LMFDB - Embedded newform 1280.4.a.bd.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,4,Mod(1,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1280.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1280.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:	$75.5224448073$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 21x^{4} + 38x^{3} + 93x^{2} - 108x - 60 $ x^6 - 2x^5 - 21x^4 + 38x^3 + 93x^2 - 108*x - 60
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 40)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.430422$ of defining polynomial
Character	$\chi$	$=$	1280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.57890 q^{3} +5.00000 q^{5} +21.5703 q^{7} +64.7554 q^{9} +37.2871 q^{11} +24.1097 q^{13} +47.8945 q^{15} -14.4940 q^{17} +26.6887 q^{19} +206.620 q^{21} -8.36366 q^{23} +25.0000 q^{25} +361.655 q^{27} -104.553 q^{29} -204.288 q^{31} +357.170 q^{33} +107.851 q^{35} -130.923 q^{37} +230.945 q^{39} +437.963 q^{41} -308.764 q^{43} +323.777 q^{45} -97.1081 q^{47} +122.277 q^{49} -138.836 q^{51} -154.502 q^{53} +186.436 q^{55} +255.648 q^{57} -544.088 q^{59} -374.601 q^{61} +1396.79 q^{63} +120.549 q^{65} +19.3982 q^{67} -80.1147 q^{69} -955.725 q^{71} -127.417 q^{73} +239.473 q^{75} +804.293 q^{77} -973.871 q^{79} +1715.87 q^{81} -55.0210 q^{83} -72.4698 q^{85} -1001.50 q^{87} +919.812 q^{89} +520.054 q^{91} -1956.85 q^{93} +133.443 q^{95} +297.986 q^{97} +2414.54 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} + 30 q^{5} + 14 q^{7} + 54 q^{9} + 44 q^{11} + 30 q^{15} + 152 q^{19} + 4 q^{21} + 302 q^{23} + 150 q^{25} + 216 q^{27} + 132 q^{31} - 116 q^{33} + 70 q^{35} - 68 q^{37} + 300 q^{39} - 20 q^{41}+ \cdots + 5516 q^{99}+O(q^{100}) $ 6 * q + 6 * q^3 + 30 * q^5 + 14 * q^7 + 54 * q^9 + 44 * q^11 + 30 * q^15 + 152 * q^19 + 4 * q^21 + 302 * q^23 + 150 * q^25 + 216 * q^27 + 132 * q^31 - 116 * q^33 + 70 * q^35 - 68 * q^37 + 300 * q^39 - 20 * q^41 + 602 * q^43 + 270 * q^45 + 470 * q^47 + 654 * q^49 + 612 * q^51 + 528 * q^53 + 220 * q^55 + 340 * q^57 + 472 * q^59 - 476 * q^61 + 650 * q^63 + 1206 * q^67 - 980 * q^69 - 796 * q^71 - 216 * q^73 + 150 * q^75 + 412 * q^77 - 1008 * q^79 + 1254 * q^81 + 1778 * q^83 - 984 * q^87 + 212 * q^89 + 3652 * q^91 - 1392 * q^93 + 760 * q^95 - 792 * q^97 + 5516 * q^99

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
$3$	9.57890	1.84346	0.921730	−	0.387831i	$-0.126776\pi$
$3$	9.57890	1.84346	0.921730	+	0.387831i	$0.126776\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	21.5703	1.16469	0.582343	−	0.812943i	$-0.302136\pi$
$7$	21.5703	1.16469	0.582343	+	0.812943i	$0.302136\pi$
$8$	0	0
$9$	64.7554	2.39835
$10$	0	0
$11$	37.2871	1.02204	0.511022	−	0.859568i	$-0.329267\pi$
$11$	37.2871	1.02204	0.511022	+	0.859568i	$0.329267\pi$
$12$	0	0
$13$	24.1097	0.514372	0.257186	−	0.966362i	$-0.417205\pi$
$13$	24.1097	0.514372	0.257186	+	0.966362i	$0.417205\pi$
$14$	0	0
$15$	47.8945	0.824421
$16$	0	0
$17$	−14.4940	−0.206783	−0.103391	−	0.994641i	$-0.532969\pi$
$17$	−14.4940	−0.206783	−0.103391	+	0.994641i	$0.532969\pi$
$18$	0	0
$19$	26.6887	0.322253	0.161126	−	0.986934i	$-0.448487\pi$
$19$	26.6887	0.322253	0.161126	+	0.986934i	$0.448487\pi$
$20$	0	0
$21$	206.620	2.14705
$22$	0	0
$23$	−8.36366	−0.0758236	−0.0379118	−	0.999281i	$-0.512071\pi$
$23$	−8.36366	−0.0758236	−0.0379118	+	0.999281i	$0.512071\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	361.655	2.57780
$28$	0	0
$29$	−104.553	−0.669481	−0.334741	−	0.942310i	$-0.608649\pi$
$29$	−104.553	−0.669481	−0.334741	+	0.942310i	$0.608649\pi$
$30$	0	0
$31$	−204.288	−1.18359	−0.591793	−	0.806090i	$-0.701580\pi$
$31$	−204.288	−1.18359	−0.591793	+	0.806090i	$0.701580\pi$
$32$	0	0
$33$	357.170	1.88410
$34$	0	0
$35$	107.851	0.520863
$36$	0	0
$37$	−130.923	−0.581720	−0.290860	−	0.956766i	$-0.593941\pi$
$37$	−130.923	−0.581720	−0.290860	+	0.956766i	$0.593941\pi$
$38$	0	0
$39$	230.945	0.948225
$40$	0	0
$41$	437.963	1.66825	0.834126	−	0.551574i	$-0.185973\pi$
$41$	437.963	1.66825	0.834126	+	0.551574i	$0.185973\pi$
$42$	0	0
$43$	−308.764	−1.09503	−0.547513	−	0.836797i	$-0.684425\pi$
$43$	−308.764	−1.09503	−0.547513	+	0.836797i	$0.684425\pi$
$44$	0	0
$45$	323.777	1.07257
$46$	0	0
$47$	−97.1081	−0.301376	−0.150688	−	0.988581i	$-0.548149\pi$
$47$	−97.1081	−0.301376	−0.150688	+	0.988581i	$0.548149\pi$
$48$	0	0
$49$	122.277	0.356492
$50$	0	0
$51$	−138.836	−0.381196
$52$	0	0
$53$	−154.502	−0.400425	−0.200212	−	0.979753i	$-0.564163\pi$
$53$	−154.502	−0.400425	−0.200212	+	0.979753i	$0.564163\pi$
$54$	0	0
$55$	186.436	0.457072
$56$	0	0
$57$	255.648	0.594060
$58$	0	0
$59$	−544.088	−1.20058	−0.600290	−	0.799782i	$-0.704948\pi$
$59$	−544.088	−1.20058	−0.600290	+	0.799782i	$0.704948\pi$
$60$	0	0
$61$	−374.601	−0.786275	−0.393137	−	0.919480i	$-0.628610\pi$
$61$	−374.601	−0.786275	−0.393137	+	0.919480i	$0.628610\pi$
$62$	0	0
$63$	1396.79	2.79332
$64$	0	0
$65$	120.549	0.230034
$66$	0	0
$67$	19.3982	0.0353711	0.0176855	−	0.999844i	$-0.494370\pi$
$67$	19.3982	0.0353711	0.0176855	+	0.999844i	$0.494370\pi$
$68$	0	0
$69$	−80.1147	−0.139778
$70$	0	0
$71$	−955.725	−1.59752	−0.798759	−	0.601651i	$-0.794510\pi$
$71$	−955.725	−1.59752	−0.798759	+	0.601651i	$0.794510\pi$
$72$	0	0
$73$	−127.417	−0.204287	−0.102144	−	0.994770i	$-0.532570\pi$
$73$	−127.417	−0.204287	−0.102144	+	0.994770i	$0.532570\pi$
$74$	0	0
$75$	239.473	0.368692
$76$	0	0
$77$	804.293	1.19036
$78$	0	0
$79$	−973.871	−1.38695	−0.693475	−	0.720481i	$-0.743921\pi$
$79$	−973.871	−1.38695	−0.693475	+	0.720481i	$0.743921\pi$
$80$	0	0
$81$	1715.87	2.35373
$82$	0	0
$83$	−55.0210	−0.0727631	−0.0363816	−	0.999338i	$-0.511583\pi$
$83$	−55.0210	−0.0727631	−0.0363816	+	0.999338i	$0.511583\pi$
$84$	0	0
$85$	−72.4698	−0.0924760
$86$	0	0
$87$	−1001.50	−1.23416
$88$	0	0
$89$	919.812	1.09550	0.547752	−	0.836641i	$-0.315484\pi$
$89$	919.812	1.09550	0.547752	+	0.836641i	$0.315484\pi$
$90$	0	0
$91$	520.054	0.599082
$92$	0	0
$93$	−1956.85	−2.18189
$94$	0	0
$95$	133.443	0.144116
$96$	0	0
$97$	297.986	0.311916	0.155958	−	0.987764i	$-0.450154\pi$
$97$	297.986	0.311916	0.155958	+	0.987764i	$0.450154\pi$
$98$	0	0
$99$	2414.54	2.45122
$100$	0	0
$101$	−1793.53	−1.76696	−0.883480	−	0.468469i	$-0.844806\pi$
$101$	−1793.53	−1.76696	−0.883480	+	0.468469i	$0.844806\pi$
$102$	0	0
$103$	2084.11	1.99372	0.996862	−	0.0791611i	$-0.0252241\pi$
$103$	2084.11	1.99372	0.996862	+	0.0791611i	$0.0252241\pi$
$104$	0	0
$105$	1033.10	0.960191
$106$	0	0
$107$	67.2542	0.0607637	0.0303818	−	0.999538i	$-0.490328\pi$
$107$	67.2542	0.0607637	0.0303818	+	0.999538i	$0.490328\pi$
$108$	0	0
$109$	1459.15	1.28222	0.641108	−	0.767451i	$-0.278475\pi$
$109$	1459.15	1.28222	0.641108	+	0.767451i	$0.278475\pi$
$110$	0	0
$111$	−1254.10	−1.07238
$112$	0	0
$113$	458.263	0.381502	0.190751	−	0.981638i	$-0.438908\pi$
$113$	458.263	0.381502	0.190751	+	0.981638i	$0.438908\pi$
$114$	0	0
$115$	−41.8183	−0.0339093
$116$	0	0
$117$	1561.24	1.23364
$118$	0	0
$119$	−312.639	−0.240837
$120$	0	0
$121$	59.3283	0.0445743
$122$	0	0
$123$	4195.21	3.07536
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−208.685	−0.145809	−0.0729047	−	0.997339i	$-0.523227\pi$
$127$	−208.685	−0.145809	−0.0729047	+	0.997339i	$0.523227\pi$
$128$	0	0
$129$	−2957.62	−2.01864
$130$	0	0
$131$	1784.69	1.19030	0.595148	−	0.803616i	$-0.297094\pi$
$131$	1784.69	1.19030	0.595148	+	0.803616i	$0.297094\pi$
$132$	0	0
$133$	575.682	0.375323
$134$	0	0
$135$	1808.28	1.15283
$136$	0	0
$137$	−2380.60	−1.48459	−0.742293	−	0.670075i	$-0.766262\pi$
$137$	−2380.60	−1.48459	−0.742293	+	0.670075i	$0.766262\pi$
$138$	0	0
$139$	2858.52	1.74429	0.872146	−	0.489245i	$-0.162728\pi$
$139$	2858.52	1.74429	0.872146	+	0.489245i	$0.162728\pi$
$140$	0	0
$141$	−930.189	−0.555575
$142$	0	0
$143$	898.983	0.525711
$144$	0	0
$145$	−522.764	−0.299401
$146$	0	0
$147$	1171.28	0.657179
$148$	0	0
$149$	897.894	0.493680	0.246840	−	0.969056i	$-0.420608\pi$
$149$	897.894	0.493680	0.246840	+	0.969056i	$0.420608\pi$
$150$	0	0
$151$	−3431.32	−1.84925	−0.924625	−	0.380878i	$-0.875622\pi$
$151$	−3431.32	−1.84925	−0.924625	+	0.380878i	$0.875622\pi$
$152$	0	0
$153$	−938.563	−0.495937
$154$	0	0
$155$	−1021.44	−0.529316
$156$	0	0
$157$	−571.369	−0.290447	−0.145224	−	0.989399i	$-0.546390\pi$
$157$	−571.369	−0.290447	−0.145224	+	0.989399i	$0.546390\pi$
$158$	0	0
$159$	−1479.96	−0.738167
$160$	0	0
$161$	−180.406	−0.0883106
$162$	0	0
$163$	−236.962	−0.113867	−0.0569333	−	0.998378i	$-0.518132\pi$
$163$	−236.962	−0.113867	−0.0569333	+	0.998378i	$0.518132\pi$
$164$	0	0
$165$	1785.85	0.842594
$166$	0	0
$167$	1448.46	0.671169	0.335585	−	0.942010i	$-0.391066\pi$
$167$	1448.46	0.671169	0.335585	+	0.942010i	$0.391066\pi$
$168$	0	0
$169$	−1615.72	−0.735421
$170$	0	0
$171$	1728.24	0.772874
$172$	0	0
$173$	−2529.74	−1.11175	−0.555875	−	0.831266i	$-0.687617\pi$
$173$	−2529.74	−1.11175	−0.555875	+	0.831266i	$0.687617\pi$
$174$	0	0
$175$	539.257	0.232937
$176$	0	0
$177$	−5211.77	−2.21322
$178$	0	0
$179$	−141.524	−0.0590949	−0.0295475	−	0.999563i	$-0.509407\pi$
$179$	−141.524	−0.0590949	−0.0295475	+	0.999563i	$0.509407\pi$
$180$	0	0
$181$	3457.79	1.41997	0.709987	−	0.704215i	$-0.248701\pi$
$181$	3457.79	1.41997	0.709987	+	0.704215i	$0.248701\pi$
$182$	0	0
$183$	−3588.27	−1.44947
$184$	0	0
$185$	−654.616	−0.260153
$186$	0	0
$187$	−540.438	−0.211341
$188$	0	0
$189$	7801.01	3.00233
$190$	0	0
$191$	1416.99	0.536804	0.268402	−	0.963307i	$-0.413504\pi$
$191$	1416.99	0.536804	0.268402	+	0.963307i	$0.413504\pi$
$192$	0	0
$193$	−2681.12	−0.999955	−0.499978	−	0.866038i	$-0.666658\pi$
$193$	−2681.12	−0.999955	−0.499978	+	0.866038i	$0.666658\pi$
$194$	0	0
$195$	1154.72	0.424059
$196$	0	0
$197$	−2697.74	−0.975664	−0.487832	−	0.872938i	$-0.662212\pi$
$197$	−2697.74	−0.975664	−0.487832	+	0.872938i	$0.662212\pi$
$198$	0	0
$199$	3543.25	1.26218	0.631091	−	0.775709i	$-0.282607\pi$
$199$	3543.25	1.26218	0.631091	+	0.775709i	$0.282607\pi$
$200$	0	0
$201$	185.813	0.0652052
$202$	0	0
$203$	−2255.23	−0.779735
$204$	0	0
$205$	2189.81	0.746065
$206$	0	0
$207$	−541.592	−0.181851
$208$	0	0
$209$	995.143	0.329356
$210$	0	0
$211$	−152.205	−0.0496598	−0.0248299	−	0.999692i	$-0.507904\pi$
$211$	−152.205	−0.0496598	−0.0248299	+	0.999692i	$0.507904\pi$
$212$	0	0
$213$	−9154.80	−2.94496
$214$	0	0
$215$	−1543.82	−0.489710
$216$	0	0
$217$	−4406.54	−1.37851
$218$	0	0
$219$	−1220.51	−0.376596
$220$	0	0
$221$	−349.446	−0.106363
$222$	0	0
$223$	2065.40	0.620222	0.310111	−	0.950700i	$-0.399634\pi$
$223$	2065.40	0.620222	0.310111	+	0.950700i	$0.399634\pi$
$224$	0	0
$225$	1618.89	0.479670
$226$	0	0
$227$	2940.45	0.859757	0.429878	−	0.902887i	$-0.358556\pi$
$227$	2940.45	0.859757	0.429878	+	0.902887i	$0.358556\pi$
$228$	0	0
$229$	4949.77	1.42834	0.714171	−	0.699971i	$-0.246804\pi$
$229$	4949.77	1.42834	0.714171	+	0.699971i	$0.246804\pi$
$230$	0	0
$231$	7704.25	2.19438
$232$	0	0
$233$	−5193.31	−1.46019	−0.730096	−	0.683344i	$-0.760525\pi$
$233$	−5193.31	−1.46019	−0.730096	+	0.683344i	$0.760525\pi$
$234$	0	0
$235$	−485.540	−0.134779
$236$	0	0
$237$	−9328.62	−2.55679
$238$	0	0
$239$	2066.30	0.559238	0.279619	−	0.960111i	$-0.409792\pi$
$239$	2066.30	0.559238	0.279619	+	0.960111i	$0.409792\pi$
$240$	0	0
$241$	−2309.08	−0.617181	−0.308591	−	0.951195i	$-0.599857\pi$
$241$	−2309.08	−0.617181	−0.308591	+	0.951195i	$0.599857\pi$
$242$	0	0
$243$	6671.42	1.76120
$244$	0	0
$245$	611.384	0.159428
$246$	0	0
$247$	643.457	0.165758
$248$	0	0
$249$	−527.041	−0.134136
$250$	0	0
$251$	3802.60	0.956246	0.478123	−	0.878293i	$-0.341317\pi$
$251$	3802.60	0.956246	0.478123	+	0.878293i	$0.341317\pi$
$252$	0	0
$253$	−311.857	−0.0774951
$254$	0	0
$255$	−694.182	−0.170476
$256$	0	0
$257$	6168.56	1.49721	0.748607	−	0.663013i	$-0.230723\pi$
$257$	6168.56	1.49721	0.748607	+	0.663013i	$0.230723\pi$
$258$	0	0
$259$	−2824.05	−0.677521
$260$	0	0
$261$	−6770.36	−1.60565
$262$	0	0
$263$	−1636.11	−0.383599	−0.191800	−	0.981434i	$-0.561432\pi$
$263$	−1636.11	−0.383599	−0.191800	+	0.981434i	$0.561432\pi$
$264$	0	0
$265$	−772.511	−0.179075
$266$	0	0
$267$	8810.79	2.01952
$268$	0	0
$269$	2262.24	0.512756	0.256378	−	0.966577i	$-0.417471\pi$
$269$	2262.24	0.512756	0.256378	+	0.966577i	$0.417471\pi$
$270$	0	0
$271$	−41.6380	−0.00933332	−0.00466666	−	0.999989i	$-0.501485\pi$
$271$	−41.6380	−0.00933332	−0.00466666	+	0.999989i	$0.501485\pi$
$272$	0	0
$273$	4981.55	1.10438
$274$	0	0
$275$	932.178	0.204409
$276$	0	0
$277$	−2139.00	−0.463971	−0.231986	−	0.972719i	$-0.574522\pi$
$277$	−2139.00	−0.463971	−0.231986	+	0.972719i	$0.574522\pi$
$278$	0	0
$279$	−13228.7	−2.83865
$280$	0	0
$281$	1493.53	0.317069	0.158534	−	0.987353i	$-0.449323\pi$
$281$	1493.53	0.317069	0.158534	+	0.987353i	$0.449323\pi$
$282$	0	0
$283$	1090.20	0.228995	0.114498	−	0.993424i	$-0.463474\pi$
$283$	1090.20	0.228995	0.114498	+	0.993424i	$0.463474\pi$
$284$	0	0
$285$	1278.24	0.265672
$286$	0	0
$287$	9446.98	1.94299
$288$	0	0
$289$	−4702.92	−0.957241
$290$	0	0
$291$	2854.38	0.575005
$292$	0	0
$293$	−4067.51	−0.811011	−0.405506	−	0.914093i	$-0.632905\pi$
$293$	−4067.51	−0.811011	−0.405506	+	0.914093i	$0.632905\pi$
$294$	0	0
$295$	−2720.44	−0.536916
$296$	0	0
$297$	13485.1	2.63463
$298$	0	0
$299$	−201.646	−0.0390016
$300$	0	0
$301$	−6660.13	−1.27536
$302$	0	0
$303$	−17180.1	−3.25732
$304$	0	0
$305$	−1873.01	−0.351633
$306$	0	0
$307$	6736.05	1.25227	0.626135	−	0.779715i	$-0.284636\pi$
$307$	6736.05	1.25227	0.626135	+	0.779715i	$0.284636\pi$
$308$	0	0
$309$	19963.5	3.67535
$310$	0	0
$311$	3274.35	0.597014	0.298507	−	0.954407i	$-0.403511\pi$
$311$	3274.35	0.597014	0.298507	+	0.954407i	$0.403511\pi$
$312$	0	0
$313$	9357.60	1.68985	0.844925	−	0.534885i	$-0.179645\pi$
$313$	9357.60	1.68985	0.844925	+	0.534885i	$0.179645\pi$
$314$	0	0
$315$	6983.96	1.24921
$316$	0	0
$317$	9064.68	1.60607	0.803033	−	0.595934i	$-0.203218\pi$
$317$	9064.68	1.60607	0.803033	+	0.595934i	$0.203218\pi$
$318$	0	0
$319$	−3898.47	−0.684239
$320$	0	0
$321$	644.222	0.112015
$322$	0	0
$323$	−386.825	−0.0666362
$324$	0	0
$325$	602.744	0.102874
$326$	0	0
$327$	13977.1	2.36371
$328$	0	0
$329$	−2094.65	−0.351008
$330$	0	0
$331$	−280.089	−0.0465108	−0.0232554	−	0.999730i	$-0.507403\pi$
$331$	−280.089	−0.0465108	−0.0232554	+	0.999730i	$0.507403\pi$
$332$	0	0
$333$	−8477.98	−1.39517
$334$	0	0
$335$	96.9908	0.0158184
$336$	0	0
$337$	−5748.75	−0.929242	−0.464621	−	0.885510i	$-0.653809\pi$
$337$	−5748.75	−0.929242	−0.464621	+	0.885510i	$0.653809\pi$
$338$	0	0
$339$	4389.65	0.703284
$340$	0	0
$341$	−7617.30	−1.20968
$342$	0	0
$343$	−4761.06	−0.749484
$344$	0	0
$345$	−400.573	−0.0625106
$346$	0	0
$347$	−2911.52	−0.450428	−0.225214	−	0.974309i	$-0.572308\pi$
$347$	−2911.52	−0.450428	−0.225214	+	0.974309i	$0.572308\pi$
$348$	0	0
$349$	5665.17	0.868910	0.434455	−	0.900693i	$-0.356941\pi$
$349$	5665.17	0.868910	0.434455	+	0.900693i	$0.356941\pi$
$350$	0	0
$351$	8719.42	1.32595
$352$	0	0
$353$	−9428.18	−1.42156	−0.710781	−	0.703414i	$-0.751658\pi$
$353$	−9428.18	−1.42156	−0.710781	+	0.703414i	$0.751658\pi$
$354$	0	0
$355$	−4778.63	−0.714432
$356$	0	0
$357$	−2994.74	−0.443973
$358$	0	0
$359$	4573.16	0.672318	0.336159	−	0.941805i	$-0.390872\pi$
$359$	4573.16	0.672318	0.336159	+	0.941805i	$0.390872\pi$
$360$	0	0
$361$	−6146.71	−0.896153
$362$	0	0
$363$	568.300	0.0821709
$364$	0	0
$365$	−637.083	−0.0913601
$366$	0	0
$367$	2717.97	0.386585	0.193293	−	0.981141i	$-0.438083\pi$
$367$	2717.97	0.386585	0.193293	+	0.981141i	$0.438083\pi$
$368$	0	0
$369$	28360.5	4.00105
$370$	0	0
$371$	−3332.65	−0.466369
$372$	0	0
$373$	5627.51	0.781183	0.390592	−	0.920564i	$-0.372270\pi$
$373$	5627.51	0.781183	0.390592	+	0.920564i	$0.372270\pi$
$374$	0	0
$375$	1197.36	0.164884
$376$	0	0
$377$	−2520.74	−0.344363
$378$	0	0
$379$	8066.74	1.09330	0.546650	−	0.837361i	$-0.315903\pi$
$379$	8066.74	1.09330	0.546650	+	0.837361i	$0.315903\pi$
$380$	0	0
$381$	−1998.97	−0.268794
$382$	0	0
$383$	−10607.8	−1.41523	−0.707613	−	0.706600i	$-0.750228\pi$
$383$	−10607.8	−1.41523	−0.707613	+	0.706600i	$0.750228\pi$
$384$	0	0
$385$	4021.47	0.532345
$386$	0	0
$387$	−19994.1	−2.62625
$388$	0	0
$389$	8992.58	1.17209	0.586044	−	0.810280i	$-0.300685\pi$
$389$	8992.58	1.17209	0.586044	+	0.810280i	$0.300685\pi$
$390$	0	0
$391$	121.223	0.0156790
$392$	0	0
$393$	17095.3	2.19426
$394$	0	0
$395$	−4869.36	−0.620263
$396$	0	0
$397$	8977.17	1.13489	0.567445	−	0.823411i	$-0.307932\pi$
$397$	8977.17	1.13489	0.567445	+	0.823411i	$0.307932\pi$
$398$	0	0
$399$	5514.40	0.691893
$400$	0	0
$401$	−3165.72	−0.394236	−0.197118	−	0.980380i	$-0.563158\pi$
$401$	−3165.72	−0.394236	−0.197118	+	0.980380i	$0.563158\pi$
$402$	0	0
$403$	−4925.33	−0.608804
$404$	0	0
$405$	8579.33	1.05262
$406$	0	0
$407$	−4881.75	−0.594543
$408$	0	0
$409$	5417.61	0.654972	0.327486	−	0.944856i	$-0.393798\pi$
$409$	5417.61	0.654972	0.327486	+	0.944856i	$0.393798\pi$
$410$	0	0
$411$	−22803.5	−2.73678
$412$	0	0
$413$	−11736.1	−1.39830
$414$	0	0
$415$	−275.105	−0.0325407
$416$	0	0
$417$	27381.5	3.21554
$418$	0	0
$419$	−6198.14	−0.722670	−0.361335	−	0.932436i	$-0.617679\pi$
$419$	−6198.14	−0.722670	−0.361335	+	0.932436i	$0.617679\pi$
$420$	0	0
$421$	−4344.57	−0.502949	−0.251474	−	0.967864i	$-0.580915\pi$
$421$	−4344.57	−0.502949	−0.251474	+	0.967864i	$0.580915\pi$
$422$	0	0
$423$	−6288.27	−0.722804
$424$	0	0
$425$	−362.349	−0.0413565
$426$	0	0
$427$	−8080.25	−0.915763
$428$	0	0
$429$	8611.27	0.969128
$430$	0	0
$431$	−4750.74	−0.530940	−0.265470	−	0.964119i	$-0.585527\pi$
$431$	−4750.74	−0.530940	−0.265470	+	0.964119i	$0.585527\pi$
$432$	0	0
$433$	4210.26	0.467280	0.233640	−	0.972323i	$-0.424936\pi$
$433$	4210.26	0.467280	0.233640	+	0.972323i	$0.424936\pi$
$434$	0	0
$435$	−5007.50	−0.551934
$436$	0	0
$437$	−223.215	−0.0244344
$438$	0	0
$439$	−5776.50	−0.628012	−0.314006	−	0.949421i	$-0.601671\pi$
$439$	−5776.50	−0.628012	−0.314006	+	0.949421i	$0.601671\pi$
$440$	0	0
$441$	7918.08	0.854992
$442$	0	0
$443$	−11981.6	−1.28502	−0.642510	−	0.766278i	$-0.722107\pi$
$443$	−11981.6	−1.28502	−0.642510	+	0.766278i	$0.722107\pi$
$444$	0	0
$445$	4599.06	0.489924
$446$	0	0
$447$	8600.84	0.910080
$448$	0	0
$449$	−5567.58	−0.585190	−0.292595	−	0.956236i	$-0.594519\pi$
$449$	−5567.58	−0.585190	−0.292595	+	0.956236i	$0.594519\pi$
$450$	0	0
$451$	16330.4	1.70503
$452$	0	0
$453$	−32868.3	−3.40902
$454$	0	0
$455$	2600.27	0.267918
$456$	0	0
$457$	7283.56	0.745538	0.372769	−	0.927924i	$-0.378408\pi$
$457$	7283.56	0.745538	0.372769	+	0.927924i	$0.378408\pi$
$458$	0	0
$459$	−5241.82	−0.533044
$460$	0	0
$461$	11901.1	1.20236	0.601181	−	0.799113i	$-0.294697\pi$
$461$	11901.1	1.20236	0.601181	+	0.799113i	$0.294697\pi$
$462$	0	0
$463$	−4915.73	−0.493419	−0.246710	−	0.969089i	$-0.579349\pi$
$463$	−4915.73	−0.493419	−0.246710	+	0.969089i	$0.579349\pi$
$464$	0	0
$465$	−9784.27	−0.975773
$466$	0	0
$467$	−3110.15	−0.308181	−0.154091	−	0.988057i	$-0.549245\pi$
$467$	−3110.15	−0.308181	−0.154091	+	0.988057i	$0.549245\pi$
$468$	0	0
$469$	418.424	0.0411962
$470$	0	0
$471$	−5473.09	−0.535428
$472$	0	0
$473$	−11512.9	−1.11916
$474$	0	0
$475$	667.217	0.0644505
$476$	0	0
$477$	−10004.8	−0.960358
$478$	0	0
$479$	−6590.43	−0.628652	−0.314326	−	0.949315i	$-0.601779\pi$
$479$	−6590.43	−0.628652	−0.314326	+	0.949315i	$0.601779\pi$
$480$	0	0
$481$	−3156.52	−0.299221
$482$	0	0
$483$	−1728.10	−0.162797
$484$	0	0
$485$	1489.93	0.139493
$486$	0	0
$487$	3139.99	0.292169	0.146085	−	0.989272i	$-0.453333\pi$
$487$	3139.99	0.292169	0.146085	+	0.989272i	$0.453333\pi$
$488$	0	0
$489$	−2269.83	−0.209909
$490$	0	0
$491$	12057.6	1.10825	0.554126	−	0.832433i	$-0.313053\pi$
$491$	12057.6	1.10825	0.554126	+	0.832433i	$0.313053\pi$
$492$	0	0
$493$	1515.38	0.138437
$494$	0	0
$495$	12072.7	1.09622
$496$	0	0
$497$	−20615.3	−1.86061
$498$	0	0
$499$	8458.64	0.758839	0.379420	−	0.925225i	$-0.376124\pi$
$499$	8458.64	0.758839	0.379420	+	0.925225i	$0.376124\pi$
$500$	0	0
$501$	13874.7	1.23727
$502$	0	0
$503$	16896.2	1.49774	0.748869	−	0.662718i	$-0.230597\pi$
$503$	16896.2	1.49774	0.748869	+	0.662718i	$0.230597\pi$
$504$	0	0
$505$	−8967.65	−0.790208
$506$	0	0
$507$	−15476.8	−1.35572
$508$	0	0
$509$	−6849.13	−0.596429	−0.298214	−	0.954499i	$-0.596391\pi$
$509$	−6849.13	−0.596429	−0.298214	+	0.954499i	$0.596391\pi$
$510$	0	0
$511$	−2748.41	−0.237931
$512$	0	0
$513$	9652.10	0.830703
$514$	0	0
$515$	10420.6	0.891620
$516$	0	0
$517$	−3620.88	−0.308019
$518$	0	0
$519$	−24232.1	−2.04947
$520$	0	0
$521$	−11044.1	−0.928696	−0.464348	−	0.885653i	$-0.653711\pi$
$521$	−11044.1	−0.928696	−0.464348	+	0.885653i	$0.653711\pi$
$522$	0	0
$523$	−3115.75	−0.260502	−0.130251	−	0.991481i	$-0.541578\pi$
$523$	−3115.75	−0.260502	−0.130251	+	0.991481i	$0.541578\pi$
$524$	0	0
$525$	5165.49	0.429410
$526$	0	0
$527$	2960.94	0.244745
$528$	0	0
$529$	−12097.0	−0.994251
$530$	0	0
$531$	−35232.6	−2.87941
$532$	0	0
$533$	10559.2	0.858103
$534$	0	0
$535$	336.271	0.0271743
$536$	0	0
$537$	−1355.64	−0.108939
$538$	0	0
$539$	4559.35	0.364350
$540$	0	0
$541$	−14840.6	−1.17938	−0.589691	−	0.807629i	$-0.700751\pi$
$541$	−14840.6	−1.17938	−0.589691	+	0.807629i	$0.700751\pi$
$542$	0	0
$543$	33121.8	2.61767
$544$	0	0
$545$	7295.76	0.573424
$546$	0	0
$547$	12622.8	0.986679	0.493340	−	0.869837i	$-0.335776\pi$
$547$	12622.8	0.986679	0.493340	+	0.869837i	$0.335776\pi$
$548$	0	0
$549$	−24257.5	−1.88576
$550$	0	0
$551$	−2790.37	−0.215742
$552$	0	0
$553$	−21006.7	−1.61536
$554$	0	0
$555$	−6270.50	−0.479582
$556$	0	0
$557$	−13382.1	−1.01798	−0.508992	−	0.860771i	$-0.669982\pi$
$557$	−13382.1	−1.01798	−0.508992	+	0.860771i	$0.669982\pi$
$558$	0	0
$559$	−7444.23	−0.563251
$560$	0	0
$561$	−5176.80	−0.389599
$562$	0	0
$563$	−153.895	−0.0115203	−0.00576014	−	0.999983i	$-0.501834\pi$
$563$	−153.895	−0.0115203	−0.00576014	+	0.999983i	$0.501834\pi$
$564$	0	0
$565$	2291.31	0.170613
$566$	0	0
$567$	37011.7	2.74135
$568$	0	0
$569$	5395.71	0.397539	0.198770	−	0.980046i	$-0.436305\pi$
$569$	5395.71	0.397539	0.198770	+	0.980046i	$0.436305\pi$
$570$	0	0
$571$	22390.5	1.64100	0.820500	−	0.571646i	$-0.193695\pi$
$571$	22390.5	1.64100	0.820500	+	0.571646i	$0.193695\pi$
$572$	0	0
$573$	13573.2	0.989578
$574$	0	0
$575$	−209.091	−0.0151647
$576$	0	0
$577$	−1935.38	−0.139638	−0.0698188	−	0.997560i	$-0.522242\pi$
$577$	−1935.38	−0.139638	−0.0698188	+	0.997560i	$0.522242\pi$
$578$	0	0
$579$	−25682.2	−1.84338
$580$	0	0
$581$	−1186.82	−0.0847461
$582$	0	0
$583$	−5760.94	−0.409252
$584$	0	0
$585$	7806.18	0.551702
$586$	0	0
$587$	7905.44	0.555865	0.277932	−	0.960601i	$-0.410351\pi$
$587$	7905.44	0.555865	0.277932	+	0.960601i	$0.410351\pi$
$588$	0	0
$589$	−5452.17	−0.381414
$590$	0	0
$591$	−25841.4	−1.79860
$592$	0	0
$593$	−11141.6	−0.771552	−0.385776	−	0.922592i	$-0.626066\pi$
$593$	−11141.6	−0.771552	−0.385776	+	0.922592i	$0.626066\pi$
$594$	0	0
$595$	−1563.19	−0.107705
$596$	0	0
$597$	33940.4	2.32678
$598$	0	0
$599$	278.847	0.0190207	0.00951033	−	0.999955i	$-0.496973\pi$
$599$	278.847	0.0190207	0.00951033	+	0.999955i	$0.496973\pi$
$600$	0	0
$601$	−18890.7	−1.28214	−0.641071	−	0.767482i	$-0.721510\pi$
$601$	−18890.7	−1.28214	−0.641071	+	0.767482i	$0.721510\pi$
$602$	0	0
$603$	1256.14	0.0848321
$604$	0	0
$605$	296.642	0.0199342
$606$	0	0
$607$	−5398.05	−0.360956	−0.180478	−	0.983579i	$-0.557764\pi$
$607$	−5398.05	−0.360956	−0.180478	+	0.983579i	$0.557764\pi$
$608$	0	0
$609$	−21602.6	−1.43741
$610$	0	0
$611$	−2341.25	−0.155019
$612$	0	0
$613$	−14412.8	−0.949635	−0.474817	−	0.880084i	$-0.657486\pi$
$613$	−14412.8	−0.949635	−0.474817	+	0.880084i	$0.657486\pi$
$614$	0	0
$615$	20976.0	1.37534
$616$	0	0
$617$	−13226.3	−0.863001	−0.431500	−	0.902113i	$-0.642016\pi$
$617$	−13226.3	−0.863001	−0.431500	+	0.902113i	$0.642016\pi$
$618$	0	0
$619$	−14179.4	−0.920705	−0.460352	−	0.887736i	$-0.652277\pi$
$619$	−14179.4	−0.920705	−0.460352	+	0.887736i	$0.652277\pi$
$620$	0	0
$621$	−3024.76	−0.195458
$622$	0	0
$623$	19840.6	1.27592
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	9532.38	0.607156
$628$	0	0
$629$	1897.60	0.120290
$630$	0	0
$631$	1533.79	0.0967657	0.0483828	−	0.998829i	$-0.484593\pi$
$631$	1533.79	0.0967657	0.0483828	+	0.998829i	$0.484593\pi$
$632$	0	0
$633$	−1457.96	−0.0915459
$634$	0	0
$635$	−1043.42	−0.0652079
$636$	0	0
$637$	2948.06	0.183370
$638$	0	0
$639$	−61888.4	−3.83140
$640$	0	0
$641$	−26682.6	−1.64415	−0.822074	−	0.569381i	$-0.807183\pi$
$641$	−26682.6	−1.64415	−0.822074	+	0.569381i	$0.807183\pi$
$642$	0	0
$643$	−16498.3	−1.01186	−0.505931	−	0.862574i	$-0.668851\pi$
$643$	−16498.3	−1.01186	−0.505931	+	0.862574i	$0.668851\pi$
$644$	0	0
$645$	−14788.1	−0.902762
$646$	0	0
$647$	−20127.5	−1.22302	−0.611510	−	0.791236i	$-0.709438\pi$
$647$	−20127.5	−1.22302	−0.611510	+	0.791236i	$0.709438\pi$
$648$	0	0
$649$	−20287.5	−1.22705
$650$	0	0
$651$	−42209.9	−2.54122
$652$	0	0
$653$	−21447.4	−1.28530	−0.642650	−	0.766160i	$-0.722165\pi$
$653$	−21447.4	−1.28530	−0.642650	+	0.766160i	$0.722165\pi$
$654$	0	0
$655$	8923.43	0.532316
$656$	0	0
$657$	−8250.91	−0.489953
$658$	0	0
$659$	−19402.5	−1.14691	−0.573455	−	0.819237i	$-0.694397\pi$
$659$	−19402.5	−1.14691	−0.573455	+	0.819237i	$0.694397\pi$
$660$	0	0
$661$	−9326.38	−0.548796	−0.274398	−	0.961616i	$-0.588479\pi$
$661$	−9326.38	−0.548796	−0.274398	+	0.961616i	$0.588479\pi$
$662$	0	0
$663$	−3347.31	−0.196076
$664$	0	0
$665$	2878.41	0.167850
$666$	0	0
$667$	874.443	0.0507625
$668$	0	0
$669$	19784.3	1.14336
$670$	0	0
$671$	−13967.8	−0.803608
$672$	0	0
$673$	−15027.4	−0.860718	−0.430359	−	0.902658i	$-0.641613\pi$
$673$	−15027.4	−0.860718	−0.430359	+	0.902658i	$0.641613\pi$
$674$	0	0
$675$	9041.38	0.515560
$676$	0	0
$677$	9717.31	0.551649	0.275825	−	0.961208i	$-0.411049\pi$
$677$	9717.31	0.551649	0.275825	+	0.961208i	$0.411049\pi$
$678$	0	0
$679$	6427.63	0.363284
$680$	0	0
$681$	28166.3	1.58493
$682$	0	0
$683$	−11473.9	−0.642804	−0.321402	−	0.946943i	$-0.604154\pi$
$683$	−11473.9	−0.642804	−0.321402	+	0.946943i	$0.604154\pi$
$684$	0	0
$685$	−11903.0	−0.663927
$686$	0	0
$687$	47413.4	2.63309
$688$	0	0
$689$	−3725.01	−0.205967
$690$	0	0
$691$	−28412.6	−1.56421	−0.782103	−	0.623149i	$-0.785853\pi$
$691$	−28412.6	−1.56421	−0.782103	+	0.623149i	$0.785853\pi$
$692$	0	0
$693$	52082.3	2.85490
$694$	0	0
$695$	14292.6	0.780071
$696$	0	0
$697$	−6347.82	−0.344965
$698$	0	0
$699$	−49746.2	−2.69181
$700$	0	0
$701$	12291.3	0.662250	0.331125	−	0.943587i	$-0.392572\pi$
$701$	12291.3	0.662250	0.331125	+	0.943587i	$0.392572\pi$
$702$	0	0
$703$	−3494.17	−0.187461
$704$	0	0
$705$	−4650.94	−0.248461
$706$	0	0
$707$	−38686.9	−2.05795
$708$	0	0
$709$	−12774.2	−0.676650	−0.338325	−	0.941029i	$-0.609860\pi$
$709$	−12774.2	−0.676650	−0.338325	+	0.941029i	$0.609860\pi$
$710$	0	0
$711$	−63063.4	−3.32639
$712$	0	0
$713$	1708.59	0.0897438
$714$	0	0
$715$	4494.91	0.235105
$716$	0	0
$717$	19792.9	1.03093
$718$	0	0
$719$	15748.6	0.816861	0.408430	−	0.912790i	$-0.366076\pi$
$719$	15748.6	0.816861	0.408430	+	0.912790i	$0.366076\pi$
$720$	0	0
$721$	44954.8	2.32206
$722$	0	0
$723$	−22118.4	−1.13775
$724$	0	0
$725$	−2613.82	−0.133896
$726$	0	0
$727$	9536.44	0.486502	0.243251	−	0.969963i	$-0.421786\pi$
$727$	9536.44	0.486502	0.243251	+	0.969963i	$0.421786\pi$
$728$	0	0
$729$	17576.5	0.892980
$730$	0	0
$731$	4475.22	0.226432
$732$	0	0
$733$	34913.5	1.75929	0.879645	−	0.475631i	$-0.157780\pi$
$733$	34913.5	1.75929	0.879645	+	0.475631i	$0.157780\pi$
$734$	0	0
$735$	5856.38	0.293899
$736$	0	0
$737$	723.301	0.0361508
$738$	0	0
$739$	19324.3	0.961918	0.480959	−	0.876743i	$-0.340289\pi$
$739$	19324.3	0.961918	0.480959	+	0.876743i	$0.340289\pi$
$740$	0	0
$741$	6163.61	0.305568
$742$	0	0
$743$	−7053.57	−0.348278	−0.174139	−	0.984721i	$-0.555714\pi$
$743$	−7053.57	−0.348278	−0.174139	+	0.984721i	$0.555714\pi$
$744$	0	0
$745$	4489.47	0.220781
$746$	0	0
$747$	−3562.91	−0.174511
$748$	0	0
$749$	1450.69	0.0707705
$750$	0	0
$751$	36810.6	1.78860	0.894299	−	0.447471i	$-0.147675\pi$
$751$	36810.6	1.78860	0.894299	+	0.447471i	$0.147675\pi$
$752$	0	0
$753$	36424.7	1.76280
$754$	0	0
$755$	−17156.6	−0.827010
$756$	0	0
$757$	−28515.3	−1.36910	−0.684549	−	0.728967i	$-0.740001\pi$
$757$	−28515.3	−1.36910	−0.684549	+	0.728967i	$0.740001\pi$
$758$	0	0
$759$	−2987.24	−0.142859
$760$	0	0
$761$	−1517.92	−0.0723057	−0.0361528	−	0.999346i	$-0.511510\pi$
$761$	−1517.92	−0.0723057	−0.0361528	+	0.999346i	$0.511510\pi$
$762$	0	0
$763$	31474.3	1.49338
$764$	0	0
$765$	−4692.81	−0.221790
$766$	0	0
$767$	−13117.8	−0.617545
$768$	0	0
$769$	−10413.9	−0.488341	−0.244171	−	0.969732i	$-0.578516\pi$
$769$	−10413.9	−0.488341	−0.244171	+	0.969732i	$0.578516\pi$
$770$	0	0
$771$	59088.0	2.76006
$772$	0	0
$773$	32909.8	1.53129	0.765643	−	0.643266i	$-0.222421\pi$
$773$	32909.8	1.53129	0.765643	+	0.643266i	$0.222421\pi$
$774$	0	0
$775$	−5107.19	−0.236717
$776$	0	0
$777$	−27051.3	−1.24898
$778$	0	0
$779$	11688.6	0.537599
$780$	0	0
$781$	−35636.2	−1.63273
$782$	0	0
$783$	−37812.1	−1.72579
$784$	0	0
$785$	−2856.84	−0.129892
$786$	0	0
$787$	35511.6	1.60845	0.804226	−	0.594324i	$-0.202580\pi$
$787$	35511.6	1.60845	0.804226	+	0.594324i	$0.202580\pi$
$788$	0	0
$789$	−15672.1	−0.707150
$790$	0	0
$791$	9884.85	0.444330
$792$	0	0
$793$	−9031.54	−0.404438
$794$	0	0
$795$	−7399.81	−0.330118
$796$	0	0
$797$	22692.3	1.00853	0.504267	−	0.863548i	$-0.331763\pi$
$797$	22692.3	1.00853	0.504267	+	0.863548i	$0.331763\pi$
$798$	0	0
$799$	1407.48	0.0623193
$800$	0	0
$801$	59562.8	2.62740
$802$	0	0
$803$	−4751.00	−0.208791
$804$	0	0
$805$	−902.032	−0.0394937
$806$	0	0
$807$	21669.8	0.945246
$808$	0	0
$809$	22716.0	0.987209	0.493604	−	0.869687i	$-0.335679\pi$
$809$	22716.0	0.987209	0.493604	+	0.869687i	$0.335679\pi$
$810$	0	0
$811$	−17237.0	−0.746327	−0.373164	−	0.927766i	$-0.621727\pi$
$811$	−17237.0	−0.746327	−0.373164	+	0.927766i	$0.621727\pi$
$812$	0	0
$813$	−398.847	−0.0172056
$814$	0	0
$815$	−1184.81	−0.0509227
$816$	0	0
$817$	−8240.51	−0.352875
$818$	0	0
$819$	33676.3	1.43681
$820$	0	0
$821$	15695.7	0.667217	0.333609	−	0.942712i	$-0.391734\pi$
$821$	15695.7	0.667217	0.333609	+	0.942712i	$0.391734\pi$
$822$	0	0
$823$	16856.0	0.713927	0.356963	−	0.934118i	$-0.383812\pi$
$823$	16856.0	0.713927	0.356963	+	0.934118i	$0.383812\pi$
$824$	0	0
$825$	8929.24	0.376820
$826$	0	0
$827$	−3712.67	−0.156109	−0.0780544	−	0.996949i	$-0.524871\pi$
$827$	−3712.67	−0.156109	−0.0780544	+	0.996949i	$0.524871\pi$
$828$	0	0
$829$	−22469.6	−0.941376	−0.470688	−	0.882300i	$-0.655994\pi$
$829$	−22469.6	−0.941376	−0.470688	+	0.882300i	$0.655994\pi$
$830$	0	0
$831$	−20489.3	−0.855313
$832$	0	0
$833$	−1772.27	−0.0737163
$834$	0	0
$835$	7242.30	0.300156
$836$	0	0
$837$	−73881.8	−3.05105
$838$	0	0
$839$	−31935.2	−1.31409	−0.657046	−	0.753850i	$-0.728194\pi$
$839$	−31935.2	−1.31409	−0.657046	+	0.753850i	$0.728194\pi$
$840$	0	0
$841$	−13457.7	−0.551795
$842$	0	0
$843$	14306.3	0.584504
$844$	0	0
$845$	−8078.60	−0.328890
$846$	0	0
$847$	1279.73	0.0519150
$848$	0	0
$849$	10442.9	0.422144
$850$	0	0
$851$	1095.00	0.0441081
$852$	0	0
$853$	14782.2	0.593355	0.296678	−	0.954978i	$-0.404121\pi$
$853$	14782.2	0.593355	0.296678	+	0.954978i	$0.404121\pi$
$854$	0	0
$855$	8641.18	0.345640
$856$	0	0
$857$	7222.91	0.287900	0.143950	−	0.989585i	$-0.454020\pi$
$857$	7222.91	0.287900	0.143950	+	0.989585i	$0.454020\pi$
$858$	0	0
$859$	−4646.96	−0.184578	−0.0922889	−	0.995732i	$-0.529418\pi$
$859$	−4646.96	−0.184578	−0.0922889	+	0.995732i	$0.529418\pi$
$860$	0	0
$861$	90491.7	3.58182
$862$	0	0
$863$	−8122.19	−0.320374	−0.160187	−	0.987087i	$-0.551210\pi$
$863$	−8122.19	−0.320374	−0.160187	+	0.987087i	$0.551210\pi$
$864$	0	0
$865$	−12648.7	−0.497189
$866$	0	0
$867$	−45048.9	−1.76464
$868$	0	0
$869$	−36312.8	−1.41752
$870$	0	0
$871$	467.685	0.0181939
$872$	0	0
$873$	19296.2	0.748083
$874$	0	0
$875$	2696.28	0.104173
$876$	0	0
$877$	−31986.1	−1.23158	−0.615790	−	0.787910i	$-0.711163\pi$
$877$	−31986.1	−1.23158	−0.615790	+	0.787910i	$0.711163\pi$
$878$	0	0
$879$	−38962.3	−1.49507
$880$	0	0
$881$	−29450.6	−1.12624	−0.563118	−	0.826376i	$-0.690398\pi$
$881$	−29450.6	−1.12624	−0.563118	+	0.826376i	$0.690398\pi$
$882$	0	0
$883$	46156.5	1.75911	0.879553	−	0.475801i	$-0.157842\pi$
$883$	46156.5	1.75911	0.879553	+	0.475801i	$0.157842\pi$
$884$	0	0
$885$	−26058.8	−0.989783
$886$	0	0
$887$	41051.2	1.55396	0.776982	−	0.629523i	$-0.216750\pi$
$887$	41051.2	1.55396	0.776982	+	0.629523i	$0.216750\pi$
$888$	0	0
$889$	−4501.39	−0.169822
$890$	0	0
$891$	63979.7	2.40561
$892$	0	0
$893$	−2591.69	−0.0971192
$894$	0	0
$895$	−707.620	−0.0264281
$896$	0	0
$897$	−1931.54	−0.0718979
$898$	0	0
$899$	21358.8	0.792389
$900$	0	0
$901$	2239.35	0.0828008
$902$	0	0
$903$	−63796.7	−2.35108
$904$	0	0
$905$	17288.9	0.635031
$906$	0	0
$907$	27915.8	1.02197	0.510987	−	0.859588i	$-0.329280\pi$
$907$	27915.8	1.02197	0.510987	+	0.859588i	$0.329280\pi$
$908$	0	0
$909$	−116141.	−4.23778
$910$	0	0
$911$	6557.22	0.238475	0.119237	−	0.992866i	$-0.461955\pi$
$911$	6557.22	0.238475	0.119237	+	0.992866i	$0.461955\pi$
$912$	0	0
$913$	−2051.57	−0.0743671
$914$	0	0
$915$	−17941.3	−0.648221
$916$	0	0
$917$	38496.2	1.38632
$918$	0	0
$919$	−11493.0	−0.412534	−0.206267	−	0.978496i	$-0.566131\pi$
$919$	−11493.0	−0.412534	−0.206267	+	0.978496i	$0.566131\pi$
$920$	0	0
$921$	64524.0	2.30851
$922$	0	0
$923$	−23042.3	−0.821719
$924$	0	0
$925$	−3273.08	−0.116344
$926$	0	0
$927$	134957.	4.78164
$928$	0	0
$929$	56597.2	1.99881	0.999405	−	0.0345020i	$-0.0109845\pi$
$929$	56597.2	1.99881	0.999405	+	0.0345020i	$0.0109845\pi$
$930$	0	0
$931$	3263.40	0.114880
$932$	0	0
$933$	31364.7	1.10057
$934$	0	0
$935$	−2702.19	−0.0945145
$936$	0	0
$937$	37901.5	1.32144	0.660720	−	0.750633i	$-0.270251\pi$
$937$	37901.5	1.32144	0.660720	+	0.750633i	$0.270251\pi$
$938$	0	0
$939$	89635.5	3.11517
$940$	0	0
$941$	4449.85	0.154156	0.0770781	−	0.997025i	$-0.475441\pi$
$941$	4449.85	0.154156	0.0770781	+	0.997025i	$0.475441\pi$
$942$	0	0
$943$	−3662.97	−0.126493
$944$	0	0
$945$	39005.0	1.34268
$946$	0	0
$947$	225.160	0.00772620	0.00386310	−	0.999993i	$-0.498770\pi$
$947$	225.160	0.00772620	0.00386310	+	0.999993i	$0.498770\pi$
$948$	0	0
$949$	−3071.98	−0.105080
$950$	0	0
$951$	86829.7	2.96072
$952$	0	0
$953$	−4845.81	−0.164713	−0.0823563	−	0.996603i	$-0.526245\pi$
$953$	−4845.81	−0.164713	−0.0823563	+	0.996603i	$0.526245\pi$
$954$	0	0
$955$	7084.94	0.240066
$956$	0	0
$957$	−37343.1	−1.26137
$958$	0	0
$959$	−51350.2	−1.72908
$960$	0	0
$961$	11942.5	0.400876
$962$	0	0
$963$	4355.07	0.145732
$964$	0	0
$965$	−13405.6	−0.447194
$966$	0	0
$967$	40394.8	1.34334	0.671670	−	0.740851i	$-0.265577\pi$
$967$	40394.8	1.34334	0.671670	+	0.740851i	$0.265577\pi$
$968$	0	0
$969$	−3705.36	−0.122841
$970$	0	0
$971$	−29066.2	−0.960637	−0.480318	−	0.877094i	$-0.659479\pi$
$971$	−29066.2	−0.960637	−0.480318	+	0.877094i	$0.659479\pi$
$972$	0	0
$973$	61659.1	2.03155
$974$	0	0
$975$	5773.62	0.189645
$976$	0	0
$977$	42580.5	1.39434	0.697170	−	0.716906i	$-0.254442\pi$
$977$	42580.5	1.39434	0.697170	+	0.716906i	$0.254442\pi$
$978$	0	0
$979$	34297.1	1.11965
$980$	0	0
$981$	94488.0	3.07520
$982$	0	0
$983$	−40586.0	−1.31688	−0.658440	−	0.752633i	$-0.728783\pi$
$983$	−40586.0	−1.31688	−0.658440	+	0.752633i	$0.728783\pi$
$984$	0	0
$985$	−13488.7	−0.436330
$986$	0	0
$987$	−20064.4	−0.647070
$988$	0	0
$989$	2582.40	0.0830288
$990$	0	0
$991$	50370.4	1.61460	0.807299	−	0.590142i	$-0.200928\pi$
$991$	50370.4	1.61460	0.807299	+	0.590142i	$0.200928\pi$
$992$	0	0
$993$	−2682.95	−0.0857409
$994$	0	0
$995$	17716.2	0.564465
$996$	0	0
$997$	22467.5	0.713694	0.356847	−	0.934163i	$-0.383852\pi$
$997$	22467.5	0.713694	0.356847	+	0.934163i	$0.383852\pi$
$998$	0	0
$999$	−47349.1	−1.49956

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1280.4.a.bd.1.6		6
4.3	odd	2		1280.4.a.bb.1.1		6
8.3	odd	2		1280.4.a.bc.1.6		6
8.5	even	2		1280.4.a.ba.1.1		6
16.3	odd	4		40.4.d.a.21.10	yes	12
16.5	even	4		160.4.d.a.81.1		12
16.11	odd	4		40.4.d.a.21.9	✓	12
16.13	even	4		160.4.d.a.81.12		12
48.5	odd	4		1440.4.k.c.721.2		12
48.11	even	4		360.4.k.c.181.4		12
48.29	odd	4		1440.4.k.c.721.8		12
48.35	even	4		360.4.k.c.181.3		12
80.3	even	4		200.4.f.c.149.9		12
80.13	odd	4		800.4.f.b.49.2		12
80.19	odd	4		200.4.d.b.101.3		12
80.27	even	4		200.4.f.c.149.10		12
80.29	even	4		800.4.d.d.401.1		12
80.37	odd	4		800.4.f.b.49.1		12
80.43	even	4		200.4.f.b.149.3		12
80.53	odd	4		800.4.f.c.49.12		12
80.59	odd	4		200.4.d.b.101.4		12
80.67	even	4		200.4.f.b.149.4		12
80.69	even	4		800.4.d.d.401.12		12
80.77	odd	4		800.4.f.c.49.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.4.d.a.21.9	✓	12	16.11	odd	4
40.4.d.a.21.10	yes	12	16.3	odd	4
160.4.d.a.81.1		12	16.5	even	4
160.4.d.a.81.12		12	16.13	even	4
200.4.d.b.101.3		12	80.19	odd	4
200.4.d.b.101.4		12	80.59	odd	4
200.4.f.b.149.3		12	80.43	even	4
200.4.f.b.149.4		12	80.67	even	4
200.4.f.c.149.9		12	80.3	even	4
200.4.f.c.149.10		12	80.27	even	4
360.4.k.c.181.3		12	48.35	even	4
360.4.k.c.181.4		12	48.11	even	4
800.4.d.d.401.1		12	80.29	even	4
800.4.d.d.401.12		12	80.69	even	4
800.4.f.b.49.1		12	80.37	odd	4
800.4.f.b.49.2		12	80.13	odd	4
800.4.f.c.49.11		12	80.77	odd	4
800.4.f.c.49.12		12	80.53	odd	4
1280.4.a.ba.1.1		6	8.5	even	2
1280.4.a.bb.1.1		6	4.3	odd	2
1280.4.a.bc.1.6		6	8.3	odd	2
1280.4.a.bd.1.6		6	1.1	even	1	trivial
1440.4.k.c.721.2		12	48.5	odd	4
1440.4.k.c.721.8		12	48.29	odd	4

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:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1280.a (trivial)

\(f(q)\)	\(=\)	\(q+9.57890 q^{3} +5.00000 q^{5} +21.5703 q^{7} +64.7554 q^{9} +37.2871 q^{11} +24.1097 q^{13} +47.8945 q^{15} -14.4940 q^{17} +26.6887 q^{19} +206.620 q^{21} -8.36366 q^{23} +25.0000 q^{25} +361.655 q^{27} -104.553 q^{29} -204.288 q^{31} +357.170 q^{33} +107.851 q^{35} -130.923 q^{37} +230.945 q^{39} +437.963 q^{41} -308.764 q^{43} +323.777 q^{45} -97.1081 q^{47} +122.277 q^{49} -138.836 q^{51} -154.502 q^{53} +186.436 q^{55} +255.648 q^{57} -544.088 q^{59} -374.601 q^{61} +1396.79 q^{63} +120.549 q^{65} +19.3982 q^{67} -80.1147 q^{69} -955.725 q^{71} -127.417 q^{73} +239.473 q^{75} +804.293 q^{77} -973.871 q^{79} +1715.87 q^{81} -55.0210 q^{83} -72.4698 q^{85} -1001.50 q^{87} +919.812 q^{89} +520.054 q^{91} -1956.85 q^{93} +133.443 q^{95} +297.986 q^{97} +2414.54 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} + 30 q^{5} + 14 q^{7} + 54 q^{9} + 44 q^{11} + 30 q^{15} + 152 q^{19} + 4 q^{21} + 302 q^{23} + 150 q^{25} + 216 q^{27} + 132 q^{31} - 116 q^{33} + 70 q^{35} - 68 q^{37} + 300 q^{39} - 20 q^{41}+ \cdots + 5516 q^{99}+O(q^{100}) \) 6 * q + 6 * q^3 + 30 * q^5 + 14 * q^7 + 54 * q^9 + 44 * q^11 + 30 * q^15 + 152 * q^19 + 4 * q^21 + 302 * q^23 + 150 * q^25 + 216 * q^27 + 132 * q^31 - 116 * q^33 + 70 * q^35 - 68 * q^37 + 300 * q^39 - 20 * q^41 + 602 * q^43 + 270 * q^45 + 470 * q^47 + 654 * q^49 + 612 * q^51 + 528 * q^53 + 220 * q^55 + 340 * q^57 + 472 * q^59 - 476 * q^61 + 650 * q^63 + 1206 * q^67 - 980 * q^69 - 796 * q^71 - 216 * q^73 + 150 * q^75 + 412 * q^77 - 1008 * q^79 + 1254 * q^81 + 1778 * q^83 - 984 * q^87 + 212 * q^89 + 3652 * q^91 - 1392 * q^93 + 760 * q^95 - 792 * q^97 + 5516 * q^99

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