# Properties

 Label 7616.2.a.z.1.1 Level $7616$ Weight $2$ Character 7616.1 Self dual yes Analytic conductor $60.814$ 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] = [7616,2,Mod(1,7616)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7616, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7616.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7616 = 2^{6} \cdot 7 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7616.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: $$60.8140661794$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 476) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.30278$$ of defining polynomial Character $$\chi$$ $$=$$ 7616.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.302776 q^{3} +3.30278 q^{5} +1.00000 q^{7} -2.90833 q^{9} +O(q^{10})$$ $$q-0.302776 q^{3} +3.30278 q^{5} +1.00000 q^{7} -2.90833 q^{9} -4.60555 q^{11} +6.60555 q^{13} -1.00000 q^{15} -1.00000 q^{17} +6.00000 q^{19} -0.302776 q^{21} -2.60555 q^{23} +5.90833 q^{25} +1.78890 q^{27} +8.60555 q^{29} -6.69722 q^{31} +1.39445 q^{33} +3.30278 q^{35} -7.21110 q^{37} -2.00000 q^{39} +9.51388 q^{41} +4.30278 q^{43} -9.60555 q^{45} +10.0000 q^{47} +1.00000 q^{49} +0.302776 q^{51} -0.697224 q^{53} -15.2111 q^{55} -1.81665 q^{57} -5.21110 q^{59} +4.30278 q^{61} -2.90833 q^{63} +21.8167 q^{65} -2.69722 q^{67} +0.788897 q^{69} -2.00000 q^{71} -13.5139 q^{73} -1.78890 q^{75} -4.60555 q^{77} -6.00000 q^{79} +8.18335 q^{81} -9.21110 q^{83} -3.30278 q^{85} -2.60555 q^{87} -2.00000 q^{89} +6.60555 q^{91} +2.02776 q^{93} +19.8167 q^{95} -2.09167 q^{97} +13.3944 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{3} + 3 q^{5} + 2 q^{7} + 5 q^{9}+O(q^{10})$$ 2 * q + 3 * q^3 + 3 * q^5 + 2 * q^7 + 5 * q^9 $$2 q + 3 q^{3} + 3 q^{5} + 2 q^{7} + 5 q^{9} - 2 q^{11} + 6 q^{13} - 2 q^{15} - 2 q^{17} + 12 q^{19} + 3 q^{21} + 2 q^{23} + q^{25} + 18 q^{27} + 10 q^{29} - 17 q^{31} + 10 q^{33} + 3 q^{35} - 4 q^{39} + q^{41} + 5 q^{43} - 12 q^{45} + 20 q^{47} + 2 q^{49} - 3 q^{51} - 5 q^{53} - 16 q^{55} + 18 q^{57} + 4 q^{59} + 5 q^{61} + 5 q^{63} + 22 q^{65} - 9 q^{67} + 16 q^{69} - 4 q^{71} - 9 q^{73} - 18 q^{75} - 2 q^{77} - 12 q^{79} + 38 q^{81} - 4 q^{83} - 3 q^{85} + 2 q^{87} - 4 q^{89} + 6 q^{91} - 32 q^{93} + 18 q^{95} - 15 q^{97} + 34 q^{99}+O(q^{100})$$ 2 * q + 3 * q^3 + 3 * q^5 + 2 * q^7 + 5 * q^9 - 2 * q^11 + 6 * q^13 - 2 * q^15 - 2 * q^17 + 12 * q^19 + 3 * q^21 + 2 * q^23 + q^25 + 18 * q^27 + 10 * q^29 - 17 * q^31 + 10 * q^33 + 3 * q^35 - 4 * q^39 + q^41 + 5 * q^43 - 12 * q^45 + 20 * q^47 + 2 * q^49 - 3 * q^51 - 5 * q^53 - 16 * q^55 + 18 * q^57 + 4 * q^59 + 5 * q^61 + 5 * q^63 + 22 * q^65 - 9 * q^67 + 16 * q^69 - 4 * q^71 - 9 * q^73 - 18 * q^75 - 2 * q^77 - 12 * q^79 + 38 * q^81 - 4 * q^83 - 3 * q^85 + 2 * q^87 - 4 * q^89 + 6 * q^91 - 32 * q^93 + 18 * q^95 - 15 * q^97 + 34 * q^99

## 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
$$3$$ −0.302776 −0.174808 −0.0874038 0.996173i $$-0.527857\pi$$
−0.0874038 + 0.996173i $$0.527857\pi$$
$$4$$ 0 0
$$5$$ 3.30278 1.47705 0.738523 0.674228i $$-0.235524\pi$$
0.738523 + 0.674228i $$0.235524\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ −2.90833 −0.969442
$$10$$ 0 0
$$11$$ −4.60555 −1.38863 −0.694313 0.719673i $$-0.744292\pi$$
−0.694313 + 0.719673i $$0.744292\pi$$
$$12$$ 0 0
$$13$$ 6.60555 1.83205 0.916025 0.401121i $$-0.131379\pi$$
0.916025 + 0.401121i $$0.131379\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 0 0
$$17$$ −1.00000 −0.242536
$$18$$ 0 0
$$19$$ 6.00000 1.37649 0.688247 0.725476i $$-0.258380\pi$$
0.688247 + 0.725476i $$0.258380\pi$$
$$20$$ 0 0
$$21$$ −0.302776 −0.0660711
$$22$$ 0 0
$$23$$ −2.60555 −0.543295 −0.271647 0.962397i $$-0.587568\pi$$
−0.271647 + 0.962397i $$0.587568\pi$$
$$24$$ 0 0
$$25$$ 5.90833 1.18167
$$26$$ 0 0
$$27$$ 1.78890 0.344273
$$28$$ 0 0
$$29$$ 8.60555 1.59801 0.799005 0.601324i $$-0.205360\pi$$
0.799005 + 0.601324i $$0.205360\pi$$
$$30$$ 0 0
$$31$$ −6.69722 −1.20286 −0.601429 0.798927i $$-0.705402\pi$$
−0.601429 + 0.798927i $$0.705402\pi$$
$$32$$ 0 0
$$33$$ 1.39445 0.242742
$$34$$ 0 0
$$35$$ 3.30278 0.558271
$$36$$ 0 0
$$37$$ −7.21110 −1.18550 −0.592749 0.805387i $$-0.701957\pi$$
−0.592749 + 0.805387i $$0.701957\pi$$
$$38$$ 0 0
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ 9.51388 1.48582 0.742909 0.669392i $$-0.233445\pi$$
0.742909 + 0.669392i $$0.233445\pi$$
$$42$$ 0 0
$$43$$ 4.30278 0.656167 0.328084 0.944649i $$-0.393597\pi$$
0.328084 + 0.944649i $$0.393597\pi$$
$$44$$ 0 0
$$45$$ −9.60555 −1.43191
$$46$$ 0 0
$$47$$ 10.0000 1.45865 0.729325 0.684167i $$-0.239834\pi$$
0.729325 + 0.684167i $$0.239834\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 0.302776 0.0423971
$$52$$ 0 0
$$53$$ −0.697224 −0.0957711 −0.0478856 0.998853i $$-0.515248\pi$$
−0.0478856 + 0.998853i $$0.515248\pi$$
$$54$$ 0 0
$$55$$ −15.2111 −2.05106
$$56$$ 0 0
$$57$$ −1.81665 −0.240622
$$58$$ 0 0
$$59$$ −5.21110 −0.678428 −0.339214 0.940709i $$-0.610161\pi$$
−0.339214 + 0.940709i $$0.610161\pi$$
$$60$$ 0 0
$$61$$ 4.30278 0.550914 0.275457 0.961313i $$-0.411171\pi$$
0.275457 + 0.961313i $$0.411171\pi$$
$$62$$ 0 0
$$63$$ −2.90833 −0.366415
$$64$$ 0 0
$$65$$ 21.8167 2.70602
$$66$$ 0 0
$$67$$ −2.69722 −0.329518 −0.164759 0.986334i $$-0.552685\pi$$
−0.164759 + 0.986334i $$0.552685\pi$$
$$68$$ 0 0
$$69$$ 0.788897 0.0949721
$$70$$ 0 0
$$71$$ −2.00000 −0.237356 −0.118678 0.992933i $$-0.537866\pi$$
−0.118678 + 0.992933i $$0.537866\pi$$
$$72$$ 0 0
$$73$$ −13.5139 −1.58168 −0.790840 0.612023i $$-0.790356\pi$$
−0.790840 + 0.612023i $$0.790356\pi$$
$$74$$ 0 0
$$75$$ −1.78890 −0.206564
$$76$$ 0 0
$$77$$ −4.60555 −0.524851
$$78$$ 0 0
$$79$$ −6.00000 −0.675053 −0.337526 0.941316i $$-0.609590\pi$$
−0.337526 + 0.941316i $$0.609590\pi$$
$$80$$ 0 0
$$81$$ 8.18335 0.909261
$$82$$ 0 0
$$83$$ −9.21110 −1.01105 −0.505525 0.862812i $$-0.668701\pi$$
−0.505525 + 0.862812i $$0.668701\pi$$
$$84$$ 0 0
$$85$$ −3.30278 −0.358236
$$86$$ 0 0
$$87$$ −2.60555 −0.279344
$$88$$ 0 0
$$89$$ −2.00000 −0.212000 −0.106000 0.994366i $$-0.533804\pi$$
−0.106000 + 0.994366i $$0.533804\pi$$
$$90$$ 0 0
$$91$$ 6.60555 0.692450
$$92$$ 0 0
$$93$$ 2.02776 0.210269
$$94$$ 0 0
$$95$$ 19.8167 2.03315
$$96$$ 0 0
$$97$$ −2.09167 −0.212377 −0.106189 0.994346i $$-0.533865\pi$$
−0.106189 + 0.994346i $$0.533865\pi$$
$$98$$ 0 0
$$99$$ 13.3944 1.34619
$$100$$ 0 0
$$101$$ 12.4222 1.23606 0.618028 0.786156i $$-0.287932\pi$$
0.618028 + 0.786156i $$0.287932\pi$$
$$102$$ 0 0
$$103$$ 13.2111 1.30173 0.650864 0.759194i $$-0.274407\pi$$
0.650864 + 0.759194i $$0.274407\pi$$
$$104$$ 0 0
$$105$$ −1.00000 −0.0975900
$$106$$ 0 0
$$107$$ 9.39445 0.908196 0.454098 0.890952i $$-0.349962\pi$$
0.454098 + 0.890952i $$0.349962\pi$$
$$108$$ 0 0
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ 2.18335 0.207234
$$112$$ 0 0
$$113$$ 19.2111 1.80723 0.903614 0.428347i $$-0.140904\pi$$
0.903614 + 0.428347i $$0.140904\pi$$
$$114$$ 0 0
$$115$$ −8.60555 −0.802472
$$116$$ 0 0
$$117$$ −19.2111 −1.77607
$$118$$ 0 0
$$119$$ −1.00000 −0.0916698
$$120$$ 0 0
$$121$$ 10.2111 0.928282
$$122$$ 0 0
$$123$$ −2.88057 −0.259732
$$124$$ 0 0
$$125$$ 3.00000 0.268328
$$126$$ 0 0
$$127$$ −5.11943 −0.454276 −0.227138 0.973863i $$-0.572937\pi$$
−0.227138 + 0.973863i $$0.572937\pi$$
$$128$$ 0 0
$$129$$ −1.30278 −0.114703
$$130$$ 0 0
$$131$$ 10.4222 0.910592 0.455296 0.890340i $$-0.349533\pi$$
0.455296 + 0.890340i $$0.349533\pi$$
$$132$$ 0 0
$$133$$ 6.00000 0.520266
$$134$$ 0 0
$$135$$ 5.90833 0.508508
$$136$$ 0 0
$$137$$ −3.09167 −0.264139 −0.132070 0.991240i $$-0.542162\pi$$
−0.132070 + 0.991240i $$0.542162\pi$$
$$138$$ 0 0
$$139$$ 1.48612 0.126051 0.0630256 0.998012i $$-0.479925\pi$$
0.0630256 + 0.998012i $$0.479925\pi$$
$$140$$ 0 0
$$141$$ −3.02776 −0.254983
$$142$$ 0 0
$$143$$ −30.4222 −2.54403
$$144$$ 0 0
$$145$$ 28.4222 2.36034
$$146$$ 0 0
$$147$$ −0.302776 −0.0249725
$$148$$ 0 0
$$149$$ −4.90833 −0.402106 −0.201053 0.979580i $$-0.564436\pi$$
−0.201053 + 0.979580i $$0.564436\pi$$
$$150$$ 0 0
$$151$$ 14.1194 1.14902 0.574511 0.818497i $$-0.305192\pi$$
0.574511 + 0.818497i $$0.305192\pi$$
$$152$$ 0 0
$$153$$ 2.90833 0.235124
$$154$$ 0 0
$$155$$ −22.1194 −1.77668
$$156$$ 0 0
$$157$$ 0.183346 0.0146326 0.00731631 0.999973i $$-0.497671\pi$$
0.00731631 + 0.999973i $$0.497671\pi$$
$$158$$ 0 0
$$159$$ 0.211103 0.0167415
$$160$$ 0 0
$$161$$ −2.60555 −0.205346
$$162$$ 0 0
$$163$$ 21.8167 1.70881 0.854406 0.519606i $$-0.173921\pi$$
0.854406 + 0.519606i $$0.173921\pi$$
$$164$$ 0 0
$$165$$ 4.60555 0.358542
$$166$$ 0 0
$$167$$ −2.11943 −0.164006 −0.0820032 0.996632i $$-0.526132\pi$$
−0.0820032 + 0.996632i $$0.526132\pi$$
$$168$$ 0 0
$$169$$ 30.6333 2.35641
$$170$$ 0 0
$$171$$ −17.4500 −1.33443
$$172$$ 0 0
$$173$$ −4.90833 −0.373173 −0.186587 0.982439i $$-0.559742\pi$$
−0.186587 + 0.982439i $$0.559742\pi$$
$$174$$ 0 0
$$175$$ 5.90833 0.446628
$$176$$ 0 0
$$177$$ 1.57779 0.118594
$$178$$ 0 0
$$179$$ −11.0917 −0.829031 −0.414515 0.910042i $$-0.636049\pi$$
−0.414515 + 0.910042i $$0.636049\pi$$
$$180$$ 0 0
$$181$$ 21.6333 1.60799 0.803996 0.594635i $$-0.202704\pi$$
0.803996 + 0.594635i $$0.202704\pi$$
$$182$$ 0 0
$$183$$ −1.30278 −0.0963039
$$184$$ 0 0
$$185$$ −23.8167 −1.75104
$$186$$ 0 0
$$187$$ 4.60555 0.336791
$$188$$ 0 0
$$189$$ 1.78890 0.130123
$$190$$ 0 0
$$191$$ −23.7250 −1.71668 −0.858340 0.513082i $$-0.828504\pi$$
−0.858340 + 0.513082i $$0.828504\pi$$
$$192$$ 0 0
$$193$$ −5.39445 −0.388301 −0.194150 0.980972i $$-0.562195\pi$$
−0.194150 + 0.980972i $$0.562195\pi$$
$$194$$ 0 0
$$195$$ −6.60555 −0.473033
$$196$$ 0 0
$$197$$ 23.0278 1.64066 0.820330 0.571891i $$-0.193790\pi$$
0.820330 + 0.571891i $$0.193790\pi$$
$$198$$ 0 0
$$199$$ 13.5139 0.957973 0.478987 0.877822i $$-0.341004\pi$$
0.478987 + 0.877822i $$0.341004\pi$$
$$200$$ 0 0
$$201$$ 0.816654 0.0576023
$$202$$ 0 0
$$203$$ 8.60555 0.603991
$$204$$ 0 0
$$205$$ 31.4222 2.19462
$$206$$ 0 0
$$207$$ 7.57779 0.526693
$$208$$ 0 0
$$209$$ −27.6333 −1.91144
$$210$$ 0 0
$$211$$ 27.0278 1.86067 0.930334 0.366714i $$-0.119517\pi$$
0.930334 + 0.366714i $$0.119517\pi$$
$$212$$ 0 0
$$213$$ 0.605551 0.0414917
$$214$$ 0 0
$$215$$ 14.2111 0.969189
$$216$$ 0 0
$$217$$ −6.69722 −0.454637
$$218$$ 0 0
$$219$$ 4.09167 0.276490
$$220$$ 0 0
$$221$$ −6.60555 −0.444337
$$222$$ 0 0
$$223$$ −21.6333 −1.44867 −0.724337 0.689446i $$-0.757854\pi$$
−0.724337 + 0.689446i $$0.757854\pi$$
$$224$$ 0 0
$$225$$ −17.1833 −1.14556
$$226$$ 0 0
$$227$$ −0.0916731 −0.00608456 −0.00304228 0.999995i $$-0.500968\pi$$
−0.00304228 + 0.999995i $$0.500968\pi$$
$$228$$ 0 0
$$229$$ 5.81665 0.384375 0.192188 0.981358i $$-0.438442\pi$$
0.192188 + 0.981358i $$0.438442\pi$$
$$230$$ 0 0
$$231$$ 1.39445 0.0917480
$$232$$ 0 0
$$233$$ 13.2111 0.865488 0.432744 0.901517i $$-0.357545\pi$$
0.432744 + 0.901517i $$0.357545\pi$$
$$234$$ 0 0
$$235$$ 33.0278 2.15449
$$236$$ 0 0
$$237$$ 1.81665 0.118004
$$238$$ 0 0
$$239$$ −12.3028 −0.795800 −0.397900 0.917429i $$-0.630261\pi$$
−0.397900 + 0.917429i $$0.630261\pi$$
$$240$$ 0 0
$$241$$ 13.9083 0.895914 0.447957 0.894055i $$-0.352152\pi$$
0.447957 + 0.894055i $$0.352152\pi$$
$$242$$ 0 0
$$243$$ −7.84441 −0.503219
$$244$$ 0 0
$$245$$ 3.30278 0.211007
$$246$$ 0 0
$$247$$ 39.6333 2.52181
$$248$$ 0 0
$$249$$ 2.78890 0.176739
$$250$$ 0 0
$$251$$ −8.78890 −0.554750 −0.277375 0.960762i $$-0.589464\pi$$
−0.277375 + 0.960762i $$0.589464\pi$$
$$252$$ 0 0
$$253$$ 12.0000 0.754434
$$254$$ 0 0
$$255$$ 1.00000 0.0626224
$$256$$ 0 0
$$257$$ −13.8167 −0.861859 −0.430930 0.902386i $$-0.641814\pi$$
−0.430930 + 0.902386i $$0.641814\pi$$
$$258$$ 0 0
$$259$$ −7.21110 −0.448076
$$260$$ 0 0
$$261$$ −25.0278 −1.54918
$$262$$ 0 0
$$263$$ 22.4222 1.38261 0.691306 0.722562i $$-0.257036\pi$$
0.691306 + 0.722562i $$0.257036\pi$$
$$264$$ 0 0
$$265$$ −2.30278 −0.141458
$$266$$ 0 0
$$267$$ 0.605551 0.0370591
$$268$$ 0 0
$$269$$ 6.00000 0.365826 0.182913 0.983129i $$-0.441447\pi$$
0.182913 + 0.983129i $$0.441447\pi$$
$$270$$ 0 0
$$271$$ −31.8167 −1.93272 −0.966362 0.257186i $$-0.917205\pi$$
−0.966362 + 0.257186i $$0.917205\pi$$
$$272$$ 0 0
$$273$$ −2.00000 −0.121046
$$274$$ 0 0
$$275$$ −27.2111 −1.64089
$$276$$ 0 0
$$277$$ −25.0278 −1.50377 −0.751886 0.659293i $$-0.770856\pi$$
−0.751886 + 0.659293i $$0.770856\pi$$
$$278$$ 0 0
$$279$$ 19.4777 1.16610
$$280$$ 0 0
$$281$$ −16.6972 −0.996073 −0.498036 0.867156i $$-0.665945\pi$$
−0.498036 + 0.867156i $$0.665945\pi$$
$$282$$ 0 0
$$283$$ −6.69722 −0.398109 −0.199054 0.979988i $$-0.563787\pi$$
−0.199054 + 0.979988i $$0.563787\pi$$
$$284$$ 0 0
$$285$$ −6.00000 −0.355409
$$286$$ 0 0
$$287$$ 9.51388 0.561586
$$288$$ 0 0
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ 0.633308 0.0371252
$$292$$ 0 0
$$293$$ −1.81665 −0.106130 −0.0530650 0.998591i $$-0.516899\pi$$
−0.0530650 + 0.998591i $$0.516899\pi$$
$$294$$ 0 0
$$295$$ −17.2111 −1.00207
$$296$$ 0 0
$$297$$ −8.23886 −0.478067
$$298$$ 0 0
$$299$$ −17.2111 −0.995344
$$300$$ 0 0
$$301$$ 4.30278 0.248008
$$302$$ 0 0
$$303$$ −3.76114 −0.216072
$$304$$ 0 0
$$305$$ 14.2111 0.813725
$$306$$ 0 0
$$307$$ 26.0000 1.48390 0.741949 0.670456i $$-0.233902\pi$$
0.741949 + 0.670456i $$0.233902\pi$$
$$308$$ 0 0
$$309$$ −4.00000 −0.227552
$$310$$ 0 0
$$311$$ −15.7250 −0.891682 −0.445841 0.895112i $$-0.647095\pi$$
−0.445841 + 0.895112i $$0.647095\pi$$
$$312$$ 0 0
$$313$$ −5.72498 −0.323595 −0.161798 0.986824i $$-0.551729\pi$$
−0.161798 + 0.986824i $$0.551729\pi$$
$$314$$ 0 0
$$315$$ −9.60555 −0.541212
$$316$$ 0 0
$$317$$ 12.6056 0.707998 0.353999 0.935246i $$-0.384822\pi$$
0.353999 + 0.935246i $$0.384822\pi$$
$$318$$ 0 0
$$319$$ −39.6333 −2.21904
$$320$$ 0 0
$$321$$ −2.84441 −0.158759
$$322$$ 0 0
$$323$$ −6.00000 −0.333849
$$324$$ 0 0
$$325$$ 39.0278 2.16487
$$326$$ 0 0
$$327$$ −0.605551 −0.0334871
$$328$$ 0 0
$$329$$ 10.0000 0.551318
$$330$$ 0 0
$$331$$ 0.275019 0.0151164 0.00755821 0.999971i $$-0.497594\pi$$
0.00755821 + 0.999971i $$0.497594\pi$$
$$332$$ 0 0
$$333$$ 20.9722 1.14927
$$334$$ 0 0
$$335$$ −8.90833 −0.486714
$$336$$ 0 0
$$337$$ 10.7889 0.587709 0.293854 0.955850i $$-0.405062\pi$$
0.293854 + 0.955850i $$0.405062\pi$$
$$338$$ 0 0
$$339$$ −5.81665 −0.315917
$$340$$ 0 0
$$341$$ 30.8444 1.67032
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ 2.60555 0.140278
$$346$$ 0 0
$$347$$ −6.00000 −0.322097 −0.161048 0.986947i $$-0.551488\pi$$
−0.161048 + 0.986947i $$0.551488\pi$$
$$348$$ 0 0
$$349$$ −19.2111 −1.02835 −0.514173 0.857686i $$-0.671901\pi$$
−0.514173 + 0.857686i $$0.671901\pi$$
$$350$$ 0 0
$$351$$ 11.8167 0.630726
$$352$$ 0 0
$$353$$ 4.60555 0.245129 0.122564 0.992461i $$-0.460888\pi$$
0.122564 + 0.992461i $$0.460888\pi$$
$$354$$ 0 0
$$355$$ −6.60555 −0.350586
$$356$$ 0 0
$$357$$ 0.302776 0.0160246
$$358$$ 0 0
$$359$$ −17.4861 −0.922882 −0.461441 0.887171i $$-0.652667\pi$$
−0.461441 + 0.887171i $$0.652667\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ 0 0
$$363$$ −3.09167 −0.162271
$$364$$ 0 0
$$365$$ −44.6333 −2.33621
$$366$$ 0 0
$$367$$ 26.7250 1.39503 0.697516 0.716569i $$-0.254288\pi$$
0.697516 + 0.716569i $$0.254288\pi$$
$$368$$ 0 0
$$369$$ −27.6695 −1.44041
$$370$$ 0 0
$$371$$ −0.697224 −0.0361981
$$372$$ 0 0
$$373$$ −12.3028 −0.637014 −0.318507 0.947921i $$-0.603181\pi$$
−0.318507 + 0.947921i $$0.603181\pi$$
$$374$$ 0 0
$$375$$ −0.908327 −0.0469058
$$376$$ 0 0
$$377$$ 56.8444 2.92764
$$378$$ 0 0
$$379$$ 28.2389 1.45053 0.725266 0.688468i $$-0.241717\pi$$
0.725266 + 0.688468i $$0.241717\pi$$
$$380$$ 0 0
$$381$$ 1.55004 0.0794109
$$382$$ 0 0
$$383$$ −3.02776 −0.154711 −0.0773556 0.997004i $$-0.524648\pi$$
−0.0773556 + 0.997004i $$0.524648\pi$$
$$384$$ 0 0
$$385$$ −15.2111 −0.775230
$$386$$ 0 0
$$387$$ −12.5139 −0.636116
$$388$$ 0 0
$$389$$ 14.0917 0.714476 0.357238 0.934013i $$-0.383718\pi$$
0.357238 + 0.934013i $$0.383718\pi$$
$$390$$ 0 0
$$391$$ 2.60555 0.131768
$$392$$ 0 0
$$393$$ −3.15559 −0.159178
$$394$$ 0 0
$$395$$ −19.8167 −0.997084
$$396$$ 0 0
$$397$$ 6.09167 0.305732 0.152866 0.988247i $$-0.451150\pi$$
0.152866 + 0.988247i $$0.451150\pi$$
$$398$$ 0 0
$$399$$ −1.81665 −0.0909464
$$400$$ 0 0
$$401$$ 24.0000 1.19850 0.599251 0.800561i $$-0.295465\pi$$
0.599251 + 0.800561i $$0.295465\pi$$
$$402$$ 0 0
$$403$$ −44.2389 −2.20369
$$404$$ 0 0
$$405$$ 27.0278 1.34302
$$406$$ 0 0
$$407$$ 33.2111 1.64621
$$408$$ 0 0
$$409$$ −3.02776 −0.149713 −0.0748565 0.997194i $$-0.523850\pi$$
−0.0748565 + 0.997194i $$0.523850\pi$$
$$410$$ 0 0
$$411$$ 0.936083 0.0461736
$$412$$ 0 0
$$413$$ −5.21110 −0.256422
$$414$$ 0 0
$$415$$ −30.4222 −1.49337
$$416$$ 0 0
$$417$$ −0.449961 −0.0220347
$$418$$ 0 0
$$419$$ 1.30278 0.0636448 0.0318224 0.999494i $$-0.489869\pi$$
0.0318224 + 0.999494i $$0.489869\pi$$
$$420$$ 0 0
$$421$$ 0.908327 0.0442691 0.0221346 0.999755i $$-0.492954\pi$$
0.0221346 + 0.999755i $$0.492954\pi$$
$$422$$ 0 0
$$423$$ −29.0833 −1.41408
$$424$$ 0 0
$$425$$ −5.90833 −0.286596
$$426$$ 0 0
$$427$$ 4.30278 0.208226
$$428$$ 0 0
$$429$$ 9.21110 0.444716
$$430$$ 0 0
$$431$$ 18.8444 0.907703 0.453852 0.891077i $$-0.350050\pi$$
0.453852 + 0.891077i $$0.350050\pi$$
$$432$$ 0 0
$$433$$ 11.6333 0.559061 0.279531 0.960137i $$-0.409821\pi$$
0.279531 + 0.960137i $$0.409821\pi$$
$$434$$ 0 0
$$435$$ −8.60555 −0.412605
$$436$$ 0 0
$$437$$ −15.6333 −0.747843
$$438$$ 0 0
$$439$$ −0.302776 −0.0144507 −0.00722535 0.999974i $$-0.502300\pi$$
−0.00722535 + 0.999974i $$0.502300\pi$$
$$440$$ 0 0
$$441$$ −2.90833 −0.138492
$$442$$ 0 0
$$443$$ 10.4222 0.495174 0.247587 0.968866i $$-0.420362\pi$$
0.247587 + 0.968866i $$0.420362\pi$$
$$444$$ 0 0
$$445$$ −6.60555 −0.313133
$$446$$ 0 0
$$447$$ 1.48612 0.0702911
$$448$$ 0 0
$$449$$ −32.2389 −1.52145 −0.760723 0.649077i $$-0.775155\pi$$
−0.760723 + 0.649077i $$0.775155\pi$$
$$450$$ 0 0
$$451$$ −43.8167 −2.06325
$$452$$ 0 0
$$453$$ −4.27502 −0.200858
$$454$$ 0 0
$$455$$ 21.8167 1.02278
$$456$$ 0 0
$$457$$ 6.09167 0.284956 0.142478 0.989798i $$-0.454493\pi$$
0.142478 + 0.989798i $$0.454493\pi$$
$$458$$ 0 0
$$459$$ −1.78890 −0.0834986
$$460$$ 0 0
$$461$$ −9.21110 −0.429004 −0.214502 0.976724i $$-0.568813\pi$$
−0.214502 + 0.976724i $$0.568813\pi$$
$$462$$ 0 0
$$463$$ 36.5416 1.69823 0.849117 0.528205i $$-0.177135\pi$$
0.849117 + 0.528205i $$0.177135\pi$$
$$464$$ 0 0
$$465$$ 6.69722 0.310576
$$466$$ 0 0
$$467$$ −17.2111 −0.796435 −0.398217 0.917291i $$-0.630371\pi$$
−0.398217 + 0.917291i $$0.630371\pi$$
$$468$$ 0 0
$$469$$ −2.69722 −0.124546
$$470$$ 0 0
$$471$$ −0.0555128 −0.00255789
$$472$$ 0 0
$$473$$ −19.8167 −0.911171
$$474$$ 0 0
$$475$$ 35.4500 1.62656
$$476$$ 0 0
$$477$$ 2.02776 0.0928446
$$478$$ 0 0
$$479$$ −24.3028 −1.11042 −0.555211 0.831709i $$-0.687363\pi$$
−0.555211 + 0.831709i $$0.687363\pi$$
$$480$$ 0 0
$$481$$ −47.6333 −2.17189
$$482$$ 0 0
$$483$$ 0.788897 0.0358961
$$484$$ 0 0
$$485$$ −6.90833 −0.313691
$$486$$ 0 0
$$487$$ 13.6333 0.617784 0.308892 0.951097i $$-0.400042\pi$$
0.308892 + 0.951097i $$0.400042\pi$$
$$488$$ 0 0
$$489$$ −6.60555 −0.298713
$$490$$ 0 0
$$491$$ −30.6972 −1.38535 −0.692673 0.721252i $$-0.743567\pi$$
−0.692673 + 0.721252i $$0.743567\pi$$
$$492$$ 0 0
$$493$$ −8.60555 −0.387575
$$494$$ 0 0
$$495$$ 44.2389 1.98839
$$496$$ 0 0
$$497$$ −2.00000 −0.0897123
$$498$$ 0 0
$$499$$ 6.00000 0.268597 0.134298 0.990941i $$-0.457122\pi$$
0.134298 + 0.990941i $$0.457122\pi$$
$$500$$ 0 0
$$501$$ 0.641712 0.0286696
$$502$$ 0 0
$$503$$ 10.8806 0.485141 0.242570 0.970134i $$-0.422009\pi$$
0.242570 + 0.970134i $$0.422009\pi$$
$$504$$ 0 0
$$505$$ 41.0278 1.82571
$$506$$ 0 0
$$507$$ −9.27502 −0.411918
$$508$$ 0 0
$$509$$ −3.39445 −0.150456 −0.0752281 0.997166i $$-0.523969\pi$$
−0.0752281 + 0.997166i $$0.523969\pi$$
$$510$$ 0 0
$$511$$ −13.5139 −0.597819
$$512$$ 0 0
$$513$$ 10.7334 0.473891
$$514$$ 0 0
$$515$$ 43.6333 1.92271
$$516$$ 0 0
$$517$$ −46.0555 −2.02552
$$518$$ 0 0
$$519$$ 1.48612 0.0652335
$$520$$ 0 0
$$521$$ 7.33053 0.321156 0.160578 0.987023i $$-0.448664\pi$$
0.160578 + 0.987023i $$0.448664\pi$$
$$522$$ 0 0
$$523$$ −17.0278 −0.744572 −0.372286 0.928118i $$-0.621426\pi$$
−0.372286 + 0.928118i $$0.621426\pi$$
$$524$$ 0 0
$$525$$ −1.78890 −0.0780739
$$526$$ 0 0
$$527$$ 6.69722 0.291736
$$528$$ 0 0
$$529$$ −16.2111 −0.704831
$$530$$ 0 0
$$531$$ 15.1556 0.657697
$$532$$ 0 0
$$533$$ 62.8444 2.72209
$$534$$ 0 0
$$535$$ 31.0278 1.34145
$$536$$ 0 0
$$537$$ 3.35829 0.144921
$$538$$ 0 0
$$539$$ −4.60555 −0.198375
$$540$$ 0 0
$$541$$ −3.81665 −0.164091 −0.0820454 0.996629i $$-0.526145\pi$$
−0.0820454 + 0.996629i $$0.526145\pi$$
$$542$$ 0 0
$$543$$ −6.55004 −0.281089
$$544$$ 0 0
$$545$$ 6.60555 0.282951
$$546$$ 0 0
$$547$$ 39.4500 1.68676 0.843379 0.537319i $$-0.180563\pi$$
0.843379 + 0.537319i $$0.180563\pi$$
$$548$$ 0 0
$$549$$ −12.5139 −0.534079
$$550$$ 0 0
$$551$$ 51.6333 2.19965
$$552$$ 0 0
$$553$$ −6.00000 −0.255146
$$554$$ 0 0
$$555$$ 7.21110 0.306094
$$556$$ 0 0
$$557$$ −11.2111 −0.475030 −0.237515 0.971384i $$-0.576333\pi$$
−0.237515 + 0.971384i $$0.576333\pi$$
$$558$$ 0 0
$$559$$ 28.4222 1.20213
$$560$$ 0 0
$$561$$ −1.39445 −0.0588737
$$562$$ 0 0
$$563$$ −32.2389 −1.35871 −0.679353 0.733812i $$-0.737739\pi$$
−0.679353 + 0.733812i $$0.737739\pi$$
$$564$$ 0 0
$$565$$ 63.4500 2.66936
$$566$$ 0 0
$$567$$ 8.18335 0.343668
$$568$$ 0 0
$$569$$ −41.5139 −1.74035 −0.870176 0.492741i $$-0.835995\pi$$
−0.870176 + 0.492741i $$0.835995\pi$$
$$570$$ 0 0
$$571$$ 33.6333 1.40751 0.703755 0.710443i $$-0.251505\pi$$
0.703755 + 0.710443i $$0.251505\pi$$
$$572$$ 0 0
$$573$$ 7.18335 0.300089
$$574$$ 0 0
$$575$$ −15.3944 −0.641993
$$576$$ 0 0
$$577$$ 39.0278 1.62475 0.812373 0.583138i $$-0.198175\pi$$
0.812373 + 0.583138i $$0.198175\pi$$
$$578$$ 0 0
$$579$$ 1.63331 0.0678779
$$580$$ 0 0
$$581$$ −9.21110 −0.382141
$$582$$ 0 0
$$583$$ 3.21110 0.132990
$$584$$ 0 0
$$585$$ −63.4500 −2.62333
$$586$$ 0 0
$$587$$ 17.2111 0.710378 0.355189 0.934794i $$-0.384416\pi$$
0.355189 + 0.934794i $$0.384416\pi$$
$$588$$ 0 0
$$589$$ −40.1833 −1.65573
$$590$$ 0 0
$$591$$ −6.97224 −0.286800
$$592$$ 0 0
$$593$$ 9.63331 0.395593 0.197796 0.980243i $$-0.436622\pi$$
0.197796 + 0.980243i $$0.436622\pi$$
$$594$$ 0 0
$$595$$ −3.30278 −0.135401
$$596$$ 0 0
$$597$$ −4.09167 −0.167461
$$598$$ 0 0
$$599$$ −32.5416 −1.32962 −0.664808 0.747015i $$-0.731486\pi$$
−0.664808 + 0.747015i $$0.731486\pi$$
$$600$$ 0 0
$$601$$ 19.2111 0.783637 0.391819 0.920042i $$-0.371846\pi$$
0.391819 + 0.920042i $$0.371846\pi$$
$$602$$ 0 0
$$603$$ 7.84441 0.319449
$$604$$ 0 0
$$605$$ 33.7250 1.37112
$$606$$ 0 0
$$607$$ −16.0917 −0.653141 −0.326570 0.945173i $$-0.605893\pi$$
−0.326570 + 0.945173i $$0.605893\pi$$
$$608$$ 0 0
$$609$$ −2.60555 −0.105582
$$610$$ 0 0
$$611$$ 66.0555 2.67232
$$612$$ 0 0
$$613$$ 30.9361 1.24950 0.624748 0.780826i $$-0.285202\pi$$
0.624748 + 0.780826i $$0.285202\pi$$
$$614$$ 0 0
$$615$$ −9.51388 −0.383637
$$616$$ 0 0
$$617$$ 4.60555 0.185413 0.0927063 0.995694i $$-0.470448\pi$$
0.0927063 + 0.995694i $$0.470448\pi$$
$$618$$ 0 0
$$619$$ 8.00000 0.321547 0.160774 0.986991i $$-0.448601\pi$$
0.160774 + 0.986991i $$0.448601\pi$$
$$620$$ 0 0
$$621$$ −4.66106 −0.187042
$$622$$ 0 0
$$623$$ −2.00000 −0.0801283
$$624$$ 0 0
$$625$$ −19.6333 −0.785332
$$626$$ 0 0
$$627$$ 8.36669 0.334134
$$628$$ 0 0
$$629$$ 7.21110 0.287525
$$630$$ 0 0
$$631$$ 13.1194 0.522276 0.261138 0.965301i $$-0.415902\pi$$
0.261138 + 0.965301i $$0.415902\pi$$
$$632$$ 0 0
$$633$$ −8.18335 −0.325259
$$634$$ 0 0
$$635$$ −16.9083 −0.670986
$$636$$ 0 0
$$637$$ 6.60555 0.261721
$$638$$ 0 0
$$639$$ 5.81665 0.230103
$$640$$ 0 0
$$641$$ −7.57779 −0.299305 −0.149652 0.988739i $$-0.547815\pi$$
−0.149652 + 0.988739i $$0.547815\pi$$
$$642$$ 0 0
$$643$$ −27.3305 −1.07781 −0.538905 0.842366i $$-0.681162\pi$$
−0.538905 + 0.842366i $$0.681162\pi$$
$$644$$ 0 0
$$645$$ −4.30278 −0.169422
$$646$$ 0 0
$$647$$ −27.3944 −1.07699 −0.538493 0.842630i $$-0.681006\pi$$
−0.538493 + 0.842630i $$0.681006\pi$$
$$648$$ 0 0
$$649$$ 24.0000 0.942082
$$650$$ 0 0
$$651$$ 2.02776 0.0794740
$$652$$ 0 0
$$653$$ −2.00000 −0.0782660 −0.0391330 0.999234i $$-0.512460\pi$$
−0.0391330 + 0.999234i $$0.512460\pi$$
$$654$$ 0 0
$$655$$ 34.4222 1.34499
$$656$$ 0 0
$$657$$ 39.3028 1.53335
$$658$$ 0 0
$$659$$ −7.54163 −0.293780 −0.146890 0.989153i $$-0.546926\pi$$
−0.146890 + 0.989153i $$0.546926\pi$$
$$660$$ 0 0
$$661$$ −15.6333 −0.608065 −0.304033 0.952662i $$-0.598333\pi$$
−0.304033 + 0.952662i $$0.598333\pi$$
$$662$$ 0 0
$$663$$ 2.00000 0.0776736
$$664$$ 0 0
$$665$$ 19.8167 0.768457
$$666$$ 0 0
$$667$$ −22.4222 −0.868191
$$668$$ 0 0
$$669$$ 6.55004 0.253239
$$670$$ 0 0
$$671$$ −19.8167 −0.765013
$$672$$ 0 0
$$673$$ −20.8444 −0.803493 −0.401746 0.915751i $$-0.631597\pi$$
−0.401746 + 0.915751i $$0.631597\pi$$
$$674$$ 0 0
$$675$$ 10.5694 0.406816
$$676$$ 0 0
$$677$$ −22.0000 −0.845529 −0.422764 0.906240i $$-0.638940\pi$$
−0.422764 + 0.906240i $$0.638940\pi$$
$$678$$ 0 0
$$679$$ −2.09167 −0.0802710
$$680$$ 0 0
$$681$$ 0.0277564 0.00106363
$$682$$ 0 0
$$683$$ 11.0278 0.421965 0.210983 0.977490i $$-0.432334\pi$$
0.210983 + 0.977490i $$0.432334\pi$$
$$684$$ 0 0
$$685$$ −10.2111 −0.390146
$$686$$ 0 0
$$687$$ −1.76114 −0.0671917
$$688$$ 0 0
$$689$$ −4.60555 −0.175458
$$690$$ 0 0
$$691$$ −4.30278 −0.163685 −0.0818426 0.996645i $$-0.526080\pi$$
−0.0818426 + 0.996645i $$0.526080\pi$$
$$692$$ 0 0
$$693$$ 13.3944 0.508813
$$694$$ 0 0
$$695$$ 4.90833 0.186183
$$696$$ 0 0
$$697$$ −9.51388 −0.360364
$$698$$ 0 0
$$699$$ −4.00000 −0.151294
$$700$$ 0 0
$$701$$ 11.2111 0.423437 0.211719 0.977331i $$-0.432094\pi$$
0.211719 + 0.977331i $$0.432094\pi$$
$$702$$ 0 0
$$703$$ −43.2666 −1.63183
$$704$$ 0 0
$$705$$ −10.0000 −0.376622
$$706$$ 0 0
$$707$$ 12.4222 0.467185
$$708$$ 0 0
$$709$$ 24.1833 0.908225 0.454112 0.890944i $$-0.349956\pi$$
0.454112 + 0.890944i $$0.349956\pi$$
$$710$$ 0 0
$$711$$ 17.4500 0.654425
$$712$$ 0 0
$$713$$ 17.4500 0.653506
$$714$$ 0 0
$$715$$ −100.478 −3.75765
$$716$$ 0 0
$$717$$ 3.72498 0.139112
$$718$$ 0 0
$$719$$ −2.48612 −0.0927167 −0.0463583 0.998925i $$-0.514762\pi$$
−0.0463583 + 0.998925i $$0.514762\pi$$
$$720$$ 0 0
$$721$$ 13.2111 0.492007
$$722$$ 0 0
$$723$$ −4.21110 −0.156613
$$724$$ 0 0
$$725$$ 50.8444 1.88831
$$726$$ 0 0
$$727$$ 49.2666 1.82720 0.913599 0.406617i $$-0.133292\pi$$
0.913599 + 0.406617i $$0.133292\pi$$
$$728$$ 0 0
$$729$$ −22.1749 −0.821294
$$730$$ 0 0
$$731$$ −4.30278 −0.159144
$$732$$ 0 0
$$733$$ 15.6333 0.577429 0.288715 0.957415i $$-0.406772\pi$$
0.288715 + 0.957415i $$0.406772\pi$$
$$734$$ 0 0
$$735$$ −1.00000 −0.0368856
$$736$$ 0 0
$$737$$ 12.4222 0.457578
$$738$$ 0 0
$$739$$ 24.9083 0.916268 0.458134 0.888883i $$-0.348518\pi$$
0.458134 + 0.888883i $$0.348518\pi$$
$$740$$ 0 0
$$741$$ −12.0000 −0.440831
$$742$$ 0 0
$$743$$ −33.3944 −1.22512 −0.612562 0.790423i $$-0.709861\pi$$
−0.612562 + 0.790423i $$0.709861\pi$$
$$744$$ 0 0
$$745$$ −16.2111 −0.593929
$$746$$ 0 0
$$747$$ 26.7889 0.980155
$$748$$ 0 0
$$749$$ 9.39445 0.343266
$$750$$ 0 0
$$751$$ −51.2111 −1.86872 −0.934360 0.356331i $$-0.884028\pi$$
−0.934360 + 0.356331i $$0.884028\pi$$
$$752$$ 0 0
$$753$$ 2.66106 0.0969746
$$754$$ 0 0
$$755$$ 46.6333 1.69716
$$756$$ 0 0
$$757$$ −23.0917 −0.839281 −0.419641 0.907690i $$-0.637844\pi$$
−0.419641 + 0.907690i $$0.637844\pi$$
$$758$$ 0 0
$$759$$ −3.63331 −0.131881
$$760$$ 0 0
$$761$$ 46.6056 1.68945 0.844725 0.535201i $$-0.179764\pi$$
0.844725 + 0.535201i $$0.179764\pi$$
$$762$$ 0 0
$$763$$ 2.00000 0.0724049
$$764$$ 0 0
$$765$$ 9.60555 0.347289
$$766$$ 0 0
$$767$$ −34.4222 −1.24291
$$768$$ 0 0
$$769$$ 11.8167 0.426119 0.213060 0.977039i $$-0.431657\pi$$
0.213060 + 0.977039i $$0.431657\pi$$
$$770$$ 0 0
$$771$$ 4.18335 0.150660
$$772$$ 0 0
$$773$$ 6.84441 0.246176 0.123088 0.992396i $$-0.460720\pi$$
0.123088 + 0.992396i $$0.460720\pi$$
$$774$$ 0 0
$$775$$ −39.5694 −1.42137
$$776$$ 0 0
$$777$$ 2.18335 0.0783271
$$778$$ 0 0
$$779$$ 57.0833 2.04522
$$780$$ 0 0
$$781$$ 9.21110 0.329599
$$782$$ 0 0
$$783$$ 15.3944 0.550153
$$784$$ 0 0
$$785$$ 0.605551 0.0216131
$$786$$ 0 0
$$787$$ 14.4222 0.514096 0.257048 0.966399i $$-0.417250\pi$$
0.257048 + 0.966399i $$0.417250\pi$$
$$788$$ 0 0
$$789$$ −6.78890 −0.241691
$$790$$ 0 0
$$791$$ 19.2111 0.683068
$$792$$ 0 0
$$793$$ 28.4222 1.00930
$$794$$ 0 0
$$795$$ 0.697224 0.0247280
$$796$$ 0 0
$$797$$ −14.7889 −0.523850 −0.261925 0.965088i $$-0.584357\pi$$
−0.261925 + 0.965088i $$0.584357\pi$$
$$798$$ 0 0
$$799$$ −10.0000 −0.353775
$$800$$ 0 0
$$801$$ 5.81665 0.205521
$$802$$ 0 0
$$803$$ 62.2389 2.19636
$$804$$ 0 0
$$805$$ −8.60555 −0.303306
$$806$$ 0 0
$$807$$ −1.81665 −0.0639492
$$808$$ 0 0
$$809$$ 45.4500 1.59794 0.798968 0.601374i $$-0.205380\pi$$
0.798968 + 0.601374i $$0.205380\pi$$
$$810$$ 0 0
$$811$$ 38.3583 1.34694 0.673471 0.739214i $$-0.264803\pi$$
0.673471 + 0.739214i $$0.264803\pi$$
$$812$$ 0 0
$$813$$ 9.63331 0.337855
$$814$$ 0 0
$$815$$ 72.0555 2.52399
$$816$$ 0 0
$$817$$ 25.8167 0.903210
$$818$$ 0 0
$$819$$ −19.2111 −0.671290
$$820$$ 0 0
$$821$$ −40.4222 −1.41074 −0.705372 0.708837i $$-0.749220\pi$$
−0.705372 + 0.708837i $$0.749220\pi$$
$$822$$ 0 0
$$823$$ −5.02776 −0.175257 −0.0876283 0.996153i $$-0.527929\pi$$
−0.0876283 + 0.996153i $$0.527929\pi$$
$$824$$ 0 0
$$825$$ 8.23886 0.286840
$$826$$ 0 0
$$827$$ −18.8444 −0.655284 −0.327642 0.944802i $$-0.606254\pi$$
−0.327642 + 0.944802i $$0.606254\pi$$
$$828$$ 0 0
$$829$$ 29.3944 1.02091 0.510456 0.859904i $$-0.329477\pi$$
0.510456 + 0.859904i $$0.329477\pi$$
$$830$$ 0 0
$$831$$ 7.57779 0.262871
$$832$$ 0 0
$$833$$ −1.00000 −0.0346479
$$834$$ 0 0
$$835$$ −7.00000 −0.242245
$$836$$ 0 0
$$837$$ −11.9806 −0.414112
$$838$$ 0 0
$$839$$ 21.2111 0.732289 0.366144 0.930558i $$-0.380678\pi$$
0.366144 + 0.930558i $$0.380678\pi$$
$$840$$ 0 0
$$841$$ 45.0555 1.55364
$$842$$ 0 0
$$843$$ 5.05551 0.174121
$$844$$ 0 0
$$845$$ 101.175 3.48052
$$846$$ 0 0
$$847$$ 10.2111 0.350858
$$848$$ 0 0
$$849$$ 2.02776 0.0695924
$$850$$ 0 0
$$851$$ 18.7889 0.644075
$$852$$ 0 0
$$853$$ 19.2111 0.657776 0.328888 0.944369i $$-0.393326\pi$$
0.328888 + 0.944369i $$0.393326\pi$$
$$854$$ 0 0
$$855$$ −57.6333 −1.97102
$$856$$ 0 0
$$857$$ 12.5139 0.427466 0.213733 0.976892i $$-0.431438\pi$$
0.213733 + 0.976892i $$0.431438\pi$$
$$858$$ 0 0
$$859$$ 45.6333 1.55699 0.778494 0.627652i $$-0.215984\pi$$
0.778494 + 0.627652i $$0.215984\pi$$
$$860$$ 0 0
$$861$$ −2.88057 −0.0981696
$$862$$ 0 0
$$863$$ 8.93608 0.304188 0.152094 0.988366i $$-0.451398\pi$$
0.152094 + 0.988366i $$0.451398\pi$$
$$864$$ 0 0
$$865$$ −16.2111 −0.551194
$$866$$ 0 0
$$867$$ −0.302776 −0.0102828
$$868$$ 0 0
$$869$$ 27.6333 0.937396
$$870$$ 0 0
$$871$$ −17.8167 −0.603694
$$872$$ 0 0
$$873$$ 6.08327 0.205887
$$874$$ 0 0
$$875$$ 3.00000 0.101419
$$876$$ 0 0
$$877$$ 6.18335 0.208797 0.104398 0.994536i $$-0.466708\pi$$
0.104398 + 0.994536i $$0.466708\pi$$
$$878$$ 0 0
$$879$$ 0.550039 0.0185523
$$880$$ 0 0
$$881$$ −7.11943 −0.239860 −0.119930 0.992782i $$-0.538267\pi$$
−0.119930 + 0.992782i $$0.538267\pi$$
$$882$$ 0 0
$$883$$ −55.7250 −1.87529 −0.937647 0.347588i $$-0.887001\pi$$
−0.937647 + 0.347588i $$0.887001\pi$$
$$884$$ 0 0
$$885$$ 5.21110 0.175169
$$886$$ 0 0
$$887$$ −12.2750 −0.412155 −0.206077 0.978536i $$-0.566070\pi$$
−0.206077 + 0.978536i $$0.566070\pi$$
$$888$$ 0 0
$$889$$ −5.11943 −0.171700
$$890$$ 0 0
$$891$$ −37.6888 −1.26262
$$892$$ 0 0
$$893$$ 60.0000 2.00782
$$894$$ 0 0
$$895$$ −36.6333 −1.22452
$$896$$ 0 0
$$897$$ 5.21110 0.173994
$$898$$ 0 0
$$899$$ −57.6333 −1.92218
$$900$$ 0 0
$$901$$ 0.697224 0.0232279
$$902$$ 0 0
$$903$$ −1.30278 −0.0433537
$$904$$ 0 0
$$905$$ 71.4500 2.37508
$$906$$ 0 0
$$907$$ 3.57779 0.118799 0.0593994 0.998234i $$-0.481081\pi$$
0.0593994 + 0.998234i $$0.481081\pi$$
$$908$$ 0 0
$$909$$ −36.1278 −1.19828
$$910$$ 0 0
$$911$$ −10.4222 −0.345303 −0.172652 0.984983i $$-0.555233\pi$$
−0.172652 + 0.984983i $$0.555233\pi$$
$$912$$ 0 0
$$913$$ 42.4222 1.40397
$$914$$ 0 0
$$915$$ −4.30278 −0.142245
$$916$$ 0 0
$$917$$ 10.4222 0.344172
$$918$$ 0 0
$$919$$ −10.4861 −0.345905 −0.172953 0.984930i $$-0.555331\pi$$
−0.172953 + 0.984930i $$0.555331\pi$$
$$920$$ 0 0
$$921$$ −7.87217 −0.259397
$$922$$ 0 0
$$923$$ −13.2111 −0.434849
$$924$$ 0 0
$$925$$ −42.6056 −1.40086
$$926$$ 0 0
$$927$$ −38.4222 −1.26195
$$928$$ 0 0
$$929$$ −24.6972 −0.810290 −0.405145 0.914253i $$-0.632779\pi$$
−0.405145 + 0.914253i $$0.632779\pi$$
$$930$$ 0 0
$$931$$ 6.00000 0.196642
$$932$$ 0 0
$$933$$ 4.76114 0.155873
$$934$$ 0 0
$$935$$ 15.2111 0.497456
$$936$$ 0 0
$$937$$ −25.2111 −0.823611 −0.411805 0.911272i $$-0.635102\pi$$
−0.411805 + 0.911272i $$0.635102\pi$$
$$938$$ 0 0
$$939$$ 1.73338 0.0565669
$$940$$ 0 0
$$941$$ 0.669468 0.0218240 0.0109120 0.999940i $$-0.496527\pi$$
0.0109120 + 0.999940i $$0.496527\pi$$
$$942$$ 0 0
$$943$$ −24.7889 −0.807238
$$944$$ 0 0
$$945$$ 5.90833 0.192198
$$946$$ 0 0
$$947$$ −20.2389 −0.657675 −0.328837 0.944387i $$-0.606657\pi$$
−0.328837 + 0.944387i $$0.606657\pi$$
$$948$$ 0 0
$$949$$ −89.2666 −2.89772
$$950$$ 0 0
$$951$$ −3.81665 −0.123763
$$952$$ 0 0
$$953$$ 26.7527 0.866606 0.433303 0.901248i $$-0.357348\pi$$
0.433303 + 0.901248i $$0.357348\pi$$
$$954$$ 0 0
$$955$$ −78.3583 −2.53561
$$956$$ 0 0
$$957$$ 12.0000 0.387905
$$958$$ 0 0
$$959$$ −3.09167 −0.0998353
$$960$$ 0 0
$$961$$ 13.8528 0.446865
$$962$$ 0 0
$$963$$ −27.3221 −0.880443
$$964$$ 0 0
$$965$$ −17.8167 −0.573538
$$966$$ 0 0
$$967$$ −20.3028 −0.652893 −0.326447 0.945216i $$-0.605851\pi$$
−0.326447 + 0.945216i $$0.605851\pi$$
$$968$$ 0 0
$$969$$ 1.81665 0.0583593
$$970$$ 0 0
$$971$$ 20.2389 0.649496 0.324748 0.945801i $$-0.394721\pi$$
0.324748 + 0.945801i $$0.394721\pi$$
$$972$$ 0 0
$$973$$ 1.48612 0.0476429
$$974$$ 0 0
$$975$$ −11.8167 −0.378436
$$976$$ 0 0
$$977$$ 12.5139 0.400354 0.200177 0.979760i $$-0.435848\pi$$
0.200177 + 0.979760i $$0.435848\pi$$
$$978$$ 0 0
$$979$$ 9.21110 0.294388
$$980$$ 0 0
$$981$$ −5.81665 −0.185711
$$982$$ 0 0
$$983$$ 14.5139 0.462921 0.231460 0.972844i $$-0.425650\pi$$
0.231460 + 0.972844i $$0.425650\pi$$
$$984$$ 0 0
$$985$$ 76.0555 2.42333
$$986$$ 0 0
$$987$$ −3.02776 −0.0963745
$$988$$ 0 0
$$989$$ −11.2111 −0.356492
$$990$$ 0 0
$$991$$ 6.00000 0.190596 0.0952981 0.995449i $$-0.469620\pi$$
0.0952981 + 0.995449i $$0.469620\pi$$
$$992$$ 0 0
$$993$$ −0.0832691 −0.00264247
$$994$$ 0 0
$$995$$ 44.6333 1.41497
$$996$$ 0 0
$$997$$ −32.9083 −1.04222 −0.521109 0.853490i $$-0.674481\pi$$
−0.521109 + 0.853490i $$0.674481\pi$$
$$998$$ 0 0
$$999$$ −12.8999 −0.408136
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 7616.2.a.z.1.1 2
4.3 odd 2 7616.2.a.m.1.2 2
8.3 odd 2 1904.2.a.l.1.1 2
8.5 even 2 476.2.a.a.1.2 2
24.5 odd 2 4284.2.a.p.1.2 2
56.13 odd 2 3332.2.a.n.1.1 2
136.101 even 2 8092.2.a.n.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
476.2.a.a.1.2 2 8.5 even 2
1904.2.a.l.1.1 2 8.3 odd 2
3332.2.a.n.1.1 2 56.13 odd 2
4284.2.a.p.1.2 2 24.5 odd 2
7616.2.a.m.1.2 2 4.3 odd 2
7616.2.a.z.1.1 2 1.1 even 1 trivial
8092.2.a.n.1.1 2 136.101 even 2