LMFDB - Embedded newform 8044.2.a.b.1.19

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			1.19
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)

$f(q)$	$=$	$q-1.89591 q^{3} +2.40728 q^{5} +3.56479 q^{7} +0.594467 q^{9} +O(q^{10})$	$q-1.89591 q^{3} +2.40728 q^{5} +3.56479 q^{7} +0.594467 q^{9} +2.17996 q^{11} +4.09162 q^{13} -4.56398 q^{15} -2.29636 q^{17} -0.788797 q^{19} -6.75852 q^{21} -0.456722 q^{23} +0.794998 q^{25} +4.56067 q^{27} +0.948023 q^{29} -6.80461 q^{31} -4.13301 q^{33} +8.58145 q^{35} +4.56613 q^{37} -7.75734 q^{39} -10.6544 q^{41} +3.63154 q^{43} +1.43105 q^{45} +8.37457 q^{47} +5.70775 q^{49} +4.35369 q^{51} +11.6348 q^{53} +5.24778 q^{55} +1.49549 q^{57} +6.54775 q^{59} +2.97102 q^{61} +2.11915 q^{63} +9.84968 q^{65} +10.1972 q^{67} +0.865902 q^{69} -12.8018 q^{71} +10.2878 q^{73} -1.50724 q^{75} +7.77112 q^{77} -0.928707 q^{79} -10.4300 q^{81} +3.81342 q^{83} -5.52799 q^{85} -1.79736 q^{87} +9.71536 q^{89} +14.5858 q^{91} +12.9009 q^{93} -1.89886 q^{95} -10.4177 q^{97} +1.29592 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.89591	−1.09460	−0.547301	−	0.836936i	$-0.684345\pi$
$3$	−1.89591	−1.09460	−0.547301	+	0.836936i	$0.684345\pi$
$4$	0	0
$5$	2.40728	1.07657	0.538284	−	0.842763i	$-0.319073\pi$
$5$	2.40728	1.07657	0.538284	+	0.842763i	$0.319073\pi$
$6$	0	0
$7$	3.56479	1.34736	0.673682	−	0.739021i	$-0.264712\pi$
$7$	3.56479	1.34736	0.673682	+	0.739021i	$0.264712\pi$
$8$	0	0
$9$	0.594467	0.198156
$10$	0	0
$11$	2.17996	0.657284	0.328642	−	0.944455i	$-0.393409\pi$
$11$	2.17996	0.657284	0.328642	+	0.944455i	$0.393409\pi$
$12$	0	0
$13$	4.09162	1.13481	0.567406	−	0.823438i	$-0.307947\pi$
$13$	4.09162	1.13481	0.567406	+	0.823438i	$0.307947\pi$
$14$	0	0
$15$	−4.56398	−1.17841
$16$	0	0
$17$	−2.29636	−0.556950	−0.278475	−	0.960444i	$-0.589829\pi$
$17$	−2.29636	−0.556950	−0.278475	+	0.960444i	$0.589829\pi$
$18$	0	0
$19$	−0.788797	−0.180962	−0.0904812	−	0.995898i	$-0.528841\pi$
$19$	−0.788797	−0.180962	−0.0904812	+	0.995898i	$0.528841\pi$
$20$	0	0
$21$	−6.75852	−1.47483
$22$	0	0
$23$	−0.456722	−0.0952331	−0.0476165	−	0.998866i	$-0.515163\pi$
$23$	−0.456722	−0.0952331	−0.0476165	+	0.998866i	$0.515163\pi$
$24$	0	0
$25$	0.794998	0.159000
$26$	0	0
$27$	4.56067	0.877701
$28$	0	0
$29$	0.948023	0.176043	0.0880217	−	0.996119i	$-0.471946\pi$
$29$	0.948023	0.176043	0.0880217	+	0.996119i	$0.471946\pi$
$30$	0	0
$31$	−6.80461	−1.22214	−0.611072	−	0.791575i	$-0.709262\pi$
$31$	−6.80461	−1.22214	−0.611072	+	0.791575i	$0.709262\pi$
$32$	0	0
$33$	−4.13301	−0.719465
$34$	0	0
$35$	8.58145	1.45053
$36$	0	0
$37$	4.56613	0.750666	0.375333	−	0.926890i	$-0.377528\pi$
$37$	4.56613	0.750666	0.375333	+	0.926890i	$0.377528\pi$
$38$	0	0
$39$	−7.75734	−1.24217
$40$	0	0
$41$	−10.6544	−1.66394	−0.831972	−	0.554817i	$-0.812788\pi$
$41$	−10.6544	−1.66394	−0.831972	+	0.554817i	$0.812788\pi$
$42$	0	0
$43$	3.63154	0.553804	0.276902	−	0.960898i	$-0.410692\pi$
$43$	3.63154	0.553804	0.276902	+	0.960898i	$0.410692\pi$
$44$	0	0
$45$	1.43105	0.213328
$46$	0	0
$47$	8.37457	1.22156	0.610779	−	0.791801i	$-0.290857\pi$
$47$	8.37457	1.22156	0.610779	+	0.791801i	$0.290857\pi$
$48$	0	0
$49$	5.70775	0.815392
$50$	0	0
$51$	4.35369	0.609639
$52$	0	0
$53$	11.6348	1.59817	0.799083	−	0.601221i	$-0.205319\pi$
$53$	11.6348	1.59817	0.799083	+	0.601221i	$0.205319\pi$
$54$	0	0
$55$	5.24778	0.707611
$56$	0	0
$57$	1.49549	0.198082
$58$	0	0
$59$	6.54775	0.852444	0.426222	−	0.904619i	$-0.359844\pi$
$59$	6.54775	0.852444	0.426222	+	0.904619i	$0.359844\pi$
$60$	0	0
$61$	2.97102	0.380400	0.190200	−	0.981745i	$-0.439086\pi$
$61$	2.97102	0.380400	0.190200	+	0.981745i	$0.439086\pi$
$62$	0	0
$63$	2.11915	0.266988
$64$	0	0
$65$	9.84968	1.22170
$66$	0	0
$67$	10.1972	1.24579	0.622896	−	0.782305i	$-0.285956\pi$
$67$	10.1972	1.24579	0.622896	+	0.782305i	$0.285956\pi$
$68$	0	0
$69$	0.865902	0.104242
$70$	0	0
$71$	−12.8018	−1.51929	−0.759647	−	0.650335i	$-0.774628\pi$
$71$	−12.8018	−1.51929	−0.759647	+	0.650335i	$0.774628\pi$
$72$	0	0
$73$	10.2878	1.20410	0.602049	−	0.798459i	$-0.294351\pi$
$73$	10.2878	1.20410	0.602049	+	0.798459i	$0.294351\pi$
$74$	0	0
$75$	−1.50724	−0.174041
$76$	0	0
$77$	7.77112	0.885601
$78$	0	0
$79$	−0.928707	−0.104488	−0.0522438	−	0.998634i	$-0.516637\pi$
$79$	−0.928707	−0.104488	−0.0522438	+	0.998634i	$0.516637\pi$
$80$	0	0
$81$	−10.4300	−1.15889
$82$	0	0
$83$	3.81342	0.418577	0.209289	−	0.977854i	$-0.432885\pi$
$83$	3.81342	0.418577	0.209289	+	0.977854i	$0.432885\pi$
$84$	0	0
$85$	−5.52799	−0.599594
$86$	0	0
$87$	−1.79736	−0.192698
$88$	0	0
$89$	9.71536	1.02983	0.514913	−	0.857242i	$-0.327824\pi$
$89$	9.71536	1.02983	0.514913	+	0.857242i	$0.327824\pi$
$90$	0	0
$91$	14.5858	1.52901
$92$	0	0
$93$	12.9009	1.33776
$94$	0	0
$95$	−1.89886	−0.194818
$96$	0	0
$97$	−10.4177	−1.05775	−0.528877	−	0.848698i	$-0.677387\pi$
$97$	−10.4177	−1.05775	−0.528877	+	0.848698i	$0.677387\pi$
$98$	0	0
$99$	1.29592	0.130244
$100$	0	0
$101$	6.64129	0.660833	0.330416	−	0.943835i	$-0.392811\pi$
$101$	6.64129	0.660833	0.330416	+	0.943835i	$0.392811\pi$
$102$	0	0
$103$	−1.47483	−0.145319	−0.0726597	−	0.997357i	$-0.523149\pi$
$103$	−1.47483	−0.145319	−0.0726597	+	0.997357i	$0.523149\pi$
$104$	0	0
$105$	−16.2696	−1.58776
$106$	0	0
$107$	7.62899	0.737522	0.368761	−	0.929524i	$-0.379782\pi$
$107$	7.62899	0.737522	0.368761	+	0.929524i	$0.379782\pi$
$108$	0	0
$109$	−10.4944	−1.00518	−0.502589	−	0.864525i	$-0.667619\pi$
$109$	−10.4944	−1.00518	−0.502589	+	0.864525i	$0.667619\pi$
$110$	0	0
$111$	−8.65695	−0.821682
$112$	0	0
$113$	7.63449	0.718193	0.359096	−	0.933300i	$-0.383085\pi$
$113$	7.63449	0.718193	0.359096	+	0.933300i	$0.383085\pi$
$114$	0	0
$115$	−1.09946	−0.102525
$116$	0	0
$117$	2.43233	0.224869
$118$	0	0
$119$	−8.18605	−0.750414
$120$	0	0
$121$	−6.24776	−0.567978
$122$	0	0
$123$	20.1998	1.82136
$124$	0	0
$125$	−10.1226	−0.905394
$126$	0	0
$127$	−3.50979	−0.311443	−0.155722	−	0.987801i	$-0.549770\pi$
$127$	−3.50979	−0.311443	−0.155722	+	0.987801i	$0.549770\pi$
$128$	0	0
$129$	−6.88506	−0.606196
$130$	0	0
$131$	−12.8516	−1.12285	−0.561423	−	0.827529i	$-0.689746\pi$
$131$	−12.8516	−1.12285	−0.561423	+	0.827529i	$0.689746\pi$
$132$	0	0
$133$	−2.81190	−0.243822
$134$	0	0
$135$	10.9788	0.944906
$136$	0	0
$137$	10.8641	0.928183	0.464092	−	0.885787i	$-0.346381\pi$
$137$	10.8641	0.928183	0.464092	+	0.885787i	$0.346381\pi$
$138$	0	0
$139$	7.96634	0.675696	0.337848	−	0.941201i	$-0.390301\pi$
$139$	7.96634	0.675696	0.337848	+	0.941201i	$0.390301\pi$
$140$	0	0
$141$	−15.8774	−1.33712
$142$	0	0
$143$	8.91959	0.745893
$144$	0	0
$145$	2.28216	0.189523
$146$	0	0
$147$	−10.8214	−0.892531
$148$	0	0
$149$	−17.0580	−1.39744	−0.698722	−	0.715393i	$-0.746248\pi$
$149$	−17.0580	−1.39744	−0.698722	+	0.715393i	$0.746248\pi$
$150$	0	0
$151$	3.42415	0.278653	0.139327	−	0.990246i	$-0.455506\pi$
$151$	3.42415	0.278653	0.139327	+	0.990246i	$0.455506\pi$
$152$	0	0
$153$	−1.36511	−0.110363
$154$	0	0
$155$	−16.3806	−1.31572
$156$	0	0
$157$	−0.677925	−0.0541043	−0.0270522	−	0.999634i	$-0.508612\pi$
$157$	−0.677925	−0.0541043	−0.0270522	+	0.999634i	$0.508612\pi$
$158$	0	0
$159$	−22.0585	−1.74936
$160$	0	0
$161$	−1.62812	−0.128314
$162$	0	0
$163$	4.34226	0.340112	0.170056	−	0.985434i	$-0.445605\pi$
$163$	4.34226	0.340112	0.170056	+	0.985434i	$0.445605\pi$
$164$	0	0
$165$	−9.94932	−0.774553
$166$	0	0
$167$	4.93108	0.381578	0.190789	−	0.981631i	$-0.438895\pi$
$167$	4.93108	0.381578	0.190789	+	0.981631i	$0.438895\pi$
$168$	0	0
$169$	3.74137	0.287797
$170$	0	0
$171$	−0.468913	−0.0358587
$172$	0	0
$173$	13.5608	1.03101	0.515503	−	0.856888i	$-0.327605\pi$
$173$	13.5608	1.03101	0.515503	+	0.856888i	$0.327605\pi$
$174$	0	0
$175$	2.83400	0.214230
$176$	0	0
$177$	−12.4139	−0.933088
$178$	0	0
$179$	7.52594	0.562515	0.281258	−	0.959632i	$-0.409248\pi$
$179$	7.52594	0.562515	0.281258	+	0.959632i	$0.409248\pi$
$180$	0	0
$181$	10.9758	0.815825	0.407913	−	0.913021i	$-0.366257\pi$
$181$	10.9758	0.815825	0.407913	+	0.913021i	$0.366257\pi$
$182$	0	0
$183$	−5.63278	−0.416387
$184$	0	0
$185$	10.9919	0.808144
$186$	0	0
$187$	−5.00599	−0.366074
$188$	0	0
$189$	16.2578	1.18258
$190$	0	0
$191$	24.9946	1.80855	0.904273	−	0.426955i	$-0.140414\pi$
$191$	24.9946	1.80855	0.904273	+	0.426955i	$0.140414\pi$
$192$	0	0
$193$	−10.2408	−0.737147	−0.368573	−	0.929599i	$-0.620154\pi$
$193$	−10.2408	−0.737147	−0.368573	+	0.929599i	$0.620154\pi$
$194$	0	0
$195$	−18.6741	−1.33728
$196$	0	0
$197$	3.56560	0.254039	0.127019	−	0.991900i	$-0.459459\pi$
$197$	3.56560	0.254039	0.127019	+	0.991900i	$0.459459\pi$
$198$	0	0
$199$	−18.2076	−1.29070	−0.645351	−	0.763886i	$-0.723289\pi$
$199$	−18.2076	−1.29070	−0.645351	+	0.763886i	$0.723289\pi$
$200$	0	0
$201$	−19.3330	−1.36365
$202$	0	0
$203$	3.37950	0.237195
$204$	0	0
$205$	−25.6482	−1.79135
$206$	0	0
$207$	−0.271506	−0.0188710
$208$	0	0
$209$	−1.71955	−0.118944
$210$	0	0
$211$	9.94828	0.684868	0.342434	−	0.939542i	$-0.388749\pi$
$211$	9.94828	0.684868	0.342434	+	0.939542i	$0.388749\pi$
$212$	0	0
$213$	24.2710	1.66302
$214$	0	0
$215$	8.74213	0.596208
$216$	0	0
$217$	−24.2570	−1.64667
$218$	0	0
$219$	−19.5048	−1.31801
$220$	0	0
$221$	−9.39584	−0.632033
$222$	0	0
$223$	24.1967	1.62033	0.810167	−	0.586200i	$-0.199377\pi$
$223$	24.1967	1.62033	0.810167	+	0.586200i	$0.199377\pi$
$224$	0	0
$225$	0.472600	0.0315066
$226$	0	0
$227$	20.4013	1.35408	0.677041	−	0.735945i	$-0.263262\pi$
$227$	20.4013	1.35408	0.677041	+	0.735945i	$0.263262\pi$
$228$	0	0
$229$	−5.47847	−0.362027	−0.181014	−	0.983481i	$-0.557938\pi$
$229$	−5.47847	−0.362027	−0.181014	+	0.983481i	$0.557938\pi$
$230$	0	0
$231$	−14.7333	−0.969382
$232$	0	0
$233$	−17.6848	−1.15857	−0.579286	−	0.815124i	$-0.696669\pi$
$233$	−17.6848	−1.15857	−0.579286	+	0.815124i	$0.696669\pi$
$234$	0	0
$235$	20.1599	1.31509
$236$	0	0
$237$	1.76074	0.114372
$238$	0	0
$239$	7.85448	0.508064	0.254032	−	0.967196i	$-0.418243\pi$
$239$	7.85448	0.508064	0.254032	+	0.967196i	$0.418243\pi$
$240$	0	0
$241$	5.72364	0.368692	0.184346	−	0.982861i	$-0.440983\pi$
$241$	5.72364	0.368692	0.184346	+	0.982861i	$0.440983\pi$
$242$	0	0
$243$	6.09233	0.390823
$244$	0	0
$245$	13.7401	0.877826
$246$	0	0
$247$	−3.22746	−0.205358
$248$	0	0
$249$	−7.22989	−0.458176
$250$	0	0
$251$	16.5122	1.04224	0.521120	−	0.853484i	$-0.325515\pi$
$251$	16.5122	1.04224	0.521120	+	0.853484i	$0.325515\pi$
$252$	0	0
$253$	−0.995637	−0.0625952
$254$	0	0
$255$	10.4806	0.656318
$256$	0	0
$257$	−30.9370	−1.92980	−0.964900	−	0.262618i	$-0.915414\pi$
$257$	−30.9370	−1.92980	−0.964900	+	0.262618i	$0.915414\pi$
$258$	0	0
$259$	16.2773	1.01142
$260$	0	0
$261$	0.563568	0.0348840
$262$	0	0
$263$	9.35423	0.576806	0.288403	−	0.957509i	$-0.406876\pi$
$263$	9.35423	0.576806	0.288403	+	0.957509i	$0.406876\pi$
$264$	0	0
$265$	28.0083	1.72053
$266$	0	0
$267$	−18.4194	−1.12725
$268$	0	0
$269$	−25.8603	−1.57673	−0.788367	−	0.615206i	$-0.789073\pi$
$269$	−25.8603	−1.57673	−0.788367	+	0.615206i	$0.789073\pi$
$270$	0	0
$271$	−19.0467	−1.15700	−0.578501	−	0.815682i	$-0.696362\pi$
$271$	−19.0467	−1.15700	−0.578501	+	0.815682i	$0.696362\pi$
$272$	0	0
$273$	−27.6533	−1.67365
$274$	0	0
$275$	1.73307	0.104508
$276$	0	0
$277$	2.22542	0.133712	0.0668562	−	0.997763i	$-0.478703\pi$
$277$	2.22542	0.133712	0.0668562	+	0.997763i	$0.478703\pi$
$278$	0	0
$279$	−4.04512	−0.242175
$280$	0	0
$281$	−20.2186	−1.20614	−0.603072	−	0.797687i	$-0.706057\pi$
$281$	−20.2186	−1.20614	−0.603072	+	0.797687i	$0.706057\pi$
$282$	0	0
$283$	7.96606	0.473533	0.236767	−	0.971567i	$-0.423912\pi$
$283$	7.96606	0.473533	0.236767	+	0.971567i	$0.423912\pi$
$284$	0	0
$285$	3.60005	0.213249
$286$	0	0
$287$	−37.9809	−2.24194
$288$	0	0
$289$	−11.7267	−0.689807
$290$	0	0
$291$	19.7509	1.15782
$292$	0	0
$293$	−15.5442	−0.908104	−0.454052	−	0.890975i	$-0.650022\pi$
$293$	−15.5442	−0.908104	−0.454052	+	0.890975i	$0.650022\pi$
$294$	0	0
$295$	15.7623	0.917714
$296$	0	0
$297$	9.94210	0.576899
$298$	0	0
$299$	−1.86873	−0.108072
$300$	0	0
$301$	12.9457	0.746177
$302$	0	0
$303$	−12.5913	−0.723349
$304$	0	0
$305$	7.15207	0.409526
$306$	0	0
$307$	6.31817	0.360597	0.180299	−	0.983612i	$-0.442294\pi$
$307$	6.31817	0.360597	0.180299	+	0.983612i	$0.442294\pi$
$308$	0	0
$309$	2.79614	0.159067
$310$	0	0
$311$	6.70290	0.380087	0.190043	−	0.981776i	$-0.439137\pi$
$311$	6.70290	0.380087	0.190043	+	0.981776i	$0.439137\pi$
$312$	0	0
$313$	2.95254	0.166887	0.0834437	−	0.996512i	$-0.473408\pi$
$313$	2.95254	0.166887	0.0834437	+	0.996512i	$0.473408\pi$
$314$	0	0
$315$	5.10139	0.287431
$316$	0	0
$317$	−11.7507	−0.659986	−0.329993	−	0.943983i	$-0.607046\pi$
$317$	−11.7507	−0.659986	−0.329993	+	0.943983i	$0.607046\pi$
$318$	0	0
$319$	2.06666	0.115710
$320$	0	0
$321$	−14.4639	−0.807294
$322$	0	0
$323$	1.81136	0.100787
$324$	0	0
$325$	3.25283	0.180435
$326$	0	0
$327$	19.8964	1.10027
$328$	0	0
$329$	29.8536	1.64588
$330$	0	0
$331$	−19.1480	−1.05247	−0.526234	−	0.850340i	$-0.676396\pi$
$331$	−19.1480	−1.05247	−0.526234	+	0.850340i	$0.676396\pi$
$332$	0	0
$333$	2.71441	0.148749
$334$	0	0
$335$	24.5476	1.34118
$336$	0	0
$337$	−6.04708	−0.329405	−0.164703	−	0.986343i	$-0.552666\pi$
$337$	−6.04708	−0.329405	−0.164703	+	0.986343i	$0.552666\pi$
$338$	0	0
$339$	−14.4743	−0.786136
$340$	0	0
$341$	−14.8338	−0.803296
$342$	0	0
$343$	−4.60661	−0.248734
$344$	0	0
$345$	2.08447	0.112224
$346$	0	0
$347$	−0.333819	−0.0179203	−0.00896017	−	0.999960i	$-0.502852\pi$
$347$	−0.333819	−0.0179203	−0.00896017	+	0.999960i	$0.502852\pi$
$348$	0	0
$349$	−21.9679	−1.17591	−0.587956	−	0.808893i	$-0.700067\pi$
$349$	−21.9679	−1.17591	−0.587956	+	0.808893i	$0.700067\pi$
$350$	0	0
$351$	18.6605	0.996026
$352$	0	0
$353$	19.2457	1.02434	0.512172	−	0.858883i	$-0.328841\pi$
$353$	19.2457	1.02434	0.512172	+	0.858883i	$0.328841\pi$
$354$	0	0
$355$	−30.8175	−1.63562
$356$	0	0
$357$	15.5200	0.821406
$358$	0	0
$359$	9.43360	0.497886	0.248943	−	0.968518i	$-0.419917\pi$
$359$	9.43360	0.497886	0.248943	+	0.968518i	$0.419917\pi$
$360$	0	0
$361$	−18.3778	−0.967253
$362$	0	0
$363$	11.8452	0.621710
$364$	0	0
$365$	24.7657	1.29629
$366$	0	0
$367$	4.19646	0.219054	0.109527	−	0.993984i	$-0.465066\pi$
$367$	4.19646	0.219054	0.109527	+	0.993984i	$0.465066\pi$
$368$	0	0
$369$	−6.33371	−0.329720
$370$	0	0
$371$	41.4757	2.15331
$372$	0	0
$373$	15.3412	0.794339	0.397169	−	0.917745i	$-0.369993\pi$
$373$	15.3412	0.794339	0.397169	+	0.917745i	$0.369993\pi$
$374$	0	0
$375$	19.1916	0.991047
$376$	0	0
$377$	3.87895	0.199776
$378$	0	0
$379$	−29.5354	−1.51713	−0.758565	−	0.651597i	$-0.774099\pi$
$379$	−29.5354	−1.51713	−0.758565	+	0.651597i	$0.774099\pi$
$380$	0	0
$381$	6.65423	0.340907
$382$	0	0
$383$	−14.5624	−0.744104	−0.372052	−	0.928212i	$-0.621346\pi$
$383$	−14.5624	−0.744104	−0.372052	+	0.928212i	$0.621346\pi$
$384$	0	0
$385$	18.7073	0.953410
$386$	0	0
$387$	2.15883	0.109739
$388$	0	0
$389$	−3.40892	−0.172839	−0.0864195	−	0.996259i	$-0.527543\pi$
$389$	−3.40892	−0.172839	−0.0864195	+	0.996259i	$0.527543\pi$
$390$	0	0
$391$	1.04880	0.0530400
$392$	0	0
$393$	24.3654	1.22907
$394$	0	0
$395$	−2.23566	−0.112488
$396$	0	0
$397$	−0.398950	−0.0200227	−0.0100114	−	0.999950i	$-0.503187\pi$
$397$	−0.398950	−0.0200227	−0.0100114	+	0.999950i	$0.503187\pi$
$398$	0	0
$399$	5.33110	0.266889
$400$	0	0
$401$	7.26135	0.362615	0.181307	−	0.983426i	$-0.441967\pi$
$401$	7.26135	0.362615	0.181307	+	0.983426i	$0.441967\pi$
$402$	0	0
$403$	−27.8419	−1.38690
$404$	0	0
$405$	−25.1080	−1.24762
$406$	0	0
$407$	9.95399	0.493401
$408$	0	0
$409$	16.6107	0.821347	0.410674	−	0.911782i	$-0.365294\pi$
$409$	16.6107	0.821347	0.410674	+	0.911782i	$0.365294\pi$
$410$	0	0
$411$	−20.5973	−1.01599
$412$	0	0
$413$	23.3414	1.14855
$414$	0	0
$415$	9.17997	0.450627
$416$	0	0
$417$	−15.1035	−0.739619
$418$	0	0
$419$	21.2117	1.03626	0.518129	−	0.855303i	$-0.326629\pi$
$419$	21.2117	1.03626	0.518129	+	0.855303i	$0.326629\pi$
$420$	0	0
$421$	24.2454	1.18165	0.590824	−	0.806800i	$-0.298803\pi$
$421$	24.2454	1.18165	0.590824	+	0.806800i	$0.298803\pi$
$422$	0	0
$423$	4.97840	0.242058
$424$	0	0
$425$	−1.82560	−0.0885548
$426$	0	0
$427$	10.5911	0.512537
$428$	0	0
$429$	−16.9107	−0.816457
$430$	0	0
$431$	28.8888	1.39153	0.695763	−	0.718271i	$-0.255066\pi$
$431$	28.8888	1.39153	0.695763	+	0.718271i	$0.255066\pi$
$432$	0	0
$433$	18.9686	0.911572	0.455786	−	0.890089i	$-0.349358\pi$
$433$	18.9686	0.911572	0.455786	+	0.890089i	$0.349358\pi$
$434$	0	0
$435$	−4.32676	−0.207452
$436$	0	0
$437$	0.360261	0.0172336
$438$	0	0
$439$	21.4172	1.02219	0.511093	−	0.859526i	$-0.329241\pi$
$439$	21.4172	1.02219	0.511093	+	0.859526i	$0.329241\pi$
$440$	0	0
$441$	3.39306	0.161575
$442$	0	0
$443$	16.6969	0.793296	0.396648	−	0.917971i	$-0.370173\pi$
$443$	16.6969	0.793296	0.396648	+	0.917971i	$0.370173\pi$
$444$	0	0
$445$	23.3876	1.10868
$446$	0	0
$447$	32.3404	1.52965
$448$	0	0
$449$	32.5174	1.53459	0.767296	−	0.641293i	$-0.221602\pi$
$449$	32.5174	1.53459	0.767296	+	0.641293i	$0.221602\pi$
$450$	0	0
$451$	−23.2263	−1.09368
$452$	0	0
$453$	−6.49187	−0.305015
$454$	0	0
$455$	35.1121	1.64608
$456$	0	0
$457$	−0.833467	−0.0389879	−0.0194940	−	0.999810i	$-0.506206\pi$
$457$	−0.833467	−0.0389879	−0.0194940	+	0.999810i	$0.506206\pi$
$458$	0	0
$459$	−10.4729	−0.488835
$460$	0	0
$461$	34.1796	1.59190	0.795951	−	0.605361i	$-0.206971\pi$
$461$	34.1796	1.59190	0.795951	+	0.605361i	$0.206971\pi$
$462$	0	0
$463$	5.51791	0.256439	0.128219	−	0.991746i	$-0.459074\pi$
$463$	5.51791	0.256439	0.128219	+	0.991746i	$0.459074\pi$
$464$	0	0
$465$	31.0561	1.44019
$466$	0	0
$467$	12.2146	0.565222	0.282611	−	0.959235i	$-0.408799\pi$
$467$	12.2146	0.565222	0.282611	+	0.959235i	$0.408799\pi$
$468$	0	0
$469$	36.3511	1.67854
$470$	0	0
$471$	1.28528	0.0592228
$472$	0	0
$473$	7.91662	0.364007
$474$	0	0
$475$	−0.627092	−0.0287730
$476$	0	0
$477$	6.91651	0.316685
$478$	0	0
$479$	−6.18972	−0.282816	−0.141408	−	0.989951i	$-0.545163\pi$
$479$	−6.18972	−0.282816	−0.141408	+	0.989951i	$0.545163\pi$
$480$	0	0
$481$	18.6829	0.851865
$482$	0	0
$483$	3.08676	0.140453
$484$	0	0
$485$	−25.0782	−1.13874
$486$	0	0
$487$	11.2512	0.509839	0.254920	−	0.966962i	$-0.417951\pi$
$487$	11.2512	0.509839	0.254920	+	0.966962i	$0.417951\pi$
$488$	0	0
$489$	−8.23252	−0.372288
$490$	0	0
$491$	17.5565	0.792314	0.396157	−	0.918183i	$-0.370344\pi$
$491$	17.5565	0.792314	0.396157	+	0.918183i	$0.370344\pi$
$492$	0	0
$493$	−2.17700	−0.0980473
$494$	0	0
$495$	3.11963	0.140217
$496$	0	0
$497$	−45.6358	−2.04704
$498$	0	0
$499$	6.59779	0.295358	0.147679	−	0.989035i	$-0.452820\pi$
$499$	6.59779	0.295358	0.147679	+	0.989035i	$0.452820\pi$
$500$	0	0
$501$	−9.34887	−0.417677
$502$	0	0
$503$	−5.71726	−0.254920	−0.127460	−	0.991844i	$-0.540682\pi$
$503$	−5.71726	−0.254920	−0.127460	+	0.991844i	$0.540682\pi$
$504$	0	0
$505$	15.9874	0.711432
$506$	0	0
$507$	−7.09329	−0.315024
$508$	0	0
$509$	−18.6126	−0.824991	−0.412495	−	0.910960i	$-0.635343\pi$
$509$	−18.6126	−0.824991	−0.412495	+	0.910960i	$0.635343\pi$
$510$	0	0
$511$	36.6739	1.62236
$512$	0	0
$513$	−3.59744	−0.158831
$514$	0	0
$515$	−3.55033	−0.156446
$516$	0	0
$517$	18.2563	0.802910
$518$	0	0
$519$	−25.7100	−1.12854
$520$	0	0
$521$	−8.02414	−0.351544	−0.175772	−	0.984431i	$-0.556242\pi$
$521$	−8.02414	−0.351544	−0.175772	+	0.984431i	$0.556242\pi$
$522$	0	0
$523$	−12.0749	−0.528000	−0.264000	−	0.964523i	$-0.585042\pi$
$523$	−12.0749	−0.528000	−0.264000	+	0.964523i	$0.585042\pi$
$524$	0	0
$525$	−5.37301	−0.234497
$526$	0	0
$527$	15.6259	0.680673
$528$	0	0
$529$	−22.7914	−0.990931
$530$	0	0
$531$	3.89242	0.168917
$532$	0	0
$533$	−43.5940	−1.88826
$534$	0	0
$535$	18.3651	0.793993
$536$	0	0
$537$	−14.2685	−0.615731
$538$	0	0
$539$	12.4427	0.535944
$540$	0	0
$541$	−32.0044	−1.37598	−0.687989	−	0.725721i	$-0.741506\pi$
$541$	−32.0044	−1.37598	−0.687989	+	0.725721i	$0.741506\pi$
$542$	0	0
$543$	−20.8091	−0.893005
$544$	0	0
$545$	−25.2629	−1.08214
$546$	0	0
$547$	10.9484	0.468120	0.234060	−	0.972222i	$-0.424799\pi$
$547$	10.9484	0.468120	0.234060	+	0.972222i	$0.424799\pi$
$548$	0	0
$549$	1.76617	0.0753783
$550$	0	0
$551$	−0.747797	−0.0318572
$552$	0	0
$553$	−3.31065	−0.140783
$554$	0	0
$555$	−20.8397	−0.884597
$556$	0	0
$557$	−19.6366	−0.832028	−0.416014	−	0.909358i	$-0.636573\pi$
$557$	−19.6366	−0.832028	−0.416014	+	0.909358i	$0.636573\pi$
$558$	0	0
$559$	14.8589	0.628464
$560$	0	0
$561$	9.49089	0.400706
$562$	0	0
$563$	27.1284	1.14333	0.571664	−	0.820488i	$-0.306298\pi$
$563$	27.1284	1.14333	0.571664	+	0.820488i	$0.306298\pi$
$564$	0	0
$565$	18.3784	0.773184
$566$	0	0
$567$	−37.1808	−1.56145
$568$	0	0
$569$	21.8058	0.914147	0.457073	−	0.889429i	$-0.348898\pi$
$569$	21.8058	0.914147	0.457073	+	0.889429i	$0.348898\pi$
$570$	0	0
$571$	6.56002	0.274528	0.137264	−	0.990534i	$-0.456169\pi$
$571$	6.56002	0.274528	0.137264	+	0.990534i	$0.456169\pi$
$572$	0	0
$573$	−47.3875	−1.97964
$574$	0	0
$575$	−0.363093	−0.0151420
$576$	0	0
$577$	22.9602	0.955846	0.477923	−	0.878402i	$-0.341390\pi$
$577$	22.9602	0.955846	0.477923	+	0.878402i	$0.341390\pi$
$578$	0	0
$579$	19.4156	0.806883
$580$	0	0
$581$	13.5941	0.563976
$582$	0	0
$583$	25.3635	1.05045
$584$	0	0
$585$	5.85530	0.242087
$586$	0	0
$587$	−18.4307	−0.760717	−0.380358	−	0.924839i	$-0.624199\pi$
$587$	−18.4307	−0.760717	−0.380358	+	0.924839i	$0.624199\pi$
$588$	0	0
$589$	5.36746	0.221162
$590$	0	0
$591$	−6.76005	−0.278071
$592$	0	0
$593$	40.9341	1.68096	0.840480	−	0.541843i	$-0.182273\pi$
$593$	40.9341	1.68096	0.840480	+	0.541843i	$0.182273\pi$
$594$	0	0
$595$	−19.7061	−0.807872
$596$	0	0
$597$	34.5199	1.41281
$598$	0	0
$599$	−5.46754	−0.223397	−0.111699	−	0.993742i	$-0.535629\pi$
$599$	−5.46754	−0.223397	−0.111699	+	0.993742i	$0.535629\pi$
$600$	0	0
$601$	26.1810	1.06795	0.533973	−	0.845502i	$-0.320699\pi$
$601$	26.1810	1.06795	0.533973	+	0.845502i	$0.320699\pi$
$602$	0	0
$603$	6.06192	0.246860
$604$	0	0
$605$	−15.0401	−0.611467
$606$	0	0
$607$	27.6491	1.12224	0.561121	−	0.827734i	$-0.310370\pi$
$607$	27.6491	1.12224	0.561121	+	0.827734i	$0.310370\pi$
$608$	0	0
$609$	−6.40723	−0.259634
$610$	0	0
$611$	34.2656	1.38624
$612$	0	0
$613$	−24.7554	−0.999861	−0.499930	−	0.866066i	$-0.666641\pi$
$613$	−24.7554	−0.999861	−0.499930	+	0.866066i	$0.666641\pi$
$614$	0	0
$615$	48.6267	1.96082
$616$	0	0
$617$	−29.0222	−1.16839	−0.584194	−	0.811614i	$-0.698589\pi$
$617$	−29.0222	−1.16839	−0.584194	+	0.811614i	$0.698589\pi$
$618$	0	0
$619$	14.5145	0.583386	0.291693	−	0.956512i	$-0.405781\pi$
$619$	14.5145	0.583386	0.291693	+	0.956512i	$0.405781\pi$
$620$	0	0
$621$	−2.08296	−0.0835862
$622$	0	0
$623$	34.6333	1.38755
$624$	0	0
$625$	−28.3430	−1.13372
$626$	0	0
$627$	3.26011	0.130196
$628$	0	0
$629$	−10.4855	−0.418083
$630$	0	0
$631$	−14.2444	−0.567061	−0.283530	−	0.958963i	$-0.591506\pi$
$631$	−14.2444	−0.567061	−0.283530	+	0.958963i	$0.591506\pi$
$632$	0	0
$633$	−18.8610	−0.749658
$634$	0	0
$635$	−8.44904	−0.335290
$636$	0	0
$637$	23.3539	0.925317
$638$	0	0
$639$	−7.61024	−0.301057
$640$	0	0
$641$	28.4831	1.12501	0.562507	−	0.826792i	$-0.309837\pi$
$641$	28.4831	1.12501	0.562507	+	0.826792i	$0.309837\pi$
$642$	0	0
$643$	−29.7330	−1.17255	−0.586277	−	0.810111i	$-0.699407\pi$
$643$	−29.7330	−1.17255	−0.586277	+	0.810111i	$0.699407\pi$
$644$	0	0
$645$	−16.5743	−0.652611
$646$	0	0
$647$	−23.4806	−0.923119	−0.461560	−	0.887109i	$-0.652710\pi$
$647$	−23.4806	−0.923119	−0.461560	+	0.887109i	$0.652710\pi$
$648$	0	0
$649$	14.2739	0.560298
$650$	0	0
$651$	45.9891	1.80246
$652$	0	0
$653$	11.4673	0.448748	0.224374	−	0.974503i	$-0.427966\pi$
$653$	11.4673	0.448748	0.224374	+	0.974503i	$0.427966\pi$
$654$	0	0
$655$	−30.9373	−1.20882
$656$	0	0
$657$	6.11576	0.238599
$658$	0	0
$659$	−16.7577	−0.652786	−0.326393	−	0.945234i	$-0.605833\pi$
$659$	−16.7577	−0.652786	−0.326393	+	0.945234i	$0.605833\pi$
$660$	0	0
$661$	−29.1175	−1.13254	−0.566269	−	0.824221i	$-0.691614\pi$
$661$	−29.1175	−1.13254	−0.566269	+	0.824221i	$0.691614\pi$
$662$	0	0
$663$	17.8137	0.691825
$664$	0	0
$665$	−6.76902	−0.262492
$666$	0	0
$667$	−0.432983	−0.0167652
$668$	0	0
$669$	−45.8748	−1.77362
$670$	0	0
$671$	6.47671	0.250031
$672$	0	0
$673$	13.0014	0.501168	0.250584	−	0.968095i	$-0.419377\pi$
$673$	13.0014	0.501168	0.250584	+	0.968095i	$0.419377\pi$
$674$	0	0
$675$	3.62572	0.139554
$676$	0	0
$677$	42.5331	1.63468	0.817340	−	0.576155i	$-0.195448\pi$
$677$	42.5331	1.63468	0.817340	+	0.576155i	$0.195448\pi$
$678$	0	0
$679$	−37.1368	−1.42518
$680$	0	0
$681$	−38.6790	−1.48218
$682$	0	0
$683$	4.32510	0.165495	0.0827477	−	0.996571i	$-0.473630\pi$
$683$	4.32510	0.165495	0.0827477	+	0.996571i	$0.473630\pi$
$684$	0	0
$685$	26.1529	0.999253
$686$	0	0
$687$	10.3867	0.396276
$688$	0	0
$689$	47.6053	1.81362
$690$	0	0
$691$	−9.31644	−0.354414	−0.177207	−	0.984174i	$-0.556706\pi$
$691$	−9.31644	−0.354414	−0.177207	+	0.984174i	$0.556706\pi$
$692$	0	0
$693$	4.61967	0.175487
$694$	0	0
$695$	19.1772	0.727433
$696$	0	0
$697$	24.4665	0.926733
$698$	0	0
$699$	33.5288	1.26818
$700$	0	0
$701$	−35.0094	−1.32229	−0.661144	−	0.750259i	$-0.729929\pi$
$701$	−35.0094	−1.32229	−0.661144	+	0.750259i	$0.729929\pi$
$702$	0	0
$703$	−3.60175	−0.135842
$704$	0	0
$705$	−38.2214	−1.43950
$706$	0	0
$707$	23.6748	0.890383
$708$	0	0
$709$	36.5459	1.37251	0.686255	−	0.727361i	$-0.259253\pi$
$709$	36.5459	1.37251	0.686255	+	0.727361i	$0.259253\pi$
$710$	0	0
$711$	−0.552085	−0.0207048
$712$	0	0
$713$	3.10781	0.116389
$714$	0	0
$715$	21.4719	0.803005
$716$	0	0
$717$	−14.8914	−0.556128
$718$	0	0
$719$	7.96347	0.296987	0.148494	−	0.988913i	$-0.452558\pi$
$719$	7.96347	0.296987	0.148494	+	0.988913i	$0.452558\pi$
$720$	0	0
$721$	−5.25747	−0.195798
$722$	0	0
$723$	−10.8515	−0.403571
$724$	0	0
$725$	0.753676	0.0279908
$726$	0	0
$727$	−28.7878	−1.06768	−0.533839	−	0.845586i	$-0.679251\pi$
$727$	−28.7878	−1.06768	−0.533839	+	0.845586i	$0.679251\pi$
$728$	0	0
$729$	19.7395	0.731094
$730$	0	0
$731$	−8.33932	−0.308441
$732$	0	0
$733$	−31.5289	−1.16455	−0.582273	−	0.812993i	$-0.697837\pi$
$733$	−31.5289	−1.16455	−0.582273	+	0.812993i	$0.697837\pi$
$734$	0	0
$735$	−26.0501	−0.960871
$736$	0	0
$737$	22.2296	0.818839
$738$	0	0
$739$	−19.3797	−0.712895	−0.356447	−	0.934315i	$-0.616012\pi$
$739$	−19.3797	−0.712895	−0.356447	+	0.934315i	$0.616012\pi$
$740$	0	0
$741$	6.11896	0.224786
$742$	0	0
$743$	10.3888	0.381128	0.190564	−	0.981675i	$-0.438968\pi$
$743$	10.3888	0.381128	0.190564	+	0.981675i	$0.438968\pi$
$744$	0	0
$745$	−41.0633	−1.50444
$746$	0	0
$747$	2.26695	0.0829434
$748$	0	0
$749$	27.1958	0.993712
$750$	0	0
$751$	28.6823	1.04663	0.523315	−	0.852139i	$-0.324695\pi$
$751$	28.6823	1.04663	0.523315	+	0.852139i	$0.324695\pi$
$752$	0	0
$753$	−31.3056	−1.14084
$754$	0	0
$755$	8.24288	0.299989
$756$	0	0
$757$	40.4813	1.47132	0.735658	−	0.677353i	$-0.236873\pi$
$757$	40.4813	1.47132	0.735658	+	0.677353i	$0.236873\pi$
$758$	0	0
$759$	1.88764	0.0685168
$760$	0	0
$761$	−12.0485	−0.436757	−0.218379	−	0.975864i	$-0.570077\pi$
$761$	−12.0485	−0.436757	−0.218379	+	0.975864i	$0.570077\pi$
$762$	0	0
$763$	−37.4102	−1.35434
$764$	0	0
$765$	−3.28620	−0.118813
$766$	0	0
$767$	26.7909	0.967364
$768$	0	0
$769$	−23.5781	−0.850247	−0.425124	−	0.905135i	$-0.639769\pi$
$769$	−23.5781	−0.850247	−0.425124	+	0.905135i	$0.639769\pi$
$770$	0	0
$771$	58.6538	2.11236
$772$	0	0
$773$	7.35133	0.264409	0.132205	−	0.991222i	$-0.457794\pi$
$773$	7.35133	0.264409	0.132205	+	0.991222i	$0.457794\pi$
$774$	0	0
$775$	−5.40965	−0.194321
$776$	0	0
$777$	−30.8602	−1.10711
$778$	0	0
$779$	8.40419	0.301111
$780$	0	0
$781$	−27.9075	−0.998608
$782$	0	0
$783$	4.32362	0.154514
$784$	0	0
$785$	−1.63196	−0.0582470
$786$	0	0
$787$	4.45109	0.158664	0.0793321	−	0.996848i	$-0.474721\pi$
$787$	4.45109	0.158664	0.0793321	+	0.996848i	$0.474721\pi$
$788$	0	0
$789$	−17.7348	−0.631374
$790$	0	0
$791$	27.2154	0.967668
$792$	0	0
$793$	12.1563	0.431682
$794$	0	0
$795$	−53.1011	−1.88330
$796$	0	0
$797$	48.8759	1.73127	0.865637	−	0.500673i	$-0.166914\pi$
$797$	48.8759	1.73127	0.865637	+	0.500673i	$0.166914\pi$
$798$	0	0
$799$	−19.2311	−0.680346
$800$	0	0
$801$	5.77546	0.204066
$802$	0	0
$803$	22.4271	0.791434
$804$	0	0
$805$	−3.91934	−0.138138
$806$	0	0
$807$	49.0288	1.72590
$808$	0	0
$809$	1.57471	0.0553639	0.0276819	−	0.999617i	$-0.491187\pi$
$809$	1.57471	0.0553639	0.0276819	+	0.999617i	$0.491187\pi$
$810$	0	0
$811$	14.4450	0.507232	0.253616	−	0.967305i	$-0.418380\pi$
$811$	14.4450	0.507232	0.253616	+	0.967305i	$0.418380\pi$
$812$	0	0
$813$	36.1107	1.26646
$814$	0	0
$815$	10.4530	0.366154
$816$	0	0
$817$	−2.86455	−0.100218
$818$	0	0
$819$	8.67076	0.302981
$820$	0	0
$821$	18.1173	0.632299	0.316150	−	0.948709i	$-0.397610\pi$
$821$	18.1173	0.632299	0.316150	+	0.948709i	$0.397610\pi$
$822$	0	0
$823$	18.6558	0.650300	0.325150	−	0.945662i	$-0.394585\pi$
$823$	18.6558	0.650300	0.325150	+	0.945662i	$0.394585\pi$
$824$	0	0
$825$	−3.28574	−0.114395
$826$	0	0
$827$	16.4604	0.572385	0.286193	−	0.958172i	$-0.407610\pi$
$827$	16.4604	0.572385	0.286193	+	0.958172i	$0.407610\pi$
$828$	0	0
$829$	1.89445	0.0657971	0.0328985	−	0.999459i	$-0.489526\pi$
$829$	1.89445	0.0657971	0.0328985	+	0.999459i	$0.489526\pi$
$830$	0	0
$831$	−4.21919	−0.146362
$832$	0	0
$833$	−13.1071	−0.454132
$834$	0	0
$835$	11.8705	0.410795
$836$	0	0
$837$	−31.0336	−1.07268
$838$	0	0
$839$	−29.2600	−1.01017	−0.505083	−	0.863071i	$-0.668538\pi$
$839$	−29.2600	−1.01017	−0.505083	+	0.863071i	$0.668538\pi$
$840$	0	0
$841$	−28.1013	−0.969009
$842$	0	0
$843$	38.3327	1.32025
$844$	0	0
$845$	9.00652	0.309834
$846$	0	0
$847$	−22.2720	−0.765274
$848$	0	0
$849$	−15.1029	−0.518331
$850$	0	0
$851$	−2.08545	−0.0714883
$852$	0	0
$853$	33.3428	1.14164	0.570818	−	0.821077i	$-0.306626\pi$
$853$	33.3428	1.14164	0.570818	+	0.821077i	$0.306626\pi$
$854$	0	0
$855$	−1.12881	−0.0386043
$856$	0	0
$857$	−39.8572	−1.36149	−0.680747	−	0.732518i	$-0.738345\pi$
$857$	−39.8572	−1.36149	−0.680747	+	0.732518i	$0.738345\pi$
$858$	0	0
$859$	−50.8169	−1.73385	−0.866925	−	0.498439i	$-0.833907\pi$
$859$	−50.8169	−1.73385	−0.866925	+	0.498439i	$0.833907\pi$
$860$	0	0
$861$	72.0083	2.45403
$862$	0	0
$863$	36.2340	1.23342	0.616710	−	0.787190i	$-0.288465\pi$
$863$	36.2340	1.23342	0.616710	+	0.787190i	$0.288465\pi$
$864$	0	0
$865$	32.6446	1.10995
$866$	0	0
$867$	22.2328	0.755065
$868$	0	0
$869$	−2.02455	−0.0686781
$870$	0	0
$871$	41.7233	1.41374
$872$	0	0
$873$	−6.19296	−0.209600
$874$	0	0
$875$	−36.0850	−1.21990
$876$	0	0
$877$	12.5575	0.424036	0.212018	−	0.977266i	$-0.431996\pi$
$877$	12.5575	0.424036	0.212018	+	0.977266i	$0.431996\pi$
$878$	0	0
$879$	29.4704	0.994013
$880$	0	0
$881$	−5.85119	−0.197132	−0.0985659	−	0.995131i	$-0.531426\pi$
$881$	−5.85119	−0.197132	−0.0985659	+	0.995131i	$0.531426\pi$
$882$	0	0
$883$	−2.73555	−0.0920586	−0.0460293	−	0.998940i	$-0.514657\pi$
$883$	−2.73555	−0.0920586	−0.0460293	+	0.998940i	$0.514657\pi$
$884$	0	0
$885$	−29.8838	−1.00453
$886$	0	0
$887$	7.72095	0.259244	0.129622	−	0.991563i	$-0.458624\pi$
$887$	7.72095	0.259244	0.129622	+	0.991563i	$0.458624\pi$
$888$	0	0
$889$	−12.5117	−0.419628
$890$	0	0
$891$	−22.7370	−0.761720
$892$	0	0
$893$	−6.60584	−0.221056
$894$	0	0
$895$	18.1171	0.605586
$896$	0	0
$897$	3.54294	0.118295
$898$	0	0
$899$	−6.45093	−0.215150
$900$	0	0
$901$	−26.7178	−0.890097
$902$	0	0
$903$	−24.5438	−0.816767
$904$	0	0
$905$	26.4218	0.878292
$906$	0	0
$907$	30.9706	1.02836	0.514182	−	0.857681i	$-0.328096\pi$
$907$	30.9706	1.02836	0.514182	+	0.857681i	$0.328096\pi$
$908$	0	0
$909$	3.94802	0.130948
$910$	0	0
$911$	42.2043	1.39829	0.699145	−	0.714980i	$-0.253564\pi$
$911$	42.2043	1.39829	0.699145	+	0.714980i	$0.253564\pi$
$912$	0	0
$913$	8.31312	0.275124
$914$	0	0
$915$	−13.5597	−0.448269
$916$	0	0
$917$	−45.8132	−1.51288
$918$	0	0
$919$	−5.93931	−0.195920	−0.0979599	−	0.995190i	$-0.531232\pi$
$919$	−5.93931	−0.195920	−0.0979599	+	0.995190i	$0.531232\pi$
$920$	0	0
$921$	−11.9787	−0.394711
$922$	0	0
$923$	−52.3801	−1.72411
$924$	0	0
$925$	3.63006	0.119356
$926$	0	0
$927$	−0.876738	−0.0287958
$928$	0	0
$929$	−20.6365	−0.677063	−0.338531	−	0.940955i	$-0.609930\pi$
$929$	−20.6365	−0.677063	−0.338531	+	0.940955i	$0.609930\pi$
$930$	0	0
$931$	−4.50225	−0.147555
$932$	0	0
$933$	−12.7081	−0.416044
$934$	0	0
$935$	−12.0508	−0.394104
$936$	0	0
$937$	−44.7982	−1.46349	−0.731746	−	0.681577i	$-0.761294\pi$
$937$	−44.7982	−1.46349	−0.731746	+	0.681577i	$0.761294\pi$
$938$	0	0
$939$	−5.59774	−0.182675
$940$	0	0
$941$	−23.3059	−0.759750	−0.379875	−	0.925038i	$-0.624033\pi$
$941$	−23.3059	−0.759750	−0.379875	+	0.925038i	$0.624033\pi$
$942$	0	0
$943$	4.86612	0.158463
$944$	0	0
$945$	39.1372	1.27313
$946$	0	0
$947$	−1.29061	−0.0419392	−0.0209696	−	0.999780i	$-0.506675\pi$
$947$	−1.29061	−0.0419392	−0.0209696	+	0.999780i	$0.506675\pi$
$948$	0	0
$949$	42.0939	1.36642
$950$	0	0
$951$	22.2783	0.722422
$952$	0	0
$953$	−13.2290	−0.428528	−0.214264	−	0.976776i	$-0.568735\pi$
$953$	−13.2290	−0.428528	−0.214264	+	0.976776i	$0.568735\pi$
$954$	0	0
$955$	60.1690	1.94702
$956$	0	0
$957$	−3.91819	−0.126657
$958$	0	0
$959$	38.7283	1.25060
$960$	0	0
$961$	15.3028	0.493638
$962$	0	0
$963$	4.53518	0.146144
$964$	0	0
$965$	−24.6524	−0.793589
$966$	0	0
$967$	10.8748	0.349711	0.174856	−	0.984594i	$-0.444054\pi$
$967$	10.8748	0.349711	0.174856	+	0.984594i	$0.444054\pi$
$968$	0	0
$969$	−3.43418	−0.110322
$970$	0	0
$971$	−23.4820	−0.753573	−0.376787	−	0.926300i	$-0.622971\pi$
$971$	−23.4820	−0.753573	−0.376787	+	0.926300i	$0.622971\pi$
$972$	0	0
$973$	28.3984	0.910410
$974$	0	0
$975$	−6.16707	−0.197504
$976$	0	0
$977$	−36.6374	−1.17213	−0.586067	−	0.810262i	$-0.699325\pi$
$977$	−36.6374	−1.17213	−0.586067	+	0.810262i	$0.699325\pi$
$978$	0	0
$979$	21.1791	0.676888
$980$	0	0
$981$	−6.23855	−0.199182
$982$	0	0
$983$	0.347103	0.0110709	0.00553543	−	0.999985i	$-0.498238\pi$
$983$	0.347103	0.0110709	0.00553543	+	0.999985i	$0.498238\pi$
$984$	0	0
$985$	8.58340	0.273490
$986$	0	0
$987$	−56.5997	−1.80159
$988$	0	0
$989$	−1.65860	−0.0527405
$990$	0	0
$991$	−37.1601	−1.18043	−0.590214	−	0.807247i	$-0.700957\pi$
$991$	−37.1601	−1.18043	−0.590214	+	0.807247i	$0.700957\pi$
$992$	0	0
$993$	36.3028	1.15203
$994$	0	0
$995$	−43.8308	−1.38953
$996$	0	0
$997$	13.3289	0.422131	0.211066	−	0.977472i	$-0.432307\pi$
$997$	13.3289	0.422131	0.211066	+	0.977472i	$0.432307\pi$
$998$	0	0
$999$	20.8246	0.658861

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8044.2.a.b.1.19	✓	87

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8044.2.a.b.1.19	✓	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)

\(f(q)\)	\(=\)	\(q-1.89591 q^{3} +2.40728 q^{5} +3.56479 q^{7} +0.594467 q^{9} +O(q^{10})\)	\(q-1.89591 q^{3} +2.40728 q^{5} +3.56479 q^{7} +0.594467 q^{9} +2.17996 q^{11} +4.09162 q^{13} -4.56398 q^{15} -2.29636 q^{17} -0.788797 q^{19} -6.75852 q^{21} -0.456722 q^{23} +0.794998 q^{25} +4.56067 q^{27} +0.948023 q^{29} -6.80461 q^{31} -4.13301 q^{33} +8.58145 q^{35} +4.56613 q^{37} -7.75734 q^{39} -10.6544 q^{41} +3.63154 q^{43} +1.43105 q^{45} +8.37457 q^{47} +5.70775 q^{49} +4.35369 q^{51} +11.6348 q^{53} +5.24778 q^{55} +1.49549 q^{57} +6.54775 q^{59} +2.97102 q^{61} +2.11915 q^{63} +9.84968 q^{65} +10.1972 q^{67} +0.865902 q^{69} -12.8018 q^{71} +10.2878 q^{73} -1.50724 q^{75} +7.77112 q^{77} -0.928707 q^{79} -10.4300 q^{81} +3.81342 q^{83} -5.52799 q^{85} -1.79736 q^{87} +9.71536 q^{89} +14.5858 q^{91} +12.9009 q^{93} -1.89886 q^{95} -10.4177 q^{97} +1.29592 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 87 q + 13 q^{3} - 2 q^{5} + 8 q^{7} + 98 q^{9}+O(q^{10}) \) 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

L-functions

Modular forms

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Database

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