LMFDB - Embedded newform 8044.2.a.a.1.9

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			1.9
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)

$f(q)$	$=$	$q-2.69065 q^{3} -1.88521 q^{5} -0.388125 q^{7} +4.23960 q^{9} +O(q^{10})$	$q-2.69065 q^{3} -1.88521 q^{5} -0.388125 q^{7} +4.23960 q^{9} +1.57157 q^{11} -3.19486 q^{13} +5.07243 q^{15} -4.51588 q^{17} +1.24830 q^{19} +1.04431 q^{21} +4.37933 q^{23} -1.44600 q^{25} -3.33532 q^{27} -6.24449 q^{29} +1.40011 q^{31} -4.22855 q^{33} +0.731696 q^{35} +10.1774 q^{37} +8.59624 q^{39} -0.537273 q^{41} +0.900344 q^{43} -7.99251 q^{45} -3.26002 q^{47} -6.84936 q^{49} +12.1506 q^{51} +0.979484 q^{53} -2.96273 q^{55} -3.35875 q^{57} +3.68126 q^{59} +7.19572 q^{61} -1.64549 q^{63} +6.02296 q^{65} -1.10948 q^{67} -11.7832 q^{69} -8.48832 q^{71} -0.940867 q^{73} +3.89068 q^{75} -0.609967 q^{77} +10.3152 q^{79} -3.74461 q^{81} -1.27477 q^{83} +8.51336 q^{85} +16.8017 q^{87} +5.69173 q^{89} +1.24000 q^{91} -3.76722 q^{93} -2.35331 q^{95} +14.8860 q^{97} +6.66283 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.69065	−1.55345	−0.776724	−	0.629841i	$-0.783120\pi$
$3$	−2.69065	−1.55345	−0.776724	+	0.629841i	$0.783120\pi$
$4$	0	0
$5$	−1.88521	−0.843090	−0.421545	−	0.906808i	$-0.638512\pi$
$5$	−1.88521	−0.843090	−0.421545	+	0.906808i	$0.638512\pi$
$6$	0	0
$7$	−0.388125	−0.146698	−0.0733488	−	0.997306i	$-0.523369\pi$
$7$	−0.388125	−0.146698	−0.0733488	+	0.997306i	$0.523369\pi$
$8$	0	0
$9$	4.23960	1.41320
$10$	0	0
$11$	1.57157	0.473847	0.236923	−	0.971528i	$-0.423861\pi$
$11$	1.57157	0.473847	0.236923	+	0.971528i	$0.423861\pi$
$12$	0	0
$13$	−3.19486	−0.886093	−0.443047	−	0.896499i	$-0.646102\pi$
$13$	−3.19486	−0.886093	−0.443047	+	0.896499i	$0.646102\pi$
$14$	0	0
$15$	5.07243	1.30970
$16$	0	0
$17$	−4.51588	−1.09526	−0.547631	−	0.836720i	$-0.684470\pi$
$17$	−4.51588	−1.09526	−0.547631	+	0.836720i	$0.684470\pi$
$18$	0	0
$19$	1.24830	0.286381	0.143190	−	0.989695i	$-0.454264\pi$
$19$	1.24830	0.286381	0.143190	+	0.989695i	$0.454264\pi$
$20$	0	0
$21$	1.04431	0.227887
$22$	0	0
$23$	4.37933	0.913152	0.456576	−	0.889684i	$-0.349076\pi$
$23$	4.37933	0.913152	0.456576	+	0.889684i	$0.349076\pi$
$24$	0	0
$25$	−1.44600	−0.289200
$26$	0	0
$27$	−3.33532	−0.641883
$28$	0	0
$29$	−6.24449	−1.15957	−0.579786	−	0.814769i	$-0.696864\pi$
$29$	−6.24449	−1.15957	−0.579786	+	0.814769i	$0.696864\pi$
$30$	0	0
$31$	1.40011	0.251468	0.125734	−	0.992064i	$-0.459871\pi$
$31$	1.40011	0.251468	0.125734	+	0.992064i	$0.459871\pi$
$32$	0	0
$33$	−4.22855	−0.736096
$34$	0	0
$35$	0.731696	0.123679
$36$	0	0
$37$	10.1774	1.67316	0.836580	−	0.547846i	$-0.184552\pi$
$37$	10.1774	1.67316	0.836580	+	0.547846i	$0.184552\pi$
$38$	0	0
$39$	8.59624	1.37650
$40$	0	0
$41$	−0.537273	−0.0839079	−0.0419540	−	0.999120i	$-0.513358\pi$
$41$	−0.537273	−0.0839079	−0.0419540	+	0.999120i	$0.513358\pi$
$42$	0	0
$43$	0.900344	0.137301	0.0686506	−	0.997641i	$-0.478131\pi$
$43$	0.900344	0.137301	0.0686506	+	0.997641i	$0.478131\pi$
$44$	0	0
$45$	−7.99251	−1.19145
$46$	0	0
$47$	−3.26002	−0.475523	−0.237761	−	0.971324i	$-0.576414\pi$
$47$	−3.26002	−0.475523	−0.237761	+	0.971324i	$0.576414\pi$
$48$	0	0
$49$	−6.84936	−0.978480
$50$	0	0
$51$	12.1506	1.70143
$52$	0	0
$53$	0.979484	0.134543	0.0672713	−	0.997735i	$-0.478571\pi$
$53$	0.979484	0.134543	0.0672713	+	0.997735i	$0.478571\pi$
$54$	0	0
$55$	−2.96273	−0.399495
$56$	0	0
$57$	−3.35875	−0.444877
$58$	0	0
$59$	3.68126	0.479259	0.239629	−	0.970864i	$-0.422974\pi$
$59$	3.68126	0.479259	0.239629	+	0.970864i	$0.422974\pi$
$60$	0	0
$61$	7.19572	0.921318	0.460659	−	0.887577i	$-0.347613\pi$
$61$	7.19572	0.921318	0.460659	+	0.887577i	$0.347613\pi$
$62$	0	0
$63$	−1.64549	−0.207313
$64$	0	0
$65$	6.02296	0.747056
$66$	0	0
$67$	−1.10948	−0.135544	−0.0677722	−	0.997701i	$-0.521589\pi$
$67$	−1.10948	−0.135544	−0.0677722	+	0.997701i	$0.521589\pi$
$68$	0	0
$69$	−11.7832	−1.41853
$70$	0	0
$71$	−8.48832	−1.00738	−0.503689	−	0.863885i	$-0.668024\pi$
$71$	−8.48832	−1.00738	−0.503689	+	0.863885i	$0.668024\pi$
$72$	0	0
$73$	−0.940867	−0.110120	−0.0550601	−	0.998483i	$-0.517535\pi$
$73$	−0.940867	−0.110120	−0.0550601	+	0.998483i	$0.517535\pi$
$74$	0	0
$75$	3.89068	0.449257
$76$	0	0
$77$	−0.609967	−0.0695122
$78$	0	0
$79$	10.3152	1.16055	0.580276	−	0.814420i	$-0.302945\pi$
$79$	10.3152	1.16055	0.580276	+	0.814420i	$0.302945\pi$
$80$	0	0
$81$	−3.74461	−0.416068
$82$	0	0
$83$	−1.27477	−0.139924	−0.0699621	−	0.997550i	$-0.522288\pi$
$83$	−1.27477	−0.139924	−0.0699621	+	0.997550i	$0.522288\pi$
$84$	0	0
$85$	8.51336	0.923404
$86$	0	0
$87$	16.8017	1.80134
$88$	0	0
$89$	5.69173	0.603322	0.301661	−	0.953415i	$-0.402459\pi$
$89$	5.69173	0.603322	0.301661	+	0.953415i	$0.402459\pi$
$90$	0	0
$91$	1.24000	0.129988
$92$	0	0
$93$	−3.76722	−0.390642
$94$	0	0
$95$	−2.35331	−0.241445
$96$	0	0
$97$	14.8860	1.51144	0.755720	−	0.654895i	$-0.227287\pi$
$97$	14.8860	1.51144	0.755720	+	0.654895i	$0.227287\pi$
$98$	0	0
$99$	6.66283	0.669639
$100$	0	0
$101$	−9.18765	−0.914205	−0.457103	−	0.889414i	$-0.651113\pi$
$101$	−9.18765	−0.914205	−0.457103	+	0.889414i	$0.651113\pi$
$102$	0	0
$103$	−8.41105	−0.828766	−0.414383	−	0.910103i	$-0.636003\pi$
$103$	−8.41105	−0.828766	−0.414383	+	0.910103i	$0.636003\pi$
$104$	0	0
$105$	−1.96874	−0.192129
$106$	0	0
$107$	1.38830	0.134212	0.0671059	−	0.997746i	$-0.478623\pi$
$107$	1.38830	0.134212	0.0671059	+	0.997746i	$0.478623\pi$
$108$	0	0
$109$	12.6445	1.21112	0.605562	−	0.795798i	$-0.292948\pi$
$109$	12.6445	1.21112	0.605562	+	0.795798i	$0.292948\pi$
$110$	0	0
$111$	−27.3839	−2.59916
$112$	0	0
$113$	16.9602	1.59548	0.797742	−	0.602999i	$-0.206028\pi$
$113$	16.9602	1.59548	0.797742	+	0.602999i	$0.206028\pi$
$114$	0	0
$115$	−8.25593	−0.769869
$116$	0	0
$117$	−13.5449	−1.25223
$118$	0	0
$119$	1.75273	0.160672
$120$	0	0
$121$	−8.53016	−0.775469
$122$	0	0
$123$	1.44561	0.130347
$124$	0	0
$125$	12.1520	1.08691
$126$	0	0
$127$	22.0110	1.95316	0.976580	−	0.215154i	$-0.0690252\pi$
$127$	22.0110	1.95316	0.976580	+	0.215154i	$0.0690252\pi$
$128$	0	0
$129$	−2.42251	−0.213290
$130$	0	0
$131$	−13.2357	−1.15641	−0.578204	−	0.815892i	$-0.696246\pi$
$131$	−13.2357	−1.15641	−0.578204	+	0.815892i	$0.696246\pi$
$132$	0	0
$133$	−0.484499	−0.0420114
$134$	0	0
$135$	6.28776	0.541164
$136$	0	0
$137$	12.4996	1.06791	0.533957	−	0.845512i	$-0.320705\pi$
$137$	12.4996	1.06791	0.533957	+	0.845512i	$0.320705\pi$
$138$	0	0
$139$	−4.41559	−0.374525	−0.187263	−	0.982310i	$-0.559962\pi$
$139$	−4.41559	−0.374525	−0.187263	+	0.982310i	$0.559962\pi$
$140$	0	0
$141$	8.77157	0.738700
$142$	0	0
$143$	−5.02094	−0.419872
$144$	0	0
$145$	11.7721	0.977624
$146$	0	0
$147$	18.4292	1.52002
$148$	0	0
$149$	12.9124	1.05782	0.528911	−	0.848677i	$-0.322601\pi$
$149$	12.9124	1.05782	0.528911	+	0.848677i	$0.322601\pi$
$150$	0	0
$151$	−23.2637	−1.89317	−0.946585	−	0.322456i	$-0.895492\pi$
$151$	−23.2637	−1.89317	−0.946585	+	0.322456i	$0.895492\pi$
$152$	0	0
$153$	−19.1455	−1.54782
$154$	0	0
$155$	−2.63950	−0.212010
$156$	0	0
$157$	−0.419713	−0.0334968	−0.0167484	−	0.999860i	$-0.505331\pi$
$157$	−0.419713	−0.0334968	−0.0167484	+	0.999860i	$0.505331\pi$
$158$	0	0
$159$	−2.63545	−0.209005
$160$	0	0
$161$	−1.69973	−0.133957
$162$	0	0
$163$	12.3015	0.963525	0.481763	−	0.876302i	$-0.339997\pi$
$163$	12.3015	0.963525	0.481763	+	0.876302i	$0.339997\pi$
$164$	0	0
$165$	7.97168	0.620595
$166$	0	0
$167$	3.19985	0.247612	0.123806	−	0.992306i	$-0.460490\pi$
$167$	3.19985	0.247612	0.123806	+	0.992306i	$0.460490\pi$
$168$	0	0
$169$	−2.79290	−0.214839
$170$	0	0
$171$	5.29231	0.404713
$172$	0	0
$173$	9.04668	0.687806	0.343903	−	0.939005i	$-0.388251\pi$
$173$	9.04668	0.687806	0.343903	+	0.939005i	$0.388251\pi$
$174$	0	0
$175$	0.561229	0.0424250
$176$	0	0
$177$	−9.90497	−0.744503
$178$	0	0
$179$	−1.81534	−0.135685	−0.0678423	−	0.997696i	$-0.521611\pi$
$179$	−1.81534	−0.135685	−0.0678423	+	0.997696i	$0.521611\pi$
$180$	0	0
$181$	24.4679	1.81869	0.909343	−	0.416048i	$-0.136585\pi$
$181$	24.4679	1.81869	0.909343	+	0.416048i	$0.136585\pi$
$182$	0	0
$183$	−19.3612	−1.43122
$184$	0	0
$185$	−19.1865	−1.41062
$186$	0	0
$187$	−7.09703	−0.518986
$188$	0	0
$189$	1.29452	0.0941626
$190$	0	0
$191$	−0.520932	−0.0376933	−0.0188467	−	0.999822i	$-0.505999\pi$
$191$	−0.520932	−0.0376933	−0.0188467	+	0.999822i	$0.505999\pi$
$192$	0	0
$193$	−2.45118	−0.176440	−0.0882198	−	0.996101i	$-0.528118\pi$
$193$	−2.45118	−0.176440	−0.0882198	+	0.996101i	$0.528118\pi$
$194$	0	0
$195$	−16.2057	−1.16051
$196$	0	0
$197$	14.5328	1.03542	0.517710	−	0.855556i	$-0.326785\pi$
$197$	14.5328	1.03542	0.517710	+	0.855556i	$0.326785\pi$
$198$	0	0
$199$	−6.23213	−0.441784	−0.220892	−	0.975298i	$-0.570897\pi$
$199$	−6.23213	−0.441784	−0.220892	+	0.975298i	$0.570897\pi$
$200$	0	0
$201$	2.98522	0.210561
$202$	0	0
$203$	2.42365	0.170107
$204$	0	0
$205$	1.01287	0.0707419
$206$	0	0
$207$	18.5666	1.29047
$208$	0	0
$209$	1.96180	0.135701
$210$	0	0
$211$	24.7994	1.70726	0.853632	−	0.520876i	$-0.174395\pi$
$211$	24.7994	1.70726	0.853632	+	0.520876i	$0.174395\pi$
$212$	0	0
$213$	22.8391	1.56491
$214$	0	0
$215$	−1.69733	−0.115757
$216$	0	0
$217$	−0.543420	−0.0368897
$218$	0	0
$219$	2.53154	0.171066
$220$	0	0
$221$	14.4276	0.970504
$222$	0	0
$223$	−15.3114	−1.02533	−0.512664	−	0.858589i	$-0.671341\pi$
$223$	−15.3114	−1.02533	−0.512664	+	0.858589i	$0.671341\pi$
$224$	0	0
$225$	−6.13046	−0.408697
$226$	0	0
$227$	5.20394	0.345398	0.172699	−	0.984975i	$-0.444751\pi$
$227$	5.20394	0.345398	0.172699	+	0.984975i	$0.444751\pi$
$228$	0	0
$229$	−18.9745	−1.25387	−0.626935	−	0.779071i	$-0.715691\pi$
$229$	−18.9745	−1.25387	−0.626935	+	0.779071i	$0.715691\pi$
$230$	0	0
$231$	1.64121	0.107983
$232$	0	0
$233$	26.2973	1.72280	0.861398	−	0.507930i	$-0.169589\pi$
$233$	26.2973	1.72280	0.861398	+	0.507930i	$0.169589\pi$
$234$	0	0
$235$	6.14581	0.400908
$236$	0	0
$237$	−27.7546	−1.80286
$238$	0	0
$239$	−24.6748	−1.59608	−0.798039	−	0.602606i	$-0.794129\pi$
$239$	−24.6748	−1.59608	−0.798039	+	0.602606i	$0.794129\pi$
$240$	0	0
$241$	−7.92902	−0.510753	−0.255376	−	0.966842i	$-0.582199\pi$
$241$	−7.92902	−0.510753	−0.255376	+	0.966842i	$0.582199\pi$
$242$	0	0
$243$	20.0814	1.28822
$244$	0	0
$245$	12.9124	0.824946
$246$	0	0
$247$	−3.98815	−0.253760
$248$	0	0
$249$	3.42996	0.217365
$250$	0	0
$251$	−8.21503	−0.518528	−0.259264	−	0.965806i	$-0.583480\pi$
$251$	−8.21503	−0.518528	−0.259264	+	0.965806i	$0.583480\pi$
$252$	0	0
$253$	6.88242	0.432694
$254$	0	0
$255$	−22.9065	−1.43446
$256$	0	0
$257$	−13.9308	−0.868980	−0.434490	−	0.900677i	$-0.643071\pi$
$257$	−13.9308	−0.868980	−0.434490	+	0.900677i	$0.643071\pi$
$258$	0	0
$259$	−3.95012	−0.245448
$260$	0	0
$261$	−26.4741	−1.63871
$262$	0	0
$263$	−25.5864	−1.57773	−0.788863	−	0.614568i	$-0.789330\pi$
$263$	−25.5864	−1.57773	−0.788863	+	0.614568i	$0.789330\pi$
$264$	0	0
$265$	−1.84653	−0.113431
$266$	0	0
$267$	−15.3144	−0.937229
$268$	0	0
$269$	−16.4385	−1.00228	−0.501138	−	0.865367i	$-0.667085\pi$
$269$	−16.4385	−1.00228	−0.501138	+	0.865367i	$0.667085\pi$
$270$	0	0
$271$	−3.21794	−0.195476	−0.0977381	−	0.995212i	$-0.531161\pi$
$271$	−3.21794	−0.195476	−0.0977381	+	0.995212i	$0.531161\pi$
$272$	0	0
$273$	−3.33642	−0.201929
$274$	0	0
$275$	−2.27249	−0.137036
$276$	0	0
$277$	−14.1743	−0.851648	−0.425824	−	0.904806i	$-0.640016\pi$
$277$	−14.1743	−0.851648	−0.425824	+	0.904806i	$0.640016\pi$
$278$	0	0
$279$	5.93592	0.355374
$280$	0	0
$281$	28.8946	1.72371	0.861853	−	0.507158i	$-0.169304\pi$
$281$	28.8946	1.72371	0.861853	+	0.507158i	$0.169304\pi$
$282$	0	0
$283$	−7.15623	−0.425394	−0.212697	−	0.977118i	$-0.568225\pi$
$283$	−7.15623	−0.425394	−0.212697	+	0.977118i	$0.568225\pi$
$284$	0	0
$285$	6.33193	0.375071
$286$	0	0
$287$	0.208529	0.0123091
$288$	0	0
$289$	3.39316	0.199598
$290$	0	0
$291$	−40.0529	−2.34794
$292$	0	0
$293$	−15.4492	−0.902554	−0.451277	−	0.892384i	$-0.649031\pi$
$293$	−15.4492	−0.902554	−0.451277	+	0.892384i	$0.649031\pi$
$294$	0	0
$295$	−6.93993	−0.404058
$296$	0	0
$297$	−5.24169	−0.304154
$298$	0	0
$299$	−13.9913	−0.809138
$300$	0	0
$301$	−0.349446	−0.0201418
$302$	0	0
$303$	24.7208	1.42017
$304$	0	0
$305$	−13.5654	−0.776753
$306$	0	0
$307$	−4.59942	−0.262503	−0.131251	−	0.991349i	$-0.541900\pi$
$307$	−4.59942	−0.262503	−0.131251	+	0.991349i	$0.541900\pi$
$308$	0	0
$309$	22.6312	1.28744
$310$	0	0
$311$	−0.886771	−0.0502842	−0.0251421	−	0.999684i	$-0.508004\pi$
$311$	−0.886771	−0.0502842	−0.0251421	+	0.999684i	$0.508004\pi$
$312$	0	0
$313$	−6.10759	−0.345222	−0.172611	−	0.984990i	$-0.555220\pi$
$313$	−6.10759	−0.345222	−0.172611	+	0.984990i	$0.555220\pi$
$314$	0	0
$315$	3.10210	0.174783
$316$	0	0
$317$	−22.8571	−1.28378	−0.641890	−	0.766797i	$-0.721849\pi$
$317$	−22.8571	−1.28378	−0.641890	+	0.766797i	$0.721849\pi$
$318$	0	0
$319$	−9.81366	−0.549460
$320$	0	0
$321$	−3.73542	−0.208491
$322$	0	0
$323$	−5.63719	−0.313662
$324$	0	0
$325$	4.61976	0.256258
$326$	0	0
$327$	−34.0219	−1.88142
$328$	0	0
$329$	1.26530	0.0697581
$330$	0	0
$331$	18.4759	1.01552	0.507762	−	0.861497i	$-0.330473\pi$
$331$	18.4759	1.01552	0.507762	+	0.861497i	$0.330473\pi$
$332$	0	0
$333$	43.1482	2.36451
$334$	0	0
$335$	2.09160	0.114276
$336$	0	0
$337$	−1.90339	−0.103684	−0.0518422	−	0.998655i	$-0.516509\pi$
$337$	−1.90339	−0.103684	−0.0518422	+	0.998655i	$0.516509\pi$
$338$	0	0
$339$	−45.6340	−2.47850
$340$	0	0
$341$	2.20038	0.119157
$342$	0	0
$343$	5.37529	0.290238
$344$	0	0
$345$	22.2138	1.19595
$346$	0	0
$347$	6.06702	0.325695	0.162847	−	0.986651i	$-0.447932\pi$
$347$	6.06702	0.325695	0.162847	+	0.986651i	$0.447932\pi$
$348$	0	0
$349$	−7.04959	−0.377356	−0.188678	−	0.982039i	$-0.560420\pi$
$349$	−7.04959	−0.377356	−0.188678	+	0.982039i	$0.560420\pi$
$350$	0	0
$351$	10.6559	0.568768
$352$	0	0
$353$	10.2471	0.545396	0.272698	−	0.962100i	$-0.412084\pi$
$353$	10.2471	0.545396	0.272698	+	0.962100i	$0.412084\pi$
$354$	0	0
$355$	16.0022	0.849310
$356$	0	0
$357$	−4.71597	−0.249596
$358$	0	0
$359$	−19.7676	−1.04329	−0.521647	−	0.853161i	$-0.674682\pi$
$359$	−19.7676	−1.04329	−0.521647	+	0.853161i	$0.674682\pi$
$360$	0	0
$361$	−17.4417	−0.917986
$362$	0	0
$363$	22.9517	1.20465
$364$	0	0
$365$	1.77373	0.0928412
$366$	0	0
$367$	−11.3342	−0.591642	−0.295821	−	0.955243i	$-0.595593\pi$
$367$	−11.3342	−0.591642	−0.295821	+	0.955243i	$0.595593\pi$
$368$	0	0
$369$	−2.27782	−0.118579
$370$	0	0
$371$	−0.380163	−0.0197371
$372$	0	0
$373$	−2.45849	−0.127296	−0.0636478	−	0.997972i	$-0.520273\pi$
$373$	−2.45849	−0.127296	−0.0636478	+	0.997972i	$0.520273\pi$
$374$	0	0
$375$	−32.6969	−1.68846
$376$	0	0
$377$	19.9502	1.02749
$378$	0	0
$379$	−15.0083	−0.770922	−0.385461	−	0.922724i	$-0.625958\pi$
$379$	−15.0083	−0.770922	−0.385461	+	0.922724i	$0.625958\pi$
$380$	0	0
$381$	−59.2239	−3.03413
$382$	0	0
$383$	13.5922	0.694527	0.347264	−	0.937768i	$-0.387111\pi$
$383$	13.5922	0.694527	0.347264	+	0.937768i	$0.387111\pi$
$384$	0	0
$385$	1.14991	0.0586050
$386$	0	0
$387$	3.81710	0.194034
$388$	0	0
$389$	−32.3678	−1.64111	−0.820556	−	0.571566i	$-0.806336\pi$
$389$	−32.3678	−1.64111	−0.820556	+	0.571566i	$0.806336\pi$
$390$	0	0
$391$	−19.7765	−1.00014
$392$	0	0
$393$	35.6126	1.79642
$394$	0	0
$395$	−19.4463	−0.978449
$396$	0	0
$397$	−0.0900745	−0.00452071	−0.00226035	−	0.999997i	$-0.500719\pi$
$397$	−0.0900745	−0.00452071	−0.00226035	+	0.999997i	$0.500719\pi$
$398$	0	0
$399$	1.30362	0.0652624
$400$	0	0
$401$	−4.62497	−0.230960	−0.115480	−	0.993310i	$-0.536841\pi$
$401$	−4.62497	−0.230960	−0.115480	+	0.993310i	$0.536841\pi$
$402$	0	0
$403$	−4.47316	−0.222824
$404$	0	0
$405$	7.05936	0.350783
$406$	0	0
$407$	15.9946	0.792821
$408$	0	0
$409$	10.4748	0.517948	0.258974	−	0.965884i	$-0.416616\pi$
$409$	10.4748	0.517948	0.258974	+	0.965884i	$0.416616\pi$
$410$	0	0
$411$	−33.6321	−1.65895
$412$	0	0
$413$	−1.42879	−0.0703061
$414$	0	0
$415$	2.40320	0.117969
$416$	0	0
$417$	11.8808	0.581805
$418$	0	0
$419$	−40.2564	−1.96665	−0.983327	−	0.181847i	$-0.941792\pi$
$419$	−40.2564	−1.96665	−0.983327	+	0.181847i	$0.941792\pi$
$420$	0	0
$421$	2.94642	0.143600	0.0717998	−	0.997419i	$-0.477126\pi$
$421$	2.94642	0.143600	0.0717998	+	0.997419i	$0.477126\pi$
$422$	0	0
$423$	−13.8212	−0.672008
$424$	0	0
$425$	6.52996	0.316750
$426$	0	0
$427$	−2.79284	−0.135155
$428$	0	0
$429$	13.5096	0.652250
$430$	0	0
$431$	−20.6361	−0.994005	−0.497003	−	0.867749i	$-0.665566\pi$
$431$	−20.6361	−0.994005	−0.497003	+	0.867749i	$0.665566\pi$
$432$	0	0
$433$	−12.1504	−0.583913	−0.291956	−	0.956432i	$-0.594306\pi$
$433$	−12.1504	−0.583913	−0.291956	+	0.956432i	$0.594306\pi$
$434$	0	0
$435$	−31.6747	−1.51869
$436$	0	0
$437$	5.46673	0.261509
$438$	0	0
$439$	25.4668	1.21546	0.607732	−	0.794143i	$-0.292080\pi$
$439$	25.4668	1.21546	0.607732	+	0.794143i	$0.292080\pi$
$440$	0	0
$441$	−29.0385	−1.38279
$442$	0	0
$443$	4.07898	0.193798	0.0968990	−	0.995294i	$-0.469108\pi$
$443$	4.07898	0.193798	0.0968990	+	0.995294i	$0.469108\pi$
$444$	0	0
$445$	−10.7301	−0.508654
$446$	0	0
$447$	−34.7426	−1.64327
$448$	0	0
$449$	−25.4712	−1.20206	−0.601030	−	0.799226i	$-0.705243\pi$
$449$	−25.4712	−1.20206	−0.601030	+	0.799226i	$0.705243\pi$
$450$	0	0
$451$	−0.844363	−0.0397595
$452$	0	0
$453$	62.5943	2.94094
$454$	0	0
$455$	−2.33766	−0.109591
$456$	0	0
$457$	−26.2913	−1.22986	−0.614928	−	0.788583i	$-0.710815\pi$
$457$	−26.2913	−1.22986	−0.614928	+	0.788583i	$0.710815\pi$
$458$	0	0
$459$	15.0619	0.703029
$460$	0	0
$461$	11.9829	0.558101	0.279050	−	0.960276i	$-0.409980\pi$
$461$	11.9829	0.558101	0.279050	+	0.960276i	$0.409980\pi$
$462$	0	0
$463$	−11.5764	−0.538000	−0.269000	−	0.963140i	$-0.586693\pi$
$463$	−11.5764	−0.538000	−0.269000	+	0.963140i	$0.586693\pi$
$464$	0	0
$465$	7.10197	0.329346
$466$	0	0
$467$	16.3523	0.756694	0.378347	−	0.925664i	$-0.376493\pi$
$467$	16.3523	0.756694	0.378347	+	0.925664i	$0.376493\pi$
$468$	0	0
$469$	0.430617	0.0198841
$470$	0	0
$471$	1.12930	0.0520355
$472$	0	0
$473$	1.41496	0.0650597
$474$	0	0
$475$	−1.80505	−0.0828213
$476$	0	0
$477$	4.15262	0.190135
$478$	0	0
$479$	−42.3237	−1.93382	−0.966911	−	0.255115i	$-0.917887\pi$
$479$	−42.3237	−1.93382	−0.966911	+	0.255115i	$0.917887\pi$
$480$	0	0
$481$	−32.5154	−1.48258
$482$	0	0
$483$	4.57337	0.208096
$484$	0	0
$485$	−28.0631	−1.27428
$486$	0	0
$487$	−5.19707	−0.235502	−0.117751	−	0.993043i	$-0.537568\pi$
$487$	−5.19707	−0.235502	−0.117751	+	0.993043i	$0.537568\pi$
$488$	0	0
$489$	−33.0989	−1.49679
$490$	0	0
$491$	−1.70903	−0.0771273	−0.0385636	−	0.999256i	$-0.512278\pi$
$491$	−1.70903	−0.0771273	−0.0385636	+	0.999256i	$0.512278\pi$
$492$	0	0
$493$	28.1994	1.27004
$494$	0	0
$495$	−12.5608	−0.564566
$496$	0	0
$497$	3.29453	0.147780
$498$	0	0
$499$	13.4829	0.603576	0.301788	−	0.953375i	$-0.402417\pi$
$499$	13.4829	0.603576	0.301788	+	0.953375i	$0.402417\pi$
$500$	0	0
$501$	−8.60967	−0.384652
$502$	0	0
$503$	−20.1192	−0.897071	−0.448535	−	0.893765i	$-0.648054\pi$
$503$	−20.1192	−0.897071	−0.448535	+	0.893765i	$0.648054\pi$
$504$	0	0
$505$	17.3206	0.770757
$506$	0	0
$507$	7.51472	0.333740
$508$	0	0
$509$	23.7346	1.05202	0.526008	−	0.850479i	$-0.323688\pi$
$509$	23.7346	1.05202	0.526008	+	0.850479i	$0.323688\pi$
$510$	0	0
$511$	0.365174	0.0161544
$512$	0	0
$513$	−4.16349	−0.183823
$514$	0	0
$515$	15.8566	0.698724
$516$	0	0
$517$	−5.12335	−0.225325
$518$	0	0
$519$	−24.3414	−1.06847
$520$	0	0
$521$	−15.5725	−0.682244	−0.341122	−	0.940019i	$-0.610807\pi$
$521$	−15.5725	−0.682244	−0.341122	+	0.940019i	$0.610807\pi$
$522$	0	0
$523$	3.34558	0.146292	0.0731460	−	0.997321i	$-0.476696\pi$
$523$	3.34558	0.146292	0.0731460	+	0.997321i	$0.476696\pi$
$524$	0	0
$525$	−1.51007	−0.0659049
$526$	0	0
$527$	−6.32274	−0.275423
$528$	0	0
$529$	−3.82151	−0.166153
$530$	0	0
$531$	15.6070	0.677288
$532$	0	0
$533$	1.71651	0.0743503
$534$	0	0
$535$	−2.61723	−0.113153
$536$	0	0
$537$	4.88443	0.210779
$538$	0	0
$539$	−10.7643	−0.463649
$540$	0	0
$541$	9.34207	0.401647	0.200823	−	0.979627i	$-0.435638\pi$
$541$	9.34207	0.401647	0.200823	+	0.979627i	$0.435638\pi$
$542$	0	0
$543$	−65.8346	−2.82523
$544$	0	0
$545$	−23.8375	−1.02109
$546$	0	0
$547$	−28.0985	−1.20141	−0.600703	−	0.799472i	$-0.705113\pi$
$547$	−28.0985	−1.20141	−0.600703	+	0.799472i	$0.705113\pi$
$548$	0	0
$549$	30.5070	1.30201
$550$	0	0
$551$	−7.79503	−0.332079
$552$	0	0
$553$	−4.00360	−0.170250
$554$	0	0
$555$	51.6243	2.19133
$556$	0	0
$557$	22.9033	0.970445	0.485222	−	0.874391i	$-0.338739\pi$
$557$	22.9033	0.970445	0.485222	+	0.874391i	$0.338739\pi$
$558$	0	0
$559$	−2.87647	−0.121662
$560$	0	0
$561$	19.0956	0.806217
$562$	0	0
$563$	19.0756	0.803942	0.401971	−	0.915652i	$-0.368325\pi$
$563$	19.0756	0.803942	0.401971	+	0.915652i	$0.368325\pi$
$564$	0	0
$565$	−31.9735	−1.34514
$566$	0	0
$567$	1.45338	0.0610362
$568$	0	0
$569$	12.8632	0.539252	0.269626	−	0.962965i	$-0.413100\pi$
$569$	12.8632	0.539252	0.269626	+	0.962965i	$0.413100\pi$
$570$	0	0
$571$	1.77101	0.0741145	0.0370572	−	0.999313i	$-0.488202\pi$
$571$	1.77101	0.0741145	0.0370572	+	0.999313i	$0.488202\pi$
$572$	0	0
$573$	1.40165	0.0585546
$574$	0	0
$575$	−6.33251	−0.264084
$576$	0	0
$577$	6.20767	0.258429	0.129214	−	0.991617i	$-0.458755\pi$
$577$	6.20767	0.258429	0.129214	+	0.991617i	$0.458755\pi$
$578$	0	0
$579$	6.59526	0.274089
$580$	0	0
$581$	0.494771	0.0205265
$582$	0	0
$583$	1.53933	0.0637525
$584$	0	0
$585$	25.5349	1.05574
$586$	0	0
$587$	−36.7308	−1.51604	−0.758021	−	0.652230i	$-0.773834\pi$
$587$	−36.7308	−1.51604	−0.758021	+	0.652230i	$0.773834\pi$
$588$	0	0
$589$	1.74777	0.0720155
$590$	0	0
$591$	−39.1027	−1.60847
$592$	0	0
$593$	19.8468	0.815009	0.407505	−	0.913203i	$-0.366399\pi$
$593$	19.8468	0.815009	0.407505	+	0.913203i	$0.366399\pi$
$594$	0	0
$595$	−3.30425	−0.135461
$596$	0	0
$597$	16.7685	0.686288
$598$	0	0
$599$	4.15468	0.169756	0.0848778	−	0.996391i	$-0.472950\pi$
$599$	4.15468	0.169756	0.0848778	+	0.996391i	$0.472950\pi$
$600$	0	0
$601$	−42.2339	−1.72276	−0.861378	−	0.507965i	$-0.830398\pi$
$601$	−42.2339	−1.72276	−0.861378	+	0.507965i	$0.830398\pi$
$602$	0	0
$603$	−4.70375	−0.191551
$604$	0	0
$605$	16.0811	0.653790
$606$	0	0
$607$	25.2958	1.02673	0.513363	−	0.858171i	$-0.328399\pi$
$607$	25.2958	1.02673	0.513363	+	0.858171i	$0.328399\pi$
$608$	0	0
$609$	−6.52118	−0.264252
$610$	0	0
$611$	10.4153	0.421358
$612$	0	0
$613$	14.2515	0.575615	0.287807	−	0.957688i	$-0.407074\pi$
$613$	14.2515	0.575615	0.287807	+	0.957688i	$0.407074\pi$
$614$	0	0
$615$	−2.72528	−0.109894
$616$	0	0
$617$	−46.7596	−1.88247	−0.941236	−	0.337749i	$-0.890334\pi$
$617$	−46.7596	−1.88247	−0.941236	+	0.337749i	$0.890334\pi$
$618$	0	0
$619$	−13.4756	−0.541629	−0.270815	−	0.962632i	$-0.587293\pi$
$619$	−13.4756	−0.541629	−0.270815	+	0.962632i	$0.587293\pi$
$620$	0	0
$621$	−14.6064	−0.586137
$622$	0	0
$623$	−2.20910	−0.0885059
$624$	0	0
$625$	−15.6791	−0.627163
$626$	0	0
$627$	−5.27852	−0.210804
$628$	0	0
$629$	−45.9600	−1.83255
$630$	0	0
$631$	−32.8950	−1.30953	−0.654765	−	0.755833i	$-0.727232\pi$
$631$	−32.8950	−1.30953	−0.654765	+	0.755833i	$0.727232\pi$
$632$	0	0
$633$	−66.7266	−2.65215
$634$	0	0
$635$	−41.4953	−1.64669
$636$	0	0
$637$	21.8827	0.867024
$638$	0	0
$639$	−35.9871	−1.42363
$640$	0	0
$641$	−26.1505	−1.03288	−0.516441	−	0.856323i	$-0.672743\pi$
$641$	−26.1505	−1.03288	−0.516441	+	0.856323i	$0.672743\pi$
$642$	0	0
$643$	20.0805	0.791898	0.395949	−	0.918273i	$-0.370416\pi$
$643$	20.0805	0.791898	0.395949	+	0.918273i	$0.370416\pi$
$644$	0	0
$645$	4.56693	0.179823
$646$	0	0
$647$	4.25280	0.167195	0.0835974	−	0.996500i	$-0.473359\pi$
$647$	4.25280	0.167195	0.0835974	+	0.996500i	$0.473359\pi$
$648$	0	0
$649$	5.78536	0.227095
$650$	0	0
$651$	1.46215	0.0573062
$652$	0	0
$653$	10.6050	0.415004	0.207502	−	0.978235i	$-0.433467\pi$
$653$	10.6050	0.415004	0.207502	+	0.978235i	$0.433467\pi$
$654$	0	0
$655$	24.9520	0.974956
$656$	0	0
$657$	−3.98890	−0.155622
$658$	0	0
$659$	44.6528	1.73943	0.869714	−	0.493557i	$-0.164303\pi$
$659$	44.6528	1.73943	0.869714	+	0.493557i	$0.164303\pi$
$660$	0	0
$661$	−3.90672	−0.151954	−0.0759768	−	0.997110i	$-0.524207\pi$
$661$	−3.90672	−0.151954	−0.0759768	+	0.997110i	$0.524207\pi$
$662$	0	0
$663$	−38.8196	−1.50763
$664$	0	0
$665$	0.913380	0.0354193
$666$	0	0
$667$	−27.3467	−1.05887
$668$	0	0
$669$	41.1977	1.59279
$670$	0	0
$671$	11.3086	0.436563
$672$	0	0
$673$	−42.3659	−1.63308	−0.816542	−	0.577286i	$-0.804112\pi$
$673$	−42.3659	−1.63308	−0.816542	+	0.577286i	$0.804112\pi$
$674$	0	0
$675$	4.82287	0.185632
$676$	0	0
$677$	−1.87717	−0.0721453	−0.0360727	−	0.999349i	$-0.511485\pi$
$677$	−1.87717	−0.0721453	−0.0360727	+	0.999349i	$0.511485\pi$
$678$	0	0
$679$	−5.77762	−0.221725
$680$	0	0
$681$	−14.0020	−0.536557
$682$	0	0
$683$	−47.6664	−1.82390	−0.911952	−	0.410296i	$-0.865425\pi$
$683$	−47.6664	−1.82390	−0.911952	+	0.410296i	$0.865425\pi$
$684$	0	0
$685$	−23.5643	−0.900347
$686$	0	0
$687$	51.0537	1.94782
$688$	0	0
$689$	−3.12931	−0.119217
$690$	0	0
$691$	−23.6462	−0.899542	−0.449771	−	0.893144i	$-0.648494\pi$
$691$	−23.6462	−0.899542	−0.449771	+	0.893144i	$0.648494\pi$
$692$	0	0
$693$	−2.58601	−0.0982345
$694$	0	0
$695$	8.32429	0.315758
$696$	0	0
$697$	2.42626	0.0919011
$698$	0	0
$699$	−70.7569	−2.67627
$700$	0	0
$701$	13.6305	0.514817	0.257409	−	0.966303i	$-0.417131\pi$
$701$	13.6305	0.514817	0.257409	+	0.966303i	$0.417131\pi$
$702$	0	0
$703$	12.7045	0.479160
$704$	0	0
$705$	−16.5362	−0.622790
$706$	0	0
$707$	3.56596	0.134112
$708$	0	0
$709$	18.2615	0.685826	0.342913	−	0.939367i	$-0.388586\pi$
$709$	18.2615	0.685826	0.342913	+	0.939367i	$0.388586\pi$
$710$	0	0
$711$	43.7323	1.64009
$712$	0	0
$713$	6.13155	0.229628
$714$	0	0
$715$	9.46551	0.353990
$716$	0	0
$717$	66.3911	2.47942
$718$	0	0
$719$	4.04766	0.150952	0.0754761	−	0.997148i	$-0.475952\pi$
$719$	4.04766	0.150952	0.0754761	+	0.997148i	$0.475952\pi$
$720$	0	0
$721$	3.26454	0.121578
$722$	0	0
$723$	21.3342	0.793428
$724$	0	0
$725$	9.02953	0.335349
$726$	0	0
$727$	3.12087	0.115746	0.0578732	−	0.998324i	$-0.481568\pi$
$727$	3.12087	0.115746	0.0578732	+	0.998324i	$0.481568\pi$
$728$	0	0
$729$	−42.7982	−1.58512
$730$	0	0
$731$	−4.06585	−0.150381
$732$	0	0
$733$	45.1859	1.66898	0.834490	−	0.551023i	$-0.185762\pi$
$733$	45.1859	1.66898	0.834490	+	0.551023i	$0.185762\pi$
$734$	0	0
$735$	−34.7429	−1.28151
$736$	0	0
$737$	−1.74363	−0.0642273
$738$	0	0
$739$	12.9216	0.475330	0.237665	−	0.971347i	$-0.423618\pi$
$739$	12.9216	0.475330	0.237665	+	0.971347i	$0.423618\pi$
$740$	0	0
$741$	10.7307	0.394203
$742$	0	0
$743$	−21.1337	−0.775321	−0.387661	−	0.921802i	$-0.626717\pi$
$743$	−21.1337	−0.775321	−0.387661	+	0.921802i	$0.626717\pi$
$744$	0	0
$745$	−24.3424	−0.891838
$746$	0	0
$747$	−5.40451	−0.197741
$748$	0	0
$749$	−0.538833	−0.0196885
$750$	0	0
$751$	22.3446	0.815368	0.407684	−	0.913123i	$-0.366337\pi$
$751$	22.3446	0.815368	0.407684	+	0.913123i	$0.366337\pi$
$752$	0	0
$753$	22.1038	0.805506
$754$	0	0
$755$	43.8568	1.59611
$756$	0	0
$757$	51.6060	1.87565	0.937826	−	0.347105i	$-0.112835\pi$
$757$	51.6060	1.87565	0.937826	+	0.347105i	$0.112835\pi$
$758$	0	0
$759$	−18.5182	−0.672168
$760$	0	0
$761$	−11.7254	−0.425044	−0.212522	−	0.977156i	$-0.568168\pi$
$761$	−11.7254	−0.425044	−0.212522	+	0.977156i	$0.568168\pi$
$762$	0	0
$763$	−4.90766	−0.177669
$764$	0	0
$765$	36.0932	1.30495
$766$	0	0
$767$	−11.7611	−0.424668
$768$	0	0
$769$	25.8137	0.930864	0.465432	−	0.885084i	$-0.345899\pi$
$769$	25.8137	0.930864	0.465432	+	0.885084i	$0.345899\pi$
$770$	0	0
$771$	37.4829	1.34991
$772$	0	0
$773$	−52.8758	−1.90181	−0.950905	−	0.309484i	$-0.899844\pi$
$773$	−52.8758	−1.90181	−0.950905	+	0.309484i	$0.899844\pi$
$774$	0	0
$775$	−2.02456	−0.0727245
$776$	0	0
$777$	10.6284	0.381291
$778$	0	0
$779$	−0.670680	−0.0240296
$780$	0	0
$781$	−13.3400	−0.477343
$782$	0	0
$783$	20.8274	0.744310
$784$	0	0
$785$	0.791246	0.0282408
$786$	0	0
$787$	26.2200	0.934642	0.467321	−	0.884088i	$-0.345219\pi$
$787$	26.2200	0.934642	0.467321	+	0.884088i	$0.345219\pi$
$788$	0	0
$789$	68.8441	2.45092
$790$	0	0
$791$	−6.58269	−0.234054
$792$	0	0
$793$	−22.9893	−0.816374
$794$	0	0
$795$	4.96836	0.176210
$796$	0	0
$797$	30.1854	1.06922	0.534612	−	0.845098i	$-0.320458\pi$
$797$	30.1854	1.06922	0.534612	+	0.845098i	$0.320458\pi$
$798$	0	0
$799$	14.7219	0.520822
$800$	0	0
$801$	24.1306	0.852614
$802$	0	0
$803$	−1.47864	−0.0521801
$804$	0	0
$805$	3.20434	0.112938
$806$	0	0
$807$	44.2304	1.55698
$808$	0	0
$809$	1.62259	0.0570471	0.0285235	−	0.999593i	$-0.490919\pi$
$809$	1.62259	0.0570471	0.0285235	+	0.999593i	$0.490919\pi$
$810$	0	0
$811$	−29.7356	−1.04416	−0.522079	−	0.852897i	$-0.674843\pi$
$811$	−29.7356	−1.04416	−0.522079	+	0.852897i	$0.674843\pi$
$812$	0	0
$813$	8.65836	0.303662
$814$	0	0
$815$	−23.1908	−0.812338
$816$	0	0
$817$	1.12390	0.0393204
$818$	0	0
$819$	5.25712	0.183699
$820$	0	0
$821$	36.3534	1.26874	0.634372	−	0.773028i	$-0.281259\pi$
$821$	36.3534	1.26874	0.634372	+	0.773028i	$0.281259\pi$
$822$	0	0
$823$	1.94869	0.0679270	0.0339635	−	0.999423i	$-0.489187\pi$
$823$	1.94869	0.0679270	0.0339635	+	0.999423i	$0.489187\pi$
$824$	0	0
$825$	6.11448	0.212879
$826$	0	0
$827$	−4.26915	−0.148453	−0.0742264	−	0.997241i	$-0.523649\pi$
$827$	−4.26915	−0.148453	−0.0742264	+	0.997241i	$0.523649\pi$
$828$	0	0
$829$	−0.0167257	−0.000580907 0	−0.000290453	−	1.00000i	$-0.500092\pi$
$829$	−0.0167257	−0.000580907 0	−0.000290453		1.00000i	$0.500092\pi$
$830$	0	0
$831$	38.1380	1.32299
$832$	0	0
$833$	30.9309	1.07169
$834$	0	0
$835$	−6.03237	−0.208759
$836$	0	0
$837$	−4.66983	−0.161413
$838$	0	0
$839$	19.9429	0.688505	0.344252	−	0.938877i	$-0.388132\pi$
$839$	19.9429	0.688505	0.344252	+	0.938877i	$0.388132\pi$
$840$	0	0
$841$	9.99366	0.344609
$842$	0	0
$843$	−77.7452	−2.67769
$844$	0	0
$845$	5.26519	0.181128
$846$	0	0
$847$	3.31077	0.113759
$848$	0	0
$849$	19.2549	0.660826
$850$	0	0
$851$	44.5703	1.52785
$852$	0	0
$853$	−34.1489	−1.16924	−0.584618	−	0.811309i	$-0.698756\pi$
$853$	−34.1489	−1.16924	−0.584618	+	0.811309i	$0.698756\pi$
$854$	0	0
$855$	−9.97709	−0.341209
$856$	0	0
$857$	43.2240	1.47650	0.738251	−	0.674526i	$-0.235652\pi$
$857$	43.2240	1.47650	0.738251	+	0.674526i	$0.235652\pi$
$858$	0	0
$859$	41.3258	1.41002	0.705009	−	0.709199i	$-0.250943\pi$
$859$	41.3258	1.41002	0.705009	+	0.709199i	$0.250943\pi$
$860$	0	0
$861$	−0.561079	−0.0191215
$862$	0	0
$863$	−50.6855	−1.72535	−0.862677	−	0.505756i	$-0.831214\pi$
$863$	−50.6855	−1.72535	−0.862677	+	0.505756i	$0.831214\pi$
$864$	0	0
$865$	−17.0548	−0.579882
$866$	0	0
$867$	−9.12981	−0.310065
$868$	0	0
$869$	16.2111	0.549924
$870$	0	0
$871$	3.54463	0.120105
$872$	0	0
$873$	63.1104	2.13596
$874$	0	0
$875$	−4.71651	−0.159447
$876$	0	0
$877$	−5.68293	−0.191899	−0.0959494	−	0.995386i	$-0.530589\pi$
$877$	−5.68293	−0.191899	−0.0959494	+	0.995386i	$0.530589\pi$
$878$	0	0
$879$	41.5685	1.40207
$880$	0	0
$881$	−0.730587	−0.0246141	−0.0123071	−	0.999924i	$-0.503918\pi$
$881$	−0.730587	−0.0246141	−0.0123071	+	0.999924i	$0.503918\pi$
$882$	0	0
$883$	17.4552	0.587413	0.293707	−	0.955896i	$-0.405111\pi$
$883$	17.4552	0.587413	0.293707	+	0.955896i	$0.405111\pi$
$884$	0	0
$885$	18.6729	0.627683
$886$	0	0
$887$	−17.5254	−0.588446	−0.294223	−	0.955737i	$-0.595061\pi$
$887$	−17.5254	−0.588446	−0.294223	+	0.955737i	$0.595061\pi$
$888$	0	0
$889$	−8.54303	−0.286524
$890$	0	0
$891$	−5.88492	−0.197152
$892$	0	0
$893$	−4.06950	−0.136181
$894$	0	0
$895$	3.42228	0.114394
$896$	0	0
$897$	37.6457	1.25695
$898$	0	0
$899$	−8.74300	−0.291595
$900$	0	0
$901$	−4.42323	−0.147359
$902$	0	0
$903$	0.940238	0.0312892
$904$	0	0
$905$	−46.1270	−1.53331
$906$	0	0
$907$	7.23655	0.240286	0.120143	−	0.992757i	$-0.461665\pi$
$907$	7.23655	0.240286	0.120143	+	0.992757i	$0.461665\pi$
$908$	0	0
$909$	−38.9519	−1.29195
$910$	0	0
$911$	10.0763	0.333841	0.166921	−	0.985970i	$-0.446618\pi$
$911$	10.0763	0.333841	0.166921	+	0.985970i	$0.446618\pi$
$912$	0	0
$913$	−2.00339	−0.0663026
$914$	0	0
$915$	36.4998	1.20665
$916$	0	0
$917$	5.13711	0.169642
$918$	0	0
$919$	29.7093	0.980020	0.490010	−	0.871717i	$-0.336993\pi$
$919$	29.7093	0.980020	0.490010	+	0.871717i	$0.336993\pi$
$920$	0	0
$921$	12.3754	0.407784
$922$	0	0
$923$	27.1190	0.892631
$924$	0	0
$925$	−14.7166	−0.483878
$926$	0	0
$927$	−35.6595	−1.17121
$928$	0	0
$929$	−29.4106	−0.964929	−0.482465	−	0.875915i	$-0.660258\pi$
$929$	−29.4106	−0.964929	−0.482465	+	0.875915i	$0.660258\pi$
$930$	0	0
$931$	−8.55009	−0.280218
$932$	0	0
$933$	2.38599	0.0781138
$934$	0	0
$935$	13.3794	0.437552
$936$	0	0
$937$	−25.0627	−0.818762	−0.409381	−	0.912364i	$-0.634255\pi$
$937$	−25.0627	−0.818762	−0.409381	+	0.912364i	$0.634255\pi$
$938$	0	0
$939$	16.4334	0.536283
$940$	0	0
$941$	−27.6562	−0.901568	−0.450784	−	0.892633i	$-0.648856\pi$
$941$	−27.6562	−0.901568	−0.450784	+	0.892633i	$0.648856\pi$
$942$	0	0
$943$	−2.35289	−0.0766207
$944$	0	0
$945$	−2.44044	−0.0793875
$946$	0	0
$947$	21.4672	0.697592	0.348796	−	0.937199i	$-0.386591\pi$
$947$	21.4672	0.697592	0.348796	+	0.937199i	$0.386591\pi$
$948$	0	0
$949$	3.00593	0.0975768
$950$	0	0
$951$	61.5003	1.99429
$952$	0	0
$953$	−3.00410	−0.0973124	−0.0486562	−	0.998816i	$-0.515494\pi$
$953$	−3.00410	−0.0973124	−0.0486562	+	0.998816i	$0.515494\pi$
$954$	0	0
$955$	0.982065	0.0317789
$956$	0	0
$957$	26.4051	0.853557
$958$	0	0
$959$	−4.85141	−0.156660
$960$	0	0
$961$	−29.0397	−0.936764
$962$	0	0
$963$	5.88582	0.189668
$964$	0	0
$965$	4.62097	0.148754
$966$	0	0
$967$	36.4154	1.17104	0.585520	−	0.810658i	$-0.300890\pi$
$967$	36.4154	1.17104	0.585520	+	0.810658i	$0.300890\pi$
$968$	0	0
$969$	15.1677	0.487257
$970$	0	0
$971$	−30.9648	−0.993709	−0.496855	−	0.867834i	$-0.665512\pi$
$971$	−30.9648	−0.993709	−0.496855	+	0.867834i	$0.665512\pi$
$972$	0	0
$973$	1.71380	0.0549420
$974$	0	0
$975$	−12.4302	−0.398084
$976$	0	0
$977$	9.27510	0.296737	0.148368	−	0.988932i	$-0.452598\pi$
$977$	9.27510	0.296737	0.148368	+	0.988932i	$0.452598\pi$
$978$	0	0
$979$	8.94496	0.285882
$980$	0	0
$981$	53.6076	1.71156
$982$	0	0
$983$	16.7303	0.533614	0.266807	−	0.963750i	$-0.414031\pi$
$983$	16.7303	0.533614	0.266807	+	0.963750i	$0.414031\pi$
$984$	0	0
$985$	−27.3974	−0.872952
$986$	0	0
$987$	−3.40447	−0.108365
$988$	0	0
$989$	3.94290	0.125377
$990$	0	0
$991$	7.15798	0.227381	0.113690	−	0.993516i	$-0.463733\pi$
$991$	7.15798	0.227381	0.113690	+	0.993516i	$0.463733\pi$
$992$	0	0
$993$	−49.7121	−1.57756
$994$	0	0
$995$	11.7488	0.372464
$996$	0	0
$997$	36.1850	1.14599	0.572996	−	0.819558i	$-0.305781\pi$
$997$	36.1850	1.14599	0.572996	+	0.819558i	$0.305781\pi$
$998$	0	0
$999$	−33.9450	−1.07397

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8044.2.a.a.1.9	✓	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8044.2.a.a.1.9	✓	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)

\(f(q)\)	\(=\)	\(q-2.69065 q^{3} -1.88521 q^{5} -0.388125 q^{7} +4.23960 q^{9} +O(q^{10})\)	\(q-2.69065 q^{3} -1.88521 q^{5} -0.388125 q^{7} +4.23960 q^{9} +1.57157 q^{11} -3.19486 q^{13} +5.07243 q^{15} -4.51588 q^{17} +1.24830 q^{19} +1.04431 q^{21} +4.37933 q^{23} -1.44600 q^{25} -3.33532 q^{27} -6.24449 q^{29} +1.40011 q^{31} -4.22855 q^{33} +0.731696 q^{35} +10.1774 q^{37} +8.59624 q^{39} -0.537273 q^{41} +0.900344 q^{43} -7.99251 q^{45} -3.26002 q^{47} -6.84936 q^{49} +12.1506 q^{51} +0.979484 q^{53} -2.96273 q^{55} -3.35875 q^{57} +3.68126 q^{59} +7.19572 q^{61} -1.64549 q^{63} +6.02296 q^{65} -1.10948 q^{67} -11.7832 q^{69} -8.48832 q^{71} -0.940867 q^{73} +3.89068 q^{75} -0.609967 q^{77} +10.3152 q^{79} -3.74461 q^{81} -1.27477 q^{83} +8.51336 q^{85} +16.8017 q^{87} +5.69173 q^{89} +1.24000 q^{91} -3.76722 q^{93} -2.35331 q^{95} +14.8860 q^{97} +6.66283 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9}+O(q^{10}) \) 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

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