# Properties

 Label 380.2.a.c.1.2 Level $380$ Weight $2$ Character 380.1 Self dual yes Analytic conductor $3.034$ Analytic rank $1$ 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] = [380,2,Mod(1,380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(380, 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("380.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$380 = 2^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 380.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: $$3.03431527681$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 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 $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 380.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.585786 q^{3} +1.00000 q^{5} -4.82843 q^{7} -2.65685 q^{9} +O(q^{10})$$ $$q-0.585786 q^{3} +1.00000 q^{5} -4.82843 q^{7} -2.65685 q^{9} -2.00000 q^{11} +2.24264 q^{13} -0.585786 q^{15} -4.82843 q^{17} -1.00000 q^{19} +2.82843 q^{21} -6.00000 q^{23} +1.00000 q^{25} +3.31371 q^{27} +10.4853 q^{29} -1.17157 q^{31} +1.17157 q^{33} -4.82843 q^{35} -10.2426 q^{37} -1.31371 q^{39} -7.65685 q^{41} -0.828427 q^{43} -2.65685 q^{45} +0.828427 q^{47} +16.3137 q^{49} +2.82843 q^{51} -5.07107 q^{53} -2.00000 q^{55} +0.585786 q^{57} +2.34315 q^{59} -2.34315 q^{61} +12.8284 q^{63} +2.24264 q^{65} -0.585786 q^{67} +3.51472 q^{69} +10.8284 q^{71} +14.4853 q^{73} -0.585786 q^{75} +9.65685 q^{77} -9.65685 q^{79} +6.02944 q^{81} +9.31371 q^{83} -4.82843 q^{85} -6.14214 q^{87} +10.4853 q^{89} -10.8284 q^{91} +0.686292 q^{93} -1.00000 q^{95} -1.75736 q^{97} +5.31371 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{3} + 2 q^{5} - 4 q^{7} + 6 q^{9}+O(q^{10})$$ 2 * q - 4 * q^3 + 2 * q^5 - 4 * q^7 + 6 * q^9 $$2 q - 4 q^{3} + 2 q^{5} - 4 q^{7} + 6 q^{9} - 4 q^{11} - 4 q^{13} - 4 q^{15} - 4 q^{17} - 2 q^{19} - 12 q^{23} + 2 q^{25} - 16 q^{27} + 4 q^{29} - 8 q^{31} + 8 q^{33} - 4 q^{35} - 12 q^{37} + 20 q^{39} - 4 q^{41} + 4 q^{43} + 6 q^{45} - 4 q^{47} + 10 q^{49} + 4 q^{53} - 4 q^{55} + 4 q^{57} + 16 q^{59} - 16 q^{61} + 20 q^{63} - 4 q^{65} - 4 q^{67} + 24 q^{69} + 16 q^{71} + 12 q^{73} - 4 q^{75} + 8 q^{77} - 8 q^{79} + 46 q^{81} - 4 q^{83} - 4 q^{85} + 16 q^{87} + 4 q^{89} - 16 q^{91} + 24 q^{93} - 2 q^{95} - 12 q^{97} - 12 q^{99}+O(q^{100})$$ 2 * q - 4 * q^3 + 2 * q^5 - 4 * q^7 + 6 * q^9 - 4 * q^11 - 4 * q^13 - 4 * q^15 - 4 * q^17 - 2 * q^19 - 12 * q^23 + 2 * q^25 - 16 * q^27 + 4 * q^29 - 8 * q^31 + 8 * q^33 - 4 * q^35 - 12 * q^37 + 20 * q^39 - 4 * q^41 + 4 * q^43 + 6 * q^45 - 4 * q^47 + 10 * q^49 + 4 * q^53 - 4 * q^55 + 4 * q^57 + 16 * q^59 - 16 * q^61 + 20 * q^63 - 4 * q^65 - 4 * q^67 + 24 * q^69 + 16 * q^71 + 12 * q^73 - 4 * q^75 + 8 * q^77 - 8 * q^79 + 46 * q^81 - 4 * q^83 - 4 * q^85 + 16 * q^87 + 4 * q^89 - 16 * q^91 + 24 * q^93 - 2 * q^95 - 12 * q^97 - 12 * q^99

## 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
$$3$$ −0.585786 −0.338204 −0.169102 0.985599i $$-0.554087\pi$$
−0.169102 + 0.985599i $$0.554087\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −4.82843 −1.82497 −0.912487 0.409106i $$-0.865841\pi$$
−0.912487 + 0.409106i $$0.865841\pi$$
$$8$$ 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
$$13$$ 2.24264 0.621997 0.310998 0.950410i $$-0.399337\pi$$
0.310998 + 0.950410i $$0.399337\pi$$
$$14$$ 0 0
$$15$$ −0.585786 −0.151249
$$16$$ 0 0
$$17$$ −4.82843 −1.17107 −0.585533 0.810649i $$-0.699115\pi$$
−0.585533 + 0.810649i $$0.699115\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 2.82843 0.617213
$$22$$ 0 0
$$23$$ −6.00000 −1.25109 −0.625543 0.780189i $$-0.715123\pi$$
−0.625543 + 0.780189i $$0.715123\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 3.31371 0.637723
$$28$$ 0 0
$$29$$ 10.4853 1.94707 0.973534 0.228543i $$-0.0733960\pi$$
0.973534 + 0.228543i $$0.0733960\pi$$
$$30$$ 0 0
$$31$$ −1.17157 −0.210421 −0.105210 0.994450i $$-0.533552\pi$$
−0.105210 + 0.994450i $$0.533552\pi$$
$$32$$ 0 0
$$33$$ 1.17157 0.203945
$$34$$ 0 0
$$35$$ −4.82843 −0.816153
$$36$$ 0 0
$$37$$ −10.2426 −1.68388 −0.841940 0.539571i $$-0.818586\pi$$
−0.841940 + 0.539571i $$0.818586\pi$$
$$38$$ 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
$$43$$ −0.828427 −0.126334 −0.0631670 0.998003i $$-0.520120\pi$$
−0.0631670 + 0.998003i $$0.520120\pi$$
$$44$$ 0 0
$$45$$ −2.65685 −0.396060
$$46$$ 0 0
$$47$$ 0.828427 0.120839 0.0604193 0.998173i $$-0.480756\pi$$
0.0604193 + 0.998173i $$0.480756\pi$$
$$48$$ 0 0
$$49$$ 16.3137 2.33053
$$50$$ 0 0
$$51$$ 2.82843 0.396059
$$52$$ 0 0
$$53$$ −5.07107 −0.696565 −0.348282 0.937390i $$-0.613235\pi$$
−0.348282 + 0.937390i $$0.613235\pi$$
$$54$$ 0 0
$$55$$ −2.00000 −0.269680
$$56$$ 0 0
$$57$$ 0.585786 0.0775893
$$58$$ 0 0
$$59$$ 2.34315 0.305052 0.152526 0.988299i $$-0.451259\pi$$
0.152526 + 0.988299i $$0.451259\pi$$
$$60$$ 0 0
$$61$$ −2.34315 −0.300009 −0.150005 0.988685i $$-0.547929\pi$$
−0.150005 + 0.988685i $$0.547929\pi$$
$$62$$ 0 0
$$63$$ 12.8284 1.61623
$$64$$ 0 0
$$65$$ 2.24264 0.278165
$$66$$ 0 0
$$67$$ −0.585786 −0.0715652 −0.0357826 0.999360i $$-0.511392\pi$$
−0.0357826 + 0.999360i $$0.511392\pi$$
$$68$$ 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
$$73$$ 14.4853 1.69537 0.847687 0.530497i $$-0.177995\pi$$
0.847687 + 0.530497i $$0.177995\pi$$
$$74$$ 0 0
$$75$$ −0.585786 −0.0676408
$$76$$ 0 0
$$77$$ 9.65685 1.10050
$$78$$ 0 0
$$79$$ −9.65685 −1.08648 −0.543240 0.839577i $$-0.682803\pi$$
−0.543240 + 0.839577i $$0.682803\pi$$
$$80$$ 0 0
$$81$$ 6.02944 0.669937
$$82$$ 0 0
$$83$$ 9.31371 1.02231 0.511156 0.859488i $$-0.329217\pi$$
0.511156 + 0.859488i $$0.329217\pi$$
$$84$$ 0 0
$$85$$ −4.82843 −0.523716
$$86$$ 0 0
$$87$$ −6.14214 −0.658506
$$88$$ 0 0
$$89$$ 10.4853 1.11144 0.555719 0.831370i $$-0.312443\pi$$
0.555719 + 0.831370i $$0.312443\pi$$
$$90$$ 0 0
$$91$$ −10.8284 −1.13513
$$92$$ 0 0
$$93$$ 0.686292 0.0711651
$$94$$ 0 0
$$95$$ −1.00000 −0.102598
$$96$$ 0 0
$$97$$ −1.75736 −0.178433 −0.0892164 0.996012i $$-0.528436\pi$$
−0.0892164 + 0.996012i $$0.528436\pi$$
$$98$$ 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
$$103$$ −15.8995 −1.56662 −0.783312 0.621629i $$-0.786471\pi$$
−0.783312 + 0.621629i $$0.786471\pi$$
$$104$$ 0 0
$$105$$ 2.82843 0.276026
$$106$$ 0 0
$$107$$ 4.58579 0.443325 0.221662 0.975123i $$-0.428852\pi$$
0.221662 + 0.975123i $$0.428852\pi$$
$$108$$ 0 0
$$109$$ −8.82843 −0.845610 −0.422805 0.906221i $$-0.638954\pi$$
−0.422805 + 0.906221i $$0.638954\pi$$
$$110$$ 0 0
$$111$$ 6.00000 0.569495
$$112$$ 0 0
$$113$$ 5.07107 0.477046 0.238523 0.971137i $$-0.423337\pi$$
0.238523 + 0.971137i $$0.423337\pi$$
$$114$$ 0 0
$$115$$ −6.00000 −0.559503
$$116$$ 0 0
$$117$$ −5.95837 −0.550851
$$118$$ 0 0
$$119$$ 23.3137 2.13716
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ 4.48528 0.404424
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −21.5563 −1.91282 −0.956408 0.292033i $$-0.905668\pi$$
−0.956408 + 0.292033i $$0.905668\pi$$
$$128$$ 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
$$133$$ 4.82843 0.418678
$$134$$ 0 0
$$135$$ 3.31371 0.285199
$$136$$ 0 0
$$137$$ −20.1421 −1.72086 −0.860429 0.509570i $$-0.829805\pi$$
−0.860429 + 0.509570i $$0.829805\pi$$
$$138$$ 0 0
$$139$$ −17.3137 −1.46853 −0.734265 0.678863i $$-0.762473\pi$$
−0.734265 + 0.678863i $$0.762473\pi$$
$$140$$ 0 0
$$141$$ −0.485281 −0.0408681
$$142$$ 0 0
$$143$$ −4.48528 −0.375078
$$144$$ 0 0
$$145$$ 10.4853 0.870755
$$146$$ 0 0
$$147$$ −9.55635 −0.788194
$$148$$ 0 0
$$149$$ −8.00000 −0.655386 −0.327693 0.944784i $$-0.606271\pi$$
−0.327693 + 0.944784i $$0.606271\pi$$
$$150$$ 0 0
$$151$$ 18.1421 1.47639 0.738193 0.674590i $$-0.235679\pi$$
0.738193 + 0.674590i $$0.235679\pi$$
$$152$$ 0 0
$$153$$ 12.8284 1.03712
$$154$$ 0 0
$$155$$ −1.17157 −0.0941030
$$156$$ 0 0
$$157$$ 3.17157 0.253119 0.126560 0.991959i $$-0.459607\pi$$
0.126560 + 0.991959i $$0.459607\pi$$
$$158$$ 0 0
$$159$$ 2.97056 0.235581
$$160$$ 0 0
$$161$$ 28.9706 2.28320
$$162$$ 0 0
$$163$$ −2.48528 −0.194662 −0.0973311 0.995252i $$-0.531031\pi$$
−0.0973311 + 0.995252i $$0.531031\pi$$
$$164$$ 0 0
$$165$$ 1.17157 0.0912068
$$166$$ 0 0
$$167$$ −10.7279 −0.830152 −0.415076 0.909787i $$-0.636245\pi$$
−0.415076 + 0.909787i $$0.636245\pi$$
$$168$$ 0 0
$$169$$ −7.97056 −0.613120
$$170$$ 0 0
$$171$$ 2.65685 0.203175
$$172$$ 0 0
$$173$$ −3.89949 −0.296473 −0.148237 0.988952i $$-0.547360\pi$$
−0.148237 + 0.988952i $$0.547360\pi$$
$$174$$ 0 0
$$175$$ −4.82843 −0.364995
$$176$$ 0 0
$$177$$ −1.37258 −0.103170
$$178$$ 0 0
$$179$$ 21.6569 1.61871 0.809355 0.587320i $$-0.199817\pi$$
0.809355 + 0.587320i $$0.199817\pi$$
$$180$$ 0 0
$$181$$ −18.0000 −1.33793 −0.668965 0.743294i $$-0.733262\pi$$
−0.668965 + 0.743294i $$0.733262\pi$$
$$182$$ 0 0
$$183$$ 1.37258 0.101464
$$184$$ 0 0
$$185$$ −10.2426 −0.753054
$$186$$ 0 0
$$187$$ 9.65685 0.706179
$$188$$ 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
$$193$$ 0.585786 0.0421658 0.0210829 0.999778i $$-0.493289\pi$$
0.0210829 + 0.999778i $$0.493289\pi$$
$$194$$ 0 0
$$195$$ −1.31371 −0.0940766
$$196$$ 0 0
$$197$$ −5.31371 −0.378586 −0.189293 0.981921i $$-0.560620\pi$$
−0.189293 + 0.981921i $$0.560620\pi$$
$$198$$ 0 0
$$199$$ −10.3431 −0.733206 −0.366603 0.930377i $$-0.619479\pi$$
−0.366603 + 0.930377i $$0.619479\pi$$
$$200$$ 0 0
$$201$$ 0.343146 0.0242036
$$202$$ 0 0
$$203$$ −50.6274 −3.55335
$$204$$ 0 0
$$205$$ −7.65685 −0.534778
$$206$$ 0 0
$$207$$ 15.9411 1.10798
$$208$$ 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
$$213$$ −6.34315 −0.434625
$$214$$ 0 0
$$215$$ −0.828427 −0.0564983
$$216$$ 0 0
$$217$$ 5.65685 0.384012
$$218$$ 0 0
$$219$$ −8.48528 −0.573382
$$220$$ 0 0
$$221$$ −10.8284 −0.728399
$$222$$ 0 0
$$223$$ 19.2132 1.28661 0.643306 0.765609i $$-0.277562\pi$$
0.643306 + 0.765609i $$0.277562\pi$$
$$224$$ 0 0
$$225$$ −2.65685 −0.177124
$$226$$ 0 0
$$227$$ 20.5858 1.36633 0.683163 0.730266i $$-0.260604\pi$$
0.683163 + 0.730266i $$0.260604\pi$$
$$228$$ 0 0
$$229$$ −21.6569 −1.43113 −0.715563 0.698549i $$-0.753830\pi$$
−0.715563 + 0.698549i $$0.753830\pi$$
$$230$$ 0 0
$$231$$ −5.65685 −0.372194
$$232$$ 0 0
$$233$$ 10.0000 0.655122 0.327561 0.944830i $$-0.393773\pi$$
0.327561 + 0.944830i $$0.393773\pi$$
$$234$$ 0 0
$$235$$ 0.828427 0.0540406
$$236$$ 0 0
$$237$$ 5.65685 0.367452
$$238$$ 0 0
$$239$$ 10.3431 0.669042 0.334521 0.942388i $$-0.391425\pi$$
0.334521 + 0.942388i $$0.391425\pi$$
$$240$$ 0 0
$$241$$ −30.2843 −1.95078 −0.975391 0.220484i $$-0.929236\pi$$
−0.975391 + 0.220484i $$0.929236\pi$$
$$242$$ 0 0
$$243$$ −13.4731 −0.864299
$$244$$ 0 0
$$245$$ 16.3137 1.04224
$$246$$ 0 0
$$247$$ −2.24264 −0.142696
$$248$$ 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
$$253$$ 12.0000 0.754434
$$254$$ 0 0
$$255$$ 2.82843 0.177123
$$256$$ 0 0
$$257$$ −2.24264 −0.139892 −0.0699460 0.997551i $$-0.522283\pi$$
−0.0699460 + 0.997551i $$0.522283\pi$$
$$258$$ 0 0
$$259$$ 49.4558 3.07304
$$260$$ 0 0
$$261$$ −27.8579 −1.72436
$$262$$ 0 0
$$263$$ 3.65685 0.225491 0.112746 0.993624i $$-0.464035\pi$$
0.112746 + 0.993624i $$0.464035\pi$$
$$264$$ 0 0
$$265$$ −5.07107 −0.311513
$$266$$ 0 0
$$267$$ −6.14214 −0.375893
$$268$$ 0 0
$$269$$ −22.4853 −1.37095 −0.685476 0.728095i $$-0.740406\pi$$
−0.685476 + 0.728095i $$0.740406\pi$$
$$270$$ 0 0
$$271$$ 23.6569 1.43705 0.718526 0.695500i $$-0.244817\pi$$
0.718526 + 0.695500i $$0.244817\pi$$
$$272$$ 0 0
$$273$$ 6.34315 0.383905
$$274$$ 0 0
$$275$$ −2.00000 −0.120605
$$276$$ 0 0
$$277$$ 16.8284 1.01112 0.505561 0.862791i $$-0.331286\pi$$
0.505561 + 0.862791i $$0.331286\pi$$
$$278$$ 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
$$283$$ −2.68629 −0.159683 −0.0798417 0.996808i $$-0.525441\pi$$
−0.0798417 + 0.996808i $$0.525441\pi$$
$$284$$ 0 0
$$285$$ 0.585786 0.0346990
$$286$$ 0 0
$$287$$ 36.9706 2.18230
$$288$$ 0 0
$$289$$ 6.31371 0.371395
$$290$$ 0 0
$$291$$ 1.02944 0.0603467
$$292$$ 0 0
$$293$$ −23.4142 −1.36787 −0.683936 0.729542i $$-0.739733\pi$$
−0.683936 + 0.729542i $$0.739733\pi$$
$$294$$ 0 0
$$295$$ 2.34315 0.136423
$$296$$ 0 0
$$297$$ −6.62742 −0.384562
$$298$$ 0 0
$$299$$ −13.4558 −0.778172
$$300$$ 0 0
$$301$$ 4.00000 0.230556
$$302$$ 0 0
$$303$$ −2.34315 −0.134610
$$304$$ 0 0
$$305$$ −2.34315 −0.134168
$$306$$ 0 0
$$307$$ 7.89949 0.450848 0.225424 0.974261i $$-0.427623\pi$$
0.225424 + 0.974261i $$0.427623\pi$$
$$308$$ 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
$$313$$ 18.0000 1.01742 0.508710 0.860938i $$-0.330123\pi$$
0.508710 + 0.860938i $$0.330123\pi$$
$$314$$ 0 0
$$315$$ 12.8284 0.722800
$$316$$ 0 0
$$317$$ −5.75736 −0.323366 −0.161683 0.986843i $$-0.551692\pi$$
−0.161683 + 0.986843i $$0.551692\pi$$
$$318$$ 0 0
$$319$$ −20.9706 −1.17413
$$320$$ 0 0
$$321$$ −2.68629 −0.149934
$$322$$ 0 0
$$323$$ 4.82843 0.268661
$$324$$ 0 0
$$325$$ 2.24264 0.124399
$$326$$ 0 0
$$327$$ 5.17157 0.285989
$$328$$ 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
$$333$$ 27.2132 1.49127
$$334$$ 0 0
$$335$$ −0.585786 −0.0320049
$$336$$ 0 0
$$337$$ 17.7574 0.967305 0.483652 0.875260i $$-0.339310\pi$$
0.483652 + 0.875260i $$0.339310\pi$$
$$338$$ 0 0
$$339$$ −2.97056 −0.161339
$$340$$ 0 0
$$341$$ 2.34315 0.126888
$$342$$ 0 0
$$343$$ −44.9706 −2.42818
$$344$$ 0 0
$$345$$ 3.51472 0.189226
$$346$$ 0 0
$$347$$ 22.9706 1.23312 0.616562 0.787306i $$-0.288525\pi$$
0.616562 + 0.787306i $$0.288525\pi$$
$$348$$ 0 0
$$349$$ −0.343146 −0.0183682 −0.00918409 0.999958i $$-0.502923\pi$$
−0.00918409 + 0.999958i $$0.502923\pi$$
$$350$$ 0 0
$$351$$ 7.43146 0.396662
$$352$$ 0 0
$$353$$ −3.85786 −0.205333 −0.102667 0.994716i $$-0.532738\pi$$
−0.102667 + 0.994716i $$0.532738\pi$$
$$354$$ 0 0
$$355$$ 10.8284 0.574713
$$356$$ 0 0
$$357$$ −13.6569 −0.722797
$$358$$ 0 0
$$359$$ −21.3137 −1.12489 −0.562447 0.826833i $$-0.690140\pi$$
−0.562447 + 0.826833i $$0.690140\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 4.10051 0.215221
$$364$$ 0 0
$$365$$ 14.4853 0.758194
$$366$$ 0 0
$$367$$ −28.1421 −1.46901 −0.734504 0.678605i $$-0.762585\pi$$
−0.734504 + 0.678605i $$0.762585\pi$$
$$368$$ 0 0
$$369$$ 20.3431 1.05902
$$370$$ 0 0
$$371$$ 24.4853 1.27121
$$372$$ 0 0
$$373$$ 9.55635 0.494809 0.247405 0.968912i $$-0.420422\pi$$
0.247405 + 0.968912i $$0.420422\pi$$
$$374$$ 0 0
$$375$$ −0.585786 −0.0302499
$$376$$ 0 0
$$377$$ 23.5147 1.21107
$$378$$ 0 0
$$379$$ 22.3431 1.14769 0.573845 0.818964i $$-0.305451\pi$$
0.573845 + 0.818964i $$0.305451\pi$$
$$380$$ 0 0
$$381$$ 12.6274 0.646922
$$382$$ 0 0
$$383$$ 2.92893 0.149661 0.0748307 0.997196i $$-0.476158\pi$$
0.0748307 + 0.997196i $$0.476158\pi$$
$$384$$ 0 0
$$385$$ 9.65685 0.492159
$$386$$ 0 0
$$387$$ 2.20101 0.111884
$$388$$ 0 0
$$389$$ −28.6274 −1.45147 −0.725734 0.687976i $$-0.758500\pi$$
−0.725734 + 0.687976i $$0.758500\pi$$
$$390$$ 0 0
$$391$$ 28.9706 1.46510
$$392$$ 0 0
$$393$$ 3.31371 0.167154
$$394$$ 0 0
$$395$$ −9.65685 −0.485889
$$396$$ 0 0
$$397$$ −32.1421 −1.61317 −0.806584 0.591120i $$-0.798686\pi$$
−0.806584 + 0.591120i $$0.798686\pi$$
$$398$$ 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
$$403$$ −2.62742 −0.130881
$$404$$ 0 0
$$405$$ 6.02944 0.299605
$$406$$ 0 0
$$407$$ 20.4853 1.01542
$$408$$ 0 0
$$409$$ 9.51472 0.470473 0.235236 0.971938i $$-0.424414\pi$$
0.235236 + 0.971938i $$0.424414\pi$$
$$410$$ 0 0
$$411$$ 11.7990 0.582001
$$412$$ 0 0
$$413$$ −11.3137 −0.556711
$$414$$ 0 0
$$415$$ 9.31371 0.457192
$$416$$ 0 0
$$417$$ 10.1421 0.496663
$$418$$ 0 0
$$419$$ 9.65685 0.471768 0.235884 0.971781i $$-0.424201\pi$$
0.235884 + 0.971781i $$0.424201\pi$$
$$420$$ 0 0
$$421$$ −8.34315 −0.406620 −0.203310 0.979114i $$-0.565170\pi$$
−0.203310 + 0.979114i $$0.565170\pi$$
$$422$$ 0 0
$$423$$ −2.20101 −0.107017
$$424$$ 0 0
$$425$$ −4.82843 −0.234213
$$426$$ 0 0
$$427$$ 11.3137 0.547509
$$428$$ 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
$$433$$ −15.2132 −0.731100 −0.365550 0.930792i $$-0.619119\pi$$
−0.365550 + 0.930792i $$0.619119\pi$$
$$434$$ 0 0
$$435$$ −6.14214 −0.294493
$$436$$ 0 0
$$437$$ 6.00000 0.287019
$$438$$ 0 0
$$439$$ −29.6569 −1.41544 −0.707722 0.706491i $$-0.750277\pi$$
−0.707722 + 0.706491i $$0.750277\pi$$
$$440$$ 0 0
$$441$$ −43.3431 −2.06396
$$442$$ 0 0
$$443$$ 34.2843 1.62889 0.814447 0.580237i $$-0.197040\pi$$
0.814447 + 0.580237i $$0.197040\pi$$
$$444$$ 0 0
$$445$$ 10.4853 0.497050
$$446$$ 0 0
$$447$$ 4.68629 0.221654
$$448$$ 0 0
$$449$$ −13.5147 −0.637799 −0.318900 0.947789i $$-0.603313\pi$$
−0.318900 + 0.947789i $$0.603313\pi$$
$$450$$ 0 0
$$451$$ 15.3137 0.721094
$$452$$ 0 0
$$453$$ −10.6274 −0.499320
$$454$$ 0 0
$$455$$ −10.8284 −0.507644
$$456$$ 0 0
$$457$$ 18.9706 0.887405 0.443703 0.896174i $$-0.353665\pi$$
0.443703 + 0.896174i $$0.353665\pi$$
$$458$$ 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
$$463$$ −9.31371 −0.432845 −0.216422 0.976300i $$-0.569439\pi$$
−0.216422 + 0.976300i $$0.569439\pi$$
$$464$$ 0 0
$$465$$ 0.686292 0.0318260
$$466$$ 0 0
$$467$$ 9.31371 0.430987 0.215494 0.976505i $$-0.430864\pi$$
0.215494 + 0.976505i $$0.430864\pi$$
$$468$$ 0 0
$$469$$ 2.82843 0.130605
$$470$$ 0 0
$$471$$ −1.85786 −0.0856059
$$472$$ 0 0
$$473$$ 1.65685 0.0761822
$$474$$ 0 0
$$475$$ −1.00000 −0.0458831
$$476$$ 0 0
$$477$$ 13.4731 0.616890
$$478$$ 0 0
$$479$$ −10.0000 −0.456912 −0.228456 0.973554i $$-0.573368\pi$$
−0.228456 + 0.973554i $$0.573368\pi$$
$$480$$ 0 0
$$481$$ −22.9706 −1.04737
$$482$$ 0 0
$$483$$ −16.9706 −0.772187
$$484$$ 0 0
$$485$$ −1.75736 −0.0797976
$$486$$ 0 0
$$487$$ 24.3848 1.10498 0.552490 0.833520i $$-0.313678\pi$$
0.552490 + 0.833520i $$0.313678\pi$$
$$488$$ 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
$$493$$ −50.6274 −2.28014
$$494$$ 0 0
$$495$$ 5.31371 0.238833
$$496$$ 0 0
$$497$$ −52.2843 −2.34527
$$498$$ 0 0
$$499$$ −26.9706 −1.20737 −0.603684 0.797224i $$-0.706301\pi$$
−0.603684 + 0.797224i $$0.706301\pi$$
$$500$$ 0 0
$$501$$ 6.28427 0.280761
$$502$$ 0 0
$$503$$ 2.97056 0.132451 0.0662254 0.997805i $$-0.478904\pi$$
0.0662254 + 0.997805i $$0.478904\pi$$
$$504$$ 0 0
$$505$$ 4.00000 0.177998
$$506$$ 0 0
$$507$$ 4.66905 0.207360
$$508$$ 0 0
$$509$$ −33.7990 −1.49811 −0.749057 0.662506i $$-0.769493\pi$$
−0.749057 + 0.662506i $$0.769493\pi$$
$$510$$ 0 0
$$511$$ −69.9411 −3.09401
$$512$$ 0 0
$$513$$ −3.31371 −0.146304
$$514$$ 0 0
$$515$$ −15.8995 −0.700615
$$516$$ 0 0
$$517$$ −1.65685 −0.0728684
$$518$$ 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
$$523$$ −6.72792 −0.294191 −0.147096 0.989122i $$-0.546993\pi$$
−0.147096 + 0.989122i $$0.546993\pi$$
$$524$$ 0 0
$$525$$ 2.82843 0.123443
$$526$$ 0 0
$$527$$ 5.65685 0.246416
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ −6.22540 −0.270159
$$532$$ 0 0
$$533$$ −17.1716 −0.743783
$$534$$ 0 0
$$535$$ 4.58579 0.198261
$$536$$ 0 0
$$537$$ −12.6863 −0.547454
$$538$$ 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
$$543$$ 10.5442 0.452493
$$544$$ 0 0
$$545$$ −8.82843 −0.378168
$$546$$ 0 0
$$547$$ 36.3848 1.55570 0.777850 0.628450i $$-0.216310\pi$$
0.777850 + 0.628450i $$0.216310\pi$$
$$548$$ 0 0
$$549$$ 6.22540 0.265693
$$550$$ 0 0
$$551$$ −10.4853 −0.446688
$$552$$ 0 0
$$553$$ 46.6274 1.98280
$$554$$ 0 0
$$555$$ 6.00000 0.254686
$$556$$ 0 0
$$557$$ 41.7990 1.77108 0.885540 0.464563i $$-0.153789\pi$$
0.885540 + 0.464563i $$0.153789\pi$$
$$558$$ 0 0
$$559$$ −1.85786 −0.0785793
$$560$$ 0 0
$$561$$ −5.65685 −0.238833
$$562$$ 0 0
$$563$$ −11.4142 −0.481052 −0.240526 0.970643i $$-0.577320\pi$$
−0.240526 + 0.970643i $$0.577320\pi$$
$$564$$ 0 0
$$565$$ 5.07107 0.213341
$$566$$ 0 0
$$567$$ −29.1127 −1.22262
$$568$$ 0 0
$$569$$ 0.828427 0.0347295 0.0173647 0.999849i $$-0.494472\pi$$
0.0173647 + 0.999849i $$0.494472\pi$$
$$570$$ 0 0
$$571$$ −14.6863 −0.614602 −0.307301 0.951612i $$-0.599426\pi$$
−0.307301 + 0.951612i $$0.599426\pi$$
$$572$$ 0 0
$$573$$ −2.34315 −0.0978863
$$574$$ 0 0
$$575$$ −6.00000 −0.250217
$$576$$ 0 0
$$577$$ −6.00000 −0.249783 −0.124892 0.992170i $$-0.539858\pi$$
−0.124892 + 0.992170i $$0.539858\pi$$
$$578$$ 0 0
$$579$$ −0.343146 −0.0142607
$$580$$ 0 0
$$581$$ −44.9706 −1.86569
$$582$$ 0 0
$$583$$ 10.1421 0.420044
$$584$$ 0 0
$$585$$ −5.95837 −0.246348
$$586$$ 0 0
$$587$$ −14.9706 −0.617901 −0.308951 0.951078i $$-0.599978\pi$$
−0.308951 + 0.951078i $$0.599978\pi$$
$$588$$ 0 0
$$589$$ 1.17157 0.0482738
$$590$$ 0 0
$$591$$ 3.11270 0.128039
$$592$$ 0 0
$$593$$ 8.62742 0.354286 0.177143 0.984185i $$-0.443315\pi$$
0.177143 + 0.984185i $$0.443315\pi$$
$$594$$ 0 0
$$595$$ 23.3137 0.955769
$$596$$ 0 0
$$597$$ 6.05887 0.247973
$$598$$ 0 0
$$599$$ −4.68629 −0.191477 −0.0957383 0.995407i $$-0.530521\pi$$
−0.0957383 + 0.995407i $$0.530521\pi$$
$$600$$ 0 0
$$601$$ −18.0000 −0.734235 −0.367118 0.930175i $$-0.619655\pi$$
−0.367118 + 0.930175i $$0.619655\pi$$
$$602$$ 0 0
$$603$$ 1.55635 0.0633794
$$604$$ 0 0
$$605$$ −7.00000 −0.284590
$$606$$ 0 0
$$607$$ 5.07107 0.205828 0.102914 0.994690i $$-0.467183\pi$$
0.102914 + 0.994690i $$0.467183\pi$$
$$608$$ 0 0
$$609$$ 29.6569 1.20176
$$610$$ 0 0
$$611$$ 1.85786 0.0751611
$$612$$ 0 0
$$613$$ 17.5147 0.707413 0.353706 0.935356i $$-0.384921\pi$$
0.353706 + 0.935356i $$0.384921\pi$$
$$614$$ 0 0
$$615$$ 4.48528 0.180864
$$616$$ 0 0
$$617$$ −6.20101 −0.249643 −0.124822 0.992179i $$-0.539836\pi$$
−0.124822 + 0.992179i $$0.539836\pi$$
$$618$$ 0 0
$$619$$ 18.0000 0.723481 0.361741 0.932279i $$-0.382183\pi$$
0.361741 + 0.932279i $$0.382183\pi$$
$$620$$ 0 0
$$621$$ −19.8823 −0.797847
$$622$$ 0 0
$$623$$ −50.6274 −2.02834
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −1.17157 −0.0467881
$$628$$ 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
$$633$$ 6.74517 0.268096
$$634$$ 0 0
$$635$$ −21.5563 −0.855438
$$636$$ 0 0
$$637$$ 36.5858 1.44958
$$638$$ 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
$$643$$ 24.6274 0.971211 0.485605 0.874178i $$-0.338599\pi$$
0.485605 + 0.874178i $$0.338599\pi$$
$$644$$ 0 0
$$645$$ 0.485281 0.0191079
$$646$$ 0 0
$$647$$ 6.68629 0.262865 0.131433 0.991325i $$-0.458042\pi$$
0.131433 + 0.991325i $$0.458042\pi$$
$$648$$ 0 0
$$649$$ −4.68629 −0.183953
$$650$$ 0 0
$$651$$ −3.31371 −0.129874
$$652$$ 0 0
$$653$$ 13.7990 0.539996 0.269998 0.962861i $$-0.412977\pi$$
0.269998 + 0.962861i $$0.412977\pi$$
$$654$$ 0 0
$$655$$ −5.65685 −0.221032
$$656$$ 0 0
$$657$$ −38.4853 −1.50145
$$658$$ 0 0
$$659$$ −34.6274 −1.34889 −0.674446 0.738324i $$-0.735618\pi$$
−0.674446 + 0.738324i $$0.735618\pi$$
$$660$$ 0 0
$$661$$ −12.6274 −0.491150 −0.245575 0.969378i $$-0.578977\pi$$
−0.245575 + 0.969378i $$0.578977\pi$$
$$662$$ 0 0
$$663$$ 6.34315 0.246347
$$664$$ 0 0
$$665$$ 4.82843 0.187238
$$666$$ 0 0
$$667$$ −62.9117 −2.43595
$$668$$ 0 0
$$669$$ −11.2548 −0.435137
$$670$$ 0 0
$$671$$ 4.68629 0.180912
$$672$$ 0 0
$$673$$ 10.2426 0.394825 0.197412 0.980321i $$-0.436746\pi$$
0.197412 + 0.980321i $$0.436746\pi$$
$$674$$ 0 0
$$675$$ 3.31371 0.127545
$$676$$ 0 0
$$677$$ 25.5563 0.982210 0.491105 0.871100i $$-0.336593\pi$$
0.491105 + 0.871100i $$0.336593\pi$$
$$678$$ 0 0
$$679$$ 8.48528 0.325635
$$680$$ 0 0
$$681$$ −12.0589 −0.462097
$$682$$ 0 0
$$683$$ −38.7279 −1.48188 −0.740941 0.671570i $$-0.765620\pi$$
−0.740941 + 0.671570i $$0.765620\pi$$
$$684$$ 0 0
$$685$$ −20.1421 −0.769591
$$686$$ 0 0
$$687$$ 12.6863 0.484012
$$688$$ 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
$$693$$ −25.6569 −0.974623
$$694$$ 0 0
$$695$$ −17.3137 −0.656746
$$696$$ 0 0
$$697$$ 36.9706 1.40036
$$698$$ 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
$$703$$ 10.2426 0.386309
$$704$$ 0 0
$$705$$ −0.485281 −0.0182768
$$706$$ 0 0
$$707$$ −19.3137 −0.726367
$$708$$ 0 0
$$709$$ 18.9706 0.712454 0.356227 0.934399i $$-0.384063\pi$$
0.356227 + 0.934399i $$0.384063\pi$$
$$710$$ 0 0
$$711$$ 25.6569 0.962207
$$712$$ 0 0
$$713$$ 7.02944 0.263254
$$714$$ 0 0
$$715$$ −4.48528 −0.167740
$$716$$ 0 0
$$717$$ −6.05887 −0.226273
$$718$$ 0 0
$$719$$ 38.2843 1.42776 0.713881 0.700267i $$-0.246936\pi$$
0.713881 + 0.700267i $$0.246936\pi$$
$$720$$ 0 0
$$721$$ 76.7696 2.85905
$$722$$ 0 0
$$723$$ 17.7401 0.659762
$$724$$ 0 0
$$725$$ 10.4853 0.389414
$$726$$ 0 0
$$727$$ −29.5147 −1.09464 −0.547320 0.836923i $$-0.684352\pi$$
−0.547320 + 0.836923i $$0.684352\pi$$
$$728$$ 0 0
$$729$$ −10.1960 −0.377628
$$730$$ 0 0
$$731$$ 4.00000 0.147945
$$732$$ 0 0
$$733$$ 39.6569 1.46476 0.732380 0.680896i $$-0.238410\pi$$
0.732380 + 0.680896i $$0.238410\pi$$
$$734$$ 0 0
$$735$$ −9.55635 −0.352491
$$736$$ 0 0
$$737$$ 1.17157 0.0431554
$$738$$ 0 0
$$739$$ 34.6274 1.27379 0.636895 0.770951i $$-0.280218\pi$$
0.636895 + 0.770951i $$0.280218\pi$$
$$740$$ 0 0
$$741$$ 1.31371 0.0482603
$$742$$ 0 0
$$743$$ 14.0416 0.515137 0.257569 0.966260i $$-0.417079\pi$$
0.257569 + 0.966260i $$0.417079\pi$$
$$744$$ 0 0
$$745$$ −8.00000 −0.293097
$$746$$ 0 0
$$747$$ −24.7452 −0.905378
$$748$$ 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
$$753$$ 8.00000 0.291536
$$754$$ 0 0
$$755$$ 18.1421 0.660260
$$756$$ 0 0
$$757$$ 19.6569 0.714441 0.357220 0.934020i $$-0.383725\pi$$
0.357220 + 0.934020i $$0.383725\pi$$
$$758$$ 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
$$763$$ 42.6274 1.54322
$$764$$ 0 0
$$765$$ 12.8284 0.463813
$$766$$ 0 0
$$767$$ 5.25483 0.189741
$$768$$ 0 0
$$769$$ 38.6274 1.39294 0.696470 0.717586i $$-0.254753\pi$$
0.696470 + 0.717586i $$0.254753\pi$$
$$770$$ 0 0
$$771$$ 1.31371 0.0473121
$$772$$ 0 0
$$773$$ −0.100505 −0.00361492 −0.00180746 0.999998i $$-0.500575\pi$$
−0.00180746 + 0.999998i $$0.500575\pi$$
$$774$$ 0 0
$$775$$ −1.17157 −0.0420841
$$776$$ 0 0
$$777$$ −28.9706 −1.03931
$$778$$ 0 0
$$779$$ 7.65685 0.274335
$$780$$ 0 0
$$781$$ −21.6569 −0.774943
$$782$$ 0 0
$$783$$ 34.7452 1.24169
$$784$$ 0 0
$$785$$ 3.17157 0.113198
$$786$$ 0 0
$$787$$ −41.8406 −1.49146 −0.745729 0.666250i $$-0.767898\pi$$
−0.745729 + 0.666250i $$0.767898\pi$$
$$788$$ 0 0
$$789$$ −2.14214 −0.0762620
$$790$$ 0 0
$$791$$ −24.4853 −0.870596
$$792$$ 0 0
$$793$$ −5.25483 −0.186605
$$794$$ 0 0
$$795$$ 2.97056 0.105355
$$796$$ 0 0
$$797$$ 11.8995 0.421502 0.210751 0.977540i $$-0.432409\pi$$
0.210751 + 0.977540i $$0.432409\pi$$
$$798$$ 0 0
$$799$$ −4.00000 −0.141510
$$800$$ 0 0
$$801$$ −27.8579 −0.984309
$$802$$ 0 0
$$803$$ −28.9706 −1.02235
$$804$$ 0 0
$$805$$ 28.9706 1.02108
$$806$$ 0 0
$$807$$ 13.1716 0.463661
$$808$$ 0 0
$$809$$ −39.2548 −1.38013 −0.690063 0.723749i $$-0.742417\pi$$
−0.690063 + 0.723749i $$0.742417\pi$$
$$810$$ 0 0
$$811$$ 19.7990 0.695237 0.347618 0.937636i $$-0.386991\pi$$
0.347618 + 0.937636i $$0.386991\pi$$
$$812$$ 0 0
$$813$$ −13.8579 −0.486017
$$814$$ 0 0
$$815$$ −2.48528 −0.0870556
$$816$$ 0 0
$$817$$ 0.828427 0.0289830
$$818$$ 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
$$823$$ −15.1716 −0.528848 −0.264424 0.964407i $$-0.585182\pi$$
−0.264424 + 0.964407i $$0.585182\pi$$
$$824$$ 0 0
$$825$$ 1.17157 0.0407889
$$826$$ 0 0
$$827$$ −29.0711 −1.01090 −0.505450 0.862856i $$-0.668674\pi$$
−0.505450 + 0.862856i $$0.668674\pi$$
$$828$$ 0 0
$$829$$ −40.4264 −1.40407 −0.702034 0.712144i $$-0.747724\pi$$
−0.702034 + 0.712144i $$0.747724\pi$$
$$830$$ 0 0
$$831$$ −9.85786 −0.341966
$$832$$ 0 0
$$833$$ −78.7696 −2.72920
$$834$$ 0 0
$$835$$ −10.7279 −0.371255
$$836$$ 0 0
$$837$$ −3.88225 −0.134190
$$838$$ 0 0
$$839$$ 43.5980 1.50517 0.752585 0.658495i $$-0.228807\pi$$
0.752585 + 0.658495i $$0.228807\pi$$
$$840$$ 0 0
$$841$$ 80.9411 2.79107
$$842$$ 0 0
$$843$$ −11.1127 −0.382742
$$844$$ 0 0
$$845$$ −7.97056 −0.274196
$$846$$ 0 0
$$847$$ 33.7990 1.16135
$$848$$ 0 0
$$849$$ 1.57359 0.0540056
$$850$$ 0 0
$$851$$ 61.4558 2.10668
$$852$$ 0 0
$$853$$ 42.0000 1.43805 0.719026 0.694983i $$-0.244588\pi$$
0.719026 + 0.694983i $$0.244588\pi$$
$$854$$ 0 0
$$855$$ 2.65685 0.0908625
$$856$$ 0 0
$$857$$ −27.2132 −0.929585 −0.464793 0.885420i $$-0.653871\pi$$
−0.464793 + 0.885420i $$0.653871\pi$$
$$858$$ 0 0
$$859$$ −24.2843 −0.828569 −0.414284 0.910148i $$-0.635968\pi$$
−0.414284 + 0.910148i $$0.635968\pi$$
$$860$$ 0 0
$$861$$ −21.6569 −0.738064
$$862$$ 0 0
$$863$$ −3.89949 −0.132740 −0.0663702 0.997795i $$-0.521142\pi$$
−0.0663702 + 0.997795i $$0.521142\pi$$
$$864$$ 0 0
$$865$$ −3.89949 −0.132587
$$866$$ 0 0
$$867$$ −3.69848 −0.125607
$$868$$ 0 0
$$869$$ 19.3137 0.655173
$$870$$ 0 0
$$871$$ −1.31371 −0.0445133
$$872$$ 0 0
$$873$$ 4.66905 0.158023
$$874$$ 0 0
$$875$$ −4.82843 −0.163231
$$876$$ 0 0
$$877$$ −37.0711 −1.25180 −0.625901 0.779903i $$-0.715268\pi$$
−0.625901 + 0.779903i $$0.715268\pi$$
$$878$$ 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
$$883$$ −39.1716 −1.31823 −0.659114 0.752043i $$-0.729069\pi$$
−0.659114 + 0.752043i $$0.729069\pi$$
$$884$$ 0 0
$$885$$ −1.37258 −0.0461389
$$886$$ 0 0
$$887$$ −49.0711 −1.64765 −0.823823 0.566848i $$-0.808163\pi$$
−0.823823 + 0.566848i $$0.808163\pi$$
$$888$$ 0 0
$$889$$ 104.083 3.49084
$$890$$ 0 0
$$891$$ −12.0589 −0.403987
$$892$$ 0 0
$$893$$ −0.828427 −0.0277223
$$894$$ 0 0
$$895$$ 21.6569 0.723909
$$896$$ 0 0
$$897$$ 7.88225 0.263181
$$898$$ 0 0
$$899$$ −12.2843 −0.409703
$$900$$ 0 0
$$901$$ 24.4853 0.815723
$$902$$ 0 0
$$903$$ −2.34315 −0.0779750
$$904$$ 0 0
$$905$$ −18.0000 −0.598340
$$906$$ 0 0
$$907$$ −20.3848 −0.676865 −0.338433 0.940991i $$-0.609897\pi$$
−0.338433 + 0.940991i $$0.609897\pi$$
$$908$$ 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
$$913$$ −18.6274 −0.616478
$$914$$ 0 0
$$915$$ 1.37258 0.0453762
$$916$$ 0 0
$$917$$ 27.3137 0.901978
$$918$$ 0 0
$$919$$ 35.3137 1.16489 0.582446 0.812869i $$-0.302096\pi$$
0.582446 + 0.812869i $$0.302096\pi$$
$$920$$ 0 0
$$921$$ −4.62742 −0.152479
$$922$$ 0 0
$$923$$ 24.2843 0.799327
$$924$$ 0 0
$$925$$ −10.2426 −0.336776
$$926$$ 0 0
$$927$$ 42.2426 1.38743
$$928$$ 0 0
$$929$$ 6.68629 0.219370 0.109685 0.993966i $$-0.465016\pi$$
0.109685 + 0.993966i $$0.465016\pi$$
$$930$$ 0 0
$$931$$ −16.3137 −0.534660
$$932$$ 0 0
$$933$$ 8.20101 0.268489
$$934$$ 0 0
$$935$$ 9.65685 0.315813
$$936$$ 0 0
$$937$$ −40.1421 −1.31139 −0.655693 0.755027i $$-0.727624\pi$$
−0.655693 + 0.755027i $$0.727624\pi$$
$$938$$ 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
$$943$$ 45.9411 1.49605
$$944$$ 0 0
$$945$$ −16.0000 −0.520480
$$946$$ 0 0
$$947$$ −8.34315 −0.271116 −0.135558 0.990769i $$-0.543283\pi$$
−0.135558 + 0.990769i $$0.543283\pi$$
$$948$$ 0 0
$$949$$ 32.4853 1.05452
$$950$$ 0 0
$$951$$ 3.37258 0.109363
$$952$$ 0 0
$$953$$ 5.75736 0.186499 0.0932496 0.995643i $$-0.470275\pi$$
0.0932496 + 0.995643i $$0.470275\pi$$
$$954$$ 0 0
$$955$$ 4.00000 0.129437
$$956$$ 0 0
$$957$$ 12.2843 0.397094
$$958$$ 0 0
$$959$$ 97.2548 3.14052
$$960$$ 0 0
$$961$$ −29.6274 −0.955723
$$962$$ 0 0
$$963$$ −12.1838 −0.392616
$$964$$ 0 0
$$965$$ 0.585786 0.0188571
$$966$$ 0 0
$$967$$ 20.6274 0.663333 0.331667 0.943397i $$-0.392389\pi$$
0.331667 + 0.943397i $$0.392389\pi$$
$$968$$ 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
$$973$$ 83.5980 2.68003
$$974$$ 0 0
$$975$$ −1.31371 −0.0420723
$$976$$ 0 0
$$977$$ 9.27208 0.296640 0.148320 0.988939i $$-0.452613\pi$$
0.148320 + 0.988939i $$0.452613\pi$$
$$978$$ 0 0
$$979$$ −20.9706 −0.670222
$$980$$ 0 0
$$981$$ 23.4558 0.748887
$$982$$ 0 0
$$983$$ −23.4142 −0.746797 −0.373399 0.927671i $$-0.621808\pi$$
−0.373399 + 0.927671i $$0.621808\pi$$
$$984$$ 0 0
$$985$$ −5.31371 −0.169309
$$986$$ 0 0
$$987$$ 2.34315 0.0745832
$$988$$ 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
$$993$$ −0.686292 −0.0217788
$$994$$ 0 0
$$995$$ −10.3431 −0.327900
$$996$$ 0 0
$$997$$ −13.5147 −0.428015 −0.214008 0.976832i $$-0.568652\pi$$
−0.214008 + 0.976832i $$0.568652\pi$$
$$998$$ 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 380.2.a.c.1.2 2
3.2 odd 2 3420.2.a.g.1.1 2
4.3 odd 2 1520.2.a.o.1.1 2
5.2 odd 4 1900.2.c.d.1749.3 4
5.3 odd 4 1900.2.c.d.1749.2 4
5.4 even 2 1900.2.a.e.1.1 2
8.3 odd 2 6080.2.a.y.1.2 2
8.5 even 2 6080.2.a.bl.1.1 2
19.18 odd 2 7220.2.a.m.1.1 2
20.19 odd 2 7600.2.a.u.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.a.c.1.2 2 1.1 even 1 trivial
1520.2.a.o.1.1 2 4.3 odd 2
1900.2.a.e.1.1 2 5.4 even 2
1900.2.c.d.1749.2 4 5.3 odd 4
1900.2.c.d.1749.3 4 5.2 odd 4
3420.2.a.g.1.1 2 3.2 odd 2
6080.2.a.y.1.2 2 8.3 odd 2
6080.2.a.bl.1.1 2 8.5 even 2
7220.2.a.m.1.1 2 19.18 odd 2
7600.2.a.u.1.2 2 20.19 odd 2