# Properties

 Label 6080.2.a.bx.1.1 Level $6080$ Weight $2$ Character 6080.1 Self dual yes Analytic conductor $48.549$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6080,2,Mod(1,6080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6080, 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("6080.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6080 = 2^{6} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6080.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: $$48.5490444289$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.316.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4x + 2$$ x^3 - x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 760) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.81361$$ of defining polynomial Character $$\chi$$ $$=$$ 6080.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.81361 q^{3} +1.00000 q^{5} -4.91638 q^{7} +0.289169 q^{9} +O(q^{10})$$ $$q-1.81361 q^{3} +1.00000 q^{5} -4.91638 q^{7} +0.289169 q^{9} -0.578337 q^{11} +6.39194 q^{13} -1.81361 q^{15} -0.710831 q^{17} -1.00000 q^{19} +8.91638 q^{21} -2.71083 q^{23} +1.00000 q^{25} +4.91638 q^{27} -6.54359 q^{29} +1.42166 q^{31} +1.04888 q^{33} -4.91638 q^{35} +9.10278 q^{37} -11.5925 q^{39} -11.0489 q^{41} -5.83276 q^{43} +0.289169 q^{45} -1.15667 q^{47} +17.1708 q^{49} +1.28917 q^{51} -13.2736 q^{53} -0.578337 q^{55} +1.81361 q^{57} -11.3869 q^{59} +9.04888 q^{61} -1.42166 q^{63} +6.39194 q^{65} -2.97028 q^{67} +4.91638 q^{69} -9.38692 q^{73} -1.81361 q^{75} +2.84333 q^{77} +4.37279 q^{79} -9.78389 q^{81} +0.372787 q^{83} -0.710831 q^{85} +11.8675 q^{87} -16.6167 q^{89} -31.4252 q^{91} -2.57834 q^{93} -1.00000 q^{95} +3.94610 q^{97} -0.167237 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{3} + 3 q^{5} - q^{7}+O(q^{10})$$ 3 * q + q^3 + 3 * q^5 - q^7 $$3 q + q^{3} + 3 q^{5} - q^{7} + 11 q^{13} + q^{15} - 3 q^{17} - 3 q^{19} + 13 q^{21} - 9 q^{23} + 3 q^{25} + q^{27} + 7 q^{29} + 6 q^{31} - 8 q^{33} - q^{35} + 20 q^{37} + 3 q^{39} - 22 q^{41} + 10 q^{43} + 12 q^{49} + 3 q^{51} + 7 q^{53} - q^{57} - 11 q^{59} + 16 q^{61} - 6 q^{63} + 11 q^{65} + q^{67} + q^{69} - 5 q^{73} + q^{75} + 12 q^{77} + 26 q^{79} - 13 q^{81} + 14 q^{83} - 3 q^{85} + 33 q^{87} - 6 q^{89} - 29 q^{91} - 6 q^{93} - 3 q^{95} + 8 q^{97} - 28 q^{99}+O(q^{100})$$ 3 * q + q^3 + 3 * q^5 - q^7 + 11 * q^13 + q^15 - 3 * q^17 - 3 * q^19 + 13 * q^21 - 9 * q^23 + 3 * q^25 + q^27 + 7 * q^29 + 6 * q^31 - 8 * q^33 - q^35 + 20 * q^37 + 3 * q^39 - 22 * q^41 + 10 * q^43 + 12 * q^49 + 3 * q^51 + 7 * q^53 - q^57 - 11 * q^59 + 16 * q^61 - 6 * q^63 + 11 * q^65 + q^67 + q^69 - 5 * q^73 + q^75 + 12 * q^77 + 26 * q^79 - 13 * q^81 + 14 * q^83 - 3 * q^85 + 33 * q^87 - 6 * q^89 - 29 * q^91 - 6 * q^93 - 3 * q^95 + 8 * q^97 - 28 * 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
$$3$$ −1.81361 −1.04709 −0.523543 0.851999i $$-0.675390\pi$$
−0.523543 + 0.851999i $$0.675390\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −4.91638 −1.85822 −0.929109 0.369807i $$-0.879424\pi$$
−0.929109 + 0.369807i $$0.879424\pi$$
$$8$$ 0 0
$$9$$ 0.289169 0.0963895
$$10$$ 0 0
$$11$$ −0.578337 −0.174375 −0.0871876 0.996192i $$-0.527788\pi$$
−0.0871876 + 0.996192i $$0.527788\pi$$
$$12$$ 0 0
$$13$$ 6.39194 1.77281 0.886403 0.462914i $$-0.153196\pi$$
0.886403 + 0.462914i $$0.153196\pi$$
$$14$$ 0 0
$$15$$ −1.81361 −0.468271
$$16$$ 0 0
$$17$$ −0.710831 −0.172402 −0.0862010 0.996278i $$-0.527473\pi$$
−0.0862010 + 0.996278i $$0.527473\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 8.91638 1.94571
$$22$$ 0 0
$$23$$ −2.71083 −0.565247 −0.282624 0.959231i $$-0.591205\pi$$
−0.282624 + 0.959231i $$0.591205\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 4.91638 0.946158
$$28$$ 0 0
$$29$$ −6.54359 −1.21512 −0.607558 0.794276i $$-0.707851\pi$$
−0.607558 + 0.794276i $$0.707851\pi$$
$$30$$ 0 0
$$31$$ 1.42166 0.255338 0.127669 0.991817i $$-0.459250\pi$$
0.127669 + 0.991817i $$0.459250\pi$$
$$32$$ 0 0
$$33$$ 1.04888 0.182586
$$34$$ 0 0
$$35$$ −4.91638 −0.831020
$$36$$ 0 0
$$37$$ 9.10278 1.49649 0.748243 0.663424i $$-0.230897\pi$$
0.748243 + 0.663424i $$0.230897\pi$$
$$38$$ 0 0
$$39$$ −11.5925 −1.85628
$$40$$ 0 0
$$41$$ −11.0489 −1.72554 −0.862772 0.505593i $$-0.831274\pi$$
−0.862772 + 0.505593i $$0.831274\pi$$
$$42$$ 0 0
$$43$$ −5.83276 −0.889488 −0.444744 0.895658i $$-0.646705\pi$$
−0.444744 + 0.895658i $$0.646705\pi$$
$$44$$ 0 0
$$45$$ 0.289169 0.0431067
$$46$$ 0 0
$$47$$ −1.15667 −0.168718 −0.0843591 0.996435i $$-0.526884\pi$$
−0.0843591 + 0.996435i $$0.526884\pi$$
$$48$$ 0 0
$$49$$ 17.1708 2.45297
$$50$$ 0 0
$$51$$ 1.28917 0.180520
$$52$$ 0 0
$$53$$ −13.2736 −1.82327 −0.911633 0.411004i $$-0.865178\pi$$
−0.911633 + 0.411004i $$0.865178\pi$$
$$54$$ 0 0
$$55$$ −0.578337 −0.0779830
$$56$$ 0 0
$$57$$ 1.81361 0.240218
$$58$$ 0 0
$$59$$ −11.3869 −1.48245 −0.741225 0.671256i $$-0.765755\pi$$
−0.741225 + 0.671256i $$0.765755\pi$$
$$60$$ 0 0
$$61$$ 9.04888 1.15859 0.579295 0.815118i $$-0.303328\pi$$
0.579295 + 0.815118i $$0.303328\pi$$
$$62$$ 0 0
$$63$$ −1.42166 −0.179113
$$64$$ 0 0
$$65$$ 6.39194 0.792823
$$66$$ 0 0
$$67$$ −2.97028 −0.362878 −0.181439 0.983402i $$-0.558075\pi$$
−0.181439 + 0.983402i $$0.558075\pi$$
$$68$$ 0 0
$$69$$ 4.91638 0.591863
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ −9.38692 −1.09866 −0.549328 0.835607i $$-0.685116\pi$$
−0.549328 + 0.835607i $$0.685116\pi$$
$$74$$ 0 0
$$75$$ −1.81361 −0.209417
$$76$$ 0 0
$$77$$ 2.84333 0.324027
$$78$$ 0 0
$$79$$ 4.37279 0.491977 0.245988 0.969273i $$-0.420887\pi$$
0.245988 + 0.969273i $$0.420887\pi$$
$$80$$ 0 0
$$81$$ −9.78389 −1.08710
$$82$$ 0 0
$$83$$ 0.372787 0.0409187 0.0204593 0.999791i $$-0.493487\pi$$
0.0204593 + 0.999791i $$0.493487\pi$$
$$84$$ 0 0
$$85$$ −0.710831 −0.0771005
$$86$$ 0 0
$$87$$ 11.8675 1.27233
$$88$$ 0 0
$$89$$ −16.6167 −1.76136 −0.880681 0.473710i $$-0.842914\pi$$
−0.880681 + 0.473710i $$0.842914\pi$$
$$90$$ 0 0
$$91$$ −31.4252 −3.29426
$$92$$ 0 0
$$93$$ −2.57834 −0.267361
$$94$$ 0 0
$$95$$ −1.00000 −0.102598
$$96$$ 0 0
$$97$$ 3.94610 0.400666 0.200333 0.979728i $$-0.435798\pi$$
0.200333 + 0.979728i $$0.435798\pi$$
$$98$$ 0 0
$$99$$ −0.167237 −0.0168079
$$100$$ 0 0
$$101$$ 6.20555 0.617475 0.308738 0.951147i $$-0.400093\pi$$
0.308738 + 0.951147i $$0.400093\pi$$
$$102$$ 0 0
$$103$$ 8.15165 0.803206 0.401603 0.915814i $$-0.368453\pi$$
0.401603 + 0.915814i $$0.368453\pi$$
$$104$$ 0 0
$$105$$ 8.91638 0.870150
$$106$$ 0 0
$$107$$ 13.0680 1.26333 0.631667 0.775240i $$-0.282371\pi$$
0.631667 + 0.775240i $$0.282371\pi$$
$$108$$ 0 0
$$109$$ 11.5925 1.11036 0.555179 0.831731i $$-0.312650\pi$$
0.555179 + 0.831731i $$0.312650\pi$$
$$110$$ 0 0
$$111$$ −16.5089 −1.56695
$$112$$ 0 0
$$113$$ −14.9355 −1.40502 −0.702509 0.711675i $$-0.747937\pi$$
−0.702509 + 0.711675i $$0.747937\pi$$
$$114$$ 0 0
$$115$$ −2.71083 −0.252786
$$116$$ 0 0
$$117$$ 1.84835 0.170880
$$118$$ 0 0
$$119$$ 3.49472 0.320360
$$120$$ 0 0
$$121$$ −10.6655 −0.969593
$$122$$ 0 0
$$123$$ 20.0383 1.80679
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 11.8867 1.05477 0.527385 0.849626i $$-0.323172\pi$$
0.527385 + 0.849626i $$0.323172\pi$$
$$128$$ 0 0
$$129$$ 10.5783 0.931371
$$130$$ 0 0
$$131$$ 9.15667 0.800022 0.400011 0.916510i $$-0.369006\pi$$
0.400011 + 0.916510i $$0.369006\pi$$
$$132$$ 0 0
$$133$$ 4.91638 0.426304
$$134$$ 0 0
$$135$$ 4.91638 0.423135
$$136$$ 0 0
$$137$$ 13.3869 1.14372 0.571861 0.820351i $$-0.306222\pi$$
0.571861 + 0.820351i $$0.306222\pi$$
$$138$$ 0 0
$$139$$ −3.42166 −0.290222 −0.145111 0.989415i $$-0.546354\pi$$
−0.145111 + 0.989415i $$0.546354\pi$$
$$140$$ 0 0
$$141$$ 2.09775 0.176663
$$142$$ 0 0
$$143$$ −3.69670 −0.309133
$$144$$ 0 0
$$145$$ −6.54359 −0.543416
$$146$$ 0 0
$$147$$ −31.1411 −2.56847
$$148$$ 0 0
$$149$$ 3.36222 0.275444 0.137722 0.990471i $$-0.456022\pi$$
0.137722 + 0.990471i $$0.456022\pi$$
$$150$$ 0 0
$$151$$ 21.1950 1.72482 0.862412 0.506207i $$-0.168953\pi$$
0.862412 + 0.506207i $$0.168953\pi$$
$$152$$ 0 0
$$153$$ −0.205550 −0.0166177
$$154$$ 0 0
$$155$$ 1.42166 0.114191
$$156$$ 0 0
$$157$$ 11.7944 0.941300 0.470650 0.882320i $$-0.344020\pi$$
0.470650 + 0.882320i $$0.344020\pi$$
$$158$$ 0 0
$$159$$ 24.0731 1.90912
$$160$$ 0 0
$$161$$ 13.3275 1.05035
$$162$$ 0 0
$$163$$ 14.2439 1.11567 0.557833 0.829953i $$-0.311633\pi$$
0.557833 + 0.829953i $$0.311633\pi$$
$$164$$ 0 0
$$165$$ 1.04888 0.0816549
$$166$$ 0 0
$$167$$ −7.47556 −0.578476 −0.289238 0.957257i $$-0.593402\pi$$
−0.289238 + 0.957257i $$0.593402\pi$$
$$168$$ 0 0
$$169$$ 27.8569 2.14284
$$170$$ 0 0
$$171$$ −0.289169 −0.0221133
$$172$$ 0 0
$$173$$ −4.05390 −0.308212 −0.154106 0.988054i $$-0.549250\pi$$
−0.154106 + 0.988054i $$0.549250\pi$$
$$174$$ 0 0
$$175$$ −4.91638 −0.371644
$$176$$ 0 0
$$177$$ 20.6514 1.55225
$$178$$ 0 0
$$179$$ −2.95112 −0.220577 −0.110289 0.993900i $$-0.535178\pi$$
−0.110289 + 0.993900i $$0.535178\pi$$
$$180$$ 0 0
$$181$$ −9.66553 −0.718433 −0.359216 0.933254i $$-0.616956\pi$$
−0.359216 + 0.933254i $$0.616956\pi$$
$$182$$ 0 0
$$183$$ −16.4111 −1.21314
$$184$$ 0 0
$$185$$ 9.10278 0.669249
$$186$$ 0 0
$$187$$ 0.411100 0.0300626
$$188$$ 0 0
$$189$$ −24.1708 −1.75817
$$190$$ 0 0
$$191$$ −15.9058 −1.15090 −0.575452 0.817835i $$-0.695174\pi$$
−0.575452 + 0.817835i $$0.695174\pi$$
$$192$$ 0 0
$$193$$ 15.6116 1.12375 0.561875 0.827222i $$-0.310080\pi$$
0.561875 + 0.827222i $$0.310080\pi$$
$$194$$ 0 0
$$195$$ −11.5925 −0.830154
$$196$$ 0 0
$$197$$ 13.6655 0.973628 0.486814 0.873506i $$-0.338159\pi$$
0.486814 + 0.873506i $$0.338159\pi$$
$$198$$ 0 0
$$199$$ 8.91638 0.632066 0.316033 0.948748i $$-0.397649\pi$$
0.316033 + 0.948748i $$0.397649\pi$$
$$200$$ 0 0
$$201$$ 5.38692 0.379964
$$202$$ 0 0
$$203$$ 32.1708 2.25795
$$204$$ 0 0
$$205$$ −11.0489 −0.771687
$$206$$ 0 0
$$207$$ −0.783887 −0.0544839
$$208$$ 0 0
$$209$$ 0.578337 0.0400044
$$210$$ 0 0
$$211$$ −19.7980 −1.36295 −0.681476 0.731840i $$-0.738662\pi$$
−0.681476 + 0.731840i $$0.738662\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −5.83276 −0.397791
$$216$$ 0 0
$$217$$ −6.98944 −0.474474
$$218$$ 0 0
$$219$$ 17.0242 1.15039
$$220$$ 0 0
$$221$$ −4.54359 −0.305635
$$222$$ 0 0
$$223$$ −1.57331 −0.105357 −0.0526784 0.998612i $$-0.516776\pi$$
−0.0526784 + 0.998612i $$0.516776\pi$$
$$224$$ 0 0
$$225$$ 0.289169 0.0192779
$$226$$ 0 0
$$227$$ 28.0575 1.86224 0.931120 0.364713i $$-0.118833\pi$$
0.931120 + 0.364713i $$0.118833\pi$$
$$228$$ 0 0
$$229$$ −2.47054 −0.163258 −0.0816289 0.996663i $$-0.526012\pi$$
−0.0816289 + 0.996663i $$0.526012\pi$$
$$230$$ 0 0
$$231$$ −5.15667 −0.339284
$$232$$ 0 0
$$233$$ 3.15667 0.206801 0.103400 0.994640i $$-0.467028\pi$$
0.103400 + 0.994640i $$0.467028\pi$$
$$234$$ 0 0
$$235$$ −1.15667 −0.0754531
$$236$$ 0 0
$$237$$ −7.93051 −0.515142
$$238$$ 0 0
$$239$$ −2.74914 −0.177827 −0.0889137 0.996039i $$-0.528340\pi$$
−0.0889137 + 0.996039i $$0.528340\pi$$
$$240$$ 0 0
$$241$$ 8.88164 0.572117 0.286058 0.958212i $$-0.407655\pi$$
0.286058 + 0.958212i $$0.407655\pi$$
$$242$$ 0 0
$$243$$ 2.99498 0.192128
$$244$$ 0 0
$$245$$ 17.1708 1.09700
$$246$$ 0 0
$$247$$ −6.39194 −0.406710
$$248$$ 0 0
$$249$$ −0.676089 −0.0428454
$$250$$ 0 0
$$251$$ −6.31335 −0.398495 −0.199248 0.979949i $$-0.563850\pi$$
−0.199248 + 0.979949i $$0.563850\pi$$
$$252$$ 0 0
$$253$$ 1.56777 0.0985651
$$254$$ 0 0
$$255$$ 1.28917 0.0807309
$$256$$ 0 0
$$257$$ 7.83830 0.488940 0.244470 0.969657i $$-0.421386\pi$$
0.244470 + 0.969657i $$0.421386\pi$$
$$258$$ 0 0
$$259$$ −44.7527 −2.78080
$$260$$ 0 0
$$261$$ −1.89220 −0.117124
$$262$$ 0 0
$$263$$ 13.8711 0.855327 0.427664 0.903938i $$-0.359337\pi$$
0.427664 + 0.903938i $$0.359337\pi$$
$$264$$ 0 0
$$265$$ −13.2736 −0.815390
$$266$$ 0 0
$$267$$ 30.1361 1.84430
$$268$$ 0 0
$$269$$ 14.0000 0.853595 0.426798 0.904347i $$-0.359642\pi$$
0.426798 + 0.904347i $$0.359642\pi$$
$$270$$ 0 0
$$271$$ −4.07306 −0.247421 −0.123710 0.992318i $$-0.539479\pi$$
−0.123710 + 0.992318i $$0.539479\pi$$
$$272$$ 0 0
$$273$$ 56.9930 3.44937
$$274$$ 0 0
$$275$$ −0.578337 −0.0348750
$$276$$ 0 0
$$277$$ −16.1361 −0.969522 −0.484761 0.874647i $$-0.661093\pi$$
−0.484761 + 0.874647i $$0.661093\pi$$
$$278$$ 0 0
$$279$$ 0.411100 0.0246119
$$280$$ 0 0
$$281$$ −11.4600 −0.683645 −0.341822 0.939765i $$-0.611044\pi$$
−0.341822 + 0.939765i $$0.611044\pi$$
$$282$$ 0 0
$$283$$ 21.7250 1.29142 0.645708 0.763585i $$-0.276563\pi$$
0.645708 + 0.763585i $$0.276563\pi$$
$$284$$ 0 0
$$285$$ 1.81361 0.107429
$$286$$ 0 0
$$287$$ 54.3205 3.20644
$$288$$ 0 0
$$289$$ −16.4947 −0.970278
$$290$$ 0 0
$$291$$ −7.15667 −0.419532
$$292$$ 0 0
$$293$$ 5.91136 0.345345 0.172673 0.984979i $$-0.444760\pi$$
0.172673 + 0.984979i $$0.444760\pi$$
$$294$$ 0 0
$$295$$ −11.3869 −0.662972
$$296$$ 0 0
$$297$$ −2.84333 −0.164986
$$298$$ 0 0
$$299$$ −17.3275 −1.00207
$$300$$ 0 0
$$301$$ 28.6761 1.65286
$$302$$ 0 0
$$303$$ −11.2544 −0.646550
$$304$$ 0 0
$$305$$ 9.04888 0.518137
$$306$$ 0 0
$$307$$ 29.7194 1.69618 0.848089 0.529854i $$-0.177753\pi$$
0.848089 + 0.529854i $$0.177753\pi$$
$$308$$ 0 0
$$309$$ −14.7839 −0.841026
$$310$$ 0 0
$$311$$ −0.407530 −0.0231089 −0.0115544 0.999933i $$-0.503678\pi$$
−0.0115544 + 0.999933i $$0.503678\pi$$
$$312$$ 0 0
$$313$$ −0.338044 −0.0191074 −0.00955370 0.999954i $$-0.503041\pi$$
−0.00955370 + 0.999954i $$0.503041\pi$$
$$314$$ 0 0
$$315$$ −1.42166 −0.0801016
$$316$$ 0 0
$$317$$ 8.75468 0.491712 0.245856 0.969306i $$-0.420931\pi$$
0.245856 + 0.969306i $$0.420931\pi$$
$$318$$ 0 0
$$319$$ 3.78440 0.211886
$$320$$ 0 0
$$321$$ −23.7003 −1.32282
$$322$$ 0 0
$$323$$ 0.710831 0.0395517
$$324$$ 0 0
$$325$$ 6.39194 0.354561
$$326$$ 0 0
$$327$$ −21.0242 −1.16264
$$328$$ 0 0
$$329$$ 5.68665 0.313515
$$330$$ 0 0
$$331$$ −23.8675 −1.31188 −0.655938 0.754814i $$-0.727727\pi$$
−0.655938 + 0.754814i $$0.727727\pi$$
$$332$$ 0 0
$$333$$ 2.63224 0.144246
$$334$$ 0 0
$$335$$ −2.97028 −0.162284
$$336$$ 0 0
$$337$$ −10.0439 −0.547124 −0.273562 0.961854i $$-0.588202\pi$$
−0.273562 + 0.961854i $$0.588202\pi$$
$$338$$ 0 0
$$339$$ 27.0872 1.47117
$$340$$ 0 0
$$341$$ −0.822200 −0.0445246
$$342$$ 0 0
$$343$$ −50.0036 −2.69994
$$344$$ 0 0
$$345$$ 4.91638 0.264689
$$346$$ 0 0
$$347$$ 15.6272 0.838913 0.419456 0.907775i $$-0.362221\pi$$
0.419456 + 0.907775i $$0.362221\pi$$
$$348$$ 0 0
$$349$$ −17.2544 −0.923608 −0.461804 0.886982i $$-0.652798\pi$$
−0.461804 + 0.886982i $$0.652798\pi$$
$$350$$ 0 0
$$351$$ 31.4252 1.67735
$$352$$ 0 0
$$353$$ 27.2197 1.44876 0.724379 0.689402i $$-0.242127\pi$$
0.724379 + 0.689402i $$0.242127\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −6.33804 −0.335445
$$358$$ 0 0
$$359$$ −7.18137 −0.379018 −0.189509 0.981879i $$-0.560690\pi$$
−0.189509 + 0.981879i $$0.560690\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 19.3431 1.01525
$$364$$ 0 0
$$365$$ −9.38692 −0.491334
$$366$$ 0 0
$$367$$ 9.15667 0.477975 0.238987 0.971023i $$-0.423185\pi$$
0.238987 + 0.971023i $$0.423185\pi$$
$$368$$ 0 0
$$369$$ −3.19499 −0.166324
$$370$$ 0 0
$$371$$ 65.2580 3.38803
$$372$$ 0 0
$$373$$ 8.75468 0.453300 0.226650 0.973976i $$-0.427223\pi$$
0.226650 + 0.973976i $$0.427223\pi$$
$$374$$ 0 0
$$375$$ −1.81361 −0.0936542
$$376$$ 0 0
$$377$$ −41.8263 −2.15416
$$378$$ 0 0
$$379$$ 12.9164 0.663470 0.331735 0.943373i $$-0.392366\pi$$
0.331735 + 0.943373i $$0.392366\pi$$
$$380$$ 0 0
$$381$$ −21.5577 −1.10444
$$382$$ 0 0
$$383$$ −1.64280 −0.0839431 −0.0419716 0.999119i $$-0.513364\pi$$
−0.0419716 + 0.999119i $$0.513364\pi$$
$$384$$ 0 0
$$385$$ 2.84333 0.144909
$$386$$ 0 0
$$387$$ −1.68665 −0.0857373
$$388$$ 0 0
$$389$$ 11.8328 0.599945 0.299972 0.953948i $$-0.403023\pi$$
0.299972 + 0.953948i $$0.403023\pi$$
$$390$$ 0 0
$$391$$ 1.92694 0.0974498
$$392$$ 0 0
$$393$$ −16.6066 −0.837692
$$394$$ 0 0
$$395$$ 4.37279 0.220019
$$396$$ 0 0
$$397$$ 24.1361 1.21135 0.605677 0.795710i $$-0.292902\pi$$
0.605677 + 0.795710i $$0.292902\pi$$
$$398$$ 0 0
$$399$$ −8.91638 −0.446377
$$400$$ 0 0
$$401$$ 19.9789 0.997697 0.498849 0.866689i $$-0.333756\pi$$
0.498849 + 0.866689i $$0.333756\pi$$
$$402$$ 0 0
$$403$$ 9.08719 0.452665
$$404$$ 0 0
$$405$$ −9.78389 −0.486165
$$406$$ 0 0
$$407$$ −5.26447 −0.260950
$$408$$ 0 0
$$409$$ 5.66553 0.280142 0.140071 0.990141i $$-0.455267\pi$$
0.140071 + 0.990141i $$0.455267\pi$$
$$410$$ 0 0
$$411$$ −24.2786 −1.19758
$$412$$ 0 0
$$413$$ 55.9824 2.75472
$$414$$ 0 0
$$415$$ 0.372787 0.0182994
$$416$$ 0 0
$$417$$ 6.20555 0.303887
$$418$$ 0 0
$$419$$ −22.6550 −1.10677 −0.553384 0.832926i $$-0.686664\pi$$
−0.553384 + 0.832926i $$0.686664\pi$$
$$420$$ 0 0
$$421$$ 25.5330 1.24440 0.622202 0.782857i $$-0.286238\pi$$
0.622202 + 0.782857i $$0.286238\pi$$
$$422$$ 0 0
$$423$$ −0.334474 −0.0162627
$$424$$ 0 0
$$425$$ −0.710831 −0.0344804
$$426$$ 0 0
$$427$$ −44.4877 −2.15291
$$428$$ 0 0
$$429$$ 6.70436 0.323689
$$430$$ 0 0
$$431$$ −13.7944 −0.664455 −0.332228 0.943199i $$-0.607800\pi$$
−0.332228 + 0.943199i $$0.607800\pi$$
$$432$$ 0 0
$$433$$ −29.4444 −1.41501 −0.707504 0.706710i $$-0.750179\pi$$
−0.707504 + 0.706710i $$0.750179\pi$$
$$434$$ 0 0
$$435$$ 11.8675 0.569003
$$436$$ 0 0
$$437$$ 2.71083 0.129677
$$438$$ 0 0
$$439$$ −34.5472 −1.64885 −0.824423 0.565974i $$-0.808500\pi$$
−0.824423 + 0.565974i $$0.808500\pi$$
$$440$$ 0 0
$$441$$ 4.96526 0.236441
$$442$$ 0 0
$$443$$ 23.5577 1.11926 0.559631 0.828742i $$-0.310943\pi$$
0.559631 + 0.828742i $$0.310943\pi$$
$$444$$ 0 0
$$445$$ −16.6167 −0.787705
$$446$$ 0 0
$$447$$ −6.09775 −0.288414
$$448$$ 0 0
$$449$$ −7.49115 −0.353529 −0.176765 0.984253i $$-0.556563\pi$$
−0.176765 + 0.984253i $$0.556563\pi$$
$$450$$ 0 0
$$451$$ 6.38997 0.300892
$$452$$ 0 0
$$453$$ −38.4394 −1.80604
$$454$$ 0 0
$$455$$ −31.4252 −1.47324
$$456$$ 0 0
$$457$$ 38.5819 1.80479 0.902393 0.430914i $$-0.141809\pi$$
0.902393 + 0.430914i $$0.141809\pi$$
$$458$$ 0 0
$$459$$ −3.49472 −0.163119
$$460$$ 0 0
$$461$$ 31.9789 1.48940 0.744702 0.667397i $$-0.232591\pi$$
0.744702 + 0.667397i $$0.232591\pi$$
$$462$$ 0 0
$$463$$ 30.8122 1.43196 0.715981 0.698120i $$-0.245980\pi$$
0.715981 + 0.698120i $$0.245980\pi$$
$$464$$ 0 0
$$465$$ −2.57834 −0.119568
$$466$$ 0 0
$$467$$ −41.4600 −1.91854 −0.959269 0.282493i $$-0.908839\pi$$
−0.959269 + 0.282493i $$0.908839\pi$$
$$468$$ 0 0
$$469$$ 14.6030 0.674305
$$470$$ 0 0
$$471$$ −21.3905 −0.985622
$$472$$ 0 0
$$473$$ 3.37330 0.155105
$$474$$ 0 0
$$475$$ −1.00000 −0.0458831
$$476$$ 0 0
$$477$$ −3.83830 −0.175744
$$478$$ 0 0
$$479$$ −9.93051 −0.453737 −0.226868 0.973925i $$-0.572849\pi$$
−0.226868 + 0.973925i $$0.572849\pi$$
$$480$$ 0 0
$$481$$ 58.1844 2.65298
$$482$$ 0 0
$$483$$ −24.1708 −1.09981
$$484$$ 0 0
$$485$$ 3.94610 0.179183
$$486$$ 0 0
$$487$$ −19.7789 −0.896266 −0.448133 0.893967i $$-0.647911\pi$$
−0.448133 + 0.893967i $$0.647911\pi$$
$$488$$ 0 0
$$489$$ −25.8328 −1.16820
$$490$$ 0 0
$$491$$ 16.1461 0.728664 0.364332 0.931269i $$-0.381297\pi$$
0.364332 + 0.931269i $$0.381297\pi$$
$$492$$ 0 0
$$493$$ 4.65139 0.209488
$$494$$ 0 0
$$495$$ −0.167237 −0.00751674
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 38.8222 1.73792 0.868960 0.494882i $$-0.164789\pi$$
0.868960 + 0.494882i $$0.164789\pi$$
$$500$$ 0 0
$$501$$ 13.5577 0.605715
$$502$$ 0 0
$$503$$ 12.5436 0.559291 0.279646 0.960103i $$-0.409783\pi$$
0.279646 + 0.960103i $$0.409783\pi$$
$$504$$ 0 0
$$505$$ 6.20555 0.276143
$$506$$ 0 0
$$507$$ −50.5215 −2.24374
$$508$$ 0 0
$$509$$ −25.0278 −1.10934 −0.554668 0.832072i $$-0.687155\pi$$
−0.554668 + 0.832072i $$0.687155\pi$$
$$510$$ 0 0
$$511$$ 46.1497 2.04154
$$512$$ 0 0
$$513$$ −4.91638 −0.217064
$$514$$ 0 0
$$515$$ 8.15165 0.359205
$$516$$ 0 0
$$517$$ 0.668948 0.0294203
$$518$$ 0 0
$$519$$ 7.35218 0.322725
$$520$$ 0 0
$$521$$ −16.6550 −0.729667 −0.364834 0.931073i $$-0.618874\pi$$
−0.364834 + 0.931073i $$0.618874\pi$$
$$522$$ 0 0
$$523$$ −24.3608 −1.06522 −0.532611 0.846360i $$-0.678789\pi$$
−0.532611 + 0.846360i $$0.678789\pi$$
$$524$$ 0 0
$$525$$ 8.91638 0.389143
$$526$$ 0 0
$$527$$ −1.01056 −0.0440208
$$528$$ 0 0
$$529$$ −15.6514 −0.680495
$$530$$ 0 0
$$531$$ −3.29274 −0.142893
$$532$$ 0 0
$$533$$ −70.6238 −3.05906
$$534$$ 0 0
$$535$$ 13.0680 0.564980
$$536$$ 0 0
$$537$$ 5.35218 0.230964
$$538$$ 0 0
$$539$$ −9.93051 −0.427738
$$540$$ 0 0
$$541$$ −15.3622 −0.660474 −0.330237 0.943898i $$-0.607129\pi$$
−0.330237 + 0.943898i $$0.607129\pi$$
$$542$$ 0 0
$$543$$ 17.5295 0.752261
$$544$$ 0 0
$$545$$ 11.5925 0.496567
$$546$$ 0 0
$$547$$ 26.5527 1.13531 0.567656 0.823266i $$-0.307850\pi$$
0.567656 + 0.823266i $$0.307850\pi$$
$$548$$ 0 0
$$549$$ 2.61665 0.111676
$$550$$ 0 0
$$551$$ 6.54359 0.278767
$$552$$ 0 0
$$553$$ −21.4983 −0.914200
$$554$$ 0 0
$$555$$ −16.5089 −0.700762
$$556$$ 0 0
$$557$$ 28.9583 1.22700 0.613501 0.789694i $$-0.289761\pi$$
0.613501 + 0.789694i $$0.289761\pi$$
$$558$$ 0 0
$$559$$ −37.2827 −1.57689
$$560$$ 0 0
$$561$$ −0.745574 −0.0314782
$$562$$ 0 0
$$563$$ 1.64280 0.0692357 0.0346179 0.999401i $$-0.488979\pi$$
0.0346179 + 0.999401i $$0.488979\pi$$
$$564$$ 0 0
$$565$$ −14.9355 −0.628343
$$566$$ 0 0
$$567$$ 48.1013 2.02007
$$568$$ 0 0
$$569$$ 14.3416 0.601232 0.300616 0.953745i $$-0.402808\pi$$
0.300616 + 0.953745i $$0.402808\pi$$
$$570$$ 0 0
$$571$$ 39.0177 1.63284 0.816420 0.577458i $$-0.195955\pi$$
0.816420 + 0.577458i $$0.195955\pi$$
$$572$$ 0 0
$$573$$ 28.8469 1.20510
$$574$$ 0 0
$$575$$ −2.71083 −0.113049
$$576$$ 0 0
$$577$$ 25.4947 1.06136 0.530680 0.847573i $$-0.321937\pi$$
0.530680 + 0.847573i $$0.321937\pi$$
$$578$$ 0 0
$$579$$ −28.3133 −1.17666
$$580$$ 0 0
$$581$$ −1.83276 −0.0760358
$$582$$ 0 0
$$583$$ 7.67661 0.317933
$$584$$ 0 0
$$585$$ 1.84835 0.0764198
$$586$$ 0 0
$$587$$ 9.72496 0.401392 0.200696 0.979654i $$-0.435680\pi$$
0.200696 + 0.979654i $$0.435680\pi$$
$$588$$ 0 0
$$589$$ −1.42166 −0.0585786
$$590$$ 0 0
$$591$$ −24.7839 −1.01947
$$592$$ 0 0
$$593$$ −13.0388 −0.535441 −0.267720 0.963497i $$-0.586270\pi$$
−0.267720 + 0.963497i $$0.586270\pi$$
$$594$$ 0 0
$$595$$ 3.49472 0.143269
$$596$$ 0 0
$$597$$ −16.1708 −0.661827
$$598$$ 0 0
$$599$$ −18.6861 −0.763495 −0.381747 0.924267i $$-0.624678\pi$$
−0.381747 + 0.924267i $$0.624678\pi$$
$$600$$ 0 0
$$601$$ 12.7355 0.519493 0.259747 0.965677i $$-0.416361\pi$$
0.259747 + 0.965677i $$0.416361\pi$$
$$602$$ 0 0
$$603$$ −0.858912 −0.0349776
$$604$$ 0 0
$$605$$ −10.6655 −0.433615
$$606$$ 0 0
$$607$$ 23.0645 0.936158 0.468079 0.883687i $$-0.344946\pi$$
0.468079 + 0.883687i $$0.344946\pi$$
$$608$$ 0 0
$$609$$ −58.3452 −2.36427
$$610$$ 0 0
$$611$$ −7.39340 −0.299105
$$612$$ 0 0
$$613$$ −25.7038 −1.03817 −0.519084 0.854723i $$-0.673727\pi$$
−0.519084 + 0.854723i $$0.673727\pi$$
$$614$$ 0 0
$$615$$ 20.0383 0.808023
$$616$$ 0 0
$$617$$ −0.205550 −0.00827514 −0.00413757 0.999991i $$-0.501317\pi$$
−0.00413757 + 0.999991i $$0.501317\pi$$
$$618$$ 0 0
$$619$$ 11.0872 0.445632 0.222816 0.974861i $$-0.428475\pi$$
0.222816 + 0.974861i $$0.428475\pi$$
$$620$$ 0 0
$$621$$ −13.3275 −0.534813
$$622$$ 0 0
$$623$$ 81.6938 3.27299
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −1.04888 −0.0418881
$$628$$ 0 0
$$629$$ −6.47054 −0.257997
$$630$$ 0 0
$$631$$ 10.4806 0.417226 0.208613 0.977998i $$-0.433105\pi$$
0.208613 + 0.977998i $$0.433105\pi$$
$$632$$ 0 0
$$633$$ 35.9058 1.42713
$$634$$ 0 0
$$635$$ 11.8867 0.471708
$$636$$ 0 0
$$637$$ 109.755 4.34864
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 37.3694 1.47600 0.738001 0.674800i $$-0.235770\pi$$
0.738001 + 0.674800i $$0.235770\pi$$
$$642$$ 0 0
$$643$$ −42.0172 −1.65700 −0.828498 0.559992i $$-0.810804\pi$$
−0.828498 + 0.559992i $$0.810804\pi$$
$$644$$ 0 0
$$645$$ 10.5783 0.416522
$$646$$ 0 0
$$647$$ 23.4635 0.922447 0.461224 0.887284i $$-0.347411\pi$$
0.461224 + 0.887284i $$0.347411\pi$$
$$648$$ 0 0
$$649$$ 6.58548 0.258503
$$650$$ 0 0
$$651$$ 12.6761 0.496815
$$652$$ 0 0
$$653$$ −33.6272 −1.31593 −0.657967 0.753047i $$-0.728583\pi$$
−0.657967 + 0.753047i $$0.728583\pi$$
$$654$$ 0 0
$$655$$ 9.15667 0.357781
$$656$$ 0 0
$$657$$ −2.71440 −0.105899
$$658$$ 0 0
$$659$$ −17.9653 −0.699827 −0.349914 0.936782i $$-0.613789\pi$$
−0.349914 + 0.936782i $$0.613789\pi$$
$$660$$ 0 0
$$661$$ −15.5542 −0.604987 −0.302493 0.953152i $$-0.597819\pi$$
−0.302493 + 0.953152i $$0.597819\pi$$
$$662$$ 0 0
$$663$$ 8.24029 0.320026
$$664$$ 0 0
$$665$$ 4.91638 0.190649
$$666$$ 0 0
$$667$$ 17.7386 0.686841
$$668$$ 0 0
$$669$$ 2.85337 0.110318
$$670$$ 0 0
$$671$$ −5.23330 −0.202029
$$672$$ 0 0
$$673$$ 12.2111 0.470703 0.235351 0.971910i $$-0.424376\pi$$
0.235351 + 0.971910i $$0.424376\pi$$
$$674$$ 0 0
$$675$$ 4.91638 0.189232
$$676$$ 0 0
$$677$$ 43.0680 1.65524 0.827619 0.561290i $$-0.189695\pi$$
0.827619 + 0.561290i $$0.189695\pi$$
$$678$$ 0 0
$$679$$ −19.4005 −0.744524
$$680$$ 0 0
$$681$$ −50.8852 −1.94993
$$682$$ 0 0
$$683$$ 45.3850 1.73661 0.868303 0.496033i $$-0.165211\pi$$
0.868303 + 0.496033i $$0.165211\pi$$
$$684$$ 0 0
$$685$$ 13.3869 0.511488
$$686$$ 0 0
$$687$$ 4.48059 0.170945
$$688$$ 0 0
$$689$$ −84.8440 −3.23230
$$690$$ 0 0
$$691$$ 0.508852 0.0193576 0.00967882 0.999953i $$-0.496919\pi$$
0.00967882 + 0.999953i $$0.496919\pi$$
$$692$$ 0 0
$$693$$ 0.822200 0.0312328
$$694$$ 0 0
$$695$$ −3.42166 −0.129791
$$696$$ 0 0
$$697$$ 7.85389 0.297487
$$698$$ 0 0
$$699$$ −5.72496 −0.216538
$$700$$ 0 0
$$701$$ 40.1844 1.51774 0.758872 0.651239i $$-0.225751\pi$$
0.758872 + 0.651239i $$0.225751\pi$$
$$702$$ 0 0
$$703$$ −9.10278 −0.343318
$$704$$ 0 0
$$705$$ 2.09775 0.0790059
$$706$$ 0 0
$$707$$ −30.5089 −1.14740
$$708$$ 0 0
$$709$$ −20.0978 −0.754787 −0.377393 0.926053i $$-0.623180\pi$$
−0.377393 + 0.926053i $$0.623180\pi$$
$$710$$ 0 0
$$711$$ 1.26447 0.0474214
$$712$$ 0 0
$$713$$ −3.85389 −0.144329
$$714$$ 0 0
$$715$$ −3.69670 −0.138249
$$716$$ 0 0
$$717$$ 4.98587 0.186201
$$718$$ 0 0
$$719$$ 35.5925 1.32738 0.663688 0.748010i $$-0.268990\pi$$
0.663688 + 0.748010i $$0.268990\pi$$
$$720$$ 0 0
$$721$$ −40.0766 −1.49253
$$722$$ 0 0
$$723$$ −16.1078 −0.599055
$$724$$ 0 0
$$725$$ −6.54359 −0.243023
$$726$$ 0 0
$$727$$ −20.1708 −0.748094 −0.374047 0.927410i $$-0.622030\pi$$
−0.374047 + 0.927410i $$0.622030\pi$$
$$728$$ 0 0
$$729$$ 23.9200 0.885924
$$730$$ 0 0
$$731$$ 4.14611 0.153349
$$732$$ 0 0
$$733$$ −30.1461 −1.11347 −0.556736 0.830689i $$-0.687947\pi$$
−0.556736 + 0.830689i $$0.687947\pi$$
$$734$$ 0 0
$$735$$ −31.1411 −1.14866
$$736$$ 0 0
$$737$$ 1.71782 0.0632768
$$738$$ 0 0
$$739$$ −26.0283 −0.957465 −0.478733 0.877961i $$-0.658904\pi$$
−0.478733 + 0.877961i $$0.658904\pi$$
$$740$$ 0 0
$$741$$ 11.5925 0.425860
$$742$$ 0 0
$$743$$ 0.221136 0.00811269 0.00405635 0.999992i $$-0.498709\pi$$
0.00405635 + 0.999992i $$0.498709\pi$$
$$744$$ 0 0
$$745$$ 3.36222 0.123182
$$746$$ 0 0
$$747$$ 0.107798 0.00394413
$$748$$ 0 0
$$749$$ −64.2474 −2.34755
$$750$$ 0 0
$$751$$ 41.9406 1.53043 0.765216 0.643773i $$-0.222632\pi$$
0.765216 + 0.643773i $$0.222632\pi$$
$$752$$ 0 0
$$753$$ 11.4499 0.417259
$$754$$ 0 0
$$755$$ 21.1950 0.771365
$$756$$ 0 0
$$757$$ 26.7456 0.972084 0.486042 0.873935i $$-0.338440\pi$$
0.486042 + 0.873935i $$0.338440\pi$$
$$758$$ 0 0
$$759$$ −2.84333 −0.103206
$$760$$ 0 0
$$761$$ −11.6519 −0.422381 −0.211191 0.977445i $$-0.567734\pi$$
−0.211191 + 0.977445i $$0.567734\pi$$
$$762$$ 0 0
$$763$$ −56.9930 −2.06329
$$764$$ 0 0
$$765$$ −0.205550 −0.00743168
$$766$$ 0 0
$$767$$ −72.7846 −2.62810
$$768$$ 0 0
$$769$$ −9.28917 −0.334976 −0.167488 0.985874i $$-0.553566\pi$$
−0.167488 + 0.985874i $$0.553566\pi$$
$$770$$ 0 0
$$771$$ −14.2156 −0.511962
$$772$$ 0 0
$$773$$ 10.3225 0.371273 0.185637 0.982618i $$-0.440565\pi$$
0.185637 + 0.982618i $$0.440565\pi$$
$$774$$ 0 0
$$775$$ 1.42166 0.0510676
$$776$$ 0 0
$$777$$ 81.1638 2.91174
$$778$$ 0 0
$$779$$ 11.0489 0.395867
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −32.1708 −1.14969
$$784$$ 0 0
$$785$$ 11.7944 0.420962
$$786$$ 0 0
$$787$$ −49.8902 −1.77839 −0.889197 0.457524i $$-0.848736\pi$$
−0.889197 + 0.457524i $$0.848736\pi$$
$$788$$ 0 0
$$789$$ −25.1567 −0.895601
$$790$$ 0 0
$$791$$ 73.4288 2.61083
$$792$$ 0 0
$$793$$ 57.8399 2.05396
$$794$$ 0 0
$$795$$ 24.0731 0.853783
$$796$$ 0 0
$$797$$ −28.7436 −1.01815 −0.509075 0.860722i $$-0.670013\pi$$
−0.509075 + 0.860722i $$0.670013\pi$$
$$798$$ 0 0
$$799$$ 0.822200 0.0290874
$$800$$ 0 0
$$801$$ −4.80501 −0.169777
$$802$$ 0 0
$$803$$ 5.42880 0.191578
$$804$$ 0 0
$$805$$ 13.3275 0.469732
$$806$$ 0 0
$$807$$ −25.3905 −0.893788
$$808$$ 0 0
$$809$$ 38.5124 1.35402 0.677012 0.735972i $$-0.263274\pi$$
0.677012 + 0.735972i $$0.263274\pi$$
$$810$$ 0 0
$$811$$ 28.1013 0.986771 0.493385 0.869811i $$-0.335759\pi$$
0.493385 + 0.869811i $$0.335759\pi$$
$$812$$ 0 0
$$813$$ 7.38692 0.259071
$$814$$ 0 0
$$815$$ 14.2439 0.498941
$$816$$ 0 0
$$817$$ 5.83276 0.204063
$$818$$ 0 0
$$819$$ −9.08719 −0.317532
$$820$$ 0 0
$$821$$ −22.2650 −0.777053 −0.388527 0.921437i $$-0.627016\pi$$
−0.388527 + 0.921437i $$0.627016\pi$$
$$822$$ 0 0
$$823$$ 39.8363 1.38861 0.694304 0.719682i $$-0.255712\pi$$
0.694304 + 0.719682i $$0.255712\pi$$
$$824$$ 0 0
$$825$$ 1.04888 0.0365172
$$826$$ 0 0
$$827$$ 48.5874 1.68955 0.844776 0.535121i $$-0.179734\pi$$
0.844776 + 0.535121i $$0.179734\pi$$
$$828$$ 0 0
$$829$$ 45.9058 1.59437 0.797187 0.603732i $$-0.206320\pi$$
0.797187 + 0.603732i $$0.206320\pi$$
$$830$$ 0 0
$$831$$ 29.2645 1.01517
$$832$$ 0 0
$$833$$ −12.2056 −0.422897
$$834$$ 0 0
$$835$$ −7.47556 −0.258702
$$836$$ 0 0
$$837$$ 6.98944 0.241590
$$838$$ 0 0
$$839$$ −27.9688 −0.965591 −0.482796 0.875733i $$-0.660379\pi$$
−0.482796 + 0.875733i $$0.660379\pi$$
$$840$$ 0 0
$$841$$ 13.8186 0.476504
$$842$$ 0 0
$$843$$ 20.7839 0.715835
$$844$$ 0 0
$$845$$ 27.8569 0.958308
$$846$$ 0 0
$$847$$ 52.4358 1.80172
$$848$$ 0 0
$$849$$ −39.4005 −1.35222
$$850$$ 0 0
$$851$$ −24.6761 −0.845885
$$852$$ 0 0
$$853$$ −7.56777 −0.259116 −0.129558 0.991572i $$-0.541356\pi$$
−0.129558 + 0.991572i $$0.541356\pi$$
$$854$$ 0 0
$$855$$ −0.289169 −0.00988936
$$856$$ 0 0
$$857$$ 21.4756 0.733591 0.366796 0.930302i $$-0.380455\pi$$
0.366796 + 0.930302i $$0.380455\pi$$
$$858$$ 0 0
$$859$$ 33.0177 1.12655 0.563275 0.826270i $$-0.309541\pi$$
0.563275 + 0.826270i $$0.309541\pi$$
$$860$$ 0 0
$$861$$ −98.5160 −3.35742
$$862$$ 0 0
$$863$$ 18.6605 0.635211 0.317605 0.948223i $$-0.397121\pi$$
0.317605 + 0.948223i $$0.397121\pi$$
$$864$$ 0 0
$$865$$ −4.05390 −0.137837
$$866$$ 0 0
$$867$$ 29.9149 1.01596
$$868$$ 0 0
$$869$$ −2.52894 −0.0857886
$$870$$ 0 0
$$871$$ −18.9859 −0.643312
$$872$$ 0 0
$$873$$ 1.14109 0.0386200
$$874$$ 0 0
$$875$$ −4.91638 −0.166204
$$876$$ 0 0
$$877$$ 57.7925 1.95151 0.975757 0.218858i $$-0.0702332\pi$$
0.975757 + 0.218858i $$0.0702332\pi$$
$$878$$ 0 0
$$879$$ −10.7209 −0.361606
$$880$$ 0 0
$$881$$ 36.2338 1.22075 0.610374 0.792113i $$-0.291019\pi$$
0.610374 + 0.792113i $$0.291019\pi$$
$$882$$ 0 0
$$883$$ 2.98944 0.100603 0.0503013 0.998734i $$-0.483982\pi$$
0.0503013 + 0.998734i $$0.483982\pi$$
$$884$$ 0 0
$$885$$ 20.6514 0.694189
$$886$$ 0 0
$$887$$ −42.2494 −1.41860 −0.709298 0.704909i $$-0.750988\pi$$
−0.709298 + 0.704909i $$0.750988\pi$$
$$888$$ 0 0
$$889$$ −58.4394 −1.95999
$$890$$ 0 0
$$891$$ 5.65838 0.189563
$$892$$ 0 0
$$893$$ 1.15667 0.0387066
$$894$$ 0 0
$$895$$ −2.95112 −0.0986452
$$896$$ 0 0
$$897$$ 31.4252 1.04926
$$898$$ 0 0
$$899$$ −9.30279 −0.310265
$$900$$ 0 0
$$901$$ 9.43528 0.314335
$$902$$ 0 0
$$903$$ −52.0071 −1.73069
$$904$$ 0 0
$$905$$ −9.66553 −0.321293
$$906$$ 0 0
$$907$$ −1.97080 −0.0654392 −0.0327196 0.999465i $$-0.510417\pi$$
−0.0327196 + 0.999465i $$0.510417\pi$$
$$908$$ 0 0
$$909$$ 1.79445 0.0595181
$$910$$ 0 0
$$911$$ 9.98995 0.330982 0.165491 0.986211i $$-0.447079\pi$$
0.165491 + 0.986211i $$0.447079\pi$$
$$912$$ 0 0
$$913$$ −0.215597 −0.00713520
$$914$$ 0 0
$$915$$ −16.4111 −0.542534
$$916$$ 0 0
$$917$$ −45.0177 −1.48662
$$918$$ 0 0
$$919$$ 4.24029 0.139874 0.0699372 0.997551i $$-0.477720\pi$$
0.0699372 + 0.997551i $$0.477720\pi$$
$$920$$ 0 0
$$921$$ −53.8993 −1.77604
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 9.10278 0.299297
$$926$$ 0 0
$$927$$ 2.35720 0.0774206
$$928$$ 0 0
$$929$$ −13.1531 −0.431539 −0.215770 0.976444i $$-0.569226\pi$$
−0.215770 + 0.976444i $$0.569226\pi$$
$$930$$ 0 0
$$931$$ −17.1708 −0.562750
$$932$$ 0 0
$$933$$ 0.739098 0.0241970
$$934$$ 0 0
$$935$$ 0.411100 0.0134444
$$936$$ 0 0
$$937$$ −38.5436 −1.25916 −0.629582 0.776934i $$-0.716774\pi$$
−0.629582 + 0.776934i $$0.716774\pi$$
$$938$$ 0 0
$$939$$ 0.613080 0.0200071
$$940$$ 0 0
$$941$$ 2.91638 0.0950713 0.0475357 0.998870i $$-0.484863\pi$$
0.0475357 + 0.998870i $$0.484863\pi$$
$$942$$ 0 0
$$943$$ 29.9516 0.975360
$$944$$ 0 0
$$945$$ −24.1708 −0.786276
$$946$$ 0 0
$$947$$ −58.4011 −1.89778 −0.948890 0.315608i $$-0.897792\pi$$
−0.948890 + 0.315608i $$0.897792\pi$$
$$948$$ 0 0
$$949$$ −60.0007 −1.94770
$$950$$ 0 0
$$951$$ −15.8776 −0.514865
$$952$$ 0 0
$$953$$ −36.8378 −1.19329 −0.596646 0.802504i $$-0.703501\pi$$
−0.596646 + 0.802504i $$0.703501\pi$$
$$954$$ 0 0
$$955$$ −15.9058 −0.514700
$$956$$ 0 0
$$957$$ −6.86342 −0.221863
$$958$$ 0 0
$$959$$ −65.8152 −2.12528
$$960$$ 0 0
$$961$$ −28.9789 −0.934802
$$962$$ 0 0
$$963$$ 3.77886 0.121772
$$964$$ 0 0
$$965$$ 15.6116 0.502556
$$966$$ 0 0
$$967$$ −43.0278 −1.38368 −0.691840 0.722051i $$-0.743199\pi$$
−0.691840 + 0.722051i $$0.743199\pi$$
$$968$$ 0 0
$$969$$ −1.28917 −0.0414141
$$970$$ 0 0
$$971$$ 49.0177 1.57305 0.786526 0.617557i $$-0.211877\pi$$
0.786526 + 0.617557i $$0.211877\pi$$
$$972$$ 0 0
$$973$$ 16.8222 0.539295
$$974$$ 0 0
$$975$$ −11.5925 −0.371256
$$976$$ 0 0
$$977$$ 20.2383 0.647481 0.323741 0.946146i $$-0.395059\pi$$
0.323741 + 0.946146i $$0.395059\pi$$
$$978$$ 0 0
$$979$$ 9.61003 0.307138
$$980$$ 0 0
$$981$$ 3.35218 0.107027
$$982$$ 0 0
$$983$$ 25.6811 0.819100 0.409550 0.912288i $$-0.365686\pi$$
0.409550 + 0.912288i $$0.365686\pi$$
$$984$$ 0 0
$$985$$ 13.6655 0.435420
$$986$$ 0 0
$$987$$ −10.3133 −0.328277
$$988$$ 0 0
$$989$$ 15.8116 0.502781
$$990$$ 0 0
$$991$$ 8.56829 0.272181 0.136090 0.990696i $$-0.456546\pi$$
0.136090 + 0.990696i $$0.456546\pi$$
$$992$$ 0 0
$$993$$ 43.2863 1.37365
$$994$$ 0 0
$$995$$ 8.91638 0.282668
$$996$$ 0 0
$$997$$ −29.9688 −0.949122 −0.474561 0.880223i $$-0.657393\pi$$
−0.474561 + 0.880223i $$0.657393\pi$$
$$998$$ 0 0
$$999$$ 44.7527 1.41591
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 6080.2.a.bx.1.1 3
4.3 odd 2 6080.2.a.br.1.3 3
8.3 odd 2 1520.2.a.q.1.1 3
8.5 even 2 760.2.a.i.1.3 3
24.5 odd 2 6840.2.a.bm.1.1 3
40.13 odd 4 3800.2.d.n.3649.5 6
40.19 odd 2 7600.2.a.bp.1.3 3
40.29 even 2 3800.2.a.w.1.1 3
40.37 odd 4 3800.2.d.n.3649.2 6

By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.i.1.3 3 8.5 even 2
1520.2.a.q.1.1 3 8.3 odd 2
3800.2.a.w.1.1 3 40.29 even 2
3800.2.d.n.3649.2 6 40.37 odd 4
3800.2.d.n.3649.5 6 40.13 odd 4
6080.2.a.br.1.3 3 4.3 odd 2
6080.2.a.bx.1.1 3 1.1 even 1 trivial
6840.2.a.bm.1.1 3 24.5 odd 2
7600.2.a.bp.1.3 3 40.19 odd 2