LMFDB - Embedded newform 135.2.a.d.1.1

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			1.1
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

Display 100 coefficients 10 coefficients

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 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \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	$-0.652366\pi$
$2$	−1.30278	−0.921201	−0.460601	+	0.887607i	$0.652366\pi$
$3$	0	0
$3$	0	0
$4$	−0.302776	−0.151388
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.60555	1.74073	0.870367	−	0.492403i	$-0.163881\pi$
$7$	4.60555	1.74073	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	$-0.371506\pi$
$11$	2.60555	0.785603	0.392802	+	0.919623i	$0.371506\pi$
$12$	0	0
$13$	−0.605551	−0.167950	−0.0839749	−	0.996468i	$-0.526762\pi$
$13$	−0.605551	−0.167950	−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	$-0.262078\pi$
$17$	5.60555	1.35955	0.679773	+	0.733423i	$0.262078\pi$
$18$	0	0
$19$	−3.60555	−0.827170	−0.413585	−	0.910465i	$-0.635724\pi$
$19$	−3.60555	−0.827170	−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	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	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	$-0.794640\pi$
$29$	−8.60555	−1.59801	−0.799005	+	0.601324i	$0.794640\pi$
$30$	0	0
$31$	1.60555	0.288366	0.144183	−	0.989551i	$-0.453945\pi$
$31$	1.60555	0.288366	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	$-0.447432\pi$
$37$	2.00000	0.328798	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	$-0.565218\pi$
$41$	−2.60555	−0.406919	−0.203459	+	0.979083i	$0.565218\pi$
$42$	0	0
$43$	−6.60555	−1.00734	−0.503669	−	0.863897i	$-0.668017\pi$
$43$	−6.60555	−1.00734	−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	$-0.624096\pi$
$47$	−5.21110	−0.760117	−0.380059	+	0.924962i	$0.624096\pi$
$48$	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	$-0.374205\pi$
$53$	5.60555	0.769982	0.384991	+	0.922920i	$0.374205\pi$
$54$	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	$-0.310734\pi$
$59$	8.60555	1.12035	0.560174	+	0.828375i	$0.310734\pi$
$60$	0	0
$61$	10.2111	1.30740	0.653699	−	0.756755i	$-0.273216\pi$
$61$	10.2111	1.30740	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	$-0.879473\pi$
$67$	−15.2111	−1.85833	−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	$-0.833751\pi$
$71$	−14.6056	−1.73336	−0.866680	+	0.498864i	$0.833751\pi$
$72$	0	0
$73$	5.39445	0.631372	0.315686	−	0.948864i	$-0.397765\pi$
$73$	5.39445	0.631372	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	$-0.579513\pi$
$79$	−4.39445	−0.494414	−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	$-0.552648\pi$
$83$	−3.00000	−0.329293	−0.164646	+	0.986353i	$0.552648\pi$
$84$	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	$-0.635967\pi$
$89$	−7.81665	−0.828564	−0.414282	+	0.910149i	$0.635967\pi$
$90$	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	$-0.366875\pi$
$97$	8.00000	0.812277	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	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−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	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−7.00000	−0.670478	−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000	−0.670478	−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	$-0.488186\pi$
$113$	0.788897	0.0742132	0.0371066	+	0.999311i	$0.488186\pi$
$114$	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	$-0.568152\pi$
$127$	−4.78890	−0.424946	−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	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	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	$-0.434034\pi$
$137$	4.81665	0.411515	0.205757	+	0.978603i	$0.434034\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−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	$-0.320825\pi$
$149$	13.0278	1.06728	0.533638	+	0.845713i	$0.320825\pi$
$150$	0	0
$151$	−14.4222	−1.17366	−0.586831	−	0.809709i	$-0.699625\pi$
$151$	−14.4222	−1.17366	−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	$-0.451332\pi$
$157$	3.81665	0.304602	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	$-0.475042\pi$
$163$	2.00000	0.156652	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	$-0.462969\pi$
$167$	3.00000	0.232147	0.116073	+	0.993241i	$0.462969\pi$
$168$	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	$-0.634886\pi$
$173$	−10.8167	−0.822375	−0.411187	+	0.911551i	$0.634886\pi$
$174$	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	$-0.418348\pi$
$179$	6.78890	0.507426	0.253713	+	0.967280i	$0.418348\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−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	$-0.702505\pi$
$191$	−16.4222	−1.18827	−0.594135	+	0.804366i	$0.702505\pi$
$192$	0	0
$193$	21.8167	1.57040	0.785199	−	0.619244i	$-0.212561\pi$
$193$	21.8167	1.57040	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	$-0.486578\pi$
$197$	1.18335	0.0843099	0.0421550	+	0.999111i	$0.486578\pi$
$198$	0	0
$199$	13.2111	0.936510	0.468255	−	0.883593i	$-0.344883\pi$
$199$	13.2111	0.936510	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	$-0.354564\pi$
$211$	12.8167	0.882335	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	$-0.608677\pi$
$223$	−10.0000	−0.669650	−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	$-0.164218\pi$
$227$	26.2111	1.73969	0.869846	+	0.493323i	$0.164218\pi$
$228$	0	0
$229$	−6.21110	−0.410441	−0.205221	−	0.978716i	$-0.565791\pi$
$229$	−6.21110	−0.410441	−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	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	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	$-0.491878\pi$
$239$	0.788897	0.0510295	0.0255148	+	0.999674i	$0.491878\pi$
$240$	0	0
$241$	28.2111	1.81724	0.908618	−	0.417627i	$-0.137138\pi$
$241$	28.2111	1.81724	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	$-0.335760\pi$
$251$	15.6333	0.986766	0.493383	+	0.869812i	$0.335760\pi$
$252$	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	$-0.752044\pi$
$257$	−22.8167	−1.42326	−0.711632	+	0.702553i	$0.752044\pi$
$258$	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	$-0.321951\pi$
$263$	17.2111	1.06128	0.530641	+	0.847597i	$0.321951\pi$
$264$	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	$-0.611029\pi$
$269$	−11.2111	−0.683553	−0.341776	+	0.939781i	$0.611029\pi$
$270$	0	0
$271$	−19.2389	−1.16868	−0.584339	−	0.811510i	$-0.698646\pi$
$271$	−19.2389	−1.16868	−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	$-0.837213\pi$
$277$	−29.0278	−1.74411	−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	$-0.517256\pi$
$281$	−1.81665	−0.108372	−0.0541862	+	0.998531i	$0.517256\pi$
$282$	0	0
$283$	10.6056	0.630435	0.315217	−	0.949020i	$-0.397923\pi$
$283$	10.6056	0.630435	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	$-0.181530\pi$
$293$	28.8167	1.68349	0.841743	+	0.539878i	$0.181530\pi$
$294$	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	$-0.698034\pi$
$307$	−20.4222	−1.16556	−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	$-0.628125\pi$
$311$	−13.8167	−0.783471	−0.391735	+	0.920078i	$0.628125\pi$
$312$	0	0
$313$	23.6333	1.33583	0.667917	−	0.744236i	$-0.267186\pi$
$313$	23.6333	1.33583	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	$-0.496474\pi$
$317$	0.394449	0.0221544	0.0110772	+	0.999939i	$0.496474\pi$
$318$	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	$-0.366772\pi$
$331$	14.7889	0.812871	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	$-0.505250\pi$
$337$	−0.605551	−0.0329865	−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	$-0.513485\pi$
$347$	−1.57779	−0.0847005	−0.0423502	+	0.999103i	$0.513485\pi$
$348$	0	0
$349$	25.8444	1.38342	0.691710	−	0.722176i	$-0.256858\pi$
$349$	25.8444	1.38342	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	$-0.304725\pi$
$353$	21.6333	1.15142	0.575712	+	0.817652i	$0.304725\pi$
$354$	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	$-0.847594\pi$
$359$	−33.6333	−1.77510	−0.887549	+	0.460713i	$0.847594\pi$
$360$	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	$-0.461645\pi$
$367$	4.60555	0.240408	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	$-0.590107\pi$
$373$	−10.7889	−0.558628	−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	$-0.379462\pi$
$379$	14.3944	0.739393	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	$-0.342065\pi$
$383$	18.6333	0.952118	0.476059	+	0.879413i	$0.342065\pi$
$384$	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	$-0.533821\pi$
$389$	−4.18335	−0.212104	−0.106052	+	0.994361i	$0.533821\pi$
$390$	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	$-0.602450\pi$
$397$	−12.6056	−0.632654	−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	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	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	$-0.460551\pi$
$409$	5.00000	0.247234	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	$-0.396914\pi$
$419$	13.0278	0.636448	0.318224	+	0.948016i	$0.396914\pi$
$420$	0	0
$421$	−23.4222	−1.14153	−0.570764	−	0.821114i	$-0.693353\pi$
$421$	−23.4222	−1.14153	−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	$-0.713587\pi$
$431$	−25.8167	−1.24354	−0.621772	+	0.783198i	$0.713587\pi$
$432$	0	0
$433$	−28.2389	−1.35707	−0.678536	−	0.734567i	$-0.737385\pi$
$433$	−28.2389	−1.35707	−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	$-0.338205\pi$
$439$	20.3944	0.973374	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	$-0.645961\pi$
$443$	−18.6333	−0.885295	−0.442648	+	0.896696i	$0.645961\pi$
$444$	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	$-0.406746\pi$
$449$	12.2389	0.577587	0.288794	+	0.957391i	$0.406746\pi$
$450$	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	$-0.490982\pi$
$457$	1.21110	0.0566530	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	$-0.331942\pi$
$461$	21.6333	1.00756	0.503782	+	0.863831i	$0.331942\pi$
$462$	0	0
$463$	−15.2111	−0.706920	−0.353460	−	0.935450i	$-0.614995\pi$
$463$	−15.2111	−0.706920	−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	$-0.483709\pi$
$467$	2.21110	0.102318	0.0511588	+	0.998691i	$0.483709\pi$
$468$	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	$-0.379454\pi$
$479$	16.1833	0.739436	0.369718	+	0.929144i	$0.379454\pi$
$480$	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	$-0.559362\pi$
$487$	−8.18335	−0.370823	−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	$-0.593182\pi$
$491$	−12.7889	−0.577155	−0.288577	+	0.957457i	$0.593182\pi$
$492$	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	$-0.712015\pi$
$499$	−27.6056	−1.23579	−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	$-0.747050\pi$
$503$	−31.4222	−1.40105	−0.700523	+	0.713629i	$0.747050\pi$
$504$	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	$-0.759504\pi$
$509$	−32.8444	−1.45580	−0.727901	+	0.685682i	$0.759504\pi$
$510$	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	$-0.344739\pi$
$521$	21.3944	0.937308	0.468654	+	0.883382i	$0.344739\pi$
$522$	0	0
$523$	23.3944	1.02297	0.511484	−	0.859293i	$-0.329096\pi$
$523$	23.3944	1.02297	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	$-0.580064\pi$
$541$	−11.5778	−0.497768	−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	$-0.545101\pi$
$547$	−6.60555	−0.282433	−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	$-0.247543\pi$
$557$	33.6333	1.42509	0.712544	+	0.701627i	$0.247543\pi$
$558$	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	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	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	$-0.791314\pi$
$569$	−37.8167	−1.58536	−0.792678	+	0.609640i	$0.791314\pi$
$570$	0	0
$571$	−36.4500	−1.52538	−0.762692	−	0.646762i	$-0.776123\pi$
$571$	−36.4500	−1.52538	−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	$-0.303439\pi$
$577$	27.8167	1.15802	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	$-0.642680\pi$
$587$	−21.0000	−0.866763	−0.433381	+	0.901211i	$0.642680\pi$
$588$	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	$-0.643636\pi$
$593$	−21.2389	−0.872175	−0.436088	+	0.899904i	$0.643636\pi$
$594$	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	$-0.398163\pi$
$599$	15.3944	0.629000	0.314500	+	0.949257i	$0.398163\pi$
$600$	0	0
$601$	32.6333	1.33114	0.665570	−	0.746335i	$-0.268188\pi$
$601$	32.6333	1.33114	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	$-0.385158\pi$
$607$	17.3944	0.706019	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	$-0.302072\pi$
$613$	28.8444	1.16501	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	$-0.321284\pi$
$617$	26.4500	1.06484	0.532418	+	0.846482i	$0.321284\pi$
$618$	0	0
$619$	−7.63331	−0.306809	−0.153404	−	0.988164i	$-0.549024\pi$
$619$	−7.63331	−0.306809	−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	$-0.296084\pi$
$631$	30.0278	1.19539	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	$-0.342820\pi$
$641$	24.0000	0.947943	0.473972	+	0.880540i	$0.342820\pi$
$642$	0	0
$643$	−23.0278	−0.908126	−0.454063	−	0.890970i	$-0.650026\pi$
$643$	−23.0278	−0.908126	−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	$-0.770484\pi$
$647$	−38.2111	−1.50223	−0.751117	+	0.660169i	$0.770484\pi$
$648$	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	$-0.678924\pi$
$653$	−27.2389	−1.06594	−0.532969	+	0.846134i	$0.678924\pi$
$654$	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	$-0.368545\pi$
$659$	20.6056	0.802678	0.401339	+	0.915930i	$0.368545\pi$
$660$	0	0
$661$	22.8444	0.888545	0.444272	−	0.895892i	$-0.353462\pi$
$661$	22.8444	0.888545	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	$-0.568175\pi$
$673$	−11.0278	−0.425089	−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	$-0.363530\pi$
$677$	21.6333	0.831436	0.415718	+	0.909494i	$0.363530\pi$
$678$	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	$-0.444916\pi$
$683$	9.00000	0.344375	0.172188	+	0.985064i	$0.444916\pi$
$684$	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	$-0.485498\pi$
$691$	2.39445	0.0910891	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	$-0.776453\pi$
$701$	−40.4222	−1.52673	−0.763363	+	0.645970i	$0.776453\pi$
$702$	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	$-0.272961\pi$
$709$	34.8444	1.30861	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	$-0.129267\pi$
$719$	49.2666	1.83733	0.918667	+	0.395032i	$0.129267\pi$
$720$	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	$-0.545209\pi$
$727$	−7.63331	−0.283104	−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	$-0.298744\pi$
$733$	32.0000	1.18195	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	$-0.313752\pi$
$739$	30.0278	1.10459	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	$-0.282475\pi$
$743$	34.4222	1.26283	0.631414	+	0.775446i	$0.282475\pi$
$744$	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	$-0.464922\pi$
$751$	6.02776	0.219956	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	$-0.308029\pi$
$757$	31.2111	1.13439	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	$-0.0797479\pi$
$761$	53.4500	1.93756	0.968780	+	0.247923i	$0.0797479\pi$
$762$	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	$-0.235181\pi$
$769$	41.0000	1.47850	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	$-0.597797\pi$
$773$	−16.8167	−0.604853	−0.302426	+	0.953173i	$0.597797\pi$
$774$	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	$-0.516870\pi$
$787$	−2.97224	−0.105949	−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	$-0.267573\pi$
$797$	37.6611	1.33402	0.667012	+	0.745047i	$0.267573\pi$
$798$	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	$-0.849016\pi$
$809$	−50.6056	−1.77920	−0.889598	+	0.456744i	$0.849016\pi$
$810$	0	0
$811$	42.4222	1.48965	0.744823	−	0.667263i	$-0.232534\pi$
$811$	42.4222	1.48965	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	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	0	0
$823$	3.81665	0.133040	0.0665201	−	0.997785i	$-0.478810\pi$
$823$	3.81665	0.133040	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	$-0.300121\pi$
$827$	33.7889	1.17496	0.587478	+	0.809240i	$0.300121\pi$
$828$	0	0
$829$	−27.2111	−0.945081	−0.472540	−	0.881309i	$-0.656663\pi$
$829$	−27.2111	−0.945081	−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	$-0.605140\pi$
$839$	−18.7889	−0.648665	−0.324332	+	0.945943i	$0.605140\pi$
$840$	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	$-0.418523\pi$
$853$	14.7889	0.506362	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	$-0.218923\pi$
$857$	45.2389	1.54533	0.772665	+	0.634814i	$0.218923\pi$
$858$	0	0
$859$	6.02776	0.205664	0.102832	−	0.994699i	$-0.467210\pi$
$859$	6.02776	0.205664	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	$-0.586604\pi$
$863$	−15.7889	−0.537460	−0.268730	+	0.963216i	$0.586604\pi$
$864$	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	$-0.905937\pi$
$877$	−56.6611	−1.91331	−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	$-0.314912\pi$
$881$	32.6056	1.09851	0.549254	+	0.835655i	$0.314912\pi$
$882$	0	0
$883$	−5.81665	−0.195746	−0.0978730	−	0.995199i	$-0.531204\pi$
$883$	−5.81665	−0.195746	−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	$-0.568028\pi$
$887$	−12.6333	−0.424185	−0.212092	+	0.977250i	$0.568028\pi$
$888$	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	$-0.331464\pi$
$907$	30.4222	1.01015	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	$-0.671836\pi$
$911$	−31.0278	−1.02800	−0.513998	+	0.857792i	$0.671836\pi$
$912$	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	$-0.512720\pi$
$919$	−2.42221	−0.0799012	−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	$-0.643764\pi$
$929$	−26.6056	−0.872900	−0.436450	+	0.899729i	$0.643764\pi$
$930$	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	$-0.355837\pi$
$937$	26.7889	0.875155	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	$-0.653324\pi$
$941$	−28.4222	−0.926537	−0.463269	+	0.886218i	$0.653324\pi$
$942$	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	$-0.718450\pi$
$947$	−39.0000	−1.26733	−0.633665	+	0.773608i	$0.718450\pi$
$948$	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	$-0.643177\pi$
$953$	−26.8444	−0.869576	−0.434788	+	0.900533i	$0.643177\pi$
$954$	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	$-0.202730\pi$
$967$	50.0000	1.60789	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	$-0.406736\pi$
$971$	18.0000	0.577647	0.288824	+	0.957382i	$0.406736\pi$
$972$	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	$-0.495983\pi$
$977$	0.788897	0.0252391	0.0126195	+	0.999920i	$0.495983\pi$
$978$	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	$-0.503215\pi$
$983$	−0.633308	−0.0201994	−0.0100997	+	0.999949i	$0.503215\pi$
$984$	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	$-0.798976\pi$
$991$	−50.8167	−1.61424	−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	$-0.174735\pi$
$997$	53.8722	1.70615	0.853074	+	0.521789i	$0.174735\pi$
$998$	35.9638	1.13842
$999$	0	0

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	135.a (trivial)

\(f(q)\)	\(=\)	\(q-1.30278 q^{2} -0.302776 q^{4} -1.00000 q^{5} +4.60555 q^{7} +3.00000 q^{8} +O(q^{10})\)	\(q-1.30278 q^{2} -0.302776 q^{4} -1.00000 q^{5} +4.60555 q^{7} +3.00000 q^{8} +1.30278 q^{10} +2.60555 q^{11} -0.605551 q^{13} -6.00000 q^{14} -3.30278 q^{16} +5.60555 q^{17} -3.60555 q^{19} +0.302776 q^{20} -3.39445 q^{22} +3.00000 q^{23} +1.00000 q^{25} +0.788897 q^{26} -1.39445 q^{28} -8.60555 q^{29} +1.60555 q^{31} -1.69722 q^{32} -7.30278 q^{34} -4.60555 q^{35} +2.00000 q^{37} +4.69722 q^{38} -3.00000 q^{40} -2.60555 q^{41} -6.60555 q^{43} -0.788897 q^{44} -3.90833 q^{46} -5.21110 q^{47} +14.2111 q^{49} -1.30278 q^{50} +0.183346 q^{52} +5.60555 q^{53} -2.60555 q^{55} +13.8167 q^{56} +11.2111 q^{58} +8.60555 q^{59} +10.2111 q^{61} -2.09167 q^{62} +8.81665 q^{64} +0.605551 q^{65} -15.2111 q^{67} -1.69722 q^{68} +6.00000 q^{70} -14.6056 q^{71} +5.39445 q^{73} -2.60555 q^{74} +1.09167 q^{76} +12.0000 q^{77} -4.39445 q^{79} +3.30278 q^{80} +3.39445 q^{82} -3.00000 q^{83} -5.60555 q^{85} +8.60555 q^{86} +7.81665 q^{88} -7.81665 q^{89} -2.78890 q^{91} -0.908327 q^{92} +6.78890 q^{94} +3.60555 q^{95} +8.00000 q^{97} -18.5139 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 3 q^{4} - 2 q^{5} + 2 q^{7} + 6 q^{8}+O(q^{10}) \) 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

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