LMFDB - Embedded newform 8036.2.a.j.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8036,2,Mod(1,8036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8036, 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("8036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8036 = 2^{2} \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8036.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:	$64.1677830643$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.1935333.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 8x^{3} + 10x^{2} + 13x - 11 $ x^5 - 2x^4 - 8x^3 + 10x^2 + 13x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1148)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.10409$ of defining polynomial
Character	$\chi$	$=$	8036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.10409 q^{3} +1.26228 q^{5} +1.42721 q^{9} +O(q^{10})$	$q+2.10409 q^{3} +1.26228 q^{5} +1.42721 q^{9} -3.91823 q^{11} -2.38693 q^{13} +2.65595 q^{15} +6.14698 q^{17} -5.49103 q^{19} -0.104093 q^{23} -3.40665 q^{25} -3.30930 q^{27} -2.65595 q^{29} -3.26228 q^{31} -8.24433 q^{33} +0.993257 q^{37} -5.02233 q^{39} -1.00000 q^{41} +8.14698 q^{43} +1.80154 q^{45} -1.90302 q^{47} +12.9338 q^{51} +1.26228 q^{53} -4.94591 q^{55} -11.5536 q^{57} -8.24433 q^{59} -6.89507 q^{61} -3.01298 q^{65} +8.69660 q^{67} -0.219022 q^{69} +0.827998 q^{71} -8.44279 q^{73} -7.16791 q^{75} -5.27079 q^{79} -11.2447 q^{81} -6.82800 q^{83} +7.75921 q^{85} -5.58838 q^{87} -10.9212 q^{89} -6.86414 q^{93} -6.93121 q^{95} -8.31228 q^{97} -5.59214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{3} - 3 q^{5} + 5 q^{9}+O(q^{10}) $ 5 * q - 2 * q^3 - 3 * q^5 + 5 * q^9	$ 5 q - 2 q^{3} - 3 q^{5} + 5 q^{9} - 7 q^{13} + 3 q^{15} + 3 q^{17} - 10 q^{19} + 12 q^{23} + 2 q^{25} - 14 q^{27} - 3 q^{29} - 7 q^{31} + 3 q^{33} + q^{37} + 7 q^{39} - 5 q^{41} + 13 q^{43} + 3 q^{45} - 9 q^{47} + 9 q^{51} - 3 q^{53} - 9 q^{55} - 11 q^{57} + 3 q^{59} - 16 q^{61} + 3 q^{65} + 19 q^{67} - 24 q^{69} - 12 q^{71} - 4 q^{73} - 8 q^{75} + 28 q^{79} + 5 q^{81} - 18 q^{83} - 15 q^{85} + 6 q^{87} + q^{93} + 3 q^{95} - 4 q^{97} - 33 q^{99}+O(q^{100}) $ 5 * q - 2 * q^3 - 3 * q^5 + 5 * q^9 - 7 * q^13 + 3 * q^15 + 3 * q^17 - 10 * q^19 + 12 * q^23 + 2 * q^25 - 14 * q^27 - 3 * q^29 - 7 * q^31 + 3 * q^33 + q^37 + 7 * q^39 - 5 * q^41 + 13 * q^43 + 3 * q^45 - 9 * q^47 + 9 * q^51 - 3 * q^53 - 9 * q^55 - 11 * q^57 + 3 * q^59 - 16 * q^61 + 3 * q^65 + 19 * q^67 - 24 * q^69 - 12 * q^71 - 4 * q^73 - 8 * q^75 + 28 * q^79 + 5 * q^81 - 18 * q^83 - 15 * q^85 + 6 * q^87 + q^93 + 3 * q^95 - 4 * q^97 - 33 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.10409	1.21480	0.607399	−	0.794397i	$-0.292213\pi$
$3$	2.10409	1.21480	0.607399	+	0.794397i	$0.292213\pi$
$4$	0	0
$5$	1.26228	0.564509	0.282254	−	0.959340i	$-0.408918\pi$
$5$	1.26228	0.564509	0.282254	+	0.959340i	$0.408918\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.42721	0.475736
$10$	0	0
$11$	−3.91823	−1.18139	−0.590696	−	0.806894i	$-0.701147\pi$
$11$	−3.91823	−1.18139	−0.590696	+	0.806894i	$0.701147\pi$
$12$	0	0
$13$	−2.38693	−0.662016	−0.331008	−	0.943628i	$-0.607389\pi$
$13$	−2.38693	−0.662016	−0.331008	+	0.943628i	$0.607389\pi$
$14$	0	0
$15$	2.65595	0.685764
$16$	0	0
$17$	6.14698	1.49086	0.745431	−	0.666583i	$-0.232244\pi$
$17$	6.14698	1.49086	0.745431	+	0.666583i	$0.232244\pi$
$18$	0	0
$19$	−5.49103	−1.25973	−0.629864	−	0.776706i	$-0.716889\pi$
$19$	−5.49103	−1.25973	−0.629864	+	0.776706i	$0.716889\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.104093	−0.0217050	−0.0108525	−	0.999941i	$-0.503455\pi$
$23$	−0.104093	−0.0217050	−0.0108525	+	0.999941i	$0.503455\pi$
$24$	0	0
$25$	−3.40665	−0.681330
$26$	0	0
$27$	−3.30930	−0.636875
$28$	0	0
$29$	−2.65595	−0.493198	−0.246599	−	0.969118i	$-0.579313\pi$
$29$	−2.65595	−0.493198	−0.246599	+	0.969118i	$0.579313\pi$
$30$	0	0
$31$	−3.26228	−0.585923	−0.292961	−	0.956124i	$-0.594641\pi$
$31$	−3.26228	−0.585923	−0.292961	+	0.956124i	$0.594641\pi$
$32$	0	0
$33$	−8.24433	−1.43515
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.993257	0.163290	0.0816452	−	0.996661i	$-0.473983\pi$
$37$	0.993257	0.163290	0.0816452	+	0.996661i	$0.473983\pi$
$38$	0	0
$39$	−5.02233	−0.804216
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	8.14698	1.24240	0.621201	−	0.783651i	$-0.286645\pi$
$43$	8.14698	1.24240	0.621201	+	0.783651i	$0.286645\pi$
$44$	0	0
$45$	1.80154	0.268557
$46$	0	0
$47$	−1.90302	−0.277584	−0.138792	−	0.990322i	$-0.544322\pi$
$47$	−1.90302	−0.277584	−0.138792	+	0.990322i	$0.544322\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	12.9338	1.81110
$52$	0	0
$53$	1.26228	0.173387	0.0866937	−	0.996235i	$-0.472370\pi$
$53$	1.26228	0.173387	0.0866937	+	0.996235i	$0.472370\pi$
$54$	0	0
$55$	−4.94591	−0.666906
$56$	0	0
$57$	−11.5536	−1.53032
$58$	0	0
$59$	−8.24433	−1.07332	−0.536660	−	0.843798i	$-0.680314\pi$
$59$	−8.24433	−1.07332	−0.536660	+	0.843798i	$0.680314\pi$
$60$	0	0
$61$	−6.89507	−0.882823	−0.441411	−	0.897305i	$-0.645522\pi$
$61$	−6.89507	−0.882823	−0.441411	+	0.897305i	$0.645522\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.01298	−0.373714
$66$	0	0
$67$	8.69660	1.06246	0.531230	−	0.847228i	$-0.321730\pi$
$67$	8.69660	1.06246	0.531230	+	0.847228i	$0.321730\pi$
$68$	0	0
$69$	−0.219022	−0.0263672
$70$	0	0
$71$	0.827998	0.0982653	0.0491326	−	0.998792i	$-0.484354\pi$
$71$	0.827998	0.0982653	0.0491326	+	0.998792i	$0.484354\pi$
$72$	0	0
$73$	−8.44279	−0.988154	−0.494077	−	0.869418i	$-0.664494\pi$
$73$	−8.44279	−0.988154	−0.494077	+	0.869418i	$0.664494\pi$
$74$	0	0
$75$	−7.16791	−0.827679
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−5.27079	−0.593010	−0.296505	−	0.955031i	$-0.595821\pi$
$79$	−5.27079	−0.593010	−0.296505	+	0.955031i	$0.595821\pi$
$80$	0	0
$81$	−11.2447	−1.24941
$82$	0	0
$83$	−6.82800	−0.749470	−0.374735	−	0.927132i	$-0.622266\pi$
$83$	−6.82800	−0.749470	−0.374735	+	0.927132i	$0.622266\pi$
$84$	0	0
$85$	7.75921	0.841604
$86$	0	0
$87$	−5.58838	−0.599137
$88$	0	0
$89$	−10.9212	−1.15765	−0.578823	−	0.815453i	$-0.696488\pi$
$89$	−10.9212	−1.15765	−0.578823	+	0.815453i	$0.696488\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−6.86414	−0.711778
$94$	0	0
$95$	−6.93121	−0.711127
$96$	0	0
$97$	−8.31228	−0.843984	−0.421992	−	0.906600i	$-0.638669\pi$
$97$	−8.31228	−0.843984	−0.421992	+	0.906600i	$0.638669\pi$
$98$	0	0
$99$	−5.59214	−0.562031
$100$	0	0
$101$	−6.13902	−0.610856	−0.305428	−	0.952215i	$-0.598799\pi$
$101$	−6.13902	−0.610856	−0.305428	+	0.952215i	$0.598799\pi$
$102$	0	0
$103$	2.28424	0.225072	0.112536	−	0.993648i	$-0.464103\pi$
$103$	2.28424	0.225072	0.112536	+	0.993648i	$0.464103\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.4995	1.49840	0.749198	−	0.662346i	$-0.230439\pi$
$107$	15.4995	1.49840	0.749198	+	0.662346i	$0.230439\pi$
$108$	0	0
$109$	1.10372	0.105717	0.0528587	−	0.998602i	$-0.483167\pi$
$109$	1.10372	0.105717	0.0528587	+	0.998602i	$0.483167\pi$
$110$	0	0
$111$	2.08991	0.198365
$112$	0	0
$113$	−6.61893	−0.622656	−0.311328	−	0.950302i	$-0.600774\pi$
$113$	−6.61893	−0.622656	−0.311328	+	0.950302i	$0.600774\pi$
$114$	0	0
$115$	−0.131395	−0.0122526
$116$	0	0
$117$	−3.40665	−0.314945
$118$	0	0
$119$	0	0
$120$	0	0
$121$	4.35256	0.395687
$122$	0	0
$123$	−2.10409	−0.189720
$124$	0	0
$125$	−10.6115	−0.949125
$126$	0	0
$127$	−8.58163	−0.761497	−0.380748	−	0.924679i	$-0.624334\pi$
$127$	−8.58163	−0.761497	−0.380748	+	0.924679i	$0.624334\pi$
$128$	0	0
$129$	17.1420	1.50927
$130$	0	0
$131$	−0.751206	−0.0656332	−0.0328166	−	0.999461i	$-0.510448\pi$
$131$	−0.751206	−0.0656332	−0.0328166	+	0.999461i	$0.510448\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−4.17726	−0.359521
$136$	0	0
$137$	−1.17651	−0.100516	−0.0502579	−	0.998736i	$-0.516004\pi$
$137$	−1.17651	−0.100516	−0.0502579	+	0.998736i	$0.516004\pi$
$138$	0	0
$139$	16.8173	1.42643	0.713213	−	0.700948i	$-0.247239\pi$
$139$	16.8173	1.42643	0.713213	+	0.700948i	$0.247239\pi$
$140$	0	0
$141$	−4.00413	−0.337209
$142$	0	0
$143$	9.35256	0.782100
$144$	0	0
$145$	−3.35256	−0.278415
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−17.3431	−1.42080	−0.710400	−	0.703798i	$-0.751486\pi$
$149$	−17.3431	−1.42080	−0.710400	+	0.703798i	$0.751486\pi$
$150$	0	0
$151$	18.5151	1.50674	0.753368	−	0.657599i	$-0.228428\pi$
$151$	18.5151	1.50674	0.753368	+	0.657599i	$0.228428\pi$
$152$	0	0
$153$	8.77302	0.709257
$154$	0	0
$155$	−4.11791	−0.330758
$156$	0	0
$157$	22.3128	1.78076	0.890379	−	0.455220i	$-0.150439\pi$
$157$	22.3128	1.78076	0.890379	+	0.455220i	$0.150439\pi$
$158$	0	0
$159$	2.65595	0.210631
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−0.444887	−0.0348462	−0.0174231	−	0.999848i	$-0.505546\pi$
$163$	−0.444887	−0.0348462	−0.0174231	+	0.999848i	$0.505546\pi$
$164$	0	0
$165$	−10.4067	−0.810157
$166$	0	0
$167$	−12.0924	−0.935736	−0.467868	−	0.883798i	$-0.654978\pi$
$167$	−12.0924	−0.935736	−0.467868	+	0.883798i	$0.654978\pi$
$168$	0	0
$169$	−7.30256	−0.561735
$170$	0	0
$171$	−7.83684	−0.599298
$172$	0	0
$173$	−14.3598	−1.09176	−0.545878	−	0.837865i	$-0.683804\pi$
$173$	−14.3598	−1.09176	−0.545878	+	0.837865i	$0.683804\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−17.3468	−1.30387
$178$	0	0
$179$	13.3740	0.999620	0.499810	−	0.866135i	$-0.333403\pi$
$179$	13.3740	0.999620	0.499810	+	0.866135i	$0.333403\pi$
$180$	0	0
$181$	10.5977	0.787723	0.393861	−	0.919170i	$-0.371139\pi$
$181$	10.5977	0.787723	0.393861	+	0.919170i	$0.371139\pi$
$182$	0	0
$183$	−14.5079	−1.07245
$184$	0	0
$185$	1.25377	0.0921789
$186$	0	0
$187$	−24.0853	−1.76129
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−6.53879	−0.473130	−0.236565	−	0.971616i	$-0.576022\pi$
$191$	−6.53879	−0.473130	−0.236565	+	0.971616i	$0.576022\pi$
$192$	0	0
$193$	−26.3398	−1.89598	−0.947991	−	0.318297i	$-0.896889\pi$
$193$	−26.3398	−1.89598	−0.947991	+	0.318297i	$0.896889\pi$
$194$	0	0
$195$	−6.33958	−0.453987
$196$	0	0
$197$	3.21409	0.228994	0.114497	−	0.993424i	$-0.463474\pi$
$197$	3.21409	0.228994	0.114497	+	0.993424i	$0.463474\pi$
$198$	0	0
$199$	5.75419	0.407903	0.203952	−	0.978981i	$-0.434621\pi$
$199$	5.75419	0.407903	0.203952	+	0.978981i	$0.434621\pi$
$200$	0	0
$201$	18.2985	1.29067
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1.26228	−0.0881614
$206$	0	0
$207$	−0.148563	−0.0103258
$208$	0	0
$209$	21.5151	1.48823
$210$	0	0
$211$	12.3573	0.850709	0.425355	−	0.905027i	$-0.360149\pi$
$211$	12.3573	0.850709	0.425355	+	0.905027i	$0.360149\pi$
$212$	0	0
$213$	1.74218	0.119373
$214$	0	0
$215$	10.2838	0.701347
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−17.7644	−1.20041
$220$	0	0
$221$	−14.6724	−0.986974
$222$	0	0
$223$	−21.7226	−1.45465	−0.727325	−	0.686293i	$-0.759237\pi$
$223$	−21.7226	−1.45465	−0.727325	+	0.686293i	$0.759237\pi$
$224$	0	0
$225$	−4.86200	−0.324133
$226$	0	0
$227$	1.38846	0.0921551	0.0460775	−	0.998938i	$-0.485328\pi$
$227$	1.38846	0.0921551	0.0460775	+	0.998938i	$0.485328\pi$
$228$	0	0
$229$	−14.8059	−0.978402	−0.489201	−	0.872171i	$-0.662712\pi$
$229$	−14.8059	−0.978402	−0.489201	+	0.872171i	$0.662712\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.48963	−0.163101	−0.0815505	−	0.996669i	$-0.525987\pi$
$233$	−2.48963	−0.163101	−0.0815505	+	0.996669i	$0.525987\pi$
$234$	0	0
$235$	−2.40214	−0.156699
$236$	0	0
$237$	−11.0902	−0.720388
$238$	0	0
$239$	−8.69781	−0.562615	−0.281307	−	0.959618i	$-0.590768\pi$
$239$	−8.69781	−0.562615	−0.281307	+	0.959618i	$0.590768\pi$
$240$	0	0
$241$	16.0312	1.03266	0.516329	−	0.856390i	$-0.327298\pi$
$241$	16.0312	1.03266	0.516329	+	0.856390i	$0.327298\pi$
$242$	0	0
$243$	−13.7320	−0.880908
$244$	0	0
$245$	0	0
$246$	0	0
$247$	13.1067	0.833959
$248$	0	0
$249$	−14.3667	−0.910455
$250$	0	0
$251$	−4.04065	−0.255043	−0.127522	−	0.991836i	$-0.540702\pi$
$251$	−4.04065	−0.255043	−0.127522	+	0.991836i	$0.540702\pi$
$252$	0	0
$253$	0.407862	0.0256421
$254$	0	0
$255$	16.3261	1.02238
$256$	0	0
$257$	−27.2076	−1.69716	−0.848581	−	0.529065i	$-0.822543\pi$
$257$	−27.2076	−1.69716	−0.848581	+	0.529065i	$0.822543\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−3.79060	−0.234632
$262$	0	0
$263$	3.66066	0.225726	0.112863	−	0.993611i	$-0.463998\pi$
$263$	3.66066	0.225726	0.112863	+	0.993611i	$0.463998\pi$
$264$	0	0
$265$	1.59335	0.0978787
$266$	0	0
$267$	−22.9793	−1.40631
$268$	0	0
$269$	4.87763	0.297394	0.148697	−	0.988883i	$-0.452492\pi$
$269$	4.87763	0.297394	0.148697	+	0.988883i	$0.452492\pi$
$270$	0	0
$271$	−6.18102	−0.375470	−0.187735	−	0.982220i	$-0.560115\pi$
$271$	−6.18102	−0.375470	−0.187735	+	0.982220i	$0.560115\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	13.3481	0.804918
$276$	0	0
$277$	−2.69122	−0.161699	−0.0808497	−	0.996726i	$-0.525763\pi$
$277$	−2.69122	−0.161699	−0.0808497	+	0.996726i	$0.525763\pi$
$278$	0	0
$279$	−4.65595	−0.278745
$280$	0	0
$281$	−22.2070	−1.32476	−0.662379	−	0.749169i	$-0.730453\pi$
$281$	−22.2070	−1.32476	−0.662379	+	0.749169i	$0.730453\pi$
$282$	0	0
$283$	17.1955	1.02216	0.511082	−	0.859532i	$-0.329245\pi$
$283$	17.1955	1.02216	0.511082	+	0.859532i	$0.329245\pi$
$284$	0	0
$285$	−14.5839	−0.863876
$286$	0	0
$287$	0	0
$288$	0	0
$289$	20.7854	1.22267
$290$	0	0
$291$	−17.4898	−1.02527
$292$	0	0
$293$	−29.3426	−1.71421	−0.857107	−	0.515139i	$-0.827740\pi$
$293$	−29.3426	−1.71421	−0.857107	+	0.515139i	$0.827740\pi$
$294$	0	0
$295$	−10.4067	−0.605899
$296$	0	0
$297$	12.9666	0.752399
$298$	0	0
$299$	0.248464	0.0143690
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−12.9171	−0.742067
$304$	0	0
$305$	−8.70350	−0.498361
$306$	0	0
$307$	−22.4351	−1.28044	−0.640219	−	0.768193i	$-0.721156\pi$
$307$	−22.4351	−1.28044	−0.640219	+	0.768193i	$0.721156\pi$
$308$	0	0
$309$	4.80624	0.273418
$310$	0	0
$311$	32.1337	1.82214	0.911069	−	0.412255i	$-0.135259\pi$
$311$	32.1337	1.82214	0.911069	+	0.412255i	$0.135259\pi$
$312$	0	0
$313$	17.5982	0.994711	0.497356	−	0.867547i	$-0.334304\pi$
$313$	17.5982	0.994711	0.497356	+	0.867547i	$0.334304\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	21.6884	1.21814	0.609070	−	0.793116i	$-0.291543\pi$
$317$	21.6884	1.21814	0.609070	+	0.793116i	$0.291543\pi$
$318$	0	0
$319$	10.4067	0.582661
$320$	0	0
$321$	32.6125	1.82025
$322$	0	0
$323$	−33.7532	−1.87808
$324$	0	0
$325$	8.13144	0.451051
$326$	0	0
$327$	2.32233	0.128425
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−19.3102	−1.06139	−0.530693	−	0.847564i	$-0.678068\pi$
$331$	−19.3102	−1.06139	−0.530693	+	0.847564i	$0.678068\pi$
$332$	0	0
$333$	1.41759	0.0776832
$334$	0	0
$335$	10.9775	0.599767
$336$	0	0
$337$	10.2520	0.558460	0.279230	−	0.960224i	$-0.409921\pi$
$337$	10.2520	0.558460	0.279230	+	0.960224i	$0.409921\pi$
$338$	0	0
$339$	−13.9268	−0.756402
$340$	0	0
$341$	12.7824	0.692204
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−0.276467	−0.0148845
$346$	0	0
$347$	20.3334	1.09155	0.545777	−	0.837931i	$-0.316235\pi$
$347$	20.3334	1.09155	0.545777	+	0.837931i	$0.316235\pi$
$348$	0	0
$349$	−19.4518	−1.04123	−0.520616	−	0.853791i	$-0.674298\pi$
$349$	−19.4518	−1.04123	−0.520616	+	0.853791i	$0.674298\pi$
$350$	0	0
$351$	7.89907	0.421621
$352$	0	0
$353$	19.5318	1.03957	0.519787	−	0.854296i	$-0.326011\pi$
$353$	19.5318	1.03957	0.519787	+	0.854296i	$0.326011\pi$
$354$	0	0
$355$	1.04516	0.0554716
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10.5680	0.557755	0.278878	−	0.960327i	$-0.410038\pi$
$359$	10.5680	0.557755	0.278878	+	0.960327i	$0.410038\pi$
$360$	0	0
$361$	11.1514	0.586914
$362$	0	0
$363$	9.15819	0.480680
$364$	0	0
$365$	−10.6572	−0.557822
$366$	0	0
$367$	−25.9662	−1.35542	−0.677712	−	0.735328i	$-0.737028\pi$
$367$	−25.9662	−1.35542	−0.677712	+	0.735328i	$0.737028\pi$
$368$	0	0
$369$	−1.42721	−0.0742975
$370$	0	0
$371$	0	0
$372$	0	0
$373$	26.6035	1.37748	0.688739	−	0.725010i	$-0.258165\pi$
$373$	26.6035	1.37748	0.688739	+	0.725010i	$0.258165\pi$
$374$	0	0
$375$	−22.3277	−1.15300
$376$	0	0
$377$	6.33958	0.326505
$378$	0	0
$379$	10.3871	0.533550	0.266775	−	0.963759i	$-0.414042\pi$
$379$	10.3871	0.533550	0.266775	+	0.963759i	$0.414042\pi$
$380$	0	0
$381$	−18.0566	−0.925065
$382$	0	0
$383$	28.9970	1.48168	0.740838	−	0.671684i	$-0.234429\pi$
$383$	28.9970	1.48168	0.740838	+	0.671684i	$0.234429\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	11.6274	0.591056
$388$	0	0
$389$	4.94763	0.250855	0.125428	−	0.992103i	$-0.459970\pi$
$389$	4.94763	0.250855	0.125428	+	0.992103i	$0.459970\pi$
$390$	0	0
$391$	−0.639860	−0.0323591
$392$	0	0
$393$	−1.58061	−0.0797311
$394$	0	0
$395$	−6.65321	−0.334759
$396$	0	0
$397$	11.1987	0.562049	0.281025	−	0.959701i	$-0.409326\pi$
$397$	11.1987	0.562049	0.281025	+	0.959701i	$0.409326\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	24.8886	1.24288	0.621439	−	0.783463i	$-0.286548\pi$
$401$	24.8886	1.24288	0.621439	+	0.783463i	$0.286548\pi$
$402$	0	0
$403$	7.78684	0.387890
$404$	0	0
$405$	−14.1940	−0.705303
$406$	0	0
$407$	−3.89181	−0.192910
$408$	0	0
$409$	−24.3927	−1.20614	−0.603070	−	0.797688i	$-0.706056\pi$
$409$	−24.3927	−1.20614	−0.603070	+	0.797688i	$0.706056\pi$
$410$	0	0
$411$	−2.47548	−0.122107
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−8.61884	−0.423082
$416$	0	0
$417$	35.3852	1.73282
$418$	0	0
$419$	8.48070	0.414309	0.207155	−	0.978308i	$-0.433580\pi$
$419$	8.48070	0.414309	0.207155	+	0.978308i	$0.433580\pi$
$420$	0	0
$421$	−2.47164	−0.120460	−0.0602301	−	0.998185i	$-0.519183\pi$
$421$	−2.47164	−0.120460	−0.0602301	+	0.998185i	$0.519183\pi$
$422$	0	0
$423$	−2.71601	−0.132057
$424$	0	0
$425$	−20.9406	−1.01577
$426$	0	0
$427$	0	0
$428$	0	0
$429$	19.6787	0.950094
$430$	0	0
$431$	29.3337	1.41295	0.706477	−	0.707736i	$-0.250283\pi$
$431$	29.3337	1.41295	0.706477	+	0.707736i	$0.250283\pi$
$432$	0	0
$433$	−1.80925	−0.0869471	−0.0434736	−	0.999055i	$-0.513842\pi$
$433$	−1.80925	−0.0869471	−0.0434736	+	0.999055i	$0.513842\pi$
$434$	0	0
$435$	−7.05409	−0.338218
$436$	0	0
$437$	0.571579	0.0273423
$438$	0	0
$439$	25.0164	1.19397	0.596984	−	0.802253i	$-0.296365\pi$
$439$	25.0164	1.19397	0.596984	+	0.802253i	$0.296365\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−22.8846	−1.08728	−0.543639	−	0.839319i	$-0.682954\pi$
$443$	−22.8846	−1.08728	−0.543639	+	0.839319i	$0.682954\pi$
$444$	0	0
$445$	−13.7856	−0.653501
$446$	0	0
$447$	−36.4915	−1.72599
$448$	0	0
$449$	30.6953	1.44860	0.724300	−	0.689484i	$-0.242163\pi$
$449$	30.6953	1.44860	0.724300	+	0.689484i	$0.242163\pi$
$450$	0	0
$451$	3.91823	0.184502
$452$	0	0
$453$	38.9575	1.83038
$454$	0	0
$455$	0	0
$456$	0	0
$457$	37.0577	1.73349	0.866743	−	0.498756i	$-0.166210\pi$
$457$	37.0577	1.73349	0.866743	+	0.498756i	$0.166210\pi$
$458$	0	0
$459$	−20.3422	−0.949492
$460$	0	0
$461$	10.5555	0.491620	0.245810	−	0.969318i	$-0.420946\pi$
$461$	10.5555	0.491620	0.245810	+	0.969318i	$0.420946\pi$
$462$	0	0
$463$	4.95884	0.230457	0.115228	−	0.993339i	$-0.463240\pi$
$463$	4.95884	0.230457	0.115228	+	0.993339i	$0.463240\pi$
$464$	0	0
$465$	−8.66447	−0.401805
$466$	0	0
$467$	4.65999	0.215639	0.107819	−	0.994171i	$-0.465613\pi$
$467$	4.65999	0.215639	0.107819	+	0.994171i	$0.465613\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	46.9483	2.16326
$472$	0	0
$473$	−31.9218	−1.46776
$474$	0	0
$475$	18.7060	0.858290
$476$	0	0
$477$	1.80154	0.0824867
$478$	0	0
$479$	36.2487	1.65625	0.828124	−	0.560545i	$-0.189408\pi$
$479$	36.2487	1.65625	0.828124	+	0.560545i	$0.189408\pi$
$480$	0	0
$481$	−2.37084	−0.108101
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−10.4924	−0.476436
$486$	0	0
$487$	34.0988	1.54516	0.772582	−	0.634915i	$-0.218965\pi$
$487$	34.0988	1.54516	0.772582	+	0.634915i	$0.218965\pi$
$488$	0	0
$489$	−0.936084	−0.0423312
$490$	0	0
$491$	−27.6348	−1.24714	−0.623570	−	0.781767i	$-0.714318\pi$
$491$	−27.6348	−1.24714	−0.623570	+	0.781767i	$0.714318\pi$
$492$	0	0
$493$	−16.3261	−0.735290
$494$	0	0
$495$	−7.05884	−0.317271
$496$	0	0
$497$	0	0
$498$	0	0
$499$	11.2386	0.503109	0.251554	−	0.967843i	$-0.419058\pi$
$499$	11.2386	0.503109	0.251554	+	0.967843i	$0.419058\pi$
$500$	0	0
$501$	−25.4435	−1.13673
$502$	0	0
$503$	−13.9283	−0.621032	−0.310516	−	0.950568i	$-0.600502\pi$
$503$	−13.9283	−0.621032	−0.310516	+	0.950568i	$0.600502\pi$
$504$	0	0
$505$	−7.74917	−0.344833
$506$	0	0
$507$	−15.3653	−0.682395
$508$	0	0
$509$	−8.50363	−0.376917	−0.188458	−	0.982081i	$-0.560349\pi$
$509$	−8.50363	−0.376917	−0.188458	+	0.982081i	$0.560349\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	18.1714	0.802289
$514$	0	0
$515$	2.88334	0.127055
$516$	0	0
$517$	7.45648	0.327936
$518$	0	0
$519$	−30.2144	−1.32626
$520$	0	0
$521$	−16.1843	−0.709046	−0.354523	−	0.935047i	$-0.615357\pi$
$521$	−16.1843	−0.709046	−0.354523	+	0.935047i	$0.615357\pi$
$522$	0	0
$523$	1.38373	0.0605061	0.0302531	−	0.999542i	$-0.490369\pi$
$523$	1.38373	0.0605061	0.0302531	+	0.999542i	$0.490369\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−20.0532	−0.873530
$528$	0	0
$529$	−22.9892	−0.999529
$530$	0	0
$531$	−11.7664	−0.510617
$532$	0	0
$533$	2.38693	0.103389
$534$	0	0
$535$	19.5647	0.845858
$536$	0	0
$537$	28.1401	1.21434
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−44.9258	−1.93151	−0.965756	−	0.259451i	$-0.916458\pi$
$541$	−44.9258	−1.93151	−0.965756	+	0.259451i	$0.916458\pi$
$542$	0	0
$543$	22.2986	0.956925
$544$	0	0
$545$	1.39321	0.0596784
$546$	0	0
$547$	−43.3783	−1.85472	−0.927360	−	0.374170i	$-0.877928\pi$
$547$	−43.3783	−1.85472	−0.927360	+	0.374170i	$0.877928\pi$
$548$	0	0
$549$	−9.84070	−0.419991
$550$	0	0
$551$	14.5839	0.621296
$552$	0	0
$553$	0	0
$554$	0	0
$555$	2.63805	0.111979
$556$	0	0
$557$	−0.217167	−0.00920166	−0.00460083	−	0.999989i	$-0.501464\pi$
$557$	−0.217167	−0.00920166	−0.00460083	+	0.999989i	$0.501464\pi$
$558$	0	0
$559$	−19.4463	−0.822490
$560$	0	0
$561$	−50.6777	−2.13962
$562$	0	0
$563$	4.67810	0.197158	0.0985791	−	0.995129i	$-0.468570\pi$
$563$	4.67810	0.197158	0.0985791	+	0.995129i	$0.468570\pi$
$564$	0	0
$565$	−8.35494	−0.351495
$566$	0	0
$567$	0	0
$568$	0	0
$569$	11.6044	0.486483	0.243241	−	0.969966i	$-0.421789\pi$
$569$	11.6044	0.486483	0.243241	+	0.969966i	$0.421789\pi$
$570$	0	0
$571$	−1.45201	−0.0607646	−0.0303823	−	0.999538i	$-0.509672\pi$
$571$	−1.45201	−0.0607646	−0.0303823	+	0.999538i	$0.509672\pi$
$572$	0	0
$573$	−13.7582	−0.574758
$574$	0	0
$575$	0.354610	0.0147882
$576$	0	0
$577$	−32.2837	−1.34399	−0.671994	−	0.740557i	$-0.734562\pi$
$577$	−32.2837	−1.34399	−0.671994	+	0.740557i	$0.734562\pi$
$578$	0	0
$579$	−55.4215	−2.30324
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−4.94591	−0.204839
$584$	0	0
$585$	−4.30015	−0.177789
$586$	0	0
$587$	−46.4884	−1.91878	−0.959391	−	0.282081i	$-0.908975\pi$
$587$	−46.4884	−1.91878	−0.959391	+	0.282081i	$0.908975\pi$
$588$	0	0
$589$	17.9133	0.738103
$590$	0	0
$591$	6.76274	0.278182
$592$	0	0
$593$	9.63781	0.395777	0.197889	−	0.980225i	$-0.436592\pi$
$593$	9.63781	0.395777	0.197889	+	0.980225i	$0.436592\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	12.1073	0.495521
$598$	0	0
$599$	34.4339	1.40693	0.703466	−	0.710729i	$-0.251635\pi$
$599$	34.4339	1.40693	0.703466	+	0.710729i	$0.251635\pi$
$600$	0	0
$601$	29.4941	1.20309	0.601545	−	0.798839i	$-0.294552\pi$
$601$	29.4941	1.20309	0.601545	+	0.798839i	$0.294552\pi$
$602$	0	0
$603$	12.4119	0.505450
$604$	0	0
$605$	5.49414	0.223369
$606$	0	0
$607$	−11.6734	−0.473809	−0.236904	−	0.971533i	$-0.576133\pi$
$607$	−11.6734	−0.473809	−0.236904	+	0.971533i	$0.576133\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.54238	0.183765
$612$	0	0
$613$	31.0109	1.25252	0.626259	−	0.779615i	$-0.284585\pi$
$613$	31.0109	1.25252	0.626259	+	0.779615i	$0.284585\pi$
$614$	0	0
$615$	−2.65595	−0.107098
$616$	0	0
$617$	17.2455	0.694277	0.347139	−	0.937814i	$-0.387153\pi$
$617$	17.2455	0.694277	0.347139	+	0.937814i	$0.387153\pi$
$618$	0	0
$619$	3.24833	0.130561	0.0652806	−	0.997867i	$-0.479206\pi$
$619$	3.24833	0.130561	0.0652806	+	0.997867i	$0.479206\pi$
$620$	0	0
$621$	0.344476	0.0138233
$622$	0	0
$623$	0	0
$624$	0	0
$625$	3.63851	0.145541
$626$	0	0
$627$	45.2698	1.80790
$628$	0	0
$629$	6.10553	0.243443
$630$	0	0
$631$	4.36377	0.173719	0.0868596	−	0.996221i	$-0.472317\pi$
$631$	4.36377	0.173719	0.0868596	+	0.996221i	$0.472317\pi$
$632$	0	0
$633$	26.0008	1.03344
$634$	0	0
$635$	−10.8324	−0.429872
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1.18173	0.0467483
$640$	0	0
$641$	31.2389	1.23386	0.616931	−	0.787017i	$-0.288376\pi$
$641$	31.2389	1.23386	0.616931	+	0.787017i	$0.288376\pi$
$642$	0	0
$643$	14.5096	0.572202	0.286101	−	0.958199i	$-0.407641\pi$
$643$	14.5096	0.572202	0.286101	+	0.958199i	$0.407641\pi$
$644$	0	0
$645$	21.6380	0.851996
$646$	0	0
$647$	−13.1309	−0.516228	−0.258114	−	0.966114i	$-0.583101\pi$
$647$	−13.1309	−0.516228	−0.258114	+	0.966114i	$0.583101\pi$
$648$	0	0
$649$	32.3032	1.26801
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−43.0021	−1.68280	−0.841401	−	0.540411i	$-0.818269\pi$
$653$	−43.0021	−1.68280	−0.841401	+	0.540411i	$0.818269\pi$
$654$	0	0
$655$	−0.948232	−0.0370505
$656$	0	0
$657$	−12.0496	−0.470101
$658$	0	0
$659$	−28.7392	−1.11952	−0.559760	−	0.828655i	$-0.689107\pi$
$659$	−28.7392	−1.11952	−0.559760	+	0.828655i	$0.689107\pi$
$660$	0	0
$661$	−7.01795	−0.272967	−0.136483	−	0.990642i	$-0.543580\pi$
$661$	−7.01795	−0.272967	−0.136483	+	0.990642i	$0.543580\pi$
$662$	0	0
$663$	−30.8721	−1.19897
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.276467	0.0107049
$668$	0	0
$669$	−45.7063	−1.76711
$670$	0	0
$671$	27.0165	1.04296
$672$	0	0
$673$	−7.21078	−0.277955	−0.138978	−	0.990296i	$-0.544382\pi$
$673$	−7.21078	−0.277955	−0.138978	+	0.990296i	$0.544382\pi$
$674$	0	0
$675$	11.2736	0.433922
$676$	0	0
$677$	19.1112	0.734504	0.367252	−	0.930122i	$-0.380299\pi$
$677$	19.1112	0.734504	0.367252	+	0.930122i	$0.380299\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	2.92144	0.111950
$682$	0	0
$683$	5.29963	0.202785	0.101392	−	0.994847i	$-0.467670\pi$
$683$	5.29963	0.202785	0.101392	+	0.994847i	$0.467670\pi$
$684$	0	0
$685$	−1.48508	−0.0567421
$686$	0	0
$687$	−31.1530	−1.18856
$688$	0	0
$689$	−3.01298	−0.114785
$690$	0	0
$691$	−9.71386	−0.369532	−0.184766	−	0.982782i	$-0.559153\pi$
$691$	−9.71386	−0.369532	−0.184766	+	0.982782i	$0.559153\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	21.2281	0.805229
$696$	0	0
$697$	−6.14698	−0.232833
$698$	0	0
$699$	−5.23841	−0.198135
$700$	0	0
$701$	12.1661	0.459506	0.229753	−	0.973249i	$-0.426208\pi$
$701$	12.1661	0.459506	0.229753	+	0.973249i	$0.426208\pi$
$702$	0	0
$703$	−5.45400	−0.205702
$704$	0	0
$705$	−5.05434	−0.190357
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−26.2297	−0.985076	−0.492538	−	0.870291i	$-0.663931\pi$
$709$	−26.2297	−0.985076	−0.492538	+	0.870291i	$0.663931\pi$
$710$	0	0
$711$	−7.52252	−0.282117
$712$	0	0
$713$	0.339582	0.0127174
$714$	0	0
$715$	11.8055	0.441502
$716$	0	0
$717$	−18.3010	−0.683464
$718$	0	0
$719$	49.3001	1.83858	0.919292	−	0.393576i	$-0.128762\pi$
$719$	49.3001	1.83858	0.919292	+	0.393576i	$0.128762\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	33.7311	1.25447
$724$	0	0
$725$	9.04791	0.336031
$726$	0	0
$727$	−36.6173	−1.35806	−0.679030	−	0.734111i	$-0.737599\pi$
$727$	−36.6173	−1.35806	−0.679030	+	0.734111i	$0.737599\pi$
$728$	0	0
$729$	4.84069	0.179285
$730$	0	0
$731$	50.0793	1.85225
$732$	0	0
$733$	−45.6769	−1.68712	−0.843558	−	0.537038i	$-0.819543\pi$
$733$	−45.6769	−1.68712	−0.843558	+	0.537038i	$0.819543\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−34.0753	−1.25518
$738$	0	0
$739$	−5.34456	−0.196603	−0.0983014	−	0.995157i	$-0.531341\pi$
$739$	−5.34456	−0.196603	−0.0983014	+	0.995157i	$0.531341\pi$
$740$	0	0
$741$	27.5777	1.01309
$742$	0	0
$743$	32.7658	1.20206	0.601031	−	0.799226i	$-0.294757\pi$
$743$	32.7658	1.20206	0.601031	+	0.799226i	$0.294757\pi$
$744$	0	0
$745$	−21.8918	−0.802054
$746$	0	0
$747$	−9.74498	−0.356550
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−46.7261	−1.70506	−0.852529	−	0.522680i	$-0.824932\pi$
$751$	−46.7261	−1.70506	−0.852529	+	0.522680i	$0.824932\pi$
$752$	0	0
$753$	−8.50190	−0.309827
$754$	0	0
$755$	23.3712	0.850565
$756$	0	0
$757$	17.0105	0.618257	0.309129	−	0.951020i	$-0.399963\pi$
$757$	17.0105	0.618257	0.309129	+	0.951020i	$0.399963\pi$
$758$	0	0
$759$	0.858180	0.0311500
$760$	0	0
$761$	0.547716	0.0198547	0.00992735	−	0.999951i	$-0.496840\pi$
$761$	0.547716	0.0198547	0.00992735	+	0.999951i	$0.496840\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	11.0740	0.400382
$766$	0	0
$767$	19.6787	0.710555
$768$	0	0
$769$	2.04223	0.0736447	0.0368224	−	0.999322i	$-0.488276\pi$
$769$	2.04223	0.0736447	0.0368224	+	0.999322i	$0.488276\pi$
$770$	0	0
$771$	−57.2473	−2.06171
$772$	0	0
$773$	−29.5340	−1.06226	−0.531132	−	0.847289i	$-0.678233\pi$
$773$	−29.5340	−1.06226	−0.531132	+	0.847289i	$0.678233\pi$
$774$	0	0
$775$	11.1134	0.399207
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5.49103	0.196736
$780$	0	0
$781$	−3.24429	−0.116090
$782$	0	0
$783$	8.78935	0.314106
$784$	0	0
$785$	28.1650	1.00525
$786$	0	0
$787$	−36.4181	−1.29817	−0.649083	−	0.760717i	$-0.724847\pi$
$787$	−36.4181	−1.29817	−0.649083	+	0.760717i	$0.724847\pi$
$788$	0	0
$789$	7.70237	0.274212
$790$	0	0
$791$	0	0
$792$	0	0
$793$	16.4581	0.584443
$794$	0	0
$795$	3.35256	0.118903
$796$	0	0
$797$	−32.2369	−1.14189	−0.570944	−	0.820989i	$-0.693423\pi$
$797$	−32.2369	−1.14189	−0.570944	+	0.820989i	$0.693423\pi$
$798$	0	0
$799$	−11.6978	−0.413840
$800$	0	0
$801$	−15.5869	−0.550734
$802$	0	0
$803$	33.0808	1.16740
$804$	0	0
$805$	0	0
$806$	0	0
$807$	10.2630	0.361274
$808$	0	0
$809$	−4.23985	−0.149065	−0.0745327	−	0.997219i	$-0.523747\pi$
$809$	−4.23985	−0.149065	−0.0745327	+	0.997219i	$0.523747\pi$
$810$	0	0
$811$	−25.2040	−0.885032	−0.442516	−	0.896761i	$-0.645914\pi$
$811$	−25.2040	−0.885032	−0.442516	+	0.896761i	$0.645914\pi$
$812$	0	0
$813$	−13.0055	−0.456121
$814$	0	0
$815$	−0.561572	−0.0196710
$816$	0	0
$817$	−44.7353	−1.56509
$818$	0	0
$819$	0	0
$820$	0	0
$821$	13.9364	0.486383	0.243191	−	0.969978i	$-0.421806\pi$
$821$	13.9364	0.486383	0.243191	+	0.969978i	$0.421806\pi$
$822$	0	0
$823$	21.9956	0.766718	0.383359	−	0.923599i	$-0.374767\pi$
$823$	21.9956	0.766718	0.383359	+	0.923599i	$0.374767\pi$
$824$	0	0
$825$	28.0855	0.977813
$826$	0	0
$827$	9.75191	0.339107	0.169554	−	0.985521i	$-0.445767\pi$
$827$	9.75191	0.339107	0.169554	+	0.985521i	$0.445767\pi$
$828$	0	0
$829$	5.58019	0.193808	0.0969040	−	0.995294i	$-0.469106\pi$
$829$	5.58019	0.193808	0.0969040	+	0.995294i	$0.469106\pi$
$830$	0	0
$831$	−5.66257	−0.196432
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−15.2640	−0.528231
$836$	0	0
$837$	10.7959	0.373160
$838$	0	0
$839$	39.0940	1.34967	0.674837	−	0.737967i	$-0.264214\pi$
$839$	39.0940	1.34967	0.674837	+	0.737967i	$0.264214\pi$
$840$	0	0
$841$	−21.9459	−0.756755
$842$	0	0
$843$	−46.7255	−1.60931
$844$	0	0
$845$	−9.21787	−0.317104
$846$	0	0
$847$	0	0
$848$	0	0
$849$	36.1808	1.24172
$850$	0	0
$851$	−0.103391	−0.00354421
$852$	0	0
$853$	−33.1285	−1.13430	−0.567149	−	0.823615i	$-0.691953\pi$
$853$	−33.1285	−1.13430	−0.567149	+	0.823615i	$0.691953\pi$
$854$	0	0
$855$	−9.89228	−0.338309
$856$	0	0
$857$	21.1806	0.723515	0.361758	−	0.932272i	$-0.382177\pi$
$857$	21.1806	0.723515	0.361758	+	0.932272i	$0.382177\pi$
$858$	0	0
$859$	−25.2122	−0.860228	−0.430114	−	0.902775i	$-0.641527\pi$
$859$	−25.2122	−0.860228	−0.430114	+	0.902775i	$0.641527\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−2.69042	−0.0915828	−0.0457914	−	0.998951i	$-0.514581\pi$
$863$	−2.69042	−0.0915828	−0.0457914	+	0.998951i	$0.514581\pi$
$864$	0	0
$865$	−18.1261	−0.616306
$866$	0	0
$867$	43.7343	1.48530
$868$	0	0
$869$	20.6522	0.700578
$870$	0	0
$871$	−20.7582	−0.703365
$872$	0	0
$873$	−11.8634	−0.401514
$874$	0	0
$875$	0	0
$876$	0	0
$877$	5.60941	0.189416	0.0947081	−	0.995505i	$-0.469808\pi$
$877$	5.60941	0.189416	0.0947081	+	0.995505i	$0.469808\pi$
$878$	0	0
$879$	−61.7396	−2.08242
$880$	0	0
$881$	−27.6042	−0.930009	−0.465004	−	0.885308i	$-0.653947\pi$
$881$	−27.6042	−0.930009	−0.465004	+	0.885308i	$0.653947\pi$
$882$	0	0
$883$	−1.26479	−0.0425634	−0.0212817	−	0.999774i	$-0.506775\pi$
$883$	−1.26479	−0.0425634	−0.0212817	+	0.999774i	$0.506775\pi$
$884$	0	0
$885$	−21.8966	−0.736045
$886$	0	0
$887$	−59.3033	−1.99121	−0.995605	−	0.0936522i	$-0.970146\pi$
$887$	−59.3033	−1.99121	−0.995605	+	0.0936522i	$0.970146\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	44.0594	1.47604
$892$	0	0
$893$	10.4495	0.349680
$894$	0	0
$895$	16.8817	0.564294
$896$	0	0
$897$	0.522791	0.0174555
$898$	0	0
$899$	8.66447	0.288976
$900$	0	0
$901$	7.75921	0.258497
$902$	0	0
$903$	0	0
$904$	0	0
$905$	13.3773	0.444676
$906$	0	0
$907$	41.5275	1.37890	0.689450	−	0.724334i	$-0.257852\pi$
$907$	41.5275	1.37890	0.689450	+	0.724334i	$0.257852\pi$
$908$	0	0
$909$	−8.76167	−0.290606
$910$	0	0
$911$	−45.8723	−1.51982	−0.759908	−	0.650030i	$-0.774756\pi$
$911$	−45.8723	−1.51982	−0.759908	+	0.650030i	$0.774756\pi$
$912$	0	0
$913$	26.7537	0.885418
$914$	0	0
$915$	−18.3130	−0.605408
$916$	0	0
$917$	0	0
$918$	0	0
$919$	12.3960	0.408905	0.204452	−	0.978876i	$-0.434459\pi$
$919$	12.3960	0.408905	0.204452	+	0.978876i	$0.434459\pi$
$920$	0	0
$921$	−47.2055	−1.55547
$922$	0	0
$923$	−1.97637	−0.0650531
$924$	0	0
$925$	−3.38368	−0.111255
$926$	0	0
$927$	3.26008	0.107075
$928$	0	0
$929$	27.3588	0.897614	0.448807	−	0.893629i	$-0.351849\pi$
$929$	27.3588	0.897614	0.448807	+	0.893629i	$0.351849\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	67.6124	2.21353
$934$	0	0
$935$	−30.4024	−0.994264
$936$	0	0
$937$	50.5181	1.65035	0.825177	−	0.564874i	$-0.191075\pi$
$937$	50.5181	1.65035	0.825177	+	0.564874i	$0.191075\pi$
$938$	0	0
$939$	37.0283	1.20837
$940$	0	0
$941$	59.8655	1.95156	0.975780	−	0.218752i	$-0.0701987\pi$
$941$	59.8655	1.95156	0.975780	+	0.218752i	$0.0701987\pi$
$942$	0	0
$943$	0.104093	0.00338975
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−39.6573	−1.28869	−0.644345	−	0.764735i	$-0.722870\pi$
$947$	−39.6573	−1.28869	−0.644345	+	0.764735i	$0.722870\pi$
$948$	0	0
$949$	20.1524	0.654174
$950$	0	0
$951$	45.6344	1.47980
$952$	0	0
$953$	−16.9581	−0.549326	−0.274663	−	0.961541i	$-0.588566\pi$
$953$	−16.9581	−0.549326	−0.274663	+	0.961541i	$0.588566\pi$
$954$	0	0
$955$	−8.25378	−0.267086
$956$	0	0
$957$	21.8966	0.707815
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−20.3575	−0.656695
$962$	0	0
$963$	22.1211	0.712842
$964$	0	0
$965$	−33.2482	−1.07030
$966$	0	0
$967$	−26.8632	−0.863861	−0.431931	−	0.901907i	$-0.642167\pi$
$967$	−26.8632	−0.863861	−0.431931	+	0.901907i	$0.642167\pi$
$968$	0	0
$969$	−71.0199	−2.28149
$970$	0	0
$971$	45.7252	1.46739	0.733696	−	0.679478i	$-0.237794\pi$
$971$	45.7252	1.46739	0.733696	+	0.679478i	$0.237794\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	17.1093	0.547936
$976$	0	0
$977$	−18.7120	−0.598651	−0.299326	−	0.954151i	$-0.596762\pi$
$977$	−18.7120	−0.598651	−0.299326	+	0.954151i	$0.596762\pi$
$978$	0	0
$979$	42.7919	1.36763
$980$	0	0
$981$	1.57524	0.0502936
$982$	0	0
$983$	19.7136	0.628766	0.314383	−	0.949296i	$-0.398202\pi$
$983$	19.7136	0.628766	0.314383	+	0.949296i	$0.398202\pi$
$984$	0	0
$985$	4.05708	0.129269
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−0.848046	−0.0269663
$990$	0	0
$991$	60.8049	1.93153	0.965766	−	0.259414i	$-0.0835294\pi$
$991$	60.8049	1.93153	0.965766	+	0.259414i	$0.0835294\pi$
$992$	0	0
$993$	−40.6305	−1.28937
$994$	0	0
$995$	7.26339	0.230265
$996$	0	0
$997$	26.5781	0.841738	0.420869	−	0.907121i	$-0.361725\pi$
$997$	26.5781	0.841738	0.420869	+	0.907121i	$0.361725\pi$
$998$	0	0
$999$	−3.28699	−0.103996

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.j.1.5		5
7.6	odd	2		1148.2.a.e.1.1	✓	5
28.27	even	2		4592.2.a.bd.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.a.e.1.1	✓	5	7.6	odd	2
4592.2.a.bd.1.5		5	28.27	even	2
8036.2.a.j.1.5		5	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8036.a (trivial)

\(f(q)\)	\(=\)	\(q+2.10409 q^{3} +1.26228 q^{5} +1.42721 q^{9} +O(q^{10})\)	\(q+2.10409 q^{3} +1.26228 q^{5} +1.42721 q^{9} -3.91823 q^{11} -2.38693 q^{13} +2.65595 q^{15} +6.14698 q^{17} -5.49103 q^{19} -0.104093 q^{23} -3.40665 q^{25} -3.30930 q^{27} -2.65595 q^{29} -3.26228 q^{31} -8.24433 q^{33} +0.993257 q^{37} -5.02233 q^{39} -1.00000 q^{41} +8.14698 q^{43} +1.80154 q^{45} -1.90302 q^{47} +12.9338 q^{51} +1.26228 q^{53} -4.94591 q^{55} -11.5536 q^{57} -8.24433 q^{59} -6.89507 q^{61} -3.01298 q^{65} +8.69660 q^{67} -0.219022 q^{69} +0.827998 q^{71} -8.44279 q^{73} -7.16791 q^{75} -5.27079 q^{79} -11.2447 q^{81} -6.82800 q^{83} +7.75921 q^{85} -5.58838 q^{87} -10.9212 q^{89} -6.86414 q^{93} -6.93121 q^{95} -8.31228 q^{97} -5.59214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{3} - 3 q^{5} + 5 q^{9}+O(q^{10}) \) 5 * q - 2 * q^3 - 3 * q^5 + 5 * q^9	\( 5 q - 2 q^{3} - 3 q^{5} + 5 q^{9} - 7 q^{13} + 3 q^{15} + 3 q^{17} - 10 q^{19} + 12 q^{23} + 2 q^{25} - 14 q^{27} - 3 q^{29} - 7 q^{31} + 3 q^{33} + q^{37} + 7 q^{39} - 5 q^{41} + 13 q^{43} + 3 q^{45} - 9 q^{47} + 9 q^{51} - 3 q^{53} - 9 q^{55} - 11 q^{57} + 3 q^{59} - 16 q^{61} + 3 q^{65} + 19 q^{67} - 24 q^{69} - 12 q^{71} - 4 q^{73} - 8 q^{75} + 28 q^{79} + 5 q^{81} - 18 q^{83} - 15 q^{85} + 6 q^{87} + q^{93} + 3 q^{95} - 4 q^{97} - 33 q^{99}+O(q^{100}) \) 5 * q - 2 * q^3 - 3 * q^5 + 5 * q^9 - 7 * q^13 + 3 * q^15 + 3 * q^17 - 10 * q^19 + 12 * q^23 + 2 * q^25 - 14 * q^27 - 3 * q^29 - 7 * q^31 + 3 * q^33 + q^37 + 7 * q^39 - 5 * q^41 + 13 * q^43 + 3 * q^45 - 9 * q^47 + 9 * q^51 - 3 * q^53 - 9 * q^55 - 11 * q^57 + 3 * q^59 - 16 * q^61 + 3 * q^65 + 19 * q^67 - 24 * q^69 - 12 * q^71 - 4 * q^73 - 8 * q^75 + 28 * q^79 + 5 * q^81 - 18 * q^83 - 15 * q^85 + 6 * q^87 + q^93 + 3 * q^95 - 4 * q^97 - 33 * q^99

Introduction

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