LMFDB - Embedded newform 1672.2.a.i.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1672,2,Mod(1,1672)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1672, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1672.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1672 = 2^{3} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1672.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:	$13.3509872180$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.106392688.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 9x^{4} + 12x^{3} + 25x^{2} - 10x - 16 $ x^6 - 2x^5 - 9x^4 + 12x^3 + 25x^2 - 10*x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.80207$ of defining polynomial
Character	$\chi$	$=$	1672.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.80207 q^{3} +0.702208 q^{5} -1.70221 q^{7} +0.247456 q^{9} +O(q^{10})$	$q+1.80207 q^{3} +0.702208 q^{5} -1.70221 q^{7} +0.247456 q^{9} -1.00000 q^{11} -6.29079 q^{13} +1.26543 q^{15} -6.35325 q^{17} +1.00000 q^{19} -3.06750 q^{21} -6.86158 q^{23} -4.50690 q^{25} -4.96028 q^{27} +1.53107 q^{29} +1.26543 q^{31} -1.80207 q^{33} -1.19530 q^{35} +11.7097 q^{37} -11.3364 q^{39} +4.31414 q^{41} -8.90848 q^{43} +0.173766 q^{45} +7.70319 q^{47} -4.10249 q^{49} -11.4490 q^{51} +3.91192 q^{53} -0.702208 q^{55} +1.80207 q^{57} +0.424525 q^{59} +6.09905 q^{61} -0.421222 q^{63} -4.41744 q^{65} +7.02309 q^{67} -12.3650 q^{69} +0.952290 q^{71} -5.42777 q^{73} -8.12176 q^{75} +1.70221 q^{77} +8.61270 q^{79} -9.68113 q^{81} -4.35338 q^{83} -4.46130 q^{85} +2.75909 q^{87} +10.3417 q^{89} +10.7082 q^{91} +2.28039 q^{93} +0.702208 q^{95} -3.39099 q^{97} -0.247456 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{3} - 3 q^{5} - 3 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q - 4 * q^3 - 3 * q^5 - 3 * q^7 + 6 * q^9	$ 6 q - 4 q^{3} - 3 q^{5} - 3 q^{7} + 6 q^{9} - 6 q^{11} - 3 q^{13} - q^{15} - q^{17} + 6 q^{19} + 5 q^{21} - 4 q^{23} + 5 q^{25} - 16 q^{27} - 7 q^{29} - q^{31} + 4 q^{33} - 32 q^{35} + 5 q^{37} - 13 q^{39} - 6 q^{41} - 16 q^{43} - 4 q^{47} - 7 q^{49} - 35 q^{51} - 11 q^{53} + 3 q^{55} - 4 q^{57} - 18 q^{59} + 16 q^{61} - 6 q^{63} + 10 q^{65} - 18 q^{67} - 2 q^{69} - 7 q^{71} - 21 q^{73} - 23 q^{75} + 3 q^{77} - 22 q^{79} - 10 q^{81} - 16 q^{83} - 3 q^{87} - 13 q^{89} - 7 q^{91} - 9 q^{93} - 3 q^{95} - 15 q^{97} - 6 q^{99}+O(q^{100}) $ 6 * q - 4 * q^3 - 3 * q^5 - 3 * q^7 + 6 * q^9 - 6 * q^11 - 3 * q^13 - q^15 - q^17 + 6 * q^19 + 5 * q^21 - 4 * q^23 + 5 * q^25 - 16 * q^27 - 7 * q^29 - q^31 + 4 * q^33 - 32 * q^35 + 5 * q^37 - 13 * q^39 - 6 * q^41 - 16 * q^43 - 4 * q^47 - 7 * q^49 - 35 * q^51 - 11 * q^53 + 3 * q^55 - 4 * q^57 - 18 * q^59 + 16 * q^61 - 6 * q^63 + 10 * q^65 - 18 * q^67 - 2 * q^69 - 7 * q^71 - 21 * q^73 - 23 * q^75 + 3 * q^77 - 22 * q^79 - 10 * q^81 - 16 * q^83 - 3 * q^87 - 13 * q^89 - 7 * q^91 - 9 * q^93 - 3 * q^95 - 15 * q^97 - 6 * 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
$3$	1.80207	1.04043	0.520213	−	0.854037i	$-0.325853\pi$
$3$	1.80207	1.04043	0.520213	+	0.854037i	$0.325853\pi$
$4$	0	0
$5$	0.702208	0.314037	0.157018	−	0.987596i	$-0.449812\pi$
$5$	0.702208	0.314037	0.157018	+	0.987596i	$0.449812\pi$
$6$	0	0
$7$	−1.70221	−0.643374	−0.321687	−	0.946846i	$-0.604250\pi$
$7$	−1.70221	−0.643374	−0.321687	+	0.946846i	$0.604250\pi$
$8$	0	0
$9$	0.247456	0.0824853
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−6.29079	−1.74475	−0.872375	−	0.488836i	$-0.837422\pi$
$13$	−6.29079	−1.74475	−0.872375	+	0.488836i	$0.837422\pi$
$14$	0	0
$15$	1.26543	0.326732
$16$	0	0
$17$	−6.35325	−1.54089	−0.770444	−	0.637507i	$-0.779966\pi$
$17$	−6.35325	−1.54089	−0.770444	+	0.637507i	$0.779966\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−3.06750	−0.669383
$22$	0	0
$23$	−6.86158	−1.43074	−0.715369	−	0.698747i	$-0.753741\pi$
$23$	−6.86158	−1.43074	−0.715369	+	0.698747i	$0.753741\pi$
$24$	0	0
$25$	−4.50690	−0.901381
$26$	0	0
$27$	−4.96028	−0.954606
$28$	0	0
$29$	1.53107	0.284312	0.142156	−	0.989844i	$-0.454596\pi$
$29$	1.53107	0.284312	0.142156	+	0.989844i	$0.454596\pi$
$30$	0	0
$31$	1.26543	0.227278	0.113639	−	0.993522i	$-0.463749\pi$
$31$	1.26543	0.227278	0.113639	+	0.993522i	$0.463749\pi$
$32$	0	0
$33$	−1.80207	−0.313700
$34$	0	0
$35$	−1.19530	−0.202043
$36$	0	0
$37$	11.7097	1.92507	0.962535	−	0.271158i	$-0.0874067\pi$
$37$	11.7097	1.92507	0.962535	+	0.271158i	$0.0874067\pi$
$38$	0	0
$39$	−11.3364	−1.81528
$40$	0	0
$41$	4.31414	0.673756	0.336878	−	0.941548i	$-0.390629\pi$
$41$	4.31414	0.673756	0.336878	+	0.941548i	$0.390629\pi$
$42$	0	0
$43$	−8.90848	−1.35853	−0.679265	−	0.733893i	$-0.737701\pi$
$43$	−8.90848	−1.35853	−0.679265	+	0.733893i	$0.737701\pi$
$44$	0	0
$45$	0.173766	0.0259034
$46$	0	0
$47$	7.70319	1.12363	0.561813	−	0.827264i	$-0.310104\pi$
$47$	7.70319	1.12363	0.561813	+	0.827264i	$0.310104\pi$
$48$	0	0
$49$	−4.10249	−0.586070
$50$	0	0
$51$	−11.4490	−1.60318
$52$	0	0
$53$	3.91192	0.537343	0.268672	−	0.963232i	$-0.413415\pi$
$53$	3.91192	0.537343	0.268672	+	0.963232i	$0.413415\pi$
$54$	0	0
$55$	−0.702208	−0.0946857
$56$	0	0
$57$	1.80207	0.238690
$58$	0	0
$59$	0.424525	0.0552684	0.0276342	−	0.999618i	$-0.491203\pi$
$59$	0.424525	0.0552684	0.0276342	+	0.999618i	$0.491203\pi$
$60$	0	0
$61$	6.09905	0.780904	0.390452	−	0.920623i	$-0.372319\pi$
$61$	6.09905	0.780904	0.390452	+	0.920623i	$0.372319\pi$
$62$	0	0
$63$	−0.421222	−0.0530689
$64$	0	0
$65$	−4.41744	−0.547916
$66$	0	0
$67$	7.02309	0.858007	0.429004	−	0.903303i	$-0.358865\pi$
$67$	7.02309	0.858007	0.429004	+	0.903303i	$0.358865\pi$
$68$	0	0
$69$	−12.3650	−1.48858
$70$	0	0
$71$	0.952290	0.113016	0.0565080	−	0.998402i	$-0.482003\pi$
$71$	0.952290	0.113016	0.0565080	+	0.998402i	$0.482003\pi$
$72$	0	0
$73$	−5.42777	−0.635272	−0.317636	−	0.948213i	$-0.602889\pi$
$73$	−5.42777	−0.635272	−0.317636	+	0.948213i	$0.602889\pi$
$74$	0	0
$75$	−8.12176	−0.937820
$76$	0	0
$77$	1.70221	0.193985
$78$	0	0
$79$	8.61270	0.969004	0.484502	−	0.874790i	$-0.339001\pi$
$79$	8.61270	0.969004	0.484502	+	0.874790i	$0.339001\pi$
$80$	0	0
$81$	−9.68113	−1.07568
$82$	0	0
$83$	−4.35338	−0.477846	−0.238923	−	0.971039i	$-0.576794\pi$
$83$	−4.35338	−0.477846	−0.238923	+	0.971039i	$0.576794\pi$
$84$	0	0
$85$	−4.46130	−0.483896
$86$	0	0
$87$	2.75909	0.295806
$88$	0	0
$89$	10.3417	1.09622	0.548109	−	0.836407i	$-0.315348\pi$
$89$	10.3417	1.09622	0.548109	+	0.836407i	$0.315348\pi$
$90$	0	0
$91$	10.7082	1.12253
$92$	0	0
$93$	2.28039	0.236465
$94$	0	0
$95$	0.702208	0.0720450
$96$	0	0
$97$	−3.39099	−0.344303	−0.172152	−	0.985070i	$-0.555072\pi$
$97$	−3.39099	−0.344303	−0.172152	+	0.985070i	$0.555072\pi$
$98$	0	0
$99$	−0.247456	−0.0248703
$100$	0	0
$101$	−4.03112	−0.401111	−0.200556	−	0.979682i	$-0.564275\pi$
$101$	−4.03112	−0.401111	−0.200556	+	0.979682i	$0.564275\pi$
$102$	0	0
$103$	−5.23801	−0.516116	−0.258058	−	0.966129i	$-0.583083\pi$
$103$	−5.23801	−0.516116	−0.258058	+	0.966129i	$0.583083\pi$
$104$	0	0
$105$	−2.15402	−0.210211
$106$	0	0
$107$	−14.2526	−1.37785	−0.688924	−	0.724833i	$-0.741917\pi$
$107$	−14.2526	−1.37785	−0.688924	+	0.724833i	$0.741917\pi$
$108$	0	0
$109$	17.6462	1.69020	0.845101	−	0.534607i	$-0.179540\pi$
$109$	17.6462	1.69020	0.845101	+	0.534607i	$0.179540\pi$
$110$	0	0
$111$	21.1018	2.00289
$112$	0	0
$113$	8.58975	0.808055	0.404028	−	0.914747i	$-0.367610\pi$
$113$	8.58975	0.808055	0.404028	+	0.914747i	$0.367610\pi$
$114$	0	0
$115$	−4.81826	−0.449305
$116$	0	0
$117$	−1.55669	−0.143916
$118$	0	0
$119$	10.8145	0.991368
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	7.77439	0.700993
$124$	0	0
$125$	−6.67582	−0.597104
$126$	0	0
$127$	−16.3430	−1.45021	−0.725103	−	0.688640i	$-0.758208\pi$
$127$	−16.3430	−1.45021	−0.725103	+	0.688640i	$0.758208\pi$
$128$	0	0
$129$	−16.0537	−1.41345
$130$	0	0
$131$	5.06687	0.442694	0.221347	−	0.975195i	$-0.428955\pi$
$131$	5.06687	0.442694	0.221347	+	0.975195i	$0.428955\pi$
$132$	0	0
$133$	−1.70221	−0.147600
$134$	0	0
$135$	−3.48315	−0.299781
$136$	0	0
$137$	−9.56620	−0.817296	−0.408648	−	0.912692i	$-0.634000\pi$
$137$	−9.56620	−0.817296	−0.408648	+	0.912692i	$0.634000\pi$
$138$	0	0
$139$	−8.60970	−0.730266	−0.365133	−	0.930955i	$-0.618976\pi$
$139$	−8.60970	−0.730266	−0.365133	+	0.930955i	$0.618976\pi$
$140$	0	0
$141$	13.8817	1.16905
$142$	0	0
$143$	6.29079	0.526062
$144$	0	0
$145$	1.07513	0.0892846
$146$	0	0
$147$	−7.39297	−0.609762
$148$	0	0
$149$	15.5796	1.27633	0.638166	−	0.769898i	$-0.279693\pi$
$149$	15.5796	1.27633	0.638166	+	0.769898i	$0.279693\pi$
$150$	0	0
$151$	−18.0919	−1.47230	−0.736150	−	0.676819i	$-0.763358\pi$
$151$	−18.0919	−1.47230	−0.736150	+	0.676819i	$0.763358\pi$
$152$	0	0
$153$	−1.57215	−0.127101
$154$	0	0
$155$	0.888593	0.0713735
$156$	0	0
$157$	−20.4692	−1.63362	−0.816811	−	0.576906i	$-0.804260\pi$
$157$	−20.4692	−1.63362	−0.816811	+	0.576906i	$0.804260\pi$
$158$	0	0
$159$	7.04955	0.559066
$160$	0	0
$161$	11.6798	0.920500
$162$	0	0
$163$	0.524409	0.0410749	0.0205374	−	0.999789i	$-0.493462\pi$
$163$	0.524409	0.0410749	0.0205374	+	0.999789i	$0.493462\pi$
$164$	0	0
$165$	−1.26543	−0.0985134
$166$	0	0
$167$	−25.2266	−1.95210	−0.976048	−	0.217557i	$-0.930191\pi$
$167$	−25.2266	−1.95210	−0.976048	+	0.217557i	$0.930191\pi$
$168$	0	0
$169$	26.5740	2.04416
$170$	0	0
$171$	0.247456	0.0189234
$172$	0	0
$173$	10.8618	0.825804	0.412902	−	0.910775i	$-0.364515\pi$
$173$	10.8618	0.825804	0.412902	+	0.910775i	$0.364515\pi$
$174$	0	0
$175$	7.67169	0.579925
$176$	0	0
$177$	0.765023	0.0575027
$178$	0	0
$179$	11.0533	0.826164	0.413082	−	0.910694i	$-0.364452\pi$
$179$	11.0533	0.826164	0.413082	+	0.910694i	$0.364452\pi$
$180$	0	0
$181$	7.36675	0.547566	0.273783	−	0.961791i	$-0.411725\pi$
$181$	7.36675	0.547566	0.273783	+	0.961791i	$0.411725\pi$
$182$	0	0
$183$	10.9909	0.812472
$184$	0	0
$185$	8.22267	0.604543
$186$	0	0
$187$	6.35325	0.464595
$188$	0	0
$189$	8.44342	0.614169
$190$	0	0
$191$	−22.8999	−1.65698	−0.828490	−	0.560004i	$-0.810800\pi$
$191$	−22.8999	−1.65698	−0.828490	+	0.560004i	$0.810800\pi$
$192$	0	0
$193$	−15.6653	−1.12762	−0.563808	−	0.825906i	$-0.690664\pi$
$193$	−15.6653	−1.12762	−0.563808	+	0.825906i	$0.690664\pi$
$194$	0	0
$195$	−7.96054	−0.570066
$196$	0	0
$197$	−7.76105	−0.552952	−0.276476	−	0.961021i	$-0.589167\pi$
$197$	−7.76105	−0.552952	−0.276476	+	0.961021i	$0.589167\pi$
$198$	0	0
$199$	3.28201	0.232656	0.116328	−	0.993211i	$-0.462888\pi$
$199$	3.28201	0.232656	0.116328	+	0.993211i	$0.462888\pi$
$200$	0	0
$201$	12.6561	0.892693
$202$	0	0
$203$	−2.60620	−0.182919
$204$	0	0
$205$	3.02943	0.211584
$206$	0	0
$207$	−1.69794	−0.118015
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−24.9296	−1.71622	−0.858111	−	0.513464i	$-0.828362\pi$
$211$	−24.9296	−1.71622	−0.858111	+	0.513464i	$0.828362\pi$
$212$	0	0
$213$	1.71609	0.117585
$214$	0	0
$215$	−6.25561	−0.426629
$216$	0	0
$217$	−2.15402	−0.146224
$218$	0	0
$219$	−9.78122	−0.660954
$220$	0	0
$221$	39.9669	2.68847
$222$	0	0
$223$	17.3802	1.16387	0.581933	−	0.813237i	$-0.302297\pi$
$223$	17.3802	1.16387	0.581933	+	0.813237i	$0.302297\pi$
$224$	0	0
$225$	−1.11526	−0.0743507
$226$	0	0
$227$	0.00934827	0.000620466 0	0.000310233	−	1.00000i	$-0.499901\pi$
$227$	0.00934827	0.000620466 0	0.000310233		1.00000i	$0.499901\pi$
$228$	0	0
$229$	5.53526	0.365780	0.182890	−	0.983133i	$-0.441455\pi$
$229$	5.53526	0.365780	0.182890	+	0.983133i	$0.441455\pi$
$230$	0	0
$231$	3.06750	0.201827
$232$	0	0
$233$	−11.5316	−0.755458	−0.377729	−	0.925916i	$-0.623295\pi$
$233$	−11.5316	−0.755458	−0.377729	+	0.925916i	$0.623295\pi$
$234$	0	0
$235$	5.40924	0.352860
$236$	0	0
$237$	15.5207	1.00818
$238$	0	0
$239$	−14.6298	−0.946321	−0.473161	−	0.880976i	$-0.656887\pi$
$239$	−14.6298	−0.946321	−0.473161	+	0.880976i	$0.656887\pi$
$240$	0	0
$241$	19.8974	1.28171	0.640853	−	0.767664i	$-0.278581\pi$
$241$	19.8974	1.28171	0.640853	+	0.767664i	$0.278581\pi$
$242$	0	0
$243$	−2.56525	−0.164561
$244$	0	0
$245$	−2.88080	−0.184048
$246$	0	0
$247$	−6.29079	−0.400273
$248$	0	0
$249$	−7.84510	−0.497163
$250$	0	0
$251$	−27.7680	−1.75270	−0.876352	−	0.481672i	$-0.840030\pi$
$251$	−27.7680	−1.75270	−0.876352	+	0.481672i	$0.840030\pi$
$252$	0	0
$253$	6.86158	0.431384
$254$	0	0
$255$	−8.03958	−0.503458
$256$	0	0
$257$	−30.5983	−1.90867	−0.954334	−	0.298743i	$-0.903433\pi$
$257$	−30.5983	−1.90867	−0.954334	+	0.298743i	$0.903433\pi$
$258$	0	0
$259$	−19.9324	−1.23854
$260$	0	0
$261$	0.378872	0.0234516
$262$	0	0
$263$	−13.5937	−0.838227	−0.419113	−	0.907934i	$-0.637659\pi$
$263$	−13.5937	−0.838227	−0.419113	+	0.907934i	$0.637659\pi$
$264$	0	0
$265$	2.74698	0.168746
$266$	0	0
$267$	18.6365	1.14053
$268$	0	0
$269$	−5.37492	−0.327715	−0.163857	−	0.986484i	$-0.552394\pi$
$269$	−5.37492	−0.327715	−0.163857	+	0.986484i	$0.552394\pi$
$270$	0	0
$271$	9.64658	0.585988	0.292994	−	0.956114i	$-0.405348\pi$
$271$	9.64658	0.585988	0.292994	+	0.956114i	$0.405348\pi$
$272$	0	0
$273$	19.2970	1.16791
$274$	0	0
$275$	4.50690	0.271777
$276$	0	0
$277$	0.994203	0.0597359	0.0298679	−	0.999554i	$-0.490491\pi$
$277$	0.994203	0.0597359	0.0298679	+	0.999554i	$0.490491\pi$
$278$	0	0
$279$	0.313138	0.0187471
$280$	0	0
$281$	−31.0311	−1.85116	−0.925579	−	0.378554i	$-0.876422\pi$
$281$	−31.0311	−1.85116	−0.925579	+	0.378554i	$0.876422\pi$
$282$	0	0
$283$	−10.1143	−0.601232	−0.300616	−	0.953745i	$-0.597192\pi$
$283$	−10.1143	−0.601232	−0.300616	+	0.953745i	$0.597192\pi$
$284$	0	0
$285$	1.26543	0.0749575
$286$	0	0
$287$	−7.34357	−0.433477
$288$	0	0
$289$	23.3638	1.37434
$290$	0	0
$291$	−6.11081	−0.358222
$292$	0	0
$293$	8.17773	0.477748	0.238874	−	0.971051i	$-0.423222\pi$
$293$	8.17773	0.477748	0.238874	+	0.971051i	$0.423222\pi$
$294$	0	0
$295$	0.298105	0.0173563
$296$	0	0
$297$	4.96028	0.287824
$298$	0	0
$299$	43.1648	2.49628
$300$	0	0
$301$	15.1641	0.874043
$302$	0	0
$303$	−7.26436	−0.417326
$304$	0	0
$305$	4.28280	0.245233
$306$	0	0
$307$	4.25758	0.242993	0.121496	−	0.992592i	$-0.461231\pi$
$307$	4.25758	0.242993	0.121496	+	0.992592i	$0.461231\pi$
$308$	0	0
$309$	−9.43926	−0.536981
$310$	0	0
$311$	−1.26758	−0.0718777	−0.0359389	−	0.999354i	$-0.511442\pi$
$311$	−1.26758	−0.0718777	−0.0359389	+	0.999354i	$0.511442\pi$
$312$	0	0
$313$	6.73769	0.380837	0.190418	−	0.981703i	$-0.439016\pi$
$313$	6.73769	0.380837	0.190418	+	0.981703i	$0.439016\pi$
$314$	0	0
$315$	−0.295785	−0.0166656
$316$	0	0
$317$	−28.5503	−1.60355	−0.801773	−	0.597628i	$-0.796110\pi$
$317$	−28.5503	−1.60355	−0.801773	+	0.597628i	$0.796110\pi$
$318$	0	0
$319$	−1.53107	−0.0857234
$320$	0	0
$321$	−25.6841	−1.43355
$322$	0	0
$323$	−6.35325	−0.353504
$324$	0	0
$325$	28.3520	1.57269
$326$	0	0
$327$	31.7997	1.75853
$328$	0	0
$329$	−13.1124	−0.722912
$330$	0	0
$331$	−28.7861	−1.58223	−0.791114	−	0.611669i	$-0.790499\pi$
$331$	−28.7861	−1.58223	−0.791114	+	0.611669i	$0.790499\pi$
$332$	0	0
$333$	2.89765	0.158790
$334$	0	0
$335$	4.93167	0.269446
$336$	0	0
$337$	26.3296	1.43426	0.717132	−	0.696937i	$-0.245454\pi$
$337$	26.3296	1.43426	0.717132	+	0.696937i	$0.245454\pi$
$338$	0	0
$339$	15.4793	0.840722
$340$	0	0
$341$	−1.26543	−0.0685268
$342$	0	0
$343$	18.8987	1.02044
$344$	0	0
$345$	−8.68283	−0.467468
$346$	0	0
$347$	27.4283	1.47243	0.736215	−	0.676748i	$-0.236611\pi$
$347$	27.4283	1.47243	0.736215	+	0.676748i	$0.236611\pi$
$348$	0	0
$349$	−24.3945	−1.30581	−0.652905	−	0.757440i	$-0.726450\pi$
$349$	−24.3945	−1.30581	−0.652905	+	0.757440i	$0.726450\pi$
$350$	0	0
$351$	31.2041	1.66555
$352$	0	0
$353$	10.7352	0.571375	0.285687	−	0.958323i	$-0.407778\pi$
$353$	10.7352	0.571375	0.285687	+	0.958323i	$0.407778\pi$
$354$	0	0
$355$	0.668706	0.0354912
$356$	0	0
$357$	19.4886	1.03144
$358$	0	0
$359$	6.22396	0.328488	0.164244	−	0.986420i	$-0.447482\pi$
$359$	6.22396	0.328488	0.164244	+	0.986420i	$0.447482\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	1.80207	0.0945841
$364$	0	0
$365$	−3.81142	−0.199499
$366$	0	0
$367$	2.13049	0.111211	0.0556054	−	0.998453i	$-0.482291\pi$
$367$	2.13049	0.111211	0.0556054	+	0.998453i	$0.482291\pi$
$368$	0	0
$369$	1.06756	0.0555750
$370$	0	0
$371$	−6.65890	−0.345713
$372$	0	0
$373$	26.2252	1.35789	0.678944	−	0.734190i	$-0.262438\pi$
$373$	26.2252	1.35789	0.678944	+	0.734190i	$0.262438\pi$
$374$	0	0
$375$	−12.0303	−0.621242
$376$	0	0
$377$	−9.63163	−0.496054
$378$	0	0
$379$	−5.02721	−0.258230	−0.129115	−	0.991630i	$-0.541214\pi$
$379$	−5.02721	−0.258230	−0.129115	+	0.991630i	$0.541214\pi$
$380$	0	0
$381$	−29.4512	−1.50883
$382$	0	0
$383$	15.2243	0.777923	0.388962	−	0.921254i	$-0.372834\pi$
$383$	15.2243	0.777923	0.388962	+	0.921254i	$0.372834\pi$
$384$	0	0
$385$	1.19530	0.0609183
$386$	0	0
$387$	−2.20446	−0.112059
$388$	0	0
$389$	−27.6173	−1.40025	−0.700127	−	0.714019i	$-0.746873\pi$
$389$	−27.6173	−1.40025	−0.700127	+	0.714019i	$0.746873\pi$
$390$	0	0
$391$	43.5933	2.20461
$392$	0	0
$393$	9.13085	0.460591
$394$	0	0
$395$	6.04790	0.304303
$396$	0	0
$397$	27.3216	1.37123	0.685616	−	0.727964i	$-0.259533\pi$
$397$	27.3216	1.37123	0.685616	+	0.727964i	$0.259533\pi$
$398$	0	0
$399$	−3.06750	−0.153567
$400$	0	0
$401$	−0.892819	−0.0445852	−0.0222926	−	0.999751i	$-0.507097\pi$
$401$	−0.892819	−0.0445852	−0.0222926	+	0.999751i	$0.507097\pi$
$402$	0	0
$403$	−7.96054	−0.396543
$404$	0	0
$405$	−6.79817	−0.337804
$406$	0	0
$407$	−11.7097	−0.580430
$408$	0	0
$409$	21.6146	1.06877	0.534387	−	0.845240i	$-0.320543\pi$
$409$	21.6146	1.06877	0.534387	+	0.845240i	$0.320543\pi$
$410$	0	0
$411$	−17.2390	−0.850335
$412$	0	0
$413$	−0.722629	−0.0355583
$414$	0	0
$415$	−3.05698	−0.150061
$416$	0	0
$417$	−15.5153	−0.759787
$418$	0	0
$419$	−18.3531	−0.896606	−0.448303	−	0.893882i	$-0.647971\pi$
$419$	−18.3531	−0.896606	−0.448303	+	0.893882i	$0.647971\pi$
$420$	0	0
$421$	25.3409	1.23504	0.617521	−	0.786555i	$-0.288137\pi$
$421$	25.3409	1.23504	0.617521	+	0.786555i	$0.288137\pi$
$422$	0	0
$423$	1.90620	0.0926826
$424$	0	0
$425$	28.6335	1.38893
$426$	0	0
$427$	−10.3819	−0.502413
$428$	0	0
$429$	11.3364	0.547329
$430$	0	0
$431$	−4.76336	−0.229443	−0.114722	−	0.993398i	$-0.536598\pi$
$431$	−4.76336	−0.229443	−0.114722	+	0.993398i	$0.536598\pi$
$432$	0	0
$433$	32.9887	1.58534	0.792668	−	0.609653i	$-0.208691\pi$
$433$	32.9887	1.58534	0.792668	+	0.609653i	$0.208691\pi$
$434$	0	0
$435$	1.93746	0.0928939
$436$	0	0
$437$	−6.86158	−0.328234
$438$	0	0
$439$	4.00329	0.191066	0.0955332	−	0.995426i	$-0.469544\pi$
$439$	4.00329	0.191066	0.0955332	+	0.995426i	$0.469544\pi$
$440$	0	0
$441$	−1.01519	−0.0483422
$442$	0	0
$443$	9.44334	0.448666	0.224333	−	0.974512i	$-0.427980\pi$
$443$	9.44334	0.448666	0.224333	+	0.974512i	$0.427980\pi$
$444$	0	0
$445$	7.26202	0.344253
$446$	0	0
$447$	28.0756	1.32793
$448$	0	0
$449$	15.3635	0.725047	0.362523	−	0.931975i	$-0.381915\pi$
$449$	15.3635	0.725047	0.362523	+	0.931975i	$0.381915\pi$
$450$	0	0
$451$	−4.31414	−0.203145
$452$	0	0
$453$	−32.6029	−1.53182
$454$	0	0
$455$	7.51940	0.352515
$456$	0	0
$457$	2.32292	0.108662	0.0543308	−	0.998523i	$-0.482697\pi$
$457$	2.32292	0.108662	0.0543308	+	0.998523i	$0.482697\pi$
$458$	0	0
$459$	31.5139	1.47094
$460$	0	0
$461$	14.8685	0.692495	0.346247	−	0.938143i	$-0.387456\pi$
$461$	14.8685	0.692495	0.346247	+	0.938143i	$0.387456\pi$
$462$	0	0
$463$	8.89544	0.413406	0.206703	−	0.978404i	$-0.433727\pi$
$463$	8.89544	0.413406	0.206703	+	0.978404i	$0.433727\pi$
$464$	0	0
$465$	1.60131	0.0742589
$466$	0	0
$467$	−20.3976	−0.943889	−0.471944	−	0.881628i	$-0.656448\pi$
$467$	−20.3976	−0.943889	−0.471944	+	0.881628i	$0.656448\pi$
$468$	0	0
$469$	−11.9548	−0.552020
$470$	0	0
$471$	−36.8870	−1.69966
$472$	0	0
$473$	8.90848	0.409612
$474$	0	0
$475$	−4.50690	−0.206791
$476$	0	0
$477$	0.968027	0.0443229
$478$	0	0
$479$	14.9259	0.681983	0.340992	−	0.940066i	$-0.389237\pi$
$479$	14.9259	0.681983	0.340992	+	0.940066i	$0.389237\pi$
$480$	0	0
$481$	−73.6635	−3.35877
$482$	0	0
$483$	21.0479	0.957712
$484$	0	0
$485$	−2.38118	−0.108124
$486$	0	0
$487$	34.3362	1.55592	0.777961	−	0.628312i	$-0.216254\pi$
$487$	34.3362	1.55592	0.777961	+	0.628312i	$0.216254\pi$
$488$	0	0
$489$	0.945021	0.0427353
$490$	0	0
$491$	−29.7975	−1.34474	−0.672371	−	0.740214i	$-0.734724\pi$
$491$	−29.7975	−1.34474	−0.672371	+	0.740214i	$0.734724\pi$
$492$	0	0
$493$	−9.72726	−0.438094
$494$	0	0
$495$	−0.173766	−0.00781018
$496$	0	0
$497$	−1.62100	−0.0727116
$498$	0	0
$499$	6.65166	0.297769	0.148885	−	0.988855i	$-0.452432\pi$
$499$	6.65166	0.297769	0.148885	+	0.988855i	$0.452432\pi$
$500$	0	0
$501$	−45.4601	−2.03101
$502$	0	0
$503$	25.7511	1.14819	0.574093	−	0.818790i	$-0.305355\pi$
$503$	25.7511	1.14819	0.574093	+	0.818790i	$0.305355\pi$
$504$	0	0
$505$	−2.83068	−0.125964
$506$	0	0
$507$	47.8883	2.12679
$508$	0	0
$509$	15.4332	0.684063	0.342031	−	0.939689i	$-0.388885\pi$
$509$	15.4332	0.684063	0.342031	+	0.939689i	$0.388885\pi$
$510$	0	0
$511$	9.23919	0.408718
$512$	0	0
$513$	−4.96028	−0.219002
$514$	0	0
$515$	−3.67817	−0.162080
$516$	0	0
$517$	−7.70319	−0.338786
$518$	0	0
$519$	19.5736	0.859187
$520$	0	0
$521$	−21.7970	−0.954946	−0.477473	−	0.878646i	$-0.658447\pi$
$521$	−21.7970	−0.954946	−0.477473	+	0.878646i	$0.658447\pi$
$522$	0	0
$523$	−31.9749	−1.39817	−0.699083	−	0.715040i	$-0.746408\pi$
$523$	−31.9749	−1.39817	−0.699083	+	0.715040i	$0.746408\pi$
$524$	0	0
$525$	13.8249	0.603369
$526$	0	0
$527$	−8.03958	−0.350209
$528$	0	0
$529$	24.0813	1.04701
$530$	0	0
$531$	0.105051	0.00455883
$532$	0	0
$533$	−27.1394	−1.17554
$534$	0	0
$535$	−10.0083	−0.432695
$536$	0	0
$537$	19.9189	0.859562
$538$	0	0
$539$	4.10249	0.176707
$540$	0	0
$541$	−2.91147	−0.125174	−0.0625870	−	0.998040i	$-0.519935\pi$
$541$	−2.91147	−0.125174	−0.0625870	+	0.998040i	$0.519935\pi$
$542$	0	0
$543$	13.2754	0.569702
$544$	0	0
$545$	12.3913	0.530786
$546$	0	0
$547$	11.1190	0.475416	0.237708	−	0.971337i	$-0.423604\pi$
$547$	11.1190	0.475416	0.237708	+	0.971337i	$0.423604\pi$
$548$	0	0
$549$	1.50925	0.0644131
$550$	0	0
$551$	1.53107	0.0652257
$552$	0	0
$553$	−14.6606	−0.623432
$554$	0	0
$555$	14.8178	0.628982
$556$	0	0
$557$	−5.06544	−0.214630	−0.107315	−	0.994225i	$-0.534225\pi$
$557$	−5.06544	−0.214630	−0.107315	+	0.994225i	$0.534225\pi$
$558$	0	0
$559$	56.0414	2.37030
$560$	0	0
$561$	11.4490	0.483377
$562$	0	0
$563$	25.3188	1.06706	0.533531	−	0.845781i	$-0.320865\pi$
$563$	25.3188	1.06706	0.533531	+	0.845781i	$0.320865\pi$
$564$	0	0
$565$	6.03179	0.253759
$566$	0	0
$567$	16.4793	0.692066
$568$	0	0
$569$	−24.5225	−1.02804	−0.514019	−	0.857779i	$-0.671844\pi$
$569$	−24.5225	−1.02804	−0.514019	+	0.857779i	$0.671844\pi$
$570$	0	0
$571$	−45.0237	−1.88418	−0.942092	−	0.335355i	$-0.891144\pi$
$571$	−45.0237	−1.88418	−0.942092	+	0.335355i	$0.891144\pi$
$572$	0	0
$573$	−41.2673	−1.72396
$574$	0	0
$575$	30.9245	1.28964
$576$	0	0
$577$	−47.0179	−1.95738	−0.978691	−	0.205338i	$-0.934171\pi$
$577$	−47.0179	−1.95738	−0.978691	+	0.205338i	$0.934171\pi$
$578$	0	0
$579$	−28.2300	−1.17320
$580$	0	0
$581$	7.41036	0.307433
$582$	0	0
$583$	−3.91192	−0.162015
$584$	0	0
$585$	−1.09312	−0.0451950
$586$	0	0
$587$	6.71061	0.276976	0.138488	−	0.990364i	$-0.455776\pi$
$587$	6.71061	0.276976	0.138488	+	0.990364i	$0.455776\pi$
$588$	0	0
$589$	1.26543	0.0521410
$590$	0	0
$591$	−13.9859	−0.575305
$592$	0	0
$593$	24.1321	0.990988	0.495494	−	0.868611i	$-0.334987\pi$
$593$	24.1321	0.990988	0.495494	+	0.868611i	$0.334987\pi$
$594$	0	0
$595$	7.59406	0.311326
$596$	0	0
$597$	5.91441	0.242061
$598$	0	0
$599$	−29.5383	−1.20690	−0.603450	−	0.797400i	$-0.706208\pi$
$599$	−29.5383	−1.20690	−0.603450	+	0.797400i	$0.706208\pi$
$600$	0	0
$601$	18.6296	0.759917	0.379959	−	0.925003i	$-0.375938\pi$
$601$	18.6296	0.759917	0.379959	+	0.925003i	$0.375938\pi$
$602$	0	0
$603$	1.73791	0.0707730
$604$	0	0
$605$	0.702208	0.0285488
$606$	0	0
$607$	12.1050	0.491327	0.245664	−	0.969355i	$-0.420994\pi$
$607$	12.1050	0.491327	0.245664	+	0.969355i	$0.420994\pi$
$608$	0	0
$609$	−4.69655	−0.190314
$610$	0	0
$611$	−48.4592	−1.96045
$612$	0	0
$613$	−4.76374	−0.192406	−0.0962029	−	0.995362i	$-0.530670\pi$
$613$	−4.76374	−0.192406	−0.0962029	+	0.995362i	$0.530670\pi$
$614$	0	0
$615$	5.45924	0.220138
$616$	0	0
$617$	2.44418	0.0983991	0.0491995	−	0.998789i	$-0.484333\pi$
$617$	2.44418	0.0983991	0.0491995	+	0.998789i	$0.484333\pi$
$618$	0	0
$619$	11.2057	0.450396	0.225198	−	0.974313i	$-0.427697\pi$
$619$	11.2057	0.450396	0.225198	+	0.974313i	$0.427697\pi$
$620$	0	0
$621$	34.0353	1.36579
$622$	0	0
$623$	−17.6037	−0.705278
$624$	0	0
$625$	17.8467	0.713868
$626$	0	0
$627$	−1.80207	−0.0719677
$628$	0	0
$629$	−74.3949	−2.96632
$630$	0	0
$631$	28.2410	1.12426	0.562129	−	0.827050i	$-0.309983\pi$
$631$	28.2410	1.12426	0.562129	+	0.827050i	$0.309983\pi$
$632$	0	0
$633$	−44.9248	−1.78560
$634$	0	0
$635$	−11.4762	−0.455418
$636$	0	0
$637$	25.8079	1.02255
$638$	0	0
$639$	0.235650	0.00932216
$640$	0	0
$641$	33.6122	1.32760	0.663801	−	0.747909i	$-0.268942\pi$
$641$	33.6122	1.32760	0.663801	+	0.747909i	$0.268942\pi$
$642$	0	0
$643$	11.5888	0.457017	0.228508	−	0.973542i	$-0.426615\pi$
$643$	11.5888	0.457017	0.228508	+	0.973542i	$0.426615\pi$
$644$	0	0
$645$	−11.2730	−0.443875
$646$	0	0
$647$	−41.7435	−1.64110	−0.820552	−	0.571571i	$-0.806334\pi$
$647$	−41.7435	−1.64110	−0.820552	+	0.571571i	$0.806334\pi$
$648$	0	0
$649$	−0.424525	−0.0166641
$650$	0	0
$651$	−3.88170	−0.152136
$652$	0	0
$653$	−22.5719	−0.883307	−0.441653	−	0.897186i	$-0.645608\pi$
$653$	−22.5719	−0.883307	−0.441653	+	0.897186i	$0.645608\pi$
$654$	0	0
$655$	3.55800	0.139022
$656$	0	0
$657$	−1.34313	−0.0524007
$658$	0	0
$659$	−1.48433	−0.0578211	−0.0289106	−	0.999582i	$-0.509204\pi$
$659$	−1.48433	−0.0578211	−0.0289106	+	0.999582i	$0.509204\pi$
$660$	0	0
$661$	17.8119	0.692801	0.346400	−	0.938087i	$-0.387404\pi$
$661$	17.8119	0.692801	0.346400	+	0.938087i	$0.387404\pi$
$662$	0	0
$663$	72.0232	2.79715
$664$	0	0
$665$	−1.19530	−0.0463519
$666$	0	0
$667$	−10.5056	−0.406777
$668$	0	0
$669$	31.3204	1.21092
$670$	0	0
$671$	−6.09905	−0.235451
$672$	0	0
$673$	23.9373	0.922717	0.461358	−	0.887214i	$-0.347362\pi$
$673$	23.9373	0.922717	0.461358	+	0.887214i	$0.347362\pi$
$674$	0	0
$675$	22.3555	0.860463
$676$	0	0
$677$	−5.77350	−0.221894	−0.110947	−	0.993826i	$-0.535388\pi$
$677$	−5.77350	−0.221894	−0.110947	+	0.993826i	$0.535388\pi$
$678$	0	0
$679$	5.77218	0.221516
$680$	0	0
$681$	0.0168462	0.000645549 0
$682$	0	0
$683$	13.3560	0.511053	0.255527	−	0.966802i	$-0.417751\pi$
$683$	13.3560	0.511053	0.255527	+	0.966802i	$0.417751\pi$
$684$	0	0
$685$	−6.71746	−0.256661
$686$	0	0
$687$	9.97493	0.380567
$688$	0	0
$689$	−24.6090	−0.937530
$690$	0	0
$691$	−41.6190	−1.58326	−0.791631	−	0.611000i	$-0.790768\pi$
$691$	−41.6190	−1.58326	−0.791631	+	0.611000i	$0.790768\pi$
$692$	0	0
$693$	0.421222	0.0160009
$694$	0	0
$695$	−6.04580	−0.229330
$696$	0	0
$697$	−27.4088	−1.03818
$698$	0	0
$699$	−20.7807	−0.785998
$700$	0	0
$701$	−8.25801	−0.311901	−0.155950	−	0.987765i	$-0.549844\pi$
$701$	−8.25801	−0.311901	−0.155950	+	0.987765i	$0.549844\pi$
$702$	0	0
$703$	11.7097	0.441641
$704$	0	0
$705$	9.74783	0.367125
$706$	0	0
$707$	6.86180	0.258065
$708$	0	0
$709$	−23.9299	−0.898707	−0.449353	−	0.893354i	$-0.648346\pi$
$709$	−23.9299	−0.898707	−0.449353	+	0.893354i	$0.648346\pi$
$710$	0	0
$711$	2.13126	0.0799286
$712$	0	0
$713$	−8.68283	−0.325175
$714$	0	0
$715$	4.41744	0.165203
$716$	0	0
$717$	−26.3639	−0.984577
$718$	0	0
$719$	45.0442	1.67986	0.839932	−	0.542691i	$-0.182595\pi$
$719$	45.0442	1.67986	0.839932	+	0.542691i	$0.182595\pi$
$720$	0	0
$721$	8.91618	0.332056
$722$	0	0
$723$	35.8565	1.33352
$724$	0	0
$725$	−6.90038	−0.256274
$726$	0	0
$727$	−47.6080	−1.76568	−0.882841	−	0.469673i	$-0.844372\pi$
$727$	−47.6080	−1.76568	−0.882841	+	0.469673i	$0.844372\pi$
$728$	0	0
$729$	24.4206	0.904468
$730$	0	0
$731$	56.5978	2.09334
$732$	0	0
$733$	−33.1137	−1.22308	−0.611541	−	0.791213i	$-0.709450\pi$
$733$	−33.1137	−1.22308	−0.611541	+	0.791213i	$0.709450\pi$
$734$	0	0
$735$	−5.19140	−0.191488
$736$	0	0
$737$	−7.02309	−0.258699
$738$	0	0
$739$	29.5198	1.08590	0.542951	−	0.839764i	$-0.317307\pi$
$739$	29.5198	1.08590	0.542951	+	0.839764i	$0.317307\pi$
$740$	0	0
$741$	−11.3364	−0.416455
$742$	0	0
$743$	7.03244	0.257995	0.128998	−	0.991645i	$-0.458824\pi$
$743$	7.03244	0.257995	0.128998	+	0.991645i	$0.458824\pi$
$744$	0	0
$745$	10.9401	0.400816
$746$	0	0
$747$	−1.07727	−0.0394152
$748$	0	0
$749$	24.2608	0.886472
$750$	0	0
$751$	43.0934	1.57250	0.786250	−	0.617909i	$-0.212020\pi$
$751$	43.0934	1.57250	0.786250	+	0.617909i	$0.212020\pi$
$752$	0	0
$753$	−50.0400	−1.82356
$754$	0	0
$755$	−12.7043	−0.462356
$756$	0	0
$757$	46.3724	1.68544	0.842718	−	0.538356i	$-0.180954\pi$
$757$	46.3724	1.68544	0.842718	+	0.538356i	$0.180954\pi$
$758$	0	0
$759$	12.3650	0.448823
$760$	0	0
$761$	5.21015	0.188868	0.0944339	−	0.995531i	$-0.469896\pi$
$761$	5.21015	0.188868	0.0944339	+	0.995531i	$0.469896\pi$
$762$	0	0
$763$	−30.0375	−1.08743
$764$	0	0
$765$	−1.10398	−0.0399143
$766$	0	0
$767$	−2.67060	−0.0964296
$768$	0	0
$769$	23.8860	0.861351	0.430676	−	0.902507i	$-0.358275\pi$
$769$	23.8860	0.861351	0.430676	+	0.902507i	$0.358275\pi$
$770$	0	0
$771$	−55.1402	−1.98583
$772$	0	0
$773$	−49.6694	−1.78648	−0.893242	−	0.449575i	$-0.851575\pi$
$773$	−49.6694	−1.78648	−0.893242	+	0.449575i	$0.851575\pi$
$774$	0	0
$775$	−5.70316	−0.204864
$776$	0	0
$777$	−35.9196	−1.28861
$778$	0	0
$779$	4.31414	0.154570
$780$	0	0
$781$	−0.952290	−0.0340756
$782$	0	0
$783$	−7.59452	−0.271406
$784$	0	0
$785$	−14.3736	−0.513017
$786$	0	0
$787$	−35.7446	−1.27416	−0.637079	−	0.770799i	$-0.719857\pi$
$787$	−35.7446	−1.27416	−0.637079	+	0.770799i	$0.719857\pi$
$788$	0	0
$789$	−24.4969	−0.872112
$790$	0	0
$791$	−14.6215	−0.519882
$792$	0	0
$793$	−38.3678	−1.36248
$794$	0	0
$795$	4.95025	0.175567
$796$	0	0
$797$	−36.3068	−1.28605	−0.643027	−	0.765844i	$-0.722322\pi$
$797$	−36.3068	−1.28605	−0.643027	+	0.765844i	$0.722322\pi$
$798$	0	0
$799$	−48.9403	−1.73138
$800$	0	0
$801$	2.55911	0.0904218
$802$	0	0
$803$	5.42777	0.191542
$804$	0	0
$805$	8.20167	0.289071
$806$	0	0
$807$	−9.68598	−0.340963
$808$	0	0
$809$	10.9696	0.385672	0.192836	−	0.981231i	$-0.438232\pi$
$809$	10.9696	0.385672	0.192836	+	0.981231i	$0.438232\pi$
$810$	0	0
$811$	−4.26695	−0.149833	−0.0749164	−	0.997190i	$-0.523869\pi$
$811$	−4.26695	−0.149833	−0.0749164	+	0.997190i	$0.523869\pi$
$812$	0	0
$813$	17.3838	0.609677
$814$	0	0
$815$	0.368244	0.0128990
$816$	0	0
$817$	−8.90848	−0.311668
$818$	0	0
$819$	2.64982	0.0925921
$820$	0	0
$821$	−52.9986	−1.84966	−0.924832	−	0.380377i	$-0.875794\pi$
$821$	−52.9986	−1.84966	−0.924832	+	0.380377i	$0.875794\pi$
$822$	0	0
$823$	42.1222	1.46829	0.734143	−	0.678995i	$-0.237584\pi$
$823$	42.1222	1.46829	0.734143	+	0.678995i	$0.237584\pi$
$824$	0	0
$825$	8.12176	0.282763
$826$	0	0
$827$	−12.5326	−0.435802	−0.217901	−	0.975971i	$-0.569921\pi$
$827$	−12.5326	−0.435802	−0.217901	+	0.975971i	$0.569921\pi$
$828$	0	0
$829$	−51.9201	−1.80326	−0.901629	−	0.432510i	$-0.857628\pi$
$829$	−51.9201	−1.80326	−0.901629	+	0.432510i	$0.857628\pi$
$830$	0	0
$831$	1.79162	0.0621507
$832$	0	0
$833$	26.0641	0.903068
$834$	0	0
$835$	−17.7143	−0.613030
$836$	0	0
$837$	−6.27687	−0.216960
$838$	0	0
$839$	0.491736	0.0169766	0.00848831	−	0.999964i	$-0.497298\pi$
$839$	0.491736	0.0169766	0.00848831	+	0.999964i	$0.497298\pi$
$840$	0	0
$841$	−26.6558	−0.919167
$842$	0	0
$843$	−55.9202	−1.92599
$844$	0	0
$845$	18.6605	0.641940
$846$	0	0
$847$	−1.70221	−0.0584886
$848$	0	0
$849$	−18.2267	−0.625537
$850$	0	0
$851$	−80.3473	−2.75427
$852$	0	0
$853$	28.0105	0.959062	0.479531	−	0.877525i	$-0.340807\pi$
$853$	28.0105	0.959062	0.479531	+	0.877525i	$0.340807\pi$
$854$	0	0
$855$	0.173766	0.00594266
$856$	0	0
$857$	−16.6711	−0.569475	−0.284738	−	0.958605i	$-0.591906\pi$
$857$	−16.6711	−0.569475	−0.284738	+	0.958605i	$0.591906\pi$
$858$	0	0
$859$	−12.3968	−0.422974	−0.211487	−	0.977381i	$-0.567831\pi$
$859$	−12.3968	−0.422974	−0.211487	+	0.977381i	$0.567831\pi$
$860$	0	0
$861$	−13.2336	−0.451001
$862$	0	0
$863$	0.524206	0.0178442	0.00892209	−	0.999960i	$-0.497160\pi$
$863$	0.524206	0.0178442	0.00892209	+	0.999960i	$0.497160\pi$
$864$	0	0
$865$	7.62721	0.259333
$866$	0	0
$867$	42.1031	1.42990
$868$	0	0
$869$	−8.61270	−0.292166
$870$	0	0
$871$	−44.1808	−1.49701
$872$	0	0
$873$	−0.839122	−0.0284000
$874$	0	0
$875$	11.3636	0.384161
$876$	0	0
$877$	−33.8573	−1.14328	−0.571640	−	0.820504i	$-0.693693\pi$
$877$	−33.8573	−1.14328	−0.571640	+	0.820504i	$0.693693\pi$
$878$	0	0
$879$	14.7368	0.497061
$880$	0	0
$881$	19.8780	0.669705	0.334853	−	0.942270i	$-0.391313\pi$
$881$	19.8780	0.669705	0.334853	+	0.942270i	$0.391313\pi$
$882$	0	0
$883$	−21.9159	−0.737529	−0.368765	−	0.929523i	$-0.620219\pi$
$883$	−21.9159	−0.737529	−0.368765	+	0.929523i	$0.620219\pi$
$884$	0	0
$885$	0.537205	0.0180580
$886$	0	0
$887$	−30.3105	−1.01773	−0.508863	−	0.860847i	$-0.669934\pi$
$887$	−30.3105	−1.01773	−0.508863	+	0.860847i	$0.669934\pi$
$888$	0	0
$889$	27.8192	0.933025
$890$	0	0
$891$	9.68113	0.324330
$892$	0	0
$893$	7.70319	0.257777
$894$	0	0
$895$	7.76173	0.259446
$896$	0	0
$897$	77.7859	2.59720
$898$	0	0
$899$	1.93746	0.0646178
$900$	0	0
$901$	−24.8534	−0.827986
$902$	0	0
$903$	27.3267	0.909377
$904$	0	0
$905$	5.17299	0.171956
$906$	0	0
$907$	−0.921098	−0.0305845	−0.0152923	−	0.999883i	$-0.504868\pi$
$907$	−0.921098	−0.0305845	−0.0152923	+	0.999883i	$0.504868\pi$
$908$	0	0
$909$	−0.997524	−0.0330858
$910$	0	0
$911$	−36.2900	−1.20234	−0.601171	−	0.799121i	$-0.705299\pi$
$911$	−36.2900	−1.20234	−0.601171	+	0.799121i	$0.705299\pi$
$912$	0	0
$913$	4.35338	0.144076
$914$	0	0
$915$	7.71791	0.255146
$916$	0	0
$917$	−8.62487	−0.284818
$918$	0	0
$919$	10.8109	0.356618	0.178309	−	0.983975i	$-0.442937\pi$
$919$	10.8109	0.356618	0.178309	+	0.983975i	$0.442937\pi$
$920$	0	0
$921$	7.67246	0.252816
$922$	0	0
$923$	−5.99066	−0.197185
$924$	0	0
$925$	−52.7747	−1.73522
$926$	0	0
$927$	−1.29618	−0.0425720
$928$	0	0
$929$	−11.2971	−0.370647	−0.185324	−	0.982678i	$-0.559333\pi$
$929$	−11.2971	−0.370647	−0.185324	+	0.982678i	$0.559333\pi$
$930$	0	0
$931$	−4.10249	−0.134454
$932$	0	0
$933$	−2.28426	−0.0747834
$934$	0	0
$935$	4.46130	0.145900
$936$	0	0
$937$	−10.2407	−0.334548	−0.167274	−	0.985910i	$-0.553496\pi$
$937$	−10.2407	−0.334548	−0.167274	+	0.985910i	$0.553496\pi$
$938$	0	0
$939$	12.1418	0.396232
$940$	0	0
$941$	45.2384	1.47473	0.737365	−	0.675494i	$-0.236070\pi$
$941$	45.2384	1.47473	0.737365	+	0.675494i	$0.236070\pi$
$942$	0	0
$943$	−29.6018	−0.963969
$944$	0	0
$945$	5.92904	0.192872
$946$	0	0
$947$	41.9969	1.36472	0.682358	−	0.731018i	$-0.260955\pi$
$947$	41.9969	1.36472	0.682358	+	0.731018i	$0.260955\pi$
$948$	0	0
$949$	34.1450	1.10839
$950$	0	0
$951$	−51.4497	−1.66837
$952$	0	0
$953$	−58.5331	−1.89607	−0.948037	−	0.318160i	$-0.896935\pi$
$953$	−58.5331	−1.89607	−0.948037	+	0.318160i	$0.896935\pi$
$954$	0	0
$955$	−16.0805	−0.520353
$956$	0	0
$957$	−2.75909	−0.0891888
$958$	0	0
$959$	16.2837	0.525827
$960$	0	0
$961$	−29.3987	−0.948345
$962$	0	0
$963$	−3.52688	−0.113652
$964$	0	0
$965$	−11.0003	−0.354113
$966$	0	0
$967$	−25.2213	−0.811062	−0.405531	−	0.914081i	$-0.632913\pi$
$967$	−25.2213	−0.811062	−0.405531	+	0.914081i	$0.632913\pi$
$968$	0	0
$969$	−11.4490	−0.367795
$970$	0	0
$971$	−24.4601	−0.784961	−0.392480	−	0.919760i	$-0.628383\pi$
$971$	−24.4601	−0.784961	−0.392480	+	0.919760i	$0.628383\pi$
$972$	0	0
$973$	14.6555	0.469834
$974$	0	0
$975$	51.0923	1.63626
$976$	0	0
$977$	−17.3563	−0.555276	−0.277638	−	0.960686i	$-0.589552\pi$
$977$	−17.3563	−0.555276	−0.277638	+	0.960686i	$0.589552\pi$
$978$	0	0
$979$	−10.3417	−0.330522
$980$	0	0
$981$	4.36666	0.139417
$982$	0	0
$983$	29.2564	0.933135	0.466567	−	0.884486i	$-0.345491\pi$
$983$	29.2564	0.933135	0.466567	+	0.884486i	$0.345491\pi$
$984$	0	0
$985$	−5.44987	−0.173647
$986$	0	0
$987$	−23.6295	−0.752136
$988$	0	0
$989$	61.1263	1.94370
$990$	0	0
$991$	24.0972	0.765474	0.382737	−	0.923857i	$-0.374982\pi$
$991$	24.0972	0.765474	0.382737	+	0.923857i	$0.374982\pi$
$992$	0	0
$993$	−51.8746	−1.64619
$994$	0	0
$995$	2.30465	0.0730624
$996$	0	0
$997$	1.12836	0.0357357	0.0178678	−	0.999840i	$-0.494312\pi$
$997$	1.12836	0.0357357	0.0178678	+	0.999840i	$0.494312\pi$
$998$	0	0
$999$	−58.0836	−1.83768

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1672.2.a.i.1.6	✓	6
4.3	odd	2		3344.2.a.z.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1672.2.a.i.1.6	✓	6	1.1	even	1	trivial
3344.2.a.z.1.1		6	4.3	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:	\( N \)	\(=\)	\( 1672 = 2^{3} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1672.a (trivial)

\(f(q)\)	\(=\)	\(q+1.80207 q^{3} +0.702208 q^{5} -1.70221 q^{7} +0.247456 q^{9} +O(q^{10})\)	\(q+1.80207 q^{3} +0.702208 q^{5} -1.70221 q^{7} +0.247456 q^{9} -1.00000 q^{11} -6.29079 q^{13} +1.26543 q^{15} -6.35325 q^{17} +1.00000 q^{19} -3.06750 q^{21} -6.86158 q^{23} -4.50690 q^{25} -4.96028 q^{27} +1.53107 q^{29} +1.26543 q^{31} -1.80207 q^{33} -1.19530 q^{35} +11.7097 q^{37} -11.3364 q^{39} +4.31414 q^{41} -8.90848 q^{43} +0.173766 q^{45} +7.70319 q^{47} -4.10249 q^{49} -11.4490 q^{51} +3.91192 q^{53} -0.702208 q^{55} +1.80207 q^{57} +0.424525 q^{59} +6.09905 q^{61} -0.421222 q^{63} -4.41744 q^{65} +7.02309 q^{67} -12.3650 q^{69} +0.952290 q^{71} -5.42777 q^{73} -8.12176 q^{75} +1.70221 q^{77} +8.61270 q^{79} -9.68113 q^{81} -4.35338 q^{83} -4.46130 q^{85} +2.75909 q^{87} +10.3417 q^{89} +10.7082 q^{91} +2.28039 q^{93} +0.702208 q^{95} -3.39099 q^{97} -0.247456 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} - 3 q^{5} - 3 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q - 4 * q^3 - 3 * q^5 - 3 * q^7 + 6 * q^9	\( 6 q - 4 q^{3} - 3 q^{5} - 3 q^{7} + 6 q^{9} - 6 q^{11} - 3 q^{13} - q^{15} - q^{17} + 6 q^{19} + 5 q^{21} - 4 q^{23} + 5 q^{25} - 16 q^{27} - 7 q^{29} - q^{31} + 4 q^{33} - 32 q^{35} + 5 q^{37} - 13 q^{39} - 6 q^{41} - 16 q^{43} - 4 q^{47} - 7 q^{49} - 35 q^{51} - 11 q^{53} + 3 q^{55} - 4 q^{57} - 18 q^{59} + 16 q^{61} - 6 q^{63} + 10 q^{65} - 18 q^{67} - 2 q^{69} - 7 q^{71} - 21 q^{73} - 23 q^{75} + 3 q^{77} - 22 q^{79} - 10 q^{81} - 16 q^{83} - 3 q^{87} - 13 q^{89} - 7 q^{91} - 9 q^{93} - 3 q^{95} - 15 q^{97} - 6 q^{99}+O(q^{100}) \) 6 * q - 4 * q^3 - 3 * q^5 - 3 * q^7 + 6 * q^9 - 6 * q^11 - 3 * q^13 - q^15 - q^17 + 6 * q^19 + 5 * q^21 - 4 * q^23 + 5 * q^25 - 16 * q^27 - 7 * q^29 - q^31 + 4 * q^33 - 32 * q^35 + 5 * q^37 - 13 * q^39 - 6 * q^41 - 16 * q^43 - 4 * q^47 - 7 * q^49 - 35 * q^51 - 11 * q^53 + 3 * q^55 - 4 * q^57 - 18 * q^59 + 16 * q^61 - 6 * q^63 + 10 * q^65 - 18 * q^67 - 2 * q^69 - 7 * q^71 - 21 * q^73 - 23 * q^75 + 3 * q^77 - 22 * q^79 - 10 * q^81 - 16 * q^83 - 3 * q^87 - 13 * q^89 - 7 * q^91 - 9 * q^93 - 3 * q^95 - 15 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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