LMFDB - Embedded newform 6728.2.a.z.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6728,2,Mod(1,6728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6728, 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("6728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6728 = 2^{3} \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6728.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:	$53.7233504799$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 18 x^{10} + 83 x^{9} + 83 x^{8} - 577 x^{7} + 121 x^{6} + 1416 x^{5} - 1289 x^{4} + \cdots + 64 $ x^12 - 4x^11 - 18x^10 + 83x^9 + 83x^8 - 577x^7 + 121x^6 + 1416x^5 - 1289x^4 - 465x^3 + 1056x^2 - 464*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 232)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-2.76023$ of defining polynomial
Character	$\chi$	$=$	6728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.76023 q^{3} -3.62598 q^{5} -2.23808 q^{7} +4.61885 q^{9} +O(q^{10})$	$q+2.76023 q^{3} -3.62598 q^{5} -2.23808 q^{7} +4.61885 q^{9} -3.57200 q^{11} +6.35276 q^{13} -10.0085 q^{15} -0.547824 q^{17} -0.184266 q^{19} -6.17762 q^{21} +7.21184 q^{23} +8.14773 q^{25} +4.46838 q^{27} -6.99718 q^{31} -9.85953 q^{33} +8.11525 q^{35} -3.06149 q^{37} +17.5350 q^{39} +4.05926 q^{41} -4.94254 q^{43} -16.7478 q^{45} -0.0885994 q^{47} -1.99098 q^{49} -1.51212 q^{51} -10.4237 q^{53} +12.9520 q^{55} -0.508615 q^{57} +10.9493 q^{59} -9.48762 q^{61} -10.3374 q^{63} -23.0350 q^{65} +2.51095 q^{67} +19.9063 q^{69} -9.11716 q^{71} +4.13408 q^{73} +22.4896 q^{75} +7.99444 q^{77} -4.53014 q^{79} -1.52280 q^{81} +9.17753 q^{83} +1.98640 q^{85} -5.39180 q^{89} -14.2180 q^{91} -19.3138 q^{93} +0.668143 q^{95} -16.0530 q^{97} -16.4985 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{3} - 4 q^{5} - q^{7} + 16 q^{9}+O(q^{10}) $ 12 * q - 4 * q^3 - 4 * q^5 - q^7 + 16 * q^9	$ 12 q - 4 q^{3} - 4 q^{5} - q^{7} + 16 q^{9} - 3 q^{11} - 3 q^{13} + 3 q^{15} - 8 q^{17} + 2 q^{19} - 6 q^{21} + 2 q^{23} + 12 q^{25} - 7 q^{27} - 29 q^{31} - 46 q^{33} + 17 q^{35} - 38 q^{37} + 10 q^{39} - 11 q^{41} - 9 q^{43} - 54 q^{45} - 34 q^{47} + 33 q^{49} + 17 q^{51} - 15 q^{53} - 2 q^{55} - q^{57} + 57 q^{59} - 37 q^{61} + 9 q^{63} - 59 q^{65} + 33 q^{67} + 21 q^{69} - 21 q^{71} - 13 q^{73} - 13 q^{75} + 3 q^{77} - 32 q^{79} + 36 q^{81} + 48 q^{83} - 17 q^{85} - 20 q^{89} - 2 q^{91} - 37 q^{93} - 7 q^{97} + 16 q^{99}+O(q^{100}) $ 12 * q - 4 * q^3 - 4 * q^5 - q^7 + 16 * q^9 - 3 * q^11 - 3 * q^13 + 3 * q^15 - 8 * q^17 + 2 * q^19 - 6 * q^21 + 2 * q^23 + 12 * q^25 - 7 * q^27 - 29 * q^31 - 46 * q^33 + 17 * q^35 - 38 * q^37 + 10 * q^39 - 11 * q^41 - 9 * q^43 - 54 * q^45 - 34 * q^47 + 33 * q^49 + 17 * q^51 - 15 * q^53 - 2 * q^55 - q^57 + 57 * q^59 - 37 * q^61 + 9 * q^63 - 59 * q^65 + 33 * q^67 + 21 * q^69 - 21 * q^71 - 13 * q^73 - 13 * q^75 + 3 * q^77 - 32 * q^79 + 36 * q^81 + 48 * q^83 - 17 * q^85 - 20 * q^89 - 2 * q^91 - 37 * q^93 - 7 * q^97 + 16 * 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
$3$	2.76023	1.59362	0.796809	−	0.604232i	$-0.206520\pi$
$3$	2.76023	1.59362	0.796809	+	0.604232i	$0.206520\pi$
$4$	0	0
$5$	−3.62598	−1.62159	−0.810794	−	0.585332i	$-0.800964\pi$
$5$	−3.62598	−1.62159	−0.810794	+	0.585332i	$0.800964\pi$
$6$	0	0
$7$	−2.23808	−0.845916	−0.422958	−	0.906149i	$-0.639008\pi$
$7$	−2.23808	−0.845916	−0.422958	+	0.906149i	$0.639008\pi$
$8$	0	0
$9$	4.61885	1.53962
$10$	0	0
$11$	−3.57200	−1.07700	−0.538499	−	0.842626i	$-0.681009\pi$
$11$	−3.57200	−1.07700	−0.538499	+	0.842626i	$0.681009\pi$
$12$	0	0
$13$	6.35276	1.76194	0.880969	−	0.473174i	$-0.156892\pi$
$13$	6.35276	1.76194	0.880969	+	0.473174i	$0.156892\pi$
$14$	0	0
$15$	−10.0085	−2.58419
$16$	0	0
$17$	−0.547824	−0.132867	−0.0664334	−	0.997791i	$-0.521162\pi$
$17$	−0.547824	−0.132867	−0.0664334	+	0.997791i	$0.521162\pi$
$18$	0	0
$19$	−0.184266	−0.0422734	−0.0211367	−	0.999777i	$-0.506729\pi$
$19$	−0.184266	−0.0422734	−0.0211367	+	0.999777i	$0.506729\pi$
$20$	0	0
$21$	−6.17762	−1.34807
$22$	0	0
$23$	7.21184	1.50377	0.751886	−	0.659293i	$-0.229144\pi$
$23$	7.21184	1.50377	0.751886	+	0.659293i	$0.229144\pi$
$24$	0	0
$25$	8.14773	1.62955
$26$	0	0
$27$	4.46838	0.859941
$28$	0	0
$29$	0	0
$29$	0	0
$30$	0	0
$31$	−6.99718	−1.25673	−0.628365	−	0.777919i	$-0.716276\pi$
$31$	−6.99718	−1.25673	−0.628365	+	0.777919i	$0.716276\pi$
$32$	0	0
$33$	−9.85953	−1.71632
$34$	0	0
$35$	8.11525	1.37173
$36$	0	0
$37$	−3.06149	−0.503306	−0.251653	−	0.967818i	$-0.580974\pi$
$37$	−3.06149	−0.503306	−0.251653	+	0.967818i	$0.580974\pi$
$38$	0	0
$39$	17.5350	2.80785
$40$	0	0
$41$	4.05926	0.633950	0.316975	−	0.948434i	$-0.397333\pi$
$41$	4.05926	0.633950	0.316975	+	0.948434i	$0.397333\pi$
$42$	0	0
$43$	−4.94254	−0.753730	−0.376865	−	0.926268i	$-0.622998\pi$
$43$	−4.94254	−0.753730	−0.376865	+	0.926268i	$0.622998\pi$
$44$	0	0
$45$	−16.7478	−2.49662
$46$	0	0
$47$	−0.0885994	−0.0129235	−0.00646177	−	0.999979i	$-0.502057\pi$
$47$	−0.0885994	−0.0129235	−0.00646177	+	0.999979i	$0.502057\pi$
$48$	0	0
$49$	−1.99098	−0.284426
$50$	0	0
$51$	−1.51212	−0.211739
$52$	0	0
$53$	−10.4237	−1.43180	−0.715902	−	0.698201i	$-0.753984\pi$
$53$	−10.4237	−1.43180	−0.715902	+	0.698201i	$0.753984\pi$
$54$	0	0
$55$	12.9520	1.74645
$56$	0	0
$57$	−0.508615	−0.0673677
$58$	0	0
$59$	10.9493	1.42548	0.712742	−	0.701426i	$-0.247453\pi$
$59$	10.9493	1.42548	0.712742	+	0.701426i	$0.247453\pi$
$60$	0	0
$61$	−9.48762	−1.21477	−0.607383	−	0.794409i	$-0.707780\pi$
$61$	−9.48762	−1.21477	−0.607383	+	0.794409i	$0.707780\pi$
$62$	0	0
$63$	−10.3374	−1.30239
$64$	0	0
$65$	−23.0350	−2.85714
$66$	0	0
$67$	2.51095	0.306761	0.153381	−	0.988167i	$-0.450984\pi$
$67$	2.51095	0.306761	0.153381	+	0.988167i	$0.450984\pi$
$68$	0	0
$69$	19.9063	2.39644
$70$	0	0
$71$	−9.11716	−1.08201	−0.541004	−	0.841020i	$-0.681956\pi$
$71$	−9.11716	−1.08201	−0.541004	+	0.841020i	$0.681956\pi$
$72$	0	0
$73$	4.13408	0.483857	0.241929	−	0.970294i	$-0.422220\pi$
$73$	4.13408	0.483857	0.241929	+	0.970294i	$0.422220\pi$
$74$	0	0
$75$	22.4896	2.59687
$76$	0	0
$77$	7.99444	0.911051
$78$	0	0
$79$	−4.53014	−0.509681	−0.254840	−	0.966983i	$-0.582023\pi$
$79$	−4.53014	−0.509681	−0.254840	+	0.966983i	$0.582023\pi$
$80$	0	0
$81$	−1.52280	−0.169200
$82$	0	0
$83$	9.17753	1.00737	0.503683	−	0.863889i	$-0.331978\pi$
$83$	9.17753	1.00737	0.503683	+	0.863889i	$0.331978\pi$
$84$	0	0
$85$	1.98640	0.215455
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−5.39180	−0.571529	−0.285765	−	0.958300i	$-0.592248\pi$
$89$	−5.39180	−0.571529	−0.285765	+	0.958300i	$0.592248\pi$
$90$	0	0
$91$	−14.2180	−1.49045
$92$	0	0
$93$	−19.3138	−2.00275
$94$	0	0
$95$	0.668143	0.0685501
$96$	0	0
$97$	−16.0530	−1.62994	−0.814968	−	0.579506i	$-0.803246\pi$
$97$	−16.0530	−1.62994	−0.814968	+	0.579506i	$0.803246\pi$
$98$	0	0
$99$	−16.4985	−1.65816
$100$	0	0
$101$	−8.08176	−0.804165	−0.402083	−	0.915603i	$-0.631714\pi$
$101$	−8.08176	−0.804165	−0.402083	+	0.915603i	$0.631714\pi$
$102$	0	0
$103$	−2.23535	−0.220255	−0.110128	−	0.993917i	$-0.535126\pi$
$103$	−2.23535	−0.220255	−0.110128	+	0.993917i	$0.535126\pi$
$104$	0	0
$105$	22.3999	2.18601
$106$	0	0
$107$	−1.87014	−0.180793	−0.0903964	−	0.995906i	$-0.528813\pi$
$107$	−1.87014	−0.180793	−0.0903964	+	0.995906i	$0.528813\pi$
$108$	0	0
$109$	−11.5903	−1.11015	−0.555075	−	0.831800i	$-0.687310\pi$
$109$	−11.5903	−1.11015	−0.555075	+	0.831800i	$0.687310\pi$
$110$	0	0
$111$	−8.45041	−0.802077
$112$	0	0
$113$	10.9718	1.03214	0.516071	−	0.856546i	$-0.327394\pi$
$113$	10.9718	1.03214	0.516071	+	0.856546i	$0.327394\pi$
$114$	0	0
$115$	−26.1500	−2.43850
$116$	0	0
$117$	29.3424	2.71271
$118$	0	0
$119$	1.22608	0.112394
$120$	0	0
$121$	1.75919	0.159927
$122$	0	0
$123$	11.2045	1.01027
$124$	0	0
$125$	−11.4136	−1.02087
$126$	0	0
$127$	−13.4610	−1.19447	−0.597235	−	0.802067i	$-0.703734\pi$
$127$	−13.4610	−1.19447	−0.597235	+	0.802067i	$0.703734\pi$
$128$	0	0
$129$	−13.6425	−1.20116
$130$	0	0
$131$	8.40177	0.734066	0.367033	−	0.930208i	$-0.380374\pi$
$131$	8.40177	0.734066	0.367033	+	0.930208i	$0.380374\pi$
$132$	0	0
$133$	0.412402	0.0357598
$134$	0	0
$135$	−16.2023	−1.39447
$136$	0	0
$137$	−5.13987	−0.439129	−0.219564	−	0.975598i	$-0.570464\pi$
$137$	−5.13987	−0.439129	−0.219564	+	0.975598i	$0.570464\pi$
$138$	0	0
$139$	5.49697	0.466247	0.233123	−	0.972447i	$-0.425105\pi$
$139$	5.49697	0.466247	0.233123	+	0.972447i	$0.425105\pi$
$140$	0	0
$141$	−0.244554	−0.0205952
$142$	0	0
$143$	−22.6921	−1.89760
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−5.49556	−0.453266
$148$	0	0
$149$	−15.5787	−1.27625	−0.638127	−	0.769931i	$-0.720291\pi$
$149$	−15.5787	−1.27625	−0.638127	+	0.769931i	$0.720291\pi$
$150$	0	0
$151$	−20.6627	−1.68150	−0.840751	−	0.541421i	$-0.817886\pi$
$151$	−20.6627	−1.68150	−0.840751	+	0.541421i	$0.817886\pi$
$152$	0	0
$153$	−2.53032	−0.204564
$154$	0	0
$155$	25.3716	2.03790
$156$	0	0
$157$	−11.1694	−0.891413	−0.445706	−	0.895179i	$-0.647047\pi$
$157$	−11.1694	−0.891413	−0.445706	+	0.895179i	$0.647047\pi$
$158$	0	0
$159$	−28.7717	−2.28175
$160$	0	0
$161$	−16.1407	−1.27207
$162$	0	0
$163$	−4.63112	−0.362737	−0.181369	−	0.983415i	$-0.558053\pi$
$163$	−4.63112	−0.362737	−0.181369	+	0.983415i	$0.558053\pi$
$164$	0	0
$165$	35.7505	2.78317
$166$	0	0
$167$	−5.94831	−0.460294	−0.230147	−	0.973156i	$-0.573921\pi$
$167$	−5.94831	−0.460294	−0.230147	+	0.973156i	$0.573921\pi$
$168$	0	0
$169$	27.3575	2.10442
$170$	0	0
$171$	−0.851094	−0.0650848
$172$	0	0
$173$	21.1968	1.61156	0.805780	−	0.592215i	$-0.201746\pi$
$173$	21.1968	1.61156	0.805780	+	0.592215i	$0.201746\pi$
$174$	0	0
$175$	−18.2353	−1.37846
$176$	0	0
$177$	30.2227	2.27168
$178$	0	0
$179$	12.8432	0.959949	0.479974	−	0.877283i	$-0.340646\pi$
$179$	12.8432	0.959949	0.479974	+	0.877283i	$0.340646\pi$
$180$	0	0
$181$	−5.45326	−0.405338	−0.202669	−	0.979247i	$-0.564962\pi$
$181$	−5.45326	−0.405338	−0.202669	+	0.979247i	$0.564962\pi$
$182$	0	0
$183$	−26.1880	−1.93587
$184$	0	0
$185$	11.1009	0.816155
$186$	0	0
$187$	1.95683	0.143097
$188$	0	0
$189$	−10.0006	−0.727438
$190$	0	0
$191$	−7.74697	−0.560551	−0.280275	−	0.959920i	$-0.590426\pi$
$191$	−7.74697	−0.560551	−0.280275	+	0.959920i	$0.590426\pi$
$192$	0	0
$193$	12.8513	0.925057	0.462528	−	0.886604i	$-0.346942\pi$
$193$	12.8513	0.925057	0.462528	+	0.886604i	$0.346942\pi$
$194$	0	0
$195$	−63.5817	−4.55318
$196$	0	0
$197$	2.89771	0.206453	0.103227	−	0.994658i	$-0.467083\pi$
$197$	2.89771	0.206453	0.103227	+	0.994658i	$0.467083\pi$
$198$	0	0
$199$	21.9859	1.55854	0.779271	−	0.626687i	$-0.215589\pi$
$199$	21.9859	1.55854	0.779271	+	0.626687i	$0.215589\pi$
$200$	0	0
$201$	6.93079	0.488860
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−14.7188	−1.02800
$206$	0	0
$207$	33.3104	2.31523
$208$	0	0
$209$	0.658197	0.0455284
$210$	0	0
$211$	−0.885009	−0.0609266	−0.0304633	−	0.999536i	$-0.509698\pi$
$211$	−0.885009	−0.0609266	−0.0304633	+	0.999536i	$0.509698\pi$
$212$	0	0
$213$	−25.1654	−1.72431
$214$	0	0
$215$	17.9215	1.22224
$216$	0	0
$217$	15.6603	1.06309
$218$	0	0
$219$	11.4110	0.771083
$220$	0	0
$221$	−3.48019	−0.234103
$222$	0	0
$223$	−22.1185	−1.48116	−0.740582	−	0.671965i	$-0.765450\pi$
$223$	−22.1185	−1.48116	−0.740582	+	0.671965i	$0.765450\pi$
$224$	0	0
$225$	37.6331	2.50888
$226$	0	0
$227$	1.13318	0.0752117	0.0376059	−	0.999293i	$-0.488027\pi$
$227$	1.13318	0.0752117	0.0376059	+	0.999293i	$0.488027\pi$
$228$	0	0
$229$	−9.50035	−0.627801	−0.313901	−	0.949456i	$-0.601636\pi$
$229$	−9.50035	−0.627801	−0.313901	+	0.949456i	$0.601636\pi$
$230$	0	0
$231$	22.0665	1.45187
$232$	0	0
$233$	−22.0978	−1.44767	−0.723837	−	0.689971i	$-0.757623\pi$
$233$	−22.0978	−1.44767	−0.723837	+	0.689971i	$0.757623\pi$
$234$	0	0
$235$	0.321260	0.0209567
$236$	0	0
$237$	−12.5042	−0.812236
$238$	0	0
$239$	0.748317	0.0484046	0.0242023	−	0.999707i	$-0.492295\pi$
$239$	0.748317	0.0484046	0.0242023	+	0.999707i	$0.492295\pi$
$240$	0	0
$241$	−14.7691	−0.951359	−0.475680	−	0.879619i	$-0.657798\pi$
$241$	−14.7691	−0.951359	−0.475680	+	0.879619i	$0.657798\pi$
$242$	0	0
$243$	−17.6084	−1.12958
$244$	0	0
$245$	7.21926	0.461222
$246$	0	0
$247$	−1.17059	−0.0744831
$248$	0	0
$249$	25.3321	1.60535
$250$	0	0
$251$	−24.3716	−1.53832	−0.769162	−	0.639054i	$-0.779326\pi$
$251$	−24.3716	−1.53832	−0.769162	+	0.639054i	$0.779326\pi$
$252$	0	0
$253$	−25.7607	−1.61956
$254$	0	0
$255$	5.48291	0.343353
$256$	0	0
$257$	11.9248	0.743848	0.371924	−	0.928263i	$-0.378698\pi$
$257$	11.9248	0.743848	0.371924	+	0.928263i	$0.378698\pi$
$258$	0	0
$259$	6.85188	0.425755
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−10.3930	−0.640861	−0.320431	−	0.947272i	$-0.603828\pi$
$263$	−10.3930	−0.640861	−0.320431	+	0.947272i	$0.603828\pi$
$264$	0	0
$265$	37.7961	2.32180
$266$	0	0
$267$	−14.8826	−0.910799
$268$	0	0
$269$	18.9183	1.15347	0.576734	−	0.816932i	$-0.304327\pi$
$269$	18.9183	1.15347	0.576734	+	0.816932i	$0.304327\pi$
$270$	0	0
$271$	−13.6247	−0.827643	−0.413822	−	0.910358i	$-0.635806\pi$
$271$	−13.6247	−0.827643	−0.413822	+	0.910358i	$0.635806\pi$
$272$	0	0
$273$	−39.2449	−2.37521
$274$	0	0
$275$	−29.1037	−1.75502
$276$	0	0
$277$	−9.43389	−0.566827	−0.283414	−	0.958998i	$-0.591467\pi$
$277$	−9.43389	−0.566827	−0.283414	+	0.958998i	$0.591467\pi$
$278$	0	0
$279$	−32.3189	−1.93488
$280$	0	0
$281$	−13.8331	−0.825214	−0.412607	−	0.910909i	$-0.635382\pi$
$281$	−13.8331	−0.825214	−0.412607	+	0.910909i	$0.635382\pi$
$282$	0	0
$283$	−0.329435	−0.0195829	−0.00979143	−	0.999952i	$-0.503117\pi$
$283$	−0.329435	−0.0195829	−0.00979143	+	0.999952i	$0.503117\pi$
$284$	0	0
$285$	1.84423	0.109243
$286$	0	0
$287$	−9.08496	−0.536268
$288$	0	0
$289$	−16.6999	−0.982346
$290$	0	0
$291$	−44.3099	−2.59749
$292$	0	0
$293$	−2.04031	−0.119196	−0.0595980	−	0.998222i	$-0.518982\pi$
$293$	−2.04031	−0.119196	−0.0595980	+	0.998222i	$0.518982\pi$
$294$	0	0
$295$	−39.7021	−2.31155
$296$	0	0
$297$	−15.9611	−0.926155
$298$	0	0
$299$	45.8151	2.64955
$300$	0	0
$301$	11.0618	0.637592
$302$	0	0
$303$	−22.3075	−1.28153
$304$	0	0
$305$	34.4019	1.96985
$306$	0	0
$307$	−20.9203	−1.19398	−0.596992	−	0.802247i	$-0.703638\pi$
$307$	−20.9203	−1.19398	−0.596992	+	0.802247i	$0.703638\pi$
$308$	0	0
$309$	−6.17006	−0.351002
$310$	0	0
$311$	28.0656	1.59145	0.795726	−	0.605656i	$-0.207089\pi$
$311$	28.0656	1.59145	0.795726	+	0.605656i	$0.207089\pi$
$312$	0	0
$313$	10.4509	0.590718	0.295359	−	0.955386i	$-0.404561\pi$
$313$	10.4509	0.590718	0.295359	+	0.955386i	$0.404561\pi$
$314$	0	0
$315$	37.4831	2.11193
$316$	0	0
$317$	−21.0900	−1.18453	−0.592266	−	0.805742i	$-0.701767\pi$
$317$	−21.0900	−1.18453	−0.592266	+	0.805742i	$0.701767\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−5.16200	−0.288115
$322$	0	0
$323$	0.100945	0.00561674
$324$	0	0
$325$	51.7606	2.87116
$326$	0	0
$327$	−31.9919	−1.76915
$328$	0	0
$329$	0.198293	0.0109322
$330$	0	0
$331$	−7.40886	−0.407228	−0.203614	−	0.979051i	$-0.565269\pi$
$331$	−7.40886	−0.407228	−0.203614	+	0.979051i	$0.565269\pi$
$332$	0	0
$333$	−14.1406	−0.774898
$334$	0	0
$335$	−9.10465	−0.497440
$336$	0	0
$337$	13.8889	0.756574	0.378287	−	0.925688i	$-0.376513\pi$
$337$	13.8889	0.756574	0.378287	+	0.925688i	$0.376513\pi$
$338$	0	0
$339$	30.2847	1.64484
$340$	0	0
$341$	24.9939	1.35350
$342$	0	0
$343$	20.1226	1.08652
$344$	0	0
$345$	−72.1799	−3.88603
$346$	0	0
$347$	17.5793	0.943706	0.471853	−	0.881677i	$-0.343585\pi$
$347$	17.5793	0.943706	0.471853	+	0.881677i	$0.343585\pi$
$348$	0	0
$349$	−8.17043	−0.437353	−0.218677	−	0.975797i	$-0.570174\pi$
$349$	−8.17043	−0.437353	−0.218677	+	0.975797i	$0.570174\pi$
$350$	0	0
$351$	28.3865	1.51516
$352$	0	0
$353$	13.7990	0.734448	0.367224	−	0.930133i	$-0.380308\pi$
$353$	13.7990	0.734448	0.367224	+	0.930133i	$0.380308\pi$
$354$	0	0
$355$	33.0586	1.75457
$356$	0	0
$357$	3.38425	0.179113
$358$	0	0
$359$	6.42385	0.339038	0.169519	−	0.985527i	$-0.445779\pi$
$359$	6.42385	0.339038	0.169519	+	0.985527i	$0.445779\pi$
$360$	0	0
$361$	−18.9660	−0.998213
$362$	0	0
$363$	4.85577	0.254862
$364$	0	0
$365$	−14.9901	−0.784617
$366$	0	0
$367$	9.66734	0.504631	0.252316	−	0.967645i	$-0.418808\pi$
$367$	9.66734	0.504631	0.252316	+	0.967645i	$0.418808\pi$
$368$	0	0
$369$	18.7491	0.976039
$370$	0	0
$371$	23.3291	1.21119
$372$	0	0
$373$	−14.4916	−0.750345	−0.375172	−	0.926955i	$-0.622417\pi$
$373$	−14.4916	−0.750345	−0.375172	+	0.926955i	$0.622417\pi$
$374$	0	0
$375$	−31.5042	−1.62687
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−21.4050	−1.09950	−0.549750	−	0.835329i	$-0.685277\pi$
$379$	−21.4050	−1.09950	−0.549750	+	0.835329i	$0.685277\pi$
$380$	0	0
$381$	−37.1554	−1.90353
$382$	0	0
$383$	16.3806	0.837011	0.418506	−	0.908214i	$-0.362554\pi$
$383$	16.3806	0.837011	0.418506	+	0.908214i	$0.362554\pi$
$384$	0	0
$385$	−28.9877	−1.47735
$386$	0	0
$387$	−22.8288	−1.16045
$388$	0	0
$389$	1.13328	0.0574596	0.0287298	−	0.999587i	$-0.490854\pi$
$389$	1.13328	0.0574596	0.0287298	+	0.999587i	$0.490854\pi$
$390$	0	0
$391$	−3.95082	−0.199802
$392$	0	0
$393$	23.1908	1.16982
$394$	0	0
$395$	16.4262	0.826492
$396$	0	0
$397$	6.26651	0.314507	0.157254	−	0.987558i	$-0.449736\pi$
$397$	6.26651	0.314507	0.157254	+	0.987558i	$0.449736\pi$
$398$	0	0
$399$	1.13832	0.0569874
$400$	0	0
$401$	−16.9139	−0.844638	−0.422319	−	0.906447i	$-0.638784\pi$
$401$	−16.9139	−0.844638	−0.422319	+	0.906447i	$0.638784\pi$
$402$	0	0
$403$	−44.4514	−2.21428
$404$	0	0
$405$	5.52163	0.274372
$406$	0	0
$407$	10.9357	0.542060
$408$	0	0
$409$	−29.1422	−1.44099	−0.720494	−	0.693461i	$-0.756085\pi$
$409$	−29.1422	−1.44099	−0.720494	+	0.693461i	$0.756085\pi$
$410$	0	0
$411$	−14.1872	−0.699803
$412$	0	0
$413$	−24.5056	−1.20584
$414$	0	0
$415$	−33.2776	−1.63353
$416$	0	0
$417$	15.1729	0.743019
$418$	0	0
$419$	−1.78493	−0.0871995	−0.0435998	−	0.999049i	$-0.513883\pi$
$419$	−1.78493	−0.0871995	−0.0435998	+	0.999049i	$0.513883\pi$
$420$	0	0
$421$	27.2359	1.32740	0.663699	−	0.748000i	$-0.268986\pi$
$421$	27.2359	1.32740	0.663699	+	0.748000i	$0.268986\pi$
$422$	0	0
$423$	−0.409227	−0.0198973
$424$	0	0
$425$	−4.46353	−0.216513
$426$	0	0
$427$	21.2341	1.02759
$428$	0	0
$429$	−62.6352	−3.02406
$430$	0	0
$431$	−16.8742	−0.812804	−0.406402	−	0.913694i	$-0.633217\pi$
$431$	−16.8742	−0.812804	−0.406402	+	0.913694i	$0.633217\pi$
$432$	0	0
$433$	−1.36659	−0.0656740	−0.0328370	−	0.999461i	$-0.510454\pi$
$433$	−1.36659	−0.0656740	−0.0328370	+	0.999461i	$0.510454\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.32889	−0.0635696
$438$	0	0
$439$	−24.2484	−1.15731	−0.578657	−	0.815571i	$-0.696423\pi$
$439$	−24.2484	−1.15731	−0.578657	+	0.815571i	$0.696423\pi$
$440$	0	0
$441$	−9.19604	−0.437906
$442$	0	0
$443$	8.22526	0.390794	0.195397	−	0.980724i	$-0.437400\pi$
$443$	8.22526	0.390794	0.195397	+	0.980724i	$0.437400\pi$
$444$	0	0
$445$	19.5505	0.926785
$446$	0	0
$447$	−43.0007	−2.03386
$448$	0	0
$449$	3.25222	0.153482	0.0767409	−	0.997051i	$-0.475549\pi$
$449$	3.25222	0.153482	0.0767409	+	0.997051i	$0.475549\pi$
$450$	0	0
$451$	−14.4997	−0.682763
$452$	0	0
$453$	−57.0336	−2.67967
$454$	0	0
$455$	51.5542	2.41690
$456$	0	0
$457$	20.6267	0.964876	0.482438	−	0.875930i	$-0.339751\pi$
$457$	20.6267	0.964876	0.482438	+	0.875930i	$0.339751\pi$
$458$	0	0
$459$	−2.44789	−0.114258
$460$	0	0
$461$	1.61470	0.0752042	0.0376021	−	0.999293i	$-0.488028\pi$
$461$	1.61470	0.0752042	0.0376021	+	0.999293i	$0.488028\pi$
$462$	0	0
$463$	30.5397	1.41930	0.709651	−	0.704553i	$-0.248853\pi$
$463$	30.5397	1.41930	0.709651	+	0.704553i	$0.248853\pi$
$464$	0	0
$465$	70.0314	3.24763
$466$	0	0
$467$	31.4134	1.45364	0.726820	−	0.686828i	$-0.240997\pi$
$467$	31.4134	1.45364	0.726820	+	0.686828i	$0.240997\pi$
$468$	0	0
$469$	−5.61971	−0.259494
$470$	0	0
$471$	−30.8300	−1.42057
$472$	0	0
$473$	17.6548	0.811766
$474$	0	0
$475$	−1.50135	−0.0688865
$476$	0	0
$477$	−48.1454	−2.20443
$478$	0	0
$479$	16.9065	0.772477	0.386239	−	0.922399i	$-0.373774\pi$
$479$	16.9065	0.772477	0.386239	+	0.922399i	$0.373774\pi$
$480$	0	0
$481$	−19.4489	−0.886794
$482$	0	0
$483$	−44.5520	−2.02719
$484$	0	0
$485$	58.2079	2.64308
$486$	0	0
$487$	−0.139377	−0.00631578	−0.00315789	−	0.999995i	$-0.501005\pi$
$487$	−0.139377	−0.00631578	−0.00315789	+	0.999995i	$0.501005\pi$
$488$	0	0
$489$	−12.7829	−0.578064
$490$	0	0
$491$	9.35056	0.421985	0.210992	−	0.977488i	$-0.432330\pi$
$491$	9.35056	0.421985	0.210992	+	0.977488i	$0.432330\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	59.8233	2.68886
$496$	0	0
$497$	20.4050	0.915288
$498$	0	0
$499$	−6.37759	−0.285500	−0.142750	−	0.989759i	$-0.545595\pi$
$499$	−6.37759	−0.285500	−0.142750	+	0.989759i	$0.545595\pi$
$500$	0	0
$501$	−16.4187	−0.733533
$502$	0	0
$503$	32.3703	1.44332	0.721659	−	0.692248i	$-0.243380\pi$
$503$	32.3703	1.44332	0.721659	+	0.692248i	$0.243380\pi$
$504$	0	0
$505$	29.3043	1.30402
$506$	0	0
$507$	75.5129	3.35365
$508$	0	0
$509$	−26.6644	−1.18188	−0.590940	−	0.806715i	$-0.701243\pi$
$509$	−26.6644	−1.18188	−0.590940	+	0.806715i	$0.701243\pi$
$510$	0	0
$511$	−9.25242	−0.409303
$512$	0	0
$513$	−0.823369	−0.0363526
$514$	0	0
$515$	8.10532	0.357163
$516$	0	0
$517$	0.316477	0.0139186
$518$	0	0
$519$	58.5079	2.56821
$520$	0	0
$521$	−34.2014	−1.49839	−0.749196	−	0.662349i	$-0.769560\pi$
$521$	−34.2014	−1.49839	−0.749196	+	0.662349i	$0.769560\pi$
$522$	0	0
$523$	−2.89329	−0.126515	−0.0632574	−	0.997997i	$-0.520149\pi$
$523$	−2.89329	−0.126515	−0.0632574	+	0.997997i	$0.520149\pi$
$524$	0	0
$525$	−50.3336	−2.19674
$526$	0	0
$527$	3.83322	0.166978
$528$	0	0
$529$	29.0106	1.26133
$530$	0	0
$531$	50.5734	2.19470
$532$	0	0
$533$	25.7875	1.11698
$534$	0	0
$535$	6.78108	0.293172
$536$	0	0
$537$	35.4502	1.52979
$538$	0	0
$539$	7.11179	0.306326
$540$	0	0
$541$	−19.4694	−0.837055	−0.418527	−	0.908204i	$-0.637454\pi$
$541$	−19.4694	−0.837055	−0.418527	+	0.908204i	$0.637454\pi$
$542$	0	0
$543$	−15.0522	−0.645953
$544$	0	0
$545$	42.0262	1.80021
$546$	0	0
$547$	27.6332	1.18151	0.590755	−	0.806851i	$-0.298830\pi$
$547$	27.6332	1.18151	0.590755	+	0.806851i	$0.298830\pi$
$548$	0	0
$549$	−43.8219	−1.87027
$550$	0	0
$551$	0	0
$552$	0	0
$553$	10.1388	0.431147
$554$	0	0
$555$	30.6410	1.30064
$556$	0	0
$557$	−0.430582	−0.0182443	−0.00912217	−	0.999958i	$-0.502904\pi$
$557$	−0.430582	−0.0182443	−0.00912217	+	0.999958i	$0.502904\pi$
$558$	0	0
$559$	−31.3987	−1.32803
$560$	0	0
$561$	5.40129	0.228043
$562$	0	0
$563$	2.47224	0.104192	0.0520962	−	0.998642i	$-0.483410\pi$
$563$	2.47224	0.104192	0.0520962	+	0.998642i	$0.483410\pi$
$564$	0	0
$565$	−39.7836	−1.67371
$566$	0	0
$567$	3.40814	0.143129
$568$	0	0
$569$	−41.5539	−1.74203	−0.871014	−	0.491258i	$-0.836537\pi$
$569$	−41.5539	−1.74203	−0.871014	+	0.491258i	$0.836537\pi$
$570$	0	0
$571$	34.0401	1.42453	0.712267	−	0.701909i	$-0.247669\pi$
$571$	34.0401	1.42453	0.712267	+	0.701909i	$0.247669\pi$
$572$	0	0
$573$	−21.3834	−0.893304
$574$	0	0
$575$	58.7601	2.45047
$576$	0	0
$577$	−19.1001	−0.795146	−0.397573	−	0.917571i	$-0.630147\pi$
$577$	−19.1001	−0.795146	−0.397573	+	0.917571i	$0.630147\pi$
$578$	0	0
$579$	35.4725	1.47419
$580$	0	0
$581$	−20.5401	−0.852147
$582$	0	0
$583$	37.2334	1.54205
$584$	0	0
$585$	−106.395	−4.39889
$586$	0	0
$587$	40.0415	1.65269	0.826344	−	0.563166i	$-0.190417\pi$
$587$	40.0415	1.65269	0.826344	+	0.563166i	$0.190417\pi$
$588$	0	0
$589$	1.28934	0.0531263
$590$	0	0
$591$	7.99833	0.329007
$592$	0	0
$593$	−6.37870	−0.261942	−0.130971	−	0.991386i	$-0.541809\pi$
$593$	−6.37870	−0.261942	−0.130971	+	0.991386i	$0.541809\pi$
$594$	0	0
$595$	−4.44573	−0.182257
$596$	0	0
$597$	60.6862	2.48372
$598$	0	0
$599$	14.1668	0.578838	0.289419	−	0.957202i	$-0.406538\pi$
$599$	14.1668	0.578838	0.289419	+	0.957202i	$0.406538\pi$
$600$	0	0
$601$	1.62234	0.0661766	0.0330883	−	0.999452i	$-0.489466\pi$
$601$	1.62234	0.0661766	0.0330883	+	0.999452i	$0.489466\pi$
$602$	0	0
$603$	11.5977	0.472294
$604$	0	0
$605$	−6.37880	−0.259335
$606$	0	0
$607$	37.1185	1.50659	0.753296	−	0.657682i	$-0.228463\pi$
$607$	37.1185	1.50659	0.753296	+	0.657682i	$0.228463\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−0.562850	−0.0227705
$612$	0	0
$613$	20.0101	0.808199	0.404099	−	0.914715i	$-0.367585\pi$
$613$	20.0101	0.808199	0.404099	+	0.914715i	$0.367585\pi$
$614$	0	0
$615$	−40.6272	−1.63825
$616$	0	0
$617$	12.7605	0.513720	0.256860	−	0.966449i	$-0.417312\pi$
$617$	12.7605	0.513720	0.256860	+	0.966449i	$0.417312\pi$
$618$	0	0
$619$	25.3729	1.01982	0.509911	−	0.860227i	$-0.329678\pi$
$619$	25.3729	1.01982	0.509911	+	0.860227i	$0.329678\pi$
$620$	0	0
$621$	32.2253	1.29315
$622$	0	0
$623$	12.0673	0.483466
$624$	0	0
$625$	0.646906	0.0258763
$626$	0	0
$627$	1.81677	0.0725549
$628$	0	0
$629$	1.67716	0.0668727
$630$	0	0
$631$	3.88627	0.154710	0.0773549	−	0.997004i	$-0.475353\pi$
$631$	3.88627	0.154710	0.0773549	+	0.997004i	$0.475353\pi$
$632$	0	0
$633$	−2.44283	−0.0970936
$634$	0	0
$635$	48.8093	1.93694
$636$	0	0
$637$	−12.6482	−0.501141
$638$	0	0
$639$	−42.1108	−1.66588
$640$	0	0
$641$	40.6230	1.60451	0.802256	−	0.596980i	$-0.203633\pi$
$641$	40.6230	1.60451	0.802256	+	0.596980i	$0.203633\pi$
$642$	0	0
$643$	20.4273	0.805573	0.402787	−	0.915294i	$-0.368042\pi$
$643$	20.4273	0.805573	0.402787	+	0.915294i	$0.368042\pi$
$644$	0	0
$645$	49.4675	1.94778
$646$	0	0
$647$	22.6812	0.891692	0.445846	−	0.895110i	$-0.352903\pi$
$647$	22.6812	0.891692	0.445846	+	0.895110i	$0.352903\pi$
$648$	0	0
$649$	−39.1111	−1.53524
$650$	0	0
$651$	43.2259	1.69416
$652$	0	0
$653$	30.0467	1.17582	0.587909	−	0.808927i	$-0.299951\pi$
$653$	30.0467	1.17582	0.587909	+	0.808927i	$0.299951\pi$
$654$	0	0
$655$	−30.4647	−1.19035
$656$	0	0
$657$	19.0947	0.744954
$658$	0	0
$659$	−21.2950	−0.829535	−0.414768	−	0.909927i	$-0.636137\pi$
$659$	−21.2950	−0.829535	−0.414768	+	0.909927i	$0.636137\pi$
$660$	0	0
$661$	−40.0853	−1.55914	−0.779569	−	0.626316i	$-0.784562\pi$
$661$	−40.0853	−1.55914	−0.779569	+	0.626316i	$0.784562\pi$
$662$	0	0
$663$	−9.60612	−0.373071
$664$	0	0
$665$	−1.49536	−0.0579876
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−61.0521	−2.36041
$670$	0	0
$671$	33.8898	1.30830
$672$	0	0
$673$	15.8180	0.609741	0.304870	−	0.952394i	$-0.401387\pi$
$673$	15.8180	0.609741	0.304870	+	0.952394i	$0.401387\pi$
$674$	0	0
$675$	36.4072	1.40131
$676$	0	0
$677$	32.9482	1.26630	0.633151	−	0.774028i	$-0.281761\pi$
$677$	32.9482	1.26630	0.633151	+	0.774028i	$0.281761\pi$
$678$	0	0
$679$	35.9280	1.37879
$680$	0	0
$681$	3.12783	0.119859
$682$	0	0
$683$	−23.2965	−0.891414	−0.445707	−	0.895179i	$-0.647048\pi$
$683$	−23.2965	−0.891414	−0.445707	+	0.895179i	$0.647048\pi$
$684$	0	0
$685$	18.6371	0.712086
$686$	0	0
$687$	−26.2231	−1.00047
$688$	0	0
$689$	−66.2192	−2.52275
$690$	0	0
$691$	43.3586	1.64944	0.824720	−	0.565541i	$-0.191333\pi$
$691$	43.3586	1.64944	0.824720	+	0.565541i	$0.191333\pi$
$692$	0	0
$693$	36.9251	1.40267
$694$	0	0
$695$	−19.9319	−0.756060
$696$	0	0
$697$	−2.22376	−0.0842309
$698$	0	0
$699$	−60.9948	−2.30704
$700$	0	0
$701$	48.8002	1.84316	0.921580	−	0.388189i	$-0.126899\pi$
$701$	48.8002	1.84316	0.921580	+	0.388189i	$0.126899\pi$
$702$	0	0
$703$	0.564128	0.0212765
$704$	0	0
$705$	0.886749	0.0333969
$706$	0	0
$707$	18.0877	0.680256
$708$	0	0
$709$	26.9523	1.01222	0.506108	−	0.862470i	$-0.331084\pi$
$709$	26.9523	1.01222	0.506108	+	0.862470i	$0.331084\pi$
$710$	0	0
$711$	−20.9240	−0.784713
$712$	0	0
$713$	−50.4625	−1.88984
$714$	0	0
$715$	82.2809	3.07713
$716$	0	0
$717$	2.06552	0.0771384
$718$	0	0
$719$	−31.8556	−1.18801	−0.594007	−	0.804460i	$-0.702455\pi$
$719$	−31.8556	−1.18801	−0.594007	+	0.804460i	$0.702455\pi$
$720$	0	0
$721$	5.00289	0.186317
$722$	0	0
$723$	−40.7660	−1.51610
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−8.26570	−0.306558	−0.153279	−	0.988183i	$-0.548983\pi$
$727$	−8.26570	−0.306558	−0.153279	+	0.988183i	$0.548983\pi$
$728$	0	0
$729$	−44.0348	−1.63092
$730$	0	0
$731$	2.70764	0.100146
$732$	0	0
$733$	27.0689	0.999814	0.499907	−	0.866079i	$-0.333368\pi$
$733$	27.0689	0.999814	0.499907	+	0.866079i	$0.333368\pi$
$734$	0	0
$735$	19.9268	0.735010
$736$	0	0
$737$	−8.96911	−0.330381
$738$	0	0
$739$	45.6437	1.67903	0.839516	−	0.543335i	$-0.182839\pi$
$739$	45.6437	1.67903	0.839516	+	0.543335i	$0.182839\pi$
$740$	0	0
$741$	−3.23110	−0.118698
$742$	0	0
$743$	1.54539	0.0566948	0.0283474	−	0.999598i	$-0.490976\pi$
$743$	1.54539	0.0566948	0.0283474	+	0.999598i	$0.490976\pi$
$744$	0	0
$745$	56.4880	2.06956
$746$	0	0
$747$	42.3896	1.55096
$748$	0	0
$749$	4.18552	0.152936
$750$	0	0
$751$	0.893292	0.0325967	0.0162983	−	0.999867i	$-0.494812\pi$
$751$	0.893292	0.0325967	0.0162983	+	0.999867i	$0.494812\pi$
$752$	0	0
$753$	−67.2712	−2.45150
$754$	0	0
$755$	74.9224	2.72670
$756$	0	0
$757$	32.1925	1.17006	0.585028	−	0.811013i	$-0.301083\pi$
$757$	32.1925	1.17006	0.585028	+	0.811013i	$0.301083\pi$
$758$	0	0
$759$	−71.1053	−2.58096
$760$	0	0
$761$	33.7861	1.22475	0.612373	−	0.790569i	$-0.290215\pi$
$761$	33.7861	1.22475	0.612373	+	0.790569i	$0.290215\pi$
$762$	0	0
$763$	25.9401	0.939094
$764$	0	0
$765$	9.17488	0.331718
$766$	0	0
$767$	69.5585	2.51161
$768$	0	0
$769$	−38.3902	−1.38439	−0.692194	−	0.721712i	$-0.743356\pi$
$769$	−38.3902	−1.38439	−0.692194	+	0.721712i	$0.743356\pi$
$770$	0	0
$771$	32.9151	1.18541
$772$	0	0
$773$	−6.24386	−0.224576	−0.112288	−	0.993676i	$-0.535818\pi$
$773$	−6.24386	−0.224576	−0.112288	+	0.993676i	$0.535818\pi$
$774$	0	0
$775$	−57.0112	−2.04790
$776$	0	0
$777$	18.9127	0.678490
$778$	0	0
$779$	−0.747982	−0.0267992
$780$	0	0
$781$	32.5665	1.16532
$782$	0	0
$783$	0	0
$784$	0	0
$785$	40.4999	1.44550
$786$	0	0
$787$	−6.56571	−0.234042	−0.117021	−	0.993129i	$-0.537335\pi$
$787$	−6.56571	−0.234042	−0.117021	+	0.993129i	$0.537335\pi$
$788$	0	0
$789$	−28.6871	−1.02129
$790$	0	0
$791$	−24.5559	−0.873106
$792$	0	0
$793$	−60.2725	−2.14034
$794$	0	0
$795$	104.326	3.70005
$796$	0	0
$797$	−23.9048	−0.846753	−0.423376	−	0.905954i	$-0.639155\pi$
$797$	−23.9048	−0.846753	−0.423376	+	0.905954i	$0.639155\pi$
$798$	0	0
$799$	0.0485369	0.00171711
$800$	0	0
$801$	−24.9039	−0.879935
$802$	0	0
$803$	−14.7669	−0.521114
$804$	0	0
$805$	58.5259	2.06277
$806$	0	0
$807$	52.2187	1.83819
$808$	0	0
$809$	16.6391	0.585001	0.292501	−	0.956265i	$-0.405513\pi$
$809$	16.6391	0.585001	0.292501	+	0.956265i	$0.405513\pi$
$810$	0	0
$811$	10.1207	0.355387	0.177694	−	0.984086i	$-0.443136\pi$
$811$	10.1207	0.355387	0.177694	+	0.984086i	$0.443136\pi$
$812$	0	0
$813$	−37.6073	−1.31895
$814$	0	0
$815$	16.7923	0.588210
$816$	0	0
$817$	0.910740	0.0318627
$818$	0	0
$819$	−65.6708	−2.29472
$820$	0	0
$821$	−30.2789	−1.05674	−0.528371	−	0.849014i	$-0.677197\pi$
$821$	−30.2789	−1.05674	−0.528371	+	0.849014i	$0.677197\pi$
$822$	0	0
$823$	−6.97251	−0.243046	−0.121523	−	0.992589i	$-0.538778\pi$
$823$	−6.97251	−0.243046	−0.121523	+	0.992589i	$0.538778\pi$
$824$	0	0
$825$	−80.3328	−2.79683
$826$	0	0
$827$	−52.0902	−1.81135	−0.905676	−	0.423970i	$-0.860636\pi$
$827$	−52.0902	−1.81135	−0.905676	+	0.423970i	$0.860636\pi$
$828$	0	0
$829$	49.6496	1.72440	0.862201	−	0.506566i	$-0.169085\pi$
$829$	49.6496	1.72440	0.862201	+	0.506566i	$0.169085\pi$
$830$	0	0
$831$	−26.0397	−0.903306
$832$	0	0
$833$	1.09071	0.0377908
$834$	0	0
$835$	21.5685	0.746407
$836$	0	0
$837$	−31.2661	−1.08071
$838$	0	0
$839$	−39.3643	−1.35901	−0.679503	−	0.733673i	$-0.737804\pi$
$839$	−39.3643	−1.35901	−0.679503	+	0.733673i	$0.737804\pi$
$840$	0	0
$841$	0	0
$842$	0	0
$843$	−38.1825	−1.31508
$844$	0	0
$845$	−99.1978	−3.41251
$846$	0	0
$847$	−3.93722	−0.135285
$848$	0	0
$849$	−0.909314	−0.0312076
$850$	0	0
$851$	−22.0790	−0.756858
$852$	0	0
$853$	−28.5200	−0.976506	−0.488253	−	0.872702i	$-0.662366\pi$
$853$	−28.5200	−0.976506	−0.488253	+	0.872702i	$0.662366\pi$
$854$	0	0
$855$	3.08605	0.105541
$856$	0	0
$857$	3.38863	0.115754	0.0578768	−	0.998324i	$-0.481567\pi$
$857$	3.38863	0.115754	0.0578768	+	0.998324i	$0.481567\pi$
$858$	0	0
$859$	−57.7404	−1.97008	−0.985038	−	0.172335i	$-0.944869\pi$
$859$	−57.7404	−1.97008	−0.985038	+	0.172335i	$0.944869\pi$
$860$	0	0
$861$	−25.0765	−0.854606
$862$	0	0
$863$	−46.2155	−1.57319	−0.786597	−	0.617467i	$-0.788159\pi$
$863$	−46.2155	−1.57319	−0.786597	+	0.617467i	$0.788159\pi$
$864$	0	0
$865$	−76.8591	−2.61329
$866$	0	0
$867$	−46.0955	−1.56548
$868$	0	0
$869$	16.1817	0.548926
$870$	0	0
$871$	15.9514	0.540494
$872$	0	0
$873$	−74.1464	−2.50947
$874$	0	0
$875$	25.5446	0.863567
$876$	0	0
$877$	22.0143	0.743369	0.371684	−	0.928359i	$-0.378780\pi$
$877$	22.0143	0.743369	0.371684	+	0.928359i	$0.378780\pi$
$878$	0	0
$879$	−5.63171	−0.189953
$880$	0	0
$881$	−22.0973	−0.744476	−0.372238	−	0.928137i	$-0.621409\pi$
$881$	−22.0973	−0.744476	−0.372238	+	0.928137i	$0.621409\pi$
$882$	0	0
$883$	9.72516	0.327278	0.163639	−	0.986520i	$-0.447677\pi$
$883$	9.72516	0.327278	0.163639	+	0.986520i	$0.447677\pi$
$884$	0	0
$885$	−109.587	−3.68372
$886$	0	0
$887$	−8.11638	−0.272521	−0.136261	−	0.990673i	$-0.543508\pi$
$887$	−8.11638	−0.272521	−0.136261	+	0.990673i	$0.543508\pi$
$888$	0	0
$889$	30.1268	1.01042
$890$	0	0
$891$	5.43943	0.182228
$892$	0	0
$893$	0.0163258	0.000546323 0
$894$	0	0
$895$	−46.5693	−1.55664
$896$	0	0
$897$	126.460	4.22237
$898$	0	0
$899$	0	0
$900$	0	0
$901$	5.71035	0.190239
$902$	0	0
$903$	30.5331	1.01608
$904$	0	0
$905$	19.7734	0.657291
$906$	0	0
$907$	−1.74403	−0.0579094	−0.0289547	−	0.999581i	$-0.509218\pi$
$907$	−1.74403	−0.0579094	−0.0289547	+	0.999581i	$0.509218\pi$
$908$	0	0
$909$	−37.3284	−1.23811
$910$	0	0
$911$	−31.9766	−1.05943	−0.529716	−	0.848175i	$-0.677702\pi$
$911$	−31.9766	−1.05943	−0.529716	+	0.848175i	$0.677702\pi$
$912$	0	0
$913$	−32.7822	−1.08493
$914$	0	0
$915$	94.9571	3.13918
$916$	0	0
$917$	−18.8039	−0.620959
$918$	0	0
$919$	−0.731778	−0.0241391	−0.0120696	−	0.999927i	$-0.503842\pi$
$919$	−0.731778	−0.0241391	−0.0120696	+	0.999927i	$0.503842\pi$
$920$	0	0
$921$	−57.7447	−1.90275
$922$	0	0
$923$	−57.9191	−1.90643
$924$	0	0
$925$	−24.9442	−0.820161
$926$	0	0
$927$	−10.3247	−0.339108
$928$	0	0
$929$	28.6849	0.941121	0.470561	−	0.882368i	$-0.344052\pi$
$929$	28.6849	0.941121	0.470561	+	0.882368i	$0.344052\pi$
$930$	0	0
$931$	0.366869	0.0120237
$932$	0	0
$933$	77.4673	2.53617
$934$	0	0
$935$	−7.09542	−0.232045
$936$	0	0
$937$	−13.8406	−0.452152	−0.226076	−	0.974110i	$-0.572590\pi$
$937$	−13.8406	−0.452152	−0.226076	+	0.974110i	$0.572590\pi$
$938$	0	0
$939$	28.8468	0.941379
$940$	0	0
$941$	−22.2051	−0.723866	−0.361933	−	0.932204i	$-0.617883\pi$
$941$	−22.2051	−0.723866	−0.361933	+	0.932204i	$0.617883\pi$
$942$	0	0
$943$	29.2747	0.953316
$944$	0	0
$945$	36.2620	1.17960
$946$	0	0
$947$	−14.0312	−0.455953	−0.227976	−	0.973667i	$-0.573211\pi$
$947$	−14.0312	−0.455953	−0.227976	+	0.973667i	$0.573211\pi$
$948$	0	0
$949$	26.2628	0.852526
$950$	0	0
$951$	−58.2131	−1.88769
$952$	0	0
$953$	24.9662	0.808736	0.404368	−	0.914596i	$-0.367492\pi$
$953$	24.9662	0.808736	0.404368	+	0.914596i	$0.367492\pi$
$954$	0	0
$955$	28.0904	0.908983
$956$	0	0
$957$	0	0
$958$	0	0
$959$	11.5035	0.371466
$960$	0	0
$961$	17.9605	0.579371
$962$	0	0
$963$	−8.63787	−0.278352
$964$	0	0
$965$	−46.5986	−1.50006
$966$	0	0
$967$	−1.74472	−0.0561064	−0.0280532	−	0.999606i	$-0.508931\pi$
$967$	−1.74472	−0.0561064	−0.0280532	+	0.999606i	$0.508931\pi$
$968$	0	0
$969$	0.278631	0.00895093
$970$	0	0
$971$	−16.6211	−0.533398	−0.266699	−	0.963780i	$-0.585933\pi$
$971$	−16.6211	−0.533398	−0.266699	+	0.963780i	$0.585933\pi$
$972$	0	0
$973$	−12.3027	−0.394406
$974$	0	0
$975$	142.871	4.57553
$976$	0	0
$977$	−1.90583	−0.0609729	−0.0304864	−	0.999535i	$-0.509706\pi$
$977$	−1.90583	−0.0609729	−0.0304864	+	0.999535i	$0.509706\pi$
$978$	0	0
$979$	19.2595	0.615536
$980$	0	0
$981$	−53.5338	−1.70920
$982$	0	0
$983$	−27.7600	−0.885407	−0.442704	−	0.896668i	$-0.645981\pi$
$983$	−27.7600	−0.885407	−0.442704	+	0.896668i	$0.645981\pi$
$984$	0	0
$985$	−10.5070	−0.334782
$986$	0	0
$987$	0.547333	0.0174218
$988$	0	0
$989$	−35.6448	−1.13344
$990$	0	0
$991$	36.1929	1.14971	0.574853	−	0.818257i	$-0.305059\pi$
$991$	36.1929	1.14971	0.574853	+	0.818257i	$0.305059\pi$
$992$	0	0
$993$	−20.4501	−0.648965
$994$	0	0
$995$	−79.7206	−2.52731
$996$	0	0
$997$	−45.7372	−1.44851	−0.724255	−	0.689532i	$-0.757816\pi$
$997$	−45.7372	−1.44851	−0.724255	+	0.689532i	$0.757816\pi$
$998$	0	0
$999$	−13.6799	−0.432813

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6728.2.a.z.1.11		12
29.16	even	7		232.2.m.d.169.1	yes	24
29.20	even	7		232.2.m.d.81.1	✓	24
29.28	even	2		6728.2.a.bb.1.2		12
116.103	odd	14		464.2.u.i.401.4		24
116.107	odd	14		464.2.u.i.81.4		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
232.2.m.d.81.1	✓	24	29.20	even	7
232.2.m.d.169.1	yes	24	29.16	even	7
464.2.u.i.81.4		24	116.107	odd	14
464.2.u.i.401.4		24	116.103	odd	14
6728.2.a.z.1.11		12	1.1	even	1	trivial
6728.2.a.bb.1.2		12	29.28	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:	\( N \)	\(=\)	\( 6728 = 2^{3} \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6728.a (trivial)

\(f(q)\)	\(=\)	\(q+2.76023 q^{3} -3.62598 q^{5} -2.23808 q^{7} +4.61885 q^{9} +O(q^{10})\)	\(q+2.76023 q^{3} -3.62598 q^{5} -2.23808 q^{7} +4.61885 q^{9} -3.57200 q^{11} +6.35276 q^{13} -10.0085 q^{15} -0.547824 q^{17} -0.184266 q^{19} -6.17762 q^{21} +7.21184 q^{23} +8.14773 q^{25} +4.46838 q^{27} -6.99718 q^{31} -9.85953 q^{33} +8.11525 q^{35} -3.06149 q^{37} +17.5350 q^{39} +4.05926 q^{41} -4.94254 q^{43} -16.7478 q^{45} -0.0885994 q^{47} -1.99098 q^{49} -1.51212 q^{51} -10.4237 q^{53} +12.9520 q^{55} -0.508615 q^{57} +10.9493 q^{59} -9.48762 q^{61} -10.3374 q^{63} -23.0350 q^{65} +2.51095 q^{67} +19.9063 q^{69} -9.11716 q^{71} +4.13408 q^{73} +22.4896 q^{75} +7.99444 q^{77} -4.53014 q^{79} -1.52280 q^{81} +9.17753 q^{83} +1.98640 q^{85} -5.39180 q^{89} -14.2180 q^{91} -19.3138 q^{93} +0.668143 q^{95} -16.0530 q^{97} -16.4985 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{3} - 4 q^{5} - q^{7} + 16 q^{9}+O(q^{10}) \) 12 * q - 4 * q^3 - 4 * q^5 - q^7 + 16 * q^9	\( 12 q - 4 q^{3} - 4 q^{5} - q^{7} + 16 q^{9} - 3 q^{11} - 3 q^{13} + 3 q^{15} - 8 q^{17} + 2 q^{19} - 6 q^{21} + 2 q^{23} + 12 q^{25} - 7 q^{27} - 29 q^{31} - 46 q^{33} + 17 q^{35} - 38 q^{37} + 10 q^{39} - 11 q^{41} - 9 q^{43} - 54 q^{45} - 34 q^{47} + 33 q^{49} + 17 q^{51} - 15 q^{53} - 2 q^{55} - q^{57} + 57 q^{59} - 37 q^{61} + 9 q^{63} - 59 q^{65} + 33 q^{67} + 21 q^{69} - 21 q^{71} - 13 q^{73} - 13 q^{75} + 3 q^{77} - 32 q^{79} + 36 q^{81} + 48 q^{83} - 17 q^{85} - 20 q^{89} - 2 q^{91} - 37 q^{93} - 7 q^{97} + 16 q^{99}+O(q^{100}) \) 12 * q - 4 * q^3 - 4 * q^5 - q^7 + 16 * q^9 - 3 * q^11 - 3 * q^13 + 3 * q^15 - 8 * q^17 + 2 * q^19 - 6 * q^21 + 2 * q^23 + 12 * q^25 - 7 * q^27 - 29 * q^31 - 46 * q^33 + 17 * q^35 - 38 * q^37 + 10 * q^39 - 11 * q^41 - 9 * q^43 - 54 * q^45 - 34 * q^47 + 33 * q^49 + 17 * q^51 - 15 * q^53 - 2 * q^55 - q^57 + 57 * q^59 - 37 * q^61 + 9 * q^63 - 59 * q^65 + 33 * q^67 + 21 * q^69 - 21 * q^71 - 13 * q^73 - 13 * q^75 + 3 * q^77 - 32 * q^79 + 36 * q^81 + 48 * q^83 - 17 * q^85 - 20 * q^89 - 2 * q^91 - 37 * q^93 - 7 * q^97 + 16 * q^99

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