# Properties

 Label 135.2.a.c.1.2 Level $135$ Weight $2$ Character 135.1 Self dual yes Analytic conductor $1.078$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [135,2,Mod(1,135)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(135, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("135.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$135 = 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 135.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: $$1.07798042729$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.30278$$ of defining polynomial Character $$\chi$$ $$=$$ 135.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.30278 q^{2} -0.302776 q^{4} +1.00000 q^{5} +4.60555 q^{7} -3.00000 q^{8} +O(q^{10})$$ $$q+1.30278 q^{2} -0.302776 q^{4} +1.00000 q^{5} +4.60555 q^{7} -3.00000 q^{8} +1.30278 q^{10} -2.60555 q^{11} -0.605551 q^{13} +6.00000 q^{14} -3.30278 q^{16} -5.60555 q^{17} -3.60555 q^{19} -0.302776 q^{20} -3.39445 q^{22} -3.00000 q^{23} +1.00000 q^{25} -0.788897 q^{26} -1.39445 q^{28} +8.60555 q^{29} +1.60555 q^{31} +1.69722 q^{32} -7.30278 q^{34} +4.60555 q^{35} +2.00000 q^{37} -4.69722 q^{38} -3.00000 q^{40} +2.60555 q^{41} -6.60555 q^{43} +0.788897 q^{44} -3.90833 q^{46} +5.21110 q^{47} +14.2111 q^{49} +1.30278 q^{50} +0.183346 q^{52} -5.60555 q^{53} -2.60555 q^{55} -13.8167 q^{56} +11.2111 q^{58} -8.60555 q^{59} +10.2111 q^{61} +2.09167 q^{62} +8.81665 q^{64} -0.605551 q^{65} -15.2111 q^{67} +1.69722 q^{68} +6.00000 q^{70} +14.6056 q^{71} +5.39445 q^{73} +2.60555 q^{74} +1.09167 q^{76} -12.0000 q^{77} -4.39445 q^{79} -3.30278 q^{80} +3.39445 q^{82} +3.00000 q^{83} -5.60555 q^{85} -8.60555 q^{86} +7.81665 q^{88} +7.81665 q^{89} -2.78890 q^{91} +0.908327 q^{92} +6.78890 q^{94} -3.60555 q^{95} +8.00000 q^{97} +18.5139 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 3 q^{4} + 2 q^{5} + 2 q^{7} - 6 q^{8}+O(q^{10})$$ 2 * q - q^2 + 3 * q^4 + 2 * q^5 + 2 * q^7 - 6 * q^8 $$2 q - q^{2} + 3 q^{4} + 2 q^{5} + 2 q^{7} - 6 q^{8} - q^{10} + 2 q^{11} + 6 q^{13} + 12 q^{14} - 3 q^{16} - 4 q^{17} + 3 q^{20} - 14 q^{22} - 6 q^{23} + 2 q^{25} - 16 q^{26} - 10 q^{28} + 10 q^{29} - 4 q^{31} + 7 q^{32} - 11 q^{34} + 2 q^{35} + 4 q^{37} - 13 q^{38} - 6 q^{40} - 2 q^{41} - 6 q^{43} + 16 q^{44} + 3 q^{46} - 4 q^{47} + 14 q^{49} - q^{50} + 22 q^{52} - 4 q^{53} + 2 q^{55} - 6 q^{56} + 8 q^{58} - 10 q^{59} + 6 q^{61} + 15 q^{62} - 4 q^{64} + 6 q^{65} - 16 q^{67} + 7 q^{68} + 12 q^{70} + 22 q^{71} + 18 q^{73} - 2 q^{74} + 13 q^{76} - 24 q^{77} - 16 q^{79} - 3 q^{80} + 14 q^{82} + 6 q^{83} - 4 q^{85} - 10 q^{86} - 6 q^{88} - 6 q^{89} - 20 q^{91} - 9 q^{92} + 28 q^{94} + 16 q^{97} + 19 q^{98}+O(q^{100})$$ 2 * q - q^2 + 3 * q^4 + 2 * q^5 + 2 * q^7 - 6 * q^8 - q^10 + 2 * q^11 + 6 * q^13 + 12 * q^14 - 3 * q^16 - 4 * q^17 + 3 * q^20 - 14 * q^22 - 6 * q^23 + 2 * q^25 - 16 * q^26 - 10 * q^28 + 10 * q^29 - 4 * q^31 + 7 * q^32 - 11 * q^34 + 2 * q^35 + 4 * q^37 - 13 * q^38 - 6 * q^40 - 2 * q^41 - 6 * q^43 + 16 * q^44 + 3 * q^46 - 4 * q^47 + 14 * q^49 - q^50 + 22 * q^52 - 4 * q^53 + 2 * q^55 - 6 * q^56 + 8 * q^58 - 10 * q^59 + 6 * q^61 + 15 * q^62 - 4 * q^64 + 6 * q^65 - 16 * q^67 + 7 * q^68 + 12 * q^70 + 22 * q^71 + 18 * q^73 - 2 * q^74 + 13 * q^76 - 24 * q^77 - 16 * q^79 - 3 * q^80 + 14 * q^82 + 6 * q^83 - 4 * q^85 - 10 * q^86 - 6 * q^88 - 6 * q^89 - 20 * q^91 - 9 * q^92 + 28 * q^94 + 16 * q^97 + 19 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.30278 0.921201 0.460601 0.887607i $$-0.347634\pi$$
0.460601 + 0.887607i $$0.347634\pi$$
$$3$$ 0 0
$$4$$ −0.302776 −0.151388
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 4.60555 1.74073 0.870367 0.492403i $$-0.163881\pi$$
0.870367 + 0.492403i $$0.163881\pi$$
$$8$$ −3.00000 −1.06066
$$9$$ 0 0
$$10$$ 1.30278 0.411974
$$11$$ −2.60555 −0.785603 −0.392802 0.919623i $$-0.628494\pi$$
−0.392802 + 0.919623i $$0.628494\pi$$
$$12$$ 0 0
$$13$$ −0.605551 −0.167950 −0.0839749 0.996468i $$-0.526762\pi$$
−0.0839749 + 0.996468i $$0.526762\pi$$
$$14$$ 6.00000 1.60357
$$15$$ 0 0
$$16$$ −3.30278 −0.825694
$$17$$ −5.60555 −1.35955 −0.679773 0.733423i $$-0.737922\pi$$
−0.679773 + 0.733423i $$0.737922\pi$$
$$18$$ 0 0
$$19$$ −3.60555 −0.827170 −0.413585 0.910465i $$-0.635724\pi$$
−0.413585 + 0.910465i $$0.635724\pi$$
$$20$$ −0.302776 −0.0677027
$$21$$ 0 0
$$22$$ −3.39445 −0.723699
$$23$$ −3.00000 −0.625543 −0.312772 0.949828i $$-0.601257\pi$$
−0.312772 + 0.949828i $$0.601257\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ −0.788897 −0.154716
$$27$$ 0 0
$$28$$ −1.39445 −0.263526
$$29$$ 8.60555 1.59801 0.799005 0.601324i $$-0.205360\pi$$
0.799005 + 0.601324i $$0.205360\pi$$
$$30$$ 0 0
$$31$$ 1.60555 0.288366 0.144183 0.989551i $$-0.453945\pi$$
0.144183 + 0.989551i $$0.453945\pi$$
$$32$$ 1.69722 0.300030
$$33$$ 0 0
$$34$$ −7.30278 −1.25242
$$35$$ 4.60555 0.778480
$$36$$ 0 0
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ −4.69722 −0.761990
$$39$$ 0 0
$$40$$ −3.00000 −0.474342
$$41$$ 2.60555 0.406919 0.203459 0.979083i $$-0.434782\pi$$
0.203459 + 0.979083i $$0.434782\pi$$
$$42$$ 0 0
$$43$$ −6.60555 −1.00734 −0.503669 0.863897i $$-0.668017\pi$$
−0.503669 + 0.863897i $$0.668017\pi$$
$$44$$ 0.788897 0.118931
$$45$$ 0 0
$$46$$ −3.90833 −0.576251
$$47$$ 5.21110 0.760117 0.380059 0.924962i $$-0.375904\pi$$
0.380059 + 0.924962i $$0.375904\pi$$
$$48$$ 0 0
$$49$$ 14.2111 2.03016
$$50$$ 1.30278 0.184240
$$51$$ 0 0
$$52$$ 0.183346 0.0254255
$$53$$ −5.60555 −0.769982 −0.384991 0.922920i $$-0.625795\pi$$
−0.384991 + 0.922920i $$0.625795\pi$$
$$54$$ 0 0
$$55$$ −2.60555 −0.351332
$$56$$ −13.8167 −1.84633
$$57$$ 0 0
$$58$$ 11.2111 1.47209
$$59$$ −8.60555 −1.12035 −0.560174 0.828375i $$-0.689266\pi$$
−0.560174 + 0.828375i $$0.689266\pi$$
$$60$$ 0 0
$$61$$ 10.2111 1.30740 0.653699 0.756755i $$-0.273216\pi$$
0.653699 + 0.756755i $$0.273216\pi$$
$$62$$ 2.09167 0.265643
$$63$$ 0 0
$$64$$ 8.81665 1.10208
$$65$$ −0.605551 −0.0751094
$$66$$ 0 0
$$67$$ −15.2111 −1.85833 −0.929166 0.369663i $$-0.879473\pi$$
−0.929166 + 0.369663i $$0.879473\pi$$
$$68$$ 1.69722 0.205819
$$69$$ 0 0
$$70$$ 6.00000 0.717137
$$71$$ 14.6056 1.73336 0.866680 0.498864i $$-0.166249\pi$$
0.866680 + 0.498864i $$0.166249\pi$$
$$72$$ 0 0
$$73$$ 5.39445 0.631372 0.315686 0.948864i $$-0.397765\pi$$
0.315686 + 0.948864i $$0.397765\pi$$
$$74$$ 2.60555 0.302889
$$75$$ 0 0
$$76$$ 1.09167 0.125223
$$77$$ −12.0000 −1.36753
$$78$$ 0 0
$$79$$ −4.39445 −0.494414 −0.247207 0.968963i $$-0.579513\pi$$
−0.247207 + 0.968963i $$0.579513\pi$$
$$80$$ −3.30278 −0.369262
$$81$$ 0 0
$$82$$ 3.39445 0.374854
$$83$$ 3.00000 0.329293 0.164646 0.986353i $$-0.447352\pi$$
0.164646 + 0.986353i $$0.447352\pi$$
$$84$$ 0 0
$$85$$ −5.60555 −0.608007
$$86$$ −8.60555 −0.927960
$$87$$ 0 0
$$88$$ 7.81665 0.833258
$$89$$ 7.81665 0.828564 0.414282 0.910149i $$-0.364033\pi$$
0.414282 + 0.910149i $$0.364033\pi$$
$$90$$ 0 0
$$91$$ −2.78890 −0.292356
$$92$$ 0.908327 0.0946996
$$93$$ 0 0
$$94$$ 6.78890 0.700221
$$95$$ −3.60555 −0.369922
$$96$$ 0 0
$$97$$ 8.00000 0.812277 0.406138 0.913812i $$-0.366875\pi$$
0.406138 + 0.913812i $$0.366875\pi$$
$$98$$ 18.5139 1.87018
$$99$$ 0 0
$$100$$ −0.302776 −0.0302776
$$101$$ 12.0000 1.19404 0.597022 0.802225i $$-0.296350\pi$$
0.597022 + 0.802225i $$0.296350\pi$$
$$102$$ 0 0
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ 1.81665 0.178138
$$105$$ 0 0
$$106$$ −7.30278 −0.709308
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ −7.00000 −0.670478 −0.335239 0.942133i $$-0.608817\pi$$
−0.335239 + 0.942133i $$0.608817\pi$$
$$110$$ −3.39445 −0.323648
$$111$$ 0 0
$$112$$ −15.2111 −1.43731
$$113$$ −0.788897 −0.0742132 −0.0371066 0.999311i $$-0.511814\pi$$
−0.0371066 + 0.999311i $$0.511814\pi$$
$$114$$ 0 0
$$115$$ −3.00000 −0.279751
$$116$$ −2.60555 −0.241919
$$117$$ 0 0
$$118$$ −11.2111 −1.03207
$$119$$ −25.8167 −2.36661
$$120$$ 0 0
$$121$$ −4.21110 −0.382828
$$122$$ 13.3028 1.20438
$$123$$ 0 0
$$124$$ −0.486122 −0.0436550
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −4.78890 −0.424946 −0.212473 0.977167i $$-0.568152\pi$$
−0.212473 + 0.977167i $$0.568152\pi$$
$$128$$ 8.09167 0.715210
$$129$$ 0 0
$$130$$ −0.788897 −0.0691909
$$131$$ 6.00000 0.524222 0.262111 0.965038i $$-0.415581\pi$$
0.262111 + 0.965038i $$0.415581\pi$$
$$132$$ 0 0
$$133$$ −16.6056 −1.43988
$$134$$ −19.8167 −1.71190
$$135$$ 0 0
$$136$$ 16.8167 1.44202
$$137$$ −4.81665 −0.411515 −0.205757 0.978603i $$-0.565966\pi$$
−0.205757 + 0.978603i $$0.565966\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ −1.39445 −0.117852
$$141$$ 0 0
$$142$$ 19.0278 1.59677
$$143$$ 1.57779 0.131942
$$144$$ 0 0
$$145$$ 8.60555 0.714652
$$146$$ 7.02776 0.581621
$$147$$ 0 0
$$148$$ −0.605551 −0.0497760
$$149$$ −13.0278 −1.06728 −0.533638 0.845713i $$-0.679175\pi$$
−0.533638 + 0.845713i $$0.679175\pi$$
$$150$$ 0 0
$$151$$ −14.4222 −1.17366 −0.586831 0.809709i $$-0.699625\pi$$
−0.586831 + 0.809709i $$0.699625\pi$$
$$152$$ 10.8167 0.877346
$$153$$ 0 0
$$154$$ −15.6333 −1.25977
$$155$$ 1.60555 0.128961
$$156$$ 0 0
$$157$$ 3.81665 0.304602 0.152301 0.988334i $$-0.451332\pi$$
0.152301 + 0.988334i $$0.451332\pi$$
$$158$$ −5.72498 −0.455455
$$159$$ 0 0
$$160$$ 1.69722 0.134177
$$161$$ −13.8167 −1.08890
$$162$$ 0 0
$$163$$ 2.00000 0.156652 0.0783260 0.996928i $$-0.475042\pi$$
0.0783260 + 0.996928i $$0.475042\pi$$
$$164$$ −0.788897 −0.0616025
$$165$$ 0 0
$$166$$ 3.90833 0.303345
$$167$$ −3.00000 −0.232147 −0.116073 0.993241i $$-0.537031\pi$$
−0.116073 + 0.993241i $$0.537031\pi$$
$$168$$ 0 0
$$169$$ −12.6333 −0.971793
$$170$$ −7.30278 −0.560097
$$171$$ 0 0
$$172$$ 2.00000 0.152499
$$173$$ 10.8167 0.822375 0.411187 0.911551i $$-0.365114\pi$$
0.411187 + 0.911551i $$0.365114\pi$$
$$174$$ 0 0
$$175$$ 4.60555 0.348147
$$176$$ 8.60555 0.648668
$$177$$ 0 0
$$178$$ 10.1833 0.763274
$$179$$ −6.78890 −0.507426 −0.253713 0.967280i $$-0.581652\pi$$
−0.253713 + 0.967280i $$0.581652\pi$$
$$180$$ 0 0
$$181$$ −7.00000 −0.520306 −0.260153 0.965567i $$-0.583773\pi$$
−0.260153 + 0.965567i $$0.583773\pi$$
$$182$$ −3.63331 −0.269319
$$183$$ 0 0
$$184$$ 9.00000 0.663489
$$185$$ 2.00000 0.147043
$$186$$ 0 0
$$187$$ 14.6056 1.06806
$$188$$ −1.57779 −0.115073
$$189$$ 0 0
$$190$$ −4.69722 −0.340772
$$191$$ 16.4222 1.18827 0.594135 0.804366i $$-0.297495\pi$$
0.594135 + 0.804366i $$0.297495\pi$$
$$192$$ 0 0
$$193$$ 21.8167 1.57040 0.785199 0.619244i $$-0.212561\pi$$
0.785199 + 0.619244i $$0.212561\pi$$
$$194$$ 10.4222 0.748271
$$195$$ 0 0
$$196$$ −4.30278 −0.307341
$$197$$ −1.18335 −0.0843099 −0.0421550 0.999111i $$-0.513422\pi$$
−0.0421550 + 0.999111i $$0.513422\pi$$
$$198$$ 0 0
$$199$$ 13.2111 0.936510 0.468255 0.883593i $$-0.344883\pi$$
0.468255 + 0.883593i $$0.344883\pi$$
$$200$$ −3.00000 −0.212132
$$201$$ 0 0
$$202$$ 15.6333 1.09996
$$203$$ 39.6333 2.78171
$$204$$ 0 0
$$205$$ 2.60555 0.181980
$$206$$ −5.21110 −0.363075
$$207$$ 0 0
$$208$$ 2.00000 0.138675
$$209$$ 9.39445 0.649828
$$210$$ 0 0
$$211$$ 12.8167 0.882335 0.441167 0.897425i $$-0.354564\pi$$
0.441167 + 0.897425i $$0.354564\pi$$
$$212$$ 1.69722 0.116566
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −6.60555 −0.450495
$$216$$ 0 0
$$217$$ 7.39445 0.501968
$$218$$ −9.11943 −0.617646
$$219$$ 0 0
$$220$$ 0.788897 0.0531875
$$221$$ 3.39445 0.228335
$$222$$ 0 0
$$223$$ −10.0000 −0.669650 −0.334825 0.942280i $$-0.608677\pi$$
−0.334825 + 0.942280i $$0.608677\pi$$
$$224$$ 7.81665 0.522272
$$225$$ 0 0
$$226$$ −1.02776 −0.0683653
$$227$$ −26.2111 −1.73969 −0.869846 0.493323i $$-0.835782\pi$$
−0.869846 + 0.493323i $$0.835782\pi$$
$$228$$ 0 0
$$229$$ −6.21110 −0.410441 −0.205221 0.978716i $$-0.565791\pi$$
−0.205221 + 0.978716i $$0.565791\pi$$
$$230$$ −3.90833 −0.257707
$$231$$ 0 0
$$232$$ −25.8167 −1.69495
$$233$$ −18.0000 −1.17922 −0.589610 0.807688i $$-0.700718\pi$$
−0.589610 + 0.807688i $$0.700718\pi$$
$$234$$ 0 0
$$235$$ 5.21110 0.339935
$$236$$ 2.60555 0.169607
$$237$$ 0 0
$$238$$ −33.6333 −2.18012
$$239$$ −0.788897 −0.0510295 −0.0255148 0.999674i $$-0.508122\pi$$
−0.0255148 + 0.999674i $$0.508122\pi$$
$$240$$ 0 0
$$241$$ 28.2111 1.81724 0.908618 0.417627i $$-0.137138\pi$$
0.908618 + 0.417627i $$0.137138\pi$$
$$242$$ −5.48612 −0.352661
$$243$$ 0 0
$$244$$ −3.09167 −0.197924
$$245$$ 14.2111 0.907914
$$246$$ 0 0
$$247$$ 2.18335 0.138923
$$248$$ −4.81665 −0.305858
$$249$$ 0 0
$$250$$ 1.30278 0.0823948
$$251$$ −15.6333 −0.986766 −0.493383 0.869812i $$-0.664240\pi$$
−0.493383 + 0.869812i $$0.664240\pi$$
$$252$$ 0 0
$$253$$ 7.81665 0.491429
$$254$$ −6.23886 −0.391461
$$255$$ 0 0
$$256$$ −7.09167 −0.443230
$$257$$ 22.8167 1.42326 0.711632 0.702553i $$-0.247956\pi$$
0.711632 + 0.702553i $$0.247956\pi$$
$$258$$ 0 0
$$259$$ 9.21110 0.572350
$$260$$ 0.183346 0.0113706
$$261$$ 0 0
$$262$$ 7.81665 0.482914
$$263$$ −17.2111 −1.06128 −0.530641 0.847597i $$-0.678049\pi$$
−0.530641 + 0.847597i $$0.678049\pi$$
$$264$$ 0 0
$$265$$ −5.60555 −0.344346
$$266$$ −21.6333 −1.32642
$$267$$ 0 0
$$268$$ 4.60555 0.281329
$$269$$ 11.2111 0.683553 0.341776 0.939781i $$-0.388971\pi$$
0.341776 + 0.939781i $$0.388971\pi$$
$$270$$ 0 0
$$271$$ −19.2389 −1.16868 −0.584339 0.811510i $$-0.698646\pi$$
−0.584339 + 0.811510i $$0.698646\pi$$
$$272$$ 18.5139 1.12257
$$273$$ 0 0
$$274$$ −6.27502 −0.379088
$$275$$ −2.60555 −0.157121
$$276$$ 0 0
$$277$$ −29.0278 −1.74411 −0.872054 0.489409i $$-0.837213\pi$$
−0.872054 + 0.489409i $$0.837213\pi$$
$$278$$ −5.21110 −0.312541
$$279$$ 0 0
$$280$$ −13.8167 −0.825703
$$281$$ 1.81665 0.108372 0.0541862 0.998531i $$-0.482744\pi$$
0.0541862 + 0.998531i $$0.482744\pi$$
$$282$$ 0 0
$$283$$ 10.6056 0.630435 0.315217 0.949020i $$-0.397923\pi$$
0.315217 + 0.949020i $$0.397923\pi$$
$$284$$ −4.42221 −0.262410
$$285$$ 0 0
$$286$$ 2.05551 0.121545
$$287$$ 12.0000 0.708338
$$288$$ 0 0
$$289$$ 14.4222 0.848365
$$290$$ 11.2111 0.658339
$$291$$ 0 0
$$292$$ −1.63331 −0.0955821
$$293$$ −28.8167 −1.68349 −0.841743 0.539878i $$-0.818470\pi$$
−0.841743 + 0.539878i $$0.818470\pi$$
$$294$$ 0 0
$$295$$ −8.60555 −0.501035
$$296$$ −6.00000 −0.348743
$$297$$ 0 0
$$298$$ −16.9722 −0.983176
$$299$$ 1.81665 0.105060
$$300$$ 0 0
$$301$$ −30.4222 −1.75351
$$302$$ −18.7889 −1.08118
$$303$$ 0 0
$$304$$ 11.9083 0.682989
$$305$$ 10.2111 0.584686
$$306$$ 0 0
$$307$$ −20.4222 −1.16556 −0.582778 0.812631i $$-0.698034\pi$$
−0.582778 + 0.812631i $$0.698034\pi$$
$$308$$ 3.63331 0.207027
$$309$$ 0 0
$$310$$ 2.09167 0.118799
$$311$$ 13.8167 0.783471 0.391735 0.920078i $$-0.371875\pi$$
0.391735 + 0.920078i $$0.371875\pi$$
$$312$$ 0 0
$$313$$ 23.6333 1.33583 0.667917 0.744236i $$-0.267186\pi$$
0.667917 + 0.744236i $$0.267186\pi$$
$$314$$ 4.97224 0.280600
$$315$$ 0 0
$$316$$ 1.33053 0.0748483
$$317$$ −0.394449 −0.0221544 −0.0110772 0.999939i $$-0.503526\pi$$
−0.0110772 + 0.999939i $$0.503526\pi$$
$$318$$ 0 0
$$319$$ −22.4222 −1.25540
$$320$$ 8.81665 0.492866
$$321$$ 0 0
$$322$$ −18.0000 −1.00310
$$323$$ 20.2111 1.12458
$$324$$ 0 0
$$325$$ −0.605551 −0.0335899
$$326$$ 2.60555 0.144308
$$327$$ 0 0
$$328$$ −7.81665 −0.431603
$$329$$ 24.0000 1.32316
$$330$$ 0 0
$$331$$ 14.7889 0.812871 0.406436 0.913679i $$-0.366772\pi$$
0.406436 + 0.913679i $$0.366772\pi$$
$$332$$ −0.908327 −0.0498509
$$333$$ 0 0
$$334$$ −3.90833 −0.213854
$$335$$ −15.2111 −0.831071
$$336$$ 0 0
$$337$$ −0.605551 −0.0329865 −0.0164932 0.999864i $$-0.505250\pi$$
−0.0164932 + 0.999864i $$0.505250\pi$$
$$338$$ −16.4584 −0.895217
$$339$$ 0 0
$$340$$ 1.69722 0.0920449
$$341$$ −4.18335 −0.226541
$$342$$ 0 0
$$343$$ 33.2111 1.79323
$$344$$ 19.8167 1.06844
$$345$$ 0 0
$$346$$ 14.0917 0.757573
$$347$$ 1.57779 0.0847005 0.0423502 0.999103i $$-0.486515\pi$$
0.0423502 + 0.999103i $$0.486515\pi$$
$$348$$ 0 0
$$349$$ 25.8444 1.38342 0.691710 0.722176i $$-0.256858\pi$$
0.691710 + 0.722176i $$0.256858\pi$$
$$350$$ 6.00000 0.320713
$$351$$ 0 0
$$352$$ −4.42221 −0.235704
$$353$$ −21.6333 −1.15142 −0.575712 0.817652i $$-0.695275\pi$$
−0.575712 + 0.817652i $$0.695275\pi$$
$$354$$ 0 0
$$355$$ 14.6056 0.775182
$$356$$ −2.36669 −0.125434
$$357$$ 0 0
$$358$$ −8.84441 −0.467442
$$359$$ 33.6333 1.77510 0.887549 0.460713i $$-0.152406\pi$$
0.887549 + 0.460713i $$0.152406\pi$$
$$360$$ 0 0
$$361$$ −6.00000 −0.315789
$$362$$ −9.11943 −0.479307
$$363$$ 0 0
$$364$$ 0.844410 0.0442591
$$365$$ 5.39445 0.282358
$$366$$ 0 0
$$367$$ 4.60555 0.240408 0.120204 0.992749i $$-0.461645\pi$$
0.120204 + 0.992749i $$0.461645\pi$$
$$368$$ 9.90833 0.516507
$$369$$ 0 0
$$370$$ 2.60555 0.135456
$$371$$ −25.8167 −1.34033
$$372$$ 0 0
$$373$$ −10.7889 −0.558628 −0.279314 0.960200i $$-0.590107\pi$$
−0.279314 + 0.960200i $$0.590107\pi$$
$$374$$ 19.0278 0.983902
$$375$$ 0 0
$$376$$ −15.6333 −0.806226
$$377$$ −5.21110 −0.268385
$$378$$ 0 0
$$379$$ 14.3944 0.739393 0.369697 0.929153i $$-0.379462\pi$$
0.369697 + 0.929153i $$0.379462\pi$$
$$380$$ 1.09167 0.0560016
$$381$$ 0 0
$$382$$ 21.3944 1.09464
$$383$$ −18.6333 −0.952118 −0.476059 0.879413i $$-0.657935\pi$$
−0.476059 + 0.879413i $$0.657935\pi$$
$$384$$ 0 0
$$385$$ −12.0000 −0.611577
$$386$$ 28.4222 1.44665
$$387$$ 0 0
$$388$$ −2.42221 −0.122969
$$389$$ 4.18335 0.212104 0.106052 0.994361i $$-0.466179\pi$$
0.106052 + 0.994361i $$0.466179\pi$$
$$390$$ 0 0
$$391$$ 16.8167 0.850455
$$392$$ −42.6333 −2.15331
$$393$$ 0 0
$$394$$ −1.54163 −0.0776664
$$395$$ −4.39445 −0.221109
$$396$$ 0 0
$$397$$ −12.6056 −0.632654 −0.316327 0.948650i $$-0.602450\pi$$
−0.316327 + 0.948650i $$0.602450\pi$$
$$398$$ 17.2111 0.862715
$$399$$ 0 0
$$400$$ −3.30278 −0.165139
$$401$$ −30.0000 −1.49813 −0.749064 0.662497i $$-0.769497\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ 0 0
$$403$$ −0.972244 −0.0484309
$$404$$ −3.63331 −0.180764
$$405$$ 0 0
$$406$$ 51.6333 2.56252
$$407$$ −5.21110 −0.258305
$$408$$ 0 0
$$409$$ 5.00000 0.247234 0.123617 0.992330i $$-0.460551\pi$$
0.123617 + 0.992330i $$0.460551\pi$$
$$410$$ 3.39445 0.167640
$$411$$ 0 0
$$412$$ 1.21110 0.0596667
$$413$$ −39.6333 −1.95023
$$414$$ 0 0
$$415$$ 3.00000 0.147264
$$416$$ −1.02776 −0.0503899
$$417$$ 0 0
$$418$$ 12.2389 0.598622
$$419$$ −13.0278 −0.636448 −0.318224 0.948016i $$-0.603086\pi$$
−0.318224 + 0.948016i $$0.603086\pi$$
$$420$$ 0 0
$$421$$ −23.4222 −1.14153 −0.570764 0.821114i $$-0.693353\pi$$
−0.570764 + 0.821114i $$0.693353\pi$$
$$422$$ 16.6972 0.812808
$$423$$ 0 0
$$424$$ 16.8167 0.816689
$$425$$ −5.60555 −0.271909
$$426$$ 0 0
$$427$$ 47.0278 2.27583
$$428$$ 0 0
$$429$$ 0 0
$$430$$ −8.60555 −0.414997
$$431$$ 25.8167 1.24354 0.621772 0.783198i $$-0.286413\pi$$
0.621772 + 0.783198i $$0.286413\pi$$
$$432$$ 0 0
$$433$$ −28.2389 −1.35707 −0.678536 0.734567i $$-0.737385\pi$$
−0.678536 + 0.734567i $$0.737385\pi$$
$$434$$ 9.63331 0.462414
$$435$$ 0 0
$$436$$ 2.11943 0.101502
$$437$$ 10.8167 0.517431
$$438$$ 0 0
$$439$$ 20.3944 0.973374 0.486687 0.873576i $$-0.338205\pi$$
0.486687 + 0.873576i $$0.338205\pi$$
$$440$$ 7.81665 0.372644
$$441$$ 0 0
$$442$$ 4.42221 0.210343
$$443$$ 18.6333 0.885295 0.442648 0.896696i $$-0.354039\pi$$
0.442648 + 0.896696i $$0.354039\pi$$
$$444$$ 0 0
$$445$$ 7.81665 0.370545
$$446$$ −13.0278 −0.616882
$$447$$ 0 0
$$448$$ 40.6056 1.91843
$$449$$ −12.2389 −0.577587 −0.288794 0.957391i $$-0.593254\pi$$
−0.288794 + 0.957391i $$0.593254\pi$$
$$450$$ 0 0
$$451$$ −6.78890 −0.319677
$$452$$ 0.238859 0.0112350
$$453$$ 0 0
$$454$$ −34.1472 −1.60261
$$455$$ −2.78890 −0.130746
$$456$$ 0 0
$$457$$ 1.21110 0.0566530 0.0283265 0.999599i $$-0.490982\pi$$
0.0283265 + 0.999599i $$0.490982\pi$$
$$458$$ −8.09167 −0.378099
$$459$$ 0 0
$$460$$ 0.908327 0.0423510
$$461$$ −21.6333 −1.00756 −0.503782 0.863831i $$-0.668058\pi$$
−0.503782 + 0.863831i $$0.668058\pi$$
$$462$$ 0 0
$$463$$ −15.2111 −0.706920 −0.353460 0.935450i $$-0.614995\pi$$
−0.353460 + 0.935450i $$0.614995\pi$$
$$464$$ −28.4222 −1.31947
$$465$$ 0 0
$$466$$ −23.4500 −1.08630
$$467$$ −2.21110 −0.102318 −0.0511588 0.998691i $$-0.516291\pi$$
−0.0511588 + 0.998691i $$0.516291\pi$$
$$468$$ 0 0
$$469$$ −70.0555 −3.23486
$$470$$ 6.78890 0.313148
$$471$$ 0 0
$$472$$ 25.8167 1.18831
$$473$$ 17.2111 0.791367
$$474$$ 0 0
$$475$$ −3.60555 −0.165434
$$476$$ 7.81665 0.358276
$$477$$ 0 0
$$478$$ −1.02776 −0.0470085
$$479$$ −16.1833 −0.739436 −0.369718 0.929144i $$-0.620546\pi$$
−0.369718 + 0.929144i $$0.620546\pi$$
$$480$$ 0 0
$$481$$ −1.21110 −0.0552215
$$482$$ 36.7527 1.67404
$$483$$ 0 0
$$484$$ 1.27502 0.0579554
$$485$$ 8.00000 0.363261
$$486$$ 0 0
$$487$$ −8.18335 −0.370823 −0.185411 0.982661i $$-0.559362\pi$$
−0.185411 + 0.982661i $$0.559362\pi$$
$$488$$ −30.6333 −1.38670
$$489$$ 0 0
$$490$$ 18.5139 0.836372
$$491$$ 12.7889 0.577155 0.288577 0.957457i $$-0.406818\pi$$
0.288577 + 0.957457i $$0.406818\pi$$
$$492$$ 0 0
$$493$$ −48.2389 −2.17257
$$494$$ 2.84441 0.127976
$$495$$ 0 0
$$496$$ −5.30278 −0.238102
$$497$$ 67.2666 3.01732
$$498$$ 0 0
$$499$$ −27.6056 −1.23579 −0.617897 0.786259i $$-0.712015\pi$$
−0.617897 + 0.786259i $$0.712015\pi$$
$$500$$ −0.302776 −0.0135405
$$501$$ 0 0
$$502$$ −20.3667 −0.909010
$$503$$ 31.4222 1.40105 0.700523 0.713629i $$-0.252950\pi$$
0.700523 + 0.713629i $$0.252950\pi$$
$$504$$ 0 0
$$505$$ 12.0000 0.533993
$$506$$ 10.1833 0.452705
$$507$$ 0 0
$$508$$ 1.44996 0.0643316
$$509$$ 32.8444 1.45580 0.727901 0.685682i $$-0.240496\pi$$
0.727901 + 0.685682i $$0.240496\pi$$
$$510$$ 0 0
$$511$$ 24.8444 1.09905
$$512$$ −25.4222 −1.12351
$$513$$ 0 0
$$514$$ 29.7250 1.31111
$$515$$ −4.00000 −0.176261
$$516$$ 0 0
$$517$$ −13.5778 −0.597151
$$518$$ 12.0000 0.527250
$$519$$ 0 0
$$520$$ 1.81665 0.0796655
$$521$$ −21.3944 −0.937308 −0.468654 0.883382i $$-0.655261\pi$$
−0.468654 + 0.883382i $$0.655261\pi$$
$$522$$ 0 0
$$523$$ 23.3944 1.02297 0.511484 0.859293i $$-0.329096\pi$$
0.511484 + 0.859293i $$0.329096\pi$$
$$524$$ −1.81665 −0.0793609
$$525$$ 0 0
$$526$$ −22.4222 −0.977655
$$527$$ −9.00000 −0.392046
$$528$$ 0 0
$$529$$ −14.0000 −0.608696
$$530$$ −7.30278 −0.317212
$$531$$ 0 0
$$532$$ 5.02776 0.217981
$$533$$ −1.57779 −0.0683419
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 45.6333 1.97106
$$537$$ 0 0
$$538$$ 14.6056 0.629690
$$539$$ −37.0278 −1.59490
$$540$$ 0 0
$$541$$ −11.5778 −0.497768 −0.248884 0.968533i $$-0.580064\pi$$
−0.248884 + 0.968533i $$0.580064\pi$$
$$542$$ −25.0639 −1.07659
$$543$$ 0 0
$$544$$ −9.51388 −0.407904
$$545$$ −7.00000 −0.299847
$$546$$ 0 0
$$547$$ −6.60555 −0.282433 −0.141216 0.989979i $$-0.545101\pi$$
−0.141216 + 0.989979i $$0.545101\pi$$
$$548$$ 1.45837 0.0622983
$$549$$ 0 0
$$550$$ −3.39445 −0.144740
$$551$$ −31.0278 −1.32183
$$552$$ 0 0
$$553$$ −20.2389 −0.860644
$$554$$ −37.8167 −1.60668
$$555$$ 0 0
$$556$$ 1.21110 0.0513622
$$557$$ −33.6333 −1.42509 −0.712544 0.701627i $$-0.752457\pi$$
−0.712544 + 0.701627i $$0.752457\pi$$
$$558$$ 0 0
$$559$$ 4.00000 0.169182
$$560$$ −15.2111 −0.642786
$$561$$ 0 0
$$562$$ 2.36669 0.0998329
$$563$$ 24.0000 1.01148 0.505740 0.862686i $$-0.331220\pi$$
0.505740 + 0.862686i $$0.331220\pi$$
$$564$$ 0 0
$$565$$ −0.788897 −0.0331892
$$566$$ 13.8167 0.580757
$$567$$ 0 0
$$568$$ −43.8167 −1.83851
$$569$$ 37.8167 1.58536 0.792678 0.609640i $$-0.208686\pi$$
0.792678 + 0.609640i $$0.208686\pi$$
$$570$$ 0 0
$$571$$ −36.4500 −1.52538 −0.762692 0.646762i $$-0.776123\pi$$
−0.762692 + 0.646762i $$0.776123\pi$$
$$572$$ −0.477718 −0.0199744
$$573$$ 0 0
$$574$$ 15.6333 0.652522
$$575$$ −3.00000 −0.125109
$$576$$ 0 0
$$577$$ 27.8167 1.15802 0.579011 0.815320i $$-0.303439\pi$$
0.579011 + 0.815320i $$0.303439\pi$$
$$578$$ 18.7889 0.781515
$$579$$ 0 0
$$580$$ −2.60555 −0.108190
$$581$$ 13.8167 0.573211
$$582$$ 0 0
$$583$$ 14.6056 0.604900
$$584$$ −16.1833 −0.669672
$$585$$ 0 0
$$586$$ −37.5416 −1.55083
$$587$$ 21.0000 0.866763 0.433381 0.901211i $$-0.357320\pi$$
0.433381 + 0.901211i $$0.357320\pi$$
$$588$$ 0 0
$$589$$ −5.78890 −0.238527
$$590$$ −11.2111 −0.461554
$$591$$ 0 0
$$592$$ −6.60555 −0.271486
$$593$$ 21.2389 0.872175 0.436088 0.899904i $$-0.356364\pi$$
0.436088 + 0.899904i $$0.356364\pi$$
$$594$$ 0 0
$$595$$ −25.8167 −1.05838
$$596$$ 3.94449 0.161572
$$597$$ 0 0
$$598$$ 2.36669 0.0967812
$$599$$ −15.3944 −0.629000 −0.314500 0.949257i $$-0.601837\pi$$
−0.314500 + 0.949257i $$0.601837\pi$$
$$600$$ 0 0
$$601$$ 32.6333 1.33114 0.665570 0.746335i $$-0.268188\pi$$
0.665570 + 0.746335i $$0.268188\pi$$
$$602$$ −39.6333 −1.61533
$$603$$ 0 0
$$604$$ 4.36669 0.177678
$$605$$ −4.21110 −0.171206
$$606$$ 0 0
$$607$$ 17.3944 0.706019 0.353009 0.935620i $$-0.385158\pi$$
0.353009 + 0.935620i $$0.385158\pi$$
$$608$$ −6.11943 −0.248176
$$609$$ 0 0
$$610$$ 13.3028 0.538614
$$611$$ −3.15559 −0.127661
$$612$$ 0 0
$$613$$ 28.8444 1.16501 0.582507 0.812825i $$-0.302072\pi$$
0.582507 + 0.812825i $$0.302072\pi$$
$$614$$ −26.6056 −1.07371
$$615$$ 0 0
$$616$$ 36.0000 1.45048
$$617$$ −26.4500 −1.06484 −0.532418 0.846482i $$-0.678716\pi$$
−0.532418 + 0.846482i $$0.678716\pi$$
$$618$$ 0 0
$$619$$ −7.63331 −0.306809 −0.153404 0.988164i $$-0.549024\pi$$
−0.153404 + 0.988164i $$0.549024\pi$$
$$620$$ −0.486122 −0.0195231
$$621$$ 0 0
$$622$$ 18.0000 0.721734
$$623$$ 36.0000 1.44231
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 30.7889 1.23057
$$627$$ 0 0
$$628$$ −1.15559 −0.0461131
$$629$$ −11.2111 −0.447016
$$630$$ 0 0
$$631$$ 30.0278 1.19539 0.597693 0.801725i $$-0.296084\pi$$
0.597693 + 0.801725i $$0.296084\pi$$
$$632$$ 13.1833 0.524405
$$633$$ 0 0
$$634$$ −0.513878 −0.0204087
$$635$$ −4.78890 −0.190042
$$636$$ 0 0
$$637$$ −8.60555 −0.340964
$$638$$ −29.2111 −1.15648
$$639$$ 0 0
$$640$$ 8.09167 0.319851
$$641$$ −24.0000 −0.947943 −0.473972 0.880540i $$-0.657180\pi$$
−0.473972 + 0.880540i $$0.657180\pi$$
$$642$$ 0 0
$$643$$ −23.0278 −0.908126 −0.454063 0.890970i $$-0.650026\pi$$
−0.454063 + 0.890970i $$0.650026\pi$$
$$644$$ 4.18335 0.164847
$$645$$ 0 0
$$646$$ 26.3305 1.03596
$$647$$ 38.2111 1.50223 0.751117 0.660169i $$-0.229516\pi$$
0.751117 + 0.660169i $$0.229516\pi$$
$$648$$ 0 0
$$649$$ 22.4222 0.880149
$$650$$ −0.788897 −0.0309431
$$651$$ 0 0
$$652$$ −0.605551 −0.0237152
$$653$$ 27.2389 1.06594 0.532969 0.846134i $$-0.321076\pi$$
0.532969 + 0.846134i $$0.321076\pi$$
$$654$$ 0 0
$$655$$ 6.00000 0.234439
$$656$$ −8.60555 −0.335990
$$657$$ 0 0
$$658$$ 31.2666 1.21890
$$659$$ −20.6056 −0.802678 −0.401339 0.915930i $$-0.631455\pi$$
−0.401339 + 0.915930i $$0.631455\pi$$
$$660$$ 0 0
$$661$$ 22.8444 0.888545 0.444272 0.895892i $$-0.353462\pi$$
0.444272 + 0.895892i $$0.353462\pi$$
$$662$$ 19.2666 0.748818
$$663$$ 0 0
$$664$$ −9.00000 −0.349268
$$665$$ −16.6056 −0.643936
$$666$$ 0 0
$$667$$ −25.8167 −0.999625
$$668$$ 0.908327 0.0351442
$$669$$ 0 0
$$670$$ −19.8167 −0.765584
$$671$$ −26.6056 −1.02710
$$672$$ 0 0
$$673$$ −11.0278 −0.425089 −0.212544 0.977151i $$-0.568175\pi$$
−0.212544 + 0.977151i $$0.568175\pi$$
$$674$$ −0.788897 −0.0303872
$$675$$ 0 0
$$676$$ 3.82506 0.147118
$$677$$ −21.6333 −0.831436 −0.415718 0.909494i $$-0.636470\pi$$
−0.415718 + 0.909494i $$0.636470\pi$$
$$678$$ 0 0
$$679$$ 36.8444 1.41396
$$680$$ 16.8167 0.644889
$$681$$ 0 0
$$682$$ −5.44996 −0.208690
$$683$$ −9.00000 −0.344375 −0.172188 0.985064i $$-0.555084\pi$$
−0.172188 + 0.985064i $$0.555084\pi$$
$$684$$ 0 0
$$685$$ −4.81665 −0.184035
$$686$$ 43.2666 1.65193
$$687$$ 0 0
$$688$$ 21.8167 0.831752
$$689$$ 3.39445 0.129318
$$690$$ 0 0
$$691$$ 2.39445 0.0910891 0.0455446 0.998962i $$-0.485498\pi$$
0.0455446 + 0.998962i $$0.485498\pi$$
$$692$$ −3.27502 −0.124498
$$693$$ 0 0
$$694$$ 2.05551 0.0780262
$$695$$ −4.00000 −0.151729
$$696$$ 0 0
$$697$$ −14.6056 −0.553225
$$698$$ 33.6695 1.27441
$$699$$ 0 0
$$700$$ −1.39445 −0.0527052
$$701$$ 40.4222 1.52673 0.763363 0.645970i $$-0.223547\pi$$
0.763363 + 0.645970i $$0.223547\pi$$
$$702$$ 0 0
$$703$$ −7.21110 −0.271972
$$704$$ −22.9722 −0.865799
$$705$$ 0 0
$$706$$ −28.1833 −1.06069
$$707$$ 55.2666 2.07851
$$708$$ 0 0
$$709$$ 34.8444 1.30861 0.654305 0.756231i $$-0.272961\pi$$
0.654305 + 0.756231i $$0.272961\pi$$
$$710$$ 19.0278 0.714099
$$711$$ 0 0
$$712$$ −23.4500 −0.878824
$$713$$ −4.81665 −0.180385
$$714$$ 0 0
$$715$$ 1.57779 0.0590062
$$716$$ 2.05551 0.0768181
$$717$$ 0 0
$$718$$ 43.8167 1.63522
$$719$$ −49.2666 −1.83733 −0.918667 0.395032i $$-0.870733\pi$$
−0.918667 + 0.395032i $$0.870733\pi$$
$$720$$ 0 0
$$721$$ −18.4222 −0.686079
$$722$$ −7.81665 −0.290906
$$723$$ 0 0
$$724$$ 2.11943 0.0787680
$$725$$ 8.60555 0.319602
$$726$$ 0 0
$$727$$ −7.63331 −0.283104 −0.141552 0.989931i $$-0.545209\pi$$
−0.141552 + 0.989931i $$0.545209\pi$$
$$728$$ 8.36669 0.310090
$$729$$ 0 0
$$730$$ 7.02776 0.260109
$$731$$ 37.0278 1.36952
$$732$$ 0 0
$$733$$ 32.0000 1.18195 0.590973 0.806691i $$-0.298744\pi$$
0.590973 + 0.806691i $$0.298744\pi$$
$$734$$ 6.00000 0.221464
$$735$$ 0 0
$$736$$ −5.09167 −0.187682
$$737$$ 39.6333 1.45991
$$738$$ 0 0
$$739$$ 30.0278 1.10459 0.552294 0.833649i $$-0.313752\pi$$
0.552294 + 0.833649i $$0.313752\pi$$
$$740$$ −0.605551 −0.0222605
$$741$$ 0 0
$$742$$ −33.6333 −1.23472
$$743$$ −34.4222 −1.26283 −0.631414 0.775446i $$-0.717525\pi$$
−0.631414 + 0.775446i $$0.717525\pi$$
$$744$$ 0 0
$$745$$ −13.0278 −0.477300
$$746$$ −14.0555 −0.514609
$$747$$ 0 0
$$748$$ −4.42221 −0.161692
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 6.02776 0.219956 0.109978 0.993934i $$-0.464922\pi$$
0.109978 + 0.993934i $$0.464922\pi$$
$$752$$ −17.2111 −0.627624
$$753$$ 0 0
$$754$$ −6.78890 −0.247237
$$755$$ −14.4222 −0.524878
$$756$$ 0 0
$$757$$ 31.2111 1.13439 0.567193 0.823585i $$-0.308029\pi$$
0.567193 + 0.823585i $$0.308029\pi$$
$$758$$ 18.7527 0.681130
$$759$$ 0 0
$$760$$ 10.8167 0.392361
$$761$$ −53.4500 −1.93756 −0.968780 0.247923i $$-0.920252\pi$$
−0.968780 + 0.247923i $$0.920252\pi$$
$$762$$ 0 0
$$763$$ −32.2389 −1.16713
$$764$$ −4.97224 −0.179889
$$765$$ 0 0
$$766$$ −24.2750 −0.877092
$$767$$ 5.21110 0.188162
$$768$$ 0 0
$$769$$ 41.0000 1.47850 0.739249 0.673432i $$-0.235181\pi$$
0.739249 + 0.673432i $$0.235181\pi$$
$$770$$ −15.6333 −0.563385
$$771$$ 0 0
$$772$$ −6.60555 −0.237739
$$773$$ 16.8167 0.604853 0.302426 0.953173i $$-0.402203\pi$$
0.302426 + 0.953173i $$0.402203\pi$$
$$774$$ 0 0
$$775$$ 1.60555 0.0576731
$$776$$ −24.0000 −0.861550
$$777$$ 0 0
$$778$$ 5.44996 0.195391
$$779$$ −9.39445 −0.336591
$$780$$ 0 0
$$781$$ −38.0555 −1.36173
$$782$$ 21.9083 0.783440
$$783$$ 0 0
$$784$$ −46.9361 −1.67629
$$785$$ 3.81665 0.136222
$$786$$ 0 0
$$787$$ −2.97224 −0.105949 −0.0529745 0.998596i $$-0.516870\pi$$
−0.0529745 + 0.998596i $$0.516870\pi$$
$$788$$ 0.358288 0.0127635
$$789$$ 0 0
$$790$$ −5.72498 −0.203686
$$791$$ −3.63331 −0.129186
$$792$$ 0 0
$$793$$ −6.18335 −0.219577
$$794$$ −16.4222 −0.582802
$$795$$ 0 0
$$796$$ −4.00000 −0.141776
$$797$$ −37.6611 −1.33402 −0.667012 0.745047i $$-0.732427\pi$$
−0.667012 + 0.745047i $$0.732427\pi$$
$$798$$ 0 0
$$799$$ −29.2111 −1.03341
$$800$$ 1.69722 0.0600059
$$801$$ 0 0
$$802$$ −39.0833 −1.38008
$$803$$ −14.0555 −0.496008
$$804$$ 0 0
$$805$$ −13.8167 −0.486973
$$806$$ −1.26662 −0.0446146
$$807$$ 0 0
$$808$$ −36.0000 −1.26648
$$809$$ 50.6056 1.77920 0.889598 0.456744i $$-0.150984\pi$$
0.889598 + 0.456744i $$0.150984\pi$$
$$810$$ 0 0
$$811$$ 42.4222 1.48965 0.744823 0.667263i $$-0.232534\pi$$
0.744823 + 0.667263i $$0.232534\pi$$
$$812$$ −12.0000 −0.421117
$$813$$ 0 0
$$814$$ −6.78890 −0.237951
$$815$$ 2.00000 0.0700569
$$816$$ 0 0
$$817$$ 23.8167 0.833239
$$818$$ 6.51388 0.227752
$$819$$ 0 0
$$820$$ −0.788897 −0.0275495
$$821$$ 30.0000 1.04701 0.523504 0.852023i $$-0.324625\pi$$
0.523504 + 0.852023i $$0.324625\pi$$
$$822$$ 0 0
$$823$$ 3.81665 0.133040 0.0665201 0.997785i $$-0.478810\pi$$
0.0665201 + 0.997785i $$0.478810\pi$$
$$824$$ 12.0000 0.418040
$$825$$ 0 0
$$826$$ −51.6333 −1.79655
$$827$$ −33.7889 −1.17496 −0.587478 0.809240i $$-0.699879\pi$$
−0.587478 + 0.809240i $$0.699879\pi$$
$$828$$ 0 0
$$829$$ −27.2111 −0.945081 −0.472540 0.881309i $$-0.656663\pi$$
−0.472540 + 0.881309i $$0.656663\pi$$
$$830$$ 3.90833 0.135660
$$831$$ 0 0
$$832$$ −5.33894 −0.185094
$$833$$ −79.6611 −2.76009
$$834$$ 0 0
$$835$$ −3.00000 −0.103819
$$836$$ −2.84441 −0.0983760
$$837$$ 0 0
$$838$$ −16.9722 −0.586296
$$839$$ 18.7889 0.648665 0.324332 0.945943i $$-0.394860\pi$$
0.324332 + 0.945943i $$0.394860\pi$$
$$840$$ 0 0
$$841$$ 45.0555 1.55364
$$842$$ −30.5139 −1.05158
$$843$$ 0 0
$$844$$ −3.88057 −0.133575
$$845$$ −12.6333 −0.434599
$$846$$ 0 0
$$847$$ −19.3944 −0.666401
$$848$$ 18.5139 0.635769
$$849$$ 0 0
$$850$$ −7.30278 −0.250483
$$851$$ −6.00000 −0.205677
$$852$$ 0 0
$$853$$ 14.7889 0.506362 0.253181 0.967419i $$-0.418523\pi$$
0.253181 + 0.967419i $$0.418523\pi$$
$$854$$ 61.2666 2.09650
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −45.2389 −1.54533 −0.772665 0.634814i $$-0.781077\pi$$
−0.772665 + 0.634814i $$0.781077\pi$$
$$858$$ 0 0
$$859$$ 6.02776 0.205664 0.102832 0.994699i $$-0.467210\pi$$
0.102832 + 0.994699i $$0.467210\pi$$
$$860$$ 2.00000 0.0681994
$$861$$ 0 0
$$862$$ 33.6333 1.14556
$$863$$ 15.7889 0.537460 0.268730 0.963216i $$-0.413396\pi$$
0.268730 + 0.963216i $$0.413396\pi$$
$$864$$ 0 0
$$865$$ 10.8167 0.367777
$$866$$ −36.7889 −1.25014
$$867$$ 0 0
$$868$$ −2.23886 −0.0759918
$$869$$ 11.4500 0.388413
$$870$$ 0 0
$$871$$ 9.21110 0.312106
$$872$$ 21.0000 0.711150
$$873$$ 0 0
$$874$$ 14.0917 0.476658
$$875$$ 4.60555 0.155696
$$876$$ 0 0
$$877$$ −56.6611 −1.91331 −0.956654 0.291227i $$-0.905937\pi$$
−0.956654 + 0.291227i $$0.905937\pi$$
$$878$$ 26.5694 0.896674
$$879$$ 0 0
$$880$$ 8.60555 0.290093
$$881$$ −32.6056 −1.09851 −0.549254 0.835655i $$-0.685088\pi$$
−0.549254 + 0.835655i $$0.685088\pi$$
$$882$$ 0 0
$$883$$ −5.81665 −0.195746 −0.0978730 0.995199i $$-0.531204\pi$$
−0.0978730 + 0.995199i $$0.531204\pi$$
$$884$$ −1.02776 −0.0345672
$$885$$ 0 0
$$886$$ 24.2750 0.815535
$$887$$ 12.6333 0.424185 0.212092 0.977250i $$-0.431972\pi$$
0.212092 + 0.977250i $$0.431972\pi$$
$$888$$ 0 0
$$889$$ −22.0555 −0.739718
$$890$$ 10.1833 0.341347
$$891$$ 0 0
$$892$$ 3.02776 0.101377
$$893$$ −18.7889 −0.628746
$$894$$ 0 0
$$895$$ −6.78890 −0.226928
$$896$$ 37.2666 1.24499
$$897$$ 0 0
$$898$$ −15.9445 −0.532074
$$899$$ 13.8167 0.460811
$$900$$ 0 0
$$901$$ 31.4222 1.04683
$$902$$ −8.84441 −0.294487
$$903$$ 0 0
$$904$$ 2.36669 0.0787150
$$905$$ −7.00000 −0.232688
$$906$$ 0 0
$$907$$ 30.4222 1.01015 0.505076 0.863075i $$-0.331464\pi$$
0.505076 + 0.863075i $$0.331464\pi$$
$$908$$ 7.93608 0.263368
$$909$$ 0 0
$$910$$ −3.63331 −0.120443
$$911$$ 31.0278 1.02800 0.513998 0.857792i $$-0.328164\pi$$
0.513998 + 0.857792i $$0.328164\pi$$
$$912$$ 0 0
$$913$$ −7.81665 −0.258693
$$914$$ 1.57779 0.0521888
$$915$$ 0 0
$$916$$ 1.88057 0.0621358
$$917$$ 27.6333 0.912532
$$918$$ 0 0
$$919$$ −2.42221 −0.0799012 −0.0399506 0.999202i $$-0.512720\pi$$
−0.0399506 + 0.999202i $$0.512720\pi$$
$$920$$ 9.00000 0.296721
$$921$$ 0 0
$$922$$ −28.1833 −0.928169
$$923$$ −8.84441 −0.291117
$$924$$ 0 0
$$925$$ 2.00000 0.0657596
$$926$$ −19.8167 −0.651216
$$927$$ 0 0
$$928$$ 14.6056 0.479451
$$929$$ 26.6056 0.872900 0.436450 0.899729i $$-0.356236\pi$$
0.436450 + 0.899729i $$0.356236\pi$$
$$930$$ 0 0
$$931$$ −51.2389 −1.67929
$$932$$ 5.44996 0.178519
$$933$$ 0 0
$$934$$ −2.88057 −0.0942551
$$935$$ 14.6056 0.477653
$$936$$ 0 0
$$937$$ 26.7889 0.875155 0.437578 0.899181i $$-0.355837\pi$$
0.437578 + 0.899181i $$0.355837\pi$$
$$938$$ −91.2666 −2.97996
$$939$$ 0 0
$$940$$ −1.57779 −0.0514620
$$941$$ 28.4222 0.926537 0.463269 0.886218i $$-0.346676\pi$$
0.463269 + 0.886218i $$0.346676\pi$$
$$942$$ 0 0
$$943$$ −7.81665 −0.254545
$$944$$ 28.4222 0.925064
$$945$$ 0 0
$$946$$ 22.4222 0.729009
$$947$$ 39.0000 1.26733 0.633665 0.773608i $$-0.281550\pi$$
0.633665 + 0.773608i $$0.281550\pi$$
$$948$$ 0 0
$$949$$ −3.26662 −0.106039
$$950$$ −4.69722 −0.152398
$$951$$ 0 0
$$952$$ 77.4500 2.51017
$$953$$ 26.8444 0.869576 0.434788 0.900533i $$-0.356823\pi$$
0.434788 + 0.900533i $$0.356823\pi$$
$$954$$ 0 0
$$955$$ 16.4222 0.531410
$$956$$ 0.238859 0.00772525
$$957$$ 0 0
$$958$$ −21.0833 −0.681170
$$959$$ −22.1833 −0.716338
$$960$$ 0 0
$$961$$ −28.4222 −0.916845
$$962$$ −1.57779 −0.0508701
$$963$$ 0 0
$$964$$ −8.54163 −0.275108
$$965$$ 21.8167 0.702303
$$966$$ 0 0
$$967$$ 50.0000 1.60789 0.803946 0.594703i $$-0.202730\pi$$
0.803946 + 0.594703i $$0.202730\pi$$
$$968$$ 12.6333 0.406050
$$969$$ 0 0
$$970$$ 10.4222 0.334637
$$971$$ −18.0000 −0.577647 −0.288824 0.957382i $$-0.593264\pi$$
−0.288824 + 0.957382i $$0.593264\pi$$
$$972$$ 0 0
$$973$$ −18.4222 −0.590589
$$974$$ −10.6611 −0.341603
$$975$$ 0 0
$$976$$ −33.7250 −1.07951
$$977$$ −0.788897 −0.0252391 −0.0126195 0.999920i $$-0.504017\pi$$
−0.0126195 + 0.999920i $$0.504017\pi$$
$$978$$ 0 0
$$979$$ −20.3667 −0.650922
$$980$$ −4.30278 −0.137447
$$981$$ 0 0
$$982$$ 16.6611 0.531676
$$983$$ 0.633308 0.0201994 0.0100997 0.999949i $$-0.496785\pi$$
0.0100997 + 0.999949i $$0.496785\pi$$
$$984$$ 0 0
$$985$$ −1.18335 −0.0377045
$$986$$ −62.8444 −2.00137
$$987$$ 0 0
$$988$$ −0.661064 −0.0210312
$$989$$ 19.8167 0.630133
$$990$$ 0 0
$$991$$ −50.8167 −1.61424 −0.807122 0.590385i $$-0.798976\pi$$
−0.807122 + 0.590385i $$0.798976\pi$$
$$992$$ 2.72498 0.0865182
$$993$$ 0 0
$$994$$ 87.6333 2.77956
$$995$$ 13.2111 0.418820
$$996$$ 0 0
$$997$$ 53.8722 1.70615 0.853074 0.521789i $$-0.174735\pi$$
0.853074 + 0.521789i $$0.174735\pi$$
$$998$$ −35.9638 −1.13842
$$999$$ 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 135.2.a.c.1.2 2
3.2 odd 2 135.2.a.d.1.1 yes 2
4.3 odd 2 2160.2.a.ba.1.1 2
5.2 odd 4 675.2.b.i.649.3 4
5.3 odd 4 675.2.b.i.649.2 4
5.4 even 2 675.2.a.p.1.1 2
7.6 odd 2 6615.2.a.p.1.2 2
8.3 odd 2 8640.2.a.ck.1.1 2
8.5 even 2 8640.2.a.cr.1.2 2
9.2 odd 6 405.2.e.j.271.2 4
9.4 even 3 405.2.e.k.136.1 4
9.5 odd 6 405.2.e.j.136.2 4
9.7 even 3 405.2.e.k.271.1 4
12.11 even 2 2160.2.a.y.1.1 2
15.2 even 4 675.2.b.h.649.2 4
15.8 even 4 675.2.b.h.649.3 4
15.14 odd 2 675.2.a.k.1.2 2
21.20 even 2 6615.2.a.v.1.1 2
24.5 odd 2 8640.2.a.df.1.2 2
24.11 even 2 8640.2.a.cy.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
135.2.a.c.1.2 2 1.1 even 1 trivial
135.2.a.d.1.1 yes 2 3.2 odd 2
405.2.e.j.136.2 4 9.5 odd 6
405.2.e.j.271.2 4 9.2 odd 6
405.2.e.k.136.1 4 9.4 even 3
405.2.e.k.271.1 4 9.7 even 3
675.2.a.k.1.2 2 15.14 odd 2
675.2.a.p.1.1 2 5.4 even 2
675.2.b.h.649.2 4 15.2 even 4
675.2.b.h.649.3 4 15.8 even 4
675.2.b.i.649.2 4 5.3 odd 4
675.2.b.i.649.3 4 5.2 odd 4
2160.2.a.y.1.1 2 12.11 even 2
2160.2.a.ba.1.1 2 4.3 odd 2
6615.2.a.p.1.2 2 7.6 odd 2
6615.2.a.v.1.1 2 21.20 even 2
8640.2.a.ck.1.1 2 8.3 odd 2
8640.2.a.cr.1.2 2 8.5 even 2
8640.2.a.cy.1.1 2 24.11 even 2
8640.2.a.df.1.2 2 24.5 odd 2