# Properties

 Label 4840.2.a.m.1.1 Level $4840$ Weight $2$ Character 4840.1 Self dual yes Analytic conductor $38.648$ 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] = [4840,2,Mod(1,4840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4840, 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("4840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4840 = 2^{3} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4840.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: $$38.6475945783$$ Analytic rank: $$0$$ 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: no (minimal twist has level 440) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 4840.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.56155 q^{3} -1.00000 q^{5} -0.438447 q^{7} -0.561553 q^{9} +O(q^{10})$$ $$q-1.56155 q^{3} -1.00000 q^{5} -0.438447 q^{7} -0.561553 q^{9} -7.12311 q^{13} +1.56155 q^{15} -4.68466 q^{17} -5.56155 q^{19} +0.684658 q^{21} -7.12311 q^{23} +1.00000 q^{25} +5.56155 q^{27} -4.43845 q^{29} -5.56155 q^{31} +0.438447 q^{35} +11.5616 q^{37} +11.1231 q^{39} -4.24621 q^{41} -5.12311 q^{43} +0.561553 q^{45} +13.3693 q^{47} -6.80776 q^{49} +7.31534 q^{51} -2.68466 q^{53} +8.68466 q^{57} -7.12311 q^{59} +8.43845 q^{61} +0.246211 q^{63} +7.12311 q^{65} +11.1231 q^{69} -8.68466 q^{71} +7.12311 q^{73} -1.56155 q^{75} -13.3693 q^{79} -7.00000 q^{81} +6.00000 q^{83} +4.68466 q^{85} +6.93087 q^{87} -2.68466 q^{89} +3.12311 q^{91} +8.68466 q^{93} +5.56155 q^{95} -13.1231 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - 2 q^{5} - 5 q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q + q^3 - 2 * q^5 - 5 * q^7 + 3 * q^9 $$2 q + q^{3} - 2 q^{5} - 5 q^{7} + 3 q^{9} - 6 q^{13} - q^{15} + 3 q^{17} - 7 q^{19} - 11 q^{21} - 6 q^{23} + 2 q^{25} + 7 q^{27} - 13 q^{29} - 7 q^{31} + 5 q^{35} + 19 q^{37} + 14 q^{39} + 8 q^{41} - 2 q^{43} - 3 q^{45} + 2 q^{47} + 7 q^{49} + 27 q^{51} + 7 q^{53} + 5 q^{57} - 6 q^{59} + 21 q^{61} - 16 q^{63} + 6 q^{65} + 14 q^{69} - 5 q^{71} + 6 q^{73} + q^{75} - 2 q^{79} - 14 q^{81} + 12 q^{83} - 3 q^{85} - 15 q^{87} + 7 q^{89} - 2 q^{91} + 5 q^{93} + 7 q^{95} - 18 q^{97}+O(q^{100})$$ 2 * q + q^3 - 2 * q^5 - 5 * q^7 + 3 * q^9 - 6 * q^13 - q^15 + 3 * q^17 - 7 * q^19 - 11 * q^21 - 6 * q^23 + 2 * q^25 + 7 * q^27 - 13 * q^29 - 7 * q^31 + 5 * q^35 + 19 * q^37 + 14 * q^39 + 8 * q^41 - 2 * q^43 - 3 * q^45 + 2 * q^47 + 7 * q^49 + 27 * q^51 + 7 * q^53 + 5 * q^57 - 6 * q^59 + 21 * q^61 - 16 * q^63 + 6 * q^65 + 14 * q^69 - 5 * q^71 + 6 * q^73 + q^75 - 2 * q^79 - 14 * q^81 + 12 * q^83 - 3 * q^85 - 15 * q^87 + 7 * q^89 - 2 * q^91 + 5 * q^93 + 7 * q^95 - 18 * 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
$$3$$ −1.56155 −0.901563 −0.450781 0.892634i $$-0.648855\pi$$
−0.450781 + 0.892634i $$0.648855\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −0.438447 −0.165717 −0.0828587 0.996561i $$-0.526405\pi$$
−0.0828587 + 0.996561i $$0.526405\pi$$
$$8$$ 0 0
$$9$$ −0.561553 −0.187184
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 0 0
$$13$$ −7.12311 −1.97559 −0.987797 0.155747i $$-0.950222\pi$$
−0.987797 + 0.155747i $$0.950222\pi$$
$$14$$ 0 0
$$15$$ 1.56155 0.403191
$$16$$ 0 0
$$17$$ −4.68466 −1.13620 −0.568098 0.822961i $$-0.692321\pi$$
−0.568098 + 0.822961i $$0.692321\pi$$
$$18$$ 0 0
$$19$$ −5.56155 −1.27591 −0.637954 0.770075i $$-0.720219\pi$$
−0.637954 + 0.770075i $$0.720219\pi$$
$$20$$ 0 0
$$21$$ 0.684658 0.149405
$$22$$ 0 0
$$23$$ −7.12311 −1.48527 −0.742635 0.669696i $$-0.766424\pi$$
−0.742635 + 0.669696i $$0.766424\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 5.56155 1.07032
$$28$$ 0 0
$$29$$ −4.43845 −0.824199 −0.412099 0.911139i $$-0.635204\pi$$
−0.412099 + 0.911139i $$0.635204\pi$$
$$30$$ 0 0
$$31$$ −5.56155 −0.998884 −0.499442 0.866347i $$-0.666462\pi$$
−0.499442 + 0.866347i $$0.666462\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0.438447 0.0741111
$$36$$ 0 0
$$37$$ 11.5616 1.90071 0.950354 0.311171i $$-0.100721\pi$$
0.950354 + 0.311171i $$0.100721\pi$$
$$38$$ 0 0
$$39$$ 11.1231 1.78112
$$40$$ 0 0
$$41$$ −4.24621 −0.663147 −0.331573 0.943429i $$-0.607579\pi$$
−0.331573 + 0.943429i $$0.607579\pi$$
$$42$$ 0 0
$$43$$ −5.12311 −0.781266 −0.390633 0.920546i $$-0.627744\pi$$
−0.390633 + 0.920546i $$0.627744\pi$$
$$44$$ 0 0
$$45$$ 0.561553 0.0837114
$$46$$ 0 0
$$47$$ 13.3693 1.95012 0.975058 0.221952i $$-0.0712428\pi$$
0.975058 + 0.221952i $$0.0712428\pi$$
$$48$$ 0 0
$$49$$ −6.80776 −0.972538
$$50$$ 0 0
$$51$$ 7.31534 1.02435
$$52$$ 0 0
$$53$$ −2.68466 −0.368766 −0.184383 0.982854i $$-0.559029\pi$$
−0.184383 + 0.982854i $$0.559029\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 8.68466 1.15031
$$58$$ 0 0
$$59$$ −7.12311 −0.927349 −0.463675 0.886006i $$-0.653469\pi$$
−0.463675 + 0.886006i $$0.653469\pi$$
$$60$$ 0 0
$$61$$ 8.43845 1.08043 0.540216 0.841526i $$-0.318342\pi$$
0.540216 + 0.841526i $$0.318342\pi$$
$$62$$ 0 0
$$63$$ 0.246211 0.0310197
$$64$$ 0 0
$$65$$ 7.12311 0.883513
$$66$$ 0 0
$$67$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$68$$ 0 0
$$69$$ 11.1231 1.33906
$$70$$ 0 0
$$71$$ −8.68466 −1.03068 −0.515340 0.856986i $$-0.672334\pi$$
−0.515340 + 0.856986i $$0.672334\pi$$
$$72$$ 0 0
$$73$$ 7.12311 0.833696 0.416848 0.908976i $$-0.363135\pi$$
0.416848 + 0.908976i $$0.363135\pi$$
$$74$$ 0 0
$$75$$ −1.56155 −0.180313
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −13.3693 −1.50417 −0.752083 0.659069i $$-0.770951\pi$$
−0.752083 + 0.659069i $$0.770951\pi$$
$$80$$ 0 0
$$81$$ −7.00000 −0.777778
$$82$$ 0 0
$$83$$ 6.00000 0.658586 0.329293 0.944228i $$-0.393190\pi$$
0.329293 + 0.944228i $$0.393190\pi$$
$$84$$ 0 0
$$85$$ 4.68466 0.508123
$$86$$ 0 0
$$87$$ 6.93087 0.743067
$$88$$ 0 0
$$89$$ −2.68466 −0.284573 −0.142287 0.989825i $$-0.545445\pi$$
−0.142287 + 0.989825i $$0.545445\pi$$
$$90$$ 0 0
$$91$$ 3.12311 0.327390
$$92$$ 0 0
$$93$$ 8.68466 0.900557
$$94$$ 0 0
$$95$$ 5.56155 0.570603
$$96$$ 0 0
$$97$$ −13.1231 −1.33245 −0.666225 0.745751i $$-0.732091\pi$$
−0.666225 + 0.745751i $$0.732091\pi$$
$$98$$ 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
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ 0 0
$$105$$ −0.684658 −0.0668158
$$106$$ 0 0
$$107$$ 1.12311 0.108575 0.0542874 0.998525i $$-0.482711\pi$$
0.0542874 + 0.998525i $$0.482711\pi$$
$$108$$ 0 0
$$109$$ −12.2462 −1.17297 −0.586487 0.809959i $$-0.699490\pi$$
−0.586487 + 0.809959i $$0.699490\pi$$
$$110$$ 0 0
$$111$$ −18.0540 −1.71361
$$112$$ 0 0
$$113$$ 9.12311 0.858230 0.429115 0.903250i $$-0.358826\pi$$
0.429115 + 0.903250i $$0.358826\pi$$
$$114$$ 0 0
$$115$$ 7.12311 0.664233
$$116$$ 0 0
$$117$$ 4.00000 0.369800
$$118$$ 0 0
$$119$$ 2.05398 0.188288
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0 0
$$123$$ 6.63068 0.597869
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −13.1231 −1.16449 −0.582244 0.813014i $$-0.697825\pi$$
−0.582244 + 0.813014i $$0.697825\pi$$
$$128$$ 0 0
$$129$$ 8.00000 0.704361
$$130$$ 0 0
$$131$$ −3.80776 −0.332686 −0.166343 0.986068i $$-0.553196\pi$$
−0.166343 + 0.986068i $$0.553196\pi$$
$$132$$ 0 0
$$133$$ 2.43845 0.211440
$$134$$ 0 0
$$135$$ −5.56155 −0.478662
$$136$$ 0 0
$$137$$ 1.12311 0.0959534 0.0479767 0.998848i $$-0.484723\pi$$
0.0479767 + 0.998848i $$0.484723\pi$$
$$138$$ 0 0
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ −20.8769 −1.75815
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 4.43845 0.368593
$$146$$ 0 0
$$147$$ 10.6307 0.876804
$$148$$ 0 0
$$149$$ 10.6847 0.875321 0.437661 0.899140i $$-0.355807\pi$$
0.437661 + 0.899140i $$0.355807\pi$$
$$150$$ 0 0
$$151$$ −15.1231 −1.23070 −0.615350 0.788254i $$-0.710985\pi$$
−0.615350 + 0.788254i $$0.710985\pi$$
$$152$$ 0 0
$$153$$ 2.63068 0.212678
$$154$$ 0 0
$$155$$ 5.56155 0.446715
$$156$$ 0 0
$$157$$ 9.31534 0.743445 0.371723 0.928344i $$-0.378767\pi$$
0.371723 + 0.928344i $$0.378767\pi$$
$$158$$ 0 0
$$159$$ 4.19224 0.332466
$$160$$ 0 0
$$161$$ 3.12311 0.246135
$$162$$ 0 0
$$163$$ 0.192236 0.0150571 0.00752854 0.999972i $$-0.497604\pi$$
0.00752854 + 0.999972i $$0.497604\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −10.6847 −0.826804 −0.413402 0.910549i $$-0.635660\pi$$
−0.413402 + 0.910549i $$0.635660\pi$$
$$168$$ 0 0
$$169$$ 37.7386 2.90297
$$170$$ 0 0
$$171$$ 3.12311 0.238830
$$172$$ 0 0
$$173$$ −19.1231 −1.45390 −0.726951 0.686689i $$-0.759063\pi$$
−0.726951 + 0.686689i $$0.759063\pi$$
$$174$$ 0 0
$$175$$ −0.438447 −0.0331435
$$176$$ 0 0
$$177$$ 11.1231 0.836064
$$178$$ 0 0
$$179$$ −2.24621 −0.167890 −0.0839449 0.996470i $$-0.526752\pi$$
−0.0839449 + 0.996470i $$0.526752\pi$$
$$180$$ 0 0
$$181$$ 8.24621 0.612936 0.306468 0.951881i $$-0.400853\pi$$
0.306468 + 0.951881i $$0.400853\pi$$
$$182$$ 0 0
$$183$$ −13.1771 −0.974078
$$184$$ 0 0
$$185$$ −11.5616 −0.850022
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −2.43845 −0.177371
$$190$$ 0 0
$$191$$ −6.24621 −0.451960 −0.225980 0.974132i $$-0.572558\pi$$
−0.225980 + 0.974132i $$0.572558\pi$$
$$192$$ 0 0
$$193$$ 25.5616 1.83996 0.919980 0.391964i $$-0.128204\pi$$
0.919980 + 0.391964i $$0.128204\pi$$
$$194$$ 0 0
$$195$$ −11.1231 −0.796542
$$196$$ 0 0
$$197$$ 26.7386 1.90505 0.952524 0.304462i $$-0.0984768\pi$$
0.952524 + 0.304462i $$0.0984768\pi$$
$$198$$ 0 0
$$199$$ −16.6847 −1.18274 −0.591372 0.806399i $$-0.701414\pi$$
−0.591372 + 0.806399i $$0.701414\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 1.94602 0.136584
$$204$$ 0 0
$$205$$ 4.24621 0.296568
$$206$$ 0 0
$$207$$ 4.00000 0.278019
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −7.31534 −0.503609 −0.251804 0.967778i $$-0.581024\pi$$
−0.251804 + 0.967778i $$0.581024\pi$$
$$212$$ 0 0
$$213$$ 13.5616 0.929222
$$214$$ 0 0
$$215$$ 5.12311 0.349393
$$216$$ 0 0
$$217$$ 2.43845 0.165533
$$218$$ 0 0
$$219$$ −11.1231 −0.751630
$$220$$ 0 0
$$221$$ 33.3693 2.24466
$$222$$ 0 0
$$223$$ 16.8769 1.13016 0.565080 0.825036i $$-0.308845\pi$$
0.565080 + 0.825036i $$0.308845\pi$$
$$224$$ 0 0
$$225$$ −0.561553 −0.0374369
$$226$$ 0 0
$$227$$ 8.24621 0.547320 0.273660 0.961826i $$-0.411766\pi$$
0.273660 + 0.961826i $$0.411766\pi$$
$$228$$ 0 0
$$229$$ −0.246211 −0.0162701 −0.00813505 0.999967i $$-0.502589\pi$$
−0.00813505 + 0.999967i $$0.502589\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 10.0540 0.658658 0.329329 0.944215i $$-0.393178\pi$$
0.329329 + 0.944215i $$0.393178\pi$$
$$234$$ 0 0
$$235$$ −13.3693 −0.872118
$$236$$ 0 0
$$237$$ 20.8769 1.35610
$$238$$ 0 0
$$239$$ −19.6155 −1.26882 −0.634412 0.772995i $$-0.718758\pi$$
−0.634412 + 0.772995i $$0.718758\pi$$
$$240$$ 0 0
$$241$$ 14.4924 0.933539 0.466769 0.884379i $$-0.345418\pi$$
0.466769 + 0.884379i $$0.345418\pi$$
$$242$$ 0 0
$$243$$ −5.75379 −0.369106
$$244$$ 0 0
$$245$$ 6.80776 0.434932
$$246$$ 0 0
$$247$$ 39.6155 2.52068
$$248$$ 0 0
$$249$$ −9.36932 −0.593756
$$250$$ 0 0
$$251$$ −2.63068 −0.166047 −0.0830236 0.996548i $$-0.526458\pi$$
−0.0830236 + 0.996548i $$0.526458\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ −7.31534 −0.458104
$$256$$ 0 0
$$257$$ −26.4924 −1.65255 −0.826276 0.563266i $$-0.809545\pi$$
−0.826276 + 0.563266i $$0.809545\pi$$
$$258$$ 0 0
$$259$$ −5.06913 −0.314980
$$260$$ 0 0
$$261$$ 2.49242 0.154277
$$262$$ 0 0
$$263$$ −8.05398 −0.496629 −0.248315 0.968679i $$-0.579877\pi$$
−0.248315 + 0.968679i $$0.579877\pi$$
$$264$$ 0 0
$$265$$ 2.68466 0.164917
$$266$$ 0 0
$$267$$ 4.19224 0.256561
$$268$$ 0 0
$$269$$ 9.12311 0.556246 0.278123 0.960546i $$-0.410288\pi$$
0.278123 + 0.960546i $$0.410288\pi$$
$$270$$ 0 0
$$271$$ 4.00000 0.242983 0.121491 0.992592i $$-0.461232\pi$$
0.121491 + 0.992592i $$0.461232\pi$$
$$272$$ 0 0
$$273$$ −4.87689 −0.295163
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 5.75379 0.345712 0.172856 0.984947i $$-0.444701\pi$$
0.172856 + 0.984947i $$0.444701\pi$$
$$278$$ 0 0
$$279$$ 3.12311 0.186975
$$280$$ 0 0
$$281$$ 8.24621 0.491928 0.245964 0.969279i $$-0.420896\pi$$
0.245964 + 0.969279i $$0.420896\pi$$
$$282$$ 0 0
$$283$$ 6.00000 0.356663 0.178331 0.983970i $$-0.442930\pi$$
0.178331 + 0.983970i $$0.442930\pi$$
$$284$$ 0 0
$$285$$ −8.68466 −0.514435
$$286$$ 0 0
$$287$$ 1.86174 0.109895
$$288$$ 0 0
$$289$$ 4.94602 0.290943
$$290$$ 0 0
$$291$$ 20.4924 1.20129
$$292$$ 0 0
$$293$$ −8.87689 −0.518594 −0.259297 0.965798i $$-0.583491\pi$$
−0.259297 + 0.965798i $$0.583491\pi$$
$$294$$ 0 0
$$295$$ 7.12311 0.414723
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 50.7386 2.93429
$$300$$ 0 0
$$301$$ 2.24621 0.129469
$$302$$ 0 0
$$303$$ −3.12311 −0.179418
$$304$$ 0 0
$$305$$ −8.43845 −0.483184
$$306$$ 0 0
$$307$$ −18.0000 −1.02731 −0.513657 0.857996i $$-0.671710\pi$$
−0.513657 + 0.857996i $$0.671710\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −13.5616 −0.769005 −0.384503 0.923124i $$-0.625627\pi$$
−0.384503 + 0.923124i $$0.625627\pi$$
$$312$$ 0 0
$$313$$ −24.7386 −1.39831 −0.699155 0.714970i $$-0.746440\pi$$
−0.699155 + 0.714970i $$0.746440\pi$$
$$314$$ 0 0
$$315$$ −0.246211 −0.0138724
$$316$$ 0 0
$$317$$ 18.6847 1.04943 0.524717 0.851276i $$-0.324171\pi$$
0.524717 + 0.851276i $$0.324171\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −1.75379 −0.0978869
$$322$$ 0 0
$$323$$ 26.0540 1.44968
$$324$$ 0 0
$$325$$ −7.12311 −0.395119
$$326$$ 0 0
$$327$$ 19.1231 1.05751
$$328$$ 0 0
$$329$$ −5.86174 −0.323168
$$330$$ 0 0
$$331$$ 30.7386 1.68955 0.844774 0.535123i $$-0.179735\pi$$
0.844774 + 0.535123i $$0.179735\pi$$
$$332$$ 0 0
$$333$$ −6.49242 −0.355783
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −15.8078 −0.861104 −0.430552 0.902566i $$-0.641681\pi$$
−0.430552 + 0.902566i $$0.641681\pi$$
$$338$$ 0 0
$$339$$ −14.2462 −0.773748
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 6.05398 0.326884
$$344$$ 0 0
$$345$$ −11.1231 −0.598848
$$346$$ 0 0
$$347$$ 14.0000 0.751559 0.375780 0.926709i $$-0.377375\pi$$
0.375780 + 0.926709i $$0.377375\pi$$
$$348$$ 0 0
$$349$$ 18.4924 0.989877 0.494938 0.868928i $$-0.335191\pi$$
0.494938 + 0.868928i $$0.335191\pi$$
$$350$$ 0 0
$$351$$ −39.6155 −2.11452
$$352$$ 0 0
$$353$$ 10.0000 0.532246 0.266123 0.963939i $$-0.414257\pi$$
0.266123 + 0.963939i $$0.414257\pi$$
$$354$$ 0 0
$$355$$ 8.68466 0.460934
$$356$$ 0 0
$$357$$ −3.20739 −0.169753
$$358$$ 0 0
$$359$$ 30.7386 1.62232 0.811162 0.584822i $$-0.198836\pi$$
0.811162 + 0.584822i $$0.198836\pi$$
$$360$$ 0 0
$$361$$ 11.9309 0.627941
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −7.12311 −0.372840
$$366$$ 0 0
$$367$$ −16.4924 −0.860897 −0.430449 0.902615i $$-0.641645\pi$$
−0.430449 + 0.902615i $$0.641645\pi$$
$$368$$ 0 0
$$369$$ 2.38447 0.124131
$$370$$ 0 0
$$371$$ 1.17708 0.0611110
$$372$$ 0 0
$$373$$ 1.36932 0.0709005 0.0354503 0.999371i $$-0.488713\pi$$
0.0354503 + 0.999371i $$0.488713\pi$$
$$374$$ 0 0
$$375$$ 1.56155 0.0806382
$$376$$ 0 0
$$377$$ 31.6155 1.62828
$$378$$ 0 0
$$379$$ −23.1231 −1.18775 −0.593877 0.804556i $$-0.702403\pi$$
−0.593877 + 0.804556i $$0.702403\pi$$
$$380$$ 0 0
$$381$$ 20.4924 1.04986
$$382$$ 0 0
$$383$$ −25.3693 −1.29631 −0.648156 0.761508i $$-0.724459\pi$$
−0.648156 + 0.761508i $$0.724459\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 2.87689 0.146241
$$388$$ 0 0
$$389$$ 18.4924 0.937603 0.468802 0.883304i $$-0.344686\pi$$
0.468802 + 0.883304i $$0.344686\pi$$
$$390$$ 0 0
$$391$$ 33.3693 1.68756
$$392$$ 0 0
$$393$$ 5.94602 0.299937
$$394$$ 0 0
$$395$$ 13.3693 0.672683
$$396$$ 0 0
$$397$$ −14.0000 −0.702640 −0.351320 0.936255i $$-0.614267\pi$$
−0.351320 + 0.936255i $$0.614267\pi$$
$$398$$ 0 0
$$399$$ −3.80776 −0.190627
$$400$$ 0 0
$$401$$ 10.1922 0.508976 0.254488 0.967076i $$-0.418093\pi$$
0.254488 + 0.967076i $$0.418093\pi$$
$$402$$ 0 0
$$403$$ 39.6155 1.97339
$$404$$ 0 0
$$405$$ 7.00000 0.347833
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 7.36932 0.364389 0.182195 0.983262i $$-0.441680\pi$$
0.182195 + 0.983262i $$0.441680\pi$$
$$410$$ 0 0
$$411$$ −1.75379 −0.0865080
$$412$$ 0 0
$$413$$ 3.12311 0.153678
$$414$$ 0 0
$$415$$ −6.00000 −0.294528
$$416$$ 0 0
$$417$$ −6.24621 −0.305878
$$418$$ 0 0
$$419$$ −32.4924 −1.58736 −0.793679 0.608336i $$-0.791837\pi$$
−0.793679 + 0.608336i $$0.791837\pi$$
$$420$$ 0 0
$$421$$ −13.6155 −0.663580 −0.331790 0.943353i $$-0.607653\pi$$
−0.331790 + 0.943353i $$0.607653\pi$$
$$422$$ 0 0
$$423$$ −7.50758 −0.365031
$$424$$ 0 0
$$425$$ −4.68466 −0.227239
$$426$$ 0 0
$$427$$ −3.69981 −0.179047
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −8.00000 −0.385346 −0.192673 0.981263i $$-0.561716\pi$$
−0.192673 + 0.981263i $$0.561716\pi$$
$$432$$ 0 0
$$433$$ −13.1231 −0.630656 −0.315328 0.948983i $$-0.602115\pi$$
−0.315328 + 0.948983i $$0.602115\pi$$
$$434$$ 0 0
$$435$$ −6.93087 −0.332310
$$436$$ 0 0
$$437$$ 39.6155 1.89507
$$438$$ 0 0
$$439$$ 6.24621 0.298115 0.149058 0.988828i $$-0.452376\pi$$
0.149058 + 0.988828i $$0.452376\pi$$
$$440$$ 0 0
$$441$$ 3.82292 0.182044
$$442$$ 0 0
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ 0 0
$$445$$ 2.68466 0.127265
$$446$$ 0 0
$$447$$ −16.6847 −0.789157
$$448$$ 0 0
$$449$$ −16.2462 −0.766706 −0.383353 0.923602i $$-0.625231\pi$$
−0.383353 + 0.923602i $$0.625231\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 23.6155 1.10955
$$454$$ 0 0
$$455$$ −3.12311 −0.146413
$$456$$ 0 0
$$457$$ 8.68466 0.406251 0.203126 0.979153i $$-0.434890\pi$$
0.203126 + 0.979153i $$0.434890\pi$$
$$458$$ 0 0
$$459$$ −26.0540 −1.21610
$$460$$ 0 0
$$461$$ −16.0540 −0.747708 −0.373854 0.927488i $$-0.621964\pi$$
−0.373854 + 0.927488i $$0.621964\pi$$
$$462$$ 0 0
$$463$$ −2.63068 −0.122258 −0.0611291 0.998130i $$-0.519470\pi$$
−0.0611291 + 0.998130i $$0.519470\pi$$
$$464$$ 0 0
$$465$$ −8.68466 −0.402741
$$466$$ 0 0
$$467$$ 12.1922 0.564189 0.282095 0.959387i $$-0.408971\pi$$
0.282095 + 0.959387i $$0.408971\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −14.5464 −0.670263
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −5.56155 −0.255182
$$476$$ 0 0
$$477$$ 1.50758 0.0690272
$$478$$ 0 0
$$479$$ 2.24621 0.102632 0.0513160 0.998682i $$-0.483658\pi$$
0.0513160 + 0.998682i $$0.483658\pi$$
$$480$$ 0 0
$$481$$ −82.3542 −3.75503
$$482$$ 0 0
$$483$$ −4.87689 −0.221906
$$484$$ 0 0
$$485$$ 13.1231 0.595890
$$486$$ 0 0
$$487$$ −13.7538 −0.623244 −0.311622 0.950206i $$-0.600872\pi$$
−0.311622 + 0.950206i $$0.600872\pi$$
$$488$$ 0 0
$$489$$ −0.300187 −0.0135749
$$490$$ 0 0
$$491$$ 28.6847 1.29452 0.647260 0.762269i $$-0.275915\pi$$
0.647260 + 0.762269i $$0.275915\pi$$
$$492$$ 0 0
$$493$$ 20.7926 0.936452
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 3.80776 0.170802
$$498$$ 0 0
$$499$$ 30.7386 1.37605 0.688025 0.725687i $$-0.258478\pi$$
0.688025 + 0.725687i $$0.258478\pi$$
$$500$$ 0 0
$$501$$ 16.6847 0.745416
$$502$$ 0 0
$$503$$ −5.12311 −0.228428 −0.114214 0.993456i $$-0.536435\pi$$
−0.114214 + 0.993456i $$0.536435\pi$$
$$504$$ 0 0
$$505$$ −2.00000 −0.0889988
$$506$$ 0 0
$$507$$ −58.9309 −2.61721
$$508$$ 0 0
$$509$$ −41.6155 −1.84458 −0.922288 0.386504i $$-0.873683\pi$$
−0.922288 + 0.386504i $$0.873683\pi$$
$$510$$ 0 0
$$511$$ −3.12311 −0.138158
$$512$$ 0 0
$$513$$ −30.9309 −1.36563
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 29.8617 1.31078
$$520$$ 0 0
$$521$$ 2.49242 0.109195 0.0545975 0.998508i $$-0.482612\pi$$
0.0545975 + 0.998508i $$0.482612\pi$$
$$522$$ 0 0
$$523$$ −33.1231 −1.44837 −0.724186 0.689605i $$-0.757784\pi$$
−0.724186 + 0.689605i $$0.757784\pi$$
$$524$$ 0 0
$$525$$ 0.684658 0.0298809
$$526$$ 0 0
$$527$$ 26.0540 1.13493
$$528$$ 0 0
$$529$$ 27.7386 1.20603
$$530$$ 0 0
$$531$$ 4.00000 0.173585
$$532$$ 0 0
$$533$$ 30.2462 1.31011
$$534$$ 0 0
$$535$$ −1.12311 −0.0485561
$$536$$ 0 0
$$537$$ 3.50758 0.151363
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −20.9309 −0.899888 −0.449944 0.893057i $$-0.648556\pi$$
−0.449944 + 0.893057i $$0.648556\pi$$
$$542$$ 0 0
$$543$$ −12.8769 −0.552600
$$544$$ 0 0
$$545$$ 12.2462 0.524570
$$546$$ 0 0
$$547$$ 35.3693 1.51228 0.756141 0.654408i $$-0.227082\pi$$
0.756141 + 0.654408i $$0.227082\pi$$
$$548$$ 0 0
$$549$$ −4.73863 −0.202240
$$550$$ 0 0
$$551$$ 24.6847 1.05160
$$552$$ 0 0
$$553$$ 5.86174 0.249267
$$554$$ 0 0
$$555$$ 18.0540 0.766349
$$556$$ 0 0
$$557$$ −41.3693 −1.75287 −0.876437 0.481516i $$-0.840086\pi$$
−0.876437 + 0.481516i $$0.840086\pi$$
$$558$$ 0 0
$$559$$ 36.4924 1.54347
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −22.9848 −0.968696 −0.484348 0.874876i $$-0.660943\pi$$
−0.484348 + 0.874876i $$0.660943\pi$$
$$564$$ 0 0
$$565$$ −9.12311 −0.383812
$$566$$ 0 0
$$567$$ 3.06913 0.128891
$$568$$ 0 0
$$569$$ 37.6155 1.57692 0.788462 0.615083i $$-0.210877\pi$$
0.788462 + 0.615083i $$0.210877\pi$$
$$570$$ 0 0
$$571$$ 28.3002 1.18433 0.592163 0.805818i $$-0.298274\pi$$
0.592163 + 0.805818i $$0.298274\pi$$
$$572$$ 0 0
$$573$$ 9.75379 0.407470
$$574$$ 0 0
$$575$$ −7.12311 −0.297054
$$576$$ 0 0
$$577$$ −32.2462 −1.34243 −0.671214 0.741264i $$-0.734227\pi$$
−0.671214 + 0.741264i $$0.734227\pi$$
$$578$$ 0 0
$$579$$ −39.9157 −1.65884
$$580$$ 0 0
$$581$$ −2.63068 −0.109139
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ −4.00000 −0.165380
$$586$$ 0 0
$$587$$ 34.0540 1.40556 0.702779 0.711408i $$-0.251942\pi$$
0.702779 + 0.711408i $$0.251942\pi$$
$$588$$ 0 0
$$589$$ 30.9309 1.27448
$$590$$ 0 0
$$591$$ −41.7538 −1.71752
$$592$$ 0 0
$$593$$ 19.6155 0.805513 0.402757 0.915307i $$-0.368052\pi$$
0.402757 + 0.915307i $$0.368052\pi$$
$$594$$ 0 0
$$595$$ −2.05398 −0.0842048
$$596$$ 0 0
$$597$$ 26.0540 1.06632
$$598$$ 0 0
$$599$$ −1.06913 −0.0436835 −0.0218417 0.999761i $$-0.506953\pi$$
−0.0218417 + 0.999761i $$0.506953\pi$$
$$600$$ 0 0
$$601$$ −38.9848 −1.59022 −0.795112 0.606462i $$-0.792588\pi$$
−0.795112 + 0.606462i $$0.792588\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −5.80776 −0.235730 −0.117865 0.993030i $$-0.537605\pi$$
−0.117865 + 0.993030i $$0.537605\pi$$
$$608$$ 0 0
$$609$$ −3.03882 −0.123139
$$610$$ 0 0
$$611$$ −95.2311 −3.85264
$$612$$ 0 0
$$613$$ 11.1231 0.449258 0.224629 0.974444i $$-0.427883\pi$$
0.224629 + 0.974444i $$0.427883\pi$$
$$614$$ 0 0
$$615$$ −6.63068 −0.267375
$$616$$ 0 0
$$617$$ 13.6155 0.548141 0.274070 0.961710i $$-0.411630\pi$$
0.274070 + 0.961710i $$0.411630\pi$$
$$618$$ 0 0
$$619$$ 14.7386 0.592396 0.296198 0.955127i $$-0.404281\pi$$
0.296198 + 0.955127i $$0.404281\pi$$
$$620$$ 0 0
$$621$$ −39.6155 −1.58972
$$622$$ 0 0
$$623$$ 1.17708 0.0471588
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −54.1619 −2.15958
$$630$$ 0 0
$$631$$ −12.1922 −0.485365 −0.242683 0.970106i $$-0.578027\pi$$
−0.242683 + 0.970106i $$0.578027\pi$$
$$632$$ 0 0
$$633$$ 11.4233 0.454035
$$634$$ 0 0
$$635$$ 13.1231 0.520775
$$636$$ 0 0
$$637$$ 48.4924 1.92134
$$638$$ 0 0
$$639$$ 4.87689 0.192927
$$640$$ 0 0
$$641$$ 16.4384 0.649280 0.324640 0.945838i $$-0.394757\pi$$
0.324640 + 0.945838i $$0.394757\pi$$
$$642$$ 0 0
$$643$$ −28.3002 −1.11605 −0.558025 0.829824i $$-0.688441\pi$$
−0.558025 + 0.829824i $$0.688441\pi$$
$$644$$ 0 0
$$645$$ −8.00000 −0.315000
$$646$$ 0 0
$$647$$ 36.8769 1.44978 0.724890 0.688864i $$-0.241890\pi$$
0.724890 + 0.688864i $$0.241890\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −3.80776 −0.149238
$$652$$ 0 0
$$653$$ 24.4384 0.956350 0.478175 0.878264i $$-0.341298\pi$$
0.478175 + 0.878264i $$0.341298\pi$$
$$654$$ 0 0
$$655$$ 3.80776 0.148782
$$656$$ 0 0
$$657$$ −4.00000 −0.156055
$$658$$ 0 0
$$659$$ 5.06913 0.197465 0.0987326 0.995114i $$-0.468521\pi$$
0.0987326 + 0.995114i $$0.468521\pi$$
$$660$$ 0 0
$$661$$ −30.4924 −1.18602 −0.593009 0.805196i $$-0.702060\pi$$
−0.593009 + 0.805196i $$0.702060\pi$$
$$662$$ 0 0
$$663$$ −52.1080 −2.02371
$$664$$ 0 0
$$665$$ −2.43845 −0.0945589
$$666$$ 0 0
$$667$$ 31.6155 1.22416
$$668$$ 0 0
$$669$$ −26.3542 −1.01891
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 13.0691 0.503778 0.251889 0.967756i $$-0.418948\pi$$
0.251889 + 0.967756i $$0.418948\pi$$
$$674$$ 0 0
$$675$$ 5.56155 0.214064
$$676$$ 0 0
$$677$$ −29.7538 −1.14353 −0.571765 0.820417i $$-0.693741\pi$$
−0.571765 + 0.820417i $$0.693741\pi$$
$$678$$ 0 0
$$679$$ 5.75379 0.220810
$$680$$ 0 0
$$681$$ −12.8769 −0.493444
$$682$$ 0 0
$$683$$ −5.17708 −0.198095 −0.0990477 0.995083i $$-0.531580\pi$$
−0.0990477 + 0.995083i $$0.531580\pi$$
$$684$$ 0 0
$$685$$ −1.12311 −0.0429117
$$686$$ 0 0
$$687$$ 0.384472 0.0146685
$$688$$ 0 0
$$689$$ 19.1231 0.728532
$$690$$ 0 0
$$691$$ −35.6155 −1.35488 −0.677439 0.735579i $$-0.736910\pi$$
−0.677439 + 0.735579i $$0.736910\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −4.00000 −0.151729
$$696$$ 0 0
$$697$$ 19.8920 0.753465
$$698$$ 0 0
$$699$$ −15.6998 −0.593821
$$700$$ 0 0
$$701$$ 16.4384 0.620872 0.310436 0.950594i $$-0.399525\pi$$
0.310436 + 0.950594i $$0.399525\pi$$
$$702$$ 0 0
$$703$$ −64.3002 −2.42513
$$704$$ 0 0
$$705$$ 20.8769 0.786269
$$706$$ 0 0
$$707$$ −0.876894 −0.0329790
$$708$$ 0 0
$$709$$ −15.7538 −0.591646 −0.295823 0.955243i $$-0.595594\pi$$
−0.295823 + 0.955243i $$0.595594\pi$$
$$710$$ 0 0
$$711$$ 7.50758 0.281556
$$712$$ 0 0
$$713$$ 39.6155 1.48361
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 30.6307 1.14392
$$718$$ 0 0
$$719$$ −2.82292 −0.105277 −0.0526386 0.998614i $$-0.516763\pi$$
−0.0526386 + 0.998614i $$0.516763\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −22.6307 −0.841644
$$724$$ 0 0
$$725$$ −4.43845 −0.164840
$$726$$ 0 0
$$727$$ −19.1231 −0.709237 −0.354618 0.935011i $$-0.615389\pi$$
−0.354618 + 0.935011i $$0.615389\pi$$
$$728$$ 0 0
$$729$$ 29.9848 1.11055
$$730$$ 0 0
$$731$$ 24.0000 0.887672
$$732$$ 0 0
$$733$$ 30.2462 1.11717 0.558585 0.829448i $$-0.311345\pi$$
0.558585 + 0.829448i $$0.311345\pi$$
$$734$$ 0 0
$$735$$ −10.6307 −0.392119
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −5.75379 −0.211657 −0.105828 0.994384i $$-0.533749\pi$$
−0.105828 + 0.994384i $$0.533749\pi$$
$$740$$ 0 0
$$741$$ −61.8617 −2.27255
$$742$$ 0 0
$$743$$ −16.4384 −0.603068 −0.301534 0.953455i $$-0.597499\pi$$
−0.301534 + 0.953455i $$0.597499\pi$$
$$744$$ 0 0
$$745$$ −10.6847 −0.391456
$$746$$ 0 0
$$747$$ −3.36932 −0.123277
$$748$$ 0 0
$$749$$ −0.492423 −0.0179927
$$750$$ 0 0
$$751$$ −51.8078 −1.89049 −0.945246 0.326358i $$-0.894178\pi$$
−0.945246 + 0.326358i $$0.894178\pi$$
$$752$$ 0 0
$$753$$ 4.10795 0.149702
$$754$$ 0 0
$$755$$ 15.1231 0.550386
$$756$$ 0 0
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −24.2462 −0.878924 −0.439462 0.898261i $$-0.644831\pi$$
−0.439462 + 0.898261i $$0.644831\pi$$
$$762$$ 0 0
$$763$$ 5.36932 0.194382
$$764$$ 0 0
$$765$$ −2.63068 −0.0951125
$$766$$ 0 0
$$767$$ 50.7386 1.83207
$$768$$ 0 0
$$769$$ −20.2462 −0.730097 −0.365049 0.930988i $$-0.618948\pi$$
−0.365049 + 0.930988i $$0.618948\pi$$
$$770$$ 0 0
$$771$$ 41.3693 1.48988
$$772$$ 0 0
$$773$$ 15.0691 0.541999 0.270999 0.962579i $$-0.412646\pi$$
0.270999 + 0.962579i $$0.412646\pi$$
$$774$$ 0 0
$$775$$ −5.56155 −0.199777
$$776$$ 0 0
$$777$$ 7.91571 0.283975
$$778$$ 0 0
$$779$$ 23.6155 0.846114
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −24.6847 −0.882158
$$784$$ 0 0
$$785$$ −9.31534 −0.332479
$$786$$ 0 0
$$787$$ −39.8617 −1.42092 −0.710459 0.703739i $$-0.751513\pi$$
−0.710459 + 0.703739i $$0.751513\pi$$
$$788$$ 0 0
$$789$$ 12.5767 0.447743
$$790$$ 0 0
$$791$$ −4.00000 −0.142224
$$792$$ 0 0
$$793$$ −60.1080 −2.13450
$$794$$ 0 0
$$795$$ −4.19224 −0.148683
$$796$$ 0 0
$$797$$ 3.75379 0.132966 0.0664830 0.997788i $$-0.478822\pi$$
0.0664830 + 0.997788i $$0.478822\pi$$
$$798$$ 0 0
$$799$$ −62.6307 −2.21571
$$800$$ 0 0
$$801$$ 1.50758 0.0532676
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −3.12311 −0.110075
$$806$$ 0 0
$$807$$ −14.2462 −0.501490
$$808$$ 0 0
$$809$$ −8.73863 −0.307234 −0.153617 0.988130i $$-0.549092\pi$$
−0.153617 + 0.988130i $$0.549092\pi$$
$$810$$ 0 0
$$811$$ −26.9309 −0.945671 −0.472835 0.881151i $$-0.656769\pi$$
−0.472835 + 0.881151i $$0.656769\pi$$
$$812$$ 0 0
$$813$$ −6.24621 −0.219064
$$814$$ 0 0
$$815$$ −0.192236 −0.00673373
$$816$$ 0 0
$$817$$ 28.4924 0.996824
$$818$$ 0 0
$$819$$ −1.75379 −0.0612823
$$820$$ 0 0
$$821$$ 18.4924 0.645390 0.322695 0.946503i $$-0.395411\pi$$
0.322695 + 0.946503i $$0.395411\pi$$
$$822$$ 0 0
$$823$$ −20.4924 −0.714321 −0.357160 0.934043i $$-0.616255\pi$$
−0.357160 + 0.934043i $$0.616255\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 27.8617 0.968848 0.484424 0.874833i $$-0.339029\pi$$
0.484424 + 0.874833i $$0.339029\pi$$
$$828$$ 0 0
$$829$$ −41.1231 −1.42826 −0.714132 0.700011i $$-0.753178\pi$$
−0.714132 + 0.700011i $$0.753178\pi$$
$$830$$ 0 0
$$831$$ −8.98485 −0.311681
$$832$$ 0 0
$$833$$ 31.8920 1.10499
$$834$$ 0 0
$$835$$ 10.6847 0.369758
$$836$$ 0 0
$$837$$ −30.9309 −1.06913
$$838$$ 0 0
$$839$$ 52.4924 1.81224 0.906120 0.423021i $$-0.139030\pi$$
0.906120 + 0.423021i $$0.139030\pi$$
$$840$$ 0 0
$$841$$ −9.30019 −0.320696
$$842$$ 0 0
$$843$$ −12.8769 −0.443504
$$844$$ 0 0
$$845$$ −37.7386 −1.29825
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −9.36932 −0.321554
$$850$$ 0 0
$$851$$ −82.3542 −2.82306
$$852$$ 0 0
$$853$$ 28.8769 0.988726 0.494363 0.869256i $$-0.335401\pi$$
0.494363 + 0.869256i $$0.335401\pi$$
$$854$$ 0 0
$$855$$ −3.12311 −0.106808
$$856$$ 0 0
$$857$$ 7.80776 0.266708 0.133354 0.991068i $$-0.457425\pi$$
0.133354 + 0.991068i $$0.457425\pi$$
$$858$$ 0 0
$$859$$ −33.8617 −1.15535 −0.577674 0.816268i $$-0.696039\pi$$
−0.577674 + 0.816268i $$0.696039\pi$$
$$860$$ 0 0
$$861$$ −2.90720 −0.0990773
$$862$$ 0 0
$$863$$ 20.8769 0.710658 0.355329 0.934741i $$-0.384369\pi$$
0.355329 + 0.934741i $$0.384369\pi$$
$$864$$ 0 0
$$865$$ 19.1231 0.650205
$$866$$ 0 0
$$867$$ −7.72348 −0.262303
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 7.36932 0.249414
$$874$$ 0 0
$$875$$ 0.438447 0.0148222
$$876$$ 0 0
$$877$$ −15.6155 −0.527299 −0.263649 0.964619i $$-0.584926\pi$$
−0.263649 + 0.964619i $$0.584926\pi$$
$$878$$ 0 0
$$879$$ 13.8617 0.467545
$$880$$ 0 0
$$881$$ 38.4924 1.29684 0.648421 0.761282i $$-0.275430\pi$$
0.648421 + 0.761282i $$0.275430\pi$$
$$882$$ 0 0
$$883$$ 42.5464 1.43180 0.715900 0.698203i $$-0.246017\pi$$
0.715900 + 0.698203i $$0.246017\pi$$
$$884$$ 0 0
$$885$$ −11.1231 −0.373899
$$886$$ 0 0
$$887$$ 18.8769 0.633824 0.316912 0.948455i $$-0.397354\pi$$
0.316912 + 0.948455i $$0.397354\pi$$
$$888$$ 0 0
$$889$$ 5.75379 0.192976
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −74.3542 −2.48817
$$894$$ 0 0
$$895$$ 2.24621 0.0750826
$$896$$ 0 0
$$897$$ −79.2311 −2.64545
$$898$$ 0 0
$$899$$ 24.6847 0.823279
$$900$$ 0 0
$$901$$ 12.5767 0.418991
$$902$$ 0 0
$$903$$ −3.50758 −0.116725
$$904$$ 0 0
$$905$$ −8.24621 −0.274113
$$906$$ 0 0
$$907$$ −31.3153 −1.03981 −0.519904 0.854224i $$-0.674032\pi$$
−0.519904 + 0.854224i $$0.674032\pi$$
$$908$$ 0 0
$$909$$ −1.12311 −0.0372511
$$910$$ 0 0
$$911$$ 45.5616 1.50952 0.754761 0.656000i $$-0.227753\pi$$
0.754761 + 0.656000i $$0.227753\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 13.1771 0.435621
$$916$$ 0 0
$$917$$ 1.66950 0.0551319
$$918$$ 0 0
$$919$$ −47.2311 −1.55801 −0.779004 0.627018i $$-0.784275\pi$$
−0.779004 + 0.627018i $$0.784275\pi$$
$$920$$ 0 0
$$921$$ 28.1080 0.926188
$$922$$ 0 0
$$923$$ 61.8617 2.03620
$$924$$ 0 0
$$925$$ 11.5616 0.380142
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −36.5464 −1.19905 −0.599524 0.800357i $$-0.704643\pi$$
−0.599524 + 0.800357i $$0.704643\pi$$
$$930$$ 0 0
$$931$$ 37.8617 1.24087
$$932$$ 0 0
$$933$$ 21.1771 0.693307
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 8.38447 0.273909 0.136954 0.990577i $$-0.456269\pi$$
0.136954 + 0.990577i $$0.456269\pi$$
$$938$$ 0 0
$$939$$ 38.6307 1.26066
$$940$$ 0 0
$$941$$ 28.0540 0.914533 0.457267 0.889330i $$-0.348828\pi$$
0.457267 + 0.889330i $$0.348828\pi$$
$$942$$ 0 0
$$943$$ 30.2462 0.984952
$$944$$ 0 0
$$945$$ 2.43845 0.0793227
$$946$$ 0 0
$$947$$ 50.0540 1.62654 0.813268 0.581890i $$-0.197686\pi$$
0.813268 + 0.581890i $$0.197686\pi$$
$$948$$ 0 0
$$949$$ −50.7386 −1.64705
$$950$$ 0 0
$$951$$ −29.1771 −0.946132
$$952$$ 0 0
$$953$$ 2.43845 0.0789891 0.0394945 0.999220i $$-0.487425\pi$$
0.0394945 + 0.999220i $$0.487425\pi$$
$$954$$ 0 0
$$955$$ 6.24621 0.202123
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −0.492423 −0.0159012
$$960$$ 0 0
$$961$$ −0.0691303 −0.00223001
$$962$$ 0 0
$$963$$ −0.630683 −0.0203235
$$964$$ 0 0
$$965$$ −25.5616 −0.822855
$$966$$ 0 0
$$967$$ 8.43845 0.271362 0.135681 0.990753i $$-0.456678\pi$$
0.135681 + 0.990753i $$0.456678\pi$$
$$968$$ 0 0
$$969$$ −40.6847 −1.30698
$$970$$ 0 0
$$971$$ −28.0000 −0.898563 −0.449281 0.893390i $$-0.648320\pi$$
−0.449281 + 0.893390i $$0.648320\pi$$
$$972$$ 0 0
$$973$$ −1.75379 −0.0562239
$$974$$ 0 0
$$975$$ 11.1231 0.356224
$$976$$ 0 0
$$977$$ 35.8617 1.14732 0.573659 0.819094i $$-0.305523\pi$$
0.573659 + 0.819094i $$0.305523\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 6.87689 0.219562
$$982$$ 0 0
$$983$$ −41.3693 −1.31948 −0.659738 0.751496i $$-0.729333\pi$$
−0.659738 + 0.751496i $$0.729333\pi$$
$$984$$ 0 0
$$985$$ −26.7386 −0.851964
$$986$$ 0 0
$$987$$ 9.15342 0.291356
$$988$$ 0 0
$$989$$ 36.4924 1.16039
$$990$$ 0 0
$$991$$ −32.0000 −1.01651 −0.508257 0.861206i $$-0.669710\pi$$
−0.508257 + 0.861206i $$0.669710\pi$$
$$992$$ 0 0
$$993$$ −48.0000 −1.52323
$$994$$ 0 0
$$995$$ 16.6847 0.528939
$$996$$ 0 0
$$997$$ −50.7386 −1.60691 −0.803454 0.595366i $$-0.797007\pi$$
−0.803454 + 0.595366i $$0.797007\pi$$
$$998$$ 0 0
$$999$$ 64.3002 2.03437
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 4840.2.a.m.1.1 2
4.3 odd 2 9680.2.a.bm.1.2 2
11.10 odd 2 440.2.a.g.1.1 2
33.32 even 2 3960.2.a.bf.1.1 2
44.43 even 2 880.2.a.k.1.2 2
55.32 even 4 2200.2.b.f.1849.3 4
55.43 even 4 2200.2.b.f.1849.2 4
55.54 odd 2 2200.2.a.l.1.2 2
88.21 odd 2 3520.2.a.bm.1.2 2
88.43 even 2 3520.2.a.br.1.1 2
132.131 odd 2 7920.2.a.by.1.2 2
220.43 odd 4 4400.2.b.w.4049.3 4
220.87 odd 4 4400.2.b.w.4049.2 4
220.219 even 2 4400.2.a.bt.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.a.g.1.1 2 11.10 odd 2
880.2.a.k.1.2 2 44.43 even 2
2200.2.a.l.1.2 2 55.54 odd 2
2200.2.b.f.1849.2 4 55.43 even 4
2200.2.b.f.1849.3 4 55.32 even 4
3520.2.a.bm.1.2 2 88.21 odd 2
3520.2.a.br.1.1 2 88.43 even 2
3960.2.a.bf.1.1 2 33.32 even 2
4400.2.a.bt.1.1 2 220.219 even 2
4400.2.b.w.4049.2 4 220.87 odd 4
4400.2.b.w.4049.3 4 220.43 odd 4
4840.2.a.m.1.1 2 1.1 even 1 trivial
7920.2.a.by.1.2 2 132.131 odd 2
9680.2.a.bm.1.2 2 4.3 odd 2