968969\pi$$
0.995252 0.0973311i $$-0.0310306\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 10.7279i 0.830152i 0.909787 + 0.415076i $$0.136245\pi$$
−0.909787 + 0.415076i $$0.863755\pi$$
$$168$$ 0 0
$$169$$ 7.97056 0.613120
$$170$$ 0 0
$$171$$ 2.65685 0.203175
$$172$$ 0 0
$$173$$ − 3.89949i − 0.296473i −0.988952 0.148237i $$-0.952640\pi$$
0.988952 0.148237i $$-0.0473597\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 1.37258i 0.103170i
$$178$$ 0 0
$$179$$ −21.6569 −1.61871 −0.809355 0.587320i $$-0.800183\pi$$
−0.809355 + 0.587320i $$0.800183\pi$$
$$180$$ 0 0
$$181$$ −18.0000 −1.33793 −0.668965 0.743294i $$-0.733262\pi$$
−0.668965 + 0.743294i $$0.733262\pi$$
$$182$$ 0 0
$$183$$ 1.37258i 0.101464i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 9.65685i − 0.706179i
$$188$$ 0 0
$$189$$ 16.0000 1.16383
$$190$$ 0 0
$$191$$ 4.00000 0.289430 0.144715 0.989473i $$-0.453773\pi$$
0.144715 + 0.989473i $$0.453773\pi$$
$$192$$ 0 0
$$193$$ 0.585786i 0.0421658i 0.999778 + 0.0210829i $$0.00671140\pi$$
−0.999778 + 0.0210829i $$0.993289\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 5.31371i 0.378586i 0.981921 + 0.189293i $$0.0606196\pi$$
−0.981921 + 0.189293i $$0.939380\pi$$
$$198$$ 0 0
$$199$$ 10.3431 0.733206 0.366603 0.930377i $$-0.380521\pi$$
0.366603 + 0.930377i $$0.380521\pi$$
$$200$$ 0 0
$$201$$ 0.343146 0.0242036
$$202$$ 0 0
$$203$$ − 50.6274i − 3.55335i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 15.9411i − 1.10798i
$$208$$ 0 0
$$209$$ −2.00000 −0.138343
$$210$$ 0 0
$$211$$ −11.5147 −0.792706 −0.396353 0.918098i $$-0.629724\pi$$
−0.396353 + 0.918098i $$0.629724\pi$$
$$212$$ 0 0
$$213$$ − 6.34315i − 0.434625i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 5.65685i − 0.384012i
$$218$$ 0 0
$$219$$ 8.48528 0.573382
$$220$$ 0 0
$$221$$ −10.8284 −0.728399
$$222$$ 0 0
$$223$$ 19.2132i 1.28661i 0.765609 + 0.643306i $$0.222438\pi$$
−0.765609 + 0.643306i $$0.777562\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 20.5858i − 1.36633i −0.730266 0.683163i $$-0.760604\pi$$
0.730266 0.683163i $$-0.239396\pi$$
$$228$$ 0 0
$$229$$ 21.6569 1.43113 0.715563 0.698549i $$-0.246170\pi$$
0.715563 + 0.698549i $$0.246170\pi$$
$$230$$ 0 0
$$231$$ −5.65685 −0.372194
$$232$$ 0 0
$$233$$ 10.0000i 0.655122i 0.944830 + 0.327561i $$0.106227\pi$$
−0.944830 + 0.327561i $$0.893773\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 5.65685i − 0.367452i
$$238$$ 0 0
$$239$$ −10.3431 −0.669042 −0.334521 0.942388i $$-0.608575\pi$$
−0.334521 + 0.942388i $$0.608575\pi$$
$$240$$ 0 0
$$241$$ −30.2843 −1.95078 −0.975391 0.220484i $$-0.929236\pi$$
−0.975391 + 0.220484i $$0.929236\pi$$
$$242$$ 0 0
$$243$$ − 13.4731i − 0.864299i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 2.24264i 0.142696i
$$248$$ 0 0
$$249$$ 5.45584 0.345750
$$250$$ 0 0
$$251$$ −13.6569 −0.862013 −0.431006 0.902349i $$-0.641841\pi$$
−0.431006 + 0.902349i $$0.641841\pi$$
$$252$$ 0 0
$$253$$ 12.0000i 0.754434i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 2.24264i 0.139892i 0.997551 + 0.0699460i $$0.0222827\pi$$
−0.997551 + 0.0699460i $$0.977717\pi$$
$$258$$ 0 0
$$259$$ −49.4558 −3.07304
$$260$$ 0 0
$$261$$ −27.8579 −1.72436
$$262$$ 0 0
$$263$$ 3.65685i 0.225491i 0.993624 + 0.112746i $$0.0359645\pi$$
−0.993624 + 0.112746i $$0.964035\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 6.14214i 0.375893i
$$268$$ 0 0
$$269$$ 22.4853 1.37095 0.685476 0.728095i $$-0.259594\pi$$
0.685476 + 0.728095i $$0.259594\pi$$
$$270$$ 0 0
$$271$$ 23.6569 1.43705 0.718526 0.695500i $$-0.244817\pi$$
0.718526 + 0.695500i $$0.244817\pi$$
$$272$$ 0 0
$$273$$ 6.34315i 0.383905i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 16.8284i − 1.01112i −0.862791 0.505561i $$-0.831286\pi$$
0.862791 0.505561i $$-0.168714\pi$$
$$278$$ 0 0
$$279$$ −3.11270 −0.186352
$$280$$ 0 0
$$281$$ 18.9706 1.13169 0.565844 0.824512i $$-0.308550\pi$$
0.565844 + 0.824512i $$0.308550\pi$$
$$282$$ 0 0
$$283$$ − 2.68629i − 0.159683i −0.996808 0.0798417i $$-0.974559\pi$$
0.996808 0.0798417i $$-0.0254415\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 36.9706i − 2.18230i
$$288$$ 0 0
$$289$$ −6.31371 −0.371395
$$290$$ 0 0
$$291$$ 1.02944 0.0603467
$$292$$ 0 0
$$293$$ − 23.4142i − 1.36787i −0.729542 0.683936i $$-0.760267\pi$$
0.729542 0.683936i $$-0.239733\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 6.62742i 0.384562i
$$298$$ 0 0
$$299$$ 13.4558 0.778172
$$300$$ 0 0
$$301$$ 4.00000 0.230556
$$302$$ 0 0
$$303$$ − 2.34315i − 0.134610i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 7.89949i − 0.450848i −0.974261 0.225424i $$-0.927623\pi$$
0.974261 0.225424i $$-0.0723767\pi$$
$$308$$ 0 0
$$309$$ −9.31371 −0.529838
$$310$$ 0 0
$$311$$ −14.0000 −0.793867 −0.396934 0.917847i $$-0.629926\pi$$
−0.396934 + 0.917847i $$0.629926\pi$$
$$312$$ 0 0
$$313$$ 18.0000i 1.01742i 0.860938 + 0.508710i $$0.169877\pi$$
−0.860938 + 0.508710i $$0.830123\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 5.75736i 0.323366i 0.986843 + 0.161683i $$0.0516921\pi$$
−0.986843 + 0.161683i $$0.948308\pi$$
$$318$$ 0 0
$$319$$ 20.9706 1.17413
$$320$$ 0 0
$$321$$ −2.68629 −0.149934
$$322$$ 0 0
$$323$$ 4.82843i 0.268661i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 5.17157i − 0.285989i
$$328$$ 0 0
$$329$$ 4.00000 0.220527
$$330$$ 0 0
$$331$$ 1.17157 0.0643955 0.0321977 0.999482i $$-0.489749\pi$$
0.0321977 + 0.999482i $$0.489749\pi$$
$$332$$ 0 0
$$333$$ 27.2132i 1.49127i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 17.7574i − 0.967305i −0.875260 0.483652i $$-0.839310\pi$$
0.875260 0.483652i $$-0.160690\pi$$
$$338$$ 0 0
$$339$$ 2.97056 0.161339
$$340$$ 0 0
$$341$$ 2.34315 0.126888
$$342$$ 0 0
$$343$$ − 44.9706i − 2.42818i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 22.9706i − 1.23312i −0.787306 0.616562i $$-0.788525\pi$$
0.787306 0.616562i $$-0.211475\pi$$
$$348$$ 0 0
$$349$$ 0.343146 0.0183682 0.00918409 0.999958i $$-0.497077\pi$$
0.00918409 + 0.999958i $$0.497077\pi$$
$$350$$ 0 0
$$351$$ 7.43146 0.396662
$$352$$ 0 0
$$353$$ − 3.85786i − 0.205333i −0.994716 0.102667i $$-0.967262\pi$$
0.994716 0.102667i $$-0.0327375\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 13.6569i 0.722797i
$$358$$ 0 0
$$359$$ 21.3137 1.12489 0.562447 0.826833i $$-0.309860\pi$$
0.562447 + 0.826833i $$0.309860\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 4.10051i 0.215221i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 28.1421i 1.46901i 0.678605 + 0.734504i $$0.262585\pi$$
−0.678605 + 0.734504i $$0.737415\pi$$
$$368$$ 0 0
$$369$$ −20.3431 −1.05902
$$370$$ 0 0
$$371$$ 24.4853 1.27121
$$372$$ 0 0
$$373$$ 9.55635i 0.494809i 0.968912 + 0.247405i $$0.0795776\pi$$
−0.968912 + 0.247405i $$0.920422\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 23.5147i − 1.21107i
$$378$$ 0 0
$$379$$ −22.3431 −1.14769 −0.573845 0.818964i $$-0.694549\pi$$
−0.573845 + 0.818964i $$0.694549\pi$$
$$380$$ 0 0
$$381$$ 12.6274 0.646922
$$382$$ 0 0
$$383$$ 2.92893i 0.149661i 0.997196 + 0.0748307i $$0.0238416\pi$$
−0.997196 + 0.0748307i $$0.976158\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 2.20101i − 0.111884i
$$388$$ 0 0
$$389$$ 28.6274 1.45147 0.725734 0.687976i $$-0.241500\pi$$
0.725734 + 0.687976i $$0.241500\pi$$
$$390$$ 0 0
$$391$$ 28.9706 1.46510
$$392$$ 0 0
$$393$$ 3.31371i 0.167154i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 32.1421i 1.61317i 0.591120 + 0.806584i $$0.298686\pi$$
−0.591120 + 0.806584i $$0.701314\pi$$
$$398$$ 0 0
$$399$$ 2.82843 0.141598
$$400$$ 0 0
$$401$$ −6.68629 −0.333897 −0.166949 0.985966i $$-0.553391\pi$$
−0.166949 + 0.985966i $$0.553391\pi$$
$$402$$ 0 0
$$403$$ − 2.62742i − 0.130881i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 20.4853i − 1.01542i
$$408$$ 0 0
$$409$$ −9.51472 −0.470473 −0.235236 0.971938i $$-0.575586\pi$$
−0.235236 + 0.971938i $$0.575586\pi$$
$$410$$ 0 0
$$411$$ 11.7990 0.582001
$$412$$ 0 0
$$413$$ − 11.3137i − 0.556711i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 10.1421i − 0.496663i
$$418$$ 0 0
$$419$$ −9.65685 −0.471768 −0.235884 0.971781i $$-0.575799\pi$$
−0.235884 + 0.971781i $$0.575799\pi$$
$$420$$ 0 0
$$421$$ −8.34315 −0.406620 −0.203310 0.979114i $$-0.565170\pi$$
−0.203310 + 0.979114i $$0.565170\pi$$
$$422$$ 0 0
$$423$$ − 2.20101i − 0.107017i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 11.3137i − 0.547509i
$$428$$ 0 0
$$429$$ −2.62742 −0.126853
$$430$$ 0 0
$$431$$ 33.4558 1.61151 0.805756 0.592248i $$-0.201759\pi$$
0.805756 + 0.592248i $$0.201759\pi$$
$$432$$ 0 0
$$433$$ − 15.2132i − 0.731100i −0.930792 0.365550i $$-0.880881\pi$$
0.930792 0.365550i $$-0.119119\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 6.00000i − 0.287019i
$$438$$ 0 0
$$439$$ 29.6569 1.41544 0.707722 0.706491i $$-0.249723\pi$$
0.707722 + 0.706491i $$0.249723\pi$$
$$440$$ 0 0
$$441$$ −43.3431 −2.06396
$$442$$ 0 0
$$443$$ 34.2843i 1.62889i 0.580237 + 0.814447i $$0.302960\pi$$
−0.580237 + 0.814447i $$0.697040\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 4.68629i − 0.221654i
$$448$$ 0 0
$$449$$ 13.5147 0.637799 0.318900 0.947789i $$-0.396687\pi$$
0.318900 + 0.947789i $$0.396687\pi$$
$$450$$ 0 0
$$451$$ 15.3137 0.721094
$$452$$ 0 0
$$453$$ − 10.6274i − 0.499320i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 18.9706i − 0.887405i −0.896174 0.443703i $$-0.853665\pi$$
0.896174 0.443703i $$-0.146335\pi$$
$$458$$ 0 0
$$459$$ 16.0000 0.746816
$$460$$ 0 0
$$461$$ −5.31371 −0.247484 −0.123742 0.992314i $$-0.539490\pi$$
−0.123742 + 0.992314i $$0.539490\pi$$
$$462$$ 0 0
$$463$$ − 9.31371i − 0.432845i −0.976300 0.216422i $$-0.930561\pi$$
0.976300 0.216422i $$-0.0694388\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 9.31371i − 0.430987i −0.976505 0.215494i $$-0.930864\pi$$
0.976505 0.215494i $$-0.0691360\pi$$
$$468$$ 0 0
$$469$$ −2.82843 −0.130605
$$470$$ 0 0
$$471$$ −1.85786 −0.0856059
$$472$$ 0 0
$$473$$ 1.65685i 0.0761822i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 13.4731i − 0.616890i
$$478$$ 0 0
$$479$$ 10.0000 0.456912 0.228456 0.973554i $$-0.426632\pi$$
0.228456 + 0.973554i $$0.426632\pi$$
$$480$$ 0 0
$$481$$ −22.9706 −1.04737
$$482$$ 0 0
$$483$$ − 16.9706i − 0.772187i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 24.3848i − 1.10498i −0.833520 0.552490i $$-0.813678\pi$$
0.833520 0.552490i $$-0.186322\pi$$
$$488$$ 0 0
$$489$$ −1.45584 −0.0658355
$$490$$ 0 0
$$491$$ 35.3137 1.59369 0.796843 0.604187i $$-0.206502\pi$$
0.796843 + 0.604187i $$0.206502\pi$$
$$492$$ 0 0
$$493$$ − 50.6274i − 2.28014i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 52.2843i 2.34527i
$$498$$ 0 0
$$499$$ 26.9706 1.20737 0.603684 0.797224i $$-0.293699\pi$$
0.603684 + 0.797224i $$0.293699\pi$$
$$500$$ 0 0
$$501$$ 6.28427 0.280761
$$502$$ 0 0
$$503$$ 2.97056i 0.132451i 0.997805 + 0.0662254i $$0.0210956\pi$$
−0.997805 + 0.0662254i $$0.978904\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 4.66905i − 0.207360i
$$508$$ 0 0
$$509$$ 33.7990 1.49811 0.749057 0.662506i $$-0.230507\pi$$
0.749057 + 0.662506i $$0.230507\pi$$
$$510$$ 0 0
$$511$$ −69.9411 −3.09401
$$512$$ 0 0
$$513$$ − 3.31371i − 0.146304i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 1.65685i 0.0728684i
$$518$$ 0 0
$$519$$ −2.28427 −0.100268
$$520$$ 0 0
$$521$$ 28.3431 1.24174 0.620868 0.783915i $$-0.286780\pi$$
0.620868 + 0.783915i $$0.286780\pi$$
$$522$$ 0 0
$$523$$ − 6.72792i − 0.294191i −0.989122 0.147096i $$-0.953007\pi$$
0.989122 0.147096i $$-0.0469925\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 5.65685i − 0.246416i
$$528$$ 0 0
$$529$$ −13.0000 −0.565217
$$530$$ 0 0
$$531$$ −6.22540 −0.270159
$$532$$ 0 0
$$533$$ − 17.1716i − 0.743783i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 12.6863i 0.547454i
$$538$$ 0 0
$$539$$ 32.6274 1.40536
$$540$$ 0 0
$$541$$ 11.3137 0.486414 0.243207 0.969974i $$-0.421801\pi$$
0.243207 + 0.969974i $$0.421801\pi$$
$$542$$ 0 0
$$543$$ 10.5442i 0.452493i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 36.3848i − 1.55570i −0.628450 0.777850i $$-0.716310\pi$$
0.628450 0.777850i $$-0.283690\pi$$
$$548$$ 0 0
$$549$$ −6.22540 −0.265693
$$550$$ 0 0
$$551$$ −10.4853 −0.446688
$$552$$ 0 0
$$553$$ 46.6274i 1.98280i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 41.7990i − 1.77108i −0.464563 0.885540i $$-0.653789\pi$$
0.464563 0.885540i $$-0.346211\pi$$
$$558$$ 0 0
$$559$$ 1.85786 0.0785793
$$560$$ 0 0
$$561$$ −5.65685 −0.238833
$$562$$ 0 0
$$563$$ − 11.4142i − 0.481052i −0.970643 0.240526i $$-0.922680\pi$$
0.970643 0.240526i $$-0.0773199\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 29.1127i 1.22262i
$$568$$ 0 0
$$569$$ −0.828427 −0.0347295 −0.0173647 0.999849i $$-0.505528\pi$$
−0.0173647 + 0.999849i $$0.505528\pi$$
$$570$$ 0 0
$$571$$ −14.6863 −0.614602 −0.307301 0.951612i $$-0.599426\pi$$
−0.307301 + 0.951612i $$0.599426\pi$$
$$572$$ 0 0
$$573$$ − 2.34315i − 0.0978863i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 6.00000i 0.249783i 0.992170 + 0.124892i $$0.0398583\pi$$
−0.992170 + 0.124892i $$0.960142\pi$$
$$578$$ 0 0
$$579$$ 0.343146 0.0142607
$$580$$ 0 0
$$581$$ −44.9706 −1.86569
$$582$$ 0 0
$$583$$ 10.1421i 0.420044i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 14.9706i 0.617901i 0.951078 + 0.308951i $$0.0999778\pi$$
−0.951078 + 0.308951i $$0.900022\pi$$
$$588$$ 0 0
$$589$$ −1.17157 −0.0482738
$$590$$ 0 0
$$591$$ 3.11270 0.128039
$$592$$ 0 0
$$593$$ 8.62742i 0.354286i 0.984185 + 0.177143i $$0.0566854\pi$$
−0.984185 + 0.177143i $$0.943315\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ − 6.05887i − 0.247973i
$$598$$ 0 0
$$599$$ 4.68629 0.191477 0.0957383 0.995407i $$-0.469479\pi$$
0.0957383 + 0.995407i $$0.469479\pi$$
$$600$$ 0 0
$$601$$ −18.0000 −0.734235 −0.367118 0.930175i $$-0.619655\pi$$
−0.367118 + 0.930175i $$0.619655\pi$$
$$602$$ 0 0
$$603$$ 1.55635i 0.0633794i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 5.07107i − 0.205828i −0.994690 0.102914i $$-0.967183\pi$$
0.994690 0.102914i $$-0.0328167\pi$$
$$608$$ 0 0
$$609$$ −29.6569 −1.20176
$$610$$ 0 0
$$611$$ 1.85786 0.0751611
$$612$$ 0 0
$$613$$ 17.5147i 0.707413i 0.935356 + 0.353706i $$0.115079\pi$$
−0.935356 + 0.353706i $$0.884921\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 6.20101i 0.249643i 0.992179 + 0.124822i $$0.0398359\pi$$
−0.992179 + 0.124822i $$0.960164\pi$$
$$618$$ 0 0
$$619$$ −18.0000 −0.723481 −0.361741 0.932279i $$-0.617817\pi$$
−0.361741 + 0.932279i $$0.617817\pi$$
$$620$$ 0 0
$$621$$ −19.8823 −0.797847
$$622$$ 0 0
$$623$$ − 50.6274i − 2.02834i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 1.17157i 0.0467881i
$$628$$ 0 0
$$629$$ −49.4558 −1.97193
$$630$$ 0 0
$$631$$ −3.65685 −0.145577 −0.0727885 0.997347i $$-0.523190\pi$$
−0.0727885 + 0.997347i $$0.523190\pi$$
$$632$$ 0 0
$$633$$ 6.74517i 0.268096i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 36.5858i − 1.44958i
$$638$$ 0 0
$$639$$ 28.7696 1.13811
$$640$$ 0 0
$$641$$ −21.3137 −0.841841 −0.420920 0.907098i $$-0.638293\pi$$
−0.420920 + 0.907098i $$0.638293\pi$$
$$642$$ 0 0
$$643$$ 24.6274i 0.971211i 0.874178 + 0.485605i $$0.161401\pi$$
−0.874178 + 0.485605i $$0.838599\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 6.68629i − 0.262865i −0.991325 0.131433i $$-0.958042\pi$$
0.991325 0.131433i $$-0.0419577\pi$$
$$648$$ 0 0
$$649$$ 4.68629 0.183953
$$650$$ 0 0
$$651$$ −3.31371 −0.129874
$$652$$ 0 0
$$653$$ 13.7990i 0.539996i 0.962861 + 0.269998i $$0.0870231\pi$$
−0.962861 + 0.269998i $$0.912977\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 38.4853i 1.50145i
$$658$$ 0 0
$$659$$ 34.6274 1.34889 0.674446 0.738324i $$-0.264382\pi$$
0.674446 + 0.738324i $$0.264382\pi$$
$$660$$ 0 0
$$661$$ −12.6274 −0.491150 −0.245575 0.969378i $$-0.578977\pi$$
−0.245575 + 0.969378i $$0.578977\pi$$
$$662$$ 0 0
$$663$$ 6.34315i 0.246347i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 62.9117i 2.43595i
$$668$$ 0 0
$$669$$ 11.2548 0.435137
$$670$$ 0 0
$$671$$ 4.68629 0.180912
$$672$$ 0 0
$$673$$ 10.2426i 0.394825i 0.980321 + 0.197412i $$0.0632538\pi$$
−0.980321 + 0.197412i $$0.936746\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 25.5563i − 0.982210i −0.871100 0.491105i $$-0.836593\pi$$
0.871100 0.491105i $$-0.163407\pi$$
$$678$$ 0 0
$$679$$ −8.48528 −0.325635
$$680$$ 0 0
$$681$$ −12.0589 −0.462097
$$682$$ 0 0
$$683$$ − 38.7279i − 1.48188i −0.671570 0.740941i $$-0.734380\pi$$
0.671570 0.740941i $$-0.265620\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 12.6863i − 0.484012i
$$688$$ 0 0
$$689$$ 11.3726 0.433261
$$690$$ 0 0
$$691$$ −15.6569 −0.595615 −0.297807 0.954626i $$-0.596255\pi$$
−0.297807 + 0.954626i $$0.596255\pi$$
$$692$$ 0 0
$$693$$ − 25.6569i − 0.974623i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 36.9706i − 1.40036i
$$698$$ 0 0
$$699$$ 5.85786 0.221565
$$700$$ 0 0
$$701$$ 49.6569 1.87551 0.937757 0.347293i $$-0.112899\pi$$
0.937757 + 0.347293i $$0.112899\pi$$
$$702$$ 0 0
$$703$$ 10.2426i 0.386309i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 19.3137i 0.726367i
$$708$$ 0 0
$$709$$ −18.9706 −0.712454 −0.356227 0.934399i $$-0.615937\pi$$
−0.356227 + 0.934399i $$0.615937\pi$$
$$710$$ 0 0
$$711$$ 25.6569 0.962207
$$712$$ 0 0
$$713$$ 7.02944i 0.263254i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 6.05887i 0.226273i
$$718$$ 0 0
$$719$$ −38.2843 −1.42776 −0.713881 0.700267i $$-0.753064\pi$$
−0.713881 + 0.700267i $$0.753064\pi$$
$$720$$ 0 0
$$721$$ 76.7696 2.85905
$$722$$ 0 0
$$723$$ 17.7401i 0.659762i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 29.5147i 1.09464i 0.836923 + 0.547320i $$0.184352\pi$$
−0.836923 + 0.547320i $$0.815648\pi$$
$$728$$ 0 0
$$729$$ 10.1960 0.377628
$$730$$ 0 0
$$731$$ 4.00000 0.147945
$$732$$ 0 0
$$733$$ 39.6569i 1.46476i 0.680896 + 0.732380i $$0.261590\pi$$
−0.680896 + 0.732380i $$0.738410\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 1.17157i − 0.0431554i
$$738$$ 0 0
$$739$$ −34.6274 −1.27379 −0.636895 0.770951i $$-0.719782\pi$$
−0.636895 + 0.770951i $$0.719782\pi$$
$$740$$ 0 0
$$741$$ 1.31371 0.0482603
$$742$$ 0 0
$$743$$ 14.0416i 0.515137i 0.966260 + 0.257569i $$0.0829214\pi$$
−0.966260 + 0.257569i $$0.917079\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 24.7452i 0.905378i
$$748$$ 0 0
$$749$$ 22.1421 0.809056
$$750$$ 0 0
$$751$$ 14.1421 0.516054 0.258027 0.966138i $$-0.416928\pi$$
0.258027 + 0.966138i $$0.416928\pi$$
$$752$$ 0 0
$$753$$ 8.00000i 0.291536i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 19.6569i − 0.714441i −0.934020 0.357220i $$-0.883725\pi$$
0.934020 0.357220i $$-0.116275\pi$$
$$758$$ 0 0
$$759$$ 7.02944 0.255152
$$760$$ 0 0
$$761$$ 30.6274 1.11024 0.555121 0.831769i $$-0.312672\pi$$
0.555121 + 0.831769i $$0.312672\pi$$
$$762$$ 0 0
$$763$$ 42.6274i 1.54322i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 5.25483i − 0.189741i
$$768$$ 0 0
$$769$$ −38.6274 −1.39294 −0.696470 0.717586i $$-0.745247\pi$$
−0.696470 + 0.717586i $$0.745247\pi$$
$$770$$ 0 0
$$771$$ 1.31371 0.0473121
$$772$$ 0 0
$$773$$ − 0.100505i − 0.00361492i −0.999998 0.00180746i $$-0.999425\pi$$
0.999998 0.00180746i $$-0.000575332\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 28.9706i 1.03931i
$$778$$ 0 0
$$779$$ −7.65685 −0.274335
$$780$$ 0 0
$$781$$ −21.6569 −0.774943
$$782$$ 0 0
$$783$$ 34.7452i 1.24169i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 41.8406i 1.49146i 0.666250 + 0.745729i $$0.267898\pi$$
−0.666250 + 0.745729i $$0.732102\pi$$
$$788$$ 0 0
$$789$$ 2.14214 0.0762620
$$790$$ 0 0
$$791$$ −24.4853 −0.870596
$$792$$ 0 0
$$793$$ − 5.25483i − 0.186605i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 11.8995i − 0.421502i −0.977540 0.210751i $$-0.932409\pi$$
0.977540 0.210751i $$-0.0675909\pi$$
$$798$$ 0 0
$$799$$ 4.00000 0.141510
$$800$$ 0 0
$$801$$ −27.8579 −0.984309
$$802$$ 0 0
$$803$$ − 28.9706i − 1.02235i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 13.1716i − 0.463661i
$$808$$ 0 0
$$809$$ 39.2548 1.38013 0.690063 0.723749i $$-0.257583\pi$$
0.690063 + 0.723749i $$0.257583\pi$$
$$810$$ 0 0
$$811$$ 19.7990 0.695237 0.347618 0.937636i $$-0.386991\pi$$
0.347618 + 0.937636i $$0.386991\pi$$
$$812$$ 0 0
$$813$$ − 13.8579i − 0.486017i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 0.828427i − 0.0289830i
$$818$$ 0 0
$$819$$ −28.7696 −1.00529
$$820$$ 0 0
$$821$$ 10.0000 0.349002 0.174501 0.984657i $$-0.444169\pi$$
0.174501 + 0.984657i $$0.444169\pi$$
$$822$$ 0 0
$$823$$ − 15.1716i − 0.528848i −0.964407 0.264424i $$-0.914818\pi$$
0.964407 0.264424i $$-0.0851818\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 29.0711i 1.01090i 0.862856 + 0.505450i $$0.168674\pi$$
−0.862856 + 0.505450i $$0.831326\pi$$
$$828$$ 0 0
$$829$$ 40.4264 1.40407 0.702034 0.712144i $$-0.252276\pi$$
0.702034 + 0.712144i $$0.252276\pi$$
$$830$$ 0 0
$$831$$ −9.85786 −0.341966
$$832$$ 0 0
$$833$$ − 78.7696i − 2.72920i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 3.88225i 0.134190i
$$838$$ 0 0
$$839$$ −43.5980 −1.50517 −0.752585 0.658495i $$-0.771193\pi$$
−0.752585 + 0.658495i $$0.771193\pi$$
$$840$$ 0 0
$$841$$ 80.9411 2.79107
$$842$$ 0 0
$$843$$ − 11.1127i − 0.382742i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 33.7990i − 1.16135i
$$848$$ 0 0
$$849$$ −1.57359 −0.0540056
$$850$$ 0 0
$$851$$ 61.4558 2.10668
$$852$$ 0 0
$$853$$ 42.0000i 1.43805i 0.694983 + 0.719026i $$0.255412\pi$$
−0.694983 + 0.719026i $$0.744588\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 27.2132i 0.929585i 0.885420 + 0.464793i $$0.153871\pi$$
−0.885420 + 0.464793i $$0.846129\pi$$
$$858$$ 0 0
$$859$$ 24.2843 0.828569 0.414284 0.910148i $$-0.364032\pi$$
0.414284 + 0.910148i $$0.364032\pi$$
$$860$$ 0 0
$$861$$ −21.6569 −0.738064
$$862$$ 0 0
$$863$$ − 3.89949i − 0.132740i −0.997795 0.0663702i $$-0.978858\pi$$
0.997795 0.0663702i $$-0.0211418\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 3.69848i 0.125607i
$$868$$ 0 0
$$869$$ −19.3137 −0.655173
$$870$$ 0 0
$$871$$ −1.31371 −0.0445133
$$872$$ 0 0
$$873$$ 4.66905i 0.158023i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 37.0711i 1.25180i 0.779903 + 0.625901i $$0.215268\pi$$
−0.779903 + 0.625901i $$0.784732\pi$$
$$878$$ 0 0
$$879$$ −13.7157 −0.462620
$$880$$ 0 0
$$881$$ 44.2843 1.49198 0.745988 0.665960i $$-0.231978\pi$$
0.745988 + 0.665960i $$0.231978\pi$$
$$882$$ 0 0
$$883$$ − 39.1716i − 1.31823i −0.752043 0.659114i $$-0.770931\pi$$
0.752043 0.659114i $$-0.229069\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 49.0711i 1.64765i 0.566848 + 0.823823i $$0.308163\pi$$
−0.566848 + 0.823823i $$0.691837\pi$$
$$888$$ 0 0
$$889$$ −104.083 −3.49084
$$890$$ 0 0
$$891$$ −12.0589 −0.403987
$$892$$ 0 0
$$893$$ − 0.828427i − 0.0277223i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 7.88225i − 0.263181i
$$898$$ 0 0
$$899$$ 12.2843 0.409703
$$900$$ 0 0
$$901$$ 24.4853 0.815723
$$902$$ 0 0
$$903$$ − 2.34315i − 0.0779750i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 20.3848i 0.676865i 0.940991 + 0.338433i $$0.109897\pi$$
−0.940991 + 0.338433i $$0.890103\pi$$
$$908$$ 0 0
$$909$$ 10.6274 0.352489
$$910$$ 0 0
$$911$$ 4.20101 0.139186 0.0695928 0.997575i $$-0.477830\pi$$
0.0695928 + 0.997575i $$0.477830\pi$$
$$912$$ 0 0
$$913$$ − 18.6274i − 0.616478i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 27.3137i − 0.901978i
$$918$$ 0 0
$$919$$ −35.3137 −1.16489 −0.582446 0.812869i $$-0.697904\pi$$
−0.582446 + 0.812869i $$0.697904\pi$$
$$920$$ 0 0
$$921$$ −4.62742 −0.152479
$$922$$ 0 0
$$923$$ 24.2843i 0.799327i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 42.2426i − 1.38743i
$$928$$ 0 0
$$929$$ −6.68629 −0.219370 −0.109685 0.993966i $$-0.534984\pi$$
−0.109685 + 0.993966i $$0.534984\pi$$
$$930$$ 0 0
$$931$$ −16.3137 −0.534660
$$932$$ 0 0
$$933$$ 8.20101i 0.268489i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 40.1421i 1.31139i 0.755027 + 0.655693i $$0.227624\pi$$
−0.755027 + 0.655693i $$0.772376\pi$$
$$938$$ 0 0
$$939$$ 10.5442 0.344096
$$940$$ 0 0
$$941$$ −30.2843 −0.987239 −0.493620 0.869678i $$-0.664326\pi$$
−0.493620 + 0.869678i $$0.664326\pi$$
$$942$$ 0 0
$$943$$ 45.9411i 1.49605i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 8.34315i 0.271116i 0.990769 + 0.135558i $$0.0432827\pi$$
−0.990769 + 0.135558i $$0.956717\pi$$
$$948$$ 0 0
$$949$$ −32.4853 −1.05452
$$950$$ 0 0
$$951$$ 3.37258 0.109363
$$952$$ 0 0
$$953$$ 5.75736i 0.186499i 0.995643 + 0.0932496i $$0.0297254\pi$$
−0.995643 + 0.0932496i $$0.970275\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 12.2843i − 0.397094i
$$958$$ 0 0
$$959$$ −97.2548 −3.14052
$$960$$ 0 0
$$961$$ −29.6274 −0.955723
$$962$$ 0 0
$$963$$ − 12.1838i − 0.392616i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 20.6274i − 0.663333i −0.943397 0.331667i $$-0.892389\pi$$
0.943397 0.331667i $$-0.107611\pi$$
$$968$$ 0 0
$$969$$ 2.82843 0.0908622
$$970$$ 0 0
$$971$$ −35.1127 −1.12682 −0.563410 0.826177i $$-0.690511\pi$$
−0.563410 + 0.826177i $$0.690511\pi$$
$$972$$ 0 0
$$973$$ 83.5980i 2.68003i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 9.27208i − 0.296640i −0.988939 0.148320i $$-0.952613\pi$$
0.988939 0.148320i $$-0.0473866\pi$$
$$978$$ 0 0
$$979$$ 20.9706 0.670222
$$980$$ 0 0
$$981$$ 23.4558 0.748887
$$982$$ 0 0
$$983$$ − 23.4142i − 0.746797i −0.927671 0.373399i $$-0.878192\pi$$
0.927671 0.373399i $$-0.121808\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 2.34315i − 0.0745832i
$$988$$ 0 0
$$989$$ −4.97056 −0.158055
$$990$$ 0 0
$$991$$ −51.1127 −1.62365 −0.811824 0.583902i $$-0.801525\pi$$
−0.811824 + 0.583902i $$0.801525\pi$$
$$992$$ 0 0
$$993$$ − 0.686292i − 0.0217788i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 13.5147i 0.428015i 0.976832 + 0.214008i $$0.0686518\pi$$
−0.976832 + 0.214008i $$0.931348\pi$$
$$998$$ 0 0
$$999$$ 33.9411 1.07385
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 1900.2.c.d.1749.2 4
5.2 odd 4 380.2.a.c.1.2 2
5.3 odd 4 1900.2.a.e.1.1 2
5.4 even 2 inner 1900.2.c.d.1749.3 4
15.2 even 4 3420.2.a.g.1.1 2
20.3 even 4 7600.2.a.u.1.2 2
20.7 even 4 1520.2.a.o.1.1 2
40.27 even 4 6080.2.a.y.1.2 2
40.37 odd 4 6080.2.a.bl.1.1 2
95.37 even 4 7220.2.a.m.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.a.c.1.2 2 5.2 odd 4
1520.2.a.o.1.1 2 20.7 even 4
1900.2.a.e.1.1 2 5.3 odd 4
1900.2.c.d.1749.2 4 1.1 even 1 trivial
1900.2.c.d.1749.3 4 5.4 even 2 inner
3420.2.a.g.1.1 2 15.2 even 4
6080.2.a.y.1.2 2 40.27 even 4
6080.2.a.bl.1.1 2 40.37 odd 4
7220.2.a.m.1.1 2 95.37 even 4
7600.2.a.u.1.2 2 20.3 even 4