LMFDB - Embedded newform 3584.2.a.m.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3584,2,Mod(1,3584)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3584, 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("3584.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3584 = 2^{9} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3584.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:	$28.6183840844$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.184365350912.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 14x^{6} + 35x^{4} - 16x^{2} + 2 $ x^8 - 14x^6 + 35x^4 - 16*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$0.481620$ of defining polynomial
Character	$\chi$	$=$	3584.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.93637 q^{3} -2.25526 q^{5} -1.00000 q^{7} +5.62226 q^{9} +O(q^{10})$	$q+2.93637 q^{3} -2.25526 q^{5} -1.00000 q^{7} +5.62226 q^{9} +3.47154 q^{11} -5.44467 q^{13} -6.62226 q^{15} -6.53686 q^{17} -3.54175 q^{19} -2.93637 q^{21} -0.0853971 q^{23} +0.0861758 q^{25} +7.69992 q^{27} -2.32547 q^{29} -1.17157 q^{31} +10.1937 q^{33} +2.25526 q^{35} +4.05336 q^{37} -15.9876 q^{39} -11.2445 q^{41} +6.76025 q^{43} -12.6796 q^{45} +6.41609 q^{47} +1.00000 q^{49} -19.1946 q^{51} -13.2148 q^{53} -7.82921 q^{55} -10.3999 q^{57} +1.10918 q^{59} -4.83929 q^{61} -5.62226 q^{63} +12.2791 q^{65} -2.40120 q^{67} -0.250757 q^{69} +14.6098 q^{71} -6.53686 q^{73} +0.253044 q^{75} -3.47154 q^{77} -14.5369 q^{79} +5.74303 q^{81} -11.7478 q^{83} +14.7423 q^{85} -6.82843 q^{87} -12.6061 q^{89} +5.44467 q^{91} -3.44017 q^{93} +7.98755 q^{95} -14.1245 q^{97} +19.5179 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{7} + 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^7 + 8 * q^9	$ 8 q - 8 q^{7} + 8 q^{9} - 16 q^{15} - 8 q^{17} - 8 q^{23} + 16 q^{25} - 32 q^{31} - 8 q^{33} - 24 q^{39} - 16 q^{41} + 8 q^{49} - 48 q^{55} - 8 q^{57} - 8 q^{63} + 16 q^{65} - 24 q^{71} - 8 q^{73} - 72 q^{79} + 16 q^{81} - 32 q^{87} - 40 q^{89} - 40 q^{95} - 40 q^{97}+O(q^{100}) $ 8 * q - 8 * q^7 + 8 * q^9 - 16 * q^15 - 8 * q^17 - 8 * q^23 + 16 * q^25 - 32 * q^31 - 8 * q^33 - 24 * q^39 - 16 * q^41 + 8 * q^49 - 48 * q^55 - 8 * q^57 - 8 * q^63 + 16 * q^65 - 24 * q^71 - 8 * q^73 - 72 * q^79 + 16 * q^81 - 32 * q^87 - 40 * q^89 - 40 * q^95 - 40 * q^97

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
$3$	2.93637	1.69531	0.847657	−	0.530545i	$-0.178013\pi$
$3$	2.93637	1.69531	0.847657	+	0.530545i	$0.178013\pi$
$4$	0	0
$5$	−2.25526	−1.00858	−0.504290	−	0.863534i	$-0.668246\pi$
$5$	−2.25526	−1.00858	−0.504290	+	0.863534i	$0.668246\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	5.62226	1.87409
$10$	0	0
$11$	3.47154	1.04671	0.523354	−	0.852115i	$-0.324680\pi$
$11$	3.47154	1.04671	0.523354	+	0.852115i	$0.324680\pi$
$12$	0	0
$13$	−5.44467	−1.51008	−0.755040	−	0.655679i	$-0.772382\pi$
$13$	−5.44467	−1.51008	−0.755040	+	0.655679i	$0.772382\pi$
$14$	0	0
$15$	−6.62226	−1.70986
$16$	0	0
$17$	−6.53686	−1.58542	−0.792711	−	0.609597i	$-0.791331\pi$
$17$	−6.53686	−1.58542	−0.792711	+	0.609597i	$0.791331\pi$
$18$	0	0
$19$	−3.54175	−0.812533	−0.406267	−	0.913755i	$-0.633170\pi$
$19$	−3.54175	−0.812533	−0.406267	+	0.913755i	$0.633170\pi$
$20$	0	0
$21$	−2.93637	−0.640768
$22$	0	0
$23$	−0.0853971	−0.0178065	−0.00890326	−	0.999960i	$-0.502834\pi$
$23$	−0.0853971	−0.0178065	−0.00890326	+	0.999960i	$0.502834\pi$
$24$	0	0
$25$	0.0861758	0.0172352
$26$	0	0
$27$	7.69992	1.48185
$28$	0	0
$29$	−2.32547	−0.431828	−0.215914	−	0.976412i	$-0.569273\pi$
$29$	−2.32547	−0.431828	−0.215914	+	0.976412i	$0.569273\pi$
$30$	0	0
$31$	−1.17157	−0.210421	−0.105210	−	0.994450i	$-0.533552\pi$
$31$	−1.17157	−0.210421	−0.105210	+	0.994450i	$0.533552\pi$
$32$	0	0
$33$	10.1937	1.77450
$34$	0	0
$35$	2.25526	0.381208
$36$	0	0
$37$	4.05336	0.666368	0.333184	−	0.942862i	$-0.391877\pi$
$37$	4.05336	0.666368	0.333184	+	0.942862i	$0.391877\pi$
$38$	0	0
$39$	−15.9876	−2.56006
$40$	0	0
$41$	−11.2445	−1.75610	−0.878050	−	0.478570i	$-0.841155\pi$
$41$	−11.2445	−1.75610	−0.878050	+	0.478570i	$0.841155\pi$
$42$	0	0
$43$	6.76025	1.03093	0.515464	−	0.856911i	$-0.327620\pi$
$43$	6.76025	1.03093	0.515464	+	0.856911i	$0.327620\pi$
$44$	0	0
$45$	−12.6796	−1.89017
$46$	0	0
$47$	6.41609	0.935883	0.467942	−	0.883759i	$-0.344996\pi$
$47$	6.41609	0.935883	0.467942	+	0.883759i	$0.344996\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−19.1946	−2.68779
$52$	0	0
$53$	−13.2148	−1.81519	−0.907596	−	0.419844i	$-0.862085\pi$
$53$	−13.2148	−1.81519	−0.907596	+	0.419844i	$0.862085\pi$
$54$	0	0
$55$	−7.82921	−1.05569
$56$	0	0
$57$	−10.3999	−1.37750
$58$	0	0
$59$	1.10918	0.144403	0.0722017	−	0.997390i	$-0.476997\pi$
$59$	1.10918	0.144403	0.0722017	+	0.997390i	$0.476997\pi$
$60$	0	0
$61$	−4.83929	−0.619607	−0.309804	−	0.950801i	$-0.600263\pi$
$61$	−4.83929	−0.619607	−0.309804	+	0.950801i	$0.600263\pi$
$62$	0	0
$63$	−5.62226	−0.708338
$64$	0	0
$65$	12.2791	1.52304
$66$	0	0
$67$	−2.40120	−0.293353	−0.146677	−	0.989185i	$-0.546858\pi$
$67$	−2.40120	−0.293353	−0.146677	+	0.989185i	$0.546858\pi$
$68$	0	0
$69$	−0.250757	−0.0301876
$70$	0	0
$71$	14.6098	1.73387	0.866933	−	0.498425i	$-0.166088\pi$
$71$	14.6098	1.73387	0.866933	+	0.498425i	$0.166088\pi$
$72$	0	0
$73$	−6.53686	−0.765082	−0.382541	−	0.923939i	$-0.624951\pi$
$73$	−6.53686	−0.765082	−0.382541	+	0.923939i	$0.624951\pi$
$74$	0	0
$75$	0.253044	0.0292190
$76$	0	0
$77$	−3.47154	−0.395619
$78$	0	0
$79$	−14.5369	−1.63552	−0.817762	−	0.575556i	$-0.804786\pi$
$79$	−14.5369	−1.63552	−0.817762	+	0.575556i	$0.804786\pi$
$80$	0	0
$81$	5.74303	0.638114
$82$	0	0
$83$	−11.7478	−1.28948	−0.644742	−	0.764400i	$-0.723035\pi$
$83$	−11.7478	−1.28948	−0.644742	+	0.764400i	$0.723035\pi$
$84$	0	0
$85$	14.7423	1.59903
$86$	0	0
$87$	−6.82843	−0.732084
$88$	0	0
$89$	−12.6061	−1.33624	−0.668119	−	0.744054i	$-0.732900\pi$
$89$	−12.6061	−1.33624	−0.668119	+	0.744054i	$0.732900\pi$
$90$	0	0
$91$	5.44467	0.570756
$92$	0	0
$93$	−3.44017	−0.356729
$94$	0	0
$95$	7.98755	0.819505
$96$	0	0
$97$	−14.1245	−1.43413	−0.717064	−	0.697007i	$-0.754515\pi$
$97$	−14.1245	−1.43413	−0.717064	+	0.697007i	$0.754515\pi$
$98$	0	0
$99$	19.5179	1.96162
$100$	0	0
$101$	6.30081	0.626954	0.313477	−	0.949596i	$-0.398506\pi$
$101$	6.30081	0.626954	0.313477	+	0.949596i	$0.398506\pi$
$102$	0	0
$103$	17.5590	1.73014	0.865070	−	0.501651i	$-0.167274\pi$
$103$	17.5590	1.73014	0.865070	+	0.501651i	$0.167274\pi$
$104$	0	0
$105$	6.62226	0.646266
$106$	0	0
$107$	12.4344	1.20208	0.601039	−	0.799220i	$-0.294754\pi$
$107$	12.4344	1.20208	0.601039	+	0.799220i	$0.294754\pi$
$108$	0	0
$109$	14.2851	1.36827	0.684134	−	0.729356i	$-0.260180\pi$
$109$	14.2851	1.36827	0.684134	+	0.729356i	$0.260180\pi$
$110$	0	0
$111$	11.9022	1.12970
$112$	0	0
$113$	15.6444	1.47170	0.735851	−	0.677144i	$-0.236782\pi$
$113$	15.6444	1.47170	0.735851	+	0.677144i	$0.236782\pi$
$114$	0	0
$115$	0.192592	0.0179593
$116$	0	0
$117$	−30.6113	−2.83002
$118$	0	0
$119$	6.53686	0.599233
$120$	0	0
$121$	1.05158	0.0955983
$122$	0	0
$123$	−33.0181	−2.97714
$124$	0	0
$125$	11.0819	0.991198
$126$	0	0
$127$	−3.15536	−0.279993	−0.139997	−	0.990152i	$-0.544709\pi$
$127$	−3.15536	−0.279993	−0.139997	+	0.990152i	$0.544709\pi$
$128$	0	0
$129$	19.8506	1.74775
$130$	0	0
$131$	−11.8992	−1.03964	−0.519820	−	0.854276i	$-0.674001\pi$
$131$	−11.8992	−1.03964	−0.519820	+	0.854276i	$0.674001\pi$
$132$	0	0
$133$	3.54175	0.307109
$134$	0	0
$135$	−17.3653	−1.49457
$136$	0	0
$137$	11.3653	0.971002	0.485501	−	0.874236i	$-0.338637\pi$
$137$	11.3653	0.971002	0.485501	+	0.874236i	$0.338637\pi$
$138$	0	0
$139$	15.7933	1.33957	0.669786	−	0.742555i	$-0.266386\pi$
$139$	15.7933	1.33957	0.669786	+	0.742555i	$0.266386\pi$
$140$	0	0
$141$	18.8400	1.58662
$142$	0	0
$143$	−18.9014	−1.58061
$144$	0	0
$145$	5.24452	0.435534
$146$	0	0
$147$	2.93637	0.242188
$148$	0	0
$149$	−5.12370	−0.419750	−0.209875	−	0.977728i	$-0.567306\pi$
$149$	−5.12370	−0.419750	−0.209875	+	0.977728i	$0.567306\pi$
$150$	0	0
$151$	−8.08540	−0.657980	−0.328990	−	0.944333i	$-0.606708\pi$
$151$	−8.08540	−0.657980	−0.328990	+	0.944333i	$0.606708\pi$
$152$	0	0
$153$	−36.7519	−2.97122
$154$	0	0
$155$	2.64220	0.212226
$156$	0	0
$157$	−9.23946	−0.737389	−0.368695	−	0.929551i	$-0.620195\pi$
$157$	−9.23946	−0.737389	−0.368695	+	0.929551i	$0.620195\pi$
$158$	0	0
$159$	−38.8035	−3.07732
$160$	0	0
$161$	0.0853971	0.00673023
$162$	0	0
$163$	0.829343	0.0649592	0.0324796	−	0.999472i	$-0.489660\pi$
$163$	0.829343	0.0649592	0.0324796	+	0.999472i	$0.489660\pi$
$164$	0	0
$165$	−22.9894	−1.78972
$166$	0	0
$167$	−12.0729	−0.934233	−0.467116	−	0.884196i	$-0.654707\pi$
$167$	−12.0729	−0.934233	−0.467116	+	0.884196i	$0.654707\pi$
$168$	0	0
$169$	16.6444	1.28034
$170$	0	0
$171$	−19.9126	−1.52276
$172$	0	0
$173$	−3.57635	−0.271905	−0.135953	−	0.990715i	$-0.543409\pi$
$173$	−3.57635	−0.271905	−0.135953	+	0.990715i	$0.543409\pi$
$174$	0	0
$175$	−0.0861758	−0.00651428
$176$	0	0
$177$	3.25697	0.244809
$178$	0	0
$179$	−11.0566	−0.826406	−0.413203	−	0.910639i	$-0.635590\pi$
$179$	−11.0566	−0.826406	−0.413203	+	0.910639i	$0.635590\pi$
$180$	0	0
$181$	4.97971	0.370139	0.185069	−	0.982725i	$-0.440749\pi$
$181$	4.97971	0.370139	0.185069	+	0.982725i	$0.440749\pi$
$182$	0	0
$183$	−14.2099	−1.05043
$184$	0	0
$185$	−9.14136	−0.672086
$186$	0	0
$187$	−22.6930	−1.65947
$188$	0	0
$189$	−7.69992	−0.560087
$190$	0	0
$191$	−4.75924	−0.344366	−0.172183	−	0.985065i	$-0.555082\pi$
$191$	−4.75924	−0.344366	−0.172183	+	0.985065i	$0.555082\pi$
$192$	0	0
$193$	−6.03913	−0.434706	−0.217353	−	0.976093i	$-0.569742\pi$
$193$	−6.03913	−0.434706	−0.217353	+	0.976093i	$0.569742\pi$
$194$	0	0
$195$	36.0560	2.58202
$196$	0	0
$197$	−5.40454	−0.385058	−0.192529	−	0.981291i	$-0.561669\pi$
$197$	−5.40454	−0.385058	−0.192529	+	0.981291i	$0.561669\pi$
$198$	0	0
$199$	−27.9022	−1.97793	−0.988966	−	0.148145i	$-0.952670\pi$
$199$	−27.9022	−1.97793	−0.988966	+	0.148145i	$0.952670\pi$
$200$	0	0
$201$	−7.05080	−0.497325
$202$	0	0
$203$	2.32547	0.163216
$204$	0	0
$205$	25.3593	1.77117
$206$	0	0
$207$	−0.480125	−0.0333710
$208$	0	0
$209$	−12.2953	−0.850485
$210$	0	0
$211$	8.78002	0.604442	0.302221	−	0.953238i	$-0.402272\pi$
$211$	8.78002	0.604442	0.302221	+	0.953238i	$0.402272\pi$
$212$	0	0
$213$	42.8998	2.93945
$214$	0	0
$215$	−15.2461	−1.03977
$216$	0	0
$217$	1.17157	0.0795315
$218$	0	0
$219$	−19.1946	−1.29705
$220$	0	0
$221$	35.5910	2.39411
$222$	0	0
$223$	−13.8276	−0.925968	−0.462984	−	0.886367i	$-0.653221\pi$
$223$	−13.8276	−0.925968	−0.462984	+	0.886367i	$0.653221\pi$
$224$	0	0
$225$	0.484503	0.0323002
$226$	0	0
$227$	19.7977	1.31402	0.657011	−	0.753881i	$-0.271821\pi$
$227$	19.7977	1.31402	0.657011	+	0.753881i	$0.271821\pi$
$228$	0	0
$229$	21.8411	1.44330	0.721649	−	0.692259i	$-0.243384\pi$
$229$	21.8411	1.44330	0.721649	+	0.692259i	$0.243384\pi$
$230$	0	0
$231$	−10.1937	−0.670697
$232$	0	0
$233$	4.29156	0.281150	0.140575	−	0.990070i	$-0.455105\pi$
$233$	4.29156	0.281150	0.140575	+	0.990070i	$0.455105\pi$
$234$	0	0
$235$	−14.4699	−0.943914
$236$	0	0
$237$	−42.6856	−2.77273
$238$	0	0
$239$	−10.3307	−0.668237	−0.334119	−	0.942531i	$-0.608439\pi$
$239$	−10.3307	−0.668237	−0.334119	+	0.942531i	$0.608439\pi$
$240$	0	0
$241$	−4.36451	−0.281143	−0.140571	−	0.990071i	$-0.544894\pi$
$241$	−4.36451	−0.281143	−0.140571	+	0.990071i	$0.544894\pi$
$242$	0	0
$243$	−6.23612	−0.400047
$244$	0	0
$245$	−2.25526	−0.144083
$246$	0	0
$247$	19.2837	1.22699
$248$	0	0
$249$	−34.4958	−2.18608
$250$	0	0
$251$	18.6281	1.17580	0.587898	−	0.808935i	$-0.299956\pi$
$251$	18.6281	1.17580	0.587898	+	0.808935i	$0.299956\pi$
$252$	0	0
$253$	−0.296459	−0.0186382
$254$	0	0
$255$	43.2888	2.71085
$256$	0	0
$257$	−17.4169	−1.08643	−0.543217	−	0.839592i	$-0.682794\pi$
$257$	−17.4169	−1.08643	−0.543217	+	0.839592i	$0.682794\pi$
$258$	0	0
$259$	−4.05336	−0.251863
$260$	0	0
$261$	−13.0744	−0.809284
$262$	0	0
$263$	−15.4345	−0.951731	−0.475865	−	0.879518i	$-0.657865\pi$
$263$	−15.4345	−0.951731	−0.475865	+	0.879518i	$0.657865\pi$
$264$	0	0
$265$	29.8027	1.83077
$266$	0	0
$267$	−37.0160	−2.26534
$268$	0	0
$269$	18.0463	1.10030	0.550151	−	0.835065i	$-0.314570\pi$
$269$	18.0463	1.10030	0.550151	+	0.835065i	$0.314570\pi$
$270$	0	0
$271$	−19.4845	−1.18360	−0.591800	−	0.806085i	$-0.701582\pi$
$271$	−19.4845	−1.18360	−0.591800	+	0.806085i	$0.701582\pi$
$272$	0	0
$273$	15.9876	0.967611
$274$	0	0
$275$	0.299163	0.0180402
$276$	0	0
$277$	21.8230	1.31122	0.655609	−	0.755100i	$-0.272412\pi$
$277$	21.8230	1.31122	0.655609	+	0.755100i	$0.272412\pi$
$278$	0	0
$279$	−6.58689	−0.394347
$280$	0	0
$281$	−6.23998	−0.372246	−0.186123	−	0.982526i	$-0.559592\pi$
$281$	−6.23998	−0.372246	−0.186123	+	0.982526i	$0.559592\pi$
$282$	0	0
$283$	17.1255	1.01800	0.509001	−	0.860766i	$-0.330015\pi$
$283$	17.1255	1.01800	0.509001	+	0.860766i	$0.330015\pi$
$284$	0	0
$285$	23.4544	1.38932
$286$	0	0
$287$	11.2445	0.663743
$288$	0	0
$289$	25.7306	1.51356
$290$	0	0
$291$	−41.4748	−2.43130
$292$	0	0
$293$	30.5592	1.78529	0.892643	−	0.450765i	$-0.148849\pi$
$293$	30.5592	1.78529	0.892643	+	0.450765i	$0.148849\pi$
$294$	0	0
$295$	−2.50149	−0.145642
$296$	0	0
$297$	26.7306	1.55107
$298$	0	0
$299$	0.464959	0.0268893
$300$	0	0
$301$	−6.76025	−0.389654
$302$	0	0
$303$	18.5015	1.06288
$304$	0	0
$305$	10.9138	0.624924
$306$	0	0
$307$	17.3130	0.988105	0.494053	−	0.869432i	$-0.335515\pi$
$307$	17.3130	0.988105	0.494053	+	0.869432i	$0.335515\pi$
$308$	0	0
$309$	51.5597	2.93313
$310$	0	0
$311$	7.41687	0.420572	0.210286	−	0.977640i	$-0.432560\pi$
$311$	7.41687	0.420572	0.210286	+	0.977640i	$0.432560\pi$
$312$	0	0
$313$	−11.8276	−0.668538	−0.334269	−	0.942478i	$-0.608489\pi$
$313$	−11.8276	−0.668538	−0.334269	+	0.942478i	$0.608489\pi$
$314$	0	0
$315$	12.6796	0.714416
$316$	0	0
$317$	3.39581	0.190728	0.0953638	−	0.995442i	$-0.469599\pi$
$317$	3.39581	0.190728	0.0953638	+	0.995442i	$0.469599\pi$
$318$	0	0
$319$	−8.07295	−0.451998
$320$	0	0
$321$	36.5120	2.03790
$322$	0	0
$323$	23.1519	1.28821
$324$	0	0
$325$	−0.469198	−0.0260264
$326$	0	0
$327$	41.9464	2.31964
$328$	0	0
$329$	−6.41609	−0.353731
$330$	0	0
$331$	−4.33872	−0.238478	−0.119239	−	0.992866i	$-0.538045\pi$
$331$	−4.33872	−0.238478	−0.119239	+	0.992866i	$0.538045\pi$
$332$	0	0
$333$	22.7890	1.24883
$334$	0	0
$335$	5.41531	0.295870
$336$	0	0
$337$	−17.7136	−0.964921	−0.482460	−	0.875918i	$-0.660257\pi$
$337$	−17.7136	−0.964921	−0.482460	+	0.875918i	$0.660257\pi$
$338$	0	0
$339$	45.9377	2.49500
$340$	0	0
$341$	−4.06716	−0.220249
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0.565522	0.0304467
$346$	0	0
$347$	−6.19599	−0.332618	−0.166309	−	0.986074i	$-0.553185\pi$
$347$	−6.19599	−0.332618	−0.166309	+	0.986074i	$0.553185\pi$
$348$	0	0
$349$	−26.9726	−1.44381	−0.721905	−	0.691993i	$-0.756733\pi$
$349$	−26.9726	−1.44381	−0.721905	+	0.691993i	$0.756733\pi$
$350$	0	0
$351$	−41.9235	−2.23771
$352$	0	0
$353$	15.4861	0.824240	0.412120	−	0.911130i	$-0.364788\pi$
$353$	15.4861	0.824240	0.412120	+	0.911130i	$0.364788\pi$
$354$	0	0
$355$	−32.9489	−1.74874
$356$	0	0
$357$	19.1946	1.01589
$358$	0	0
$359$	2.42854	0.128174	0.0640868	−	0.997944i	$-0.479587\pi$
$359$	2.42854	0.128174	0.0640868	+	0.997944i	$0.479587\pi$
$360$	0	0
$361$	−6.45600	−0.339790
$362$	0	0
$363$	3.08783	0.162069
$364$	0	0
$365$	14.7423	0.771647
$366$	0	0
$367$	−0.412334	−0.0215236	−0.0107618	−	0.999942i	$-0.503426\pi$
$367$	−0.412334	−0.0215236	−0.0107618	+	0.999942i	$0.503426\pi$
$368$	0	0
$369$	−63.2196	−3.29108
$370$	0	0
$371$	13.2148	0.686078
$372$	0	0
$373$	−25.1589	−1.30268	−0.651338	−	0.758787i	$-0.725792\pi$
$373$	−25.1589	−1.30268	−0.651338	+	0.758787i	$0.725792\pi$
$374$	0	0
$375$	32.5406	1.68039
$376$	0	0
$377$	12.6614	0.652095
$378$	0	0
$379$	−1.75926	−0.0903671	−0.0451836	−	0.998979i	$-0.514387\pi$
$379$	−1.75926	−0.0903671	−0.0451836	+	0.998979i	$0.514387\pi$
$380$	0	0
$381$	−9.26531	−0.474676
$382$	0	0
$383$	11.1024	0.567305	0.283653	−	0.958927i	$-0.408454\pi$
$383$	11.1024	0.567305	0.283653	+	0.958927i	$0.408454\pi$
$384$	0	0
$385$	7.82921	0.399013
$386$	0	0
$387$	38.0079	1.93205
$388$	0	0
$389$	−35.5467	−1.80229	−0.901144	−	0.433519i	$-0.857272\pi$
$389$	−35.5467	−1.80229	−0.901144	+	0.433519i	$0.857272\pi$
$390$	0	0
$391$	0.558229	0.0282309
$392$	0	0
$393$	−34.9405	−1.76252
$394$	0	0
$395$	32.7843	1.64956
$396$	0	0
$397$	12.0742	0.605989	0.302995	−	0.952992i	$-0.402014\pi$
$397$	12.0742	0.605989	0.302995	+	0.952992i	$0.402014\pi$
$398$	0	0
$399$	10.3999	0.520645
$400$	0	0
$401$	12.2791	0.613190	0.306595	−	0.951840i	$-0.400810\pi$
$401$	12.2791	0.613190	0.306595	+	0.951840i	$0.400810\pi$
$402$	0	0
$403$	6.37882	0.317752
$404$	0	0
$405$	−12.9520	−0.643590
$406$	0	0
$407$	14.0714	0.697493
$408$	0	0
$409$	−5.65841	−0.279790	−0.139895	−	0.990166i	$-0.544677\pi$
$409$	−5.65841	−0.279790	−0.139895	+	0.990166i	$0.544677\pi$
$410$	0	0
$411$	33.3727	1.64615
$412$	0	0
$413$	−1.10918	−0.0545793
$414$	0	0
$415$	26.4942	1.30055
$416$	0	0
$417$	46.3750	2.27099
$418$	0	0
$419$	−10.7983	−0.527533	−0.263766	−	0.964587i	$-0.584965\pi$
$419$	−10.7983	−0.527533	−0.263766	+	0.964587i	$0.584965\pi$
$420$	0	0
$421$	−25.5873	−1.24705	−0.623524	−	0.781805i	$-0.714299\pi$
$421$	−25.5873	−1.24705	−0.623524	+	0.781805i	$0.714299\pi$
$422$	0	0
$423$	36.0729	1.75393
$424$	0	0
$425$	−0.563319	−0.0273250
$426$	0	0
$427$	4.83929	0.234189
$428$	0	0
$429$	−55.5014	−2.67963
$430$	0	0
$431$	16.9830	0.818043	0.409021	−	0.912525i	$-0.365870\pi$
$431$	16.9830	0.818043	0.409021	+	0.912525i	$0.365870\pi$
$432$	0	0
$433$	19.1275	0.919210	0.459605	−	0.888124i	$-0.347991\pi$
$433$	19.1275	0.919210	0.459605	+	0.888124i	$0.347991\pi$
$434$	0	0
$435$	15.3998	0.738366
$436$	0	0
$437$	0.302455	0.0144684
$438$	0	0
$439$	−41.4936	−1.98038	−0.990190	−	0.139726i	$-0.955378\pi$
$439$	−41.4936	−1.98038	−0.990190	+	0.139726i	$0.955378\pi$
$440$	0	0
$441$	5.62226	0.267727
$442$	0	0
$443$	−7.20359	−0.342253	−0.171127	−	0.985249i	$-0.554741\pi$
$443$	−7.20359	−0.342253	−0.171127	+	0.985249i	$0.554741\pi$
$444$	0	0
$445$	28.4299	1.34770
$446$	0	0
$447$	−15.0451	−0.711607
$448$	0	0
$449$	−13.7069	−0.646868	−0.323434	−	0.946251i	$-0.604837\pi$
$449$	−13.7069	−0.646868	−0.323434	+	0.946251i	$0.604837\pi$
$450$	0	0
$451$	−39.0358	−1.83812
$452$	0	0
$453$	−23.7417	−1.11548
$454$	0	0
$455$	−12.2791	−0.575654
$456$	0	0
$457$	3.22455	0.150838	0.0754191	−	0.997152i	$-0.475971\pi$
$457$	3.22455	0.150838	0.0754191	+	0.997152i	$0.475971\pi$
$458$	0	0
$459$	−50.3333	−2.34936
$460$	0	0
$461$	−0.0258489	−0.00120390	−0.000601952	−	1.00000i	$-0.500192\pi$
$461$	−0.0258489	−0.00120390	−0.000601952		1.00000i	$0.500192\pi$
$462$	0	0
$463$	34.2204	1.59036	0.795178	−	0.606375i	$-0.207377\pi$
$463$	34.2204	1.59036	0.795178	+	0.606375i	$0.207377\pi$
$464$	0	0
$465$	7.75846	0.359790
$466$	0	0
$467$	18.5915	0.860314	0.430157	−	0.902754i	$-0.358458\pi$
$467$	18.5915	0.860314	0.430157	+	0.902754i	$0.358458\pi$
$468$	0	0
$469$	2.40120	0.110877
$470$	0	0
$471$	−27.1305	−1.25011
$472$	0	0
$473$	23.4685	1.07908
$474$	0	0
$475$	−0.305213	−0.0140041
$476$	0	0
$477$	−74.2971	−3.40183
$478$	0	0
$479$	8.38212	0.382989	0.191494	−	0.981494i	$-0.438667\pi$
$479$	8.38212	0.382989	0.191494	+	0.981494i	$0.438667\pi$
$480$	0	0
$481$	−22.0692	−1.00627
$482$	0	0
$483$	0.250757	0.0114099
$484$	0	0
$485$	31.8544	1.44643
$486$	0	0
$487$	15.2607	0.691530	0.345765	−	0.938321i	$-0.387619\pi$
$487$	15.2607	0.691530	0.345765	+	0.938321i	$0.387619\pi$
$488$	0	0
$489$	2.43526	0.110126
$490$	0	0
$491$	38.9111	1.75603	0.878017	−	0.478629i	$-0.158866\pi$
$491$	38.9111	1.75603	0.878017	+	0.478629i	$0.158866\pi$
$492$	0	0
$493$	15.2013	0.684630
$494$	0	0
$495$	−44.0178	−1.97845
$496$	0	0
$497$	−14.6098	−0.655340
$498$	0	0
$499$	−20.8912	−0.935217	−0.467608	−	0.883936i	$-0.654884\pi$
$499$	−20.8912	−0.935217	−0.467608	+	0.883936i	$0.654884\pi$
$500$	0	0
$501$	−35.4506	−1.58382
$502$	0	0
$503$	22.0692	0.984016	0.492008	−	0.870591i	$-0.336263\pi$
$503$	22.0692	0.984016	0.492008	+	0.870591i	$0.336263\pi$
$504$	0	0
$505$	−14.2099	−0.632333
$506$	0	0
$507$	48.8741	2.17058
$508$	0	0
$509$	9.98527	0.442589	0.221295	−	0.975207i	$-0.428972\pi$
$509$	9.98527	0.442589	0.221295	+	0.975207i	$0.428972\pi$
$510$	0	0
$511$	6.53686	0.289174
$512$	0	0
$513$	−27.2712	−1.20405
$514$	0	0
$515$	−39.6000	−1.74499
$516$	0	0
$517$	22.2737	0.979597
$518$	0	0
$519$	−10.5015	−0.460964
$520$	0	0
$521$	9.89684	0.433588	0.216794	−	0.976217i	$-0.430440\pi$
$521$	9.89684	0.433588	0.216794	+	0.976217i	$0.430440\pi$
$522$	0	0
$523$	−6.27724	−0.274485	−0.137242	−	0.990537i	$-0.543824\pi$
$523$	−6.27724	−0.274485	−0.137242	+	0.990537i	$0.543824\pi$
$524$	0	0
$525$	−0.253044	−0.0110437
$526$	0	0
$527$	7.65841	0.333606
$528$	0	0
$529$	−22.9927	−0.999683
$530$	0	0
$531$	6.23612	0.270624
$532$	0	0
$533$	61.2227	2.65185
$534$	0	0
$535$	−28.0427	−1.21239
$536$	0	0
$537$	−32.4661	−1.40102
$538$	0	0
$539$	3.47154	0.149530
$540$	0	0
$541$	−6.85159	−0.294573	−0.147286	−	0.989094i	$-0.547054\pi$
$541$	−6.85159	−0.294573	−0.147286	+	0.989094i	$0.547054\pi$
$542$	0	0
$543$	14.6223	0.627501
$544$	0	0
$545$	−32.2166	−1.38001
$546$	0	0
$547$	−23.4557	−1.00289	−0.501446	−	0.865189i	$-0.667198\pi$
$547$	−23.4557	−1.00289	−0.501446	+	0.865189i	$0.667198\pi$
$548$	0	0
$549$	−27.2077	−1.16120
$550$	0	0
$551$	8.23622	0.350875
$552$	0	0
$553$	14.5369	0.618170
$554$	0	0
$555$	−26.8424	−1.13940
$556$	0	0
$557$	−13.7163	−0.581179	−0.290589	−	0.956848i	$-0.593851\pi$
$557$	−13.7163	−0.581179	−0.290589	+	0.956848i	$0.593851\pi$
$558$	0	0
$559$	−36.8073	−1.55678
$560$	0	0
$561$	−66.6349	−2.81333
$562$	0	0
$563$	32.8473	1.38435	0.692174	−	0.721731i	$-0.256653\pi$
$563$	32.8473	1.38435	0.692174	+	0.721731i	$0.256653\pi$
$564$	0	0
$565$	−35.2821	−1.48433
$566$	0	0
$567$	−5.74303	−0.241185
$568$	0	0
$569$	−36.3058	−1.52202	−0.761009	−	0.648741i	$-0.775296\pi$
$569$	−36.3058	−1.52202	−0.761009	+	0.648741i	$0.775296\pi$
$570$	0	0
$571$	−16.2828	−0.681413	−0.340707	−	0.940170i	$-0.610666\pi$
$571$	−16.2828	−0.681413	−0.340707	+	0.940170i	$0.610666\pi$
$572$	0	0
$573$	−13.9749	−0.583809
$574$	0	0
$575$	−0.00735916	−0.000306898 0
$576$	0	0
$577$	−7.72760	−0.321704	−0.160852	−	0.986979i	$-0.551424\pi$
$577$	−7.72760	−0.321704	−0.160852	+	0.986979i	$0.551424\pi$
$578$	0	0
$579$	−17.7331	−0.736963
$580$	0	0
$581$	11.7478	0.487379
$582$	0	0
$583$	−45.8757	−1.89998
$584$	0	0
$585$	69.0364	2.85430
$586$	0	0
$587$	−7.61995	−0.314509	−0.157255	−	0.987558i	$-0.550264\pi$
$587$	−7.61995	−0.314509	−0.157255	+	0.987558i	$0.550264\pi$
$588$	0	0
$589$	4.14942	0.170974
$590$	0	0
$591$	−15.8697	−0.652794
$592$	0	0
$593$	12.7306	0.522782	0.261391	−	0.965233i	$-0.415819\pi$
$593$	12.7306	0.522782	0.261391	+	0.965233i	$0.415819\pi$
$594$	0	0
$595$	−14.7423	−0.604375
$596$	0	0
$597$	−81.9310	−3.35321
$598$	0	0
$599$	7.65841	0.312914	0.156457	−	0.987685i	$-0.449993\pi$
$599$	7.65841	0.312914	0.156457	+	0.987685i	$0.449993\pi$
$600$	0	0
$601$	36.7519	1.49914	0.749572	−	0.661923i	$-0.230260\pi$
$601$	36.7519	1.49914	0.749572	+	0.661923i	$0.230260\pi$
$602$	0	0
$603$	−13.5002	−0.549769
$604$	0	0
$605$	−2.37159	−0.0964187
$606$	0	0
$607$	−16.5831	−0.673088	−0.336544	−	0.941668i	$-0.609258\pi$
$607$	−16.5831	−0.673088	−0.336544	+	0.941668i	$0.609258\pi$
$608$	0	0
$609$	6.82843	0.276702
$610$	0	0
$611$	−34.9335	−1.41326
$612$	0	0
$613$	−2.73826	−0.110597	−0.0552986	−	0.998470i	$-0.517611\pi$
$613$	−2.73826	−0.110597	−0.0552986	+	0.998470i	$0.517611\pi$
$614$	0	0
$615$	74.4641	3.00268
$616$	0	0
$617$	37.4987	1.50964	0.754821	−	0.655931i	$-0.227724\pi$
$617$	37.4987	1.50964	0.754821	+	0.655931i	$0.227724\pi$
$618$	0	0
$619$	33.1608	1.33284	0.666422	−	0.745575i	$-0.267825\pi$
$619$	33.1608	1.33284	0.666422	+	0.745575i	$0.267825\pi$
$620$	0	0
$621$	−0.657551	−0.0263866
$622$	0	0
$623$	12.6061	0.505051
$624$	0	0
$625$	−25.4235	−1.01694
$626$	0	0
$627$	−36.1036	−1.44184
$628$	0	0
$629$	−26.4962	−1.05647
$630$	0	0
$631$	16.9706	0.675587	0.337794	−	0.941220i	$-0.390319\pi$
$631$	16.9706	0.675587	0.337794	+	0.941220i	$0.390319\pi$
$632$	0	0
$633$	25.7814	1.02472
$634$	0	0
$635$	7.11615	0.282396
$636$	0	0
$637$	−5.44467	−0.215726
$638$	0	0
$639$	82.1402	3.24941
$640$	0	0
$641$	10.8962	0.430375	0.215187	−	0.976573i	$-0.430964\pi$
$641$	10.8962	0.430375	0.215187	+	0.976573i	$0.430964\pi$
$642$	0	0
$643$	−26.2347	−1.03460	−0.517298	−	0.855805i	$-0.673062\pi$
$643$	−26.2347	−1.03460	−0.517298	+	0.855805i	$0.673062\pi$
$644$	0	0
$645$	−44.7681	−1.76274
$646$	0	0
$647$	−8.93003	−0.351076	−0.175538	−	0.984473i	$-0.556166\pi$
$647$	−8.93003	−0.351076	−0.175538	+	0.984473i	$0.556166\pi$
$648$	0	0
$649$	3.85057	0.151148
$650$	0	0
$651$	3.44017	0.134831
$652$	0	0
$653$	12.2764	0.480413	0.240206	−	0.970722i	$-0.422785\pi$
$653$	12.2764	0.480413	0.240206	+	0.970722i	$0.422785\pi$
$654$	0	0
$655$	26.8358	1.04856
$656$	0	0
$657$	−36.7519	−1.43383
$658$	0	0
$659$	−15.3841	−0.599279	−0.299640	−	0.954052i	$-0.596866\pi$
$659$	−15.3841	−0.599279	−0.299640	+	0.954052i	$0.596866\pi$
$660$	0	0
$661$	39.3660	1.53116	0.765580	−	0.643341i	$-0.222452\pi$
$661$	39.3660	1.53116	0.765580	+	0.643341i	$0.222452\pi$
$662$	0	0
$663$	104.508	4.05877
$664$	0	0
$665$	−7.98755	−0.309744
$666$	0	0
$667$	0.198588	0.00768936
$668$	0	0
$669$	−40.6031	−1.56981
$670$	0	0
$671$	−16.7998	−0.648548
$672$	0	0
$673$	−9.65685	−0.372244	−0.186122	−	0.982527i	$-0.559592\pi$
$673$	−9.65685	−0.372244	−0.186122	+	0.982527i	$0.559592\pi$
$674$	0	0
$675$	0.663547	0.0255399
$676$	0	0
$677$	11.8451	0.455244	0.227622	−	0.973750i	$-0.426905\pi$
$677$	11.8451	0.455244	0.227622	+	0.973750i	$0.426905\pi$
$678$	0	0
$679$	14.1245	0.542050
$680$	0	0
$681$	58.1334	2.22768
$682$	0	0
$683$	8.11143	0.310375	0.155188	−	0.987885i	$-0.450402\pi$
$683$	8.11143	0.310375	0.155188	+	0.987885i	$0.450402\pi$
$684$	0	0
$685$	−25.6316	−0.979334
$686$	0	0
$687$	64.1334	2.44684
$688$	0	0
$689$	71.9502	2.74108
$690$	0	0
$691$	−17.3130	−0.658617	−0.329309	−	0.944222i	$-0.606816\pi$
$691$	−17.3130	−0.658617	−0.329309	+	0.944222i	$0.606816\pi$
$692$	0	0
$693$	−19.5179	−0.741424
$694$	0	0
$695$	−35.6179	−1.35107
$696$	0	0
$697$	73.5039	2.78416
$698$	0	0
$699$	12.6016	0.476637
$700$	0	0
$701$	10.1513	0.383411	0.191705	−	0.981453i	$-0.438598\pi$
$701$	10.1513	0.383411	0.191705	+	0.981453i	$0.438598\pi$
$702$	0	0
$703$	−14.3560	−0.541446
$704$	0	0
$705$	−42.4890	−1.60023
$706$	0	0
$707$	−6.30081	−0.236966
$708$	0	0
$709$	−3.34411	−0.125591	−0.0627953	−	0.998026i	$-0.520002\pi$
$709$	−3.34411	−0.125591	−0.0627953	+	0.998026i	$0.520002\pi$
$710$	0	0
$711$	−81.7300	−3.06512
$712$	0	0
$713$	0.100049	0.00374686
$714$	0	0
$715$	42.6274	1.59418
$716$	0	0
$717$	−30.3347	−1.13287
$718$	0	0
$719$	10.2777	0.383294	0.191647	−	0.981464i	$-0.438617\pi$
$719$	10.2777	0.383294	0.191647	+	0.981464i	$0.438617\pi$
$720$	0	0
$721$	−17.5590	−0.653932
$722$	0	0
$723$	−12.8158	−0.476625
$724$	0	0
$725$	−0.200399	−0.00744263
$726$	0	0
$727$	49.2926	1.82816	0.914080	−	0.405534i	$-0.132914\pi$
$727$	49.2926	1.82816	0.914080	+	0.405534i	$0.132914\pi$
$728$	0	0
$729$	−35.5406	−1.31632
$730$	0	0
$731$	−44.1908	−1.63446
$732$	0	0
$733$	44.6600	1.64955	0.824777	−	0.565458i	$-0.191300\pi$
$733$	44.6600	1.64955	0.824777	+	0.565458i	$0.191300\pi$
$734$	0	0
$735$	−6.62226	−0.244266
$736$	0	0
$737$	−8.33585	−0.307055
$738$	0	0
$739$	−7.04109	−0.259011	−0.129505	−	0.991579i	$-0.541339\pi$
$739$	−7.04109	−0.259011	−0.129505	+	0.991579i	$0.541339\pi$
$740$	0	0
$741$	56.6239	2.08013
$742$	0	0
$743$	24.2562	0.889873	0.444937	−	0.895562i	$-0.353226\pi$
$743$	24.2562	0.889873	0.444937	+	0.895562i	$0.353226\pi$
$744$	0	0
$745$	11.5552	0.423352
$746$	0	0
$747$	−66.0490	−2.41661
$748$	0	0
$749$	−12.4344	−0.454343
$750$	0	0
$751$	−31.1629	−1.13715	−0.568575	−	0.822632i	$-0.692505\pi$
$751$	−31.1629	−1.13715	−0.568575	+	0.822632i	$0.692505\pi$
$752$	0	0
$753$	54.6990	1.99334
$754$	0	0
$755$	18.2346	0.663626
$756$	0	0
$757$	−34.1438	−1.24098	−0.620489	−	0.784215i	$-0.713066\pi$
$757$	−34.1438	−1.24098	−0.620489	+	0.784215i	$0.713066\pi$
$758$	0	0
$759$	−0.870514	−0.0315977
$760$	0	0
$761$	−45.6335	−1.65421	−0.827107	−	0.562045i	$-0.810015\pi$
$761$	−45.6335	−1.65421	−0.827107	+	0.562045i	$0.810015\pi$
$762$	0	0
$763$	−14.2851	−0.517157
$764$	0	0
$765$	82.8850	2.99671
$766$	0	0
$767$	−6.03913	−0.218060
$768$	0	0
$769$	24.6136	0.887588	0.443794	−	0.896129i	$-0.353632\pi$
$769$	24.6136	0.887588	0.443794	+	0.896129i	$0.353632\pi$
$770$	0	0
$771$	−51.1424	−1.84185
$772$	0	0
$773$	−25.3029	−0.910080	−0.455040	−	0.890471i	$-0.650375\pi$
$773$	−25.3029	−0.910080	−0.455040	+	0.890471i	$0.650375\pi$
$774$	0	0
$775$	−0.100961	−0.00362663
$776$	0	0
$777$	−11.9022	−0.426987
$778$	0	0
$779$	39.8253	1.42689
$780$	0	0
$781$	50.7185	1.81485
$782$	0	0
$783$	−17.9059	−0.639905
$784$	0	0
$785$	20.8373	0.743717
$786$	0	0
$787$	−1.55253	−0.0553418	−0.0276709	−	0.999617i	$-0.508809\pi$
$787$	−1.55253	−0.0553418	−0.0276709	+	0.999617i	$0.508809\pi$
$788$	0	0
$789$	−45.3213	−1.61348
$790$	0	0
$791$	−15.6444	−0.556251
$792$	0	0
$793$	26.3483	0.935656
$794$	0	0
$795$	87.5119	3.10373
$796$	0	0
$797$	30.7955	1.09083	0.545415	−	0.838166i	$-0.316372\pi$
$797$	30.7955	1.09083	0.545415	+	0.838166i	$0.316372\pi$
$798$	0	0
$799$	−41.9411	−1.48377
$800$	0	0
$801$	−70.8745	−2.50423
$802$	0	0
$803$	−22.6930	−0.800818
$804$	0	0
$805$	−0.192592	−0.00678798
$806$	0	0
$807$	52.9905	1.86536
$808$	0	0
$809$	−13.3337	−0.468787	−0.234394	−	0.972142i	$-0.575310\pi$
$809$	−13.3337	−0.468787	−0.234394	+	0.972142i	$0.575310\pi$
$810$	0	0
$811$	54.9310	1.92889	0.964443	−	0.264289i	$-0.0851374\pi$
$811$	54.9310	1.92889	0.964443	+	0.264289i	$0.0851374\pi$
$812$	0	0
$813$	−57.2137	−2.00657
$814$	0	0
$815$	−1.87038	−0.0655166
$816$	0	0
$817$	−23.9431	−0.837663
$818$	0	0
$819$	30.6113	1.06965
$820$	0	0
$821$	55.2649	1.92876	0.964379	−	0.264523i	$-0.0852144\pi$
$821$	55.2649	1.92876	0.964379	+	0.264523i	$0.0852144\pi$
$822$	0	0
$823$	−35.6835	−1.24385	−0.621925	−	0.783077i	$-0.713649\pi$
$823$	−35.6835	−1.24385	−0.621925	+	0.783077i	$0.713649\pi$
$824$	0	0
$825$	0.878452	0.0305838
$826$	0	0
$827$	33.2831	1.15737	0.578684	−	0.815552i	$-0.303566\pi$
$827$	33.2831	1.15737	0.578684	+	0.815552i	$0.303566\pi$
$828$	0	0
$829$	−34.6509	−1.20348	−0.601738	−	0.798694i	$-0.705525\pi$
$829$	−34.6509	−1.20348	−0.601738	+	0.798694i	$0.705525\pi$
$830$	0	0
$831$	64.0805	2.22293
$832$	0	0
$833$	−6.53686	−0.226489
$834$	0	0
$835$	27.2276	0.942249
$836$	0	0
$837$	−9.02102	−0.311812
$838$	0	0
$839$	10.9316	0.377400	0.188700	−	0.982035i	$-0.439573\pi$
$839$	10.9316	0.377400	0.188700	+	0.982035i	$0.439573\pi$
$840$	0	0
$841$	−23.5922	−0.813524
$842$	0	0
$843$	−18.3229	−0.631074
$844$	0	0
$845$	−37.5374	−1.29133
$846$	0	0
$847$	−1.05158	−0.0361328
$848$	0	0
$849$	50.2866	1.72583
$850$	0	0
$851$	−0.346145	−0.0118657
$852$	0	0
$853$	−41.9075	−1.43488	−0.717442	−	0.696618i	$-0.754687\pi$
$853$	−41.9075	−1.43488	−0.717442	+	0.696618i	$0.754687\pi$
$854$	0	0
$855$	44.9081	1.53582
$856$	0	0
$857$	3.51096	0.119932	0.0599660	−	0.998200i	$-0.480901\pi$
$857$	3.51096	0.119932	0.0599660	+	0.998200i	$0.480901\pi$
$858$	0	0
$859$	−15.4493	−0.527122	−0.263561	−	0.964643i	$-0.584897\pi$
$859$	−15.4493	−0.527122	−0.263561	+	0.964643i	$0.584897\pi$
$860$	0	0
$861$	33.0181	1.12525
$862$	0	0
$863$	27.7282	0.943880	0.471940	−	0.881631i	$-0.343554\pi$
$863$	27.7282	0.943880	0.471940	+	0.881631i	$0.343554\pi$
$864$	0	0
$865$	8.06559	0.274238
$866$	0	0
$867$	75.5545	2.56596
$868$	0	0
$869$	−50.4653	−1.71192
$870$	0	0
$871$	13.0737	0.442986
$872$	0	0
$873$	−79.4118	−2.68768
$874$	0	0
$875$	−11.0819	−0.374638
$876$	0	0
$877$	18.9361	0.639426	0.319713	−	0.947514i	$-0.396414\pi$
$877$	18.9361	0.639426	0.319713	+	0.947514i	$0.396414\pi$
$878$	0	0
$879$	89.7330	3.02662
$880$	0	0
$881$	41.0473	1.38292	0.691459	−	0.722416i	$-0.256968\pi$
$881$	41.0473	1.38292	0.691459	+	0.722416i	$0.256968\pi$
$882$	0	0
$883$	36.4073	1.22520	0.612602	−	0.790391i	$-0.290123\pi$
$883$	36.4073	1.22520	0.612602	+	0.790391i	$0.290123\pi$
$884$	0	0
$885$	−7.34530	−0.246910
$886$	0	0
$887$	45.2850	1.52052	0.760262	−	0.649617i	$-0.225071\pi$
$887$	45.2850	1.52052	0.760262	+	0.649617i	$0.225071\pi$
$888$	0	0
$889$	3.15536	0.105828
$890$	0	0
$891$	19.9372	0.667920
$892$	0	0
$893$	−22.7242	−0.760436
$894$	0	0
$895$	24.9354	0.833497
$896$	0	0
$897$	1.36529	0.0455857
$898$	0	0
$899$	2.72445	0.0908656
$900$	0	0
$901$	86.3834	2.87785
$902$	0	0
$903$	−19.8506	−0.660586
$904$	0	0
$905$	−11.2305	−0.373315
$906$	0	0
$907$	−49.6129	−1.64737	−0.823685	−	0.567048i	$-0.808085\pi$
$907$	−49.6129	−1.64737	−0.823685	+	0.567048i	$0.808085\pi$
$908$	0	0
$909$	35.4248	1.17497
$910$	0	0
$911$	−6.67540	−0.221166	−0.110583	−	0.993867i	$-0.535272\pi$
$911$	−6.67540	−0.221166	−0.110583	+	0.993867i	$0.535272\pi$
$912$	0	0
$913$	−40.7828	−1.34971
$914$	0	0
$915$	32.0470	1.05944
$916$	0	0
$917$	11.8992	0.392947
$918$	0	0
$919$	−42.1651	−1.39090	−0.695448	−	0.718576i	$-0.744794\pi$
$919$	−42.1651	−1.39090	−0.695448	+	0.718576i	$0.744794\pi$
$920$	0	0
$921$	50.8373	1.67515
$922$	0	0
$923$	−79.5456	−2.61827
$924$	0	0
$925$	0.349301	0.0114850
$926$	0	0
$927$	98.7213	3.24243
$928$	0	0
$929$	−10.0463	−0.329607	−0.164804	−	0.986326i	$-0.552699\pi$
$929$	−10.0463	−0.329607	−0.164804	+	0.986326i	$0.552699\pi$
$930$	0	0
$931$	−3.54175	−0.116076
$932$	0	0
$933$	21.7787	0.713002
$934$	0	0
$935$	51.1784	1.67371
$936$	0	0
$937$	−54.0996	−1.76736	−0.883679	−	0.468093i	$-0.844941\pi$
$937$	−54.0996	−1.76736	−0.883679	+	0.468093i	$0.844941\pi$
$938$	0	0
$939$	−34.7303	−1.13338
$940$	0	0
$941$	−2.38120	−0.0776250	−0.0388125	−	0.999247i	$-0.512358\pi$
$941$	−2.38120	−0.0776250	−0.0388125	+	0.999247i	$0.512358\pi$
$942$	0	0
$943$	0.960249	0.0312700
$944$	0	0
$945$	17.3653	0.564893
$946$	0	0
$947$	−12.5748	−0.408627	−0.204313	−	0.978906i	$-0.565496\pi$
$947$	−12.5748	−0.408627	−0.204313	+	0.978906i	$0.565496\pi$
$948$	0	0
$949$	35.5910	1.15533
$950$	0	0
$951$	9.97134	0.323343
$952$	0	0
$953$	12.4890	0.404560	0.202280	−	0.979328i	$-0.435165\pi$
$953$	12.4890	0.404560	0.202280	+	0.979328i	$0.435165\pi$
$954$	0	0
$955$	10.7333	0.347321
$956$	0	0
$957$	−23.7051	−0.766279
$958$	0	0
$959$	−11.3653	−0.367004
$960$	0	0
$961$	−29.6274	−0.955723
$962$	0	0
$963$	69.9094	2.25280
$964$	0	0
$965$	13.6198	0.438436
$966$	0	0
$967$	−42.0621	−1.35262	−0.676312	−	0.736615i	$-0.736423\pi$
$967$	−42.0621	−1.35262	−0.676312	+	0.736615i	$0.736423\pi$
$968$	0	0
$969$	67.9826	2.18392
$970$	0	0
$971$	−2.57575	−0.0826597	−0.0413298	−	0.999146i	$-0.513159\pi$
$971$	−2.57575	−0.0826597	−0.0413298	+	0.999146i	$0.513159\pi$
$972$	0	0
$973$	−15.7933	−0.506310
$974$	0	0
$975$	−1.37774	−0.0441230
$976$	0	0
$977$	−22.6422	−0.724389	−0.362195	−	0.932103i	$-0.617972\pi$
$977$	−22.6422	−0.724389	−0.362195	+	0.932103i	$0.617972\pi$
$978$	0	0
$979$	−43.7624	−1.39865
$980$	0	0
$981$	80.3148	2.56425
$982$	0	0
$983$	14.0038	0.446651	0.223325	−	0.974744i	$-0.428309\pi$
$983$	14.0038	0.446651	0.223325	+	0.974744i	$0.428309\pi$
$984$	0	0
$985$	12.1886	0.388362
$986$	0	0
$987$	−18.8400	−0.599684
$988$	0	0
$989$	−0.577305	−0.0183572
$990$	0	0
$991$	−55.5074	−1.76325	−0.881626	−	0.471949i	$-0.843551\pi$
$991$	−55.5074	−1.76325	−0.881626	+	0.471949i	$0.843551\pi$
$992$	0	0
$993$	−12.7401	−0.404294
$994$	0	0
$995$	62.9265	1.99490
$996$	0	0
$997$	−32.8258	−1.03960	−0.519802	−	0.854287i	$-0.673994\pi$
$997$	−32.8258	−1.03960	−0.519802	+	0.854287i	$0.673994\pi$
$998$	0	0
$999$	31.2105	0.987458

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3584.2.a.m.1.8	yes	8
4.3	odd	2		3584.2.a.n.1.1	yes	8
8.3	odd	2		3584.2.a.n.1.8	yes	8
8.5	even	2	inner	3584.2.a.m.1.1	✓	8
16.3	odd	4		3584.2.b.e.1793.1		8
16.5	even	4		3584.2.b.f.1793.1		8
16.11	odd	4		3584.2.b.e.1793.8		8
16.13	even	4		3584.2.b.f.1793.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3584.2.a.m.1.1	✓	8	8.5	even	2	inner
3584.2.a.m.1.8	yes	8	1.1	even	1	trivial
3584.2.a.n.1.1	yes	8	4.3	odd	2
3584.2.a.n.1.8	yes	8	8.3	odd	2
3584.2.b.e.1793.1		8	16.3	odd	4
3584.2.b.e.1793.8		8	16.11	odd	4
3584.2.b.f.1793.1		8	16.5	even	4
3584.2.b.f.1793.8		8	16.13	even	4

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:	\( N \)	\(=\)	\( 3584 = 2^{9} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3584.a (trivial)

\(f(q)\)	\(=\)	\(q+2.93637 q^{3} -2.25526 q^{5} -1.00000 q^{7} +5.62226 q^{9} +O(q^{10})\)	\(q+2.93637 q^{3} -2.25526 q^{5} -1.00000 q^{7} +5.62226 q^{9} +3.47154 q^{11} -5.44467 q^{13} -6.62226 q^{15} -6.53686 q^{17} -3.54175 q^{19} -2.93637 q^{21} -0.0853971 q^{23} +0.0861758 q^{25} +7.69992 q^{27} -2.32547 q^{29} -1.17157 q^{31} +10.1937 q^{33} +2.25526 q^{35} +4.05336 q^{37} -15.9876 q^{39} -11.2445 q^{41} +6.76025 q^{43} -12.6796 q^{45} +6.41609 q^{47} +1.00000 q^{49} -19.1946 q^{51} -13.2148 q^{53} -7.82921 q^{55} -10.3999 q^{57} +1.10918 q^{59} -4.83929 q^{61} -5.62226 q^{63} +12.2791 q^{65} -2.40120 q^{67} -0.250757 q^{69} +14.6098 q^{71} -6.53686 q^{73} +0.253044 q^{75} -3.47154 q^{77} -14.5369 q^{79} +5.74303 q^{81} -11.7478 q^{83} +14.7423 q^{85} -6.82843 q^{87} -12.6061 q^{89} +5.44467 q^{91} -3.44017 q^{93} +7.98755 q^{95} -14.1245 q^{97} +19.5179 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{7} + 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^7 + 8 * q^9	\( 8 q - 8 q^{7} + 8 q^{9} - 16 q^{15} - 8 q^{17} - 8 q^{23} + 16 q^{25} - 32 q^{31} - 8 q^{33} - 24 q^{39} - 16 q^{41} + 8 q^{49} - 48 q^{55} - 8 q^{57} - 8 q^{63} + 16 q^{65} - 24 q^{71} - 8 q^{73} - 72 q^{79} + 16 q^{81} - 32 q^{87} - 40 q^{89} - 40 q^{95} - 40 q^{97}+O(q^{100}) \) 8 * q - 8 * q^7 + 8 * q^9 - 16 * q^15 - 8 * q^17 - 8 * q^23 + 16 * q^25 - 32 * q^31 - 8 * q^33 - 24 * q^39 - 16 * q^41 + 8 * q^49 - 48 * q^55 - 8 * q^57 - 8 * q^63 + 16 * q^65 - 24 * q^71 - 8 * q^73 - 72 * q^79 + 16 * q^81 - 32 * q^87 - 40 * q^89 - 40 * q^95 - 40 * q^97

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