LMFDB - Embedded newform 8048.2.a.x.1.17

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:	$33$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
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)

$f(q)$	$=$	$q-0.410265 q^{3} -4.21207 q^{5} -3.36092 q^{7} -2.83168 q^{9} +O(q^{10})$	$q-0.410265 q^{3} -4.21207 q^{5} -3.36092 q^{7} -2.83168 q^{9} -4.13893 q^{11} -3.22425 q^{13} +1.72807 q^{15} +4.83674 q^{17} +7.13255 q^{19} +1.37887 q^{21} +0.264477 q^{23} +12.7416 q^{25} +2.39254 q^{27} -6.92998 q^{29} -1.07140 q^{31} +1.69806 q^{33} +14.1564 q^{35} +5.79815 q^{37} +1.32280 q^{39} +1.24932 q^{41} -3.97547 q^{43} +11.9273 q^{45} +7.96976 q^{47} +4.29578 q^{49} -1.98435 q^{51} +13.9969 q^{53} +17.4335 q^{55} -2.92624 q^{57} -1.88127 q^{59} -13.0212 q^{61} +9.51706 q^{63} +13.5808 q^{65} -7.34403 q^{67} -0.108506 q^{69} -13.5632 q^{71} -2.49323 q^{73} -5.22742 q^{75} +13.9106 q^{77} +8.07702 q^{79} +7.51347 q^{81} +10.1672 q^{83} -20.3727 q^{85} +2.84313 q^{87} +3.18164 q^{89} +10.8364 q^{91} +0.439559 q^{93} -30.0428 q^{95} +7.00896 q^{97} +11.7201 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q - 10 q^{3} - 12 q^{7} + 47 q^{9}+O(q^{10}) $ 33 * q - 10 * q^3 - 12 * q^7 + 47 * q^9	$ 33 q - 10 q^{3} - 12 q^{7} + 47 q^{9} - 22 q^{11} - 17 q^{13} - 22 q^{15} + 9 q^{17} - 16 q^{19} + 6 q^{21} - 36 q^{23} + 47 q^{25} - 34 q^{27} + 13 q^{29} - 21 q^{31} + 14 q^{33} - 33 q^{35} - 55 q^{37} - 37 q^{39} + 42 q^{41} - 23 q^{43} + 5 q^{45} - 20 q^{47} + 55 q^{49} - 53 q^{51} - 32 q^{53} - 35 q^{55} + 21 q^{57} - 20 q^{59} - 15 q^{61} - 48 q^{63} + 34 q^{65} - 66 q^{67} - 4 q^{69} - 61 q^{71} + 19 q^{73} - 59 q^{75} + 2 q^{77} - 62 q^{79} + 77 q^{81} - 36 q^{83} - 14 q^{85} - 43 q^{87} + 34 q^{89} - 41 q^{91} - 11 q^{93} - 61 q^{95} - 8 q^{97} - 98 q^{99}+O(q^{100}) $ 33 * q - 10 * q^3 - 12 * q^7 + 47 * q^9 - 22 * q^11 - 17 * q^13 - 22 * q^15 + 9 * q^17 - 16 * q^19 + 6 * q^21 - 36 * q^23 + 47 * q^25 - 34 * q^27 + 13 * q^29 - 21 * q^31 + 14 * q^33 - 33 * q^35 - 55 * q^37 - 37 * q^39 + 42 * q^41 - 23 * q^43 + 5 * q^45 - 20 * q^47 + 55 * q^49 - 53 * q^51 - 32 * q^53 - 35 * q^55 + 21 * q^57 - 20 * q^59 - 15 * q^61 - 48 * q^63 + 34 * q^65 - 66 * q^67 - 4 * q^69 - 61 * q^71 + 19 * q^73 - 59 * q^75 + 2 * q^77 - 62 * q^79 + 77 * q^81 - 36 * q^83 - 14 * q^85 - 43 * q^87 + 34 * q^89 - 41 * q^91 - 11 * q^93 - 61 * q^95 - 8 * q^97 - 98 * 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
$3$	−0.410265	−0.236867	−0.118433	−	0.992962i	$-0.537787\pi$
$3$	−0.410265	−0.236867	−0.118433	+	0.992962i	$0.537787\pi$
$4$	0	0
$5$	−4.21207	−1.88370	−0.941848	−	0.336038i	$-0.890913\pi$
$5$	−4.21207	−1.88370	−0.941848	+	0.336038i	$0.890913\pi$
$6$	0	0
$7$	−3.36092	−1.27031	−0.635154	−	0.772385i	$-0.719063\pi$
$7$	−3.36092	−1.27031	−0.635154	+	0.772385i	$0.719063\pi$
$8$	0	0
$9$	−2.83168	−0.943894
$10$	0	0
$11$	−4.13893	−1.24793	−0.623967	−	0.781451i	$-0.714480\pi$
$11$	−4.13893	−1.24793	−0.623967	+	0.781451i	$0.714480\pi$
$12$	0	0
$13$	−3.22425	−0.894245	−0.447122	−	0.894473i	$-0.647551\pi$
$13$	−3.22425	−0.894245	−0.447122	+	0.894473i	$0.647551\pi$
$14$	0	0
$15$	1.72807	0.446185
$16$	0	0
$17$	4.83674	1.17308	0.586541	−	0.809919i	$-0.300489\pi$
$17$	4.83674	1.17308	0.586541	+	0.809919i	$0.300489\pi$
$18$	0	0
$19$	7.13255	1.63632	0.818159	−	0.574991i	$-0.194995\pi$
$19$	7.13255	1.63632	0.818159	+	0.574991i	$0.194995\pi$
$20$	0	0
$21$	1.37887	0.300894
$22$	0	0
$23$	0.264477	0.0551472	0.0275736	−	0.999620i	$-0.491222\pi$
$23$	0.264477	0.0551472	0.0275736	+	0.999620i	$0.491222\pi$
$24$	0	0
$25$	12.7416	2.54831
$26$	0	0
$27$	2.39254	0.460444
$28$	0	0
$29$	−6.92998	−1.28686	−0.643432	−	0.765503i	$-0.722490\pi$
$29$	−6.92998	−1.28686	−0.643432	+	0.765503i	$0.722490\pi$
$30$	0	0
$31$	−1.07140	−0.192429	−0.0962147	−	0.995361i	$-0.530674\pi$
$31$	−1.07140	−0.192429	−0.0962147	+	0.995361i	$0.530674\pi$
$32$	0	0
$33$	1.69806	0.295594
$34$	0	0
$35$	14.1564	2.39288
$36$	0	0
$37$	5.79815	0.953210	0.476605	−	0.879117i	$-0.341867\pi$
$37$	5.79815	0.953210	0.476605	+	0.879117i	$0.341867\pi$
$38$	0	0
$39$	1.32280	0.211817
$40$	0	0
$41$	1.24932	0.195111	0.0975553	−	0.995230i	$-0.468898\pi$
$41$	1.24932	0.195111	0.0975553	+	0.995230i	$0.468898\pi$
$42$	0	0
$43$	−3.97547	−0.606253	−0.303126	−	0.952950i	$-0.598030\pi$
$43$	−3.97547	−0.606253	−0.303126	+	0.952950i	$0.598030\pi$
$44$	0	0
$45$	11.9273	1.77801
$46$	0	0
$47$	7.96976	1.16251	0.581255	−	0.813722i	$-0.302562\pi$
$47$	7.96976	1.16251	0.581255	+	0.813722i	$0.302562\pi$
$48$	0	0
$49$	4.29578	0.613684
$50$	0	0
$51$	−1.98435	−0.277864
$52$	0	0
$53$	13.9969	1.92263	0.961313	−	0.275460i	$-0.0888301\pi$
$53$	13.9969	1.92263	0.961313	+	0.275460i	$0.0888301\pi$
$54$	0	0
$55$	17.4335	2.35073
$56$	0	0
$57$	−2.92624	−0.387590
$58$	0	0
$59$	−1.88127	−0.244921	−0.122460	−	0.992473i	$-0.539078\pi$
$59$	−1.88127	−0.244921	−0.122460	+	0.992473i	$0.539078\pi$
$60$	0	0
$61$	−13.0212	−1.66719	−0.833596	−	0.552374i	$-0.813722\pi$
$61$	−13.0212	−1.66719	−0.833596	+	0.552374i	$0.813722\pi$
$62$	0	0
$63$	9.51706	1.19904
$64$	0	0
$65$	13.5808	1.68449
$66$	0	0
$67$	−7.34403	−0.897215	−0.448608	−	0.893729i	$-0.648080\pi$
$67$	−7.34403	−0.897215	−0.448608	+	0.893729i	$0.648080\pi$
$68$	0	0
$69$	−0.108506	−0.0130625
$70$	0	0
$71$	−13.5632	−1.60965	−0.804826	−	0.593510i	$-0.797742\pi$
$71$	−13.5632	−1.60965	−0.804826	+	0.593510i	$0.797742\pi$
$72$	0	0
$73$	−2.49323	−0.291810	−0.145905	−	0.989299i	$-0.546609\pi$
$73$	−2.49323	−0.291810	−0.145905	+	0.989299i	$0.546609\pi$
$74$	0	0
$75$	−5.22742	−0.603611
$76$	0	0
$77$	13.9106	1.58526
$78$	0	0
$79$	8.07702	0.908736	0.454368	−	0.890814i	$-0.349865\pi$
$79$	8.07702	0.908736	0.454368	+	0.890814i	$0.349865\pi$
$80$	0	0
$81$	7.51347	0.834830
$82$	0	0
$83$	10.1672	1.11599	0.557997	−	0.829843i	$-0.311570\pi$
$83$	10.1672	1.11599	0.557997	+	0.829843i	$0.311570\pi$
$84$	0	0
$85$	−20.3727	−2.20973
$86$	0	0
$87$	2.84313	0.304816
$88$	0	0
$89$	3.18164	0.337253	0.168627	−	0.985680i	$-0.446067\pi$
$89$	3.18164	0.337253	0.168627	+	0.985680i	$0.446067\pi$
$90$	0	0
$91$	10.8364	1.13597
$92$	0	0
$93$	0.439559	0.0455801
$94$	0	0
$95$	−30.0428	−3.08233
$96$	0	0
$97$	7.00896	0.711652	0.355826	−	0.934552i	$-0.384200\pi$
$97$	7.00896	0.711652	0.355826	+	0.934552i	$0.384200\pi$
$98$	0	0
$99$	11.7201	1.17792
$100$	0	0
$101$	7.13363	0.709823	0.354911	−	0.934900i	$-0.384511\pi$
$101$	7.13363	0.709823	0.354911	+	0.934900i	$0.384511\pi$
$102$	0	0
$103$	−8.64606	−0.851922	−0.425961	−	0.904742i	$-0.640064\pi$
$103$	−8.64606	−0.851922	−0.425961	+	0.904742i	$0.640064\pi$
$104$	0	0
$105$	−5.80790	−0.566793
$106$	0	0
$107$	−2.00657	−0.193982	−0.0969912	−	0.995285i	$-0.530922\pi$
$107$	−2.00657	−0.193982	−0.0969912	+	0.995285i	$0.530922\pi$
$108$	0	0
$109$	−3.48178	−0.333494	−0.166747	−	0.986000i	$-0.553326\pi$
$109$	−3.48178	−0.333494	−0.166747	+	0.986000i	$0.553326\pi$
$110$	0	0
$111$	−2.37878	−0.225784
$112$	0	0
$113$	17.5080	1.64701	0.823505	−	0.567308i	$-0.192015\pi$
$113$	17.5080	1.64701	0.823505	+	0.567308i	$0.192015\pi$
$114$	0	0
$115$	−1.11400	−0.103881
$116$	0	0
$117$	9.13004	0.844072
$118$	0	0
$119$	−16.2559	−1.49018
$120$	0	0
$121$	6.13072	0.557338
$122$	0	0
$123$	−0.512552	−0.0462152
$124$	0	0
$125$	−32.6081	−2.91655
$126$	0	0
$127$	−2.74768	−0.243817	−0.121909	−	0.992541i	$-0.538901\pi$
$127$	−2.74768	−0.243817	−0.121909	+	0.992541i	$0.538901\pi$
$128$	0	0
$129$	1.63100	0.143601
$130$	0	0
$131$	−9.58654	−0.837580	−0.418790	−	0.908083i	$-0.637546\pi$
$131$	−9.58654	−0.837580	−0.418790	+	0.908083i	$0.637546\pi$
$132$	0	0
$133$	−23.9719	−2.07863
$134$	0	0
$135$	−10.0775	−0.867337
$136$	0	0
$137$	14.1551	1.20935	0.604677	−	0.796471i	$-0.293302\pi$
$137$	14.1551	1.20935	0.604677	+	0.796471i	$0.293302\pi$
$138$	0	0
$139$	−3.03212	−0.257181	−0.128591	−	0.991698i	$-0.541045\pi$
$139$	−3.03212	−0.257181	−0.128591	+	0.991698i	$0.541045\pi$
$140$	0	0
$141$	−3.26972	−0.275360
$142$	0	0
$143$	13.3449	1.11596
$144$	0	0
$145$	29.1896	2.42406
$146$	0	0
$147$	−1.76241	−0.145361
$148$	0	0
$149$	9.35096	0.766061	0.383030	−	0.923736i	$-0.374880\pi$
$149$	9.35096	0.766061	0.383030	+	0.923736i	$0.374880\pi$
$150$	0	0
$151$	12.0486	0.980504	0.490252	−	0.871581i	$-0.336905\pi$
$151$	12.0486	0.980504	0.490252	+	0.871581i	$0.336905\pi$
$152$	0	0
$153$	−13.6961	−1.10727
$154$	0	0
$155$	4.51282	0.362479
$156$	0	0
$157$	−2.85173	−0.227593	−0.113796	−	0.993504i	$-0.536301\pi$
$157$	−2.85173	−0.227593	−0.113796	+	0.993504i	$0.536301\pi$
$158$	0	0
$159$	−5.74245	−0.455406
$160$	0	0
$161$	−0.888885	−0.0700539
$162$	0	0
$163$	−12.7867	−1.00153	−0.500764	−	0.865584i	$-0.666948\pi$
$163$	−12.7867	−1.00153	−0.500764	+	0.865584i	$0.666948\pi$
$164$	0	0
$165$	−7.15235	−0.556810
$166$	0	0
$167$	−16.3443	−1.26476	−0.632381	−	0.774658i	$-0.717922\pi$
$167$	−16.3443	−1.26476	−0.632381	+	0.774658i	$0.717922\pi$
$168$	0	0
$169$	−2.60424	−0.200326
$170$	0	0
$171$	−20.1971	−1.54451
$172$	0	0
$173$	3.23063	0.245620	0.122810	−	0.992430i	$-0.460809\pi$
$173$	3.23063	0.245620	0.122810	+	0.992430i	$0.460809\pi$
$174$	0	0
$175$	−42.8234	−3.23714
$176$	0	0
$177$	0.771821	0.0580136
$178$	0	0
$179$	20.9167	1.56339	0.781694	−	0.623662i	$-0.214356\pi$
$179$	20.9167	1.56339	0.781694	+	0.623662i	$0.214356\pi$
$180$	0	0
$181$	−16.1263	−1.19866	−0.599328	−	0.800504i	$-0.704565\pi$
$181$	−16.1263	−1.19866	−0.599328	+	0.800504i	$0.704565\pi$
$182$	0	0
$183$	5.34214	0.394903
$184$	0	0
$185$	−24.4222	−1.79556
$186$	0	0
$187$	−20.0189	−1.46393
$188$	0	0
$189$	−8.04113	−0.584906
$190$	0	0
$191$	−15.1057	−1.09301	−0.546506	−	0.837455i	$-0.684042\pi$
$191$	−15.1057	−1.09301	−0.546506	+	0.837455i	$0.684042\pi$
$192$	0	0
$193$	16.5469	1.19107	0.595535	−	0.803329i	$-0.296940\pi$
$193$	16.5469	1.19107	0.595535	+	0.803329i	$0.296940\pi$
$194$	0	0
$195$	−5.57172	−0.398999
$196$	0	0
$197$	15.0779	1.07426	0.537129	−	0.843500i	$-0.319509\pi$
$197$	15.0779	1.07426	0.537129	+	0.843500i	$0.319509\pi$
$198$	0	0
$199$	−3.05087	−0.216271	−0.108135	−	0.994136i	$-0.534488\pi$
$199$	−3.05087	−0.216271	−0.108135	+	0.994136i	$0.534488\pi$
$200$	0	0
$201$	3.01300	0.212521
$202$	0	0
$203$	23.2911	1.63472
$204$	0	0
$205$	−5.26222	−0.367529
$206$	0	0
$207$	−0.748914	−0.0520531
$208$	0	0
$209$	−29.5211	−2.04202
$210$	0	0
$211$	−22.4233	−1.54368	−0.771840	−	0.635816i	$-0.780664\pi$
$211$	−22.4233	−1.54368	−0.771840	+	0.635816i	$0.780664\pi$
$212$	0	0
$213$	5.56450	0.381273
$214$	0	0
$215$	16.7450	1.14200
$216$	0	0
$217$	3.60089	0.244445
$218$	0	0
$219$	1.02289	0.0691202
$220$	0	0
$221$	−15.5949	−1.04902
$222$	0	0
$223$	3.37993	0.226337	0.113169	−	0.993576i	$-0.463900\pi$
$223$	3.37993	0.226337	0.113169	+	0.993576i	$0.463900\pi$
$224$	0	0
$225$	−36.0801	−2.40534
$226$	0	0
$227$	−24.3974	−1.61931	−0.809655	−	0.586907i	$-0.800346\pi$
$227$	−24.3974	−1.61931	−0.809655	+	0.586907i	$0.800346\pi$
$228$	0	0
$229$	22.6787	1.49865	0.749326	−	0.662202i	$-0.230378\pi$
$229$	22.6787	1.49865	0.749326	+	0.662202i	$0.230378\pi$
$230$	0	0
$231$	−5.70704	−0.375496
$232$	0	0
$233$	24.2114	1.58614	0.793070	−	0.609130i	$-0.208481\pi$
$233$	24.2114	1.58614	0.793070	+	0.609130i	$0.208481\pi$
$234$	0	0
$235$	−33.5692	−2.18982
$236$	0	0
$237$	−3.31372	−0.215249
$238$	0	0
$239$	1.85452	0.119959	0.0599795	−	0.998200i	$-0.480896\pi$
$239$	1.85452	0.119959	0.0599795	+	0.998200i	$0.480896\pi$
$240$	0	0
$241$	10.1215	0.651985	0.325993	−	0.945372i	$-0.394302\pi$
$241$	10.1215	0.651985	0.325993	+	0.945372i	$0.394302\pi$
$242$	0	0
$243$	−10.2601	−0.658188
$244$	0	0
$245$	−18.0942	−1.15599
$246$	0	0
$247$	−22.9971	−1.46327
$248$	0	0
$249$	−4.17125	−0.264342
$250$	0	0
$251$	−4.41727	−0.278816	−0.139408	−	0.990235i	$-0.544520\pi$
$251$	−4.41727	−0.278816	−0.139408	+	0.990235i	$0.544520\pi$
$252$	0	0
$253$	−1.09465	−0.0688200
$254$	0	0
$255$	8.35822	0.523412
$256$	0	0
$257$	19.8163	1.23611	0.618055	−	0.786135i	$-0.287921\pi$
$257$	19.8163	1.23611	0.618055	+	0.786135i	$0.287921\pi$
$258$	0	0
$259$	−19.4871	−1.21087
$260$	0	0
$261$	19.6235	1.21466
$262$	0	0
$263$	−28.1818	−1.73776	−0.868882	−	0.495019i	$-0.835161\pi$
$263$	−28.1818	−1.73776	−0.868882	+	0.495019i	$0.835161\pi$
$264$	0	0
$265$	−58.9561	−3.62164
$266$	0	0
$267$	−1.30532	−0.0798842
$268$	0	0
$269$	6.37959	0.388971	0.194485	−	0.980905i	$-0.437696\pi$
$269$	6.37959	0.388971	0.194485	+	0.980905i	$0.437696\pi$
$270$	0	0
$271$	−11.1955	−0.680076	−0.340038	−	0.940412i	$-0.610440\pi$
$271$	−11.1955	−0.680076	−0.340038	+	0.940412i	$0.610440\pi$
$272$	0	0
$273$	−4.44581	−0.269073
$274$	0	0
$275$	−52.7364	−3.18013
$276$	0	0
$277$	−26.1314	−1.57008	−0.785042	−	0.619443i	$-0.787359\pi$
$277$	−26.1314	−1.57008	−0.785042	+	0.619443i	$0.787359\pi$
$278$	0	0
$279$	3.03387	0.181633
$280$	0	0
$281$	−4.54850	−0.271341	−0.135670	−	0.990754i	$-0.543319\pi$
$281$	−4.54850	−0.271341	−0.135670	+	0.990754i	$0.543319\pi$
$282$	0	0
$283$	−19.4187	−1.15432	−0.577162	−	0.816630i	$-0.695840\pi$
$283$	−19.4187	−1.15432	−0.577162	+	0.816630i	$0.695840\pi$
$284$	0	0
$285$	12.3255	0.730101
$286$	0	0
$287$	−4.19886	−0.247851
$288$	0	0
$289$	6.39410	0.376123
$290$	0	0
$291$	−2.87553	−0.168567
$292$	0	0
$293$	8.94508	0.522577	0.261289	−	0.965261i	$-0.415853\pi$
$293$	8.94508	0.522577	0.261289	+	0.965261i	$0.415853\pi$
$294$	0	0
$295$	7.92406	0.461357
$296$	0	0
$297$	−9.90254	−0.574604
$298$	0	0
$299$	−0.852738	−0.0493151
$300$	0	0
$301$	13.3612	0.770128
$302$	0	0
$303$	−2.92668	−0.168133
$304$	0	0
$305$	54.8462	3.14049
$306$	0	0
$307$	22.9511	1.30989	0.654943	−	0.755678i	$-0.272693\pi$
$307$	22.9511	1.30989	0.654943	+	0.755678i	$0.272693\pi$
$308$	0	0
$309$	3.54718	0.201792
$310$	0	0
$311$	33.3402	1.89055	0.945276	−	0.326272i	$-0.105793\pi$
$311$	33.3402	1.89055	0.945276	+	0.326272i	$0.105793\pi$
$312$	0	0
$313$	−14.3601	−0.811680	−0.405840	−	0.913944i	$-0.633021\pi$
$313$	−14.3601	−0.811680	−0.405840	+	0.913944i	$0.633021\pi$
$314$	0	0
$315$	−40.0866	−2.25862
$316$	0	0
$317$	−1.42229	−0.0798840	−0.0399420	−	0.999202i	$-0.512717\pi$
$317$	−1.42229	−0.0798840	−0.0399420	+	0.999202i	$0.512717\pi$
$318$	0	0
$319$	28.6827	1.60592
$320$	0	0
$321$	0.823226	0.0459480
$322$	0	0
$323$	34.4983	1.91954
$324$	0	0
$325$	−41.0820	−2.27882
$326$	0	0
$327$	1.42845	0.0789936
$328$	0	0
$329$	−26.7857	−1.47675
$330$	0	0
$331$	0.937586	0.0515344	0.0257672	−	0.999668i	$-0.491797\pi$
$331$	0.937586	0.0515344	0.0257672	+	0.999668i	$0.491797\pi$
$332$	0	0
$333$	−16.4185	−0.899730
$334$	0	0
$335$	30.9336	1.69008
$336$	0	0
$337$	16.3871	0.892662	0.446331	−	0.894868i	$-0.352730\pi$
$337$	16.3871	0.892662	0.446331	+	0.894868i	$0.352730\pi$
$338$	0	0
$339$	−7.18291	−0.390122
$340$	0	0
$341$	4.43445	0.240139
$342$	0	0
$343$	9.08865	0.490741
$344$	0	0
$345$	0.457034	0.0246059
$346$	0	0
$347$	21.5142	1.15494	0.577470	−	0.816412i	$-0.304040\pi$
$347$	21.5142	1.15494	0.577470	+	0.816412i	$0.304040\pi$
$348$	0	0
$349$	−35.9721	−1.92554	−0.962770	−	0.270320i	$-0.912870\pi$
$349$	−35.9721	−1.92554	−0.962770	+	0.270320i	$0.912870\pi$
$350$	0	0
$351$	−7.71413	−0.411750
$352$	0	0
$353$	−31.5436	−1.67890	−0.839449	−	0.543438i	$-0.817122\pi$
$353$	−31.5436	−1.67890	−0.839449	+	0.543438i	$0.817122\pi$
$354$	0	0
$355$	57.1291	3.03210
$356$	0	0
$357$	6.66924	0.352973
$358$	0	0
$359$	36.2750	1.91452	0.957261	−	0.289227i	$-0.0933982\pi$
$359$	36.2750	1.91452	0.957261	+	0.289227i	$0.0933982\pi$
$360$	0	0
$361$	31.8732	1.67754
$362$	0	0
$363$	−2.51522	−0.132015
$364$	0	0
$365$	10.5017	0.549682
$366$	0	0
$367$	−18.3867	−0.959776	−0.479888	−	0.877330i	$-0.659323\pi$
$367$	−18.3867	−0.959776	−0.479888	+	0.877330i	$0.659323\pi$
$368$	0	0
$369$	−3.53767	−0.184164
$370$	0	0
$371$	−47.0425	−2.44233
$372$	0	0
$373$	24.2082	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$373$	24.2082	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$374$	0	0
$375$	13.3780	0.690835
$376$	0	0
$377$	22.3440	1.15077
$378$	0	0
$379$	1.23158	0.0632622	0.0316311	−	0.999500i	$-0.489930\pi$
$379$	1.23158	0.0632622	0.0316311	+	0.999500i	$0.489930\pi$
$380$	0	0
$381$	1.12728	0.0577522
$382$	0	0
$383$	22.7396	1.16194	0.580969	−	0.813926i	$-0.302674\pi$
$383$	22.7396	1.16194	0.580969	+	0.813926i	$0.302674\pi$
$384$	0	0
$385$	−58.5925	−2.98615
$386$	0	0
$387$	11.2573	0.572239
$388$	0	0
$389$	−15.0769	−0.764429	−0.382214	−	0.924074i	$-0.624838\pi$
$389$	−15.0769	−0.764429	−0.382214	+	0.924074i	$0.624838\pi$
$390$	0	0
$391$	1.27921	0.0646922
$392$	0	0
$393$	3.93302	0.198395
$394$	0	0
$395$	−34.0210	−1.71178
$396$	0	0
$397$	10.8639	0.545246	0.272623	−	0.962121i	$-0.412109\pi$
$397$	10.8639	0.545246	0.272623	+	0.962121i	$0.412109\pi$
$398$	0	0
$399$	9.83485	0.492358
$400$	0	0
$401$	−11.8865	−0.593581	−0.296791	−	0.954943i	$-0.595916\pi$
$401$	−11.8865	−0.593581	−0.296791	+	0.954943i	$0.595916\pi$
$402$	0	0
$403$	3.45446	0.172079
$404$	0	0
$405$	−31.6473	−1.57257
$406$	0	0
$407$	−23.9981	−1.18954
$408$	0	0
$409$	−20.2652	−1.00205	−0.501026	−	0.865432i	$-0.667044\pi$
$409$	−20.2652	−1.00205	−0.501026	+	0.865432i	$0.667044\pi$
$410$	0	0
$411$	−5.80736	−0.286456
$412$	0	0
$413$	6.32281	0.311125
$414$	0	0
$415$	−42.8250	−2.10219
$416$	0	0
$417$	1.24398	0.0609177
$418$	0	0
$419$	5.46963	0.267209	0.133604	−	0.991035i	$-0.457345\pi$
$419$	5.46963	0.267209	0.133604	+	0.991035i	$0.457345\pi$
$420$	0	0
$421$	34.3527	1.67425	0.837123	−	0.547015i	$-0.184236\pi$
$421$	34.3527	1.67425	0.837123	+	0.547015i	$0.184236\pi$
$422$	0	0
$423$	−22.5678	−1.09729
$424$	0	0
$425$	61.6277	2.98938
$426$	0	0
$427$	43.7632	2.11785
$428$	0	0
$429$	−5.47496	−0.264333
$430$	0	0
$431$	5.85713	0.282128	0.141064	−	0.990000i	$-0.454948\pi$
$431$	5.85713	0.282128	0.141064	+	0.990000i	$0.454948\pi$
$432$	0	0
$433$	2.13150	0.102434	0.0512168	−	0.998688i	$-0.483690\pi$
$433$	2.13150	0.102434	0.0512168	+	0.998688i	$0.483690\pi$
$434$	0	0
$435$	−11.9755	−0.574180
$436$	0	0
$437$	1.88639	0.0902384
$438$	0	0
$439$	25.8927	1.23579	0.617896	−	0.786259i	$-0.287985\pi$
$439$	25.8927	1.23579	0.617896	+	0.786259i	$0.287985\pi$
$440$	0	0
$441$	−12.1643	−0.579252
$442$	0	0
$443$	11.3441	0.538974	0.269487	−	0.963004i	$-0.413146\pi$
$443$	11.3441	0.538974	0.269487	+	0.963004i	$0.413146\pi$
$444$	0	0
$445$	−13.4013	−0.635283
$446$	0	0
$447$	−3.83638	−0.181454
$448$	0	0
$449$	29.4795	1.39123	0.695613	−	0.718417i	$-0.255133\pi$
$449$	29.4795	1.39123	0.695613	+	0.718417i	$0.255133\pi$
$450$	0	0
$451$	−5.17084	−0.243485
$452$	0	0
$453$	−4.94314	−0.232249
$454$	0	0
$455$	−45.6439	−2.13982
$456$	0	0
$457$	−36.6285	−1.71341	−0.856705	−	0.515806i	$-0.827492\pi$
$457$	−36.6285	−1.71341	−0.856705	+	0.515806i	$0.827492\pi$
$458$	0	0
$459$	11.5721	0.540139
$460$	0	0
$461$	25.4126	1.18358	0.591791	−	0.806091i	$-0.298421\pi$
$461$	25.4126	1.18358	0.591791	+	0.806091i	$0.298421\pi$
$462$	0	0
$463$	24.3241	1.13044	0.565218	−	0.824942i	$-0.308792\pi$
$463$	24.3241	1.13044	0.565218	+	0.824942i	$0.308792\pi$
$464$	0	0
$465$	−1.85145	−0.0858592
$466$	0	0
$467$	−27.9330	−1.29258	−0.646292	−	0.763090i	$-0.723681\pi$
$467$	−27.9330	−1.29258	−0.646292	+	0.763090i	$0.723681\pi$
$468$	0	0
$469$	24.6827	1.13974
$470$	0	0
$471$	1.16997	0.0539092
$472$	0	0
$473$	16.4542	0.756563
$474$	0	0
$475$	90.8799	4.16985
$476$	0	0
$477$	−39.6348	−1.81475
$478$	0	0
$479$	−31.8937	−1.45726	−0.728631	−	0.684907i	$-0.759843\pi$
$479$	−31.8937	−1.45726	−0.728631	+	0.684907i	$0.759843\pi$
$480$	0	0
$481$	−18.6947	−0.852403
$482$	0	0
$483$	0.364679	0.0165935
$484$	0	0
$485$	−29.5223	−1.34054
$486$	0	0
$487$	−9.97200	−0.451875	−0.225937	−	0.974142i	$-0.572544\pi$
$487$	−9.97200	−0.451875	−0.225937	+	0.974142i	$0.572544\pi$
$488$	0	0
$489$	5.24592	0.237229
$490$	0	0
$491$	12.5310	0.565515	0.282758	−	0.959191i	$-0.408751\pi$
$491$	12.5310	0.565515	0.282758	+	0.959191i	$0.408751\pi$
$492$	0	0
$493$	−33.5185	−1.50960
$494$	0	0
$495$	−49.3660	−2.21884
$496$	0	0
$497$	45.5847	2.04476
$498$	0	0
$499$	24.7196	1.10660	0.553301	−	0.832981i	$-0.313368\pi$
$499$	24.7196	1.10660	0.553301	+	0.832981i	$0.313368\pi$
$500$	0	0
$501$	6.70551	0.299580
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−30.0474	−1.33709
$506$	0	0
$507$	1.06843	0.0474506
$508$	0	0
$509$	−13.3757	−0.592866	−0.296433	−	0.955054i	$-0.595797\pi$
$509$	−13.3757	−0.592866	−0.296433	+	0.955054i	$0.595797\pi$
$510$	0	0
$511$	8.37954	0.370689
$512$	0	0
$513$	17.0649	0.753433
$514$	0	0
$515$	36.4178	1.60476
$516$	0	0
$517$	−32.9863	−1.45073
$518$	0	0
$519$	−1.32541	−0.0581792
$520$	0	0
$521$	−3.01024	−0.131881	−0.0659406	−	0.997824i	$-0.521005\pi$
$521$	−3.01024	−0.131881	−0.0659406	+	0.997824i	$0.521005\pi$
$522$	0	0
$523$	−0.371149	−0.0162292	−0.00811462	−	0.999967i	$-0.502583\pi$
$523$	−0.371149	−0.0162292	−0.00811462	+	0.999967i	$0.502583\pi$
$524$	0	0
$525$	17.5690	0.766772
$526$	0	0
$527$	−5.18209	−0.225736
$528$	0	0
$529$	−22.9301	−0.996959
$530$	0	0
$531$	5.32717	0.231179
$532$	0	0
$533$	−4.02811	−0.174477
$534$	0	0
$535$	8.45182	0.365404
$536$	0	0
$537$	−8.58140	−0.370315
$538$	0	0
$539$	−17.7799	−0.765836
$540$	0	0
$541$	−12.2639	−0.527266	−0.263633	−	0.964623i	$-0.584921\pi$
$541$	−12.2639	−0.527266	−0.263633	+	0.964623i	$0.584921\pi$
$542$	0	0
$543$	6.61605	0.283922
$544$	0	0
$545$	14.6655	0.628201
$546$	0	0
$547$	36.4868	1.56006	0.780031	−	0.625741i	$-0.215203\pi$
$547$	36.4868	1.56006	0.780031	+	0.625741i	$0.215203\pi$
$548$	0	0
$549$	36.8719	1.57365
$550$	0	0
$551$	−49.4284	−2.10572
$552$	0	0
$553$	−27.1462	−1.15437
$554$	0	0
$555$	10.0196	0.425308
$556$	0	0
$557$	26.0934	1.10561	0.552807	−	0.833310i	$-0.313557\pi$
$557$	26.0934	1.10561	0.552807	+	0.833310i	$0.313557\pi$
$558$	0	0
$559$	12.8179	0.542139
$560$	0	0
$561$	8.21308	0.346756
$562$	0	0
$563$	−12.5584	−0.529274	−0.264637	−	0.964348i	$-0.585252\pi$
$563$	−12.5584	−0.529274	−0.264637	+	0.964348i	$0.585252\pi$
$564$	0	0
$565$	−73.7448	−3.10247
$566$	0	0
$567$	−25.2522	−1.06049
$568$	0	0
$569$	−13.7961	−0.578364	−0.289182	−	0.957274i	$-0.593383\pi$
$569$	−13.7961	−0.578364	−0.289182	+	0.957274i	$0.593383\pi$
$570$	0	0
$571$	−12.9291	−0.541067	−0.270534	−	0.962711i	$-0.587200\pi$
$571$	−12.9291	−0.541067	−0.270534	+	0.962711i	$0.587200\pi$
$572$	0	0
$573$	6.19736	0.258898
$574$	0	0
$575$	3.36985	0.140532
$576$	0	0
$577$	38.2876	1.59393	0.796967	−	0.604023i	$-0.206436\pi$
$577$	38.2876	1.59393	0.796967	+	0.604023i	$0.206436\pi$
$578$	0	0
$579$	−6.78861	−0.282125
$580$	0	0
$581$	−34.1711	−1.41766
$582$	0	0
$583$	−57.9322	−2.39931
$584$	0	0
$585$	−38.4564	−1.58998
$586$	0	0
$587$	−41.9521	−1.73155	−0.865775	−	0.500434i	$-0.833174\pi$
$587$	−41.9521	−1.73155	−0.865775	+	0.500434i	$0.833174\pi$
$588$	0	0
$589$	−7.64182	−0.314876
$590$	0	0
$591$	−6.18595	−0.254456
$592$	0	0
$593$	2.09406	0.0859928	0.0429964	−	0.999075i	$-0.486310\pi$
$593$	2.09406	0.0859928	0.0429964	+	0.999075i	$0.486310\pi$
$594$	0	0
$595$	68.4711	2.80704
$596$	0	0
$597$	1.25167	0.0512274
$598$	0	0
$599$	−9.46204	−0.386608	−0.193304	−	0.981139i	$-0.561920\pi$
$599$	−9.46204	−0.386608	−0.193304	+	0.981139i	$0.561920\pi$
$600$	0	0
$601$	34.1000	1.39097	0.695484	−	0.718542i	$-0.255190\pi$
$601$	34.1000	1.39097	0.695484	+	0.718542i	$0.255190\pi$
$602$	0	0
$603$	20.7959	0.846876
$604$	0	0
$605$	−25.8230	−1.04986
$606$	0	0
$607$	34.3332	1.39354	0.696770	−	0.717294i	$-0.254620\pi$
$607$	34.3332	1.39354	0.696770	+	0.717294i	$0.254620\pi$
$608$	0	0
$609$	−9.55554	−0.387210
$610$	0	0
$611$	−25.6965	−1.03957
$612$	0	0
$613$	−10.9176	−0.440957	−0.220479	−	0.975392i	$-0.570762\pi$
$613$	−10.9176	−0.440957	−0.220479	+	0.975392i	$0.570762\pi$
$614$	0	0
$615$	2.15891	0.0870555
$616$	0	0
$617$	−37.3923	−1.50536	−0.752678	−	0.658389i	$-0.771238\pi$
$617$	−37.3923	−1.50536	−0.752678	+	0.658389i	$0.771238\pi$
$618$	0	0
$619$	10.0717	0.404817	0.202409	−	0.979301i	$-0.435123\pi$
$619$	10.0717	0.404817	0.202409	+	0.979301i	$0.435123\pi$
$620$	0	0
$621$	0.632770	0.0253922
$622$	0	0
$623$	−10.6932	−0.428416
$624$	0	0
$625$	73.6397	2.94559
$626$	0	0
$627$	12.1115	0.483686
$628$	0	0
$629$	28.0442	1.11819
$630$	0	0
$631$	2.11890	0.0843520	0.0421760	−	0.999110i	$-0.486571\pi$
$631$	2.11890	0.0843520	0.0421760	+	0.999110i	$0.486571\pi$
$632$	0	0
$633$	9.19949	0.365647
$634$	0	0
$635$	11.5734	0.459277
$636$	0	0
$637$	−13.8507	−0.548783
$638$	0	0
$639$	38.4066	1.51934
$640$	0	0
$641$	−4.78686	−0.189070	−0.0945348	−	0.995522i	$-0.530136\pi$
$641$	−4.78686	−0.189070	−0.0945348	+	0.995522i	$0.530136\pi$
$642$	0	0
$643$	−31.6313	−1.24742	−0.623709	−	0.781656i	$-0.714375\pi$
$643$	−31.6313	−1.24742	−0.623709	+	0.781656i	$0.714375\pi$
$644$	0	0
$645$	−6.86988	−0.270501
$646$	0	0
$647$	13.7434	0.540310	0.270155	−	0.962817i	$-0.412925\pi$
$647$	13.7434	0.540310	0.270155	+	0.962817i	$0.412925\pi$
$648$	0	0
$649$	7.78645	0.305645
$650$	0	0
$651$	−1.47732	−0.0579008
$652$	0	0
$653$	−0.0587916	−0.00230069	−0.00115035	−	0.999999i	$-0.500366\pi$
$653$	−0.0587916	−0.00230069	−0.00115035	+	0.999999i	$0.500366\pi$
$654$	0	0
$655$	40.3792	1.57775
$656$	0	0
$657$	7.06003	0.275438
$658$	0	0
$659$	−14.1013	−0.549308	−0.274654	−	0.961543i	$-0.588563\pi$
$659$	−14.1013	−0.549308	−0.274654	+	0.961543i	$0.588563\pi$
$660$	0	0
$661$	2.47912	0.0964265	0.0482133	−	0.998837i	$-0.484647\pi$
$661$	2.47912	0.0964265	0.0482133	+	0.998837i	$0.484647\pi$
$662$	0	0
$663$	6.39803	0.248479
$664$	0	0
$665$	100.972	3.91551
$666$	0	0
$667$	−1.83282	−0.0709670
$668$	0	0
$669$	−1.38667	−0.0536117
$670$	0	0
$671$	53.8938	2.08055
$672$	0	0
$673$	1.88041	0.0724845	0.0362423	−	0.999343i	$-0.488461\pi$
$673$	1.88041	0.0724845	0.0362423	+	0.999343i	$0.488461\pi$
$674$	0	0
$675$	30.4847	1.17336
$676$	0	0
$677$	11.6258	0.446815	0.223407	−	0.974725i	$-0.428282\pi$
$677$	11.6258	0.446815	0.223407	+	0.974725i	$0.428282\pi$
$678$	0	0
$679$	−23.5565	−0.904017
$680$	0	0
$681$	10.0094	0.383561
$682$	0	0
$683$	−0.350145	−0.0133979	−0.00669897	−	0.999978i	$-0.502132\pi$
$683$	−0.350145	−0.0133979	−0.00669897	+	0.999978i	$0.502132\pi$
$684$	0	0
$685$	−59.6225	−2.27806
$686$	0	0
$687$	−9.30429	−0.354981
$688$	0	0
$689$	−45.1295	−1.71930
$690$	0	0
$691$	12.9419	0.492335	0.246167	−	0.969227i	$-0.420829\pi$
$691$	12.9419	0.492335	0.246167	+	0.969227i	$0.420829\pi$
$692$	0	0
$693$	−39.3904	−1.49632
$694$	0	0
$695$	12.7715	0.484452
$696$	0	0
$697$	6.04263	0.228881
$698$	0	0
$699$	−9.93309	−0.375704
$700$	0	0
$701$	−42.8216	−1.61735	−0.808674	−	0.588256i	$-0.799815\pi$
$701$	−42.8216	−1.61735	−0.808674	+	0.588256i	$0.799815\pi$
$702$	0	0
$703$	41.3556	1.55976
$704$	0	0
$705$	13.7723	0.518695
$706$	0	0
$707$	−23.9756	−0.901694
$708$	0	0
$709$	−47.8206	−1.79594	−0.897971	−	0.440055i	$-0.854959\pi$
$709$	−47.8206	−1.79594	−0.897971	+	0.440055i	$0.854959\pi$
$710$	0	0
$711$	−22.8716	−0.857751
$712$	0	0
$713$	−0.283361	−0.0106119
$714$	0	0
$715$	−56.2098	−2.10213
$716$	0	0
$717$	−0.760845	−0.0284143
$718$	0	0
$719$	1.69842	0.0633402	0.0316701	−	0.999498i	$-0.489917\pi$
$719$	1.69842	0.0633402	0.0316701	+	0.999498i	$0.489917\pi$
$720$	0	0
$721$	29.0587	1.08220
$722$	0	0
$723$	−4.15252	−0.154434
$724$	0	0
$725$	−88.2988	−3.27934
$726$	0	0
$727$	−50.3164	−1.86613	−0.933066	−	0.359706i	$-0.882877\pi$
$727$	−50.3164	−1.86613	−0.933066	+	0.359706i	$0.882877\pi$
$728$	0	0
$729$	−18.3310	−0.678927
$730$	0	0
$731$	−19.2283	−0.711185
$732$	0	0
$733$	4.24192	0.156679	0.0783395	−	0.996927i	$-0.475038\pi$
$733$	4.24192	0.156679	0.0783395	+	0.996927i	$0.475038\pi$
$734$	0	0
$735$	7.42341	0.273817
$736$	0	0
$737$	30.3964	1.11967
$738$	0	0
$739$	−19.1496	−0.704428	−0.352214	−	0.935920i	$-0.614571\pi$
$739$	−19.1496	−0.704428	−0.352214	+	0.935920i	$0.614571\pi$
$740$	0	0
$741$	9.43491	0.346600
$742$	0	0
$743$	−42.3558	−1.55388	−0.776942	−	0.629572i	$-0.783230\pi$
$743$	−42.3558	−1.55388	−0.776942	+	0.629572i	$0.783230\pi$
$744$	0	0
$745$	−39.3870	−1.44303
$746$	0	0
$747$	−28.7903	−1.05338
$748$	0	0
$749$	6.74392	0.246417
$750$	0	0
$751$	27.0770	0.988053	0.494027	−	0.869447i	$-0.335525\pi$
$751$	27.0770	0.988053	0.494027	+	0.869447i	$0.335525\pi$
$752$	0	0
$753$	1.81225	0.0660422
$754$	0	0
$755$	−50.7497	−1.84697
$756$	0	0
$757$	−37.8455	−1.37552	−0.687759	−	0.725939i	$-0.741405\pi$
$757$	−37.8455	−1.37552	−0.687759	+	0.725939i	$0.741405\pi$
$758$	0	0
$759$	0.449097	0.0163012
$760$	0	0
$761$	35.5873	1.29004	0.645019	−	0.764166i	$-0.276849\pi$
$761$	35.5873	1.29004	0.645019	+	0.764166i	$0.276849\pi$
$762$	0	0
$763$	11.7020	0.423640
$764$	0	0
$765$	57.6891	2.08575
$766$	0	0
$767$	6.06569	0.219019
$768$	0	0
$769$	37.9803	1.36960	0.684802	−	0.728729i	$-0.259889\pi$
$769$	37.9803	1.36960	0.684802	+	0.728729i	$0.259889\pi$
$770$	0	0
$771$	−8.12996	−0.292793
$772$	0	0
$773$	16.7036	0.600786	0.300393	−	0.953816i	$-0.402882\pi$
$773$	16.7036	0.600786	0.300393	+	0.953816i	$0.402882\pi$
$774$	0	0
$775$	−13.6513	−0.490370
$776$	0	0
$777$	7.99489	0.286815
$778$	0	0
$779$	8.91082	0.319263
$780$	0	0
$781$	56.1370	2.00874
$782$	0	0
$783$	−16.5802	−0.592529
$784$	0	0
$785$	12.0117	0.428716
$786$	0	0
$787$	−7.94706	−0.283282	−0.141641	−	0.989918i	$-0.545238\pi$
$787$	−7.94706	−0.283282	−0.141641	+	0.989918i	$0.545238\pi$
$788$	0	0
$789$	11.5620	0.411619
$790$	0	0
$791$	−58.8429	−2.09221
$792$	0	0
$793$	41.9835	1.49088
$794$	0	0
$795$	24.1876	0.857847
$796$	0	0
$797$	−8.76611	−0.310511	−0.155256	−	0.987874i	$-0.549620\pi$
$797$	−8.76611	−0.310511	−0.155256	+	0.987874i	$0.549620\pi$
$798$	0	0
$799$	38.5477	1.36372
$800$	0	0
$801$	−9.00940	−0.318332
$802$	0	0
$803$	10.3193	0.364160
$804$	0	0
$805$	3.74405	0.131960
$806$	0	0
$807$	−2.61733	−0.0921343
$808$	0	0
$809$	7.02651	0.247039	0.123519	−	0.992342i	$-0.460582\pi$
$809$	7.02651	0.247039	0.123519	+	0.992342i	$0.460582\pi$
$810$	0	0
$811$	−9.74637	−0.342241	−0.171121	−	0.985250i	$-0.554739\pi$
$811$	−9.74637	−0.342241	−0.171121	+	0.985250i	$0.554739\pi$
$812$	0	0
$813$	4.59311	0.161087
$814$	0	0
$815$	53.8584	1.88658
$816$	0	0
$817$	−28.3552	−0.992023
$818$	0	0
$819$	−30.6853	−1.07223
$820$	0	0
$821$	−3.40971	−0.119000	−0.0594998	−	0.998228i	$-0.518951\pi$
$821$	−3.40971	−0.119000	−0.0594998	+	0.998228i	$0.518951\pi$
$822$	0	0
$823$	6.84363	0.238554	0.119277	−	0.992861i	$-0.461942\pi$
$823$	6.84363	0.238554	0.119277	+	0.992861i	$0.461942\pi$
$824$	0	0
$825$	21.6359	0.753266
$826$	0	0
$827$	−34.8563	−1.21207	−0.606036	−	0.795437i	$-0.707241\pi$
$827$	−34.8563	−1.21207	−0.606036	+	0.795437i	$0.707241\pi$
$828$	0	0
$829$	27.9587	0.971047	0.485523	−	0.874224i	$-0.338629\pi$
$829$	27.9587	0.971047	0.485523	+	0.874224i	$0.338629\pi$
$830$	0	0
$831$	10.7208	0.371901
$832$	0	0
$833$	20.7776	0.719902
$834$	0	0
$835$	68.8435	2.38243
$836$	0	0
$837$	−2.56337	−0.0886030
$838$	0	0
$839$	51.8984	1.79173	0.895865	−	0.444326i	$-0.146557\pi$
$839$	51.8984	1.79173	0.895865	+	0.444326i	$0.146557\pi$
$840$	0	0
$841$	19.0246	0.656021
$842$	0	0
$843$	1.86609	0.0642716
$844$	0	0
$845$	10.9693	0.377354
$846$	0	0
$847$	−20.6049	−0.707991
$848$	0	0
$849$	7.96683	0.273421
$850$	0	0
$851$	1.53348	0.0525669
$852$	0	0
$853$	−14.5182	−0.497094	−0.248547	−	0.968620i	$-0.579953\pi$
$853$	−14.5182	−0.497094	−0.248547	+	0.968620i	$0.579953\pi$
$854$	0	0
$855$	85.0717	2.90939
$856$	0	0
$857$	−6.51805	−0.222652	−0.111326	−	0.993784i	$-0.535510\pi$
$857$	−6.51805	−0.222652	−0.111326	+	0.993784i	$0.535510\pi$
$858$	0	0
$859$	18.9016	0.644913	0.322456	−	0.946584i	$-0.395491\pi$
$859$	18.9016	0.644913	0.322456	+	0.946584i	$0.395491\pi$
$860$	0	0
$861$	1.72265	0.0587076
$862$	0	0
$863$	40.4662	1.37749	0.688743	−	0.725005i	$-0.258163\pi$
$863$	40.4662	1.37749	0.688743	+	0.725005i	$0.258163\pi$
$864$	0	0
$865$	−13.6076	−0.462674
$866$	0	0
$867$	−2.62328	−0.0890911
$868$	0	0
$869$	−33.4302	−1.13404
$870$	0	0
$871$	23.6789	0.802330
$872$	0	0
$873$	−19.8471	−0.671724
$874$	0	0
$875$	109.593	3.70492
$876$	0	0
$877$	−38.3146	−1.29379	−0.646896	−	0.762578i	$-0.723933\pi$
$877$	−38.3146	−1.29379	−0.646896	+	0.762578i	$0.723933\pi$
$878$	0	0
$879$	−3.66986	−0.123781
$880$	0	0
$881$	−48.1919	−1.62363	−0.811813	−	0.583917i	$-0.801519\pi$
$881$	−48.1919	−1.62363	−0.811813	+	0.583917i	$0.801519\pi$
$882$	0	0
$883$	−24.0449	−0.809176	−0.404588	−	0.914499i	$-0.632585\pi$
$883$	−24.0449	−0.809176	−0.404588	+	0.914499i	$0.632585\pi$
$884$	0	0
$885$	−3.25097	−0.109280
$886$	0	0
$887$	−3.95712	−0.132867	−0.0664336	−	0.997791i	$-0.521162\pi$
$887$	−3.95712	−0.132867	−0.0664336	+	0.997791i	$0.521162\pi$
$888$	0	0
$889$	9.23473	0.309723
$890$	0	0
$891$	−31.0977	−1.04181
$892$	0	0
$893$	56.8447	1.90224
$894$	0	0
$895$	−88.1027	−2.94495
$896$	0	0
$897$	0.349849	0.0116811
$898$	0	0
$899$	7.42479	0.247631
$900$	0	0
$901$	67.6995	2.25540
$902$	0	0
$903$	−5.48165	−0.182418
$904$	0	0
$905$	67.9250	2.25790
$906$	0	0
$907$	−46.3721	−1.53976	−0.769880	−	0.638189i	$-0.779684\pi$
$907$	−46.3721	−1.53976	−0.769880	+	0.638189i	$0.779684\pi$
$908$	0	0
$909$	−20.2002	−0.669998
$910$	0	0
$911$	13.0731	0.433133	0.216566	−	0.976268i	$-0.430514\pi$
$911$	13.0731	0.433133	0.216566	+	0.976268i	$0.430514\pi$
$912$	0	0
$913$	−42.0813	−1.39269
$914$	0	0
$915$	−22.5015	−0.743877
$916$	0	0
$917$	32.2196	1.06398
$918$	0	0
$919$	−55.9346	−1.84511	−0.922556	−	0.385864i	$-0.873903\pi$
$919$	−55.9346	−1.84511	−0.922556	+	0.385864i	$0.873903\pi$
$920$	0	0
$921$	−9.41603	−0.310269
$922$	0	0
$923$	43.7310	1.43942
$924$	0	0
$925$	73.8776	2.42908
$926$	0	0
$927$	24.4829	0.804124
$928$	0	0
$929$	20.7469	0.680684	0.340342	−	0.940302i	$-0.389457\pi$
$929$	20.7469	0.680684	0.340342	+	0.940302i	$0.389457\pi$
$930$	0	0
$931$	30.6399	1.00418
$932$	0	0
$933$	−13.6783	−0.447809
$934$	0	0
$935$	84.3212	2.75760
$936$	0	0
$937$	−21.5805	−0.705006	−0.352503	−	0.935811i	$-0.614669\pi$
$937$	−21.5805	−0.705006	−0.352503	+	0.935811i	$0.614669\pi$
$938$	0	0
$939$	5.89145	0.192260
$940$	0	0
$941$	53.5707	1.74636	0.873178	−	0.487401i	$-0.162055\pi$
$941$	53.5707	1.74636	0.873178	+	0.487401i	$0.162055\pi$
$942$	0	0
$943$	0.330415	0.0107598
$944$	0	0
$945$	33.8698	1.10179
$946$	0	0
$947$	−7.27841	−0.236516	−0.118258	−	0.992983i	$-0.537731\pi$
$947$	−7.27841	−0.236516	−0.118258	+	0.992983i	$0.537731\pi$
$948$	0	0
$949$	8.03878	0.260950
$950$	0	0
$951$	0.583518	0.0189219
$952$	0	0
$953$	33.6483	1.08998	0.544988	−	0.838444i	$-0.316534\pi$
$953$	33.6483	1.08998	0.544988	+	0.838444i	$0.316534\pi$
$954$	0	0
$955$	63.6265	2.05890
$956$	0	0
$957$	−11.7675	−0.380390
$958$	0	0
$959$	−47.5743	−1.53625
$960$	0	0
$961$	−29.8521	−0.962971
$962$	0	0
$963$	5.68196	0.183099
$964$	0	0
$965$	−69.6966	−2.24361
$966$	0	0
$967$	−12.9918	−0.417788	−0.208894	−	0.977938i	$-0.566986\pi$
$967$	−12.9918	−0.417788	−0.208894	+	0.977938i	$0.566986\pi$
$968$	0	0
$969$	−14.1535	−0.454675
$970$	0	0
$971$	−6.30177	−0.202233	−0.101117	−	0.994875i	$-0.532242\pi$
$971$	−6.30177	−0.202233	−0.101117	+	0.994875i	$0.532242\pi$
$972$	0	0
$973$	10.1907	0.326700
$974$	0	0
$975$	16.8545	0.539776
$976$	0	0
$977$	−42.7142	−1.36655	−0.683275	−	0.730161i	$-0.739445\pi$
$977$	−42.7142	−1.36655	−0.683275	+	0.730161i	$0.739445\pi$
$978$	0	0
$979$	−13.1686	−0.420870
$980$	0	0
$981$	9.85929	0.314783
$982$	0	0
$983$	0.161818	0.00516121	0.00258060	−	0.999997i	$-0.499179\pi$
$983$	0.161818	0.00516121	0.00258060	+	0.999997i	$0.499179\pi$
$984$	0	0
$985$	−63.5093	−2.02358
$986$	0	0
$987$	10.9893	0.349792
$988$	0	0
$989$	−1.05142	−0.0334331
$990$	0	0
$991$	−8.47381	−0.269179	−0.134590	−	0.990901i	$-0.542972\pi$
$991$	−8.47381	−0.269179	−0.134590	+	0.990901i	$0.542972\pi$
$992$	0	0
$993$	−0.384659	−0.0122068
$994$	0	0
$995$	12.8505	0.407389
$996$	0	0
$997$	−33.6467	−1.06560	−0.532800	−	0.846241i	$-0.678860\pi$
$997$	−33.6467	−1.06560	−0.532800	+	0.846241i	$0.678860\pi$
$998$	0	0
$999$	13.8723	0.438900

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.x.1.17		33
4.3	odd	2		4024.2.a.g.1.17	✓	33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.g.1.17	✓	33	4.3	odd	2
8048.2.a.x.1.17		33	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)

\(f(q)\)	\(=\)	\(q-0.410265 q^{3} -4.21207 q^{5} -3.36092 q^{7} -2.83168 q^{9} +O(q^{10})\)	\(q-0.410265 q^{3} -4.21207 q^{5} -3.36092 q^{7} -2.83168 q^{9} -4.13893 q^{11} -3.22425 q^{13} +1.72807 q^{15} +4.83674 q^{17} +7.13255 q^{19} +1.37887 q^{21} +0.264477 q^{23} +12.7416 q^{25} +2.39254 q^{27} -6.92998 q^{29} -1.07140 q^{31} +1.69806 q^{33} +14.1564 q^{35} +5.79815 q^{37} +1.32280 q^{39} +1.24932 q^{41} -3.97547 q^{43} +11.9273 q^{45} +7.96976 q^{47} +4.29578 q^{49} -1.98435 q^{51} +13.9969 q^{53} +17.4335 q^{55} -2.92624 q^{57} -1.88127 q^{59} -13.0212 q^{61} +9.51706 q^{63} +13.5808 q^{65} -7.34403 q^{67} -0.108506 q^{69} -13.5632 q^{71} -2.49323 q^{73} -5.22742 q^{75} +13.9106 q^{77} +8.07702 q^{79} +7.51347 q^{81} +10.1672 q^{83} -20.3727 q^{85} +2.84313 q^{87} +3.18164 q^{89} +10.8364 q^{91} +0.439559 q^{93} -30.0428 q^{95} +7.00896 q^{97} +11.7201 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q - 10 q^{3} - 12 q^{7} + 47 q^{9}+O(q^{10}) \) 33 * q - 10 * q^3 - 12 * q^7 + 47 * q^9	\( 33 q - 10 q^{3} - 12 q^{7} + 47 q^{9} - 22 q^{11} - 17 q^{13} - 22 q^{15} + 9 q^{17} - 16 q^{19} + 6 q^{21} - 36 q^{23} + 47 q^{25} - 34 q^{27} + 13 q^{29} - 21 q^{31} + 14 q^{33} - 33 q^{35} - 55 q^{37} - 37 q^{39} + 42 q^{41} - 23 q^{43} + 5 q^{45} - 20 q^{47} + 55 q^{49} - 53 q^{51} - 32 q^{53} - 35 q^{55} + 21 q^{57} - 20 q^{59} - 15 q^{61} - 48 q^{63} + 34 q^{65} - 66 q^{67} - 4 q^{69} - 61 q^{71} + 19 q^{73} - 59 q^{75} + 2 q^{77} - 62 q^{79} + 77 q^{81} - 36 q^{83} - 14 q^{85} - 43 q^{87} + 34 q^{89} - 41 q^{91} - 11 q^{93} - 61 q^{95} - 8 q^{97} - 98 q^{99}+O(q^{100}) \) 33 * q - 10 * q^3 - 12 * q^7 + 47 * q^9 - 22 * q^11 - 17 * q^13 - 22 * q^15 + 9 * q^17 - 16 * q^19 + 6 * q^21 - 36 * q^23 + 47 * q^25 - 34 * q^27 + 13 * q^29 - 21 * q^31 + 14 * q^33 - 33 * q^35 - 55 * q^37 - 37 * q^39 + 42 * q^41 - 23 * q^43 + 5 * q^45 - 20 * q^47 + 55 * q^49 - 53 * q^51 - 32 * q^53 - 35 * q^55 + 21 * q^57 - 20 * q^59 - 15 * q^61 - 48 * q^63 + 34 * q^65 - 66 * q^67 - 4 * q^69 - 61 * q^71 + 19 * q^73 - 59 * q^75 + 2 * q^77 - 62 * q^79 + 77 * q^81 - 36 * q^83 - 14 * q^85 - 43 * q^87 + 34 * q^89 - 41 * q^91 - 11 * q^93 - 61 * q^95 - 8 * q^97 - 98 * q^99

Introduction

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