# Properties

 Label 275.2.b.d.199.4 Level $275$ Weight $2$ Character 275.199 Analytic conductor $2.196$ 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] = [275,2,Mod(199,275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(275, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("275.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$275 = 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 275.b (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: $$2.19588605559$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 55) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 199.4 Root $$0.707107 + 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 275.199 Dual form 275.2.b.d.199.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.41421i q^{2} +2.82843i q^{3} -3.82843 q^{4} -6.82843 q^{6} -2.00000i q^{7} -4.41421i q^{8} -5.00000 q^{9} +O(q^{10})$$ $$q+2.41421i q^{2} +2.82843i q^{3} -3.82843 q^{4} -6.82843 q^{6} -2.00000i q^{7} -4.41421i q^{8} -5.00000 q^{9} +1.00000 q^{11} -10.8284i q^{12} +1.17157i q^{13} +4.82843 q^{14} +3.00000 q^{16} +6.82843i q^{17} -12.0711i q^{18} +5.65685 q^{21} +2.41421i q^{22} +2.82843i q^{23} +12.4853 q^{24} -2.82843 q^{26} -5.65685i q^{27} +7.65685i q^{28} +3.65685 q^{29} -1.58579i q^{32} +2.82843i q^{33} -16.4853 q^{34} +19.1421 q^{36} -7.65685i q^{37} -3.31371 q^{39} +6.00000 q^{41} +13.6569i q^{42} +6.00000i q^{43} -3.82843 q^{44} -6.82843 q^{46} +2.82843i q^{47} +8.48528i q^{48} +3.00000 q^{49} -19.3137 q^{51} -4.48528i q^{52} -11.6569i q^{53} +13.6569 q^{54} -8.82843 q^{56} +8.82843i q^{58} -1.65685 q^{59} -9.31371 q^{61} +10.0000i q^{63} +9.82843 q^{64} -6.82843 q^{66} +12.4853i q^{67} -26.1421i q^{68} -8.00000 q^{69} +11.3137 q^{71} +22.0711i q^{72} +1.17157i q^{73} +18.4853 q^{74} -2.00000i q^{77} -8.00000i q^{78} -4.00000 q^{79} +1.00000 q^{81} +14.4853i q^{82} +6.00000i q^{83} -21.6569 q^{84} -14.4853 q^{86} +10.3431i q^{87} -4.41421i q^{88} +13.3137 q^{89} +2.34315 q^{91} -10.8284i q^{92} -6.82843 q^{94} +4.48528 q^{96} +3.65685i q^{97} +7.24264i q^{98} -5.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} - 16 q^{6} - 20 q^{9}+O(q^{10})$$ 4 * q - 4 * q^4 - 16 * q^6 - 20 * q^9 $$4 q - 4 q^{4} - 16 q^{6} - 20 q^{9} + 4 q^{11} + 8 q^{14} + 12 q^{16} + 16 q^{24} - 8 q^{29} - 32 q^{34} + 20 q^{36} + 32 q^{39} + 24 q^{41} - 4 q^{44} - 16 q^{46} + 12 q^{49} - 32 q^{51} + 32 q^{54} - 24 q^{56} + 16 q^{59} + 8 q^{61} + 28 q^{64} - 16 q^{66} - 32 q^{69} + 40 q^{74} - 16 q^{79} + 4 q^{81} - 64 q^{84} - 24 q^{86} + 8 q^{89} + 32 q^{91} - 16 q^{94} - 16 q^{96} - 20 q^{99}+O(q^{100})$$ 4 * q - 4 * q^4 - 16 * q^6 - 20 * q^9 + 4 * q^11 + 8 * q^14 + 12 * q^16 + 16 * q^24 - 8 * q^29 - 32 * q^34 + 20 * q^36 + 32 * q^39 + 24 * q^41 - 4 * q^44 - 16 * q^46 + 12 * q^49 - 32 * q^51 + 32 * q^54 - 24 * q^56 + 16 * q^59 + 8 * q^61 + 28 * q^64 - 16 * q^66 - 32 * q^69 + 40 * q^74 - 16 * q^79 + 4 * q^81 - 64 * q^84 - 24 * q^86 + 8 * q^89 + 32 * q^91 - 16 * q^94 - 16 * q^96 - 20 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/275\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$0.325557\pi$$
−0.521005 + 0.853553i $$0.674443\pi$$
$$3$$ 2.82843i 1.63299i 0.577350 + 0.816497i $$0.304087\pi$$
−0.577350 + 0.816497i $$0.695913\pi$$
$$4$$ −3.82843 −1.91421
$$5$$ 0 0
$$6$$ −6.82843 −2.78769
$$7$$ − 2.00000i − 0.755929i −0.925820 0.377964i $$-0.876624\pi$$
0.925820 0.377964i $$-0.123376\pi$$
$$8$$ − 4.41421i − 1.56066i
$$9$$ −5.00000 −1.66667
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ − 10.8284i − 3.12590i
$$13$$ 1.17157i 0.324936i 0.986714 + 0.162468i $$0.0519454\pi$$
−0.986714 + 0.162468i $$0.948055\pi$$
$$14$$ 4.82843 1.29045
$$15$$ 0 0
$$16$$ 3.00000 0.750000
$$17$$ 6.82843i 1.65614i 0.560627 + 0.828068i $$0.310560\pi$$
−0.560627 + 0.828068i $$0.689440\pi$$
$$18$$ − 12.0711i − 2.84518i
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ 5.65685 1.23443
$$22$$ 2.41421i 0.514712i
$$23$$ 2.82843i 0.589768i 0.955533 + 0.294884i $$0.0952810\pi$$
−0.955533 + 0.294884i $$0.904719\pi$$
$$24$$ 12.4853 2.54855
$$25$$ 0 0
$$26$$ −2.82843 −0.554700
$$27$$ − 5.65685i − 1.08866i
$$28$$ 7.65685i 1.44701i
$$29$$ 3.65685 0.679061 0.339530 0.940595i $$-0.389732\pi$$
0.339530 + 0.940595i $$0.389732\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ − 1.58579i − 0.280330i
$$33$$ 2.82843i 0.492366i
$$34$$ −16.4853 −2.82720
$$35$$ 0 0
$$36$$ 19.1421 3.19036
$$37$$ − 7.65685i − 1.25878i −0.777090 0.629390i $$-0.783305\pi$$
0.777090 0.629390i $$-0.216695\pi$$
$$38$$ 0 0
$$39$$ −3.31371 −0.530618
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 13.6569i 2.10730i
$$43$$ 6.00000i 0.914991i 0.889212 + 0.457496i $$0.151253\pi$$
−0.889212 + 0.457496i $$0.848747\pi$$
$$44$$ −3.82843 −0.577157
$$45$$ 0 0
$$46$$ −6.82843 −1.00680
$$47$$ 2.82843i 0.412568i 0.978492 + 0.206284i $$0.0661372\pi$$
−0.978492 + 0.206284i $$0.933863\pi$$
$$48$$ 8.48528i 1.22474i
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ −19.3137 −2.70446
$$52$$ − 4.48528i − 0.621997i
$$53$$ − 11.6569i − 1.60119i −0.599204 0.800596i $$-0.704516\pi$$
0.599204 0.800596i $$-0.295484\pi$$
$$54$$ 13.6569 1.85846
$$55$$ 0 0
$$56$$ −8.82843 −1.17975
$$57$$ 0 0
$$58$$ 8.82843i 1.15923i
$$59$$ −1.65685 −0.215704 −0.107852 0.994167i $$-0.534397\pi$$
−0.107852 + 0.994167i $$0.534397\pi$$
$$60$$ 0 0
$$61$$ −9.31371 −1.19250 −0.596249 0.802799i $$-0.703343\pi$$
−0.596249 + 0.802799i $$0.703343\pi$$
$$62$$ 0 0
$$63$$ 10.0000i 1.25988i
$$64$$ 9.82843 1.22855
$$65$$ 0 0
$$66$$ −6.82843 −0.840521
$$67$$ 12.4853i 1.52532i 0.646800 + 0.762660i $$0.276107\pi$$
−0.646800 + 0.762660i $$0.723893\pi$$
$$68$$ − 26.1421i − 3.17020i
$$69$$ −8.00000 −0.963087
$$70$$ 0 0
$$71$$ 11.3137 1.34269 0.671345 0.741145i $$-0.265717\pi$$
0.671345 + 0.741145i $$0.265717\pi$$
$$72$$ 22.0711i 2.60110i
$$73$$ 1.17157i 0.137122i 0.997647 + 0.0685611i $$0.0218408\pi$$
−0.997647 + 0.0685611i $$0.978159\pi$$
$$74$$ 18.4853 2.14887
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 2.00000i − 0.227921i
$$78$$ − 8.00000i − 0.905822i
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 14.4853i 1.59963i
$$83$$ 6.00000i 0.658586i 0.944228 + 0.329293i $$0.106810\pi$$
−0.944228 + 0.329293i $$0.893190\pi$$
$$84$$ −21.6569 −2.36296
$$85$$ 0 0
$$86$$ −14.4853 −1.56199
$$87$$ 10.3431i 1.10890i
$$88$$ − 4.41421i − 0.470557i
$$89$$ 13.3137 1.41125 0.705625 0.708585i $$-0.250666\pi$$
0.705625 + 0.708585i $$0.250666\pi$$
$$90$$ 0 0
$$91$$ 2.34315 0.245628
$$92$$ − 10.8284i − 1.12894i
$$93$$ 0 0
$$94$$ −6.82843 −0.704298
$$95$$ 0 0
$$96$$ 4.48528 0.457777
$$97$$ 3.65685i 0.371297i 0.982616 + 0.185649i $$0.0594386\pi$$
−0.982616 + 0.185649i $$0.940561\pi$$
$$98$$ 7.24264i 0.731617i
$$99$$ −5.00000 −0.502519
$$100$$ 0 0
$$101$$ 9.31371 0.926749 0.463374 0.886163i $$-0.346639\pi$$
0.463374 + 0.886163i $$0.346639\pi$$
$$102$$ − 46.6274i − 4.61680i
$$103$$ − 6.82843i − 0.672825i −0.941715 0.336412i $$-0.890786\pi$$
0.941715 0.336412i $$-0.109214\pi$$
$$104$$ 5.17157 0.507114
$$105$$ 0 0
$$106$$ 28.1421 2.73341
$$107$$ 7.65685i 0.740216i 0.928989 + 0.370108i $$0.120679\pi$$
−0.928989 + 0.370108i $$0.879321\pi$$
$$108$$ 21.6569i 2.08393i
$$109$$ 7.65685 0.733394 0.366697 0.930341i $$-0.380489\pi$$
0.366697 + 0.930341i $$0.380489\pi$$
$$110$$ 0 0
$$111$$ 21.6569 2.05558
$$112$$ − 6.00000i − 0.566947i
$$113$$ − 19.6569i − 1.84916i −0.380986 0.924581i $$-0.624416\pi$$
0.380986 0.924581i $$-0.375584\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −14.0000 −1.29987
$$117$$ − 5.85786i − 0.541560i
$$118$$ − 4.00000i − 0.368230i
$$119$$ 13.6569 1.25192
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ − 22.4853i − 2.03572i
$$123$$ 16.9706i 1.53018i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ −24.1421 −2.15075
$$127$$ 4.34315i 0.385392i 0.981259 + 0.192696i $$0.0617231\pi$$
−0.981259 + 0.192696i $$0.938277\pi$$
$$128$$ 20.5563i 1.81694i
$$129$$ −16.9706 −1.49417
$$130$$ 0 0
$$131$$ −11.3137 −0.988483 −0.494242 0.869325i $$-0.664554\pi$$
−0.494242 + 0.869325i $$0.664554\pi$$
$$132$$ − 10.8284i − 0.942494i
$$133$$ 0 0
$$134$$ −30.1421 −2.60388
$$135$$ 0 0
$$136$$ 30.1421 2.58467
$$137$$ − 10.9706i − 0.937278i −0.883390 0.468639i $$-0.844744\pi$$
0.883390 0.468639i $$-0.155256\pi$$
$$138$$ − 19.3137i − 1.64409i
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ −8.00000 −0.673722
$$142$$ 27.3137i 2.29212i
$$143$$ 1.17157i 0.0979718i
$$144$$ −15.0000 −1.25000
$$145$$ 0 0
$$146$$ −2.82843 −0.234082
$$147$$ 8.48528i 0.699854i
$$148$$ 29.3137i 2.40957i
$$149$$ −0.343146 −0.0281116 −0.0140558 0.999901i $$-0.504474\pi$$
−0.0140558 + 0.999901i $$0.504474\pi$$
$$150$$ 0 0
$$151$$ −12.0000 −0.976546 −0.488273 0.872691i $$-0.662373\pi$$
−0.488273 + 0.872691i $$0.662373\pi$$
$$152$$ 0 0
$$153$$ − 34.1421i − 2.76023i
$$154$$ 4.82843 0.389086
$$155$$ 0 0
$$156$$ 12.6863 1.01572
$$157$$ − 14.0000i − 1.11732i −0.829396 0.558661i $$-0.811315\pi$$
0.829396 0.558661i $$-0.188685\pi$$
$$158$$ − 9.65685i − 0.768258i
$$159$$ 32.9706 2.61474
$$160$$ 0 0
$$161$$ 5.65685 0.445823
$$162$$ 2.41421i 0.189679i
$$163$$ − 16.4853i − 1.29123i −0.763664 0.645613i $$-0.776602\pi$$
0.763664 0.645613i $$-0.223398\pi$$
$$164$$ −22.9706 −1.79370
$$165$$ 0 0
$$166$$ −14.4853 −1.12428
$$167$$ − 22.9706i − 1.77752i −0.458377 0.888758i $$-0.651569\pi$$
0.458377 0.888758i $$-0.348431\pi$$
$$168$$ − 24.9706i − 1.92652i
$$169$$ 11.6274 0.894417
$$170$$ 0 0
$$171$$ 0 0
$$172$$ − 22.9706i − 1.75149i
$$173$$ 22.1421i 1.68344i 0.539918 + 0.841718i $$0.318455\pi$$
−0.539918 + 0.841718i $$0.681545\pi$$
$$174$$ −24.9706 −1.89301
$$175$$ 0 0
$$176$$ 3.00000 0.226134
$$177$$ − 4.68629i − 0.352243i
$$178$$ 32.1421i 2.40915i
$$179$$ −9.65685 −0.721787 −0.360894 0.932607i $$-0.617528\pi$$
−0.360894 + 0.932607i $$0.617528\pi$$
$$180$$ 0 0
$$181$$ 21.3137 1.58424 0.792118 0.610368i $$-0.208979\pi$$
0.792118 + 0.610368i $$0.208979\pi$$
$$182$$ 5.65685i 0.419314i
$$183$$ − 26.3431i − 1.94734i
$$184$$ 12.4853 0.920427
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 6.82843i 0.499344i
$$188$$ − 10.8284i − 0.789744i
$$189$$ −11.3137 −0.822951
$$190$$ 0 0
$$191$$ 3.31371 0.239772 0.119886 0.992788i $$-0.461747\pi$$
0.119886 + 0.992788i $$0.461747\pi$$
$$192$$ 27.7990i 2.00622i
$$193$$ 1.17157i 0.0843317i 0.999111 + 0.0421658i $$0.0134258\pi$$
−0.999111 + 0.0421658i $$0.986574\pi$$
$$194$$ −8.82843 −0.633844
$$195$$ 0 0
$$196$$ −11.4853 −0.820377
$$197$$ − 10.8284i − 0.771493i −0.922605 0.385747i $$-0.873944\pi$$
0.922605 0.385747i $$-0.126056\pi$$
$$198$$ − 12.0711i − 0.857853i
$$199$$ −10.3431 −0.733206 −0.366603 0.930377i $$-0.619479\pi$$
−0.366603 + 0.930377i $$0.619479\pi$$
$$200$$ 0 0
$$201$$ −35.3137 −2.49084
$$202$$ 22.4853i 1.58206i
$$203$$ − 7.31371i − 0.513322i
$$204$$ 73.9411 5.17691
$$205$$ 0 0
$$206$$ 16.4853 1.14858
$$207$$ − 14.1421i − 0.982946i
$$208$$ 3.51472i 0.243702i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −16.0000 −1.10149 −0.550743 0.834675i $$-0.685655\pi$$
−0.550743 + 0.834675i $$0.685655\pi$$
$$212$$ 44.6274i 3.06502i
$$213$$ 32.0000i 2.19260i
$$214$$ −18.4853 −1.26363
$$215$$ 0 0
$$216$$ −24.9706 −1.69903
$$217$$ 0 0
$$218$$ 18.4853i 1.25198i
$$219$$ −3.31371 −0.223920
$$220$$ 0 0
$$221$$ −8.00000 −0.538138
$$222$$ 52.2843i 3.50909i
$$223$$ 10.8284i 0.725125i 0.931959 + 0.362563i $$0.118098\pi$$
−0.931959 + 0.362563i $$0.881902\pi$$
$$224$$ −3.17157 −0.211910
$$225$$ 0 0
$$226$$ 47.4558 3.15672
$$227$$ 25.3137i 1.68013i 0.542486 + 0.840065i $$0.317483\pi$$
−0.542486 + 0.840065i $$0.682517\pi$$
$$228$$ 0 0
$$229$$ −1.31371 −0.0868123 −0.0434062 0.999058i $$-0.513821\pi$$
−0.0434062 + 0.999058i $$0.513821\pi$$
$$230$$ 0 0
$$231$$ 5.65685 0.372194
$$232$$ − 16.1421i − 1.05978i
$$233$$ 6.14214i 0.402385i 0.979552 + 0.201192i $$0.0644816\pi$$
−0.979552 + 0.201192i $$0.935518\pi$$
$$234$$ 14.1421 0.924500
$$235$$ 0 0
$$236$$ 6.34315 0.412904
$$237$$ − 11.3137i − 0.734904i
$$238$$ 32.9706i 2.13716i
$$239$$ 23.3137 1.50804 0.754019 0.656852i $$-0.228113\pi$$
0.754019 + 0.656852i $$0.228113\pi$$
$$240$$ 0 0
$$241$$ 6.00000 0.386494 0.193247 0.981150i $$-0.438098\pi$$
0.193247 + 0.981150i $$0.438098\pi$$
$$242$$ 2.41421i 0.155192i
$$243$$ − 14.1421i − 0.907218i
$$244$$ 35.6569 2.28270
$$245$$ 0 0
$$246$$ −40.9706 −2.61219
$$247$$ 0 0
$$248$$ 0 0
$$249$$ −16.9706 −1.07547
$$250$$ 0 0
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ − 38.2843i − 2.41168i
$$253$$ 2.82843i 0.177822i
$$254$$ −10.4853 −0.657905
$$255$$ 0 0
$$256$$ −29.9706 −1.87316
$$257$$ − 9.31371i − 0.580973i −0.956879 0.290487i $$-0.906183\pi$$
0.956879 0.290487i $$-0.0938172\pi$$
$$258$$ − 40.9706i − 2.55072i
$$259$$ −15.3137 −0.951548
$$260$$ 0 0
$$261$$ −18.2843 −1.13177
$$262$$ − 27.3137i − 1.68745i
$$263$$ 10.9706i 0.676474i 0.941061 + 0.338237i $$0.109831\pi$$
−0.941061 + 0.338237i $$0.890169\pi$$
$$264$$ 12.4853 0.768416
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 37.6569i 2.30456i
$$268$$ − 47.7990i − 2.91979i
$$269$$ −17.3137 −1.05564 −0.527818 0.849358i $$-0.676990\pi$$
−0.527818 + 0.849358i $$0.676990\pi$$
$$270$$ 0 0
$$271$$ 7.31371 0.444276 0.222138 0.975015i $$-0.428696\pi$$
0.222138 + 0.975015i $$0.428696\pi$$
$$272$$ 20.4853i 1.24210i
$$273$$ 6.62742i 0.401110i
$$274$$ 26.4853 1.60003
$$275$$ 0 0
$$276$$ 30.6274 1.84355
$$277$$ 6.82843i 0.410280i 0.978733 + 0.205140i $$0.0657650\pi$$
−0.978733 + 0.205140i $$0.934235\pi$$
$$278$$ 9.65685i 0.579180i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 17.3137 1.03285 0.516425 0.856333i $$-0.327263\pi$$
0.516425 + 0.856333i $$0.327263\pi$$
$$282$$ − 19.3137i − 1.15011i
$$283$$ − 32.6274i − 1.93950i −0.244103 0.969749i $$-0.578493\pi$$
0.244103 0.969749i $$-0.421507\pi$$
$$284$$ −43.3137 −2.57020
$$285$$ 0 0
$$286$$ −2.82843 −0.167248
$$287$$ − 12.0000i − 0.708338i
$$288$$ 7.92893i 0.467217i
$$289$$ −29.6274 −1.74279
$$290$$ 0 0
$$291$$ −10.3431 −0.606326
$$292$$ − 4.48528i − 0.262481i
$$293$$ 9.17157i 0.535809i 0.963445 + 0.267905i $$0.0863312\pi$$
−0.963445 + 0.267905i $$0.913669\pi$$
$$294$$ −20.4853 −1.19473
$$295$$ 0 0
$$296$$ −33.7990 −1.96453
$$297$$ − 5.65685i − 0.328244i
$$298$$ − 0.828427i − 0.0479895i
$$299$$ −3.31371 −0.191637
$$300$$ 0 0
$$301$$ 12.0000 0.691669
$$302$$ − 28.9706i − 1.66707i
$$303$$ 26.3431i 1.51337i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 82.4264 4.71200
$$307$$ − 16.3431i − 0.932753i −0.884586 0.466376i $$-0.845559\pi$$
0.884586 0.466376i $$-0.154441\pi$$
$$308$$ 7.65685i 0.436290i
$$309$$ 19.3137 1.09872
$$310$$ 0 0
$$311$$ 4.68629 0.265735 0.132868 0.991134i $$-0.457581\pi$$
0.132868 + 0.991134i $$0.457581\pi$$
$$312$$ 14.6274i 0.828114i
$$313$$ 1.31371i 0.0742552i 0.999311 + 0.0371276i $$0.0118208\pi$$
−0.999311 + 0.0371276i $$0.988179\pi$$
$$314$$ 33.7990 1.90739
$$315$$ 0 0
$$316$$ 15.3137 0.861463
$$317$$ − 1.31371i − 0.0737852i −0.999319 0.0368926i $$-0.988254\pi$$
0.999319 0.0368926i $$-0.0117459\pi$$
$$318$$ 79.5980i 4.46363i
$$319$$ 3.65685 0.204745
$$320$$ 0 0
$$321$$ −21.6569 −1.20877
$$322$$ 13.6569i 0.761067i
$$323$$ 0 0
$$324$$ −3.82843 −0.212690
$$325$$ 0 0
$$326$$ 39.7990 2.20426
$$327$$ 21.6569i 1.19763i
$$328$$ − 26.4853i − 1.46241i
$$329$$ 5.65685 0.311872
$$330$$ 0 0
$$331$$ −7.31371 −0.401998 −0.200999 0.979591i $$-0.564419\pi$$
−0.200999 + 0.979591i $$0.564419\pi$$
$$332$$ − 22.9706i − 1.26067i
$$333$$ 38.2843i 2.09797i
$$334$$ 55.4558 3.03441
$$335$$ 0 0
$$336$$ 16.9706 0.925820
$$337$$ − 20.4853i − 1.11590i −0.829873 0.557952i $$-0.811587\pi$$
0.829873 0.557952i $$-0.188413\pi$$
$$338$$ 28.0711i 1.52686i
$$339$$ 55.5980 3.01967
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ − 20.0000i − 1.07990i
$$344$$ 26.4853 1.42799
$$345$$ 0 0
$$346$$ −53.4558 −2.87380
$$347$$ 10.9706i 0.588931i 0.955662 + 0.294465i $$0.0951416\pi$$
−0.955662 + 0.294465i $$0.904858\pi$$
$$348$$ − 39.5980i − 2.12267i
$$349$$ −26.9706 −1.44370 −0.721851 0.692049i $$-0.756708\pi$$
−0.721851 + 0.692049i $$0.756708\pi$$
$$350$$ 0 0
$$351$$ 6.62742 0.353745
$$352$$ − 1.58579i − 0.0845227i
$$353$$ − 21.3137i − 1.13441i −0.823575 0.567207i $$-0.808024\pi$$
0.823575 0.567207i $$-0.191976\pi$$
$$354$$ 11.3137 0.601317
$$355$$ 0 0
$$356$$ −50.9706 −2.70143
$$357$$ 38.6274i 2.04438i
$$358$$ − 23.3137i − 1.23217i
$$359$$ −0.686292 −0.0362211 −0.0181105 0.999836i $$-0.505765\pi$$
−0.0181105 + 0.999836i $$0.505765\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ 51.4558i 2.70446i
$$363$$ 2.82843i 0.148454i
$$364$$ −8.97056 −0.470185
$$365$$ 0 0
$$366$$ 63.5980 3.32432
$$367$$ − 8.48528i − 0.442928i −0.975169 0.221464i $$-0.928916\pi$$
0.975169 0.221464i $$-0.0710835\pi$$
$$368$$ 8.48528i 0.442326i
$$369$$ −30.0000 −1.56174
$$370$$ 0 0
$$371$$ −23.3137 −1.21039
$$372$$ 0 0
$$373$$ − 35.7990i − 1.85360i −0.375554 0.926801i $$-0.622547\pi$$
0.375554 0.926801i $$-0.377453\pi$$
$$374$$ −16.4853 −0.852434
$$375$$ 0 0
$$376$$ 12.4853 0.643879
$$377$$ 4.28427i 0.220651i
$$378$$ − 27.3137i − 1.40487i
$$379$$ −33.6569 −1.72884 −0.864418 0.502773i $$-0.832313\pi$$
−0.864418 + 0.502773i $$0.832313\pi$$
$$380$$ 0 0
$$381$$ −12.2843 −0.629342
$$382$$ 8.00000i 0.409316i
$$383$$ 5.85786i 0.299323i 0.988737 + 0.149661i $$0.0478184\pi$$
−0.988737 + 0.149661i $$0.952182\pi$$
$$384$$ −58.1421 −2.96705
$$385$$ 0 0
$$386$$ −2.82843 −0.143963
$$387$$ − 30.0000i − 1.52499i
$$388$$ − 14.0000i − 0.710742i
$$389$$ −20.6274 −1.04585 −0.522926 0.852378i $$-0.675160\pi$$
−0.522926 + 0.852378i $$0.675160\pi$$
$$390$$ 0 0
$$391$$ −19.3137 −0.976736
$$392$$ − 13.2426i − 0.668854i
$$393$$ − 32.0000i − 1.61419i
$$394$$ 26.1421 1.31702
$$395$$ 0 0
$$396$$ 19.1421 0.961929
$$397$$ − 9.31371i − 0.467442i −0.972304 0.233721i $$-0.924910\pi$$
0.972304 0.233721i $$-0.0750902\pi$$
$$398$$ − 24.9706i − 1.25166i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −5.31371 −0.265354 −0.132677 0.991159i $$-0.542357\pi$$
−0.132677 + 0.991159i $$0.542357\pi$$
$$402$$ − 85.2548i − 4.25212i
$$403$$ 0 0
$$404$$ −35.6569 −1.77399
$$405$$ 0 0
$$406$$ 17.6569 0.876295
$$407$$ − 7.65685i − 0.379536i
$$408$$ 85.2548i 4.22074i
$$409$$ −1.02944 −0.0509024 −0.0254512 0.999676i $$-0.508102\pi$$
−0.0254512 + 0.999676i $$0.508102\pi$$
$$410$$ 0 0
$$411$$ 31.0294 1.53057
$$412$$ 26.1421i 1.28793i
$$413$$ 3.31371i 0.163057i
$$414$$ 34.1421 1.67799
$$415$$ 0 0
$$416$$ 1.85786 0.0910893
$$417$$ 11.3137i 0.554035i
$$418$$ 0 0
$$419$$ 25.6569 1.25342 0.626710 0.779253i $$-0.284401\pi$$
0.626710 + 0.779253i $$0.284401\pi$$
$$420$$ 0 0
$$421$$ −6.00000 −0.292422 −0.146211 0.989253i $$-0.546708\pi$$
−0.146211 + 0.989253i $$0.546708\pi$$
$$422$$ − 38.6274i − 1.88035i
$$423$$ − 14.1421i − 0.687614i
$$424$$ −51.4558 −2.49892
$$425$$ 0 0
$$426$$ −77.2548 −3.74301
$$427$$ 18.6274i 0.901444i
$$428$$ − 29.3137i − 1.41693i
$$429$$ −3.31371 −0.159987
$$430$$ 0 0
$$431$$ −11.3137 −0.544962 −0.272481 0.962161i $$-0.587844\pi$$
−0.272481 + 0.962161i $$0.587844\pi$$
$$432$$ − 16.9706i − 0.816497i
$$433$$ 7.65685i 0.367965i 0.982930 + 0.183982i $$0.0588990\pi$$
−0.982930 + 0.183982i $$0.941101\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −29.3137 −1.40387
$$437$$ 0 0
$$438$$ − 8.00000i − 0.382255i
$$439$$ 16.0000 0.763638 0.381819 0.924237i $$-0.375298\pi$$
0.381819 + 0.924237i $$0.375298\pi$$
$$440$$ 0 0
$$441$$ −15.0000 −0.714286
$$442$$ − 19.3137i − 0.918659i
$$443$$ 26.8284i 1.27466i 0.770592 + 0.637329i $$0.219961\pi$$
−0.770592 + 0.637329i $$0.780039\pi$$
$$444$$ −82.9117 −3.93481
$$445$$ 0 0
$$446$$ −26.1421 −1.23787
$$447$$ − 0.970563i − 0.0459060i
$$448$$ − 19.6569i − 0.928699i
$$449$$ −28.6274 −1.35101 −0.675506 0.737355i $$-0.736075\pi$$
−0.675506 + 0.737355i $$0.736075\pi$$
$$450$$ 0 0
$$451$$ 6.00000 0.282529
$$452$$ 75.2548i 3.53969i
$$453$$ − 33.9411i − 1.59469i
$$454$$ −61.1127 −2.86816
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 0.485281i 0.0227005i 0.999936 + 0.0113503i $$0.00361298\pi$$
−0.999936 + 0.0113503i $$0.996387\pi$$
$$458$$ − 3.17157i − 0.148198i
$$459$$ 38.6274 1.80297
$$460$$ 0 0
$$461$$ 12.6274 0.588117 0.294059 0.955787i $$-0.404994\pi$$
0.294059 + 0.955787i $$0.404994\pi$$
$$462$$ 13.6569i 0.635374i
$$463$$ 6.14214i 0.285449i 0.989762 + 0.142725i $$0.0455863\pi$$
−0.989762 + 0.142725i $$0.954414\pi$$
$$464$$ 10.9706 0.509296
$$465$$ 0 0
$$466$$ −14.8284 −0.686914
$$467$$ − 14.8284i − 0.686178i −0.939303 0.343089i $$-0.888527\pi$$
0.939303 0.343089i $$-0.111473\pi$$
$$468$$ 22.4264i 1.03666i
$$469$$ 24.9706 1.15303
$$470$$ 0 0
$$471$$ 39.5980 1.82458
$$472$$ 7.31371i 0.336641i
$$473$$ 6.00000i 0.275880i
$$474$$ 27.3137 1.25456
$$475$$ 0 0
$$476$$ −52.2843 −2.39645
$$477$$ 58.2843i 2.66865i
$$478$$ 56.2843i 2.57438i
$$479$$ 36.0000 1.64488 0.822441 0.568850i $$-0.192612\pi$$
0.822441 + 0.568850i $$0.192612\pi$$
$$480$$ 0 0
$$481$$ 8.97056 0.409022
$$482$$ 14.4853i 0.659786i
$$483$$ 16.0000i 0.728025i
$$484$$ −3.82843 −0.174019
$$485$$ 0 0
$$486$$ 34.1421 1.54872
$$487$$ − 24.4853i − 1.10953i −0.832006 0.554767i $$-0.812807\pi$$
0.832006 0.554767i $$-0.187193\pi$$
$$488$$ 41.1127i 1.86108i
$$489$$ 46.6274 2.10856
$$490$$ 0 0
$$491$$ −0.686292 −0.0309719 −0.0154860 0.999880i $$-0.504930\pi$$
−0.0154860 + 0.999880i $$0.504930\pi$$
$$492$$ − 64.9706i − 2.92910i
$$493$$ 24.9706i 1.12462i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 22.6274i − 1.01498i
$$498$$ − 40.9706i − 1.83593i
$$499$$ −9.65685 −0.432300 −0.216150 0.976360i $$-0.569350\pi$$
−0.216150 + 0.976360i $$0.569350\pi$$
$$500$$ 0 0
$$501$$ 64.9706 2.90267
$$502$$ 28.9706i 1.29302i
$$503$$ − 16.6274i − 0.741380i −0.928757 0.370690i $$-0.879121\pi$$
0.928757 0.370690i $$-0.120879\pi$$
$$504$$ 44.1421 1.96625
$$505$$ 0 0
$$506$$ −6.82843 −0.303561
$$507$$ 32.8873i 1.46058i
$$508$$ − 16.6274i − 0.737722i
$$509$$ 13.3137 0.590120 0.295060 0.955479i $$-0.404660\pi$$
0.295060 + 0.955479i $$0.404660\pi$$
$$510$$ 0 0
$$511$$ 2.34315 0.103655
$$512$$ − 31.2426i − 1.38074i
$$513$$ 0 0
$$514$$ 22.4853 0.991783
$$515$$ 0 0
$$516$$ 64.9706 2.86017
$$517$$ 2.82843i 0.124394i
$$518$$ − 36.9706i − 1.62439i
$$519$$ −62.6274 −2.74904
$$520$$ 0 0
$$521$$ 25.3137 1.10901 0.554507 0.832179i $$-0.312907\pi$$
0.554507 + 0.832179i $$0.312907\pi$$
$$522$$ − 44.1421i − 1.93205i
$$523$$ 41.5980i 1.81895i 0.415756 + 0.909476i $$0.363517\pi$$
−0.415756 + 0.909476i $$0.636483\pi$$
$$524$$ 43.3137 1.89217
$$525$$ 0 0
$$526$$ −26.4853 −1.15481
$$527$$ 0 0
$$528$$ 8.48528i 0.369274i
$$529$$ 15.0000 0.652174
$$530$$ 0 0
$$531$$ 8.28427 0.359507
$$532$$ 0 0
$$533$$ 7.02944i 0.304479i
$$534$$ −90.9117 −3.93413
$$535$$ 0 0
$$536$$ 55.1127 2.38051
$$537$$ − 27.3137i − 1.17867i
$$538$$ − 41.7990i − 1.80208i
$$539$$ 3.00000 0.129219
$$540$$ 0 0
$$541$$ 6.00000 0.257960 0.128980 0.991647i $$-0.458830\pi$$
0.128980 + 0.991647i $$0.458830\pi$$
$$542$$ 17.6569i 0.758427i
$$543$$ 60.2843i 2.58705i
$$544$$ 10.8284 0.464265
$$545$$ 0 0
$$546$$ −16.0000 −0.684737
$$547$$ − 34.0000i − 1.45374i −0.686778 0.726868i $$-0.740975\pi$$
0.686778 0.726868i $$-0.259025\pi$$
$$548$$ 42.0000i 1.79415i
$$549$$ 46.5685 1.98750
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 35.3137i 1.50305i
$$553$$ 8.00000i 0.340195i
$$554$$ −16.4853 −0.700392
$$555$$ 0 0
$$556$$ −15.3137 −0.649446
$$557$$ 9.85786i 0.417691i 0.977949 + 0.208846i $$0.0669706\pi$$
−0.977949 + 0.208846i $$0.933029\pi$$
$$558$$ 0 0
$$559$$ −7.02944 −0.297314
$$560$$ 0 0
$$561$$ −19.3137 −0.815425
$$562$$ 41.7990i 1.76318i
$$563$$ − 0.343146i − 0.0144619i −0.999974 0.00723093i $$-0.997698\pi$$
0.999974 0.00723093i $$-0.00230170\pi$$
$$564$$ 30.6274 1.28965
$$565$$ 0 0
$$566$$ 78.7696 3.31093
$$567$$ − 2.00000i − 0.0839921i
$$568$$ − 49.9411i − 2.09548i
$$569$$ −31.6569 −1.32712 −0.663562 0.748121i $$-0.730956\pi$$
−0.663562 + 0.748121i $$0.730956\pi$$
$$570$$ 0 0
$$571$$ −21.9411 −0.918208 −0.459104 0.888383i $$-0.651829\pi$$
−0.459104 + 0.888383i $$0.651829\pi$$
$$572$$ − 4.48528i − 0.187539i
$$573$$ 9.37258i 0.391545i
$$574$$ 28.9706 1.20921
$$575$$ 0 0
$$576$$ −49.1421 −2.04759
$$577$$ − 26.9706i − 1.12280i −0.827545 0.561400i $$-0.810263\pi$$
0.827545 0.561400i $$-0.189737\pi$$
$$578$$ − 71.5269i − 2.97513i
$$579$$ −3.31371 −0.137713
$$580$$ 0 0
$$581$$ 12.0000 0.497844
$$582$$ − 24.9706i − 1.03506i
$$583$$ − 11.6569i − 0.482778i
$$584$$ 5.17157 0.214001
$$585$$ 0 0
$$586$$ −22.1421 −0.914683
$$587$$ − 2.14214i − 0.0884154i −0.999022 0.0442077i $$-0.985924\pi$$
0.999022 0.0442077i $$-0.0140763\pi$$
$$588$$ − 32.4853i − 1.33967i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 30.6274 1.25984
$$592$$ − 22.9706i − 0.944084i
$$593$$ − 3.51472i − 0.144332i −0.997393 0.0721661i $$-0.977009\pi$$
0.997393 0.0721661i $$-0.0229912\pi$$
$$594$$ 13.6569 0.560348
$$595$$ 0 0
$$596$$ 1.31371 0.0538116
$$597$$ − 29.2548i − 1.19732i
$$598$$ − 8.00000i − 0.327144i
$$599$$ 5.65685 0.231133 0.115566 0.993300i $$-0.463132\pi$$
0.115566 + 0.993300i $$0.463132\pi$$
$$600$$ 0 0
$$601$$ 23.9411 0.976579 0.488289 0.872682i $$-0.337621\pi$$
0.488289 + 0.872682i $$0.337621\pi$$
$$602$$ 28.9706i 1.18075i
$$603$$ − 62.4264i − 2.54220i
$$604$$ 45.9411 1.86932
$$605$$ 0 0
$$606$$ −63.5980 −2.58349
$$607$$ 38.2843i 1.55391i 0.629556 + 0.776955i $$0.283237\pi$$
−0.629556 + 0.776955i $$0.716763\pi$$
$$608$$ 0 0
$$609$$ 20.6863 0.838251
$$610$$ 0 0
$$611$$ −3.31371 −0.134058
$$612$$ 130.711i 5.28367i
$$613$$ 25.4558i 1.02815i 0.857745 + 0.514076i $$0.171865\pi$$
−0.857745 + 0.514076i $$0.828135\pi$$
$$614$$ 39.4558 1.59231
$$615$$ 0 0
$$616$$ −8.82843 −0.355707
$$617$$ 0.343146i 0.0138145i 0.999976 + 0.00690726i $$0.00219867\pi$$
−0.999976 + 0.00690726i $$0.997801\pi$$
$$618$$ 46.6274i 1.87563i
$$619$$ 14.3431 0.576500 0.288250 0.957555i $$-0.406927\pi$$
0.288250 + 0.957555i $$0.406927\pi$$
$$620$$ 0 0
$$621$$ 16.0000 0.642058
$$622$$ 11.3137i 0.453638i
$$623$$ − 26.6274i − 1.06680i
$$624$$ −9.94113 −0.397964
$$625$$ 0 0
$$626$$ −3.17157 −0.126762
$$627$$ 0 0
$$628$$ 53.5980i 2.13879i
$$629$$ 52.2843 2.08471
$$630$$ 0 0
$$631$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$632$$ 17.6569i 0.702352i
$$633$$ − 45.2548i − 1.79872i
$$634$$ 3.17157 0.125959
$$635$$ 0 0
$$636$$ −126.225 −5.00516
$$637$$ 3.51472i 0.139258i
$$638$$ 8.82843i 0.349521i
$$639$$ −56.5685 −2.23782
$$640$$ 0 0
$$641$$ 30.0000 1.18493 0.592464 0.805597i $$-0.298155\pi$$
0.592464 + 0.805597i $$0.298155\pi$$
$$642$$ − 52.2843i − 2.06350i
$$643$$ 1.45584i 0.0574129i 0.999588 + 0.0287064i $$0.00913880\pi$$
−0.999588 + 0.0287064i $$0.990861\pi$$
$$644$$ −21.6569 −0.853400
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 27.1127i 1.06591i 0.846144 + 0.532955i $$0.178919\pi$$
−0.846144 + 0.532955i $$0.821081\pi$$
$$648$$ − 4.41421i − 0.173407i
$$649$$ −1.65685 −0.0650372
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 63.1127i 2.47168i
$$653$$ − 11.6569i − 0.456168i −0.973641 0.228084i $$-0.926754\pi$$
0.973641 0.228084i $$-0.0732461\pi$$
$$654$$ −52.2843 −2.04448
$$655$$ 0 0
$$656$$ 18.0000 0.702782
$$657$$ − 5.85786i − 0.228537i
$$658$$ 13.6569i 0.532400i
$$659$$ −45.9411 −1.78961 −0.894806 0.446455i $$-0.852686\pi$$
−0.894806 + 0.446455i $$0.852686\pi$$
$$660$$ 0 0
$$661$$ 44.6274 1.73581 0.867903 0.496734i $$-0.165468\pi$$
0.867903 + 0.496734i $$0.165468\pi$$
$$662$$ − 17.6569i − 0.686253i
$$663$$ − 22.6274i − 0.878776i
$$664$$ 26.4853 1.02783
$$665$$ 0 0
$$666$$ −92.4264 −3.58145
$$667$$ 10.3431i 0.400488i
$$668$$ 87.9411i 3.40254i
$$669$$ −30.6274 −1.18412
$$670$$ 0 0
$$671$$ −9.31371 −0.359552
$$672$$ − 8.97056i − 0.346047i
$$673$$ 12.4853i 0.481272i 0.970615 + 0.240636i $$0.0773560\pi$$
−0.970615 + 0.240636i $$0.922644\pi$$
$$674$$ 49.4558 1.90497
$$675$$ 0 0
$$676$$ −44.5147 −1.71210
$$677$$ 22.8284i 0.877368i 0.898641 + 0.438684i $$0.144555\pi$$
−0.898641 + 0.438684i $$0.855445\pi$$
$$678$$ 134.225i 5.15490i
$$679$$ 7.31371 0.280674
$$680$$ 0 0
$$681$$ −71.5980 −2.74364
$$682$$ 0 0
$$683$$ 7.79899i 0.298420i 0.988806 + 0.149210i $$0.0476731\pi$$
−0.988806 + 0.149210i $$0.952327\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 48.2843 1.84350
$$687$$ − 3.71573i − 0.141764i
$$688$$ 18.0000i 0.686244i
$$689$$ 13.6569 0.520285
$$690$$ 0 0
$$691$$ −39.3137 −1.49556 −0.747782 0.663944i $$-0.768881\pi$$
−0.747782 + 0.663944i $$0.768881\pi$$
$$692$$ − 84.7696i − 3.22245i
$$693$$ 10.0000i 0.379869i
$$694$$ −26.4853 −1.00537
$$695$$ 0 0
$$696$$ 45.6569 1.73062
$$697$$ 40.9706i 1.55187i
$$698$$ − 65.1127i − 2.46455i
$$699$$ −17.3726 −0.657091
$$700$$ 0 0
$$701$$ −12.6274 −0.476931 −0.238465 0.971151i $$-0.576644\pi$$
−0.238465 + 0.971151i $$0.576644\pi$$
$$702$$ 16.0000i 0.603881i
$$703$$ 0 0
$$704$$ 9.82843 0.370423
$$705$$ 0 0
$$706$$ 51.4558 1.93657
$$707$$ − 18.6274i − 0.700556i
$$708$$ 17.9411i 0.674269i
$$709$$ −24.6274 −0.924902 −0.462451 0.886645i $$-0.653030\pi$$
−0.462451 + 0.886645i $$0.653030\pi$$
$$710$$ 0 0
$$711$$ 20.0000 0.750059
$$712$$ − 58.7696i − 2.20248i
$$713$$ 0 0
$$714$$ −93.2548 −3.48997
$$715$$ 0 0
$$716$$ 36.9706 1.38165
$$717$$ 65.9411i 2.46262i
$$718$$ − 1.65685i − 0.0618333i
$$719$$ −18.3431 −0.684084 −0.342042 0.939685i $$-0.611118\pi$$
−0.342042 + 0.939685i $$0.611118\pi$$
$$720$$ 0 0
$$721$$ −13.6569 −0.508608
$$722$$ − 45.8701i − 1.70711i
$$723$$ 16.9706i 0.631142i
$$724$$ −81.5980 −3.03257
$$725$$ 0 0
$$726$$ −6.82843 −0.253427
$$727$$ − 19.5147i − 0.723761i −0.932225 0.361880i $$-0.882135\pi$$
0.932225 0.361880i $$-0.117865\pi$$
$$728$$ − 10.3431i − 0.383342i
$$729$$ 43.0000 1.59259
$$730$$ 0 0
$$731$$ −40.9706 −1.51535
$$732$$ 100.853i 3.72763i
$$733$$ 17.4558i 0.644746i 0.946613 + 0.322373i $$0.104481\pi$$
−0.946613 + 0.322373i $$0.895519\pi$$
$$734$$ 20.4853 0.756126
$$735$$ 0 0
$$736$$ 4.48528 0.165330
$$737$$ 12.4853i 0.459901i
$$738$$ − 72.4264i − 2.66605i
$$739$$ −29.9411 −1.10140 −0.550701 0.834703i $$-0.685640\pi$$
−0.550701 + 0.834703i $$0.685640\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ − 56.2843i − 2.06626i
$$743$$ 49.5980i 1.81957i 0.415076 + 0.909787i $$0.363755\pi$$
−0.415076 + 0.909787i $$0.636245\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 86.4264 3.16430
$$747$$ − 30.0000i − 1.09764i
$$748$$ − 26.1421i − 0.955851i
$$749$$ 15.3137 0.559551
$$750$$ 0 0
$$751$$ −16.0000 −0.583848 −0.291924 0.956441i $$-0.594295\pi$$
−0.291924 + 0.956441i $$0.594295\pi$$
$$752$$ 8.48528i 0.309426i
$$753$$ 33.9411i 1.23688i
$$754$$ −10.3431 −0.376675
$$755$$ 0 0
$$756$$ 43.3137 1.57530
$$757$$ 13.3137i 0.483895i 0.970289 + 0.241947i $$0.0777862\pi$$
−0.970289 + 0.241947i $$0.922214\pi$$
$$758$$ − 81.2548i − 2.95131i
$$759$$ −8.00000 −0.290382
$$760$$ 0 0
$$761$$ −30.0000 −1.08750 −0.543750 0.839248i $$-0.682996\pi$$
−0.543750 + 0.839248i $$0.682996\pi$$
$$762$$ − 29.6569i − 1.07435i
$$763$$ − 15.3137i − 0.554393i
$$764$$ −12.6863 −0.458974
$$765$$ 0 0
$$766$$ −14.1421 −0.510976
$$767$$ − 1.94113i − 0.0700900i
$$768$$ − 84.7696i − 3.05886i
$$769$$ 18.9706 0.684096 0.342048 0.939682i $$-0.388879\pi$$
0.342048 + 0.939682i $$0.388879\pi$$
$$770$$ 0 0
$$771$$ 26.3431 0.948725
$$772$$ − 4.48528i − 0.161429i
$$773$$ − 26.2843i − 0.945380i −0.881229 0.472690i $$-0.843283\pi$$
0.881229 0.472690i $$-0.156717\pi$$
$$774$$ 72.4264 2.60331
$$775$$ 0 0
$$776$$ 16.1421 0.579469
$$777$$ − 43.3137i − 1.55387i
$$778$$ − 49.7990i − 1.78538i
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 11.3137 0.404836
$$782$$ − 46.6274i − 1.66739i
$$783$$ − 20.6863i − 0.739268i
$$784$$ 9.00000 0.321429
$$785$$ 0 0
$$786$$ 77.2548 2.75559
$$787$$ − 14.9706i − 0.533643i −0.963746 0.266821i $$-0.914027\pi$$
0.963746 0.266821i $$-0.0859734\pi$$
$$788$$ 41.4558i 1.47680i
$$789$$ −31.0294 −1.10468
$$790$$ 0 0
$$791$$ −39.3137 −1.39783
$$792$$ 22.0711i 0.784261i
$$793$$ − 10.9117i − 0.387485i
$$794$$ 22.4853 0.797973
$$795$$ 0 0
$$796$$ 39.5980 1.40351
$$797$$ 32.6274i 1.15572i 0.816135 + 0.577861i $$0.196113\pi$$
−0.816135 + 0.577861i $$0.803887\pi$$
$$798$$ 0 0
$$799$$ −19.3137 −0.683270
$$800$$ 0 0
$$801$$ −66.5685 −2.35208
$$802$$ − 12.8284i − 0.452988i
$$803$$ 1.17157i 0.0413439i
$$804$$ 135.196 4.76799
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 48.9706i − 1.72385i
$$808$$ − 41.1127i − 1.44634i
$$809$$ 10.9706 0.385704 0.192852 0.981228i $$-0.438226\pi$$
0.192852 + 0.981228i $$0.438226\pi$$
$$810$$ 0 0
$$811$$ 53.9411 1.89413 0.947065 0.321043i $$-0.104033\pi$$
0.947065 + 0.321043i $$0.104033\pi$$
$$812$$ 28.0000i 0.982607i
$$813$$ 20.6863i 0.725500i
$$814$$ 18.4853 0.647909
$$815$$ 0 0
$$816$$ −57.9411 −2.02835
$$817$$ 0 0
$$818$$ − 2.48528i − 0.0868958i
$$819$$ −11.7157 −0.409381
$$820$$ 0 0
$$821$$ −41.3137 −1.44186 −0.720929 0.693009i $$-0.756285\pi$$
−0.720929 + 0.693009i $$0.756285\pi$$
$$822$$ 74.9117i 2.61285i
$$823$$ − 19.5147i − 0.680240i −0.940382 0.340120i $$-0.889532\pi$$
0.940382 0.340120i $$-0.110468\pi$$
$$824$$ −30.1421 −1.05005
$$825$$ 0 0
$$826$$ −8.00000 −0.278356
$$827$$ 22.2843i 0.774900i 0.921891 + 0.387450i $$0.126644\pi$$
−0.921891 + 0.387450i $$0.873356\pi$$
$$828$$ 54.1421i 1.88157i
$$829$$ −18.0000 −0.625166 −0.312583 0.949890i $$-0.601194\pi$$
−0.312583 + 0.949890i $$0.601194\pi$$
$$830$$ 0 0
$$831$$ −19.3137 −0.669985
$$832$$ 11.5147i 0.399201i
$$833$$ 20.4853i 0.709773i
$$834$$ −27.3137 −0.945796
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 61.9411i 2.13972i
$$839$$ −26.3431 −0.909466 −0.454733 0.890628i $$-0.650265\pi$$
−0.454733 + 0.890628i $$0.650265\pi$$
$$840$$ 0 0
$$841$$ −15.6274 −0.538876
$$842$$ − 14.4853i − 0.499196i
$$843$$ 48.9706i 1.68664i
$$844$$ 61.2548 2.10848
$$845$$ 0 0
$$846$$ 34.1421 1.17383
$$847$$ − 2.00000i − 0.0687208i
$$848$$ − 34.9706i − 1.20089i
$$849$$ 92.2843 3.16719
$$850$$ 0 0
$$851$$ 21.6569 0.742387
$$852$$ − 122.510i − 4.19711i
$$853$$ 15.5147i 0.531214i 0.964081 + 0.265607i $$0.0855723\pi$$
−0.964081 + 0.265607i $$0.914428\pi$$
$$854$$ −44.9706 −1.53886
$$855$$ 0 0
$$856$$ 33.7990 1.15523
$$857$$ 24.7696i 0.846112i 0.906104 + 0.423056i $$0.139043\pi$$
−0.906104 + 0.423056i $$0.860957\pi$$
$$858$$ − 8.00000i − 0.273115i
$$859$$ −24.2843 −0.828569 −0.414284 0.910148i $$-0.635968\pi$$
−0.414284 + 0.910148i $$0.635968\pi$$
$$860$$ 0 0
$$861$$ 33.9411 1.15671
$$862$$ − 27.3137i − 0.930309i
$$863$$ 9.17157i 0.312204i 0.987741 + 0.156102i $$0.0498929\pi$$
−0.987741 + 0.156102i $$0.950107\pi$$
$$864$$ −8.97056 −0.305185
$$865$$ 0 0
$$866$$ −18.4853 −0.628155
$$867$$ − 83.7990i − 2.84596i
$$868$$ 0 0
$$869$$ −4.00000 −0.135691
$$870$$ 0 0
$$871$$ −14.6274 −0.495631
$$872$$ − 33.7990i − 1.14458i
$$873$$ − 18.2843i − 0.618829i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 12.6863 0.428630
$$877$$ − 49.4558i − 1.67001i −0.550246 0.835003i $$-0.685466\pi$$
0.550246 0.835003i $$-0.314534\pi$$
$$878$$ 38.6274i 1.30361i
$$879$$ −25.9411 −0.874972
$$880$$ 0 0
$$881$$ −7.37258 −0.248389 −0.124194 0.992258i $$-0.539635\pi$$
−0.124194 + 0.992258i $$0.539635\pi$$
$$882$$ − 36.2132i − 1.21936i
$$883$$ − 37.1716i − 1.25092i −0.780255 0.625462i $$-0.784911\pi$$
0.780255 0.625462i $$-0.215089\pi$$
$$884$$ 30.6274 1.03011
$$885$$ 0 0
$$886$$ −64.7696 −2.17598
$$887$$ 38.2843i 1.28546i 0.766093 + 0.642730i $$0.222198\pi$$
−0.766093 + 0.642730i $$0.777802\pi$$
$$888$$ − 95.5980i − 3.20806i
$$889$$ 8.68629 0.291329
$$890$$ 0 0
$$891$$ 1.00000 0.0335013
$$892$$ − 41.4558i − 1.38804i
$$893$$ 0 0
$$894$$ 2.34315 0.0783665
$$895$$ 0 0
$$896$$ 41.1127 1.37348
$$897$$ − 9.37258i − 0.312941i
$$898$$ − 69.1127i − 2.30632i
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 79.5980 2.65179
$$902$$ 14.4853i 0.482307i
$$903$$ 33.9411i 1.12949i
$$904$$ −86.7696 −2.88591
$$905$$ 0 0
$$906$$ 81.9411 2.72231
$$907$$ − 27.5147i − 0.913611i −0.889567 0.456806i $$-0.848993\pi$$
0.889567 0.456806i $$-0.151007\pi$$
$$908$$ − 96.9117i − 3.21613i
$$909$$ −46.5685 −1.54458
$$910$$ 0 0
$$911$$ −9.94113 −0.329364 −0.164682 0.986347i $$-0.552660\pi$$
−0.164682 + 0.986347i $$0.552660\pi$$
$$912$$ 0 0
$$913$$ 6.00000i 0.198571i
$$914$$ −1.17157 −0.0387522
$$915$$ 0 0
$$916$$ 5.02944 0.166177
$$917$$ 22.6274i 0.747223i
$$918$$ 93.2548i 3.07787i
$$919$$ 32.0000 1.05558 0.527791 0.849374i $$-0.323020\pi$$
0.527791 + 0.849374i $$0.323020\pi$$
$$920$$ 0 0
$$921$$ 46.2254 1.52318
$$922$$ 30.4853i 1.00398i
$$923$$ 13.2548i 0.436288i
$$924$$ −21.6569 −0.712458
$$925$$ 0 0
$$926$$ −14.8284 −0.487292
$$927$$ 34.1421i 1.12137i
$$928$$ − 5.79899i − 0.190361i
$$929$$ −5.31371 −0.174337 −0.0871686 0.996194i $$-0.527782\pi$$
−0.0871686 + 0.996194i $$0.527782\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ − 23.5147i − 0.770250i
$$933$$ 13.2548i 0.433944i
$$934$$ 35.7990 1.17138
$$935$$ 0 0
$$936$$ −25.8579 −0.845191
$$937$$ − 1.45584i − 0.0475604i −0.999717 0.0237802i $$-0.992430\pi$$
0.999717 0.0237802i $$-0.00757018\pi$$
$$938$$ 60.2843i 1.96835i
$$939$$ −3.71573 −0.121258
$$940$$ 0 0
$$941$$ −6.68629 −0.217967 −0.108983 0.994044i $$-0.534760\pi$$
−0.108983 + 0.994044i $$0.534760\pi$$
$$942$$ 95.5980i 3.11475i
$$943$$ 16.9706i 0.552638i
$$944$$ −4.97056 −0.161778
$$945$$ 0 0
$$946$$ −14.4853 −0.470957
$$947$$ 41.1716i 1.33790i 0.743309 + 0.668948i $$0.233255\pi$$
−0.743309 + 0.668948i $$0.766745\pi$$
$$948$$ 43.3137i 1.40676i
$$949$$ −1.37258 −0.0445559
$$950$$ 0 0
$$951$$ 3.71573 0.120491
$$952$$ − 60.2843i − 1.95382i
$$953$$ − 53.1716i − 1.72240i −0.508269 0.861198i $$-0.669715\pi$$
0.508269 0.861198i $$-0.330285\pi$$
$$954$$ −140.711 −4.55568
$$955$$ 0 0
$$956$$ −89.2548 −2.88671
$$957$$ 10.3431i 0.334346i
$$958$$ 86.9117i 2.80799i
$$959$$ −21.9411 −0.708516
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 21.6569i 0.698245i
$$963$$ − 38.2843i − 1.23369i
$$964$$ −22.9706 −0.739832
$$965$$ 0 0
$$966$$ −38.6274 −1.24282
$$967$$ − 14.9706i − 0.481421i −0.970597 0.240710i $$-0.922620\pi$$
0.970597 0.240710i $$-0.0773804\pi$$
$$968$$ − 4.41421i − 0.141878i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −8.68629 −0.278756 −0.139378 0.990239i $$-0.544510\pi$$
−0.139378 + 0.990239i $$0.544510\pi$$
$$972$$ 54.1421i 1.73661i
$$973$$ − 8.00000i − 0.256468i
$$974$$ 59.1127 1.89409
$$975$$ 0 0
$$976$$ −27.9411 −0.894374
$$977$$ 32.3431i 1.03475i 0.855759 + 0.517374i $$0.173091\pi$$
−0.855759 + 0.517374i $$0.826909\pi$$
$$978$$ 112.569i 3.59955i
$$979$$ 13.3137 0.425508
$$980$$ 0 0
$$981$$ −38.2843 −1.22232
$$982$$ − 1.65685i − 0.0528723i
$$983$$ 21.8579i 0.697158i 0.937279 + 0.348579i $$0.113336\pi$$
−0.937279 + 0.348579i $$0.886664\pi$$
$$984$$ 74.9117 2.38810
$$985$$ 0 0
$$986$$ −60.2843 −1.91984
$$987$$ 16.0000i 0.509286i
$$988$$ 0 0
$$989$$ −16.9706 −0.539633
$$990$$ 0 0
$$991$$ −57.9411 −1.84056 −0.920280 0.391260i $$-0.872039\pi$$
−0.920280 + 0.391260i $$0.872039\pi$$
$$992$$ 0 0
$$993$$ − 20.6863i − 0.656460i
$$994$$ 54.6274 1.73268
$$995$$ 0 0
$$996$$ 64.9706 2.05867
$$997$$ − 41.4558i − 1.31292i −0.754361 0.656460i $$-0.772053\pi$$
0.754361 0.656460i $$-0.227947\pi$$
$$998$$ − 23.3137i − 0.737983i
$$999$$ −43.3137 −1.37039
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 275.2.b.d.199.4 4
3.2 odd 2 2475.2.c.l.199.1 4
4.3 odd 2 4400.2.b.q.4049.2 4
5.2 odd 4 275.2.a.c.1.1 2
5.3 odd 4 55.2.a.b.1.2 2
5.4 even 2 inner 275.2.b.d.199.1 4
15.2 even 4 2475.2.a.x.1.2 2
15.8 even 4 495.2.a.b.1.1 2
15.14 odd 2 2475.2.c.l.199.4 4
20.3 even 4 880.2.a.m.1.2 2
20.7 even 4 4400.2.a.bn.1.1 2
20.19 odd 2 4400.2.b.q.4049.3 4
35.13 even 4 2695.2.a.f.1.2 2
40.3 even 4 3520.2.a.bo.1.1 2
40.13 odd 4 3520.2.a.bn.1.2 2
55.3 odd 20 605.2.g.f.251.2 8
55.8 even 20 605.2.g.l.251.1 8
55.13 even 20 605.2.g.l.81.2 8
55.18 even 20 605.2.g.l.511.1 8
55.28 even 20 605.2.g.l.366.2 8
55.32 even 4 3025.2.a.o.1.2 2
55.38 odd 20 605.2.g.f.366.1 8
55.43 even 4 605.2.a.d.1.1 2
55.48 odd 20 605.2.g.f.511.2 8
55.53 odd 20 605.2.g.f.81.1 8
60.23 odd 4 7920.2.a.ch.1.2 2
65.38 odd 4 9295.2.a.g.1.1 2
165.98 odd 4 5445.2.a.y.1.2 2
220.43 odd 4 9680.2.a.bn.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
55.2.a.b.1.2 2 5.3 odd 4
275.2.a.c.1.1 2 5.2 odd 4
275.2.b.d.199.1 4 5.4 even 2 inner
275.2.b.d.199.4 4 1.1 even 1 trivial
495.2.a.b.1.1 2 15.8 even 4
605.2.a.d.1.1 2 55.43 even 4
605.2.g.f.81.1 8 55.53 odd 20
605.2.g.f.251.2 8 55.3 odd 20
605.2.g.f.366.1 8 55.38 odd 20
605.2.g.f.511.2 8 55.48 odd 20
605.2.g.l.81.2 8 55.13 even 20
605.2.g.l.251.1 8 55.8 even 20
605.2.g.l.366.2 8 55.28 even 20
605.2.g.l.511.1 8 55.18 even 20
880.2.a.m.1.2 2 20.3 even 4
2475.2.a.x.1.2 2 15.2 even 4
2475.2.c.l.199.1 4 3.2 odd 2
2475.2.c.l.199.4 4 15.14 odd 2
2695.2.a.f.1.2 2 35.13 even 4
3025.2.a.o.1.2 2 55.32 even 4
3520.2.a.bn.1.2 2 40.13 odd 4
3520.2.a.bo.1.1 2 40.3 even 4
4400.2.a.bn.1.1 2 20.7 even 4
4400.2.b.q.4049.2 4 4.3 odd 2
4400.2.b.q.4049.3 4 20.19 odd 2
5445.2.a.y.1.2 2 165.98 odd 4
7920.2.a.ch.1.2 2 60.23 odd 4
9295.2.a.g.1.1 2 65.38 odd 4
9680.2.a.bn.1.2 2 220.43 odd 4