# Properties

 Label 1305.2.a.s.1.4 Level $1305$ Weight $2$ Character 1305.1 Self dual yes Analytic conductor $10.420$ Analytic rank $0$ Dimension $7$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1305,2,Mod(1,1305)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1305, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1305.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1305 = 3^{2} \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1305.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: $$10.4204774638$$ Analytic rank: $$0$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{7} - x^{6} - 13x^{5} + 12x^{4} + 47x^{3} - 37x^{2} - 35x + 18$$ x^7 - x^6 - 13*x^5 + 12*x^4 + 47*x^3 - 37*x^2 - 35*x + 18 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$0.431560$$ of defining polynomial Character $$\chi$$ $$=$$ 1305.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.431560 q^{2} -1.81376 q^{4} -1.00000 q^{5} -2.42505 q^{7} +1.64586 q^{8} +O(q^{10})$$ $$q-0.431560 q^{2} -1.81376 q^{4} -1.00000 q^{5} -2.42505 q^{7} +1.64586 q^{8} +0.431560 q^{10} +0.00755077 q^{11} -2.73098 q^{13} +1.04655 q^{14} +2.91722 q^{16} -4.73098 q^{17} +2.74388 q^{19} +1.81376 q^{20} -0.00325861 q^{22} -3.68872 q^{23} +1.00000 q^{25} +1.17858 q^{26} +4.39845 q^{28} +1.00000 q^{29} +1.25612 q^{31} -4.55068 q^{32} +2.04170 q^{34} +2.42505 q^{35} +6.60420 q^{37} -1.18415 q^{38} -1.64586 q^{40} -0.160198 q^{41} +11.0103 q^{43} -0.0136953 q^{44} +1.59190 q^{46} -10.3736 q^{47} -1.11912 q^{49} -0.431560 q^{50} +4.95333 q^{52} +11.1099 q^{53} -0.00755077 q^{55} -3.99130 q^{56} -0.431560 q^{58} +8.85010 q^{59} +13.5237 q^{61} -0.542089 q^{62} -3.87056 q^{64} +2.73098 q^{65} +4.69881 q^{67} +8.58085 q^{68} -1.04655 q^{70} -1.48777 q^{71} +2.75695 q^{73} -2.85010 q^{74} -4.97674 q^{76} -0.0183110 q^{77} +2.72878 q^{79} -2.91722 q^{80} +0.0691351 q^{82} +7.81988 q^{83} +4.73098 q^{85} -4.75160 q^{86} +0.0124275 q^{88} -7.59619 q^{89} +6.62277 q^{91} +6.69044 q^{92} +4.47682 q^{94} -2.74388 q^{95} -7.02540 q^{97} +0.482968 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q - q^{2} + 13 q^{4} - 7 q^{5} + 10 q^{7}+O(q^{10})$$ 7 * q - q^2 + 13 * q^4 - 7 * q^5 + 10 * q^7 $$7 q - q^{2} + 13 q^{4} - 7 q^{5} + 10 q^{7} + q^{10} + 3 q^{11} + 6 q^{13} + 9 q^{14} + 21 q^{16} - 8 q^{17} + 10 q^{19} - 13 q^{20} + 9 q^{22} - 11 q^{23} + 7 q^{25} - 3 q^{26} + 25 q^{28} + 7 q^{29} + 18 q^{31} - q^{32} - q^{34} - 10 q^{35} + 13 q^{37} - 12 q^{38} + 13 q^{41} + 9 q^{43} + 37 q^{44} - 8 q^{46} - 2 q^{47} + 21 q^{49} - q^{50} - q^{52} - 5 q^{53} - 3 q^{55} + 30 q^{56} - q^{58} + 8 q^{59} + 14 q^{61} + 8 q^{62} + 8 q^{64} - 6 q^{65} + 14 q^{67} - 27 q^{68} - 9 q^{70} + 8 q^{71} + 3 q^{73} + 34 q^{74} + 4 q^{76} + 28 q^{77} + 4 q^{79} - 21 q^{80} - 20 q^{82} - 17 q^{83} + 8 q^{85} - 4 q^{86} + 26 q^{88} + 20 q^{89} + 12 q^{91} - 60 q^{92} - 21 q^{94} - 10 q^{95} + 13 q^{97} + 20 q^{98}+O(q^{100})$$ 7 * q - q^2 + 13 * q^4 - 7 * q^5 + 10 * q^7 + q^10 + 3 * q^11 + 6 * q^13 + 9 * q^14 + 21 * q^16 - 8 * q^17 + 10 * q^19 - 13 * q^20 + 9 * q^22 - 11 * q^23 + 7 * q^25 - 3 * q^26 + 25 * q^28 + 7 * q^29 + 18 * q^31 - q^32 - q^34 - 10 * q^35 + 13 * q^37 - 12 * q^38 + 13 * q^41 + 9 * q^43 + 37 * q^44 - 8 * q^46 - 2 * q^47 + 21 * q^49 - q^50 - q^52 - 5 * q^53 - 3 * q^55 + 30 * q^56 - q^58 + 8 * q^59 + 14 * q^61 + 8 * q^62 + 8 * q^64 - 6 * q^65 + 14 * q^67 - 27 * q^68 - 9 * q^70 + 8 * q^71 + 3 * q^73 + 34 * q^74 + 4 * q^76 + 28 * q^77 + 4 * q^79 - 21 * q^80 - 20 * q^82 - 17 * q^83 + 8 * q^85 - 4 * q^86 + 26 * q^88 + 20 * q^89 + 12 * q^91 - 60 * q^92 - 21 * q^94 - 10 * q^95 + 13 * q^97 + 20 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.431560 −0.305159 −0.152579 0.988291i $$-0.548758\pi$$
−0.152579 + 0.988291i $$0.548758\pi$$
$$3$$ 0 0
$$4$$ −1.81376 −0.906878
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −2.42505 −0.916583 −0.458292 0.888802i $$-0.651539\pi$$
−0.458292 + 0.888802i $$0.651539\pi$$
$$8$$ 1.64586 0.581901
$$9$$ 0 0
$$10$$ 0.431560 0.136471
$$11$$ 0.00755077 0.00227664 0.00113832 0.999999i $$-0.499638\pi$$
0.00113832 + 0.999999i $$0.499638\pi$$
$$12$$ 0 0
$$13$$ −2.73098 −0.757438 −0.378719 0.925512i $$-0.623635\pi$$
−0.378719 + 0.925512i $$0.623635\pi$$
$$14$$ 1.04655 0.279703
$$15$$ 0 0
$$16$$ 2.91722 0.729306
$$17$$ −4.73098 −1.14743 −0.573716 0.819055i $$-0.694499\pi$$
−0.573716 + 0.819055i $$0.694499\pi$$
$$18$$ 0 0
$$19$$ 2.74388 0.629490 0.314745 0.949176i $$-0.398081\pi$$
0.314745 + 0.949176i $$0.398081\pi$$
$$20$$ 1.81376 0.405568
$$21$$ 0 0
$$22$$ −0.00325861 −0.000694738 0
$$23$$ −3.68872 −0.769151 −0.384575 0.923094i $$-0.625652\pi$$
−0.384575 + 0.923094i $$0.625652\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 1.17858 0.231139
$$27$$ 0 0
$$28$$ 4.39845 0.831230
$$29$$ 1.00000 0.185695
$$30$$ 0 0
$$31$$ 1.25612 0.225605 0.112803 0.993617i $$-0.464017\pi$$
0.112803 + 0.993617i $$0.464017\pi$$
$$32$$ −4.55068 −0.804455
$$33$$ 0 0
$$34$$ 2.04170 0.350149
$$35$$ 2.42505 0.409909
$$36$$ 0 0
$$37$$ 6.60420 1.08572 0.542861 0.839822i $$-0.317341\pi$$
0.542861 + 0.839822i $$0.317341\pi$$
$$38$$ −1.18415 −0.192094
$$39$$ 0 0
$$40$$ −1.64586 −0.260234
$$41$$ −0.160198 −0.0250188 −0.0125094 0.999922i $$-0.503982\pi$$
−0.0125094 + 0.999922i $$0.503982\pi$$
$$42$$ 0 0
$$43$$ 11.0103 1.67906 0.839528 0.543317i $$-0.182832\pi$$
0.839528 + 0.543317i $$0.182832\pi$$
$$44$$ −0.0136953 −0.00206464
$$45$$ 0 0
$$46$$ 1.59190 0.234713
$$47$$ −10.3736 −1.51314 −0.756572 0.653910i $$-0.773127\pi$$
−0.756572 + 0.653910i $$0.773127\pi$$
$$48$$ 0 0
$$49$$ −1.11912 −0.159875
$$50$$ −0.431560 −0.0610317
$$51$$ 0 0
$$52$$ 4.95333 0.686904
$$53$$ 11.1099 1.52607 0.763033 0.646360i $$-0.223709\pi$$
0.763033 + 0.646360i $$0.223709\pi$$
$$54$$ 0 0
$$55$$ −0.00755077 −0.00101815
$$56$$ −3.99130 −0.533360
$$57$$ 0 0
$$58$$ −0.431560 −0.0566666
$$59$$ 8.85010 1.15219 0.576093 0.817384i $$-0.304577\pi$$
0.576093 + 0.817384i $$0.304577\pi$$
$$60$$ 0 0
$$61$$ 13.5237 1.73153 0.865766 0.500449i $$-0.166832\pi$$
0.865766 + 0.500449i $$0.166832\pi$$
$$62$$ −0.542089 −0.0688454
$$63$$ 0 0
$$64$$ −3.87056 −0.483820
$$65$$ 2.73098 0.338736
$$66$$ 0 0
$$67$$ 4.69881 0.574051 0.287026 0.957923i $$-0.407333\pi$$
0.287026 + 0.957923i $$0.407333\pi$$
$$68$$ 8.58085 1.04058
$$69$$ 0 0
$$70$$ −1.04655 −0.125087
$$71$$ −1.48777 −0.176566 −0.0882828 0.996095i $$-0.528138\pi$$
−0.0882828 + 0.996095i $$0.528138\pi$$
$$72$$ 0 0
$$73$$ 2.75695 0.322677 0.161339 0.986899i $$-0.448419\pi$$
0.161339 + 0.986899i $$0.448419\pi$$
$$74$$ −2.85010 −0.331318
$$75$$ 0 0
$$76$$ −4.97674 −0.570871
$$77$$ −0.0183110 −0.00208673
$$78$$ 0 0
$$79$$ 2.72878 0.307012 0.153506 0.988148i $$-0.450944\pi$$
0.153506 + 0.988148i $$0.450944\pi$$
$$80$$ −2.91722 −0.326156
$$81$$ 0 0
$$82$$ 0.0691351 0.00763470
$$83$$ 7.81988 0.858343 0.429172 0.903223i $$-0.358806\pi$$
0.429172 + 0.903223i $$0.358806\pi$$
$$84$$ 0 0
$$85$$ 4.73098 0.513147
$$86$$ −4.75160 −0.512378
$$87$$ 0 0
$$88$$ 0.0124275 0.00132478
$$89$$ −7.59619 −0.805194 −0.402597 0.915377i $$-0.631892\pi$$
−0.402597 + 0.915377i $$0.631892\pi$$
$$90$$ 0 0
$$91$$ 6.62277 0.694255
$$92$$ 6.69044 0.697526
$$93$$ 0 0
$$94$$ 4.47682 0.461749
$$95$$ −2.74388 −0.281517
$$96$$ 0 0
$$97$$ −7.02540 −0.713322 −0.356661 0.934234i $$-0.616085\pi$$
−0.356661 + 0.934234i $$0.616085\pi$$
$$98$$ 0.482968 0.0487872
$$99$$ 0 0
$$100$$ −1.81376 −0.181376
$$101$$ 15.4826 1.54058 0.770289 0.637695i $$-0.220112\pi$$
0.770289 + 0.637695i $$0.220112\pi$$
$$102$$ 0 0
$$103$$ 7.00119 0.689847 0.344924 0.938631i $$-0.387905\pi$$
0.344924 + 0.938631i $$0.387905\pi$$
$$104$$ −4.49482 −0.440753
$$105$$ 0 0
$$106$$ −4.79460 −0.465692
$$107$$ −11.3367 −1.09596 −0.547979 0.836492i $$-0.684603\pi$$
−0.547979 + 0.836492i $$0.684603\pi$$
$$108$$ 0 0
$$109$$ 15.5028 1.48490 0.742450 0.669901i $$-0.233664\pi$$
0.742450 + 0.669901i $$0.233664\pi$$
$$110$$ 0.00325861 0.000310696 0
$$111$$ 0 0
$$112$$ −7.07442 −0.668470
$$113$$ 0.883740 0.0831353 0.0415676 0.999136i $$-0.486765\pi$$
0.0415676 + 0.999136i $$0.486765\pi$$
$$114$$ 0 0
$$115$$ 3.68872 0.343975
$$116$$ −1.81376 −0.168403
$$117$$ 0 0
$$118$$ −3.81935 −0.351599
$$119$$ 11.4729 1.05172
$$120$$ 0 0
$$121$$ −10.9999 −0.999995
$$122$$ −5.83628 −0.528392
$$123$$ 0 0
$$124$$ −2.27829 −0.204596
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 11.0254 0.978346 0.489173 0.872187i $$-0.337299\pi$$
0.489173 + 0.872187i $$0.337299\pi$$
$$128$$ 10.7717 0.952096
$$129$$ 0 0
$$130$$ −1.17858 −0.103368
$$131$$ −19.6037 −1.71278 −0.856391 0.516328i $$-0.827298\pi$$
−0.856391 + 0.516328i $$0.827298\pi$$
$$132$$ 0 0
$$133$$ −6.65406 −0.576980
$$134$$ −2.02782 −0.175177
$$135$$ 0 0
$$136$$ −7.78655 −0.667691
$$137$$ −7.12387 −0.608633 −0.304316 0.952571i $$-0.598428\pi$$
−0.304316 + 0.952571i $$0.598428\pi$$
$$138$$ 0 0
$$139$$ −9.68606 −0.821561 −0.410781 0.911734i $$-0.634744\pi$$
−0.410781 + 0.911734i $$0.634744\pi$$
$$140$$ −4.39845 −0.371737
$$141$$ 0 0
$$142$$ 0.642061 0.0538805
$$143$$ −0.0206210 −0.00172442
$$144$$ 0 0
$$145$$ −1.00000 −0.0830455
$$146$$ −1.18979 −0.0984677
$$147$$ 0 0
$$148$$ −11.9784 −0.984618
$$149$$ 17.7002 1.45006 0.725029 0.688719i $$-0.241826\pi$$
0.725029 + 0.688719i $$0.241826\pi$$
$$150$$ 0 0
$$151$$ 20.7409 1.68787 0.843937 0.536443i $$-0.180232\pi$$
0.843937 + 0.536443i $$0.180232\pi$$
$$152$$ 4.51606 0.366301
$$153$$ 0 0
$$154$$ 0.00790230 0.000636785 0
$$155$$ −1.25612 −0.100894
$$156$$ 0 0
$$157$$ −6.65849 −0.531406 −0.265703 0.964055i $$-0.585604\pi$$
−0.265703 + 0.964055i $$0.585604\pi$$
$$158$$ −1.17763 −0.0936874
$$159$$ 0 0
$$160$$ 4.55068 0.359763
$$161$$ 8.94533 0.704991
$$162$$ 0 0
$$163$$ −5.89592 −0.461804 −0.230902 0.972977i $$-0.574168\pi$$
−0.230902 + 0.972977i $$0.574168\pi$$
$$164$$ 0.290561 0.0226890
$$165$$ 0 0
$$166$$ −3.37474 −0.261931
$$167$$ −10.1921 −0.788688 −0.394344 0.918963i $$-0.629028\pi$$
−0.394344 + 0.918963i $$0.629028\pi$$
$$168$$ 0 0
$$169$$ −5.54174 −0.426288
$$170$$ −2.04170 −0.156591
$$171$$ 0 0
$$172$$ −19.9700 −1.52270
$$173$$ −10.8564 −0.825394 −0.412697 0.910868i $$-0.635413\pi$$
−0.412697 + 0.910868i $$0.635413\pi$$
$$174$$ 0 0
$$175$$ −2.42505 −0.183317
$$176$$ 0.0220273 0.00166037
$$177$$ 0 0
$$178$$ 3.27821 0.245712
$$179$$ 20.7170 1.54846 0.774230 0.632904i $$-0.218137\pi$$
0.774230 + 0.632904i $$0.218137\pi$$
$$180$$ 0 0
$$181$$ −0.0561768 −0.00417559 −0.00208779 0.999998i $$-0.500665\pi$$
−0.00208779 + 0.999998i $$0.500665\pi$$
$$182$$ −2.85812 −0.211858
$$183$$ 0 0
$$184$$ −6.07113 −0.447569
$$185$$ −6.60420 −0.485550
$$186$$ 0 0
$$187$$ −0.0357226 −0.00261229
$$188$$ 18.8152 1.37224
$$189$$ 0 0
$$190$$ 1.18415 0.0859072
$$191$$ 3.24218 0.234596 0.117298 0.993097i $$-0.462577\pi$$
0.117298 + 0.993097i $$0.462577\pi$$
$$192$$ 0 0
$$193$$ −10.2522 −0.737967 −0.368983 0.929436i $$-0.620294\pi$$
−0.368983 + 0.929436i $$0.620294\pi$$
$$194$$ 3.03188 0.217676
$$195$$ 0 0
$$196$$ 2.02982 0.144987
$$197$$ −10.7593 −0.766570 −0.383285 0.923630i $$-0.625207\pi$$
−0.383285 + 0.923630i $$0.625207\pi$$
$$198$$ 0 0
$$199$$ 8.13396 0.576601 0.288301 0.957540i $$-0.406910\pi$$
0.288301 + 0.957540i $$0.406910\pi$$
$$200$$ 1.64586 0.116380
$$201$$ 0 0
$$202$$ −6.68167 −0.470121
$$203$$ −2.42505 −0.170205
$$204$$ 0 0
$$205$$ 0.160198 0.0111887
$$206$$ −3.02143 −0.210513
$$207$$ 0 0
$$208$$ −7.96688 −0.552404
$$209$$ 0.0207184 0.00143313
$$210$$ 0 0
$$211$$ 2.42349 0.166840 0.0834199 0.996514i $$-0.473416\pi$$
0.0834199 + 0.996514i $$0.473416\pi$$
$$212$$ −20.1507 −1.38396
$$213$$ 0 0
$$214$$ 4.89246 0.334441
$$215$$ −11.0103 −0.750896
$$216$$ 0 0
$$217$$ −3.04615 −0.206786
$$218$$ −6.69039 −0.453130
$$219$$ 0 0
$$220$$ 0.0136953 0.000923334 0
$$221$$ 12.9202 0.869108
$$222$$ 0 0
$$223$$ 16.1406 1.08085 0.540427 0.841391i $$-0.318263\pi$$
0.540427 + 0.841391i $$0.318263\pi$$
$$224$$ 11.0356 0.737350
$$225$$ 0 0
$$226$$ −0.381387 −0.0253695
$$227$$ 19.7831 1.31305 0.656527 0.754303i $$-0.272025\pi$$
0.656527 + 0.754303i $$0.272025\pi$$
$$228$$ 0 0
$$229$$ −1.99560 −0.131873 −0.0659366 0.997824i $$-0.521004\pi$$
−0.0659366 + 0.997824i $$0.521004\pi$$
$$230$$ −1.59190 −0.104967
$$231$$ 0 0
$$232$$ 1.64586 0.108056
$$233$$ −3.98606 −0.261136 −0.130568 0.991439i $$-0.541680\pi$$
−0.130568 + 0.991439i $$0.541680\pi$$
$$234$$ 0 0
$$235$$ 10.3736 0.676699
$$236$$ −16.0519 −1.04489
$$237$$ 0 0
$$238$$ −4.95123 −0.320941
$$239$$ 29.5973 1.91449 0.957245 0.289279i $$-0.0934155\pi$$
0.957245 + 0.289279i $$0.0934155\pi$$
$$240$$ 0 0
$$241$$ −13.1622 −0.847854 −0.423927 0.905696i $$-0.639349\pi$$
−0.423927 + 0.905696i $$0.639349\pi$$
$$242$$ 4.74713 0.305157
$$243$$ 0 0
$$244$$ −24.5287 −1.57029
$$245$$ 1.11912 0.0714982
$$246$$ 0 0
$$247$$ −7.49350 −0.476800
$$248$$ 2.06739 0.131280
$$249$$ 0 0
$$250$$ 0.431560 0.0272942
$$251$$ −6.84372 −0.431972 −0.215986 0.976397i $$-0.569296\pi$$
−0.215986 + 0.976397i $$0.569296\pi$$
$$252$$ 0 0
$$253$$ −0.0278527 −0.00175108
$$254$$ −4.75812 −0.298551
$$255$$ 0 0
$$256$$ 3.09247 0.193279
$$257$$ 8.42123 0.525302 0.262651 0.964891i $$-0.415403\pi$$
0.262651 + 0.964891i $$0.415403\pi$$
$$258$$ 0 0
$$259$$ −16.0155 −0.995156
$$260$$ −4.95333 −0.307193
$$261$$ 0 0
$$262$$ 8.46016 0.522670
$$263$$ 2.28447 0.140866 0.0704331 0.997517i $$-0.477562\pi$$
0.0704331 + 0.997517i $$0.477562\pi$$
$$264$$ 0 0
$$265$$ −11.1099 −0.682477
$$266$$ 2.87162 0.176071
$$267$$ 0 0
$$268$$ −8.52250 −0.520595
$$269$$ 1.62731 0.0992186 0.0496093 0.998769i $$-0.484202\pi$$
0.0496093 + 0.998769i $$0.484202\pi$$
$$270$$ 0 0
$$271$$ 6.18126 0.375485 0.187742 0.982218i $$-0.439883\pi$$
0.187742 + 0.982218i $$0.439883\pi$$
$$272$$ −13.8013 −0.836829
$$273$$ 0 0
$$274$$ 3.07437 0.185730
$$275$$ 0.00755077 0.000455329 0
$$276$$ 0 0
$$277$$ −12.5435 −0.753668 −0.376834 0.926281i $$-0.622987\pi$$
−0.376834 + 0.926281i $$0.622987\pi$$
$$278$$ 4.18011 0.250707
$$279$$ 0 0
$$280$$ 3.99130 0.238526
$$281$$ 3.78236 0.225637 0.112818 0.993616i $$-0.464012\pi$$
0.112818 + 0.993616i $$0.464012\pi$$
$$282$$ 0 0
$$283$$ −20.7316 −1.23237 −0.616183 0.787603i $$-0.711322\pi$$
−0.616183 + 0.787603i $$0.711322\pi$$
$$284$$ 2.69845 0.160124
$$285$$ 0 0
$$286$$ 0.00889920 0.000526221 0
$$287$$ 0.388489 0.0229318
$$288$$ 0 0
$$289$$ 5.38218 0.316599
$$290$$ 0.431560 0.0253421
$$291$$ 0 0
$$292$$ −5.00044 −0.292629
$$293$$ −15.3438 −0.896394 −0.448197 0.893935i $$-0.647934\pi$$
−0.448197 + 0.893935i $$0.647934\pi$$
$$294$$ 0 0
$$295$$ −8.85010 −0.515273
$$296$$ 10.8696 0.631783
$$297$$ 0 0
$$298$$ −7.63869 −0.442498
$$299$$ 10.0738 0.582584
$$300$$ 0 0
$$301$$ −26.7006 −1.53899
$$302$$ −8.95095 −0.515069
$$303$$ 0 0
$$304$$ 8.00453 0.459091
$$305$$ −13.5237 −0.774365
$$306$$ 0 0
$$307$$ −6.67480 −0.380951 −0.190476 0.981692i $$-0.561003\pi$$
−0.190476 + 0.981692i $$0.561003\pi$$
$$308$$ 0.0332117 0.00189241
$$309$$ 0 0
$$310$$ 0.542089 0.0307886
$$311$$ 4.07460 0.231049 0.115525 0.993305i $$-0.463145\pi$$
0.115525 + 0.993305i $$0.463145\pi$$
$$312$$ 0 0
$$313$$ 10.8923 0.615672 0.307836 0.951439i $$-0.400395\pi$$
0.307836 + 0.951439i $$0.400395\pi$$
$$314$$ 2.87354 0.162163
$$315$$ 0 0
$$316$$ −4.94935 −0.278423
$$317$$ −21.5993 −1.21314 −0.606569 0.795031i $$-0.707455\pi$$
−0.606569 + 0.795031i $$0.707455\pi$$
$$318$$ 0 0
$$319$$ 0.00755077 0.000422762 0
$$320$$ 3.87056 0.216371
$$321$$ 0 0
$$322$$ −3.86044 −0.215134
$$323$$ −12.9813 −0.722297
$$324$$ 0 0
$$325$$ −2.73098 −0.151488
$$326$$ 2.54444 0.140924
$$327$$ 0 0
$$328$$ −0.263665 −0.0145584
$$329$$ 25.1565 1.38692
$$330$$ 0 0
$$331$$ 15.1834 0.834558 0.417279 0.908778i $$-0.362984\pi$$
0.417279 + 0.908778i $$0.362984\pi$$
$$332$$ −14.1834 −0.778413
$$333$$ 0 0
$$334$$ 4.39850 0.240675
$$335$$ −4.69881 −0.256724
$$336$$ 0 0
$$337$$ 11.0960 0.604437 0.302219 0.953239i $$-0.402273\pi$$
0.302219 + 0.953239i $$0.402273\pi$$
$$338$$ 2.39159 0.130086
$$339$$ 0 0
$$340$$ −8.58085 −0.465362
$$341$$ 0.00948465 0.000513622 0
$$342$$ 0 0
$$343$$ 19.6893 1.06312
$$344$$ 18.1215 0.977043
$$345$$ 0 0
$$346$$ 4.68517 0.251876
$$347$$ −13.9866 −0.750838 −0.375419 0.926855i $$-0.622501\pi$$
−0.375419 + 0.926855i $$0.622501\pi$$
$$348$$ 0 0
$$349$$ −23.8346 −1.27584 −0.637918 0.770104i $$-0.720204\pi$$
−0.637918 + 0.770104i $$0.720204\pi$$
$$350$$ 1.04655 0.0559407
$$351$$ 0 0
$$352$$ −0.0343612 −0.00183146
$$353$$ 20.0478 1.06704 0.533518 0.845789i $$-0.320870\pi$$
0.533518 + 0.845789i $$0.320870\pi$$
$$354$$ 0 0
$$355$$ 1.48777 0.0789626
$$356$$ 13.7776 0.730213
$$357$$ 0 0
$$358$$ −8.94061 −0.472526
$$359$$ −22.5167 −1.18839 −0.594193 0.804322i $$-0.702528\pi$$
−0.594193 + 0.804322i $$0.702528\pi$$
$$360$$ 0 0
$$361$$ −11.4711 −0.603742
$$362$$ 0.0242436 0.00127422
$$363$$ 0 0
$$364$$ −12.0121 −0.629605
$$365$$ −2.75695 −0.144306
$$366$$ 0 0
$$367$$ 35.5683 1.85665 0.928324 0.371772i $$-0.121250\pi$$
0.928324 + 0.371772i $$0.121250\pi$$
$$368$$ −10.7608 −0.560947
$$369$$ 0 0
$$370$$ 2.85010 0.148170
$$371$$ −26.9422 −1.39877
$$372$$ 0 0
$$373$$ 6.61603 0.342565 0.171283 0.985222i $$-0.445209\pi$$
0.171283 + 0.985222i $$0.445209\pi$$
$$374$$ 0.0154164 0.000797164 0
$$375$$ 0 0
$$376$$ −17.0735 −0.880499
$$377$$ −2.73098 −0.140653
$$378$$ 0 0
$$379$$ 16.6561 0.855567 0.427784 0.903881i $$-0.359295\pi$$
0.427784 + 0.903881i $$0.359295\pi$$
$$380$$ 4.97674 0.255301
$$381$$ 0 0
$$382$$ −1.39919 −0.0715890
$$383$$ −16.6700 −0.851796 −0.425898 0.904771i $$-0.640042\pi$$
−0.425898 + 0.904771i $$0.640042\pi$$
$$384$$ 0 0
$$385$$ 0.0183110 0.000933216 0
$$386$$ 4.42442 0.225197
$$387$$ 0 0
$$388$$ 12.7424 0.646896
$$389$$ −37.3519 −1.89382 −0.946908 0.321505i $$-0.895811\pi$$
−0.946908 + 0.321505i $$0.895811\pi$$
$$390$$ 0 0
$$391$$ 17.4513 0.882548
$$392$$ −1.84192 −0.0930312
$$393$$ 0 0
$$394$$ 4.64329 0.233926
$$395$$ −2.72878 −0.137300
$$396$$ 0 0
$$397$$ 20.0742 1.00750 0.503748 0.863850i $$-0.331954\pi$$
0.503748 + 0.863850i $$0.331954\pi$$
$$398$$ −3.51029 −0.175955
$$399$$ 0 0
$$400$$ 2.91722 0.145861
$$401$$ 12.4073 0.619591 0.309795 0.950803i $$-0.399739\pi$$
0.309795 + 0.950803i $$0.399739\pi$$
$$402$$ 0 0
$$403$$ −3.43043 −0.170882
$$404$$ −28.0817 −1.39712
$$405$$ 0 0
$$406$$ 1.04655 0.0519396
$$407$$ 0.0498668 0.00247180
$$408$$ 0 0
$$409$$ −16.9907 −0.840138 −0.420069 0.907492i $$-0.637994\pi$$
−0.420069 + 0.907492i $$0.637994\pi$$
$$410$$ −0.0691351 −0.00341434
$$411$$ 0 0
$$412$$ −12.6984 −0.625608
$$413$$ −21.4620 −1.05607
$$414$$ 0 0
$$415$$ −7.81988 −0.383863
$$416$$ 12.4278 0.609324
$$417$$ 0 0
$$418$$ −0.00894125 −0.000437331 0
$$419$$ 8.07891 0.394680 0.197340 0.980335i $$-0.436770\pi$$
0.197340 + 0.980335i $$0.436770\pi$$
$$420$$ 0 0
$$421$$ 9.72062 0.473754 0.236877 0.971540i $$-0.423876\pi$$
0.236877 + 0.971540i $$0.423876\pi$$
$$422$$ −1.04588 −0.0509126
$$423$$ 0 0
$$424$$ 18.2854 0.888019
$$425$$ −4.73098 −0.229486
$$426$$ 0 0
$$427$$ −32.7957 −1.58709
$$428$$ 20.5620 0.993901
$$429$$ 0 0
$$430$$ 4.75160 0.229143
$$431$$ 38.4501 1.85208 0.926038 0.377429i $$-0.123192\pi$$
0.926038 + 0.377429i $$0.123192\pi$$
$$432$$ 0 0
$$433$$ 1.54078 0.0740452 0.0370226 0.999314i $$-0.488213\pi$$
0.0370226 + 0.999314i $$0.488213\pi$$
$$434$$ 1.31459 0.0631025
$$435$$ 0 0
$$436$$ −28.1183 −1.34662
$$437$$ −10.1214 −0.484173
$$438$$ 0 0
$$439$$ 13.0277 0.621778 0.310889 0.950446i $$-0.399373\pi$$
0.310889 + 0.950446i $$0.399373\pi$$
$$440$$ −0.0124275 −0.000592460 0
$$441$$ 0 0
$$442$$ −5.57584 −0.265216
$$443$$ 4.37921 0.208063 0.104031 0.994574i $$-0.466826\pi$$
0.104031 + 0.994574i $$0.466826\pi$$
$$444$$ 0 0
$$445$$ 7.59619 0.360094
$$446$$ −6.96562 −0.329832
$$447$$ 0 0
$$448$$ 9.38630 0.443461
$$449$$ 19.0425 0.898673 0.449336 0.893363i $$-0.351661\pi$$
0.449336 + 0.893363i $$0.351661\pi$$
$$450$$ 0 0
$$451$$ −0.00120962 −5.69588e−5 0
$$452$$ −1.60289 −0.0753936
$$453$$ 0 0
$$454$$ −8.53761 −0.400690
$$455$$ −6.62277 −0.310480
$$456$$ 0 0
$$457$$ −12.5512 −0.587122 −0.293561 0.955940i $$-0.594840\pi$$
−0.293561 + 0.955940i $$0.594840\pi$$
$$458$$ 0.861222 0.0402423
$$459$$ 0 0
$$460$$ −6.69044 −0.311943
$$461$$ 29.0335 1.35223 0.676113 0.736798i $$-0.263663\pi$$
0.676113 + 0.736798i $$0.263663\pi$$
$$462$$ 0 0
$$463$$ 2.74254 0.127457 0.0637284 0.997967i $$-0.479701\pi$$
0.0637284 + 0.997967i $$0.479701\pi$$
$$464$$ 2.91722 0.135429
$$465$$ 0 0
$$466$$ 1.72022 0.0796878
$$467$$ 7.53116 0.348501 0.174250 0.984701i $$-0.444250\pi$$
0.174250 + 0.984701i $$0.444250\pi$$
$$468$$ 0 0
$$469$$ −11.3949 −0.526166
$$470$$ −4.47682 −0.206501
$$471$$ 0 0
$$472$$ 14.5661 0.670457
$$473$$ 0.0831363 0.00382261
$$474$$ 0 0
$$475$$ 2.74388 0.125898
$$476$$ −20.8090 −0.953779
$$477$$ 0 0
$$478$$ −12.7730 −0.584223
$$479$$ −25.4783 −1.16414 −0.582068 0.813140i $$-0.697756\pi$$
−0.582068 + 0.813140i $$0.697756\pi$$
$$480$$ 0 0
$$481$$ −18.0359 −0.822368
$$482$$ 5.68029 0.258730
$$483$$ 0 0
$$484$$ 19.9512 0.906873
$$485$$ 7.02540 0.319007
$$486$$ 0 0
$$487$$ 2.38516 0.108082 0.0540409 0.998539i $$-0.482790\pi$$
0.0540409 + 0.998539i $$0.482790\pi$$
$$488$$ 22.2582 1.00758
$$489$$ 0 0
$$490$$ −0.482968 −0.0218183
$$491$$ 32.4925 1.46637 0.733184 0.680031i $$-0.238034\pi$$
0.733184 + 0.680031i $$0.238034\pi$$
$$492$$ 0 0
$$493$$ −4.73098 −0.213073
$$494$$ 3.23389 0.145500
$$495$$ 0 0
$$496$$ 3.66437 0.164535
$$497$$ 3.60792 0.161837
$$498$$ 0 0
$$499$$ −8.56521 −0.383431 −0.191716 0.981451i $$-0.561405\pi$$
−0.191716 + 0.981451i $$0.561405\pi$$
$$500$$ 1.81376 0.0811136
$$501$$ 0 0
$$502$$ 2.95347 0.131820
$$503$$ 3.39783 0.151502 0.0757510 0.997127i $$-0.475865\pi$$
0.0757510 + 0.997127i $$0.475865\pi$$
$$504$$ 0 0
$$505$$ −15.4826 −0.688968
$$506$$ 0.0120201 0.000534358 0
$$507$$ 0 0
$$508$$ −19.9974 −0.887241
$$509$$ 29.0172 1.28616 0.643082 0.765797i $$-0.277655\pi$$
0.643082 + 0.765797i $$0.277655\pi$$
$$510$$ 0 0
$$511$$ −6.68576 −0.295761
$$512$$ −22.8781 −1.01108
$$513$$ 0 0
$$514$$ −3.63426 −0.160301
$$515$$ −7.00119 −0.308509
$$516$$ 0 0
$$517$$ −0.0783287 −0.00344489
$$518$$ 6.91165 0.303680
$$519$$ 0 0
$$520$$ 4.49482 0.197111
$$521$$ −10.8098 −0.473588 −0.236794 0.971560i $$-0.576097\pi$$
−0.236794 + 0.971560i $$0.576097\pi$$
$$522$$ 0 0
$$523$$ 24.7294 1.08134 0.540671 0.841234i $$-0.318170\pi$$
0.540671 + 0.841234i $$0.318170\pi$$
$$524$$ 35.5563 1.55328
$$525$$ 0 0
$$526$$ −0.985884 −0.0429866
$$527$$ −5.94266 −0.258866
$$528$$ 0 0
$$529$$ −9.39336 −0.408407
$$530$$ 4.79460 0.208264
$$531$$ 0 0
$$532$$ 12.0688 0.523251
$$533$$ 0.437499 0.0189502
$$534$$ 0 0
$$535$$ 11.3367 0.490128
$$536$$ 7.73360 0.334041
$$537$$ 0 0
$$538$$ −0.702279 −0.0302774
$$539$$ −0.00845025 −0.000363978 0
$$540$$ 0 0
$$541$$ −10.2403 −0.440264 −0.220132 0.975470i $$-0.570649\pi$$
−0.220132 + 0.975470i $$0.570649\pi$$
$$542$$ −2.66758 −0.114582
$$543$$ 0 0
$$544$$ 21.5292 0.923057
$$545$$ −15.5028 −0.664068
$$546$$ 0 0
$$547$$ −1.68008 −0.0718349 −0.0359175 0.999355i $$-0.511435\pi$$
−0.0359175 + 0.999355i $$0.511435\pi$$
$$548$$ 12.9210 0.551956
$$549$$ 0 0
$$550$$ −0.00325861 −0.000138948 0
$$551$$ 2.74388 0.116893
$$552$$ 0 0
$$553$$ −6.61744 −0.281402
$$554$$ 5.41329 0.229988
$$555$$ 0 0
$$556$$ 17.5682 0.745056
$$557$$ −17.6075 −0.746053 −0.373026 0.927821i $$-0.621680\pi$$
−0.373026 + 0.927821i $$0.621680\pi$$
$$558$$ 0 0
$$559$$ −30.0689 −1.27178
$$560$$ 7.07442 0.298949
$$561$$ 0 0
$$562$$ −1.63231 −0.0688550
$$563$$ −38.3553 −1.61648 −0.808241 0.588852i $$-0.799580\pi$$
−0.808241 + 0.588852i $$0.799580\pi$$
$$564$$ 0 0
$$565$$ −0.883740 −0.0371792
$$566$$ 8.94692 0.376067
$$567$$ 0 0
$$568$$ −2.44866 −0.102744
$$569$$ −5.63212 −0.236111 −0.118055 0.993007i $$-0.537666\pi$$
−0.118055 + 0.993007i $$0.537666\pi$$
$$570$$ 0 0
$$571$$ 13.6537 0.571389 0.285695 0.958321i $$-0.407776\pi$$
0.285695 + 0.958321i $$0.407776\pi$$
$$572$$ 0.0374015 0.00156384
$$573$$ 0 0
$$574$$ −0.167656 −0.00699784
$$575$$ −3.68872 −0.153830
$$576$$ 0 0
$$577$$ 12.1911 0.507521 0.253760 0.967267i $$-0.418333\pi$$
0.253760 + 0.967267i $$0.418333\pi$$
$$578$$ −2.32273 −0.0966129
$$579$$ 0 0
$$580$$ 1.81376 0.0753121
$$581$$ −18.9636 −0.786743
$$582$$ 0 0
$$583$$ 0.0838886 0.00347431
$$584$$ 4.53757 0.187766
$$585$$ 0 0
$$586$$ 6.62176 0.273543
$$587$$ −16.0556 −0.662687 −0.331343 0.943510i $$-0.607502\pi$$
−0.331343 + 0.943510i $$0.607502\pi$$
$$588$$ 0 0
$$589$$ 3.44664 0.142016
$$590$$ 3.81935 0.157240
$$591$$ 0 0
$$592$$ 19.2659 0.791824
$$593$$ −18.6072 −0.764108 −0.382054 0.924140i $$-0.624783\pi$$
−0.382054 + 0.924140i $$0.624783\pi$$
$$594$$ 0 0
$$595$$ −11.4729 −0.470342
$$596$$ −32.1039 −1.31503
$$597$$ 0 0
$$598$$ −4.34745 −0.177781
$$599$$ 28.3057 1.15654 0.578269 0.815846i $$-0.303728\pi$$
0.578269 + 0.815846i $$0.303728\pi$$
$$600$$ 0 0
$$601$$ 30.1736 1.23081 0.615404 0.788212i $$-0.288993\pi$$
0.615404 + 0.788212i $$0.288993\pi$$
$$602$$ 11.5229 0.469638
$$603$$ 0 0
$$604$$ −37.6190 −1.53070
$$605$$ 10.9999 0.447211
$$606$$ 0 0
$$607$$ −3.99729 −0.162245 −0.0811224 0.996704i $$-0.525850\pi$$
−0.0811224 + 0.996704i $$0.525850\pi$$
$$608$$ −12.4865 −0.506396
$$609$$ 0 0
$$610$$ 5.83628 0.236304
$$611$$ 28.3301 1.14611
$$612$$ 0 0
$$613$$ 38.3571 1.54923 0.774614 0.632434i $$-0.217944\pi$$
0.774614 + 0.632434i $$0.217944\pi$$
$$614$$ 2.88058 0.116251
$$615$$ 0 0
$$616$$ −0.0301374 −0.00121427
$$617$$ −16.7381 −0.673852 −0.336926 0.941531i $$-0.609387\pi$$
−0.336926 + 0.941531i $$0.609387\pi$$
$$618$$ 0 0
$$619$$ 47.4248 1.90616 0.953081 0.302714i $$-0.0978929\pi$$
0.953081 + 0.302714i $$0.0978929\pi$$
$$620$$ 2.27829 0.0914982
$$621$$ 0 0
$$622$$ −1.75843 −0.0705067
$$623$$ 18.4211 0.738028
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −4.70070 −0.187878
$$627$$ 0 0
$$628$$ 12.0769 0.481920
$$629$$ −31.2443 −1.24579
$$630$$ 0 0
$$631$$ 21.1985 0.843898 0.421949 0.906620i $$-0.361346\pi$$
0.421949 + 0.906620i $$0.361346\pi$$
$$632$$ 4.49120 0.178650
$$633$$ 0 0
$$634$$ 9.32139 0.370200
$$635$$ −11.0254 −0.437530
$$636$$ 0 0
$$637$$ 3.05630 0.121095
$$638$$ −0.00325861 −0.000129010 0
$$639$$ 0 0
$$640$$ −10.7717 −0.425790
$$641$$ 9.76078 0.385528 0.192764 0.981245i $$-0.438255\pi$$
0.192764 + 0.981245i $$0.438255\pi$$
$$642$$ 0 0
$$643$$ −3.87214 −0.152702 −0.0763511 0.997081i $$-0.524327\pi$$
−0.0763511 + 0.997081i $$0.524327\pi$$
$$644$$ −16.2247 −0.639341
$$645$$ 0 0
$$646$$ 5.60219 0.220415
$$647$$ 11.7270 0.461037 0.230518 0.973068i $$-0.425958\pi$$
0.230518 + 0.973068i $$0.425958\pi$$
$$648$$ 0 0
$$649$$ 0.0668251 0.00262312
$$650$$ 1.17858 0.0462278
$$651$$ 0 0
$$652$$ 10.6938 0.418800
$$653$$ 27.0409 1.05819 0.529097 0.848561i $$-0.322531\pi$$
0.529097 + 0.848561i $$0.322531\pi$$
$$654$$ 0 0
$$655$$ 19.6037 0.765979
$$656$$ −0.467334 −0.0182463
$$657$$ 0 0
$$658$$ −10.8565 −0.423232
$$659$$ 27.3773 1.06647 0.533235 0.845967i $$-0.320976\pi$$
0.533235 + 0.845967i $$0.320976\pi$$
$$660$$ 0 0
$$661$$ 42.6188 1.65768 0.828839 0.559487i $$-0.189002\pi$$
0.828839 + 0.559487i $$0.189002\pi$$
$$662$$ −6.55256 −0.254673
$$663$$ 0 0
$$664$$ 12.8704 0.499470
$$665$$ 6.65406 0.258033
$$666$$ 0 0
$$667$$ −3.68872 −0.142828
$$668$$ 18.4860 0.715244
$$669$$ 0 0
$$670$$ 2.02782 0.0783414
$$671$$ 0.102114 0.00394208
$$672$$ 0 0
$$673$$ 30.4014 1.17189 0.585943 0.810352i $$-0.300724\pi$$
0.585943 + 0.810352i $$0.300724\pi$$
$$674$$ −4.78858 −0.184449
$$675$$ 0 0
$$676$$ 10.0514 0.386591
$$677$$ 39.1078 1.50303 0.751517 0.659714i $$-0.229323\pi$$
0.751517 + 0.659714i $$0.229323\pi$$
$$678$$ 0 0
$$679$$ 17.0370 0.653819
$$680$$ 7.78655 0.298600
$$681$$ 0 0
$$682$$ −0.00409319 −0.000156736 0
$$683$$ 22.2515 0.851431 0.425716 0.904857i $$-0.360022\pi$$
0.425716 + 0.904857i $$0.360022\pi$$
$$684$$ 0 0
$$685$$ 7.12387 0.272189
$$686$$ −8.49710 −0.324421
$$687$$ 0 0
$$688$$ 32.1195 1.22455
$$689$$ −30.3410 −1.15590
$$690$$ 0 0
$$691$$ −19.8899 −0.756647 −0.378323 0.925674i $$-0.623499\pi$$
−0.378323 + 0.925674i $$0.623499\pi$$
$$692$$ 19.6908 0.748531
$$693$$ 0 0
$$694$$ 6.03603 0.229125
$$695$$ 9.68606 0.367413
$$696$$ 0 0
$$697$$ 0.757895 0.0287073
$$698$$ 10.2861 0.389333
$$699$$ 0 0
$$700$$ 4.39845 0.166246
$$701$$ 35.7640 1.35079 0.675393 0.737458i $$-0.263974\pi$$
0.675393 + 0.737458i $$0.263974\pi$$
$$702$$ 0 0
$$703$$ 18.1211 0.683452
$$704$$ −0.0292257 −0.00110149
$$705$$ 0 0
$$706$$ −8.65182 −0.325615
$$707$$ −37.5462 −1.41207
$$708$$ 0 0
$$709$$ 36.8609 1.38434 0.692171 0.721734i $$-0.256654\pi$$
0.692171 + 0.721734i $$0.256654\pi$$
$$710$$ −0.642061 −0.0240961
$$711$$ 0 0
$$712$$ −12.5023 −0.468543
$$713$$ −4.63346 −0.173524
$$714$$ 0 0
$$715$$ 0.0206210 0.000771182 0
$$716$$ −37.5756 −1.40426
$$717$$ 0 0
$$718$$ 9.71730 0.362646
$$719$$ 5.42783 0.202424 0.101212 0.994865i $$-0.467728\pi$$
0.101212 + 0.994865i $$0.467728\pi$$
$$720$$ 0 0
$$721$$ −16.9782 −0.632303
$$722$$ 4.95046 0.184237
$$723$$ 0 0
$$724$$ 0.101891 0.00378675
$$725$$ 1.00000 0.0371391
$$726$$ 0 0
$$727$$ 24.6244 0.913267 0.456633 0.889655i $$-0.349055\pi$$
0.456633 + 0.889655i $$0.349055\pi$$
$$728$$ 10.9002 0.403987
$$729$$ 0 0
$$730$$ 1.18979 0.0440361
$$731$$ −52.0895 −1.92660
$$732$$ 0 0
$$733$$ −41.7803 −1.54319 −0.771594 0.636115i $$-0.780540\pi$$
−0.771594 + 0.636115i $$0.780540\pi$$
$$734$$ −15.3498 −0.566572
$$735$$ 0 0
$$736$$ 16.7862 0.618747
$$737$$ 0.0354797 0.00130691
$$738$$ 0 0
$$739$$ −1.46856 −0.0540217 −0.0270108 0.999635i $$-0.508599\pi$$
−0.0270108 + 0.999635i $$0.508599\pi$$
$$740$$ 11.9784 0.440335
$$741$$ 0 0
$$742$$ 11.6271 0.426846
$$743$$ 42.7834 1.56957 0.784785 0.619768i $$-0.212773\pi$$
0.784785 + 0.619768i $$0.212773\pi$$
$$744$$ 0 0
$$745$$ −17.7002 −0.648485
$$746$$ −2.85521 −0.104537
$$747$$ 0 0
$$748$$ 0.0647920 0.00236903
$$749$$ 27.4921 1.00454
$$750$$ 0 0
$$751$$ −32.9769 −1.20335 −0.601673 0.798743i $$-0.705499\pi$$
−0.601673 + 0.798743i $$0.705499\pi$$
$$752$$ −30.2621 −1.10355
$$753$$ 0 0
$$754$$ 1.17858 0.0429214
$$755$$ −20.7409 −0.754840
$$756$$ 0 0
$$757$$ 43.4442 1.57900 0.789502 0.613748i $$-0.210339\pi$$
0.789502 + 0.613748i $$0.210339\pi$$
$$758$$ −7.18811 −0.261084
$$759$$ 0 0
$$760$$ −4.51606 −0.163815
$$761$$ −12.9239 −0.468492 −0.234246 0.972177i $$-0.575262\pi$$
−0.234246 + 0.972177i $$0.575262\pi$$
$$762$$ 0 0
$$763$$ −37.5951 −1.36103
$$764$$ −5.88052 −0.212750
$$765$$ 0 0
$$766$$ 7.19409 0.259933
$$767$$ −24.1695 −0.872709
$$768$$ 0 0
$$769$$ 33.6451 1.21327 0.606636 0.794979i $$-0.292518\pi$$
0.606636 + 0.794979i $$0.292518\pi$$
$$770$$ −0.00790230 −0.000284779 0
$$771$$ 0 0
$$772$$ 18.5949 0.669246
$$773$$ −47.5209 −1.70921 −0.854603 0.519281i $$-0.826200\pi$$
−0.854603 + 0.519281i $$0.826200\pi$$
$$774$$ 0 0
$$775$$ 1.25612 0.0451210
$$776$$ −11.5629 −0.415082
$$777$$ 0 0
$$778$$ 16.1196 0.577914
$$779$$ −0.439566 −0.0157491
$$780$$ 0 0
$$781$$ −0.0112338 −0.000401977 0
$$782$$ −7.53126 −0.269317
$$783$$ 0 0
$$784$$ −3.26473 −0.116598
$$785$$ 6.65849 0.237652
$$786$$ 0 0
$$787$$ −2.20720 −0.0786783 −0.0393391 0.999226i $$-0.512525\pi$$
−0.0393391 + 0.999226i $$0.512525\pi$$
$$788$$ 19.5148 0.695186
$$789$$ 0 0
$$790$$ 1.17763 0.0418983
$$791$$ −2.14312 −0.0762004
$$792$$ 0 0
$$793$$ −36.9330 −1.31153
$$794$$ −8.66323 −0.307446
$$795$$ 0 0
$$796$$ −14.7530 −0.522907
$$797$$ 35.5149 1.25800 0.629001 0.777405i $$-0.283464\pi$$
0.629001 + 0.777405i $$0.283464\pi$$
$$798$$ 0 0
$$799$$ 49.0773 1.73623
$$800$$ −4.55068 −0.160891
$$801$$ 0 0
$$802$$ −5.35449 −0.189074
$$803$$ 0.0208171 0.000734621 0
$$804$$ 0 0
$$805$$ −8.94533 −0.315282
$$806$$ 1.48043 0.0521461
$$807$$ 0 0
$$808$$ 25.4823 0.896463
$$809$$ −16.1284 −0.567044 −0.283522 0.958966i $$-0.591503\pi$$
−0.283522 + 0.958966i $$0.591503\pi$$
$$810$$ 0 0
$$811$$ −21.5983 −0.758417 −0.379209 0.925311i $$-0.623804\pi$$
−0.379209 + 0.925311i $$0.623804\pi$$
$$812$$ 4.39845 0.154355
$$813$$ 0 0
$$814$$ −0.0215205 −0.000754293 0
$$815$$ 5.89592 0.206525
$$816$$ 0 0
$$817$$ 30.2110 1.05695
$$818$$ 7.33252 0.256376
$$819$$ 0 0
$$820$$ −0.290561 −0.0101468
$$821$$ −35.7338 −1.24712 −0.623559 0.781777i $$-0.714314\pi$$
−0.623559 + 0.781777i $$0.714314\pi$$
$$822$$ 0 0
$$823$$ 19.2184 0.669911 0.334956 0.942234i $$-0.391279\pi$$
0.334956 + 0.942234i $$0.391279\pi$$
$$824$$ 11.5230 0.401423
$$825$$ 0 0
$$826$$ 9.26212 0.322270
$$827$$ 15.1283 0.526061 0.263031 0.964787i $$-0.415278\pi$$
0.263031 + 0.964787i $$0.415278\pi$$
$$828$$ 0 0
$$829$$ 20.8410 0.723837 0.361918 0.932210i $$-0.382122\pi$$
0.361918 + 0.932210i $$0.382122\pi$$
$$830$$ 3.37474 0.117139
$$831$$ 0 0
$$832$$ 10.5704 0.366463
$$833$$ 5.29455 0.183445
$$834$$ 0 0
$$835$$ 10.1921 0.352712
$$836$$ −0.0375782 −0.00129967
$$837$$ 0 0
$$838$$ −3.48653 −0.120440
$$839$$ −17.3275 −0.598213 −0.299107 0.954220i $$-0.596689\pi$$
−0.299107 + 0.954220i $$0.596689\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ −4.19503 −0.144570
$$843$$ 0 0
$$844$$ −4.39562 −0.151303
$$845$$ 5.54174 0.190642
$$846$$ 0 0
$$847$$ 26.6754 0.916579
$$848$$ 32.4102 1.11297
$$849$$ 0 0
$$850$$ 2.04170 0.0700297
$$851$$ −24.3610 −0.835085
$$852$$ 0 0
$$853$$ −33.5458 −1.14859 −0.574293 0.818650i $$-0.694723\pi$$
−0.574293 + 0.818650i $$0.694723\pi$$
$$854$$ 14.1533 0.484316
$$855$$ 0 0
$$856$$ −18.6586 −0.637739
$$857$$ 55.0914 1.88189 0.940944 0.338563i $$-0.109941\pi$$
0.940944 + 0.338563i $$0.109941\pi$$
$$858$$ 0 0
$$859$$ 4.34649 0.148300 0.0741501 0.997247i $$-0.476376\pi$$
0.0741501 + 0.997247i $$0.476376\pi$$
$$860$$ 19.9700 0.680972
$$861$$ 0 0
$$862$$ −16.5935 −0.565177
$$863$$ −39.8846 −1.35769 −0.678844 0.734282i $$-0.737519\pi$$
−0.678844 + 0.734282i $$0.737519\pi$$
$$864$$ 0 0
$$865$$ 10.8564 0.369127
$$866$$ −0.664939 −0.0225955
$$867$$ 0 0
$$868$$ 5.52497 0.187530
$$869$$ 0.0206044 0.000698957 0
$$870$$ 0 0
$$871$$ −12.8324 −0.434808
$$872$$ 25.5155 0.864064
$$873$$ 0 0
$$874$$ 4.36799 0.147750
$$875$$ 2.42505 0.0819817
$$876$$ 0 0
$$877$$ −41.1414 −1.38925 −0.694623 0.719374i $$-0.744429\pi$$
−0.694623 + 0.719374i $$0.744429\pi$$
$$878$$ −5.62223 −0.189741
$$879$$ 0 0
$$880$$ −0.0220273 −0.000742540 0
$$881$$ 19.1383 0.644784 0.322392 0.946606i $$-0.395513\pi$$
0.322392 + 0.946606i $$0.395513\pi$$
$$882$$ 0 0
$$883$$ 10.3573 0.348550 0.174275 0.984697i $$-0.444242\pi$$
0.174275 + 0.984697i $$0.444242\pi$$
$$884$$ −23.4341 −0.788175
$$885$$ 0 0
$$886$$ −1.88989 −0.0634921
$$887$$ 11.5233 0.386915 0.193457 0.981109i $$-0.438030\pi$$
0.193457 + 0.981109i $$0.438030\pi$$
$$888$$ 0 0
$$889$$ −26.7372 −0.896736
$$890$$ −3.27821 −0.109886
$$891$$ 0 0
$$892$$ −29.2751 −0.980202
$$893$$ −28.4639 −0.952510
$$894$$ 0 0
$$895$$ −20.7170 −0.692492
$$896$$ −26.1220 −0.872676
$$897$$ 0 0
$$898$$ −8.21799 −0.274238
$$899$$ 1.25612 0.0418938
$$900$$ 0 0
$$901$$ −52.5609 −1.75106
$$902$$ 0.000522024 0 1.73815e−5 0
$$903$$ 0 0
$$904$$ 1.45452 0.0483765
$$905$$ 0.0561768 0.00186738
$$906$$ 0 0
$$907$$ 49.6527 1.64869 0.824345 0.566088i $$-0.191544\pi$$
0.824345 + 0.566088i $$0.191544\pi$$
$$908$$ −35.8818 −1.19078
$$909$$ 0 0
$$910$$ 2.85812 0.0947458
$$911$$ −46.1515 −1.52907 −0.764534 0.644583i $$-0.777031\pi$$
−0.764534 + 0.644583i $$0.777031\pi$$
$$912$$ 0 0
$$913$$ 0.0590461 0.00195414
$$914$$ 5.41660 0.179165
$$915$$ 0 0
$$916$$ 3.61954 0.119593
$$917$$ 47.5399 1.56991
$$918$$ 0 0
$$919$$ −49.4890 −1.63249 −0.816245 0.577706i $$-0.803948\pi$$
−0.816245 + 0.577706i $$0.803948\pi$$
$$920$$ 6.07113 0.200159
$$921$$ 0 0
$$922$$ −12.5297 −0.412643
$$923$$ 4.06307 0.133737
$$924$$ 0 0
$$925$$ 6.60420 0.217145
$$926$$ −1.18357 −0.0388946
$$927$$ 0 0
$$928$$ −4.55068 −0.149383
$$929$$ −26.7056 −0.876182 −0.438091 0.898931i $$-0.644345\pi$$
−0.438091 + 0.898931i $$0.644345\pi$$
$$930$$ 0 0
$$931$$ −3.07074 −0.100640
$$932$$ 7.22975 0.236818
$$933$$ 0 0
$$934$$ −3.25014 −0.106348
$$935$$ 0.0357226 0.00116825
$$936$$ 0 0
$$937$$ −35.2391 −1.15121 −0.575605 0.817728i $$-0.695233\pi$$
−0.575605 + 0.817728i $$0.695233\pi$$
$$938$$ 4.91756 0.160564
$$939$$ 0 0
$$940$$ −18.8152 −0.613683
$$941$$ 9.14156 0.298006 0.149003 0.988837i $$-0.452394\pi$$
0.149003 + 0.988837i $$0.452394\pi$$
$$942$$ 0 0
$$943$$ 0.590927 0.0192432
$$944$$ 25.8177 0.840296
$$945$$ 0 0
$$946$$ −0.0358783 −0.00116650
$$947$$ 19.2419 0.625277 0.312638 0.949872i $$-0.398787\pi$$
0.312638 + 0.949872i $$0.398787\pi$$
$$948$$ 0 0
$$949$$ −7.52919 −0.244408
$$950$$ −1.18415 −0.0384189
$$951$$ 0 0
$$952$$ 18.8828 0.611994
$$953$$ 22.3736 0.724751 0.362376 0.932032i $$-0.381966\pi$$
0.362376 + 0.932032i $$0.381966\pi$$
$$954$$ 0 0
$$955$$ −3.24218 −0.104914
$$956$$ −53.6823 −1.73621
$$957$$ 0 0
$$958$$ 10.9954 0.355246
$$959$$ 17.2757 0.557863
$$960$$ 0 0
$$961$$ −29.4222 −0.949102
$$962$$ 7.78358 0.250953
$$963$$ 0 0
$$964$$ 23.8731 0.768900
$$965$$ 10.2522 0.330029
$$966$$ 0 0
$$967$$ 14.7362 0.473885 0.236943 0.971524i $$-0.423855\pi$$
0.236943 + 0.971524i $$0.423855\pi$$
$$968$$ −18.1044 −0.581898
$$969$$ 0 0
$$970$$ −3.03188 −0.0973478
$$971$$ 21.7758 0.698819 0.349410 0.936970i $$-0.386382\pi$$
0.349410 + 0.936970i $$0.386382\pi$$
$$972$$ 0 0
$$973$$ 23.4892 0.753029
$$974$$ −1.02934 −0.0329821
$$975$$ 0 0
$$976$$ 39.4517 1.26282
$$977$$ −2.59126 −0.0829018 −0.0414509 0.999141i $$-0.513198\pi$$
−0.0414509 + 0.999141i $$0.513198\pi$$
$$978$$ 0 0
$$979$$ −0.0573571 −0.00183314
$$980$$ −2.02982 −0.0648401
$$981$$ 0 0
$$982$$ −14.0225 −0.447475
$$983$$ 41.5671 1.32578 0.662892 0.748715i $$-0.269329\pi$$
0.662892 + 0.748715i $$0.269329\pi$$
$$984$$ 0 0
$$985$$ 10.7593 0.342821
$$986$$ 2.04170 0.0650210
$$987$$ 0 0
$$988$$ 13.5914 0.432399
$$989$$ −40.6139 −1.29145
$$990$$ 0 0
$$991$$ 12.6958 0.403297 0.201648 0.979458i $$-0.435370\pi$$
0.201648 + 0.979458i $$0.435370\pi$$
$$992$$ −5.71618 −0.181489
$$993$$ 0 0
$$994$$ −1.55703 −0.0493860
$$995$$ −8.13396 −0.257864
$$996$$ 0 0
$$997$$ −54.4560 −1.72464 −0.862320 0.506364i $$-0.830989\pi$$
−0.862320 + 0.506364i $$0.830989\pi$$
$$998$$ 3.69640 0.117007
$$999$$ 0 0
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 1305.2.a.s.1.4 7
3.2 odd 2 1305.2.a.t.1.4 yes 7
5.4 even 2 6525.2.a.bw.1.4 7
15.14 odd 2 6525.2.a.bv.1.4 7

By twisted newform
Twist Min Dim Char Parity Ord Type
1305.2.a.s.1.4 7 1.1 even 1 trivial
1305.2.a.t.1.4 yes 7 3.2 odd 2
6525.2.a.bv.1.4 7 15.14 odd 2
6525.2.a.bw.1.4 7 5.4 even 2