LMFDB - Embedded newform 8044.2.a.b.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8044,2,Mod(1,8044)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8044, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8044.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8044 = 2^{2} \cdot 2011 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8044.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:	$64.2316633859$
Analytic rank:	$0$
Dimension:	$87$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	8044.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.01797 q^{3} +3.94367 q^{5} +1.03981 q^{7} +6.10812 q^{9} +O(q^{10})$	$q-3.01797 q^{3} +3.94367 q^{5} +1.03981 q^{7} +6.10812 q^{9} +2.33777 q^{11} -0.742233 q^{13} -11.9019 q^{15} -7.70341 q^{17} +6.29759 q^{19} -3.13810 q^{21} +0.614708 q^{23} +10.5526 q^{25} -9.38019 q^{27} -8.29213 q^{29} -2.70266 q^{31} -7.05532 q^{33} +4.10065 q^{35} +2.04189 q^{37} +2.24004 q^{39} +0.788974 q^{41} +0.200598 q^{43} +24.0884 q^{45} +4.71293 q^{47} -5.91881 q^{49} +23.2486 q^{51} +0.703196 q^{53} +9.21941 q^{55} -19.0059 q^{57} -0.238843 q^{59} +2.48162 q^{61} +6.35125 q^{63} -2.92713 q^{65} +5.29490 q^{67} -1.85517 q^{69} +9.57977 q^{71} -6.63382 q^{73} -31.8473 q^{75} +2.43083 q^{77} +8.01269 q^{79} +9.98475 q^{81} +8.53477 q^{83} -30.3798 q^{85} +25.0254 q^{87} +16.1392 q^{89} -0.771778 q^{91} +8.15653 q^{93} +24.8357 q^{95} -3.71329 q^{97} +14.2794 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 87 q + 13 q^{3} - 2 q^{5} + 8 q^{7} + 98 q^{9}+O(q^{10}) $ 87 * q + 13 * q^3 - 2 * q^5 + 8 * q^7 + 98 * q^9	$ 87 q + 13 q^{3} - 2 q^{5} + 8 q^{7} + 98 q^{9} + 36 q^{11} - q^{13} + 16 q^{15} + 31 q^{17} + 35 q^{19} - 3 q^{21} + 39 q^{23} + 93 q^{25} + 55 q^{27} - 5 q^{29} + 46 q^{31} + 25 q^{33} + 68 q^{35} - 11 q^{37} + 54 q^{39} + 83 q^{41} + 28 q^{43} - 14 q^{45} + 48 q^{47} + 103 q^{49} + 77 q^{51} + 3 q^{53} + 35 q^{55} + 14 q^{57} + 122 q^{59} - 13 q^{61} + 39 q^{63} + 41 q^{65} + 32 q^{67} - 10 q^{69} + 100 q^{71} + 34 q^{73} + 97 q^{75} + 4 q^{77} + 52 q^{79} + 131 q^{81} + 67 q^{83} - 2 q^{85} + 89 q^{87} + 68 q^{89} + 75 q^{91} + 138 q^{95} + 36 q^{97} + 107 q^{99}+O(q^{100}) $ 87 * q + 13 * q^3 - 2 * q^5 + 8 * q^7 + 98 * q^9 + 36 * q^11 - q^13 + 16 * q^15 + 31 * q^17 + 35 * q^19 - 3 * q^21 + 39 * q^23 + 93 * q^25 + 55 * q^27 - 5 * q^29 + 46 * q^31 + 25 * q^33 + 68 * q^35 - 11 * q^37 + 54 * q^39 + 83 * q^41 + 28 * q^43 - 14 * q^45 + 48 * q^47 + 103 * q^49 + 77 * q^51 + 3 * q^53 + 35 * q^55 + 14 * q^57 + 122 * q^59 - 13 * q^61 + 39 * q^63 + 41 * q^65 + 32 * q^67 - 10 * q^69 + 100 * q^71 + 34 * q^73 + 97 * q^75 + 4 * q^77 + 52 * q^79 + 131 * q^81 + 67 * q^83 - 2 * q^85 + 89 * q^87 + 68 * q^89 + 75 * q^91 + 138 * q^95 + 36 * q^97 + 107 * 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
$3$	−3.01797	−1.74242	−0.871212	−	0.490907i	$-0.836665\pi$
$3$	−3.01797	−1.74242	−0.871212	+	0.490907i	$0.836665\pi$
$4$	0	0
$5$	3.94367	1.76366	0.881832	−	0.471563i	$-0.156310\pi$
$5$	3.94367	1.76366	0.881832	+	0.471563i	$0.156310\pi$
$6$	0	0
$7$	1.03981	0.393009	0.196505	−	0.980503i	$-0.437041\pi$
$7$	1.03981	0.393009	0.196505	+	0.980503i	$0.437041\pi$
$8$	0	0
$9$	6.10812	2.03604
$10$	0	0
$11$	2.33777	0.704865	0.352432	−	0.935837i	$-0.385355\pi$
$11$	2.33777	0.704865	0.352432	+	0.935837i	$0.385355\pi$
$12$	0	0
$13$	−0.742233	−0.205859	−0.102929	−	0.994689i	$-0.532822\pi$
$13$	−0.742233	−0.205859	−0.102929	+	0.994689i	$0.532822\pi$
$14$	0	0
$15$	−11.9019	−3.07305
$16$	0	0
$17$	−7.70341	−1.86835	−0.934176	−	0.356812i	$-0.883864\pi$
$17$	−7.70341	−1.86835	−0.934176	+	0.356812i	$0.883864\pi$
$18$	0	0
$19$	6.29759	1.44477	0.722383	−	0.691493i	$-0.243047\pi$
$19$	6.29759	1.44477	0.722383	+	0.691493i	$0.243047\pi$
$20$	0	0
$21$	−3.13810	−0.684789
$22$	0	0
$23$	0.614708	0.128176	0.0640878	−	0.997944i	$-0.479586\pi$
$23$	0.614708	0.128176	0.0640878	+	0.997944i	$0.479586\pi$
$24$	0	0
$25$	10.5526	2.11051
$26$	0	0
$27$	−9.38019	−1.80522
$28$	0	0
$29$	−8.29213	−1.53981	−0.769905	−	0.638159i	$-0.779696\pi$
$29$	−8.29213	−1.53981	−0.769905	+	0.638159i	$0.779696\pi$
$30$	0	0
$31$	−2.70266	−0.485412	−0.242706	−	0.970100i	$-0.578035\pi$
$31$	−2.70266	−0.485412	−0.242706	+	0.970100i	$0.578035\pi$
$32$	0	0
$33$	−7.05532	−1.22817
$34$	0	0
$35$	4.10065	0.693137
$36$	0	0
$37$	2.04189	0.335684	0.167842	−	0.985814i	$-0.446320\pi$
$37$	2.04189	0.335684	0.167842	+	0.985814i	$0.446320\pi$
$38$	0	0
$39$	2.24004	0.358693
$40$	0	0
$41$	0.788974	0.123217	0.0616085	−	0.998100i	$-0.480377\pi$
$41$	0.788974	0.123217	0.0616085	+	0.998100i	$0.480377\pi$
$42$	0	0
$43$	0.200598	0.0305909	0.0152955	−	0.999883i	$-0.495131\pi$
$43$	0.200598	0.0305909	0.0152955	+	0.999883i	$0.495131\pi$
$44$	0	0
$45$	24.0884	3.59089
$46$	0	0
$47$	4.71293	0.687451	0.343725	−	0.939070i	$-0.388311\pi$
$47$	4.71293	0.687451	0.343725	+	0.939070i	$0.388311\pi$
$48$	0	0
$49$	−5.91881	−0.845544
$50$	0	0
$51$	23.2486	3.25546
$52$	0	0
$53$	0.703196	0.0965914	0.0482957	−	0.998833i	$-0.484621\pi$
$53$	0.703196	0.0965914	0.0482957	+	0.998833i	$0.484621\pi$
$54$	0	0
$55$	9.21941	1.24314
$56$	0	0
$57$	−19.0059	−2.51740
$58$	0	0
$59$	−0.238843	−0.0310947	−0.0155473	−	0.999879i	$-0.504949\pi$
$59$	−0.238843	−0.0310947	−0.0155473	+	0.999879i	$0.504949\pi$
$60$	0	0
$61$	2.48162	0.317739	0.158870	−	0.987300i	$-0.449215\pi$
$61$	2.48162	0.317739	0.158870	+	0.987300i	$0.449215\pi$
$62$	0	0
$63$	6.35125	0.800183
$64$	0	0
$65$	−2.92713	−0.363065
$66$	0	0
$67$	5.29490	0.646874	0.323437	−	0.946250i	$-0.395162\pi$
$67$	5.29490	0.646874	0.323437	+	0.946250i	$0.395162\pi$
$68$	0	0
$69$	−1.85517	−0.223336
$70$	0	0
$71$	9.57977	1.13691	0.568455	−	0.822714i	$-0.307541\pi$
$71$	9.57977	1.13691	0.568455	+	0.822714i	$0.307541\pi$
$72$	0	0
$73$	−6.63382	−0.776430	−0.388215	−	0.921569i	$-0.626908\pi$
$73$	−6.63382	−0.776430	−0.388215	+	0.921569i	$0.626908\pi$
$74$	0	0
$75$	−31.8473	−3.67741
$76$	0	0
$77$	2.43083	0.277018
$78$	0	0
$79$	8.01269	0.901498	0.450749	−	0.892651i	$-0.351157\pi$
$79$	8.01269	0.901498	0.450749	+	0.892651i	$0.351157\pi$
$80$	0	0
$81$	9.98475	1.10942
$82$	0	0
$83$	8.53477	0.936813	0.468406	−	0.883513i	$-0.344828\pi$
$83$	8.53477	0.936813	0.468406	+	0.883513i	$0.344828\pi$
$84$	0	0
$85$	−30.3798	−3.29515
$86$	0	0
$87$	25.0254	2.68300
$88$	0	0
$89$	16.1392	1.71075	0.855375	−	0.518009i	$-0.173327\pi$
$89$	16.1392	1.71075	0.855375	+	0.518009i	$0.173327\pi$
$90$	0	0
$91$	−0.771778	−0.0809043
$92$	0	0
$93$	8.15653	0.845792
$94$	0	0
$95$	24.8357	2.54808
$96$	0	0
$97$	−3.71329	−0.377027	−0.188514	−	0.982071i	$-0.560367\pi$
$97$	−3.71329	−0.377027	−0.188514	+	0.982071i	$0.560367\pi$
$98$	0	0
$99$	14.2794	1.43513
$100$	0	0
$101$	1.48038	0.147303	0.0736515	−	0.997284i	$-0.476535\pi$
$101$	1.48038	0.147303	0.0736515	+	0.997284i	$0.476535\pi$
$102$	0	0
$103$	10.1671	1.00180	0.500898	−	0.865506i	$-0.333003\pi$
$103$	10.1671	1.00180	0.500898	+	0.865506i	$0.333003\pi$
$104$	0	0
$105$	−12.3756	−1.20774
$106$	0	0
$107$	7.02746	0.679370	0.339685	−	0.940539i	$-0.389680\pi$
$107$	7.02746	0.679370	0.339685	+	0.940539i	$0.389680\pi$
$108$	0	0
$109$	15.5967	1.49389	0.746946	−	0.664885i	$-0.231519\pi$
$109$	15.5967	1.49389	0.746946	+	0.664885i	$0.231519\pi$
$110$	0	0
$111$	−6.16235	−0.584904
$112$	0	0
$113$	6.30525	0.593148	0.296574	−	0.955010i	$-0.404156\pi$
$113$	6.30525	0.593148	0.296574	+	0.955010i	$0.404156\pi$
$114$	0	0
$115$	2.42421	0.226059
$116$	0	0
$117$	−4.53365	−0.419136
$118$	0	0
$119$	−8.01005	−0.734280
$120$	0	0
$121$	−5.53482	−0.503166
$122$	0	0
$123$	−2.38110	−0.214696
$124$	0	0
$125$	21.8975	1.95857
$126$	0	0
$127$	6.98080	0.619446	0.309723	−	0.950827i	$-0.399764\pi$
$127$	6.98080	0.619446	0.309723	+	0.950827i	$0.399764\pi$
$128$	0	0
$129$	−0.605399	−0.0533024
$130$	0	0
$131$	16.0519	1.40246	0.701232	−	0.712933i	$-0.252634\pi$
$131$	16.0519	1.40246	0.701232	+	0.712933i	$0.252634\pi$
$132$	0	0
$133$	6.54827	0.567807
$134$	0	0
$135$	−36.9924	−3.18380
$136$	0	0
$137$	−18.6345	−1.59205	−0.796025	−	0.605264i	$-0.793068\pi$
$137$	−18.6345	−1.59205	−0.796025	+	0.605264i	$0.793068\pi$
$138$	0	0
$139$	3.69346	0.313275	0.156637	−	0.987656i	$-0.449935\pi$
$139$	3.69346	0.313275	0.156637	+	0.987656i	$0.449935\pi$
$140$	0	0
$141$	−14.2234	−1.19783
$142$	0	0
$143$	−1.73517	−0.145102
$144$	0	0
$145$	−32.7014	−2.71571
$146$	0	0
$147$	17.8628	1.47329
$148$	0	0
$149$	14.0462	1.15071	0.575354	−	0.817905i	$-0.304864\pi$
$149$	14.0462	1.15071	0.575354	+	0.817905i	$0.304864\pi$
$150$	0	0
$151$	4.93010	0.401205	0.200603	−	0.979673i	$-0.435710\pi$
$151$	4.93010	0.401205	0.200603	+	0.979673i	$0.435710\pi$
$152$	0	0
$153$	−47.0534	−3.80404
$154$	0	0
$155$	−10.6584	−0.856103
$156$	0	0
$157$	0.452914	0.0361465	0.0180732	−	0.999837i	$-0.494247\pi$
$157$	0.452914	0.0361465	0.0180732	+	0.999837i	$0.494247\pi$
$158$	0	0
$159$	−2.12222	−0.168303
$160$	0	0
$161$	0.639177	0.0503742
$162$	0	0
$163$	−20.1917	−1.58153	−0.790767	−	0.612118i	$-0.790318\pi$
$163$	−20.1917	−1.58153	−0.790767	+	0.612118i	$0.790318\pi$
$164$	0	0
$165$	−27.8239	−2.16608
$166$	0	0
$167$	2.50790	0.194067	0.0970334	−	0.995281i	$-0.469065\pi$
$167$	2.50790	0.194067	0.0970334	+	0.995281i	$0.469065\pi$
$168$	0	0
$169$	−12.4491	−0.957622
$170$	0	0
$171$	38.4664	2.94160
$172$	0	0
$173$	−1.59572	−0.121320	−0.0606600	−	0.998158i	$-0.519321\pi$
$173$	−1.59572	−0.121320	−0.0606600	+	0.998158i	$0.519321\pi$
$174$	0	0
$175$	10.9726	0.829451
$176$	0	0
$177$	0.720819	0.0541801
$178$	0	0
$179$	−2.98534	−0.223135	−0.111567	−	0.993757i	$-0.535587\pi$
$179$	−2.98534	−0.223135	−0.111567	+	0.993757i	$0.535587\pi$
$180$	0	0
$181$	−22.9753	−1.70774	−0.853872	−	0.520484i	$-0.825752\pi$
$181$	−22.9753	−1.70774	−0.853872	+	0.520484i	$0.825752\pi$
$182$	0	0
$183$	−7.48946	−0.553637
$184$	0	0
$185$	8.05254	0.592035
$186$	0	0
$187$	−18.0088	−1.31694
$188$	0	0
$189$	−9.75358	−0.709468
$190$	0	0
$191$	−22.6093	−1.63595	−0.817974	−	0.575255i	$-0.804903\pi$
$191$	−22.6093	−1.63595	−0.817974	+	0.575255i	$0.804903\pi$
$192$	0	0
$193$	13.5383	0.974509	0.487255	−	0.873260i	$-0.337998\pi$
$193$	13.5383	0.974509	0.487255	+	0.873260i	$0.337998\pi$
$194$	0	0
$195$	8.83397	0.632614
$196$	0	0
$197$	−12.1000	−0.862086	−0.431043	−	0.902331i	$-0.641854\pi$
$197$	−12.1000	−0.862086	−0.431043	+	0.902331i	$0.641854\pi$
$198$	0	0
$199$	−19.9719	−1.41577	−0.707886	−	0.706327i	$-0.750351\pi$
$199$	−19.9719	−1.41577	−0.707886	+	0.706327i	$0.750351\pi$
$200$	0	0
$201$	−15.9798	−1.12713
$202$	0	0
$203$	−8.62220	−0.605160
$204$	0	0
$205$	3.11146	0.217314
$206$	0	0
$207$	3.75471	0.260970
$208$	0	0
$209$	14.7223	1.01837
$210$	0	0
$211$	−9.69887	−0.667698	−0.333849	−	0.942627i	$-0.608348\pi$
$211$	−9.69887	−0.667698	−0.333849	+	0.942627i	$0.608348\pi$
$212$	0	0
$213$	−28.9114	−1.98098
$214$	0	0
$215$	0.791094	0.0539522
$216$	0	0
$217$	−2.81024	−0.190771
$218$	0	0
$219$	20.0207	1.35287
$220$	0	0
$221$	5.71773	0.384616
$222$	0	0
$223$	−21.5830	−1.44530	−0.722651	−	0.691213i	$-0.757077\pi$
$223$	−21.5830	−1.44530	−0.722651	+	0.691213i	$0.757077\pi$
$224$	0	0
$225$	64.4563	4.29709
$226$	0	0
$227$	17.9660	1.19244	0.596222	−	0.802820i	$-0.296668\pi$
$227$	17.9660	1.19244	0.596222	+	0.802820i	$0.296668\pi$
$228$	0	0
$229$	27.0209	1.78559	0.892796	−	0.450461i	$-0.148741\pi$
$229$	27.0209	1.78559	0.892796	+	0.450461i	$0.148741\pi$
$230$	0	0
$231$	−7.33615	−0.482683
$232$	0	0
$233$	−4.87919	−0.319646	−0.159823	−	0.987146i	$-0.551092\pi$
$233$	−4.87919	−0.319646	−0.159823	+	0.987146i	$0.551092\pi$
$234$	0	0
$235$	18.5862	1.21243
$236$	0	0
$237$	−24.1820	−1.57079
$238$	0	0
$239$	−3.72308	−0.240826	−0.120413	−	0.992724i	$-0.538422\pi$
$239$	−3.72308	−0.240826	−0.120413	+	0.992724i	$0.538422\pi$
$240$	0	0
$241$	16.7962	1.08194	0.540969	−	0.841042i	$-0.318057\pi$
$241$	16.7962	1.08194	0.540969	+	0.841042i	$0.318057\pi$
$242$	0	0
$243$	−1.99306	−0.127855
$244$	0	0
$245$	−23.3418	−1.49126
$246$	0	0
$247$	−4.67428	−0.297418
$248$	0	0
$249$	−25.7576	−1.63232
$250$	0	0
$251$	12.4423	0.785348	0.392674	−	0.919678i	$-0.371550\pi$
$251$	12.4423	0.785348	0.392674	+	0.919678i	$0.371550\pi$
$252$	0	0
$253$	1.43705	0.0903464
$254$	0	0
$255$	91.6851	5.74154
$256$	0	0
$257$	19.5501	1.21950	0.609750	−	0.792594i	$-0.291270\pi$
$257$	19.5501	1.21950	0.609750	+	0.792594i	$0.291270\pi$
$258$	0	0
$259$	2.12317	0.131927
$260$	0	0
$261$	−50.6493	−3.13511
$262$	0	0
$263$	9.00372	0.555193	0.277597	−	0.960698i	$-0.410462\pi$
$263$	9.00372	0.555193	0.277597	+	0.960698i	$0.410462\pi$
$264$	0	0
$265$	2.77317	0.170355
$266$	0	0
$267$	−48.7075	−2.98085
$268$	0	0
$269$	13.6401	0.831654	0.415827	−	0.909444i	$-0.363492\pi$
$269$	13.6401	0.831654	0.415827	+	0.909444i	$0.363492\pi$
$270$	0	0
$271$	6.99815	0.425107	0.212554	−	0.977149i	$-0.431822\pi$
$271$	6.99815	0.425107	0.212554	+	0.977149i	$0.431822\pi$
$272$	0	0
$273$	2.32920	0.140970
$274$	0	0
$275$	24.6695	1.48763
$276$	0	0
$277$	11.3737	0.683379	0.341690	−	0.939813i	$-0.389001\pi$
$277$	11.3737	0.683379	0.341690	+	0.939813i	$0.389001\pi$
$278$	0	0
$279$	−16.5081	−0.988317
$280$	0	0
$281$	−8.27576	−0.493691	−0.246845	−	0.969055i	$-0.579394\pi$
$281$	−8.27576	−0.493691	−0.246845	+	0.969055i	$0.579394\pi$
$282$	0	0
$283$	−25.9365	−1.54176	−0.770882	−	0.636978i	$-0.780184\pi$
$283$	−25.9365	−1.54176	−0.770882	+	0.636978i	$0.780184\pi$
$284$	0	0
$285$	−74.9531	−4.43984
$286$	0	0
$287$	0.820379	0.0484255
$288$	0	0
$289$	42.3426	2.49074
$290$	0	0
$291$	11.2066	0.656941
$292$	0	0
$293$	6.67954	0.390223	0.195112	−	0.980781i	$-0.437493\pi$
$293$	6.67954	0.390223	0.195112	+	0.980781i	$0.437493\pi$
$294$	0	0
$295$	−0.941918	−0.0548406
$296$	0	0
$297$	−21.9288	−1.27244
$298$	0	0
$299$	−0.456257	−0.0263860
$300$	0	0
$301$	0.208583	0.0120225
$302$	0	0
$303$	−4.46772	−0.256664
$304$	0	0
$305$	9.78672	0.560386
$306$	0	0
$307$	6.72372	0.383743	0.191871	−	0.981420i	$-0.438544\pi$
$307$	6.72372	0.383743	0.191871	+	0.981420i	$0.438544\pi$
$308$	0	0
$309$	−30.6840	−1.74555
$310$	0	0
$311$	2.84714	0.161446	0.0807232	−	0.996737i	$-0.474277\pi$
$311$	2.84714	0.161446	0.0807232	+	0.996737i	$0.474277\pi$
$312$	0	0
$313$	15.2502	0.861991	0.430996	−	0.902354i	$-0.358162\pi$
$313$	15.2502	0.861991	0.430996	+	0.902354i	$0.358162\pi$
$314$	0	0
$315$	25.0473	1.41125
$316$	0	0
$317$	18.7496	1.05308	0.526541	−	0.850150i	$-0.323489\pi$
$317$	18.7496	1.05308	0.526541	+	0.850150i	$0.323489\pi$
$318$	0	0
$319$	−19.3851	−1.08536
$320$	0	0
$321$	−21.2086	−1.18375
$322$	0	0
$323$	−48.5130	−2.69933
$324$	0	0
$325$	−7.83247	−0.434467
$326$	0	0
$327$	−47.0703	−2.60299
$328$	0	0
$329$	4.90052	0.270175
$330$	0	0
$331$	23.9057	1.31398	0.656988	−	0.753901i	$-0.271830\pi$
$331$	23.9057	1.31398	0.656988	+	0.753901i	$0.271830\pi$
$332$	0	0
$333$	12.4721	0.683466
$334$	0	0
$335$	20.8813	1.14087
$336$	0	0
$337$	−14.8582	−0.809376	−0.404688	−	0.914455i	$-0.632620\pi$
$337$	−14.8582	−0.809376	−0.404688	+	0.914455i	$0.632620\pi$
$338$	0	0
$339$	−19.0290	−1.03352
$340$	0	0
$341$	−6.31820	−0.342149
$342$	0	0
$343$	−13.4330	−0.725316
$344$	0	0
$345$	−7.31618	−0.393890
$346$	0	0
$347$	−17.5811	−0.943805	−0.471903	−	0.881651i	$-0.656433\pi$
$347$	−17.5811	−0.943805	−0.471903	+	0.881651i	$0.656433\pi$
$348$	0	0
$349$	4.25625	0.227832	0.113916	−	0.993490i	$-0.463661\pi$
$349$	4.25625	0.227832	0.113916	+	0.993490i	$0.463661\pi$
$350$	0	0
$351$	6.96229	0.371620
$352$	0	0
$353$	−6.74807	−0.359164	−0.179582	−	0.983743i	$-0.557474\pi$
$353$	−6.74807	−0.359164	−0.179582	+	0.983743i	$0.557474\pi$
$354$	0	0
$355$	37.7795	2.00513
$356$	0	0
$357$	24.1741	1.27943
$358$	0	0
$359$	15.9082	0.839600	0.419800	−	0.907617i	$-0.362100\pi$
$359$	15.9082	0.839600	0.419800	+	0.907617i	$0.362100\pi$
$360$	0	0
$361$	20.6597	1.08735
$362$	0	0
$363$	16.7039	0.876728
$364$	0	0
$365$	−26.1616	−1.36936
$366$	0	0
$367$	27.0114	1.40998	0.704991	−	0.709216i	$-0.250951\pi$
$367$	27.0114	1.40998	0.704991	+	0.709216i	$0.250951\pi$
$368$	0	0
$369$	4.81915	0.250875
$370$	0	0
$371$	0.731187	0.0379613
$372$	0	0
$373$	2.83150	0.146610	0.0733048	−	0.997310i	$-0.476645\pi$
$373$	2.83150	0.146610	0.0733048	+	0.997310i	$0.476645\pi$
$374$	0	0
$375$	−66.0859	−3.41266
$376$	0	0
$377$	6.15469	0.316983
$378$	0	0
$379$	19.4042	0.996727	0.498363	−	0.866968i	$-0.333935\pi$
$379$	19.4042	0.996727	0.498363	+	0.866968i	$0.333935\pi$
$380$	0	0
$381$	−21.0678	−1.07934
$382$	0	0
$383$	−7.47163	−0.381783	−0.190891	−	0.981611i	$-0.561138\pi$
$383$	−7.47163	−0.381783	−0.190891	+	0.981611i	$0.561138\pi$
$384$	0	0
$385$	9.58639	0.488568
$386$	0	0
$387$	1.22528	0.0622844
$388$	0	0
$389$	8.88635	0.450556	0.225278	−	0.974295i	$-0.427671\pi$
$389$	8.88635	0.450556	0.225278	+	0.974295i	$0.427671\pi$
$390$	0	0
$391$	−4.73535	−0.239477
$392$	0	0
$393$	−48.4442	−2.44369
$394$	0	0
$395$	31.5994	1.58994
$396$	0	0
$397$	−32.7467	−1.64351	−0.821755	−	0.569841i	$-0.807005\pi$
$397$	−32.7467	−1.64351	−0.821755	+	0.569841i	$0.807005\pi$
$398$	0	0
$399$	−19.7625	−0.989360
$400$	0	0
$401$	8.88462	0.443677	0.221838	−	0.975083i	$-0.428794\pi$
$401$	8.88462	0.443677	0.221838	+	0.975083i	$0.428794\pi$
$402$	0	0
$403$	2.00600	0.0999261
$404$	0	0
$405$	39.3766	1.95664
$406$	0	0
$407$	4.77347	0.236612
$408$	0	0
$409$	−6.31144	−0.312081	−0.156040	−	0.987751i	$-0.549873\pi$
$409$	−6.31144	−0.312081	−0.156040	+	0.987751i	$0.549873\pi$
$410$	0	0
$411$	56.2382	2.77402
$412$	0	0
$413$	−0.248350	−0.0122205
$414$	0	0
$415$	33.6583	1.65222
$416$	0	0
$417$	−11.1467	−0.545858
$418$	0	0
$419$	14.2314	0.695250	0.347625	−	0.937634i	$-0.386988\pi$
$419$	14.2314	0.695250	0.347625	+	0.937634i	$0.386988\pi$
$420$	0	0
$421$	20.5349	1.00081	0.500405	−	0.865792i	$-0.333185\pi$
$421$	20.5349	1.00081	0.500405	+	0.865792i	$0.333185\pi$
$422$	0	0
$423$	28.7871	1.39968
$424$	0	0
$425$	−81.2908	−3.94318
$426$	0	0
$427$	2.58041	0.124875
$428$	0	0
$429$	5.23669	0.252830
$430$	0	0
$431$	−3.72578	−0.179465	−0.0897324	−	0.995966i	$-0.528601\pi$
$431$	−3.72578	−0.179465	−0.0897324	+	0.995966i	$0.528601\pi$
$432$	0	0
$433$	11.4207	0.548842	0.274421	−	0.961610i	$-0.411514\pi$
$433$	11.4207	0.548842	0.274421	+	0.961610i	$0.411514\pi$
$434$	0	0
$435$	98.6918	4.73191
$436$	0	0
$437$	3.87118	0.185184
$438$	0	0
$439$	−8.39766	−0.400799	−0.200399	−	0.979714i	$-0.564224\pi$
$439$	−8.39766	−0.400799	−0.200399	+	0.979714i	$0.564224\pi$
$440$	0	0
$441$	−36.1528	−1.72156
$442$	0	0
$443$	9.12589	0.433584	0.216792	−	0.976218i	$-0.430441\pi$
$443$	9.12589	0.433584	0.216792	+	0.976218i	$0.430441\pi$
$444$	0	0
$445$	63.6477	3.01719
$446$	0	0
$447$	−42.3909	−2.00502
$448$	0	0
$449$	2.42159	0.114282	0.0571409	−	0.998366i	$-0.481802\pi$
$449$	2.42159	0.114282	0.0571409	+	0.998366i	$0.481802\pi$
$450$	0	0
$451$	1.84444	0.0868514
$452$	0	0
$453$	−14.8789	−0.699070
$454$	0	0
$455$	−3.04364	−0.142688
$456$	0	0
$457$	5.19183	0.242864	0.121432	−	0.992600i	$-0.461251\pi$
$457$	5.19183	0.242864	0.121432	+	0.992600i	$0.461251\pi$
$458$	0	0
$459$	72.2595	3.37279
$460$	0	0
$461$	22.5637	1.05090	0.525448	−	0.850826i	$-0.323898\pi$
$461$	22.5637	1.05090	0.525448	+	0.850826i	$0.323898\pi$
$462$	0	0
$463$	−24.3307	−1.13074	−0.565371	−	0.824836i	$-0.691267\pi$
$463$	−24.3307	−1.13074	−0.565371	+	0.824836i	$0.691267\pi$
$464$	0	0
$465$	32.1667	1.49169
$466$	0	0
$467$	5.54742	0.256704	0.128352	−	0.991729i	$-0.459031\pi$
$467$	5.54742	0.256704	0.128352	+	0.991729i	$0.459031\pi$
$468$	0	0
$469$	5.50566	0.254228
$470$	0	0
$471$	−1.36688	−0.0629825
$472$	0	0
$473$	0.468953	0.0215625
$474$	0	0
$475$	66.4557	3.04920
$476$	0	0
$477$	4.29520	0.196664
$478$	0	0
$479$	31.5749	1.44269	0.721346	−	0.692575i	$-0.243524\pi$
$479$	31.5749	1.44269	0.721346	+	0.692575i	$0.243524\pi$
$480$	0	0
$481$	−1.51556	−0.0691035
$482$	0	0
$483$	−1.92901	−0.0877732
$484$	0	0
$485$	−14.6440	−0.664950
$486$	0	0
$487$	−18.9419	−0.858339	−0.429169	−	0.903224i	$-0.641194\pi$
$487$	−18.9419	−0.858339	−0.429169	+	0.903224i	$0.641194\pi$
$488$	0	0
$489$	60.9378	2.75570
$490$	0	0
$491$	16.6827	0.752878	0.376439	−	0.926441i	$-0.377148\pi$
$491$	16.6827	0.752878	0.376439	+	0.926441i	$0.377148\pi$
$492$	0	0
$493$	63.8777	2.87691
$494$	0	0
$495$	56.3132	2.53109
$496$	0	0
$497$	9.96110	0.446816
$498$	0	0
$499$	26.7591	1.19790	0.598952	−	0.800785i	$-0.295584\pi$
$499$	26.7591	1.19790	0.598952	+	0.800785i	$0.295584\pi$
$500$	0	0
$501$	−7.56874	−0.338147
$502$	0	0
$503$	15.0080	0.669172	0.334586	−	0.942365i	$-0.391404\pi$
$503$	15.0080	0.669172	0.334586	+	0.942365i	$0.391404\pi$
$504$	0	0
$505$	5.83812	0.259793
$506$	0	0
$507$	37.5709	1.66858
$508$	0	0
$509$	−43.6273	−1.93375	−0.966873	−	0.255259i	$-0.917839\pi$
$509$	−43.6273	−1.93375	−0.966873	+	0.255259i	$0.917839\pi$
$510$	0	0
$511$	−6.89788	−0.305144
$512$	0	0
$513$	−59.0726	−2.60812
$514$	0	0
$515$	40.0958	1.76683
$516$	0	0
$517$	11.0177	0.484560
$518$	0	0
$519$	4.81581	0.211391
$520$	0	0
$521$	28.9432	1.26803	0.634013	−	0.773323i	$-0.281407\pi$
$521$	28.9432	1.26803	0.634013	+	0.773323i	$0.281407\pi$
$522$	0	0
$523$	4.29967	0.188011	0.0940056	−	0.995572i	$-0.470033\pi$
$523$	4.29967	0.188011	0.0940056	+	0.995572i	$0.470033\pi$
$524$	0	0
$525$	−33.1150	−1.44526
$526$	0	0
$527$	20.8197	0.906920
$528$	0	0
$529$	−22.6221	−0.983571
$530$	0	0
$531$	−1.45888	−0.0633100
$532$	0	0
$533$	−0.585603	−0.0253653
$534$	0	0
$535$	27.7140	1.19818
$536$	0	0
$537$	9.00966	0.388795
$538$	0	0
$539$	−13.8368	−0.595994
$540$	0	0
$541$	2.95064	0.126858	0.0634289	−	0.997986i	$-0.479796\pi$
$541$	2.95064	0.126858	0.0634289	+	0.997986i	$0.479796\pi$
$542$	0	0
$543$	69.3388	2.97561
$544$	0	0
$545$	61.5083	2.63472
$546$	0	0
$547$	−41.3119	−1.76637	−0.883184	−	0.469027i	$-0.844605\pi$
$547$	−41.3119	−1.76637	−0.883184	+	0.469027i	$0.844605\pi$
$548$	0	0
$549$	15.1581	0.646930
$550$	0	0
$551$	−52.2204	−2.22467
$552$	0	0
$553$	8.33164	0.354297
$554$	0	0
$555$	−24.3023	−1.03157
$556$	0	0
$557$	−32.6660	−1.38410	−0.692052	−	0.721848i	$-0.743293\pi$
$557$	−32.6660	−1.38410	−0.692052	+	0.721848i	$0.743293\pi$
$558$	0	0
$559$	−0.148891	−0.00629741
$560$	0	0
$561$	54.3500	2.29466
$562$	0	0
$563$	−13.9593	−0.588314	−0.294157	−	0.955757i	$-0.595039\pi$
$563$	−13.9593	−0.588314	−0.294157	+	0.955757i	$0.595039\pi$
$564$	0	0
$565$	24.8659	1.04611
$566$	0	0
$567$	10.3822	0.436011
$568$	0	0
$569$	−28.1415	−1.17975	−0.589877	−	0.807493i	$-0.700824\pi$
$569$	−28.1415	−1.17975	−0.589877	+	0.807493i	$0.700824\pi$
$570$	0	0
$571$	4.38726	0.183601	0.0918006	−	0.995777i	$-0.470738\pi$
$571$	4.38726	0.183601	0.0918006	+	0.995777i	$0.470738\pi$
$572$	0	0
$573$	68.2340	2.85051
$574$	0	0
$575$	6.48675	0.270516
$576$	0	0
$577$	33.2340	1.38355	0.691775	−	0.722113i	$-0.256829\pi$
$577$	33.2340	1.38355	0.691775	+	0.722113i	$0.256829\pi$
$578$	0	0
$579$	−40.8582	−1.69801
$580$	0	0
$581$	8.87450	0.368176
$582$	0	0
$583$	1.64391	0.0680838
$584$	0	0
$585$	−17.8792	−0.739215
$586$	0	0
$587$	−38.0806	−1.57176	−0.785878	−	0.618382i	$-0.787789\pi$
$587$	−38.0806	−1.57176	−0.785878	+	0.618382i	$0.787789\pi$
$588$	0	0
$589$	−17.0202	−0.701307
$590$	0	0
$591$	36.5173	1.50212
$592$	0	0
$593$	−17.8566	−0.733283	−0.366641	−	0.930362i	$-0.619492\pi$
$593$	−17.8566	−0.733283	−0.366641	+	0.930362i	$0.619492\pi$
$594$	0	0
$595$	−31.5890	−1.29502
$596$	0	0
$597$	60.2746	2.46687
$598$	0	0
$599$	43.0882	1.76054	0.880268	−	0.474478i	$-0.157363\pi$
$599$	43.0882	1.76054	0.880268	+	0.474478i	$0.157363\pi$
$600$	0	0
$601$	24.9312	1.01696	0.508482	−	0.861073i	$-0.330207\pi$
$601$	24.9312	1.01696	0.508482	+	0.861073i	$0.330207\pi$
$602$	0	0
$603$	32.3418	1.31706
$604$	0	0
$605$	−21.8275	−0.887416
$606$	0	0
$607$	−10.9578	−0.444764	−0.222382	−	0.974960i	$-0.571383\pi$
$607$	−10.9578	−0.444764	−0.222382	+	0.974960i	$0.571383\pi$
$608$	0	0
$609$	26.0215	1.05444
$610$	0	0
$611$	−3.49809	−0.141518
$612$	0	0
$613$	6.85359	0.276814	0.138407	−	0.990375i	$-0.455802\pi$
$613$	6.85359	0.276814	0.138407	+	0.990375i	$0.455802\pi$
$614$	0	0
$615$	−9.39027	−0.378652
$616$	0	0
$617$	−32.6871	−1.31593	−0.657966	−	0.753047i	$-0.728583\pi$
$617$	−32.6871	−1.31593	−0.657966	+	0.753047i	$0.728583\pi$
$618$	0	0
$619$	34.2749	1.37762	0.688812	−	0.724940i	$-0.258133\pi$
$619$	34.2749	1.37762	0.688812	+	0.724940i	$0.258133\pi$
$620$	0	0
$621$	−5.76608	−0.231385
$622$	0	0
$623$	16.7816	0.672341
$624$	0	0
$625$	33.5938	1.34375
$626$	0	0
$627$	−44.4315	−1.77442
$628$	0	0
$629$	−15.7295	−0.627176
$630$	0	0
$631$	7.66630	0.305190	0.152595	−	0.988289i	$-0.451237\pi$
$631$	7.66630	0.305190	0.152595	+	0.988289i	$0.451237\pi$
$632$	0	0
$633$	29.2709	1.16341
$634$	0	0
$635$	27.5300	1.09249
$636$	0	0
$637$	4.39314	0.174062
$638$	0	0
$639$	58.5144	2.31479
$640$	0	0
$641$	25.1882	0.994876	0.497438	−	0.867500i	$-0.334274\pi$
$641$	25.1882	0.994876	0.497438	+	0.867500i	$0.334274\pi$
$642$	0	0
$643$	−2.67779	−0.105602	−0.0528010	−	0.998605i	$-0.516815\pi$
$643$	−2.67779	−0.105602	−0.0528010	+	0.998605i	$0.516815\pi$
$644$	0	0
$645$	−2.38750	−0.0940075
$646$	0	0
$647$	41.1642	1.61833	0.809166	−	0.587580i	$-0.199919\pi$
$647$	41.1642	1.61833	0.809166	+	0.587580i	$0.199919\pi$
$648$	0	0
$649$	−0.558360	−0.0219175
$650$	0	0
$651$	8.48120	0.332404
$652$	0	0
$653$	12.8364	0.502327	0.251163	−	0.967945i	$-0.419187\pi$
$653$	12.8364	0.502327	0.251163	+	0.967945i	$0.419187\pi$
$654$	0	0
$655$	63.3036	2.47348
$656$	0	0
$657$	−40.5202	−1.58084
$658$	0	0
$659$	−29.5576	−1.15140	−0.575700	−	0.817661i	$-0.695270\pi$
$659$	−29.5576	−1.15140	−0.575700	+	0.817661i	$0.695270\pi$
$660$	0	0
$661$	−30.3569	−1.18075	−0.590373	−	0.807131i	$-0.701019\pi$
$661$	−30.3569	−1.18075	−0.590373	+	0.807131i	$0.701019\pi$
$662$	0	0
$663$	−17.2559	−0.670164
$664$	0	0
$665$	25.8242	1.00142
$666$	0	0
$667$	−5.09724	−0.197366
$668$	0	0
$669$	65.1367	2.51833
$670$	0	0
$671$	5.80147	0.223963
$672$	0	0
$673$	43.6491	1.68255	0.841275	−	0.540608i	$-0.181806\pi$
$673$	43.6491	1.68255	0.841275	+	0.540608i	$0.181806\pi$
$674$	0	0
$675$	−98.9851	−3.80994
$676$	0	0
$677$	−36.7113	−1.41093	−0.705465	−	0.708744i	$-0.749262\pi$
$677$	−36.7113	−1.41093	−0.705465	+	0.708744i	$0.749262\pi$
$678$	0	0
$679$	−3.86110	−0.148175
$680$	0	0
$681$	−54.2207	−2.07774
$682$	0	0
$683$	−32.2895	−1.23552	−0.617762	−	0.786365i	$-0.711961\pi$
$683$	−32.2895	−1.23552	−0.617762	+	0.786365i	$0.711961\pi$
$684$	0	0
$685$	−73.4883	−2.80784
$686$	0	0
$687$	−81.5482	−3.11126
$688$	0	0
$689$	−0.521935	−0.0198842
$690$	0	0
$691$	27.7401	1.05528	0.527642	−	0.849467i	$-0.323076\pi$
$691$	27.7401	1.05528	0.527642	+	0.849467i	$0.323076\pi$
$692$	0	0
$693$	14.8478	0.564021
$694$	0	0
$695$	14.5658	0.552512
$696$	0	0
$697$	−6.07779	−0.230213
$698$	0	0
$699$	14.7252	0.556959
$700$	0	0
$701$	33.2360	1.25531	0.627653	−	0.778493i	$-0.284016\pi$
$701$	33.2360	1.25531	0.627653	+	0.778493i	$0.284016\pi$
$702$	0	0
$703$	12.8590	0.484986
$704$	0	0
$705$	−56.0926	−2.11257
$706$	0	0
$707$	1.53930	0.0578914
$708$	0	0
$709$	−23.7375	−0.891479	−0.445739	−	0.895163i	$-0.647059\pi$
$709$	−23.7375	−0.891479	−0.445739	+	0.895163i	$0.647059\pi$
$710$	0	0
$711$	48.9425	1.83549
$712$	0	0
$713$	−1.66135	−0.0622179
$714$	0	0
$715$	−6.84295	−0.255912
$716$	0	0
$717$	11.2361	0.419621
$718$	0	0
$719$	1.58319	0.0590428	0.0295214	−	0.999564i	$-0.490602\pi$
$719$	1.58319	0.0590428	0.0295214	+	0.999564i	$0.490602\pi$
$720$	0	0
$721$	10.5718	0.393715
$722$	0	0
$723$	−50.6904	−1.88520
$724$	0	0
$725$	−87.5032	−3.24979
$726$	0	0
$727$	−44.5539	−1.65241	−0.826207	−	0.563367i	$-0.809506\pi$
$727$	−44.5539	−1.65241	−0.826207	+	0.563367i	$0.809506\pi$
$728$	0	0
$729$	−23.9393	−0.886640
$730$	0	0
$731$	−1.54529	−0.0571547
$732$	0	0
$733$	−44.2362	−1.63390	−0.816950	−	0.576709i	$-0.804337\pi$
$733$	−44.2362	−1.63390	−0.816950	+	0.576709i	$0.804337\pi$
$734$	0	0
$735$	70.4449	2.59840
$736$	0	0
$737$	12.3783	0.455959
$738$	0	0
$739$	−25.4857	−0.937508	−0.468754	−	0.883329i	$-0.655297\pi$
$739$	−25.4857	−0.937508	−0.468754	+	0.883329i	$0.655297\pi$
$740$	0	0
$741$	14.1068	0.518227
$742$	0	0
$743$	2.43037	0.0891616	0.0445808	−	0.999006i	$-0.485805\pi$
$743$	2.43037	0.0891616	0.0445808	+	0.999006i	$0.485805\pi$
$744$	0	0
$745$	55.3936	2.02946
$746$	0	0
$747$	52.1314	1.90739
$748$	0	0
$749$	7.30719	0.266999
$750$	0	0
$751$	7.29913	0.266349	0.133174	−	0.991093i	$-0.457483\pi$
$751$	7.29913	0.266349	0.133174	+	0.991093i	$0.457483\pi$
$752$	0	0
$753$	−37.5503	−1.36841
$754$	0	0
$755$	19.4427	0.707592
$756$	0	0
$757$	−24.0332	−0.873500	−0.436750	−	0.899583i	$-0.643871\pi$
$757$	−24.0332	−0.873500	−0.436750	+	0.899583i	$0.643871\pi$
$758$	0	0
$759$	−4.33696	−0.157422
$760$	0	0
$761$	−28.6982	−1.04031	−0.520154	−	0.854072i	$-0.674126\pi$
$761$	−28.6982	−1.04031	−0.520154	+	0.854072i	$0.674126\pi$
$762$	0	0
$763$	16.2175	0.587114
$764$	0	0
$765$	−185.563	−6.70905
$766$	0	0
$767$	0.177277	0.00640111
$768$	0	0
$769$	16.2773	0.586975	0.293487	−	0.955963i	$-0.405184\pi$
$769$	16.2773	0.586975	0.293487	+	0.955963i	$0.405184\pi$
$770$	0	0
$771$	−59.0014	−2.12488
$772$	0	0
$773$	−52.7934	−1.89885	−0.949423	−	0.313999i	$-0.898331\pi$
$773$	−52.7934	−1.89885	−0.949423	+	0.313999i	$0.898331\pi$
$774$	0	0
$775$	−28.5200	−1.02447
$776$	0	0
$777$	−6.40764	−0.229873
$778$	0	0
$779$	4.96864	0.178020
$780$	0	0
$781$	22.3953	0.801368
$782$	0	0
$783$	77.7818	2.77969
$784$	0	0
$785$	1.78615	0.0637503
$786$	0	0
$787$	−3.96188	−0.141226	−0.0706130	−	0.997504i	$-0.522496\pi$
$787$	−3.96188	−0.141226	−0.0706130	+	0.997504i	$0.522496\pi$
$788$	0	0
$789$	−27.1729	−0.967382
$790$	0	0
$791$	6.55624	0.233113
$792$	0	0
$793$	−1.84194	−0.0654094
$794$	0	0
$795$	−8.36935	−0.296830
$796$	0	0
$797$	9.36585	0.331756	0.165878	−	0.986146i	$-0.446954\pi$
$797$	9.36585	0.331756	0.165878	+	0.986146i	$0.446954\pi$
$798$	0	0
$799$	−36.3056	−1.28440
$800$	0	0
$801$	98.5801	3.48315
$802$	0	0
$803$	−15.5084	−0.547278
$804$	0	0
$805$	2.52071	0.0888432
$806$	0	0
$807$	−41.1655	−1.44909
$808$	0	0
$809$	5.56473	0.195645	0.0978227	−	0.995204i	$-0.468812\pi$
$809$	5.56473	0.195645	0.0978227	+	0.995204i	$0.468812\pi$
$810$	0	0
$811$	0.515335	0.0180958	0.00904792	−	0.999959i	$-0.497120\pi$
$811$	0.515335	0.0180958	0.00904792	+	0.999959i	$0.497120\pi$
$812$	0	0
$813$	−21.1202	−0.740717
$814$	0	0
$815$	−79.6293	−2.78929
$816$	0	0
$817$	1.26329	0.0441968
$818$	0	0
$819$	−4.71411	−0.164724
$820$	0	0
$821$	−37.1502	−1.29655	−0.648275	−	0.761406i	$-0.724509\pi$
$821$	−37.1502	−1.29655	−0.648275	+	0.761406i	$0.724509\pi$
$822$	0	0
$823$	5.11425	0.178272	0.0891358	−	0.996019i	$-0.471589\pi$
$823$	5.11425	0.178272	0.0891358	+	0.996019i	$0.471589\pi$
$824$	0	0
$825$	−74.4517	−2.59207
$826$	0	0
$827$	−14.6376	−0.509000	−0.254500	−	0.967073i	$-0.581911\pi$
$827$	−14.6376	−0.509000	−0.254500	+	0.967073i	$0.581911\pi$
$828$	0	0
$829$	5.85919	0.203498	0.101749	−	0.994810i	$-0.467556\pi$
$829$	5.85919	0.203498	0.101749	+	0.994810i	$0.467556\pi$
$830$	0	0
$831$	−34.3254	−1.19074
$832$	0	0
$833$	45.5950	1.57977
$834$	0	0
$835$	9.89032	0.342269
$836$	0	0
$837$	25.3515	0.876274
$838$	0	0
$839$	18.7825	0.648443	0.324221	−	0.945981i	$-0.394898\pi$
$839$	18.7825	0.648443	0.324221	+	0.945981i	$0.394898\pi$
$840$	0	0
$841$	39.7594	1.37101
$842$	0	0
$843$	24.9760	0.860218
$844$	0	0
$845$	−49.0951	−1.68892
$846$	0	0
$847$	−5.75514	−0.197749
$848$	0	0
$849$	78.2755	2.68641
$850$	0	0
$851$	1.25517	0.0430265
$852$	0	0
$853$	32.8748	1.12561	0.562806	−	0.826589i	$-0.309722\pi$
$853$	32.8748	1.12561	0.562806	+	0.826589i	$0.309722\pi$
$854$	0	0
$855$	151.699	5.18800
$856$	0	0
$857$	43.0184	1.46948	0.734741	−	0.678348i	$-0.237304\pi$
$857$	43.0184	1.46948	0.734741	+	0.678348i	$0.237304\pi$
$858$	0	0
$859$	13.6003	0.464037	0.232019	−	0.972711i	$-0.425467\pi$
$859$	13.6003	0.464037	0.232019	+	0.972711i	$0.425467\pi$
$860$	0	0
$861$	−2.47588	−0.0843777
$862$	0	0
$863$	−0.0404551	−0.00137711	−0.000688554	−	1.00000i	$-0.500219\pi$
$863$	−0.0404551	−0.00137711	−0.000688554		1.00000i	$0.500219\pi$
$864$	0	0
$865$	−6.29298	−0.213968
$866$	0	0
$867$	−127.788	−4.33992
$868$	0	0
$869$	18.7318	0.635434
$870$	0	0
$871$	−3.93005	−0.133165
$872$	0	0
$873$	−22.6812	−0.767642
$874$	0	0
$875$	22.7691	0.769737
$876$	0	0
$877$	6.66425	0.225036	0.112518	−	0.993650i	$-0.464108\pi$
$877$	6.66425	0.225036	0.112518	+	0.993650i	$0.464108\pi$
$878$	0	0
$879$	−20.1586	−0.679934
$880$	0	0
$881$	46.9411	1.58149	0.790743	−	0.612148i	$-0.209695\pi$
$881$	46.9411	1.58149	0.790743	+	0.612148i	$0.209695\pi$
$882$	0	0
$883$	5.74018	0.193172	0.0965862	−	0.995325i	$-0.469208\pi$
$883$	5.74018	0.193172	0.0965862	+	0.995325i	$0.469208\pi$
$884$	0	0
$885$	2.84268	0.0955555
$886$	0	0
$887$	14.0945	0.473247	0.236623	−	0.971601i	$-0.423959\pi$
$887$	14.0945	0.473247	0.236623	+	0.971601i	$0.423959\pi$
$888$	0	0
$889$	7.25867	0.243448
$890$	0	0
$891$	23.3421	0.781989
$892$	0	0
$893$	29.6801	0.993206
$894$	0	0
$895$	−11.7732	−0.393535
$896$	0	0
$897$	1.37697	0.0459756
$898$	0	0
$899$	22.4108	0.747441
$900$	0	0
$901$	−5.41701	−0.180467
$902$	0	0
$903$	−0.629497	−0.0209483
$904$	0	0
$905$	−90.6072	−3.01189
$906$	0	0
$907$	−33.6416	−1.11705	−0.558525	−	0.829488i	$-0.688633\pi$
$907$	−33.6416	−1.11705	−0.558525	+	0.829488i	$0.688633\pi$
$908$	0	0
$909$	9.04231	0.299915
$910$	0	0
$911$	−16.5664	−0.548871	−0.274435	−	0.961606i	$-0.588491\pi$
$911$	−16.5664	−0.548871	−0.274435	+	0.961606i	$0.588491\pi$
$912$	0	0
$913$	19.9523	0.660326
$914$	0	0
$915$	−29.5360	−0.976429
$916$	0	0
$917$	16.6909	0.551182
$918$	0	0
$919$	−26.8094	−0.884362	−0.442181	−	0.896926i	$-0.645795\pi$
$919$	−26.8094	−0.884362	−0.442181	+	0.896926i	$0.645795\pi$
$920$	0	0
$921$	−20.2919	−0.668642
$922$	0	0
$923$	−7.11043	−0.234043
$924$	0	0
$925$	21.5472	0.708466
$926$	0	0
$927$	62.1019	2.03970
$928$	0	0
$929$	13.7593	0.451426	0.225713	−	0.974194i	$-0.427529\pi$
$929$	13.7593	0.451426	0.225713	+	0.974194i	$0.427529\pi$
$930$	0	0
$931$	−37.2742	−1.22161
$932$	0	0
$933$	−8.59257	−0.281308
$934$	0	0
$935$	−71.0209	−2.32263
$936$	0	0
$937$	−47.8705	−1.56386	−0.781931	−	0.623365i	$-0.785765\pi$
$937$	−47.8705	−1.56386	−0.781931	+	0.623365i	$0.785765\pi$
$938$	0	0
$939$	−46.0245	−1.50195
$940$	0	0
$941$	41.5851	1.35564	0.677818	−	0.735230i	$-0.262926\pi$
$941$	41.5851	1.35564	0.677818	+	0.735230i	$0.262926\pi$
$942$	0	0
$943$	0.484989	0.0157934
$944$	0	0
$945$	−38.4649	−1.25126
$946$	0	0
$947$	34.7978	1.13078	0.565388	−	0.824825i	$-0.308726\pi$
$947$	34.7978	1.13078	0.565388	+	0.824825i	$0.308726\pi$
$948$	0	0
$949$	4.92385	0.159835
$950$	0	0
$951$	−56.5856	−1.83491
$952$	0	0
$953$	40.7391	1.31967	0.659835	−	0.751410i	$-0.270626\pi$
$953$	40.7391	1.31967	0.659835	+	0.751410i	$0.270626\pi$
$954$	0	0
$955$	−89.1635	−2.88526
$956$	0	0
$957$	58.5036	1.89115
$958$	0	0
$959$	−19.3762	−0.625691
$960$	0	0
$961$	−23.6956	−0.764376
$962$	0	0
$963$	42.9246	1.38322
$964$	0	0
$965$	53.3907	1.71871
$966$	0	0
$967$	50.3605	1.61948	0.809742	−	0.586786i	$-0.199607\pi$
$967$	50.3605	1.61948	0.809742	+	0.586786i	$0.199607\pi$
$968$	0	0
$969$	146.410	4.70338
$970$	0	0
$971$	7.55070	0.242314	0.121157	−	0.992633i	$-0.461340\pi$
$971$	7.55070	0.242314	0.121157	+	0.992633i	$0.461340\pi$
$972$	0	0
$973$	3.84048	0.123120
$974$	0	0
$975$	23.6381	0.757026
$976$	0	0
$977$	−25.9190	−0.829221	−0.414610	−	0.909999i	$-0.636082\pi$
$977$	−25.9190	−0.829221	−0.414610	+	0.909999i	$0.636082\pi$
$978$	0	0
$979$	37.7297	1.20585
$980$	0	0
$981$	95.2664	3.04162
$982$	0	0
$983$	31.0668	0.990878	0.495439	−	0.868643i	$-0.335007\pi$
$983$	31.0668	0.990878	0.495439	+	0.868643i	$0.335007\pi$
$984$	0	0
$985$	−47.7183	−1.52043
$986$	0	0
$987$	−14.7896	−0.470759
$988$	0	0
$989$	0.123309	0.00392101
$990$	0	0
$991$	12.9663	0.411887	0.205944	−	0.978564i	$-0.433974\pi$
$991$	12.9663	0.411887	0.205944	+	0.978564i	$0.433974\pi$
$992$	0	0
$993$	−72.1466	−2.28950
$994$	0	0
$995$	−78.7628	−2.49695
$996$	0	0
$997$	−21.2297	−0.672352	−0.336176	−	0.941799i	$-0.609134\pi$
$997$	−21.2297	−0.672352	−0.336176	+	0.941799i	$0.609134\pi$
$998$	0	0
$999$	−19.1533	−0.605984

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8044.2.a.b.1.4	✓	87

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8044.2.a.b.1.4	✓	87	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:	\( N \)	\(=\)	\( 8044 = 2^{2} \cdot 2011 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8044.a (trivial)

\(f(q)\)	\(=\)	\(q-3.01797 q^{3} +3.94367 q^{5} +1.03981 q^{7} +6.10812 q^{9} +O(q^{10})\)	\(q-3.01797 q^{3} +3.94367 q^{5} +1.03981 q^{7} +6.10812 q^{9} +2.33777 q^{11} -0.742233 q^{13} -11.9019 q^{15} -7.70341 q^{17} +6.29759 q^{19} -3.13810 q^{21} +0.614708 q^{23} +10.5526 q^{25} -9.38019 q^{27} -8.29213 q^{29} -2.70266 q^{31} -7.05532 q^{33} +4.10065 q^{35} +2.04189 q^{37} +2.24004 q^{39} +0.788974 q^{41} +0.200598 q^{43} +24.0884 q^{45} +4.71293 q^{47} -5.91881 q^{49} +23.2486 q^{51} +0.703196 q^{53} +9.21941 q^{55} -19.0059 q^{57} -0.238843 q^{59} +2.48162 q^{61} +6.35125 q^{63} -2.92713 q^{65} +5.29490 q^{67} -1.85517 q^{69} +9.57977 q^{71} -6.63382 q^{73} -31.8473 q^{75} +2.43083 q^{77} +8.01269 q^{79} +9.98475 q^{81} +8.53477 q^{83} -30.3798 q^{85} +25.0254 q^{87} +16.1392 q^{89} -0.771778 q^{91} +8.15653 q^{93} +24.8357 q^{95} -3.71329 q^{97} +14.2794 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 87 q + 13 q^{3} - 2 q^{5} + 8 q^{7} + 98 q^{9}+O(q^{10}) \) 87 * q + 13 * q^3 - 2 * q^5 + 8 * q^7 + 98 * q^9	\( 87 q + 13 q^{3} - 2 q^{5} + 8 q^{7} + 98 q^{9} + 36 q^{11} - q^{13} + 16 q^{15} + 31 q^{17} + 35 q^{19} - 3 q^{21} + 39 q^{23} + 93 q^{25} + 55 q^{27} - 5 q^{29} + 46 q^{31} + 25 q^{33} + 68 q^{35} - 11 q^{37} + 54 q^{39} + 83 q^{41} + 28 q^{43} - 14 q^{45} + 48 q^{47} + 103 q^{49} + 77 q^{51} + 3 q^{53} + 35 q^{55} + 14 q^{57} + 122 q^{59} - 13 q^{61} + 39 q^{63} + 41 q^{65} + 32 q^{67} - 10 q^{69} + 100 q^{71} + 34 q^{73} + 97 q^{75} + 4 q^{77} + 52 q^{79} + 131 q^{81} + 67 q^{83} - 2 q^{85} + 89 q^{87} + 68 q^{89} + 75 q^{91} + 138 q^{95} + 36 q^{97} + 107 q^{99}+O(q^{100}) \) 87 * q + 13 * q^3 - 2 * q^5 + 8 * q^7 + 98 * q^9 + 36 * q^11 - q^13 + 16 * q^15 + 31 * q^17 + 35 * q^19 - 3 * q^21 + 39 * q^23 + 93 * q^25 + 55 * q^27 - 5 * q^29 + 46 * q^31 + 25 * q^33 + 68 * q^35 - 11 * q^37 + 54 * q^39 + 83 * q^41 + 28 * q^43 - 14 * q^45 + 48 * q^47 + 103 * q^49 + 77 * q^51 + 3 * q^53 + 35 * q^55 + 14 * q^57 + 122 * q^59 - 13 * q^61 + 39 * q^63 + 41 * q^65 + 32 * q^67 - 10 * q^69 + 100 * q^71 + 34 * q^73 + 97 * q^75 + 4 * q^77 + 52 * q^79 + 131 * q^81 + 67 * q^83 - 2 * q^85 + 89 * q^87 + 68 * q^89 + 75 * q^91 + 138 * q^95 + 36 * q^97 + 107 * q^99

Introduction

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