LMFDB - Embedded newform 80.3.b.a.31.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [80,3,Mod(31,80)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(80, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0]))

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

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

chi := DirichletCharacter("80.31");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 80 = 2^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	80.b (of order $2$, degree $1$, 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:	$2.17984211488$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 2x^{2} + x + 1 $ x^4 - x^3 + 2*x^2 + x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			31.1
Root			$0.809017 - 1.40126i$ of defining polynomial
Character	$\chi$	$=$	80.31
Dual form			80.3.b.a.31.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.60503i q^{3} -2.23607 q^{5} +1.32317i q^{7} -22.4164 q^{9} -11.2101i q^{11} +17.4164 q^{13} +12.5332i q^{15} +18.0000 q^{17} -5.29268i q^{19} +7.41641 q^{21} -15.1796i q^{23} +5.00000 q^{25} +75.1994i q^{27} +8.83282 q^{29} +42.1939i q^{31} -62.8328 q^{33} -2.95870i q^{35} +33.4164 q^{37} -97.6196i q^{39} -28.2492 q^{41} +25.3788i q^{43} +50.1246 q^{45} -10.5116i q^{47} +47.2492 q^{49} -100.891i q^{51} -28.2492 q^{53} +25.0665i q^{55} -29.6656 q^{57} -44.8403i q^{59} -77.4164 q^{61} -29.6607i q^{63} -38.9443 q^{65} +36.5889i q^{67} -85.0820 q^{69} +97.6196i q^{71} +15.6656 q^{73} -28.0252i q^{75} +14.8328 q^{77} +112.101i q^{79} +219.748 q^{81} -92.0145i q^{83} -40.2492 q^{85} -49.5082i q^{87} -59.6656 q^{89} +23.0449i q^{91} +236.498 q^{93} +11.8348i q^{95} +108.164 q^{97} +251.289i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 36 q^{9} + 16 q^{13} + 72 q^{17} - 24 q^{21} + 20 q^{25} - 72 q^{29} - 144 q^{33} + 80 q^{37} + 48 q^{41} + 120 q^{45} + 28 q^{49} + 48 q^{53} + 96 q^{57} - 256 q^{61} - 120 q^{65} - 72 q^{69} - 152 q^{73}+ \cdots - 104 q^{97}+O(q^{100}) $ 4 * q - 36 * q^9 + 16 * q^13 + 72 * q^17 - 24 * q^21 + 20 * q^25 - 72 * q^29 - 144 * q^33 + 80 * q^37 + 48 * q^41 + 120 * q^45 + 28 * q^49 + 48 * q^53 + 96 * q^57 - 256 * q^61 - 120 * q^65 - 72 * q^69 - 152 * q^73 - 48 * q^77 + 396 * q^81 - 24 * q^89 + 624 * q^93 - 104 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/80\mathbb{Z}\right)^\times$.

$n$	$17$	$21$	$31$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 5.60503i	− 1.86834i	−0.356822	−	0.934172i	$-0.616140\pi$
$3$	− 5.60503i	− 1.86834i	0.356822	−	0.934172i	$-0.383860\pi$
$4$	0	0
$5$	−2.23607	−0.447214
$5$	−2.23607	−0.447214
$6$	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
$11$	− 11.2101i	− 1.01910i	−0.860442	−	0.509549i	$-0.829812\pi$
$11$	− 11.2101i	− 1.01910i	0.860442	−	0.509549i	$-0.170188\pi$
$12$	0	0
$13$	17.4164	1.33972	0.669862	−	0.742486i	$-0.266353\pi$
$13$	17.4164	1.33972	0.669862	+	0.742486i	$0.266353\pi$
$14$	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
$19$	− 5.29268i	− 0.278562i	−0.990253	−	0.139281i	$-0.955521\pi$
$19$	− 5.29268i	− 0.278562i	0.990253	−	0.139281i	$-0.0444791\pi$
$20$	0	0
$21$	7.41641	0.353162
$22$	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
$27$	75.1994i	2.78516i
$28$	0	0
$29$	8.83282	0.304580	0.152290	−	0.988336i	$-0.451335\pi$
$29$	8.83282	0.304580	0.152290	+	0.988336i	$0.451335\pi$
$30$	0	0
$31$	42.1939i	1.36109i	0.732704	+	0.680547i	$0.238258\pi$
$31$	42.1939i	1.36109i	−0.732704	+	0.680547i	$0.761742\pi$
$32$	0	0
$33$	−62.8328	−1.90402
$34$	0	0
$35$	− 2.95870i	− 0.0845342i
$36$	0	0
$37$	33.4164	0.903146	0.451573	−	0.892234i	$-0.350863\pi$
$37$	33.4164	0.903146	0.451573	+	0.892234i	$0.350863\pi$
$38$	0	0
$39$	− 97.6196i	− 2.50307i
$40$	0	0
$41$	−28.2492	−0.689005	−0.344503	−	0.938785i	$-0.611952\pi$
$41$	−28.2492	−0.689005	−0.344503	+	0.938785i	$0.611952\pi$
$42$	0	0
$43$	25.3788i	0.590205i	0.955466	+	0.295103i	$0.0953539\pi$
$43$	25.3788i	0.590205i	−0.955466	+	0.295103i	$0.904646\pi$
$44$	0	0
$45$	50.1246	1.11388
$46$	0	0
$47$	− 10.5116i	− 0.223651i	−0.993728	−	0.111826i	$-0.964330\pi$
$47$	− 10.5116i	− 0.223651i	0.993728	−	0.111826i	$-0.0356698\pi$
$48$	0	0
$49$	47.2492	0.964270
$50$	0	0
$51$	− 100.891i	− 1.97825i
$52$	0	0
$53$	−28.2492	−0.533004	−0.266502	−	0.963834i	$-0.585868\pi$
$53$	−28.2492	−0.533004	−0.266502	+	0.963834i	$0.585868\pi$
$54$	0	0
$55$	25.0665i	0.455754i
$56$	0	0
$57$	−29.6656	−0.520450
$58$	0	0
$59$	− 44.8403i	− 0.760005i	−0.924986	−	0.380002i	$-0.875923\pi$
$59$	− 44.8403i	− 0.760005i	0.924986	−	0.380002i	$-0.124077\pi$
$60$	0	0
$61$	−77.4164	−1.26912	−0.634561	−	0.772873i	$-0.718819\pi$
$61$	−77.4164	−1.26912	−0.634561	+	0.772873i	$0.718819\pi$
$62$	0	0
$63$	− 29.6607i	− 0.470805i
$64$	0	0
$65$	−38.9443	−0.599143
$66$	0	0
$67$	36.5889i	0.546103i	0.961999	+	0.273051i	$0.0880329\pi$
$67$	36.5889i	0.546103i	−0.961999	+	0.273051i	$0.911967\pi$
$68$	0	0
$69$	−85.0820	−1.23307
$70$	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	$-0.465780\pi$
$73$	15.6656	0.214598	0.107299	+	0.994227i	$0.465780\pi$
$74$	0	0
$75$	− 28.0252i	− 0.373669i
$76$	0	0
$77$	14.8328	0.192634
$78$	0	0
$79$	112.101i	1.41900i	0.704707	+	0.709498i	$0.251078\pi$
$79$	112.101i	1.41900i	−0.704707	+	0.709498i	$0.748922\pi$
$80$	0	0
$81$	219.748	2.71293
$82$	0	0
$83$	− 92.0145i	− 1.10861i	−0.832314	−	0.554304i	$-0.812984\pi$
$83$	− 92.0145i	− 1.10861i	0.832314	−	0.554304i	$-0.187016\pi$
$84$	0	0
$85$	−40.2492	−0.473520
$86$	0	0
$87$	− 49.5082i	− 0.569060i
$88$	0	0
$89$	−59.6656	−0.670400	−0.335200	−	0.942147i	$-0.608804\pi$
$89$	−59.6656	−0.670400	−0.335200	+	0.942147i	$0.608804\pi$
$90$	0	0
$91$	23.0449i	0.253240i
$92$	0	0
$93$	236.498	2.54299
$94$	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
$99$	251.289i	2.53828i
$100$	0	0
$101$	−83.6656	−0.828373	−0.414186	−	0.910192i	$-0.635934\pi$
$101$	−83.6656	−0.828373	−0.414186	+	0.910192i	$0.635934\pi$
$102$	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
$107$	− 122.374i	− 1.14368i	−0.820365	−	0.571840i	$-0.806230\pi$
$107$	− 122.374i	− 1.14368i	0.820365	−	0.571840i	$-0.193770\pi$
$108$	0	0
$109$	2.58359	0.0237027	0.0118513	−	0.999930i	$-0.496228\pi$
$109$	2.58359	0.0237027	0.0118513	+	0.999930i	$0.496228\pi$
$110$	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
$115$	33.9426i	0.295153i
$116$	0	0
$117$	−390.413	−3.33687
$118$	0	0
$119$	23.8170i	0.200143i
$120$	0	0
$121$	−4.66563	−0.0385589
$122$	0	0
$123$	158.338i	1.28730i
$124$	0	0
$125$	−11.1803	−0.0894427
$126$	0	0
$127$	− 128.677i	− 1.01321i	−0.862179	−	0.506603i	$-0.830901\pi$
$127$	− 128.677i	− 1.01321i	0.862179	−	0.506603i	$-0.169099\pi$
$128$	0	0
$129$	142.249	1.10271
$130$	0	0
$131$	222.327i	1.69715i	0.529072	+	0.848577i	$0.322540\pi$
$131$	222.327i	1.69715i	−0.529072	+	0.848577i	$0.677460\pi$
$132$	0	0
$133$	7.00311	0.0526549
$134$	0	0
$135$	− 168.151i	− 1.24556i
$136$	0	0
$137$	128.833	0.940386	0.470193	−	0.882564i	$-0.344184\pi$
$137$	128.833	0.940386	0.470193	+	0.882564i	$0.344184\pi$
$138$	0	0
$139$	− 230.743i	− 1.66002i	−0.557745	−	0.830012i	$-0.688333\pi$
$139$	− 230.743i	− 1.66002i	0.557745	−	0.830012i	$-0.311667\pi$
$140$	0	0
$141$	−58.9180	−0.417858
$142$	0	0
$143$	− 195.239i	− 1.36531i
$144$	0	0
$145$	−19.7508	−0.136212
$146$	0	0
$147$	− 264.834i	− 1.80159i
$148$	0	0
$149$	285.915	1.91889	0.959446	−	0.281893i	$-0.0909623\pi$
$149$	285.915	1.91889	0.959446	+	0.281893i	$0.0909623\pi$
$150$	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
$155$	− 94.3485i	− 0.608700i
$156$	0	0
$157$	−14.5836	−0.0928891	−0.0464446	−	0.998921i	$-0.514789\pi$
$157$	−14.5836	−0.0928891	−0.0464446	+	0.998921i	$0.514789\pi$
$158$	0	0
$159$	158.338i	0.995836i
$160$	0	0
$161$	20.0851	0.124752
$162$	0	0
$163$	20.8583i	0.127965i	0.997951	+	0.0639827i	$0.0203802\pi$
$163$	20.8583i	0.127965i	−0.997951	+	0.0639827i	$0.979620\pi$
$164$	0	0
$165$	140.498	0.851506
$166$	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
$171$	118.643i	0.693817i
$172$	0	0
$173$	−46.5836	−0.269269	−0.134635	−	0.990895i	$-0.542986\pi$
$173$	−46.5836	−0.269269	−0.134635	+	0.990895i	$0.542986\pi$
$174$	0	0
$175$	6.61585i	0.0378048i
$176$	0	0
$177$	−251.331	−1.41995
$178$	0	0
$179$	6.54211i	0.0365481i	0.999833	+	0.0182740i	$0.00581713\pi$
$179$	6.54211i	0.0365481i	−0.999833	+	0.0182740i	$0.994183\pi$
$180$	0	0
$181$	−241.331	−1.33332	−0.666661	−	0.745361i	$-0.732277\pi$
$181$	−241.331	−1.33332	−0.666661	+	0.745361i	$0.732277\pi$
$182$	0	0
$183$	433.922i	2.37116i
$184$	0	0
$185$	−74.7214	−0.403899
$186$	0	0
$187$	− 201.781i	− 1.07904i
$188$	0	0
$189$	−99.5016	−0.526463
$190$	0	0
$191$	− 59.3214i	− 0.310583i	−0.987869	−	0.155292i	$-0.950368\pi$
$191$	− 59.3214i	− 0.310583i	0.987869	−	0.155292i	$-0.0496317\pi$
$192$	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
$195$	218.284i	1.11940i
$196$	0	0
$197$	230.912	1.17214	0.586070	−	0.810260i	$-0.300674\pi$
$197$	230.912	1.17214	0.586070	+	0.810260i	$0.300674\pi$
$198$	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
$203$	11.6873i	0.0575729i
$204$	0	0
$205$	63.1672	0.308133
$206$	0	0
$207$	340.272i	1.64382i
$208$	0	0
$209$	−59.3313	−0.283882
$210$	0	0
$211$	− 93.0991i	− 0.441228i	−0.975361	−	0.220614i	$-0.929194\pi$
$211$	− 93.0991i	− 0.441228i	0.975361	−	0.220614i	$-0.0708061\pi$
$212$	0	0
$213$	547.161	2.56883
$214$	0	0
$215$	− 56.7488i	− 0.263948i
$216$	0	0
$217$	−55.8297	−0.257280
$218$	0	0
$219$	− 87.8064i	− 0.400942i
$220$	0	0
$221$	313.495	1.41853
$222$	0	0
$223$	− 307.891i	− 1.38068i	−0.723487	−	0.690338i	$-0.757462\pi$
$223$	− 307.891i	− 1.38068i	0.723487	−	0.690338i	$-0.242538\pi$
$224$	0	0
$225$	−112.082	−0.498142
$226$	0	0
$227$	255.020i	1.12344i	0.827328	+	0.561719i	$0.189860\pi$
$227$	255.020i	1.12344i	−0.827328	+	0.561719i	$0.810140\pi$
$228$	0	0
$229$	−411.495	−1.79692	−0.898461	−	0.439052i	$-0.855314\pi$
$229$	−411.495	−1.79692	−0.898461	+	0.439052i	$0.855314\pi$
$230$	0	0
$231$	− 83.1384i	− 0.359907i
$232$	0	0
$233$	−332.833	−1.42847	−0.714234	−	0.699907i	$-0.753225\pi$
$233$	−332.833	−1.42847	−0.714234	+	0.699907i	$0.753225\pi$
$234$	0	0
$235$	23.5047i	0.100020i
$236$	0	0
$237$	628.328	2.65117
$238$	0	0
$239$	266.248i	1.11401i	0.830510	+	0.557004i	$0.188049\pi$
$239$	266.248i	1.11401i	−0.830510	+	0.557004i	$0.811951\pi$
$240$	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
$243$	− 554.898i	− 2.28353i
$244$	0	0
$245$	−105.652	−0.431235
$246$	0	0
$247$	− 92.1794i	− 0.373196i
$248$	0	0
$249$	−515.745	−2.07126
$250$	0	0
$251$	404.482i	1.61148i	0.592268	+	0.805741i	$0.298233\pi$
$251$	404.482i	1.61148i	−0.592268	+	0.805741i	$0.701767\pi$
$252$	0	0
$253$	−170.164	−0.672585
$254$	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
$259$	44.2156i	0.170716i
$260$	0	0
$261$	−198.000	−0.758621
$262$	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
$267$	334.428i	1.25254i
$268$	0	0
$269$	−205.416	−0.763630	−0.381815	−	0.924239i	$-0.624701\pi$
$269$	−205.416	−0.763630	−0.381815	+	0.924239i	$0.624701\pi$
$270$	0	0
$271$	− 418.044i	− 1.54260i	−0.636474	−	0.771298i	$-0.719608\pi$
$271$	− 418.044i	− 1.54260i	0.636474	−	0.771298i	$-0.280392\pi$
$272$	0	0
$273$	129.167	0.473140
$274$	0	0
$275$	− 56.0503i	− 0.203819i
$276$	0	0
$277$	−511.410	−1.84625	−0.923123	−	0.384505i	$-0.874372\pi$
$277$	−511.410	−1.84625	−0.923123	+	0.384505i	$0.874372\pi$
$278$	0	0
$279$	− 945.836i	− 3.39009i
$280$	0	0
$281$	115.751	0.411924	0.205962	−	0.978560i	$-0.433968\pi$
$281$	115.751	0.411924	0.205962	+	0.978560i	$0.433968\pi$
$282$	0	0
$283$	402.001i	1.42050i	0.703951	+	0.710249i	$0.251417\pi$
$283$	402.001i	1.42050i	−0.703951	+	0.710249i	$0.748583\pi$
$284$	0	0
$285$	66.3344	0.232752
$286$	0	0
$287$	− 37.3785i	− 0.130239i
$288$	0	0
$289$	35.0000	0.121107
$290$	0	0
$291$	− 606.263i	− 2.08338i
$292$	0	0
$293$	90.4133	0.308578	0.154289	−	0.988026i	$-0.450691\pi$
$293$	90.4133	0.308578	0.154289	+	0.988026i	$0.450691\pi$
$294$	0	0
$295$	100.266i	0.339884i
$296$	0	0
$297$	842.991	2.83835
$298$	0	0
$299$	− 264.374i	− 0.884193i
$300$	0	0
$301$	−33.5805	−0.111563
$302$	0	0
$303$	468.949i	1.54769i
$304$	0	0
$305$	173.108	0.567568
$306$	0	0
$307$	275.419i	0.897130i	0.893750	+	0.448565i	$0.148065\pi$
$307$	275.419i	0.897130i	−0.893750	+	0.448565i	$0.851935\pi$
$308$	0	0
$309$	757.909	2.45278
$310$	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	$-0.498983\pi$
$313$	2.00000	0.00638978	0.00319489	+	0.999995i	$0.498983\pi$
$314$	0	0
$315$	66.3233i	0.210550i
$316$	0	0
$317$	−394.741	−1.24524	−0.622621	−	0.782524i	$-0.713932\pi$
$317$	−394.741	−1.24524	−0.622621	+	0.782524i	$0.713932\pi$
$318$	0	0
$319$	− 99.0165i	− 0.310396i
$320$	0	0
$321$	−685.909	−2.13679
$322$	0	0
$323$	− 95.2682i	− 0.294948i
$324$	0	0
$325$	87.0820	0.267945
$326$	0	0
$327$	− 14.4811i	− 0.0442848i
$328$	0	0
$329$	13.9086	0.0422755
$330$	0	0
$331$	− 296.130i	− 0.894652i	−0.894371	−	0.447326i	$-0.852376\pi$
$331$	− 296.130i	− 0.894652i	0.894371	−	0.447326i	$-0.147624\pi$
$332$	0	0
$333$	−749.076	−2.24948
$334$	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
$339$	187.777i	0.553915i
$340$	0	0
$341$	472.997	1.38709
$342$	0	0
$343$	127.354i	0.371294i
$344$	0	0
$345$	190.249	0.551447
$346$	0	0
$347$	546.959i	1.57625i	0.615514	+	0.788126i	$0.288948\pi$
$347$	546.959i	1.57625i	−0.615514	+	0.788126i	$0.711052\pi$
$348$	0	0
$349$	40.8328	0.116999	0.0584997	−	0.998287i	$-0.481368\pi$
$349$	40.8328	0.116999	0.0584997	+	0.998287i	$0.481368\pi$
$350$	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
$355$	− 218.284i	− 0.614884i
$356$	0	0
$357$	133.495	0.373937
$358$	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
$363$	26.1510i	0.0720414i
$364$	0	0
$365$	−35.0294	−0.0959710
$366$	0	0
$367$	13.6352i	0.0371531i	0.999827	+	0.0185766i	$0.00591344\pi$
$367$	13.6352i	0.0371531i	−0.999827	+	0.0185766i	$0.994087\pi$
$368$	0	0
$369$	633.246	1.71611
$370$	0	0
$371$	− 37.3785i	− 0.100751i
$372$	0	0
$373$	−136.748	−0.366616	−0.183308	−	0.983056i	$-0.558681\pi$
$373$	−136.748	−0.366616	−0.183308	+	0.983056i	$0.558681\pi$
$374$	0	0
$375$	62.6662i	0.167110i
$376$	0	0
$377$	153.836	0.408053
$378$	0	0
$379$	− 186.198i	− 0.491288i	−0.969360	−	0.245644i	$-0.921001\pi$
$379$	− 186.198i	− 0.491288i	0.969360	−	0.245644i	$-0.0789994\pi$
$380$	0	0
$381$	−721.240	−1.89302
$382$	0	0
$383$	− 475.712i	− 1.24207i	−0.783784	−	0.621034i	$-0.786713\pi$
$383$	− 475.712i	− 1.24207i	0.783784	−	0.621034i	$-0.213287\pi$
$384$	0	0
$385$	−33.1672	−0.0861485
$386$	0	0
$387$	− 568.902i	− 1.47003i
$388$	0	0
$389$	11.9211	0.0306454	0.0153227	−	0.999883i	$-0.495122\pi$
$389$	11.9211	0.0306454	0.0153227	+	0.999883i	$0.495122\pi$
$390$	0	0
$391$	− 273.232i	− 0.698804i
$392$	0	0
$393$	1246.15	3.17087
$394$	0	0
$395$	− 250.665i	− 0.634594i
$396$	0	0
$397$	250.754	0.631622	0.315811	−	0.948822i	$-0.397723\pi$
$397$	250.754	0.631622	0.315811	+	0.948822i	$0.397723\pi$
$398$	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
$403$	734.867i	1.82349i
$404$	0	0
$405$	−491.371	−1.21326
$406$	0	0
$407$	− 374.600i	− 0.920394i
$408$	0	0
$409$	−27.0820	−0.0662153	−0.0331076	−	0.999452i	$-0.510540\pi$
$409$	−27.0820	−0.0662153	−0.0331076	+	0.999452i	$0.510540\pi$
$410$	0	0
$411$	− 722.112i	− 1.75696i
$412$	0	0
$413$	59.3313	0.143659
$414$	0	0
$415$	205.751i	0.495785i
$416$	0	0
$417$	−1293.33	−3.10150
$418$	0	0
$419$	198.033i	0.472632i	0.971676	+	0.236316i	$0.0759401\pi$
$419$	198.033i	0.472632i	−0.971676	+	0.236316i	$0.924060\pi$
$420$	0	0
$421$	475.580	1.12964	0.564822	−	0.825212i	$-0.308945\pi$
$421$	475.580	1.12964	0.564822	+	0.825212i	$0.308945\pi$
$422$	0	0
$423$	235.633i	0.557051i
$424$	0	0
$425$	90.0000	0.211765
$426$	0	0
$427$	− 102.435i	− 0.239895i
$428$	0	0
$429$	−1094.32	−2.55087
$430$	0	0
$431$	− 489.052i	− 1.13469i	−0.823479	−	0.567346i	$-0.807970\pi$
$431$	− 489.052i	− 1.13469i	0.823479	−	0.567346i	$-0.192030\pi$
$432$	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
$435$	110.704i	0.254491i
$436$	0	0
$437$	−80.3406	−0.183846
$438$	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
$443$	− 668.873i	− 1.50987i	−0.655799	−	0.754936i	$-0.727668\pi$
$443$	− 668.873i	− 1.50987i	0.655799	−	0.754936i	$-0.272332\pi$
$444$	0	0
$445$	133.416	0.299812
$446$	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
$451$	316.676i	0.702163i
$452$	0	0
$453$	−362.164	−0.799479
$454$	0	0
$455$	− 51.5299i	− 0.113252i
$456$	0	0
$457$	−320.334	−0.700950	−0.350475	−	0.936572i	$-0.613980\pi$
$457$	−320.334	−0.700950	−0.350475	+	0.936572i	$0.613980\pi$
$458$	0	0
$459$	1353.59i	2.94900i
$460$	0	0
$461$	163.337	0.354311	0.177156	−	0.984183i	$-0.443310\pi$
$461$	163.337	0.354311	0.177156	+	0.984183i	$0.443310\pi$
$462$	0	0
$463$	− 108.131i	− 0.233545i	−0.993159	−	0.116772i	$-0.962745\pi$
$463$	− 108.131i	− 0.233545i	0.993159	−	0.116772i	$-0.0372548\pi$
$464$	0	0
$465$	−528.827	−1.13726
$466$	0	0
$467$	207.386i	0.444082i	0.975037	+	0.222041i	$0.0712719\pi$
$467$	207.386i	0.444082i	−0.975037	+	0.222041i	$0.928728\pi$
$468$	0	0
$469$	−48.4133	−0.103227
$470$	0	0
$471$	81.7415i	0.173549i
$472$	0	0
$473$	284.498	0.601477
$474$	0	0
$475$	− 26.4634i	− 0.0557124i
$476$	0	0
$477$	633.246	1.32756
$478$	0	0
$479$	− 233.537i	− 0.487552i	−0.969832	−	0.243776i	$-0.921614\pi$
$479$	− 233.537i	− 0.487552i	0.969832	−	0.243776i	$-0.0783861\pi$
$480$	0	0
$481$	581.994	1.20997
$482$	0	0
$483$	− 112.578i	− 0.233081i
$484$	0	0
$485$	−241.862	−0.498685
$486$	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
$491$	187.777i	0.382439i	0.981547	+	0.191219i	$0.0612442\pi$
$491$	187.777i	0.382439i	−0.981547	+	0.191219i	$0.938756\pi$
$492$	0	0
$493$	158.991	0.322496
$494$	0	0
$495$	− 561.900i	− 1.13515i
$496$	0	0
$497$	−129.167	−0.259894
$498$	0	0
$499$	231.038i	0.463003i	0.972835	+	0.231501i	$0.0743638\pi$
$499$	231.038i	0.463003i	−0.972835	+	0.231501i	$0.925636\pi$
$500$	0	0
$501$	1216.07	2.42729
$502$	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
$507$	− 752.931i	− 1.48507i
$508$	0	0
$509$	363.167	0.713492	0.356746	−	0.934202i	$-0.383886\pi$
$509$	363.167	0.713492	0.356746	+	0.934202i	$0.383886\pi$
$510$	0	0
$511$	20.7283i	0.0405641i
$512$	0	0
$513$	398.006	0.775841
$514$	0	0
$515$	− 302.359i	− 0.587106i
$516$	0	0
$517$	−117.836	−0.227922
$518$	0	0
$519$	261.103i	0.503088i
$520$	0	0
$521$	−491.666	−0.943696	−0.471848	−	0.881680i	$-0.656413\pi$
$521$	−491.666	−0.943696	−0.471848	+	0.881680i	$0.656413\pi$
$522$	0	0
$523$	133.914i	0.256049i	0.991771	+	0.128024i	$0.0408636\pi$
$523$	133.914i	0.256049i	−0.991771	+	0.128024i	$0.959136\pi$
$524$	0	0
$525$	37.0820	0.0706325
$526$	0	0
$527$	759.491i	1.44116i
$528$	0	0
$529$	298.580	0.564424
$530$	0	0
$531$	1005.16i	1.89295i
$532$	0	0
$533$	−492.000	−0.923077
$534$	0	0
$535$	273.636i	0.511469i
$536$	0	0
$537$	36.6687	0.0682844
$538$	0	0
$539$	− 529.667i	− 0.982685i
$540$	0	0
$541$	−303.337	−0.560698	−0.280349	−	0.959898i	$-0.590450\pi$
$541$	−303.337	−0.560698	−0.280349	+	0.959898i	$0.590450\pi$
$542$	0	0
$543$	1352.67i	2.49110i
$544$	0	0
$545$	−5.77709	−0.0106002
$546$	0	0
$547$	− 89.6631i	− 0.163918i	−0.996636	−	0.0819590i	$-0.973882\pi$
$547$	− 89.6631i	− 0.163918i	0.996636	−	0.0819590i	$-0.0261177\pi$
$548$	0	0
$549$	1735.40	3.16102
$550$	0	0
$551$	− 46.7492i	− 0.0848443i
$552$	0	0
$553$	−148.328	−0.268225
$554$	0	0
$555$	418.816i	0.754623i
$556$	0	0
$557$	−490.741	−0.881044	−0.440522	−	0.897742i	$-0.645207\pi$
$557$	−490.741	−0.881044	−0.440522	+	0.897742i	$0.645207\pi$
$558$	0	0
$559$	442.008i	0.790712i
$560$	0	0
$561$	−1130.99	−2.01603
$562$	0	0
$563$	987.423i	1.75386i	0.480618	+	0.876930i	$0.340412\pi$
$563$	987.423i	1.75386i	−0.480618	+	0.876930i	$0.659588\pi$
$564$	0	0
$565$	74.9117	0.132587
$566$	0	0
$567$	290.763i	0.512810i
$568$	0	0
$569$	−475.082	−0.834942	−0.417471	−	0.908690i	$-0.637083\pi$
$569$	−475.082	−0.834942	−0.417471	+	0.908690i	$0.637083\pi$
$570$	0	0
$571$	182.485i	0.319588i	0.987150	+	0.159794i	$0.0510830\pi$
$571$	182.485i	0.319588i	−0.987150	+	0.159794i	$0.948917\pi$
$572$	0	0
$573$	−332.498	−0.580277
$574$	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
$579$	− 1317.17i	− 2.27490i
$580$	0	0
$581$	121.751	0.209554
$582$	0	0
$583$	316.676i	0.543183i
$584$	0	0
$585$	872.991	1.49229
$586$	0	0
$587$	− 904.762i	− 1.54133i	−0.637239	−	0.770666i	$-0.719924\pi$
$587$	− 904.762i	− 1.54133i	0.637239	−	0.770666i	$-0.280076\pi$
$588$	0	0
$589$	223.319	0.379149
$590$	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
$595$	− 53.2565i	− 0.0895068i
$596$	0	0
$597$	−155.331	−0.260186
$598$	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	$-0.723817\pi$
$601$	−777.234	−1.29323	−0.646617	+	0.762815i	$0.723817\pi$
$602$	0	0
$603$	− 820.192i	− 1.36019i
$604$	0	0
$605$	10.4327	0.0172441
$606$	0	0
$607$	699.436i	1.15228i	0.817350	+	0.576142i	$0.195442\pi$
$607$	699.436i	1.15228i	−0.817350	+	0.576142i	$0.804558\pi$
$608$	0	0
$609$	65.5078	0.107566
$610$	0	0
$611$	− 183.075i	− 0.299631i
$612$	0	0
$613$	−11.0820	−0.0180784	−0.00903918	−	0.999959i	$-0.502877\pi$
$613$	−11.0820	−0.0180784	−0.00903918	+	0.999959i	$0.502877\pi$
$614$	0	0
$615$	− 354.054i	− 0.575698i
$616$	0	0
$617$	1189.16	1.92733	0.963664	−	0.267119i	$-0.0860715\pi$
$617$	1189.16	1.92733	0.963664	+	0.267119i	$0.0860715\pi$
$618$	0	0
$619$	− 325.717i	− 0.526198i	−0.964769	−	0.263099i	$-0.915255\pi$
$619$	− 325.717i	− 0.526198i	0.964769	−	0.263099i	$-0.0847446\pi$
$620$	0	0
$621$	1141.50	1.83816
$622$	0	0
$623$	− 78.9477i	− 0.126722i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	332.554i	0.530389i
$628$	0	0
$629$	601.495	0.956272
$630$	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
$635$	287.731i	0.453120i
$636$	0	0
$637$	822.912	1.29186
$638$	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
$643$	− 408.690i	− 0.635599i	−0.948158	−	0.317800i	$-0.897056\pi$
$643$	− 408.690i	− 0.635599i	0.948158	−	0.317800i	$-0.102944\pi$
$644$	0	0
$645$	−318.079	−0.493146
$646$	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
$651$	312.927i	0.480687i
$652$	0	0
$653$	−164.420	−0.251791	−0.125895	−	0.992044i	$-0.540180\pi$
$653$	−164.420	−0.251791	−0.125895	+	0.992044i	$0.540180\pi$
$654$	0	0
$655$	− 497.139i	− 0.758990i
$656$	0	0
$657$	−351.167	−0.534501
$658$	0	0
$659$	− 743.613i	− 1.12840i	−0.825640	−	0.564198i	$-0.809186\pi$
$659$	− 743.613i	− 1.12840i	0.825640	−	0.564198i	$-0.190814\pi$
$660$	0	0
$661$	−758.073	−1.14686	−0.573429	−	0.819255i	$-0.694387\pi$
$661$	−758.073	−1.14686	−0.573429	+	0.819255i	$0.694387\pi$
$662$	0	0
$663$	− 1757.15i	− 2.65030i
$664$	0	0
$665$	−15.6594	−0.0235480
$666$	0	0
$667$	− 134.078i	− 0.201017i
$668$	0	0
$669$	−1725.74	−2.57958
$670$	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
$675$	375.997i	0.557033i
$676$	0	0
$677$	−708.079	−1.04591	−0.522953	−	0.852361i	$-0.675170\pi$
$677$	−708.079	−1.04591	−0.522953	+	0.852361i	$0.675170\pi$
$678$	0	0
$679$	143.119i	0.210780i
$680$	0	0
$681$	1429.40	2.09897
$682$	0	0
$683$	− 355.469i	− 0.520452i	−0.965548	−	0.260226i	$-0.916203\pi$
$683$	− 355.469i	− 0.520452i	0.965548	−	0.260226i	$-0.0837970\pi$
$684$	0	0
$685$	−288.079	−0.420553
$686$	0	0
$687$	2306.45i	3.35727i
$688$	0	0
$689$	−492.000	−0.714078
$690$	0	0
$691$	116.769i	0.168985i	0.996424	+	0.0844925i	$0.0269269\pi$
$691$	116.769i	0.168985i	−0.996424	+	0.0844925i	$0.973073\pi$
$692$	0	0
$693$	−332.498	−0.479796
$694$	0	0
$695$	515.958i	0.742386i
$696$	0	0
$697$	−508.486	−0.729535
$698$	0	0
$699$	1865.54i	2.66887i
$700$	0	0
$701$	−246.413	−0.351517	−0.175758	−	0.984433i	$-0.556238\pi$
$701$	−246.413	−0.351517	−0.175758	+	0.984433i	$0.556238\pi$
$702$	0	0
$703$	− 176.862i	− 0.251582i
$704$	0	0
$705$	131.745	0.186872
$706$	0	0
$707$	− 110.704i	− 0.156582i
$708$	0	0
$709$	−552.820	−0.779718	−0.389859	−	0.920874i	$-0.627476\pi$
$709$	−552.820	−0.779718	−0.389859	+	0.920874i	$0.627476\pi$
$710$	0	0
$711$	− 2512.89i	− 3.53431i
$712$	0	0
$713$	640.486	0.898297
$714$	0	0
$715$	436.568i	0.610585i
$716$	0	0
$717$	1492.33	2.08135
$718$	0	0
$719$	99.9709i	0.139042i	0.997581	+	0.0695208i	$0.0221470\pi$
$719$	99.9709i	0.139042i	−0.997581	+	0.0695208i	$0.977853\pi$
$720$	0	0
$721$	−178.918	−0.248153
$722$	0	0
$723$	440.464i	0.609217i
$724$	0	0
$725$	44.1641	0.0609160
$726$	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
$731$	456.819i	0.624923i
$732$	0	0
$733$	380.407	0.518973	0.259486	−	0.965747i	$-0.416447\pi$
$733$	380.407	0.518973	0.259486	+	0.965747i	$0.416447\pi$
$734$	0	0
$735$	592.186i	0.805695i
$736$	0	0
$737$	410.164	0.556532
$738$	0	0
$739$	− 1381.60i	− 1.86955i	−0.355240	−	0.934775i	$-0.615601\pi$
$739$	− 1381.60i	− 1.86955i	0.355240	−	0.934775i	$-0.384399\pi$
$740$	0	0
$741$	−516.669	−0.697259
$742$	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
$747$	2062.64i	2.76123i
$748$	0	0
$749$	161.921	0.216183
$750$	0	0
$751$	− 338.653i	− 0.450937i	−0.974250	−	0.225468i	$-0.927609\pi$
$751$	− 338.653i	− 0.450937i	0.974250	−	0.225468i	$-0.0723912\pi$
$752$	0	0
$753$	2267.14	3.01080
$754$	0	0
$755$	144.481i	0.191366i
$756$	0	0
$757$	1247.91	1.64849	0.824246	−	0.566232i	$-0.191599\pi$
$757$	1247.91	1.64849	0.824246	+	0.566232i	$0.191599\pi$
$758$	0	0
$759$	953.775i	1.25662i
$760$	0	0
$761$	−14.3406	−0.0188444	−0.00942219	−	0.999956i	$-0.502999\pi$
$761$	−14.3406	−0.0188444	−0.00942219	+	0.999956i	$0.502999\pi$
$762$	0	0
$763$	3.41853i	0.00448038i
$764$	0	0
$765$	902.243	1.17940
$766$	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
$771$	481.078i	0.623967i
$772$	0	0
$773$	1197.91	1.54970	0.774848	−	0.632148i	$-0.217826\pi$
$773$	1197.91	1.54970	0.774848	+	0.632148i	$0.217826\pi$
$774$	0	0
$775$	210.970i	0.272219i
$776$	0	0
$777$	247.830	0.318957
$778$	0	0
$779$	149.514i	0.191931i
$780$	0	0
$781$	1094.32	1.40118
$782$	0	0
$783$	664.223i	0.848305i
$784$	0	0
$785$	32.6099	0.0415413
$786$	0	0
$787$	− 99.1814i	− 0.126025i	−0.998013	−	0.0630123i	$-0.979929\pi$
$787$	− 99.1814i	− 0.126025i	0.998013	−	0.0630123i	$-0.0200707\pi$
$788$	0	0
$789$	−2470.05	−3.13061
$790$	0	0
$791$	− 44.3282i	− 0.0560407i
$792$	0	0
$793$	−1348.32	−1.70027
$794$	0	0
$795$	− 354.054i	− 0.445351i
$796$	0	0
$797$	139.240	0.174705	0.0873525	−	0.996177i	$-0.472159\pi$
$797$	139.240	0.174705	0.0873525	+	0.996177i	$0.472159\pi$
$798$	0	0
$799$	− 189.209i	− 0.236807i
$800$	0	0
$801$	1337.49	1.66977
$802$	0	0
$803$	− 175.613i	− 0.218696i
$804$	0	0
$805$	−44.9117	−0.0557910
$806$	0	0
$807$	1151.37i	1.42672i
$808$	0	0
$809$	403.653	0.498953	0.249477	−	0.968381i	$-0.419741\pi$
$809$	403.653	0.498953	0.249477	+	0.968381i	$0.419741\pi$
$810$	0	0
$811$	− 384.561i	− 0.474181i	−0.971488	−	0.237091i	$-0.923806\pi$
$811$	− 384.561i	− 0.474181i	0.971488	−	0.237091i	$-0.0761938\pi$
$812$	0	0
$813$	−2343.15	−2.88210
$814$	0	0
$815$	− 46.6407i	− 0.0572278i
$816$	0	0
$817$	134.322	0.164409
$818$	0	0
$819$	− 516.583i	− 0.630748i
$820$	0	0
$821$	−997.732	−1.21526	−0.607632	−	0.794219i	$-0.707881\pi$
$821$	−997.732	−1.21526	−0.607632	+	0.794219i	$0.707881\pi$
$822$	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
$827$	213.451i	0.258103i	0.991638	+	0.129051i	$0.0411932\pi$
$827$	213.451i	0.258103i	−0.991638	+	0.129051i	$0.958807\pi$
$828$	0	0
$829$	−73.4288	−0.0885752	−0.0442876	−	0.999019i	$-0.514102\pi$
$829$	−73.4288	−0.0885752	−0.0442876	+	0.999019i	$0.514102\pi$
$830$	0	0
$831$	2866.47i	3.44942i
$832$	0	0
$833$	850.486	1.02099
$834$	0	0
$835$	− 485.139i	− 0.581005i
$836$	0	0
$837$	−3172.96	−3.79087
$838$	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
$843$	− 648.787i	− 0.769617i
$844$	0	0
$845$	−300.374	−0.355472
$846$	0	0
$847$	− 6.17342i	− 0.00728857i
$848$	0	0
$849$	2253.23	2.65398
$850$	0	0
$851$	− 507.247i	− 0.596060i
$852$	0	0
$853$	−351.410	−0.411970	−0.205985	−	0.978555i	$-0.566040\pi$
$853$	−351.410	−0.411970	−0.205985	+	0.978555i	$0.566040\pi$
$854$	0	0
$855$	− 265.293i	− 0.310285i
$856$	0	0
$857$	1326.16	1.54744	0.773721	−	0.633526i	$-0.218393\pi$
$857$	1326.16	1.54744	0.773721	+	0.633526i	$0.218393\pi$
$858$	0	0
$859$	343.729i	0.400150i	0.979781	+	0.200075i	$0.0641186\pi$
$859$	343.729i	0.400150i	−0.979781	+	0.200075i	$0.935881\pi$
$860$	0	0
$861$	−209.508	−0.243331
$862$	0	0
$863$	1136.15i	1.31651i	0.752793	+	0.658257i	$0.228706\pi$
$863$	1136.15i	1.31651i	−0.752793	+	0.658257i	$0.771294\pi$
$864$	0	0
$865$	104.164	0.120421
$866$	0	0
$867$	− 196.176i	− 0.226270i
$868$	0	0
$869$	1256.66	1.44609
$870$	0	0
$871$	637.247i	0.731627i
$872$	0	0
$873$	−2424.65	−2.77738
$874$	0	0
$875$	− 14.7935i	− 0.0169068i
$876$	0	0
$877$	769.076	0.876939	0.438470	−	0.898746i	$-0.355521\pi$
$877$	769.076	0.876939	0.438470	+	0.898746i	$0.355521\pi$
$878$	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
$883$	− 1066.19i	− 1.20746i	−0.797188	−	0.603731i	$-0.793680\pi$
$883$	− 1066.19i	− 1.20746i	0.797188	−	0.603731i	$-0.206320\pi$
$884$	0	0
$885$	561.994	0.635021
$886$	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
$891$	− 2463.39i	− 2.76474i
$892$	0	0
$893$	−55.6346	−0.0623008
$894$	0	0
$895$	− 14.6286i	− 0.0163448i
$896$	0	0
$897$	−1481.82	−1.65198
$898$	0	0
$899$	372.691i	0.414562i
$900$	0	0
$901$	−508.486	−0.564357
$902$	0	0
$903$	188.220i	0.208438i
$904$	0	0
$905$	539.633	0.596280
$906$	0	0
$907$	− 1021.02i	− 1.12571i	−0.826556	−	0.562855i	$-0.809703\pi$
$907$	− 1021.02i	− 1.12571i	0.826556	−	0.562855i	$-0.190297\pi$
$908$	0	0
$909$	1875.48	2.06324
$910$	0	0
$911$	− 1118.66i	− 1.22794i	−0.789328	−	0.613971i	$-0.789571\pi$
$911$	− 1118.66i	− 1.22794i	0.789328	−	0.613971i	$-0.210429\pi$
$912$	0	0
$913$	−1031.49	−1.12978
$914$	0	0
$915$	− 970.278i	− 1.06041i
$916$	0	0
$917$	−294.177	−0.320803
$918$	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
$923$	1700.18i	1.84202i
$924$	0	0
$925$	167.082	0.180629
$926$	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
$931$	− 250.075i	− 0.268609i
$932$	0	0
$933$	2720.15	2.91548
$934$	0	0
$935$	451.197i	0.482563i
$936$	0	0
$937$	−332.152	−0.354484	−0.177242	−	0.984167i	$-0.556718\pi$
$937$	−332.152	−0.354484	−0.177242	+	0.984167i	$0.556718\pi$
$938$	0	0
$939$	− 11.2101i	− 0.0119383i
$940$	0	0
$941$	−1468.66	−1.56075	−0.780373	−	0.625314i	$-0.784971\pi$
$941$	−1468.66	−1.56075	−0.780373	+	0.625314i	$0.784971\pi$
$942$	0	0
$943$	428.811i	0.454731i
$944$	0	0
$945$	222.492	0.235442
$946$	0	0
$947$	574.082i	0.606211i	0.952957	+	0.303106i	$0.0980235\pi$
$947$	574.082i	0.606211i	−0.952957	+	0.303106i	$0.901976\pi$
$948$	0	0
$949$	272.839	0.287502
$950$	0	0
$951$	2212.54i	2.32654i
$952$	0	0
$953$	280.663	0.294504	0.147252	−	0.989099i	$-0.452957\pi$
$953$	280.663	0.294504	0.147252	+	0.989099i	$0.452957\pi$
$954$	0	0
$955$	132.647i	0.138897i
$956$	0	0
$957$	−554.991	−0.579928
$958$	0	0
$959$	170.468i	0.177756i
$960$	0	0
$961$	−819.328	−0.852579
$962$	0	0
$963$	2743.18i	2.84858i
$964$	0	0
$965$	−525.469	−0.544527
$966$	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
$971$	− 117.723i	− 0.121239i	−0.998161	−	0.0606195i	$-0.980692\pi$
$971$	− 117.723i	− 0.121239i	0.998161	−	0.0606195i	$-0.0193076\pi$
$972$	0	0
$973$	305.313	0.313785
$974$	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
$979$	668.856i	0.683203i
$980$	0	0
$981$	−57.9149	−0.0590365
$982$	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
$987$	− 77.9584i	− 0.0789852i
$988$	0	0
$989$	385.240	0.389525
$990$	0	0
$991$	408.413i	0.412122i	0.978539	+	0.206061i	$0.0660645\pi$
$991$	408.413i	0.412122i	−0.978539	+	0.206061i	$0.933935\pi$
$992$	0	0
$993$	−1659.82	−1.67152
$994$	0	0
$995$	61.9677i	0.0622791i
$996$	0	0
$997$	−1554.91	−1.55959	−0.779795	−	0.626035i	$-0.784677\pi$
$997$	−1554.91	−1.55959	−0.779795	+	0.626035i	$0.784677\pi$
$998$	0	0
$999$	2512.89i	2.51541i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.3.b.a.31.1	✓	4
3.2	odd	2		720.3.e.c.271.4		4
4.3	odd	2	inner	80.3.b.a.31.4	yes	4
5.2	odd	4		400.3.h.d.399.1		8
5.3	odd	4		400.3.h.d.399.7		8
5.4	even	2		400.3.b.g.351.4		4
8.3	odd	2		320.3.b.a.191.1		4
8.5	even	2		320.3.b.a.191.4		4
12.11	even	2		720.3.e.c.271.3		4
15.2	even	4		3600.3.j.k.1999.4		8
15.8	even	4		3600.3.j.k.1999.6		8
15.14	odd	2		3600.3.e.bb.3151.2		4
16.3	odd	4		1280.3.g.f.1151.2		8
16.5	even	4		1280.3.g.f.1151.1		8
16.11	odd	4		1280.3.g.f.1151.7		8
16.13	even	4		1280.3.g.f.1151.8		8
20.3	even	4		400.3.h.d.399.2		8
20.7	even	4		400.3.h.d.399.8		8
20.19	odd	2		400.3.b.g.351.1		4
24.5	odd	2		2880.3.e.b.2431.2		4
24.11	even	2		2880.3.e.b.2431.1		4
40.3	even	4		1600.3.h.p.1599.7		8
40.13	odd	4		1600.3.h.p.1599.2		8
40.19	odd	2		1600.3.b.k.1151.4		4
40.27	even	4		1600.3.h.p.1599.1		8
40.29	even	2		1600.3.b.k.1151.1		4
40.37	odd	4		1600.3.h.p.1599.8		8
60.23	odd	4		3600.3.j.k.1999.3		8
60.47	odd	4		3600.3.j.k.1999.5		8
60.59	even	2		3600.3.e.bb.3151.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.3.b.a.31.1	✓	4	1.1	even	1	trivial
80.3.b.a.31.4	yes	4	4.3	odd	2	inner
320.3.b.a.191.1		4	8.3	odd	2
320.3.b.a.191.4		4	8.5	even	2
400.3.b.g.351.1		4	20.19	odd	2
400.3.b.g.351.4		4	5.4	even	2
400.3.h.d.399.1		8	5.2	odd	4
400.3.h.d.399.2		8	20.3	even	4
400.3.h.d.399.7		8	5.3	odd	4
400.3.h.d.399.8		8	20.7	even	4
720.3.e.c.271.3		4	12.11	even	2
720.3.e.c.271.4		4	3.2	odd	2
1280.3.g.f.1151.1		8	16.5	even	4
1280.3.g.f.1151.2		8	16.3	odd	4
1280.3.g.f.1151.7		8	16.11	odd	4
1280.3.g.f.1151.8		8	16.13	even	4
1600.3.b.k.1151.1		4	40.29	even	2
1600.3.b.k.1151.4		4	40.19	odd	2
1600.3.h.p.1599.1		8	40.27	even	4
1600.3.h.p.1599.2		8	40.13	odd	4
1600.3.h.p.1599.7		8	40.3	even	4
1600.3.h.p.1599.8		8	40.37	odd	4
2880.3.e.b.2431.1		4	24.11	even	2
2880.3.e.b.2431.2		4	24.5	odd	2
3600.3.e.bb.3151.2		4	15.14	odd	2
3600.3.e.bb.3151.3		4	60.59	even	2
3600.3.j.k.1999.3		8	60.23	odd	4
3600.3.j.k.1999.4		8	15.2	even	4
3600.3.j.k.1999.5		8	60.47	odd	4
3600.3.j.k.1999.6		8	15.8	even	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:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	80.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.60503i q^{3} -2.23607 q^{5} +1.32317i q^{7} -22.4164 q^{9} -11.2101i q^{11} +17.4164 q^{13} +12.5332i q^{15} +18.0000 q^{17} -5.29268i q^{19} +7.41641 q^{21} -15.1796i q^{23} +5.00000 q^{25} +75.1994i q^{27} +8.83282 q^{29} +42.1939i q^{31} -62.8328 q^{33} -2.95870i q^{35} +33.4164 q^{37} -97.6196i q^{39} -28.2492 q^{41} +25.3788i q^{43} +50.1246 q^{45} -10.5116i q^{47} +47.2492 q^{49} -100.891i q^{51} -28.2492 q^{53} +25.0665i q^{55} -29.6656 q^{57} -44.8403i q^{59} -77.4164 q^{61} -29.6607i q^{63} -38.9443 q^{65} +36.5889i q^{67} -85.0820 q^{69} +97.6196i q^{71} +15.6656 q^{73} -28.0252i q^{75} +14.8328 q^{77} +112.101i q^{79} +219.748 q^{81} -92.0145i q^{83} -40.2492 q^{85} -49.5082i q^{87} -59.6656 q^{89} +23.0449i q^{91} +236.498 q^{93} +11.8348i q^{95} +108.164 q^{97} +251.289i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 36 q^{9} + 16 q^{13} + 72 q^{17} - 24 q^{21} + 20 q^{25} - 72 q^{29} - 144 q^{33} + 80 q^{37} + 48 q^{41} + 120 q^{45} + 28 q^{49} + 48 q^{53} + 96 q^{57} - 256 q^{61} - 120 q^{65} - 72 q^{69} - 152 q^{73}+ \cdots - 104 q^{97}+O(q^{100}) \) 4 * q - 36 * q^9 + 16 * q^13 + 72 * q^17 - 24 * q^21 + 20 * q^25 - 72 * q^29 - 144 * q^33 + 80 * q^37 + 48 * q^41 + 120 * q^45 + 28 * q^49 + 48 * q^53 + 96 * q^57 - 256 * q^61 - 120 * q^65 - 72 * q^69 - 152 * q^73 - 48 * q^77 + 396 * q^81 - 24 * q^89 + 624 * q^93 - 104 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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