# Properties

 Label 275.2.b.d.199.3 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 199.3 Root $$-0.707107 + 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 275.199 Dual form 275.2.b.d.199.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.414214i q^{2} +2.82843i q^{3} +1.82843 q^{4} -1.17157 q^{6} +2.00000i q^{7} +1.58579i q^{8} -5.00000 q^{9} +O(q^{10})$$ $$q+0.414214i q^{2} +2.82843i q^{3} +1.82843 q^{4} -1.17157 q^{6} +2.00000i q^{7} +1.58579i q^{8} -5.00000 q^{9} +1.00000 q^{11} +5.17157i q^{12} -6.82843i q^{13} -0.828427 q^{14} +3.00000 q^{16} -1.17157i q^{17} -2.07107i q^{18} -5.65685 q^{21} +0.414214i q^{22} +2.82843i q^{23} -4.48528 q^{24} +2.82843 q^{26} -5.65685i q^{27} +3.65685i q^{28} -7.65685 q^{29} +4.41421i q^{32} +2.82843i q^{33} +0.485281 q^{34} -9.14214 q^{36} -3.65685i q^{37} +19.3137 q^{39} +6.00000 q^{41} -2.34315i q^{42} -6.00000i q^{43} +1.82843 q^{44} -1.17157 q^{46} +2.82843i q^{47} +8.48528i q^{48} +3.00000 q^{49} +3.31371 q^{51} -12.4853i q^{52} +0.343146i q^{53} +2.34315 q^{54} -3.17157 q^{56} -3.17157i q^{58} +9.65685 q^{59} +13.3137 q^{61} -10.0000i q^{63} +4.17157 q^{64} -1.17157 q^{66} +4.48528i q^{67} -2.14214i q^{68} -8.00000 q^{69} -11.3137 q^{71} -7.92893i q^{72} -6.82843i q^{73} +1.51472 q^{74} +2.00000i q^{77} +8.00000i q^{78} -4.00000 q^{79} +1.00000 q^{81} +2.48528i q^{82} -6.00000i q^{83} -10.3431 q^{84} +2.48528 q^{86} -21.6569i q^{87} +1.58579i q^{88} -9.31371 q^{89} +13.6569 q^{91} +5.17157i q^{92} -1.17157 q^{94} -12.4853 q^{96} +7.65685i q^{97} +1.24264i q^{98} -5.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} - 16 q^{6} - 20 q^{9}+O(q^{10})$$ 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.414214i 0.292893i 0.989219 + 0.146447i $$0.0467837\pi$$
−0.989219 + 0.146447i $$0.953216\pi$$
$$3$$ 2.82843i 1.63299i 0.577350 + 0.816497i $$0.304087\pi$$
−0.577350 + 0.816497i $$0.695913\pi$$
$$4$$ 1.82843 0.914214
$$5$$ 0 0
$$6$$ −1.17157 −0.478293
$$7$$ 2.00000i 0.755929i 0.925820 + 0.377964i $$0.123376\pi$$
−0.925820 + 0.377964i $$0.876624\pi$$
$$8$$ 1.58579i 0.560660i
$$9$$ −5.00000 −1.66667
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 5.17157i 1.49290i
$$13$$ − 6.82843i − 1.89386i −0.321433 0.946932i $$-0.604164\pi$$
0.321433 0.946932i $$-0.395836\pi$$
$$14$$ −0.828427 −0.221406
$$15$$ 0 0
$$16$$ 3.00000 0.750000
$$17$$ − 1.17157i − 0.284148i −0.989856 0.142074i $$-0.954623\pi$$
0.989856 0.142074i $$-0.0453771\pi$$
$$18$$ − 2.07107i − 0.488155i
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ −5.65685 −1.23443
$$22$$ 0.414214i 0.0883106i
$$23$$ 2.82843i 0.589768i 0.955533 + 0.294884i $$0.0952810\pi$$
−0.955533 + 0.294884i $$0.904719\pi$$
$$24$$ −4.48528 −0.915554
$$25$$ 0 0
$$26$$ 2.82843 0.554700
$$27$$ − 5.65685i − 1.08866i
$$28$$ 3.65685i 0.691080i
$$29$$ −7.65685 −1.42184 −0.710921 0.703272i $$-0.751722\pi$$
−0.710921 + 0.703272i $$0.751722\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 4.41421i 0.780330i
$$33$$ 2.82843i 0.492366i
$$34$$ 0.485281 0.0832251
$$35$$ 0 0
$$36$$ −9.14214 −1.52369
$$37$$ − 3.65685i − 0.601183i −0.953753 0.300592i $$-0.902816\pi$$
0.953753 0.300592i $$-0.0971841\pi$$
$$38$$ 0 0
$$39$$ 19.3137 3.09267
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ − 2.34315i − 0.361555i
$$43$$ − 6.00000i − 0.914991i −0.889212 0.457496i $$-0.848747\pi$$
0.889212 0.457496i $$-0.151253\pi$$
$$44$$ 1.82843 0.275646
$$45$$ 0 0
$$46$$ −1.17157 −0.172739
$$47$$ 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
$$51$$ 3.31371 0.464012
$$52$$ − 12.4853i − 1.73140i
$$53$$ 0.343146i 0.0471347i 0.999722 + 0.0235673i $$0.00750241\pi$$
−0.999722 + 0.0235673i $$0.992498\pi$$
$$54$$ 2.34315 0.318862
$$55$$ 0 0
$$56$$ −3.17157 −0.423819
$$57$$ 0 0
$$58$$ − 3.17157i − 0.416448i
$$59$$ 9.65685 1.25722 0.628608 0.777723i $$-0.283625\pi$$
0.628608 + 0.777723i $$0.283625\pi$$
$$60$$ 0 0
$$61$$ 13.3137 1.70465 0.852323 0.523016i $$-0.175193\pi$$
0.852323 + 0.523016i $$0.175193\pi$$
$$62$$ 0 0
$$63$$ − 10.0000i − 1.25988i
$$64$$ 4.17157 0.521447
$$65$$ 0 0
$$66$$ −1.17157 −0.144211
$$67$$ 4.48528i 0.547964i 0.961735 + 0.273982i $$0.0883409\pi$$
−0.961735 + 0.273982i $$0.911659\pi$$
$$68$$ − 2.14214i − 0.259772i
$$69$$ −8.00000 −0.963087
$$70$$ 0 0
$$71$$ −11.3137 −1.34269 −0.671345 0.741145i $$-0.734283\pi$$
−0.671345 + 0.741145i $$0.734283\pi$$
$$72$$ − 7.92893i − 0.934434i
$$73$$ − 6.82843i − 0.799207i −0.916688 0.399603i $$-0.869148\pi$$
0.916688 0.399603i $$-0.130852\pi$$
$$74$$ 1.51472 0.176082
$$75$$ 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
$$82$$ 2.48528i 0.274453i
$$83$$ − 6.00000i − 0.658586i −0.944228 0.329293i $$-0.893190\pi$$
0.944228 0.329293i $$-0.106810\pi$$
$$84$$ −10.3431 −1.12853
$$85$$ 0 0
$$86$$ 2.48528 0.267995
$$87$$ − 21.6569i − 2.32186i
$$88$$ 1.58579i 0.169045i
$$89$$ −9.31371 −0.987251 −0.493626 0.869675i $$-0.664329\pi$$
−0.493626 + 0.869675i $$0.664329\pi$$
$$90$$ 0 0
$$91$$ 13.6569 1.43163
$$92$$ 5.17157i 0.539174i
$$93$$ 0 0
$$94$$ −1.17157 −0.120839
$$95$$ 0 0
$$96$$ −12.4853 −1.27427
$$97$$ 7.65685i 0.777436i 0.921357 + 0.388718i $$0.127082\pi$$
−0.921357 + 0.388718i $$0.872918\pi$$
$$98$$ 1.24264i 0.125526i
$$99$$ −5.00000 −0.502519
$$100$$ 0 0
$$101$$ −13.3137 −1.32476 −0.662382 0.749166i $$-0.730454\pi$$
−0.662382 + 0.749166i $$0.730454\pi$$
$$102$$ 1.37258i 0.135906i
$$103$$ 1.17157i 0.115439i 0.998333 + 0.0577193i $$0.0183828\pi$$
−0.998333 + 0.0577193i $$0.981617\pi$$
$$104$$ 10.8284 1.06181
$$105$$ 0 0
$$106$$ −0.142136 −0.0138054
$$107$$ 3.65685i 0.353521i 0.984254 + 0.176761i $$0.0565619\pi$$
−0.984254 + 0.176761i $$0.943438\pi$$
$$108$$ − 10.3431i − 0.995270i
$$109$$ −3.65685 −0.350263 −0.175132 0.984545i $$-0.556035\pi$$
−0.175132 + 0.984545i $$0.556035\pi$$
$$110$$ 0 0
$$111$$ 10.3431 0.981728
$$112$$ 6.00000i 0.566947i
$$113$$ 8.34315i 0.784857i 0.919782 + 0.392429i $$0.128365\pi$$
−0.919782 + 0.392429i $$0.871635\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −14.0000 −1.29987
$$117$$ 34.1421i 3.15644i
$$118$$ 4.00000i 0.368230i
$$119$$ 2.34315 0.214796
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 5.51472i 0.499279i
$$123$$ 16.9706i 1.53018i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 4.14214 0.369011
$$127$$ − 15.6569i − 1.38932i −0.719338 0.694661i $$-0.755555\pi$$
0.719338 0.694661i $$-0.244445\pi$$
$$128$$ 10.5563i 0.933058i
$$129$$ 16.9706 1.49417
$$130$$ 0 0
$$131$$ 11.3137 0.988483 0.494242 0.869325i $$-0.335446\pi$$
0.494242 + 0.869325i $$0.335446\pi$$
$$132$$ 5.17157i 0.450128i
$$133$$ 0 0
$$134$$ −1.85786 −0.160495
$$135$$ 0 0
$$136$$ 1.85786 0.159311
$$137$$ − 22.9706i − 1.96251i −0.192720 0.981254i $$-0.561731\pi$$
0.192720 0.981254i $$-0.438269\pi$$
$$138$$ − 3.31371i − 0.282082i
$$139$$ 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
$$142$$ − 4.68629i − 0.393265i
$$143$$ − 6.82843i − 0.571022i
$$144$$ −15.0000 −1.25000
$$145$$ 0 0
$$146$$ 2.82843 0.234082
$$147$$ 8.48528i 0.699854i
$$148$$ − 6.68629i − 0.549610i
$$149$$ −11.6569 −0.954967 −0.477483 0.878641i $$-0.658451\pi$$
−0.477483 + 0.878641i $$0.658451\pi$$
$$150$$ 0 0
$$151$$ −12.0000 −0.976546 −0.488273 0.872691i $$-0.662373\pi$$
−0.488273 + 0.872691i $$0.662373\pi$$
$$152$$ 0 0
$$153$$ 5.85786i 0.473580i
$$154$$ −0.828427 −0.0667566
$$155$$ 0 0
$$156$$ 35.3137 2.82736
$$157$$ 14.0000i 1.11732i 0.829396 + 0.558661i $$0.188685\pi$$
−0.829396 + 0.558661i $$0.811315\pi$$
$$158$$ − 1.65685i − 0.131812i
$$159$$ −0.970563 −0.0769706
$$160$$ 0 0
$$161$$ −5.65685 −0.445823
$$162$$ 0.414214i 0.0325437i
$$163$$ − 0.485281i − 0.0380102i −0.999819 0.0190051i $$-0.993950\pi$$
0.999819 0.0190051i $$-0.00604987\pi$$
$$164$$ 10.9706 0.856657
$$165$$ 0 0
$$166$$ 2.48528 0.192895
$$167$$ − 10.9706i − 0.848928i −0.905445 0.424464i $$-0.860463\pi$$
0.905445 0.424464i $$-0.139537\pi$$
$$168$$ − 8.97056i − 0.692094i
$$169$$ −33.6274 −2.58672
$$170$$ 0 0
$$171$$ 0 0
$$172$$ − 10.9706i − 0.836498i
$$173$$ 6.14214i 0.466978i 0.972359 + 0.233489i $$0.0750143\pi$$
−0.972359 + 0.233489i $$0.924986\pi$$
$$174$$ 8.97056 0.680057
$$175$$ 0 0
$$176$$ 3.00000 0.226134
$$177$$ 27.3137i 2.05302i
$$178$$ − 3.85786i − 0.289159i
$$179$$ 1.65685 0.123839 0.0619196 0.998081i $$-0.480278\pi$$
0.0619196 + 0.998081i $$0.480278\pi$$
$$180$$ 0 0
$$181$$ −1.31371 −0.0976472 −0.0488236 0.998807i $$-0.515547\pi$$
−0.0488236 + 0.998807i $$0.515547\pi$$
$$182$$ 5.65685i 0.419314i
$$183$$ 37.6569i 2.78367i
$$184$$ −4.48528 −0.330659
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 1.17157i − 0.0856739i
$$188$$ 5.17157i 0.377176i
$$189$$ 11.3137 0.822951
$$190$$ 0 0
$$191$$ −19.3137 −1.39749 −0.698745 0.715370i $$-0.746258\pi$$
−0.698745 + 0.715370i $$0.746258\pi$$
$$192$$ 11.7990i 0.851519i
$$193$$ − 6.82843i − 0.491521i −0.969331 0.245760i $$-0.920962\pi$$
0.969331 0.245760i $$-0.0790377\pi$$
$$194$$ −3.17157 −0.227706
$$195$$ 0 0
$$196$$ 5.48528 0.391806
$$197$$ 5.17157i 0.368459i 0.982883 + 0.184230i $$0.0589790\pi$$
−0.982883 + 0.184230i $$0.941021\pi$$
$$198$$ − 2.07107i − 0.147184i
$$199$$ −21.6569 −1.53521 −0.767607 0.640921i $$-0.778553\pi$$
−0.767607 + 0.640921i $$0.778553\pi$$
$$200$$ 0 0
$$201$$ −12.6863 −0.894822
$$202$$ − 5.51472i − 0.388014i
$$203$$ − 15.3137i − 1.07481i
$$204$$ 6.05887 0.424206
$$205$$ 0 0
$$206$$ −0.485281 −0.0338112
$$207$$ − 14.1421i − 0.982946i
$$208$$ − 20.4853i − 1.42040i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −16.0000 −1.10149 −0.550743 0.834675i $$-0.685655\pi$$
−0.550743 + 0.834675i $$0.685655\pi$$
$$212$$ 0.627417i 0.0430912i
$$213$$ − 32.0000i − 2.19260i
$$214$$ −1.51472 −0.103544
$$215$$ 0 0
$$216$$ 8.97056 0.610369
$$217$$ 0 0
$$218$$ − 1.51472i − 0.102590i
$$219$$ 19.3137 1.30510
$$220$$ 0 0
$$221$$ −8.00000 −0.538138
$$222$$ 4.28427i 0.287541i
$$223$$ − 5.17157i − 0.346314i −0.984894 0.173157i $$-0.944603\pi$$
0.984894 0.173157i $$-0.0553968\pi$$
$$224$$ −8.82843 −0.589874
$$225$$ 0 0
$$226$$ −3.45584 −0.229879
$$227$$ − 2.68629i − 0.178295i −0.996018 0.0891477i $$-0.971586\pi$$
0.996018 0.0891477i $$-0.0284143\pi$$
$$228$$ 0 0
$$229$$ 21.3137 1.40845 0.704225 0.709977i $$-0.251295\pi$$
0.704225 + 0.709977i $$0.251295\pi$$
$$230$$ 0 0
$$231$$ −5.65685 −0.372194
$$232$$ − 12.1421i − 0.797170i
$$233$$ 22.1421i 1.45058i 0.688444 + 0.725290i $$0.258294\pi$$
−0.688444 + 0.725290i $$0.741706\pi$$
$$234$$ −14.1421 −0.924500
$$235$$ 0 0
$$236$$ 17.6569 1.14936
$$237$$ − 11.3137i − 0.734904i
$$238$$ 0.970563i 0.0629122i
$$239$$ 0.686292 0.0443925 0.0221963 0.999754i $$-0.492934\pi$$
0.0221963 + 0.999754i $$0.492934\pi$$
$$240$$ 0 0
$$241$$ 6.00000 0.386494 0.193247 0.981150i $$-0.438098\pi$$
0.193247 + 0.981150i $$0.438098\pi$$
$$242$$ 0.414214i 0.0266267i
$$243$$ − 14.1421i − 0.907218i
$$244$$ 24.3431 1.55841
$$245$$ 0 0
$$246$$ −7.02944 −0.448181
$$247$$ 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 $$0.376365\pi$$
$$252$$ − 18.2843i − 1.15180i
$$253$$ 2.82843i 0.177822i
$$254$$ 6.48528 0.406923
$$255$$ 0 0
$$256$$ 3.97056 0.248160
$$257$$ − 13.3137i − 0.830486i −0.909710 0.415243i $$-0.863696\pi$$
0.909710 0.415243i $$-0.136304\pi$$
$$258$$ 7.02944i 0.437634i
$$259$$ 7.31371 0.454452
$$260$$ 0 0
$$261$$ 38.2843 2.36974
$$262$$ 4.68629i 0.289520i
$$263$$ 22.9706i 1.41643i 0.705999 + 0.708213i $$0.250498\pi$$
−0.705999 + 0.708213i $$0.749502\pi$$
$$264$$ −4.48528 −0.276050
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 26.3431i − 1.61217i
$$268$$ 8.20101i 0.500956i
$$269$$ 5.31371 0.323983 0.161991 0.986792i $$-0.448208\pi$$
0.161991 + 0.986792i $$0.448208\pi$$
$$270$$ 0 0
$$271$$ −15.3137 −0.930242 −0.465121 0.885247i $$-0.653989\pi$$
−0.465121 + 0.885247i $$0.653989\pi$$
$$272$$ − 3.51472i − 0.213111i
$$273$$ 38.6274i 2.33784i
$$274$$ 9.51472 0.574805
$$275$$ 0 0
$$276$$ −14.6274 −0.880467
$$277$$ − 1.17157i − 0.0703930i −0.999380 0.0351965i $$-0.988794\pi$$
0.999380 0.0351965i $$-0.0112057\pi$$
$$278$$ 1.65685i 0.0993715i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −5.31371 −0.316989 −0.158495 0.987360i $$-0.550664\pi$$
−0.158495 + 0.987360i $$0.550664\pi$$
$$282$$ − 3.31371i − 0.197328i
$$283$$ − 12.6274i − 0.750622i −0.926899 0.375311i $$-0.877536\pi$$
0.926899 0.375311i $$-0.122464\pi$$
$$284$$ −20.6863 −1.22751
$$285$$ 0 0
$$286$$ 2.82843 0.167248
$$287$$ 12.0000i 0.708338i
$$288$$ − 22.0711i − 1.30055i
$$289$$ 15.6274 0.919260
$$290$$ 0 0
$$291$$ −21.6569 −1.26955
$$292$$ − 12.4853i − 0.730646i
$$293$$ − 14.8284i − 0.866286i −0.901325 0.433143i $$-0.857405\pi$$
0.901325 0.433143i $$-0.142595\pi$$
$$294$$ −3.51472 −0.204983
$$295$$ 0 0
$$296$$ 5.79899 0.337059
$$297$$ − 5.65685i − 0.328244i
$$298$$ − 4.82843i − 0.279703i
$$299$$ 19.3137 1.11694
$$300$$ 0 0
$$301$$ 12.0000 0.691669
$$302$$ − 4.97056i − 0.286024i
$$303$$ − 37.6569i − 2.16333i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ −2.42641 −0.138708
$$307$$ 27.6569i 1.57846i 0.614098 + 0.789230i $$0.289520\pi$$
−0.614098 + 0.789230i $$0.710480\pi$$
$$308$$ 3.65685i 0.208369i
$$309$$ −3.31371 −0.188510
$$310$$ 0 0
$$311$$ 27.3137 1.54882 0.774409 0.632685i $$-0.218047\pi$$
0.774409 + 0.632685i $$0.218047\pi$$
$$312$$ 30.6274i 1.73394i
$$313$$ 21.3137i 1.20472i 0.798224 + 0.602361i $$0.205773\pi$$
−0.798224 + 0.602361i $$0.794227\pi$$
$$314$$ −5.79899 −0.327256
$$315$$ 0 0
$$316$$ −7.31371 −0.411428
$$317$$ − 21.3137i − 1.19710i −0.801087 0.598549i $$-0.795744\pi$$
0.801087 0.598549i $$-0.204256\pi$$
$$318$$ − 0.402020i − 0.0225442i
$$319$$ −7.65685 −0.428702
$$320$$ 0 0
$$321$$ −10.3431 −0.577298
$$322$$ − 2.34315i − 0.130578i
$$323$$ 0 0
$$324$$ 1.82843 0.101579
$$325$$ 0 0
$$326$$ 0.201010 0.0111329
$$327$$ − 10.3431i − 0.571977i
$$328$$ 9.51472i 0.525362i
$$329$$ −5.65685 −0.311872
$$330$$ 0 0
$$331$$ 15.3137 0.841718 0.420859 0.907126i $$-0.361729\pi$$
0.420859 + 0.907126i $$0.361729\pi$$
$$332$$ − 10.9706i − 0.602088i
$$333$$ 18.2843i 1.00197i
$$334$$ 4.54416 0.248645
$$335$$ 0 0
$$336$$ −16.9706 −0.925820
$$337$$ 3.51472i 0.191459i 0.995407 + 0.0957295i $$0.0305184\pi$$
−0.995407 + 0.0957295i $$0.969482\pi$$
$$338$$ − 13.9289i − 0.757634i
$$339$$ −23.5980 −1.28167
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 20.0000i 1.07990i
$$344$$ 9.51472 0.512999
$$345$$ 0 0
$$346$$ −2.54416 −0.136775
$$347$$ 22.9706i 1.23312i 0.787306 + 0.616562i $$0.211475\pi$$
−0.787306 + 0.616562i $$0.788525\pi$$
$$348$$ − 39.5980i − 2.12267i
$$349$$ 6.97056 0.373126 0.186563 0.982443i $$-0.440265\pi$$
0.186563 + 0.982443i $$0.440265\pi$$
$$350$$ 0 0
$$351$$ −38.6274 −2.06178
$$352$$ 4.41421i 0.235278i
$$353$$ − 1.31371i − 0.0699216i −0.999389 0.0349608i $$-0.988869\pi$$
0.999389 0.0349608i $$-0.0111306\pi$$
$$354$$ −11.3137 −0.601317
$$355$$ 0 0
$$356$$ −17.0294 −0.902558
$$357$$ 6.62742i 0.350760i
$$358$$ 0.686292i 0.0362716i
$$359$$ −23.3137 −1.23045 −0.615225 0.788351i $$-0.710935\pi$$
−0.615225 + 0.788351i $$0.710935\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ − 0.544156i − 0.0286002i
$$363$$ 2.82843i 0.148454i
$$364$$ 24.9706 1.30881
$$365$$ 0 0
$$366$$ −15.5980 −0.815319
$$367$$ − 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
$$371$$ −0.686292 −0.0356305
$$372$$ 0 0
$$373$$ − 3.79899i − 0.196704i −0.995152 0.0983521i $$-0.968643\pi$$
0.995152 0.0983521i $$-0.0313571\pi$$
$$374$$ 0.485281 0.0250933
$$375$$ 0 0
$$376$$ −4.48528 −0.231311
$$377$$ 52.2843i 2.69278i
$$378$$ 4.68629i 0.241037i
$$379$$ −22.3431 −1.14769 −0.573845 0.818964i $$-0.694549\pi$$
−0.573845 + 0.818964i $$0.694549\pi$$
$$380$$ 0 0
$$381$$ 44.2843 2.26875
$$382$$ − 8.00000i − 0.409316i
$$383$$ − 34.1421i − 1.74458i −0.488987 0.872291i $$-0.662634\pi$$
0.488987 0.872291i $$-0.337366\pi$$
$$384$$ −29.8579 −1.52368
$$385$$ 0 0
$$386$$ 2.82843 0.143963
$$387$$ 30.0000i 1.52499i
$$388$$ 14.0000i 0.710742i
$$389$$ 24.6274 1.24866 0.624330 0.781161i $$-0.285372\pi$$
0.624330 + 0.781161i $$0.285372\pi$$
$$390$$ 0 0
$$391$$ 3.31371 0.167581
$$392$$ 4.75736i 0.240283i
$$393$$ 32.0000i 1.61419i
$$394$$ −2.14214 −0.107919
$$395$$ 0 0
$$396$$ −9.14214 −0.459410
$$397$$ − 13.3137i − 0.668196i −0.942538 0.334098i $$-0.891568\pi$$
0.942538 0.334098i $$-0.108432\pi$$
$$398$$ − 8.97056i − 0.449654i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 17.3137 0.864605 0.432303 0.901729i $$-0.357701\pi$$
0.432303 + 0.901729i $$0.357701\pi$$
$$402$$ − 5.25483i − 0.262087i
$$403$$ 0 0
$$404$$ −24.3431 −1.21112
$$405$$ 0 0
$$406$$ 6.34315 0.314805
$$407$$ − 3.65685i − 0.181264i
$$408$$ 5.25483i 0.260153i
$$409$$ −34.9706 −1.72918 −0.864592 0.502475i $$-0.832423\pi$$
−0.864592 + 0.502475i $$0.832423\pi$$
$$410$$ 0 0
$$411$$ 64.9706 3.20476
$$412$$ 2.14214i 0.105535i
$$413$$ 19.3137i 0.950365i
$$414$$ 5.85786 0.287898
$$415$$ 0 0
$$416$$ 30.1421 1.47784
$$417$$ 11.3137i 0.554035i
$$418$$ 0 0
$$419$$ 14.3431 0.700709 0.350354 0.936617i $$-0.386061\pi$$
0.350354 + 0.936617i $$0.386061\pi$$
$$420$$ 0 0
$$421$$ −6.00000 −0.292422 −0.146211 0.989253i $$-0.546708\pi$$
−0.146211 + 0.989253i $$0.546708\pi$$
$$422$$ − 6.62742i − 0.322618i
$$423$$ − 14.1421i − 0.687614i
$$424$$ −0.544156 −0.0264265
$$425$$ 0 0
$$426$$ 13.2548 0.642199
$$427$$ 26.6274i 1.28859i
$$428$$ 6.68629i 0.323194i
$$429$$ 19.3137 0.932475
$$430$$ 0 0
$$431$$ 11.3137 0.544962 0.272481 0.962161i $$-0.412156\pi$$
0.272481 + 0.962161i $$0.412156\pi$$
$$432$$ − 16.9706i − 0.816497i
$$433$$ 3.65685i 0.175737i 0.996132 + 0.0878686i $$0.0280056\pi$$
−0.996132 + 0.0878686i $$0.971994\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −6.68629 −0.320215
$$437$$ 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
$$442$$ − 3.31371i − 0.157617i
$$443$$ − 21.1716i − 1.00589i −0.864318 0.502946i $$-0.832249\pi$$
0.864318 0.502946i $$-0.167751\pi$$
$$444$$ 18.9117 0.897509
$$445$$ 0 0
$$446$$ 2.14214 0.101433
$$447$$ − 32.9706i − 1.55945i
$$448$$ 8.34315i 0.394177i
$$449$$ 16.6274 0.784696 0.392348 0.919817i $$-0.371663\pi$$
0.392348 + 0.919817i $$0.371663\pi$$
$$450$$ 0 0
$$451$$ 6.00000 0.282529
$$452$$ 15.2548i 0.717527i
$$453$$ − 33.9411i − 1.59469i
$$454$$ 1.11270 0.0522215
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 16.4853i 0.771149i 0.922677 + 0.385574i $$0.125997\pi$$
−0.922677 + 0.385574i $$0.874003\pi$$
$$458$$ 8.82843i 0.412525i
$$459$$ −6.62742 −0.309341
$$460$$ 0 0
$$461$$ −32.6274 −1.51961 −0.759805 0.650151i $$-0.774706\pi$$
−0.759805 + 0.650151i $$0.774706\pi$$
$$462$$ − 2.34315i − 0.109013i
$$463$$ 22.1421i 1.02903i 0.857481 + 0.514516i $$0.172028\pi$$
−0.857481 + 0.514516i $$0.827972\pi$$
$$464$$ −22.9706 −1.06638
$$465$$ 0 0
$$466$$ −9.17157 −0.424865
$$467$$ 9.17157i 0.424410i 0.977225 + 0.212205i $$0.0680644\pi$$
−0.977225 + 0.212205i $$0.931936\pi$$
$$468$$ 62.4264i 2.88566i
$$469$$ −8.97056 −0.414222
$$470$$ 0 0
$$471$$ −39.5980 −1.82458
$$472$$ 15.3137i 0.704871i
$$473$$ − 6.00000i − 0.275880i
$$474$$ 4.68629 0.215248
$$475$$ 0 0
$$476$$ 4.28427 0.196369
$$477$$ − 1.71573i − 0.0785578i
$$478$$ 0.284271i 0.0130023i
$$479$$ 36.0000 1.64488 0.822441 0.568850i $$-0.192612\pi$$
0.822441 + 0.568850i $$0.192612\pi$$
$$480$$ 0 0
$$481$$ −24.9706 −1.13856
$$482$$ 2.48528i 0.113201i
$$483$$ − 16.0000i − 0.728025i
$$484$$ 1.82843 0.0831103
$$485$$ 0 0
$$486$$ 5.85786 0.265718
$$487$$ 7.51472i 0.340524i 0.985399 + 0.170262i $$0.0544615\pi$$
−0.985399 + 0.170262i $$0.945539\pi$$
$$488$$ 21.1127i 0.955727i
$$489$$ 1.37258 0.0620703
$$490$$ 0 0
$$491$$ −23.3137 −1.05213 −0.526066 0.850443i $$-0.676334\pi$$
−0.526066 + 0.850443i $$0.676334\pi$$
$$492$$ 31.0294i 1.39892i
$$493$$ 8.97056i 0.404014i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 22.6274i − 1.01498i
$$498$$ 7.02944i 0.314997i
$$499$$ 1.65685 0.0741710 0.0370855 0.999312i $$-0.488193\pi$$
0.0370855 + 0.999312i $$0.488193\pi$$
$$500$$ 0 0
$$501$$ 31.0294 1.38629
$$502$$ 4.97056i 0.221847i
$$503$$ − 28.6274i − 1.27643i −0.769857 0.638217i $$-0.779672\pi$$
0.769857 0.638217i $$-0.220328\pi$$
$$504$$ 15.8579 0.706365
$$505$$ 0 0
$$506$$ −1.17157 −0.0520828
$$507$$ − 95.1127i − 4.22410i
$$508$$ − 28.6274i − 1.27014i
$$509$$ −9.31371 −0.412823 −0.206411 0.978465i $$-0.566179\pi$$
−0.206411 + 0.978465i $$0.566179\pi$$
$$510$$ 0 0
$$511$$ 13.6569 0.604144
$$512$$ 22.7574i 1.00574i
$$513$$ 0 0
$$514$$ 5.51472 0.243244
$$515$$ 0 0
$$516$$ 31.0294 1.36599
$$517$$ 2.82843i 0.124394i
$$518$$ 3.02944i 0.133106i
$$519$$ −17.3726 −0.762572
$$520$$ 0 0
$$521$$ 2.68629 0.117689 0.0588443 0.998267i $$-0.481258\pi$$
0.0588443 + 0.998267i $$0.481258\pi$$
$$522$$ 15.8579i 0.694080i
$$523$$ 37.5980i 1.64404i 0.569455 + 0.822022i $$0.307154\pi$$
−0.569455 + 0.822022i $$0.692846\pi$$
$$524$$ 20.6863 0.903685
$$525$$ 0 0
$$526$$ −9.51472 −0.414861
$$527$$ 0 0
$$528$$ 8.48528i 0.369274i
$$529$$ 15.0000 0.652174
$$530$$ 0 0
$$531$$ −48.2843 −2.09536
$$532$$ 0 0
$$533$$ − 40.9706i − 1.77463i
$$534$$ 10.9117 0.472195
$$535$$ 0 0
$$536$$ −7.11270 −0.307222
$$537$$ 4.68629i 0.202228i
$$538$$ 2.20101i 0.0948923i
$$539$$ 3.00000 0.129219
$$540$$ 0 0
$$541$$ 6.00000 0.257960 0.128980 0.991647i $$-0.458830\pi$$
0.128980 + 0.991647i $$0.458830\pi$$
$$542$$ − 6.34315i − 0.272461i
$$543$$ − 3.71573i − 0.159457i
$$544$$ 5.17157 0.221729
$$545$$ 0 0
$$546$$ −16.0000 −0.684737
$$547$$ 34.0000i 1.45374i 0.686778 + 0.726868i $$0.259025\pi$$
−0.686778 + 0.726868i $$0.740975\pi$$
$$548$$ − 42.0000i − 1.79415i
$$549$$ −66.5685 −2.84108
$$550$$ 0 0
$$551$$ 0 0
$$552$$ − 12.6863i − 0.539964i
$$553$$ − 8.00000i − 0.340195i
$$554$$ 0.485281 0.0206176
$$555$$ 0 0
$$556$$ 7.31371 0.310170
$$557$$ − 38.1421i − 1.61613i −0.589090 0.808067i $$-0.700514\pi$$
0.589090 0.808067i $$-0.299486\pi$$
$$558$$ 0 0
$$559$$ −40.9706 −1.73287
$$560$$ 0 0
$$561$$ 3.31371 0.139905
$$562$$ − 2.20101i − 0.0928440i
$$563$$ 11.6569i 0.491278i 0.969361 + 0.245639i $$0.0789977\pi$$
−0.969361 + 0.245639i $$0.921002\pi$$
$$564$$ −14.6274 −0.615925
$$565$$ 0 0
$$566$$ 5.23045 0.219852
$$567$$ 2.00000i 0.0839921i
$$568$$ − 17.9411i − 0.752793i
$$569$$ −20.3431 −0.852829 −0.426415 0.904528i $$-0.640224\pi$$
−0.426415 + 0.904528i $$0.640224\pi$$
$$570$$ 0 0
$$571$$ 45.9411 1.92258 0.961288 0.275545i $$-0.0888584\pi$$
0.961288 + 0.275545i $$0.0888584\pi$$
$$572$$ − 12.4853i − 0.522036i
$$573$$ − 54.6274i − 2.28209i
$$574$$ −4.97056 −0.207467
$$575$$ 0 0
$$576$$ −20.8579 −0.869078
$$577$$ − 6.97056i − 0.290188i −0.989418 0.145094i $$-0.953651\pi$$
0.989418 0.145094i $$-0.0463485\pi$$
$$578$$ 6.47309i 0.269245i
$$579$$ 19.3137 0.802650
$$580$$ 0 0
$$581$$ 12.0000 0.497844
$$582$$ − 8.97056i − 0.371842i
$$583$$ 0.343146i 0.0142116i
$$584$$ 10.8284 0.448084
$$585$$ 0 0
$$586$$ 6.14214 0.253729
$$587$$ − 26.1421i − 1.07900i −0.841985 0.539501i $$-0.818613\pi$$
0.841985 0.539501i $$-0.181387\pi$$
$$588$$ 15.5147i 0.639816i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −14.6274 −0.601692
$$592$$ − 10.9706i − 0.450887i
$$593$$ 20.4853i 0.841230i 0.907239 + 0.420615i $$0.138186\pi$$
−0.907239 + 0.420615i $$0.861814\pi$$
$$594$$ 2.34315 0.0961404
$$595$$ 0 0
$$596$$ −21.3137 −0.873044
$$597$$ − 61.2548i − 2.50699i
$$598$$ 8.00000i 0.327144i
$$599$$ −5.65685 −0.231133 −0.115566 0.993300i $$-0.536868\pi$$
−0.115566 + 0.993300i $$0.536868\pi$$
$$600$$ 0 0
$$601$$ −43.9411 −1.79240 −0.896198 0.443654i $$-0.853682\pi$$
−0.896198 + 0.443654i $$0.853682\pi$$
$$602$$ 4.97056i 0.202585i
$$603$$ − 22.4264i − 0.913274i
$$604$$ −21.9411 −0.892772
$$605$$ 0 0
$$606$$ 15.5980 0.633625
$$607$$ 18.2843i 0.742136i 0.928606 + 0.371068i $$0.121008\pi$$
−0.928606 + 0.371068i $$0.878992\pi$$
$$608$$ 0 0
$$609$$ 43.3137 1.75516
$$610$$ 0 0
$$611$$ 19.3137 0.781349
$$612$$ 10.7107i 0.432954i
$$613$$ 25.4558i 1.02815i 0.857745 + 0.514076i $$0.171865\pi$$
−0.857745 + 0.514076i $$0.828135\pi$$
$$614$$ −11.4558 −0.462320
$$615$$ 0 0
$$616$$ −3.17157 −0.127786
$$617$$ − 11.6569i − 0.469287i −0.972081 0.234644i $$-0.924608\pi$$
0.972081 0.234644i $$-0.0753923\pi$$
$$618$$ − 1.37258i − 0.0552134i
$$619$$ 25.6569 1.03124 0.515618 0.856819i $$-0.327562\pi$$
0.515618 + 0.856819i $$0.327562\pi$$
$$620$$ 0 0
$$621$$ 16.0000 0.642058
$$622$$ 11.3137i 0.453638i
$$623$$ − 18.6274i − 0.746292i
$$624$$ 57.9411 2.31950
$$625$$ 0 0
$$626$$ −8.82843 −0.352855
$$627$$ 0 0
$$628$$ 25.5980i 1.02147i
$$629$$ −4.28427 −0.170825
$$630$$ 0 0
$$631$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$632$$ − 6.34315i − 0.252317i
$$633$$ − 45.2548i − 1.79872i
$$634$$ 8.82843 0.350622
$$635$$ 0 0
$$636$$ −1.77460 −0.0703676
$$637$$ − 20.4853i − 0.811656i
$$638$$ − 3.17157i − 0.125564i
$$639$$ 56.5685 2.23782
$$640$$ 0 0
$$641$$ 30.0000 1.18493 0.592464 0.805597i $$-0.298155\pi$$
0.592464 + 0.805597i $$0.298155\pi$$
$$642$$ − 4.28427i − 0.169087i
$$643$$ 49.4558i 1.95035i 0.221440 + 0.975174i $$0.428924\pi$$
−0.221440 + 0.975174i $$0.571076\pi$$
$$644$$ −10.3431 −0.407577
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 35.1127i 1.38042i 0.723608 + 0.690211i $$0.242482\pi$$
−0.723608 + 0.690211i $$0.757518\pi$$
$$648$$ 1.58579i 0.0622956i
$$649$$ 9.65685 0.379065
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 0.887302i − 0.0347494i
$$653$$ 0.343146i 0.0134283i 0.999977 + 0.00671417i $$0.00213720\pi$$
−0.999977 + 0.00671417i $$0.997863\pi$$
$$654$$ 4.28427 0.167528
$$655$$ 0 0
$$656$$ 18.0000 0.702782
$$657$$ 34.1421i 1.33201i
$$658$$ − 2.34315i − 0.0913453i
$$659$$ 21.9411 0.854705 0.427352 0.904085i $$-0.359446\pi$$
0.427352 + 0.904085i $$0.359446\pi$$
$$660$$ 0 0
$$661$$ −0.627417 −0.0244037 −0.0122018 0.999926i $$-0.503884\pi$$
−0.0122018 + 0.999926i $$0.503884\pi$$
$$662$$ 6.34315i 0.246533i
$$663$$ − 22.6274i − 0.878776i
$$664$$ 9.51472 0.369243
$$665$$ 0 0
$$666$$ −7.57359 −0.293471
$$667$$ − 21.6569i − 0.838557i
$$668$$ − 20.0589i − 0.776101i
$$669$$ 14.6274 0.565529
$$670$$ 0 0
$$671$$ 13.3137 0.513970
$$672$$ − 24.9706i − 0.963260i
$$673$$ 4.48528i 0.172895i 0.996256 + 0.0864474i $$0.0275515\pi$$
−0.996256 + 0.0864474i $$0.972449\pi$$
$$674$$ −1.45584 −0.0560770
$$675$$ 0 0
$$676$$ −61.4853 −2.36482
$$677$$ − 17.1716i − 0.659957i −0.943988 0.329979i $$-0.892958\pi$$
0.943988 0.329979i $$-0.107042\pi$$
$$678$$ − 9.77460i − 0.375391i
$$679$$ −15.3137 −0.587686
$$680$$ 0 0
$$681$$ 7.59798 0.291155
$$682$$ 0 0
$$683$$ 31.7990i 1.21675i 0.793648 + 0.608377i $$0.208179\pi$$
−0.793648 + 0.608377i $$0.791821\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −8.28427 −0.316295
$$687$$ 60.2843i 2.29999i
$$688$$ − 18.0000i − 0.686244i
$$689$$ 2.34315 0.0892667
$$690$$ 0 0
$$691$$ −16.6863 −0.634776 −0.317388 0.948296i $$-0.602806\pi$$
−0.317388 + 0.948296i $$0.602806\pi$$
$$692$$ 11.2304i 0.426918i
$$693$$ − 10.0000i − 0.379869i
$$694$$ −9.51472 −0.361174
$$695$$ 0 0
$$696$$ 34.3431 1.30177
$$697$$ − 7.02944i − 0.266259i
$$698$$ 2.88730i 0.109286i
$$699$$ −62.6274 −2.36879
$$700$$ 0 0
$$701$$ 32.6274 1.23232 0.616160 0.787621i $$-0.288687\pi$$
0.616160 + 0.787621i $$0.288687\pi$$
$$702$$ − 16.0000i − 0.603881i
$$703$$ 0 0
$$704$$ 4.17157 0.157222
$$705$$ 0 0
$$706$$ 0.544156 0.0204796
$$707$$ − 26.6274i − 1.00143i
$$708$$ 49.9411i 1.87690i
$$709$$ 20.6274 0.774679 0.387339 0.921937i $$-0.373394\pi$$
0.387339 + 0.921937i $$0.373394\pi$$
$$710$$ 0 0
$$711$$ 20.0000 0.750059
$$712$$ − 14.7696i − 0.553512i
$$713$$ 0 0
$$714$$ −2.74517 −0.102735
$$715$$ 0 0
$$716$$ 3.02944 0.113215
$$717$$ 1.94113i 0.0724927i
$$718$$ − 9.65685i − 0.360391i
$$719$$ −29.6569 −1.10601 −0.553007 0.833177i $$-0.686520\pi$$
−0.553007 + 0.833177i $$0.686520\pi$$
$$720$$ 0 0
$$721$$ −2.34315 −0.0872633
$$722$$ − 7.87006i − 0.292893i
$$723$$ 16.9706i 0.631142i
$$724$$ −2.40202 −0.0892704
$$725$$ 0 0
$$726$$ −1.17157 −0.0434811
$$727$$ 36.4853i 1.35316i 0.736367 + 0.676582i $$0.236540\pi$$
−0.736367 + 0.676582i $$0.763460\pi$$
$$728$$ 21.6569i 0.802656i
$$729$$ 43.0000 1.59259
$$730$$ 0 0
$$731$$ −7.02944 −0.259993
$$732$$ 68.8528i 2.54487i
$$733$$ 33.4558i 1.23572i 0.786288 + 0.617860i $$0.212000\pi$$
−0.786288 + 0.617860i $$0.788000\pi$$
$$734$$ 3.51472 0.129731
$$735$$ 0 0
$$736$$ −12.4853 −0.460214
$$737$$ 4.48528i 0.165217i
$$738$$ − 12.4264i − 0.457422i
$$739$$ 37.9411 1.39569 0.697843 0.716250i $$-0.254143\pi$$
0.697843 + 0.716250i $$0.254143\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ − 0.284271i − 0.0104359i
$$743$$ 29.5980i 1.08584i 0.839783 + 0.542922i $$0.182682\pi$$
−0.839783 + 0.542922i $$0.817318\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 1.57359 0.0576133
$$747$$ 30.0000i 1.09764i
$$748$$ − 2.14214i − 0.0783242i
$$749$$ −7.31371 −0.267237
$$750$$ 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
$$754$$ −21.6569 −0.788696
$$755$$ 0 0
$$756$$ 20.6863 0.752353
$$757$$ 9.31371i 0.338512i 0.985572 + 0.169256i $$0.0541365\pi$$
−0.985572 + 0.169256i $$0.945863\pi$$
$$758$$ − 9.25483i − 0.336151i
$$759$$ −8.00000 −0.290382
$$760$$ 0 0
$$761$$ −30.0000 −1.08750 −0.543750 0.839248i $$-0.682996\pi$$
−0.543750 + 0.839248i $$0.682996\pi$$
$$762$$ 18.3431i 0.664502i
$$763$$ − 7.31371i − 0.264774i
$$764$$ −35.3137 −1.27761
$$765$$ 0 0
$$766$$ 14.1421 0.510976
$$767$$ − 65.9411i − 2.38100i
$$768$$ 11.2304i 0.405244i
$$769$$ −14.9706 −0.539852 −0.269926 0.962881i $$-0.586999\pi$$
−0.269926 + 0.962881i $$0.586999\pi$$
$$770$$ 0 0
$$771$$ 37.6569 1.35618
$$772$$ − 12.4853i − 0.449355i
$$773$$ − 30.2843i − 1.08925i −0.838680 0.544625i $$-0.816672\pi$$
0.838680 0.544625i $$-0.183328\pi$$
$$774$$ −12.4264 −0.446658
$$775$$ 0 0
$$776$$ −12.1421 −0.435877
$$777$$ 20.6863i 0.742117i
$$778$$ 10.2010i 0.365724i
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −11.3137 −0.404836
$$782$$ 1.37258i 0.0490835i
$$783$$ 43.3137i 1.54791i
$$784$$ 9.00000 0.321429
$$785$$ 0 0
$$786$$ −13.2548 −0.472784
$$787$$ − 18.9706i − 0.676228i −0.941105 0.338114i $$-0.890211\pi$$
0.941105 0.338114i $$-0.109789\pi$$
$$788$$ 9.45584i 0.336850i
$$789$$ −64.9706 −2.31301
$$790$$ 0 0
$$791$$ −16.6863 −0.593296
$$792$$ − 7.92893i − 0.281742i
$$793$$ − 90.9117i − 3.22837i
$$794$$ 5.51472 0.195710
$$795$$ 0 0
$$796$$ −39.5980 −1.40351
$$797$$ 12.6274i 0.447286i 0.974671 + 0.223643i $$0.0717950\pi$$
−0.974671 + 0.223643i $$0.928205\pi$$
$$798$$ 0 0
$$799$$ 3.31371 0.117231
$$800$$ 0 0
$$801$$ 46.5685 1.64542
$$802$$ 7.17157i 0.253237i
$$803$$ − 6.82843i − 0.240970i
$$804$$ −23.1960 −0.818058
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 15.0294i 0.529061i
$$808$$ − 21.1127i − 0.742742i
$$809$$ −22.9706 −0.807602 −0.403801 0.914847i $$-0.632311\pi$$
−0.403801 + 0.914847i $$0.632311\pi$$
$$810$$ 0 0
$$811$$ −13.9411 −0.489539 −0.244770 0.969581i $$-0.578712\pi$$
−0.244770 + 0.969581i $$0.578712\pi$$
$$812$$ − 28.0000i − 0.982607i
$$813$$ − 43.3137i − 1.51908i
$$814$$ 1.51472 0.0530909
$$815$$ 0 0
$$816$$ 9.94113 0.348009
$$817$$ 0 0
$$818$$ − 14.4853i − 0.506466i
$$819$$ −68.2843 −2.38605
$$820$$ 0 0
$$821$$ −18.6863 −0.652156 −0.326078 0.945343i $$-0.605727\pi$$
−0.326078 + 0.945343i $$0.605727\pi$$
$$822$$ 26.9117i 0.938653i
$$823$$ 36.4853i 1.27180i 0.771773 + 0.635898i $$0.219370\pi$$
−0.771773 + 0.635898i $$0.780630\pi$$
$$824$$ −1.85786 −0.0647218
$$825$$ 0 0
$$826$$ −8.00000 −0.278356
$$827$$ 34.2843i 1.19218i 0.802917 + 0.596090i $$0.203280\pi$$
−0.802917 + 0.596090i $$0.796720\pi$$
$$828$$ − 25.8579i − 0.898623i
$$829$$ −18.0000 −0.625166 −0.312583 0.949890i $$-0.601194\pi$$
−0.312583 + 0.949890i $$0.601194\pi$$
$$830$$ 0 0
$$831$$ 3.31371 0.114951
$$832$$ − 28.4853i − 0.987549i
$$833$$ − 3.51472i − 0.121778i
$$834$$ −4.68629 −0.162273
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 5.94113i 0.205233i
$$839$$ −37.6569 −1.30006 −0.650029 0.759909i $$-0.725243\pi$$
−0.650029 + 0.759909i $$0.725243\pi$$
$$840$$ 0 0
$$841$$ 29.6274 1.02164
$$842$$ − 2.48528i − 0.0856485i
$$843$$ − 15.0294i − 0.517641i
$$844$$ −29.2548 −1.00699
$$845$$ 0 0
$$846$$ 5.85786 0.201398
$$847$$ 2.00000i 0.0687208i
$$848$$ 1.02944i 0.0353510i
$$849$$ 35.7157 1.22576
$$850$$ 0 0
$$851$$ 10.3431 0.354558
$$852$$ − 58.5097i − 2.00451i
$$853$$ − 32.4853i − 1.11227i −0.831090 0.556137i $$-0.812283\pi$$
0.831090 0.556137i $$-0.187717\pi$$
$$854$$ −11.0294 −0.377420
$$855$$ 0 0
$$856$$ −5.79899 −0.198205
$$857$$ 48.7696i 1.66594i 0.553321 + 0.832968i $$0.313360\pi$$
−0.553321 + 0.832968i $$0.686640\pi$$
$$858$$ 8.00000i 0.273115i
$$859$$ 32.2843 1.10153 0.550763 0.834662i $$-0.314337\pi$$
0.550763 + 0.834662i $$0.314337\pi$$
$$860$$ 0 0
$$861$$ −33.9411 −1.15671
$$862$$ 4.68629i 0.159616i
$$863$$ − 14.8284i − 0.504766i −0.967627 0.252383i $$-0.918786\pi$$
0.967627 0.252383i $$-0.0812142\pi$$
$$864$$ 24.9706 0.849516
$$865$$ 0 0
$$866$$ −1.51472 −0.0514722
$$867$$ 44.2010i 1.50115i
$$868$$ 0 0
$$869$$ −4.00000 −0.135691
$$870$$ 0 0
$$871$$ 30.6274 1.03777
$$872$$ − 5.79899i − 0.196379i
$$873$$ − 38.2843i − 1.29573i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 35.3137 1.19314
$$877$$ − 1.45584i − 0.0491604i −0.999698 0.0245802i $$-0.992175\pi$$
0.999698 0.0245802i $$-0.00782490\pi$$
$$878$$ 6.62742i 0.223664i
$$879$$ 41.9411 1.41464
$$880$$ 0 0
$$881$$ −52.6274 −1.77306 −0.886531 0.462668i $$-0.846892\pi$$
−0.886531 + 0.462668i $$0.846892\pi$$
$$882$$ − 6.21320i − 0.209209i
$$883$$ 42.8284i 1.44129i 0.693304 + 0.720646i $$0.256155\pi$$
−0.693304 + 0.720646i $$0.743845\pi$$
$$884$$ −14.6274 −0.491973
$$885$$ 0 0
$$886$$ 8.76955 0.294619
$$887$$ 18.2843i 0.613926i 0.951721 + 0.306963i $$0.0993127\pi$$
−0.951721 + 0.306963i $$0.900687\pi$$
$$888$$ 16.4020i 0.550416i
$$889$$ 31.3137 1.05023
$$890$$ 0 0
$$891$$ 1.00000 0.0335013
$$892$$ − 9.45584i − 0.316605i
$$893$$ 0 0
$$894$$ 13.6569 0.456754
$$895$$ 0 0
$$896$$ −21.1127 −0.705326
$$897$$ 54.6274i 1.82396i
$$898$$ 6.88730i 0.229832i
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 0.402020 0.0133932
$$902$$ 2.48528i 0.0827508i
$$903$$ 33.9411i 1.12949i
$$904$$ −13.2304 −0.440038
$$905$$ 0 0
$$906$$ 14.0589 0.467075
$$907$$ 44.4853i 1.47711i 0.674193 + 0.738555i $$0.264491\pi$$
−0.674193 + 0.738555i $$0.735509\pi$$
$$908$$ − 4.91169i − 0.163000i
$$909$$ 66.5685 2.20794
$$910$$ 0 0
$$911$$ 57.9411 1.91968 0.959838 0.280556i $$-0.0905189\pi$$
0.959838 + 0.280556i $$0.0905189\pi$$
$$912$$ 0 0
$$913$$ − 6.00000i − 0.198571i
$$914$$ −6.82843 −0.225864
$$915$$ 0 0
$$916$$ 38.9706 1.28762
$$917$$ 22.6274i 0.747223i
$$918$$ − 2.74517i − 0.0906040i
$$919$$ 32.0000 1.05558 0.527791 0.849374i $$-0.323020\pi$$
0.527791 + 0.849374i $$0.323020\pi$$
$$920$$ 0 0
$$921$$ −78.2254 −2.57761
$$922$$ − 13.5147i − 0.445084i
$$923$$ 77.2548i 2.54287i
$$924$$ −10.3431 −0.340265
$$925$$ 0 0
$$926$$ −9.17157 −0.301397
$$927$$ − 5.85786i − 0.192398i
$$928$$ − 33.7990i − 1.10951i
$$929$$ 17.3137 0.568044 0.284022 0.958818i $$-0.408331\pi$$
0.284022 + 0.958818i $$0.408331\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 40.4853i 1.32614i
$$933$$ 77.2548i 2.52921i
$$934$$ −3.79899 −0.124307
$$935$$ 0 0
$$936$$ −54.1421 −1.76969
$$937$$ − 49.4558i − 1.61565i −0.589421 0.807826i $$-0.700644\pi$$
0.589421 0.807826i $$-0.299356\pi$$
$$938$$ − 3.71573i − 0.121323i
$$939$$ −60.2843 −1.96730
$$940$$ 0 0
$$941$$ −29.3137 −0.955600 −0.477800 0.878469i $$-0.658566\pi$$
−0.477800 + 0.878469i $$0.658566\pi$$
$$942$$ − 16.4020i − 0.534407i
$$943$$ 16.9706i 0.552638i
$$944$$ 28.9706 0.942912
$$945$$ 0 0
$$946$$ 2.48528 0.0808035
$$947$$ − 46.8284i − 1.52172i −0.648916 0.760860i $$-0.724778\pi$$
0.648916 0.760860i $$-0.275222\pi$$
$$948$$ − 20.6863i − 0.671860i
$$949$$ −46.6274 −1.51359
$$950$$ 0 0
$$951$$ 60.2843 1.95485
$$952$$ 3.71573i 0.120427i
$$953$$ 58.8284i 1.90564i 0.303536 + 0.952820i $$0.401833\pi$$
−0.303536 + 0.952820i $$0.598167\pi$$
$$954$$ 0.710678 0.0230091
$$955$$ 0 0
$$956$$ 1.25483 0.0405842
$$957$$ − 21.6569i − 0.700067i
$$958$$ 14.9117i 0.481775i
$$959$$ 45.9411 1.48352
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ − 10.3431i − 0.333476i
$$963$$ − 18.2843i − 0.589202i
$$964$$ 10.9706 0.353338
$$965$$ 0 0
$$966$$ 6.62742 0.213234
$$967$$ − 18.9706i − 0.610052i −0.952344 0.305026i $$-0.901335\pi$$
0.952344 0.305026i $$-0.0986652\pi$$
$$968$$ 1.58579i 0.0509691i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −31.3137 −1.00490 −0.502452 0.864605i $$-0.667569\pi$$
−0.502452 + 0.864605i $$0.667569\pi$$
$$972$$ − 25.8579i − 0.829391i
$$973$$ 8.00000i 0.256468i
$$974$$ −3.11270 −0.0997373
$$975$$ 0 0
$$976$$ 39.9411 1.27848
$$977$$ − 43.6569i − 1.39671i −0.715753 0.698353i $$-0.753916\pi$$
0.715753 0.698353i $$-0.246084\pi$$
$$978$$ 0.568542i 0.0181800i
$$979$$ −9.31371 −0.297667
$$980$$ 0 0
$$981$$ 18.2843 0.583772
$$982$$ − 9.65685i − 0.308163i
$$983$$ − 50.1421i − 1.59929i −0.600476 0.799643i $$-0.705022\pi$$
0.600476 0.799643i $$-0.294978\pi$$
$$984$$ −26.9117 −0.857913
$$985$$ 0 0
$$986$$ −3.71573 −0.118333
$$987$$ − 16.0000i − 0.509286i
$$988$$ 0 0
$$989$$ 16.9706 0.539633
$$990$$ 0 0
$$991$$ 9.94113 0.315790 0.157895 0.987456i $$-0.449529\pi$$
0.157895 + 0.987456i $$0.449529\pi$$
$$992$$ 0 0
$$993$$ 43.3137i 1.37452i
$$994$$ 9.37258 0.297280
$$995$$ 0 0
$$996$$ 31.0294 0.983205
$$997$$ − 9.45584i − 0.299470i −0.988726 0.149735i $$-0.952158\pi$$
0.988726 0.149735i $$-0.0478420\pi$$
$$998$$ 0.686292i 0.0217242i
$$999$$ −20.6863 −0.654485
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.199.3 4
3.2 odd 2 2475.2.c.l.199.2 4
4.3 odd 2 4400.2.b.q.4049.1 4
5.2 odd 4 55.2.a.b.1.1 2
5.3 odd 4 275.2.a.c.1.2 2
5.4 even 2 inner 275.2.b.d.199.2 4
15.2 even 4 495.2.a.b.1.2 2
15.8 even 4 2475.2.a.x.1.1 2
15.14 odd 2 2475.2.c.l.199.3 4
20.3 even 4 4400.2.a.bn.1.2 2
20.7 even 4 880.2.a.m.1.1 2
20.19 odd 2 4400.2.b.q.4049.4 4
35.27 even 4 2695.2.a.f.1.1 2
40.27 even 4 3520.2.a.bo.1.2 2
40.37 odd 4 3520.2.a.bn.1.1 2
55.2 even 20 605.2.g.l.81.1 8
55.7 even 20 605.2.g.l.511.2 8
55.17 even 20 605.2.g.l.366.1 8
55.27 odd 20 605.2.g.f.366.2 8
55.32 even 4 605.2.a.d.1.2 2
55.37 odd 20 605.2.g.f.511.1 8
55.42 odd 20 605.2.g.f.81.2 8
55.43 even 4 3025.2.a.o.1.1 2
55.47 odd 20 605.2.g.f.251.1 8
55.52 even 20 605.2.g.l.251.2 8
60.47 odd 4 7920.2.a.ch.1.1 2
65.12 odd 4 9295.2.a.g.1.2 2
165.32 odd 4 5445.2.a.y.1.1 2
220.87 odd 4 9680.2.a.bn.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
55.2.a.b.1.1 2 5.2 odd 4
275.2.a.c.1.2 2 5.3 odd 4
275.2.b.d.199.2 4 5.4 even 2 inner
275.2.b.d.199.3 4 1.1 even 1 trivial
495.2.a.b.1.2 2 15.2 even 4
605.2.a.d.1.2 2 55.32 even 4
605.2.g.f.81.2 8 55.42 odd 20
605.2.g.f.251.1 8 55.47 odd 20
605.2.g.f.366.2 8 55.27 odd 20
605.2.g.f.511.1 8 55.37 odd 20
605.2.g.l.81.1 8 55.2 even 20
605.2.g.l.251.2 8 55.52 even 20
605.2.g.l.366.1 8 55.17 even 20
605.2.g.l.511.2 8 55.7 even 20
880.2.a.m.1.1 2 20.7 even 4
2475.2.a.x.1.1 2 15.8 even 4
2475.2.c.l.199.2 4 3.2 odd 2
2475.2.c.l.199.3 4 15.14 odd 2
2695.2.a.f.1.1 2 35.27 even 4
3025.2.a.o.1.1 2 55.43 even 4
3520.2.a.bn.1.1 2 40.37 odd 4
3520.2.a.bo.1.2 2 40.27 even 4
4400.2.a.bn.1.2 2 20.3 even 4
4400.2.b.q.4049.1 4 4.3 odd 2
4400.2.b.q.4049.4 4 20.19 odd 2
5445.2.a.y.1.1 2 165.32 odd 4
7920.2.a.ch.1.1 2 60.47 odd 4
9295.2.a.g.1.2 2 65.12 odd 4
9680.2.a.bn.1.1 2 220.87 odd 4