LMFDB - Embedded newform 1280.3.g.f.1151.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,3,Mod(1151,1280)]

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([1, 1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1280.1151");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$N$	$=$	$1280 = 2^{8} \cdot 5$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	1280.g (of order $2$ , degree $1$ , not minimal)

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:	no
Analytic conductor:	$34.8774738381$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.12960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1$ x^8 - 3x^6 + 8x^4 - 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$2^{16}$
Twist minimal:	no (minimal twist has level 80)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.2
Root			$1.40126 + 0.809017i$ of defining polynomial
Character	$\chi$	$=$	1280.1151
Dual form			1280.3.g.f.1151.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.60503 q^{3} +2.23607i q^{5} +1.32317i q^{7} +22.4164 q^{9} +11.2101 q^{11} +17.4164i q^{13} -12.5332i q^{15} +18.0000 q^{17} -5.29268 q^{19} -7.41641i q^{21} -15.1796i q^{23} -5.00000 q^{25} -75.1994 q^{27} +8.83282i q^{29} -42.1939i q^{31} -62.8328 q^{33} -2.95870 q^{35} -33.4164i q^{37} -97.6196i q^{39} +28.2492 q^{41} -25.3788 q^{43} +50.1246i q^{45} +10.5116i q^{47} +47.2492 q^{49} -100.891 q^{51} +28.2492i q^{53} +25.0665i q^{55} +29.6656 q^{57} +44.8403 q^{59} -77.4164i q^{61} +29.6607i q^{63} -38.9443 q^{65} +36.5889 q^{67} +85.0820i q^{69} +97.6196i q^{71} -15.6656 q^{73} +28.0252 q^{75} +14.8328i q^{77} -112.101i q^{79} +219.748 q^{81} -92.0145 q^{83} +40.2492i q^{85} -49.5082i q^{87} +59.6656 q^{89} -23.0449 q^{91} +236.498i q^{93} -11.8348i q^{95} +108.164 q^{97} +251.289 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$8 q + 72 q^{9} + 144 q^{17} - 40 q^{25} - 288 q^{33} - 96 q^{41} + 56 q^{49} - 192 q^{57} - 240 q^{65} + 304 q^{73} + 792 q^{81} + 48 q^{89} - 208 q^{97}+O(q^{100})$ 8 * q + 72 * q^9 + 144 * q^17 - 40 * q^25 - 288 * q^33 - 96 * q^41 + 56 * q^49 - 192 * q^57 - 240 * q^65 + 304 * q^73 + 792 * q^81 + 48 * q^89 - 208 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$ .

$n$	$257$	$261$	$511$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	−5.60503	−1.86834	−0.934172	−	0.356822i	$-0.883860\pi$
$3$	−5.60503	−1.86834	−0.934172	+	0.356822i	$0.883860\pi$
$4$	0	0
$5$	2.23607i	0.447214i
$5$	2.23607i	0.447214i
$6$	0	0
$7$	1.32317i	0.189024i	0.995524	+	0.0945121i	$0.0301291\pi$
$7$	1.32317i	0.189024i	−0.995524	+	0.0945121i	$0.969871\pi$
$8$	0	0
$9$	22.4164	2.49071
$10$	0	0
$11$	11.2101	1.01910	0.509549	−	0.860442i	$-0.329812\pi$
$11$	11.2101	1.01910	0.509549	+	0.860442i	$0.329812\pi$
$12$	0	0
$13$	17.4164i	1.33972i	0.742486	+	0.669862i	$0.233647\pi$
$13$	17.4164i	1.33972i	−0.742486	+	0.669862i	$0.766353\pi$
$14$	0	0
$15$	− 12.5332i	− 0.835549i
$16$	0	0
$17$	18.0000	1.05882	0.529412	−	0.848365i	$-0.322413\pi$
$17$	18.0000	1.05882	0.529412	+	0.848365i	$0.322413\pi$
$18$	0	0
$19$	−5.29268	−0.278562	−0.139281	−	0.990253i	$-0.544479\pi$
$19$	−5.29268	−0.278562	−0.139281	+	0.990253i	$0.544479\pi$
$20$	0	0
$21$	− 7.41641i	− 0.353162i
$22$	0	0
$23$	− 15.1796i	− 0.659982i	−0.943984	−	0.329991i	$-0.892954\pi$
$23$	− 15.1796i	− 0.659982i	0.943984	−	0.329991i	$-0.107046\pi$
$24$	0	0
$25$	−5.00000	−0.200000
$26$	0	0
$27$	−75.1994	−2.78516
$28$	0	0
$29$	8.83282i	0.304580i	0.988336	+	0.152290i	$0.0486647\pi$
$29$	8.83282i	0.304580i	−0.988336	+	0.152290i	$0.951335\pi$
$30$	0	0
$31$	− 42.1939i	− 1.36109i	−0.732704	−	0.680547i	$-0.761742\pi$
$31$	− 42.1939i	− 1.36109i	0.732704	−	0.680547i	$-0.238258\pi$
$32$	0	0
$33$	−62.8328	−1.90402
$34$	0	0
$35$	−2.95870	−0.0845342
$36$	0	0
$37$	− 33.4164i	− 0.903146i	−0.892234	−	0.451573i	$-0.850863\pi$
$37$	− 33.4164i	− 0.903146i	0.892234	−	0.451573i	$-0.149137\pi$
$38$	0	0
$39$	− 97.6196i	− 2.50307i
$40$	0	0
$41$	28.2492	0.689005	0.344503	−	0.938785i	$-0.388048\pi$
$41$	28.2492	0.689005	0.344503	+	0.938785i	$0.388048\pi$
$42$	0	0
$43$	−25.3788	−0.590205	−0.295103	−	0.955466i	$-0.595354\pi$
$43$	−25.3788	−0.590205	−0.295103	+	0.955466i	$0.595354\pi$
$44$	0	0
$45$	50.1246i	1.11388i
$46$	0	0
$47$	10.5116i	0.223651i	0.993728	+	0.111826i	$0.0356698\pi$
$47$	10.5116i	0.223651i	−0.993728	+	0.111826i	$0.964330\pi$
$48$	0	0
$49$	47.2492	0.964270
$50$	0	0
$51$	−100.891	−1.97825
$52$	0	0
$53$	28.2492i	0.533004i	0.963834	+	0.266502i	$0.0858679\pi$
$53$	28.2492i	0.533004i	−0.963834	+	0.266502i	$0.914132\pi$
$54$	0	0
$55$	25.0665i	0.455754i
$56$	0	0
$57$	29.6656	0.520450
$58$	0	0
$59$	44.8403	0.760005	0.380002	−	0.924986i	$-0.375923\pi$
$59$	44.8403	0.760005	0.380002	+	0.924986i	$0.375923\pi$
$60$	0	0
$61$	− 77.4164i	− 1.26912i	−0.772873	−	0.634561i	$-0.781181\pi$
$61$	− 77.4164i	− 1.26912i	0.772873	−	0.634561i	$-0.218819\pi$
$62$	0	0
$63$	29.6607i	0.470805i
$64$	0	0
$65$	−38.9443	−0.599143
$66$	0	0
$67$	36.5889	0.546103	0.273051	−	0.961999i	$-0.411967\pi$
$67$	36.5889	0.546103	0.273051	+	0.961999i	$0.411967\pi$
$68$	0	0
$69$	85.0820i	1.23307i
$70$	0	0
$71$	97.6196i	1.37492i	0.726221	+	0.687462i	$0.241275\pi$
$71$	97.6196i	1.37492i	−0.726221	+	0.687462i	$0.758725\pi$
$72$	0	0
$73$	−15.6656	−0.214598	−0.107299	−	0.994227i	$-0.534220\pi$
$73$	−15.6656	−0.214598	−0.107299	+	0.994227i	$0.534220\pi$
$74$	0	0
$75$	28.0252	0.373669
$76$	0	0
$77$	14.8328i	0.192634i
$78$	0	0
$79$	− 112.101i	− 1.41900i	−0.704707	−	0.709498i	$-0.748922\pi$
$79$	− 112.101i	− 1.41900i	0.704707	−	0.709498i	$-0.251078\pi$
$80$	0	0
$81$	219.748	2.71293
$82$	0	0
$83$	−92.0145	−1.10861	−0.554304	−	0.832314i	$-0.687016\pi$
$83$	−92.0145	−1.10861	−0.554304	+	0.832314i	$0.687016\pi$
$84$	0	0
$85$	40.2492i	0.473520i
$86$	0	0
$87$	− 49.5082i	− 0.569060i
$88$	0	0
$89$	59.6656	0.670400	0.335200	−	0.942147i	$-0.391196\pi$
$89$	59.6656	0.670400	0.335200	+	0.942147i	$0.391196\pi$
$90$	0	0
$91$	−23.0449	−0.253240
$92$	0	0
$93$	236.498i	2.54299i
$94$	0	0
$95$	− 11.8348i	− 0.124577i
$96$	0	0
$97$	108.164	1.11509	0.557547	−	0.830146i	$-0.311743\pi$
$97$	108.164	1.11509	0.557547	+	0.830146i	$0.311743\pi$
$98$	0	0
$99$	251.289	2.53828
$100$	0	0
$101$	83.6656i	0.828373i	0.910192	+	0.414186i	$0.135934\pi$
$101$	83.6656i	0.828373i	−0.910192	+	0.414186i	$0.864066\pi$
$102$	0	0
$103$	135.219i	1.31281i	0.754409	+	0.656404i	$0.227923\pi$
$103$	135.219i	1.31281i	−0.754409	+	0.656404i	$0.772077\pi$
$104$	0	0
$105$	16.5836	0.157939
$106$	0	0
$107$	122.374	1.14368	0.571840	−	0.820365i	$-0.306230\pi$
$107$	122.374	1.14368	0.571840	+	0.820365i	$0.306230\pi$
$108$	0	0
$109$	2.58359i	0.0237027i	0.999930	+	0.0118513i	$0.00377249\pi$
$109$	2.58359i	0.0237027i	−0.999930	+	0.0118513i	$0.996228\pi$
$110$	0	0
$111$	187.300i	1.68739i
$112$	0	0
$113$	−33.5016	−0.296474	−0.148237	−	0.988952i	$-0.547360\pi$
$113$	−33.5016	−0.296474	−0.148237	+	0.988952i	$0.547360\pi$
$114$	0	0
$115$	33.9426	0.295153
$116$	0	0
$117$	390.413i	3.33687i
$118$	0	0
$119$	23.8170i	0.200143i
$120$	0	0
$121$	4.66563	0.0385589
$122$	0	0
$123$	−158.338	−1.28730
$124$	0	0
$125$	− 11.1803i	− 0.0894427i
$126$	0	0
$127$	128.677i	1.01321i	0.862179	+	0.506603i	$0.169099\pi$
$127$	128.677i	1.01321i	−0.862179	+	0.506603i	$0.830901\pi$
$128$	0	0
$129$	142.249	1.10271
$130$	0	0
$131$	222.327	1.69715	0.848577	−	0.529072i	$-0.177460\pi$
$131$	222.327	1.69715	0.848577	+	0.529072i	$0.177460\pi$
$132$	0	0
$133$	− 7.00311i	− 0.0526549i
$134$	0	0
$135$	− 168.151i	− 1.24556i
$136$	0	0
$137$	−128.833	−0.940386	−0.470193	−	0.882564i	$-0.655816\pi$
$137$	−128.833	−0.940386	−0.470193	+	0.882564i	$0.655816\pi$
$138$	0	0
$139$	230.743	1.66002	0.830012	−	0.557745i	$-0.188333\pi$
$139$	230.743	1.66002	0.830012	+	0.557745i	$0.188333\pi$
$140$	0	0
$141$	− 58.9180i	− 0.417858i
$142$	0	0
$143$	195.239i	1.36531i
$144$	0	0
$145$	−19.7508	−0.136212
$146$	0	0
$147$	−264.834	−1.80159
$148$	0	0
$149$	− 285.915i	− 1.91889i	−0.281893	−	0.959446i	$-0.590962\pi$
$149$	− 285.915i	− 1.91889i	0.281893	−	0.959446i	$-0.409038\pi$
$150$	0	0
$151$	− 64.6141i	− 0.427908i	−0.976844	−	0.213954i	$-0.931366\pi$
$151$	− 64.6141i	− 0.427908i	0.976844	−	0.213954i	$-0.0686342\pi$
$152$	0	0
$153$	403.495	2.63722
$154$	0	0
$155$	94.3485	0.608700
$156$	0	0
$157$	− 14.5836i	− 0.0928891i	−0.998921	−	0.0464446i	$-0.985211\pi$
$157$	− 14.5836i	− 0.0928891i	0.998921	−	0.0464446i	$-0.0147891\pi$
$158$	0	0
$159$	− 158.338i	− 0.995836i
$160$	0	0
$161$	20.0851	0.124752
$162$	0	0
$163$	20.8583	0.127965	0.0639827	−	0.997951i	$-0.479620\pi$
$163$	20.8583	0.127965	0.0639827	+	0.997951i	$0.479620\pi$
$164$	0	0
$165$	− 140.498i	− 0.851506i
$166$	0	0
$167$	216.961i	1.29917i	0.760290	+	0.649583i	$0.225057\pi$
$167$	216.961i	1.29917i	−0.760290	+	0.649583i	$0.774943\pi$
$168$	0	0
$169$	−134.331	−0.794860
$170$	0	0
$171$	−118.643	−0.693817
$172$	0	0
$173$	− 46.5836i	− 0.269269i	−0.990895	−	0.134635i	$-0.957014\pi$
$173$	− 46.5836i	− 0.269269i	0.990895	−	0.134635i	$-0.0429861\pi$
$174$	0	0
$175$	− 6.61585i	− 0.0378048i
$176$	0	0
$177$	−251.331	−1.41995
$178$	0	0
$179$	6.54211	0.0365481	0.0182740	−	0.999833i	$-0.494183\pi$
$179$	6.54211	0.0365481	0.0182740	+	0.999833i	$0.494183\pi$
$180$	0	0
$181$	241.331i	1.33332i	0.745361	+	0.666661i	$0.232277\pi$
$181$	241.331i	1.33332i	−0.745361	+	0.666661i	$0.767723\pi$
$182$	0	0
$183$	433.922i	2.37116i
$184$	0	0
$185$	74.7214	0.403899
$186$	0	0
$187$	201.781	1.07904
$188$	0	0
$189$	− 99.5016i	− 0.526463i
$190$	0	0
$191$	59.3214i	0.310583i	0.987869	+	0.155292i	$0.0496317\pi$
$191$	59.3214i	0.310583i	−0.987869	+	0.155292i	$0.950368\pi$
$192$	0	0
$193$	234.997	1.21760	0.608800	−	0.793324i	$-0.291651\pi$
$193$	234.997	1.21760	0.608800	+	0.793324i	$0.291651\pi$
$194$	0	0
$195$	218.284	1.11940
$196$	0	0
$197$	− 230.912i	− 1.17214i	−0.810260	−	0.586070i	$-0.800674\pi$
$197$	− 230.912i	− 1.17214i	0.810260	−	0.586070i	$-0.199326\pi$
$198$	0	0
$199$	− 27.7128i	− 0.139260i	−0.997573	−	0.0696302i	$-0.977818\pi$
$199$	− 27.7128i	− 0.139260i	0.997573	−	0.0696302i	$-0.0221819\pi$
$200$	0	0
$201$	−205.082	−1.02031
$202$	0	0
$203$	−11.6873	−0.0575729
$204$	0	0
$205$	63.1672i	0.308133i
$206$	0	0
$207$	− 340.272i	− 1.64382i
$208$	0	0
$209$	−59.3313	−0.283882
$210$	0	0
$211$	−93.0991	−0.441228	−0.220614	−	0.975361i	$-0.570806\pi$
$211$	−93.0991	−0.441228	−0.220614	+	0.975361i	$0.570806\pi$
$212$	0	0
$213$	− 547.161i	− 2.56883i
$214$	0	0
$215$	− 56.7488i	− 0.263948i
$216$	0	0
$217$	55.8297	0.257280
$218$	0	0
$219$	87.8064	0.400942
$220$	0	0
$221$	313.495i	1.41853i
$222$	0	0
$223$	307.891i	1.38068i	0.723487	+	0.690338i	$0.242538\pi$
$223$	307.891i	1.38068i	−0.723487	+	0.690338i	$0.757462\pi$
$224$	0	0
$225$	−112.082	−0.498142
$226$	0	0
$227$	255.020	1.12344	0.561719	−	0.827328i	$-0.310140\pi$
$227$	255.020	1.12344	0.561719	+	0.827328i	$0.310140\pi$
$228$	0	0
$229$	411.495i	1.79692i	0.439052	+	0.898461i	$0.355314\pi$
$229$	411.495i	1.79692i	−0.439052	+	0.898461i	$0.644686\pi$
$230$	0	0
$231$	− 83.1384i	− 0.359907i
$232$	0	0
$233$	332.833	1.42847	0.714234	−	0.699907i	$-0.246775\pi$
$233$	332.833	1.42847	0.714234	+	0.699907i	$0.246775\pi$
$234$	0	0
$235$	−23.5047	−0.100020
$236$	0	0
$237$	628.328i	2.65117i
$238$	0	0
$239$	− 266.248i	− 1.11401i	−0.830510	−	0.557004i	$-0.811951\pi$
$239$	− 266.248i	− 1.11401i	0.830510	−	0.557004i	$-0.188049\pi$
$240$	0	0
$241$	−78.5836	−0.326073	−0.163036	−	0.986620i	$-0.552129\pi$
$241$	−78.5836	−0.326073	−0.163036	+	0.986620i	$0.552129\pi$
$242$	0	0
$243$	−554.898	−2.28353
$244$	0	0
$245$	105.652i	0.431235i
$246$	0	0
$247$	− 92.1794i	− 0.373196i
$248$	0	0
$249$	515.745	2.07126
$250$	0	0
$251$	−404.482	−1.61148	−0.805741	−	0.592268i	$-0.798233\pi$
$251$	−404.482	−1.61148	−0.805741	+	0.592268i	$0.798233\pi$
$252$	0	0
$253$	− 170.164i	− 0.672585i
$254$	0	0
$255$	− 225.598i	− 0.884699i
$256$	0	0
$257$	−85.8297	−0.333968	−0.166984	−	0.985960i	$-0.553403\pi$
$257$	−85.8297	−0.333968	−0.166984	+	0.985960i	$0.553403\pi$
$258$	0	0
$259$	44.2156	0.170716
$260$	0	0
$261$	198.000i	0.758621i
$262$	0	0
$263$	− 440.685i	− 1.67561i	−0.545971	−	0.837804i	$-0.683839\pi$
$263$	− 440.685i	− 1.67561i	0.545971	−	0.837804i	$-0.316161\pi$
$264$	0	0
$265$	−63.1672	−0.238367
$266$	0	0
$267$	−334.428	−1.25254
$268$	0	0
$269$	− 205.416i	− 0.763630i	−0.924239	−	0.381815i	$-0.875299\pi$
$269$	− 205.416i	− 0.763630i	0.924239	−	0.381815i	$-0.124701\pi$
$270$	0	0
$271$	418.044i	1.54260i	0.636474	+	0.771298i	$0.280392\pi$
$271$	418.044i	1.54260i	−0.636474	+	0.771298i	$0.719608\pi$
$272$	0	0
$273$	129.167	0.473140
$274$	0	0
$275$	−56.0503	−0.203819
$276$	0	0
$277$	511.410i	1.84625i	0.384505	+	0.923123i	$0.374372\pi$
$277$	511.410i	1.84625i	−0.384505	+	0.923123i	$0.625628\pi$
$278$	0	0
$279$	− 945.836i	− 3.39009i
$280$	0	0
$281$	−115.751	−0.411924	−0.205962	−	0.978560i	$-0.566032\pi$
$281$	−115.751	−0.411924	−0.205962	+	0.978560i	$0.566032\pi$
$282$	0	0
$283$	−402.001	−1.42050	−0.710249	−	0.703951i	$-0.751417\pi$
$283$	−402.001	−1.42050	−0.710249	+	0.703951i	$0.751417\pi$
$284$	0	0
$285$	66.3344i	0.232752i
$286$	0	0
$287$	37.3785i	0.130239i
$288$	0	0
$289$	35.0000	0.121107
$290$	0	0
$291$	−606.263	−2.08338
$292$	0	0
$293$	− 90.4133i	− 0.308578i	−0.988026	−	0.154289i	$-0.950691\pi$
$293$	− 90.4133i	− 0.308578i	0.988026	−	0.154289i	$-0.0493087\pi$
$294$	0	0
$295$	100.266i	0.339884i
$296$	0	0
$297$	−842.991	−2.83835
$298$	0	0
$299$	264.374	0.884193
$300$	0	0
$301$	− 33.5805i	− 0.111563i
$302$	0	0
$303$	− 468.949i	− 1.54769i
$304$	0	0
$305$	173.108	0.567568
$306$	0	0
$307$	275.419	0.897130	0.448565	−	0.893750i	$-0.351935\pi$
$307$	275.419	0.897130	0.448565	+	0.893750i	$0.351935\pi$
$308$	0	0
$309$	− 757.909i	− 2.45278i
$310$	0	0
$311$	485.304i	1.56046i	0.625491	+	0.780231i	$0.284899\pi$
$311$	485.304i	1.56046i	−0.625491	+	0.780231i	$0.715101\pi$
$312$	0	0
$313$	−2.00000	−0.00638978	−0.00319489	−	0.999995i	$-0.501017\pi$
$313$	−2.00000	−0.00638978	−0.00319489	+	0.999995i	$0.501017\pi$
$314$	0	0
$315$	−66.3233	−0.210550
$316$	0	0
$317$	− 394.741i	− 1.24524i	−0.782524	−	0.622621i	$-0.786068\pi$
$317$	− 394.741i	− 1.24524i	0.782524	−	0.622621i	$-0.213932\pi$
$318$	0	0
$319$	99.0165i	0.310396i
$320$	0	0
$321$	−685.909	−2.13679
$322$	0	0
$323$	−95.2682	−0.294948
$324$	0	0
$325$	− 87.0820i	− 0.267945i
$326$	0	0
$327$	− 14.4811i	− 0.0442848i
$328$	0	0
$329$	−13.9086	−0.0422755
$330$	0	0
$331$	296.130	0.894652	0.447326	−	0.894371i	$-0.352376\pi$
$331$	296.130	0.894652	0.447326	+	0.894371i	$0.352376\pi$
$332$	0	0
$333$	− 749.076i	− 2.24948i
$334$	0	0
$335$	81.8153i	0.244225i
$336$	0	0
$337$	19.9938	0.0593288	0.0296644	−	0.999560i	$-0.490556\pi$
$337$	19.9938	0.0593288	0.0296644	+	0.999560i	$0.490556\pi$
$338$	0	0
$339$	187.777	0.553915
$340$	0	0
$341$	− 472.997i	− 1.38709i
$342$	0	0
$343$	127.354i	0.371294i
$344$	0	0
$345$	−190.249	−0.551447
$346$	0	0
$347$	−546.959	−1.57625	−0.788126	−	0.615514i	$-0.788948\pi$
$347$	−546.959	−1.57625	−0.788126	+	0.615514i	$0.788948\pi$
$348$	0	0
$349$	40.8328i	0.116999i	0.998287	+	0.0584997i	$0.0186317\pi$
$349$	40.8328i	0.116999i	−0.998287	+	0.0584997i	$0.981368\pi$
$350$	0	0
$351$	− 1309.70i	− 3.73135i
$352$	0	0
$353$	−555.325	−1.57316	−0.786579	−	0.617489i	$-0.788150\pi$
$353$	−555.325	−1.57316	−0.786579	+	0.617489i	$0.788150\pi$
$354$	0	0
$355$	−218.284	−0.614884
$356$	0	0
$357$	− 133.495i	− 0.373937i
$358$	0	0
$359$	326.012i	0.908110i	0.890974	+	0.454055i	$0.150023\pi$
$359$	326.012i	0.908110i	−0.890974	+	0.454055i	$0.849977\pi$
$360$	0	0
$361$	−332.988	−0.922403
$362$	0	0
$363$	−26.1510	−0.0720414
$364$	0	0
$365$	− 35.0294i	− 0.0959710i
$366$	0	0
$367$	− 13.6352i	− 0.0371531i	−0.999827	−	0.0185766i	$-0.994087\pi$
$367$	− 13.6352i	− 0.0371531i	0.999827	−	0.0185766i	$-0.00591344\pi$
$368$	0	0
$369$	633.246	1.71611
$370$	0	0
$371$	−37.3785	−0.100751
$372$	0	0
$373$	136.748i	0.366616i	0.983056	+	0.183308i	$0.0586805\pi$
$373$	136.748i	0.366616i	−0.983056	+	0.183308i	$0.941319\pi$
$374$	0	0
$375$	62.6662i	0.167110i
$376$	0	0
$377$	−153.836	−0.408053
$378$	0	0
$379$	186.198	0.491288	0.245644	−	0.969360i	$-0.421001\pi$
$379$	186.198	0.491288	0.245644	+	0.969360i	$0.421001\pi$
$380$	0	0
$381$	− 721.240i	− 1.89302i
$382$	0	0
$383$	475.712i	1.24207i	0.783784	+	0.621034i	$0.213287\pi$
$383$	475.712i	1.24207i	−0.783784	+	0.621034i	$0.786713\pi$
$384$	0	0
$385$	−33.1672	−0.0861485
$386$	0	0
$387$	−568.902	−1.47003
$388$	0	0
$389$	− 11.9211i	− 0.0306454i	−0.999883	−	0.0153227i	$-0.995122\pi$
$389$	− 11.9211i	− 0.0306454i	0.999883	−	0.0153227i	$-0.00487756\pi$
$390$	0	0
$391$	− 273.232i	− 0.698804i
$392$	0	0
$393$	−1246.15	−3.17087
$394$	0	0
$395$	250.665	0.634594
$396$	0	0
$397$	250.754i	0.631622i	0.948822	+	0.315811i	$0.102277\pi$
$397$	250.754i	0.631622i	−0.948822	+	0.315811i	$0.897723\pi$
$398$	0	0
$399$	39.2526i	0.0983776i
$400$	0	0
$401$	550.328	1.37239	0.686195	−	0.727418i	$-0.259280\pi$
$401$	550.328	1.37239	0.686195	+	0.727418i	$0.259280\pi$
$402$	0	0
$403$	734.867	1.82349
$404$	0	0
$405$	491.371i	1.21326i
$406$	0	0
$407$	− 374.600i	− 0.920394i
$408$	0	0
$409$	27.0820	0.0662153	0.0331076	−	0.999452i	$-0.489460\pi$
$409$	27.0820	0.0662153	0.0331076	+	0.999452i	$0.489460\pi$
$410$	0	0
$411$	722.112	1.75696
$412$	0	0
$413$	59.3313i	0.143659i
$414$	0	0
$415$	− 205.751i	− 0.495785i
$416$	0	0
$417$	−1293.33	−3.10150
$418$	0	0
$419$	198.033	0.472632	0.236316	−	0.971676i	$-0.424060\pi$
$419$	198.033	0.472632	0.236316	+	0.971676i	$0.424060\pi$
$420$	0	0
$421$	− 475.580i	− 1.12964i	−0.825212	−	0.564822i	$-0.808945\pi$
$421$	− 475.580i	− 1.12964i	0.825212	−	0.564822i	$-0.191055\pi$
$422$	0	0
$423$	235.633i	0.557051i
$424$	0	0
$425$	−90.0000	−0.211765
$426$	0	0
$427$	102.435	0.239895
$428$	0	0
$429$	− 1094.32i	− 2.55087i
$430$	0	0
$431$	489.052i	1.13469i	0.823479	+	0.567346i	$0.192030\pi$
$431$	489.052i	1.13469i	−0.823479	+	0.567346i	$0.807970\pi$
$432$	0	0
$433$	622.498	1.43764	0.718820	−	0.695196i	$-0.244682\pi$
$433$	622.498	1.43764	0.718820	+	0.695196i	$0.244682\pi$
$434$	0	0
$435$	110.704	0.254491
$436$	0	0
$437$	80.3406i	0.183846i
$438$	0	0
$439$	552.712i	1.25902i	0.776990	+	0.629512i	$0.216745\pi$
$439$	552.712i	1.25902i	−0.776990	+	0.629512i	$0.783255\pi$
$440$	0	0
$441$	1059.16	2.40172
$442$	0	0
$443$	668.873	1.50987	0.754936	−	0.655799i	$-0.227668\pi$
$443$	668.873	1.50987	0.754936	+	0.655799i	$0.227668\pi$
$444$	0	0
$445$	133.416i	0.299812i
$446$	0	0
$447$	1602.56i	3.58515i
$448$	0	0
$449$	−280.407	−0.624515	−0.312257	−	0.949998i	$-0.601085\pi$
$449$	−280.407	−0.624515	−0.312257	+	0.949998i	$0.601085\pi$
$450$	0	0
$451$	316.676	0.702163
$452$	0	0
$453$	362.164i	0.799479i
$454$	0	0
$455$	− 51.5299i	− 0.113252i
$456$	0	0
$457$	320.334	0.700950	0.350475	−	0.936572i	$-0.386020\pi$
$457$	320.334	0.700950	0.350475	+	0.936572i	$0.386020\pi$
$458$	0	0
$459$	−1353.59	−2.94900
$460$	0	0
$461$	163.337i	0.354311i	0.984183	+	0.177156i	$0.0566896\pi$
$461$	163.337i	0.354311i	−0.984183	+	0.177156i	$0.943310\pi$
$462$	0	0
$463$	108.131i	0.233545i	0.993159	+	0.116772i	$0.0372548\pi$
$463$	108.131i	0.233545i	−0.993159	+	0.116772i	$0.962745\pi$
$464$	0	0
$465$	−528.827	−1.13726
$466$	0	0
$467$	207.386	0.444082	0.222041	−	0.975037i	$-0.428728\pi$
$467$	207.386	0.444082	0.222041	+	0.975037i	$0.428728\pi$
$468$	0	0
$469$	48.4133i	0.103227i
$470$	0	0
$471$	81.7415i	0.173549i
$472$	0	0
$473$	−284.498	−0.601477
$474$	0	0
$475$	26.4634	0.0557124
$476$	0	0
$477$	633.246i	1.32756i
$478$	0	0
$479$	233.537i	0.487552i	0.969832	+	0.243776i	$0.0783861\pi$
$479$	233.537i	0.487552i	−0.969832	+	0.243776i	$0.921614\pi$
$480$	0	0
$481$	581.994	1.20997
$482$	0	0
$483$	−112.578	−0.233081
$484$	0	0
$485$	241.862i	0.498685i
$486$	0	0
$487$	308.368i	0.633199i	0.948559	+	0.316600i	$0.102541\pi$
$487$	308.368i	0.633199i	−0.948559	+	0.316600i	$0.897459\pi$
$488$	0	0
$489$	−116.912	−0.239083
$490$	0	0
$491$	−187.777	−0.382439	−0.191219	−	0.981547i	$-0.561244\pi$
$491$	−187.777	−0.382439	−0.191219	+	0.981547i	$0.561244\pi$
$492$	0	0
$493$	158.991i	0.322496i
$494$	0	0
$495$	561.900i	1.13515i
$496$	0	0
$497$	−129.167	−0.259894
$498$	0	0
$499$	231.038	0.463003	0.231501	−	0.972835i	$-0.425636\pi$
$499$	231.038	0.463003	0.231501	+	0.972835i	$0.425636\pi$
$500$	0	0
$501$	− 1216.07i	− 2.42729i
$502$	0	0
$503$	− 389.780i	− 0.774910i	−0.921889	−	0.387455i	$-0.873354\pi$
$503$	− 389.780i	− 0.774910i	0.921889	−	0.387455i	$-0.126646\pi$
$504$	0	0
$505$	−187.082	−0.370459
$506$	0	0
$507$	752.931	1.48507
$508$	0	0
$509$	363.167i	0.713492i	0.934202	+	0.356746i	$0.116114\pi$
$509$	363.167i	0.713492i	−0.934202	+	0.356746i	$0.883886\pi$
$510$	0	0
$511$	− 20.7283i	− 0.0405641i
$512$	0	0
$513$	398.006	0.775841
$514$	0	0
$515$	−302.359	−0.587106
$516$	0	0
$517$	117.836i	0.227922i
$518$	0	0
$519$	261.103i	0.503088i
$520$	0	0
$521$	491.666	0.943696	0.471848	−	0.881680i	$-0.343587\pi$
$521$	491.666	0.943696	0.471848	+	0.881680i	$0.343587\pi$
$522$	0	0
$523$	−133.914	−0.256049	−0.128024	−	0.991771i	$-0.540864\pi$
$523$	−133.914	−0.256049	−0.128024	+	0.991771i	$0.540864\pi$
$524$	0	0
$525$	37.0820i	0.0706325i
$526$	0	0
$527$	− 759.491i	− 1.44116i
$528$	0	0
$529$	298.580	0.564424
$530$	0	0
$531$	1005.16	1.89295
$532$	0	0
$533$	492.000i	0.923077i
$534$	0	0
$535$	273.636i	0.511469i
$536$	0	0
$537$	−36.6687	−0.0682844
$538$	0	0
$539$	529.667	0.982685
$540$	0	0
$541$	− 303.337i	− 0.560698i	−0.959898	−	0.280349i	$-0.909550\pi$
$541$	− 303.337i	− 0.560698i	0.959898	−	0.280349i	$-0.0904502\pi$
$542$	0	0
$543$	− 1352.67i	− 2.49110i
$544$	0	0
$545$	−5.77709	−0.0106002
$546$	0	0
$547$	−89.6631	−0.163918	−0.0819590	−	0.996636i	$-0.526118\pi$
$547$	−89.6631	−0.163918	−0.0819590	+	0.996636i	$0.526118\pi$
$548$	0	0
$549$	− 1735.40i	− 3.16102i
$550$	0	0
$551$	− 46.7492i	− 0.0848443i
$552$	0	0
$553$	148.328	0.268225
$554$	0	0
$555$	−418.816	−0.754623
$556$	0	0
$557$	− 490.741i	− 0.881044i	−0.897742	−	0.440522i	$-0.854793\pi$
$557$	− 490.741i	− 0.881044i	0.897742	−	0.440522i	$-0.145207\pi$
$558$	0	0
$559$	− 442.008i	− 0.790712i
$560$	0	0
$561$	−1130.99	−2.01603
$562$	0	0
$563$	987.423	1.75386	0.876930	−	0.480618i	$-0.159588\pi$
$563$	987.423	1.75386	0.876930	+	0.480618i	$0.159588\pi$
$564$	0	0
$565$	− 74.9117i	− 0.132587i
$566$	0	0
$567$	290.763i	0.512810i
$568$	0	0
$569$	475.082	0.834942	0.417471	−	0.908690i	$-0.362917\pi$
$569$	475.082	0.834942	0.417471	+	0.908690i	$0.362917\pi$
$570$	0	0
$571$	−182.485	−0.319588	−0.159794	−	0.987150i	$-0.551083\pi$
$571$	−182.485	−0.319588	−0.159794	+	0.987150i	$0.551083\pi$
$572$	0	0
$573$	− 332.498i	− 0.580277i
$574$	0	0
$575$	75.8979i	0.131996i
$576$	0	0
$577$	−319.167	−0.553149	−0.276575	−	0.960992i	$-0.589199\pi$
$577$	−319.167	−0.553149	−0.276575	+	0.960992i	$0.589199\pi$
$578$	0	0
$579$	−1317.17	−2.27490
$580$	0	0
$581$	− 121.751i	− 0.209554i
$582$	0	0
$583$	316.676i	0.543183i
$584$	0	0
$585$	−872.991	−1.49229
$586$	0	0
$587$	904.762	1.54133	0.770666	−	0.637239i	$-0.219924\pi$
$587$	904.762	1.54133	0.770666	+	0.637239i	$0.219924\pi$
$588$	0	0
$589$	223.319i	0.379149i
$590$	0	0
$591$	1294.27i	2.18996i
$592$	0	0
$593$	−747.325	−1.26024	−0.630122	−	0.776496i	$-0.716995\pi$
$593$	−747.325	−1.26024	−0.630122	+	0.776496i	$0.716995\pi$
$594$	0	0
$595$	−53.2565	−0.0895068
$596$	0	0
$597$	155.331i	0.260186i
$598$	0	0
$599$	874.385i	1.45974i	0.683585	+	0.729871i	$0.260420\pi$
$599$	874.385i	1.45974i	−0.683585	+	0.729871i	$0.739580\pi$
$600$	0	0
$601$	777.234	1.29323	0.646617	−	0.762815i	$-0.276183\pi$
$601$	777.234	1.29323	0.646617	+	0.762815i	$0.276183\pi$
$602$	0	0
$603$	820.192	1.36019
$604$	0	0
$605$	10.4327i	0.0172441i
$606$	0	0
$607$	− 699.436i	− 1.15228i	−0.817350	−	0.576142i	$-0.804558\pi$
$607$	− 699.436i	− 1.15228i	0.817350	−	0.576142i	$-0.195442\pi$
$608$	0	0
$609$	65.5078	0.107566
$610$	0	0
$611$	−183.075	−0.299631
$612$	0	0
$613$	11.0820i	0.0180784i	0.999959	+	0.00903918i	$0.00287730\pi$
$613$	11.0820i	0.0180784i	−0.999959	+	0.00903918i	$0.997123\pi$
$614$	0	0
$615$	− 354.054i	− 0.575698i
$616$	0	0
$617$	−1189.16	−1.92733	−0.963664	−	0.267119i	$-0.913928\pi$
$617$	−1189.16	−1.92733	−0.963664	+	0.267119i	$0.913928\pi$
$618$	0	0
$619$	325.717	0.526198	0.263099	−	0.964769i	$-0.415255\pi$
$619$	325.717	0.526198	0.263099	+	0.964769i	$0.415255\pi$
$620$	0	0
$621$	1141.50i	1.83816i
$622$	0	0
$623$	78.9477i	0.126722i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	332.554	0.530389
$628$	0	0
$629$	− 601.495i	− 0.956272i
$630$	0	0
$631$	316.823i	0.502097i	0.967975	+	0.251048i	$0.0807754\pi$
$631$	316.823i	0.502097i	−0.967975	+	0.251048i	$0.919225\pi$
$632$	0	0
$633$	521.823	0.824366
$634$	0	0
$635$	−287.731	−0.453120
$636$	0	0
$637$	822.912i	1.29186i
$638$	0	0
$639$	2188.28i	3.42454i
$640$	0	0
$641$	−149.587	−0.233365	−0.116682	−	0.993169i	$-0.537226\pi$
$641$	−149.587	−0.233365	−0.116682	+	0.993169i	$0.537226\pi$
$642$	0	0
$643$	−408.690	−0.635599	−0.317800	−	0.948158i	$-0.602944\pi$
$643$	−408.690	−0.635599	−0.317800	+	0.948158i	$0.602944\pi$
$644$	0	0
$645$	318.079i	0.493146i
$646$	0	0
$647$	104.383i	0.161334i	0.996741	+	0.0806668i	$0.0257050\pi$
$647$	104.383i	0.161334i	−0.996741	+	0.0806668i	$0.974295\pi$
$648$	0	0
$649$	502.663	0.774519
$650$	0	0
$651$	−312.927	−0.480687
$652$	0	0
$653$	− 164.420i	− 0.251791i	−0.992044	−	0.125895i	$-0.959820\pi$
$653$	− 164.420i	− 0.251791i	0.992044	−	0.125895i	$-0.0401804\pi$
$654$	0	0
$655$	497.139i	0.758990i
$656$	0	0
$657$	−351.167	−0.534501
$658$	0	0
$659$	−743.613	−1.12840	−0.564198	−	0.825640i	$-0.690814\pi$
$659$	−743.613	−1.12840	−0.564198	+	0.825640i	$0.690814\pi$
$660$	0	0
$661$	758.073i	1.14686i	0.819255	+	0.573429i	$0.194387\pi$
$661$	758.073i	1.14686i	−0.819255	+	0.573429i	$0.805613\pi$
$662$	0	0
$663$	− 1757.15i	− 2.65030i
$664$	0	0
$665$	15.6594	0.0235480
$666$	0	0
$667$	134.078	0.201017
$668$	0	0
$669$	− 1725.74i	− 2.57958i
$670$	0	0
$671$	− 867.843i	− 1.29336i
$672$	0	0
$673$	364.164	0.541106	0.270553	−	0.962705i	$-0.412794\pi$
$673$	364.164	0.541106	0.270553	+	0.962705i	$0.412794\pi$
$674$	0	0
$675$	375.997	0.557033
$676$	0	0
$677$	708.079i	1.04591i	0.852361	+	0.522953i	$0.175170\pi$
$677$	708.079i	1.04591i	−0.852361	+	0.522953i	$0.824830\pi$
$678$	0	0
$679$	143.119i	0.210780i
$680$	0	0
$681$	−1429.40	−2.09897
$682$	0	0
$683$	355.469	0.520452	0.260226	−	0.965548i	$-0.416203\pi$
$683$	355.469	0.520452	0.260226	+	0.965548i	$0.416203\pi$
$684$	0	0
$685$	− 288.079i	− 0.420553i
$686$	0	0
$687$	− 2306.45i	− 3.35727i
$688$	0	0
$689$	−492.000	−0.714078
$690$	0	0
$691$	116.769	0.168985	0.0844925	−	0.996424i	$-0.473073\pi$
$691$	116.769	0.168985	0.0844925	+	0.996424i	$0.473073\pi$
$692$	0	0
$693$	332.498i	0.479796i
$694$	0	0
$695$	515.958i	0.742386i
$696$	0	0
$697$	508.486	0.729535
$698$	0	0
$699$	−1865.54	−2.66887
$700$	0	0
$701$	− 246.413i	− 0.351517i	−0.984433	−	0.175758i	$-0.943762\pi$
$701$	− 246.413i	− 0.351517i	0.984433	−	0.175758i	$-0.0562378\pi$
$702$	0	0
$703$	176.862i	0.251582i
$704$	0	0
$705$	131.745	0.186872
$706$	0	0
$707$	−110.704	−0.156582
$708$	0	0
$709$	552.820i	0.779718i	0.920874	+	0.389859i	$0.127476\pi$
$709$	552.820i	0.779718i	−0.920874	+	0.389859i	$0.872524\pi$
$710$	0	0
$711$	− 2512.89i	− 3.53431i
$712$	0	0
$713$	−640.486	−0.898297
$714$	0	0
$715$	−436.568	−0.610585
$716$	0	0
$717$	1492.33i	2.08135i
$718$	0	0
$719$	− 99.9709i	− 0.139042i	−0.997581	−	0.0695208i	$-0.977853\pi$
$719$	− 99.9709i	− 0.139042i	0.997581	−	0.0695208i	$-0.0221470\pi$
$720$	0	0
$721$	−178.918	−0.248153
$722$	0	0
$723$	440.464	0.609217
$724$	0	0
$725$	− 44.1641i	− 0.0609160i
$726$	0	0
$727$	− 1081.39i	− 1.48746i	−0.668479	−	0.743731i	$-0.733054\pi$
$727$	− 1081.39i	− 1.48746i	0.668479	−	0.743731i	$-0.266946\pi$
$728$	0	0
$729$	1132.50	1.55349
$730$	0	0
$731$	−456.819	−0.624923
$732$	0	0
$733$	380.407i	0.518973i	0.965747	+	0.259486i	$0.0835533\pi$
$733$	380.407i	0.518973i	−0.965747	+	0.259486i	$0.916447\pi$
$734$	0	0
$735$	− 592.186i	− 0.805695i
$736$	0	0
$737$	410.164	0.556532
$738$	0	0
$739$	−1381.60	−1.86955	−0.934775	−	0.355240i	$-0.884399\pi$
$739$	−1381.60	−1.86955	−0.934775	+	0.355240i	$0.884399\pi$
$740$	0	0
$741$	516.669i	0.697259i
$742$	0	0
$743$	267.424i	0.359924i	0.983674	+	0.179962i	$0.0575975\pi$
$743$	267.424i	0.359924i	−0.983674	+	0.179962i	$0.942403\pi$
$744$	0	0
$745$	639.325	0.858154
$746$	0	0
$747$	−2062.64	−2.76123
$748$	0	0
$749$	161.921i	0.216183i
$750$	0	0
$751$	338.653i	0.450937i	0.974250	+	0.225468i	$0.0723912\pi$
$751$	338.653i	0.450937i	−0.974250	+	0.225468i	$0.927609\pi$
$752$	0	0
$753$	2267.14	3.01080
$754$	0	0
$755$	144.481	0.191366
$756$	0	0
$757$	− 1247.91i	− 1.64849i	−0.566232	−	0.824246i	$-0.691599\pi$
$757$	− 1247.91i	− 1.64849i	0.566232	−	0.824246i	$-0.308401\pi$
$758$	0	0
$759$	953.775i	1.25662i
$760$	0	0
$761$	14.3406	0.0188444	0.00942219	−	0.999956i	$-0.497001\pi$
$761$	14.3406	0.0188444	0.00942219	+	0.999956i	$0.497001\pi$
$762$	0	0
$763$	−3.41853	−0.00448038
$764$	0	0
$765$	902.243i	1.17940i
$766$	0	0
$767$	780.956i	1.01820i
$768$	0	0
$769$	1036.99	1.34849	0.674246	−	0.738507i	$-0.264469\pi$
$769$	1036.99	1.34849	0.674246	+	0.738507i	$0.264469\pi$
$770$	0	0
$771$	481.078	0.623967
$772$	0	0
$773$	− 1197.91i	− 1.54970i	−0.632148	−	0.774848i	$-0.717826\pi$
$773$	− 1197.91i	− 1.54970i	0.632148	−	0.774848i	$-0.282174\pi$
$774$	0	0
$775$	210.970i	0.272219i
$776$	0	0
$777$	−247.830	−0.318957
$778$	0	0
$779$	−149.514	−0.191931
$780$	0	0
$781$	1094.32i	1.40118i
$782$	0	0
$783$	− 664.223i	− 0.848305i
$784$	0	0
$785$	32.6099	0.0415413
$786$	0	0
$787$	−99.1814	−0.126025	−0.0630123	−	0.998013i	$-0.520071\pi$
$787$	−99.1814	−0.126025	−0.0630123	+	0.998013i	$0.520071\pi$
$788$	0	0
$789$	2470.05i	3.13061i
$790$	0	0
$791$	− 44.3282i	− 0.0560407i
$792$	0	0
$793$	1348.32	1.70027
$794$	0	0
$795$	354.054	0.445351
$796$	0	0
$797$	139.240i	0.174705i	0.996177	+	0.0873525i	$0.0278407\pi$
$797$	139.240i	0.174705i	−0.996177	+	0.0873525i	$0.972159\pi$
$798$	0	0
$799$	189.209i	0.236807i
$800$	0	0
$801$	1337.49	1.66977
$802$	0	0
$803$	−175.613	−0.218696
$804$	0	0
$805$	44.9117i	0.0557910i
$806$	0	0
$807$	1151.37i	1.42672i
$808$	0	0
$809$	−403.653	−0.498953	−0.249477	−	0.968381i	$-0.580259\pi$
$809$	−403.653	−0.498953	−0.249477	+	0.968381i	$0.580259\pi$
$810$	0	0
$811$	384.561	0.474181	0.237091	−	0.971488i	$-0.423806\pi$
$811$	384.561	0.474181	0.237091	+	0.971488i	$0.423806\pi$
$812$	0	0
$813$	− 2343.15i	− 2.88210i
$814$	0	0
$815$	46.6407i	0.0572278i
$816$	0	0
$817$	134.322	0.164409
$818$	0	0
$819$	−516.583	−0.630748
$820$	0	0
$821$	997.732i	1.21526i	0.794219	+	0.607632i	$0.207881\pi$
$821$	997.732i	1.21526i	−0.794219	+	0.607632i	$0.792119\pi$
$822$	0	0
$823$	− 955.281i	− 1.16073i	−0.814356	−	0.580365i	$-0.802910\pi$
$823$	− 955.281i	− 1.16073i	0.814356	−	0.580365i	$-0.197090\pi$
$824$	0	0
$825$	314.164	0.380805
$826$	0	0
$827$	−213.451	−0.258103	−0.129051	−	0.991638i	$-0.541193\pi$
$827$	−213.451	−0.258103	−0.129051	+	0.991638i	$0.541193\pi$
$828$	0	0
$829$	− 73.4288i	− 0.0885752i	−0.999019	−	0.0442876i	$-0.985898\pi$
$829$	− 73.4288i	− 0.0885752i	0.999019	−	0.0442876i	$-0.0141018\pi$
$830$	0	0
$831$	− 2866.47i	− 3.44942i
$832$	0	0
$833$	850.486	1.02099
$834$	0	0
$835$	−485.139	−0.581005
$836$	0	0
$837$	3172.96i	3.79087i
$838$	0	0
$839$	− 643.642i	− 0.767154i	−0.923509	−	0.383577i	$-0.874692\pi$
$839$	− 643.642i	− 0.767154i	0.923509	−	0.383577i	$-0.125308\pi$
$840$	0	0
$841$	762.981	0.907231
$842$	0	0
$843$	648.787	0.769617
$844$	0	0
$845$	− 300.374i	− 0.355472i
$846$	0	0
$847$	6.17342i	0.00728857i
$848$	0	0
$849$	2253.23	2.65398
$850$	0	0
$851$	−507.247	−0.596060
$852$	0	0
$853$	351.410i	0.411970i	0.978555	+	0.205985i	$0.0660398\pi$
$853$	351.410i	0.411970i	−0.978555	+	0.205985i	$0.933960\pi$
$854$	0	0
$855$	− 265.293i	− 0.310285i
$856$	0	0
$857$	−1326.16	−1.54744	−0.773721	−	0.633526i	$-0.781607\pi$
$857$	−1326.16	−1.54744	−0.773721	+	0.633526i	$0.781607\pi$
$858$	0	0
$859$	−343.729	−0.400150	−0.200075	−	0.979781i	$-0.564119\pi$
$859$	−343.729	−0.400150	−0.200075	+	0.979781i	$0.564119\pi$
$860$	0	0
$861$	− 209.508i	− 0.243331i
$862$	0	0
$863$	− 1136.15i	− 1.31651i	−0.752793	−	0.658257i	$-0.771294\pi$
$863$	− 1136.15i	− 1.31651i	0.752793	−	0.658257i	$-0.228706\pi$
$864$	0	0
$865$	104.164	0.120421
$866$	0	0
$867$	−196.176	−0.226270
$868$	0	0
$869$	− 1256.66i	− 1.44609i
$870$	0	0
$871$	637.247i	0.731627i
$872$	0	0
$873$	2424.65	2.77738
$874$	0	0
$875$	14.7935	0.0169068
$876$	0	0
$877$	769.076i	0.876939i	0.898746	+	0.438470i	$0.144479\pi$
$877$	769.076i	0.876939i	−0.898746	+	0.438470i	$0.855521\pi$
$878$	0	0
$879$	506.770i	0.576530i
$880$	0	0
$881$	−731.568	−0.830384	−0.415192	−	0.909734i	$-0.636286\pi$
$881$	−731.568	−0.830384	−0.415192	+	0.909734i	$0.636286\pi$
$882$	0	0
$883$	−1066.19	−1.20746	−0.603731	−	0.797188i	$-0.706320\pi$
$883$	−1066.19	−1.20746	−0.603731	+	0.797188i	$0.706320\pi$
$884$	0	0
$885$	− 561.994i	− 0.635021i
$886$	0	0
$887$	− 397.242i	− 0.447848i	−0.974607	−	0.223924i	$-0.928113\pi$
$887$	− 397.242i	− 0.447848i	0.974607	−	0.223924i	$-0.0718868\pi$
$888$	0	0
$889$	−170.262	−0.191520
$890$	0	0
$891$	2463.39	2.76474
$892$	0	0
$893$	− 55.6346i	− 0.0623008i
$894$	0	0
$895$	14.6286i	0.0163448i
$896$	0	0
$897$	−1481.82	−1.65198
$898$	0	0
$899$	372.691	0.414562
$900$	0	0
$901$	508.486i	0.564357i
$902$	0	0
$903$	188.220i	0.208438i
$904$	0	0
$905$	−539.633	−0.596280
$906$	0	0
$907$	1021.02	1.12571	0.562855	−	0.826556i	$-0.309703\pi$
$907$	1021.02	1.12571	0.562855	+	0.826556i	$0.309703\pi$
$908$	0	0
$909$	1875.48i	2.06324i
$910$	0	0
$911$	1118.66i	1.22794i	0.789328	+	0.613971i	$0.210429\pi$
$911$	1118.66i	1.22794i	−0.789328	+	0.613971i	$0.789571\pi$
$912$	0	0
$913$	−1031.49	−1.12978
$914$	0	0
$915$	−970.278	−1.06041
$916$	0	0
$917$	294.177i	0.320803i
$918$	0	0
$919$	− 1177.68i	− 1.28148i	−0.767757	−	0.640741i	$-0.778627\pi$
$919$	− 1177.68i	− 1.28148i	0.767757	−	0.640741i	$-0.221373\pi$
$920$	0	0
$921$	−1543.73	−1.67615
$922$	0	0
$923$	−1700.18	−1.84202
$924$	0	0
$925$	167.082i	0.180629i
$926$	0	0
$927$	3031.13i	3.26983i
$928$	0	0
$929$	−699.423	−0.752877	−0.376438	−	0.926442i	$-0.622851\pi$
$929$	−699.423	−0.752877	−0.376438	+	0.926442i	$0.622851\pi$
$930$	0	0
$931$	−250.075	−0.268609
$932$	0	0
$933$	− 2720.15i	− 2.91548i
$934$	0	0
$935$	451.197i	0.482563i
$936$	0	0
$937$	332.152	0.354484	0.177242	−	0.984167i	$-0.443282\pi$
$937$	332.152	0.354484	0.177242	+	0.984167i	$0.443282\pi$
$938$	0	0
$939$	11.2101	0.0119383
$940$	0	0
$941$	− 1468.66i	− 1.56075i	−0.625314	−	0.780373i	$-0.715029\pi$
$941$	− 1468.66i	− 1.56075i	0.625314	−	0.780373i	$-0.284971\pi$
$942$	0	0
$943$	− 428.811i	− 0.454731i
$944$	0	0
$945$	222.492	0.235442
$946$	0	0
$947$	574.082	0.606211	0.303106	−	0.952957i	$-0.401976\pi$
$947$	574.082	0.606211	0.303106	+	0.952957i	$0.401976\pi$
$948$	0	0
$949$	− 272.839i	− 0.287502i
$950$	0	0
$951$	2212.54i	2.32654i
$952$	0	0
$953$	−280.663	−0.294504	−0.147252	−	0.989099i	$-0.547043\pi$
$953$	−280.663	−0.294504	−0.147252	+	0.989099i	$0.547043\pi$
$954$	0	0
$955$	−132.647	−0.138897
$956$	0	0
$957$	− 554.991i	− 0.579928i
$958$	0	0
$959$	− 170.468i	− 0.177756i
$960$	0	0
$961$	−819.328	−0.852579
$962$	0	0
$963$	2743.18	2.84858
$964$	0	0
$965$	525.469i	0.544527i
$966$	0	0
$967$	− 1278.65i	− 1.32228i	−0.750262	−	0.661141i	$-0.770073\pi$
$967$	− 1278.65i	− 1.32228i	0.750262	−	0.661141i	$-0.229927\pi$
$968$	0	0
$969$	533.981	0.551064
$970$	0	0
$971$	117.723	0.121239	0.0606195	−	0.998161i	$-0.480692\pi$
$971$	117.723	0.121239	0.0606195	+	0.998161i	$0.480692\pi$
$972$	0	0
$973$	305.313i	0.313785i
$974$	0	0
$975$	488.098i	0.500613i
$976$	0	0
$977$	983.155	1.00630	0.503150	−	0.864199i	$-0.332174\pi$
$977$	983.155	1.00630	0.503150	+	0.864199i	$0.332174\pi$
$978$	0	0
$979$	668.856	0.683203
$980$	0	0
$981$	57.9149i	0.0590365i
$982$	0	0
$983$	607.404i	0.617909i	0.951077	+	0.308954i	$0.0999790\pi$
$983$	607.404i	0.617909i	−0.951077	+	0.308954i	$0.900021\pi$
$984$	0	0
$985$	516.334	0.524197
$986$	0	0
$987$	77.9584	0.0789852
$988$	0	0
$989$	385.240i	0.389525i
$990$	0	0
$991$	− 408.413i	− 0.412122i	−0.978539	−	0.206061i	$-0.933935\pi$
$991$	− 408.413i	− 0.412122i	0.978539	−	0.206061i	$-0.0660645\pi$
$992$	0	0
$993$	−1659.82	−1.67152
$994$	0	0
$995$	61.9677	0.0622791
$996$	0	0
$997$	1554.91i	1.55959i	0.626035	+	0.779795i	$0.284677\pi$
$997$	1554.91i	1.55959i	−0.626035	+	0.779795i	$0.715323\pi$
$998$	0	0
$999$	2512.89i	2.51541i

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1280.3.g.f.1151.2		8
4.3	odd	2	inner	1280.3.g.f.1151.8		8
8.3	odd	2	inner	1280.3.g.f.1151.1		8
8.5	even	2	inner	1280.3.g.f.1151.7		8
16.3	odd	4		320.3.b.a.191.4		4
16.5	even	4		80.3.b.a.31.4	yes	4
16.11	odd	4		80.3.b.a.31.1	✓	4
16.13	even	4		320.3.b.a.191.1		4
48.5	odd	4		720.3.e.c.271.3		4
48.11	even	4		720.3.e.c.271.4		4
48.29	odd	4		2880.3.e.b.2431.1		4
48.35	even	4		2880.3.e.b.2431.2		4
80.3	even	4		1600.3.h.p.1599.2		8
80.13	odd	4		1600.3.h.p.1599.7		8
80.19	odd	4		1600.3.b.k.1151.1		4
80.27	even	4		400.3.h.d.399.1		8
80.29	even	4		1600.3.b.k.1151.4		4
80.37	odd	4		400.3.h.d.399.8		8
80.43	even	4		400.3.h.d.399.7		8
80.53	odd	4		400.3.h.d.399.2		8
80.59	odd	4		400.3.b.g.351.4		4
80.67	even	4		1600.3.h.p.1599.8		8
80.69	even	4		400.3.b.g.351.1		4
80.77	odd	4		1600.3.h.p.1599.1		8
240.53	even	4		3600.3.j.k.1999.3		8
240.59	even	4		3600.3.e.bb.3151.2		4
240.107	odd	4		3600.3.j.k.1999.4		8
240.149	odd	4		3600.3.e.bb.3151.3		4
240.197	even	4		3600.3.j.k.1999.5		8
240.203	odd	4		3600.3.j.k.1999.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.3.b.a.31.1	✓	4	16.11	odd	4
80.3.b.a.31.4	yes	4	16.5	even	4
320.3.b.a.191.1		4	16.13	even	4
320.3.b.a.191.4		4	16.3	odd	4
400.3.b.g.351.1		4	80.69	even	4
400.3.b.g.351.4		4	80.59	odd	4
400.3.h.d.399.1		8	80.27	even	4
400.3.h.d.399.2		8	80.53	odd	4
400.3.h.d.399.7		8	80.43	even	4
400.3.h.d.399.8		8	80.37	odd	4
720.3.e.c.271.3		4	48.5	odd	4
720.3.e.c.271.4		4	48.11	even	4
1280.3.g.f.1151.1		8	8.3	odd	2	inner
1280.3.g.f.1151.2		8	1.1	even	1	trivial
1280.3.g.f.1151.7		8	8.5	even	2	inner
1280.3.g.f.1151.8		8	4.3	odd	2	inner
1600.3.b.k.1151.1		4	80.19	odd	4
1600.3.b.k.1151.4		4	80.29	even	4
1600.3.h.p.1599.1		8	80.77	odd	4
1600.3.h.p.1599.2		8	80.3	even	4
1600.3.h.p.1599.7		8	80.13	odd	4
1600.3.h.p.1599.8		8	80.67	even	4
2880.3.e.b.2431.1		4	48.29	odd	4
2880.3.e.b.2431.2		4	48.35	even	4
3600.3.e.bb.3151.2		4	240.59	even	4
3600.3.e.bb.3151.3		4	240.149	odd	4
3600.3.j.k.1999.3		8	240.53	even	4
3600.3.j.k.1999.4		8	240.107	odd	4
3600.3.j.k.1999.5		8	240.197	even	4
3600.3.j.k.1999.6		8	240.203	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1280.3.g.f.1151.2

Level	$1280$
Weight	$3$
Character	1280.1151
Analytic conductor	$34.877$
Analytic rank	$0$
Dimension	$8$
Inner twists	$4$