Properties

 Label 4368.2.a.br.1.4 Level $4368$ Weight $2$ Character 4368.1 Self dual yes Analytic conductor $34.879$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(1,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4368.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: $$34.8786556029$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.17428.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 6x^{2} + 4x + 6$$ x^4 - x^3 - 6*x^2 + 4*x + 6 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 273) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.4 Root $$1.52616$$ of defining polynomial Character $$\chi$$ $$=$$ 4368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} +2.54997 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} +2.54997 q^{5} -1.00000 q^{7} +1.00000 q^{9} +3.05232 q^{11} +1.00000 q^{13} -2.54997 q^{15} +1.34169 q^{17} -7.60228 q^{19} +1.00000 q^{21} -1.84403 q^{23} +1.50235 q^{25} -1.00000 q^{27} -5.60228 q^{29} -10.2857 q^{31} -3.05232 q^{33} -2.54997 q^{35} -9.20457 q^{37} -1.00000 q^{39} -8.04762 q^{41} -1.49765 q^{43} +2.54997 q^{45} +3.89166 q^{47} +1.00000 q^{49} -1.34169 q^{51} +0.502345 q^{53} +7.78331 q^{55} +7.60228 q^{57} +4.28937 q^{59} -0.683372 q^{61} -1.00000 q^{63} +2.54997 q^{65} +7.68806 q^{67} +1.84403 q^{69} +14.2569 q^{71} -12.0189 q^{73} -1.50235 q^{75} -3.05232 q^{77} -4.91891 q^{79} +1.00000 q^{81} +1.20828 q^{83} +3.42126 q^{85} +5.60228 q^{87} +13.7545 q^{89} -1.00000 q^{91} +10.2857 q^{93} -19.3856 q^{95} -7.18572 q^{97} +3.05232 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^3 - 3 * q^5 - 4 * q^7 + 4 * q^9 $$4 q - 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9} + 2 q^{11} + 4 q^{13} + 3 q^{15} - 2 q^{17} - 7 q^{19} + 4 q^{21} - 3 q^{23} + 9 q^{25} - 4 q^{27} + q^{29} - 3 q^{31} - 2 q^{33} + 3 q^{35} + 10 q^{37} - 4 q^{39} - 16 q^{41} - 3 q^{43} - 3 q^{45} - 5 q^{47} + 4 q^{49} + 2 q^{51} + 5 q^{53} - 10 q^{55} + 7 q^{57} + 20 q^{59} + 12 q^{61} - 4 q^{63} - 3 q^{65} + 22 q^{67} + 3 q^{69} - 13 q^{73} - 9 q^{75} - 2 q^{77} - 11 q^{79} + 4 q^{81} - q^{83} + 8 q^{85} - q^{87} - 5 q^{89} - 4 q^{91} + 3 q^{93} - 13 q^{95} - 17 q^{97} + 2 q^{99}+O(q^{100})$$ 4 * q - 4 * q^3 - 3 * q^5 - 4 * q^7 + 4 * q^9 + 2 * q^11 + 4 * q^13 + 3 * q^15 - 2 * q^17 - 7 * q^19 + 4 * q^21 - 3 * q^23 + 9 * q^25 - 4 * q^27 + q^29 - 3 * q^31 - 2 * q^33 + 3 * q^35 + 10 * q^37 - 4 * q^39 - 16 * q^41 - 3 * q^43 - 3 * q^45 - 5 * q^47 + 4 * q^49 + 2 * q^51 + 5 * q^53 - 10 * q^55 + 7 * q^57 + 20 * q^59 + 12 * q^61 - 4 * q^63 - 3 * q^65 + 22 * q^67 + 3 * q^69 - 13 * q^73 - 9 * q^75 - 2 * q^77 - 11 * q^79 + 4 * q^81 - q^83 + 8 * q^85 - q^87 - 5 * q^89 - 4 * q^91 + 3 * q^93 - 13 * q^95 - 17 * q^97 + 2 * 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
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ 2.54997 1.14038 0.570191 0.821512i $$-0.306869\pi$$
0.570191 + 0.821512i $$0.306869\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 3.05232 0.920308 0.460154 0.887839i $$-0.347794\pi$$
0.460154 + 0.887839i $$0.347794\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350
$$14$$ 0 0
$$15$$ −2.54997 −0.658399
$$16$$ 0 0
$$17$$ 1.34169 0.325407 0.162703 0.986675i $$-0.447979\pi$$
0.162703 + 0.986675i $$0.447979\pi$$
$$18$$ 0 0
$$19$$ −7.60228 −1.74408 −0.872042 0.489431i $$-0.837204\pi$$
−0.872042 + 0.489431i $$0.837204\pi$$
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ 0 0
$$23$$ −1.84403 −0.384507 −0.192254 0.981345i $$-0.561580\pi$$
−0.192254 + 0.981345i $$0.561580\pi$$
$$24$$ 0 0
$$25$$ 1.50235 0.300469
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −5.60228 −1.04032 −0.520159 0.854069i $$-0.674127\pi$$
−0.520159 + 0.854069i $$0.674127\pi$$
$$30$$ 0 0
$$31$$ −10.2857 −1.84736 −0.923679 0.383167i $$-0.874834\pi$$
−0.923679 + 0.383167i $$0.874834\pi$$
$$32$$ 0 0
$$33$$ −3.05232 −0.531340
$$34$$ 0 0
$$35$$ −2.54997 −0.431024
$$36$$ 0 0
$$37$$ −9.20457 −1.51322 −0.756611 0.653865i $$-0.773146\pi$$
−0.756611 + 0.653865i $$0.773146\pi$$
$$38$$ 0 0
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ −8.04762 −1.25683 −0.628414 0.777879i $$-0.716296\pi$$
−0.628414 + 0.777879i $$0.716296\pi$$
$$42$$ 0 0
$$43$$ −1.49765 −0.228390 −0.114195 0.993458i $$-0.536429\pi$$
−0.114195 + 0.993458i $$0.536429\pi$$
$$44$$ 0 0
$$45$$ 2.54997 0.380127
$$46$$ 0 0
$$47$$ 3.89166 0.567656 0.283828 0.958875i $$-0.408395\pi$$
0.283828 + 0.958875i $$0.408395\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −1.34169 −0.187874
$$52$$ 0 0
$$53$$ 0.502345 0.0690025 0.0345012 0.999405i $$-0.489016\pi$$
0.0345012 + 0.999405i $$0.489016\pi$$
$$54$$ 0 0
$$55$$ 7.78331 1.04950
$$56$$ 0 0
$$57$$ 7.60228 1.00695
$$58$$ 0 0
$$59$$ 4.28937 0.558429 0.279214 0.960229i $$-0.409926\pi$$
0.279214 + 0.960229i $$0.409926\pi$$
$$60$$ 0 0
$$61$$ −0.683372 −0.0874968 −0.0437484 0.999043i $$-0.513930\pi$$
−0.0437484 + 0.999043i $$0.513930\pi$$
$$62$$ 0 0
$$63$$ −1.00000 −0.125988
$$64$$ 0 0
$$65$$ 2.54997 0.316285
$$66$$ 0 0
$$67$$ 7.68806 0.939246 0.469623 0.882867i $$-0.344390\pi$$
0.469623 + 0.882867i $$0.344390\pi$$
$$68$$ 0 0
$$69$$ 1.84403 0.221995
$$70$$ 0 0
$$71$$ 14.2569 1.69198 0.845990 0.533198i $$-0.179010\pi$$
0.845990 + 0.533198i $$0.179010\pi$$
$$72$$ 0 0
$$73$$ −12.0189 −1.40670 −0.703350 0.710844i $$-0.748313\pi$$
−0.703350 + 0.710844i $$0.748313\pi$$
$$74$$ 0 0
$$75$$ −1.50235 −0.173476
$$76$$ 0 0
$$77$$ −3.05232 −0.347844
$$78$$ 0 0
$$79$$ −4.91891 −0.553421 −0.276710 0.960953i $$-0.589244\pi$$
−0.276710 + 0.960953i $$0.589244\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 1.20828 0.132626 0.0663132 0.997799i $$-0.478876\pi$$
0.0663132 + 0.997799i $$0.478876\pi$$
$$84$$ 0 0
$$85$$ 3.42126 0.371088
$$86$$ 0 0
$$87$$ 5.60228 0.600628
$$88$$ 0 0
$$89$$ 13.7545 1.45798 0.728989 0.684525i $$-0.239990\pi$$
0.728989 + 0.684525i $$0.239990\pi$$
$$90$$ 0 0
$$91$$ −1.00000 −0.104828
$$92$$ 0 0
$$93$$ 10.2857 1.06657
$$94$$ 0 0
$$95$$ −19.3856 −1.98892
$$96$$ 0 0
$$97$$ −7.18572 −0.729599 −0.364800 0.931086i $$-0.618862\pi$$
−0.364800 + 0.931086i $$0.618862\pi$$
$$98$$ 0 0
$$99$$ 3.05232 0.306769
$$100$$ 0 0
$$101$$ −18.4416 −1.83501 −0.917505 0.397724i $$-0.869800\pi$$
−0.917505 + 0.397724i $$0.869800\pi$$
$$102$$ 0 0
$$103$$ −1.89537 −0.186756 −0.0933782 0.995631i $$-0.529767\pi$$
−0.0933782 + 0.995631i $$0.529767\pi$$
$$104$$ 0 0
$$105$$ 2.54997 0.248852
$$106$$ 0 0
$$107$$ −18.5463 −1.79293 −0.896467 0.443110i $$-0.853875\pi$$
−0.896467 + 0.443110i $$0.853875\pi$$
$$108$$ 0 0
$$109$$ −1.42126 −0.136132 −0.0680659 0.997681i $$-0.521683\pi$$
−0.0680659 + 0.997681i $$0.521683\pi$$
$$110$$ 0 0
$$111$$ 9.20457 0.873659
$$112$$ 0 0
$$113$$ 5.60228 0.527019 0.263509 0.964657i $$-0.415120\pi$$
0.263509 + 0.964657i $$0.415120\pi$$
$$114$$ 0 0
$$115$$ −4.70222 −0.438485
$$116$$ 0 0
$$117$$ 1.00000 0.0924500
$$118$$ 0 0
$$119$$ −1.34169 −0.122992
$$120$$ 0 0
$$121$$ −1.68337 −0.153034
$$122$$ 0 0
$$123$$ 8.04762 0.725630
$$124$$ 0 0
$$125$$ −8.91891 −0.797732
$$126$$ 0 0
$$127$$ −14.1046 −1.25158 −0.625792 0.779990i $$-0.715224\pi$$
−0.625792 + 0.779990i $$0.715224\pi$$
$$128$$ 0 0
$$129$$ 1.49765 0.131861
$$130$$ 0 0
$$131$$ 8.78800 0.767811 0.383906 0.923372i $$-0.374579\pi$$
0.383906 + 0.923372i $$0.374579\pi$$
$$132$$ 0 0
$$133$$ 7.60228 0.659202
$$134$$ 0 0
$$135$$ −2.54997 −0.219466
$$136$$ 0 0
$$137$$ −1.97743 −0.168944 −0.0844718 0.996426i $$-0.526920\pi$$
−0.0844718 + 0.996426i $$0.526920\pi$$
$$138$$ 0 0
$$139$$ −13.0999 −1.11112 −0.555561 0.831476i $$-0.687497\pi$$
−0.555561 + 0.831476i $$0.687497\pi$$
$$140$$ 0 0
$$141$$ −3.89166 −0.327737
$$142$$ 0 0
$$143$$ 3.05232 0.255247
$$144$$ 0 0
$$145$$ −14.2857 −1.18636
$$146$$ 0 0
$$147$$ −1.00000 −0.0824786
$$148$$ 0 0
$$149$$ 14.5939 1.19558 0.597789 0.801654i $$-0.296046\pi$$
0.597789 + 0.801654i $$0.296046\pi$$
$$150$$ 0 0
$$151$$ 5.41188 0.440412 0.220206 0.975453i $$-0.429327\pi$$
0.220206 + 0.975453i $$0.429327\pi$$
$$152$$ 0 0
$$153$$ 1.34169 0.108469
$$154$$ 0 0
$$155$$ −26.2281 −2.10669
$$156$$ 0 0
$$157$$ 23.4044 1.86788 0.933939 0.357432i $$-0.116348\pi$$
0.933939 + 0.357432i $$0.116348\pi$$
$$158$$ 0 0
$$159$$ −0.502345 −0.0398386
$$160$$ 0 0
$$161$$ 1.84403 0.145330
$$162$$ 0 0
$$163$$ 10.1046 0.791456 0.395728 0.918368i $$-0.370492\pi$$
0.395728 + 0.918368i $$0.370492\pi$$
$$164$$ 0 0
$$165$$ −7.78331 −0.605930
$$166$$ 0 0
$$167$$ 3.09623 0.239593 0.119797 0.992798i $$-0.461776\pi$$
0.119797 + 0.992798i $$0.461776\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ −7.60228 −0.581361
$$172$$ 0 0
$$173$$ 2.75356 0.209349 0.104675 0.994507i $$-0.466620\pi$$
0.104675 + 0.994507i $$0.466620\pi$$
$$174$$ 0 0
$$175$$ −1.50235 −0.113567
$$176$$ 0 0
$$177$$ −4.28937 −0.322409
$$178$$ 0 0
$$179$$ 1.57723 0.117887 0.0589437 0.998261i $$-0.481227\pi$$
0.0589437 + 0.998261i $$0.481227\pi$$
$$180$$ 0 0
$$181$$ 9.51651 0.707356 0.353678 0.935367i $$-0.384931\pi$$
0.353678 + 0.935367i $$0.384931\pi$$
$$182$$ 0 0
$$183$$ 0.683372 0.0505163
$$184$$ 0 0
$$185$$ −23.4714 −1.72565
$$186$$ 0 0
$$187$$ 4.09525 0.299474
$$188$$ 0 0
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ 22.8508 1.65342 0.826712 0.562626i $$-0.190209\pi$$
0.826712 + 0.562626i $$0.190209\pi$$
$$192$$ 0 0
$$193$$ 19.5760 1.40911 0.704556 0.709649i $$-0.251146\pi$$
0.704556 + 0.709649i $$0.251146\pi$$
$$194$$ 0 0
$$195$$ −2.54997 −0.182607
$$196$$ 0 0
$$197$$ −21.2271 −1.51237 −0.756185 0.654357i $$-0.772939\pi$$
−0.756185 + 0.654357i $$0.772939\pi$$
$$198$$ 0 0
$$199$$ 6.68337 0.473772 0.236886 0.971537i $$-0.423873\pi$$
0.236886 + 0.971537i $$0.423873\pi$$
$$200$$ 0 0
$$201$$ −7.68806 −0.542274
$$202$$ 0 0
$$203$$ 5.60228 0.393203
$$204$$ 0 0
$$205$$ −20.5212 −1.43326
$$206$$ 0 0
$$207$$ −1.84403 −0.128169
$$208$$ 0 0
$$209$$ −23.2046 −1.60509
$$210$$ 0 0
$$211$$ 4.60698 0.317157 0.158579 0.987346i $$-0.449309\pi$$
0.158579 + 0.987346i $$0.449309\pi$$
$$212$$ 0 0
$$213$$ −14.2569 −0.976866
$$214$$ 0 0
$$215$$ −3.81897 −0.260452
$$216$$ 0 0
$$217$$ 10.2857 0.698236
$$218$$ 0 0
$$219$$ 12.0189 0.812159
$$220$$ 0 0
$$221$$ 1.34169 0.0902516
$$222$$ 0 0
$$223$$ −8.60698 −0.576366 −0.288183 0.957575i $$-0.593051\pi$$
−0.288183 + 0.957575i $$0.593051\pi$$
$$224$$ 0 0
$$225$$ 1.50235 0.100156
$$226$$ 0 0
$$227$$ 14.4892 0.961685 0.480843 0.876807i $$-0.340331\pi$$
0.480843 + 0.876807i $$0.340331\pi$$
$$228$$ 0 0
$$229$$ −20.1999 −1.33485 −0.667423 0.744679i $$-0.732603\pi$$
−0.667423 + 0.744679i $$0.732603\pi$$
$$230$$ 0 0
$$231$$ 3.05232 0.200828
$$232$$ 0 0
$$233$$ 7.28097 0.476992 0.238496 0.971143i $$-0.423346\pi$$
0.238496 + 0.971143i $$0.423346\pi$$
$$234$$ 0 0
$$235$$ 9.92360 0.647345
$$236$$ 0 0
$$237$$ 4.91891 0.319518
$$238$$ 0 0
$$239$$ −27.6688 −1.78974 −0.894872 0.446323i $$-0.852733\pi$$
−0.894872 + 0.446323i $$0.852733\pi$$
$$240$$ 0 0
$$241$$ −22.2187 −1.43123 −0.715617 0.698493i $$-0.753854\pi$$
−0.715617 + 0.698493i $$0.753854\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 2.54997 0.162912
$$246$$ 0 0
$$247$$ −7.60228 −0.483722
$$248$$ 0 0
$$249$$ −1.20828 −0.0765719
$$250$$ 0 0
$$251$$ −17.3092 −1.09255 −0.546274 0.837607i $$-0.683954\pi$$
−0.546274 + 0.837607i $$0.683954\pi$$
$$252$$ 0 0
$$253$$ −5.62857 −0.353865
$$254$$ 0 0
$$255$$ −3.42126 −0.214248
$$256$$ 0 0
$$257$$ −5.55837 −0.346722 −0.173361 0.984858i $$-0.555463\pi$$
−0.173361 + 0.984858i $$0.555463\pi$$
$$258$$ 0 0
$$259$$ 9.20457 0.571944
$$260$$ 0 0
$$261$$ −5.60228 −0.346773
$$262$$ 0 0
$$263$$ −25.0486 −1.54456 −0.772281 0.635281i $$-0.780884\pi$$
−0.772281 + 0.635281i $$0.780884\pi$$
$$264$$ 0 0
$$265$$ 1.28097 0.0786891
$$266$$ 0 0
$$267$$ −13.7545 −0.841764
$$268$$ 0 0
$$269$$ −4.55368 −0.277643 −0.138821 0.990317i $$-0.544331\pi$$
−0.138821 + 0.990317i $$0.544331\pi$$
$$270$$ 0 0
$$271$$ −8.88325 −0.539619 −0.269810 0.962914i $$-0.586961\pi$$
−0.269810 + 0.962914i $$0.586961\pi$$
$$272$$ 0 0
$$273$$ 1.00000 0.0605228
$$274$$ 0 0
$$275$$ 4.58563 0.276524
$$276$$ 0 0
$$277$$ −5.38560 −0.323589 −0.161795 0.986824i $$-0.551728\pi$$
−0.161795 + 0.986824i $$0.551728\pi$$
$$278$$ 0 0
$$279$$ −10.2857 −0.615786
$$280$$ 0 0
$$281$$ −12.9727 −0.773889 −0.386944 0.922103i $$-0.626469\pi$$
−0.386944 + 0.922103i $$0.626469\pi$$
$$282$$ 0 0
$$283$$ 1.52589 0.0907047 0.0453523 0.998971i $$-0.485559\pi$$
0.0453523 + 0.998971i $$0.485559\pi$$
$$284$$ 0 0
$$285$$ 19.3856 1.14830
$$286$$ 0 0
$$287$$ 8.04762 0.475036
$$288$$ 0 0
$$289$$ −15.1999 −0.894111
$$290$$ 0 0
$$291$$ 7.18572 0.421234
$$292$$ 0 0
$$293$$ 18.8545 1.10149 0.550745 0.834673i $$-0.314344\pi$$
0.550745 + 0.834673i $$0.314344\pi$$
$$294$$ 0 0
$$295$$ 10.9378 0.636821
$$296$$ 0 0
$$297$$ −3.05232 −0.177113
$$298$$ 0 0
$$299$$ −1.84403 −0.106643
$$300$$ 0 0
$$301$$ 1.49765 0.0863234
$$302$$ 0 0
$$303$$ 18.4416 1.05944
$$304$$ 0 0
$$305$$ −1.74258 −0.0997797
$$306$$ 0 0
$$307$$ −15.3856 −0.878102 −0.439051 0.898462i $$-0.644685\pi$$
−0.439051 + 0.898462i $$0.644685\pi$$
$$308$$ 0 0
$$309$$ 1.89537 0.107824
$$310$$ 0 0
$$311$$ 17.0999 0.969649 0.484824 0.874612i $$-0.338884\pi$$
0.484824 + 0.874612i $$0.338884\pi$$
$$312$$ 0 0
$$313$$ 4.89263 0.276548 0.138274 0.990394i $$-0.455845\pi$$
0.138274 + 0.990394i $$0.455845\pi$$
$$314$$ 0 0
$$315$$ −2.54997 −0.143675
$$316$$ 0 0
$$317$$ 1.44382 0.0810933 0.0405466 0.999178i $$-0.487090\pi$$
0.0405466 + 0.999178i $$0.487090\pi$$
$$318$$ 0 0
$$319$$ −17.0999 −0.957413
$$320$$ 0 0
$$321$$ 18.5463 1.03515
$$322$$ 0 0
$$323$$ −10.1999 −0.567536
$$324$$ 0 0
$$325$$ 1.50235 0.0833351
$$326$$ 0 0
$$327$$ 1.42126 0.0785958
$$328$$ 0 0
$$329$$ −3.89166 −0.214554
$$330$$ 0 0
$$331$$ 14.9953 0.824217 0.412108 0.911135i $$-0.364793\pi$$
0.412108 + 0.911135i $$0.364793\pi$$
$$332$$ 0 0
$$333$$ −9.20457 −0.504407
$$334$$ 0 0
$$335$$ 19.6043 1.07110
$$336$$ 0 0
$$337$$ −14.9115 −0.812280 −0.406140 0.913811i $$-0.633126\pi$$
−0.406140 + 0.913811i $$0.633126\pi$$
$$338$$ 0 0
$$339$$ −5.60228 −0.304274
$$340$$ 0 0
$$341$$ −31.3951 −1.70014
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 4.70222 0.253159
$$346$$ 0 0
$$347$$ −7.07488 −0.379800 −0.189900 0.981803i $$-0.560816\pi$$
−0.189900 + 0.981803i $$0.560816\pi$$
$$348$$ 0 0
$$349$$ 21.1188 1.13046 0.565232 0.824932i $$-0.308787\pi$$
0.565232 + 0.824932i $$0.308787\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ 0 0
$$353$$ −17.3643 −0.924206 −0.462103 0.886826i $$-0.652905\pi$$
−0.462103 + 0.886826i $$0.652905\pi$$
$$354$$ 0 0
$$355$$ 36.3546 1.92950
$$356$$ 0 0
$$357$$ 1.34169 0.0710096
$$358$$ 0 0
$$359$$ −18.3521 −0.968589 −0.484294 0.874905i $$-0.660924\pi$$
−0.484294 + 0.874905i $$0.660924\pi$$
$$360$$ 0 0
$$361$$ 38.7947 2.04183
$$362$$ 0 0
$$363$$ 1.68337 0.0883541
$$364$$ 0 0
$$365$$ −30.6477 −1.60417
$$366$$ 0 0
$$367$$ −8.40719 −0.438852 −0.219426 0.975629i $$-0.570418\pi$$
−0.219426 + 0.975629i $$0.570418\pi$$
$$368$$ 0 0
$$369$$ −8.04762 −0.418943
$$370$$ 0 0
$$371$$ −0.502345 −0.0260805
$$372$$ 0 0
$$373$$ 27.0925 1.40280 0.701399 0.712769i $$-0.252559\pi$$
0.701399 + 0.712769i $$0.252559\pi$$
$$374$$ 0 0
$$375$$ 8.91891 0.460571
$$376$$ 0 0
$$377$$ −5.60228 −0.288532
$$378$$ 0 0
$$379$$ 6.41657 0.329597 0.164798 0.986327i $$-0.447303\pi$$
0.164798 + 0.986327i $$0.447303\pi$$
$$380$$ 0 0
$$381$$ 14.1046 0.722602
$$382$$ 0 0
$$383$$ −32.5939 −1.66547 −0.832735 0.553672i $$-0.813226\pi$$
−0.832735 + 0.553672i $$0.813226\pi$$
$$384$$ 0 0
$$385$$ −7.78331 −0.396674
$$386$$ 0 0
$$387$$ −1.49765 −0.0761301
$$388$$ 0 0
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 0 0
$$391$$ −2.47411 −0.125121
$$392$$ 0 0
$$393$$ −8.78800 −0.443296
$$394$$ 0 0
$$395$$ −12.5431 −0.631111
$$396$$ 0 0
$$397$$ 15.7520 0.790573 0.395286 0.918558i $$-0.370645\pi$$
0.395286 + 0.918558i $$0.370645\pi$$
$$398$$ 0 0
$$399$$ −7.60228 −0.380590
$$400$$ 0 0
$$401$$ 26.1867 1.30770 0.653851 0.756624i $$-0.273152\pi$$
0.653851 + 0.756624i $$0.273152\pi$$
$$402$$ 0 0
$$403$$ −10.2857 −0.512365
$$404$$ 0 0
$$405$$ 2.54997 0.126709
$$406$$ 0 0
$$407$$ −28.0952 −1.39263
$$408$$ 0 0
$$409$$ −17.3856 −0.859662 −0.429831 0.902909i $$-0.641427\pi$$
−0.429831 + 0.902909i $$0.641427\pi$$
$$410$$ 0 0
$$411$$ 1.97743 0.0975396
$$412$$ 0 0
$$413$$ −4.28937 −0.211066
$$414$$ 0 0
$$415$$ 3.08109 0.151245
$$416$$ 0 0
$$417$$ 13.0999 0.641507
$$418$$ 0 0
$$419$$ 12.4761 0.609496 0.304748 0.952433i $$-0.401428\pi$$
0.304748 + 0.952433i $$0.401428\pi$$
$$420$$ 0 0
$$421$$ 9.57405 0.466611 0.233305 0.972404i $$-0.425046\pi$$
0.233305 + 0.972404i $$0.425046\pi$$
$$422$$ 0 0
$$423$$ 3.89166 0.189219
$$424$$ 0 0
$$425$$ 2.01568 0.0977746
$$426$$ 0 0
$$427$$ 0.683372 0.0330707
$$428$$ 0 0
$$429$$ −3.05232 −0.147367
$$430$$ 0 0
$$431$$ −5.20208 −0.250575 −0.125288 0.992120i $$-0.539985\pi$$
−0.125288 + 0.992120i $$0.539985\pi$$
$$432$$ 0 0
$$433$$ −17.2547 −0.829207 −0.414604 0.910002i $$-0.636080\pi$$
−0.414604 + 0.910002i $$0.636080\pi$$
$$434$$ 0 0
$$435$$ 14.2857 0.684945
$$436$$ 0 0
$$437$$ 14.0189 0.670613
$$438$$ 0 0
$$439$$ 22.1925 1.05919 0.529594 0.848251i $$-0.322344\pi$$
0.529594 + 0.848251i $$0.322344\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ 37.9319 1.80220 0.901098 0.433615i $$-0.142762\pi$$
0.901098 + 0.433615i $$0.142762\pi$$
$$444$$ 0 0
$$445$$ 35.0737 1.66265
$$446$$ 0 0
$$447$$ −14.5939 −0.690267
$$448$$ 0 0
$$449$$ −33.3150 −1.57223 −0.786115 0.618080i $$-0.787911\pi$$
−0.786115 + 0.618080i $$0.787911\pi$$
$$450$$ 0 0
$$451$$ −24.5639 −1.15667
$$452$$ 0 0
$$453$$ −5.41188 −0.254272
$$454$$ 0 0
$$455$$ −2.54997 −0.119544
$$456$$ 0 0
$$457$$ −7.79269 −0.364527 −0.182263 0.983250i $$-0.558342\pi$$
−0.182263 + 0.983250i $$0.558342\pi$$
$$458$$ 0 0
$$459$$ −1.34169 −0.0626245
$$460$$ 0 0
$$461$$ 30.4568 1.41851 0.709256 0.704951i $$-0.249031\pi$$
0.709256 + 0.704951i $$0.249031\pi$$
$$462$$ 0 0
$$463$$ −39.4639 −1.83405 −0.917023 0.398835i $$-0.869415\pi$$
−0.917023 + 0.398835i $$0.869415\pi$$
$$464$$ 0 0
$$465$$ 26.2281 1.21630
$$466$$ 0 0
$$467$$ 9.93307 0.459648 0.229824 0.973232i $$-0.426185\pi$$
0.229824 + 0.973232i $$0.426185\pi$$
$$468$$ 0 0
$$469$$ −7.68806 −0.355002
$$470$$ 0 0
$$471$$ −23.4044 −1.07842
$$472$$ 0 0
$$473$$ −4.57131 −0.210189
$$474$$ 0 0
$$475$$ −11.4213 −0.524043
$$476$$ 0 0
$$477$$ 0.502345 0.0230008
$$478$$ 0 0
$$479$$ 0.941479 0.0430173 0.0215086 0.999769i $$-0.493153\pi$$
0.0215086 + 0.999769i $$0.493153\pi$$
$$480$$ 0 0
$$481$$ −9.20457 −0.419692
$$482$$ 0 0
$$483$$ −1.84403 −0.0839063
$$484$$ 0 0
$$485$$ −18.3234 −0.832021
$$486$$ 0 0
$$487$$ 39.3499 1.78312 0.891558 0.452907i $$-0.149613\pi$$
0.891558 + 0.452907i $$0.149613\pi$$
$$488$$ 0 0
$$489$$ −10.1046 −0.456947
$$490$$ 0 0
$$491$$ −5.45374 −0.246124 −0.123062 0.992399i $$-0.539271\pi$$
−0.123062 + 0.992399i $$0.539271\pi$$
$$492$$ 0 0
$$493$$ −7.51651 −0.338526
$$494$$ 0 0
$$495$$ 7.78331 0.349834
$$496$$ 0 0
$$497$$ −14.2569 −0.639509
$$498$$ 0 0
$$499$$ 31.3574 1.40375 0.701874 0.712301i $$-0.252347\pi$$
0.701874 + 0.712301i $$0.252347\pi$$
$$500$$ 0 0
$$501$$ −3.09623 −0.138329
$$502$$ 0 0
$$503$$ −43.9926 −1.96153 −0.980766 0.195188i $$-0.937468\pi$$
−0.980766 + 0.195188i $$0.937468\pi$$
$$504$$ 0 0
$$505$$ −47.0256 −2.09261
$$506$$ 0 0
$$507$$ −1.00000 −0.0444116
$$508$$ 0 0
$$509$$ −41.8498 −1.85496 −0.927480 0.373874i $$-0.878029\pi$$
−0.927480 + 0.373874i $$0.878029\pi$$
$$510$$ 0 0
$$511$$ 12.0189 0.531683
$$512$$ 0 0
$$513$$ 7.60228 0.335649
$$514$$ 0 0
$$515$$ −4.83314 −0.212973
$$516$$ 0 0
$$517$$ 11.8786 0.522418
$$518$$ 0 0
$$519$$ −2.75356 −0.120868
$$520$$ 0 0
$$521$$ 8.32957 0.364925 0.182462 0.983213i $$-0.441593\pi$$
0.182462 + 0.983213i $$0.441593\pi$$
$$522$$ 0 0
$$523$$ −25.3092 −1.10669 −0.553347 0.832951i $$-0.686650\pi$$
−0.553347 + 0.832951i $$0.686650\pi$$
$$524$$ 0 0
$$525$$ 1.50235 0.0655677
$$526$$ 0 0
$$527$$ −13.8001 −0.601143
$$528$$ 0 0
$$529$$ −19.5995 −0.852154
$$530$$ 0 0
$$531$$ 4.28937 0.186143
$$532$$ 0 0
$$533$$ −8.04762 −0.348581
$$534$$ 0 0
$$535$$ −47.2924 −2.04463
$$536$$ 0 0
$$537$$ −1.57723 −0.0680624
$$538$$ 0 0
$$539$$ 3.05232 0.131473
$$540$$ 0 0
$$541$$ 27.1501 1.16727 0.583636 0.812015i $$-0.301630\pi$$
0.583636 + 0.812015i $$0.301630\pi$$
$$542$$ 0 0
$$543$$ −9.51651 −0.408392
$$544$$ 0 0
$$545$$ −3.62417 −0.155242
$$546$$ 0 0
$$547$$ −27.5949 −1.17987 −0.589935 0.807450i $$-0.700847\pi$$
−0.589935 + 0.807450i $$0.700847\pi$$
$$548$$ 0 0
$$549$$ −0.683372 −0.0291656
$$550$$ 0 0
$$551$$ 42.5902 1.81440
$$552$$ 0 0
$$553$$ 4.91891 0.209173
$$554$$ 0 0
$$555$$ 23.4714 0.996304
$$556$$ 0 0
$$557$$ 7.19881 0.305024 0.152512 0.988302i $$-0.451264\pi$$
0.152512 + 0.988302i $$0.451264\pi$$
$$558$$ 0 0
$$559$$ −1.49765 −0.0633440
$$560$$ 0 0
$$561$$ −4.09525 −0.172902
$$562$$ 0 0
$$563$$ 20.2594 0.853831 0.426915 0.904292i $$-0.359600\pi$$
0.426915 + 0.904292i $$0.359600\pi$$
$$564$$ 0 0
$$565$$ 14.2857 0.601002
$$566$$ 0 0
$$567$$ −1.00000 −0.0419961
$$568$$ 0 0
$$569$$ 7.28097 0.305234 0.152617 0.988285i $$-0.451230\pi$$
0.152617 + 0.988285i $$0.451230\pi$$
$$570$$ 0 0
$$571$$ 22.1141 0.925446 0.462723 0.886503i $$-0.346872\pi$$
0.462723 + 0.886503i $$0.346872\pi$$
$$572$$ 0 0
$$573$$ −22.8508 −0.954604
$$574$$ 0 0
$$575$$ −2.77037 −0.115533
$$576$$ 0 0
$$577$$ 20.3139 0.845678 0.422839 0.906205i $$-0.361034\pi$$
0.422839 + 0.906205i $$0.361034\pi$$
$$578$$ 0 0
$$579$$ −19.5760 −0.813551
$$580$$ 0 0
$$581$$ −1.20828 −0.0501281
$$582$$ 0 0
$$583$$ 1.53332 0.0635035
$$584$$ 0 0
$$585$$ 2.54997 0.105428
$$586$$ 0 0
$$587$$ −35.8842 −1.48110 −0.740550 0.672001i $$-0.765435\pi$$
−0.740550 + 0.672001i $$0.765435\pi$$
$$588$$ 0 0
$$589$$ 78.1945 3.22195
$$590$$ 0 0
$$591$$ 21.2271 0.873168
$$592$$ 0 0
$$593$$ 20.2664 0.832239 0.416120 0.909310i $$-0.363390\pi$$
0.416120 + 0.909310i $$0.363390\pi$$
$$594$$ 0 0
$$595$$ −3.42126 −0.140258
$$596$$ 0 0
$$597$$ −6.68337 −0.273532
$$598$$ 0 0
$$599$$ −14.9365 −0.610291 −0.305145 0.952306i $$-0.598705\pi$$
−0.305145 + 0.952306i $$0.598705\pi$$
$$600$$ 0 0
$$601$$ 4.89263 0.199575 0.0997873 0.995009i $$-0.468184\pi$$
0.0997873 + 0.995009i $$0.468184\pi$$
$$602$$ 0 0
$$603$$ 7.68806 0.313082
$$604$$ 0 0
$$605$$ −4.29255 −0.174517
$$606$$ 0 0
$$607$$ −20.6164 −0.836796 −0.418398 0.908264i $$-0.637408\pi$$
−0.418398 + 0.908264i $$0.637408\pi$$
$$608$$ 0 0
$$609$$ −5.60228 −0.227016
$$610$$ 0 0
$$611$$ 3.89166 0.157440
$$612$$ 0 0
$$613$$ −9.73818 −0.393321 −0.196661 0.980472i $$-0.563010\pi$$
−0.196661 + 0.980472i $$0.563010\pi$$
$$614$$ 0 0
$$615$$ 20.5212 0.827495
$$616$$ 0 0
$$617$$ −0.763482 −0.0307366 −0.0153683 0.999882i $$-0.504892\pi$$
−0.0153683 + 0.999882i $$0.504892\pi$$
$$618$$ 0 0
$$619$$ 41.0925 1.65165 0.825824 0.563928i $$-0.190711\pi$$
0.825824 + 0.563928i $$0.190711\pi$$
$$620$$ 0 0
$$621$$ 1.84403 0.0739984
$$622$$ 0 0
$$623$$ −13.7545 −0.551064
$$624$$ 0 0
$$625$$ −30.2547 −1.21019
$$626$$ 0 0
$$627$$ 23.2046 0.926701
$$628$$ 0 0
$$629$$ −12.3496 −0.492412
$$630$$ 0 0
$$631$$ 14.4667 0.575910 0.287955 0.957644i $$-0.407025\pi$$
0.287955 + 0.957644i $$0.407025\pi$$
$$632$$ 0 0
$$633$$ −4.60698 −0.183111
$$634$$ 0 0
$$635$$ −35.9664 −1.42728
$$636$$ 0 0
$$637$$ 1.00000 0.0396214
$$638$$ 0 0
$$639$$ 14.2569 0.563994
$$640$$ 0 0
$$641$$ 35.7069 1.41034 0.705169 0.709039i $$-0.250871\pi$$
0.705169 + 0.709039i $$0.250871\pi$$
$$642$$ 0 0
$$643$$ 26.4091 1.04147 0.520737 0.853717i $$-0.325657\pi$$
0.520737 + 0.853717i $$0.325657\pi$$
$$644$$ 0 0
$$645$$ 3.81897 0.150372
$$646$$ 0 0
$$647$$ −37.3543 −1.46855 −0.734275 0.678852i $$-0.762478\pi$$
−0.734275 + 0.678852i $$0.762478\pi$$
$$648$$ 0 0
$$649$$ 13.0925 0.513926
$$650$$ 0 0
$$651$$ −10.2857 −0.403127
$$652$$ 0 0
$$653$$ −0.987881 −0.0386588 −0.0193294 0.999813i $$-0.506153\pi$$
−0.0193294 + 0.999813i $$0.506153\pi$$
$$654$$ 0 0
$$655$$ 22.4091 0.875598
$$656$$ 0 0
$$657$$ −12.0189 −0.468900
$$658$$ 0 0
$$659$$ 1.24653 0.0485578 0.0242789 0.999705i $$-0.492271\pi$$
0.0242789 + 0.999705i $$0.492271\pi$$
$$660$$ 0 0
$$661$$ −33.8994 −1.31853 −0.659266 0.751910i $$-0.729133\pi$$
−0.659266 + 0.751910i $$0.729133\pi$$
$$662$$ 0 0
$$663$$ −1.34169 −0.0521068
$$664$$ 0 0
$$665$$ 19.3856 0.751741
$$666$$ 0 0
$$667$$ 10.3308 0.400010
$$668$$ 0 0
$$669$$ 8.60698 0.332765
$$670$$ 0 0
$$671$$ −2.08587 −0.0805240
$$672$$ 0 0
$$673$$ 42.5808 1.64137 0.820684 0.571382i $$-0.193592\pi$$
0.820684 + 0.571382i $$0.193592\pi$$
$$674$$ 0 0
$$675$$ −1.50235 −0.0578253
$$676$$ 0 0
$$677$$ −33.8629 −1.30146 −0.650728 0.759311i $$-0.725536\pi$$
−0.650728 + 0.759311i $$0.725536\pi$$
$$678$$ 0 0
$$679$$ 7.18572 0.275763
$$680$$ 0 0
$$681$$ −14.4892 −0.555229
$$682$$ 0 0
$$683$$ 34.5163 1.32073 0.660364 0.750946i $$-0.270402\pi$$
0.660364 + 0.750946i $$0.270402\pi$$
$$684$$ 0 0
$$685$$ −5.04240 −0.192660
$$686$$ 0 0
$$687$$ 20.1999 0.770673
$$688$$ 0 0
$$689$$ 0.502345 0.0191378
$$690$$ 0 0
$$691$$ 7.90679 0.300789 0.150394 0.988626i $$-0.451946\pi$$
0.150394 + 0.988626i $$0.451946\pi$$
$$692$$ 0 0
$$693$$ −3.05232 −0.115948
$$694$$ 0 0
$$695$$ −33.4044 −1.26710
$$696$$ 0 0
$$697$$ −10.7974 −0.408980
$$698$$ 0 0
$$699$$ −7.28097 −0.275391
$$700$$ 0 0
$$701$$ −5.19510 −0.196216 −0.0981081 0.995176i $$-0.531279\pi$$
−0.0981081 + 0.995176i $$0.531279\pi$$
$$702$$ 0 0
$$703$$ 69.9758 2.63919
$$704$$ 0 0
$$705$$ −9.92360 −0.373745
$$706$$ 0 0
$$707$$ 18.4416 0.693569
$$708$$ 0 0
$$709$$ 33.1971 1.24674 0.623372 0.781925i $$-0.285762\pi$$
0.623372 + 0.781925i $$0.285762\pi$$
$$710$$ 0 0
$$711$$ −4.91891 −0.184474
$$712$$ 0 0
$$713$$ 18.9671 0.710323
$$714$$ 0 0
$$715$$ 7.78331 0.291079
$$716$$ 0 0
$$717$$ 27.6688 1.03331
$$718$$ 0 0
$$719$$ −30.7261 −1.14589 −0.572944 0.819594i $$-0.694199\pi$$
−0.572944 + 0.819594i $$0.694199\pi$$
$$720$$ 0 0
$$721$$ 1.89537 0.0705873
$$722$$ 0 0
$$723$$ 22.2187 0.826324
$$724$$ 0 0
$$725$$ −8.41657 −0.312583
$$726$$ 0 0
$$727$$ −1.83783 −0.0681612 −0.0340806 0.999419i $$-0.510850\pi$$
−0.0340806 + 0.999419i $$0.510850\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −2.00938 −0.0743197
$$732$$ 0 0
$$733$$ −19.1857 −0.708641 −0.354320 0.935124i $$-0.615288\pi$$
−0.354320 + 0.935124i $$0.615288\pi$$
$$734$$ 0 0
$$735$$ −2.54997 −0.0940570
$$736$$ 0 0
$$737$$ 23.4664 0.864396
$$738$$ 0 0
$$739$$ 21.4882 0.790456 0.395228 0.918583i $$-0.370666\pi$$
0.395228 + 0.918583i $$0.370666\pi$$
$$740$$ 0 0
$$741$$ 7.60228 0.279277
$$742$$ 0 0
$$743$$ −19.9430 −0.731637 −0.365819 0.930686i $$-0.619211\pi$$
−0.365819 + 0.930686i $$0.619211\pi$$
$$744$$ 0 0
$$745$$ 37.2140 1.36341
$$746$$ 0 0
$$747$$ 1.20828 0.0442088
$$748$$ 0 0
$$749$$ 18.5463 0.677665
$$750$$ 0 0
$$751$$ −38.4498 −1.40305 −0.701526 0.712644i $$-0.747498\pi$$
−0.701526 + 0.712644i $$0.747498\pi$$
$$752$$ 0 0
$$753$$ 17.3092 0.630782
$$754$$ 0 0
$$755$$ 13.8001 0.502238
$$756$$ 0 0
$$757$$ −16.2931 −0.592182 −0.296091 0.955160i $$-0.595683\pi$$
−0.296091 + 0.955160i $$0.595683\pi$$
$$758$$ 0 0
$$759$$ 5.62857 0.204304
$$760$$ 0 0
$$761$$ 35.3380 1.28100 0.640500 0.767958i $$-0.278727\pi$$
0.640500 + 0.767958i $$0.278727\pi$$
$$762$$ 0 0
$$763$$ 1.42126 0.0514530
$$764$$ 0 0
$$765$$ 3.42126 0.123696
$$766$$ 0 0
$$767$$ 4.28937 0.154880
$$768$$ 0 0
$$769$$ −11.5428 −0.416244 −0.208122 0.978103i $$-0.566735\pi$$
−0.208122 + 0.978103i $$0.566735\pi$$
$$770$$ 0 0
$$771$$ 5.55837 0.200180
$$772$$ 0 0
$$773$$ −5.89786 −0.212131 −0.106066 0.994359i $$-0.533825\pi$$
−0.106066 + 0.994359i $$0.533825\pi$$
$$774$$ 0 0
$$775$$ −15.4526 −0.555074
$$776$$ 0 0
$$777$$ −9.20457 −0.330212
$$778$$ 0 0
$$779$$ 61.1803 2.19201
$$780$$ 0 0
$$781$$ 43.5165 1.55714
$$782$$ 0 0
$$783$$ 5.60228 0.200209
$$784$$ 0 0
$$785$$ 59.6806 2.13009
$$786$$ 0 0
$$787$$ −6.33079 −0.225668 −0.112834 0.993614i $$-0.535993\pi$$
−0.112834 + 0.993614i $$0.535993\pi$$
$$788$$ 0 0
$$789$$ 25.0486 0.891754
$$790$$ 0 0
$$791$$ −5.60228 −0.199194
$$792$$ 0 0
$$793$$ −0.683372 −0.0242672
$$794$$ 0 0
$$795$$ −1.28097 −0.0454312
$$796$$ 0 0
$$797$$ −22.0721 −0.781835 −0.390918 0.920426i $$-0.627842\pi$$
−0.390918 + 0.920426i $$0.627842\pi$$
$$798$$ 0 0
$$799$$ 5.22138 0.184719
$$800$$ 0 0
$$801$$ 13.7545 0.485993
$$802$$ 0 0
$$803$$ −36.6853 −1.29460
$$804$$ 0 0
$$805$$ 4.70222 0.165732
$$806$$ 0 0
$$807$$ 4.55368 0.160297
$$808$$ 0 0
$$809$$ 36.2019 1.27279 0.636396 0.771363i $$-0.280424\pi$$
0.636396 + 0.771363i $$0.280424\pi$$
$$810$$ 0 0
$$811$$ 52.4950 1.84335 0.921674 0.387964i $$-0.126822\pi$$
0.921674 + 0.387964i $$0.126822\pi$$
$$812$$ 0 0
$$813$$ 8.88325 0.311549
$$814$$ 0 0
$$815$$ 25.7665 0.902561
$$816$$ 0 0
$$817$$ 11.3856 0.398332
$$818$$ 0 0
$$819$$ −1.00000 −0.0349428
$$820$$ 0 0
$$821$$ −0.972743 −0.0339490 −0.0169745 0.999856i $$-0.505403\pi$$
−0.0169745 + 0.999856i $$0.505403\pi$$
$$822$$ 0 0
$$823$$ 29.5572 1.03030 0.515150 0.857100i $$-0.327736\pi$$
0.515150 + 0.857100i $$0.327736\pi$$
$$824$$ 0 0
$$825$$ −4.58563 −0.159651
$$826$$ 0 0
$$827$$ −15.3191 −0.532698 −0.266349 0.963877i $$-0.585817\pi$$
−0.266349 + 0.963877i $$0.585817\pi$$
$$828$$ 0 0
$$829$$ 27.0236 0.938570 0.469285 0.883047i $$-0.344512\pi$$
0.469285 + 0.883047i $$0.344512\pi$$
$$830$$ 0 0
$$831$$ 5.38560 0.186824
$$832$$ 0 0
$$833$$ 1.34169 0.0464867
$$834$$ 0 0
$$835$$ 7.89528 0.273227
$$836$$ 0 0
$$837$$ 10.2857 0.355524
$$838$$ 0 0
$$839$$ −46.2200 −1.59569 −0.797846 0.602862i $$-0.794027\pi$$
−0.797846 + 0.602862i $$0.794027\pi$$
$$840$$ 0 0
$$841$$ 2.38560 0.0822619
$$842$$ 0 0
$$843$$ 12.9727 0.446805
$$844$$ 0 0
$$845$$ 2.54997 0.0877216
$$846$$ 0 0
$$847$$ 1.68337 0.0578413
$$848$$ 0 0
$$849$$ −1.52589 −0.0523684
$$850$$ 0 0
$$851$$ 16.9735 0.581845
$$852$$ 0 0
$$853$$ −15.1094 −0.517336 −0.258668 0.965966i $$-0.583284\pi$$
−0.258668 + 0.965966i $$0.583284\pi$$
$$854$$ 0 0
$$855$$ −19.3856 −0.662973
$$856$$ 0 0
$$857$$ −0.191631 −0.00654599 −0.00327299 0.999995i $$-0.501042\pi$$
−0.00327299 + 0.999995i $$0.501042\pi$$
$$858$$ 0 0
$$859$$ −7.06419 −0.241027 −0.120514 0.992712i $$-0.538454\pi$$
−0.120514 + 0.992712i $$0.538454\pi$$
$$860$$ 0 0
$$861$$ −8.04762 −0.274262
$$862$$ 0 0
$$863$$ −41.6142 −1.41657 −0.708283 0.705929i $$-0.750530\pi$$
−0.708283 + 0.705929i $$0.750530\pi$$
$$864$$ 0 0
$$865$$ 7.02150 0.238738
$$866$$ 0 0
$$867$$ 15.1999 0.516215
$$868$$ 0 0
$$869$$ −15.0141 −0.509318
$$870$$ 0 0
$$871$$ 7.68806 0.260500
$$872$$ 0 0
$$873$$ −7.18572 −0.243200
$$874$$ 0 0
$$875$$ 8.91891 0.301514
$$876$$ 0 0
$$877$$ −52.5471 −1.77439 −0.887194 0.461396i $$-0.847349\pi$$
−0.887194 + 0.461396i $$0.847349\pi$$
$$878$$ 0 0
$$879$$ −18.8545 −0.635946
$$880$$ 0 0
$$881$$ 0.934500 0.0314841 0.0157421 0.999876i $$-0.494989\pi$$
0.0157421 + 0.999876i $$0.494989\pi$$
$$882$$ 0 0
$$883$$ 42.6448 1.43511 0.717555 0.696501i $$-0.245261\pi$$
0.717555 + 0.696501i $$0.245261\pi$$
$$884$$ 0 0
$$885$$ −10.9378 −0.367669
$$886$$ 0 0
$$887$$ −24.8833 −0.835498 −0.417749 0.908563i $$-0.637181\pi$$
−0.417749 + 0.908563i $$0.637181\pi$$
$$888$$ 0 0
$$889$$ 14.1046 0.473054
$$890$$ 0 0
$$891$$ 3.05232 0.102256
$$892$$ 0 0
$$893$$ −29.5855 −0.990040
$$894$$ 0 0
$$895$$ 4.02188 0.134437
$$896$$ 0 0
$$897$$ 1.84403 0.0615704
$$898$$ 0 0
$$899$$ 57.6232 1.92184
$$900$$ 0 0
$$901$$ 0.673990 0.0224539
$$902$$ 0 0
$$903$$ −1.49765 −0.0498388
$$904$$ 0 0
$$905$$ 24.2668 0.806656
$$906$$ 0 0
$$907$$ −42.2207 −1.40191 −0.700957 0.713203i $$-0.747244\pi$$
−0.700957 + 0.713203i $$0.747244\pi$$
$$908$$ 0 0
$$909$$ −18.4416 −0.611670
$$910$$ 0 0
$$911$$ −14.1108 −0.467513 −0.233756 0.972295i $$-0.575102\pi$$
−0.233756 + 0.972295i $$0.575102\pi$$
$$912$$ 0 0
$$913$$ 3.68806 0.122057
$$914$$ 0 0
$$915$$ 1.74258 0.0576078
$$916$$ 0 0
$$917$$ −8.78800 −0.290205
$$918$$ 0 0
$$919$$ 18.1547 0.598870 0.299435 0.954117i $$-0.403202\pi$$
0.299435 + 0.954117i $$0.403202\pi$$
$$920$$ 0 0
$$921$$ 15.3856 0.506973
$$922$$ 0 0
$$923$$ 14.2569 0.469271
$$924$$ 0 0
$$925$$ −13.8284 −0.454676
$$926$$ 0 0
$$927$$ −1.89537 −0.0622521
$$928$$ 0 0
$$929$$ −8.24741 −0.270589 −0.135294 0.990805i $$-0.543198\pi$$
−0.135294 + 0.990805i $$0.543198\pi$$
$$930$$ 0 0
$$931$$ −7.60228 −0.249155
$$932$$ 0 0
$$933$$ −17.0999 −0.559827
$$934$$ 0 0
$$935$$ 10.4428 0.341515
$$936$$ 0 0
$$937$$ −21.6693 −0.707905 −0.353953 0.935263i $$-0.615163\pi$$
−0.353953 + 0.935263i $$0.615163\pi$$
$$938$$ 0 0
$$939$$ −4.89263 −0.159665
$$940$$ 0 0
$$941$$ 42.5238 1.38624 0.693118 0.720824i $$-0.256237\pi$$
0.693118 + 0.720824i $$0.256237\pi$$
$$942$$ 0 0
$$943$$ 14.8401 0.483259
$$944$$ 0 0
$$945$$ 2.54997 0.0829505
$$946$$ 0 0
$$947$$ 6.79792 0.220903 0.110451 0.993882i $$-0.464770\pi$$
0.110451 + 0.993882i $$0.464770\pi$$
$$948$$ 0 0
$$949$$ −12.0189 −0.390148
$$950$$ 0 0
$$951$$ −1.44382 −0.0468192
$$952$$ 0 0
$$953$$ 57.3782 1.85866 0.929331 0.369249i $$-0.120385\pi$$
0.929331 + 0.369249i $$0.120385\pi$$
$$954$$ 0 0
$$955$$ 58.2688 1.88553
$$956$$ 0 0
$$957$$ 17.0999 0.552763
$$958$$ 0 0
$$959$$ 1.97743 0.0638547
$$960$$ 0 0
$$961$$ 74.7947 2.41273
$$962$$ 0 0
$$963$$ −18.5463 −0.597645
$$964$$ 0 0
$$965$$ 49.9182 1.60692
$$966$$ 0 0
$$967$$ −7.52393 −0.241953 −0.120977 0.992655i $$-0.538603\pi$$
−0.120977 + 0.992655i $$0.538603\pi$$
$$968$$ 0 0
$$969$$ 10.1999 0.327667
$$970$$ 0 0
$$971$$ 18.5807 0.596283 0.298141 0.954522i $$-0.403633\pi$$
0.298141 + 0.954522i $$0.403633\pi$$
$$972$$ 0 0
$$973$$ 13.0999 0.419965
$$974$$ 0 0
$$975$$ −1.50235 −0.0481136
$$976$$ 0 0
$$977$$ −38.5939 −1.23473 −0.617364 0.786678i $$-0.711799\pi$$
−0.617364 + 0.786678i $$0.711799\pi$$
$$978$$ 0 0
$$979$$ 41.9832 1.34179
$$980$$ 0 0
$$981$$ −1.42126 −0.0453773
$$982$$ 0 0
$$983$$ 0.591837 0.0188767 0.00943834 0.999955i $$-0.496996\pi$$
0.00943834 + 0.999955i $$0.496996\pi$$
$$984$$ 0 0
$$985$$ −54.1286 −1.72468
$$986$$ 0 0
$$987$$ 3.89166 0.123873
$$988$$ 0 0
$$989$$ 2.76172 0.0878177
$$990$$ 0 0
$$991$$ −9.41686 −0.299136 −0.149568 0.988751i $$-0.547788\pi$$
−0.149568 + 0.988751i $$0.547788\pi$$
$$992$$ 0 0
$$993$$ −14.9953 −0.475862
$$994$$ 0 0
$$995$$ 17.0424 0.540280
$$996$$ 0 0
$$997$$ 22.1716 0.702180 0.351090 0.936342i $$-0.385811\pi$$
0.351090 + 0.936342i $$0.385811\pi$$
$$998$$ 0 0
$$999$$ 9.20457 0.291220
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 4368.2.a.br.1.4 4
4.3 odd 2 273.2.a.e.1.2 4
12.11 even 2 819.2.a.k.1.3 4
20.19 odd 2 6825.2.a.bg.1.3 4
28.27 even 2 1911.2.a.s.1.2 4
52.51 odd 2 3549.2.a.w.1.3 4
84.83 odd 2 5733.2.a.bf.1.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
273.2.a.e.1.2 4 4.3 odd 2
819.2.a.k.1.3 4 12.11 even 2
1911.2.a.s.1.2 4 28.27 even 2
3549.2.a.w.1.3 4 52.51 odd 2
4368.2.a.br.1.4 4 1.1 even 1 trivial
5733.2.a.bf.1.3 4 84.83 odd 2
6825.2.a.bg.1.3 4 20.19 odd 2