# Properties

 Label 300.2.i.b.257.2 Level $300$ Weight $2$ Character 300.257 Analytic conductor $2.396$ Analytic rank $0$ Dimension $4$ CM discriminant -3 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,2,Mod(257,300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(300, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 1]))

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

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

chi := DirichletCharacter("300.257");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 300.i (of order $$4$$, degree $$2$$, 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: $$2.39551206064$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## Embedding invariants

 Embedding label 257.2 Root $$1.22474 - 1.22474i$$ of defining polynomial Character $$\chi$$ $$=$$ 300.257 Dual form 300.2.i.b.293.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+(1.22474 - 1.22474i) q^{3} +(-2.44949 - 2.44949i) q^{7} -3.00000i q^{9} +O(q^{10})$$ $$q+(1.22474 - 1.22474i) q^{3} +(-2.44949 - 2.44949i) q^{7} -3.00000i q^{9} +(4.89898 - 4.89898i) q^{13} +8.00000i q^{19} -6.00000 q^{21} +(-3.67423 - 3.67423i) q^{27} +4.00000 q^{31} +(4.89898 + 4.89898i) q^{37} -12.0000i q^{39} +(-7.34847 + 7.34847i) q^{43} +5.00000i q^{49} +(9.79796 + 9.79796i) q^{57} +14.0000 q^{61} +(-7.34847 + 7.34847i) q^{63} +(-2.44949 - 2.44949i) q^{67} +(-9.79796 + 9.79796i) q^{73} +4.00000i q^{79} -9.00000 q^{81} -24.0000 q^{91} +(4.89898 - 4.89898i) q^{93} +(9.79796 + 9.79796i) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 24 q^{21} + 16 q^{31} + 56 q^{61} - 36 q^{81} - 96 q^{91}+O(q^{100})$$ 4 * q - 24 * q^21 + 16 * q^31 + 56 * q^61 - 36 * q^81 - 96 * q^91

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/300\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$151$$ $$277$$ $$\chi(n)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{4}\right)$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.22474 1.22474i 0.707107 0.707107i
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −2.44949 2.44949i −0.925820 0.925820i 0.0716124 0.997433i $$-0.477186\pi$$
−0.997433 + 0.0716124i $$0.977186\pi$$
$$8$$ 0 0
$$9$$ 3.00000i 1.00000i
$$10$$ 0 0
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ 0 0
$$13$$ 4.89898 4.89898i 1.35873 1.35873i 0.483250 0.875482i $$-0.339456\pi$$
0.875482 0.483250i $$-0.160544\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$18$$ 0 0
$$19$$ 8.00000i 1.83533i 0.397360 + 0.917663i $$0.369927\pi$$
−0.397360 + 0.917663i $$0.630073\pi$$
$$20$$ 0 0
$$21$$ −6.00000 −1.30931
$$22$$ 0 0
$$23$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −3.67423 3.67423i −0.707107 0.707107i
$$28$$ 0 0
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 4.89898 + 4.89898i 0.805387 + 0.805387i 0.983932 0.178545i $$-0.0571389\pi$$
−0.178545 + 0.983932i $$0.557139\pi$$
$$38$$ 0 0
$$39$$ 12.0000i 1.92154i
$$40$$ 0 0
$$41$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$42$$ 0 0
$$43$$ −7.34847 + 7.34847i −1.12063 + 1.12063i −0.128984 + 0.991647i $$0.541172\pi$$
−0.991647 + 0.128984i $$0.958828\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$48$$ 0 0
$$49$$ 5.00000i 0.714286i
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 9.79796 + 9.79796i 1.29777 + 1.29777i
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 14.0000 1.79252 0.896258 0.443533i $$-0.146275\pi$$
0.896258 + 0.443533i $$0.146275\pi$$
$$62$$ 0 0
$$63$$ −7.34847 + 7.34847i −0.925820 + 0.925820i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −2.44949 2.44949i −0.299253 0.299253i 0.541468 0.840721i $$-0.317869\pi$$
−0.840721 + 0.541468i $$0.817869\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 0 0
$$73$$ −9.79796 + 9.79796i −1.14676 + 1.14676i −0.159579 + 0.987185i $$0.551014\pi$$
−0.987185 + 0.159579i $$0.948986\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 4.00000i 0.450035i 0.974355 + 0.225018i $$0.0722440\pi$$
−0.974355 + 0.225018i $$0.927756\pi$$
$$80$$ 0 0
$$81$$ −9.00000 −1.00000
$$82$$ 0 0
$$83$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ −24.0000 −2.51588
$$92$$ 0 0
$$93$$ 4.89898 4.89898i 0.508001 0.508001i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 9.79796 + 9.79796i 0.994832 + 0.994832i 0.999987 0.00515471i $$-0.00164080\pi$$
−0.00515471 + 0.999987i $$0.501641\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$102$$ 0 0
$$103$$ 2.44949 2.44949i 0.241355 0.241355i −0.576055 0.817411i $$-0.695409\pi$$
0.817411 + 0.576055i $$0.195409\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$108$$ 0 0
$$109$$ 2.00000i 0.191565i −0.995402 0.0957826i $$-0.969465\pi$$
0.995402 0.0957826i $$-0.0305354\pi$$
$$110$$ 0 0
$$111$$ 12.0000 1.13899
$$112$$ 0 0
$$113$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −14.6969 14.6969i −1.35873 1.35873i
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 11.0000 1.00000
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −7.34847 7.34847i −0.652071 0.652071i 0.301420 0.953491i $$-0.402539\pi$$
−0.953491 + 0.301420i $$0.902539\pi$$
$$128$$ 0 0
$$129$$ 18.0000i 1.58481i
$$130$$ 0 0
$$131$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$132$$ 0 0
$$133$$ 19.5959 19.5959i 1.69918 1.69918i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$138$$ 0 0
$$139$$ 16.0000i 1.35710i −0.734553 0.678551i $$-0.762608\pi$$
0.734553 0.678551i $$-0.237392\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 6.12372 + 6.12372i 0.505076 + 0.505076i
$$148$$ 0 0
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ 0 0
$$151$$ 4.00000 0.325515 0.162758 0.986666i $$-0.447961\pi$$
0.162758 + 0.986666i $$0.447961\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −14.6969 14.6969i −1.17294 1.17294i −0.981505 0.191439i $$-0.938685\pi$$
−0.191439 0.981505i $$-0.561315\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 17.1464 17.1464i 1.34301 1.34301i 0.449966 0.893045i $$-0.351436\pi$$
0.893045 0.449966i $$-0.148564\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$168$$ 0 0
$$169$$ 35.0000i 2.69231i
$$170$$ 0 0
$$171$$ 24.0000 1.83533
$$172$$ 0 0
$$173$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ −26.0000 −1.93256 −0.966282 0.257485i $$-0.917106\pi$$
−0.966282 + 0.257485i $$0.917106\pi$$
$$182$$ 0 0
$$183$$ 17.1464 17.1464i 1.26750 1.26750i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 18.0000i 1.30931i
$$190$$ 0 0
$$191$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$192$$ 0 0
$$193$$ −19.5959 + 19.5959i −1.41055 + 1.41055i −0.654374 + 0.756171i $$0.727068\pi$$
−0.756171 + 0.654374i $$0.772932\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$198$$ 0 0
$$199$$ 28.0000i 1.98487i 0.122782 + 0.992434i $$0.460818\pi$$
−0.122782 + 0.992434i $$0.539182\pi$$
$$200$$ 0 0
$$201$$ −6.00000 −0.423207
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 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 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −9.79796 9.79796i −0.665129 0.665129i
$$218$$ 0 0
$$219$$ 24.0000i 1.62177i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −7.34847 + 7.34847i −0.492090 + 0.492090i −0.908964 0.416874i $$-0.863126\pi$$
0.416874 + 0.908964i $$0.363126\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$228$$ 0 0
$$229$$ 22.0000i 1.45380i −0.686743 0.726900i $$-0.740960\pi$$
0.686743 0.726900i $$-0.259040\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 4.89898 + 4.89898i 0.318223 + 0.318223i
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 14.0000 0.901819 0.450910 0.892570i $$-0.351100\pi$$
0.450910 + 0.892570i $$0.351100\pi$$
$$242$$ 0 0
$$243$$ −11.0227 + 11.0227i −0.707107 + 0.707107i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 39.1918 + 39.1918i 2.49372 + 2.49372i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$258$$ 0 0
$$259$$ 24.0000i 1.49129i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 0 0
$$271$$ −28.0000 −1.70088 −0.850439 0.526073i $$-0.823664\pi$$
−0.850439 + 0.526073i $$0.823664\pi$$
$$272$$ 0 0
$$273$$ −29.3939 + 29.3939i −1.77900 + 1.77900i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −14.6969 14.6969i −0.883053 0.883053i 0.110790 0.993844i $$-0.464662\pi$$
−0.993844 + 0.110790i $$0.964662\pi$$
$$278$$ 0 0
$$279$$ 12.0000i 0.718421i
$$280$$ 0 0
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ 0 0
$$283$$ −7.34847 + 7.34847i −0.436821 + 0.436821i −0.890941 0.454120i $$-0.849954\pi$$
0.454120 + 0.890941i $$0.349954\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 17.0000i 1.00000i
$$290$$ 0 0
$$291$$ 24.0000 1.40690
$$292$$ 0 0
$$293$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 36.0000 2.07501
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 22.0454 + 22.0454i 1.25820 + 1.25820i 0.951953 + 0.306245i $$0.0990727\pi$$
0.306245 + 0.951953i $$0.400927\pi$$
$$308$$ 0 0
$$309$$ 6.00000i 0.341328i
$$310$$ 0 0
$$311$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$312$$ 0 0
$$313$$ −19.5959 + 19.5959i −1.10763 + 1.10763i −0.114165 + 0.993462i $$0.536419\pi$$
−0.993462 + 0.114165i $$0.963581\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −2.44949 2.44949i −0.135457 0.135457i
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 32.0000 1.75888 0.879440 0.476011i $$-0.157918\pi$$
0.879440 + 0.476011i $$0.157918\pi$$
$$332$$ 0 0
$$333$$ 14.6969 14.6969i 0.805387 0.805387i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 9.79796 + 9.79796i 0.533729 + 0.533729i 0.921680 0.387951i $$-0.126817\pi$$
−0.387951 + 0.921680i $$0.626817\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −4.89898 + 4.89898i −0.264520 + 0.264520i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$348$$ 0 0
$$349$$ 14.0000i 0.749403i 0.927146 + 0.374701i $$0.122255\pi$$
−0.927146 + 0.374701i $$0.877745\pi$$
$$350$$ 0 0
$$351$$ −36.0000 −1.92154
$$352$$ 0 0
$$353$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ −45.0000 −2.36842
$$362$$ 0 0
$$363$$ 13.4722 13.4722i 0.707107 0.707107i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −26.9444 26.9444i −1.40649 1.40649i −0.777064 0.629421i $$-0.783292\pi$$
−0.629421 0.777064i $$-0.716708\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 4.89898 4.89898i 0.253660 0.253660i −0.568810 0.822469i $$-0.692596\pi$$
0.822469 + 0.568810i $$0.192596\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 8.00000i 0.410932i 0.978664 + 0.205466i $$0.0658711\pi$$
−0.978664 + 0.205466i $$0.934129\pi$$
$$380$$ 0 0
$$381$$ −18.0000 −0.922168
$$382$$ 0 0
$$383$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 22.0454 + 22.0454i 1.12063 + 1.12063i
$$388$$ 0 0
$$389$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −14.6969 14.6969i −0.737618 0.737618i 0.234498 0.972117i $$-0.424655\pi$$
−0.972117 + 0.234498i $$0.924655\pi$$
$$398$$ 0 0
$$399$$ 48.0000i 2.40301i
$$400$$ 0 0
$$401$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$402$$ 0 0
$$403$$ 19.5959 19.5959i 0.976142 0.976142i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 38.0000i 1.87898i 0.342578 + 0.939490i $$0.388700\pi$$
−0.342578 + 0.939490i $$0.611300\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −19.5959 19.5959i −0.959616 0.959616i
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −34.2929 34.2929i −1.65955 1.65955i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ 0 0
$$433$$ 29.3939 29.3939i 1.41258 1.41258i 0.672308 0.740271i $$-0.265303\pi$$
0.740271 0.672308i $$-0.234697\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 28.0000i 1.33637i 0.743996 + 0.668184i $$0.232928\pi$$
−0.743996 + 0.668184i $$0.767072\pi$$
$$440$$ 0 0
$$441$$ 15.0000 0.714286
$$442$$ 0 0
$$443$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 4.89898 4.89898i 0.230174 0.230174i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 29.3939 + 29.3939i 1.37499 + 1.37499i 0.852879 + 0.522108i $$0.174854\pi$$
0.522108 + 0.852879i $$0.325146\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$462$$ 0 0
$$463$$ 26.9444 26.9444i 1.25221 1.25221i 0.297486 0.954726i $$-0.403852\pi$$
0.954726 0.297486i $$-0.0961480\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$468$$ 0 0
$$469$$ 12.0000i 0.554109i
$$470$$ 0 0
$$471$$ −36.0000 −1.65879
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 48.0000 2.18861
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −2.44949 2.44949i −0.110997 0.110997i 0.649427 0.760424i $$-0.275009\pi$$
−0.760424 + 0.649427i $$0.775009\pi$$
$$488$$ 0 0
$$489$$ 42.0000i 1.89931i
$$490$$ 0 0
$$491$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 32.0000i 1.43252i −0.697835 0.716258i $$-0.745853\pi$$
0.697835 0.716258i $$-0.254147\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −42.8661 42.8661i −1.90375 1.90375i
$$508$$ 0 0
$$509$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$510$$ 0 0
$$511$$ 48.0000 2.12339
$$512$$ 0 0
$$513$$ 29.3939 29.3939i 1.29777 1.29777i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$522$$ 0 0
$$523$$ −31.8434 + 31.8434i −1.39241 + 1.39241i −0.572528 + 0.819885i $$0.694037\pi$$
−0.819885 + 0.572528i $$0.805963\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 23.0000i 1.00000i
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −46.0000 −1.97769 −0.988847 0.148933i $$-0.952416\pi$$
−0.988847 + 0.148933i $$0.952416\pi$$
$$542$$ 0 0
$$543$$ −31.8434 + 31.8434i −1.36653 + 1.36653i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 17.1464 + 17.1464i 0.733128 + 0.733128i 0.971238 0.238110i $$-0.0765278\pi$$
−0.238110 + 0.971238i $$0.576528\pi$$
$$548$$ 0 0
$$549$$ 42.0000i 1.79252i
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 9.79796 9.79796i 0.416652 0.416652i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$558$$ 0 0
$$559$$ 72.0000i 3.04528i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 22.0454 + 22.0454i 0.925820 + 0.925820i
$$568$$ 0 0
$$569$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$570$$ 0 0
$$571$$ −16.0000 −0.669579 −0.334790 0.942293i $$-0.608665\pi$$
−0.334790 + 0.942293i $$0.608665\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 9.79796 + 9.79796i 0.407894 + 0.407894i 0.881004 0.473109i $$-0.156868\pi$$
−0.473109 + 0.881004i $$0.656868\pi$$
$$578$$ 0 0
$$579$$ 48.0000i 1.99481i
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$588$$ 0 0
$$589$$ 32.0000i 1.31854i
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 34.2929 + 34.2929i 1.40351 + 1.40351i
$$598$$ 0 0
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ −26.0000 −1.06056 −0.530281 0.847822i $$-0.677914\pi$$
−0.530281 + 0.847822i $$0.677914\pi$$
$$602$$ 0 0
$$603$$ −7.34847 + 7.34847i −0.299253 + 0.299253i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −31.8434 31.8434i −1.29248 1.29248i −0.933247 0.359235i $$-0.883038\pi$$
−0.359235 0.933247i $$-0.616962\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −34.2929 + 34.2929i −1.38508 + 1.38508i −0.549739 + 0.835337i $$0.685273\pi$$
−0.835337 + 0.549739i $$0.814727\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$618$$ 0 0
$$619$$ 32.0000i 1.28619i −0.765787 0.643094i $$-0.777650\pi$$
0.765787 0.643094i $$-0.222350\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 44.0000 1.75161 0.875806 0.482663i $$-0.160330\pi$$
0.875806 + 0.482663i $$0.160330\pi$$
$$632$$ 0 0
$$633$$ −19.5959 + 19.5959i −0.778868 + 0.778868i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 24.4949 + 24.4949i 0.970523 + 0.970523i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$642$$ 0 0
$$643$$ −22.0454 + 22.0454i −0.869386 + 0.869386i −0.992404 0.123018i $$-0.960743\pi$$
0.123018 + 0.992404i $$0.460743\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −24.0000 −0.940634
$$652$$ 0 0
$$653$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 29.3939 + 29.3939i 1.14676 + 1.14676i
$$658$$ 0 0
$$659$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ 0 0
$$661$$ −38.0000 −1.47803 −0.739014 0.673690i $$-0.764708\pi$$
−0.739014 + 0.673690i $$0.764708\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 18.0000i 0.695920i
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −9.79796 + 9.79796i −0.377684 + 0.377684i −0.870266 0.492582i $$-0.836053\pi$$
0.492582 + 0.870266i $$0.336053\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$678$$ 0 0
$$679$$ 48.0000i 1.84207i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −26.9444 26.9444i −1.02799 1.02799i
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −8.00000 −0.304334 −0.152167 0.988355i $$-0.548625\pi$$
−0.152167 + 0.988355i $$0.548625\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$702$$ 0 0
$$703$$ −39.1918 + 39.1918i −1.47815 + 1.47815i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 22.0000i 0.826227i −0.910679 0.413114i $$-0.864441\pi$$
0.910679 0.413114i $$-0.135559\pi$$
$$710$$ 0 0
$$711$$ 12.0000 0.450035
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ −12.0000 −0.446903
$$722$$ 0 0
$$723$$ 17.1464 17.1464i 0.637683 0.637683i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 22.0454 + 22.0454i 0.817619 + 0.817619i 0.985763 0.168144i $$-0.0537772\pi$$
−0.168144 + 0.985763i $$0.553777\pi$$
$$728$$ 0 0
$$729$$ 27.0000i 1.00000i
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 14.6969 14.6969i 0.542844 0.542844i −0.381518 0.924362i $$-0.624598\pi$$
0.924362 + 0.381518i $$0.124598\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 16.0000i 0.588570i −0.955718 0.294285i $$-0.904919\pi$$
0.955718 0.294285i $$-0.0950814\pi$$
$$740$$ 0 0
$$741$$ 96.0000 3.52665
$$742$$ 0 0
$$743$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 52.0000 1.89751 0.948753 0.316017i $$-0.102346\pi$$
0.948753 + 0.316017i $$0.102346\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 34.2929 + 34.2929i 1.24640 + 1.24640i 0.957301 + 0.289095i $$0.0933542\pi$$
0.289095 + 0.957301i $$0.406646\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$762$$ 0 0
$$763$$ −4.89898 + 4.89898i −0.177355 + 0.177355i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 2.00000i 0.0721218i −0.999350 0.0360609i $$-0.988519\pi$$
0.999350 0.0360609i $$-0.0114810\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −29.3939 29.3939i −1.05450 1.05450i
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −2.44949 2.44949i −0.0873149 0.0873149i 0.662100 0.749415i $$-0.269665\pi$$
−0.749415 + 0.662100i $$0.769665\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 68.5857 68.5857i 2.43555 2.43555i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0