LMFDB - Embedded newform 8100.2.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8100,2,Mod(1,8100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8100, 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("8100.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8100.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:	$64.6788256372$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	8100.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.85410 q^{7} +O(q^{10})$	$q+2.85410 q^{7} -1.85410 q^{11} -0.854102 q^{13} +1.14590 q^{17} +2.00000 q^{19} -4.85410 q^{23} -3.70820 q^{29} +2.70820 q^{31} +5.85410 q^{37} +11.5623 q^{41} -0.854102 q^{43} -6.70820 q^{47} +1.14590 q^{49} +4.85410 q^{53} -1.14590 q^{59} +0.854102 q^{61} +7.00000 q^{67} -9.00000 q^{71} -2.70820 q^{73} -5.29180 q^{77} +11.7082 q^{79} +6.70820 q^{83} +12.0000 q^{89} -2.43769 q^{91} +10.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{7}+O(q^{10}) $ 2 * q - q^7	$ 2 q - q^{7} + 3 q^{11} + 5 q^{13} + 9 q^{17} + 4 q^{19} - 3 q^{23} + 6 q^{29} - 8 q^{31} + 5 q^{37} + 3 q^{41} + 5 q^{43} + 9 q^{49} + 3 q^{53} - 9 q^{59} - 5 q^{61} + 14 q^{67} - 18 q^{71} + 8 q^{73} - 24 q^{77} + 10 q^{79} + 24 q^{89} - 25 q^{91} + 20 q^{97}+O(q^{100}) $ 2 * q - q^7 + 3 * q^11 + 5 * q^13 + 9 * q^17 + 4 * q^19 - 3 * q^23 + 6 * q^29 - 8 * q^31 + 5 * q^37 + 3 * q^41 + 5 * q^43 + 9 * q^49 + 3 * q^53 - 9 * q^59 - 5 * q^61 + 14 * q^67 - 18 * q^71 + 8 * q^73 - 24 * q^77 + 10 * q^79 + 24 * q^89 - 25 * q^91 + 20 * q^97

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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.85410	1.07875	0.539375	−	0.842066i	$-0.318661\pi$
$7$	2.85410	1.07875	0.539375	+	0.842066i	$0.318661\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.85410	−0.559033	−0.279516	−	0.960141i	$-0.590174\pi$
$11$	−1.85410	−0.559033	−0.279516	+	0.960141i	$0.590174\pi$
$12$	0	0
$13$	−0.854102	−0.236885	−0.118443	−	0.992961i	$-0.537790\pi$
$13$	−0.854102	−0.236885	−0.118443	+	0.992961i	$0.537790\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.14590	0.277921	0.138961	−	0.990298i	$-0.455624\pi$
$17$	1.14590	0.277921	0.138961	+	0.990298i	$0.455624\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.85410	−1.01215	−0.506075	−	0.862489i	$-0.668904\pi$
$23$	−4.85410	−1.01215	−0.506075	+	0.862489i	$0.668904\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.70820	−0.688596	−0.344298	−	0.938860i	$-0.611883\pi$
$29$	−3.70820	−0.688596	−0.344298	+	0.938860i	$0.611883\pi$
$30$	0	0
$31$	2.70820	0.486408	0.243204	−	0.969975i	$-0.421802\pi$
$31$	2.70820	0.486408	0.243204	+	0.969975i	$0.421802\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.85410	0.962408	0.481204	−	0.876609i	$-0.340200\pi$
$37$	5.85410	0.962408	0.481204	+	0.876609i	$0.340200\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	11.5623	1.80573	0.902864	−	0.429925i	$-0.141460\pi$
$41$	11.5623	1.80573	0.902864	+	0.429925i	$0.141460\pi$
$42$	0	0
$43$	−0.854102	−0.130249	−0.0651247	−	0.997877i	$-0.520745\pi$
$43$	−0.854102	−0.130249	−0.0651247	+	0.997877i	$0.520745\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.70820	−0.978492	−0.489246	−	0.872146i	$-0.662728\pi$
$47$	−6.70820	−0.978492	−0.489246	+	0.872146i	$0.662728\pi$
$48$	0	0
$49$	1.14590	0.163700
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.85410	0.666762	0.333381	−	0.942792i	$-0.391810\pi$
$53$	4.85410	0.666762	0.333381	+	0.942792i	$0.391810\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.14590	−0.149183	−0.0745916	−	0.997214i	$-0.523765\pi$
$59$	−1.14590	−0.149183	−0.0745916	+	0.997214i	$0.523765\pi$
$60$	0	0
$61$	0.854102	0.109357	0.0546783	−	0.998504i	$-0.482587\pi$
$61$	0.854102	0.109357	0.0546783	+	0.998504i	$0.482587\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.00000	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$67$	7.00000	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−9.00000	−1.06810	−0.534052	−	0.845452i	$-0.679331\pi$
$71$	−9.00000	−1.06810	−0.534052	+	0.845452i	$0.679331\pi$
$72$	0	0
$73$	−2.70820	−0.316971	−0.158486	−	0.987361i	$-0.550661\pi$
$73$	−2.70820	−0.316971	−0.158486	+	0.987361i	$0.550661\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−5.29180	−0.603056
$78$	0	0
$79$	11.7082	1.31728	0.658638	−	0.752460i	$-0.271133\pi$
$79$	11.7082	1.31728	0.658638	+	0.752460i	$0.271133\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.70820	0.736321	0.368161	−	0.929762i	$-0.379988\pi$
$83$	6.70820	0.736321	0.368161	+	0.929762i	$0.379988\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	−2.43769	−0.255540
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−4.85410	−0.483001	−0.241501	−	0.970401i	$-0.577640\pi$
$101$	−4.85410	−0.483001	−0.241501	+	0.970401i	$0.577640\pi$
$102$	0	0
$103$	−0.145898	−0.0143758	−0.00718788	−	0.999974i	$-0.502288\pi$
$103$	−0.145898	−0.0143758	−0.00718788	+	0.999974i	$0.502288\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.7082	1.80859	0.904295	−	0.426908i	$-0.140397\pi$
$107$	18.7082	1.80859	0.904295	+	0.426908i	$0.140397\pi$
$108$	0	0
$109$	3.85410	0.369156	0.184578	−	0.982818i	$-0.440908\pi$
$109$	3.85410	0.369156	0.184578	+	0.982818i	$0.440908\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1.85410	−0.174419	−0.0872096	−	0.996190i	$-0.527795\pi$
$113$	−1.85410	−0.174419	−0.0872096	+	0.996190i	$0.527795\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.27051	0.299807
$120$	0	0
$121$	−7.56231	−0.687482
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	7.00000	0.621150	0.310575	−	0.950549i	$-0.399478\pi$
$127$	7.00000	0.621150	0.310575	+	0.950549i	$0.399478\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2.29180	0.200235	0.100118	−	0.994976i	$-0.468078\pi$
$131$	2.29180	0.200235	0.100118	+	0.994976i	$0.468078\pi$
$132$	0	0
$133$	5.70820	0.494964
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.56231	0.218913	0.109456	−	0.993992i	$-0.465089\pi$
$137$	2.56231	0.218913	0.109456	+	0.993992i	$0.465089\pi$
$138$	0	0
$139$	17.2705	1.46487	0.732433	−	0.680839i	$-0.238385\pi$
$139$	17.2705	1.46487	0.732433	+	0.680839i	$0.238385\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.58359	0.132427
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−15.0000	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	0	0
$151$	−8.14590	−0.662904	−0.331452	−	0.943472i	$-0.607538\pi$
$151$	−8.14590	−0.662904	−0.331452	+	0.943472i	$0.607538\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	16.7082	1.33346	0.666730	−	0.745299i	$-0.267693\pi$
$157$	16.7082	1.33346	0.666730	+	0.745299i	$0.267693\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−13.8541	−1.09186
$162$	0	0
$163$	14.4164	1.12918	0.564590	−	0.825371i	$-0.309034\pi$
$163$	14.4164	1.12918	0.564590	+	0.825371i	$0.309034\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−22.8541	−1.76850	−0.884252	−	0.467011i	$-0.845331\pi$
$167$	−22.8541	−1.76850	−0.884252	+	0.467011i	$0.845331\pi$
$168$	0	0
$169$	−12.2705	−0.943885
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−15.7082	−1.19427	−0.597136	−	0.802140i	$-0.703695\pi$
$173$	−15.7082	−1.19427	−0.597136	+	0.802140i	$0.703695\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	5.29180	0.395527	0.197764	−	0.980250i	$-0.436632\pi$
$179$	5.29180	0.395527	0.197764	+	0.980250i	$0.436632\pi$
$180$	0	0
$181$	−14.4164	−1.07156	−0.535782	−	0.844357i	$-0.679983\pi$
$181$	−14.4164	−1.07156	−0.535782	+	0.844357i	$0.679983\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−2.12461	−0.155367
$188$	0	0
$189$	0	0
$190$	0	0
$191$	19.4164	1.40492	0.702461	−	0.711722i	$-0.252085\pi$
$191$	19.4164	1.40492	0.702461	+	0.711722i	$0.252085\pi$
$192$	0	0
$193$	16.7082	1.20268	0.601341	−	0.798992i	$-0.294633\pi$
$193$	16.7082	1.20268	0.601341	+	0.798992i	$0.294633\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	25.8541	1.84203	0.921014	−	0.389529i	$-0.127362\pi$
$197$	25.8541	1.84203	0.921014	+	0.389529i	$0.127362\pi$
$198$	0	0
$199$	15.8541	1.12387	0.561934	−	0.827182i	$-0.310058\pi$
$199$	15.8541	1.12387	0.561934	+	0.827182i	$0.310058\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−10.5836	−0.742823
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.70820	−0.256502
$210$	0	0
$211$	−22.7082	−1.56330	−0.781649	−	0.623719i	$-0.785621\pi$
$211$	−22.7082	−1.56330	−0.781649	+	0.623719i	$0.785621\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	7.72949	0.524712
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−0.978714	−0.0658354
$222$	0	0
$223$	3.29180	0.220435	0.110217	−	0.993907i	$-0.464845\pi$
$223$	3.29180	0.220435	0.110217	+	0.993907i	$0.464845\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.00000	0.199117	0.0995585	−	0.995032i	$-0.468257\pi$
$227$	3.00000	0.199117	0.0995585	+	0.995032i	$0.468257\pi$
$228$	0	0
$229$	−8.41641	−0.556172	−0.278086	−	0.960556i	$-0.589700\pi$
$229$	−8.41641	−0.556172	−0.278086	+	0.960556i	$0.589700\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	20.1246	1.31841	0.659204	−	0.751965i	$-0.270894\pi$
$233$	20.1246	1.31841	0.659204	+	0.751965i	$0.270894\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−13.4164	−0.867835	−0.433918	−	0.900953i	$-0.642869\pi$
$239$	−13.4164	−0.867835	−0.433918	+	0.900953i	$0.642869\pi$
$240$	0	0
$241$	−17.4164	−1.12189	−0.560945	−	0.827853i	$-0.689562\pi$
$241$	−17.4164	−1.12189	−0.560945	+	0.827853i	$0.689562\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.70820	−0.108690
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−10.8541	−0.685105	−0.342552	−	0.939499i	$-0.611291\pi$
$251$	−10.8541	−0.685105	−0.342552	+	0.939499i	$0.611291\pi$
$252$	0	0
$253$	9.00000	0.565825
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−25.4164	−1.58543	−0.792716	−	0.609591i	$-0.791334\pi$
$257$	−25.4164	−1.58543	−0.792716	+	0.609591i	$0.791334\pi$
$258$	0	0
$259$	16.7082	1.03820
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2.29180	0.141318	0.0706591	−	0.997501i	$-0.477490\pi$
$263$	2.29180	0.141318	0.0706591	+	0.997501i	$0.477490\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	27.2705	1.66271	0.831356	−	0.555740i	$-0.187565\pi$
$269$	27.2705	1.66271	0.831356	+	0.555740i	$0.187565\pi$
$270$	0	0
$271$	−8.41641	−0.511260	−0.255630	−	0.966775i	$-0.582283\pi$
$271$	−8.41641	−0.511260	−0.255630	+	0.966775i	$0.582283\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	23.4164	1.40696	0.703478	−	0.710717i	$-0.251629\pi$
$277$	23.4164	1.40696	0.703478	+	0.710717i	$0.251629\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6.70820	0.400178	0.200089	−	0.979778i	$-0.435877\pi$
$281$	6.70820	0.400178	0.200089	+	0.979778i	$0.435877\pi$
$282$	0	0
$283$	11.8541	0.704653	0.352327	−	0.935877i	$-0.385391\pi$
$283$	11.8541	0.704653	0.352327	+	0.935877i	$0.385391\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	33.0000	1.94793
$288$	0	0
$289$	−15.6869	−0.922760
$290$	0	0
$291$	0	0
$292$	0	0
$293$	20.1246	1.17569	0.587846	−	0.808973i	$-0.299976\pi$
$293$	20.1246	1.17569	0.587846	+	0.808973i	$0.299976\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.14590	0.239763
$300$	0	0
$301$	−2.43769	−0.140506
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	10.0000	0.570730	0.285365	−	0.958419i	$-0.407885\pi$
$307$	10.0000	0.570730	0.285365	+	0.958419i	$0.407885\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−0.708204	−0.0401586	−0.0200793	−	0.999798i	$-0.506392\pi$
$311$	−0.708204	−0.0401586	−0.0200793	+	0.999798i	$0.506392\pi$
$312$	0	0
$313$	20.8541	1.17874	0.589372	−	0.807862i	$-0.299375\pi$
$313$	20.8541	1.17874	0.589372	+	0.807862i	$0.299375\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.58359	−0.0889434	−0.0444717	−	0.999011i	$-0.514160\pi$
$317$	−1.58359	−0.0889434	−0.0444717	+	0.999011i	$0.514160\pi$
$318$	0	0
$319$	6.87539	0.384948
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.29180	0.127519
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−19.1459	−1.05555
$330$	0	0
$331$	−17.4164	−0.957292	−0.478646	−	0.878008i	$-0.658872\pi$
$331$	−17.4164	−0.957292	−0.478646	+	0.878008i	$0.658872\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	18.2918	0.996418	0.498209	−	0.867057i	$-0.333991\pi$
$337$	18.2918	0.996418	0.498209	+	0.867057i	$0.333991\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−5.02129	−0.271918
$342$	0	0
$343$	−16.7082	−0.902158
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−35.5623	−1.90908	−0.954542	−	0.298075i	$-0.903655\pi$
$347$	−35.5623	−1.90908	−0.954542	+	0.298075i	$0.903655\pi$
$348$	0	0
$349$	23.7082	1.26907	0.634536	−	0.772894i	$-0.281191\pi$
$349$	23.7082	1.26907	0.634536	+	0.772894i	$0.281191\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−17.5623	−0.934747	−0.467374	−	0.884060i	$-0.654800\pi$
$353$	−17.5623	−0.934747	−0.467374	+	0.884060i	$0.654800\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	16.1459	0.852148	0.426074	−	0.904688i	$-0.359896\pi$
$359$	16.1459	0.852148	0.426074	+	0.904688i	$0.359896\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	9.56231	0.499148	0.249574	−	0.968356i	$-0.419709\pi$
$367$	9.56231	0.499148	0.249574	+	0.968356i	$0.419709\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	13.8541	0.719269
$372$	0	0
$373$	−16.1246	−0.834901	−0.417450	−	0.908700i	$-0.637076\pi$
$373$	−16.1246	−0.834901	−0.417450	+	0.908700i	$0.637076\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.16718	0.163118
$378$	0	0
$379$	20.2705	1.04123	0.520613	−	0.853793i	$-0.325703\pi$
$379$	20.2705	1.04123	0.520613	+	0.853793i	$0.325703\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−30.9787	−1.58294	−0.791469	−	0.611209i	$-0.790683\pi$
$383$	−30.9787	−1.58294	−0.791469	+	0.611209i	$0.790683\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−6.70820	−0.340119	−0.170060	−	0.985434i	$-0.554396\pi$
$389$	−6.70820	−0.340119	−0.170060	+	0.985434i	$0.554396\pi$
$390$	0	0
$391$	−5.56231	−0.281298
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	31.5410	1.58300	0.791499	−	0.611170i	$-0.209301\pi$
$397$	31.5410	1.58300	0.791499	+	0.611170i	$0.209301\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	35.1246	1.75404	0.877020	−	0.480454i	$-0.159528\pi$
$401$	35.1246	1.75404	0.877020	+	0.480454i	$0.159528\pi$
$402$	0	0
$403$	−2.31308	−0.115223
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−10.8541	−0.538018
$408$	0	0
$409$	14.2705	0.705631	0.352816	−	0.935693i	$-0.385224\pi$
$409$	14.2705	0.705631	0.352816	+	0.935693i	$0.385224\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−3.27051	−0.160931
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4.14590	−0.202540	−0.101270	−	0.994859i	$-0.532291\pi$
$419$	−4.14590	−0.202540	−0.101270	+	0.994859i	$0.532291\pi$
$420$	0	0
$421$	5.70820	0.278201	0.139100	−	0.990278i	$-0.455579\pi$
$421$	5.70820	0.278201	0.139100	+	0.990278i	$0.455579\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2.43769	0.117968
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−21.0000	−1.01153	−0.505767	−	0.862670i	$-0.668791\pi$
$431$	−21.0000	−1.01153	−0.505767	+	0.862670i	$0.668791\pi$
$432$	0	0
$433$	37.7082	1.81214	0.906070	−	0.423127i	$-0.139068\pi$
$433$	37.7082	1.81214	0.906070	+	0.423127i	$0.139068\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−9.70820	−0.464406
$438$	0	0
$439$	−9.56231	−0.456384	−0.228192	−	0.973616i	$-0.573281\pi$
$439$	−9.56231	−0.456384	−0.228192	+	0.973616i	$0.573281\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	19.1459	0.909649	0.454825	−	0.890581i	$-0.349702\pi$
$443$	19.1459	0.909649	0.454825	+	0.890581i	$0.349702\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	7.14590	0.337236	0.168618	−	0.985681i	$-0.446070\pi$
$449$	7.14590	0.337236	0.168618	+	0.985681i	$0.446070\pi$
$450$	0	0
$451$	−21.4377	−1.00946
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−23.5623	−1.09741	−0.548703	−	0.836017i	$-0.684878\pi$
$461$	−23.5623	−1.09741	−0.548703	+	0.836017i	$0.684878\pi$
$462$	0	0
$463$	−3.14590	−0.146202	−0.0731011	−	0.997325i	$-0.523290\pi$
$463$	−3.14590	−0.146202	−0.0731011	+	0.997325i	$0.523290\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	0	0
$469$	19.9787	0.922531
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.58359	0.0728136
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−16.1459	−0.737725	−0.368862	−	0.929484i	$-0.620253\pi$
$479$	−16.1459	−0.737725	−0.368862	+	0.929484i	$0.620253\pi$
$480$	0	0
$481$	−5.00000	−0.227980
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−3.41641	−0.154812	−0.0774061	−	0.997000i	$-0.524664\pi$
$487$	−3.41641	−0.154812	−0.0774061	+	0.997000i	$0.524664\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	17.8328	0.804784	0.402392	−	0.915468i	$-0.368179\pi$
$491$	17.8328	0.804784	0.402392	+	0.915468i	$0.368179\pi$
$492$	0	0
$493$	−4.24922	−0.191375
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−25.6869	−1.15222
$498$	0	0
$499$	5.43769	0.243425	0.121712	−	0.992565i	$-0.461161\pi$
$499$	5.43769	0.243425	0.121712	+	0.992565i	$0.461161\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	24.2705	1.08217	0.541084	−	0.840968i	$-0.318014\pi$
$503$	24.2705	1.08217	0.541084	+	0.840968i	$0.318014\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−21.0000	−0.930809	−0.465404	−	0.885098i	$-0.654091\pi$
$509$	−21.0000	−0.930809	−0.465404	+	0.885098i	$0.654091\pi$
$510$	0	0
$511$	−7.72949	−0.341933
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	12.4377	0.547009
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−35.1246	−1.53884	−0.769419	−	0.638745i	$-0.779454\pi$
$521$	−35.1246	−1.53884	−0.769419	+	0.638745i	$0.779454\pi$
$522$	0	0
$523$	−18.4164	−0.805293	−0.402647	−	0.915355i	$-0.631910\pi$
$523$	−18.4164	−0.805293	−0.402647	+	0.915355i	$0.631910\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.10333	0.135183
$528$	0	0
$529$	0.562306	0.0244481
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−9.87539	−0.427751
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.12461	−0.0915135
$540$	0	0
$541$	40.3951	1.73672	0.868361	−	0.495933i	$-0.165174\pi$
$541$	40.3951	1.73672	0.868361	+	0.495933i	$0.165174\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	38.4164	1.64257	0.821283	−	0.570520i	$-0.193258\pi$
$547$	38.4164	1.64257	0.821283	+	0.570520i	$0.193258\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−7.41641	−0.315950
$552$	0	0
$553$	33.4164	1.42101
$554$	0	0
$555$	0	0
$556$	0	0
$557$	40.2492	1.70541	0.852707	−	0.522389i	$-0.174959\pi$
$557$	40.2492	1.70541	0.852707	+	0.522389i	$0.174959\pi$
$558$	0	0
$559$	0.729490	0.0308541
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−36.7082	−1.54707	−0.773533	−	0.633756i	$-0.781512\pi$
$563$	−36.7082	−1.54707	−0.773533	+	0.633756i	$0.781512\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	3.70820	0.155456	0.0777280	−	0.996975i	$-0.475233\pi$
$569$	3.70820	0.155456	0.0777280	+	0.996975i	$0.475233\pi$
$570$	0	0
$571$	10.8328	0.453339	0.226670	−	0.973972i	$-0.427216\pi$
$571$	10.8328	0.453339	0.226670	+	0.973972i	$0.427216\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−0.416408	−0.0173353	−0.00866764	−	0.999962i	$-0.502759\pi$
$577$	−0.416408	−0.0173353	−0.00866764	+	0.999962i	$0.502759\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	19.1459	0.794306
$582$	0	0
$583$	−9.00000	−0.372742
$584$	0	0
$585$	0	0
$586$	0	0
$587$	15.0000	0.619116	0.309558	−	0.950881i	$-0.399819\pi$
$587$	15.0000	0.619116	0.309558	+	0.950881i	$0.399819\pi$
$588$	0	0
$589$	5.41641	0.223179
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−3.27051	−0.134304	−0.0671519	−	0.997743i	$-0.521391\pi$
$593$	−3.27051	−0.134304	−0.0671519	+	0.997743i	$0.521391\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	20.5623	0.840153	0.420077	−	0.907489i	$-0.362003\pi$
$599$	20.5623	0.840153	0.420077	+	0.907489i	$0.362003\pi$
$600$	0	0
$601$	9.85410	0.401957	0.200979	−	0.979596i	$-0.435588\pi$
$601$	9.85410	0.401957	0.200979	+	0.979596i	$0.435588\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	28.5410	1.15844	0.579222	−	0.815170i	$-0.303356\pi$
$607$	28.5410	1.15844	0.579222	+	0.815170i	$0.303356\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.72949	0.231790
$612$	0	0
$613$	−11.7082	−0.472890	−0.236445	−	0.971645i	$-0.575982\pi$
$613$	−11.7082	−0.472890	−0.236445	+	0.971645i	$0.575982\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	20.1246	0.810186	0.405093	−	0.914275i	$-0.367239\pi$
$617$	20.1246	0.810186	0.405093	+	0.914275i	$0.367239\pi$
$618$	0	0
$619$	33.4164	1.34312	0.671559	−	0.740951i	$-0.265625\pi$
$619$	33.4164	1.34312	0.671559	+	0.740951i	$0.265625\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	34.2492	1.37217
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6.70820	0.267474
$630$	0	0
$631$	0.145898	0.00580811	0.00290405	−	0.999996i	$-0.499076\pi$
$631$	0.145898	0.00580811	0.00290405	+	0.999996i	$0.499076\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−0.978714	−0.0387781
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−11.5623	−0.456684	−0.228342	−	0.973581i	$-0.573330\pi$
$641$	−11.5623	−0.456684	−0.228342	+	0.973581i	$0.573330\pi$
$642$	0	0
$643$	−37.5623	−1.48131	−0.740656	−	0.671884i	$-0.765485\pi$
$643$	−37.5623	−1.48131	−0.740656	+	0.671884i	$0.765485\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24.7082	0.971380	0.485690	−	0.874131i	$-0.338568\pi$
$647$	24.7082	0.971380	0.485690	+	0.874131i	$0.338568\pi$
$648$	0	0
$649$	2.12461	0.0833983
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.14590	0.162242	0.0811208	−	0.996704i	$-0.474150\pi$
$653$	4.14590	0.162242	0.0811208	+	0.996704i	$0.474150\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−18.2705	−0.711718	−0.355859	−	0.934540i	$-0.615812\pi$
$659$	−18.2705	−0.711718	−0.355859	+	0.934540i	$0.615812\pi$
$660$	0	0
$661$	25.8328	1.00478	0.502390	−	0.864641i	$-0.332454\pi$
$661$	25.8328	1.00478	0.502390	+	0.864641i	$0.332454\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	18.0000	0.696963
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−1.58359	−0.0611339
$672$	0	0
$673$	−2.70820	−0.104394	−0.0521968	−	0.998637i	$-0.516622\pi$
$673$	−2.70820	−0.104394	−0.0521968	+	0.998637i	$0.516622\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−16.4164	−0.630934	−0.315467	−	0.948937i	$-0.602161\pi$
$677$	−16.4164	−0.630934	−0.315467	+	0.948937i	$0.602161\pi$
$678$	0	0
$679$	28.5410	1.09530
$680$	0	0
$681$	0	0
$682$	0	0
$683$	29.3951	1.12477	0.562387	−	0.826874i	$-0.309883\pi$
$683$	29.3951	1.12477	0.562387	+	0.826874i	$0.309883\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−4.14590	−0.157946
$690$	0	0
$691$	10.1246	0.385158	0.192579	−	0.981281i	$-0.438315\pi$
$691$	10.1246	0.385158	0.192579	+	0.981281i	$0.438315\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	13.2492	0.501850
$698$	0	0
$699$	0	0
$700$	0	0
$701$	15.7082	0.593291	0.296645	−	0.954988i	$-0.404132\pi$
$701$	15.7082	0.593291	0.296645	+	0.954988i	$0.404132\pi$
$702$	0	0
$703$	11.7082	0.441583
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−13.8541	−0.521037
$708$	0	0
$709$	−19.0000	−0.713560	−0.356780	−	0.934188i	$-0.616125\pi$
$709$	−19.0000	−0.713560	−0.356780	+	0.934188i	$0.616125\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−13.1459	−0.492318
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	20.5623	0.766845	0.383422	−	0.923573i	$-0.374745\pi$
$719$	20.5623	0.766845	0.383422	+	0.923573i	$0.374745\pi$
$720$	0	0
$721$	−0.416408	−0.0155078
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−37.5623	−1.39311	−0.696554	−	0.717504i	$-0.745285\pi$
$727$	−37.5623	−1.39311	−0.696554	+	0.717504i	$0.745285\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−0.978714	−0.0361990
$732$	0	0
$733$	36.5623	1.35046	0.675230	−	0.737607i	$-0.264044\pi$
$733$	36.5623	1.35046	0.675230	+	0.737607i	$0.264044\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−12.9787	−0.478077
$738$	0	0
$739$	5.43769	0.200029	0.100014	−	0.994986i	$-0.468111\pi$
$739$	5.43769	0.200029	0.100014	+	0.994986i	$0.468111\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−0.875388	−0.0321149	−0.0160574	−	0.999871i	$-0.505111\pi$
$743$	−0.875388	−0.0321149	−0.0160574	+	0.999871i	$0.505111\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	53.3951	1.95102
$750$	0	0
$751$	−40.9787	−1.49533	−0.747667	−	0.664074i	$-0.768826\pi$
$751$	−40.9787	−1.49533	−0.747667	+	0.664074i	$0.768826\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	34.2705	1.24558	0.622791	−	0.782388i	$-0.285999\pi$
$757$	34.2705	1.24558	0.622791	+	0.782388i	$0.285999\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	34.8541	1.26346	0.631730	−	0.775188i	$-0.282345\pi$
$761$	34.8541	1.26346	0.631730	+	0.775188i	$0.282345\pi$
$762$	0	0
$763$	11.0000	0.398227
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0.978714	0.0353393
$768$	0	0
$769$	−26.1459	−0.942845	−0.471423	−	0.881907i	$-0.656259\pi$
$769$	−26.1459	−0.942845	−0.471423	+	0.881907i	$0.656259\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−33.5410	−1.20639	−0.603193	−	0.797595i	$-0.706105\pi$
$773$	−33.5410	−1.20639	−0.603193	+	0.797595i	$0.706105\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	23.1246	0.828525
$780$	0	0
$781$	16.6869	0.597105
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	2.14590	0.0764930	0.0382465	−	0.999268i	$-0.487823\pi$
$787$	2.14590	0.0764930	0.0382465	+	0.999268i	$0.487823\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−5.29180	−0.188155
$792$	0	0
$793$	−0.729490	−0.0259050
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−9.27051	−0.328378	−0.164189	−	0.986429i	$-0.552501\pi$
$797$	−9.27051	−0.328378	−0.164189	+	0.986429i	$0.552501\pi$
$798$	0	0
$799$	−7.68692	−0.271944
$800$	0	0
$801$	0	0
$802$	0	0
$803$	5.02129	0.177197
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−8.12461	−0.285646	−0.142823	−	0.989748i	$-0.545618\pi$
$809$	−8.12461	−0.285646	−0.142823	+	0.989748i	$0.545618\pi$
$810$	0	0
$811$	−16.9787	−0.596203	−0.298102	−	0.954534i	$-0.596353\pi$
$811$	−16.9787	−0.596203	−0.298102	+	0.954534i	$0.596353\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−1.70820	−0.0597625
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−4.41641	−0.154134	−0.0770668	−	0.997026i	$-0.524555\pi$
$821$	−4.41641	−0.154134	−0.0770668	+	0.997026i	$0.524555\pi$
$822$	0	0
$823$	−17.5410	−0.611442	−0.305721	−	0.952121i	$-0.598897\pi$
$823$	−17.5410	−0.611442	−0.305721	+	0.952121i	$0.598897\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−44.3951	−1.54377	−0.771885	−	0.635762i	$-0.780686\pi$
$827$	−44.3951	−1.54377	−0.771885	+	0.635762i	$0.780686\pi$
$828$	0	0
$829$	−5.41641	−0.188120	−0.0940598	−	0.995567i	$-0.529984\pi$
$829$	−5.41641	−0.188120	−0.0940598	+	0.995567i	$0.529984\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.31308	0.0454956
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−53.1246	−1.83407	−0.917033	−	0.398812i	$-0.869423\pi$
$839$	−53.1246	−1.83407	−0.917033	+	0.398812i	$0.869423\pi$
$840$	0	0
$841$	−15.2492	−0.525835
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−21.5836	−0.741621
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−28.4164	−0.974102
$852$	0	0
$853$	24.1246	0.826011	0.413005	−	0.910729i	$-0.364479\pi$
$853$	24.1246	0.826011	0.413005	+	0.910729i	$0.364479\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−4.14590	−0.141621	−0.0708106	−	0.997490i	$-0.522559\pi$
$857$	−4.14590	−0.141621	−0.0708106	+	0.997490i	$0.522559\pi$
$858$	0	0
$859$	9.58359	0.326988	0.163494	−	0.986544i	$-0.447724\pi$
$859$	9.58359	0.326988	0.163494	+	0.986544i	$0.447724\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	15.2705	0.519814	0.259907	−	0.965634i	$-0.416308\pi$
$863$	15.2705	0.519814	0.259907	+	0.965634i	$0.416308\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−21.7082	−0.736400
$870$	0	0
$871$	−5.97871	−0.202581
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−14.2705	−0.481881	−0.240940	−	0.970540i	$-0.577456\pi$
$877$	−14.2705	−0.481881	−0.240940	+	0.970540i	$0.577456\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−26.1246	−0.880161	−0.440080	−	0.897958i	$-0.645050\pi$
$881$	−26.1246	−0.880161	−0.440080	+	0.897958i	$0.645050\pi$
$882$	0	0
$883$	−31.1246	−1.04743	−0.523713	−	0.851895i	$-0.675454\pi$
$883$	−31.1246	−1.04743	−0.523713	+	0.851895i	$0.675454\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−0.978714	−0.0328620	−0.0164310	−	0.999865i	$-0.505230\pi$
$887$	−0.978714	−0.0328620	−0.0164310	+	0.999865i	$0.505230\pi$
$888$	0	0
$889$	19.9787	0.670065
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−13.4164	−0.448963
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−10.0426	−0.334939
$900$	0	0
$901$	5.56231	0.185307
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	13.9787	0.464156	0.232078	−	0.972697i	$-0.425448\pi$
$907$	13.9787	0.464156	0.232078	+	0.972697i	$0.425448\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	26.1246	0.865547	0.432774	−	0.901503i	$-0.357535\pi$
$911$	26.1246	0.865547	0.432774	+	0.901503i	$0.357535\pi$
$912$	0	0
$913$	−12.4377	−0.411628
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.54102	0.216003
$918$	0	0
$919$	−24.5623	−0.810236	−0.405118	−	0.914264i	$-0.632770\pi$
$919$	−24.5623	−0.810236	−0.405118	+	0.914264i	$0.632770\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	7.68692	0.253018
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	41.2918	1.35474	0.677370	−	0.735643i	$-0.263120\pi$
$929$	41.2918	1.35474	0.677370	+	0.735643i	$0.263120\pi$
$930$	0	0
$931$	2.29180	0.0751106
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−43.8328	−1.43196	−0.715978	−	0.698123i	$-0.754019\pi$
$937$	−43.8328	−1.43196	−0.715978	+	0.698123i	$0.754019\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−29.3951	−0.958254	−0.479127	−	0.877746i	$-0.659047\pi$
$941$	−29.3951	−0.958254	−0.479127	+	0.877746i	$0.659047\pi$
$942$	0	0
$943$	−56.1246	−1.82767
$944$	0	0
$945$	0	0
$946$	0	0
$947$	46.4164	1.50833	0.754165	−	0.656685i	$-0.228042\pi$
$947$	46.4164	1.50833	0.754165	+	0.656685i	$0.228042\pi$
$948$	0	0
$949$	2.31308	0.0750858
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−3.16718	−0.102595	−0.0512976	−	0.998683i	$-0.516336\pi$
$953$	−3.16718	−0.102595	−0.0512976	+	0.998683i	$0.516336\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	7.31308	0.236152
$960$	0	0
$961$	−23.6656	−0.763407
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−43.8328	−1.40957	−0.704784	−	0.709422i	$-0.748956\pi$
$967$	−43.8328	−1.40957	−0.704784	+	0.709422i	$0.748956\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	21.9787	0.705330	0.352665	−	0.935750i	$-0.385275\pi$
$971$	21.9787	0.705330	0.352665	+	0.935750i	$0.385275\pi$
$972$	0	0
$973$	49.2918	1.58022
$974$	0	0
$975$	0	0
$976$	0	0
$977$	37.6869	1.20571	0.602856	−	0.797850i	$-0.294029\pi$
$977$	37.6869	1.20571	0.602856	+	0.797850i	$0.294029\pi$
$978$	0	0
$979$	−22.2492	−0.711088
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−9.54102	−0.304311	−0.152156	−	0.988357i	$-0.548621\pi$
$983$	−9.54102	−0.304311	−0.152156	+	0.988357i	$0.548621\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4.14590	0.131832
$990$	0	0
$991$	−23.1459	−0.735254	−0.367627	−	0.929973i	$-0.619830\pi$
$991$	−23.1459	−0.735254	−0.367627	+	0.929973i	$0.619830\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	3.12461	0.0989574	0.0494787	−	0.998775i	$-0.484244\pi$
$997$	3.12461	0.0989574	0.0494787	+	0.998775i	$0.484244\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8100.2.a.p.1.2	yes	2
3.2	odd	2		8100.2.a.o.1.2	✓	2
5.2	odd	4		8100.2.d.n.649.3		4
5.3	odd	4		8100.2.d.n.649.2		4
5.4	even	2		8100.2.a.r.1.1	yes	2
15.2	even	4		8100.2.d.k.649.3		4
15.8	even	4		8100.2.d.k.649.2		4
15.14	odd	2		8100.2.a.q.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8100.2.a.o.1.2	✓	2	3.2	odd	2
8100.2.a.p.1.2	yes	2	1.1	even	1	trivial
8100.2.a.q.1.1	yes	2	15.14	odd	2
8100.2.a.r.1.1	yes	2	5.4	even	2
8100.2.d.k.649.2		4	15.8	even	4
8100.2.d.k.649.3		4	15.2	even	4
8100.2.d.n.649.2		4	5.3	odd	4
8100.2.d.n.649.3		4	5.2	odd	4

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:	\( N \)	\(=\)	\( 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8100.a (trivial)

\(f(q)\)	\(=\)	\(q+2.85410 q^{7} +O(q^{10})\)	\(q+2.85410 q^{7} -1.85410 q^{11} -0.854102 q^{13} +1.14590 q^{17} +2.00000 q^{19} -4.85410 q^{23} -3.70820 q^{29} +2.70820 q^{31} +5.85410 q^{37} +11.5623 q^{41} -0.854102 q^{43} -6.70820 q^{47} +1.14590 q^{49} +4.85410 q^{53} -1.14590 q^{59} +0.854102 q^{61} +7.00000 q^{67} -9.00000 q^{71} -2.70820 q^{73} -5.29180 q^{77} +11.7082 q^{79} +6.70820 q^{83} +12.0000 q^{89} -2.43769 q^{91} +10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{7}+O(q^{10}) \) 2 * q - q^7	\( 2 q - q^{7} + 3 q^{11} + 5 q^{13} + 9 q^{17} + 4 q^{19} - 3 q^{23} + 6 q^{29} - 8 q^{31} + 5 q^{37} + 3 q^{41} + 5 q^{43} + 9 q^{49} + 3 q^{53} - 9 q^{59} - 5 q^{61} + 14 q^{67} - 18 q^{71} + 8 q^{73} - 24 q^{77} + 10 q^{79} + 24 q^{89} - 25 q^{91} + 20 q^{97}+O(q^{100}) \) 2 * q - q^7 + 3 * q^11 + 5 * q^13 + 9 * q^17 + 4 * q^19 - 3 * q^23 + 6 * q^29 - 8 * q^31 + 5 * q^37 + 3 * q^41 + 5 * q^43 + 9 * q^49 + 3 * q^53 - 9 * q^59 - 5 * q^61 + 14 * q^67 - 18 * q^71 + 8 * q^73 - 24 * q^77 + 10 * q^79 + 24 * q^89 - 25 * q^91 + 20 * q^97

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