LMFDB - Embedded newform 8048.2.a.v.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8048,2,Mod(1,8048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8048, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8048 = 2^{4} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8048.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:	$64.2636035467$
Analytic rank:	$1$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	8048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.667261 q^{3} +0.307121 q^{5} -4.25072 q^{7} -2.55476 q^{9} +O(q^{10})$	$q+0.667261 q^{3} +0.307121 q^{5} -4.25072 q^{7} -2.55476 q^{9} -0.721001 q^{11} -4.50964 q^{13} +0.204930 q^{15} +7.96970 q^{17} +6.29804 q^{19} -2.83634 q^{21} +4.88211 q^{23} -4.90568 q^{25} -3.70648 q^{27} +7.38483 q^{29} +8.57254 q^{31} -0.481096 q^{33} -1.30549 q^{35} -0.903054 q^{37} -3.00911 q^{39} -9.48154 q^{41} +1.86056 q^{43} -0.784621 q^{45} +0.432654 q^{47} +11.0686 q^{49} +5.31787 q^{51} -13.1315 q^{53} -0.221434 q^{55} +4.20244 q^{57} -3.80194 q^{59} -6.98011 q^{61} +10.8596 q^{63} -1.38501 q^{65} -4.59763 q^{67} +3.25764 q^{69} -1.98521 q^{71} +3.07898 q^{73} -3.27337 q^{75} +3.06477 q^{77} -1.39903 q^{79} +5.19110 q^{81} +8.69377 q^{83} +2.44766 q^{85} +4.92761 q^{87} +3.04116 q^{89} +19.1692 q^{91} +5.72013 q^{93} +1.93426 q^{95} -5.67187 q^{97} +1.84199 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) $ 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9	$ 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9} + 14 q^{11} - 31 q^{13} + 2 q^{15} - 9 q^{17} + 8 q^{19} - 28 q^{21} + 4 q^{23} + 22 q^{25} + 4 q^{27} - 47 q^{29} + 5 q^{31} - 26 q^{33} + 13 q^{35} - 67 q^{37} + 9 q^{39} - 28 q^{41} - 15 q^{43} - 57 q^{45} + 10 q^{47} + 20 q^{49} + 11 q^{51} - 58 q^{53} - 15 q^{55} - 31 q^{57} + 32 q^{59} - 55 q^{61} + 16 q^{63} - 44 q^{65} - 22 q^{67} - 44 q^{69} + 47 q^{71} - 5 q^{73} + 25 q^{75} - 50 q^{77} + 14 q^{79} - 28 q^{81} + 16 q^{83} - 78 q^{85} + 11 q^{87} - 20 q^{89} + 15 q^{91} - 83 q^{93} + 27 q^{95} - 8 q^{97} + 70 q^{99}+O(q^{100}) $ 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9 + 14 * q^11 - 31 * q^13 + 2 * q^15 - 9 * q^17 + 8 * q^19 - 28 * q^21 + 4 * q^23 + 22 * q^25 + 4 * q^27 - 47 * q^29 + 5 * q^31 - 26 * q^33 + 13 * q^35 - 67 * q^37 + 9 * q^39 - 28 * q^41 - 15 * q^43 - 57 * q^45 + 10 * q^47 + 20 * q^49 + 11 * q^51 - 58 * q^53 - 15 * q^55 - 31 * q^57 + 32 * q^59 - 55 * q^61 + 16 * q^63 - 44 * q^65 - 22 * q^67 - 44 * q^69 + 47 * q^71 - 5 * q^73 + 25 * q^75 - 50 * q^77 + 14 * q^79 - 28 * q^81 + 16 * q^83 - 78 * q^85 + 11 * q^87 - 20 * q^89 + 15 * q^91 - 83 * q^93 + 27 * q^95 - 8 * q^97 + 70 * 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
$3$	0.667261	0.385244	0.192622	−	0.981273i	$-0.438301\pi$
$3$	0.667261	0.385244	0.192622	+	0.981273i	$0.438301\pi$
$4$	0	0
$5$	0.307121	0.137349	0.0686743	−	0.997639i	$-0.478123\pi$
$5$	0.307121	0.137349	0.0686743	+	0.997639i	$0.478123\pi$
$6$	0	0
$7$	−4.25072	−1.60662	−0.803311	−	0.595560i	$-0.796930\pi$
$7$	−4.25072	−1.60662	−0.803311	+	0.595560i	$0.796930\pi$
$8$	0	0
$9$	−2.55476	−0.851587
$10$	0	0
$11$	−0.721001	−0.217390	−0.108695	−	0.994075i	$-0.534667\pi$
$11$	−0.721001	−0.217390	−0.108695	+	0.994075i	$0.534667\pi$
$12$	0	0
$13$	−4.50964	−1.25075	−0.625375	−	0.780325i	$-0.715054\pi$
$13$	−4.50964	−1.25075	−0.625375	+	0.780325i	$0.715054\pi$
$14$	0	0
$15$	0.204930	0.0529127
$16$	0	0
$17$	7.96970	1.93294	0.966468	−	0.256786i	$-0.0826635\pi$
$17$	7.96970	1.93294	0.966468	+	0.256786i	$0.0826635\pi$
$18$	0	0
$19$	6.29804	1.44487	0.722435	−	0.691439i	$-0.243023\pi$
$19$	6.29804	1.44487	0.722435	+	0.691439i	$0.243023\pi$
$20$	0	0
$21$	−2.83634	−0.618941
$22$	0	0
$23$	4.88211	1.01799	0.508995	−	0.860769i	$-0.330017\pi$
$23$	4.88211	1.01799	0.508995	+	0.860769i	$0.330017\pi$
$24$	0	0
$25$	−4.90568	−0.981135
$26$	0	0
$27$	−3.70648	−0.713312
$28$	0	0
$29$	7.38483	1.37133	0.685664	−	0.727918i	$-0.259512\pi$
$29$	7.38483	1.37133	0.685664	+	0.727918i	$0.259512\pi$
$30$	0	0
$31$	8.57254	1.53967	0.769837	−	0.638241i	$-0.220337\pi$
$31$	8.57254	1.53967	0.769837	+	0.638241i	$0.220337\pi$
$32$	0	0
$33$	−0.481096	−0.0837480
$34$	0	0
$35$	−1.30549	−0.220667
$36$	0	0
$37$	−0.903054	−0.148461	−0.0742306	−	0.997241i	$-0.523650\pi$
$37$	−0.903054	−0.148461	−0.0742306	+	0.997241i	$0.523650\pi$
$38$	0	0
$39$	−3.00911	−0.481843
$40$	0	0
$41$	−9.48154	−1.48077	−0.740384	−	0.672184i	$-0.765356\pi$
$41$	−9.48154	−1.48077	−0.740384	+	0.672184i	$0.765356\pi$
$42$	0	0
$43$	1.86056	0.283732	0.141866	−	0.989886i	$-0.454690\pi$
$43$	1.86056	0.283732	0.141866	+	0.989886i	$0.454690\pi$
$44$	0	0
$45$	−0.784621	−0.116964
$46$	0	0
$47$	0.432654	0.0631090	0.0315545	−	0.999502i	$-0.489954\pi$
$47$	0.432654	0.0631090	0.0315545	+	0.999502i	$0.489954\pi$
$48$	0	0
$49$	11.0686	1.58123
$50$	0	0
$51$	5.31787	0.744651
$52$	0	0
$53$	−13.1315	−1.80375	−0.901876	−	0.431994i	$-0.857810\pi$
$53$	−13.1315	−1.80375	−0.901876	+	0.431994i	$0.857810\pi$
$54$	0	0
$55$	−0.221434	−0.0298582
$56$	0	0
$57$	4.20244	0.556627
$58$	0	0
$59$	−3.80194	−0.494971	−0.247485	−	0.968892i	$-0.579604\pi$
$59$	−3.80194	−0.494971	−0.247485	+	0.968892i	$0.579604\pi$
$60$	0	0
$61$	−6.98011	−0.893712	−0.446856	−	0.894606i	$-0.647456\pi$
$61$	−6.98011	−0.893712	−0.446856	+	0.894606i	$0.647456\pi$
$62$	0	0
$63$	10.8596	1.36818
$64$	0	0
$65$	−1.38501	−0.171789
$66$	0	0
$67$	−4.59763	−0.561690	−0.280845	−	0.959753i	$-0.590615\pi$
$67$	−4.59763	−0.561690	−0.280845	+	0.959753i	$0.590615\pi$
$68$	0	0
$69$	3.25764	0.392174
$70$	0	0
$71$	−1.98521	−0.235601	−0.117800	−	0.993037i	$-0.537584\pi$
$71$	−1.98521	−0.235601	−0.117800	+	0.993037i	$0.537584\pi$
$72$	0	0
$73$	3.07898	0.360368	0.180184	−	0.983633i	$-0.442331\pi$
$73$	3.07898	0.360368	0.180184	+	0.983633i	$0.442331\pi$
$74$	0	0
$75$	−3.27337	−0.377976
$76$	0	0
$77$	3.06477	0.349263
$78$	0	0
$79$	−1.39903	−0.157403	−0.0787014	−	0.996898i	$-0.525077\pi$
$79$	−1.39903	−0.157403	−0.0787014	+	0.996898i	$0.525077\pi$
$80$	0	0
$81$	5.19110	0.576789
$82$	0	0
$83$	8.69377	0.954265	0.477132	−	0.878831i	$-0.341676\pi$
$83$	8.69377	0.954265	0.477132	+	0.878831i	$0.341676\pi$
$84$	0	0
$85$	2.44766	0.265486
$86$	0	0
$87$	4.92761	0.528296
$88$	0	0
$89$	3.04116	0.322362	0.161181	−	0.986925i	$-0.448470\pi$
$89$	3.04116	0.322362	0.161181	+	0.986925i	$0.448470\pi$
$90$	0	0
$91$	19.1692	2.00948
$92$	0	0
$93$	5.72013	0.593149
$94$	0	0
$95$	1.93426	0.198451
$96$	0	0
$97$	−5.67187	−0.575891	−0.287945	−	0.957647i	$-0.592972\pi$
$97$	−5.67187	−0.575891	−0.287945	+	0.957647i	$0.592972\pi$
$98$	0	0
$99$	1.84199	0.185126
$100$	0	0
$101$	6.91983	0.688548	0.344274	−	0.938869i	$-0.388125\pi$
$101$	6.91983	0.688548	0.344274	+	0.938869i	$0.388125\pi$
$102$	0	0
$103$	1.40902	0.138835	0.0694174	−	0.997588i	$-0.477886\pi$
$103$	1.40902	0.138835	0.0694174	+	0.997588i	$0.477886\pi$
$104$	0	0
$105$	−0.871100	−0.0850106
$106$	0	0
$107$	3.70772	0.358439	0.179220	−	0.983809i	$-0.442643\pi$
$107$	3.70772	0.358439	0.179220	+	0.983809i	$0.442643\pi$
$108$	0	0
$109$	−6.48992	−0.621622	−0.310811	−	0.950472i	$-0.600601\pi$
$109$	−6.48992	−0.621622	−0.310811	+	0.950472i	$0.600601\pi$
$110$	0	0
$111$	−0.602573	−0.0571937
$112$	0	0
$113$	−10.0417	−0.944645	−0.472322	−	0.881426i	$-0.656584\pi$
$113$	−10.0417	−0.944645	−0.472322	+	0.881426i	$0.656584\pi$
$114$	0	0
$115$	1.49940	0.139820
$116$	0	0
$117$	11.5211	1.06512
$118$	0	0
$119$	−33.8770	−3.10550
$120$	0	0
$121$	−10.4802	−0.952742
$122$	0	0
$123$	−6.32667	−0.570456
$124$	0	0
$125$	−3.04224	−0.272106
$126$	0	0
$127$	−19.2654	−1.70953	−0.854764	−	0.519018i	$-0.826298\pi$
$127$	−19.2654	−1.70953	−0.854764	+	0.519018i	$0.826298\pi$
$128$	0	0
$129$	1.24148	0.109306
$130$	0	0
$131$	−20.1144	−1.75740	−0.878701	−	0.477372i	$-0.841589\pi$
$131$	−20.1144	−1.75740	−0.878701	+	0.477372i	$0.841589\pi$
$132$	0	0
$133$	−26.7712	−2.32136
$134$	0	0
$135$	−1.13834	−0.0979725
$136$	0	0
$137$	16.4500	1.40542	0.702711	−	0.711476i	$-0.251973\pi$
$137$	16.4500	1.40542	0.702711	+	0.711476i	$0.251973\pi$
$138$	0	0
$139$	9.60624	0.814791	0.407395	−	0.913252i	$-0.366437\pi$
$139$	9.60624	0.814791	0.407395	+	0.913252i	$0.366437\pi$
$140$	0	0
$141$	0.288693	0.0243124
$142$	0	0
$143$	3.25145	0.271900
$144$	0	0
$145$	2.26804	0.188350
$146$	0	0
$147$	7.38567	0.609160
$148$	0	0
$149$	−22.3862	−1.83395	−0.916974	−	0.398947i	$-0.869376\pi$
$149$	−22.3862	−1.83395	−0.916974	+	0.398947i	$0.869376\pi$
$150$	0	0
$151$	−11.8741	−0.966297	−0.483149	−	0.875538i	$-0.660507\pi$
$151$	−11.8741	−0.966297	−0.483149	+	0.875538i	$0.660507\pi$
$152$	0	0
$153$	−20.3607	−1.64606
$154$	0	0
$155$	2.63281	0.211472
$156$	0	0
$157$	−18.8085	−1.50108	−0.750540	−	0.660825i	$-0.770207\pi$
$157$	−18.8085	−1.50108	−0.750540	+	0.660825i	$0.770207\pi$
$158$	0	0
$159$	−8.76215	−0.694884
$160$	0	0
$161$	−20.7525	−1.63552
$162$	0	0
$163$	23.9259	1.87402	0.937012	−	0.349297i	$-0.113580\pi$
$163$	23.9259	1.87402	0.937012	+	0.349297i	$0.113580\pi$
$164$	0	0
$165$	−0.147755	−0.0115027
$166$	0	0
$167$	−15.9339	−1.23301	−0.616503	−	0.787353i	$-0.711451\pi$
$167$	−15.9339	−1.23301	−0.616503	+	0.787353i	$0.711451\pi$
$168$	0	0
$169$	7.33686	0.564374
$170$	0	0
$171$	−16.0900	−1.23043
$172$	0	0
$173$	−6.77409	−0.515024	−0.257512	−	0.966275i	$-0.582903\pi$
$173$	−6.77409	−0.515024	−0.257512	+	0.966275i	$0.582903\pi$
$174$	0	0
$175$	20.8527	1.57631
$176$	0	0
$177$	−2.53689	−0.190684
$178$	0	0
$179$	10.7988	0.807142	0.403571	−	0.914948i	$-0.367769\pi$
$179$	10.7988	0.807142	0.403571	+	0.914948i	$0.367769\pi$
$180$	0	0
$181$	12.9864	0.965269	0.482635	−	0.875822i	$-0.339680\pi$
$181$	12.9864	0.965269	0.482635	+	0.875822i	$0.339680\pi$
$182$	0	0
$183$	−4.65756	−0.344297
$184$	0	0
$185$	−0.277347	−0.0203909
$186$	0	0
$187$	−5.74616	−0.420201
$188$	0	0
$189$	15.7552	1.14602
$190$	0	0
$191$	13.7261	0.993188	0.496594	−	0.867983i	$-0.334584\pi$
$191$	13.7261	0.993188	0.496594	+	0.867983i	$0.334584\pi$
$192$	0	0
$193$	13.4246	0.966324	0.483162	−	0.875531i	$-0.339488\pi$
$193$	13.4246	0.966324	0.483162	+	0.875531i	$0.339488\pi$
$194$	0	0
$195$	−0.924160	−0.0661805
$196$	0	0
$197$	−8.99175	−0.640635	−0.320318	−	0.947310i	$-0.603790\pi$
$197$	−8.99175	−0.640635	−0.320318	+	0.947310i	$0.603790\pi$
$198$	0	0
$199$	−14.5387	−1.03062	−0.515311	−	0.857003i	$-0.672324\pi$
$199$	−14.5387	−1.03062	−0.515311	+	0.857003i	$0.672324\pi$
$200$	0	0
$201$	−3.06782	−0.216387
$202$	0	0
$203$	−31.3909	−2.20321
$204$	0	0
$205$	−2.91198	−0.203382
$206$	0	0
$207$	−12.4726	−0.866908
$208$	0	0
$209$	−4.54089	−0.314100
$210$	0	0
$211$	1.55378	0.106966	0.0534832	−	0.998569i	$-0.482968\pi$
$211$	1.55378	0.106966	0.0534832	+	0.998569i	$0.482968\pi$
$212$	0	0
$213$	−1.32465	−0.0907636
$214$	0	0
$215$	0.571416	0.0389702
$216$	0	0
$217$	−36.4395	−2.47367
$218$	0	0
$219$	2.05449	0.138829
$220$	0	0
$221$	−35.9405	−2.41762
$222$	0	0
$223$	−15.7987	−1.05796	−0.528978	−	0.848635i	$-0.677425\pi$
$223$	−15.7987	−1.05796	−0.528978	+	0.848635i	$0.677425\pi$
$224$	0	0
$225$	12.5328	0.835523
$226$	0	0
$227$	20.1142	1.33502	0.667512	−	0.744599i	$-0.267359\pi$
$227$	20.1142	1.33502	0.667512	+	0.744599i	$0.267359\pi$
$228$	0	0
$229$	−0.478210	−0.0316010	−0.0158005	−	0.999875i	$-0.505030\pi$
$229$	−0.478210	−0.0316010	−0.0158005	+	0.999875i	$0.505030\pi$
$230$	0	0
$231$	2.04500	0.134551
$232$	0	0
$233$	−26.9502	−1.76557	−0.882783	−	0.469781i	$-0.844333\pi$
$233$	−26.9502	−1.76557	−0.882783	+	0.469781i	$0.844333\pi$
$234$	0	0
$235$	0.132877	0.00866794
$236$	0	0
$237$	−0.933516	−0.0606384
$238$	0	0
$239$	−2.77549	−0.179532	−0.0897659	−	0.995963i	$-0.528612\pi$
$239$	−2.77549	−0.179532	−0.0897659	+	0.995963i	$0.528612\pi$
$240$	0	0
$241$	4.65974	0.300160	0.150080	−	0.988674i	$-0.452047\pi$
$241$	4.65974	0.300160	0.150080	+	0.988674i	$0.452047\pi$
$242$	0	0
$243$	14.5833	0.935516
$244$	0	0
$245$	3.39941	0.217180
$246$	0	0
$247$	−28.4019	−1.80717
$248$	0	0
$249$	5.80101	0.367624
$250$	0	0
$251$	−20.0865	−1.26785	−0.633926	−	0.773394i	$-0.718558\pi$
$251$	−20.0865	−1.26785	−0.633926	+	0.773394i	$0.718558\pi$
$252$	0	0
$253$	−3.52000	−0.221301
$254$	0	0
$255$	1.63323	0.102277
$256$	0	0
$257$	−12.3177	−0.768358	−0.384179	−	0.923259i	$-0.625515\pi$
$257$	−12.3177	−0.768358	−0.384179	+	0.923259i	$0.625515\pi$
$258$	0	0
$259$	3.83863	0.238521
$260$	0	0
$261$	−18.8665	−1.16781
$262$	0	0
$263$	−5.98111	−0.368811	−0.184406	−	0.982850i	$-0.559036\pi$
$263$	−5.98111	−0.368811	−0.184406	+	0.982850i	$0.559036\pi$
$264$	0	0
$265$	−4.03296	−0.247743
$266$	0	0
$267$	2.02925	0.124188
$268$	0	0
$269$	−17.6549	−1.07644	−0.538218	−	0.842806i	$-0.680902\pi$
$269$	−17.6549	−1.07644	−0.538218	+	0.842806i	$0.680902\pi$
$270$	0	0
$271$	16.3089	0.990694	0.495347	−	0.868695i	$-0.335041\pi$
$271$	16.3089	0.990694	0.495347	+	0.868695i	$0.335041\pi$
$272$	0	0
$273$	12.7909	0.774139
$274$	0	0
$275$	3.53700	0.213289
$276$	0	0
$277$	−8.18714	−0.491917	−0.245959	−	0.969280i	$-0.579103\pi$
$277$	−8.18714	−0.491917	−0.245959	+	0.969280i	$0.579103\pi$
$278$	0	0
$279$	−21.9008	−1.31117
$280$	0	0
$281$	−20.6880	−1.23414	−0.617070	−	0.786908i	$-0.711681\pi$
$281$	−20.6880	−1.23414	−0.617070	+	0.786908i	$0.711681\pi$
$282$	0	0
$283$	−0.937311	−0.0557174	−0.0278587	−	0.999612i	$-0.508869\pi$
$283$	−0.937311	−0.0557174	−0.0278587	+	0.999612i	$0.508869\pi$
$284$	0	0
$285$	1.29066	0.0764520
$286$	0	0
$287$	40.3034	2.37903
$288$	0	0
$289$	46.5162	2.73624
$290$	0	0
$291$	−3.78462	−0.221858
$292$	0	0
$293$	−15.9140	−0.929707	−0.464854	−	0.885387i	$-0.653893\pi$
$293$	−15.9140	−0.929707	−0.464854	+	0.885387i	$0.653893\pi$
$294$	0	0
$295$	−1.16766	−0.0679836
$296$	0	0
$297$	2.67237	0.155067
$298$	0	0
$299$	−22.0166	−1.27325
$300$	0	0
$301$	−7.90871	−0.455850
$302$	0	0
$303$	4.61733	0.265259
$304$	0	0
$305$	−2.14374	−0.122750
$306$	0	0
$307$	−12.1293	−0.692258	−0.346129	−	0.938187i	$-0.612504\pi$
$307$	−12.1293	−0.692258	−0.346129	+	0.938187i	$0.612504\pi$
$308$	0	0
$309$	0.940184	0.0534852
$310$	0	0
$311$	9.73181	0.551840	0.275920	−	0.961181i	$-0.411017\pi$
$311$	9.73181	0.551840	0.275920	+	0.961181i	$0.411017\pi$
$312$	0	0
$313$	−8.94322	−0.505501	−0.252750	−	0.967532i	$-0.581335\pi$
$313$	−8.94322	−0.505501	−0.252750	+	0.967532i	$0.581335\pi$
$314$	0	0
$315$	3.33520	0.187917
$316$	0	0
$317$	−31.7836	−1.78515	−0.892573	−	0.450902i	$-0.851102\pi$
$317$	−31.7836	−1.78515	−0.892573	+	0.450902i	$0.851102\pi$
$318$	0	0
$319$	−5.32447	−0.298113
$320$	0	0
$321$	2.47402	0.138086
$322$	0	0
$323$	50.1935	2.79284
$324$	0	0
$325$	22.1228	1.22715
$326$	0	0
$327$	−4.33047	−0.239476
$328$	0	0
$329$	−1.83909	−0.101392
$330$	0	0
$331$	−5.60674	−0.308174	−0.154087	−	0.988057i	$-0.549244\pi$
$331$	−5.60674	−0.308174	−0.154087	+	0.988057i	$0.549244\pi$
$332$	0	0
$333$	2.30709	0.126428
$334$	0	0
$335$	−1.41203	−0.0771474
$336$	0	0
$337$	14.7356	0.802701	0.401350	−	0.915925i	$-0.368541\pi$
$337$	14.7356	0.802701	0.401350	+	0.915925i	$0.368541\pi$
$338$	0	0
$339$	−6.70044	−0.363918
$340$	0	0
$341$	−6.18081	−0.334710
$342$	0	0
$343$	−17.2946	−0.933821
$344$	0	0
$345$	1.00049	0.0538646
$346$	0	0
$347$	21.2441	1.14044	0.570222	−	0.821491i	$-0.306857\pi$
$347$	21.2441	1.14044	0.570222	+	0.821491i	$0.306857\pi$
$348$	0	0
$349$	5.58047	0.298716	0.149358	−	0.988783i	$-0.452279\pi$
$349$	5.58047	0.298716	0.149358	+	0.988783i	$0.452279\pi$
$350$	0	0
$351$	16.7149	0.892175
$352$	0	0
$353$	−23.5258	−1.25215	−0.626075	−	0.779763i	$-0.715340\pi$
$353$	−23.5258	−1.25215	−0.626075	+	0.779763i	$0.715340\pi$
$354$	0	0
$355$	−0.609698	−0.0323594
$356$	0	0
$357$	−22.6048	−1.19637
$358$	0	0
$359$	−3.93140	−0.207492	−0.103746	−	0.994604i	$-0.533083\pi$
$359$	−3.93140	−0.207492	−0.103746	+	0.994604i	$0.533083\pi$
$360$	0	0
$361$	20.6654	1.08765
$362$	0	0
$363$	−6.99300	−0.367038
$364$	0	0
$365$	0.945621	0.0494960
$366$	0	0
$367$	33.7574	1.76212	0.881061	−	0.473003i	$-0.156830\pi$
$367$	33.7574	1.76212	0.881061	+	0.473003i	$0.156830\pi$
$368$	0	0
$369$	24.2231	1.26100
$370$	0	0
$371$	55.8184	2.89795
$372$	0	0
$373$	21.9721	1.13767	0.568836	−	0.822451i	$-0.307394\pi$
$373$	21.9721	1.13767	0.568836	+	0.822451i	$0.307394\pi$
$374$	0	0
$375$	−2.02997	−0.104827
$376$	0	0
$377$	−33.3029	−1.71519
$378$	0	0
$379$	36.9373	1.89734	0.948672	−	0.316262i	$-0.102428\pi$
$379$	36.9373	1.89734	0.948672	+	0.316262i	$0.102428\pi$
$380$	0	0
$381$	−12.8551	−0.658584
$382$	0	0
$383$	−38.3731	−1.96077	−0.980387	−	0.197084i	$-0.936853\pi$
$383$	−38.3731	−1.96077	−0.980387	+	0.197084i	$0.936853\pi$
$384$	0	0
$385$	0.941256	0.0479708
$386$	0	0
$387$	−4.75328	−0.241623
$388$	0	0
$389$	3.13829	0.159117	0.0795587	−	0.996830i	$-0.474649\pi$
$389$	3.13829	0.159117	0.0795587	+	0.996830i	$0.474649\pi$
$390$	0	0
$391$	38.9090	1.96771
$392$	0	0
$393$	−13.4216	−0.677028
$394$	0	0
$395$	−0.429670	−0.0216191
$396$	0	0
$397$	−12.6226	−0.633509	−0.316755	−	0.948508i	$-0.602593\pi$
$397$	−12.6226	−0.633509	−0.316755	+	0.948508i	$0.602593\pi$
$398$	0	0
$399$	−17.8634	−0.894289
$400$	0	0
$401$	−13.1940	−0.658874	−0.329437	−	0.944177i	$-0.606859\pi$
$401$	−13.1940	−0.658874	−0.329437	+	0.944177i	$0.606859\pi$
$402$	0	0
$403$	−38.6591	−1.92575
$404$	0	0
$405$	1.59429	0.0792211
$406$	0	0
$407$	0.651103	0.0322740
$408$	0	0
$409$	18.6687	0.923110	0.461555	−	0.887112i	$-0.347292\pi$
$409$	18.6687	0.923110	0.461555	+	0.887112i	$0.347292\pi$
$410$	0	0
$411$	10.9765	0.541430
$412$	0	0
$413$	16.1610	0.795231
$414$	0	0
$415$	2.67004	0.131067
$416$	0	0
$417$	6.40987	0.313893
$418$	0	0
$419$	−23.8992	−1.16755	−0.583777	−	0.811914i	$-0.698426\pi$
$419$	−23.8992	−1.16755	−0.583777	+	0.811914i	$0.698426\pi$
$420$	0	0
$421$	12.2272	0.595915	0.297958	−	0.954579i	$-0.403695\pi$
$421$	12.2272	0.595915	0.297958	+	0.954579i	$0.403695\pi$
$422$	0	0
$423$	−1.10533	−0.0537429
$424$	0	0
$425$	−39.0968	−1.89647
$426$	0	0
$427$	29.6705	1.43586
$428$	0	0
$429$	2.16957	0.104748
$430$	0	0
$431$	−12.6741	−0.610491	−0.305246	−	0.952274i	$-0.598739\pi$
$431$	−12.6741	−0.610491	−0.305246	+	0.952274i	$0.598739\pi$
$432$	0	0
$433$	−34.0284	−1.63530	−0.817651	−	0.575714i	$-0.804724\pi$
$433$	−34.0284	−1.63530	−0.817651	+	0.575714i	$0.804724\pi$
$434$	0	0
$435$	1.51337	0.0725607
$436$	0	0
$437$	30.7477	1.47086
$438$	0	0
$439$	24.6920	1.17849	0.589244	−	0.807956i	$-0.299426\pi$
$439$	24.6920	1.17849	0.589244	+	0.807956i	$0.299426\pi$
$440$	0	0
$441$	−28.2777	−1.34656
$442$	0	0
$443$	24.5906	1.16833	0.584167	−	0.811633i	$-0.301421\pi$
$443$	24.5906	1.16833	0.584167	+	0.811633i	$0.301421\pi$
$444$	0	0
$445$	0.934002	0.0442760
$446$	0	0
$447$	−14.9374	−0.706517
$448$	0	0
$449$	−7.17674	−0.338691	−0.169346	−	0.985557i	$-0.554165\pi$
$449$	−7.17674	−0.338691	−0.169346	+	0.985557i	$0.554165\pi$
$450$	0	0
$451$	6.83620	0.321904
$452$	0	0
$453$	−7.92310	−0.372260
$454$	0	0
$455$	5.88727	0.275999
$456$	0	0
$457$	23.1927	1.08491	0.542455	−	0.840085i	$-0.317495\pi$
$457$	23.1927	1.08491	0.542455	+	0.840085i	$0.317495\pi$
$458$	0	0
$459$	−29.5395	−1.37879
$460$	0	0
$461$	−5.75291	−0.267940	−0.133970	−	0.990985i	$-0.542773\pi$
$461$	−5.75291	−0.267940	−0.133970	+	0.990985i	$0.542773\pi$
$462$	0	0
$463$	−19.8464	−0.922342	−0.461171	−	0.887311i	$-0.652571\pi$
$463$	−19.8464	−0.922342	−0.461171	+	0.887311i	$0.652571\pi$
$464$	0	0
$465$	1.75677	0.0814683
$466$	0	0
$467$	13.3506	0.617792	0.308896	−	0.951096i	$-0.400040\pi$
$467$	13.3506	0.617792	0.308896	+	0.951096i	$0.400040\pi$
$468$	0	0
$469$	19.5432	0.902423
$470$	0	0
$471$	−12.5502	−0.578282
$472$	0	0
$473$	−1.34146	−0.0616805
$474$	0	0
$475$	−30.8962	−1.41761
$476$	0	0
$477$	33.5479	1.53605
$478$	0	0
$479$	8.17837	0.373679	0.186840	−	0.982390i	$-0.440176\pi$
$479$	8.17837	0.373679	0.186840	+	0.982390i	$0.440176\pi$
$480$	0	0
$481$	4.07245	0.185688
$482$	0	0
$483$	−13.8473	−0.630075
$484$	0	0
$485$	−1.74195	−0.0790978
$486$	0	0
$487$	−32.0826	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$487$	−32.0826	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$488$	0	0
$489$	15.9649	0.721956
$490$	0	0
$491$	−39.3077	−1.77393	−0.886965	−	0.461836i	$-0.847191\pi$
$491$	−39.3077	−1.77393	−0.886965	+	0.461836i	$0.847191\pi$
$492$	0	0
$493$	58.8549	2.65069
$494$	0	0
$495$	0.565712	0.0254269
$496$	0	0
$497$	8.43855	0.378521
$498$	0	0
$499$	21.2872	0.952947	0.476474	−	0.879189i	$-0.341915\pi$
$499$	21.2872	0.952947	0.476474	+	0.879189i	$0.341915\pi$
$500$	0	0
$501$	−10.6321	−0.475008
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	2.12522	0.0945712
$506$	0	0
$507$	4.89560	0.217421
$508$	0	0
$509$	32.9814	1.46187	0.730937	−	0.682445i	$-0.239083\pi$
$509$	32.9814	1.46187	0.730937	+	0.682445i	$0.239083\pi$
$510$	0	0
$511$	−13.0879	−0.578975
$512$	0	0
$513$	−23.3436	−1.03064
$514$	0	0
$515$	0.432739	0.0190688
$516$	0	0
$517$	−0.311944	−0.0137193
$518$	0	0
$519$	−4.52009	−0.198410
$520$	0	0
$521$	43.6614	1.91284	0.956421	−	0.291990i	$-0.0943175\pi$
$521$	43.6614	1.91284	0.956421	+	0.291990i	$0.0943175\pi$
$522$	0	0
$523$	−17.6035	−0.769748	−0.384874	−	0.922969i	$-0.625755\pi$
$523$	−17.6035	−0.769748	−0.384874	+	0.922969i	$0.625755\pi$
$524$	0	0
$525$	13.9142	0.607264
$526$	0	0
$527$	68.3206	2.97609
$528$	0	0
$529$	0.834992	0.0363040
$530$	0	0
$531$	9.71306	0.421511
$532$	0	0
$533$	42.7584	1.85207
$534$	0	0
$535$	1.13872	0.0492311
$536$	0	0
$537$	7.20564	0.310946
$538$	0	0
$539$	−7.98049	−0.343744
$540$	0	0
$541$	−12.7313	−0.547361	−0.273681	−	0.961821i	$-0.588241\pi$
$541$	−12.7313	−0.547361	−0.273681	+	0.961821i	$0.588241\pi$
$542$	0	0
$543$	8.66530	0.371864
$544$	0	0
$545$	−1.99319	−0.0853789
$546$	0	0
$547$	−8.99668	−0.384670	−0.192335	−	0.981329i	$-0.561606\pi$
$547$	−8.99668	−0.384670	−0.192335	+	0.981329i	$0.561606\pi$
$548$	0	0
$549$	17.8325	0.761074
$550$	0	0
$551$	46.5100	1.98139
$552$	0	0
$553$	5.94687	0.252887
$554$	0	0
$555$	−0.185063	−0.00785548
$556$	0	0
$557$	15.6045	0.661182	0.330591	−	0.943774i	$-0.392752\pi$
$557$	15.6045	0.661182	0.330591	+	0.943774i	$0.392752\pi$
$558$	0	0
$559$	−8.39044	−0.354878
$560$	0	0
$561$	−3.83419	−0.161880
$562$	0	0
$563$	−30.8071	−1.29837	−0.649183	−	0.760632i	$-0.724889\pi$
$563$	−30.8071	−1.29837	−0.649183	+	0.760632i	$0.724889\pi$
$564$	0	0
$565$	−3.08402	−0.129746
$566$	0	0
$567$	−22.0659	−0.926681
$568$	0	0
$569$	12.3008	0.515677	0.257838	−	0.966188i	$-0.416990\pi$
$569$	12.3008	0.515677	0.257838	+	0.966188i	$0.416990\pi$
$570$	0	0
$571$	−35.1173	−1.46961	−0.734807	−	0.678276i	$-0.762727\pi$
$571$	−35.1173	−1.46961	−0.734807	+	0.678276i	$0.762727\pi$
$572$	0	0
$573$	9.15892	0.382619
$574$	0	0
$575$	−23.9501	−0.998786
$576$	0	0
$577$	−15.6086	−0.649796	−0.324898	−	0.945749i	$-0.605330\pi$
$577$	−15.6086	−0.649796	−0.324898	+	0.945749i	$0.605330\pi$
$578$	0	0
$579$	8.95772	0.372270
$580$	0	0
$581$	−36.9548	−1.53314
$582$	0	0
$583$	9.46783	0.392118
$584$	0	0
$585$	3.53836	0.146293
$586$	0	0
$587$	−14.0930	−0.581681	−0.290841	−	0.956771i	$-0.593935\pi$
$587$	−14.0930	−0.581681	−0.290841	+	0.956771i	$0.593935\pi$
$588$	0	0
$589$	53.9902	2.22463
$590$	0	0
$591$	−5.99984	−0.246801
$592$	0	0
$593$	−3.74557	−0.153812	−0.0769060	−	0.997038i	$-0.524504\pi$
$593$	−3.74557	−0.153812	−0.0769060	+	0.997038i	$0.524504\pi$
$594$	0	0
$595$	−10.4043	−0.426536
$596$	0	0
$597$	−9.70112	−0.397040
$598$	0	0
$599$	−4.31104	−0.176144	−0.0880722	−	0.996114i	$-0.528071\pi$
$599$	−4.31104	−0.176144	−0.0880722	+	0.996114i	$0.528071\pi$
$600$	0	0
$601$	−9.45071	−0.385503	−0.192751	−	0.981248i	$-0.561741\pi$
$601$	−9.45071	−0.385503	−0.192751	+	0.981248i	$0.561741\pi$
$602$	0	0
$603$	11.7459	0.478328
$604$	0	0
$605$	−3.21868	−0.130858
$606$	0	0
$607$	−41.0664	−1.66683	−0.833417	−	0.552645i	$-0.813619\pi$
$607$	−41.0664	−1.66683	−0.833417	+	0.552645i	$0.813619\pi$
$608$	0	0
$609$	−20.9459	−0.848771
$610$	0	0
$611$	−1.95111	−0.0789336
$612$	0	0
$613$	−6.03276	−0.243661	−0.121831	−	0.992551i	$-0.538876\pi$
$613$	−6.03276	−0.243661	−0.121831	+	0.992551i	$0.538876\pi$
$614$	0	0
$615$	−1.94305	−0.0783514
$616$	0	0
$617$	−3.88785	−0.156519	−0.0782595	−	0.996933i	$-0.524936\pi$
$617$	−3.88785	−0.156519	−0.0782595	+	0.996933i	$0.524936\pi$
$618$	0	0
$619$	−17.1723	−0.690213	−0.345107	−	0.938563i	$-0.612157\pi$
$619$	−17.1723	−0.690213	−0.345107	+	0.938563i	$0.612157\pi$
$620$	0	0
$621$	−18.0954	−0.726145
$622$	0	0
$623$	−12.9271	−0.517913
$624$	0	0
$625$	23.5940	0.943762
$626$	0	0
$627$	−3.02996	−0.121005
$628$	0	0
$629$	−7.19707	−0.286966
$630$	0	0
$631$	−15.9300	−0.634165	−0.317082	−	0.948398i	$-0.602703\pi$
$631$	−15.9300	−0.634165	−0.317082	+	0.948398i	$0.602703\pi$
$632$	0	0
$633$	1.03677	0.0412081
$634$	0	0
$635$	−5.91681	−0.234801
$636$	0	0
$637$	−49.9155	−1.97773
$638$	0	0
$639$	5.07173	0.200634
$640$	0	0
$641$	39.4598	1.55857	0.779284	−	0.626671i	$-0.215583\pi$
$641$	39.4598	1.55857	0.779284	+	0.626671i	$0.215583\pi$
$642$	0	0
$643$	18.1011	0.713836	0.356918	−	0.934136i	$-0.383828\pi$
$643$	18.1011	0.713836	0.356918	+	0.934136i	$0.383828\pi$
$644$	0	0
$645$	0.381284	0.0150130
$646$	0	0
$647$	49.6770	1.95301	0.976503	−	0.215504i	$-0.0691394\pi$
$647$	49.6770	1.95301	0.976503	+	0.215504i	$0.0691394\pi$
$648$	0	0
$649$	2.74120	0.107602
$650$	0	0
$651$	−24.3147	−0.952967
$652$	0	0
$653$	−4.61404	−0.180561	−0.0902807	−	0.995916i	$-0.528776\pi$
$653$	−4.61404	−0.180561	−0.0902807	+	0.995916i	$0.528776\pi$
$654$	0	0
$655$	−6.17755	−0.241377
$656$	0	0
$657$	−7.86607	−0.306885
$658$	0	0
$659$	−20.5534	−0.800646	−0.400323	−	0.916374i	$-0.631102\pi$
$659$	−20.5534	−0.800646	−0.400323	+	0.916374i	$0.631102\pi$
$660$	0	0
$661$	19.5211	0.759282	0.379641	−	0.925134i	$-0.376047\pi$
$661$	19.5211	0.759282	0.379641	+	0.925134i	$0.376047\pi$
$662$	0	0
$663$	−23.9817	−0.931372
$664$	0	0
$665$	−8.22200	−0.318836
$666$	0	0
$667$	36.0536	1.39600
$668$	0	0
$669$	−10.5418	−0.407571
$670$	0	0
$671$	5.03267	0.194284
$672$	0	0
$673$	40.6149	1.56559	0.782794	−	0.622281i	$-0.213794\pi$
$673$	40.6149	1.56559	0.782794	+	0.622281i	$0.213794\pi$
$674$	0	0
$675$	18.1828	0.699856
$676$	0	0
$677$	18.7650	0.721199	0.360599	−	0.932721i	$-0.382572\pi$
$677$	18.7650	0.721199	0.360599	+	0.932721i	$0.382572\pi$
$678$	0	0
$679$	24.1095	0.925238
$680$	0	0
$681$	13.4214	0.514310
$682$	0	0
$683$	−16.4215	−0.628353	−0.314176	−	0.949365i	$-0.601728\pi$
$683$	−16.4215	−0.628353	−0.314176	+	0.949365i	$0.601728\pi$
$684$	0	0
$685$	5.05215	0.193033
$686$	0	0
$687$	−0.319091	−0.0121741
$688$	0	0
$689$	59.2184	2.25604
$690$	0	0
$691$	11.3456	0.431606	0.215803	−	0.976437i	$-0.430763\pi$
$691$	11.3456	0.431606	0.215803	+	0.976437i	$0.430763\pi$
$692$	0	0
$693$	−7.82977	−0.297428
$694$	0	0
$695$	2.95028	0.111910
$696$	0	0
$697$	−75.5651	−2.86223
$698$	0	0
$699$	−17.9828	−0.680173
$700$	0	0
$701$	2.66016	0.100473	0.0502364	−	0.998737i	$-0.484003\pi$
$701$	2.66016	0.100473	0.0502364	+	0.998737i	$0.484003\pi$
$702$	0	0
$703$	−5.68747	−0.214507
$704$	0	0
$705$	0.0886637	0.00333927
$706$	0	0
$707$	−29.4142	−1.10624
$708$	0	0
$709$	−39.7288	−1.49205	−0.746024	−	0.665919i	$-0.768040\pi$
$709$	−39.7288	−1.49205	−0.746024	+	0.665919i	$0.768040\pi$
$710$	0	0
$711$	3.57418	0.134042
$712$	0	0
$713$	41.8521	1.56737
$714$	0	0
$715$	0.998590	0.0373451
$716$	0	0
$717$	−1.85198	−0.0691635
$718$	0	0
$719$	12.3195	0.459441	0.229721	−	0.973257i	$-0.426219\pi$
$719$	12.3195	0.459441	0.229721	+	0.973257i	$0.426219\pi$
$720$	0	0
$721$	−5.98935	−0.223055
$722$	0	0
$723$	3.10927	0.115635
$724$	0	0
$725$	−36.2276	−1.34546
$726$	0	0
$727$	−25.3154	−0.938897	−0.469449	−	0.882960i	$-0.655547\pi$
$727$	−25.3154	−0.938897	−0.469449	+	0.882960i	$0.655547\pi$
$728$	0	0
$729$	−5.84245	−0.216387
$730$	0	0
$731$	14.8281	0.548436
$732$	0	0
$733$	−28.5561	−1.05474	−0.527372	−	0.849634i	$-0.676823\pi$
$733$	−28.5561	−1.05474	−0.527372	+	0.849634i	$0.676823\pi$
$734$	0	0
$735$	2.26829	0.0836672
$736$	0	0
$737$	3.31490	0.122106
$738$	0	0
$739$	24.0882	0.886100	0.443050	−	0.896497i	$-0.353896\pi$
$739$	24.0882	0.886100	0.443050	+	0.896497i	$0.353896\pi$
$740$	0	0
$741$	−18.9515	−0.696201
$742$	0	0
$743$	32.2680	1.18380	0.591899	−	0.806012i	$-0.298378\pi$
$743$	32.2680	1.18380	0.591899	+	0.806012i	$0.298378\pi$
$744$	0	0
$745$	−6.87527	−0.251890
$746$	0	0
$747$	−22.2105	−0.812640
$748$	0	0
$749$	−15.7605	−0.575876
$750$	0	0
$751$	−27.9809	−1.02104	−0.510519	−	0.859867i	$-0.670547\pi$
$751$	−27.9809	−1.02104	−0.510519	+	0.859867i	$0.670547\pi$
$752$	0	0
$753$	−13.4030	−0.488432
$754$	0	0
$755$	−3.64677	−0.132720
$756$	0	0
$757$	8.00964	0.291115	0.145558	−	0.989350i	$-0.453502\pi$
$757$	8.00964	0.291115	0.145558	+	0.989350i	$0.453502\pi$
$758$	0	0
$759$	−2.34876	−0.0852547
$760$	0	0
$761$	24.9041	0.902775	0.451387	−	0.892328i	$-0.350929\pi$
$761$	24.9041	0.902775	0.451387	+	0.892328i	$0.350929\pi$
$762$	0	0
$763$	27.5868	0.998710
$764$	0	0
$765$	−6.25320	−0.226085
$766$	0	0
$767$	17.1454	0.619085
$768$	0	0
$769$	−33.8461	−1.22052	−0.610261	−	0.792200i	$-0.708935\pi$
$769$	−33.8461	−1.22052	−0.610261	+	0.792200i	$0.708935\pi$
$770$	0	0
$771$	−8.21914	−0.296005
$772$	0	0
$773$	−45.6404	−1.64157	−0.820786	−	0.571236i	$-0.806464\pi$
$773$	−45.6404	−1.64157	−0.820786	+	0.571236i	$0.806464\pi$
$774$	0	0
$775$	−42.0541	−1.51063
$776$	0	0
$777$	2.56137	0.0918886
$778$	0	0
$779$	−59.7152	−2.13952
$780$	0	0
$781$	1.43133	0.0512172
$782$	0	0
$783$	−27.3717	−0.978185
$784$	0	0
$785$	−5.77648	−0.206171
$786$	0	0
$787$	−42.2047	−1.50444	−0.752218	−	0.658914i	$-0.771016\pi$
$787$	−42.2047	−1.50444	−0.752218	+	0.658914i	$0.771016\pi$
$788$	0	0
$789$	−3.99097	−0.142082
$790$	0	0
$791$	42.6845	1.51769
$792$	0	0
$793$	31.4778	1.11781
$794$	0	0
$795$	−2.69104	−0.0954414
$796$	0	0
$797$	−24.9132	−0.882471	−0.441236	−	0.897391i	$-0.645460\pi$
$797$	−24.9132	−0.882471	−0.441236	+	0.897391i	$0.645460\pi$
$798$	0	0
$799$	3.44812	0.121986
$800$	0	0
$801$	−7.76943	−0.274519
$802$	0	0
$803$	−2.21995	−0.0783403
$804$	0	0
$805$	−6.37352	−0.224637
$806$	0	0
$807$	−11.7804	−0.414690
$808$	0	0
$809$	−48.3047	−1.69830	−0.849152	−	0.528148i	$-0.822887\pi$
$809$	−48.3047	−1.69830	−0.849152	+	0.528148i	$0.822887\pi$
$810$	0	0
$811$	4.98459	0.175033	0.0875163	−	0.996163i	$-0.472107\pi$
$811$	4.98459	0.175033	0.0875163	+	0.996163i	$0.472107\pi$
$812$	0	0
$813$	10.8823	0.381658
$814$	0	0
$815$	7.34815	0.257395
$816$	0	0
$817$	11.7179	0.409956
$818$	0	0
$819$	−48.9728	−1.71125
$820$	0	0
$821$	20.6862	0.721954	0.360977	−	0.932575i	$-0.382443\pi$
$821$	20.6862	0.721954	0.360977	+	0.932575i	$0.382443\pi$
$822$	0	0
$823$	−12.0734	−0.420853	−0.210427	−	0.977610i	$-0.567485\pi$
$823$	−12.0734	−0.420853	−0.210427	+	0.977610i	$0.567485\pi$
$824$	0	0
$825$	2.36010	0.0821682
$826$	0	0
$827$	−34.2071	−1.18950	−0.594748	−	0.803912i	$-0.702748\pi$
$827$	−34.2071	−1.18950	−0.594748	+	0.803912i	$0.702748\pi$
$828$	0	0
$829$	−47.2677	−1.64168	−0.820838	−	0.571161i	$-0.806493\pi$
$829$	−47.2677	−1.64168	−0.820838	+	0.571161i	$0.806493\pi$
$830$	0	0
$831$	−5.46296	−0.189508
$832$	0	0
$833$	88.2137	3.05642
$834$	0	0
$835$	−4.89365	−0.169352
$836$	0	0
$837$	−31.7739	−1.09827
$838$	0	0
$839$	36.6906	1.26670	0.633350	−	0.773865i	$-0.281679\pi$
$839$	36.6906	1.26670	0.633350	+	0.773865i	$0.281679\pi$
$840$	0	0
$841$	25.5357	0.880543
$842$	0	0
$843$	−13.8043	−0.475445
$844$	0	0
$845$	2.25330	0.0775160
$846$	0	0
$847$	44.5482	1.53070
$848$	0	0
$849$	−0.625431	−0.0214648
$850$	0	0
$851$	−4.40881	−0.151132
$852$	0	0
$853$	16.9228	0.579427	0.289713	−	0.957113i	$-0.406440\pi$
$853$	16.9228	0.579427	0.289713	+	0.957113i	$0.406440\pi$
$854$	0	0
$855$	−4.94158	−0.168998
$856$	0	0
$857$	32.3955	1.10661	0.553305	−	0.832979i	$-0.313367\pi$
$857$	32.3955	1.10661	0.553305	+	0.832979i	$0.313367\pi$
$858$	0	0
$859$	−42.2728	−1.44233	−0.721164	−	0.692764i	$-0.756393\pi$
$859$	−42.2728	−1.44233	−0.721164	+	0.692764i	$0.756393\pi$
$860$	0	0
$861$	26.8929	0.916508
$862$	0	0
$863$	16.5450	0.563198	0.281599	−	0.959532i	$-0.409135\pi$
$863$	16.5450	0.563198	0.281599	+	0.959532i	$0.409135\pi$
$864$	0	0
$865$	−2.08046	−0.0707379
$866$	0	0
$867$	31.0384	1.05412
$868$	0	0
$869$	1.00870	0.0342178
$870$	0	0
$871$	20.7337	0.702534
$872$	0	0
$873$	14.4903	0.490421
$874$	0	0
$875$	12.9317	0.437172
$876$	0	0
$877$	7.41710	0.250458	0.125229	−	0.992128i	$-0.460033\pi$
$877$	7.41710	0.250458	0.125229	+	0.992128i	$0.460033\pi$
$878$	0	0
$879$	−10.6188	−0.358164
$880$	0	0
$881$	36.5860	1.23261	0.616307	−	0.787506i	$-0.288628\pi$
$881$	36.5860	1.23261	0.616307	+	0.787506i	$0.288628\pi$
$882$	0	0
$883$	−4.76867	−0.160479	−0.0802393	−	0.996776i	$-0.525568\pi$
$883$	−4.76867	−0.160479	−0.0802393	+	0.996776i	$0.525568\pi$
$884$	0	0
$885$	−0.779132	−0.0261902
$886$	0	0
$887$	−14.5077	−0.487120	−0.243560	−	0.969886i	$-0.578315\pi$
$887$	−14.5077	−0.487120	−0.243560	+	0.969886i	$0.578315\pi$
$888$	0	0
$889$	81.8918	2.74656
$890$	0	0
$891$	−3.74278	−0.125388
$892$	0	0
$893$	2.72487	0.0911844
$894$	0	0
$895$	3.31655	0.110860
$896$	0	0
$897$	−14.6908	−0.490512
$898$	0	0
$899$	63.3068	2.11140
$900$	0	0
$901$	−104.654	−3.48654
$902$	0	0
$903$	−5.27717	−0.175613
$904$	0	0
$905$	3.98839	0.132578
$906$	0	0
$907$	−5.98338	−0.198675	−0.0993374	−	0.995054i	$-0.531672\pi$
$907$	−5.98338	−0.198675	−0.0993374	+	0.995054i	$0.531672\pi$
$908$	0	0
$909$	−17.6785	−0.586359
$910$	0	0
$911$	−22.6582	−0.750701	−0.375350	−	0.926883i	$-0.622478\pi$
$911$	−22.6582	−0.750701	−0.375350	+	0.926883i	$0.622478\pi$
$912$	0	0
$913$	−6.26821	−0.207448
$914$	0	0
$915$	−1.43043	−0.0472887
$916$	0	0
$917$	85.5006	2.82348
$918$	0	0
$919$	8.10778	0.267451	0.133726	−	0.991018i	$-0.457306\pi$
$919$	8.10778	0.267451	0.133726	+	0.991018i	$0.457306\pi$
$920$	0	0
$921$	−8.09344	−0.266688
$922$	0	0
$923$	8.95256	0.294677
$924$	0	0
$925$	4.43009	0.145660
$926$	0	0
$927$	−3.59971	−0.118230
$928$	0	0
$929$	9.57905	0.314278	0.157139	−	0.987576i	$-0.449773\pi$
$929$	9.57905	0.314278	0.157139	+	0.987576i	$0.449773\pi$
$930$	0	0
$931$	69.7107	2.28468
$932$	0	0
$933$	6.49366	0.212593
$934$	0	0
$935$	−1.76477	−0.0577140
$936$	0	0
$937$	0.196050	0.00640467	0.00320234	−	0.999995i	$-0.498981\pi$
$937$	0.196050	0.00640467	0.00320234	+	0.999995i	$0.498981\pi$
$938$	0	0
$939$	−5.96746	−0.194741
$940$	0	0
$941$	0.893569	0.0291295	0.0145648	−	0.999894i	$-0.495364\pi$
$941$	0.893569	0.0291295	0.0145648	+	0.999894i	$0.495364\pi$
$942$	0	0
$943$	−46.2899	−1.50741
$944$	0	0
$945$	4.83875	0.157405
$946$	0	0
$947$	34.2934	1.11439	0.557193	−	0.830383i	$-0.311878\pi$
$947$	34.2934	1.11439	0.557193	+	0.830383i	$0.311878\pi$
$948$	0	0
$949$	−13.8851	−0.450730
$950$	0	0
$951$	−21.2080	−0.687716
$952$	0	0
$953$	14.0568	0.455346	0.227673	−	0.973738i	$-0.426888\pi$
$953$	14.0568	0.455346	0.227673	+	0.973738i	$0.426888\pi$
$954$	0	0
$955$	4.21558	0.136413
$956$	0	0
$957$	−3.55281	−0.114846
$958$	0	0
$959$	−69.9245	−2.25798
$960$	0	0
$961$	42.4885	1.37060
$962$	0	0
$963$	−9.47235	−0.305242
$964$	0	0
$965$	4.12298	0.132723
$966$	0	0
$967$	−36.9849	−1.18935	−0.594677	−	0.803965i	$-0.702720\pi$
$967$	−36.9849	−1.18935	−0.594677	+	0.803965i	$0.702720\pi$
$968$	0	0
$969$	33.4922	1.07592
$970$	0	0
$971$	31.2185	1.00185	0.500924	−	0.865491i	$-0.332993\pi$
$971$	31.2185	1.00185	0.500924	+	0.865491i	$0.332993\pi$
$972$	0	0
$973$	−40.8334	−1.30906
$974$	0	0
$975$	14.7617	0.472753
$976$	0	0
$977$	19.1478	0.612593	0.306297	−	0.951936i	$-0.400910\pi$
$977$	19.1478	0.612593	0.306297	+	0.951936i	$0.400910\pi$
$978$	0	0
$979$	−2.19268	−0.0700782
$980$	0	0
$981$	16.5802	0.529365
$982$	0	0
$983$	16.2033	0.516805	0.258402	−	0.966037i	$-0.416804\pi$
$983$	16.2033	0.516805	0.258402	+	0.966037i	$0.416804\pi$
$984$	0	0
$985$	−2.76155	−0.0879904
$986$	0	0
$987$	−1.22715	−0.0390607
$988$	0	0
$989$	9.08344	0.288837
$990$	0	0
$991$	44.9646	1.42835	0.714174	−	0.699968i	$-0.246802\pi$
$991$	44.9646	1.42835	0.714174	+	0.699968i	$0.246802\pi$
$992$	0	0
$993$	−3.74116	−0.118722
$994$	0	0
$995$	−4.46514	−0.141555
$996$	0	0
$997$	48.1546	1.52507	0.762536	−	0.646946i	$-0.223954\pi$
$997$	48.1546	1.52507	0.762536	+	0.646946i	$0.223954\pi$
$998$	0	0
$999$	3.34715	0.105899

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.v.1.18		28
4.3	odd	2		4024.2.a.d.1.11	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.d.1.11	✓	28	4.3	odd	2
8048.2.a.v.1.18		28	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:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q+0.667261 q^{3} +0.307121 q^{5} -4.25072 q^{7} -2.55476 q^{9} +O(q^{10})\)	\(q+0.667261 q^{3} +0.307121 q^{5} -4.25072 q^{7} -2.55476 q^{9} -0.721001 q^{11} -4.50964 q^{13} +0.204930 q^{15} +7.96970 q^{17} +6.29804 q^{19} -2.83634 q^{21} +4.88211 q^{23} -4.90568 q^{25} -3.70648 q^{27} +7.38483 q^{29} +8.57254 q^{31} -0.481096 q^{33} -1.30549 q^{35} -0.903054 q^{37} -3.00911 q^{39} -9.48154 q^{41} +1.86056 q^{43} -0.784621 q^{45} +0.432654 q^{47} +11.0686 q^{49} +5.31787 q^{51} -13.1315 q^{53} -0.221434 q^{55} +4.20244 q^{57} -3.80194 q^{59} -6.98011 q^{61} +10.8596 q^{63} -1.38501 q^{65} -4.59763 q^{67} +3.25764 q^{69} -1.98521 q^{71} +3.07898 q^{73} -3.27337 q^{75} +3.06477 q^{77} -1.39903 q^{79} +5.19110 q^{81} +8.69377 q^{83} +2.44766 q^{85} +4.92761 q^{87} +3.04116 q^{89} +19.1692 q^{91} +5.72013 q^{93} +1.93426 q^{95} -5.67187 q^{97} +1.84199 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) \) 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9	\( 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9} + 14 q^{11} - 31 q^{13} + 2 q^{15} - 9 q^{17} + 8 q^{19} - 28 q^{21} + 4 q^{23} + 22 q^{25} + 4 q^{27} - 47 q^{29} + 5 q^{31} - 26 q^{33} + 13 q^{35} - 67 q^{37} + 9 q^{39} - 28 q^{41} - 15 q^{43} - 57 q^{45} + 10 q^{47} + 20 q^{49} + 11 q^{51} - 58 q^{53} - 15 q^{55} - 31 q^{57} + 32 q^{59} - 55 q^{61} + 16 q^{63} - 44 q^{65} - 22 q^{67} - 44 q^{69} + 47 q^{71} - 5 q^{73} + 25 q^{75} - 50 q^{77} + 14 q^{79} - 28 q^{81} + 16 q^{83} - 78 q^{85} + 11 q^{87} - 20 q^{89} + 15 q^{91} - 83 q^{93} + 27 q^{95} - 8 q^{97} + 70 q^{99}+O(q^{100}) \) 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9 + 14 * q^11 - 31 * q^13 + 2 * q^15 - 9 * q^17 + 8 * q^19 - 28 * q^21 + 4 * q^23 + 22 * q^25 + 4 * q^27 - 47 * q^29 + 5 * q^31 - 26 * q^33 + 13 * q^35 - 67 * q^37 + 9 * q^39 - 28 * q^41 - 15 * q^43 - 57 * q^45 + 10 * q^47 + 20 * q^49 + 11 * q^51 - 58 * q^53 - 15 * q^55 - 31 * q^57 + 32 * q^59 - 55 * q^61 + 16 * q^63 - 44 * q^65 - 22 * q^67 - 44 * q^69 + 47 * q^71 - 5 * q^73 + 25 * q^75 - 50 * q^77 + 14 * q^79 - 28 * q^81 + 16 * q^83 - 78 * q^85 + 11 * q^87 - 20 * q^89 + 15 * q^91 - 83 * q^93 + 27 * q^95 - 8 * q^97 + 70 * q^99

Introduction

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