# Properties

 Label 6552.2.a.bb.1.1 Level $6552$ Weight $2$ Character 6552.1 Self dual yes Analytic conductor $52.318$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6552,2,Mod(1,6552)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6552, 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("6552.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6552 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6552.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: $$52.3179834043$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$3.37228$$ of defining polynomial Character $$\chi$$ $$=$$ 6552.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-4.37228 q^{5} -1.00000 q^{7} +O(q^{10})$$ $$q-4.37228 q^{5} -1.00000 q^{7} +0.372281 q^{11} -1.00000 q^{13} -0.372281 q^{17} +1.62772 q^{19} +4.37228 q^{23} +14.1168 q^{25} -10.3723 q^{29} +0.744563 q^{31} +4.37228 q^{35} -0.372281 q^{37} +6.00000 q^{41} +1.62772 q^{43} +13.4891 q^{47} +1.00000 q^{49} -8.74456 q^{53} -1.62772 q^{55} -8.74456 q^{59} +5.11684 q^{61} +4.37228 q^{65} +12.0000 q^{67} -10.0000 q^{71} +3.62772 q^{73} -0.372281 q^{77} -3.25544 q^{79} +12.7446 q^{83} +1.62772 q^{85} +7.48913 q^{89} +1.00000 q^{91} -7.11684 q^{95} +7.48913 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{5} - 2 q^{7}+O(q^{10})$$ 2 * q - 3 * q^5 - 2 * q^7 $$2 q - 3 q^{5} - 2 q^{7} - 5 q^{11} - 2 q^{13} + 5 q^{17} + 9 q^{19} + 3 q^{23} + 11 q^{25} - 15 q^{29} - 10 q^{31} + 3 q^{35} + 5 q^{37} + 12 q^{41} + 9 q^{43} + 4 q^{47} + 2 q^{49} - 6 q^{53} - 9 q^{55} - 6 q^{59} - 7 q^{61} + 3 q^{65} + 24 q^{67} - 20 q^{71} + 13 q^{73} + 5 q^{77} - 18 q^{79} + 14 q^{83} + 9 q^{85} - 8 q^{89} + 2 q^{91} + 3 q^{95} - 8 q^{97}+O(q^{100})$$ 2 * q - 3 * q^5 - 2 * q^7 - 5 * q^11 - 2 * q^13 + 5 * q^17 + 9 * q^19 + 3 * q^23 + 11 * q^25 - 15 * q^29 - 10 * q^31 + 3 * q^35 + 5 * q^37 + 12 * q^41 + 9 * q^43 + 4 * q^47 + 2 * q^49 - 6 * q^53 - 9 * q^55 - 6 * q^59 - 7 * q^61 + 3 * q^65 + 24 * q^67 - 20 * q^71 + 13 * q^73 + 5 * q^77 - 18 * q^79 + 14 * q^83 + 9 * q^85 - 8 * q^89 + 2 * q^91 + 3 * q^95 - 8 * 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
$$5$$ −4.37228 −1.95534 −0.977672 0.210138i $$-0.932609\pi$$
−0.977672 + 0.210138i $$0.932609\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 0.372281 0.112247 0.0561235 0.998424i $$-0.482126\pi$$
0.0561235 + 0.998424i $$0.482126\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −0.372281 −0.0902915 −0.0451457 0.998980i $$-0.514375\pi$$
−0.0451457 + 0.998980i $$0.514375\pi$$
$$18$$ 0 0
$$19$$ 1.62772 0.373424 0.186712 0.982415i $$-0.440217\pi$$
0.186712 + 0.982415i $$0.440217\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.37228 0.911684 0.455842 0.890061i $$-0.349338\pi$$
0.455842 + 0.890061i $$0.349338\pi$$
$$24$$ 0 0
$$25$$ 14.1168 2.82337
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −10.3723 −1.92608 −0.963042 0.269351i $$-0.913191\pi$$
−0.963042 + 0.269351i $$0.913191\pi$$
$$30$$ 0 0
$$31$$ 0.744563 0.133727 0.0668637 0.997762i $$-0.478701\pi$$
0.0668637 + 0.997762i $$0.478701\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 4.37228 0.739050
$$36$$ 0 0
$$37$$ −0.372281 −0.0612027 −0.0306013 0.999532i $$-0.509742\pi$$
−0.0306013 + 0.999532i $$0.509742\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ 1.62772 0.248225 0.124112 0.992268i $$-0.460392\pi$$
0.124112 + 0.992268i $$0.460392\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 13.4891 1.96759 0.983796 0.179294i $$-0.0573813\pi$$
0.983796 + 0.179294i $$0.0573813\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −8.74456 −1.20116 −0.600579 0.799565i $$-0.705063\pi$$
−0.600579 + 0.799565i $$0.705063\pi$$
$$54$$ 0 0
$$55$$ −1.62772 −0.219482
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −8.74456 −1.13845 −0.569223 0.822183i $$-0.692756\pi$$
−0.569223 + 0.822183i $$0.692756\pi$$
$$60$$ 0 0
$$61$$ 5.11684 0.655145 0.327572 0.944826i $$-0.393769\pi$$
0.327572 + 0.944826i $$0.393769\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 4.37228 0.542315
$$66$$ 0 0
$$67$$ 12.0000 1.46603 0.733017 0.680211i $$-0.238112\pi$$
0.733017 + 0.680211i $$0.238112\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −10.0000 −1.18678 −0.593391 0.804914i $$-0.702211\pi$$
−0.593391 + 0.804914i $$0.702211\pi$$
$$72$$ 0 0
$$73$$ 3.62772 0.424592 0.212296 0.977205i $$-0.431906\pi$$
0.212296 + 0.977205i $$0.431906\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −0.372281 −0.0424254
$$78$$ 0 0
$$79$$ −3.25544 −0.366265 −0.183133 0.983088i $$-0.558624\pi$$
−0.183133 + 0.983088i $$0.558624\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 12.7446 1.39890 0.699449 0.714683i $$-0.253429\pi$$
0.699449 + 0.714683i $$0.253429\pi$$
$$84$$ 0 0
$$85$$ 1.62772 0.176551
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 7.48913 0.793846 0.396923 0.917852i $$-0.370078\pi$$
0.396923 + 0.917852i $$0.370078\pi$$
$$90$$ 0 0
$$91$$ 1.00000 0.104828
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −7.11684 −0.730173
$$96$$ 0 0
$$97$$ 7.48913 0.760405 0.380203 0.924903i $$-0.375854\pi$$
0.380203 + 0.924903i $$0.375854\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ 0 0
$$103$$ 11.1168 1.09538 0.547688 0.836683i $$-0.315508\pi$$
0.547688 + 0.836683i $$0.315508\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −10.0000 −0.966736 −0.483368 0.875417i $$-0.660587\pi$$
−0.483368 + 0.875417i $$0.660587\pi$$
$$108$$ 0 0
$$109$$ −13.1168 −1.25637 −0.628183 0.778066i $$-0.716201\pi$$
−0.628183 + 0.778066i $$0.716201\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −9.48913 −0.892662 −0.446331 0.894868i $$-0.647270\pi$$
−0.446331 + 0.894868i $$0.647270\pi$$
$$114$$ 0 0
$$115$$ −19.1168 −1.78265
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0.372281 0.0341270
$$120$$ 0 0
$$121$$ −10.8614 −0.987401
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −39.8614 −3.56531
$$126$$ 0 0
$$127$$ 8.00000 0.709885 0.354943 0.934888i $$-0.384500\pi$$
0.354943 + 0.934888i $$0.384500\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −18.3723 −1.60519 −0.802597 0.596522i $$-0.796549\pi$$
−0.802597 + 0.596522i $$0.796549\pi$$
$$132$$ 0 0
$$133$$ −1.62772 −0.141141
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −15.8614 −1.35513 −0.677566 0.735462i $$-0.736965\pi$$
−0.677566 + 0.735462i $$0.736965\pi$$
$$138$$ 0 0
$$139$$ 14.2337 1.20729 0.603643 0.797255i $$-0.293715\pi$$
0.603643 + 0.797255i $$0.293715\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −0.372281 −0.0311317
$$144$$ 0 0
$$145$$ 45.3505 3.76616
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 5.48913 0.449687 0.224843 0.974395i $$-0.427813\pi$$
0.224843 + 0.974395i $$0.427813\pi$$
$$150$$ 0 0
$$151$$ −18.3723 −1.49512 −0.747558 0.664197i $$-0.768774\pi$$
−0.747558 + 0.664197i $$0.768774\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −3.25544 −0.261483
$$156$$ 0 0
$$157$$ 11.6277 0.927993 0.463996 0.885837i $$-0.346415\pi$$
0.463996 + 0.885837i $$0.346415\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −4.37228 −0.344584
$$162$$ 0 0
$$163$$ 8.74456 0.684927 0.342464 0.939531i $$-0.388739\pi$$
0.342464 + 0.939531i $$0.388739\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 19.1168 1.47931 0.739653 0.672989i $$-0.234990\pi$$
0.739653 + 0.672989i $$0.234990\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −22.0000 −1.67263 −0.836315 0.548250i $$-0.815294\pi$$
−0.836315 + 0.548250i $$0.815294\pi$$
$$174$$ 0 0
$$175$$ −14.1168 −1.06713
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 8.23369 0.615415 0.307707 0.951481i $$-0.400438\pi$$
0.307707 + 0.951481i $$0.400438\pi$$
$$180$$ 0 0
$$181$$ −23.4891 −1.74593 −0.872966 0.487780i $$-0.837807\pi$$
−0.872966 + 0.487780i $$0.837807\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 1.62772 0.119672
$$186$$ 0 0
$$187$$ −0.138593 −0.0101350
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −18.6060 −1.34628 −0.673140 0.739515i $$-0.735055\pi$$
−0.673140 + 0.739515i $$0.735055\pi$$
$$192$$ 0 0
$$193$$ −19.4891 −1.40286 −0.701429 0.712739i $$-0.747454\pi$$
−0.701429 + 0.712739i $$0.747454\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −7.25544 −0.516929 −0.258464 0.966021i $$-0.583216\pi$$
−0.258464 + 0.966021i $$0.583216\pi$$
$$198$$ 0 0
$$199$$ −5.62772 −0.398938 −0.199469 0.979904i $$-0.563922\pi$$
−0.199469 + 0.979904i $$0.563922\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 10.3723 0.727991
$$204$$ 0 0
$$205$$ −26.2337 −1.83224
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0.605969 0.0419158
$$210$$ 0 0
$$211$$ −11.1168 −0.765315 −0.382658 0.923890i $$-0.624991\pi$$
−0.382658 + 0.923890i $$0.624991\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −7.11684 −0.485365
$$216$$ 0 0
$$217$$ −0.744563 −0.0505442
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0.372281 0.0250424
$$222$$ 0 0
$$223$$ −17.4891 −1.17116 −0.585579 0.810615i $$-0.699133\pi$$
−0.585579 + 0.810615i $$0.699133\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −15.2554 −1.01254 −0.506269 0.862375i $$-0.668976\pi$$
−0.506269 + 0.862375i $$0.668976\pi$$
$$228$$ 0 0
$$229$$ 19.4891 1.28788 0.643939 0.765077i $$-0.277299\pi$$
0.643939 + 0.765077i $$0.277299\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −12.0000 −0.786146 −0.393073 0.919507i $$-0.628588\pi$$
−0.393073 + 0.919507i $$0.628588\pi$$
$$234$$ 0 0
$$235$$ −58.9783 −3.84732
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 14.7446 0.953746 0.476873 0.878972i $$-0.341770\pi$$
0.476873 + 0.878972i $$0.341770\pi$$
$$240$$ 0 0
$$241$$ −3.48913 −0.224754 −0.112377 0.993666i $$-0.535846\pi$$
−0.112377 + 0.993666i $$0.535846\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −4.37228 −0.279335
$$246$$ 0 0
$$247$$ −1.62772 −0.103569
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 19.1168 1.20664 0.603322 0.797498i $$-0.293843\pi$$
0.603322 + 0.797498i $$0.293843\pi$$
$$252$$ 0 0
$$253$$ 1.62772 0.102334
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −0.510875 −0.0318675 −0.0159337 0.999873i $$-0.505072\pi$$
−0.0159337 + 0.999873i $$0.505072\pi$$
$$258$$ 0 0
$$259$$ 0.372281 0.0231324
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −9.25544 −0.570715 −0.285357 0.958421i $$-0.592112\pi$$
−0.285357 + 0.958421i $$0.592112\pi$$
$$264$$ 0 0
$$265$$ 38.2337 2.34868
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 15.4891 0.944389 0.472194 0.881494i $$-0.343462\pi$$
0.472194 + 0.881494i $$0.343462\pi$$
$$270$$ 0 0
$$271$$ −26.2337 −1.59358 −0.796792 0.604254i $$-0.793471\pi$$
−0.796792 + 0.604254i $$0.793471\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 5.25544 0.316915
$$276$$ 0 0
$$277$$ 18.7446 1.12625 0.563126 0.826371i $$-0.309599\pi$$
0.563126 + 0.826371i $$0.309599\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 26.2337 1.56497 0.782485 0.622669i $$-0.213952\pi$$
0.782485 + 0.622669i $$0.213952\pi$$
$$282$$ 0 0
$$283$$ −22.9783 −1.36592 −0.682958 0.730458i $$-0.739307\pi$$
−0.682958 + 0.730458i $$0.739307\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −6.00000 −0.354169
$$288$$ 0 0
$$289$$ −16.8614 −0.991847
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −7.48913 −0.437519 −0.218760 0.975779i $$-0.570201\pi$$
−0.218760 + 0.975779i $$0.570201\pi$$
$$294$$ 0 0
$$295$$ 38.2337 2.22605
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −4.37228 −0.252856
$$300$$ 0 0
$$301$$ −1.62772 −0.0938201
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −22.3723 −1.28103
$$306$$ 0 0
$$307$$ 4.00000 0.228292 0.114146 0.993464i $$-0.463587\pi$$
0.114146 + 0.993464i $$0.463587\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −4.00000 −0.226819 −0.113410 0.993548i $$-0.536177\pi$$
−0.113410 + 0.993548i $$0.536177\pi$$
$$312$$ 0 0
$$313$$ 2.00000 0.113047 0.0565233 0.998401i $$-0.481998\pi$$
0.0565233 + 0.998401i $$0.481998\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −18.2337 −1.02411 −0.512053 0.858954i $$-0.671115\pi$$
−0.512053 + 0.858954i $$0.671115\pi$$
$$318$$ 0 0
$$319$$ −3.86141 −0.216197
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −0.605969 −0.0337170
$$324$$ 0 0
$$325$$ −14.1168 −0.783062
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −13.4891 −0.743680
$$330$$ 0 0
$$331$$ −10.2337 −0.562494 −0.281247 0.959635i $$-0.590748\pi$$
−0.281247 + 0.959635i $$0.590748\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −52.4674 −2.86660
$$336$$ 0 0
$$337$$ −12.3723 −0.673961 −0.336981 0.941512i $$-0.609406\pi$$
−0.336981 + 0.941512i $$0.609406\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0.277187 0.0150105
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −17.2554 −0.926320 −0.463160 0.886275i $$-0.653285\pi$$
−0.463160 + 0.886275i $$0.653285\pi$$
$$348$$ 0 0
$$349$$ 28.2337 1.51131 0.755657 0.654967i $$-0.227318\pi$$
0.755657 + 0.654967i $$0.227318\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 11.4891 0.611504 0.305752 0.952111i $$-0.401092\pi$$
0.305752 + 0.952111i $$0.401092\pi$$
$$354$$ 0 0
$$355$$ 43.7228 2.32057
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 4.51087 0.238075 0.119037 0.992890i $$-0.462019\pi$$
0.119037 + 0.992890i $$0.462019\pi$$
$$360$$ 0 0
$$361$$ −16.3505 −0.860554
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −15.8614 −0.830224
$$366$$ 0 0
$$367$$ −18.9783 −0.990657 −0.495328 0.868706i $$-0.664952\pi$$
−0.495328 + 0.868706i $$0.664952\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 8.74456 0.453995
$$372$$ 0 0
$$373$$ 8.97825 0.464876 0.232438 0.972611i $$-0.425330\pi$$
0.232438 + 0.972611i $$0.425330\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 10.3723 0.534200
$$378$$ 0 0
$$379$$ −10.2337 −0.525669 −0.262835 0.964841i $$-0.584657\pi$$
−0.262835 + 0.964841i $$0.584657\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −19.8614 −1.01487 −0.507435 0.861690i $$-0.669406\pi$$
−0.507435 + 0.861690i $$0.669406\pi$$
$$384$$ 0 0
$$385$$ 1.62772 0.0829562
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −3.25544 −0.165057 −0.0825286 0.996589i $$-0.526300\pi$$
−0.0825286 + 0.996589i $$0.526300\pi$$
$$390$$ 0 0
$$391$$ −1.62772 −0.0823173
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 14.2337 0.716175
$$396$$ 0 0
$$397$$ −8.23369 −0.413237 −0.206618 0.978422i $$-0.566246\pi$$
−0.206618 + 0.978422i $$0.566246\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 10.2337 0.511046 0.255523 0.966803i $$-0.417752\pi$$
0.255523 + 0.966803i $$0.417752\pi$$
$$402$$ 0 0
$$403$$ −0.744563 −0.0370893
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −0.138593 −0.00686982
$$408$$ 0 0
$$409$$ 8.37228 0.413983 0.206991 0.978343i $$-0.433633\pi$$
0.206991 + 0.978343i $$0.433633\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 8.74456 0.430292
$$414$$ 0 0
$$415$$ −55.7228 −2.73533
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −17.6277 −0.861170 −0.430585 0.902550i $$-0.641693\pi$$
−0.430585 + 0.902550i $$0.641693\pi$$
$$420$$ 0 0
$$421$$ 32.9783 1.60726 0.803631 0.595128i $$-0.202899\pi$$
0.803631 + 0.595128i $$0.202899\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −5.25544 −0.254926
$$426$$ 0 0
$$427$$ −5.11684 −0.247621
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −8.51087 −0.409954 −0.204977 0.978767i $$-0.565712\pi$$
−0.204977 + 0.978767i $$0.565712\pi$$
$$432$$ 0 0
$$433$$ −24.9783 −1.20038 −0.600189 0.799858i $$-0.704908\pi$$
−0.600189 + 0.799858i $$0.704908\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 7.11684 0.340445
$$438$$ 0 0
$$439$$ −26.3723 −1.25868 −0.629340 0.777130i $$-0.716675\pi$$
−0.629340 + 0.777130i $$0.716675\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −30.0000 −1.42534 −0.712672 0.701498i $$-0.752515\pi$$
−0.712672 + 0.701498i $$0.752515\pi$$
$$444$$ 0 0
$$445$$ −32.7446 −1.55224
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 15.1168 0.713408 0.356704 0.934217i $$-0.383900\pi$$
0.356704 + 0.934217i $$0.383900\pi$$
$$450$$ 0 0
$$451$$ 2.23369 0.105180
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −4.37228 −0.204976
$$456$$ 0 0
$$457$$ 1.25544 0.0587269 0.0293634 0.999569i $$-0.490652\pi$$
0.0293634 + 0.999569i $$0.490652\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −5.11684 −0.238315 −0.119158 0.992875i $$-0.538019\pi$$
−0.119158 + 0.992875i $$0.538019\pi$$
$$462$$ 0 0
$$463$$ 15.1168 0.702539 0.351270 0.936274i $$-0.385750\pi$$
0.351270 + 0.936274i $$0.385750\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −33.3505 −1.54328 −0.771639 0.636060i $$-0.780563\pi$$
−0.771639 + 0.636060i $$0.780563\pi$$
$$468$$ 0 0
$$469$$ −12.0000 −0.554109
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0.605969 0.0278625
$$474$$ 0 0
$$475$$ 22.9783 1.05431
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −28.6060 −1.30704 −0.653520 0.756909i $$-0.726708\pi$$
−0.653520 + 0.756909i $$0.726708\pi$$
$$480$$ 0 0
$$481$$ 0.372281 0.0169746
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −32.7446 −1.48685
$$486$$ 0 0
$$487$$ −8.00000 −0.362515 −0.181257 0.983436i $$-0.558017\pi$$
−0.181257 + 0.983436i $$0.558017\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 30.7446 1.38748 0.693741 0.720224i $$-0.255961\pi$$
0.693741 + 0.720224i $$0.255961\pi$$
$$492$$ 0 0
$$493$$ 3.86141 0.173909
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 10.0000 0.448561
$$498$$ 0 0
$$499$$ 14.9783 0.670519 0.335259 0.942126i $$-0.391176\pi$$
0.335259 + 0.942126i $$0.391176\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 18.9783 0.846198 0.423099 0.906083i $$-0.360942\pi$$
0.423099 + 0.906083i $$0.360942\pi$$
$$504$$ 0 0
$$505$$ −8.74456 −0.389128
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 0.372281 0.0165011 0.00825054 0.999966i $$-0.497374\pi$$
0.00825054 + 0.999966i $$0.497374\pi$$
$$510$$ 0 0
$$511$$ −3.62772 −0.160481
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −48.6060 −2.14183
$$516$$ 0 0
$$517$$ 5.02175 0.220856
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 26.8832 1.17777 0.588886 0.808216i $$-0.299567\pi$$
0.588886 + 0.808216i $$0.299567\pi$$
$$522$$ 0 0
$$523$$ 23.7228 1.03733 0.518663 0.854979i $$-0.326430\pi$$
0.518663 + 0.854979i $$0.326430\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −0.277187 −0.0120744
$$528$$ 0 0
$$529$$ −3.88316 −0.168833
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −6.00000 −0.259889
$$534$$ 0 0
$$535$$ 43.7228 1.89030
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0.372281 0.0160353
$$540$$ 0 0
$$541$$ 0.372281 0.0160056 0.00800281 0.999968i $$-0.497453\pi$$
0.00800281 + 0.999968i $$0.497453\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 57.3505 2.45663
$$546$$ 0 0
$$547$$ 14.9783 0.640424 0.320212 0.947346i $$-0.396246\pi$$
0.320212 + 0.947346i $$0.396246\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −16.8832 −0.719247
$$552$$ 0 0
$$553$$ 3.25544 0.138435
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 8.74456 0.370519 0.185260 0.982690i $$-0.440687\pi$$
0.185260 + 0.982690i $$0.440687\pi$$
$$558$$ 0 0
$$559$$ −1.62772 −0.0688452
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 11.1168 0.468519 0.234260 0.972174i $$-0.424733\pi$$
0.234260 + 0.972174i $$0.424733\pi$$
$$564$$ 0 0
$$565$$ 41.4891 1.74546
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 8.74456 0.366591 0.183296 0.983058i $$-0.441323\pi$$
0.183296 + 0.983058i $$0.441323\pi$$
$$570$$ 0 0
$$571$$ −12.0000 −0.502184 −0.251092 0.967963i $$-0.580790\pi$$
−0.251092 + 0.967963i $$0.580790\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 61.7228 2.57402
$$576$$ 0 0
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −12.7446 −0.528734
$$582$$ 0 0
$$583$$ −3.25544 −0.134826
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −19.7228 −0.814048 −0.407024 0.913418i $$-0.633433\pi$$
−0.407024 + 0.913418i $$0.633433\pi$$
$$588$$ 0 0
$$589$$ 1.21194 0.0499371
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −2.74456 −0.112706 −0.0563528 0.998411i $$-0.517947\pi$$
−0.0563528 + 0.998411i $$0.517947\pi$$
$$594$$ 0 0
$$595$$ −1.62772 −0.0667300
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 45.1168 1.84342 0.921712 0.387875i $$-0.126791\pi$$
0.921712 + 0.387875i $$0.126791\pi$$
$$600$$ 0 0
$$601$$ 16.5109 0.673493 0.336746 0.941595i $$-0.390674\pi$$
0.336746 + 0.941595i $$0.390674\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 47.4891 1.93071
$$606$$ 0 0
$$607$$ 7.11684 0.288864 0.144432 0.989515i $$-0.453865\pi$$
0.144432 + 0.989515i $$0.453865\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −13.4891 −0.545712
$$612$$ 0 0
$$613$$ 0.372281 0.0150363 0.00751815 0.999972i $$-0.497607\pi$$
0.00751815 + 0.999972i $$0.497607\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 15.8614 0.638556 0.319278 0.947661i $$-0.396560\pi$$
0.319278 + 0.947661i $$0.396560\pi$$
$$618$$ 0 0
$$619$$ −2.37228 −0.0953500 −0.0476750 0.998863i $$-0.515181\pi$$
−0.0476750 + 0.998863i $$0.515181\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −7.48913 −0.300045
$$624$$ 0 0
$$625$$ 103.701 4.14804
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0.138593 0.00552608
$$630$$ 0 0
$$631$$ 16.6060 0.661073 0.330537 0.943793i $$-0.392770\pi$$
0.330537 + 0.943793i $$0.392770\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −34.9783 −1.38807
$$636$$ 0 0
$$637$$ −1.00000 −0.0396214
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 8.74456 0.345389 0.172695 0.984975i $$-0.444753\pi$$
0.172695 + 0.984975i $$0.444753\pi$$
$$642$$ 0 0
$$643$$ −27.1168 −1.06938 −0.534692 0.845047i $$-0.679572\pi$$
−0.534692 + 0.845047i $$0.679572\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −40.4674 −1.59094 −0.795468 0.605995i $$-0.792775\pi$$
−0.795468 + 0.605995i $$0.792775\pi$$
$$648$$ 0 0
$$649$$ −3.25544 −0.127787
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 38.8397 1.51991 0.759957 0.649974i $$-0.225220\pi$$
0.759957 + 0.649974i $$0.225220\pi$$
$$654$$ 0 0
$$655$$ 80.3288 3.13871
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −17.7228 −0.690383 −0.345191 0.938532i $$-0.612186\pi$$
−0.345191 + 0.938532i $$0.612186\pi$$
$$660$$ 0 0
$$661$$ −24.2337 −0.942581 −0.471291 0.881978i $$-0.656212\pi$$
−0.471291 + 0.881978i $$0.656212\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 7.11684 0.275979
$$666$$ 0 0
$$667$$ −45.3505 −1.75598
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 1.90491 0.0735381
$$672$$ 0 0
$$673$$ 45.1168 1.73913 0.869563 0.493822i $$-0.164400\pi$$
0.869563 + 0.493822i $$0.164400\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 2.00000 0.0768662 0.0384331 0.999261i $$-0.487763\pi$$
0.0384331 + 0.999261i $$0.487763\pi$$
$$678$$ 0 0
$$679$$ −7.48913 −0.287406
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 2.60597 0.0997146 0.0498573 0.998756i $$-0.484123\pi$$
0.0498573 + 0.998756i $$0.484123\pi$$
$$684$$ 0 0
$$685$$ 69.3505 2.64975
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 8.74456 0.333141
$$690$$ 0 0
$$691$$ 10.9783 0.417632 0.208816 0.977955i $$-0.433039\pi$$
0.208816 + 0.977955i $$0.433039\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −62.2337 −2.36066
$$696$$ 0 0
$$697$$ −2.23369 −0.0846070
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −36.7446 −1.38782 −0.693911 0.720060i $$-0.744114\pi$$
−0.693911 + 0.720060i $$0.744114\pi$$
$$702$$ 0 0
$$703$$ −0.605969 −0.0228546
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −2.00000 −0.0752177
$$708$$ 0 0
$$709$$ 35.4891 1.33282 0.666411 0.745585i $$-0.267830\pi$$
0.666411 + 0.745585i $$0.267830\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 3.25544 0.121917
$$714$$ 0 0
$$715$$ 1.62772 0.0608732
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 34.9783 1.30447 0.652234 0.758017i $$-0.273832\pi$$
0.652234 + 0.758017i $$0.273832\pi$$
$$720$$ 0 0
$$721$$ −11.1168 −0.414013
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −146.424 −5.43805
$$726$$ 0 0
$$727$$ −49.3505 −1.83031 −0.915155 0.403102i $$-0.867932\pi$$
−0.915155 + 0.403102i $$0.867932\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −0.605969 −0.0224126
$$732$$ 0 0
$$733$$ 46.7446 1.72655 0.863275 0.504734i $$-0.168409\pi$$
0.863275 + 0.504734i $$0.168409\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 4.46738 0.164558
$$738$$ 0 0
$$739$$ −16.7446 −0.615959 −0.307979 0.951393i $$-0.599653\pi$$
−0.307979 + 0.951393i $$0.599653\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 26.7446 0.981163 0.490581 0.871395i $$-0.336784\pi$$
0.490581 + 0.871395i $$0.336784\pi$$
$$744$$ 0 0
$$745$$ −24.0000 −0.879292
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 10.0000 0.365392
$$750$$ 0 0
$$751$$ 10.9783 0.400602 0.200301 0.979734i $$-0.435808\pi$$
0.200301 + 0.979734i $$0.435808\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 80.3288 2.92346
$$756$$ 0 0
$$757$$ −45.7228 −1.66182 −0.830912 0.556404i $$-0.812181\pi$$
−0.830912 + 0.556404i $$0.812181\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 45.7228 1.65745 0.828725 0.559656i $$-0.189067\pi$$
0.828725 + 0.559656i $$0.189067\pi$$
$$762$$ 0 0
$$763$$ 13.1168 0.474862
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 8.74456 0.315748
$$768$$ 0 0
$$769$$ 32.8397 1.18423 0.592114 0.805854i $$-0.298293\pi$$
0.592114 + 0.805854i $$0.298293\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −10.1386 −0.364660 −0.182330 0.983237i $$-0.558364\pi$$
−0.182330 + 0.983237i $$0.558364\pi$$
$$774$$ 0 0
$$775$$ 10.5109 0.377562
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 9.76631 0.349914
$$780$$ 0 0
$$781$$ −3.72281 −0.133213
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −50.8397 −1.81455
$$786$$ 0 0
$$787$$ −42.3723 −1.51041 −0.755204 0.655489i $$-0.772462\pi$$
−0.755204 + 0.655489i $$0.772462\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 9.48913 0.337394
$$792$$ 0 0
$$793$$ −5.11684 −0.181704
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 3.76631 0.133410 0.0667048 0.997773i $$-0.478751\pi$$
0.0667048 + 0.997773i $$0.478751\pi$$
$$798$$ 0 0
$$799$$ −5.02175 −0.177657
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 1.35053 0.0476592
$$804$$ 0 0
$$805$$ 19.1168 0.673780
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −33.2119 −1.16767 −0.583835 0.811872i $$-0.698448\pi$$
−0.583835 + 0.811872i $$0.698448\pi$$
$$810$$ 0 0
$$811$$ −13.3505 −0.468801 −0.234400 0.972140i $$-0.575313\pi$$
−0.234400 + 0.972140i $$0.575313\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −38.2337 −1.33927
$$816$$ 0 0
$$817$$ 2.64947 0.0926932
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 9.02175 0.314861 0.157431 0.987530i $$-0.449679\pi$$
0.157431 + 0.987530i $$0.449679\pi$$
$$822$$ 0 0
$$823$$ −23.7228 −0.826925 −0.413463 0.910521i $$-0.635681\pi$$
−0.413463 + 0.910521i $$0.635681\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 27.3505 0.951071 0.475536 0.879696i $$-0.342254\pi$$
0.475536 + 0.879696i $$0.342254\pi$$
$$828$$ 0 0
$$829$$ −56.3723 −1.95789 −0.978945 0.204124i $$-0.934566\pi$$
−0.978945 + 0.204124i $$0.934566\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −0.372281 −0.0128988
$$834$$ 0 0
$$835$$ −83.5842 −2.89255
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −13.4891 −0.465696 −0.232848 0.972513i $$-0.574805\pi$$
−0.232848 + 0.972513i $$0.574805\pi$$
$$840$$ 0 0
$$841$$ 78.5842 2.70980
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −4.37228 −0.150411
$$846$$ 0 0
$$847$$ 10.8614 0.373202
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −1.62772 −0.0557975
$$852$$ 0 0
$$853$$ −17.7228 −0.606818 −0.303409 0.952860i $$-0.598125\pi$$
−0.303409 + 0.952860i $$0.598125\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 32.9783 1.12652 0.563258 0.826281i $$-0.309548\pi$$
0.563258 + 0.826281i $$0.309548\pi$$
$$858$$ 0 0
$$859$$ 26.2337 0.895082 0.447541 0.894263i $$-0.352300\pi$$
0.447541 + 0.894263i $$0.352300\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −39.2119 −1.33479 −0.667395 0.744704i $$-0.732591\pi$$
−0.667395 + 0.744704i $$0.732591\pi$$
$$864$$ 0 0
$$865$$ 96.1902 3.27056
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −1.21194 −0.0411122
$$870$$ 0 0
$$871$$ −12.0000 −0.406604
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 39.8614 1.34756
$$876$$ 0 0
$$877$$ −36.9783 −1.24867 −0.624333 0.781158i $$-0.714629\pi$$
−0.624333 + 0.781158i $$0.714629\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −32.3723 −1.09065 −0.545325 0.838225i $$-0.683594\pi$$
−0.545325 + 0.838225i $$0.683594\pi$$
$$882$$ 0 0
$$883$$ −7.86141 −0.264557 −0.132279 0.991213i $$-0.542229\pi$$
−0.132279 + 0.991213i $$0.542229\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 19.7228 0.662227 0.331114 0.943591i $$-0.392576\pi$$
0.331114 + 0.943591i $$0.392576\pi$$
$$888$$ 0 0
$$889$$ −8.00000 −0.268311
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 21.9565 0.734746
$$894$$ 0 0
$$895$$ −36.0000 −1.20335
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −7.72281 −0.257570
$$900$$ 0 0
$$901$$ 3.25544 0.108454
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 102.701 3.41390
$$906$$ 0 0
$$907$$ 20.0000 0.664089 0.332045 0.943264i $$-0.392262\pi$$
0.332045 + 0.943264i $$0.392262\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −20.3723 −0.674964 −0.337482 0.941332i $$-0.609575\pi$$
−0.337482 + 0.941332i $$0.609575\pi$$
$$912$$ 0 0
$$913$$ 4.74456 0.157022
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 18.3723 0.606706
$$918$$ 0 0
$$919$$ 17.7663 0.586057 0.293028 0.956104i $$-0.405337\pi$$
0.293028 + 0.956104i $$0.405337\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 10.0000 0.329154
$$924$$ 0 0
$$925$$ −5.25544 −0.172798
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 15.4891 0.508182 0.254091 0.967180i $$-0.418224\pi$$
0.254091 + 0.967180i $$0.418224\pi$$
$$930$$ 0 0
$$931$$ 1.62772 0.0533463
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0.605969 0.0198173
$$936$$ 0 0
$$937$$ −4.51087 −0.147364 −0.0736819 0.997282i $$-0.523475\pi$$
−0.0736819 + 0.997282i $$0.523475\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −18.0000 −0.586783 −0.293392 0.955992i $$-0.594784\pi$$
−0.293392 + 0.955992i $$0.594784\pi$$
$$942$$ 0 0
$$943$$ 26.2337 0.854286
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −18.6060 −0.604613 −0.302306 0.953211i $$-0.597757\pi$$
−0.302306 + 0.953211i $$0.597757\pi$$
$$948$$ 0 0
$$949$$ −3.62772 −0.117761
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −45.9565 −1.48868 −0.744339 0.667802i $$-0.767235\pi$$
−0.744339 + 0.667802i $$0.767235\pi$$
$$954$$ 0 0
$$955$$ 81.3505 2.63244
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 15.8614 0.512192
$$960$$ 0 0
$$961$$ −30.4456 −0.982117
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 85.2119 2.74307
$$966$$ 0 0
$$967$$ −47.1168 −1.51518 −0.757588 0.652733i $$-0.773622\pi$$
−0.757588 + 0.652733i $$0.773622\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −48.0000 −1.54039 −0.770197 0.637806i $$-0.779842\pi$$
−0.770197 + 0.637806i $$0.779842\pi$$
$$972$$ 0 0
$$973$$ −14.2337 −0.456311
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −3.11684 −0.0997167 −0.0498583 0.998756i $$-0.515877\pi$$
−0.0498583 + 0.998756i $$0.515877\pi$$
$$978$$ 0 0
$$979$$ 2.78806 0.0891068
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −21.6277 −0.689817 −0.344909 0.938636i $$-0.612090\pi$$
−0.344909 + 0.938636i $$0.612090\pi$$
$$984$$ 0 0
$$985$$ 31.7228 1.01077
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 7.11684 0.226302
$$990$$ 0 0
$$991$$ −50.7011 −1.61057 −0.805286 0.592886i $$-0.797988\pi$$
−0.805286 + 0.592886i $$0.797988\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 24.6060 0.780062
$$996$$ 0 0
$$997$$ −38.4674 −1.21827 −0.609137 0.793065i $$-0.708484\pi$$
−0.609137 + 0.793065i $$0.708484\pi$$
$$998$$ 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 6552.2.a.bb.1.1 2
3.2 odd 2 6552.2.a.bm.1.2 yes 2

By twisted newform
Twist Min Dim Char Parity Ord Type
6552.2.a.bb.1.1 2 1.1 even 1 trivial
6552.2.a.bm.1.2 yes 2 3.2 odd 2