LMFDB - Embedded newform 8044.2.a.a.1.6

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:	$1$
Dimension:	$80$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
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)

$f(q)$	$=$	$q-3.01017 q^{3} -0.450076 q^{5} +4.39052 q^{7} +6.06114 q^{9} +O(q^{10})$	$q-3.01017 q^{3} -0.450076 q^{5} +4.39052 q^{7} +6.06114 q^{9} -3.35779 q^{11} +3.20878 q^{13} +1.35481 q^{15} -5.73094 q^{17} +2.89632 q^{19} -13.2162 q^{21} +8.12441 q^{23} -4.79743 q^{25} -9.21456 q^{27} +8.58091 q^{29} -4.22971 q^{31} +10.1075 q^{33} -1.97607 q^{35} +0.640533 q^{37} -9.65899 q^{39} -11.1589 q^{41} -1.93361 q^{43} -2.72797 q^{45} -9.57255 q^{47} +12.2766 q^{49} +17.2511 q^{51} -1.94657 q^{53} +1.51126 q^{55} -8.71842 q^{57} -12.7989 q^{59} +3.73375 q^{61} +26.6115 q^{63} -1.44420 q^{65} -1.08064 q^{67} -24.4559 q^{69} +9.29175 q^{71} -8.08619 q^{73} +14.4411 q^{75} -14.7424 q^{77} -16.5537 q^{79} +9.55399 q^{81} +4.50046 q^{83} +2.57936 q^{85} -25.8300 q^{87} -12.1007 q^{89} +14.0882 q^{91} +12.7322 q^{93} -1.30356 q^{95} -9.80010 q^{97} -20.3520 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9}+O(q^{10}) $ 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9	$ 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9} - 34 q^{11} - q^{13} - 24 q^{15} - 35 q^{17} - 31 q^{19} - 3 q^{21} - 43 q^{23} + 58 q^{25} - 49 q^{27} - 5 q^{29} - 56 q^{31} - 23 q^{33} - 72 q^{35} - 11 q^{37} - 74 q^{39} - 81 q^{41} - 34 q^{43} - 14 q^{45} - 64 q^{47} + 40 q^{49} - 59 q^{51} + 3 q^{53} - 53 q^{55} - 34 q^{57} - 116 q^{59} - 13 q^{61} - 61 q^{63} - 55 q^{65} - 22 q^{67} - 10 q^{69} - 86 q^{71} - 32 q^{73} - 85 q^{75} + 4 q^{77} - 88 q^{79} + 12 q^{81} - 83 q^{83} - 2 q^{85} - 87 q^{87} - 72 q^{89} - 49 q^{91} - 102 q^{95} - 34 q^{97} - 103 q^{99}+O(q^{100}) $ 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9 - 34 * q^11 - q^13 - 24 * q^15 - 35 * q^17 - 31 * q^19 - 3 * q^21 - 43 * q^23 + 58 * q^25 - 49 * q^27 - 5 * q^29 - 56 * q^31 - 23 * q^33 - 72 * q^35 - 11 * q^37 - 74 * q^39 - 81 * q^41 - 34 * q^43 - 14 * q^45 - 64 * q^47 + 40 * q^49 - 59 * q^51 + 3 * q^53 - 53 * q^55 - 34 * q^57 - 116 * q^59 - 13 * q^61 - 61 * q^63 - 55 * q^65 - 22 * q^67 - 10 * q^69 - 86 * q^71 - 32 * q^73 - 85 * q^75 + 4 * q^77 - 88 * q^79 + 12 * q^81 - 83 * q^83 - 2 * q^85 - 87 * q^87 - 72 * q^89 - 49 * q^91 - 102 * q^95 - 34 * q^97 - 103 * 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
$3$	−3.01017	−1.73792	−0.868962	−	0.494879i	$-0.835212\pi$
$3$	−3.01017	−1.73792	−0.868962	+	0.494879i	$0.835212\pi$
$4$	0	0
$5$	−0.450076	−0.201280	−0.100640	−	0.994923i	$-0.532089\pi$
$5$	−0.450076	−0.201280	−0.100640	+	0.994923i	$0.532089\pi$
$6$	0	0
$7$	4.39052	1.65946	0.829730	−	0.558165i	$-0.188495\pi$
$7$	4.39052	1.65946	0.829730	+	0.558165i	$0.188495\pi$
$8$	0	0
$9$	6.06114	2.02038
$10$	0	0
$11$	−3.35779	−1.01241	−0.506206	−	0.862413i	$-0.668952\pi$
$11$	−3.35779	−1.01241	−0.506206	+	0.862413i	$0.668952\pi$
$12$	0	0
$13$	3.20878	0.889956	0.444978	−	0.895541i	$-0.353211\pi$
$13$	3.20878	0.889956	0.444978	+	0.895541i	$0.353211\pi$
$14$	0	0
$15$	1.35481	0.349810
$16$	0	0
$17$	−5.73094	−1.38996	−0.694978	−	0.719031i	$-0.744586\pi$
$17$	−5.73094	−1.38996	−0.694978	+	0.719031i	$0.744586\pi$
$18$	0	0
$19$	2.89632	0.664461	0.332231	−	0.943198i	$-0.392199\pi$
$19$	2.89632	0.664461	0.332231	+	0.943198i	$0.392199\pi$
$20$	0	0
$21$	−13.2162	−2.88401
$22$	0	0
$23$	8.12441	1.69406	0.847028	−	0.531548i	$-0.178390\pi$
$23$	8.12441	1.69406	0.847028	+	0.531548i	$0.178390\pi$
$24$	0	0
$25$	−4.79743	−0.959486
$26$	0	0
$27$	−9.21456	−1.77334
$28$	0	0
$29$	8.58091	1.59344	0.796718	−	0.604352i	$-0.206568\pi$
$29$	8.58091	1.59344	0.796718	+	0.604352i	$0.206568\pi$
$30$	0	0
$31$	−4.22971	−0.759678	−0.379839	−	0.925053i	$-0.624021\pi$
$31$	−4.22971	−0.759678	−0.379839	+	0.925053i	$0.624021\pi$
$32$	0	0
$33$	10.1075	1.75950
$34$	0	0
$35$	−1.97607	−0.334016
$36$	0	0
$37$	0.640533	0.105303	0.0526515	−	0.998613i	$-0.483233\pi$
$37$	0.640533	0.105303	0.0526515	+	0.998613i	$0.483233\pi$
$38$	0	0
$39$	−9.65899	−1.54668
$40$	0	0
$41$	−11.1589	−1.74273	−0.871367	−	0.490632i	$-0.836766\pi$
$41$	−11.1589	−1.74273	−0.871367	+	0.490632i	$0.836766\pi$
$42$	0	0
$43$	−1.93361	−0.294872	−0.147436	−	0.989072i	$-0.547102\pi$
$43$	−1.93361	−0.294872	−0.147436	+	0.989072i	$0.547102\pi$
$44$	0	0
$45$	−2.72797	−0.406662
$46$	0	0
$47$	−9.57255	−1.39630	−0.698150	−	0.715952i	$-0.745993\pi$
$47$	−9.57255	−1.39630	−0.698150	+	0.715952i	$0.745993\pi$
$48$	0	0
$49$	12.2766	1.75381
$50$	0	0
$51$	17.2511	2.41564
$52$	0	0
$53$	−1.94657	−0.267382	−0.133691	−	0.991023i	$-0.542683\pi$
$53$	−1.94657	−0.267382	−0.133691	+	0.991023i	$0.542683\pi$
$54$	0	0
$55$	1.51126	0.203778
$56$	0	0
$57$	−8.71842	−1.15478
$58$	0	0
$59$	−12.7989	−1.66628	−0.833138	−	0.553065i	$-0.813458\pi$
$59$	−12.7989	−1.66628	−0.833138	+	0.553065i	$0.813458\pi$
$60$	0	0
$61$	3.73375	0.478057	0.239029	−	0.971012i	$-0.423171\pi$
$61$	3.73375	0.478057	0.239029	+	0.971012i	$0.423171\pi$
$62$	0	0
$63$	26.6115	3.35274
$64$	0	0
$65$	−1.44420	−0.179130
$66$	0	0
$67$	−1.08064	−0.132022	−0.0660109	−	0.997819i	$-0.521027\pi$
$67$	−1.08064	−0.132022	−0.0660109	+	0.997819i	$0.521027\pi$
$68$	0	0
$69$	−24.4559	−2.94414
$70$	0	0
$71$	9.29175	1.10273	0.551364	−	0.834265i	$-0.314108\pi$
$71$	9.29175	1.10273	0.551364	+	0.834265i	$0.314108\pi$
$72$	0	0
$73$	−8.08619	−0.946417	−0.473208	−	0.880950i	$-0.656904\pi$
$73$	−8.08619	−0.946417	−0.473208	+	0.880950i	$0.656904\pi$
$74$	0	0
$75$	14.4411	1.66751
$76$	0	0
$77$	−14.7424	−1.68006
$78$	0	0
$79$	−16.5537	−1.86243	−0.931216	−	0.364467i	$-0.881251\pi$
$79$	−16.5537	−1.86243	−0.931216	+	0.364467i	$0.881251\pi$
$80$	0	0
$81$	9.55399	1.06155
$82$	0	0
$83$	4.50046	0.493989	0.246995	−	0.969017i	$-0.420557\pi$
$83$	4.50046	0.493989	0.246995	+	0.969017i	$0.420557\pi$
$84$	0	0
$85$	2.57936	0.279771
$86$	0	0
$87$	−25.8300	−2.76927
$88$	0	0
$89$	−12.1007	−1.28267	−0.641336	−	0.767260i	$-0.721620\pi$
$89$	−12.1007	−1.28267	−0.641336	+	0.767260i	$0.721620\pi$
$90$	0	0
$91$	14.0882	1.47685
$92$	0	0
$93$	12.7322	1.32026
$94$	0	0
$95$	−1.30356	−0.133743
$96$	0	0
$97$	−9.80010	−0.995049	−0.497525	−	0.867450i	$-0.665758\pi$
$97$	−9.80010	−0.995049	−0.497525	+	0.867450i	$0.665758\pi$
$98$	0	0
$99$	−20.3520	−2.04546
$100$	0	0
$101$	7.53246	0.749508	0.374754	−	0.927124i	$-0.377727\pi$
$101$	7.53246	0.749508	0.374754	+	0.927124i	$0.377727\pi$
$102$	0	0
$103$	−4.91631	−0.484418	−0.242209	−	0.970224i	$-0.577872\pi$
$103$	−4.91631	−0.484418	−0.242209	+	0.970224i	$0.577872\pi$
$104$	0	0
$105$	5.94830	0.580495
$106$	0	0
$107$	−14.9220	−1.44256	−0.721280	−	0.692644i	$-0.756446\pi$
$107$	−14.9220	−1.44256	−0.721280	+	0.692644i	$0.756446\pi$
$108$	0	0
$109$	18.0996	1.73363	0.866814	−	0.498631i	$-0.166164\pi$
$109$	18.0996	1.73363	0.866814	+	0.498631i	$0.166164\pi$
$110$	0	0
$111$	−1.92811	−0.183008
$112$	0	0
$113$	16.1123	1.51572	0.757858	−	0.652420i	$-0.226246\pi$
$113$	16.1123	1.51572	0.757858	+	0.652420i	$0.226246\pi$
$114$	0	0
$115$	−3.65660	−0.340980
$116$	0	0
$117$	19.4489	1.79805
$118$	0	0
$119$	−25.1618	−2.30658
$120$	0	0
$121$	0.274761	0.0249783
$122$	0	0
$123$	33.5903	3.02874
$124$	0	0
$125$	4.40959	0.394406
$126$	0	0
$127$	4.06286	0.360520	0.180260	−	0.983619i	$-0.442306\pi$
$127$	4.06286	0.360520	0.180260	+	0.983619i	$0.442306\pi$
$128$	0	0
$129$	5.82049	0.512466
$130$	0	0
$131$	1.83735	0.160530	0.0802649	−	0.996774i	$-0.474423\pi$
$131$	1.83735	0.160530	0.0802649	+	0.996774i	$0.474423\pi$
$132$	0	0
$133$	12.7163	1.10265
$134$	0	0
$135$	4.14725	0.356939
$136$	0	0
$137$	5.93298	0.506889	0.253445	−	0.967350i	$-0.418436\pi$
$137$	5.93298	0.506889	0.253445	+	0.967350i	$0.418436\pi$
$138$	0	0
$139$	13.3847	1.13527	0.567637	−	0.823279i	$-0.307858\pi$
$139$	13.3847	1.13527	0.567637	+	0.823279i	$0.307858\pi$
$140$	0	0
$141$	28.8150	2.42666
$142$	0	0
$143$	−10.7744	−0.901002
$144$	0	0
$145$	−3.86206	−0.320727
$146$	0	0
$147$	−36.9548	−3.04798
$148$	0	0
$149$	18.5994	1.52372	0.761862	−	0.647740i	$-0.224286\pi$
$149$	18.5994	1.52372	0.761862	+	0.647740i	$0.224286\pi$
$150$	0	0
$151$	8.65640	0.704448	0.352224	−	0.935916i	$-0.385426\pi$
$151$	8.65640	0.704448	0.352224	+	0.935916i	$0.385426\pi$
$152$	0	0
$153$	−34.7360	−2.80824
$154$	0	0
$155$	1.90369	0.152908
$156$	0	0
$157$	20.9913	1.67529	0.837645	−	0.546215i	$-0.183932\pi$
$157$	20.9913	1.67529	0.837645	+	0.546215i	$0.183932\pi$
$158$	0	0
$159$	5.85952	0.464690
$160$	0	0
$161$	35.6704	2.81122
$162$	0	0
$163$	−5.32080	−0.416757	−0.208379	−	0.978048i	$-0.566819\pi$
$163$	−5.32080	−0.416757	−0.208379	+	0.978048i	$0.566819\pi$
$164$	0	0
$165$	−4.54916	−0.354151
$166$	0	0
$167$	−5.52779	−0.427753	−0.213877	−	0.976861i	$-0.568609\pi$
$167$	−5.52779	−0.427753	−0.213877	+	0.976861i	$0.568609\pi$
$168$	0	0
$169$	−2.70371	−0.207978
$170$	0	0
$171$	17.5550	1.34246
$172$	0	0
$173$	−12.5602	−0.954934	−0.477467	−	0.878650i	$-0.658445\pi$
$173$	−12.5602	−0.954934	−0.477467	+	0.878650i	$0.658445\pi$
$174$	0	0
$175$	−21.0632	−1.59223
$176$	0	0
$177$	38.5269	2.89586
$178$	0	0
$179$	−22.4795	−1.68019	−0.840097	−	0.542436i	$-0.817502\pi$
$179$	−22.4795	−1.68019	−0.840097	+	0.542436i	$0.817502\pi$
$180$	0	0
$181$	−1.33073	−0.0989124	−0.0494562	−	0.998776i	$-0.515749\pi$
$181$	−1.33073	−0.0989124	−0.0494562	+	0.998776i	$0.515749\pi$
$182$	0	0
$183$	−11.2392	−0.830827
$184$	0	0
$185$	−0.288288	−0.0211954
$186$	0	0
$187$	19.2433	1.40721
$188$	0	0
$189$	−40.4567	−2.94279
$190$	0	0
$191$	17.4114	1.25984	0.629921	−	0.776659i	$-0.283087\pi$
$191$	17.4114	1.25984	0.629921	+	0.776659i	$0.283087\pi$
$192$	0	0
$193$	1.93552	0.139322	0.0696610	−	0.997571i	$-0.477808\pi$
$193$	1.93552	0.139322	0.0696610	+	0.997571i	$0.477808\pi$
$194$	0	0
$195$	4.34728	0.311315
$196$	0	0
$197$	−20.9619	−1.49348	−0.746738	−	0.665119i	$-0.768381\pi$
$197$	−20.9619	−1.49348	−0.746738	+	0.665119i	$0.768381\pi$
$198$	0	0
$199$	−13.3988	−0.949817	−0.474908	−	0.880035i	$-0.657519\pi$
$199$	−13.3988	−0.949817	−0.474908	+	0.880035i	$0.657519\pi$
$200$	0	0
$201$	3.25293	0.229444
$202$	0	0
$203$	37.6746	2.64424
$204$	0	0
$205$	5.02237	0.350778
$206$	0	0
$207$	49.2432	3.42264
$208$	0	0
$209$	−9.72524	−0.672709
$210$	0	0
$211$	−15.6085	−1.07453	−0.537267	−	0.843412i	$-0.680543\pi$
$211$	−15.6085	−1.07453	−0.537267	+	0.843412i	$0.680543\pi$
$212$	0	0
$213$	−27.9698	−1.91646
$214$	0	0
$215$	0.870270	0.0593519
$216$	0	0
$217$	−18.5706	−1.26066
$218$	0	0
$219$	24.3408	1.64480
$220$	0	0
$221$	−18.3893	−1.23700
$222$	0	0
$223$	1.82686	0.122336	0.0611679	−	0.998127i	$-0.480517\pi$
$223$	1.82686	0.122336	0.0611679	+	0.998127i	$0.480517\pi$
$224$	0	0
$225$	−29.0779	−1.93853
$226$	0	0
$227$	−19.9449	−1.32379	−0.661895	−	0.749597i	$-0.730247\pi$
$227$	−19.9449	−1.32379	−0.661895	+	0.749597i	$0.730247\pi$
$228$	0	0
$229$	4.85806	0.321030	0.160515	−	0.987033i	$-0.448685\pi$
$229$	4.85806	0.321030	0.160515	+	0.987033i	$0.448685\pi$
$230$	0	0
$231$	44.3773	2.91981
$232$	0	0
$233$	28.5223	1.86856	0.934279	−	0.356543i	$-0.116045\pi$
$233$	28.5223	1.86856	0.934279	+	0.356543i	$0.116045\pi$
$234$	0	0
$235$	4.30837	0.281047
$236$	0	0
$237$	49.8294	3.23677
$238$	0	0
$239$	5.10134	0.329978	0.164989	−	0.986295i	$-0.447241\pi$
$239$	5.10134	0.329978	0.164989	+	0.986295i	$0.447241\pi$
$240$	0	0
$241$	−7.14512	−0.460258	−0.230129	−	0.973160i	$-0.573915\pi$
$241$	−7.14512	−0.460258	−0.230129	+	0.973160i	$0.573915\pi$
$242$	0	0
$243$	−1.11548	−0.0715583
$244$	0	0
$245$	−5.52542	−0.353006
$246$	0	0
$247$	9.29366	0.591341
$248$	0	0
$249$	−13.5472	−0.858516
$250$	0	0
$251$	2.70982	0.171042	0.0855211	−	0.996336i	$-0.472745\pi$
$251$	2.70982	0.171042	0.0855211	+	0.996336i	$0.472745\pi$
$252$	0	0
$253$	−27.2801	−1.71508
$254$	0	0
$255$	−7.76431	−0.486220
$256$	0	0
$257$	−26.3936	−1.64639	−0.823194	−	0.567760i	$-0.807810\pi$
$257$	−26.3936	−1.64639	−0.823194	+	0.567760i	$0.807810\pi$
$258$	0	0
$259$	2.81227	0.174746
$260$	0	0
$261$	52.0101	3.21934
$262$	0	0
$263$	−22.3239	−1.37655	−0.688277	−	0.725448i	$-0.741632\pi$
$263$	−22.3239	−1.37655	−0.688277	+	0.725448i	$0.741632\pi$
$264$	0	0
$265$	0.876106	0.0538188
$266$	0	0
$267$	36.4252	2.22919
$268$	0	0
$269$	18.8974	1.15220	0.576098	−	0.817381i	$-0.304575\pi$
$269$	18.8974	1.15220	0.576098	+	0.817381i	$0.304575\pi$
$270$	0	0
$271$	−8.69373	−0.528106	−0.264053	−	0.964508i	$-0.585059\pi$
$271$	−8.69373	−0.528106	−0.264053	+	0.964508i	$0.585059\pi$
$272$	0	0
$273$	−42.4080	−2.56665
$274$	0	0
$275$	16.1088	0.971396
$276$	0	0
$277$	24.0005	1.44205	0.721025	−	0.692909i	$-0.243671\pi$
$277$	24.0005	1.44205	0.721025	+	0.692909i	$0.243671\pi$
$278$	0	0
$279$	−25.6369	−1.53484
$280$	0	0
$281$	8.45970	0.504663	0.252332	−	0.967641i	$-0.418803\pi$
$281$	8.45970	0.504663	0.252332	+	0.967641i	$0.418803\pi$
$282$	0	0
$283$	3.25134	0.193272	0.0966361	−	0.995320i	$-0.469192\pi$
$283$	3.25134	0.193272	0.0966361	+	0.995320i	$0.469192\pi$
$284$	0	0
$285$	3.92395	0.232435
$286$	0	0
$287$	−48.9935	−2.89200
$288$	0	0
$289$	15.8436	0.931979
$290$	0	0
$291$	29.5000	1.72932
$292$	0	0
$293$	−20.6040	−1.20370	−0.601851	−	0.798609i	$-0.705570\pi$
$293$	−20.6040	−1.20370	−0.601851	+	0.798609i	$0.705570\pi$
$294$	0	0
$295$	5.76048	0.335388
$296$	0	0
$297$	30.9406	1.79535
$298$	0	0
$299$	26.0695	1.50764
$300$	0	0
$301$	−8.48953	−0.489329
$302$	0	0
$303$	−22.6740	−1.30259
$304$	0	0
$305$	−1.68047	−0.0962235
$306$	0	0
$307$	3.26209	0.186177	0.0930885	−	0.995658i	$-0.470326\pi$
$307$	3.26209	0.186177	0.0930885	+	0.995658i	$0.470326\pi$
$308$	0	0
$309$	14.7989	0.841882
$310$	0	0
$311$	−26.2764	−1.49000	−0.745000	−	0.667064i	$-0.767551\pi$
$311$	−26.2764	−1.49000	−0.745000	+	0.667064i	$0.767551\pi$
$312$	0	0
$313$	21.5260	1.21672	0.608362	−	0.793660i	$-0.291827\pi$
$313$	21.5260	1.21672	0.608362	+	0.793660i	$0.291827\pi$
$314$	0	0
$315$	−11.9772	−0.674839
$316$	0	0
$317$	−6.45550	−0.362577	−0.181289	−	0.983430i	$-0.558027\pi$
$317$	−6.45550	−0.362577	−0.181289	+	0.983430i	$0.558027\pi$
$318$	0	0
$319$	−28.8129	−1.61321
$320$	0	0
$321$	44.9177	2.50706
$322$	0	0
$323$	−16.5986	−0.923572
$324$	0	0
$325$	−15.3939	−0.853901
$326$	0	0
$327$	−54.4830	−3.01291
$328$	0	0
$329$	−42.0284	−2.31710
$330$	0	0
$331$	9.20498	0.505951	0.252976	−	0.967473i	$-0.418591\pi$
$331$	9.20498	0.505951	0.252976	+	0.967473i	$0.418591\pi$
$332$	0	0
$333$	3.88236	0.212752
$334$	0	0
$335$	0.486372	0.0265734
$336$	0	0
$337$	−10.9547	−0.596740	−0.298370	−	0.954450i	$-0.596443\pi$
$337$	−10.9547	−0.596740	−0.298370	+	0.954450i	$0.596443\pi$
$338$	0	0
$339$	−48.5007	−2.63420
$340$	0	0
$341$	14.2025	0.769108
$342$	0	0
$343$	23.1672	1.25091
$344$	0	0
$345$	11.0070	0.592597
$346$	0	0
$347$	−27.8709	−1.49619	−0.748093	−	0.663594i	$-0.769030\pi$
$347$	−27.8709	−1.49619	−0.748093	+	0.663594i	$0.769030\pi$
$348$	0	0
$349$	24.1439	1.29239	0.646197	−	0.763171i	$-0.276359\pi$
$349$	24.1439	1.29239	0.646197	+	0.763171i	$0.276359\pi$
$350$	0	0
$351$	−29.5675	−1.57820
$352$	0	0
$353$	−9.57597	−0.509677	−0.254839	−	0.966984i	$-0.582022\pi$
$353$	−9.57597	−0.509677	−0.254839	+	0.966984i	$0.582022\pi$
$354$	0	0
$355$	−4.18199	−0.221957
$356$	0	0
$357$	75.7413	4.00865
$358$	0	0
$359$	−20.1085	−1.06129	−0.530643	−	0.847596i	$-0.678049\pi$
$359$	−20.1085	−1.06129	−0.530643	+	0.847596i	$0.678049\pi$
$360$	0	0
$361$	−10.6113	−0.558491
$362$	0	0
$363$	−0.827077	−0.0434103
$364$	0	0
$365$	3.63940	0.190495
$366$	0	0
$367$	19.7710	1.03204	0.516020	−	0.856577i	$-0.327413\pi$
$367$	19.7710	1.03204	0.516020	+	0.856577i	$0.327413\pi$
$368$	0	0
$369$	−67.6359	−3.52098
$370$	0	0
$371$	−8.54647	−0.443710
$372$	0	0
$373$	16.6441	0.861800	0.430900	−	0.902400i	$-0.358196\pi$
$373$	16.6441	0.861800	0.430900	+	0.902400i	$0.358196\pi$
$374$	0	0
$375$	−13.2736	−0.685447
$376$	0	0
$377$	27.5343	1.41809
$378$	0	0
$379$	−0.139931	−0.00718775	−0.00359387	−	0.999994i	$-0.501144\pi$
$379$	−0.139931	−0.00718775	−0.00359387	+	0.999994i	$0.501144\pi$
$380$	0	0
$381$	−12.2299	−0.626557
$382$	0	0
$383$	−5.81622	−0.297195	−0.148597	−	0.988898i	$-0.547476\pi$
$383$	−5.81622	−0.297195	−0.148597	+	0.988898i	$0.547476\pi$
$384$	0	0
$385$	6.63522	0.338162
$386$	0	0
$387$	−11.7199	−0.595754
$388$	0	0
$389$	−21.5752	−1.09391	−0.546954	−	0.837163i	$-0.684213\pi$
$389$	−21.5752	−1.09391	−0.546954	+	0.837163i	$0.684213\pi$
$390$	0	0
$391$	−46.5605	−2.35466
$392$	0	0
$393$	−5.53074	−0.278989
$394$	0	0
$395$	7.45041	0.374871
$396$	0	0
$397$	−2.94941	−0.148027	−0.0740134	−	0.997257i	$-0.523581\pi$
$397$	−2.94941	−0.148027	−0.0740134	+	0.997257i	$0.523581\pi$
$398$	0	0
$399$	−38.2784	−1.91632
$400$	0	0
$401$	−3.64023	−0.181785	−0.0908923	−	0.995861i	$-0.528972\pi$
$401$	−3.64023	−0.181785	−0.0908923	+	0.995861i	$0.528972\pi$
$402$	0	0
$403$	−13.5722	−0.676080
$404$	0	0
$405$	−4.30002	−0.213670
$406$	0	0
$407$	−2.15077	−0.106610
$408$	0	0
$409$	−12.1541	−0.600981	−0.300491	−	0.953785i	$-0.597150\pi$
$409$	−12.1541	−0.600981	−0.300491	+	0.953785i	$0.597150\pi$
$410$	0	0
$411$	−17.8593	−0.880935
$412$	0	0
$413$	−56.1938	−2.76512
$414$	0	0
$415$	−2.02555	−0.0994302
$416$	0	0
$417$	−40.2902	−1.97302
$418$	0	0
$419$	−9.97175	−0.487152	−0.243576	−	0.969882i	$-0.578320\pi$
$419$	−9.97175	−0.487152	−0.243576	+	0.969882i	$0.578320\pi$
$420$	0	0
$421$	−16.5786	−0.807990	−0.403995	−	0.914761i	$-0.632379\pi$
$421$	−16.5786	−0.807990	−0.403995	+	0.914761i	$0.632379\pi$
$422$	0	0
$423$	−58.0205	−2.82106
$424$	0	0
$425$	27.4938	1.33364
$426$	0	0
$427$	16.3931	0.793317
$428$	0	0
$429$	32.4329	1.56587
$430$	0	0
$431$	−6.88640	−0.331706	−0.165853	−	0.986150i	$-0.553038\pi$
$431$	−6.88640	−0.331706	−0.165853	+	0.986150i	$0.553038\pi$
$432$	0	0
$433$	6.46070	0.310481	0.155241	−	0.987877i	$-0.450385\pi$
$433$	6.46070	0.310481	0.155241	+	0.987877i	$0.450385\pi$
$434$	0	0
$435$	11.6255	0.557399
$436$	0	0
$437$	23.5309	1.12563
$438$	0	0
$439$	18.1125	0.864462	0.432231	−	0.901763i	$-0.357727\pi$
$439$	18.1125	0.864462	0.432231	+	0.901763i	$0.357727\pi$
$440$	0	0
$441$	74.4104	3.54335
$442$	0	0
$443$	−4.08092	−0.193890	−0.0969451	−	0.995290i	$-0.530907\pi$
$443$	−4.08092	−0.193890	−0.0969451	+	0.995290i	$0.530907\pi$
$444$	0	0
$445$	5.44624	0.258177
$446$	0	0
$447$	−55.9875	−2.64812
$448$	0	0
$449$	−6.03707	−0.284907	−0.142454	−	0.989801i	$-0.545499\pi$
$449$	−6.03707	−0.284907	−0.142454	+	0.989801i	$0.545499\pi$
$450$	0	0
$451$	37.4694	1.76436
$452$	0	0
$453$	−26.0572	−1.22428
$454$	0	0
$455$	−6.34077	−0.297260
$456$	0	0
$457$	−16.6493	−0.778819	−0.389410	−	0.921065i	$-0.627321\pi$
$457$	−16.6493	−0.778819	−0.389410	+	0.921065i	$0.627321\pi$
$458$	0	0
$459$	52.8080	2.46487
$460$	0	0
$461$	−32.6410	−1.52025	−0.760123	−	0.649780i	$-0.774861\pi$
$461$	−32.6410	−1.52025	−0.760123	+	0.649780i	$0.774861\pi$
$462$	0	0
$463$	−37.5436	−1.74480	−0.872401	−	0.488792i	$-0.837438\pi$
$463$	−37.5436	−1.74480	−0.872401	+	0.488792i	$0.837438\pi$
$464$	0	0
$465$	−5.73044	−0.265743
$466$	0	0
$467$	−8.16645	−0.377898	−0.188949	−	0.981987i	$-0.560508\pi$
$467$	−8.16645	−0.377898	−0.188949	+	0.981987i	$0.560508\pi$
$468$	0	0
$469$	−4.74459	−0.219085
$470$	0	0
$471$	−63.1875	−2.91153
$472$	0	0
$473$	6.49265	0.298532
$474$	0	0
$475$	−13.8949	−0.637542
$476$	0	0
$477$	−11.7985	−0.540214
$478$	0	0
$479$	−25.1550	−1.14936	−0.574681	−	0.818377i	$-0.694874\pi$
$479$	−25.1550	−1.14936	−0.574681	+	0.818377i	$0.694874\pi$
$480$	0	0
$481$	2.05533	0.0937150
$482$	0	0
$483$	−107.374	−4.88568
$484$	0	0
$485$	4.41079	0.200284
$486$	0	0
$487$	4.57066	0.207117	0.103558	−	0.994623i	$-0.466977\pi$
$487$	4.57066	0.207117	0.103558	+	0.994623i	$0.466977\pi$
$488$	0	0
$489$	16.0165	0.724292
$490$	0	0
$491$	−38.8080	−1.75138	−0.875690	−	0.482873i	$-0.839593\pi$
$491$	−38.8080	−1.75138	−0.875690	+	0.482873i	$0.839593\pi$
$492$	0	0
$493$	−49.1767	−2.21481
$494$	0	0
$495$	9.15996	0.411710
$496$	0	0
$497$	40.7956	1.82993
$498$	0	0
$499$	−10.9256	−0.489096	−0.244548	−	0.969637i	$-0.578640\pi$
$499$	−10.9256	−0.489096	−0.244548	+	0.969637i	$0.578640\pi$
$500$	0	0
$501$	16.6396	0.743403
$502$	0	0
$503$	−9.22075	−0.411133	−0.205566	−	0.978643i	$-0.565904\pi$
$503$	−9.22075	−0.411133	−0.205566	+	0.978643i	$0.565904\pi$
$504$	0	0
$505$	−3.39018	−0.150861
$506$	0	0
$507$	8.13865	0.361450
$508$	0	0
$509$	14.6081	0.647492	0.323746	−	0.946144i	$-0.395058\pi$
$509$	14.6081	0.647492	0.323746	+	0.946144i	$0.395058\pi$
$510$	0	0
$511$	−35.5025	−1.57054
$512$	0	0
$513$	−26.6883	−1.17832
$514$	0	0
$515$	2.21271	0.0975037
$516$	0	0
$517$	32.1426	1.41363
$518$	0	0
$519$	37.8083	1.65960
$520$	0	0
$521$	−33.6755	−1.47535	−0.737676	−	0.675155i	$-0.764077\pi$
$521$	−33.6755	−1.47535	−0.737676	+	0.675155i	$0.764077\pi$
$522$	0	0
$523$	4.91803	0.215051	0.107525	−	0.994202i	$-0.465707\pi$
$523$	4.91803	0.215051	0.107525	+	0.994202i	$0.465707\pi$
$524$	0	0
$525$	63.4039	2.76717
$526$	0	0
$527$	24.2402	1.05592
$528$	0	0
$529$	43.0060	1.86983
$530$	0	0
$531$	−77.5760	−3.36651
$532$	0	0
$533$	−35.8066	−1.55096
$534$	0	0
$535$	6.71601	0.290359
$536$	0	0
$537$	67.6671	2.92005
$538$	0	0
$539$	−41.2224	−1.77557
$540$	0	0
$541$	26.0973	1.12201	0.561006	−	0.827812i	$-0.310415\pi$
$541$	26.0973	1.12201	0.561006	+	0.827812i	$0.310415\pi$
$542$	0	0
$543$	4.00573	0.171902
$544$	0	0
$545$	−8.14620	−0.348945
$546$	0	0
$547$	42.8068	1.83028	0.915142	−	0.403131i	$-0.132078\pi$
$547$	42.8068	1.83028	0.915142	+	0.403131i	$0.132078\pi$
$548$	0	0
$549$	22.6308	0.965858
$550$	0	0
$551$	24.8531	1.05878
$552$	0	0
$553$	−72.6791	−3.09063
$554$	0	0
$555$	0.867798	0.0368360
$556$	0	0
$557$	5.42959	0.230059	0.115030	−	0.993362i	$-0.463304\pi$
$557$	5.42959	0.230059	0.115030	+	0.993362i	$0.463304\pi$
$558$	0	0
$559$	−6.20452	−0.262423
$560$	0	0
$561$	−57.9256	−2.44562
$562$	0	0
$563$	37.1754	1.56676	0.783378	−	0.621546i	$-0.213495\pi$
$563$	37.1754	1.56676	0.783378	+	0.621546i	$0.213495\pi$
$564$	0	0
$565$	−7.25175	−0.305083
$566$	0	0
$567$	41.9469	1.76161
$568$	0	0
$569$	26.4962	1.11078	0.555388	−	0.831591i	$-0.312570\pi$
$569$	26.4962	1.11078	0.555388	+	0.831591i	$0.312570\pi$
$570$	0	0
$571$	25.1439	1.05224	0.526120	−	0.850410i	$-0.323646\pi$
$571$	25.1439	1.05224	0.526120	+	0.850410i	$0.323646\pi$
$572$	0	0
$573$	−52.4112	−2.18951
$574$	0	0
$575$	−38.9763	−1.62542
$576$	0	0
$577$	28.6636	1.19328	0.596642	−	0.802508i	$-0.296501\pi$
$577$	28.6636	1.19328	0.596642	+	0.802508i	$0.296501\pi$
$578$	0	0
$579$	−5.82626	−0.242131
$580$	0	0
$581$	19.7593	0.819755
$582$	0	0
$583$	6.53619	0.270701
$584$	0	0
$585$	−8.75347	−0.361912
$586$	0	0
$587$	−35.4983	−1.46517	−0.732586	−	0.680675i	$-0.761687\pi$
$587$	−35.4983	−1.46517	−0.732586	+	0.680675i	$0.761687\pi$
$588$	0	0
$589$	−12.2506	−0.504777
$590$	0	0
$591$	63.0990	2.59555
$592$	0	0
$593$	28.9607	1.18927	0.594637	−	0.803994i	$-0.297296\pi$
$593$	28.9607	1.18927	0.594637	+	0.803994i	$0.297296\pi$
$594$	0	0
$595$	11.3247	0.464268
$596$	0	0
$597$	40.3327	1.65071
$598$	0	0
$599$	1.74896	0.0714607	0.0357303	−	0.999361i	$-0.488624\pi$
$599$	1.74896	0.0714607	0.0357303	+	0.999361i	$0.488624\pi$
$600$	0	0
$601$	37.7461	1.53969	0.769847	−	0.638228i	$-0.220332\pi$
$601$	37.7461	1.53969	0.769847	+	0.638228i	$0.220332\pi$
$602$	0	0
$603$	−6.54994	−0.266734
$604$	0	0
$605$	−0.123663	−0.00502763
$606$	0	0
$607$	−46.8173	−1.90025	−0.950127	−	0.311864i	$-0.899047\pi$
$607$	−46.8173	−1.90025	−0.950127	+	0.311864i	$0.899047\pi$
$608$	0	0
$609$	−113.407	−4.59549
$610$	0	0
$611$	−30.7162	−1.24265
$612$	0	0
$613$	−26.8388	−1.08401	−0.542005	−	0.840375i	$-0.682334\pi$
$613$	−26.8388	−1.08401	−0.542005	+	0.840375i	$0.682334\pi$
$614$	0	0
$615$	−15.1182	−0.609625
$616$	0	0
$617$	−32.5303	−1.30962	−0.654810	−	0.755794i	$-0.727251\pi$
$617$	−32.5303	−1.30962	−0.654810	+	0.755794i	$0.727251\pi$
$618$	0	0
$619$	11.4178	0.458919	0.229459	−	0.973318i	$-0.426304\pi$
$619$	11.4178	0.458919	0.229459	+	0.973318i	$0.426304\pi$
$620$	0	0
$621$	−74.8628	−3.00414
$622$	0	0
$623$	−53.1284	−2.12854
$624$	0	0
$625$	22.0025	0.880100
$626$	0	0
$627$	29.2746	1.16912
$628$	0	0
$629$	−3.67085	−0.146366
$630$	0	0
$631$	−8.34575	−0.332239	−0.166119	−	0.986106i	$-0.553124\pi$
$631$	−8.34575	−0.332239	−0.166119	+	0.986106i	$0.553124\pi$
$632$	0	0
$633$	46.9843	1.86746
$634$	0	0
$635$	−1.82860	−0.0725656
$636$	0	0
$637$	39.3931	1.56081
$638$	0	0
$639$	56.3186	2.22793
$640$	0	0
$641$	−25.4813	−1.00645	−0.503226	−	0.864155i	$-0.667854\pi$
$641$	−25.4813	−1.00645	−0.503226	+	0.864155i	$0.667854\pi$
$642$	0	0
$643$	−27.1671	−1.07137	−0.535683	−	0.844419i	$-0.679946\pi$
$643$	−27.1671	−1.07137	−0.535683	+	0.844419i	$0.679946\pi$
$644$	0	0
$645$	−2.61966	−0.103149
$646$	0	0
$647$	−8.86149	−0.348381	−0.174191	−	0.984712i	$-0.555731\pi$
$647$	−8.86149	−0.348381	−0.174191	+	0.984712i	$0.555731\pi$
$648$	0	0
$649$	42.9761	1.68696
$650$	0	0
$651$	55.9008	2.19092
$652$	0	0
$653$	−26.4266	−1.03415	−0.517076	−	0.855940i	$-0.672979\pi$
$653$	−26.4266	−1.03415	−0.517076	+	0.855940i	$0.672979\pi$
$654$	0	0
$655$	−0.826947	−0.0323115
$656$	0	0
$657$	−49.0115	−1.91212
$658$	0	0
$659$	−37.4896	−1.46039	−0.730194	−	0.683240i	$-0.760570\pi$
$659$	−37.4896	−1.46039	−0.730194	+	0.683240i	$0.760570\pi$
$660$	0	0
$661$	2.50290	0.0973516	0.0486758	−	0.998815i	$-0.484500\pi$
$661$	2.50290	0.0973516	0.0486758	+	0.998815i	$0.484500\pi$
$662$	0	0
$663$	55.3551	2.14981
$664$	0	0
$665$	−5.72332	−0.221941
$666$	0	0
$667$	69.7148	2.69937
$668$	0	0
$669$	−5.49918	−0.212610
$670$	0	0
$671$	−12.5371	−0.483991
$672$	0	0
$673$	−10.2853	−0.396469	−0.198234	−	0.980155i	$-0.563521\pi$
$673$	−10.2853	−0.396469	−0.198234	+	0.980155i	$0.563521\pi$
$674$	0	0
$675$	44.2062	1.70150
$676$	0	0
$677$	25.0946	0.964463	0.482231	−	0.876044i	$-0.339826\pi$
$677$	25.0946	0.964463	0.482231	+	0.876044i	$0.339826\pi$
$678$	0	0
$679$	−43.0275	−1.65124
$680$	0	0
$681$	60.0376	2.30065
$682$	0	0
$683$	24.3508	0.931758	0.465879	−	0.884849i	$-0.345738\pi$
$683$	24.3508	0.931758	0.465879	+	0.884849i	$0.345738\pi$
$684$	0	0
$685$	−2.67029	−0.102027
$686$	0	0
$687$	−14.6236	−0.557926
$688$	0	0
$689$	−6.24613	−0.237959
$690$	0	0
$691$	−49.3708	−1.87816	−0.939078	−	0.343705i	$-0.888318\pi$
$691$	−49.3708	−1.87816	−0.939078	+	0.343705i	$0.888318\pi$
$692$	0	0
$693$	−89.3560	−3.39435
$694$	0	0
$695$	−6.02413	−0.228508
$696$	0	0
$697$	63.9512	2.42232
$698$	0	0
$699$	−85.8570	−3.24741
$700$	0	0
$701$	−35.9162	−1.35654	−0.678268	−	0.734814i	$-0.737269\pi$
$701$	−35.9162	−1.35654	−0.678268	+	0.734814i	$0.737269\pi$
$702$	0	0
$703$	1.85519	0.0699697
$704$	0	0
$705$	−12.9690	−0.488439
$706$	0	0
$707$	33.0714	1.24378
$708$	0	0
$709$	−42.3210	−1.58940	−0.794699	−	0.607004i	$-0.792371\pi$
$709$	−42.3210	−1.58940	−0.794699	+	0.607004i	$0.792371\pi$
$710$	0	0
$711$	−100.334	−3.76282
$712$	0	0
$713$	−34.3639	−1.28694
$714$	0	0
$715$	4.84931	0.181354
$716$	0	0
$717$	−15.3559	−0.573477
$718$	0	0
$719$	−1.26184	−0.0470588	−0.0235294	−	0.999723i	$-0.507490\pi$
$719$	−1.26184	−0.0470588	−0.0235294	+	0.999723i	$0.507490\pi$
$720$	0	0
$721$	−21.5851	−0.803872
$722$	0	0
$723$	21.5080	0.799893
$724$	0	0
$725$	−41.1663	−1.52888
$726$	0	0
$727$	29.5432	1.09570	0.547848	−	0.836578i	$-0.315447\pi$
$727$	29.5432	1.09570	0.547848	+	0.836578i	$0.315447\pi$
$728$	0	0
$729$	−25.3042	−0.937191
$730$	0	0
$731$	11.0814	0.409860
$732$	0	0
$733$	−23.0295	−0.850615	−0.425308	−	0.905049i	$-0.639834\pi$
$733$	−23.0295	−0.850615	−0.425308	+	0.905049i	$0.639834\pi$
$734$	0	0
$735$	16.6325	0.613498
$736$	0	0
$737$	3.62858	0.133660
$738$	0	0
$739$	1.18228	0.0434907	0.0217454	−	0.999764i	$-0.493078\pi$
$739$	1.18228	0.0434907	0.0217454	+	0.999764i	$0.493078\pi$
$740$	0	0
$741$	−27.9755	−1.02771
$742$	0	0
$743$	25.1796	0.923750	0.461875	−	0.886945i	$-0.347177\pi$
$743$	25.1796	0.923750	0.461875	+	0.886945i	$0.347177\pi$
$744$	0	0
$745$	−8.37115	−0.306695
$746$	0	0
$747$	27.2779	0.998046
$748$	0	0
$749$	−65.5151	−2.39387
$750$	0	0
$751$	12.4160	0.453065	0.226533	−	0.974004i	$-0.427261\pi$
$751$	12.4160	0.453065	0.226533	+	0.974004i	$0.427261\pi$
$752$	0	0
$753$	−8.15702	−0.297258
$754$	0	0
$755$	−3.89604	−0.141791
$756$	0	0
$757$	−12.9934	−0.472252	−0.236126	−	0.971722i	$-0.575878\pi$
$757$	−12.9934	−0.472252	−0.236126	+	0.971722i	$0.575878\pi$
$758$	0	0
$759$	82.1177	2.98068
$760$	0	0
$761$	36.6962	1.33024	0.665119	−	0.746738i	$-0.268381\pi$
$761$	36.6962	1.33024	0.665119	+	0.746738i	$0.268381\pi$
$762$	0	0
$763$	79.4667	2.87689
$764$	0	0
$765$	15.6338	0.565243
$766$	0	0
$767$	−41.0689	−1.48291
$768$	0	0
$769$	−27.0573	−0.975710	−0.487855	−	0.872925i	$-0.662221\pi$
$769$	−27.0573	−0.975710	−0.487855	+	0.872925i	$0.662221\pi$
$770$	0	0
$771$	79.4493	2.86130
$772$	0	0
$773$	16.0077	0.575757	0.287878	−	0.957667i	$-0.407050\pi$
$773$	16.0077	0.575757	0.287878	+	0.957667i	$0.407050\pi$
$774$	0	0
$775$	20.2917	0.728901
$776$	0	0
$777$	−8.46542	−0.303695
$778$	0	0
$779$	−32.3199	−1.15798
$780$	0	0
$781$	−31.1998	−1.11642
$782$	0	0
$783$	−79.0693	−2.82571
$784$	0	0
$785$	−9.44769	−0.337202
$786$	0	0
$787$	13.1883	0.470113	0.235057	−	0.971982i	$-0.424472\pi$
$787$	13.1883	0.470113	0.235057	+	0.971982i	$0.424472\pi$
$788$	0	0
$789$	67.1989	2.39234
$790$	0	0
$791$	70.7412	2.51527
$792$	0	0
$793$	11.9808	0.425450
$794$	0	0
$795$	−2.63723	−0.0935329
$796$	0	0
$797$	−19.7362	−0.699091	−0.349546	−	0.936919i	$-0.613664\pi$
$797$	−19.7362	−0.699091	−0.349546	+	0.936919i	$0.613664\pi$
$798$	0	0
$799$	54.8597	1.94080
$800$	0	0
$801$	−73.3441	−2.59149
$802$	0	0
$803$	27.1517	0.958164
$804$	0	0
$805$	−16.0544	−0.565842
$806$	0	0
$807$	−56.8845	−2.00243
$808$	0	0
$809$	−52.2766	−1.83795	−0.918973	−	0.394320i	$-0.870980\pi$
$809$	−52.2766	−1.83795	−0.918973	+	0.394320i	$0.870980\pi$
$810$	0	0
$811$	−55.0798	−1.93411	−0.967057	−	0.254559i	$-0.918070\pi$
$811$	−55.0798	−1.93411	−0.967057	+	0.254559i	$0.918070\pi$
$812$	0	0
$813$	26.1696	0.917809
$814$	0	0
$815$	2.39476	0.0838849
$816$	0	0
$817$	−5.60034	−0.195931
$818$	0	0
$819$	85.3906	2.98379
$820$	0	0
$821$	34.5869	1.20709	0.603546	−	0.797328i	$-0.293754\pi$
$821$	34.5869	1.20709	0.603546	+	0.797328i	$0.293754\pi$
$822$	0	0
$823$	10.9908	0.383115	0.191558	−	0.981481i	$-0.438646\pi$
$823$	10.9908	0.383115	0.191558	+	0.981481i	$0.438646\pi$
$824$	0	0
$825$	−48.4902	−1.68821
$826$	0	0
$827$	−7.43741	−0.258624	−0.129312	−	0.991604i	$-0.541277\pi$
$827$	−7.43741	−0.258624	−0.129312	+	0.991604i	$0.541277\pi$
$828$	0	0
$829$	−17.6233	−0.612081	−0.306041	−	0.952018i	$-0.599004\pi$
$829$	−17.6233	−0.612081	−0.306041	+	0.952018i	$0.599004\pi$
$830$	0	0
$831$	−72.2456	−2.50617
$832$	0	0
$833$	−70.3566	−2.43771
$834$	0	0
$835$	2.48793	0.0860983
$836$	0	0
$837$	38.9749	1.34717
$838$	0	0
$839$	−37.5894	−1.29773	−0.648865	−	0.760903i	$-0.724756\pi$
$839$	−37.5894	−1.29773	−0.648865	+	0.760903i	$0.724756\pi$
$840$	0	0
$841$	44.6321	1.53904
$842$	0	0
$843$	−25.4652	−0.877067
$844$	0	0
$845$	1.21688	0.0418618
$846$	0	0
$847$	1.20634	0.0414504
$848$	0	0
$849$	−9.78710	−0.335892
$850$	0	0
$851$	5.20395	0.178389
$852$	0	0
$853$	43.0032	1.47240	0.736200	−	0.676764i	$-0.236618\pi$
$853$	43.0032	1.47240	0.736200	+	0.676764i	$0.236618\pi$
$854$	0	0
$855$	−7.90108	−0.270211
$856$	0	0
$857$	−12.2767	−0.419364	−0.209682	−	0.977770i	$-0.567243\pi$
$857$	−12.2767	−0.419364	−0.209682	+	0.977770i	$0.567243\pi$
$858$	0	0
$859$	30.8503	1.05260	0.526299	−	0.850300i	$-0.323579\pi$
$859$	30.8503	1.05260	0.526299	+	0.850300i	$0.323579\pi$
$860$	0	0
$861$	147.479	5.02607
$862$	0	0
$863$	3.48991	0.118798	0.0593990	−	0.998234i	$-0.481082\pi$
$863$	3.48991	0.118798	0.0593990	+	0.998234i	$0.481082\pi$
$864$	0	0
$865$	5.65304	0.192209
$866$	0	0
$867$	−47.6921	−1.61971
$868$	0	0
$869$	55.5837	1.88555
$870$	0	0
$871$	−3.46755	−0.117494
$872$	0	0
$873$	−59.3998	−2.01038
$874$	0	0
$875$	19.3604	0.654500
$876$	0	0
$877$	−3.75855	−0.126917	−0.0634587	−	0.997984i	$-0.520213\pi$
$877$	−3.75855	−0.126917	−0.0634587	+	0.997984i	$0.520213\pi$
$878$	0	0
$879$	62.0217	2.09194
$880$	0	0
$881$	−3.67412	−0.123784	−0.0618920	−	0.998083i	$-0.519713\pi$
$881$	−3.67412	−0.123784	−0.0618920	+	0.998083i	$0.519713\pi$
$882$	0	0
$883$	28.7227	0.966597	0.483299	−	0.875456i	$-0.339439\pi$
$883$	28.7227	0.966597	0.483299	+	0.875456i	$0.339439\pi$
$884$	0	0
$885$	−17.3400	−0.582879
$886$	0	0
$887$	24.8372	0.833953	0.416977	−	0.908917i	$-0.363090\pi$
$887$	24.8372	0.833953	0.416977	+	0.908917i	$0.363090\pi$
$888$	0	0
$889$	17.8380	0.598269
$890$	0	0
$891$	−32.0803	−1.07473
$892$	0	0
$893$	−27.7252	−0.927787
$894$	0	0
$895$	10.1175	0.338190
$896$	0	0
$897$	−78.4736	−2.62016
$898$	0	0
$899$	−36.2948	−1.21050
$900$	0	0
$901$	11.1557	0.371650
$902$	0	0
$903$	25.5550	0.850416
$904$	0	0
$905$	0.598930	0.0199091
$906$	0	0
$907$	−43.6030	−1.44781	−0.723907	−	0.689898i	$-0.757655\pi$
$907$	−43.6030	−1.44781	−0.723907	+	0.689898i	$0.757655\pi$
$908$	0	0
$909$	45.6553	1.51429
$910$	0	0
$911$	19.4615	0.644790	0.322395	−	0.946605i	$-0.395512\pi$
$911$	19.4615	0.644790	0.322395	+	0.946605i	$0.395512\pi$
$912$	0	0
$913$	−15.1116	−0.500121
$914$	0	0
$915$	5.05851	0.167229
$916$	0	0
$917$	8.06691	0.266393
$918$	0	0
$919$	33.5780	1.10764	0.553818	−	0.832638i	$-0.313170\pi$
$919$	33.5780	1.10764	0.553818	+	0.832638i	$0.313170\pi$
$920$	0	0
$921$	−9.81944	−0.323562
$922$	0	0
$923$	29.8152	0.981379
$924$	0	0
$925$	−3.07291	−0.101037
$926$	0	0
$927$	−29.7984	−0.978708
$928$	0	0
$929$	42.3990	1.39106	0.695532	−	0.718495i	$-0.255169\pi$
$929$	42.3990	1.39106	0.695532	+	0.718495i	$0.255169\pi$
$930$	0	0
$931$	35.5571	1.16534
$932$	0	0
$933$	79.0966	2.58951
$934$	0	0
$935$	−8.66094	−0.283243
$936$	0	0
$937$	35.7814	1.16893	0.584464	−	0.811420i	$-0.301305\pi$
$937$	35.7814	1.16893	0.584464	+	0.811420i	$0.301305\pi$
$938$	0	0
$939$	−64.7971	−2.11457
$940$	0	0
$941$	11.1341	0.362961	0.181481	−	0.983395i	$-0.441911\pi$
$941$	11.1341	0.362961	0.181481	+	0.983395i	$0.441911\pi$
$942$	0	0
$943$	−90.6598	−2.95229
$944$	0	0
$945$	18.2086	0.592325
$946$	0	0
$947$	−5.36303	−0.174275	−0.0871375	−	0.996196i	$-0.527772\pi$
$947$	−5.36303	−0.174275	−0.0871375	+	0.996196i	$0.527772\pi$
$948$	0	0
$949$	−25.9468	−0.842270
$950$	0	0
$951$	19.4322	0.630132
$952$	0	0
$953$	12.2690	0.397432	0.198716	−	0.980057i	$-0.436323\pi$
$953$	12.2690	0.397432	0.198716	+	0.980057i	$0.436323\pi$
$954$	0	0
$955$	−7.83644	−0.253581
$956$	0	0
$957$	86.7318	2.80364
$958$	0	0
$959$	26.0489	0.841162
$960$	0	0
$961$	−13.1096	−0.422889
$962$	0	0
$963$	−90.4440	−2.91452
$964$	0	0
$965$	−0.871132	−0.0280427
$966$	0	0
$967$	−17.3489	−0.557903	−0.278952	−	0.960305i	$-0.589987\pi$
$967$	−17.3489	−0.557903	−0.278952	+	0.960305i	$0.589987\pi$
$968$	0	0
$969$	49.9647	1.60510
$970$	0	0
$971$	37.7567	1.21167	0.605835	−	0.795590i	$-0.292839\pi$
$971$	37.7567	1.21167	0.605835	+	0.795590i	$0.292839\pi$
$972$	0	0
$973$	58.7657	1.88394
$974$	0	0
$975$	46.3383	1.48401
$976$	0	0
$977$	−46.9810	−1.50306	−0.751528	−	0.659701i	$-0.770683\pi$
$977$	−46.9810	−1.50306	−0.751528	+	0.659701i	$0.770683\pi$
$978$	0	0
$979$	40.6317	1.29859
$980$	0	0
$981$	109.704	3.50259
$982$	0	0
$983$	32.4081	1.03366	0.516829	−	0.856088i	$-0.327112\pi$
$983$	32.4081	1.03366	0.516829	+	0.856088i	$0.327112\pi$
$984$	0	0
$985$	9.43446	0.300607
$986$	0	0
$987$	126.513	4.02695
$988$	0	0
$989$	−15.7094	−0.499530
$990$	0	0
$991$	4.84748	0.153985	0.0769926	−	0.997032i	$-0.475468\pi$
$991$	4.84748	0.153985	0.0769926	+	0.997032i	$0.475468\pi$
$992$	0	0
$993$	−27.7086	−0.879305
$994$	0	0
$995$	6.03048	0.191179
$996$	0	0
$997$	−37.2160	−1.17864	−0.589322	−	0.807898i	$-0.700605\pi$
$997$	−37.2160	−1.17864	−0.589322	+	0.807898i	$0.700605\pi$
$998$	0	0
$999$	−5.90222	−0.186738

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8044.2.a.a.1.6	✓	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8044.2.a.a.1.6	✓	80	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)

\(f(q)\)	\(=\)	\(q-3.01017 q^{3} -0.450076 q^{5} +4.39052 q^{7} +6.06114 q^{9} +O(q^{10})\)	\(q-3.01017 q^{3} -0.450076 q^{5} +4.39052 q^{7} +6.06114 q^{9} -3.35779 q^{11} +3.20878 q^{13} +1.35481 q^{15} -5.73094 q^{17} +2.89632 q^{19} -13.2162 q^{21} +8.12441 q^{23} -4.79743 q^{25} -9.21456 q^{27} +8.58091 q^{29} -4.22971 q^{31} +10.1075 q^{33} -1.97607 q^{35} +0.640533 q^{37} -9.65899 q^{39} -11.1589 q^{41} -1.93361 q^{43} -2.72797 q^{45} -9.57255 q^{47} +12.2766 q^{49} +17.2511 q^{51} -1.94657 q^{53} +1.51126 q^{55} -8.71842 q^{57} -12.7989 q^{59} +3.73375 q^{61} +26.6115 q^{63} -1.44420 q^{65} -1.08064 q^{67} -24.4559 q^{69} +9.29175 q^{71} -8.08619 q^{73} +14.4411 q^{75} -14.7424 q^{77} -16.5537 q^{79} +9.55399 q^{81} +4.50046 q^{83} +2.57936 q^{85} -25.8300 q^{87} -12.1007 q^{89} +14.0882 q^{91} +12.7322 q^{93} -1.30356 q^{95} -9.80010 q^{97} -20.3520 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9}+O(q^{10}) \) 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9	\( 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9} - 34 q^{11} - q^{13} - 24 q^{15} - 35 q^{17} - 31 q^{19} - 3 q^{21} - 43 q^{23} + 58 q^{25} - 49 q^{27} - 5 q^{29} - 56 q^{31} - 23 q^{33} - 72 q^{35} - 11 q^{37} - 74 q^{39} - 81 q^{41} - 34 q^{43} - 14 q^{45} - 64 q^{47} + 40 q^{49} - 59 q^{51} + 3 q^{53} - 53 q^{55} - 34 q^{57} - 116 q^{59} - 13 q^{61} - 61 q^{63} - 55 q^{65} - 22 q^{67} - 10 q^{69} - 86 q^{71} - 32 q^{73} - 85 q^{75} + 4 q^{77} - 88 q^{79} + 12 q^{81} - 83 q^{83} - 2 q^{85} - 87 q^{87} - 72 q^{89} - 49 q^{91} - 102 q^{95} - 34 q^{97} - 103 q^{99}+O(q^{100}) \) 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9 - 34 * q^11 - q^13 - 24 * q^15 - 35 * q^17 - 31 * q^19 - 3 * q^21 - 43 * q^23 + 58 * q^25 - 49 * q^27 - 5 * q^29 - 56 * q^31 - 23 * q^33 - 72 * q^35 - 11 * q^37 - 74 * q^39 - 81 * q^41 - 34 * q^43 - 14 * q^45 - 64 * q^47 + 40 * q^49 - 59 * q^51 + 3 * q^53 - 53 * q^55 - 34 * q^57 - 116 * q^59 - 13 * q^61 - 61 * q^63 - 55 * q^65 - 22 * q^67 - 10 * q^69 - 86 * q^71 - 32 * q^73 - 85 * q^75 + 4 * q^77 - 88 * q^79 + 12 * q^81 - 83 * q^83 - 2 * q^85 - 87 * q^87 - 72 * q^89 - 49 * q^91 - 102 * q^95 - 34 * q^97 - 103 * q^99

Introduction

L-functions

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