LMFDB - Embedded newform 8044.2.a.b.1.13

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			1.13
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)

$f(q)$	$=$	$q-2.39922 q^{3} -4.31592 q^{5} -4.16363 q^{7} +2.75627 q^{9} +O(q^{10})$	$q-2.39922 q^{3} -4.31592 q^{5} -4.16363 q^{7} +2.75627 q^{9} +5.53109 q^{11} -1.86792 q^{13} +10.3549 q^{15} +4.49755 q^{17} -1.21613 q^{19} +9.98948 q^{21} -1.47938 q^{23} +13.6271 q^{25} +0.584756 q^{27} -8.11550 q^{29} +2.10985 q^{31} -13.2703 q^{33} +17.9699 q^{35} -1.53384 q^{37} +4.48155 q^{39} -2.27595 q^{41} -6.63787 q^{43} -11.8958 q^{45} -3.35069 q^{47} +10.3358 q^{49} -10.7906 q^{51} -7.59794 q^{53} -23.8717 q^{55} +2.91776 q^{57} +4.88615 q^{59} +5.43627 q^{61} -11.4761 q^{63} +8.06177 q^{65} -15.8307 q^{67} +3.54937 q^{69} -11.6385 q^{71} -4.40824 q^{73} -32.6946 q^{75} -23.0294 q^{77} -8.44498 q^{79} -9.67178 q^{81} +0.645620 q^{83} -19.4110 q^{85} +19.4709 q^{87} -6.00235 q^{89} +7.77731 q^{91} -5.06200 q^{93} +5.24871 q^{95} +10.8315 q^{97} +15.2452 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.39922	−1.38519	−0.692596	−	0.721326i	$-0.743533\pi$
$3$	−2.39922	−1.38519	−0.692596	+	0.721326i	$0.743533\pi$
$4$	0	0
$5$	−4.31592	−1.93014	−0.965069	−	0.261998i	$-0.915619\pi$
$5$	−4.31592	−1.93014	−0.965069	+	0.261998i	$0.915619\pi$
$6$	0	0
$7$	−4.16363	−1.57370	−0.786852	−	0.617141i	$-0.788291\pi$
$7$	−4.16363	−1.57370	−0.786852	+	0.617141i	$0.788291\pi$
$8$	0	0
$9$	2.75627	0.918758
$10$	0	0
$11$	5.53109	1.66768	0.833842	−	0.552002i	$-0.186136\pi$
$11$	5.53109	1.66768	0.833842	+	0.552002i	$0.186136\pi$
$12$	0	0
$13$	−1.86792	−0.518067	−0.259033	−	0.965868i	$-0.583404\pi$
$13$	−1.86792	−0.518067	−0.259033	+	0.965868i	$0.583404\pi$
$14$	0	0
$15$	10.3549	2.67361
$16$	0	0
$17$	4.49755	1.09082	0.545408	−	0.838171i	$-0.316375\pi$
$17$	4.49755	1.09082	0.545408	+	0.838171i	$0.316375\pi$
$18$	0	0
$19$	−1.21613	−0.278999	−0.139499	−	0.990222i	$-0.544549\pi$
$19$	−1.21613	−0.278999	−0.139499	+	0.990222i	$0.544549\pi$
$20$	0	0
$21$	9.98948	2.17988
$22$	0	0
$23$	−1.47938	−0.308473	−0.154236	−	0.988034i	$-0.549292\pi$
$23$	−1.47938	−0.308473	−0.154236	+	0.988034i	$0.549292\pi$
$24$	0	0
$25$	13.6271	2.72543
$26$	0	0
$27$	0.584756	0.112536
$28$	0	0
$29$	−8.11550	−1.50701	−0.753505	−	0.657442i	$-0.771639\pi$
$29$	−8.11550	−1.50701	−0.753505	+	0.657442i	$0.771639\pi$
$30$	0	0
$31$	2.10985	0.378940	0.189470	−	0.981887i	$-0.439323\pi$
$31$	2.10985	0.378940	0.189470	+	0.981887i	$0.439323\pi$
$32$	0	0
$33$	−13.2703	−2.31006
$34$	0	0
$35$	17.9699	3.03747
$36$	0	0
$37$	−1.53384	−0.252161	−0.126081	−	0.992020i	$-0.540240\pi$
$37$	−1.53384	−0.252161	−0.126081	+	0.992020i	$0.540240\pi$
$38$	0	0
$39$	4.48155	0.717622
$40$	0	0
$41$	−2.27595	−0.355443	−0.177721	−	0.984081i	$-0.556873\pi$
$41$	−2.27595	−0.355443	−0.177721	+	0.984081i	$0.556873\pi$
$42$	0	0
$43$	−6.63787	−1.01227	−0.506133	−	0.862456i	$-0.668925\pi$
$43$	−6.63787	−1.01227	−0.506133	+	0.862456i	$0.668925\pi$
$44$	0	0
$45$	−11.8958	−1.77333
$46$	0	0
$47$	−3.35069	−0.488748	−0.244374	−	0.969681i	$-0.578582\pi$
$47$	−3.35069	−0.488748	−0.244374	+	0.969681i	$0.578582\pi$
$48$	0	0
$49$	10.3358	1.47655
$50$	0	0
$51$	−10.7906	−1.51099
$52$	0	0
$53$	−7.59794	−1.04366	−0.521828	−	0.853050i	$-0.674750\pi$
$53$	−7.59794	−1.04366	−0.521828	+	0.853050i	$0.674750\pi$
$54$	0	0
$55$	−23.8717	−3.21886
$56$	0	0
$57$	2.91776	0.386467
$58$	0	0
$59$	4.88615	0.636123	0.318062	−	0.948070i	$-0.396968\pi$
$59$	4.88615	0.636123	0.318062	+	0.948070i	$0.396968\pi$
$60$	0	0
$61$	5.43627	0.696043	0.348021	−	0.937487i	$-0.386854\pi$
$61$	5.43627	0.696043	0.348021	+	0.937487i	$0.386854\pi$
$62$	0	0
$63$	−11.4761	−1.44585
$64$	0	0
$65$	8.06177	0.999940
$66$	0	0
$67$	−15.8307	−1.93403	−0.967013	−	0.254729i	$-0.918014\pi$
$67$	−15.8307	−1.93403	−0.967013	+	0.254729i	$0.918014\pi$
$68$	0	0
$69$	3.54937	0.427294
$70$	0	0
$71$	−11.6385	−1.38123	−0.690616	−	0.723221i	$-0.742661\pi$
$71$	−11.6385	−1.38123	−0.690616	+	0.723221i	$0.742661\pi$
$72$	0	0
$73$	−4.40824	−0.515945	−0.257973	−	0.966152i	$-0.583054\pi$
$73$	−4.40824	−0.515945	−0.257973	+	0.966152i	$0.583054\pi$
$74$	0	0
$75$	−32.6946	−3.77524
$76$	0	0
$77$	−23.0294	−2.62444
$78$	0	0
$79$	−8.44498	−0.950135	−0.475067	−	0.879949i	$-0.657576\pi$
$79$	−8.44498	−0.950135	−0.475067	+	0.879949i	$0.657576\pi$
$80$	0	0
$81$	−9.67178	−1.07464
$82$	0	0
$83$	0.645620	0.0708660	0.0354330	−	0.999372i	$-0.488719\pi$
$83$	0.645620	0.0708660	0.0354330	+	0.999372i	$0.488719\pi$
$84$	0	0
$85$	−19.4110	−2.10542
$86$	0	0
$87$	19.4709	2.08750
$88$	0	0
$89$	−6.00235	−0.636248	−0.318124	−	0.948049i	$-0.603053\pi$
$89$	−6.00235	−0.636248	−0.318124	+	0.948049i	$0.603053\pi$
$90$	0	0
$91$	7.77731	0.815284
$92$	0	0
$93$	−5.06200	−0.524905
$94$	0	0
$95$	5.24871	0.538506
$96$	0	0
$97$	10.8315	1.09977	0.549884	−	0.835241i	$-0.314672\pi$
$97$	10.8315	1.09977	0.549884	+	0.835241i	$0.314672\pi$
$98$	0	0
$99$	15.2452	1.53220
$100$	0	0
$101$	0.529193	0.0526567	0.0263284	−	0.999653i	$-0.491618\pi$
$101$	0.529193	0.0526567	0.0263284	+	0.999653i	$0.491618\pi$
$102$	0	0
$103$	19.1901	1.89085	0.945427	−	0.325834i	$-0.105645\pi$
$103$	19.1901	1.89085	0.945427	+	0.325834i	$0.105645\pi$
$104$	0	0
$105$	−43.1138	−4.20747
$106$	0	0
$107$	2.34224	0.226433	0.113216	−	0.993570i	$-0.463885\pi$
$107$	2.34224	0.226433	0.113216	+	0.993570i	$0.463885\pi$
$108$	0	0
$109$	4.52347	0.433270	0.216635	−	0.976253i	$-0.430492\pi$
$109$	4.52347	0.433270	0.216635	+	0.976253i	$0.430492\pi$
$110$	0	0
$111$	3.68002	0.349292
$112$	0	0
$113$	−1.59898	−0.150420	−0.0752099	−	0.997168i	$-0.523963\pi$
$113$	−1.59898	−0.150420	−0.0752099	+	0.997168i	$0.523963\pi$
$114$	0	0
$115$	6.38490	0.595395
$116$	0	0
$117$	−5.14849	−0.475978
$118$	0	0
$119$	−18.7261	−1.71662
$120$	0	0
$121$	19.5929	1.78117
$122$	0	0
$123$	5.46050	0.492357
$124$	0	0
$125$	−37.2341	−3.33031
$126$	0	0
$127$	−13.2857	−1.17891	−0.589457	−	0.807800i	$-0.700658\pi$
$127$	−13.2857	−1.17891	−0.589457	+	0.807800i	$0.700658\pi$
$128$	0	0
$129$	15.9257	1.40218
$130$	0	0
$131$	−3.13295	−0.273727	−0.136863	−	0.990590i	$-0.543702\pi$
$131$	−3.13295	−0.273727	−0.136863	+	0.990590i	$0.543702\pi$
$132$	0	0
$133$	5.06351	0.439062
$134$	0	0
$135$	−2.52376	−0.217211
$136$	0	0
$137$	−12.8005	−1.09362	−0.546810	−	0.837256i	$-0.684158\pi$
$137$	−12.8005	−1.09362	−0.546810	+	0.837256i	$0.684158\pi$
$138$	0	0
$139$	−7.82322	−0.663557	−0.331779	−	0.943357i	$-0.607649\pi$
$139$	−7.82322	−0.663557	−0.331779	+	0.943357i	$0.607649\pi$
$140$	0	0
$141$	8.03904	0.677010
$142$	0	0
$143$	−10.3316	−0.863972
$144$	0	0
$145$	35.0258	2.90874
$146$	0	0
$147$	−24.7979	−2.04530
$148$	0	0
$149$	−0.222528	−0.0182302	−0.00911508	−	0.999958i	$-0.502901\pi$
$149$	−0.222528	−0.0182302	−0.00911508	+	0.999958i	$0.502901\pi$
$150$	0	0
$151$	11.3680	0.925112	0.462556	−	0.886590i	$-0.346932\pi$
$151$	11.3680	0.925112	0.462556	+	0.886590i	$0.346932\pi$
$152$	0	0
$153$	12.3965	1.00219
$154$	0	0
$155$	−9.10593	−0.731406
$156$	0	0
$157$	−18.8222	−1.50217	−0.751087	−	0.660204i	$-0.770470\pi$
$157$	−18.8222	−1.50217	−0.751087	+	0.660204i	$0.770470\pi$
$158$	0	0
$159$	18.2291	1.44567
$160$	0	0
$161$	6.15961	0.485445
$162$	0	0
$163$	−12.6418	−0.990181	−0.495090	−	0.868841i	$-0.664865\pi$
$163$	−12.6418	−0.990181	−0.495090	+	0.868841i	$0.664865\pi$
$164$	0	0
$165$	57.2736	4.45874
$166$	0	0
$167$	7.33167	0.567341	0.283671	−	0.958922i	$-0.408448\pi$
$167$	7.33167	0.567341	0.283671	+	0.958922i	$0.408448\pi$
$168$	0	0
$169$	−9.51089	−0.731607
$170$	0	0
$171$	−3.35198	−0.256332
$172$	0	0
$173$	10.6545	0.810045	0.405022	−	0.914307i	$-0.367264\pi$
$173$	10.6545	0.810045	0.405022	+	0.914307i	$0.367264\pi$
$174$	0	0
$175$	−56.7384	−4.28902
$176$	0	0
$177$	−11.7230	−0.881153
$178$	0	0
$179$	10.5151	0.785935	0.392967	−	0.919552i	$-0.371449\pi$
$179$	10.5151	0.785935	0.392967	+	0.919552i	$0.371449\pi$
$180$	0	0
$181$	−10.0545	−0.747342	−0.373671	−	0.927561i	$-0.621901\pi$
$181$	−10.0545	−0.747342	−0.373671	+	0.927561i	$0.621901\pi$
$182$	0	0
$183$	−13.0428	−0.964153
$184$	0	0
$185$	6.61992	0.486706
$186$	0	0
$187$	24.8763	1.81914
$188$	0	0
$189$	−2.43471	−0.177099
$190$	0	0
$191$	−14.1023	−1.02041	−0.510204	−	0.860053i	$-0.670430\pi$
$191$	−14.1023	−1.02041	−0.510204	+	0.860053i	$0.670430\pi$
$192$	0	0
$193$	−19.8420	−1.42826	−0.714130	−	0.700014i	$-0.753177\pi$
$193$	−19.8420	−1.42826	−0.714130	+	0.700014i	$0.753177\pi$
$194$	0	0
$195$	−19.3420	−1.38511
$196$	0	0
$197$	−7.06904	−0.503648	−0.251824	−	0.967773i	$-0.581030\pi$
$197$	−7.06904	−0.503648	−0.251824	+	0.967773i	$0.581030\pi$
$198$	0	0
$199$	22.8256	1.61806	0.809032	−	0.587765i	$-0.199992\pi$
$199$	22.8256	1.61806	0.809032	+	0.587765i	$0.199992\pi$
$200$	0	0
$201$	37.9813	2.67900
$202$	0	0
$203$	33.7899	2.37159
$204$	0	0
$205$	9.82279	0.686054
$206$	0	0
$207$	−4.07759	−0.283412
$208$	0	0
$209$	−6.72651	−0.465282
$210$	0	0
$211$	11.4930	0.791214	0.395607	−	0.918420i	$-0.370534\pi$
$211$	11.4930	0.791214	0.395607	+	0.918420i	$0.370534\pi$
$212$	0	0
$213$	27.9233	1.91327
$214$	0	0
$215$	28.6485	1.95381
$216$	0	0
$217$	−8.78463	−0.596340
$218$	0	0
$219$	10.5763	0.714683
$220$	0	0
$221$	−8.40104	−0.565115
$222$	0	0
$223$	−12.6256	−0.845474	−0.422737	−	0.906252i	$-0.638931\pi$
$223$	−12.6256	−0.845474	−0.422737	+	0.906252i	$0.638931\pi$
$224$	0	0
$225$	37.5601	2.50401
$226$	0	0
$227$	−25.1687	−1.67051	−0.835253	−	0.549865i	$-0.814679\pi$
$227$	−25.1687	−1.67051	−0.835253	+	0.549865i	$0.814679\pi$
$228$	0	0
$229$	−21.1357	−1.39668	−0.698342	−	0.715764i	$-0.746079\pi$
$229$	−21.1357	−1.39668	−0.698342	+	0.715764i	$0.746079\pi$
$230$	0	0
$231$	55.2527	3.63536
$232$	0	0
$233$	1.56022	0.102213	0.0511065	−	0.998693i	$-0.483725\pi$
$233$	1.56022	0.102213	0.0511065	+	0.998693i	$0.483725\pi$
$234$	0	0
$235$	14.4613	0.943350
$236$	0	0
$237$	20.2614	1.31612
$238$	0	0
$239$	−6.08128	−0.393365	−0.196683	−	0.980467i	$-0.563017\pi$
$239$	−6.08128	−0.393365	−0.196683	+	0.980467i	$0.563017\pi$
$240$	0	0
$241$	10.7526	0.692639	0.346319	−	0.938117i	$-0.387431\pi$
$241$	10.7526	0.692639	0.346319	+	0.938117i	$0.387431\pi$
$242$	0	0
$243$	21.4505	1.37605
$244$	0	0
$245$	−44.6085	−2.84994
$246$	0	0
$247$	2.27163	0.144540
$248$	0	0
$249$	−1.54899	−0.0981630
$250$	0	0
$251$	21.0829	1.33074	0.665371	−	0.746513i	$-0.268273\pi$
$251$	21.0829	1.33074	0.665371	+	0.746513i	$0.268273\pi$
$252$	0	0
$253$	−8.18260	−0.514436
$254$	0	0
$255$	46.5714	2.91642
$256$	0	0
$257$	−14.4296	−0.900093	−0.450046	−	0.893005i	$-0.648593\pi$
$257$	−14.4296	−0.900093	−0.450046	+	0.893005i	$0.648593\pi$
$258$	0	0
$259$	6.38633	0.396827
$260$	0	0
$261$	−22.3685	−1.38458
$262$	0	0
$263$	21.1946	1.30691	0.653457	−	0.756964i	$-0.273318\pi$
$263$	21.1946	1.30691	0.653457	+	0.756964i	$0.273318\pi$
$264$	0	0
$265$	32.7921	2.01440
$266$	0	0
$267$	14.4010	0.881326
$268$	0	0
$269$	−13.9865	−0.852772	−0.426386	−	0.904541i	$-0.640213\pi$
$269$	−13.9865	−0.852772	−0.426386	+	0.904541i	$0.640213\pi$
$270$	0	0
$271$	20.5896	1.25073	0.625364	−	0.780333i	$-0.284950\pi$
$271$	20.5896	1.25073	0.625364	+	0.780333i	$0.284950\pi$
$272$	0	0
$273$	−18.6595	−1.12933
$274$	0	0
$275$	75.3729	4.54516
$276$	0	0
$277$	−2.86823	−0.172335	−0.0861677	−	0.996281i	$-0.527462\pi$
$277$	−2.86823	−0.172335	−0.0861677	+	0.996281i	$0.527462\pi$
$278$	0	0
$279$	5.81532	0.348154
$280$	0	0
$281$	25.7071	1.53356	0.766778	−	0.641912i	$-0.221859\pi$
$281$	25.7071	1.53356	0.766778	+	0.641912i	$0.221859\pi$
$282$	0	0
$283$	5.57608	0.331463	0.165732	−	0.986171i	$-0.447001\pi$
$283$	5.57608	0.331463	0.165732	+	0.986171i	$0.447001\pi$
$284$	0	0
$285$	−12.5928	−0.745935
$286$	0	0
$287$	9.47620	0.559362
$288$	0	0
$289$	3.22793	0.189878
$290$	0	0
$291$	−25.9871	−1.52339
$292$	0	0
$293$	−23.6091	−1.37926	−0.689628	−	0.724164i	$-0.742226\pi$
$293$	−23.6091	−1.37926	−0.689628	+	0.724164i	$0.742226\pi$
$294$	0	0
$295$	−21.0882	−1.22780
$296$	0	0
$297$	3.23434	0.187675
$298$	0	0
$299$	2.76337	0.159810
$300$	0	0
$301$	27.6376	1.59301
$302$	0	0
$303$	−1.26965	−0.0729397
$304$	0	0
$305$	−23.4625	−1.34346
$306$	0	0
$307$	−4.01264	−0.229013	−0.114507	−	0.993422i	$-0.536529\pi$
$307$	−4.01264	−0.229013	−0.114507	+	0.993422i	$0.536529\pi$
$308$	0	0
$309$	−46.0413	−2.61920
$310$	0	0
$311$	25.7552	1.46045	0.730223	−	0.683209i	$-0.239416\pi$
$311$	25.7552	1.46045	0.730223	+	0.683209i	$0.239416\pi$
$312$	0	0
$313$	12.5071	0.706944	0.353472	−	0.935445i	$-0.385001\pi$
$313$	12.5071	0.706944	0.353472	+	0.935445i	$0.385001\pi$
$314$	0	0
$315$	49.5299	2.79069
$316$	0	0
$317$	29.2876	1.64495	0.822477	−	0.568798i	$-0.192591\pi$
$317$	29.2876	1.64495	0.822477	+	0.568798i	$0.192591\pi$
$318$	0	0
$319$	−44.8875	−2.51322
$320$	0	0
$321$	−5.61956	−0.313653
$322$	0	0
$323$	−5.46959	−0.304336
$324$	0	0
$325$	−25.4544	−1.41195
$326$	0	0
$327$	−10.8528	−0.600162
$328$	0	0
$329$	13.9510	0.769144
$330$	0	0
$331$	2.95366	0.162348	0.0811740	−	0.996700i	$-0.474133\pi$
$331$	2.95366	0.162348	0.0811740	+	0.996700i	$0.474133\pi$
$332$	0	0
$333$	−4.22767	−0.231675
$334$	0	0
$335$	68.3239	3.73293
$336$	0	0
$337$	−25.1453	−1.36975	−0.684877	−	0.728659i	$-0.740144\pi$
$337$	−25.1453	−1.36975	−0.684877	+	0.728659i	$0.740144\pi$
$338$	0	0
$339$	3.83632	0.208360
$340$	0	0
$341$	11.6698	0.631953
$342$	0	0
$343$	−13.8891	−0.749942
$344$	0	0
$345$	−15.3188	−0.824736
$346$	0	0
$347$	−33.5752	−1.80241	−0.901206	−	0.433390i	$-0.857317\pi$
$347$	−33.5752	−1.80241	−0.901206	+	0.433390i	$0.857317\pi$
$348$	0	0
$349$	−7.03154	−0.376390	−0.188195	−	0.982132i	$-0.560264\pi$
$349$	−7.03154	−0.376390	−0.188195	+	0.982132i	$0.560264\pi$
$350$	0	0
$351$	−1.09228	−0.0583014
$352$	0	0
$353$	−25.2623	−1.34458	−0.672288	−	0.740290i	$-0.734688\pi$
$353$	−25.2623	−1.34458	−0.672288	+	0.740290i	$0.734688\pi$
$354$	0	0
$355$	50.2307	2.66597
$356$	0	0
$357$	44.9282	2.37785
$358$	0	0
$359$	4.73723	0.250022	0.125011	−	0.992155i	$-0.460103\pi$
$359$	4.73723	0.250022	0.125011	+	0.992155i	$0.460103\pi$
$360$	0	0
$361$	−17.5210	−0.922160
$362$	0	0
$363$	−47.0078	−2.46727
$364$	0	0
$365$	19.0256	0.995845
$366$	0	0
$367$	1.26125	0.0658366	0.0329183	−	0.999458i	$-0.489520\pi$
$367$	1.26125	0.0658366	0.0329183	+	0.999458i	$0.489520\pi$
$368$	0	0
$369$	−6.27313	−0.326566
$370$	0	0
$371$	31.6350	1.64241
$372$	0	0
$373$	8.96267	0.464070	0.232035	−	0.972707i	$-0.425462\pi$
$373$	8.96267	0.464070	0.232035	+	0.972707i	$0.425462\pi$
$374$	0	0
$375$	89.3328	4.61313
$376$	0	0
$377$	15.1591	0.780732
$378$	0	0
$379$	−0.481773	−0.0247470	−0.0123735	−	0.999923i	$-0.503939\pi$
$379$	−0.481773	−0.0247470	−0.0123735	+	0.999923i	$0.503939\pi$
$380$	0	0
$381$	31.8753	1.63302
$382$	0	0
$383$	−33.9761	−1.73610	−0.868048	−	0.496480i	$-0.834626\pi$
$383$	−33.9761	−1.73610	−0.868048	+	0.496480i	$0.834626\pi$
$384$	0	0
$385$	99.3930	5.06554
$386$	0	0
$387$	−18.2958	−0.930026
$388$	0	0
$389$	−13.8552	−0.702484	−0.351242	−	0.936285i	$-0.614241\pi$
$389$	−13.8552	−0.702484	−0.351242	+	0.936285i	$0.614241\pi$
$390$	0	0
$391$	−6.65360	−0.336487
$392$	0	0
$393$	7.51664	0.379164
$394$	0	0
$395$	36.4478	1.83389
$396$	0	0
$397$	36.6710	1.84047	0.920233	−	0.391372i	$-0.127999\pi$
$397$	36.6710	1.84047	0.920233	+	0.391372i	$0.127999\pi$
$398$	0	0
$399$	−12.1485	−0.608185
$400$	0	0
$401$	16.8614	0.842016	0.421008	−	0.907057i	$-0.361676\pi$
$401$	16.8614	0.842016	0.421008	+	0.907057i	$0.361676\pi$
$402$	0	0
$403$	−3.94102	−0.196316
$404$	0	0
$405$	41.7426	2.07421
$406$	0	0
$407$	−8.48379	−0.420526
$408$	0	0
$409$	−30.8892	−1.52737	−0.763687	−	0.645587i	$-0.776613\pi$
$409$	−30.8892	−1.52737	−0.763687	+	0.645587i	$0.776613\pi$
$410$	0	0
$411$	30.7113	1.51488
$412$	0	0
$413$	−20.3441	−1.00107
$414$	0	0
$415$	−2.78644	−0.136781
$416$	0	0
$417$	18.7697	0.919154
$418$	0	0
$419$	40.4804	1.97759	0.988797	−	0.149264i	$-0.0476905\pi$
$419$	40.4804	1.97759	0.988797	+	0.149264i	$0.0476905\pi$
$420$	0	0
$421$	−19.5284	−0.951756	−0.475878	−	0.879511i	$-0.657870\pi$
$421$	−19.5284	−0.951756	−0.475878	+	0.879511i	$0.657870\pi$
$422$	0	0
$423$	−9.23540	−0.449041
$424$	0	0
$425$	61.2887	2.97294
$426$	0	0
$427$	−22.6346	−1.09537
$428$	0	0
$429$	24.7878	1.19677
$430$	0	0
$431$	−1.76118	−0.0848329	−0.0424164	−	0.999100i	$-0.513506\pi$
$431$	−1.76118	−0.0848329	−0.0424164	+	0.999100i	$0.513506\pi$
$432$	0	0
$433$	9.94135	0.477751	0.238875	−	0.971050i	$-0.423221\pi$
$433$	9.94135	0.477751	0.238875	+	0.971050i	$0.423221\pi$
$434$	0	0
$435$	−84.0348	−4.02916
$436$	0	0
$437$	1.79912	0.0860636
$438$	0	0
$439$	−33.3688	−1.59261	−0.796304	−	0.604897i	$-0.793214\pi$
$439$	−33.3688	−1.59261	−0.796304	+	0.604897i	$0.793214\pi$
$440$	0	0
$441$	28.4883	1.35659
$442$	0	0
$443$	21.3725	1.01544	0.507719	−	0.861523i	$-0.330489\pi$
$443$	21.3725	1.01544	0.507719	+	0.861523i	$0.330489\pi$
$444$	0	0
$445$	25.9057	1.22805
$446$	0	0
$447$	0.533893	0.0252523
$448$	0	0
$449$	−32.4838	−1.53301	−0.766503	−	0.642241i	$-0.778005\pi$
$449$	−32.4838	−1.53301	−0.766503	+	0.642241i	$0.778005\pi$
$450$	0	0
$451$	−12.5884	−0.592767
$452$	0	0
$453$	−27.2743	−1.28146
$454$	0	0
$455$	−33.5662	−1.57361
$456$	0	0
$457$	10.4199	0.487423	0.243712	−	0.969848i	$-0.421635\pi$
$457$	10.4199	0.487423	0.243712	+	0.969848i	$0.421635\pi$
$458$	0	0
$459$	2.62997	0.122756
$460$	0	0
$461$	−11.2076	−0.521992	−0.260996	−	0.965340i	$-0.584051\pi$
$461$	−11.2076	−0.521992	−0.260996	+	0.965340i	$0.584051\pi$
$462$	0	0
$463$	4.31552	0.200559	0.100280	−	0.994959i	$-0.468026\pi$
$463$	4.31552	0.200559	0.100280	+	0.994959i	$0.468026\pi$
$464$	0	0
$465$	21.8472	1.01314
$466$	0	0
$467$	−32.5739	−1.50734	−0.753670	−	0.657253i	$-0.771718\pi$
$467$	−32.5739	−1.50734	−0.753670	+	0.657253i	$0.771718\pi$
$468$	0	0
$469$	65.9131	3.04358
$470$	0	0
$471$	45.1586	2.08080
$472$	0	0
$473$	−36.7146	−1.68814
$474$	0	0
$475$	−16.5724	−0.760392
$476$	0	0
$477$	−20.9420	−0.958868
$478$	0	0
$479$	−4.46882	−0.204185	−0.102093	−	0.994775i	$-0.532554\pi$
$479$	−4.46882	−0.204185	−0.102093	+	0.994775i	$0.532554\pi$
$480$	0	0
$481$	2.86508	0.130636
$482$	0	0
$483$	−14.7783	−0.672435
$484$	0	0
$485$	−46.7477	−2.12270
$486$	0	0
$487$	−8.85162	−0.401105	−0.200553	−	0.979683i	$-0.564274\pi$
$487$	−8.85162	−0.401105	−0.200553	+	0.979683i	$0.564274\pi$
$488$	0	0
$489$	30.3305	1.37159
$490$	0	0
$491$	−21.9221	−0.989331	−0.494665	−	0.869083i	$-0.664709\pi$
$491$	−21.9221	−0.989331	−0.494665	+	0.869083i	$0.664709\pi$
$492$	0	0
$493$	−36.4998	−1.64387
$494$	0	0
$495$	−65.7969	−2.95735
$496$	0	0
$497$	48.4583	2.17365
$498$	0	0
$499$	−1.00107	−0.0448142	−0.0224071	−	0.999749i	$-0.507133\pi$
$499$	−1.00107	−0.0448142	−0.0224071	+	0.999749i	$0.507133\pi$
$500$	0	0
$501$	−17.5903	−0.785877
$502$	0	0
$503$	34.5593	1.54092	0.770460	−	0.637488i	$-0.220026\pi$
$503$	34.5593	1.54092	0.770460	+	0.637488i	$0.220026\pi$
$504$	0	0
$505$	−2.28395	−0.101635
$506$	0	0
$507$	22.8187	1.01342
$508$	0	0
$509$	0.860390	0.0381361	0.0190680	−	0.999818i	$-0.493930\pi$
$509$	0.860390	0.0381361	0.0190680	+	0.999818i	$0.493930\pi$
$510$	0	0
$511$	18.3543	0.811945
$512$	0	0
$513$	−0.711139	−0.0313975
$514$	0	0
$515$	−82.8228	−3.64961
$516$	0	0
$517$	−18.5329	−0.815077
$518$	0	0
$519$	−25.5625	−1.12207
$520$	0	0
$521$	−22.1586	−0.970785	−0.485392	−	0.874296i	$-0.661323\pi$
$521$	−22.1586	−0.970785	−0.485392	+	0.874296i	$0.661323\pi$
$522$	0	0
$523$	−1.75395	−0.0766948	−0.0383474	−	0.999264i	$-0.512209\pi$
$523$	−1.75395	−0.0766948	−0.0383474	+	0.999264i	$0.512209\pi$
$524$	0	0
$525$	136.128	5.94112
$526$	0	0
$527$	9.48914	0.413354
$528$	0	0
$529$	−20.8114	−0.904845
$530$	0	0
$531$	13.4676	0.584443
$532$	0	0
$533$	4.25128	0.184143
$534$	0	0
$535$	−10.1089	−0.437046
$536$	0	0
$537$	−25.2280	−1.08867
$538$	0	0
$539$	57.1683	2.46241
$540$	0	0
$541$	−12.4582	−0.535620	−0.267810	−	0.963472i	$-0.586300\pi$
$541$	−12.4582	−0.535620	−0.267810	+	0.963472i	$0.586300\pi$
$542$	0	0
$543$	24.1229	1.03521
$544$	0	0
$545$	−19.5229	−0.836270
$546$	0	0
$547$	5.96447	0.255022	0.127511	−	0.991837i	$-0.459301\pi$
$547$	5.96447	0.255022	0.127511	+	0.991837i	$0.459301\pi$
$548$	0	0
$549$	14.9838	0.639495
$550$	0	0
$551$	9.86949	0.420454
$552$	0	0
$553$	35.1618	1.49523
$554$	0	0
$555$	−15.8827	−0.674181
$556$	0	0
$557$	0.821897	0.0348249	0.0174125	−	0.999848i	$-0.494457\pi$
$557$	0.821897	0.0348249	0.0174125	+	0.999848i	$0.494457\pi$
$558$	0	0
$559$	12.3990	0.524421
$560$	0	0
$561$	−59.6838	−2.51985
$562$	0	0
$563$	−35.4857	−1.49554	−0.747772	−	0.663955i	$-0.768876\pi$
$563$	−35.4857	−1.49554	−0.747772	+	0.663955i	$0.768876\pi$
$564$	0	0
$565$	6.90109	0.290331
$566$	0	0
$567$	40.2697	1.69117
$568$	0	0
$569$	36.5218	1.53107	0.765537	−	0.643392i	$-0.222474\pi$
$569$	36.5218	1.53107	0.765537	+	0.643392i	$0.222474\pi$
$570$	0	0
$571$	−26.1034	−1.09239	−0.546196	−	0.837657i	$-0.683925\pi$
$571$	−26.1034	−1.09239	−0.546196	+	0.837657i	$0.683925\pi$
$572$	0	0
$573$	33.8346	1.41346
$574$	0	0
$575$	−20.1598	−0.840721
$576$	0	0
$577$	22.0454	0.917761	0.458881	−	0.888498i	$-0.348251\pi$
$577$	22.0454	0.917761	0.458881	+	0.888498i	$0.348251\pi$
$578$	0	0
$579$	47.6054	1.97841
$580$	0	0
$581$	−2.68812	−0.111522
$582$	0	0
$583$	−42.0248	−1.74049
$584$	0	0
$585$	22.2204	0.918702
$586$	0	0
$587$	−28.3374	−1.16961	−0.584806	−	0.811173i	$-0.698829\pi$
$587$	−28.3374	−1.16961	−0.584806	+	0.811173i	$0.698829\pi$
$588$	0	0
$589$	−2.56585	−0.105724
$590$	0	0
$591$	16.9602	0.697650
$592$	0	0
$593$	−29.2000	−1.19910	−0.599549	−	0.800338i	$-0.704654\pi$
$593$	−29.2000	−1.19910	−0.599549	+	0.800338i	$0.704654\pi$
$594$	0	0
$595$	80.8204	3.31331
$596$	0	0
$597$	−54.7637	−2.24133
$598$	0	0
$599$	35.7222	1.45957	0.729784	−	0.683678i	$-0.239621\pi$
$599$	35.7222	1.45957	0.729784	+	0.683678i	$0.239621\pi$
$600$	0	0
$601$	21.0509	0.858686	0.429343	−	0.903142i	$-0.358745\pi$
$601$	21.0509	0.858686	0.429343	+	0.903142i	$0.358745\pi$
$602$	0	0
$603$	−43.6337	−1.77690
$604$	0	0
$605$	−84.5614	−3.43791
$606$	0	0
$607$	45.7539	1.85709	0.928547	−	0.371215i	$-0.121059\pi$
$607$	45.7539	1.85709	0.928547	+	0.371215i	$0.121059\pi$
$608$	0	0
$609$	−81.0696	−3.28511
$610$	0	0
$611$	6.25880	0.253204
$612$	0	0
$613$	25.6630	1.03652	0.518259	−	0.855224i	$-0.326580\pi$
$613$	25.6630	1.03652	0.518259	+	0.855224i	$0.326580\pi$
$614$	0	0
$615$	−23.5671	−0.950316
$616$	0	0
$617$	2.12068	0.0853753	0.0426877	−	0.999088i	$-0.486408\pi$
$617$	2.12068	0.0853753	0.0426877	+	0.999088i	$0.486408\pi$
$618$	0	0
$619$	19.5534	0.785917	0.392959	−	0.919556i	$-0.371452\pi$
$619$	19.5534	0.785917	0.392959	+	0.919556i	$0.371452\pi$
$620$	0	0
$621$	−0.865079	−0.0347144
$622$	0	0
$623$	24.9916	1.00127
$624$	0	0
$625$	92.5634	3.70253
$626$	0	0
$627$	16.1384	0.644506
$628$	0	0
$629$	−6.89851	−0.275061
$630$	0	0
$631$	22.4921	0.895396	0.447698	−	0.894185i	$-0.352244\pi$
$631$	22.4921	0.895396	0.447698	+	0.894185i	$0.352244\pi$
$632$	0	0
$633$	−27.5744	−1.09598
$634$	0	0
$635$	57.3400	2.27547
$636$	0	0
$637$	−19.3064	−0.764949
$638$	0	0
$639$	−32.0788	−1.26902
$640$	0	0
$641$	35.2232	1.39123	0.695616	−	0.718414i	$-0.255132\pi$
$641$	35.2232	1.39123	0.695616	+	0.718414i	$0.255132\pi$
$642$	0	0
$643$	1.95035	0.0769141	0.0384571	−	0.999260i	$-0.487756\pi$
$643$	1.95035	0.0769141	0.0384571	+	0.999260i	$0.487756\pi$
$644$	0	0
$645$	−68.7341	−2.70640
$646$	0	0
$647$	−41.5461	−1.63335	−0.816673	−	0.577101i	$-0.804184\pi$
$647$	−41.5461	−1.63335	−0.816673	+	0.577101i	$0.804184\pi$
$648$	0	0
$649$	27.0257	1.06085
$650$	0	0
$651$	21.0763	0.826045
$652$	0	0
$653$	−20.7446	−0.811799	−0.405900	−	0.913918i	$-0.633042\pi$
$653$	−20.7446	−0.811799	−0.405900	+	0.913918i	$0.633042\pi$
$654$	0	0
$655$	13.5215	0.528331
$656$	0	0
$657$	−12.1503	−0.474029
$658$	0	0
$659$	25.7961	1.00487	0.502436	−	0.864615i	$-0.332437\pi$
$659$	25.7961	1.00487	0.502436	+	0.864615i	$0.332437\pi$
$660$	0	0
$661$	−7.56773	−0.294351	−0.147175	−	0.989110i	$-0.547018\pi$
$661$	−7.56773	−0.294351	−0.147175	+	0.989110i	$0.547018\pi$
$662$	0	0
$663$	20.1560	0.782793
$664$	0	0
$665$	−21.8537	−0.847450
$666$	0	0
$667$	12.0059	0.464872
$668$	0	0
$669$	30.2917	1.17114
$670$	0	0
$671$	30.0685	1.16078
$672$	0	0
$673$	2.89475	0.111584	0.0557922	−	0.998442i	$-0.482232\pi$
$673$	2.89475	0.111584	0.0557922	+	0.998442i	$0.482232\pi$
$674$	0	0
$675$	7.96856	0.306710
$676$	0	0
$677$	−13.9179	−0.534908	−0.267454	−	0.963571i	$-0.586182\pi$
$677$	−13.9179	−0.534908	−0.267454	+	0.963571i	$0.586182\pi$
$678$	0	0
$679$	−45.0982	−1.73071
$680$	0	0
$681$	60.3854	2.31397
$682$	0	0
$683$	39.3231	1.50466	0.752328	−	0.658789i	$-0.228931\pi$
$683$	39.3231	1.50466	0.752328	+	0.658789i	$0.228931\pi$
$684$	0	0
$685$	55.2459	2.11084
$686$	0	0
$687$	50.7092	1.93468
$688$	0	0
$689$	14.1923	0.540684
$690$	0	0
$691$	23.0536	0.876999	0.438499	−	0.898731i	$-0.355510\pi$
$691$	23.0536	0.876999	0.438499	+	0.898731i	$0.355510\pi$
$692$	0	0
$693$	−63.4753	−2.41123
$694$	0	0
$695$	33.7644	1.28076
$696$	0	0
$697$	−10.2362	−0.387723
$698$	0	0
$699$	−3.74330	−0.141585
$700$	0	0
$701$	−1.17685	−0.0444492	−0.0222246	−	0.999753i	$-0.507075\pi$
$701$	−1.17685	−0.0444492	−0.0222246	+	0.999753i	$0.507075\pi$
$702$	0	0
$703$	1.86534	0.0703528
$704$	0	0
$705$	−34.6958	−1.30672
$706$	0	0
$707$	−2.20337	−0.0828661
$708$	0	0
$709$	−39.8192	−1.49544	−0.747721	−	0.664013i	$-0.768852\pi$
$709$	−39.8192	−1.49544	−0.747721	+	0.664013i	$0.768852\pi$
$710$	0	0
$711$	−23.2767	−0.872943
$712$	0	0
$713$	−3.12128	−0.116893
$714$	0	0
$715$	44.5904	1.66758
$716$	0	0
$717$	14.5903	0.544887
$718$	0	0
$719$	29.3035	1.09284	0.546419	−	0.837512i	$-0.315991\pi$
$719$	29.3035	1.09284	0.546419	+	0.837512i	$0.315991\pi$
$720$	0	0
$721$	−79.9004	−2.97565
$722$	0	0
$723$	−25.7980	−0.959438
$724$	0	0
$725$	−110.591	−4.10725
$726$	0	0
$727$	36.9464	1.37027	0.685133	−	0.728418i	$-0.259744\pi$
$727$	36.9464	1.37027	0.685133	+	0.728418i	$0.259744\pi$
$728$	0	0
$729$	−22.4492	−0.831451
$730$	0	0
$731$	−29.8541	−1.10419
$732$	0	0
$733$	−26.7604	−0.988418	−0.494209	−	0.869343i	$-0.664542\pi$
$733$	−26.7604	−0.988418	−0.494209	+	0.869343i	$0.664542\pi$
$734$	0	0
$735$	107.026	3.94771
$736$	0	0
$737$	−87.5608	−3.22534
$738$	0	0
$739$	−44.1344	−1.62351	−0.811755	−	0.583998i	$-0.801488\pi$
$739$	−44.1344	−1.62351	−0.811755	+	0.583998i	$0.801488\pi$
$740$	0	0
$741$	−5.45014	−0.200216
$742$	0	0
$743$	−35.7393	−1.31115	−0.655574	−	0.755131i	$-0.727573\pi$
$743$	−35.7393	−1.31115	−0.655574	+	0.755131i	$0.727573\pi$
$744$	0	0
$745$	0.960411	0.0351867
$746$	0	0
$747$	1.77950	0.0651086
$748$	0	0
$749$	−9.75222	−0.356338
$750$	0	0
$751$	−23.7050	−0.865007	−0.432503	−	0.901632i	$-0.642370\pi$
$751$	−23.7050	−0.865007	−0.432503	+	0.901632i	$0.642370\pi$
$752$	0	0
$753$	−50.5826	−1.84333
$754$	0	0
$755$	−49.0632	−1.78559
$756$	0	0
$757$	−20.7204	−0.753095	−0.376547	−	0.926397i	$-0.622889\pi$
$757$	−20.7204	−0.753095	−0.376547	+	0.926397i	$0.622889\pi$
$758$	0	0
$759$	19.6319	0.712592
$760$	0	0
$761$	39.0677	1.41620	0.708102	−	0.706110i	$-0.249552\pi$
$761$	39.0677	1.41620	0.708102	+	0.706110i	$0.249552\pi$
$762$	0	0
$763$	−18.8341	−0.681839
$764$	0	0
$765$	−53.5021	−1.93437
$766$	0	0
$767$	−9.12693	−0.329554
$768$	0	0
$769$	−44.8559	−1.61755	−0.808773	−	0.588121i	$-0.799868\pi$
$769$	−44.8559	−1.61755	−0.808773	+	0.588121i	$0.799868\pi$
$770$	0	0
$771$	34.6198	1.24680
$772$	0	0
$773$	−5.94714	−0.213904	−0.106952	−	0.994264i	$-0.534109\pi$
$773$	−5.94714	−0.213904	−0.106952	+	0.994264i	$0.534109\pi$
$774$	0	0
$775$	28.7512	1.03277
$776$	0	0
$777$	−15.3222	−0.549682
$778$	0	0
$779$	2.76784	0.0991682
$780$	0	0
$781$	−64.3734	−2.30346
$782$	0	0
$783$	−4.74559	−0.169594
$784$	0	0
$785$	81.2350	2.89940
$786$	0	0
$787$	2.27769	0.0811907	0.0405954	−	0.999176i	$-0.487075\pi$
$787$	2.27769	0.0811907	0.0405954	+	0.999176i	$0.487075\pi$
$788$	0	0
$789$	−50.8505	−1.81033
$790$	0	0
$791$	6.65758	0.236716
$792$	0	0
$793$	−10.1545	−0.360597
$794$	0	0
$795$	−78.6755	−2.79033
$796$	0	0
$797$	12.4325	0.440381	0.220191	−	0.975457i	$-0.429332\pi$
$797$	12.4325	0.440381	0.220191	+	0.975457i	$0.429332\pi$
$798$	0	0
$799$	−15.0699	−0.533133
$800$	0	0
$801$	−16.5441	−0.584558
$802$	0	0
$803$	−24.3823	−0.860434
$804$	0	0
$805$	−26.5844	−0.936976
$806$	0	0
$807$	33.5567	1.18125
$808$	0	0
$809$	9.15384	0.321832	0.160916	−	0.986968i	$-0.448555\pi$
$809$	9.15384	0.321832	0.160916	+	0.986968i	$0.448555\pi$
$810$	0	0
$811$	31.9360	1.12142	0.560712	−	0.828011i	$-0.310527\pi$
$811$	31.9360	1.12142	0.560712	+	0.828011i	$0.310527\pi$
$812$	0	0
$813$	−49.3990	−1.73250
$814$	0	0
$815$	54.5609	1.91118
$816$	0	0
$817$	8.07250	0.282421
$818$	0	0
$819$	21.4364	0.749048
$820$	0	0
$821$	49.3900	1.72372	0.861862	−	0.507143i	$-0.169298\pi$
$821$	49.3900	1.72372	0.861862	+	0.507143i	$0.169298\pi$
$822$	0	0
$823$	−5.41011	−0.188585	−0.0942923	−	0.995545i	$-0.530059\pi$
$823$	−5.41011	−0.188585	−0.0942923	+	0.995545i	$0.530059\pi$
$824$	0	0
$825$	−180.836	−6.29592
$826$	0	0
$827$	54.7940	1.90537	0.952687	−	0.303955i	$-0.0983071\pi$
$827$	54.7940	1.90537	0.952687	+	0.303955i	$0.0983071\pi$
$828$	0	0
$829$	−26.8421	−0.932263	−0.466132	−	0.884715i	$-0.654353\pi$
$829$	−26.8421	−0.932263	−0.466132	+	0.884715i	$0.654353\pi$
$830$	0	0
$831$	6.88153	0.238718
$832$	0	0
$833$	46.4858	1.61064
$834$	0	0
$835$	−31.6429	−1.09505
$836$	0	0
$837$	1.23375	0.0426445
$838$	0	0
$839$	43.9395	1.51696	0.758480	−	0.651697i	$-0.225943\pi$
$839$	43.9395	1.51696	0.758480	+	0.651697i	$0.225943\pi$
$840$	0	0
$841$	36.8613	1.27108
$842$	0	0
$843$	−61.6771	−2.12427
$844$	0	0
$845$	41.0482	1.41210
$846$	0	0
$847$	−81.5776	−2.80304
$848$	0	0
$849$	−13.3783	−0.459140
$850$	0	0
$851$	2.26913	0.0777849
$852$	0	0
$853$	−6.17269	−0.211349	−0.105675	−	0.994401i	$-0.533700\pi$
$853$	−6.17269	−0.211349	−0.105675	+	0.994401i	$0.533700\pi$
$854$	0	0
$855$	14.4669	0.494757
$856$	0	0
$857$	−2.08961	−0.0713798	−0.0356899	−	0.999363i	$-0.511363\pi$
$857$	−2.08961	−0.0713798	−0.0356899	+	0.999363i	$0.511363\pi$
$858$	0	0
$859$	3.72665	0.127152	0.0635759	−	0.997977i	$-0.479750\pi$
$859$	3.72665	0.127152	0.0635759	+	0.997977i	$0.479750\pi$
$860$	0	0
$861$	−22.7355	−0.774824
$862$	0	0
$863$	−41.4478	−1.41090	−0.705449	−	0.708761i	$-0.749255\pi$
$863$	−41.4478	−1.41090	−0.705449	+	0.708761i	$0.749255\pi$
$864$	0	0
$865$	−45.9838	−1.56350
$866$	0	0
$867$	−7.74451	−0.263018
$868$	0	0
$869$	−46.7099	−1.58453
$870$	0	0
$871$	29.5704	1.00195
$872$	0	0
$873$	29.8545	1.01042
$874$	0	0
$875$	155.029	5.24093
$876$	0	0
$877$	−30.3070	−1.02340	−0.511698	−	0.859166i	$-0.670983\pi$
$877$	−30.3070	−1.02340	−0.511698	+	0.859166i	$0.670983\pi$
$878$	0	0
$879$	56.6434	1.91053
$880$	0	0
$881$	26.0271	0.876876	0.438438	−	0.898762i	$-0.355532\pi$
$881$	26.0271	0.876876	0.438438	+	0.898762i	$0.355532\pi$
$882$	0	0
$883$	17.7307	0.596686	0.298343	−	0.954459i	$-0.403566\pi$
$883$	17.7307	0.596686	0.298343	+	0.954459i	$0.403566\pi$
$884$	0	0
$885$	50.5954	1.70075
$886$	0	0
$887$	25.1187	0.843404	0.421702	−	0.906735i	$-0.361433\pi$
$887$	25.1187	0.843404	0.421702	+	0.906735i	$0.361433\pi$
$888$	0	0
$889$	55.3167	1.85526
$890$	0	0
$891$	−53.4954	−1.79216
$892$	0	0
$893$	4.07486	0.136360
$894$	0	0
$895$	−45.3823	−1.51696
$896$	0	0
$897$	−6.62993	−0.221367
$898$	0	0
$899$	−17.1225	−0.571067
$900$	0	0
$901$	−34.1721	−1.13844
$902$	0	0
$903$	−66.3088	−2.20662
$904$	0	0
$905$	43.3942	1.44247
$906$	0	0
$907$	8.42708	0.279816	0.139908	−	0.990164i	$-0.455319\pi$
$907$	8.42708	0.279816	0.139908	+	0.990164i	$0.455319\pi$
$908$	0	0
$909$	1.45860	0.0483787
$910$	0	0
$911$	55.1917	1.82858	0.914291	−	0.405058i	$-0.132749\pi$
$911$	55.1917	1.82858	0.914291	+	0.405058i	$0.132749\pi$
$912$	0	0
$913$	3.57098	0.118182
$914$	0	0
$915$	56.2918	1.86095
$916$	0	0
$917$	13.0444	0.430765
$918$	0	0
$919$	−9.94728	−0.328130	−0.164065	−	0.986450i	$-0.552461\pi$
$919$	−9.94728	−0.328130	−0.164065	+	0.986450i	$0.552461\pi$
$920$	0	0
$921$	9.62721	0.317227
$922$	0	0
$923$	21.7397	0.715571
$924$	0	0
$925$	−20.9018	−0.687248
$926$	0	0
$927$	52.8931	1.73724
$928$	0	0
$929$	−18.3648	−0.602529	−0.301265	−	0.953541i	$-0.597409\pi$
$929$	−18.3648	−0.602529	−0.301265	+	0.953541i	$0.597409\pi$
$930$	0	0
$931$	−12.5697	−0.411955
$932$	0	0
$933$	−61.7925	−2.02300
$934$	0	0
$935$	−107.364	−3.51118
$936$	0	0
$937$	50.6917	1.65603	0.828013	−	0.560709i	$-0.189471\pi$
$937$	50.6917	1.65603	0.828013	+	0.560709i	$0.189471\pi$
$938$	0	0
$939$	−30.0074	−0.979253
$940$	0	0
$941$	−16.5404	−0.539201	−0.269600	−	0.962972i	$-0.586892\pi$
$941$	−16.5404	−0.539201	−0.269600	+	0.962972i	$0.586892\pi$
$942$	0	0
$943$	3.36700	0.109644
$944$	0	0
$945$	10.5080	0.341825
$946$	0	0
$947$	0.636122	0.0206712	0.0103356	−	0.999947i	$-0.496710\pi$
$947$	0.636122	0.0206712	0.0103356	+	0.999947i	$0.496710\pi$
$948$	0	0
$949$	8.23422	0.267294
$950$	0	0
$951$	−70.2675	−2.27858
$952$	0	0
$953$	24.7565	0.801943	0.400971	−	0.916091i	$-0.368673\pi$
$953$	24.7565	0.801943	0.400971	+	0.916091i	$0.368673\pi$
$954$	0	0
$955$	60.8645	1.96953
$956$	0	0
$957$	107.695	3.48129
$958$	0	0
$959$	53.2966	1.72104
$960$	0	0
$961$	−26.5485	−0.856404
$962$	0	0
$963$	6.45585	0.208037
$964$	0	0
$965$	85.6365	2.75674
$966$	0	0
$967$	−7.51994	−0.241825	−0.120913	−	0.992663i	$-0.538582\pi$
$967$	−7.51994	−0.241825	−0.120913	+	0.992663i	$0.538582\pi$
$968$	0	0
$969$	13.1228	0.421564
$970$	0	0
$971$	−39.8191	−1.27786	−0.638928	−	0.769266i	$-0.720622\pi$
$971$	−39.8191	−1.27786	−0.638928	+	0.769266i	$0.720622\pi$
$972$	0	0
$973$	32.5730	1.04424
$974$	0	0
$975$	61.0707	1.95583
$976$	0	0
$977$	12.8039	0.409633	0.204817	−	0.978800i	$-0.434340\pi$
$977$	12.8039	0.409633	0.204817	+	0.978800i	$0.434340\pi$
$978$	0	0
$979$	−33.1995	−1.06106
$980$	0	0
$981$	12.4679	0.398070
$982$	0	0
$983$	57.6441	1.83856	0.919280	−	0.393604i	$-0.128772\pi$
$983$	57.6441	1.83856	0.919280	+	0.393604i	$0.128772\pi$
$984$	0	0
$985$	30.5094	0.972110
$986$	0	0
$987$	−33.4716	−1.06541
$988$	0	0
$989$	9.81995	0.312256
$990$	0	0
$991$	−14.9623	−0.475294	−0.237647	−	0.971352i	$-0.576376\pi$
$991$	−14.9623	−0.475294	−0.237647	+	0.971352i	$0.576376\pi$
$992$	0	0
$993$	−7.08650	−0.224883
$994$	0	0
$995$	−98.5134	−3.12308
$996$	0	0
$997$	−10.8507	−0.343645	−0.171822	−	0.985128i	$-0.554966\pi$
$997$	−10.8507	−0.343645	−0.171822	+	0.985128i	$0.554966\pi$
$998$	0	0
$999$	−0.896921	−0.0283773

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8044.2.a.b.1.13	✓	87

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8044.2.a.b.1.13	✓	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)

\(f(q)\)	\(=\)	\(q-2.39922 q^{3} -4.31592 q^{5} -4.16363 q^{7} +2.75627 q^{9} +O(q^{10})\)	\(q-2.39922 q^{3} -4.31592 q^{5} -4.16363 q^{7} +2.75627 q^{9} +5.53109 q^{11} -1.86792 q^{13} +10.3549 q^{15} +4.49755 q^{17} -1.21613 q^{19} +9.98948 q^{21} -1.47938 q^{23} +13.6271 q^{25} +0.584756 q^{27} -8.11550 q^{29} +2.10985 q^{31} -13.2703 q^{33} +17.9699 q^{35} -1.53384 q^{37} +4.48155 q^{39} -2.27595 q^{41} -6.63787 q^{43} -11.8958 q^{45} -3.35069 q^{47} +10.3358 q^{49} -10.7906 q^{51} -7.59794 q^{53} -23.8717 q^{55} +2.91776 q^{57} +4.88615 q^{59} +5.43627 q^{61} -11.4761 q^{63} +8.06177 q^{65} -15.8307 q^{67} +3.54937 q^{69} -11.6385 q^{71} -4.40824 q^{73} -32.6946 q^{75} -23.0294 q^{77} -8.44498 q^{79} -9.67178 q^{81} +0.645620 q^{83} -19.4110 q^{85} +19.4709 q^{87} -6.00235 q^{89} +7.77731 q^{91} -5.06200 q^{93} +5.24871 q^{95} +10.8315 q^{97} +15.2452 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 87 q + 13 q^{3} - 2 q^{5} + 8 q^{7} + 98 q^{9}+O(q^{10}) \) 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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