# Properties

 Label 1425.2.c.j.799.3 Level $1425$ Weight $2$ Character 1425.799 Analytic conductor $11.379$ Analytic rank $0$ Dimension $4$ CM no 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 799.3 Root $$-0.707107 + 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 1425.799 Dual form 1425.2.c.j.799.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.414214i q^{2} -1.00000i q^{3} +1.82843 q^{4} +0.414214 q^{6} -0.585786i q^{7} +1.58579i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+0.414214i q^{2} -1.00000i q^{3} +1.82843 q^{4} +0.414214 q^{6} -0.585786i q^{7} +1.58579i q^{8} -1.00000 q^{9} +1.41421 q^{11} -1.82843i q^{12} +5.41421i q^{13} +0.242641 q^{14} +3.00000 q^{16} +1.17157i q^{17} -0.414214i q^{18} -1.00000 q^{19} -0.585786 q^{21} +0.585786i q^{22} +7.65685i q^{23} +1.58579 q^{24} -2.24264 q^{26} +1.00000i q^{27} -1.07107i q^{28} +9.07107 q^{29} +6.48528 q^{31} +4.41421i q^{32} -1.41421i q^{33} -0.485281 q^{34} -1.82843 q^{36} -11.0711i q^{37} -0.414214i q^{38} +5.41421 q^{39} -7.41421 q^{41} -0.242641i q^{42} +0.585786i q^{43} +2.58579 q^{44} -3.17157 q^{46} -0.343146i q^{47} -3.00000i q^{48} +6.65685 q^{49} +1.17157 q^{51} +9.89949i q^{52} +4.00000i q^{53} -0.414214 q^{54} +0.928932 q^{56} +1.00000i q^{57} +3.75736i q^{58} -8.48528 q^{59} +5.65685 q^{61} +2.68629i q^{62} +0.585786i q^{63} +4.17157 q^{64} +0.585786 q^{66} -12.0000i q^{67} +2.14214i q^{68} +7.65685 q^{69} -4.48528 q^{71} -1.58579i q^{72} -2.00000i q^{73} +4.58579 q^{74} -1.82843 q^{76} -0.828427i q^{77} +2.24264i q^{78} +11.3137 q^{79} +1.00000 q^{81} -3.07107i q^{82} -10.4853i q^{83} -1.07107 q^{84} -0.242641 q^{86} -9.07107i q^{87} +2.24264i q^{88} -10.7279 q^{89} +3.17157 q^{91} +14.0000i q^{92} -6.48528i q^{93} +0.142136 q^{94} +4.41421 q^{96} +4.24264i q^{97} +2.75736i q^{98} -1.41421 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} - 4 q^{6} - 4 q^{9}+O(q^{10})$$ 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.414214i 0.292893i 0.989219 + 0.146447i $$0.0467837\pi$$
−0.989219 + 0.146447i $$0.953216\pi$$
$$3$$ − 1.00000i − 0.577350i
$$4$$ 1.82843 0.914214
$$5$$ 0 0
$$6$$ 0.414214 0.169102
$$7$$ − 0.585786i − 0.221406i −0.993854 0.110703i $$-0.964690\pi$$
0.993854 0.110703i $$-0.0353103\pi$$
$$8$$ 1.58579i 0.560660i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 1.41421 0.426401 0.213201 0.977008i $$-0.431611\pi$$
0.213201 + 0.977008i $$0.431611\pi$$
$$12$$ − 1.82843i − 0.527821i
$$13$$ 5.41421i 1.50163i 0.660511 + 0.750816i $$0.270340\pi$$
−0.660511 + 0.750816i $$0.729660\pi$$
$$14$$ 0.242641 0.0648485
$$15$$ 0 0
$$16$$ 3.00000 0.750000
$$17$$ 1.17157i 0.284148i 0.989856 + 0.142074i $$0.0453771\pi$$
−0.989856 + 0.142074i $$0.954623\pi$$
$$18$$ − 0.414214i − 0.0976311i
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −0.585786 −0.127829
$$22$$ 0.585786i 0.124890i
$$23$$ 7.65685i 1.59656i 0.602284 + 0.798282i $$0.294258\pi$$
−0.602284 + 0.798282i $$0.705742\pi$$
$$24$$ 1.58579 0.323697
$$25$$ 0 0
$$26$$ −2.24264 −0.439818
$$27$$ 1.00000i 0.192450i
$$28$$ − 1.07107i − 0.202413i
$$29$$ 9.07107 1.68446 0.842228 0.539122i $$-0.181244\pi$$
0.842228 + 0.539122i $$0.181244\pi$$
$$30$$ 0 0
$$31$$ 6.48528 1.16479 0.582395 0.812906i $$-0.302116\pi$$
0.582395 + 0.812906i $$0.302116\pi$$
$$32$$ 4.41421i 0.780330i
$$33$$ − 1.41421i − 0.246183i
$$34$$ −0.485281 −0.0832251
$$35$$ 0 0
$$36$$ −1.82843 −0.304738
$$37$$ − 11.0711i − 1.82007i −0.414529 0.910036i $$-0.636054\pi$$
0.414529 0.910036i $$-0.363946\pi$$
$$38$$ − 0.414214i − 0.0671943i
$$39$$ 5.41421 0.866968
$$40$$ 0 0
$$41$$ −7.41421 −1.15791 −0.578953 0.815361i $$-0.696538\pi$$
−0.578953 + 0.815361i $$0.696538\pi$$
$$42$$ − 0.242641i − 0.0374403i
$$43$$ 0.585786i 0.0893316i 0.999002 + 0.0446658i $$0.0142223\pi$$
−0.999002 + 0.0446658i $$0.985778\pi$$
$$44$$ 2.58579 0.389822
$$45$$ 0 0
$$46$$ −3.17157 −0.467623
$$47$$ − 0.343146i − 0.0500530i −0.999687 0.0250265i $$-0.992033\pi$$
0.999687 0.0250265i $$-0.00796701\pi$$
$$48$$ − 3.00000i − 0.433013i
$$49$$ 6.65685 0.950979
$$50$$ 0 0
$$51$$ 1.17157 0.164053
$$52$$ 9.89949i 1.37281i
$$53$$ 4.00000i 0.549442i 0.961524 + 0.274721i $$0.0885855\pi$$
−0.961524 + 0.274721i $$0.911414\pi$$
$$54$$ −0.414214 −0.0563673
$$55$$ 0 0
$$56$$ 0.928932 0.124134
$$57$$ 1.00000i 0.132453i
$$58$$ 3.75736i 0.493365i
$$59$$ −8.48528 −1.10469 −0.552345 0.833616i $$-0.686267\pi$$
−0.552345 + 0.833616i $$0.686267\pi$$
$$60$$ 0 0
$$61$$ 5.65685 0.724286 0.362143 0.932123i $$-0.382045\pi$$
0.362143 + 0.932123i $$0.382045\pi$$
$$62$$ 2.68629i 0.341159i
$$63$$ 0.585786i 0.0738022i
$$64$$ 4.17157 0.521447
$$65$$ 0 0
$$66$$ 0.585786 0.0721053
$$67$$ − 12.0000i − 1.46603i −0.680211 0.733017i $$-0.738112\pi$$
0.680211 0.733017i $$-0.261888\pi$$
$$68$$ 2.14214i 0.259772i
$$69$$ 7.65685 0.921777
$$70$$ 0 0
$$71$$ −4.48528 −0.532305 −0.266152 0.963931i $$-0.585752\pi$$
−0.266152 + 0.963931i $$0.585752\pi$$
$$72$$ − 1.58579i − 0.186887i
$$73$$ − 2.00000i − 0.234082i −0.993127 0.117041i $$-0.962659\pi$$
0.993127 0.117041i $$-0.0373409\pi$$
$$74$$ 4.58579 0.533087
$$75$$ 0 0
$$76$$ −1.82843 −0.209735
$$77$$ − 0.828427i − 0.0944080i
$$78$$ 2.24264i 0.253929i
$$79$$ 11.3137 1.27289 0.636446 0.771321i $$-0.280404\pi$$
0.636446 + 0.771321i $$0.280404\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 3.07107i − 0.339143i
$$83$$ − 10.4853i − 1.15091i −0.817834 0.575455i $$-0.804825\pi$$
0.817834 0.575455i $$-0.195175\pi$$
$$84$$ −1.07107 −0.116863
$$85$$ 0 0
$$86$$ −0.242641 −0.0261646
$$87$$ − 9.07107i − 0.972521i
$$88$$ 2.24264i 0.239066i
$$89$$ −10.7279 −1.13716 −0.568579 0.822629i $$-0.692507\pi$$
−0.568579 + 0.822629i $$0.692507\pi$$
$$90$$ 0 0
$$91$$ 3.17157 0.332471
$$92$$ 14.0000i 1.45960i
$$93$$ − 6.48528i − 0.672492i
$$94$$ 0.142136 0.0146602
$$95$$ 0 0
$$96$$ 4.41421 0.450524
$$97$$ 4.24264i 0.430775i 0.976529 + 0.215387i $$0.0691014\pi$$
−0.976529 + 0.215387i $$0.930899\pi$$
$$98$$ 2.75736i 0.278535i
$$99$$ −1.41421 −0.142134
$$100$$ 0 0
$$101$$ −4.82843 −0.480446 −0.240223 0.970718i $$-0.577221\pi$$
−0.240223 + 0.970718i $$0.577221\pi$$
$$102$$ 0.485281i 0.0480500i
$$103$$ − 1.65685i − 0.163255i −0.996663 0.0816274i $$-0.973988\pi$$
0.996663 0.0816274i $$-0.0260117\pi$$
$$104$$ −8.58579 −0.841906
$$105$$ 0 0
$$106$$ −1.65685 −0.160928
$$107$$ − 8.00000i − 0.773389i −0.922208 0.386695i $$-0.873617\pi$$
0.922208 0.386695i $$-0.126383\pi$$
$$108$$ 1.82843i 0.175940i
$$109$$ −3.17157 −0.303782 −0.151891 0.988397i $$-0.548536\pi$$
−0.151891 + 0.988397i $$0.548536\pi$$
$$110$$ 0 0
$$111$$ −11.0711 −1.05082
$$112$$ − 1.75736i − 0.166055i
$$113$$ 12.4853i 1.17452i 0.809400 + 0.587258i $$0.199793\pi$$
−0.809400 + 0.587258i $$0.800207\pi$$
$$114$$ −0.414214 −0.0387947
$$115$$ 0 0
$$116$$ 16.5858 1.53995
$$117$$ − 5.41421i − 0.500544i
$$118$$ − 3.51472i − 0.323556i
$$119$$ 0.686292 0.0629122
$$120$$ 0 0
$$121$$ −9.00000 −0.818182
$$122$$ 2.34315i 0.212138i
$$123$$ 7.41421i 0.668517i
$$124$$ 11.8579 1.06487
$$125$$ 0 0
$$126$$ −0.242641 −0.0216162
$$127$$ − 8.00000i − 0.709885i −0.934888 0.354943i $$-0.884500\pi$$
0.934888 0.354943i $$-0.115500\pi$$
$$128$$ 10.5563i 0.933058i
$$129$$ 0.585786 0.0515756
$$130$$ 0 0
$$131$$ 16.7279 1.46153 0.730763 0.682632i $$-0.239165\pi$$
0.730763 + 0.682632i $$0.239165\pi$$
$$132$$ − 2.58579i − 0.225064i
$$133$$ 0.585786i 0.0507941i
$$134$$ 4.97056 0.429391
$$135$$ 0 0
$$136$$ −1.85786 −0.159311
$$137$$ 14.0000i 1.19610i 0.801459 + 0.598050i $$0.204058\pi$$
−0.801459 + 0.598050i $$0.795942\pi$$
$$138$$ 3.17157i 0.269982i
$$139$$ −1.17157 −0.0993715 −0.0496858 0.998765i $$-0.515822\pi$$
−0.0496858 + 0.998765i $$0.515822\pi$$
$$140$$ 0 0
$$141$$ −0.343146 −0.0288981
$$142$$ − 1.85786i − 0.155909i
$$143$$ 7.65685i 0.640298i
$$144$$ −3.00000 −0.250000
$$145$$ 0 0
$$146$$ 0.828427 0.0685611
$$147$$ − 6.65685i − 0.549048i
$$148$$ − 20.2426i − 1.66393i
$$149$$ 3.65685 0.299581 0.149791 0.988718i $$-0.452140\pi$$
0.149791 + 0.988718i $$0.452140\pi$$
$$150$$ 0 0
$$151$$ 17.7990 1.44846 0.724231 0.689558i $$-0.242195\pi$$
0.724231 + 0.689558i $$0.242195\pi$$
$$152$$ − 1.58579i − 0.128624i
$$153$$ − 1.17157i − 0.0947161i
$$154$$ 0.343146 0.0276515
$$155$$ 0 0
$$156$$ 9.89949 0.792594
$$157$$ 21.7990i 1.73975i 0.493273 + 0.869874i $$0.335800\pi$$
−0.493273 + 0.869874i $$0.664200\pi$$
$$158$$ 4.68629i 0.372821i
$$159$$ 4.00000 0.317221
$$160$$ 0 0
$$161$$ 4.48528 0.353490
$$162$$ 0.414214i 0.0325437i
$$163$$ 7.89949i 0.618736i 0.950942 + 0.309368i $$0.100118\pi$$
−0.950942 + 0.309368i $$0.899882\pi$$
$$164$$ −13.5563 −1.05857
$$165$$ 0 0
$$166$$ 4.34315 0.337093
$$167$$ − 10.0000i − 0.773823i −0.922117 0.386912i $$-0.873542\pi$$
0.922117 0.386912i $$-0.126458\pi$$
$$168$$ − 0.928932i − 0.0716687i
$$169$$ −16.3137 −1.25490
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ 1.07107i 0.0816682i
$$173$$ − 6.14214i − 0.466978i −0.972359 0.233489i $$-0.924986\pi$$
0.972359 0.233489i $$-0.0750143\pi$$
$$174$$ 3.75736 0.284845
$$175$$ 0 0
$$176$$ 4.24264 0.319801
$$177$$ 8.48528i 0.637793i
$$178$$ − 4.44365i − 0.333066i
$$179$$ −17.1716 −1.28346 −0.641732 0.766929i $$-0.721784\pi$$
−0.641732 + 0.766929i $$0.721784\pi$$
$$180$$ 0 0
$$181$$ −19.1716 −1.42501 −0.712506 0.701666i $$-0.752440\pi$$
−0.712506 + 0.701666i $$0.752440\pi$$
$$182$$ 1.31371i 0.0973786i
$$183$$ − 5.65685i − 0.418167i
$$184$$ −12.1421 −0.895130
$$185$$ 0 0
$$186$$ 2.68629 0.196968
$$187$$ 1.65685i 0.121161i
$$188$$ − 0.627417i − 0.0457591i
$$189$$ 0.585786 0.0426097
$$190$$ 0 0
$$191$$ 1.89949 0.137443 0.0687213 0.997636i $$-0.478108\pi$$
0.0687213 + 0.997636i $$0.478108\pi$$
$$192$$ − 4.17157i − 0.301057i
$$193$$ − 15.0711i − 1.08484i −0.840108 0.542420i $$-0.817508\pi$$
0.840108 0.542420i $$-0.182492\pi$$
$$194$$ −1.75736 −0.126171
$$195$$ 0 0
$$196$$ 12.1716 0.869398
$$197$$ 14.8284i 1.05648i 0.849095 + 0.528241i $$0.177148\pi$$
−0.849095 + 0.528241i $$0.822852\pi$$
$$198$$ − 0.585786i − 0.0416300i
$$199$$ 16.4853 1.16861 0.584305 0.811534i $$-0.301367\pi$$
0.584305 + 0.811534i $$0.301367\pi$$
$$200$$ 0 0
$$201$$ −12.0000 −0.846415
$$202$$ − 2.00000i − 0.140720i
$$203$$ − 5.31371i − 0.372949i
$$204$$ 2.14214 0.149979
$$205$$ 0 0
$$206$$ 0.686292 0.0478162
$$207$$ − 7.65685i − 0.532188i
$$208$$ 16.2426i 1.12622i
$$209$$ −1.41421 −0.0978232
$$210$$ 0 0
$$211$$ −15.3137 −1.05424 −0.527120 0.849791i $$-0.676728\pi$$
−0.527120 + 0.849791i $$0.676728\pi$$
$$212$$ 7.31371i 0.502308i
$$213$$ 4.48528i 0.307326i
$$214$$ 3.31371 0.226520
$$215$$ 0 0
$$216$$ −1.58579 −0.107899
$$217$$ − 3.79899i − 0.257892i
$$218$$ − 1.31371i − 0.0889756i
$$219$$ −2.00000 −0.135147
$$220$$ 0 0
$$221$$ −6.34315 −0.426686
$$222$$ − 4.58579i − 0.307778i
$$223$$ 6.34315i 0.424768i 0.977186 + 0.212384i $$0.0681228\pi$$
−0.977186 + 0.212384i $$0.931877\pi$$
$$224$$ 2.58579 0.172770
$$225$$ 0 0
$$226$$ −5.17157 −0.344008
$$227$$ 18.9706i 1.25912i 0.776952 + 0.629560i $$0.216765\pi$$
−0.776952 + 0.629560i $$0.783235\pi$$
$$228$$ 1.82843i 0.121091i
$$229$$ 1.65685 0.109488 0.0547440 0.998500i $$-0.482566\pi$$
0.0547440 + 0.998500i $$0.482566\pi$$
$$230$$ 0 0
$$231$$ −0.828427 −0.0545065
$$232$$ 14.3848i 0.944407i
$$233$$ − 8.34315i − 0.546578i −0.961932 0.273289i $$-0.911889\pi$$
0.961932 0.273289i $$-0.0881115\pi$$
$$234$$ 2.24264 0.146606
$$235$$ 0 0
$$236$$ −15.5147 −1.00992
$$237$$ − 11.3137i − 0.734904i
$$238$$ 0.284271i 0.0184266i
$$239$$ −2.58579 −0.167261 −0.0836303 0.996497i $$-0.526651\pi$$
−0.0836303 + 0.996497i $$0.526651\pi$$
$$240$$ 0 0
$$241$$ −14.9706 −0.964339 −0.482169 0.876078i $$-0.660151\pi$$
−0.482169 + 0.876078i $$0.660151\pi$$
$$242$$ − 3.72792i − 0.239640i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 10.3431 0.662152
$$245$$ 0 0
$$246$$ −3.07107 −0.195804
$$247$$ − 5.41421i − 0.344498i
$$248$$ 10.2843i 0.653052i
$$249$$ −10.4853 −0.664478
$$250$$ 0 0
$$251$$ 12.9289 0.816067 0.408033 0.912967i $$-0.366215\pi$$
0.408033 + 0.912967i $$0.366215\pi$$
$$252$$ 1.07107i 0.0674709i
$$253$$ 10.8284i 0.680777i
$$254$$ 3.31371 0.207921
$$255$$ 0 0
$$256$$ 3.97056 0.248160
$$257$$ − 1.17157i − 0.0730807i −0.999332 0.0365404i $$-0.988366\pi$$
0.999332 0.0365404i $$-0.0116337\pi$$
$$258$$ 0.242641i 0.0151061i
$$259$$ −6.48528 −0.402976
$$260$$ 0 0
$$261$$ −9.07107 −0.561485
$$262$$ 6.92893i 0.428071i
$$263$$ − 32.1421i − 1.98197i −0.133977 0.990984i $$-0.542775\pi$$
0.133977 0.990984i $$-0.457225\pi$$
$$264$$ 2.24264 0.138025
$$265$$ 0 0
$$266$$ −0.242641 −0.0148773
$$267$$ 10.7279i 0.656538i
$$268$$ − 21.9411i − 1.34027i
$$269$$ −8.38478 −0.511229 −0.255614 0.966779i $$-0.582278\pi$$
−0.255614 + 0.966779i $$0.582278\pi$$
$$270$$ 0 0
$$271$$ −30.8284 −1.87269 −0.936347 0.351076i $$-0.885816\pi$$
−0.936347 + 0.351076i $$0.885816\pi$$
$$272$$ 3.51472i 0.213111i
$$273$$ − 3.17157i − 0.191952i
$$274$$ −5.79899 −0.350330
$$275$$ 0 0
$$276$$ 14.0000 0.842701
$$277$$ − 10.9706i − 0.659157i −0.944128 0.329579i $$-0.893093\pi$$
0.944128 0.329579i $$-0.106907\pi$$
$$278$$ − 0.485281i − 0.0291052i
$$279$$ −6.48528 −0.388264
$$280$$ 0 0
$$281$$ −14.7279 −0.878594 −0.439297 0.898342i $$-0.644772\pi$$
−0.439297 + 0.898342i $$0.644772\pi$$
$$282$$ − 0.142136i − 0.00846405i
$$283$$ 6.24264i 0.371086i 0.982636 + 0.185543i $$0.0594045\pi$$
−0.982636 + 0.185543i $$0.940596\pi$$
$$284$$ −8.20101 −0.486640
$$285$$ 0 0
$$286$$ −3.17157 −0.187539
$$287$$ 4.34315i 0.256368i
$$288$$ − 4.41421i − 0.260110i
$$289$$ 15.6274 0.919260
$$290$$ 0 0
$$291$$ 4.24264 0.248708
$$292$$ − 3.65685i − 0.214001i
$$293$$ − 31.7990i − 1.85772i −0.370435 0.928858i $$-0.620791\pi$$
0.370435 0.928858i $$-0.379209\pi$$
$$294$$ 2.75736 0.160812
$$295$$ 0 0
$$296$$ 17.5563 1.02044
$$297$$ 1.41421i 0.0820610i
$$298$$ 1.51472i 0.0877453i
$$299$$ −41.4558 −2.39745
$$300$$ 0 0
$$301$$ 0.343146 0.0197786
$$302$$ 7.37258i 0.424244i
$$303$$ 4.82843i 0.277386i
$$304$$ −3.00000 −0.172062
$$305$$ 0 0
$$306$$ 0.485281 0.0277417
$$307$$ − 7.79899i − 0.445112i −0.974920 0.222556i $$-0.928560\pi$$
0.974920 0.222556i $$-0.0714400\pi$$
$$308$$ − 1.51472i − 0.0863091i
$$309$$ −1.65685 −0.0942551
$$310$$ 0 0
$$311$$ −32.2426 −1.82831 −0.914156 0.405362i $$-0.867145\pi$$
−0.914156 + 0.405362i $$0.867145\pi$$
$$312$$ 8.58579i 0.486074i
$$313$$ − 9.51472i − 0.537804i −0.963168 0.268902i $$-0.913339\pi$$
0.963168 0.268902i $$-0.0866607\pi$$
$$314$$ −9.02944 −0.509561
$$315$$ 0 0
$$316$$ 20.6863 1.16369
$$317$$ − 11.3137i − 0.635441i −0.948184 0.317721i $$-0.897083\pi$$
0.948184 0.317721i $$-0.102917\pi$$
$$318$$ 1.65685i 0.0929118i
$$319$$ 12.8284 0.718254
$$320$$ 0 0
$$321$$ −8.00000 −0.446516
$$322$$ 1.85786i 0.103535i
$$323$$ − 1.17157i − 0.0651881i
$$324$$ 1.82843 0.101579
$$325$$ 0 0
$$326$$ −3.27208 −0.181224
$$327$$ 3.17157i 0.175388i
$$328$$ − 11.7574i − 0.649192i
$$329$$ −0.201010 −0.0110820
$$330$$ 0 0
$$331$$ 7.17157 0.394185 0.197093 0.980385i $$-0.436850\pi$$
0.197093 + 0.980385i $$0.436850\pi$$
$$332$$ − 19.1716i − 1.05218i
$$333$$ 11.0711i 0.606691i
$$334$$ 4.14214 0.226648
$$335$$ 0 0
$$336$$ −1.75736 −0.0958718
$$337$$ − 28.2426i − 1.53847i −0.638963 0.769237i $$-0.720636\pi$$
0.638963 0.769237i $$-0.279364\pi$$
$$338$$ − 6.75736i − 0.367552i
$$339$$ 12.4853 0.678107
$$340$$ 0 0
$$341$$ 9.17157 0.496669
$$342$$ 0.414214i 0.0223981i
$$343$$ − 8.00000i − 0.431959i
$$344$$ −0.928932 −0.0500847
$$345$$ 0 0
$$346$$ 2.54416 0.136775
$$347$$ − 10.4853i − 0.562879i −0.959579 0.281440i $$-0.909188\pi$$
0.959579 0.281440i $$-0.0908119\pi$$
$$348$$ − 16.5858i − 0.889091i
$$349$$ −29.3137 −1.56913 −0.784563 0.620049i $$-0.787113\pi$$
−0.784563 + 0.620049i $$0.787113\pi$$
$$350$$ 0 0
$$351$$ −5.41421 −0.288989
$$352$$ 6.24264i 0.332734i
$$353$$ 3.65685i 0.194635i 0.995253 + 0.0973174i $$0.0310262\pi$$
−0.995253 + 0.0973174i $$0.968974\pi$$
$$354$$ −3.51472 −0.186805
$$355$$ 0 0
$$356$$ −19.6152 −1.03960
$$357$$ − 0.686292i − 0.0363224i
$$358$$ − 7.11270i − 0.375918i
$$359$$ −9.89949 −0.522475 −0.261238 0.965275i $$-0.584131\pi$$
−0.261238 + 0.965275i $$0.584131\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ − 7.94113i − 0.417376i
$$363$$ 9.00000i 0.472377i
$$364$$ 5.79899 0.303950
$$365$$ 0 0
$$366$$ 2.34315 0.122478
$$367$$ 19.4142i 1.01341i 0.862118 + 0.506707i $$0.169137\pi$$
−0.862118 + 0.506707i $$0.830863\pi$$
$$368$$ 22.9706i 1.19742i
$$369$$ 7.41421 0.385969
$$370$$ 0 0
$$371$$ 2.34315 0.121650
$$372$$ − 11.8579i − 0.614802i
$$373$$ − 9.89949i − 0.512576i −0.966600 0.256288i $$-0.917500\pi$$
0.966600 0.256288i $$-0.0824996\pi$$
$$374$$ −0.686292 −0.0354873
$$375$$ 0 0
$$376$$ 0.544156 0.0280627
$$377$$ 49.1127i 2.52943i
$$378$$ 0.242641i 0.0124801i
$$379$$ −24.1421 −1.24010 −0.620049 0.784563i $$-0.712887\pi$$
−0.620049 + 0.784563i $$0.712887\pi$$
$$380$$ 0 0
$$381$$ −8.00000 −0.409852
$$382$$ 0.786797i 0.0402560i
$$383$$ 12.0000i 0.613171i 0.951843 + 0.306586i $$0.0991866\pi$$
−0.951843 + 0.306586i $$0.900813\pi$$
$$384$$ 10.5563 0.538701
$$385$$ 0 0
$$386$$ 6.24264 0.317742
$$387$$ − 0.585786i − 0.0297772i
$$388$$ 7.75736i 0.393820i
$$389$$ −14.9706 −0.759038 −0.379519 0.925184i $$-0.623910\pi$$
−0.379519 + 0.925184i $$0.623910\pi$$
$$390$$ 0 0
$$391$$ −8.97056 −0.453661
$$392$$ 10.5563i 0.533176i
$$393$$ − 16.7279i − 0.843812i
$$394$$ −6.14214 −0.309436
$$395$$ 0 0
$$396$$ −2.58579 −0.129941
$$397$$ − 35.6569i − 1.78957i −0.446501 0.894783i $$-0.647330\pi$$
0.446501 0.894783i $$-0.352670\pi$$
$$398$$ 6.82843i 0.342278i
$$399$$ 0.585786 0.0293260
$$400$$ 0 0
$$401$$ 30.0416 1.50021 0.750104 0.661320i $$-0.230004\pi$$
0.750104 + 0.661320i $$0.230004\pi$$
$$402$$ − 4.97056i − 0.247909i
$$403$$ 35.1127i 1.74909i
$$404$$ −8.82843 −0.439231
$$405$$ 0 0
$$406$$ 2.20101 0.109234
$$407$$ − 15.6569i − 0.776081i
$$408$$ 1.85786i 0.0919780i
$$409$$ −9.51472 −0.470473 −0.235236 0.971938i $$-0.575586\pi$$
−0.235236 + 0.971938i $$0.575586\pi$$
$$410$$ 0 0
$$411$$ 14.0000 0.690569
$$412$$ − 3.02944i − 0.149250i
$$413$$ 4.97056i 0.244585i
$$414$$ 3.17157 0.155874
$$415$$ 0 0
$$416$$ −23.8995 −1.17177
$$417$$ 1.17157i 0.0573722i
$$418$$ − 0.585786i − 0.0286518i
$$419$$ 34.8701 1.70351 0.851757 0.523937i $$-0.175537\pi$$
0.851757 + 0.523937i $$0.175537\pi$$
$$420$$ 0 0
$$421$$ −14.6863 −0.715766 −0.357883 0.933766i $$-0.616501\pi$$
−0.357883 + 0.933766i $$0.616501\pi$$
$$422$$ − 6.34315i − 0.308780i
$$423$$ 0.343146i 0.0166843i
$$424$$ −6.34315 −0.308050
$$425$$ 0 0
$$426$$ −1.85786 −0.0900138
$$427$$ − 3.31371i − 0.160362i
$$428$$ − 14.6274i − 0.707043i
$$429$$ 7.65685 0.369676
$$430$$ 0 0
$$431$$ 3.51472 0.169298 0.0846490 0.996411i $$-0.473023\pi$$
0.0846490 + 0.996411i $$0.473023\pi$$
$$432$$ 3.00000i 0.144338i
$$433$$ 0.928932i 0.0446416i 0.999751 + 0.0223208i $$0.00710553\pi$$
−0.999751 + 0.0223208i $$0.992894\pi$$
$$434$$ 1.57359 0.0755349
$$435$$ 0 0
$$436$$ −5.79899 −0.277721
$$437$$ − 7.65685i − 0.366277i
$$438$$ − 0.828427i − 0.0395838i
$$439$$ −0.970563 −0.0463224 −0.0231612 0.999732i $$-0.507373\pi$$
−0.0231612 + 0.999732i $$0.507373\pi$$
$$440$$ 0 0
$$441$$ −6.65685 −0.316993
$$442$$ − 2.62742i − 0.124973i
$$443$$ − 1.31371i − 0.0624162i −0.999513 0.0312081i $$-0.990065\pi$$
0.999513 0.0312081i $$-0.00993546\pi$$
$$444$$ −20.2426 −0.960673
$$445$$ 0 0
$$446$$ −2.62742 −0.124412
$$447$$ − 3.65685i − 0.172963i
$$448$$ − 2.44365i − 0.115452i
$$449$$ −7.89949 −0.372800 −0.186400 0.982474i $$-0.559682\pi$$
−0.186400 + 0.982474i $$0.559682\pi$$
$$450$$ 0 0
$$451$$ −10.4853 −0.493733
$$452$$ 22.8284i 1.07376i
$$453$$ − 17.7990i − 0.836269i
$$454$$ −7.85786 −0.368788
$$455$$ 0 0
$$456$$ −1.58579 −0.0742613
$$457$$ − 28.8284i − 1.34854i −0.738486 0.674268i $$-0.764459\pi$$
0.738486 0.674268i $$-0.235541\pi$$
$$458$$ 0.686292i 0.0320683i
$$459$$ −1.17157 −0.0546843
$$460$$ 0 0
$$461$$ 10.6863 0.497710 0.248855 0.968541i $$-0.419946\pi$$
0.248855 + 0.968541i $$0.419946\pi$$
$$462$$ − 0.343146i − 0.0159646i
$$463$$ − 8.10051i − 0.376462i −0.982125 0.188231i $$-0.939725\pi$$
0.982125 0.188231i $$-0.0602754\pi$$
$$464$$ 27.2132 1.26334
$$465$$ 0 0
$$466$$ 3.45584 0.160089
$$467$$ − 16.3431i − 0.756271i −0.925750 0.378135i $$-0.876565\pi$$
0.925750 0.378135i $$-0.123435\pi$$
$$468$$ − 9.89949i − 0.457604i
$$469$$ −7.02944 −0.324589
$$470$$ 0 0
$$471$$ 21.7990 1.00444
$$472$$ − 13.4558i − 0.619355i
$$473$$ 0.828427i 0.0380911i
$$474$$ 4.68629 0.215248
$$475$$ 0 0
$$476$$ 1.25483 0.0575152
$$477$$ − 4.00000i − 0.183147i
$$478$$ − 1.07107i − 0.0489895i
$$479$$ 38.1838 1.74466 0.872330 0.488917i $$-0.162608\pi$$
0.872330 + 0.488917i $$0.162608\pi$$
$$480$$ 0 0
$$481$$ 59.9411 2.73308
$$482$$ − 6.20101i − 0.282448i
$$483$$ − 4.48528i − 0.204087i
$$484$$ −16.4558 −0.747993
$$485$$ 0 0
$$486$$ 0.414214 0.0187891
$$487$$ 2.82843i 0.128168i 0.997944 + 0.0640841i $$0.0204126\pi$$
−0.997944 + 0.0640841i $$0.979587\pi$$
$$488$$ 8.97056i 0.406078i
$$489$$ 7.89949 0.357228
$$490$$ 0 0
$$491$$ 21.8995 0.988310 0.494155 0.869374i $$-0.335477\pi$$
0.494155 + 0.869374i $$0.335477\pi$$
$$492$$ 13.5563i 0.611167i
$$493$$ 10.6274i 0.478635i
$$494$$ 2.24264 0.100901
$$495$$ 0 0
$$496$$ 19.4558 0.873593
$$497$$ 2.62742i 0.117856i
$$498$$ − 4.34315i − 0.194621i
$$499$$ 23.7990 1.06539 0.532695 0.846308i $$-0.321179\pi$$
0.532695 + 0.846308i $$0.321179\pi$$
$$500$$ 0 0
$$501$$ −10.0000 −0.446767
$$502$$ 5.35534i 0.239020i
$$503$$ 0.828427i 0.0369377i 0.999829 + 0.0184689i $$0.00587916\pi$$
−0.999829 + 0.0184689i $$0.994121\pi$$
$$504$$ −0.928932 −0.0413779
$$505$$ 0 0
$$506$$ −4.48528 −0.199395
$$507$$ 16.3137i 0.724517i
$$508$$ − 14.6274i − 0.648987i
$$509$$ −32.3848 −1.43543 −0.717715 0.696337i $$-0.754812\pi$$
−0.717715 + 0.696337i $$0.754812\pi$$
$$510$$ 0 0
$$511$$ −1.17157 −0.0518273
$$512$$ 22.7574i 1.00574i
$$513$$ − 1.00000i − 0.0441511i
$$514$$ 0.485281 0.0214048
$$515$$ 0 0
$$516$$ 1.07107 0.0471511
$$517$$ − 0.485281i − 0.0213427i
$$518$$ − 2.68629i − 0.118029i
$$519$$ −6.14214 −0.269610
$$520$$ 0 0
$$521$$ −24.3848 −1.06832 −0.534158 0.845385i $$-0.679371\pi$$
−0.534158 + 0.845385i $$0.679371\pi$$
$$522$$ − 3.75736i − 0.164455i
$$523$$ − 15.7990i − 0.690842i −0.938448 0.345421i $$-0.887736\pi$$
0.938448 0.345421i $$-0.112264\pi$$
$$524$$ 30.5858 1.33615
$$525$$ 0 0
$$526$$ 13.3137 0.580505
$$527$$ 7.59798i 0.330973i
$$528$$ − 4.24264i − 0.184637i
$$529$$ −35.6274 −1.54902
$$530$$ 0 0
$$531$$ 8.48528 0.368230
$$532$$ 1.07107i 0.0464367i
$$533$$ − 40.1421i − 1.73875i
$$534$$ −4.44365 −0.192296
$$535$$ 0 0
$$536$$ 19.0294 0.821946
$$537$$ 17.1716i 0.741008i
$$538$$ − 3.47309i − 0.149735i
$$539$$ 9.41421 0.405499
$$540$$ 0 0
$$541$$ −32.6274 −1.40276 −0.701381 0.712786i $$-0.747433\pi$$
−0.701381 + 0.712786i $$0.747433\pi$$
$$542$$ − 12.7696i − 0.548499i
$$543$$ 19.1716i 0.822731i
$$544$$ −5.17157 −0.221729
$$545$$ 0 0
$$546$$ 1.31371 0.0562215
$$547$$ 34.1421i 1.45981i 0.683547 + 0.729906i $$0.260436\pi$$
−0.683547 + 0.729906i $$0.739564\pi$$
$$548$$ 25.5980i 1.09349i
$$549$$ −5.65685 −0.241429
$$550$$ 0 0
$$551$$ −9.07107 −0.386440
$$552$$ 12.1421i 0.516804i
$$553$$ − 6.62742i − 0.281826i
$$554$$ 4.54416 0.193063
$$555$$ 0 0
$$556$$ −2.14214 −0.0908468
$$557$$ − 10.0000i − 0.423714i −0.977301 0.211857i $$-0.932049\pi$$
0.977301 0.211857i $$-0.0679510\pi$$
$$558$$ − 2.68629i − 0.113720i
$$559$$ −3.17157 −0.134143
$$560$$ 0 0
$$561$$ 1.65685 0.0699524
$$562$$ − 6.10051i − 0.257334i
$$563$$ 42.2843i 1.78207i 0.453935 + 0.891035i $$0.350020\pi$$
−0.453935 + 0.891035i $$0.649980\pi$$
$$564$$ −0.627417 −0.0264190
$$565$$ 0 0
$$566$$ −2.58579 −0.108689
$$567$$ − 0.585786i − 0.0246007i
$$568$$ − 7.11270i − 0.298442i
$$569$$ 6.72792 0.282049 0.141025 0.990006i $$-0.454960\pi$$
0.141025 + 0.990006i $$0.454960\pi$$
$$570$$ 0 0
$$571$$ 19.7990 0.828562 0.414281 0.910149i $$-0.364033\pi$$
0.414281 + 0.910149i $$0.364033\pi$$
$$572$$ 14.0000i 0.585369i
$$573$$ − 1.89949i − 0.0793525i
$$574$$ −1.79899 −0.0750884
$$575$$ 0 0
$$576$$ −4.17157 −0.173816
$$577$$ 37.7990i 1.57359i 0.617213 + 0.786796i $$0.288262\pi$$
−0.617213 + 0.786796i $$0.711738\pi$$
$$578$$ 6.47309i 0.269245i
$$579$$ −15.0711 −0.626332
$$580$$ 0 0
$$581$$ −6.14214 −0.254819
$$582$$ 1.75736i 0.0728449i
$$583$$ 5.65685i 0.234283i
$$584$$ 3.17157 0.131241
$$585$$ 0 0
$$586$$ 13.1716 0.544113
$$587$$ − 11.6569i − 0.481130i −0.970633 0.240565i $$-0.922667\pi$$
0.970633 0.240565i $$-0.0773327\pi$$
$$588$$ − 12.1716i − 0.501947i
$$589$$ −6.48528 −0.267221
$$590$$ 0 0
$$591$$ 14.8284 0.609960
$$592$$ − 33.2132i − 1.36505i
$$593$$ − 29.3137i − 1.20377i −0.798583 0.601885i $$-0.794417\pi$$
0.798583 0.601885i $$-0.205583\pi$$
$$594$$ −0.585786 −0.0240351
$$595$$ 0 0
$$596$$ 6.68629 0.273881
$$597$$ − 16.4853i − 0.674698i
$$598$$ − 17.1716i − 0.702198i
$$599$$ 21.9411 0.896490 0.448245 0.893911i $$-0.352049\pi$$
0.448245 + 0.893911i $$0.352049\pi$$
$$600$$ 0 0
$$601$$ −28.8284 −1.17594 −0.587968 0.808884i $$-0.700072\pi$$
−0.587968 + 0.808884i $$0.700072\pi$$
$$602$$ 0.142136i 0.00579302i
$$603$$ 12.0000i 0.488678i
$$604$$ 32.5442 1.32420
$$605$$ 0 0
$$606$$ −2.00000 −0.0812444
$$607$$ 2.14214i 0.0869466i 0.999055 + 0.0434733i $$0.0138423\pi$$
−0.999055 + 0.0434733i $$0.986158\pi$$
$$608$$ − 4.41421i − 0.179020i
$$609$$ −5.31371 −0.215322
$$610$$ 0 0
$$611$$ 1.85786 0.0751611
$$612$$ − 2.14214i − 0.0865907i
$$613$$ 25.1127i 1.01429i 0.861860 + 0.507146i $$0.169300\pi$$
−0.861860 + 0.507146i $$0.830700\pi$$
$$614$$ 3.23045 0.130370
$$615$$ 0 0
$$616$$ 1.31371 0.0529308
$$617$$ − 10.1421i − 0.408307i −0.978939 0.204154i $$-0.934556\pi$$
0.978939 0.204154i $$-0.0654442\pi$$
$$618$$ − 0.686292i − 0.0276067i
$$619$$ 43.7990 1.76043 0.880215 0.474575i $$-0.157398\pi$$
0.880215 + 0.474575i $$0.157398\pi$$
$$620$$ 0 0
$$621$$ −7.65685 −0.307259
$$622$$ − 13.3553i − 0.535500i
$$623$$ 6.28427i 0.251774i
$$624$$ 16.2426 0.650226
$$625$$ 0 0
$$626$$ 3.94113 0.157519
$$627$$ 1.41421i 0.0564782i
$$628$$ 39.8579i 1.59050i
$$629$$ 12.9706 0.517170
$$630$$ 0 0
$$631$$ 22.6274 0.900783 0.450392 0.892831i $$-0.351284\pi$$
0.450392 + 0.892831i $$0.351284\pi$$
$$632$$ 17.9411i 0.713660i
$$633$$ 15.3137i 0.608665i
$$634$$ 4.68629 0.186116
$$635$$ 0 0
$$636$$ 7.31371 0.290007
$$637$$ 36.0416i 1.42802i
$$638$$ 5.31371i 0.210372i
$$639$$ 4.48528 0.177435
$$640$$ 0 0
$$641$$ −8.58579 −0.339118 −0.169559 0.985520i $$-0.554234\pi$$
−0.169559 + 0.985520i $$0.554234\pi$$
$$642$$ − 3.31371i − 0.130782i
$$643$$ 6.04163i 0.238259i 0.992879 + 0.119129i $$0.0380103\pi$$
−0.992879 + 0.119129i $$0.961990\pi$$
$$644$$ 8.20101 0.323165
$$645$$ 0 0
$$646$$ 0.485281 0.0190931
$$647$$ − 45.1127i − 1.77356i −0.462189 0.886782i $$-0.652936\pi$$
0.462189 0.886782i $$-0.347064\pi$$
$$648$$ 1.58579i 0.0622956i
$$649$$ −12.0000 −0.471041
$$650$$ 0 0
$$651$$ −3.79899 −0.148894
$$652$$ 14.4437i 0.565657i
$$653$$ 42.4264i 1.66027i 0.557560 + 0.830137i $$0.311738\pi$$
−0.557560 + 0.830137i $$0.688262\pi$$
$$654$$ −1.31371 −0.0513701
$$655$$ 0 0
$$656$$ −22.2426 −0.868429
$$657$$ 2.00000i 0.0780274i
$$658$$ − 0.0832611i − 0.00324586i
$$659$$ 20.0000 0.779089 0.389545 0.921008i $$-0.372632\pi$$
0.389545 + 0.921008i $$0.372632\pi$$
$$660$$ 0 0
$$661$$ 18.4853 0.718994 0.359497 0.933146i $$-0.382948\pi$$
0.359497 + 0.933146i $$0.382948\pi$$
$$662$$ 2.97056i 0.115454i
$$663$$ 6.34315i 0.246347i
$$664$$ 16.6274 0.645269
$$665$$ 0 0
$$666$$ −4.58579 −0.177696
$$667$$ 69.4558i 2.68934i
$$668$$ − 18.2843i − 0.707440i
$$669$$ 6.34315 0.245240
$$670$$ 0 0
$$671$$ 8.00000 0.308837
$$672$$ − 2.58579i − 0.0997489i
$$673$$ 21.8995i 0.844163i 0.906558 + 0.422082i $$0.138700\pi$$
−0.906558 + 0.422082i $$0.861300\pi$$
$$674$$ 11.6985 0.450609
$$675$$ 0 0
$$676$$ −29.8284 −1.14725
$$677$$ 44.9706i 1.72836i 0.503184 + 0.864180i $$0.332162\pi$$
−0.503184 + 0.864180i $$0.667838\pi$$
$$678$$ 5.17157i 0.198613i
$$679$$ 2.48528 0.0953763
$$680$$ 0 0
$$681$$ 18.9706 0.726954
$$682$$ 3.79899i 0.145471i
$$683$$ 5.65685i 0.216454i 0.994126 + 0.108227i $$0.0345173\pi$$
−0.994126 + 0.108227i $$0.965483\pi$$
$$684$$ 1.82843 0.0699117
$$685$$ 0 0
$$686$$ 3.31371 0.126518
$$687$$ − 1.65685i − 0.0632129i
$$688$$ 1.75736i 0.0669987i
$$689$$ −21.6569 −0.825060
$$690$$ 0 0
$$691$$ −23.1127 −0.879248 −0.439624 0.898182i $$-0.644888\pi$$
−0.439624 + 0.898182i $$0.644888\pi$$
$$692$$ − 11.2304i − 0.426918i
$$693$$ 0.828427i 0.0314693i
$$694$$ 4.34315 0.164864
$$695$$ 0 0
$$696$$ 14.3848 0.545254
$$697$$ − 8.68629i − 0.329017i
$$698$$ − 12.1421i − 0.459587i
$$699$$ −8.34315 −0.315567
$$700$$ 0 0
$$701$$ −0.343146 −0.0129604 −0.00648022 0.999979i $$-0.502063\pi$$
−0.00648022 + 0.999979i $$0.502063\pi$$
$$702$$ − 2.24264i − 0.0846430i
$$703$$ 11.0711i 0.417553i
$$704$$ 5.89949 0.222346
$$705$$ 0 0
$$706$$ −1.51472 −0.0570072
$$707$$ 2.82843i 0.106374i
$$708$$ 15.5147i 0.583079i
$$709$$ 35.3137 1.32623 0.663117 0.748516i $$-0.269233\pi$$
0.663117 + 0.748516i $$0.269233\pi$$
$$710$$ 0 0
$$711$$ −11.3137 −0.424297
$$712$$ − 17.0122i − 0.637559i
$$713$$ 49.6569i 1.85966i
$$714$$ 0.284271 0.0106386
$$715$$ 0 0
$$716$$ −31.3970 −1.17336
$$717$$ 2.58579i 0.0965680i
$$718$$ − 4.10051i − 0.153029i
$$719$$ 16.4437 0.613245 0.306622 0.951831i $$-0.400801\pi$$
0.306622 + 0.951831i $$0.400801\pi$$
$$720$$ 0 0
$$721$$ −0.970563 −0.0361456
$$722$$ 0.414214i 0.0154154i
$$723$$ 14.9706i 0.556761i
$$724$$ −35.0538 −1.30277
$$725$$ 0 0
$$726$$ −3.72792 −0.138356
$$727$$ 4.58579i 0.170077i 0.996378 + 0.0850387i $$0.0271014\pi$$
−0.996378 + 0.0850387i $$0.972899\pi$$
$$728$$ 5.02944i 0.186403i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −0.686292 −0.0253834
$$732$$ − 10.3431i − 0.382294i
$$733$$ − 1.31371i − 0.0485229i −0.999706 0.0242615i $$-0.992277\pi$$
0.999706 0.0242615i $$-0.00772342\pi$$
$$734$$ −8.04163 −0.296822
$$735$$ 0 0
$$736$$ −33.7990 −1.24585
$$737$$ − 16.9706i − 0.625119i
$$738$$ 3.07107i 0.113048i
$$739$$ 9.65685 0.355233 0.177617 0.984100i $$-0.443161\pi$$
0.177617 + 0.984100i $$0.443161\pi$$
$$740$$ 0 0
$$741$$ −5.41421 −0.198896
$$742$$ 0.970563i 0.0356305i
$$743$$ − 15.3137i − 0.561805i −0.959736 0.280903i $$-0.909366\pi$$
0.959736 0.280903i $$-0.0906338\pi$$
$$744$$ 10.2843 0.377040
$$745$$ 0 0
$$746$$ 4.10051 0.150130
$$747$$ 10.4853i 0.383636i
$$748$$ 3.02944i 0.110767i
$$749$$ −4.68629 −0.171233
$$750$$ 0 0
$$751$$ −37.1127 −1.35426 −0.677131 0.735863i $$-0.736777\pi$$
−0.677131 + 0.735863i $$0.736777\pi$$
$$752$$ − 1.02944i − 0.0375397i
$$753$$ − 12.9289i − 0.471156i
$$754$$ −20.3431 −0.740854
$$755$$ 0 0
$$756$$ 1.07107 0.0389544
$$757$$ − 16.8284i − 0.611640i −0.952089 0.305820i $$-0.901069\pi$$
0.952089 0.305820i $$-0.0989305\pi$$
$$758$$ − 10.0000i − 0.363216i
$$759$$ 10.8284 0.393047
$$760$$ 0 0
$$761$$ −10.2843 −0.372805 −0.186402 0.982474i $$-0.559683\pi$$
−0.186402 + 0.982474i $$0.559683\pi$$
$$762$$ − 3.31371i − 0.120043i
$$763$$ 1.85786i 0.0672592i
$$764$$ 3.47309 0.125652
$$765$$ 0 0
$$766$$ −4.97056 −0.179594
$$767$$ − 45.9411i − 1.65884i
$$768$$ − 3.97056i − 0.143275i
$$769$$ 35.6569 1.28582 0.642910 0.765942i $$-0.277727\pi$$
0.642910 + 0.765942i $$0.277727\pi$$
$$770$$ 0 0
$$771$$ −1.17157 −0.0421932
$$772$$ − 27.5563i − 0.991775i
$$773$$ − 32.9706i − 1.18587i −0.805251 0.592934i $$-0.797969\pi$$
0.805251 0.592934i $$-0.202031\pi$$
$$774$$ 0.242641 0.00872154
$$775$$ 0 0
$$776$$ −6.72792 −0.241518
$$777$$ 6.48528i 0.232658i
$$778$$ − 6.20101i − 0.222317i
$$779$$ 7.41421 0.265642
$$780$$ 0 0
$$781$$ −6.34315 −0.226976
$$782$$ − 3.71573i − 0.132874i
$$783$$ 9.07107i 0.324174i
$$784$$ 19.9706 0.713234
$$785$$ 0 0
$$786$$ 6.92893 0.247147
$$787$$ − 13.4558i − 0.479649i −0.970816 0.239825i $$-0.922910\pi$$
0.970816 0.239825i $$-0.0770899\pi$$
$$788$$ 27.1127i 0.965850i
$$789$$ −32.1421 −1.14429
$$790$$ 0 0
$$791$$ 7.31371 0.260046
$$792$$ − 2.24264i − 0.0796888i
$$793$$ 30.6274i 1.08761i
$$794$$ 14.7696 0.524152
$$795$$ 0 0
$$796$$ 30.1421 1.06836
$$797$$ − 10.8284i − 0.383563i −0.981438 0.191781i $$-0.938574\pi$$
0.981438 0.191781i $$-0.0614264\pi$$
$$798$$ 0.242641i 0.00858939i
$$799$$ 0.402020 0.0142225
$$800$$ 0 0
$$801$$ 10.7279 0.379052
$$802$$ 12.4437i 0.439401i
$$803$$ − 2.82843i − 0.0998130i
$$804$$ −21.9411 −0.773804
$$805$$ 0 0
$$806$$ −14.5442 −0.512296
$$807$$ 8.38478i 0.295158i
$$808$$ − 7.65685i − 0.269367i
$$809$$ −49.3137 −1.73378 −0.866889 0.498502i $$-0.833884\pi$$
−0.866889 + 0.498502i $$0.833884\pi$$
$$810$$ 0 0
$$811$$ −15.3137 −0.537737 −0.268869 0.963177i $$-0.586650\pi$$
−0.268869 + 0.963177i $$0.586650\pi$$
$$812$$ − 9.71573i − 0.340955i
$$813$$ 30.8284i 1.08120i
$$814$$ 6.48528 0.227309
$$815$$ 0 0
$$816$$ 3.51472 0.123040
$$817$$ − 0.585786i − 0.0204941i
$$818$$ − 3.94113i − 0.137798i
$$819$$ −3.17157 −0.110824
$$820$$ 0 0
$$821$$ 51.4558 1.79582 0.897911 0.440178i $$-0.145085\pi$$
0.897911 + 0.440178i $$0.145085\pi$$
$$822$$ 5.79899i 0.202263i
$$823$$ − 2.72792i − 0.0950894i −0.998869 0.0475447i $$-0.984860\pi$$
0.998869 0.0475447i $$-0.0151397\pi$$
$$824$$ 2.62742 0.0915304
$$825$$ 0 0
$$826$$ −2.05887 −0.0716374
$$827$$ − 48.6274i − 1.69094i −0.534022 0.845470i $$-0.679320\pi$$
0.534022 0.845470i $$-0.320680\pi$$
$$828$$ − 14.0000i − 0.486534i
$$829$$ 38.4853 1.33665 0.668325 0.743870i $$-0.267012\pi$$
0.668325 + 0.743870i $$0.267012\pi$$
$$830$$ 0 0
$$831$$ −10.9706 −0.380565
$$832$$ 22.5858i 0.783021i
$$833$$ 7.79899i 0.270219i
$$834$$ −0.485281 −0.0168039
$$835$$ 0 0
$$836$$ −2.58579 −0.0894313
$$837$$ 6.48528i 0.224164i
$$838$$ 14.4437i 0.498948i
$$839$$ −27.1127 −0.936034 −0.468017 0.883719i $$-0.655031\pi$$
−0.468017 + 0.883719i $$0.655031\pi$$
$$840$$ 0 0
$$841$$ 53.2843 1.83739
$$842$$ − 6.08326i − 0.209643i
$$843$$ 14.7279i 0.507257i
$$844$$ −28.0000 −0.963800
$$845$$ 0 0
$$846$$ −0.142136 −0.00488672
$$847$$ 5.27208i 0.181151i
$$848$$ 12.0000i 0.412082i
$$849$$ 6.24264 0.214247
$$850$$ 0 0
$$851$$ 84.7696 2.90586
$$852$$ 8.20101i 0.280962i
$$853$$ − 9.51472i − 0.325778i −0.986644 0.162889i $$-0.947919\pi$$
0.986644 0.162889i $$-0.0520812\pi$$
$$854$$ 1.37258 0.0469688
$$855$$ 0 0
$$856$$ 12.6863 0.433609
$$857$$ − 37.9411i − 1.29604i −0.761622 0.648022i $$-0.775596\pi$$
0.761622 0.648022i $$-0.224404\pi$$
$$858$$ 3.17157i 0.108276i
$$859$$ −25.9411 −0.885100 −0.442550 0.896744i $$-0.645926\pi$$
−0.442550 + 0.896744i $$0.645926\pi$$
$$860$$ 0 0
$$861$$ 4.34315 0.148014
$$862$$ 1.45584i 0.0495862i
$$863$$ − 31.3137i − 1.06593i −0.846137 0.532966i $$-0.821078\pi$$
0.846137 0.532966i $$-0.178922\pi$$
$$864$$ −4.41421 −0.150175
$$865$$ 0 0
$$866$$ −0.384776 −0.0130752
$$867$$ − 15.6274i − 0.530735i
$$868$$ − 6.94618i − 0.235769i
$$869$$ 16.0000 0.542763
$$870$$ 0 0
$$871$$ 64.9706 2.20144
$$872$$ − 5.02944i − 0.170318i
$$873$$ − 4.24264i − 0.143592i
$$874$$ 3.17157 0.107280
$$875$$ 0 0
$$876$$ −3.65685 −0.123554
$$877$$ − 17.8995i − 0.604423i −0.953241 0.302211i $$-0.902275\pi$$
0.953241 0.302211i $$-0.0977249\pi$$
$$878$$ − 0.402020i − 0.0135675i
$$879$$ −31.7990 −1.07255
$$880$$ 0 0
$$881$$ −47.4558 −1.59883 −0.799414 0.600781i $$-0.794857\pi$$
−0.799414 + 0.600781i $$0.794857\pi$$
$$882$$ − 2.75736i − 0.0928451i
$$883$$ 46.5269i 1.56576i 0.622176 + 0.782878i $$0.286249\pi$$
−0.622176 + 0.782878i $$0.713751\pi$$
$$884$$ −11.5980 −0.390082
$$885$$ 0 0
$$886$$ 0.544156 0.0182813
$$887$$ 8.00000i 0.268614i 0.990940 + 0.134307i $$0.0428808\pi$$
−0.990940 + 0.134307i $$0.957119\pi$$
$$888$$ − 17.5563i − 0.589153i
$$889$$ −4.68629 −0.157173
$$890$$ 0 0
$$891$$ 1.41421 0.0473779
$$892$$ 11.5980i 0.388329i
$$893$$ 0.343146i 0.0114829i
$$894$$ 1.51472 0.0506598
$$895$$ 0 0
$$896$$ 6.18377 0.206585
$$897$$ 41.4558i 1.38417i
$$898$$ − 3.27208i − 0.109191i
$$899$$ 58.8284 1.96204
$$900$$ 0 0
$$901$$ −4.68629 −0.156123
$$902$$ − 4.34315i − 0.144611i
$$903$$ − 0.343146i − 0.0114192i
$$904$$ −19.7990 −0.658505
$$905$$ 0 0
$$906$$ 7.37258 0.244938
$$907$$ − 38.1421i − 1.26649i −0.773952 0.633244i $$-0.781723\pi$$
0.773952 0.633244i $$-0.218277\pi$$
$$908$$ 34.6863i 1.15111i
$$909$$ 4.82843 0.160149
$$910$$ 0 0
$$911$$ −23.3137 −0.772418 −0.386209 0.922411i $$-0.626216\pi$$
−0.386209 + 0.922411i $$0.626216\pi$$
$$912$$ 3.00000i 0.0993399i
$$913$$ − 14.8284i − 0.490749i
$$914$$ 11.9411 0.394977
$$915$$ 0 0
$$916$$ 3.02944 0.100095
$$917$$ − 9.79899i − 0.323591i
$$918$$ − 0.485281i − 0.0160167i
$$919$$ 12.0000 0.395843 0.197922 0.980218i $$-0.436581\pi$$
0.197922 + 0.980218i $$0.436581\pi$$
$$920$$ 0 0
$$921$$ −7.79899 −0.256985
$$922$$ 4.42641i 0.145776i
$$923$$ − 24.2843i − 0.799327i
$$924$$ −1.51472 −0.0498306
$$925$$ 0 0
$$926$$ 3.35534 0.110263
$$927$$ 1.65685i 0.0544182i
$$928$$ 40.0416i 1.31443i
$$929$$ 51.4558 1.68821 0.844106 0.536177i $$-0.180132\pi$$
0.844106 + 0.536177i $$0.180132\pi$$
$$930$$ 0 0
$$931$$ −6.65685 −0.218170
$$932$$ − 15.2548i − 0.499689i
$$933$$ 32.2426i 1.05558i
$$934$$ 6.76955 0.221507
$$935$$ 0 0
$$936$$ 8.58579 0.280635
$$937$$ 18.7696i 0.613175i 0.951843 + 0.306587i $$0.0991871\pi$$
−0.951843 + 0.306587i $$0.900813\pi$$
$$938$$ − 2.91169i − 0.0950700i
$$939$$ −9.51472 −0.310501
$$940$$ 0 0
$$941$$ −17.5563 −0.572321 −0.286160 0.958182i $$-0.592379\pi$$
−0.286160 + 0.958182i $$0.592379\pi$$
$$942$$ 9.02944i 0.294195i
$$943$$ − 56.7696i − 1.84867i
$$944$$ −25.4558 −0.828517
$$945$$ 0 0
$$946$$ −0.343146 −0.0111566
$$947$$ − 12.8284i − 0.416868i −0.978036 0.208434i $$-0.933163\pi$$
0.978036 0.208434i $$-0.0668366\pi$$
$$948$$ − 20.6863i − 0.671860i
$$949$$ 10.8284 0.351506
$$950$$ 0 0
$$951$$ −11.3137 −0.366872
$$952$$ 1.08831i 0.0352724i
$$953$$ − 5.85786i − 0.189755i −0.995489 0.0948774i $$-0.969754\pi$$
0.995489 0.0948774i $$-0.0302459\pi$$
$$954$$ 1.65685 0.0536426
$$955$$ 0 0
$$956$$ −4.72792 −0.152912
$$957$$ − 12.8284i − 0.414684i
$$958$$ 15.8162i 0.510999i
$$959$$ 8.20101 0.264824
$$960$$ 0 0
$$961$$ 11.0589 0.356738
$$962$$ 24.8284i 0.800501i
$$963$$ 8.00000i 0.257796i
$$964$$ −27.3726 −0.881612
$$965$$ 0 0
$$966$$ 1.85786 0.0597758
$$967$$ 27.8995i 0.897187i 0.893736 + 0.448594i $$0.148075\pi$$
−0.893736 + 0.448594i $$0.851925\pi$$
$$968$$ − 14.2721i − 0.458722i
$$969$$ −1.17157 −0.0376363
$$970$$ 0 0
$$971$$ 17.6569 0.566635 0.283318 0.959026i $$-0.408565\pi$$
0.283318 + 0.959026i $$0.408565\pi$$
$$972$$ − 1.82843i − 0.0586468i
$$973$$ 0.686292i 0.0220015i
$$974$$ −1.17157 −0.0375396
$$975$$ 0 0
$$976$$ 16.9706 0.543214
$$977$$ − 39.5980i − 1.26685i −0.773803 0.633426i $$-0.781648\pi$$
0.773803 0.633426i $$-0.218352\pi$$
$$978$$ 3.27208i 0.104630i
$$979$$ −15.1716 −0.484886
$$980$$ 0 0
$$981$$ 3.17157 0.101261
$$982$$ 9.07107i 0.289469i
$$983$$ 31.9411i 1.01876i 0.860541 + 0.509382i $$0.170126\pi$$
−0.860541 + 0.509382i $$0.829874\pi$$
$$984$$ −11.7574 −0.374811
$$985$$ 0 0
$$986$$ −4.40202 −0.140189
$$987$$ 0.201010i 0.00639822i
$$988$$ − 9.89949i − 0.314945i
$$989$$ −4.48528 −0.142624
$$990$$ 0 0
$$991$$ −61.6569 −1.95859 −0.979297 0.202427i $$-0.935117\pi$$
−0.979297 + 0.202427i $$0.935117\pi$$
$$992$$ 28.6274i 0.908921i
$$993$$ − 7.17157i − 0.227583i
$$994$$ −1.08831 −0.0345192
$$995$$ 0 0
$$996$$ −19.1716 −0.607475
$$997$$ 50.4853i 1.59888i 0.600743 + 0.799442i $$0.294872\pi$$
−0.600743 + 0.799442i $$0.705128\pi$$
$$998$$ 9.85786i 0.312045i
$$999$$ 11.0711 0.350273
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.799.3 4
5.2 odd 4 285.2.a.f.1.1 2
5.3 odd 4 1425.2.a.l.1.2 2
5.4 even 2 inner 1425.2.c.j.799.2 4
15.2 even 4 855.2.a.e.1.2 2
15.8 even 4 4275.2.a.x.1.1 2
20.7 even 4 4560.2.a.bj.1.2 2
95.37 even 4 5415.2.a.p.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.f.1.1 2 5.2 odd 4
855.2.a.e.1.2 2 15.2 even 4
1425.2.a.l.1.2 2 5.3 odd 4
1425.2.c.j.799.2 4 5.4 even 2 inner
1425.2.c.j.799.3 4 1.1 even 1 trivial
4275.2.a.x.1.1 2 15.8 even 4
4560.2.a.bj.1.2 2 20.7 even 4
5415.2.a.p.1.2 2 95.37 even 4