# Properties

 Label 768.2.a.l.1.2 Level $768$ Weight $2$ Character 768.1 Self dual yes Analytic conductor $6.133$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,2,Mod(1,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, 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("768.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 768.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: $$6.13251087523$$ Analytic rank: $$0$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 384) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 768.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +2.82843 q^{5} +2.82843 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +2.82843 q^{5} +2.82843 q^{7} +1.00000 q^{9} +4.00000 q^{11} -5.65685 q^{13} +2.82843 q^{15} -2.00000 q^{17} +4.00000 q^{19} +2.82843 q^{21} -5.65685 q^{23} +3.00000 q^{25} +1.00000 q^{27} -2.82843 q^{29} -8.48528 q^{31} +4.00000 q^{33} +8.00000 q^{35} -5.65685 q^{39} -10.0000 q^{41} +12.0000 q^{43} +2.82843 q^{45} +5.65685 q^{47} +1.00000 q^{49} -2.00000 q^{51} -2.82843 q^{53} +11.3137 q^{55} +4.00000 q^{57} -4.00000 q^{59} +11.3137 q^{61} +2.82843 q^{63} -16.0000 q^{65} +4.00000 q^{67} -5.65685 q^{69} +5.65685 q^{71} +2.00000 q^{73} +3.00000 q^{75} +11.3137 q^{77} -8.48528 q^{79} +1.00000 q^{81} -4.00000 q^{83} -5.65685 q^{85} -2.82843 q^{87} -6.00000 q^{89} -16.0000 q^{91} -8.48528 q^{93} +11.3137 q^{95} +14.0000 q^{97} +4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^9 $$2 q + 2 q^{3} + 2 q^{9} + 8 q^{11} - 4 q^{17} + 8 q^{19} + 6 q^{25} + 2 q^{27} + 8 q^{33} + 16 q^{35} - 20 q^{41} + 24 q^{43} + 2 q^{49} - 4 q^{51} + 8 q^{57} - 8 q^{59} - 32 q^{65} + 8 q^{67} + 4 q^{73} + 6 q^{75} + 2 q^{81} - 8 q^{83} - 12 q^{89} - 32 q^{91} + 28 q^{97} + 8 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^9 + 8 * q^11 - 4 * q^17 + 8 * q^19 + 6 * q^25 + 2 * q^27 + 8 * q^33 + 16 * q^35 - 20 * q^41 + 24 * q^43 + 2 * q^49 - 4 * q^51 + 8 * q^57 - 8 * q^59 - 32 * q^65 + 8 * q^67 + 4 * q^73 + 6 * q^75 + 2 * q^81 - 8 * q^83 - 12 * q^89 - 32 * q^91 + 28 * q^97 + 8 * 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
$$5$$ 2.82843 1.26491 0.632456 0.774597i $$-0.282047\pi$$
0.632456 + 0.774597i $$0.282047\pi$$
$$6$$ 0 0
$$7$$ 2.82843 1.06904 0.534522 0.845154i $$-0.320491\pi$$
0.534522 + 0.845154i $$0.320491\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 0 0
$$13$$ −5.65685 −1.56893 −0.784465 0.620174i $$-0.787062\pi$$
−0.784465 + 0.620174i $$0.787062\pi$$
$$14$$ 0 0
$$15$$ 2.82843 0.730297
$$16$$ 0 0
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 0 0
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ 2.82843 0.617213
$$22$$ 0 0
$$23$$ −5.65685 −1.17954 −0.589768 0.807573i $$-0.700781\pi$$
−0.589768 + 0.807573i $$0.700781\pi$$
$$24$$ 0 0
$$25$$ 3.00000 0.600000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −2.82843 −0.525226 −0.262613 0.964901i $$-0.584584\pi$$
−0.262613 + 0.964901i $$0.584584\pi$$
$$30$$ 0 0
$$31$$ −8.48528 −1.52400 −0.762001 0.647576i $$-0.775783\pi$$
−0.762001 + 0.647576i $$0.775783\pi$$
$$32$$ 0 0
$$33$$ 4.00000 0.696311
$$34$$ 0 0
$$35$$ 8.00000 1.35225
$$36$$ 0 0
$$37$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$38$$ 0 0
$$39$$ −5.65685 −0.905822
$$40$$ 0 0
$$41$$ −10.0000 −1.56174 −0.780869 0.624695i $$-0.785223\pi$$
−0.780869 + 0.624695i $$0.785223\pi$$
$$42$$ 0 0
$$43$$ 12.0000 1.82998 0.914991 0.403473i $$-0.132197\pi$$
0.914991 + 0.403473i $$0.132197\pi$$
$$44$$ 0 0
$$45$$ 2.82843 0.421637
$$46$$ 0 0
$$47$$ 5.65685 0.825137 0.412568 0.910927i $$-0.364632\pi$$
0.412568 + 0.910927i $$0.364632\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −2.00000 −0.280056
$$52$$ 0 0
$$53$$ −2.82843 −0.388514 −0.194257 0.980951i $$-0.562230\pi$$
−0.194257 + 0.980951i $$0.562230\pi$$
$$54$$ 0 0
$$55$$ 11.3137 1.52554
$$56$$ 0 0
$$57$$ 4.00000 0.529813
$$58$$ 0 0
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ 11.3137 1.44857 0.724286 0.689500i $$-0.242170\pi$$
0.724286 + 0.689500i $$0.242170\pi$$
$$62$$ 0 0
$$63$$ 2.82843 0.356348
$$64$$ 0 0
$$65$$ −16.0000 −1.98456
$$66$$ 0 0
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 0 0
$$69$$ −5.65685 −0.681005
$$70$$ 0 0
$$71$$ 5.65685 0.671345 0.335673 0.941979i $$-0.391036\pi$$
0.335673 + 0.941979i $$0.391036\pi$$
$$72$$ 0 0
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ 0 0
$$75$$ 3.00000 0.346410
$$76$$ 0 0
$$77$$ 11.3137 1.28932
$$78$$ 0 0
$$79$$ −8.48528 −0.954669 −0.477334 0.878722i $$-0.658397\pi$$
−0.477334 + 0.878722i $$0.658397\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −4.00000 −0.439057 −0.219529 0.975606i $$-0.570452\pi$$
−0.219529 + 0.975606i $$0.570452\pi$$
$$84$$ 0 0
$$85$$ −5.65685 −0.613572
$$86$$ 0 0
$$87$$ −2.82843 −0.303239
$$88$$ 0 0
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ −16.0000 −1.67726
$$92$$ 0 0
$$93$$ −8.48528 −0.879883
$$94$$ 0 0
$$95$$ 11.3137 1.16076
$$96$$ 0 0
$$97$$ 14.0000 1.42148 0.710742 0.703452i $$-0.248359\pi$$
0.710742 + 0.703452i $$0.248359\pi$$
$$98$$ 0 0
$$99$$ 4.00000 0.402015
$$100$$ 0 0
$$101$$ −14.1421 −1.40720 −0.703598 0.710599i $$-0.748424\pi$$
−0.703598 + 0.710599i $$0.748424\pi$$
$$102$$ 0 0
$$103$$ −2.82843 −0.278693 −0.139347 0.990244i $$-0.544500\pi$$
−0.139347 + 0.990244i $$0.544500\pi$$
$$104$$ 0 0
$$105$$ 8.00000 0.780720
$$106$$ 0 0
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 0 0
$$109$$ 5.65685 0.541828 0.270914 0.962604i $$-0.412674\pi$$
0.270914 + 0.962604i $$0.412674\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 2.00000 0.188144 0.0940721 0.995565i $$-0.470012\pi$$
0.0940721 + 0.995565i $$0.470012\pi$$
$$114$$ 0 0
$$115$$ −16.0000 −1.49201
$$116$$ 0 0
$$117$$ −5.65685 −0.522976
$$118$$ 0 0
$$119$$ −5.65685 −0.518563
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 0 0
$$123$$ −10.0000 −0.901670
$$124$$ 0 0
$$125$$ −5.65685 −0.505964
$$126$$ 0 0
$$127$$ 8.48528 0.752947 0.376473 0.926427i $$-0.377137\pi$$
0.376473 + 0.926427i $$0.377137\pi$$
$$128$$ 0 0
$$129$$ 12.0000 1.05654
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ 11.3137 0.981023
$$134$$ 0 0
$$135$$ 2.82843 0.243432
$$136$$ 0 0
$$137$$ −10.0000 −0.854358 −0.427179 0.904167i $$-0.640493\pi$$
−0.427179 + 0.904167i $$0.640493\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 5.65685 0.476393
$$142$$ 0 0
$$143$$ −22.6274 −1.89220
$$144$$ 0 0
$$145$$ −8.00000 −0.664364
$$146$$ 0 0
$$147$$ 1.00000 0.0824786
$$148$$ 0 0
$$149$$ 14.1421 1.15857 0.579284 0.815125i $$-0.303332\pi$$
0.579284 + 0.815125i $$0.303332\pi$$
$$150$$ 0 0
$$151$$ −8.48528 −0.690522 −0.345261 0.938507i $$-0.612210\pi$$
−0.345261 + 0.938507i $$0.612210\pi$$
$$152$$ 0 0
$$153$$ −2.00000 −0.161690
$$154$$ 0 0
$$155$$ −24.0000 −1.92773
$$156$$ 0 0
$$157$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$158$$ 0 0
$$159$$ −2.82843 −0.224309
$$160$$ 0 0
$$161$$ −16.0000 −1.26098
$$162$$ 0 0
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ 0 0
$$165$$ 11.3137 0.880771
$$166$$ 0 0
$$167$$ −11.3137 −0.875481 −0.437741 0.899101i $$-0.644221\pi$$
−0.437741 + 0.899101i $$0.644221\pi$$
$$168$$ 0 0
$$169$$ 19.0000 1.46154
$$170$$ 0 0
$$171$$ 4.00000 0.305888
$$172$$ 0 0
$$173$$ −19.7990 −1.50529 −0.752645 0.658427i $$-0.771222\pi$$
−0.752645 + 0.658427i $$0.771222\pi$$
$$174$$ 0 0
$$175$$ 8.48528 0.641427
$$176$$ 0 0
$$177$$ −4.00000 −0.300658
$$178$$ 0 0
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 5.65685 0.420471 0.210235 0.977651i $$-0.432577\pi$$
0.210235 + 0.977651i $$0.432577\pi$$
$$182$$ 0 0
$$183$$ 11.3137 0.836333
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −8.00000 −0.585018
$$188$$ 0 0
$$189$$ 2.82843 0.205738
$$190$$ 0 0
$$191$$ −11.3137 −0.818631 −0.409316 0.912393i $$-0.634232\pi$$
−0.409316 + 0.912393i $$0.634232\pi$$
$$192$$ 0 0
$$193$$ 6.00000 0.431889 0.215945 0.976406i $$-0.430717\pi$$
0.215945 + 0.976406i $$0.430717\pi$$
$$194$$ 0 0
$$195$$ −16.0000 −1.14578
$$196$$ 0 0
$$197$$ 8.48528 0.604551 0.302276 0.953221i $$-0.402254\pi$$
0.302276 + 0.953221i $$0.402254\pi$$
$$198$$ 0 0
$$199$$ −2.82843 −0.200502 −0.100251 0.994962i $$-0.531965\pi$$
−0.100251 + 0.994962i $$0.531965\pi$$
$$200$$ 0 0
$$201$$ 4.00000 0.282138
$$202$$ 0 0
$$203$$ −8.00000 −0.561490
$$204$$ 0 0
$$205$$ −28.2843 −1.97546
$$206$$ 0 0
$$207$$ −5.65685 −0.393179
$$208$$ 0 0
$$209$$ 16.0000 1.10674
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ 0 0
$$213$$ 5.65685 0.387601
$$214$$ 0 0
$$215$$ 33.9411 2.31477
$$216$$ 0 0
$$217$$ −24.0000 −1.62923
$$218$$ 0 0
$$219$$ 2.00000 0.135147
$$220$$ 0 0
$$221$$ 11.3137 0.761042
$$222$$ 0 0
$$223$$ 19.7990 1.32584 0.662919 0.748691i $$-0.269317\pi$$
0.662919 + 0.748691i $$0.269317\pi$$
$$224$$ 0 0
$$225$$ 3.00000 0.200000
$$226$$ 0 0
$$227$$ 12.0000 0.796468 0.398234 0.917284i $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ 0 0
$$229$$ −5.65685 −0.373815 −0.186908 0.982377i $$-0.559847\pi$$
−0.186908 + 0.982377i $$0.559847\pi$$
$$230$$ 0 0
$$231$$ 11.3137 0.744387
$$232$$ 0 0
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ 0 0
$$235$$ 16.0000 1.04372
$$236$$ 0 0
$$237$$ −8.48528 −0.551178
$$238$$ 0 0
$$239$$ 22.6274 1.46365 0.731823 0.681495i $$-0.238670\pi$$
0.731823 + 0.681495i $$0.238670\pi$$
$$240$$ 0 0
$$241$$ 6.00000 0.386494 0.193247 0.981150i $$-0.438098\pi$$
0.193247 + 0.981150i $$0.438098\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 2.82843 0.180702
$$246$$ 0 0
$$247$$ −22.6274 −1.43975
$$248$$ 0 0
$$249$$ −4.00000 −0.253490
$$250$$ 0 0
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 0 0
$$253$$ −22.6274 −1.42257
$$254$$ 0 0
$$255$$ −5.65685 −0.354246
$$256$$ 0 0
$$257$$ −30.0000 −1.87135 −0.935674 0.352865i $$-0.885208\pi$$
−0.935674 + 0.352865i $$0.885208\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −2.82843 −0.175075
$$262$$ 0 0
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ −8.00000 −0.491436
$$266$$ 0 0
$$267$$ −6.00000 −0.367194
$$268$$ 0 0
$$269$$ 8.48528 0.517357 0.258678 0.965964i $$-0.416713\pi$$
0.258678 + 0.965964i $$0.416713\pi$$
$$270$$ 0 0
$$271$$ 19.7990 1.20270 0.601351 0.798985i $$-0.294629\pi$$
0.601351 + 0.798985i $$0.294629\pi$$
$$272$$ 0 0
$$273$$ −16.0000 −0.968364
$$274$$ 0 0
$$275$$ 12.0000 0.723627
$$276$$ 0 0
$$277$$ −16.9706 −1.01966 −0.509831 0.860274i $$-0.670292\pi$$
−0.509831 + 0.860274i $$0.670292\pi$$
$$278$$ 0 0
$$279$$ −8.48528 −0.508001
$$280$$ 0 0
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ 0 0
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ 0 0
$$285$$ 11.3137 0.670166
$$286$$ 0 0
$$287$$ −28.2843 −1.66957
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 14.0000 0.820695
$$292$$ 0 0
$$293$$ 19.7990 1.15667 0.578335 0.815800i $$-0.303703\pi$$
0.578335 + 0.815800i $$0.303703\pi$$
$$294$$ 0 0
$$295$$ −11.3137 −0.658710
$$296$$ 0 0
$$297$$ 4.00000 0.232104
$$298$$ 0 0
$$299$$ 32.0000 1.85061
$$300$$ 0 0
$$301$$ 33.9411 1.95633
$$302$$ 0 0
$$303$$ −14.1421 −0.812444
$$304$$ 0 0
$$305$$ 32.0000 1.83231
$$306$$ 0 0
$$307$$ 4.00000 0.228292 0.114146 0.993464i $$-0.463587\pi$$
0.114146 + 0.993464i $$0.463587\pi$$
$$308$$ 0 0
$$309$$ −2.82843 −0.160904
$$310$$ 0 0
$$311$$ −11.3137 −0.641542 −0.320771 0.947157i $$-0.603942\pi$$
−0.320771 + 0.947157i $$0.603942\pi$$
$$312$$ 0 0
$$313$$ 14.0000 0.791327 0.395663 0.918396i $$-0.370515\pi$$
0.395663 + 0.918396i $$0.370515\pi$$
$$314$$ 0 0
$$315$$ 8.00000 0.450749
$$316$$ 0 0
$$317$$ 8.48528 0.476581 0.238290 0.971194i $$-0.423413\pi$$
0.238290 + 0.971194i $$0.423413\pi$$
$$318$$ 0 0
$$319$$ −11.3137 −0.633446
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ −8.00000 −0.445132
$$324$$ 0 0
$$325$$ −16.9706 −0.941357
$$326$$ 0 0
$$327$$ 5.65685 0.312825
$$328$$ 0 0
$$329$$ 16.0000 0.882109
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 11.3137 0.618134
$$336$$ 0 0
$$337$$ −22.0000 −1.19842 −0.599208 0.800593i $$-0.704518\pi$$
−0.599208 + 0.800593i $$0.704518\pi$$
$$338$$ 0 0
$$339$$ 2.00000 0.108625
$$340$$ 0 0
$$341$$ −33.9411 −1.83801
$$342$$ 0 0
$$343$$ −16.9706 −0.916324
$$344$$ 0 0
$$345$$ −16.0000 −0.861411
$$346$$ 0 0
$$347$$ 36.0000 1.93258 0.966291 0.257454i $$-0.0828835\pi$$
0.966291 + 0.257454i $$0.0828835\pi$$
$$348$$ 0 0
$$349$$ 11.3137 0.605609 0.302804 0.953053i $$-0.402077\pi$$
0.302804 + 0.953053i $$0.402077\pi$$
$$350$$ 0 0
$$351$$ −5.65685 −0.301941
$$352$$ 0 0
$$353$$ 18.0000 0.958043 0.479022 0.877803i $$-0.340992\pi$$
0.479022 + 0.877803i $$0.340992\pi$$
$$354$$ 0 0
$$355$$ 16.0000 0.849192
$$356$$ 0 0
$$357$$ −5.65685 −0.299392
$$358$$ 0 0
$$359$$ 28.2843 1.49279 0.746393 0.665505i $$-0.231784\pi$$
0.746393 + 0.665505i $$0.231784\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 0 0
$$363$$ 5.00000 0.262432
$$364$$ 0 0
$$365$$ 5.65685 0.296093
$$366$$ 0 0
$$367$$ −8.48528 −0.442928 −0.221464 0.975169i $$-0.571084\pi$$
−0.221464 + 0.975169i $$0.571084\pi$$
$$368$$ 0 0
$$369$$ −10.0000 −0.520579
$$370$$ 0 0
$$371$$ −8.00000 −0.415339
$$372$$ 0 0
$$373$$ 33.9411 1.75740 0.878702 0.477370i $$-0.158410\pi$$
0.878702 + 0.477370i $$0.158410\pi$$
$$374$$ 0 0
$$375$$ −5.65685 −0.292119
$$376$$ 0 0
$$377$$ 16.0000 0.824042
$$378$$ 0 0
$$379$$ 28.0000 1.43826 0.719132 0.694874i $$-0.244540\pi$$
0.719132 + 0.694874i $$0.244540\pi$$
$$380$$ 0 0
$$381$$ 8.48528 0.434714
$$382$$ 0 0
$$383$$ 11.3137 0.578103 0.289052 0.957313i $$-0.406660\pi$$
0.289052 + 0.957313i $$0.406660\pi$$
$$384$$ 0 0
$$385$$ 32.0000 1.63087
$$386$$ 0 0
$$387$$ 12.0000 0.609994
$$388$$ 0 0
$$389$$ −31.1127 −1.57748 −0.788738 0.614729i $$-0.789265\pi$$
−0.788738 + 0.614729i $$0.789265\pi$$
$$390$$ 0 0
$$391$$ 11.3137 0.572159
$$392$$ 0 0
$$393$$ −12.0000 −0.605320
$$394$$ 0 0
$$395$$ −24.0000 −1.20757
$$396$$ 0 0
$$397$$ 33.9411 1.70346 0.851728 0.523984i $$-0.175555\pi$$
0.851728 + 0.523984i $$0.175555\pi$$
$$398$$ 0 0
$$399$$ 11.3137 0.566394
$$400$$ 0 0
$$401$$ 30.0000 1.49813 0.749064 0.662497i $$-0.230503\pi$$
0.749064 + 0.662497i $$0.230503\pi$$
$$402$$ 0 0
$$403$$ 48.0000 2.39105
$$404$$ 0 0
$$405$$ 2.82843 0.140546
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −10.0000 −0.494468 −0.247234 0.968956i $$-0.579522\pi$$
−0.247234 + 0.968956i $$0.579522\pi$$
$$410$$ 0 0
$$411$$ −10.0000 −0.493264
$$412$$ 0 0
$$413$$ −11.3137 −0.556711
$$414$$ 0 0
$$415$$ −11.3137 −0.555368
$$416$$ 0 0
$$417$$ −4.00000 −0.195881
$$418$$ 0 0
$$419$$ 28.0000 1.36789 0.683945 0.729534i $$-0.260263\pi$$
0.683945 + 0.729534i $$0.260263\pi$$
$$420$$ 0 0
$$421$$ −16.9706 −0.827095 −0.413547 0.910483i $$-0.635710\pi$$
−0.413547 + 0.910483i $$0.635710\pi$$
$$422$$ 0 0
$$423$$ 5.65685 0.275046
$$424$$ 0 0
$$425$$ −6.00000 −0.291043
$$426$$ 0 0
$$427$$ 32.0000 1.54859
$$428$$ 0 0
$$429$$ −22.6274 −1.09246
$$430$$ 0 0
$$431$$ −28.2843 −1.36241 −0.681203 0.732095i $$-0.738543\pi$$
−0.681203 + 0.732095i $$0.738543\pi$$
$$432$$ 0 0
$$433$$ −34.0000 −1.63394 −0.816968 0.576683i $$-0.804347\pi$$
−0.816968 + 0.576683i $$0.804347\pi$$
$$434$$ 0 0
$$435$$ −8.00000 −0.383571
$$436$$ 0 0
$$437$$ −22.6274 −1.08242
$$438$$ 0 0
$$439$$ 31.1127 1.48493 0.742464 0.669886i $$-0.233657\pi$$
0.742464 + 0.669886i $$0.233657\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ 20.0000 0.950229 0.475114 0.879924i $$-0.342407\pi$$
0.475114 + 0.879924i $$0.342407\pi$$
$$444$$ 0 0
$$445$$ −16.9706 −0.804482
$$446$$ 0 0
$$447$$ 14.1421 0.668900
$$448$$ 0 0
$$449$$ −18.0000 −0.849473 −0.424736 0.905317i $$-0.639633\pi$$
−0.424736 + 0.905317i $$0.639633\pi$$
$$450$$ 0 0
$$451$$ −40.0000 −1.88353
$$452$$ 0 0
$$453$$ −8.48528 −0.398673
$$454$$ 0 0
$$455$$ −45.2548 −2.12158
$$456$$ 0 0
$$457$$ 6.00000 0.280668 0.140334 0.990104i $$-0.455182\pi$$
0.140334 + 0.990104i $$0.455182\pi$$
$$458$$ 0 0
$$459$$ −2.00000 −0.0933520
$$460$$ 0 0
$$461$$ −8.48528 −0.395199 −0.197599 0.980283i $$-0.563315\pi$$
−0.197599 + 0.980283i $$0.563315\pi$$
$$462$$ 0 0
$$463$$ 19.7990 0.920137 0.460069 0.887883i $$-0.347825\pi$$
0.460069 + 0.887883i $$0.347825\pi$$
$$464$$ 0 0
$$465$$ −24.0000 −1.11297
$$466$$ 0 0
$$467$$ 28.0000 1.29569 0.647843 0.761774i $$-0.275671\pi$$
0.647843 + 0.761774i $$0.275671\pi$$
$$468$$ 0 0
$$469$$ 11.3137 0.522419
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 48.0000 2.20704
$$474$$ 0 0
$$475$$ 12.0000 0.550598
$$476$$ 0 0
$$477$$ −2.82843 −0.129505
$$478$$ 0 0
$$479$$ 28.2843 1.29234 0.646171 0.763193i $$-0.276369\pi$$
0.646171 + 0.763193i $$0.276369\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ −16.0000 −0.728025
$$484$$ 0 0
$$485$$ 39.5980 1.79805
$$486$$ 0 0
$$487$$ 14.1421 0.640841 0.320421 0.947275i $$-0.396176\pi$$
0.320421 + 0.947275i $$0.396176\pi$$
$$488$$ 0 0
$$489$$ 4.00000 0.180886
$$490$$ 0 0
$$491$$ −20.0000 −0.902587 −0.451294 0.892375i $$-0.649037\pi$$
−0.451294 + 0.892375i $$0.649037\pi$$
$$492$$ 0 0
$$493$$ 5.65685 0.254772
$$494$$ 0 0
$$495$$ 11.3137 0.508513
$$496$$ 0 0
$$497$$ 16.0000 0.717698
$$498$$ 0 0
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 0 0
$$501$$ −11.3137 −0.505459
$$502$$ 0 0
$$503$$ 5.65685 0.252227 0.126113 0.992016i $$-0.459750\pi$$
0.126113 + 0.992016i $$0.459750\pi$$
$$504$$ 0 0
$$505$$ −40.0000 −1.77998
$$506$$ 0 0
$$507$$ 19.0000 0.843820
$$508$$ 0 0
$$509$$ −2.82843 −0.125368 −0.0626839 0.998033i $$-0.519966\pi$$
−0.0626839 + 0.998033i $$0.519966\pi$$
$$510$$ 0 0
$$511$$ 5.65685 0.250244
$$512$$ 0 0
$$513$$ 4.00000 0.176604
$$514$$ 0 0
$$515$$ −8.00000 −0.352522
$$516$$ 0 0
$$517$$ 22.6274 0.995153
$$518$$ 0 0
$$519$$ −19.7990 −0.869079
$$520$$ 0 0
$$521$$ 6.00000 0.262865 0.131432 0.991325i $$-0.458042\pi$$
0.131432 + 0.991325i $$0.458042\pi$$
$$522$$ 0 0
$$523$$ −20.0000 −0.874539 −0.437269 0.899331i $$-0.644054\pi$$
−0.437269 + 0.899331i $$0.644054\pi$$
$$524$$ 0 0
$$525$$ 8.48528 0.370328
$$526$$ 0 0
$$527$$ 16.9706 0.739249
$$528$$ 0 0
$$529$$ 9.00000 0.391304
$$530$$ 0 0
$$531$$ −4.00000 −0.173585
$$532$$ 0 0
$$533$$ 56.5685 2.45026
$$534$$ 0 0
$$535$$ 33.9411 1.46740
$$536$$ 0 0
$$537$$ −12.0000 −0.517838
$$538$$ 0 0
$$539$$ 4.00000 0.172292
$$540$$ 0 0
$$541$$ −16.9706 −0.729621 −0.364811 0.931082i $$-0.618866\pi$$
−0.364811 + 0.931082i $$0.618866\pi$$
$$542$$ 0 0
$$543$$ 5.65685 0.242759
$$544$$ 0 0
$$545$$ 16.0000 0.685365
$$546$$ 0 0
$$547$$ 20.0000 0.855138 0.427569 0.903983i $$-0.359370\pi$$
0.427569 + 0.903983i $$0.359370\pi$$
$$548$$ 0 0
$$549$$ 11.3137 0.482857
$$550$$ 0 0
$$551$$ −11.3137 −0.481980
$$552$$ 0 0
$$553$$ −24.0000 −1.02058
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −8.48528 −0.359533 −0.179766 0.983709i $$-0.557534\pi$$
−0.179766 + 0.983709i $$0.557534\pi$$
$$558$$ 0 0
$$559$$ −67.8823 −2.87111
$$560$$ 0 0
$$561$$ −8.00000 −0.337760
$$562$$ 0 0
$$563$$ −4.00000 −0.168580 −0.0842900 0.996441i $$-0.526862\pi$$
−0.0842900 + 0.996441i $$0.526862\pi$$
$$564$$ 0 0
$$565$$ 5.65685 0.237986
$$566$$ 0 0
$$567$$ 2.82843 0.118783
$$568$$ 0 0
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ 0 0
$$573$$ −11.3137 −0.472637
$$574$$ 0 0
$$575$$ −16.9706 −0.707721
$$576$$ 0 0
$$577$$ −22.0000 −0.915872 −0.457936 0.888985i $$-0.651411\pi$$
−0.457936 + 0.888985i $$0.651411\pi$$
$$578$$ 0 0
$$579$$ 6.00000 0.249351
$$580$$ 0 0
$$581$$ −11.3137 −0.469372
$$582$$ 0 0
$$583$$ −11.3137 −0.468566
$$584$$ 0 0
$$585$$ −16.0000 −0.661519
$$586$$ 0 0
$$587$$ −36.0000 −1.48588 −0.742940 0.669359i $$-0.766569\pi$$
−0.742940 + 0.669359i $$0.766569\pi$$
$$588$$ 0 0
$$589$$ −33.9411 −1.39852
$$590$$ 0 0
$$591$$ 8.48528 0.349038
$$592$$ 0 0
$$593$$ 34.0000 1.39621 0.698106 0.715994i $$-0.254026\pi$$
0.698106 + 0.715994i $$0.254026\pi$$
$$594$$ 0 0
$$595$$ −16.0000 −0.655936
$$596$$ 0 0
$$597$$ −2.82843 −0.115760
$$598$$ 0 0
$$599$$ −39.5980 −1.61793 −0.808965 0.587857i $$-0.799972\pi$$
−0.808965 + 0.587857i $$0.799972\pi$$
$$600$$ 0 0
$$601$$ 34.0000 1.38689 0.693444 0.720510i $$-0.256092\pi$$
0.693444 + 0.720510i $$0.256092\pi$$
$$602$$ 0 0
$$603$$ 4.00000 0.162893
$$604$$ 0 0
$$605$$ 14.1421 0.574960
$$606$$ 0 0
$$607$$ −14.1421 −0.574012 −0.287006 0.957929i $$-0.592660\pi$$
−0.287006 + 0.957929i $$0.592660\pi$$
$$608$$ 0 0
$$609$$ −8.00000 −0.324176
$$610$$ 0 0
$$611$$ −32.0000 −1.29458
$$612$$ 0 0
$$613$$ 11.3137 0.456956 0.228478 0.973549i $$-0.426625\pi$$
0.228478 + 0.973549i $$0.426625\pi$$
$$614$$ 0 0
$$615$$ −28.2843 −1.14053
$$616$$ 0 0
$$617$$ −38.0000 −1.52982 −0.764911 0.644136i $$-0.777217\pi$$
−0.764911 + 0.644136i $$0.777217\pi$$
$$618$$ 0 0
$$619$$ −36.0000 −1.44696 −0.723481 0.690344i $$-0.757459\pi$$
−0.723481 + 0.690344i $$0.757459\pi$$
$$620$$ 0 0
$$621$$ −5.65685 −0.227002
$$622$$ 0 0
$$623$$ −16.9706 −0.679911
$$624$$ 0 0
$$625$$ −31.0000 −1.24000
$$626$$ 0 0
$$627$$ 16.0000 0.638978
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −8.48528 −0.337794 −0.168897 0.985634i $$-0.554020\pi$$
−0.168897 + 0.985634i $$0.554020\pi$$
$$632$$ 0 0
$$633$$ −12.0000 −0.476957
$$634$$ 0 0
$$635$$ 24.0000 0.952411
$$636$$ 0 0
$$637$$ −5.65685 −0.224133
$$638$$ 0 0
$$639$$ 5.65685 0.223782
$$640$$ 0 0
$$641$$ −34.0000 −1.34292 −0.671460 0.741041i $$-0.734332\pi$$
−0.671460 + 0.741041i $$0.734332\pi$$
$$642$$ 0 0
$$643$$ −44.0000 −1.73519 −0.867595 0.497271i $$-0.834335\pi$$
−0.867595 + 0.497271i $$0.834335\pi$$
$$644$$ 0 0
$$645$$ 33.9411 1.33643
$$646$$ 0 0
$$647$$ 39.5980 1.55676 0.778379 0.627795i $$-0.216042\pi$$
0.778379 + 0.627795i $$0.216042\pi$$
$$648$$ 0 0
$$649$$ −16.0000 −0.628055
$$650$$ 0 0
$$651$$ −24.0000 −0.940634
$$652$$ 0 0
$$653$$ −42.4264 −1.66027 −0.830137 0.557560i $$-0.811738\pi$$
−0.830137 + 0.557560i $$0.811738\pi$$
$$654$$ 0 0
$$655$$ −33.9411 −1.32619
$$656$$ 0 0
$$657$$ 2.00000 0.0780274
$$658$$ 0 0
$$659$$ 4.00000 0.155818 0.0779089 0.996960i $$-0.475176\pi$$
0.0779089 + 0.996960i $$0.475176\pi$$
$$660$$ 0 0
$$661$$ −45.2548 −1.76021 −0.880105 0.474780i $$-0.842528\pi$$
−0.880105 + 0.474780i $$0.842528\pi$$
$$662$$ 0 0
$$663$$ 11.3137 0.439388
$$664$$ 0 0
$$665$$ 32.0000 1.24091
$$666$$ 0 0
$$667$$ 16.0000 0.619522
$$668$$ 0 0
$$669$$ 19.7990 0.765473
$$670$$ 0 0
$$671$$ 45.2548 1.74704
$$672$$ 0 0
$$673$$ 14.0000 0.539660 0.269830 0.962908i $$-0.413032\pi$$
0.269830 + 0.962908i $$0.413032\pi$$
$$674$$ 0 0
$$675$$ 3.00000 0.115470
$$676$$ 0 0
$$677$$ −19.7990 −0.760937 −0.380468 0.924794i $$-0.624237\pi$$
−0.380468 + 0.924794i $$0.624237\pi$$
$$678$$ 0 0
$$679$$ 39.5980 1.51963
$$680$$ 0 0
$$681$$ 12.0000 0.459841
$$682$$ 0 0
$$683$$ 20.0000 0.765279 0.382639 0.923898i $$-0.375015\pi$$
0.382639 + 0.923898i $$0.375015\pi$$
$$684$$ 0 0
$$685$$ −28.2843 −1.08069
$$686$$ 0 0
$$687$$ −5.65685 −0.215822
$$688$$ 0 0
$$689$$ 16.0000 0.609551
$$690$$ 0 0
$$691$$ −28.0000 −1.06517 −0.532585 0.846376i $$-0.678779\pi$$
−0.532585 + 0.846376i $$0.678779\pi$$
$$692$$ 0 0
$$693$$ 11.3137 0.429772
$$694$$ 0 0
$$695$$ −11.3137 −0.429153
$$696$$ 0 0
$$697$$ 20.0000 0.757554
$$698$$ 0 0
$$699$$ −6.00000 −0.226941
$$700$$ 0 0
$$701$$ 19.7990 0.747798 0.373899 0.927470i $$-0.378021\pi$$
0.373899 + 0.927470i $$0.378021\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 16.0000 0.602595
$$706$$ 0 0
$$707$$ −40.0000 −1.50435
$$708$$ 0 0
$$709$$ 16.9706 0.637343 0.318671 0.947865i $$-0.396763\pi$$
0.318671 + 0.947865i $$0.396763\pi$$
$$710$$ 0 0
$$711$$ −8.48528 −0.318223
$$712$$ 0 0
$$713$$ 48.0000 1.79761
$$714$$ 0 0
$$715$$ −64.0000 −2.39346
$$716$$ 0 0
$$717$$ 22.6274 0.845036
$$718$$ 0 0
$$719$$ 16.9706 0.632895 0.316448 0.948610i $$-0.397510\pi$$
0.316448 + 0.948610i $$0.397510\pi$$
$$720$$ 0 0
$$721$$ −8.00000 −0.297936
$$722$$ 0 0
$$723$$ 6.00000 0.223142
$$724$$ 0 0
$$725$$ −8.48528 −0.315135
$$726$$ 0 0
$$727$$ 25.4558 0.944105 0.472052 0.881570i $$-0.343513\pi$$
0.472052 + 0.881570i $$0.343513\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −24.0000 −0.887672
$$732$$ 0 0
$$733$$ 16.9706 0.626822 0.313411 0.949618i $$-0.398528\pi$$
0.313411 + 0.949618i $$0.398528\pi$$
$$734$$ 0 0
$$735$$ 2.82843 0.104328
$$736$$ 0 0
$$737$$ 16.0000 0.589368
$$738$$ 0 0
$$739$$ 36.0000 1.32428 0.662141 0.749380i $$-0.269648\pi$$
0.662141 + 0.749380i $$0.269648\pi$$
$$740$$ 0 0
$$741$$ −22.6274 −0.831239
$$742$$ 0 0
$$743$$ 22.6274 0.830119 0.415060 0.909794i $$-0.363761\pi$$
0.415060 + 0.909794i $$0.363761\pi$$
$$744$$ 0 0
$$745$$ 40.0000 1.46549
$$746$$ 0 0
$$747$$ −4.00000 −0.146352
$$748$$ 0 0
$$749$$ 33.9411 1.24018
$$750$$ 0 0
$$751$$ 48.0833 1.75458 0.877292 0.479958i $$-0.159348\pi$$
0.877292 + 0.479958i $$0.159348\pi$$
$$752$$ 0 0
$$753$$ −12.0000 −0.437304
$$754$$ 0 0
$$755$$ −24.0000 −0.873449
$$756$$ 0 0
$$757$$ −28.2843 −1.02801 −0.514005 0.857787i $$-0.671839\pi$$
−0.514005 + 0.857787i $$0.671839\pi$$
$$758$$ 0 0
$$759$$ −22.6274 −0.821323
$$760$$ 0 0
$$761$$ −42.0000 −1.52250 −0.761249 0.648459i $$-0.775414\pi$$
−0.761249 + 0.648459i $$0.775414\pi$$
$$762$$ 0 0
$$763$$ 16.0000 0.579239
$$764$$ 0 0
$$765$$ −5.65685 −0.204524
$$766$$ 0 0
$$767$$ 22.6274 0.817029
$$768$$ 0 0
$$769$$ 38.0000 1.37032 0.685158 0.728395i $$-0.259733\pi$$
0.685158 + 0.728395i $$0.259733\pi$$
$$770$$ 0 0
$$771$$ −30.0000 −1.08042
$$772$$ 0 0
$$773$$ −2.82843 −0.101731 −0.0508657 0.998706i $$-0.516198\pi$$
−0.0508657 + 0.998706i $$0.516198\pi$$
$$774$$ 0 0
$$775$$ −25.4558 −0.914401
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −40.0000 −1.43315
$$780$$ 0 0
$$781$$ 22.6274 0.809673
$$782$$ 0 0
$$783$$ −2.82843 −0.101080
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 20.0000 0.712923 0.356462 0.934310i $$-0.383983\pi$$
0.356462 + 0.934310i $$0.383983\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 5.65685 0.201135
$$792$$ 0 0
$$793$$ −64.0000 −2.27271
$$794$$ 0 0
$$795$$ −8.00000 −0.283731
$$796$$ 0 0
$$797$$ −19.7990 −0.701316 −0.350658 0.936504i $$-0.614042\pi$$
−0.350658 + 0.936504i $$0.614042\pi$$
$$798$$ 0 0
$$799$$ −11.3137 −0.400250
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 0 0
$$803$$ 8.00000 0.282314
$$804$$ 0 0
$$805$$ −45.2548 −1.59502
$$806$$ 0 0
$$807$$ 8.48528 0.298696
$$808$$ 0 0
$$809$$ −10.0000 −0.351581 −0.175791 0.984428i $$-0.556248\pi$$
−0.175791 + 0.984428i $$0.556248\pi$$
$$810$$ 0 0
$$811$$ −20.0000 −0.702295 −0.351147 0.936320i $$-0.614208\pi$$
−0.351147 + 0.936320i $$0.614208\pi$$
$$812$$ 0 0
$$813$$ 19.7990 0.694381
$$814$$ 0 0
$$815$$ 11.3137 0.396302
$$816$$ 0 0
$$817$$ 48.0000 1.67931
$$818$$ 0 0
$$819$$ −16.0000 −0.559085
$$820$$ 0 0
$$821$$ 31.1127 1.08584 0.542920 0.839784i $$-0.317319\pi$$
0.542920 + 0.839784i $$0.317319\pi$$
$$822$$ 0 0
$$823$$ −19.7990 −0.690149 −0.345075 0.938575i $$-0.612146\pi$$
−0.345075 + 0.938575i $$0.612146\pi$$
$$824$$ 0 0
$$825$$ 12.0000 0.417786
$$826$$ 0 0
$$827$$ 12.0000 0.417281 0.208640 0.977992i $$-0.433096\pi$$
0.208640 + 0.977992i $$0.433096\pi$$
$$828$$ 0 0
$$829$$ 50.9117 1.76824 0.884118 0.467264i $$-0.154760\pi$$
0.884118 + 0.467264i $$0.154760\pi$$
$$830$$ 0 0
$$831$$ −16.9706 −0.588702
$$832$$ 0 0
$$833$$ −2.00000 −0.0692959
$$834$$ 0 0
$$835$$ −32.0000 −1.10741
$$836$$ 0 0
$$837$$ −8.48528 −0.293294
$$838$$ 0 0
$$839$$ −5.65685 −0.195296 −0.0976481 0.995221i $$-0.531132\pi$$
−0.0976481 + 0.995221i $$0.531132\pi$$
$$840$$ 0 0
$$841$$ −21.0000 −0.724138
$$842$$ 0 0
$$843$$ 10.0000 0.344418
$$844$$ 0 0
$$845$$ 53.7401 1.84872
$$846$$ 0 0
$$847$$ 14.1421 0.485930
$$848$$ 0 0
$$849$$ −4.00000 −0.137280
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 22.6274 0.774748 0.387374 0.921923i $$-0.373382\pi$$
0.387374 + 0.921923i $$0.373382\pi$$
$$854$$ 0 0
$$855$$ 11.3137 0.386921
$$856$$ 0 0
$$857$$ 6.00000 0.204956 0.102478 0.994735i $$-0.467323\pi$$
0.102478 + 0.994735i $$0.467323\pi$$
$$858$$ 0 0
$$859$$ 44.0000 1.50126 0.750630 0.660722i $$-0.229750\pi$$
0.750630 + 0.660722i $$0.229750\pi$$
$$860$$ 0 0
$$861$$ −28.2843 −0.963925
$$862$$ 0 0
$$863$$ −56.5685 −1.92562 −0.962808 0.270187i $$-0.912914\pi$$
−0.962808 + 0.270187i $$0.912914\pi$$
$$864$$ 0 0
$$865$$ −56.0000 −1.90406
$$866$$ 0 0
$$867$$ −13.0000 −0.441503
$$868$$ 0 0
$$869$$ −33.9411 −1.15137
$$870$$ 0 0
$$871$$ −22.6274 −0.766701
$$872$$ 0 0
$$873$$ 14.0000 0.473828
$$874$$ 0 0
$$875$$ −16.0000 −0.540899
$$876$$ 0 0
$$877$$ −11.3137 −0.382037 −0.191018 0.981586i $$-0.561179\pi$$
−0.191018 + 0.981586i $$0.561179\pi$$
$$878$$ 0 0
$$879$$ 19.7990 0.667803
$$880$$ 0 0
$$881$$ −46.0000 −1.54978 −0.774890 0.632096i $$-0.782195\pi$$
−0.774890 + 0.632096i $$0.782195\pi$$
$$882$$ 0 0
$$883$$ −12.0000 −0.403832 −0.201916 0.979403i $$-0.564717\pi$$
−0.201916 + 0.979403i $$0.564717\pi$$
$$884$$ 0 0
$$885$$ −11.3137 −0.380306
$$886$$ 0 0
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 0 0
$$889$$ 24.0000 0.804934
$$890$$ 0 0
$$891$$ 4.00000 0.134005
$$892$$ 0 0
$$893$$ 22.6274 0.757198
$$894$$ 0 0
$$895$$ −33.9411 −1.13453
$$896$$ 0 0
$$897$$ 32.0000 1.06845
$$898$$ 0 0
$$899$$ 24.0000 0.800445
$$900$$ 0 0
$$901$$ 5.65685 0.188457
$$902$$ 0 0
$$903$$ 33.9411 1.12949
$$904$$ 0 0
$$905$$ 16.0000 0.531858
$$906$$ 0 0
$$907$$ 44.0000 1.46100 0.730498 0.682915i $$-0.239288\pi$$
0.730498 + 0.682915i $$0.239288\pi$$
$$908$$ 0 0
$$909$$ −14.1421 −0.469065
$$910$$ 0 0
$$911$$ −22.6274 −0.749680 −0.374840 0.927090i $$-0.622302\pi$$
−0.374840 + 0.927090i $$0.622302\pi$$
$$912$$ 0 0
$$913$$ −16.0000 −0.529523
$$914$$ 0 0
$$915$$ 32.0000 1.05789
$$916$$ 0 0
$$917$$ −33.9411 −1.12083
$$918$$ 0 0
$$919$$ 2.82843 0.0933012 0.0466506 0.998911i $$-0.485145\pi$$
0.0466506 + 0.998911i $$0.485145\pi$$
$$920$$ 0 0
$$921$$ 4.00000 0.131804
$$922$$ 0 0
$$923$$ −32.0000 −1.05329
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −2.82843 −0.0928977
$$928$$ 0 0
$$929$$ −18.0000 −0.590561 −0.295280 0.955411i $$-0.595413\pi$$
−0.295280 + 0.955411i $$0.595413\pi$$
$$930$$ 0 0
$$931$$ 4.00000 0.131095
$$932$$ 0 0
$$933$$ −11.3137 −0.370394
$$934$$ 0 0
$$935$$ −22.6274 −0.739996
$$936$$ 0 0
$$937$$ 42.0000 1.37208 0.686040 0.727564i $$-0.259347\pi$$
0.686040 + 0.727564i $$0.259347\pi$$
$$938$$ 0 0
$$939$$ 14.0000 0.456873
$$940$$ 0 0
$$941$$ 14.1421 0.461020 0.230510 0.973070i $$-0.425960\pi$$
0.230510 + 0.973070i $$0.425960\pi$$
$$942$$ 0 0
$$943$$ 56.5685 1.84213
$$944$$ 0 0
$$945$$ 8.00000 0.260240
$$946$$ 0 0
$$947$$ 36.0000 1.16984 0.584921 0.811090i $$-0.301125\pi$$
0.584921 + 0.811090i $$0.301125\pi$$
$$948$$ 0 0
$$949$$ −11.3137 −0.367259
$$950$$ 0 0
$$951$$ 8.48528 0.275154
$$952$$ 0 0
$$953$$ −42.0000 −1.36051 −0.680257 0.732974i $$-0.738132\pi$$
−0.680257 + 0.732974i $$0.738132\pi$$
$$954$$ 0 0
$$955$$ −32.0000 −1.03550
$$956$$ 0 0
$$957$$ −11.3137 −0.365720
$$958$$ 0 0
$$959$$ −28.2843 −0.913347
$$960$$ 0 0
$$961$$ 41.0000 1.32258
$$962$$ 0 0
$$963$$ 12.0000 0.386695
$$964$$ 0 0
$$965$$ 16.9706 0.546302
$$966$$ 0 0
$$967$$ −14.1421 −0.454780 −0.227390 0.973804i $$-0.573019\pi$$
−0.227390 + 0.973804i $$0.573019\pi$$
$$968$$ 0 0
$$969$$ −8.00000 −0.256997
$$970$$ 0 0
$$971$$ −12.0000 −0.385098 −0.192549 0.981287i $$-0.561675\pi$$
−0.192549 + 0.981287i $$0.561675\pi$$
$$972$$ 0 0
$$973$$ −11.3137 −0.362701
$$974$$ 0 0
$$975$$ −16.9706 −0.543493
$$976$$ 0 0
$$977$$ −2.00000 −0.0639857 −0.0319928 0.999488i $$-0.510185\pi$$
−0.0319928 + 0.999488i $$0.510185\pi$$
$$978$$ 0 0
$$979$$ −24.0000 −0.767043
$$980$$ 0 0
$$981$$ 5.65685 0.180609
$$982$$ 0 0
$$983$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$984$$ 0 0
$$985$$ 24.0000 0.764704
$$986$$ 0 0
$$987$$ 16.0000 0.509286
$$988$$ 0 0
$$989$$ −67.8823 −2.15853
$$990$$ 0 0
$$991$$ 25.4558 0.808632 0.404316 0.914619i $$-0.367510\pi$$
0.404316 + 0.914619i $$0.367510\pi$$
$$992$$ 0 0
$$993$$ −20.0000 −0.634681
$$994$$ 0 0
$$995$$ −8.00000 −0.253617
$$996$$ 0 0
$$997$$ 33.9411 1.07493 0.537463 0.843287i $$-0.319383\pi$$
0.537463 + 0.843287i $$0.319383\pi$$
$$998$$ 0 0
$$999$$ 0 0
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 768.2.a.l.1.2 2
3.2 odd 2 2304.2.a.r.1.1 2
4.3 odd 2 768.2.a.i.1.2 2
8.3 odd 2 inner 768.2.a.l.1.1 2
8.5 even 2 768.2.a.i.1.1 2
12.11 even 2 2304.2.a.x.1.1 2
16.3 odd 4 384.2.d.c.193.1 4
16.5 even 4 384.2.d.c.193.2 yes 4
16.11 odd 4 384.2.d.c.193.4 yes 4
16.13 even 4 384.2.d.c.193.3 yes 4
24.5 odd 2 2304.2.a.x.1.2 2
24.11 even 2 2304.2.a.r.1.2 2
48.5 odd 4 1152.2.d.h.577.1 4
48.11 even 4 1152.2.d.h.577.2 4
48.29 odd 4 1152.2.d.h.577.3 4
48.35 even 4 1152.2.d.h.577.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
384.2.d.c.193.1 4 16.3 odd 4
384.2.d.c.193.2 yes 4 16.5 even 4
384.2.d.c.193.3 yes 4 16.13 even 4
384.2.d.c.193.4 yes 4 16.11 odd 4
768.2.a.i.1.1 2 8.5 even 2
768.2.a.i.1.2 2 4.3 odd 2
768.2.a.l.1.1 2 8.3 odd 2 inner
768.2.a.l.1.2 2 1.1 even 1 trivial
1152.2.d.h.577.1 4 48.5 odd 4
1152.2.d.h.577.2 4 48.11 even 4
1152.2.d.h.577.3 4 48.29 odd 4
1152.2.d.h.577.4 4 48.35 even 4
2304.2.a.r.1.1 2 3.2 odd 2
2304.2.a.r.1.2 2 24.11 even 2
2304.2.a.x.1.1 2 12.11 even 2
2304.2.a.x.1.2 2 24.5 odd 2