# Properties

 Label 1305.2.a.q.1.2 Level $1305$ Weight $2$ Character 1305.1 Self dual yes Analytic conductor $10.420$ Analytic rank $1$ 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] = [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: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.469.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 5x + 4$$ x^3 - x^2 - 5*x + 4 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 435) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$0.772866$$ 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)

 $$f(q)$$ $$=$$ $$q-0.772866 q^{2} -1.40268 q^{4} -1.00000 q^{5} +2.17554 q^{7} +2.62981 q^{8} +O(q^{10})$$ $$q-0.772866 q^{2} -1.40268 q^{4} -1.00000 q^{5} +2.17554 q^{7} +2.62981 q^{8} +0.772866 q^{10} -3.00000 q^{11} -0.629813 q^{13} -1.68140 q^{14} +0.772866 q^{16} -4.17554 q^{17} +4.80536 q^{19} +1.40268 q^{20} +2.31860 q^{22} -2.08408 q^{23} +1.00000 q^{25} +0.486761 q^{26} -3.05159 q^{28} +1.00000 q^{29} -5.85695 q^{32} +3.22713 q^{34} -2.17554 q^{35} +6.08408 q^{37} -3.71390 q^{38} -2.62981 q^{40} -0.824456 q^{41} -8.72128 q^{43} +4.20804 q^{44} +1.61072 q^{46} +8.98090 q^{47} -2.26701 q^{49} -0.772866 q^{50} +0.883426 q^{52} -6.88944 q^{53} +3.00000 q^{55} +5.72128 q^{56} -0.772866 q^{58} -6.45427 q^{59} -2.80536 q^{61} +2.98090 q^{64} +0.629813 q^{65} -11.0841 q^{67} +5.85695 q^{68} +1.68140 q^{70} -2.63719 q^{71} -14.7863 q^{73} -4.70218 q^{74} -6.74037 q^{76} -6.52663 q^{77} -12.0650 q^{79} -0.772866 q^{80} +0.637193 q^{82} +7.62981 q^{83} +4.17554 q^{85} +6.74037 q^{86} -7.88944 q^{88} -17.8894 q^{89} -1.37019 q^{91} +2.92330 q^{92} -6.94103 q^{94} -4.80536 q^{95} +0.538351 q^{97} +1.75209 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} + 5 q^{4} - 3 q^{5} - 4 q^{7}+O(q^{10})$$ 3 * q - q^2 + 5 * q^4 - 3 * q^5 - 4 * q^7 $$3 q - q^{2} + 5 q^{4} - 3 q^{5} - 4 q^{7} + q^{10} - 9 q^{11} + 6 q^{13} - 9 q^{14} + q^{16} - 2 q^{17} - 4 q^{19} - 5 q^{20} + 3 q^{22} - q^{23} + 3 q^{25} - 13 q^{26} - 21 q^{28} + 3 q^{29} - 11 q^{32} + 11 q^{34} + 4 q^{35} + 13 q^{37} + 2 q^{38} - 13 q^{41} - 13 q^{43} - 15 q^{44} - 32 q^{46} - 2 q^{47} + 9 q^{49} - q^{50} + 25 q^{52} + 3 q^{53} + 9 q^{55} + 4 q^{56} - q^{58} - 22 q^{59} + 10 q^{61} - 20 q^{64} - 6 q^{65} - 28 q^{67} + 11 q^{68} + 9 q^{70} + 3 q^{73} + 28 q^{74} - 36 q^{76} + 12 q^{77} - 2 q^{79} - q^{80} - 6 q^{82} + 15 q^{83} + 2 q^{85} + 36 q^{86} - 30 q^{89} - 12 q^{91} + 14 q^{92} - 9 q^{94} + 4 q^{95} - q^{97} + 50 q^{98}+O(q^{100})$$ 3 * q - q^2 + 5 * q^4 - 3 * q^5 - 4 * q^7 + q^10 - 9 * q^11 + 6 * q^13 - 9 * q^14 + q^16 - 2 * q^17 - 4 * q^19 - 5 * q^20 + 3 * q^22 - q^23 + 3 * q^25 - 13 * q^26 - 21 * q^28 + 3 * q^29 - 11 * q^32 + 11 * q^34 + 4 * q^35 + 13 * q^37 + 2 * q^38 - 13 * q^41 - 13 * q^43 - 15 * q^44 - 32 * q^46 - 2 * q^47 + 9 * q^49 - q^50 + 25 * q^52 + 3 * q^53 + 9 * q^55 + 4 * q^56 - q^58 - 22 * q^59 + 10 * q^61 - 20 * q^64 - 6 * q^65 - 28 * q^67 + 11 * q^68 + 9 * q^70 + 3 * q^73 + 28 * q^74 - 36 * q^76 + 12 * q^77 - 2 * q^79 - q^80 - 6 * q^82 + 15 * q^83 + 2 * q^85 + 36 * q^86 - 30 * q^89 - 12 * q^91 + 14 * q^92 - 9 * q^94 + 4 * q^95 - q^97 + 50 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.772866 −0.546498 −0.273249 0.961943i $$-0.588098\pi$$
−0.273249 + 0.961943i $$0.588098\pi$$
$$3$$ 0 0
$$4$$ −1.40268 −0.701339
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 2.17554 0.822278 0.411139 0.911573i $$-0.365131\pi$$
0.411139 + 0.911573i $$0.365131\pi$$
$$8$$ 2.62981 0.929779
$$9$$ 0 0
$$10$$ 0.772866 0.244402
$$11$$ −3.00000 −0.904534 −0.452267 0.891883i $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ 0 0
$$13$$ −0.629813 −0.174679 −0.0873394 0.996179i $$-0.527836\pi$$
−0.0873394 + 0.996179i $$0.527836\pi$$
$$14$$ −1.68140 −0.449374
$$15$$ 0 0
$$16$$ 0.772866 0.193216
$$17$$ −4.17554 −1.01272 −0.506359 0.862323i $$-0.669009\pi$$
−0.506359 + 0.862323i $$0.669009\pi$$
$$18$$ 0 0
$$19$$ 4.80536 1.10242 0.551212 0.834365i $$-0.314165\pi$$
0.551212 + 0.834365i $$0.314165\pi$$
$$20$$ 1.40268 0.313649
$$21$$ 0 0
$$22$$ 2.31860 0.494326
$$23$$ −2.08408 −0.434561 −0.217281 0.976109i $$-0.569719\pi$$
−0.217281 + 0.976109i $$0.569719\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0.486761 0.0954617
$$27$$ 0 0
$$28$$ −3.05159 −0.576696
$$29$$ 1.00000 0.185695
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ −5.85695 −1.03537
$$33$$ 0 0
$$34$$ 3.22713 0.553449
$$35$$ −2.17554 −0.367734
$$36$$ 0 0
$$37$$ 6.08408 1.00022 0.500108 0.865963i $$-0.333293\pi$$
0.500108 + 0.865963i $$0.333293\pi$$
$$38$$ −3.71390 −0.602473
$$39$$ 0 0
$$40$$ −2.62981 −0.415810
$$41$$ −0.824456 −0.128758 −0.0643792 0.997926i $$-0.520507\pi$$
−0.0643792 + 0.997926i $$0.520507\pi$$
$$42$$ 0 0
$$43$$ −8.72128 −1.32998 −0.664991 0.746851i $$-0.731565\pi$$
−0.664991 + 0.746851i $$0.731565\pi$$
$$44$$ 4.20804 0.634385
$$45$$ 0 0
$$46$$ 1.61072 0.237487
$$47$$ 8.98090 1.31000 0.655000 0.755629i $$-0.272669\pi$$
0.655000 + 0.755629i $$0.272669\pi$$
$$48$$ 0 0
$$49$$ −2.26701 −0.323858
$$50$$ −0.772866 −0.109300
$$51$$ 0 0
$$52$$ 0.883426 0.122509
$$53$$ −6.88944 −0.946337 −0.473169 0.880972i $$-0.656890\pi$$
−0.473169 + 0.880972i $$0.656890\pi$$
$$54$$ 0 0
$$55$$ 3.00000 0.404520
$$56$$ 5.72128 0.764538
$$57$$ 0 0
$$58$$ −0.772866 −0.101482
$$59$$ −6.45427 −0.840274 −0.420137 0.907461i $$-0.638018\pi$$
−0.420137 + 0.907461i $$0.638018\pi$$
$$60$$ 0 0
$$61$$ −2.80536 −0.359189 −0.179595 0.983741i $$-0.557479\pi$$
−0.179595 + 0.983741i $$0.557479\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 2.98090 0.372613
$$65$$ 0.629813 0.0781187
$$66$$ 0 0
$$67$$ −11.0841 −1.35414 −0.677068 0.735920i $$-0.736750\pi$$
−0.677068 + 0.735920i $$0.736750\pi$$
$$68$$ 5.85695 0.710259
$$69$$ 0 0
$$70$$ 1.68140 0.200966
$$71$$ −2.63719 −0.312977 −0.156489 0.987680i $$-0.550017\pi$$
−0.156489 + 0.987680i $$0.550017\pi$$
$$72$$ 0 0
$$73$$ −14.7863 −1.73060 −0.865300 0.501254i $$-0.832872\pi$$
−0.865300 + 0.501254i $$0.832872\pi$$
$$74$$ −4.70218 −0.546617
$$75$$ 0 0
$$76$$ −6.74037 −0.773174
$$77$$ −6.52663 −0.743779
$$78$$ 0 0
$$79$$ −12.0650 −1.35742 −0.678708 0.734408i $$-0.737460\pi$$
−0.678708 + 0.734408i $$0.737460\pi$$
$$80$$ −0.772866 −0.0864090
$$81$$ 0 0
$$82$$ 0.637193 0.0703662
$$83$$ 7.62981 0.837481 0.418740 0.908106i $$-0.362472\pi$$
0.418740 + 0.908106i $$0.362472\pi$$
$$84$$ 0 0
$$85$$ 4.17554 0.452901
$$86$$ 6.74037 0.726833
$$87$$ 0 0
$$88$$ −7.88944 −0.841017
$$89$$ −17.8894 −1.89628 −0.948138 0.317858i $$-0.897037\pi$$
−0.948138 + 0.317858i $$0.897037\pi$$
$$90$$ 0 0
$$91$$ −1.37019 −0.143635
$$92$$ 2.92330 0.304775
$$93$$ 0 0
$$94$$ −6.94103 −0.715913
$$95$$ −4.80536 −0.493019
$$96$$ 0 0
$$97$$ 0.538351 0.0546613 0.0273306 0.999626i $$-0.491299\pi$$
0.0273306 + 0.999626i $$0.491299\pi$$
$$98$$ 1.75209 0.176988
$$99$$ 0 0
$$100$$ −1.40268 −0.140268
$$101$$ −13.8054 −1.37368 −0.686842 0.726807i $$-0.741004\pi$$
−0.686842 + 0.726807i $$0.741004\pi$$
$$102$$ 0 0
$$103$$ −13.4426 −1.32453 −0.662267 0.749268i $$-0.730406\pi$$
−0.662267 + 0.749268i $$0.730406\pi$$
$$104$$ −1.65629 −0.162413
$$105$$ 0 0
$$106$$ 5.32461 0.517172
$$107$$ 9.25963 0.895162 0.447581 0.894243i $$-0.352286\pi$$
0.447581 + 0.894243i $$0.352286\pi$$
$$108$$ 0 0
$$109$$ 8.61072 0.824757 0.412378 0.911013i $$-0.364698\pi$$
0.412378 + 0.911013i $$0.364698\pi$$
$$110$$ −2.31860 −0.221070
$$111$$ 0 0
$$112$$ 1.68140 0.158878
$$113$$ −11.2670 −1.05991 −0.529955 0.848026i $$-0.677791\pi$$
−0.529955 + 0.848026i $$0.677791\pi$$
$$114$$ 0 0
$$115$$ 2.08408 0.194342
$$116$$ −1.40268 −0.130235
$$117$$ 0 0
$$118$$ 4.98828 0.459209
$$119$$ −9.08408 −0.832736
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 2.16816 0.196296
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 16.7863 1.48954 0.744770 0.667321i $$-0.232559\pi$$
0.744770 + 0.667321i $$0.232559\pi$$
$$128$$ 9.41006 0.831740
$$129$$ 0 0
$$130$$ −0.486761 −0.0426918
$$131$$ −11.2670 −0.984403 −0.492201 0.870481i $$-0.663808\pi$$
−0.492201 + 0.870481i $$0.663808\pi$$
$$132$$ 0 0
$$133$$ 10.4543 0.906500
$$134$$ 8.56651 0.740033
$$135$$ 0 0
$$136$$ −10.9809 −0.941605
$$137$$ 18.8703 1.61220 0.806101 0.591778i $$-0.201574\pi$$
0.806101 + 0.591778i $$0.201574\pi$$
$$138$$ 0 0
$$139$$ 3.80536 0.322766 0.161383 0.986892i $$-0.448405\pi$$
0.161383 + 0.986892i $$0.448405\pi$$
$$140$$ 3.05159 0.257906
$$141$$ 0 0
$$142$$ 2.03820 0.171042
$$143$$ 1.88944 0.158003
$$144$$ 0 0
$$145$$ −1.00000 −0.0830455
$$146$$ 11.4278 0.945771
$$147$$ 0 0
$$148$$ −8.53401 −0.701492
$$149$$ −4.45427 −0.364908 −0.182454 0.983214i $$-0.558404\pi$$
−0.182454 + 0.983214i $$0.558404\pi$$
$$150$$ 0 0
$$151$$ 1.17554 0.0956644 0.0478322 0.998855i $$-0.484769\pi$$
0.0478322 + 0.998855i $$0.484769\pi$$
$$152$$ 12.6372 1.02501
$$153$$ 0 0
$$154$$ 5.04421 0.406474
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −0.805358 −0.0642745 −0.0321373 0.999483i $$-0.510231\pi$$
−0.0321373 + 0.999483i $$0.510231\pi$$
$$158$$ 9.32461 0.741826
$$159$$ 0 0
$$160$$ 5.85695 0.463032
$$161$$ −4.53401 −0.357330
$$162$$ 0 0
$$163$$ −1.11056 −0.0869858 −0.0434929 0.999054i $$-0.513849\pi$$
−0.0434929 + 0.999054i $$0.513849\pi$$
$$164$$ 1.15645 0.0903033
$$165$$ 0 0
$$166$$ −5.89682 −0.457682
$$167$$ 8.70218 0.673395 0.336697 0.941613i $$-0.390690\pi$$
0.336697 + 0.941613i $$0.390690\pi$$
$$168$$ 0 0
$$169$$ −12.6033 −0.969487
$$170$$ −3.22713 −0.247510
$$171$$ 0 0
$$172$$ 12.2331 0.932769
$$173$$ 7.69480 0.585025 0.292512 0.956262i $$-0.405509\pi$$
0.292512 + 0.956262i $$0.405509\pi$$
$$174$$ 0 0
$$175$$ 2.17554 0.164456
$$176$$ −2.31860 −0.174771
$$177$$ 0 0
$$178$$ 13.8261 1.03631
$$179$$ 6.06498 0.453318 0.226659 0.973974i $$-0.427220\pi$$
0.226659 + 0.973974i $$0.427220\pi$$
$$180$$ 0 0
$$181$$ 4.09146 0.304116 0.152058 0.988372i $$-0.451410\pi$$
0.152058 + 0.988372i $$0.451410\pi$$
$$182$$ 1.05897 0.0784961
$$183$$ 0 0
$$184$$ −5.48075 −0.404046
$$185$$ −6.08408 −0.447311
$$186$$ 0 0
$$187$$ 12.5266 0.916038
$$188$$ −12.5973 −0.918754
$$189$$ 0 0
$$190$$ 3.71390 0.269434
$$191$$ 0.267007 0.0193199 0.00965996 0.999953i $$-0.496925\pi$$
0.00965996 + 0.999953i $$0.496925\pi$$
$$192$$ 0 0
$$193$$ 10.7022 0.770360 0.385180 0.922842i $$-0.374139\pi$$
0.385180 + 0.922842i $$0.374139\pi$$
$$194$$ −0.416073 −0.0298723
$$195$$ 0 0
$$196$$ 3.17988 0.227134
$$197$$ −8.08408 −0.575967 −0.287984 0.957635i $$-0.592985\pi$$
−0.287984 + 0.957635i $$0.592985\pi$$
$$198$$ 0 0
$$199$$ −15.8054 −1.12041 −0.560206 0.828353i $$-0.689278\pi$$
−0.560206 + 0.828353i $$0.689278\pi$$
$$200$$ 2.62981 0.185956
$$201$$ 0 0
$$202$$ 10.6697 0.750716
$$203$$ 2.17554 0.152693
$$204$$ 0 0
$$205$$ 0.824456 0.0575825
$$206$$ 10.3893 0.723856
$$207$$ 0 0
$$208$$ −0.486761 −0.0337508
$$209$$ −14.4161 −0.997181
$$210$$ 0 0
$$211$$ −25.2214 −1.73631 −0.868157 0.496289i $$-0.834696\pi$$
−0.868157 + 0.496289i $$0.834696\pi$$
$$212$$ 9.66367 0.663704
$$213$$ 0 0
$$214$$ −7.15645 −0.489205
$$215$$ 8.72128 0.594786
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −6.65493 −0.450728
$$219$$ 0 0
$$220$$ −4.20804 −0.283706
$$221$$ 2.62981 0.176900
$$222$$ 0 0
$$223$$ −12.5266 −0.838845 −0.419423 0.907791i $$-0.637767\pi$$
−0.419423 + 0.907791i $$0.637767\pi$$
$$224$$ −12.7420 −0.851364
$$225$$ 0 0
$$226$$ 8.70788 0.579240
$$227$$ 7.46165 0.495247 0.247624 0.968856i $$-0.420350\pi$$
0.247624 + 0.968856i $$0.420350\pi$$
$$228$$ 0 0
$$229$$ 21.7936 1.44016 0.720082 0.693889i $$-0.244104\pi$$
0.720082 + 0.693889i $$0.244104\pi$$
$$230$$ −1.61072 −0.106207
$$231$$ 0 0
$$232$$ 2.62981 0.172656
$$233$$ 9.69480 0.635127 0.317564 0.948237i $$-0.397135\pi$$
0.317564 + 0.948237i $$0.397135\pi$$
$$234$$ 0 0
$$235$$ −8.98090 −0.585849
$$236$$ 9.05327 0.589317
$$237$$ 0 0
$$238$$ 7.02077 0.455089
$$239$$ 10.7022 0.692266 0.346133 0.938185i $$-0.387495\pi$$
0.346133 + 0.938185i $$0.387495\pi$$
$$240$$ 0 0
$$241$$ 11.1682 0.719405 0.359702 0.933067i $$-0.382878\pi$$
0.359702 + 0.933067i $$0.382878\pi$$
$$242$$ 1.54573 0.0993634
$$243$$ 0 0
$$244$$ 3.93502 0.251914
$$245$$ 2.26701 0.144834
$$246$$ 0 0
$$247$$ −3.02648 −0.192570
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0.772866 0.0488803
$$251$$ −14.3585 −0.906299 −0.453149 0.891435i $$-0.649700\pi$$
−0.453149 + 0.891435i $$0.649700\pi$$
$$252$$ 0 0
$$253$$ 6.25225 0.393075
$$254$$ −12.9735 −0.814031
$$255$$ 0 0
$$256$$ −13.2345 −0.827157
$$257$$ 7.11056 0.443545 0.221772 0.975098i $$-0.428816\pi$$
0.221772 + 0.975098i $$0.428816\pi$$
$$258$$ 0 0
$$259$$ 13.2362 0.822457
$$260$$ −0.883426 −0.0547877
$$261$$ 0 0
$$262$$ 8.70788 0.537975
$$263$$ −19.2214 −1.18524 −0.592622 0.805481i $$-0.701907\pi$$
−0.592622 + 0.805481i $$0.701907\pi$$
$$264$$ 0 0
$$265$$ 6.88944 0.423215
$$266$$ −8.07974 −0.495401
$$267$$ 0 0
$$268$$ 15.5474 0.949709
$$269$$ 9.88944 0.602970 0.301485 0.953471i $$-0.402518\pi$$
0.301485 + 0.953471i $$0.402518\pi$$
$$270$$ 0 0
$$271$$ −13.7936 −0.837904 −0.418952 0.908008i $$-0.637602\pi$$
−0.418952 + 0.908008i $$0.637602\pi$$
$$272$$ −3.22713 −0.195674
$$273$$ 0 0
$$274$$ −14.5842 −0.881066
$$275$$ −3.00000 −0.180907
$$276$$ 0 0
$$277$$ −14.9661 −0.899228 −0.449614 0.893223i $$-0.648439\pi$$
−0.449614 + 0.893223i $$0.648439\pi$$
$$278$$ −2.94103 −0.176391
$$279$$ 0 0
$$280$$ −5.72128 −0.341912
$$281$$ −11.7287 −0.699673 −0.349836 0.936811i $$-0.613763\pi$$
−0.349836 + 0.936811i $$0.613763\pi$$
$$282$$ 0 0
$$283$$ −3.96180 −0.235505 −0.117752 0.993043i $$-0.537569\pi$$
−0.117752 + 0.993043i $$0.537569\pi$$
$$284$$ 3.69914 0.219503
$$285$$ 0 0
$$286$$ −1.46028 −0.0863483
$$287$$ −1.79364 −0.105875
$$288$$ 0 0
$$289$$ 0.435171 0.0255983
$$290$$ 0.772866 0.0453842
$$291$$ 0 0
$$292$$ 20.7404 1.21374
$$293$$ 3.26701 0.190861 0.0954303 0.995436i $$-0.469577\pi$$
0.0954303 + 0.995436i $$0.469577\pi$$
$$294$$ 0 0
$$295$$ 6.45427 0.375782
$$296$$ 16.0000 0.929981
$$297$$ 0 0
$$298$$ 3.44255 0.199422
$$299$$ 1.31258 0.0759086
$$300$$ 0 0
$$301$$ −18.9735 −1.09362
$$302$$ −0.908538 −0.0522805
$$303$$ 0 0
$$304$$ 3.71390 0.213007
$$305$$ 2.80536 0.160634
$$306$$ 0 0
$$307$$ 20.4352 1.16630 0.583148 0.812366i $$-0.301821\pi$$
0.583148 + 0.812366i $$0.301821\pi$$
$$308$$ 9.15477 0.521641
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −11.0000 −0.623753 −0.311876 0.950123i $$-0.600957\pi$$
−0.311876 + 0.950123i $$0.600957\pi$$
$$312$$ 0 0
$$313$$ 27.6683 1.56391 0.781953 0.623337i $$-0.214224\pi$$
0.781953 + 0.623337i $$0.214224\pi$$
$$314$$ 0.622433 0.0351259
$$315$$ 0 0
$$316$$ 16.9233 0.952010
$$317$$ 12.8777 0.723285 0.361642 0.932317i $$-0.382216\pi$$
0.361642 + 0.932317i $$0.382216\pi$$
$$318$$ 0 0
$$319$$ −3.00000 −0.167968
$$320$$ −2.98090 −0.166637
$$321$$ 0 0
$$322$$ 3.50418 0.195280
$$323$$ −20.0650 −1.11645
$$324$$ 0 0
$$325$$ −0.629813 −0.0349358
$$326$$ 0.858314 0.0475376
$$327$$ 0 0
$$328$$ −2.16816 −0.119717
$$329$$ 19.5384 1.07718
$$330$$ 0 0
$$331$$ 24.6874 1.35694 0.678472 0.734627i $$-0.262643\pi$$
0.678472 + 0.734627i $$0.262643\pi$$
$$332$$ −10.7022 −0.587358
$$333$$ 0 0
$$334$$ −6.72561 −0.368009
$$335$$ 11.0841 0.605588
$$336$$ 0 0
$$337$$ −21.0915 −1.14893 −0.574463 0.818531i $$-0.694789\pi$$
−0.574463 + 0.818531i $$0.694789\pi$$
$$338$$ 9.74068 0.529823
$$339$$ 0 0
$$340$$ −5.85695 −0.317638
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −20.1608 −1.08858
$$344$$ −22.9353 −1.23659
$$345$$ 0 0
$$346$$ −5.94704 −0.319715
$$347$$ 18.8245 1.01055 0.505275 0.862958i $$-0.331391\pi$$
0.505275 + 0.862958i $$0.331391\pi$$
$$348$$ 0 0
$$349$$ 17.4884 0.936135 0.468067 0.883693i $$-0.344950\pi$$
0.468067 + 0.883693i $$0.344950\pi$$
$$350$$ −1.68140 −0.0898748
$$351$$ 0 0
$$352$$ 17.5708 0.936529
$$353$$ 8.38928 0.446517 0.223258 0.974759i $$-0.428331\pi$$
0.223258 + 0.974759i $$0.428331\pi$$
$$354$$ 0 0
$$355$$ 2.63719 0.139968
$$356$$ 25.0931 1.32993
$$357$$ 0 0
$$358$$ −4.68742 −0.247738
$$359$$ 27.6948 1.46168 0.730838 0.682551i $$-0.239130\pi$$
0.730838 + 0.682551i $$0.239130\pi$$
$$360$$ 0 0
$$361$$ 4.09146 0.215340
$$362$$ −3.16215 −0.166199
$$363$$ 0 0
$$364$$ 1.92193 0.100737
$$365$$ 14.7863 0.773948
$$366$$ 0 0
$$367$$ 9.00738 0.470181 0.235091 0.971973i $$-0.424461\pi$$
0.235091 + 0.971973i $$0.424461\pi$$
$$368$$ −1.61072 −0.0839643
$$369$$ 0 0
$$370$$ 4.70218 0.244455
$$371$$ −14.9883 −0.778153
$$372$$ 0 0
$$373$$ −25.0533 −1.29721 −0.648604 0.761126i $$-0.724647\pi$$
−0.648604 + 0.761126i $$0.724647\pi$$
$$374$$ −9.68140 −0.500613
$$375$$ 0 0
$$376$$ 23.6181 1.21801
$$377$$ −0.629813 −0.0324370
$$378$$ 0 0
$$379$$ −27.5725 −1.41631 −0.708153 0.706059i $$-0.750471\pi$$
−0.708153 + 0.706059i $$0.750471\pi$$
$$380$$ 6.74037 0.345774
$$381$$ 0 0
$$382$$ −0.206360 −0.0105583
$$383$$ 13.0724 0.667967 0.333983 0.942579i $$-0.391607\pi$$
0.333983 + 0.942579i $$0.391607\pi$$
$$384$$ 0 0
$$385$$ 6.52663 0.332628
$$386$$ −8.27134 −0.421000
$$387$$ 0 0
$$388$$ −0.755134 −0.0383361
$$389$$ 23.0797 1.17019 0.585095 0.810965i $$-0.301057\pi$$
0.585095 + 0.810965i $$0.301057\pi$$
$$390$$ 0 0
$$391$$ 8.70218 0.440088
$$392$$ −5.96180 −0.301117
$$393$$ 0 0
$$394$$ 6.24791 0.314765
$$395$$ 12.0650 0.607055
$$396$$ 0 0
$$397$$ −12.3129 −0.617966 −0.308983 0.951067i $$-0.599989\pi$$
−0.308983 + 0.951067i $$0.599989\pi$$
$$398$$ 12.2154 0.612304
$$399$$ 0 0
$$400$$ 0.772866 0.0386433
$$401$$ 36.4308 1.81927 0.909634 0.415410i $$-0.136362\pi$$
0.909634 + 0.415410i $$0.136362\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 19.3645 0.963419
$$405$$ 0 0
$$406$$ −1.68140 −0.0834466
$$407$$ −18.2522 −0.904730
$$408$$ 0 0
$$409$$ −22.5842 −1.11672 −0.558359 0.829599i $$-0.688569\pi$$
−0.558359 + 0.829599i $$0.688569\pi$$
$$410$$ −0.637193 −0.0314687
$$411$$ 0 0
$$412$$ 18.8556 0.928948
$$413$$ −14.0415 −0.690939
$$414$$ 0 0
$$415$$ −7.62981 −0.374533
$$416$$ 3.68878 0.180857
$$417$$ 0 0
$$418$$ 11.1417 0.544958
$$419$$ 2.90854 0.142091 0.0710457 0.997473i $$-0.477366\pi$$
0.0710457 + 0.997473i $$0.477366\pi$$
$$420$$ 0 0
$$421$$ 27.1183 1.32166 0.660831 0.750534i $$-0.270204\pi$$
0.660831 + 0.750534i $$0.270204\pi$$
$$422$$ 19.4928 0.948893
$$423$$ 0 0
$$424$$ −18.1179 −0.879885
$$425$$ −4.17554 −0.202544
$$426$$ 0 0
$$427$$ −6.10318 −0.295354
$$428$$ −12.9883 −0.627812
$$429$$ 0 0
$$430$$ −6.74037 −0.325050
$$431$$ −19.5960 −0.943904 −0.471952 0.881624i $$-0.656450\pi$$
−0.471952 + 0.881624i $$0.656450\pi$$
$$432$$ 0 0
$$433$$ 24.1638 1.16124 0.580620 0.814175i $$-0.302810\pi$$
0.580620 + 0.814175i $$0.302810\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −12.0781 −0.578435
$$437$$ −10.0148 −0.479071
$$438$$ 0 0
$$439$$ −30.4235 −1.45203 −0.726016 0.687678i $$-0.758630\pi$$
−0.726016 + 0.687678i $$0.758630\pi$$
$$440$$ 7.88944 0.376114
$$441$$ 0 0
$$442$$ −2.03249 −0.0966758
$$443$$ 32.3853 1.53867 0.769335 0.638846i $$-0.220588\pi$$
0.769335 + 0.638846i $$0.220588\pi$$
$$444$$ 0 0
$$445$$ 17.8894 0.848041
$$446$$ 9.68140 0.458428
$$447$$ 0 0
$$448$$ 6.48508 0.306391
$$449$$ 5.62243 0.265339 0.132670 0.991160i $$-0.457645\pi$$
0.132670 + 0.991160i $$0.457645\pi$$
$$450$$ 0 0
$$451$$ 2.47337 0.116466
$$452$$ 15.8040 0.743357
$$453$$ 0 0
$$454$$ −5.76685 −0.270652
$$455$$ 1.37019 0.0642353
$$456$$ 0 0
$$457$$ 3.33199 0.155864 0.0779320 0.996959i $$-0.475168\pi$$
0.0779320 + 0.996959i $$0.475168\pi$$
$$458$$ −16.8436 −0.787048
$$459$$ 0 0
$$460$$ −2.92330 −0.136299
$$461$$ −23.3437 −1.08722 −0.543612 0.839336i $$-0.682944\pi$$
−0.543612 + 0.839336i $$0.682944\pi$$
$$462$$ 0 0
$$463$$ 4.13735 0.192279 0.0961394 0.995368i $$-0.469351\pi$$
0.0961394 + 0.995368i $$0.469351\pi$$
$$464$$ 0.772866 0.0358794
$$465$$ 0 0
$$466$$ −7.49278 −0.347096
$$467$$ −35.7554 −1.65456 −0.827282 0.561786i $$-0.810114\pi$$
−0.827282 + 0.561786i $$0.810114\pi$$
$$468$$ 0 0
$$469$$ −24.1139 −1.11348
$$470$$ 6.94103 0.320166
$$471$$ 0 0
$$472$$ −16.9735 −0.781270
$$473$$ 26.1638 1.20301
$$474$$ 0 0
$$475$$ 4.80536 0.220485
$$476$$ 12.7420 0.584031
$$477$$ 0 0
$$478$$ −8.27134 −0.378322
$$479$$ 29.6107 1.35295 0.676474 0.736466i $$-0.263507\pi$$
0.676474 + 0.736466i $$0.263507\pi$$
$$480$$ 0 0
$$481$$ −3.83184 −0.174717
$$482$$ −8.63149 −0.393154
$$483$$ 0 0
$$484$$ 2.80536 0.127516
$$485$$ −0.538351 −0.0244453
$$486$$ 0 0
$$487$$ −4.33633 −0.196498 −0.0982489 0.995162i $$-0.531324\pi$$
−0.0982489 + 0.995162i $$0.531324\pi$$
$$488$$ −7.37757 −0.333967
$$489$$ 0 0
$$490$$ −1.75209 −0.0791514
$$491$$ −22.5193 −1.01628 −0.508140 0.861275i $$-0.669667\pi$$
−0.508140 + 0.861275i $$0.669667\pi$$
$$492$$ 0 0
$$493$$ −4.17554 −0.188057
$$494$$ 2.33906 0.105239
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −5.73733 −0.257354
$$498$$ 0 0
$$499$$ −41.1638 −1.84275 −0.921373 0.388680i $$-0.872931\pi$$
−0.921373 + 0.388680i $$0.872931\pi$$
$$500$$ 1.40268 0.0627297
$$501$$ 0 0
$$502$$ 11.0972 0.495291
$$503$$ −31.5149 −1.40518 −0.702590 0.711595i $$-0.747973\pi$$
−0.702590 + 0.711595i $$0.747973\pi$$
$$504$$ 0 0
$$505$$ 13.8054 0.614330
$$506$$ −4.83215 −0.214815
$$507$$ 0 0
$$508$$ −23.5457 −1.04467
$$509$$ −22.6224 −1.00272 −0.501361 0.865238i $$-0.667167\pi$$
−0.501361 + 0.865238i $$0.667167\pi$$
$$510$$ 0 0
$$511$$ −32.1682 −1.42304
$$512$$ −8.59162 −0.379699
$$513$$ 0 0
$$514$$ −5.49551 −0.242396
$$515$$ 13.4426 0.592350
$$516$$ 0 0
$$517$$ −26.9427 −1.18494
$$518$$ −10.2298 −0.449471
$$519$$ 0 0
$$520$$ 1.65629 0.0726332
$$521$$ 2.06498 0.0904686 0.0452343 0.998976i $$-0.485597\pi$$
0.0452343 + 0.998976i $$0.485597\pi$$
$$522$$ 0 0
$$523$$ 24.5266 1.07247 0.536237 0.844067i $$-0.319845\pi$$
0.536237 + 0.844067i $$0.319845\pi$$
$$524$$ 15.8040 0.690401
$$525$$ 0 0
$$526$$ 14.8556 0.647734
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −18.6566 −0.811157
$$530$$ −5.32461 −0.231286
$$531$$ 0 0
$$532$$ −14.6640 −0.635764
$$533$$ 0.519253 0.0224913
$$534$$ 0 0
$$535$$ −9.25963 −0.400329
$$536$$ −29.1491 −1.25905
$$537$$ 0 0
$$538$$ −7.64321 −0.329522
$$539$$ 6.80102 0.292941
$$540$$ 0 0
$$541$$ 0.389285 0.0167367 0.00836833 0.999965i $$-0.497336\pi$$
0.00836833 + 0.999965i $$0.497336\pi$$
$$542$$ 10.6606 0.457913
$$543$$ 0 0
$$544$$ 24.4559 1.04854
$$545$$ −8.61072 −0.368843
$$546$$ 0 0
$$547$$ 21.4352 0.916502 0.458251 0.888823i $$-0.348476\pi$$
0.458251 + 0.888823i $$0.348476\pi$$
$$548$$ −26.4690 −1.13070
$$549$$ 0 0
$$550$$ 2.31860 0.0988653
$$551$$ 4.80536 0.204715
$$552$$ 0 0
$$553$$ −26.2479 −1.11617
$$554$$ 11.5668 0.491427
$$555$$ 0 0
$$556$$ −5.33769 −0.226369
$$557$$ 19.7598 0.837249 0.418624 0.908159i $$-0.362512\pi$$
0.418624 + 0.908159i $$0.362512\pi$$
$$558$$ 0 0
$$559$$ 5.49278 0.232320
$$560$$ −1.68140 −0.0710523
$$561$$ 0 0
$$562$$ 9.06467 0.382370
$$563$$ −6.46165 −0.272326 −0.136163 0.990686i $$-0.543477\pi$$
−0.136163 + 0.990686i $$0.543477\pi$$
$$564$$ 0 0
$$565$$ 11.2670 0.474007
$$566$$ 3.06194 0.128703
$$567$$ 0 0
$$568$$ −6.93533 −0.291000
$$569$$ 14.9427 0.626431 0.313215 0.949682i $$-0.398594\pi$$
0.313215 + 0.949682i $$0.398594\pi$$
$$570$$ 0 0
$$571$$ 35.1521 1.47107 0.735535 0.677487i $$-0.236931\pi$$
0.735535 + 0.677487i $$0.236931\pi$$
$$572$$ −2.65028 −0.110814
$$573$$ 0 0
$$574$$ 1.38624 0.0578606
$$575$$ −2.08408 −0.0869122
$$576$$ 0 0
$$577$$ −0.740373 −0.0308222 −0.0154111 0.999881i $$-0.504906\pi$$
−0.0154111 + 0.999881i $$0.504906\pi$$
$$578$$ −0.336329 −0.0139894
$$579$$ 0 0
$$580$$ 1.40268 0.0582431
$$581$$ 16.5990 0.688642
$$582$$ 0 0
$$583$$ 20.6683 0.855994
$$584$$ −38.8851 −1.60908
$$585$$ 0 0
$$586$$ −2.52496 −0.104305
$$587$$ 18.5842 0.767054 0.383527 0.923530i $$-0.374709\pi$$
0.383527 + 0.923530i $$0.374709\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ −4.98828 −0.205364
$$591$$ 0 0
$$592$$ 4.70218 0.193258
$$593$$ −24.0268 −0.986662 −0.493331 0.869842i $$-0.664221\pi$$
−0.493331 + 0.869842i $$0.664221\pi$$
$$594$$ 0 0
$$595$$ 9.08408 0.372411
$$596$$ 6.24791 0.255924
$$597$$ 0 0
$$598$$ −1.01445 −0.0414839
$$599$$ −22.3585 −0.913542 −0.456771 0.889584i $$-0.650994\pi$$
−0.456771 + 0.889584i $$0.650994\pi$$
$$600$$ 0 0
$$601$$ −20.3511 −0.830138 −0.415069 0.909790i $$-0.636243\pi$$
−0.415069 + 0.909790i $$0.636243\pi$$
$$602$$ 14.6640 0.597659
$$603$$ 0 0
$$604$$ −1.64891 −0.0670932
$$605$$ 2.00000 0.0813116
$$606$$ 0 0
$$607$$ 19.4928 0.791187 0.395594 0.918426i $$-0.370539\pi$$
0.395594 + 0.918426i $$0.370539\pi$$
$$608$$ −28.1447 −1.14142
$$609$$ 0 0
$$610$$ −2.16816 −0.0877864
$$611$$ −5.65629 −0.228829
$$612$$ 0 0
$$613$$ 39.1638 1.58181 0.790906 0.611938i $$-0.209610\pi$$
0.790906 + 0.611938i $$0.209610\pi$$
$$614$$ −15.7936 −0.637379
$$615$$ 0 0
$$616$$ −17.1638 −0.691550
$$617$$ −26.8321 −1.08022 −0.540111 0.841594i $$-0.681618\pi$$
−0.540111 + 0.841594i $$0.681618\pi$$
$$618$$ 0 0
$$619$$ −33.4278 −1.34358 −0.671788 0.740743i $$-0.734473\pi$$
−0.671788 + 0.740743i $$0.734473\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 8.50152 0.340880
$$623$$ −38.9193 −1.55927
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −21.3839 −0.854672
$$627$$ 0 0
$$628$$ 1.12966 0.0450783
$$629$$ −25.4044 −1.01294
$$630$$ 0 0
$$631$$ −32.3705 −1.28865 −0.644325 0.764752i $$-0.722861\pi$$
−0.644325 + 0.764752i $$0.722861\pi$$
$$632$$ −31.7287 −1.26210
$$633$$ 0 0
$$634$$ −9.95275 −0.395274
$$635$$ −16.7863 −0.666142
$$636$$ 0 0
$$637$$ 1.42779 0.0565711
$$638$$ 2.31860 0.0917941
$$639$$ 0 0
$$640$$ −9.41006 −0.371965
$$641$$ −33.0415 −1.30506 −0.652531 0.757762i $$-0.726293\pi$$
−0.652531 + 0.757762i $$0.726293\pi$$
$$642$$ 0 0
$$643$$ 27.4734 1.08344 0.541722 0.840558i $$-0.317773\pi$$
0.541722 + 0.840558i $$0.317773\pi$$
$$644$$ 6.35976 0.250610
$$645$$ 0 0
$$646$$ 15.5075 0.610136
$$647$$ 23.2023 0.912178 0.456089 0.889934i $$-0.349250\pi$$
0.456089 + 0.889934i $$0.349250\pi$$
$$648$$ 0 0
$$649$$ 19.3628 0.760057
$$650$$ 0.486761 0.0190923
$$651$$ 0 0
$$652$$ 1.55776 0.0610066
$$653$$ 28.9159 1.13157 0.565784 0.824554i $$-0.308574\pi$$
0.565784 + 0.824554i $$0.308574\pi$$
$$654$$ 0 0
$$655$$ 11.2670 0.440238
$$656$$ −0.637193 −0.0248782
$$657$$ 0 0
$$658$$ −15.1005 −0.588680
$$659$$ −36.1832 −1.40950 −0.704749 0.709456i $$-0.748941\pi$$
−0.704749 + 0.709456i $$0.748941\pi$$
$$660$$ 0 0
$$661$$ 45.0918 1.75387 0.876933 0.480612i $$-0.159585\pi$$
0.876933 + 0.480612i $$0.159585\pi$$
$$662$$ −19.0801 −0.741567
$$663$$ 0 0
$$664$$ 20.0650 0.778672
$$665$$ −10.4543 −0.405399
$$666$$ 0 0
$$667$$ −2.08408 −0.0806960
$$668$$ −12.2064 −0.472278
$$669$$ 0 0
$$670$$ −8.56651 −0.330953
$$671$$ 8.41607 0.324899
$$672$$ 0 0
$$673$$ −23.1491 −0.892331 −0.446165 0.894950i $$-0.647211\pi$$
−0.446165 + 0.894950i $$0.647211\pi$$
$$674$$ 16.3009 0.627886
$$675$$ 0 0
$$676$$ 17.6784 0.679940
$$677$$ −38.1521 −1.46630 −0.733152 0.680064i $$-0.761952\pi$$
−0.733152 + 0.680064i $$0.761952\pi$$
$$678$$ 0 0
$$679$$ 1.17121 0.0449468
$$680$$ 10.9809 0.421098
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0.228811 0.00875520 0.00437760 0.999990i $$-0.498607\pi$$
0.00437760 + 0.999990i $$0.498607\pi$$
$$684$$ 0 0
$$685$$ −18.8703 −0.720999
$$686$$ 15.5816 0.594907
$$687$$ 0 0
$$688$$ −6.74037 −0.256974
$$689$$ 4.33906 0.165305
$$690$$ 0 0
$$691$$ 16.2258 0.617257 0.308629 0.951183i $$-0.400130\pi$$
0.308629 + 0.951183i $$0.400130\pi$$
$$692$$ −10.7933 −0.410301
$$693$$ 0 0
$$694$$ −14.5488 −0.552264
$$695$$ −3.80536 −0.144345
$$696$$ 0 0
$$697$$ 3.44255 0.130396
$$698$$ −13.5162 −0.511596
$$699$$ 0 0
$$700$$ −3.05159 −0.115339
$$701$$ −9.69914 −0.366331 −0.183166 0.983082i $$-0.558634\pi$$
−0.183166 + 0.983082i $$0.558634\pi$$
$$702$$ 0 0
$$703$$ 29.2362 1.10266
$$704$$ −8.94271 −0.337041
$$705$$ 0 0
$$706$$ −6.48379 −0.244021
$$707$$ −30.0342 −1.12955
$$708$$ 0 0
$$709$$ 49.4884 1.85858 0.929289 0.369354i $$-0.120421\pi$$
0.929289 + 0.369354i $$0.120421\pi$$
$$710$$ −2.03820 −0.0764921
$$711$$ 0 0
$$712$$ −47.0459 −1.76312
$$713$$ 0 0
$$714$$ 0 0
$$715$$ −1.88944 −0.0706610
$$716$$ −8.50722 −0.317930
$$717$$ 0 0
$$718$$ −21.4044 −0.798803
$$719$$ −15.7139 −0.586029 −0.293015 0.956108i $$-0.594658\pi$$
−0.293015 + 0.956108i $$0.594658\pi$$
$$720$$ 0 0
$$721$$ −29.2449 −1.08914
$$722$$ −3.16215 −0.117683
$$723$$ 0 0
$$724$$ −5.73901 −0.213289
$$725$$ 1.00000 0.0371391
$$726$$ 0 0
$$727$$ 6.51925 0.241786 0.120893 0.992666i $$-0.461424\pi$$
0.120893 + 0.992666i $$0.461424\pi$$
$$728$$ −3.60334 −0.133548
$$729$$ 0 0
$$730$$ −11.4278 −0.422962
$$731$$ 36.4161 1.34690
$$732$$ 0 0
$$733$$ −6.76716 −0.249951 −0.124975 0.992160i $$-0.539885\pi$$
−0.124975 + 0.992160i $$0.539885\pi$$
$$734$$ −6.96149 −0.256953
$$735$$ 0 0
$$736$$ 12.2064 0.449932
$$737$$ 33.2522 1.22486
$$738$$ 0 0
$$739$$ −41.1183 −1.51256 −0.756280 0.654248i $$-0.772985\pi$$
−0.756280 + 0.654248i $$0.772985\pi$$
$$740$$ 8.53401 0.313717
$$741$$ 0 0
$$742$$ 11.5839 0.425259
$$743$$ −36.5534 −1.34101 −0.670507 0.741903i $$-0.733924\pi$$
−0.670507 + 0.741903i $$0.733924\pi$$
$$744$$ 0 0
$$745$$ 4.45427 0.163192
$$746$$ 19.3628 0.708923
$$747$$ 0 0
$$748$$ −17.5708 −0.642454
$$749$$ 20.1447 0.736072
$$750$$ 0 0
$$751$$ −45.1980 −1.64930 −0.824649 0.565645i $$-0.808627\pi$$
−0.824649 + 0.565645i $$0.808627\pi$$
$$752$$ 6.94103 0.253113
$$753$$ 0 0
$$754$$ 0.486761 0.0177268
$$755$$ −1.17554 −0.0427824
$$756$$ 0 0
$$757$$ 53.4737 1.94353 0.971767 0.235943i $$-0.0758178\pi$$
0.971767 + 0.235943i $$0.0758178\pi$$
$$758$$ 21.3099 0.774009
$$759$$ 0 0
$$760$$ −12.6372 −0.458399
$$761$$ 49.4693 1.79326 0.896631 0.442778i $$-0.146007\pi$$
0.896631 + 0.442778i $$0.146007\pi$$
$$762$$ 0 0
$$763$$ 18.7330 0.678180
$$764$$ −0.374525 −0.0135498
$$765$$ 0 0
$$766$$ −10.1032 −0.365043
$$767$$ 4.06498 0.146778
$$768$$ 0 0
$$769$$ −14.2331 −0.513260 −0.256630 0.966510i $$-0.582612\pi$$
−0.256630 + 0.966510i $$0.582612\pi$$
$$770$$ −5.04421 −0.181781
$$771$$ 0 0
$$772$$ −15.0117 −0.540284
$$773$$ −21.7936 −0.783863 −0.391931 0.919994i $$-0.628193\pi$$
−0.391931 + 0.919994i $$0.628193\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 1.41576 0.0508229
$$777$$ 0 0
$$778$$ −17.8375 −0.639507
$$779$$ −3.96180 −0.141946
$$780$$ 0 0
$$781$$ 7.91158 0.283099
$$782$$ −6.72561 −0.240507
$$783$$ 0 0
$$784$$ −1.75209 −0.0625747
$$785$$ 0.805358 0.0287444
$$786$$ 0 0
$$787$$ −14.1829 −0.505567 −0.252783 0.967523i $$-0.581346\pi$$
−0.252783 + 0.967523i $$0.581346\pi$$
$$788$$ 11.3394 0.403948
$$789$$ 0 0
$$790$$ −9.32461 −0.331755
$$791$$ −24.5119 −0.871542
$$792$$ 0 0
$$793$$ 1.76685 0.0627427
$$794$$ 9.51621 0.337718
$$795$$ 0 0
$$796$$ 22.1698 0.785789
$$797$$ 2.00000 0.0708436 0.0354218 0.999372i $$-0.488723\pi$$
0.0354218 + 0.999372i $$0.488723\pi$$
$$798$$ 0 0
$$799$$ −37.5002 −1.32666
$$800$$ −5.85695 −0.207074
$$801$$ 0 0
$$802$$ −28.1561 −0.994228
$$803$$ 44.3588 1.56539
$$804$$ 0 0
$$805$$ 4.53401 0.159803
$$806$$ 0 0
$$807$$ 0 0
$$808$$ −36.3055 −1.27722
$$809$$ −30.6757 −1.07850 −0.539250 0.842146i $$-0.681292\pi$$
−0.539250 + 0.842146i $$0.681292\pi$$
$$810$$ 0 0
$$811$$ 36.6609 1.28734 0.643670 0.765303i $$-0.277411\pi$$
0.643670 + 0.765303i $$0.277411\pi$$
$$812$$ −3.05159 −0.107090
$$813$$ 0 0
$$814$$ 14.1065 0.494434
$$815$$ 1.11056 0.0389012
$$816$$ 0 0
$$817$$ −41.9088 −1.46620
$$818$$ 17.4546 0.610285
$$819$$ 0 0
$$820$$ −1.15645 −0.0403849
$$821$$ 36.1447 1.26146 0.630730 0.776002i $$-0.282756\pi$$
0.630730 + 0.776002i $$0.282756\pi$$
$$822$$ 0 0
$$823$$ −10.4395 −0.363898 −0.181949 0.983308i $$-0.558241\pi$$
−0.181949 + 0.983308i $$0.558241\pi$$
$$824$$ −35.3514 −1.23152
$$825$$ 0 0
$$826$$ 10.8522 0.377597
$$827$$ 34.1447 1.18733 0.593664 0.804713i $$-0.297681\pi$$
0.593664 + 0.804713i $$0.297681\pi$$
$$828$$ 0 0
$$829$$ 42.6874 1.48260 0.741298 0.671176i $$-0.234211\pi$$
0.741298 + 0.671176i $$0.234211\pi$$
$$830$$ 5.89682 0.204682
$$831$$ 0 0
$$832$$ −1.87741 −0.0650875
$$833$$ 9.46599 0.327977
$$834$$ 0 0
$$835$$ −8.70218 −0.301151
$$836$$ 20.2211 0.699362
$$837$$ 0 0
$$838$$ −2.24791 −0.0776527
$$839$$ −13.9926 −0.483079 −0.241539 0.970391i $$-0.577652\pi$$
−0.241539 + 0.970391i $$0.577652\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ −20.9588 −0.722287
$$843$$ 0 0
$$844$$ 35.3776 1.21775
$$845$$ 12.6033 0.433568
$$846$$ 0 0
$$847$$ −4.35109 −0.149505
$$848$$ −5.32461 −0.182848
$$849$$ 0 0
$$850$$ 3.22713 0.110690
$$851$$ −12.6797 −0.434655
$$852$$ 0 0
$$853$$ 34.6978 1.18803 0.594016 0.804453i $$-0.297542\pi$$
0.594016 + 0.804453i $$0.297542\pi$$
$$854$$ 4.71694 0.161410
$$855$$ 0 0
$$856$$ 24.3511 0.832303
$$857$$ −48.0077 −1.63991 −0.819956 0.572427i $$-0.806002\pi$$
−0.819956 + 0.572427i $$0.806002\pi$$
$$858$$ 0 0
$$859$$ 16.2627 0.554875 0.277438 0.960744i $$-0.410515\pi$$
0.277438 + 0.960744i $$0.410515\pi$$
$$860$$ −12.2331 −0.417147
$$861$$ 0 0
$$862$$ 15.1450 0.515842
$$863$$ −13.5605 −0.461604 −0.230802 0.973001i $$-0.574135\pi$$
−0.230802 + 0.973001i $$0.574135\pi$$
$$864$$ 0 0
$$865$$ −7.69480 −0.261631
$$866$$ −18.6754 −0.634616
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 36.1950 1.22783
$$870$$ 0 0
$$871$$ 6.98090 0.236539
$$872$$ 22.6446 0.766842
$$873$$ 0 0
$$874$$ 7.74006 0.261812
$$875$$ −2.17554 −0.0735468
$$876$$ 0 0
$$877$$ −8.01476 −0.270639 −0.135320 0.990802i $$-0.543206\pi$$
−0.135320 + 0.990802i $$0.543206\pi$$
$$878$$ 23.5132 0.793533
$$879$$ 0 0
$$880$$ 2.31860 0.0781599
$$881$$ 44.1564 1.48767 0.743834 0.668364i $$-0.233005\pi$$
0.743834 + 0.668364i $$0.233005\pi$$
$$882$$ 0 0
$$883$$ −20.7022 −0.696684 −0.348342 0.937368i $$-0.613255\pi$$
−0.348342 + 0.937368i $$0.613255\pi$$
$$884$$ −3.68878 −0.124067
$$885$$ 0 0
$$886$$ −25.0294 −0.840881
$$887$$ −48.5387 −1.62977 −0.814884 0.579624i $$-0.803200\pi$$
−0.814884 + 0.579624i $$0.803200\pi$$
$$888$$ 0 0
$$889$$ 36.5193 1.22482
$$890$$ −13.8261 −0.463453
$$891$$ 0 0
$$892$$ 17.5708 0.588315
$$893$$ 43.1564 1.44418
$$894$$ 0 0
$$895$$ −6.06498 −0.202730
$$896$$ 20.4720 0.683922
$$897$$ 0 0
$$898$$ −4.34538 −0.145007
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 28.7672 0.958373
$$902$$ −1.91158 −0.0636487
$$903$$ 0 0
$$904$$ −29.6301 −0.985483
$$905$$ −4.09146 −0.136005
$$906$$ 0 0
$$907$$ −26.5149 −0.880413 −0.440207 0.897896i $$-0.645095\pi$$
−0.440207 + 0.897896i $$0.645095\pi$$
$$908$$ −10.4663 −0.347336
$$909$$ 0 0
$$910$$ −1.05897 −0.0351045
$$911$$ −53.8469 −1.78403 −0.892014 0.452008i $$-0.850708\pi$$
−0.892014 + 0.452008i $$0.850708\pi$$
$$912$$ 0 0
$$913$$ −22.8894 −0.757530
$$914$$ −2.57518 −0.0851794
$$915$$ 0 0
$$916$$ −30.5695 −1.01004
$$917$$ −24.5119 −0.809453
$$918$$ 0 0
$$919$$ −11.4084 −0.376328 −0.188164 0.982138i $$-0.560254\pi$$
−0.188164 + 0.982138i $$0.560254\pi$$
$$920$$ 5.48075 0.180695
$$921$$ 0 0
$$922$$ 18.0415 0.594167
$$923$$ 1.66094 0.0546705
$$924$$ 0 0
$$925$$ 6.08408 0.200043
$$926$$ −3.19761 −0.105080
$$927$$ 0 0
$$928$$ −5.85695 −0.192264
$$929$$ −13.7554 −0.451301 −0.225651 0.974208i $$-0.572451\pi$$
−0.225651 + 0.974208i $$0.572451\pi$$
$$930$$ 0 0
$$931$$ −10.8938 −0.357029
$$932$$ −13.5987 −0.445440
$$933$$ 0 0
$$934$$ 27.6342 0.904217
$$935$$ −12.5266 −0.409665
$$936$$ 0 0
$$937$$ 1.29379 0.0422664 0.0211332 0.999777i $$-0.493273\pi$$
0.0211332 + 0.999777i $$0.493273\pi$$
$$938$$ 18.6368 0.608514
$$939$$ 0 0
$$940$$ 12.5973 0.410879
$$941$$ 49.8586 1.62534 0.812672 0.582721i $$-0.198012\pi$$
0.812672 + 0.582721i $$0.198012\pi$$
$$942$$ 0 0
$$943$$ 1.71823 0.0559534
$$944$$ −4.98828 −0.162355
$$945$$ 0 0
$$946$$ −20.2211 −0.657445
$$947$$ 21.1109 0.686011 0.343006 0.939333i $$-0.388555\pi$$
0.343006 + 0.939333i $$0.388555\pi$$
$$948$$ 0 0
$$949$$ 9.31258 0.302299
$$950$$ −3.71390 −0.120495
$$951$$ 0 0
$$952$$ −23.8894 −0.774261
$$953$$ 42.7672 1.38536 0.692682 0.721243i $$-0.256429\pi$$
0.692682 + 0.721243i $$0.256429\pi$$
$$954$$ 0 0
$$955$$ −0.267007 −0.00864013
$$956$$ −15.0117 −0.485514
$$957$$ 0 0
$$958$$ −22.8851 −0.739384
$$959$$ 41.0533 1.32568
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 2.96149 0.0954824
$$963$$ 0 0
$$964$$ −15.6653 −0.504547
$$965$$ −10.7022 −0.344515
$$966$$ 0 0
$$967$$ −33.0342 −1.06231 −0.531154 0.847276i $$-0.678241\pi$$
−0.531154 + 0.847276i $$0.678241\pi$$
$$968$$ −5.25963 −0.169051
$$969$$ 0 0
$$970$$ 0.416073 0.0133593
$$971$$ 26.0841 0.837078 0.418539 0.908199i $$-0.362542\pi$$
0.418539 + 0.908199i $$0.362542\pi$$
$$972$$ 0 0
$$973$$ 8.27872 0.265404
$$974$$ 3.35140 0.107386
$$975$$ 0 0
$$976$$ −2.16816 −0.0694012
$$977$$ 43.6948 1.39792 0.698960 0.715161i $$-0.253646\pi$$
0.698960 + 0.715161i $$0.253646\pi$$
$$978$$ 0 0
$$979$$ 53.6683 1.71525
$$980$$ −3.17988 −0.101578
$$981$$ 0 0
$$982$$ 17.4044 0.555395
$$983$$ −2.55745 −0.0815700 −0.0407850 0.999168i $$-0.512986\pi$$
−0.0407850 + 0.999168i $$0.512986\pi$$
$$984$$ 0 0
$$985$$ 8.08408 0.257580
$$986$$ 3.22713 0.102773
$$987$$ 0 0
$$988$$ 4.24518 0.135057
$$989$$ 18.1759 0.577959
$$990$$ 0 0
$$991$$ 14.7287 0.467871 0.233936 0.972252i $$-0.424840\pi$$
0.233936 + 0.972252i $$0.424840\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 4.43419 0.140644
$$995$$ 15.8054 0.501064
$$996$$ 0 0
$$997$$ 33.0992 1.04826 0.524130 0.851638i $$-0.324390\pi$$
0.524130 + 0.851638i $$0.324390\pi$$
$$998$$ 31.8141 1.00706
$$999$$ 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.q.1.2 3
3.2 odd 2 435.2.a.i.1.2 3
5.4 even 2 6525.2.a.bf.1.2 3
12.11 even 2 6960.2.a.cl.1.1 3
15.2 even 4 2175.2.c.m.349.4 6
15.8 even 4 2175.2.c.m.349.3 6
15.14 odd 2 2175.2.a.u.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.i.1.2 3 3.2 odd 2
1305.2.a.q.1.2 3 1.1 even 1 trivial
2175.2.a.u.1.2 3 15.14 odd 2
2175.2.c.m.349.3 6 15.8 even 4
2175.2.c.m.349.4 6 15.2 even 4
6525.2.a.bf.1.2 3 5.4 even 2
6960.2.a.cl.1.1 3 12.11 even 2