# Properties

 Label 920.2.a.j.1.5 Level $920$ Weight $2$ Character 920.1 Self dual yes Analytic conductor $7.346$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [920,2,Mod(1,920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(920, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$920 = 2^{3} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 920.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$7.34623698596$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.13955077.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - 14x^{3} - x^{2} + 32x + 16$$ x^5 - 14*x^3 - x^2 + 32*x + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.5 Root $$3.30649$$ of defining polynomial Character $$\chi$$ $$=$$ 920.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.30649 q^{3} -1.00000 q^{5} -2.55040 q^{7} +7.93288 q^{9} +O(q^{10})$$ $$q+3.30649 q^{3} -1.00000 q^{5} -2.55040 q^{7} +7.93288 q^{9} -2.72314 q^{11} +7.12637 q^{13} -3.30649 q^{15} +0.924010 q^{17} +7.51623 q^{19} -8.43286 q^{21} -1.00000 q^{23} +1.00000 q^{25} +16.3105 q^{27} -2.38248 q^{29} +0.866248 q^{31} -9.00402 q^{33} +2.55040 q^{35} +0.352855 q^{37} +23.5633 q^{39} +4.34066 q^{41} -7.93288 q^{45} -13.3239 q^{47} -0.495474 q^{49} +3.05523 q^{51} +3.99262 q^{53} +2.72314 q^{55} +24.8523 q^{57} -3.84064 q^{59} -9.14262 q^{61} -20.2320 q^{63} -7.12637 q^{65} -3.15933 q^{67} -3.30649 q^{69} -6.07883 q^{71} -11.3239 q^{73} +3.30649 q^{75} +6.94508 q^{77} -12.0593 q^{79} +30.1319 q^{81} -6.35285 q^{83} -0.924010 q^{85} -7.87765 q^{87} -9.71377 q^{89} -18.1751 q^{91} +2.86424 q^{93} -7.51623 q^{95} +8.76465 q^{97} -21.6023 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 5 q^{5} - 2 q^{7} + 13 q^{9}+O(q^{10})$$ 5 * q - 5 * q^5 - 2 * q^7 + 13 * q^9 $$5 q - 5 q^{5} - 2 q^{7} + 13 q^{9} - q^{11} + 4 q^{13} + 4 q^{17} + 7 q^{19} + 6 q^{21} - 5 q^{23} + 5 q^{25} + 3 q^{27} + 4 q^{29} + 19 q^{31} + 17 q^{33} + 2 q^{35} + 15 q^{37} + 19 q^{39} + 25 q^{41} - 13 q^{45} - 11 q^{47} + 25 q^{49} + 19 q^{51} + 3 q^{53} + q^{55} + 48 q^{57} - q^{59} - 5 q^{61} - 41 q^{63} - 4 q^{65} + 9 q^{67} + q^{71} - q^{73} + 18 q^{77} - 2 q^{79} + 57 q^{81} - 45 q^{83} - 4 q^{85} - 9 q^{87} + 6 q^{89} + 11 q^{91} - 39 q^{93} - 7 q^{95} + 25 q^{97} - 65 q^{99}+O(q^{100})$$ 5 * q - 5 * q^5 - 2 * q^7 + 13 * q^9 - q^11 + 4 * q^13 + 4 * q^17 + 7 * q^19 + 6 * q^21 - 5 * q^23 + 5 * q^25 + 3 * q^27 + 4 * q^29 + 19 * q^31 + 17 * q^33 + 2 * q^35 + 15 * q^37 + 19 * q^39 + 25 * q^41 - 13 * q^45 - 11 * q^47 + 25 * q^49 + 19 * q^51 + 3 * q^53 + q^55 + 48 * q^57 - q^59 - 5 * q^61 - 41 * q^63 - 4 * q^65 + 9 * q^67 + q^71 - q^73 + 18 * q^77 - 2 * q^79 + 57 * q^81 - 45 * q^83 - 4 * q^85 - 9 * q^87 + 6 * q^89 + 11 * q^91 - 39 * q^93 - 7 * q^95 + 25 * q^97 - 65 * 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
$$3$$ 3.30649 1.90900 0.954501 0.298206i $$-0.0963883\pi$$
0.954501 + 0.298206i $$0.0963883\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −2.55040 −0.963960 −0.481980 0.876182i $$-0.660082\pi$$
−0.481980 + 0.876182i $$0.660082\pi$$
$$8$$ 0 0
$$9$$ 7.93288 2.64429
$$10$$ 0 0
$$11$$ −2.72314 −0.821056 −0.410528 0.911848i $$-0.634656\pi$$
−0.410528 + 0.911848i $$0.634656\pi$$
$$12$$ 0 0
$$13$$ 7.12637 1.97650 0.988250 0.152845i $$-0.0488435\pi$$
0.988250 + 0.152845i $$0.0488435\pi$$
$$14$$ 0 0
$$15$$ −3.30649 −0.853732
$$16$$ 0 0
$$17$$ 0.924010 0.224105 0.112053 0.993702i $$-0.464257\pi$$
0.112053 + 0.993702i $$0.464257\pi$$
$$18$$ 0 0
$$19$$ 7.51623 1.72434 0.862171 0.506617i $$-0.169104\pi$$
0.862171 + 0.506617i $$0.169104\pi$$
$$20$$ 0 0
$$21$$ −8.43286 −1.84020
$$22$$ 0 0
$$23$$ −1.00000 −0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 16.3105 3.13896
$$28$$ 0 0
$$29$$ −2.38248 −0.442415 −0.221208 0.975227i $$-0.571000\pi$$
−0.221208 + 0.975227i $$0.571000\pi$$
$$30$$ 0 0
$$31$$ 0.866248 0.155583 0.0777913 0.996970i $$-0.475213\pi$$
0.0777913 + 0.996970i $$0.475213\pi$$
$$32$$ 0 0
$$33$$ −9.00402 −1.56740
$$34$$ 0 0
$$35$$ 2.55040 0.431096
$$36$$ 0 0
$$37$$ 0.352855 0.0580089 0.0290045 0.999579i $$-0.490766\pi$$
0.0290045 + 0.999579i $$0.490766\pi$$
$$38$$ 0 0
$$39$$ 23.5633 3.77315
$$40$$ 0 0
$$41$$ 4.34066 0.677896 0.338948 0.940805i $$-0.389929\pi$$
0.338948 + 0.940805i $$0.389929\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ 0 0
$$45$$ −7.93288 −1.18256
$$46$$ 0 0
$$47$$ −13.3239 −1.94349 −0.971746 0.236027i $$-0.924155\pi$$
−0.971746 + 0.236027i $$0.924155\pi$$
$$48$$ 0 0
$$49$$ −0.495474 −0.0707819
$$50$$ 0 0
$$51$$ 3.05523 0.427818
$$52$$ 0 0
$$53$$ 3.99262 0.548429 0.274214 0.961669i $$-0.411582\pi$$
0.274214 + 0.961669i $$0.411582\pi$$
$$54$$ 0 0
$$55$$ 2.72314 0.367187
$$56$$ 0 0
$$57$$ 24.8523 3.29177
$$58$$ 0 0
$$59$$ −3.84064 −0.500009 −0.250004 0.968245i $$-0.580432\pi$$
−0.250004 + 0.968245i $$0.580432\pi$$
$$60$$ 0 0
$$61$$ −9.14262 −1.17059 −0.585296 0.810820i $$-0.699022\pi$$
−0.585296 + 0.810820i $$0.699022\pi$$
$$62$$ 0 0
$$63$$ −20.2320 −2.54899
$$64$$ 0 0
$$65$$ −7.12637 −0.883918
$$66$$ 0 0
$$67$$ −3.15933 −0.385974 −0.192987 0.981201i $$-0.561817\pi$$
−0.192987 + 0.981201i $$0.561817\pi$$
$$68$$ 0 0
$$69$$ −3.30649 −0.398055
$$70$$ 0 0
$$71$$ −6.07883 −0.721424 −0.360712 0.932677i $$-0.617466\pi$$
−0.360712 + 0.932677i $$0.617466\pi$$
$$72$$ 0 0
$$73$$ −11.3239 −1.32536 −0.662682 0.748901i $$-0.730582\pi$$
−0.662682 + 0.748901i $$0.730582\pi$$
$$74$$ 0 0
$$75$$ 3.30649 0.381801
$$76$$ 0 0
$$77$$ 6.94508 0.791465
$$78$$ 0 0
$$79$$ −12.0593 −1.35677 −0.678386 0.734706i $$-0.737320\pi$$
−0.678386 + 0.734706i $$0.737320\pi$$
$$80$$ 0 0
$$81$$ 30.1319 3.34799
$$82$$ 0 0
$$83$$ −6.35285 −0.697316 −0.348658 0.937250i $$-0.613363\pi$$
−0.348658 + 0.937250i $$0.613363\pi$$
$$84$$ 0 0
$$85$$ −0.924010 −0.100223
$$86$$ 0 0
$$87$$ −7.87765 −0.844572
$$88$$ 0 0
$$89$$ −9.71377 −1.02966 −0.514829 0.857293i $$-0.672145\pi$$
−0.514829 + 0.857293i $$0.672145\pi$$
$$90$$ 0 0
$$91$$ −18.1751 −1.90527
$$92$$ 0 0
$$93$$ 2.86424 0.297008
$$94$$ 0 0
$$95$$ −7.51623 −0.771149
$$96$$ 0 0
$$97$$ 8.76465 0.889916 0.444958 0.895552i $$-0.353219\pi$$
0.444958 + 0.895552i $$0.353219\pi$$
$$98$$ 0 0
$$99$$ −21.6023 −2.17111
$$100$$ 0 0
$$101$$ 4.74794 0.472438 0.236219 0.971700i $$-0.424092\pi$$
0.236219 + 0.971700i $$0.424092\pi$$
$$102$$ 0 0
$$103$$ 12.6304 1.24451 0.622255 0.782814i $$-0.286217\pi$$
0.622255 + 0.782814i $$0.286217\pi$$
$$104$$ 0 0
$$105$$ 8.43286 0.822963
$$106$$ 0 0
$$107$$ −12.9658 −1.25345 −0.626727 0.779239i $$-0.715606\pi$$
−0.626727 + 0.779239i $$0.715606\pi$$
$$108$$ 0 0
$$109$$ −15.0040 −1.43712 −0.718562 0.695463i $$-0.755199\pi$$
−0.718562 + 0.695463i $$0.755199\pi$$
$$110$$ 0 0
$$111$$ 1.16671 0.110739
$$112$$ 0 0
$$113$$ 2.13496 0.200840 0.100420 0.994945i $$-0.467981\pi$$
0.100420 + 0.994945i $$0.467981\pi$$
$$114$$ 0 0
$$115$$ 1.00000 0.0932505
$$116$$ 0 0
$$117$$ 56.5326 5.22644
$$118$$ 0 0
$$119$$ −2.35659 −0.216029
$$120$$ 0 0
$$121$$ −3.58454 −0.325867
$$122$$ 0 0
$$123$$ 14.3523 1.29411
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −3.54181 −0.314285 −0.157142 0.987576i $$-0.550228\pi$$
−0.157142 + 0.987576i $$0.550228\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 19.5499 1.70808 0.854042 0.520205i $$-0.174144\pi$$
0.854042 + 0.520205i $$0.174144\pi$$
$$132$$ 0 0
$$133$$ −19.1694 −1.66220
$$134$$ 0 0
$$135$$ −16.3105 −1.40379
$$136$$ 0 0
$$137$$ 4.41544 0.377236 0.188618 0.982051i $$-0.439599\pi$$
0.188618 + 0.982051i $$0.439599\pi$$
$$138$$ 0 0
$$139$$ 21.9954 1.86563 0.932814 0.360358i $$-0.117345\pi$$
0.932814 + 0.360358i $$0.117345\pi$$
$$140$$ 0 0
$$141$$ −44.0554 −3.71013
$$142$$ 0 0
$$143$$ −19.4061 −1.62282
$$144$$ 0 0
$$145$$ 2.38248 0.197854
$$146$$ 0 0
$$147$$ −1.63828 −0.135123
$$148$$ 0 0
$$149$$ 19.0780 1.56293 0.781466 0.623947i $$-0.214472\pi$$
0.781466 + 0.623947i $$0.214472\pi$$
$$150$$ 0 0
$$151$$ −7.75273 −0.630908 −0.315454 0.948941i $$-0.602157\pi$$
−0.315454 + 0.948941i $$0.602157\pi$$
$$152$$ 0 0
$$153$$ 7.33006 0.592600
$$154$$ 0 0
$$155$$ −0.866248 −0.0695787
$$156$$ 0 0
$$157$$ 17.5129 1.39768 0.698841 0.715277i $$-0.253700\pi$$
0.698841 + 0.715277i $$0.253700\pi$$
$$158$$ 0 0
$$159$$ 13.2016 1.04695
$$160$$ 0 0
$$161$$ 2.55040 0.200999
$$162$$ 0 0
$$163$$ −2.55491 −0.200116 −0.100058 0.994982i $$-0.531903\pi$$
−0.100058 + 0.994982i $$0.531903\pi$$
$$164$$ 0 0
$$165$$ 9.00402 0.700962
$$166$$ 0 0
$$167$$ −3.61301 −0.279583 −0.139791 0.990181i $$-0.544643\pi$$
−0.139791 + 0.990181i $$0.544643\pi$$
$$168$$ 0 0
$$169$$ 37.7852 2.90655
$$170$$ 0 0
$$171$$ 59.6253 4.55966
$$172$$ 0 0
$$173$$ 12.7824 0.971827 0.485913 0.874007i $$-0.338487\pi$$
0.485913 + 0.874007i $$0.338487\pi$$
$$174$$ 0 0
$$175$$ −2.55040 −0.192792
$$176$$ 0 0
$$177$$ −12.6990 −0.954519
$$178$$ 0 0
$$179$$ 10.0149 0.748546 0.374273 0.927318i $$-0.377892\pi$$
0.374273 + 0.927318i $$0.377892\pi$$
$$180$$ 0 0
$$181$$ 9.96248 0.740505 0.370252 0.928931i $$-0.379271\pi$$
0.370252 + 0.928931i $$0.379271\pi$$
$$182$$ 0 0
$$183$$ −30.2300 −2.23466
$$184$$ 0 0
$$185$$ −0.352855 −0.0259424
$$186$$ 0 0
$$187$$ −2.51621 −0.184003
$$188$$ 0 0
$$189$$ −41.5983 −3.02583
$$190$$ 0 0
$$191$$ −6.61298 −0.478498 −0.239249 0.970958i $$-0.576901\pi$$
−0.239249 + 0.970958i $$0.576901\pi$$
$$192$$ 0 0
$$193$$ −19.0540 −1.37154 −0.685768 0.727820i $$-0.740534\pi$$
−0.685768 + 0.727820i $$0.740534\pi$$
$$194$$ 0 0
$$195$$ −23.5633 −1.68740
$$196$$ 0 0
$$197$$ −11.0886 −0.790030 −0.395015 0.918675i $$-0.629261\pi$$
−0.395015 + 0.918675i $$0.629261\pi$$
$$198$$ 0 0
$$199$$ 3.71377 0.263263 0.131631 0.991299i $$-0.457979\pi$$
0.131631 + 0.991299i $$0.457979\pi$$
$$200$$ 0 0
$$201$$ −10.4463 −0.736825
$$202$$ 0 0
$$203$$ 6.07627 0.426471
$$204$$ 0 0
$$205$$ −4.34066 −0.303165
$$206$$ 0 0
$$207$$ −7.93288 −0.551373
$$208$$ 0 0
$$209$$ −20.4677 −1.41578
$$210$$ 0 0
$$211$$ −21.7656 −1.49841 −0.749204 0.662339i $$-0.769564\pi$$
−0.749204 + 0.662339i $$0.769564\pi$$
$$212$$ 0 0
$$213$$ −20.0996 −1.37720
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −2.20928 −0.149975
$$218$$ 0 0
$$219$$ −37.4424 −2.53012
$$220$$ 0 0
$$221$$ 6.58484 0.442944
$$222$$ 0 0
$$223$$ 6.90730 0.462547 0.231273 0.972889i $$-0.425711\pi$$
0.231273 + 0.972889i $$0.425711\pi$$
$$224$$ 0 0
$$225$$ 7.93288 0.528858
$$226$$ 0 0
$$227$$ −23.0325 −1.52872 −0.764359 0.644791i $$-0.776945\pi$$
−0.764359 + 0.644791i $$0.776945\pi$$
$$228$$ 0 0
$$229$$ 12.9665 0.856854 0.428427 0.903576i $$-0.359068\pi$$
0.428427 + 0.903576i $$0.359068\pi$$
$$230$$ 0 0
$$231$$ 22.9638 1.51091
$$232$$ 0 0
$$233$$ −4.80934 −0.315071 −0.157535 0.987513i $$-0.550355\pi$$
−0.157535 + 0.987513i $$0.550355\pi$$
$$234$$ 0 0
$$235$$ 13.3239 0.869156
$$236$$ 0 0
$$237$$ −39.8738 −2.59008
$$238$$ 0 0
$$239$$ −20.4661 −1.32384 −0.661922 0.749573i $$-0.730259\pi$$
−0.661922 + 0.749573i $$0.730259\pi$$
$$240$$ 0 0
$$241$$ 26.8657 1.73057 0.865287 0.501277i $$-0.167136\pi$$
0.865287 + 0.501277i $$0.167136\pi$$
$$242$$ 0 0
$$243$$ 50.6993 3.25236
$$244$$ 0 0
$$245$$ 0.495474 0.0316546
$$246$$ 0 0
$$247$$ 53.5635 3.40816
$$248$$ 0 0
$$249$$ −21.0057 −1.33118
$$250$$ 0 0
$$251$$ −15.9625 −1.00754 −0.503771 0.863837i $$-0.668055\pi$$
−0.503771 + 0.863837i $$0.668055\pi$$
$$252$$ 0 0
$$253$$ 2.72314 0.171202
$$254$$ 0 0
$$255$$ −3.05523 −0.191326
$$256$$ 0 0
$$257$$ 3.33867 0.208261 0.104130 0.994564i $$-0.466794\pi$$
0.104130 + 0.994564i $$0.466794\pi$$
$$258$$ 0 0
$$259$$ −0.899919 −0.0559183
$$260$$ 0 0
$$261$$ −18.8999 −1.16988
$$262$$ 0 0
$$263$$ −23.3880 −1.44217 −0.721083 0.692849i $$-0.756355\pi$$
−0.721083 + 0.692849i $$0.756355\pi$$
$$264$$ 0 0
$$265$$ −3.99262 −0.245265
$$266$$ 0 0
$$267$$ −32.1185 −1.96562
$$268$$ 0 0
$$269$$ 14.1378 0.861995 0.430998 0.902353i $$-0.358162\pi$$
0.430998 + 0.902353i $$0.358162\pi$$
$$270$$ 0 0
$$271$$ 25.9283 1.57503 0.787517 0.616293i $$-0.211366\pi$$
0.787517 + 0.616293i $$0.211366\pi$$
$$272$$ 0 0
$$273$$ −60.0957 −3.63716
$$274$$ 0 0
$$275$$ −2.72314 −0.164211
$$276$$ 0 0
$$277$$ 1.07117 0.0643603 0.0321802 0.999482i $$-0.489755\pi$$
0.0321802 + 0.999482i $$0.489755\pi$$
$$278$$ 0 0
$$279$$ 6.87184 0.411406
$$280$$ 0 0
$$281$$ 12.2260 0.729341 0.364671 0.931137i $$-0.381182\pi$$
0.364671 + 0.931137i $$0.381182\pi$$
$$282$$ 0 0
$$283$$ −30.5454 −1.81573 −0.907867 0.419259i $$-0.862290\pi$$
−0.907867 + 0.419259i $$0.862290\pi$$
$$284$$ 0 0
$$285$$ −24.8523 −1.47213
$$286$$ 0 0
$$287$$ −11.0704 −0.653465
$$288$$ 0 0
$$289$$ −16.1462 −0.949777
$$290$$ 0 0
$$291$$ 28.9802 1.69885
$$292$$ 0 0
$$293$$ −8.46838 −0.494728 −0.247364 0.968923i $$-0.579564\pi$$
−0.247364 + 0.968923i $$0.579564\pi$$
$$294$$ 0 0
$$295$$ 3.84064 0.223611
$$296$$ 0 0
$$297$$ −44.4157 −2.57726
$$298$$ 0 0
$$299$$ −7.12637 −0.412129
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 15.6990 0.901885
$$304$$ 0 0
$$305$$ 9.14262 0.523505
$$306$$ 0 0
$$307$$ −1.32802 −0.0757942 −0.0378971 0.999282i $$-0.512066\pi$$
−0.0378971 + 0.999282i $$0.512066\pi$$
$$308$$ 0 0
$$309$$ 41.7623 2.37578
$$310$$ 0 0
$$311$$ 24.2971 1.37776 0.688882 0.724874i $$-0.258102\pi$$
0.688882 + 0.724874i $$0.258102\pi$$
$$312$$ 0 0
$$313$$ 20.3478 1.15013 0.575063 0.818109i $$-0.304977\pi$$
0.575063 + 0.818109i $$0.304977\pi$$
$$314$$ 0 0
$$315$$ 20.2320 1.13994
$$316$$ 0 0
$$317$$ −8.57309 −0.481513 −0.240756 0.970586i $$-0.577396\pi$$
−0.240756 + 0.970586i $$0.577396\pi$$
$$318$$ 0 0
$$319$$ 6.48781 0.363248
$$320$$ 0 0
$$321$$ −42.8714 −2.39285
$$322$$ 0 0
$$323$$ 6.94508 0.386434
$$324$$ 0 0
$$325$$ 7.12637 0.395300
$$326$$ 0 0
$$327$$ −49.6106 −2.74347
$$328$$ 0 0
$$329$$ 33.9813 1.87345
$$330$$ 0 0
$$331$$ −20.9767 −1.15299 −0.576493 0.817102i $$-0.695579\pi$$
−0.576493 + 0.817102i $$0.695579\pi$$
$$332$$ 0 0
$$333$$ 2.79915 0.153393
$$334$$ 0 0
$$335$$ 3.15933 0.172613
$$336$$ 0 0
$$337$$ −2.17314 −0.118379 −0.0591894 0.998247i $$-0.518852\pi$$
−0.0591894 + 0.998247i $$0.518852\pi$$
$$338$$ 0 0
$$339$$ 7.05922 0.383404
$$340$$ 0 0
$$341$$ −2.35891 −0.127742
$$342$$ 0 0
$$343$$ 19.1164 1.03219
$$344$$ 0 0
$$345$$ 3.30649 0.178015
$$346$$ 0 0
$$347$$ −32.1335 −1.72502 −0.862509 0.506042i $$-0.831108\pi$$
−0.862509 + 0.506042i $$0.831108\pi$$
$$348$$ 0 0
$$349$$ 13.5157 0.723479 0.361740 0.932279i $$-0.382183\pi$$
0.361740 + 0.932279i $$0.382183\pi$$
$$350$$ 0 0
$$351$$ 116.235 6.20415
$$352$$ 0 0
$$353$$ −28.0645 −1.49372 −0.746861 0.664981i $$-0.768440\pi$$
−0.746861 + 0.664981i $$0.768440\pi$$
$$354$$ 0 0
$$355$$ 6.07883 0.322631
$$356$$ 0 0
$$357$$ −7.79205 −0.412399
$$358$$ 0 0
$$359$$ 20.2340 1.06791 0.533956 0.845513i $$-0.320705\pi$$
0.533956 + 0.845513i $$0.320705\pi$$
$$360$$ 0 0
$$361$$ 37.4937 1.97336
$$362$$ 0 0
$$363$$ −11.8522 −0.622081
$$364$$ 0 0
$$365$$ 11.3239 0.592721
$$366$$ 0 0
$$367$$ 4.76666 0.248818 0.124409 0.992231i $$-0.460297\pi$$
0.124409 + 0.992231i $$0.460297\pi$$
$$368$$ 0 0
$$369$$ 34.4339 1.79256
$$370$$ 0 0
$$371$$ −10.1828 −0.528663
$$372$$ 0 0
$$373$$ −29.1510 −1.50938 −0.754690 0.656082i $$-0.772213\pi$$
−0.754690 + 0.656082i $$0.772213\pi$$
$$374$$ 0 0
$$375$$ −3.30649 −0.170746
$$376$$ 0 0
$$377$$ −16.9784 −0.874434
$$378$$ 0 0
$$379$$ 4.20856 0.216179 0.108090 0.994141i $$-0.465527\pi$$
0.108090 + 0.994141i $$0.465527\pi$$
$$380$$ 0 0
$$381$$ −11.7110 −0.599971
$$382$$ 0 0
$$383$$ 3.55206 0.181502 0.0907508 0.995874i $$-0.471073\pi$$
0.0907508 + 0.995874i $$0.471073\pi$$
$$384$$ 0 0
$$385$$ −6.94508 −0.353954
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −29.2515 −1.48311 −0.741554 0.670893i $$-0.765911\pi$$
−0.741554 + 0.670893i $$0.765911\pi$$
$$390$$ 0 0
$$391$$ −0.924010 −0.0467292
$$392$$ 0 0
$$393$$ 64.6416 3.26074
$$394$$ 0 0
$$395$$ 12.0593 0.606767
$$396$$ 0 0
$$397$$ −17.8000 −0.893356 −0.446678 0.894695i $$-0.647393\pi$$
−0.446678 + 0.894695i $$0.647393\pi$$
$$398$$ 0 0
$$399$$ −63.3834 −3.17314
$$400$$ 0 0
$$401$$ −11.6210 −0.580327 −0.290164 0.956977i $$-0.593710\pi$$
−0.290164 + 0.956977i $$0.593710\pi$$
$$402$$ 0 0
$$403$$ 6.17320 0.307509
$$404$$ 0 0
$$405$$ −30.1319 −1.49727
$$406$$ 0 0
$$407$$ −0.960871 −0.0476286
$$408$$ 0 0
$$409$$ −13.5710 −0.671041 −0.335521 0.942033i $$-0.608912\pi$$
−0.335521 + 0.942033i $$0.608912\pi$$
$$410$$ 0 0
$$411$$ 14.5996 0.720145
$$412$$ 0 0
$$413$$ 9.79516 0.481988
$$414$$ 0 0
$$415$$ 6.35285 0.311849
$$416$$ 0 0
$$417$$ 72.7277 3.56149
$$418$$ 0 0
$$419$$ 13.4959 0.659317 0.329658 0.944100i $$-0.393066\pi$$
0.329658 + 0.944100i $$0.393066\pi$$
$$420$$ 0 0
$$421$$ 37.6495 1.83492 0.917462 0.397824i $$-0.130234\pi$$
0.917462 + 0.397824i $$0.130234\pi$$
$$422$$ 0 0
$$423$$ −105.697 −5.13916
$$424$$ 0 0
$$425$$ 0.924010 0.0448211
$$426$$ 0 0
$$427$$ 23.3173 1.12840
$$428$$ 0 0
$$429$$ −64.1660 −3.09796
$$430$$ 0 0
$$431$$ 10.0568 0.484421 0.242210 0.970224i $$-0.422128\pi$$
0.242210 + 0.970224i $$0.422128\pi$$
$$432$$ 0 0
$$433$$ 2.30001 0.110531 0.0552657 0.998472i $$-0.482399\pi$$
0.0552657 + 0.998472i $$0.482399\pi$$
$$434$$ 0 0
$$435$$ 7.87765 0.377704
$$436$$ 0 0
$$437$$ −7.51623 −0.359550
$$438$$ 0 0
$$439$$ 12.4273 0.593124 0.296562 0.955014i $$-0.404160\pi$$
0.296562 + 0.955014i $$0.404160\pi$$
$$440$$ 0 0
$$441$$ −3.93053 −0.187168
$$442$$ 0 0
$$443$$ 2.18268 0.103702 0.0518510 0.998655i $$-0.483488\pi$$
0.0518510 + 0.998655i $$0.483488\pi$$
$$444$$ 0 0
$$445$$ 9.71377 0.460477
$$446$$ 0 0
$$447$$ 63.0813 2.98364
$$448$$ 0 0
$$449$$ 3.09603 0.146111 0.0730554 0.997328i $$-0.476725\pi$$
0.0730554 + 0.997328i $$0.476725\pi$$
$$450$$ 0 0
$$451$$ −11.8202 −0.556591
$$452$$ 0 0
$$453$$ −25.6343 −1.20441
$$454$$ 0 0
$$455$$ 18.1751 0.852061
$$456$$ 0 0
$$457$$ −0.320363 −0.0149860 −0.00749298 0.999972i $$-0.502385\pi$$
−0.00749298 + 0.999972i $$0.502385\pi$$
$$458$$ 0 0
$$459$$ 15.0711 0.703458
$$460$$ 0 0
$$461$$ −15.3239 −0.713706 −0.356853 0.934161i $$-0.616150\pi$$
−0.356853 + 0.934161i $$0.616150\pi$$
$$462$$ 0 0
$$463$$ 6.81556 0.316746 0.158373 0.987379i $$-0.449375\pi$$
0.158373 + 0.987379i $$0.449375\pi$$
$$464$$ 0 0
$$465$$ −2.86424 −0.132826
$$466$$ 0 0
$$467$$ 27.2444 1.26072 0.630360 0.776303i $$-0.282907\pi$$
0.630360 + 0.776303i $$0.282907\pi$$
$$468$$ 0 0
$$469$$ 8.05755 0.372063
$$470$$ 0 0
$$471$$ 57.9062 2.66818
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 7.51623 0.344868
$$476$$ 0 0
$$477$$ 31.6730 1.45021
$$478$$ 0 0
$$479$$ 23.5031 1.07388 0.536942 0.843619i $$-0.319579\pi$$
0.536942 + 0.843619i $$0.319579\pi$$
$$480$$ 0 0
$$481$$ 2.51457 0.114655
$$482$$ 0 0
$$483$$ 8.43286 0.383709
$$484$$ 0 0
$$485$$ −8.76465 −0.397982
$$486$$ 0 0
$$487$$ 7.63058 0.345774 0.172887 0.984942i $$-0.444690\pi$$
0.172887 + 0.984942i $$0.444690\pi$$
$$488$$ 0 0
$$489$$ −8.44779 −0.382022
$$490$$ 0 0
$$491$$ −19.1206 −0.862902 −0.431451 0.902136i $$-0.641998\pi$$
−0.431451 + 0.902136i $$0.641998\pi$$
$$492$$ 0 0
$$493$$ −2.20144 −0.0991477
$$494$$ 0 0
$$495$$ 21.6023 0.970951
$$496$$ 0 0
$$497$$ 15.5034 0.695424
$$498$$ 0 0
$$499$$ −25.6581 −1.14861 −0.574306 0.818641i $$-0.694728\pi$$
−0.574306 + 0.818641i $$0.694728\pi$$
$$500$$ 0 0
$$501$$ −11.9464 −0.533725
$$502$$ 0 0
$$503$$ −8.42282 −0.375555 −0.187777 0.982212i $$-0.560128\pi$$
−0.187777 + 0.982212i $$0.560128\pi$$
$$504$$ 0 0
$$505$$ −4.74794 −0.211281
$$506$$ 0 0
$$507$$ 124.936 5.54862
$$508$$ 0 0
$$509$$ 18.2328 0.808153 0.404076 0.914725i $$-0.367593\pi$$
0.404076 + 0.914725i $$0.367593\pi$$
$$510$$ 0 0
$$511$$ 28.8805 1.27760
$$512$$ 0 0
$$513$$ 122.594 5.41264
$$514$$ 0 0
$$515$$ −12.6304 −0.556562
$$516$$ 0 0
$$517$$ 36.2828 1.59572
$$518$$ 0 0
$$519$$ 42.2648 1.85522
$$520$$ 0 0
$$521$$ 19.8738 0.870689 0.435345 0.900264i $$-0.356627\pi$$
0.435345 + 0.900264i $$0.356627\pi$$
$$522$$ 0 0
$$523$$ −19.7228 −0.862418 −0.431209 0.902252i $$-0.641913\pi$$
−0.431209 + 0.902252i $$0.641913\pi$$
$$524$$ 0 0
$$525$$ −8.43286 −0.368040
$$526$$ 0 0
$$527$$ 0.800422 0.0348669
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −30.4673 −1.32217
$$532$$ 0 0
$$533$$ 30.9331 1.33986
$$534$$ 0 0
$$535$$ 12.9658 0.560562
$$536$$ 0 0
$$537$$ 33.1141 1.42898
$$538$$ 0 0
$$539$$ 1.34924 0.0581160
$$540$$ 0 0
$$541$$ −16.4662 −0.707938 −0.353969 0.935257i $$-0.615168\pi$$
−0.353969 + 0.935257i $$0.615168\pi$$
$$542$$ 0 0
$$543$$ 32.9408 1.41363
$$544$$ 0 0
$$545$$ 15.0040 0.642702
$$546$$ 0 0
$$547$$ −38.3286 −1.63881 −0.819405 0.573215i $$-0.805696\pi$$
−0.819405 + 0.573215i $$0.805696\pi$$
$$548$$ 0 0
$$549$$ −72.5273 −3.09539
$$550$$ 0 0
$$551$$ −17.9073 −0.762875
$$552$$ 0 0
$$553$$ 30.7559 1.30787
$$554$$ 0 0
$$555$$ −1.16671 −0.0495241
$$556$$ 0 0
$$557$$ −20.3973 −0.864263 −0.432132 0.901811i $$-0.642238\pi$$
−0.432132 + 0.901811i $$0.642238\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −8.31981 −0.351263
$$562$$ 0 0
$$563$$ −1.72270 −0.0726032 −0.0363016 0.999341i $$-0.511558\pi$$
−0.0363016 + 0.999341i $$0.511558\pi$$
$$564$$ 0 0
$$565$$ −2.13496 −0.0898184
$$566$$ 0 0
$$567$$ −76.8483 −3.22733
$$568$$ 0 0
$$569$$ 25.3283 1.06182 0.530909 0.847429i $$-0.321850\pi$$
0.530909 + 0.847429i $$0.321850\pi$$
$$570$$ 0 0
$$571$$ −5.68968 −0.238106 −0.119053 0.992888i $$-0.537986\pi$$
−0.119053 + 0.992888i $$0.537986\pi$$
$$572$$ 0 0
$$573$$ −21.8658 −0.913455
$$574$$ 0 0
$$575$$ −1.00000 −0.0417029
$$576$$ 0 0
$$577$$ −18.0377 −0.750918 −0.375459 0.926839i $$-0.622515\pi$$
−0.375459 + 0.926839i $$0.622515\pi$$
$$578$$ 0 0
$$579$$ −63.0019 −2.61827
$$580$$ 0 0
$$581$$ 16.2023 0.672185
$$582$$ 0 0
$$583$$ −10.8724 −0.450291
$$584$$ 0 0
$$585$$ −56.5326 −2.33734
$$586$$ 0 0
$$587$$ 13.1894 0.544383 0.272192 0.962243i $$-0.412252\pi$$
0.272192 + 0.962243i $$0.412252\pi$$
$$588$$ 0 0
$$589$$ 6.51092 0.268278
$$590$$ 0 0
$$591$$ −36.6643 −1.50817
$$592$$ 0 0
$$593$$ −0.326755 −0.0134182 −0.00670911 0.999977i $$-0.502136\pi$$
−0.00670911 + 0.999977i $$0.502136\pi$$
$$594$$ 0 0
$$595$$ 2.35659 0.0966109
$$596$$ 0 0
$$597$$ 12.2796 0.502569
$$598$$ 0 0
$$599$$ −7.49619 −0.306286 −0.153143 0.988204i $$-0.548939\pi$$
−0.153143 + 0.988204i $$0.548939\pi$$
$$600$$ 0 0
$$601$$ 0.121724 0.00496523 0.00248262 0.999997i $$-0.499210\pi$$
0.00248262 + 0.999997i $$0.499210\pi$$
$$602$$ 0 0
$$603$$ −25.0626 −1.02063
$$604$$ 0 0
$$605$$ 3.58454 0.145732
$$606$$ 0 0
$$607$$ 1.52992 0.0620975 0.0310488 0.999518i $$-0.490115\pi$$
0.0310488 + 0.999518i $$0.490115\pi$$
$$608$$ 0 0
$$609$$ 20.0911 0.814134
$$610$$ 0 0
$$611$$ −94.9512 −3.84131
$$612$$ 0 0
$$613$$ 24.3927 0.985211 0.492605 0.870253i $$-0.336045\pi$$
0.492605 + 0.870253i $$0.336045\pi$$
$$614$$ 0 0
$$615$$ −14.3523 −0.578742
$$616$$ 0 0
$$617$$ 27.8906 1.12283 0.561416 0.827534i $$-0.310257\pi$$
0.561416 + 0.827534i $$0.310257\pi$$
$$618$$ 0 0
$$619$$ 12.8617 0.516957 0.258478 0.966017i $$-0.416779\pi$$
0.258478 + 0.966017i $$0.416779\pi$$
$$620$$ 0 0
$$621$$ −16.3105 −0.654518
$$622$$ 0 0
$$623$$ 24.7740 0.992549
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −67.6763 −2.70273
$$628$$ 0 0
$$629$$ 0.326041 0.0130001
$$630$$ 0 0
$$631$$ −11.9518 −0.475794 −0.237897 0.971290i $$-0.576458\pi$$
−0.237897 + 0.971290i $$0.576458\pi$$
$$632$$ 0 0
$$633$$ −71.9679 −2.86047
$$634$$ 0 0
$$635$$ 3.54181 0.140552
$$636$$ 0 0
$$637$$ −3.53093 −0.139901
$$638$$ 0 0
$$639$$ −48.2226 −1.90766
$$640$$ 0 0
$$641$$ −23.9139 −0.944544 −0.472272 0.881453i $$-0.656566\pi$$
−0.472272 + 0.881453i $$0.656566\pi$$
$$642$$ 0 0
$$643$$ 3.87313 0.152741 0.0763707 0.997079i $$-0.475667\pi$$
0.0763707 + 0.997079i $$0.475667\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −10.3330 −0.406231 −0.203115 0.979155i $$-0.565107\pi$$
−0.203115 + 0.979155i $$0.565107\pi$$
$$648$$ 0 0
$$649$$ 10.4586 0.410535
$$650$$ 0 0
$$651$$ −7.30495 −0.286303
$$652$$ 0 0
$$653$$ 21.6319 0.846522 0.423261 0.906008i $$-0.360885\pi$$
0.423261 + 0.906008i $$0.360885\pi$$
$$654$$ 0 0
$$655$$ −19.5499 −0.763878
$$656$$ 0 0
$$657$$ −89.8312 −3.50465
$$658$$ 0 0
$$659$$ 15.7168 0.612238 0.306119 0.951993i $$-0.400970\pi$$
0.306119 + 0.951993i $$0.400970\pi$$
$$660$$ 0 0
$$661$$ −40.3002 −1.56750 −0.783749 0.621078i $$-0.786695\pi$$
−0.783749 + 0.621078i $$0.786695\pi$$
$$662$$ 0 0
$$663$$ 21.7727 0.845582
$$664$$ 0 0
$$665$$ 19.1694 0.743357
$$666$$ 0 0
$$667$$ 2.38248 0.0922500
$$668$$ 0 0
$$669$$ 22.8389 0.883003
$$670$$ 0 0
$$671$$ 24.8966 0.961122
$$672$$ 0 0
$$673$$ −6.69888 −0.258223 −0.129111 0.991630i $$-0.541212\pi$$
−0.129111 + 0.991630i $$0.541212\pi$$
$$674$$ 0 0
$$675$$ 16.3105 0.627792
$$676$$ 0 0
$$677$$ −22.5812 −0.867866 −0.433933 0.900945i $$-0.642875\pi$$
−0.433933 + 0.900945i $$0.642875\pi$$
$$678$$ 0 0
$$679$$ −22.3533 −0.857843
$$680$$ 0 0
$$681$$ −76.1566 −2.91833
$$682$$ 0 0
$$683$$ −1.42797 −0.0546398 −0.0273199 0.999627i $$-0.508697\pi$$
−0.0273199 + 0.999627i $$0.508697\pi$$
$$684$$ 0 0
$$685$$ −4.41544 −0.168705
$$686$$ 0 0
$$687$$ 42.8738 1.63574
$$688$$ 0 0
$$689$$ 28.4529 1.08397
$$690$$ 0 0
$$691$$ −1.50255 −0.0571595 −0.0285798 0.999592i $$-0.509098\pi$$
−0.0285798 + 0.999592i $$0.509098\pi$$
$$692$$ 0 0
$$693$$ 55.0944 2.09286
$$694$$ 0 0
$$695$$ −21.9954 −0.834334
$$696$$ 0 0
$$697$$ 4.01081 0.151920
$$698$$ 0 0
$$699$$ −15.9020 −0.601471
$$700$$ 0 0
$$701$$ 18.7341 0.707577 0.353789 0.935325i $$-0.384893\pi$$
0.353789 + 0.935325i $$0.384893\pi$$
$$702$$ 0 0
$$703$$ 2.65214 0.100027
$$704$$ 0 0
$$705$$ 44.0554 1.65922
$$706$$ 0 0
$$707$$ −12.1091 −0.455411
$$708$$ 0 0
$$709$$ −13.5052 −0.507199 −0.253599 0.967309i $$-0.581615\pi$$
−0.253599 + 0.967309i $$0.581615\pi$$
$$710$$ 0 0
$$711$$ −95.6646 −3.58770
$$712$$ 0 0
$$713$$ −0.866248 −0.0324412
$$714$$ 0 0
$$715$$ 19.4061 0.725746
$$716$$ 0 0
$$717$$ −67.6711 −2.52722
$$718$$ 0 0
$$719$$ 31.7496 1.18406 0.592030 0.805916i $$-0.298327\pi$$
0.592030 + 0.805916i $$0.298327\pi$$
$$720$$ 0 0
$$721$$ −32.2126 −1.19966
$$722$$ 0 0
$$723$$ 88.8313 3.30367
$$724$$ 0 0
$$725$$ −2.38248 −0.0884831
$$726$$ 0 0
$$727$$ 21.6531 0.803070 0.401535 0.915844i $$-0.368477\pi$$
0.401535 + 0.915844i $$0.368477\pi$$
$$728$$ 0 0
$$729$$ 77.2411 2.86078
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0.640507 0.0236577 0.0118288 0.999930i $$-0.496235\pi$$
0.0118288 + 0.999930i $$0.496235\pi$$
$$734$$ 0 0
$$735$$ 1.63828 0.0604288
$$736$$ 0 0
$$737$$ 8.60329 0.316906
$$738$$ 0 0
$$739$$ 16.8191 0.618700 0.309350 0.950948i $$-0.399889\pi$$
0.309350 + 0.950948i $$0.399889\pi$$
$$740$$ 0 0
$$741$$ 177.107 6.50619
$$742$$ 0 0
$$743$$ −15.9691 −0.585851 −0.292925 0.956135i $$-0.594629\pi$$
−0.292925 + 0.956135i $$0.594629\pi$$
$$744$$ 0 0
$$745$$ −19.0780 −0.698965
$$746$$ 0 0
$$747$$ −50.3964 −1.84391
$$748$$ 0 0
$$749$$ 33.0680 1.20828
$$750$$ 0 0
$$751$$ −8.09841 −0.295515 −0.147758 0.989024i $$-0.547206\pi$$
−0.147758 + 0.989024i $$0.547206\pi$$
$$752$$ 0 0
$$753$$ −52.7798 −1.92340
$$754$$ 0 0
$$755$$ 7.75273 0.282151
$$756$$ 0 0
$$757$$ 20.5691 0.747598 0.373799 0.927510i $$-0.378055\pi$$
0.373799 + 0.927510i $$0.378055\pi$$
$$758$$ 0 0
$$759$$ 9.00402 0.326825
$$760$$ 0 0
$$761$$ −17.1840 −0.622919 −0.311459 0.950259i $$-0.600818\pi$$
−0.311459 + 0.950259i $$0.600818\pi$$
$$762$$ 0 0
$$763$$ 38.2662 1.38533
$$764$$ 0 0
$$765$$ −7.33006 −0.265019
$$766$$ 0 0
$$767$$ −27.3698 −0.988268
$$768$$ 0 0
$$769$$ 41.8648 1.50968 0.754841 0.655908i $$-0.227714\pi$$
0.754841 + 0.655908i $$0.227714\pi$$
$$770$$ 0 0
$$771$$ 11.0393 0.397570
$$772$$ 0 0
$$773$$ −35.7639 −1.28634 −0.643170 0.765724i $$-0.722381\pi$$
−0.643170 + 0.765724i $$0.722381\pi$$
$$774$$ 0 0
$$775$$ 0.866248 0.0311165
$$776$$ 0 0
$$777$$ −2.97557 −0.106748
$$778$$ 0 0
$$779$$ 32.6254 1.16893
$$780$$ 0 0
$$781$$ 16.5535 0.592330
$$782$$ 0 0
$$783$$ −38.8595 −1.38872
$$784$$ 0 0
$$785$$ −17.5129 −0.625062
$$786$$ 0 0
$$787$$ 26.9835 0.961858 0.480929 0.876759i $$-0.340299\pi$$
0.480929 + 0.876759i $$0.340299\pi$$
$$788$$ 0 0
$$789$$ −77.3321 −2.75310
$$790$$ 0 0
$$791$$ −5.44500 −0.193602
$$792$$ 0 0
$$793$$ −65.1537 −2.31368
$$794$$ 0 0
$$795$$ −13.2016 −0.468211
$$796$$ 0 0
$$797$$ 26.7226 0.946561 0.473281 0.880912i $$-0.343070\pi$$
0.473281 + 0.880912i $$0.343070\pi$$
$$798$$ 0 0
$$799$$ −12.3114 −0.435547
$$800$$ 0 0
$$801$$ −77.0582 −2.72272
$$802$$ 0 0
$$803$$ 30.8366 1.08820
$$804$$ 0 0
$$805$$ −2.55040 −0.0898897
$$806$$ 0 0
$$807$$ 46.7464 1.64555
$$808$$ 0 0
$$809$$ 50.0319 1.75903 0.879513 0.475874i $$-0.157868\pi$$
0.879513 + 0.475874i $$0.157868\pi$$
$$810$$ 0 0
$$811$$ 33.5774 1.17906 0.589531 0.807746i $$-0.299313\pi$$
0.589531 + 0.807746i $$0.299313\pi$$
$$812$$ 0 0
$$813$$ 85.7318 3.00675
$$814$$ 0 0
$$815$$ 2.55491 0.0894946
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ −144.181 −5.03808
$$820$$ 0 0
$$821$$ 27.1429 0.947295 0.473647 0.880715i $$-0.342937\pi$$
0.473647 + 0.880715i $$0.342937\pi$$
$$822$$ 0 0
$$823$$ 32.6392 1.13773 0.568866 0.822431i $$-0.307382\pi$$
0.568866 + 0.822431i $$0.307382\pi$$
$$824$$ 0 0
$$825$$ −9.00402 −0.313480
$$826$$ 0 0
$$827$$ −37.0828 −1.28949 −0.644747 0.764396i $$-0.723037\pi$$
−0.644747 + 0.764396i $$0.723037\pi$$
$$828$$ 0 0
$$829$$ 33.9405 1.17880 0.589401 0.807841i $$-0.299364\pi$$
0.589401 + 0.807841i $$0.299364\pi$$
$$830$$ 0 0
$$831$$ 3.54181 0.122864
$$832$$ 0 0
$$833$$ −0.457823 −0.0158626
$$834$$ 0 0
$$835$$ 3.61301 0.125033
$$836$$ 0 0
$$837$$ 14.1289 0.488368
$$838$$ 0 0
$$839$$ −1.36929 −0.0472730 −0.0236365 0.999721i $$-0.507524\pi$$
−0.0236365 + 0.999721i $$0.507524\pi$$
$$840$$ 0 0
$$841$$ −23.3238 −0.804269
$$842$$ 0 0
$$843$$ 40.4251 1.39231
$$844$$ 0 0
$$845$$ −37.7852 −1.29985
$$846$$ 0 0
$$847$$ 9.14199 0.314122
$$848$$ 0 0
$$849$$ −100.998 −3.46624
$$850$$ 0 0
$$851$$ −0.352855 −0.0120957
$$852$$ 0 0
$$853$$ 9.21382 0.315475 0.157738 0.987481i $$-0.449580\pi$$
0.157738 + 0.987481i $$0.449580\pi$$
$$854$$ 0 0
$$855$$ −59.6253 −2.03914
$$856$$ 0 0
$$857$$ 14.3564 0.490405 0.245202 0.969472i $$-0.421146\pi$$
0.245202 + 0.969472i $$0.421146\pi$$
$$858$$ 0 0
$$859$$ 31.3758 1.07053 0.535264 0.844685i $$-0.320212\pi$$
0.535264 + 0.844685i $$0.320212\pi$$
$$860$$ 0 0
$$861$$ −36.6042 −1.24747
$$862$$ 0 0
$$863$$ 26.1214 0.889182 0.444591 0.895734i $$-0.353349\pi$$
0.444591 + 0.895734i $$0.353349\pi$$
$$864$$ 0 0
$$865$$ −12.7824 −0.434614
$$866$$ 0 0
$$867$$ −53.3873 −1.81313
$$868$$ 0 0
$$869$$ 32.8390 1.11399
$$870$$ 0 0
$$871$$ −22.5146 −0.762877
$$872$$ 0 0
$$873$$ 69.5289 2.35320
$$874$$ 0 0
$$875$$ 2.55040 0.0862192
$$876$$ 0 0
$$877$$ −11.5530 −0.390118 −0.195059 0.980792i $$-0.562490\pi$$
−0.195059 + 0.980792i $$0.562490\pi$$
$$878$$ 0 0
$$879$$ −28.0006 −0.944437
$$880$$ 0 0
$$881$$ −34.3559 −1.15748 −0.578740 0.815512i $$-0.696456\pi$$
−0.578740 + 0.815512i $$0.696456\pi$$
$$882$$ 0 0
$$883$$ 12.8316 0.431818 0.215909 0.976414i $$-0.430729\pi$$
0.215909 + 0.976414i $$0.430729\pi$$
$$884$$ 0 0
$$885$$ 12.6990 0.426874
$$886$$ 0 0
$$887$$ −1.46724 −0.0492652 −0.0246326 0.999697i $$-0.507842\pi$$
−0.0246326 + 0.999697i $$0.507842\pi$$
$$888$$ 0 0
$$889$$ 9.03303 0.302958
$$890$$ 0 0
$$891$$ −82.0533 −2.74889
$$892$$ 0 0
$$893$$ −100.146 −3.35125
$$894$$ 0 0
$$895$$ −10.0149 −0.334760
$$896$$ 0 0
$$897$$ −23.5633 −0.786755
$$898$$ 0 0
$$899$$ −2.06382 −0.0688322
$$900$$ 0 0
$$901$$ 3.68922 0.122906
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −9.96248 −0.331164
$$906$$ 0 0
$$907$$ 4.44558 0.147613 0.0738066 0.997273i $$-0.476485\pi$$
0.0738066 + 0.997273i $$0.476485\pi$$
$$908$$ 0 0
$$909$$ 37.6648 1.24926
$$910$$ 0 0
$$911$$ 40.1697 1.33088 0.665440 0.746451i $$-0.268244\pi$$
0.665440 + 0.746451i $$0.268244\pi$$
$$912$$ 0 0
$$913$$ 17.2997 0.572536
$$914$$ 0 0
$$915$$ 30.2300 0.999372
$$916$$ 0 0
$$917$$ −49.8600 −1.64652
$$918$$ 0 0
$$919$$ 46.8063 1.54400 0.771999 0.635624i $$-0.219257\pi$$
0.771999 + 0.635624i $$0.219257\pi$$
$$920$$ 0 0
$$921$$ −4.39109 −0.144691
$$922$$ 0 0
$$923$$ −43.3200 −1.42590
$$924$$ 0 0
$$925$$ 0.352855 0.0116018
$$926$$ 0 0
$$927$$ 100.195 3.29085
$$928$$ 0 0
$$929$$ 8.92145 0.292703 0.146352 0.989233i $$-0.453247\pi$$
0.146352 + 0.989233i $$0.453247\pi$$
$$930$$ 0 0
$$931$$ −3.72409 −0.122052
$$932$$ 0 0
$$933$$ 80.3382 2.63016
$$934$$ 0 0
$$935$$ 2.51621 0.0822887
$$936$$ 0 0
$$937$$ 7.70070 0.251571 0.125785 0.992057i $$-0.459855\pi$$
0.125785 + 0.992057i $$0.459855\pi$$
$$938$$ 0 0
$$939$$ 67.2799 2.19560
$$940$$ 0 0
$$941$$ −16.8979 −0.550856 −0.275428 0.961322i $$-0.588820\pi$$
−0.275428 + 0.961322i $$0.588820\pi$$
$$942$$ 0 0
$$943$$ −4.34066 −0.141351
$$944$$ 0 0
$$945$$ 41.5983 1.35319
$$946$$ 0 0
$$947$$ −43.7150 −1.42055 −0.710273 0.703927i $$-0.751428\pi$$
−0.710273 + 0.703927i $$0.751428\pi$$
$$948$$ 0 0
$$949$$ −80.6985 −2.61958
$$950$$ 0 0
$$951$$ −28.3469 −0.919210
$$952$$ 0 0
$$953$$ 30.2353 0.979418 0.489709 0.871886i $$-0.337103\pi$$
0.489709 + 0.871886i $$0.337103\pi$$
$$954$$ 0 0
$$955$$ 6.61298 0.213991
$$956$$ 0 0
$$957$$ 21.4519 0.693441
$$958$$ 0 0
$$959$$ −11.2611 −0.363641
$$960$$ 0 0
$$961$$ −30.2496 −0.975794
$$962$$ 0 0
$$963$$ −102.856 −3.31450
$$964$$ 0 0
$$965$$ 19.0540 0.613370
$$966$$ 0 0
$$967$$ 16.2727 0.523296 0.261648 0.965163i $$-0.415734\pi$$
0.261648 + 0.965163i $$0.415734\pi$$
$$968$$ 0 0
$$969$$ 22.9638 0.737704
$$970$$ 0 0
$$971$$ 6.21424 0.199424 0.0997122 0.995016i $$-0.468208\pi$$
0.0997122 + 0.995016i $$0.468208\pi$$
$$972$$ 0 0
$$973$$ −56.0971 −1.79839
$$974$$ 0 0
$$975$$ 23.5633 0.754629
$$976$$ 0 0
$$977$$ 25.6705 0.821271 0.410636 0.911800i $$-0.365307\pi$$
0.410636 + 0.911800i $$0.365307\pi$$
$$978$$ 0 0
$$979$$ 26.4519 0.845407
$$980$$ 0 0
$$981$$ −119.025 −3.80018
$$982$$ 0 0
$$983$$ −30.7124 −0.979574 −0.489787 0.871842i $$-0.662925\pi$$
−0.489787 + 0.871842i $$0.662925\pi$$
$$984$$ 0 0
$$985$$ 11.0886 0.353312
$$986$$ 0 0
$$987$$ 112.359 3.57642
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 29.4052 0.934086 0.467043 0.884235i $$-0.345319\pi$$
0.467043 + 0.884235i $$0.345319\pi$$
$$992$$ 0 0
$$993$$ −69.3594 −2.20105
$$994$$ 0 0
$$995$$ −3.71377 −0.117735
$$996$$ 0 0
$$997$$ 5.59825 0.177298 0.0886492 0.996063i $$-0.471745\pi$$
0.0886492 + 0.996063i $$0.471745\pi$$
$$998$$ 0 0
$$999$$ 5.75524 0.182088
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 920.2.a.j.1.5 5
3.2 odd 2 8280.2.a.bs.1.2 5
4.3 odd 2 1840.2.a.v.1.1 5
5.2 odd 4 4600.2.e.u.4049.2 10
5.3 odd 4 4600.2.e.u.4049.9 10
5.4 even 2 4600.2.a.be.1.1 5
8.3 odd 2 7360.2.a.cp.1.5 5
8.5 even 2 7360.2.a.co.1.1 5
20.19 odd 2 9200.2.a.cu.1.5 5

By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.j.1.5 5 1.1 even 1 trivial
1840.2.a.v.1.1 5 4.3 odd 2
4600.2.a.be.1.1 5 5.4 even 2
4600.2.e.u.4049.2 10 5.2 odd 4
4600.2.e.u.4049.9 10 5.3 odd 4
7360.2.a.co.1.1 5 8.5 even 2
7360.2.a.cp.1.5 5 8.3 odd 2
8280.2.a.bs.1.2 5 3.2 odd 2
9200.2.a.cu.1.5 5 20.19 odd 2