LMFDB - Embedded newform 8550.2.a.bt.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8550,2,Mod(1,8550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8550, 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("8550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8550.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:	$68.2720937282$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2850)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	8550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -0.732051 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -0.732051 q^{7} -1.00000 q^{8} +5.19615 q^{11} +1.26795 q^{13} +0.732051 q^{14} +1.00000 q^{16} -0.732051 q^{17} +1.00000 q^{19} -5.19615 q^{22} -1.53590 q^{23} -1.26795 q^{26} -0.732051 q^{28} -1.19615 q^{29} +7.92820 q^{31} -1.00000 q^{32} +0.732051 q^{34} +4.92820 q^{37} -1.00000 q^{38} +5.26795 q^{43} +5.19615 q^{44} +1.53590 q^{46} -3.46410 q^{47} -6.46410 q^{49} +1.26795 q^{52} -12.6603 q^{53} +0.732051 q^{56} +1.19615 q^{58} +8.19615 q^{59} +11.7321 q^{61} -7.92820 q^{62} +1.00000 q^{64} +5.19615 q^{67} -0.732051 q^{68} -0.196152 q^{71} +7.53590 q^{73} -4.92820 q^{74} +1.00000 q^{76} -3.80385 q^{77} -7.92820 q^{79} +0.660254 q^{83} -5.26795 q^{86} -5.19615 q^{88} +9.92820 q^{89} -0.928203 q^{91} -1.53590 q^{92} +3.46410 q^{94} -7.12436 q^{97} +6.46410 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8} + 6 q^{13} - 2 q^{14} + 2 q^{16} + 2 q^{17} + 2 q^{19} - 10 q^{23} - 6 q^{26} + 2 q^{28} + 8 q^{29} + 2 q^{31} - 2 q^{32} - 2 q^{34} - 4 q^{37} - 2 q^{38} + 14 q^{43} + 10 q^{46} - 6 q^{49} + 6 q^{52} - 8 q^{53} - 2 q^{56} - 8 q^{58} + 6 q^{59} + 20 q^{61} - 2 q^{62} + 2 q^{64} + 2 q^{68} + 10 q^{71} + 22 q^{73} + 4 q^{74} + 2 q^{76} - 18 q^{77} - 2 q^{79} - 16 q^{83} - 14 q^{86} + 6 q^{89} + 12 q^{91} - 10 q^{92} + 10 q^{97} + 6 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8 + 6 * q^13 - 2 * q^14 + 2 * q^16 + 2 * q^17 + 2 * q^19 - 10 * q^23 - 6 * q^26 + 2 * q^28 + 8 * q^29 + 2 * q^31 - 2 * q^32 - 2 * q^34 - 4 * q^37 - 2 * q^38 + 14 * q^43 + 10 * q^46 - 6 * q^49 + 6 * q^52 - 8 * q^53 - 2 * q^56 - 8 * q^58 + 6 * q^59 + 20 * q^61 - 2 * q^62 + 2 * q^64 + 2 * q^68 + 10 * q^71 + 22 * q^73 + 4 * q^74 + 2 * q^76 - 18 * q^77 - 2 * q^79 - 16 * q^83 - 14 * q^86 + 6 * q^89 + 12 * q^91 - 10 * q^92 + 10 * q^97 + 6 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.732051	−0.276689	−0.138345	−	0.990384i	$-0.544178\pi$
$7$	−0.732051	−0.276689	−0.138345	+	0.990384i	$0.544178\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	5.19615	1.56670	0.783349	−	0.621582i	$-0.213510\pi$
$11$	5.19615	1.56670	0.783349	+	0.621582i	$0.213510\pi$
$12$	0	0
$13$	1.26795	0.351666	0.175833	−	0.984420i	$-0.443738\pi$
$13$	1.26795	0.351666	0.175833	+	0.984420i	$0.443738\pi$
$14$	0.732051	0.195649
$15$	0	0
$16$	1.00000	0.250000
$17$	−0.732051	−0.177548	−0.0887742	−	0.996052i	$-0.528295\pi$
$17$	−0.732051	−0.177548	−0.0887742	+	0.996052i	$0.528295\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	−5.19615	−1.10782
$23$	−1.53590	−0.320257	−0.160128	−	0.987096i	$-0.551191\pi$
$23$	−1.53590	−0.320257	−0.160128	+	0.987096i	$0.551191\pi$
$24$	0	0
$25$	0	0
$26$	−1.26795	−0.248665
$27$	0	0
$28$	−0.732051	−0.138345
$29$	−1.19615	−0.222120	−0.111060	−	0.993814i	$-0.535425\pi$
$29$	−1.19615	−0.222120	−0.111060	+	0.993814i	$0.535425\pi$
$30$	0	0
$31$	7.92820	1.42395	0.711974	−	0.702206i	$-0.247802\pi$
$31$	7.92820	1.42395	0.711974	+	0.702206i	$0.247802\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0.732051	0.125546
$35$	0	0
$36$	0	0
$37$	4.92820	0.810192	0.405096	−	0.914274i	$-0.367238\pi$
$37$	4.92820	0.810192	0.405096	+	0.914274i	$0.367238\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	5.26795	0.803355	0.401677	−	0.915781i	$-0.368427\pi$
$43$	5.26795	0.803355	0.401677	+	0.915781i	$0.368427\pi$
$44$	5.19615	0.783349
$45$	0	0
$46$	1.53590	0.226456
$47$	−3.46410	−0.505291	−0.252646	−	0.967559i	$-0.581301\pi$
$47$	−3.46410	−0.505291	−0.252646	+	0.967559i	$0.581301\pi$
$48$	0	0
$49$	−6.46410	−0.923443
$50$	0	0
$51$	0	0
$52$	1.26795	0.175833
$53$	−12.6603	−1.73902	−0.869510	−	0.493916i	$-0.835565\pi$
$53$	−12.6603	−1.73902	−0.869510	+	0.493916i	$0.835565\pi$
$54$	0	0
$55$	0	0
$56$	0.732051	0.0978244
$57$	0	0
$58$	1.19615	0.157063
$59$	8.19615	1.06705	0.533524	−	0.845785i	$-0.320867\pi$
$59$	8.19615	1.06705	0.533524	+	0.845785i	$0.320867\pi$
$60$	0	0
$61$	11.7321	1.50214	0.751068	−	0.660225i	$-0.229539\pi$
$61$	11.7321	1.50214	0.751068	+	0.660225i	$0.229539\pi$
$62$	−7.92820	−1.00688
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	5.19615	0.634811	0.317406	−	0.948290i	$-0.397188\pi$
$67$	5.19615	0.634811	0.317406	+	0.948290i	$0.397188\pi$
$68$	−0.732051	−0.0887742
$69$	0	0
$70$	0	0
$71$	−0.196152	−0.0232790	−0.0116395	−	0.999932i	$-0.503705\pi$
$71$	−0.196152	−0.0232790	−0.0116395	+	0.999932i	$0.503705\pi$
$72$	0	0
$73$	7.53590	0.882010	0.441005	−	0.897505i	$-0.354622\pi$
$73$	7.53590	0.882010	0.441005	+	0.897505i	$0.354622\pi$
$74$	−4.92820	−0.572892
$75$	0	0
$76$	1.00000	0.114708
$77$	−3.80385	−0.433489
$78$	0	0
$79$	−7.92820	−0.891993	−0.445996	−	0.895035i	$-0.647151\pi$
$79$	−7.92820	−0.891993	−0.445996	+	0.895035i	$0.647151\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0.660254	0.0724723	0.0362361	−	0.999343i	$-0.488463\pi$
$83$	0.660254	0.0724723	0.0362361	+	0.999343i	$0.488463\pi$
$84$	0	0
$85$	0	0
$86$	−5.26795	−0.568058
$87$	0	0
$88$	−5.19615	−0.553912
$89$	9.92820	1.05239	0.526194	−	0.850365i	$-0.323619\pi$
$89$	9.92820	1.05239	0.526194	+	0.850365i	$0.323619\pi$
$90$	0	0
$91$	−0.928203	−0.0973021
$92$	−1.53590	−0.160128
$93$	0	0
$94$	3.46410	0.357295
$95$	0	0
$96$	0	0
$97$	−7.12436	−0.723369	−0.361684	−	0.932301i	$-0.617798\pi$
$97$	−7.12436	−0.723369	−0.361684	+	0.932301i	$0.617798\pi$
$98$	6.46410	0.652973
$99$	0	0
$100$	0	0
$101$	−5.66025	−0.563216	−0.281608	−	0.959529i	$-0.590868\pi$
$101$	−5.66025	−0.563216	−0.281608	+	0.959529i	$0.590868\pi$
$102$	0	0
$103$	12.8564	1.26678	0.633390	−	0.773833i	$-0.281663\pi$
$103$	12.8564	1.26678	0.633390	+	0.773833i	$0.281663\pi$
$104$	−1.26795	−0.124333
$105$	0	0
$106$	12.6603	1.22967
$107$	6.19615	0.599005	0.299502	−	0.954096i	$-0.403179\pi$
$107$	6.19615	0.599005	0.299502	+	0.954096i	$0.403179\pi$
$108$	0	0
$109$	10.3923	0.995402	0.497701	−	0.867349i	$-0.334178\pi$
$109$	10.3923	0.995402	0.497701	+	0.867349i	$0.334178\pi$
$110$	0	0
$111$	0	0
$112$	−0.732051	−0.0691723
$113$	−20.3205	−1.91159	−0.955796	−	0.294030i	$-0.905004\pi$
$113$	−20.3205	−1.91159	−0.955796	+	0.294030i	$0.905004\pi$
$114$	0	0
$115$	0	0
$116$	−1.19615	−0.111060
$117$	0	0
$118$	−8.19615	−0.754517
$119$	0.535898	0.0491257
$120$	0	0
$121$	16.0000	1.45455
$122$	−11.7321	−1.06217
$123$	0	0
$124$	7.92820	0.711974
$125$	0	0
$126$	0	0
$127$	11.3923	1.01090	0.505452	−	0.862855i	$-0.331326\pi$
$127$	11.3923	1.01090	0.505452	+	0.862855i	$0.331326\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	0.803848	0.0702325	0.0351162	−	0.999383i	$-0.488820\pi$
$131$	0.803848	0.0702325	0.0351162	+	0.999383i	$0.488820\pi$
$132$	0	0
$133$	−0.732051	−0.0634769
$134$	−5.19615	−0.448879
$135$	0	0
$136$	0.732051	0.0627728
$137$	4.39230	0.375260	0.187630	−	0.982240i	$-0.439919\pi$
$137$	4.39230	0.375260	0.187630	+	0.982240i	$0.439919\pi$
$138$	0	0
$139$	−14.1962	−1.20410	−0.602051	−	0.798458i	$-0.705650\pi$
$139$	−14.1962	−1.20410	−0.602051	+	0.798458i	$0.705650\pi$
$140$	0	0
$141$	0	0
$142$	0.196152	0.0164607
$143$	6.58846	0.550954
$144$	0	0
$145$	0	0
$146$	−7.53590	−0.623675
$147$	0	0
$148$	4.92820	0.405096
$149$	−5.85641	−0.479776	−0.239888	−	0.970801i	$-0.577111\pi$
$149$	−5.85641	−0.479776	−0.239888	+	0.970801i	$0.577111\pi$
$150$	0	0
$151$	−10.3923	−0.845714	−0.422857	−	0.906196i	$-0.638973\pi$
$151$	−10.3923	−0.845714	−0.422857	+	0.906196i	$0.638973\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	3.80385	0.306523
$155$	0	0
$156$	0	0
$157$	−5.85641	−0.467392	−0.233696	−	0.972310i	$-0.575082\pi$
$157$	−5.85641	−0.467392	−0.233696	+	0.972310i	$0.575082\pi$
$158$	7.92820	0.630734
$159$	0	0
$160$	0	0
$161$	1.12436	0.0886116
$162$	0	0
$163$	−10.0000	−0.783260	−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000	−0.783260	−0.391630	+	0.920123i	$0.628089\pi$
$164$	0	0
$165$	0	0
$166$	−0.660254	−0.0512457
$167$	−2.00000	−0.154765	−0.0773823	−	0.997001i	$-0.524656\pi$
$167$	−2.00000	−0.154765	−0.0773823	+	0.997001i	$0.524656\pi$
$168$	0	0
$169$	−11.3923	−0.876331
$170$	0	0
$171$	0	0
$172$	5.26795	0.401677
$173$	−5.33975	−0.405973	−0.202987	−	0.979181i	$-0.565065\pi$
$173$	−5.33975	−0.405973	−0.202987	+	0.979181i	$0.565065\pi$
$174$	0	0
$175$	0	0
$176$	5.19615	0.391675
$177$	0	0
$178$	−9.92820	−0.744150
$179$	−17.8564	−1.33465	−0.667325	−	0.744766i	$-0.732561\pi$
$179$	−17.8564	−1.33465	−0.667325	+	0.744766i	$0.732561\pi$
$180$	0	0
$181$	22.7321	1.68966	0.844830	−	0.535035i	$-0.179702\pi$
$181$	22.7321	1.68966	0.844830	+	0.535035i	$0.179702\pi$
$182$	0.928203	0.0688030
$183$	0	0
$184$	1.53590	0.113228
$185$	0	0
$186$	0	0
$187$	−3.80385	−0.278165
$188$	−3.46410	−0.252646
$189$	0	0
$190$	0	0
$191$	17.5359	1.26885	0.634427	−	0.772983i	$-0.281236\pi$
$191$	17.5359	1.26885	0.634427	+	0.772983i	$0.281236\pi$
$192$	0	0
$193$	0.196152	0.0141194	0.00705968	−	0.999975i	$-0.497753\pi$
$193$	0.196152	0.0141194	0.00705968	+	0.999975i	$0.497753\pi$
$194$	7.12436	0.511499
$195$	0	0
$196$	−6.46410	−0.461722
$197$	−17.6603	−1.25824	−0.629121	−	0.777308i	$-0.716585\pi$
$197$	−17.6603	−1.25824	−0.629121	+	0.777308i	$0.716585\pi$
$198$	0	0
$199$	16.9282	1.20001	0.600004	−	0.799997i	$-0.295166\pi$
$199$	16.9282	1.20001	0.600004	+	0.799997i	$0.295166\pi$
$200$	0	0
$201$	0	0
$202$	5.66025	0.398254
$203$	0.875644	0.0614582
$204$	0	0
$205$	0	0
$206$	−12.8564	−0.895748
$207$	0	0
$208$	1.26795	0.0879165
$209$	5.19615	0.359425
$210$	0	0
$211$	−3.19615	−0.220032	−0.110016	−	0.993930i	$-0.535090\pi$
$211$	−3.19615	−0.220032	−0.110016	+	0.993930i	$0.535090\pi$
$212$	−12.6603	−0.869510
$213$	0	0
$214$	−6.19615	−0.423560
$215$	0	0
$216$	0	0
$217$	−5.80385	−0.393991
$218$	−10.3923	−0.703856
$219$	0	0
$220$	0	0
$221$	−0.928203	−0.0624377
$222$	0	0
$223$	4.07180	0.272668	0.136334	−	0.990663i	$-0.456468\pi$
$223$	4.07180	0.272668	0.136334	+	0.990663i	$0.456468\pi$
$224$	0.732051	0.0489122
$225$	0	0
$226$	20.3205	1.35170
$227$	−2.00000	−0.132745	−0.0663723	−	0.997795i	$-0.521143\pi$
$227$	−2.00000	−0.132745	−0.0663723	+	0.997795i	$0.521143\pi$
$228$	0	0
$229$	−5.19615	−0.343371	−0.171686	−	0.985152i	$-0.554921\pi$
$229$	−5.19615	−0.343371	−0.171686	+	0.985152i	$0.554921\pi$
$230$	0	0
$231$	0	0
$232$	1.19615	0.0785313
$233$	0.339746	0.0222575	0.0111287	−	0.999938i	$-0.496458\pi$
$233$	0.339746	0.0222575	0.0111287	+	0.999938i	$0.496458\pi$
$234$	0	0
$235$	0	0
$236$	8.19615	0.533524
$237$	0	0
$238$	−0.535898	−0.0347371
$239$	20.9282	1.35373	0.676866	−	0.736106i	$-0.263337\pi$
$239$	20.9282	1.35373	0.676866	+	0.736106i	$0.263337\pi$
$240$	0	0
$241$	18.1962	1.17212	0.586059	−	0.810269i	$-0.300679\pi$
$241$	18.1962	1.17212	0.586059	+	0.810269i	$0.300679\pi$
$242$	−16.0000	−1.02852
$243$	0	0
$244$	11.7321	0.751068
$245$	0	0
$246$	0	0
$247$	1.26795	0.0806777
$248$	−7.92820	−0.503441
$249$	0	0
$250$	0	0
$251$	3.46410	0.218652	0.109326	−	0.994006i	$-0.465131\pi$
$251$	3.46410	0.218652	0.109326	+	0.994006i	$0.465131\pi$
$252$	0	0
$253$	−7.98076	−0.501746
$254$	−11.3923	−0.714817
$255$	0	0
$256$	1.00000	0.0625000
$257$	2.07180	0.129235	0.0646176	−	0.997910i	$-0.479417\pi$
$257$	2.07180	0.129235	0.0646176	+	0.997910i	$0.479417\pi$
$258$	0	0
$259$	−3.60770	−0.224171
$260$	0	0
$261$	0	0
$262$	−0.803848	−0.0496619
$263$	−5.53590	−0.341358	−0.170679	−	0.985327i	$-0.554596\pi$
$263$	−5.53590	−0.341358	−0.170679	+	0.985327i	$0.554596\pi$
$264$	0	0
$265$	0	0
$266$	0.732051	0.0448849
$267$	0	0
$268$	5.19615	0.317406
$269$	−8.53590	−0.520443	−0.260221	−	0.965549i	$-0.583796\pi$
$269$	−8.53590	−0.520443	−0.260221	+	0.965549i	$0.583796\pi$
$270$	0	0
$271$	−0.875644	−0.0531916	−0.0265958	−	0.999646i	$-0.508467\pi$
$271$	−0.875644	−0.0531916	−0.0265958	+	0.999646i	$0.508467\pi$
$272$	−0.732051	−0.0443871
$273$	0	0
$274$	−4.39230	−0.265349
$275$	0	0
$276$	0	0
$277$	20.5167	1.23273	0.616363	−	0.787462i	$-0.288605\pi$
$277$	20.5167	1.23273	0.616363	+	0.787462i	$0.288605\pi$
$278$	14.1962	0.851429
$279$	0	0
$280$	0	0
$281$	11.0000	0.656205	0.328102	−	0.944642i	$-0.393591\pi$
$281$	11.0000	0.656205	0.328102	+	0.944642i	$0.393591\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−0.196152	−0.0116395
$285$	0	0
$286$	−6.58846	−0.389584
$287$	0	0
$288$	0	0
$289$	−16.4641	−0.968477
$290$	0	0
$291$	0	0
$292$	7.53590	0.441005
$293$	9.33975	0.545634	0.272817	−	0.962066i	$-0.412045\pi$
$293$	9.33975	0.545634	0.272817	+	0.962066i	$0.412045\pi$
$294$	0	0
$295$	0	0
$296$	−4.92820	−0.286446
$297$	0	0
$298$	5.85641	0.339253
$299$	−1.94744	−0.112623
$300$	0	0
$301$	−3.85641	−0.222280
$302$	10.3923	0.598010
$303$	0	0
$304$	1.00000	0.0573539
$305$	0	0
$306$	0	0
$307$	−12.6603	−0.722559	−0.361279	−	0.932458i	$-0.617660\pi$
$307$	−12.6603	−0.722559	−0.361279	+	0.932458i	$0.617660\pi$
$308$	−3.80385	−0.216744
$309$	0	0
$310$	0	0
$311$	8.53590	0.484026	0.242013	−	0.970273i	$-0.422192\pi$
$311$	8.53590	0.484026	0.242013	+	0.970273i	$0.422192\pi$
$312$	0	0
$313$	22.8564	1.29192	0.645960	−	0.763371i	$-0.276457\pi$
$313$	22.8564	1.29192	0.645960	+	0.763371i	$0.276457\pi$
$314$	5.85641	0.330496
$315$	0	0
$316$	−7.92820	−0.445996
$317$	15.0526	0.845436	0.422718	−	0.906261i	$-0.361076\pi$
$317$	15.0526	0.845436	0.422718	+	0.906261i	$0.361076\pi$
$318$	0	0
$319$	−6.21539	−0.347995
$320$	0	0
$321$	0	0
$322$	−1.12436	−0.0626579
$323$	−0.732051	−0.0407324
$324$	0	0
$325$	0	0
$326$	10.0000	0.553849
$327$	0	0
$328$	0	0
$329$	2.53590	0.139809
$330$	0	0
$331$	8.80385	0.483903	0.241952	−	0.970288i	$-0.422212\pi$
$331$	8.80385	0.483903	0.241952	+	0.970288i	$0.422212\pi$
$332$	0.660254	0.0362361
$333$	0	0
$334$	2.00000	0.109435
$335$	0	0
$336$	0	0
$337$	−19.3205	−1.05246	−0.526228	−	0.850344i	$-0.676394\pi$
$337$	−19.3205	−1.05246	−0.526228	+	0.850344i	$0.676394\pi$
$338$	11.3923	0.619660
$339$	0	0
$340$	0	0
$341$	41.1962	2.23090
$342$	0	0
$343$	9.85641	0.532196
$344$	−5.26795	−0.284029
$345$	0	0
$346$	5.33975	0.287067
$347$	−14.9282	−0.801388	−0.400694	−	0.916212i	$-0.631231\pi$
$347$	−14.9282	−0.801388	−0.400694	+	0.916212i	$0.631231\pi$
$348$	0	0
$349$	17.5885	0.941489	0.470744	−	0.882270i	$-0.343985\pi$
$349$	17.5885	0.941489	0.470744	+	0.882270i	$0.343985\pi$
$350$	0	0
$351$	0	0
$352$	−5.19615	−0.276956
$353$	34.5885	1.84096	0.920479	−	0.390792i	$-0.127799\pi$
$353$	34.5885	1.84096	0.920479	+	0.390792i	$0.127799\pi$
$354$	0	0
$355$	0	0
$356$	9.92820	0.526194
$357$	0	0
$358$	17.8564	0.943740
$359$	7.85641	0.414645	0.207323	−	0.978273i	$-0.433525\pi$
$359$	7.85641	0.414645	0.207323	+	0.978273i	$0.433525\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−22.7321	−1.19477
$363$	0	0
$364$	−0.928203	−0.0486511
$365$	0	0
$366$	0	0
$367$	−15.5167	−0.809963	−0.404982	−	0.914325i	$-0.632722\pi$
$367$	−15.5167	−0.809963	−0.404982	+	0.914325i	$0.632722\pi$
$368$	−1.53590	−0.0800642
$369$	0	0
$370$	0	0
$371$	9.26795	0.481168
$372$	0	0
$373$	10.3397	0.535372	0.267686	−	0.963506i	$-0.413741\pi$
$373$	10.3397	0.535372	0.267686	+	0.963506i	$0.413741\pi$
$374$	3.80385	0.196692
$375$	0	0
$376$	3.46410	0.178647
$377$	−1.51666	−0.0781120
$378$	0	0
$379$	33.1769	1.70418	0.852092	−	0.523392i	$-0.175334\pi$
$379$	33.1769	1.70418	0.852092	+	0.523392i	$0.175334\pi$
$380$	0	0
$381$	0	0
$382$	−17.5359	−0.897215
$383$	34.0526	1.74000	0.870002	−	0.493048i	$-0.164117\pi$
$383$	34.0526	1.74000	0.870002	+	0.493048i	$0.164117\pi$
$384$	0	0
$385$	0	0
$386$	−0.196152	−0.00998390
$387$	0	0
$388$	−7.12436	−0.361684
$389$	0.928203	0.0470618	0.0235309	−	0.999723i	$-0.492509\pi$
$389$	0.928203	0.0470618	0.0235309	+	0.999723i	$0.492509\pi$
$390$	0	0
$391$	1.12436	0.0568611
$392$	6.46410	0.326486
$393$	0	0
$394$	17.6603	0.889711
$395$	0	0
$396$	0	0
$397$	−9.87564	−0.495644	−0.247822	−	0.968806i	$-0.579715\pi$
$397$	−9.87564	−0.495644	−0.247822	+	0.968806i	$0.579715\pi$
$398$	−16.9282	−0.848534
$399$	0	0
$400$	0	0
$401$	36.7128	1.83335	0.916675	−	0.399633i	$-0.130862\pi$
$401$	36.7128	1.83335	0.916675	+	0.399633i	$0.130862\pi$
$402$	0	0
$403$	10.0526	0.500754
$404$	−5.66025	−0.281608
$405$	0	0
$406$	−0.875644	−0.0434575
$407$	25.6077	1.26933
$408$	0	0
$409$	−17.3205	−0.856444	−0.428222	−	0.903674i	$-0.640860\pi$
$409$	−17.3205	−0.856444	−0.428222	+	0.903674i	$0.640860\pi$
$410$	0	0
$411$	0	0
$412$	12.8564	0.633390
$413$	−6.00000	−0.295241
$414$	0	0
$415$	0	0
$416$	−1.26795	−0.0621663
$417$	0	0
$418$	−5.19615	−0.254152
$419$	−21.8564	−1.06776	−0.533878	−	0.845562i	$-0.679266\pi$
$419$	−21.8564	−1.06776	−0.533878	+	0.845562i	$0.679266\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	3.19615	0.155586
$423$	0	0
$424$	12.6603	0.614836
$425$	0	0
$426$	0	0
$427$	−8.58846	−0.415625
$428$	6.19615	0.299502
$429$	0	0
$430$	0	0
$431$	22.3923	1.07860	0.539300	−	0.842114i	$-0.318689\pi$
$431$	22.3923	1.07860	0.539300	+	0.842114i	$0.318689\pi$
$432$	0	0
$433$	16.1962	0.778337	0.389169	−	0.921166i	$-0.372762\pi$
$433$	16.1962	0.778337	0.389169	+	0.921166i	$0.372762\pi$
$434$	5.80385	0.278594
$435$	0	0
$436$	10.3923	0.497701
$437$	−1.53590	−0.0734720
$438$	0	0
$439$	−17.3923	−0.830089	−0.415045	−	0.909801i	$-0.636234\pi$
$439$	−17.3923	−0.830089	−0.415045	+	0.909801i	$0.636234\pi$
$440$	0	0
$441$	0	0
$442$	0.928203	0.0441501
$443$	−24.5167	−1.16482	−0.582411	−	0.812895i	$-0.697890\pi$
$443$	−24.5167	−1.16482	−0.582411	+	0.812895i	$0.697890\pi$
$444$	0	0
$445$	0	0
$446$	−4.07180	−0.192805
$447$	0	0
$448$	−0.732051	−0.0345861
$449$	6.46410	0.305060	0.152530	−	0.988299i	$-0.451258\pi$
$449$	6.46410	0.305060	0.152530	+	0.988299i	$0.451258\pi$
$450$	0	0
$451$	0	0
$452$	−20.3205	−0.955796
$453$	0	0
$454$	2.00000	0.0938647
$455$	0	0
$456$	0	0
$457$	−32.0000	−1.49690	−0.748448	−	0.663193i	$-0.769201\pi$
$457$	−32.0000	−1.49690	−0.748448	+	0.663193i	$0.769201\pi$
$458$	5.19615	0.242800
$459$	0	0
$460$	0	0
$461$	−25.8564	−1.20425	−0.602126	−	0.798401i	$-0.705680\pi$
$461$	−25.8564	−1.20425	−0.602126	+	0.798401i	$0.705680\pi$
$462$	0	0
$463$	32.3923	1.50540	0.752699	−	0.658365i	$-0.228752\pi$
$463$	32.3923	1.50540	0.752699	+	0.658365i	$0.228752\pi$
$464$	−1.19615	−0.0555300
$465$	0	0
$466$	−0.339746	−0.0157384
$467$	0.660254	0.0305529	0.0152765	−	0.999883i	$-0.495137\pi$
$467$	0.660254	0.0305529	0.0152765	+	0.999883i	$0.495137\pi$
$468$	0	0
$469$	−3.80385	−0.175645
$470$	0	0
$471$	0	0
$472$	−8.19615	−0.377258
$473$	27.3731	1.25861
$474$	0	0
$475$	0	0
$476$	0.535898	0.0245629
$477$	0	0
$478$	−20.9282	−0.957234
$479$	37.3923	1.70850	0.854249	−	0.519864i	$-0.174017\pi$
$479$	37.3923	1.70850	0.854249	+	0.519864i	$0.174017\pi$
$480$	0	0
$481$	6.24871	0.284917
$482$	−18.1962	−0.828812
$483$	0	0
$484$	16.0000	0.727273
$485$	0	0
$486$	0	0
$487$	−26.9282	−1.22023	−0.610117	−	0.792312i	$-0.708877\pi$
$487$	−26.9282	−1.22023	−0.610117	+	0.792312i	$0.708877\pi$
$488$	−11.7321	−0.531085
$489$	0	0
$490$	0	0
$491$	21.0718	0.950957	0.475478	−	0.879727i	$-0.342275\pi$
$491$	21.0718	0.950957	0.475478	+	0.879727i	$0.342275\pi$
$492$	0	0
$493$	0.875644	0.0394370
$494$	−1.26795	−0.0570477
$495$	0	0
$496$	7.92820	0.355987
$497$	0.143594	0.00644105
$498$	0	0
$499$	−11.1244	−0.497995	−0.248997	−	0.968504i	$-0.580101\pi$
$499$	−11.1244	−0.497995	−0.248997	+	0.968504i	$0.580101\pi$
$500$	0	0
$501$	0	0
$502$	−3.46410	−0.154610
$503$	−28.7846	−1.28344	−0.641721	−	0.766938i	$-0.721779\pi$
$503$	−28.7846	−1.28344	−0.641721	+	0.766938i	$0.721779\pi$
$504$	0	0
$505$	0	0
$506$	7.98076	0.354788
$507$	0	0
$508$	11.3923	0.505452
$509$	−3.73205	−0.165420	−0.0827101	−	0.996574i	$-0.526358\pi$
$509$	−3.73205	−0.165420	−0.0827101	+	0.996574i	$0.526358\pi$
$510$	0	0
$511$	−5.51666	−0.244043
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−2.07180	−0.0913830
$515$	0	0
$516$	0	0
$517$	−18.0000	−0.791639
$518$	3.60770	0.158513
$519$	0	0
$520$	0	0
$521$	−38.3205	−1.67885	−0.839426	−	0.543474i	$-0.817109\pi$
$521$	−38.3205	−1.67885	−0.839426	+	0.543474i	$0.817109\pi$
$522$	0	0
$523$	−9.85641	−0.430991	−0.215495	−	0.976505i	$-0.569137\pi$
$523$	−9.85641	−0.430991	−0.215495	+	0.976505i	$0.569137\pi$
$524$	0.803848	0.0351162
$525$	0	0
$526$	5.53590	0.241377
$527$	−5.80385	−0.252820
$528$	0	0
$529$	−20.6410	−0.897435
$530$	0	0
$531$	0	0
$532$	−0.732051	−0.0317384
$533$	0	0
$534$	0	0
$535$	0	0
$536$	−5.19615	−0.224440
$537$	0	0
$538$	8.53590	0.368009
$539$	−33.5885	−1.44676
$540$	0	0
$541$	−8.66025	−0.372333	−0.186167	−	0.982518i	$-0.559606\pi$
$541$	−8.66025	−0.372333	−0.186167	+	0.982518i	$0.559606\pi$
$542$	0.875644	0.0376121
$543$	0	0
$544$	0.732051	0.0313864
$545$	0	0
$546$	0	0
$547$	−27.5885	−1.17960	−0.589799	−	0.807550i	$-0.700793\pi$
$547$	−27.5885	−1.17960	−0.589799	+	0.807550i	$0.700793\pi$
$548$	4.39230	0.187630
$549$	0	0
$550$	0	0
$551$	−1.19615	−0.0509578
$552$	0	0
$553$	5.80385	0.246805
$554$	−20.5167	−0.871669
$555$	0	0
$556$	−14.1962	−0.602051
$557$	7.41154	0.314037	0.157019	−	0.987596i	$-0.449812\pi$
$557$	7.41154	0.314037	0.157019	+	0.987596i	$0.449812\pi$
$558$	0	0
$559$	6.67949	0.282512
$560$	0	0
$561$	0	0
$562$	−11.0000	−0.464007
$563$	21.8038	0.918923	0.459461	−	0.888198i	$-0.348042\pi$
$563$	21.8038	0.918923	0.459461	+	0.888198i	$0.348042\pi$
$564$	0	0
$565$	0	0
$566$	4.00000	0.168133
$567$	0	0
$568$	0.196152	0.00823037
$569$	41.7128	1.74869	0.874346	−	0.485303i	$-0.161291\pi$
$569$	41.7128	1.74869	0.874346	+	0.485303i	$0.161291\pi$
$570$	0	0
$571$	−7.26795	−0.304154	−0.152077	−	0.988369i	$-0.548596\pi$
$571$	−7.26795	−0.304154	−0.152077	+	0.988369i	$0.548596\pi$
$572$	6.58846	0.275477
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	14.3205	0.596171	0.298085	−	0.954539i	$-0.403652\pi$
$577$	14.3205	0.596171	0.298085	+	0.954539i	$0.403652\pi$
$578$	16.4641	0.684816
$579$	0	0
$580$	0	0
$581$	−0.483340	−0.0200523
$582$	0	0
$583$	−65.7846	−2.72452
$584$	−7.53590	−0.311838
$585$	0	0
$586$	−9.33975	−0.385821
$587$	21.7321	0.896978	0.448489	−	0.893788i	$-0.351962\pi$
$587$	21.7321	0.896978	0.448489	+	0.893788i	$0.351962\pi$
$588$	0	0
$589$	7.92820	0.326676
$590$	0	0
$591$	0	0
$592$	4.92820	0.202548
$593$	−18.2487	−0.749385	−0.374692	−	0.927149i	$-0.622252\pi$
$593$	−18.2487	−0.749385	−0.374692	+	0.927149i	$0.622252\pi$
$594$	0	0
$595$	0	0
$596$	−5.85641	−0.239888
$597$	0	0
$598$	1.94744	0.0796368
$599$	−34.9808	−1.42928	−0.714638	−	0.699495i	$-0.753408\pi$
$599$	−34.9808	−1.42928	−0.714638	+	0.699495i	$0.753408\pi$
$600$	0	0
$601$	36.1962	1.47647	0.738236	−	0.674543i	$-0.235659\pi$
$601$	36.1962	1.47647	0.738236	+	0.674543i	$0.235659\pi$
$602$	3.85641	0.157175
$603$	0	0
$604$	−10.3923	−0.422857
$605$	0	0
$606$	0	0
$607$	41.1051	1.66841	0.834203	−	0.551458i	$-0.185928\pi$
$607$	41.1051	1.66841	0.834203	+	0.551458i	$0.185928\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	0	0
$611$	−4.39230	−0.177694
$612$	0	0
$613$	20.9282	0.845282	0.422641	−	0.906297i	$-0.361103\pi$
$613$	20.9282	0.845282	0.422641	+	0.906297i	$0.361103\pi$
$614$	12.6603	0.510926
$615$	0	0
$616$	3.80385	0.153261
$617$	33.3205	1.34143	0.670717	−	0.741714i	$-0.265987\pi$
$617$	33.3205	1.34143	0.670717	+	0.741714i	$0.265987\pi$
$618$	0	0
$619$	−24.9808	−1.00406	−0.502031	−	0.864850i	$-0.667414\pi$
$619$	−24.9808	−1.00406	−0.502031	+	0.864850i	$0.667414\pi$
$620$	0	0
$621$	0	0
$622$	−8.53590	−0.342258
$623$	−7.26795	−0.291184
$624$	0	0
$625$	0	0
$626$	−22.8564	−0.913526
$627$	0	0
$628$	−5.85641	−0.233696
$629$	−3.60770	−0.143848
$630$	0	0
$631$	10.9282	0.435045	0.217522	−	0.976055i	$-0.430202\pi$
$631$	10.9282	0.435045	0.217522	+	0.976055i	$0.430202\pi$
$632$	7.92820	0.315367
$633$	0	0
$634$	−15.0526	−0.597813
$635$	0	0
$636$	0	0
$637$	−8.19615	−0.324743
$638$	6.21539	0.246070
$639$	0	0
$640$	0	0
$641$	−8.39230	−0.331476	−0.165738	−	0.986170i	$-0.553001\pi$
$641$	−8.39230	−0.331476	−0.165738	+	0.986170i	$0.553001\pi$
$642$	0	0
$643$	19.0718	0.752118	0.376059	−	0.926596i	$-0.377279\pi$
$643$	19.0718	0.752118	0.376059	+	0.926596i	$0.377279\pi$
$644$	1.12436	0.0443058
$645$	0	0
$646$	0.732051	0.0288022
$647$	0.0717968	0.00282262	0.00141131	−	0.999999i	$-0.499551\pi$
$647$	0.0717968	0.00282262	0.00141131	+	0.999999i	$0.499551\pi$
$648$	0	0
$649$	42.5885	1.67174
$650$	0	0
$651$	0	0
$652$	−10.0000	−0.391630
$653$	37.1769	1.45485	0.727423	−	0.686190i	$-0.240718\pi$
$653$	37.1769	1.45485	0.727423	+	0.686190i	$0.240718\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	−2.53590	−0.0988596
$659$	10.7321	0.418061	0.209031	−	0.977909i	$-0.432969\pi$
$659$	10.7321	0.418061	0.209031	+	0.977909i	$0.432969\pi$
$660$	0	0
$661$	13.0718	0.508434	0.254217	−	0.967147i	$-0.418182\pi$
$661$	13.0718	0.508434	0.254217	+	0.967147i	$0.418182\pi$
$662$	−8.80385	−0.342171
$663$	0	0
$664$	−0.660254	−0.0256228
$665$	0	0
$666$	0	0
$667$	1.83717	0.0711355
$668$	−2.00000	−0.0773823
$669$	0	0
$670$	0	0
$671$	60.9615	2.35339
$672$	0	0
$673$	−32.7321	−1.26173	−0.630864	−	0.775893i	$-0.717299\pi$
$673$	−32.7321	−1.26173	−0.630864	+	0.775893i	$0.717299\pi$
$674$	19.3205	0.744198
$675$	0	0
$676$	−11.3923	−0.438166
$677$	48.1244	1.84957	0.924785	−	0.380491i	$-0.124245\pi$
$677$	48.1244	1.84957	0.924785	+	0.380491i	$0.124245\pi$
$678$	0	0
$679$	5.21539	0.200148
$680$	0	0
$681$	0	0
$682$	−41.1962	−1.57748
$683$	45.1244	1.72664	0.863318	−	0.504661i	$-0.168382\pi$
$683$	45.1244	1.72664	0.863318	+	0.504661i	$0.168382\pi$
$684$	0	0
$685$	0	0
$686$	−9.85641	−0.376319
$687$	0	0
$688$	5.26795	0.200839
$689$	−16.0526	−0.611554
$690$	0	0
$691$	−5.12436	−0.194940	−0.0974698	−	0.995238i	$-0.531075\pi$
$691$	−5.12436	−0.194940	−0.0974698	+	0.995238i	$0.531075\pi$
$692$	−5.33975	−0.202987
$693$	0	0
$694$	14.9282	0.566667
$695$	0	0
$696$	0	0
$697$	0	0
$698$	−17.5885	−0.665733
$699$	0	0
$700$	0	0
$701$	−8.67949	−0.327820	−0.163910	−	0.986475i	$-0.552411\pi$
$701$	−8.67949	−0.327820	−0.163910	+	0.986475i	$0.552411\pi$
$702$	0	0
$703$	4.92820	0.185871
$704$	5.19615	0.195837
$705$	0	0
$706$	−34.5885	−1.30175
$707$	4.14359	0.155836
$708$	0	0
$709$	29.4449	1.10583	0.552913	−	0.833239i	$-0.313516\pi$
$709$	29.4449	1.10583	0.552913	+	0.833239i	$0.313516\pi$
$710$	0	0
$711$	0	0
$712$	−9.92820	−0.372075
$713$	−12.1769	−0.456029
$714$	0	0
$715$	0	0
$716$	−17.8564	−0.667325
$717$	0	0
$718$	−7.85641	−0.293198
$719$	−42.5692	−1.58756	−0.793782	−	0.608202i	$-0.791891\pi$
$719$	−42.5692	−1.58756	−0.793782	+	0.608202i	$0.791891\pi$
$720$	0	0
$721$	−9.41154	−0.350504
$722$	−1.00000	−0.0372161
$723$	0	0
$724$	22.7321	0.844830
$725$	0	0
$726$	0	0
$727$	3.46410	0.128476	0.0642382	−	0.997935i	$-0.479538\pi$
$727$	3.46410	0.128476	0.0642382	+	0.997935i	$0.479538\pi$
$728$	0.928203	0.0344015
$729$	0	0
$730$	0	0
$731$	−3.85641	−0.142634
$732$	0	0
$733$	−8.26795	−0.305384	−0.152692	−	0.988274i	$-0.548794\pi$
$733$	−8.26795	−0.305384	−0.152692	+	0.988274i	$0.548794\pi$
$734$	15.5167	0.572730
$735$	0	0
$736$	1.53590	0.0566140
$737$	27.0000	0.994558
$738$	0	0
$739$	26.2487	0.965574	0.482787	−	0.875738i	$-0.339624\pi$
$739$	26.2487	0.965574	0.482787	+	0.875738i	$0.339624\pi$
$740$	0	0
$741$	0	0
$742$	−9.26795	−0.340237
$743$	−24.7846	−0.909259	−0.454630	−	0.890681i	$-0.650228\pi$
$743$	−24.7846	−0.909259	−0.454630	+	0.890681i	$0.650228\pi$
$744$	0	0
$745$	0	0
$746$	−10.3397	−0.378565
$747$	0	0
$748$	−3.80385	−0.139082
$749$	−4.53590	−0.165738
$750$	0	0
$751$	24.5359	0.895328	0.447664	−	0.894202i	$-0.352256\pi$
$751$	24.5359	0.895328	0.447664	+	0.894202i	$0.352256\pi$
$752$	−3.46410	−0.126323
$753$	0	0
$754$	1.51666	0.0552335
$755$	0	0
$756$	0	0
$757$	−39.9808	−1.45313	−0.726563	−	0.687100i	$-0.758883\pi$
$757$	−39.9808	−1.45313	−0.726563	+	0.687100i	$0.758883\pi$
$758$	−33.1769	−1.20504
$759$	0	0
$760$	0	0
$761$	−23.5167	−0.852478	−0.426239	−	0.904611i	$-0.640162\pi$
$761$	−23.5167	−0.852478	−0.426239	+	0.904611i	$0.640162\pi$
$762$	0	0
$763$	−7.60770	−0.275417
$764$	17.5359	0.634427
$765$	0	0
$766$	−34.0526	−1.23037
$767$	10.3923	0.375244
$768$	0	0
$769$	38.0718	1.37290	0.686452	−	0.727175i	$-0.259167\pi$
$769$	38.0718	1.37290	0.686452	+	0.727175i	$0.259167\pi$
$770$	0	0
$771$	0	0
$772$	0.196152	0.00705968
$773$	21.6077	0.777175	0.388587	−	0.921412i	$-0.372963\pi$
$773$	21.6077	0.777175	0.388587	+	0.921412i	$0.372963\pi$
$774$	0	0
$775$	0	0
$776$	7.12436	0.255749
$777$	0	0
$778$	−0.928203	−0.0332777
$779$	0	0
$780$	0	0
$781$	−1.01924	−0.0364712
$782$	−1.12436	−0.0402069
$783$	0	0
$784$	−6.46410	−0.230861
$785$	0	0
$786$	0	0
$787$	−9.98076	−0.355776	−0.177888	−	0.984051i	$-0.556926\pi$
$787$	−9.98076	−0.355776	−0.177888	+	0.984051i	$0.556926\pi$
$788$	−17.6603	−0.629121
$789$	0	0
$790$	0	0
$791$	14.8756	0.528917
$792$	0	0
$793$	14.8756	0.528250
$794$	9.87564	0.350474
$795$	0	0
$796$	16.9282	0.600004
$797$	53.5692	1.89752	0.948760	−	0.315999i	$-0.102340\pi$
$797$	53.5692	1.89752	0.948760	+	0.315999i	$0.102340\pi$
$798$	0	0
$799$	2.53590	0.0897136
$800$	0	0
$801$	0	0
$802$	−36.7128	−1.29637
$803$	39.1577	1.38184
$804$	0	0
$805$	0	0
$806$	−10.0526	−0.354086
$807$	0	0
$808$	5.66025	0.199127
$809$	−5.12436	−0.180163	−0.0900814	−	0.995934i	$-0.528713\pi$
$809$	−5.12436	−0.180163	−0.0900814	+	0.995934i	$0.528713\pi$
$810$	0	0
$811$	26.5167	0.931126	0.465563	−	0.885015i	$-0.345852\pi$
$811$	26.5167	0.931126	0.465563	+	0.885015i	$0.345852\pi$
$812$	0.875644	0.0307291
$813$	0	0
$814$	−25.6077	−0.897549
$815$	0	0
$816$	0	0
$817$	5.26795	0.184302
$818$	17.3205	0.605597
$819$	0	0
$820$	0	0
$821$	−25.8038	−0.900560	−0.450280	−	0.892887i	$-0.648676\pi$
$821$	−25.8038	−0.900560	−0.450280	+	0.892887i	$0.648676\pi$
$822$	0	0
$823$	29.7128	1.03572	0.517862	−	0.855464i	$-0.326728\pi$
$823$	29.7128	1.03572	0.517862	+	0.855464i	$0.326728\pi$
$824$	−12.8564	−0.447874
$825$	0	0
$826$	6.00000	0.208767
$827$	−9.26795	−0.322278	−0.161139	−	0.986932i	$-0.551517\pi$
$827$	−9.26795	−0.322278	−0.161139	+	0.986932i	$0.551517\pi$
$828$	0	0
$829$	−3.26795	−0.113501	−0.0567503	−	0.998388i	$-0.518074\pi$
$829$	−3.26795	−0.113501	−0.0567503	+	0.998388i	$0.518074\pi$
$830$	0	0
$831$	0	0
$832$	1.26795	0.0439582
$833$	4.73205	0.163956
$834$	0	0
$835$	0	0
$836$	5.19615	0.179713
$837$	0	0
$838$	21.8564	0.755017
$839$	−29.9090	−1.03257	−0.516286	−	0.856416i	$-0.672686\pi$
$839$	−29.9090	−1.03257	−0.516286	+	0.856416i	$0.672686\pi$
$840$	0	0
$841$	−27.5692	−0.950663
$842$	2.00000	0.0689246
$843$	0	0
$844$	−3.19615	−0.110016
$845$	0	0
$846$	0	0
$847$	−11.7128	−0.402457
$848$	−12.6603	−0.434755
$849$	0	0
$850$	0	0
$851$	−7.56922	−0.259469
$852$	0	0
$853$	51.1769	1.75226	0.876132	−	0.482071i	$-0.160115\pi$
$853$	51.1769	1.75226	0.876132	+	0.482071i	$0.160115\pi$
$854$	8.58846	0.293891
$855$	0	0
$856$	−6.19615	−0.211780
$857$	13.8564	0.473326	0.236663	−	0.971592i	$-0.423946\pi$
$857$	13.8564	0.473326	0.236663	+	0.971592i	$0.423946\pi$
$858$	0	0
$859$	10.7846	0.367966	0.183983	−	0.982929i	$-0.441101\pi$
$859$	10.7846	0.367966	0.183983	+	0.982929i	$0.441101\pi$
$860$	0	0
$861$	0	0
$862$	−22.3923	−0.762685
$863$	44.3923	1.51113	0.755566	−	0.655073i	$-0.227362\pi$
$863$	44.3923	1.51113	0.755566	+	0.655073i	$0.227362\pi$
$864$	0	0
$865$	0	0
$866$	−16.1962	−0.550368
$867$	0	0
$868$	−5.80385	−0.196995
$869$	−41.1962	−1.39748
$870$	0	0
$871$	6.58846	0.223241
$872$	−10.3923	−0.351928
$873$	0	0
$874$	1.53590	0.0519525
$875$	0	0
$876$	0	0
$877$	44.5359	1.50387	0.751935	−	0.659237i	$-0.229121\pi$
$877$	44.5359	1.50387	0.751935	+	0.659237i	$0.229121\pi$
$878$	17.3923	0.586962
$879$	0	0
$880$	0	0
$881$	−40.3923	−1.36085	−0.680426	−	0.732817i	$-0.738205\pi$
$881$	−40.3923	−1.36085	−0.680426	+	0.732817i	$0.738205\pi$
$882$	0	0
$883$	−21.9090	−0.737295	−0.368648	−	0.929569i	$-0.620179\pi$
$883$	−21.9090	−0.737295	−0.368648	+	0.929569i	$0.620179\pi$
$884$	−0.928203	−0.0312189
$885$	0	0
$886$	24.5167	0.823653
$887$	−5.46410	−0.183467	−0.0917333	−	0.995784i	$-0.529241\pi$
$887$	−5.46410	−0.183467	−0.0917333	+	0.995784i	$0.529241\pi$
$888$	0	0
$889$	−8.33975	−0.279706
$890$	0	0
$891$	0	0
$892$	4.07180	0.136334
$893$	−3.46410	−0.115922
$894$	0	0
$895$	0	0
$896$	0.732051	0.0244561
$897$	0	0
$898$	−6.46410	−0.215710
$899$	−9.48334	−0.316287
$900$	0	0
$901$	9.26795	0.308760
$902$	0	0
$903$	0	0
$904$	20.3205	0.675850
$905$	0	0
$906$	0	0
$907$	41.3205	1.37202	0.686012	−	0.727590i	$-0.259360\pi$
$907$	41.3205	1.37202	0.686012	+	0.727590i	$0.259360\pi$
$908$	−2.00000	−0.0663723
$909$	0	0
$910$	0	0
$911$	38.3923	1.27199	0.635997	−	0.771692i	$-0.280589\pi$
$911$	38.3923	1.27199	0.635997	+	0.771692i	$0.280589\pi$
$912$	0	0
$913$	3.43078	0.113542
$914$	32.0000	1.05847
$915$	0	0
$916$	−5.19615	−0.171686
$917$	−0.588457	−0.0194326
$918$	0	0
$919$	−42.8372	−1.41307	−0.706534	−	0.707679i	$-0.749742\pi$
$919$	−42.8372	−1.41307	−0.706534	+	0.707679i	$0.749742\pi$
$920$	0	0
$921$	0	0
$922$	25.8564	0.851535
$923$	−0.248711	−0.00818643
$924$	0	0
$925$	0	0
$926$	−32.3923	−1.06448
$927$	0	0
$928$	1.19615	0.0392656
$929$	−21.0718	−0.691343	−0.345672	−	0.938356i	$-0.612349\pi$
$929$	−21.0718	−0.691343	−0.345672	+	0.938356i	$0.612349\pi$
$930$	0	0
$931$	−6.46410	−0.211852
$932$	0.339746	0.0111287
$933$	0	0
$934$	−0.660254	−0.0216042
$935$	0	0
$936$	0	0
$937$	46.7846	1.52839	0.764193	−	0.644987i	$-0.223137\pi$
$937$	46.7846	1.52839	0.764193	+	0.644987i	$0.223137\pi$
$938$	3.80385	0.124200
$939$	0	0
$940$	0	0
$941$	4.26795	0.139131	0.0695656	−	0.997577i	$-0.477839\pi$
$941$	4.26795	0.139131	0.0695656	+	0.997577i	$0.477839\pi$
$942$	0	0
$943$	0	0
$944$	8.19615	0.266762
$945$	0	0
$946$	−27.3731	−0.889975
$947$	2.14359	0.0696574	0.0348287	−	0.999393i	$-0.488911\pi$
$947$	2.14359	0.0696574	0.0348287	+	0.999393i	$0.488911\pi$
$948$	0	0
$949$	9.55514	0.310173
$950$	0	0
$951$	0	0
$952$	−0.535898	−0.0173686
$953$	−13.3923	−0.433819	−0.216910	−	0.976192i	$-0.569598\pi$
$953$	−13.3923	−0.433819	−0.216910	+	0.976192i	$0.569598\pi$
$954$	0	0
$955$	0	0
$956$	20.9282	0.676866
$957$	0	0
$958$	−37.3923	−1.20809
$959$	−3.21539	−0.103830
$960$	0	0
$961$	31.8564	1.02763
$962$	−6.24871	−0.201467
$963$	0	0
$964$	18.1962	0.586059
$965$	0	0
$966$	0	0
$967$	41.0333	1.31954	0.659771	−	0.751466i	$-0.270653\pi$
$967$	41.0333	1.31954	0.659771	+	0.751466i	$0.270653\pi$
$968$	−16.0000	−0.514259
$969$	0	0
$970$	0	0
$971$	44.1962	1.41832	0.709161	−	0.705047i	$-0.249074\pi$
$971$	44.1962	1.41832	0.709161	+	0.705047i	$0.249074\pi$
$972$	0	0
$973$	10.3923	0.333162
$974$	26.9282	0.862835
$975$	0	0
$976$	11.7321	0.375534
$977$	9.46410	0.302783	0.151392	−	0.988474i	$-0.451625\pi$
$977$	9.46410	0.302783	0.151392	+	0.988474i	$0.451625\pi$
$978$	0	0
$979$	51.5885	1.64877
$980$	0	0
$981$	0	0
$982$	−21.0718	−0.672428
$983$	−5.66025	−0.180534	−0.0902670	−	0.995918i	$-0.528772\pi$
$983$	−5.66025	−0.180534	−0.0902670	+	0.995918i	$0.528772\pi$
$984$	0	0
$985$	0	0
$986$	−0.875644	−0.0278862
$987$	0	0
$988$	1.26795	0.0403388
$989$	−8.09103	−0.257280
$990$	0	0
$991$	−27.7846	−0.882607	−0.441304	−	0.897358i	$-0.645484\pi$
$991$	−27.7846	−0.882607	−0.441304	+	0.897358i	$0.645484\pi$
$992$	−7.92820	−0.251721
$993$	0	0
$994$	−0.143594	−0.00455451
$995$	0	0
$996$	0	0
$997$	40.6603	1.28772	0.643862	−	0.765142i	$-0.277331\pi$
$997$	40.6603	1.28772	0.643862	+	0.765142i	$0.277331\pi$
$998$	11.1244	0.352135
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8550.2.a.bt.1.1		2
3.2	odd	2		2850.2.a.bh.1.1	yes	2
5.4	even	2		8550.2.a.bz.1.2		2
15.2	even	4		2850.2.d.u.799.3		4
15.8	even	4		2850.2.d.u.799.2		4
15.14	odd	2		2850.2.a.be.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2850.2.a.be.1.2	✓	2	15.14	odd	2
2850.2.a.bh.1.1	yes	2	3.2	odd	2
2850.2.d.u.799.2		4	15.8	even	4
2850.2.d.u.799.3		4	15.2	even	4
8550.2.a.bt.1.1		2	1.1	even	1	trivial
8550.2.a.bz.1.2		2	5.4	even	2

Overview	Random
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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:	\( N \)	\(=\)	\( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8550.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -0.732051 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -0.732051 q^{7} -1.00000 q^{8} +5.19615 q^{11} +1.26795 q^{13} +0.732051 q^{14} +1.00000 q^{16} -0.732051 q^{17} +1.00000 q^{19} -5.19615 q^{22} -1.53590 q^{23} -1.26795 q^{26} -0.732051 q^{28} -1.19615 q^{29} +7.92820 q^{31} -1.00000 q^{32} +0.732051 q^{34} +4.92820 q^{37} -1.00000 q^{38} +5.26795 q^{43} +5.19615 q^{44} +1.53590 q^{46} -3.46410 q^{47} -6.46410 q^{49} +1.26795 q^{52} -12.6603 q^{53} +0.732051 q^{56} +1.19615 q^{58} +8.19615 q^{59} +11.7321 q^{61} -7.92820 q^{62} +1.00000 q^{64} +5.19615 q^{67} -0.732051 q^{68} -0.196152 q^{71} +7.53590 q^{73} -4.92820 q^{74} +1.00000 q^{76} -3.80385 q^{77} -7.92820 q^{79} +0.660254 q^{83} -5.26795 q^{86} -5.19615 q^{88} +9.92820 q^{89} -0.928203 q^{91} -1.53590 q^{92} +3.46410 q^{94} -7.12436 q^{97} +6.46410 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8} + 6 q^{13} - 2 q^{14} + 2 q^{16} + 2 q^{17} + 2 q^{19} - 10 q^{23} - 6 q^{26} + 2 q^{28} + 8 q^{29} + 2 q^{31} - 2 q^{32} - 2 q^{34} - 4 q^{37} - 2 q^{38} + 14 q^{43} + 10 q^{46} - 6 q^{49} + 6 q^{52} - 8 q^{53} - 2 q^{56} - 8 q^{58} + 6 q^{59} + 20 q^{61} - 2 q^{62} + 2 q^{64} + 2 q^{68} + 10 q^{71} + 22 q^{73} + 4 q^{74} + 2 q^{76} - 18 q^{77} - 2 q^{79} - 16 q^{83} - 14 q^{86} + 6 q^{89} + 12 q^{91} - 10 q^{92} + 10 q^{97} + 6 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8 + 6 * q^13 - 2 * q^14 + 2 * q^16 + 2 * q^17 + 2 * q^19 - 10 * q^23 - 6 * q^26 + 2 * q^28 + 8 * q^29 + 2 * q^31 - 2 * q^32 - 2 * q^34 - 4 * q^37 - 2 * q^38 + 14 * q^43 + 10 * q^46 - 6 * q^49 + 6 * q^52 - 8 * q^53 - 2 * q^56 - 8 * q^58 + 6 * q^59 + 20 * q^61 - 2 * q^62 + 2 * q^64 + 2 * q^68 + 10 * q^71 + 22 * q^73 + 4 * q^74 + 2 * q^76 - 18 * q^77 - 2 * q^79 - 16 * q^83 - 14 * q^86 + 6 * q^89 + 12 * q^91 - 10 * q^92 + 10 * q^97 + 6 * q^98

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