# Properties

 Label 1425.2.c.j.799.1 Level $1425$ Weight $2$ Character 1425.799 Analytic conductor $11.379$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1425,2,Mod(799,1425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1425, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1425.799");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1425 = 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1425.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

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

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

 Self dual: no Analytic conductor: $$11.3786822880$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 285) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 799.1 Root $$0.707107 - 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 1425.799 Dual form 1425.2.c.j.799.4

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.41421i q^{2} -1.00000i q^{3} -3.82843 q^{4} -2.41421 q^{6} -3.41421i q^{7} +4.41421i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-2.41421i q^{2} -1.00000i q^{3} -3.82843 q^{4} -2.41421 q^{6} -3.41421i q^{7} +4.41421i q^{8} -1.00000 q^{9} -1.41421 q^{11} +3.82843i q^{12} +2.58579i q^{13} -8.24264 q^{14} +3.00000 q^{16} +6.82843i q^{17} +2.41421i q^{18} -1.00000 q^{19} -3.41421 q^{21} +3.41421i q^{22} -3.65685i q^{23} +4.41421 q^{24} +6.24264 q^{26} +1.00000i q^{27} +13.0711i q^{28} -5.07107 q^{29} -10.4853 q^{31} +1.58579i q^{32} +1.41421i q^{33} +16.4853 q^{34} +3.82843 q^{36} +3.07107i q^{37} +2.41421i q^{38} +2.58579 q^{39} -4.58579 q^{41} +8.24264i q^{42} +3.41421i q^{43} +5.41421 q^{44} -8.82843 q^{46} -11.6569i q^{47} -3.00000i q^{48} -4.65685 q^{49} +6.82843 q^{51} -9.89949i q^{52} +4.00000i q^{53} +2.41421 q^{54} +15.0711 q^{56} +1.00000i q^{57} +12.2426i q^{58} +8.48528 q^{59} -5.65685 q^{61} +25.3137i q^{62} +3.41421i q^{63} +9.82843 q^{64} +3.41421 q^{66} -12.0000i q^{67} -26.1421i q^{68} -3.65685 q^{69} +12.4853 q^{71} -4.41421i q^{72} -2.00000i q^{73} +7.41421 q^{74} +3.82843 q^{76} +4.82843i q^{77} -6.24264i q^{78} -11.3137 q^{79} +1.00000 q^{81} +11.0711i q^{82} +6.48528i q^{83} +13.0711 q^{84} +8.24264 q^{86} +5.07107i q^{87} -6.24264i q^{88} +14.7279 q^{89} +8.82843 q^{91} +14.0000i q^{92} +10.4853i q^{93} -28.1421 q^{94} +1.58579 q^{96} -4.24264i q^{97} +11.2426i q^{98} +1.41421 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} - 4 q^{6} - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^4 - 4 * q^6 - 4 * q^9 $$4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} - 16 q^{14} + 12 q^{16} - 4 q^{19} - 8 q^{21} + 12 q^{24} + 8 q^{26} + 8 q^{29} - 8 q^{31} + 32 q^{34} + 4 q^{36} + 16 q^{39} - 24 q^{41} + 16 q^{44} - 24 q^{46} + 4 q^{49} + 16 q^{51} + 4 q^{54} + 32 q^{56} + 28 q^{64} + 8 q^{66} + 8 q^{69} + 16 q^{71} + 24 q^{74} + 4 q^{76} + 4 q^{81} + 24 q^{84} + 16 q^{86} + 8 q^{89} + 24 q^{91} - 56 q^{94} + 12 q^{96}+O(q^{100})$$ 4 * q - 4 * q^4 - 4 * q^6 - 4 * q^9 - 16 * q^14 + 12 * q^16 - 4 * q^19 - 8 * q^21 + 12 * q^24 + 8 * q^26 + 8 * q^29 - 8 * q^31 + 32 * q^34 + 4 * q^36 + 16 * q^39 - 24 * q^41 + 16 * q^44 - 24 * q^46 + 4 * q^49 + 16 * q^51 + 4 * q^54 + 32 * q^56 + 28 * q^64 + 8 * q^66 + 8 * q^69 + 16 * q^71 + 24 * q^74 + 4 * q^76 + 4 * q^81 + 24 * q^84 + 16 * q^86 + 8 * q^89 + 24 * q^91 - 56 * q^94 + 12 * q^96

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times$$.

 $$n$$ $$476$$ $$1027$$ $$1351$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 2.41421i − 1.70711i −0.521005 0.853553i $$-0.674443\pi$$
0.521005 0.853553i $$-0.325557\pi$$
$$3$$ − 1.00000i − 0.577350i
$$4$$ −3.82843 −1.91421
$$5$$ 0 0
$$6$$ −2.41421 −0.985599
$$7$$ − 3.41421i − 1.29045i −0.763992 0.645226i $$-0.776763\pi$$
0.763992 0.645226i $$-0.223237\pi$$
$$8$$ 4.41421i 1.56066i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −1.41421 −0.426401 −0.213201 0.977008i $$-0.568389\pi$$
−0.213201 + 0.977008i $$0.568389\pi$$
$$12$$ 3.82843i 1.10517i
$$13$$ 2.58579i 0.717168i 0.933497 + 0.358584i $$0.116740\pi$$
−0.933497 + 0.358584i $$0.883260\pi$$
$$14$$ −8.24264 −2.20294
$$15$$ 0 0
$$16$$ 3.00000 0.750000
$$17$$ 6.82843i 1.65614i 0.560627 + 0.828068i $$0.310560\pi$$
−0.560627 + 0.828068i $$0.689440\pi$$
$$18$$ 2.41421i 0.569036i
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −3.41421 −0.745042
$$22$$ 3.41421i 0.727913i
$$23$$ − 3.65685i − 0.762507i −0.924471 0.381253i $$-0.875493\pi$$
0.924471 0.381253i $$-0.124507\pi$$
$$24$$ 4.41421 0.901048
$$25$$ 0 0
$$26$$ 6.24264 1.22428
$$27$$ 1.00000i 0.192450i
$$28$$ 13.0711i 2.47020i
$$29$$ −5.07107 −0.941674 −0.470837 0.882220i $$-0.656048\pi$$
−0.470837 + 0.882220i $$0.656048\pi$$
$$30$$ 0 0
$$31$$ −10.4853 −1.88321 −0.941606 0.336717i $$-0.890684\pi$$
−0.941606 + 0.336717i $$0.890684\pi$$
$$32$$ 1.58579i 0.280330i
$$33$$ 1.41421i 0.246183i
$$34$$ 16.4853 2.82720
$$35$$ 0 0
$$36$$ 3.82843 0.638071
$$37$$ 3.07107i 0.504880i 0.967612 + 0.252440i $$0.0812331\pi$$
−0.967612 + 0.252440i $$0.918767\pi$$
$$38$$ 2.41421i 0.391637i
$$39$$ 2.58579 0.414057
$$40$$ 0 0
$$41$$ −4.58579 −0.716180 −0.358090 0.933687i $$-0.616572\pi$$
−0.358090 + 0.933687i $$0.616572\pi$$
$$42$$ 8.24264i 1.27187i
$$43$$ 3.41421i 0.520663i 0.965519 + 0.260331i $$0.0838318\pi$$
−0.965519 + 0.260331i $$0.916168\pi$$
$$44$$ 5.41421 0.816223
$$45$$ 0 0
$$46$$ −8.82843 −1.30168
$$47$$ − 11.6569i − 1.70033i −0.526519 0.850163i $$-0.676503\pi$$
0.526519 0.850163i $$-0.323497\pi$$
$$48$$ − 3.00000i − 0.433013i
$$49$$ −4.65685 −0.665265
$$50$$ 0 0
$$51$$ 6.82843 0.956171
$$52$$ − 9.89949i − 1.37281i
$$53$$ 4.00000i 0.549442i 0.961524 + 0.274721i $$0.0885855\pi$$
−0.961524 + 0.274721i $$0.911414\pi$$
$$54$$ 2.41421 0.328533
$$55$$ 0 0
$$56$$ 15.0711 2.01396
$$57$$ 1.00000i 0.132453i
$$58$$ 12.2426i 1.60754i
$$59$$ 8.48528 1.10469 0.552345 0.833616i $$-0.313733\pi$$
0.552345 + 0.833616i $$0.313733\pi$$
$$60$$ 0 0
$$61$$ −5.65685 −0.724286 −0.362143 0.932123i $$-0.617955\pi$$
−0.362143 + 0.932123i $$0.617955\pi$$
$$62$$ 25.3137i 3.21484i
$$63$$ 3.41421i 0.430150i
$$64$$ 9.82843 1.22855
$$65$$ 0 0
$$66$$ 3.41421 0.420261
$$67$$ − 12.0000i − 1.46603i −0.680211 0.733017i $$-0.738112\pi$$
0.680211 0.733017i $$-0.261888\pi$$
$$68$$ − 26.1421i − 3.17020i
$$69$$ −3.65685 −0.440234
$$70$$ 0 0
$$71$$ 12.4853 1.48173 0.740865 0.671654i $$-0.234416\pi$$
0.740865 + 0.671654i $$0.234416\pi$$
$$72$$ − 4.41421i − 0.520220i
$$73$$ − 2.00000i − 0.234082i −0.993127 0.117041i $$-0.962659\pi$$
0.993127 0.117041i $$-0.0373409\pi$$
$$74$$ 7.41421 0.861885
$$75$$ 0 0
$$76$$ 3.82843 0.439151
$$77$$ 4.82843i 0.550250i
$$78$$ − 6.24264i − 0.706840i
$$79$$ −11.3137 −1.27289 −0.636446 0.771321i $$-0.719596\pi$$
−0.636446 + 0.771321i $$0.719596\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 11.0711i 1.22259i
$$83$$ 6.48528i 0.711852i 0.934514 + 0.355926i $$0.115835\pi$$
−0.934514 + 0.355926i $$0.884165\pi$$
$$84$$ 13.0711 1.42617
$$85$$ 0 0
$$86$$ 8.24264 0.888827
$$87$$ 5.07107i 0.543676i
$$88$$ − 6.24264i − 0.665468i
$$89$$ 14.7279 1.56116 0.780578 0.625058i $$-0.214925\pi$$
0.780578 + 0.625058i $$0.214925\pi$$
$$90$$ 0 0
$$91$$ 8.82843 0.925471
$$92$$ 14.0000i 1.45960i
$$93$$ 10.4853i 1.08727i
$$94$$ −28.1421 −2.90264
$$95$$ 0 0
$$96$$ 1.58579 0.161849
$$97$$ − 4.24264i − 0.430775i −0.976529 0.215387i $$-0.930899\pi$$
0.976529 0.215387i $$-0.0691014\pi$$
$$98$$ 11.2426i 1.13568i
$$99$$ 1.41421 0.142134
$$100$$ 0 0
$$101$$ 0.828427 0.0824316 0.0412158 0.999150i $$-0.486877\pi$$
0.0412158 + 0.999150i $$0.486877\pi$$
$$102$$ − 16.4853i − 1.63229i
$$103$$ 9.65685i 0.951518i 0.879576 + 0.475759i $$0.157827\pi$$
−0.879576 + 0.475759i $$0.842173\pi$$
$$104$$ −11.4142 −1.11926
$$105$$ 0 0
$$106$$ 9.65685 0.937957
$$107$$ − 8.00000i − 0.773389i −0.922208 0.386695i $$-0.873617\pi$$
0.922208 0.386695i $$-0.126383\pi$$
$$108$$ − 3.82843i − 0.368391i
$$109$$ −8.82843 −0.845610 −0.422805 0.906221i $$-0.638954\pi$$
−0.422805 + 0.906221i $$0.638954\pi$$
$$110$$ 0 0
$$111$$ 3.07107 0.291493
$$112$$ − 10.2426i − 0.967839i
$$113$$ − 4.48528i − 0.421940i −0.977493 0.210970i $$-0.932338\pi$$
0.977493 0.210970i $$-0.0676622\pi$$
$$114$$ 2.41421 0.226112
$$115$$ 0 0
$$116$$ 19.4142 1.80256
$$117$$ − 2.58579i − 0.239056i
$$118$$ − 20.4853i − 1.88582i
$$119$$ 23.3137 2.13716
$$120$$ 0 0
$$121$$ −9.00000 −0.818182
$$122$$ 13.6569i 1.23643i
$$123$$ 4.58579i 0.413486i
$$124$$ 40.1421 3.60487
$$125$$ 0 0
$$126$$ 8.24264 0.734313
$$127$$ − 8.00000i − 0.709885i −0.934888 0.354943i $$-0.884500\pi$$
0.934888 0.354943i $$-0.115500\pi$$
$$128$$ − 20.5563i − 1.81694i
$$129$$ 3.41421 0.300605
$$130$$ 0 0
$$131$$ −8.72792 −0.762562 −0.381281 0.924459i $$-0.624517\pi$$
−0.381281 + 0.924459i $$0.624517\pi$$
$$132$$ − 5.41421i − 0.471247i
$$133$$ 3.41421i 0.296050i
$$134$$ −28.9706 −2.50268
$$135$$ 0 0
$$136$$ −30.1421 −2.58467
$$137$$ 14.0000i 1.19610i 0.801459 + 0.598050i $$0.204058\pi$$
−0.801459 + 0.598050i $$0.795942\pi$$
$$138$$ 8.82843i 0.751526i
$$139$$ −6.82843 −0.579180 −0.289590 0.957151i $$-0.593519\pi$$
−0.289590 + 0.957151i $$0.593519\pi$$
$$140$$ 0 0
$$141$$ −11.6569 −0.981684
$$142$$ − 30.1421i − 2.52947i
$$143$$ − 3.65685i − 0.305802i
$$144$$ −3.00000 −0.250000
$$145$$ 0 0
$$146$$ −4.82843 −0.399603
$$147$$ 4.65685i 0.384091i
$$148$$ − 11.7574i − 0.966449i
$$149$$ −7.65685 −0.627274 −0.313637 0.949543i $$-0.601547\pi$$
−0.313637 + 0.949543i $$0.601547\pi$$
$$150$$ 0 0
$$151$$ −21.7990 −1.77398 −0.886988 0.461792i $$-0.847207\pi$$
−0.886988 + 0.461792i $$0.847207\pi$$
$$152$$ − 4.41421i − 0.358040i
$$153$$ − 6.82843i − 0.552046i
$$154$$ 11.6569 0.939336
$$155$$ 0 0
$$156$$ −9.89949 −0.792594
$$157$$ − 17.7990i − 1.42051i −0.703942 0.710257i $$-0.748579\pi$$
0.703942 0.710257i $$-0.251421\pi$$
$$158$$ 27.3137i 2.17296i
$$159$$ 4.00000 0.317221
$$160$$ 0 0
$$161$$ −12.4853 −0.983978
$$162$$ − 2.41421i − 0.189679i
$$163$$ − 11.8995i − 0.932040i −0.884774 0.466020i $$-0.845687\pi$$
0.884774 0.466020i $$-0.154313\pi$$
$$164$$ 17.5563 1.37092
$$165$$ 0 0
$$166$$ 15.6569 1.21521
$$167$$ − 10.0000i − 0.773823i −0.922117 0.386912i $$-0.873542\pi$$
0.922117 0.386912i $$-0.126458\pi$$
$$168$$ − 15.0711i − 1.16276i
$$169$$ 6.31371 0.485670
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ − 13.0711i − 0.996660i
$$173$$ 22.1421i 1.68344i 0.539918 + 0.841718i $$0.318455\pi$$
−0.539918 + 0.841718i $$0.681545\pi$$
$$174$$ 12.2426 0.928112
$$175$$ 0 0
$$176$$ −4.24264 −0.319801
$$177$$ − 8.48528i − 0.637793i
$$178$$ − 35.5563i − 2.66506i
$$179$$ −22.8284 −1.70628 −0.853138 0.521685i $$-0.825304\pi$$
−0.853138 + 0.521685i $$0.825304\pi$$
$$180$$ 0 0
$$181$$ −24.8284 −1.84548 −0.922741 0.385420i $$-0.874057\pi$$
−0.922741 + 0.385420i $$0.874057\pi$$
$$182$$ − 21.3137i − 1.57988i
$$183$$ 5.65685i 0.418167i
$$184$$ 16.1421 1.19001
$$185$$ 0 0
$$186$$ 25.3137 1.85609
$$187$$ − 9.65685i − 0.706179i
$$188$$ 44.6274i 3.25479i
$$189$$ 3.41421 0.248347
$$190$$ 0 0
$$191$$ −17.8995 −1.29516 −0.647581 0.761997i $$-0.724219\pi$$
−0.647581 + 0.761997i $$0.724219\pi$$
$$192$$ − 9.82843i − 0.709306i
$$193$$ − 0.928932i − 0.0668660i −0.999441 0.0334330i $$-0.989356\pi$$
0.999441 0.0334330i $$-0.0106440\pi$$
$$194$$ −10.2426 −0.735379
$$195$$ 0 0
$$196$$ 17.8284 1.27346
$$197$$ 9.17157i 0.653448i 0.945120 + 0.326724i $$0.105945\pi$$
−0.945120 + 0.326724i $$0.894055\pi$$
$$198$$ − 3.41421i − 0.242638i
$$199$$ −0.485281 −0.0344007 −0.0172003 0.999852i $$-0.505475\pi$$
−0.0172003 + 0.999852i $$0.505475\pi$$
$$200$$ 0 0
$$201$$ −12.0000 −0.846415
$$202$$ − 2.00000i − 0.140720i
$$203$$ 17.3137i 1.21518i
$$204$$ −26.1421 −1.83032
$$205$$ 0 0
$$206$$ 23.3137 1.62434
$$207$$ 3.65685i 0.254169i
$$208$$ 7.75736i 0.537876i
$$209$$ 1.41421 0.0978232
$$210$$ 0 0
$$211$$ 7.31371 0.503496 0.251748 0.967793i $$-0.418995\pi$$
0.251748 + 0.967793i $$0.418995\pi$$
$$212$$ − 15.3137i − 1.05175i
$$213$$ − 12.4853i − 0.855477i
$$214$$ −19.3137 −1.32026
$$215$$ 0 0
$$216$$ −4.41421 −0.300349
$$217$$ 35.7990i 2.43019i
$$218$$ 21.3137i 1.44355i
$$219$$ −2.00000 −0.135147
$$220$$ 0 0
$$221$$ −17.6569 −1.18773
$$222$$ − 7.41421i − 0.497609i
$$223$$ 17.6569i 1.18239i 0.806529 + 0.591195i $$0.201344\pi$$
−0.806529 + 0.591195i $$0.798656\pi$$
$$224$$ 5.41421 0.361752
$$225$$ 0 0
$$226$$ −10.8284 −0.720296
$$227$$ − 14.9706i − 0.993631i −0.867856 0.496816i $$-0.834503\pi$$
0.867856 0.496816i $$-0.165497\pi$$
$$228$$ − 3.82843i − 0.253544i
$$229$$ −9.65685 −0.638143 −0.319071 0.947731i $$-0.603371\pi$$
−0.319071 + 0.947731i $$0.603371\pi$$
$$230$$ 0 0
$$231$$ 4.82843 0.317687
$$232$$ − 22.3848i − 1.46963i
$$233$$ − 19.6569i − 1.28776i −0.765125 0.643882i $$-0.777323\pi$$
0.765125 0.643882i $$-0.222677\pi$$
$$234$$ −6.24264 −0.408094
$$235$$ 0 0
$$236$$ −32.4853 −2.11461
$$237$$ 11.3137i 0.734904i
$$238$$ − 56.2843i − 3.64837i
$$239$$ −5.41421 −0.350216 −0.175108 0.984549i $$-0.556028\pi$$
−0.175108 + 0.984549i $$0.556028\pi$$
$$240$$ 0 0
$$241$$ 18.9706 1.22200 0.611001 0.791630i $$-0.290767\pi$$
0.611001 + 0.791630i $$0.290767\pi$$
$$242$$ 21.7279i 1.39672i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 21.6569 1.38644
$$245$$ 0 0
$$246$$ 11.0711 0.705866
$$247$$ − 2.58579i − 0.164530i
$$248$$ − 46.2843i − 2.93905i
$$249$$ 6.48528 0.410988
$$250$$ 0 0
$$251$$ 27.0711 1.70871 0.854355 0.519689i $$-0.173952\pi$$
0.854355 + 0.519689i $$0.173952\pi$$
$$252$$ − 13.0711i − 0.823400i
$$253$$ 5.17157i 0.325134i
$$254$$ −19.3137 −1.21185
$$255$$ 0 0
$$256$$ −29.9706 −1.87316
$$257$$ − 6.82843i − 0.425946i −0.977058 0.212973i $$-0.931685\pi$$
0.977058 0.212973i $$-0.0683146\pi$$
$$258$$ − 8.24264i − 0.513164i
$$259$$ 10.4853 0.651524
$$260$$ 0 0
$$261$$ 5.07107 0.313891
$$262$$ 21.0711i 1.30177i
$$263$$ − 3.85786i − 0.237886i −0.992901 0.118943i $$-0.962049\pi$$
0.992901 0.118943i $$-0.0379506\pi$$
$$264$$ −6.24264 −0.384208
$$265$$ 0 0
$$266$$ 8.24264 0.505389
$$267$$ − 14.7279i − 0.901334i
$$268$$ 45.9411i 2.80630i
$$269$$ 28.3848 1.73065 0.865325 0.501211i $$-0.167112\pi$$
0.865325 + 0.501211i $$0.167112\pi$$
$$270$$ 0 0
$$271$$ −25.1716 −1.52906 −0.764532 0.644586i $$-0.777030\pi$$
−0.764532 + 0.644586i $$0.777030\pi$$
$$272$$ 20.4853i 1.24210i
$$273$$ − 8.82843i − 0.534321i
$$274$$ 33.7990 2.04187
$$275$$ 0 0
$$276$$ 14.0000 0.842701
$$277$$ 22.9706i 1.38017i 0.723730 + 0.690084i $$0.242426\pi$$
−0.723730 + 0.690084i $$0.757574\pi$$
$$278$$ 16.4853i 0.988721i
$$279$$ 10.4853 0.627737
$$280$$ 0 0
$$281$$ 10.7279 0.639974 0.319987 0.947422i $$-0.396321\pi$$
0.319987 + 0.947422i $$0.396321\pi$$
$$282$$ 28.1421i 1.67584i
$$283$$ − 2.24264i − 0.133311i −0.997776 0.0666556i $$-0.978767\pi$$
0.997776 0.0666556i $$-0.0212329\pi$$
$$284$$ −47.7990 −2.83635
$$285$$ 0 0
$$286$$ −8.82843 −0.522036
$$287$$ 15.6569i 0.924195i
$$288$$ − 1.58579i − 0.0934434i
$$289$$ −29.6274 −1.74279
$$290$$ 0 0
$$291$$ −4.24264 −0.248708
$$292$$ 7.65685i 0.448084i
$$293$$ 7.79899i 0.455622i 0.973705 + 0.227811i $$0.0731568\pi$$
−0.973705 + 0.227811i $$0.926843\pi$$
$$294$$ 11.2426 0.655684
$$295$$ 0 0
$$296$$ −13.5563 −0.787947
$$297$$ − 1.41421i − 0.0820610i
$$298$$ 18.4853i 1.07082i
$$299$$ 9.45584 0.546846
$$300$$ 0 0
$$301$$ 11.6569 0.671890
$$302$$ 52.6274i 3.02837i
$$303$$ − 0.828427i − 0.0475919i
$$304$$ −3.00000 −0.172062
$$305$$ 0 0
$$306$$ −16.4853 −0.942401
$$307$$ 31.7990i 1.81486i 0.420199 + 0.907432i $$0.361960\pi$$
−0.420199 + 0.907432i $$0.638040\pi$$
$$308$$ − 18.4853i − 1.05330i
$$309$$ 9.65685 0.549359
$$310$$ 0 0
$$311$$ −23.7574 −1.34716 −0.673578 0.739116i $$-0.735244\pi$$
−0.673578 + 0.739116i $$0.735244\pi$$
$$312$$ 11.4142i 0.646203i
$$313$$ − 26.4853i − 1.49704i −0.663114 0.748518i $$-0.730766\pi$$
0.663114 0.748518i $$-0.269234\pi$$
$$314$$ −42.9706 −2.42497
$$315$$ 0 0
$$316$$ 43.3137 2.43659
$$317$$ 11.3137i 0.635441i 0.948184 + 0.317721i $$0.102917\pi$$
−0.948184 + 0.317721i $$0.897083\pi$$
$$318$$ − 9.65685i − 0.541529i
$$319$$ 7.17157 0.401531
$$320$$ 0 0
$$321$$ −8.00000 −0.446516
$$322$$ 30.1421i 1.67976i
$$323$$ − 6.82843i − 0.379944i
$$324$$ −3.82843 −0.212690
$$325$$ 0 0
$$326$$ −28.7279 −1.59109
$$327$$ 8.82843i 0.488213i
$$328$$ − 20.2426i − 1.11771i
$$329$$ −39.7990 −2.19419
$$330$$ 0 0
$$331$$ 12.8284 0.705114 0.352557 0.935790i $$-0.385312\pi$$
0.352557 + 0.935790i $$0.385312\pi$$
$$332$$ − 24.8284i − 1.36264i
$$333$$ − 3.07107i − 0.168293i
$$334$$ −24.1421 −1.32100
$$335$$ 0 0
$$336$$ −10.2426 −0.558782
$$337$$ − 19.7574i − 1.07625i −0.842864 0.538126i $$-0.819132\pi$$
0.842864 0.538126i $$-0.180868\pi$$
$$338$$ − 15.2426i − 0.829090i
$$339$$ −4.48528 −0.243607
$$340$$ 0 0
$$341$$ 14.8284 0.803004
$$342$$ − 2.41421i − 0.130546i
$$343$$ − 8.00000i − 0.431959i
$$344$$ −15.0711 −0.812578
$$345$$ 0 0
$$346$$ 53.4558 2.87380
$$347$$ 6.48528i 0.348148i 0.984733 + 0.174074i $$0.0556932\pi$$
−0.984733 + 0.174074i $$0.944307\pi$$
$$348$$ − 19.4142i − 1.04071i
$$349$$ −6.68629 −0.357909 −0.178954 0.983857i $$-0.557271\pi$$
−0.178954 + 0.983857i $$0.557271\pi$$
$$350$$ 0 0
$$351$$ −2.58579 −0.138019
$$352$$ − 2.24264i − 0.119533i
$$353$$ − 7.65685i − 0.407533i −0.979019 0.203767i $$-0.934682\pi$$
0.979019 0.203767i $$-0.0653184\pi$$
$$354$$ −20.4853 −1.08878
$$355$$ 0 0
$$356$$ −56.3848 −2.98839
$$357$$ − 23.3137i − 1.23389i
$$358$$ 55.1127i 2.91280i
$$359$$ 9.89949 0.522475 0.261238 0.965275i $$-0.415869\pi$$
0.261238 + 0.965275i $$0.415869\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 59.9411i 3.15044i
$$363$$ 9.00000i 0.472377i
$$364$$ −33.7990 −1.77155
$$365$$ 0 0
$$366$$ 13.6569 0.713855
$$367$$ 16.5858i 0.865771i 0.901449 + 0.432886i $$0.142505\pi$$
−0.901449 + 0.432886i $$0.857495\pi$$
$$368$$ − 10.9706i − 0.571880i
$$369$$ 4.58579 0.238727
$$370$$ 0 0
$$371$$ 13.6569 0.709029
$$372$$ − 40.1421i − 2.08127i
$$373$$ 9.89949i 0.512576i 0.966600 + 0.256288i $$0.0824996\pi$$
−0.966600 + 0.256288i $$0.917500\pi$$
$$374$$ −23.3137 −1.20552
$$375$$ 0 0
$$376$$ 51.4558 2.65363
$$377$$ − 13.1127i − 0.675338i
$$378$$ − 8.24264i − 0.423956i
$$379$$ 4.14214 0.212767 0.106384 0.994325i $$-0.466073\pi$$
0.106384 + 0.994325i $$0.466073\pi$$
$$380$$ 0 0
$$381$$ −8.00000 −0.409852
$$382$$ 43.2132i 2.21098i
$$383$$ 12.0000i 0.613171i 0.951843 + 0.306586i $$0.0991866\pi$$
−0.951843 + 0.306586i $$0.900813\pi$$
$$384$$ −20.5563 −1.04901
$$385$$ 0 0
$$386$$ −2.24264 −0.114147
$$387$$ − 3.41421i − 0.173554i
$$388$$ 16.2426i 0.824595i
$$389$$ 18.9706 0.961846 0.480923 0.876763i $$-0.340302\pi$$
0.480923 + 0.876763i $$0.340302\pi$$
$$390$$ 0 0
$$391$$ 24.9706 1.26282
$$392$$ − 20.5563i − 1.03825i
$$393$$ 8.72792i 0.440265i
$$394$$ 22.1421 1.11550
$$395$$ 0 0
$$396$$ −5.41421 −0.272074
$$397$$ − 24.3431i − 1.22175i −0.791728 0.610874i $$-0.790818\pi$$
0.791728 0.610874i $$-0.209182\pi$$
$$398$$ 1.17157i 0.0587256i
$$399$$ 3.41421 0.170924
$$400$$ 0 0
$$401$$ −18.0416 −0.900956 −0.450478 0.892788i $$-0.648746\pi$$
−0.450478 + 0.892788i $$0.648746\pi$$
$$402$$ 28.9706i 1.44492i
$$403$$ − 27.1127i − 1.35058i
$$404$$ −3.17157 −0.157792
$$405$$ 0 0
$$406$$ 41.7990 2.07445
$$407$$ − 4.34315i − 0.215282i
$$408$$ 30.1421i 1.49226i
$$409$$ −26.4853 −1.30961 −0.654806 0.755797i $$-0.727250\pi$$
−0.654806 + 0.755797i $$0.727250\pi$$
$$410$$ 0 0
$$411$$ 14.0000 0.690569
$$412$$ − 36.9706i − 1.82141i
$$413$$ − 28.9706i − 1.42555i
$$414$$ 8.82843 0.433894
$$415$$ 0 0
$$416$$ −4.10051 −0.201044
$$417$$ 6.82843i 0.334390i
$$418$$ − 3.41421i − 0.166995i
$$419$$ −18.8701 −0.921863 −0.460931 0.887436i $$-0.652485\pi$$
−0.460931 + 0.887436i $$0.652485\pi$$
$$420$$ 0 0
$$421$$ −37.3137 −1.81856 −0.909279 0.416186i $$-0.863366\pi$$
−0.909279 + 0.416186i $$0.863366\pi$$
$$422$$ − 17.6569i − 0.859522i
$$423$$ 11.6569i 0.566776i
$$424$$ −17.6569 −0.857493
$$425$$ 0 0
$$426$$ −30.1421 −1.46039
$$427$$ 19.3137i 0.934656i
$$428$$ 30.6274i 1.48043i
$$429$$ −3.65685 −0.176555
$$430$$ 0 0
$$431$$ 20.4853 0.986741 0.493371 0.869819i $$-0.335765\pi$$
0.493371 + 0.869819i $$0.335765\pi$$
$$432$$ 3.00000i 0.144338i
$$433$$ 15.0711i 0.724269i 0.932126 + 0.362135i $$0.117952\pi$$
−0.932126 + 0.362135i $$0.882048\pi$$
$$434$$ 86.4264 4.14860
$$435$$ 0 0
$$436$$ 33.7990 1.61868
$$437$$ 3.65685i 0.174931i
$$438$$ 4.82843i 0.230711i
$$439$$ 32.9706 1.57360 0.786800 0.617209i $$-0.211737\pi$$
0.786800 + 0.617209i $$0.211737\pi$$
$$440$$ 0 0
$$441$$ 4.65685 0.221755
$$442$$ 42.6274i 2.02758i
$$443$$ 21.3137i 1.01264i 0.862344 + 0.506322i $$0.168995\pi$$
−0.862344 + 0.506322i $$0.831005\pi$$
$$444$$ −11.7574 −0.557980
$$445$$ 0 0
$$446$$ 42.6274 2.01847
$$447$$ 7.65685i 0.362157i
$$448$$ − 33.5563i − 1.58539i
$$449$$ 11.8995 0.561572 0.280786 0.959770i $$-0.409405\pi$$
0.280786 + 0.959770i $$0.409405\pi$$
$$450$$ 0 0
$$451$$ 6.48528 0.305380
$$452$$ 17.1716i 0.807683i
$$453$$ 21.7990i 1.02421i
$$454$$ −36.1421 −1.69623
$$455$$ 0 0
$$456$$ −4.41421 −0.206714
$$457$$ − 23.1716i − 1.08392i −0.840404 0.541960i $$-0.817682\pi$$
0.840404 0.541960i $$-0.182318\pi$$
$$458$$ 23.3137i 1.08938i
$$459$$ −6.82843 −0.318724
$$460$$ 0 0
$$461$$ 33.3137 1.55157 0.775787 0.630995i $$-0.217353\pi$$
0.775787 + 0.630995i $$0.217353\pi$$
$$462$$ − 11.6569i − 0.542326i
$$463$$ − 27.8995i − 1.29660i −0.761385 0.648300i $$-0.775480\pi$$
0.761385 0.648300i $$-0.224520\pi$$
$$464$$ −15.2132 −0.706255
$$465$$ 0 0
$$466$$ −47.4558 −2.19835
$$467$$ − 27.6569i − 1.27981i −0.768455 0.639903i $$-0.778974\pi$$
0.768455 0.639903i $$-0.221026\pi$$
$$468$$ 9.89949i 0.457604i
$$469$$ −40.9706 −1.89184
$$470$$ 0 0
$$471$$ −17.7990 −0.820134
$$472$$ 37.4558i 1.72404i
$$473$$ − 4.82843i − 0.222011i
$$474$$ 27.3137 1.25456
$$475$$ 0 0
$$476$$ −89.2548 −4.09099
$$477$$ − 4.00000i − 0.183147i
$$478$$ 13.0711i 0.597857i
$$479$$ −38.1838 −1.74466 −0.872330 0.488917i $$-0.837392\pi$$
−0.872330 + 0.488917i $$0.837392\pi$$
$$480$$ 0 0
$$481$$ −7.94113 −0.362084
$$482$$ − 45.7990i − 2.08609i
$$483$$ 12.4853i 0.568100i
$$484$$ 34.4558 1.56617
$$485$$ 0 0
$$486$$ −2.41421 −0.109511
$$487$$ − 2.82843i − 0.128168i −0.997944 0.0640841i $$-0.979587\pi$$
0.997944 0.0640841i $$-0.0204126\pi$$
$$488$$ − 24.9706i − 1.13036i
$$489$$ −11.8995 −0.538114
$$490$$ 0 0
$$491$$ 2.10051 0.0947945 0.0473972 0.998876i $$-0.484907\pi$$
0.0473972 + 0.998876i $$0.484907\pi$$
$$492$$ − 17.5563i − 0.791501i
$$493$$ − 34.6274i − 1.55954i
$$494$$ −6.24264 −0.280870
$$495$$ 0 0
$$496$$ −31.4558 −1.41241
$$497$$ − 42.6274i − 1.91210i
$$498$$ − 15.6569i − 0.701600i
$$499$$ −15.7990 −0.707260 −0.353630 0.935385i $$-0.615053\pi$$
−0.353630 + 0.935385i $$0.615053\pi$$
$$500$$ 0 0
$$501$$ −10.0000 −0.446767
$$502$$ − 65.3553i − 2.91695i
$$503$$ − 4.82843i − 0.215289i −0.994189 0.107644i $$-0.965669\pi$$
0.994189 0.107644i $$-0.0343308\pi$$
$$504$$ −15.0711 −0.671319
$$505$$ 0 0
$$506$$ 12.4853 0.555038
$$507$$ − 6.31371i − 0.280402i
$$508$$ 30.6274i 1.35887i
$$509$$ 4.38478 0.194352 0.0971759 0.995267i $$-0.469019\pi$$
0.0971759 + 0.995267i $$0.469019\pi$$
$$510$$ 0 0
$$511$$ −6.82843 −0.302072
$$512$$ 31.2426i 1.38074i
$$513$$ − 1.00000i − 0.0441511i
$$514$$ −16.4853 −0.727135
$$515$$ 0 0
$$516$$ −13.0711 −0.575422
$$517$$ 16.4853i 0.725022i
$$518$$ − 25.3137i − 1.11222i
$$519$$ 22.1421 0.971932
$$520$$ 0 0
$$521$$ 12.3848 0.542587 0.271293 0.962497i $$-0.412549\pi$$
0.271293 + 0.962497i $$0.412549\pi$$
$$522$$ − 12.2426i − 0.535846i
$$523$$ 23.7990i 1.04066i 0.853966 + 0.520329i $$0.174191\pi$$
−0.853966 + 0.520329i $$0.825809\pi$$
$$524$$ 33.4142 1.45971
$$525$$ 0 0
$$526$$ −9.31371 −0.406097
$$527$$ − 71.5980i − 3.11886i
$$528$$ 4.24264i 0.184637i
$$529$$ 9.62742 0.418583
$$530$$ 0 0
$$531$$ −8.48528 −0.368230
$$532$$ − 13.0711i − 0.566703i
$$533$$ − 11.8579i − 0.513621i
$$534$$ −35.5563 −1.53867
$$535$$ 0 0
$$536$$ 52.9706 2.28798
$$537$$ 22.8284i 0.985119i
$$538$$ − 68.5269i − 2.95440i
$$539$$ 6.58579 0.283670
$$540$$ 0 0
$$541$$ 12.6274 0.542895 0.271448 0.962453i $$-0.412498\pi$$
0.271448 + 0.962453i $$0.412498\pi$$
$$542$$ 60.7696i 2.61028i
$$543$$ 24.8284i 1.06549i
$$544$$ −10.8284 −0.464265
$$545$$ 0 0
$$546$$ −21.3137 −0.912143
$$547$$ 5.85786i 0.250464i 0.992127 + 0.125232i $$0.0399676\pi$$
−0.992127 + 0.125232i $$0.960032\pi$$
$$548$$ − 53.5980i − 2.28959i
$$549$$ 5.65685 0.241429
$$550$$ 0 0
$$551$$ 5.07107 0.216035
$$552$$ − 16.1421i − 0.687055i
$$553$$ 38.6274i 1.64260i
$$554$$ 55.4558 2.35609
$$555$$ 0 0
$$556$$ 26.1421 1.10867
$$557$$ − 10.0000i − 0.423714i −0.977301 0.211857i $$-0.932049\pi$$
0.977301 0.211857i $$-0.0679510\pi$$
$$558$$ − 25.3137i − 1.07161i
$$559$$ −8.82843 −0.373403
$$560$$ 0 0
$$561$$ −9.65685 −0.407713
$$562$$ − 25.8995i − 1.09250i
$$563$$ − 14.2843i − 0.602010i −0.953623 0.301005i $$-0.902678\pi$$
0.953623 0.301005i $$-0.0973221\pi$$
$$564$$ 44.6274 1.87915
$$565$$ 0 0
$$566$$ −5.41421 −0.227576
$$567$$ − 3.41421i − 0.143383i
$$568$$ 55.1127i 2.31248i
$$569$$ −18.7279 −0.785115 −0.392558 0.919727i $$-0.628410\pi$$
−0.392558 + 0.919727i $$0.628410\pi$$
$$570$$ 0 0
$$571$$ −19.7990 −0.828562 −0.414281 0.910149i $$-0.635967\pi$$
−0.414281 + 0.910149i $$0.635967\pi$$
$$572$$ 14.0000i 0.585369i
$$573$$ 17.8995i 0.747762i
$$574$$ 37.7990 1.57770
$$575$$ 0 0
$$576$$ −9.82843 −0.409518
$$577$$ − 1.79899i − 0.0748929i −0.999299 0.0374465i $$-0.988078\pi$$
0.999299 0.0374465i $$-0.0119224\pi$$
$$578$$ 71.5269i 2.97513i
$$579$$ −0.928932 −0.0386051
$$580$$ 0 0
$$581$$ 22.1421 0.918611
$$582$$ 10.2426i 0.424571i
$$583$$ − 5.65685i − 0.234283i
$$584$$ 8.82843 0.365323
$$585$$ 0 0
$$586$$ 18.8284 0.777795
$$587$$ − 0.343146i − 0.0141631i −0.999975 0.00708157i $$-0.997746\pi$$
0.999975 0.00708157i $$-0.00225415\pi$$
$$588$$ − 17.8284i − 0.735232i
$$589$$ 10.4853 0.432038
$$590$$ 0 0
$$591$$ 9.17157 0.377268
$$592$$ 9.21320i 0.378660i
$$593$$ − 6.68629i − 0.274573i −0.990531 0.137287i $$-0.956162\pi$$
0.990531 0.137287i $$-0.0438381\pi$$
$$594$$ −3.41421 −0.140087
$$595$$ 0 0
$$596$$ 29.3137 1.20074
$$597$$ 0.485281i 0.0198612i
$$598$$ − 22.8284i − 0.933524i
$$599$$ −45.9411 −1.87710 −0.938552 0.345139i $$-0.887832\pi$$
−0.938552 + 0.345139i $$0.887832\pi$$
$$600$$ 0 0
$$601$$ −23.1716 −0.945188 −0.472594 0.881280i $$-0.656682\pi$$
−0.472594 + 0.881280i $$0.656682\pi$$
$$602$$ − 28.1421i − 1.14699i
$$603$$ 12.0000i 0.488678i
$$604$$ 83.4558 3.39577
$$605$$ 0 0
$$606$$ −2.00000 −0.0812444
$$607$$ − 26.1421i − 1.06108i −0.847661 0.530538i $$-0.821990\pi$$
0.847661 0.530538i $$-0.178010\pi$$
$$608$$ − 1.58579i − 0.0643121i
$$609$$ 17.3137 0.701587
$$610$$ 0 0
$$611$$ 30.1421 1.21942
$$612$$ 26.1421i 1.05673i
$$613$$ − 37.1127i − 1.49897i −0.662023 0.749484i $$-0.730302\pi$$
0.662023 0.749484i $$-0.269698\pi$$
$$614$$ 76.7696 3.09817
$$615$$ 0 0
$$616$$ −21.3137 −0.858754
$$617$$ 18.1421i 0.730375i 0.930934 + 0.365187i $$0.118995\pi$$
−0.930934 + 0.365187i $$0.881005\pi$$
$$618$$ − 23.3137i − 0.937815i
$$619$$ 4.20101 0.168853 0.0844264 0.996430i $$-0.473094\pi$$
0.0844264 + 0.996430i $$0.473094\pi$$
$$620$$ 0 0
$$621$$ 3.65685 0.146745
$$622$$ 57.3553i 2.29974i
$$623$$ − 50.2843i − 2.01460i
$$624$$ 7.75736 0.310543
$$625$$ 0 0
$$626$$ −63.9411 −2.55560
$$627$$ − 1.41421i − 0.0564782i
$$628$$ 68.1421i 2.71917i
$$629$$ −20.9706 −0.836151
$$630$$ 0 0
$$631$$ −22.6274 −0.900783 −0.450392 0.892831i $$-0.648716\pi$$
−0.450392 + 0.892831i $$0.648716\pi$$
$$632$$ − 49.9411i − 1.98655i
$$633$$ − 7.31371i − 0.290694i
$$634$$ 27.3137 1.08477
$$635$$ 0 0
$$636$$ −15.3137 −0.607228
$$637$$ − 12.0416i − 0.477107i
$$638$$ − 17.3137i − 0.685456i
$$639$$ −12.4853 −0.493910
$$640$$ 0 0
$$641$$ −11.4142 −0.450834 −0.225417 0.974262i $$-0.572375\pi$$
−0.225417 + 0.974262i $$0.572375\pi$$
$$642$$ 19.3137i 0.762251i
$$643$$ − 42.0416i − 1.65796i −0.559278 0.828980i $$-0.688922\pi$$
0.559278 0.828980i $$-0.311078\pi$$
$$644$$ 47.7990 1.88354
$$645$$ 0 0
$$646$$ −16.4853 −0.648605
$$647$$ 17.1127i 0.672770i 0.941725 + 0.336385i $$0.109204\pi$$
−0.941725 + 0.336385i $$0.890796\pi$$
$$648$$ 4.41421i 0.173407i
$$649$$ −12.0000 −0.471041
$$650$$ 0 0
$$651$$ 35.7990 1.40307
$$652$$ 45.5563i 1.78412i
$$653$$ − 42.4264i − 1.66027i −0.557560 0.830137i $$-0.688262\pi$$
0.557560 0.830137i $$-0.311738\pi$$
$$654$$ 21.3137 0.833432
$$655$$ 0 0
$$656$$ −13.7574 −0.537135
$$657$$ 2.00000i 0.0780274i
$$658$$ 96.0833i 3.74572i
$$659$$ 20.0000 0.779089 0.389545 0.921008i $$-0.372632\pi$$
0.389545 + 0.921008i $$0.372632\pi$$
$$660$$ 0 0
$$661$$ 1.51472 0.0589157 0.0294579 0.999566i $$-0.490622\pi$$
0.0294579 + 0.999566i $$0.490622\pi$$
$$662$$ − 30.9706i − 1.20371i
$$663$$ 17.6569i 0.685735i
$$664$$ −28.6274 −1.11096
$$665$$ 0 0
$$666$$ −7.41421 −0.287295
$$667$$ 18.5442i 0.718033i
$$668$$ 38.2843i 1.48126i
$$669$$ 17.6569 0.682653
$$670$$ 0 0
$$671$$ 8.00000 0.308837
$$672$$ − 5.41421i − 0.208858i
$$673$$ 2.10051i 0.0809685i 0.999180 + 0.0404843i $$0.0128901\pi$$
−0.999180 + 0.0404843i $$0.987110\pi$$
$$674$$ −47.6985 −1.83728
$$675$$ 0 0
$$676$$ −24.1716 −0.929676
$$677$$ 11.0294i 0.423896i 0.977281 + 0.211948i $$0.0679807\pi$$
−0.977281 + 0.211948i $$0.932019\pi$$
$$678$$ 10.8284i 0.415863i
$$679$$ −14.4853 −0.555894
$$680$$ 0 0
$$681$$ −14.9706 −0.573673
$$682$$ − 35.7990i − 1.37081i
$$683$$ − 5.65685i − 0.216454i −0.994126 0.108227i $$-0.965483\pi$$
0.994126 0.108227i $$-0.0345173\pi$$
$$684$$ −3.82843 −0.146384
$$685$$ 0 0
$$686$$ −19.3137 −0.737401
$$687$$ 9.65685i 0.368432i
$$688$$ 10.2426i 0.390497i
$$689$$ −10.3431 −0.394042
$$690$$ 0 0
$$691$$ 39.1127 1.48792 0.743959 0.668226i $$-0.232946\pi$$
0.743959 + 0.668226i $$0.232946\pi$$
$$692$$ − 84.7696i − 3.22245i
$$693$$ − 4.82843i − 0.183417i
$$694$$ 15.6569 0.594326
$$695$$ 0 0
$$696$$ −22.3848 −0.848493
$$697$$ − 31.3137i − 1.18609i
$$698$$ 16.1421i 0.610989i
$$699$$ −19.6569 −0.743491
$$700$$ 0 0
$$701$$ −11.6569 −0.440273 −0.220137 0.975469i $$-0.570650\pi$$
−0.220137 + 0.975469i $$0.570650\pi$$
$$702$$ 6.24264i 0.235613i
$$703$$ − 3.07107i − 0.115828i
$$704$$ −13.8995 −0.523857
$$705$$ 0 0
$$706$$ −18.4853 −0.695703
$$707$$ − 2.82843i − 0.106374i
$$708$$ 32.4853i 1.22087i
$$709$$ 12.6863 0.476444 0.238222 0.971211i $$-0.423435\pi$$
0.238222 + 0.971211i $$0.423435\pi$$
$$710$$ 0 0
$$711$$ 11.3137 0.424297
$$712$$ 65.0122i 2.43643i
$$713$$ 38.3431i 1.43596i
$$714$$ −56.2843 −2.10639
$$715$$ 0 0
$$716$$ 87.3970 3.26618
$$717$$ 5.41421i 0.202198i
$$718$$ − 23.8995i − 0.891921i
$$719$$ 47.5563 1.77355 0.886776 0.462199i $$-0.152939\pi$$
0.886776 + 0.462199i $$0.152939\pi$$
$$720$$ 0 0
$$721$$ 32.9706 1.22789
$$722$$ − 2.41421i − 0.0898477i
$$723$$ − 18.9706i − 0.705523i
$$724$$ 95.0538 3.53265
$$725$$ 0 0
$$726$$ 21.7279 0.806399
$$727$$ 7.41421i 0.274978i 0.990503 + 0.137489i $$0.0439032\pi$$
−0.990503 + 0.137489i $$0.956097\pi$$
$$728$$ 38.9706i 1.44435i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −23.3137 −0.862289
$$732$$ − 21.6569i − 0.800460i
$$733$$ 21.3137i 0.787240i 0.919273 + 0.393620i $$0.128777\pi$$
−0.919273 + 0.393620i $$0.871223\pi$$
$$734$$ 40.0416 1.47796
$$735$$ 0 0
$$736$$ 5.79899 0.213754
$$737$$ 16.9706i 0.625119i
$$738$$ − 11.0711i − 0.407532i
$$739$$ −1.65685 −0.0609484 −0.0304742 0.999536i $$-0.509702\pi$$
−0.0304742 + 0.999536i $$0.509702\pi$$
$$740$$ 0 0
$$741$$ −2.58579 −0.0949912
$$742$$ − 32.9706i − 1.21039i
$$743$$ 7.31371i 0.268314i 0.990960 + 0.134157i $$0.0428326\pi$$
−0.990960 + 0.134157i $$0.957167\pi$$
$$744$$ −46.2843 −1.69686
$$745$$ 0 0
$$746$$ 23.8995 0.875023
$$747$$ − 6.48528i − 0.237284i
$$748$$ 36.9706i 1.35178i
$$749$$ −27.3137 −0.998021
$$750$$ 0 0
$$751$$ 25.1127 0.916375 0.458188 0.888855i $$-0.348499\pi$$
0.458188 + 0.888855i $$0.348499\pi$$
$$752$$ − 34.9706i − 1.27525i
$$753$$ − 27.0711i − 0.986525i
$$754$$ −31.6569 −1.15287
$$755$$ 0 0
$$756$$ −13.0711 −0.475390
$$757$$ − 11.1716i − 0.406038i −0.979175 0.203019i $$-0.934925\pi$$
0.979175 0.203019i $$-0.0650753\pi$$
$$758$$ − 10.0000i − 0.363216i
$$759$$ 5.17157 0.187716
$$760$$ 0 0
$$761$$ 46.2843 1.67780 0.838902 0.544283i $$-0.183198\pi$$
0.838902 + 0.544283i $$0.183198\pi$$
$$762$$ 19.3137i 0.699662i
$$763$$ 30.1421i 1.09122i
$$764$$ 68.5269 2.47922
$$765$$ 0 0
$$766$$ 28.9706 1.04675
$$767$$ 21.9411i 0.792248i
$$768$$ 29.9706i 1.08147i
$$769$$ 24.3431 0.877836 0.438918 0.898527i $$-0.355362\pi$$
0.438918 + 0.898527i $$0.355362\pi$$
$$770$$ 0 0
$$771$$ −6.82843 −0.245920
$$772$$ 3.55635i 0.127996i
$$773$$ 0.970563i 0.0349087i 0.999848 + 0.0174544i $$0.00555618\pi$$
−0.999848 + 0.0174544i $$0.994444\pi$$
$$774$$ −8.24264 −0.296276
$$775$$ 0 0
$$776$$ 18.7279 0.672293
$$777$$ − 10.4853i − 0.376157i
$$778$$ − 45.7990i − 1.64197i
$$779$$ 4.58579 0.164303
$$780$$ 0 0
$$781$$ −17.6569 −0.631812
$$782$$ − 60.2843i − 2.15576i
$$783$$ − 5.07107i − 0.181225i
$$784$$ −13.9706 −0.498949
$$785$$ 0 0
$$786$$ 21.0711 0.751580
$$787$$ 37.4558i 1.33516i 0.744540 + 0.667578i $$0.232669\pi$$
−0.744540 + 0.667578i $$0.767331\pi$$
$$788$$ − 35.1127i − 1.25084i
$$789$$ −3.85786 −0.137344
$$790$$ 0 0
$$791$$ −15.3137 −0.544493
$$792$$ 6.24264i 0.221823i
$$793$$ − 14.6274i − 0.519435i
$$794$$ −58.7696 −2.08565
$$795$$ 0 0
$$796$$ 1.85786 0.0658503
$$797$$ − 5.17157i − 0.183187i −0.995797 0.0915933i $$-0.970804\pi$$
0.995797 0.0915933i $$-0.0291960\pi$$
$$798$$ − 8.24264i − 0.291786i
$$799$$ 79.5980 2.81597
$$800$$ 0 0
$$801$$ −14.7279 −0.520386
$$802$$ 43.5563i 1.53803i
$$803$$ 2.82843i 0.0998130i
$$804$$ 45.9411 1.62022
$$805$$ 0 0
$$806$$ −65.4558 −2.30558
$$807$$ − 28.3848i − 0.999191i
$$808$$ 3.65685i 0.128648i
$$809$$ −26.6863 −0.938240 −0.469120 0.883134i $$-0.655429\pi$$
−0.469120 + 0.883134i $$0.655429\pi$$
$$810$$ 0 0
$$811$$ 7.31371 0.256819 0.128410 0.991721i $$-0.459013\pi$$
0.128410 + 0.991721i $$0.459013\pi$$
$$812$$ − 66.2843i − 2.32612i
$$813$$ 25.1716i 0.882806i
$$814$$ −10.4853 −0.367509
$$815$$ 0 0
$$816$$ 20.4853 0.717128
$$817$$ − 3.41421i − 0.119448i
$$818$$ 63.9411i 2.23565i
$$819$$ −8.82843 −0.308490
$$820$$ 0 0
$$821$$ 0.544156 0.0189912 0.00949559 0.999955i $$-0.496977\pi$$
0.00949559 + 0.999955i $$0.496977\pi$$
$$822$$ − 33.7990i − 1.17888i
$$823$$ 22.7279i 0.792246i 0.918198 + 0.396123i $$0.129645\pi$$
−0.918198 + 0.396123i $$0.870355\pi$$
$$824$$ −42.6274 −1.48500
$$825$$ 0 0
$$826$$ −69.9411 −2.43356
$$827$$ − 3.37258i − 0.117276i −0.998279 0.0586381i $$-0.981324\pi$$
0.998279 0.0586381i $$-0.0186758\pi$$
$$828$$ − 14.0000i − 0.486534i
$$829$$ 21.5147 0.747237 0.373619 0.927582i $$-0.378117\pi$$
0.373619 + 0.927582i $$0.378117\pi$$
$$830$$ 0 0
$$831$$ 22.9706 0.796840
$$832$$ 25.4142i 0.881079i
$$833$$ − 31.7990i − 1.10177i
$$834$$ 16.4853 0.570839
$$835$$ 0 0
$$836$$ −5.41421 −0.187254
$$837$$ − 10.4853i − 0.362424i
$$838$$ 45.5563i 1.57372i
$$839$$ 35.1127 1.21222 0.606112 0.795379i $$-0.292728\pi$$
0.606112 + 0.795379i $$0.292728\pi$$
$$840$$ 0 0
$$841$$ −3.28427 −0.113251
$$842$$ 90.0833i 3.10447i
$$843$$ − 10.7279i − 0.369489i
$$844$$ −28.0000 −0.963800
$$845$$ 0 0
$$846$$ 28.1421 0.967547
$$847$$ 30.7279i 1.05582i
$$848$$ 12.0000i 0.412082i
$$849$$ −2.24264 −0.0769672
$$850$$ 0 0
$$851$$ 11.2304 0.384975
$$852$$ 47.7990i 1.63757i
$$853$$ − 26.4853i − 0.906839i −0.891297 0.453419i $$-0.850204\pi$$
0.891297 0.453419i $$-0.149796\pi$$
$$854$$ 46.6274 1.59556
$$855$$ 0 0
$$856$$ 35.3137 1.20700
$$857$$ 29.9411i 1.02277i 0.859352 + 0.511385i $$0.170867\pi$$
−0.859352 + 0.511385i $$0.829133\pi$$
$$858$$ 8.82843i 0.301398i
$$859$$ 41.9411 1.43101 0.715506 0.698606i $$-0.246196\pi$$
0.715506 + 0.698606i $$0.246196\pi$$
$$860$$ 0 0
$$861$$ 15.6569 0.533584
$$862$$ − 49.4558i − 1.68447i
$$863$$ − 8.68629i − 0.295685i −0.989011 0.147842i $$-0.952767\pi$$
0.989011 0.147842i $$-0.0472328\pi$$
$$864$$ −1.58579 −0.0539496
$$865$$ 0 0
$$866$$ 36.3848 1.23641
$$867$$ 29.6274i 1.00620i
$$868$$ − 137.054i − 4.65191i
$$869$$ 16.0000 0.542763
$$870$$ 0 0
$$871$$ 31.0294 1.05139
$$872$$ − 38.9706i − 1.31971i
$$873$$ 4.24264i 0.143592i
$$874$$ 8.82843 0.298626
$$875$$ 0 0
$$876$$ 7.65685 0.258701
$$877$$ 1.89949i 0.0641414i 0.999486 + 0.0320707i $$0.0102102\pi$$
−0.999486 + 0.0320707i $$0.989790\pi$$
$$878$$ − 79.5980i − 2.68630i
$$879$$ 7.79899 0.263053
$$880$$ 0 0
$$881$$ 3.45584 0.116430 0.0582152 0.998304i $$-0.481459\pi$$
0.0582152 + 0.998304i $$0.481459\pi$$
$$882$$ − 11.2426i − 0.378559i
$$883$$ − 18.5269i − 0.623480i −0.950167 0.311740i $$-0.899088\pi$$
0.950167 0.311740i $$-0.100912\pi$$
$$884$$ 67.5980 2.27357
$$885$$ 0 0
$$886$$ 51.4558 1.72869
$$887$$ 8.00000i 0.268614i 0.990940 + 0.134307i $$0.0428808\pi$$
−0.990940 + 0.134307i $$0.957119\pi$$
$$888$$ 13.5563i 0.454921i
$$889$$ −27.3137 −0.916072
$$890$$ 0 0
$$891$$ −1.41421 −0.0473779
$$892$$ − 67.5980i − 2.26335i
$$893$$ 11.6569i 0.390082i
$$894$$ 18.4853 0.618240
$$895$$ 0 0
$$896$$ −70.1838 −2.34468
$$897$$ − 9.45584i − 0.315721i
$$898$$ − 28.7279i − 0.958663i
$$899$$ 53.1716 1.77337
$$900$$ 0 0
$$901$$ −27.3137 −0.909952
$$902$$ − 15.6569i − 0.521316i
$$903$$ − 11.6569i − 0.387916i
$$904$$ 19.7990 0.658505
$$905$$ 0 0
$$906$$ 52.6274 1.74843
$$907$$ − 9.85786i − 0.327325i −0.986516 0.163663i $$-0.947669\pi$$
0.986516 0.163663i $$-0.0523308\pi$$
$$908$$ 57.3137i 1.90202i
$$909$$ −0.828427 −0.0274772
$$910$$ 0 0
$$911$$ −0.686292 −0.0227379 −0.0113689 0.999935i $$-0.503619\pi$$
−0.0113689 + 0.999935i $$0.503619\pi$$
$$912$$ 3.00000i 0.0993399i
$$913$$ − 9.17157i − 0.303535i
$$914$$ −55.9411 −1.85037
$$915$$ 0 0
$$916$$ 36.9706 1.22154
$$917$$ 29.7990i 0.984049i
$$918$$ 16.4853i 0.544095i
$$919$$ 12.0000 0.395843 0.197922 0.980218i $$-0.436581\pi$$
0.197922 + 0.980218i $$0.436581\pi$$
$$920$$ 0 0
$$921$$ 31.7990 1.04781
$$922$$ − 80.4264i − 2.64870i
$$923$$ 32.2843i 1.06265i
$$924$$ −18.4853 −0.608121
$$925$$ 0 0
$$926$$ −67.3553 −2.21343
$$927$$ − 9.65685i − 0.317173i
$$928$$ − 8.04163i − 0.263979i
$$929$$ 0.544156 0.0178532 0.00892659 0.999960i $$-0.497159\pi$$
0.00892659 + 0.999960i $$0.497159\pi$$
$$930$$ 0 0
$$931$$ 4.65685 0.152622
$$932$$ 75.2548i 2.46505i
$$933$$ 23.7574i 0.777781i
$$934$$ −66.7696 −2.18477
$$935$$ 0 0
$$936$$ 11.4142 0.373085
$$937$$ − 54.7696i − 1.78924i −0.446824 0.894622i $$-0.647445\pi$$
0.446824 0.894622i $$-0.352555\pi$$
$$938$$ 98.9117i 3.22958i
$$939$$ −26.4853 −0.864314
$$940$$ 0 0
$$941$$ 13.5563 0.441924 0.220962 0.975282i $$-0.429080\pi$$
0.220962 + 0.975282i $$0.429080\pi$$
$$942$$ 42.9706i 1.40006i
$$943$$ 16.7696i 0.546092i
$$944$$ 25.4558 0.828517
$$945$$ 0 0
$$946$$ −11.6569 −0.378997
$$947$$ − 7.17157i − 0.233045i −0.993188 0.116522i $$-0.962825\pi$$
0.993188 0.116522i $$-0.0371747\pi$$
$$948$$ − 43.3137i − 1.40676i
$$949$$ 5.17157 0.167876
$$950$$ 0 0
$$951$$ 11.3137 0.366872
$$952$$ 102.912i 3.33539i
$$953$$ − 34.1421i − 1.10597i −0.833190 0.552986i $$-0.813488\pi$$
0.833190 0.552986i $$-0.186512\pi$$
$$954$$ −9.65685 −0.312652
$$955$$ 0 0
$$956$$ 20.7279 0.670389
$$957$$ − 7.17157i − 0.231824i
$$958$$ 92.1838i 2.97832i
$$959$$ 47.7990 1.54351
$$960$$ 0 0
$$961$$ 78.9411 2.54649
$$962$$ 19.1716i 0.618116i
$$963$$ 8.00000i 0.257796i
$$964$$ −72.6274 −2.33917
$$965$$ 0 0
$$966$$ 30.1421 0.969807
$$967$$ 8.10051i 0.260495i 0.991482 + 0.130247i $$0.0415771\pi$$
−0.991482 + 0.130247i $$0.958423\pi$$
$$968$$ − 39.7279i − 1.27690i
$$969$$ −6.82843 −0.219361
$$970$$ 0 0
$$971$$ 6.34315 0.203561 0.101781 0.994807i $$-0.467546\pi$$
0.101781 + 0.994807i $$0.467546\pi$$
$$972$$ 3.82843i 0.122797i
$$973$$ 23.3137i 0.747403i
$$974$$ −6.82843 −0.218797
$$975$$ 0 0
$$976$$ −16.9706 −0.543214
$$977$$ 39.5980i 1.26685i 0.773803 + 0.633426i $$0.218352\pi$$
−0.773803 + 0.633426i $$0.781648\pi$$
$$978$$ 28.7279i 0.918618i
$$979$$ −20.8284 −0.665679
$$980$$ 0 0
$$981$$ 8.82843 0.281870
$$982$$ − 5.07107i − 0.161824i
$$983$$ − 35.9411i − 1.14634i −0.819435 0.573172i $$-0.805713\pi$$
0.819435 0.573172i $$-0.194287\pi$$
$$984$$ −20.2426 −0.645312
$$985$$ 0 0
$$986$$ −83.5980 −2.66230
$$987$$ 39.7990i 1.26682i
$$988$$ 9.89949i 0.314945i
$$989$$ 12.4853 0.397009
$$990$$ 0 0
$$991$$ −50.3431 −1.59920 −0.799601 0.600531i $$-0.794956\pi$$
−0.799601 + 0.600531i $$0.794956\pi$$
$$992$$ − 16.6274i − 0.527921i
$$993$$ − 12.8284i − 0.407098i
$$994$$ −102.912 −3.26416
$$995$$ 0 0
$$996$$ −24.8284 −0.786719
$$997$$ 33.5147i 1.06142i 0.847553 + 0.530711i $$0.178075\pi$$
−0.847553 + 0.530711i $$0.821925\pi$$
$$998$$ 38.1421i 1.20737i
$$999$$ −3.07107 −0.0971643
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 1425.2.c.j.799.1 4
5.2 odd 4 285.2.a.f.1.2 2
5.3 odd 4 1425.2.a.l.1.1 2
5.4 even 2 inner 1425.2.c.j.799.4 4
15.2 even 4 855.2.a.e.1.1 2
15.8 even 4 4275.2.a.x.1.2 2
20.7 even 4 4560.2.a.bj.1.1 2
95.37 even 4 5415.2.a.p.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.f.1.2 2 5.2 odd 4
855.2.a.e.1.1 2 15.2 even 4
1425.2.a.l.1.1 2 5.3 odd 4
1425.2.c.j.799.1 4 1.1 even 1 trivial
1425.2.c.j.799.4 4 5.4 even 2 inner
4275.2.a.x.1.2 2 15.8 even 4
4560.2.a.bj.1.1 2 20.7 even 4
5415.2.a.p.1.1 2 95.37 even 4