# Properties

 Label 9576.2.a.by.1.1 Level $9576$ Weight $2$ Character 9576.1 Self dual yes Analytic conductor $76.465$ Analytic rank $0$ 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] = [9576,2,Mod(1,9576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9576, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("9576.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9576.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: $$76.4647449756$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3192) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 9576.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.585786 q^{5} -1.00000 q^{7} +O(q^{10})$$ $$q+0.585786 q^{5} -1.00000 q^{7} -2.82843 q^{11} -5.65685 q^{13} +6.24264 q^{17} +1.00000 q^{19} -8.48528 q^{23} -4.65685 q^{25} -1.75736 q^{29} +3.17157 q^{31} -0.585786 q^{35} -6.00000 q^{37} +3.17157 q^{41} -3.17157 q^{43} +13.4142 q^{47} +1.00000 q^{49} -9.07107 q^{53} -1.65685 q^{55} +13.6569 q^{59} +2.00000 q^{61} -3.31371 q^{65} -14.8284 q^{67} +12.7279 q^{71} -3.17157 q^{73} +2.82843 q^{77} +13.6569 q^{79} +3.75736 q^{83} +3.65685 q^{85} +11.1716 q^{89} +5.65685 q^{91} +0.585786 q^{95} -11.6569 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{5} - 2 q^{7}+O(q^{10})$$ 2 * q + 4 * q^5 - 2 * q^7 $$2 q + 4 q^{5} - 2 q^{7} + 4 q^{17} + 2 q^{19} + 2 q^{25} - 12 q^{29} + 12 q^{31} - 4 q^{35} - 12 q^{37} + 12 q^{41} - 12 q^{43} + 24 q^{47} + 2 q^{49} - 4 q^{53} + 8 q^{55} + 16 q^{59} + 4 q^{61} + 16 q^{65} - 24 q^{67} - 12 q^{73} + 16 q^{79} + 16 q^{83} - 4 q^{85} + 28 q^{89} + 4 q^{95} - 12 q^{97}+O(q^{100})$$ 2 * q + 4 * q^5 - 2 * q^7 + 4 * q^17 + 2 * q^19 + 2 * q^25 - 12 * q^29 + 12 * q^31 - 4 * q^35 - 12 * q^37 + 12 * q^41 - 12 * q^43 + 24 * q^47 + 2 * q^49 - 4 * q^53 + 8 * q^55 + 16 * q^59 + 4 * q^61 + 16 * q^65 - 24 * q^67 - 12 * q^73 + 16 * q^79 + 16 * q^83 - 4 * q^85 + 28 * q^89 + 4 * q^95 - 12 * 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 0.585786 0.261972 0.130986 0.991384i $$-0.458186\pi$$
0.130986 + 0.991384i $$0.458186\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −2.82843 −0.852803 −0.426401 0.904534i $$-0.640219\pi$$
−0.426401 + 0.904534i $$0.640219\pi$$
$$12$$ 0 0
$$13$$ −5.65685 −1.56893 −0.784465 0.620174i $$-0.787062\pi$$
−0.784465 + 0.620174i $$0.787062\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 6.24264 1.51406 0.757031 0.653379i $$-0.226649\pi$$
0.757031 + 0.653379i $$0.226649\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −8.48528 −1.76930 −0.884652 0.466252i $$-0.845604\pi$$
−0.884652 + 0.466252i $$0.845604\pi$$
$$24$$ 0 0
$$25$$ −4.65685 −0.931371
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −1.75736 −0.326333 −0.163167 0.986599i $$-0.552171\pi$$
−0.163167 + 0.986599i $$0.552171\pi$$
$$30$$ 0 0
$$31$$ 3.17157 0.569631 0.284816 0.958582i $$-0.408068\pi$$
0.284816 + 0.958582i $$0.408068\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −0.585786 −0.0990160
$$36$$ 0 0
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 3.17157 0.495316 0.247658 0.968847i $$-0.420339\pi$$
0.247658 + 0.968847i $$0.420339\pi$$
$$42$$ 0 0
$$43$$ −3.17157 −0.483660 −0.241830 0.970319i $$-0.577748\pi$$
−0.241830 + 0.970319i $$0.577748\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 13.4142 1.95666 0.978332 0.207042i $$-0.0663836\pi$$
0.978332 + 0.207042i $$0.0663836\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −9.07107 −1.24601 −0.623003 0.782219i $$-0.714088\pi$$
−0.623003 + 0.782219i $$0.714088\pi$$
$$54$$ 0 0
$$55$$ −1.65685 −0.223410
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 13.6569 1.77797 0.888985 0.457935i $$-0.151411\pi$$
0.888985 + 0.457935i $$0.151411\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −3.31371 −0.411015
$$66$$ 0 0
$$67$$ −14.8284 −1.81158 −0.905790 0.423726i $$-0.860722\pi$$
−0.905790 + 0.423726i $$0.860722\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 12.7279 1.51053 0.755263 0.655422i $$-0.227509\pi$$
0.755263 + 0.655422i $$0.227509\pi$$
$$72$$ 0 0
$$73$$ −3.17157 −0.371205 −0.185602 0.982625i $$-0.559424\pi$$
−0.185602 + 0.982625i $$0.559424\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 2.82843 0.322329
$$78$$ 0 0
$$79$$ 13.6569 1.53652 0.768258 0.640140i $$-0.221124\pi$$
0.768258 + 0.640140i $$0.221124\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 3.75736 0.412424 0.206212 0.978507i $$-0.433886\pi$$
0.206212 + 0.978507i $$0.433886\pi$$
$$84$$ 0 0
$$85$$ 3.65685 0.396642
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 11.1716 1.18418 0.592092 0.805870i $$-0.298302\pi$$
0.592092 + 0.805870i $$0.298302\pi$$
$$90$$ 0 0
$$91$$ 5.65685 0.592999
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0.585786 0.0601004
$$96$$ 0 0
$$97$$ −11.6569 −1.18357 −0.591787 0.806094i $$-0.701577\pi$$
−0.591787 + 0.806094i $$0.701577\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 5.07107 0.504590 0.252295 0.967650i $$-0.418815\pi$$
0.252295 + 0.967650i $$0.418815\pi$$
$$102$$ 0 0
$$103$$ 8.00000 0.788263 0.394132 0.919054i $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −8.72792 −0.843760 −0.421880 0.906652i $$-0.638630\pi$$
−0.421880 + 0.906652i $$0.638630\pi$$
$$108$$ 0 0
$$109$$ 12.1421 1.16301 0.581503 0.813544i $$-0.302465\pi$$
0.581503 + 0.813544i $$0.302465\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −11.4142 −1.07376 −0.536879 0.843659i $$-0.680397\pi$$
−0.536879 + 0.843659i $$0.680397\pi$$
$$114$$ 0 0
$$115$$ −4.97056 −0.463507
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −6.24264 −0.572262
$$120$$ 0 0
$$121$$ −3.00000 −0.272727
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −5.65685 −0.505964
$$126$$ 0 0
$$127$$ 8.48528 0.752947 0.376473 0.926427i $$-0.377137\pi$$
0.376473 + 0.926427i $$0.377137\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 9.89949 0.864923 0.432461 0.901652i $$-0.357645\pi$$
0.432461 + 0.901652i $$0.357645\pi$$
$$132$$ 0 0
$$133$$ −1.00000 −0.0867110
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0.343146 0.0293169 0.0146585 0.999893i $$-0.495334\pi$$
0.0146585 + 0.999893i $$0.495334\pi$$
$$138$$ 0 0
$$139$$ 4.48528 0.380437 0.190218 0.981742i $$-0.439080\pi$$
0.190218 + 0.981742i $$0.439080\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 16.0000 1.33799
$$144$$ 0 0
$$145$$ −1.02944 −0.0854901
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −1.51472 −0.124091 −0.0620453 0.998073i $$-0.519762\pi$$
−0.0620453 + 0.998073i $$0.519762\pi$$
$$150$$ 0 0
$$151$$ −8.48528 −0.690522 −0.345261 0.938507i $$-0.612210\pi$$
−0.345261 + 0.938507i $$0.612210\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 1.85786 0.149227
$$156$$ 0 0
$$157$$ 3.65685 0.291849 0.145924 0.989296i $$-0.453384\pi$$
0.145924 + 0.989296i $$0.453384\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 8.48528 0.668734
$$162$$ 0 0
$$163$$ −18.4853 −1.44788 −0.723939 0.689863i $$-0.757671\pi$$
−0.723939 + 0.689863i $$0.757671\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 17.1716 1.32878 0.664388 0.747388i $$-0.268692\pi$$
0.664388 + 0.747388i $$0.268692\pi$$
$$168$$ 0 0
$$169$$ 19.0000 1.46154
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 8.82843 0.671213 0.335606 0.942002i $$-0.391059\pi$$
0.335606 + 0.942002i $$0.391059\pi$$
$$174$$ 0 0
$$175$$ 4.65685 0.352025
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 11.0711 0.827490 0.413745 0.910393i $$-0.364220\pi$$
0.413745 + 0.910393i $$0.364220\pi$$
$$180$$ 0 0
$$181$$ 13.3137 0.989600 0.494800 0.869007i $$-0.335241\pi$$
0.494800 + 0.869007i $$0.335241\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −3.51472 −0.258407
$$186$$ 0 0
$$187$$ −17.6569 −1.29120
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 9.17157 0.663632 0.331816 0.943344i $$-0.392339\pi$$
0.331816 + 0.943344i $$0.392339\pi$$
$$192$$ 0 0
$$193$$ −11.6569 −0.839079 −0.419539 0.907737i $$-0.637808\pi$$
−0.419539 + 0.907737i $$0.637808\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 18.9706 1.35160 0.675798 0.737087i $$-0.263799\pi$$
0.675798 + 0.737087i $$0.263799\pi$$
$$198$$ 0 0
$$199$$ 14.1421 1.00251 0.501255 0.865300i $$-0.332872\pi$$
0.501255 + 0.865300i $$0.332872\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 1.75736 0.123342
$$204$$ 0 0
$$205$$ 1.85786 0.129759
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −2.82843 −0.195646
$$210$$ 0 0
$$211$$ −4.97056 −0.342188 −0.171094 0.985255i $$-0.554730\pi$$
−0.171094 + 0.985255i $$0.554730\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −1.85786 −0.126705
$$216$$ 0 0
$$217$$ −3.17157 −0.215300
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −35.3137 −2.37546
$$222$$ 0 0
$$223$$ 4.82843 0.323335 0.161668 0.986845i $$-0.448313\pi$$
0.161668 + 0.986845i $$0.448313\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −28.4853 −1.89063 −0.945317 0.326152i $$-0.894248\pi$$
−0.945317 + 0.326152i $$0.894248\pi$$
$$228$$ 0 0
$$229$$ −6.48528 −0.428559 −0.214280 0.976772i $$-0.568740\pi$$
−0.214280 + 0.976772i $$0.568740\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 6.48528 0.424865 0.212432 0.977176i $$-0.431861\pi$$
0.212432 + 0.977176i $$0.431861\pi$$
$$234$$ 0 0
$$235$$ 7.85786 0.512591
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 13.6569 0.883388 0.441694 0.897166i $$-0.354378\pi$$
0.441694 + 0.897166i $$0.354378\pi$$
$$240$$ 0 0
$$241$$ 14.0000 0.901819 0.450910 0.892570i $$-0.351100\pi$$
0.450910 + 0.892570i $$0.351100\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0.585786 0.0374245
$$246$$ 0 0
$$247$$ −5.65685 −0.359937
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −4.24264 −0.267793 −0.133897 0.990995i $$-0.542749\pi$$
−0.133897 + 0.990995i $$0.542749\pi$$
$$252$$ 0 0
$$253$$ 24.0000 1.50887
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 14.0000 0.873296 0.436648 0.899632i $$-0.356166\pi$$
0.436648 + 0.899632i $$0.356166\pi$$
$$258$$ 0 0
$$259$$ 6.00000 0.372822
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 9.65685 0.595467 0.297734 0.954649i $$-0.403769\pi$$
0.297734 + 0.954649i $$0.403769\pi$$
$$264$$ 0 0
$$265$$ −5.31371 −0.326419
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 0.343146 0.0209220 0.0104610 0.999945i $$-0.496670\pi$$
0.0104610 + 0.999945i $$0.496670\pi$$
$$270$$ 0 0
$$271$$ 16.9706 1.03089 0.515444 0.856923i $$-0.327627\pi$$
0.515444 + 0.856923i $$0.327627\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 13.1716 0.794276
$$276$$ 0 0
$$277$$ 11.3137 0.679775 0.339887 0.940466i $$-0.389611\pi$$
0.339887 + 0.940466i $$0.389611\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −10.2426 −0.611025 −0.305512 0.952188i $$-0.598828\pi$$
−0.305512 + 0.952188i $$0.598828\pi$$
$$282$$ 0 0
$$283$$ −14.1421 −0.840663 −0.420331 0.907371i $$-0.638086\pi$$
−0.420331 + 0.907371i $$0.638086\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −3.17157 −0.187212
$$288$$ 0 0
$$289$$ 21.9706 1.29239
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −18.4853 −1.07992 −0.539961 0.841690i $$-0.681561\pi$$
−0.539961 + 0.841690i $$0.681561\pi$$
$$294$$ 0 0
$$295$$ 8.00000 0.465778
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 48.0000 2.77591
$$300$$ 0 0
$$301$$ 3.17157 0.182806
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 1.17157 0.0670841
$$306$$ 0 0
$$307$$ 12.1421 0.692988 0.346494 0.938052i $$-0.387372\pi$$
0.346494 + 0.938052i $$0.387372\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −20.0416 −1.13646 −0.568228 0.822871i $$-0.692371\pi$$
−0.568228 + 0.822871i $$0.692371\pi$$
$$312$$ 0 0
$$313$$ −30.9706 −1.75056 −0.875280 0.483617i $$-0.839323\pi$$
−0.875280 + 0.483617i $$0.839323\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 15.2132 0.854459 0.427229 0.904143i $$-0.359490\pi$$
0.427229 + 0.904143i $$0.359490\pi$$
$$318$$ 0 0
$$319$$ 4.97056 0.278298
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 6.24264 0.347350
$$324$$ 0 0
$$325$$ 26.3431 1.46125
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −13.4142 −0.739550
$$330$$ 0 0
$$331$$ 14.1421 0.777322 0.388661 0.921381i $$-0.372938\pi$$
0.388661 + 0.921381i $$0.372938\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −8.68629 −0.474583
$$336$$ 0 0
$$337$$ −3.17157 −0.172767 −0.0863833 0.996262i $$-0.527531\pi$$
−0.0863833 + 0.996262i $$0.527531\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −8.97056 −0.485783
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 12.9706 0.696296 0.348148 0.937440i $$-0.386811\pi$$
0.348148 + 0.937440i $$0.386811\pi$$
$$348$$ 0 0
$$349$$ 6.68629 0.357909 0.178954 0.983857i $$-0.442729\pi$$
0.178954 + 0.983857i $$0.442729\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −19.8995 −1.05914 −0.529572 0.848265i $$-0.677647\pi$$
−0.529572 + 0.848265i $$0.677647\pi$$
$$354$$ 0 0
$$355$$ 7.45584 0.395715
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 12.4853 0.658948 0.329474 0.944165i $$-0.393129\pi$$
0.329474 + 0.944165i $$0.393129\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −1.85786 −0.0972451
$$366$$ 0 0
$$367$$ 34.1421 1.78220 0.891102 0.453802i $$-0.149933\pi$$
0.891102 + 0.453802i $$0.149933\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 9.07107 0.470946
$$372$$ 0 0
$$373$$ 35.6569 1.84624 0.923121 0.384510i $$-0.125629\pi$$
0.923121 + 0.384510i $$0.125629\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 9.94113 0.511994
$$378$$ 0 0
$$379$$ 9.17157 0.471112 0.235556 0.971861i $$-0.424309\pi$$
0.235556 + 0.971861i $$0.424309\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 3.51472 0.179594 0.0897969 0.995960i $$-0.471378\pi$$
0.0897969 + 0.995960i $$0.471378\pi$$
$$384$$ 0 0
$$385$$ 1.65685 0.0844411
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 15.4558 0.783642 0.391821 0.920041i $$-0.371845\pi$$
0.391821 + 0.920041i $$0.371845\pi$$
$$390$$ 0 0
$$391$$ −52.9706 −2.67884
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 8.00000 0.402524
$$396$$ 0 0
$$397$$ −21.3137 −1.06970 −0.534852 0.844946i $$-0.679633\pi$$
−0.534852 + 0.844946i $$0.679633\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −24.3848 −1.21772 −0.608859 0.793279i $$-0.708372\pi$$
−0.608859 + 0.793279i $$0.708372\pi$$
$$402$$ 0 0
$$403$$ −17.9411 −0.893711
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 16.9706 0.841200
$$408$$ 0 0
$$409$$ 26.6274 1.31664 0.658321 0.752738i $$-0.271267\pi$$
0.658321 + 0.752738i $$0.271267\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −13.6569 −0.672010
$$414$$ 0 0
$$415$$ 2.20101 0.108043
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −20.7279 −1.01263 −0.506313 0.862350i $$-0.668992\pi$$
−0.506313 + 0.862350i $$0.668992\pi$$
$$420$$ 0 0
$$421$$ −18.4853 −0.900917 −0.450459 0.892797i $$-0.648740\pi$$
−0.450459 + 0.892797i $$0.648740\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −29.0711 −1.41015
$$426$$ 0 0
$$427$$ −2.00000 −0.0967868
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 30.5858 1.47327 0.736633 0.676293i $$-0.236415\pi$$
0.736633 + 0.676293i $$0.236415\pi$$
$$432$$ 0 0
$$433$$ −40.6274 −1.95243 −0.976215 0.216807i $$-0.930436\pi$$
−0.976215 + 0.216807i $$0.930436\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −8.48528 −0.405906
$$438$$ 0 0
$$439$$ −2.20101 −0.105048 −0.0525242 0.998620i $$-0.516727\pi$$
−0.0525242 + 0.998620i $$0.516727\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −23.3137 −1.10767 −0.553834 0.832627i $$-0.686836\pi$$
−0.553834 + 0.832627i $$0.686836\pi$$
$$444$$ 0 0
$$445$$ 6.54416 0.310223
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 36.8701 1.74001 0.870003 0.493047i $$-0.164117\pi$$
0.870003 + 0.493047i $$0.164117\pi$$
$$450$$ 0 0
$$451$$ −8.97056 −0.422407
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 3.31371 0.155349
$$456$$ 0 0
$$457$$ −4.68629 −0.219215 −0.109608 0.993975i $$-0.534959\pi$$
−0.109608 + 0.993975i $$0.534959\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −26.5269 −1.23548 −0.617741 0.786382i $$-0.711952\pi$$
−0.617741 + 0.786382i $$0.711952\pi$$
$$462$$ 0 0
$$463$$ −24.0000 −1.11537 −0.557687 0.830051i $$-0.688311\pi$$
−0.557687 + 0.830051i $$0.688311\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 9.89949 0.458094 0.229047 0.973415i $$-0.426439\pi$$
0.229047 + 0.973415i $$0.426439\pi$$
$$468$$ 0 0
$$469$$ 14.8284 0.684713
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 8.97056 0.412467
$$474$$ 0 0
$$475$$ −4.65685 −0.213671
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −24.7279 −1.12985 −0.564924 0.825143i $$-0.691094\pi$$
−0.564924 + 0.825143i $$0.691094\pi$$
$$480$$ 0 0
$$481$$ 33.9411 1.54758
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −6.82843 −0.310063
$$486$$ 0 0
$$487$$ 12.0000 0.543772 0.271886 0.962329i $$-0.412353\pi$$
0.271886 + 0.962329i $$0.412353\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −1.65685 −0.0747728 −0.0373864 0.999301i $$-0.511903\pi$$
−0.0373864 + 0.999301i $$0.511903\pi$$
$$492$$ 0 0
$$493$$ −10.9706 −0.494089
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −12.7279 −0.570925
$$498$$ 0 0
$$499$$ −35.5980 −1.59358 −0.796792 0.604253i $$-0.793471\pi$$
−0.796792 + 0.604253i $$0.793471\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 34.1838 1.52418 0.762089 0.647472i $$-0.224174\pi$$
0.762089 + 0.647472i $$0.224174\pi$$
$$504$$ 0 0
$$505$$ 2.97056 0.132188
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −18.9706 −0.840855 −0.420428 0.907326i $$-0.638120\pi$$
−0.420428 + 0.907326i $$0.638120\pi$$
$$510$$ 0 0
$$511$$ 3.17157 0.140302
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 4.68629 0.206503
$$516$$ 0 0
$$517$$ −37.9411 −1.66865
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 5.02944 0.220344 0.110172 0.993913i $$-0.464860\pi$$
0.110172 + 0.993913i $$0.464860\pi$$
$$522$$ 0 0
$$523$$ 28.1421 1.23057 0.615285 0.788305i $$-0.289041\pi$$
0.615285 + 0.788305i $$0.289041\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 19.7990 0.862458
$$528$$ 0 0
$$529$$ 49.0000 2.13043
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −17.9411 −0.777116
$$534$$ 0 0
$$535$$ −5.11270 −0.221041
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −2.82843 −0.121829
$$540$$ 0 0
$$541$$ −22.0000 −0.945854 −0.472927 0.881102i $$-0.656803\pi$$
−0.472927 + 0.881102i $$0.656803\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 7.11270 0.304675
$$546$$ 0 0
$$547$$ 43.5980 1.86412 0.932058 0.362310i $$-0.118012\pi$$
0.932058 + 0.362310i $$0.118012\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −1.75736 −0.0748660
$$552$$ 0 0
$$553$$ −13.6569 −0.580749
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −8.82843 −0.374072 −0.187036 0.982353i $$-0.559888\pi$$
−0.187036 + 0.982353i $$0.559888\pi$$
$$558$$ 0 0
$$559$$ 17.9411 0.758829
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 7.51472 0.316708 0.158354 0.987382i $$-0.449381\pi$$
0.158354 + 0.987382i $$0.449381\pi$$
$$564$$ 0 0
$$565$$ −6.68629 −0.281294
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −46.5269 −1.95051 −0.975255 0.221084i $$-0.929041\pi$$
−0.975255 + 0.221084i $$0.929041\pi$$
$$570$$ 0 0
$$571$$ 20.9706 0.877591 0.438795 0.898587i $$-0.355405\pi$$
0.438795 + 0.898587i $$0.355405\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 39.5147 1.64788
$$576$$ 0 0
$$577$$ 15.6569 0.651803 0.325902 0.945404i $$-0.394332\pi$$
0.325902 + 0.945404i $$0.394332\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −3.75736 −0.155882
$$582$$ 0 0
$$583$$ 25.6569 1.06260
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 15.0711 0.622050 0.311025 0.950402i $$-0.399328\pi$$
0.311025 + 0.950402i $$0.399328\pi$$
$$588$$ 0 0
$$589$$ 3.17157 0.130682
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 11.6152 0.476980 0.238490 0.971145i $$-0.423348\pi$$
0.238490 + 0.971145i $$0.423348\pi$$
$$594$$ 0 0
$$595$$ −3.65685 −0.149916
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −41.8995 −1.71197 −0.855983 0.517003i $$-0.827048\pi$$
−0.855983 + 0.517003i $$0.827048\pi$$
$$600$$ 0 0
$$601$$ 13.3137 0.543077 0.271539 0.962428i $$-0.412467\pi$$
0.271539 + 0.962428i $$0.412467\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −1.75736 −0.0714468
$$606$$ 0 0
$$607$$ −41.9411 −1.70234 −0.851169 0.524892i $$-0.824106\pi$$
−0.851169 + 0.524892i $$0.824106\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −75.8823 −3.06987
$$612$$ 0 0
$$613$$ 44.2843 1.78862 0.894312 0.447443i $$-0.147665\pi$$
0.894312 + 0.447443i $$0.147665\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −31.9411 −1.28590 −0.642951 0.765908i $$-0.722290\pi$$
−0.642951 + 0.765908i $$0.722290\pi$$
$$618$$ 0 0
$$619$$ 28.4853 1.14492 0.572460 0.819933i $$-0.305989\pi$$
0.572460 + 0.819933i $$0.305989\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −11.1716 −0.447580
$$624$$ 0 0
$$625$$ 19.9706 0.798823
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −37.4558 −1.49346
$$630$$ 0 0
$$631$$ −35.4558 −1.41147 −0.705737 0.708473i $$-0.749384\pi$$
−0.705737 + 0.708473i $$0.749384\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 4.97056 0.197251
$$636$$ 0 0
$$637$$ −5.65685 −0.224133
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 11.6985 0.462062 0.231031 0.972946i $$-0.425790\pi$$
0.231031 + 0.972946i $$0.425790\pi$$
$$642$$ 0 0
$$643$$ 26.8284 1.05801 0.529005 0.848619i $$-0.322565\pi$$
0.529005 + 0.848619i $$0.322565\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 24.7279 0.972155 0.486077 0.873916i $$-0.338427\pi$$
0.486077 + 0.873916i $$0.338427\pi$$
$$648$$ 0 0
$$649$$ −38.6274 −1.51626
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 41.5980 1.62785 0.813927 0.580967i $$-0.197325\pi$$
0.813927 + 0.580967i $$0.197325\pi$$
$$654$$ 0 0
$$655$$ 5.79899 0.226585
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −7.75736 −0.302184 −0.151092 0.988520i $$-0.548279\pi$$
−0.151092 + 0.988520i $$0.548279\pi$$
$$660$$ 0 0
$$661$$ 12.0000 0.466746 0.233373 0.972387i $$-0.425024\pi$$
0.233373 + 0.972387i $$0.425024\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −0.585786 −0.0227158
$$666$$ 0 0
$$667$$ 14.9117 0.577383
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −5.65685 −0.218380
$$672$$ 0 0
$$673$$ −1.31371 −0.0506397 −0.0253199 0.999679i $$-0.508060\pi$$
−0.0253199 + 0.999679i $$0.508060\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 27.9411 1.07386 0.536932 0.843625i $$-0.319583\pi$$
0.536932 + 0.843625i $$0.319583\pi$$
$$678$$ 0 0
$$679$$ 11.6569 0.447349
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 2.10051 0.0803736 0.0401868 0.999192i $$-0.487205\pi$$
0.0401868 + 0.999192i $$0.487205\pi$$
$$684$$ 0 0
$$685$$ 0.201010 0.00768020
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 51.3137 1.95490
$$690$$ 0 0
$$691$$ −4.97056 −0.189089 −0.0945446 0.995521i $$-0.530139\pi$$
−0.0945446 + 0.995521i $$0.530139\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 2.62742 0.0996636
$$696$$ 0 0
$$697$$ 19.7990 0.749940
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −50.0000 −1.88847 −0.944237 0.329267i $$-0.893198\pi$$
−0.944237 + 0.329267i $$0.893198\pi$$
$$702$$ 0 0
$$703$$ −6.00000 −0.226294
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −5.07107 −0.190717
$$708$$ 0 0
$$709$$ 9.65685 0.362671 0.181335 0.983421i $$-0.441958\pi$$
0.181335 + 0.983421i $$0.441958\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −26.9117 −1.00785
$$714$$ 0 0
$$715$$ 9.37258 0.350515
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 30.3848 1.13316 0.566580 0.824006i $$-0.308266\pi$$
0.566580 + 0.824006i $$0.308266\pi$$
$$720$$ 0 0
$$721$$ −8.00000 −0.297936
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 8.18377 0.303937
$$726$$ 0 0
$$727$$ 18.6274 0.690853 0.345426 0.938446i $$-0.387734\pi$$
0.345426 + 0.938446i $$0.387734\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −19.7990 −0.732292
$$732$$ 0 0
$$733$$ −15.1716 −0.560375 −0.280187 0.959945i $$-0.590397\pi$$
−0.280187 + 0.959945i $$0.590397\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 41.9411 1.54492
$$738$$ 0 0
$$739$$ 1.79899 0.0661769 0.0330885 0.999452i $$-0.489466\pi$$
0.0330885 + 0.999452i $$0.489466\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 24.9289 0.914554 0.457277 0.889324i $$-0.348825\pi$$
0.457277 + 0.889324i $$0.348825\pi$$
$$744$$ 0 0
$$745$$ −0.887302 −0.0325082
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 8.72792 0.318911
$$750$$ 0 0
$$751$$ 51.7990 1.89017 0.945086 0.326822i $$-0.105978\pi$$
0.945086 + 0.326822i $$0.105978\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −4.97056 −0.180897
$$756$$ 0 0
$$757$$ 15.6569 0.569058 0.284529 0.958667i $$-0.408163\pi$$
0.284529 + 0.958667i $$0.408163\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −4.78680 −0.173521 −0.0867606 0.996229i $$-0.527652\pi$$
−0.0867606 + 0.996229i $$0.527652\pi$$
$$762$$ 0 0
$$763$$ −12.1421 −0.439575
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −77.2548 −2.78951
$$768$$ 0 0
$$769$$ 34.9706 1.26107 0.630535 0.776161i $$-0.282835\pi$$
0.630535 + 0.776161i $$0.282835\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 30.4853 1.09648 0.548240 0.836321i $$-0.315298\pi$$
0.548240 + 0.836321i $$0.315298\pi$$
$$774$$ 0 0
$$775$$ −14.7696 −0.530538
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 3.17157 0.113633
$$780$$ 0 0
$$781$$ −36.0000 −1.28818
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 2.14214 0.0764561
$$786$$ 0 0
$$787$$ −12.1421 −0.432820 −0.216410 0.976303i $$-0.569435\pi$$
−0.216410 + 0.976303i $$0.569435\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 11.4142 0.405843
$$792$$ 0 0
$$793$$ −11.3137 −0.401762
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 10.4853 0.371408 0.185704 0.982606i $$-0.440543\pi$$
0.185704 + 0.982606i $$0.440543\pi$$
$$798$$ 0 0
$$799$$ 83.7401 2.96251
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 8.97056 0.316564
$$804$$ 0 0
$$805$$ 4.97056 0.175189
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −6.97056 −0.245072 −0.122536 0.992464i $$-0.539103\pi$$
−0.122536 + 0.992464i $$0.539103\pi$$
$$810$$ 0 0
$$811$$ −11.0294 −0.387296 −0.193648 0.981071i $$-0.562032\pi$$
−0.193648 + 0.981071i $$0.562032\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −10.8284 −0.379303
$$816$$ 0 0
$$817$$ −3.17157 −0.110959
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 11.1716 0.389891 0.194945 0.980814i $$-0.437547\pi$$
0.194945 + 0.980814i $$0.437547\pi$$
$$822$$ 0 0
$$823$$ 15.8579 0.552770 0.276385 0.961047i $$-0.410863\pi$$
0.276385 + 0.961047i $$0.410863\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 2.10051 0.0730417 0.0365209 0.999333i $$-0.488372\pi$$
0.0365209 + 0.999333i $$0.488372\pi$$
$$828$$ 0 0
$$829$$ −6.28427 −0.218262 −0.109131 0.994027i $$-0.534807\pi$$
−0.109131 + 0.994027i $$0.534807\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 6.24264 0.216295
$$834$$ 0 0
$$835$$ 10.0589 0.348102
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 49.2548 1.70047 0.850233 0.526407i $$-0.176461\pi$$
0.850233 + 0.526407i $$0.176461\pi$$
$$840$$ 0 0
$$841$$ −25.9117 −0.893506
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 11.1299 0.382882
$$846$$ 0 0
$$847$$ 3.00000 0.103081
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 50.9117 1.74523
$$852$$ 0 0
$$853$$ 34.9706 1.19737 0.598685 0.800985i $$-0.295690\pi$$
0.598685 + 0.800985i $$0.295690\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 9.51472 0.325017 0.162508 0.986707i $$-0.448042\pi$$
0.162508 + 0.986707i $$0.448042\pi$$
$$858$$ 0 0
$$859$$ 42.3431 1.44473 0.722365 0.691512i $$-0.243055\pi$$
0.722365 + 0.691512i $$0.243055\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −36.2426 −1.23371 −0.616857 0.787075i $$-0.711594\pi$$
−0.616857 + 0.787075i $$0.711594\pi$$
$$864$$ 0 0
$$865$$ 5.17157 0.175839
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −38.6274 −1.31035
$$870$$ 0 0
$$871$$ 83.8823 2.84224
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 5.65685 0.191237
$$876$$ 0 0
$$877$$ 41.3137 1.39506 0.697532 0.716553i $$-0.254281\pi$$
0.697532 + 0.716553i $$0.254281\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 16.3848 0.552017 0.276009 0.961155i $$-0.410988\pi$$
0.276009 + 0.961155i $$0.410988\pi$$
$$882$$ 0 0
$$883$$ −44.0000 −1.48072 −0.740359 0.672212i $$-0.765344\pi$$
−0.740359 + 0.672212i $$0.765344\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −42.4264 −1.42454 −0.712270 0.701906i $$-0.752333\pi$$
−0.712270 + 0.701906i $$0.752333\pi$$
$$888$$ 0 0
$$889$$ −8.48528 −0.284587
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 13.4142 0.448890
$$894$$ 0 0
$$895$$ 6.48528 0.216779
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −5.57359 −0.185890
$$900$$ 0 0
$$901$$ −56.6274 −1.88653
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 7.79899 0.259247
$$906$$ 0 0
$$907$$ 47.5980 1.58046 0.790232 0.612807i $$-0.209960\pi$$
0.790232 + 0.612807i $$0.209960\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 47.5563 1.57561 0.787806 0.615923i $$-0.211217\pi$$
0.787806 + 0.615923i $$0.211217\pi$$
$$912$$ 0 0
$$913$$ −10.6274 −0.351716
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −9.89949 −0.326910
$$918$$ 0 0
$$919$$ −24.0000 −0.791687 −0.395843 0.918318i $$-0.629548\pi$$
−0.395843 + 0.918318i $$0.629548\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −72.0000 −2.36991
$$924$$ 0 0
$$925$$ 27.9411 0.918699
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −0.786797 −0.0258140 −0.0129070 0.999917i $$-0.504109\pi$$
−0.0129070 + 0.999917i $$0.504109\pi$$
$$930$$ 0 0
$$931$$ 1.00000 0.0327737
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −10.3431 −0.338257
$$936$$ 0 0
$$937$$ 58.9706 1.92648 0.963242 0.268635i $$-0.0865724\pi$$
0.963242 + 0.268635i $$0.0865724\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −26.2843 −0.856843 −0.428421 0.903579i $$-0.640930\pi$$
−0.428421 + 0.903579i $$0.640930\pi$$
$$942$$ 0 0
$$943$$ −26.9117 −0.876365
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 12.4853 0.405717 0.202859 0.979208i $$-0.434977\pi$$
0.202859 + 0.979208i $$0.434977\pi$$
$$948$$ 0 0
$$949$$ 17.9411 0.582394
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 1.95837 0.0634378 0.0317189 0.999497i $$-0.489902\pi$$
0.0317189 + 0.999497i $$0.489902\pi$$
$$954$$ 0 0
$$955$$ 5.37258 0.173853
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −0.343146 −0.0110808
$$960$$ 0 0
$$961$$ −20.9411 −0.675520
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −6.82843 −0.219815
$$966$$ 0 0
$$967$$ −19.1716 −0.616516 −0.308258 0.951303i $$-0.599746\pi$$
−0.308258 + 0.951303i $$0.599746\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 35.3137 1.13327 0.566635 0.823969i $$-0.308245\pi$$
0.566635 + 0.823969i $$0.308245\pi$$
$$972$$ 0 0
$$973$$ −4.48528 −0.143792
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 49.1543 1.57259 0.786293 0.617854i $$-0.211998\pi$$
0.786293 + 0.617854i $$0.211998\pi$$
$$978$$ 0 0
$$979$$ −31.5980 −1.00988
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 28.2843 0.902128 0.451064 0.892492i $$-0.351045\pi$$
0.451064 + 0.892492i $$0.351045\pi$$
$$984$$ 0 0
$$985$$ 11.1127 0.354080
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 26.9117 0.855742
$$990$$ 0 0
$$991$$ −32.2843 −1.02554 −0.512772 0.858525i $$-0.671381\pi$$
−0.512772 + 0.858525i $$0.671381\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 8.28427 0.262629
$$996$$ 0 0
$$997$$ 8.34315 0.264230 0.132115 0.991234i $$-0.457823\pi$$
0.132115 + 0.991234i $$0.457823\pi$$
$$998$$ 0 0
$$999$$ 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 9576.2.a.by.1.1 2
3.2 odd 2 3192.2.a.r.1.2 2
12.11 even 2 6384.2.a.bh.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
3192.2.a.r.1.2 2 3.2 odd 2
6384.2.a.bh.1.2 2 12.11 even 2
9576.2.a.by.1.1 2 1.1 even 1 trivial