# Properties

 Label 4840.2.a.ba.1.3 Level $4840$ Weight $2$ Character 4840.1 Self dual yes Analytic conductor $38.648$ Analytic rank $1$ Dimension $6$ CM no Inner twists $1$

# Learn more

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: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.25903625.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 3x^{5} - 7x^{4} + 17x^{3} + 16x^{2} - 20x - 5$$ x^6 - 3*x^5 - 7*x^4 + 17*x^3 + 16*x^2 - 20*x - 5 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 1.3 Root $$1.03795$$ 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)

 $$f(q)$$ $$=$$ $$q-1.03795 q^{3} -1.00000 q^{5} +2.60210 q^{7} -1.92266 q^{9} +O(q^{10})$$ $$q-1.03795 q^{3} -1.00000 q^{5} +2.60210 q^{7} -1.92266 q^{9} +2.87756 q^{13} +1.03795 q^{15} -0.810293 q^{17} -5.15069 q^{19} -2.70085 q^{21} +4.98262 q^{23} +1.00000 q^{25} +5.10948 q^{27} -6.72933 q^{29} -4.14084 q^{31} -2.60210 q^{35} +4.05133 q^{37} -2.98677 q^{39} -9.76082 q^{41} +5.92912 q^{43} +1.92266 q^{45} -0.967904 q^{47} -0.229090 q^{49} +0.841046 q^{51} +4.47445 q^{53} +5.34617 q^{57} +6.80756 q^{59} +2.61307 q^{61} -5.00294 q^{63} -2.87756 q^{65} -15.8408 q^{67} -5.17172 q^{69} -1.41346 q^{71} +13.1429 q^{73} -1.03795 q^{75} -5.69123 q^{79} +0.464568 q^{81} -7.96967 q^{83} +0.810293 q^{85} +6.98472 q^{87} -14.5196 q^{89} +7.48768 q^{91} +4.29799 q^{93} +5.15069 q^{95} -2.99325 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 3 q^{3} - 6 q^{5} - 7 q^{7} + 5 q^{9}+O(q^{10})$$ 6 * q - 3 * q^3 - 6 * q^5 - 7 * q^7 + 5 * q^9 $$6 q - 3 q^{3} - 6 q^{5} - 7 q^{7} + 5 q^{9} + q^{13} + 3 q^{15} + 6 q^{17} - 7 q^{19} + 14 q^{21} - 9 q^{23} + 6 q^{25} - 21 q^{27} + 10 q^{29} + q^{31} + 7 q^{35} + 3 q^{37} - q^{39} - 6 q^{41} + 18 q^{43} - 5 q^{45} - 3 q^{47} + 17 q^{49} + 15 q^{51} - 23 q^{53} + 9 q^{57} - 2 q^{59} + 6 q^{61} - 49 q^{63} - q^{65} - 22 q^{67} + 2 q^{69} - 13 q^{71} + 10 q^{73} - 3 q^{75} - 22 q^{79} + 10 q^{81} + 10 q^{83} - 6 q^{85} - 3 q^{87} - 25 q^{89} + 12 q^{91} + 19 q^{93} + 7 q^{95} - 33 q^{97}+O(q^{100})$$ 6 * q - 3 * q^3 - 6 * q^5 - 7 * q^7 + 5 * q^9 + q^13 + 3 * q^15 + 6 * q^17 - 7 * q^19 + 14 * q^21 - 9 * q^23 + 6 * q^25 - 21 * q^27 + 10 * q^29 + q^31 + 7 * q^35 + 3 * q^37 - q^39 - 6 * q^41 + 18 * q^43 - 5 * q^45 - 3 * q^47 + 17 * q^49 + 15 * q^51 - 23 * q^53 + 9 * q^57 - 2 * q^59 + 6 * q^61 - 49 * q^63 - q^65 - 22 * q^67 + 2 * q^69 - 13 * q^71 + 10 * q^73 - 3 * q^75 - 22 * q^79 + 10 * q^81 + 10 * q^83 - 6 * q^85 - 3 * q^87 - 25 * q^89 + 12 * q^91 + 19 * q^93 + 7 * q^95 - 33 * 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
$$3$$ −1.03795 −0.599262 −0.299631 0.954055i $$-0.596864\pi$$
−0.299631 + 0.954055i $$0.596864\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 2.60210 0.983500 0.491750 0.870736i $$-0.336357\pi$$
0.491750 + 0.870736i $$0.336357\pi$$
$$8$$ 0 0
$$9$$ −1.92266 −0.640885
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 0 0
$$13$$ 2.87756 0.798091 0.399045 0.916931i $$-0.369342\pi$$
0.399045 + 0.916931i $$0.369342\pi$$
$$14$$ 0 0
$$15$$ 1.03795 0.267998
$$16$$ 0 0
$$17$$ −0.810293 −0.196525 −0.0982625 0.995161i $$-0.531328\pi$$
−0.0982625 + 0.995161i $$0.531328\pi$$
$$18$$ 0 0
$$19$$ −5.15069 −1.18165 −0.590824 0.806800i $$-0.701197\pi$$
−0.590824 + 0.806800i $$0.701197\pi$$
$$20$$ 0 0
$$21$$ −2.70085 −0.589374
$$22$$ 0 0
$$23$$ 4.98262 1.03895 0.519474 0.854486i $$-0.326128\pi$$
0.519474 + 0.854486i $$0.326128\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 5.10948 0.983320
$$28$$ 0 0
$$29$$ −6.72933 −1.24961 −0.624803 0.780783i $$-0.714821\pi$$
−0.624803 + 0.780783i $$0.714821\pi$$
$$30$$ 0 0
$$31$$ −4.14084 −0.743717 −0.371858 0.928290i $$-0.621279\pi$$
−0.371858 + 0.928290i $$0.621279\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −2.60210 −0.439835
$$36$$ 0 0
$$37$$ 4.05133 0.666034 0.333017 0.942921i $$-0.391933\pi$$
0.333017 + 0.942921i $$0.391933\pi$$
$$38$$ 0 0
$$39$$ −2.98677 −0.478266
$$40$$ 0 0
$$41$$ −9.76082 −1.52438 −0.762192 0.647351i $$-0.775877\pi$$
−0.762192 + 0.647351i $$0.775877\pi$$
$$42$$ 0 0
$$43$$ 5.92912 0.904182 0.452091 0.891972i $$-0.350678\pi$$
0.452091 + 0.891972i $$0.350678\pi$$
$$44$$ 0 0
$$45$$ 1.92266 0.286613
$$46$$ 0 0
$$47$$ −0.967904 −0.141183 −0.0705917 0.997505i $$-0.522489\pi$$
−0.0705917 + 0.997505i $$0.522489\pi$$
$$48$$ 0 0
$$49$$ −0.229090 −0.0327272
$$50$$ 0 0
$$51$$ 0.841046 0.117770
$$52$$ 0 0
$$53$$ 4.47445 0.614613 0.307307 0.951611i $$-0.400572\pi$$
0.307307 + 0.951611i $$0.400572\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 5.34617 0.708117
$$58$$ 0 0
$$59$$ 6.80756 0.886269 0.443135 0.896455i $$-0.353866\pi$$
0.443135 + 0.896455i $$0.353866\pi$$
$$60$$ 0 0
$$61$$ 2.61307 0.334569 0.167285 0.985909i $$-0.446500\pi$$
0.167285 + 0.985909i $$0.446500\pi$$
$$62$$ 0 0
$$63$$ −5.00294 −0.630311
$$64$$ 0 0
$$65$$ −2.87756 −0.356917
$$66$$ 0 0
$$67$$ −15.8408 −1.93526 −0.967629 0.252378i $$-0.918788\pi$$
−0.967629 + 0.252378i $$0.918788\pi$$
$$68$$ 0 0
$$69$$ −5.17172 −0.622602
$$70$$ 0 0
$$71$$ −1.41346 −0.167746 −0.0838732 0.996476i $$-0.526729\pi$$
−0.0838732 + 0.996476i $$0.526729\pi$$
$$72$$ 0 0
$$73$$ 13.1429 1.53826 0.769131 0.639091i $$-0.220689\pi$$
0.769131 + 0.639091i $$0.220689\pi$$
$$74$$ 0 0
$$75$$ −1.03795 −0.119852
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −5.69123 −0.640314 −0.320157 0.947365i $$-0.603736\pi$$
−0.320157 + 0.947365i $$0.603736\pi$$
$$80$$ 0 0
$$81$$ 0.464568 0.0516187
$$82$$ 0 0
$$83$$ −7.96967 −0.874785 −0.437392 0.899271i $$-0.644098\pi$$
−0.437392 + 0.899271i $$0.644098\pi$$
$$84$$ 0 0
$$85$$ 0.810293 0.0878887
$$86$$ 0 0
$$87$$ 6.98472 0.748841
$$88$$ 0 0
$$89$$ −14.5196 −1.53907 −0.769535 0.638605i $$-0.779512\pi$$
−0.769535 + 0.638605i $$0.779512\pi$$
$$90$$ 0 0
$$91$$ 7.48768 0.784923
$$92$$ 0 0
$$93$$ 4.29799 0.445681
$$94$$ 0 0
$$95$$ 5.15069 0.528449
$$96$$ 0 0
$$97$$ −2.99325 −0.303919 −0.151959 0.988387i $$-0.548558\pi$$
−0.151959 + 0.988387i $$0.548558\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 15.3700 1.52938 0.764688 0.644401i $$-0.222893\pi$$
0.764688 + 0.644401i $$0.222893\pi$$
$$102$$ 0 0
$$103$$ 5.49309 0.541251 0.270625 0.962685i $$-0.412770\pi$$
0.270625 + 0.962685i $$0.412770\pi$$
$$104$$ 0 0
$$105$$ 2.70085 0.263576
$$106$$ 0 0
$$107$$ 6.23672 0.602927 0.301463 0.953478i $$-0.402525\pi$$
0.301463 + 0.953478i $$0.402525\pi$$
$$108$$ 0 0
$$109$$ 19.1460 1.83385 0.916927 0.399056i $$-0.130662\pi$$
0.916927 + 0.399056i $$0.130662\pi$$
$$110$$ 0 0
$$111$$ −4.20508 −0.399129
$$112$$ 0 0
$$113$$ −10.8681 −1.02238 −0.511192 0.859467i $$-0.670796\pi$$
−0.511192 + 0.859467i $$0.670796\pi$$
$$114$$ 0 0
$$115$$ −4.98262 −0.464632
$$116$$ 0 0
$$117$$ −5.53255 −0.511485
$$118$$ 0 0
$$119$$ −2.10846 −0.193282
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0 0
$$123$$ 10.1313 0.913505
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −18.9400 −1.68065 −0.840325 0.542083i $$-0.817636\pi$$
−0.840325 + 0.542083i $$0.817636\pi$$
$$128$$ 0 0
$$129$$ −6.15414 −0.541842
$$130$$ 0 0
$$131$$ −14.5409 −1.27045 −0.635224 0.772328i $$-0.719092\pi$$
−0.635224 + 0.772328i $$0.719092\pi$$
$$132$$ 0 0
$$133$$ −13.4026 −1.16215
$$134$$ 0 0
$$135$$ −5.10948 −0.439754
$$136$$ 0 0
$$137$$ −19.9840 −1.70735 −0.853675 0.520806i $$-0.825632\pi$$
−0.853675 + 0.520806i $$0.825632\pi$$
$$138$$ 0 0
$$139$$ 2.07071 0.175635 0.0878176 0.996137i $$-0.472011\pi$$
0.0878176 + 0.996137i $$0.472011\pi$$
$$140$$ 0 0
$$141$$ 1.00464 0.0846058
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 6.72933 0.558840
$$146$$ 0 0
$$147$$ 0.237785 0.0196122
$$148$$ 0 0
$$149$$ −0.504694 −0.0413461 −0.0206731 0.999786i $$-0.506581\pi$$
−0.0206731 + 0.999786i $$0.506581\pi$$
$$150$$ 0 0
$$151$$ −12.1188 −0.986216 −0.493108 0.869968i $$-0.664139\pi$$
−0.493108 + 0.869968i $$0.664139\pi$$
$$152$$ 0 0
$$153$$ 1.55791 0.125950
$$154$$ 0 0
$$155$$ 4.14084 0.332600
$$156$$ 0 0
$$157$$ −11.8853 −0.948548 −0.474274 0.880377i $$-0.657289\pi$$
−0.474274 + 0.880377i $$0.657289\pi$$
$$158$$ 0 0
$$159$$ −4.64427 −0.368314
$$160$$ 0 0
$$161$$ 12.9653 1.02181
$$162$$ 0 0
$$163$$ 0.0631196 0.00494391 0.00247195 0.999997i $$-0.499213\pi$$
0.00247195 + 0.999997i $$0.499213\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 7.54352 0.583735 0.291868 0.956459i $$-0.405723\pi$$
0.291868 + 0.956459i $$0.405723\pi$$
$$168$$ 0 0
$$169$$ −4.71966 −0.363051
$$170$$ 0 0
$$171$$ 9.90300 0.757301
$$172$$ 0 0
$$173$$ 1.41262 0.107400 0.0536998 0.998557i $$-0.482899\pi$$
0.0536998 + 0.998557i $$0.482899\pi$$
$$174$$ 0 0
$$175$$ 2.60210 0.196700
$$176$$ 0 0
$$177$$ −7.06593 −0.531108
$$178$$ 0 0
$$179$$ −15.8217 −1.18257 −0.591285 0.806463i $$-0.701379\pi$$
−0.591285 + 0.806463i $$0.701379\pi$$
$$180$$ 0 0
$$181$$ 18.9773 1.41057 0.705287 0.708922i $$-0.250818\pi$$
0.705287 + 0.708922i $$0.250818\pi$$
$$182$$ 0 0
$$183$$ −2.71224 −0.200495
$$184$$ 0 0
$$185$$ −4.05133 −0.297860
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 13.2954 0.967096
$$190$$ 0 0
$$191$$ 10.3666 0.750098 0.375049 0.927005i $$-0.377626\pi$$
0.375049 + 0.927005i $$0.377626\pi$$
$$192$$ 0 0
$$193$$ 0.608652 0.0438117 0.0219059 0.999760i $$-0.493027\pi$$
0.0219059 + 0.999760i $$0.493027\pi$$
$$194$$ 0 0
$$195$$ 2.98677 0.213887
$$196$$ 0 0
$$197$$ −11.1521 −0.794553 −0.397277 0.917699i $$-0.630045\pi$$
−0.397277 + 0.917699i $$0.630045\pi$$
$$198$$ 0 0
$$199$$ 8.91481 0.631954 0.315977 0.948767i $$-0.397668\pi$$
0.315977 + 0.948767i $$0.397668\pi$$
$$200$$ 0 0
$$201$$ 16.4420 1.15973
$$202$$ 0 0
$$203$$ −17.5104 −1.22899
$$204$$ 0 0
$$205$$ 9.76082 0.681725
$$206$$ 0 0
$$207$$ −9.57987 −0.665847
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −17.4046 −1.19818 −0.599092 0.800680i $$-0.704472\pi$$
−0.599092 + 0.800680i $$0.704472\pi$$
$$212$$ 0 0
$$213$$ 1.46710 0.100524
$$214$$ 0 0
$$215$$ −5.92912 −0.404362
$$216$$ 0 0
$$217$$ −10.7749 −0.731445
$$218$$ 0 0
$$219$$ −13.6417 −0.921822
$$220$$ 0 0
$$221$$ −2.33167 −0.156845
$$222$$ 0 0
$$223$$ −1.53670 −0.102905 −0.0514524 0.998675i $$-0.516385\pi$$
−0.0514524 + 0.998675i $$0.516385\pi$$
$$224$$ 0 0
$$225$$ −1.92266 −0.128177
$$226$$ 0 0
$$227$$ −10.0981 −0.670231 −0.335116 0.942177i $$-0.608775\pi$$
−0.335116 + 0.942177i $$0.608775\pi$$
$$228$$ 0 0
$$229$$ 2.39276 0.158118 0.0790590 0.996870i $$-0.474808\pi$$
0.0790590 + 0.996870i $$0.474808\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 8.28726 0.542916 0.271458 0.962450i $$-0.412494\pi$$
0.271458 + 0.962450i $$0.412494\pi$$
$$234$$ 0 0
$$235$$ 0.967904 0.0631391
$$236$$ 0 0
$$237$$ 5.90723 0.383716
$$238$$ 0 0
$$239$$ −4.56206 −0.295095 −0.147548 0.989055i $$-0.547138\pi$$
−0.147548 + 0.989055i $$0.547138\pi$$
$$240$$ 0 0
$$241$$ −23.4148 −1.50828 −0.754139 0.656715i $$-0.771946\pi$$
−0.754139 + 0.656715i $$0.771946\pi$$
$$242$$ 0 0
$$243$$ −15.8106 −1.01425
$$244$$ 0 0
$$245$$ 0.229090 0.0146361
$$246$$ 0 0
$$247$$ −14.8214 −0.943063
$$248$$ 0 0
$$249$$ 8.27214 0.524225
$$250$$ 0 0
$$251$$ −27.7446 −1.75122 −0.875611 0.483017i $$-0.839541\pi$$
−0.875611 + 0.483017i $$0.839541\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ −0.841046 −0.0526683
$$256$$ 0 0
$$257$$ −25.4843 −1.58966 −0.794832 0.606829i $$-0.792441\pi$$
−0.794832 + 0.606829i $$0.792441\pi$$
$$258$$ 0 0
$$259$$ 10.5419 0.655045
$$260$$ 0 0
$$261$$ 12.9382 0.800853
$$262$$ 0 0
$$263$$ −16.1271 −0.994442 −0.497221 0.867624i $$-0.665646\pi$$
−0.497221 + 0.867624i $$0.665646\pi$$
$$264$$ 0 0
$$265$$ −4.47445 −0.274863
$$266$$ 0 0
$$267$$ 15.0706 0.922306
$$268$$ 0 0
$$269$$ 17.5236 1.06843 0.534216 0.845348i $$-0.320607\pi$$
0.534216 + 0.845348i $$0.320607\pi$$
$$270$$ 0 0
$$271$$ −7.33746 −0.445719 −0.222860 0.974851i $$-0.571539\pi$$
−0.222860 + 0.974851i $$0.571539\pi$$
$$272$$ 0 0
$$273$$ −7.77186 −0.470374
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 17.4068 1.04588 0.522938 0.852371i $$-0.324836\pi$$
0.522938 + 0.852371i $$0.324836\pi$$
$$278$$ 0 0
$$279$$ 7.96140 0.476637
$$280$$ 0 0
$$281$$ −13.7827 −0.822206 −0.411103 0.911589i $$-0.634856\pi$$
−0.411103 + 0.911589i $$0.634856\pi$$
$$282$$ 0 0
$$283$$ −11.5904 −0.688980 −0.344490 0.938790i $$-0.611948\pi$$
−0.344490 + 0.938790i $$0.611948\pi$$
$$284$$ 0 0
$$285$$ −5.34617 −0.316680
$$286$$ 0 0
$$287$$ −25.3986 −1.49923
$$288$$ 0 0
$$289$$ −16.3434 −0.961378
$$290$$ 0 0
$$291$$ 3.10685 0.182127
$$292$$ 0 0
$$293$$ 16.1449 0.943197 0.471598 0.881813i $$-0.343677\pi$$
0.471598 + 0.881813i $$0.343677\pi$$
$$294$$ 0 0
$$295$$ −6.80756 −0.396352
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 14.3378 0.829176
$$300$$ 0 0
$$301$$ 15.4281 0.889263
$$302$$ 0 0
$$303$$ −15.9534 −0.916497
$$304$$ 0 0
$$305$$ −2.61307 −0.149624
$$306$$ 0 0
$$307$$ 14.3107 0.816753 0.408377 0.912814i $$-0.366095\pi$$
0.408377 + 0.912814i $$0.366095\pi$$
$$308$$ 0 0
$$309$$ −5.70157 −0.324351
$$310$$ 0 0
$$311$$ −22.3744 −1.26873 −0.634367 0.773032i $$-0.718739\pi$$
−0.634367 + 0.773032i $$0.718739\pi$$
$$312$$ 0 0
$$313$$ 11.5368 0.652100 0.326050 0.945352i $$-0.394282\pi$$
0.326050 + 0.945352i $$0.394282\pi$$
$$314$$ 0 0
$$315$$ 5.00294 0.281883
$$316$$ 0 0
$$317$$ −25.3314 −1.42275 −0.711376 0.702812i $$-0.751928\pi$$
−0.711376 + 0.702812i $$0.751928\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −6.47342 −0.361311
$$322$$ 0 0
$$323$$ 4.17357 0.232224
$$324$$ 0 0
$$325$$ 2.87756 0.159618
$$326$$ 0 0
$$327$$ −19.8726 −1.09896
$$328$$ 0 0
$$329$$ −2.51858 −0.138854
$$330$$ 0 0
$$331$$ −20.9669 −1.15244 −0.576221 0.817294i $$-0.695473\pi$$
−0.576221 + 0.817294i $$0.695473\pi$$
$$332$$ 0 0
$$333$$ −7.78931 −0.426851
$$334$$ 0 0
$$335$$ 15.8408 0.865474
$$336$$ 0 0
$$337$$ 18.1430 0.988311 0.494156 0.869373i $$-0.335477\pi$$
0.494156 + 0.869373i $$0.335477\pi$$
$$338$$ 0 0
$$339$$ 11.2806 0.612675
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −18.8108 −1.01569
$$344$$ 0 0
$$345$$ 5.17172 0.278436
$$346$$ 0 0
$$347$$ −18.9900 −1.01944 −0.509718 0.860342i $$-0.670250\pi$$
−0.509718 + 0.860342i $$0.670250\pi$$
$$348$$ 0 0
$$349$$ −16.1172 −0.862732 −0.431366 0.902177i $$-0.641968\pi$$
−0.431366 + 0.902177i $$0.641968\pi$$
$$350$$ 0 0
$$351$$ 14.7028 0.784779
$$352$$ 0 0
$$353$$ 28.0252 1.49163 0.745816 0.666152i $$-0.232060\pi$$
0.745816 + 0.666152i $$0.232060\pi$$
$$354$$ 0 0
$$355$$ 1.41346 0.0750185
$$356$$ 0 0
$$357$$ 2.18848 0.115827
$$358$$ 0 0
$$359$$ −5.23896 −0.276502 −0.138251 0.990397i $$-0.544148\pi$$
−0.138251 + 0.990397i $$0.544148\pi$$
$$360$$ 0 0
$$361$$ 7.52958 0.396294
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −13.1429 −0.687932
$$366$$ 0 0
$$367$$ −26.3079 −1.37326 −0.686632 0.727005i $$-0.740911\pi$$
−0.686632 + 0.727005i $$0.740911\pi$$
$$368$$ 0 0
$$369$$ 18.7667 0.976955
$$370$$ 0 0
$$371$$ 11.6430 0.604472
$$372$$ 0 0
$$373$$ 36.0081 1.86443 0.932215 0.361905i $$-0.117873\pi$$
0.932215 + 0.361905i $$0.117873\pi$$
$$374$$ 0 0
$$375$$ 1.03795 0.0535996
$$376$$ 0 0
$$377$$ −19.3640 −0.997298
$$378$$ 0 0
$$379$$ −14.0945 −0.723986 −0.361993 0.932181i $$-0.617904\pi$$
−0.361993 + 0.932181i $$0.617904\pi$$
$$380$$ 0 0
$$381$$ 19.6588 1.00715
$$382$$ 0 0
$$383$$ 2.14120 0.109410 0.0547052 0.998503i $$-0.482578\pi$$
0.0547052 + 0.998503i $$0.482578\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −11.3996 −0.579477
$$388$$ 0 0
$$389$$ −29.2825 −1.48468 −0.742340 0.670023i $$-0.766284\pi$$
−0.742340 + 0.670023i $$0.766284\pi$$
$$390$$ 0 0
$$391$$ −4.03739 −0.204179
$$392$$ 0 0
$$393$$ 15.0928 0.761331
$$394$$ 0 0
$$395$$ 5.69123 0.286357
$$396$$ 0 0
$$397$$ 9.09620 0.456525 0.228263 0.973600i $$-0.426696\pi$$
0.228263 + 0.973600i $$0.426696\pi$$
$$398$$ 0 0
$$399$$ 13.9112 0.696434
$$400$$ 0 0
$$401$$ 32.3614 1.61605 0.808027 0.589146i $$-0.200536\pi$$
0.808027 + 0.589146i $$0.200536\pi$$
$$402$$ 0 0
$$403$$ −11.9155 −0.593553
$$404$$ 0 0
$$405$$ −0.464568 −0.0230846
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 20.8620 1.03156 0.515779 0.856722i $$-0.327502\pi$$
0.515779 + 0.856722i $$0.327502\pi$$
$$410$$ 0 0
$$411$$ 20.7425 1.02315
$$412$$ 0 0
$$413$$ 17.7139 0.871646
$$414$$ 0 0
$$415$$ 7.96967 0.391216
$$416$$ 0 0
$$417$$ −2.14930 −0.105251
$$418$$ 0 0
$$419$$ −36.4444 −1.78043 −0.890213 0.455545i $$-0.849444\pi$$
−0.890213 + 0.455545i $$0.849444\pi$$
$$420$$ 0 0
$$421$$ −32.4285 −1.58047 −0.790235 0.612804i $$-0.790041\pi$$
−0.790235 + 0.612804i $$0.790041\pi$$
$$422$$ 0 0
$$423$$ 1.86095 0.0904823
$$424$$ 0 0
$$425$$ −0.810293 −0.0393050
$$426$$ 0 0
$$427$$ 6.79946 0.329049
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −28.8732 −1.39077 −0.695387 0.718636i $$-0.744767\pi$$
−0.695387 + 0.718636i $$0.744767\pi$$
$$432$$ 0 0
$$433$$ 7.55542 0.363090 0.181545 0.983383i $$-0.441890\pi$$
0.181545 + 0.983383i $$0.441890\pi$$
$$434$$ 0 0
$$435$$ −6.98472 −0.334892
$$436$$ 0 0
$$437$$ −25.6639 −1.22767
$$438$$ 0 0
$$439$$ 1.96072 0.0935801 0.0467900 0.998905i $$-0.485101\pi$$
0.0467900 + 0.998905i $$0.485101\pi$$
$$440$$ 0 0
$$441$$ 0.440462 0.0209744
$$442$$ 0 0
$$443$$ 6.03304 0.286638 0.143319 0.989677i $$-0.454222\pi$$
0.143319 + 0.989677i $$0.454222\pi$$
$$444$$ 0 0
$$445$$ 14.5196 0.688293
$$446$$ 0 0
$$447$$ 0.523848 0.0247772
$$448$$ 0 0
$$449$$ 9.82399 0.463623 0.231811 0.972761i $$-0.425535\pi$$
0.231811 + 0.972761i $$0.425535\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 12.5788 0.591002
$$454$$ 0 0
$$455$$ −7.48768 −0.351028
$$456$$ 0 0
$$457$$ 7.82880 0.366216 0.183108 0.983093i $$-0.441384\pi$$
0.183108 + 0.983093i $$0.441384\pi$$
$$458$$ 0 0
$$459$$ −4.14018 −0.193247
$$460$$ 0 0
$$461$$ 3.19972 0.149026 0.0745130 0.997220i $$-0.476260\pi$$
0.0745130 + 0.997220i $$0.476260\pi$$
$$462$$ 0 0
$$463$$ −21.2096 −0.985696 −0.492848 0.870115i $$-0.664044\pi$$
−0.492848 + 0.870115i $$0.664044\pi$$
$$464$$ 0 0
$$465$$ −4.29799 −0.199315
$$466$$ 0 0
$$467$$ −31.2967 −1.44824 −0.724118 0.689676i $$-0.757753\pi$$
−0.724118 + 0.689676i $$0.757753\pi$$
$$468$$ 0 0
$$469$$ −41.2192 −1.90333
$$470$$ 0 0
$$471$$ 12.3363 0.568429
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −5.15069 −0.236330
$$476$$ 0 0
$$477$$ −8.60283 −0.393896
$$478$$ 0 0
$$479$$ −18.0022 −0.822542 −0.411271 0.911513i $$-0.634915\pi$$
−0.411271 + 0.911513i $$0.634915\pi$$
$$480$$ 0 0
$$481$$ 11.6579 0.531556
$$482$$ 0 0
$$483$$ −13.4573 −0.612330
$$484$$ 0 0
$$485$$ 2.99325 0.135917
$$486$$ 0 0
$$487$$ 21.0873 0.955558 0.477779 0.878480i $$-0.341442\pi$$
0.477779 + 0.878480i $$0.341442\pi$$
$$488$$ 0 0
$$489$$ −0.0655151 −0.00296270
$$490$$ 0 0
$$491$$ 22.0057 0.993104 0.496552 0.868007i $$-0.334599\pi$$
0.496552 + 0.868007i $$0.334599\pi$$
$$492$$ 0 0
$$493$$ 5.45273 0.245579
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −3.67795 −0.164979
$$498$$ 0 0
$$499$$ 16.1255 0.721877 0.360939 0.932590i $$-0.382456\pi$$
0.360939 + 0.932590i $$0.382456\pi$$
$$500$$ 0 0
$$501$$ −7.82982 −0.349810
$$502$$ 0 0
$$503$$ −3.16694 −0.141207 −0.0706035 0.997504i $$-0.522493\pi$$
−0.0706035 + 0.997504i $$0.522493\pi$$
$$504$$ 0 0
$$505$$ −15.3700 −0.683957
$$506$$ 0 0
$$507$$ 4.89878 0.217563
$$508$$ 0 0
$$509$$ 28.9843 1.28470 0.642352 0.766409i $$-0.277959\pi$$
0.642352 + 0.766409i $$0.277959\pi$$
$$510$$ 0 0
$$511$$ 34.1992 1.51288
$$512$$ 0 0
$$513$$ −26.3173 −1.16194
$$514$$ 0 0
$$515$$ −5.49309 −0.242055
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −1.46623 −0.0643605
$$520$$ 0 0
$$521$$ −15.8497 −0.694388 −0.347194 0.937793i $$-0.612865\pi$$
−0.347194 + 0.937793i $$0.612865\pi$$
$$522$$ 0 0
$$523$$ 24.0929 1.05351 0.526754 0.850018i $$-0.323409\pi$$
0.526754 + 0.850018i $$0.323409\pi$$
$$524$$ 0 0
$$525$$ −2.70085 −0.117875
$$526$$ 0 0
$$527$$ 3.35529 0.146159
$$528$$ 0 0
$$529$$ 1.82653 0.0794143
$$530$$ 0 0
$$531$$ −13.0886 −0.567997
$$532$$ 0 0
$$533$$ −28.0873 −1.21660
$$534$$ 0 0
$$535$$ −6.23672 −0.269637
$$536$$ 0 0
$$537$$ 16.4222 0.708669
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 21.3207 0.916648 0.458324 0.888785i $$-0.348450\pi$$
0.458324 + 0.888785i $$0.348450\pi$$
$$542$$ 0 0
$$543$$ −19.6976 −0.845303
$$544$$ 0 0
$$545$$ −19.1460 −0.820124
$$546$$ 0 0
$$547$$ 29.3180 1.25355 0.626775 0.779201i $$-0.284375\pi$$
0.626775 + 0.779201i $$0.284375\pi$$
$$548$$ 0 0
$$549$$ −5.02403 −0.214420
$$550$$ 0 0
$$551$$ 34.6607 1.47659
$$552$$ 0 0
$$553$$ −14.8091 −0.629749
$$554$$ 0 0
$$555$$ 4.20508 0.178496
$$556$$ 0 0
$$557$$ 1.68279 0.0713019 0.0356510 0.999364i $$-0.488650\pi$$
0.0356510 + 0.999364i $$0.488650\pi$$
$$558$$ 0 0
$$559$$ 17.0614 0.721619
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −1.14833 −0.0483963 −0.0241981 0.999707i $$-0.507703\pi$$
−0.0241981 + 0.999707i $$0.507703\pi$$
$$564$$ 0 0
$$565$$ 10.8681 0.457224
$$566$$ 0 0
$$567$$ 1.20885 0.0507670
$$568$$ 0 0
$$569$$ 6.68468 0.280237 0.140118 0.990135i $$-0.455252\pi$$
0.140118 + 0.990135i $$0.455252\pi$$
$$570$$ 0 0
$$571$$ −17.7964 −0.744756 −0.372378 0.928081i $$-0.621458\pi$$
−0.372378 + 0.928081i $$0.621458\pi$$
$$572$$ 0 0
$$573$$ −10.7600 −0.449505
$$574$$ 0 0
$$575$$ 4.98262 0.207790
$$576$$ 0 0
$$577$$ −45.9807 −1.91420 −0.957102 0.289752i $$-0.906427\pi$$
−0.957102 + 0.289752i $$0.906427\pi$$
$$578$$ 0 0
$$579$$ −0.631752 −0.0262547
$$580$$ 0 0
$$581$$ −20.7379 −0.860351
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 5.53255 0.228743
$$586$$ 0 0
$$587$$ 4.72240 0.194914 0.0974571 0.995240i $$-0.468929\pi$$
0.0974571 + 0.995240i $$0.468929\pi$$
$$588$$ 0 0
$$589$$ 21.3282 0.878812
$$590$$ 0 0
$$591$$ 11.5753 0.476146
$$592$$ 0 0
$$593$$ −3.67545 −0.150932 −0.0754662 0.997148i $$-0.524045\pi$$
−0.0754662 + 0.997148i $$0.524045\pi$$
$$594$$ 0 0
$$595$$ 2.10846 0.0864385
$$596$$ 0 0
$$597$$ −9.25315 −0.378706
$$598$$ 0 0
$$599$$ −39.8714 −1.62910 −0.814550 0.580093i $$-0.803016\pi$$
−0.814550 + 0.580093i $$0.803016\pi$$
$$600$$ 0 0
$$601$$ 12.3476 0.503670 0.251835 0.967770i $$-0.418966\pi$$
0.251835 + 0.967770i $$0.418966\pi$$
$$602$$ 0 0
$$603$$ 30.4563 1.24028
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 19.7067 0.799869 0.399935 0.916544i $$-0.369033\pi$$
0.399935 + 0.916544i $$0.369033\pi$$
$$608$$ 0 0
$$609$$ 18.1749 0.736485
$$610$$ 0 0
$$611$$ −2.78520 −0.112677
$$612$$ 0 0
$$613$$ −37.7646 −1.52530 −0.762650 0.646812i $$-0.776102\pi$$
−0.762650 + 0.646812i $$0.776102\pi$$
$$614$$ 0 0
$$615$$ −10.1313 −0.408532
$$616$$ 0 0
$$617$$ −3.49353 −0.140644 −0.0703222 0.997524i $$-0.522403\pi$$
−0.0703222 + 0.997524i $$0.522403\pi$$
$$618$$ 0 0
$$619$$ 6.48840 0.260791 0.130395 0.991462i $$-0.458375\pi$$
0.130395 + 0.991462i $$0.458375\pi$$
$$620$$ 0 0
$$621$$ 25.4586 1.02162
$$622$$ 0 0
$$623$$ −37.7813 −1.51368
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −3.28276 −0.130892
$$630$$ 0 0
$$631$$ 32.7333 1.30309 0.651547 0.758609i $$-0.274120\pi$$
0.651547 + 0.758609i $$0.274120\pi$$
$$632$$ 0 0
$$633$$ 18.0652 0.718026
$$634$$ 0 0
$$635$$ 18.9400 0.751609
$$636$$ 0 0
$$637$$ −0.659221 −0.0261193
$$638$$ 0 0
$$639$$ 2.71759 0.107506
$$640$$ 0 0
$$641$$ −7.19375 −0.284136 −0.142068 0.989857i $$-0.545375\pi$$
−0.142068 + 0.989857i $$0.545375\pi$$
$$642$$ 0 0
$$643$$ 8.21093 0.323808 0.161904 0.986807i $$-0.448237\pi$$
0.161904 + 0.986807i $$0.448237\pi$$
$$644$$ 0 0
$$645$$ 6.15414 0.242319
$$646$$ 0 0
$$647$$ −49.0008 −1.92642 −0.963210 0.268750i $$-0.913389\pi$$
−0.963210 + 0.268750i $$0.913389\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 11.1838 0.438327
$$652$$ 0 0
$$653$$ −43.8455 −1.71581 −0.857903 0.513811i $$-0.828233\pi$$
−0.857903 + 0.513811i $$0.828233\pi$$
$$654$$ 0 0
$$655$$ 14.5409 0.568161
$$656$$ 0 0
$$657$$ −25.2693 −0.985850
$$658$$ 0 0
$$659$$ 37.2905 1.45263 0.726315 0.687362i $$-0.241231\pi$$
0.726315 + 0.687362i $$0.241231\pi$$
$$660$$ 0 0
$$661$$ 35.5579 1.38304 0.691520 0.722357i $$-0.256941\pi$$
0.691520 + 0.722357i $$0.256941\pi$$
$$662$$ 0 0
$$663$$ 2.42016 0.0939912
$$664$$ 0 0
$$665$$ 13.4026 0.519730
$$666$$ 0 0
$$667$$ −33.5297 −1.29828
$$668$$ 0 0
$$669$$ 1.59502 0.0616669
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 13.6219 0.525084 0.262542 0.964921i $$-0.415439\pi$$
0.262542 + 0.964921i $$0.415439\pi$$
$$674$$ 0 0
$$675$$ 5.10948 0.196664
$$676$$ 0 0
$$677$$ −6.89247 −0.264899 −0.132449 0.991190i $$-0.542284\pi$$
−0.132449 + 0.991190i $$0.542284\pi$$
$$678$$ 0 0
$$679$$ −7.78874 −0.298904
$$680$$ 0 0
$$681$$ 10.4813 0.401644
$$682$$ 0 0
$$683$$ 8.29642 0.317454 0.158727 0.987323i $$-0.449261\pi$$
0.158727 + 0.987323i $$0.449261\pi$$
$$684$$ 0 0
$$685$$ 19.9840 0.763550
$$686$$ 0 0
$$687$$ −2.48357 −0.0947541
$$688$$ 0 0
$$689$$ 12.8755 0.490517
$$690$$ 0 0
$$691$$ 18.3562 0.698303 0.349151 0.937066i $$-0.386470\pi$$
0.349151 + 0.937066i $$0.386470\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −2.07071 −0.0785464
$$696$$ 0 0
$$697$$ 7.90913 0.299580
$$698$$ 0 0
$$699$$ −8.60178 −0.325349
$$700$$ 0 0
$$701$$ 28.0624 1.05990 0.529951 0.848028i $$-0.322210\pi$$
0.529951 + 0.848028i $$0.322210\pi$$
$$702$$ 0 0
$$703$$ −20.8671 −0.787018
$$704$$ 0 0
$$705$$ −1.00464 −0.0378369
$$706$$ 0 0
$$707$$ 39.9943 1.50414
$$708$$ 0 0
$$709$$ −16.2588 −0.610612 −0.305306 0.952254i $$-0.598759\pi$$
−0.305306 + 0.952254i $$0.598759\pi$$
$$710$$ 0 0
$$711$$ 10.9423 0.410368
$$712$$ 0 0
$$713$$ −20.6322 −0.772683
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 4.73520 0.176839
$$718$$ 0 0
$$719$$ −3.75469 −0.140026 −0.0700132 0.997546i $$-0.522304\pi$$
−0.0700132 + 0.997546i $$0.522304\pi$$
$$720$$ 0 0
$$721$$ 14.2936 0.532320
$$722$$ 0 0
$$723$$ 24.3034 0.903854
$$724$$ 0 0
$$725$$ −6.72933 −0.249921
$$726$$ 0 0
$$727$$ 35.1985 1.30544 0.652720 0.757599i $$-0.273628\pi$$
0.652720 + 0.757599i $$0.273628\pi$$
$$728$$ 0 0
$$729$$ 15.0170 0.556185
$$730$$ 0 0
$$731$$ −4.80432 −0.177694
$$732$$ 0 0
$$733$$ 20.2869 0.749313 0.374656 0.927164i $$-0.377761\pi$$
0.374656 + 0.927164i $$0.377761\pi$$
$$734$$ 0 0
$$735$$ −0.237785 −0.00877083
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −15.6336 −0.575093 −0.287546 0.957767i $$-0.592840\pi$$
−0.287546 + 0.957767i $$0.592840\pi$$
$$740$$ 0 0
$$741$$ 15.3839 0.565142
$$742$$ 0 0
$$743$$ 19.9836 0.733129 0.366564 0.930393i $$-0.380534\pi$$
0.366564 + 0.930393i $$0.380534\pi$$
$$744$$ 0 0
$$745$$ 0.504694 0.0184905
$$746$$ 0 0
$$747$$ 15.3229 0.560637
$$748$$ 0 0
$$749$$ 16.2286 0.592979
$$750$$ 0 0
$$751$$ 31.4457 1.14747 0.573734 0.819041i $$-0.305494\pi$$
0.573734 + 0.819041i $$0.305494\pi$$
$$752$$ 0 0
$$753$$ 28.7975 1.04944
$$754$$ 0 0
$$755$$ 12.1188 0.441049
$$756$$ 0 0
$$757$$ 45.3850 1.64955 0.824773 0.565463i $$-0.191303\pi$$
0.824773 + 0.565463i $$0.191303\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −35.6102 −1.29087 −0.645435 0.763816i $$-0.723324\pi$$
−0.645435 + 0.763816i $$0.723324\pi$$
$$762$$ 0 0
$$763$$ 49.8197 1.80360
$$764$$ 0 0
$$765$$ −1.55791 −0.0563265
$$766$$ 0 0
$$767$$ 19.5892 0.707324
$$768$$ 0 0
$$769$$ 18.9907 0.684821 0.342411 0.939550i $$-0.388757\pi$$
0.342411 + 0.939550i $$0.388757\pi$$
$$770$$ 0 0
$$771$$ 26.4514 0.952625
$$772$$ 0 0
$$773$$ 43.6808 1.57109 0.785544 0.618805i $$-0.212383\pi$$
0.785544 + 0.618805i $$0.212383\pi$$
$$774$$ 0 0
$$775$$ −4.14084 −0.148743
$$776$$ 0 0
$$777$$ −10.9420 −0.392543
$$778$$ 0 0
$$779$$ 50.2749 1.80129
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −34.3834 −1.22876
$$784$$ 0 0
$$785$$ 11.8853 0.424204
$$786$$ 0 0
$$787$$ 50.8854 1.81387 0.906934 0.421272i $$-0.138416\pi$$
0.906934 + 0.421272i $$0.138416\pi$$
$$788$$ 0 0
$$789$$ 16.7392 0.595931
$$790$$ 0 0
$$791$$ −28.2798 −1.00551
$$792$$ 0 0
$$793$$ 7.51925 0.267017
$$794$$ 0 0
$$795$$ 4.64427 0.164715
$$796$$ 0 0
$$797$$ −18.9156 −0.670025 −0.335013 0.942214i $$-0.608741\pi$$
−0.335013 + 0.942214i $$0.608741\pi$$
$$798$$ 0 0
$$799$$ 0.784286 0.0277461
$$800$$ 0 0
$$801$$ 27.9161 0.986367
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −12.9653 −0.456966
$$806$$ 0 0
$$807$$ −18.1887 −0.640271
$$808$$ 0 0
$$809$$ −20.8204 −0.732005 −0.366002 0.930614i $$-0.619274\pi$$
−0.366002 + 0.930614i $$0.619274\pi$$
$$810$$ 0 0
$$811$$ 36.0642 1.26639 0.633193 0.773994i $$-0.281744\pi$$
0.633193 + 0.773994i $$0.281744\pi$$
$$812$$ 0 0
$$813$$ 7.61594 0.267103
$$814$$ 0 0
$$815$$ −0.0631196 −0.00221098
$$816$$ 0 0
$$817$$ −30.5390 −1.06843
$$818$$ 0 0
$$819$$ −14.3962 −0.503045
$$820$$ 0 0
$$821$$ 37.9278 1.32369 0.661845 0.749641i $$-0.269774\pi$$
0.661845 + 0.749641i $$0.269774\pi$$
$$822$$ 0 0
$$823$$ −47.3524 −1.65060 −0.825300 0.564695i $$-0.808994\pi$$
−0.825300 + 0.564695i $$0.808994\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 32.7566 1.13906 0.569530 0.821970i $$-0.307125\pi$$
0.569530 + 0.821970i $$0.307125\pi$$
$$828$$ 0 0
$$829$$ −24.3065 −0.844199 −0.422099 0.906550i $$-0.638707\pi$$
−0.422099 + 0.906550i $$0.638707\pi$$
$$830$$ 0 0
$$831$$ −18.0675 −0.626753
$$832$$ 0 0
$$833$$ 0.185630 0.00643171
$$834$$ 0 0
$$835$$ −7.54352 −0.261054
$$836$$ 0 0
$$837$$ −21.1575 −0.731311
$$838$$ 0 0
$$839$$ −38.6436 −1.33413 −0.667063 0.745001i $$-0.732449\pi$$
−0.667063 + 0.745001i $$0.732449\pi$$
$$840$$ 0 0
$$841$$ 16.2839 0.561513
$$842$$ 0 0
$$843$$ 14.3058 0.492717
$$844$$ 0 0
$$845$$ 4.71966 0.162361
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 12.0303 0.412879
$$850$$ 0 0
$$851$$ 20.1862 0.691975
$$852$$ 0 0
$$853$$ −28.7982 −0.986033 −0.493017 0.870020i $$-0.664106\pi$$
−0.493017 + 0.870020i $$0.664106\pi$$
$$854$$ 0 0
$$855$$ −9.90300 −0.338675
$$856$$ 0 0
$$857$$ 44.6116 1.52390 0.761952 0.647633i $$-0.224241\pi$$
0.761952 + 0.647633i $$0.224241\pi$$
$$858$$ 0 0
$$859$$ 46.4941 1.58636 0.793180 0.608988i $$-0.208424\pi$$
0.793180 + 0.608988i $$0.208424\pi$$
$$860$$ 0 0
$$861$$ 26.3625 0.898433
$$862$$ 0 0
$$863$$ 15.6647 0.533231 0.266616 0.963803i $$-0.414095\pi$$
0.266616 + 0.963803i $$0.414095\pi$$
$$864$$ 0 0
$$865$$ −1.41262 −0.0480306
$$866$$ 0 0
$$867$$ 16.9637 0.576117
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −45.5827 −1.54451
$$872$$ 0 0
$$873$$ 5.75499 0.194777
$$874$$ 0 0
$$875$$ −2.60210 −0.0879669
$$876$$ 0 0
$$877$$ 33.6285 1.13555 0.567777 0.823182i $$-0.307804\pi$$
0.567777 + 0.823182i $$0.307804\pi$$
$$878$$ 0 0
$$879$$ −16.7577 −0.565222
$$880$$ 0 0
$$881$$ 42.4530 1.43028 0.715139 0.698982i $$-0.246363\pi$$
0.715139 + 0.698982i $$0.246363\pi$$
$$882$$ 0 0
$$883$$ −8.63358 −0.290543 −0.145272 0.989392i $$-0.546406\pi$$
−0.145272 + 0.989392i $$0.546406\pi$$
$$884$$ 0 0
$$885$$ 7.06593 0.237519
$$886$$ 0 0
$$887$$ 4.32578 0.145246 0.0726228 0.997359i $$-0.476863\pi$$
0.0726228 + 0.997359i $$0.476863\pi$$
$$888$$ 0 0
$$889$$ −49.2836 −1.65292
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 4.98537 0.166829
$$894$$ 0 0
$$895$$ 15.8217 0.528862
$$896$$ 0 0
$$897$$ −14.8819 −0.496893
$$898$$ 0 0
$$899$$ 27.8651 0.929352
$$900$$ 0 0
$$901$$ −3.62562 −0.120787
$$902$$ 0 0
$$903$$ −16.0137 −0.532902
$$904$$ 0 0
$$905$$ −18.9773 −0.630828
$$906$$ 0 0
$$907$$ −47.4842 −1.57669 −0.788344 0.615234i $$-0.789061\pi$$
−0.788344 + 0.615234i $$0.789061\pi$$
$$908$$ 0 0
$$909$$ −29.5513 −0.980154
$$910$$ 0 0
$$911$$ −22.7476 −0.753662 −0.376831 0.926282i $$-0.622986\pi$$
−0.376831 + 0.926282i $$0.622986\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 2.71224 0.0896639
$$916$$ 0 0
$$917$$ −37.8369 −1.24949
$$918$$ 0 0
$$919$$ −37.0964 −1.22370 −0.611849 0.790975i $$-0.709574\pi$$
−0.611849 + 0.790975i $$0.709574\pi$$
$$920$$ 0 0
$$921$$ −14.8538 −0.489449
$$922$$ 0 0
$$923$$ −4.06730 −0.133877
$$924$$ 0 0
$$925$$ 4.05133 0.133207
$$926$$ 0 0
$$927$$ −10.5613 −0.346879
$$928$$ 0 0
$$929$$ 43.0548 1.41258 0.706290 0.707922i $$-0.250367\pi$$
0.706290 + 0.707922i $$0.250367\pi$$
$$930$$ 0 0
$$931$$ 1.17997 0.0386721
$$932$$ 0 0
$$933$$ 23.2235 0.760304
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −23.5075 −0.767956 −0.383978 0.923342i $$-0.625446\pi$$
−0.383978 + 0.923342i $$0.625446\pi$$
$$938$$ 0 0
$$939$$ −11.9747 −0.390779
$$940$$ 0 0
$$941$$ 37.1410 1.21076 0.605382 0.795935i $$-0.293021\pi$$
0.605382 + 0.795935i $$0.293021\pi$$
$$942$$ 0 0
$$943$$ −48.6345 −1.58376
$$944$$ 0 0
$$945$$ −13.2954 −0.432498
$$946$$ 0 0
$$947$$ −16.3128 −0.530094 −0.265047 0.964235i $$-0.585388\pi$$
−0.265047 + 0.964235i $$0.585388\pi$$
$$948$$ 0 0
$$949$$ 37.8195 1.22767
$$950$$ 0 0
$$951$$ 26.2928 0.852601
$$952$$ 0 0
$$953$$ −58.7967 −1.90461 −0.952306 0.305146i $$-0.901295\pi$$
−0.952306 + 0.305146i $$0.901295\pi$$
$$954$$ 0 0
$$955$$ −10.3666 −0.335454
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −52.0004 −1.67918
$$960$$ 0 0
$$961$$ −13.8535 −0.446886
$$962$$ 0 0
$$963$$ −11.9911 −0.386407
$$964$$ 0 0
$$965$$ −0.608652 −0.0195932
$$966$$ 0 0
$$967$$ 60.4542 1.94408 0.972039 0.234821i $$-0.0754502\pi$$
0.972039 + 0.234821i $$0.0754502\pi$$
$$968$$ 0 0
$$969$$ −4.33196 −0.139163
$$970$$ 0 0
$$971$$ 41.6267 1.33586 0.667932 0.744222i $$-0.267180\pi$$
0.667932 + 0.744222i $$0.267180\pi$$
$$972$$ 0 0
$$973$$ 5.38818 0.172737
$$974$$ 0 0
$$975$$ −2.98677 −0.0956531
$$976$$ 0 0
$$977$$ 47.7290 1.52699 0.763493 0.645816i $$-0.223483\pi$$
0.763493 + 0.645816i $$0.223483\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −36.8111 −1.17529
$$982$$ 0 0
$$983$$ −5.92744 −0.189056 −0.0945280 0.995522i $$-0.530134\pi$$
−0.0945280 + 0.995522i $$0.530134\pi$$
$$984$$ 0 0
$$985$$ 11.1521 0.355335
$$986$$ 0 0
$$987$$ 2.61417 0.0832098
$$988$$ 0 0
$$989$$ 29.5425 0.939398
$$990$$ 0 0
$$991$$ 34.5440 1.09733 0.548663 0.836044i $$-0.315137\pi$$
0.548663 + 0.836044i $$0.315137\pi$$
$$992$$ 0 0
$$993$$ 21.7626 0.690615
$$994$$ 0 0
$$995$$ −8.91481 −0.282618
$$996$$ 0 0
$$997$$ −34.4542 −1.09117 −0.545587 0.838054i $$-0.683693\pi$$
−0.545587 + 0.838054i $$0.683693\pi$$
$$998$$ 0 0
$$999$$ 20.7002 0.654925
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.ba.1.3 6
4.3 odd 2 9680.2.a.dd.1.4 6
11.2 odd 10 440.2.y.c.81.2 12
11.6 odd 10 440.2.y.c.201.2 yes 12
11.10 odd 2 4840.2.a.bb.1.3 6
44.35 even 10 880.2.bo.i.81.2 12
44.39 even 10 880.2.bo.i.641.2 12
44.43 even 2 9680.2.a.dc.1.4 6

By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.c.81.2 12 11.2 odd 10
440.2.y.c.201.2 yes 12 11.6 odd 10
880.2.bo.i.81.2 12 44.35 even 10
880.2.bo.i.641.2 12 44.39 even 10
4840.2.a.ba.1.3 6 1.1 even 1 trivial
4840.2.a.bb.1.3 6 11.10 odd 2
9680.2.a.dc.1.4 6 44.43 even 2
9680.2.a.dd.1.4 6 4.3 odd 2