Properties

 Label 520.2.a.c.1.2 Level $520$ Weight $2$ Character 520.1 Self dual yes Analytic conductor $4.152$ 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] = [520,2,Mod(1,520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(520, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$520 = 2^{3} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 520.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: $$4.15222090511$$ 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 $$\chi$$ $$=$$ 520.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.585786 q^{3} +1.00000 q^{5} -2.00000 q^{7} -2.65685 q^{9} +O(q^{10})$$ $$q-0.585786 q^{3} +1.00000 q^{5} -2.00000 q^{7} -2.65685 q^{9} -4.24264 q^{11} -1.00000 q^{13} -0.585786 q^{15} +0.828427 q^{17} +0.242641 q^{19} +1.17157 q^{21} -9.07107 q^{23} +1.00000 q^{25} +3.31371 q^{27} +1.65685 q^{29} +1.41421 q^{31} +2.48528 q^{33} -2.00000 q^{35} -6.82843 q^{37} +0.585786 q^{39} +4.82843 q^{41} -10.2426 q^{43} -2.65685 q^{45} +2.00000 q^{47} -3.00000 q^{49} -0.485281 q^{51} -8.82843 q^{53} -4.24264 q^{55} -0.142136 q^{57} -2.58579 q^{59} +15.3137 q^{61} +5.31371 q^{63} -1.00000 q^{65} -4.82843 q^{67} +5.31371 q^{69} +9.89949 q^{71} +1.17157 q^{73} -0.585786 q^{75} +8.48528 q^{77} +1.17157 q^{79} +6.02944 q^{81} -2.00000 q^{83} +0.828427 q^{85} -0.970563 q^{87} -10.0000 q^{89} +2.00000 q^{91} -0.828427 q^{93} +0.242641 q^{95} +11.6569 q^{97} +11.2721 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{3} + 2 q^{5} - 4 q^{7} + 6 q^{9}+O(q^{10})$$ 2 * q - 4 * q^3 + 2 * q^5 - 4 * q^7 + 6 * q^9 $$2 q - 4 q^{3} + 2 q^{5} - 4 q^{7} + 6 q^{9} - 2 q^{13} - 4 q^{15} - 4 q^{17} - 8 q^{19} + 8 q^{21} - 4 q^{23} + 2 q^{25} - 16 q^{27} - 8 q^{29} - 12 q^{33} - 4 q^{35} - 8 q^{37} + 4 q^{39} + 4 q^{41} - 12 q^{43} + 6 q^{45} + 4 q^{47} - 6 q^{49} + 16 q^{51} - 12 q^{53} + 28 q^{57} - 8 q^{59} + 8 q^{61} - 12 q^{63} - 2 q^{65} - 4 q^{67} - 12 q^{69} + 8 q^{73} - 4 q^{75} + 8 q^{79} + 46 q^{81} - 4 q^{83} - 4 q^{85} + 32 q^{87} - 20 q^{89} + 4 q^{91} + 4 q^{93} - 8 q^{95} + 12 q^{97} + 48 q^{99}+O(q^{100})$$ 2 * q - 4 * q^3 + 2 * q^5 - 4 * q^7 + 6 * q^9 - 2 * q^13 - 4 * q^15 - 4 * q^17 - 8 * q^19 + 8 * q^21 - 4 * q^23 + 2 * q^25 - 16 * q^27 - 8 * q^29 - 12 * q^33 - 4 * q^35 - 8 * q^37 + 4 * q^39 + 4 * q^41 - 12 * q^43 + 6 * q^45 + 4 * q^47 - 6 * q^49 + 16 * q^51 - 12 * q^53 + 28 * q^57 - 8 * q^59 + 8 * q^61 - 12 * q^63 - 2 * q^65 - 4 * q^67 - 12 * q^69 + 8 * q^73 - 4 * q^75 + 8 * q^79 + 46 * q^81 - 4 * q^83 - 4 * q^85 + 32 * q^87 - 20 * q^89 + 4 * q^91 + 4 * q^93 - 8 * q^95 + 12 * q^97 + 48 * 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
$$7$$ −2.00000 −0.755929 −0.377964 0.925820i $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ 0 0
$$9$$ −2.65685 −0.885618
$$10$$ 0 0
$$11$$ −4.24264 −1.27920 −0.639602 0.768706i $$-0.720901\pi$$
−0.639602 + 0.768706i $$0.720901\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ −0.585786 −0.151249
$$16$$ 0 0
$$17$$ 0.828427 0.200923 0.100462 0.994941i $$-0.467968\pi$$
0.100462 + 0.994941i $$0.467968\pi$$
$$18$$ 0 0
$$19$$ 0.242641 0.0556656 0.0278328 0.999613i $$-0.491139\pi$$
0.0278328 + 0.999613i $$0.491139\pi$$
$$20$$ 0 0
$$21$$ 1.17157 0.255658
$$22$$ 0 0
$$23$$ −9.07107 −1.89145 −0.945724 0.324970i $$-0.894646\pi$$
−0.945724 + 0.324970i $$0.894646\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 3.31371 0.637723
$$28$$ 0 0
$$29$$ 1.65685 0.307670 0.153835 0.988097i $$-0.450838\pi$$
0.153835 + 0.988097i $$0.450838\pi$$
$$30$$ 0 0
$$31$$ 1.41421 0.254000 0.127000 0.991903i $$-0.459465\pi$$
0.127000 + 0.991903i $$0.459465\pi$$
$$32$$ 0 0
$$33$$ 2.48528 0.432632
$$34$$ 0 0
$$35$$ −2.00000 −0.338062
$$36$$ 0 0
$$37$$ −6.82843 −1.12259 −0.561293 0.827617i $$-0.689696\pi$$
−0.561293 + 0.827617i $$0.689696\pi$$
$$38$$ 0 0
$$39$$ 0.585786 0.0938009
$$40$$ 0 0
$$41$$ 4.82843 0.754074 0.377037 0.926198i $$-0.376943\pi$$
0.377037 + 0.926198i $$0.376943\pi$$
$$42$$ 0 0
$$43$$ −10.2426 −1.56199 −0.780994 0.624538i $$-0.785287\pi$$
−0.780994 + 0.624538i $$0.785287\pi$$
$$44$$ 0 0
$$45$$ −2.65685 −0.396060
$$46$$ 0 0
$$47$$ 2.00000 0.291730 0.145865 0.989305i $$-0.453403\pi$$
0.145865 + 0.989305i $$0.453403\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ −0.485281 −0.0679530
$$52$$ 0 0
$$53$$ −8.82843 −1.21268 −0.606339 0.795206i $$-0.707362\pi$$
−0.606339 + 0.795206i $$0.707362\pi$$
$$54$$ 0 0
$$55$$ −4.24264 −0.572078
$$56$$ 0 0
$$57$$ −0.142136 −0.0188263
$$58$$ 0 0
$$59$$ −2.58579 −0.336641 −0.168320 0.985732i $$-0.553834\pi$$
−0.168320 + 0.985732i $$0.553834\pi$$
$$60$$ 0 0
$$61$$ 15.3137 1.96072 0.980360 0.197218i $$-0.0631906\pi$$
0.980360 + 0.197218i $$0.0631906\pi$$
$$62$$ 0 0
$$63$$ 5.31371 0.669464
$$64$$ 0 0
$$65$$ −1.00000 −0.124035
$$66$$ 0 0
$$67$$ −4.82843 −0.589886 −0.294943 0.955515i $$-0.595301\pi$$
−0.294943 + 0.955515i $$0.595301\pi$$
$$68$$ 0 0
$$69$$ 5.31371 0.639695
$$70$$ 0 0
$$71$$ 9.89949 1.17485 0.587427 0.809277i $$-0.300141\pi$$
0.587427 + 0.809277i $$0.300141\pi$$
$$72$$ 0 0
$$73$$ 1.17157 0.137122 0.0685611 0.997647i $$-0.478159\pi$$
0.0685611 + 0.997647i $$0.478159\pi$$
$$74$$ 0 0
$$75$$ −0.585786 −0.0676408
$$76$$ 0 0
$$77$$ 8.48528 0.966988
$$78$$ 0 0
$$79$$ 1.17157 0.131812 0.0659061 0.997826i $$-0.479006\pi$$
0.0659061 + 0.997826i $$0.479006\pi$$
$$80$$ 0 0
$$81$$ 6.02944 0.669937
$$82$$ 0 0
$$83$$ −2.00000 −0.219529 −0.109764 0.993958i $$-0.535010\pi$$
−0.109764 + 0.993958i $$0.535010\pi$$
$$84$$ 0 0
$$85$$ 0.828427 0.0898555
$$86$$ 0 0
$$87$$ −0.970563 −0.104055
$$88$$ 0 0
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ 0 0
$$93$$ −0.828427 −0.0859039
$$94$$ 0 0
$$95$$ 0.242641 0.0248944
$$96$$ 0 0
$$97$$ 11.6569 1.18357 0.591787 0.806094i $$-0.298423\pi$$
0.591787 + 0.806094i $$0.298423\pi$$
$$98$$ 0 0
$$99$$ 11.2721 1.13289
$$100$$ 0 0
$$101$$ −3.65685 −0.363871 −0.181935 0.983311i $$-0.558236\pi$$
−0.181935 + 0.983311i $$0.558236\pi$$
$$102$$ 0 0
$$103$$ 11.4142 1.12468 0.562338 0.826908i $$-0.309902\pi$$
0.562338 + 0.826908i $$0.309902\pi$$
$$104$$ 0 0
$$105$$ 1.17157 0.114334
$$106$$ 0 0
$$107$$ 5.07107 0.490239 0.245119 0.969493i $$-0.421173\pi$$
0.245119 + 0.969493i $$0.421173\pi$$
$$108$$ 0 0
$$109$$ 17.3137 1.65835 0.829176 0.558987i $$-0.188810\pi$$
0.829176 + 0.558987i $$0.188810\pi$$
$$110$$ 0 0
$$111$$ 4.00000 0.379663
$$112$$ 0 0
$$113$$ −8.82843 −0.830509 −0.415254 0.909705i $$-0.636307\pi$$
−0.415254 + 0.909705i $$0.636307\pi$$
$$114$$ 0 0
$$115$$ −9.07107 −0.845881
$$116$$ 0 0
$$117$$ 2.65685 0.245626
$$118$$ 0 0
$$119$$ −1.65685 −0.151884
$$120$$ 0 0
$$121$$ 7.00000 0.636364
$$122$$ 0 0
$$123$$ −2.82843 −0.255031
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 15.8995 1.41085 0.705426 0.708784i $$-0.250756\pi$$
0.705426 + 0.708784i $$0.250756\pi$$
$$128$$ 0 0
$$129$$ 6.00000 0.528271
$$130$$ 0 0
$$131$$ −19.3137 −1.68745 −0.843723 0.536778i $$-0.819641\pi$$
−0.843723 + 0.536778i $$0.819641\pi$$
$$132$$ 0 0
$$133$$ −0.485281 −0.0420792
$$134$$ 0 0
$$135$$ 3.31371 0.285199
$$136$$ 0 0
$$137$$ 2.00000 0.170872 0.0854358 0.996344i $$-0.472772\pi$$
0.0854358 + 0.996344i $$0.472772\pi$$
$$138$$ 0 0
$$139$$ −8.48528 −0.719712 −0.359856 0.933008i $$-0.617174\pi$$
−0.359856 + 0.933008i $$0.617174\pi$$
$$140$$ 0 0
$$141$$ −1.17157 −0.0986642
$$142$$ 0 0
$$143$$ 4.24264 0.354787
$$144$$ 0 0
$$145$$ 1.65685 0.137594
$$146$$ 0 0
$$147$$ 1.75736 0.144945
$$148$$ 0 0
$$149$$ −22.9706 −1.88182 −0.940911 0.338654i $$-0.890028\pi$$
−0.940911 + 0.338654i $$0.890028\pi$$
$$150$$ 0 0
$$151$$ 15.0711 1.22647 0.613233 0.789902i $$-0.289869\pi$$
0.613233 + 0.789902i $$0.289869\pi$$
$$152$$ 0 0
$$153$$ −2.20101 −0.177941
$$154$$ 0 0
$$155$$ 1.41421 0.113592
$$156$$ 0 0
$$157$$ −17.3137 −1.38178 −0.690892 0.722958i $$-0.742782\pi$$
−0.690892 + 0.722958i $$0.742782\pi$$
$$158$$ 0 0
$$159$$ 5.17157 0.410132
$$160$$ 0 0
$$161$$ 18.1421 1.42980
$$162$$ 0 0
$$163$$ −5.51472 −0.431946 −0.215973 0.976399i $$-0.569292\pi$$
−0.215973 + 0.976399i $$0.569292\pi$$
$$164$$ 0 0
$$165$$ 2.48528 0.193479
$$166$$ 0 0
$$167$$ 9.31371 0.720716 0.360358 0.932814i $$-0.382654\pi$$
0.360358 + 0.932814i $$0.382654\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ −0.644661 −0.0492985
$$172$$ 0 0
$$173$$ −15.1716 −1.15347 −0.576737 0.816930i $$-0.695674\pi$$
−0.576737 + 0.816930i $$0.695674\pi$$
$$174$$ 0 0
$$175$$ −2.00000 −0.151186
$$176$$ 0 0
$$177$$ 1.51472 0.113853
$$178$$ 0 0
$$179$$ 11.3137 0.845626 0.422813 0.906217i $$-0.361043\pi$$
0.422813 + 0.906217i $$0.361043\pi$$
$$180$$ 0 0
$$181$$ 0.686292 0.0510116 0.0255058 0.999675i $$-0.491880\pi$$
0.0255058 + 0.999675i $$0.491880\pi$$
$$182$$ 0 0
$$183$$ −8.97056 −0.663123
$$184$$ 0 0
$$185$$ −6.82843 −0.502036
$$186$$ 0 0
$$187$$ −3.51472 −0.257022
$$188$$ 0 0
$$189$$ −6.62742 −0.482074
$$190$$ 0 0
$$191$$ −10.3431 −0.748404 −0.374202 0.927347i $$-0.622083\pi$$
−0.374202 + 0.927347i $$0.622083\pi$$
$$192$$ 0 0
$$193$$ −19.6569 −1.41493 −0.707466 0.706748i $$-0.750162\pi$$
−0.707466 + 0.706748i $$0.750162\pi$$
$$194$$ 0 0
$$195$$ 0.585786 0.0419490
$$196$$ 0 0
$$197$$ −11.6569 −0.830516 −0.415258 0.909704i $$-0.636309\pi$$
−0.415258 + 0.909704i $$0.636309\pi$$
$$198$$ 0 0
$$199$$ 17.6569 1.25166 0.625831 0.779959i $$-0.284760\pi$$
0.625831 + 0.779959i $$0.284760\pi$$
$$200$$ 0 0
$$201$$ 2.82843 0.199502
$$202$$ 0 0
$$203$$ −3.31371 −0.232577
$$204$$ 0 0
$$205$$ 4.82843 0.337232
$$206$$ 0 0
$$207$$ 24.1005 1.67510
$$208$$ 0 0
$$209$$ −1.02944 −0.0712077
$$210$$ 0 0
$$211$$ −5.65685 −0.389434 −0.194717 0.980859i $$-0.562379\pi$$
−0.194717 + 0.980859i $$0.562379\pi$$
$$212$$ 0 0
$$213$$ −5.79899 −0.397340
$$214$$ 0 0
$$215$$ −10.2426 −0.698542
$$216$$ 0 0
$$217$$ −2.82843 −0.192006
$$218$$ 0 0
$$219$$ −0.686292 −0.0463753
$$220$$ 0 0
$$221$$ −0.828427 −0.0557260
$$222$$ 0 0
$$223$$ 2.68629 0.179887 0.0899437 0.995947i $$-0.471331\pi$$
0.0899437 + 0.995947i $$0.471331\pi$$
$$224$$ 0 0
$$225$$ −2.65685 −0.177124
$$226$$ 0 0
$$227$$ 15.1716 1.00697 0.503486 0.864003i $$-0.332050\pi$$
0.503486 + 0.864003i $$0.332050\pi$$
$$228$$ 0 0
$$229$$ 25.7990 1.70485 0.852423 0.522853i $$-0.175132\pi$$
0.852423 + 0.522853i $$0.175132\pi$$
$$230$$ 0 0
$$231$$ −4.97056 −0.327039
$$232$$ 0 0
$$233$$ −22.0000 −1.44127 −0.720634 0.693316i $$-0.756149\pi$$
−0.720634 + 0.693316i $$0.756149\pi$$
$$234$$ 0 0
$$235$$ 2.00000 0.130466
$$236$$ 0 0
$$237$$ −0.686292 −0.0445794
$$238$$ 0 0
$$239$$ −16.7279 −1.08204 −0.541020 0.841010i $$-0.681962\pi$$
−0.541020 + 0.841010i $$0.681962\pi$$
$$240$$ 0 0
$$241$$ 12.8284 0.826352 0.413176 0.910651i $$-0.364419\pi$$
0.413176 + 0.910651i $$0.364419\pi$$
$$242$$ 0 0
$$243$$ −13.4731 −0.864299
$$244$$ 0 0
$$245$$ −3.00000 −0.191663
$$246$$ 0 0
$$247$$ −0.242641 −0.0154389
$$248$$ 0 0
$$249$$ 1.17157 0.0742454
$$250$$ 0 0
$$251$$ −18.1421 −1.14512 −0.572561 0.819862i $$-0.694050\pi$$
−0.572561 + 0.819862i $$0.694050\pi$$
$$252$$ 0 0
$$253$$ 38.4853 2.41955
$$254$$ 0 0
$$255$$ −0.485281 −0.0303895
$$256$$ 0 0
$$257$$ −18.9706 −1.18335 −0.591676 0.806176i $$-0.701533\pi$$
−0.591676 + 0.806176i $$0.701533\pi$$
$$258$$ 0 0
$$259$$ 13.6569 0.848596
$$260$$ 0 0
$$261$$ −4.40202 −0.272478
$$262$$ 0 0
$$263$$ 22.2426 1.37154 0.685770 0.727818i $$-0.259466\pi$$
0.685770 + 0.727818i $$0.259466\pi$$
$$264$$ 0 0
$$265$$ −8.82843 −0.542326
$$266$$ 0 0
$$267$$ 5.85786 0.358495
$$268$$ 0 0
$$269$$ −28.6274 −1.74544 −0.872722 0.488217i $$-0.837647\pi$$
−0.872722 + 0.488217i $$0.837647\pi$$
$$270$$ 0 0
$$271$$ −9.89949 −0.601351 −0.300676 0.953726i $$-0.597212\pi$$
−0.300676 + 0.953726i $$0.597212\pi$$
$$272$$ 0 0
$$273$$ −1.17157 −0.0709068
$$274$$ 0 0
$$275$$ −4.24264 −0.255841
$$276$$ 0 0
$$277$$ −24.1421 −1.45056 −0.725280 0.688454i $$-0.758290\pi$$
−0.725280 + 0.688454i $$0.758290\pi$$
$$278$$ 0 0
$$279$$ −3.75736 −0.224947
$$280$$ 0 0
$$281$$ −5.51472 −0.328981 −0.164490 0.986379i $$-0.552598\pi$$
−0.164490 + 0.986379i $$0.552598\pi$$
$$282$$ 0 0
$$283$$ −15.8995 −0.945127 −0.472563 0.881297i $$-0.656671\pi$$
−0.472563 + 0.881297i $$0.656671\pi$$
$$284$$ 0 0
$$285$$ −0.142136 −0.00841939
$$286$$ 0 0
$$287$$ −9.65685 −0.570026
$$288$$ 0 0
$$289$$ −16.3137 −0.959630
$$290$$ 0 0
$$291$$ −6.82843 −0.400289
$$292$$ 0 0
$$293$$ −2.82843 −0.165238 −0.0826192 0.996581i $$-0.526329\pi$$
−0.0826192 + 0.996581i $$0.526329\pi$$
$$294$$ 0 0
$$295$$ −2.58579 −0.150550
$$296$$ 0 0
$$297$$ −14.0589 −0.815779
$$298$$ 0 0
$$299$$ 9.07107 0.524593
$$300$$ 0 0
$$301$$ 20.4853 1.18075
$$302$$ 0 0
$$303$$ 2.14214 0.123062
$$304$$ 0 0
$$305$$ 15.3137 0.876860
$$306$$ 0 0
$$307$$ 4.34315 0.247876 0.123938 0.992290i $$-0.460448\pi$$
0.123938 + 0.992290i $$0.460448\pi$$
$$308$$ 0 0
$$309$$ −6.68629 −0.380370
$$310$$ 0 0
$$311$$ 18.1421 1.02875 0.514373 0.857567i $$-0.328025\pi$$
0.514373 + 0.857567i $$0.328025\pi$$
$$312$$ 0 0
$$313$$ −12.8284 −0.725106 −0.362553 0.931963i $$-0.618095\pi$$
−0.362553 + 0.931963i $$0.618095\pi$$
$$314$$ 0 0
$$315$$ 5.31371 0.299394
$$316$$ 0 0
$$317$$ 7.51472 0.422069 0.211034 0.977479i $$-0.432317\pi$$
0.211034 + 0.977479i $$0.432317\pi$$
$$318$$ 0 0
$$319$$ −7.02944 −0.393573
$$320$$ 0 0
$$321$$ −2.97056 −0.165801
$$322$$ 0 0
$$323$$ 0.201010 0.0111845
$$324$$ 0 0
$$325$$ −1.00000 −0.0554700
$$326$$ 0 0
$$327$$ −10.1421 −0.560861
$$328$$ 0 0
$$329$$ −4.00000 −0.220527
$$330$$ 0 0
$$331$$ −15.0711 −0.828381 −0.414190 0.910190i $$-0.635935\pi$$
−0.414190 + 0.910190i $$0.635935\pi$$
$$332$$ 0 0
$$333$$ 18.1421 0.994183
$$334$$ 0 0
$$335$$ −4.82843 −0.263805
$$336$$ 0 0
$$337$$ −7.85786 −0.428045 −0.214023 0.976829i $$-0.568657\pi$$
−0.214023 + 0.976829i $$0.568657\pi$$
$$338$$ 0 0
$$339$$ 5.17157 0.280881
$$340$$ 0 0
$$341$$ −6.00000 −0.324918
$$342$$ 0 0
$$343$$ 20.0000 1.07990
$$344$$ 0 0
$$345$$ 5.31371 0.286080
$$346$$ 0 0
$$347$$ −32.3848 −1.73851 −0.869253 0.494368i $$-0.835400\pi$$
−0.869253 + 0.494368i $$0.835400\pi$$
$$348$$ 0 0
$$349$$ −15.4558 −0.827332 −0.413666 0.910429i $$-0.635752\pi$$
−0.413666 + 0.910429i $$0.635752\pi$$
$$350$$ 0 0
$$351$$ −3.31371 −0.176873
$$352$$ 0 0
$$353$$ 10.8284 0.576339 0.288170 0.957579i $$-0.406953\pi$$
0.288170 + 0.957579i $$0.406953\pi$$
$$354$$ 0 0
$$355$$ 9.89949 0.525411
$$356$$ 0 0
$$357$$ 0.970563 0.0513676
$$358$$ 0 0
$$359$$ 3.55635 0.187697 0.0938485 0.995586i $$-0.470083\pi$$
0.0938485 + 0.995586i $$0.470083\pi$$
$$360$$ 0 0
$$361$$ −18.9411 −0.996901
$$362$$ 0 0
$$363$$ −4.10051 −0.215221
$$364$$ 0 0
$$365$$ 1.17157 0.0613229
$$366$$ 0 0
$$367$$ 2.72792 0.142396 0.0711982 0.997462i $$-0.477318\pi$$
0.0711982 + 0.997462i $$0.477318\pi$$
$$368$$ 0 0
$$369$$ −12.8284 −0.667821
$$370$$ 0 0
$$371$$ 17.6569 0.916698
$$372$$ 0 0
$$373$$ 10.0000 0.517780 0.258890 0.965907i $$-0.416643\pi$$
0.258890 + 0.965907i $$0.416643\pi$$
$$374$$ 0 0
$$375$$ −0.585786 −0.0302499
$$376$$ 0 0
$$377$$ −1.65685 −0.0853323
$$378$$ 0 0
$$379$$ 7.75736 0.398469 0.199234 0.979952i $$-0.436154\pi$$
0.199234 + 0.979952i $$0.436154\pi$$
$$380$$ 0 0
$$381$$ −9.31371 −0.477156
$$382$$ 0 0
$$383$$ 22.0000 1.12415 0.562074 0.827087i $$-0.310004\pi$$
0.562074 + 0.827087i $$0.310004\pi$$
$$384$$ 0 0
$$385$$ 8.48528 0.432450
$$386$$ 0 0
$$387$$ 27.2132 1.38332
$$388$$ 0 0
$$389$$ 9.31371 0.472224 0.236112 0.971726i $$-0.424127\pi$$
0.236112 + 0.971726i $$0.424127\pi$$
$$390$$ 0 0
$$391$$ −7.51472 −0.380036
$$392$$ 0 0
$$393$$ 11.3137 0.570701
$$394$$ 0 0
$$395$$ 1.17157 0.0589482
$$396$$ 0 0
$$397$$ 6.82843 0.342709 0.171354 0.985209i $$-0.445186\pi$$
0.171354 + 0.985209i $$0.445186\pi$$
$$398$$ 0 0
$$399$$ 0.284271 0.0142314
$$400$$ 0 0
$$401$$ −16.6274 −0.830334 −0.415167 0.909745i $$-0.636277\pi$$
−0.415167 + 0.909745i $$0.636277\pi$$
$$402$$ 0 0
$$403$$ −1.41421 −0.0704470
$$404$$ 0 0
$$405$$ 6.02944 0.299605
$$406$$ 0 0
$$407$$ 28.9706 1.43602
$$408$$ 0 0
$$409$$ −0.828427 −0.0409631 −0.0204815 0.999790i $$-0.506520\pi$$
−0.0204815 + 0.999790i $$0.506520\pi$$
$$410$$ 0 0
$$411$$ −1.17157 −0.0577894
$$412$$ 0 0
$$413$$ 5.17157 0.254476
$$414$$ 0 0
$$415$$ −2.00000 −0.0981761
$$416$$ 0 0
$$417$$ 4.97056 0.243410
$$418$$ 0 0
$$419$$ −5.85786 −0.286175 −0.143088 0.989710i $$-0.545703\pi$$
−0.143088 + 0.989710i $$0.545703\pi$$
$$420$$ 0 0
$$421$$ 24.3431 1.18641 0.593206 0.805051i $$-0.297862\pi$$
0.593206 + 0.805051i $$0.297862\pi$$
$$422$$ 0 0
$$423$$ −5.31371 −0.258361
$$424$$ 0 0
$$425$$ 0.828427 0.0401846
$$426$$ 0 0
$$427$$ −30.6274 −1.48216
$$428$$ 0 0
$$429$$ −2.48528 −0.119991
$$430$$ 0 0
$$431$$ −29.2132 −1.40715 −0.703575 0.710621i $$-0.748414\pi$$
−0.703575 + 0.710621i $$0.748414\pi$$
$$432$$ 0 0
$$433$$ 26.9706 1.29612 0.648061 0.761588i $$-0.275580\pi$$
0.648061 + 0.761588i $$0.275580\pi$$
$$434$$ 0 0
$$435$$ −0.970563 −0.0465349
$$436$$ 0 0
$$437$$ −2.20101 −0.105289
$$438$$ 0 0
$$439$$ −28.2843 −1.34993 −0.674967 0.737848i $$-0.735842\pi$$
−0.674967 + 0.737848i $$0.735842\pi$$
$$440$$ 0 0
$$441$$ 7.97056 0.379551
$$442$$ 0 0
$$443$$ −0.585786 −0.0278316 −0.0139158 0.999903i $$-0.504430\pi$$
−0.0139158 + 0.999903i $$0.504430\pi$$
$$444$$ 0 0
$$445$$ −10.0000 −0.474045
$$446$$ 0 0
$$447$$ 13.4558 0.636440
$$448$$ 0 0
$$449$$ 36.1421 1.70565 0.852826 0.522195i $$-0.174886\pi$$
0.852826 + 0.522195i $$0.174886\pi$$
$$450$$ 0 0
$$451$$ −20.4853 −0.964614
$$452$$ 0 0
$$453$$ −8.82843 −0.414796
$$454$$ 0 0
$$455$$ 2.00000 0.0937614
$$456$$ 0 0
$$457$$ 12.6274 0.590686 0.295343 0.955391i $$-0.404566\pi$$
0.295343 + 0.955391i $$0.404566\pi$$
$$458$$ 0 0
$$459$$ 2.74517 0.128133
$$460$$ 0 0
$$461$$ 8.14214 0.379217 0.189609 0.981860i $$-0.439278\pi$$
0.189609 + 0.981860i $$0.439278\pi$$
$$462$$ 0 0
$$463$$ 5.51472 0.256291 0.128145 0.991755i $$-0.459098\pi$$
0.128145 + 0.991755i $$0.459098\pi$$
$$464$$ 0 0
$$465$$ −0.828427 −0.0384174
$$466$$ 0 0
$$467$$ 5.75736 0.266419 0.133209 0.991088i $$-0.457472\pi$$
0.133209 + 0.991088i $$0.457472\pi$$
$$468$$ 0 0
$$469$$ 9.65685 0.445912
$$470$$ 0 0
$$471$$ 10.1421 0.467325
$$472$$ 0 0
$$473$$ 43.4558 1.99810
$$474$$ 0 0
$$475$$ 0.242641 0.0111331
$$476$$ 0 0
$$477$$ 23.4558 1.07397
$$478$$ 0 0
$$479$$ −23.7574 −1.08550 −0.542751 0.839894i $$-0.682617\pi$$
−0.542751 + 0.839894i $$0.682617\pi$$
$$480$$ 0 0
$$481$$ 6.82843 0.311349
$$482$$ 0 0
$$483$$ −10.6274 −0.483564
$$484$$ 0 0
$$485$$ 11.6569 0.529310
$$486$$ 0 0
$$487$$ −10.4853 −0.475133 −0.237567 0.971371i $$-0.576350\pi$$
−0.237567 + 0.971371i $$0.576350\pi$$
$$488$$ 0 0
$$489$$ 3.23045 0.146086
$$490$$ 0 0
$$491$$ 43.1127 1.94565 0.972824 0.231544i $$-0.0743777\pi$$
0.972824 + 0.231544i $$0.0743777\pi$$
$$492$$ 0 0
$$493$$ 1.37258 0.0618180
$$494$$ 0 0
$$495$$ 11.2721 0.506642
$$496$$ 0 0
$$497$$ −19.7990 −0.888106
$$498$$ 0 0
$$499$$ −29.4142 −1.31676 −0.658381 0.752685i $$-0.728758\pi$$
−0.658381 + 0.752685i $$0.728758\pi$$
$$500$$ 0 0
$$501$$ −5.45584 −0.243749
$$502$$ 0 0
$$503$$ −24.5858 −1.09623 −0.548113 0.836404i $$-0.684654\pi$$
−0.548113 + 0.836404i $$0.684654\pi$$
$$504$$ 0 0
$$505$$ −3.65685 −0.162728
$$506$$ 0 0
$$507$$ −0.585786 −0.0260157
$$508$$ 0 0
$$509$$ 34.4853 1.52853 0.764267 0.644900i $$-0.223101\pi$$
0.764267 + 0.644900i $$0.223101\pi$$
$$510$$ 0 0
$$511$$ −2.34315 −0.103655
$$512$$ 0 0
$$513$$ 0.804041 0.0354993
$$514$$ 0 0
$$515$$ 11.4142 0.502970
$$516$$ 0 0
$$517$$ −8.48528 −0.373182
$$518$$ 0 0
$$519$$ 8.88730 0.390109
$$520$$ 0 0
$$521$$ −35.3137 −1.54712 −0.773561 0.633722i $$-0.781526\pi$$
−0.773561 + 0.633722i $$0.781526\pi$$
$$522$$ 0 0
$$523$$ 35.6985 1.56099 0.780493 0.625165i $$-0.214968\pi$$
0.780493 + 0.625165i $$0.214968\pi$$
$$524$$ 0 0
$$525$$ 1.17157 0.0511316
$$526$$ 0 0
$$527$$ 1.17157 0.0510345
$$528$$ 0 0
$$529$$ 59.2843 2.57758
$$530$$ 0 0
$$531$$ 6.87006 0.298135
$$532$$ 0 0
$$533$$ −4.82843 −0.209142
$$534$$ 0 0
$$535$$ 5.07107 0.219241
$$536$$ 0 0
$$537$$ −6.62742 −0.285994
$$538$$ 0 0
$$539$$ 12.7279 0.548230
$$540$$ 0 0
$$541$$ −2.48528 −0.106851 −0.0534253 0.998572i $$-0.517014\pi$$
−0.0534253 + 0.998572i $$0.517014\pi$$
$$542$$ 0 0
$$543$$ −0.402020 −0.0172523
$$544$$ 0 0
$$545$$ 17.3137 0.741638
$$546$$ 0 0
$$547$$ −18.9289 −0.809343 −0.404671 0.914462i $$-0.632614\pi$$
−0.404671 + 0.914462i $$0.632614\pi$$
$$548$$ 0 0
$$549$$ −40.6863 −1.73645
$$550$$ 0 0
$$551$$ 0.402020 0.0171266
$$552$$ 0 0
$$553$$ −2.34315 −0.0996407
$$554$$ 0 0
$$555$$ 4.00000 0.169791
$$556$$ 0 0
$$557$$ 13.8579 0.587177 0.293588 0.955932i $$-0.405151\pi$$
0.293588 + 0.955932i $$0.405151\pi$$
$$558$$ 0 0
$$559$$ 10.2426 0.433218
$$560$$ 0 0
$$561$$ 2.05887 0.0869257
$$562$$ 0 0
$$563$$ −13.5563 −0.571332 −0.285666 0.958329i $$-0.592215\pi$$
−0.285666 + 0.958329i $$0.592215\pi$$
$$564$$ 0 0
$$565$$ −8.82843 −0.371415
$$566$$ 0 0
$$567$$ −12.0589 −0.506425
$$568$$ 0 0
$$569$$ −8.68629 −0.364148 −0.182074 0.983285i $$-0.558281\pi$$
−0.182074 + 0.983285i $$0.558281\pi$$
$$570$$ 0 0
$$571$$ −29.1716 −1.22079 −0.610396 0.792096i $$-0.708990\pi$$
−0.610396 + 0.792096i $$0.708990\pi$$
$$572$$ 0 0
$$573$$ 6.05887 0.253113
$$574$$ 0 0
$$575$$ −9.07107 −0.378290
$$576$$ 0 0
$$577$$ 5.85786 0.243866 0.121933 0.992538i $$-0.461091\pi$$
0.121933 + 0.992538i $$0.461091\pi$$
$$578$$ 0 0
$$579$$ 11.5147 0.478535
$$580$$ 0 0
$$581$$ 4.00000 0.165948
$$582$$ 0 0
$$583$$ 37.4558 1.55126
$$584$$ 0 0
$$585$$ 2.65685 0.109847
$$586$$ 0 0
$$587$$ 2.48528 0.102579 0.0512893 0.998684i $$-0.483667\pi$$
0.0512893 + 0.998684i $$0.483667\pi$$
$$588$$ 0 0
$$589$$ 0.343146 0.0141391
$$590$$ 0 0
$$591$$ 6.82843 0.280884
$$592$$ 0 0
$$593$$ −26.0000 −1.06769 −0.533846 0.845582i $$-0.679254\pi$$
−0.533846 + 0.845582i $$0.679254\pi$$
$$594$$ 0 0
$$595$$ −1.65685 −0.0679244
$$596$$ 0 0
$$597$$ −10.3431 −0.423317
$$598$$ 0 0
$$599$$ −11.5147 −0.470479 −0.235239 0.971937i $$-0.575587\pi$$
−0.235239 + 0.971937i $$0.575587\pi$$
$$600$$ 0 0
$$601$$ 35.9411 1.46607 0.733035 0.680191i $$-0.238103\pi$$
0.733035 + 0.680191i $$0.238103\pi$$
$$602$$ 0 0
$$603$$ 12.8284 0.522414
$$604$$ 0 0
$$605$$ 7.00000 0.284590
$$606$$ 0 0
$$607$$ −6.44365 −0.261540 −0.130770 0.991413i $$-0.541745\pi$$
−0.130770 + 0.991413i $$0.541745\pi$$
$$608$$ 0 0
$$609$$ 1.94113 0.0786584
$$610$$ 0 0
$$611$$ −2.00000 −0.0809113
$$612$$ 0 0
$$613$$ 22.0000 0.888572 0.444286 0.895885i $$-0.353457\pi$$
0.444286 + 0.895885i $$0.353457\pi$$
$$614$$ 0 0
$$615$$ −2.82843 −0.114053
$$616$$ 0 0
$$617$$ −3.65685 −0.147219 −0.0736097 0.997287i $$-0.523452\pi$$
−0.0736097 + 0.997287i $$0.523452\pi$$
$$618$$ 0 0
$$619$$ 36.0416 1.44864 0.724318 0.689466i $$-0.242155\pi$$
0.724318 + 0.689466i $$0.242155\pi$$
$$620$$ 0 0
$$621$$ −30.0589 −1.20622
$$622$$ 0 0
$$623$$ 20.0000 0.801283
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0.603030 0.0240827
$$628$$ 0 0
$$629$$ −5.65685 −0.225554
$$630$$ 0 0
$$631$$ −12.7279 −0.506691 −0.253345 0.967376i $$-0.581531\pi$$
−0.253345 + 0.967376i $$0.581531\pi$$
$$632$$ 0 0
$$633$$ 3.31371 0.131708
$$634$$ 0 0
$$635$$ 15.8995 0.630952
$$636$$ 0 0
$$637$$ 3.00000 0.118864
$$638$$ 0 0
$$639$$ −26.3015 −1.04047
$$640$$ 0 0
$$641$$ 28.3431 1.11949 0.559743 0.828666i $$-0.310900\pi$$
0.559743 + 0.828666i $$0.310900\pi$$
$$642$$ 0 0
$$643$$ −13.3137 −0.525041 −0.262521 0.964926i $$-0.584554\pi$$
−0.262521 + 0.964926i $$0.584554\pi$$
$$644$$ 0 0
$$645$$ 6.00000 0.236250
$$646$$ 0 0
$$647$$ −37.3553 −1.46859 −0.734295 0.678831i $$-0.762487\pi$$
−0.734295 + 0.678831i $$0.762487\pi$$
$$648$$ 0 0
$$649$$ 10.9706 0.430632
$$650$$ 0 0
$$651$$ 1.65685 0.0649372
$$652$$ 0 0
$$653$$ −9.02944 −0.353349 −0.176675 0.984269i $$-0.556534\pi$$
−0.176675 + 0.984269i $$0.556534\pi$$
$$654$$ 0 0
$$655$$ −19.3137 −0.754649
$$656$$ 0 0
$$657$$ −3.11270 −0.121438
$$658$$ 0 0
$$659$$ 4.48528 0.174722 0.0873609 0.996177i $$-0.472157\pi$$
0.0873609 + 0.996177i $$0.472157\pi$$
$$660$$ 0 0
$$661$$ −9.02944 −0.351204 −0.175602 0.984461i $$-0.556187\pi$$
−0.175602 + 0.984461i $$0.556187\pi$$
$$662$$ 0 0
$$663$$ 0.485281 0.0188468
$$664$$ 0 0
$$665$$ −0.485281 −0.0188184
$$666$$ 0 0
$$667$$ −15.0294 −0.581942
$$668$$ 0 0
$$669$$ −1.57359 −0.0608386
$$670$$ 0 0
$$671$$ −64.9706 −2.50816
$$672$$ 0 0
$$673$$ 33.7990 1.30286 0.651428 0.758711i $$-0.274170\pi$$
0.651428 + 0.758711i $$0.274170\pi$$
$$674$$ 0 0
$$675$$ 3.31371 0.127545
$$676$$ 0 0
$$677$$ −20.1421 −0.774125 −0.387063 0.922053i $$-0.626510\pi$$
−0.387063 + 0.922053i $$0.626510\pi$$
$$678$$ 0 0
$$679$$ −23.3137 −0.894698
$$680$$ 0 0
$$681$$ −8.88730 −0.340562
$$682$$ 0 0
$$683$$ 37.1127 1.42008 0.710039 0.704162i $$-0.248677\pi$$
0.710039 + 0.704162i $$0.248677\pi$$
$$684$$ 0 0
$$685$$ 2.00000 0.0764161
$$686$$ 0 0
$$687$$ −15.1127 −0.576585
$$688$$ 0 0
$$689$$ 8.82843 0.336336
$$690$$ 0 0
$$691$$ −49.4142 −1.87981 −0.939903 0.341443i $$-0.889085\pi$$
−0.939903 + 0.341443i $$0.889085\pi$$
$$692$$ 0 0
$$693$$ −22.5442 −0.856382
$$694$$ 0 0
$$695$$ −8.48528 −0.321865
$$696$$ 0 0
$$697$$ 4.00000 0.151511
$$698$$ 0 0
$$699$$ 12.8873 0.487443
$$700$$ 0 0
$$701$$ −20.6274 −0.779087 −0.389543 0.921008i $$-0.627367\pi$$
−0.389543 + 0.921008i $$0.627367\pi$$
$$702$$ 0 0
$$703$$ −1.65685 −0.0624894
$$704$$ 0 0
$$705$$ −1.17157 −0.0441240
$$706$$ 0 0
$$707$$ 7.31371 0.275060
$$708$$ 0 0
$$709$$ −19.4558 −0.730680 −0.365340 0.930874i $$-0.619047\pi$$
−0.365340 + 0.930874i $$0.619047\pi$$
$$710$$ 0 0
$$711$$ −3.11270 −0.116735
$$712$$ 0 0
$$713$$ −12.8284 −0.480428
$$714$$ 0 0
$$715$$ 4.24264 0.158666
$$716$$ 0 0
$$717$$ 9.79899 0.365950
$$718$$ 0 0
$$719$$ −18.6274 −0.694685 −0.347343 0.937738i $$-0.612916\pi$$
−0.347343 + 0.937738i $$0.612916\pi$$
$$720$$ 0 0
$$721$$ −22.8284 −0.850175
$$722$$ 0 0
$$723$$ −7.51472 −0.279475
$$724$$ 0 0
$$725$$ 1.65685 0.0615340
$$726$$ 0 0
$$727$$ −2.24264 −0.0831749 −0.0415875 0.999135i $$-0.513242\pi$$
−0.0415875 + 0.999135i $$0.513242\pi$$
$$728$$ 0 0
$$729$$ −10.1960 −0.377628
$$730$$ 0 0
$$731$$ −8.48528 −0.313839
$$732$$ 0 0
$$733$$ 32.6274 1.20512 0.602561 0.798073i $$-0.294147\pi$$
0.602561 + 0.798073i $$0.294147\pi$$
$$734$$ 0 0
$$735$$ 1.75736 0.0648212
$$736$$ 0 0
$$737$$ 20.4853 0.754585
$$738$$ 0 0
$$739$$ 18.5858 0.683689 0.341845 0.939756i $$-0.388948\pi$$
0.341845 + 0.939756i $$0.388948\pi$$
$$740$$ 0 0
$$741$$ 0.142136 0.00522148
$$742$$ 0 0
$$743$$ 30.9706 1.13620 0.568100 0.822960i $$-0.307679\pi$$
0.568100 + 0.822960i $$0.307679\pi$$
$$744$$ 0 0
$$745$$ −22.9706 −0.841576
$$746$$ 0 0
$$747$$ 5.31371 0.194418
$$748$$ 0 0
$$749$$ −10.1421 −0.370586
$$750$$ 0 0
$$751$$ 8.48528 0.309632 0.154816 0.987943i $$-0.450521\pi$$
0.154816 + 0.987943i $$0.450521\pi$$
$$752$$ 0 0
$$753$$ 10.6274 0.387285
$$754$$ 0 0
$$755$$ 15.0711 0.548492
$$756$$ 0 0
$$757$$ −8.14214 −0.295931 −0.147965 0.988993i $$-0.547272\pi$$
−0.147965 + 0.988993i $$0.547272\pi$$
$$758$$ 0 0
$$759$$ −22.5442 −0.818301
$$760$$ 0 0
$$761$$ −41.3137 −1.49762 −0.748810 0.662784i $$-0.769375\pi$$
−0.748810 + 0.662784i $$0.769375\pi$$
$$762$$ 0 0
$$763$$ −34.6274 −1.25360
$$764$$ 0 0
$$765$$ −2.20101 −0.0795777
$$766$$ 0 0
$$767$$ 2.58579 0.0933673
$$768$$ 0 0
$$769$$ −19.6569 −0.708844 −0.354422 0.935086i $$-0.615322\pi$$
−0.354422 + 0.935086i $$0.615322\pi$$
$$770$$ 0 0
$$771$$ 11.1127 0.400214
$$772$$ 0 0
$$773$$ 18.1421 0.652527 0.326264 0.945279i $$-0.394210\pi$$
0.326264 + 0.945279i $$0.394210\pi$$
$$774$$ 0 0
$$775$$ 1.41421 0.0508001
$$776$$ 0 0
$$777$$ −8.00000 −0.286998
$$778$$ 0 0
$$779$$ 1.17157 0.0419760
$$780$$ 0 0
$$781$$ −42.0000 −1.50288
$$782$$ 0 0
$$783$$ 5.49033 0.196208
$$784$$ 0 0
$$785$$ −17.3137 −0.617953
$$786$$ 0 0
$$787$$ −38.0000 −1.35455 −0.677277 0.735728i $$-0.736840\pi$$
−0.677277 + 0.735728i $$0.736840\pi$$
$$788$$ 0 0
$$789$$ −13.0294 −0.463860
$$790$$ 0 0
$$791$$ 17.6569 0.627805
$$792$$ 0 0
$$793$$ −15.3137 −0.543806
$$794$$ 0 0
$$795$$ 5.17157 0.183417
$$796$$ 0 0
$$797$$ 49.5980 1.75685 0.878425 0.477880i $$-0.158595\pi$$
0.878425 + 0.477880i $$0.158595\pi$$
$$798$$ 0 0
$$799$$ 1.65685 0.0586153
$$800$$ 0 0
$$801$$ 26.5685 0.938753
$$802$$ 0 0
$$803$$ −4.97056 −0.175407
$$804$$ 0 0
$$805$$ 18.1421 0.639426
$$806$$ 0 0
$$807$$ 16.7696 0.590316
$$808$$ 0 0
$$809$$ −41.6569 −1.46458 −0.732289 0.680995i $$-0.761548\pi$$
−0.732289 + 0.680995i $$0.761548\pi$$
$$810$$ 0 0
$$811$$ 9.89949 0.347618 0.173809 0.984779i $$-0.444392\pi$$
0.173809 + 0.984779i $$0.444392\pi$$
$$812$$ 0 0
$$813$$ 5.79899 0.203379
$$814$$ 0 0
$$815$$ −5.51472 −0.193172
$$816$$ 0 0
$$817$$ −2.48528 −0.0869490
$$818$$ 0 0
$$819$$ −5.31371 −0.185676
$$820$$ 0 0
$$821$$ 6.68629 0.233353 0.116677 0.993170i $$-0.462776\pi$$
0.116677 + 0.993170i $$0.462776\pi$$
$$822$$ 0 0
$$823$$ −16.8701 −0.588053 −0.294027 0.955797i $$-0.594995\pi$$
−0.294027 + 0.955797i $$0.594995\pi$$
$$824$$ 0 0
$$825$$ 2.48528 0.0865264
$$826$$ 0 0
$$827$$ 20.6274 0.717286 0.358643 0.933475i $$-0.383240\pi$$
0.358643 + 0.933475i $$0.383240\pi$$
$$828$$ 0 0
$$829$$ 14.3431 0.498158 0.249079 0.968483i $$-0.419872\pi$$
0.249079 + 0.968483i $$0.419872\pi$$
$$830$$ 0 0
$$831$$ 14.1421 0.490585
$$832$$ 0 0
$$833$$ −2.48528 −0.0861099
$$834$$ 0 0
$$835$$ 9.31371 0.322314
$$836$$ 0 0
$$837$$ 4.68629 0.161982
$$838$$ 0 0
$$839$$ −37.8995 −1.30844 −0.654218 0.756306i $$-0.727002\pi$$
−0.654218 + 0.756306i $$0.727002\pi$$
$$840$$ 0 0
$$841$$ −26.2548 −0.905339
$$842$$ 0 0
$$843$$ 3.23045 0.111263
$$844$$ 0 0
$$845$$ 1.00000 0.0344010
$$846$$ 0 0
$$847$$ −14.0000 −0.481046
$$848$$ 0 0
$$849$$ 9.31371 0.319646
$$850$$ 0 0
$$851$$ 61.9411 2.12331
$$852$$ 0 0
$$853$$ −47.1127 −1.61311 −0.806554 0.591160i $$-0.798670\pi$$
−0.806554 + 0.591160i $$0.798670\pi$$
$$854$$ 0 0
$$855$$ −0.644661 −0.0220469
$$856$$ 0 0
$$857$$ −52.9117 −1.80743 −0.903714 0.428136i $$-0.859170\pi$$
−0.903714 + 0.428136i $$0.859170\pi$$
$$858$$ 0 0
$$859$$ 54.8284 1.87072 0.935361 0.353695i $$-0.115075\pi$$
0.935361 + 0.353695i $$0.115075\pi$$
$$860$$ 0 0
$$861$$ 5.65685 0.192785
$$862$$ 0 0
$$863$$ 53.5980 1.82450 0.912248 0.409638i $$-0.134345\pi$$
0.912248 + 0.409638i $$0.134345\pi$$
$$864$$ 0 0
$$865$$ −15.1716 −0.515849
$$866$$ 0 0
$$867$$ 9.55635 0.324551
$$868$$ 0 0
$$869$$ −4.97056 −0.168615
$$870$$ 0 0
$$871$$ 4.82843 0.163605
$$872$$ 0 0
$$873$$ −30.9706 −1.04819
$$874$$ 0 0
$$875$$ −2.00000 −0.0676123
$$876$$ 0 0
$$877$$ −29.3137 −0.989854 −0.494927 0.868935i $$-0.664805\pi$$
−0.494927 + 0.868935i $$0.664805\pi$$
$$878$$ 0 0
$$879$$ 1.65685 0.0558843
$$880$$ 0 0
$$881$$ 25.9411 0.873979 0.436989 0.899467i $$-0.356045\pi$$
0.436989 + 0.899467i $$0.356045\pi$$
$$882$$ 0 0
$$883$$ −18.2426 −0.613914 −0.306957 0.951723i $$-0.599311\pi$$
−0.306957 + 0.951723i $$0.599311\pi$$
$$884$$ 0 0
$$885$$ 1.51472 0.0509167
$$886$$ 0 0
$$887$$ 30.0416 1.00870 0.504350 0.863500i $$-0.331732\pi$$
0.504350 + 0.863500i $$0.331732\pi$$
$$888$$ 0 0
$$889$$ −31.7990 −1.06650
$$890$$ 0 0
$$891$$ −25.5807 −0.856987
$$892$$ 0 0
$$893$$ 0.485281 0.0162393
$$894$$ 0 0
$$895$$ 11.3137 0.378176
$$896$$ 0 0
$$897$$ −5.31371 −0.177420
$$898$$ 0 0
$$899$$ 2.34315 0.0781483
$$900$$ 0 0
$$901$$ −7.31371 −0.243655
$$902$$ 0 0
$$903$$ −12.0000 −0.399335
$$904$$ 0 0
$$905$$ 0.686292 0.0228131
$$906$$ 0 0
$$907$$ 1.07107 0.0355642 0.0177821 0.999842i $$-0.494339\pi$$
0.0177821 + 0.999842i $$0.494339\pi$$
$$908$$ 0 0
$$909$$ 9.71573 0.322250
$$910$$ 0 0
$$911$$ 51.5980 1.70952 0.854759 0.519026i $$-0.173705\pi$$
0.854759 + 0.519026i $$0.173705\pi$$
$$912$$ 0 0
$$913$$ 8.48528 0.280822
$$914$$ 0 0
$$915$$ −8.97056 −0.296558
$$916$$ 0 0
$$917$$ 38.6274 1.27559
$$918$$ 0 0
$$919$$ −3.11270 −0.102678 −0.0513392 0.998681i $$-0.516349\pi$$
−0.0513392 + 0.998681i $$0.516349\pi$$
$$920$$ 0 0
$$921$$ −2.54416 −0.0838328
$$922$$ 0 0
$$923$$ −9.89949 −0.325846
$$924$$ 0 0
$$925$$ −6.82843 −0.224517
$$926$$ 0 0
$$927$$ −30.3259 −0.996033
$$928$$ 0 0
$$929$$ −41.1127 −1.34886 −0.674432 0.738337i $$-0.735611\pi$$
−0.674432 + 0.738337i $$0.735611\pi$$
$$930$$ 0 0
$$931$$ −0.727922 −0.0238567
$$932$$ 0 0
$$933$$ −10.6274 −0.347926
$$934$$ 0 0
$$935$$ −3.51472 −0.114944
$$936$$ 0 0
$$937$$ 46.2843 1.51204 0.756021 0.654548i $$-0.227141\pi$$
0.756021 + 0.654548i $$0.227141\pi$$
$$938$$ 0 0
$$939$$ 7.51472 0.245234
$$940$$ 0 0
$$941$$ −18.4853 −0.602603 −0.301301 0.953529i $$-0.597421\pi$$
−0.301301 + 0.953529i $$0.597421\pi$$
$$942$$ 0 0
$$943$$ −43.7990 −1.42629
$$944$$ 0 0
$$945$$ −6.62742 −0.215590
$$946$$ 0 0
$$947$$ 46.9706 1.52634 0.763169 0.646199i $$-0.223642\pi$$
0.763169 + 0.646199i $$0.223642\pi$$
$$948$$ 0 0
$$949$$ −1.17157 −0.0380309
$$950$$ 0 0
$$951$$ −4.40202 −0.142745
$$952$$ 0 0
$$953$$ −19.9411 −0.645956 −0.322978 0.946406i $$-0.604684\pi$$
−0.322978 + 0.946406i $$0.604684\pi$$
$$954$$ 0 0
$$955$$ −10.3431 −0.334696
$$956$$ 0 0
$$957$$ 4.11775 0.133108
$$958$$ 0 0
$$959$$ −4.00000 −0.129167
$$960$$ 0 0
$$961$$ −29.0000 −0.935484
$$962$$ 0 0
$$963$$ −13.4731 −0.434164
$$964$$ 0 0
$$965$$ −19.6569 −0.632777
$$966$$ 0 0
$$967$$ −44.8284 −1.44159 −0.720793 0.693151i $$-0.756222\pi$$
−0.720793 + 0.693151i $$0.756222\pi$$
$$968$$ 0 0
$$969$$ −0.117749 −0.00378264
$$970$$ 0 0
$$971$$ −6.62742 −0.212684 −0.106342 0.994330i $$-0.533914\pi$$
−0.106342 + 0.994330i $$0.533914\pi$$
$$972$$ 0 0
$$973$$ 16.9706 0.544051
$$974$$ 0 0
$$975$$ 0.585786 0.0187602
$$976$$ 0 0
$$977$$ 62.4264 1.99720 0.998599 0.0529182i $$-0.0168523\pi$$
0.998599 + 0.0529182i $$0.0168523\pi$$
$$978$$ 0 0
$$979$$ 42.4264 1.35595
$$980$$ 0 0
$$981$$ −46.0000 −1.46867
$$982$$ 0 0
$$983$$ −34.4853 −1.09991 −0.549955 0.835194i $$-0.685355\pi$$
−0.549955 + 0.835194i $$0.685355\pi$$
$$984$$ 0 0
$$985$$ −11.6569 −0.371418
$$986$$ 0 0
$$987$$ 2.34315 0.0745832
$$988$$ 0 0
$$989$$ 92.9117 2.95442
$$990$$ 0 0
$$991$$ 55.5980 1.76613 0.883064 0.469253i $$-0.155477\pi$$
0.883064 + 0.469253i $$0.155477\pi$$
$$992$$ 0 0
$$993$$ 8.82843 0.280162
$$994$$ 0 0
$$995$$ 17.6569 0.559760
$$996$$ 0 0
$$997$$ −3.17157 −0.100445 −0.0502224 0.998738i $$-0.515993\pi$$
−0.0502224 + 0.998738i $$0.515993\pi$$
$$998$$ 0 0
$$999$$ −22.6274 −0.715900
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 520.2.a.c.1.2 2
3.2 odd 2 4680.2.a.w.1.2 2
4.3 odd 2 1040.2.a.n.1.1 2
5.2 odd 4 2600.2.d.i.1249.3 4
5.3 odd 4 2600.2.d.i.1249.2 4
5.4 even 2 2600.2.a.w.1.1 2
8.3 odd 2 4160.2.a.u.1.2 2
8.5 even 2 4160.2.a.bn.1.1 2
12.11 even 2 9360.2.a.ck.1.1 2
13.12 even 2 6760.2.a.n.1.2 2
20.19 odd 2 5200.2.a.bl.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
520.2.a.c.1.2 2 1.1 even 1 trivial
1040.2.a.n.1.1 2 4.3 odd 2
2600.2.a.w.1.1 2 5.4 even 2
2600.2.d.i.1249.2 4 5.3 odd 4
2600.2.d.i.1249.3 4 5.2 odd 4
4160.2.a.u.1.2 2 8.3 odd 2
4160.2.a.bn.1.1 2 8.5 even 2
4680.2.a.w.1.2 2 3.2 odd 2
5200.2.a.bl.1.2 2 20.19 odd 2
6760.2.a.n.1.2 2 13.12 even 2
9360.2.a.ck.1.1 2 12.11 even 2