LMFDB - Embedded newform 8100.2.a.o.1.1

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			1.1
Root			$1.61803$ 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)

$f(q)$	$=$	$q-3.85410 q^{7} +O(q^{10})$	$q-3.85410 q^{7} -4.85410 q^{11} +5.85410 q^{13} -7.85410 q^{17} +2.00000 q^{19} -1.85410 q^{23} -9.70820 q^{29} -10.7082 q^{31} -0.854102 q^{37} +8.56231 q^{41} +5.85410 q^{43} -6.70820 q^{47} +7.85410 q^{49} +1.85410 q^{53} +7.85410 q^{59} -5.85410 q^{61} +7.00000 q^{67} +9.00000 q^{71} +10.7082 q^{73} +18.7082 q^{77} -1.70820 q^{79} +6.70820 q^{83} -12.0000 q^{89} -22.5623 q^{91} +10.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−3.85410	−1.45671	−0.728357	−	0.685198i	$-0.759716\pi$
$7$	−3.85410	−1.45671	−0.728357	+	0.685198i	$0.759716\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.85410	−1.46357	−0.731783	−	0.681537i	$-0.761312\pi$
$11$	−4.85410	−1.46357	−0.731783	+	0.681537i	$0.761312\pi$
$12$	0	0
$13$	5.85410	1.62364	0.811818	−	0.583911i	$-0.198478\pi$
$13$	5.85410	1.62364	0.811818	+	0.583911i	$0.198478\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.85410	−1.90490	−0.952450	−	0.304696i	$-0.901445\pi$
$17$	−7.85410	−1.90490	−0.952450	+	0.304696i	$0.901445\pi$
$18$	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
$23$	−1.85410	−0.386607	−0.193303	−	0.981139i	$-0.561920\pi$
$23$	−1.85410	−0.386607	−0.193303	+	0.981139i	$0.561920\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−9.70820	−1.80277	−0.901384	−	0.433020i	$-0.857448\pi$
$29$	−9.70820	−1.80277	−0.901384	+	0.433020i	$0.857448\pi$
$30$	0	0
$31$	−10.7082	−1.92325	−0.961625	−	0.274367i	$-0.911532\pi$
$31$	−10.7082	−1.92325	−0.961625	+	0.274367i	$0.911532\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.854102	−0.140413	−0.0702067	−	0.997532i	$-0.522366\pi$
$37$	−0.854102	−0.140413	−0.0702067	+	0.997532i	$0.522366\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.56231	1.33721	0.668604	−	0.743619i	$-0.266892\pi$
$41$	8.56231	1.33721	0.668604	+	0.743619i	$0.266892\pi$
$42$	0	0
$43$	5.85410	0.892742	0.446371	−	0.894848i	$-0.352716\pi$
$43$	5.85410	0.892742	0.446371	+	0.894848i	$0.352716\pi$
$44$	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
$49$	7.85410	1.12201
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.85410	0.254680	0.127340	−	0.991859i	$-0.459356\pi$
$53$	1.85410	0.254680	0.127340	+	0.991859i	$0.459356\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.85410	1.02252	0.511258	−	0.859427i	$-0.329179\pi$
$59$	7.85410	1.02252	0.511258	+	0.859427i	$0.329179\pi$
$60$	0	0
$61$	−5.85410	−0.749541	−0.374770	−	0.927118i	$-0.622278\pi$
$61$	−5.85410	−0.749541	−0.374770	+	0.927118i	$0.622278\pi$
$62$	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	$-0.320669\pi$
$71$	9.00000	1.06810	0.534052	+	0.845452i	$0.320669\pi$
$72$	0	0
$73$	10.7082	1.25330	0.626650	−	0.779301i	$-0.284425\pi$
$73$	10.7082	1.25330	0.626650	+	0.779301i	$0.284425\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	18.7082	2.13200
$78$	0	0
$79$	−1.70820	−0.192188	−0.0960940	−	0.995372i	$-0.530635\pi$
$79$	−1.70820	−0.192188	−0.0960940	+	0.995372i	$0.530635\pi$
$80$	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	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	0	0
$91$	−22.5623	−2.36517
$92$	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
$101$	−1.85410	−0.184490	−0.0922450	−	0.995736i	$-0.529404\pi$
$101$	−1.85410	−0.184490	−0.0922450	+	0.995736i	$0.529404\pi$
$102$	0	0
$103$	−6.85410	−0.675355	−0.337677	−	0.941262i	$-0.609641\pi$
$103$	−6.85410	−0.675355	−0.337677	+	0.941262i	$0.609641\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.29180	−0.511577	−0.255789	−	0.966733i	$-0.582335\pi$
$107$	−5.29180	−0.511577	−0.255789	+	0.966733i	$0.582335\pi$
$108$	0	0
$109$	−2.85410	−0.273373	−0.136687	−	0.990614i	$-0.543645\pi$
$109$	−2.85410	−0.273373	−0.136687	+	0.990614i	$0.543645\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−4.85410	−0.456636	−0.228318	−	0.973587i	$-0.573323\pi$
$113$	−4.85410	−0.456636	−0.228318	+	0.973587i	$0.573323\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	30.2705	2.77489
$120$	0	0
$121$	12.5623	1.14203
$122$	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
$131$	−15.7082	−1.37243	−0.686216	−	0.727398i	$-0.740730\pi$
$131$	−15.7082	−1.37243	−0.686216	+	0.727398i	$0.740730\pi$
$132$	0	0
$133$	−7.70820	−0.668386
$134$	0	0
$135$	0	0
$136$	0	0
$137$	17.5623	1.50045	0.750225	−	0.661183i	$-0.229945\pi$
$137$	17.5623	1.50045	0.750225	+	0.661183i	$0.229945\pi$
$138$	0	0
$139$	−16.2705	−1.38005	−0.690023	−	0.723787i	$-0.742400\pi$
$139$	−16.2705	−1.38005	−0.690023	+	0.723787i	$0.742400\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−28.4164	−2.37630
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	15.0000	1.22885	0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000	1.22885	0.614424	+	0.788976i	$0.289388\pi$
$150$	0	0
$151$	−14.8541	−1.20881	−0.604405	−	0.796677i	$-0.706589\pi$
$151$	−14.8541	−1.20881	−0.604405	+	0.796677i	$0.706589\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3.29180	0.262714	0.131357	−	0.991335i	$-0.458067\pi$
$157$	3.29180	0.262714	0.131357	+	0.991335i	$0.458067\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.14590	0.563176
$162$	0	0
$163$	−12.4164	−0.972528	−0.486264	−	0.873812i	$-0.661641\pi$
$163$	−12.4164	−0.972528	−0.486264	+	0.873812i	$0.661641\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.1459	1.24941	0.624704	−	0.780862i	$-0.285220\pi$
$167$	16.1459	1.24941	0.624704	+	0.780862i	$0.285220\pi$
$168$	0	0
$169$	21.2705	1.63619
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.29180	0.174242	0.0871210	−	0.996198i	$-0.472233\pi$
$173$	2.29180	0.174242	0.0871210	+	0.996198i	$0.472233\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−18.7082	−1.39832	−0.699158	−	0.714967i	$-0.746442\pi$
$179$	−18.7082	−1.39832	−0.699158	+	0.714967i	$0.746442\pi$
$180$	0	0
$181$	12.4164	0.922904	0.461452	−	0.887165i	$-0.347329\pi$
$181$	12.4164	0.922904	0.461452	+	0.887165i	$0.347329\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	38.1246	2.78795
$188$	0	0
$189$	0	0
$190$	0	0
$191$	7.41641	0.536632	0.268316	−	0.963331i	$-0.413533\pi$
$191$	7.41641	0.536632	0.268316	+	0.963331i	$0.413533\pi$
$192$	0	0
$193$	3.29180	0.236949	0.118474	−	0.992957i	$-0.462200\pi$
$193$	3.29180	0.236949	0.118474	+	0.992957i	$0.462200\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−19.1459	−1.36409	−0.682044	−	0.731311i	$-0.738909\pi$
$197$	−19.1459	−1.36409	−0.682044	+	0.731311i	$0.738909\pi$
$198$	0	0
$199$	9.14590	0.648336	0.324168	−	0.946000i	$-0.394916\pi$
$199$	9.14590	0.648336	0.324168	+	0.946000i	$0.394916\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	37.4164	2.62612
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−9.70820	−0.671531
$210$	0	0
$211$	−9.29180	−0.639674	−0.319837	−	0.947473i	$-0.603628\pi$
$211$	−9.29180	−0.639674	−0.319837	+	0.947473i	$0.603628\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	41.2705	2.80162
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−45.9787	−3.09286
$222$	0	0
$223$	16.7082	1.11886	0.559432	−	0.828876i	$-0.311019\pi$
$223$	16.7082	1.11886	0.559432	+	0.828876i	$0.311019\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3.00000	−0.199117	−0.0995585	−	0.995032i	$-0.531743\pi$
$227$	−3.00000	−0.199117	−0.0995585	+	0.995032i	$0.531743\pi$
$228$	0	0
$229$	18.4164	1.21699	0.608495	−	0.793558i	$-0.291773\pi$
$229$	18.4164	1.21699	0.608495	+	0.793558i	$0.291773\pi$
$230$	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
$241$	9.41641	0.606564	0.303282	−	0.952901i	$-0.401918\pi$
$241$	9.41641	0.606564	0.303282	+	0.952901i	$0.401918\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	11.7082	0.744975
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4.14590	0.261687	0.130843	−	0.991403i	$-0.458231\pi$
$251$	4.14590	0.261687	0.130843	+	0.991403i	$0.458231\pi$
$252$	0	0
$253$	9.00000	0.565825
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.41641	−0.0883531	−0.0441765	−	0.999024i	$-0.514066\pi$
$257$	−1.41641	−0.0883531	−0.0441765	+	0.999024i	$0.514066\pi$
$258$	0	0
$259$	3.29180	0.204542
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−15.7082	−0.968609	−0.484305	−	0.874899i	$-0.660927\pi$
$263$	−15.7082	−0.968609	−0.484305	+	0.874899i	$0.660927\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	6.27051	0.382320	0.191160	−	0.981559i	$-0.438775\pi$
$269$	6.27051	0.382320	0.191160	+	0.981559i	$0.438775\pi$
$270$	0	0
$271$	18.4164	1.11872	0.559359	−	0.828926i	$-0.311048\pi$
$271$	18.4164	1.11872	0.559359	+	0.828926i	$0.311048\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−3.41641	−0.205272	−0.102636	−	0.994719i	$-0.532728\pi$
$277$	−3.41641	−0.205272	−0.102636	+	0.994719i	$0.532728\pi$
$278$	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
$283$	5.14590	0.305892	0.152946	−	0.988235i	$-0.451124\pi$
$283$	5.14590	0.305892	0.152946	+	0.988235i	$0.451124\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−33.0000	−1.94793
$288$	0	0
$289$	44.6869	2.62864
$290$	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
$299$	−10.8541	−0.627709
$300$	0	0
$301$	−22.5623	−1.30047
$302$	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
$311$	−12.7082	−0.720616	−0.360308	−	0.932833i	$-0.617328\pi$
$311$	−12.7082	−0.720616	−0.360308	+	0.932833i	$0.617328\pi$
$312$	0	0
$313$	14.1459	0.799573	0.399787	−	0.916608i	$-0.369084\pi$
$313$	14.1459	0.799573	0.399787	+	0.916608i	$0.369084\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	28.4164	1.59602	0.798012	−	0.602641i	$-0.205885\pi$
$317$	28.4164	1.59602	0.798012	+	0.602641i	$0.205885\pi$
$318$	0	0
$319$	47.1246	2.63847
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−15.7082	−0.874028
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	25.8541	1.42538
$330$	0	0
$331$	9.41641	0.517573	0.258786	−	0.965935i	$-0.416677\pi$
$331$	9.41641	0.517573	0.258786	+	0.965935i	$0.416677\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	31.7082	1.72726	0.863628	−	0.504130i	$-0.168187\pi$
$337$	31.7082	1.72726	0.863628	+	0.504130i	$0.168187\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	51.9787	2.81481
$342$	0	0
$343$	−3.29180	−0.177740
$344$	0	0
$345$	0	0
$346$	0	0
$347$	15.4377	0.828739	0.414369	−	0.910109i	$-0.364002\pi$
$347$	15.4377	0.828739	0.414369	+	0.910109i	$0.364002\pi$
$348$	0	0
$349$	10.2918	0.550907	0.275454	−	0.961314i	$-0.411172\pi$
$349$	10.2918	0.550907	0.275454	+	0.961314i	$0.411172\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−2.56231	−0.136378	−0.0681889	−	0.997672i	$-0.521722\pi$
$353$	−2.56231	−0.136378	−0.0681889	+	0.997672i	$0.521722\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−22.8541	−1.20619	−0.603097	−	0.797668i	$-0.706067\pi$
$359$	−22.8541	−1.20619	−0.603097	+	0.797668i	$0.706067\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−10.5623	−0.551348	−0.275674	−	0.961251i	$-0.588901\pi$
$367$	−10.5623	−0.551348	−0.275674	+	0.961251i	$0.588901\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−7.14590	−0.370997
$372$	0	0
$373$	24.1246	1.24913	0.624563	−	0.780975i	$-0.285277\pi$
$373$	24.1246	1.24913	0.624563	+	0.780975i	$0.285277\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−56.8328	−2.92704
$378$	0	0
$379$	−13.2705	−0.681660	−0.340830	−	0.940125i	$-0.610708\pi$
$379$	−13.2705	−0.681660	−0.340830	+	0.940125i	$0.610708\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−15.9787	−0.816474	−0.408237	−	0.912876i	$-0.633856\pi$
$383$	−15.9787	−0.816474	−0.408237	+	0.912876i	$0.633856\pi$
$384$	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
$391$	14.5623	0.736447
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−35.5410	−1.78375	−0.891876	−	0.452279i	$-0.850611\pi$
$397$	−35.5410	−1.78375	−0.891876	+	0.452279i	$0.850611\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5.12461	0.255911	0.127955	−	0.991780i	$-0.459159\pi$
$401$	5.12461	0.255911	0.127955	+	0.991780i	$0.459159\pi$
$402$	0	0
$403$	−62.6869	−3.12266
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.14590	0.205505
$408$	0	0
$409$	−19.2705	−0.952865	−0.476433	−	0.879211i	$-0.658070\pi$
$409$	−19.2705	−0.952865	−0.476433	+	0.879211i	$0.658070\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−30.2705	−1.48951
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	10.8541	0.530258	0.265129	−	0.964213i	$-0.414586\pi$
$419$	10.8541	0.530258	0.265129	+	0.964213i	$0.414586\pi$
$420$	0	0
$421$	−7.70820	−0.375675	−0.187837	−	0.982200i	$-0.560148\pi$
$421$	−7.70820	−0.375675	−0.187837	+	0.982200i	$0.560148\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	22.5623	1.09187
$428$	0	0
$429$	0	0
$430$	0	0
$431$	21.0000	1.01153	0.505767	−	0.862670i	$-0.331209\pi$
$431$	21.0000	1.01153	0.505767	+	0.862670i	$0.331209\pi$
$432$	0	0
$433$	24.2918	1.16739	0.583695	−	0.811973i	$-0.301607\pi$
$433$	24.2918	1.16739	0.583695	+	0.811973i	$0.301607\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.70820	−0.177387
$438$	0	0
$439$	10.5623	0.504111	0.252056	−	0.967713i	$-0.418893\pi$
$439$	10.5623	0.504111	0.252056	+	0.967713i	$0.418893\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−25.8541	−1.22837	−0.614183	−	0.789164i	$-0.710514\pi$
$443$	−25.8541	−1.22837	−0.614183	+	0.789164i	$0.710514\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−13.8541	−0.653815	−0.326908	−	0.945056i	$-0.606007\pi$
$449$	−13.8541	−0.653815	−0.326908	+	0.945056i	$0.606007\pi$
$450$	0	0
$451$	−41.5623	−1.95709
$452$	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
$461$	3.43769	0.160109	0.0800547	−	0.996790i	$-0.474491\pi$
$461$	3.43769	0.160109	0.0800547	+	0.996790i	$0.474491\pi$
$462$	0	0
$463$	−9.85410	−0.457959	−0.228979	−	0.973431i	$-0.573539\pi$
$463$	−9.85410	−0.457959	−0.228979	+	0.973431i	$0.573539\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	0	0
$469$	−26.9787	−1.24576
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−28.4164	−1.30659
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	22.8541	1.04423	0.522115	−	0.852875i	$-0.325143\pi$
$479$	22.8541	1.04423	0.522115	+	0.852875i	$0.325143\pi$
$480$	0	0
$481$	−5.00000	−0.227980
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	23.4164	1.06110	0.530549	−	0.847654i	$-0.321986\pi$
$487$	23.4164	1.06110	0.530549	+	0.847654i	$0.321986\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	35.8328	1.61711	0.808556	−	0.588419i	$-0.200249\pi$
$491$	35.8328	1.61711	0.808556	+	0.588419i	$0.200249\pi$
$492$	0	0
$493$	76.2492	3.43409
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−34.6869	−1.55592
$498$	0	0
$499$	25.5623	1.14433	0.572163	−	0.820140i	$-0.306104\pi$
$499$	25.5623	1.14433	0.572163	+	0.820140i	$0.306104\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	9.27051	0.413352	0.206676	−	0.978409i	$-0.433735\pi$
$503$	9.27051	0.413352	0.206676	+	0.978409i	$0.433735\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	21.0000	0.930809	0.465404	−	0.885098i	$-0.345909\pi$
$509$	21.0000	0.930809	0.465404	+	0.885098i	$0.345909\pi$
$510$	0	0
$511$	−41.2705	−1.82570
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	32.5623	1.43209
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−5.12461	−0.224513	−0.112257	−	0.993679i	$-0.535808\pi$
$521$	−5.12461	−0.224513	−0.112257	+	0.993679i	$0.535808\pi$
$522$	0	0
$523$	8.41641	0.368024	0.184012	−	0.982924i	$-0.441091\pi$
$523$	8.41641	0.368024	0.184012	+	0.982924i	$0.441091\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	84.1033	3.66360
$528$	0	0
$529$	−19.5623	−0.850535
$530$	0	0
$531$	0	0
$532$	0	0
$533$	50.1246	2.17114
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−38.1246	−1.64214
$540$	0	0
$541$	−33.3951	−1.43577	−0.717884	−	0.696163i	$-0.754889\pi$
$541$	−33.3951	−1.43577	−0.717884	+	0.696163i	$0.754889\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	11.5836	0.495279	0.247639	−	0.968852i	$-0.420345\pi$
$547$	11.5836	0.495279	0.247639	+	0.968852i	$0.420345\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−19.4164	−0.827167
$552$	0	0
$553$	6.58359	0.279963
$554$	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
$559$	34.2705	1.44949
$560$	0	0
$561$	0	0
$562$	0	0
$563$	23.2918	0.981632	0.490816	−	0.871263i	$-0.336699\pi$
$563$	23.2918	0.981632	0.490816	+	0.871263i	$0.336699\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	9.70820	0.406989	0.203495	−	0.979076i	$-0.434770\pi$
$569$	9.70820	0.406989	0.203495	+	0.979076i	$0.434770\pi$
$570$	0	0
$571$	−42.8328	−1.79250	−0.896249	−	0.443552i	$-0.853718\pi$
$571$	−42.8328	−1.79250	−0.896249	+	0.443552i	$0.853718\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	26.4164	1.09973	0.549865	−	0.835254i	$-0.314679\pi$
$577$	26.4164	1.09973	0.549865	+	0.835254i	$0.314679\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−25.8541	−1.07261
$582$	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	$-0.600181\pi$
$587$	−15.0000	−0.619116	−0.309558	+	0.950881i	$0.600181\pi$
$588$	0	0
$589$	−21.4164	−0.882448
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−30.2705	−1.24306	−0.621530	−	0.783390i	$-0.713489\pi$
$593$	−30.2705	−1.24306	−0.621530	+	0.783390i	$0.713489\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−0.437694	−0.0178837	−0.00894185	−	0.999960i	$-0.502846\pi$
$599$	−0.437694	−0.0178837	−0.00894185	+	0.999960i	$0.502846\pi$
$600$	0	0
$601$	3.14590	0.128324	0.0641619	−	0.997940i	$-0.479563\pi$
$601$	3.14590	0.128324	0.0641619	+	0.997940i	$0.479563\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−38.5410	−1.56433	−0.782166	−	0.623070i	$-0.785885\pi$
$607$	−38.5410	−1.56433	−0.782166	+	0.623070i	$0.785885\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−39.2705	−1.58871
$612$	0	0
$613$	1.70820	0.0689937	0.0344969	−	0.999405i	$-0.489017\pi$
$613$	1.70820	0.0689937	0.0344969	+	0.999405i	$0.489017\pi$
$614$	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
$619$	6.58359	0.264617	0.132308	−	0.991209i	$-0.457761\pi$
$619$	6.58359	0.264617	0.132308	+	0.991209i	$0.457761\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	46.2492	1.85294
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6.70820	0.267474
$630$	0	0
$631$	6.85410	0.272857	0.136429	−	0.990650i	$-0.456438\pi$
$631$	6.85410	0.272857	0.136429	+	0.990650i	$0.456438\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	45.9787	1.82174
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−8.56231	−0.338191	−0.169095	−	0.985600i	$-0.554085\pi$
$641$	−8.56231	−0.338191	−0.169095	+	0.985600i	$0.554085\pi$
$642$	0	0
$643$	−17.4377	−0.687676	−0.343838	−	0.939029i	$-0.611727\pi$
$643$	−17.4377	−0.687676	−0.343838	+	0.939029i	$0.611727\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−11.2918	−0.443926	−0.221963	−	0.975055i	$-0.571246\pi$
$647$	−11.2918	−0.443926	−0.221963	+	0.975055i	$0.571246\pi$
$648$	0	0
$649$	−38.1246	−1.49652
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−10.8541	−0.424754	−0.212377	−	0.977188i	$-0.568120\pi$
$653$	−10.8541	−0.424754	−0.212377	+	0.977188i	$0.568120\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−15.2705	−0.594855	−0.297427	−	0.954744i	$-0.596129\pi$
$659$	−15.2705	−0.594855	−0.297427	+	0.954744i	$0.596129\pi$
$660$	0	0
$661$	−27.8328	−1.08257	−0.541286	−	0.840839i	$-0.682062\pi$
$661$	−27.8328	−1.08257	−0.541286	+	0.840839i	$0.682062\pi$
$662$	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
$671$	28.4164	1.09700
$672$	0	0
$673$	10.7082	0.412771	0.206385	−	0.978471i	$-0.433830\pi$
$673$	10.7082	0.412771	0.206385	+	0.978471i	$0.433830\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−10.4164	−0.400335	−0.200168	−	0.979762i	$-0.564149\pi$
$677$	−10.4164	−0.400335	−0.200168	+	0.979762i	$0.564149\pi$
$678$	0	0
$679$	−38.5410	−1.47907
$680$	0	0
$681$	0	0
$682$	0	0
$683$	44.3951	1.69873	0.849366	−	0.527804i	$-0.176985\pi$
$683$	44.3951	1.69873	0.849366	+	0.527804i	$0.176985\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	10.8541	0.413508
$690$	0	0
$691$	−30.1246	−1.14599	−0.572997	−	0.819557i	$-0.694219\pi$
$691$	−30.1246	−1.14599	−0.572997	+	0.819557i	$0.694219\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−67.2492	−2.54725
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−2.29180	−0.0865599	−0.0432800	−	0.999063i	$-0.513781\pi$
$701$	−2.29180	−0.0865599	−0.0432800	+	0.999063i	$0.513781\pi$
$702$	0	0
$703$	−1.70820	−0.0644261
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7.14590	0.268749
$708$	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
$713$	19.8541	0.743542
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−0.437694	−0.0163232	−0.00816162	−	0.999967i	$-0.502598\pi$
$719$	−0.437694	−0.0163232	−0.00816162	+	0.999967i	$0.502598\pi$
$720$	0	0
$721$	26.4164	0.983798
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−17.4377	−0.646728	−0.323364	−	0.946275i	$-0.604814\pi$
$727$	−17.4377	−0.646728	−0.323364	+	0.946275i	$0.604814\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−45.9787	−1.70058
$732$	0	0
$733$	16.4377	0.607140	0.303570	−	0.952809i	$-0.401821\pi$
$733$	16.4377	0.607140	0.303570	+	0.952809i	$0.401821\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−33.9787	−1.25162
$738$	0	0
$739$	25.5623	0.940325	0.470162	−	0.882580i	$-0.344195\pi$
$739$	25.5623	0.940325	0.470162	+	0.882580i	$0.344195\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	41.1246	1.50872	0.754358	−	0.656463i	$-0.227948\pi$
$743$	41.1246	1.50872	0.754358	+	0.656463i	$0.227948\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	20.3951	0.745222
$750$	0	0
$751$	5.97871	0.218166	0.109083	−	0.994033i	$-0.465209\pi$
$751$	5.97871	0.218166	0.109083	+	0.994033i	$0.465209\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0.729490	0.0265138	0.0132569	−	0.999912i	$-0.495780\pi$
$757$	0.729490	0.0265138	0.0132569	+	0.999912i	$0.495780\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−28.1459	−1.02029	−0.510144	−	0.860089i	$-0.670408\pi$
$761$	−28.1459	−1.02029	−0.510144	+	0.860089i	$0.670408\pi$
$762$	0	0
$763$	11.0000	0.398227
$764$	0	0
$765$	0	0
$766$	0	0
$767$	45.9787	1.66020
$768$	0	0
$769$	−32.8541	−1.18475	−0.592375	−	0.805663i	$-0.701809\pi$
$769$	−32.8541	−1.18475	−0.592375	+	0.805663i	$0.701809\pi$
$770$	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
$779$	17.1246	0.613553
$780$	0	0
$781$	−43.6869	−1.56324
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	8.85410	0.315615	0.157807	−	0.987470i	$-0.449558\pi$
$787$	8.85410	0.315615	0.157807	+	0.987470i	$0.449558\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	18.7082	0.665187
$792$	0	0
$793$	−34.2705	−1.21698
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−24.2705	−0.859706	−0.429853	−	0.902899i	$-0.641435\pi$
$797$	−24.2705	−0.859706	−0.429853	+	0.902899i	$0.641435\pi$
$798$	0	0
$799$	52.6869	1.86393
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−51.9787	−1.83429
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−32.1246	−1.12944	−0.564721	−	0.825282i	$-0.691016\pi$
$809$	−32.1246	−1.12944	−0.564721	+	0.825282i	$0.691016\pi$
$810$	0	0
$811$	29.9787	1.05270	0.526348	−	0.850270i	$-0.323561\pi$
$811$	29.9787	1.05270	0.526348	+	0.850270i	$0.323561\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	11.7082	0.409618
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−22.4164	−0.782338	−0.391169	−	0.920319i	$-0.627929\pi$
$821$	−22.4164	−0.782338	−0.391169	+	0.920319i	$0.627929\pi$
$822$	0	0
$823$	49.5410	1.72689	0.863446	−	0.504442i	$-0.168302\pi$
$823$	49.5410	1.72689	0.863446	+	0.504442i	$0.168302\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−29.3951	−1.02217	−0.511084	−	0.859531i	$-0.670756\pi$
$827$	−29.3951	−1.02217	−0.511084	+	0.859531i	$0.670756\pi$
$828$	0	0
$829$	21.4164	0.743823	0.371911	−	0.928268i	$-0.378703\pi$
$829$	21.4164	0.743823	0.371911	+	0.928268i	$0.378703\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−61.6869	−2.13733
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	12.8754	0.444508	0.222254	−	0.974989i	$-0.428659\pi$
$839$	12.8754	0.444508	0.222254	+	0.974989i	$0.428659\pi$
$840$	0	0
$841$	65.2492	2.24997
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−48.4164	−1.66361
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.58359	0.0542848
$852$	0	0
$853$	−16.1246	−0.552096	−0.276048	−	0.961144i	$-0.589025\pi$
$853$	−16.1246	−0.552096	−0.276048	+	0.961144i	$0.589025\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	10.8541	0.370769	0.185385	−	0.982666i	$-0.440647\pi$
$857$	10.8541	0.370769	0.185385	+	0.982666i	$0.440647\pi$
$858$	0	0
$859$	36.4164	1.24251	0.621256	−	0.783608i	$-0.286623\pi$
$859$	36.4164	1.24251	0.621256	+	0.783608i	$0.286623\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	18.2705	0.621935	0.310968	−	0.950420i	$-0.399347\pi$
$863$	18.2705	0.621935	0.310968	+	0.950420i	$0.399347\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	8.29180	0.281280
$870$	0	0
$871$	40.9787	1.38851
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	19.2705	0.650719	0.325359	−	0.945590i	$-0.394515\pi$
$877$	19.2705	0.650719	0.325359	+	0.945590i	$0.394515\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−14.1246	−0.475870	−0.237935	−	0.971281i	$-0.576471\pi$
$881$	−14.1246	−0.475870	−0.237935	+	0.971281i	$0.576471\pi$
$882$	0	0
$883$	9.12461	0.307068	0.153534	−	0.988143i	$-0.450935\pi$
$883$	9.12461	0.307068	0.153534	+	0.988143i	$0.450935\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−45.9787	−1.54381	−0.771907	−	0.635735i	$-0.780697\pi$
$887$	−45.9787	−1.54381	−0.771907	+	0.635735i	$0.780697\pi$
$888$	0	0
$889$	−26.9787	−0.904837
$890$	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
$899$	103.957	3.46717
$900$	0	0
$901$	−14.5623	−0.485141
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−32.9787	−1.09504	−0.547520	−	0.836793i	$-0.684428\pi$
$907$	−32.9787	−1.09504	−0.547520	+	0.836793i	$0.684428\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	14.1246	0.467969	0.233985	−	0.972240i	$-0.424823\pi$
$911$	14.1246	0.467969	0.233985	+	0.972240i	$0.424823\pi$
$912$	0	0
$913$	−32.5623	−1.07766
$914$	0	0
$915$	0	0
$916$	0	0
$917$	60.5410	1.99924
$918$	0	0
$919$	−4.43769	−0.146386	−0.0731930	−	0.997318i	$-0.523319\pi$
$919$	−4.43769	−0.146386	−0.0731930	+	0.997318i	$0.523319\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	52.6869	1.73421
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−54.7082	−1.79492	−0.897459	−	0.441098i	$-0.854589\pi$
$929$	−54.7082	−1.79492	−0.897459	+	0.441098i	$0.854589\pi$
$930$	0	0
$931$	15.7082	0.514816
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	9.83282	0.321224	0.160612	−	0.987018i	$-0.448653\pi$
$937$	9.83282	0.321224	0.160612	+	0.987018i	$0.448653\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−44.3951	−1.44724	−0.723620	−	0.690199i	$-0.757523\pi$
$941$	−44.3951	−1.44724	−0.723620	+	0.690199i	$0.757523\pi$
$942$	0	0
$943$	−15.8754	−0.516974
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−19.5836	−0.636381	−0.318191	−	0.948027i	$-0.603075\pi$
$947$	−19.5836	−0.636381	−0.318191	+	0.948027i	$0.603075\pi$
$948$	0	0
$949$	62.6869	2.03490
$950$	0	0
$951$	0	0
$952$	0	0
$953$	56.8328	1.84100	0.920498	−	0.390748i	$-0.127784\pi$
$953$	56.8328	1.84100	0.920498	+	0.390748i	$0.127784\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−67.6869	−2.18572
$960$	0	0
$961$	83.6656	2.69889
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	9.83282	0.316202	0.158101	−	0.987423i	$-0.449463\pi$
$967$	9.83282	0.316202	0.158101	+	0.987423i	$0.449463\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	24.9787	0.801605	0.400803	−	0.916164i	$-0.368731\pi$
$971$	24.9787	0.801605	0.400803	+	0.916164i	$0.368731\pi$
$972$	0	0
$973$	62.7082	2.01033
$974$	0	0
$975$	0	0
$976$	0	0
$977$	22.6869	0.725819	0.362909	−	0.931824i	$-0.381783\pi$
$977$	22.6869	0.725819	0.362909	+	0.931824i	$0.381783\pi$
$978$	0	0
$979$	58.2492	1.86165
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−57.5410	−1.83527	−0.917637	−	0.397420i	$-0.869906\pi$
$983$	−57.5410	−1.83527	−0.917637	+	0.397420i	$0.869906\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−10.8541	−0.345140
$990$	0	0
$991$	−29.8541	−0.948347	−0.474173	−	0.880431i	$-0.657253\pi$
$991$	−29.8541	−0.948347	−0.474173	+	0.880431i	$0.657253\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−37.1246	−1.17575	−0.587874	−	0.808952i	$-0.700035\pi$
$997$	−37.1246	−1.17575	−0.587874	+	0.808952i	$0.700035\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8100.2.a.o.1.1	✓	2
3.2	odd	2		8100.2.a.p.1.1	yes	2
5.2	odd	4		8100.2.d.k.649.1		4
5.3	odd	4		8100.2.d.k.649.4		4
5.4	even	2		8100.2.a.q.1.2	yes	2
15.2	even	4		8100.2.d.n.649.1		4
15.8	even	4		8100.2.d.n.649.4		4
15.14	odd	2		8100.2.a.r.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8100.2.a.o.1.1	✓	2	1.1	even	1	trivial
8100.2.a.p.1.1	yes	2	3.2	odd	2
8100.2.a.q.1.2	yes	2	5.4	even	2
8100.2.a.r.1.2	yes	2	15.14	odd	2
8100.2.d.k.649.1		4	5.2	odd	4
8100.2.d.k.649.4		4	5.3	odd	4
8100.2.d.n.649.1		4	15.2	even	4
8100.2.d.n.649.4		4	15.8	even	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)

\(f(q)\)	\(=\)	\(q-3.85410 q^{7} +O(q^{10})\)	\(q-3.85410 q^{7} -4.85410 q^{11} +5.85410 q^{13} -7.85410 q^{17} +2.00000 q^{19} -1.85410 q^{23} -9.70820 q^{29} -10.7082 q^{31} -0.854102 q^{37} +8.56231 q^{41} +5.85410 q^{43} -6.70820 q^{47} +7.85410 q^{49} +1.85410 q^{53} +7.85410 q^{59} -5.85410 q^{61} +7.00000 q^{67} +9.00000 q^{71} +10.7082 q^{73} +18.7082 q^{77} -1.70820 q^{79} +6.70820 q^{83} -12.0000 q^{89} -22.5623 q^{91} +10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{7}+O(q^{10}) \) 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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