LMFDB - Embedded newform 8048.2.a.u.1.22

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:	$0$
Dimension:	$26$
Twist minimal:	no (minimal twist has level 503)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.22
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)

$f(q)$	$=$	$q+2.39019 q^{3} -3.16539 q^{5} +3.22876 q^{7} +2.71299 q^{9} +O(q^{10})$	$q+2.39019 q^{3} -3.16539 q^{5} +3.22876 q^{7} +2.71299 q^{9} +3.58671 q^{11} -7.07014 q^{13} -7.56588 q^{15} +6.97702 q^{17} +5.21579 q^{19} +7.71734 q^{21} +3.09603 q^{23} +5.01972 q^{25} -0.686000 q^{27} -3.23446 q^{29} +3.98968 q^{31} +8.57290 q^{33} -10.2203 q^{35} +7.10689 q^{37} -16.8990 q^{39} -2.86132 q^{41} -9.31372 q^{43} -8.58769 q^{45} +11.6543 q^{47} +3.42490 q^{49} +16.6764 q^{51} -11.3201 q^{53} -11.3533 q^{55} +12.4667 q^{57} +11.1359 q^{59} -9.12777 q^{61} +8.75961 q^{63} +22.3798 q^{65} +3.40992 q^{67} +7.40009 q^{69} +1.17221 q^{71} +3.17531 q^{73} +11.9981 q^{75} +11.5806 q^{77} -3.67810 q^{79} -9.77865 q^{81} +0.597645 q^{83} -22.0850 q^{85} -7.73097 q^{87} +3.80922 q^{89} -22.8278 q^{91} +9.53607 q^{93} -16.5100 q^{95} +12.6081 q^{97} +9.73071 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9}+O(q^{10}) $ 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9	$ 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9} + 17 q^{11} + 14 q^{13} - 18 q^{15} + 17 q^{17} + 22 q^{19} - 16 q^{21} - 27 q^{23} + 93 q^{25} - 31 q^{27} + 13 q^{29} - 26 q^{31} + 6 q^{33} + 22 q^{35} + 55 q^{37} + 15 q^{39} + 24 q^{41} - 20 q^{43} - 8 q^{45} + 25 q^{47} + 65 q^{49} - 7 q^{51} + 30 q^{53} - 25 q^{55} + 9 q^{57} + 26 q^{59} + 15 q^{61} + 19 q^{63} + 20 q^{65} + 20 q^{67} - 27 q^{69} + 35 q^{71} + 38 q^{73} - 2 q^{75} - 6 q^{77} - 21 q^{79} + 70 q^{81} + 48 q^{83} + 6 q^{85} + 9 q^{87} - 5 q^{89} + 24 q^{91} - 8 q^{93} - 43 q^{95} + 142 q^{97} + 50 q^{99}+O(q^{100}) $ 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9 + 17 * q^11 + 14 * q^13 - 18 * q^15 + 17 * q^17 + 22 * q^19 - 16 * q^21 - 27 * q^23 + 93 * q^25 - 31 * q^27 + 13 * q^29 - 26 * q^31 + 6 * q^33 + 22 * q^35 + 55 * q^37 + 15 * q^39 + 24 * q^41 - 20 * q^43 - 8 * q^45 + 25 * q^47 + 65 * q^49 - 7 * q^51 + 30 * q^53 - 25 * q^55 + 9 * q^57 + 26 * q^59 + 15 * q^61 + 19 * q^63 + 20 * q^65 + 20 * q^67 - 27 * q^69 + 35 * q^71 + 38 * q^73 - 2 * q^75 - 6 * q^77 - 21 * q^79 + 70 * q^81 + 48 * q^83 + 6 * q^85 + 9 * q^87 - 5 * q^89 + 24 * q^91 - 8 * q^93 - 43 * q^95 + 142 * q^97 + 50 * 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
$3$	2.39019	1.37998	0.689988	−	0.723821i	$-0.257616\pi$
$3$	2.39019	1.37998	0.689988	+	0.723821i	$0.257616\pi$
$4$	0	0
$5$	−3.16539	−1.41561	−0.707804	−	0.706409i	$-0.750314\pi$
$5$	−3.16539	−1.41561	−0.707804	+	0.706409i	$0.750314\pi$
$6$	0	0
$7$	3.22876	1.22036	0.610178	−	0.792264i	$-0.291098\pi$
$7$	3.22876	1.22036	0.610178	+	0.792264i	$0.291098\pi$
$8$	0	0
$9$	2.71299	0.904331
$10$	0	0
$11$	3.58671	1.08143	0.540716	−	0.841205i	$-0.318153\pi$
$11$	3.58671	1.08143	0.540716	+	0.841205i	$0.318153\pi$
$12$	0	0
$13$	−7.07014	−1.96091	−0.980453	−	0.196756i	$-0.936959\pi$
$13$	−7.07014	−1.96091	−0.980453	+	0.196756i	$0.936959\pi$
$14$	0	0
$15$	−7.56588	−1.95350
$16$	0	0
$17$	6.97702	1.69218	0.846088	−	0.533043i	$-0.178952\pi$
$17$	6.97702	1.69218	0.846088	+	0.533043i	$0.178952\pi$
$18$	0	0
$19$	5.21579	1.19659	0.598293	−	0.801278i	$-0.295846\pi$
$19$	5.21579	1.19659	0.598293	+	0.801278i	$0.295846\pi$
$20$	0	0
$21$	7.71734	1.68406
$22$	0	0
$23$	3.09603	0.645567	0.322784	−	0.946473i	$-0.395381\pi$
$23$	3.09603	0.645567	0.322784	+	0.946473i	$0.395381\pi$
$24$	0	0
$25$	5.01972	1.00394
$26$	0	0
$27$	−0.686000	−0.132021
$28$	0	0
$29$	−3.23446	−0.600625	−0.300312	−	0.953841i	$-0.597091\pi$
$29$	−3.23446	−0.600625	−0.300312	+	0.953841i	$0.597091\pi$
$30$	0	0
$31$	3.98968	0.716567	0.358284	−	0.933613i	$-0.383362\pi$
$31$	3.98968	0.716567	0.358284	+	0.933613i	$0.383362\pi$
$32$	0	0
$33$	8.57290	1.49235
$34$	0	0
$35$	−10.2203	−1.72755
$36$	0	0
$37$	7.10689	1.16837	0.584183	−	0.811622i	$-0.301415\pi$
$37$	7.10689	1.16837	0.584183	+	0.811622i	$0.301415\pi$
$38$	0	0
$39$	−16.8990	−2.70600
$40$	0	0
$41$	−2.86132	−0.446863	−0.223432	−	0.974720i	$-0.571726\pi$
$41$	−2.86132	−0.446863	−0.223432	+	0.974720i	$0.571726\pi$
$42$	0	0
$43$	−9.31372	−1.42033	−0.710164	−	0.704036i	$-0.751379\pi$
$43$	−9.31372	−1.42033	−0.710164	+	0.704036i	$0.751379\pi$
$44$	0	0
$45$	−8.58769	−1.28018
$46$	0	0
$47$	11.6543	1.69995	0.849976	−	0.526822i	$-0.176617\pi$
$47$	11.6543	1.69995	0.849976	+	0.526822i	$0.176617\pi$
$48$	0	0
$49$	3.42490	0.489271
$50$	0	0
$51$	16.6764	2.33516
$52$	0	0
$53$	−11.3201	−1.55494	−0.777468	−	0.628923i	$-0.783496\pi$
$53$	−11.3201	−1.55494	−0.777468	+	0.628923i	$0.783496\pi$
$54$	0	0
$55$	−11.3533	−1.53088
$56$	0	0
$57$	12.4667	1.65126
$58$	0	0
$59$	11.1359	1.44977	0.724885	−	0.688869i	$-0.241893\pi$
$59$	11.1359	1.44977	0.724885	+	0.688869i	$0.241893\pi$
$60$	0	0
$61$	−9.12777	−1.16869	−0.584345	−	0.811505i	$-0.698649\pi$
$61$	−9.12777	−1.16869	−0.584345	+	0.811505i	$0.698649\pi$
$62$	0	0
$63$	8.75961	1.10361
$64$	0	0
$65$	22.3798	2.77587
$66$	0	0
$67$	3.40992	0.416588	0.208294	−	0.978066i	$-0.433209\pi$
$67$	3.40992	0.416588	0.208294	+	0.978066i	$0.433209\pi$
$68$	0	0
$69$	7.40009	0.890867
$70$	0	0
$71$	1.17221	0.139116	0.0695581	−	0.997578i	$-0.477841\pi$
$71$	1.17221	0.139116	0.0695581	+	0.997578i	$0.477841\pi$
$72$	0	0
$73$	3.17531	0.371642	0.185821	−	0.982584i	$-0.440505\pi$
$73$	3.17531	0.371642	0.185821	+	0.982584i	$0.440505\pi$
$74$	0	0
$75$	11.9981	1.38542
$76$	0	0
$77$	11.5806	1.31973
$78$	0	0
$79$	−3.67810	−0.413819	−0.206909	−	0.978360i	$-0.566341\pi$
$79$	−3.67810	−0.413819	−0.206909	+	0.978360i	$0.566341\pi$
$80$	0	0
$81$	−9.77865	−1.08652
$82$	0	0
$83$	0.597645	0.0656000	0.0328000	−	0.999462i	$-0.489558\pi$
$83$	0.597645	0.0656000	0.0328000	+	0.999462i	$0.489558\pi$
$84$	0	0
$85$	−22.0850	−2.39546
$86$	0	0
$87$	−7.73097	−0.828847
$88$	0	0
$89$	3.80922	0.403776	0.201888	−	0.979409i	$-0.435292\pi$
$89$	3.80922	0.403776	0.201888	+	0.979409i	$0.435292\pi$
$90$	0	0
$91$	−22.8278	−2.39300
$92$	0	0
$93$	9.53607	0.988845
$94$	0	0
$95$	−16.5100	−1.69389
$96$	0	0
$97$	12.6081	1.28016	0.640078	−	0.768310i	$-0.278902\pi$
$97$	12.6081	1.28016	0.640078	+	0.768310i	$0.278902\pi$
$98$	0	0
$99$	9.73071	0.977973
$100$	0	0
$101$	−6.14953	−0.611901	−0.305950	−	0.952047i	$-0.598974\pi$
$101$	−6.14953	−0.611901	−0.305950	+	0.952047i	$0.598974\pi$
$102$	0	0
$103$	0.189729	0.0186945	0.00934727	−	0.999956i	$-0.497025\pi$
$103$	0.189729	0.0186945	0.00934727	+	0.999956i	$0.497025\pi$
$104$	0	0
$105$	−24.4284	−2.38397
$106$	0	0
$107$	−0.836977	−0.0809136	−0.0404568	−	0.999181i	$-0.512881\pi$
$107$	−0.836977	−0.0809136	−0.0404568	+	0.999181i	$0.512881\pi$
$108$	0	0
$109$	−4.41825	−0.423192	−0.211596	−	0.977357i	$-0.567866\pi$
$109$	−4.41825	−0.423192	−0.211596	+	0.977357i	$0.567866\pi$
$110$	0	0
$111$	16.9868	1.61231
$112$	0	0
$113$	4.48584	0.421993	0.210996	−	0.977487i	$-0.432329\pi$
$113$	4.48584	0.421993	0.210996	+	0.977487i	$0.432329\pi$
$114$	0	0
$115$	−9.80016	−0.913870
$116$	0	0
$117$	−19.1813	−1.77331
$118$	0	0
$119$	22.5271	2.06506
$120$	0	0
$121$	1.86446	0.169497
$122$	0	0
$123$	−6.83909	−0.616660
$124$	0	0
$125$	−0.0624343	−0.00558429
$126$	0	0
$127$	−1.65924	−0.147234	−0.0736169	−	0.997287i	$-0.523454\pi$
$127$	−1.65924	−0.147234	−0.0736169	+	0.997287i	$0.523454\pi$
$128$	0	0
$129$	−22.2615	−1.96002
$130$	0	0
$131$	12.3239	1.07675	0.538373	−	0.842706i	$-0.319039\pi$
$131$	12.3239	1.07675	0.538373	+	0.842706i	$0.319039\pi$
$132$	0	0
$133$	16.8406	1.46026
$134$	0	0
$135$	2.17146	0.186890
$136$	0	0
$137$	−8.58468	−0.733438	−0.366719	−	0.930332i	$-0.619519\pi$
$137$	−8.58468	−0.733438	−0.366719	+	0.930332i	$0.619519\pi$
$138$	0	0
$139$	5.88638	0.499277	0.249638	−	0.968339i	$-0.419688\pi$
$139$	5.88638	0.499277	0.249638	+	0.968339i	$0.419688\pi$
$140$	0	0
$141$	27.8559	2.34589
$142$	0	0
$143$	−25.3585	−2.12059
$144$	0	0
$145$	10.2383	0.850249
$146$	0	0
$147$	8.18615	0.675182
$148$	0	0
$149$	10.9494	0.897010	0.448505	−	0.893780i	$-0.351957\pi$
$149$	10.9494	0.897010	0.448505	+	0.893780i	$0.351957\pi$
$150$	0	0
$151$	15.0748	1.22677	0.613385	−	0.789784i	$-0.289808\pi$
$151$	15.0748	1.22677	0.613385	+	0.789784i	$0.289808\pi$
$152$	0	0
$153$	18.9286	1.53029
$154$	0	0
$155$	−12.6289	−1.01438
$156$	0	0
$157$	11.3813	0.908324	0.454162	−	0.890919i	$-0.349939\pi$
$157$	11.3813	0.908324	0.454162	+	0.890919i	$0.349939\pi$
$158$	0	0
$159$	−27.0572	−2.14577
$160$	0	0
$161$	9.99635	0.787822
$162$	0	0
$163$	11.9678	0.937392	0.468696	−	0.883359i	$-0.344724\pi$
$163$	11.9678	0.937392	0.468696	+	0.883359i	$0.344724\pi$
$164$	0	0
$165$	−27.1366	−2.11258
$166$	0	0
$167$	−4.72961	−0.365988	−0.182994	−	0.983114i	$-0.558579\pi$
$167$	−4.72961	−0.365988	−0.182994	+	0.983114i	$0.558579\pi$
$168$	0	0
$169$	36.9869	2.84515
$170$	0	0
$171$	14.1504	1.08211
$172$	0	0
$173$	−5.08381	−0.386515	−0.193257	−	0.981148i	$-0.561905\pi$
$173$	−5.08381	−0.386515	−0.193257	+	0.981148i	$0.561905\pi$
$174$	0	0
$175$	16.2075	1.22517
$176$	0	0
$177$	26.6169	2.00065
$178$	0	0
$179$	1.24986	0.0934190	0.0467095	−	0.998909i	$-0.485126\pi$
$179$	1.24986	0.0934190	0.0467095	+	0.998909i	$0.485126\pi$
$180$	0	0
$181$	21.7700	1.61815	0.809077	−	0.587702i	$-0.199967\pi$
$181$	21.7700	1.61815	0.809077	+	0.587702i	$0.199967\pi$
$182$	0	0
$183$	−21.8171	−1.61276
$184$	0	0
$185$	−22.4961	−1.65395
$186$	0	0
$187$	25.0245	1.82997
$188$	0	0
$189$	−2.21493	−0.161112
$190$	0	0
$191$	13.1029	0.948090	0.474045	−	0.880501i	$-0.342793\pi$
$191$	13.1029	0.948090	0.474045	+	0.880501i	$0.342793\pi$
$192$	0	0
$193$	14.8814	1.07119	0.535594	−	0.844476i	$-0.320088\pi$
$193$	14.8814	1.07119	0.535594	+	0.844476i	$0.320088\pi$
$194$	0	0
$195$	53.4919	3.83063
$196$	0	0
$197$	−4.97496	−0.354451	−0.177226	−	0.984170i	$-0.556712\pi$
$197$	−4.97496	−0.354451	−0.177226	+	0.984170i	$0.556712\pi$
$198$	0	0
$199$	24.7765	1.75636	0.878179	−	0.478333i	$-0.158759\pi$
$199$	24.7765	1.75636	0.878179	+	0.478333i	$0.158759\pi$
$200$	0	0
$201$	8.15035	0.574881
$202$	0	0
$203$	−10.4433	−0.732976
$204$	0	0
$205$	9.05721	0.632583
$206$	0	0
$207$	8.39951	0.583807
$208$	0	0
$209$	18.7075	1.29403
$210$	0	0
$211$	−24.9775	−1.71952	−0.859759	−	0.510699i	$-0.829386\pi$
$211$	−24.9775	−1.71952	−0.859759	+	0.510699i	$0.829386\pi$
$212$	0	0
$213$	2.80181	0.191977
$214$	0	0
$215$	29.4816	2.01063
$216$	0	0
$217$	12.8817	0.874468
$218$	0	0
$219$	7.58959	0.512857
$220$	0	0
$221$	−49.3285	−3.31820
$222$	0	0
$223$	5.02422	0.336447	0.168223	−	0.985749i	$-0.446197\pi$
$223$	5.02422	0.336447	0.168223	+	0.985749i	$0.446197\pi$
$224$	0	0
$225$	13.6185	0.907898
$226$	0	0
$227$	3.38467	0.224648	0.112324	−	0.993672i	$-0.464170\pi$
$227$	3.38467	0.224648	0.112324	+	0.993672i	$0.464170\pi$
$228$	0	0
$229$	8.70517	0.575254	0.287627	−	0.957743i	$-0.407134\pi$
$229$	8.70517	0.575254	0.287627	+	0.957743i	$0.407134\pi$
$230$	0	0
$231$	27.6798	1.82120
$232$	0	0
$233$	−19.8019	−1.29726	−0.648632	−	0.761102i	$-0.724659\pi$
$233$	−19.8019	−1.29726	−0.648632	+	0.761102i	$0.724659\pi$
$234$	0	0
$235$	−36.8904	−2.40646
$236$	0	0
$237$	−8.79135	−0.571060
$238$	0	0
$239$	−23.5458	−1.52305	−0.761527	−	0.648134i	$-0.775550\pi$
$239$	−23.5458	−1.52305	−0.761527	+	0.648134i	$0.775550\pi$
$240$	0	0
$241$	−7.88272	−0.507771	−0.253885	−	0.967234i	$-0.581709\pi$
$241$	−7.88272	−0.507771	−0.253885	+	0.967234i	$0.581709\pi$
$242$	0	0
$243$	−21.3148	−1.36734
$244$	0	0
$245$	−10.8412	−0.692616
$246$	0	0
$247$	−36.8764	−2.34639
$248$	0	0
$249$	1.42848	0.0905264
$250$	0	0
$251$	1.26104	0.0795964	0.0397982	−	0.999208i	$-0.487328\pi$
$251$	1.26104	0.0795964	0.0397982	+	0.999208i	$0.487328\pi$
$252$	0	0
$253$	11.1046	0.698138
$254$	0	0
$255$	−52.7873	−3.30567
$256$	0	0
$257$	1.67299	0.104358	0.0521792	−	0.998638i	$-0.483383\pi$
$257$	1.67299	0.104358	0.0521792	+	0.998638i	$0.483383\pi$
$258$	0	0
$259$	22.9464	1.42582
$260$	0	0
$261$	−8.77507	−0.543163
$262$	0	0
$263$	3.12527	0.192712	0.0963562	−	0.995347i	$-0.469281\pi$
$263$	3.12527	0.192712	0.0963562	+	0.995347i	$0.469281\pi$
$264$	0	0
$265$	35.8326	2.20118
$266$	0	0
$267$	9.10474	0.557201
$268$	0	0
$269$	−12.4312	−0.757944	−0.378972	−	0.925408i	$-0.623722\pi$
$269$	−12.4312	−0.757944	−0.378972	+	0.925408i	$0.623722\pi$
$270$	0	0
$271$	5.81546	0.353264	0.176632	−	0.984277i	$-0.443480\pi$
$271$	5.81546	0.353264	0.176632	+	0.984277i	$0.443480\pi$
$272$	0	0
$273$	−54.5627	−3.30229
$274$	0	0
$275$	18.0043	1.08570
$276$	0	0
$277$	20.1014	1.20777	0.603887	−	0.797070i	$-0.293618\pi$
$277$	20.1014	1.20777	0.603887	+	0.797070i	$0.293618\pi$
$278$	0	0
$279$	10.8240	0.648014
$280$	0	0
$281$	7.21011	0.430119	0.215060	−	0.976601i	$-0.431005\pi$
$281$	7.21011	0.430119	0.215060	+	0.976601i	$0.431005\pi$
$282$	0	0
$283$	−12.1481	−0.722129	−0.361064	−	0.932541i	$-0.617587\pi$
$283$	−12.1481	−0.722129	−0.361064	+	0.932541i	$0.617587\pi$
$284$	0	0
$285$	−39.4621	−2.33753
$286$	0	0
$287$	−9.23852	−0.545333
$288$	0	0
$289$	31.6788	1.86346
$290$	0	0
$291$	30.1357	1.76658
$292$	0	0
$293$	−30.6600	−1.79118	−0.895589	−	0.444883i	$-0.853245\pi$
$293$	−30.6600	−1.79118	−0.895589	+	0.444883i	$0.853245\pi$
$294$	0	0
$295$	−35.2495	−2.05231
$296$	0	0
$297$	−2.46048	−0.142772
$298$	0	0
$299$	−21.8894	−1.26590
$300$	0	0
$301$	−30.0718	−1.73331
$302$	0	0
$303$	−14.6985	−0.844408
$304$	0	0
$305$	28.8930	1.65441
$306$	0	0
$307$	20.4325	1.16614	0.583071	−	0.812421i	$-0.301851\pi$
$307$	20.4325	1.16614	0.583071	+	0.812421i	$0.301851\pi$
$308$	0	0
$309$	0.453487	0.0257980
$310$	0	0
$311$	10.4987	0.595328	0.297664	−	0.954671i	$-0.403792\pi$
$311$	10.4987	0.595328	0.297664	+	0.954671i	$0.403792\pi$
$312$	0	0
$313$	−8.89190	−0.502600	−0.251300	−	0.967909i	$-0.580858\pi$
$313$	−8.89190	−0.502600	−0.251300	+	0.967909i	$0.580858\pi$
$314$	0	0
$315$	−27.7276	−1.56227
$316$	0	0
$317$	28.6880	1.61128	0.805640	−	0.592405i	$-0.201822\pi$
$317$	28.6880	1.61128	0.805640	+	0.592405i	$0.201822\pi$
$318$	0	0
$319$	−11.6011	−0.649535
$320$	0	0
$321$	−2.00053	−0.111659
$322$	0	0
$323$	36.3907	2.02483
$324$	0	0
$325$	−35.4902	−1.96864
$326$	0	0
$327$	−10.5604	−0.583994
$328$	0	0
$329$	37.6289	2.07455
$330$	0	0
$331$	25.8590	1.42134	0.710670	−	0.703526i	$-0.248392\pi$
$331$	25.8590	1.42134	0.710670	+	0.703526i	$0.248392\pi$
$332$	0	0
$333$	19.2809	1.05659
$334$	0	0
$335$	−10.7937	−0.589725
$336$	0	0
$337$	−9.87578	−0.537968	−0.268984	−	0.963145i	$-0.586688\pi$
$337$	−9.87578	−0.537968	−0.268984	+	0.963145i	$0.586688\pi$
$338$	0	0
$339$	10.7220	0.582339
$340$	0	0
$341$	14.3098	0.774919
$342$	0	0
$343$	−11.5432	−0.623272
$344$	0	0
$345$	−23.4242	−1.26112
$346$	0	0
$347$	−11.8982	−0.638732	−0.319366	−	0.947632i	$-0.603470\pi$
$347$	−11.8982	−0.638732	−0.319366	+	0.947632i	$0.603470\pi$
$348$	0	0
$349$	−28.2833	−1.51397	−0.756985	−	0.653432i	$-0.773329\pi$
$349$	−28.2833	−1.51397	−0.756985	+	0.653432i	$0.773329\pi$
$350$	0	0
$351$	4.85012	0.258880
$352$	0	0
$353$	−23.9596	−1.27524	−0.637619	−	0.770352i	$-0.720081\pi$
$353$	−23.9596	−1.27524	−0.637619	+	0.770352i	$0.720081\pi$
$354$	0	0
$355$	−3.71052	−0.196934
$356$	0	0
$357$	53.8441	2.84973
$358$	0	0
$359$	4.05462	0.213995	0.106997	−	0.994259i	$-0.465876\pi$
$359$	4.05462	0.213995	0.106997	+	0.994259i	$0.465876\pi$
$360$	0	0
$361$	8.20450	0.431816
$362$	0	0
$363$	4.45642	0.233901
$364$	0	0
$365$	−10.0511	−0.526100
$366$	0	0
$367$	26.1108	1.36297	0.681486	−	0.731831i	$-0.261334\pi$
$367$	26.1108	1.36297	0.681486	+	0.731831i	$0.261334\pi$
$368$	0	0
$369$	−7.76275	−0.404112
$370$	0	0
$371$	−36.5499	−1.89758
$372$	0	0
$373$	−22.5682	−1.16854	−0.584269	−	0.811560i	$-0.698619\pi$
$373$	−22.5682	−1.16854	−0.584269	+	0.811560i	$0.698619\pi$
$374$	0	0
$375$	−0.149230	−0.00770618
$376$	0	0
$377$	22.8681	1.17777
$378$	0	0
$379$	4.94459	0.253986	0.126993	−	0.991904i	$-0.459467\pi$
$379$	4.94459	0.253986	0.126993	+	0.991904i	$0.459467\pi$
$380$	0	0
$381$	−3.96589	−0.203179
$382$	0	0
$383$	29.5314	1.50899	0.754493	−	0.656308i	$-0.227883\pi$
$383$	29.5314	1.50899	0.754493	+	0.656308i	$0.227883\pi$
$384$	0	0
$385$	−36.6572	−1.86823
$386$	0	0
$387$	−25.2681	−1.28445
$388$	0	0
$389$	10.3624	0.525393	0.262697	−	0.964878i	$-0.415388\pi$
$389$	10.3624	0.525393	0.262697	+	0.964878i	$0.415388\pi$
$390$	0	0
$391$	21.6011	1.09241
$392$	0	0
$393$	29.4565	1.48588
$394$	0	0
$395$	11.6426	0.585805
$396$	0	0
$397$	33.3455	1.67356	0.836781	−	0.547538i	$-0.184435\pi$
$397$	33.3455	1.67356	0.836781	+	0.547538i	$0.184435\pi$
$398$	0	0
$399$	40.2521	2.01512
$400$	0	0
$401$	−14.0781	−0.703026	−0.351513	−	0.936183i	$-0.614333\pi$
$401$	−14.0781	−0.703026	−0.351513	+	0.936183i	$0.614333\pi$
$402$	0	0
$403$	−28.2076	−1.40512
$404$	0	0
$405$	30.9533	1.53808
$406$	0	0
$407$	25.4903	1.26351
$408$	0	0
$409$	−30.8734	−1.52659	−0.763297	−	0.646048i	$-0.776420\pi$
$409$	−30.8734	−1.52659	−0.763297	+	0.646048i	$0.776420\pi$
$410$	0	0
$411$	−20.5190	−1.01213
$412$	0	0
$413$	35.9552	1.76924
$414$	0	0
$415$	−1.89178	−0.0928639
$416$	0	0
$417$	14.0696	0.688989
$418$	0	0
$419$	−17.3641	−0.848292	−0.424146	−	0.905594i	$-0.639426\pi$
$419$	−17.3641	−0.848292	−0.424146	+	0.905594i	$0.639426\pi$
$420$	0	0
$421$	22.5477	1.09891	0.549454	−	0.835524i	$-0.314836\pi$
$421$	22.5477	1.09891	0.549454	+	0.835524i	$0.314836\pi$
$422$	0	0
$423$	31.6180	1.53732
$424$	0	0
$425$	35.0227	1.69885
$426$	0	0
$427$	−29.4714	−1.42622
$428$	0	0
$429$	−60.6116	−2.92636
$430$	0	0
$431$	22.9956	1.10766	0.553829	−	0.832631i	$-0.313166\pi$
$431$	22.9956	1.10766	0.553829	+	0.832631i	$0.313166\pi$
$432$	0	0
$433$	13.7593	0.661231	0.330615	−	0.943766i	$-0.392744\pi$
$433$	13.7593	0.661231	0.330615	+	0.943766i	$0.392744\pi$
$434$	0	0
$435$	24.4716	1.17332
$436$	0	0
$437$	16.1483	0.772476
$438$	0	0
$439$	−11.3452	−0.541476	−0.270738	−	0.962653i	$-0.587268\pi$
$439$	−11.3452	−0.541476	−0.270738	+	0.962653i	$0.587268\pi$
$440$	0	0
$441$	9.29172	0.442463
$442$	0	0
$443$	3.46821	0.164779	0.0823897	−	0.996600i	$-0.473745\pi$
$443$	3.46821	0.164779	0.0823897	+	0.996600i	$0.473745\pi$
$444$	0	0
$445$	−12.0577	−0.571589
$446$	0	0
$447$	26.1711	1.23785
$448$	0	0
$449$	−36.0913	−1.70325	−0.851626	−	0.524150i	$-0.824383\pi$
$449$	−36.0913	−1.70325	−0.851626	+	0.524150i	$0.824383\pi$
$450$	0	0
$451$	−10.2627	−0.483253
$452$	0	0
$453$	36.0316	1.69291
$454$	0	0
$455$	72.2590	3.38755
$456$	0	0
$457$	20.2170	0.945712	0.472856	−	0.881140i	$-0.343223\pi$
$457$	20.2170	0.945712	0.472856	+	0.881140i	$0.343223\pi$
$458$	0	0
$459$	−4.78624	−0.223402
$460$	0	0
$461$	−38.2129	−1.77975	−0.889876	−	0.456203i	$-0.849209\pi$
$461$	−38.2129	−1.77975	−0.889876	+	0.456203i	$0.849209\pi$
$462$	0	0
$463$	−23.0399	−1.07075	−0.535377	−	0.844613i	$-0.679830\pi$
$463$	−23.0399	−1.07075	−0.535377	+	0.844613i	$0.679830\pi$
$464$	0	0
$465$	−30.1854	−1.39982
$466$	0	0
$467$	−25.3907	−1.17494	−0.587471	−	0.809245i	$-0.699876\pi$
$467$	−25.3907	−1.17494	−0.587471	+	0.809245i	$0.699876\pi$
$468$	0	0
$469$	11.0098	0.508386
$470$	0	0
$471$	27.2034	1.25346
$472$	0	0
$473$	−33.4056	−1.53599
$474$	0	0
$475$	26.1818	1.20131
$476$	0	0
$477$	−30.7114	−1.40618
$478$	0	0
$479$	−29.2413	−1.33607	−0.668034	−	0.744131i	$-0.732864\pi$
$479$	−29.2413	−1.33607	−0.668034	+	0.744131i	$0.732864\pi$
$480$	0	0
$481$	−50.2467	−2.29105
$482$	0	0
$483$	23.8931	1.08718
$484$	0	0
$485$	−39.9095	−1.81220
$486$	0	0
$487$	18.6135	0.843457	0.421729	−	0.906722i	$-0.361423\pi$
$487$	18.6135	0.843457	0.421729	+	0.906722i	$0.361423\pi$
$488$	0	0
$489$	28.6053	1.29358
$490$	0	0
$491$	13.3830	0.603964	0.301982	−	0.953314i	$-0.402352\pi$
$491$	13.3830	0.603964	0.301982	+	0.953314i	$0.402352\pi$
$492$	0	0
$493$	−22.5669	−1.01636
$494$	0	0
$495$	−30.8015	−1.38443
$496$	0	0
$497$	3.78480	0.169772
$498$	0	0
$499$	29.2568	1.30971	0.654857	−	0.755753i	$-0.272729\pi$
$499$	29.2568	1.30971	0.654857	+	0.755753i	$0.272729\pi$
$500$	0	0
$501$	−11.3047	−0.505055
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	19.4657	0.866211
$506$	0	0
$507$	88.4057	3.92623
$508$	0	0
$509$	−5.62413	−0.249285	−0.124643	−	0.992202i	$-0.539778\pi$
$509$	−5.62413	−0.249285	−0.124643	+	0.992202i	$0.539778\pi$
$510$	0	0
$511$	10.2523	0.453536
$512$	0	0
$513$	−3.57803	−0.157974
$514$	0	0
$515$	−0.600567	−0.0264641
$516$	0	0
$517$	41.8005	1.83838
$518$	0	0
$519$	−12.1512	−0.533381
$520$	0	0
$521$	−7.91529	−0.346775	−0.173388	−	0.984854i	$-0.555471\pi$
$521$	−7.91529	−0.346775	−0.173388	+	0.984854i	$0.555471\pi$
$522$	0	0
$523$	6.77268	0.296148	0.148074	−	0.988976i	$-0.452693\pi$
$523$	6.77268	0.296148	0.148074	+	0.988976i	$0.452693\pi$
$524$	0	0
$525$	38.7389	1.69071
$526$	0	0
$527$	27.8361	1.21256
$528$	0	0
$529$	−13.4146	−0.583243
$530$	0	0
$531$	30.2116	1.31107
$532$	0	0
$533$	20.2300	0.876257
$534$	0	0
$535$	2.64936	0.114542
$536$	0	0
$537$	2.98740	0.128916
$538$	0	0
$539$	12.2841	0.529114
$540$	0	0
$541$	−15.4668	−0.664969	−0.332485	−	0.943109i	$-0.607887\pi$
$541$	−15.4668	−0.664969	−0.332485	+	0.943109i	$0.607887\pi$
$542$	0	0
$543$	52.0345	2.23301
$544$	0	0
$545$	13.9855	0.599074
$546$	0	0
$547$	−1.21528	−0.0519618	−0.0259809	−	0.999662i	$-0.508271\pi$
$547$	−1.21528	−0.0519618	−0.0259809	+	0.999662i	$0.508271\pi$
$548$	0	0
$549$	−24.7636	−1.05688
$550$	0	0
$551$	−16.8703	−0.718698
$552$	0	0
$553$	−11.8757	−0.505007
$554$	0	0
$555$	−53.7699	−2.28241
$556$	0	0
$557$	−15.0015	−0.635636	−0.317818	−	0.948152i	$-0.602950\pi$
$557$	−15.0015	−0.635636	−0.317818	+	0.948152i	$0.602950\pi$
$558$	0	0
$559$	65.8493	2.78513
$560$	0	0
$561$	59.8133	2.52532
$562$	0	0
$563$	5.32702	0.224507	0.112254	−	0.993680i	$-0.464193\pi$
$563$	5.32702	0.224507	0.112254	+	0.993680i	$0.464193\pi$
$564$	0	0
$565$	−14.1995	−0.597376
$566$	0	0
$567$	−31.5729	−1.32594
$568$	0	0
$569$	8.50166	0.356408	0.178204	−	0.983994i	$-0.442971\pi$
$569$	8.50166	0.356408	0.178204	+	0.983994i	$0.442971\pi$
$570$	0	0
$571$	−26.1366	−1.09378	−0.546890	−	0.837204i	$-0.684189\pi$
$571$	−26.1366	−1.09378	−0.546890	+	0.837204i	$0.684189\pi$
$572$	0	0
$573$	31.3183	1.30834
$574$	0	0
$575$	15.5412	0.648114
$576$	0	0
$577$	−16.6489	−0.693101	−0.346550	−	0.938031i	$-0.612647\pi$
$577$	−16.6489	−0.693101	−0.346550	+	0.938031i	$0.612647\pi$
$578$	0	0
$579$	35.5694	1.47821
$580$	0	0
$581$	1.92965	0.0800555
$582$	0	0
$583$	−40.6019	−1.68156
$584$	0	0
$585$	60.7162	2.51031
$586$	0	0
$587$	38.7592	1.59976	0.799881	−	0.600158i	$-0.204896\pi$
$587$	38.7592	1.59976	0.799881	+	0.600158i	$0.204896\pi$
$588$	0	0
$589$	20.8093	0.857434
$590$	0	0
$591$	−11.8911	−0.489134
$592$	0	0
$593$	3.37913	0.138764	0.0693820	−	0.997590i	$-0.477897\pi$
$593$	3.37913	0.138764	0.0693820	+	0.997590i	$0.477897\pi$
$594$	0	0
$595$	−71.3073	−2.92331
$596$	0	0
$597$	59.2204	2.42373
$598$	0	0
$599$	−21.1088	−0.862482	−0.431241	−	0.902237i	$-0.641924\pi$
$599$	−21.1088	−0.862482	−0.431241	+	0.902237i	$0.641924\pi$
$600$	0	0
$601$	21.7704	0.888034	0.444017	−	0.896018i	$-0.353553\pi$
$601$	21.7704	0.888034	0.444017	+	0.896018i	$0.353553\pi$
$602$	0	0
$603$	9.25109	0.376733
$604$	0	0
$605$	−5.90176	−0.239941
$606$	0	0
$607$	−29.0514	−1.17916	−0.589580	−	0.807710i	$-0.700707\pi$
$607$	−29.0514	−1.17916	−0.589580	+	0.807710i	$0.700707\pi$
$608$	0	0
$609$	−24.9615	−1.01149
$610$	0	0
$611$	−82.3974	−3.33344
$612$	0	0
$613$	0.642478	0.0259495	0.0129747	−	0.999916i	$-0.495870\pi$
$613$	0.642478	0.0259495	0.0129747	+	0.999916i	$0.495870\pi$
$614$	0	0
$615$	21.6484	0.872949
$616$	0	0
$617$	10.5550	0.424928	0.212464	−	0.977169i	$-0.431851\pi$
$617$	10.5550	0.424928	0.212464	+	0.977169i	$0.431851\pi$
$618$	0	0
$619$	8.89869	0.357669	0.178834	−	0.983879i	$-0.442767\pi$
$619$	8.89869	0.357669	0.178834	+	0.983879i	$0.442767\pi$
$620$	0	0
$621$	−2.12388	−0.0852283
$622$	0	0
$623$	12.2991	0.492751
$624$	0	0
$625$	−24.9010	−0.996040
$626$	0	0
$627$	44.7145	1.78572
$628$	0	0
$629$	49.5849	1.97708
$630$	0	0
$631$	−5.97191	−0.237738	−0.118869	−	0.992910i	$-0.537927\pi$
$631$	−5.97191	−0.237738	−0.118869	+	0.992910i	$0.537927\pi$
$632$	0	0
$633$	−59.7008	−2.37289
$634$	0	0
$635$	5.25215	0.208425
$636$	0	0
$637$	−24.2145	−0.959414
$638$	0	0
$639$	3.18021	0.125807
$640$	0	0
$641$	−2.58488	−0.102097	−0.0510484	−	0.998696i	$-0.516256\pi$
$641$	−2.58488	−0.102097	−0.0510484	+	0.998696i	$0.516256\pi$
$642$	0	0
$643$	−19.2473	−0.759040	−0.379520	−	0.925183i	$-0.623911\pi$
$643$	−19.2473	−0.759040	−0.379520	+	0.925183i	$0.623911\pi$
$644$	0	0
$645$	70.4665	2.77462
$646$	0	0
$647$	−37.5360	−1.47569	−0.737845	−	0.674970i	$-0.764157\pi$
$647$	−37.5360	−1.47569	−0.737845	+	0.674970i	$0.764157\pi$
$648$	0	0
$649$	39.9412	1.56783
$650$	0	0
$651$	30.7897	1.20674
$652$	0	0
$653$	−21.4495	−0.839382	−0.419691	−	0.907667i	$-0.637862\pi$
$653$	−21.4495	−0.839382	−0.419691	+	0.907667i	$0.637862\pi$
$654$	0	0
$655$	−39.0101	−1.52425
$656$	0	0
$657$	8.61461	0.336088
$658$	0	0
$659$	−22.5061	−0.876715	−0.438357	−	0.898801i	$-0.644440\pi$
$659$	−22.5061	−0.876715	−0.438357	+	0.898801i	$0.644440\pi$
$660$	0	0
$661$	19.3775	0.753699	0.376850	−	0.926274i	$-0.377007\pi$
$661$	19.3775	0.753699	0.376850	+	0.926274i	$0.377007\pi$
$662$	0	0
$663$	−117.904	−4.57903
$664$	0	0
$665$	−53.3070	−2.06716
$666$	0	0
$667$	−10.0140	−0.387744
$668$	0	0
$669$	12.0088	0.464288
$670$	0	0
$671$	−32.7386	−1.26386
$672$	0	0
$673$	6.70805	0.258576	0.129288	−	0.991607i	$-0.458731\pi$
$673$	6.70805	0.258576	0.129288	+	0.991607i	$0.458731\pi$
$674$	0	0
$675$	−3.44353	−0.132542
$676$	0	0
$677$	−13.1924	−0.507027	−0.253513	−	0.967332i	$-0.581586\pi$
$677$	−13.1924	−0.507027	−0.253513	+	0.967332i	$0.581586\pi$
$678$	0	0
$679$	40.7085	1.56225
$680$	0	0
$681$	8.08999	0.310009
$682$	0	0
$683$	12.7404	0.487496	0.243748	−	0.969839i	$-0.421623\pi$
$683$	12.7404	0.487496	0.243748	+	0.969839i	$0.421623\pi$
$684$	0	0
$685$	27.1739	1.03826
$686$	0	0
$687$	20.8070	0.793835
$688$	0	0
$689$	80.0348	3.04908
$690$	0	0
$691$	29.2318	1.11203	0.556015	−	0.831172i	$-0.312330\pi$
$691$	29.2318	1.11203	0.556015	+	0.831172i	$0.312330\pi$
$692$	0	0
$693$	31.4181	1.19348
$694$	0	0
$695$	−18.6327	−0.706780
$696$	0	0
$697$	−19.9635	−0.756172
$698$	0	0
$699$	−47.3302	−1.79019
$700$	0	0
$701$	−32.5774	−1.23043	−0.615216	−	0.788359i	$-0.710931\pi$
$701$	−32.5774	−1.23043	−0.615216	+	0.788359i	$0.710931\pi$
$702$	0	0
$703$	37.0681	1.39805
$704$	0	0
$705$	−88.1749	−3.32086
$706$	0	0
$707$	−19.8553	−0.746737
$708$	0	0
$709$	0.0854425	0.00320886	0.00160443	−	0.999999i	$-0.499489\pi$
$709$	0.0854425	0.00320886	0.00160443	+	0.999999i	$0.499489\pi$
$710$	0	0
$711$	−9.97867	−0.374229
$712$	0	0
$713$	12.3522	0.462592
$714$	0	0
$715$	80.2698	3.00192
$716$	0	0
$717$	−56.2789	−2.10178
$718$	0	0
$719$	−4.75131	−0.177194	−0.0885969	−	0.996068i	$-0.528238\pi$
$719$	−4.75131	−0.177194	−0.0885969	+	0.996068i	$0.528238\pi$
$720$	0	0
$721$	0.612589	0.0228140
$722$	0	0
$723$	−18.8412	−0.700711
$724$	0	0
$725$	−16.2361	−0.602994
$726$	0	0
$727$	9.75499	0.361792	0.180896	−	0.983502i	$-0.442100\pi$
$727$	9.75499	0.361792	0.180896	+	0.983502i	$0.442100\pi$
$728$	0	0
$729$	−21.6104	−0.800385
$730$	0	0
$731$	−64.9820	−2.40345
$732$	0	0
$733$	11.7959	0.435693	0.217846	−	0.975983i	$-0.430097\pi$
$733$	11.7959	0.435693	0.217846	+	0.975983i	$0.430097\pi$
$734$	0	0
$735$	−25.9124	−0.955793
$736$	0	0
$737$	12.2304	0.450512
$738$	0	0
$739$	−5.31136	−0.195382	−0.0976908	−	0.995217i	$-0.531146\pi$
$739$	−5.31136	−0.195382	−0.0976908	+	0.995217i	$0.531146\pi$
$740$	0	0
$741$	−88.1415	−3.23796
$742$	0	0
$743$	14.5358	0.533268	0.266634	−	0.963798i	$-0.414088\pi$
$743$	14.5358	0.533268	0.266634	+	0.963798i	$0.414088\pi$
$744$	0	0
$745$	−34.6592	−1.26981
$746$	0	0
$747$	1.62141	0.0593242
$748$	0	0
$749$	−2.70240	−0.0987435
$750$	0	0
$751$	−7.93562	−0.289575	−0.144787	−	0.989463i	$-0.546250\pi$
$751$	−7.93562	−0.289575	−0.144787	+	0.989463i	$0.546250\pi$
$752$	0	0
$753$	3.01413	0.109841
$754$	0	0
$755$	−47.7177	−1.73662
$756$	0	0
$757$	42.8394	1.55702	0.778512	−	0.627630i	$-0.215975\pi$
$757$	42.8394	1.55702	0.778512	+	0.627630i	$0.215975\pi$
$758$	0	0
$759$	26.5420	0.963412
$760$	0	0
$761$	−28.9585	−1.04975	−0.524873	−	0.851181i	$-0.675887\pi$
$761$	−28.9585	−1.04975	−0.524873	+	0.851181i	$0.675887\pi$
$762$	0	0
$763$	−14.2655	−0.516445
$764$	0	0
$765$	−59.9165	−2.16629
$766$	0	0
$767$	−78.7324	−2.84286
$768$	0	0
$769$	12.2687	0.442422	0.221211	−	0.975226i	$-0.428999\pi$
$769$	12.2687	0.442422	0.221211	+	0.975226i	$0.428999\pi$
$770$	0	0
$771$	3.99876	0.144012
$772$	0	0
$773$	51.4816	1.85166	0.925832	−	0.377936i	$-0.123366\pi$
$773$	51.4816	1.85166	0.925832	+	0.377936i	$0.123366\pi$
$774$	0	0
$775$	20.0271	0.719394
$776$	0	0
$777$	54.8463	1.96760
$778$	0	0
$779$	−14.9241	−0.534710
$780$	0	0
$781$	4.20439	0.150445
$782$	0	0
$783$	2.21884	0.0792949
$784$	0	0
$785$	−36.0262	−1.28583
$786$	0	0
$787$	−17.8944	−0.637865	−0.318933	−	0.947777i	$-0.603324\pi$
$787$	−17.8944	−0.637865	−0.318933	+	0.947777i	$0.603324\pi$
$788$	0	0
$789$	7.46998	0.265938
$790$	0	0
$791$	14.4837	0.514982
$792$	0	0
$793$	64.5346	2.29169
$794$	0	0
$795$	85.6466	3.03757
$796$	0	0
$797$	−27.0833	−0.959339	−0.479670	−	0.877449i	$-0.659243\pi$
$797$	−27.0833	−0.959339	−0.479670	+	0.877449i	$0.659243\pi$
$798$	0	0
$799$	81.3122	2.87662
$800$	0	0
$801$	10.3344	0.365147
$802$	0	0
$803$	11.3889	0.401906
$804$	0	0
$805$	−31.6424	−1.11525
$806$	0	0
$807$	−29.7129	−1.04594
$808$	0	0
$809$	10.4944	0.368962	0.184481	−	0.982836i	$-0.440940\pi$
$809$	10.4944	0.368962	0.184481	+	0.982836i	$0.440940\pi$
$810$	0	0
$811$	−16.7434	−0.587939	−0.293970	−	0.955815i	$-0.594976\pi$
$811$	−16.7434	−0.587939	−0.293970	+	0.955815i	$0.594976\pi$
$812$	0	0
$813$	13.9000	0.487496
$814$	0	0
$815$	−37.8829	−1.32698
$816$	0	0
$817$	−48.5784	−1.69954
$818$	0	0
$819$	−61.9317	−2.16407
$820$	0	0
$821$	34.6673	1.20990	0.604949	−	0.796264i	$-0.293194\pi$
$821$	34.6673	1.20990	0.604949	+	0.796264i	$0.293194\pi$
$822$	0	0
$823$	−52.6783	−1.83625	−0.918125	−	0.396291i	$-0.870297\pi$
$823$	−52.6783	−1.83625	−0.918125	+	0.396291i	$0.870297\pi$
$824$	0	0
$825$	43.0336	1.49824
$826$	0	0
$827$	−0.0394049	−0.00137024	−0.000685121	−	1.00000i	$-0.500218\pi$
$827$	−0.0394049	−0.00137024	−0.000685121		1.00000i	$0.500218\pi$
$828$	0	0
$829$	−21.4503	−0.745001	−0.372501	−	0.928032i	$-0.621500\pi$
$829$	−21.4503	−0.745001	−0.372501	+	0.928032i	$0.621500\pi$
$830$	0	0
$831$	48.0460	1.66670
$832$	0	0
$833$	23.8956	0.827933
$834$	0	0
$835$	14.9711	0.518096
$836$	0	0
$837$	−2.73692	−0.0946017
$838$	0	0
$839$	11.6878	0.403509	0.201755	−	0.979436i	$-0.435336\pi$
$839$	11.6878	0.403509	0.201755	+	0.979436i	$0.435336\pi$
$840$	0	0
$841$	−18.5383	−0.639250
$842$	0	0
$843$	17.2335	0.593554
$844$	0	0
$845$	−117.078	−4.02761
$846$	0	0
$847$	6.01991	0.206847
$848$	0	0
$849$	−29.0362	−0.996519
$850$	0	0
$851$	22.0032	0.754258
$852$	0	0
$853$	−1.86247	−0.0637697	−0.0318849	−	0.999492i	$-0.510151\pi$
$853$	−1.86247	−0.0637697	−0.0318849	+	0.999492i	$0.510151\pi$
$854$	0	0
$855$	−44.7916	−1.53184
$856$	0	0
$857$	20.2321	0.691116	0.345558	−	0.938397i	$-0.387690\pi$
$857$	20.2321	0.691116	0.345558	+	0.938397i	$0.387690\pi$
$858$	0	0
$859$	−8.90924	−0.303979	−0.151990	−	0.988382i	$-0.548568\pi$
$859$	−8.90924	−0.303979	−0.151990	+	0.988382i	$0.548568\pi$
$860$	0	0
$861$	−22.0818	−0.752546
$862$	0	0
$863$	−11.4265	−0.388964	−0.194482	−	0.980906i	$-0.562303\pi$
$863$	−11.4265	−0.388964	−0.194482	+	0.980906i	$0.562303\pi$
$864$	0	0
$865$	16.0923	0.547153
$866$	0	0
$867$	75.7183	2.57153
$868$	0	0
$869$	−13.1923	−0.447517
$870$	0	0
$871$	−24.1086	−0.816890
$872$	0	0
$873$	34.2056	1.15768
$874$	0	0
$875$	−0.201585	−0.00681483
$876$	0	0
$877$	14.8559	0.501648	0.250824	−	0.968033i	$-0.419298\pi$
$877$	14.8559	0.501648	0.250824	+	0.968033i	$0.419298\pi$
$878$	0	0
$879$	−73.2832	−2.47178
$880$	0	0
$881$	−17.0556	−0.574616	−0.287308	−	0.957838i	$-0.592760\pi$
$881$	−17.0556	−0.574616	−0.287308	+	0.957838i	$0.592760\pi$
$882$	0	0
$883$	−26.0563	−0.876866	−0.438433	−	0.898764i	$-0.644466\pi$
$883$	−26.0563	−0.876866	−0.438433	+	0.898764i	$0.644466\pi$
$884$	0	0
$885$	−84.2529	−2.83213
$886$	0	0
$887$	−11.4064	−0.382988	−0.191494	−	0.981494i	$-0.561333\pi$
$887$	−11.4064	−0.382988	−0.191494	+	0.981494i	$0.561333\pi$
$888$	0	0
$889$	−5.35729	−0.179678
$890$	0	0
$891$	−35.0731	−1.17499
$892$	0	0
$893$	60.7863	2.03414
$894$	0	0
$895$	−3.95630	−0.132245
$896$	0	0
$897$	−52.3197	−1.74690
$898$	0	0
$899$	−12.9045	−0.430388
$900$	0	0
$901$	−78.9806	−2.63123
$902$	0	0
$903$	−71.8772	−2.39192
$904$	0	0
$905$	−68.9108	−2.29067
$906$	0	0
$907$	6.95027	0.230780	0.115390	−	0.993320i	$-0.463188\pi$
$907$	6.95027	0.230780	0.115390	+	0.993320i	$0.463188\pi$
$908$	0	0
$909$	−16.6836	−0.553361
$910$	0	0
$911$	−36.9253	−1.22339	−0.611695	−	0.791094i	$-0.709512\pi$
$911$	−36.9253	−1.22339	−0.611695	+	0.791094i	$0.709512\pi$
$912$	0	0
$913$	2.14358	0.0709420
$914$	0	0
$915$	69.0596	2.28304
$916$	0	0
$917$	39.7910	1.31402
$918$	0	0
$919$	−30.8285	−1.01694	−0.508470	−	0.861080i	$-0.669789\pi$
$919$	−30.8285	−1.01694	−0.508470	+	0.861080i	$0.669789\pi$
$920$	0	0
$921$	48.8374	1.60925
$922$	0	0
$923$	−8.28773	−0.272794
$924$	0	0
$925$	35.6746	1.17297
$926$	0	0
$927$	0.514733	0.0169060
$928$	0	0
$929$	−20.5575	−0.674471	−0.337235	−	0.941420i	$-0.609492\pi$
$929$	−20.5575	−0.674471	−0.337235	+	0.941420i	$0.609492\pi$
$930$	0	0
$931$	17.8636	0.585454
$932$	0	0
$933$	25.0939	0.821538
$934$	0	0
$935$	−79.2125	−2.59053
$936$	0	0
$937$	−30.9158	−1.00998	−0.504988	−	0.863126i	$-0.668503\pi$
$937$	−30.9158	−1.00998	−0.504988	+	0.863126i	$0.668503\pi$
$938$	0	0
$939$	−21.2533	−0.693576
$940$	0	0
$941$	22.1553	0.722243	0.361121	−	0.932519i	$-0.382394\pi$
$941$	22.1553	0.722243	0.361121	+	0.932519i	$0.382394\pi$
$942$	0	0
$943$	−8.85874	−0.288480
$944$	0	0
$945$	7.01113	0.228072
$946$	0	0
$947$	24.1496	0.784755	0.392378	−	0.919804i	$-0.371653\pi$
$947$	24.1496	0.784755	0.392378	+	0.919804i	$0.371653\pi$
$948$	0	0
$949$	−22.4499	−0.728755
$950$	0	0
$951$	68.5697	2.22353
$952$	0	0
$953$	−22.2620	−0.721138	−0.360569	−	0.932733i	$-0.617417\pi$
$953$	−22.2620	−0.721138	−0.360569	+	0.932733i	$0.617417\pi$
$954$	0	0
$955$	−41.4757	−1.34212
$956$	0	0
$957$	−27.7287	−0.896342
$958$	0	0
$959$	−27.7179	−0.895057
$960$	0	0
$961$	−15.0825	−0.486531
$962$	0	0
$963$	−2.27071	−0.0731727
$964$	0	0
$965$	−47.1056	−1.51638
$966$	0	0
$967$	−20.1965	−0.649476	−0.324738	−	0.945804i	$-0.605276\pi$
$967$	−20.1965	−0.649476	−0.324738	+	0.945804i	$0.605276\pi$
$968$	0	0
$969$	86.9806	2.79422
$970$	0	0
$971$	−44.7407	−1.43580	−0.717898	−	0.696148i	$-0.754896\pi$
$971$	−44.7407	−1.43580	−0.717898	+	0.696148i	$0.754896\pi$
$972$	0	0
$973$	19.0057	0.609296
$974$	0	0
$975$	−84.8281	−2.71667
$976$	0	0
$977$	54.9870	1.75919	0.879594	−	0.475725i	$-0.157814\pi$
$977$	54.9870	1.75919	0.879594	+	0.475725i	$0.157814\pi$
$978$	0	0
$979$	13.6625	0.436657
$980$	0	0
$981$	−11.9867	−0.382706
$982$	0	0
$983$	−29.3295	−0.935465	−0.467732	−	0.883870i	$-0.654929\pi$
$983$	−29.3295	−0.935465	−0.467732	+	0.883870i	$0.654929\pi$
$984$	0	0
$985$	15.7477	0.501764
$986$	0	0
$987$	89.9401	2.86282
$988$	0	0
$989$	−28.8356	−0.916918
$990$	0	0
$991$	31.6756	1.00621	0.503104	−	0.864226i	$-0.332191\pi$
$991$	31.6756	1.00621	0.503104	+	0.864226i	$0.332191\pi$
$992$	0	0
$993$	61.8078	1.96141
$994$	0	0
$995$	−78.4273	−2.48631
$996$	0	0
$997$	−27.9057	−0.883782	−0.441891	−	0.897069i	$-0.645692\pi$
$997$	−27.9057	−0.883782	−0.441891	+	0.897069i	$0.645692\pi$
$998$	0	0
$999$	−4.87533	−0.154248

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.u.1.22		26
4.3	odd	2		503.2.a.f.1.13	✓	26
12.11	even	2		4527.2.a.o.1.14		26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.a.f.1.13	✓	26	4.3	odd	2
4527.2.a.o.1.14		26	12.11	even	2
8048.2.a.u.1.22		26	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)

\(f(q)\)	\(=\)	\(q+2.39019 q^{3} -3.16539 q^{5} +3.22876 q^{7} +2.71299 q^{9} +O(q^{10})\)	\(q+2.39019 q^{3} -3.16539 q^{5} +3.22876 q^{7} +2.71299 q^{9} +3.58671 q^{11} -7.07014 q^{13} -7.56588 q^{15} +6.97702 q^{17} +5.21579 q^{19} +7.71734 q^{21} +3.09603 q^{23} +5.01972 q^{25} -0.686000 q^{27} -3.23446 q^{29} +3.98968 q^{31} +8.57290 q^{33} -10.2203 q^{35} +7.10689 q^{37} -16.8990 q^{39} -2.86132 q^{41} -9.31372 q^{43} -8.58769 q^{45} +11.6543 q^{47} +3.42490 q^{49} +16.6764 q^{51} -11.3201 q^{53} -11.3533 q^{55} +12.4667 q^{57} +11.1359 q^{59} -9.12777 q^{61} +8.75961 q^{63} +22.3798 q^{65} +3.40992 q^{67} +7.40009 q^{69} +1.17221 q^{71} +3.17531 q^{73} +11.9981 q^{75} +11.5806 q^{77} -3.67810 q^{79} -9.77865 q^{81} +0.597645 q^{83} -22.0850 q^{85} -7.73097 q^{87} +3.80922 q^{89} -22.8278 q^{91} +9.53607 q^{93} -16.5100 q^{95} +12.6081 q^{97} +9.73071 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9}+O(q^{10}) \) 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9	\( 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9} + 17 q^{11} + 14 q^{13} - 18 q^{15} + 17 q^{17} + 22 q^{19} - 16 q^{21} - 27 q^{23} + 93 q^{25} - 31 q^{27} + 13 q^{29} - 26 q^{31} + 6 q^{33} + 22 q^{35} + 55 q^{37} + 15 q^{39} + 24 q^{41} - 20 q^{43} - 8 q^{45} + 25 q^{47} + 65 q^{49} - 7 q^{51} + 30 q^{53} - 25 q^{55} + 9 q^{57} + 26 q^{59} + 15 q^{61} + 19 q^{63} + 20 q^{65} + 20 q^{67} - 27 q^{69} + 35 q^{71} + 38 q^{73} - 2 q^{75} - 6 q^{77} - 21 q^{79} + 70 q^{81} + 48 q^{83} + 6 q^{85} + 9 q^{87} - 5 q^{89} + 24 q^{91} - 8 q^{93} - 43 q^{95} + 142 q^{97} + 50 q^{99}+O(q^{100}) \) 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9 + 17 * q^11 + 14 * q^13 - 18 * q^15 + 17 * q^17 + 22 * q^19 - 16 * q^21 - 27 * q^23 + 93 * q^25 - 31 * q^27 + 13 * q^29 - 26 * q^31 + 6 * q^33 + 22 * q^35 + 55 * q^37 + 15 * q^39 + 24 * q^41 - 20 * q^43 - 8 * q^45 + 25 * q^47 + 65 * q^49 - 7 * q^51 + 30 * q^53 - 25 * q^55 + 9 * q^57 + 26 * q^59 + 15 * q^61 + 19 * q^63 + 20 * q^65 + 20 * q^67 - 27 * q^69 + 35 * q^71 + 38 * q^73 - 2 * q^75 - 6 * q^77 - 21 * q^79 + 70 * q^81 + 48 * q^83 + 6 * q^85 + 9 * q^87 - 5 * q^89 + 24 * q^91 - 8 * q^93 - 43 * q^95 + 142 * q^97 + 50 * q^99

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