LMFDB - Embedded newform 1648.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1648,2,Mod(1,1648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1648, 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("1648.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1648 = 2^{4} \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1648.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:	$13.1593462531$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 824)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	1648.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.61803 q^{5} -0.236068 q^{7} -2.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.61803 q^{5} -0.236068 q^{7} -2.00000 q^{9} +1.38197 q^{11} -4.85410 q^{13} +1.61803 q^{15} -1.61803 q^{17} +1.38197 q^{19} +0.236068 q^{21} +2.00000 q^{23} -2.38197 q^{25} +5.00000 q^{27} +7.70820 q^{29} +2.23607 q^{31} -1.38197 q^{33} +0.381966 q^{35} +7.47214 q^{37} +4.85410 q^{39} -8.94427 q^{41} +6.70820 q^{43} +3.23607 q^{45} -4.85410 q^{47} -6.94427 q^{49} +1.61803 q^{51} +4.61803 q^{53} -2.23607 q^{55} -1.38197 q^{57} -5.32624 q^{59} +0.145898 q^{61} +0.472136 q^{63} +7.85410 q^{65} +13.4721 q^{67} -2.00000 q^{69} +13.7984 q^{71} +9.79837 q^{73} +2.38197 q^{75} -0.326238 q^{77} +3.61803 q^{79} +1.00000 q^{81} -10.7984 q^{83} +2.61803 q^{85} -7.70820 q^{87} +5.70820 q^{89} +1.14590 q^{91} -2.23607 q^{93} -2.23607 q^{95} +3.23607 q^{97} -2.76393 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - q^{5} + 4 q^{7} - 4 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - q^5 + 4 * q^7 - 4 * q^9	$ 2 q - 2 q^{3} - q^{5} + 4 q^{7} - 4 q^{9} + 5 q^{11} - 3 q^{13} + q^{15} - q^{17} + 5 q^{19} - 4 q^{21} + 4 q^{23} - 7 q^{25} + 10 q^{27} + 2 q^{29} - 5 q^{33} + 3 q^{35} + 6 q^{37} + 3 q^{39} + 2 q^{45} - 3 q^{47} + 4 q^{49} + q^{51} + 7 q^{53} - 5 q^{57} + 5 q^{59} + 7 q^{61} - 8 q^{63} + 9 q^{65} + 18 q^{67} - 4 q^{69} + 3 q^{71} - 5 q^{73} + 7 q^{75} + 15 q^{77} + 5 q^{79} + 2 q^{81} + 3 q^{83} + 3 q^{85} - 2 q^{87} - 2 q^{89} + 9 q^{91} + 2 q^{97} - 10 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - q^5 + 4 * q^7 - 4 * q^9 + 5 * q^11 - 3 * q^13 + q^15 - q^17 + 5 * q^19 - 4 * q^21 + 4 * q^23 - 7 * q^25 + 10 * q^27 + 2 * q^29 - 5 * q^33 + 3 * q^35 + 6 * q^37 + 3 * q^39 + 2 * q^45 - 3 * q^47 + 4 * q^49 + q^51 + 7 * q^53 - 5 * q^57 + 5 * q^59 + 7 * q^61 - 8 * q^63 + 9 * q^65 + 18 * q^67 - 4 * q^69 + 3 * q^71 - 5 * q^73 + 7 * q^75 + 15 * q^77 + 5 * q^79 + 2 * q^81 + 3 * q^83 + 3 * q^85 - 2 * q^87 - 2 * q^89 + 9 * q^91 + 2 * q^97 - 10 * 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
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	0	0
$5$	−1.61803	−0.723607	−0.361803	−	0.932254i	$-0.617839\pi$
$5$	−1.61803	−0.723607	−0.361803	+	0.932254i	$0.617839\pi$
$6$	0	0
$7$	−0.236068	−0.0892253	−0.0446127	−	0.999004i	$-0.514205\pi$
$7$	−0.236068	−0.0892253	−0.0446127	+	0.999004i	$0.514205\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	1.38197	0.416678	0.208339	−	0.978057i	$-0.433194\pi$
$11$	1.38197	0.416678	0.208339	+	0.978057i	$0.433194\pi$
$12$	0	0
$13$	−4.85410	−1.34629	−0.673143	−	0.739512i	$-0.735056\pi$
$13$	−4.85410	−1.34629	−0.673143	+	0.739512i	$0.735056\pi$
$14$	0	0
$15$	1.61803	0.417775
$16$	0	0
$17$	−1.61803	−0.392431	−0.196215	−	0.980561i	$-0.562865\pi$
$17$	−1.61803	−0.392431	−0.196215	+	0.980561i	$0.562865\pi$
$18$	0	0
$19$	1.38197	0.317045	0.158522	−	0.987355i	$-0.449327\pi$
$19$	1.38197	0.317045	0.158522	+	0.987355i	$0.449327\pi$
$20$	0	0
$21$	0.236068	0.0515143
$22$	0	0
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	0	0
$25$	−2.38197	−0.476393
$26$	0	0
$27$	5.00000	0.962250
$28$	0	0
$29$	7.70820	1.43138	0.715689	−	0.698419i	$-0.246113\pi$
$29$	7.70820	1.43138	0.715689	+	0.698419i	$0.246113\pi$
$30$	0	0
$31$	2.23607	0.401610	0.200805	−	0.979631i	$-0.435644\pi$
$31$	2.23607	0.401610	0.200805	+	0.979631i	$0.435644\pi$
$32$	0	0
$33$	−1.38197	−0.240569
$34$	0	0
$35$	0.381966	0.0645640
$36$	0	0
$37$	7.47214	1.22841	0.614206	−	0.789146i	$-0.289476\pi$
$37$	7.47214	1.22841	0.614206	+	0.789146i	$0.289476\pi$
$38$	0	0
$39$	4.85410	0.777278
$40$	0	0
$41$	−8.94427	−1.39686	−0.698430	−	0.715678i	$-0.746118\pi$
$41$	−8.94427	−1.39686	−0.698430	+	0.715678i	$0.746118\pi$
$42$	0	0
$43$	6.70820	1.02299	0.511496	−	0.859286i	$-0.329092\pi$
$43$	6.70820	1.02299	0.511496	+	0.859286i	$0.329092\pi$
$44$	0	0
$45$	3.23607	0.482405
$46$	0	0
$47$	−4.85410	−0.708044	−0.354022	−	0.935237i	$-0.615186\pi$
$47$	−4.85410	−0.708044	−0.354022	+	0.935237i	$0.615186\pi$
$48$	0	0
$49$	−6.94427	−0.992039
$50$	0	0
$51$	1.61803	0.226570
$52$	0	0
$53$	4.61803	0.634336	0.317168	−	0.948369i	$-0.397268\pi$
$53$	4.61803	0.634336	0.317168	+	0.948369i	$0.397268\pi$
$54$	0	0
$55$	−2.23607	−0.301511
$56$	0	0
$57$	−1.38197	−0.183046
$58$	0	0
$59$	−5.32624	−0.693417	−0.346709	−	0.937973i	$-0.612701\pi$
$59$	−5.32624	−0.693417	−0.346709	+	0.937973i	$0.612701\pi$
$60$	0	0
$61$	0.145898	0.0186803	0.00934016	−	0.999956i	$-0.497027\pi$
$61$	0.145898	0.0186803	0.00934016	+	0.999956i	$0.497027\pi$
$62$	0	0
$63$	0.472136	0.0594835
$64$	0	0
$65$	7.85410	0.974181
$66$	0	0
$67$	13.4721	1.64588	0.822942	−	0.568126i	$-0.192331\pi$
$67$	13.4721	1.64588	0.822942	+	0.568126i	$0.192331\pi$
$68$	0	0
$69$	−2.00000	−0.240772
$70$	0	0
$71$	13.7984	1.63757	0.818783	−	0.574103i	$-0.194649\pi$
$71$	13.7984	1.63757	0.818783	+	0.574103i	$0.194649\pi$
$72$	0	0
$73$	9.79837	1.14681	0.573406	−	0.819271i	$-0.305622\pi$
$73$	9.79837	1.14681	0.573406	+	0.819271i	$0.305622\pi$
$74$	0	0
$75$	2.38197	0.275046
$76$	0	0
$77$	−0.326238	−0.0371783
$78$	0	0
$79$	3.61803	0.407061	0.203530	−	0.979069i	$-0.434758\pi$
$79$	3.61803	0.407061	0.203530	+	0.979069i	$0.434758\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.7984	−1.18528	−0.592638	−	0.805469i	$-0.701913\pi$
$83$	−10.7984	−1.18528	−0.592638	+	0.805469i	$0.701913\pi$
$84$	0	0
$85$	2.61803	0.283966
$86$	0	0
$87$	−7.70820	−0.826406
$88$	0	0
$89$	5.70820	0.605068	0.302534	−	0.953139i	$-0.402167\pi$
$89$	5.70820	0.605068	0.302534	+	0.953139i	$0.402167\pi$
$90$	0	0
$91$	1.14590	0.120123
$92$	0	0
$93$	−2.23607	−0.231869
$94$	0	0
$95$	−2.23607	−0.229416
$96$	0	0
$97$	3.23607	0.328573	0.164286	−	0.986413i	$-0.447468\pi$
$97$	3.23607	0.328573	0.164286	+	0.986413i	$0.447468\pi$
$98$	0	0
$99$	−2.76393	−0.277786
$100$	0	0
$101$	13.6180	1.35505	0.677523	−	0.735502i	$-0.263054\pi$
$101$	13.6180	1.35505	0.677523	+	0.735502i	$0.263054\pi$
$102$	0	0
$103$	1.00000	0.0985329
$103$	1.00000	0.0985329
$104$	0	0
$105$	−0.381966	−0.0372761
$106$	0	0
$107$	15.8541	1.53267	0.766337	−	0.642439i	$-0.222077\pi$
$107$	15.8541	1.53267	0.766337	+	0.642439i	$0.222077\pi$
$108$	0	0
$109$	−9.09017	−0.870680	−0.435340	−	0.900266i	$-0.643372\pi$
$109$	−9.09017	−0.870680	−0.435340	+	0.900266i	$0.643372\pi$
$110$	0	0
$111$	−7.47214	−0.709224
$112$	0	0
$113$	9.76393	0.918513	0.459257	−	0.888304i	$-0.348116\pi$
$113$	9.76393	0.918513	0.459257	+	0.888304i	$0.348116\pi$
$114$	0	0
$115$	−3.23607	−0.301765
$116$	0	0
$117$	9.70820	0.897524
$118$	0	0
$119$	0.381966	0.0350148
$120$	0	0
$121$	−9.09017	−0.826379
$122$	0	0
$123$	8.94427	0.806478
$124$	0	0
$125$	11.9443	1.06833
$126$	0	0
$127$	10.3262	0.916305	0.458153	−	0.888873i	$-0.348511\pi$
$127$	10.3262	0.916305	0.458153	+	0.888873i	$0.348511\pi$
$128$	0	0
$129$	−6.70820	−0.590624
$130$	0	0
$131$	3.47214	0.303362	0.151681	−	0.988430i	$-0.451531\pi$
$131$	3.47214	0.303362	0.151681	+	0.988430i	$0.451531\pi$
$132$	0	0
$133$	−0.326238	−0.0282884
$134$	0	0
$135$	−8.09017	−0.696291
$136$	0	0
$137$	−6.23607	−0.532783	−0.266392	−	0.963865i	$-0.585831\pi$
$137$	−6.23607	−0.532783	−0.266392	+	0.963865i	$0.585831\pi$
$138$	0	0
$139$	−5.32624	−0.451766	−0.225883	−	0.974154i	$-0.572527\pi$
$139$	−5.32624	−0.451766	−0.225883	+	0.974154i	$0.572527\pi$
$140$	0	0
$141$	4.85410	0.408789
$142$	0	0
$143$	−6.70820	−0.560968
$144$	0	0
$145$	−12.4721	−1.03575
$146$	0	0
$147$	6.94427	0.572754
$148$	0	0
$149$	8.52786	0.698630	0.349315	−	0.937005i	$-0.386414\pi$
$149$	8.52786	0.698630	0.349315	+	0.937005i	$0.386414\pi$
$150$	0	0
$151$	−3.94427	−0.320980	−0.160490	−	0.987037i	$-0.551307\pi$
$151$	−3.94427	−0.320980	−0.160490	+	0.987037i	$0.551307\pi$
$152$	0	0
$153$	3.23607	0.261621
$154$	0	0
$155$	−3.61803	−0.290607
$156$	0	0
$157$	−5.94427	−0.474405	−0.237202	−	0.971460i	$-0.576230\pi$
$157$	−5.94427	−0.474405	−0.237202	+	0.971460i	$0.576230\pi$
$158$	0	0
$159$	−4.61803	−0.366234
$160$	0	0
$161$	−0.472136	−0.0372095
$162$	0	0
$163$	−8.52786	−0.667954	−0.333977	−	0.942581i	$-0.608391\pi$
$163$	−8.52786	−0.667954	−0.333977	+	0.942581i	$0.608391\pi$
$164$	0	0
$165$	2.23607	0.174078
$166$	0	0
$167$	12.2361	0.946855	0.473428	−	0.880833i	$-0.343017\pi$
$167$	12.2361	0.946855	0.473428	+	0.880833i	$0.343017\pi$
$168$	0	0
$169$	10.5623	0.812485
$170$	0	0
$171$	−2.76393	−0.211363
$172$	0	0
$173$	6.61803	0.503160	0.251580	−	0.967837i	$-0.419050\pi$
$173$	6.61803	0.503160	0.251580	+	0.967837i	$0.419050\pi$
$174$	0	0
$175$	0.562306	0.0425063
$176$	0	0
$177$	5.32624	0.400345
$178$	0	0
$179$	6.14590	0.459366	0.229683	−	0.973265i	$-0.426231\pi$
$179$	6.14590	0.459366	0.229683	+	0.973265i	$0.426231\pi$
$180$	0	0
$181$	1.85410	0.137814	0.0689072	−	0.997623i	$-0.478049\pi$
$181$	1.85410	0.137814	0.0689072	+	0.997623i	$0.478049\pi$
$182$	0	0
$183$	−0.145898	−0.0107851
$184$	0	0
$185$	−12.0902	−0.888887
$186$	0	0
$187$	−2.23607	−0.163517
$188$	0	0
$189$	−1.18034	−0.0858571
$190$	0	0
$191$	−14.3820	−1.04064	−0.520321	−	0.853971i	$-0.674188\pi$
$191$	−14.3820	−1.04064	−0.520321	+	0.853971i	$0.674188\pi$
$192$	0	0
$193$	−13.0000	−0.935760	−0.467880	−	0.883792i	$-0.654982\pi$
$193$	−13.0000	−0.935760	−0.467880	+	0.883792i	$0.654982\pi$
$194$	0	0
$195$	−7.85410	−0.562444
$196$	0	0
$197$	14.2361	1.01428	0.507139	−	0.861864i	$-0.330703\pi$
$197$	14.2361	1.01428	0.507139	+	0.861864i	$0.330703\pi$
$198$	0	0
$199$	3.05573	0.216615	0.108307	−	0.994117i	$-0.465457\pi$
$199$	3.05573	0.216615	0.108307	+	0.994117i	$0.465457\pi$
$200$	0	0
$201$	−13.4721	−0.950251
$202$	0	0
$203$	−1.81966	−0.127715
$204$	0	0
$205$	14.4721	1.01078
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	1.90983	0.132106
$210$	0	0
$211$	−27.0344	−1.86113	−0.930564	−	0.366130i	$-0.880683\pi$
$211$	−27.0344	−1.86113	−0.930564	+	0.366130i	$0.880683\pi$
$212$	0	0
$213$	−13.7984	−0.945449
$214$	0	0
$215$	−10.8541	−0.740244
$216$	0	0
$217$	−0.527864	−0.0358337
$218$	0	0
$219$	−9.79837	−0.662113
$220$	0	0
$221$	7.85410	0.528324
$222$	0	0
$223$	−2.76393	−0.185087	−0.0925433	−	0.995709i	$-0.529500\pi$
$223$	−2.76393	−0.185087	−0.0925433	+	0.995709i	$0.529500\pi$
$224$	0	0
$225$	4.76393	0.317595
$226$	0	0
$227$	23.8885	1.58554	0.792769	−	0.609522i	$-0.208639\pi$
$227$	23.8885	1.58554	0.792769	+	0.609522i	$0.208639\pi$
$228$	0	0
$229$	−9.29180	−0.614019	−0.307010	−	0.951706i	$-0.599328\pi$
$229$	−9.29180	−0.614019	−0.307010	+	0.951706i	$0.599328\pi$
$230$	0	0
$231$	0.326238	0.0214649
$232$	0	0
$233$	9.65248	0.632355	0.316177	−	0.948700i	$-0.397601\pi$
$233$	9.65248	0.632355	0.316177	+	0.948700i	$0.397601\pi$
$234$	0	0
$235$	7.85410	0.512345
$236$	0	0
$237$	−3.61803	−0.235017
$238$	0	0
$239$	−7.61803	−0.492770	−0.246385	−	0.969172i	$-0.579243\pi$
$239$	−7.61803	−0.492770	−0.246385	+	0.969172i	$0.579243\pi$
$240$	0	0
$241$	−7.61803	−0.490721	−0.245360	−	0.969432i	$-0.578906\pi$
$241$	−7.61803	−0.490721	−0.245360	+	0.969432i	$0.578906\pi$
$242$	0	0
$243$	−16.0000	−1.02640
$244$	0	0
$245$	11.2361	0.717846
$246$	0	0
$247$	−6.70820	−0.426833
$248$	0	0
$249$	10.7984	0.684319
$250$	0	0
$251$	−2.29180	−0.144657	−0.0723284	−	0.997381i	$-0.523043\pi$
$251$	−2.29180	−0.144657	−0.0723284	+	0.997381i	$0.523043\pi$
$252$	0	0
$253$	2.76393	0.173767
$254$	0	0
$255$	−2.61803	−0.163948
$256$	0	0
$257$	0.236068	0.0147255	0.00736276	−	0.999973i	$-0.497656\pi$
$257$	0.236068	0.0147255	0.00736276	+	0.999973i	$0.497656\pi$
$258$	0	0
$259$	−1.76393	−0.109605
$260$	0	0
$261$	−15.4164	−0.954252
$262$	0	0
$263$	0.326238	0.0201167	0.0100583	−	0.999949i	$-0.496798\pi$
$263$	0.326238	0.0201167	0.0100583	+	0.999949i	$0.496798\pi$
$264$	0	0
$265$	−7.47214	−0.459010
$266$	0	0
$267$	−5.70820	−0.349336
$268$	0	0
$269$	26.2148	1.59834	0.799172	−	0.601103i	$-0.205272\pi$
$269$	26.2148	1.59834	0.799172	+	0.601103i	$0.205272\pi$
$270$	0	0
$271$	−15.3607	−0.933095	−0.466547	−	0.884496i	$-0.654502\pi$
$271$	−15.3607	−0.933095	−0.466547	+	0.884496i	$0.654502\pi$
$272$	0	0
$273$	−1.14590	−0.0693529
$274$	0	0
$275$	−3.29180	−0.198503
$276$	0	0
$277$	−0.0557281	−0.00334838	−0.00167419	−	0.999999i	$-0.500533\pi$
$277$	−0.0557281	−0.00334838	−0.00167419	+	0.999999i	$0.500533\pi$
$278$	0	0
$279$	−4.47214	−0.267740
$280$	0	0
$281$	−22.7082	−1.35466	−0.677329	−	0.735680i	$-0.736863\pi$
$281$	−22.7082	−1.35466	−0.677329	+	0.735680i	$0.736863\pi$
$282$	0	0
$283$	2.18034	0.129608	0.0648039	−	0.997898i	$-0.479358\pi$
$283$	2.18034	0.129608	0.0648039	+	0.997898i	$0.479358\pi$
$284$	0	0
$285$	2.23607	0.132453
$286$	0	0
$287$	2.11146	0.124635
$288$	0	0
$289$	−14.3820	−0.845998
$290$	0	0
$291$	−3.23607	−0.189702
$292$	0	0
$293$	17.0000	0.993151	0.496575	−	0.867994i	$-0.334591\pi$
$293$	17.0000	0.993151	0.496575	+	0.867994i	$0.334591\pi$
$294$	0	0
$295$	8.61803	0.501761
$296$	0	0
$297$	6.90983	0.400949
$298$	0	0
$299$	−9.70820	−0.561440
$300$	0	0
$301$	−1.58359	−0.0912767
$302$	0	0
$303$	−13.6180	−0.782336
$304$	0	0
$305$	−0.236068	−0.0135172
$306$	0	0
$307$	15.9787	0.911953	0.455977	−	0.889992i	$-0.349290\pi$
$307$	15.9787	0.911953	0.455977	+	0.889992i	$0.349290\pi$
$308$	0	0
$309$	−1.00000	−0.0568880
$310$	0	0
$311$	24.5967	1.39475	0.697377	−	0.716705i	$-0.254350\pi$
$311$	24.5967	1.39475	0.697377	+	0.716705i	$0.254350\pi$
$312$	0	0
$313$	−21.2361	−1.20033	−0.600167	−	0.799875i	$-0.704899\pi$
$313$	−21.2361	−1.20033	−0.600167	+	0.799875i	$0.704899\pi$
$314$	0	0
$315$	−0.763932	−0.0430427
$316$	0	0
$317$	−19.3607	−1.08740	−0.543702	−	0.839278i	$-0.682978\pi$
$317$	−19.3607	−1.08740	−0.543702	+	0.839278i	$0.682978\pi$
$318$	0	0
$319$	10.6525	0.596424
$320$	0	0
$321$	−15.8541	−0.884890
$322$	0	0
$323$	−2.23607	−0.124418
$324$	0	0
$325$	11.5623	0.641361
$326$	0	0
$327$	9.09017	0.502688
$328$	0	0
$329$	1.14590	0.0631754
$330$	0	0
$331$	−2.03444	−0.111823	−0.0559115	−	0.998436i	$-0.517806\pi$
$331$	−2.03444	−0.111823	−0.0559115	+	0.998436i	$0.517806\pi$
$332$	0	0
$333$	−14.9443	−0.818941
$334$	0	0
$335$	−21.7984	−1.19097
$336$	0	0
$337$	12.6180	0.687348	0.343674	−	0.939089i	$-0.388328\pi$
$337$	12.6180	0.687348	0.343674	+	0.939089i	$0.388328\pi$
$338$	0	0
$339$	−9.76393	−0.530304
$340$	0	0
$341$	3.09017	0.167342
$342$	0	0
$343$	3.29180	0.177740
$344$	0	0
$345$	3.23607	0.174224
$346$	0	0
$347$	−4.70820	−0.252750	−0.126375	−	0.991983i	$-0.540334\pi$
$347$	−4.70820	−0.252750	−0.126375	+	0.991983i	$0.540334\pi$
$348$	0	0
$349$	−32.9443	−1.76347	−0.881733	−	0.471748i	$-0.843623\pi$
$349$	−32.9443	−1.76347	−0.881733	+	0.471748i	$0.843623\pi$
$350$	0	0
$351$	−24.2705	−1.29546
$352$	0	0
$353$	−1.32624	−0.0705885	−0.0352943	−	0.999377i	$-0.511237\pi$
$353$	−1.32624	−0.0705885	−0.0352943	+	0.999377i	$0.511237\pi$
$354$	0	0
$355$	−22.3262	−1.18495
$356$	0	0
$357$	−0.381966	−0.0202158
$358$	0	0
$359$	37.2148	1.96412	0.982061	−	0.188566i	$-0.0603839\pi$
$359$	37.2148	1.96412	0.982061	+	0.188566i	$0.0603839\pi$
$360$	0	0
$361$	−17.0902	−0.899483
$362$	0	0
$363$	9.09017	0.477110
$364$	0	0
$365$	−15.8541	−0.829842
$366$	0	0
$367$	16.9098	0.882686	0.441343	−	0.897338i	$-0.354502\pi$
$367$	16.9098	0.882686	0.441343	+	0.897338i	$0.354502\pi$
$368$	0	0
$369$	17.8885	0.931240
$370$	0	0
$371$	−1.09017	−0.0565988
$372$	0	0
$373$	−2.90983	−0.150665	−0.0753326	−	0.997158i	$-0.524002\pi$
$373$	−2.90983	−0.150665	−0.0753326	+	0.997158i	$0.524002\pi$
$374$	0	0
$375$	−11.9443	−0.616800
$376$	0	0
$377$	−37.4164	−1.92704
$378$	0	0
$379$	30.8885	1.58664	0.793319	−	0.608806i	$-0.208351\pi$
$379$	30.8885	1.58664	0.793319	+	0.608806i	$0.208351\pi$
$380$	0	0
$381$	−10.3262	−0.529029
$382$	0	0
$383$	15.6525	0.799804	0.399902	−	0.916558i	$-0.369044\pi$
$383$	15.6525	0.799804	0.399902	+	0.916558i	$0.369044\pi$
$384$	0	0
$385$	0.527864	0.0269024
$386$	0	0
$387$	−13.4164	−0.681994
$388$	0	0
$389$	−33.3050	−1.68863	−0.844314	−	0.535849i	$-0.819992\pi$
$389$	−33.3050	−1.68863	−0.844314	+	0.535849i	$0.819992\pi$
$390$	0	0
$391$	−3.23607	−0.163655
$392$	0	0
$393$	−3.47214	−0.175146
$394$	0	0
$395$	−5.85410	−0.294552
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	0	0
$399$	0.326238	0.0163323
$400$	0	0
$401$	−9.05573	−0.452221	−0.226111	−	0.974102i	$-0.572601\pi$
$401$	−9.05573	−0.452221	−0.226111	+	0.974102i	$0.572601\pi$
$402$	0	0
$403$	−10.8541	−0.540681
$404$	0	0
$405$	−1.61803	−0.0804008
$406$	0	0
$407$	10.3262	0.511853
$408$	0	0
$409$	−30.7082	−1.51842	−0.759211	−	0.650844i	$-0.774415\pi$
$409$	−30.7082	−1.51842	−0.759211	+	0.650844i	$0.774415\pi$
$410$	0	0
$411$	6.23607	0.307603
$412$	0	0
$413$	1.25735	0.0618704
$414$	0	0
$415$	17.4721	0.857673
$416$	0	0
$417$	5.32624	0.260827
$418$	0	0
$419$	36.5623	1.78619	0.893093	−	0.449873i	$-0.148531\pi$
$419$	36.5623	1.78619	0.893093	+	0.449873i	$0.148531\pi$
$420$	0	0
$421$	19.3607	0.943582	0.471791	−	0.881710i	$-0.343608\pi$
$421$	19.3607	0.943582	0.471791	+	0.881710i	$0.343608\pi$
$422$	0	0
$423$	9.70820	0.472029
$424$	0	0
$425$	3.85410	0.186951
$426$	0	0
$427$	−0.0344419	−0.00166676
$428$	0	0
$429$	6.70820	0.323875
$430$	0	0
$431$	−0.472136	−0.0227420	−0.0113710	−	0.999935i	$-0.503620\pi$
$431$	−0.472136	−0.0227420	−0.0113710	+	0.999935i	$0.503620\pi$
$432$	0	0
$433$	18.5967	0.893703	0.446851	−	0.894608i	$-0.352545\pi$
$433$	18.5967	0.893703	0.446851	+	0.894608i	$0.352545\pi$
$434$	0	0
$435$	12.4721	0.597993
$436$	0	0
$437$	2.76393	0.132217
$438$	0	0
$439$	13.5623	0.647294	0.323647	−	0.946178i	$-0.395091\pi$
$439$	13.5623	0.647294	0.323647	+	0.946178i	$0.395091\pi$
$440$	0	0
$441$	13.8885	0.661359
$442$	0	0
$443$	−31.2148	−1.48306	−0.741530	−	0.670920i	$-0.765899\pi$
$443$	−31.2148	−1.48306	−0.741530	+	0.670920i	$0.765899\pi$
$444$	0	0
$445$	−9.23607	−0.437832
$446$	0	0
$447$	−8.52786	−0.403354
$448$	0	0
$449$	1.29180	0.0609636	0.0304818	−	0.999535i	$-0.490296\pi$
$449$	1.29180	0.0609636	0.0304818	+	0.999535i	$0.490296\pi$
$450$	0	0
$451$	−12.3607	−0.582042
$452$	0	0
$453$	3.94427	0.185318
$454$	0	0
$455$	−1.85410	−0.0869216
$456$	0	0
$457$	−3.27051	−0.152988	−0.0764940	−	0.997070i	$-0.524373\pi$
$457$	−3.27051	−0.152988	−0.0764940	+	0.997070i	$0.524373\pi$
$458$	0	0
$459$	−8.09017	−0.377617
$460$	0	0
$461$	21.7426	1.01266	0.506328	−	0.862341i	$-0.331003\pi$
$461$	21.7426	1.01266	0.506328	+	0.862341i	$0.331003\pi$
$462$	0	0
$463$	−33.7771	−1.56975	−0.784877	−	0.619651i	$-0.787274\pi$
$463$	−33.7771	−1.56975	−0.784877	+	0.619651i	$0.787274\pi$
$464$	0	0
$465$	3.61803	0.167782
$466$	0	0
$467$	−23.8328	−1.10285	−0.551426	−	0.834224i	$-0.685916\pi$
$467$	−23.8328	−1.10285	−0.551426	+	0.834224i	$0.685916\pi$
$468$	0	0
$469$	−3.18034	−0.146854
$470$	0	0
$471$	5.94427	0.273898
$472$	0	0
$473$	9.27051	0.426258
$474$	0	0
$475$	−3.29180	−0.151038
$476$	0	0
$477$	−9.23607	−0.422891
$478$	0	0
$479$	7.23607	0.330624	0.165312	−	0.986241i	$-0.447137\pi$
$479$	7.23607	0.330624	0.165312	+	0.986241i	$0.447137\pi$
$480$	0	0
$481$	−36.2705	−1.65379
$482$	0	0
$483$	0.472136	0.0214829
$484$	0	0
$485$	−5.23607	−0.237758
$486$	0	0
$487$	26.5279	1.20209	0.601046	−	0.799214i	$-0.294751\pi$
$487$	26.5279	1.20209	0.601046	+	0.799214i	$0.294751\pi$
$488$	0	0
$489$	8.52786	0.385643
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	−12.4721	−0.561717
$494$	0	0
$495$	4.47214	0.201008
$496$	0	0
$497$	−3.25735	−0.146112
$498$	0	0
$499$	32.6869	1.46327	0.731634	−	0.681698i	$-0.238758\pi$
$499$	32.6869	1.46327	0.731634	+	0.681698i	$0.238758\pi$
$500$	0	0
$501$	−12.2361	−0.546667
$502$	0	0
$503$	8.23607	0.367228	0.183614	−	0.982998i	$-0.441220\pi$
$503$	8.23607	0.367228	0.183614	+	0.982998i	$0.441220\pi$
$504$	0	0
$505$	−22.0344	−0.980520
$506$	0	0
$507$	−10.5623	−0.469088
$508$	0	0
$509$	4.50658	0.199751	0.0998753	−	0.995000i	$-0.468156\pi$
$509$	4.50658	0.199751	0.0998753	+	0.995000i	$0.468156\pi$
$510$	0	0
$511$	−2.31308	−0.102325
$512$	0	0
$513$	6.90983	0.305076
$514$	0	0
$515$	−1.61803	−0.0712991
$516$	0	0
$517$	−6.70820	−0.295026
$518$	0	0
$519$	−6.61803	−0.290499
$520$	0	0
$521$	9.36068	0.410099	0.205049	−	0.978752i	$-0.434264\pi$
$521$	9.36068	0.410099	0.205049	+	0.978752i	$0.434264\pi$
$522$	0	0
$523$	−23.5279	−1.02880	−0.514401	−	0.857550i	$-0.671986\pi$
$523$	−23.5279	−1.02880	−0.514401	+	0.857550i	$0.671986\pi$
$524$	0	0
$525$	−0.562306	−0.0245410
$526$	0	0
$527$	−3.61803	−0.157604
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	10.6525	0.462278
$532$	0	0
$533$	43.4164	1.88057
$534$	0	0
$535$	−25.6525	−1.10905
$536$	0	0
$537$	−6.14590	−0.265215
$538$	0	0
$539$	−9.59675	−0.413361
$540$	0	0
$541$	−1.32624	−0.0570194	−0.0285097	−	0.999594i	$-0.509076\pi$
$541$	−1.32624	−0.0570194	−0.0285097	+	0.999594i	$0.509076\pi$
$542$	0	0
$543$	−1.85410	−0.0795671
$544$	0	0
$545$	14.7082	0.630030
$546$	0	0
$547$	8.27051	0.353621	0.176811	−	0.984245i	$-0.443422\pi$
$547$	8.27051	0.353621	0.176811	+	0.984245i	$0.443422\pi$
$548$	0	0
$549$	−0.291796	−0.0124536
$550$	0	0
$551$	10.6525	0.453811
$552$	0	0
$553$	−0.854102	−0.0363201
$554$	0	0
$555$	12.0902	0.513199
$556$	0	0
$557$	3.79837	0.160942	0.0804711	−	0.996757i	$-0.474358\pi$
$557$	3.79837	0.160942	0.0804711	+	0.996757i	$0.474358\pi$
$558$	0	0
$559$	−32.5623	−1.37724
$560$	0	0
$561$	2.23607	0.0944069
$562$	0	0
$563$	−1.20163	−0.0506425	−0.0253213	−	0.999679i	$-0.508061\pi$
$563$	−1.20163	−0.0506425	−0.0253213	+	0.999679i	$0.508061\pi$
$564$	0	0
$565$	−15.7984	−0.664643
$566$	0	0
$567$	−0.236068	−0.00991392
$568$	0	0
$569$	26.6180	1.11589	0.557943	−	0.829879i	$-0.311591\pi$
$569$	26.6180	1.11589	0.557943	+	0.829879i	$0.311591\pi$
$570$	0	0
$571$	38.4508	1.60912	0.804559	−	0.593873i	$-0.202402\pi$
$571$	38.4508	1.60912	0.804559	+	0.593873i	$0.202402\pi$
$572$	0	0
$573$	14.3820	0.600815
$574$	0	0
$575$	−4.76393	−0.198670
$576$	0	0
$577$	−10.3607	−0.431321	−0.215660	−	0.976468i	$-0.569190\pi$
$577$	−10.3607	−0.431321	−0.215660	+	0.976468i	$0.569190\pi$
$578$	0	0
$579$	13.0000	0.540262
$580$	0	0
$581$	2.54915	0.105757
$582$	0	0
$583$	6.38197	0.264314
$584$	0	0
$585$	−15.7082	−0.649454
$586$	0	0
$587$	−8.41641	−0.347382	−0.173691	−	0.984800i	$-0.555569\pi$
$587$	−8.41641	−0.347382	−0.173691	+	0.984800i	$0.555569\pi$
$588$	0	0
$589$	3.09017	0.127328
$590$	0	0
$591$	−14.2361	−0.585594
$592$	0	0
$593$	−15.7082	−0.645059	−0.322529	−	0.946559i	$-0.604533\pi$
$593$	−15.7082	−0.645059	−0.322529	+	0.946559i	$0.604533\pi$
$594$	0	0
$595$	−0.618034	−0.0253369
$596$	0	0
$597$	−3.05573	−0.125063
$598$	0	0
$599$	12.3262	0.503636	0.251818	−	0.967775i	$-0.418971\pi$
$599$	12.3262	0.503636	0.251818	+	0.967775i	$0.418971\pi$
$600$	0	0
$601$	−2.43769	−0.0994356	−0.0497178	−	0.998763i	$-0.515832\pi$
$601$	−2.43769	−0.0994356	−0.0497178	+	0.998763i	$0.515832\pi$
$602$	0	0
$603$	−26.9443	−1.09726
$604$	0	0
$605$	14.7082	0.597974
$606$	0	0
$607$	36.5410	1.48315	0.741577	−	0.670868i	$-0.234078\pi$
$607$	36.5410	1.48315	0.741577	+	0.670868i	$0.234078\pi$
$608$	0	0
$609$	1.81966	0.0737363
$610$	0	0
$611$	23.5623	0.953229
$612$	0	0
$613$	−15.0000	−0.605844	−0.302922	−	0.953015i	$-0.597962\pi$
$613$	−15.0000	−0.605844	−0.302922	+	0.953015i	$0.597962\pi$
$614$	0	0
$615$	−14.4721	−0.583573
$616$	0	0
$617$	14.4377	0.581240	0.290620	−	0.956839i	$-0.406139\pi$
$617$	14.4377	0.581240	0.290620	+	0.956839i	$0.406139\pi$
$618$	0	0
$619$	8.14590	0.327411	0.163706	−	0.986509i	$-0.447655\pi$
$619$	8.14590	0.327411	0.163706	+	0.986509i	$0.447655\pi$
$620$	0	0
$621$	10.0000	0.401286
$622$	0	0
$623$	−1.34752	−0.0539874
$624$	0	0
$625$	−7.41641	−0.296656
$626$	0	0
$627$	−1.90983	−0.0762713
$628$	0	0
$629$	−12.0902	−0.482067
$630$	0	0
$631$	32.6738	1.30072	0.650361	−	0.759625i	$-0.274618\pi$
$631$	32.6738	1.30072	0.650361	+	0.759625i	$0.274618\pi$
$632$	0	0
$633$	27.0344	1.07452
$634$	0	0
$635$	−16.7082	−0.663045
$636$	0	0
$637$	33.7082	1.33557
$638$	0	0
$639$	−27.5967	−1.09171
$640$	0	0
$641$	−2.63932	−0.104247	−0.0521234	−	0.998641i	$-0.516599\pi$
$641$	−2.63932	−0.104247	−0.0521234	+	0.998641i	$0.516599\pi$
$642$	0	0
$643$	−31.5410	−1.24386	−0.621928	−	0.783074i	$-0.713650\pi$
$643$	−31.5410	−1.24386	−0.621928	+	0.783074i	$0.713650\pi$
$644$	0	0
$645$	10.8541	0.427380
$646$	0	0
$647$	4.96556	0.195216	0.0976081	−	0.995225i	$-0.468881\pi$
$647$	4.96556	0.195216	0.0976081	+	0.995225i	$0.468881\pi$
$648$	0	0
$649$	−7.36068	−0.288932
$650$	0	0
$651$	0.527864	0.0206886
$652$	0	0
$653$	6.18034	0.241855	0.120928	−	0.992661i	$-0.461413\pi$
$653$	6.18034	0.241855	0.120928	+	0.992661i	$0.461413\pi$
$654$	0	0
$655$	−5.61803	−0.219515
$656$	0	0
$657$	−19.5967	−0.764542
$658$	0	0
$659$	−30.0557	−1.17080	−0.585402	−	0.810743i	$-0.699063\pi$
$659$	−30.0557	−1.17080	−0.585402	+	0.810743i	$0.699063\pi$
$660$	0	0
$661$	10.4377	0.405979	0.202990	−	0.979181i	$-0.434934\pi$
$661$	10.4377	0.405979	0.202990	+	0.979181i	$0.434934\pi$
$662$	0	0
$663$	−7.85410	−0.305028
$664$	0	0
$665$	0.527864	0.0204697
$666$	0	0
$667$	15.4164	0.596926
$668$	0	0
$669$	2.76393	0.106860
$670$	0	0
$671$	0.201626	0.00778369
$672$	0	0
$673$	48.0689	1.85292	0.926460	−	0.376394i	$-0.122836\pi$
$673$	48.0689	1.85292	0.926460	+	0.376394i	$0.122836\pi$
$674$	0	0
$675$	−11.9098	−0.458410
$676$	0	0
$677$	29.5623	1.13617	0.568086	−	0.822969i	$-0.307684\pi$
$677$	29.5623	1.13617	0.568086	+	0.822969i	$0.307684\pi$
$678$	0	0
$679$	−0.763932	−0.0293170
$680$	0	0
$681$	−23.8885	−0.915411
$682$	0	0
$683$	46.6525	1.78511	0.892554	−	0.450941i	$-0.148912\pi$
$683$	46.6525	1.78511	0.892554	+	0.450941i	$0.148912\pi$
$684$	0	0
$685$	10.0902	0.385526
$686$	0	0
$687$	9.29180	0.354504
$688$	0	0
$689$	−22.4164	−0.853997
$690$	0	0
$691$	−24.5066	−0.932274	−0.466137	−	0.884713i	$-0.654355\pi$
$691$	−24.5066	−0.932274	−0.466137	+	0.884713i	$0.654355\pi$
$692$	0	0
$693$	0.652476	0.0247855
$694$	0	0
$695$	8.61803	0.326901
$696$	0	0
$697$	14.4721	0.548171
$698$	0	0
$699$	−9.65248	−0.365090
$700$	0	0
$701$	−9.79837	−0.370079	−0.185040	−	0.982731i	$-0.559241\pi$
$701$	−9.79837	−0.370079	−0.185040	+	0.982731i	$0.559241\pi$
$702$	0	0
$703$	10.3262	0.389461
$704$	0	0
$705$	−7.85410	−0.295803
$706$	0	0
$707$	−3.21478	−0.120904
$708$	0	0
$709$	46.0344	1.72886	0.864430	−	0.502753i	$-0.167680\pi$
$709$	46.0344	1.72886	0.864430	+	0.502753i	$0.167680\pi$
$710$	0	0
$711$	−7.23607	−0.271374
$712$	0	0
$713$	4.47214	0.167483
$714$	0	0
$715$	10.8541	0.405920
$716$	0	0
$717$	7.61803	0.284501
$718$	0	0
$719$	−51.5066	−1.92087	−0.960436	−	0.278502i	$-0.910162\pi$
$719$	−51.5066	−1.92087	−0.960436	+	0.278502i	$0.910162\pi$
$720$	0	0
$721$	−0.236068	−0.00879163
$722$	0	0
$723$	7.61803	0.283318
$724$	0	0
$725$	−18.3607	−0.681899
$726$	0	0
$727$	23.2705	0.863055	0.431528	−	0.902100i	$-0.357975\pi$
$727$	23.2705	0.863055	0.431528	+	0.902100i	$0.357975\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−10.8541	−0.401453
$732$	0	0
$733$	5.94427	0.219557	0.109778	−	0.993956i	$-0.464986\pi$
$733$	5.94427	0.219557	0.109778	+	0.993956i	$0.464986\pi$
$734$	0	0
$735$	−11.2361	−0.414449
$736$	0	0
$737$	18.6180	0.685804
$738$	0	0
$739$	−4.47214	−0.164510	−0.0822551	−	0.996611i	$-0.526212\pi$
$739$	−4.47214	−0.164510	−0.0822551	+	0.996611i	$0.526212\pi$
$740$	0	0
$741$	6.70820	0.246432
$742$	0	0
$743$	−4.70820	−0.172727	−0.0863636	−	0.996264i	$-0.527525\pi$
$743$	−4.70820	−0.172727	−0.0863636	+	0.996264i	$0.527525\pi$
$744$	0	0
$745$	−13.7984	−0.505533
$746$	0	0
$747$	21.5967	0.790184
$748$	0	0
$749$	−3.74265	−0.136753
$750$	0	0
$751$	46.3050	1.68969	0.844846	−	0.535010i	$-0.179692\pi$
$751$	46.3050	1.68969	0.844846	+	0.535010i	$0.179692\pi$
$752$	0	0
$753$	2.29180	0.0835177
$754$	0	0
$755$	6.38197	0.232264
$756$	0	0
$757$	5.18034	0.188283	0.0941413	−	0.995559i	$-0.469989\pi$
$757$	5.18034	0.188283	0.0941413	+	0.995559i	$0.469989\pi$
$758$	0	0
$759$	−2.76393	−0.100324
$760$	0	0
$761$	−20.2361	−0.733557	−0.366778	−	0.930308i	$-0.619539\pi$
$761$	−20.2361	−0.733557	−0.366778	+	0.930308i	$0.619539\pi$
$762$	0	0
$763$	2.14590	0.0776867
$764$	0	0
$765$	−5.23607	−0.189310
$766$	0	0
$767$	25.8541	0.933538
$768$	0	0
$769$	36.6312	1.32095	0.660477	−	0.750846i	$-0.270354\pi$
$769$	36.6312	1.32095	0.660477	+	0.750846i	$0.270354\pi$
$770$	0	0
$771$	−0.236068	−0.00850178
$772$	0	0
$773$	21.0000	0.755318	0.377659	−	0.925945i	$-0.376729\pi$
$773$	21.0000	0.755318	0.377659	+	0.925945i	$0.376729\pi$
$774$	0	0
$775$	−5.32624	−0.191324
$776$	0	0
$777$	1.76393	0.0632807
$778$	0	0
$779$	−12.3607	−0.442867
$780$	0	0
$781$	19.0689	0.682338
$782$	0	0
$783$	38.5410	1.37734
$784$	0	0
$785$	9.61803	0.343282
$786$	0	0
$787$	−41.5279	−1.48031	−0.740154	−	0.672437i	$-0.765248\pi$
$787$	−41.5279	−1.48031	−0.740154	+	0.672437i	$0.765248\pi$
$788$	0	0
$789$	−0.326238	−0.0116144
$790$	0	0
$791$	−2.30495	−0.0819546
$792$	0	0
$793$	−0.708204	−0.0251491
$794$	0	0
$795$	7.47214	0.265009
$796$	0	0
$797$	32.7082	1.15858	0.579292	−	0.815120i	$-0.303329\pi$
$797$	32.7082	1.15858	0.579292	+	0.815120i	$0.303329\pi$
$798$	0	0
$799$	7.85410	0.277858
$800$	0	0
$801$	−11.4164	−0.403379
$802$	0	0
$803$	13.5410	0.477852
$804$	0	0
$805$	0.763932	0.0269251
$806$	0	0
$807$	−26.2148	−0.922804
$808$	0	0
$809$	10.0000	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$810$	0	0
$811$	16.5967	0.582791	0.291395	−	0.956603i	$-0.405880\pi$
$811$	16.5967	0.582791	0.291395	+	0.956603i	$0.405880\pi$
$812$	0	0
$813$	15.3607	0.538723
$814$	0	0
$815$	13.7984	0.483336
$816$	0	0
$817$	9.27051	0.324334
$818$	0	0
$819$	−2.29180	−0.0800818
$820$	0	0
$821$	−15.9443	−0.556459	−0.278229	−	0.960515i	$-0.589748\pi$
$821$	−15.9443	−0.556459	−0.278229	+	0.960515i	$0.589748\pi$
$822$	0	0
$823$	2.85410	0.0994877	0.0497439	−	0.998762i	$-0.484159\pi$
$823$	2.85410	0.0994877	0.0497439	+	0.998762i	$0.484159\pi$
$824$	0	0
$825$	3.29180	0.114606
$826$	0	0
$827$	47.0689	1.63675	0.818373	−	0.574688i	$-0.194877\pi$
$827$	47.0689	1.63675	0.818373	+	0.574688i	$0.194877\pi$
$828$	0	0
$829$	30.5623	1.06147	0.530736	−	0.847537i	$-0.321915\pi$
$829$	30.5623	1.06147	0.530736	+	0.847537i	$0.321915\pi$
$830$	0	0
$831$	0.0557281	0.00193319
$832$	0	0
$833$	11.2361	0.389307
$834$	0	0
$835$	−19.7984	−0.685151
$836$	0	0
$837$	11.1803	0.386449
$838$	0	0
$839$	−17.6180	−0.608242	−0.304121	−	0.952633i	$-0.598363\pi$
$839$	−17.6180	−0.608242	−0.304121	+	0.952633i	$0.598363\pi$
$840$	0	0
$841$	30.4164	1.04884
$842$	0	0
$843$	22.7082	0.782112
$844$	0	0
$845$	−17.0902	−0.587920
$846$	0	0
$847$	2.14590	0.0737339
$848$	0	0
$849$	−2.18034	−0.0748291
$850$	0	0
$851$	14.9443	0.512283
$852$	0	0
$853$	26.5623	0.909476	0.454738	−	0.890625i	$-0.349733\pi$
$853$	26.5623	0.909476	0.454738	+	0.890625i	$0.349733\pi$
$854$	0	0
$855$	4.47214	0.152944
$856$	0	0
$857$	0.819660	0.0279991	0.0139995	−	0.999902i	$-0.495544\pi$
$857$	0.819660	0.0279991	0.0139995	+	0.999902i	$0.495544\pi$
$858$	0	0
$859$	−7.90983	−0.269880	−0.134940	−	0.990854i	$-0.543084\pi$
$859$	−7.90983	−0.269880	−0.134940	+	0.990854i	$0.543084\pi$
$860$	0	0
$861$	−2.11146	−0.0719582
$862$	0	0
$863$	13.1246	0.446767	0.223383	−	0.974731i	$-0.428290\pi$
$863$	13.1246	0.446767	0.223383	+	0.974731i	$0.428290\pi$
$864$	0	0
$865$	−10.7082	−0.364090
$866$	0	0
$867$	14.3820	0.488437
$868$	0	0
$869$	5.00000	0.169613
$870$	0	0
$871$	−65.3951	−2.21583
$872$	0	0
$873$	−6.47214	−0.219049
$874$	0	0
$875$	−2.81966	−0.0953219
$876$	0	0
$877$	−31.2918	−1.05665	−0.528324	−	0.849043i	$-0.677180\pi$
$877$	−31.2918	−1.05665	−0.528324	+	0.849043i	$0.677180\pi$
$878$	0	0
$879$	−17.0000	−0.573396
$880$	0	0
$881$	−8.00000	−0.269527	−0.134763	−	0.990878i	$-0.543027\pi$
$881$	−8.00000	−0.269527	−0.134763	+	0.990878i	$0.543027\pi$
$882$	0	0
$883$	−51.8328	−1.74431	−0.872157	−	0.489227i	$-0.837279\pi$
$883$	−51.8328	−1.74431	−0.872157	+	0.489227i	$0.837279\pi$
$884$	0	0
$885$	−8.61803	−0.289692
$886$	0	0
$887$	−51.5410	−1.73058	−0.865289	−	0.501273i	$-0.832865\pi$
$887$	−51.5410	−1.73058	−0.865289	+	0.501273i	$0.832865\pi$
$888$	0	0
$889$	−2.43769	−0.0817576
$890$	0	0
$891$	1.38197	0.0462976
$892$	0	0
$893$	−6.70820	−0.224481
$894$	0	0
$895$	−9.94427	−0.332400
$896$	0	0
$897$	9.70820	0.324147
$898$	0	0
$899$	17.2361	0.574855
$900$	0	0
$901$	−7.47214	−0.248933
$902$	0	0
$903$	1.58359	0.0526986
$904$	0	0
$905$	−3.00000	−0.0997234
$906$	0	0
$907$	29.2361	0.970768	0.485384	−	0.874301i	$-0.338680\pi$
$907$	29.2361	0.970768	0.485384	+	0.874301i	$0.338680\pi$
$908$	0	0
$909$	−27.2361	−0.903363
$910$	0	0
$911$	−34.0344	−1.12761	−0.563806	−	0.825907i	$-0.690663\pi$
$911$	−34.0344	−1.12761	−0.563806	+	0.825907i	$0.690663\pi$
$912$	0	0
$913$	−14.9230	−0.493879
$914$	0	0
$915$	0.236068	0.00780417
$916$	0	0
$917$	−0.819660	−0.0270676
$918$	0	0
$919$	37.1033	1.22393	0.611963	−	0.790886i	$-0.290380\pi$
$919$	37.1033	1.22393	0.611963	+	0.790886i	$0.290380\pi$
$920$	0	0
$921$	−15.9787	−0.526517
$922$	0	0
$923$	−66.9787	−2.20463
$924$	0	0
$925$	−17.7984	−0.585207
$926$	0	0
$927$	−2.00000	−0.0656886
$928$	0	0
$929$	−10.0000	−0.328089	−0.164045	−	0.986453i	$-0.552454\pi$
$929$	−10.0000	−0.328089	−0.164045	+	0.986453i	$0.552454\pi$
$930$	0	0
$931$	−9.59675	−0.314521
$932$	0	0
$933$	−24.5967	−0.805261
$934$	0	0
$935$	3.61803	0.118322
$936$	0	0
$937$	53.1803	1.73733	0.868663	−	0.495403i	$-0.164980\pi$
$937$	53.1803	1.73733	0.868663	+	0.495403i	$0.164980\pi$
$938$	0	0
$939$	21.2361	0.693013
$940$	0	0
$941$	−23.9787	−0.781684	−0.390842	−	0.920458i	$-0.627816\pi$
$941$	−23.9787	−0.781684	−0.390842	+	0.920458i	$0.627816\pi$
$942$	0	0
$943$	−17.8885	−0.582531
$944$	0	0
$945$	1.90983	0.0621268
$946$	0	0
$947$	14.5410	0.472520	0.236260	−	0.971690i	$-0.424078\pi$
$947$	14.5410	0.472520	0.236260	+	0.971690i	$0.424078\pi$
$948$	0	0
$949$	−47.5623	−1.54394
$950$	0	0
$951$	19.3607	0.627813
$952$	0	0
$953$	19.1115	0.619081	0.309540	−	0.950886i	$-0.399825\pi$
$953$	19.1115	0.619081	0.309540	+	0.950886i	$0.399825\pi$
$954$	0	0
$955$	23.2705	0.753016
$956$	0	0
$957$	−10.6525	−0.344346
$958$	0	0
$959$	1.47214	0.0475377
$960$	0	0
$961$	−26.0000	−0.838710
$962$	0	0
$963$	−31.7082	−1.02178
$964$	0	0
$965$	21.0344	0.677123
$966$	0	0
$967$	−7.94427	−0.255471	−0.127735	−	0.991808i	$-0.540771\pi$
$967$	−7.94427	−0.255471	−0.127735	+	0.991808i	$0.540771\pi$
$968$	0	0
$969$	2.23607	0.0718329
$970$	0	0
$971$	16.6525	0.534403	0.267202	−	0.963641i	$-0.413901\pi$
$971$	16.6525	0.534403	0.267202	+	0.963641i	$0.413901\pi$
$972$	0	0
$973$	1.25735	0.0403089
$974$	0	0
$975$	−11.5623	−0.370290
$976$	0	0
$977$	39.1033	1.25103	0.625513	−	0.780214i	$-0.284890\pi$
$977$	39.1033	1.25103	0.625513	+	0.780214i	$0.284890\pi$
$978$	0	0
$979$	7.88854	0.252119
$980$	0	0
$981$	18.1803	0.580454
$982$	0	0
$983$	6.29180	0.200677	0.100339	−	0.994953i	$-0.468007\pi$
$983$	6.29180	0.200677	0.100339	+	0.994953i	$0.468007\pi$
$984$	0	0
$985$	−23.0344	−0.733938
$986$	0	0
$987$	−1.14590	−0.0364743
$988$	0	0
$989$	13.4164	0.426617
$990$	0	0
$991$	13.7295	0.436132	0.218066	−	0.975934i	$-0.430025\pi$
$991$	13.7295	0.436132	0.218066	+	0.975934i	$0.430025\pi$
$992$	0	0
$993$	2.03444	0.0645611
$994$	0	0
$995$	−4.94427	−0.156744
$996$	0	0
$997$	15.3607	0.486478	0.243239	−	0.969966i	$-0.421790\pi$
$997$	15.3607	0.486478	0.243239	+	0.969966i	$0.421790\pi$
$998$	0	0
$999$	37.3607	1.18204

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1648.2.a.b.1.1		2
4.3	odd	2		824.2.a.d.1.1	✓	2
8.3	odd	2		6592.2.a.e.1.2		2
8.5	even	2		6592.2.a.r.1.2		2
12.11	even	2		7416.2.a.g.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
824.2.a.d.1.1	✓	2	4.3	odd	2
1648.2.a.b.1.1		2	1.1	even	1	trivial
6592.2.a.e.1.2		2	8.3	odd	2
6592.2.a.r.1.2		2	8.5	even	2
7416.2.a.g.1.2		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1648 = 2^{4} \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1648.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.61803 q^{5} -0.236068 q^{7} -2.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.61803 q^{5} -0.236068 q^{7} -2.00000 q^{9} +1.38197 q^{11} -4.85410 q^{13} +1.61803 q^{15} -1.61803 q^{17} +1.38197 q^{19} +0.236068 q^{21} +2.00000 q^{23} -2.38197 q^{25} +5.00000 q^{27} +7.70820 q^{29} +2.23607 q^{31} -1.38197 q^{33} +0.381966 q^{35} +7.47214 q^{37} +4.85410 q^{39} -8.94427 q^{41} +6.70820 q^{43} +3.23607 q^{45} -4.85410 q^{47} -6.94427 q^{49} +1.61803 q^{51} +4.61803 q^{53} -2.23607 q^{55} -1.38197 q^{57} -5.32624 q^{59} +0.145898 q^{61} +0.472136 q^{63} +7.85410 q^{65} +13.4721 q^{67} -2.00000 q^{69} +13.7984 q^{71} +9.79837 q^{73} +2.38197 q^{75} -0.326238 q^{77} +3.61803 q^{79} +1.00000 q^{81} -10.7984 q^{83} +2.61803 q^{85} -7.70820 q^{87} +5.70820 q^{89} +1.14590 q^{91} -2.23607 q^{93} -2.23607 q^{95} +3.23607 q^{97} -2.76393 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - q^{5} + 4 q^{7} - 4 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - q^5 + 4 * q^7 - 4 * q^9	\( 2 q - 2 q^{3} - q^{5} + 4 q^{7} - 4 q^{9} + 5 q^{11} - 3 q^{13} + q^{15} - q^{17} + 5 q^{19} - 4 q^{21} + 4 q^{23} - 7 q^{25} + 10 q^{27} + 2 q^{29} - 5 q^{33} + 3 q^{35} + 6 q^{37} + 3 q^{39} + 2 q^{45} - 3 q^{47} + 4 q^{49} + q^{51} + 7 q^{53} - 5 q^{57} + 5 q^{59} + 7 q^{61} - 8 q^{63} + 9 q^{65} + 18 q^{67} - 4 q^{69} + 3 q^{71} - 5 q^{73} + 7 q^{75} + 15 q^{77} + 5 q^{79} + 2 q^{81} + 3 q^{83} + 3 q^{85} - 2 q^{87} - 2 q^{89} + 9 q^{91} + 2 q^{97} - 10 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - q^5 + 4 * q^7 - 4 * q^9 + 5 * q^11 - 3 * q^13 + q^15 - q^17 + 5 * q^19 - 4 * q^21 + 4 * q^23 - 7 * q^25 + 10 * q^27 + 2 * q^29 - 5 * q^33 + 3 * q^35 + 6 * q^37 + 3 * q^39 + 2 * q^45 - 3 * q^47 + 4 * q^49 + q^51 + 7 * q^53 - 5 * q^57 + 5 * q^59 + 7 * q^61 - 8 * q^63 + 9 * q^65 + 18 * q^67 - 4 * q^69 + 3 * q^71 - 5 * q^73 + 7 * q^75 + 15 * q^77 + 5 * q^79 + 2 * q^81 + 3 * q^83 + 3 * q^85 - 2 * q^87 - 2 * q^89 + 9 * q^91 + 2 * q^97 - 10 * q^99

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