LMFDB - Embedded newform 860.2.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [860,2,Mod(1,860)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(860, 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("860.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 860 = 2^{2} \cdot 5 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	860.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:	$6.86713457383$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.65057.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + 5x + 12 $ x^4 - x^3 - 8x^2 + 5x + 12
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.07193$ of defining polynomial
Character	$\chi$	$=$	860.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.07193 q^{3} +1.00000 q^{5} +4.85096 q^{7} -1.85096 q^{9} +O(q^{10})$	$q-1.07193 q^{3} +1.00000 q^{5} +4.85096 q^{7} -1.85096 q^{9} -0.779027 q^{11} +4.90700 q^{13} -1.07193 q^{15} -1.34894 q^{17} +2.00000 q^{19} -5.19990 q^{21} -7.05086 q^{23} +1.00000 q^{25} +5.19990 q^{27} +6.19990 q^{31} +0.835064 q^{33} +4.85096 q^{35} -0.127971 q^{37} -5.25997 q^{39} +0.420877 q^{41} +1.00000 q^{43} -1.85096 q^{45} +12.5437 q^{47} +16.5318 q^{49} +1.44598 q^{51} -8.60892 q^{53} -0.779027 q^{55} -2.14387 q^{57} +0.815992 q^{59} -1.41419 q^{61} -8.97893 q^{63} +4.90700 q^{65} +8.93879 q^{67} +7.55805 q^{69} +4.14387 q^{71} -5.76982 q^{73} -1.07193 q^{75} -3.77903 q^{77} +11.6860 q^{79} -0.0210696 q^{81} +14.3108 q^{83} -1.34894 q^{85} +8.39981 q^{89} +23.8036 q^{91} -6.64588 q^{93} +2.00000 q^{95} -8.35298 q^{97} +1.44195 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{3} + 4 q^{5} + 7 q^{7} + 5 q^{9}+O(q^{10}) $ 4 * q + q^3 + 4 * q^5 + 7 * q^7 + 5 * q^9	$ 4 q + q^{3} + 4 q^{5} + 7 q^{7} + 5 q^{9} + 4 q^{11} + q^{13} + q^{15} - q^{17} + 8 q^{19} - 4 q^{21} + q^{23} + 4 q^{25} + 4 q^{27} + 8 q^{31} - 10 q^{33} + 7 q^{35} + 11 q^{37} + 18 q^{39} - 8 q^{41} + 4 q^{43} + 5 q^{45} + 6 q^{47} + 5 q^{49} + 4 q^{51} + 9 q^{53} + 4 q^{55} + 2 q^{57} + 21 q^{59} - 2 q^{61} - 12 q^{63} + q^{65} + 19 q^{67} + 16 q^{69} + 6 q^{71} - 9 q^{73} + q^{75} - 8 q^{77} + 21 q^{79} - 24 q^{81} - 11 q^{83} - q^{85} + 12 q^{91} - 8 q^{93} + 8 q^{95} - 13 q^{97} + 20 q^{99}+O(q^{100}) $ 4 * q + q^3 + 4 * q^5 + 7 * q^7 + 5 * q^9 + 4 * q^11 + q^13 + q^15 - q^17 + 8 * q^19 - 4 * q^21 + q^23 + 4 * q^25 + 4 * q^27 + 8 * q^31 - 10 * q^33 + 7 * q^35 + 11 * q^37 + 18 * q^39 - 8 * q^41 + 4 * q^43 + 5 * q^45 + 6 * q^47 + 5 * q^49 + 4 * q^51 + 9 * q^53 + 4 * q^55 + 2 * q^57 + 21 * q^59 - 2 * q^61 - 12 * q^63 + q^65 + 19 * q^67 + 16 * q^69 + 6 * q^71 - 9 * q^73 + q^75 - 8 * q^77 + 21 * q^79 - 24 * q^81 - 11 * q^83 - q^85 + 12 * q^91 - 8 * q^93 + 8 * q^95 - 13 * q^97 + 20 * q^99

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
$3$	−1.07193	−0.618881	−0.309440	−	0.950919i	$-0.600142\pi$
$3$	−1.07193	−0.618881	−0.309440	+	0.950919i	$0.600142\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.85096	1.83349	0.916745	−	0.399472i	$-0.130807\pi$
$7$	4.85096	1.83349	0.916745	+	0.399472i	$0.130807\pi$
$8$	0	0
$9$	−1.85096	−0.616987
$10$	0	0
$11$	−0.779027	−0.234885	−0.117443	−	0.993080i	$-0.537470\pi$
$11$	−0.779027	−0.234885	−0.117443	+	0.993080i	$0.537470\pi$
$12$	0	0
$13$	4.90700	1.36096	0.680478	−	0.732768i	$-0.261772\pi$
$13$	4.90700	1.36096	0.680478	+	0.732768i	$0.261772\pi$
$14$	0	0
$15$	−1.07193	−0.276772
$16$	0	0
$17$	−1.34894	−0.327167	−0.163584	−	0.986529i	$-0.552305\pi$
$17$	−1.34894	−0.327167	−0.163584	+	0.986529i	$0.552305\pi$
$18$	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
$21$	−5.19990	−1.13471
$22$	0	0
$23$	−7.05086	−1.47021	−0.735103	−	0.677955i	$-0.762866\pi$
$23$	−7.05086	−1.47021	−0.735103	+	0.677955i	$0.762866\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.19990	1.00072
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	6.19990	1.11354	0.556768	−	0.830668i	$-0.312041\pi$
$31$	6.19990	1.11354	0.556768	+	0.830668i	$0.312041\pi$
$32$	0	0
$33$	0.835064	0.145366
$34$	0	0
$35$	4.85096	0.819962
$36$	0	0
$37$	−0.127971	−0.0210383	−0.0105191	−	0.999945i	$-0.503348\pi$
$37$	−0.127971	−0.0210383	−0.0105191	+	0.999945i	$0.503348\pi$
$38$	0	0
$39$	−5.25997	−0.842270
$40$	0	0
$41$	0.420877	0.0657300	0.0328650	−	0.999460i	$-0.489537\pi$
$41$	0.420877	0.0657300	0.0328650	+	0.999460i	$0.489537\pi$
$42$	0	0
$43$	1.00000	0.152499
$43$	1.00000	0.152499
$44$	0	0
$45$	−1.85096	−0.275925
$46$	0	0
$47$	12.5437	1.82968	0.914841	−	0.403813i	$-0.132315\pi$
$47$	12.5437	1.82968	0.914841	+	0.403813i	$0.132315\pi$
$48$	0	0
$49$	16.5318	2.36169
$50$	0	0
$51$	1.44598	0.202477
$52$	0	0
$53$	−8.60892	−1.18253	−0.591263	−	0.806479i	$-0.701370\pi$
$53$	−8.60892	−1.18253	−0.591263	+	0.806479i	$0.701370\pi$
$54$	0	0
$55$	−0.779027	−0.105044
$56$	0	0
$57$	−2.14387	−0.283962
$58$	0	0
$59$	0.815992	0.106233	0.0531165	−	0.998588i	$-0.483085\pi$
$59$	0.815992	0.106233	0.0531165	+	0.998588i	$0.483085\pi$
$60$	0	0
$61$	−1.41419	−0.181068	−0.0905341	−	0.995893i	$-0.528857\pi$
$61$	−1.41419	−0.181068	−0.0905341	+	0.995893i	$0.528857\pi$
$62$	0	0
$63$	−8.97893	−1.13124
$64$	0	0
$65$	4.90700	0.608638
$66$	0	0
$67$	8.93879	1.09205	0.546023	−	0.837770i	$-0.316141\pi$
$67$	8.93879	1.09205	0.546023	+	0.837770i	$0.316141\pi$
$68$	0	0
$69$	7.55805	0.909883
$70$	0	0
$71$	4.14387	0.491786	0.245893	−	0.969297i	$-0.420919\pi$
$71$	4.14387	0.491786	0.245893	+	0.969297i	$0.420919\pi$
$72$	0	0
$73$	−5.76982	−0.675307	−0.337653	−	0.941271i	$-0.609633\pi$
$73$	−5.76982	−0.675307	−0.337653	+	0.941271i	$0.609633\pi$
$74$	0	0
$75$	−1.07193	−0.123776
$76$	0	0
$77$	−3.77903	−0.430660
$78$	0	0
$79$	11.6860	1.31478	0.657390	−	0.753550i	$-0.271660\pi$
$79$	11.6860	1.31478	0.657390	+	0.753550i	$0.271660\pi$
$80$	0	0
$81$	−0.0210696	−0.00234107
$82$	0	0
$83$	14.3108	1.57082	0.785409	−	0.618977i	$-0.212453\pi$
$83$	14.3108	1.57082	0.785409	+	0.618977i	$0.212453\pi$
$84$	0	0
$85$	−1.34894	−0.146314
$86$	0	0
$87$	0	0
$88$	0	0
$89$	8.39981	0.890378	0.445189	−	0.895437i	$-0.353137\pi$
$89$	8.39981	0.890378	0.445189	+	0.895437i	$0.353137\pi$
$90$	0	0
$91$	23.8036	2.49530
$92$	0	0
$93$	−6.64588	−0.689146
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	−8.35298	−0.848116	−0.424058	−	0.905635i	$-0.639395\pi$
$97$	−8.35298	−0.848116	−0.424058	+	0.905635i	$0.639395\pi$
$98$	0	0
$99$	1.44195	0.144921
$100$	0	0
$101$	−5.94396	−0.591446	−0.295723	−	0.955274i	$-0.595561\pi$
$101$	−5.94396	−0.591446	−0.295723	+	0.955274i	$0.595561\pi$
$102$	0	0
$103$	−18.3108	−1.80422	−0.902110	−	0.431506i	$-0.857982\pi$
$103$	−18.3108	−1.80422	−0.902110	+	0.431506i	$0.857982\pi$
$104$	0	0
$105$	−5.19990	−0.507459
$106$	0	0
$107$	−3.30211	−0.319227	−0.159614	−	0.987180i	$-0.551025\pi$
$107$	−3.30211	−0.319227	−0.159614	+	0.987180i	$0.551025\pi$
$108$	0	0
$109$	−6.45585	−0.618358	−0.309179	−	0.951004i	$-0.600054\pi$
$109$	−6.45585	−0.618358	−0.309179	+	0.951004i	$0.600054\pi$
$110$	0	0
$111$	0.137176	0.0130202
$112$	0	0
$113$	−16.1109	−1.51559	−0.757795	−	0.652493i	$-0.773723\pi$
$113$	−16.1109	−1.51559	−0.757795	+	0.652493i	$0.773723\pi$
$114$	0	0
$115$	−7.05086	−0.657496
$116$	0	0
$117$	−9.08265	−0.839692
$118$	0	0
$119$	−6.54367	−0.599858
$120$	0	0
$121$	−10.3931	−0.944829
$122$	0	0
$123$	−0.451152	−0.0406790
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−19.4507	−1.72597	−0.862984	−	0.505231i	$-0.831407\pi$
$127$	−19.4507	−1.72597	−0.862984	+	0.505231i	$0.831407\pi$
$128$	0	0
$129$	−1.07193	−0.0943784
$130$	0	0
$131$	17.1058	1.49454	0.747268	−	0.664522i	$-0.231365\pi$
$131$	17.1058	1.49454	0.747268	+	0.664522i	$0.231365\pi$
$132$	0	0
$133$	9.70192	0.841263
$134$	0	0
$135$	5.19990	0.447536
$136$	0	0
$137$	−11.0719	−0.945939	−0.472970	−	0.881079i	$-0.656818\pi$
$137$	−11.0719	−0.945939	−0.472970	+	0.881079i	$0.656818\pi$
$138$	0	0
$139$	20.2349	1.71630	0.858150	−	0.513400i	$-0.171614\pi$
$139$	20.2349	1.71630	0.858150	+	0.513400i	$0.171614\pi$
$140$	0	0
$141$	−13.4460	−1.13236
$142$	0	0
$143$	−3.82268	−0.319669
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−17.7210	−1.46160
$148$	0	0
$149$	−11.8140	−0.967840	−0.483920	−	0.875112i	$-0.660787\pi$
$149$	−11.8140	−0.967840	−0.483920	+	0.875112i	$0.660787\pi$
$150$	0	0
$151$	−15.1058	−1.22929	−0.614645	−	0.788804i	$-0.710701\pi$
$151$	−15.1058	−1.22929	−0.614645	+	0.788804i	$0.710701\pi$
$152$	0	0
$153$	2.49684	0.201858
$154$	0	0
$155$	6.19990	0.497988
$156$	0	0
$157$	−6.18804	−0.493859	−0.246930	−	0.969033i	$-0.579422\pi$
$157$	−6.18804	−0.493859	−0.246930	+	0.969033i	$0.579422\pi$
$158$	0	0
$159$	9.22818	0.731842
$160$	0	0
$161$	−34.2035	−2.69561
$162$	0	0
$163$	−4.15976	−0.325818	−0.162909	−	0.986641i	$-0.552088\pi$
$163$	−4.15976	−0.325818	−0.162909	+	0.986641i	$0.552088\pi$
$164$	0	0
$165$	0.835064	0.0650097
$166$	0	0
$167$	−8.19070	−0.633815	−0.316908	−	0.948456i	$-0.602645\pi$
$167$	−8.19070	−0.633815	−0.316908	+	0.948456i	$0.602645\pi$
$168$	0	0
$169$	11.0786	0.852202
$170$	0	0
$171$	−3.70192	−0.283093
$172$	0	0
$173$	21.0636	1.60144	0.800719	−	0.599041i	$-0.204451\pi$
$173$	21.0636	1.60144	0.800719	+	0.599041i	$0.204451\pi$
$174$	0	0
$175$	4.85096	0.366698
$176$	0	0
$177$	−0.874688	−0.0657456
$178$	0	0
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	4.07596	0.302964	0.151482	−	0.988460i	$-0.451595\pi$
$181$	4.07596	0.302964	0.151482	+	0.988460i	$0.451595\pi$
$182$	0	0
$183$	1.51591	0.112060
$184$	0	0
$185$	−0.127971	−0.00940860
$186$	0	0
$187$	1.05086	0.0768467
$188$	0	0
$189$	25.2245	1.83481
$190$	0	0
$191$	−13.9897	−1.01226	−0.506128	−	0.862458i	$-0.668924\pi$
$191$	−13.9897	−1.01226	−0.506128	+	0.862458i	$0.668924\pi$
$192$	0	0
$193$	−13.2051	−0.950522	−0.475261	−	0.879845i	$-0.657646\pi$
$193$	−13.2051	−0.950522	−0.475261	+	0.879845i	$0.657646\pi$
$194$	0	0
$195$	−5.25997	−0.376674
$196$	0	0
$197$	6.95383	0.495440	0.247720	−	0.968832i	$-0.420319\pi$
$197$	6.95383	0.495440	0.247720	+	0.968832i	$0.420319\pi$
$198$	0	0
$199$	−7.73371	−0.548228	−0.274114	−	0.961697i	$-0.588385\pi$
$199$	−7.73371	−0.548228	−0.274114	+	0.961697i	$0.588385\pi$
$200$	0	0
$201$	−9.58178	−0.675847
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0.420877	0.0293953
$206$	0	0
$207$	13.0509	0.907098
$208$	0	0
$209$	−1.55805	−0.107773
$210$	0	0
$211$	−1.11611	−0.0768359	−0.0384180	−	0.999262i	$-0.512232\pi$
$211$	−1.11611	−0.0768359	−0.0384180	+	0.999262i	$0.512232\pi$
$212$	0	0
$213$	−4.44195	−0.304357
$214$	0	0
$215$	1.00000	0.0681994
$216$	0	0
$217$	30.0755	2.04166
$218$	0	0
$219$	6.18486	0.417934
$220$	0	0
$221$	−6.61927	−0.445260
$222$	0	0
$223$	−21.2178	−1.42085	−0.710426	−	0.703772i	$-0.751498\pi$
$223$	−21.2178	−1.42085	−0.710426	+	0.703772i	$0.751498\pi$
$224$	0	0
$225$	−1.85096	−0.123397
$226$	0	0
$227$	−26.6650	−1.76982	−0.884908	−	0.465767i	$-0.845779\pi$
$227$	−26.6650	−1.76982	−0.884908	+	0.465767i	$0.845779\pi$
$228$	0	0
$229$	7.22097	0.477175	0.238588	−	0.971121i	$-0.423316\pi$
$229$	7.22097	0.477175	0.238588	+	0.971121i	$0.423316\pi$
$230$	0	0
$231$	4.05086	0.266527
$232$	0	0
$233$	3.64740	0.238949	0.119474	−	0.992837i	$-0.461879\pi$
$233$	3.64740	0.238949	0.119474	+	0.992837i	$0.461879\pi$
$234$	0	0
$235$	12.5437	0.818259
$236$	0	0
$237$	−12.5266	−0.813692
$238$	0	0
$239$	10.4090	0.673303	0.336652	−	0.941629i	$-0.390706\pi$
$239$	10.4090	0.673303	0.336652	+	0.941629i	$0.390706\pi$
$240$	0	0
$241$	6.36802	0.410200	0.205100	−	0.978741i	$-0.434248\pi$
$241$	6.36802	0.410200	0.205100	+	0.978741i	$0.434248\pi$
$242$	0	0
$243$	−15.5771	−0.999273
$244$	0	0
$245$	16.5318	1.05618
$246$	0	0
$247$	9.81399	0.624450
$248$	0	0
$249$	−15.3403	−0.972149
$250$	0	0
$251$	−11.1947	−0.706605	−0.353303	−	0.935509i	$-0.614941\pi$
$251$	−11.1947	−0.706605	−0.353303	+	0.935509i	$0.614941\pi$
$252$	0	0
$253$	5.49281	0.345330
$254$	0	0
$255$	1.44598	0.0905506
$256$	0	0
$257$	1.71781	0.107154	0.0535772	−	0.998564i	$-0.482938\pi$
$257$	1.71781	0.107154	0.0535772	+	0.998564i	$0.482938\pi$
$258$	0	0
$259$	−0.620781	−0.0385734
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2.85765	0.176210	0.0881051	−	0.996111i	$-0.471919\pi$
$263$	2.85765	0.176210	0.0881051	+	0.996111i	$0.471919\pi$
$264$	0	0
$265$	−8.60892	−0.528842
$266$	0	0
$267$	−9.00403	−0.551038
$268$	0	0
$269$	−26.8808	−1.63895	−0.819474	−	0.573116i	$-0.805734\pi$
$269$	−26.8808	−1.63895	−0.819474	+	0.573116i	$0.805734\pi$
$270$	0	0
$271$	−14.9963	−0.910963	−0.455481	−	0.890245i	$-0.650533\pi$
$271$	−14.9963	−0.910963	−0.455481	+	0.890245i	$0.650533\pi$
$272$	0	0
$273$	−25.5159	−1.54429
$274$	0	0
$275$	−0.779027	−0.0469771
$276$	0	0
$277$	32.5107	1.95338	0.976691	−	0.214652i	$-0.0688619\pi$
$277$	32.5107	1.95338	0.976691	+	0.214652i	$0.0688619\pi$
$278$	0	0
$279$	−11.4758	−0.687036
$280$	0	0
$281$	9.29091	0.554249	0.277125	−	0.960834i	$-0.410619\pi$
$281$	9.29091	0.554249	0.277125	+	0.960834i	$0.410619\pi$
$282$	0	0
$283$	−25.1526	−1.49517	−0.747583	−	0.664168i	$-0.768786\pi$
$283$	−25.1526	−1.49517	−0.747583	+	0.664168i	$0.768786\pi$
$284$	0	0
$285$	−2.14387	−0.126992
$286$	0	0
$287$	2.04166	0.120515
$288$	0	0
$289$	−15.1803	−0.892962
$290$	0	0
$291$	8.95383	0.524883
$292$	0	0
$293$	19.5795	1.14385	0.571923	−	0.820307i	$-0.306198\pi$
$293$	19.5795	1.14385	0.571923	+	0.820307i	$0.306198\pi$
$294$	0	0
$295$	0.815992	0.0475089
$296$	0	0
$297$	−4.05086	−0.235055
$298$	0	0
$299$	−34.5986	−2.00089
$300$	0	0
$301$	4.85096	0.279605
$302$	0	0
$303$	6.37153	0.366035
$304$	0	0
$305$	−1.41419	−0.0809761
$306$	0	0
$307$	32.8863	1.87692	0.938460	−	0.345388i	$-0.112253\pi$
$307$	32.8863	1.87692	0.938460	+	0.345388i	$0.112253\pi$
$308$	0	0
$309$	19.6280	1.11660
$310$	0	0
$311$	7.43526	0.421615	0.210807	−	0.977528i	$-0.432391\pi$
$311$	7.43526	0.421615	0.210807	+	0.977528i	$0.432391\pi$
$312$	0	0
$313$	−14.4090	−0.814446	−0.407223	−	0.913329i	$-0.633503\pi$
$313$	−14.4090	−0.814446	−0.407223	+	0.913329i	$0.633503\pi$
$314$	0	0
$315$	−8.97893	−0.505905
$316$	0	0
$317$	18.0231	1.01228	0.506139	−	0.862452i	$-0.331072\pi$
$317$	18.0231	1.01228	0.506139	+	0.862452i	$0.331072\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	3.53964	0.197564
$322$	0	0
$323$	−2.69789	−0.150115
$324$	0	0
$325$	4.90700	0.272191
$326$	0	0
$327$	6.92023	0.382690
$328$	0	0
$329$	60.8489	3.35471
$330$	0	0
$331$	1.75441	0.0964309	0.0482155	−	0.998837i	$-0.484647\pi$
$331$	1.75441	0.0964309	0.0482155	+	0.998837i	$0.484647\pi$
$332$	0	0
$333$	0.236869	0.0129803
$334$	0	0
$335$	8.93879	0.488378
$336$	0	0
$337$	−10.3346	−0.562960	−0.281480	−	0.959567i	$-0.590825\pi$
$337$	−10.3346	−0.562960	−0.281480	+	0.959567i	$0.590825\pi$
$338$	0	0
$339$	17.2698	0.937969
$340$	0	0
$341$	−4.82989	−0.261553
$342$	0	0
$343$	46.2384	2.49664
$344$	0	0
$345$	7.55805	0.406912
$346$	0	0
$347$	−7.70975	−0.413881	−0.206941	−	0.978354i	$-0.566351\pi$
$347$	−7.70975	−0.413881	−0.206941	+	0.978354i	$0.566351\pi$
$348$	0	0
$349$	−1.22818	−0.0657431	−0.0328715	−	0.999460i	$-0.510465\pi$
$349$	−1.22818	−0.0657431	−0.0328715	+	0.999460i	$0.510465\pi$
$350$	0	0
$351$	25.5159	1.36194
$352$	0	0
$353$	−16.0549	−0.854516	−0.427258	−	0.904130i	$-0.640520\pi$
$353$	−16.0549	−0.854516	−0.427258	+	0.904130i	$0.640520\pi$
$354$	0	0
$355$	4.14387	0.219934
$356$	0	0
$357$	7.01438	0.371240
$358$	0	0
$359$	−5.38073	−0.283984	−0.141992	−	0.989868i	$-0.545351\pi$
$359$	−5.38073	−0.283984	−0.141992	+	0.989868i	$0.545351\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	11.1407	0.584736
$364$	0	0
$365$	−5.76982	−0.302006
$366$	0	0
$367$	11.7123	0.611375	0.305688	−	0.952132i	$-0.401114\pi$
$367$	11.7123	0.611375	0.305688	+	0.952132i	$0.401114\pi$
$368$	0	0
$369$	−0.779027	−0.0405545
$370$	0	0
$371$	−41.7615	−2.16815
$372$	0	0
$373$	8.68868	0.449883	0.224941	−	0.974372i	$-0.427781\pi$
$373$	8.68868	0.449883	0.224941	+	0.974372i	$0.427781\pi$
$374$	0	0
$375$	−1.07193	−0.0553544
$376$	0	0
$377$	0	0
$378$	0	0
$379$	15.5087	0.796629	0.398314	−	0.917249i	$-0.369595\pi$
$379$	15.5087	0.796629	0.398314	+	0.917249i	$0.369595\pi$
$380$	0	0
$381$	20.8498	1.06817
$382$	0	0
$383$	1.95583	0.0999381	0.0499690	−	0.998751i	$-0.484088\pi$
$383$	1.95583	0.0999381	0.0499690	+	0.998751i	$0.484088\pi$
$384$	0	0
$385$	−3.77903	−0.192597
$386$	0	0
$387$	−1.85096	−0.0940896
$388$	0	0
$389$	−12.6557	−0.641672	−0.320836	−	0.947135i	$-0.603964\pi$
$389$	−12.6557	−0.641672	−0.320836	+	0.947135i	$0.603964\pi$
$390$	0	0
$391$	9.51122	0.481003
$392$	0	0
$393$	−18.3362	−0.924940
$394$	0	0
$395$	11.6860	0.587988
$396$	0	0
$397$	−29.3196	−1.47151	−0.735753	−	0.677250i	$-0.763172\pi$
$397$	−29.3196	−1.47151	−0.735753	+	0.677250i	$0.763172\pi$
$398$	0	0
$399$	−10.3998	−0.520642
$400$	0	0
$401$	−30.3231	−1.51426	−0.757131	−	0.653263i	$-0.773400\pi$
$401$	−30.3231	−1.51426	−0.757131	+	0.653263i	$0.773400\pi$
$402$	0	0
$403$	30.4229	1.51547
$404$	0	0
$405$	−0.0210696	−0.00104696
$406$	0	0
$407$	0.0996926	0.00494158
$408$	0	0
$409$	−5.16128	−0.255209	−0.127604	−	0.991825i	$-0.540729\pi$
$409$	−5.16128	−0.255209	−0.127604	+	0.991825i	$0.540729\pi$
$410$	0	0
$411$	11.8684	0.585424
$412$	0	0
$413$	3.95834	0.194777
$414$	0	0
$415$	14.3108	0.702491
$416$	0	0
$417$	−21.6904	−1.06218
$418$	0	0
$419$	33.8839	1.65534	0.827669	−	0.561216i	$-0.189666\pi$
$419$	33.8839	1.65534	0.827669	+	0.561216i	$0.189666\pi$
$420$	0	0
$421$	28.3339	1.38091	0.690456	−	0.723375i	$-0.257410\pi$
$421$	28.3339	1.38091	0.690456	+	0.723375i	$0.257410\pi$
$422$	0	0
$423$	−23.2178	−1.12889
$424$	0	0
$425$	−1.34894	−0.0654334
$426$	0	0
$427$	−6.86016	−0.331987
$428$	0	0
$429$	4.09766	0.197837
$430$	0	0
$431$	19.2626	0.927848	0.463924	−	0.885875i	$-0.346441\pi$
$431$	19.2626	0.927848	0.463924	+	0.885875i	$0.346441\pi$
$432$	0	0
$433$	−4.57798	−0.220004	−0.110002	−	0.993931i	$-0.535086\pi$
$433$	−4.57798	−0.220004	−0.110002	+	0.993931i	$0.535086\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−14.1017	−0.674577
$438$	0	0
$439$	−24.5961	−1.17391	−0.586953	−	0.809621i	$-0.699673\pi$
$439$	−24.5961	−1.17391	−0.586953	+	0.809621i	$0.699673\pi$
$440$	0	0
$441$	−30.5997	−1.45713
$442$	0	0
$443$	−32.9649	−1.56621	−0.783105	−	0.621889i	$-0.786365\pi$
$443$	−32.9649	−1.56621	−0.783105	+	0.621889i	$0.786365\pi$
$444$	0	0
$445$	8.39981	0.398189
$446$	0	0
$447$	12.6638	0.598978
$448$	0	0
$449$	−38.3473	−1.80972	−0.904861	−	0.425707i	$-0.860025\pi$
$449$	−38.3473	−1.80972	−0.904861	+	0.425707i	$0.860025\pi$
$450$	0	0
$451$	−0.327874	−0.0154390
$452$	0	0
$453$	16.1924	0.760783
$454$	0	0
$455$	23.8036	1.11593
$456$	0	0
$457$	26.7421	1.25094	0.625470	−	0.780248i	$-0.284907\pi$
$457$	26.7421	1.25094	0.625470	+	0.780248i	$0.284907\pi$
$458$	0	0
$459$	−7.01438	−0.327403
$460$	0	0
$461$	−10.4157	−0.485108	−0.242554	−	0.970138i	$-0.577985\pi$
$461$	−10.4157	−0.485108	−0.242554	+	0.970138i	$0.577985\pi$
$462$	0	0
$463$	19.9217	0.925842	0.462921	−	0.886399i	$-0.346801\pi$
$463$	19.9217	0.925842	0.462921	+	0.886399i	$0.346801\pi$
$464$	0	0
$465$	−6.64588	−0.308195
$466$	0	0
$467$	−26.9382	−1.24655	−0.623275	−	0.782003i	$-0.714198\pi$
$467$	−26.9382	−1.24655	−0.623275	+	0.782003i	$0.714198\pi$
$468$	0	0
$469$	43.3617	2.00226
$470$	0	0
$471$	6.63316	0.305640
$472$	0	0
$473$	−0.779027	−0.0358197
$474$	0	0
$475$	2.00000	0.0917663
$476$	0	0
$477$	15.9348	0.729602
$478$	0	0
$479$	21.1197	0.964982	0.482491	−	0.875901i	$-0.339732\pi$
$479$	21.1197	0.964982	0.482491	+	0.875901i	$0.339732\pi$
$480$	0	0
$481$	−0.627952	−0.0286322
$482$	0	0
$483$	36.6638	1.66826
$484$	0	0
$485$	−8.35298	−0.379289
$486$	0	0
$487$	8.59616	0.389529	0.194765	−	0.980850i	$-0.437606\pi$
$487$	8.59616	0.389529	0.194765	+	0.980850i	$0.437606\pi$
$488$	0	0
$489$	4.45899	0.201642
$490$	0	0
$491$	7.98159	0.360204	0.180102	−	0.983648i	$-0.442357\pi$
$491$	7.98159	0.360204	0.180102	+	0.983648i	$0.442357\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	1.44195	0.0648107
$496$	0	0
$497$	20.1017	0.901686
$498$	0	0
$499$	3.31246	0.148286	0.0741430	−	0.997248i	$-0.476378\pi$
$499$	3.31246	0.148286	0.0741430	+	0.997248i	$0.476378\pi$
$500$	0	0
$501$	8.77988	0.392256
$502$	0	0
$503$	−2.02259	−0.0901826	−0.0450913	−	0.998983i	$-0.514358\pi$
$503$	−2.02259	−0.0901826	−0.0450913	+	0.998983i	$0.514358\pi$
$504$	0	0
$505$	−5.94396	−0.264503
$506$	0	0
$507$	−11.8755	−0.527411
$508$	0	0
$509$	−22.9376	−1.01669	−0.508347	−	0.861153i	$-0.669743\pi$
$509$	−22.9376	−1.01669	−0.508347	+	0.861153i	$0.669743\pi$
$510$	0	0
$511$	−27.9892	−1.23817
$512$	0	0
$513$	10.3998	0.459163
$514$	0	0
$515$	−18.3108	−0.806872
$516$	0	0
$517$	−9.77186	−0.429766
$518$	0	0
$519$	−22.5788	−0.991099
$520$	0	0
$521$	21.4276	0.938759	0.469379	−	0.882997i	$-0.344478\pi$
$521$	21.4276	0.938759	0.469379	+	0.882997i	$0.344478\pi$
$522$	0	0
$523$	−10.5174	−0.459895	−0.229948	−	0.973203i	$-0.573855\pi$
$523$	−10.5174	−0.459895	−0.229948	+	0.973203i	$0.573855\pi$
$524$	0	0
$525$	−5.19990	−0.226942
$526$	0	0
$527$	−8.36332	−0.364312
$528$	0	0
$529$	26.7147	1.16151
$530$	0	0
$531$	−1.51037	−0.0655444
$532$	0	0
$533$	2.06524	0.0894556
$534$	0	0
$535$	−3.30211	−0.142763
$536$	0	0
$537$	−6.43160	−0.277544
$538$	0	0
$539$	−12.8787	−0.554726
$540$	0	0
$541$	−17.5469	−0.754398	−0.377199	−	0.926132i	$-0.623113\pi$
$541$	−17.5469	−0.754398	−0.377199	+	0.926132i	$0.623113\pi$
$542$	0	0
$543$	−4.36916	−0.187499
$544$	0	0
$545$	−6.45585	−0.276538
$546$	0	0
$547$	0.128824	0.00550812	0.00275406	−	0.999996i	$-0.499123\pi$
$547$	0.128824	0.00550812	0.00275406	+	0.999996i	$0.499123\pi$
$548$	0	0
$549$	2.61760	0.111717
$550$	0	0
$551$	0	0
$552$	0	0
$553$	56.6884	2.41064
$554$	0	0
$555$	0.137176	0.00582280
$556$	0	0
$557$	−5.97690	−0.253249	−0.126625	−	0.991951i	$-0.540414\pi$
$557$	−5.97690	−0.253249	−0.126625	+	0.991951i	$0.540414\pi$
$558$	0	0
$559$	4.90700	0.207544
$560$	0	0
$561$	−1.12646	−0.0475590
$562$	0	0
$563$	26.8331	1.13088	0.565439	−	0.824790i	$-0.308707\pi$
$563$	26.8331	1.13088	0.565439	+	0.824790i	$0.308707\pi$
$564$	0	0
$565$	−16.1109	−0.677792
$566$	0	0
$567$	−0.102208	−0.00429233
$568$	0	0
$569$	20.5760	0.862590	0.431295	−	0.902211i	$-0.358057\pi$
$569$	20.5760	0.862590	0.431295	+	0.902211i	$0.358057\pi$
$570$	0	0
$571$	−2.54064	−0.106323	−0.0531613	−	0.998586i	$-0.516930\pi$
$571$	−2.54064	−0.106323	−0.0531613	+	0.998586i	$0.516930\pi$
$572$	0	0
$573$	14.9960	0.626466
$574$	0	0
$575$	−7.05086	−0.294041
$576$	0	0
$577$	36.4357	1.51684	0.758418	−	0.651768i	$-0.225972\pi$
$577$	36.4357	1.51684	0.758418	+	0.651768i	$0.225972\pi$
$578$	0	0
$579$	14.1550	0.588260
$580$	0	0
$581$	69.4213	2.88008
$582$	0	0
$583$	6.70658	0.277758
$584$	0	0
$585$	−9.08265	−0.375522
$586$	0	0
$587$	−36.1912	−1.49377	−0.746886	−	0.664952i	$-0.768452\pi$
$587$	−36.1912	−1.49377	−0.746886	+	0.664952i	$0.768452\pi$
$588$	0	0
$589$	12.3998	0.510925
$590$	0	0
$591$	−7.45404	−0.306618
$592$	0	0
$593$	5.05807	0.207710	0.103855	−	0.994592i	$-0.466882\pi$
$593$	5.05807	0.207710	0.103855	+	0.994592i	$0.466882\pi$
$594$	0	0
$595$	−6.54367	−0.268264
$596$	0	0
$597$	8.29002	0.339288
$598$	0	0
$599$	−10.9081	−0.445695	−0.222847	−	0.974853i	$-0.571535\pi$
$599$	−10.9081	−0.445695	−0.222847	+	0.974853i	$0.571535\pi$
$600$	0	0
$601$	34.3473	1.40106	0.700528	−	0.713625i	$-0.252948\pi$
$601$	34.3473	1.40106	0.700528	+	0.713625i	$0.252948\pi$
$602$	0	0
$603$	−16.5453	−0.673778
$604$	0	0
$605$	−10.3931	−0.422540
$606$	0	0
$607$	−25.6722	−1.04200	−0.521000	−	0.853556i	$-0.674441\pi$
$607$	−25.6722	−1.04200	−0.521000	+	0.853556i	$0.674441\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	61.5518	2.49012
$612$	0	0
$613$	−44.0389	−1.77871	−0.889357	−	0.457213i	$-0.848848\pi$
$613$	−44.0389	−1.77871	−0.889357	+	0.457213i	$0.848848\pi$
$614$	0	0
$615$	−0.451152	−0.0181922
$616$	0	0
$617$	−37.9388	−1.52736	−0.763680	−	0.645595i	$-0.776609\pi$
$617$	−37.9388	−1.52736	−0.763680	+	0.645595i	$0.776609\pi$
$618$	0	0
$619$	−8.70709	−0.349968	−0.174984	−	0.984571i	$-0.555987\pi$
$619$	−8.70709	−0.349968	−0.174984	+	0.984571i	$0.555987\pi$
$620$	0	0
$621$	−36.6638	−1.47127
$622$	0	0
$623$	40.7471	1.63250
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	1.67013	0.0666985
$628$	0	0
$629$	0.172625	0.00688302
$630$	0	0
$631$	16.3092	0.649258	0.324629	−	0.945841i	$-0.394761\pi$
$631$	16.3092	0.649258	0.324629	+	0.945841i	$0.394761\pi$
$632$	0	0
$633$	1.19639	0.0475523
$634$	0	0
$635$	−19.4507	−0.771876
$636$	0	0
$637$	81.1215	3.21415
$638$	0	0
$639$	−7.67013	−0.303426
$640$	0	0
$641$	33.5558	1.32537	0.662687	−	0.748897i	$-0.269416\pi$
$641$	33.5558	1.32537	0.662687	+	0.748897i	$0.269416\pi$
$642$	0	0
$643$	−19.0255	−0.750291	−0.375146	−	0.926966i	$-0.622407\pi$
$643$	−19.0255	−0.750291	−0.375146	+	0.926966i	$0.622407\pi$
$644$	0	0
$645$	−1.07193	−0.0422073
$646$	0	0
$647$	−50.0688	−1.96841	−0.984204	−	0.177040i	$-0.943348\pi$
$647$	−50.0688	−1.96841	−0.984204	+	0.177040i	$0.943348\pi$
$648$	0	0
$649$	−0.635679	−0.0249526
$650$	0	0
$651$	−32.2389	−1.26354
$652$	0	0
$653$	2.60537	0.101956	0.0509779	−	0.998700i	$-0.483766\pi$
$653$	2.60537	0.101956	0.0509779	+	0.998700i	$0.483766\pi$
$654$	0	0
$655$	17.1058	0.668377
$656$	0	0
$657$	10.6797	0.416655
$658$	0	0
$659$	17.8247	0.694352	0.347176	−	0.937800i	$-0.387141\pi$
$659$	17.8247	0.694352	0.347176	+	0.937800i	$0.387141\pi$
$660$	0	0
$661$	−27.4880	−1.06916	−0.534580	−	0.845118i	$-0.679530\pi$
$661$	−27.4880	−1.06916	−0.534580	+	0.845118i	$0.679530\pi$
$662$	0	0
$663$	7.09541	0.275563
$664$	0	0
$665$	9.70192	0.376224
$666$	0	0
$667$	0	0
$668$	0	0
$669$	22.7441	0.879338
$670$	0	0
$671$	1.10169	0.0425302
$672$	0	0
$673$	−12.6716	−0.488456	−0.244228	−	0.969718i	$-0.578535\pi$
$673$	−12.6716	−0.488456	−0.244228	+	0.969718i	$0.578535\pi$
$674$	0	0
$675$	5.19990	0.200144
$676$	0	0
$677$	11.4381	0.439604	0.219802	−	0.975545i	$-0.429459\pi$
$677$	11.4381	0.439604	0.219802	+	0.975545i	$0.429459\pi$
$678$	0	0
$679$	−40.5199	−1.55501
$680$	0	0
$681$	28.5830	1.09530
$682$	0	0
$683$	−22.9100	−0.876628	−0.438314	−	0.898822i	$-0.644424\pi$
$683$	−22.9100	−0.876628	−0.438314	+	0.898822i	$0.644424\pi$
$684$	0	0
$685$	−11.0719	−0.423037
$686$	0	0
$687$	−7.74040	−0.295315
$688$	0	0
$689$	−42.2439	−1.60937
$690$	0	0
$691$	36.1295	1.37443	0.687216	−	0.726453i	$-0.258833\pi$
$691$	36.1295	1.37443	0.687216	+	0.726453i	$0.258833\pi$
$692$	0	0
$693$	6.99483	0.265711
$694$	0	0
$695$	20.2349	0.767552
$696$	0	0
$697$	−0.567740	−0.0215047
$698$	0	0
$699$	−3.90977	−0.147881
$700$	0	0
$701$	18.3839	0.694351	0.347175	−	0.937800i	$-0.387141\pi$
$701$	18.3839	0.694351	0.347175	+	0.937800i	$0.387141\pi$
$702$	0	0
$703$	−0.255941	−0.00965302
$704$	0	0
$705$	−13.4460	−0.506405
$706$	0	0
$707$	−28.8339	−1.08441
$708$	0	0
$709$	−15.6006	−0.585892	−0.292946	−	0.956129i	$-0.594636\pi$
$709$	−15.6006	−0.585892	−0.292946	+	0.956129i	$0.594636\pi$
$710$	0	0
$711$	−21.6304	−0.811202
$712$	0	0
$713$	−43.7147	−1.63713
$714$	0	0
$715$	−3.82268	−0.142960
$716$	0	0
$717$	−11.1578	−0.416694
$718$	0	0
$719$	−29.0661	−1.08398	−0.541992	−	0.840384i	$-0.682330\pi$
$719$	−29.0661	−1.08398	−0.541992	+	0.840384i	$0.682330\pi$
$720$	0	0
$721$	−88.8251	−3.30802
$722$	0	0
$723$	−6.82609	−0.253865
$724$	0	0
$725$	0	0
$726$	0	0
$727$	46.8514	1.73762	0.868811	−	0.495145i	$-0.164885\pi$
$727$	46.8514	1.73762	0.868811	+	0.495145i	$0.164885\pi$
$728$	0	0
$729$	16.7608	0.620772
$730$	0	0
$731$	−1.34894	−0.0498925
$732$	0	0
$733$	37.8221	1.39699	0.698495	−	0.715615i	$-0.253854\pi$
$733$	37.8221	1.39699	0.698495	+	0.715615i	$0.253854\pi$
$734$	0	0
$735$	−17.7210	−0.653649
$736$	0	0
$737$	−6.96355	−0.256506
$738$	0	0
$739$	−29.2814	−1.07713	−0.538567	−	0.842583i	$-0.681034\pi$
$739$	−29.2814	−1.07713	−0.538567	+	0.842583i	$0.681034\pi$
$740$	0	0
$741$	−10.5199	−0.386460
$742$	0	0
$743$	10.9074	0.400153	0.200076	−	0.979780i	$-0.435881\pi$
$743$	10.9074	0.400153	0.200076	+	0.979780i	$0.435881\pi$
$744$	0	0
$745$	−11.8140	−0.432831
$746$	0	0
$747$	−26.4888	−0.969174
$748$	0	0
$749$	−16.0184	−0.585300
$750$	0	0
$751$	21.4460	0.782575	0.391287	−	0.920269i	$-0.372030\pi$
$751$	21.4460	0.782575	0.391287	+	0.920269i	$0.372030\pi$
$752$	0	0
$753$	12.0000	0.437304
$754$	0	0
$755$	−15.1058	−0.549755
$756$	0	0
$757$	−18.3883	−0.668335	−0.334167	−	0.942514i	$-0.608455\pi$
$757$	−18.3883	−0.668335	−0.334167	+	0.942514i	$0.608455\pi$
$758$	0	0
$759$	−5.88792	−0.213718
$760$	0	0
$761$	−6.97527	−0.252853	−0.126427	−	0.991976i	$-0.540351\pi$
$761$	−6.97527	−0.252853	−0.126427	+	0.991976i	$0.540351\pi$
$762$	0	0
$763$	−31.3170	−1.13375
$764$	0	0
$765$	2.49684	0.0902735
$766$	0	0
$767$	4.00407	0.144579
$768$	0	0
$769$	−3.66092	−0.132016	−0.0660081	−	0.997819i	$-0.521026\pi$
$769$	−3.66092	−0.132016	−0.0660081	+	0.997819i	$0.521026\pi$
$770$	0	0
$771$	−1.84138	−0.0663158
$772$	0	0
$773$	−23.0123	−0.827694	−0.413847	−	0.910346i	$-0.635815\pi$
$773$	−23.0123	−0.827694	−0.413847	+	0.910346i	$0.635815\pi$
$774$	0	0
$775$	6.19990	0.222707
$776$	0	0
$777$	0.665436	0.0238724
$778$	0	0
$779$	0.841754	0.0301590
$780$	0	0
$781$	−3.22818	−0.115513
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−6.18804	−0.220861
$786$	0	0
$787$	−8.35131	−0.297692	−0.148846	−	0.988860i	$-0.547556\pi$
$787$	−8.35131	−0.297692	−0.148846	+	0.988860i	$0.547556\pi$
$788$	0	0
$789$	−3.06321	−0.109053
$790$	0	0
$791$	−78.1535	−2.77882
$792$	0	0
$793$	−6.93941	−0.246426
$794$	0	0
$795$	9.22818	0.327290
$796$	0	0
$797$	49.4016	1.74989	0.874946	−	0.484220i	$-0.160897\pi$
$797$	49.4016	1.74989	0.874946	+	0.484220i	$0.160897\pi$
$798$	0	0
$799$	−16.9207	−0.598612
$800$	0	0
$801$	−15.5477	−0.549351
$802$	0	0
$803$	4.49484	0.158620
$804$	0	0
$805$	−34.2035	−1.20551
$806$	0	0
$807$	28.8144	1.01431
$808$	0	0
$809$	11.0194	0.387423	0.193712	−	0.981059i	$-0.437947\pi$
$809$	11.0194	0.387423	0.193712	+	0.981059i	$0.437947\pi$
$810$	0	0
$811$	53.1891	1.86772	0.933860	−	0.357637i	$-0.116418\pi$
$811$	53.1891	1.86772	0.933860	+	0.357637i	$0.116418\pi$
$812$	0	0
$813$	16.0751	0.563777
$814$	0	0
$815$	−4.15976	−0.145710
$816$	0	0
$817$	2.00000	0.0699711
$818$	0	0
$819$	−44.0596	−1.53957
$820$	0	0
$821$	46.6163	1.62692	0.813459	−	0.581622i	$-0.197582\pi$
$821$	46.6163	1.62692	0.813459	+	0.581622i	$0.197582\pi$
$822$	0	0
$823$	−28.3292	−0.987495	−0.493748	−	0.869605i	$-0.664373\pi$
$823$	−28.3292	−0.987495	−0.493748	+	0.869605i	$0.664373\pi$
$824$	0	0
$825$	0.835064	0.0290732
$826$	0	0
$827$	3.11611	0.108358	0.0541788	−	0.998531i	$-0.482746\pi$
$827$	3.11611	0.108358	0.0541788	+	0.998531i	$0.482746\pi$
$828$	0	0
$829$	42.9793	1.49273	0.746367	−	0.665535i	$-0.231796\pi$
$829$	42.9793	1.49273	0.746367	+	0.665535i	$0.231796\pi$
$830$	0	0
$831$	−34.8493	−1.20891
$832$	0	0
$833$	−22.3005	−0.772666
$834$	0	0
$835$	−8.19070	−0.283451
$836$	0	0
$837$	32.2389	1.11434
$838$	0	0
$839$	−47.0666	−1.62492	−0.812461	−	0.583016i	$-0.801872\pi$
$839$	−47.0666	−1.62492	−0.812461	+	0.583016i	$0.801872\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	−9.95923	−0.343014
$844$	0	0
$845$	11.0786	0.381116
$846$	0	0
$847$	−50.4166	−1.73233
$848$	0	0
$849$	26.9619	0.925330
$850$	0	0
$851$	0.902304	0.0309306
$852$	0	0
$853$	−36.6766	−1.25578	−0.627891	−	0.778301i	$-0.716082\pi$
$853$	−36.6766	−1.25578	−0.627891	+	0.778301i	$0.716082\pi$
$854$	0	0
$855$	−3.70192	−0.126603
$856$	0	0
$857$	−22.4801	−0.767905	−0.383953	−	0.923353i	$-0.625437\pi$
$857$	−22.4801	−0.767905	−0.383953	+	0.923353i	$0.625437\pi$
$858$	0	0
$859$	−16.0515	−0.547671	−0.273835	−	0.961777i	$-0.588292\pi$
$859$	−16.0515	−0.547671	−0.273835	+	0.961777i	$0.588292\pi$
$860$	0	0
$861$	−2.18852	−0.0745846
$862$	0	0
$863$	−15.1099	−0.514348	−0.257174	−	0.966365i	$-0.582791\pi$
$863$	−15.1099	−0.514348	−0.257174	+	0.966365i	$0.582791\pi$
$864$	0	0
$865$	21.0636	0.716185
$866$	0	0
$867$	16.2723	0.552637
$868$	0	0
$869$	−9.10372	−0.308823
$870$	0	0
$871$	43.8626	1.48623
$872$	0	0
$873$	15.4610	0.523276
$874$	0	0
$875$	4.85096	0.163992
$876$	0	0
$877$	−17.8792	−0.603739	−0.301869	−	0.953349i	$-0.597611\pi$
$877$	−17.8792	−0.603739	−0.301869	+	0.953349i	$0.597611\pi$
$878$	0	0
$879$	−20.9879	−0.707905
$880$	0	0
$881$	45.6171	1.53688	0.768440	−	0.639922i	$-0.221033\pi$
$881$	45.6171	1.53688	0.768440	+	0.639922i	$0.221033\pi$
$882$	0	0
$883$	34.1115	1.14794	0.573971	−	0.818876i	$-0.305402\pi$
$883$	34.1115	1.14794	0.573971	+	0.818876i	$0.305402\pi$
$884$	0	0
$885$	−0.874688	−0.0294023
$886$	0	0
$887$	28.6217	0.961022	0.480511	−	0.876989i	$-0.340451\pi$
$887$	28.6217	0.961022	0.480511	+	0.876989i	$0.340451\pi$
$888$	0	0
$889$	−94.3544	−3.16455
$890$	0	0
$891$	0.0164138	0.000549883 0
$892$	0	0
$893$	25.0873	0.839516
$894$	0	0
$895$	6.00000	0.200558
$896$	0	0
$897$	37.0873	1.23831
$898$	0	0
$899$	0	0
$900$	0	0
$901$	11.6129	0.386883
$902$	0	0
$903$	−5.19990	−0.173042
$904$	0	0
$905$	4.07596	0.135490
$906$	0	0
$907$	35.7568	1.18729	0.593643	−	0.804729i	$-0.297689\pi$
$907$	35.7568	1.18729	0.593643	+	0.804729i	$0.297689\pi$
$908$	0	0
$909$	11.0020	0.364914
$910$	0	0
$911$	27.4356	0.908983	0.454491	−	0.890751i	$-0.349821\pi$
$911$	27.4356	0.908983	0.454491	+	0.890751i	$0.349821\pi$
$912$	0	0
$913$	−11.1485	−0.368962
$914$	0	0
$915$	1.51591	0.0501146
$916$	0	0
$917$	82.9793	2.74022
$918$	0	0
$919$	18.0067	0.593987	0.296994	−	0.954879i	$-0.404016\pi$
$919$	18.0067	0.593987	0.296994	+	0.954879i	$0.404016\pi$
$920$	0	0
$921$	−35.2519	−1.16159
$922$	0	0
$923$	20.3339	0.669300
$924$	0	0
$925$	−0.127971	−0.00420765
$926$	0	0
$927$	33.8926	1.11318
$928$	0	0
$929$	−38.5548	−1.26494	−0.632470	−	0.774585i	$-0.717959\pi$
$929$	−38.5548	−1.26494	−0.632470	+	0.774585i	$0.717959\pi$
$930$	0	0
$931$	33.0636	1.08362
$932$	0	0
$933$	−7.97010	−0.260929
$934$	0	0
$935$	1.05086	0.0343669
$936$	0	0
$937$	−49.8591	−1.62882	−0.814412	−	0.580287i	$-0.802940\pi$
$937$	−49.8591	−1.62882	−0.814412	+	0.580287i	$0.802940\pi$
$938$	0	0
$939$	15.4455	0.504045
$940$	0	0
$941$	−22.8308	−0.744262	−0.372131	−	0.928180i	$-0.621373\pi$
$941$	−22.8308	−0.744262	−0.372131	+	0.928180i	$0.621373\pi$
$942$	0	0
$943$	−2.96755	−0.0966366
$944$	0	0
$945$	25.2245	0.820554
$946$	0	0
$947$	−56.2583	−1.82815	−0.914075	−	0.405544i	$-0.867082\pi$
$947$	−56.2583	−1.82815	−0.914075	+	0.405544i	$0.867082\pi$
$948$	0	0
$949$	−28.3125	−0.919063
$950$	0	0
$951$	−19.3196	−0.626480
$952$	0	0
$953$	41.2641	1.33668	0.668338	−	0.743857i	$-0.267006\pi$
$953$	41.2641	1.33668	0.668338	+	0.743857i	$0.267006\pi$
$954$	0	0
$955$	−13.9897	−0.452694
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−53.7095	−1.73437
$960$	0	0
$961$	7.43881	0.239962
$962$	0	0
$963$	6.11208	0.196959
$964$	0	0
$965$	−13.2051	−0.425087
$966$	0	0
$967$	−3.96255	−0.127427	−0.0637136	−	0.997968i	$-0.520294\pi$
$967$	−3.96255	−0.127427	−0.0637136	+	0.997968i	$0.520294\pi$
$968$	0	0
$969$	2.89196	0.0929030
$970$	0	0
$971$	36.4957	1.17120	0.585601	−	0.810599i	$-0.300858\pi$
$971$	36.4957	1.17120	0.585601	+	0.810599i	$0.300858\pi$
$972$	0	0
$973$	98.1585	3.14682
$974$	0	0
$975$	−5.25997	−0.168454
$976$	0	0
$977$	55.9753	1.79081	0.895404	−	0.445254i	$-0.146887\pi$
$977$	55.9753	1.79081	0.895404	+	0.445254i	$0.146887\pi$
$978$	0	0
$979$	−6.54367	−0.209137
$980$	0	0
$981$	11.9495	0.381518
$982$	0	0
$983$	−7.39023	−0.235712	−0.117856	−	0.993031i	$-0.537602\pi$
$983$	−7.39023	−0.235712	−0.117856	+	0.993031i	$0.537602\pi$
$984$	0	0
$985$	6.95383	0.221567
$986$	0	0
$987$	−65.2259	−2.07616
$988$	0	0
$989$	−7.05086	−0.224204
$990$	0	0
$991$	−24.9515	−0.792612	−0.396306	−	0.918119i	$-0.629708\pi$
$991$	−24.9515	−0.792612	−0.396306	+	0.918119i	$0.629708\pi$
$992$	0	0
$993$	−1.88061	−0.0596793
$994$	0	0
$995$	−7.73371	−0.245175
$996$	0	0
$997$	−54.2090	−1.71682	−0.858408	−	0.512967i	$-0.828546\pi$
$997$	−54.2090	−1.71682	−0.858408	+	0.512967i	$0.828546\pi$
$998$	0	0
$999$	−0.665436	−0.0210534

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	860.2.a.d.1.2	✓	4
3.2	odd	2		7740.2.a.m.1.4		4
4.3	odd	2		3440.2.a.q.1.3		4
5.2	odd	4		4300.2.d.g.1549.5		8
5.3	odd	4		4300.2.d.g.1549.4		8
5.4	even	2		4300.2.a.g.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
860.2.a.d.1.2	✓	4	1.1	even	1	trivial
3440.2.a.q.1.3		4	4.3	odd	2
4300.2.a.g.1.3		4	5.4	even	2
4300.2.d.g.1549.4		8	5.3	odd	4
4300.2.d.g.1549.5		8	5.2	odd	4
7740.2.a.m.1.4		4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 860 = 2^{2} \cdot 5 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	860.a (trivial)

\(f(q)\)	\(=\)	\(q-1.07193 q^{3} +1.00000 q^{5} +4.85096 q^{7} -1.85096 q^{9} +O(q^{10})\)	\(q-1.07193 q^{3} +1.00000 q^{5} +4.85096 q^{7} -1.85096 q^{9} -0.779027 q^{11} +4.90700 q^{13} -1.07193 q^{15} -1.34894 q^{17} +2.00000 q^{19} -5.19990 q^{21} -7.05086 q^{23} +1.00000 q^{25} +5.19990 q^{27} +6.19990 q^{31} +0.835064 q^{33} +4.85096 q^{35} -0.127971 q^{37} -5.25997 q^{39} +0.420877 q^{41} +1.00000 q^{43} -1.85096 q^{45} +12.5437 q^{47} +16.5318 q^{49} +1.44598 q^{51} -8.60892 q^{53} -0.779027 q^{55} -2.14387 q^{57} +0.815992 q^{59} -1.41419 q^{61} -8.97893 q^{63} +4.90700 q^{65} +8.93879 q^{67} +7.55805 q^{69} +4.14387 q^{71} -5.76982 q^{73} -1.07193 q^{75} -3.77903 q^{77} +11.6860 q^{79} -0.0210696 q^{81} +14.3108 q^{83} -1.34894 q^{85} +8.39981 q^{89} +23.8036 q^{91} -6.64588 q^{93} +2.00000 q^{95} -8.35298 q^{97} +1.44195 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{3} + 4 q^{5} + 7 q^{7} + 5 q^{9}+O(q^{10}) \) 4 * q + q^3 + 4 * q^5 + 7 * q^7 + 5 * q^9	\( 4 q + q^{3} + 4 q^{5} + 7 q^{7} + 5 q^{9} + 4 q^{11} + q^{13} + q^{15} - q^{17} + 8 q^{19} - 4 q^{21} + q^{23} + 4 q^{25} + 4 q^{27} + 8 q^{31} - 10 q^{33} + 7 q^{35} + 11 q^{37} + 18 q^{39} - 8 q^{41} + 4 q^{43} + 5 q^{45} + 6 q^{47} + 5 q^{49} + 4 q^{51} + 9 q^{53} + 4 q^{55} + 2 q^{57} + 21 q^{59} - 2 q^{61} - 12 q^{63} + q^{65} + 19 q^{67} + 16 q^{69} + 6 q^{71} - 9 q^{73} + q^{75} - 8 q^{77} + 21 q^{79} - 24 q^{81} - 11 q^{83} - q^{85} + 12 q^{91} - 8 q^{93} + 8 q^{95} - 13 q^{97} + 20 q^{99}+O(q^{100}) \) 4 * q + q^3 + 4 * q^5 + 7 * q^7 + 5 * q^9 + 4 * q^11 + q^13 + q^15 - q^17 + 8 * q^19 - 4 * q^21 + q^23 + 4 * q^25 + 4 * q^27 + 8 * q^31 - 10 * q^33 + 7 * q^35 + 11 * q^37 + 18 * q^39 - 8 * q^41 + 4 * q^43 + 5 * q^45 + 6 * q^47 + 5 * q^49 + 4 * q^51 + 9 * q^53 + 4 * q^55 + 2 * q^57 + 21 * q^59 - 2 * q^61 - 12 * q^63 + q^65 + 19 * q^67 + 16 * q^69 + 6 * q^71 - 9 * q^73 + q^75 - 8 * q^77 + 21 * q^79 - 24 * q^81 - 11 * q^83 - q^85 + 12 * q^91 - 8 * q^93 + 8 * q^95 - 13 * q^97 + 20 * q^99

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