LMFDB - Embedded newform 804.2.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [804,2,Mod(1,804)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(804, 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("804.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 804 = 2^{2} \cdot 3 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	804.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$6.41997232251$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1076.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 8x - 6 $ x^3 - 8*x - 6
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			1.3
Root			$-2.32887$ of defining polynomial
Character	$\chi$	$=$	804.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.32887 q^{5} +3.08139 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.32887 q^{5} +3.08139 q^{7} +1.00000 q^{9} +1.42365 q^{11} +0.576354 q^{13} -1.32887 q^{15} +0.576354 q^{17} -6.08139 q^{19} -3.08139 q^{21} +5.32887 q^{23} -3.23410 q^{25} -1.00000 q^{27} +8.00000 q^{29} -4.50504 q^{31} -1.42365 q^{33} +4.09477 q^{35} +11.0814 q^{37} -0.576354 q^{39} +2.09477 q^{41} +9.16278 q^{43} +1.32887 q^{45} -8.73913 q^{47} +2.49496 q^{49} -0.576354 q^{51} +8.25755 q^{53} +1.89184 q^{55} +6.08139 q^{57} -4.25755 q^{59} +4.92868 q^{61} +3.08139 q^{63} +0.765901 q^{65} -1.00000 q^{67} -5.32887 q^{69} +13.8918 q^{71} -7.84729 q^{73} +3.23410 q^{75} +4.38681 q^{77} -11.5050 q^{79} +1.00000 q^{81} -2.25755 q^{83} +0.765901 q^{85} -8.00000 q^{87} +8.92868 q^{89} +1.77597 q^{91} +4.50504 q^{93} -8.08139 q^{95} -0.0813900 q^{97} +1.42365 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 3 q^{5} - 5 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - 3 * q^5 - 5 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} - 3 q^{5} - 5 q^{7} + 3 q^{9} + 4 q^{11} + 2 q^{13} + 3 q^{15} + 2 q^{17} - 4 q^{19} + 5 q^{21} + 9 q^{23} + 4 q^{25} - 3 q^{27} + 24 q^{29} + q^{31} - 4 q^{33} + 19 q^{35} + 19 q^{37} - 2 q^{39} + 13 q^{41} - q^{43} - 3 q^{45} + 2 q^{47} + 22 q^{49} - 2 q^{51} + 3 q^{53} - 22 q^{55} + 4 q^{57} + 9 q^{59} - 5 q^{63} + 16 q^{65} - 3 q^{67} - 9 q^{69} + 14 q^{71} - 23 q^{73} - 4 q^{75} - 20 q^{79} + 3 q^{81} + 15 q^{83} + 16 q^{85} - 24 q^{87} + 12 q^{89} - 10 q^{91} - q^{93} - 10 q^{95} + 14 q^{97} + 4 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^5 - 5 * q^7 + 3 * q^9 + 4 * q^11 + 2 * q^13 + 3 * q^15 + 2 * q^17 - 4 * q^19 + 5 * q^21 + 9 * q^23 + 4 * q^25 - 3 * q^27 + 24 * q^29 + q^31 - 4 * q^33 + 19 * q^35 + 19 * q^37 - 2 * q^39 + 13 * q^41 - q^43 - 3 * q^45 + 2 * q^47 + 22 * q^49 - 2 * q^51 + 3 * q^53 - 22 * q^55 + 4 * q^57 + 9 * q^59 - 5 * q^63 + 16 * q^65 - 3 * q^67 - 9 * q^69 + 14 * q^71 - 23 * q^73 - 4 * q^75 - 20 * q^79 + 3 * q^81 + 15 * q^83 + 16 * q^85 - 24 * q^87 + 12 * q^89 - 10 * q^91 - q^93 - 10 * q^95 + 14 * q^97 + 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.32887	0.594290	0.297145	−	0.954832i	$-0.403966\pi$
$5$	1.32887	0.594290	0.297145	+	0.954832i	$0.403966\pi$
$6$	0	0
$7$	3.08139	1.16466	0.582328	−	0.812954i	$-0.302142\pi$
$7$	3.08139	1.16466	0.582328	+	0.812954i	$0.302142\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.42365	0.429245	0.214623	−	0.976697i	$-0.431148\pi$
$11$	1.42365	0.429245	0.214623	+	0.976697i	$0.431148\pi$
$12$	0	0
$13$	0.576354	0.159852	0.0799260	−	0.996801i	$-0.474532\pi$
$13$	0.576354	0.159852	0.0799260	+	0.996801i	$0.474532\pi$
$14$	0	0
$15$	−1.32887	−0.343113
$16$	0	0
$17$	0.576354	0.139786	0.0698932	−	0.997554i	$-0.477734\pi$
$17$	0.576354	0.139786	0.0698932	+	0.997554i	$0.477734\pi$
$18$	0	0
$19$	−6.08139	−1.39517	−0.697583	−	0.716504i	$-0.745741\pi$
$19$	−6.08139	−1.39517	−0.697583	+	0.716504i	$0.745741\pi$
$20$	0	0
$21$	−3.08139	−0.672414
$22$	0	0
$23$	5.32887	1.11115	0.555573	−	0.831468i	$-0.312499\pi$
$23$	5.32887	1.11115	0.555573	+	0.831468i	$0.312499\pi$
$24$	0	0
$25$	−3.23410	−0.646820
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	0	0
$31$	−4.50504	−0.809128	−0.404564	−	0.914510i	$-0.632577\pi$
$31$	−4.50504	−0.809128	−0.404564	+	0.914510i	$0.632577\pi$
$32$	0	0
$33$	−1.42365	−0.247825
$34$	0	0
$35$	4.09477	0.692143
$36$	0	0
$37$	11.0814	1.82177	0.910885	−	0.412661i	$-0.135401\pi$
$37$	11.0814	1.82177	0.910885	+	0.412661i	$0.135401\pi$
$38$	0	0
$39$	−0.576354	−0.0922906
$40$	0	0
$41$	2.09477	0.327149	0.163574	−	0.986531i	$-0.447698\pi$
$41$	2.09477	0.327149	0.163574	+	0.986531i	$0.447698\pi$
$42$	0	0
$43$	9.16278	1.39731	0.698655	−	0.715458i	$-0.253782\pi$
$43$	9.16278	1.39731	0.698655	+	0.715458i	$0.253782\pi$
$44$	0	0
$45$	1.32887	0.198097
$46$	0	0
$47$	−8.73913	−1.27473	−0.637367	−	0.770560i	$-0.719976\pi$
$47$	−8.73913	−1.27473	−0.637367	+	0.770560i	$0.719976\pi$
$48$	0	0
$49$	2.49496	0.356423
$50$	0	0
$51$	−0.576354	−0.0807058
$52$	0	0
$53$	8.25755	1.13426	0.567131	−	0.823628i	$-0.308053\pi$
$53$	8.25755	1.13426	0.567131	+	0.823628i	$0.308053\pi$
$54$	0	0
$55$	1.89184	0.255096
$56$	0	0
$57$	6.08139	0.805500
$58$	0	0
$59$	−4.25755	−0.554286	−0.277143	−	0.960829i	$-0.589388\pi$
$59$	−4.25755	−0.554286	−0.277143	+	0.960829i	$0.589388\pi$
$60$	0	0
$61$	4.92868	0.631053	0.315526	−	0.948917i	$-0.397819\pi$
$61$	4.92868	0.631053	0.315526	+	0.948917i	$0.397819\pi$
$62$	0	0
$63$	3.08139	0.388219
$64$	0	0
$65$	0.765901	0.0949984
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	0	0
$69$	−5.32887	−0.641521
$70$	0	0
$71$	13.8918	1.64866	0.824329	−	0.566111i	$-0.191553\pi$
$71$	13.8918	1.64866	0.824329	+	0.566111i	$0.191553\pi$
$72$	0	0
$73$	−7.84729	−0.918456	−0.459228	−	0.888318i	$-0.651874\pi$
$73$	−7.84729	−0.918456	−0.459228	+	0.888318i	$0.651874\pi$
$74$	0	0
$75$	3.23410	0.373442
$76$	0	0
$77$	4.38681	0.499923
$78$	0	0
$79$	−11.5050	−1.29442	−0.647209	−	0.762313i	$-0.724064\pi$
$79$	−11.5050	−1.29442	−0.647209	+	0.762313i	$0.724064\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−2.25755	−0.247799	−0.123899	−	0.992295i	$-0.539540\pi$
$83$	−2.25755	−0.247799	−0.123899	+	0.992295i	$0.539540\pi$
$84$	0	0
$85$	0.765901	0.0830737
$86$	0	0
$87$	−8.00000	−0.857690
$88$	0	0
$89$	8.92868	0.946438	0.473219	−	0.880945i	$-0.343092\pi$
$89$	8.92868	0.946438	0.473219	+	0.880945i	$0.343092\pi$
$90$	0	0
$91$	1.77597	0.186173
$92$	0	0
$93$	4.50504	0.467150
$94$	0	0
$95$	−8.08139	−0.829133
$96$	0	0
$97$	−0.0813900	−0.00826390	−0.00413195	−	0.999991i	$-0.501315\pi$
$97$	−0.0813900	−0.00826390	−0.00413195	+	0.999991i	$0.501315\pi$
$98$	0	0
$99$	1.42365	0.143082
$100$	0	0
$101$	19.3969	1.93006	0.965031	−	0.262137i	$-0.0844271\pi$
$101$	19.3969	1.93006	0.965031	+	0.262137i	$0.0844271\pi$
$102$	0	0
$103$	−13.2341	−1.30399	−0.651997	−	0.758221i	$-0.726069\pi$
$103$	−13.2341	−1.30399	−0.651997	+	0.758221i	$0.726069\pi$
$104$	0	0
$105$	−4.09477	−0.399609
$106$	0	0
$107$	−8.24417	−0.796994	−0.398497	−	0.917170i	$-0.630468\pi$
$107$	−8.24417	−0.796994	−0.398497	+	0.917170i	$0.630468\pi$
$108$	0	0
$109$	−2.08139	−0.199361	−0.0996805	−	0.995019i	$-0.531782\pi$
$109$	−2.08139	−0.199361	−0.0996805	+	0.995019i	$0.531782\pi$
$110$	0	0
$111$	−11.0814	−1.05180
$112$	0	0
$113$	−4.27094	−0.401776	−0.200888	−	0.979614i	$-0.564383\pi$
$113$	−4.27094	−0.401776	−0.200888	+	0.979614i	$0.564383\pi$
$114$	0	0
$115$	7.08139	0.660343
$116$	0	0
$117$	0.576354	0.0532840
$118$	0	0
$119$	1.77597	0.162803
$120$	0	0
$121$	−8.97323	−0.815748
$122$	0	0
$123$	−2.09477	−0.188879
$124$	0	0
$125$	−10.9421	−0.978688
$126$	0	0
$127$	−9.23410	−0.819394	−0.409697	−	0.912222i	$-0.634366\pi$
$127$	−9.23410	−0.819394	−0.409697	+	0.912222i	$0.634366\pi$
$128$	0	0
$129$	−9.16278	−0.806738
$130$	0	0
$131$	−1.49165	−0.130326	−0.0651631	−	0.997875i	$-0.520757\pi$
$131$	−1.49165	−0.130326	−0.0651631	+	0.997875i	$0.520757\pi$
$132$	0	0
$133$	−18.7391	−1.62489
$134$	0	0
$135$	−1.32887	−0.114371
$136$	0	0
$137$	−2.56297	−0.218969	−0.109485	−	0.993988i	$-0.534920\pi$
$137$	−2.56297	−0.218969	−0.109485	+	0.993988i	$0.534920\pi$
$138$	0	0
$139$	−10.5864	−0.897929	−0.448964	−	0.893550i	$-0.648207\pi$
$139$	−10.5864	−0.897929	−0.448964	+	0.893550i	$0.648207\pi$
$140$	0	0
$141$	8.73913	0.735968
$142$	0	0
$143$	0.820524	0.0686157
$144$	0	0
$145$	10.6310	0.882855
$146$	0	0
$147$	−2.49496	−0.205781
$148$	0	0
$149$	−1.15271	−0.0944336	−0.0472168	−	0.998885i	$-0.515035\pi$
$149$	−1.15271	−0.0944336	−0.0472168	+	0.998885i	$0.515035\pi$
$150$	0	0
$151$	6.90191	0.561670	0.280835	−	0.959756i	$-0.409389\pi$
$151$	6.90191	0.561670	0.280835	+	0.959756i	$0.409389\pi$
$152$	0	0
$153$	0.576354	0.0465955
$154$	0	0
$155$	−5.98662	−0.480857
$156$	0	0
$157$	5.35233	0.427162	0.213581	−	0.976925i	$-0.431487\pi$
$157$	5.35233	0.427162	0.213581	+	0.976925i	$0.431487\pi$
$158$	0	0
$159$	−8.25755	−0.654867
$160$	0	0
$161$	16.4203	1.29410
$162$	0	0
$163$	−13.9732	−1.09447	−0.547234	−	0.836980i	$-0.684319\pi$
$163$	−13.9732	−1.09447	−0.547234	+	0.836980i	$0.684319\pi$
$164$	0	0
$165$	−1.89184	−0.147280
$166$	0	0
$167$	18.5017	1.43171	0.715853	−	0.698251i	$-0.246038\pi$
$167$	18.5017	1.43171	0.715853	+	0.698251i	$0.246038\pi$
$168$	0	0
$169$	−12.6678	−0.974447
$170$	0	0
$171$	−6.08139	−0.465056
$172$	0	0
$173$	−22.0546	−1.67678	−0.838391	−	0.545069i	$-0.816503\pi$
$173$	−22.0546	−1.67678	−0.838391	+	0.545069i	$0.816503\pi$
$174$	0	0
$175$	−9.96552	−0.753322
$176$	0	0
$177$	4.25755	0.320017
$178$	0	0
$179$	−2.76590	−0.206733	−0.103367	−	0.994643i	$-0.532961\pi$
$179$	−2.76590	−0.206733	−0.103367	+	0.994643i	$0.532961\pi$
$180$	0	0
$181$	1.57635	0.117169	0.0585847	−	0.998282i	$-0.481341\pi$
$181$	1.57635	0.117169	0.0585847	+	0.998282i	$0.481341\pi$
$182$	0	0
$183$	−4.92868	−0.364339
$184$	0	0
$185$	14.7258	1.08266
$186$	0	0
$187$	0.820524	0.0600027
$188$	0	0
$189$	−3.08139	−0.224138
$190$	0	0
$191$	8.16278	0.590638	0.295319	−	0.955399i	$-0.404574\pi$
$191$	8.16278	0.590638	0.295319	+	0.955399i	$0.404574\pi$
$192$	0	0
$193$	0.342256	0.0246361	0.0123180	−	0.999924i	$-0.496079\pi$
$193$	0.342256	0.0246361	0.0123180	+	0.999924i	$0.496079\pi$
$194$	0	0
$195$	−0.765901	−0.0548473
$196$	0	0
$197$	−26.3936	−1.88046	−0.940232	−	0.340535i	$-0.889392\pi$
$197$	−26.3936	−1.88046	−0.940232	+	0.340535i	$0.889392\pi$
$198$	0	0
$199$	−11.5050	−0.815570	−0.407785	−	0.913078i	$-0.633699\pi$
$199$	−11.5050	−0.815570	−0.407785	+	0.913078i	$0.633699\pi$
$200$	0	0
$201$	1.00000	0.0705346
$202$	0	0
$203$	24.6511	1.73017
$204$	0	0
$205$	2.78369	0.194421
$206$	0	0
$207$	5.32887	0.370382
$208$	0	0
$209$	−8.65774	−0.598869
$210$	0	0
$211$	−2.82052	−0.194173	−0.0970865	−	0.995276i	$-0.530952\pi$
$211$	−2.82052	−0.194173	−0.0970865	+	0.995276i	$0.530952\pi$
$212$	0	0
$213$	−13.8918	−0.951853
$214$	0	0
$215$	12.1762	0.830407
$216$	0	0
$217$	−13.8818	−0.942356
$218$	0	0
$219$	7.84729	0.530271
$220$	0	0
$221$	0.332184	0.0223451
$222$	0	0
$223$	17.0101	1.13908	0.569539	−	0.821964i	$-0.307122\pi$
$223$	17.0101	1.13908	0.569539	+	0.821964i	$0.307122\pi$
$224$	0	0
$225$	−3.23410	−0.215607
$226$	0	0
$227$	−4.75252	−0.315436	−0.157718	−	0.987484i	$-0.550414\pi$
$227$	−4.75252	−0.315436	−0.157718	+	0.987484i	$0.550414\pi$
$228$	0	0
$229$	19.8105	1.30911	0.654556	−	0.756014i	$-0.272856\pi$
$229$	19.8105	1.30911	0.654556	+	0.756014i	$0.272856\pi$
$230$	0	0
$231$	−4.38681	−0.288631
$232$	0	0
$233$	−16.3389	−1.07040	−0.535200	−	0.844725i	$-0.679764\pi$
$233$	−16.3389	−1.07040	−0.535200	+	0.844725i	$0.679764\pi$
$234$	0	0
$235$	−11.6132	−0.757561
$236$	0	0
$237$	11.5050	0.747332
$238$	0	0
$239$	−13.5050	−0.873568	−0.436784	−	0.899566i	$-0.643883\pi$
$239$	−13.5050	−0.873568	−0.436784	+	0.899566i	$0.643883\pi$
$240$	0	0
$241$	0.260866	0.0168038	0.00840192	−	0.999965i	$-0.497326\pi$
$241$	0.260866	0.0168038	0.00840192	+	0.999965i	$0.497326\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	3.31549	0.211819
$246$	0	0
$247$	−3.50504	−0.223020
$248$	0	0
$249$	2.25755	0.143067
$250$	0	0
$251$	4.98993	0.314962	0.157481	−	0.987522i	$-0.449663\pi$
$251$	4.98993	0.314962	0.157481	+	0.987522i	$0.449663\pi$
$252$	0	0
$253$	7.58643	0.476954
$254$	0	0
$255$	−0.765901	−0.0479626
$256$	0	0
$257$	−24.3256	−1.51739	−0.758693	−	0.651448i	$-0.774162\pi$
$257$	−24.3256	−1.51739	−0.758693	+	0.651448i	$0.774162\pi$
$258$	0	0
$259$	34.1461	2.12173
$260$	0	0
$261$	8.00000	0.495188
$262$	0	0
$263$	10.7525	0.663029	0.331514	−	0.943450i	$-0.392440\pi$
$263$	10.7525	0.663029	0.331514	+	0.943450i	$0.392440\pi$
$264$	0	0
$265$	10.9732	0.674080
$266$	0	0
$267$	−8.92868	−0.546426
$268$	0	0
$269$	11.0915	0.676258	0.338129	−	0.941100i	$-0.390206\pi$
$269$	11.0915	0.676258	0.338129	+	0.941100i	$0.390206\pi$
$270$	0	0
$271$	−19.8306	−1.20462	−0.602311	−	0.798261i	$-0.705753\pi$
$271$	−19.8306	−1.20462	−0.602311	+	0.798261i	$0.705753\pi$
$272$	0	0
$273$	−1.77597	−0.107487
$274$	0	0
$275$	−4.60421	−0.277644
$276$	0	0
$277$	1.49496	0.0898237	0.0449119	−	0.998991i	$-0.485699\pi$
$277$	1.49496	0.0898237	0.0449119	+	0.998991i	$0.485699\pi$
$278$	0	0
$279$	−4.50504	−0.269709
$280$	0	0
$281$	7.23410	0.431550	0.215775	−	0.976443i	$-0.430772\pi$
$281$	7.23410	0.431550	0.215775	+	0.976443i	$0.430772\pi$
$282$	0	0
$283$	−12.3868	−0.736319	−0.368160	−	0.929763i	$-0.620012\pi$
$283$	−12.3868	−0.736319	−0.368160	+	0.929763i	$0.620012\pi$
$284$	0	0
$285$	8.08139	0.478700
$286$	0	0
$287$	6.45481	0.381016
$288$	0	0
$289$	−16.6678	−0.980460
$290$	0	0
$291$	0.0813900	0.00477117
$292$	0	0
$293$	23.3624	1.36485	0.682423	−	0.730958i	$-0.260926\pi$
$293$	23.3624	1.36485	0.682423	+	0.730958i	$0.260926\pi$
$294$	0	0
$295$	−5.65774	−0.329407
$296$	0	0
$297$	−1.42365	−0.0826083
$298$	0	0
$299$	3.07132	0.177619
$300$	0	0
$301$	28.2341	1.62739
$302$	0	0
$303$	−19.3969	−1.11432
$304$	0	0
$305$	6.54959	0.375028
$306$	0	0
$307$	−5.31549	−0.303371	−0.151685	−	0.988429i	$-0.548470\pi$
$307$	−5.31549	−0.303371	−0.151685	+	0.988429i	$0.548470\pi$
$308$	0	0
$309$	13.2341	0.752862
$310$	0	0
$311$	−24.9019	−1.41206	−0.706029	−	0.708183i	$-0.749515\pi$
$311$	−24.9019	−1.41206	−0.706029	+	0.708183i	$0.749515\pi$
$312$	0	0
$313$	11.3969	0.644190	0.322095	−	0.946707i	$-0.395613\pi$
$313$	11.3969	0.644190	0.322095	+	0.946707i	$0.395613\pi$
$314$	0	0
$315$	4.09477	0.230714
$316$	0	0
$317$	−25.0647	−1.40777	−0.703887	−	0.710312i	$-0.748554\pi$
$317$	−25.0647	−1.40777	−0.703887	+	0.710312i	$0.748554\pi$
$318$	0	0
$319$	11.3892	0.637671
$320$	0	0
$321$	8.24417	0.460145
$322$	0	0
$323$	−3.50504	−0.195025
$324$	0	0
$325$	−1.86399	−0.103395
$326$	0	0
$327$	2.08139	0.115101
$328$	0	0
$329$	−26.9287	−1.48463
$330$	0	0
$331$	34.2887	1.88468	0.942339	−	0.334659i	$-0.108621\pi$
$331$	34.2887	1.88468	0.942339	+	0.334659i	$0.108621\pi$
$332$	0	0
$333$	11.0814	0.607256
$334$	0	0
$335$	−1.32887	−0.0726040
$336$	0	0
$337$	15.9388	0.868239	0.434120	−	0.900855i	$-0.357060\pi$
$337$	15.9388	0.868239	0.434120	+	0.900855i	$0.357060\pi$
$338$	0	0
$339$	4.27094	0.231965
$340$	0	0
$341$	−6.41357	−0.347315
$342$	0	0
$343$	−13.8818	−0.749545
$344$	0	0
$345$	−7.08139	−0.381249
$346$	0	0
$347$	16.1628	0.867663	0.433832	−	0.900994i	$-0.357161\pi$
$347$	16.1628	0.867663	0.433832	+	0.900994i	$0.357161\pi$
$348$	0	0
$349$	3.10816	0.166376	0.0831879	−	0.996534i	$-0.473490\pi$
$349$	3.10816	0.166376	0.0831879	+	0.996534i	$0.473490\pi$
$350$	0	0
$351$	−0.576354	−0.0307635
$352$	0	0
$353$	−16.8071	−0.894554	−0.447277	−	0.894396i	$-0.647606\pi$
$353$	−16.8071	−0.894554	−0.447277	+	0.894396i	$0.647606\pi$
$354$	0	0
$355$	18.4605	0.979781
$356$	0	0
$357$	−1.77597	−0.0939944
$358$	0	0
$359$	18.8071	0.992603	0.496301	−	0.868150i	$-0.334691\pi$
$359$	18.8071	0.992603	0.496301	+	0.868150i	$0.334691\pi$
$360$	0	0
$361$	17.9833	0.946490
$362$	0	0
$363$	8.97323	0.470973
$364$	0	0
$365$	−10.4280	−0.545829
$366$	0	0
$367$	−24.9833	−1.30412	−0.652059	−	0.758168i	$-0.726095\pi$
$367$	−24.9833	−1.30412	−0.652059	+	0.758168i	$0.726095\pi$
$368$	0	0
$369$	2.09477	0.109050
$370$	0	0
$371$	25.4447	1.32103
$372$	0	0
$373$	23.3969	1.21144	0.605722	−	0.795676i	$-0.292884\pi$
$373$	23.3969	1.21144	0.605722	+	0.795676i	$0.292884\pi$
$374$	0	0
$375$	10.9421	0.565046
$376$	0	0
$377$	4.61083	0.237470
$378$	0	0
$379$	−18.0915	−0.929296	−0.464648	−	0.885495i	$-0.653819\pi$
$379$	−18.0915	−0.929296	−0.464648	+	0.885495i	$0.653819\pi$
$380$	0	0
$381$	9.23410	0.473077
$382$	0	0
$383$	−13.6477	−0.697364	−0.348682	−	0.937241i	$-0.613371\pi$
$383$	−13.6477	−0.697364	−0.348682	+	0.937241i	$0.613371\pi$
$384$	0	0
$385$	5.82951	0.297099
$386$	0	0
$387$	9.16278	0.465770
$388$	0	0
$389$	−36.5965	−1.85552	−0.927758	−	0.373182i	$-0.878267\pi$
$389$	−36.5965	−1.85552	−0.927758	+	0.373182i	$0.878267\pi$
$390$	0	0
$391$	3.07132	0.155323
$392$	0	0
$393$	1.49165	0.0752439
$394$	0	0
$395$	−15.2887	−0.769259
$396$	0	0
$397$	0.847291	0.0425243	0.0212622	−	0.999774i	$-0.493232\pi$
$397$	0.847291	0.0425243	0.0212622	+	0.999774i	$0.493232\pi$
$398$	0	0
$399$	18.7391	0.938130
$400$	0	0
$401$	18.7727	0.937462	0.468731	−	0.883341i	$-0.344711\pi$
$401$	18.7727	0.937462	0.468731	+	0.883341i	$0.344711\pi$
$402$	0	0
$403$	−2.59650	−0.129341
$404$	0	0
$405$	1.32887	0.0660322
$406$	0	0
$407$	15.7760	0.781986
$408$	0	0
$409$	12.1628	0.601411	0.300705	−	0.953717i	$-0.402778\pi$
$409$	12.1628	0.601411	0.300705	+	0.953717i	$0.402778\pi$
$410$	0	0
$411$	2.56297	0.126422
$412$	0	0
$413$	−13.1192	−0.645553
$414$	0	0
$415$	−3.00000	−0.147264
$416$	0	0
$417$	10.5864	0.518419
$418$	0	0
$419$	26.4750	1.29339	0.646693	−	0.762750i	$-0.276151\pi$
$419$	26.4750	1.29339	0.646693	+	0.762750i	$0.276151\pi$
$420$	0	0
$421$	−8.17285	−0.398320	−0.199160	−	0.979967i	$-0.563821\pi$
$421$	−8.17285	−0.398320	−0.199160	+	0.979967i	$0.563821\pi$
$422$	0	0
$423$	−8.73913	−0.424911
$424$	0	0
$425$	−1.86399	−0.0904166
$426$	0	0
$427$	15.1872	0.734960
$428$	0	0
$429$	−0.820524	−0.0396153
$430$	0	0
$431$	−30.5563	−1.47185	−0.735924	−	0.677064i	$-0.763252\pi$
$431$	−30.5563	−1.47185	−0.735924	+	0.677064i	$0.763252\pi$
$432$	0	0
$433$	−22.0201	−1.05822	−0.529110	−	0.848553i	$-0.677474\pi$
$433$	−22.0201	−1.05822	−0.529110	+	0.848553i	$0.677474\pi$
$434$	0	0
$435$	−10.6310	−0.509716
$436$	0	0
$437$	−32.4070	−1.55023
$438$	0	0
$439$	−31.9934	−1.52696	−0.763480	−	0.645831i	$-0.776511\pi$
$439$	−31.9934	−1.52696	−0.763480	+	0.645831i	$0.776511\pi$
$440$	0	0
$441$	2.49496	0.118808
$442$	0	0
$443$	−2.22403	−0.105667	−0.0528334	−	0.998603i	$-0.516825\pi$
$443$	−2.22403	−0.105667	−0.0528334	+	0.998603i	$0.516825\pi$
$444$	0	0
$445$	11.8651	0.562459
$446$	0	0
$447$	1.15271	0.0545213
$448$	0	0
$449$	3.72906	0.175985	0.0879927	−	0.996121i	$-0.471955\pi$
$449$	3.72906	0.175985	0.0879927	+	0.996121i	$0.471955\pi$
$450$	0	0
$451$	2.98222	0.140427
$452$	0	0
$453$	−6.90191	−0.324280
$454$	0	0
$455$	2.36004	0.110640
$456$	0	0
$457$	−4.68451	−0.219132	−0.109566	−	0.993980i	$-0.534946\pi$
$457$	−4.68451	−0.219132	−0.109566	+	0.993980i	$0.534946\pi$
$458$	0	0
$459$	−0.576354	−0.0269019
$460$	0	0
$461$	−1.19962	−0.0558718	−0.0279359	−	0.999610i	$-0.508893\pi$
$461$	−1.19962	−0.0558718	−0.0279359	+	0.999610i	$0.508893\pi$
$462$	0	0
$463$	−7.57635	−0.352103	−0.176052	−	0.984381i	$-0.556333\pi$
$463$	−7.57635	−0.352103	−0.176052	+	0.984381i	$0.556333\pi$
$464$	0	0
$465$	5.98662	0.277623
$466$	0	0
$467$	25.8918	1.19813	0.599066	−	0.800700i	$-0.295539\pi$
$467$	25.8918	1.19813	0.599066	+	0.800700i	$0.295539\pi$
$468$	0	0
$469$	−3.08139	−0.142285
$470$	0	0
$471$	−5.35233	−0.246622
$472$	0	0
$473$	13.0446	0.599789
$474$	0	0
$475$	19.6678	0.902421
$476$	0	0
$477$	8.25755	0.378087
$478$	0	0
$479$	7.76259	0.354682	0.177341	−	0.984149i	$-0.443250\pi$
$479$	7.76259	0.354682	0.177341	+	0.984149i	$0.443250\pi$
$480$	0	0
$481$	6.38681	0.291213
$482$	0	0
$483$	−16.4203	−0.747151
$484$	0	0
$485$	−0.108157	−0.00491115
$486$	0	0
$487$	25.6310	1.16145	0.580725	−	0.814100i	$-0.302769\pi$
$487$	25.6310	1.16145	0.580725	+	0.814100i	$0.302769\pi$
$488$	0	0
$489$	13.9732	0.631891
$490$	0	0
$491$	28.5362	1.28782	0.643910	−	0.765101i	$-0.277311\pi$
$491$	28.5362	1.28782	0.643910	+	0.765101i	$0.277311\pi$
$492$	0	0
$493$	4.61083	0.207662
$494$	0	0
$495$	1.89184	0.0850320
$496$	0	0
$497$	42.8062	1.92012
$498$	0	0
$499$	37.9732	1.69992	0.849958	−	0.526851i	$-0.176627\pi$
$499$	37.9732	1.69992	0.849958	+	0.526851i	$0.176627\pi$
$500$	0	0
$501$	−18.5017	−0.826596
$502$	0	0
$503$	11.2810	0.502995	0.251498	−	0.967858i	$-0.419077\pi$
$503$	11.2810	0.502995	0.251498	+	0.967858i	$0.419077\pi$
$504$	0	0
$505$	25.7760	1.14702
$506$	0	0
$507$	12.6678	0.562597
$508$	0	0
$509$	−18.3868	−0.814981	−0.407490	−	0.913209i	$-0.633596\pi$
$509$	−18.3868	−0.814981	−0.407490	+	0.913209i	$0.633596\pi$
$510$	0	0
$511$	−24.1806	−1.06969
$512$	0	0
$513$	6.08139	0.268500
$514$	0	0
$515$	−17.5864	−0.774951
$516$	0	0
$517$	−12.4414	−0.547173
$518$	0	0
$519$	22.0546	0.968091
$520$	0	0
$521$	35.5597	1.55790	0.778948	−	0.627088i	$-0.215753\pi$
$521$	35.5597	1.55790	0.778948	+	0.627088i	$0.215753\pi$
$522$	0	0
$523$	−14.2240	−0.621973	−0.310987	−	0.950414i	$-0.600659\pi$
$523$	−14.2240	−0.621973	−0.310987	+	0.950414i	$0.600659\pi$
$524$	0	0
$525$	9.96552	0.434931
$526$	0	0
$527$	−2.59650	−0.113105
$528$	0	0
$529$	5.39688	0.234647
$530$	0	0
$531$	−4.25755	−0.184762
$532$	0	0
$533$	1.20733	0.0522953
$534$	0	0
$535$	−10.9554	−0.473645
$536$	0	0
$537$	2.76590	0.119357
$538$	0	0
$539$	3.55195	0.152993
$540$	0	0
$541$	0.189547	0.00814926	0.00407463	−	0.999992i	$-0.498703\pi$
$541$	0.189547	0.00814926	0.00407463	+	0.999992i	$0.498703\pi$
$542$	0	0
$543$	−1.57635	−0.0676478
$544$	0	0
$545$	−2.76590	−0.118478
$546$	0	0
$547$	−17.7391	−0.758471	−0.379235	−	0.925300i	$-0.623813\pi$
$547$	−17.7391	−0.758471	−0.379235	+	0.925300i	$0.623813\pi$
$548$	0	0
$549$	4.92868	0.210351
$550$	0	0
$551$	−48.6511	−2.07261
$552$	0	0
$553$	−35.4515	−1.50755
$554$	0	0
$555$	−14.7258	−0.625073
$556$	0	0
$557$	9.39688	0.398159	0.199079	−	0.979983i	$-0.436205\pi$
$557$	9.39688	0.398159	0.199079	+	0.979983i	$0.436205\pi$
$558$	0	0
$559$	5.28101	0.223363
$560$	0	0
$561$	−0.820524	−0.0346426
$562$	0	0
$563$	−15.9388	−0.671738	−0.335869	−	0.941909i	$-0.609030\pi$
$563$	−15.9388	−0.671738	−0.335869	+	0.941909i	$0.609030\pi$
$564$	0	0
$565$	−5.67553	−0.238771
$566$	0	0
$567$	3.08139	0.129406
$568$	0	0
$569$	−14.4136	−0.604248	−0.302124	−	0.953269i	$-0.597696\pi$
$569$	−14.4136	−0.604248	−0.302124	+	0.953269i	$0.597696\pi$
$570$	0	0
$571$	17.8651	0.747630	0.373815	−	0.927503i	$-0.378049\pi$
$571$	17.8651	0.747630	0.373815	+	0.927503i	$0.378049\pi$
$572$	0	0
$573$	−8.16278	−0.341005
$574$	0	0
$575$	−17.2341	−0.718712
$576$	0	0
$577$	−7.64767	−0.318377	−0.159188	−	0.987248i	$-0.550888\pi$
$577$	−7.64767	−0.318377	−0.159188	+	0.987248i	$0.550888\pi$
$578$	0	0
$579$	−0.342256	−0.0142237
$580$	0	0
$581$	−6.95640	−0.288600
$582$	0	0
$583$	11.7558	0.486877
$584$	0	0
$585$	0.765901	0.0316661
$586$	0	0
$587$	22.0201	0.908869	0.454434	−	0.890780i	$-0.349841\pi$
$587$	22.0201	0.908869	0.454434	+	0.890780i	$0.349841\pi$
$588$	0	0
$589$	27.3969	1.12887
$590$	0	0
$591$	26.3936	1.08569
$592$	0	0
$593$	−21.5463	−0.884799	−0.442400	−	0.896818i	$-0.645873\pi$
$593$	−21.5463	−0.884799	−0.442400	+	0.896818i	$0.645873\pi$
$594$	0	0
$595$	2.36004	0.0967522
$596$	0	0
$597$	11.5050	0.470870
$598$	0	0
$599$	0.386807	0.0158045	0.00790226	−	0.999969i	$-0.497485\pi$
$599$	0.386807	0.0158045	0.00790226	+	0.999969i	$0.497485\pi$
$600$	0	0
$601$	35.3624	1.44246	0.721231	−	0.692694i	$-0.243577\pi$
$601$	35.3624	1.44246	0.721231	+	0.692694i	$0.243577\pi$
$602$	0	0
$603$	−1.00000	−0.0407231
$604$	0	0
$605$	−11.9243	−0.484791
$606$	0	0
$607$	43.1116	1.74985	0.874923	−	0.484262i	$-0.160912\pi$
$607$	43.1116	1.74985	0.874923	+	0.484262i	$0.160912\pi$
$608$	0	0
$609$	−24.6511	−0.998914
$610$	0	0
$611$	−5.03684	−0.203769
$612$	0	0
$613$	−22.6678	−0.915544	−0.457772	−	0.889070i	$-0.651352\pi$
$613$	−22.6678	−0.915544	−0.457772	+	0.889070i	$0.651352\pi$
$614$	0	0
$615$	−2.78369	−0.112249
$616$	0	0
$617$	2.44034	0.0982444	0.0491222	−	0.998793i	$-0.484358\pi$
$617$	2.44034	0.0982444	0.0491222	+	0.998793i	$0.484358\pi$
$618$	0	0
$619$	1.11823	0.0449454	0.0224727	−	0.999747i	$-0.492846\pi$
$619$	1.11823	0.0449454	0.0224727	+	0.999747i	$0.492846\pi$
$620$	0	0
$621$	−5.32887	−0.213840
$622$	0	0
$623$	27.5127	1.10228
$624$	0	0
$625$	1.62989	0.0651955
$626$	0	0
$627$	8.65774	0.345757
$628$	0	0
$629$	6.38681	0.254659
$630$	0	0
$631$	24.1628	0.961905	0.480953	−	0.876747i	$-0.340291\pi$
$631$	24.1628	0.961905	0.480953	+	0.876747i	$0.340291\pi$
$632$	0	0
$633$	2.82052	0.112106
$634$	0	0
$635$	−12.2709	−0.486957
$636$	0	0
$637$	1.43798	0.0569750
$638$	0	0
$639$	13.8918	0.549553
$640$	0	0
$641$	−31.6277	−1.24922	−0.624609	−	0.780938i	$-0.714742\pi$
$641$	−31.6277	−1.24922	−0.624609	+	0.780938i	$0.714742\pi$
$642$	0	0
$643$	−15.9655	−0.629619	−0.314809	−	0.949155i	$-0.601941\pi$
$643$	−15.9655	−0.629619	−0.314809	+	0.949155i	$0.601941\pi$
$644$	0	0
$645$	−12.1762	−0.479436
$646$	0	0
$647$	−50.1092	−1.97000	−0.984999	−	0.172561i	$-0.944796\pi$
$647$	−50.1092	−1.97000	−0.984999	+	0.172561i	$0.944796\pi$
$648$	0	0
$649$	−6.06125	−0.237925
$650$	0	0
$651$	13.8818	0.544070
$652$	0	0
$653$	−25.8172	−1.01031	−0.505153	−	0.863030i	$-0.668564\pi$
$653$	−25.8172	−1.01031	−0.505153	+	0.863030i	$0.668564\pi$
$654$	0	0
$655$	−1.98222	−0.0774516
$656$	0	0
$657$	−7.84729	−0.306152
$658$	0	0
$659$	31.9866	1.24602	0.623011	−	0.782213i	$-0.285909\pi$
$659$	31.9866	1.24602	0.623011	+	0.782213i	$0.285909\pi$
$660$	0	0
$661$	−14.3322	−0.557457	−0.278729	−	0.960370i	$-0.589913\pi$
$661$	−14.3322	−0.557457	−0.278729	+	0.960370i	$0.589913\pi$
$662$	0	0
$663$	−0.332184	−0.0129010
$664$	0	0
$665$	−24.9019	−0.965655
$666$	0	0
$667$	42.6310	1.65068
$668$	0	0
$669$	−17.0101	−0.657647
$670$	0	0
$671$	7.01670	0.270877
$672$	0	0
$673$	−9.72906	−0.375028	−0.187514	−	0.982262i	$-0.560043\pi$
$673$	−9.72906	−0.375028	−0.187514	+	0.982262i	$0.560043\pi$
$674$	0	0
$675$	3.23410	0.124481
$676$	0	0
$677$	−46.1494	−1.77367	−0.886833	−	0.462091i	$-0.847099\pi$
$677$	−46.1494	−1.77367	−0.886833	+	0.462091i	$0.847099\pi$
$678$	0	0
$679$	−0.250794	−0.00962460
$680$	0	0
$681$	4.75252	0.182117
$682$	0	0
$683$	−41.0647	−1.57130	−0.785648	−	0.618673i	$-0.787671\pi$
$683$	−41.0647	−1.57130	−0.785648	+	0.618673i	$0.787671\pi$
$684$	0	0
$685$	−3.40586	−0.130131
$686$	0	0
$687$	−19.8105	−0.755816
$688$	0	0
$689$	4.75928	0.181314
$690$	0	0
$691$	−30.1092	−1.14541	−0.572705	−	0.819762i	$-0.694106\pi$
$691$	−30.1092	−1.14541	−0.572705	+	0.819762i	$0.694106\pi$
$692$	0	0
$693$	4.38681	0.166641
$694$	0	0
$695$	−14.0680	−0.533630
$696$	0	0
$697$	1.20733	0.0457310
$698$	0	0
$699$	16.3389	0.617996
$700$	0	0
$701$	−0.292034	−0.0110300	−0.00551499	−	0.999985i	$-0.501755\pi$
$701$	−0.292034	−0.0110300	−0.00551499	+	0.999985i	$0.501755\pi$
$702$	0	0
$703$	−67.3903	−2.54167
$704$	0	0
$705$	11.6132	0.437378
$706$	0	0
$707$	59.7693	2.24786
$708$	0	0
$709$	−41.1829	−1.54666	−0.773329	−	0.634005i	$-0.781410\pi$
$709$	−41.1829	−1.54666	−0.773329	+	0.634005i	$0.781410\pi$
$710$	0	0
$711$	−11.5050	−0.431473
$712$	0	0
$713$	−24.0068	−0.899060
$714$	0	0
$715$	1.09037	0.0407776
$716$	0	0
$717$	13.5050	0.504355
$718$	0	0
$719$	−9.41026	−0.350944	−0.175472	−	0.984484i	$-0.556145\pi$
$719$	−9.41026	−0.350944	−0.175472	+	0.984484i	$0.556145\pi$
$720$	0	0
$721$	−40.7794	−1.51870
$722$	0	0
$723$	−0.260866	−0.00970170
$724$	0	0
$725$	−25.8728	−0.960891
$726$	0	0
$727$	18.4515	0.684328	0.342164	−	0.939640i	$-0.388840\pi$
$727$	18.4515	0.684328	0.342164	+	0.939640i	$0.388840\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	5.28101	0.195325
$732$	0	0
$733$	15.3155	0.565690	0.282845	−	0.959166i	$-0.408722\pi$
$733$	15.3155	0.565690	0.282845	+	0.959166i	$0.408722\pi$
$734$	0	0
$735$	−3.31549	−0.122294
$736$	0	0
$737$	−1.42365	−0.0524407
$738$	0	0
$739$	5.13601	0.188931	0.0944656	−	0.995528i	$-0.469886\pi$
$739$	5.13601	0.188931	0.0944656	+	0.995528i	$0.469886\pi$
$740$	0	0
$741$	3.50504	0.128761
$742$	0	0
$743$	−6.34557	−0.232796	−0.116398	−	0.993203i	$-0.537135\pi$
$743$	−6.34557	−0.232796	−0.116398	+	0.993203i	$0.537135\pi$
$744$	0	0
$745$	−1.53180	−0.0561209
$746$	0	0
$747$	−2.25755	−0.0825996
$748$	0	0
$749$	−25.4035	−0.928224
$750$	0	0
$751$	47.6880	1.74016	0.870079	−	0.492912i	$-0.164068\pi$
$751$	47.6880	1.74016	0.870079	+	0.492912i	$0.164068\pi$
$752$	0	0
$753$	−4.98993	−0.181843
$754$	0	0
$755$	9.17176	0.333795
$756$	0	0
$757$	−2.16278	−0.0786076	−0.0393038	−	0.999227i	$-0.512514\pi$
$757$	−2.16278	−0.0786076	−0.0393038	+	0.999227i	$0.512514\pi$
$758$	0	0
$759$	−7.58643	−0.275370
$760$	0	0
$761$	−22.8205	−0.827243	−0.413622	−	0.910449i	$-0.635736\pi$
$761$	−22.8205	−0.827243	−0.413622	+	0.910449i	$0.635736\pi$
$762$	0	0
$763$	−6.41357	−0.232187
$764$	0	0
$765$	0.765901	0.0276912
$766$	0	0
$767$	−2.45386	−0.0886037
$768$	0	0
$769$	10.8128	0.389920	0.194960	−	0.980811i	$-0.437542\pi$
$769$	10.8128	0.389920	0.194960	+	0.980811i	$0.437542\pi$
$770$	0	0
$771$	24.3256	0.876064
$772$	0	0
$773$	−30.8473	−1.10950	−0.554750	−	0.832017i	$-0.687186\pi$
$773$	−30.8473	−1.10950	−0.554750	+	0.832017i	$0.687186\pi$
$774$	0	0
$775$	14.5697	0.523360
$776$	0	0
$777$	−34.1461	−1.22498
$778$	0	0
$779$	−12.7391	−0.456427
$780$	0	0
$781$	19.7771	0.707679
$782$	0	0
$783$	−8.00000	−0.285897
$784$	0	0
$785$	7.11256	0.253858
$786$	0	0
$787$	−0.0368382	−0.00131314	−0.000656570	−	1.00000i	$-0.500209\pi$
$787$	−0.0368382	−0.00131314	−0.000656570		1.00000i	$0.500209\pi$
$788$	0	0
$789$	−10.7525	−0.382800
$790$	0	0
$791$	−13.1604	−0.467931
$792$	0	0
$793$	2.84067	0.100875
$794$	0	0
$795$	−10.9732	−0.389180
$796$	0	0
$797$	16.0077	0.567022	0.283511	−	0.958969i	$-0.408501\pi$
$797$	16.0077	0.567022	0.283511	+	0.958969i	$0.408501\pi$
$798$	0	0
$799$	−5.03684	−0.178191
$800$	0	0
$801$	8.92868	0.315479
$802$	0	0
$803$	−11.1718	−0.394243
$804$	0	0
$805$	21.8205	0.769072
$806$	0	0
$807$	−11.0915	−0.390438
$808$	0	0
$809$	3.92537	0.138009	0.0690043	−	0.997616i	$-0.478018\pi$
$809$	3.92537	0.138009	0.0690043	+	0.997616i	$0.478018\pi$
$810$	0	0
$811$	−12.1996	−0.428387	−0.214193	−	0.976791i	$-0.568712\pi$
$811$	−12.1996	−0.428387	−0.214193	+	0.976791i	$0.568712\pi$
$812$	0	0
$813$	19.8306	0.695489
$814$	0	0
$815$	−18.5686	−0.650431
$816$	0	0
$817$	−55.7224	−1.94948
$818$	0	0
$819$	1.77597	0.0620575
$820$	0	0
$821$	−2.84729	−0.0993712	−0.0496856	−	0.998765i	$-0.515822\pi$
$821$	−2.84729	−0.0993712	−0.0496856	+	0.998765i	$0.515822\pi$
$822$	0	0
$823$	−8.27094	−0.288307	−0.144153	−	0.989555i	$-0.546046\pi$
$823$	−8.27094	−0.288307	−0.144153	+	0.989555i	$0.546046\pi$
$824$	0	0
$825$	4.60421	0.160298
$826$	0	0
$827$	19.0446	0.662244	0.331122	−	0.943588i	$-0.392573\pi$
$827$	19.0446	0.662244	0.331122	+	0.943588i	$0.392573\pi$
$828$	0	0
$829$	−12.9632	−0.450229	−0.225115	−	0.974332i	$-0.572276\pi$
$829$	−12.9632	−0.450229	−0.225115	+	0.974332i	$0.572276\pi$
$830$	0	0
$831$	−1.49496	−0.0518597
$832$	0	0
$833$	1.43798	0.0498232
$834$	0	0
$835$	24.5864	0.850848
$836$	0	0
$837$	4.50504	0.155717
$838$	0	0
$839$	33.2275	1.14714	0.573570	−	0.819157i	$-0.305558\pi$
$839$	33.2275	1.14714	0.573570	+	0.819157i	$0.305558\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	0	0
$843$	−7.23410	−0.249156
$844$	0	0
$845$	−16.8339	−0.579104
$846$	0	0
$847$	−27.6500	−0.950066
$848$	0	0
$849$	12.3868	0.425114
$850$	0	0
$851$	59.0513	2.02425
$852$	0	0
$853$	32.0201	1.09635	0.548174	−	0.836364i	$-0.315323\pi$
$853$	32.0201	1.09635	0.548174	+	0.836364i	$0.315323\pi$
$854$	0	0
$855$	−8.08139	−0.276378
$856$	0	0
$857$	−14.3389	−0.489809	−0.244904	−	0.969547i	$-0.578757\pi$
$857$	−14.3389	−0.489809	−0.244904	+	0.969547i	$0.578757\pi$
$858$	0	0
$859$	−2.25079	−0.0767961	−0.0383981	−	0.999263i	$-0.512225\pi$
$859$	−2.25079	−0.0767961	−0.0383981	+	0.999263i	$0.512225\pi$
$860$	0	0
$861$	−6.45481	−0.219979
$862$	0	0
$863$	9.59981	0.326781	0.163391	−	0.986561i	$-0.447757\pi$
$863$	9.59981	0.326781	0.163391	+	0.986561i	$0.447757\pi$
$864$	0	0
$865$	−29.3078	−0.996494
$866$	0	0
$867$	16.6678	0.566069
$868$	0	0
$869$	−16.3791	−0.555623
$870$	0	0
$871$	−0.576354	−0.0195290
$872$	0	0
$873$	−0.0813900	−0.00275463
$874$	0	0
$875$	−33.7168	−1.13983
$876$	0	0
$877$	20.3054	0.685665	0.342833	−	0.939397i	$-0.388614\pi$
$877$	20.3054	0.685665	0.342833	+	0.939397i	$0.388614\pi$
$878$	0	0
$879$	−23.3624	−0.787994
$880$	0	0
$881$	53.1461	1.79054	0.895269	−	0.445527i	$-0.146984\pi$
$881$	53.1461	1.79054	0.895269	+	0.445527i	$0.146984\pi$
$882$	0	0
$883$	−48.0101	−1.61567	−0.807834	−	0.589410i	$-0.799360\pi$
$883$	−48.0101	−1.61567	−0.807834	+	0.589410i	$0.799360\pi$
$884$	0	0
$885$	5.65774	0.190183
$886$	0	0
$887$	−49.8718	−1.67453	−0.837266	−	0.546796i	$-0.815848\pi$
$887$	−49.8718	−1.67453	−0.837266	+	0.546796i	$0.815848\pi$
$888$	0	0
$889$	−28.4539	−0.954312
$890$	0	0
$891$	1.42365	0.0476939
$892$	0	0
$893$	53.1461	1.77847
$894$	0	0
$895$	−3.67553	−0.122859
$896$	0	0
$897$	−3.07132	−0.102548
$898$	0	0
$899$	−36.0403	−1.20201
$900$	0	0
$901$	4.75928	0.158555
$902$	0	0
$903$	−28.2341	−0.939572
$904$	0	0
$905$	2.09477	0.0696326
$906$	0	0
$907$	49.3624	1.63905	0.819526	−	0.573042i	$-0.194237\pi$
$907$	49.3624	1.63905	0.819526	+	0.573042i	$0.194237\pi$
$908$	0	0
$909$	19.3969	0.643354
$910$	0	0
$911$	40.6166	1.34569	0.672845	−	0.739784i	$-0.265072\pi$
$911$	40.6166	1.34569	0.672845	+	0.739784i	$0.265072\pi$
$912$	0	0
$913$	−3.21396	−0.106366
$914$	0	0
$915$	−6.54959	−0.216523
$916$	0	0
$917$	−4.59636	−0.151785
$918$	0	0
$919$	−24.4716	−0.807245	−0.403623	−	0.914926i	$-0.632249\pi$
$919$	−24.4716	−0.807245	−0.403623	+	0.914926i	$0.632249\pi$
$920$	0	0
$921$	5.31549	0.175151
$922$	0	0
$923$	8.00662	0.263541
$924$	0	0
$925$	−35.8383	−1.17836
$926$	0	0
$927$	−13.2341	−0.434665
$928$	0	0
$929$	45.2542	1.48474	0.742372	−	0.669988i	$-0.233701\pi$
$929$	45.2542	1.48474	0.742372	+	0.669988i	$0.233701\pi$
$930$	0	0
$931$	−15.1729	−0.497270
$932$	0	0
$933$	24.9019	0.815252
$934$	0	0
$935$	1.09037	0.0356590
$936$	0	0
$937$	−37.9120	−1.23853	−0.619265	−	0.785182i	$-0.712569\pi$
$937$	−37.9120	−1.23853	−0.619265	+	0.785182i	$0.712569\pi$
$938$	0	0
$939$	−11.3969	−0.371923
$940$	0	0
$941$	5.53952	0.180583	0.0902915	−	0.995915i	$-0.471220\pi$
$941$	5.53952	0.180583	0.0902915	+	0.995915i	$0.471220\pi$
$942$	0	0
$943$	11.1628	0.363510
$944$	0	0
$945$	−4.09477	−0.133203
$946$	0	0
$947$	10.5285	0.342130	0.171065	−	0.985260i	$-0.445279\pi$
$947$	10.5285	0.342130	0.171065	+	0.985260i	$0.445279\pi$
$948$	0	0
$949$	−4.52282	−0.146817
$950$	0	0
$951$	25.0647	0.812778
$952$	0	0
$953$	−43.8306	−1.41981	−0.709906	−	0.704296i	$-0.751263\pi$
$953$	−43.8306	−1.41981	−0.709906	+	0.704296i	$0.751263\pi$
$954$	0	0
$955$	10.8473	0.351010
$956$	0	0
$957$	−11.3892	−0.368159
$958$	0	0
$959$	−7.89751	−0.255024
$960$	0	0
$961$	−10.7047	−0.345311
$962$	0	0
$963$	−8.24417	−0.265665
$964$	0	0
$965$	0.454814	0.0146410
$966$	0	0
$967$	36.7124	1.18059	0.590295	−	0.807188i	$-0.299011\pi$
$967$	36.7124	1.18059	0.590295	+	0.807188i	$0.299011\pi$
$968$	0	0
$969$	3.50504	0.112598
$970$	0	0
$971$	51.8585	1.66422	0.832108	−	0.554613i	$-0.187134\pi$
$971$	51.8585	1.66422	0.832108	+	0.554613i	$0.187134\pi$
$972$	0	0
$973$	−32.6209	−1.04578
$974$	0	0
$975$	1.86399	0.0596954
$976$	0	0
$977$	9.17285	0.293466	0.146733	−	0.989176i	$-0.453124\pi$
$977$	9.17285	0.293466	0.146733	+	0.989176i	$0.453124\pi$
$978$	0	0
$979$	12.7113	0.406254
$980$	0	0
$981$	−2.08139	−0.0664537
$982$	0	0
$983$	34.4682	1.09936	0.549682	−	0.835374i	$-0.314749\pi$
$983$	34.4682	1.09936	0.549682	+	0.835374i	$0.314749\pi$
$984$	0	0
$985$	−35.0737	−1.11754
$986$	0	0
$987$	26.9287	0.857149
$988$	0	0
$989$	48.8273	1.55262
$990$	0	0
$991$	−37.6856	−1.19712	−0.598561	−	0.801077i	$-0.704261\pi$
$991$	−37.6856	−1.19712	−0.598561	+	0.801077i	$0.704261\pi$
$992$	0	0
$993$	−34.2887	−1.08812
$994$	0	0
$995$	−15.2887	−0.484685
$996$	0	0
$997$	−12.8449	−0.406803	−0.203402	−	0.979095i	$-0.565200\pi$
$997$	−12.8449	−0.406803	−0.203402	+	0.979095i	$0.565200\pi$
$998$	0	0
$999$	−11.0814	−0.350600

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.a.e.1.3	✓	3
3.2	odd	2		2412.2.a.h.1.1		3
4.3	odd	2		3216.2.a.t.1.3		3
12.11	even	2		9648.2.a.br.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.a.e.1.3	✓	3	1.1	even	1	trivial
2412.2.a.h.1.1		3	3.2	odd	2
3216.2.a.t.1.3		3	4.3	odd	2
9648.2.a.br.1.1		3	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	804.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.32887 q^{5} +3.08139 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.32887 q^{5} +3.08139 q^{7} +1.00000 q^{9} +1.42365 q^{11} +0.576354 q^{13} -1.32887 q^{15} +0.576354 q^{17} -6.08139 q^{19} -3.08139 q^{21} +5.32887 q^{23} -3.23410 q^{25} -1.00000 q^{27} +8.00000 q^{29} -4.50504 q^{31} -1.42365 q^{33} +4.09477 q^{35} +11.0814 q^{37} -0.576354 q^{39} +2.09477 q^{41} +9.16278 q^{43} +1.32887 q^{45} -8.73913 q^{47} +2.49496 q^{49} -0.576354 q^{51} +8.25755 q^{53} +1.89184 q^{55} +6.08139 q^{57} -4.25755 q^{59} +4.92868 q^{61} +3.08139 q^{63} +0.765901 q^{65} -1.00000 q^{67} -5.32887 q^{69} +13.8918 q^{71} -7.84729 q^{73} +3.23410 q^{75} +4.38681 q^{77} -11.5050 q^{79} +1.00000 q^{81} -2.25755 q^{83} +0.765901 q^{85} -8.00000 q^{87} +8.92868 q^{89} +1.77597 q^{91} +4.50504 q^{93} -8.08139 q^{95} -0.0813900 q^{97} +1.42365 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{5} - 5 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - 3 * q^5 - 5 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} - 3 q^{5} - 5 q^{7} + 3 q^{9} + 4 q^{11} + 2 q^{13} + 3 q^{15} + 2 q^{17} - 4 q^{19} + 5 q^{21} + 9 q^{23} + 4 q^{25} - 3 q^{27} + 24 q^{29} + q^{31} - 4 q^{33} + 19 q^{35} + 19 q^{37} - 2 q^{39} + 13 q^{41} - q^{43} - 3 q^{45} + 2 q^{47} + 22 q^{49} - 2 q^{51} + 3 q^{53} - 22 q^{55} + 4 q^{57} + 9 q^{59} - 5 q^{63} + 16 q^{65} - 3 q^{67} - 9 q^{69} + 14 q^{71} - 23 q^{73} - 4 q^{75} - 20 q^{79} + 3 q^{81} + 15 q^{83} + 16 q^{85} - 24 q^{87} + 12 q^{89} - 10 q^{91} - q^{93} - 10 q^{95} + 14 q^{97} + 4 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^5 - 5 * q^7 + 3 * q^9 + 4 * q^11 + 2 * q^13 + 3 * q^15 + 2 * q^17 - 4 * q^19 + 5 * q^21 + 9 * q^23 + 4 * q^25 - 3 * q^27 + 24 * q^29 + q^31 - 4 * q^33 + 19 * q^35 + 19 * q^37 - 2 * q^39 + 13 * q^41 - q^43 - 3 * q^45 + 2 * q^47 + 22 * q^49 - 2 * q^51 + 3 * q^53 - 22 * q^55 + 4 * q^57 + 9 * q^59 - 5 * q^63 + 16 * q^65 - 3 * q^67 - 9 * q^69 + 14 * q^71 - 23 * q^73 - 4 * q^75 - 20 * q^79 + 3 * q^81 + 15 * q^83 + 16 * q^85 - 24 * q^87 + 12 * q^89 - 10 * q^91 - q^93 - 10 * q^95 + 14 * q^97 + 4 * q^99

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