LMFDB - Embedded newform 800.2.f.e.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,2,Mod(49,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("800.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	800.f (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:	$6.38803216170$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	800.49
Dual form			800.2.f.e.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.732051 q^{3} +2.73205i q^{7} -2.46410 q^{9} +O(q^{10})$	$q-0.732051 q^{3} +2.73205i q^{7} -2.46410 q^{9} -2.00000i q^{11} -3.46410 q^{13} -3.46410i q^{17} -7.46410i q^{19} -2.00000i q^{21} +4.19615i q^{23} +4.00000 q^{27} -6.92820i q^{29} -1.46410 q^{31} +1.46410i q^{33} -2.00000 q^{37} +2.53590 q^{39} -5.46410 q^{41} -8.73205 q^{43} -6.73205i q^{47} -0.464102 q^{49} +2.53590i q^{51} -4.53590 q^{53} +5.46410i q^{57} +0.535898i q^{59} -4.92820i q^{61} -6.73205i q^{63} -7.26795 q^{67} -3.07180i q^{69} +1.46410 q^{71} -0.535898i q^{73} +5.46410 q^{77} -14.9282 q^{79} +4.46410 q^{81} +4.73205 q^{83} +5.07180i q^{87} +4.92820 q^{89} -9.46410i q^{91} +1.07180 q^{93} +6.39230i q^{97} +4.92820i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 4 * q^9	$ 4 q + 4 q^{3} + 4 q^{9} + 16 q^{27} + 8 q^{31} - 8 q^{37} + 24 q^{39} - 8 q^{41} - 28 q^{43} + 12 q^{49} - 32 q^{53} - 36 q^{67} - 8 q^{71} + 8 q^{77} - 32 q^{79} + 4 q^{81} + 12 q^{83} - 8 q^{89} + 32 q^{93}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^9 + 16 * q^27 + 8 * q^31 - 8 * q^37 + 24 * q^39 - 8 * q^41 - 28 * q^43 + 12 * q^49 - 32 * q^53 - 36 * q^67 - 8 * q^71 + 8 * q^77 - 32 * q^79 + 4 * q^81 + 12 * q^83 - 8 * q^89 + 32 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	−0.732051	−0.422650	−0.211325	−	0.977416i	$-0.567778\pi$
$3$	−0.732051	−0.422650	−0.211325	+	0.977416i	$0.567778\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.73205i	1.03262i	0.856402	+	0.516309i	$0.172694\pi$
$7$	2.73205i	1.03262i	−0.856402	+	0.516309i	$0.827306\pi$
$8$	0	0
$9$	−2.46410	−0.821367
$10$	0	0
$11$	− 2.00000i	− 0.603023i	−0.953463	−	0.301511i	$-0.902509\pi$
$11$	− 2.00000i	− 0.603023i	0.953463	−	0.301511i	$-0.0974911\pi$
$12$	0	0
$13$	−3.46410	−0.960769	−0.480384	−	0.877058i	$-0.659503\pi$
$13$	−3.46410	−0.960769	−0.480384	+	0.877058i	$0.659503\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 3.46410i	− 0.840168i	−0.907485	−	0.420084i	$-0.862001\pi$
$17$	− 3.46410i	− 0.840168i	0.907485	−	0.420084i	$-0.137999\pi$
$18$	0	0
$19$	− 7.46410i	− 1.71238i	−0.516659	−	0.856191i	$-0.672825\pi$
$19$	− 7.46410i	− 1.71238i	0.516659	−	0.856191i	$-0.327175\pi$
$20$	0	0
$21$	− 2.00000i	− 0.436436i
$22$	0	0
$23$	4.19615i	0.874958i	0.899229	+	0.437479i	$0.144129\pi$
$23$	4.19615i	0.874958i	−0.899229	+	0.437479i	$0.855871\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	− 6.92820i	− 1.28654i	−0.765641	−	0.643268i	$-0.777578\pi$
$29$	− 6.92820i	− 1.28654i	0.765641	−	0.643268i	$-0.222422\pi$
$30$	0	0
$31$	−1.46410	−0.262960	−0.131480	−	0.991319i	$-0.541973\pi$
$31$	−1.46410	−0.262960	−0.131480	+	0.991319i	$0.541973\pi$
$32$	0	0
$33$	1.46410i	0.254867i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	2.53590	0.406069
$40$	0	0
$41$	−5.46410	−0.853349	−0.426675	−	0.904405i	$-0.640315\pi$
$41$	−5.46410	−0.853349	−0.426675	+	0.904405i	$0.640315\pi$
$42$	0	0
$43$	−8.73205	−1.33163	−0.665813	−	0.746119i	$-0.731915\pi$
$43$	−8.73205	−1.33163	−0.665813	+	0.746119i	$0.731915\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 6.73205i	− 0.981971i	−0.871168	−	0.490985i	$-0.836637\pi$
$47$	− 6.73205i	− 0.981971i	0.871168	−	0.490985i	$-0.163363\pi$
$48$	0	0
$49$	−0.464102	−0.0663002
$50$	0	0
$51$	2.53590i	0.355097i
$52$	0	0
$53$	−4.53590	−0.623054	−0.311527	−	0.950237i	$-0.600840\pi$
$53$	−4.53590	−0.623054	−0.311527	+	0.950237i	$0.600840\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.46410i	0.723738i
$58$	0	0
$59$	0.535898i	0.0697680i	0.999391	+	0.0348840i	$0.0111062\pi$
$59$	0.535898i	0.0697680i	−0.999391	+	0.0348840i	$0.988894\pi$
$60$	0	0
$61$	− 4.92820i	− 0.630992i	−0.948927	−	0.315496i	$-0.897829\pi$
$61$	− 4.92820i	− 0.630992i	0.948927	−	0.315496i	$-0.102171\pi$
$62$	0	0
$63$	− 6.73205i	− 0.848159i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−7.26795	−0.887921	−0.443961	−	0.896046i	$-0.646427\pi$
$67$	−7.26795	−0.887921	−0.443961	+	0.896046i	$0.646427\pi$
$68$	0	0
$69$	− 3.07180i	− 0.369801i
$70$	0	0
$71$	1.46410	0.173757	0.0868784	−	0.996219i	$-0.472311\pi$
$71$	1.46410	0.173757	0.0868784	+	0.996219i	$0.472311\pi$
$72$	0	0
$73$	− 0.535898i	− 0.0627222i	−0.999508	−	0.0313611i	$-0.990016\pi$
$73$	− 0.535898i	− 0.0627222i	0.999508	−	0.0313611i	$-0.00998418\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.46410	0.622692
$78$	0	0
$79$	−14.9282	−1.67955	−0.839777	−	0.542931i	$-0.817314\pi$
$79$	−14.9282	−1.67955	−0.839777	+	0.542931i	$0.817314\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	0	0
$83$	4.73205	0.519410	0.259705	−	0.965688i	$-0.416375\pi$
$83$	4.73205	0.519410	0.259705	+	0.965688i	$0.416375\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	5.07180i	0.543754i
$88$	0	0
$89$	4.92820	0.522388	0.261194	−	0.965286i	$-0.415884\pi$
$89$	4.92820	0.522388	0.261194	+	0.965286i	$0.415884\pi$
$90$	0	0
$91$	− 9.46410i	− 0.992107i
$92$	0	0
$93$	1.07180	0.111140
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.39230i	0.649040i	0.945879	+	0.324520i	$0.105203\pi$
$97$	6.39230i	0.649040i	−0.945879	+	0.324520i	$0.894797\pi$
$98$	0	0
$99$	4.92820i	0.495303i
$100$	0	0
$101$	10.9282i	1.08740i	0.839281	+	0.543698i	$0.182976\pi$
$101$	10.9282i	1.08740i	−0.839281	+	0.543698i	$0.817024\pi$
$102$	0	0
$103$	− 1.66025i	− 0.163590i	−0.996649	−	0.0817948i	$-0.973935\pi$
$103$	− 1.66025i	− 0.163590i	0.996649	−	0.0817948i	$-0.0260652\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.732051	0.0707700	0.0353850	−	0.999374i	$-0.488734\pi$
$107$	0.732051	0.0707700	0.0353850	+	0.999374i	$0.488734\pi$
$108$	0	0
$109$	3.07180i	0.294225i	0.989120	+	0.147112i	$0.0469979\pi$
$109$	3.07180i	0.294225i	−0.989120	+	0.147112i	$0.953002\pi$
$110$	0	0
$111$	1.46410	0.138966
$112$	0	0
$113$	− 0.928203i	− 0.0873180i	−0.999046	−	0.0436590i	$-0.986098\pi$
$113$	− 0.928203i	− 0.0873180i	0.999046	−	0.0436590i	$-0.0139015\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	8.53590	0.789144
$118$	0	0
$119$	9.46410	0.867573
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	4.00000	0.360668
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 13.2679i	− 1.17734i	−0.808373	−	0.588670i	$-0.799652\pi$
$127$	− 13.2679i	− 1.17734i	0.808373	−	0.588670i	$-0.200348\pi$
$128$	0	0
$129$	6.39230	0.562811
$130$	0	0
$131$	7.85641i	0.686417i	0.939259	+	0.343209i	$0.111514\pi$
$131$	7.85641i	0.686417i	−0.939259	+	0.343209i	$0.888486\pi$
$132$	0	0
$133$	20.3923	1.76824
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 8.92820i	− 0.762788i	−0.924413	−	0.381394i	$-0.875444\pi$
$137$	− 8.92820i	− 0.762788i	0.924413	−	0.381394i	$-0.124556\pi$
$138$	0	0
$139$	7.46410i	0.633097i	0.948576	+	0.316548i	$0.102524\pi$
$139$	7.46410i	0.633097i	−0.948576	+	0.316548i	$0.897476\pi$
$140$	0	0
$141$	4.92820i	0.415030i
$142$	0	0
$143$	6.92820i	0.579365i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0.339746	0.0280218
$148$	0	0
$149$	19.8564i	1.62670i	0.581775	+	0.813350i	$0.302359\pi$
$149$	19.8564i	1.62670i	−0.581775	+	0.813350i	$0.697641\pi$
$150$	0	0
$151$	−8.39230	−0.682956	−0.341478	−	0.939890i	$-0.610927\pi$
$151$	−8.39230	−0.682956	−0.341478	+	0.939890i	$0.610927\pi$
$152$	0	0
$153$	8.53590i	0.690086i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−16.9282	−1.35102	−0.675509	−	0.737352i	$-0.736076\pi$
$157$	−16.9282	−1.35102	−0.675509	+	0.737352i	$0.736076\pi$
$158$	0	0
$159$	3.32051	0.263333
$160$	0	0
$161$	−11.4641	−0.903498
$162$	0	0
$163$	10.1962	0.798624	0.399312	−	0.916815i	$-0.369249\pi$
$163$	10.1962	0.798624	0.399312	+	0.916815i	$0.369249\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	20.1962i	1.56283i	0.624015	+	0.781413i	$0.285501\pi$
$167$	20.1962i	1.56283i	−0.624015	+	0.781413i	$0.714499\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	18.3923i	1.40649i
$172$	0	0
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 0.392305i	− 0.0294874i
$178$	0	0
$179$	15.4641i	1.15584i	0.816093	+	0.577921i	$0.196136\pi$
$179$	15.4641i	1.15584i	−0.816093	+	0.577921i	$0.803864\pi$
$180$	0	0
$181$	− 16.0000i	− 1.18927i	−0.803996	−	0.594635i	$-0.797296\pi$
$181$	− 16.0000i	− 1.18927i	0.803996	−	0.594635i	$-0.202704\pi$
$182$	0	0
$183$	3.60770i	0.266688i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−6.92820	−0.506640
$188$	0	0
$189$	10.9282i	0.794910i
$190$	0	0
$191$	19.3205	1.39798	0.698991	−	0.715130i	$-0.253633\pi$
$191$	19.3205	1.39798	0.698991	+	0.715130i	$0.253633\pi$
$192$	0	0
$193$	− 7.46410i	− 0.537278i	−0.963241	−	0.268639i	$-0.913426\pi$
$193$	− 7.46410i	− 0.537278i	0.963241	−	0.268639i	$-0.0865738\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−12.5359	−0.893146	−0.446573	−	0.894747i	$-0.647356\pi$
$197$	−12.5359	−0.893146	−0.446573	+	0.894747i	$0.647356\pi$
$198$	0	0
$199$	25.8564	1.83291	0.916456	−	0.400135i	$-0.131037\pi$
$199$	25.8564	1.83291	0.916456	+	0.400135i	$0.131037\pi$
$200$	0	0
$201$	5.32051	0.375280
$202$	0	0
$203$	18.9282	1.32850
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 10.3397i	− 0.718662i
$208$	0	0
$209$	−14.9282	−1.03261
$210$	0	0
$211$	− 14.7846i	− 1.01781i	−0.860821	−	0.508907i	$-0.830050\pi$
$211$	− 14.7846i	− 1.01781i	0.860821	−	0.508907i	$-0.169950\pi$
$212$	0	0
$213$	−1.07180	−0.0734383
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 4.00000i	− 0.271538i
$218$	0	0
$219$	0.392305i	0.0265095i
$220$	0	0
$221$	12.0000i	0.807207i
$222$	0	0
$223$	− 16.1962i	− 1.08457i	−0.840193	−	0.542287i	$-0.817558\pi$
$223$	− 16.1962i	− 1.08457i	0.840193	−	0.542287i	$-0.182442\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	28.0526	1.86191	0.930957	−	0.365129i	$-0.118975\pi$
$227$	28.0526	1.86191	0.930957	+	0.365129i	$0.118975\pi$
$228$	0	0
$229$	− 4.00000i	− 0.264327i	−0.991228	−	0.132164i	$-0.957808\pi$
$229$	− 4.00000i	− 0.264327i	0.991228	−	0.132164i	$-0.0421925\pi$
$230$	0	0
$231$	−4.00000	−0.263181
$232$	0	0
$233$	− 29.3205i	− 1.92085i	−0.278538	−	0.960425i	$-0.589850\pi$
$233$	− 29.3205i	− 1.92085i	0.278538	−	0.960425i	$-0.410150\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	10.9282	0.709863
$238$	0	0
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	0	0
$241$	−4.39230	−0.282933	−0.141467	−	0.989943i	$-0.545182\pi$
$241$	−4.39230	−0.282933	−0.141467	+	0.989943i	$0.545182\pi$
$242$	0	0
$243$	−15.2679	−0.979439
$244$	0	0
$245$	0	0
$246$	0	0
$247$	25.8564i	1.64520i
$248$	0	0
$249$	−3.46410	−0.219529
$250$	0	0
$251$	11.0718i	0.698846i	0.936965	+	0.349423i	$0.113622\pi$
$251$	11.0718i	0.698846i	−0.936965	+	0.349423i	$0.886378\pi$
$252$	0	0
$253$	8.39230	0.527620
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 2.00000i	− 0.124757i	−0.998053	−	0.0623783i	$-0.980131\pi$
$257$	− 2.00000i	− 0.124757i	0.998053	−	0.0623783i	$-0.0198685\pi$
$258$	0	0
$259$	− 5.46410i	− 0.339523i
$260$	0	0
$261$	17.0718i	1.05672i
$262$	0	0
$263$	5.66025i	0.349026i	0.984655	+	0.174513i	$0.0558351\pi$
$263$	5.66025i	0.349026i	−0.984655	+	0.174513i	$0.944165\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−3.60770	−0.220787
$268$	0	0
$269$	4.92820i	0.300478i	0.988650	+	0.150239i	$0.0480043\pi$
$269$	4.92820i	0.300478i	−0.988650	+	0.150239i	$0.951996\pi$
$270$	0	0
$271$	−15.3205	−0.930655	−0.465327	−	0.885139i	$-0.654063\pi$
$271$	−15.3205	−0.930655	−0.465327	+	0.885139i	$0.654063\pi$
$272$	0	0
$273$	6.92820i	0.419314i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	0	0
$279$	3.60770	0.215987
$280$	0	0
$281$	17.4641	1.04182	0.520910	−	0.853611i	$-0.325593\pi$
$281$	17.4641	1.04182	0.520910	+	0.853611i	$0.325593\pi$
$282$	0	0
$283$	7.66025	0.455355	0.227677	−	0.973737i	$-0.426887\pi$
$283$	7.66025	0.455355	0.227677	+	0.973737i	$0.426887\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 14.9282i	− 0.881184i
$288$	0	0
$289$	5.00000	0.294118
$290$	0	0
$291$	− 4.67949i	− 0.274317i
$292$	0	0
$293$	−11.8564	−0.692659	−0.346329	−	0.938113i	$-0.612572\pi$
$293$	−11.8564	−0.692659	−0.346329	+	0.938113i	$0.612572\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 8.00000i	− 0.464207i
$298$	0	0
$299$	− 14.5359i	− 0.840633i
$300$	0	0
$301$	− 23.8564i	− 1.37506i
$302$	0	0
$303$	− 8.00000i	− 0.459588i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−26.9808	−1.53987	−0.769937	−	0.638120i	$-0.779712\pi$
$307$	−26.9808	−1.53987	−0.769937	+	0.638120i	$0.779712\pi$
$308$	0	0
$309$	1.21539i	0.0691411i
$310$	0	0
$311$	3.32051	0.188289	0.0941444	−	0.995559i	$-0.469988\pi$
$311$	3.32051	0.188289	0.0941444	+	0.995559i	$0.469988\pi$
$312$	0	0
$313$	31.8564i	1.80063i	0.435238	+	0.900315i	$0.356664\pi$
$313$	31.8564i	1.80063i	−0.435238	+	0.900315i	$0.643336\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	15.4641	0.868550	0.434275	−	0.900780i	$-0.357005\pi$
$317$	15.4641	0.868550	0.434275	+	0.900780i	$0.357005\pi$
$318$	0	0
$319$	−13.8564	−0.775810
$320$	0	0
$321$	−0.535898	−0.0299109
$322$	0	0
$323$	−25.8564	−1.43869
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 2.24871i	− 0.124354i
$328$	0	0
$329$	18.3923	1.01400
$330$	0	0
$331$	− 14.0000i	− 0.769510i	−0.923019	−	0.384755i	$-0.874286\pi$
$331$	− 14.0000i	− 0.769510i	0.923019	−	0.384755i	$-0.125714\pi$
$332$	0	0
$333$	4.92820	0.270064
$334$	0	0
$335$	0	0
$336$	0	0
$337$	7.85641i	0.427966i	0.976837	+	0.213983i	$0.0686437\pi$
$337$	7.85641i	0.427966i	−0.976837	+	0.213983i	$0.931356\pi$
$338$	0	0
$339$	0.679492i	0.0369049i
$340$	0	0
$341$	2.92820i	0.158571i
$342$	0	0
$343$	17.8564i	0.964155i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−15.6603	−0.840686	−0.420343	−	0.907365i	$-0.638090\pi$
$347$	−15.6603	−0.840686	−0.420343	+	0.907365i	$0.638090\pi$
$348$	0	0
$349$	− 28.0000i	− 1.49881i	−0.662114	−	0.749403i	$-0.730341\pi$
$349$	− 28.0000i	− 1.49881i	0.662114	−	0.749403i	$-0.269659\pi$
$350$	0	0
$351$	−13.8564	−0.739600
$352$	0	0
$353$	0.928203i	0.0494033i	0.999695	+	0.0247016i	$0.00786357\pi$
$353$	0.928203i	0.0494033i	−0.999695	+	0.0247016i	$0.992136\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−6.92820	−0.366679
$358$	0	0
$359$	5.07180	0.267679	0.133840	−	0.991003i	$-0.457269\pi$
$359$	5.07180	0.267679	0.133840	+	0.991003i	$0.457269\pi$
$360$	0	0
$361$	−36.7128	−1.93225
$362$	0	0
$363$	−5.12436	−0.268959
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 27.1244i	− 1.41588i	−0.706273	−	0.707940i	$-0.749625\pi$
$367$	− 27.1244i	− 1.41588i	0.706273	−	0.707940i	$-0.250375\pi$
$368$	0	0
$369$	13.4641	0.700913
$370$	0	0
$371$	− 12.3923i	− 0.643376i
$372$	0	0
$373$	−29.7128	−1.53847	−0.769236	−	0.638965i	$-0.779363\pi$
$373$	−29.7128	−1.53847	−0.769236	+	0.638965i	$0.779363\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	24.0000i	1.23606i
$378$	0	0
$379$	− 12.2487i	− 0.629174i	−0.949229	−	0.314587i	$-0.898134\pi$
$379$	− 12.2487i	− 0.629174i	0.949229	−	0.314587i	$-0.101866\pi$
$380$	0	0
$381$	9.71281i	0.497602i
$382$	0	0
$383$	− 3.12436i	− 0.159647i	−0.996809	−	0.0798236i	$-0.974564\pi$
$383$	− 3.12436i	− 0.159647i	0.996809	−	0.0798236i	$-0.0254357\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	21.5167	1.09375
$388$	0	0
$389$	− 34.7846i	− 1.76365i	−0.471577	−	0.881825i	$-0.656315\pi$
$389$	− 34.7846i	− 1.76365i	0.471577	−	0.881825i	$-0.343685\pi$
$390$	0	0
$391$	14.5359	0.735112
$392$	0	0
$393$	− 5.75129i	− 0.290114i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−16.2487	−0.815499	−0.407750	−	0.913094i	$-0.633686\pi$
$397$	−16.2487	−0.815499	−0.407750	+	0.913094i	$0.633686\pi$
$398$	0	0
$399$	−14.9282	−0.747345
$400$	0	0
$401$	19.8564	0.991582	0.495791	−	0.868442i	$-0.334878\pi$
$401$	19.8564	0.991582	0.495791	+	0.868442i	$0.334878\pi$
$402$	0	0
$403$	5.07180	0.252644
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.00000i	0.198273i
$408$	0	0
$409$	−23.3205	−1.15312	−0.576562	−	0.817053i	$-0.695606\pi$
$409$	−23.3205	−1.15312	−0.576562	+	0.817053i	$0.695606\pi$
$410$	0	0
$411$	6.53590i	0.322392i
$412$	0	0
$413$	−1.46410	−0.0720437
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 5.46410i	− 0.267578i
$418$	0	0
$419$	2.39230i	0.116872i	0.998291	+	0.0584359i	$0.0186113\pi$
$419$	2.39230i	0.116872i	−0.998291	+	0.0584359i	$0.981389\pi$
$420$	0	0
$421$	− 27.8564i	− 1.35764i	−0.734306	−	0.678819i	$-0.762492\pi$
$421$	− 27.8564i	− 1.35764i	0.734306	−	0.678819i	$-0.237508\pi$
$422$	0	0
$423$	16.5885i	0.806558i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	13.4641	0.651574
$428$	0	0
$429$	− 5.07180i	− 0.244869i
$430$	0	0
$431$	14.5359	0.700170	0.350085	−	0.936718i	$-0.386153\pi$
$431$	14.5359	0.700170	0.350085	+	0.936718i	$0.386153\pi$
$432$	0	0
$433$	12.5359i	0.602437i	0.953555	+	0.301218i	$0.0973933\pi$
$433$	12.5359i	0.602437i	−0.953555	+	0.301218i	$0.902607\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	31.3205	1.49826
$438$	0	0
$439$	−0.784610	−0.0374474	−0.0187237	−	0.999825i	$-0.505960\pi$
$439$	−0.784610	−0.0374474	−0.0187237	+	0.999825i	$0.505960\pi$
$440$	0	0
$441$	1.14359	0.0544568
$442$	0	0
$443$	−30.9808	−1.47194	−0.735970	−	0.677014i	$-0.763274\pi$
$443$	−30.9808	−1.47194	−0.735970	+	0.677014i	$0.763274\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 14.5359i	− 0.687524i
$448$	0	0
$449$	−11.3205	−0.534248	−0.267124	−	0.963662i	$-0.586073\pi$
$449$	−11.3205	−0.534248	−0.267124	+	0.963662i	$0.586073\pi$
$450$	0	0
$451$	10.9282i	0.514589i
$452$	0	0
$453$	6.14359	0.288651
$454$	0	0
$455$	0	0
$456$	0	0
$457$	14.7846i	0.691595i	0.938309	+	0.345797i	$0.112392\pi$
$457$	14.7846i	0.691595i	−0.938309	+	0.345797i	$0.887608\pi$
$458$	0	0
$459$	− 13.8564i	− 0.646762i
$460$	0	0
$461$	− 2.92820i	− 0.136380i	−0.997672	−	0.0681900i	$-0.978278\pi$
$461$	− 2.92820i	− 0.136380i	0.997672	−	0.0681900i	$-0.0217224\pi$
$462$	0	0
$463$	− 14.7321i	− 0.684656i	−0.939580	−	0.342328i	$-0.888785\pi$
$463$	− 14.7321i	− 0.684656i	0.939580	−	0.342328i	$-0.111215\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.33975	0.385917	0.192959	−	0.981207i	$-0.438192\pi$
$467$	8.33975	0.385917	0.192959	+	0.981207i	$0.438192\pi$
$468$	0	0
$469$	− 19.8564i	− 0.916884i
$470$	0	0
$471$	12.3923	0.571007
$472$	0	0
$473$	17.4641i	0.803000i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	11.1769	0.511756
$478$	0	0
$479$	−21.8564	−0.998645	−0.499322	−	0.866416i	$-0.666418\pi$
$479$	−21.8564	−0.998645	−0.499322	+	0.866416i	$0.666418\pi$
$480$	0	0
$481$	6.92820	0.315899
$482$	0	0
$483$	8.39230	0.381863
$484$	0	0
$485$	0	0
$486$	0	0
$487$	24.5885i	1.11421i	0.830442	+	0.557105i	$0.188088\pi$
$487$	24.5885i	1.11421i	−0.830442	+	0.557105i	$0.811912\pi$
$488$	0	0
$489$	−7.46410	−0.337538
$490$	0	0
$491$	− 3.07180i	− 0.138628i	−0.997595	−	0.0693141i	$-0.977919\pi$
$491$	− 3.07180i	− 0.138628i	0.997595	−	0.0693141i	$-0.0220811\pi$
$492$	0	0
$493$	−24.0000	−1.08091
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.00000i	0.179425i
$498$	0	0
$499$	− 24.5359i	− 1.09838i	−0.835698	−	0.549189i	$-0.814937\pi$
$499$	− 24.5359i	− 1.09838i	0.835698	−	0.549189i	$-0.185063\pi$
$500$	0	0
$501$	− 14.7846i	− 0.660528i
$502$	0	0
$503$	− 17.6603i	− 0.787432i	−0.919232	−	0.393716i	$-0.871189\pi$
$503$	− 17.6603i	− 0.787432i	0.919232	−	0.393716i	$-0.128811\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0.732051	0.0325115
$508$	0	0
$509$	25.8564i	1.14607i	0.819533	+	0.573033i	$0.194233\pi$
$509$	25.8564i	1.14607i	−0.819533	+	0.573033i	$0.805767\pi$
$510$	0	0
$511$	1.46410	0.0647680
$512$	0	0
$513$	− 29.8564i	− 1.31819i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−13.4641	−0.592151
$518$	0	0
$519$	1.46410	0.0642669
$520$	0	0
$521$	−16.1436	−0.707264	−0.353632	−	0.935385i	$-0.615053\pi$
$521$	−16.1436	−0.707264	−0.353632	+	0.935385i	$0.615053\pi$
$522$	0	0
$523$	−22.1962	−0.970570	−0.485285	−	0.874356i	$-0.661284\pi$
$523$	−22.1962	−0.970570	−0.485285	+	0.874356i	$0.661284\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.07180i	0.220931i
$528$	0	0
$529$	5.39230	0.234448
$530$	0	0
$531$	− 1.32051i	− 0.0573052i
$532$	0	0
$533$	18.9282	0.819871
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 11.3205i	− 0.488516i
$538$	0	0
$539$	0.928203i	0.0399805i
$540$	0	0
$541$	13.0718i	0.562000i	0.959708	+	0.281000i	$0.0906662\pi$
$541$	13.0718i	0.562000i	−0.959708	+	0.281000i	$0.909334\pi$
$542$	0	0
$543$	11.7128i	0.502645i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−36.7321	−1.57055	−0.785275	−	0.619148i	$-0.787478\pi$
$547$	−36.7321	−1.57055	−0.785275	+	0.619148i	$0.787478\pi$
$548$	0	0
$549$	12.1436i	0.518276i
$550$	0	0
$551$	−51.7128	−2.20304
$552$	0	0
$553$	− 40.7846i	− 1.73434i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−26.7846	−1.13490	−0.567450	−	0.823408i	$-0.692070\pi$
$557$	−26.7846	−1.13490	−0.567450	+	0.823408i	$0.692070\pi$
$558$	0	0
$559$	30.2487	1.27938
$560$	0	0
$561$	5.07180	0.214131
$562$	0	0
$563$	16.0526	0.676535	0.338267	−	0.941050i	$-0.390159\pi$
$563$	16.0526	0.676535	0.338267	+	0.941050i	$0.390159\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	12.1962i	0.512190i
$568$	0	0
$569$	6.53590	0.273999	0.137000	−	0.990571i	$-0.456254\pi$
$569$	6.53590	0.273999	0.137000	+	0.990571i	$0.456254\pi$
$570$	0	0
$571$	− 34.7846i	− 1.45569i	−0.685741	−	0.727845i	$-0.740522\pi$
$571$	− 34.7846i	− 1.45569i	0.685741	−	0.727845i	$-0.259478\pi$
$572$	0	0
$573$	−14.1436	−0.590857
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 43.5692i	− 1.81381i	−0.421335	−	0.906905i	$-0.638438\pi$
$577$	− 43.5692i	− 1.81381i	0.421335	−	0.906905i	$-0.361562\pi$
$578$	0	0
$579$	5.46410i	0.227080i
$580$	0	0
$581$	12.9282i	0.536352i
$582$	0	0
$583$	9.07180i	0.375715i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14.1962	0.585938	0.292969	−	0.956122i	$-0.405357\pi$
$587$	14.1962	0.585938	0.292969	+	0.956122i	$0.405357\pi$
$588$	0	0
$589$	10.9282i	0.450289i
$590$	0	0
$591$	9.17691	0.377488
$592$	0	0
$593$	36.6410i	1.50467i	0.658783	+	0.752333i	$0.271072\pi$
$593$	36.6410i	1.50467i	−0.658783	+	0.752333i	$0.728928\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−18.9282	−0.774680
$598$	0	0
$599$	34.6410	1.41539	0.707697	−	0.706516i	$-0.249734\pi$
$599$	34.6410	1.41539	0.707697	+	0.706516i	$0.249734\pi$
$600$	0	0
$601$	25.4641	1.03870	0.519351	−	0.854561i	$-0.326174\pi$
$601$	25.4641	1.03870	0.519351	+	0.854561i	$0.326174\pi$
$602$	0	0
$603$	17.9090	0.729309
$604$	0	0
$605$	0	0
$606$	0	0
$607$	20.9808i	0.851583i	0.904821	+	0.425791i	$0.140004\pi$
$607$	20.9808i	0.851583i	−0.904821	+	0.425791i	$0.859996\pi$
$608$	0	0
$609$	−13.8564	−0.561490
$610$	0	0
$611$	23.3205i	0.943447i
$612$	0	0
$613$	−5.60770	−0.226493	−0.113246	−	0.993567i	$-0.536125\pi$
$613$	−5.60770	−0.226493	−0.113246	+	0.993567i	$0.536125\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 27.4641i	− 1.10566i	−0.833293	−	0.552832i	$-0.813547\pi$
$617$	− 27.4641i	− 1.10566i	0.833293	−	0.552832i	$-0.186453\pi$
$618$	0	0
$619$	33.3205i	1.33926i	0.742693	+	0.669632i	$0.233548\pi$
$619$	33.3205i	1.33926i	−0.742693	+	0.669632i	$0.766452\pi$
$620$	0	0
$621$	16.7846i	0.673543i
$622$	0	0
$623$	13.4641i	0.539428i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	10.9282	0.436430
$628$	0	0
$629$	6.92820i	0.276246i
$630$	0	0
$631$	−11.3205	−0.450662	−0.225331	−	0.974282i	$-0.572346\pi$
$631$	−11.3205	−0.450662	−0.225331	+	0.974282i	$0.572346\pi$
$632$	0	0
$633$	10.8231i	0.430179i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.60770	0.0636992
$638$	0	0
$639$	−3.60770	−0.142718
$640$	0	0
$641$	−20.3923	−0.805448	−0.402724	−	0.915322i	$-0.631936\pi$
$641$	−20.3923	−0.805448	−0.402724	+	0.915322i	$0.631936\pi$
$642$	0	0
$643$	14.8756	0.586638	0.293319	−	0.956015i	$-0.405240\pi$
$643$	14.8756	0.586638	0.293319	+	0.956015i	$0.405240\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 13.2679i	− 0.521617i	−0.965391	−	0.260808i	$-0.916011\pi$
$647$	− 13.2679i	− 0.521617i	0.965391	−	0.260808i	$-0.0839891\pi$
$648$	0	0
$649$	1.07180	0.0420717
$650$	0	0
$651$	2.92820i	0.114765i
$652$	0	0
$653$	36.2487	1.41852	0.709261	−	0.704946i	$-0.249029\pi$
$653$	36.2487	1.41852	0.709261	+	0.704946i	$0.249029\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	1.32051i	0.0515179i
$658$	0	0
$659$	− 17.3205i	− 0.674711i	−0.941377	−	0.337356i	$-0.890468\pi$
$659$	− 17.3205i	− 0.674711i	0.941377	−	0.337356i	$-0.109532\pi$
$660$	0	0
$661$	35.8564i	1.39465i	0.716754	+	0.697326i	$0.245627\pi$
$661$	35.8564i	1.39465i	−0.716754	+	0.697326i	$0.754373\pi$
$662$	0	0
$663$	− 8.78461i	− 0.341166i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	29.0718	1.12566
$668$	0	0
$669$	11.8564i	0.458395i
$670$	0	0
$671$	−9.85641	−0.380502
$672$	0	0
$673$	− 19.4641i	− 0.750286i	−0.926967	−	0.375143i	$-0.877594\pi$
$673$	− 19.4641i	− 0.750286i	0.926967	−	0.375143i	$-0.122406\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	38.3923	1.47554	0.737768	−	0.675054i	$-0.235880\pi$
$677$	38.3923	1.47554	0.737768	+	0.675054i	$0.235880\pi$
$678$	0	0
$679$	−17.4641	−0.670211
$680$	0	0
$681$	−20.5359	−0.786937
$682$	0	0
$683$	34.9808	1.33850	0.669251	−	0.743037i	$-0.266615\pi$
$683$	34.9808	1.33850	0.669251	+	0.743037i	$0.266615\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.92820i	0.111718i
$688$	0	0
$689$	15.7128	0.598610
$690$	0	0
$691$	18.0000i	0.684752i	0.939563	+	0.342376i	$0.111232\pi$
$691$	18.0000i	0.684752i	−0.939563	+	0.342376i	$0.888768\pi$
$692$	0	0
$693$	−13.4641	−0.511459
$694$	0	0
$695$	0	0
$696$	0	0
$697$	18.9282i	0.716957i
$698$	0	0
$699$	21.4641i	0.811847i
$700$	0	0
$701$	− 32.9282i	− 1.24368i	−0.783144	−	0.621841i	$-0.786385\pi$
$701$	− 32.9282i	− 1.24368i	0.783144	−	0.621841i	$-0.213615\pi$
$702$	0	0
$703$	14.9282i	0.563028i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−29.8564	−1.12287
$708$	0	0
$709$	28.7846i	1.08103i	0.841335	+	0.540514i	$0.181770\pi$
$709$	28.7846i	1.08103i	−0.841335	+	0.540514i	$0.818230\pi$
$710$	0	0
$711$	36.7846	1.37953
$712$	0	0
$713$	− 6.14359i	− 0.230079i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	14.6410	0.546779
$718$	0	0
$719$	25.8564	0.964281	0.482141	−	0.876094i	$-0.339859\pi$
$719$	25.8564	0.964281	0.482141	+	0.876094i	$0.339859\pi$
$720$	0	0
$721$	4.53590	0.168926
$722$	0	0
$723$	3.21539	0.119582
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 14.0526i	− 0.521181i	−0.965449	−	0.260590i	$-0.916083\pi$
$727$	− 14.0526i	− 0.521181i	0.965449	−	0.260590i	$-0.0839172\pi$
$728$	0	0
$729$	−2.21539	−0.0820515
$730$	0	0
$731$	30.2487i	1.11879i
$732$	0	0
$733$	48.9282	1.80720	0.903602	−	0.428373i	$-0.140913\pi$
$733$	48.9282	1.80720	0.903602	+	0.428373i	$0.140913\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	14.5359i	0.535437i
$738$	0	0
$739$	5.32051i	0.195718i	0.995200	+	0.0978590i	$0.0311994\pi$
$739$	5.32051i	0.195718i	−0.995200	+	0.0978590i	$0.968801\pi$
$740$	0	0
$741$	− 18.9282i	− 0.695345i
$742$	0	0
$743$	− 40.9808i	− 1.50344i	−0.659483	−	0.751719i	$-0.729225\pi$
$743$	− 40.9808i	− 1.50344i	0.659483	−	0.751719i	$-0.270775\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−11.6603	−0.426626
$748$	0	0
$749$	2.00000i	0.0730784i
$750$	0	0
$751$	22.2487	0.811867	0.405934	−	0.913903i	$-0.366946\pi$
$751$	22.2487	0.811867	0.405934	+	0.913903i	$0.366946\pi$
$752$	0	0
$753$	− 8.10512i	− 0.295367i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−32.9282	−1.19680	−0.598398	−	0.801199i	$-0.704196\pi$
$757$	−32.9282	−1.19680	−0.598398	+	0.801199i	$0.704196\pi$
$758$	0	0
$759$	−6.14359	−0.222998
$760$	0	0
$761$	49.7128	1.80209	0.901044	−	0.433728i	$-0.142802\pi$
$761$	49.7128	1.80209	0.901044	+	0.433728i	$0.142802\pi$
$762$	0	0
$763$	−8.39230	−0.303822
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1.85641i	− 0.0670310i
$768$	0	0
$769$	0.928203	0.0334719	0.0167359	−	0.999860i	$-0.494673\pi$
$769$	0.928203	0.0334719	0.0167359	+	0.999860i	$0.494673\pi$
$770$	0	0
$771$	1.46410i	0.0527283i
$772$	0	0
$773$	1.60770	0.0578248	0.0289124	−	0.999582i	$-0.490796\pi$
$773$	1.60770	0.0578248	0.0289124	+	0.999582i	$0.490796\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	4.00000i	0.143499i
$778$	0	0
$779$	40.7846i	1.46126i
$780$	0	0
$781$	− 2.92820i	− 0.104779i
$782$	0	0
$783$	− 27.7128i	− 0.990375i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	14.5885	0.520022	0.260011	−	0.965606i	$-0.416274\pi$
$787$	14.5885	0.520022	0.260011	+	0.965606i	$0.416274\pi$
$788$	0	0
$789$	− 4.14359i	− 0.147516i
$790$	0	0
$791$	2.53590	0.0901662
$792$	0	0
$793$	17.0718i	0.606237i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−26.1051	−0.924691	−0.462345	−	0.886700i	$-0.652992\pi$
$797$	−26.1051	−0.924691	−0.462345	+	0.886700i	$0.652992\pi$
$798$	0	0
$799$	−23.3205	−0.825020
$800$	0	0
$801$	−12.1436	−0.429073
$802$	0	0
$803$	−1.07180	−0.0378229
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 3.60770i	− 0.126997i
$808$	0	0
$809$	3.85641	0.135584	0.0677920	−	0.997699i	$-0.478405\pi$
$809$	3.85641	0.135584	0.0677920	+	0.997699i	$0.478405\pi$
$810$	0	0
$811$	− 15.0718i	− 0.529242i	−0.964352	−	0.264621i	$-0.914753\pi$
$811$	− 15.0718i	− 0.529242i	0.964352	−	0.264621i	$-0.0852469\pi$
$812$	0	0
$813$	11.2154	0.393341
$814$	0	0
$815$	0	0
$816$	0	0
$817$	65.1769i	2.28025i
$818$	0	0
$819$	23.3205i	0.814885i
$820$	0	0
$821$	6.78461i	0.236785i	0.992967	+	0.118392i	$0.0377740\pi$
$821$	6.78461i	0.236785i	−0.992967	+	0.118392i	$0.962226\pi$
$822$	0	0
$823$	15.1244i	0.527202i	0.964632	+	0.263601i	$0.0849102\pi$
$823$	15.1244i	0.527202i	−0.964632	+	0.263601i	$0.915090\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.12436	0.0390977	0.0195488	−	0.999809i	$-0.493777\pi$
$827$	1.12436	0.0390977	0.0195488	+	0.999809i	$0.493777\pi$
$828$	0	0
$829$	15.0718i	0.523465i	0.965140	+	0.261733i	$0.0842938\pi$
$829$	15.0718i	0.523465i	−0.965140	+	0.261733i	$0.915706\pi$
$830$	0	0
$831$	−1.46410	−0.0507891
$832$	0	0
$833$	1.60770i	0.0557033i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−5.85641	−0.202427
$838$	0	0
$839$	−16.7846	−0.579469	−0.289735	−	0.957107i	$-0.593567\pi$
$839$	−16.7846	−0.579469	−0.289735	+	0.957107i	$0.593567\pi$
$840$	0	0
$841$	−19.0000	−0.655172
$842$	0	0
$843$	−12.7846	−0.440325
$844$	0	0
$845$	0	0
$846$	0	0
$847$	19.1244i	0.657121i
$848$	0	0
$849$	−5.60770	−0.192456
$850$	0	0
$851$	− 8.39230i	− 0.287685i
$852$	0	0
$853$	42.3923	1.45148	0.725742	−	0.687967i	$-0.241496\pi$
$853$	42.3923	1.45148	0.725742	+	0.687967i	$0.241496\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	7.85641i	0.268370i	0.990956	+	0.134185i	$0.0428416\pi$
$857$	7.85641i	0.268370i	−0.990956	+	0.134185i	$0.957158\pi$
$858$	0	0
$859$	20.2487i	0.690877i	0.938441	+	0.345439i	$0.112270\pi$
$859$	20.2487i	0.690877i	−0.938441	+	0.345439i	$0.887730\pi$
$860$	0	0
$861$	10.9282i	0.372432i
$862$	0	0
$863$	30.3397i	1.03278i	0.856354	+	0.516388i	$0.172724\pi$
$863$	30.3397i	1.03278i	−0.856354	+	0.516388i	$0.827276\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−3.66025	−0.124309
$868$	0	0
$869$	29.8564i	1.01281i
$870$	0	0
$871$	25.1769	0.853087
$872$	0	0
$873$	− 15.7513i	− 0.533100i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	53.7128	1.81375	0.906876	−	0.421397i	$-0.138460\pi$
$877$	53.7128	1.81375	0.906876	+	0.421397i	$0.138460\pi$
$878$	0	0
$879$	8.67949	0.292752
$880$	0	0
$881$	2.53590	0.0854366	0.0427183	−	0.999087i	$-0.486398\pi$
$881$	2.53590	0.0854366	0.0427183	+	0.999087i	$0.486398\pi$
$882$	0	0
$883$	37.9090	1.27574	0.637869	−	0.770145i	$-0.279816\pi$
$883$	37.9090	1.27574	0.637869	+	0.770145i	$0.279816\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 51.9090i	− 1.74293i	−0.490455	−	0.871466i	$-0.663170\pi$
$887$	− 51.9090i	− 1.74293i	0.490455	−	0.871466i	$-0.336830\pi$
$888$	0	0
$889$	36.2487	1.21574
$890$	0	0
$891$	− 8.92820i	− 0.299106i
$892$	0	0
$893$	−50.2487	−1.68151
$894$	0	0
$895$	0	0
$896$	0	0
$897$	10.6410i	0.355293i
$898$	0	0
$899$	10.1436i	0.338308i
$900$	0	0
$901$	15.7128i	0.523470i
$902$	0	0
$903$	17.4641i	0.581169i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−29.1244	−0.967058	−0.483529	−	0.875328i	$-0.660645\pi$
$907$	−29.1244	−0.967058	−0.483529	+	0.875328i	$0.660645\pi$
$908$	0	0
$909$	− 26.9282i	− 0.893152i
$910$	0	0
$911$	−13.1769	−0.436571	−0.218285	−	0.975885i	$-0.570046\pi$
$911$	−13.1769	−0.436571	−0.218285	+	0.975885i	$0.570046\pi$
$912$	0	0
$913$	− 9.46410i	− 0.313216i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−21.4641	−0.708807
$918$	0	0
$919$	−25.0718	−0.827042	−0.413521	−	0.910495i	$-0.635701\pi$
$919$	−25.0718	−0.827042	−0.413521	+	0.910495i	$0.635701\pi$
$920$	0	0
$921$	19.7513	0.650827
$922$	0	0
$923$	−5.07180	−0.166940
$924$	0	0
$925$	0	0
$926$	0	0
$927$	4.09103i	0.134367i
$928$	0	0
$929$	10.5359	0.345672	0.172836	−	0.984951i	$-0.444707\pi$
$929$	10.5359	0.345672	0.172836	+	0.984951i	$0.444707\pi$
$930$	0	0
$931$	3.46410i	0.113531i
$932$	0	0
$933$	−2.43078	−0.0795802
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 44.2487i	− 1.44554i	−0.691087	−	0.722771i	$-0.742868\pi$
$937$	− 44.2487i	− 1.44554i	0.691087	−	0.722771i	$-0.257132\pi$
$938$	0	0
$939$	− 23.3205i	− 0.761036i
$940$	0	0
$941$	32.0000i	1.04317i	0.853199	+	0.521585i	$0.174659\pi$
$941$	32.0000i	1.04317i	−0.853199	+	0.521585i	$0.825341\pi$
$942$	0	0
$943$	− 22.9282i	− 0.746645i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−21.1244	−0.686449	−0.343225	−	0.939253i	$-0.611519\pi$
$947$	−21.1244	−0.686449	−0.343225	+	0.939253i	$0.611519\pi$
$948$	0	0
$949$	1.85641i	0.0602615i
$950$	0	0
$951$	−11.3205	−0.367093
$952$	0	0
$953$	58.7846i	1.90422i	0.305755	+	0.952110i	$0.401091\pi$
$953$	58.7846i	1.90422i	−0.305755	+	0.952110i	$0.598909\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	10.1436	0.327896
$958$	0	0
$959$	24.3923	0.787669
$960$	0	0
$961$	−28.8564	−0.930852
$962$	0	0
$963$	−1.80385	−0.0581282
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 33.6603i	− 1.08244i	−0.840881	−	0.541220i	$-0.817962\pi$
$967$	− 33.6603i	− 1.08244i	0.840881	−	0.541220i	$-0.182038\pi$
$968$	0	0
$969$	18.9282	0.608061
$970$	0	0
$971$	23.0718i	0.740409i	0.928950	+	0.370205i	$0.120712\pi$
$971$	23.0718i	0.740409i	−0.928950	+	0.370205i	$0.879288\pi$
$972$	0	0
$973$	−20.3923	−0.653747
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 31.4641i	− 1.00663i	−0.864104	−	0.503313i	$-0.832114\pi$
$977$	− 31.4641i	− 1.00663i	0.864104	−	0.503313i	$-0.167886\pi$
$978$	0	0
$979$	− 9.85641i	− 0.315012i
$980$	0	0
$981$	− 7.56922i	− 0.241667i
$982$	0	0
$983$	− 45.2679i	− 1.44382i	−0.691985	−	0.721912i	$-0.743264\pi$
$983$	− 45.2679i	− 1.44382i	0.691985	−	0.721912i	$-0.256736\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−13.4641	−0.428567
$988$	0	0
$989$	− 36.6410i	− 1.16512i
$990$	0	0
$991$	−34.5359	−1.09707	−0.548534	−	0.836128i	$-0.684814\pi$
$991$	−34.5359	−1.09707	−0.548534	+	0.836128i	$0.684814\pi$
$992$	0	0
$993$	10.2487i	0.325233i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	51.1769	1.62079	0.810395	−	0.585884i	$-0.199253\pi$
$997$	51.1769	1.62079	0.810395	+	0.585884i	$0.199253\pi$
$998$	0	0
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.2.f.e.49.2		4
3.2	odd	2		7200.2.d.n.2449.4		4
4.3	odd	2		200.2.f.e.149.4		4
5.2	odd	4		800.2.d.e.401.3		4
5.3	odd	4		160.2.d.a.81.2		4
5.4	even	2		800.2.f.c.49.3		4
8.3	odd	2		200.2.f.c.149.2		4
8.5	even	2		800.2.f.c.49.4		4
12.11	even	2		1800.2.d.l.1549.1		4
15.2	even	4		7200.2.k.j.3601.2		4
15.8	even	4		1440.2.k.e.721.2		4
15.14	odd	2		7200.2.d.o.2449.1		4
20.3	even	4		40.2.d.a.21.3	✓	4
20.7	even	4		200.2.d.f.101.2		4
20.19	odd	2		200.2.f.c.149.1		4
24.5	odd	2		7200.2.d.o.2449.4		4
24.11	even	2		1800.2.d.p.1549.3		4
40.3	even	4		40.2.d.a.21.4	yes	4
40.13	odd	4		160.2.d.a.81.3		4
40.19	odd	2		200.2.f.e.149.3		4
40.27	even	4		200.2.d.f.101.1		4
40.29	even	2	inner	800.2.f.e.49.1		4
40.37	odd	4		800.2.d.e.401.2		4
60.23	odd	4		360.2.k.e.181.2		4
60.47	odd	4		1800.2.k.j.901.3		4
60.59	even	2		1800.2.d.p.1549.4		4
80.3	even	4		1280.2.a.o.1.1		2
80.13	odd	4		1280.2.a.d.1.2		2
80.27	even	4		6400.2.a.ce.1.1		2
80.37	odd	4		6400.2.a.be.1.2		2
80.43	even	4		1280.2.a.a.1.2		2
80.53	odd	4		1280.2.a.n.1.1		2
80.67	even	4		6400.2.a.z.1.2		2
80.77	odd	4		6400.2.a.cj.1.1		2
120.29	odd	2		7200.2.d.n.2449.1		4
120.53	even	4		1440.2.k.e.721.4		4
120.59	even	2		1800.2.d.l.1549.2		4
120.77	even	4		7200.2.k.j.3601.1		4
120.83	odd	4		360.2.k.e.181.1		4
120.107	odd	4		1800.2.k.j.901.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.2.d.a.21.3	✓	4	20.3	even	4
40.2.d.a.21.4	yes	4	40.3	even	4
160.2.d.a.81.2		4	5.3	odd	4
160.2.d.a.81.3		4	40.13	odd	4
200.2.d.f.101.1		4	40.27	even	4
200.2.d.f.101.2		4	20.7	even	4
200.2.f.c.149.1		4	20.19	odd	2
200.2.f.c.149.2		4	8.3	odd	2
200.2.f.e.149.3		4	40.19	odd	2
200.2.f.e.149.4		4	4.3	odd	2
360.2.k.e.181.1		4	120.83	odd	4
360.2.k.e.181.2		4	60.23	odd	4
800.2.d.e.401.2		4	40.37	odd	4
800.2.d.e.401.3		4	5.2	odd	4
800.2.f.c.49.3		4	5.4	even	2
800.2.f.c.49.4		4	8.5	even	2
800.2.f.e.49.1		4	40.29	even	2	inner
800.2.f.e.49.2		4	1.1	even	1	trivial
1280.2.a.a.1.2		2	80.43	even	4
1280.2.a.d.1.2		2	80.13	odd	4
1280.2.a.n.1.1		2	80.53	odd	4
1280.2.a.o.1.1		2	80.3	even	4
1440.2.k.e.721.2		4	15.8	even	4
1440.2.k.e.721.4		4	120.53	even	4
1800.2.d.l.1549.1		4	12.11	even	2
1800.2.d.l.1549.2		4	120.59	even	2
1800.2.d.p.1549.3		4	24.11	even	2
1800.2.d.p.1549.4		4	60.59	even	2
1800.2.k.j.901.3		4	60.47	odd	4
1800.2.k.j.901.4		4	120.107	odd	4
6400.2.a.z.1.2		2	80.67	even	4
6400.2.a.be.1.2		2	80.37	odd	4
6400.2.a.ce.1.1		2	80.27	even	4
6400.2.a.cj.1.1		2	80.77	odd	4
7200.2.d.n.2449.1		4	120.29	odd	2
7200.2.d.n.2449.4		4	3.2	odd	2
7200.2.d.o.2449.1		4	15.14	odd	2
7200.2.d.o.2449.4		4	24.5	odd	2
7200.2.k.j.3601.1		4	120.77	even	4
7200.2.k.j.3601.2		4	15.2	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:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	800.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.732051 q^{3} +2.73205i q^{7} -2.46410 q^{9} +O(q^{10})\)	\(q-0.732051 q^{3} +2.73205i q^{7} -2.46410 q^{9} -2.00000i q^{11} -3.46410 q^{13} -3.46410i q^{17} -7.46410i q^{19} -2.00000i q^{21} +4.19615i q^{23} +4.00000 q^{27} -6.92820i q^{29} -1.46410 q^{31} +1.46410i q^{33} -2.00000 q^{37} +2.53590 q^{39} -5.46410 q^{41} -8.73205 q^{43} -6.73205i q^{47} -0.464102 q^{49} +2.53590i q^{51} -4.53590 q^{53} +5.46410i q^{57} +0.535898i q^{59} -4.92820i q^{61} -6.73205i q^{63} -7.26795 q^{67} -3.07180i q^{69} +1.46410 q^{71} -0.535898i q^{73} +5.46410 q^{77} -14.9282 q^{79} +4.46410 q^{81} +4.73205 q^{83} +5.07180i q^{87} +4.92820 q^{89} -9.46410i q^{91} +1.07180 q^{93} +6.39230i q^{97} +4.92820i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 4 * q^9	\( 4 q + 4 q^{3} + 4 q^{9} + 16 q^{27} + 8 q^{31} - 8 q^{37} + 24 q^{39} - 8 q^{41} - 28 q^{43} + 12 q^{49} - 32 q^{53} - 36 q^{67} - 8 q^{71} + 8 q^{77} - 32 q^{79} + 4 q^{81} + 12 q^{83} - 8 q^{89} + 32 q^{93}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^9 + 16 * q^27 + 8 * q^31 - 8 * q^37 + 24 * q^39 - 8 * q^41 - 28 * q^43 + 12 * q^49 - 32 * q^53 - 36 * q^67 - 8 * q^71 + 8 * q^77 - 32 * q^79 + 4 * q^81 + 12 * q^83 - 8 * q^89 + 32 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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