# Properties

 Label 1600.2.a.bb.1.1 Level $1600$ Weight $2$ Character 1600.1 Self dual yes Analytic conductor $12.776$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,2,Mod(1,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1600.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: $$12.7760643234$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 800) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 1600.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.23607 q^{3} +4.47214 q^{7} +2.00000 q^{9} +O(q^{10})$$ $$q-2.23607 q^{3} +4.47214 q^{7} +2.00000 q^{9} +2.23607 q^{11} +4.00000 q^{13} +7.00000 q^{17} -6.70820 q^{19} -10.0000 q^{21} -4.47214 q^{23} +2.23607 q^{27} -4.47214 q^{31} -5.00000 q^{33} +2.00000 q^{37} -8.94427 q^{39} +5.00000 q^{41} -8.94427 q^{47} +13.0000 q^{49} -15.6525 q^{51} +6.00000 q^{53} +15.0000 q^{57} +8.94427 q^{59} -10.0000 q^{61} +8.94427 q^{63} +2.23607 q^{67} +10.0000 q^{69} +8.94427 q^{71} +9.00000 q^{73} +10.0000 q^{77} +4.47214 q^{79} -11.0000 q^{81} +11.1803 q^{83} -5.00000 q^{89} +17.8885 q^{91} +10.0000 q^{93} -2.00000 q^{97} +4.47214 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{9}+O(q^{10})$$ 2 * q + 4 * q^9 $$2 q + 4 q^{9} + 8 q^{13} + 14 q^{17} - 20 q^{21} - 10 q^{33} + 4 q^{37} + 10 q^{41} + 26 q^{49} + 12 q^{53} + 30 q^{57} - 20 q^{61} + 20 q^{69} + 18 q^{73} + 20 q^{77} - 22 q^{81} - 10 q^{89} + 20 q^{93} - 4 q^{97}+O(q^{100})$$ 2 * q + 4 * q^9 + 8 * q^13 + 14 * q^17 - 20 * q^21 - 10 * q^33 + 4 * q^37 + 10 * q^41 + 26 * q^49 + 12 * q^53 + 30 * q^57 - 20 * q^61 + 20 * q^69 + 18 * q^73 + 20 * q^77 - 22 * q^81 - 10 * q^89 + 20 * q^93 - 4 * q^97

## 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
$$3$$ −2.23607 −1.29099 −0.645497 0.763763i $$-0.723350\pi$$
−0.645497 + 0.763763i $$0.723350\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 4.47214 1.69031 0.845154 0.534522i $$-0.179509\pi$$
0.845154 + 0.534522i $$0.179509\pi$$
$$8$$ 0 0
$$9$$ 2.00000 0.666667
$$10$$ 0 0
$$11$$ 2.23607 0.674200 0.337100 0.941469i $$-0.390554\pi$$
0.337100 + 0.941469i $$0.390554\pi$$
$$12$$ 0 0
$$13$$ 4.00000 1.10940 0.554700 0.832050i $$-0.312833\pi$$
0.554700 + 0.832050i $$0.312833\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 7.00000 1.69775 0.848875 0.528594i $$-0.177281\pi$$
0.848875 + 0.528594i $$0.177281\pi$$
$$18$$ 0 0
$$19$$ −6.70820 −1.53897 −0.769484 0.638666i $$-0.779486\pi$$
−0.769484 + 0.638666i $$0.779486\pi$$
$$20$$ 0 0
$$21$$ −10.0000 −2.18218
$$22$$ 0 0
$$23$$ −4.47214 −0.932505 −0.466252 0.884652i $$-0.654396\pi$$
−0.466252 + 0.884652i $$0.654396\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 2.23607 0.430331
$$28$$ 0 0
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ −4.47214 −0.803219 −0.401610 0.915811i $$-0.631549\pi$$
−0.401610 + 0.915811i $$0.631549\pi$$
$$32$$ 0 0
$$33$$ −5.00000 −0.870388
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ 0 0
$$39$$ −8.94427 −1.43223
$$40$$ 0 0
$$41$$ 5.00000 0.780869 0.390434 0.920631i $$-0.372325\pi$$
0.390434 + 0.920631i $$0.372325\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −8.94427 −1.30466 −0.652328 0.757937i $$-0.726208\pi$$
−0.652328 + 0.757937i $$0.726208\pi$$
$$48$$ 0 0
$$49$$ 13.0000 1.85714
$$50$$ 0 0
$$51$$ −15.6525 −2.19179
$$52$$ 0 0
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 15.0000 1.98680
$$58$$ 0 0
$$59$$ 8.94427 1.16445 0.582223 0.813029i $$-0.302183\pi$$
0.582223 + 0.813029i $$0.302183\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 0 0
$$63$$ 8.94427 1.12687
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 2.23607 0.273179 0.136590 0.990628i $$-0.456386\pi$$
0.136590 + 0.990628i $$0.456386\pi$$
$$68$$ 0 0
$$69$$ 10.0000 1.20386
$$70$$ 0 0
$$71$$ 8.94427 1.06149 0.530745 0.847532i $$-0.321912\pi$$
0.530745 + 0.847532i $$0.321912\pi$$
$$72$$ 0 0
$$73$$ 9.00000 1.05337 0.526685 0.850060i $$-0.323435\pi$$
0.526685 + 0.850060i $$0.323435\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 10.0000 1.13961
$$78$$ 0 0
$$79$$ 4.47214 0.503155 0.251577 0.967837i $$-0.419051\pi$$
0.251577 + 0.967837i $$0.419051\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 0 0
$$83$$ 11.1803 1.22720 0.613601 0.789616i $$-0.289720\pi$$
0.613601 + 0.789616i $$0.289720\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −5.00000 −0.529999 −0.264999 0.964249i $$-0.585372\pi$$
−0.264999 + 0.964249i $$0.585372\pi$$
$$90$$ 0 0
$$91$$ 17.8885 1.87523
$$92$$ 0 0
$$93$$ 10.0000 1.03695
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −2.00000 −0.203069 −0.101535 0.994832i $$-0.532375\pi$$
−0.101535 + 0.994832i $$0.532375\pi$$
$$98$$ 0 0
$$99$$ 4.47214 0.449467
$$100$$ 0 0
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ 0 0
$$103$$ −8.94427 −0.881305 −0.440653 0.897678i $$-0.645253\pi$$
−0.440653 + 0.897678i $$0.645253\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −2.23607 −0.216169 −0.108084 0.994142i $$-0.534472\pi$$
−0.108084 + 0.994142i $$0.534472\pi$$
$$108$$ 0 0
$$109$$ −6.00000 −0.574696 −0.287348 0.957826i $$-0.592774\pi$$
−0.287348 + 0.957826i $$0.592774\pi$$
$$110$$ 0 0
$$111$$ −4.47214 −0.424476
$$112$$ 0 0
$$113$$ 1.00000 0.0940721 0.0470360 0.998893i $$-0.485022\pi$$
0.0470360 + 0.998893i $$0.485022\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 8.00000 0.739600
$$118$$ 0 0
$$119$$ 31.3050 2.86972
$$120$$ 0 0
$$121$$ −6.00000 −0.545455
$$122$$ 0 0
$$123$$ −11.1803 −1.00810
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 4.47214 0.396838 0.198419 0.980117i $$-0.436419\pi$$
0.198419 + 0.980117i $$0.436419\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 17.8885 1.56293 0.781465 0.623949i $$-0.214473\pi$$
0.781465 + 0.623949i $$0.214473\pi$$
$$132$$ 0 0
$$133$$ −30.0000 −2.60133
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 3.00000 0.256307 0.128154 0.991754i $$-0.459095\pi$$
0.128154 + 0.991754i $$0.459095\pi$$
$$138$$ 0 0
$$139$$ 2.23607 0.189661 0.0948304 0.995493i $$-0.469769\pi$$
0.0948304 + 0.995493i $$0.469769\pi$$
$$140$$ 0 0
$$141$$ 20.0000 1.68430
$$142$$ 0 0
$$143$$ 8.94427 0.747958
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −29.0689 −2.39756
$$148$$ 0 0
$$149$$ 16.0000 1.31077 0.655386 0.755295i $$-0.272506\pi$$
0.655386 + 0.755295i $$0.272506\pi$$
$$150$$ 0 0
$$151$$ 13.4164 1.09181 0.545906 0.837846i $$-0.316186\pi$$
0.545906 + 0.837846i $$0.316186\pi$$
$$152$$ 0 0
$$153$$ 14.0000 1.13183
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 18.0000 1.43656 0.718278 0.695756i $$-0.244931\pi$$
0.718278 + 0.695756i $$0.244931\pi$$
$$158$$ 0 0
$$159$$ −13.4164 −1.06399
$$160$$ 0 0
$$161$$ −20.0000 −1.57622
$$162$$ 0 0
$$163$$ 2.23607 0.175142 0.0875712 0.996158i $$-0.472089\pi$$
0.0875712 + 0.996158i $$0.472089\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 8.94427 0.692129 0.346064 0.938211i $$-0.387518\pi$$
0.346064 + 0.938211i $$0.387518\pi$$
$$168$$ 0 0
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ −13.4164 −1.02598
$$172$$ 0 0
$$173$$ 16.0000 1.21646 0.608229 0.793762i $$-0.291880\pi$$
0.608229 + 0.793762i $$0.291880\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −20.0000 −1.50329
$$178$$ 0 0
$$179$$ 2.23607 0.167132 0.0835658 0.996502i $$-0.473369\pi$$
0.0835658 + 0.996502i $$0.473369\pi$$
$$180$$ 0 0
$$181$$ −2.00000 −0.148659 −0.0743294 0.997234i $$-0.523682\pi$$
−0.0743294 + 0.997234i $$0.523682\pi$$
$$182$$ 0 0
$$183$$ 22.3607 1.65295
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 15.6525 1.14462
$$188$$ 0 0
$$189$$ 10.0000 0.727393
$$190$$ 0 0
$$191$$ −13.4164 −0.970777 −0.485389 0.874299i $$-0.661322\pi$$
−0.485389 + 0.874299i $$0.661322\pi$$
$$192$$ 0 0
$$193$$ −9.00000 −0.647834 −0.323917 0.946085i $$-0.605000\pi$$
−0.323917 + 0.946085i $$0.605000\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −18.0000 −1.28245 −0.641223 0.767354i $$-0.721573\pi$$
−0.641223 + 0.767354i $$0.721573\pi$$
$$198$$ 0 0
$$199$$ −8.94427 −0.634043 −0.317021 0.948418i $$-0.602683\pi$$
−0.317021 + 0.948418i $$0.602683\pi$$
$$200$$ 0 0
$$201$$ −5.00000 −0.352673
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −8.94427 −0.621670
$$208$$ 0 0
$$209$$ −15.0000 −1.03757
$$210$$ 0 0
$$211$$ −20.1246 −1.38544 −0.692718 0.721209i $$-0.743587\pi$$
−0.692718 + 0.721209i $$0.743587\pi$$
$$212$$ 0 0
$$213$$ −20.0000 −1.37038
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −20.0000 −1.35769
$$218$$ 0 0
$$219$$ −20.1246 −1.35990
$$220$$ 0 0
$$221$$ 28.0000 1.88348
$$222$$ 0 0
$$223$$ −26.8328 −1.79686 −0.898429 0.439119i $$-0.855291\pi$$
−0.898429 + 0.439119i $$0.855291\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 17.8885 1.18730 0.593652 0.804722i $$-0.297686\pi$$
0.593652 + 0.804722i $$0.297686\pi$$
$$228$$ 0 0
$$229$$ −20.0000 −1.32164 −0.660819 0.750546i $$-0.729791\pi$$
−0.660819 + 0.750546i $$0.729791\pi$$
$$230$$ 0 0
$$231$$ −22.3607 −1.47122
$$232$$ 0 0
$$233$$ 26.0000 1.70332 0.851658 0.524097i $$-0.175597\pi$$
0.851658 + 0.524097i $$0.175597\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −10.0000 −0.649570
$$238$$ 0 0
$$239$$ −17.8885 −1.15711 −0.578557 0.815642i $$-0.696384\pi$$
−0.578557 + 0.815642i $$0.696384\pi$$
$$240$$ 0 0
$$241$$ 5.00000 0.322078 0.161039 0.986948i $$-0.448515\pi$$
0.161039 + 0.986948i $$0.448515\pi$$
$$242$$ 0 0
$$243$$ 17.8885 1.14755
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −26.8328 −1.70733
$$248$$ 0 0
$$249$$ −25.0000 −1.58431
$$250$$ 0 0
$$251$$ −11.1803 −0.705697 −0.352848 0.935681i $$-0.614787\pi$$
−0.352848 + 0.935681i $$0.614787\pi$$
$$252$$ 0 0
$$253$$ −10.0000 −0.628695
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 2.00000 0.124757 0.0623783 0.998053i $$-0.480131\pi$$
0.0623783 + 0.998053i $$0.480131\pi$$
$$258$$ 0 0
$$259$$ 8.94427 0.555770
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −22.3607 −1.37882 −0.689409 0.724372i $$-0.742130\pi$$
−0.689409 + 0.724372i $$0.742130\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 11.1803 0.684226
$$268$$ 0 0
$$269$$ −24.0000 −1.46331 −0.731653 0.681677i $$-0.761251\pi$$
−0.731653 + 0.681677i $$0.761251\pi$$
$$270$$ 0 0
$$271$$ 13.4164 0.814989 0.407494 0.913208i $$-0.366403\pi$$
0.407494 + 0.913208i $$0.366403\pi$$
$$272$$ 0 0
$$273$$ −40.0000 −2.42091
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 8.00000 0.480673 0.240337 0.970690i $$-0.422742\pi$$
0.240337 + 0.970690i $$0.422742\pi$$
$$278$$ 0 0
$$279$$ −8.94427 −0.535480
$$280$$ 0 0
$$281$$ 30.0000 1.78965 0.894825 0.446417i $$-0.147300\pi$$
0.894825 + 0.446417i $$0.147300\pi$$
$$282$$ 0 0
$$283$$ −6.70820 −0.398761 −0.199381 0.979922i $$-0.563893\pi$$
−0.199381 + 0.979922i $$0.563893\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 22.3607 1.31991
$$288$$ 0 0
$$289$$ 32.0000 1.88235
$$290$$ 0 0
$$291$$ 4.47214 0.262161
$$292$$ 0 0
$$293$$ 14.0000 0.817889 0.408944 0.912559i $$-0.365897\pi$$
0.408944 + 0.912559i $$0.365897\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 5.00000 0.290129
$$298$$ 0 0
$$299$$ −17.8885 −1.03452
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −4.47214 −0.256917
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 15.6525 0.893334 0.446667 0.894700i $$-0.352611\pi$$
0.446667 + 0.894700i $$0.352611\pi$$
$$308$$ 0 0
$$309$$ 20.0000 1.13776
$$310$$ 0 0
$$311$$ 22.3607 1.26796 0.633979 0.773350i $$-0.281421\pi$$
0.633979 + 0.773350i $$0.281421\pi$$
$$312$$ 0 0
$$313$$ 6.00000 0.339140 0.169570 0.985518i $$-0.445762\pi$$
0.169570 + 0.985518i $$0.445762\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −12.0000 −0.673987 −0.336994 0.941507i $$-0.609410\pi$$
−0.336994 + 0.941507i $$0.609410\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 5.00000 0.279073
$$322$$ 0 0
$$323$$ −46.9574 −2.61278
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 13.4164 0.741929
$$328$$ 0 0
$$329$$ −40.0000 −2.20527
$$330$$ 0 0
$$331$$ 11.1803 0.614527 0.307264 0.951624i $$-0.400587\pi$$
0.307264 + 0.951624i $$0.400587\pi$$
$$332$$ 0 0
$$333$$ 4.00000 0.219199
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −27.0000 −1.47078 −0.735392 0.677642i $$-0.763002\pi$$
−0.735392 + 0.677642i $$0.763002\pi$$
$$338$$ 0 0
$$339$$ −2.23607 −0.121447
$$340$$ 0 0
$$341$$ −10.0000 −0.541530
$$342$$ 0 0
$$343$$ 26.8328 1.44884
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −20.1246 −1.08035 −0.540173 0.841554i $$-0.681641\pi$$
−0.540173 + 0.841554i $$0.681641\pi$$
$$348$$ 0 0
$$349$$ −30.0000 −1.60586 −0.802932 0.596071i $$-0.796728\pi$$
−0.802932 + 0.596071i $$0.796728\pi$$
$$350$$ 0 0
$$351$$ 8.94427 0.477410
$$352$$ 0 0
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −70.0000 −3.70479
$$358$$ 0 0
$$359$$ −31.3050 −1.65221 −0.826106 0.563515i $$-0.809449\pi$$
−0.826106 + 0.563515i $$0.809449\pi$$
$$360$$ 0 0
$$361$$ 26.0000 1.36842
$$362$$ 0 0
$$363$$ 13.4164 0.704179
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 26.8328 1.40066 0.700331 0.713818i $$-0.253036\pi$$
0.700331 + 0.713818i $$0.253036\pi$$
$$368$$ 0 0
$$369$$ 10.0000 0.520579
$$370$$ 0 0
$$371$$ 26.8328 1.39309
$$372$$ 0 0
$$373$$ −14.0000 −0.724893 −0.362446 0.932005i $$-0.618058\pi$$
−0.362446 + 0.932005i $$0.618058\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 15.6525 0.804014 0.402007 0.915637i $$-0.368313\pi$$
0.402007 + 0.915637i $$0.368313\pi$$
$$380$$ 0 0
$$381$$ −10.0000 −0.512316
$$382$$ 0 0
$$383$$ −4.47214 −0.228515 −0.114258 0.993451i $$-0.536449\pi$$
−0.114258 + 0.993451i $$0.536449\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −6.00000 −0.304212 −0.152106 0.988364i $$-0.548606\pi$$
−0.152106 + 0.988364i $$0.548606\pi$$
$$390$$ 0 0
$$391$$ −31.3050 −1.58316
$$392$$ 0 0
$$393$$ −40.0000 −2.01773
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −28.0000 −1.40528 −0.702640 0.711546i $$-0.747995\pi$$
−0.702640 + 0.711546i $$0.747995\pi$$
$$398$$ 0 0
$$399$$ 67.0820 3.35830
$$400$$ 0 0
$$401$$ 17.0000 0.848939 0.424470 0.905442i $$-0.360461\pi$$
0.424470 + 0.905442i $$0.360461\pi$$
$$402$$ 0 0
$$403$$ −17.8885 −0.891092
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 4.47214 0.221676
$$408$$ 0 0
$$409$$ 29.0000 1.43396 0.716979 0.697095i $$-0.245524\pi$$
0.716979 + 0.697095i $$0.245524\pi$$
$$410$$ 0 0
$$411$$ −6.70820 −0.330891
$$412$$ 0 0
$$413$$ 40.0000 1.96827
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −5.00000 −0.244851
$$418$$ 0 0
$$419$$ −15.6525 −0.764673 −0.382337 0.924023i $$-0.624881\pi$$
−0.382337 + 0.924023i $$0.624881\pi$$
$$420$$ 0 0
$$421$$ −20.0000 −0.974740 −0.487370 0.873195i $$-0.662044\pi$$
−0.487370 + 0.873195i $$0.662044\pi$$
$$422$$ 0 0
$$423$$ −17.8885 −0.869771
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −44.7214 −2.16422
$$428$$ 0 0
$$429$$ −20.0000 −0.965609
$$430$$ 0 0
$$431$$ 4.47214 0.215415 0.107708 0.994183i $$-0.465649\pi$$
0.107708 + 0.994183i $$0.465649\pi$$
$$432$$ 0 0
$$433$$ −1.00000 −0.0480569 −0.0240285 0.999711i $$-0.507649\pi$$
−0.0240285 + 0.999711i $$0.507649\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 30.0000 1.43509
$$438$$ 0 0
$$439$$ 35.7771 1.70755 0.853774 0.520644i $$-0.174308\pi$$
0.853774 + 0.520644i $$0.174308\pi$$
$$440$$ 0 0
$$441$$ 26.0000 1.23810
$$442$$ 0 0
$$443$$ 38.0132 1.80606 0.903030 0.429578i $$-0.141338\pi$$
0.903030 + 0.429578i $$0.141338\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −35.7771 −1.69220
$$448$$ 0 0
$$449$$ −19.0000 −0.896665 −0.448333 0.893867i $$-0.647982\pi$$
−0.448333 + 0.893867i $$0.647982\pi$$
$$450$$ 0 0
$$451$$ 11.1803 0.526462
$$452$$ 0 0
$$453$$ −30.0000 −1.40952
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 7.00000 0.327446 0.163723 0.986506i $$-0.447650\pi$$
0.163723 + 0.986506i $$0.447650\pi$$
$$458$$ 0 0
$$459$$ 15.6525 0.730595
$$460$$ 0 0
$$461$$ 12.0000 0.558896 0.279448 0.960161i $$-0.409849\pi$$
0.279448 + 0.960161i $$0.409849\pi$$
$$462$$ 0 0
$$463$$ 26.8328 1.24703 0.623513 0.781813i $$-0.285705\pi$$
0.623513 + 0.781813i $$0.285705\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 17.8885 0.827783 0.413892 0.910326i $$-0.364169\pi$$
0.413892 + 0.910326i $$0.364169\pi$$
$$468$$ 0 0
$$469$$ 10.0000 0.461757
$$470$$ 0 0
$$471$$ −40.2492 −1.85459
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 12.0000 0.549442
$$478$$ 0 0
$$479$$ −13.4164 −0.613011 −0.306506 0.951869i $$-0.599160\pi$$
−0.306506 + 0.951869i $$0.599160\pi$$
$$480$$ 0 0
$$481$$ 8.00000 0.364769
$$482$$ 0 0
$$483$$ 44.7214 2.03489
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 4.47214 0.202652 0.101326 0.994853i $$-0.467692\pi$$
0.101326 + 0.994853i $$0.467692\pi$$
$$488$$ 0 0
$$489$$ −5.00000 −0.226108
$$490$$ 0 0
$$491$$ −35.7771 −1.61460 −0.807299 0.590143i $$-0.799071\pi$$
−0.807299 + 0.590143i $$0.799071\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 40.0000 1.79425
$$498$$ 0 0
$$499$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$500$$ 0 0
$$501$$ −20.0000 −0.893534
$$502$$ 0 0
$$503$$ 26.8328 1.19642 0.598208 0.801341i $$-0.295880\pi$$
0.598208 + 0.801341i $$0.295880\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −6.70820 −0.297922
$$508$$ 0 0
$$509$$ −10.0000 −0.443242 −0.221621 0.975133i $$-0.571135\pi$$
−0.221621 + 0.975133i $$0.571135\pi$$
$$510$$ 0 0
$$511$$ 40.2492 1.78052
$$512$$ 0 0
$$513$$ −15.0000 −0.662266
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −20.0000 −0.879599
$$518$$ 0 0
$$519$$ −35.7771 −1.57044
$$520$$ 0 0
$$521$$ 13.0000 0.569540 0.284770 0.958596i $$-0.408083\pi$$
0.284770 + 0.958596i $$0.408083\pi$$
$$522$$ 0 0
$$523$$ 6.70820 0.293329 0.146665 0.989186i $$-0.453146\pi$$
0.146665 + 0.989186i $$0.453146\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −31.3050 −1.36367
$$528$$ 0 0
$$529$$ −3.00000 −0.130435
$$530$$ 0 0
$$531$$ 17.8885 0.776297
$$532$$ 0 0
$$533$$ 20.0000 0.866296
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −5.00000 −0.215766
$$538$$ 0 0
$$539$$ 29.0689 1.25209
$$540$$ 0 0
$$541$$ −8.00000 −0.343947 −0.171973 0.985102i $$-0.555014\pi$$
−0.171973 + 0.985102i $$0.555014\pi$$
$$542$$ 0 0
$$543$$ 4.47214 0.191918
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −20.1246 −0.860466 −0.430233 0.902718i $$-0.641569\pi$$
−0.430233 + 0.902718i $$0.641569\pi$$
$$548$$ 0 0
$$549$$ −20.0000 −0.853579
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 20.0000 0.850487
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −28.0000 −1.18640 −0.593199 0.805056i $$-0.702135\pi$$
−0.593199 + 0.805056i $$0.702135\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −35.0000 −1.47770
$$562$$ 0 0
$$563$$ −44.7214 −1.88478 −0.942390 0.334515i $$-0.891427\pi$$
−0.942390 + 0.334515i $$0.891427\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −49.1935 −2.06593
$$568$$ 0 0
$$569$$ 1.00000 0.0419222 0.0209611 0.999780i $$-0.493327\pi$$
0.0209611 + 0.999780i $$0.493327\pi$$
$$570$$ 0 0
$$571$$ −17.8885 −0.748612 −0.374306 0.927305i $$-0.622119\pi$$
−0.374306 + 0.927305i $$0.622119\pi$$
$$572$$ 0 0
$$573$$ 30.0000 1.25327
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −43.0000 −1.79011 −0.895057 0.445952i $$-0.852865\pi$$
−0.895057 + 0.445952i $$0.852865\pi$$
$$578$$ 0 0
$$579$$ 20.1246 0.836350
$$580$$ 0 0
$$581$$ 50.0000 2.07435
$$582$$ 0 0
$$583$$ 13.4164 0.555651
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 20.1246 0.830632 0.415316 0.909677i $$-0.363671\pi$$
0.415316 + 0.909677i $$0.363671\pi$$
$$588$$ 0 0
$$589$$ 30.0000 1.23613
$$590$$ 0 0
$$591$$ 40.2492 1.65563
$$592$$ 0 0
$$593$$ 9.00000 0.369586 0.184793 0.982777i $$-0.440839\pi$$
0.184793 + 0.982777i $$0.440839\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 20.0000 0.818546
$$598$$ 0 0
$$599$$ 4.47214 0.182727 0.0913633 0.995818i $$-0.470878\pi$$
0.0913633 + 0.995818i $$0.470878\pi$$
$$600$$ 0 0
$$601$$ −25.0000 −1.01977 −0.509886 0.860242i $$-0.670312\pi$$
−0.509886 + 0.860242i $$0.670312\pi$$
$$602$$ 0 0
$$603$$ 4.47214 0.182119
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 17.8885 0.726074 0.363037 0.931775i $$-0.381740\pi$$
0.363037 + 0.931775i $$0.381740\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −35.7771 −1.44739
$$612$$ 0 0
$$613$$ −26.0000 −1.05013 −0.525065 0.851062i $$-0.675959\pi$$
−0.525065 + 0.851062i $$0.675959\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 2.00000 0.0805170 0.0402585 0.999189i $$-0.487182\pi$$
0.0402585 + 0.999189i $$0.487182\pi$$
$$618$$ 0 0
$$619$$ 17.8885 0.719001 0.359501 0.933145i $$-0.382947\pi$$
0.359501 + 0.933145i $$0.382947\pi$$
$$620$$ 0 0
$$621$$ −10.0000 −0.401286
$$622$$ 0 0
$$623$$ −22.3607 −0.895862
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 33.5410 1.33950
$$628$$ 0 0
$$629$$ 14.0000 0.558217
$$630$$ 0 0
$$631$$ 13.4164 0.534099 0.267049 0.963683i $$-0.413951\pi$$
0.267049 + 0.963683i $$0.413951\pi$$
$$632$$ 0 0
$$633$$ 45.0000 1.78859
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 52.0000 2.06032
$$638$$ 0 0
$$639$$ 17.8885 0.707660
$$640$$ 0 0
$$641$$ 10.0000 0.394976 0.197488 0.980305i $$-0.436722\pi$$
0.197488 + 0.980305i $$0.436722\pi$$
$$642$$ 0 0
$$643$$ 17.8885 0.705455 0.352728 0.935726i $$-0.385254\pi$$
0.352728 + 0.935726i $$0.385254\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 8.94427 0.351636 0.175818 0.984423i $$-0.443743\pi$$
0.175818 + 0.984423i $$0.443743\pi$$
$$648$$ 0 0
$$649$$ 20.0000 0.785069
$$650$$ 0 0
$$651$$ 44.7214 1.75277
$$652$$ 0 0
$$653$$ −34.0000 −1.33052 −0.665261 0.746611i $$-0.731680\pi$$
−0.665261 + 0.746611i $$0.731680\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 18.0000 0.702247
$$658$$ 0 0
$$659$$ 6.70820 0.261315 0.130657 0.991428i $$-0.458291\pi$$
0.130657 + 0.991428i $$0.458291\pi$$
$$660$$ 0 0
$$661$$ −40.0000 −1.55582 −0.777910 0.628376i $$-0.783720\pi$$
−0.777910 + 0.628376i $$0.783720\pi$$
$$662$$ 0 0
$$663$$ −62.6099 −2.43157
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 60.0000 2.31973
$$670$$ 0 0
$$671$$ −22.3607 −0.863224
$$672$$ 0 0
$$673$$ −26.0000 −1.00223 −0.501113 0.865382i $$-0.667076\pi$$
−0.501113 + 0.865382i $$0.667076\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 32.0000 1.22986 0.614930 0.788582i $$-0.289184\pi$$
0.614930 + 0.788582i $$0.289184\pi$$
$$678$$ 0 0
$$679$$ −8.94427 −0.343250
$$680$$ 0 0
$$681$$ −40.0000 −1.53280
$$682$$ 0 0
$$683$$ −11.1803 −0.427804 −0.213902 0.976855i $$-0.568617\pi$$
−0.213902 + 0.976855i $$0.568617\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 44.7214 1.70623
$$688$$ 0 0
$$689$$ 24.0000 0.914327
$$690$$ 0 0
$$691$$ 38.0132 1.44609 0.723044 0.690802i $$-0.242742\pi$$
0.723044 + 0.690802i $$0.242742\pi$$
$$692$$ 0 0
$$693$$ 20.0000 0.759737
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 35.0000 1.32572
$$698$$ 0 0
$$699$$ −58.1378 −2.19897
$$700$$ 0 0
$$701$$ 20.0000 0.755390 0.377695 0.925930i $$-0.376717\pi$$
0.377695 + 0.925930i $$0.376717\pi$$
$$702$$ 0 0
$$703$$ −13.4164 −0.506009
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 8.94427 0.336384
$$708$$ 0 0
$$709$$ −20.0000 −0.751116 −0.375558 0.926799i $$-0.622549\pi$$
−0.375558 + 0.926799i $$0.622549\pi$$
$$710$$ 0 0
$$711$$ 8.94427 0.335436
$$712$$ 0 0
$$713$$ 20.0000 0.749006
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 40.0000 1.49383
$$718$$ 0 0
$$719$$ −4.47214 −0.166783 −0.0833913 0.996517i $$-0.526575\pi$$
−0.0833913 + 0.996517i $$0.526575\pi$$
$$720$$ 0 0
$$721$$ −40.0000 −1.48968
$$722$$ 0 0
$$723$$ −11.1803 −0.415801
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$728$$ 0 0
$$729$$ −7.00000 −0.259259
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −44.0000 −1.62518 −0.812589 0.582838i $$-0.801942\pi$$
−0.812589 + 0.582838i $$0.801942\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 5.00000 0.184177
$$738$$ 0 0
$$739$$ −35.7771 −1.31608 −0.658041 0.752982i $$-0.728615\pi$$
−0.658041 + 0.752982i $$0.728615\pi$$
$$740$$ 0 0
$$741$$ 60.0000 2.20416
$$742$$ 0 0
$$743$$ 31.3050 1.14847 0.574234 0.818691i $$-0.305300\pi$$
0.574234 + 0.818691i $$0.305300\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 22.3607 0.818134
$$748$$ 0 0
$$749$$ −10.0000 −0.365392
$$750$$ 0 0
$$751$$ −53.6656 −1.95829 −0.979143 0.203171i $$-0.934875\pi$$
−0.979143 + 0.203171i $$0.934875\pi$$
$$752$$ 0 0
$$753$$ 25.0000 0.911051
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 8.00000 0.290765 0.145382 0.989376i $$-0.453559\pi$$
0.145382 + 0.989376i $$0.453559\pi$$
$$758$$ 0 0
$$759$$ 22.3607 0.811641
$$760$$ 0 0
$$761$$ −43.0000 −1.55875 −0.779374 0.626559i $$-0.784463\pi$$
−0.779374 + 0.626559i $$0.784463\pi$$
$$762$$ 0 0
$$763$$ −26.8328 −0.971413
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 35.7771 1.29184
$$768$$ 0 0
$$769$$ −15.0000 −0.540914 −0.270457 0.962732i $$-0.587175\pi$$
−0.270457 + 0.962732i $$0.587175\pi$$
$$770$$ 0 0
$$771$$ −4.47214 −0.161060
$$772$$ 0 0
$$773$$ 24.0000 0.863220 0.431610 0.902060i $$-0.357946\pi$$
0.431610 + 0.902060i $$0.357946\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −20.0000 −0.717496
$$778$$ 0 0
$$779$$ −33.5410 −1.20173
$$780$$ 0 0
$$781$$ 20.0000 0.715656
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$788$$ 0 0
$$789$$ 50.0000 1.78005
$$790$$ 0 0
$$791$$ 4.47214 0.159011
$$792$$ 0 0
$$793$$ −40.0000 −1.42044
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 18.0000 0.637593 0.318796 0.947823i $$-0.396721\pi$$
0.318796 + 0.947823i $$0.396721\pi$$
$$798$$ 0 0
$$799$$ −62.6099 −2.21498
$$800$$ 0 0
$$801$$ −10.0000 −0.353333
$$802$$ 0 0
$$803$$ 20.1246 0.710182
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 53.6656 1.88912
$$808$$ 0 0
$$809$$ −30.0000 −1.05474 −0.527372 0.849635i $$-0.676823\pi$$
−0.527372 + 0.849635i $$0.676823\pi$$
$$810$$ 0 0
$$811$$ −17.8885 −0.628152 −0.314076 0.949398i $$-0.601695\pi$$
−0.314076 + 0.949398i $$0.601695\pi$$
$$812$$ 0 0
$$813$$ −30.0000 −1.05215
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 35.7771 1.25015
$$820$$ 0 0
$$821$$ −50.0000 −1.74501 −0.872506 0.488603i $$-0.837507\pi$$
−0.872506 + 0.488603i $$0.837507\pi$$
$$822$$ 0 0
$$823$$ 26.8328 0.935333 0.467667 0.883905i $$-0.345095\pi$$
0.467667 + 0.883905i $$0.345095\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −51.4296 −1.78838 −0.894191 0.447687i $$-0.852248\pi$$
−0.894191 + 0.447687i $$0.852248\pi$$
$$828$$ 0 0
$$829$$ −4.00000 −0.138926 −0.0694629 0.997585i $$-0.522129\pi$$
−0.0694629 + 0.997585i $$0.522129\pi$$
$$830$$ 0 0
$$831$$ −17.8885 −0.620547
$$832$$ 0 0
$$833$$ 91.0000 3.15296
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −10.0000 −0.345651
$$838$$ 0 0
$$839$$ 17.8885 0.617581 0.308791 0.951130i $$-0.400076\pi$$
0.308791 + 0.951130i $$0.400076\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ 0 0
$$843$$ −67.0820 −2.31043
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −26.8328 −0.921986
$$848$$ 0 0
$$849$$ 15.0000 0.514799
$$850$$ 0 0
$$851$$ −8.94427 −0.306606
$$852$$ 0 0
$$853$$ 34.0000 1.16414 0.582069 0.813139i $$-0.302243\pi$$
0.582069 + 0.813139i $$0.302243\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 3.00000 0.102478 0.0512390 0.998686i $$-0.483683\pi$$
0.0512390 + 0.998686i $$0.483683\pi$$
$$858$$ 0 0
$$859$$ −33.5410 −1.14440 −0.572202 0.820112i $$-0.693911\pi$$
−0.572202 + 0.820112i $$0.693911\pi$$
$$860$$ 0 0
$$861$$ −50.0000 −1.70400
$$862$$ 0 0
$$863$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −71.5542 −2.43011
$$868$$ 0 0
$$869$$ 10.0000 0.339227
$$870$$ 0 0
$$871$$ 8.94427 0.303065
$$872$$ 0 0
$$873$$ −4.00000 −0.135379
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 28.0000 0.945493 0.472746 0.881199i $$-0.343263\pi$$
0.472746 + 0.881199i $$0.343263\pi$$
$$878$$ 0 0
$$879$$ −31.3050 −1.05589
$$880$$ 0 0
$$881$$ 30.0000 1.01073 0.505363 0.862907i $$-0.331359\pi$$
0.505363 + 0.862907i $$0.331359\pi$$
$$882$$ 0 0
$$883$$ −2.23607 −0.0752497 −0.0376248 0.999292i $$-0.511979\pi$$
−0.0376248 + 0.999292i $$0.511979\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −8.94427 −0.300319 −0.150160 0.988662i $$-0.547979\pi$$
−0.150160 + 0.988662i $$0.547979\pi$$
$$888$$ 0 0
$$889$$ 20.0000 0.670778
$$890$$ 0 0
$$891$$ −24.5967 −0.824022
$$892$$ 0 0
$$893$$ 60.0000 2.00782
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 40.0000 1.33556
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 42.0000 1.39922
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 53.6656 1.78194 0.890969 0.454064i $$-0.150026\pi$$
0.890969 + 0.454064i $$0.150026\pi$$
$$908$$ 0 0
$$909$$ 4.00000 0.132672
$$910$$ 0 0
$$911$$ −44.7214 −1.48168 −0.740842 0.671679i $$-0.765573\pi$$
−0.740842 + 0.671679i $$0.765573\pi$$
$$912$$ 0 0
$$913$$ 25.0000 0.827379
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 80.0000 2.64183
$$918$$ 0 0
$$919$$ −13.4164 −0.442566 −0.221283 0.975210i $$-0.571025\pi$$
−0.221283 + 0.975210i $$0.571025\pi$$
$$920$$ 0 0
$$921$$ −35.0000 −1.15329
$$922$$ 0 0
$$923$$ 35.7771 1.17762
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −17.8885 −0.587537
$$928$$ 0 0
$$929$$ 14.0000 0.459325 0.229663 0.973270i $$-0.426238\pi$$
0.229663 + 0.973270i $$0.426238\pi$$
$$930$$ 0 0
$$931$$ −87.2067 −2.85808
$$932$$ 0 0
$$933$$ −50.0000 −1.63693
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 17.0000 0.555366 0.277683 0.960673i $$-0.410434\pi$$
0.277683 + 0.960673i $$0.410434\pi$$
$$938$$ 0 0
$$939$$ −13.4164 −0.437828
$$940$$ 0 0
$$941$$ 28.0000 0.912774 0.456387 0.889781i $$-0.349143\pi$$
0.456387 + 0.889781i $$0.349143\pi$$
$$942$$ 0 0
$$943$$ −22.3607 −0.728164
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −26.8328 −0.871949 −0.435975 0.899959i $$-0.643596\pi$$
−0.435975 + 0.899959i $$0.643596\pi$$
$$948$$ 0 0
$$949$$ 36.0000 1.16861
$$950$$ 0 0
$$951$$ 26.8328 0.870114
$$952$$ 0 0
$$953$$ −9.00000 −0.291539 −0.145769 0.989319i $$-0.546566\pi$$
−0.145769 + 0.989319i $$0.546566\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 13.4164 0.433238
$$960$$ 0 0
$$961$$ −11.0000 −0.354839
$$962$$ 0 0
$$963$$ −4.47214 −0.144113
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −53.6656 −1.72577 −0.862885 0.505400i $$-0.831345\pi$$
−0.862885 + 0.505400i $$0.831345\pi$$
$$968$$ 0 0
$$969$$ 105.000 3.37309
$$970$$ 0 0
$$971$$ 55.9017 1.79397 0.896985 0.442060i $$-0.145752\pi$$
0.896985 + 0.442060i $$0.145752\pi$$
$$972$$ 0 0
$$973$$ 10.0000 0.320585
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 3.00000 0.0959785 0.0479893 0.998848i $$-0.484719\pi$$
0.0479893 + 0.998848i $$0.484719\pi$$
$$978$$ 0 0
$$979$$ −11.1803 −0.357325
$$980$$ 0 0
$$981$$ −12.0000 −0.383131
$$982$$ 0 0
$$983$$ −4.47214 −0.142639 −0.0713195 0.997454i $$-0.522721\pi$$
−0.0713195 + 0.997454i $$0.522721\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 89.4427 2.84699
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −13.4164 −0.426186 −0.213093 0.977032i $$-0.568354\pi$$
−0.213093 + 0.977032i $$0.568354\pi$$
$$992$$ 0 0
$$993$$ −25.0000 −0.793351
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −52.0000 −1.64686 −0.823428 0.567420i $$-0.807941\pi$$
−0.823428 + 0.567420i $$0.807941\pi$$
$$998$$ 0 0
$$999$$ 4.47214 0.141492
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 1600.2.a.bb.1.1 2
4.3 odd 2 inner 1600.2.a.bb.1.2 2
5.2 odd 4 1600.2.c.o.449.4 4
5.3 odd 4 1600.2.c.o.449.2 4
5.4 even 2 1600.2.a.ba.1.2 2
8.3 odd 2 800.2.a.k.1.1 2
8.5 even 2 800.2.a.k.1.2 yes 2
20.3 even 4 1600.2.c.o.449.3 4
20.7 even 4 1600.2.c.o.449.1 4
20.19 odd 2 1600.2.a.ba.1.1 2
24.5 odd 2 7200.2.a.cf.1.2 2
24.11 even 2 7200.2.a.cf.1.1 2
40.3 even 4 800.2.c.g.449.2 4
40.13 odd 4 800.2.c.g.449.3 4
40.19 odd 2 800.2.a.l.1.2 yes 2
40.27 even 4 800.2.c.g.449.4 4
40.29 even 2 800.2.a.l.1.1 yes 2
40.37 odd 4 800.2.c.g.449.1 4
120.29 odd 2 7200.2.a.cn.1.1 2
120.53 even 4 7200.2.f.bg.6049.2 4
120.59 even 2 7200.2.a.cn.1.2 2
120.77 even 4 7200.2.f.bg.6049.4 4
120.83 odd 4 7200.2.f.bg.6049.3 4
120.107 odd 4 7200.2.f.bg.6049.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
800.2.a.k.1.1 2 8.3 odd 2
800.2.a.k.1.2 yes 2 8.5 even 2
800.2.a.l.1.1 yes 2 40.29 even 2
800.2.a.l.1.2 yes 2 40.19 odd 2
800.2.c.g.449.1 4 40.37 odd 4
800.2.c.g.449.2 4 40.3 even 4
800.2.c.g.449.3 4 40.13 odd 4
800.2.c.g.449.4 4 40.27 even 4
1600.2.a.ba.1.1 2 20.19 odd 2
1600.2.a.ba.1.2 2 5.4 even 2
1600.2.a.bb.1.1 2 1.1 even 1 trivial
1600.2.a.bb.1.2 2 4.3 odd 2 inner
1600.2.c.o.449.1 4 20.7 even 4
1600.2.c.o.449.2 4 5.3 odd 4
1600.2.c.o.449.3 4 20.3 even 4
1600.2.c.o.449.4 4 5.2 odd 4
7200.2.a.cf.1.1 2 24.11 even 2
7200.2.a.cf.1.2 2 24.5 odd 2
7200.2.a.cn.1.1 2 120.29 odd 2
7200.2.a.cn.1.2 2 120.59 even 2
7200.2.f.bg.6049.1 4 120.107 odd 4
7200.2.f.bg.6049.2 4 120.53 even 4
7200.2.f.bg.6049.3 4 120.83 odd 4
7200.2.f.bg.6049.4 4 120.77 even 4