# Properties

 Label 405.2.a.g.1.2 Level $405$ Weight $2$ Character 405.1 Self dual yes Analytic conductor $3.234$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,2,Mod(1,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("405.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 405.a (trivial)

## 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: yes Analytic conductor: $$3.23394128186$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 405.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.732051 q^{2} -1.46410 q^{4} +1.00000 q^{5} -4.73205 q^{7} -2.53590 q^{8} +O(q^{10})$$ $$q+0.732051 q^{2} -1.46410 q^{4} +1.00000 q^{5} -4.73205 q^{7} -2.53590 q^{8} +0.732051 q^{10} -5.73205 q^{11} +1.46410 q^{13} -3.46410 q^{14} +1.07180 q^{16} -2.73205 q^{17} +4.46410 q^{19} -1.46410 q^{20} -4.19615 q^{22} -3.46410 q^{23} +1.00000 q^{25} +1.07180 q^{26} +6.92820 q^{28} +3.19615 q^{29} -3.00000 q^{31} +5.85641 q^{32} -2.00000 q^{34} -4.73205 q^{35} -2.73205 q^{37} +3.26795 q^{38} -2.53590 q^{40} -7.19615 q^{41} +0.196152 q^{43} +8.39230 q^{44} -2.53590 q^{46} -8.73205 q^{47} +15.3923 q^{49} +0.732051 q^{50} -2.14359 q^{52} +6.73205 q^{53} -5.73205 q^{55} +12.0000 q^{56} +2.33975 q^{58} -8.26795 q^{59} +4.00000 q^{61} -2.19615 q^{62} +2.14359 q^{64} +1.46410 q^{65} +3.46410 q^{67} +4.00000 q^{68} -3.46410 q^{70} -3.73205 q^{71} -7.66025 q^{73} -2.00000 q^{74} -6.53590 q^{76} +27.1244 q^{77} +15.4641 q^{79} +1.07180 q^{80} -5.26795 q^{82} +2.19615 q^{83} -2.73205 q^{85} +0.143594 q^{86} +14.5359 q^{88} +5.19615 q^{89} -6.92820 q^{91} +5.07180 q^{92} -6.39230 q^{94} +4.46410 q^{95} -9.66025 q^{97} +11.2679 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 4 q^{4} + 2 q^{5} - 6 q^{7} - 12 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 4 * q^4 + 2 * q^5 - 6 * q^7 - 12 * q^8 $$2 q - 2 q^{2} + 4 q^{4} + 2 q^{5} - 6 q^{7} - 12 q^{8} - 2 q^{10} - 8 q^{11} - 4 q^{13} + 16 q^{16} - 2 q^{17} + 2 q^{19} + 4 q^{20} + 2 q^{22} + 2 q^{25} + 16 q^{26} - 4 q^{29} - 6 q^{31} - 16 q^{32} - 4 q^{34} - 6 q^{35} - 2 q^{37} + 10 q^{38} - 12 q^{40} - 4 q^{41} - 10 q^{43} - 4 q^{44} - 12 q^{46} - 14 q^{47} + 10 q^{49} - 2 q^{50} - 32 q^{52} + 10 q^{53} - 8 q^{55} + 24 q^{56} + 22 q^{58} - 20 q^{59} + 8 q^{61} + 6 q^{62} + 32 q^{64} - 4 q^{65} + 8 q^{68} - 4 q^{71} + 2 q^{73} - 4 q^{74} - 20 q^{76} + 30 q^{77} + 24 q^{79} + 16 q^{80} - 14 q^{82} - 6 q^{83} - 2 q^{85} + 28 q^{86} + 36 q^{88} + 24 q^{92} + 8 q^{94} + 2 q^{95} - 2 q^{97} + 26 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 4 * q^4 + 2 * q^5 - 6 * q^7 - 12 * q^8 - 2 * q^10 - 8 * q^11 - 4 * q^13 + 16 * q^16 - 2 * q^17 + 2 * q^19 + 4 * q^20 + 2 * q^22 + 2 * q^25 + 16 * q^26 - 4 * q^29 - 6 * q^31 - 16 * q^32 - 4 * q^34 - 6 * q^35 - 2 * q^37 + 10 * q^38 - 12 * q^40 - 4 * q^41 - 10 * q^43 - 4 * q^44 - 12 * q^46 - 14 * q^47 + 10 * q^49 - 2 * q^50 - 32 * q^52 + 10 * q^53 - 8 * q^55 + 24 * q^56 + 22 * q^58 - 20 * q^59 + 8 * q^61 + 6 * q^62 + 32 * q^64 - 4 * q^65 + 8 * q^68 - 4 * q^71 + 2 * q^73 - 4 * q^74 - 20 * q^76 + 30 * q^77 + 24 * q^79 + 16 * q^80 - 14 * q^82 - 6 * q^83 - 2 * q^85 + 28 * q^86 + 36 * q^88 + 24 * q^92 + 8 * q^94 + 2 * q^95 - 2 * q^97 + 26 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.732051 0.517638 0.258819 0.965926i $$-0.416667\pi$$
0.258819 + 0.965926i $$0.416667\pi$$
$$3$$ 0 0
$$4$$ −1.46410 −0.732051
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −4.73205 −1.78855 −0.894274 0.447521i $$-0.852307\pi$$
−0.894274 + 0.447521i $$0.852307\pi$$
$$8$$ −2.53590 −0.896575
$$9$$ 0 0
$$10$$ 0.732051 0.231495
$$11$$ −5.73205 −1.72828 −0.864139 0.503253i $$-0.832136\pi$$
−0.864139 + 0.503253i $$0.832136\pi$$
$$12$$ 0 0
$$13$$ 1.46410 0.406069 0.203034 0.979172i $$-0.434920\pi$$
0.203034 + 0.979172i $$0.434920\pi$$
$$14$$ −3.46410 −0.925820
$$15$$ 0 0
$$16$$ 1.07180 0.267949
$$17$$ −2.73205 −0.662620 −0.331310 0.943522i $$-0.607491\pi$$
−0.331310 + 0.943522i $$0.607491\pi$$
$$18$$ 0 0
$$19$$ 4.46410 1.02414 0.512068 0.858945i $$-0.328880\pi$$
0.512068 + 0.858945i $$0.328880\pi$$
$$20$$ −1.46410 −0.327383
$$21$$ 0 0
$$22$$ −4.19615 −0.894623
$$23$$ −3.46410 −0.722315 −0.361158 0.932505i $$-0.617618\pi$$
−0.361158 + 0.932505i $$0.617618\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 1.07180 0.210197
$$27$$ 0 0
$$28$$ 6.92820 1.30931
$$29$$ 3.19615 0.593511 0.296755 0.954954i $$-0.404095\pi$$
0.296755 + 0.954954i $$0.404095\pi$$
$$30$$ 0 0
$$31$$ −3.00000 −0.538816 −0.269408 0.963026i $$-0.586828\pi$$
−0.269408 + 0.963026i $$0.586828\pi$$
$$32$$ 5.85641 1.03528
$$33$$ 0 0
$$34$$ −2.00000 −0.342997
$$35$$ −4.73205 −0.799863
$$36$$ 0 0
$$37$$ −2.73205 −0.449146 −0.224573 0.974457i $$-0.572099\pi$$
−0.224573 + 0.974457i $$0.572099\pi$$
$$38$$ 3.26795 0.530131
$$39$$ 0 0
$$40$$ −2.53590 −0.400961
$$41$$ −7.19615 −1.12385 −0.561925 0.827188i $$-0.689939\pi$$
−0.561925 + 0.827188i $$0.689939\pi$$
$$42$$ 0 0
$$43$$ 0.196152 0.0299130 0.0149565 0.999888i $$-0.495239\pi$$
0.0149565 + 0.999888i $$0.495239\pi$$
$$44$$ 8.39230 1.26519
$$45$$ 0 0
$$46$$ −2.53590 −0.373898
$$47$$ −8.73205 −1.27370 −0.636850 0.770988i $$-0.719763\pi$$
−0.636850 + 0.770988i $$0.719763\pi$$
$$48$$ 0 0
$$49$$ 15.3923 2.19890
$$50$$ 0.732051 0.103528
$$51$$ 0 0
$$52$$ −2.14359 −0.297263
$$53$$ 6.73205 0.924718 0.462359 0.886693i $$-0.347003\pi$$
0.462359 + 0.886693i $$0.347003\pi$$
$$54$$ 0 0
$$55$$ −5.73205 −0.772910
$$56$$ 12.0000 1.60357
$$57$$ 0 0
$$58$$ 2.33975 0.307224
$$59$$ −8.26795 −1.07640 −0.538198 0.842819i $$-0.680895\pi$$
−0.538198 + 0.842819i $$0.680895\pi$$
$$60$$ 0 0
$$61$$ 4.00000 0.512148 0.256074 0.966657i $$-0.417571\pi$$
0.256074 + 0.966657i $$0.417571\pi$$
$$62$$ −2.19615 −0.278912
$$63$$ 0 0
$$64$$ 2.14359 0.267949
$$65$$ 1.46410 0.181599
$$66$$ 0 0
$$67$$ 3.46410 0.423207 0.211604 0.977356i $$-0.432131\pi$$
0.211604 + 0.977356i $$0.432131\pi$$
$$68$$ 4.00000 0.485071
$$69$$ 0 0
$$70$$ −3.46410 −0.414039
$$71$$ −3.73205 −0.442913 −0.221456 0.975170i $$-0.571081\pi$$
−0.221456 + 0.975170i $$0.571081\pi$$
$$72$$ 0 0
$$73$$ −7.66025 −0.896565 −0.448282 0.893892i $$-0.647964\pi$$
−0.448282 + 0.893892i $$0.647964\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 0 0
$$76$$ −6.53590 −0.749719
$$77$$ 27.1244 3.09111
$$78$$ 0 0
$$79$$ 15.4641 1.73985 0.869924 0.493186i $$-0.164168\pi$$
0.869924 + 0.493186i $$0.164168\pi$$
$$80$$ 1.07180 0.119831
$$81$$ 0 0
$$82$$ −5.26795 −0.581748
$$83$$ 2.19615 0.241059 0.120530 0.992710i $$-0.461541\pi$$
0.120530 + 0.992710i $$0.461541\pi$$
$$84$$ 0 0
$$85$$ −2.73205 −0.296333
$$86$$ 0.143594 0.0154841
$$87$$ 0 0
$$88$$ 14.5359 1.54953
$$89$$ 5.19615 0.550791 0.275396 0.961331i $$-0.411191\pi$$
0.275396 + 0.961331i $$0.411191\pi$$
$$90$$ 0 0
$$91$$ −6.92820 −0.726273
$$92$$ 5.07180 0.528771
$$93$$ 0 0
$$94$$ −6.39230 −0.659316
$$95$$ 4.46410 0.458007
$$96$$ 0 0
$$97$$ −9.66025 −0.980850 −0.490425 0.871483i $$-0.663158\pi$$
−0.490425 + 0.871483i $$0.663158\pi$$
$$98$$ 11.2679 1.13823
$$99$$ 0 0
$$100$$ −1.46410 −0.146410
$$101$$ −2.66025 −0.264705 −0.132353 0.991203i $$-0.542253\pi$$
−0.132353 + 0.991203i $$0.542253\pi$$
$$102$$ 0 0
$$103$$ −0.535898 −0.0528036 −0.0264018 0.999651i $$-0.508405\pi$$
−0.0264018 + 0.999651i $$0.508405\pi$$
$$104$$ −3.71281 −0.364071
$$105$$ 0 0
$$106$$ 4.92820 0.478669
$$107$$ −8.53590 −0.825196 −0.412598 0.910913i $$-0.635379\pi$$
−0.412598 + 0.910913i $$0.635379\pi$$
$$108$$ 0 0
$$109$$ −6.07180 −0.581573 −0.290786 0.956788i $$-0.593917\pi$$
−0.290786 + 0.956788i $$0.593917\pi$$
$$110$$ −4.19615 −0.400087
$$111$$ 0 0
$$112$$ −5.07180 −0.479240
$$113$$ −19.1244 −1.79907 −0.899534 0.436851i $$-0.856094\pi$$
−0.899534 + 0.436851i $$0.856094\pi$$
$$114$$ 0 0
$$115$$ −3.46410 −0.323029
$$116$$ −4.67949 −0.434480
$$117$$ 0 0
$$118$$ −6.05256 −0.557183
$$119$$ 12.9282 1.18513
$$120$$ 0 0
$$121$$ 21.8564 1.98695
$$122$$ 2.92820 0.265107
$$123$$ 0 0
$$124$$ 4.39230 0.394441
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −14.5885 −1.29452 −0.647258 0.762271i $$-0.724084\pi$$
−0.647258 + 0.762271i $$0.724084\pi$$
$$128$$ −10.1436 −0.896575
$$129$$ 0 0
$$130$$ 1.07180 0.0940028
$$131$$ 15.5885 1.36197 0.680985 0.732297i $$-0.261552\pi$$
0.680985 + 0.732297i $$0.261552\pi$$
$$132$$ 0 0
$$133$$ −21.1244 −1.83171
$$134$$ 2.53590 0.219068
$$135$$ 0 0
$$136$$ 6.92820 0.594089
$$137$$ 2.53590 0.216656 0.108328 0.994115i $$-0.465450\pi$$
0.108328 + 0.994115i $$0.465450\pi$$
$$138$$ 0 0
$$139$$ 0.607695 0.0515440 0.0257720 0.999668i $$-0.491796\pi$$
0.0257720 + 0.999668i $$0.491796\pi$$
$$140$$ 6.92820 0.585540
$$141$$ 0 0
$$142$$ −2.73205 −0.229269
$$143$$ −8.39230 −0.701800
$$144$$ 0 0
$$145$$ 3.19615 0.265426
$$146$$ −5.60770 −0.464096
$$147$$ 0 0
$$148$$ 4.00000 0.328798
$$149$$ −8.00000 −0.655386 −0.327693 0.944784i $$-0.606271\pi$$
−0.327693 + 0.944784i $$0.606271\pi$$
$$150$$ 0 0
$$151$$ 5.39230 0.438820 0.219410 0.975633i $$-0.429587\pi$$
0.219410 + 0.975633i $$0.429587\pi$$
$$152$$ −11.3205 −0.918214
$$153$$ 0 0
$$154$$ 19.8564 1.60007
$$155$$ −3.00000 −0.240966
$$156$$ 0 0
$$157$$ −19.1244 −1.52629 −0.763145 0.646227i $$-0.776346\pi$$
−0.763145 + 0.646227i $$0.776346\pi$$
$$158$$ 11.3205 0.900611
$$159$$ 0 0
$$160$$ 5.85641 0.462990
$$161$$ 16.3923 1.29189
$$162$$ 0 0
$$163$$ 12.7321 0.997251 0.498626 0.866817i $$-0.333838\pi$$
0.498626 + 0.866817i $$0.333838\pi$$
$$164$$ 10.5359 0.822715
$$165$$ 0 0
$$166$$ 1.60770 0.124781
$$167$$ −17.6603 −1.36659 −0.683296 0.730142i $$-0.739454\pi$$
−0.683296 + 0.730142i $$0.739454\pi$$
$$168$$ 0 0
$$169$$ −10.8564 −0.835108
$$170$$ −2.00000 −0.153393
$$171$$ 0 0
$$172$$ −0.287187 −0.0218978
$$173$$ 8.53590 0.648972 0.324486 0.945890i $$-0.394809\pi$$
0.324486 + 0.945890i $$0.394809\pi$$
$$174$$ 0 0
$$175$$ −4.73205 −0.357709
$$176$$ −6.14359 −0.463091
$$177$$ 0 0
$$178$$ 3.80385 0.285110
$$179$$ 8.12436 0.607243 0.303621 0.952793i $$-0.401804\pi$$
0.303621 + 0.952793i $$0.401804\pi$$
$$180$$ 0 0
$$181$$ 26.4641 1.96706 0.983531 0.180742i $$-0.0578498\pi$$
0.983531 + 0.180742i $$0.0578498\pi$$
$$182$$ −5.07180 −0.375947
$$183$$ 0 0
$$184$$ 8.78461 0.647610
$$185$$ −2.73205 −0.200864
$$186$$ 0 0
$$187$$ 15.6603 1.14519
$$188$$ 12.7846 0.932413
$$189$$ 0 0
$$190$$ 3.26795 0.237082
$$191$$ −8.12436 −0.587858 −0.293929 0.955827i $$-0.594963\pi$$
−0.293929 + 0.955827i $$0.594963\pi$$
$$192$$ 0 0
$$193$$ −5.26795 −0.379195 −0.189598 0.981862i $$-0.560718\pi$$
−0.189598 + 0.981862i $$0.560718\pi$$
$$194$$ −7.07180 −0.507725
$$195$$ 0 0
$$196$$ −22.5359 −1.60971
$$197$$ −13.8564 −0.987228 −0.493614 0.869681i $$-0.664324\pi$$
−0.493614 + 0.869681i $$0.664324\pi$$
$$198$$ 0 0
$$199$$ −2.00000 −0.141776 −0.0708881 0.997484i $$-0.522583\pi$$
−0.0708881 + 0.997484i $$0.522583\pi$$
$$200$$ −2.53590 −0.179315
$$201$$ 0 0
$$202$$ −1.94744 −0.137021
$$203$$ −15.1244 −1.06152
$$204$$ 0 0
$$205$$ −7.19615 −0.502601
$$206$$ −0.392305 −0.0273332
$$207$$ 0 0
$$208$$ 1.56922 0.108806
$$209$$ −25.5885 −1.76999
$$210$$ 0 0
$$211$$ −8.85641 −0.609700 −0.304850 0.952400i $$-0.598606\pi$$
−0.304850 + 0.952400i $$0.598606\pi$$
$$212$$ −9.85641 −0.676941
$$213$$ 0 0
$$214$$ −6.24871 −0.427153
$$215$$ 0.196152 0.0133775
$$216$$ 0 0
$$217$$ 14.1962 0.963698
$$218$$ −4.44486 −0.301044
$$219$$ 0 0
$$220$$ 8.39230 0.565809
$$221$$ −4.00000 −0.269069
$$222$$ 0 0
$$223$$ 16.7846 1.12398 0.561990 0.827144i $$-0.310036\pi$$
0.561990 + 0.827144i $$0.310036\pi$$
$$224$$ −27.7128 −1.85164
$$225$$ 0 0
$$226$$ −14.0000 −0.931266
$$227$$ −18.0526 −1.19819 −0.599095 0.800678i $$-0.704473\pi$$
−0.599095 + 0.800678i $$0.704473\pi$$
$$228$$ 0 0
$$229$$ −12.0000 −0.792982 −0.396491 0.918039i $$-0.629772\pi$$
−0.396491 + 0.918039i $$0.629772\pi$$
$$230$$ −2.53590 −0.167212
$$231$$ 0 0
$$232$$ −8.10512 −0.532127
$$233$$ −28.0526 −1.83778 −0.918892 0.394509i $$-0.870915\pi$$
−0.918892 + 0.394509i $$0.870915\pi$$
$$234$$ 0 0
$$235$$ −8.73205 −0.569616
$$236$$ 12.1051 0.787976
$$237$$ 0 0
$$238$$ 9.46410 0.613467
$$239$$ −0.535898 −0.0346644 −0.0173322 0.999850i $$-0.505517\pi$$
−0.0173322 + 0.999850i $$0.505517\pi$$
$$240$$ 0 0
$$241$$ −16.3205 −1.05130 −0.525648 0.850702i $$-0.676177\pi$$
−0.525648 + 0.850702i $$0.676177\pi$$
$$242$$ 16.0000 1.02852
$$243$$ 0 0
$$244$$ −5.85641 −0.374918
$$245$$ 15.3923 0.983378
$$246$$ 0 0
$$247$$ 6.53590 0.415869
$$248$$ 7.60770 0.483089
$$249$$ 0 0
$$250$$ 0.732051 0.0462990
$$251$$ −10.3923 −0.655956 −0.327978 0.944685i $$-0.606367\pi$$
−0.327978 + 0.944685i $$0.606367\pi$$
$$252$$ 0 0
$$253$$ 19.8564 1.24836
$$254$$ −10.6795 −0.670091
$$255$$ 0 0
$$256$$ −11.7128 −0.732051
$$257$$ 10.3923 0.648254 0.324127 0.946014i $$-0.394929\pi$$
0.324127 + 0.946014i $$0.394929\pi$$
$$258$$ 0 0
$$259$$ 12.9282 0.803319
$$260$$ −2.14359 −0.132940
$$261$$ 0 0
$$262$$ 11.4115 0.705007
$$263$$ 21.3205 1.31468 0.657339 0.753595i $$-0.271682\pi$$
0.657339 + 0.753595i $$0.271682\pi$$
$$264$$ 0 0
$$265$$ 6.73205 0.413547
$$266$$ −15.4641 −0.948165
$$267$$ 0 0
$$268$$ −5.07180 −0.309809
$$269$$ 10.6603 0.649967 0.324984 0.945720i $$-0.394641\pi$$
0.324984 + 0.945720i $$0.394641\pi$$
$$270$$ 0 0
$$271$$ −2.92820 −0.177876 −0.0889378 0.996037i $$-0.528347\pi$$
−0.0889378 + 0.996037i $$0.528347\pi$$
$$272$$ −2.92820 −0.177548
$$273$$ 0 0
$$274$$ 1.85641 0.112150
$$275$$ −5.73205 −0.345656
$$276$$ 0 0
$$277$$ 3.80385 0.228551 0.114276 0.993449i $$-0.463545\pi$$
0.114276 + 0.993449i $$0.463545\pi$$
$$278$$ 0.444864 0.0266812
$$279$$ 0 0
$$280$$ 12.0000 0.717137
$$281$$ 15.4641 0.922511 0.461255 0.887267i $$-0.347399\pi$$
0.461255 + 0.887267i $$0.347399\pi$$
$$282$$ 0 0
$$283$$ −29.3205 −1.74292 −0.871462 0.490464i $$-0.836827\pi$$
−0.871462 + 0.490464i $$0.836827\pi$$
$$284$$ 5.46410 0.324235
$$285$$ 0 0
$$286$$ −6.14359 −0.363278
$$287$$ 34.0526 2.01006
$$288$$ 0 0
$$289$$ −9.53590 −0.560935
$$290$$ 2.33975 0.137395
$$291$$ 0 0
$$292$$ 11.2154 0.656331
$$293$$ 28.7321 1.67854 0.839272 0.543712i $$-0.182981\pi$$
0.839272 + 0.543712i $$0.182981\pi$$
$$294$$ 0 0
$$295$$ −8.26795 −0.481379
$$296$$ 6.92820 0.402694
$$297$$ 0 0
$$298$$ −5.85641 −0.339253
$$299$$ −5.07180 −0.293310
$$300$$ 0 0
$$301$$ −0.928203 −0.0535007
$$302$$ 3.94744 0.227150
$$303$$ 0 0
$$304$$ 4.78461 0.274416
$$305$$ 4.00000 0.229039
$$306$$ 0 0
$$307$$ 14.0526 0.802022 0.401011 0.916073i $$-0.368659\pi$$
0.401011 + 0.916073i $$0.368659\pi$$
$$308$$ −39.7128 −2.26285
$$309$$ 0 0
$$310$$ −2.19615 −0.124733
$$311$$ 19.7321 1.11890 0.559451 0.828863i $$-0.311012\pi$$
0.559451 + 0.828863i $$0.311012\pi$$
$$312$$ 0 0
$$313$$ −9.07180 −0.512768 −0.256384 0.966575i $$-0.582531\pi$$
−0.256384 + 0.966575i $$0.582531\pi$$
$$314$$ −14.0000 −0.790066
$$315$$ 0 0
$$316$$ −22.6410 −1.27366
$$317$$ 6.19615 0.348011 0.174005 0.984745i $$-0.444329\pi$$
0.174005 + 0.984745i $$0.444329\pi$$
$$318$$ 0 0
$$319$$ −18.3205 −1.02575
$$320$$ 2.14359 0.119831
$$321$$ 0 0
$$322$$ 12.0000 0.668734
$$323$$ −12.1962 −0.678612
$$324$$ 0 0
$$325$$ 1.46410 0.0812137
$$326$$ 9.32051 0.516215
$$327$$ 0 0
$$328$$ 18.2487 1.00762
$$329$$ 41.3205 2.27807
$$330$$ 0 0
$$331$$ −0.464102 −0.0255093 −0.0127547 0.999919i $$-0.504060\pi$$
−0.0127547 + 0.999919i $$0.504060\pi$$
$$332$$ −3.21539 −0.176467
$$333$$ 0 0
$$334$$ −12.9282 −0.707400
$$335$$ 3.46410 0.189264
$$336$$ 0 0
$$337$$ 27.3205 1.48824 0.744121 0.668044i $$-0.232868\pi$$
0.744121 + 0.668044i $$0.232868\pi$$
$$338$$ −7.94744 −0.432284
$$339$$ 0 0
$$340$$ 4.00000 0.216930
$$341$$ 17.1962 0.931224
$$342$$ 0 0
$$343$$ −39.7128 −2.14429
$$344$$ −0.497423 −0.0268192
$$345$$ 0 0
$$346$$ 6.24871 0.335933
$$347$$ −28.5885 −1.53471 −0.767354 0.641223i $$-0.778427\pi$$
−0.767354 + 0.641223i $$0.778427\pi$$
$$348$$ 0 0
$$349$$ −18.8564 −1.00936 −0.504680 0.863306i $$-0.668390\pi$$
−0.504680 + 0.863306i $$0.668390\pi$$
$$350$$ −3.46410 −0.185164
$$351$$ 0 0
$$352$$ −33.5692 −1.78925
$$353$$ 25.5167 1.35811 0.679057 0.734085i $$-0.262389\pi$$
0.679057 + 0.734085i $$0.262389\pi$$
$$354$$ 0 0
$$355$$ −3.73205 −0.198077
$$356$$ −7.60770 −0.403207
$$357$$ 0 0
$$358$$ 5.94744 0.314332
$$359$$ −6.12436 −0.323231 −0.161616 0.986854i $$-0.551670\pi$$
−0.161616 + 0.986854i $$0.551670\pi$$
$$360$$ 0 0
$$361$$ 0.928203 0.0488528
$$362$$ 19.3731 1.01823
$$363$$ 0 0
$$364$$ 10.1436 0.531669
$$365$$ −7.66025 −0.400956
$$366$$ 0 0
$$367$$ 31.1769 1.62742 0.813711 0.581270i $$-0.197444\pi$$
0.813711 + 0.581270i $$0.197444\pi$$
$$368$$ −3.71281 −0.193544
$$369$$ 0 0
$$370$$ −2.00000 −0.103975
$$371$$ −31.8564 −1.65390
$$372$$ 0 0
$$373$$ 20.0526 1.03828 0.519141 0.854689i $$-0.326252\pi$$
0.519141 + 0.854689i $$0.326252\pi$$
$$374$$ 11.4641 0.592795
$$375$$ 0 0
$$376$$ 22.1436 1.14197
$$377$$ 4.67949 0.241006
$$378$$ 0 0
$$379$$ 2.39230 0.122884 0.0614422 0.998111i $$-0.480430\pi$$
0.0614422 + 0.998111i $$0.480430\pi$$
$$380$$ −6.53590 −0.335285
$$381$$ 0 0
$$382$$ −5.94744 −0.304298
$$383$$ 2.53590 0.129578 0.0647892 0.997899i $$-0.479363\pi$$
0.0647892 + 0.997899i $$0.479363\pi$$
$$384$$ 0 0
$$385$$ 27.1244 1.38239
$$386$$ −3.85641 −0.196286
$$387$$ 0 0
$$388$$ 14.1436 0.718032
$$389$$ 27.4641 1.39249 0.696243 0.717807i $$-0.254854\pi$$
0.696243 + 0.717807i $$0.254854\pi$$
$$390$$ 0 0
$$391$$ 9.46410 0.478620
$$392$$ −39.0333 −1.97148
$$393$$ 0 0
$$394$$ −10.1436 −0.511027
$$395$$ 15.4641 0.778083
$$396$$ 0 0
$$397$$ 14.3923 0.722329 0.361165 0.932502i $$-0.382379\pi$$
0.361165 + 0.932502i $$0.382379\pi$$
$$398$$ −1.46410 −0.0733888
$$399$$ 0 0
$$400$$ 1.07180 0.0535898
$$401$$ 24.9282 1.24486 0.622428 0.782677i $$-0.286147\pi$$
0.622428 + 0.782677i $$0.286147\pi$$
$$402$$ 0 0
$$403$$ −4.39230 −0.218796
$$404$$ 3.89488 0.193778
$$405$$ 0 0
$$406$$ −11.0718 −0.549484
$$407$$ 15.6603 0.776250
$$408$$ 0 0
$$409$$ −17.8564 −0.882942 −0.441471 0.897275i $$-0.645543\pi$$
−0.441471 + 0.897275i $$0.645543\pi$$
$$410$$ −5.26795 −0.260165
$$411$$ 0 0
$$412$$ 0.784610 0.0386549
$$413$$ 39.1244 1.92518
$$414$$ 0 0
$$415$$ 2.19615 0.107805
$$416$$ 8.57437 0.420393
$$417$$ 0 0
$$418$$ −18.7321 −0.916215
$$419$$ 20.3923 0.996229 0.498115 0.867111i $$-0.334026\pi$$
0.498115 + 0.867111i $$0.334026\pi$$
$$420$$ 0 0
$$421$$ 33.7846 1.64656 0.823281 0.567635i $$-0.192141\pi$$
0.823281 + 0.567635i $$0.192141\pi$$
$$422$$ −6.48334 −0.315604
$$423$$ 0 0
$$424$$ −17.0718 −0.829080
$$425$$ −2.73205 −0.132524
$$426$$ 0 0
$$427$$ −18.9282 −0.916000
$$428$$ 12.4974 0.604086
$$429$$ 0 0
$$430$$ 0.143594 0.00692470
$$431$$ −21.3397 −1.02790 −0.513950 0.857820i $$-0.671818\pi$$
−0.513950 + 0.857820i $$0.671818\pi$$
$$432$$ 0 0
$$433$$ −35.4641 −1.70430 −0.852148 0.523301i $$-0.824700\pi$$
−0.852148 + 0.523301i $$0.824700\pi$$
$$434$$ 10.3923 0.498847
$$435$$ 0 0
$$436$$ 8.88973 0.425741
$$437$$ −15.4641 −0.739748
$$438$$ 0 0
$$439$$ 5.39230 0.257361 0.128680 0.991686i $$-0.458926\pi$$
0.128680 + 0.991686i $$0.458926\pi$$
$$440$$ 14.5359 0.692972
$$441$$ 0 0
$$442$$ −2.92820 −0.139280
$$443$$ −0.339746 −0.0161418 −0.00807091 0.999967i $$-0.502569\pi$$
−0.00807091 + 0.999967i $$0.502569\pi$$
$$444$$ 0 0
$$445$$ 5.19615 0.246321
$$446$$ 12.2872 0.581815
$$447$$ 0 0
$$448$$ −10.1436 −0.479240
$$449$$ 8.12436 0.383412 0.191706 0.981452i $$-0.438598\pi$$
0.191706 + 0.981452i $$0.438598\pi$$
$$450$$ 0 0
$$451$$ 41.2487 1.94233
$$452$$ 28.0000 1.31701
$$453$$ 0 0
$$454$$ −13.2154 −0.620229
$$455$$ −6.92820 −0.324799
$$456$$ 0 0
$$457$$ 2.73205 0.127800 0.0639000 0.997956i $$-0.479646\pi$$
0.0639000 + 0.997956i $$0.479646\pi$$
$$458$$ −8.78461 −0.410478
$$459$$ 0 0
$$460$$ 5.07180 0.236474
$$461$$ −37.0526 −1.72571 −0.862855 0.505452i $$-0.831326\pi$$
−0.862855 + 0.505452i $$0.831326\pi$$
$$462$$ 0 0
$$463$$ −10.3923 −0.482971 −0.241486 0.970404i $$-0.577635\pi$$
−0.241486 + 0.970404i $$0.577635\pi$$
$$464$$ 3.42563 0.159031
$$465$$ 0 0
$$466$$ −20.5359 −0.951307
$$467$$ 37.3731 1.72942 0.864710 0.502272i $$-0.167502\pi$$
0.864710 + 0.502272i $$0.167502\pi$$
$$468$$ 0 0
$$469$$ −16.3923 −0.756926
$$470$$ −6.39230 −0.294855
$$471$$ 0 0
$$472$$ 20.9667 0.965070
$$473$$ −1.12436 −0.0516979
$$474$$ 0 0
$$475$$ 4.46410 0.204827
$$476$$ −18.9282 −0.867573
$$477$$ 0 0
$$478$$ −0.392305 −0.0179436
$$479$$ −11.8756 −0.542612 −0.271306 0.962493i $$-0.587456\pi$$
−0.271306 + 0.962493i $$0.587456\pi$$
$$480$$ 0 0
$$481$$ −4.00000 −0.182384
$$482$$ −11.9474 −0.544191
$$483$$ 0 0
$$484$$ −32.0000 −1.45455
$$485$$ −9.66025 −0.438650
$$486$$ 0 0
$$487$$ 24.3923 1.10532 0.552660 0.833407i $$-0.313613\pi$$
0.552660 + 0.833407i $$0.313613\pi$$
$$488$$ −10.1436 −0.459179
$$489$$ 0 0
$$490$$ 11.2679 0.509034
$$491$$ −13.8756 −0.626199 −0.313100 0.949720i $$-0.601367\pi$$
−0.313100 + 0.949720i $$0.601367\pi$$
$$492$$ 0 0
$$493$$ −8.73205 −0.393272
$$494$$ 4.78461 0.215270
$$495$$ 0 0
$$496$$ −3.21539 −0.144375
$$497$$ 17.6603 0.792171
$$498$$ 0 0
$$499$$ 24.3205 1.08874 0.544368 0.838847i $$-0.316770\pi$$
0.544368 + 0.838847i $$0.316770\pi$$
$$500$$ −1.46410 −0.0654766
$$501$$ 0 0
$$502$$ −7.60770 −0.339548
$$503$$ −7.32051 −0.326405 −0.163203 0.986593i $$-0.552182\pi$$
−0.163203 + 0.986593i $$0.552182\pi$$
$$504$$ 0 0
$$505$$ −2.66025 −0.118380
$$506$$ 14.5359 0.646200
$$507$$ 0 0
$$508$$ 21.3590 0.947652
$$509$$ −6.78461 −0.300723 −0.150361 0.988631i $$-0.548044\pi$$
−0.150361 + 0.988631i $$0.548044\pi$$
$$510$$ 0 0
$$511$$ 36.2487 1.60355
$$512$$ 11.7128 0.517638
$$513$$ 0 0
$$514$$ 7.60770 0.335561
$$515$$ −0.535898 −0.0236145
$$516$$ 0 0
$$517$$ 50.0526 2.20131
$$518$$ 9.46410 0.415829
$$519$$ 0 0
$$520$$ −3.71281 −0.162818
$$521$$ 19.4641 0.852738 0.426369 0.904549i $$-0.359793\pi$$
0.426369 + 0.904549i $$0.359793\pi$$
$$522$$ 0 0
$$523$$ 22.2487 0.972868 0.486434 0.873717i $$-0.338297\pi$$
0.486434 + 0.873717i $$0.338297\pi$$
$$524$$ −22.8231 −0.997031
$$525$$ 0 0
$$526$$ 15.6077 0.680528
$$527$$ 8.19615 0.357030
$$528$$ 0 0
$$529$$ −11.0000 −0.478261
$$530$$ 4.92820 0.214067
$$531$$ 0 0
$$532$$ 30.9282 1.34091
$$533$$ −10.5359 −0.456360
$$534$$ 0 0
$$535$$ −8.53590 −0.369039
$$536$$ −8.78461 −0.379437
$$537$$ 0 0
$$538$$ 7.80385 0.336448
$$539$$ −88.2295 −3.80031
$$540$$ 0 0
$$541$$ −24.4641 −1.05179 −0.525897 0.850548i $$-0.676270\pi$$
−0.525897 + 0.850548i $$0.676270\pi$$
$$542$$ −2.14359 −0.0920752
$$543$$ 0 0
$$544$$ −16.0000 −0.685994
$$545$$ −6.07180 −0.260087
$$546$$ 0 0
$$547$$ −33.8564 −1.44760 −0.723798 0.690012i $$-0.757605\pi$$
−0.723798 + 0.690012i $$0.757605\pi$$
$$548$$ −3.71281 −0.158604
$$549$$ 0 0
$$550$$ −4.19615 −0.178925
$$551$$ 14.2679 0.607835
$$552$$ 0 0
$$553$$ −73.1769 −3.11180
$$554$$ 2.78461 0.118307
$$555$$ 0 0
$$556$$ −0.889727 −0.0377328
$$557$$ −2.53590 −0.107449 −0.0537247 0.998556i $$-0.517109\pi$$
−0.0537247 + 0.998556i $$0.517109\pi$$
$$558$$ 0 0
$$559$$ 0.287187 0.0121467
$$560$$ −5.07180 −0.214323
$$561$$ 0 0
$$562$$ 11.3205 0.477527
$$563$$ −16.7321 −0.705172 −0.352586 0.935779i $$-0.614698\pi$$
−0.352586 + 0.935779i $$0.614698\pi$$
$$564$$ 0 0
$$565$$ −19.1244 −0.804568
$$566$$ −21.4641 −0.902203
$$567$$ 0 0
$$568$$ 9.46410 0.397105
$$569$$ −32.9090 −1.37962 −0.689808 0.723993i $$-0.742305\pi$$
−0.689808 + 0.723993i $$0.742305\pi$$
$$570$$ 0 0
$$571$$ −39.7846 −1.66493 −0.832467 0.554075i $$-0.813072\pi$$
−0.832467 + 0.554075i $$0.813072\pi$$
$$572$$ 12.2872 0.513753
$$573$$ 0 0
$$574$$ 24.9282 1.04048
$$575$$ −3.46410 −0.144463
$$576$$ 0 0
$$577$$ 15.2679 0.635613 0.317807 0.948156i $$-0.397054\pi$$
0.317807 + 0.948156i $$0.397054\pi$$
$$578$$ −6.98076 −0.290361
$$579$$ 0 0
$$580$$ −4.67949 −0.194305
$$581$$ −10.3923 −0.431145
$$582$$ 0 0
$$583$$ −38.5885 −1.59817
$$584$$ 19.4256 0.803838
$$585$$ 0 0
$$586$$ 21.0333 0.868878
$$587$$ −11.6603 −0.481270 −0.240635 0.970616i $$-0.577356\pi$$
−0.240635 + 0.970616i $$0.577356\pi$$
$$588$$ 0 0
$$589$$ −13.3923 −0.551820
$$590$$ −6.05256 −0.249180
$$591$$ 0 0
$$592$$ −2.92820 −0.120348
$$593$$ −0.143594 −0.00589668 −0.00294834 0.999996i $$-0.500938\pi$$
−0.00294834 + 0.999996i $$0.500938\pi$$
$$594$$ 0 0
$$595$$ 12.9282 0.530005
$$596$$ 11.7128 0.479776
$$597$$ 0 0
$$598$$ −3.71281 −0.151828
$$599$$ 27.1962 1.11120 0.555602 0.831448i $$-0.312488\pi$$
0.555602 + 0.831448i $$0.312488\pi$$
$$600$$ 0 0
$$601$$ −31.2487 −1.27466 −0.637331 0.770590i $$-0.719961\pi$$
−0.637331 + 0.770590i $$0.719961\pi$$
$$602$$ −0.679492 −0.0276940
$$603$$ 0 0
$$604$$ −7.89488 −0.321238
$$605$$ 21.8564 0.888589
$$606$$ 0 0
$$607$$ −16.1962 −0.657382 −0.328691 0.944438i $$-0.606607\pi$$
−0.328691 + 0.944438i $$0.606607\pi$$
$$608$$ 26.1436 1.06026
$$609$$ 0 0
$$610$$ 2.92820 0.118559
$$611$$ −12.7846 −0.517210
$$612$$ 0 0
$$613$$ 1.46410 0.0591345 0.0295673 0.999563i $$-0.490587\pi$$
0.0295673 + 0.999563i $$0.490587\pi$$
$$614$$ 10.2872 0.415157
$$615$$ 0 0
$$616$$ −68.7846 −2.77141
$$617$$ −6.92820 −0.278919 −0.139459 0.990228i $$-0.544536\pi$$
−0.139459 + 0.990228i $$0.544536\pi$$
$$618$$ 0 0
$$619$$ −11.8564 −0.476549 −0.238275 0.971198i $$-0.576582\pi$$
−0.238275 + 0.971198i $$0.576582\pi$$
$$620$$ 4.39230 0.176399
$$621$$ 0 0
$$622$$ 14.4449 0.579186
$$623$$ −24.5885 −0.985116
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −6.64102 −0.265428
$$627$$ 0 0
$$628$$ 28.0000 1.11732
$$629$$ 7.46410 0.297613
$$630$$ 0 0
$$631$$ −32.7128 −1.30228 −0.651138 0.758959i $$-0.725708\pi$$
−0.651138 + 0.758959i $$0.725708\pi$$
$$632$$ −39.2154 −1.55990
$$633$$ 0 0
$$634$$ 4.53590 0.180144
$$635$$ −14.5885 −0.578925
$$636$$ 0 0
$$637$$ 22.5359 0.892905
$$638$$ −13.4115 −0.530968
$$639$$ 0 0
$$640$$ −10.1436 −0.400961
$$641$$ 9.33975 0.368898 0.184449 0.982842i $$-0.440950\pi$$
0.184449 + 0.982842i $$0.440950\pi$$
$$642$$ 0 0
$$643$$ 41.6603 1.64292 0.821460 0.570266i $$-0.193160\pi$$
0.821460 + 0.570266i $$0.193160\pi$$
$$644$$ −24.0000 −0.945732
$$645$$ 0 0
$$646$$ −8.92820 −0.351275
$$647$$ 11.4641 0.450700 0.225350 0.974278i $$-0.427647\pi$$
0.225350 + 0.974278i $$0.427647\pi$$
$$648$$ 0 0
$$649$$ 47.3923 1.86031
$$650$$ 1.07180 0.0420393
$$651$$ 0 0
$$652$$ −18.6410 −0.730039
$$653$$ −17.4641 −0.683423 −0.341712 0.939805i $$-0.611007\pi$$
−0.341712 + 0.939805i $$0.611007\pi$$
$$654$$ 0 0
$$655$$ 15.5885 0.609091
$$656$$ −7.71281 −0.301135
$$657$$ 0 0
$$658$$ 30.2487 1.17922
$$659$$ −1.46410 −0.0570333 −0.0285167 0.999593i $$-0.509078\pi$$
−0.0285167 + 0.999593i $$0.509078\pi$$
$$660$$ 0 0
$$661$$ 18.3205 0.712585 0.356293 0.934374i $$-0.384041\pi$$
0.356293 + 0.934374i $$0.384041\pi$$
$$662$$ −0.339746 −0.0132046
$$663$$ 0 0
$$664$$ −5.56922 −0.216128
$$665$$ −21.1244 −0.819167
$$666$$ 0 0
$$667$$ −11.0718 −0.428702
$$668$$ 25.8564 1.00041
$$669$$ 0 0
$$670$$ 2.53590 0.0979703
$$671$$ −22.9282 −0.885133
$$672$$ 0 0
$$673$$ −10.3923 −0.400594 −0.200297 0.979735i $$-0.564191\pi$$
−0.200297 + 0.979735i $$0.564191\pi$$
$$674$$ 20.0000 0.770371
$$675$$ 0 0
$$676$$ 15.8949 0.611342
$$677$$ −18.0000 −0.691796 −0.345898 0.938272i $$-0.612426\pi$$
−0.345898 + 0.938272i $$0.612426\pi$$
$$678$$ 0 0
$$679$$ 45.7128 1.75430
$$680$$ 6.92820 0.265684
$$681$$ 0 0
$$682$$ 12.5885 0.482037
$$683$$ −19.6077 −0.750268 −0.375134 0.926971i $$-0.622403\pi$$
−0.375134 + 0.926971i $$0.622403\pi$$
$$684$$ 0 0
$$685$$ 2.53590 0.0968917
$$686$$ −29.0718 −1.10997
$$687$$ 0 0
$$688$$ 0.210236 0.00801515
$$689$$ 9.85641 0.375499
$$690$$ 0 0
$$691$$ −17.7128 −0.673827 −0.336914 0.941536i $$-0.609383\pi$$
−0.336914 + 0.941536i $$0.609383\pi$$
$$692$$ −12.4974 −0.475081
$$693$$ 0 0
$$694$$ −20.9282 −0.794424
$$695$$ 0.607695 0.0230512
$$696$$ 0 0
$$697$$ 19.6603 0.744685
$$698$$ −13.8038 −0.522483
$$699$$ 0 0
$$700$$ 6.92820 0.261861
$$701$$ −20.8038 −0.785750 −0.392875 0.919592i $$-0.628520\pi$$
−0.392875 + 0.919592i $$0.628520\pi$$
$$702$$ 0 0
$$703$$ −12.1962 −0.459987
$$704$$ −12.2872 −0.463091
$$705$$ 0 0
$$706$$ 18.6795 0.703012
$$707$$ 12.5885 0.473438
$$708$$ 0 0
$$709$$ −22.5359 −0.846353 −0.423177 0.906047i $$-0.639085\pi$$
−0.423177 + 0.906047i $$0.639085\pi$$
$$710$$ −2.73205 −0.102532
$$711$$ 0 0
$$712$$ −13.1769 −0.493826
$$713$$ 10.3923 0.389195
$$714$$ 0 0
$$715$$ −8.39230 −0.313854
$$716$$ −11.8949 −0.444533
$$717$$ 0 0
$$718$$ −4.48334 −0.167317
$$719$$ 8.41154 0.313698 0.156849 0.987623i $$-0.449866\pi$$
0.156849 + 0.987623i $$0.449866\pi$$
$$720$$ 0 0
$$721$$ 2.53590 0.0944418
$$722$$ 0.679492 0.0252881
$$723$$ 0 0
$$724$$ −38.7461 −1.43999
$$725$$ 3.19615 0.118702
$$726$$ 0 0
$$727$$ −8.39230 −0.311253 −0.155627 0.987816i $$-0.549740\pi$$
−0.155627 + 0.987816i $$0.549740\pi$$
$$728$$ 17.5692 0.651159
$$729$$ 0 0
$$730$$ −5.60770 −0.207550
$$731$$ −0.535898 −0.0198209
$$732$$ 0 0
$$733$$ 34.7846 1.28480 0.642399 0.766370i $$-0.277939\pi$$
0.642399 + 0.766370i $$0.277939\pi$$
$$734$$ 22.8231 0.842415
$$735$$ 0 0
$$736$$ −20.2872 −0.747796
$$737$$ −19.8564 −0.731420
$$738$$ 0 0
$$739$$ 22.4641 0.826355 0.413178 0.910650i $$-0.364419\pi$$
0.413178 + 0.910650i $$0.364419\pi$$
$$740$$ 4.00000 0.147043
$$741$$ 0 0
$$742$$ −23.3205 −0.856123
$$743$$ 19.9090 0.730389 0.365195 0.930931i $$-0.381002\pi$$
0.365195 + 0.930931i $$0.381002\pi$$
$$744$$ 0 0
$$745$$ −8.00000 −0.293097
$$746$$ 14.6795 0.537454
$$747$$ 0 0
$$748$$ −22.9282 −0.838338
$$749$$ 40.3923 1.47590
$$750$$ 0 0
$$751$$ 48.7846 1.78018 0.890088 0.455789i $$-0.150643\pi$$
0.890088 + 0.455789i $$0.150643\pi$$
$$752$$ −9.35898 −0.341287
$$753$$ 0 0
$$754$$ 3.42563 0.124754
$$755$$ 5.39230 0.196246
$$756$$ 0 0
$$757$$ −9.17691 −0.333541 −0.166770 0.985996i $$-0.553334\pi$$
−0.166770 + 0.985996i $$0.553334\pi$$
$$758$$ 1.75129 0.0636097
$$759$$ 0 0
$$760$$ −11.3205 −0.410638
$$761$$ 23.4449 0.849876 0.424938 0.905223i $$-0.360296\pi$$
0.424938 + 0.905223i $$0.360296\pi$$
$$762$$ 0 0
$$763$$ 28.7321 1.04017
$$764$$ 11.8949 0.430342
$$765$$ 0 0
$$766$$ 1.85641 0.0670747
$$767$$ −12.1051 −0.437090
$$768$$ 0 0
$$769$$ −9.53590 −0.343873 −0.171937 0.985108i $$-0.555002\pi$$
−0.171937 + 0.985108i $$0.555002\pi$$
$$770$$ 19.8564 0.715575
$$771$$ 0 0
$$772$$ 7.71281 0.277590
$$773$$ 1.51666 0.0545505 0.0272752 0.999628i $$-0.491317\pi$$
0.0272752 + 0.999628i $$0.491317\pi$$
$$774$$ 0 0
$$775$$ −3.00000 −0.107763
$$776$$ 24.4974 0.879406
$$777$$ 0 0
$$778$$ 20.1051 0.720803
$$779$$ −32.1244 −1.15097
$$780$$ 0 0
$$781$$ 21.3923 0.765477
$$782$$ 6.92820 0.247752
$$783$$ 0 0
$$784$$ 16.4974 0.589194
$$785$$ −19.1244 −0.682578
$$786$$ 0 0
$$787$$ −48.0526 −1.71289 −0.856444 0.516239i $$-0.827332\pi$$
−0.856444 + 0.516239i $$0.827332\pi$$
$$788$$ 20.2872 0.722701
$$789$$ 0 0
$$790$$ 11.3205 0.402766
$$791$$ 90.4974 3.21772
$$792$$ 0 0
$$793$$ 5.85641 0.207967
$$794$$ 10.5359 0.373905
$$795$$ 0 0
$$796$$ 2.92820 0.103787
$$797$$ −15.6077 −0.552853 −0.276426 0.961035i $$-0.589150\pi$$
−0.276426 + 0.961035i $$0.589150\pi$$
$$798$$ 0 0
$$799$$ 23.8564 0.843979
$$800$$ 5.85641 0.207055
$$801$$ 0 0
$$802$$ 18.2487 0.644384
$$803$$ 43.9090 1.54951
$$804$$ 0 0
$$805$$ 16.3923 0.577753
$$806$$ −3.21539 −0.113257
$$807$$ 0 0
$$808$$ 6.74613 0.237328
$$809$$ 45.4449 1.59776 0.798878 0.601493i $$-0.205427\pi$$
0.798878 + 0.601493i $$0.205427\pi$$
$$810$$ 0 0
$$811$$ −18.4641 −0.648362 −0.324181 0.945995i $$-0.605089\pi$$
−0.324181 + 0.945995i $$0.605089\pi$$
$$812$$ 22.1436 0.777088
$$813$$ 0 0
$$814$$ 11.4641 0.401817
$$815$$ 12.7321 0.445984
$$816$$ 0 0
$$817$$ 0.875644 0.0306349
$$818$$ −13.0718 −0.457045
$$819$$ 0 0
$$820$$ 10.5359 0.367930
$$821$$ −37.7321 −1.31686 −0.658429 0.752643i $$-0.728779\pi$$
−0.658429 + 0.752643i $$0.728779\pi$$
$$822$$ 0 0
$$823$$ 3.85641 0.134426 0.0672129 0.997739i $$-0.478589\pi$$
0.0672129 + 0.997739i $$0.478589\pi$$
$$824$$ 1.35898 0.0473424
$$825$$ 0 0
$$826$$ 28.6410 0.996548
$$827$$ −11.6077 −0.403639 −0.201820 0.979423i $$-0.564685\pi$$
−0.201820 + 0.979423i $$0.564685\pi$$
$$828$$ 0 0
$$829$$ −23.7846 −0.826074 −0.413037 0.910714i $$-0.635532\pi$$
−0.413037 + 0.910714i $$0.635532\pi$$
$$830$$ 1.60770 0.0558039
$$831$$ 0 0
$$832$$ 3.13844 0.108806
$$833$$ −42.0526 −1.45703
$$834$$ 0 0
$$835$$ −17.6603 −0.611158
$$836$$ 37.4641 1.29572
$$837$$ 0 0
$$838$$ 14.9282 0.515686
$$839$$ 33.1962 1.14606 0.573029 0.819535i $$-0.305768\pi$$
0.573029 + 0.819535i $$0.305768\pi$$
$$840$$ 0 0
$$841$$ −18.7846 −0.647745
$$842$$ 24.7321 0.852323
$$843$$ 0 0
$$844$$ 12.9667 0.446331
$$845$$ −10.8564 −0.373472
$$846$$ 0 0
$$847$$ −103.426 −3.55375
$$848$$ 7.21539 0.247778
$$849$$ 0 0
$$850$$ −2.00000 −0.0685994
$$851$$ 9.46410 0.324425
$$852$$ 0 0
$$853$$ 10.4833 0.358943 0.179471 0.983763i $$-0.442561\pi$$
0.179471 + 0.983763i $$0.442561\pi$$
$$854$$ −13.8564 −0.474156
$$855$$ 0 0
$$856$$ 21.6462 0.739851
$$857$$ 44.4974 1.52000 0.760002 0.649921i $$-0.225198\pi$$
0.760002 + 0.649921i $$0.225198\pi$$
$$858$$ 0 0
$$859$$ −11.0000 −0.375315 −0.187658 0.982235i $$-0.560090\pi$$
−0.187658 + 0.982235i $$0.560090\pi$$
$$860$$ −0.287187 −0.00979300
$$861$$ 0 0
$$862$$ −15.6218 −0.532080
$$863$$ −23.1244 −0.787162 −0.393581 0.919290i $$-0.628764\pi$$
−0.393581 + 0.919290i $$0.628764\pi$$
$$864$$ 0 0
$$865$$ 8.53590 0.290229
$$866$$ −25.9615 −0.882209
$$867$$ 0 0
$$868$$ −20.7846 −0.705476
$$869$$ −88.6410 −3.00694
$$870$$ 0 0
$$871$$ 5.07180 0.171851
$$872$$ 15.3975 0.521424
$$873$$ 0 0
$$874$$ −11.3205 −0.382922
$$875$$ −4.73205 −0.159973
$$876$$ 0 0
$$877$$ 33.4641 1.13000 0.565001 0.825090i $$-0.308876\pi$$
0.565001 + 0.825090i $$0.308876\pi$$
$$878$$ 3.94744 0.133220
$$879$$ 0 0
$$880$$ −6.14359 −0.207100
$$881$$ −47.0526 −1.58524 −0.792620 0.609715i $$-0.791284\pi$$
−0.792620 + 0.609715i $$0.791284\pi$$
$$882$$ 0 0
$$883$$ −30.1962 −1.01618 −0.508091 0.861304i $$-0.669649\pi$$
−0.508091 + 0.861304i $$0.669649\pi$$
$$884$$ 5.85641 0.196972
$$885$$ 0 0
$$886$$ −0.248711 −0.00835562
$$887$$ 33.8038 1.13502 0.567511 0.823366i $$-0.307906\pi$$
0.567511 + 0.823366i $$0.307906\pi$$
$$888$$ 0 0
$$889$$ 69.0333 2.31530
$$890$$ 3.80385 0.127505
$$891$$ 0 0
$$892$$ −24.5744 −0.822811
$$893$$ −38.9808 −1.30444
$$894$$ 0 0
$$895$$ 8.12436 0.271567
$$896$$ 48.0000 1.60357
$$897$$ 0 0
$$898$$ 5.94744 0.198469
$$899$$ −9.58846 −0.319793
$$900$$ 0 0
$$901$$ −18.3923 −0.612737
$$902$$ 30.1962 1.00542
$$903$$ 0 0
$$904$$ 48.4974 1.61300
$$905$$ 26.4641 0.879697
$$906$$ 0 0
$$907$$ −30.0000 −0.996134 −0.498067 0.867139i $$-0.665957\pi$$
−0.498067 + 0.867139i $$0.665957\pi$$
$$908$$ 26.4308 0.877136
$$909$$ 0 0
$$910$$ −5.07180 −0.168128
$$911$$ 23.5885 0.781520 0.390760 0.920493i $$-0.372212\pi$$
0.390760 + 0.920493i $$0.372212\pi$$
$$912$$ 0 0
$$913$$ −12.5885 −0.416617
$$914$$ 2.00000 0.0661541
$$915$$ 0 0
$$916$$ 17.5692 0.580503
$$917$$ −73.7654 −2.43595
$$918$$ 0 0
$$919$$ −56.9615 −1.87899 −0.939494 0.342566i $$-0.888704\pi$$
−0.939494 + 0.342566i $$0.888704\pi$$
$$920$$ 8.78461 0.289620
$$921$$ 0 0
$$922$$ −27.1244 −0.893293
$$923$$ −5.46410 −0.179853
$$924$$ 0 0
$$925$$ −2.73205 −0.0898293
$$926$$ −7.60770 −0.250004
$$927$$ 0 0
$$928$$ 18.7180 0.614447
$$929$$ −44.3731 −1.45583 −0.727917 0.685666i $$-0.759511\pi$$
−0.727917 + 0.685666i $$0.759511\pi$$
$$930$$ 0 0
$$931$$ 68.7128 2.25197
$$932$$ 41.0718 1.34535
$$933$$ 0 0
$$934$$ 27.3590 0.895213
$$935$$ 15.6603 0.512145
$$936$$ 0 0
$$937$$ −4.14359 −0.135365 −0.0676827 0.997707i $$-0.521561\pi$$
−0.0676827 + 0.997707i $$0.521561\pi$$
$$938$$ −12.0000 −0.391814
$$939$$ 0 0
$$940$$ 12.7846 0.416988
$$941$$ 35.1769 1.14673 0.573367 0.819298i $$-0.305637\pi$$
0.573367 + 0.819298i $$0.305637\pi$$
$$942$$ 0 0
$$943$$ 24.9282 0.811774
$$944$$ −8.86156 −0.288419
$$945$$ 0 0
$$946$$ −0.823085 −0.0267608
$$947$$ −57.7128 −1.87541 −0.937707 0.347427i $$-0.887056\pi$$
−0.937707 + 0.347427i $$0.887056\pi$$
$$948$$ 0 0
$$949$$ −11.2154 −0.364067
$$950$$ 3.26795 0.106026
$$951$$ 0 0
$$952$$ −32.7846 −1.06256
$$953$$ 36.3923 1.17886 0.589431 0.807819i $$-0.299352\pi$$
0.589431 + 0.807819i $$0.299352\pi$$
$$954$$ 0 0
$$955$$ −8.12436 −0.262898
$$956$$ 0.784610 0.0253761
$$957$$ 0 0
$$958$$ −8.69358 −0.280877
$$959$$ −12.0000 −0.387500
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ −2.92820 −0.0944091
$$963$$ 0 0
$$964$$ 23.8949 0.769602
$$965$$ −5.26795 −0.169581
$$966$$ 0 0
$$967$$ −25.8038 −0.829796 −0.414898 0.909868i $$-0.636183\pi$$
−0.414898 + 0.909868i $$0.636183\pi$$
$$968$$ −55.4256 −1.78145
$$969$$ 0 0
$$970$$ −7.07180 −0.227062
$$971$$ 17.4449 0.559832 0.279916 0.960024i $$-0.409693\pi$$
0.279916 + 0.960024i $$0.409693\pi$$
$$972$$ 0 0
$$973$$ −2.87564 −0.0921889
$$974$$ 17.8564 0.572156
$$975$$ 0 0
$$976$$ 4.28719 0.137230
$$977$$ 5.46410 0.174812 0.0874060 0.996173i $$-0.472142\pi$$
0.0874060 + 0.996173i $$0.472142\pi$$
$$978$$ 0 0
$$979$$ −29.7846 −0.951920
$$980$$ −22.5359 −0.719883
$$981$$ 0 0
$$982$$ −10.1577 −0.324144
$$983$$ 48.5885 1.54973 0.774866 0.632126i $$-0.217818\pi$$
0.774866 + 0.632126i $$0.217818\pi$$
$$984$$ 0 0
$$985$$ −13.8564 −0.441502
$$986$$ −6.39230 −0.203572
$$987$$ 0 0
$$988$$ −9.56922 −0.304437
$$989$$ −0.679492 −0.0216066
$$990$$ 0 0
$$991$$ 30.8564 0.980186 0.490093 0.871670i $$-0.336963\pi$$
0.490093 + 0.871670i $$0.336963\pi$$
$$992$$ −17.5692 −0.557823
$$993$$ 0 0
$$994$$ 12.9282 0.410058
$$995$$ −2.00000 −0.0634043
$$996$$ 0 0
$$997$$ 25.5167 0.808121 0.404060 0.914732i $$-0.367599\pi$$
0.404060 + 0.914732i $$0.367599\pi$$
$$998$$ 17.8038 0.563571
$$999$$ 0 0
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 405.2.a.g.1.2 2
3.2 odd 2 405.2.a.h.1.1 yes 2
4.3 odd 2 6480.2.a.br.1.2 2
5.2 odd 4 2025.2.b.g.649.3 4
5.3 odd 4 2025.2.b.g.649.2 4
5.4 even 2 2025.2.a.m.1.1 2
9.2 odd 6 405.2.e.i.271.2 4
9.4 even 3 405.2.e.l.136.1 4
9.5 odd 6 405.2.e.i.136.2 4
9.7 even 3 405.2.e.l.271.1 4
12.11 even 2 6480.2.a.bi.1.2 2
15.2 even 4 2025.2.b.h.649.2 4
15.8 even 4 2025.2.b.h.649.3 4
15.14 odd 2 2025.2.a.g.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
405.2.a.g.1.2 2 1.1 even 1 trivial
405.2.a.h.1.1 yes 2 3.2 odd 2
405.2.e.i.136.2 4 9.5 odd 6
405.2.e.i.271.2 4 9.2 odd 6
405.2.e.l.136.1 4 9.4 even 3
405.2.e.l.271.1 4 9.7 even 3
2025.2.a.g.1.2 2 15.14 odd 2
2025.2.a.m.1.1 2 5.4 even 2
2025.2.b.g.649.2 4 5.3 odd 4
2025.2.b.g.649.3 4 5.2 odd 4
2025.2.b.h.649.2 4 15.2 even 4
2025.2.b.h.649.3 4 15.8 even 4
6480.2.a.bi.1.2 2 12.11 even 2
6480.2.a.br.1.2 2 4.3 odd 2