# Properties

 Label 920.2.a.e.1.2 Level $920$ Weight $2$ Character 920.1 Self dual yes Analytic conductor $7.346$ 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] = [920,2,Mod(1,920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(920, 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("920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$920 = 2^{3} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 920.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: $$7.34623698596$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 920.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.56155 q^{3} -1.00000 q^{5} -2.56155 q^{7} -0.561553 q^{9} +O(q^{10})$$ $$q+1.56155 q^{3} -1.00000 q^{5} -2.56155 q^{7} -0.561553 q^{9} -2.00000 q^{11} -3.56155 q^{13} -1.56155 q^{15} +1.43845 q^{17} -2.00000 q^{19} -4.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} -5.56155 q^{27} -8.12311 q^{29} +0.123106 q^{31} -3.12311 q^{33} +2.56155 q^{35} +0.561553 q^{37} -5.56155 q^{39} +4.12311 q^{41} +6.24621 q^{43} +0.561553 q^{45} -8.68466 q^{47} -0.438447 q^{49} +2.24621 q^{51} +8.56155 q^{53} +2.00000 q^{55} -3.12311 q^{57} -1.43845 q^{59} -0.876894 q^{61} +1.43845 q^{63} +3.56155 q^{65} -7.43845 q^{67} -1.56155 q^{69} -5.00000 q^{71} +7.56155 q^{73} +1.56155 q^{75} +5.12311 q^{77} -0.876894 q^{79} -7.00000 q^{81} +7.68466 q^{83} -1.43845 q^{85} -12.6847 q^{87} +8.00000 q^{89} +9.12311 q^{91} +0.192236 q^{93} +2.00000 q^{95} +5.12311 q^{97} +1.12311 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} - 2 q^{5} - q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q - q^3 - 2 * q^5 - q^7 + 3 * q^9 $$2 q - q^{3} - 2 q^{5} - q^{7} + 3 q^{9} - 4 q^{11} - 3 q^{13} + q^{15} + 7 q^{17} - 4 q^{19} - 8 q^{21} - 2 q^{23} + 2 q^{25} - 7 q^{27} - 8 q^{29} - 8 q^{31} + 2 q^{33} + q^{35} - 3 q^{37} - 7 q^{39} - 4 q^{43} - 3 q^{45} - 5 q^{47} - 5 q^{49} - 12 q^{51} + 13 q^{53} + 4 q^{55} + 2 q^{57} - 7 q^{59} - 10 q^{61} + 7 q^{63} + 3 q^{65} - 19 q^{67} + q^{69} - 10 q^{71} + 11 q^{73} - q^{75} + 2 q^{77} - 10 q^{79} - 14 q^{81} + 3 q^{83} - 7 q^{85} - 13 q^{87} + 16 q^{89} + 10 q^{91} + 21 q^{93} + 4 q^{95} + 2 q^{97} - 6 q^{99}+O(q^{100})$$ 2 * q - q^3 - 2 * q^5 - q^7 + 3 * q^9 - 4 * q^11 - 3 * q^13 + q^15 + 7 * q^17 - 4 * q^19 - 8 * q^21 - 2 * q^23 + 2 * q^25 - 7 * q^27 - 8 * q^29 - 8 * q^31 + 2 * q^33 + q^35 - 3 * q^37 - 7 * q^39 - 4 * q^43 - 3 * q^45 - 5 * q^47 - 5 * q^49 - 12 * q^51 + 13 * q^53 + 4 * q^55 + 2 * q^57 - 7 * q^59 - 10 * q^61 + 7 * q^63 + 3 * q^65 - 19 * q^67 + q^69 - 10 * q^71 + 11 * q^73 - q^75 + 2 * q^77 - 10 * q^79 - 14 * q^81 + 3 * q^83 - 7 * q^85 - 13 * q^87 + 16 * q^89 + 10 * q^91 + 21 * q^93 + 4 * q^95 + 2 * q^97 - 6 * 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
$$3$$ 1.56155 0.901563 0.450781 0.892634i $$-0.351145\pi$$
0.450781 + 0.892634i $$0.351145\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −2.56155 −0.968176 −0.484088 0.875019i $$-0.660849\pi$$
−0.484088 + 0.875019i $$0.660849\pi$$
$$8$$ 0 0
$$9$$ −0.561553 −0.187184
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 0 0
$$13$$ −3.56155 −0.987797 −0.493899 0.869520i $$-0.664429\pi$$
−0.493899 + 0.869520i $$0.664429\pi$$
$$14$$ 0 0
$$15$$ −1.56155 −0.403191
$$16$$ 0 0
$$17$$ 1.43845 0.348875 0.174437 0.984668i $$-0.444189\pi$$
0.174437 + 0.984668i $$0.444189\pi$$
$$18$$ 0 0
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 0 0
$$21$$ −4.00000 −0.872872
$$22$$ 0 0
$$23$$ −1.00000 −0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −5.56155 −1.07032
$$28$$ 0 0
$$29$$ −8.12311 −1.50842 −0.754211 0.656632i $$-0.771981\pi$$
−0.754211 + 0.656632i $$0.771981\pi$$
$$30$$ 0 0
$$31$$ 0.123106 0.0221104 0.0110552 0.999939i $$-0.496481\pi$$
0.0110552 + 0.999939i $$0.496481\pi$$
$$32$$ 0 0
$$33$$ −3.12311 −0.543663
$$34$$ 0 0
$$35$$ 2.56155 0.432981
$$36$$ 0 0
$$37$$ 0.561553 0.0923187 0.0461594 0.998934i $$-0.485302\pi$$
0.0461594 + 0.998934i $$0.485302\pi$$
$$38$$ 0 0
$$39$$ −5.56155 −0.890561
$$40$$ 0 0
$$41$$ 4.12311 0.643921 0.321960 0.946753i $$-0.395658\pi$$
0.321960 + 0.946753i $$0.395658\pi$$
$$42$$ 0 0
$$43$$ 6.24621 0.952538 0.476269 0.879300i $$-0.341989\pi$$
0.476269 + 0.879300i $$0.341989\pi$$
$$44$$ 0 0
$$45$$ 0.561553 0.0837114
$$46$$ 0 0
$$47$$ −8.68466 −1.26679 −0.633394 0.773830i $$-0.718339\pi$$
−0.633394 + 0.773830i $$0.718339\pi$$
$$48$$ 0 0
$$49$$ −0.438447 −0.0626353
$$50$$ 0 0
$$51$$ 2.24621 0.314532
$$52$$ 0 0
$$53$$ 8.56155 1.17602 0.588010 0.808854i $$-0.299912\pi$$
0.588010 + 0.808854i $$0.299912\pi$$
$$54$$ 0 0
$$55$$ 2.00000 0.269680
$$56$$ 0 0
$$57$$ −3.12311 −0.413665
$$58$$ 0 0
$$59$$ −1.43845 −0.187270 −0.0936349 0.995607i $$-0.529849\pi$$
−0.0936349 + 0.995607i $$0.529849\pi$$
$$60$$ 0 0
$$61$$ −0.876894 −0.112275 −0.0561374 0.998423i $$-0.517878\pi$$
−0.0561374 + 0.998423i $$0.517878\pi$$
$$62$$ 0 0
$$63$$ 1.43845 0.181227
$$64$$ 0 0
$$65$$ 3.56155 0.441756
$$66$$ 0 0
$$67$$ −7.43845 −0.908751 −0.454375 0.890810i $$-0.650138\pi$$
−0.454375 + 0.890810i $$0.650138\pi$$
$$68$$ 0 0
$$69$$ −1.56155 −0.187989
$$70$$ 0 0
$$71$$ −5.00000 −0.593391 −0.296695 0.954972i $$-0.595885\pi$$
−0.296695 + 0.954972i $$0.595885\pi$$
$$72$$ 0 0
$$73$$ 7.56155 0.885013 0.442506 0.896765i $$-0.354089\pi$$
0.442506 + 0.896765i $$0.354089\pi$$
$$74$$ 0 0
$$75$$ 1.56155 0.180313
$$76$$ 0 0
$$77$$ 5.12311 0.583832
$$78$$ 0 0
$$79$$ −0.876894 −0.0986583 −0.0493292 0.998783i $$-0.515708\pi$$
−0.0493292 + 0.998783i $$0.515708\pi$$
$$80$$ 0 0
$$81$$ −7.00000 −0.777778
$$82$$ 0 0
$$83$$ 7.68466 0.843501 0.421750 0.906712i $$-0.361416\pi$$
0.421750 + 0.906712i $$0.361416\pi$$
$$84$$ 0 0
$$85$$ −1.43845 −0.156022
$$86$$ 0 0
$$87$$ −12.6847 −1.35994
$$88$$ 0 0
$$89$$ 8.00000 0.847998 0.423999 0.905663i $$-0.360626\pi$$
0.423999 + 0.905663i $$0.360626\pi$$
$$90$$ 0 0
$$91$$ 9.12311 0.956361
$$92$$ 0 0
$$93$$ 0.192236 0.0199339
$$94$$ 0 0
$$95$$ 2.00000 0.205196
$$96$$ 0 0
$$97$$ 5.12311 0.520173 0.260086 0.965585i $$-0.416249\pi$$
0.260086 + 0.965585i $$0.416249\pi$$
$$98$$ 0 0
$$99$$ 1.12311 0.112876
$$100$$ 0 0
$$101$$ 10.8078 1.07541 0.537706 0.843132i $$-0.319291\pi$$
0.537706 + 0.843132i $$0.319291\pi$$
$$102$$ 0 0
$$103$$ −14.2462 −1.40372 −0.701860 0.712314i $$-0.747647\pi$$
−0.701860 + 0.712314i $$0.747647\pi$$
$$104$$ 0 0
$$105$$ 4.00000 0.390360
$$106$$ 0 0
$$107$$ −1.68466 −0.162862 −0.0814310 0.996679i $$-0.525949\pi$$
−0.0814310 + 0.996679i $$0.525949\pi$$
$$108$$ 0 0
$$109$$ −1.12311 −0.107574 −0.0537870 0.998552i $$-0.517129\pi$$
−0.0537870 + 0.998552i $$0.517129\pi$$
$$110$$ 0 0
$$111$$ 0.876894 0.0832311
$$112$$ 0 0
$$113$$ 5.43845 0.511606 0.255803 0.966729i $$-0.417660\pi$$
0.255803 + 0.966729i $$0.417660\pi$$
$$114$$ 0 0
$$115$$ 1.00000 0.0932505
$$116$$ 0 0
$$117$$ 2.00000 0.184900
$$118$$ 0 0
$$119$$ −3.68466 −0.337772
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ 6.43845 0.580535
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −19.8078 −1.75765 −0.878827 0.477140i $$-0.841674\pi$$
−0.878827 + 0.477140i $$0.841674\pi$$
$$128$$ 0 0
$$129$$ 9.75379 0.858773
$$130$$ 0 0
$$131$$ −18.9309 −1.65400 −0.826999 0.562204i $$-0.809954\pi$$
−0.826999 + 0.562204i $$0.809954\pi$$
$$132$$ 0 0
$$133$$ 5.12311 0.444230
$$134$$ 0 0
$$135$$ 5.56155 0.478662
$$136$$ 0 0
$$137$$ 21.6155 1.84674 0.923370 0.383912i $$-0.125423\pi$$
0.923370 + 0.383912i $$0.125423\pi$$
$$138$$ 0 0
$$139$$ −10.3693 −0.879514 −0.439757 0.898117i $$-0.644935\pi$$
−0.439757 + 0.898117i $$0.644935\pi$$
$$140$$ 0 0
$$141$$ −13.5616 −1.14209
$$142$$ 0 0
$$143$$ 7.12311 0.595664
$$144$$ 0 0
$$145$$ 8.12311 0.674587
$$146$$ 0 0
$$147$$ −0.684658 −0.0564697
$$148$$ 0 0
$$149$$ 0.876894 0.0718380 0.0359190 0.999355i $$-0.488564\pi$$
0.0359190 + 0.999355i $$0.488564\pi$$
$$150$$ 0 0
$$151$$ 1.56155 0.127077 0.0635387 0.997979i $$-0.479761\pi$$
0.0635387 + 0.997979i $$0.479761\pi$$
$$152$$ 0 0
$$153$$ −0.807764 −0.0653039
$$154$$ 0 0
$$155$$ −0.123106 −0.00988808
$$156$$ 0 0
$$157$$ −15.6847 −1.25177 −0.625886 0.779915i $$-0.715262\pi$$
−0.625886 + 0.779915i $$0.715262\pi$$
$$158$$ 0 0
$$159$$ 13.3693 1.06026
$$160$$ 0 0
$$161$$ 2.56155 0.201879
$$162$$ 0 0
$$163$$ −6.68466 −0.523583 −0.261791 0.965124i $$-0.584313\pi$$
−0.261791 + 0.965124i $$0.584313\pi$$
$$164$$ 0 0
$$165$$ 3.12311 0.243133
$$166$$ 0 0
$$167$$ 8.00000 0.619059 0.309529 0.950890i $$-0.399829\pi$$
0.309529 + 0.950890i $$0.399829\pi$$
$$168$$ 0 0
$$169$$ −0.315342 −0.0242570
$$170$$ 0 0
$$171$$ 1.12311 0.0858860
$$172$$ 0 0
$$173$$ −7.12311 −0.541560 −0.270780 0.962641i $$-0.587282\pi$$
−0.270780 + 0.962641i $$0.587282\pi$$
$$174$$ 0 0
$$175$$ −2.56155 −0.193635
$$176$$ 0 0
$$177$$ −2.24621 −0.168836
$$178$$ 0 0
$$179$$ −7.31534 −0.546774 −0.273387 0.961904i $$-0.588144\pi$$
−0.273387 + 0.961904i $$0.588144\pi$$
$$180$$ 0 0
$$181$$ −8.87689 −0.659814 −0.329907 0.944013i $$-0.607017\pi$$
−0.329907 + 0.944013i $$0.607017\pi$$
$$182$$ 0 0
$$183$$ −1.36932 −0.101223
$$184$$ 0 0
$$185$$ −0.561553 −0.0412862
$$186$$ 0 0
$$187$$ −2.87689 −0.210379
$$188$$ 0 0
$$189$$ 14.2462 1.03626
$$190$$ 0 0
$$191$$ −3.12311 −0.225980 −0.112990 0.993596i $$-0.536043\pi$$
−0.112990 + 0.993596i $$0.536043\pi$$
$$192$$ 0 0
$$193$$ 18.9309 1.36267 0.681337 0.731970i $$-0.261399\pi$$
0.681337 + 0.731970i $$0.261399\pi$$
$$194$$ 0 0
$$195$$ 5.56155 0.398271
$$196$$ 0 0
$$197$$ 17.8078 1.26875 0.634375 0.773025i $$-0.281257\pi$$
0.634375 + 0.773025i $$0.281257\pi$$
$$198$$ 0 0
$$199$$ 18.0000 1.27599 0.637993 0.770042i $$-0.279765\pi$$
0.637993 + 0.770042i $$0.279765\pi$$
$$200$$ 0 0
$$201$$ −11.6155 −0.819296
$$202$$ 0 0
$$203$$ 20.8078 1.46042
$$204$$ 0 0
$$205$$ −4.12311 −0.287970
$$206$$ 0 0
$$207$$ 0.561553 0.0390306
$$208$$ 0 0
$$209$$ 4.00000 0.276686
$$210$$ 0 0
$$211$$ −9.93087 −0.683669 −0.341835 0.939760i $$-0.611048\pi$$
−0.341835 + 0.939760i $$0.611048\pi$$
$$212$$ 0 0
$$213$$ −7.80776 −0.534979
$$214$$ 0 0
$$215$$ −6.24621 −0.425988
$$216$$ 0 0
$$217$$ −0.315342 −0.0214068
$$218$$ 0 0
$$219$$ 11.8078 0.797895
$$220$$ 0 0
$$221$$ −5.12311 −0.344617
$$222$$ 0 0
$$223$$ 18.8769 1.26409 0.632045 0.774932i $$-0.282216\pi$$
0.632045 + 0.774932i $$0.282216\pi$$
$$224$$ 0 0
$$225$$ −0.561553 −0.0374369
$$226$$ 0 0
$$227$$ 18.2462 1.21104 0.605522 0.795829i $$-0.292964\pi$$
0.605522 + 0.795829i $$0.292964\pi$$
$$228$$ 0 0
$$229$$ −5.75379 −0.380221 −0.190111 0.981763i $$-0.560885\pi$$
−0.190111 + 0.981763i $$0.560885\pi$$
$$230$$ 0 0
$$231$$ 8.00000 0.526361
$$232$$ 0 0
$$233$$ 28.3002 1.85401 0.927003 0.375053i $$-0.122375\pi$$
0.927003 + 0.375053i $$0.122375\pi$$
$$234$$ 0 0
$$235$$ 8.68466 0.566525
$$236$$ 0 0
$$237$$ −1.36932 −0.0889467
$$238$$ 0 0
$$239$$ −7.87689 −0.509514 −0.254757 0.967005i $$-0.581995\pi$$
−0.254757 + 0.967005i $$0.581995\pi$$
$$240$$ 0 0
$$241$$ −18.4924 −1.19120 −0.595601 0.803281i $$-0.703086\pi$$
−0.595601 + 0.803281i $$0.703086\pi$$
$$242$$ 0 0
$$243$$ 5.75379 0.369106
$$244$$ 0 0
$$245$$ 0.438447 0.0280114
$$246$$ 0 0
$$247$$ 7.12311 0.453232
$$248$$ 0 0
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ 9.12311 0.575845 0.287923 0.957654i $$-0.407035\pi$$
0.287923 + 0.957654i $$0.407035\pi$$
$$252$$ 0 0
$$253$$ 2.00000 0.125739
$$254$$ 0 0
$$255$$ −2.24621 −0.140663
$$256$$ 0 0
$$257$$ −0.684658 −0.0427078 −0.0213539 0.999772i $$-0.506798\pi$$
−0.0213539 + 0.999772i $$0.506798\pi$$
$$258$$ 0 0
$$259$$ −1.43845 −0.0893808
$$260$$ 0 0
$$261$$ 4.56155 0.282353
$$262$$ 0 0
$$263$$ −14.3153 −0.882722 −0.441361 0.897330i $$-0.645504\pi$$
−0.441361 + 0.897330i $$0.645504\pi$$
$$264$$ 0 0
$$265$$ −8.56155 −0.525932
$$266$$ 0 0
$$267$$ 12.4924 0.764524
$$268$$ 0 0
$$269$$ −27.4924 −1.67624 −0.838121 0.545484i $$-0.816346\pi$$
−0.838121 + 0.545484i $$0.816346\pi$$
$$270$$ 0 0
$$271$$ 24.1771 1.46865 0.734327 0.678796i $$-0.237498\pi$$
0.734327 + 0.678796i $$0.237498\pi$$
$$272$$ 0 0
$$273$$ 14.2462 0.862220
$$274$$ 0 0
$$275$$ −2.00000 −0.120605
$$276$$ 0 0
$$277$$ −20.4384 −1.22803 −0.614014 0.789295i $$-0.710446\pi$$
−0.614014 + 0.789295i $$0.710446\pi$$
$$278$$ 0 0
$$279$$ −0.0691303 −0.00413872
$$280$$ 0 0
$$281$$ −9.12311 −0.544239 −0.272119 0.962263i $$-0.587725\pi$$
−0.272119 + 0.962263i $$0.587725\pi$$
$$282$$ 0 0
$$283$$ 3.93087 0.233666 0.116833 0.993152i $$-0.462726\pi$$
0.116833 + 0.993152i $$0.462726\pi$$
$$284$$ 0 0
$$285$$ 3.12311 0.184997
$$286$$ 0 0
$$287$$ −10.5616 −0.623429
$$288$$ 0 0
$$289$$ −14.9309 −0.878286
$$290$$ 0 0
$$291$$ 8.00000 0.468968
$$292$$ 0 0
$$293$$ 14.5616 0.850695 0.425347 0.905030i $$-0.360152\pi$$
0.425347 + 0.905030i $$0.360152\pi$$
$$294$$ 0 0
$$295$$ 1.43845 0.0837496
$$296$$ 0 0
$$297$$ 11.1231 0.645428
$$298$$ 0 0
$$299$$ 3.56155 0.205970
$$300$$ 0 0
$$301$$ −16.0000 −0.922225
$$302$$ 0 0
$$303$$ 16.8769 0.969552
$$304$$ 0 0
$$305$$ 0.876894 0.0502108
$$306$$ 0 0
$$307$$ −15.3693 −0.877173 −0.438587 0.898689i $$-0.644521\pi$$
−0.438587 + 0.898689i $$0.644521\pi$$
$$308$$ 0 0
$$309$$ −22.2462 −1.26554
$$310$$ 0 0
$$311$$ −4.19224 −0.237720 −0.118860 0.992911i $$-0.537924\pi$$
−0.118860 + 0.992911i $$0.537924\pi$$
$$312$$ 0 0
$$313$$ −13.9309 −0.787419 −0.393710 0.919235i $$-0.628808\pi$$
−0.393710 + 0.919235i $$0.628808\pi$$
$$314$$ 0 0
$$315$$ −1.43845 −0.0810473
$$316$$ 0 0
$$317$$ 25.3693 1.42488 0.712441 0.701732i $$-0.247589\pi$$
0.712441 + 0.701732i $$0.247589\pi$$
$$318$$ 0 0
$$319$$ 16.2462 0.909613
$$320$$ 0 0
$$321$$ −2.63068 −0.146830
$$322$$ 0 0
$$323$$ −2.87689 −0.160075
$$324$$ 0 0
$$325$$ −3.56155 −0.197559
$$326$$ 0 0
$$327$$ −1.75379 −0.0969847
$$328$$ 0 0
$$329$$ 22.2462 1.22647
$$330$$ 0 0
$$331$$ −3.49242 −0.191961 −0.0959805 0.995383i $$-0.530599\pi$$
−0.0959805 + 0.995383i $$0.530599\pi$$
$$332$$ 0 0
$$333$$ −0.315342 −0.0172806
$$334$$ 0 0
$$335$$ 7.43845 0.406406
$$336$$ 0 0
$$337$$ 13.1231 0.714861 0.357431 0.933940i $$-0.383653\pi$$
0.357431 + 0.933940i $$0.383653\pi$$
$$338$$ 0 0
$$339$$ 8.49242 0.461245
$$340$$ 0 0
$$341$$ −0.246211 −0.0133331
$$342$$ 0 0
$$343$$ 19.0540 1.02882
$$344$$ 0 0
$$345$$ 1.56155 0.0840712
$$346$$ 0 0
$$347$$ −14.2462 −0.764777 −0.382388 0.924002i $$-0.624898\pi$$
−0.382388 + 0.924002i $$0.624898\pi$$
$$348$$ 0 0
$$349$$ −4.12311 −0.220705 −0.110352 0.993893i $$-0.535198\pi$$
−0.110352 + 0.993893i $$0.535198\pi$$
$$350$$ 0 0
$$351$$ 19.8078 1.05726
$$352$$ 0 0
$$353$$ 4.19224 0.223130 0.111565 0.993757i $$-0.464414\pi$$
0.111565 + 0.993757i $$0.464414\pi$$
$$354$$ 0 0
$$355$$ 5.00000 0.265372
$$356$$ 0 0
$$357$$ −5.75379 −0.304523
$$358$$ 0 0
$$359$$ 11.1231 0.587055 0.293528 0.955951i $$-0.405171\pi$$
0.293528 + 0.955951i $$0.405171\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 0 0
$$363$$ −10.9309 −0.573722
$$364$$ 0 0
$$365$$ −7.56155 −0.395790
$$366$$ 0 0
$$367$$ −11.1922 −0.584230 −0.292115 0.956383i $$-0.594359\pi$$
−0.292115 + 0.956383i $$0.594359\pi$$
$$368$$ 0 0
$$369$$ −2.31534 −0.120532
$$370$$ 0 0
$$371$$ −21.9309 −1.13859
$$372$$ 0 0
$$373$$ 24.7386 1.28092 0.640459 0.767992i $$-0.278744\pi$$
0.640459 + 0.767992i $$0.278744\pi$$
$$374$$ 0 0
$$375$$ −1.56155 −0.0806382
$$376$$ 0 0
$$377$$ 28.9309 1.49002
$$378$$ 0 0
$$379$$ −32.9848 −1.69432 −0.847159 0.531340i $$-0.821689\pi$$
−0.847159 + 0.531340i $$0.821689\pi$$
$$380$$ 0 0
$$381$$ −30.9309 −1.58464
$$382$$ 0 0
$$383$$ −13.9309 −0.711834 −0.355917 0.934518i $$-0.615831\pi$$
−0.355917 + 0.934518i $$0.615831\pi$$
$$384$$ 0 0
$$385$$ −5.12311 −0.261098
$$386$$ 0 0
$$387$$ −3.50758 −0.178300
$$388$$ 0 0
$$389$$ −18.8769 −0.957097 −0.478548 0.878061i $$-0.658837\pi$$
−0.478548 + 0.878061i $$0.658837\pi$$
$$390$$ 0 0
$$391$$ −1.43845 −0.0727454
$$392$$ 0 0
$$393$$ −29.5616 −1.49118
$$394$$ 0 0
$$395$$ 0.876894 0.0441213
$$396$$ 0 0
$$397$$ −21.1771 −1.06285 −0.531424 0.847106i $$-0.678343\pi$$
−0.531424 + 0.847106i $$0.678343\pi$$
$$398$$ 0 0
$$399$$ 8.00000 0.400501
$$400$$ 0 0
$$401$$ −32.7386 −1.63489 −0.817445 0.576007i $$-0.804610\pi$$
−0.817445 + 0.576007i $$0.804610\pi$$
$$402$$ 0 0
$$403$$ −0.438447 −0.0218406
$$404$$ 0 0
$$405$$ 7.00000 0.347833
$$406$$ 0 0
$$407$$ −1.12311 −0.0556703
$$408$$ 0 0
$$409$$ 5.00000 0.247234 0.123617 0.992330i $$-0.460551\pi$$
0.123617 + 0.992330i $$0.460551\pi$$
$$410$$ 0 0
$$411$$ 33.7538 1.66495
$$412$$ 0 0
$$413$$ 3.68466 0.181310
$$414$$ 0 0
$$415$$ −7.68466 −0.377225
$$416$$ 0 0
$$417$$ −16.1922 −0.792937
$$418$$ 0 0
$$419$$ −21.6155 −1.05599 −0.527994 0.849248i $$-0.677056\pi$$
−0.527994 + 0.849248i $$0.677056\pi$$
$$420$$ 0 0
$$421$$ −30.8769 −1.50485 −0.752424 0.658679i $$-0.771115\pi$$
−0.752424 + 0.658679i $$0.771115\pi$$
$$422$$ 0 0
$$423$$ 4.87689 0.237123
$$424$$ 0 0
$$425$$ 1.43845 0.0697749
$$426$$ 0 0
$$427$$ 2.24621 0.108702
$$428$$ 0 0
$$429$$ 11.1231 0.537029
$$430$$ 0 0
$$431$$ 30.2462 1.45691 0.728454 0.685094i $$-0.240239\pi$$
0.728454 + 0.685094i $$0.240239\pi$$
$$432$$ 0 0
$$433$$ −13.6847 −0.657643 −0.328821 0.944392i $$-0.606651\pi$$
−0.328821 + 0.944392i $$0.606651\pi$$
$$434$$ 0 0
$$435$$ 12.6847 0.608183
$$436$$ 0 0
$$437$$ 2.00000 0.0956730
$$438$$ 0 0
$$439$$ −3.80776 −0.181735 −0.0908673 0.995863i $$-0.528964\pi$$
−0.0908673 + 0.995863i $$0.528964\pi$$
$$440$$ 0 0
$$441$$ 0.246211 0.0117243
$$442$$ 0 0
$$443$$ 22.0540 1.04782 0.523908 0.851775i $$-0.324474\pi$$
0.523908 + 0.851775i $$0.324474\pi$$
$$444$$ 0 0
$$445$$ −8.00000 −0.379236
$$446$$ 0 0
$$447$$ 1.36932 0.0647665
$$448$$ 0 0
$$449$$ 18.3153 0.864354 0.432177 0.901789i $$-0.357745\pi$$
0.432177 + 0.901789i $$0.357745\pi$$
$$450$$ 0 0
$$451$$ −8.24621 −0.388299
$$452$$ 0 0
$$453$$ 2.43845 0.114568
$$454$$ 0 0
$$455$$ −9.12311 −0.427698
$$456$$ 0 0
$$457$$ −17.9309 −0.838771 −0.419385 0.907808i $$-0.637754\pi$$
−0.419385 + 0.907808i $$0.637754\pi$$
$$458$$ 0 0
$$459$$ −8.00000 −0.373408
$$460$$ 0 0
$$461$$ −39.1771 −1.82466 −0.912329 0.409457i $$-0.865718\pi$$
−0.912329 + 0.409457i $$0.865718\pi$$
$$462$$ 0 0
$$463$$ −29.1231 −1.35347 −0.676733 0.736229i $$-0.736605\pi$$
−0.676733 + 0.736229i $$0.736605\pi$$
$$464$$ 0 0
$$465$$ −0.192236 −0.00891473
$$466$$ 0 0
$$467$$ −9.68466 −0.448153 −0.224076 0.974572i $$-0.571936\pi$$
−0.224076 + 0.974572i $$0.571936\pi$$
$$468$$ 0 0
$$469$$ 19.0540 0.879831
$$470$$ 0 0
$$471$$ −24.4924 −1.12855
$$472$$ 0 0
$$473$$ −12.4924 −0.574402
$$474$$ 0 0
$$475$$ −2.00000 −0.0917663
$$476$$ 0 0
$$477$$ −4.80776 −0.220132
$$478$$ 0 0
$$479$$ −4.00000 −0.182765 −0.0913823 0.995816i $$-0.529129\pi$$
−0.0913823 + 0.995816i $$0.529129\pi$$
$$480$$ 0 0
$$481$$ −2.00000 −0.0911922
$$482$$ 0 0
$$483$$ 4.00000 0.182006
$$484$$ 0 0
$$485$$ −5.12311 −0.232628
$$486$$ 0 0
$$487$$ 0.684658 0.0310248 0.0155124 0.999880i $$-0.495062\pi$$
0.0155124 + 0.999880i $$0.495062\pi$$
$$488$$ 0 0
$$489$$ −10.4384 −0.472043
$$490$$ 0 0
$$491$$ −12.6155 −0.569331 −0.284665 0.958627i $$-0.591882\pi$$
−0.284665 + 0.958627i $$0.591882\pi$$
$$492$$ 0 0
$$493$$ −11.6847 −0.526251
$$494$$ 0 0
$$495$$ −1.12311 −0.0504798
$$496$$ 0 0
$$497$$ 12.8078 0.574507
$$498$$ 0 0
$$499$$ −39.7386 −1.77895 −0.889473 0.456988i $$-0.848928\pi$$
−0.889473 + 0.456988i $$0.848928\pi$$
$$500$$ 0 0
$$501$$ 12.4924 0.558120
$$502$$ 0 0
$$503$$ 15.4384 0.688366 0.344183 0.938903i $$-0.388156\pi$$
0.344183 + 0.938903i $$0.388156\pi$$
$$504$$ 0 0
$$505$$ −10.8078 −0.480939
$$506$$ 0 0
$$507$$ −0.492423 −0.0218693
$$508$$ 0 0
$$509$$ −24.4384 −1.08322 −0.541608 0.840631i $$-0.682184\pi$$
−0.541608 + 0.840631i $$0.682184\pi$$
$$510$$ 0 0
$$511$$ −19.3693 −0.856848
$$512$$ 0 0
$$513$$ 11.1231 0.491097
$$514$$ 0 0
$$515$$ 14.2462 0.627763
$$516$$ 0 0
$$517$$ 17.3693 0.763902
$$518$$ 0 0
$$519$$ −11.1231 −0.488250
$$520$$ 0 0
$$521$$ 23.8617 1.04540 0.522701 0.852516i $$-0.324924\pi$$
0.522701 + 0.852516i $$0.324924\pi$$
$$522$$ 0 0
$$523$$ 10.2462 0.448036 0.224018 0.974585i $$-0.428083\pi$$
0.224018 + 0.974585i $$0.428083\pi$$
$$524$$ 0 0
$$525$$ −4.00000 −0.174574
$$526$$ 0 0
$$527$$ 0.177081 0.00771377
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 0.807764 0.0350540
$$532$$ 0 0
$$533$$ −14.6847 −0.636063
$$534$$ 0 0
$$535$$ 1.68466 0.0728341
$$536$$ 0 0
$$537$$ −11.4233 −0.492951
$$538$$ 0 0
$$539$$ 0.876894 0.0377705
$$540$$ 0 0
$$541$$ 16.0540 0.690214 0.345107 0.938563i $$-0.387843\pi$$
0.345107 + 0.938563i $$0.387843\pi$$
$$542$$ 0 0
$$543$$ −13.8617 −0.594864
$$544$$ 0 0
$$545$$ 1.12311 0.0481086
$$546$$ 0 0
$$547$$ −7.31534 −0.312781 −0.156391 0.987695i $$-0.549986\pi$$
−0.156391 + 0.987695i $$0.549986\pi$$
$$548$$ 0 0
$$549$$ 0.492423 0.0210161
$$550$$ 0 0
$$551$$ 16.2462 0.692112
$$552$$ 0 0
$$553$$ 2.24621 0.0955186
$$554$$ 0 0
$$555$$ −0.876894 −0.0372221
$$556$$ 0 0
$$557$$ −21.3002 −0.902518 −0.451259 0.892393i $$-0.649025\pi$$
−0.451259 + 0.892393i $$0.649025\pi$$
$$558$$ 0 0
$$559$$ −22.2462 −0.940914
$$560$$ 0 0
$$561$$ −4.49242 −0.189670
$$562$$ 0 0
$$563$$ −11.1922 −0.471697 −0.235848 0.971790i $$-0.575787\pi$$
−0.235848 + 0.971790i $$0.575787\pi$$
$$564$$ 0 0
$$565$$ −5.43845 −0.228797
$$566$$ 0 0
$$567$$ 17.9309 0.753026
$$568$$ 0 0
$$569$$ 44.7386 1.87554 0.937771 0.347256i $$-0.112886\pi$$
0.937771 + 0.347256i $$0.112886\pi$$
$$570$$ 0 0
$$571$$ −30.7386 −1.28637 −0.643186 0.765710i $$-0.722388\pi$$
−0.643186 + 0.765710i $$0.722388\pi$$
$$572$$ 0 0
$$573$$ −4.87689 −0.203735
$$574$$ 0 0
$$575$$ −1.00000 −0.0417029
$$576$$ 0 0
$$577$$ −7.31534 −0.304542 −0.152271 0.988339i $$-0.548659\pi$$
−0.152271 + 0.988339i $$0.548659\pi$$
$$578$$ 0 0
$$579$$ 29.5616 1.22854
$$580$$ 0 0
$$581$$ −19.6847 −0.816657
$$582$$ 0 0
$$583$$ −17.1231 −0.709167
$$584$$ 0 0
$$585$$ −2.00000 −0.0826898
$$586$$ 0 0
$$587$$ −10.3002 −0.425134 −0.212567 0.977146i $$-0.568182\pi$$
−0.212567 + 0.977146i $$0.568182\pi$$
$$588$$ 0 0
$$589$$ −0.246211 −0.0101450
$$590$$ 0 0
$$591$$ 27.8078 1.14386
$$592$$ 0 0
$$593$$ −42.3542 −1.73928 −0.869638 0.493689i $$-0.835648\pi$$
−0.869638 + 0.493689i $$0.835648\pi$$
$$594$$ 0 0
$$595$$ 3.68466 0.151056
$$596$$ 0 0
$$597$$ 28.1080 1.15038
$$598$$ 0 0
$$599$$ −16.0000 −0.653742 −0.326871 0.945069i $$-0.605994\pi$$
−0.326871 + 0.945069i $$0.605994\pi$$
$$600$$ 0 0
$$601$$ 16.8617 0.687805 0.343902 0.939005i $$-0.388251\pi$$
0.343902 + 0.939005i $$0.388251\pi$$
$$602$$ 0 0
$$603$$ 4.17708 0.170104
$$604$$ 0 0
$$605$$ 7.00000 0.284590
$$606$$ 0 0
$$607$$ 36.0000 1.46119 0.730597 0.682808i $$-0.239242\pi$$
0.730597 + 0.682808i $$0.239242\pi$$
$$608$$ 0 0
$$609$$ 32.4924 1.31666
$$610$$ 0 0
$$611$$ 30.9309 1.25133
$$612$$ 0 0
$$613$$ −9.61553 −0.388368 −0.194184 0.980965i $$-0.562206\pi$$
−0.194184 + 0.980965i $$0.562206\pi$$
$$614$$ 0 0
$$615$$ −6.43845 −0.259623
$$616$$ 0 0
$$617$$ −7.30019 −0.293894 −0.146947 0.989144i $$-0.546945\pi$$
−0.146947 + 0.989144i $$0.546945\pi$$
$$618$$ 0 0
$$619$$ 44.4924 1.78830 0.894151 0.447766i $$-0.147780\pi$$
0.894151 + 0.447766i $$0.147780\pi$$
$$620$$ 0 0
$$621$$ 5.56155 0.223177
$$622$$ 0 0
$$623$$ −20.4924 −0.821012
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 6.24621 0.249450
$$628$$ 0 0
$$629$$ 0.807764 0.0322077
$$630$$ 0 0
$$631$$ 14.7386 0.586736 0.293368 0.956000i $$-0.405224\pi$$
0.293368 + 0.956000i $$0.405224\pi$$
$$632$$ 0 0
$$633$$ −15.5076 −0.616371
$$634$$ 0 0
$$635$$ 19.8078 0.786047
$$636$$ 0 0
$$637$$ 1.56155 0.0618710
$$638$$ 0 0
$$639$$ 2.80776 0.111073
$$640$$ 0 0
$$641$$ 2.87689 0.113630 0.0568152 0.998385i $$-0.481905\pi$$
0.0568152 + 0.998385i $$0.481905\pi$$
$$642$$ 0 0
$$643$$ −22.1771 −0.874579 −0.437289 0.899321i $$-0.644061\pi$$
−0.437289 + 0.899321i $$0.644061\pi$$
$$644$$ 0 0
$$645$$ −9.75379 −0.384055
$$646$$ 0 0
$$647$$ 11.1771 0.439416 0.219708 0.975566i $$-0.429489\pi$$
0.219708 + 0.975566i $$0.429489\pi$$
$$648$$ 0 0
$$649$$ 2.87689 0.112928
$$650$$ 0 0
$$651$$ −0.492423 −0.0192996
$$652$$ 0 0
$$653$$ 29.4233 1.15142 0.575711 0.817653i $$-0.304725\pi$$
0.575711 + 0.817653i $$0.304725\pi$$
$$654$$ 0 0
$$655$$ 18.9309 0.739690
$$656$$ 0 0
$$657$$ −4.24621 −0.165660
$$658$$ 0 0
$$659$$ −29.6155 −1.15366 −0.576829 0.816865i $$-0.695710\pi$$
−0.576829 + 0.816865i $$0.695710\pi$$
$$660$$ 0 0
$$661$$ 27.7538 1.07950 0.539749 0.841826i $$-0.318519\pi$$
0.539749 + 0.841826i $$0.318519\pi$$
$$662$$ 0 0
$$663$$ −8.00000 −0.310694
$$664$$ 0 0
$$665$$ −5.12311 −0.198666
$$666$$ 0 0
$$667$$ 8.12311 0.314528
$$668$$ 0 0
$$669$$ 29.4773 1.13966
$$670$$ 0 0
$$671$$ 1.75379 0.0677043
$$672$$ 0 0
$$673$$ −16.1922 −0.624165 −0.312082 0.950055i $$-0.601027\pi$$
−0.312082 + 0.950055i $$0.601027\pi$$
$$674$$ 0 0
$$675$$ −5.56155 −0.214064
$$676$$ 0 0
$$677$$ 5.68466 0.218479 0.109240 0.994015i $$-0.465158\pi$$
0.109240 + 0.994015i $$0.465158\pi$$
$$678$$ 0 0
$$679$$ −13.1231 −0.503619
$$680$$ 0 0
$$681$$ 28.4924 1.09183
$$682$$ 0 0
$$683$$ 6.93087 0.265202 0.132601 0.991169i $$-0.457667\pi$$
0.132601 + 0.991169i $$0.457667\pi$$
$$684$$ 0 0
$$685$$ −21.6155 −0.825887
$$686$$ 0 0
$$687$$ −8.98485 −0.342793
$$688$$ 0 0
$$689$$ −30.4924 −1.16167
$$690$$ 0 0
$$691$$ −0.492423 −0.0187326 −0.00936632 0.999956i $$-0.502981\pi$$
−0.00936632 + 0.999956i $$0.502981\pi$$
$$692$$ 0 0
$$693$$ −2.87689 −0.109284
$$694$$ 0 0
$$695$$ 10.3693 0.393331
$$696$$ 0 0
$$697$$ 5.93087 0.224648
$$698$$ 0 0
$$699$$ 44.1922 1.67150
$$700$$ 0 0
$$701$$ 9.75379 0.368396 0.184198 0.982889i $$-0.441031\pi$$
0.184198 + 0.982889i $$0.441031\pi$$
$$702$$ 0 0
$$703$$ −1.12311 −0.0423587
$$704$$ 0 0
$$705$$ 13.5616 0.510758
$$706$$ 0 0
$$707$$ −27.6847 −1.04119
$$708$$ 0 0
$$709$$ −16.0000 −0.600893 −0.300446 0.953799i $$-0.597136\pi$$
−0.300446 + 0.953799i $$0.597136\pi$$
$$710$$ 0 0
$$711$$ 0.492423 0.0184673
$$712$$ 0 0
$$713$$ −0.123106 −0.00461034
$$714$$ 0 0
$$715$$ −7.12311 −0.266389
$$716$$ 0 0
$$717$$ −12.3002 −0.459359
$$718$$ 0 0
$$719$$ −3.68466 −0.137415 −0.0687073 0.997637i $$-0.521887\pi$$
−0.0687073 + 0.997637i $$0.521887\pi$$
$$720$$ 0 0
$$721$$ 36.4924 1.35905
$$722$$ 0 0
$$723$$ −28.8769 −1.07394
$$724$$ 0 0
$$725$$ −8.12311 −0.301685
$$726$$ 0 0
$$727$$ 11.4384 0.424229 0.212114 0.977245i $$-0.431965\pi$$
0.212114 + 0.977245i $$0.431965\pi$$
$$728$$ 0 0
$$729$$ 29.9848 1.11055
$$730$$ 0 0
$$731$$ 8.98485 0.332316
$$732$$ 0 0
$$733$$ 40.1771 1.48397 0.741987 0.670414i $$-0.233884\pi$$
0.741987 + 0.670414i $$0.233884\pi$$
$$734$$ 0 0
$$735$$ 0.684658 0.0252540
$$736$$ 0 0
$$737$$ 14.8769 0.547997
$$738$$ 0 0
$$739$$ 1.49242 0.0548996 0.0274498 0.999623i $$-0.491261\pi$$
0.0274498 + 0.999623i $$0.491261\pi$$
$$740$$ 0 0
$$741$$ 11.1231 0.408617
$$742$$ 0 0
$$743$$ −46.2462 −1.69661 −0.848304 0.529509i $$-0.822376\pi$$
−0.848304 + 0.529509i $$0.822376\pi$$
$$744$$ 0 0
$$745$$ −0.876894 −0.0321269
$$746$$ 0 0
$$747$$ −4.31534 −0.157890
$$748$$ 0 0
$$749$$ 4.31534 0.157679
$$750$$ 0 0
$$751$$ 44.3542 1.61851 0.809253 0.587460i $$-0.199872\pi$$
0.809253 + 0.587460i $$0.199872\pi$$
$$752$$ 0 0
$$753$$ 14.2462 0.519161
$$754$$ 0 0
$$755$$ −1.56155 −0.0568307
$$756$$ 0 0
$$757$$ −25.5464 −0.928500 −0.464250 0.885704i $$-0.653676\pi$$
−0.464250 + 0.885704i $$0.653676\pi$$
$$758$$ 0 0
$$759$$ 3.12311 0.113362
$$760$$ 0 0
$$761$$ 21.9848 0.796950 0.398475 0.917179i $$-0.369540\pi$$
0.398475 + 0.917179i $$0.369540\pi$$
$$762$$ 0 0
$$763$$ 2.87689 0.104151
$$764$$ 0 0
$$765$$ 0.807764 0.0292048
$$766$$ 0 0
$$767$$ 5.12311 0.184985
$$768$$ 0 0
$$769$$ −7.61553 −0.274623 −0.137311 0.990528i $$-0.543846\pi$$
−0.137311 + 0.990528i $$0.543846\pi$$
$$770$$ 0 0
$$771$$ −1.06913 −0.0385038
$$772$$ 0 0
$$773$$ −29.1231 −1.04749 −0.523743 0.851877i $$-0.675465\pi$$
−0.523743 + 0.851877i $$0.675465\pi$$
$$774$$ 0 0
$$775$$ 0.123106 0.00442208
$$776$$ 0 0
$$777$$ −2.24621 −0.0805824
$$778$$ 0 0
$$779$$ −8.24621 −0.295451
$$780$$ 0 0
$$781$$ 10.0000 0.357828
$$782$$ 0 0
$$783$$ 45.1771 1.61450
$$784$$ 0 0
$$785$$ 15.6847 0.559809
$$786$$ 0 0
$$787$$ −33.3002 −1.18702 −0.593512 0.804825i $$-0.702259\pi$$
−0.593512 + 0.804825i $$0.702259\pi$$
$$788$$ 0 0
$$789$$ −22.3542 −0.795829
$$790$$ 0 0
$$791$$ −13.9309 −0.495325
$$792$$ 0 0
$$793$$ 3.12311 0.110905
$$794$$ 0 0
$$795$$ −13.3693 −0.474161
$$796$$ 0 0
$$797$$ 25.1922 0.892355 0.446177 0.894945i $$-0.352785\pi$$
0.446177 + 0.894945i $$0.352785\pi$$
$$798$$ 0 0
$$799$$ −12.4924 −0.441950
$$800$$ 0 0
$$801$$ −4.49242 −0.158732
$$802$$ 0 0
$$803$$ −15.1231 −0.533683
$$804$$ 0 0
$$805$$ −2.56155 −0.0902829
$$806$$ 0 0
$$807$$ −42.9309 −1.51124
$$808$$ 0 0
$$809$$ 50.8078 1.78631 0.893153 0.449753i $$-0.148488\pi$$
0.893153 + 0.449753i $$0.148488\pi$$
$$810$$ 0 0
$$811$$ 14.3693 0.504575 0.252287 0.967652i $$-0.418817\pi$$
0.252287 + 0.967652i $$0.418817\pi$$
$$812$$ 0 0
$$813$$ 37.7538 1.32408
$$814$$ 0 0
$$815$$ 6.68466 0.234153
$$816$$ 0 0
$$817$$ −12.4924 −0.437055
$$818$$ 0 0
$$819$$ −5.12311 −0.179016
$$820$$ 0 0
$$821$$ 38.4924 1.34339 0.671697 0.740826i $$-0.265566\pi$$
0.671697 + 0.740826i $$0.265566\pi$$
$$822$$ 0 0
$$823$$ −33.1771 −1.15648 −0.578240 0.815867i $$-0.696260\pi$$
−0.578240 + 0.815867i $$0.696260\pi$$
$$824$$ 0 0
$$825$$ −3.12311 −0.108733
$$826$$ 0 0
$$827$$ 28.3153 0.984621 0.492310 0.870420i $$-0.336152\pi$$
0.492310 + 0.870420i $$0.336152\pi$$
$$828$$ 0 0
$$829$$ −20.5616 −0.714132 −0.357066 0.934079i $$-0.616223\pi$$
−0.357066 + 0.934079i $$0.616223\pi$$
$$830$$ 0 0
$$831$$ −31.9157 −1.10714
$$832$$ 0 0
$$833$$ −0.630683 −0.0218519
$$834$$ 0 0
$$835$$ −8.00000 −0.276851
$$836$$ 0 0
$$837$$ −0.684658 −0.0236653
$$838$$ 0 0
$$839$$ −33.6155 −1.16054 −0.580268 0.814425i $$-0.697052\pi$$
−0.580268 + 0.814425i $$0.697052\pi$$
$$840$$ 0 0
$$841$$ 36.9848 1.27534
$$842$$ 0 0
$$843$$ −14.2462 −0.490666
$$844$$ 0 0
$$845$$ 0.315342 0.0108481
$$846$$ 0 0
$$847$$ 17.9309 0.616112
$$848$$ 0 0
$$849$$ 6.13826 0.210665
$$850$$ 0 0
$$851$$ −0.561553 −0.0192498
$$852$$ 0 0
$$853$$ −14.4924 −0.496211 −0.248106 0.968733i $$-0.579808\pi$$
−0.248106 + 0.968733i $$0.579808\pi$$
$$854$$ 0 0
$$855$$ −1.12311 −0.0384094
$$856$$ 0 0
$$857$$ −40.5464 −1.38504 −0.692519 0.721399i $$-0.743499\pi$$
−0.692519 + 0.721399i $$0.743499\pi$$
$$858$$ 0 0
$$859$$ −0.369317 −0.0126009 −0.00630046 0.999980i $$-0.502006\pi$$
−0.00630046 + 0.999980i $$0.502006\pi$$
$$860$$ 0 0
$$861$$ −16.4924 −0.562060
$$862$$ 0 0
$$863$$ 9.31534 0.317098 0.158549 0.987351i $$-0.449318\pi$$
0.158549 + 0.987351i $$0.449318\pi$$
$$864$$ 0 0
$$865$$ 7.12311 0.242193
$$866$$ 0 0
$$867$$ −23.3153 −0.791831
$$868$$ 0 0
$$869$$ 1.75379 0.0594932
$$870$$ 0 0
$$871$$ 26.4924 0.897661
$$872$$ 0 0
$$873$$ −2.87689 −0.0973681
$$874$$ 0 0
$$875$$ 2.56155 0.0865963
$$876$$ 0 0
$$877$$ −47.4773 −1.60319 −0.801597 0.597865i $$-0.796016\pi$$
−0.801597 + 0.597865i $$0.796016\pi$$
$$878$$ 0 0
$$879$$ 22.7386 0.766955
$$880$$ 0 0
$$881$$ 22.7386 0.766084 0.383042 0.923731i $$-0.374876\pi$$
0.383042 + 0.923731i $$0.374876\pi$$
$$882$$ 0 0
$$883$$ 44.9848 1.51386 0.756930 0.653496i $$-0.226698\pi$$
0.756930 + 0.653496i $$0.226698\pi$$
$$884$$ 0 0
$$885$$ 2.24621 0.0755056
$$886$$ 0 0
$$887$$ 43.1771 1.44974 0.724872 0.688883i $$-0.241899\pi$$
0.724872 + 0.688883i $$0.241899\pi$$
$$888$$ 0 0
$$889$$ 50.7386 1.70172
$$890$$ 0 0
$$891$$ 14.0000 0.469018
$$892$$ 0 0
$$893$$ 17.3693 0.581242
$$894$$ 0 0
$$895$$ 7.31534 0.244525
$$896$$ 0 0
$$897$$ 5.56155 0.185695
$$898$$ 0 0
$$899$$ −1.00000 −0.0333519
$$900$$ 0 0
$$901$$ 12.3153 0.410284
$$902$$ 0 0
$$903$$ −24.9848 −0.831444
$$904$$ 0 0
$$905$$ 8.87689 0.295078
$$906$$ 0 0
$$907$$ −24.4233 −0.810962 −0.405481 0.914103i $$-0.632896\pi$$
−0.405481 + 0.914103i $$0.632896\pi$$
$$908$$ 0 0
$$909$$ −6.06913 −0.201300
$$910$$ 0 0
$$911$$ 23.1231 0.766103 0.383051 0.923727i $$-0.374873\pi$$
0.383051 + 0.923727i $$0.374873\pi$$
$$912$$ 0 0
$$913$$ −15.3693 −0.508650
$$914$$ 0 0
$$915$$ 1.36932 0.0452682
$$916$$ 0 0
$$917$$ 48.4924 1.60136
$$918$$ 0 0
$$919$$ −20.4924 −0.675983 −0.337991 0.941149i $$-0.609747\pi$$
−0.337991 + 0.941149i $$0.609747\pi$$
$$920$$ 0 0
$$921$$ −24.0000 −0.790827
$$922$$ 0 0
$$923$$ 17.8078 0.586150
$$924$$ 0 0
$$925$$ 0.561553 0.0184637
$$926$$ 0 0
$$927$$ 8.00000 0.262754
$$928$$ 0 0
$$929$$ 0.261366 0.00857515 0.00428757 0.999991i $$-0.498635\pi$$
0.00428757 + 0.999991i $$0.498635\pi$$
$$930$$ 0 0
$$931$$ 0.876894 0.0287391
$$932$$ 0 0
$$933$$ −6.54640 −0.214319
$$934$$ 0 0
$$935$$ 2.87689 0.0940845
$$936$$ 0 0
$$937$$ 26.0000 0.849383 0.424691 0.905338i $$-0.360383\pi$$
0.424691 + 0.905338i $$0.360383\pi$$
$$938$$ 0 0
$$939$$ −21.7538 −0.709908
$$940$$ 0 0
$$941$$ 32.4924 1.05922 0.529611 0.848240i $$-0.322338\pi$$
0.529611 + 0.848240i $$0.322338\pi$$
$$942$$ 0 0
$$943$$ −4.12311 −0.134267
$$944$$ 0 0
$$945$$ −14.2462 −0.463429
$$946$$ 0 0
$$947$$ 17.5616 0.570674 0.285337 0.958427i $$-0.407895\pi$$
0.285337 + 0.958427i $$0.407895\pi$$
$$948$$ 0 0
$$949$$ −26.9309 −0.874213
$$950$$ 0 0
$$951$$ 39.6155 1.28462
$$952$$ 0 0
$$953$$ −44.2462 −1.43328 −0.716638 0.697446i $$-0.754320\pi$$
−0.716638 + 0.697446i $$0.754320\pi$$
$$954$$ 0 0
$$955$$ 3.12311 0.101061
$$956$$ 0 0
$$957$$ 25.3693 0.820074
$$958$$ 0 0
$$959$$ −55.3693 −1.78797
$$960$$ 0 0
$$961$$ −30.9848 −0.999511
$$962$$ 0 0
$$963$$ 0.946025 0.0304852
$$964$$ 0 0
$$965$$ −18.9309 −0.609406
$$966$$ 0 0
$$967$$ 9.17708 0.295115 0.147558 0.989053i $$-0.452859\pi$$
0.147558 + 0.989053i $$0.452859\pi$$
$$968$$ 0 0
$$969$$ −4.49242 −0.144317
$$970$$ 0 0
$$971$$ −40.1080 −1.28713 −0.643563 0.765393i $$-0.722544\pi$$
−0.643563 + 0.765393i $$0.722544\pi$$
$$972$$ 0 0
$$973$$ 26.5616 0.851524
$$974$$ 0 0
$$975$$ −5.56155 −0.178112
$$976$$ 0 0
$$977$$ 53.6847 1.71752 0.858762 0.512374i $$-0.171234\pi$$
0.858762 + 0.512374i $$0.171234\pi$$
$$978$$ 0 0
$$979$$ −16.0000 −0.511362
$$980$$ 0 0
$$981$$ 0.630683 0.0201362
$$982$$ 0 0
$$983$$ 62.0388 1.97873 0.989366 0.145450i $$-0.0464631\pi$$
0.989366 + 0.145450i $$0.0464631\pi$$
$$984$$ 0 0
$$985$$ −17.8078 −0.567403
$$986$$ 0 0
$$987$$ 34.7386 1.10574
$$988$$ 0 0
$$989$$ −6.24621 −0.198618
$$990$$ 0 0
$$991$$ 22.5616 0.716691 0.358346 0.933589i $$-0.383341\pi$$
0.358346 + 0.933589i $$0.383341\pi$$
$$992$$ 0 0
$$993$$ −5.45360 −0.173065
$$994$$ 0 0
$$995$$ −18.0000 −0.570638
$$996$$ 0 0
$$997$$ 28.1080 0.890188 0.445094 0.895484i $$-0.353170\pi$$
0.445094 + 0.895484i $$0.353170\pi$$
$$998$$ 0 0
$$999$$ −3.12311 −0.0988107
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 920.2.a.e.1.2 2
3.2 odd 2 8280.2.a.bf.1.1 2
4.3 odd 2 1840.2.a.o.1.1 2
5.2 odd 4 4600.2.e.n.4049.2 4
5.3 odd 4 4600.2.e.n.4049.3 4
5.4 even 2 4600.2.a.t.1.1 2
8.3 odd 2 7360.2.a.bl.1.2 2
8.5 even 2 7360.2.a.bp.1.1 2
20.19 odd 2 9200.2.a.bq.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.e.1.2 2 1.1 even 1 trivial
1840.2.a.o.1.1 2 4.3 odd 2
4600.2.a.t.1.1 2 5.4 even 2
4600.2.e.n.4049.2 4 5.2 odd 4
4600.2.e.n.4049.3 4 5.3 odd 4
7360.2.a.bl.1.2 2 8.3 odd 2
7360.2.a.bp.1.1 2 8.5 even 2
8280.2.a.bf.1.1 2 3.2 odd 2
9200.2.a.bq.1.2 2 20.19 odd 2